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abstract: 'RXJ0501.7–0359 is a new $V\sim 17$mag eclipsing polar discovered during the ROSAT all-sky survey (RASS). Extensive follow-up observations at X-ray (pointed ROSAT PSPC and HRI observations) and optical (time-resolved photometry and spectroscopy) wavelengths were obtained which allowed us to determine a very precise orbital period and to provide constraints for the masses of the binary components and the orbital inclination. With an orbital period of $P \sim 171$min the system is placed at the upper end of the period gap. From the radial velocity amplitude of the secondary and the duration of the primary eclipse we obtain a mass ratio $q = M_{2}/M_{1} = 0.83 \pm 0.15$ and an inclination $i = 75\deg \pm 3\deg$ for the system. This yields a white dwarf mass $M_{1} = 0.43^{+0.10}_{-0.07}\,M_{\sun}$. Cyclotron humps might be present in some of our optical spectra leading to a tentative magnetic field strength of $B\sim 25$MG.'
author:
- 'Vadim Burwitz$^{1,2}$, Klaus Reinsch$^{2}$, Klaus Beuermann$^{2,1}$, and Hans-Christoph Thomas$^{3}$'
title: 'RXJ0501.7–0359: a new ROSAT discovered eclipsing polar in the period gap'
---
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Introduction
============
The ROSAT All-Sky Survey (RASS) revealed a great wealth of new objects with soft X-ray spectra. Our optical identification of a sample of these sources has led to a significant increase in the number of known polars (Beuermann 1998, Thomas et al. 1998). Detailed X-ray and optical follow-up studies allow us to determine their orbital periods, magnetic field strengths, accretion geometry, etc. This leads to a largely increased data base of known polars which is important for studying the general properties of these systems as well as their evolution. Apart from this it is important to study individual objects in great detail as many of the new systems have remarkable characteristics such as extremely short periods and high magnetic field strengths (see Burwitz et al. 1997, 1998).
Observations and discussion
===========================
RXJ0501.7-0359 (= 1RXSJ050146.2–035927) was detected as a soft (hardness ratio $HR1 = (-0.96\pm 0.03$), bright ($0.22\pm 0.03$ cts/s), and variable X-ray source in the RASS (Voges et al. 1996). As its optical counterpart, we have identified an eclipsing $V\sim 17$mag polar located at RA = $05^{\rm h}01^{\rm m}46\fs 1$, DEC = $-03\deg 59\arcmin 32\arcsec$.
Here, we present the analysis of our large amount of follow-up observations of this new object: optical V-band CCD photometry with the Dutch 0.9-m telescope and time-resolved spectroscopy with EFOSC2 at the ESO/MPI 2.2-m telescope on La Silla, Chile, infrared J, H, and K photometry with MAGIC at the 3.5-m telescope on Calar Alto, Spain, and X-ray data with ROSAT using both the PSPC and the HRI.
The optical light curves show a strong modulation which could be caused either by cyclotron beaming or by variations of the effective emitting area of the accretion stream as seen from different angles during the orbit. The light curve also features a deep total eclipse of the accreting white dwarf and the stream by the secondary star and a pre-eclipse dip, possibly due to the accretion stream crossing our line-of-sight towards the accretion region (cf. Fig. \[licu\], top panel).
From our optical, IR, and spectrophotometry, we have derived 16 mid-eclipse timings between August 1993 and January 1996. These were used to determine the orbital period of RXJ0501.7–0359 very precisely and lead to the following ephemeris: $$\label{eq.ephem}
T_{\rm mid-eclipse}({\rm HJD}) = 2449748.83782(20)+0.11896906(7)\times E.$$ From the optical spectra we have determined radial velocities of the narrow and broad components of the Balmer (H$\alpha$, H$\beta$) and the He[II]{}4686Å emission lines. The average radial velocity curve yields an amplitude K$'\!_{2}=(73.6\pm 25)$km/s for the narrow component (cf. Fig. \[rvcurve\]). Maximum redshifts of the broad and narrow components occur at orbital phases 0.02 and 0.12, respectively.
Assuming a Roche-lobe filling secondary star, Kepler’s laws define a relation between the orbital period $P$ of a CV and the mass $M_2$ and radius $R_2$ of the secondary (Eq. 1 in Beuermann et al. 1998). As RXJ0501.7-0359 is probably a system which has been born in the period gap, its secondary cannot be much evolved. Therefore, the theoretical mass-radius relation for ZAMS stars with solar metallicity (Baraffe et al. 1998 with fit parameters from Beuermann and Weichhold 1998, Eq. 5) provides a valid second relation between $M_2$ and $R_2$. Combining both equations gives the mass-period relation $$\label{eq.m}
M_2/M_{\sun} = 0.0686 \left({P_{\rm orb}/{\rm hours}}\right)^{1.59}\\$$ from which we obtain $M_2 = 0.36\,M_{\sun}$ and $R_2 = 0.33\,R_{\sun}$ for the mass and radius of the secondary in RXJ0501.7-0359, respectively. The corresponding mass-function is shown in Fig. \[masses\].
The duration of the eclipse, $\Delta T_{\rm ecl} = (13.1\pm 1.2)$min $=
(0.077\pm 0.007)\,P_{\rm orb}$, provides a second constraint for the inclination $i$ and the mass ratio $Q = M_1/M_2$ in this system (see Fig. \[masses\]). Combining both constraints the valid values for $i$ and $Q$ are restricted to the very narrow ranges $i = (75\pm 3)\deg$ and $Q = 1.20^{+0.18}_{-0.27}$ (shaded area in Fig. \[masses\]). Finally, using $M_{2}$ and $Q$ we get $M_{1} = 0.43^{+0.10}_{-0.07}\,M_{\sun}$ for the mass of the white dwarf.
Besides atomic emission lines, our optical spectra of RXJ0513.7–0359 generally show a smooth continuum (top and middle panel in Fig. \[ospec\]). Only during one observing run some hump structure might be present during the orbital phases 0.8–0.95 (bottom panel in Fig. \[ospec\]). We have tentatively identified these features as the 5th and 6th cyclotron harmonics corresponding to a magnetic field strength of $B\sim 25$MG in the accretion region. Given the weakness of the possible cyclotron signatures, the field strength of RXJ0501.7-0359, however, has still to be regarded as uncertain.
The X-ray data show a very strong modulation with a bright phase lasting for about half of the orbital cycle (phases 0.35–0.75, cf. Fig. \[licu\]). This on-off modulation is most likely caused by the accretion pole disappearing behind the limb of the white dwarf. Two component (blackbody + thermal bremsstrahlung with $kT_{\rm tb} = 20\,$keV fixed) model fits to the ROSAT PSPC X-ray spectra which cover mainly the bright phase yield a blackbody temperature $kT_{\rm bb} = 38^{+5}_{-10}$eV with an absorption column density of $N_{\rm H} = (0.78^{+0.14}_{-0.27})\,10^{21}$atoms/cm$^2$ which is slightly above the galactic value of $N_{\rm H,gal} = 0.60\,10^{21}$ (3$\sigma$ errors are given).
The accretion geometry of RXJ0501.7–0359 appears to be intriguing as the X-ray bright phase occurs around the superior conjunction of the white dwarf. This implies that the visible accreting pole must be located on the white dwarf hemisphere pointing away from the secondary star.
A more detailed analysis and discussion of our X-ray, optical, and IR data of this interesting new eclipsing polar in the period gap will be presented elsewhere (Burwitz et al., in prep.).
We thank the ROSAT team for its enduring work which resulted in the All-Sky-Survey data base and the staff at the La Silla and Calar Alto observatories for their competent assistance during our observing runs. This work was supported by the DLR under grant 50OR92105 and, in part, by the Deutsche Forschungsgemeinschaft under grant Re 1100/3-1.
Beuermann, K. 1998, High energy astronomy & astrophysics, Agrawal P. (ed.), Tata Inst.of Fund. Res., p. 100 Beuermann, K., & Burwitz, V. 1997, ASP Conf. Ser., 85, 99 Beuermann, K., Baraffe, I., Kolb, U., and Weichhold, M. 1998, , 339, 518 Beuermann, K., Thomas, H.-C. 1990, 230, 326 Beuermann, K. & Weichhold, M. 1998, these proceedings Burwitz, V., Reinsch, K., Beuermann, K., Thomas, H.-C. 1997, , 327, 183 Burwitz, V., Reinsch, K., Schwope, A.D., et al. 1998, , 331, 262 Thomas, H.-C., Beuermann, K., Reinsch, K., Schwope, A.D., Trümper, J., & Voges, W. 1998, , 335, 467 Voges, W., Aschenbach, B., Boller, Th., et al. 1996, IAUC 6420, The ROSAT All-Sky Survey Bright Source Catalogue (1RXS), http://www.rosat.mpe-garching.mpg.de/survey/rass-bsc/
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abstract: 'The spatiotemporal propagation of a momentum excitation on the finite Fermi-Pasta-Ulam lattices is investigated. The competition between the solitary wave and phonons gives rise to interesting propagation behaviors. For a moderate nonlinearity, the initially excited pulse may propagate coherently along the lattice for a long time in a solitary wave manner accompanied by phonon tails. The lifetime of the long-transient propagation state exhibits a sensitivity to the nonlinear parameter. The solitary wave decays exponentially during the final loss of stability, and the decay rate varying with the nonlinear parameter exhibits two different scaling laws. This decay is found to be related to the largest Lyapunov exponent of the corresponding Hamiltonian system, which manifests a transition from weak to strong chaos. The mean-free-path of the solitary waves is estimated in the strong chaos regime, which may be helpful to understand the origin of anomalous conductivity in the Fermi-Pasta-Ulam lattice.'
author:
- Zongqiang Yuan
- Zhigang Zheng
bibliography:
- 'References.bib'
title: 'Propagation dynamics on the Fermi-Pasta-Ulam lattices'
---
Introduction
============
The study of transport process of matter and energy is of fundamental importance in understanding numerous nonequilibrium phenomena occuring in nature. Heat conduction is one of the most important manners of energy transport. Recently, heat conduction in low-dimensional materials has attracted much attention among physicists for the reason that classical one-dimensional lattices frequently exhibit anomalous heat conduction behavior, *i.e.*, the thermal conductivity depends crucially on the size of the material [@Lepri1997; @Aoki2001; @Lepri2003; @Dhar2008c; @Li2012]. This arouse a tide of interest in the microscopic foundation of normal heat conduction, and a number of viewpoints on the relation between heat conduction and dynamical properties have been proposed, such as chaos, mixing, energy diffusion, and so on.
The manipulation of heat flow is an important and practical issue, which has been developed rapidly in recent years [@Terraneo2002; @Li2004; @Chang2007a; @Wang2007; @Wang2008a; @Wang2008]. Thermal rectifier had been experimentally realized in nanoscale systems [@Chang2006a; @Kobayashi2009]. By periodically modulating thermal baths, heat flow can even be created and controlled at strict zero thermal bias [@Ren2010]. A relevant problem is the competition between time scale of the manipulation of the thermal bath and the relaxation time scale of the heat flow along the lattice. Therefore it is important to study the propagation process of energy in nonlinear low-dimensional systems from the microscopic point of view.
In studies of heat conductions of nonlinear lattices, the thermal baths usually contact with the system by coupling the particles at two ends. The influence of the thermal bath on the lattice can be considered as a series of stochastic perturbations. These perturbations start from the ends of the lattice and propagate along the lattice with a finite speed. It is a significant topic to study the propagation dynamics and dispersion behaviors of energy pulses on the lattice. Therefore, we may explore the evolution behavior of a single pulse excited at one end of the lattice as the first step. This may give us a more profound microscopic understanding of the transfer process of heat on low-dimensional nonlinear lattices. In this aspect, energy propagation for an excitation of a single particle on infinite nonlinear lattices has been discussed [@Zavt1993; @Sarmiento1999; @Rosas2004]. Practically, systems have finite sizes, and the propagation of energy on materials usually possesses a finite time scale. Therefore, the influence of finite length of low-dimensional materials should be taken into account.
In the present paper, we investigate the propagation behavior of an energy pulse initially excited at one end of a finite Fermi-Pasta-Ulam $\beta$ (FPU-$\beta$) lattice. The initially excited pulse may propagate along the lattice in a solitary wave manner accompanied by phonon tails. For a moderate nonlinearity, the solitary wave can propagate coherently for a long time before its collapse, which is called the long-transient propagation state. The lifetime of the long-transient propagation state displays a sensitive dependence on the nonlinear parameter $\beta$. The energy of the solitary wave decreases exponentially during the collapse process, which is irrelevant to the boundary conditions. The decay rate against $\beta$ exhibits two different scaling laws which is found to be related to the largest Lyapunov exponent of the corresponding Hamiltonian system. The multiple-peak structure of the lifetime of the long-transient propagation state on $\beta$ is understood by the residual high-dimensional Kolmogorov-Arnold-Morse (KAM) tori of the Hamiltonian lattice system in the parameter regime of $\beta$ with moderate stochasticity. Our results presented in this paper may be helpful to understand miscellaneous recently studied macroscopic heat phenomena based on microscopic energy wave properties on the nonlinear lattices.
The Fermi-Pasta-Ulam model
==========================
The famous FPU model was initially introduced by Fermi, Pasta, and Ulam to investigate the energy equipartition problem and the ergodic hypothesis in nonlinear systems [@Fermi1955]. The attempt to resolve the mystery of the FPU recurrence has led to the discovery of solitons [@Zabusky1965]. Later tremendous progresses on FPU model have been extended to studies on intrinsic localized modes in perfect lattices, Bose-Einstein condensates, stochastic resonance, and so on [@Sievers1988; @Flach2005; @Villain2000; @Miloshevich2009]. Recently the FPU lattice was studied in relating to heat conductions in low-dimensional systems [@Li2005a; @Mai2007; @Dhar2008b; @Li2010]. In this paper we adopt the FPU-$\beta$ model as our prototype to study the energy transport on nonlinear lattices. The Hamiltonian of the FPU-$\beta$ lattice consisting of $N$ particles with open boundary condition can be written as $$\begin{aligned}
\label{DefineHamitonian}
H &=& \sum _{i=1}^N \frac{p_i^2}{2} + \sum _{i=1}^{N-1} V(q_{i+1},q_i), \nonumber\\
V(q_{i+1},q_i)&=& \frac{k}{2}\Bigl(q_{i+1}-q_i\Bigr)^2 + \frac{\beta}{4}\Bigl(q_{i+1}-q_i \Bigr)^4,\end{aligned}$$ where $p_i$ and $q_i$ denote the momentum and the displacement from the equilibrium position of the $i$-th particle, respectively. The local energy of the *i*-th particle can be defined as $E_i=\frac{p_i^2}{2}+\frac{1}{2}V(q_{i+1},q_i)+\frac{1}{2}V(q_i,q_{i-1})$. In the absence of the quartic term, *i.e.*, $\beta=0$, the above Hamiltonian reduces to a one-dimensional harmonic chain, which is integrable and can be analytically solved. The presence of the anharmonic terms breaks the integrability and brings forth the intermingling of regular and chaotic motions in phase space [@Antonopoulos2006].
To explore the evolution behavior of a single pulse excited at one end of the lattice, we may impart an initial momentum excitation to the first particle of an initially quiescent lattice. In our numerical simulation, we adopt the fourth-order symplectic method in order to solve the dynamics of the FPU lattice as a Hamilton dynamical system. We further fix the harmonic coefficient $k=0.5$, particles of the lattice $N=50$ and energy of the initial excitation $E=50$ (this also gives the total energy of the lattice) throughout the simulation.
The long transient propagation
==============================
Here we are concerned with the destiny of an initial pulse on a lattice with finite size. For a weak nonlinearity $\beta$, the dispersion effect dominates this weakly nonlinear system and leads to the rapid collapsing of the initial local excitation, as shown in Fig. \[Evolution\](a) for $\beta=0.001$. When we increase $\beta$, the initial excitation may excite a solitary wave. This is shown in Figs. \[Evolution\](c) for $\beta=0.05$ and \[Evolution\](e) for $\beta=0.4$. These are consistent with previous works [@Wattis1993; @Friesecke1994; @Zhang2000; @Zhang2001]. The solitary wave will propagate with no decay if the lattice is extended to infinity. However, with finite lattice, one finds two distinctively different stages as depicted in Figs. \[Evolution\](d) and \[Evolution\](f). In the first stage, the energy pulse initiated at the first particle can be coherently transferred to its neighboring and other particles, and this pulse forms a solitary wave along the lattice. The propagation of the solitary wave keeps stable for a long time. The second stage comes when the solitary wave loses its stability and collapses in a rather short duration, and the energy of the solitary wave is distributed to all particles in the lattice. This behavior is very interesting, indicating that the solitary wave can dominate by suppressing the phonon waves for a long time.
![(Color on-line) Spatiotemporal propagation behavior of the initial momentum excitation imposed on the first particle of the lattice. (a), (b) $\beta=0.001$, (c), (d) $\beta=0.05$, and (e), (f) $\beta=0.4$. The left and right columns correspond to the short and long time scales, respectively.[]{data-label="Evolution"}](Evolution.eps){width="\linewidth"}
![(Color on-line) The lifetime $\tau$ of the solitary wave. (a) and (b) corresponds to the cases of open and periodic boundary conditions, respectively. To numerically get the fractal dimension of the hierarchical multiple-peak structure for the case of open boundary condition, result of a typical point count (see text) is shown in the inset of (a).[]{data-label="Lifetime"}](Lifetime.eps){width="\linewidth"}
The lifetime $\tau$ of the solitary wave (the long-transient propagation state) as the function of the anharmonic parameter $\beta$ for the case of open boundary condition is shown in Fig. \[Lifetime\](a). The value of $\tau$ is sensitive to $\beta$, especially for $\beta \in (0.05, 0.2)$ where a hierarchical multiple-peak structure is found. By adopting the method proposed in ref. [@Forrest1979], the fractal dimension of this hierarchical structure can be estimated as follows. The lifetime $\tau$ is re-scaled and a center point on the image is picked at random, and then a series of nested circles of different sizes are placed around it and the number of points (lifetime data for $\beta$) in each circle counted. The number of points $M$ in a circle with a radius $r$ satisfies as $M \propto r^D$, where $D$ is the dimension of the measured object. We numerically get the fractal dimension to be $D \approx 1.66$. This result implies the complex dynamical stability of the transient propagation by varying the nonlinear parameter. As will be shown below, this property indicates a connection between the stability of solitary wave and the structure of the phase space of the Hamiltonian lattice system.
Phonon-soliton interaction
==========================
It is important to study the competition of various propagation modes (waves) on the lattice to understand the above results. A sufficiently large momentum excitation initially imposed on the first particle excites not only a solitary wave, but also a small-amplitude tail. The solitary wave moves along the lattice much faster than the phonon-wave tail, which disperses due to the dispersion property of the phonon modes. Moreover, the reflections of the solitary wave at both ends of the lattice can also excite additional small-amplitude tails for the case of open boundary condition. In Figs. \[EnergyProfile\](a), (c) and (e), the snapshots of propagations of the energy waves along the lattice for the open boundary condition case are plotted for different moments. It is clear that the solitary wave moves with a hierarchy of lower pulses that move slower, and the heights of these small pulses decrease during their motion along the lattice. As the solitary wave moves to the boundary, it will be bounced back with a radiation of additional phonon waves \[Fig. \[EnergyProfile\](e)\]. The multiple collisions between the solitary wave and phonon waves give rise to the instability of the solitary wave and its collapse.
![(Color on-line) Energy distribution profiles among the particles for the FPU-$\beta$ lattice with $\beta=0.05$ for different moments: (a), (b) $t=12$, (c), (d) $t=30$, and (e), (f) $t=48$. The left and right columns correspond to the cases of open and periodic boundary conditions, respectively. Because the first peak is much higher than others, we only plot the energy profile in (-0.001,0.1) to get a clearer observation of the wave tails.[]{data-label="EnergyProfile"}](EnergyProfile.eps){width="\linewidth"}
This phonon-soliton-interaction mechanism is also valid for other types of boundary conditions, *e.g.*, the periodic boundary condition. Technically a single solitary wave can be produced by initially exciting an energy pulse at one end of an open chain and then connecting both ends when the solitary wave arrives at the middle of the lattice. Different from the open boundary condition case, for the periodic boundary condition, there is no reflection of the solitary wave at the boundary. Therefore the solitary wave moves unidirectionally, and only the phonon-wave tail due to the initial excitation can be found, as shown in Figs. \[EnergyProfile\] (b), (d) and (f). Due to the lack of additional excitations of phonons at the boundary, the lifetime of the solitary wave moving on a circular topology of the lattice should be much longer than that on an open lattice, as shown in Fig. \[Lifetime\](b). However, because the solitary wave moves faster than its initial phonon tail, they will frequently collide when they meet. This interaction will eventually lead to the collapse of the solitary wave.
Decay process of the solitary wave
==================================
By resorting to the evolution of the energy of the small-amplitude tails, we now focus on the collapse process of the solitary wave due to the interaction with phonons mentioned above. For the lattice system we are studying here, the energy of the tails is defined as the residual energy of the solitary wave. Due to the spatial localization of the solitary wave, one can write the tail energy as $$\label{DefineEtails}
E_{tails} = E - \sum_{i=i_c-i_n}^{i_c+i_n} E_i(t),$$ where $E$ is the total energy of the system, $E_i$ is the local energy of the *i*-th particle, $i_c(t)$ is the center position of the solitary wave at time $t$, and $i_n$ denotes the number of the left/right neighboring particles of the center particle of the solitary wave packet. Numerically $i_n=2$ is enough due to the energy localization of the solitary wave. The increase of $E_{tails}$ corresponds to the dissipation of the solitary wave energy considering that the total energy of the system is conserved.
![(Color on-line) Energy increase of the tails with open (blue line) and periodic (black line) boundary conditions: (a-c) $\beta=0.2$, (d-f) $\beta=0.4$, (g) $\beta=0.56$, (h) $\beta=0.72$. (b), (c) and (e), (f) enlarge the final stages of the collapse processes in (a) and (d), respectively. The red lines are for guiding the eyes. (i) Decay rate $\gamma$ against $\beta$ for the cases of open (green line) and periodic (pink line) boundary conditions.[]{data-label="Collapse"}](Collapse.eps){width="\linewidth"}
We present the evolution of $E_{tails}$ for several typical values of $\beta$ in Figs. \[Collapse\](a)-(h). Although the lifetime of the solitary wave varies for different boundary conditions and different $\beta$, the collapse process of the solitary wave exhibits the same scenario, *i.e.*, the energy of the solitary wave decays exponentially during the final loss of stability. We label the exponential decay rate by $\gamma$. In Fig. \[Collapse\](i), the decay rate $\gamma$ against $\beta$ is given for both open and periodic boundary conditions. The consistency of different types of boundary conditions indicates that the final loss of stability of the solitary wave is irrelevant to the boundary conditions.
![(Color on-line) Decay rate $\gamma$ against the nonlinear parameter $\beta$ for the case of open boundary condition (blue triangles). Green circles correspond to the rescaled decay rate $\gamma_L$. The black line corresponds to the largest Lyapunov exponent $\lambda$ computed according to the analytic expression (\[formula for lamda\]). The red lines are for guiding the eyes.[]{data-label="DecayExponents"}](DecayExponents.eps){width="\linewidth"}
In Fig. \[DecayExponents\], the decay rate $\gamma$ is computed numerically in a larger scale of the nonlinear parameter $\beta$ for the case of open boundary condition. It can be seen that $\gamma$ against $\beta$ displays two different scaling laws $\gamma \propto \beta^{\kappa}$. For lower values of $\beta$, the scaling exponent $\kappa \approx 2/3$. For larger nonlinear parameter $\beta$, $\kappa \approx 1/4$. It is instructive to note that a similar result was obtained in studies of the nonlinear Hamiltonian dynamics of the FPU-$\beta$ lattice [@Gallavotti2008]. It was found that the largest Lyapunov exponent $\lambda$ of the system varying with the energy density $\epsilon=E/N$ exhibits a crossover between two scaling laws: $\lambda \propto \epsilon^2$ at low-energy density, and $\lambda \propto \epsilon^{2/3}$ at larger $\epsilon$ values, reaching on an asymptotic value at large energy of $\lambda \propto \epsilon^{1/4}$. One should note that changing the nonlinear parameter $\beta$ is equivalent to changing the energy density $\epsilon$ [@Aoki2001].
We can analytically estimate the largest Lyapunov exponent $\lambda$ of the corresponding Hamiltonian system as a function of $\beta$ following the theoretical approach of Riemannian differential geometry of Newtonian dynamics [@Casetti1995; @Casetti1996]. In the geometric approach to Hamiltonian chaos, the dynamics described by the equations of motion is equivalent to a geodesic flow on a Riemannian manifold. Dynamical instability (chaos) is related to curvature fluctuations of the manifolds and is described by means of the Jacobi-Levi-Civita equation for geodesic spread. The analytic formula for $\lambda$ is $$\label{formula for lamda}
\lambda = \frac{1}{2} \biggl ( \Lambda - \frac{4 \Omega_0}{3 \Lambda} \biggr),$$ $$\Lambda = \biggl [ 2 \sigma_{\Omega}^2 \tau + \sqrt{ \biggl( \frac{4 \Omega_0}{3} \biggr)^3 + (2 \sigma_{\Omega}^2 \tau})^2 \biggr]^{1/3},$$ $$2 \tau = \frac{\pi \sqrt{\Omega_0}}{2 \sqrt{\Omega_0(\Omega_0+\sigma_{\Omega})} + \pi \sigma_{\Omega}},$$ where $\Omega_0$ and $\sigma_{\Omega}$ corresponds to the average Ricci curvature and its fluctuation, respectively. For our Hamiltonian (\[DefineHamitonian\]), the explicit expression for the Ricci curvature $k_R$ is $$\label{kR}
k_R = 2k + \frac{6\beta}{N} \sum_{i=1}^N (q_{i+1}-q_i)^2,$$ where $k$ and $\beta$ correspond to the harmonic and anharmonic coefficients in the Hamiltonian of the FPU-$\beta$ lattice, respectively. Then the expressions for $\Omega_0$ and $\sigma_{\Omega}$ can be derived as $$\Omega_0 = 2k + \frac{3k}{\theta} \frac{D_{-3/2}(\theta)}{D_{-1/2}(\theta)},
\label{compute average of Ricci curvature of FPU beta curvature}$$ $$\sigma_{\Omega}^2 = \frac{9 k^2}{\theta^2} \biggr \{ 2 - 2 \theta \frac{ D_{-3/2} (\theta)}{D_{-1/2}(\theta)} - \biggl[ \frac{ D_{-3/2} (\theta)}{D_{-1/2}(\theta)} \biggr ]^2 \biggr \} +
F(\theta),
\label{compute average fluctuation of Ricci curvature of FPU beta curvature}$$ where $D_x$ are parabolic cylinder functions. The results are expressed in terms of the parameter $\theta=k \sqrt{\Theta /2 \beta}$, where $\Theta$ is the inverse temperature introduced by the Gibbsian weight $e^{-\Theta H}$. The additional term $F(\theta)$ is $$\label{corrective}
F(\theta) = - \frac{\Theta^2}{c_V(\theta)} \biggl ( \frac{\partial \Omega_0 (\theta)}{\partial \Theta} \biggr)^2,$$ where the derivative part is $$\frac{\partial \Omega_0}{\partial \Theta} = \frac{-3k^3}{8 \beta \theta^3} \biggr \{2 \theta - 2(\theta^2-1) \frac{D_{-3/2}(\theta)}{ D_{-1/2}^2(\theta)} - \theta \biggl[ \frac{ D_{-3/2} (\theta)}{D_{-1/2}(\theta)} \biggr ]^2 \biggr \},$$ and the specific heat per particle $c_V$ is found to be $$c_V = \frac{3}{4} + \frac{\theta^2}{8} - \frac{\theta(\theta^2-1)}{8} \frac{D_{-3/2}(\theta)}{ D_{-1/2}^2(\theta)} - \frac{\theta^2}{16} \biggl[ \frac{ D_{-3/2} (\theta)}{D_{-1/2}(\theta)} \biggr ]^2.$$ Substituting Eqs. (\[compute average of Ricci curvature of FPU beta curvature\]) and (\[compute average fluctuation of Ricci curvature of FPU beta curvature\]) into Eq. (\[formula for lamda\]) yields an analytic expression of $\lambda$ for the FPU-$\beta$ model (\[DefineHamitonian\]), valid in the thermodynamic limit $N \to \infty$. A relation between the nonlinear parameter $\beta$ and the parameter $\theta$ $$\beta(\theta)=\frac{k^2}{8 \epsilon} \biggl [ \frac{3}{\theta^2} + \frac{1}{\theta} \frac{D_{-3/2}(\theta)}{D_{-1/2}(\theta)} \biggr ]
\label{compute average fluctuation of Ricci curvature of FPU beta parameter}$$ allows one to obtain $\lambda$ as a function of $\beta$. We present $\lambda$ against $\beta$ in Fig. \[DecayExponents\]. It is clear that the rescaled decay rate $\gamma_L=L\gamma$ agree with the analytic computation of the largest Lyapunov exponent, where the scaling constant $L=50$ in Fig. \[DecayExponents\]. The crossover of the largest Lyapunov exponent indicates the existence of a threshold corresponding to the transition from weak chaos to strong chaos.
In the weak chaos regime ($\beta \to 0$), the harmonic coupling plays the dominant role and leads to phonons and their interactions.
In the transition regime, a moderate nonlinearity allows the emergence of solitary waves. The phase space of the Hamiltonian system is composed of chaotic trajectories intermingled with KAM tori. The multiple-peak structure of the lifetime of the solitary wave observed in Fig. \[Lifetime\] is found in this transition regime and can be explained by the transiently quasi-regular motions in phase space induced by moderate stochasticity, where KAM invariant tori are dominant. The long-transient solitary wave states dynamically correspond to these KAM tori, and the lifetime of the solitary wave actually implies the relative stability of the corresponding KAM torus. These results may provide useful hints in understanding the stability of KAM tori in high-dimensional Hamiltonian systems.
In the strong chaos regime, the nonlinear coupling plays the main role and leads to the dominance of solitary waves. We discuss below the relation between the macroscopic heat phenomena on low-dimensional nonlinear lattices and our microscopic results, which is the initial motivation of our present work.
The anomalous thermal conductivity of low-dimensional nonlinear lattices is not completely well theoretically understood now. Based on the effective phonon theory and a conjecture that the mean-free-path of the effective phonons is inversely proportional to the dimensionless nonlinearity $\xi$ as a ratio between the average of nonlinear potential energy and the total potential energy, temperature dependence of thermal conductivity in the FPU-$\beta$ model is well explained [@LiEPL2007].
We can directly estimate the mean-free-path of the solitary waves and the interesting thing is that our estimation is the same as the conjecture of the mean-free-path for the effective phonons at the high temperature regime [@LiEPL2007]. This indicates the similarity between the solitary waves and the effective phonons at the high temperature regime. The process of our estimation is presented as follows. The large energy scaling of the solitary wave velocity with the energy as $v \propto E^{1/4}$ is well know [@Aoki2001]. Since changing the nonlinear parameter $\beta$ is equivalent to changing the energy density, we have the scaling of the solitary wave velocity with the nonlinear parameter $\beta$, $v \propto \beta^{1/4}$. Microscopically, we show in our work that the motion of the solitary wave on the finite FPU-$\beta$ lattice may give rise to phonon tails. The interaction between the solitary wave and the phonon waves leads to the instability of the solitary wave, especially in the strong chaos regime. The existence of thermal baths on the boundaries, which are usually adopted in studies of thermal conductions of low-dimensional lattices, may drastically enhance the excitations of more phonons and the instability of the solitary wave. According to the exponential decay law of the solitary wave during the final loss of stability, the relaxation time of the solitary wave can be estimated as $\tau \propto 1/\gamma$. For the decay rate $\gamma$ obeys the scaling law $\gamma \propto \beta^{1/4}$ in the strong chaos regime, the mean-free-path of the solitary wave can be estimated to be $\l=v \tau \propto 1$.
Concluding remarks
==================
In conclusion, in this Letter we extensively explored the spatiotemporal propagation behavior of a momentum excitation traveling along the FPU-$\beta$ lattice with finite size for different nonlinear strengths and different boundary conditions. For a moderate nonlinearity, the solitary wave can coherently propagate along the lattice for a long time and then decays rapidly after this transient due to the final dominance of phonons. The lifetime of the long-transient propagation state is sensitive to the value of the anharmonic parameter $\beta$ and exhibits a fractal dependence on $\beta$ for the case of open boundary condition.
The energy of the solitary wave is found to decay exponentially during the final loss of stability, which is an intrinsic property of the lattice independence of the type of boundary conditions and the parameters of the system. The decay rate $\gamma$ of the solitary wave as the function of $\beta$ exhibits two different scaling laws, which is consistent with the scenario predicted in the largest Lyapunov exponent of the FPU-$\beta$ model. Therefore the loss of stability of the solitary wave is a manifestation of dynamical instability of orbits in the Hamiltonian system. We found that the appearance of the hierarchical multiple-peak structure of the lifetime of the long-transient propagation state is in cases of $\beta$ in the transition regime from weak chaos to strong chaos. Therefore this interesting result can be well explained by the residuals of KAM tori in the phase space of the Hamiltonian system in the parameter regime of $\beta$ with the moderate stochasticity, which induce the “stickness” effect despite of their instability in high-dimensional Hamiltonian cases. In fact, the multiple-peak of the lifetime is closely related to the structure of these unstable KAM tori in phase space.
To solve the debate about the energy carriers responsible for the heat conduction in the FPU-$\beta$ lattice, the sound velocity of energy transfer was measured to examine the properties of the energy carriers, by using both nonequilibrium and equilibrium approaches. Nevertheless, the uncertainty of the computational data is too large to distinguish between the two predictions based on soliton theory and effective phonon theory. Our discussion of mean-free-path of the solitary waves may be helpful to understand the origin of anomalous conductivity and the debate about the energy carriers in the FPU-$\beta$ lattice [@Zhang2000; @Zhang2001; @Aoki2001; @Zhao2006; @Li2006; @Li2010].
Project supported by the National Natural Science Foundation of China (Grant No. 11075016), the Fundamental Research Funds for the Central Universities of China (Grant No. 201001), and the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20100003110007).
|
---
abstract: 'In this paper we investigate the affect of various acceptance conditions on recogniser membrane systems without dissolution. We demonstrate that two particular acceptance conditions (one easier to program, the other easier to prove correctness) both characterise the same complexity class, $\NL$. We also find that by restricting the acceptance conditions we obtain a characterisation of $\L$. We obtain these results by investigating the connectivity properties of dependency graphs that model membrane system computations.'
author:
- 'Niall Murphy[^1]'
- 'Damien Woods[^2]'
title: 'On acceptance conditions for membrane systems: characterisations of [L]{} and [NL]{}'
---
Introduction
============
In the membrane systems (also known as P-systems [@Pau2002x]) computational complexity community it is common practice to explore the power of systems by allowing and prohibiting different developmental rules. This technique has yielded several interesting results such as the role of membrane dissolution in recognising $\PSPACE$-complete problems [@NJNC2006p] and the role of membrane division in recognising problems outside of $\P$ [@ZFM2000c].
In this paper we do not vary the rules permitted in membrane systems but instead we vary the acceptance conditions and observe the change (or lack of change) this makes to the computing power of the system. Our main technique is to analyse the structure, and connectivity, of dependency graphs [@NJNC2006p] that are induced by acceptance conditions. Our approach builds on previous work on dependency graphs [@NJNC2006p; @GPRR2006c] to give a number of new techniques and results. Our techniques and results should be of interest to those who wish to characterise complexity classes, those studying acceptance conditions for membrane systems, and those characterising the power of membrane systems.
This research was motivated by the realisation that in prior work [@MW2008c] we were using a seemingly more general halting condition than is used by the membrane community. Previously, we showed that $\AC^0$-uniform families[^3] of active membrane systems without dissolution, and using the acceptance conditions specified in Section \[ssec:general\_rec\_sys\], characterise $\NL$ [@MW2008c]. However, most researchers use a more restricted acceptance condition (see Section \[ssec:standard\_rec\_sys\]). We show here that this more restricted definition also characterises $\NL$. This means that the two definitions are equivalent in terms of computing power for [$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}$]{}systems. The choice of which definition to use is now mostly a matter of personal taste as we have shown that the two are equivalent under $\AC^0$ reductions, i.e. there is a (very efficient) compiler to translate one definition to another.
In Section \[ssec:strict\_rec\_sys\] we show that active membrane systems without dissolution, and using a restriction on the standard acceptance definition, characterise $\L$. This demonstrates that not all (minor) restrictions on halting definitions yield systems that characterise $\NL$.
We note here that the three definitions that we consider in Section \[sec:different\_accepting\] all characterise $\P$ if they are generalised to use $\P$-uniformity. The $\P$ lower bound of this characterisation is a trivial corollary of the fact that such membrane systems can easily embed polynomial time deterministic Turing machines, and is not related to the differences in their definitions.
Preliminaries {#sec:definitions}
=============
In this section we define membrane systems and some complexity classes. These definitions are based on those from P[ă]{}un [@Pau2002x; @Pau2001p], Sos[' i]{}k and Rodr[í]{}guez-Pat[ó]{}n [@SR2007p], Guti[é]{}rrez-Naranjo et al. [@NJNC2006p], and P[é]{}rez-Jim[é]{}nez et al. [@PRS2003p]. Previous works on complexity and membrane systems spoke of solving a problem in a “uniform way”, that is, in a manner reminiscent of how families of circuits solve a problem. Sos[' i]{}k and Rodr[í]{}guez-Pat[ó]{}n defined uniformity for membrane systems in a similar manner to circuit uniformity, this allows us to refer to uniform families of membrane systems.
Active membrane systems
-----------------------
Active membrane systems are a class of membrane systems with membrane division rules. Division rules can either only act on elementary membranes, or else on both elementary and non-elementary membranes. An elementary membrane is one which does not contain other membranes (a leaf node, in tree terminology).
\[def:membrane\] An active membrane system without charges is a tuple $\Pi = ({\ensuremath{O}\xspace}, H, \mu, w_1, \ldots, w_m, R)$ where,
1. $m \geq 1$ is the initial number of membranes;
2. [$O$]{}is the alphabet of objects;
3. $H$ is the finite set of labels for the membranes;
4. $\mu$ is a membrane structure in the form of a tree, consisting of $m$ membranes (nodes), labelled with elements of $H$. The parent of all membranes (the root node) is called the “environment” and has label $env \in H$;
5. $w_1, \ldots, w_m$ are strings over [$O$]{}, describing the multisets of objects placed in the $m$ regions of $\mu$.
6. $R$ is a finite set of developmental rules, of the following forms:
1. $[\ a\ \rightarrow\ u\ ]_h$, for $h \in H,\ a \in {\ensuremath{O}\xspace},\ u \in {\ensuremath{O}\xspace}^{*}$
2. $a[\ ]_h \rightarrow [\ b\ ]_h$, for $h \in H,\ a,b \in {\ensuremath{O}\xspace}$
3. $[\ a\ ]_h \rightarrow [\ ]_h\ b$, for $h \in H,\ a,b \in {\ensuremath{O}\xspace}$
4. $[\ a\ ]_h \rightarrow b$, for $h \in H,\ a,b \in {\ensuremath{O}\xspace}$
5. $[\ a\ ]_h \rightarrow [\ b\ ]_h\ [\ c\ ]_h$, for $h \in H,\ a,b,c \in {\ensuremath{O}\xspace}$.
6. $[\ a\ [\ ]_{h_1}\ [\ ]_{h_2}\ [\ ]_{h_3}\ ]_{h_0} \rightarrow [\ b\ [\ ]_{h_1}\ [\ ]_{h_3} ]_{h_0}\ [\ c \ [\ ]_{h_2}\ [\ ]_{h_3} ]_{h_0}$,\
for $h_0, h_1, h_2, h_3 \in H,\ a,b,c \in {\ensuremath{O}\xspace}$.
These rules are applied according to the following principles:
- All the rules are applied in a maximally parallel manner. That is, in one step, one object of a membrane is used by at most one rule (chosen in a non-deterministic way), but any object which can evolve by one rule of any form, must evolve.
- If at the same time a membrane labelled with $h$ is divided by a rule of type ($e$) or ($f$) and there are objects in this membrane which evolve by means of rules of type ($a$), then we suppose that first the evolution rules of type ($a$) are used, and then the division is produced. This process takes only one step.
- The rules associated with membranes labelled with $h$ are used for membranes with that label. At one step, a membrane can be the subject of only one rule of types ($b$)–($f$).
- Rules of type ($f$) are division rules for non-elementary membranes. These rules allow us duplicate an entire branch of the membrane structure in the following manner. If the membrane (label $h_0$) to which the non-elementary division rule is applied contains objects and child membranes then copies of those membranes and all of their contents (including their own child membranes) are found in both resulting copies of $h_0$.
Recogniser membrane systems
---------------------------
In this paper one of our goals is to unify and clarify definitions for language recognising variants of membrane systems. To achieve this, we consider three different notions of acceptance for recogniser systems, one in each of Sections \[ssec:general\_rec\_sys\] to \[ssec:strict\_rec\_sys\]. Each of these three definitions is a restriction on the general (and purposely vague) Definition \[def:vague\_recogniser\] below.
We recall from [@NJNC2006p] that a computation of the system is a sequence of configurations such that each configuration (except the initial one) is obtained from the previous one by a transition. A computation that reaches a configuration where no more rules can be applied to the existing objects and membranes is called a halting computation.
\[def:vague\_recogniser\] A [*recognizer membrane system*]{} is a membrane system with external output (that is, the results of halting computations are encoded in the environment) such that:
1. the working alphabet contains two distinguished elements [[yes]{}]{}and [[no]{}]{};
2. if $C$ is a computation of the system, then it is either an accepting or a rejecting computation.
This definition is vague since we have not defined accepting and rejecting computations. In Section \[sec:different\_accepting\] we show the set of problems that a membrane system accepts when using various notions of accepting (or rejecting) computations.
Complexity classes
------------------
Consider a decision problem $X$, i.e. a set of instances $X = \left\{x_1,x_2,\ldots \right\}$ over some finite alphabet such that to each $x_i$ there is an unique answer “yes” or “no”. We say that a [*family*]{} of membrane systems solves a decision problem if each instance of the problem is solved by some family member. We denote by $|x| = n$ the length of any instance $x \in X$. Throughout this paper, $\AC^0$ circuits are $\DLOGTIME$-uniform, polynomial sized (in input length $n$), constant depth, circuits with AND, OR and NOT gates, and unbounded fanin [@BIS1990p]. The complexity class $\L$ ($\NL$) is the set of problems solved by (non-)deterministic Turing machines using only $O(\log n)$ space, where $n$ is the length of the input instance.
\[def:uniformFam\] Let $\mathcal{D}$ be a class of membrane systems and let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a total function. The class of [*problems solved by $\AC^0$-uniform families of membrane systems*]{} of type $\mathcal{D}$ in time $f$, denoted $(\AC^0)\text{--}{\bf MC}_{\mathcal{D}}(f)$, contains all problems $X$ such that:
- There exists an $\AC^0$-[*uniform*]{} family of membrane systems, $\bm{\Pi}_X = (\Pi_X(1),\Pi_X(2),\ldots)$ of type $\mathcal{D}$: that is, there exists an $\AC^0$ circuit family such that on unary input $1^n$ the $n^{\mathrm{th}}$ member of the circuit family constructs $\Pi_X(n)$.
- There exists an $\AC^0$ circuit family such that on input $x \in X$, of length $|x|=n$, the $n^{\mathrm{th}}$ member of the family encodes $x$ as a multiset of input objects placed in the distinct input membrane of $\Pi_{X}(n)$.
- $\bm{\Pi}_X$ is [*sound*]{} and [*complete*]{} with respect to problem $X$: $\Pi_X(n)$ starting with an encoding of input $x \in X$ of length $n$ accepts iff the answer to $x$ is “yes”.
- $\bm{\Pi}_X$ is $f$-efficient: $\Pi_X(n)$ always halts in at most $f(n)$ steps.
Definition \[def:uniformFam\] describes $\AC^{0}$-uniform families and we generalise this to define [*$\AC^{0}$-semi-uniform families of membrane systems*]{} $\bm{\Pi}_X =$ $(\Pi_X(x_1);$ $\Pi_X(x_2);\ldots)$ where there exists an $\AC^0$ circuit family which, on an input $x \in X$, constructs membrane system $\Pi_X(x)$. Here a single circuit family (rather than two) is used to construct the semi-uniform membrane family, and so the problem instance is encoded using objects, membranes, and rules. In this case, for each instance of $x \in X$ we have a special membrane system which does not need a separately constructed input. The resulting class of problems is denoted by $(\AC^0)\text{--}\mathbf{MC}^{*}_{\mathcal{D}}(f)$. Obviously, $(\AC^0)\text{--}\mathbf{MC}_{\mathcal{D}}(f) \subseteq (\AC^0)\text{--}\mathbf{MC}^*_{\mathcal{D}}(f)$ for any given class $\mathcal{D}$ and a valid [@BDG1988x] complexity function $f$.
We define $(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}}\xspace}_{\mathcal{D}}$ and $(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{D}}$ as $$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}}\xspace}_{\mathcal{D}} = \bigcup\limits_{k\in\mathbb{N}}(\AC^0)\text{--}\mathbf{MC}_{\mathcal{D}}(n^{k}),$$\
and $$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{D}} = \bigcup\limits_{k\in\mathbb{N}}(\AC^0)\text{--}\mathbf{MC}^{*}_{\mathcal{D}}(n^{k}).$$ In other words, $(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}}\xspace}_{\mathcal{D}}$ (and $(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{D}}$) is the class of problems solvable by uniform (respectively semi-uniform) families of membrane systems in polynomial time. We let $\mathcal{AM}^{0}$ denote the class of membrane systems with active membranes and no charges. We let ${\ensuremath{(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}}\xspace}$ denote the class of problems solvable by $\AC^0$-semi-uniform families of membrane systems in polynomial time with no dissolution rules. In an abuse of notation, we often let [$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}$]{}refer to the class of such membrane systems (rather than problems). For brevity we often write $\Pi_X$ instead of $\Pi_{X}(n)$ or $\Pi_{X}(x)$.
\[rem:confluent\] A membrane system is [*confluent*]{} if it is both sound and complete. That is a $\Pi_X$ is [*confluent*]{} if all computations of $\Pi_X$ with the same input give the same result; either always accepting or else always rejecting.
In a confluent membrane system, given a fixed initial configuration, the system non-deterministically chooses one from a number of valid configuration sequences, but all of the reachable configuration sequences must lead to the same result, either all accepting or all rejecting.
Dependency graphs and normal forms
----------------------------------
The [*dependency graph*]{} (first introduced by Guti[é]{}rrez-Naranjo et al. [@NJNC2006p]) is an indispensable tool for characterising the computational complexity of membrane systems without dissolution. This technique is reminiscent of configuration graphs for Turing Machines. Similarly to a configuration graph, a dependency graph helps visualise a computation. However, it differs in its approach by representing a membrane system configuration as a set of nodes rather than as a single node in configuration space.
Looking at membrane systems without dissolution as dependency graphs allows us to employ the existing, mature corpse of techniques and complexity results for graph problems. As we show in this paper, this greatly simplifies the process of proving upper and lower bounds for such systems. A key technique we use in this paper is to transfer from a dependency graph to a new membrane system, $\Pi \rightarrow {\ensuremath{\mathcal{G}}}_{\Pi} \rightarrow \Pi_{{\ensuremath{\mathcal{G}}}_{\Pi}}$. This new system accepts iff the original membrane system accepts, since their dependency graphs are isomorphic. Also, the new system is considerably simplified as it uses only one membrane (the environment) and all rules are of type ($a$). This is used as a normal form for membrane systems without dissolution.
In Sections \[ssec:general\_rec\_sys\] to \[ssec:strict\_rec\_sys\] we define reachability problems for dependency graphs such that if the answer to the graph reachability problem is yes, then the membrane system it represents is an accepting system. This is because the nodes of a dependency graph represent an object being in a certain membrane, and an edge between two nodes represents a developmental rule that causes that object to be in that membrane. Thus if the object ${{\tt yes}\xspace}$ arrives in the environment (the acceptance signal) of the membrane system, then there is a directed path leading from one special node (${{\tt in}\xspace}$) to another special node (${{\tt yes}\xspace}$) in the dependency graph. For more details about how a dependency graph is constructed and its proof of correctness see Guti[é]{}rrez-Naranjo et al. [@NJNC2006p; @GPRR2006c].
The dependency graph for a membrane system $\Pi$ is a directed graph ${\ensuremath{\mathcal{G}}}= ({\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}, {\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}, {{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace})$ where ${{\tt in}\xspace}\subseteq {\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}$ represents the input multiset, and ${{\tt yes}\xspace}, {{\tt no}\xspace}\in {\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}$, represent the accepting and rejecting signals respectively. Each vertex $a \in {\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}$ is a pair $a=({\ensuremath{o}\xspace}, h) \in {\ensuremath{O}\xspace}\times H$, where ${\ensuremath{O}\xspace}$ is the set of objects in $\Pi$ and $H$ is the set of membrane labels in $\Pi$. An edge $(a,b)$ exists iff there is a developmental rule in $\Pi$ such that the left hand side of the rule has the same object-membrane pair as $a$ and the right hand side has an object-membrane pair matching $b$. Since there is no membrane dissolution allowed, the parent/child relationships of membranes does not change during the computation. This allows us to determine the correct parent and child membranes for type ($b$) and type ($c$) rules.
Previously [@NJNC2006p], the graph ${\ensuremath{\mathcal{G}}}$ was constructed from $\Pi$ in polynomial time. We make the observation that the graph ${\ensuremath{\mathcal{G}}}$ can be constructed in $\AC^0$. We use a common circuit technique known as “masking” whereby using AND gates and a desired pattern we filter out the bits of the input string that we are interested in. We take as input a binary string $x$ that encodes a membrane system, $\Pi$. To make a dependency graph from a membrane system requires a constant number of parallel steps that are as follows. First, a row of circuits identifies all type ($b$) and ($c$) rules and uses the membrane structure to determine the correct parent membranes, then writes out (a binary encoding of) edges representing these rules. Next, a row of circuits writes out all edges representing type ($e$) and ($f$) rules (see [@NJNC2006p] for more details about the representation of these rules in dependency graphs). For ($a$) rules it is possible to have polynomially many copies of polynomially many distinct objects on the right hand side of a rule. To write out edges for these rules in constant time we take advantage of the fact that we require at most one edge for each object-membrane pair in ${\ensuremath{O}\xspace}\times H$. We have a circuit for each element of $\{ o_h \mid o \in O, h \in H \}$. The circuit for $o_h$ takes as input (an encoding of) all rules in $R$ whose left hand side is of the form $ [ o ]_h $. The circuit then, in a parallel manner, masks (an encoding of) the right hand side of the rule (for example $ [ bbcdc ]_{h}$) with the encoding of each object in ${\ensuremath{O}\xspace}$, (in the example, masking for (encoded) $b$ would produce (encoded) $bb000$). All encoded objects in the string are then ORed together so that if there was at least one copy of that object in the system we obtain a single instance of it. The circuit being unique for a specific left hand side $[ o ]_h$ now writes out an encoding of the edge $(o_h, b_h)$ and an encoding of all other edges for objects that existed on the right hand side of this rule in parallel.
\[rem:dg\_to\_memsys\] Of course one can take the opposite view. We observe that to convert a dependency graph ${\ensuremath{\mathcal{G}}}=({\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}},{\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}, {{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace})$ into a new membrane system, $\Pi_{\ensuremath{\mathcal{G}}}$, we simply convert the edges of the graph into object evolution rules. The set of objects of $\Pi_{\ensuremath{\mathcal{G}}}$ is $O_{\ensuremath{\mathcal{G}}}= {\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}$. The rules of $\Pi_{\ensuremath{\mathcal{G}}}$ are$\left\{ \left[\,v \rightarrow S(v) \right]_{env} \vert\ \forall\ v \in {\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}\right\}$ where $ S(v) = \left\{ s \in {\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}\vert (v,s) \in {\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}\right\}$. The nodes ${{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace}$ become the input multiset, ${{\tt yes}\xspace}$ object, and ${{\tt no}\xspace}$ object respectively. We compute this in $\AC^0$.
This new membrane system, $\Pi_{\ensuremath{\mathcal{G}}}$, highlights some points about active membrane systems without dissolution. These give rise to significant simplifications and normal forms.
\[lem:pmcwodiss\_one\_mem\] Any ${\ensuremath{(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}}\xspace}$, $\Pi$, with $m$ membranes can be simulated by a [$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}$]{}system, $\Pi'$, that (1) has no membranes other than the environment and (2) uses only rules of type ($a$).
By [*simulate*]{} we mean that the latter system accepts on input ${{\tt in}\xspace}$ iff the former does. To see that Lemma \[lem:pmcwodiss\_one\_mem\] holds, first notice how the dependency graph represents an (object, label) pair as a single node. Also if we convert the dependency graph ${\ensuremath{\mathcal{G}}}$ into a membrane system $\Pi_{{\ensuremath{\mathcal{G}}}}$, (1) it uses a single membrane with label [*env*]{}, and each node is modelled by a single object. (2) Each edge in ${\ensuremath{\mathcal{G}}}$ becomes a rule of type ($a$). Notice that the dependency graphs of $\Pi$ and $\Pi_{{\ensuremath{\mathcal{G}}}}$ are isomorphic.
\[lem:pmcwodiss\_multi\] Any [$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}$]{}system, $\Pi$, which has, as usual, multisets of objects in each membrane can be simulated by another [$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}$]{}system, $\Pi'$, which has sets of objects in each membrane.
We verify Lemma \[lem:pmcwodiss\_multi\] by observing that in a dependency graph, ${\ensuremath{\mathcal{G}}}$, the multiset of objects is encoded as a [*set*]{} of vertices, no information is kept regarding object multiplicities. Thus when ${\ensuremath{\mathcal{G}}}$ is converted into a new membrane system, $\Pi_{{\ensuremath{\mathcal{G}}}}$, there are no rules with a right hand side with more than one instance of each object. The resulting system $\Pi_{\ensuremath{\mathcal{G}}}$ accepts iff $\Pi$ accepts since the dependency graphs of both systems are isomorphic. Thus object multiplicities do not affect whether the system accepts or rejects.
Three different acceptance conditions {#sec:different_accepting}
=====================================
Here we present three different acceptance conditions for membrane systems with active membranes and show what complexity class they characterise. We define each acceptance condition; define a graph reachability problem that models the computation of such a system; then prove both upper and lower bounds on the computational power of the system. Each of Definitions \[def:general\_recogniser\_membranes\], \[def:recogniser\_membranes\], \[def:strict\_recogniser\_membranes\], is a more concrete replacement for Definition \[def:vague\_recogniser\]. Most results in this section use reductions to and from reachability problems on membrane dependency graphs. Solving these reachability problems is equivalent to simulating such a membrane system since we translate (via $\AC^0$ reductions) from a membrane system to a corresponding reachability problem, and vice-versa.
General recogniser systems characterise $\NL$ {#ssec:general_rec_sys}
---------------------------------------------
In previous works [@MW2008c; @MW2007p] we used a definition of recogniser membrane systems that is more general than is typical of other work in the area (i.e. Section \[ssec:standard\_rec\_sys\]). In this more general definition it is possible for the membrane system to output both [[yes]{}]{}and [[no]{}]{}symbols. However, when the first of these symbols is produced we call it the accepting/rejecting step of the computation. (Note that it is forbidden for both [[yes]{}]{}and [[no]{}]{}to be produced in the same timestep.) We now define this acceptance condition and then go on to show that ${\ensuremath{(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}}\xspace}$ systems with this acceptance condition characterise $\NL$.
\[def:general\_recogniser\_membranes\] A [*general recognizer membrane system*]{}, $\Pi$, is a membrane system with external output (that is, the results of halting computations are encoded in the environment) such that:
1. the working alphabet contains two distinguished elements [[yes]{}]{}and [[no]{}]{};
2. if $C$ is a computation of the system, (i) then a [[yes]{}]{}or [[no]{}]{}object is released into the environment, (ii) but not in the same timestep. If ${{\tt yes}\xspace}$ is released before ${{\tt no}\xspace}$ then the computation is accepting, otherwise the computation is rejecting.
![An example dependency graph ${\ensuremath{\mathcal{G}}}$ for some unspecified [*general recogniser membrane system*]{} (Definition \[def:general\_recogniser\_membranes\]). Note that this represents a rejecting computation since the minimum directed path from ${{\tt in}\xspace}$ to ${{\tt no}\xspace}$ is of length 6, while the minimum directed path from ${{\tt in}\xspace}$ to ${{\tt yes}\xspace}$ is of length 7. []{data-label="fig:DG_of_general_semi_recogniser"}](genrec.pdf)
We now define the reachability problem for ${\ensuremath{(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}}\xspace}$ systems whose acceptance conditions are as in Definition \[def:general\_recogniser\_membranes\]. Solving this problem is equivalent (via a reduction) to simulating such a system.
\
**Instance:** A dependency graph ${\ensuremath{\mathcal{G}}}= \left({\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}, {\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}, {{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace}\right)$ where $\left\{ {{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace}\right\} \subseteq {\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}$, representing the rules of a general recogniser membrane system $\Pi$ as defined in Definition \[def:general\_recogniser\_membranes\].\
**Problem:** Is the shortest directed path from ${{\tt in}\xspace}$ to ${{\tt yes}\xspace}$ of length less than the shortest directed path from ${{\tt in}\xspace}$ to ${{\tt no}\xspace}$?
We also define the problem $\STCON$, the canonical $\NL$-complete problem [@Jon1975p]. This problem is also known as , , and .
\
**Instance:** A directed acyclic graph $G=(V,E, s, t)$ where $\left\{s, t\right\} \subseteq V$.\
**Problem:** Is there a directed path in $G$ from $s$ to $t$?
We now provide a result which is used to show that [$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}$]{}systems whose acceptance conditions are as in Definition \[def:general\_recogniser\_membranes\] characterise $\NL$ (this characterisation has been published elsewhere [@MW2008c], we present a shorter proof here).
\[thm:genrec\_is\_NLc\] $\GENERALREACH$ is $\NL$-complete
First we show $\STCON \leq_{\AC^0} \GENERALREACH$. Given an instance $G=\left(V, E, s, t\right)$ of $\STCON$, we construct a dependency graph ${\ensuremath{\mathcal{G}}}= \left({\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}, {\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}, {{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace}\right)$ such that ${\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}= V \cup \left\{{{\tt no}\xspace}\right\}$ and ${\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}= E$. We replace all instances of $s$ with ${{\tt in}\xspace}$, and $t$ with ${{\tt yes}\xspace}$, in ${\ensuremath{\mathcal{G}}}$. Clearly there is a path from ${{\tt in}\xspace}$ to ${{\tt yes}\xspace}$ iff there is a path from $s$ to $t$ in ${\ensuremath{\mathcal{G}}}$. We also add a directed path of length $|V|+1$ from ${{\tt in}\xspace}$ to ${{\tt no}\xspace}$ in ${\ensuremath{\mathcal{G}}}$. This ensures that if there is not a path from $s$ to $t$ in $G$, than ${{\tt no}\xspace}$ is reached after all other paths have terminated. This reduction is computed in $\AC^0$.
We now prove the correctness of the above reduction. Since $\GENERALREACH$ is defined in terms of the general recogniser membrane systems (Definition \[def:general\_recogniser\_membranes\]), we often appeal to Definition \[def:general\_recogniser\_membranes\] in the proof. Recall that, via Remark \[rem:dg\_to\_memsys\], we can translate ${\ensuremath{\mathcal{G}}}$ to a membrane system $\Pi_{\ensuremath{\mathcal{G}}}$ in $\AC^0$.
- By adding a path of length $|V|+1$ from ${{\tt in}\xspace}$ to ${{\tt no}\xspace}$ we are guaranteeing that object ${{\tt no}\xspace}$ is not produced by the membrane system $\Pi_{\ensuremath{\mathcal{G}}}$ at the same time as [*any*]{} other object, this satisfies point 2(ii) of Definition \[def:general\_recogniser\_membranes\].
- If there is a path from $s$ to $t$ in $G$ (and ${{\tt yes}\xspace}$ is evolved in $\Pi_{{\ensuremath{\mathcal{G}}}}$) the reduction ensures that a path from ${{\tt in}\xspace}$ to ${{\tt yes}\xspace}$ exists in ${\ensuremath{\mathcal{G}}}$. Also in either case a path from ${{\tt in}\xspace}$ to ${{\tt no}\xspace}$ is created by the reduction that ensures the correct output from $\Pi_{\ensuremath{\mathcal{G}}}$. Thus we satisfy point 2(i) of Definition \[def:general\_recogniser\_membranes\].
We now show that $\GENERALREACH \in \NL$. Let $M$ be a non-deterministic Turing machine with two variables $x$ and $y$. Finding the shortest path between two nodes is well known to be computable in $\NL$ via $\leq n$ iterations of a $\STCON$ algorithm. Set $x$ to be the shortest path from ${{\tt in}\xspace}$ to ${{\tt yes}\xspace}$. Set $y$ to be the shortest path from ${{\tt in}\xspace}$ to ${{\tt no}\xspace}$. If $x < y$, $M$ accepts, otherwise $M$ rejects. Thus $M$ uses a non-deterministic algorithm and two binary counters to solve $\GENERALREACH$ and so the problem is in $\NL$.
$\NL$ is characterised by ${\ensuremath{(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}}\xspace}$ using the [*general*]{} acceptance conditions from Definition \[def:general\_recogniser\_membranes\].
The proof is omitted, but can be obtained by using standard techniques along with Remark \[rem:dg\_to\_memsys\], Theorem \[thm:genrec\_is\_NLc\], and Definition \[def:general\_recogniser\_membranes\].
Standard recogniser membrane systems characterise $\NL$ {#ssec:standard_rec_sys}
-------------------------------------------------------
In this section we discuss the “standard” definition for recogniser membrane systems, i.e. the definition that most researchers use when proving results about recogniser membrane systems. On a given input, these systems produce either a [[yes]{}]{}object or a [[no]{}]{}object, but not both. Also it is assumed that this occurs in the last timestep of the computation where no other rules are applicable.
By showing an $\NL$ characterisation for such systems, we are showing that this definition has equal power to the more general definition discussed above in Section \[ssec:general\_rec\_sys\]. Furthermore, we have provided a “compiler,” via reductions, to translate a system that uses the general definition into a system that uses the standard definition. This is significant since the general definition is often easier to program, while it is often easier to prove certain properties (such as correctness) for the standard definition. We begin with a definition of standard recogniser membrane systems from Guti[é]{}rrez-Naranjo et al. [@NJNC2006p].
\[def:recogniser\_membranes\] A [*recognizer membrane system*]{}, $\Pi$, is a membrane system with external output (that is, the results of halting computations are encoded in the environment) such that:
1. the working alphabet contains two distinguished elements [[yes]{}]{}and [[no]{}]{};
2. all computations halt; and
3. if $C$ is a computation of the system, then (i) either object [[yes]{}]{}or object [[no]{}]{}(but not both) must have been released into the environment, and (ii) only in the last step of the computation. If ${{\tt yes}\xspace}$ is released then the computation is accepting, otherwise the computation is rejecting.
Definition \[def:recogniser\_membranes\] affects the dependency graph of such systems so that we can define the following subsets of the objects [$O$]{}.\
${\ensuremath{O}\xspace}_{{{\tt yes}\xspace}} = \{ {\ensuremath{o}\xspace}\ \vert\ {\ensuremath{o}\xspace}\in {\ensuremath{O}\xspace}\text{ and $o$ eventually evolves } {{\tt yes}\xspace}\}$,\
${\ensuremath{O}\xspace}_{{{\tt no}\xspace}} = \{ {\ensuremath{o}\xspace}\ \vert\ {\ensuremath{o}\xspace}\in {\ensuremath{O}\xspace}\text{ and $o$ eventually evolves } {{\tt no}\xspace}\}$, and ${\ensuremath{O}\xspace}_{\mathrm{other}} = {\ensuremath{O}\xspace}\backslash ({\ensuremath{O}\xspace}_{{{\tt yes}\xspace}} \cup {\ensuremath{O}\xspace}_{{{\tt no}\xspace}})$.
\[lem:yes\_no\_intersect\_null\] ${\ensuremath{O}\xspace}_{{{\tt yes}\xspace}} \cap {\ensuremath{O}\xspace}_{{{\tt no}\xspace}} = \emptyset $.
Assume that object ${\ensuremath{o}\xspace}\in {\ensuremath{O}\xspace}_{{{\tt yes}\xspace}} \cap {\ensuremath{O}\xspace}_{{{\tt no}\xspace}}$, this implies that both a [[yes]{}]{}and a [[no]{}]{}object are produced by the confluent system on a given input which contradicts point 3(i) of Definition \[def:recogniser\_membranes\].
These observations are illustrated in Figure \[fig:DG\_of\_standard\_semi\_recogniser\].
![An example dependency graph ${\ensuremath{\mathcal{G}}}$ for some unspecified standard recogniser membrane system (Definition \[def:recogniser\_membranes\]). Note that via Lemma \[lem:yes\_no\_intersect\_null\] there are no directed paths from ${\ensuremath{O}\xspace}_{{{\tt yes}\xspace}}$ to ${\ensuremath{O}\xspace}_{{{\tt no}\xspace}}$, they are [*weakly connected*]{}. []{data-label="fig:DG_of_standard_semi_recogniser"}](stdrec.pdf)
We now define the reachability problem for ${\ensuremath{(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}}\xspace}$ systems whose acceptance conditions are as in Definition \[def:recogniser\_membranes\]. We remind the reader that these systems are confluent via Definition \[def:uniformFam\] and Remark \[rem:confluent\].
\
**Instance:** A dependency graph ${\ensuremath{\mathcal{G}}}= \left({\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}, {\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}, {{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace}\right)$ where $\{ {{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace}\} \subseteq {\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}$, representing the rules of a [$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}$]{}recogniser membrane system $\Pi$ as defined in Definition \[def:recogniser\_membranes\].\
**Problem:** Is there a directed path from ${{\tt in}\xspace}$ to ${{\tt yes}\xspace}$?
We now provide the main result needed to show that standard [$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}$]{}characterises $\NL$.
\[thm:stdrec\_is\_NLc\] $\STANDARDREACH$ is $\NL$-complete.
First we show $\STCON \leq_{\AC^0} \STANDARDREACH$. Given an instance $G=\left(V, E, s, t\right)$ of $\STCON$, we construct a dependency graph ${\ensuremath{\mathcal{G}}}= \left({\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}, {\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}, {{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace}\right)$ such that ${\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}= V \cup \left\{ {{\tt yes}\xspace}, {{\tt no}\xspace}\right\}$ and ${\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}= E$. We replace $s$ with ${{\tt in}\xspace}$ in [$\mathcal{G}$]{}. We add a directed path of $|V|+1$ edges leading from $t$ to ${{\tt yes}\xspace}$ to ensure that all other computations have halted before [[yes]{}]{}is evolved. Clearly there is a path from ${{\tt in}\xspace}$ to ${{\tt yes}\xspace}$ in ${\ensuremath{\mathcal{G}}}$ iff there is a path from $s$ to $t$ in graph $G$.
So far, ${\ensuremath{\mathcal{G}}}$ we have shown that ${\ensuremath{(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}}\xspace}$ recogniser membrane systems, as in Definition \[def:recogniser\_membranes\], [*accept*]{} words in $\STCON$. However, the construction does not explicitly say how to reject words that are not in the language, which is a requirement of Definition \[def:recogniser\_membranes\]. We extend the proof as follows. Let $\NOTSTCON$ be the complementary problem to $\STCON$, i.e. given an acyclic graph $G'$ is there no directed path from $s'$ to $t'$? $\NOTSTCON$ is $\coNL$-complete (via the same reduction that is used to show the $\NL$-completeness of $\STCON$), and so is also $\NL$-complete (since $\NL=\coNL$ [@Imm1988p; @Sze1987p]). Now we define a third $\NL$-complete problem $\STCON\text{--}\NOTSTCON$; the set of graphs with two disjoint components $G, G'$ that are related in the following sense: $s$ eventually yields $t$ in $G$ iff $s'$ does not eventually yield $t'$ in $G'$. Now we reduce this graph to a dependency graph ${\ensuremath{\mathcal{G}}}$ in a similar manner as the above reduction. That is, we place an edge from ${{\tt in}\xspace}$ to $s$ and from ${{\tt in}\xspace}$ to $s'$. We add a directed path of $|V|+1$ edges leading from $t$ to ${{\tt yes}\xspace}$, and another directed path of $|V|+1$ edges leading from $t'$ to ${{\tt no}\xspace}$. Then the induced membrane system $\Pi_{\ensuremath{\mathcal{G}}}$ correctly decides $\STCON\text{--}\NOTSTCON$ since it answers [[yes]{}]{}iff $s$ leads to $t$, otherwise it answers [[no]{}]{}. This reduction is computed in $\AC^0$.
We now prove the correctness of the above reduction. Recall that, via Remark \[rem:dg\_to\_memsys\], we translate ${\ensuremath{\mathcal{G}}}$ to a membrane system $\Pi_{\ensuremath{\mathcal{G}}}$ in $\AC^0$.
- Since an instance of $\STCON\text{--}\NOTSTCON$ is an acyclic graph we trivially satisfy point 2 of Definition \[def:recogniser\_membranes\].
- In the induced membrane system $\Pi_{\ensuremath{\mathcal{G}}}$ the node [[in]{}]{}can only lead to one of [[yes]{}]{}or [[no]{}]{}, but not both, since the embedded $\STCON$ and $\NOTSTCON$ problems are complementary. This satisfies point 3(i) of Definition \[def:recogniser\_membranes\].
- $\Pi_{\ensuremath{\mathcal{G}}}$ outputs (either [[yes]{}]{}or [[no]{}]{}) in the last step because we add $|V|+1$ extra edges from $t$ and $t'$ so that the accepting or rejecting path is the longest in the dependency graph, satisfying point 3(ii) of Definition \[def:recogniser\_membranes\].
Now we show that ${\ensuremath{(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}}\xspace}$, as in Definition \[def:recogniser\_membranes\], can recognise no more than $\NL$ by showing that $\STANDARDREACH \leq_{\AC^0} \STCON$. We observe that an instance of $\STANDARDREACH$ is a directed acyclic graph (via point 2 of Definition \[def:recogniser\_membranes\]). Given an instance ${\ensuremath{\mathcal{G}}}= \left({\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}, {\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}, {{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace}\right)$ of , we construct $G=\left(V, E, s,t\right)$ such that $V = {\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}$ and $E = {\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}$ and replace all instances of ${{\tt in}\xspace}$ with $s$ and ${{\tt yes}\xspace}$ with $t$ in ${\ensuremath{\mathcal{G}}}$. Clearly there is a path from $s$ to $t$ in $G$ iff there is a path from ${{\tt in}\xspace}$ to ${{\tt yes}\xspace}$ in the dependency graph ${\ensuremath{\mathcal{G}}}$. This reduction is computed in $\AC^0$.
$\NL$ is characterised by ${\ensuremath{(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}}\xspace}$ using the [*standard*]{} acceptance conditions from Definition \[def:recogniser\_membranes\].
The proof is omitted, but can be obtained by using standard techniques along with Remark \[rem:dg\_to\_memsys\], Theorem \[thm:stdrec\_is\_NLc\], and Definition \[def:recogniser\_membranes\].
Restricted recogniser membrane systems characterise $\L$ {#ssec:strict_rec_sys}
--------------------------------------------------------
We now consider a restriction on the standard definition of recogniser membrane systems. Above in Section \[ssec:standard\_rec\_sys\], we forbid an object that eventually yielded a [[yes]{}]{}from also yielding a [[no]{}]{}(and vice versa). Now we further restrict the system and require that [*all*]{} descendent nodes of ${{\tt in}\xspace}$ must eventually yield [[yes]{}]{}, or all must eventually yield [[no]{}]{}. Notice that this restriction forbids objects that do not contribute to the final answer (accept or reject) and forbids rules of the form $\left[ a \rightarrow \lambda \right]$ where $\lambda$ is the empty word.
\[def:strict\_recogniser\_membranes\] A [*restricted recogniser membrane system*]{}, $\Pi$, is a membrane system with external output (that is, the results of halting computations are encoded in the environment) such that:
1. the working alphabet contains two distinguished elements [[yes]{}]{}and [[no]{}]{};
2. all computations halt;
3. if $C$ is a computation of the system, then (i) either object [[yes]{}]{}or object [[no]{}]{}(but not both) must have been released into the environment, and (ii) only in the last step of the computation. If ${{\tt yes}\xspace}$ is released then the computation is accepting, otherwise the computation is rejecting.
4. each object ${\ensuremath{o}\xspace}\in O$ must, via a sequence of zero or more developmental rules, lead to [[yes]{}]{}, or else lead to [[no]{}]{}, but not both.
The definition has the following effect on the dependency graph.
Since every object eventually yields exactly one [[yes]{}]{}, or exactly one [[no]{}]{}, the graph ${\ensuremath{\mathcal{G}}}$ consists of exactly two disjoint components.
![An example dependency graph ${\ensuremath{\mathcal{G}}}$ for some unspecified restricted recogniser membrane system (Definition \[def:strict\_recogniser\_membranes\]).[]{data-label="fig:DG_of_strict_semi_recogniser"}](resrec.pdf)
We now define a graph reachability problem for ${\ensuremath{(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}}\xspace}$ systems whose acceptance conditions are as in Definition \[def:strict\_recogniser\_membranes\].
\
**Instance:** A dependency graph ${\ensuremath{\mathcal{G}}}= \left({\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}, {\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}, {{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace}\right)$ where $\{ {{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace}\} \subseteq {\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}$, representing the rules of an [$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}$]{}recogniser membrane system $\Pi$ as defined in Definition \[def:strict\_recogniser\_membranes\].\
**Problem:** Is there a directed path from ${{\tt in}\xspace}$ to ${{\tt yes}\xspace}$?
We define the $\L$-complete problem [Directed Forest Accessibility]{} ($\DFA$) [@CMcK1987p].
\
**Instance:** An acyclic directed graph $G = (V, E, s, t)$ where $\{s,t \} \subseteq V$ and each node is of out-degree $0$ or $1$.\
**Property:** Is there a directed path from $s$ to $t$?
\[thm:rstreach\_lc\] $\RESTRICTEDREACH$ is $\L$-complete
First we show $\DFA \leq_{\AC^0} \RESTRICTEDREACH$. Given an instance $G=\left(V, E, s, t\right)$ of $\DFA$, we construct a dependency graph ${\ensuremath{\mathcal{G}}}= \left({\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}, {\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}, {{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace}\right)$ such that ${\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}= V \cup \left\{ {{\tt no}\xspace}\right\}$ and ${\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}= E \backslash \{(t,v) \vert v \in V\}$. We also replace $s$ with ${{\tt in}\xspace}$, and add a directed path of length $|V|+1$ from $t$ to ${{\tt yes}\xspace}$ in ${\ensuremath{\mathcal{G}}}$. Clearly there is a path from ${{\tt in}\xspace}$ to ${{\tt yes}\xspace}$ in ${\ensuremath{\mathcal{G}}}$ iff there is a path from $s$ to $t$ in graph $G$. Note that since we removed the edge (if it exists) leaving $t$, every computation halts (in the induced membrane system $\Pi_{{\ensuremath{\mathcal{G}}}}$) upon evolving ${{\tt yes}\xspace}$. We also add an edge from all nodes, except [[yes]{}]{}, of out-degree 0 to ${{\tt no}\xspace}$. There is now a path from ${{\tt in}\xspace}$ to ${{\tt no}\xspace}$ iff there is no path from $s$ to $t$ in $G$ because all paths that do not lead to ${{\tt yes}\xspace}$ now lead to ${{\tt no}\xspace}$. This reduction is computed in $\AC^0$.
We now prove the correctness of the above reduction. Recall that, via Remark \[rem:dg\_to\_memsys\], we translate ${\ensuremath{\mathcal{G}}}$ to a membrane system $\Pi_{\ensuremath{\mathcal{G}}}$ in $\AC^0$.
- Since $G$ (as a forest) is acyclic, our reduction ensures ${\ensuremath{\mathcal{G}}}$, and hence any computation of $\Pi_{\ensuremath{\mathcal{G}}}$, is acyclic also, satisfying point 2 of Definition \[def:strict\_recogniser\_membranes\].
- Our reduction ensures that exactly 2 nodes in ${\ensuremath{\mathcal{G}}}$ have out-degree 0, the (sink) nodes ${{\tt yes}\xspace}$ and ${{\tt no}\xspace}$, this implies that the only objects that have no applicable rules in $\Pi_{\ensuremath{\mathcal{G}}}$ are ${{\tt yes}\xspace}$ and ${{\tt no}\xspace}$. This satisfies points 1 and 3(ii) of Definition \[def:strict\_recogniser\_membranes\].
- Since every node in $G$ has out-degree 0 or 1, then every node in ${\ensuremath{\mathcal{G}}}$ has out-degree 0 or 1 (and every object in $\Pi_{\ensuremath{\mathcal{G}}}$ has 0 or 1 applicable developmental rules). Combined with the previous point, this implies that all nodes in ${\ensuremath{\mathcal{G}}}$ are on a path to either ${{\tt yes}\xspace}$ or ${{\tt no}\xspace}$, and that all objects in $\Pi_{\ensuremath{\mathcal{G}}}$ eventually yield either ${{\tt yes}\xspace}$ or ${{\tt no}\xspace}$, satisfying points 4 and 3(i) of Definition \[def:strict\_recogniser\_membranes\].
Now we show $\RESTRICTEDREACH$ is contained in $\L$ by outlining a deterministic logspace Turing machine $M$ that decides $\RESTRICTEDREACH$. The input tape of $M$ encodes an instance ${\ensuremath{\mathcal{G}}}= ({\ensuremath{V_{{\ensuremath{\mathcal{G}}}}}}, {\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}, {{\tt in}\xspace}, {{\tt yes}\xspace}, {{\tt no}\xspace})$ of $\RESTRICTEDREACH$. Starting with the input node ${{\tt in}\xspace}$, $M$ stores this node in a variable called $x$ on its work tape. If $x$ is neither ${{\tt yes}\xspace}$ nor ${{\tt no}\xspace}$ then $M$ searches the set of edges ${\ensuremath{E_{{\ensuremath{\mathcal{G}}}}}}$ on its input tape, upon finding an edge $(x,v)$, the machine sets $x$ to be $v$ (overwriting the previous value). The computation carries on in this fashion until either $x$ equals ${{\tt no}\xspace}$ causing $M$ to reject, or ${{\tt yes}\xspace}$, in which case $M$ accepts. The algorithm correctly decides $\RESTRICTEDREACH$ because each node in the data-structure has out-degree 0 or 1 and we simply trace along a path until we reach a sink. If the sink is [[yes]{}]{}, we accept, otherwise we reject. Since only one node is stored on $M$’s work tape at any time, $M$ uses $O(\log n)$ space (where $n$ is the input length). Thus $\RESTRICTEDREACH \in \L$.
$\L$ is characterised by [$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}$]{}using the [*restricted*]{} acceptance conditions from Definition \[def:strict\_recogniser\_membranes\].
The proof is omitted, but can be obtained by using standard techniques along with Remark \[rem:dg\_to\_memsys\], Theorem \[thm:rstreach\_lc\], and Definition \[def:strict\_recogniser\_membranes\].
Conclusions
===========
In this paper we have shown how the acceptance conditions of membrane systems affect the computational complexity of the system. We have presented an analysis of three different acceptance conditions and proved that they each characterise one of two logspace complexity classes, $\NL$ or $\L$.
In our previous work [@MW2008c] we used Definition \[def:general\_recogniser\_membranes\] as our acceptance condition. Systems using this definition are relatively easy to program (construct a membrane system to solve a problem) because one is not concerned with ensuring the system halts or that only ${{\tt yes}\xspace}$ or only ${{\tt no}\xspace}$ is output. However Definition \[def:recogniser\_membranes\] is the more common definition that is used when discussing active membrane systems as it is easier to prove correctness for these systems. The results in Sections \[ssec:general\_rec\_sys\] and \[ssec:standard\_rec\_sys\] reveal that when working with [$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}$]{}systems, both of Definitions \[def:general\_recogniser\_membranes\] and \[def:recogniser\_membranes\] characterise $\NL$. Our result gives an $\AC^0$ computable compiler to turn a system obeying one definition into a system obeying the other definition. This makes the choice of either definition a matter of taste and convenience.
We also have given the first complexity class defined by membrane systems that characterises $\L$. It is interesting to note that the rules of [$(\AC^0)\text{--}{\ensuremath{\mathbf{PMC}^{*}}\xspace}_{\mathcal{AM}^{0}_{-d}}$]{}systems allow for the generation of an exponential amount of objects and membranes. However these systems decide only those problems that a (non-)deterministic Turing machine uses logarithmic space to decide.
Here we looked at a number of acceptance conditions for active membrane systems and then characterised the computational complexity classes of the systems. However, it is also possible to go in the other direction, that is, to choose a complexity class and then try to engineer an acceptance condition in order to characterise the class. This technique may give rise to interesting new characterisations. Furthermore, we would hope that it may even be useful to help solve some open questions on the power of certain classes of membrane systems.
We intend to extend this research to see what effect, if any, acceptance conditions have on the complexity of [*uniform*]{} active membrane systems. The techniques may also prove useful for exploring other classes of membrane systems such as tissue P-systems.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Mario J. P[é]{}rez-Jim[é]{}nez and Agust[í]{}n Riscos-N[ú]{}[ñ]{}ez for interesting discussion and clarification on the standard definition of recogniser membrane systems. We would also like to thank Petr Sos[í]{}k for his comments on an earlier draft of this paper.
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[^1]: Funded by the Irish Research Council for Science, Engineering and Technology
[^2]: Funded by Junta de Andaluc[í]{}a grant TIC-581
[^3]: All membrane systems in this paper are $\AC^0$-uniform and run for polynomial time.
|
---
abstract: 'Nature inspired neuromorphic architectures are being explored as an alternative to imminent limitations of conventional complementary metal-oxide semiconductor (CMOS) architectures. Utilization of such architectures for practical applications like advanced pattern recognition tasks will require synaptic connections that are both reconfigurable and stable. Here, we report realization of stable atomic-switch networks (ASN), with inherent complex connectivity, self-assembled from percolating metal nanoparticles (NPs). The device conductance reflects the configuration of synapses which can be modulated via voltage stimulus. By controlling Relative Humidity (RH) and oxygen partial-pressure during NP deposition we obtain stochastic conductance switching that is stable over several months. Detailed characterization reveals signatures of electric-field induced atomic-wire formation within the tunnel-gaps of the *oxidized* percolating network. Finally we show that the synaptic structure can be reconfigured by stimulating at different repetition rates, which can be utilized as short-term to long-term memory conversion. This demonstration of stable stochastic switching in ASNs provides a promising route to hardware implementation of biological neuronal models and, as an example, we highlight possible applications in Reservoir Computing (RC).'
author:
- 'Saurabh K. Bose, Joshua B. Mallinson, Rodrigo M. Gazoni, and Simon A. Brown'
title: 'Stable Self-Assembled Atomic-Switch Networks for Neuromorphic Applications'
---
[^1]
Atomic switch networks, Clusters, Neuromorphic architecture
Introduction
============
astounding success of the von Neumann architecture for computers[@Goldstine1946], as encapsulated in Moore’s Law, is now meeting with fundamental limitations (physical transistor dimensions are approaching classical limits) and practical limitations (the exponential increase in research and development costs for every new process line)[@Taur1997; @Frank2002]. Natural information processing systems, like the biological brain, on the other hand, can perform highly complex computational tasks like navigation, recognition and decision-making with remarkable ease and with very low energy consumption[@Kandel2000]. This natural computation, processing the useful data (patterns) from a multitude of sensory information, is immediate and cannot be matched by even the most-advanced supercomputers[@Roska2005; @Wong2012]. Nature inspired architectures[@Tuma2016; @Bose2015; @Service2014; @Avizienis2012; @Stieg2012; @ohno2011; @Suri2013] are therefore currently being pursued as a disruptive alternative to the von Neumann architecture. A recent review on Neuromorphic architecture[@Mead1990a] and implementations can be found in Ref. [@Nawrocki2016].
![(a) Schematic depiction (top and side view) of the nanoparticle network between electrical contacts of our two-electrode devices showing tunnelling gaps in the Sn NP network. The outer shaded region on the NPs depicts the thin oxide layer. (b) Variation of the onset of conductance with increase in partial pressure $P_{dep}$ (over 2 orders of magnitude) during NP deposition. (c) This pressure variation results in shorter conduction onset times $t_0$ (left scale) with longer onset width $\triangle t_0$ (right scale). (d) Scanning electron micrograph of samples prepared at lowest ambient air pressure ($P_{dep}\sim$10 $\mu$Torr) shows more coalescence and larger grain-size in comparison to the highest pressure (600 $\mu$Torr) prepared samples shown in (e). (f) Use of dry synthetic air ($P_{dep}\sim$10 $\mu$Torr) produces a different microstructure with reduced sample stability as compared with ambient air. The white scale bars are 100 nm.[]{data-label="bose1"}](bose1.pdf){width="10"}
The alternative brain-inspired hardware approach must address three key issues simultaneously: mimic the complex biological network of neurons, replicate synaptic structures and allow implementation of standard computational algorithms[@Sterratt2011]. Achieving all of these goals is obviously enormously challenging and will require long-term research. Nevertheless significant progress has been made towards solving several different problems, using a variety of architectures. Proposals for non-CMOS approaches include those based on networks of memristors[@Indiveri2013; @Querlioz2013a; @Adam2017], atomic switches[@Avizienis2012; @Stieg2012] and Metal Oxide Resistive Random Access Memory (RRAM)[@Yu2011; @Park2016]. There have been interesting demonstrations of memristor-based neural networks[@Thomas2013], associative memory[@Pershin2010], synaptic emulators[@Wang2017], conditional programming[@Borghetti2009], reconfigurable logic[@Xia2009], solving mazes[@Pershin2011], pattern recognition[@Snider2007; @Alibart2013; @Chu2015; @Sheridan2014], and reservoir computing (RC)[@Konkoli2014; @Kulkarni2012a; @Sillin2013]. Even in the most heavily explored architectures (regular cross-bar arrays of memristors) [@Burr2017] there remain unsolved challenges in regard to realisation of the required properties of both individual switching elements and networks of these elements.
RC is simpler to implement than many other unconventional computation schemes since synapses do not need to be addressed individually and can be seen as an important step towards achieving other types brain-like computation. In RC a ‘reservoir’ comprising a complex network of switching elements allows the transformation of input signals into a higher dimensional space.[@Konkoli2014; @Kulkarni2012a; @Sillin2013] Training of a single ‘output layer’ then allows implementation of various time series prediction, pattern recognition and classification tasks. [@Fernando2003; @Choi2015; @Demis2016] Randomly assembled atomic switch networks (ASNs) based on sulphidised Ag nanowires [@Avizienis2012; @Stieg2012] and percolating films of nanoparticles [@Sattar2013; @Fostner2015] are immediately ammenable to RC. ASNs are also an appealing alternative to regular arrays of devices because they allow realisation of complexity similar to that of the brain and fabrication via self-assembly immediately circumvent the limitations of lithographic processing. Ag-based ASNs have recently been used to demonstrate a form of RC in which the non-linear properties of the reservoir allows generation of target waveforms, and a clear roadmap towards further implementation has been mapped out[@Demis2016].
Systems of inorganic synapses are however in the early stages of development with improvements required in production methods, reliability and actual functionality[@Burr2017]. While robust switching over 10,000 cycles has been reported for Ag-AgS nanowire systems[@Avizienis2012] a different, and a particularly important, issue for any real-world applications is long-term device stability, which is the main focus of the present work. Such stability has not been reported previously in either Ag-AgS nanowire [@Avizienis2012; @Stieg2012; @Sillin2013; @Stieg2014; @Demis2016] or percolating ASNs[@Sattar2013].
Here we report a straightforward fabrication procedure for realization of randomly connected ASNs within a percolating network of metal nanoparticles (NPs). We show that deliberate introduction of oxygen and moisture during NP deposition leads to long term device stability. Despite the presence of oxides, the switching mechanisms associated with increases(decreases) in device conductance $G_{\uparrow}$($G_{\downarrow}$) are shown to be formation (destruction) of atomic scale wires in tunnel gaps in the network. The two-terminal device conductance (*G*) quantifies the input-output electrode connectivity[@Fostner2015] and reflects the *synaptic configuration* of the network which can be reconfigured by voltage stimulation. Finally, we discuss the observed synaptic stochasticity and why it is useful for implementation of hardware analogues of the biological brain[@Rolls2010].
Experimental
============
The nano-cluster deposition system used in this study is based on magnetron sputtering to generate a vapour of the metal of interest and gas aggregation to condense the vapour into particles, and has been described in detail in previous publications[@reichel2006a; @Dunbar2006; @Ayesh2007]. The deposition scheme provides a narrow cluster size distribution[@reichel2006a] and allows precise control over the NP surface coverage near the percolation threshold[@stauffer], so that the system is poised near criticality[@Chialvo2010]. Sn NPs with mean diameter $\sim$ 8.5 nm are deposited between 50 nm thick Au/NiCr electrodes on Si$_3$N$_4$ substrates, with active area of 100 $\mu$m $\times$ 300 $\mu$m. The two-contact devices allow for a demonstration of network stability and associated dynamics, but samples with multiple contacts will be required for demonstration of RC.
The Sn NPs are deposited at room temperature which means that ordinarily the surface atoms have sufficient mobility to allow coalescence[@Yu1991]. For samples poised near the percolation threshold, the coalescence can lead to the loss of conducting pathways through the film, because neighboring particles that are initially joined by a fragile connection are pulled apart as they coalesce with other neighbor NPs, thus contributing to the short life-span ($\sim$ hours) of previous devices[@Sattar2013]. In the present work, the coalescence of the Sn NPs is controlled by partial oxidation (during NP deposition) via a controlled leak of air with a needle-valve. As will be shown, the controlled oxide formation leads to reduced coalescence and enhanced device stability. We emphasize that by ‘stability’ we do not mean that the device has a fixed conductance, but that the device is in a state in which it continues to exhibit multiple switching events in response to voltage stimuli.
![Stochastic and stable switching behaviour for sample prepared with $P_{dep} =$ 10 $\mu$Torr and high humidity (RH $\sim$ 80%) ambient air conditions, showing multi-level conductance switching, induced by application of both triangular and pulsed voltage stimulus. On the 1$^{st}$ day, immediately after sample fabrication, we use voltage-sweeps in order to check the voltage threshold for switching, as discussed in section \[Dynamics\]. On subsequent days we utilize controlled voltage pulses. As we tested the samples in a variety of ways at different times over a period of several months, the applied voltage in panel (d) is slightly different to that in the other panels. The sample exhibits qualitatively similar switching for several months of device operation.[]{data-label="bose2"}](bose2.pdf){width="10"}
Sample Fabrication
==================
Self-assembly of ASNs
---------------------
We have self-assembled interconnected and active network of atomic-switches as depicted in the schematic shown in Fig. \[bose1\](a). Deposition of the Sn NPs at Base Pressure (BP, $\sim 6 \mu$Torr) led to initial observation of a non-zero conductance (time $\emph{t}_0$) at around 800s with a sharp onset behaviour ($\triangle t_0$) $\sim$ 10s, as shown in Fig. 1(b). Here, we define the width of the onset $\triangle$t$_0$ as the time after *t*$_0$ required to reach a conductance of 6$G_0$ ($G_0 = 2e^2/h $ is the quantum of conductance[@VanWees1988]). The cluster deposition is stopped \[arrows in Fig. \[bose1\](b)\] when $G\sim$8$G_0$, which represents a nanoparticle surface coverage slightly greater than the percolation threshold, and has been found experimentally to yield optimal switching behaviour when $P_{dep}$ is in the range $\sim 10-50 \mu$Torr, as described below . As can be seen in Fig. \[bose1\](c) the increase in deposition pressure ($P_{dep}$) leads to a monotonic *decrease* in *t*$_0$ coupled with an *increase* in $\triangle$*t*$_0$. The smooth conduction onset and longer $\triangle$*t*$_0$ is in contrast to the step-wise conduction onset of samples deposited at BP (minimum oxidation) in Ref.[@Sattar2013]; those samples were dominated by few quantized conduction pathways and therefore lacked the large-scale distributed synaptic network essential for neuromorphic applications. The post-deposition scanning electron micrographs (SEM) shown in Fig. \[bose1\](d-e) reveal markedly smaller NP sizes for devices prepared at higher $P_{dep}$. This can be understood in the framework of diffusion of the tin NPs on the substrate being inhibited by the formation of tin-oxide-shell, leading to reduced coalescence at higher $P_{dep}$. Such formation of oxide-shell is known to inhibit grain-rotation-induced grain coalescence (GRIGC) [@MOldovan2002] thus favoring smaller grain-sizes. This reduced coalescence means that the percolation threshold is reached more quickly (smaller *t*$_0$) and the conductance then increases more slowly (larger $\triangle t_0$). After the deposition is stopped \[arrows in Fig. \[bose1\](b)\] the slow conductance change is primarily due a small amount of further coalescence of the NPs, decreasing $G$.
![The normalized distribution of the change in conductance for the switching events ($\lvert\Delta G\rvert$) on day 10 and day 40, for long periods of time, shows the stability of the switching dynamics. The solid lines are Gaussian fits to the data. []{data-label="bose3"}](bose3.pdf){width="10"}
Optimization of Pressure and Humidity
-------------------------------------
The samples fabricated with *high* Relative Humidity (RH $\sim$ 80%) ambient air have been stimulated with voltage sweeps and square voltage pulses over several months. Data for a typical sample are shown in Fig. \[bose2\]. The four panels show representative snapshots of the switching events on the 1$^{st}$, 10$^{th}$, 15$^{th}$ and 40$^{th}$ day. As described below, the detailed switching behaviour of the network is a complex function of applied voltage, pulse widths ($\tau_p$) and history of the inputs, but qualitatively similar conductance switching is observed for long periods of time. Application of voltage sweep or pulses induces Electric Field Induced Evaporation (EFIE) and Electric Field Induced Surface Diffusion (EFISD) of the surface atoms[@Olsen2012], resulting in atomic-wire formation in tunnel gaps in the percolating network (resulting in $G_{\uparrow}$). The electronic flow causes electromigration induced opening of the previously connected atomic-wires [@Xiang2009] resulting in $G_{\downarrow}$. The conductance thus switches between multiple conductance states with $G \sim$ 1–3$G_0$. $G$ $\rightarrow$ 0 at multiple points, but the electric field induces reconnections and results in the network configuration returning to the connected regime of non-zero conductance. Samples fabricated in high humidity conditions (RH $\sim$ 80%) with ambient air have been tested in this way for periods of several months without the sample becoming permanently open circuit. The key point is that the samples prepared with high RH oxidation exhibit stable switching behaviour without significant performance degradation. For example, Fig. \[bose3\] shows the distribution of the change in conductance for switching events ($\lvert\Delta G\rvert$) on day 10 and 40 (see Fig. \[bose2\]). Both the mean and variance of the distributions are very similar, indicating that the average switching behavior is the same. Obviously it is an onerous task to test such samples for much longer periods and further testing is required to determine the ultimate lifetime of the samples.Both the mean and variance of the distributions are very similar. Obviously it is an onerous task to test such samples for much longer periods and further testing is required to determine the ultimate lifetime of the samples. The same kind of stable switching behaviour is observed for voltages upto ($\geq$ 7-8V) but application of very high voltages ($\geq$ 10V) causes irreversible breakdown in the devices.
The oxidation of pure Sn into tin-oxides \[SnO, SnO$_2$\] is well known to be accelerated under humid conditions.[@Cho2005] Studies of oxidation of Sn in conditions similar to the present ones show that only partial surface oxides are formed.[@Sutter2014; @Lee1998] Ex-situ analysis of the present oxide-structure is obviously not feasible and so instead we have investigated in-situ the device stability over several weeks. Fig. \[bose4\] shows that the device conductance did not change measurably when the device was left for 5 days with 0.1V applied. The conductance switching is resumed when voltage pulses were applied again, which clearly indicates that the oxide structure did not evolve significantly in this period.
![Temporal retention of the network conductance tested over more than one week. Left: the device shows switching in response to 4V pulses before a constant ‘read’ voltage (0.1V) was applied for 5 days. Right: the conductance showed no measurable change and switching re-commenced on application of 4V pulses again. []{data-label="bose4"}](bose4.pdf){width="10"}
The final microstructure and the associated device stability achieved with NP oxidation depends strongly on the relative humidity during NP deposition. Therefore, in order to develop a reliable fabrication process, mitigating day-to-day variation in RH of the ambient air and to build understanding of the critical RH necessary for stabilization, a new set of samples were prepared in a more controlled environment using commercial dry synthetic air coupled with custom-built humidifier (bubbler). The deposition with dry synthetic air resulted in unstable samples with the corresponding SEM micrographs \[shown in \[bose1\](f)\] indicating a slightly more coalesced morphology in comparison to the RH $\sim$ 80% ambient air in \[bose1\](d). Although the microstructure has only subtle differences, these differences become very important as the NP system is poised near the percolation threshold and nanoscale changes can promote(inhibit) inter-NP atomic-switch formation.
Current surge protection
------------------------
The variation in conductance during the first voltage sweeps applied to the new series of samples are shown in Fig. \[bose5\]. The top panel of Fig. \[bose5\](a) shows that the atomic switch networks prepared at BP with no oxygen and no moisture are disconnected on application of very small voltages of 0.1V. Introduction of dry air at $P_{dep}= 10 \mu$Torr, no moisture \[Fig. \[bose5\](a) middle panel\] leads to samples that can only sustain small voltage sweeps. Samples prepared with the same $P_{dep}= 10 \mu$Torr and higher RH $\sim$ 55% are more stable but still do not show sustained reconnections ($G_{\uparrow}$) and become disconnected at $\sim$2V, indicating that these samples are only partially stabilized. This provides an opportunity to demonstrate an additional method for stabilizing the switching behaviour in these devices.
{width="15"}
Fig. \[bose5\](b) shows that the devices show more stable conductance switching when measured with a current-limiting resistor (1k$\Omega$) in series with the device. The series resistor limits the maximum allowed current flowing through the percolating network and hence prevents the destruction of the key connections via electromigration. The sample prepared with $P_{dep}= 10 \mu$Torr but without moisture and measured with the series resistor survived the voltage sweeps \[Fig. \[bose5\](b) middle panel\], but got disconnected in the next measurement (not shown here). In contrast, the sample prepared with oxidation with RH $\sim$ 55% and measured with the in-line resistor was stable for several weeks. Further samples prepared in synthetic air with a higher RH ($\sim$60%) exhibit stable (i.e. for months) switching behaviour as in Fig. \[bose2\], *without* a current-limiting resistor. This indicates that oxidation with a critical amount of moisture RH ($\geq$60%) creates a microstructure which incorporates a robust current-limiting resistor backbone, and thus do not require additional in-series resistor protection.
![Further conductance data of the sample described in Fig. \[bose2\] showing that switching in these NP assemblies requires application of a minimum voltage-stimulus. The 3V pulses generate very few switching events, whereas 4V pulses trigger multitude of switching events.[]{data-label="bose6"}](bose6.pdf){width="10"}
Fabrication summary
-------------------
As discussed above, the crucial fabrication parameters for ASN stability are $P_{dep}$ and relative humidity RH%. $P_{dep}$ was varied between BP (6 $\mu$Torr) and 600 $\mu$Torr, and RH% was varied from completely dry (0%) to nearly saturated (80%). The optimal fabrication parameters for stability of these Sn cluster devices are $P_{dep}$ $=$ 10-50 $\mu$Torr and relative humidity RH $=$ 60-80% when an in-series resistor of 1k$\Omega$ is used for current surge protection.
Switching mechanism and dynamics {#Dynamics}
================================
To understand the physical process underlying the switching mechanism, we present further voltage and time dependent studies. Fig. \[bose6\] presents a segment of data acquired during the long sequence of measurements on the sample used to obtain the data in Fig. \[bose2\] (ambient air, $P_{dep} = 10\mu$Torr, RH $\sim$ 80%) showing that a critical voltage (or equivalently, electric field) is required to activate the switching process. The switching dynamics is voltage polarity independent, with negative V pulses (not shown here) showing exactly the same switching dynamics as positive V pulses. This polarity-independence allows us to eliminate other possible switching mechanisms such as Coulomb charging and electrochemical redox reactions[@Avizienis2012], and further substantiates the electric-field and current induced switching mechanism described here. As shown in Fig. \[bose6\], the pulses with amplitude 3V cause almost no switching events, whereas 4V pulses induce multiple stochastic switching events. The inherently probabilistic nature of the synaptic connections are clearly visible in the snapshots shown in Fig. \[bose2\], where stimulus near the threshold voltage induces less than one switching event per pulse. Such stochastic or probabilistic dynamics of the synapses are integral to the functioning of the biological brain: e.g. the opening and closing of synaptic ion channels and associated transmission of neurotransmitter molecules is inherently stochastic[@Rolls2010] and is understood to be critical for noise-filtering[@Faisal2008], signal transmission [@Tuma2016] and reward-modulated Hebbian learning[@Hoerzer2014]. The existence of a critical stimulus strength (electric field here) for the stochastic formation (annihilation) of atomic-scale wires in tunnel gaps in the network is consistent with EFIE/EFISD (electromigration) mechanisms and provides a unique global control over the synaptic network reconfiguration.
{width="15"}
To further validate the model and estimate the effective tunnel barrier parameters associated with the gap in which the atomic wires are formed, pulsed voltage measurements were stopped when the device became open circuit (i.e. $G < 10^{-5}$$G_0$). A series of slow bipolar voltage sweeps were then applied which showed non-linear current-voltage *I*(*V*) characteristics as in Fig. \[bose7\](a–b). On the 1$^{st}$ voltage sweep, *G* jumps from $< 10^{-5}$$G_0$ to $\sim 10^{-2}G_0$, which corresponds to the formation of a tunnel gap which is sufficiently small to allow a measurable current to flow. The corresponding tunneling current through a non-ideal potential barrier with height $\Phi_{B}$ and width $\emph{d}$ is[@Simmons1963]:
$$\label{eq:Simm01}
\begin{split}
I\propto{}(\Phi_B-eV/2)\exp
\Big[-\frac{2(2m)^{1/2}}{\hbar}\alpha(\Phi_B-eV/2)^{1/2}d \Big]\\
-(\Phi_B+eV/2)\exp
\Big[-\frac{2(2m)^{1/2}}{\hbar}\alpha(\Phi_B+eV/2)^{1/2}d \Big]
\end{split}$$
with $m$ being the free electron mass and $\alpha$ being an adjustable parameter representing the non-ideal character of the tunneling barrier and effective electron mass. Fig. \[bose7\](b) shows representative *I*(*V*) characteristics for the voltage sweeps marked A, B and C in Fig. \[bose7\](a) along with the fits to (\[eq:Simm01\]) (solid lines). The associated barrier width *d*, with calculated barrier height $\Phi_{B}$ $\sim$ 2 eV, decreases monotonically with the sweep number \#, when either an ideal ($\alpha =$ 1) or highly non-ideal ($\alpha
=$ 0.5) barrier is assumed. Interestingly the resultant electric field exceeds the $\sim$ 1 Vnm$^{-1}$ threshold for EFISD but remains lower than the $\sim$ 25 Vnm$^{-1}$ required for EFIE[@Sattar2013; @Olsen2012]. As shown schematically in Fig. \[bose7\](d), the narrowing of the tunnel gap under the influence of the electric field continues, until after about 20 minutes an atomic scale wire closes the tunnel gap, i.e. a “jump to contact" occurs[@Agrait2003] leading to a conductance $G =
2 G_0$. These atomic-scale wires are similar to those formed in mechanically controlled break junctions (MCBJs)[@Agrait2003]. The wire breaks and re-forms a couple of times \[Fig. \[bose7\](a)\] and then is observed to be completely stable when subjected to further voltage sweeps. The Ohmic conductance is marked by linear *I*(*V*) behavior depicted in curve C of Fig. \[bose7\](b). Such conductance modulation with successive stimulus (electric-field) is one of the key requirements for synaptic learning capability in neuromorphic systems[@Fostner2015; @Hasegawa2010], and is similar to the sensory memory reported in Ref.[@ohno2011a].
![(a) Synaptic plasticity dependence on stimulus frequency is depicted in switching behaviour in response to a sequence of voltage pulses with fixed $V_p$=4V and variable pulse widths $\tau_p$ (1 – 30s). The conductance remains unaltered for read voltages of 0.1 V as seen in the flat section (OFF) in middle of the sequence. (b) Schematic depiction of representative synaptic pathways with only very few pathways shown for clarity. The real device ASN is much more complex. Shorter pulse widths leads to electric-field induced connections as described in Fig. \[bose7\]. Formation of additional atomic-wire connections (one depicted here) cause more potentiated synaptic pathways and thus higher conductance as seen in panel (a).[]{data-label="bose8"}](bose8.pdf){width="10"}
Realization of neuromorphic behavior in these ASNs also requires a scheme to modify the density of potentiated synaptic pathways[@Fostner2015]. In Fig. \[bose8\] we show such stimulus frequency dependent potentiation. Square voltage pulses with fixed $V_p$=4V (just above threshold voltage) with various pulse widths $\tau_p$ (1, 2, 5, 10, 20, and 30 s) are applied successively for 30 mins each (the green lines depict the pulse-width). Longer $\tau_p$ (slow pulses) leads to lower conductance whereas shorter $\tau_p$ (faster pulses) leads to additional formation of synaptic pathways as schematically depicted in Fig. \[bose8\](b), resulting in higher $G$. This variation can be understood in the light of the electric field-induced reconnections dominating over the electromigration induced disconnection of atomic wires for shorter $\tau_p$. Such stimulus rate dependent reconfiguration of network connectivity can be modeled as short-term to long-term memory conversion[@ohno2011; @Chang2011].
Conclusion
==========
In summary, we have demonstrated a unique approach for realization of self-assembled atomic switch networks with stimulus induced control of the synaptic configuration reflected in the device conductance. By controlling oxidation and humidity during NP deposition, nanoparticle coalescence is inhibited resulting in stochastic switching that is stable over several months. The atomic-wire formation in these *oxidised* nanostructures is very surprising and detailed modeling [@Onofrio2015] of the atomistic mechanisms [@Xiang2009; @Olsen2012; @Sattar2013] in the presence of oxides [@Sutter2014; @Lee1998] is required, as is atomic scale modelling of the effect of humidity [@Cho2005] on the oxidation process.
We have also highlighted the stochastic nature of the switching mechanism together with the stability of the distribution of switching events - these reflect inherently complex network dynamics that are a key requirement for neuromorphic applications. The next stage of this research will be to build devices with multiple contacts and to demonstrate that the networks exhibit the required network dynamics. The precise requirements are different for different applications[@Burr2017; @Nawrocki2016], but for RC include distributed spatio-temporal dynamics, recurrency, higher harmonic generation, switching speed, network size and type of connectivity[@Mantas2009; @Kulkarni2012a; @Konkoli2014; @Avizienis2012; @Busing2010; @Demis2016].
More generally, these complex percolating structures mimic some of the features of biological neural networks and synaptic structures and so could provide a foundation for a range of future neuromorphic architectures. Future work will focus on utilizing these structures to implement previously suggested algorithms [@Konkoli2014; @Kulkarni2012a] and hence to demonstrate utility in key applications.
Acknowledgment {#acknowledgment .unnumbered}
==============
The technical support of G. MacDonald and G. Graham are acknowledged.
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[^1]: The authors are with The MacDiarmid Institute for Advanced Materials and Nanotechnology, Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand (email:simon.brown@canterbury.ac.nz)\
The authors gratefully acknowledge financial support from the Marsden Fund, New Zealand, and the MacDiarmid Institute for Advanced Materials and Nanotechnology.\
This is a post-peer-review, pre-copyedit version of an article published in IEEE Trans. Elect. Dev. 2017. The final authenticated version is available online at: http://dx.doi.org/10.1109/TED.2017.2766063”.
|
---
abstract: 'We present an exact diagonalisation study of bilayer quantum Hall systems at filling factor $\nu=2$ in the spherical geometry. We find the high-Zeeman-coupling phase boundary of the broken symmetry canted antiferromagnet is given exactly by previous Hartree-Fock mean-field theories, but that the state’s stability at weak Zeeman coupling has been qualitatively overestimated. In the absence of interlayer tunneling, degeneracies occur between total spin multiplets due to the Hamiltonian’s invariance under independent spin-rotations in top and bottom two-dimensional electron layers.'
address: |
$^{\ast}$Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany\
$^{\dag}$Department of Physics, Indiana University, Bloomington, IN 47405
author:
- 'John Schliemann$^{\ast}$[@emadd] and A. H. MacDonald$^{\dag}$'
title: |
Bilayer Quantum Hall Systems at Filling Factor $\nu=2$:\
An Exact Diagonalisation Study
---
\#1[[$\backslash$\#1]{}]{}
In the last decade there has been an increasing interest in quantum Hall ferromagnets [@DaPi:97]. Most recently bilayer systems at a filling factor of two have become the object of intensive theoretical [@ZRS:97]–[@YaCh:99] and experimental [@PPDPPW:97]–[@KDSDHWCG:99] research. The rich phenomenology of $\nu=2$ bilayers mirrors a complex interplay between Coulomb interactions in the lowest Landau level, Fermi statistics, and the coupling of external fields to spin and layer degrees of freedom.\
Our current microscopic understanding of $\nu=2$ bilayer quantum Hall ferromagnets is based on Hartree–Fock mean–field theory calculations [@ZRS:97; @MRJ:99], and on a partially phenomenological effective spin Hamiltonian description [@DeDa:99]. Both approaches lead to the prediction of a novel broken symmetry canted–antiferromagnet ground state with finite spin–suscpetibility which interpolates, as external field parameters are varied, between a fully spin-polarized state and a spin singlet state, both of which have charge and spin gaps and zero differential spin susceptibility. In this Letter we report on the first exact diagonalisation study of finite bilayer systems at filling factor $\nu=2$. Our calculations support the predicted occurrence of a broken symmetry ground state, and indeed demonstrate that the Hartree–Fock result for its large Zeeman coupling phase boundary is exact. We find that the stability of the canted antiferromagnet state relative to the spin singlet state is overstated by the Hartree–Fock approximation, and estimate the correct position of the phase boundary.\
A bilayer system in a strong magnetic field is described in spherical geometry by the following Hamiltonian, $${\cal H}={\cal H}_{{1\rm P}}+{\cal H}_{{\rm Coul}}\quad,$$ where ${\cal H}_{{\rm Coul}}$ represents the usual Coulomb interaction within and between layers, and the single–particle Hamiltonian ${\cal H}_{{1\rm P}}$ is given by $$\begin{aligned}
{\cal H}_{{1\rm P}} & = &
-\frac{1}{2}\sum_{m}c^{+}_{\mu,\sigma,m}
\big[\Delta_{v}\tau^{z}_{\mu,\mu'} \delta_{\sigma,\sigma'}\nonumber\\
& &+\Delta_{t}\tau^{x}_{\mu,\mu'}\delta_{\sigma,\sigma'}
+\Delta_{z}\delta_{\mu,\mu'}\sigma^{z}_{\sigma,\sigma'}\big]
c_{\mu',\sigma',m}\,.\end{aligned}$$ A summation convention is understood for repeated greek indices, where $\mu,\mu'\in\{+,-\}$ run over the layer (or pseudospin) index, while $\sigma,\sigma'\in\{\uparrow,\downarrow\}$ run over the $z$–projections of the electron spin; $\vec\tau$, $\vec\sigma$ are Pauli matrices for pseudospin and electron spin, respectively. $m\in\{-N_{\phi}/2,\dots,N_{\phi}/2\}$ is the z-projection of the orbital angular momentum of each electron in the lowest Landau level, where $N_{\phi}$ is the number of flux quanta penetrating the sphere. The Hamiltonian contains bias voltage ($\Delta_{v}$), tunneling ($\Delta_{t}$), and Zeeman coupling ($\Delta_{z}$) terms. In the following we measure the interlayer separation $d$ in units of the magnetic length $l_{B}=\sqrt{{\hbar c}/{eB}}$ and all energies in units of the Coulomb energy scale $e^{2}/\epsilon l_{B}$.\
We first consider the case where all single–particle coupling constants vanish. The number of particles in each layer is then a good quantum number and, in the ground state, both layers have filling factor one. Moreover, because the Coulomb interaction is spin–independent, the Hamiltonian is invariant under independent spin rotations in either layer. It follows that the total spin in either layer is a good quantum number.
For layer separation $d \to \infty$ it is known that the ground state of each isolated layer is a quantum Hall ferromagnet with $S=N/4$, where $N$ is the total number of particles in the two layers. Since the Coulomb interaction within the layers is stronger than between them, we anticipate that this should remain true at any finite $d$. Indeed, this expectation is confirmed numerically. The upper diagram of figure \[fig1\] shows the low–lying spectrum of a double layer system at filling factor two with eight electrons in the case of vanishing single–particle couplings. The ground state as described there consists of multiplets with total spin quantum number $S$ varying from $0$ to $4$, the total spin representations of a state with good spin quantum number $S=2$ in each layer. When an interlayer tunneling term is added to the Hamiltonian, the additional symmetry responsible for these degeneracies is lost and only total spin is a good quantum number.\
The robustness of individual layer ground state spin quantum numbers against the effects of added interlayer interactions is not restricted to total filling factor two. In fact, in our finite-size numerical calculations, it holds at all combination of numbers of flux quanta and electrons we have checked.
For instance, consider the system discussed above, but with one electron either added or removed. (These two cases are equivalent by particle–hole symmetry.) In the ground state, one of the layers has a filling factor of one (a ground state with $L=0$, $S=2$), while the other one contains a hole and the low–lying states are organised in the well–known skyrmion branch having quantum numbers $L=S=1/2,3/2$ [@XiHe:96]. The spectrum of this double layer system is shown in the upper diagram of figure \[fig2\]. The ground state carries quantum numbers $L=1/2$ and $S=3/2$, $S=5/2$ and is the result of coupling the $L=0$, $S=2$ ground state of the full layer to the $L=S=1/2$ ground state of the hole system. Since either layer can carry the hole, we get two copies of the degenerate multiplets. The next higher degenerate group of multiplets is the result of coupling the ground state of the $\nu=1$ layer to the $L=S=3/2$ state of the hole layer, and again all multiplets are doubled. To confirm this interpretation we have verified that that this degeneracy doubling is lifted by applying a bias voltage, as shown in the lower diagram of figure \[fig2\]. These results demonstrate that in the absence of interlayer tunneling, bilayer states for $\nu$ near $2$ can be safely regarded as two single-layer quantum Hall ferromagnets whose coupling has only a quantitative significance, for example in changing the energies of the skyrmionic elementary charged excitations.
With this established, we now focus on changes in the nature of the ground state at $\nu=2$ as the single–particle coupling constants are varied. An arbitrarily small tunneling amplitude is enough to break the degeneracy among the different spin multiplets and make the spin singlet state with the most antiparallel electron spin structure and consequently the most parallel pseudospin structure the nondegenerate ground state. More precisely, the energy levels are found to be ordered by the total spin $S$ with the difference between neighboring levels increasing with increasing $S$, as shown in the lower diagram of figure \[fig1\]. We note that the lone maximally polarized $S=N/2$ multiplet is annihilated by the interlayer tunneling term in the Hamiltonian and has an eigenenergy which is independent of $\Delta_t$. Turning on the Zeeman coupling $\Delta_{z}$ at a given value of the tunneling $\Delta_{t}$ does not change the eigenstates themselves, but only shifts their energies and breaks the degeneracy [*within*]{} each spin multiplet.
These findings lead to the following scenario: With increasing $\Delta_{z}$ a lower critical Zeeman coupling $\Delta_{z}^{(1)}(\Delta_{t})$ is reached where the state with $S^{z}=S=1$ becomes the ground state, i. e. the system leaves the spin singlet phase. If $\Delta_{z}$ is increased further, the ground state $S^{z}=S$ quantum number increases monotonously until, at an upper critical value, $\Delta_{z}^{(2)}(\Delta_{t})$ the fully spin–polarized state $S^{z}=S=N/2$ is reached.
Finite–size spectra are shown in figure \[fig3\] for several Zeeman couplings at $\Delta_{t}=0.8$. In the top diagram the system is in the spin singlet phase, while in the bottom the ground state is fully spin–polarized. In the narrow transition area $\Delta_{z}^{(1)}\leq\Delta_{z}\leq\Delta_{z}^{(2)}$ all low–lying states with $S^{z}=0,\dots,N/2$ have energies very close to each other. This holds also for a branch of states with angular momentum $L>0$ which appears to be separated by a gap from higher-lying parts of the spectrum. We identify this transition region with the canted antiferromagnetic phase first proposed by Zheng [*et al.*]{} on the basis of the unrestricted Hartree–Fock approximation[@ZRS:97].
The mean-field state, which breaks spin–rotational symmetry around the $\hat z$ axis can be constructed as a linear combination of the nearly degenerate exact eigenstates $|S^{z}\rangle$ carrying quantum numbers $L=0$ and definite $S^{z}$ values. To analyse the spin structure perpendicular to the Zeeman axis we introduce spin operators for each layer separately, $\vec S_{\mu}$, $\mu\in\{+,-\}$, where the total spin of the bilayer system is given by $\vec S=\sum_{\mu}\vec S_{\mu}$. Since the states $|S^{z}\rangle$ belong to different multiplets, all matrix elements of $S^{\pm}$ between them are zero, which means that $$\langle S^{z}|S_{\mu}^{+}|S^{z}-1\rangle
=-\langle S^{z}|S_{-\mu}^{+}|S^{z}-1\rangle\,.$$ Thus, the matrix elements of spin components perpendicular to the Zeeman axis have the same magnitude and opposite sign in opposite layers. Any wavepacket constructed from these states will, like the mean-field state, have opposite transverse spin–polarization in the two layers.\
Next let us consider the phase diagram, i. e. the functions $\Delta_{z}^{(1)}(\Delta_{t})$ and $\Delta_{z}^{(2)}(\Delta_{t})$. In the thermodynamic limit these lines mark the boundaries between spin–polarized, spin–singlet, and canted antiferromagnet phases. They may be compared with the Hartree–Fock results obtained recently in the planar geometry for an infinite system [@MRJ:99]. To avoid unnecessary finite–size uncertainty, we have rederived the Hartree–Fock equations for the spherical geometry obtaining explicit expressions for finite systems. We find that the phase boundaries have the same form as for the infinite system [@MRJ:99]: $$\Delta_{z}^{(1)}=\sqrt{\Delta_{t}(\Delta_{t}-2F_{-})}
\label{pb1}$$ for $\Delta_{t}>2F_{-}$, otherwise $\Delta_{z}^{(1)}=0$, and $$\Delta_{z}^{(2)}=\sqrt{\Delta_{t}^{2}+F_{-}^{2}}-F_{-}\quad.
\label{pb2}$$ In the present case, however, the exchange parameter $F_{-}$ is size–dependent: $$F_{-}=\frac{e^2}{\epsilon l_{B}}\frac{N_{\phi}+1}{\sqrt{2N_{\phi}}}
\left(I(1)
-\left(\frac{1}{\alpha}\right)^{N_{\phi}+\frac{1}{2}}I(\alpha)\right)\quad,
\label{Fminus}$$ with $$I(\alpha)=\int_{0}^{\alpha}\,dx\frac{x^{N_{\phi}}}{\sqrt{1-x}}
\quad,\quad
\alpha=\frac{1}{1+\frac{1}{N_{\phi}}\frac{d^2}{2l_{B}^{2}}}\,.
\label{intpar}$$ In figure \[fig4\] we compare exact–diagonalization and Hartree–Fock finite-size phase boundaries for $N=6,8,10,12$ electrons. Not unexpectedly, the region of the canted antiferromagnetic phase turns out to be much smaller than predicted by the Hartree–Fock theory. Interestingly however, the two results for the phase boundary $\Delta_{z}^{(2)}(\Delta_{t})$ coincide within our numerical precision of at least $10^{-12}$. Since both the spin–polarized state and its elementary collective excitations, which go soft at the phase boundary, are described exactly [@Mac:85] by Hartree–Fock theory, this coincidence is not entirely unexpected. It does, however, very convincingly demonstrate that the spin-polarized to canted phase transition remains continuous when quantum fluctuations are included. We conclude that the position of this phase boundary in the thermodynamic limit can be calculated [*exactly*]{} using the expressions given in reference [@MRJ:99], or equivalently the $N_{\phi}\to\infty$ limit of equations (\[pb2\])–(\[intpar\]), adding finite–well–width and other sample–specific corrections as required.
The result for this upper boundary is in marked contrast with our findings for the lower boundary. Here the Hartree–Fock approach leads to a canted antiferromagnetic region at all system sizes even in the absence of the Zeeman coupling. A similar conclusion was reached by Demler and Das Sarma using the effective spin theory [@DeDa:99]. In our finite–size exact diagonalization calculations, on the other hand, $\Delta_{z}^{(1)}$ is always finite. For $\Delta_{t} \lesssim 0.35$, marked by a vertical line in the last panel of figure \[fig4\], $\Delta_{z}^{(1)}$ decreases monotonously with system size, while it increases monotonously for larger values of $\Delta_{t}$. These observations guarantee that the canted antiferromagnetic phase predicted by Hartree–Fock theory is actually present in the infinite system (although clearly diminished by quantum fluctuations). Moreover, our findings are consistent with a vanishing thermodynamic limit of $\Delta_{z}^{(1)}(\Delta_{t})$ and a nonzero spin susceptibility for $\Delta_{t} \lesssim 0.35$. Static correlation function calculations [@unpub] at $\Delta_{z}=0$ are also consistent with a nonzero canted order parameter for $\Delta_{t} < 0.35$. These numerical results are thus consistent with a single intermediate phase which has both a finite spin–susceptibility and canted–antiferromagnet order. For $\Delta_{t} \gtrsim 0.35$, $\Delta_{z}^{(1)}$ increases slowly with system size, presumably saturating at a finite value smaller than $\Delta_{z}^{(2)}$ and leaving a narrow canted-antiferromagnet strip in the phase diagram. The maximum $\Delta_{t}$ value at which the ordered state phase extends down to $\Delta_{z}=0$ is much smaller than the value $\Delta_{t}=0.60$, predicted by Hartree–Fock theory. These findings are illustrated along with the exact upper phase boundary in the right bottom panel of figure \[fig4\], which shows that the Hartree–Fock approximation strongly overestimates the stability of the canted antiferromagnetic phase against the spin singlet phase in finite systems and as well in the thermodynamic limit.\
JS acknowledges support from the Deutsche Forschungsgemeinschaft under grant SCHL 539/1–1. AHM was supported from the National Science Foundation under grant DMR–9714055.
e-mail: btp338@theo.phy.uni-bayreuth.de; address after January 1 2000: Department of Physics, Indiana University, Bloomington, IN 47405
For a review on experimental research on multicomponent quantum Hall systems see J. P. Eisenstein in [*Perspectives in Quantum Hall Effects*]{}, edited by S. Das Sarma and A. Pinczuk, Wiley 1997; for a review on theoretical research see S. M. Girvin and A. H. MacDonald, in the same volume. For a general discussion of quantum Hall ferromagnets see also T. Jungwirth and A. H. MacDonald (in preparation).
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A. H. MacDonald, R. Rajaraman, T. Jungwirth, Phys. Rev. B [**60**]{}, 8817 (1999)
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E. Demler, E. H. Kim, S. Das Sarma, cond-mat/9907107
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V. S. Khrapai, E. V. Deviatov, A. A. Shashkin, V. T. Dolgopolov, F. Hastreiter, A. Wixforth, K. L. Campman, A. C. Gossard, cond-mat/9903396
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See, for example, A. H. MacDonald, J. Phys. C [**18**]{}, 1003 (1985).
John Schliemann and A.H. MacDonald, unpublished.
|
---
abstract: 'Reinforcement learning augmented by the representational power of deep neural networks, has shown promising results on high-dimensional problems, such as game playing and robotic control. However, the sequential nature of these problems poses a fundamental challenge for computational efficiency. Recently, alternative approaches such as evolutionary strategies and deep neuroevolution demonstrated competitive results with faster training time on distributed CPU cores. Here, we report record training times (running at about 1 million frames per second) for Atari 2600 games using deep neuroevolution implemented on distributed FPGAs. Combined hardware implementation of the game console, image pre-processing and the neural network in an optimized pipeline, multiplied with the system level parallelism enabled the acceleration. These results are the first application demonstration on the IBM Neural Computer, which is a custom designed system that consists of 432 Xilinx FPGAs interconnected in a 3D mesh network topology. In addition to high performance, experiments also showed improvement in accuracy for all games compared to the CPU-implementation of the same algorithm.'
author:
- 'Alexis Asseman[ [](https://orcid.org/0000-0003-4482-5744)]{}'
- 'Nicolas Antoine[ [](https://orcid.org/0000-0003-0020-1278)]{}'
- 'Ahmet S. Ozcan[ [](https://orcid.org/0000-0002-4689-7971)]{}'
bibliography:
- 'references.bib'
title: '**Accelerating Deep Neuroevolution on Distributed FPGAs for Reinforcement Learning Problems**'
---
Introduction
============
In reinforcement learning (RL) [@arulkumaran2017brief][@li2017deep], an agent learns an optimal behavior by observing and interacting with the environment, which provides a reward signal back to the agent. This loop of observing, interacting and receiving rewards, applies to many problems in the real world, especially in control and robotics [@polydoros2017survey]. Video games can be easily modeled as learning environments in an RL setting [@jaderberg2019human], where the players act as agents. The most appealing part of video games for reinforcement learning research is the availability of the game score as a direct reward signal, as well as the low cost of running large amounts of virtual experiments on computers without actual consequences (e.g., crashing a car hundreds of times would not be acceptable).
Deep learning based game playing reached popularity when Deep Q-Network (DQN) [@mnih2015human] showed human-level scores for several Atari 2600 games. The most important aspect of this achievement was learning control policies directly from raw pixels in an end-to-end fashion (i.e., pixels to actions). Subsequent innovations in DQN [@zhao2016deep], and new algorithms such as the Asynchronous Advantage Actor-Critic (A3C) [@mnih2016asynchronous] and Rainbow [@hessel2018rainbow] made further progress and launched the field to an explosive growth. A comprehensive and recent review of deep learning for video game playing can be found in [@justesen2019deep].
However, gradient-based optimization algorithms, used for the training of neural networks, have performance limitations, as they do not lend themselves to parallelization, and they require heavy computations and a large amount of memory, requiring the use of specialized hardware such a Graphical Processing Units (GPU).
Compared to the gradient descent based optimization techniques mentioned above, derivative-free optimization methods such as evolutionary algorithms have recently shown great promise. One of these approaches, called deep neuroevolution, can optimize a neural network’s weights as well as its architecture. Recent work in [@such2017deep] showed that a simple genetic algorithm with a Gaussian noise mutation can successfully evolve the parameters of a neural network and achieve competitive scores across several Atari games. Training neural networks with derivative-free methods opens the door for innovations in hardware beyond GPUs. The main implications are related to precision and data flow. Rather than floating point operations, fixed point precision is sufficient [@courbariaux2014training] and data flow is only forward (i.e., inference only, no backward flow). Moreover, genetic algorithms are population-based optimization techniques, which greatly benefit from distributed parallel computation.
These observations led us to conclude that genetic algorithm–based optimization of neural networks could be accelerated (and made more efficient) by the use of hardware optimized for fast inference, and the use of multiplicity of such devices would easily take advantage of the inherent parallelism of the algorithm. Hence, we implemented our solution on the IBM Neural Computer [@narayanan2020overview], which is a custom-designed distributed FPGA system developed by IBM Research. By implementing two instances of the whole application on each of the 416 FPGAs we used (i.e., game console, image pre-processing and the neural net), we were able to run 832 instances in parallel, at an aggregated rate of 1.2 million frames per second. Our main contributions are:
- Introduction of an FPGA-accelerated *Fitness Evaluation Module* consisting of a neural network and Atari 2600 pair, for use with evolutionary algorithms.
- The first demonstration of accelerated training quantized neural networks using neuroevolution on distributed FPGAs.
- Extensive results on 59 Atari 2600 games trained for six billion frames using deep neuroevolution and performance analysis of our results on the IBM Neural Computer compared to baselines.
Related Work {#sec:relatedwork}
============
Most of the FPGA-based implementations of neural networks target inference applications due to the advantages related to energy efficiency and latency [@umuroglu2017finn] [@xu2018scaling] [@wei2017automated]. These are often based on high-level synthesis for FPGAs, while some of them utilize frameworks that convert and optimize neural network models into bitstreams. FPGA maker Xilinx recently launched a new software platform called Vitis to make it easier for software developers to convert neural network models to FPGA bitstreams.
In addition to the inference-only applications, few studies utilized FPGAs to accelerate reinforcement learning and genetic algorithms. For example [@cho2019fa3c] proposed the FA3C (FPGA-based Asynchronous Advantage Actor-Critic) platform which targets both inference and training using single-precision floating point arithmetic in the FPGA. They show that the performance and energy efficiency of FA3C is better than a high-end GPU-based implementation. Similar to our work, they chose the Atari 2600 games (only six) to demonstrate their results. However, unlike our work, their Atari 2600 environment is the Arcade Learning Environment [@bellemare2013arcade], which runs on the host CPU.
Genetic algorithms (GA) are another class of optimization methods that FPGA acceleration can help. For example, [@tang2004hardware] implemented GA on FPGA hardware and proposed designs for genetic operations, such as mutation, crossover, selection. Their approach tried to exploit parallelism and pipelining to speed up the algorithm. Experimental results were limited to the optimization of a modified Witte and Holst’s Strait Equation, $ f(x_1, x_2, x_3) = |x_1 - a| + |x_2 - b| + |x_3 - c| $, and showed about an order of magnitude speed up compared to a CPU implementation at the time.
A more recent study [@torquato2019high] proposed a parallel implementation of GA on FPGAs. They showed results for the optimization of various simple mathematical functions, which are trivial to implement and evaluate in the FPGA itself. Compared to previous studies, they report speed-up values ranging from one to four orders of magnitude.
Even though these related studies are not a complete picture of the field, our approach is fundamentally different and unique in several aspects. Rather than accelerating the optimization algorithm (e.g. RL or GA) we have taken a different approach and addressed the data generation (i.e. Atari game environment and obtaining frames). Moreover, we are pipelining the image pre-processing and neural network inference entirely within the FPGA, thus avoiding the costly external memory access, contributing significantly to our results.
Implementation {#sec:implementation}
==============
IBM Neural Computer {#sec:inc}
-------------------
![IBM Neural Computer: (a) Cage holding 16 cards (b) Card composed of 27 nodes (c) Node based on a Zynq-7045 with 1GB of dedicated RAM[]{data-label="fig:inc"}](img/inc_systems.jpg){width="\linewidth"}
The IBM Neural Computer (INC) [@narayanan2020overview] is a parallel processing system with a large number of compute nodes organized in a high bandwidth, low latency 3D mesh network. Within each node is a Zynq-7045 system-on-chip, which integrates a dual-core Cortex A9 ARM processor and an FPGA, alongside 1GB of DRAM used both by the ARM CPU and the FPGA.
The INC cage is comprised of a 3D network of $12\times12\times3 = 432$ nodes, which is obtained by connecting 16 cards through a backplane, each containing $3\times3\times3$ nodes (27 nodes per card). The total system consumes about 4kW of power. Each card has one “special” node at coordinate $(xyz)=(000)$ with supplementary control capabilities over its card, and also provides a 4-lane PCIe 2.0 connection to communicate with an external computer.
The 3D mesh network is supported by the high frequency transceivers integrated into the Zynq chip. These are entirely controlled by the FPGA, thus enabling a low level optimization of the network for the target applications. In particular, the currently implemented network protocols over the hardware network enable us to communicate from any node to any other node of the system, including reading and writing any address accessible over its AXI bus. That last point enables us to control all the Atari 2600 environment fitness evaluation modules present over all the nodes of the system, from the gateway node connected to the computer through PCIe.
We elected to use 26 out of the 27 nodes of each card, leaving the node $(xyz)=(000)$ of each card. Therefore, all the computation carried out in the experiments described herein was on a total of 416 nodes.
The Fitness Evaluation Module
-----------------------------
![Schematic representation of the fitness evaluation module carrying out the evaluation loop—entirely in FPGA.[]{data-label="fig:game_nn_loop"}](img/game_nn_loop.pdf){width="\linewidth"}
Submodule Slice LUTs BRAM Tiles DSPs
---------------------- ------------ ------------ ------
Atari 2600 1,875 9 0
Image pre-processing 677 16.5 2
Neural network 22,855 140 416
Miscellaneous 1,337 0 0
Total 26,744 165.5 418
\[tab:hw\_util\]
The Atari 2600 $\rightarrow$ image pre-processing $\rightarrow$ ANN $\rightarrow$ Atari 2600 loop is integrated in a fitness evaluation module, which can communicate with the AXI bus in order to control the operation from the outside – i.e. by reading and writing memory-mapped registers exposed on the AXI bus (see fig. \[fig:game\_nn\_loop\]).
The whole loop is pipelined together, and caching is reduced to the bare minimum to decrease the latency of the loop. Moreover, information exchange between the loop and the rest of the system is done asynchronously, such that the loop is never interrupted by external events. This enabled us to achieve 1,450 frames per second while running the Atari 2600 inside the loop described above.
The module exposes on the AXI address space:
- The Atari 2600’s block RAM containing the game ROM (write), such that games can be loaded dynamically from the outside.
- The ANN’s block RAM containing the parameters (write).
- The game identifier (write) – used by the fitness evaluation module to know where in the console’s RAM the game status as well as the score are stored.
- The status of the game (read) – Alive or Dead.
- The game’s score (read).
- A frame counter (read).
- A clock counter (read) – to deduce the wall time that passed since the game start.
- A command register (write) – to reset the whole loop (when a new game, new parameters are loaded) and start the game, or to forcibly stop the loop’s execution.
Table \[tab:hw\_util\] contains a summary of the hardware utilization of the different submodules comprising the fitness evaluation module, as reported by Xilinx’s Vivado tool.
We implemented two instances of the fitness evaluation module per INC node, which brings us to a total of 832 instances used in parallel, for a total maximum of 1,206,400 frames per second.
### Atari 2600
To obtain the highest performance, we chose to avoid software emulation of the Atari 2600 console and took advantage of the FPGA instead, which can easily implement the original hardware functionality of the console at a much higher frequency. We used an open-source VHDL implementation from the open-source MiSTer project[^1].
We ran the Atari 2600’s main clock at 150 MHz, instead of the original 3.58 MHz[@stella_guide]. As we are using it in NTSC [@pritchard1977us] picture mode, we obtain $\sim2514$ frames per second, compared to 60 frames per second when running the console at its originally intended frequency. Figure 1 shows snapshots from selected Atari 2600 games.
### Image Pre-processing
We chose to apply the same image pre-processing as in [@mnih2015human] and [@such2017deep], for the dual purpose of enabling an easier comparison with those results, as well as reducing the hardware cost of the ANN (Artificial Neural Network). The entire pre-processing stack is implemented on the FPGA in a pipelined fashion for maximum throughput. The pre-processing stack is comprised of:
- A **color conversion module** that converts the console’s 128 color palette to luminance, using the ITU BT.601 [@bt2011studio] conversion standard. This is done instead of just keeping the 3-bit luminance from the console’s NTSC signal, such that the 128 color palette is converted to a 124 grayscale color palette (4 levels are lost due to some overlaps in the conversion).
- A **frame-pooling module**. Its purpose is to eliminate sprite flickering (where sprites show on the screen in half of the frames to bypass the sprite limitations of the console). This is achieved by keeping the previous frame in memory, and for each pixel, showing the one that has the highest luminance between the current frame and the previous frame.
- A **re-scaling module**. To re-scale the image from the original $160\times210$ pixels down to $84\times84$ pixels, while applying a bilinear filter to reduce information loss.
- A **frame stacking module**. To stack the frames in groups of 4, where each of the 4 frames becomes a channel of the payload that is fed into the ANN. This has two purposes: It divides the number of inputs to the ANN by 4, and also enables the ANN to see 4 frames at a time, therefore being able to deduce motion within those 4 frames.
### ANN Model
Operation Filter size Stride Output dimensions Activation function CPF KPF
--------------- -------------- -------------- -------------------------- --------------------- ----- -----
Input image - - $84 \times 84 \times 4$ - - -
Convolution $8 \times 8$ $4 \times 4$ $20 \times 20 \times 32$ ReLU 4 32
Convolution $4 \times 4$ $2 \times 2$ $9 \times 9\times 64$ ReLU 32 4
Convolution $3 \times 3$ $1 \times 1$ $7 \times 7 \times 64$ ReLU 4 32
Inner product - - $18$ - 4 1
\[tab:cnn\]
------------------- ----
Bit-width 16
Weights radix 13
Activations radix 6
------------------- ----
: DNNBuilder Fixed-Point Numerical Precision Settings for All Layers.
\[tab:dnnbuilder\_params\]
The hardware architecture for the neural network was generated using the open-source tool DNNBuilder[^2][@zhang2018dnnbuilder]. It was chosen because it generates human-readable register transfer level (RTL) code, which describes a fully-pipelined neural network, optimized for low block RAM utilization and low latency. DNNBuilder makes this possible by implementing a Channel Parallelism Factor (CPF) and Kernel Parallelism Factor (KPF), which respectively unroll the input and output channels of an ANN layer, at the cost of higher hardware utilization. By alternating the CPF and KPF values at each stage of the ANN, caching, and therefore latency, can be reduced.
Table \[tab:cnn\] illustrates the architecture of the model, which has been implemented and trained in this study. Note that the model is similar to the one used in [@mnih2015human], but the convolutions are done without padding, and the first fully-connected layer is removed. This was necessary to bring the number of parameters from $\sim 4$ million down to 134,272, such that all the parameters can fit into block RAM for faster access by the ANN modules. Also, we are not using biases since we have not noticed any significant impact on the training performance. Table \[tab:dnnbuilder\_params\] shows the fixed-point numerical precision settings we used for DNNBuilder.
### Action Selection
The action selection submodule selects the joypad action to apply for the next 4 frames by selecting the action with the maximum reward as predicted by the ANN’s output. To introduce stochasticity into the games, we used *sticky actions* as recommended in [@machado2018revisiting], which introduces stochasticity by having a probability $\varsigma$ of maintaining the action sent to the environment at the previous frame during the current frame, instead of applying the latest selected action. We used the recommended stickiness parameter value $\varsigma = 0.25$. The randomness is sampled from a rather large maximum-length 41-bit linear feedback shift register running independently from the rest of the module.
Genetic Algorithm {#sec:geneticalgorithm}
-----------------
The Genetic Algorithm runs on an external computer, connected to the INC through a PCIe connection that connects it to node (000). The node (000) acts as a gateway to the 3D mesh network and enables us to send neural network weights, game ROMs, and start games. It also allows us to gather results from the 832 instances of the fitness evaluation module that are scattered across the 3D mesh network.
The Genetic Algorithm we describe in Algorithm \[alg:ga\] is largely based upon [@such2017deep]. It only includes mutation and selection. Each generation has a population $\mathcal{P}$ that is composed of $N$ individuals. To iterate to the next generation, the top $T$ fittest individuals are selected as parents of the next generation (truncation selection). Each offspring individual is generated from a randomly selected parent with parameters vector $\theta$, to which a vector of random noise is added (mutation) to form the offspring’s parameters vector $\theta' = \theta + \sigma \epsilon$, where $\sigma$ is a mutation power hyper-parameter, and $\epsilon$ is a standard normal random vector. Moreover, the fittest parent (elite) is preserved (i.e. unmodified) as individual for the subsequent generation.
mutation power $\sigma$, population size $N$, number of selected individuals $T$, Xavier random initialization [@glorot2010understanding] function $xi$, standard normal random vector generator function $snrv$, fitness function $F$. ${\theta}^{g=1}_i = xi()$ $k = \text{uniformRandom}(1, T)$ $\theta^{g}_i = \theta^{g-1}_{k} + \sigma * snrv() $ Evaluate $F_i = F(\theta^{g}_i)$
Sort $\theta^{g}_i$ with descending order by $F_i$ Set Elite Candidates $C \leftarrow \theta^{g=1}_{1...T}$ Set Elite Candidates $C \leftarrow \theta^g_{1...T}\cup \{\text{Elite}\}$ Set Elite $\leftarrow \operatorname*{arg\,max}_{\theta\in C} \frac{1}{5}\sum_{j=1}^{5}{F(\theta)}$ $\theta^{g} \leftarrow [\text{Elite}, \theta^{g}-\{\text{Elite}\}]$
Experiments {#sec:experiments}
===========
We chose to run the training on 59 out of the 60 games evaluated in [@machado2018revisiting], excluding Wizard Of Wor, which presented some bugs on our Atari 2600 core. The training was carried out in 5 separate experiments to measure the run-to-run variance. Moreover, because the game environment is stochastic, during each run we average the fitness scores of the $T$ fittest individuals over 5 evaluations before selecting the $E$ elites out of those. This procedure helps with generalization of the trained agents. The hyper-parameters of the Genetic Algorithm are presented in table \[tab:hyperparams\].
---------------------------------- -------------
Population size ($N$) $1000+1$
Truncation size ($T$) $20$
Number of elites ($E$) $1$
Mutation power ($\sigma$) $0.002$
Survivor re-evaluations $5$
Maximum game time per evaluation $5$ minutes
---------------------------------- -------------
: Experimental Hyper-Parameters. Most Were Chosen to Be The Same as in [@such2017deep].
\[tab:hyperparams\]
A subset of the results is summarized in Table \[tab:table\_results\_sample\], with the corresponding training plots in Fig. \[fig:training\_plots\_sample\]. The complete table of results is available in Appendix \[sec:appendix\_all\_results\] in Table \[tab:table\_results\], along with all the learning plots in Fig. \[fig:training\_plots\]. All of our performance numbers are based on the average and variance over 5 training runs, where each run’s performance is based on the average score of the best individual, which was evaluated 5 times. We are comparing with DQN (as does [@such2017deep]) experiments carried-out in [@machado2018revisiting] that use sticky actions as a source of stochasticity as we do. We are also comparing with the results from [@such2017deep], which implements very similar experiments in software, with a larger neural network, with the caveat that it uses initial no-ops as a source of stochasticity.
We are also comparing the approximate wall-clock duration needed to complete a single training experiment with the corresponding algorithms and number of frames. We have measured an evaluation speed of $\sim$ 1 million frames per second, or about 25% slower than the maximal theoretical speed derived in section \[sec:inc\]. This is despite running several experiments in parallel to maximize resource utilization and it is largely due to overheads coming from the host computer running the algorithm and communicating with the individual nodes of the INC. Indeed, the current implementation is polling the status of the nodes, and has to send a new set of weights and load the Atari with a new game ROM before starting to evaluate a new individual. This could be further optimized in the future, however, for the current work we chose to avoid the added complexity.
\[tab:table\_results\_sample\]
{width=".9\linewidth"}
Discussion
==========
The success of a simple GA algorithm in solving complex RL problems was a surprising result [@such2017deep] and attracted more research in this area including this work. One of the hypotheses is the improved exploration compared to gradient-based methods. Potentially GA can avoid being stuck in local minima unlike gradient methods which require additional tricks (e.g., momentum). The promise of GA for training deep neural networks on reinforcement learning problems also depends on the computational resources. Even though [@such2017deep] showed that the wall clock time can be an order of magnitude smaller compared to RL in learning to play Atari games, the data efficiency does not compare favorably against modern RL methods (e.g. billions of game frames for GA vs. hundreds of millions for algorithms such as A3C). In our work, we attempted to accelerate the game environment and the neural network inference in order to alleviate this bottleneck.
Distributed hardware such as CPUs in the cloud data centers or custom built systems such as ours are a naturally good fit for GA type population-based optimization methods. Depending on the application, computation vs. communication time needs to be considered carefully. For example, for game playing, a significant portion of the time is spent during the game itself, which results in a long sequence of inference of game frames and actions. Communicating game scores and updating neural network weights are sparse in comparison. Therefore, rather than accelerating the genetic algorithm, acceleration of the game environment and the inference can make a big difference as our results have shown.
The analysis of the game scores agrees with the findings of [@such2017deep] and shows that the simple approach of the GA is competitive against a basic RL model such as DQN. Our GA experiments surpass DQN on 30 out of 59 games for an equal number of 200 million training frames, while taking 3 orders of magnitude less wall clock time. When not taking data efficiency into account, GA with 6 billion training frames surpasses DQN with 200 million training frames in 36 out of 59 games, while still taking about 2 orders of magnitude less wall clock time.
Compared to [@such2017deep], which demonstrated results on thirteen games, we obtained results for 59 games up to six billion frames. Our implementation is about twice as fast as the one in [@such2017deep], which used 720 CPU cores in the cloud. In all instances, our game scores match [@such2017deep], and in some cases even surpass them. Even though the GA algorithm and the experimental hyper-parameters (e.g. population size, mutation power etc.) were identical, the neural network implementations differed. The most significant difference in our implementation is the removal of a fully connected layer and the drastic reduction in the number of weights ($\sim$134k vs. $\sim$4M). One can speculate that the reduced number of parameters was helpful for the GA optimization, however, this needs to be confirmed with an ablation study in the future. Moreover, to introduce stochasticity, we used sticky actions rather than introducing no-ops at the beginning of the game as in [@such2017deep]. Indeed, as we have observed experimentally, GA trained models using the random 30 no-op would not generalize to slight perturbations in the game environment, thus invalidating the performance of the trained model. This confirms the findings of [@machado2018revisiting] that the random 30 no-op randomization is obsolete, and supports our decision to only present results using sticky actions.
We note that GA failed at hard exploration games such as Montezuma’s Revenge or Pitfall. More interestingly, we also note that for games such as Pitfall, Tennis and Double Dunk, the failure was due to the greediness of the algorithm, where initial exploration of the game’s mechanics induces a negative score. Therefore the adopted solution is not to act on the game such that the score remains at 0. Pong and Ice Hockey were not affected because the player is not in control of the ball’s service.
Conclusion
==========
In this work, we have shown the acceleration of the fitness evaluation of neural networks playing Atari 2600 games using FPGAs. Our results were obtained on the recently built IBM Neural Computer, a large distributed FPGA system, demonstrating the advantage of whole application acceleration. We used that acceleration with a Genetic Algorithm from [@such2017deep] applied to training a deep neural network on Atari 2600 games. Compared to the CPU implementation of the neural network in [@such2017deep], the FPGA implementation used a significantly smaller network with quantized weights and activations. The improvements in the game scores compared to [@such2017deep] might be due to these differences, which is worth further investigations. Our results successfully demonstrated that the GA, as a gradient-free optimization method, is an effective way of leveraging the power of hardware that is optimized for limited precision computing and neural network inference. We hope to leverage the accelerator to pursue research on gradient-free optimization methods. Moreover, we are convinced that significant further acceleration and efficiency gains could be achieved with state of the art FPGAs (the Xilinx Zynq-7000 family was released in 2011).
Acknowledgements {#acknowledgements .unnumbered}
================
This paper and the research behind it would not have been possible without the exceptional work and dedication of Chuck Cox (IBM Research) who designed and built the INC system. The authors would also like to thank Winfried Wilcke (IBM Research) for his leadership, support and constant encouragement. Some of the early experiments were run by Miaochen Jin (University of Chicago) during his internship at IBM Research. The authors would like to acknowledge Kamil Rocki (previously at IBM Research) who contributed to the project during its conception.
Results on the 59 games {#sec:appendix_all_results}
=======================
\[tab:table\_results\]
{width="\linewidth"}
[^1]: <https://github.com/MiSTer-devel/Main_MiSTer/wiki>
[^2]: Also known as AccDNN, available at <https://github.com/IBM/AccDNN>
|
---
address:
- 'R. M.: Department of Mathematics, Stanford University, Stanford, CA 94305'
- 'A. V.: Department of Mathematics, Massachusetts Institute of Technology, MA 02139'
author:
- Rafe Mazzeo and András Vasy
bibliography:
- 'sm.bib'
date: 'November 29, 2000'
title: Resolvents and Martin boundaries of product spaces
---
macro.tex
[^1]
Introduction
============
Geometric scattering theory, as espoused in [@RBMGeo], is the study of natural operators such as the Laplacian on non-compact Riemannian manifolds $X$ with controlled asymptotic geometry using geometrically-informed, fully (i.e. to the extent it is possible) microlocal methods. The goals of this subject include the construction of an analytically useful compactification of $X$ and the definition of an appropriate class of pseudodifferential operators containing the resolvent of the Laplacian, the Schwartz kernel of which is a particularly simple distribution on this compactification. In this paper we examine the case where $X$ is a product of asymptotically hyperbolic (or conformally compact, as they are often called) spaces from this point of view. This is intended both as an initial application of these methods to higher rank symmetric spaces and their geometric generalizations, and also as a relatively simple (although still surprisingly complicated) example which should provide a guide for what to expect in further development in this area.
The results here tend to be notationally complicated, so for the purposes of this introduction we state a simple, yet representative, result concerning the asymptotics of the resolvent applied to a Schwartz function. Suppose that $(M_j,g_j)$, $j=1,2$, are (conformally compact) asymptotically hyperbolic, $\dim M_j= k_j+1$, $H_j=-\Delta_{g_j}$ positive, $\phi_{ji}$ are the $L^2$ eigenfunctions of $H_j$ with eigenvalue $\lambda_{ji} < k_j^2/4$. The usual compactification of $M_j$ as a $\Cinf$ manifold with boundary is denoted ${\overline{M}}_j$, and has boundary defining function $x_j$. Adjoining $\rho_j=
-1/\log x_j$ to the smooth structure of ${\overline{M}}_j$ yields the logarithmic blow up $({\overline{M}}_j)_{\log}$. Let $H$ be the sum of the Laplacians from the two factors: $H=H_1\otimes\Id+\Id\otimes H_2$, and $R(\mu)$ the resolvent, $(H - \mu)^{-1}$. Define $$\label{eq:def-Xt}
\Xt=[({\overline{M}}_1)_{\log}\times({\overline{M}}_2)_{\log};\pa({\overline{M}}_1)_{\log}
\times\pa({\overline{M}}_2)_{\log}],$$ which is a resolution of ${\overline{X}}={\overline{M}}_1\times{\overline{M}}_2$. By definition, each $\rho_j=-1/\log x_j$ is smooth on $({\overline{M}}_j)_{\log}$ and thus $\rho$, where $\rho^{-1}=\sqrt{\rho_1^{-2}+\rho_2^{-2}}$ is smooth on $\Xt$.
(See Theorem \[thm:cont-spec-asymp\].) Suppose $f\in\dCinf({\overline{X}})$, $\Real \ni \mu>k^2/4=(k_1^2+k_2^2)/4$. Then on $\Xt$: $$\begin{split}
& R(\mu-i0)f=x_1^{k_1/2}x_2^{k_2/2}\exp (-i\sqrt{\mu-k^2/4}/\rho)g\\
+ & \sum_{i=1}^{N_1}x_2^{k_2/2+
i\sqrt{\mu-\lambda_{1i}-k_2^2/4}}(\phi_{1i} \otimes g_{1i})
+\sum_{i=1}^{N_2}x_1^{k_1/2+
i\sqrt{\mu-\lambda_{1i}-k_2^2/4}}(g_{2i}\otimes \phi_{2i}),
\end{split}$$ $g$ polyhomogeneous on $\Xt$, $g_{1i}$ and $g_{2i}$ are polyhomogeneous on $\bar M_1$ and $\bar M_2$ respectively.
The leading term of each part in the asymptotics can be described explicitly. Since $\phi_{ji}$ decays like $x_j^{k_j/2+\sqrt{k_1^2/4-\lambda_{1i}}}$, the first term dominates the other two in the interior of the front face of the blow-up ; the second term, involving the eigenfunctions of $H_1$, is only comparable to it at the lift of ${\overline{M}}_1\times\pa{\overline{M}}_2$, while the third term, involving the eigenfunctions of $H_2$, is only comparable to it at the lift of $\pa{\overline{M}}_1\times{\overline{M}}_2$.
Similar results are valid outside the continuous spectrum, where the exponents are not pure imaginary, but care needs to be taken as to which eigenvalue terms appear in the asymptotics. Moreover, additional singularities appear along the front face of $\Xt$ when $\mu$ is real, below the spectrum of $H$, which, however, can be resolved by further real blow-ups (or understood as a type of Legendre singularity). We refer to Theorem \[thm:res-set-asymp\] for the detailed statement of the results in that case. We deduce similar results for the resolvent kernel, which we use in turn to analyze the Martin compactification ${\overline X}_M$of $M_1\times M_2$. While the latter behaves (nearly) as expected when $H_1$ and $H_2$ have no $L^2$ eigenvalues, it experiences a substantial collapse in the presence of such eigenvalues.
There is a natural continuous surjection $\Xt\to{\overline X}_M$. Moreover, the following hold.
1. If neither $H_1$ nor $H_2$ have $L^2$ eigenvalues, the restriction of this map to a neighborhood of the front face of the blow-up in is injective. In general, its injectivity on the ‘side faces’ of $\Xt$ depends on properties of the spherical functions, i.e. on $\frac{d}{d\tau}|_{\tau=0}
R_j(k_j^2/4+\tau^2)$.
2. If both $H_1$ and $H_2$ have $L^2$ eigenvalues, the Martin boundary $\pa{\overline X}_M$ is of the form $\pa {\overline{M}}_1\cup\pa{\overline{M}}_2\cup\pa {\overline{M}}_1\times\pa{\overline{M}}_2\times I$, $I$ an open interval. This set is naturally identified with a collapsed version of $\Xt$, and the map $\pa\Xt\to\pa{\overline X}_M$ factors through it.
There is an extensive literature on scattering on conformally compact manifolds, including especially the important special case of convex cocompact (and geometrically finite) hyperbolic manifolds. This contains geometric constructions of the resolvent, scattering matrix and generalized eigenfunctions, as well as trace formulæ and asymptotics for the counting function for resonances. For brevity, we list only [@Mazzeo-Melrose:Meromorphic], [@Mazzeo:Hodge], [@Mazzeo:Unique], [@Perry:Laplace-I], [@Hislop:Geometry], [@Zworski:Dimension] for references. A parallel theory for complex hyperbolic manifolds and their perturbations is initiated in [@Epstein-Melrose-Mendoza:Resolvent], and while not appearing explicitly, this theory extends to manifolds with the asymptotic structure of quaternionic hyperbolic spaces and the Cayley plane.
Concerning higher rank noncompact symmetric spaces (and in less generality, their quotients too), the compactification theory is now well-understood [@Guivarch-Ji-Taylor:Compactifications], as well as at least some aspects of the analysis of the Laplacian. However, the current methods here rely heavily on the special algebraic structure of these spaces, so extensions of these results to even relatively modest geometric perturbations of these spaces are basically not understood at all.
Alongside this is the study of the asymptotically Euclidean scattering metrics initiated in [@RBMSpec], [@RBMZw]. This theory extends the extensive classical literature, but has led to a new and detailed understanding of quantum N-body scattering where the geometry has much in common with that of flats in non-compact symmetric spaces, see e.g. [@Vasy:Structure], [@Hassell:Plane], [@Vasy:Propagation-2] and [@Vasy:Bound-States].
These different geometric settings have led to the understanding of diverse analytic phenomena which may sometimes be traced to the effects of the asymptotically flat or negative curvature. One of the attractions of studying higher rank symmetric spaces from the point of view of geometric scattering theory is to isolate the specific ways in which the flat and negatively curved directions interact with one another and affect the analysis. The simplest setting where these sorts of effects might be seen is on the product ${\overline{X}}= {\overline{M}}_1 \times {\overline{M}}_2$, where both factors $({\overline{M}}_j,g_j)$ are conformally compact metrics. Recall that $({\overline{M}},g)$ is conformally compact if ${\overline{M}}$ is a smooth compact manifold with boundary, such that for some defining function $x$ for ${\partial}{\overline{M}}$, $g$ takes the form $(dx^2 + h)/x^2$, where $h$ is a nonnegative smooth symmetric $2$-tensor which restricts to a nondegenerate metric on ${\partial}{\overline{M}}$. (Note, however, that only the conformal class of $h$ on ${\partial}{\overline{M}}$ is well-defined from $g$.) The prototype of a conformally compact manifold is hyperbolic space, and so the prototypes for the spaces we consider here are products of hyperbolic spaces. The reader should be aware, though, that ${\overline{M}}_1 \times {\overline{M}}_2$ is asymptotically like the product of hyperbolic spaces ${\mathbb H}^{n_1} \times {\mathbb H}^{n_2}$ ($n_i =
\dim M_i$) only near ${\partial}{\overline{M}}_1 \times {\partial}{\overline{M}}_2$, but elsewhere may be a rather severe metric and topological perturbation of this model space.
The basic questions we ask here concerning the product space $X$ with metric $g = g_1 + g_2$ concern the resolvent $R_X(\lambda) =
({\Delta}_g - \lambda)^{-1}$ of its Laplacian ${\Delta}_g$. We sometimes write $R_X(\lambda)$ as $R_g(\lambda)$ or simply $R(\lambda)$. This is a holomorphic family of elliptic pseudodifferential operators of order $-2$ in the resolvent set ${\mathbb C}\setminus \spec({\Delta}_g)$, which contains at least the complement of the positive real axis. The precise extent of the spectrum is straightforward to determine from the spectra on either factor, but the first substantial question is to understand the behaviour of $R(\lambda)$ as $\lambda$ approaches the (continuous part of the) spectrum. Existence of a limit, in an appropriate sense, is known as the limiting absorption principle. Even better is the existence of a meromorphic continuation of $R(\lambda)$ beyond the spectrum. Usually, when it exists, this continuation lives on some Riemann surface covering the complex plane and ramified at the thresholds of the spectrum. Meromorphic continuations of this type are known to exist for the Laplacian of conformally compact manifolds [@Mazzeo-Melrose:Meromorphic]. Further important questions concern the geometric space which is a resolution of $X \times X$ obtained by a process of real blow-up, and on which the Schwartz kernel of the resolvent lives as a polyhomogeneous (or more generally, a Legendrian) distribution. Detailed knowledge of this space is central in understanding the finer properties of the resolvent, and conversely is key in its initial geometric construction. Geometric and analytic compactifications of the space $X$ itself are quite relevant to this. There are two well-known compactifications which are closely related to our methods: the geodesic (also known as the conic) compactification, and the Martin compactification, which is defined using function theory, specifically the space of positive solutions of $({\Delta}_g - \lambda)u = 0$ for $\lambda$ real and below the bottom of $\spec({\Delta}_g)$. This Martin compactification is known in many instances, including for the product of hyperbolic spaces [@Giulini-Woess:Martin] and more recently for general symmetric spaces of noncompact type [@Guivarch-Ji-Taylor:Compactifications]. The ‘resolvent compactification’ of $X \times X$ we construct below, and on which the Schwartz kernel of the resolvent lives, has a structure on some of its hypersurface boundary faces which is inherited from the geodesic and Martin compactifications.
These questions together have led to the somewhat modest goals of the present paper. In the next section we present a contour integral formula for $R_X(\lambda)$ in terms of the resolvents of the two factors $R_{M_j}
(\lambda)$. A related expression, written as an integral over the spectral measure, derived in the context in Euclidean scattering, appears in work of Ben-Artzi and Devinatz [@Ben-Artzi-Devinatz:Resolvent]. The representation formula here holds in great generality. In §3 we specialize and review the detailed structure of the resolvent $R_{M}$ when $M$ is a conformally compact manifold; some auxiliary estimates required later are also derived here. An immediate consequence of the representation formula of §2 is the existence of an appropriate meromorphic continuation for the resolvent $R_X(\lambda)$, and we describe this in §4. After this, §5 contains a discussion of general compactification theory, specialized to this context. The main work is done in §6 and §7, where we describe the asymptotics of $R_X(\mu)f$ when $f$ is Schwartz, first when $\mu$ is in the resolvent set and then when $\mu$ is in the main sheet of continuous spectrum of $H$. The first application of this is given in §8, where we construct the ‘resolvent double space’, a resolution of $X \times X$ which carries the Schwartz kernel of $R_X(\mu)$ in as simple a fashion as possible. Finally, in §9, we use the resolvent asymptotics to determine the Martin compactification of $X$.
There are several features of this work to which we wish to draw particular attention. First, the main tool in deriving the resolvent asymptotics is stationary phase, which is in many respects local. The previous identification of the Martin compactification (at least for the product of hyperbolic spaces) in [@Giulini-Woess:Martin] relies heavily on global heat kernel bounds, which we feel are intrinsically more complicated. Note also that by using stationary phase we are taking advantage of the oscillatory nature of the resolvent, even when studying it for certain values of $\mu$ where it has been more traditional to rely on ‘positivity methods’ such as the maximum principle, the Harnack inequality, etc. Another interesting feature here is the surprisingly complicated way the existence of bound states, i.e. $L^2$ eigenvalues, for the Laplacian on either factor affects the asymptotics and the structure of the Martin boundary. Finally, we have given a detailed description of the smooth structure of the various compactifications we construct; this aspect is usually neglected in other discussions of compactification theory, but as we show, plays a significant role.
Our intention is that the investigations here will form the basis for a more thorough investigation of the geometric scattering theory for higher rank spaces.
The authors wish to thank Richard Melrose for helpful advice, and also Lizhen Ji for encouraging us to study the Martin compactification.
Resolvent formula
=================
Let $H_1$, $H_2$ be self-adjoint operators on Hilbert spaces $V_1$ and $V_2$ which are bounded below; the precise structure of their spectra will be unimportant for the present. We denote $\inf \spec (H_j) =
\lambda_0(H_j)$. Now let $$H=H_1\otimes\Id+\Id\otimes H_2$$ be the self-adjoint operator on the completed tensor product space $V_1 \hat \otimes V_2$. Then $\inf\spec(H) \equiv \lambda_0(H) =
\lambda_0(H_1) + \lambda_0(H_2)$. Let $$R_j(\mu_j) = (H_j - \mu_j)^{-1}, \quad j = 1,2, \qquad \mbox{and}\quad
R(\mu) = (H - \mu)^{-1}$$ be the resolvents of the $H_j$ and $H$, respectively. This setting has been investigated by Ben-Artzi and Devinatz in [@Ben-Artzi-Devinatz:Resolvent], mostly from the point of view of the limiting absorption princple, i.e. the existence of the boundary values $R(\mu\pm i0)$, $\mu\in[\lambda_0(H_1)+
\lambda_0(H_2),\infty)$ on suitable weighted spaces, under the assumption that the limiting absorption principle holds for the $R_j$ individually.
One of our first goals is to show that if both $R_j$ admit meromorphic continuations, then $R(\mu)$ does as well. Later we wish to obtain precise asymptotics for the Schwartz kernel of $R(\mu)$ for $\mu$ in the resolvent set and in the continuous spectrum. In this section we derive a representation of $R(\mu)$ as a contour integral which will be useful for both of these purposes.
So, fix $\mu$ in the resolvent set ${\mathbb C}\setminus [\lambda_0(H_1)+ \lambda_0
(H_2),\infty)$, and let $\gamma$ be a parametrized curve in the complex plane which is disjoint from $[\lambda_0(H_1),\infty)$ and such that $\mu-\gamma$ is disjoint from $[\lambda_0(H_2),\infty)$, or in other words, such that $\gamma$ does not intersect $[\lambda_0(H_1),\infty)\cup(\mu-[\lambda_0(H_2),
\infty))$. Suppose also that $\gamma(t)=c_\pm t$ for $\pm t\geq T>0$ with $\im c_\pm > 0$. This is illustrated in Figure \[fig:contour\]. The precise values of $c_\pm$ are unimportant for our purposes, and indeed there is considerably more leeway than this in choosing $\gamma$, but the definite separation of $\gamma$ from the spectra, ensured by $\im c_\pm>0$, is crucial. (One can allow slighly subconic separation, but this is of no interest here.) Then we claim that $$R(\mu)=\frac{1}{2\pi i}\int_{\gamma} R_1(\mu_1)
R_2(\mu-\mu_1)\,d\mu_1.
\label{eq:resform}$$
To prove this formula, first observe that since $$\|R_j(\mu_j)\|\leq |\im\mu_j|^{-1},$$ the norm of the integrand in (\[eq:resform\]) (as a bounded operator on $L^2$) is estimated by $C(1+|t|)^{-2}$, and hence the integral converges. Next, note that it suffices to show that both sides of (\[eq:resform\]) produce the same result when restricted to the range of $\chi_I(H_1)\otimes
\Id$ where $I$ is any compact interval and $\chi_I$ its characteristic function, because the union of these ranges is dense.
Fixing the interval $I$, the integrand $R_1(\mu_1)R_2(\mu - \mu_2)$ is holomorphic for $\mu_1 \notin I\cup(\mu-\spec(H_2))$. Therefore we may deform the contour to one which is the union of two curves, $\hat{\gamma}$ and $\gammat$, where $\hat{\gamma}$ agrees with $\gamma$ for $t\geq T'>0$ and intersects the real axis precisely once, somewhere in $(\sup I,\infty)$, while $\gammat$ surrounds $I$ once. Thus $$\begin{split}
\frac{1}{2\pi i}&\int_{\gamma} R_1(\mu_1)
R_2(\mu-\mu_1)\,d\mu_1\\
&= \frac{1}{2\pi i}\int_{\hat{\gamma}} R_1(\mu_1)
R_2(\mu-\mu_1)\,d\mu_1+ \frac{1}{2\pi i}\int_{\gammat} R_1(\mu_1)
R_2(\mu-\mu_1)\,d\mu_1.
\end{split}$$
But now, if $\hat{\gamma}$ is moved to infinity, the first integral on the right tends to 0, as follows by directly estimating the integral. On the other hand, letting $\gammat$ tend to $I$ and applying Stone’s theorem, the second integral on the right is the same as $$\int_I R_2(\mu-\mu_1)\,dE_1(\mu_1).$$ This is identical to $R(\mu)$ on the range of $\chi_I(H_1) \otimes \Id$ since applying $H-\mu$ to it gives (with a slight abuse of notation) $$\int_I (H_1 - \mu_1 + H_2 - (\mu - \mu_1)) R_2(\mu - \mu_1)\, dE_1(\mu_1).$$ $$= \int_I ((H_1-\mu_1)R_2(\mu-\mu_1)+\Id)\,dE_1(\mu_1)
=\int_I\,dE_1(\mu_1) = \chi_I(H_1)\otimes\Id,$$ i.e. the identity on this subspace. Thus (\[eq:resform\]) is established.
Resolvents of conformally compact manifolds {#sec:conf-compact}
===========================================
In this section we briefly collect some facts about the resolvent family $R_M(\lambda)$, which we abbreviate simply as $R(\lambda)$ for the duration of this section, when $(M,g)$ is a conformally compact manifold. These are all discussed and proved in [@Mazzeo-Melrose:Meromorphic], to which we refer for all details.
Recall that $M$ is identified with the interior of a compact smooth manifold with boundary ${\overline{M}}$. The compactification ${\overline{M}}$ is geometrically natural: ${\overline{M}}$ may also be identified with the both the geodesic and Martin compactifications, as we discuss further below. Locally, $R(\lambda)$ is a pseudodifferential operator of order $-2$, but our focus is on understanding the behavior of its Schwartz kernel $K(z,z')= K(\lambda;z,z')$, $z, z' \in M$, when one or both of the variables tend to infinity in $M$, i.e. to a point of ${\partial}M$. This can occur in various ways, of course, and the most efficient way to encode this information is to consider $K$ as a distribution on the $0$-stretched product ${\overline{M}}_0^2$ introduced in [@Mazzeo-Melrose:Meromorphic] and [@Mazzeo:Hodge]. This space is obtained from ${\overline{M}}^2$ by blowing up the boundary of the diagonal ${\partial}\Delta
\iota = {\partial}\{z = z'\}$; equivalently, ${\overline{M}}^2_0 = [{\overline{M}}^2; {\partial}\Delta
\iota]$ is the disjoint union of ${\overline{M}}^2 \setminus {\partial}\Delta \iota $ with the interior spherical normal bundle of ${\partial}\Delta \iota$ at the corner ${\partial}{\overline{M}}\times {\partial}{\overline{M}}$, this set then being endowed with the minimal ${\mathcal C}^\infty$ structure containing the lifts of smooth functions on ${\overline{M}}^2$ and polar coordinates around ${\partial}\Delta \iota$.
${\overline{M}}^2_0$ has three different boundary hypersurface faces: the left face which covers (indeed, is identified with) ${\partial}{\overline{M}}\times {\overline{M}}$, the right face which covers ${\overline{M}}\times {\partial}{\overline{M}}$, and the new front face covering ${\partial}\Delta \iota$. We denote these $B_{10}$, $B_{01}$ and $B_{11}$, and their boundary defining functions $\rho_{10}$, $\rho_{01}$ and $\rho_{11}$, respectively. Writing $z = (x,y)$ and $z' = (x',y')$ near ${\partial}{\overline{M}}$, then $$\rho_{11} = \sqrt{x^2 + (x')^2 + |y-y'|^2},
\qquad \mbox{and} \qquad
\rho_{10} = x/\rho_{11}, \quad \rho_{01} = x'/\rho_{11}.$$
If $\dim M = n = k+1$, then write $\lambda = \zeta (k - \zeta)$, where by convention the region ${\mbox{\rm Re}\,}\zeta > k/2$ corresponds to the resolvent set ${\mathbb C}\setminus [k^2/4,\infty)$ of ${\Delta}_g$. Thus for $\lambda$ in the resolvent set, $$\zeta=k/2+i\sqrt{\lambda-k^2/4},$$ where the branch of the square root is chosen so that its imaginary part is negative.
(Mazzeo and Melrose,[@Mazzeo-Melrose:Meromorphic Theorem 7.1]) For $\zeta$ in the half-plane ${\mbox{\rm Re}\,}\zeta > k/2$, the Schwartz kernel $K_\zeta$ of $R(\zeta(k - \zeta))$ is a polyhomogeneous distribution on ${\overline{M}}^2_0$ with the following properties. First, $R(\zeta(k - \zeta))$ has a decomposition $$R(\zeta(k - \zeta))=R'(\zeta(k - \zeta))+R''(\zeta(k - \zeta)),$$ where $R'$ is an element in the small calculus $\Psi_0^{-2}({\overline{M}})$ of $0$-pseudodifferential operators on $M$ and $R''$ is a residual element in the large calculus $\Psi_0^{-\infty,\zeta,\zeta}({\overline{M}})$ of $0$-pseudodifferential operators. This means that the Schwartz kernel of $R'$ has a standard polyhomogeneous singularity corresponding to pseudodifferential order $-2$ at the lifted diagonal $\Delta \iota_0$ in ${\overline{M}}_0^2$ and vanishes to infinite order along $B_{10}$ and $B_{01}$, while the Schwartz kernel of $R''$ takes the form $$\rho_{10}^\zeta \rho_{01}^\zeta F'',\quad F''\in\Cinf({\overline{M}}^2_0;\pi_R^*\Omega_0);$$ here $\pi_R^*\Omega_0$ is the lift of the 0-density bundle from the right factor (a non-vanishing section of which is given by the Riemannian density $dV_g$). $R(\zeta(k-\zeta))$ is holomorphic, both as a map into the space of bounded operators on $L^2$ and also into the space of distributions on ${\overline{M}}_0^2$, for ${\mbox{\rm Re}\,}\zeta > k/2$, and extends meromorphically, as a function with values in the space of distributions on ${\overline{M}}_0^2$, to the complex plane when $k$ is even, and to $\Cx \setminus -{\mathbb N}$ when $k$ is odd, with all poles of finite rank. Because it is constructed using only the symbol calculus, $R'$ is holomorphic in $\lambda$. Finally, the restrictions of $\rho_{10}^{-\zeta}K_\zeta$ to the left face $B_{10}$ and of $\rho_{01}^{-\zeta}K_\zeta$ to the right face $B_{01}$ are nonvanishing for all $\zeta$ with ${\mbox{\rm Re}\,}\zeta > k/2$. \[th:ccst\]
It will be convenient below to write $$K_\zeta = \rho_{10}^\zeta \rho_{01}^\zeta F$$ where $F$ is smooth on ${\overline{M}}^2_0$, apart from its conormal singularity at the lifted diagonal $\Delta \iota_0$.
For us, the main import of this theorem is its conclusion that $K_\zeta$ simple polyhomogeneous behavior on the space ${\overline{M}}^2_0$. One of our ultimate goals here is to find a resolution of the space $X^2$, where $X$ is the product of two conformally compact manifolds, on which the Schwartz kernel for the resolvent of its Laplacian is also simple.
In the next sections we shall require uniform weighted $L^2$ estimates for this resolvent $R(\lambda)$ as $\im\lambda\to\infty$, and we now show how these follow from the parametrix construction in ([@Mazzeo-Melrose:Meromorphic]). We set $H = -\Delta_g$ here.
For any $s > 0$ and all $\zeta$ with ${\mbox{\rm Re}\,}\zeta > k/2 + s$ there is a constant $C_s$ which is independent of $\zeta$ and such that $$R(\zeta(k-\zeta)): x^s L^2(M,dV_g) \longrightarrow x^s L^2(M,dV_g)
\label{eq:unifwl2o}$$ is bounded and satisfies $$\| R(\zeta(k-\zeta))\|_{\bop(x^s L^2,x^sL^2)} \leq \frac{C_s}{|\im
\zeta(k-\zeta)|}
\label{eq:unifwl2e}$$ for all $\zeta$ in this half-plane. \[th:unifwl2e\]
The proof has a few steps. Setting $\lambda = \zeta(k-\zeta)$, the parametrix $P(\lambda)$ for $H - \lambda$ constructed in [@Mazzeo-Melrose:Meromorphic] satisfies $$P(\lambda)(H-\lambda)=\Id+E(\lambda),\quad
(H-\lambda)P(\lambda)=\Id+F(\lambda),$$ where $E(\lambda)$ and $F(\lambda)$ are residual, and is related to $R(\lambda)$ by the formula $$\label{eq:parametrix-identity}
R(\lambda)=P(\lambda)-E(\lambda)P(\lambda)+E(\lambda)R(\lambda)F(\lambda).$$ The desired uniform estimate for $R(\lambda)$ then follows from the standard uniform $L^2$ estimate $$\|R(\lambda)\|_{\bop(L^2,L^2)}
\leq \frac{1}{|\im\lambda|}, \qquad\lambda\in\Cx\setminus\spec(H),$$ and from appropriate uniform estimates for $P(\lambda)$, $E(\lambda)$ and $F(\lambda)$ in weighted spaces which we now derive.
The main work involves demonstrating a uniform estimate in weighted spaces for the resolvent of the Laplacian in hyperbolic space, and so we turn to this first. Uniform boundedness of $$R(\zeta(k-\zeta)): x^s L^2({\mathbb H}^{k+1}; dV) \longrightarrow
x^s L^2({\mathbb H}^{k+1}; dV)$$ is equivalent to the uniform boundedness of the conjugated operator $x^{-s}R(\zeta(k-\zeta))x^s$ on $L^2({\mathbb H}^{k+1}; dV)$. The Schwartz kernel of this conjugated operator is $$K^s_\zeta(z,z') = K^0_\zeta(z,z')(x'/x)^s = K^0_\zeta(z,z')(\rho_{01}/
\rho_{10})^s,$$ where $K^0_\zeta(z,z')$ is the Schwartz kernel of the resolvent of the Laplacian in ${{\mathbb H}^{k+1}}$. Recall from [@Mazzeo-Melrose:Meromorphic] that there is an explicit formula for $K^0_\zeta$: $$\begin{aligned}
K^0_\zeta(\delta) =
\label{eq:skhs} \\
c_k\left(\frac{1}{\sinh \delta}
\frac{{\partial}\,}{{\partial}\delta}\right)^{\frac{k-2}{2}}
\left(\frac{1}{\sinh \delta} e^{-(\zeta - k/2)\delta}\right),
&\quad &k\ \mbox{even}, \nonumber \\
c_k\int_0^\infty e^{-(\zeta - k/2)\omega}
(\cosh \omega - \cosh \delta)_+^{-k/2}\, d\omega, &\quad &k\ \mbox{odd}.
\nonumber\end{aligned}$$
Here we use that the resolvent is a point-pair invariant, so $R(\lambda;z,z')$ only involves the Riemannian (hyperbolic) distance between $z$ and $z'$, and equivalently, may be written in terms of the elementary point-pair invariant $\delta(z,z')$ defined by $$\cosh \delta(z,z') = 1 + \frac{|z-z'|^2}{2xx'}.$$
There are two regions in $({\mathbb H}^{k+1})^2_0$ where $K^s_\zeta$ behaves slightly differently as $\zeta\to\infty$. These are one near the diagonal, say $\delta(z,z')\leq 1$, and one away from the diagonal, say $\delta(z,z')\geq 1$. We use a partition of unity to divide the Schwartz kernel into two parts. It suffices to prove $L^2$ boundedness of each piece separately. In the former region, $\delta(z,z')\leq 1$, the factor $(\rho_{01}/\rho_{10})^s$ is bounded, independent of $\zeta$, hence the uniform $L^2$ boundedness of $K^s_\zeta$, and thus can be proved just as the $L^2$ boundedness of $K^0_\zeta$, directly from properties of the Schwartz kernel. More explicitly, the result in this region may be deduced either by invoking the standard extension of the symbol calculus with spectral parameter, or else simply by direct calculation. On the other hand, in $\delta(z,z')\geq 1$, $K^s_\zeta$ decays exponentially as $\zeta\to\infty$, so the additional factor $(\rho_{01}/\rho_{10})^s$, which takes the form $e^{s(\delta(p,z)-\delta(p,z'))}$, where $p$ is some fixed point in the interior of ${\mathbb H}^{k+1}$, does not make much difference: the Schwartz lemma shows the desired bound (and in fact better bounds for this piece!).
To proceed, we recall that $P(\lambda)$ is constructed in stages, and as a sum of two terms, $P(\lambda) = P_1(\lambda) + P_2(\lambda)$. Here $P_1(\lambda)$ is in the small calculus, and is a slight modification of the operator $P_1'(\lambda)$ obtained from the use of the symbol calculus to solve away the conormal singularity along $\Delta \iota_0$. By standard methods, such a $P_1'(\lambda)$ may be constructed so as to depend holomorphically on $\zeta\in \Cx$, and to have uniformly bounded norm on $L^2$. It also acts on $x^s L^2$ for each $s$, with norm depending on $s$, but not $\zeta$. Next, $P_2(\lambda)$ is obtained after solving away the first $\ell$ terms of the Taylor series expansion of the error term $F_1(\lambda) = I - (H-\lambda)P'_1(\lambda)$, $F_1(\lambda) \in \Psi_0^{-\infty}(M)$, where $\ell$ is sufficiently large (greater than $s$). This approximate solution of $(H-\lambda)
P_2(\lambda) = F_1(\lambda)$ is found as follows: first solve the normal equation $$(N(H) - \lambda)N(P_2(\lambda)) = N(F_1(\lambda)),$$ which is the restriction of the equation to the front face $B_{11}$. But $N(H)$ is naturally identified with the Laplacian on ${\mathbb H}^{k+1}$, and so we may apply the result of the discussion above to choose $P_2'(\lambda)$ with this property, which satisfies the estimate (\[eq:unifwl2e\]), and such that $F_2'(\lambda) = F_1(\lambda)
- (H-\lambda)P_1'(\lambda)$ vanishes to first order along $B_{11}$. Thus $$(H-\lambda)(P_1'(\lambda) + P_2'(\lambda)) = I - F_2'(\lambda),
\qquad F_2'(\lambda) \in \Psi^{-\infty,\zeta+1,\zeta}_0(M).$$ Applying the first $\ell$ terms of the Neumann series for $(I -
F_2'(\lambda))^{-1}$, $$I + S^{(\ell)} = I + F_2'(\lambda) + \ldots + (F_2'(\lambda))^{\ell-1}$$ on the right on both sides of this equation yields $$(H - \lambda)(P_1(\lambda) + P_2(\lambda)) = I - F_2(\lambda),$$ where $$P_j(\lambda) = P_j'(\lambda)(I + S^{(\ell)}(\lambda)), \ \ j = 1,2,
\quad
\mbox{and} \qquad \ \ F_2(\lambda) \in \Psi_0^{-\infty,\zeta + \ell,\zeta}(M).$$ Clearly $$\|F_2(\zeta(k-\zeta))\|_{\bop(x^s L^2,L^2)} \leq \frac{C}{|{\mbox{\rm Im}\,}\zeta(k-\zeta)|},$$ with $C$ independent of $\zeta$. We also obtain a remainder term $E(\lambda)$ from applying $H-\lambda$ on the right of $P(\lambda)
= P_1(\lambda) + P_2(\lambda)$, and this satisfies uniform bounds (from $L^2$ to $x^sL^2$) of the same form.
These facts taken all together finish the proof.
Analytic continuation of the resolvent of $H$
=============================================
In this section we shall show how to use the integral formula (\[eq:resform\]) to obtain an analytic continuation for the resolvent $R_X(\mu) = (H-\mu)^{-1}$ on $X = M_1 \times M_2$ past the continuous spectrum of $H = -\Delta_g = H_1 + H_2$. For this continuation we can either regard $R_X(\mu)$ as an analytic family of bounded operators between weighted $L^2$ spaces, or else view its Schwartz kernel as an analytic function with values in an appropriate space of distributions. The key ingredient here is the existence of similar analytic continuations for the resolvents $R_j(\mu)
= (H_j - \mu)^{-1}$. Although this continuation result holds in considerably greater generality than just for products of conformally compact spaces, we shall focus exclusively on this case for the sake of being specific.
We first set up some notation. Let $\dim M_j=k_j+1$. Then it follows from Section \[sec:conf-compact\], cf. also [@Mazzeo-Melrose:Meromorphic] that the spectrum of $H_j$ decomposes into the union of a band of continuous spectrum $ [k_j^2/4,+\infty)$, as well as possibly a finite number of $L^2$ eigenvalues $\lambda_{ji}$, $i = 1, \ldots, N_j$, $j = 1,2$, in $(0, k_j^2/4)$, with the corresponding finite rank eigenprojections $\Pi_{ji}$. Next, each $R_j(\mu_j)$ continues meromorphically from the resolvent set to the Riemann surface $\Sigma$ for $\sqrt{\mu_j-k_j^2/4}$, which we think of as two copies of $\Cx$ attached in the usual way along a cut ${\mathcal C}_j$ extending from $k_j^2/4$ to $\infty$; the resolvent set is identified with the subset of $\Sigma$ where $\im\sqrt{\mu_j-k_j^2/4}<0$. Usually ${\mathcal C}_j$ is taken to be the ray along the positive real axis, but it will be convenient to choose it differently later. As already noted, this continuation of $R_j$ is either as a map into an appropriate spaces of distributions, or else for any given $s>0$, in the region $\im\sqrt{\mu_j-k_j^2/4}<s$, as bounded operators from $x^sL^2$ to $x^{-s}L^2$. In any case, its poles are of finite rank, and we denote them by $\tilde\lambda_{ji}$, with corresponding finite rank residues $\tilde\Pi_{ji}$ (the tildes are meant to distinguish these from the eigendata for $H_j$). Note, however, that these poles of $R_j$ in the nonphysical part of $\Sigma$ need not be simple.
We assume until near the end of the argument that neither $H_j$ has any $L^2$ eigenvalues. Define $k$ by $$\frac{k^2}{4} = \frac{k_1^2}{4} + \frac{k_2^2}{4}.$$ Then $R(\mu)$ is analytic in $\Cx \setminus [k^2/4,\infty)$, and we wish to show that it continues analytically past the continuous spectrum. We do this by deforming the contour of integration $\gamma$ in (\[eq:resform\]) in the following manner. First fix $\mu$ in the resolvent set, so that $R_1(\mu_1)R_2(\mu - \mu_1)$ is defined and analytic for $\mu_1 \notin {\mathcal C}_1 \cup (\mu - {\mathcal C}_2)$, i.e. outside of two horizontal rays, one extending from $k_1^2/4$ to the right and the other from $\mu - k_2^2/4$ to the left. Next, rotate these cuts to rays $\tilde{\mathcal C}_j$ by pivoting them by some angle $\alpha$ counterclockwise around their endpoints. Thus we have ‘exposed’ two sectors of angle $\alpha$ from the nonphysical portion of $\Sigma$, and at the same time concealed an equal portion of the physical part of $\Sigma$. Now it is possible to deform $\gamma$ to a new contour $\gamma'$ which lies partly in the newly uncovered sectors in $\Cx\setminus (\tilde{\mathcal C}_1\cup\tilde{\mathcal C}_2)$, as in Figure \[fig:contour3\]. Finally, $\mu$ can then be moved into the nonphysical region.
We make this a bit more explicit. Suppose that $\mu_0$ is such that $\mu^0-k_1^2/4 \neq \tilde{\lambda}_{1i}$ and $\mu^0-k_2^2/4 \neq
\tilde{\lambda}_{2i'}$ for any $i,i'$, and also $\mu^0 \neq
\tilde{\lambda}_{1i} + \tilde{\lambda}_{2i'}$. Rotate the cuts ${\mathcal C}_j$ by an angle $\alpha > \arg(\mu^0-k_j^2/4)$, $j = 1,2$. Now deform the contour $\gamma$ past $\mu^0-k_2^2/4$ to a new contour $\gamma'$, such that this point is now below $\gamma'$. During the deformation, $\gamma$ may pass through a finite number of poles of $R_1$, yielding residues $\tilde\Pi_{1i}\otimes R_2(\mu-\tilde\lambda_{1i})$. So long as $\mu$ is (sufficiently) far away from ${\mathcal C}_2$, $\mu-\mu_1$ is in the resolvent set of $H_2$, hence no poles of $R_2$ are encountered during this deformation. Thus we have a new representation $$R(\mu)=\frac{1}{2\pi i}\int_{\gamma'}R_1(\mu_1)R_2(\mu-\mu_1)\,d\mu_1
-\sum_{\text{finite}} \tilde\Pi_{1i}\otimes
R_2(\mu-\tilde\lambda_{1i}).$$ Since $\mu$ is in the resolvent set, the left hand side is still a bounded operator on $L^2$, although on the right hand side, one factor of the integrand $R_1(\mu_1)$ is not bounded on $L^2$ along the entire contour.
At this point we merely have a new representation of an operator we already know exists, namely $R(\mu)$ for $\mu$ in the resolvent set, in terms of operators which are only bounded between weighted spaces. However, fixing $\gamma'$, we can now let $\mu$ vary arbitrarily below this curve, and hence across the spectrum $[k^2/4,\infty)$, as long as the ramification point $k_2^2/4$ of $R_2$ does not lie on $\gamma'$. When $\mu-\tilde\lambda_{2i}$ crosses $\gamma'$, the residue term $-R_1(\mu-\tilde\lambda_{2i})\otimes\tilde\Pi_{1i}$ is produced. This yields $$R(\mu)=\frac{1}{2\pi i}\int_{\gamma'}R_1(\mu_1)R_2(\mu-\mu_1)\,d\mu_1$$ $$-\sum_{\text{finite}} \tilde\Pi_{1i}\otimes
R_2(\mu-\tilde\lambda_{1i})
-\sum_{\text{finite}} R_1(\mu-\tilde\lambda_{2i})\otimes\tilde\Pi_{2i};$$ Each of the residue terms here continue meromorphically in $\mu$ past the spectrum, with ramification points at $\tilde\lambda_{1i}+k_2^2/4$ (since $R_2$ is ramified at $k_2^2/4$) and similarly, with the indices $1$ and $2$ interchanged. These terms also have poles at $\lambdat_{1i}+
\lambdat_{2j}$, with finite rank residues.
We have proved
The resolvent $R(\mu)$ for $H$ on $X = M_1 \times M_2$ extends across the cut $[k^2/4,\infty)$ to a meromorphic function (with values in an appropriate space of distributions) on a Riemann surface ${\mathcal T}$ ramified at $\tilde\lambda_{1i}+k_2^2/4$ and $\tilde\lambda_{2i}+k_1^2/4$ (these points are known as Regge poles), and with finite rank poles at $\lambdat_{1i}+\lambdat_{2i}$.
When either $H_1$ or $H_2$ has $L^2$ eigenvalues, then the only difference is that $\gamma'$ must cross these as well. Thus, the analytic continuation can be written as above, but now we must include sums over the $\lambda_{ji}$ too, and there are new ramification points at $\lambda_{1i} + k_2^2/4$ and $\lambda_{2i} + k_1^2/4$.
Compactification constructions
==============================
We now turn to the other major theme of this paper, which is the various ways one might compactify the conformally compact spaces $M_j$ or their product $X = M_1 \times M_2$. The general problem of finding good compactifications of Riemannian manifolds, and in particular of (locally) symmetric spaces, has been an area of active research. We refer to [@Ji:Satake] and [@Guivarch-Ji-Taylor:Compactifications] for more discussion of this in the symmetric space setting, and to [@Anderson-Schoen:Positive] and [@Schoen-Yau:DG] and [@Freire:Martin] for some beautiful results in some general geometric settings.
There are two different compactication constructions we shall discuss here, the geodesic compactification (sometimes also called the conic compactification), as well as the Martin compactification. Each has a key role in understanding different aspects of the global geometry and function theory of a space. We define these now in turn.
When $Z$ is a complete, simply connected manifold of nonpositive curvature (i.e.a Cartan-Hadamard manifold), then the geodesic compactification ${\overline{Z}}$ is obtained by adjoining to $Z$ an ideal boundary ${\partial}{\overline{Z}}$, points of which are equivalence classes of geodesic rays. Two geodesic rays $\gamma(t)$, $\tilde{\gamma}(t)$, $t \geq 0$, are said to be equivalent if $d(\gamma(t),\tilde{\gamma}(t))$ remains bounded as $t \to \infty$. Thus in ${\mathbb R}^n$ any two parallel lines are identified, as are any two geodesics in the ball model of hyperbolic space ${\mathbb H}^n$ which converge to the same point on $S^{n-1}$. If $q \in {\partial}{\overline{Z}}$, then a neighborhood system at $q$ is given by sets of the form $(Z \setminus B_R(p')) \cap
\exp_p( [T,\infty) \times {\mathcal U})$, where ${\mathcal U}$ is an open set in the unit sphere $S_pZ$ in $T_pZ$. A point on this set is on some geodesic $\gamma(t)$ emanating from $p$ with $\gamma'(0) \in
{\mathcal U}$ and $t \geq T$ as well as in the exterior of the ball $B_R(p')$. Thus whenever $Z$ is Cartan-Hadamard, then ${\overline{Z}}$ is homeomorphic to a closed ball $\overline{B^n}$, and its boundary ${\partial}{\overline{Z}}$ is identified with the unit tangent sphere $S_p Z$ for any $p \in Z$. There are fairly obvious modifications of this construction when $Z$ is not necessarily simply connected, or only has nonpositive curvature outside a compact set, and then the structure of ${\overline{Z}}$ is more complicated, but only on account of the topology of $\mbox{int}\, Z$.
It is of substantial interest to understand when the compactification ${\overline{Z}}$ carries more structure than its initial definition as a topological space. In particular, we would like to determine when ${\overline{Z}}$ is naturally defined as a smooth manifold with boundary, or with corners. For example, When $Z = {\mathbb R}^n$ or ${\mathbb H}^n$, then as we have already stated ${\overline{Z}}\approx \overline{B^n}$, but obviously in these two examples, ${\overline{Z}}$ is ‘really’ a smooth closed ball. An interesting and underappreciated feature of this construction is that although it is possible to identify each of these compactifications with a ball, the smooth structures at the boundary are different, as we now explain.
When considering whether ${\overline{Z}}$ admits a natural smooth structure, the first key point is the regularity of the transition maps, as we now describe. Remaining within the context of Cartan-Hadamard manifolds for simplicity, the transition maps are the homeomorphisms between the unit tangent spheres at any two points $p,p' \in Z$, $$S_p Z \longrightarrow {\partial}{\overline{Z}}\longleftarrow S_{p'}Z$$ provided through the geodesic spray from these two points. When $Z$ has curvatures bounded between two negative constants $-a^2$ and $-b^2$, then from [@Anderson-Schoen:Positive] these transition maps are of Hölder class ${\mathcal C}^{0,\alpha}$, $\alpha = a/b$, and accordingly, in this generality, ${\overline{Z}}$ has only a Hölder structure. However, for either of the examples above these transition maps are smooth; this is also true for general conformally compact manifolds for suitable localized versions of these transition maps (see [@Mazzeo:Hodge]). The other key point concerns the choice of a class of defining functions for ${\partial}{\overline{Z}}$ (or, if ${\overline{Z}}$ is to be regarded as a smooth manifold with corners, then for subsets of it which are identified with the various boundary hypersurfaces). Smoothness of the transition maps and choice of defining functions together determine the ${\mathcal C}^\infty$ structure on ${\overline{Z}}$.
There is considerable flexibility in choosing (equivalence classes of) defining functions. For ${\mathbb R}^n$ it is most natural to use the radial compactification with defining function for ${\partial}\overline{{\mathbb R}^n}$, $\rho_1 =1/|z|=1/\mbox{dist}\,(z,0)$, $z\in{\mathbb R}^n$, the inverse of the polar distance variable. On the other hand, the Poincaré ball model for ${\mathbb H}^n$ suggests that the more natural choice now is $\rho_2 = \exp(-\mbox{dist}\,(z,0))$, $0$ any fixed point in ${\mathbb H}^n$. These two defining functions for ${\partial}\overline{B^n}$ are quite different, since $\rho_1 = -1/\log \rho_2$. We say that $(\overline{B^n},\rho_1)$ is the ‘log blow-up’ of $(\overline{B^n},\rho_2)$. For later reference, we shall refer to defining functions defined as the reciprocals of the Riemannian distance function or its exponential as being of polynomial or exponential type, respectively.
The difference between these polynomial and exponential type defining functions appears most clearly in the spaces of functions with polyhomogeneous behaviour at the boundary, since this involves expansions in a discrete set of complex powers of the defining function $\rho$ along with nonnegative integer powers of its logarithm, $\log \rho$. In particular, functions which are polyhomogeneous with respect to exponential-type defining functions, with all exponents having positive real part, are residual, i.e. rapidly decreasing, with respect to the log blow-up structure. The guiding principle for which defining functions to choose as primary is determined by the asymptotic behaviour of eigenfunctions of the Laplacian; one prefers eigenfunctions not to be automatically residual! The choices made above for $\overline{{\mathbb R}^n}$ and $\overline{{\mathbb H}^n}$, respectively, are vindicated by the analysis of manifolds with ‘scattering metrics’, see [@RBMGeo], and with conformally compact metrics, see [@Mazzeo-Melrose:Meromorphic] and [@Mazzeo:Hodge].
As indicated above, the geodesic compactification of the conformally compact manifold $(M,g)$ is just ${\overline{M}}$ with its usual smooth structure. It is obviously of interest to identify the geodesic compactification for the product of two such manifolds, $X = M_1 \times M_2$. In fact, ${\partial}{\overline{X}}$ is the simplicial join of $\pa {\overline{M}}_1$ and $\pa {\overline{M}}_2$. More specifically, it is obtained from ${\partial}{\overline{M}}_1\times{\partial}{\overline{M}}_2\times[0,\pi/2]_\theta$ by collapsing each ${\partial}{\overline{M}}_1 \times \{q_2\}\times \{0\}$ and $\{q_1\} \times {\partial}{\overline{M}}_2 \times \{\pi/2\}$ to a point. Alternatively, it can be described as the manifold with corners obtained by blowing up the corner ${\partial}{\overline{M}}_1 \times {\partial}{\overline{M}}_2$ in the product ${\overline{M}}_1 \times {\overline{M}}_2$, and then blowing down the ${\overline{M}}_2$ fibres in ${\partial}{\overline{M}}_1\times{\overline{M}}_2$ and the ${\overline{M}}_1$ fibres in ${\overline{M}}_1\times{\partial}{\overline{M}}_2$ in the original faces. There are three boundary hypersurfaces, ${\partial}{\overline{M}}_1 \times M_2$ and $M_1 \times {\partial}{\overline{M}}_2$, corresponding to limits of geodesics of the form $(\gamma_1(t),p_2)$ and $(p_1,\gamma_2(t))$, respectively, and also the new face ${\partial}{\overline{M}}_1 \times {\partial}{\overline{M}}_2 \times (0,\pi/2)$ corresponding to limits of geodesics $(\gamma_1(\alpha t), \gamma_2(\beta t))$, $\alpha^2 +
\beta^2 = 1$. It is less clear what defining functions to use for these faces, though since the ‘new face’ covering the corner comes from the two-dimensional flats, i.e. products of geodesics in each factor, it is reasonable that we should use a polynomial type defining function here. In fact, using the usual coordinates on each factor, we shall use $$\rho_j = -1/\log x_j, \quad j = 1,2, \qquad r = |(\rho_1,\rho_2)|.$$ In other words, the smooth structure on ${\overline{X}}$ is the one induced from the normal blow-up at the corner of the log blow-ups of the two factors ${\overline{M}}_1$ and ${\overline{M}}_2$. The reason we use the log blow-up at all faces is that as we shall prove in the next section, the asymptotic behaviour of the resolvent involves expansions in powers of these particular defining functions.
The other compactification we consider here is due to Martin [@Martin:Minimal], and uses the function theory of ${\Delta}_g$ to associate a set of ideal boundary points ${\partial}Z$ to $Z$. Notably, it may be carried out in great generality for pairs $(Z,H)$ where $H$ is a semibounded self-adjoint elliptic operator on a space $Z$, though we shall always assume here that $H$ is the Laplacian. Let $\lambda_0 = \inf\spec(H)$. Then for every $\lambda \in {\mathbb R}\setminus (\lambda_0,\infty)$, there is a compactification ${\overline{Z}}_M(\lambda)$. Actually, it follows from the construction that ${\overline{Z}}_M(\lambda)$ is identified with ${\overline{Z}}_M(\lambda')$ for any two numbers $\lambda, \lambda' < \lambda_0$, so one needs to consider only ${\overline{Z}}_M(\lambda_0)$ and any other ${\overline{Z}}_M(\lambda)$.
Fix $\lambda < \lambda_0$. By [@Sullivan:Related], the set of solutions $u$ to the equation $(H - \lambda)u = 0$ which remain everywhere positive is nonempty; we denote this ${\mathbb R}^+$-invariant set by ${\mathcal P}_+(\lambda)$. The structure of this positive cone is encoded in the slice ${\mathcal P}_+^p(\lambda) = \{u \in {\mathcal P}_+(\lambda):
u(p) = 1$ for any fixed $p \in Z$. It follows readily from the Harnack inequality and elliptic estimates that for any sequence $u_j \in
{\mathcal P}_+^p(\lambda)$ there is a subsequence $u_{j'}$ converging to a point in this space, and so ${\mathcal P}_+^p(\lambda)$ is compact. It is also obviously convex. Let ${\mathcal E}$ denote the set of its extreme points. Then for any $u \in {\mathcal P}_+^p(\lambda)$, the Krein-Milman theorem gives a measure $dm_u(e)$ supported on ${\mathcal E}$ such that $u = \int_{\mathcal E} dm_u(e)$. This is the generalized Poisson representation theorem! The set ${\mathcal E}$, or ${\mathcal E}(Z,
\lambda)$ is called the minimal Martin boundary of $Z$. If $u \in
{\mathcal E}$, then whenever $v \in {\mathcal P}_+^p(\lambda)$ and $v \leq u$ then $v = c u$ for some constant $0 < c \leq 1$, and this justifies the moniker ‘minimal’.
If $Z$ were to naturally embed in ${\mathcal P}_+^p(\lambda)$, then the closure of its image would be an obvious way to compactify it. Unfortunately, this is not the case, but instead we consider the resolvent kernel $R(\lambda;z,w)$. This is a solution of $(H_z-\lambda)R(\lambda;z,w) = 0$ when $z \neq w$, but is singular at $z = w$ and in addition, $R(\lambda;
p,w) \neq 1$. Thus we define $$u_w(z) = R(\lambda,z,w)/R(\lambda,p,w),$$ so that $u_w(p) = 1$ and $u_w$ is a regular solution of $(H - \lambda)u_w
= 0$ on $X \setminus \{w\}$. Now let $w_j$ be any sequence of points in $Z$ which leaves any compact set. Then some subsequence $u_{w_j'}$ converges to an element of ${\mathcal P}_+^p(\lambda)$. The (full) Martin boundary is the set of equivalence classes of these sequences, or equivalently, is the set of all possible functions $u$ obtained as limits in this fashion. We label the different boundary points by $q \in {\partial}_M Z(\lambda)$, and write $u_q(z)$ for the corresponding limiting solution. The Martin compactification ${\overline{Z}}_M$ (or ${\overline{Z}}_M(\lambda)$) is the union of $Z$ and ${\partial}_M Z(\lambda)$. There is a metric on the set of functions $u_w(z)$, $w \in {\overline{Z}}_M$, given by $$d_p(u_w,u_{w'}) = \int_{B_1(p)} |u_w(z) - u_{w'}(z)|\, dV_g.$$ Thus ${\overline{Z}}_M$ not only a topological space, but a metric space.
This definition must be modified when $\lambda = \lambda_0$ and there is an $L^2$ eigenfunction $u_0$ with eigenvalue $\lambda_0$, since then the resolvent kernel $R(\lambda_0,z,w)$ does not exist. In this case $u_0 > 0$, and in fact ${\mathbb R}^+ \cdot u_0 = {\mathcal P}_+^p(\lambda_0)$, i.e. the only positive solutions are positive multiples of $u_0$. Hence it is consistent to let ${\overline{Z}}_M(\lambda_0)$ be the one point compactification of $Z$ then. Otherwise, if $\lambda_0$ is not in the point spectrum, then the definition is the same as before.
If $Z = {\mathbb R}^n$, then $\lambda_0 = 0$ and it is well-known that ${\partial}_M
{\mathbb R}^n(0)$ is a single point, so that $(\overline{{\mathbb R}^n})_M(0)$ is the one-point compactification $S^n$. On the other hand, for $\lambda < 0$, the extreme points of ${\mathcal P}_+^0(\lambda)$ are the exponentials $e^{x \cdot \xi}$, $|\xi|^2 = -\lambda$, and this sphere of radius $\sqrt{-\lambda}$ is the full Martin boundary, and so $(\overline{{\mathbb R}^n})_M(\lambda) = \bar{B}^n$. If $Z = {\mathbb H}^n$ and $n = k+1$, then $\lambda_0 = k^2/4$ and ${\overline{Z}}_M(\lambda)$ is the closed ball $\overline{B^n}$ for all $\lambda \leq \lambda_0$. The minimal positive eigenfunctions are the ones of the form $x^{k/2 +
\sqrt{k^2/4 - \lambda}}$ for all possible choices of upper half-space coordinates (i.e. choice of which point on the boundary of the ball to send to infinity). From [@Mazzeo-Melrose:Meromorphic] it follows that that when $M$ is conformally compact, ${\overline{M}}_M(\lambda)$ is still equal to $\overline{M}$. To survey other cases relevant to us in which the Martin compactification is known, when $Z$ is Cartan-Hadamard with curvatures pinched between two negative constants $-a^2$ and $-b^2$, then Anderson and Schoen [@Anderson-Schoen:Positive] and Ancona [@Ancona:Negatively] proved that ${\overline{Z}}_M(\lambda) = \overline{B^n}$ (as metric spaces). There has been recent significant progress in determining the Martin compactifications for general symmetric spaces of noncompact type; definitive results are proved in the recent monograph [@Guivarch-Ji-Taylor:Compactifications], and this has an extensive bibliography of the literature on these developments.
Of particular relevance to us here is the work of Giulini and Woess [@Giulini-Woess:Martin] where the Martin compactification of the product of two hyperbolic spaces is determined. Their result is that now ${\overline{X}}_M(\lambda)$ is identified with $[\overline{{\mathbb H}^{n_1}}\times
\overline{{\mathbb H}^{n_2}};{\partial}\overline{{\mathbb H}^{n_1}}\times {\partial}\overline{{\mathbb H}^{n_2}}]$, the blow-up of the product of the two balls along the corner; recall that this space appeared as an intermediate picture in the description of the geodesic compactification. Their proof uses rather involved global heat-kernel estimates, and one of the motivations of this paper was to demonstrate how this result, and the analogous one for products of conformally compact spaces, may be obtained in a more straightforward manner using resolvent estimates and stationary phase.
To conclude this section on compactifications, we state once again that our primary interest is in obtaining a compactification of the double-space $X \times X$, where $X = M_1 \times M_2$, which is natural with respect to the resolvent. More specifically, we wish that at least for $\lambda$ in the resolvent set, $R_X(\lambda)$ should have at most polyhomogeneous singularities at the boundary hypersurfaces of this compactification (apart from its usual diagonal singularity). This compactification is determined by examining the asymptotics of $R_X(\lambda;z,w)$ as the points $z=(z_1,z_2)$ and $w=(w_1,w_2)$ diverge in all possible directions. We begin the study these asymptotics in the next section. The Martin compactification is, at best, essentially a ‘slice’ of this resolvent compactification, and in any case is easily determined from this asymptotic analysis, as we do in the final section. We shall see there that when either $H_1$ or $H_2$ has eigenvalues below the continuous spectrum, then the Martin compactification of $X$ is obtained by substantially collapsing part of the boundary of the geodesic compactification ${\overline{X}}$, and so loses a lot of information about the fine structure of the resolvent. Thus the resolvent compactification is the primary object of interest.
Asymptotics
===========
In the remainder of this paper we shall give a more detailed description of the structure of the resolvent $R(\mu)$ on $X$. As a first approach to this we adopt the more traditional viewpoint and derive the asymptotic behavior of $R(\mu)f$ when $f\in \dCinf(\Xb)$. (Functions vanishing to all orders at ${\partial}X$ are a suitable analogue of the space of Schwartz functions.) More generally, it makes absolutely no difference if we allow $f$ to be the sum of an element of $\dCinf(\Xb)$ and a distribution of compact support. In particular, all of the calculations below apply when $f = \delta_p$, $p\in X=M_1\times M_2$; indeed, this is the basis for our identification of the Martin boundary of $X$. However, we shall simply assume that $f$ is Schwartz, and also, in this section, that $\mu$ is in the resolvent set for $H$.
Recall our convention that when $\mu_j$ is in the resolvent set for $H_j$, then the imaginary part of $\sqrt{\mu_j - k_j^2/4}$ is negative, and by the results of last section, $R_j(\mu_j)f$ is then of the form $$x_j^{k_j/2+i\sqrt{\mu_j-k_j^2/4}}g_j,\ g_j\in\Cinf({\overline{M}}_j).$$ Also, if $\phi_{ji}$ is an $L^2$ eigenfunction of of $H_j$ with eigenvalue $\lambda_{ji}$, then $$\phi_{ji}=x_j^{k_j/2+i\sqrt{\lambda_{ji}-k_j^2/4}}g_j,\ g_j\in\Cinf({\overline{M}}_j);$$ the Schwartz kernel of $\Pi_{ji}$ is $\phi_{ji}(z_j)\otimes
\overline{\phi_{ji}}(z_j')$ if the eigenvalue is simple, and is a finite sum of such terms otherwise.
We start with a slightly stronger result concerning the structure of $R_{12}(\mu_1,\mu_2)=R_1(\mu_1)\otimes R_2(\mu_2)$ applied to $f\in\dCinf(\Xb)$, considered as a function of both $\mu_1$ and $\mu_2$. Although this follows directly from the corresponding statement in each factor if $f=f_1\otimes f_2$, $f_j\in\dCinf({\overline{M}}_j)$, it is convenient to prove the general result directly. Thus, the kernel of $R_{12}(\mu_1,
\mu_2)$ is polyhomogeneous on $({\overline{M}}_1)^2_0\times ({\overline{M}}_2)^2_0$. Let $\pi_L$, $\pi_R$ be the projections to the left and right factors of ${\overline{M}}_1\times{\overline{M}}_2$. That is, if $\pi_{L,j}$, resp. $\pi_{R,j}$, denote the projection of $({\overline{M}}_j)^2_0$ to its left factor, resp. right, factor, then $\pi_L=\pi_{L,1}\times\pi_{L,2}$, $\pi_R=\pi_{R,1}\times\pi_{R,2}$. Note that $\pi_L$, $\pi_R$, are b-fibrations. Then $R_{12}(\mu_1,\mu_2)$ applied to $f$ is given by the push-forward of $R_{12}(\mu_1,\mu_2)\pi_R^* f$ under the map $\pi_L$. Thus $$R_{12}(\mu_1,\mu_2)f=x_1^{k_1/2+i\sqrt{\mu_1-k_1^2/4}}
x_2^{k_2/2+i\sqrt{\mu_2-k_2^2/4}}g,\ g\in\Cinf({\overline{M}}_1\times{\overline{M}}_2).
\label{eq:jdres}$$ The dependence on $\mu_1$ and $\mu_2$ here is uniform in the strong sense that the function $g(z_1,z_2,\mu_1,\mu_2)$ appearing in (\[eq:jdres\]) satisfies $$g\in\Cinf({\overline{M}}_1\times{\overline{M}}_2\times
(\Cx\setminus\spec(H_1))_{\mu_1}\times(\Cx\setminus\spec(H_2))_{\mu_2}),$$ with natural extensions corresponding to the meromorphic extension of the $R_j$. This follows directly from the usual push-forward formula [@RBMCalcCon]. [*Although we use this result throughout the section, we usually state arguments for simplicity as if $R_1$ and $R_2$ are applied separately to $f$.*]{} In one particular case, when analyzing $R(\mu)f$ for $\mu$ real, below $\spec(H)$, and when either $H_1$ or $H_2$ have $L^2$ eigenvalues, we need a stronger result, where $f$ is not required to be Schwartz. We postpone that discussion until it is required, see the arguments preceeding Theorem \[thm:res-set-asymp\].
The asymptotic behavior of $R(\mu)f(z)$ must be analyzed in three separate regions: near ${\partial}{\overline{M}}_1 \times M_2$, near $M_1 \times {\partial}{\overline{M}}_2$ and near the corner ${\partial}{\overline{M}}_1 \times {\partial}{\overline{M}}_2$. Using coordinates $z = (z_1,z_2)$, $z_j = (x_j,y_j)$, these correspond to $x_1 \to 0$, $x_2 \geq c > 0$, or $x_1 \geq c > 0$, $x_2 \to 0$ or $x_1,x_2 \to 0$, respectively.
Asymptotics at $M_1\times\pa{\overline{M}}_2$ and at $\pa{\overline{M}}_1\times M_2$
------------------------------------------------------------------------------------
We first describe the uniform behaviour of $R(\mu)f$ on $M_1\times
{\overline{M}}_2$, i.e. at infinity in $M_2$. This is given as an asymptotic series in powers of $-1/\log x_2$; we shall think of this later as an asymptotic expansion on the logarithmic blow-up of ${\overline{M}}_2$, which simply means that we change the ${\mathcal C}^\infty$ structure of ${\overline{M}}_2$ by replacing the defining function $x_2$ for the boundary by $-1/\log x_2$.
Suppose first that $H_1$ has no $L^2$ eigenvalues. We analyze $R(\mu)f$, using the representation (\[eq:resform\]) for $R(\mu)$, by shifting $\gamma$ so that it passes through $\inf\spec(H_1)=k_1^2/4$, and so that the minimum of $\im\sqrt{\mu-k_2^2/4-\mu_1}$ along this path is attained at that point, see Figure \[fig:contour2\], and then applying (complex) stationary phase.
To set things up for stationary phase, first note that $R_2(\mu_2)$ maps $\dCinf({\overline{M}}_2)$ to $x_2^{k_2/2 + i\sqrt{\mu_2-k_2^2/4}}\Cinf({\overline{M}}_2)$. Therefore, the oscillatory factor $e^{(k_2/2 + i\sqrt{\mu_2-k_2^2/4})\log x_2}$, $\mu_2 = \mu - \mu_1$ appears in the integrand. On the other hand, $R_1(\mu_1)$ is [*not*]{} smooth at $\mu_1=k_1^2/4$, but instead has an expansion in powers of $\sqrt{\mu_1-k_1^2/4}$. Thus it decomposes into the sum of odd and even powers, respectively, $R_1(\mu_1)=R_1^{\text{odd}}(\mu_1)+R_1^{\text{even}}(\mu_1)$, and we write $R_1^{\text{odd}}(\mu_1)=\sqrt{\mu_1-k_1^2/4}\,G_1(\mu_1-k_1^2/4)$ where $G_1$ is smooth. Because the phase $\sqrt{\mu-\mu_1-k_2^2/4}\,\log x_2$ is not stationary at $k_1^2/4$, the smooth part $R_1^{\text{even}}$ contributes only terms decaying faster than any power of $-1/\log x_2$. Thus we are left only with $$\int_{\gamma} e^{(k_2/2 + i\sqrt{\mu - \mu_1 -k_2^2/4})\log x_2}
\sqrt{\mu_1-k_1^2/4}\,G_1(\mu_1-k_1^2/4) \tilde{g}(\mu_1,z,z')\, d\mu_1,$$ where $\tilde{g}$ is smooth in all variables. Actually, it suffices to take this integral only over some compact segment of $\gamma$ containing $k_1^2/4$ and where the decomposition of $R_1$ into even and odd parts is valid; using the uniform weighted $L^2$ estimates of §3, the integral over the remaining portion of $\gamma$ contributes a term vanishing like $x_2^{k_2/2 + m}$, $m>0$ can be made as large as wished by increasing the length of the compact segment, and is therefore negligible in the asymptotics. Changing the variable of integration to $\tau =
\sqrt{\mu_1-k_1^2/4}$, so that $\mu_1 = k_1^2/4 + \tau^2$, leads to $$\int e^{(k_2/2 + i\sqrt{\mu - k^2/4 - \tau^2})\log x_2}
2\tau^2G(\tau^2)\tilde{g}(k_1^2/4 + \tau^2,z,z')\, d\tau.$$ The phase function $\phi(\tau) = \sqrt{\mu - k^2/4 - \tau^2}\log x_2$ is now stationary at $\tau = 0$; furthermore, $\phi''(0) = -1/\sqrt{\mu-k^2/4}$, and so the further change of variables $\sigma = \tau \sqrt{-\log x_2}/
(\mu-k^2/4)^{1/4}$ reduces this integral to $$x_2^{k_2/2 +i\sqrt{\mu-k^2/4}}(\mu-k^2/4)^{3/4}(-1/\log x_2)^{3/2}
\int e^{i\sigma^2/2}\hat{g}(\sigma,z,z')\, d\sigma$$ where $\hat{g}$ is obtained from $\tilde{g}$ by replacing $\sigma$ by $\tau$. Applying the stationary phase for phase functions with nonnegative imaginary part, see [@Hor Theorem 7.7.5], we obtain that $$\label{eq:res-side-free-8}
R(\mu)f=
x_2^{k_2/2 + i\sqrt{\mu-k^2/4}}(-1/\log x_2)^{3/2}g,$$ where $g$ is $\Cinf$ on the logarithmic blow-up $M_1\times ({\overline{M}}_2)_{\log}$ of $M_1\times {\overline{M}}_2$, i.e. in the variables $(z_1,-1/\log x_2,y_2)$ for $x_1 \geq c > 0$. (Of course, $g$ is continuous on $M_1\times {\overline{M}}_2$; it is the lower order terms in the asymptotics that necessitate the change of the smooth structure.) In fact, $$g|_{M_1\times\pa {\overline{M}}_2}
= \left. c\, (\mu-k^2/4)^{3/4}x_2^{-k_2/2 -i\sqrt{\mu-k^2/4}}
G_1(0)R_2(\mu-k_1^2/4)f\right|_{M_1\times\pa {\overline{M}}_2},$$ where $c$ is a constant arising from the stationary phase lemma.
There is a final simplification of this formula, arising from the fact that we can identify the operators here slightly more explicitly. First, according to [@Mazzeo-Melrose:Meromorphic], for each $z_1 \in M_1$, $$\left. x_2^{-k_2/2 -i\sqrt{\mu-k^2/4}}R_2(\mu-k_2^2/4)f
\right|_{\pa {\overline{M}}_2}$$ $$= \int_{\pa {\overline{M}}_2}P_2^t(\mu-k_2^2/4;y_2,z_2')f(z_1,z_2')\, dz_2'
\equiv P_2^t(\mu-k_2^2/4)(f)(z_1,y_2),$$ where $P_2(\mu-k_2^2/4)$ is the Poisson transform on $M_2$ at the spectral parameter $\mu-k_2^2/4$, and $P_2^t$ its transpose. Next, by definition, $$G_1(0) = \left. \frac{d\,}{d\tau}\right|_{\tau=0}R_1(k_1^2/4 + \tau^2).$$ We can already see from how it was introduced that this operator arises from the non-holomorphic part of $R_1$, and so its Schwartz kernel is smooth on $({\overline{M}}_1)^2_0$ because the diagonal singularity of $R_1$, represented by $R'_1$ in Theorem \[th:ccst\], does not contribute to it. However, we can see this more directly by differentiating $$(H_1 - k_1^2/4-\tau^2)R_1(k_1^2/4 + \tau^2) = I$$ with respect to $\tau$ and setting $\tau=0$ to get $(H_1 - k_1^2/4)G(0) = 0$, and so concluding that $G_1(0)$ is smooth by elliptic regularity. The Schwartz kernel $G_1(0;z_1,z_1')$ is a familiar function on hyperbolic space: it is a non-square-integrable eigenfunction for $H_1$, with threshold eigenvalue $k_1^2/4$, invariant under the group of rotations which fix $z_1'$, and is known as a spherical function. We shall denote it by $S_1(k_1^2/4;z_1,z_1')$, and regard it as a Schwartz kernel. Altogether, we have now shown that $$g|_{M_1\times\pa {\overline{M}}_2}
= c\, (\mu-k^2/4)^{3/4}S_1(k_1^2/4)P_2^t(\mu-k_2^2/4)f.
\label{eq:x2face}$$
When $H_1$ has $L^2$ eigenvalues, then shifting the contour $\gamma$ through these gives a contribution $-\Pi_{1i}\otimes R_2(\mu-\lambda_{1i})$ from the residues of the integrand. (For simplicity we assume that the ramification point $k_1^2/4$ is not a pole of the meromorphic continuation of $R_1$ to the Riemann surface, though this can obviously be handled too by the arguments preceeding Theorem \[thm:res-set-asymp\].) Hence in this case we obtain $$\begin{split}\label{eq:pa-M_2-asymp}
R(\mu)f&=x_2^{k_2/2 + i\sqrt{\mu-k^2/4}}(-1/\log x_2)^{3/2}g\\
&\qquad+\sum_{i=1}^{N_1}x_2^{k_2/2+
i\sqrt{\mu-\lambda_{1i}-k_2^2/4}}g_i,
\qquad \mbox{where}\ \ g,g_i\in\Cinf(M_1\times ({\overline{M}}_2)_{\log}).
\end{split}$$ In fact, $g_i = \phi_{1i}\otimes \tilde g_i$, with $\tilde g_i\in\Cinf(
{\overline{M}}_2)$ and $\phi_{1i}$ an eigenfunction of $H_1$ with eigenvalue $\lambda_{1i}$; indeed, $g_i|_{M_1\times \pa{\overline{M}}_2}=-(\Pi_{1i}\otimes P_2^t(\mu-\lambda_{1i}))f$.
In this region, near $M_1\times{\partial}{\overline{M}}_2$, only the $L^2$ eigenvalues of $H_1$ play a role in the asymptotics, but perhaps surprisingly, [*not*]{} those of $H_2$. Notice that all terms in which come from the $L^2$ eigenvalues of $H_1$ dominate the term coming from the continuous spectrum since $\im\sqrt{\mu-k^2/4}<\im\sqrt{\mu-\lambda_{1i}-
k_2^2/4}$. In addition, the term corresponding to the lowest eigenvalue $\lambda_{10}$ of $H_1$ dominates all the other terms. This will be important later in the determination of the Martin boundary in the presence of bound states.
The asymptotics of $R(\mu)f$ at the other face ${\partial}{\overline{M}}_1\times M_2$ is completely analogous. The same calculations lead to the expression $$\begin{split}\label{eq:pa-M_1-asymp}
R(\mu)f&=x_1^{k_1/2 + i\sqrt{\mu-k^2/4}}(-1/\log x_1)^{3/2}g\\
&\qquad+\sum_{i=1}^{N_1}x_1^{k_1/2+
i\sqrt{\mu-\lambda_{2i}-k_1^2/4}}g_i,
\qquad \mbox{where}\ \ g,g_i\in\Cinf(({\overline{M}}_1)_{\log}\times M_2).
\end{split}$$ In addition, $$g|_{\pa {\overline{M}}_1\times M_2}
= c\, (\mu-k^2/4)^{3/4}P_1^t(\mu-k_1^2/4)S_2(k_2^2/4)f,
\label{eq:x1face}$$ where $P_1^t$ is the transpose of the Poisson transform on $M_1$ and $S_2$ is the ‘spherical function’ at eigenvalue $k_2^2/4$ for $M_2$.
We note once again that even if both $H_1$ and $H_2$ have $L^2$ eigenvalues, only those of $H_2$ contribute to the asymptotics when $x_1 \to 0$, $x_2 \geq c > 0$ and only those of $H_1$ contribute to the asymptotics when $x_2 \to 0$, $x_1 \geq c > 0$. This indicates that the asymptotics at the corner ${\partial}{\overline{M}}_1\times {\partial}{\overline{M}}_2$, where both $x_1,x_2 \to 0$ must be more complicated, at least in the presence of bound states, because it intermediates this transition.
Asymptotics at $\pa{\overline{M}}_1\times\pa{\overline{M}}_2$ in the absence of $L^2$ eigenvalues
-------------------------------------------------------------------------------------------------
We now proceed to the analysis of $R(\mu)f$ at this corner. We have already seen that necessity of logarithmic blow ups at the side faces of the product. Correspondingly, the asymptotics at the corner necessitate that we pass to the log blow-ups of both factors ${\overline{M}}_j$, which we denote $({\overline{M}}_j)_{\log}$. In terms of these, define $$\Xt=[({\overline{M}}_1)_{\log}\times({\overline{M}}_2)_{\log};\pa({\overline{M}}_1)_{\log}
\times\pa({\overline{M}}_2)_{\log}].$$ This is the correct space on which to consider asymptotics of $R(\mu)f$, as we shall now prove.
We let $$\rho_j=-1/\log x_j,\quad j=1,2,$$ be boundary defining functions of $({\overline{M}}_j)_{\log}$, and we keep denoting their pull-back to $\Xt$ with the same notation. Also let $y_1$, $y_2$ denote coordinates on $\pa({\overline{M}}_j)_{\log}$, $j=1,2$. Thus, $\Xt$ is the blow-up of $({\overline{M}}_1)_{\log}\times({\overline{M}}_2)_{\log}$ at the corner $\rho_1=\rho_2=0$. Hence valid coordinates in the region where $\rho_1/\rho_2<C$, for some $C>0$, are given by $$\rho_2,\ s=\rho_1/\rho_2,\ y_1,\Mand y_2.$$ In terms of the original boundary defining functions $x_j$ this means that in any region where $\log x_2/\log x_1<C$, for some $C>0$, we use the projective coordinate $$s = \frac{-1/\log x_1}{-1/\log x_2} = \frac{\log x_2}{\log x_1}$$ along the new ‘front’ face of $\Xt$ covering the corner, and $-1/\log x_2$ as a defining function for this face. Thus $s\to 0$ upon approach to the lift of $\pa {\overline{M}}_1\times {\overline{M}}_2$, while $s\to\infty$ on approach to the lift of ${\overline{M}}_1\times\pa {\overline{M}}_2$. We note that a [*total*]{} boundary defining function of $\Xt$ is given by $$\rho=(\rho_1^{-2}+\rho_2^{-2})^{-1/2},$$ i.e. with the usual Euclidean notation, $r_j=\rho_j^{-1}$, $r=\rho^{-1}$, $r=\sqrt{r_1^2+r_2^2}$. This explains the appearance of $\rho^{-1}$ in our results below. In the region $s<C$, $\rho_1=-1/\log x_1$ is another total boundary defining function; it will be used for the initial local calculation and then we change to $\rho$ for the global statements. The trivial identity $$x_2 = e^{\log x_2} = e^{s \log x_1} = x_1^s$$ will be used repeatedly.
The integrand $R_1(\mu_1)R_2(\mu-\mu_1)f$ in the contour integral representation of $R(\mu)f$ has the form $$x_1^{k_1/2+i\sqrt{\mu_1-k_1^2/4}}x_2^{k_2/2+i\sqrt{\mu-\mu_1-k_2^2/4}}g,$$ or equivalently, $$\label{eq:integrand-5}
\exp\left\{\left[k_1/2+s k_2/2+i\left(\sqrt{\mu_1-k_1^2/4}+
s\sqrt{\mu-\mu_1-k_2^2/4}\right)\right]\log x_1\right\}g,$$ where $g$ is $\Cinf$ in $\mu, \mu_1$ and on ${\overline{M}}_1 \times {\overline{M}}_2$. The expressions inside these square roots assume values in $\Cx\setminus
[0,\infty)$, and we assume the square roots have [*negative*]{} imaginary parts in this region.
Suppose first that neither $H_1$ nor $H_2$ has $L^2$ eigenvalues. We again do a stationary phase analysis of the integral of (\[eq:integrand-5\]), and for this it is necessary to choose the contour of integration so that, if we set $$F(\mu_1)=F_s(\mu_1) = \sqrt{\mu_1-k_1^2/4}+s\sqrt{\mu-\mu_1-k_2^2/4},$$ then the supremum of $\im F(\mu_1)$ along $\gamma$ is as negative as possible. Both $\mu$ and $s$ are parameters here, and we must choose the contour differently corresponding to the different points of the front face of $\Xt$, i.e. the different values of $s$. Since $F$ is analytic outside $[k_1^2/4,
+\infty) \cup(\mu-[k_2^2/4, +\infty))$, its critical points and those of its imaginary part are the same. In fact, $F$ has a unique critical point, located at $$\label{eq:crit-point-im}
\mu_1^0=\mu_1^0(s)=\frac{\mu-k_2^2/4+s^2 k_1^2/4}{1+s^{2}},$$ and this is a saddle point of the harmonic function $\im F(\mu_1)$. (We return to this and shall explain it in greater detail below.) Moreover, this critical point always lies on the straight line segment connecting the two ramification points $k_1^2/4$ and $\mu-k_2^2/4$, and tends to the former as $s\to+\infty$ and to the latter as $s\to 0$. Next, $$F(\mu_1^0) = \sqrt{(\mu-k^2/4)(1+s^2)}, \qquad F'(\mu_1^0) = 0,$$ and $$F''(\mu_1^0) = -\frac{1}{4}s^{-2}(1+s^2)^{5/2}(\mu-k^2/4)^{-3/2}.$$ Therefore we may choose an appropriate contour $\gamma$ so that $\im F(\gamma(t))$ attains a non-degenerate maximum when $\gamma(t) =
\mu_1^0$. The stationary phase lemma may now be applied as before, although the singular change of variables is no longer required (in $s>0$); we deduce that the asymptotics of $R(\mu)f$ have the form $$\begin{split}\label{eq:corner-asymp-free-8}
&\exp\left(\left(k_1/2+s k_2/2+i\sqrt{(\mu-k^2/4)(1+s^2)}\right)
\log x_1\right)(-1/\log x_1)^{1/2} g\\
&=x_1^{k_1/2}x_2^{k_2/2}\left(\exp\sqrt{(\log x_1)^2+(\log x_2)^2}\right)
^{-i\sqrt{\mu-k^2/4}}(-1/\log x_1)^{1/2}g.
\end{split}$$ At the points of the front face of $\Xt$ where $\log x_2/\log x_1 =s$ this coefficient function $g$ is a nonvanishing multiple of $$s(1+s^2)^{-5/4}(\mu-k^2/4)^{3/4}g(\mu_1^0(s)) = s\, (1+s^2)^{-5/4}
((\mu-k^2/4)^{-3/2})^{-1/2}$$ $$\times x_1^{-k_1/2-i\sqrt{\mu_1^0(s)-k_1^2/4}}
x_2^{-k_2/2-i\sqrt{\mu-\mu_1^0(s)-k_2^2/4}}R_1(\mu_1^0(s))R_2(\mu-\mu_1^0(s))f,$$ or, finally, $$c \, s\, (1+s^2)^{-5/4}(\mu-k^2/4)^{3/4}
P_1^t(\mu_1^0(s))P_2^t(\mu-\mu_1^0(s))f.
\label{eq:frontface}$$ Note that although this computation is done separately for each value of $s$, this final expression depends smoothly on $s$ and in fact $g$ is smooth on $\Xt$ up to the front face.
To be precise, we still need to examine what happens near and at the corner $s=0$, $-1/\log x_2=0$. However, this is hardly different from the discussion at $\pa{\overline{M}}_1\cap M_2$. Indeed, we simply introduce $\tau=\sqrt{\mu-\mu_1-k_2^2/4}$ as the new smooth variable of integration, and then stationary phase can be performed uniformly in $s$ down to $s=0$, with the limiting contour being the same as for the discussion at $\pa{\overline{M}}_1\cap M_2$, showing that $R(\mu)f$ is actually polyhomogeneous. In particular, note that the expansions (\[eq:x2face\]), (\[eq:x1face\]) and (\[eq:frontface\]) at the front and side faces of $\Xt$ match up at the corners, in agreement with the polyhomogeneity. The one point to note is that, for example, there is a factor of $(-1/\log x_1)^{1/2}$ in (\[eq:frontface\]), whereas (\[eq:x1face\]) has a factor of $(-1/\log x_1)^{3/2}$. To reconcile this, observe that (\[eq:x1face\]) has an extra factor of $s$, and $$s (-1/\log x_1)^{1/2} = \frac{-\log x_2}{-\log x_1}(-1/\log x_1)^{1/2}
= (-\log x_2)(-1/\log x_1)^{3/2},$$ so in fact the powers match up.
We remark that if the contour $\gamma$ does not go through the critical point $\mu_1^0(s)$, then it must necessarily contain points where the integrand is larger; however, at those points the phase function itself, [*not just its imaginary part*]{}, will fail to be stationary, and so stationary phase gives a decay rate $O((-1/\log x)^{\infty})$; this merely indicates that the contour has not been chosen optimally and a better result is possible.
Before proceeding to the rather more subtle discussion of asymptotics at the corner in the presence of bound states, we summarize our results in their absence.
\[prop:no-bd-state-asymp\] Suppose $f\in\dCinf({\overline{X}})$, $\mu\in \Cx\setminus[k^2/4,+\infty)$, and $H_1$, $H_2$ have no $L^2$ eigenfunctions. Then, with $\rho_i=-1/\log x_i$, $\rho^{-1}=\sqrt{\rho_1^{-2}+\rho_2^{-2}}$, $R(\mu)f$ has the following asymptotic expansion on $\Xt$: $$\begin{split}
R(\mu)f&=x_1^{k_1/2}x_2^{k_2/2}\exp (-i\sqrt{\mu-k^2/4}/\rho)h.
\end{split}$$ Here $h$ polyhomogeneous on $\Xt$, with order $1/2$ on the front face and $3/2$ on the side faces of $\Xt$. Moreover, the principal symbol of $h$ is an elliptic $f$-independent multiple of $$\begin{split}
&P_1^t(\mu_1^0(s))P_2^t(\mu-\mu_1^0(s))f\ \text{on the front face},\\
&
S_1(k_1^2/4)P_2^t(\mu-k_2^2/4)f\ \text{on the lift of}\ {\overline{M}}_1\times
\pa{\overline{M}}_2,\\
&P_1^t(\mu-k_1^2/4)S_2(k_2^2/4)f\ \text{on the lift of}\ \pa{\overline{M}}_1\times {\overline{M}}_2,
\end{split}$$ with $\mu_1^0(s)$ given by .
This discussion already indicates that $s$ plays a dual role, both as a coordinate on the space $\tilde{X}$ and also as a spectral parameter. The function $\mu_1^0(s)$ identifies the front face of $\Xt$ with the line joining the two threshold values $k_1^2/4$ and $\mu-
k_2^2/4$ in the spectral plane. This role becomes even more pronounced in determining asymptotics at the front face in the presence of bound states, for then the optimal locus for the contour $\gamma$ determines which of the residue terms corresponding to the different eigenvalues must be included in the asymptotics. In addition, for certain real values of $\mu$, there are a finite number of exceptional values of $s$ at which $\mu_1^0(s)$ equals either $\lambda_{1i}$ or $\mu - \lambda_{2i}$; then the optimal contour must pass through this pole, and this creates an additional Legendrian singularity. We explain all of this now.
Asymptotics at $\pa{\overline{M}}_1\times\pa{\overline{M}}_2$ in the presence of $L^2$ eigenvalues for $\mu\in\Cx\setminus\Real$
--------------------------------------------------------------------------------------------------------------------------------
Thus suppose that $H_1$ or $H_2$ have $L^2$ eigenvalues. As we have already seen, these correspond to poles of the integrand and the residues at these poles may contribute to the asymptotics. To see when this happens, first note that the additional terms coming from these residues have the form $$\label{eq:corner-asymp-ev-8}
x_2^{k_2/2+
i\sqrt{\mu-\lambda_{1i}-k_2^2/4}}\phi_{1i}\otimes \tilde g_{1i},\ \tilde g_{1i}
\in\Cinf({\overline{M}}_2),$$ with analogous terms when the roles of ${\overline{M}}_1$ and ${\overline{M}}_2$ have been switched. Here $$\phi_{1i}\in x_1^{k_1/2+i\sqrt{\lambda_{1i}-k_1^2/4}}\Cinf({\overline{M}}_1).$$ The first main observation is that the terms dominate the previous one coming from the continuous spectrum only when $$\im\left(\sqrt{\lambda_{1i}-k_1^2/4}+s\sqrt{\mu-\lambda_{1i}-k_2^2/4}\right)
> \im\left(\sqrt{\mu-k^2/4}\sqrt{1+s^2}\right),$$ or in other words when $$\im F(\lambda_{1i}) > \im F(\mu_1^0(s)).
\label{eq:eigineq}$$ In all other cases, these terms coming from the bound states are lower order in the expansion .
To analyze this, we must examine the geometry of the function $F$ a bit more closely. Set $$\Lambda(\mu) = \Cx \setminus \big( [k_1^2/4,\infty)\cup (\mu - [k_2^2/4,
\infty)\big),$$ and let $L \subset \Lambda(\mu)$ be the open line segment connecting the two threshold values $k_1^2/4$ and $\mu-k_2^2/4$; this segment is the image of the map $(0,\infty) \ni s \to \mu_1^0(s)$. The function $F$ depends implicitly on both $\mu$ and $s$; we shall hold $\mu$ fixed throughout, but shall study how $F$ changes as $s$ varies. We have already noted that $\im F$ has a saddle point at $\mu_1^0(s)$. To see this, observe first that this is the unique critical point. Next, $$\lim_{\mu_1 \to k_1^2/4} \im F(\mu_1) = s \im \sqrt{\mu - k^2/4},
\qquad
\lim_{\mu_1 \to \mu-k_2^2/4} \im F(\mu_1) = \im \sqrt{\mu - k^2/4},$$ and both of these values are greater than $\im F(\mu_1^0(s)) =
\sqrt{1+s^2}\im \sqrt{\mu-k^2/4}$ since $\im \sqrt{\mu-k^2/4} < 0$. Also, each of the square roots in the expression for $F$ has imaginary part vanishing along one of the slits and tending to $-\infty$ as $\mu_1$ moves horizontally away from that slit, and so $\im F$ tends to $-\infty$ as $|\mu_1| \to \infty$. Thus $\mu_1^0(s)$ is indeed a saddle point for $\im F$. We make the following definition.
\[Def:Ns-Ps\] For each $s$, let ${\mathcal N}_s$ and ${\mathcal P}_s$ denote the regions in $\Lambda(\mu)$ where $\im F(\mu_1) < \im F(\mu_1^0(s))$ and $\im F(\mu_1) > \im F(\mu_1^0(s))$, respectively.
The union of ${\mathcal N}_s$ and the slits always has one component, while ${\mathcal P}_s$ has two components, each of which has compact closure intersecting precisely one of the two slit axes and which intersect at $\mu_1^0(s)$. Label these two components ${\mathcal P}_s^\ell$ and ${\mathcal P}_s^r$, respectively, the $\ell$ and $r$ denoting whether these regions touch the slits extending toward the left or right. Finally, recall that the ‘optimal’ contour of integration $\gamma$ is any contour which remains entirely within the region ${\mathcal N}_s$, except when it passes through the saddle point.
Now consider the locations of the poles $\lambda_{1i}$ and $\lambda_{2i}$ relative to the contour $\gamma$ and these regions ${\mathcal N}_s$ and ${\mathcal P}_s$. For any $s$, these two collections of poles are separated by the initial choice of contour $\gamma$ in . However, in deforming $\gamma$ to a $\gamma'$ which lies entirely within ${\mathcal N}_s$ it may be necessary to shift past some of these poles. This is not necessary if all of the $\lambda_{1i}$ lie within ${\mathcal N}_s \cup {\mathcal P}_s^r$ and all of the $\mu-\lambda_{2i}$ lie within ${\mathcal N}_s \cup {\mathcal P}_s^\ell$. When $s$ is sufficiently small, then in fact all of the $\lambda_{1i}$ lie within ${\mathcal N}_s \cup {\mathcal P}_s^r$; all of the $\mu-\lambda_{2i}$ lie in this region too, and none lie in ${\mathcal P}_s^\ell$. On the other hand, when $s \to \infty$, both the $\lambda_{1i}$ and the $\lambda_{2i}$ lie within ${\mathcal N}_s \cup {\mathcal P}_s^\ell$. As $s$ increases from $0$ to $\infty$, the region ${\mathcal P}_s^\ell$ gradually engulfs each of the points $\lambda_{1i}$, and so the optimal contour (which must always lie in ${\mathcal N}_s$) must cross these before this happens. A similar phenomenon occurs for the $\lambda_{2i}$.
Asymptotics at $\pa{\overline{M}}_1\times\pa{\overline{M}}_2$ in the presence of $L^2$ eigenvalues for $\mu\in\Real$
--------------------------------------------------------------------------------------------------------------------
On the other hand, when $\mu \in {\mathbb R}$ and there are bound states in the interval $(\mu-k_2^2/4,k_1^2/4)$, i.e. when $$\mu < \min\big\{k_1^2/4 - \max\{\lambda_{1i}\},
k_2^2/4 - \max\{\lambda_{2i}\}\big\},$$ then this analysis must fail for certain values of $s$, namely those values where either $\mu_1^0(s) = \lambda_{1i}$ or $\mu - \mu_1^0(s) =
\lambda_{2i'}$ for some $i,i'$. For then an optimal contour would need to pass directly through a pole. It is clear that there must be some sort of jump in the behaviour at these points, because for nearby values of $s$ this phenomenon does not occur and the previous analysis may be used to get the asymptotics; the residue at the pole would be included in these asymptotics if $s$ varies slightly to one side, but are not included if it varies slightly to the other. We now prove that $R(\mu)$ has a Legendrian singularity at this front face of $\Xt$ at those values of $s$ where this occurs.
To be definite, suppose that $s_0$ satisfies $\mu_1^0(s_0) =\lambda_{1i}$. Let $\gamma$ be the original contour defining $R(\mu)$, as in Figure \[fig:contour\], and let $\gamma_s$ be an optimal contour for any value $s$ near $s_0$. We may arrange that $\gamma_s(0)=\mu_1^0(s)$, so $\gamma_{s_0}(0)=\lambda_{1i}$. When $s>s_0$ (closer to the lift of ${\overline{M}}_1\times\pa {\overline{M}}_2$) the contour has crossed past $\lambda_{1i}$, and the residue term is present, whereas when $s<s_0$ (closer to the lift of $\pa {\overline{M}}_1\times {\overline{M}}_2$) this residue term is not included.
We proceed as follows. First consider what happens when $s\to s_0$ from the left, i.e. with $s\leq s_0$. Equation describes the asymptotics when $s<s_0$, and we are interested in the the behavior in the limit. Write $$R_2(\mu-\mu_1)=R_2(\mu-\lambda_{1i})+(\mu_1-\lambda_{1i})\tilde R_2(\mu-\mu_1),$$ i.e. expand $R_2$ to first order in Taylor series in $\mu_1$ around $\mu_1^0(s_0)=\lambda_{1i}$. Thus, $$\begin{split}\label{eq:tilde-R_2-def}
\tilde R_2(\mu-\mu_1)=
\frac{R_2(\mu-\mu_1)-R_2(\mu-\lambda_{1i})}{\mu_1-\lambda_{1i}} \\
=-\int_0^1 R_2'(\sigma(\mu-\mu_1)+(1-\sigma)(\mu-\lambda_{1i}))\,d\sigma,
\end{split}$$ and in particular, $\lim_{\mu_1\to\lambda_{1i}}\tilde R_2(\mu-\mu_1)=
-R'_2(\mu-\lambda_{1i})$.
Our first remark is that the contour in $\int_{\gamma_s} R_1(\mu_1)R_2(\mu-\lambda_{1i})f\,d\mu_1$ can be shifted farther from the spectrum of $H_1$ since now the $R_2$ term is independent of $\mu_1$, so its contribution is negligible. Hence we only need to analyze $\int_{\gamma_s} R_1(\mu_1)(\mu_1-\lambda_{1i})\tilde R_2(\mu-\mu_1)\,d\mu_1$. But $$\tilde R_1(\mu_1)\tilde R_2(\mu-\mu_1)
=R_1(\mu_1)(\mu_1-\lambda_{1i})\tilde R_2(\mu-\mu_1)$$ is holomorphic for $\mu_1$ near $\lambda_{1i}$, so we may move the contour to go through $\lambda_{1i}$ when $s=s_0$. Moreover, from the usual parametrix identity , using the holomorphy of $\tilde R_1(\mu_1)
=(\mu_1-\lambda_{1i})
R_1(\mu_1)$ as a bounded operator on $L^2$ near $\mu_1=\lambda_{1i}$, $(\mu_1-\lambda_{1i})R_1(\mu_1)$ has the standard asymptotics. Note that $\tilde R_1(\lambda_{1i})=\Pi_{1i}$.
Thus, the holomorphic function $\tilde R_1(\mu_1)R_2(\mu_2)f$ has asymptotics $$\tilde R_1(\mu_1)R_2(\mu_2)f=x_1^{k_1/2}x_2^{k_2/2}x_1^{i\sqrt{\mu_1-k_1^2/4}}
x_2^{i\sqrt{\mu_2-k_2^2/4}} a(\mu_1,\mu_2),$$ where $a(\mu_1,\mu_2)$ is $\Cinf$ on $U_{\mu_1}\times U_{\mu-\mu_2}
\times {\overline{M}}_1\times{\overline{M}}_2$, and holomorphic in $\mu_1$ and $\mu_2$, where $U$ is a neighborhood of $\lambda_{1i}$. Hence $$\tilde R_1(\mu_1)\tilde R_2(\mu-\mu_1)f = \frac{\tilde R_1(\mu_1)
R_2(\mu-\mu_1) f -\tilde R_1(\mu_1)R_2(\mu-\lambda_{1i})f}{\mu_1
-\lambda_{1i}}$$ $$=x_1^{k_1/2}x_2^{k_1/2} \cdot (\mu_1-\lambda_{1i})^{-1} \cdot
\left(x_1^{i\sqrt{\mu_1-k_1^2/4}}x_2^{i\sqrt{\mu-\mu_1-k_2^2/4}}
a(\mu_1,\mu-\mu_1) \right.$$ $$\left. - x_1^{i\sqrt{\mu_1-k_1^2/4}}
x_2^{i\sqrt{\mu-\lambda_{1i}-k_2^2/4}}
a(\mu_1,\mu-\lambda_{1i}) \right)$$ Now add and subtract $x_1^{i\sqrt{\mu_1-k_1^2/4}}
x_2^{i\sqrt{\mu-\mu_1-k_2^2/4}} a(\mu_1,\mu-\lambda_{1i})$ in the numerator of this fraction on the right, and note that the term involving the holomorphic function $$\frac{a(\mu_1,\mu-\mu_1)-a(\mu_1,\mu-\lambda_{1i})}{\mu_1-\lambda_{1i}}$$ has the same asymptotics as if $\lambda_{1i}$ were not a pole; thus we only need to consider $$\int_{\gamma_s}\frac{x_1^{i\sqrt{\mu_1-k_1^2/4}}x_2^{i\sqrt{\mu-\mu_1-k_2^2/4}}
-x_1^{i\sqrt{\mu_1-k_1^2/4}}
x_2^{i\sqrt{\mu-\lambda_{1i}-k_2^2/4}}}{\mu_1-\lambda_{1i}}\,a(\mu_1,
\mu-\lambda_{1i})\,d\mu_1.$$ We can also write $a(\mu_1,\mu-\lambda_{1i})=a(\lambda_{1i},\mu-
\lambda_{1i})+(\mu_1-\lambda_{1i})a_1(\mu_1,\mu-\lambda_{1i})$, where $a_1$ is holomorphic in $\mu_1$ and smooth on ${\overline{M}}_1\times{\overline{M}}_2$; the term involving $a_1$ again yields an expression with the standard asymptotics, as if $\lambda_{1i}$ were not a pole. Hence it remains to consider $$a(\lambda_{1i},\mu-\lambda_{1i})
\int_{\gamma_s}\frac{x_1^{i\sqrt{\mu_1-k_1^2/4}}x_2^{i\sqrt{\mu-\mu_1-k_2^2/4}}
-x_1^{i\sqrt{\mu_1-k_1^2/4}}
x_2^{i\sqrt{\mu-\lambda_{1i}-k_2^2/4}}}{\mu_1-\lambda_{1i}}\,d\mu_1.$$ This depends on $f$ only via $$\begin{split}
a(\lambda_{1i},\mu-\lambda_{1i})&=-x_1^{-k_1/2-i\sqrt{\lambda_{1i}-k_1^2/4}}
x_2^{-k_2/2-
i\sqrt{\mu-\lambda_{1i}-k_2^2/4}}(\Pi_{1i}\otimes R_2(\mu-\lambda_{1i}))f\\
&=-x_1^{-k_1/2-i\sqrt{\lambda_{1i}-k_1^2/4}}
(\Pi_{1i}\otimes P_2^t(\mu-\lambda_{1i}))f.
\end{split}$$ The factor $x_1^{-i\sqrt{\lambda_{1i}-k_1^2/4}}$ here corresponds to the precise decay of the eigenfunction $\phi_{1i}$. The remaining integral is an explicit function which depends only on $s$, $x_1$ and $x_2$ (and of course $\mu$). Note that in the region $x_2>0$, it is simply a smooth function of $s$, and $x_2$ (or equivalently, $s$ and $x_1$, keeping in mind that $s=\log x_2/\log x_1$ is near $s_0\in(0,+\infty)$. Thus we only need to understand the asymptotics of this function as $x_1,x_2\to 0$.
Factor out $x_1^{iF(\mu_1)}=x_1^{iF(\mu_1^0(s))}
x_1^{i(F(\mu_1)-F(\mu_1^0(s)))}$ to get $$\label{eq:integral-77}
x_1^{iF(\mu_1^0(s))}\int_{\gamma_s}e^{-i(F(\mu_1)-F(\mu_1^0(s)))/\rho_1}
\frac{1-e^{-is(\sqrt{\mu-\lambda_{1i}-k_2^2/4}-\sqrt{\mu-\mu_1-k_2^2/4})
/\rho_1}}{\mu_1-\lambda_{1i}}\,d\mu_1,$$ where we wrote the integrand in terms of $\rho_1=-1/\log x_1\geq 0$ to keep the signs easier to follow. Note that $\im(F(\mu_1)-F(\mu_1^0(s)))\leq 0$ along $\gamma_s$.
So far [*any*]{} contour $\gamma_s$ that stays in ${\mathcal N}_s$ has been suitable for our calculations. Now we impose an additional condition, namely that there exists a fixed interval $[-\ep,\ep]$ such that for $s_0-\delta\leq s\leq s_0$ $\gamma_s$ stays in the region where $\im \sqrt{\mu-\mu_1-k_2^2/4}\geq \im \sqrt{\mu-\lambda_{1i}-k_2^2/4}$. Since this region is one side of a parabola, see Figure \[fig:contour2\], this condition can be easily satisfied. This condition ensures that the real part of the exponent in the numerator in is non-positive, hence bounded even as $\rho_1\to 0$. In addition, the fraction in is bounded by $C\rho_1^{-1}$ for some $C>0$, since dividing the denominator by $\rho_1$ yields a bounded function.
The key point is that $$F(\mu_1)-F(\mu_1^0(s))=(\mu_1-\mu_1^0(s))^2F_2(\mu_1-\mu_1^0(s),\mu_1^0(s)),$$ with $F_2$ non-zero at $(0,\mu_1^0(s))$, since $\mu_1^0(s)$ is a non-degenerate critical point of $F$, while $$\begin{split}
\sqrt{\mu-\lambda_{1i}-k_2^2/4}-\sqrt{\mu-\mu_1-k_2^2/4}
=(\mu_1-\lambda_{1i})F_1(\mu_1-\lambda_{1i})
\end{split}$$ with $F_1$ nonvanishing and holomorphic near $0$. This implies that the first exponent is a smooth function of $(\mu_1-\mu_1^0(s))^2/\rho_1$, with a negative real part of the same order of magnitude as $\rho_1\to 0$, hence Schwartz as $(\mu_1-\mu_1^0(s))^2/\rho_1\to \infty$, and the second one is a smooth function of $(\mu_1-\lambda_{1i})/\rho_1$ with a negative real part. However, we can only make the Schwartz conclusion here if the real part of the same order of magnitude as $(\mu_1-\lambda_{1i})/\rho_1$. Since $\gamma_s$ is on one side of a parabola, this is impossible to accomplish if $\gamma_s$ is to be smooth (for then it is tangent to the parabola). Hence, we need to break up $\gamma_s$ into two integrals which will yield boundary terms. These boundary terms cancel when $s\neq s_0$.
The geometry is as follows. Start with the space $Z_0=[0,1)_{\rho_1}\times(s_0-\delta,s_0]_s\times \Real_t$. We first blow up $t=0$, $\rho_1=0$ parabolically (recall that $t$ is the parameter along $\gamma_s$; $\gamma_s(0)=\mu_1^0(s)$), so that in the interior of the front face $s$, $t/\rho_1^{1/2}$ and $\rho_1$ become valid coordinates. Then $(F(\mu_1)-F(\mu_1^0(s)))/\rho_1$ is smooth near the interior of the front face, and $e^{i(F(\mu_1)-F(\mu_1^0(s)))/\rho_1}$ is smooth on the blown up space with infinite order vanishing off the front face. Away from $s=s_0$, $e^{-is(\sqrt{\mu-\lambda_{1i}-k_2^2/4}-\sqrt{\mu-\mu_1-k_2^2/4})
/\rho_1}$ is rapidly decreasing, hence the standard push-forward theorem for polyhomogeneous functions [@RBMCalcCon] yields the same result as stationary phase. In fact, we should think of integrating densities, hence we need to change $d\mu_1$, i.e. $dt$, to $\rho^{1/2}d(t/\rho^{1/2})$, giving the stationary phase asymptotics $\rho^{1/2}$ times a smooth function of $(s,\rho^{1/2})$ (multiplied by the exponential $x_1^{iF(\mu_1^0(s))}$ that we took outside the integral in ), though the terms of the form $\rho$ times a smooth function of $\rho$ and $s$ in fact cancel (these are the boundary terms of the integration). This is an alternative way of thinking about complex stationary phase when the real part of the exponent is non-positive and at least comparable to or larger than the imaginary part, as suggested to us by Richard Melrose. It simplifies our previous arguments here, although unfortunately it does not directly apply when considering asymptotics at the continuous spectrum, as we do in the next section.
At $s=s_0$ we also need to resolve the geometry of the second factor in the integrand of . Had we not performed the parabolic blow-up above, it would suffice to blow up $$F'_1=\{(\rho_1,s,t):\ t=0,\ \rho_1=0,\ s=s_0\}\subset Z_0$$ in the usual spherical (homogeneous) sense. Indeed, the exponent in the second factor as well as the denominator are smooth functions of $(\mu_1-\lambda_{1i})/\rho_1$, hence of $t/\rho_1$ and $(s-s_0)/\rho_1$, and the exponential is rapidly decreasing as $|t/\rho_1|+|(s-s_0)/\rho_1|\to
\infty$ since $$\im(\sqrt{\mu-\lambda_{1i}-k_2^2/4}-\sqrt{\mu-\mu_1-k_2^2/4})
\leq -C'|\mu_1-\lambda_{1i}|\leq -C(|t|+|s-s_0|).$$ Thus, the second factor is polyhomogeneous on this blown up space, of order $-1$ on the front face, order $0$ off the front face.
Unfortunately these two blow ups, i.e. the parabolic one of $t=0$, $\rho_1=0$, and the spherical one of $t=0$, $\rho_1=0$, $s=s_0$, conflict with each other, and we need to find a common resolution. One common resolution is the following: we blow up the boundary $\rho_1=0$ of $Z_0$ by introducing $\hat\rho=\rho_1^{1/2}$ as our new boundary defining function. We write $(Z_0)_{1/2}$ for the blown up space. The spherical blow-ups of $$F_1=\{(\hat\rho,s,t):\ \hat\rho=0,\ t=0\} \Mand
F_2=\{(\hat\rho,s,t):\ \hat\rho=0,\ t=0,\ s=s_0\}$$ commute with each other since $F_2$ is a p-submanifold of $F_1$, so $[(Z_0)_{1/2};F_1,F_2]=[(Z_0)_{1/2};F_2,F_1]$. Let $\ff_j$ denote the front face of the blow-up of $F_j$, $j=1,2$. Near the interior of the front face of the second blow-up, i.e. $\ff_2$, $t/\hat\rho$, $(s-s_0)/\hat\rho$ and $\hat\rho$ are valid coordinates. Now blow up the submanifold $$F_3=\{\hat\rho,(s-s_0)/\hat\rho,t/\hat\rho):
\ t/\hat\rho=0,\ (s-s_0)/\hat\rho=0,\ \hat\rho=0\}$$ (which is a single point) to obtain $$Z=[(Z_0)_{1/2};F_1,F_2,F_3].$$ This introduces, in particular, the front face of the spherical blow up of $$F'_1=\{(\rho_1,t,s):\ t=0,\ \rho_1=0,\ s=s_0\}\subset Z_0$$ in $Z_0$, except that the boundary defining functions differ. In other words, if we blow up the front face of $[Z_0;F'_1]$ to admit $\rho_1^{1/2}$ as a smooth function, then a neighborhood of the interior of the front face is naturally diffeomorphic to a neighborhood of the interior of the front face $\ff_3$ of the blow-up of $F_3$. This can be seen explicitly since local coordinates which are valid in this region are given by $(t/\hat\rho)/\hat\rho=t/\rho$, $((s-s_0)/\hat\rho)/\hat\rho=(s-s_0)/\rho$ and $\hat\rho=\rho^{1/2}$. Since upon blowing up $F_2$, the lifts of $F_1$ and $F_3$ are disjoint, their blow-ups commute. Thus, we can rewrite $Z$ as $[(Z_0)_{1/2};F_2,F_3,F_1]$. Since the first factor of the integrand of is polyhomogeneous on $[(Z_0)^{1/2};F_1]$, while the second factor is polyhomogeneous on $[(Z_0)_{1/2};F_2,F_3]$, we deduce that the integrand is (one-step) polyhomogeneous on $Z$.
While the geometry of the integrand only requires these blow-ups, we need further blow-ups to create a b-fibration for the push-forward. Unfortunately, with the blow-ups discussed above, all we can hope for is to create a b-fibration with base given by a double blow-up of $(Y_0)_{1/2}=[0,1)_{\hat\rho}\times(s_0-\delta,s_0]$ (which is already a blow-up, in the sense of change of the $\Cinf$ structure, of $Y_0=[0,1)_{\rho_1}\times(s_0-\delta,s_0]$). Namely, one first blows up $$G_2=\{(\hat\rho,s):\ s=s_0,\ \hat\rho=0\},$$ in $(Y_0)_{1/2}$, and then $$G_3=\{(\hat\rho,s):\ (s-s_0)/\hat\rho=0,\ \hat\rho=0\}$$ in $[(Y_0)_{1/2};G_2]$; we denote this space by $$\label{eq:Y-def}
Y=[(Y_0)_{1/2};G_2,G_3].$$ It is then straight-forward to carry out appropriate blow-ups in our resolved space $Z$ to obtain a new space $Z'$ such that $Z'\to Y$ is a b-fibration. Hence we can apply the push-forward theorem of [@RBMCalcCon], and deduce that the integral of yields a polyhomogeneous function on $Y$. More specifically, the result is a one-step polyhomogeneous function on $Y$ (keep in mind that $\hat\rho$ is the boundary defining function!) with order $1$ on the lift of $\hat\rho=0$, and order $0$ on each of the two new front faces.
The asymptotics from $s\geq s_0$, $s\to s_0$, can be seen similarly. The optimal contours in $s>s_0$ have been shifted through $\lambda_{1i}$, so there is automatically a contribution from the pole. We are interested in letting $s$ decrease to $s_0$. Since in this region the real part of the exponent of $x_1^{i\sqrt{\mu_1-k_1^2/4}}$ in the expansion of $R_1(\mu_1)$ is less than that of $x_1^{i\sqrt{\lambda_{1i}-k_1^2/4}}$, we now need to expand an operator associated to $H_1$ rather than one associated to $H_2$ around $\lambda_{1i}$. This is a little more delicate, as we discuss below. The parametrix identity gives $$R_1(\mu_1)=P_1(\mu_1)-E(\mu_1)P_1(\mu_1)+E(\mu_1)R_1(\mu_1)F(\mu_1),$$ with all terms meromorphic. The first two terms are actually holomorphic, so we can shift the integral to the optimal location, and even let it go through $\lambda_{1i}$. Thus, we only need to consider the last term. Here the kernel of $E(\mu_1)$ is polyhomogeneous on the standard double space ${\overline{M}}_1\times{\overline{M}}_1$, with rapid vanishing on the right face. We expand the holomorphic function $E_1(\mu_1)$ as $$E(\mu_1)=E_1(\lambda_{1i})+(\mu_1-\lambda_{1i})\tilde E_1(\mu_1)$$ similarly to the expansion for $R_2(\mu-\mu_1)$ above. The contour of the integral $\int_{\gamma_s}E_1(\lambda_{1i})R_1(\mu_1)F(\mu_1)R_2(\mu-\mu_1)f\,d\mu_1$ can be shifted farther from $\spec(H_1)$, hence it is negligible compared to the other terms. So it remains to deal with $$\int_{\gamma_s}\tilde E_1(\lambda_{1i})(\mu_1-\lambda_{1i})R_1(\mu_1)
F(\mu_1)R_2(\mu-\mu_1)f\,d\mu_1.$$ But the integrand above again has the standard asymptotics by virtue of the holomorphy of $(\mu_1-\lambda_{1i})R_1(\mu_1)$, hence completely analogous calculation apply as for $s\leq s_0$ above.
We have introduced these new front faces in $Y_0$ in order to understand the asymptotics in terms of polyhomogeneous expansions. We should certainly discuss whether this complicated geometry is simply an artifact of our method, or whether it is necessary in the sense that polyhomogeneous asymptotics do not hold in a simpler space. We certainly need at least one of the front faces. Indeed, the residue term is order $0$ in $s>s_0$, and is missing in $s<s_0$, where the asymptotic expansion is order $1$ (in terms of $\rho_1^{1/2}$). Thus, it is possible to obtain polyhomogeneous asymptotics through $s=s_0$ only if there is some blow-up, and hence a new boundary face, along which the term of order $0$ tends to zero on approach to the lift of $s<s_0$. In addition, the exponential in the second factor is smooth only after $F'_1$ is blown up in $Z_0$, and so it appears that the spherical blow-up of $s=s_0$, $\rho_1=0$ in $Y_0$, or at least the presence of the front face of the last blow-up in $[(Y_0)_{1/2};G_2;G_3]$, is necessary. It then remains to see whether the integral is polyhomogeneous on $[Y_0;\{(\rho_1,s):\ \rho_1=0,\ s=s_0\}]$. If it is, it must be order $0$ on the front face and order $1/2$ on the lift of $s<s_0$. Correspondingly it has order at most $1/2$ on the blow-up of the corner, which is the interior of the front face $[(Y_0)_{1/2};G_2]$, although with a different boundary defining function. Nonetheless, we would expect to see decay in the asymptotics at the front face of $[(Y_0)_{1/2};G_2]$ away from $G_3$, and there is no such decay. Indeed, the exponential in the second factor of the integrand in is rapidly decreasing in the inverse image of this region under the projection, so it can be disregarded. Thus the dominant term as $\rho_1\to 0$ is an integral of the form $\int_{\gamma_s} e^{a\mu_1^2/\rho_1}(\mu_1-\lambda_{1i})^{-1}\,d\mu_1$, $a=-iF''(\mu_1^0(s))/2>0$. Shifting the contour to be vertical, the integral becomes $
i\int_{\Real} e^{-at^2/\rho_1}(it+(\mu_1^0(s)-\lambda_{1i}))^{-1}\,dt$, with imaginary part $$\int_{\Real} e^{-at^2/\rho_1}\frac{\mu_1^0(s)-\lambda_{1i}}
{t^2+(\mu_1^0(s)-\lambda_{1i}))^2}\,dt.$$ Thus, with $\mu_1^0(s)-\lambda_{1i}=\rho_1^{1/2}S$, $S<0$, and changing variables $T=t/\rho_1^{1/2}$, $$\int_{\Real} e^{-aT^2}\frac{S}{T^2+S^2}\,dT<0,$$ independent of $\rho_1>0$. Hence there is no decay as $\rho_1\to 0$. The limit $S\to 0$ can also be analyzed; indeed, it is the distribution $(T+i0)^{-1}$ paired with $e^{-aT^2}$. The limit from $s>s_0$ is similar, but now we get $(T-i0)^{-1}$ paired with $e^{-aT^2}$. The difference, $2\pi i$, is exactly the residue corresponding to the pole.
Note moreover that the principal symbol of $R(\mu)f$ in $s< s_0$ is an elliptic $f$-independent multiple of $P_1^t(\mu_1^0(s))P_2^t(\mu-\mu_1^0(s))f$, which in turn is the restriction of $$x_1^{-k_1/2-i\sqrt{\mu_1^0(s)-k_1^2/4}}R_1(\mu_1^0(s))P_2^t(\mu-\mu_1^0(s))f$$ to $x_1=0$. That is, the top term of the asymptotics of $R(\mu)f$, up to elliptic smooth factors and after factoring out $x_1^{k_1/2}x_2^{k_2/2}e^{-i\sqrt{\mu-k^2/4}/\rho}$, is $$\rho_1^{1/2}x_1^{-k_1/2-i\sqrt{\mu_1^0(s)-k_1^2/4}}R_1(\mu_1^0(s))
P_2^t(\mu-\mu_1^0(s))f,$$ which behaves as $\frac{\rho_1^{1/2}}{\mu_1^0(s)
-\lambda_{1i}}x_1^{-k_1/2-i\sqrt{\mu_1^0(s)-k_1^2/4}}\Pi_{1i}
P_2^t(\mu-\mu_1^0(s))f$ as $s\to s_0$ (up to lower order terms). We have thus explicitly matched the coefficients from the two sides of the corner.
Altogether, we have proved the following theorem.
\[thm:res-set-asymp\] Suppose $f\in\dCinf({\overline{X}})$, $\mu\in\Cx\setminus\spec(H)$. Let $\rho_i=-1/\log x_i$, $\rho^{-1}=\sqrt{\rho_1^{-2}+\rho_2^{-2}}$, $s=\rho_1/\rho_2$. Then $R(\mu)f$ has the following asymptotic expansion on $\Xt$: $$\begin{split}
R(\mu)f & =x_1^{k_1/2}x_2^{k_2/2}\exp (-i\sqrt{\mu-k^2/4}/\rho)h \\
+\sum_{i=1}^{N_1}x_2^{k_2/2+
i\sqrt{\mu-\lambda_{1i}-k_2^2/4}}(\phi_{1i} & \otimes h_{1i})\chi_{1i}
+\sum_{i=1}^{N_2}x_1^{k_1/2+
i\sqrt{\mu-\lambda_{1i}-k_2^2/4}}(h_{2i}\otimes \phi_{2i})\chi_{2i},
\end{split}
\label{eq:asbs}$$ where the terms have the following properties. (See Definition \[Def:Ns-Ps\] for the notation, Figure \[fig:contour4\] for a picture.)
1. $h_{1i}\in\Cinf({\overline{M}}_2)$, $h_{2i}\in\Cinf({\overline{M}}_1)$ satisfy $$\begin{split}
&(\phi_{1i}\otimes h_{1i})|_{{\overline{M}}_1\times \pa{\overline{M}}_2}
=-(\Pi_{1i}\otimes P_2^t(\mu-\lambda_{1i}))f,\ \text{resp.}\\
&(h_{2i}\otimes \phi_{2i})|_{\pa{\overline{M}}_1\times {\overline{M}}_2}.
=-(P_1^t(\mu-\lambda_{2i})\otimes\Pi_{2i})f.
\end{split}$$
2. For $\mu$ not real, $\chi_{ji}$ is a function of $s$, $\chi_{ji}\in\Cinf(\Real^+)$ is identically $1$ or $0$ for sufficiently large and small $s$, with the effect that
1. If $\lambda_{1i}\in{\mathcal P}^r_s$, then $\chi_{1i}(s)=0$, and if $\lambda_{2i}\in{\mathcal P}^\ell_s$, then $\chi_{2i}(s)=0$.
2. If $\lambda_{1i}\in{\mathcal P}^\ell_s$, respectively if $\lambda_{2i}\in{\mathcal P}^r_s$, then $\chi_{1i}(s)=1$, resp. $\chi_{2i}(s)=1$.
3. If $\lambda_{ji}\in{\mathcal N}_s$, then the first term in (\[eq:asbs\]) dominates the one corresponding to $\phi_{ji}$, hence the choice of $\chi_{ji}$ is irrelevant.
3. If $\mu$ is real, $\mu<\inf\spec(H)$, then $\chi_{1i}$ is a $\Cinf$ function of $(s-s_0)/\rho$, $s_0$ given by $$\label{eq:lambda_1i-def}
\lambda_{1i}=\mu_1^0(s_0)=\frac{\mu-k_2^2/4+s_0^2 k_1^2/4}{1+s_0^2},$$ $\chi'_{1i}$ is compactly supported, $\lim_{S\to +\infty}\chi_{1i}(S)=1$, $\lim_{S\to -\infty}\chi_{1i}(S)=0$. Similarly, $\chi_{2i}$ is a function of $(s-s_0)/\rho$, $s_0$ given by $$\label{eq:lambda_2i-def}
\lambda_{2i}=\mu-\mu_1^0(s_0)=\frac{s_0^2(\mu-k_1^2/4)+k_2^2/4}{1+s_0^2},$$ $\chi'_{2i}$ is compactly supported, $\lim_{S\to +\infty}\chi_{2i}(S)=0$, $\lim_{S\to -\infty}\chi_{1i}(S)=1$.
4. If $\mu\nin\Real$, then $h$ is polyhomogeneous on $\Xt$, of order $1/2$ on the front face, $3/2$ on the side faces. The principal symbol of $h$ is an elliptic $f$-independent multiple of $$\begin{split}\label{eq:pr-symbol-88}
&P_1^t(\mu_1^0(s))P_2^t(\mu-\mu_1^0(s))f\ \text{on the front face},\\
&
S_1(k_1^2/4)P_2^t(\mu-k_2^2/4)f\ \text{on the lift of}\ {\overline{M}}_1\times \pa{\overline{M}}_2,\\
&P_1^t(\mu-k_1^2/4)S_2(k_2^2/4)f\ \text{on the lift of}\ \pa{\overline{M}}_1\times {\overline{M}}_2.
\end{split}$$
5. If $\mu\in\Real$, $\mu\nin\spec(H)$, then $g$ is polyhomogeneous on $\Xt$ doubly blown up at the finite number of submanifolds $s=s_0$, $\rho_1=0$, of the front face of $\Xt$, as in , with $s_0$ given by either of the equations -. The order at the front faces of the blow-ups is $0$, while the order on the old front face is still $1/2$. The principal symbol on the front faces depends on $f$ only via $(\Pi_{1i}\otimes P_2^t(\mu-\lambda_{1i}))f$, resp. $(P_1^t(\mu-\lambda_{2i})\otimes \Pi_{2i})f$; it is indeed an elliptic multiple of these.
Asymptotics inside the continuous spectrum
==========================================
The arguments needed to analyze the asymptotics of $R(\mu\pm i0)f$ when $f \in \dCinf(X)$ and $\mu > k^2/4$ are not substantially different, and in some senses even simpler because the contributions from the $L^2$ eigenfunctions below the continuous spectrum are always negligible away from the side-faces. Since threshold eigenvalues introduce additional complications similar to the discussion of the previous section, with the additional issue that the push-forward results of [@RBMCalcCon] are not directly applicable since the stationary phase arguments have a substantially different flavor when the phase is pure imaginary, we will assume that $k_j^2/4$ is not a pole of $R_j$. Thus, we have the following theorem.
\[thm:cont-spec-asymp\] Suppose $f\in\dCinf({\overline{X}})$, $\mu>k^2/4=(k_1^2+k_2^2)/4$, and $k_j^2/4$ is not a pole of $R_j$, $j=1,2$. Then, with $\rho_i=-1/\log x_i$, $\rho^{-1}=\sqrt{\rho_1^{-2}+\rho_2^{-2}}$, $R(\mu-i0)f$ has the following asymptotic expansion on $\Xt$: $$\begin{split}
& R(\mu-i0)f =x_1^{k_1/2}x_2^{k_2/2}\exp (-i\sqrt{\mu-k^2/4}/\rho)h \\
& +\sum_{i=1}^{N_1}x_2^{k_2/2+
i\sqrt{\mu-\lambda_{1i}-k_2^2/4}}(\phi_{1i}\otimes h_{1i})
+\sum_{i=1}^{N_2}x_1^{k_1/2+
i\sqrt{\mu-\lambda_{1i}-k_2^2/4}}(h_{2i}\otimes \phi_{2i}),
\end{split}$$ where $h$ is one-step polyhomogeneous on $\Xt$, and $h_{1i}$ and $h_{2i}$ are one-step polyhomogeneous on $\bar M_2$ and $\bar M_1$ respectively. Moreover, $h$ has order $1/2$ on the front face and $3/2$ on the side faces, while the $h_{1i}$ is in $\Cinf({\overline{M}}_2)$ (i.e. has order $0$ at $\pa{\overline{M}}_2$) and $h_{2i}\in\Cinf({\overline{M}}_1)$. The principal symbol of $h$ is given by .
Note that due to the exponential decay of $x_1^{-k_1/2}
\phi_{1i}$ for non-threshold eigenvalues $\lambda_{1i}$, namely $$x_1^{-k_1/2}\phi_{1i}
\in x_1^{\sqrt{k_1^2/4-\lambda_{1i}}}\Cinf(\bar M_1),\ k_1^2/4-\lambda_{1i}>0,$$ and the similar decay of $x_2^{-k_2/2}\phi_{2i}$, the second term is dominated by the first one everywhere but at the lift of $\bar M_1\times
\pa\bar M_2$, and similarly the third term is dominated by the first one everywhere but at the lift of $\pa\bar M_1\times
\bar M_2$. In particular, on a neigborhood $U$ of the lift of $\bar M_1\times
\pa\bar M_2$, one has the asymptotics $$R(\mu)f|_U=x_1^{k_1/2}x_2^{k_2/2}\exp (-i\sqrt{\mu-k^2/4}/\rho)g+
\sum_{i=1}^{N_1}x_2^{k_2/2+
i\sqrt{\mu-\lambda_{1i}-k_2^2/4}}(\phi_{1i}\otimes g_{1i}),$$ while on a neighborhood $U'$ of the front face which is disjoint from $\pa\bar M_1\times \bar M_2$ and $\bar M_1\times
\pa\bar M_2$ both the second and third terms are irrelevant: $$R(\mu)f|_{U'}=x_1^{k_1/2}x_2^{k_2/2}\exp (-i\sqrt{\mu-k^2/4}/\rho)g.$$
The asymptotics is unchanged away from the corners of $\Xt$ even if $k_j^2/4$ is a pole of $R_j$, $j=1,2$, in the sense that these do not contribute to the asymptotics in the interior of the front face of $\Xt$, and give the standard contribution in each appropriate side face. This follows from stationary phase arguments using the integral representation . In fact, we can always reduce the calculation to the evaluation of an integral like , though here it is convenient to rewrite the fraction appearing in that formula as an integral analogous to .
This is simply a limiting case of the argument of the previous section, but which has additional simplifying features. Thus, we simply consider a smooth contour $\gamma'$ that runs along the real axis on the interval $[k_1^2/4,\mu-k_2^2/4]$, but avoids the poles of $R_1$ as well as $\mu$ minus the poles of $R_2$, see Figure \[fig:contour5\]. Such a contour is simply the limit of the contours $\gamma'$ that we have considered before. Note, in particular, that with $$F(\mu_1) = \sqrt{\mu_1-k_1^2/4}+s\sqrt{\mu-\mu_1-k_2^2/4},
\qquad s=\frac{\rho_1}{\rho_2}$$ as in the previous section, $\im F(\mu_1)\leq 0$ everywhere, and it is strictly negative when $\mu_1\nin[k_1^2/4,\mu-k_2^2/4]$. Since the critical point of $F$ lies in $[k_1^2/4,\mu-k_2^2/4]$, the choice of the contour outside this interval is irrelevant (except that it should be far from the real axis near infinity, as before, to ensure convergence). In particular, $\lambda_{ji}\in{\mathcal N}_s$, for all $s\in[0,+\infty)$, except if $s=0$, when $\im F(\mu-\lambda_{2i})=0$ (but $\im F(\lambda_{1i})<0$); of course a similar statement holds at $s^{-1}=0$ with $\lambda_{2i}$ replaced by $\lambda_{1i}$. Thus, the residues of the poles of $R_1$ and $R_2$ also provide lower order contributions than the stationary phase term in most regions in $\Xt$, except that the poles of $R_1$ give the same order as the stationary phase term at $M_1\times\pa{\overline{M}}_2$, and similarly for $R_2$. The standard stationary phase lemma now gives the desired results, much as in the previous section. To deal with the points $\mu_1=k_1^2/4$ and $\mu_1=\mu-k_2^2/4$, i.e. the endpoints of $[k_1^2/4,\mu-k_2^2/4]$, one introduces $\tau_1=\sqrt{\mu_1-k_1^2/4}$, resp. $\tau_2=\sqrt{\mu-\mu_1-k_2^2/4}$, just as in the previous section, and notes that in $0\leq s\leq C$, the phase with respect to $\tau_1$ is never stationary, while with respect to $\tau_2$ it is only stationary at $s=0$, hence the left end point gives a rapidly decreasing contribution in this region, while the right end point gives asymptotically non-trivial terms only at $s=0$, exactly as expected.
The resolvent compactification
==============================
We now turn to a problem originally posited as one of our main goals, namely to define a resolution $X^2_{{{\mathrm{res}}}}$ of $X^2$, which we call the resolvent double space, which is a manifold with corners and on which $R(\mu)$ is polyhomogeneous, at least when $\mu \notin \spec(H)$. This space is meant to be an analogue of the $0$-double space $({\overline{M}}_j)^2_0$, for either of the conformally compact factors ${\overline{M}}_j$, which we discussed in §3. The existence of this space sets the stage for all further development of the analytic properties of the Laplacian, including the scattering theory, on $X$.
In fact, the last section contains essentially all of the requisite analysis, and what remains here is to show how those calculations may be interpreted. As usual, the starting point is the integral representation (\[eq:resform\]) for $R(\mu)$. Again we begin by assuming that neither operator $H_j$ has bound states.
First, as explained in Theorem \[th:ccst\], apart from the usual diagonal singularities, the Schwartz kernels of both $R_1(\mu_1)$ and $R_2(\mu-\mu_1)$ are polyhomogeneous at the boundary hypersurfaces of the $0$-double spaces $({\overline{M}}_j)_0^2$. Hence the Schwartz kernel of the integrand $R_1(\mu_1)
R_2(\mu-\mu_1)$ is polyhomogeneous on the product of these $0$-double spaces, $({\overline{M}}_1)^2_0 \times({\overline{M}}_2)^2_0$, again with diagonal singularities on each factor, and thus it is clear that our resolution process should start here. Our task is to see what new singularities are introduced by the contour integration, and how these may be accomodated geometrically. We shall systematically neglect the diagonal singularities in the ensuing discussion; that they produce the correct singularity upon integration is obvious in the interior, and requires only minor justification, which we omit.
This product of double spaces has six boundary hypersurfaces, namely $$\ff_1\times ({\overline{M}}_2)_0^2,\ \lf_1\times ({\overline{M}}_2)_0^2,\ \rf_1\times ({\overline{M}}_2)_0^2,
\ ({\overline{M}}_1)_0^2\times\ff_2,\ ({\overline{M}}_1)_0^2\times\lf_2,\ ({\overline{M}}_1)_0^2\times\rf_2,$$ where $\ff_j$ is the front face, and $\lf_j$ and $\rf_j$ the left and right faces of $({\overline{M}}_j)^2_0$. The kernel of $R_j(\nu)$ is the product of a smooth function on $({\overline{M}}_j)^2_0$ (albeit with a diagonal singularity) with the factor $$(\rho_{\lf_j}\rho_{\rf_j})^{k_j/2+i\sqrt{\nu-k_j^2/4}};$$ in particular, it is smooth up the $\ff_j$. It is straightforward to see that $R(\mu)$ is smooth at the two ‘front faces’ of the product, $\ff_1 \times ({\overline{M}}_2)_0^2$ and $({\overline{M}}_1)_0^2 \times \ff_2$. This is because the exponents in the expansions of $R_1(\mu_1) R_2(\mu-\mu_1)$ at these faces do not depend on $\mu_1$. It remains to analyze the behaviour at the remaining ‘side faces’.
In fact, the computations of the last section give the asymptotics of $R(\mu)$ at $(\lf_1\times ({\overline{M}}_2)_0^2)\cup(({\overline{M}}_1)_0^2\times\lf_2)$, and in particular at the corner $(\lf_1\times({\overline{M}}_2)_0^2)\cap(({\overline{M}}_1)_0^2
\times\lf_2)$. To proceed further, the salient observation is that while those stationary phase computations were motivated by the geometric picture of letting the variable $(z_1,z_2) \in M_1 \times M_2$ converge to the boundary (in one of three ways), they can also be interpreted purely analytically as a derivation of asymptotics when integrating a function of the form $x_1^{\alpha_1}x_2^{\alpha_2}F$, where $\alpha_1$ and $\alpha_2$ are the appropriate exponents depending on $\mu$ and $\mu_1$ and $F$ depends smoothly on these variables. What we are asking now is the asymptotics of an integral of almost exactly the same form, but where each $x_j$ is replaced by $\rho_{\lf_j}\rho_{\rf_j}$.
Replace each of the exponential type defining function $\rho$ at the side faces of these $0$-double spaces by the polynomial type defining function $-1/\log \rho$. In other words, logarithmically blow up each of the two side faces $\lf_j$ and $\rf_j$ of $({\overline{M}}_j)_0^2$, but [*not*]{} the front face. Call the resulting space $({\overline{M}}_j)^2_{0,\log}$. For notational convenience, set $$R_{\lf_j} = -1/\log \rho_{\lf_j},\qquad R_{\rf_j} = -1/\log \rho_{\rf_j},
\qquad j = 1,2.$$ Now perform the same calculations as in the last section. It follows that the asymptotics of $R(\mu)$ along the interiors of each of the four side faces is polyhomogeneous in terms of these new defining functions.
However, to fully resolve the singularities at the various corners, we must perform a further sequence of blow-ups. More specifically, the stationary phase calculation produces terms polyhomogeneous in $R_{\lf_j}$ and $R_{\rf_j}$, and the only other terms which arise here are polyhomogeneous in $(\log \rho_{\lf_2}\rho_{\rf_2})/(\log \rho_{\lf_1}\rho_{\rf_1})$. Thus we need to find a resolution so that this projective coordinate is $\mathcal C^\infty$ when it is bounded, and similarly for its inverse. It is not hard to do this. We can certainly assume that $\rho_{\lf_j},
\rho_{\rf_j} \leq 1/2$. Now observe that $$\label{eq:log-rho-id-9}
\frac{\log(\rho_{\lf_2}\rho_{\rf_2})}{\log (\rho_{\lf_1}\rho_{\rf_1})}=
\frac{\log\rho_{\lf_2}}{\log \rho_{\lf_1}+\log\rho_{\rf_1}}
+\frac{\log\rho_{\rf_2}}{\log \rho_{\lf_1}+\log\rho_{\rf_1}},$$ which is equal to $$\label{eq:log-rho-id-99}
\frac{R_{\lf_1}}{R_{\lf_2}} \frac{R_{\rf_1}}{R_{\lf_1}+R_{\rf_1}}
+ \frac{R_{\lf_1}}{R_{\rf_2}} \frac{R_{\rf_1}}{R_{\lf_1}+R_{\rf_1}}.$$ Both terms in this last expression are positive, and hence their sum is bounded if and only if both terms are. Clearly we require a space where all of the quotients $$R_{\lf_i}/R_{\lf_j}, \quad R_{\lf_i}/R_{\rf_j}, \quad R_{\rf_i}/R_{\rf_j}
\qquad i,j = 1,2,$$ and their inverses are smooth (when bounded).
Thus first blow up $$\label{eq:double-blow-up-1}
(\lf_1\cap\rf_1)\times ({\overline{M}}_2)^2_{0,\log},\ ({\overline{M}}_1)^2_{0,\log}
\times(\lf_2\cap\rf_2).$$ This may be done in either order since these submanifolds are transverse. Next, we must blow up the collection of submanifolds covering $$\begin{split}
&(\lf_1\times ({\overline{M}}_2)^2_{0,\log})\cap(({\overline{M}}_1)^2_{0,\log}\times\lf_2),
\ (\lf_1\times ({\overline{M}}_2)^2_{0,\log})\cap(({\overline{M}}_1)^2_{0,\log}\times\rf_2),\\
&(\rf_1\times ({\overline{M}}_2)^2_{0,\log})\cap(({\overline{M}}_1)^2_{0,\log}\times\lf_2),
\ (\rf_1\times ({\overline{M}}_2)^2_{0,\log})\cap(({\overline{M}}_1)^2_{0,\log}\times\rf_2).
\end{split}$$ These are disjoint now after the initial blow-up, and so may be blown up in any order. The manifold $X^2_{{{\mathrm{res}}}}$ obtained in this way has all the desired properties. In fact, working in a region where is bounded, we must show that both terms in are smooth. But $$\frac{R_{\lf_1}}{R_{\lf_2}} \frac{R_{\rf_1}}{R_{\lf_1}+R_{\rf_1}}
=\frac{R_{\lf_1}}{R_{\lf_2}} \frac{1}{(R_{\lf_1}/R_{\rf_1}) + 1},$$ with a similar expression for the other term. The first spherical blow up makes $R_{\rf_j}/R_{\lf_j}$ and its inverse smooth where bounded, while the prefactors $R_{\lf_1}/R_{\lf_2}$ and $R_{\lf_1}/R_{\rf_2}$ also become smooth because $(\lf_1 \times
({\overline{M}}_2)^2_{0,\log}) \cap (({\overline{M}}_1)^2_{0,\log} \times \lf_2)$ and $(\lf_1 \times ({\overline{M}}_2)^2_{0,\log}) \cap (({\overline{M}}_1)^2_{0,\log} \times \rf_2)$ have been blown up. We thus conclude that when $H_1$ and $H_2$ have no $L^2$ eigenfunctions, $R(\mu)$ is polyhomogeneous on this space $X^2_{{\mathrm{res}}}$.
To conclude this discussion, suppose that either $H_1$ or $H_2$ have bound states. Then $R(\mu)$ will have additional singularities at certain submanifolds of $X^2_{{\mathrm{res}}}$ where $\log(\rho_{\lf_2}\rho_{\rf_2})/\log (\rho_{\lf_1}\rho_{\rf_1})$ assumes the same critical values $s_0$ as already appeared for $s=\log x_2/\log x_1$ in part (v) of Theorem \[thm:res-set-asymp\]. That is, these singularities happen at the submanifolds given by $S=s_0$ [*inside $\pa X^2_{{{\mathrm{res}}}}$*]{}, where $S$ is the smooth function on $X^2_{{\mathrm{res}}}$ given by $$S=\left(\frac{R_{\lf_1}}{R_{\lf_2}}+ \frac{R_{\lf_1}}{R_{\rf_2}}\right)
\frac{1}{(R_{\lf_1}/R_{\rf_1}) + 1}=(\sigma_{l1l2}+\sigma_{l1r2})
\frac{1}{\sigma_{l1r1}+1},$$ in the region where $\sigma_{l1l2}=\frac {R_{\lf_1}}{R_{\lf_2}}$, $\sigma_{l1r2}=\frac{R_{\lf_1}}{R_{\rf_2}}$ and $\sigma_{l1r1}=R_{\lf_1}/R_{\rf_1}$ are valid coordinates, i.e. where they are bounded. Since the second factor is thus bounded from both below and from above, we conclude that, as long as $s_0=0$ is not one of the critical values, at least one of $\sigma_{l1l2}$ and $\sigma_{l1r2}$ is non-zero at any point on $S=s_0$, and in addition $\pa_{\sigma_{l1l2}}S\neq 0$, $\pa_{\sigma_{l1r2}}S\neq 0$ there, so $S=s_0$ is a codimension one p-submanifold of $X^2_{{{\mathrm{res}}}}$, and we have singularities at its intersection with $\pa X^2_{{{\mathrm{res}}}}$, which in turn is a finite union of codimension one p-submanifolds of $\pa X^2_{{{\mathrm{res}}}}$. These singularities can be resolved by the same double blow up, preceeded by the square root blow up of the boundary defining function, as on the single space.
While these blow ups may appear somewhat complicated, the only relevant part of $X^2_{{{\mathrm{res}}}}$ as far as the Martin boundary is concerned, is the inverse image $U'$ of regions $X\times U$, $\overline{U}$ compact subset of $X$, $U=U_1\times U_2\subset {\overline{M}}_1\times {\overline{M}}_2$ open, under the blow-down map $\beta:X^2_{{{\mathrm{res}}}}\to X^2$. Thus, in the second factor we always stay away from the boundary, hence any blow-ups involving $\rf_1$, $\rf_2$ can be neglected. Thus, the intersection of $U'$ with $X^2_{{{\mathrm{res}}}}$ is a subset of $$[({\overline{M}}_1)_{\log}\times U_1\times ({\overline{M}}_2)_{\log}\times U_2;
\pa ({\overline{M}}_1)_{\log}\times U_1\times\pa ({\overline{M}}_2)_{\log}\times U_2]$$ which is essentially the same as $\Xt\times U_1\times U_2$ (i.e. they agree up to the rearranging of factors). Hence, in this set the asymptotics of the kernel of $R(\mu)$ is given by Theorem \[thm:res-set-asymp\], with an extra variable on $U$ added, and references to $f$ dropped (this really corresponds to applying the resolvent to delta distributions at points $z$ and letting $z$ vary).
The Martin compactification
===========================
With the information we have collected and proved about the resolvent $R(\mu)$, it is now a simple matter to determine the full Martin compactification of $X = M_1 \times M_2$. We refer back to §5 for the general details of how this construction is to be carried out, but we briefly recall the main ideas. Fix any $\mu \in {\mathbb R}$, $\mu<\inf\spec(H)$. In addition, let $p\in X$ be fixed and $w^{(\ell)} = ({w^{(\ell)}}_1,{w^{(\ell)}}_2) \in X$ any sequence which leaves every compact set. Define $$U^{(\ell)}(z) = \frac{R(\mu;z,{w^{(\ell)}})}{R(\mu;p,{w^{(\ell)}})};$$ this function is a solution of $(H - \mu)U^\ell = 0$ for $z \neq {w^{(\ell)}}$ and is normalized so that $U^{(\ell)}(p) = 1$. Any such sequence has a subsequence $U^{\ell'}$ which converges uniformly on compact sets to a function $U$ which satisfies $(H-\mu)U = 0$ on all of $X$ and $U(p) = 1$. The Martin boundary $\overline{X}_M$ is defined to be the set of all possible limit eigenfunctions that arise in this manner.
First consider the case if neither $H_1$ nor $H_2$ has $L^2$ eigenvalues. In this case, by the asymptotics of Proposition \[prop:no-bd-state-asymp\] (see also the last paragraph of the previous section) $$\begin{split}
R(\mu)|_{\Xt\times X}&=x_1^{k_1/2}x_2^{k_2/2}
\exp (-i\sqrt{\mu-k^2/4}/\rho)h_0 g,
\end{split}$$ where $h_0$ is the product of the square root of a defining function of the front face of $\Xt$ with the $3/2$ power of a defining function of each side face of $\Xt$, and $g$ is $\Cinf$ in $\Xt\times X$ with restriction to each face being an elliptic multiple of $$\begin{split}
&P_1^t(\mu_1^0(s))P_2^t(\mu-\mu_1^0(s))\ \text{on the front face},\\
&
S_1(k_1^2/4)P_2^t(\mu-k_2^2/4)\ \text{on the lift of}\ {\overline{M}}_1\times
\pa{\overline{M}}_2\times U,\\
&P_1^t(\mu-k_1^2/4)S_2(k_2^2/4)\ \text{on the lift of}\ \pa{\overline{M}}_1\times {\overline{M}}_2
\times U.
\end{split}$$ In particular, $$U(\mu,z,w)=\frac{R(\mu;z,w)}{R(\mu;p,w)}$$ is a continuous function on $\Xt_w\times X_z$. Thus, the map $$\label{eq:Xt-to-XM}
\pa\Xt\ni w\mapsto U(\mu,.,w)\in\Cinf(X)$$ defines a map $\Xt\to\pa\overline{X}_M$ which is continuous and surjective. More explicitly, if ${w^{(\ell)}}$ is any sequence of points in $X$ converging to $w\in\pa\Xt$, the continuity of $U$ implies that $U^{(\ell)}(z)=U(\mu,z,w)$, hence $U(\mu,.,w)$ is a point in the Martin boundary. Conversely, suppose that ${w^{(\ell)}}$ is any sequence of points in $X$ which leaves every compact subset of $X$, and such that $U^{(\ell)}$ converges uniformly on compact subsets of $X$. By passing to a subsequence $w^{(\ell')}$ we may assume that $w^{(\ell')}$ converges to some $w\in\pa
\Xt$ since $\Xt$ is compact. Hence $\lim U^{(\ell')}(z)=U(\mu,w,z)$, so the map is indeed surjective.
Explicitly, $U$ is given by $$\begin{split}
&\frac{P_1^t(\mu_1^0(s);z_1,w_1)P_2^t(\mu-\mu_1^0(s);z_2,w_2)}
{P_1^t(\mu_1^0(s);p_1,w_1)P_2^t(\mu-\mu_1^0(s);p_2,w_2)}
,\ (s,w_1,w_2)\in
[0,+\infty)_s\times\pa {\overline{M}}_1\times\pa{\overline{M}}_2,\\
&
\frac{S_1(k_1^2/4;z_1,w_1)P_2^t(\mu-k_2^2/4;z_2,w_2)}
{S_1(k_1^2/4;p_1,w_1)P_2^t(\mu-k_2^2/4;p_2,w_2)},
\ (w_1,w_2)\in{\overline{M}}_1\times \pa{\overline{M}}_2,\\
&\frac{P_1^t(\mu-k_1^2/4;z_1,w_1)S_2(k_2^2/4;z_2,w_2)}
{P_1^t(\mu-k_1^2/4;p_1,w_1)S_2(k_2^2/4;p_2,w_2)},
\ (w_1,w_2)\in\pa{\overline{M}}_1\times {\overline{M}}_2.
\end{split}$$ Thus, the injectivity of the map depends on whether these are all different elements of $\Cinf(X_z)$ as $w$ varies. It is easy to see that the restriction of this map to the front face is indeed injective since $P_j^t(\nu;z_j,w_j)$ has a pole when $z_j \to w_j$ and is otherwise bounded, and so the points $w_1$ and $w_2$ are uniquely identified, as is the eigenparameter $\mu_1^0(s)$ and hence $s$ itself. On the other hand, it is not obviously true that the generalized spherical function $S_j(k_j^2/4;z_j,w_j^0)$ determines the point $w_j^0$. In case $M_j$ is hyperbolic space, this is holds because of the rotational symmetry around $w_j^0$, and so must also hold for conformally compact metrics (globally) near to the hyperbolic one. In general, these portions of the Martin boundary are identified with ${\mathcal I}_1 \times {\overline{M}}_2$ and ${\overline{M}}_1 \times {\mathcal I}_2$, where $$\mathcal I_j = \mbox{image}\big( w_j^0 \longrightarrow
S_j(k_j^2/4;z_j,w_j^0)\big).$$
Next suppose that $H_1$ has $L^2$ eigenvalues. The asymptotics of the resolvent kernel in $\Xt\times X$ are now given by Theorem \[thm:res-set-asymp\], again in the sense discussed in the last paragraph of the previous section. Let $\lambda_{11}$ be the bottom eigenvalue of $H_1$ which is hence simple, and let $s_0$ be the corresponding value of $s=\rho_1/\rho_2$ given by . In the region $s<s_0$, in particular near $\pa {\overline{M}}_1\times{\overline{M}}_2\times X$, $R(\mu)$ has the same asymptotics as before, while on the front face of the double blow up of $s=s_0$ as well as in $s>s_0$ the leading part of the asymptotics is given by elliptic multiples of $x_1^{-k_1/2-i\sqrt{\lambda_{1i}-k_1^2/4}}
\Pi_{11}\times P_2^t(\mu-\lambda_{1i})$ which are independent of $z\in X$. We again set $$U(\mu,z,w)=\frac{R(\mu;z,w)}{R(\mu;p,w)},$$ and $U$ is again continuous on $\Xt\times X$. Now, however, $w\mapsto
U(\mu,.,w)$ factors through the map $\pa\Xt\to \pa\Xt_c$, where $\pa\Xt_c$ is the collapsed boundary of $\Xt$ given by $$\pa\Xt_c=(\pa {\overline{M}}_1\times{\overline{M}}_2)\cup ([0,s_0)_s\times \pa{\overline{M}}_1\times\pa{\overline{M}}_2)
\cup \pa{\overline{M}}_2,$$ and the collapse map $\pa\Xt\to\pa\Xt_c$ is the identity on the first two sets on the right hand side and is the projection $[s_0,+\infty)
\times\pa{\overline{M}}_1\times\pa{\overline{M}}_2\to\pa{\overline{M}}_2$, resp. ${\overline{M}}_1\times\pa{\overline{M}}_2
\to\pa{\overline{M}}_2$ in the other parts of $\pa\Xt$. Thus, we obtain a continuous surjection from $\pa\Xt_c$ to $\pa\overline{X}_M$.
Again, $U$ can be written down explicitly. It is the same as in the case without $L^2$ eigenfunctions when $s<s_0$, and it is $$\frac{P_2^t(\mu-k_2^2/4;z_2,w_2)}
{P_2^t(\mu-k_2^2/4;p_2,w_2)},
\ w_2\in \pa{\overline{M}}_2.$$ Thus, the map $\pa\Xt_c\to\pa\overline{X}_M$ is injective over the collapsed part of the boundary, as can be seen by letting $z_2\to w_2$, and otherwise we are exactly in the same situation as in the case where there are no $L^2$ eigenfunctions.
Similar arguments work if $H_2$ has eigenvalues, or even if both $H_1$ and $H_2$ do. In the latter case $\Xt$ collapses near both side faces.
[^1]: R. M. partially supported by NSF grant \#DMS-99-1975; A. V. partially supported by NSF grant \#DMS-99-70607.
|
---
abstract: 'We show that the discrete anomaly constraints governing popular non-Abelian symmetries of use in (e.g.) flavoured, supersymmetric, and dark matter model building typically subdivide into two classes differentiated by the simple restrictions they impose on the *number* of fields transforming under certain irreducible representations of the relevant groups. These constraints lead us both to generic conclusions for common Beyond-the-Standard-Model constructions (including rather powerful statements for Grand Unified theories) as well as to simplified formulae that can be rapidly applied to determine whether a given field and symmetry content suffers from gauge and gravitational anomalies.'
author:
- Jim Talbert
title: 'Pocket Formulae for Non-Abelian Discrete Anomaly Freedom'
---
Introduction {#sec:intro}
============
Discrete symmetries are ubiquitous in Beyond-the-Standard-Model (BSM) constructions. Not only are they utilized ad-hoc to prevent unwanted couplings, as is often required to (e.g.) stabilize a dark matter candidate, they also have more purposeful implementations; non-Abelian discrete symmetries can explain observed patterns of fermionic mass and mixing (see e.g. [@Altarelli:2010gt; @King:2013eh]), control models of inflation [@Ross:2009hg], and may even be naturally realized as interchange symmetries of fixed-points in orbifold compactifications [@Kobayashi:2006wq; @Adulpravitchai:2009id].
Regardless of motivation, imposing a discrete symmetry on a fixed Lagrangian practically amounts to manipulating a global symmetry — no additional gauge bosons are present. However, it has long been argued that global discrete symmetries *must* be gauged in the ultra-violet (UV) in order to respect quantum gravity (wormhole) effects [@Krauss:1988zc; @Gilbert:1989nq; @Choi:2017xmg], and therefore models employing discrete symmetries should be anomaly free. Constraints for Abelian discrete symmetries were first obtained in [@Ibanez:1991hv; @Ibanez:1991wt; @Banks:1991xj] by assuming that cyclic $Z_{N}$ groups (with N the order of the group) originate from the breakdown of a gauged $U(1)$. Analogous considerations were made for non-Abelian discrete symmetries in [@Frampton:1994rk; @Luhn:2008sa]. These studies have since been generalized [@Araki:2007zza; @Araki:2008ek; @Ishimori:2010au; @Chen:2015aba] [^1] with a path-integral approach [@Fujikawa:1979ay; @Fujikawa:1980eg], with the conclusion that a fully massless spectrum in the IR is only subject to mixed non-Abelian gauge ($G$) and gravitational ($\mathcal{G}$) anomaly constraints of the form: $$D-G-G, \,\,\,\,\,\,\,\,\,\, D-\mathcal{G}-\mathcal{G}$$ where $D$ can be either an Abelian or non-Abelian discrete symmetry. Triangles like $\left[D \right]^{2} U(1)$ and $\left[ U(1)\right]^{2} D$ do not provide concrete information in the IR because the corresponding discrete charge $\alpha$ of any group element transformation is always defined modulo $N$. One can always rescale the hypercharges of the $U(1)$ symmetry groups to satisfy this modulo constraint. Also, cubic discrete anomalies ($\left[D\right]^{3}$) can be avoided by arguing charge fractionalization in the massive particle spectrum [@Ibanez:1991hv; @Ibanez:1991wt; @Banks:1991xj; @Csaki:1997aw],[^2] and indeed do not even appear in the path integral approach [@Araki:2008ek; @Lee:2011dya].
In this note we extend [@Araki:2007zza; @Araki:2008ek; @Ishimori:2010au; @Chen:2015aba] by showing that, after reorganizing discrete anomaly constraints into compact multiplicative forms, popular non-Abelian discrete symmetries are generally subject to one of two classes of constraints distinguished by the restriction they impose on the *number* of fields transforming in certain irreducible representations (irreps) of $G$ and $D$. This leads us to a host of generic conclusions that are relevant in many BSM contexts, especially Grand Unified theories, and which at the very least yield simplified ‘field equations’ that can be rapidly applied to concrete models.
The paper develops as follows: in Section \[sec:formalism\] we review the path-integral formalism developed in [@Araki:2007zza; @Araki:2008ek; @Ishimori:2010au; @Chen:2015aba], extend it to obtain the new multiplicative basis, and then specify the resulting anomaly constraints to explicit discrete symmetry groups. Then in Section \[sec:symmetries\] we place said constraints into two classes, each characterized by a ‘field equation,’ before deriving generic conclusions and simplified formulae relevant to common model building scenarios, examples of which we explore in Section \[sec:APPS\]. We conclude in Section \[sec:conclusions\].
Discrete Anomaly Constraints {#sec:formalism}
============================
Consider a set of Dirac fermions $\Psi$ living in the irreps $\bold{r}$ and $\bold{d}$ of a non-Abelian gauge $G$ and non-Abelian discrete group $D$, respectively. Then the unitary representation of a discrete transformation associated to the element $g \in D$ is given by $$\label{eq:unitrep}
U_{\bold{d}}(g) = e^{i \alpha_{\bold{d}}(g)} = e^{i \,2 \pi \,\tau_{\bold{d}}(g)/N_{g}}$$ in terms of a charge $\alpha_{\bold{d}}$ defined by $N_{g}$, the order of the element $g$, and a charge matrix $\tau_{\bold{d}}(g)$ which has integer eigenvalues. In general, chiral transformations of the fermions $\Psi$ under $U_{\bold{d}}(g)$ source a Jacobian in the path integral measure of a quantum field theory [@Fujikawa:1979ay; @Fujikawa:1980eg]: $$\mathcal{D}\Psi \mathcal{D} \bar{\Psi} \underset{U} {\longrightarrow} J^{-2}(\alpha(g)) \mathcal{D}\Psi \mathcal{D} \bar{\Psi}$$ where, if the Jacobian is found to be non-trivial ($J \neq 1$), the symmetry $D$ is *anomalous*.
There are both gauge $G$ and gravitational $\mathcal{G}$ contributions to the anomaly [@AlvarezGaume:1983ig; @AlvarezGaume:1984dr; @Fujikawa:1986hk], but we focus on the former for the moment. Consider the Jacobian for transformations on left-handed fields, $\psi_{L} \rightarrow \psi^{\prime}_{L} = e^{i \alpha_{\bold{d}}(g)}\psi_{L}$: $$\label{Jacobian}
J^{-2}_{G} = exp \left( i \int d^{4}x \, \frac{1}{16 \pi^{2}} \text{tr} \left[ \alpha_{\bold{d}}(g)F^{\mu \nu} \tilde{F}_{\mu \nu} \right] \right)$$ where the trace runs over all internal indices and the field strength tensor embeds the generators $t$ of the associated gauge group $G$, $F_{\mu \nu} = F_{\mu \nu}^{a} t_{a}(\bold{r})$. Its dual is given by $\tilde{F} ^{\mu \nu} = \frac{1}{2} \epsilon^{\mu \nu \rho\sigma} F_{\rho \sigma}$. Recalling the index theorems of [@AlvarezGaume:1983ig; @AlvarezGaume:1984dr] and defining the Dynkin index of the gauge representation $\bold{r}$ by$$\label{Dynkin}
l (\bold{r}) \delta_{ab} = tr \left[ t_{a} (\bold{r}) t_{b} (\bold{r})\right]$$ we define the function $p$ as $$\label{index}
p \equiv \int d^{4}x \frac{1}{64 \pi^{2}} \epsilon^{\mu \nu \rho \sigma} F^{a}_{\mu \nu} F^{a}_{\rho \sigma} \,\,\, \in \,\,\, \mathbb{Z}$$ (with $\mathbb{Z}$ denoting ‘integers’) and then observe that reduces to $$J^{-2}_{G} = exp \left( i \frac{2 \pi}{N_{g}} \cdot \text{tr}\left[\tau_{\bold{d}}(g) \right] \cdot 2 l(\bold{r}) \cdot p \right)$$ such that the transformation $U_{\bold{d}}(g)$ is free of $D-G-G$ anomalies if and only if [@Araki:2007zza; @Araki:2008ek; @Ishimori:2010au; @Chen:2015aba]: $$\label{zanom}
\underset{f}{\sum}\, \text{tr}\left[\tau_{\bold{d}^{(f)}}(g)\right] \cdot l(\bold{r}^{(f)}) \overset{!}{=} 0 \,\, \text{mod}\,\, \frac{N_{g}}{2}$$ where the notation in implies that the summation is only over chiral fermions $f$ living in representations that are non-trivial with respect to both $G$ and $D$. Here it is clear that $\text{tr}\left[\tau_{\bold{d}^{(f)}}(g)\right]$ mimics a $Z_{N_{g}}$ charge that can be written in terms of a (multi-valued) logarithm: $$\label{NAcharge}
\text{tr}\left[\tau_{\bold{d}^{(f)}}(g)\right] = N_{g} \frac{\text{ln}\, \text{det}\left[ U_{\bold{d}^{(f)}}(g)\right]}{2 \pi i}$$ Were we to repeat the above analysis for $D-\mathcal{G}-\mathcal{G}$ triangles, we would find the following constraint [@Araki:2008ek; @Ishimori:2010au]: $$\label{eq:Ganom}
\underset{f}{\sum}\, \text{tr}\left[\tau_{\bold{d}^{(f)}}(g)\right] \overset{!}{=} 0 \,\, \text{mod}\,\, \frac{N_{g}}{2}$$ where now it is understood that the summation is over all chiral fermions non-trivial in $D$, irrespective of $G$.[^3]
We conclude that gauge and gravitationally anomalous transformations correspond to those with $\text{det}\left[ U_{\bold{d}^{(f)}}(g)\right] \neq 1$, a condition that must be checked for all $g \in D$.
The Multiplicative Approach {#sec:Jacobian}
---------------------------
As observed in [@Chen:2015aba], one can rewrite the Jacobian using to find multiplicative anomaly constraints (here written only for $D-G-G$): $$\label{constraint2}
\underset{f}{\prod} \text{det} \left[ U_{\bold{d}^{(f)}}(g)\right]^{2\, l(\bold{r}^{(f)})} \overset{!}{=} 1$$ Like its additive equivalent , must be checked for every element $g \in D$. At least two simplifying approaches exist in the literature to do so:
1. Imagine that $D$ is generated by two elements $\lbrace h_{1}, h_{2} \rbrace \in D$, and that we have found that the Abelian transformations represented by $\lbrace h_{1}, h_{2} \rbrace$ are anomaly free. As any other element $g \in D$ can be seen as a product of $h_{1}$ and/or $h_{2}$, this implies that $g$ is itself anomaly free. Therefore, calculating for each generator $h_{i}$ of $D$ is sufficient to determine anomaly freedom [@Araki:2007zza; @Araki:2008ek].
2. Finite groups are also subdivided into conjugacy classes $C_{i}$, such that two elements $g_{1,2}$ belong to the same $C_{i}$ if and only if they are related by conjugation: $g g_{1} g^{-1} = g_{2}$ for an element $g \in D$. Since $\text{det}(gg_{i}g^{-1}) =\text{det}(g_{i})$, the determinant is constant over a conjugacy class and it is therefore sufficient to calculate for each $C_{i}$ [@Ishimori:2010au].
While these arguments were made in the context of , they are also true for . That is, the constraints $$\begin{aligned}
\label{eq:GENconstraint2}
\underset{f}{\prod} \text{det} \left[ U_{\bold{d}^{(f)}}(h_{i})\right]^{(2\, l(\bold{r}^{(f)}),\,2)} &\overset{!}{=} 1 \\
\label{eq:CJconstraint2}
\underset{f}{\prod} \text{det} \left[ U_{\bold{d}^{(f)}}(C_{i})\right]^{(2\, l(\bold{r}^{(f)}),\,2)} &\overset{!}{=} 1\end{aligned}$$ represent equivalent approaches to determining $D-(G,\mathcal{G})-(G,\mathcal{G})$ anomaly freedom. In either case, one requires the determinants over either $h$ or $C$ for each irrep of $D$.
We now observe that the left-hand-sides (LHS) of - will always be composed of a finite number of basis elements, generically denoted $x^{a_{i}}_{i}$, with both $x$ and $a$ implicitly depending on the irrep $\bold{d}^{(f)}$, and $a$ also depending on the gauge representations $\bold{r}^{(f)}$. That is, $$\label{eq:basisone}
x_{1}^{a_{1}} \cdot x_{2}^{a_{2}} \,\text{...}\, x_{M-1}^{a_{M-1}} \cdot x_{M}^{a_{M}} \overset{!}{=} 1$$ where $M$ represents the number of irreps of $D$.[^4] Restoring the dependence on the irreps, we derive alternative anomaly constraints from : $$\begin{aligned}
\label{eq:basisthree}
B^{(G,\mathcal{G})}_{(\bold{d},\bold{r})} \equiv \sum_{\bold{d}^{(f)}} \, a^{(G,\mathcal{G})}_{(\bold{d}^{(f)}, \bold{r}^{(f)})} \,\ln x_{\bold{d}^{(f)}} &\overset{!}{=} 0 \,\, \text{mod} \,\,2 \pi i \end{aligned}$$ The functions $a^{(G,\mathcal{G})}_{(\bold{d}^{(f)}, \bold{r}^{(f)})}$ depend on the number of fields transforming in irreps of $G$ and $D$ as well as any additional gauge symmetry factors, and will obviously differ between $D-G/\mathcal{G}-G/\mathcal{G}$ calculations. The basis elements $x_{\bold{d}}$ are fixed numbers that can be extracted from any finite group. Indeed, in the dependence on the (normalized) charge of any fermion under $D$ is fully factorized, a fact we will exploit in the next section.
Although computable with mathematics software equipped with finite group libraries, we now perform explicit extractions of the basis logarithms $\ln x_{\bold{d}}$ from the $D_{N}$ and $\Delta(3N^{2})$ series while leaving other groups to Table \[tab:two\].[^5]
### $D_{N}$ {#sec:DN}
The dihedral groups $D_{N}$ describe the symmetries of $N$-sided regular polygons and are composed of $Z_{N}$ cyclic rotations and $Z_{2}$ reflections; they are isomorphic to $Z_{N} \rtimes Z_{2}$. $D_{N}$ is order $2N$ and is generated by two elements. For odd $N$ the group has $(N+3)/2$ conjugacy classes and irreps, whereas for even $N$ there are $3+N/2$. Dihedral groups have applications in flavoured [@Grimus:2003kq; @Blum:2007jz; @Hagedorn:2012pg; @Varzielas:2016zuo], inflationary [@Ross:2009hg], and dark matter [@Adulpravitchai:2011ei] model-building. From Table \[tab:3N2\] one observes that for even $N$ we obtain $\lbrace (\bold{d}, \ln x_{\bold{d}}/(\pi i)) \rbrace_{a} = \lbrace (\bold{1_{-+}}, 1), (\bold{1_{--}}, 1), (\bold{2_{k}}, 1) \rbrace$ under transformations of the generator $a$ and $\lbrace (\bold{d}, \ln x_{\bold{d}}/(\pi i)) \rbrace_{ab} = \lbrace (\bold{1_{+-}}, 1), (\bold{1_{--}}, 1), (\bold{2_{k}}, 1) \rbrace$ under $ab$. In both cases $k \in \lbrace 1, (N-2)/2 \rbrace$. For odd $N$ we are only concerned with transformations under the generator $b$, $\lbrace (\bold{d}, \ln x_{\bold{d}}/(\pi i)) \rbrace_{b} = \lbrace (\bold{1_{-}}, 1), (\bold{2_{k}}, 1) \rbrace$ with $k \in \lbrace 1, (N-1)/2 \rbrace$.
[|c|c|c|c|]{}\
& $\bold{1_{k,l}}$ & $\bold{3_{[k][l]}}$\
det$\left( b \right)$ & $\omega_{3}^{k}$ & 1\
det$\left( a \right)$ & $\omega_{3}^{l}$ & 1\
det$\left( a^{\prime} \right)$ & $\omega_{3}^{l}$ & 1\
[|c|c|c|]{}\
& $\bold{1_{k}}$ & $\bold{3_{[k][l]}}$\
det$\left( b \right)$ & $\omega_{3}^{k}$ & 1\
det$\left( a \right)$ & 1 & 1\
det$\left( a^{\prime} \right)$ & 1 & 1\
\
[|c|c|c|c|]{}\
& $\bold{1_{+}}$ & $\bold{1_{-}}$ & $\bold{2_{k}}$\
det$\left( b \right)$ & 1 & -1 & -1\
det$\left( a \right)$ & 1 & 1 & 1\
[|c|c|c|c|c|c|]{}\
& $\bold{1_{++}}$ & $\bold{1_{+-}}$ &$\bold{1_{-+}}$ & $\bold{1_{--}}$ & $\bold{2_{k}}$\
det$\left( a \right)$ & 1 & 1 & -1 & -1 & -1\
det$\left( a b \right)$ & 1 & -1 & 1 & -1 & -1\
### $\Delta(3N^{2})$ {#sec:3N2}
The series $\Delta(3N^{2})$ is known (along with $\Delta(6N^{2})$) to have realistic applications in flavoured model building [@deMedeirosVarzielas:2006fc; @Ma:2006ip; @Varzielas:2015aua; @Luhn:2007uq; @deMedeirosVarzielas:2017sdv; @Lam:2012ga; @Holthausen:2012wt; @King:2013vna; @Holthausen:2013vba; @Ishimori:2014jwa; @Lavoura:2014kwa; @Talbert:2014bda; @Ishimori:2014nxa; @Yao:2015dwa]. The group is isomorphic to $\left(Z_{N} \times Z^{\prime}_{N} \right) \rtimes Z_{3}$ and can be generated by the three elements $\lbrace a, a^{\prime}, b \rbrace$ associated to $Z_{N,N^{\prime},3}$ respectively. The group is order $3N^{2}$, and when $N/3 \notin \mathbb{Z}$ there are three singlet and $(N^{2} - 1)/3$ three-dimensional representations, whereas when $N/3 \in \mathbb{Z}$ there are nine singlet and $(N^{2} - 3)/3$ three-dimensional representations. The determinants over the generators in these irreps are given in Table \[tab:3N2\]. Note that the tetrahedral group $A_{4}$, which is useful in dark matter [@Hirsch:2010ru] and flavoured [@Babu:2002dz; @Ma:2002yp; @Ma:2001dn; @Altarelli:2005yx] model building, is isomorphic to $\Delta(12)$.
For $N/3 \in \mathbb{Z}$ there are potentially anomalous transformations under all three generators $a$, $a^{\prime}$, and $b$ when fermions sit in the singlet representations. However, one notices that there are only two independent parameter sets: $\lbrace (\bold{d}, \ln x_{\bold{d}}/(\pi i)) \rbrace_{a} = \lbrace (\bold{1_{k,l}}, 2l/3) \rbrace$ and $\lbrace (\bold{d}, \ln x_{\bold{d}}/(\pi i)) \rbrace_{b} = \lbrace(\bold{1_{k,l}}, 2k/3) \rbrace$. Whenever $N/3 \notin \mathbb{Z}$, the only irrep that contributes is $\bold{1_{k}}$ for $k\neq 0$, yielding $\lbrace (\bold{d}, \ln x_{\bold{d}}/(\pi i)) \rbrace_{b} = \lbrace (\bold{1_{k}}, 2k/3) \rbrace$.
Simplified Anomaly Constraints {#sec:symmetries}
==============================
[|c|c|]{}\
Group & $ \lbrace (\bold{d},\ln x_{\bold{d}}/(\pi i)) \rbrace$\
$D_{N \in odd}$ $\star$ & $ (\bold{1_{-}},1), (\bold{2_{k}}, 1) $\
& $ (\bold{1_{-+}}, 1), (\bold{1_{--}}, 1), (\bold{2_{k}}, 1) $\
& $ (\bold{1_{+-}}, 1), (\bold{1_{--}}, 1), (\bold{2_{k}},1) $\
$S_{3}$ $\star$ & $ (\bold{1^{\prime}}, 1), (\bold{2},1) $\
$S_{4}$ $\star$ & $ (\bold{1^{\prime}}, 1), (\bold{2}, 1), (\bold{3}, 1) $\
$A_{4}$ $\star$$\star$ & $ (\bold{1^{\prime}}, 2/3), (\bold{1^{\prime\prime}}, -2/3) $\
$A_{N \ge 5}$ & $ (\text{null}, 0) $\
& $ (\bold{1_{-+}}, 1), (\bold{1_{--}}, 1), (\bold{2_{k_{e}}}, 1) $\
& $ (\bold{1_{+-}}, 1), (\bold{1_{--}}, 1), (\bold{2_{k_{e}}}, 1) $\
& $ (\bold{1_{+-}}, 1/2), (\bold{1_{-+}}, -1/2), (\bold{1_{--}}, 1), (\bold{2_{k_{e}}}, 1) $\
& $\circ$ $(\bold{1_{+-}}, 1), (\bold{1_{-+}}, 1) $\
& $(\bold{1_{+-}}, 1), (\bold{1_{--}}, 1), (\bold{2_{k_{o}}}, 1), (\bold{2_{k_{e}}}, 1)$\
& $ (\bold{1_{-+}}, 1), (\bold{1_{--}}, 1), (\bold{2_{k_{o}}}, 1) $\
$T_{N}$ $\star$$\star$ & $ (\bold{1_{1}}, 2/3), (\bold{1_{2}}, -2/3) $\
$T^{\prime}$ $\star$$\star$ & $ (\bold{1^{\prime}}, 2/3), (\bold{1^{\prime\prime}}, -2/3), (\bold{2^{\prime}}, -2/3), (\bold{2^{\prime\prime}}, 2/3)$\
& $ (\bold{1_{k,l}}, 2k/3) $\
& $ (\bold{1_{k,l}}, 2l/3) $\
$\Delta(3N^{2})_{N/3 \notin \mathbb{Z}}$ $\star$$\star$ & $ (\bold{1_{k}}, 2k/3) $\
$\Delta(6N^{2})_{3N \in \mathbb{Z}}$ $\star$ & $ (\bold{1_{1}}, 1), (\bold{2_{n}}, 1), (\bold{3_{1k}}, 1), (\bold{6_{[k][l]}}, 1) $\
$\Delta(6N^{2})_{3N \notin \mathbb{Z}}$ $\star$ & $ (\bold{1_{1}}, 1), (\bold{2}, 1), (\bold{3_{1k}}, 1), (\bold{6_{[k][l]}}, 1) $\
& $\circ$ $(\bold{1_{-n}}, 1), (\bold{2_{p,q}}, 1)$\
& $(\bold{1_{\pm n}}, 2n/N), (\bold{2_{p,q}}, 2(p+q)/N)$\
& $\circ$ $(\bold{1_{k,l}}, 2k/3)$\
& $(\bold{1_{k,l}}, 2l/N), (\bold{3_{[l][m][n]}},2(l+m+n)/N)$\
Consider the case where a non-Abelian discrete symmetry is appended to a single non-Abelian gauge group, as occurs in many $SU(5)$ and $SO(10)$ Grand Unified models. Respectively denoting the number of fields simultaneously in the $\bold{r}$ and $\bold{d}$ irreps as $\phi_{(\bold{d},\bold{r})}$, the $D-G-G$ anomaly constraint from then becomes: $$\begin{aligned}
\label{eq:anommasterGUT}
B^{G}_{(\bold{d},\bold{r})} &= \sum_{\bold{d}} \sum_{\bold{r}} 2\,l(\bold{r}) \cdot \left[ \phi_{(\bold{d},\bold{r})}\right] \cdot \ln x_{\bold{d}} \\
&\equiv \sum_{\bold{d}} K^{G}_{1}(\phi_{\bold{d}}) \cdot \ln x_{\bold{d}} \end{aligned}$$ where we have defined the field kernel $K^{G}_{1}(\phi_{\bold{d}})$ for a single gauge factor and left its dependence on $\bold{r}$ implicit.[^6] The constraint becomes more complex when two gauge symmetries are considered. Explicitly writing the gauge symmetry factor, $\hat{\bold{r}} \equiv \text{dim}(\bold{r})$, the field kernel becomes $$\begin{aligned}
\nonumber
K^{G}_{2}(\phi_{\bold{d}}) = \sum_{i} 2\, l(\bold{r}_{i}) \cdot \left[ \sum_{j}\hat{\bold{r}}_{j} \cdot \phi_{(\bold{d},\bold{r}_{i},\bold{r}_{j})} \right] \end{aligned}$$ with the subscripts on the parameters $\phi$ denoting the relevant representations under all three symmetries. It is understood that the anomaly constraint from $i \leftrightarrow j$ must be satisfied simultaneously and that $\bold{r}_{i=1} \neq \bold{r}_{j=1} = \bold{1}$. Continuing, the number of subscripts on $\phi$, sums and symmetry factors within the square brackets, and independent discrete anomaly constraints will increase by one for each additional gauge symmetry considered.
On the other hand, the structure of the analogous $D-\mathcal{G}-\mathcal{G}$ field kernel is universal: $$\begin{aligned}
\label{eq:anommasterGRAV}
K^{\mathcal{G}}(\phi_{\bold{d}}) = \sum_{\lbrace \bold{r} \rbrace}\,2\,\left[\prod_{i=1}^{m}\, \hat{\bold{r}}_{i}\,\right] \phi_{(\bold{d},\lbrace \bold{r} \rbrace)} \end{aligned}$$ where $m$ denotes the number of symmetries $G_{i}$ in the theory and the sum is over the set of unique gauge symmetry assignments $\lbrace \bold{r} \rbrace \sim \left( \bold{r}_{1},\bold{r}_{2}, ... \bold{r}_{m} \right)$, where (e.g.) $\left( \bold{1}, \bold{2}, \bold{3}, ... \right) \neq \left( \bold{2}, \bold{1}, \bold{3}, ... \right)$ and so on.
We now make the observation that the non-Abelian groups catalogued in Table \[tab:two\] are generically subject to one of two classes of constraints distinguished by the following ‘field equations’: $$\begin{aligned}
\label{eq:genfield}
D_{(1)}:& \,\,\,\,\, \sum_{\bold{d}} K^{(G,\mathcal{G})}(\phi_{\bold{d}}) \overset{!}{=} 2n, \,\,\, n \in \mathbb{Z}^{\pm}\\
\label{eq:genfield2}
D_{(2)}:& \,\,\,\,\,
\sum_{\bold{d}_{+}} K^{(G,\mathcal{G})}(\phi_{\bold{d}_{+}}) \overset{!}{=} 3n + \sum_{\bold{d}_{-}} K^{(G,\mathcal{G})}(\phi_{\bold{d}_{-}}) \end{aligned}$$ where the $\bold{d}_{\pm}$ notation indicates irreps with positive or negative $\ln x_{\bold{d}}$, and where free parameters in these basis logarithms (like $k$ or $l$ for $\Delta(3N^{2})$) are also implied in the field kernels. In Table \[tab:two\] we indicate symmetries governed by $D_{(1)}$ and $D_{(2)}$ with one or two stars ($\star$), respectively. Note that satisfying - is necessary but not sufficient to determine complete anomaly freedom for some groups, as they only have one independent discrete transformation (labeled by a ($\circ$) in Table \[tab:two\]) subject to or .
Pocket Formulae and Results {#sec:derivatives}
---------------------------
We can now derive a handful of powerful consequences from - relevant to realistic BSM scenarios:
1. \[itm:grav1D1\] Any model subject only to $D_{(1)}$ is free of gravitational anomalies.[^7]
2. \[itm:1G1D1\] Any model subject only to $D_{(1)}$ is free of gauge anomalies if $$\label{eq:integer}
l(\bold{r}) \in \mathbb{Z}^{+}\, \forall\, f$$ This is the case for (e.g.) $SO(10)$, $E_{6}$, $E_{7}$, $E_{8}$, $F_{4}$, and $G_{2}$ theories, and therefore anomaly cancellation for such theories proceeds automatically, as it does when considering models based on these continuous groups alone (see e.g. [@Slansky:1981yr]).
3. Points \[itm:grav1D1\] and \[itm:1G1D1\] are consistent with, and provide concrete examples of, those drawn in [@Chen:2015aba] regarding anomaly freedom for non-perfect finite groups, and - further imply that a condition for such groups to be generically anomaly free is, in addition to for gauge anomalies, given by $$\left(\det\left[h_{i}(\bold{d}_{j})\right]\right)^{2} = 1 \,\forall \, \lbrace i, j \rbrace$$ or equivalently for all $C_{i}(\bold{d}_{j})$.
4. For the special case of $SU(5)$ Grand Unified constructions subject to $D_{(1)}$, fermions in the $\bold{5}$ and $\bold{10}$ (and conjugates) are often the only exceptions to . Then the sum of all such fermions in irreps $\bold{d}$ must itself be even,[^8] [^9] $$\label{eq:SU5simple}
\sum_{\bold{d}} \phi_{(\bold{d},\bold{5})} + 3\,\phi_{(\bold{d},\bold{10})} \overset{!}{=} 2n$$ It is easy to extend this to include additional gauge irreps.
5. \[itm:2G1D1\] For models subject to $D_{(1)}$ and employing multiple gauge symmetries $G$, but possibly non-integer $l(\bold{r})$, anomaly freedom is only determined once the representations $\bold{r}_{i}$ under $G_{i}$ are specified. As a special but important case, consider an extension to the SM with all chiral fields transforming under the trivial or (anti-)fundamental irreps of the SM gauge groups, as is often the case in BSM flavour and dark matter models. Anomaly freedom then requires $$\begin{aligned}
\label{eq:SMD1}
\sum_{\bold{d}} \left[ \phi_{(\bold{d},\bold{3},\bold{1})} + \bold{2} \cdot \phi_{(\bold{d},\bold{3},\bold{2})} \right] &\overset{!}{=} 2n\\
\label{eq:SMD1b}
\sum_{\bold{d}} \left[ \phi_{(\bold{d},\bold{2},\bold{1})} + \bold{3} \cdot \phi_{(\bold{d},\bold{2},\bold{3})} \right] &\overset{!}{=} 2n\end{aligned}$$ sourced from $\left[SU(3) \right]^{2} D_{(1)}$ and $\left[SU(2) \right]^{2} D_{(1)}$ triangles, respectively. Note that in the former case the restriction actually reduces to $$\label{eq:RHquark}
\sum_{\bold{d}} \phi_{(\bold{d},\bold{3},\bold{1})} \overset{!}{=} 2n$$ which normally amounts to counting the number of singlet quarks non-trivially charged under $D$. Similar considerations can be made for Pati-Salam constructions.
6. \[itm:grav1D2\] Any model subject to $D_{(2)}$ and employing one $G$ suffers from gravitational anomalies if [^10] $$\label{eq:grav1D2}
\,\,\,\,\,\,\,\,\sum_{\bold{d}_{+}} \sum_{\bold{r}} \hat{\bold{r}} \cdot \phi_{(\bold{d}_{+},\bold{r})} \overset{!}{=} \frac{3}{2} n + \sum_{\bold{d}_{-}} \sum_{\bold{r}} \hat{\bold{r}} \cdot\phi_{(\bold{d}_{-},\bold{r})}$$ is not satisfied, and from gauge anomalies if $$\,\,\,\,\,\,\,\,\sum_{\bold{d}_{+}}\sum_{\bold{r}} l(\bold{r}) \cdot \phi_{(\bold{d}_{+},\bold{r})} \overset{!}{=} \frac{3}{2} n + \sum_{\bold{d}_{-}}\sum_{\bold{r}} l(\bold{r}) \cdot\phi_{(\bold{d}_{-},\bold{r})}$$ is not satisfied. For the $T_{N}$ and $A_{4}$ groups there is only one $\bold{d}_{\pm}$ irrep each.
7. \[itm:1G1D2\] Any model subject to $D_{(2)}$ and employing multiple symmetries $G$ suffers from gravitational anomalies if $$\begin{aligned}
\nonumber
&\sum_{\bold{d}_{+}}\sum_{\lbrace \bold{r} \rbrace} \left[ \prod _{i=1}^{m} \hat{\bold{r}}_{i} \right] \cdot \phi_{(\bold{d}_{+},\lbrace \bold{r} \rbrace)} \overset{!}{=} \frac{3}{2} n \\
\label{eq:grav2D2}
+ &\sum_{\bold{d}_{-}}\sum_{\lbrace \bold{r} \rbrace} \left[ \prod _{i=1}^{m} \hat{\bold{r}}_{i} \right] \cdot\phi_{(\bold{d}_{-},\lbrace \bold{r} \rbrace)}\end{aligned}$$ As an obvious point, it should be clear that the sums of *neither* or *both* $\bold{d}_{\pm}$ field kernels must be a multiple of three, a fact that in some instances may be easier to exploit.
8. \[itm:2G1D2\] For models employing multiple $G$ and subject to $D_{(2)}$ it is generally easier to determine gauge anomaly freedom by expanding for the particular discrete symmetry at hand, which normally has (at most) a handful of relevant $\bold{d}$. The structure of the field kernels will mimic the LHS of -, with $\bold{d} \rightarrow \bold{d}_{\pm}$, for the special case with SM gauge structure and fundamental or trivial irreps.
Of course, if anomalies are encountered, it may still be possible to cancel them with the discrete version of the Green-Schwarz Mechanism [@Green:1984sg; @Lee:2011dya; @Chen:2013dpa], although the phenomenology of the model may also be altered [@Chen:2015aba]. In the event one also wishes to preserve MSSM type gauge coupling unification, there is the further requirement of ‘anomaly universality’ [@Chen:2013dpa; @Chen:2012jg; @Ibanez:1994ig] which forces to be equal for all gauge groups $G_{i}$ in the theory, e.g. $$\label{eq:universal}
B^{(Z,SU(3))}_{(\bold{d},\bold{r})} = B^{(Z,SU(2))}_{(\bold{d},\bold{r})} = \rho \,\, \text{mod} \,\,2 \pi i$$ for SM constructions (with $Z$ representing an independent Abelian transform of the larger non-Abelian group $D$ and $\rho$ possibly non-zero). Hence anomaly universality forces the LHS of, e.g., - to both be equal modulo two. Similar constraints for other simplified formulae also hold.
Applications {#sec:APPS}
============
$\bold{\text{(1)}}$ $L_{\mu}$ $L_{\tau}$ $l^{c}_{\mu}$ $l^{c}_{\tau}$
--------------------- --------------------- ---------------------------- ---------------------------- ----------------------
$SU(2)$ $\bold{2}$ $\bold{2}$ $\bold{1}$ $\bold{1}$
$A_{4}$ $\bold{1^{\prime}}$ $\bold{1^{\prime \prime}}$ $\bold{1^{\prime \prime}}$ $\bold{1^{ \prime}}$
: $\bold{\text{(1)}}$: The relevant field and symmetry content from the $A_{4}$ dark matter model of [@Hirsch:2010ru]. $\bold{\text{(2)}}$: The same for the $S_{3}$ flavour model of [@Feruglio:2007hi]. $\bold{\text{(3)}}$: The same for the $D_{4}$ leptonic flavour model of [@Grimus:2003kq]. $\bold{\text{(4)}}$: The same for the $\Delta(6N^{2})$ quark flavour model of [@Ishimori:2014jwa]. $\bold{\text{(5)}}$: The same for the $S_{4}$ GUT model of [@Meloni:2011fx]. $\bold{\text{(6)}}$: The same for the $A_{4}$ GUT model of [@Bjorkeroth:2015ora]. Here $i = \lbrace 1,3, 5, 7 \rbrace$ and $j = \lbrace 2, 4, 6, 8 \rbrace$.[]{data-label="tab:models"}
\
$\bold{\text{(2)}}$ $(D_{\mu}, D_{\tau})$ $(D_{Q_{1}}, D_{Q_{2}})$
--------------------- ----------------------- --------------------------
$(SU(2),SU(3))$ $(\bold{2},\bold{1})$ $(\bold{2},\bold{3})$
$S_{3}$ $\bold{2}$ $\bold{2}$
: $\bold{\text{(1)}}$: The relevant field and symmetry content from the $A_{4}$ dark matter model of [@Hirsch:2010ru]. $\bold{\text{(2)}}$: The same for the $S_{3}$ flavour model of [@Feruglio:2007hi]. $\bold{\text{(3)}}$: The same for the $D_{4}$ leptonic flavour model of [@Grimus:2003kq]. $\bold{\text{(4)}}$: The same for the $\Delta(6N^{2})$ quark flavour model of [@Ishimori:2014jwa]. $\bold{\text{(5)}}$: The same for the $S_{4}$ GUT model of [@Meloni:2011fx]. $\bold{\text{(6)}}$: The same for the $A_{4}$ GUT model of [@Bjorkeroth:2015ora]. Here $i = \lbrace 1,3, 5, 7 \rbrace$ and $j = \lbrace 2, 4, 6, 8 \rbrace$.[]{data-label="tab:models"}
\
$\bold{\text{(3)}}$ $(D_{\mu}, D_{\tau})$
--------------------- -----------------------
$SU(2)$ $\bold{2}$
$D_{4}$ $\bold{2}$
: $\bold{\text{(1)}}$: The relevant field and symmetry content from the $A_{4}$ dark matter model of [@Hirsch:2010ru]. $\bold{\text{(2)}}$: The same for the $S_{3}$ flavour model of [@Feruglio:2007hi]. $\bold{\text{(3)}}$: The same for the $D_{4}$ leptonic flavour model of [@Grimus:2003kq]. $\bold{\text{(4)}}$: The same for the $\Delta(6N^{2})$ quark flavour model of [@Ishimori:2014jwa]. $\bold{\text{(5)}}$: The same for the $S_{4}$ GUT model of [@Meloni:2011fx]. $\bold{\text{(6)}}$: The same for the $A_{4}$ GUT model of [@Bjorkeroth:2015ora]. Here $i = \lbrace 1,3, 5, 7 \rbrace$ and $j = \lbrace 2, 4, 6, 8 \rbrace$.[]{data-label="tab:models"}
$\bold{\text{(4)}}$ $t^{c}$
--------------------- ------------------
$SU(3)$ $\bold{\bar{3}}$
$\Delta(6N^{2})$ $\bold{1_{1}}$
: $\bold{\text{(1)}}$: The relevant field and symmetry content from the $A_{4}$ dark matter model of [@Hirsch:2010ru]. $\bold{\text{(2)}}$: The same for the $S_{3}$ flavour model of [@Feruglio:2007hi]. $\bold{\text{(3)}}$: The same for the $D_{4}$ leptonic flavour model of [@Grimus:2003kq]. $\bold{\text{(4)}}$: The same for the $\Delta(6N^{2})$ quark flavour model of [@Ishimori:2014jwa]. $\bold{\text{(5)}}$: The same for the $S_{4}$ GUT model of [@Meloni:2011fx]. $\bold{\text{(6)}}$: The same for the $A_{4}$ GUT model of [@Bjorkeroth:2015ora]. Here $i = \lbrace 1,3, 5, 7 \rbrace$ and $j = \lbrace 2, 4, 6, 8 \rbrace$.[]{data-label="tab:models"}
$\bold{\text{(5)}}$ $F$
--------------------- ------------------
$SU(5)$ $\bold{\bar{5}}$
$S_{4}$ $\bold{3}$
: $\bold{\text{(1)}}$: The relevant field and symmetry content from the $A_{4}$ dark matter model of [@Hirsch:2010ru]. $\bold{\text{(2)}}$: The same for the $S_{3}$ flavour model of [@Feruglio:2007hi]. $\bold{\text{(3)}}$: The same for the $D_{4}$ leptonic flavour model of [@Grimus:2003kq]. $\bold{\text{(4)}}$: The same for the $\Delta(6N^{2})$ quark flavour model of [@Ishimori:2014jwa]. $\bold{\text{(5)}}$: The same for the $S_{4}$ GUT model of [@Meloni:2011fx]. $\bold{\text{(6)}}$: The same for the $A_{4}$ GUT model of [@Bjorkeroth:2015ora]. Here $i = \lbrace 1,3, 5, 7 \rbrace$ and $j = \lbrace 2, 4, 6, 8 \rbrace$.[]{data-label="tab:models"}
\
$\bold{\text{(6)}}$ $H_{24}$ $\Lambda_{24}$ $X_{5}$ $X_{6}$ $X_{8}$ $X_{9}$ $X_{10}$ $Z_{1,2}$ $Z_{3}$ $\Upsilon_{i}$ $\Upsilon_{j}$
--------------------- --------------------- --------------------- ---------------------------- --------------------- ----------------------------- --------------------- --------------------- ---------------------------- --------------------- --------------------- ----------------------------
$SU(5)$ $\bold{24}$ $\bold{24}$ $\bold{\bar{5}}$ $\bold{5}$ $\bold{5}$ $\bold{\bar{5}}$ $\bold{5}$ $\bold{24}$ $\bold{24}$ $\bold{24}$ $\bold{24}$
$A_{4}$ $\bold{1^{\prime}}$ $\bold{1^{\prime}}$ $\bold{1^{\prime \prime}}$ $\bold{1^{\prime}}$ $\bold{1^{\prime \prime}} $ $\bold{1^{\prime}}$ $\bold{1^{\prime}}$ $\bold{1}^{\prime \prime}$ $\bold{1}^{\prime}$ $\bold{1}^{\prime}$ $\bold{1}^{\prime \prime}$
: $\bold{\text{(1)}}$: The relevant field and symmetry content from the $A_{4}$ dark matter model of [@Hirsch:2010ru]. $\bold{\text{(2)}}$: The same for the $S_{3}$ flavour model of [@Feruglio:2007hi]. $\bold{\text{(3)}}$: The same for the $D_{4}$ leptonic flavour model of [@Grimus:2003kq]. $\bold{\text{(4)}}$: The same for the $\Delta(6N^{2})$ quark flavour model of [@Ishimori:2014jwa]. $\bold{\text{(5)}}$: The same for the $S_{4}$ GUT model of [@Meloni:2011fx]. $\bold{\text{(6)}}$: The same for the $A_{4}$ GUT model of [@Bjorkeroth:2015ora]. Here $i = \lbrace 1,3, 5, 7 \rbrace$ and $j = \lbrace 2, 4, 6, 8 \rbrace$.[]{data-label="tab:models"}
We apply the constraints found in Section \[sec:derivatives\] to a host of models representative of common BSM symmetry environments. We present only the field and symmetry content of the models required to calculate the non-Abelian discrete anomalies, and in each case we only probe the ‘easiest’ simplified constraints from Section \[sec:derivatives\] until we determine (if) the model is anomalous. We do not address the possibility that effective theories may receive anomaly contributions from additional light states in the UV, thereby changing the low-energy conclusions.
$A_{4}$ Dark Matter Model of [@Hirsch:2010ru] {#sec:A4dark}
---------------------------------------------
All chiral fermions in this model are in the trivial or fundamental irreps of (at least one of) the SM gauge groups. We observe from Table \[tab:models\] that $\phi_{(\bold{1^{\prime}},\bold{2},\bold{1})} = \phi_{(\bold{1^{\prime \prime}},\bold{2},\bold{1})} = \phi_{(\bold{1^{\prime \prime}},\bold{1},\bold{1})} =\phi_{(\bold{1^{\prime}},\bold{1},\bold{1})} = 1$ and zero for all other entries. This gives field kernels of $K^{(G,\mathcal{G})}(\phi_{\bold{d}_{+}}) = K^{(G,\mathcal{G})}(\phi_{\bold{d}_{-}})$ trivially satisfying for both gauge and gravitational constraints. The model is therefore anomaly free.
$S_{3}$ Flavour Model of [@Feruglio:2007hi] {#sec:flavS3}
-------------------------------------------
This model is subject to -. Considering $\left[SU(2) \right]^{2} D_{(1)}$ anomalies, from Table \[tab:models\] we count $\phi_{(\bold{2},\bold{2},\bold{1})} = \phi_{(\bold{2},\bold{2},\bold{3})} = 1$ and zero for all other contributions, satisfying . For $\left[SU(3) \right]^{2} D_{(1)}$, we find a lone contribution from $\phi_{(\bold{2},\bold{3},\bold{2})} = 1$, which also satisfies . As $D_{(1)}$ symmetries do not suffer from gravitational anomalies, we conclude that this model is anomaly free.
$D_{4}$ Leptonic Flavour Model of [@Grimus:2003kq] {#sec:A4dark}
--------------------------------------------------
This $D_{4}$ flavour model provides a final example of the power of -. From Table \[tab:models\] we find that the $SU(2)_{L}$ doublets provide the only anomalous contributions, giving $\phi_{(\bold{2},\bold{2},\bold{1})} = 1$. It is clear that can never be realized and thus the model suffers from $\left[SU(2) \right]^{2} D_{(1)}$ anomalies, a conclusion consistent with [@Araki:2008ek].
$\Delta(6N^{2})$ Quark Flavour Model of [@Ishimori:2014jwa] {#sec:6N2flav}
-----------------------------------------------------------
We take the symmetry assignments of additional flavons and driving superfields to be trivial under the SM gauge group, as is standard. However, from Table \[tab:models\] we immediately see that $\phi_{(\bold{1}_{1},\bold{3},\bold{1})} = 1$ and that by virtue of the model suffers from $\left[SU(3) \right]^{2} D_{(1)}$ anomalies.
$SU(5) \times S_{4}$ Grand Unified Model of [@Meloni:2011fx] {#sec:grandS4}
------------------------------------------------------------
We count that $\phi_{(\bold{3}, \bold{5})} = 1$ and zero for all other parameters relevant to gauge constraints, so can never be realized and hence the model suffers from $\left[SU(5) \right]^{2}D_{(1)}$ anomalies. Note that this model is extra-dimensional, with the field $F$ presented in Table \[tab:models\] living on the brane.
$SU(5) \times A_{4}$ Grand Unified Model of [@Bjorkeroth:2015ora] {#sec:grandA4}
-----------------------------------------------------------------
The relevant field content of [@Bjorkeroth:2015ora] sits in both the fundamentals and adjoints of $SU(5)$: $\phi_{(\bold{1^{\prime}},\bold{5})} = 3$, $\phi_{(\bold{1^{\prime \prime}},\bold{5})} = 2$, $\phi_{(\bold{1^{\prime}},\bold{24})} = 7$, and $\phi_{(\bold{1^{\prime \prime}},\bold{24})} = 6$. The quickest Type-2 constraint comes from , which gives: $$24\cdot7 + 5\cdot 3 \neq \frac{3}{2} n + 5\cdot 2 + 24 \cdot 6$$ implying that the model suffers from gravitational anomalies.
Conclusion {#sec:conclusions}
==========
We have shown that models employing non-Abelian discrete symmetries are typically subject to one of two classes of anomaly constraint, the first restricting the sum of fields charged under $G$ and $D$ to be even, and the second restricting them to be a multiple of three, upon accounting for all relevant gauge symmetry factors. These simple equations have powerful implications in realistic BSM environments, especially Grand Unified scenarios. Of course, specificity is always limited by scope, and hence it would be interesting to study the derivatives of our generic formulae when additional theoretical or phenomenological considerations are imposed on the gauge and/or discrete symmetry structure of a theory; it is likely that in specific model building environments (e.g. dark matter) even more powerful constraints on acceptable field contents arise.
Acknowledgements
================
I am grateful to Sven Krippendorf for inspiring conversations at the beginning of this work, to Graham Ross for many important insights, and to both of them for their review of the manuscript. I acknowledge research and travel support from DESY, and thank the University of Oxford for hospitality during the completion of portions of this project.
[9]{}
[^1]: We largely follow the notation of [@Chen:2015aba] in the equations that follow.
[^2]: Taking the charge fractionalization approach, anomalies following from cubic constraints can give valuable information about the ultimate order required of the $D$ groups for the model to be completely consistent. I thank G.G. Ross for this comment.
[^3]: Note that in both and additional gauge symmetry factors are left implicit.
[^4]: The determinants $\text{det} \left[ U_{\bold{d}^{(f)}}(h_{i})\right]$ and $\text{det} \left[ U_{\bold{d}^{(f)}}(C_{i})\right]$ are one-dimensional and hence the basis elements $x_{i}$ can always be rescaled to a common multiple $x$, $x_{i}^{s_{i}} = x$, such that $$\label{eq:basistwo}
x_{1}^{a_{1}} \cdot x_{2}^{a_{2}} \,\text{...}\, x_{M-1}^{a_{M-1}} \cdot x_{M}^{a_{M}} \equiv x^{A_{(\bold{d},\bold{r})}}$$ from which one can rederive the constraints catalogued in [@Ishimori:2010au] by identifying the order $\mathcal{N}$ of the basis element $x$ and scale factors $s_{i}$ required to obtain $A_{(\bold{d},\bold{r})}$.
[^5]: In what follows we use the catalogue in [@Ishimori:2010au] and maintain their notation on irreps.
[^6]: Note that sums over $f$ are now gone, as implied by the introduction of the parameters $\phi_{(\bold{d},\bold{r})}$ which are by definition $\in \mathbb{Z}^{+}$ (the positive integers including zero).
[^7]: This conclusion is consistent with the observation in [@Ishimori:2010au] that $\left[ \mathcal{G} \right]^{2}D$ anomalies are trivially satisfied by $\mathcal{O}(2)$ discrete transformations.
[^8]: Our convention is such that $l(\bold{F})$ is $\frac{1}{2}$ and $1$ for fundamentals $\bold{F}$ in $SU(N)$ and $SO(N)$ groups respectively [@Bernard:1977nr; @Yamatsu:2015npn].
[^9]: Fields in conjugate gauge irreps $\bar{\bold{r}}$ are counted within $\phi_{(\bold{d}, \bold{r}_{1},...)}$ such that, e.g., $\phi_{(\bold{d},\bold{5})} \equiv \phi_{(\bold{d},\bold{5})} + \phi_{(\bold{d},\bar{\bold{5}})}$.
[^10]: Not including free parameters like $k$ or $l$ for $\Delta(3N^{2})$...
|
---
abstract: 'The [*RXTE*]{}observed four outbursts of the accreting X-ray binary transient source, GX 304$-$1 in 2010 and 2011. We present results of detailed 3$-$100 keV spectral analysis of 69 separate observations, and report a positive correlation between cyclotron line parameters, as well as other spectral parameters, with power law flux. The cyclotron line energy, width and depth versus flux, and thus luminosity, correlations show a flattening of the relationships with increasing luminosity, which are well described by quasi-spherical or disk accretion that yield the surface magnetic field to be $\sim$60 keV. Since HEXTE cluster A was fixed aligned with the PCA field of view and cluster B was fixed viewing a background region 1.5 degrees off of the source direction during these observations near the end of the [*RXTE*]{}mission, the cluster A background was estimated from cluster B events using HEXTEBACKEST. This made possible the detection of the $\sim$55 keV cyclotron line and an accurate measurement of the continuum. Correlations of all spectral parameters with the primary 2$-$10 keV power law flux reveal it to be the primary driver of the spectral shape. The accretion is found to be in the collisionless shock braking regime.'
author:
- |
\
\
$^{1}$Center for Astrophysics and Space Sciences, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA 92093-0424, USA\
$^{2}$CRESST, Department of Physics, and Center for Space Science and Technology, UMBC, Baltimore, MD 21250, USA, and\
NASA Goddard Space Flight Center, Code 661, Greenbelt, MD 20771, USA\
$^{3}$Dr. Karl-Remeis-Sternwarte and ECAP, Sternwartstr. 7, 96049 Bamberg, Germany\
$^{4}$Institut für Astronomie und Astrophysik, Universität Tübingen, Sand 1, 72076 Tübingen, Germany\
$^{5}$Sternberg Astronomical Institute, Moscow State University, Universtetskij pr. 13, 119234 Moscow, Russia\
$^{6}$Faculty of Physics, Moscow State University, Leninskie Gory 1, 119991 Moscow, Russia\
$^{7}$Cahill Center for Astronomy and Astrophysics, California Institute of Technology, MC 290-17, 1200 E. California Blvd, Pasadena,\
CA 91125, USA
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Discovery and Modeling of a Flattening of the Positive Cyclotron Line/Luminosity Relation in GX 304$-$1 with [*RXTE*]{}'
---
\[firstpage\]
pulsars: individual (GX 304-1) – X-rays: binaries – stars: neutron – magnetic fields – X-rays individual (GX 304-1)
INTRODUCTION
============
The study of neutron star magnetic fields in accreting X-ray pulsars has progressed significantly over the past few decades through the observations of cyclotron resonance scattering features (CRSFs), or cyclotron lines. Beginning with the discovery in 1976 of such a feature in Her X$-$1 [@Trumper78], we now have identified about two dozen accreting X-ray pulsars that exhibit cyclotron line features[^1]. The fundamental line energies range from 10 to 55 keV, implying magnetic field strengths from about 1 to 5 TG. Recent work to model the accretion column emission from a physics-based point of view is based upon the accreted material passing through a radiative, radiation dominated shock and forming a thermal mound just above the surface at the magnetic poles, as first proposed by @Davidson73. Conditions in the infalling supersonic material are dominated by either radiation pressure at high luminosities or Coulomb interactions at lower luminosities before settling on the neutron star surface . At the lowest luminosities no shock is formed and the material flows unabated until reaching the mound of material piled up on the magnetic poles. At high luminosities – defined as above the critical luminosity where radiation pressure dominates over gas pressure [@Mushtukov15a]– an increase in flux causes the shock, and thus the scattering region, to rise and sample lower magnetic field strengths, giving rise to a negative correlation of the cyclotron line energy with luminosity. Physically, the structure of accretion column starts changing with decreasing mass accretion rate when the photon diffusion time across the optically thick column becomes comparable to the matter settling time from the radiative shock height, and generally can be different in different sources. First estimates [e.g. @Basko76] shows it to be around $10^{37}$ erg s$^{-1}$ if the height of the radiative shock above the neutron star surface is comparable to the accretion column radius. With further decrease in the mass accretion rate onto the neutron star magnetic poles, the accretion flow decelerates most likely due to plasma instabilities leading to the formation of a collisionless shock, as numerical calculations performed at $\dot M<10^16$ g s$^{s}$ [e.g. @BykovKrassilshchikov04] suggest. The intermediate regime (i.e. between the radiative shock at high accretion rates and collisionless shock) is the most difficult to treatment, and still is to be explored numerically with taking into account the relevant complicated microphysics. In the collisionless shock regime, the height of the the scattering region decreases with increasing mass accretion rate thus producing a positive correlation of the cyclotron line with luminosity. @Nishimura14 reproduces the same correlations with the line forming region being that between the top of the thermal mound and a height equal to twice the accretion column radius, both of which rise as the luminosity increases. @Poutanen13 have asserted a reflection model for the cyclotron line formation in which the shocked infalling matter generates X-rays that illuminate the atmosphere of the neutron star. In this case, increased accretion, and thus increased luminosity, increases the height of the X-ray emitting region and thus increases the area of the neutron star that is illuminated. This increased area contains lower values of the dipole magnetic field and thus the resulting cyclotron line has a lower value.
To date six accreting X-ray pulsars are known to have correlations of the fundamental cyclotron line energy with luminosity: one with a negative correlation, V 0332+53 [@Tsygankov06; @Klochkov11], and five with a positive correlation, Her X$-$1 [@Staubert07; @Klochkov11; @Staubert14; @Staubert16], GX 304$-$1 [@Klochkov12], A 0535+26 [@Klochkov11], the first harmonic of Vela X$-$1 [@Fuerst14], and Cep X$-$4 [@Fuerst15]. Note: 4U0115+63 is no longer deemed to have a correlation of the cyclotron line energy with luminosity [@Mueller13; @Boldin13]. @LaParola16 have recently published results from analysis of *Swift*/BAT observations of Vela X-1 where they find a positive correlation of the first harmonic cyclotron line energy with luminosity, and in addition, find a flattening of the correlation with increasing luminosity. Other spectral components, such as the power law index [e.g., @Malacaria15; @Postnov15b] and iron line flux, have been seen to vary with accretion rate as expressed by the X-ray flux.
GX 304$-$1 was first detected in a balloon flight [@McClintock71] and later by the *Uhuru* satellite as 2U1258-61 [@Giacconi72]. It is an accreting neutron star exhibiting a teraGauss magnetic field in a high mass X-ray binary system with its companion B2Vne star, V850 Cen [@Mason78; @Reig97]. The system has an orbital period of 132.1885$\pm$0.022 days [@Sugizaki15], a pulse period of $\sim$272 seconds [@McClintock77], and a distance of 2.4$\pm$0.5 kpc [@Parkes80]. After a nearly three decade period of quiescence, GX 304$-$1 emerged in 2008 [@Manousakis08], and began a series of regularly spaced outbursts in late 2009 [see fig. 1 of @Yamamoto11]. A cyclotron resonance scattering feature at $\sim$54 keV was discovered during the 2010 August outburst [@Yamamoto11], and a possible positive correlation with flux was suggested. This has been confirmed with recent [*INTEGRAL*]{}results by @Klochkov12, who found the line varying between $\sim$48keV and $\sim$55keV, and by @Malacaria15 who found the range to be 50 to 59 keV with newer [*INTEGRAL*]{}calibrations. Four outbursts in 2010 and 2011 were observed by [*RXTE*]{}until its demise in 2012 January.
In this work we present an analysis of [*RXTE*]{}data of the outbursts in 2010 March/April, 2010 August, 2010 December/2011 January, and 2011 May, which represent 72 separate observations, of which 69 were analyzed in detail. From this we determine the variations of various spectral components with respect to unabsorbed power law flux, with which all are correlated. We present the Observations and Data Reduction in Section 2, Data Analysis in Section 3, Results in Section 4, and Discussion in Section 5, and present our conclusions in Section 6. In Appendix A we give the background and analysis that is the basis for the cluster A background estimation tool, HEXTEBACKEST. In Appendix B we give the tables of best-fit spectral parameters and plot them versus unabsorbed power law flux. Also in Appendix B we present representative contour plots of the cyclotron line parameters versus various spectral components. In Appendix C we discuss tests of the HEXTE background estimation and plot the systematic normalization constants.
OBSERVATIONS AND DATA REDUCTION
===============================
Observations
------------
The *Rossi X-ray Timing Exlorer* ([*RXTE*]{}) observed GX 304$-$1 72 different times over its operational lifetime from 1996 to 2012, with three outbursts (2010 August, 2010 December, and 2011 May), numbering 69 observations, covered extensively. The outburst in 2010 March/April outburst had only 3 observations, and they are included to show consistency with the other outbursts. Three of the observations had less livetime than the GX 304-1 pulse period (Table \[tab:rxte\_all\] numbers 10, 52, and 62), and they were not included in subsequent analyses. Table \[tab:rxte\_all\] gives the dates, ObsIds, livetimes, and rates for both the Proportional Counter Array [PCA; @Jahoda06] Proportional Counter Unit 2 (PCU2) and for the High Energy X-ray Timing Experiment [HEXTE; @Rothschild98] Cluster A. Rates for PCU2 and HEXTE Cluster A are background subtracted. The sequential numbering of the individual observations is for identification in subsequent tables.
Data Reduction
--------------
PCA data were restricted to the 3$-$60 keV range of the top xenon layer of PCU2 due to the extensive calibration of this detector [@Jahoda06] that did not experience high voltage break down during the mission and thus were included in all PCA observations. The observational data were filtered to accept only observations with elevation above the Earth’s limb of greater than 10$^\circ$, observation times more than 30 minutes from the start of the previous South Atlantic Anomaly passage, and electron rate below 0.5, instead of the nominal 0.1, due to the high X-ray flux adding counts to the electron detection portions of the proportional counter. The HEXTE data utilized the PCU2 filter criteria, were restricted to the 20$-$100 keV range, and data from both clusters were included in the analyses. The PCU2 background was estimated using PCABACKEST, and the PCU2 response was generated for the specific observation day using PCARSP. Due to rocking mechanism failures in the latter stages of the [*RXTE*]{} mission, HEXTE cluster A was continuously pointed on-source (after 2006 October 20), and cluster B was continuously pointed 1.5$^\circ$ off-source (after 2009 December 12) to collect background data for all observations[^2]. The background spectrum for cluster A was then generated from that of cluster B using HEXTEBACKEST, as discussed in subsequent subsections and Appendix A. The cluster A spectral response was generated using HEXTERSP, which did not vary during the mission due to HEXTE’s automatic gain control.
The 3$-$60 keV PCU2, top layer, background subtracted, counting rates and the 20$-$100 keV HEXTE cluster A, background subtracted, counting rates for each of the three observing epochs are shown in panels a), b), and c) in Fig. \[fig:pca\_hexte\_rates\]. The HEXTE rates are multiplied by five in order to visually compare them with those of the PCU2. The 2010 August epoch observations cover from just before the maximum through decay to the beginning of a low state.
\[tab:rxte\_all\]
\# Date ObsID MJD$^a$ PCA Lvt$^b$ PCA Rate$^c$ HEXTE Lvt$^b$ HEXTE Rate$^d$
---- ------------- ----------------- ---------- ------------- ---------------- --------------- ----------------
1 2010 Mar 27 95417-01-01-00 55282.34 2880 108.5$\pm$0.2 1620 13.7$\pm$0.3
2 2010 Mar 27 95417-01-01-01 55282.61 2192 114.1$\pm$0.2 1480 12.1$\pm$0.3
3 2010 Apr 6 95417-01-02-00 55292.68 3296 195.8$\pm$0.3 2275 69.8$\pm$0.3
4 2010 Aug 13 95417-01-03-03 55421.15 2304 997.4$\pm$0.7 1399 149.0$\pm$0.4
5 2010 Aug 13 95417-01-03-00 55421.20 3712 1060.2$\pm$0.5 2300 156.1$\pm$0.3
6 2010 Aug 14 95417-01-03-01 55422.07 5408 1125.0$\pm$0.5 1480 165.7$\pm$0.4
7 2010 Aug 15 95417-01-03-02 55423.09 6096 1212.4$\pm$0.4 1961 177.3$\pm$0.4
8 2010 Aug 18 95417-01-04-00 55426.10 3328 1197.0$\pm$0.6 184 190.5$\pm$1.1
9 2010 Aug 19 95417-01-04-01 55427.08 3216 1289.0$\pm$0.6 1966 178.0$\pm$0.4
10 2010 Aug 19 95417-01-04-02 55427.99 64 1470.0$\pm$4.8 33 186.3$\pm$2.9
11 2010 Aug 20 95417-01-05-00 55428.00 3120 1175.0$\pm$0.6 1922 161.1$\pm$0.4
12 2010 Aug 21 95417-01-05-01 55429.85 992 820.6$\pm$0.9 638 103.2$\pm$0.6
13 2010 Aug 23 95417-01-05-02 55431.00 2016 693.5$\pm$0.6 1159 84.0$\pm$0.4
14 2010 Aug 24 95417-01-05-03 55432.11 3408 578.7$\pm$0.4 2072 65.1$\pm$0.3
15 2010 Aug 25 95417-01-05-04 55433.24 1184 446.2$\pm$0.6 870 49.0$\pm$0.4
16 2010 Aug 26 95417-01-05-05 55434.03 1328 397.9$\pm$0.6 770 45.0$\pm$0.4
17 2010 Aug 27 95417-01-06-00 55435.26 1696 252.2$\pm$0.4 1234 29.0$\pm$0.3
18 2010 Aug 28 95417-01-06-01 55436.03 1984 188.3$\pm$0.3 1105 22.2$\pm$0.3
19 2010 Aug 29 95417-01-06-02 55437.35 1568 85.6$\pm$0.3 1108 16.2$\pm$0.4
20 2010 Aug 30 95417-01-06-03 55438.20 2336 58.2$\pm$0.2 1648 9.7$\pm$0.3
21 2010 Aug 31 95417-01-06-04 55439.07 1440 30.8$\pm$0.2 817 8.1$\pm$0.3
22 2010 Aug 31 95417-01-06-06 55439.13 1344 34.1$\pm$0.2 828 7.2$\pm$0.3
23 2010 Sep 1 95417-01-06-05 55440.75 832 22.9$\pm$0.2 543 6.8$\pm$0.4
24 2010 Dec 17 95417-01-07-00 55547.16 16400 162.1$\pm$0.1 10680 20.4$\pm$0.1
25 2010 Dec 19 95417-01-07-01 55549.83 2944 340.3$\pm$0.4 1793 40.8$\pm$0.3
26 2010 Dec 20 95417-01-07-02 55550.22 12210 315.9$\pm$0.2 7418 37.0$\pm$0.1
27 2010 Dec 21 95417-01-07-03 55551.27 7744 457.1$\pm$0.2 4651 57.4$\pm$0.2
28 2010 Dec 22 95417-01-07-04 55552.33 2848 698.7$\pm$0.5 1738 94.4$\pm$0.3
29 2010 Dec 23 95417-01-07-05 55553.12 8880 816.4$\pm$0.3 5625 81.6$\pm$0.3
30 2010 Dec 23 95417-01-07-06 55553.30 3664 756.6$\pm$0.5 2181 103.5$\pm$0.3
31 2010 Dec 23 95417-01-07-07 55553.37 3200 799.2$\pm$0.5 1807 110.7$\pm$0.3
32 2010 Dec 24 95417-01-08-00 55554.16 3408 827.8$\pm$0.5 2156 122.4$\pm$0.3
33 2010 Dec 25 95417-01-08-01 55555.07 3520 939.0$\pm$0.5 216 127.1$\pm$0.9
34 2010 Dec 26 95417-01-08-02 55556.18 3344 775.5$\pm$0.5 2016 112.5$\pm$0.3
35 2010 Dec 27 95417-01-08-03 55557.35 768 850.1$\pm$1.1 407 110.3$\pm$0.8
36 2010 Dec 28 95417-01-08-04 55558.27 2736 684.8$\pm$0.5 1592 88.9$\pm$0.3
37 2010 Dec 28 95417-01-08-05 55558.92 5760 692.5$\pm$0.4 3812 90.5$\pm$0.2
38 2010 Dec 29 95417-01-08-06 55559.92 5136 525.8$\pm$0.3 3295 67.2$\pm$0.2
39 2010 Dec 30 95417-01-08-07 55560.95 3344 412.6$\pm$0.4 2199 51.0$\pm$0.3
40 2011 Jan 1 96369-01-01-00 55562.80 9939 272.6$\pm$0.2 6471 31.9$\pm$0.1
41 2011 Jan 5 96369-01-01-01 55566.91 2524 28.5$\pm$0.1 1673 6.3$\pm$0.2
42 2011 Jan 8 96369-01-02-00 55569.59 1744 12.8$\pm$0.1 1105 7.1$\pm$0.3
43 2011 Jan 10 96369-01-02-01 55571.66 2544 16.1$\pm$0.1 1619 5.5$\pm$0.2
44 2011 Jan 12 96369-01-02-02 55573.82 2528 20.8$\pm$0.1 1496 5.7$\pm$0.2
45 2011 May 3 96369-01-03-00 55684.49 1280 633.2$\pm$0.7 866 81.4$\pm$0.4
46 2011 May 3 96369-01-03-01 55684.76 960 656.1$\pm$0.8 660 91.1$\pm$0.6
47 2011 May 4 96369-01-03-02 55685.00 1168 605.5$\pm$0.7 774 82.7$\pm$0.5
48 2011 May 4 96369-01-04-00 55685.53 1984 594.9$\pm$0.6 1387 74.3$\pm$0.3
49 2011 May 5 96369-01-05-00 55686.31 3584 565.2$\pm$0.4 2056 87.9$\pm$0.3
50 2011 May 5 96369-01-05-01 55686.44 6272 652.1$\pm$0.3 4313 96.2$\pm$0.2
51 2011 May 5 96369-01-05-02 55686.96 1136 621.1$\pm$0.8 749 91.4$\pm$0.5
52 2011 May 6 96369-02-01-00 55687.00 32 475.7$\pm$4.0 13 65.4$\pm$3.4
53 2011 May 6 96369-02-01-000 55687.00 17730 662.4$\pm$0.2 9977 104.0$\pm$0.1
54 2011 May 6 96369-02-01-02 55787.77 1056 1012.0$\pm$1.0 714 190.3$\pm$0.7
55 2011 May 6 96369-02-01-03 55687.84 768 732.2$\pm$1.0 514 125.9$\pm$0.7
56 2011 May 6 96369-02-01-04 55687.94 1104 568.1$\pm$0.07 738 94.0$\pm$0.5
57 2011 May 7 96369-02-01-01G 55688.00 18300 792.6$\pm$0.2 10000 126.6$\pm$0.1
58 2011 May 7 96369-02-01-05 55688.54 3072 986.9$\pm$0.6 830 152.4$\pm$0.5
59 2011 May 7 96369-02-01-06 55698.68 1344 806.9$\pm$0.8 919 131.2$\pm$0.5
60 2011 May 8 96369-01-06-00 55689.26 2064 1134.0$\pm$0.7 1131 180.8$\pm$0.5
61 2011 May 8 96369-01-06-01 55689.32 2912 974.5$\pm$0.6 1643 130.5$\pm$0.3
62 2011 May 9 96369-01-06-02 55690.27 96 845.3$\pm$3.0 51 134.6$\pm$2.1
63 2011 May 10 96369-01-06-03 55691.34 656 1289.0$\pm$1.4 433 187.3$\pm$0.8
64 2011 May 10 96369-01-06-04 55691.47 1344 947.1$\pm$0.8 962 127.0$\pm$0.5
65 2011 May 10 96369-01-07-00 55691.68 1728 875.5$\pm$0.7 1170 128.1$\pm$0.5
66 2011 May 11 96369-01-07-01 55692.25 4076 795.9$\pm$0.4 2372 101.2$\pm$0.3
67 2011 May 13 96369-01-08-00 55694.31 7056 457.2$\pm$0.3 4703 53.4$\pm$0.2
68 2011 May 14 96369-01-08-01 55695.29 1136 414.9$\pm$0.6 603 46.1$\pm$0.5
69 2011 May 15 96369-01-09-00 55696.34 3760 277.5$\pm$0.3 2544 29.5$\pm$0.2
70 2011 May 16 96369-01-09-01 55617.31 832 163.9$\pm$0.5 496 17.2$\pm$0.4
71 2011 May 17 96369-01-10-00 55698.40 3104 114.5$\pm$0.2 2165 16.0$\pm$0.2
72 2011 May 19 96369-01-10-01 55700.28 480 32.9$\pm$0.3 345 7.0$\pm$0.5
: [*RXTE*]{}Observations of GX 304$-$1
\
$^a$ Start time of the observation\
$^b$ Livetime in seconds\
$^c$ 3$-$60 keV count rate in c/s\
$^d$ 20$-$100 keV count rate in c/s\
The 2010 December epoch covers a full outburst from just after the start to well into the low state, but not reaching the peak intensities of the other two epochs. [*RXTE*]{}began observing the 2011 May epoch after it was well underway, similarly to that of the 2010 August epoch, and followed it to the low state. While all three light curves show similar decreases from peak values to a low state, the third epoch shows substantial counting rate variability approaching and at the peak of the outburst. As shown below, the majority of this variability is due to large variations in column density.
Systematics of 0.5% ($<$15 keV) and 1% (15$-$60 keV) were added to the PCU2 data for observations 5, 7, 8, 26, 29, 50, 53, 57, 60, and 61 to reduce the chi-square to an acceptable range for interpretation of parameter uncertainties. Addition of similar systematic errors to the other PCU2 data would have resulted in unreasonably low chi-square values in the spectral fitting. Otherwise, no systematic uncertainties were added to PCU2 data. No systematic uncertainties were added to the HEXTE data. In addition no spectral binning of either PCU2 or HEXTE-A data was used.
![The PCU2 top layer 3$-$60 keV background subtracted, counting rates and the HEXTE cluster A 20$-$100 keV background subtracted, counting rates as a function of the observation date in Modified Julian Days. The 2010 August outburst is seen in panel a), the 2010 December outburst in panel b), and the 2011 May outburst in panel c). The PCU2 data are in black and the HEXTE data are in red. The HEXTE data have been multiplied by five. \[fig:pca\_hexte\_rates\]](f1.eps "fig:"){width="3.0in"}\
![The PCU2 top layer 3$-$60 keV background subtracted, counting rates and the HEXTE cluster A 20$-$100 keV background subtracted, counting rates as a function of the observation date in Modified Julian Days. The 2010 August outburst is seen in panel a), the 2010 December outburst in panel b), and the 2011 May outburst in panel c). The PCU2 data are in black and the HEXTE data are in red. The HEXTE data have been multiplied by five. \[fig:pca\_hexte\_rates\]](f2.eps "fig:"){width="3.0in"}\
![The PCU2 top layer 3$-$60 keV background subtracted, counting rates and the HEXTE cluster A 20$-$100 keV background subtracted, counting rates as a function of the observation date in Modified Julian Days. The 2010 August outburst is seen in panel a), the 2010 December outburst in panel b), and the 2011 May outburst in panel c). The PCU2 data are in black and the HEXTE data are in red. The HEXTE data have been multiplied by five. \[fig:pca\_hexte\_rates\]](f3.eps "fig:"){width="3.0in"}\
DATA ANALYSIS
=============
For each ObsID, the spectral histograms of PCU2 covering 3$-$60 keV and HEXTE cluster A covering 20$-$100 keV were simultaneously fit using ISIS 1.6.2-30 [@Houck00], and verified with XSPEC 12.8.2 [@Arnaud96]. For this analysis, two spectral models were utilized. The `cutoffpl` model approximated the continuum with a power law times an exponential to form a continuously steepening continuum, plus a blackbody (`CUTOFFPL + BBODY`), and the `highecut` model used a power law that abruptly changes to an exponentially falling continuum at a break energy (`POWERLAW x HIGHECUT`). Both models included low energy photoelectric absorption with interstellar abundances (`TBnew`)[^3]. The abundances of @Wilms00 and cross sections of @Verner95 were used in the analysis. The continuum was further modified by a Gaussian shaped cyclotron resonance scattering feature, or cyclotron line, (`GAUABS`) for those observations when the depth was measured, or had a lower limit, at greater than 90% confidence level. In addition, narrow ($\sigma$=10 eV), Gaussian line components were added fixed at 6.40 and 7.06 keV representing iron K$\alpha$ and K$\beta$ emission with the K$\beta$ flux set to 13% of the K$\alpha$ flux.
As presented in Appendix A, HEXTEBACKEST is based upon the channel by channel comparison of cluster A and cluster B background data for all of the observations in 2009 that included South Atlantic Anomaly passages. As such, the correlation parameters in each spectral bin are an average. Fig. \[fig:bkgex\] gives an idea of the spread in the data for two spectral channels. For any given observation, the correction factors will not give a cluster A background prediction that exactly expresses the background that would have been observed by cluster A, if it were rocking. Additionally, as the mission progressed from 2009, the satellite experienced lower and lower altitudes with the attendant increased magnetic rigidity and lesser South Atlantic Anomaly fluxes. This resulted in a somewhat lower background in the instruments. Consequently, four narrow Gaussians with fixed energies at 30.17, 39.04, 53.0, and 66.64 keV, representing corrections to the HEXTEBACKEST estimated fluxes of the four major HEXTE background lines were included in the modeling (see Appendix A for a description of HEXTEBACKEST and Appendix B for a presentation of the systematic lines versus 2$-$10 keV flux). The four energies were determined by averaging the individual fitted values during preliminary spectral analyses. The 30 keV and 67 keV lines are the strongest in the HEXTE background. While the lines at 39 keV and 53 keV are of lesser strength, they may affect the measurement of the energy of the known cyclotron line at $\sim$50-55 keV [@Yamamoto11], and were thus included in the fitting procedure.
The ‘10 keV feature’, which is seen in fits to accreting pulsar spectra [e.g., @Coburn02], was modeled by a negative Gaussian at 10.5 keV, when its inclusion reduced chi-square by 10 or more. No clear correlation was seen with respect to the detection of the 10 keV feature and power law flux. A systematic feature in the PCU2 fits occurs at about 3.88 keV in some of the observations, and it was modeled as a fixed energy, negative, narrow Gaussian, if the fitted depth was inconsistent with zero at the 90% confidence level.
The HEXTE model included the above mentioned parameters plus a constant representing the fractional difference in the response collecting area with respect to PCU2. The HEXTE constant was generally near 0.88, and was included in the variables of a given fitting procedure. Calculation of the PCU2 dead time showed that the deadtime correction was only a few percent at the highest PCU2 counting rates, and thus, no PCU2 deadtime correction was made. The HEXTE deadtime was calculated as an integral part of the data preparation using HEXTEDEAD. Since HEXTEDEAD is based upon average rates from two upper level discriminator rates [@Rothschild98], any individual observation may deviate from the average. Thus, to compensate for the few percent uncertainty of the PCA background and HEXTE background and deadtime models, the background subtractions were optimized with multiplicative parameters (RECOR in XSPEC and CORBACK in ISIS) during the fitting process. All uncertainties are expressed as 90% confidence.
The XSPEC model forms were:
F(E)=Recor\*Const\*TBnew\*(Powerlaw\*Highecut\*Gauabs + Gauss(Fe$_{K\alpha}$) + Gauss(Fe$_{K\beta}$)) + Sys
or
F(E)=Recor\*Const\*TBnew\*(Cutoffpl\*Gauabs + Bbody + Gauss(Fe$_{K\alpha}$) + Gauss(Fe$_{K\beta}$)) + Sys
where
Sys = Gauss(3.88 keV) + Gauss(10.5 keV) + Gauss (30.17 keV) + Gauss(66.37 keV) + Gauss (39.04 keV) + Gauss (53.00 keV)
The best fit continuum parameters for all observations using the `highecut` and `cutoffpl` models are given in Appendix B as Tables \[tab:best\_fit\_highecut\_cont\] and \[tab:best\_fit\_cutoffpl\_cont\]. The best fit spectral line parameters are given in Tables \[tab:best\_fit\_highecut\_lines\] and \[tab:best\_fit\_cutoffpl\_lines\]. Plots of the various continuum parameters versus unabsorbed power law (`highecut`) or unabsorbed power law times exponential (`cutoffpl`) fluxes can be found in Appendix B and plots of the `recor` parameter and the HEXTE constant can be found in Appendix C. For those fittings where the search for the depth of the cyclotron line reached zero, no values for the cyclotron line parameters were reported and only double dashes are in Tables B1 and B2. For those fittings where a lower limit on the depth was found but not an upper limit, lower limits are given and values for the cyclotron line energy and width are given. Otherwise, both high and low limits are given. Examples of correlations between the fitted cyclotron line parameters and background lines at 53 keV and 66 keV, as well as versus the cutoff energy and folding energy of the continuum, are displayed in Appendix B for high and low flux observations \#9 (12$\times10^{-9}$ ergs cm$^{-2}$ s$^{-1}$) and \#39 (4.7$\times10^{-9}$ ergs cm$^{-2}$ s$^{-1}$) . In addition the correlation between the folding and cutoff energies is shown for those examples. At the lower flux levels, the correlation contours are somewhat bimodal and that the more significant maximum occurred for the higher value of the cyclotron line parameter.
As an example, the fit to ObsID 95417-01-04-01 is shown in Figure \[fig:bright\_spectrum\]. The effects of excluding a cyclotron line component (panel b) and excluding the four HEXTE-A background lines (panel c) are shown as the ratio of the data to the model. Panel d) gives the ratio when all parameters are at their best-fit values. The reduced chi-square for this fit was 1.09 for 151 degrees of freedom. Note that the cyclotron line is clearly seen in the high energy portion of the PCU2 data (panel b), thus supporting the background estimation technique for HEXTE cluster A.
![**a)**: The PCU2 (black) and HEXTE-A (red) counts histogram for ObsID 95417-01-04-01 (2010 August 19) plotted versus energy. The best-fit model is the solid histogram in black. **b)**: The residuals of the best-fit model with the depth of the cyclotron line set to zero. The residuals are expressed as the Ratio of the data to the model. **c)**: The residuals to the best-fit HEXTE model with the four additional HEXTE-A background line fluxes set to zero. **d)**: The residuals to the best-fit model with all parameters set to their best-fit values. The solid black line in the three Ratio residuals denotes a value of 1.\[fig:bright\_spectrum\]](f4.eps){width="3.1in"}
RESULTS
=======
The two spectral models employed in the analysis generally lead to qualitatively similar results. From hereon throughout the rest of the paper, the `highecut` model results will be the subject of the discussion for two reasons. First, it has one parameter less than the `cutoffpl` model, and secondly, the continuum parameters do not influence each other to the degree that they do in the `cutoffpl` model, where the black body flux and the photon index are strongly correlated. This results in the parameters using the `highecut` model being better determined, such as the power law index and 2$-$10 keV continuum flux. The stable behavior of the column density at lower fluxes in the `highecut` model is preferred over the strong correlation with flux seen when modeling with the `cutoffpl` model. Section 4.3 gives a short discussion of the `cutoffpl` model fitting.
Peak Phase Zero Offsets
-----------------------
The orbital period of 132.1885$\pm$0.022 days [@Sugizaki15], and T$_0$=MJD 55554.75, determined from *MAXI* observations, were used to generate the respective orbital phases for each observation. The three sets of observations (now versus fitted 2$-$10 keV power law flux) have quite similar outburst decay profiles (Fig \[fig:three\_outbursts\]-a) with rise to peak flux and then decay to the lowest fluxes. By shifting the overall orbital phases slightly, the decay portions of the profiles align well (Fig. \[fig:three\_outbursts\]-b). The amounts of the peak epoch phase shifts were determined by first centering the midpoint of the peak of the 2010 December data on phase zero, since that outburst showed a relatively complete rise and fall of the flux. Then the remaining two data sets were shifted to align their falling portions to that of the 2010 December data. The resulting phase shifts are $-$0.045 for 2010 August, $-$0.010 for 2010 December, and $-$0.020 for 2011 May. These phase shifts amount to 5.9, 1.3, and 2.6 days earlier than the orbital period derived from the *MAXI* data would have suggested. This is consistent with the residual offsets from the orbital model in fig. 2 of @Sugizaki15 for these three outbursts covered by [*RXTE*]{}. This reveals that the shapes of the outbursts are quite similar once the flux drops below $\sim$10$\times 10^{-9}$ erg cm$^{-2}$ s$^{-1}$. The rising portions of the 2010 December and 2011 May outbursts also appear consistent with each other below $\sim$10$\times 10^{-9}$ erg cm$^{-2}$ s$^{-1}$. The first four 2010 August observations (black filled circles) may indicate that the 2010 August outburst exhibited an outburst with wider extent than the others, or was indicative of flaring during the rising portion of the outburst. The four 2011 May data points (red filled squares) above the common outburst trend are indicative of flaring near the peak of the 2011 May outburst. The three 2010 March/April points are not included here, since a phase shift could not be determined from so few points.
The flaring activity seen on the rising portion of the 2011 May outburst in Fig. \[fig:pca\_hexte\_rates\], is absent in Fig. \[fig:three\_outbursts\] and is attributable to the variation in column density affecting the PCU2 counting rate (see Fig. \[fig:highecut\_continuum\_flux\]a). Individual points do remain above the overall outburst trend in Fig. \[fig:three\_outbursts\], which may be considered flaring to some extent. Such flaring may be similar to the flaring activity seen on the rising portion of the 2005 August/September outburst of A0535$+$26 [@Postnov08; @Caballero08], and attributed to a low mode magnetospheric instability. These GX 304$-$1 data, however, do not show a significant change in the cyclotron line energy for any of the high flux points, whereas the A0535$+$26 data did, and other than the four earliest 2010 August outburst points, the points above the trend are at the maximum of the outbursts, and not on the rising edge, as in the 2005 August/September flares [@Postnov08; @Caballero08].
![**a**: The unobscured power law 2$-$10 keV power law flux plotted versus orbital phase for the three outbursts in 2010 August, 2010 December, and 2011 May of GX 304$-$1, as observed by [*RXTE*]{}. **b**: The same data but with the orbital phases shifted by $-$0.045, $-$0.010, and $-$0.020, respectively, to match the 2010 August and 2011 May data to the falling portion of the 2010 December outburst. The 90% uncertainties are generally less than the size of the data points.\[fig:three\_outbursts\]](f5.eps "fig:"){width="3.2in"} ![**a**: The unobscured power law 2$-$10 keV power law flux plotted versus orbital phase for the three outbursts in 2010 August, 2010 December, and 2011 May of GX 304$-$1, as observed by [*RXTE*]{}. **b**: The same data but with the orbital phases shifted by $-$0.045, $-$0.010, and $-$0.020, respectively, to match the 2010 August and 2011 May data to the falling portion of the 2010 December outburst. The 90% uncertainties are generally less than the size of the data points.\[fig:three\_outbursts\]](f6.eps "fig:"){width="3.2in"}
Variations with Power law Flux {#sec:variations}
------------------------------
Fig. \[fig:highecut\_continuum\_flux\] reveals that the `highecut` spectral parameters from the four outbursts have the same variations with power law flux and essentially the same values at any given flux level. Thus the accretion process for matter onto the neutron star was the same for all four outbursts.
A complete discussion of the column density variations is presented in Kühnel et al. (in preparation) where a large ($\times$3) column density enhancement event is detected in the 2011 May outburst (red points in Fig. \[fig:highecut\_continuum\_flux\]a) and a smaller ($\times$0.5) enhancement is seen in the 2010 December data (blue points in Fig. \[fig:highecut\_continuum\_flux\]a). These values associated with the large and small increases in column density are significant outliers from the overall trend of decreasing column density with increasing power law flux above a few 10$^{-9}$ ergs cm$^{-2}$ s$^{-1}$ and a constant value below that flux value.
The power law index has a strong positive correlation with power law flux (Fig. \[fig:highecut\_continuum\_flux\]b). The four early 2010 August points noted earlier are now indistinguishable from the overall correlation with flux, which supports the contention that the flux is the primary driver of the continuum spectral shape. The continuum cut-off break energy (Fig. \[fig:highecut\_continuum\_flux\]c) exhibits two distinct levels in the `highecut` model: $\sim$7.8 keV and $\sim$5.0$-$6.5 keV. The sharp transition from high to lower cut-off break energies appears at $\sim$6.5$\times 10^{-9}$ ergs cm$^{-2}$ s$^{-1}$, or (4.5$\pm$0.9)$\times 10^{36}$ ergs s$^{-1}$ for a distance of 2.4$\pm$0.5 kpc [@Parkes80]. The continuum folding energy shows an overall trend of decreasing energy with increasing power law flux (Fig. \[fig:highecut\_continuum\_flux\]d).
The cyclotron line energy ($E_\mathrm{cyc}$; Fig. \[fig:highecut\_continuum\_flux\]g) is found to range from 50 to 60 keV with an ever increasing value with power law flux in agreement with @Klochkov11. The widths ($W_\mathrm{cyc}$; Fig. \[fig:highecut\_continuum\_flux\]h) vary with power law flux from 4 to 12 keV, and the depths ($\tau_l$; Fig. \[fig:highecut\_continuum\_flux\]i) range from $\sim$1.1 down to $\sim$0.4, beyond which the depth is not significantly detected. For the cyclotron line energy and width, a positive correlation is clearly seen, while that for the depth or strength is less clear.
The iron line flux (Fig. \[fig:highecut\_continuum\_flux\]f) shows a relatively smooth increase with flux. The iron line equivalent width variation with power law flux (Fig. \[fig:highecut\_continuum\_flux\]e) was somewhat constant versus flux with large scatter between 2 and 4$\times 10^{-9}$ ergs cm$^{-2}$ s$^{-1}$ and at fluxes in excess of 10$\times 10^{-9}$ ergs cm$^{-2}$ s$^{-1}$.
`Cutoffpl` Fits {#sec:cutoffpl}
---------------
Fig. \[fig:cutoffpl\_continuum\_flux\] shows the variation of spectral parameters with cutoff power law flux. Due to the shape of the cutoff power law and the blackbody component, the shape of the continuum is somewhat different than that of a straight power law. Therefore the values of the column density and power law index are slightly different than those from the `highecut` model. The column density still drops with increasing cutoff power law flux above $\sim 3 \times 10^{-9}$ ergs cm$^{-2}$ s$^{-1}$ and the two column density enhancements are still above the trend. Where the `highecut` column density values leveled off at a value of $\sim 7 \times 10^{22}$ cm$^{-2}$, those for `cutoffpl` drop to $\sim 3 \times 10^{22}$ cm$^{-2}$ below $\sim 1 \times 10^{-9}$ ergs cm$^{-2}$ s$^{-1}$. Similarly for the power law index, while `highecut` values have a linear series of values over the entire power law flux range, the `cutoffpl` values exhibit an abrupt change from the linear trend of the index at $\sim 1 \times 10^{-9}$ ergs cm$^{-2}$ s$^{-1}$ to that of a constant value of $\sim$0.75 with large uncertainties. The blackbody temperature is constant at $\sim$1.1 keV from the lowest cutoff power law fluxes to $\sim 1 \times 10^{-9}$ ergs cm$^{-2}$ s$^{-1}$, beyond which it rises linearly with flux to $\sim$2.7 keV. At ever-increasing cutoff power law flux, the trend is to decrease somewhat, albeit with large uncertainties. The cyclotron line parameters and the iron line fluxes variations are quite similar to those found in the `highecut` modeling.
Color/Intensity Diagrams
------------------------
We have created the GX 304$-$1 soft (SC) and hard color (HC) versus intensity diagrams following the prescription of @Reig13 with the intensity being the PCU2 4$-$30 keV count rate, the soft color being the ratio of the PCU2 4$-$7 keV to 7$-$10 keV count rates, and the hard color being the ratio of 15$-$30 keV to 10$-$15 keV rates. Fig \[fig:sc\_flux\]a shows the SC versus intensity and Fig \[fig:sc\_flux\]b the HC versus intensity. Both show increases in the color indices with increasing intensity, as expected for a hardening of the power law flux with intensity. For the SC/intensity diagram, the 2010 August and 2010 December outbursts follow the same track throughout their observations. The 2011 May outburst also follows the same track except for the period of time when the large column density enhancement was present. The larger column density values reduce the 4$-$7 keV fluxes and therefore raise the value of the soft color ratios. The hard color/intensity plot shows overlapping tracks for the three outbursts without the large deviations at higher intensity seen in the soft color plot, except for two of the the four last observations in 2010 December. The general trend of a reduction in soft and hard color indices throughout the outbursts can be attributed to the steepening of the power law component as the power law flux decreased, and the reversal of the hard color diagram below $\sim$100 counts per second may be attributable to the hardening of the spectrum at low fluxes as expressed in the spectral fitting by the increased values of E$_{fold}$. All together the soft and hard color/intensity diagrams imply that the accretion onto the neutron star was nearly identical in all three outbursts.
![**a**: Soft color (7$-$10 keV/4$-$7 keV) plotted versus PCU2 4$-$30 keV counting rate for the three outbursts observed, following the prescription of @Reig13. **b**: Hard color (15$-$30 keV/10$-$15 keV) plotted versus PCU2 4$-$30 keV counting rate.The colors for the three outbursts are the same as Fig \[fig:three\_outbursts\].\[fig:sc\_flux\]](f7.eps "fig:"){width="2.8in"} ![**a**: Soft color (7$-$10 keV/4$-$7 keV) plotted versus PCU2 4$-$30 keV counting rate for the three outbursts observed, following the prescription of @Reig13. **b**: Hard color (15$-$30 keV/10$-$15 keV) plotted versus PCU2 4$-$30 keV counting rate.The colors for the three outbursts are the same as Fig \[fig:three\_outbursts\].\[fig:sc\_flux\]](f8.eps "fig:"){width="2.8in"}\
Variance-Weighted Averages {#sec:variance}
--------------------------
Since we have demonstrated the nearly identical spectral performance in the four outbursts through the continuum parameters’ variations versus source flux and through the overlapping color intensity diagrams, we have performed a variance-weighted average of the power law index, cyclotron line energy and width, the iron line flux, and its equivalent width in six flux bins of width 2$\times10^{-9}$ ergs cm$^{-2}$ s$^{-1}$ from zero to 12$\times10^{-9}$ ergs cm$^{-2}$ s$^{-1}$ in order to reduce the scatter in parameter values and reduce uncertainties. The residual flux and cyclotron line depth values are given without the lowest flux bin since at most only lower limits were achieved.
The average CRSF energy, $\langle E_{\mathrm{cyc},i}\rangle$, in a certain flux bin, $i$, was found by minimizing the $\chi_i^2$ defined as $$\chi_i^2 = \sum_k \frac{(E_\mathrm{cyc,k} - \langle E_{\mathrm{cyc},i}\rangle)^2}{S(\sigma_\mathrm{cyc,k}^+, \sigma_\mathrm{cyc,k}^-)^2}$$ $\quad\text{with}\quad S=\begin{cases}
\sigma_\mathrm{cyc,k}^+ \quad\text{for~} E_\mathrm{cyc,k} - \langle E_{\mathrm{cyc},i}\rangle \le 0,\\
\sigma_\mathrm{cyc,k}^- \quad\text{for~} E_\mathrm{cyc,k} - \langle E_{\mathrm{cyc},i}\rangle > 0,\\
\end{cases}$\
with the CRSF energy, $E_\mathrm{cyc,k}$, of each observation $k$ falling into the flux bin $i$, and the upper or the lower uncertainty, $\sigma_\mathrm{cyc,k}^+$ and $ \sigma_\mathrm{cyc,k}^-$, of the CRSF energy. The average CRSF width ($W_\mathrm{cyc,k}$), depth ($\tau_\mathrm{cyc,k}$), and residual flux ($r=F_\ell(E_\mathrm{cyc})/F_\mathrm{c}(E_\mathrm{cyc})$) in each flux bin was found in the same way. The residual flux is related to the line ’optical depth’, $\tau_\mathrm{\ell}$, as $r=e^{-\tau_\mathrm{\ell}}$. Note that in case of symmetric uncertainties, i.e., $\sigma_\mathrm{cyc,k}^+$ = $ \sigma_\mathrm{cyc,k}^-$, the average CRSF parameter value obtained is equivalent to the mean value weighted by the corresponding uncertainties [see, e.g., @bevington].
The results are plotted in Fig. \[fig:ecyc\_index\] where the cyclotron line parameters are in the lefthand panels and the power law index, iron line flux and equivalent width are in the righthand panels. All, except the residual flux, show positive correlations with flux, with the cyclotron line parameters gradually flattening with increasing flux. In Section \[sec:cyclo\_fits\] below, we show successful fits to the cyclotron line parameters with both disk accretion and quasi-spherical accretion models.
Comparison to Previous Observations
-----------------------------------
@Yamamoto11 presented spectral analyses of [*RXTE*]{}observations of the first two thirds of the 2010 August outburst, plus that of a *Suzaku* observation on 2010 August 13 after the second [*RXTE*]{}observation. Their analysis differed from that of the present work by only covering the 3$-$20 keV band in PCU2, using no extra Gaussians to augment `hextebackest`, ignoring the HEXTE band from 61$-$71 keV, and normalizing the PCU2 to HEXTE spectra by assuming no HEXTE flux above 150 keV. In addition, a different spectral model for the continuum, `NPEX`, was used. Nevertheless, they discovered the cyclotron line and concluded that the line had a positive correlation with overall flux or it had a bi-modal distribution. Cyclotron line energies ranged from 49$-$54 keV, albeit with large uncertainties on those values from lower luminosities. @Klochkov12 used 6 INTEGRAL observations covering the 2012 January outburst to confirm a positive correlation of the cyclotron line energy with flux employing the `cutoffpl` spectral model. The range of INTEGRAL cyclotron line energies was 48$-$ 55 keV. In the present work we have detected the cyclotron line in 54 of 69 observations, with individual energies ranging from 49 keV to 59 keV. @Jaisawal16 recently presented results from two *Suzaku* observations, one of which occurred at the time of the [*RXTE*]{}observations (4 & 5) on 2010 August 13. Their use of the `NPEX` and `CYCLABS` models for spectral fitting does not allow comparison to the present results due to the differing assumptions of spectral shapes. They did report, however, that the higher cyclotron line energies did occur for the brighter observation, as one would expect from the positive correlation with luminosity.
DISCUSSION
==========
The present work covers three outbursts of GX 304$-$1 with twenty or more observations per outburst over a range of luminosities. The detailed modeling and corrections to the PCU2 background via the `RECOR` function and to the HEXTE background utilizing additional flux from the four prominent background lines in addition to `HEXTEBACKEST` plus `RECOR` has resulted in best-fit spectral parameters from spectra covering 3$-$100 keV with significant overlap in the 20$-$60 keV band, which allows for confirming the lower energy portions the HEXTE background subtraction.
Scaling Laws of CRSF Properties {#sec:cyclo_fits}
-------------------------------
The correlations of the CRSF properties with flux during outbursts of GX 304-1 suggest that the mass accretion rate onto the neutron star poles is the driver of the CRSF changes. The CRSF formation is a very complicated problem that can be solved only numerically by taking into account the dynamics of the accretion flow near the neutron star surface coupled with the radiation in the strong magnetic field. Qualitatively, however, it is clear that at low accretion rates, when the radiation field is not very strong, the braking of the flow is mediated by Coulomb interactions in the accreting plasma (e.g. [@Nelson93]), while at high accretion rates the flow is decelerated mostly due to interactions with photons [@Davidson73]. The transition between these two extreme cases occurs gradually around some critical luminosity $\sim 10^{37}$ erg s$^{-1}$, which depends on the geometry of the flow and the structure of magnetic field near the neutron star surface and may be different in different sources (see [@Basko76], and more recent calculations in [@Becker12], [@Mushtukov15a]). At low luminosities, the CRSF energy in some sources (e.g. Her X-1) was found to positively correlate with X-ray flux, and in the simplest interpretation this can be due to a closer location of the effective site of CRSF formation with respect to the neutron star surface, where the magnetic field is stronger, with increasing mass accretion rate [@Staubert07]. Clearly, with increasing X-ray luminosity, transition to the radiation-dominated regime occurs, where the effective height of accretion column gets higher, and hence the CRSF energy is expected to decrease with increasing X-ray flux, as indeed observed in some bright transient X-ray pulsars (e.g. V0332+53, [@Tsygankov06]). @LaParola16 make similar assumptions in the fitting of the Vela X-1 first harmonic positive correlation of cyclotron line energy with luminosity.
Here we suggest a possible interpretation of the observed correlations in GX 304-1, assuming that the source, even at the highest X-ray flux in the outburst, is indeed well below the critical luminosity [@Becker12], which implies it remains in the regime where the radiation effects are subdominant in braking the accretion flow. This will enable us to use the results of detailed calculations of the (effectively one-dimensional in this case) plasma flow above the neutron star surface. In this way we will obtain simple formulae that can be used to fit the observed correlations of the CRSF energy, $E_\mathrm{cyc}$, its width, $W$, the line residual flux, $r$, and its related line optical depth $\tau_\ell$ with changing X-ray flux (see Table \[t:exponents\]).
### Physical setup
In the GX 304$-$1 case, the accretion flow decelerates in a collisionless shock [@LangerRappaport82]. The height of the collisionless shock above the neutron star surface, $H_\mathrm{s}$, is governed by energy exchange between protons (which tap most of the post-shock energy) and electrons, and the cooling of electrons and ions via bremsstrahlung and cyclotron losses; photons participate in the post-shock dynamics of the flow via resonant and non-resonant scattering on electrons in the strong magnetic field, but their density is insufficient to produce a radiation-dominated shock [@BykovKrassilshchikov04]. With increasing mass accretion rate, $H_\mathrm{s}$ decreases because the plasma density increases, and the line formation region within the cyclotron resonant layer downstream of the shock gets closer to the neutron star surface. The CRSF formation is governed by the resonance electron scattering of thermal photons produced at the base of the accretion mound where most of the free-fall energy is released. Thus, the scaling with mass accretion rate appears for the line centroid energy, its width, residual flux, and depth.
A photon with energy $\hbar\omega$ experiences resonant scattering on an electron at the fundamental cyclotron resonance frequency $\omega_\mathrm{cyc}$ in the magnetic field $B$, and $E_\mathrm{cyc}=\hbar\omega_\mathrm{cyc}=\hbar eB/(m_\mathrm{e}c)$, where $e$ is the electron charge, $m_\mathrm{e}$ is the electron mass, and $c$ is the speed of light. Therefore, in the plasma above the neutron star surface, for each photon of energy $E$ there should be the cyclotron resonance scattering radius, $r_\mathrm{res}=R_\mathrm{NS}((\hbar eB_\mathrm{NS}/m_\mathrm{e}c)/E)^{1/3}$, due to inhomogeneity of the dipole magnetic field, $B = B_\mathrm{NS}(R_\mathrm{NS}/r)^3$, where $R_\mathrm{NS}$ is the neutron star radius and $B_\mathrm{NS}$ is the surface magnetic field at the magnetic pole [@Zheleznyakov96].
![The variance weighed average cyclotron line energies (a), power law indices (b), cyclotron line widths (c), iron line flux (d), cyclotron line residual fluxes (e), iron line equivalent widths (f), and cyclotron line average depth (g) in six 2$-$10 keV, unabsorbed, power law flux bins for the energy and width of the cyclotron lines, and in five flux bins for the residual flux and the depth. \[fig:ecyc\_index\]](f9.eps){width="6.6in"}
As shown in @Zheleznyakov96, the width of the resonant layer for the assumed dipole magnetic field is $\Delta r_\mathrm{res}\sim \beta_{T_\mathrm{e}} r_\mathrm{res}/3$, where $\beta_{T_\mathrm{e}}=v_{T_\mathrm{e}}/c\sim 1/10$ is the thermal velocity of post-shock electrons; for typical temperatures $T_\mathrm{e}\sim 10$ keV and $\hbar\omega_\mathrm{cyc}\sim 50$ keV, $\Delta r_\mathrm{res}\sim 6\times 10^4$ cm can be comparable with the shock size $H_\mathrm{s}$ and thus can substantially modify the CRSF formation. Note that the post-shock electron temperature $T_\mathrm{e}$ does not vary substantially. The characteristic optical depth of the resonant layer in the inhomogeneous dipole magnetic field $B$ is [@Zheleznyakov96] $$\begin{gathered}
\label{taures}
\tau_\mathrm{res}=\frac{16}{3}\frac{\pi^2 e^2}{m_\mathrm{e}c}\frac{n_\mathrm{e}\Delta r_\mathrm{res}}{\omega_\mathrm{cyc}}
\sim \\10^4\left(\frac{n_\mathrm{e}}{10^{20}\mathrm{cm}^{-3}}\right)\myfrac{50\,\mathrm{keV}}{E_\mathrm{cyc}}^\frac{4}{3}\myfrac{R_\mathrm{NS}}{10^6\mathrm{cm}}\myfrac{B_\mathrm{NS}}{10^{12}\mathrm{G}}^\frac{1}{3}\,.\end{gathered}$$ It is also known that during the cyclotron resonance scatterings the number of scatterings of a photon in the resonant layer scales as the optical depth, $N_\mathrm{sc}\propto \tau_\mathrm{res}$, in contrast to the scaling $N_\mathrm{sc}\propto \tau^2$ for the non-resonance Thomson scattering [@Wasserman80; @Lyutinkov06; @Garasev11]. This has an important consequence for the CRSF discussed below.
Formula $x_\mathrm{d}=2/7$ $x_\mathrm{s}=2/11$
------------------------------------------------------------------------------------------------------------ -------------------------- --------------------------
$E_\mathrm{cyc}(F_\mathrm{x})=E_0(K_1 F_\mathrm{x}^{-\alpha}+1)^{-3}$ $\alpha_\mathrm{d}=5/7$ $\alpha_\mathrm{s}=9/11$
$W(F_\mathrm{x})=K_2E_\mathrm{cyc}^{1/3}(F_\mathrm{x})F_\mathrm{x}^\beta$ $\beta_\mathrm{d}=5/14$ $\beta_\mathrm{s}=9/22$
$r(F_\mathrm{x})=K_3 E_\mathrm{cyc}^{2/3}(F_\mathrm{x}) F_\mathrm{x}^{-\gamma} $ $\gamma_\mathrm{d}=5/14$ $\gamma_\mathrm{s}=9/22$
$\tau_\ell(F_\mathrm{x})=K_4 + \ln\left(E_\mathrm{cyc}^{-2/3}(F_\mathrm{x}) F_\mathrm{x}^{\gamma}\right) $ $\gamma_\mathrm{d}=5/14$ $\gamma_\mathrm{s}=9/22$
: Fitting formulae for the CRSF centre energy, $E_\mathrm{cyc}$, width $W$, the residual line flux $r$ and line optical depth $\tau_\ell$ as a function of X-ray flux $F_\mathrm{x}$ in the collisionless shock braking regime, assuming a magnetospheric radius $R_\mathrm{m}\sim M^{-x}$
\[t:exponents\]
The height of the collisionless shock is $H_\mathrm{s}\sim (v_0/4) t_\mathrm{eq}\propto 1/n_\mathrm{e}$ [here $t_{eq}$ is the electron-proton equilibration time; see e.g. @ShapiroSalpeter75]. The electron number density behind the shock can be estimated from the mass continuity equation $${}
n_\mathrm{e}= \myfrac{\dot M}{A}\myfrac{4}{v_0 m_\mathrm{p}}\,.$$ The accretion area $A$ is determined by the magnetospheric radius $R_\mathrm{m}$ and for the dipole field should vary as $A\sim (R_\mathrm{NS}\sqrt{R_\mathrm{NS}/R_\mathrm{m}})^2\propto 1/R_\mathrm{m}$. In the general case, the magnetospheric radius is inversely proportional to the mass accretion rate, $R_\mathrm{m}\propto \dot M^{-x}$, where $x=x_\mathrm{d}=2/7$ for disc accretion or Bondi quasi-spherical accretion, or $x=x_\mathrm{s}=2/11$ for quasi-spherical settling accretion [@Shakura12], where the latter may be realized in the case of GX 304-1 [@Postnov15a].
With these scalings, we find for the electron number density $n_\mathrm{e}\propto \dot M/A\propto \dot M^{1-x}$. Therefore, the characteristic shock height scales with accretion rate as $$\label{l*}
H_\mathrm{s}\propto 1/n_\mathrm{e}\propto A/\dot M \propto \dot M^{-\alpha}\,,$$ and $n_\mathrm{e}\propto \dot M^\alpha$, where $\alpha=\alpha_\mathrm{d}=1-x_\mathrm{d}=5/7$ for disc or Bondi quasi-spherical accretion and $\alpha=\alpha_\mathrm{s}=1-x_\mathrm{s}=9/11$ for quasi-spherical settling accretion.
### Cyclotron line energy scaling with X-ray flux
Consider the case where the characteristic size of the plasma region, $H_\mathrm{s}\lesssim 10^5$ cm, is comparable with the thickness of the resonant layer, $\Delta r_\mathrm{res}\sim 6 \times 10^4$ cm. The optical depth of the resonant layer is very large (see ). The CRSF is formed at some effective energy corresponding to the magnetic field at some height within the resonance layer, which is related to the shock height, $H_\mathrm{CRSF}\lesssim H_\mathrm{s}$, and hence should have the same dependence on the mass accretion rate as H$_\mathrm{s}$. The CRSF energy is $E_\mathrm{cyc}\propto B(R)\propto 1/R^3$, and noticing that $R=R_\mathrm{NS}+H_\mathrm{CRSF}$, we find, for the assumed dipole magnetic field, $${E_\mathrm{cyc}(\dot M)}=E_0\myfrac{R_\mathrm{NS}}{H_\mathrm{CRSF}(\dot M)+R_\mathrm{NS}}^3\,,$$ where $E_0$ corresponds to the line emitted from the NS surface magnetic field $B_\mathrm{NS}$. Clearly, the line dependence on the observed X-ray flux is entirely determined by how the collisionless shock height $H_\mathrm{s}$ responds to the variable mass accretion rate (see above). As the observed X-ray flux $F_\mathrm{x}$ is directly proportional to $\dot M$ and introducing the relation $H_\mathrm{CRSF}/R_\mathrm{NS}=K_1 F_\mathrm{x}^{-\alpha}$, we arrive at $$\label{e:sol1}
E_\mathrm{cyc}(F_\mathrm{x})=E_0(K_1 F_\mathrm{x}^{-\alpha}+1)^{-3}\,.$$ The constant $K_1$, which determines the CRSF location height, $H_\mathrm{CRSF}/R_\mathrm{
NS}$, can be found from fitting the observational data, $\alpha_d$=5/7 and $\alpha_s$=9/11. Generally, $K_1$ may be a function of $\dot M$ as well, but in view of lack of solid theory of CRSF formation downstream the shock we will assume $K_1=const$.
### Cyclotron line width scaling with X-ray flux
As discussed above, the resonant line is formed by multiple scatterings in a resonant layer behind the shock. In each single scattering on an electron, moving essentially in one dimension along the magnetic field lines, the energy of the resonant photon is Doppler shifted, $(\Delta E_\mathrm{cyc}/E_\mathrm{cyc})_1=\pm\beta_{T_\mathrm{e}}$, where the post-shock electron temperature $T_\mathrm{e}\sim 10$ keV does not strongly vary in the scattering region. Therefore, after many scatterings the CRSF width will be $W/E_\mathrm{cyc}
\simeq \sqrt{(\Delta E_\mathrm{cyc}/E_\mathrm{cyc})_1^2 N_\mathrm{sc}}\propto \sqrt{T_\mathrm{e} N_\mathrm{sc}}\propto \sqrt{N_\mathrm{sc}} \propto \sqrt{\tau_\mathrm{res}}$. As follows from , $\tau_\mathrm{res}\propto n_\mathrm{e}/E_\mathrm{cyc}^{4/3}$, and hence the observed CRSF width can be fitted by the following formula: $$\label{e:Wsol1}
W(F_\mathrm{x})=K_2E_\mathrm{cyc}^{1/3}(F_\mathrm{x})F_\mathrm{x}^\beta\,,$$ where $\beta=\alpha/2$, $E_\mathrm{cyc}(F_\mathrm{x})$ is determined by formula (\[e:sol1\]) and $K_2$ is a constant.
### Cyclotron line residual flux and line ‘depth’ scaling with X-ray flux
Finally, we consider how the residual flux at the line center changes with X-ray luminosity in our model. Consider the simplest case of an isothermal atmosphere with resonance scattering (the Eddington model), which can be a good first approximation for the resonant layer behind the collisionless shock front. It is easy to check that in our case with $E_\mathrm{cyc}=\hbar\omega_\mathrm{cyc}\sim 50$ keV $\gg kT_\mathrm{e}\sim$ 10 keV and with typical densities $n_\mathrm{e}\sim 10^{20}$ cm$^{-3}$, the ratio of the absorption to scattering is very small, i.e. we can neglect absorptions of scattered photons altogether. According to the theory of resonance scattering lines in an isothermal atmosphere [see, e.g., @Ivanov69 (chapter 7)], and [@Ivanov73], in the limit of high survival probability of scattered photons in the continuum and neglecting the absorption, the residual flux $r$ of a resonance line (the so-called ’$\lambda$-solution’) is determined solely by the number of scatterings of the line photons and scales as $$\label{}
r = \frac{1}{\sqrt{N_\mathrm{sc}}}\propto \frac{1}{\sqrt{\tau_\mathrm{res}}} \propto \frac{E_\mathrm{cyc}^{2/3}}{\sqrt{n_\mathrm{e}}}\,.$$ Plugging in the scaling $n_\mathrm{e}\propto \dot M^\alpha$, we can recast this expression into the convenient form: $$\label{e:r}
r(F_\mathrm{x})=K_3E_\mathrm{cyc}^{2/3}(F_\mathrm{x})F_\mathrm{x}^{-\gamma}\,,$$ where $K_3$ is a constant and $\gamma=\alpha/2$, yielding $\gamma_\mathrm{d}=5/14$ and $\gamma_\mathrm{s}=9/22$ for disc and quasi-spherical accretion, respectively.
It is also possible to introduce the line ‘optical depth’ $\tau_\ell$ defined as $r=e^{-\tau_\ell}$. It is this parameter that is usually inferred from data analysis. The application of formula in this case is straightforward: $$\label{e:taul}
\tau_\ell(F_\mathrm{x})=K_4+\ln(E_\mathrm{cyc}^{-2/3}(F_\mathrm{x})F_\mathrm{x}^{\gamma})\,,$$ where $K_4$ is the constant to be found from fitting. (Note that the fitting procedure of $\tau_\ell(F_\mathrm{x})$ should be done independently of fitting $r(F_\mathrm{x})$, since these quantities are derived independently from the data analysis.)
### Fitting the Variance-weighted Data
![Best-fits of the observed cyclotron line parameters versus X-ray flux for two possible types of accretion in GX 304-1, disc or quasi-spherical, shown by solid green line and dashed red line, respectively. **a)** Cyclotron line energy (\[e:sol1\]); **b)** cyclotron line width (\[e:Wsol1\]); **c)** cyclotron line residual flux (\[e:r\]); and **d)** cyclotron line ’depth’ (\[e:taul\]). Horizontal bars indicate the width of the flux bins inside which averaging was done. []{data-label="f:plotE"}](f10.eps){width="3.4in"}
The results of fitting the variance-weighted data (described in section \[sec:variance\] and shown in Fig. \[fig:ecyc\_index\]) by formulae , , , and are shown in Fig. \[f:plotE\] and listed in Table \[t:fits\]. We do not show formal errors in the fitting coefficients due to roughness of the model physical assumptions (constant electron temperature, approximate treatment of the cyclotron resonance scattering, etc.). It is also seen that the data do not allow us to distinguish between the two possible dependences of the magnetospheric radius on $\dot M$ for different types of accretion (disc or quasi-spherical one).
[lcc]{} Param & Disc accretion & Quasi-spherical accretion\
\
$E_0$ & $59.99$ & $58.62$\
$K_1$ & $0.1$ & $0.09$\
\
$K_2$ & $1.25$ & $1.12$\
\
$K_3$ & $0.06$ & $0.07$\
\
$K_4$ & $2.77$ & $2.66$\
\
$E_0$ is energy in keV\
$K_1$ is flux$^{\alpha}$ ($10^{-9\alpha}$ erg$^\alpha$ cm$^{-2\alpha}$ s$^{-\alpha}$)\
$K_2$ is energy$^{2/3}$ flux$^{-\beta}$ (keV$^{2/3}$ 10$^{9\beta}$ erg$^{-\beta}$ cm$^{2\beta}$ s$^{\beta}$)\
$K_3$ is energy$^{-2/3}$ flux$^{\gamma}$ (keV$^{-2/3}$ 10$^{-9\gamma}$ erg$^{\gamma}$ cm$^{-2\gamma}$ s$^{-\gamma}$)\
$K_4$ is dimensionless\
\[t:fitting\]
With further increase in accretion rate, the transition to radiation braking regime and the appearance of an optically thick accretion column should occur [@Basko76]. The critical luminosity for the transition is expected near $10^{37}$erg s$^{-1}$ (Becker et al. 2012, see also Mushtukov et al. 2015a for recent more accurate calculations). While the brightest single observation only reached $\sim(7\pm1.4)\times10^{36}$ erg s$^{-1}$ for the 2.4$\pm$0.5 kpc of Parkes et al. (1980), it would be interesting to probe the transition between different accretion regimes in transient X-ray pulsars with more powerful outbursts.
Note that an alternative explanation of the positive correlations between $E_\mathrm{cyc}-F_\mathrm{x}$ and $W-F_\mathrm{x}$ at moderate X-ray luminosities was recently proposed by @Mushtukov15b. However, that model predicts the *opposite* sign of the second derivative in the $E_\mathrm{cyc}-F_\mathrm{x}$ and $W-F_\mathrm{x}$ relations [cf. black solid lines in Fig. 6a and 6b Fig. 7a in @Mushtukov15b], while the simple physical explanation given above is consistent with observations of GX 304$-$1.
Outburst Shifts in Orbital Phase
--------------------------------
The shifts in orbital phase applied to the three outbursts can be understood in terms of changes in the size of the circumstellar disk around V850 Cen. Referring to Fig. 4 in @Postnov15a, the disk is inclined with respect to the orbital plane of the neutron star, and the neutron star passes through the disk at point A, accumulating matter that forms a temporary accretion disk [@Devasia11]. The lack of a double peak to the three outbursts implies that the circumstellar disk does not extend to the recrossing of the line of nodes at point B. Changes in the thickness of the circumstellar disk from one orbit to the next will affect the amount of matter captured in the accretion disk and thus the duration of the outburst. The few percent orbital phase shifts imply small variations in the circumstellar disk on timescales of a hundred days.
Flux Correlation in General
---------------------------
The strong positive correlations of spectral parameters with source flux, clearly indicate that the source flux, or indeed the mass accretion rate, is responsible for the overall continuum shape and that of the cyclotron line as well. This is also supported by the nearly identical soft color/intensity curves for the three outbursts and the fact that the four early 2010 August observations yield consistency with other observations when plotted versus flux as opposed to plotted versus orbital phase. @Kuhnel13 similarly found that the key driver for the continuum shape in GRO J1008$-$57 was the power law flux. They found a common spectral model based on flux independent parameters and flux correlations for three Type I outbursts and a Type II outburst, where the power law flux was the defining variable, when the source was in the subcritical state.
Soft Color versus Flux
----------------------
We find that the soft color ratio increases with increasing flux along the horizontal branch [@Reig13], with excursions from the overall track due to an extra amount of material in the line of sight over about 3 days. The hard color ratio shows a similar horizontal branch increase with intensity, but also shows a reversal of the trend at the lowest intensities. The changes in the soft and hard color ratios with intensity can be related to the overall steepening of the power law index with decreasing intensity and its hardening of the falling exponential at the lowest intensities. The flattening of spectra with increasing X-ray flux, as is seen in Fig. \[fig:sc\_flux\], could be due to increase in the optical depth inside the scattering region behind the shock and hence in the $y$-parameter in the unsaturated Comptonization regime.
CONCLUSIONS
===========
This work presents the analysis of [*RXTE*]{}observations of the accreting X-ray pulsar GX 304$-$1 that provides the finest detail to date on the correlation of the cyclotron line parameters (energy, width, depth, and residual flux) with source flux for any accreting X-ray binary system. The correlations display, for the first time, a flattening with increasing power law flux. This is successfully modeled by a rather simple one-dimensional physical treatment of both disk accretion and quasi-spherical accretion, since in this case no optically thick accretion column is assumed to form above the neutron star polar caps, and the emergent radiation is thus dynamically unimportant. The neutron star surface magnetic field is measured to be $\sim$60 keV in both models. In addition, the correlations of the power law index, break energy, and iron line flux with power law flux points strongly to the source flux, and thus the mass accretion rate, as the overarching determinant of the spectral behavior.
acknowledgements {#acknowledgements .unnumbered}
================
We thank the referee for the careful reading of the paper and the thoughtful comments proferred. We acknowledge the on-going efforts of the Magnet collaboration on accreting X-ray pulsars. Their work over the years has led to a better understanding of emission from the accretion column, and has led to production of physics-based models of both the continuum and cyclotron line shapes. We acknowledge the support of the International Space Science Institute (ISSI) in Bern, Switzerland, for workshops supporting the Magnet collaboration. The work of K. Postnov is supported by RFBF grants 14-02-00657 and 14-02-91345. The work of M. Gornostaev and N. Shakura (calculations of the scaling laws) is supported by RSF grant 14-12-00146. The work of D. Klochkov, J. Wilms, and R. Staubert was supported by DFG grants KL2734/2-1 and WI 1860/11-1. M. Kühnel acknowledges support by the Bundesministerium für Wirtschaft und Technologie under Deutsches Zentrum für Luft- und Raumfahrt grant 50OR1207.
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HEXTEBACKEST
============
[This section is based upon the poster “Estimating the HEXTE A Background Spectrum”, which was presented at the 9th AAS HEAD meeting, @Pottschmidt06]. Since the launch of *RXTE* in 1995, the HEXTE instrument mostly operated in its standard ‘rocking’ mode where the pointing direction of each of its two clusters alternated between source and background measurements in such a way that one cluster was always looking at the source while the other sampled the background. During the extraction of source light curves and spectra, each cluster uses its own background measurements for correction. This allowed HEXTE to achieve signal to background ratios of <1% for long observations ($\gtrsim$400ks) of weak sources [@Rothschild98]. Starting in 2005 December the rocking mechanism of cluster A began to display increasingly frequent interruptions and since 2006 July was permanently fixed in the on-source staring position. We have developed a FTOOL, [`HEXTEBACKEST`]{}, which for a given observation uses the background measured by cluster B to produce an estimated cluster A background spectrum. The tool uses a set of channel dependent parameters to perform a linear transformation of the count rates. We explain how these parameters were derived, compare estimated and measured cluster A backgrounds for archived rocking observations, and present examples of the application of the method. Cluster B began experiencing similar rocking interruptions in 2009 December and was permanently fixed in one of the off-source positions at the end of 2010 March. This enabled cluster B to collect background data for use with `HEXTEBACKEST` to estimate cluster A background for the rest of the [*RXTE*]{}mission.
Introduction
------------
![**Left**: Background spectrum measured by HEXTE’s clusters A (red) and B (blue) for AO9 observation ObsID 90152-01-17. Note that there is an instrumental cutoff below channel $\sim10$ and above channel $\sim246$ (starting from 0). **Right**: Cluster A versus cluster B background rates measured in channels 70 (purple) and 100 (green) for 3570 ObsIDs of AO9. The inset shows the correlation coefficient between the A and B rates for all channels based on these observations.[]{data-label="fig:bkgex"}](f11.eps "fig:"){width="3.2in"} ![**Left**: Background spectrum measured by HEXTE’s clusters A (red) and B (blue) for AO9 observation ObsID 90152-01-17. Note that there is an instrumental cutoff below channel $\sim10$ and above channel $\sim246$ (starting from 0). **Right**: Cluster A versus cluster B background rates measured in channels 70 (purple) and 100 (green) for 3570 ObsIDs of AO9. The inset shows the correlation coefficient between the A and B rates for all channels based on these observations.[]{data-label="fig:bkgex"}](f12.eps "fig:"){width="3.2in"}
Both clusters used their off-source observations to measure their individual backgrounds, which are different from each other mainly but not only due to the fact that cluster B had only 3 operating detectors after 1996 March. For an example of the measured background spectra, see top panel of Fig. \[fig:bkgex\]. The cluster A background can be estimated based on the measured cluster B background: their rates are well correlated for each detector channel (inset of bottom panel of Fig. \[fig:bkgex\], with varying correlation coefficients which become especially high in the background lines around 30 and 70keV \[detector channels $\sim$ energy channels for HEXTE\]). We extracted the background spectra of several thousand exposures performed during the ninth mission year (AO9, 2004). Fig. \[fig:bkgex\]-bottom panel demonstrates the correlation in two selected channels, one associated with a peak in the spectrum and one not.
Linear Correction Parameters
----------------------------
![**Left**:Results of linear fits, $rate\_A=m(channel) \times
rate\_B+\Delta y(channel)$, for 3570 ObsIDs. **Left, Above:** Offset $\Delta y$. **Left, Below:** Slope $m$. Both parameters have been set to $0$ for channels below 10 and above 246. **Right**: $\chi^2$ comparison of the estimated and measured cluster A backgrounds for the AO9 ObsIDs (red). The theoretical distribution is also shown (purple). The inset shows the difference between the estimated and measured cluster A backgrounds for one typical observation.[]{data-label="fig:corpar"}](f13.eps "fig:"){width="3.4in"} ![**Left**:Results of linear fits, $rate\_A=m(channel) \times
rate\_B+\Delta y(channel)$, for 3570 ObsIDs. **Left, Above:** Offset $\Delta y$. **Left, Below:** Slope $m$. Both parameters have been set to $0$ for channels below 10 and above 246. **Right**: $\chi^2$ comparison of the estimated and measured cluster A backgrounds for the AO9 ObsIDs (red). The theoretical distribution is also shown (purple). The inset shows the difference between the estimated and measured cluster A backgrounds for one typical observation.[]{data-label="fig:corpar"}](f14.eps "fig:"){width="3.4in"}
We performed linear fits to the A versus B background rates for each detector channel based on the AO9 data set using `poly_fit` in IDL and taking A and B uncertainties into account. Note that the 3570 ObsIDs are the result of pre-selection: (1) observations with high A or B rates in the lower channels have been omitted to screen against sources in the background field of view, (2) since observations performed far from the SAA show different background correlations, they have also been omitted. The top panel in Fig. \[fig:corpar\] shows the correction parameters we obtained. In the bottom panel of Fig. \[fig:corpar\], the estimated and measured cluster A spectra are compared (red) – the former based on the AO9 cluster B measurements and on the correlation parameters – using the statistic $\chi^2_\text{red}=\sum[d^2/(\sigma_\text{est}^2 +
\sigma_\text{meas}^2)]/$dof for each observation, where $d$ is the estimated minus the measured cluster A background spectrum, $\sigma_\text{est}$ and $\sigma_\text{meas}$ are the spectral uncertainties, and the number of valid channels, dof (degrees of freedom), is 236 (see @bevington [for comparing two independent data sets]). With respect to the theoretical distribution (purple) a small shift and a tail of higher $\chi^2$ values can be seen.
Applications
------------
The method outlined above is available to derive HEXTE cluster A background spectra for post-2006 July observations. Each HEASOFT release contains the FTOOL `HEXTEBACKEST`which takes an input .pha file, performs the linear correction for all channels, and writes a corrected output .pha file. A FITS file with the correction parameters is part of the calibration database (CALDB), distributed from NASA’s High Energy Science Archive and Research Center (HEASARC). As a hidden parameter of `HEXTEBACKEST` it will by default be remotely accessed. See ‘`fhelp hextebackest`’ for more details (e.g., on spectral binning). Here we show that for recent observations of bright sources the estimated cluster A background gives satisfactory results in the sense that the same source fits as with the measured cluster A backgrounds are obtained applying systematic uncertainties of 2% or less. Limited tests with spectra from AO4 and earlier show that the correction parameters are not adequate for older observations. Fig. \[fig:cyg\_bkg\] shows the comparison between the measured cluster A background and that generated by `hextebackest` for one example observation. Deviations between the two data sets are mostly seen at the peaks of the stronger background lines. `HEXTEBACKEST` was applied for the observation of a smooth continuum (Cyg X$-$1; Fig. \[fig:cyg\_hexte\]-left) and one with two cyclotron line features imposed on the continuum (V0332+653; Fig. \[fig:cyg\_hexte\]-right). In both cases the residuals to the fit are shown for the case of estimated and measured backgrounds, and they are comparable in both cases. This demonstrates that the `HEXTEBACKEST` does not introduce spurious features in the spectra.
![Estimated (red) and measured (black) cluster A background for the Cyg X-1 observation shown in Fig. \[fig:cyg\_hexte\]. As confirmed by the source fit the estimated background is a good match, however, small deviations, especially in the line peaks, remain.[]{data-label="fig:cyg_bkg"}](f15.eps){width="7.5in"}
![ **Left, Top:** HEXTE cluster A (red) and B (blue) counts spectra with the best fit `cutoffpl` model (black) for an observation of the black hole binary Cyg X-1 performed on 2004 Nov. 30. The spectrum has been averaged over 5 ObsIDs. The spectrum used for the cluster A background subtraction has been estimated based on the cluster B background and the correction parameters. **Left, Middle:** Residuals using the estimated cluster A background, best fit parameters: $\Gamma=1.53^{+0.02}_{-0.02}$, $E_\text{cut}=132^{+8}_{-7}$keV, $K=1.22^{+0.07}_{-0.07}$ photons/keV/cm$^2$/s at 1keV. **Left, Bottom:** Residuals using the measured cluster A background, best fit parameters: $\Gamma=1.54^{+0.02}_{-0.02}$, $E_\text{cut}=134^{+8}_{-7}$keV, $K=1.25^{+0.07}_{-0.07}$ photons/keV/cm$^2$/s at 1keV. The two fits thus lead to consistent results without applying additional systematics in order to take uncertainties in the cluster A background estimate into account. **Right, Top:** HEXTE cluster A (red) and B (blue) counts spectra with the best fit two-cyclotron-lines model (black) for an observation of the transient pulsar V0332$+$53 performed on 2004 Dec. 12. The spectrum used for the cluster A background subtraction has been estimated based on the cluster B background and the correction parameters. **Right, Middle:** Residuals using the estimated cluster A background, best fit parameters: $\Gamma=-0.15^{+0.75}_{-0.57}$, $E_\text{cycl1}=28.83^{+0.08}_{-0.07}$keV, $E_\text{cycl2}=51.5^{+0.6}_{-0.6}$keV. **Right, Bottom:** Residuals using the measured cluster A background, best fit parameters: $\Gamma=+0.50^{+0.51}_{-0.55}$, $E_\text{cycl1}=28.83^{+0.08}_{-0.07}$keV, $E_\text{cycl2}=51.3^{+0.6}_{-0.6}$keV. The two fits thus lead to consistent results, in this case, however, systematics of 2% had to be applied in order to take uncertainties in the cluster A background estimate into account. []{data-label="fig:cyg_hexte"}](f16.eps "fig:"){width="3.3in"} ![ **Left, Top:** HEXTE cluster A (red) and B (blue) counts spectra with the best fit `cutoffpl` model (black) for an observation of the black hole binary Cyg X-1 performed on 2004 Nov. 30. The spectrum has been averaged over 5 ObsIDs. The spectrum used for the cluster A background subtraction has been estimated based on the cluster B background and the correction parameters. **Left, Middle:** Residuals using the estimated cluster A background, best fit parameters: $\Gamma=1.53^{+0.02}_{-0.02}$, $E_\text{cut}=132^{+8}_{-7}$keV, $K=1.22^{+0.07}_{-0.07}$ photons/keV/cm$^2$/s at 1keV. **Left, Bottom:** Residuals using the measured cluster A background, best fit parameters: $\Gamma=1.54^{+0.02}_{-0.02}$, $E_\text{cut}=134^{+8}_{-7}$keV, $K=1.25^{+0.07}_{-0.07}$ photons/keV/cm$^2$/s at 1keV. The two fits thus lead to consistent results without applying additional systematics in order to take uncertainties in the cluster A background estimate into account. **Right, Top:** HEXTE cluster A (red) and B (blue) counts spectra with the best fit two-cyclotron-lines model (black) for an observation of the transient pulsar V0332$+$53 performed on 2004 Dec. 12. The spectrum used for the cluster A background subtraction has been estimated based on the cluster B background and the correction parameters. **Right, Middle:** Residuals using the estimated cluster A background, best fit parameters: $\Gamma=-0.15^{+0.75}_{-0.57}$, $E_\text{cycl1}=28.83^{+0.08}_{-0.07}$keV, $E_\text{cycl2}=51.5^{+0.6}_{-0.6}$keV. **Right, Bottom:** Residuals using the measured cluster A background, best fit parameters: $\Gamma=+0.50^{+0.51}_{-0.55}$, $E_\text{cycl1}=28.83^{+0.08}_{-0.07}$keV, $E_\text{cycl2}=51.3^{+0.6}_{-0.6}$keV. The two fits thus lead to consistent results, in this case, however, systematics of 2% had to be applied in order to take uncertainties in the cluster A background estimate into account. []{data-label="fig:cyg_hexte"}](f17.eps "fig:"){width="3.3in"}
Spectral Fit Tables and Figures {#sec:fits}
===============================
This section contains the best fit parameters from the spectral fitting of each observation with both the `highecut` and `cutoffpl` models. The tables are divided into the continuum and the line parameters. After the tables, plots of the various parameters are given.
\# $N_\mathrm{H}^a$ Index$^b$ Flux$^c$ Ecut$^d$ Efold$^d$ Ecyc$^e$ Width$^e$ Depth$^e$ $\chi^2$/dof
---- ------------------------- --------------------------- ---------------------------- ------------------------- --------------------------- -------------------------- ------------------------- ------------------------ --------------
1 $7.30^{+0.50}_{-0.60}$ $1.700^{+0.050}_{-0.060}$ $1.377^{+0.028}_{-0.030}$ $7.90^{+0.50}_{-0.60}$ $26.90^{+3.30}_{-2.90}$ $43.30^{+6.60}_{-3.00}$ $2.40^{+3.70}_{-2.20}$ $\ge 0.28$ 1.22/152
2 $6.70^{+0.60}_{-0.80}$ $1.640^{+0.060}_{-0.090}$ $1.410^{+0.040}_{-0.050}$ $7.50^{+0.70}_{-0.70}$ $25.00^{+4.00}_{-4.00}$ $--$ $--$ $--$ 1.04/155
3 $7.70^{+0.40}_{-0.40}$ $1.533^{+0.028}_{-0.031}$ $2.380^{+0.040}_{-0.040}$ $8.01^{+0.28}_{-0.30}$ $24.60^{+1.60}_{-1.50}$ $46.00^{+5.00}_{-5.00}$ $4.00^{+4.00}_{-4.00}$ $0.37^{+6.48}_{-0.19}$ 1.20/152
4 $3.10^{+0.23}_{-0.23}$ $0.848^{+0.030}_{-0.029}$ $9.280^{+0.080}_{-0.080}$ $4.96^{+0.24}_{-0.26}$ $20.30^{+1.20}_{-1.00}$ $58.80^{+2.20}_{-1.70}$ $12.90^{+1.20}_{-1.00}$ $1.18^{+0.15}_{-0.12}$ 1.18/151
5 $2.70^{+1.90}_{-0.40}$ $0.760^{+0.040}_{-0.040}$ $9.720^{+0.680}_{-0.130}$ $4.70^{+0.40}_{-2.80}$ $17.50^{+0.90}_{-0.80}$ $57.40^{+1.40}_{-1.20}$ $11.00^{+1.00}_{-0.90}$ $1.00^{+0.10}_{-0.09}$ 0.89/151
6 $2.94^{+0.21}_{-0.16}$ $0.811^{+0.031}_{-0.022}$ $10.400^{+0.090}_{-0.060}$ $5.07^{+0.15}_{-0.15}$ $19.00^{+1.30}_{-0.70}$ $60.30^{+8.00}_{-2.10}$ $\ge 12.2$ $1.16^{+0.69}_{-0.13}$ 1.24/151
7 $2.80^{+0.40}_{-0.40}$ $0.770^{+0.040}_{-0.040}$ $11.080^{+0.130}_{-0.120}$ $5.14^{+0.27}_{-0.26}$ $18.00^{+0.90}_{-0.80}$ $58.90^{+1.60}_{-1.30}$ $11.90^{+0.90}_{-0.80}$ $1.09^{+0.11}_{-0.09}$ 0.87/152
8 $2.60^{+0.60}_{-0.40}$ $0.820^{+0.090}_{-0.060}$ $11.070^{+0.260}_{-0.170}$ $5.40^{+0.60}_{-0.40}$ $17.30^{+2.00}_{-1.10}$ $59.00^{+4.10}_{-2.80}$ $10.80^{+2.00}_{-1.50}$ $1.09^{+0.35}_{-0.21}$ 1.01/153
9 $2.71^{+0.27}_{-0.29}$ $0.880^{+0.040}_{-0.040}$ $11.990^{+0.120}_{-0.140}$ $6.08^{+0.16}_{-0.19}$ $18.90^{+1.00}_{-1.00}$ $57.50^{+1.70}_{-1.40}$ $12.40^{+1.10}_{-1.00}$ $0.84^{+0.08}_{-0.07}$ 1.09/151
11 $3.36^{+0.27}_{-0.30}$ $0.940^{+0.040}_{-0.040}$ $11.260^{+0.120}_{-0.130}$ $6.20^{+0.17}_{-0.19}$ $19.40^{+1.10}_{-1.10}$ $58.10^{+2.40}_{-1.90}$ $12.30^{+1.60}_{-1.50}$ $0.77^{+0.11}_{-0.09}$ 1.25/152
12 $4.10^{+0.60}_{-0.60}$ $1.080^{+0.070}_{-0.070}$ $8.300^{+0.170}_{-0.180}$ $6.30^{+0.40}_{-0.40}$ $20.40^{+2.20}_{-1.90}$ $55.20^{+3.40}_{-2.50}$ $10.10^{+2.40}_{-2.00}$ $0.69^{+0.19}_{-0.16}$ 0.75/152
13 $4.30^{+0.40}_{-0.40}$ $1.180^{+0.050}_{-0.050}$ $7.190^{+0.120}_{-0.110}$ $6.40^{+0.31}_{-0.26}$ $21.40^{+1.80}_{-1.50}$ $52.00^{+1.90}_{-1.30}$ $7.40^{+1.90}_{-1.70}$ $0.58^{+0.14}_{-0.08}$ 0.88/152
14 $5.49^{+0.21}_{-0.23}$ $1.371^{+0.021}_{-0.027}$ $6.400^{+0.060}_{-0.060}$ $7.80^{+0.24}_{-0.30}$ $26.10^{+1.10}_{-1.20}$ $54.70^{+2.40}_{-1.70}$ $8.60^{+1.80}_{-1.60}$ $0.69^{+0.16}_{-0.13}$ 1.01/152
15 $5.70^{+0.50}_{-0.70}$ $1.320^{+0.060}_{-0.090}$ $4.930^{+0.100}_{-0.140}$ $7.50^{+0.60}_{-0.70}$ $21.50^{+1.90}_{-2.60}$ $50.90^{+2.70}_{-3.50}$ $6.10^{+2.50}_{-2.00}$ $0.62^{+0.57}_{-0.26}$ 0.93/152
16 $6.90^{+0.40}_{-0.40}$ $1.410^{+0.040}_{-0.040}$ $4.600^{+0.070}_{-0.070}$ $8.00^{+0.40}_{-0.50}$ $23.10^{+1.50}_{-1.50}$ $53.00^{+6.00}_{-4.00}$ $4.90^{+4.50}_{-2.80}$ $0.60^{+5.40}_{-0.40}$ 0.78/152
17 $7.40^{+0.50}_{-0.50}$ $1.510^{+0.040}_{-0.050}$ $3.040^{+0.050}_{-0.060}$ $7.90^{+0.40}_{-0.40}$ $23.30^{+1.80}_{-1.70}$ $52.00^{+5.00}_{-12.00}$ $4.00^{+5.00}_{-4.00}$ $\ge 0.10$ 1.01/152
18 $7.70^{+0.50}_{-0.60}$ $1.630^{+0.050}_{-0.060}$ $2.370^{+0.050}_{-0.060}$ $7.70^{+0.50}_{-0.60}$ $24.70^{+2.40}_{-2.40}$ $--$ $--$ $--$ 1.19/155
19 $7.00^{+0.80}_{-0.90}$ $1.880^{+0.070}_{-0.100}$ $1.150^{+0.040}_{-0.050}$ $7.40^{+0.80}_{-0.90}$ $37.00^{+14.00}_{-10.00}$ $48.90^{+2.50}_{-3.50}$ $1.20^{+2.10}_{-1.10}$ $\ge 0.18$ 1.23/151
20 $7.50^{+0.70}_{-0.90}$ $2.010^{+0.060}_{-0.100}$ $0.801^{+0.026}_{-0.032}$ $7.80^{+1.10}_{-1.30}$ $48.00^{+23.00}_{-14.00}$ $--$ $--$ $--$ 0.96/155
21 $7.66^{+0.00}_{-1.11}$ $2.160^{+0.060}_{-0.130}$ $0.447^{+0.000}_{-0.024}$ $7.30^{+1.80}_{-5.40}$ $\ge 44.8$ $41.00^{+7.00}_{-4.00}$ $3.30^{+2.90}_{-2.30}$ $\ge 0.67$ 1.08/153
22 $7.70^{+1.00}_{-1.40}$ $2.160^{+0.060}_{-0.200}$ $0.504^{+0.026}_{-0.037}$ $8.00^{+13.00}_{-6.00}$ $\ge 38.0$ $--$ $--$ $--$ 0.84/155
23 $6.80^{+1.70}_{-1.90}$ $2.160^{+0.100}_{-0.210}$ $0.328^{+0.027}_{-0.029}$ $7.00^{+13.00}_{-6.00}$ $\ge 32.5$ $40.00^{+8.00}_{-4.00}$ $4.00^{+4.00}_{-4.00}$ $\ge 0.21$ 0.97/153
24 $7.47^{+0.17}_{-0.18}$ $1.672^{+0.016}_{-0.018}$ $2.060^{+0.015}_{-0.016}$ $7.85^{+0.20}_{-0.21}$ $27.30^{+1.00}_{-1.00}$ $--$ $--$ $--$ 1.85/154
25 $8.40^{+0.30}_{-0.32}$ $1.388^{+0.027}_{-0.033}$ $4.050^{+0.050}_{-0.050}$ $7.89^{+0.26}_{-0.31}$ $21.70^{+1.00}_{-1.10}$ $51.20^{+1.20}_{-5.30}$ $2.60^{+3.60}_{-1.80}$ $\ge 0.15$ 0.67/152
26 $7.30^{+0.40}_{-0.40}$ $1.450^{+0.026}_{-0.028}$ $3.740^{+0.050}_{-0.050}$ $7.94^{+0.25}_{-0.27}$ $24.60^{+1.20}_{-1.20}$ $50.80^{+1.10}_{-1.70}$ $5.00^{+1.20}_{-1.00}$ $0.53^{+0.30}_{-0.18}$ 0.98/152
27 $6.72^{+0.17}_{-0.18}$ $1.371^{+0.016}_{-0.019}$ $5.170^{+0.040}_{-0.040}$ $7.83^{+0.20}_{-0.23}$ $27.60^{+0.90}_{-1.00}$ $54.30^{+1.90}_{-1.30}$ $8.40^{+1.60}_{-1.40}$ $0.61^{+0.11}_{-0.09}$ 1.06/152
28 $4.10^{+0.40}_{-0.40}$ $1.070^{+0.040}_{-0.050}$ $7.000^{+0.090}_{-0.100}$ $5.79^{+0.24}_{-0.30}$ $22.30^{+1.40}_{-1.50}$ $57.90^{+2.60}_{-2.10}$ $9.60^{+1.50}_{-1.40}$ $1.00^{+0.19}_{-0.15}$ 1.13/153
29 $4.00^{+0.40}_{-0.40}$ $0.970^{+0.050}_{-0.040}$ $8.010^{+0.120}_{-0.110}$ $5.40^{+0.40}_{-0.40}$ $20.40^{+1.30}_{-1.10}$ $56.90^{+1.20}_{-1.00}$ $10.00^{+0.90}_{-0.90}$ $0.86^{+0.08}_{-0.07}$ 0.98/152
30 $4.44^{+0.28}_{-0.30}$ $1.100^{+0.040}_{-0.040}$ $7.670^{+0.090}_{-0.100}$ $5.97^{+0.21}_{-0.24}$ $23.50^{+1.50}_{-1.50}$ $56.10^{+1.60}_{-1.40}$ $10.10^{+1.10}_{-1.10}$ $0.89^{+0.11}_{-0.10}$ 1.11/152
31 $4.60^{+0.40}_{-0.50}$ $1.090^{+0.050}_{-0.070}$ $8.110^{+0.120}_{-0.150}$ $5.70^{+0.40}_{-0.60}$ $24.10^{+2.00}_{-2.60}$ $56.70^{+1.80}_{-1.50}$ $10.80^{+1.10}_{-1.20}$ $0.86^{+0.11}_{-0.10}$ 1.11/152
32 $6.90^{+0.40}_{-0.40}$ $1.000^{+0.040}_{-0.050}$ $8.600^{+0.110}_{-0.120}$ $5.77^{+0.23}_{-0.30}$ $21.70^{+1.40}_{-1.50}$ $57.00^{+1.80}_{-1.50}$ $11.00^{+1.20}_{-1.10}$ $0.87^{+0.11}_{-0.08}$ 1.01/152
33 $5.90^{+0.40}_{-0.50}$ $0.990^{+0.050}_{-0.070}$ $9.550^{+0.150}_{-0.170}$ $5.81^{+0.26}_{-0.33}$ $21.80^{+2.20}_{-2.10}$ $54.60^{+4.00}_{-2.90}$ $11.10^{+2.10}_{-1.90}$ $0.76^{+0.24}_{-0.17}$ 1.19/151
34 $7.10^{+0.30}_{-0.32}$ $1.100^{+0.040}_{-0.050}$ $8.290^{+0.100}_{-0.110}$ $5.83^{+0.25}_{-0.30}$ $24.20^{+1.70}_{-1.70}$ $55.90^{+1.40}_{-1.20}$ $10.10^{+1.00}_{-1.00}$ $0.91^{+0.10}_{-0.09}$ 0.90/152
35 $5.60^{+0.50}_{-1.30}$ $1.300^{+0.050}_{-0.170}$ $9.180^{+0.140}_{-0.440}$ $7.60^{+0.60}_{-1.40}$ $30.00^{+4.00}_{-8.00}$ $55.50^{+4.40}_{-2.70}$ $11.70^{+2.60}_{-3.70}$ $0.85^{+0.20}_{-0.22}$ 0.85/152
36 $4.40^{+0.40}_{-0.40}$ $1.150^{+0.040}_{-0.050}$ $7.040^{+0.100}_{-0.100}$ $6.12^{+0.25}_{-0.27}$ $23.40^{+1.70}_{-1.60}$ $54.10^{+1.50}_{-1.20}$ $8.70^{+1.30}_{-1.20}$ $0.85^{+0.13}_{-0.12}$ 0.95/152
37 $4.91^{+0.23}_{-0.23}$ $1.197^{+0.028}_{-0.028}$ $7.280^{+0.070}_{-0.070}$ $6.43^{+0.20}_{-0.18}$ $24.40^{+1.40}_{-1.20}$ $55.70^{+1.70}_{-1.40}$ $10.20^{+1.20}_{-1.10}$ $0.76^{+0.09}_{-0.08}$ 1.06/152
38 $6.11^{+0.18}_{-0.19}$ $1.362^{+0.017}_{-0.020}$ $5.830^{+0.040}_{-0.050}$ $7.89^{+0.22}_{-0.27}$ $30.40^{+1.20}_{-1.20}$ $54.20^{+1.40}_{-1.10}$ $8.70^{+1.10}_{-1.00}$ $0.83^{+0.11}_{-0.10}$ 1.19/152
39 $6.79^{+0.26}_{-0.29}$ $1.395^{+0.024}_{-0.032}$ $4.730^{+0.050}_{-0.060}$ $7.82^{+0.29}_{-0.36}$ $26.60^{+1.30}_{-1.50}$ $50.90^{+1.60}_{-1.80}$ $6.80^{+1.60}_{-1.40}$ $0.57^{+0.21}_{-0.14}$ 0.78/152
40 $7.02^{+0.17}_{-0.18}$ $1.512^{+0.016}_{-0.017}$ $3.260^{+0.022}_{-0.023}$ $7.83^{+0.17}_{-0.18}$ $25.60^{+0.90}_{-0.90}$ $49.80^{+1.30}_{-4.40}$ $3.90^{+1.10}_{-3.60}$ $0.66^{+0.56}_{-0.28}$ 1.15/152
41 $7.00^{+1.00}_{-1.10}$ $2.090^{+0.080}_{-0.110}$ $0.403^{+0.019}_{-0.020}$ $7.40^{+1.30}_{-1.40}$ $\ge 33.8$ $43.00^{+8.00}_{-6.00}$ $3.60^{+4.00}_{-3.00}$ $\ge 0.56$ 1.13/153
42 $7.80^{+1.80}_{-3.10}$ $2.300^{+0.140}_{-0.590}$ $0.196^{+0.019}_{-0.033}$ $8.00^{+4.00}_{-6.00}$ $\ge 19.6$ $--$ $--$ $--$ 1.20/156
43 $7.30^{+1.40}_{-1.90}$ $2.160^{+0.110}_{-0.220}$ $0.234^{+0.016}_{-0.021}$ $7.80^{+2.20}_{-2.50}$ $\ge 24.8$ $--$ $--$ $--$ 0.88/156
44 $7.60^{+1.10}_{-1.60}$ $2.140^{+0.080}_{-0.190}$ $0.304^{+0.016}_{-0.023}$ $7.70^{+2.50}_{-5.80}$ $\ge 35.4$ $--$ $--$ $--$ 0.90/156
45 $3.70^{+0.60}_{-0.40}$ $1.000^{+0.100}_{-0.050}$ $6.300^{+0.160}_{-0.090}$ $5.10^{+0.70}_{-0.50}$ $19.20^{+2.90}_{-1.30}$ $54.80^{+3.30}_{-1.90}$ $7.60^{+2.60}_{-2.20}$ $0.72^{+0.15}_{-0.11}$ 0.96/152
46 $3.50^{+0.60}_{-0.50}$ $1.000^{+0.080}_{-0.060}$ $6.440^{+0.160}_{-0.110}$ $5.30^{+0.50}_{-0.40}$ $20.00^{+2.70}_{-1.70}$ $52.80^{+3.70}_{-1.60}$ $7.10^{+4.00}_{-2.00}$ $0.79^{+0.36}_{-0.18}$ 0.87/153
47 $4.60^{+0.60}_{-0.70}$ $1.140^{+0.060}_{-0.080}$ $6.260^{+0.130}_{-0.150}$ $6.00^{+0.40}_{-0.60}$ $22.60^{+2.50}_{-2.50}$ $56.40^{+5.20}_{-2.70}$ $9.30^{+3.30}_{-2.40}$ $0.82^{+0.25}_{-0.14}$ 1.02/153
48 $5.60^{+0.50}_{-0.50}$ $1.250^{+0.050}_{-0.060}$ $6.430^{+0.120}_{-0.120}$ $6.00^{+0.40}_{-0.50}$ $27.00^{+3.00}_{-2.70}$ $55.10^{+2.30}_{-1.70}$ $9.60^{+1.80}_{-1.80}$ $0.76^{+0.14}_{-0.08}$ 0.95/152
49 $12.70^{+0.40}_{-0.40}$ $1.110^{+0.050}_{-0.050}$ $6.720^{+0.110}_{-0.100}$ $5.60^{+0.40}_{-0.40}$ $23.00^{+2.00}_{-1.80}$ $56.10^{+1.90}_{-1.50}$ $9.60^{+1.40}_{-1.40}$ $0.78^{+0.12}_{-0.11}$ 1.16/152
50 $9.40^{+0.40}_{-0.40}$ $1.070^{+0.050}_{-0.050}$ $7.220^{+0.120}_{-0.120}$ $5.70^{+0.40}_{-0.40}$ $22.20^{+1.60}_{-1.50}$ $55.60^{+1.10}_{-1.00}$ $9.50^{+0.90}_{-0.90}$ $0.79^{+0.08}_{-0.08}$ 0.87/152
51 $10.70^{+0.60}_{-0.60}$ $1.200^{+0.060}_{-0.060}$ $7.310^{+0.150}_{-0.150}$ $6.20^{+0.40}_{-0.40}$ $25.30^{+2.80}_{-2.40}$ $56.00^{+3.50}_{-2.20}$ $8.90^{+2.10}_{-1.90}$ $0.78^{+0.21}_{-0.11}$ 0.95/152
52 $11.40^{+0.40}_{-0.40}$ $1.040^{+0.040}_{-0.040}$ $7.580^{+0.100}_{-0.100}$ $5.72^{+0.26}_{-0.30}$ $21.10^{+1.10}_{-1.00}$ $56.60^{+0.90}_{-0.80}$ $10.00^{+0.70}_{-0.70}$ $0.78^{+0.06}_{-0.06}$ 1.19/152
54 $15.10^{+0.60}_{-0.60}$ $0.860^{+0.060}_{-0.060}$ $11.640^{+0.220}_{-0.220}$ $5.80^{+0.27}_{-0.31}$ $18.90^{+1.80}_{-1.50}$ $61.00^{+5.00}_{-4.00}$ $12.60^{+2.70}_{-2.00}$ $0.97^{+0.30}_{-0.16}$ 1.12/152
55 $13.30^{+1.60}_{-0.80}$ $0.870^{+0.070}_{-0.050}$ $8.420^{+0.510}_{-0.220}$ $4.30^{+0.80}_{-2.40}$ $17.60^{+1.40}_{-1.10}$ $59.00^{+6.00}_{-4.00}$ $10.50^{+3.30}_{-2.50}$ $0.80^{+0.36}_{-0.22}$ 1.03/152
56 $14.60^{+0.70}_{-0.70}$ $1.080^{+0.060}_{-0.070}$ $6.940^{+0.160}_{-0.150}$ $5.80^{+0.40}_{-0.40}$ $21.90^{+2.40}_{-2.10}$ $60.00^{+10.00}_{-4.00}$ $11.20^{+4.30}_{-2.60}$ $1.04^{+0.79}_{-0.24}$ 0.88/152
57 $12.60^{+0.40}_{-0.40}$ $0.940^{+0.040}_{-0.040}$ $9.010^{+0.120}_{-0.110}$ $5.66^{+0.24}_{-0.27}$ $19.30^{+0.90}_{-0.80}$ $57.00^{+0.90}_{-0.80}$ $10.40^{+0.70}_{-0.70}$ $0.81^{+0.05}_{-0.05}$ 1.11/152
58 $8.60^{+0.40}_{-0.40}$ $0.930^{+0.050}_{-0.050}$ $10.410^{+0.140}_{-0.140}$ $5.77^{+0.23}_{-0.27}$ $19.90^{+1.30}_{-1.20}$ $56.50^{+1.70}_{-1.40}$ $10.10^{+1.10}_{-1.10}$ $0.93^{+0.13}_{-0.11}$ 1.34/152
59 $8.90^{+0.50}_{-0.50}$ $1.030^{+0.050}_{-0.070}$ $8.770^{+0.150}_{-0.170}$ $6.06^{+0.29}_{-0.36}$ $22.10^{+2.00}_{-2.00}$ $56.20^{+2.90}_{-2.00}$ $11.20^{+1.80}_{-1.60}$ $0.84^{+0.13}_{-0.10}$ 1.00/152
60 $3.10^{+0.50}_{-0.40}$ $0.830^{+0.050}_{-0.050}$ $10.620^{+0.170}_{-0.160}$ $5.58^{+0.30}_{-0.31}$ $17.60^{+1.10}_{-1.00}$ $58.60^{+2.40}_{-1.80}$ $10.70^{+1.50}_{-1.40}$ $0.84^{+0.13}_{-0.11}$ 0.91/153
61 $3.50^{+0.50}_{-0.50}$ $0.940^{+0.050}_{-0.060}$ $9.450^{+0.160}_{-0.170}$ $5.85^{+0.29}_{-0.34}$ $18.70^{+1.30}_{-1.30}$ $56.20^{+1.90}_{-1.50}$ $10.00^{+1.40}_{-1.30}$ $0.73^{+0.10}_{-0.07}$ 0.88/153
63 $1.90^{+0.70}_{-0.60}$ $0.740^{+0.080}_{-0.080}$ $11.620^{+0.260}_{-0.240}$ $5.60^{+0.40}_{-0.40}$ $16.50^{+1.90}_{-1.50}$ $57.90^{+4.20}_{-2.90}$ $11.80^{+2.80}_{-2.60}$ $0.72^{+0.18}_{-0.14}$ 0.93/152
64 $3.80^{+0.40}_{-0.40}$ $1.000^{+0.050}_{-0.050}$ $9.320^{+0.130}_{-0.140}$ $6.25^{+0.25}_{-0.26}$ $20.10^{+1.40}_{-1.30}$ $55.30^{+2.30}_{-1.70}$ $10.00^{+1.60}_{-1.40}$ $0.80^{+0.14}_{-0.12}$ 1.03/153
: Best-fit Highecut Continuum Spectral Parameters of GX 304$-$1\[tab:best\_fit\_highecut\_cont\]
\# $N_\mathrm{H}^a$ Index$^b$ Flux$^c$ Ecut$^d$ Efold$^d$ Ecyc$^e$ Width$^e$ Depth$^e$ $\chi^2$/dof
---- ------------------------ --------------------------- --------------------------- ------------------------- ------------------------- -------------------------- ------------------------- ------------------------ --------------
65 $4.20^{+0.40}_{-0.40}$ $1.070^{+0.050}_{-0.050}$ $8.770^{+0.130}_{-0.130}$ $6.50^{+0.33}_{-0.27}$ $22.90^{+2.10}_{-1.70}$ $56.40^{+3.10}_{-2.20}$ $11.10^{+2.10}_{-1.70}$ $0.79^{+0.15}_{-0.11}$ 1.21/152
66 $3.79^{+0.24}_{-0.25}$ $1.056^{+0.028}_{-0.029}$ $7.980^{+0.080}_{-0.080}$ $6.30^{+0.16}_{-0.16}$ $19.80^{+0.90}_{-0.80}$ $55.50^{+1.90}_{-1.40}$ $8.50^{+1.40}_{-1.30}$ $0.74^{+0.11}_{-0.10}$ 1.14/152
67 $6.16^{+0.17}_{-0.17}$ $1.367^{+0.016}_{-0.018}$ $5.140^{+0.040}_{-0.040}$ $7.86^{+0.17}_{-0.19}$ $25.50^{+0.80}_{-0.80}$ $52.20^{+1.00}_{-0.90}$ $6.90^{+1.00}_{-1.00}$ $0.75^{+0.16}_{-0.12}$ 1.23/152
68 $6.60^{+0.50}_{-0.50}$ $1.420^{+0.040}_{-0.050}$ $4.790^{+0.080}_{-0.090}$ $7.90^{+0.40}_{-0.50}$ $24.60^{+1.90}_{-1.90}$ $51.00^{+2.20}_{-3.50}$ $5.20^{+2.30}_{-2.10}$ $0.80^{+2.00}_{-0.40}$ 0.79/152
69 $6.80^{+0.29}_{-0.32}$ $1.531^{+0.028}_{-0.037}$ $3.340^{+0.040}_{-0.050}$ $7.80^{+0.40}_{-0.40}$ $25.20^{+1.50}_{-1.60}$ $50.00^{+4.00}_{-5.00}$ $5.90^{+3.20}_{-2.70}$ $0.39^{+0.88}_{-0.18}$ 0.86/152
70 $7.60^{+0.70}_{-0.90}$ $1.750^{+0.060}_{-0.110}$ $2.160^{+0.070}_{-0.090}$ $7.90^{+0.60}_{-1.10}$ $29.00^{+6.00}_{-6.00}$ $--$ $--$ $--$ 0.98/155
71 $7.30^{+0.50}_{-0.50}$ $1.830^{+0.040}_{-0.050}$ $1.521^{+0.030}_{-0.032}$ $7.60^{+0.40}_{-0.50}$ $30.00^{+5.00}_{-4.00}$ $--$ $--$ $--$ 1.43/155
72 $7.90^{+1.90}_{-2.00}$ $2.130^{+0.100}_{-0.270}$ $0.490^{+0.050}_{-0.050}$ $8.00^{+13.00}_{-6.00}$ $\ge 28.5$ $40.00^{+19.00}_{-4.00}$ $4.30^{+4.00}_{-2.50}$ $\ge 0.69$ 0.90/153
\
$^a$ Column density in 10$^{22}$ cm$^{-2}$\
$^b$ Power law photon index\
$^c$ Unabsorbed power law 2$-$10 keV flux in 10$^{-9}$ ergs cm$^{-2}$ s$^{-1}$\
$^d$ Ecut is cutoff energy in keV; Efold is folding energy in keV\
$^e$ Ecyc is CRSF energy; Width is CRSF width in keV; Depth is CRSF depth\
\# iron$^a$ iron$^b$ 10.5keV$^c$ 3.88keV$^d$ 30keV$^e$ 39keV$^f$ 53keV$^g$ 66keV$^h$
---- ------------------- ------------------ ---------------------- ---------------------- --------------------- --------------------- --------------------- ---------------------
1 $5^{+2}_{-2}$ $36^{+16}_{-17}$ $--$ $-0.7^{+0.4}_{-0.4}$ $0.9^{+0.5}_{-0.5}$ $1.3^{+0.7}_{-0.6}$ $0.7^{+0.5}_{-0.3}$ $1.4^{+0.5}_{-0.5}$
2 $5^{+3}_{-3}$ $31^{+20}_{-21}$ $--$ $-0.5^{+0.4}_{-0.4}$ $1.5^{+0.5}_{-0.5}$ $1.1^{+0.3}_{-0.3}$ $0.5^{+0.3}_{-0.3}$ $2.2^{+0.5}_{-0.5}$
3 $12^{+3}_{-3}$ $45^{+11}_{-11}$ $--$ $-1.0^{+0.4}_{-0.4}$ $\le 0.1$ $1.3^{+0.7}_{-0.5}$ $\le 1.1$ $\le 0.7$
4 $91^{+8}_{-8}$ $79^{+7}_{-7}$ $-11^{+6}_{-6}$ $-1.9^{+1.1}_{-1.2}$ $4.9^{+1.4}_{-1.3}$ $2.9^{+0.9}_{-0.9}$ $\le 1.0$ $4.1^{+0.7}_{-0.7}$
5 $90^{+13}_{-13}$ $74^{+11}_{-11}$ $-19^{+8}_{-8}$ $-2.5^{+2.3}_{-4.7}$ $3.6^{+0.9}_{-0.9}$ $2.3^{+0.7}_{-0.7}$ $\le 0.8$ $2.9^{+0.6}_{-0.6}$
6 $89^{+6}_{-6}$ $69^{+5}_{-5}$ $-12^{+4}_{-4}$ $-1.7^{+0.7}_{-0.8}$ $4.0^{+1.1}_{-2.7}$ $2.5^{+0.8}_{-1.1}$ $\le 0.9$ $3.3^{+0.6}_{-0.6}$
7 $106^{+15}_{-16}$ $75^{+12}_{-12}$ $-17^{+8}_{-8}$ $--$ $4.3^{+1.0}_{-0.9}$ $2.8^{+0.7}_{-0.7}$ $0.8^{+0.5}_{-0.5}$ $4.0^{+0.6}_{-0.6}$
8 $94^{+20}_{-25}$ $66^{+16}_{-19}$ $--$ $--$ $4.0^{+2.4}_{-2.3}$ $3.1^{+1.8}_{-1.8}$ $\le 1.7$ $3.5^{+1.4}_{-1.5}$
9 $79^{+8}_{-7}$ $51^{+6}_{-5}$ $-10^{+5}_{-5}$ $-1.3^{+1.0}_{-1.0}$ $4.8^{+1.1}_{-1.1}$ $2.6^{+0.8}_{-0.8}$ $\le 0.4$ $2.7^{+0.6}_{-0.6}$
11 $65^{+7}_{-7}$ $45^{+5}_{-5}$ $--$ $-3.2^{+1.0}_{-1.0}$ $4.6^{+1.1}_{-1.0}$ $2.4^{+0.8}_{-0.8}$ $\le 0.6$ $2.9^{+0.6}_{-0.7}$
12 $14^{+10}_{-9}$ $13^{+10}_{-9}$ $--$ $-1.8^{+1.5}_{-1.5}$ $4.2^{+1.5}_{-1.3}$ $2.1^{+1.2}_{-1.2}$ $\le 1.3$ $2.8^{+1.0}_{-1.0}$
13 $30^{+6}_{-6}$ $35^{+7}_{-7}$ $--$ $-1.1^{+1.0}_{-1.0}$ $3.4^{+0.9}_{-0.8}$ $1.4^{+0.9}_{-0.9}$ $\le 0.5$ $2.0^{+0.7}_{-0.7}$
14 $27^{+5}_{-6}$ $37^{+7}_{-8}$ $--$ $-3.0^{+0.7}_{-0.7}$ $2.6^{+0.6}_{-0.6}$ $1.2^{+0.6}_{-0.6}$ $0.7^{+0.5}_{-0.5}$ $2.9^{+0.6}_{-0.6}$
15 $18^{+10}_{-11}$ $31^{+17}_{-19}$ $--$ $-2.5^{+1.0}_{-1.0}$ $2.0^{+0.8}_{-0.8}$ $1.7^{+0.9}_{-0.8}$ $\le 1.3$ $1.4^{+0.8}_{-0.7}$
16 $22^{+7}_{-7}$ $43^{+13}_{-13}$ $--$ $-2.0^{+0.9}_{-0.9}$ $2.0^{+0.8}_{-0.8}$ $0.9^{+0.7}_{-0.6}$ $\le 1.9$ $1.8^{+0.8}_{-0.7}$
17 $12^{+5}_{-5}$ $36^{+14}_{-14}$ $--$ $-1.3^{+0.6}_{-0.6}$ $1.5^{+0.7}_{-0.7}$ $1.3^{+1.1}_{-0.5}$ $\le 1.8$ $1.6^{+0.6}_{-0.6}$
18 $8^{+4}_{-4}$ $32^{+15}_{-16}$ $--$ $-0.9^{+0.5}_{-0.5}$ $1.6^{+0.6}_{-0.6}$ $1.2^{+0.4}_{-0.4}$ $0.3^{+0.3}_{-0.3}$ $1.4^{+0.6}_{-0.6}$
19 $5^{+3}_{-3}$ $48^{+23}_{-26}$ $-2.5^{+1.7}_{-1.7}$ $-1.1^{+0.5}_{-0.5}$ $\le 0.9$ $1.3^{+0.4}_{-0.4}$ $1.2^{+0.4}_{-0.5}$ $1.8^{+0.6}_{-0.6}$
20 $4^{+2}_{-2}$ $58^{+24}_{-26}$ $--$ $-0.3^{+0.3}_{-0.3}$ $0.5^{+0.6}_{-0.6}$ $0.7^{+0.3}_{-0.3}$ $0.6^{+0.2}_{-0.2}$ $1.7^{+0.5}_{-0.5}$
21 $3^{+0}_{-2}$ $77^{+0}_{-39}$ $--$ $--$ $1.3^{+0.8}_{-0.7}$ $1.1^{+0.4}_{-0.5}$ $0.7^{+0.5}_{-0.4}$ $1.5^{+0.7}_{-0.7}$
22 $\le 3$ $\le 64$ $--$ $-0.4^{+0.3}_{-0.3}$ $0.7^{+0.7}_{-0.7}$ $0.5^{+0.4}_{-0.4}$ $0.5^{+0.4}_{-0.4}$ $1.5^{+0.7}_{-0.7}$
23 $\le 3$ $\le 99$ $--$ $--$ $1.2^{+1.1}_{-1.1}$ $1.0^{+0.5}_{-0.6}$ $0.9^{+0.5}_{-0.5}$ $2.2^{+0.9}_{-0.9}$
24 $8^{+1}_{-1}$ $37^{+6}_{-6}$ $-1.5^{+0.8}_{-0.8}$ $-1.0^{+0.2}_{-0.2}$ $1.6^{+0.2}_{-0.2}$ $1.1^{+0.1}_{-0.1}$ $0.3^{+0.1}_{-0.1}$ $1.5^{+0.2}_{-0.2}$
25 $16^{+5}_{-5}$ $35^{+10}_{-10}$ $--$ $-1.2^{+0.6}_{-0.5}$ $1.5^{+0.6}_{-0.6}$ $0.8^{+0.6}_{-0.4}$ $\le 2.0$ $1.5^{+0.5}_{-0.5}$
26 $19^{+5}_{-5}$ $45^{+11}_{-11}$ $--$ $-2.0^{+0.6}_{-0.6}$ $1.5^{+0.2}_{-0.2}$ $1.0^{+0.2}_{-0.2}$ $0.5^{+0.4}_{-0.4}$ $1.4^{+0.2}_{-0.2}$
27 $25^{+4}_{-4}$ $43^{+6}_{-6}$ $--$ $-1.8^{+0.4}_{-0.4}$ $2.3^{+0.4}_{-0.4}$ $1.0^{+0.4}_{-0.4}$ $\le 0.6$ $1.8^{+0.4}_{-0.4}$
28 $41^{+8}_{-7}$ $48^{+10}_{-8}$ $--$ $--$ $2.9^{+0.7}_{-0.7}$ $1.2^{+0.7}_{-0.7}$ $\le 1.0$ $3.5^{+0.7}_{-0.8}$
29 $56^{+13}_{-13}$ $57^{+13}_{-14}$ $--$ $-1.5^{+1.3}_{-1.4}$ $3.3^{+0.6}_{-0.5}$ $1.4^{+0.4}_{-0.4}$ $0.3^{+0.3}_{-0.3}$ $2.7^{+0.5}_{-0.5}$
30 $35^{+7}_{-6}$ $37^{+7}_{-6}$ $--$ $-1.5^{+0.7}_{-0.7}$ $3.8^{+0.8}_{-0.8}$ $1.6^{+0.6}_{-0.6}$ $\le 0.8$ $2.9^{+0.7}_{-0.7}$
31 $48^{+14}_{-8}$ $48^{+15}_{-9}$ $--$ $-2.4^{+0.8}_{-0.8}$ $3.9^{+1.0}_{-1.0}$ $2.1^{+0.7}_{-0.7}$ $\le 0.8$ $3.6^{+0.7}_{-0.8}$
32 $47^{+9}_{-8}$ $44^{+9}_{-7}$ $--$ $-1.2^{+0.7}_{-0.8}$ $4.4^{+0.9}_{-0.9}$ $2.1^{+0.7}_{-0.7}$ $\le 0.5$ $3.6^{+0.7}_{-0.7}$
33 $52^{+11}_{-8}$ $43^{+10}_{-7}$ $-9^{+5}_{-5}$ $-1.4^{+0.9}_{-0.9}$ $6.7^{+2.1}_{-2.1}$ $4.4^{+1.4}_{-1.4}$ $\le 1.8$ $2.4^{+1.6}_{-1.6}$
34 $46^{+8}_{-7}$ $46^{+9}_{-7}$ $--$ $-1.7^{+0.7}_{-0.7}$ $4.6^{+0.9}_{-0.9}$ $1.9^{+0.7}_{-0.7}$ $0.8^{+0.5}_{-0.5}$ $3.2^{+0.7}_{-0.7}$
35 $57^{+15}_{-26}$ $53^{+14}_{-24}$ $--$ $-4.7^{+2.5}_{-1.6}$ $5.3^{+2.2}_{-3.3}$ $2.2^{+1.5}_{-1.5}$ $\le 1.9$ $4.3^{+1.3}_{-2.4}$
36 $29^{+6}_{-6}$ $34^{+7}_{-7}$ $--$ $-1.7^{+0.8}_{-0.8}$ $2.9^{+0.9}_{-0.8}$ $1.6^{+0.8}_{-0.8}$ $0.8^{+0.6}_{-0.6}$ $2.4^{+0.7}_{-0.7}$
37 $23^{+4}_{-4}$ $27^{+4}_{-4}$ $--$ $-2.1^{+0.6}_{-0.6}$ $3.8^{+0.6}_{-0.6}$ $1.6^{+0.5}_{-0.5}$ $0.4^{+0.4}_{-0.4}$ $2.3^{+0.5}_{-0.5}$
38 $30^{+4}_{-5}$ $44^{+6}_{-7}$ $--$ $-2.0^{+0.5}_{-0.5}$ $3.5^{+0.6}_{-0.5}$ $1.3^{+0.5}_{-0.5}$ $0.7^{+0.4}_{-0.4}$ $2.7^{+0.5}_{-0.5}$
39 $21^{+5}_{-5}$ $39^{+9}_{-10}$ $--$ $-1.8^{+0.6}_{-0.6}$ $1.9^{+0.6}_{-0.6}$ $1.4^{+0.6}_{-0.6}$ $\le 0.8$ $1.4^{+0.5}_{-0.5}$
40 $15^{+2}_{-2}$ $41^{+6}_{-6}$ $--$ $-1.1^{+0.3}_{-0.3}$ $1.7^{+0.3}_{-0.3}$ $1.1^{+0.2}_{-0.2}$ $\le 1.1$ $1.3^{+0.2}_{-0.2}$
41 $3^{+1}_{-1}$ $80^{+40}_{-40}$ $--$ $--$ $1.1^{+0.7}_{-0.5}$ $1.1^{+0.3}_{-0.6}$ $0.6^{+0.3}_{-0.3}$ $1.8^{+0.5}_{-0.5}$
42 $\le 1$ $\le 44$ $--$ $--$ $\le 0.7$ $0.7^{+0.4}_{-0.4}$ $0.6^{+0.3}_{-0.3}$ $1.6^{+0.6}_{-0.6}$
43 $\le 1$ $\le 69$ $--$ $--$ $\le 0.9$ $0.8^{+0.3}_{-0.3}$ $0.6^{+0.2}_{-0.2}$ $1.6^{+0.5}_{-0.5}$
44 $2^{+1}_{-1}$ $70^{+40}_{-50}$ $--$ $--$ $1.2^{+0.5}_{-0.5}$ $0.5^{+0.3}_{-0.3}$ $0.4^{+0.2}_{-0.2}$ $1.5^{+0.5}_{-0.5}$
45 $41^{+9}_{-15}$ $54^{+12}_{-20}$ $--$ $-2.1^{+1.2}_{-1.4}$ $1.8^{+0.9}_{-0.9}$ $\le 1.9$ $\le 0.4$ $1.9^{+1.0}_{-0.9}$
46 $48^{+11}_{-13}$ $62^{+15}_{-16}$ $--$ $--$ $\le 2.4$ $2.0^{+1.4}_{-1.3}$ $\le 1.5$ $\le 2.1$
47 $24^{+11}_{-8}$ $31^{+14}_{-10}$ $--$ $--$ $2.7^{+1.1}_{-1.1}$ $1.0^{+1.1}_{-1.0}$ $\le 0.6$ $1.7^{+1.0}_{-1.0}$
48 $26^{+8}_{-7}$ $33^{+11}_{-9}$ $--$ $-1.3^{+0.9}_{-0.9}$ $3.5^{+1.0}_{-0.9}$ $1.2^{+0.8}_{-0.8}$ $\le 0.6$ $2.9^{+0.8}_{-0.8}$
49 $34^{+9}_{-8}$ $42^{+11}_{-10}$ $--$ $-1.0^{+0.6}_{-0.6}$ $2.8^{+0.8}_{-0.7}$ $1.4^{+0.7}_{-0.6}$ $\le 0.8$ $2.6^{+0.7}_{-0.7}$
50 $47^{+11}_{-10}$ $54^{+14}_{-12}$ $--$ $-1.5^{+1.0}_{-1.0}$ $3.2^{+0.6}_{-0.6}$ $1.1^{+0.5}_{-0.5}$ $\le 0.6$ $2.7^{+0.5}_{-0.5}$
51 $29^{+9}_{-9}$ $33^{+11}_{-10}$ $--$ $-1.7^{+1.1}_{-1.1}$ $3.2^{+1.1}_{-1.1}$ $1.0^{+1.0}_{-1.0}$ $\le 0.8$ $2.7^{+1.1}_{-1.1}$
52 $46^{+11}_{-10}$ $50^{+12}_{-11}$ $--$ $-1.3^{+0.9}_{-0.9}$ $2.9^{+0.4}_{-0.4}$ $1.4^{+0.3}_{-0.3}$ $0.2^{+0.2}_{-0.2}$ $2.4^{+0.4}_{-0.4}$
54 $87^{+17}_{-15}$ $59^{+12}_{-11}$ $--$ $-1.8^{+1.2}_{-1.3}$ $2.7^{+1.8}_{-1.7}$ $1.6^{+1.3}_{-1.3}$ $\le 0.7$ $2.6^{+0.9}_{-1.0}$
55 $83^{+11}_{-11}$ $80^{+11}_{-10}$ $--$ $-2.4^{+2.2}_{-1.7}$ $\le 2.1$ $\le 2.4$ $\le 1.5$ $\le 2.3$
56 $33^{+11}_{-10}$ $40^{+14}_{-12}$ $--$ $-1.5^{+1.0}_{-1.0}$ $2.7^{+1.3}_{-1.5}$ $1.5^{+1.1}_{-1.0}$ $\le 0.9$ $3.2^{+0.9}_{-1.0}$
57 $58^{+13}_{-12}$ $52^{+12}_{-11}$ $--$ $-1.2^{+1.0}_{-1.0}$ $3.1^{+0.4}_{-0.4}$ $1.5^{+0.3}_{-0.3}$ $0.3^{+0.2}_{-0.2}$ $2.6^{+0.4}_{-0.4}$
58 $70^{+11}_{-10}$ $53^{+9}_{-7}$ $--$ $-1.9^{+0.8}_{-0.8}$ $3.4^{+1.2}_{-1.2}$ $1.0^{+1.0}_{-1.0}$ $\le 1.1$ $1.2^{+0.9}_{-0.9}$
59 $46^{+11}_{-9}$ $43^{+11}_{-9}$ $--$ $-1.5^{+1.1}_{-1.1}$ $1.9^{+1.5}_{-1.4}$ $1.7^{+1.1}_{-1.1}$ $\le 0.6$ $2.0^{+0.9}_{-0.9}$
60 $98^{+18}_{-18}$ $73^{+15}_{-14}$ $--$ $--$ $4.2^{+1.1}_{-1.1}$ $1.8^{+1.0}_{-1.0}$ $\le 1.1$ $2.7^{+0.8}_{-0.9}$
61 $56^{+16}_{-14}$ $47^{+14}_{-12}$ $--$ $--$ $3.0^{+0.9}_{-0.8}$ $1.7^{+0.7}_{-0.8}$ $\le 0.4$ $2.0^{+0.7}_{-0.7}$
63 $98^{+23}_{-20}$ $65^{+17}_{-14}$ $-24^{+11}_{-11}$ $--$ $5.3^{+2.3}_{-2.0}$ $2.9^{+1.7}_{-1.7}$ $\le 0.7$ $2.4^{+1.2}_{-1.3}$
64 $44^{+9}_{-8}$ $38^{+8}_{-7}$ $--$ $--$ $3.6^{+1.2}_{-1.2}$ $2.5^{+1.0}_{-1.0}$ $\le 1.2$ $2.4^{+0.9}_{-0.9}$
: Best-fit Highecut Spectral Lines\[tab:best\_fit\_highecut\_lines\]
\# iron$^a$ iron$^b$ 10.5keV$^c$ 3.88keV$^d$ 30keV$^e$ 39keV$^f$ 53keV$^g$ 66keV$^h$
---- ---------------- ------------------ ------------- ---------------------- --------------------- --------------------- --------------------- ---------------------
65 $37^{+7}_{-7}$ $34^{+7}_{-7}$ $--$ $-1.8^{+1.1}_{-1.1}$ $2.2^{+1.4}_{-1.3}$ $1.3^{+1.0}_{-1.0}$ $\le 0.9$ $2.1^{+1.0}_{-1.0}$
66 $33^{+5}_{-5}$ $33^{+5}_{-5}$ $--$ $-1.3^{+0.7}_{-0.7}$ $2.9^{+0.6}_{-0.6}$ $1.1^{+0.6}_{-0.6}$ $\le 0.9$ $2.3^{+0.6}_{-0.6}$
67 $28^{+4}_{-4}$ $47^{+6}_{-6}$ $--$ $-1.8^{+0.4}_{-0.4}$ $1.7^{+0.4}_{-0.4}$ $1.1^{+0.4}_{-0.4}$ $0.7^{+0.4}_{-0.4}$ $1.6^{+0.4}_{-0.4}$
68 $27^{+8}_{-8}$ $51^{+14}_{-15}$ $--$ $-1.2^{+1.0}_{-1.0}$ $0.9^{+0.9}_{-0.9}$ $1.3^{+1.0}_{-0.9}$ $\le 2.1$ $1.5^{+0.8}_{-0.8}$
69 $13^{+4}_{-4}$ $35^{+10}_{-11}$ $--$ $-1.3^{+0.5}_{-0.5}$ $1.4^{+0.5}_{-0.5}$ $1.1^{+0.5}_{-0.5}$ $\le 0.9$ $1.5^{+0.4}_{-0.4}$
70 $6^{+5}_{-6}$ $26^{+22}_{-26}$ $--$ $-1.7^{+0.8}_{-0.8}$ $1.5^{+0.9}_{-0.8}$ $1.1^{+0.5}_{-0.5}$ $\le 0.5$ $1.6^{+0.8}_{-0.8}$
71 $6^{+2}_{-2}$ $42^{+15}_{-15}$ $--$ $-1.2^{+0.4}_{-0.4}$ $\le 0.6$ $1.2^{+0.3}_{-0.3}$ $0.5^{+0.2}_{-0.2}$ $1.5^{+0.5}_{-0.4}$
72 $3^{+3}_{-3}$ $60^{+70}_{-70}$ $--$ $--$ $\le 1.8$ $1.2^{+0.6}_{-0.6}$ $\le 1.2$ $1.5^{+1.0}_{-1.0}$
\
$^a$ iron line flux in 10$^{-4}$ photons cm$^{-2}$ s$^{-1}$\
$^b$ iron line equivalent width in eV\
$^c$ 10.5 keV negative line flux in units of 10$^{-3}$ photons cm$^{-2}$ s$^{-1}$\
$^d$ 3.88 keV line flux in units of 10$^{-3}$ photons cm$^{-2}$ s$^{-1}$\
$^e$ 30.17 keV line flux in units of 10$^{-3}$ photons cm$^{-2}$ s$^{-1}$\
$^f$ 39.04 keV line flux in units of 10$^{-3}$ photons cm$^{-2}$ s$^{-1}$\
$^g$ 53.00 keV line flux in units of 10$^{-3}$ photons cm$^{-2}$ s$^{-1}$\
$^h$ 66.64 keV line flux in units of 10$^{-3}$ photons cm$^{-2}$ s$^{-1}$
\# $N_\mathrm{H}^a$ Index$^b$ Flux$^c$ Efold$^d$ kT$^e$ Flux$_\mathrm{BB}$$^f$ Ecyc$^g$ Width$^g$ Depth$^g$ $\chi^2$/dof
---- ------------------------- --------------------------- ---------------------------- --------------------------- ------------------------ ------------------------ ------------------------- ------------------------- ------------------------ --------------
1 $4.80^{+0.80}_{-0.90}$ $1.580^{+0.080}_{-0.130}$ $0.920^{+0.080}_{-0.090}$ $\ge 55.2$ $1.86^{+0.09}_{-0.08}$ $0.31^{+0.05}_{-0.04}$ $47.30^{+3.00}_{-5.30}$ $5.70^{+2.40}_{-4.10}$ $0.84^{+0.65}_{-0.30}$ 0.89/152
2 $4.50^{+1.00}_{-1.40}$ $1.550^{+0.100}_{-0.300}$ $0.970^{+0.100}_{-0.120}$ $\ge 30.0$ $1.85^{+0.11}_{-0.16}$ $0.31^{+0.05}_{-0.05}$ $51.60^{+2.50}_{-2.80}$ $5.10^{+4.70}_{-2.90}$ $\ge 0.6$ 0.84/152
3 $6.90^{+0.50}_{-0.60}$ $1.620^{+0.060}_{-0.110}$ $1.880^{+0.070}_{-0.080}$ $\ge 49.3$ $2.32^{+0.11}_{-0.12}$ $0.44^{+0.04}_{-0.05}$ $49.40^{+2.00}_{-2.40}$ $7.50^{+2.30}_{-1.80}$ $0.83^{+0.26}_{-0.22}$ 1.00/150
4 $3.10^{+0.70}_{-0.70}$ $0.540^{+0.170}_{-0.190}$ $8.200^{+0.700}_{-0.800}$ $14.80^{+2.80}_{-2.10}$ $1.34^{+0.07}_{-0.07}$ $1.00^{+0.50}_{-0.50}$ $60.90^{+4.00}_{-2.70}$ $12.50^{+2.50}_{-2.20}$ $1.07^{+0.27}_{-0.23}$ 1.19/151
5 $3.00^{+0.70}_{-0.80}$ $0.490^{+0.150}_{-0.180}$ $8.700^{+0.700}_{-0.800}$ $14.00^{+1.80}_{-1.60}$ $1.35^{+0.08}_{-0.10}$ $1.10^{+0.60}_{-0.50}$ $57.70^{+2.10}_{-1.80}$ $10.30^{+1.50}_{-1.70}$ $0.86^{+0.15}_{-0.15}$ 0.92/151
6 $2.40^{+0.50}_{-0.40}$ $0.370^{+0.140}_{-0.120}$ $8.600^{+0.600}_{-0.500}$ $12.80^{+1.50}_{-1.00}$ $1.40^{+0.03}_{-0.03}$ $1.60^{+0.40}_{-0.40}$ $59.00^{+6.00}_{-4.00}$ $10.80^{+3.20}_{-2.50}$ $0.82^{+0.39}_{-0.18}$ 1.19/151
7 $3.20^{+0.70}_{-0.70}$ $0.560^{+0.120}_{-0.150}$ $10.100^{+0.700}_{-0.800}$ $15.00^{+1.70}_{-1.60}$ $1.43^{+0.09}_{-0.10}$ $1.10^{+0.50}_{-0.50}$ $60.30^{+2.60}_{-1.90}$ $12.10^{+1.40}_{-1.30}$ $1.03^{+0.16}_{-0.15}$ 0.89/151
8 $2.60^{+0.90}_{-0.80}$ $0.530^{+0.170}_{-0.180}$ $9.700^{+0.900}_{-1.000}$ $13.90^{+2.10}_{-1.60}$ $1.54^{+0.14}_{-0.08}$ $1.30^{+0.70}_{-0.60}$ $59.00^{+6.00}_{-4.00}$ $9.90^{+2.70}_{-2.30}$ $1.00^{+0.50}_{-0.26}$ 0.95/151
9 $1.60^{+0.60}_{-0.50}$ $0.410^{+0.130}_{-0.130}$ $9.800^{+0.600}_{-0.600}$ $12.80^{+1.40}_{-1.10}$ $1.59^{+0.08}_{-0.05}$ $1.70^{+0.50}_{-0.40}$ $57.10^{+3.30}_{-2.50}$ $10.10^{+2.60}_{-2.40}$ $0.56^{+0.16}_{-0.12}$ 0.98/151
11 $2.20^{+0.50}_{-0.50}$ $0.460^{+0.100}_{-0.100}$ $9.200^{+0.500}_{-0.500}$ $12.90^{+1.00}_{-0.80}$ $1.59^{+0.06}_{-0.04}$ $1.50^{+0.40}_{-0.40}$ $54.10^{+2.40}_{-1.40}$ $7.10^{+2.20}_{-1.60}$ $0.45^{+0.11}_{-0.09}$ 1.20/151
12 $3.80^{+0.80}_{-1.00}$ $0.880^{+0.160}_{-0.210}$ $7.400^{+0.400}_{-0.700}$ $18.00^{+6.00}_{-4.00}$ $1.91^{+0.40}_{-0.29}$ $0.75^{+0.32}_{-0.24}$ $55.00^{+5.00}_{-4.00}$ $11.00^{+4.00}_{-4.00}$ $0.65^{+0.24}_{-0.25}$ 0.77/151
13 $3.30^{+0.80}_{-0.70}$ $0.860^{+0.150}_{-0.140}$ $6.100^{+0.400}_{-0.500}$ $16.40^{+3.10}_{-1.90}$ $1.71^{+0.25}_{-0.12}$ $0.78^{+0.26}_{-0.21}$ $51.30^{+2.00}_{-1.40}$ $5.90^{+3.00}_{-2.20}$ $0.51^{+0.28}_{-0.12}$ 0.88/151
14 $3.00^{+0.60}_{-0.60}$ $0.870^{+0.120}_{-0.120}$ $4.870^{+0.270}_{-0.290}$ $16.50^{+2.40}_{-1.80}$ $1.75^{+0.13}_{-0.09}$ $0.89^{+0.17}_{-0.14}$ $54.10^{+3.60}_{-1.90}$ $6.90^{+3.40}_{-2.60}$ $0.53^{+0.35}_{-0.19}$ 1.03/151
15 $3.50^{+1.40}_{-1.10}$ $0.900^{+0.400}_{-0.230}$ $3.760^{+0.270}_{-0.380}$ $16.40^{+21.50}_{-1.90}$ $1.91^{+0.40}_{-0.18}$ $0.75^{+0.21}_{-0.15}$ $50.00^{+6.00}_{-6.00}$ $6.00^{+10.00}_{-4.00}$ $0.53^{+0.60}_{-0.30}$ 0.95/151
16 $5.70^{+0.70}_{-1.00}$ $1.370^{+0.130}_{-0.290}$ $3.530^{+0.180}_{-0.210}$ $42.00^{+22.00}_{-22.00}$ $2.26^{+0.16}_{-0.22}$ $0.85^{+0.10}_{-0.17}$ $54.00^{+8.00}_{-4.00}$ $\ge 5.4$ $0.77^{+0.23}_{-0.28}$ 0.75/151
17 $5.60^{+0.80}_{-1.10}$ $1.400^{+0.140}_{-0.250}$ $2.180^{+0.140}_{-0.150}$ $41.00^{+30.00}_{-10.00}$ $2.06^{+0.13}_{-0.18}$ $0.62^{+0.07}_{-0.09}$ $52.70^{+3.50}_{-2.50}$ $8.00^{+4.00}_{-4.00}$ $0.90^{+0.80}_{-0.40}$ 0.89/152
18 $6.30^{+0.80}_{-1.10}$ $1.620^{+0.100}_{-0.220}$ $1.760^{+0.120}_{-0.130}$ $\ge 49.5$ $2.00^{+0.12}_{-0.16}$ $0.46^{+0.06}_{-0.06}$ $49.50^{+2.80}_{-5.10}$ $5.90^{+3.30}_{-3.00}$ $0.60^{+0.80}_{-0.40}$ 1.04/152
19 $3.00^{+1.40}_{-1.50}$ $1.550^{+0.150}_{-0.190}$ $0.660^{+0.130}_{-0.130}$ $\ge 70.4$ $1.58^{+0.07}_{-0.06}$ $0.29^{+0.08}_{-0.07}$ $49.30^{+1.70}_{-3.10}$ $1.40^{+1.70}_{-1.00}$ $\ge 0.9$ 1.03/151
20 $4.40^{+1.50}_{-1.30}$ $1.750^{+0.150}_{-0.200}$ $0.540^{+0.100}_{-0.100}$ $\ge 56.2$ $1.55^{+0.16}_{-0.09}$ $0.14^{+0.05}_{-0.05}$ $45.00^{+6.00}_{-6.00}$ $0.40^{+0.80}_{-0.40}$ $\ge 0.0$ 0.87/152
21 $\le 2.1$ $1.150^{+0.290}_{-0.280}$ $0.144^{+0.069}_{-0.023}$ $\ge 31.7$ $1.40^{+0.05}_{-0.08}$ $0.16^{+0.02}_{-0.04}$ $44.00^{+4.00}_{-6.00}$ $4.50^{+2.80}_{-2.70}$ $\ge 0.7$ 0.95/152
22 $\le 1.6$ $1.300^{+0.400}_{-0.400}$ $0.200^{+0.100}_{-0.060}$ $\ge 54.0$ $1.39^{+0.07}_{-0.09}$ $0.16^{+0.04}_{-0.05}$ $48.30^{+2.10}_{-4.80}$ $1.80^{+3.50}_{-1.10}$ $\ge 0.8$ 0.76/152
23 $\le 1.0$ $0.820^{+0.230}_{-0.820}$ $0.140^{+0.050}_{-0.060}$ $\ge 27.8$ $1.33^{+0.06}_{-0.09}$ $0.15^{+0.07}_{-0.04}$ $37.00^{+4.00}_{-0.00}$ $\ge 11.1$ $1.50^{+0.80}_{-0.50}$ 0.86/152
24 $6.00^{+0.50}_{-0.40}$ $1.560^{+0.090}_{-0.080}$ $1.600^{+0.050}_{-0.050}$ $44.00^{+19.00}_{-9.00}$ $1.95^{+0.10}_{-0.08}$ $0.32^{+0.02}_{-0.02}$ $50.60^{+1.00}_{-1.80}$ $4.20^{+2.40}_{-1.30}$ $0.80^{+0.90}_{-0.40}$ 1.09/150
25 $5.60^{+0.60}_{-0.70}$ $0.840^{+0.110}_{-0.120}$ $2.940^{+0.170}_{-0.190}$ $14.80^{+1.80}_{-1.30}$ $1.82^{+0.11}_{-0.09}$ $0.66^{+0.11}_{-0.10}$ $51.20^{+1.00}_{-1.80}$ $1.00^{+0.80}_{-0.70}$ $\ge 0.4$ 0.74/152
26 $4.90^{+0.70}_{-0.70}$ $1.060^{+0.110}_{-0.110}$ $2.780^{+0.150}_{-0.160}$ $19.60^{+3.10}_{-2.20}$ $1.92^{+0.12}_{-0.09}$ $0.59^{+0.08}_{-0.08}$ $50.50^{+1.30}_{-2.10}$ $5.40^{+1.60}_{-1.30}$ $0.49^{+0.24}_{-0.17}$ 0.92/151
27 $4.60^{+0.40}_{-0.50}$ $0.930^{+0.080}_{-0.090}$ $4.160^{+0.150}_{-0.180}$ $17.60^{+1.60}_{-1.50}$ $1.77^{+0.11}_{-0.09}$ $0.58^{+0.10}_{-0.08}$ $53.00^{+1.70}_{-1.20}$ $5.70^{+2.20}_{-1.90}$ $0.51^{+0.36}_{-0.16}$ 1.10/151
28 $4.20^{+0.80}_{-0.50}$ $0.910^{+0.140}_{-0.090}$ $6.590^{+0.440}_{-0.280}$ $19.10^{+3.90}_{-1.70}$ $1.63^{+0.59}_{-0.14}$ $0.43^{+0.17}_{-0.25}$ $58.30^{+3.60}_{-2.70}$ $9.50^{+2.10}_{-1.90}$ $0.96^{+0.26}_{-0.17}$ 1.15/151
29 $4.60^{+0.70}_{-0.60}$ $0.860^{+0.100}_{-0.110}$ $7.700^{+0.500}_{-0.500}$ $18.40^{+2.00}_{-1.70}$ $1.59^{+0.35}_{-0.15}$ $0.44^{+0.28}_{-0.29}$ $57.10^{+1.40}_{-1.20}$ $10.00^{+1.10}_{-1.10}$ $0.81^{+0.09}_{-0.09}$ 1.00/151
30 $4.10^{+0.80}_{-0.50}$ $0.860^{+0.140}_{-0.090}$ $6.950^{+0.460}_{-0.300}$ $18.40^{+3.70}_{-1.60}$ $1.60^{+0.34}_{-0.10}$ $0.60^{+0.18}_{-0.26}$ $56.10^{+2.60}_{-1.70}$ $9.40^{+2.10}_{-1.50}$ $0.79^{+0.18}_{-0.11}$ 1.18/151
31 $3.80^{+0.70}_{-0.60}$ $0.740^{+0.150}_{-0.120}$ $6.900^{+0.600}_{-0.500}$ $16.60^{+2.70}_{-1.70}$ $1.45^{+0.09}_{-0.05}$ $0.93^{+0.27}_{-0.33}$ $57.30^{+3.50}_{-2.40}$ $9.90^{+2.30}_{-2.10}$ $0.69^{+0.18}_{-0.12}$ 1.00/151
32 $6.60^{+0.70}_{-0.50}$ $0.720^{+0.140}_{-0.110}$ $7.700^{+0.600}_{-0.400}$ $16.40^{+2.50}_{-1.50}$ $1.53^{+0.17}_{-0.07}$ $0.79^{+0.25}_{-0.33}$ $57.10^{+3.10}_{-2.10}$ $10.00^{+2.20}_{-1.80}$ $0.74^{+0.16}_{-0.10}$ 1.01/151
33 $4.50^{+0.50}_{-0.50}$ $0.450^{+0.120}_{-0.120}$ $7.500^{+0.500}_{-0.500}$ $13.20^{+1.20}_{-1.00}$ $1.46^{+0.03}_{-0.03}$ $1.50^{+0.40}_{-0.40}$ $51.20^{+2.70}_{-1.90}$ $6.10^{+2.00}_{-1.70}$ $0.47^{+0.21}_{-0.15}$ 1.22/151
34 $6.80^{+0.60}_{-0.70}$ $0.880^{+0.100}_{-0.130}$ $7.500^{+0.400}_{-0.500}$ $19.10^{+2.30}_{-2.30}$ $1.56^{+0.16}_{-0.10}$ $0.65^{+0.29}_{-0.22}$ $56.30^{+2.00}_{-1.60}$ $9.90^{+1.50}_{-1.60}$ $0.82^{+0.13}_{-0.13}$ 0.91/151
35 $4.30^{+0.80}_{-2.10}$ $1.020^{+0.140}_{-0.500}$ $7.800^{+0.500}_{-1.500}$ $23.00^{+7.00}_{-11.00}$ $2.00^{+0.40}_{-0.50}$ $0.90^{+0.40}_{-0.40}$ $57.00^{+8.00}_{-6.00}$ $\ge 4.2$ $0.80^{+0.40}_{-0.50}$ 0.89/151
36 $3.70^{+0.70}_{-0.70}$ $0.860^{+0.120}_{-0.140}$ $6.200^{+0.400}_{-0.500}$ $17.50^{+2.60}_{-2.10}$ $1.59^{+0.20}_{-0.10}$ $0.64^{+0.26}_{-0.23}$ $53.80^{+1.80}_{-1.40}$ $7.70^{+1.80}_{-1.70}$ $0.75^{+0.17}_{-0.15}$ 0.95/151
37 $3.30^{+0.60}_{-0.60}$ $0.760^{+0.120}_{-0.120}$ $5.900^{+0.400}_{-0.400}$ $16.00^{+2.10}_{-1.50}$ $1.58^{+0.10}_{-0.06}$ $0.93^{+0.22}_{-0.19}$ $54.40^{+3.20}_{-1.60}$ $8.20^{+2.90}_{-2.10}$ $0.53^{+0.13}_{-0.11}$ 1.03/151
38 $4.80^{+0.40}_{-0.50}$ $1.090^{+0.090}_{-0.100}$ $5.000^{+0.110}_{-0.170}$ $23.10^{+4.40}_{-3.00}$ $2.07^{+0.21}_{-0.21}$ $0.52^{+0.09}_{-0.06}$ $54.00^{+1.90}_{-1.30}$ $8.60^{+1.70}_{-1.40}$ $0.77^{+0.14}_{-0.13}$ 1.20/150
39 $5.00^{+0.70}_{-0.70}$ $1.040^{+0.150}_{-0.130}$ $3.830^{+0.170}_{-0.220}$ $19.60^{+5.60}_{-2.80}$ $1.91^{+0.24}_{-0.17}$ $0.54^{+0.11}_{-0.09}$ $50.20^{+2.00}_{-2.30}$ $6.30^{+2.60}_{-2.00}$ $0.48^{+0.24}_{-0.16}$ 0.85/151
40 $4.70^{+0.50}_{-0.50}$ $1.140^{+0.080}_{-0.090}$ $2.450^{+0.090}_{-0.100}$ $20.90^{+2.80}_{-2.20}$ $1.83^{+0.09}_{-0.08}$ $0.50^{+0.05}_{-0.05}$ $50.20^{+1.10}_{-2.10}$ $4.10^{+1.30}_{-1.00}$ $0.80^{+0.70}_{-0.40}$ 1.11/151
41 $\le 1.5$ $1.230^{+0.270}_{-0.160}$ $0.146^{+0.048}_{-0.017}$ $\ge 42.6$ $1.42^{+0.05}_{-0.06}$ $0.14^{+0.02}_{-0.03}$ $49.00^{+2.70}_{-6.90}$ $2.30^{+6.30}_{-1.50}$ $\ge 0.8$ 0.95/152
42 $\le 1.0$ $0.690^{+0.230}_{-0.700}$ $0.068^{+0.029}_{-0.027}$ $\ge 29.1$ $1.31^{+0.06}_{-0.07}$ $0.10^{+0.04}_{-0.01}$ $37.00^{+1.20}_{-0.00}$ $\ge 12.1$ $2.00^{+1.10}_{-0.70}$ 1.06/152
43 $\le 7.5$ $1.800^{+0.400}_{-0.700}$ $0.150^{+0.080}_{-0.080}$ $\ge 31.0$ $1.44^{+0.59}_{-0.14}$ $0.05^{+0.04}_{-0.04}$ $--$ $--$ $--$ 0.86/155
44 $\le 4.3$ $1.400^{+0.400}_{-0.400}$ $0.130^{+0.090}_{-0.050}$ $\ge 43.2$ $1.38^{+0.08}_{-0.10}$ $0.09^{+0.04}_{-0.05}$ $48.00^{+4.00}_{-9.00}$ $1.60^{+2.60}_{-1.30}$ $\ge 0.3$ 0.86/152
45 $5.40^{+0.60}_{-0.70}$ $1.140^{+0.120}_{-0.170}$ $6.440^{+0.190}_{-0.150}$ $25.00^{+8.00}_{-7.00}$ $2.70^{+0.40}_{-1.00}$ $0.30^{+0.19}_{-0.22}$ $55.30^{+2.50}_{-1.90}$ $9.80^{+2.00}_{-2.50}$ $0.82^{+0.12}_{-0.13}$ 0.95/151
46 $5.30^{+0.40}_{-0.40}$ $1.230^{+0.060}_{-0.090}$ $6.440^{+0.170}_{-0.150}$ $34.00^{+7.00}_{-8.00}$ $3.00^{+0.00}_{-0.21}$ $0.52^{+0.14}_{-0.19}$ $54.20^{+3.00}_{-1.70}$ $9.90^{+2.30}_{-1.80}$ $1.00^{+0.18}_{-0.17}$ 0.79/150
47 $4.00^{+1.10}_{-1.00}$ $0.830^{+0.200}_{-0.200}$ $5.500^{+0.600}_{-0.600}$ $16.50^{+4.20}_{-2.50}$ $1.54^{+0.43}_{-0.12}$ $0.60^{+0.40}_{-0.40}$ $55.20^{+6.50}_{-2.40}$ $7.30^{+4.10}_{-2.80}$ $0.66^{+0.36}_{-0.18}$ 1.06/152
48 $5.00^{+1.00}_{-1.20}$ $1.000^{+0.190}_{-0.260}$ $5.800^{+0.600}_{-0.800}$ $20.00^{+7.00}_{-5.00}$ $1.50^{+0.47}_{-0.14}$ $0.49^{+0.44}_{-0.28}$ $54.60^{+3.40}_{-2.20}$ $8.00^{+4.00}_{-4.00}$ $0.68^{+0.74}_{-0.12}$ 0.99/151
49 $12.50^{+0.70}_{-0.70}$ $0.860^{+0.130}_{-0.140}$ $6.100^{+0.500}_{-0.500}$ $17.80^{+2.60}_{-2.10}$ $1.46^{+0.15}_{-0.08}$ $0.57^{+0.26}_{-0.25}$ $55.90^{+2.60}_{-1.90}$ $8.60^{+2.00}_{-2.30}$ $0.67^{+0.15}_{-0.14}$ 1.15/151
50 $9.90^{+0.60}_{-0.70}$ $0.970^{+0.080}_{-0.120}$ $7.000^{+0.400}_{-0.500}$ $20.50^{+2.30}_{-2.40}$ $1.78^{+0.46}_{-0.24}$ $0.33^{+0.26}_{-0.18}$ $55.70^{+1.20}_{-1.00}$ $9.60^{+1.00}_{-1.10}$ $0.77^{+0.09}_{-0.10}$ 0.90/151
51 $9.30^{+1.10}_{-1.00}$ $0.820^{+0.200}_{-0.200}$ $5.900^{+0.700}_{-0.600}$ $17.60^{+4.30}_{-2.80}$ $1.58^{+0.19}_{-0.09}$ $1.00^{+0.40}_{-0.40}$ $55.90^{+6.00}_{-2.60}$ $7.90^{+3.70}_{-2.70}$ $0.68^{+0.26}_{-0.16}$ 0.96/151
52 $11.40^{+0.70}_{-0.60}$ $0.830^{+0.100}_{-0.090}$ $6.900^{+0.500}_{-0.400}$ $17.70^{+1.80}_{-1.30}$ $1.63^{+0.16}_{-0.09}$ $0.67^{+0.21}_{-0.26}$ $57.00^{+1.30}_{-1.10}$ $10.10^{+1.00}_{-0.90}$ $0.70^{+0.08}_{-0.07}$ 1.15/151
54 $15.30^{+0.90}_{-0.80}$ $0.660^{+0.120}_{-0.160}$ $10.800^{+0.800}_{-0.900}$ $16.00^{+3.30}_{-2.10}$ $1.70^{+0.46}_{-0.17}$ $0.90^{+0.50}_{-0.50}$ $62.00^{+6.00}_{-5.00}$ $12.00^{+4.00}_{-4.00}$ $0.90^{+0.35}_{-0.24}$ 1.20/151
55 $14.40^{+0.80}_{-1.00}$ $0.920^{+0.150}_{-0.190}$ $8.500^{+0.400}_{-0.800}$ $19.00^{+6.00}_{-4.00}$ $2.00^{+1.00}_{-1.40}$ $0.24^{+0.41}_{-0.25}$ $59.00^{+7.00}_{-4.00}$ $11.70^{+3.80}_{-2.80}$ $0.84^{+0.36}_{-0.22}$ 1.04/151
56 $15.00^{+1.00}_{-1.00}$ $0.970^{+0.160}_{-0.180}$ $6.600^{+0.500}_{-0.600}$ $20.00^{+7.00}_{-4.00}$ $1.78^{+0.76}_{-0.30}$ $0.39^{+0.32}_{-0.23}$ $61.00^{+9.00}_{-5.00}$ $\ge 8.4$ $1.05^{+0.95}_{-0.28}$ 0.92/151
57 $12.70^{+0.60}_{-0.60}$ $0.720^{+0.100}_{-0.100}$ $8.100^{+0.500}_{-0.500}$ $16.40^{+1.40}_{-1.20}$ $1.64^{+0.12}_{-0.09}$ $0.86^{+0.28}_{-0.29}$ $57.60^{+1.30}_{-1.10}$ $10.50^{+0.90}_{-1.00}$ $0.73^{+0.08}_{-0.08}$ 1.08/151
58 $7.70^{+0.50}_{-0.50}$ $0.480^{+0.110}_{-0.110}$ $8.500^{+0.500}_{-0.500}$ $13.60^{+1.10}_{-1.00}$ $1.51^{+0.04}_{-0.04}$ $1.50^{+0.40}_{-0.40}$ $55.90^{+2.00}_{-1.30}$ $7.70^{+1.50}_{-1.30}$ $0.77^{+0.15}_{-0.14}$ 1.21/151
59 $9.10^{+0.60}_{-0.70}$ $0.900^{+0.100}_{-0.130}$ $8.200^{+0.400}_{-0.500}$ $20.00^{+4.00}_{-4.00}$ $1.96^{+0.39}_{-0.29}$ $0.60^{+0.23}_{-0.24}$ $56.70^{+3.60}_{-2.30}$ $11.80^{+2.20}_{-2.10}$ $0.85^{+0.15}_{-0.16}$ 0.99/151
60 $2.10^{+0.80}_{-0.70}$ $0.310^{+0.180}_{-0.170}$ $8.200^{+0.900}_{-0.800}$ $12.20^{+1.50}_{-1.10}$ $1.54^{+0.06}_{-0.06}$ $2.00^{+0.60}_{-0.60}$ $57.00^{+5.00}_{-4.00}$ $8.00^{+4.00}_{-4.00}$ $0.58^{+0.19}_{-0.14}$ 0.81/151
61 $3.40^{+0.90}_{-0.80}$ $0.690^{+0.160}_{-0.150}$ $8.500^{+0.800}_{-0.700}$ $15.30^{+2.50}_{-1.70}$ $1.65^{+0.30}_{-0.12}$ $0.90^{+0.50}_{-0.50}$ $56.10^{+2.60}_{-1.90}$ $9.20^{+2.10}_{-1.90}$ $0.64^{+0.13}_{-0.11}$ 0.90/151
63 $3.30^{+0.60}_{-1.30}$ $0.740^{+0.130}_{-0.300}$ $11.600^{+0.500}_{-1.200}$ $17.30^{+2.70}_{-4.40}$ $2.50^{+0.60}_{-1.00}$ $0.60^{+0.70}_{-0.40}$ $57.60^{+4.20}_{-2.60}$ $12.20^{+2.70}_{-2.90}$ $0.78^{+0.18}_{-0.28}$ 0.97/150
64 $3.40^{+0.80}_{-0.70}$ $0.750^{+0.150}_{-0.150}$ $8.300^{+0.500}_{-0.600}$ $16.50^{+3.50}_{-2.20}$ $1.83^{+0.36}_{-0.18}$ $0.88^{+0.31}_{-0.28}$ $55.30^{+3.20}_{-2.20}$ $9.90^{+2.50}_{-2.20}$ $0.71^{+0.21}_{-0.17}$ 0.99/151
: Best-fit Cutoffpl Continuum Spectral Parameters of GX 304$-$1\[tab:best\_fit\_cutoffpl\_cont\]
\# $N_\mathrm{H}^a$ Index$^b$ Flux$^c$ Efold$^d$ kT$^e$ Flux$_\mathrm{BB}$$^f$ Ecyc$^g$ Width$^g$ Depth$^g$ $\chi^2$/dof
---- ------------------------ --------------------------- --------------------------- --------------------------- ------------------------ ------------------------ -------------------------- ------------------------ ------------------------ --------------
65 $3.30^{+0.60}_{-0.90}$ $0.810^{+0.110}_{-0.230}$ $7.500^{+0.400}_{-0.600}$ $19.00^{+4.00}_{-5.00}$ $1.86^{+0.21}_{-0.24}$ $0.98^{+0.25}_{-0.23}$ $58.00^{+6.00}_{-6.00}$ $\ge 7.6$ $0.79^{+0.29}_{-0.33}$ 1.27/151
66 $2.90^{+0.50}_{-0.50}$ $0.730^{+0.100}_{-0.090}$ $6.770^{+0.300}_{-0.310}$ $15.40^{+1.50}_{-1.20}$ $1.74^{+0.13}_{-0.09}$ $0.91^{+0.19}_{-0.18}$ $54.90^{+2.40}_{-1.40}$ $7.30^{+2.20}_{-1.90}$ $0.64^{+0.17}_{-0.13}$ 1.13/151
67 $3.90^{+0.50}_{-0.50}$ $0.920^{+0.100}_{-0.090}$ $3.970^{+0.150}_{-0.160}$ $17.50^{+2.20}_{-1.60}$ $1.84^{+0.12}_{-0.09}$ $0.70^{+0.09}_{-0.08}$ $51.80^{+1.10}_{-1.10}$ $6.00^{+1.60}_{-1.40}$ $0.67^{+0.25}_{-0.16}$ 1.32/151
68 $5.50^{+0.70}_{-0.70}$ $1.400^{+0.100}_{-0.160}$ $3.710^{+0.170}_{-0.190}$ $49.00^{+25.00}_{-19.00}$ $2.26^{+0.13}_{-0.15}$ $0.87^{+0.10}_{-0.12}$ $51.70^{+2.50}_{-1.90}$ $9.00^{+2.20}_{-2.10}$ $0.94^{+0.23}_{-0.21}$ 0.71/152
69 $4.80^{+1.10}_{-0.80}$ $1.170^{+0.310}_{-0.170}$ $2.640^{+0.140}_{-0.180}$ $19.30^{+23.80}_{-2.60}$ $1.85^{+0.31}_{-0.18}$ $0.42^{+0.09}_{-0.07}$ $49.00^{+8.00}_{-7.00}$ $\ge 1.5$ $0.33^{+5.69}_{-0.20}$ 0.87/151
70 $5.40^{+1.30}_{-1.70}$ $1.650^{+0.120}_{-0.330}$ $1.510^{+0.200}_{-0.240}$ $\ge 29.8$ $1.83^{+0.15}_{-0.17}$ $0.43^{+0.11}_{-0.10}$ $52.00^{+6.00}_{-7.00}$ $6.00^{+6.00}_{-4.00}$ $0.90^{+2.70}_{-0.60}$ 0.88/151
71 $3.90^{+0.90}_{-1.50}$ $1.600^{+0.090}_{-0.250}$ $0.930^{+0.110}_{-0.150}$ $\ge 80.1$ $1.70^{+0.06}_{-0.09}$ $0.37^{+0.08}_{-0.06}$ $43.10^{+3.60}_{-2.60}$ $6.50^{+8.70}_{-1.90}$ $0.51^{+0.28}_{-0.18}$ 0.86/151
72 $\le 2.2$ $0.800^{+1.100}_{-0.800}$ $0.140^{+0.110}_{-0.060}$ $\ge 22.7$ $1.38^{+0.10}_{-0.14}$ $0.20^{+0.11}_{-0.14}$ $42.00^{+12.00}_{-6.00}$ $\ge 2.8$ $\ge 0.9$ 0.87/152
\
$^a$ Column density in 10$^{22}$ cm$^{-2}$\
$^b$ Power law photon index\
$^c$ Unabsorbed power law 2$-$10 keV flux in 10$^{-9}$ ergs cm$^{-2}$ s$^{-1}$\
$^d$ Efold is folding energy in keV\
$^e$ Blackbody temperature in keV\
$^f$ Blackbody flux in 10$^{-9}$ ergs cm$^{-2}$ s$^{-1}$\
$^g$ Ecyc is CRSF energy; Width is CRSF width in keV; Depth is CRSF depth\
\# iron$^a$ iron$^b$ 10.5keV$^c$ 3.88keV$^d$ 30keV$^e$ 39keV$^f$ 53keV$^g$ 66keV$^h$
---- ------------------- ------------------ ---------------------- ---------------------- --------------------- --------------------- --------------------- ---------------------
1 $3^{+2}_{-2}$ $19^{+15}_{-14}$ $--$ $--$ $1.5^{+0.6}_{-0.7}$ $1.0^{+0.6}_{-0.6}$ $\le 1.2$ $1.9^{+0.6}_{-0.6}$
2 $3^{+2}_{-2}$ $23^{+16}_{-16}$ $--$ $--$ $2.0^{+0.7}_{-1.0}$ $0.7^{+0.7}_{-0.5}$ $1.1^{+0.7}_{-0.7}$ $2.8^{+0.7}_{-0.9}$
3 $9^{+2}_{-2}$ $36^{+10}_{-10}$ $-5.9^{+2.0}_{-2.0}$ $-0.7^{+0.4}_{-0.4}$ $\le 1.6$ $1.2^{+0.6}_{-0.6}$ $\le 0.8$ $\le 1.1$
4 $88^{+7}_{-7}$ $76^{+6}_{-6}$ $--$ $-3.8^{+1.0}_{-1.0}$ $2.7^{+1.5}_{-1.1}$ $2.3^{+1.0}_{-0.9}$ $\le 0.8$ $2.7^{+0.8}_{-0.8}$
5 $83^{+12}_{-12}$ $67^{+10}_{-10}$ $--$ $-4.3^{+1.8}_{-1.8}$ $2.5^{+0.9}_{-0.8}$ $2.1^{+0.7}_{-0.7}$ $\le 0.7$ $2.0^{+0.7}_{-0.8}$
6 $85^{+5}_{-5}$ $66^{+4}_{-4}$ $--$ $-3.7^{+0.7}_{-0.7}$ $2.1^{+0.9}_{-0.9}$ $2.0^{+0.8}_{-0.7}$ $\le 0.8$ $1.7^{+0.8}_{-0.9}$
7 $102^{+13}_{-13}$ $72^{+10}_{-10}$ $--$ $-4.0^{+1.9}_{-1.9}$ $3.1^{+1.0}_{-1.0}$ $2.6^{+0.7}_{-0.7}$ $0.6^{+0.5}_{-0.5}$ $3.2^{+0.7}_{-0.7}$
8 $92^{+14}_{-14}$ $66^{+10}_{-10}$ $--$ $-3.9^{+2.1}_{-2.1}$ $3.0^{+2.2}_{-2.2}$ $2.8^{+1.7}_{-1.7}$ $\le 1.7$ $2.6^{+1.5}_{-1.7}$
9 $109^{+6}_{-6}$ $73^{+4}_{-4}$ $--$ $-2.9^{+1.0}_{-1.0}$ $2.4^{+0.9}_{-0.9}$ $2.1^{+0.8}_{-0.8}$ $\le 0.3$ $1.1^{+0.7}_{-0.7}$
11 $94^{+6}_{-6}$ $68^{+5}_{-5}$ $--$ $-4.4^{+0.9}_{-0.9}$ $2.5^{+0.8}_{-0.8}$ $1.6^{+0.8}_{-0.7}$ $\le 0.8$ $1.0^{+0.7}_{-0.6}$
12 $35^{+9}_{-9}$ $35^{+9}_{-9}$ $--$ $-2.7^{+1.4}_{-1.4}$ $3.8^{+2.1}_{-1.5}$ $2.1^{+1.2}_{-1.2}$ $\le 1.2$ $2.6^{+1.3}_{-1.1}$
13 $47^{+6}_{-6}$ $56^{+7}_{-7}$ $--$ $-1.3^{+1.0}_{-1.0}$ $2.7^{+0.8}_{-0.8}$ $1.1^{+1.0}_{-0.8}$ $\le 0.7$ $1.7^{+0.7}_{-0.6}$
14 $24^{+4}_{-4}$ $34^{+6}_{-6}$ $--$ $-1.7^{+0.7}_{-0.7}$ $1.7^{+0.6}_{-0.6}$ $1.0^{+0.6}_{-0.6}$ $0.6^{+0.7}_{-0.6}$ $2.2^{+0.6}_{-0.6}$
15 $20^{+6}_{-6}$ $36^{+11}_{-11}$ $--$ $-1.7^{+1.0}_{-1.0}$ $1.4^{+4.2}_{-0.8}$ $1.6^{+1.6}_{-0.9}$ $\le 1.2$ $1.2^{+2.7}_{-0.7}$
16 $19^{+6}_{-6}$ $37^{+11}_{-11}$ $--$ $-1.2^{+0.9}_{-0.9}$ $3.7^{+2.0}_{-2.6}$ $1.3^{+1.0}_{-1.3}$ $\le 1.1$ $3.5^{+1.3}_{-2.0}$
17 $9^{+4}_{-4}$ $29^{+12}_{-12}$ $--$ $--$ $1.7^{+1.4}_{-1.2}$ $0.9^{+0.8}_{-0.8}$ $0.8^{+0.6}_{-0.6}$ $2.2^{+1.1}_{-1.0}$
18 $7^{+4}_{-4}$ $26^{+13}_{-13}$ $--$ $--$ $2.1^{+1.0}_{-1.1}$ $0.8^{+0.7}_{-0.6}$ $\le 1.1$ $1.9^{+0.9}_{-0.9}$
19 $4^{+2}_{-2}$ $32^{+22}_{-22}$ $--$ $-0.6^{+0.5}_{-0.5}$ $\le 1.3$ $0.6^{+0.4}_{-0.5}$ $1.0^{+0.6}_{-0.9}$ $1.8^{+0.6}_{-0.6}$
20 $3^{+2}_{-2}$ $38^{+23}_{-23}$ $--$ $--$ $0.6^{+0.6}_{-0.6}$ $0.5^{+1.1}_{-0.3}$ $0.4^{+0.5}_{-0.3}$ $1.8^{+0.5}_{-0.5}$
21 $\le 3$ $\le 75$ $--$ $--$ $1.6^{+0.8}_{-0.9}$ $1.3^{+0.4}_{-1.3}$ $1.0^{+0.5}_{-1.0}$ $1.7^{+0.7}_{-0.7}$
22 $\le 2$ $\le 38$ $--$ $--$ $0.8^{+0.8}_{-0.7}$ $\le 1.0$ $0.7^{+0.5}_{-0.5}$ $1.7^{+0.7}_{-0.7}$
23 $\le 3$ $\le 74$ $--$ $--$ $5.0^{+2.4}_{-2.1}$ $1.3^{+0.9}_{-0.7}$ $0.9^{+0.6}_{-0.6}$ $3.9^{+1.2}_{-1.3}$
24 $6^{+1}_{-1}$ $28^{+5}_{-5}$ $-2.0^{+0.9}_{-0.9}$ $-0.5^{+0.2}_{-0.2}$ $1.5^{+0.4}_{-0.2}$ $0.6^{+0.3}_{-0.2}$ $0.5^{+0.4}_{-0.4}$ $1.6^{+0.3}_{-0.2}$
25 $14^{+4}_{-4}$ $31^{+8}_{-8}$ $--$ $--$ $1.1^{+0.6}_{-0.6}$ $1.0^{+0.5}_{-0.5}$ $1.3^{+0.7}_{-0.9}$ $1.6^{+0.5}_{-0.5}$
26 $15^{+4}_{-4}$ $37^{+10}_{-10}$ $--$ $-1.0^{+0.6}_{-0.6}$ $1.0^{+0.3}_{-0.3}$ $0.8^{+0.3}_{-0.3}$ $0.4^{+0.4}_{-0.4}$ $1.4^{+0.2}_{-0.2}$
27 $24^{+2}_{-2}$ $41^{+5}_{-5}$ $--$ $-1.0^{+0.4}_{-0.4}$ $1.4^{+0.4}_{-0.4}$ $0.8^{+0.4}_{-0.4}$ $\le 0.8$ $1.1^{+0.4}_{-0.4}$
28 $50^{+6}_{-5}$ $59^{+7}_{-6}$ $--$ $-1.9^{+0.8}_{-1.0}$ $2.3^{+0.9}_{-0.7}$ $1.1^{+0.7}_{-0.7}$ $\le 1.0$ $3.0^{+1.0}_{-0.8}$
29 $59^{+10}_{-9}$ $60^{+10}_{-10}$ $--$ $-3.4^{+1.4}_{-1.5}$ $2.9^{+0.7}_{-0.6}$ $1.4^{+0.4}_{-0.4}$ $\le 0.5$ $2.4^{+0.6}_{-0.6}$
30 $47^{+5}_{-5}$ $52^{+6}_{-5}$ $--$ $-2.5^{+0.7}_{-0.9}$ $2.6^{+1.1}_{-0.7}$ $1.3^{+0.8}_{-0.6}$ $\le 0.7$ $2.0^{+1.1}_{-0.7}$
31 $54^{+5}_{-5}$ $56^{+5}_{-6}$ $--$ $-3.4^{+0.8}_{-0.8}$ $2.2^{+0.9}_{-0.7}$ $1.8^{+0.7}_{-0.7}$ $\le 0.6$ $2.4^{+0.8}_{-0.8}$
32 $58^{+6}_{-5}$ $55^{+5}_{-5}$ $--$ $-2.4^{+0.7}_{-0.8}$ $3.0^{+1.0}_{-0.7}$ $1.9^{+0.7}_{-0.7}$ $\le 0.4$ $2.6^{+0.8}_{-0.7}$
33 $65^{+5}_{-6}$ $56^{+5}_{-5}$ $--$ $-2.4^{+0.8}_{-0.8}$ $3.6^{+1.8}_{-1.8}$ $3.4^{+1.3}_{-1.3}$ $\le 1.9$ $\le 1.7$
34 $56^{+5}_{-5}$ $57^{+5}_{-6}$ $--$ $-2.7^{+0.8}_{-0.8}$ $3.3^{+1.0}_{-0.9}$ $1.7^{+0.7}_{-0.7}$ $0.5^{+0.5}_{-0.5}$ $2.3^{+0.8}_{-0.9}$
35 $59^{+11}_{-11}$ $56^{+10}_{-11}$ $--$ $-3.9^{+1.7}_{-1.6}$ $4.0^{+4.0}_{-5.0}$ $\le 3.6$ $\le 1.6$ $3.7^{+1.9}_{-3.2}$
36 $42^{+5}_{-5}$ $50^{+6}_{-6}$ $--$ $-2.4^{+0.8}_{-0.8}$ $1.9^{+0.9}_{-0.8}$ $1.3^{+0.8}_{-0.7}$ $0.6^{+0.6}_{-0.6}$ $1.6^{+0.8}_{-0.7}$
37 $37^{+4}_{-4}$ $43^{+5}_{-4}$ $--$ $-2.0^{+0.6}_{-0.6}$ $2.3^{+0.7}_{-0.6}$ $1.3^{+0.5}_{-0.5}$ $\le 0.6$ $1.3^{+0.7}_{-0.5}$
38 $28^{+4}_{-4}$ $42^{+5}_{-5}$ $-6.1^{+2.9}_{-2.9}$ $-1.4^{+0.6}_{-0.5}$ $2.5^{+1.0}_{-0.7}$ $1.2^{+0.5}_{-0.5}$ $0.6^{+0.4}_{-0.4}$ $2.2^{+0.8}_{-0.6}$
39 $20^{+4}_{-4}$ $38^{+7}_{-7}$ $--$ $-1.1^{+0.6}_{-0.6}$ $1.2^{+0.9}_{-0.6}$ $1.3^{+0.6}_{-0.6}$ $\le 0.7$ $1.2^{+0.6}_{-0.5}$
40 $12^{+2}_{-2}$ $35^{+5}_{-5}$ $--$ $-0.3^{+0.3}_{-0.3}$ $1.3^{+0.3}_{-0.3}$ $0.8^{+0.3}_{-0.2}$ $0.7^{+0.5}_{-0.5}$ $1.3^{+0.2}_{-0.2}$
41 $1^{+1}_{-1}$ $40^{+40}_{-40}$ $--$ $--$ $1.3^{+0.5}_{-0.6}$ $0.5^{+0.9}_{-0.3}$ $0.7^{+0.4}_{-0.6}$ $2.0^{+0.5}_{-0.5}$
42 $\le 1$ $\le 30$ $--$ $--$ $3.5^{+1.6}_{-1.4}$ $1.3^{+0.5}_{-0.5}$ $0.8^{+0.4}_{-0.4}$ $2.9^{+0.8}_{-0.9}$
43 $\le 1$ $\le 47$ $--$ $--$ $\le 0.9$ $0.7^{+0.3}_{-0.3}$ $0.5^{+0.2}_{-0.3}$ $1.6^{+0.5}_{-0.5}$
44 $\le 2$ $\le 78$ $--$ $--$ $1.3^{+0.5}_{-0.5}$ $0.4^{+1.0}_{-0.3}$ $0.5^{+0.4}_{-0.4}$ $1.6^{+0.5}_{-0.5}$
45 $40^{+7}_{-7}$ $51^{+9}_{-9}$ $--$ $-3.8^{+1.1}_{-1.1}$ $2.9^{+2.1}_{-1.6}$ $1.3^{+1.1}_{-1.0}$ $\le 0.4$ $3.0^{+1.4}_{-1.3}$
46 $49^{+8}_{-8}$ $61^{+10}_{-10}$ $-13^{+7}_{-7}$ $-3.2^{+1.2}_{-1.2}$ $4.3^{+2.1}_{-2.3}$ $2.9^{+1.3}_{-1.3}$ $\le 1.3$ $2.6^{+1.3}_{-1.5}$
47 $36^{+7}_{-7}$ $49^{+10}_{-10}$ $--$ $--$ $2.1^{+1.0}_{-1.0}$ $1.0^{+1.0}_{-1.0}$ $\le 0.8$ $0.9^{+1.2}_{-0.9}$
48 $34^{+6}_{-6}$ $46^{+8}_{-8}$ $--$ $-1.9^{+0.9}_{-1.0}$ $2.3^{+1.5}_{-0.9}$ $\le 1.8$ $\le 1.4$ $2.0^{+1.3}_{-1.1}$
49 $39^{+5}_{-5}$ $48^{+6}_{-6}$ $--$ $-1.8^{+0.6}_{-0.6}$ $2.0^{+0.7}_{-0.7}$ $1.2^{+0.6}_{-0.6}$ $\le 0.8$ $1.9^{+0.7}_{-0.7}$
50 $55^{+9}_{-9}$ $63^{+10}_{-10}$ $--$ $-2.7^{+1.1}_{-1.0}$ $2.9^{+0.7}_{-0.7}$ $1.0^{+0.5}_{-0.5}$ $\le 0.5$ $2.5^{+0.6}_{-0.6}$
51 $39^{+8}_{-8}$ $46^{+10}_{-10}$ $--$ $-1.7^{+1.1}_{-1.1}$ $1.9^{+1.2}_{-1.1}$ $\le 1.8$ $\le 0.7$ $2.0^{+1.3}_{-1.1}$
52 $52^{+9}_{-8}$ $57^{+10}_{-9}$ $--$ $-2.1^{+0.9}_{-1.0}$ $2.3^{+0.5}_{-0.4}$ $1.4^{+0.3}_{-0.3}$ $\le 0.3$ $1.9^{+0.4}_{-0.4}$
54 $105^{+13}_{-11}$ $72^{+8}_{-8}$ $--$ $-3.2^{+1.2}_{-1.3}$ $1.7^{+1.9}_{-1.6}$ $1.4^{+1.4}_{-1.3}$ $\le 0.7$ $2.3^{+1.0}_{-1.1}$
55 $73^{+12}_{-12}$ $69^{+11}_{-11}$ $--$ $-2.1^{+1.4}_{-1.3}$ $\le 2.5$ $\le 2.7$ $\le 1.4$ $1.3^{+1.5}_{-1.2}$
56 $43^{+9}_{-9}$ $51^{+11}_{-11}$ $--$ $-2.3^{+1.0}_{-1.0}$ $2.3^{+1.9}_{-1.7}$ $1.5^{+1.1}_{-1.0}$ $\le 0.9$ $3.0^{+1.2}_{-1.0}$
57 $65^{+10}_{-10}$ $58^{+10}_{-9}$ $--$ $-2.3^{+1.0}_{-1.0}$ $2.4^{+0.5}_{-0.4}$ $1.4^{+0.3}_{-0.3}$ $\le 0.4$ $2.2^{+0.4}_{-0.4}$
58 $82^{+6}_{-6}$ $64^{+5}_{-5}$ $--$ $-3.1^{+0.8}_{-0.8}$ $1.9^{+1.0}_{-1.0}$ $\le 1.4$ $\le 1.4$ $\le 0.6$
59 $64^{+9}_{-8}$ $60^{+8}_{-8}$ $--$ $-2.6^{+1.1}_{-1.2}$ $\le 3.3$ $1.8^{+1.0}_{-1.1}$ $\le 0.5$ $2.0^{+0.9}_{-1.0}$
60 $100^{+14}_{-14}$ $75^{+11}_{-11}$ $--$ $-3.0^{+2.0}_{-2.1}$ $3.4^{+1.1}_{-1.1}$ $1.7^{+1.0}_{-1.0}$ $\le 1.4$ $1.2^{+1.0}_{-1.0}$
61 $70^{+12}_{-12}$ $60^{+10}_{-10}$ $--$ $-2.6^{+1.7}_{-1.9}$ $2.2^{+0.9}_{-0.8}$ $1.6^{+0.8}_{-0.8}$ $\le 0.3$ $1.4^{+0.9}_{-0.8}$
63 $116^{+14}_{-14}$ $77^{+9}_{-10}$ $-31^{+14}_{-13}$ $-5.1^{+2.1}_{-2.0}$ $5.7^{+3.2}_{-2.4}$ $3.0^{+1.7}_{-1.7}$ $\le 0.8$ $2.8^{+1.9}_{-2.0}$
64 $67^{+9}_{-8}$ $59^{+8}_{-7}$ $--$ $-1.7^{+1.3}_{-1.4}$ $2.6^{+1.7}_{-1.2}$ $2.3^{+1.1}_{-1.0}$ $\le 1.0$ $1.8^{+1.3}_{-1.0}$
: Best-fit Cutoffpl Spectral Lines of GX 304$-$1\[tab:best\_fit\_cutoffpl\_lines\]
\# iron$^a$ iron$^b$ 10.5keV$^c$ 3.88keV$^d$ 30keV$^e$ 39keV$^f$ 53keV$^g$ 66keV$^h$
---- ---------------- ------------------ ------------- ---------------------- --------------------- --------------------- --------------------- ---------------------
65 $56^{+7}_{-7}$ $54^{+7}_{-7}$ $--$ $-2.1^{+1.1}_{-1.1}$ $\le 3.2$ $1.5^{+1.0}_{-1.1}$ $\le 0.6$ $2.0^{+0.9}_{-1.4}$
66 $53^{+5}_{-5}$ $55^{+5}_{-5}$ $--$ $-1.9^{+0.7}_{-0.7}$ $2.2^{+0.6}_{-0.6}$ $1.0^{+0.6}_{-0.6}$ $\le 1.0$ $1.7^{+0.7}_{-0.6}$
67 $25^{+3}_{-2}$ $43^{+5}_{-5}$ $--$ $-0.9^{+0.4}_{-0.5}$ $1.0^{+0.4}_{-0.4}$ $1.0^{+0.4}_{-0.4}$ $0.6^{+0.5}_{-0.4}$ $1.3^{+0.4}_{-0.4}$
68 $26^{+6}_{-6}$ $48^{+12}_{-12}$ $--$ $--$ $3.0^{+2.1}_{-2.3}$ $1.9^{+1.1}_{-1.2}$ $1.0^{+0.9}_{-0.9}$ $3.5^{+1.6}_{-1.7}$
69 $12^{+3}_{-3}$ $33^{+8}_{-8}$ $--$ $-0.7^{+0.5}_{-0.5}$ $1.1^{+2.4}_{-0.5}$ $1.1^{+0.9}_{-0.6}$ $\le 1.2$ $1.5^{+1.1}_{-0.4}$
70 $\le 6$ $\le 30$ $--$ $-1.0^{+0.8}_{-0.8}$ $2.1^{+0.9}_{-1.4}$ $\le 1.5$ $\le 1.5$ $2.4^{+1.1}_{-1.3}$
71 $3^{+2}_{-2}$ $23^{+14}_{-13}$ $--$ $-0.5^{+0.4}_{-0.4}$ $0.9^{+1.9}_{-0.7}$ $1.3^{+0.6}_{-0.6}$ $\le 0.3$ $1.5^{+0.4}_{-0.4}$
72 $\le 5$ $\le 98$ $--$ $--$ $\le 7.0$ $1.6^{+1.0}_{-1.0}$ $1.0^{+0.9}_{-0.9}$ $2.8^{+1.6}_{-2.1}$
\
$^a$ iron line flux in 10$^{-4}$ photons cm$^{-2}$ s$^{-1}$\
$^b$ iron line equivalent width in eV\
$^c$ 10.5 keV negative line flux in units of 10$^{-3}$ photons cm$^{-2}$ s$^{-1}$\
$^d$ 3.88 keV line flux in units of 10$^{-3}$ photons cm$^{-2}$ s$^{-1}$\
$^e$ 30.17 keV line flux in units of 10$^{-3}$ photons cm$^{-2}$ s$^{-1}$\
$^f$ 39.04 keV line flux in units of 10$^{-3}$ photons cm$^{-2}$ s$^{-1}$\
$^g$ 53.00 keV line flux in units of 10$^{-3}$ photons cm$^{-2}$ s$^{-1}$\
$^h$ 66.64 keV line flux in units of 10$^{-3}$ photons cm$^{-2}$ s$^{-1}$
![The continuum parameters for the `highecut` model plotted versus 2$-$10 keV unabsorbed flux in units of ergs cm$^{-2}$ s$^{-1}$. Data from 2010 March/April are in green, 2010 August are in black, 2010 December are in blue, and 2011 May are in red.\[fig:highecut\_continuum\_flux\]](f18.eps){width="7.0in"}
![The continuum parameters for the `cutoffpl` model plotted versus 2$-$10 keV unabsorbed flux in units of ergs cm$^{-2}$ s$^{-1}$. The blackbody flux (Flux$_{BB}$) is in units of L$_{39}$/D$^2$, where L$_{39}$ is the flux in units of 10$^{39}$ ergs s$^{-1}$, and D is the distance to the source in units of 10 kpc. Data from 2010 March/April are in green, 2010 August are in black, 2010 December are in blue, and 2011 May are in red.\[fig:cutoffpl\_continuum\_flux\]](f19.eps){width="7.0in"}
![The various line fluxes in units of 10$^{-3}$ cm$^{-2}$ s$^{-1}$ for the `highecut` model plotted versus 2$-$10 keV unabsorbed flux in units of ergs cm$^{-2}$ s$^{-1}$. Data from 2010 March/April are in green, 2010 August are in black, 2010 December are in blue, and 2011 May are in red.\[fig:highecut\_line\_flux\]](f20.eps){width="6.6in"}
![The various line fluxes in units of 10$^{-3}$ cm$^{-2}$ s$^{-1}$ for the `cutoffpl` model plotted versus 2$-$10 keV unabsorbed flux in units of ergs cm$^{-2}$ s$^{-1}$. Data from 2010 March/April are in green, 2010 August are in black, 2010 December are in blue, and 2011 May are in red.\[fig:cutoffpl\_line\_flux\]](f21.eps){width="6.6in"}
Correlations between the fitted cyclotron line parameters and background lines at 53 keV and 66 keV, as well as versus the cutoff energy and folding energy of the continuum, are displayed in Fig. \[obs9\] and \[obs39\] for high and low flux observations \#9 and \#39, respectively. In addition the correlation between the folding and cutoff energies is shown for those observations.
![Contours of the cyclotron line fitted parameters versus the background lines at 66 keV, 53 keV, and the continuum parameters Ecut and Efold, plus the contours for Ecut versus Efold for observation 9. The red, green, and blue contours represent the 68%, 90%, and 99% significance levels.\[obs9\]](obs9_contours.eps){width="6.in"}
![Contours of the cyclotron line fitted parameters versus the background lines at 66 keV, 53 keV, and the continuum parameters Ecut and Efold, plus the contours for Ecut versus Efold for observation 39. The red, green, and blue contours represent the 68%, 90%, and 99% significance levels.\[obs39\]](obs39_contours.eps){width="6.in"}
Test of HEXTE Background Estimation for GX 304$-$1
==================================================
Counts and Rates
----------------
One test of the HEXTE background estimation method described above is whether or not the total number of counts in the background-subtracted HEXTE spectrum was linearly proportional to that in the background-subtracted PCU2 spectrum. Fig. \[fig:total\_counts\]-Left shows the product of the counting rate in the spectral band (3$-$60 keV PCU2; 20$-$100 keV HEXTE) times the lifetime per observation. The linear relationship is clearly followed with the exception of 6 observations where the HEXTE total counts are low. Since the 6 outliers are not evident in the rate plot (Fig. \[fig:total\_counts\]-Right), the HEXTE spectral data, from which the rates were extracted using the `SHOW RATE` command in XSPEC, are not suspect, and the outliers appear to be due to abnormally low lifetimes in the spectral extraction (as compared to that expected from the value of the PCU2 livetime) that resulted from missing HEXTE data. This can also be seen when one calculates the ratio of PCU2 to HEXTE livetimes.
Fig. \[fig:total\_counts\]-Right shows the HEXTE 20$-$100 keV counting rate versus the PCU2 3-60 keV rate. The HEXTE and PCU2 counting rates again are linearly correlated until about 500 counts/s in the PCU2. The deviation from linearity at higher rates is due to the change in the column density above a flux of $\sim 4 \times 10^{-9}$ ergs cm$^{-2}$ s$^{-1}$, and the added column density during the 2011 May column density enhancement. The column density variations will affect the 3$-$60 keV PCU2 rate while leaving the 20$-$100 keV HEXTE rate unaffected.
![**Left**: Total PCU2 counts 3$-$60 keV versus total HEXTE-A counts 20$-$100 keV. Total counts are calculated as lifetime times count rate. The six outliers result from significantly lower HEXTE A lifetime than expected. **Right**: Comparison of PCU2 and HEXTE A counting rates when the HEXTE background estimation method is used. The deviations from the linear relation are due to the variation in the column depth at the higher flux levels and the column density enhancement events (see Fig. \[fig:highecut\_continuum\_flux\]).\[fig:total\_counts\]](f22.eps "fig:"){width="3.3in"} ![**Left**: Total PCU2 counts 3$-$60 keV versus total HEXTE-A counts 20$-$100 keV. Total counts are calculated as lifetime times count rate. The six outliers result from significantly lower HEXTE A lifetime than expected. **Right**: Comparison of PCU2 and HEXTE A counting rates when the HEXTE background estimation method is used. The deviations from the linear relation are due to the variation in the column depth at the higher flux levels and the column density enhancement events (see Fig. \[fig:highecut\_continuum\_flux\]).\[fig:total\_counts\]](f23.eps "fig:"){width="3.3in"}
Lines and Normalizations
------------------------
The various line fluxes’ variations with power law flux from the `highecut` and `cutoffpl` model fittings are shown in Figs. \[fig:highecut\_line\_flux\] and \[fig:cutoffpl\_line\_flux\]. The 10.5 keV feature (panel a) appears at the very highest fluxes in the `highecut` model, while being rare in the `cutoffpl` modeling. The 3.88 PCU2 systematic feature (panel b) shows a stronger correlation with the power law flux for the `cutoffpl` model than that for `highecut`. This could be due to the more curved shape of the `cutoffpl` model as compared to the straight power law in `highecut`. The 30 keV HEXTE background line (panel c) appears stronger at high power law flux levels in the `highecut` model. The other HEXTE background lines (panels d, e, & f) show similar behaviors with increasing flux in both models.
Fig. \[fig:recor\] gives the values of the `recor` parameter for PCU2 and HEXTE as well as the HEXTE constant with respect to the PCU2 flux. The top panels are for the `highecut` model and the bottom panels are for the `cutoffpl` model. In the fitting process, the `corback` function found in ISIS and `recor` function in XSPEC are used to optimize the background subtraction by adjusting the background live time as part of the fitting process. PCU2 background estimates are based upon the observed background as a function of certain instrument charged particle average counting rates, and as such, may not reflect the exact background experienced during a given observation. The spectral shape of the background is assumed to remain the same and just its intensity is adjusted via the live time. A similar estimation is done for both the effects of the averages associated with HEXTE deadtime and HEXTEBACKEST. The `recor/corback` free parameter is the fraction of the estimated background to be added or subtracted. , The originally determined background normalizations have to be reduced by increasing amounts for increasing source fluxes, since true X-rays can contaminate the average charged particle counting rates for sources at high fluxes in the PCU2 and HEXTE. These counting rates are the basis for the background estimates. The effect is larger in the `highecut` models as compared to that in the `cutoffpl` models.
The relative normalization between the PCU2 and HEXTE instruments is plotted in Fig. \[fig:recor\]-Top/Bottom panel c. The HEXTE normalization constant is around 0.88 except at the lower power law fluxes where it increases with ever larger uncertainties, and at higher fluxes when the column density enhancements affect the 2$-$10 keV PCU2 fluxes.
![The variation of the recor normalization versus power law continuum flux in units of ergs cm$^{-2}$ s$^{-1}$ is shown for the PCU2 (a) and HEXTE (b), plus the relative normalization constant for the HEXTE cluster A with respect to the PCU2 normalization (c). The values resulting from `highecut` are plotted above those from `cutoffpl`.\[fig:recor\]](f24.eps "fig:"){width="7.0in"}\
![The variation of the recor normalization versus power law continuum flux in units of ergs cm$^{-2}$ s$^{-1}$ is shown for the PCU2 (a) and HEXTE (b), plus the relative normalization constant for the HEXTE cluster A with respect to the PCU2 normalization (c). The values resulting from `highecut` are plotted above those from `cutoffpl`.\[fig:recor\]](f25.eps "fig:"){width="7.0in"}\
\[lastpage\]
[^1]: http://www.sternwarte.uni-erlangen.de/wiki/doku.php?id=cyclo:start
[^2]: see http://heasarc.gsfc.nasa.gov/docs/xte/whatsnew/big.html for details of HEXTE rocking.
[^3]: This is a revised version of the absorption model `TBABS` of @Wilms00
|
---
abstract: 'In this survey paper we discuss recent advances on short interest rate models which can be formulated in terms of a stochastic differential equation for the instantaneous interest rate (also called short rate) or a system of such equations in case the short rate is assumed to depend also on other stochastic factors. Our focus is on convergence models, which explain the evolution of interest rate in connection with the adoption of Euro currency. Here, the domestic short rate depends on a stochastic European short rate. In short rate models, the bond prices, which determine the term structure of interest rate, are obtained as solutions to partial differential equations. Analytical solutions are available only in special cases; therefore we consider the question of obtaining their approximations. We use both analytical and numerical methods to get an approximate solution to the partial differential equation for bond prices.'
author:
- Zuzana Bučková
- Beáta Stehlíková
- 'Daniel Ševčovič [^1]'
title: Numerical and analytical methods for bond pricing in short rate convergence models of interest rates
---
Introduction {#sec:introduction}
============
An interest rate model is a description of interest rates’ evolution[^2] and their dependence on maturity[^3]; the dependence of the interest rate on maturity is called the term structure of interest rates. Given the state of the market today, the future interest rates cannot be predicted exactly; the models gives their probability distribution. However, since the interest rates are interconnected, often only some underlying processes are modeled, which in turn determine the interest rates.
We deal with so-called short rate models, which are based on a theoretical quantity, the *short rate*. It is a rate of interest for a default-free investment with infinitely small maturity. The other investments, with other maturities, include some risk: the evolution of the interest rates during the “life” of this investment can increase or decrease their value. Therefore it may not surprising that, besides the probabilistic description of the short rate evolution, there is another input - called market price of risk - needed in order to compute the term structure of interest rates; cf. [@fabozzi pp. 29-31] for a further intuition following these ideas.
Mathematical models can be described by solutions to linear parabolic differential equations, which degenerate to the hyperbolic ones at the boundary. Applying the Fichera theory to interest rates models one can treat the boundary conditions in a proper way. Correct treatment of boundary conditions is important in numerical schemes.
We propose an approximate analytical solution for a class of one-factor models and derive the order of its accuracy. These models can be used to model the European short rate in convergence models. We show an example of a convergence model of this kind and the analytical approximation formula for domestic bond prices, together with the derivation of its accuracy.
In some cases, a one-factor model is not sufficient to fit the European interest rates and we need a two-factor model to model the European short rate. Therefore, we also investigate a three-factor convergence model.
Which model for term structures should one use?
===============================================
This is the title of the paper [@which], in the beginning of which the author presents several criteria which a suitable model should have:
Our work is mainly concerned with the point (b). Approximate analytical formulae enlarge the set of models for which *one can compute answers in reasonable time*, as required above. Moreover, an easy computation of the observed quantities can significantly simplify a calibration of the model. Note that calibration of the model based on a comparison of market prices and theoretical prices given by the model often requires many evaluations of theoretical prices for different sets of parameters, as well as times to maturity and the short rate levels. Hence it is useful also to establish whether the point (e) above holds or not.
Basic concepts of stochastic calculus
======================================
In this section we briefly present the basic definitions and theorems of stochastic calculus which will be needed to formulate models considered here. For more details see, e.g., [@oksendal], [@karatzas-shreve].
[@oksendal Definition 2.1.4] A [stochastic process]{} is a parametrized collection of random variables $\{ X_t \}_{t \in \mathcal{T}}$ defined on a probability space $(\Omega, \mathcal{F}, \mathcal{P})$ and assuming values in $\mathbb{R}^n$.
An important stochastic process, used as a building block for other, more complicated processes, is a Wiener process.
[@sevcovic Definition 2.1] \[def:Wiener\] A stochastic process $\{w(t), t \geq 0\}$ is called a [Wiener process]{}, if it satisfies the following properties:
(i) [$w(0)=0$ with probability 1;]{}
(ii) [every increment $w(t+\Delta t)-w(t)$ has the normal distribution $N(0,\Delta t)$;]{}
(iii) [the increments $w(t_n)-w(t_{n-1})$, $w(t_{n-1})-w(t_{n-2})$, $\dots$, $w(t_2)-w(t_1)$ for $0 \leq t_1 < \dots < t_n$ are independent.]{}
Existence of such a process can be asserted using the Kolmogorov extension theorem, which builds a stochastic process from its finite dimensional distributions, cf. [@oksendal Chapters 2.1 and 2.2], [@karatzas-shreve Chapter 2.2].
Using a Wiener process, we are able to define new processes. It would be useful to be able to use some kind of “noise” in the ordinary differential equations and a Wiener process provides a way of doing so. This leads to so called stochastic integrals and stochastic differential equations. Again, we follow the main ideas of [@oksendal].
The first idea might be to consider an equation of the form $$\frac{d X}{d t} = b(t,X_t) + \sigma(X_t,t) \, u_t, \label{ito1}$$ where the term $u$ denotes some “noise”, which should be stationary, with values at different time being independent and having a zero expected value. However, there is no continuous process satisfying these conditions. Moreover, as a function on $[0,\infty) \times \Omega$ it cannot be even measurable (considering Borel-measurable sets on $[0,\infty)$), see [@oksendal pp. 21-22] and references therein. Therefore, another approach is used. We write (\[ito1\]) in a discrete form as $$X_{t_{k+1}} = X_{t_k} + b(t,X_t) (t_{k+1} - t_k) + \sigma(X_t,t) \, u_{t_k} (t_{k+1} - t_k),$$ where $0=t_0 < t_1 < \dots < t_m=t$ is a partition of the interval $[0,t]$. Recalling the desirable properties of the noise, the term $ u_{t_k} (t_{k+1} - t_k)$ should have stationary independent increments, which suggests using a Wiener process $w_{t_k}$. Then we have an equation $$X_{t_{k+1}} = X_{0} + \sum_{j=0}^{k-1} b(t,X_{t_j}) (t_{j+1} - t_j) + \sum_{j=0}^{k-1} \sigma(X_{t_k},t)
\,(w_{t_{k+1}} - w_{t_{k}})$$ and if we are able to make a limit of the last sum in some “reasonable way”, by denoting it by $\int_0^t \sigma(s,X_s) \, d w_s$ we can write $$X_t = X_0 + \int_0^t b(s,X_s) \, d s + \int_0^t \sigma(s,X_s) \, d w_s. \label{ito2}$$ This can indeed be done; in several ways, in fact, which leads to different kinds of stochastic integrals (Itō vs. Stratonovich). We use [Itō integral]{}, see the cited references [@oksendal] for details on its construction.
Finally, let us note that the equation (\[ito2\]) is often written in a “differential form” $$d X_t = b(t,X_t) \, d t + \sigma(t,X_t) \, d w_t \label{eq:sde}$$ which is called a [stochastic differential equation]{}.
The computation of the “differential” $d Y_t$, where $Y_t$ is defined as $Y_t=f(t,X_t)$, where $f$ is a smooth function and $X$ satisfies the stochastic differential equation (\[eq:sde\]) is performed via a stochastic generalization of the chain rule known from calculus. This can be done precisely using the integral representation of the stochastic processes (cf. [@karatzas-shreve pp. 150-153]) and results in the famous [Itō lemma]{}. We provide its formulation for the case of a one-dimensional process from [@oksendal].
[@oksendal Theorem 4.1.6] Let $X_t$ be an Itō process given by $$d X_t = u(t, X_t) \: d t + v(t, X_t) \: d w.$$ Let $g(t,x) \in C^2([0,\infty) \times \mathbb{R})$. Then $Y_t=f(t,X_t)$ is again an Itō process and $$d Y_t = \frac{\partial g}{\partial t}(t,X_t) d t + \frac{\partial g}{\partial x}(t, X_t) d X_t +
\frac{1}{2} \frac{\partial^2 g}{\partial x^2}(t,X_t) (d X_t)^2,$$ where $(d X_t)^2 = (d X_t)(d X_t)$ is computed according to the “rules” $$d t\, d t = d t \, d w_t = d w_t \, d t=0, d w_t \, d w_t = d t.$$
A multidimensional formulation can be found for example in [@oksendal Theorem 4.2.1], [@karatzas-shreve Theorem 3.6] or in the original paper by Kiyoshi Itō [@ito Theorem 6].
In order to illustrate Itō’s process, we present an example of a stochastic differential equation which will be useful later. It describes the evolution of a so called [Ornstein-Uhlenbeck process]{}: $$d x = \kappa (\theta - x) \, d t + \sigma \, d w,$$ where $\kappa, \theta$ and $\sigma$ are positive constants. Without the stochastic $d w$ term, it would be an ordinary differential equation with the solution $x_t = x_0 e^{-\kappa t} + \theta (1 - e^{-\kappa t}),$ where $x_0$ is the value of the process at time $t=0$. With the stochastic term included, the solution becomes a random variable and can be written in an explicit form $$x_t = x_0 e^{-\kappa t} + \theta (1 - e^{-\kappa t}) + \sigma \int_0^t d w.$$ The trend, reversion to the equilibrium level $\theta$, whose speed depends on $\kappa$, is preserved (processes with this property are called [mean-reversion processes]{}). Furthermore, there are random fluctuations around this trend; their impact depends on the parameter $\sigma$. Sample trajectory of an Ornstein-Uhlenbeck process is presented in Figure \[fig:ou\].
![Sample path of an Ornstein-Uhlenbeck process.[]{data-label="fig:ou"}](figures/habilitacia/ornstein-uhlenbeck){width="50.00000%"}
Similarly as in the case of ordinary differential equations, a closed-form solution is not always available, but numerical approximations are still possible. The simplest one is the [Euler-Maruyama]{} scheme, which is a generalization of the Euler method known from numerical methods for ordinary differential equations. It consists of replacing the differentials in (\[eq:sde\]) by finite differences and simulating the increments of a Wiener process: $$\begin{aligned}
X_0 &=& x_0, \nonumber \\
X_{t+\Delta t} &=& X_t + b(t,X_t) \, \Delta t + \sigma(t,X_t) \, \Delta w_t, \nonumber\end{aligned}$$ where $\Delta w$ are independent realizations from $\mathcal{N}(0,\Delta t)$ distribution. There are also other methods, which have a higher precision (for example, Milstein scheme, Runge-Kutta methods, cf. [@seydel] for an introduction or [@kloeden-platen] for more details).
Short rate models
==================
Short rate models are formulated in terms of a stochastic differential equation (one-factor models) a system of stochastic differential equations (multi-factor models) determining the short rate, see Figure \[fig:short-rate-example\] for an example of market data which - being interest rates with short maturities - can be thought of as approximations of the theoretical short rate.
![Euro interest rates with short maturities - possible approximations of short rate. Data source: *http://www.emmi-benchmarks.eu*[]{data-label="fig:short-rate-example"}](figures/habilitacia/obrazky1){width="50.00000%"}
We start with a simple stochastic differential equation which describes some popular features of the market rates. Then, seeing the shortcomings of the models, we move to more complicated ones. Each of them addresses a specific feature and the choice of the model needs to take this into account. For selected stochastic processes we explain the motivation that leads to considering them as a model for the short rate.
We also discuss bond prices. A zero-coupon bond is a financial security that pays a unit amount money to its holder at the specified time of maturity. The bond prices $P=P(t,T,\mathbf{x})$ (where $t$ is time, $T$ is time to maturity and $\mathbf{x}$ is a vector of factors determining the short rate) are then connected with interest rates $R=R(t,T,\mathbf{x})$ through the formula $$P(t,T,\mathbf{x}) = e^{-R(t,T,\mathbf{x})(T-t)}, \; \textrm{ i.e., } \; R(t,T,\mathbf{x}) = -
\frac{ \textrm{ln}\,P(t,T,\mathbf{x})}{T-t}.$$ Examples of interest rates with different maturities can be seen in Figure \[fig:term-structure-example\].
![Euro interest rates - examples of term structures. Data source: *http://www.emmi-benchmarks.eu*[]{data-label="fig:term-structure-example"}](figures/habilitacia/hist_EURIBOR_2013){width="50.00000%"}
In short rate models, the prices of bonds (as well as other interest rate derivatives) are solutions to a parabolic partial differential equation. Even in a case of a derivative with such a simple payoff, as it is the case of a bond, closed form solution is available only in very special cases. The later topics presented in this paper are then connected by a pursuit of finding approximations of the bond prices (and hence also term structures) in those cases when they are not known in a closed form.
One-factor models
-----------------
When speaking of one-factor short rate models, the term *one-factor* refers to the fact that there is *one* Wiener process used in the definition of the short rate process, i.e., there is *one* source of randomness.
Hence, there is a scalar stochastic differential equation for the short rate $r$, which can be written in a general form as $$d r = \mu(r,t) d t + \sigma(r,t) d w,$$ where $w$ is a Wiener process. Recall from the section on stochastic processes that the function $\mu(r,t)$ determines the trend of the process, while the function $\sigma(r,t)$ determines the nature of the random fluctuations. Specifying the functions $\mu(r,t)$ and $\sigma(r,t)$ characterizes the short rate model.
Let $P=P(r,t)$ be the price of a derivative at time $t$ when the current level of the short rate is $r$, which pays a given payoff at time $T$. We consider a construction of a portfolio consisting of derivatives with two different maturities, continuously rebalanced so that the risk coming from the Wiener process is eliminated[^4]. Then, to eliminate a possibility of an arbitrage, the return of such a portfolio has to be equal to the current short rate, which leads to a partial differential equation for the derivative price $P$, which reads as $$\partial_t P + (\mu(r,t) - \lambda(r,t) \sigma(r,t)) \partial_r P + \frac{1}{2} \sigma(r,t)^2 \partial^2_{r} P = 0$$ for all admissible values of $r$ and for all $t \in [0,T)$. We refer to [@kwok], [@sevcovic], for more details on the derivation of the partial differential equation. Here and after we denote by $\partial_t P, \partial_r P$ the first partial derivatives of $P$ with respect to $t$, $r$ and the second derivative $\partial^2_r P$ of $P$ with respect to $r$. Note that the equation includes a new function $\lambda(r,t)$. It appears during the derivation of the equation, when it turns out that a certain quantity, measuring the rise of the expected return for one unit of risk, has to be independent of the maturity $T$. It is denoted by $\lambda(r,t)$ and because of its interpretation it is called market price of risk. It is necessary to include it into the specification of a model when we want to price derivatives, in addition to talking about the short rate evolution. Note that the equation holds for any derivative, the specific derivative determines the terminal condition $P(r,T)$ which equals the security payoff.
If we consider only Markov models, i.e., $\mu$, $\sigma$ and $\lambda$ are functions only of the variable $r$ and do not explicitly depend on time $t$ (which will be the case for the models studied in this thesis), it is convenient to introduce a new variable $\tau=T-t$ denoting time remaining to maturity. For the bond price we obtain the partial differential equation (PDE) $$\begin{aligned}
-\partial_{\tau} P + (\mu(r) - \lambda(r) \sigma(r)) \partial_r P + \frac{1}{2} \sigma(r)^2 \partial^2_{r} P &=& 0\,\,\, \textrm{ for all } r \textrm{ and } \tau \in (0,T], \label{eq:pde-bond-1f} \\
P(r,0)&=&1 \,\,\, \textrm{ for all } r. \label{eq:pde-bond-1f-}\end{aligned}$$
Alternatively, a model can be formulated in the so-called risk-neutral measure $\mathbb{Q}$, which is an equivalent probability measure to $\mathbb{P}$, in which the process is physically observed. In the risk-neutral measure, the prices of the securities can be expressed in a form of expected values. The change of measure is related to the market price of risk from the partial derivative approach above and mathematically it is based on Girsanov theorem (cf. [@oksendal Section 8.6]). The general model above in the risk neutral model reads as $$\mathrm{d}r= \tilde{\mu}(r) \mathrm{d}t + \tilde{\sigma}(r) \mathrm{d}w^{\mathbb{Q}},$$ where $w^{\mathbb{Q}}$ is a Wiener process under the risk neutral measure, while the risk-neutral drift and volatility are given by $$\tilde{\mu}(r) = \mu(r) - \lambda(r) \sigma(r), \tilde{\sigma}(r) = {\sigma}(r),$$ cf. [@kwok2 Section 7.2]. Comparing this with (\[eq:pde-bond-1f\]) we can see that the risk-neutral formulation contains all information needed to write the valuation PDE. Therefore, when dealing with pricing bonds or other derivatives, the model is often formulated in the risk-neutral form. Finally, let us note that the two alternative expressions for the prices - expected values under the risk-neutral measure and solutions to partial differential equations - are related also via so-called Feynman-Kac formula, cf. [@oksendal Theorem 8.2.1].
Vasicek and Cox-Ingersoll-Ross models
-------------------------------------
Recall that the Ornstein-Uhlenbeck process is a stochastic process given by $$d r= \kappa(\theta -r) \: d t + \sigma \: d w,$$ where $\kappa, \theta, \sigma >0$ are given constants and $w$ is a Wiener process. This process can be used as a simple model for the short rate, known as Vasicek model, as it has been suggested in [@vasicek] by Oldřich Vašíček. He defined the market price of risk to be equal to a constant $\lambda$, which results in the partial differential equation for the bond prices that reads as (recall its general form (\[eq:pde-bond-1f\])-(\[eq:pde-bond-1f-\])) $$-\partial_{\tau} P + (\kappa(\theta-r)- \lambda \sigma ) \partial_r P + \frac{1}{2} \sigma^2 \partial^2_{r} P = 0 \label{eq:pdr:vas}$$ for all $r$ and $\tau \in (0,T]$, and $P(r,0)=1$ for all $r$. This differential equation can be solved explicitly; its solution has the form $$P(r,\tau)=A(\tau)e^{-B(\tau)r} \label{eq:vasicek-form}$$ and the functions $A,B$ are given by (see [@vasicek]) $$\ln A(\tau) = \left( -\theta + \frac{\lambda \sigma}{\kappa} + \frac{\sigma^2}{2 \kappa^2} \right)
\left(-\frac{1-e^{-\kappa \tau}}{\kappa}+\tau \right)
-\frac{\sigma^2}{4 \kappa^3} (1-e^{-\kappa \tau})^2, \; B(\tau) = \frac{1-e^{-\kappa \tau}}{\kappa}.
\label{vasicek-bond}$$
One of the consequences of the constant volatility is a conditional normal distribution of the future interest rates and thus a possibility of negative interest rates. Historically, this was been one of the motivations for proposing other short rate models. Note, however, that while some of the interest rates observed in these days can be indeed negative, the negative values of the Ornstein-Uhlenbeck stochastic process is not consistent with absence of arbitrage in the context of default intensity models [@kane] which leads to solving exactly the same parabolic PDEs. A popular alternative is the Cox-Ingersoll-Ross model [@cir] (usually abbreviated as CIR model) which does not allow negative interest rates, while it preserves analytical tractability of bond prices. The stochastic differential equation for the short rate is given by $$d r= \kappa(\theta -r) \: d t + \sigma \sqrt{r} \: d w, \label{eq:cir_sde}$$ with $\kappa, \theta, \sigma >0$ being constants. The difference from the Vasicek model lies in the volatility, which is now equal to $\sigma \sqrt{r}$. Intuitively, if the short rate $r$ is small, then also the volatility is small; if short rate hits zero, the volatility becomes zero as well and the positive drift pushes the short rate to a positive value. It can be shown that the process is indeed nonnegative for all times and, moreover, if the condition $2 \kappa \theta > \sigma^2$ is satisfied, the process remains strictly positive. If the market price of risk is chosen to be $\lambda \sqrt{r}$, the equation (\[eq:pde-bond-1f\]) with initial condition (\[eq:pde-bond-1f-\]) becomes $$-\partial_{\tau} P + (\kappa(\theta-r)- \lambda \sigma r) \partial_r P + \frac{1}{2} \sigma^2 r \partial^2_{r} P = 0 \label{eq:pdr:cir}$$ for all $r$ and $\tau \in (0,T]$, and $P(r,0)=1$ for all $r$. Again, it can be solved in a closed form, assuming the solution (\[eq:vasicek-form\]), inserting it into the partial differential equation and obtaining a system of ordinary differential equations for the functions $A(\tau), B(\tau)$. This system can be solved explicitly, see [@cir] for the exact formulae.
Chan-Karolyi-Longstaff-Sanders short rate model
-----------------------------------------------
As we have seen, changing the constant volatility from the Vasicek model to $\sigma \sqrt{r}$ in CIR model prevents the short rate from becoming negative. However, the same reasoning applies to any volatility of the form $\sigma r^{\gamma}$ with $\gamma >0$. Models with general $\gamma$ may perform better when applied to real data and the hypothesis of $\gamma=1/2$ is actually often rejected by statistical tests.
The pioneering paper [@ckls] by Chan, Karolyi, Longstaff and Sanders started the discussion on the correct form of the volatility. Authors used proxy for the short rate process and considered a general short rate model expressed in terms of a single stochastic differential equation $$d r=(\alpha+\beta r)\,d t+\sigma r^{\gamma} \,d w, \label{eq-ckls-}$$ which has become known as the CKLS model. It includes Vasicek ($\gamma=0$) and CIR ($\gamma=1/2$) models as special cases (and thus allows testing them as statistical hypotheses on the model parameters), as well as several other models, see Table \[table:ckls-1\]. Chan *et al.* estimated the parameters using the generalized method of moments. They found the parameter $\gamma$ to be significantly different from the values indicated by Vasicek and CIR models, see Table \[table:ckls-2\].
Model: Equation for the short rate:
----------------------------------------------------------------------------- -------------------------------------------
Merton [@irmodels-merton] $dr=\alpha dt + \sigma dw$
Vasicek [@vasicek] $dr=(\alpha+\beta r)dt+\sigma dw$
Cox-Ingersoll-Ross (1985) [@cir] $dr=(\alpha+\beta r)dt+\sigma r^{1/2} dw$
Dothan [@irmodels-dothan], [@irmodels-br-sch-77] $dr=\sigma r dw$
Geometrical Brownian motion [@irmodels-marsh] $dr=\beta r dt + \sigma r dw$
Brennan-Schwartz [@irmodels-br-sch-80], [@irmodels-crank-nicolson-numerics] $dr=(\alpha + \beta r) dt + \sigma r dw$
Cox-Ingersoll-Ross (1980) [@irmodels-cir-80] $dr=\sigma r^{3/2} dw$
Constant elasticity of variance [@irmodels-marsh] $dr = \beta r dt + \sigma r{^\gamma} dw$
: One-factor short rate models considered in [@ckls] as special cases of the stochastic process (\[eq-ckls-\]).
\[table:ckls-1\]
Model: $\alpha$ $\beta$ $\sigma^2$ $\gamma$ P-value
-------------- ---------- --------- ------------ ---------- ---------
unrestricted 0.0408 -0.5921 1.6704 1.4999 -
Vasicek 0.0154 -0.1779 0.0004 0 0.0029
CIR 0.0189 -0.2339 0.0073 1/2 0.0131
: Parameters estimates and results from testing the hypotheses given by Vasicek and CIR models in [@ckls].
\[table:ckls-2\]
A modification of the generalized method of moments (so called robust generalized method of moments), which is robust to a presence of outliers, was developed in [@aquilla]. Another contribution to this class of estimators is for example indirect robust estimation by [@czellar]. Another popular method for parameter estimation are Nowman’s Gaussian estimates [@nowman], based on approximating the likelihood function. They were used in [@episcopos] for a wide range of interest rate markets. There are several other calibration methods for the short rate process, such as quasi maximum likelihood, maximum likelihood based on series expansion of likelihood function by Aït-Sahalia [@ait-sahalia-transition], Bayesian methods such as Markov chain Monte Carlo and others.
A common feature of these approaches is taking a certain market rate as a proxy to the short rate and using the econometric techniques of time series analysis to estimate the parameters of the model. These parameters can be used afterwards to price the bonds and other derivatives. For example in [@nowman-2], the parameters of the CKLS process were first estimated using the Nowman’s methodology and afterwards derivatives prices were computed by numerically solving the partial differential equation using the Box method. For more results of this kind see [@nowman-3], [@nowman-4].
An alternative would be using the derivatives prices to calibrate the parameters of the model. This, however, requires a quick computation of the prices, since they have to be computed many times with different parameters during the calibration procedure. Exact solution to the bond pricing equation available for Vasicek and CIR model made this possible in the case of these two models, cf. [@sevcovic-csajkova-2], [@sevcovic-csajkova-1]. In general, when the exact solution is not available, approximate analytical solution provides a convenient alternative.
Other one-factor models
-----------------------
Modifying the constant volatility is not the only way for ensuring positivity of short rate. Another simple way is defining short rate as a positive function whose argument is a stochastic process. In particular, Black-Karasinski model [@black-karasinski] (also called exponential Vasicek because of its construction, cf. [@brigo-mercurio Section 3.2.5]) defined the short rate as $r=e^x$, where $x$ follows an Ornstein-Uhlenbeck process $$d x = \kappa(\theta-x)\, d t + \sigma \, d w. \label{eq:bk-1}$$
Note that in the case of Black-Karasinski model, the stochastic differential equation for the short rate $r$ reads as $$d r= r ( \kappa \theta + \frac{1}{2} \sigma^2 - \kappa \textrm{ ln}\,r) \, d t + \sigma r \, d w,$$ which means that the short rate does not have a linear drift, common to the previously considered models.
Another nonlinear-drift model has been suggested by Aït-Sahalia in [@ait-sahalia-drift] to produce very little mean reversion while the interest rates remain in the middle part of their domain, and strong nonlinear mean reversion at either end of the domain. This property is achieved by the stochastic differential equation $$d r = (\alpha_{-1} r^{-1} + \alpha_0 + \alpha_1 r + \alpha_2 r^2) \: d t + \sigma r^{\gamma} d w,$$ see Figure \[fig:ait-sahalia-drift\] for a plot of the drift function for $\alpha_{-1} = 0.000693, \alpha_0 = -0.0347, \alpha_1 = 0.676, \alpha_2 = -4.059$ which are taken from [@ait-sahalia-transition].
![Nonlinear drift of the Aït-Sahalia model [@ait-sahalia-drift] for parameters $\alpha_{-1} = 0.000693, \alpha_0 = -0.0347, \alpha_1 = 0.676, \alpha_2 = -4.059$, taken from [@ait-sahalia-transition][]{data-label="fig:ait-sahalia-drift"}](figures/habilitacia/nonlineardrift){width="60.00000%"}
Short rate as a sum of multiple factors
---------------------------------------
One of the consequences of using a one-factor short rate model is the bond price of the form $P=P(\tau,r)$. This means that the bond price with a given maturity is uniquely determined by the short rate level. Translating this into the language of term structures: the term structure is uniquely determined by its beginning (interest rate for infinitesimally small maturity, i.e., the short rate). While this might not be an unreasonable property of the interest rates in certain time periods, it clearly does not hold in others, as demonstrated in Figure \[fig:multifactor-motivation\].
![Euro interest rates - examples of term structure starting from the same point. Data source: *http://www.emmi-benchmarks.eu*[]{data-label="fig:multifactor-motivation"}](figures/habilitacia/obrazky2){width="50.00000%"}
If we define the short rate as a function of more factors, i.e., $r=r(x_1,\dots,x_n)$, then the bond price has the form $P=P(\tau,x_1,\dots,x_n)$. If the same short rate level can be achieved for several combinations of the factors $x_1,\dots,x_n$, these can produce different bond prices and, consequently, term structures - such as those seen in Figure \[fig:multifactor-motivation\]. Moreover, the factors determining the short rate may have a plausible interpretation on their own.
In [@babbs-nowman] the authors propose the model for the short rate $r$ to be $$r = \mu - \sum_{j=1}^n x_i,$$ where $\mu$ is interpreted as the long-run average rate and $x_1, \dots,, x_n$ represent the current effect of $n$ streams of economic “news”, among which they include rumors about central bank decisions, economic statistics, etc. The arrival of each of these news is modeled by the process $$d x_i = \xi_i x_i \, d t + \sigma_i \, d w_i$$ with negative constants $\xi_i$ and possibly correlated Wiener processes $w_i$. Thus, the impact of any news dies away exponentially. If the market prices of risk are taken to be constant, it is possible to express the bond prices in a closed form.
A multi-factor version of a one-factor CIR model is formulated in [@chen-scott], where the short rate $r$ is a sum of $n$ components, i.e., $$r=\sum_{j=1}^n r_i, \label{eq:n-CIR-1}$$ with each $x_i$ following a Bessel square root process $$d r_i = \kappa(\theta - r_i) \, d t + \sigma_i \sqrt{r_i} \, d w_i, \label{eq:n-CIR-2}$$ assuming independent Wiener processes. Their independence and the choice of market prices of risk to be $\lambda_i \sqrt{r_i}$ again allows analytical expressions for the prices of bonds. In Figure \[fig:2fCIR\] we show sample trajectories of a two-factor CIR model with parameters equal to $\kappa_1=0.7298, \theta_1=0.04013, \sigma_1=0.16885$, $\kappa_2=0.021185, \theta_2=0.022543, \sigma_2=0.054415$, which are taken from [@chen-scott].
The equations (\[eq:n-CIR-1\])-(\[eq:n-CIR-2\]) can be generalized to general CKLS processes (\[eq-ckls-\]) and correlated Wiener processes. However, with the exception of the special cases above, the closed form formulae for bond prices are not available and, therefore, their approximations are necessary.
![Two-factor CIR model: sample trajectories of the factors and the short rate for parameters $\kappa_1=0.7298, \theta_1=0.04013, \sigma_1=0.16885$, $\kappa_2=0.021185, \theta_2=0.022543, \sigma_2=0.054415$, taken from [@chen-scott]. []{data-label="fig:2fCIR"}](figures/habilitacia/2f-cir){width="50.00000%"}
Stochastic volatility multiple-factor interest rate models
----------------------------------------------------------
Non-constant volatility is known especially from the market of stocks and the derived options. The most famous index measuring the volatility is arguably VIX, CBOE Volatility Index. It is a key measure of market expectations of near-term volatility conveyed by S&P 500 stock index option prices. Since its introduction in 1993, it has been considered by many to be a barometer of investor sentiment and market volatility[^5]. We present its evolution in Figure \[fig:vix\].
![VIX, CBOE Volatility Index. Data source: *http://www.cboe.com/micro/VIX/* []{data-label="fig:vix"}](figures/habilitacia/data_volatilita){width="60.00000%"}
Moreover, besides the volatility being non-constant and stochastic, there is an evidence that it evolves in a different time scale than the stock price, see a concise book [@sircar1] by Jean-Pierre Fouque, George Papanicolaou and K. Ronnie Sircar summarizing their work this area of using perturbation methods for the partial differential equation for the option prices in models incorporating this feature.
Approximately ten years later, in 2011, the same authors and in addition Knut Solna, published a new book [@sircar2] with a broader content, *Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives*, thus featuring the topic of interest rates already in the title. The randomness of volatility and interest in its measurement can be seen also from the fact, that CBOE has started to calculate also volatility indices related to interest rates market: CBOE/CBOT 10-year U.S. Treasury Note Volatility Index[^6] and CBOE Interest Rate Swap Volatility Index[^7]. As an example, let us consider stochastic volatility Vasicek model, as given in [@sircar1]. It differs from the ordinary Vasicek model by its volatility. Instead of a constant, it is a nonnegative function $f$ evaluated in the value of a stochastic process $y$, following an Ornstein-Uhlenbeck type: $$\begin{aligned}
d r &=& \kappa_1 (\theta_1 - r) \: d t + f(y) \: d w_1, \nonumber \\
d y &=& \kappa_2 (\theta_2 - y) \: d t + v \: d w_2, \nonumber \end{aligned}$$ where the correlation between the increments $d w_1$ and $d w_2$ is $\rho \in (-1,1)$. Empirical data suggest $\rho>0$, see, e.g., [@sircar1 p. 177].
Another example of a stochastic volatility short rate model has been proposed by Fong and Vasicek in [@fong-vasicek] by the following system of stochastic differential equations: $$\begin{aligned}
d r &=& \kappa_1 (\theta_1 - r) \: d t + \sqrt{y} \: d w_1, \nonumber \\
d y &=& \kappa_2 (\theta_2 - y) \: d t + v \sqrt{y} \: d w_2, \nonumber \end{aligned}$$ where again the Wiener processes can be correlated and the correlation between the increments $d w_1$ and $d w_2$ is $\rho \in (-1,1)$. If the market prices of risk are given by[^8] $\lambda_1 \sqrt{y}$ (market price of risk of short rate) and $\lambda_2 \sqrt{y}$ (market price of risk of volatility), then the partial differential equation for the bond price can be split into solving a system of three ordinary differential equation.
Convergence multiple-factor models modeling entry to a monetary union
---------------------------------------------------------------------
The basic convergence model of interest rates is suggested by Corzo and Schwarz in [@corzo-schwarz], where they model the interest rates before the formation of the European monetary union. Participating countries fixed their exchange rate to Euro in January 1999. With fixed exchange rate, the interest rates have to be the same across the countries. However, already before fixing the exchange rate, the convergence of the interest rates in participating countries was possible to be observed. This motivates the following model for the European short rate $r_e$ and the domestic short rate $r_d$: $$\begin{aligned}
d r_d &=& (a + b(r_e - r_d)) \, d t + \sigma_d \, d w_d, \label{conv-vas-1} \\
d r_e &=& c(d - r_e) \, d t + \sigma_e \, d w_e, \label{conv-vas-2} \end{aligned}$$ where the Wiener processes are, in general, correlated: $\textrm{cov}(d w_d, d w_e)=\rho \, d t$. Note that the equation (\[conv-vas-2\]) is a Vasicek model for the European rate, while (\[conv-vas-1\]) models a reversion of the domestic rate to the European rate, with a possible minor divergence given by $a$. Figure \[fig:conv\] shows sample trajectories for the parameters $c=0.2087, d=0.035, \sigma_e=0.016$ for the European rate, $a=0.0938, b=3.67, \sigma_d =0.032$ for the domestic rate and the correlation $\rho=0.219$, taken from [@corzo-schwarz]. Note that in the case of nonzero $a$, the instantaneous drift from (\[conv-vas-1\]) forces the domestic rate to revert not exactly to the European rate $r_e$, but the value $r_e + a/b$. For the given set of the parameters, the “divergence term” $a/b$ equals to approximately 0.02, which can be observed in Figure \[fig:conv\]. However, with fixed exchange rate, economically plausible value of $a$ is zero. Indeed, when the original model was estimated using the last 3.5 years before entering the European Monetary Union (EMU) in [@corzo-schwarz], this coefficient turned to be highly insignificant. We also note that negative value of the parameter $a$ would, interestingly, cause also mathematical problems in the generalizations of the model (related to the short rate evolution as well as the bond prices), see [@lacko-dp].
![Sample paths of the European and the domestic short rate in Corzo-Schwarz convergence model with parameters $a=0.0938, b=3.67, \sigma_d =0.032$, $c=0.2087, d=0.035, \sigma_e=0.016$, $\rho=0.219$, taken from [@corzo-schwarz]. []{data-label="fig:conv"}](figures/habilitacia/convergenceCS){width="60.00000%"}
In the market prices of risk are constant, there is an explicit solution for the domestic bond prices[^9] of the form $$P(\tau,r_d,r_e) = A(\tau)e^{-B(\tau) r_d - C(\tau) r_e}. \label{eq:conv-form}$$ In [@corzo-schwarz] authors claim that the same analysis can be done for the CIR-type convergence model; this has been studied by Lacko in [@lacko-dp].
If the correlation between $d w_d$ and $d w_e$ is zero, then the solution can be again written in the form (\[eq:conv-form\]) and the functions can be found numerically by solving a system of ordinary differential equations. In the general correlated case, the solution cannot be written in the separated form (\[eq:conv-form\]). This is true also for another natural generalization, where the European rate is modeled by a CKLS-type process (\[eq-ckls-\]) and we allow a general form of volatility $\sigma_d r^{\gamma_d}$ also in the equation (\[conv-vas-1\]) describing the behavior of the domestic rate. An analytical approximation formula for bond prices the CKLS-type model is studied by Zíková and Stehlíková in [@zikova-stehlikova].
Approximate analytical solutions in selected bond pricing problems
==================================================================
Let us consider an example of market interest rates and Euribor rates in particular. Panel banks provide daily quotes of the rate, rounded to two decimal places, that each panel bank believes one prime bank is quoting to another prime bank for interbank term deposits within the Euro zone. Then, after collecting the data from panel banks: The calculation agent shall, for each maturity, eliminate the highest and lowest 15% of all the quotes collected. The remaining rates will be averaged and rounded to three decimal places. These rates are quoted in percentage points. After dividing them by 100, we obtain them as decimal numbers which are used as the variable $r$ in the models described in the previous chapter. It follows that the value, e.g., 0.123 percentage points from the market data is not an exact figure, but, in terms of decimal numbers, can represent anything from the interval $[0.001225, 0.001235)$. On the other hand, any two numbers from this interval obtained from models would be in practice indistinguishable. Therefore, going above a certain precision in the computations does not bring any practical advantage when analyzing the market interest rates. In other words, two approximative results that coincide to certain decimal points are practically equally useful and therefore comparing their computational complexity is in place. Approximate analytical solutions, which we deal with, are very convenient in this regard.
Chan-Karolyi-Longstaff-Sanders model
------------------------------------
In this section we consider the Chan-Karolyi-Longstaff-Sanders (CKLS hereafter) model in the risk neutral measure $$d r=(\alpha+\beta r) \, d t+ \sigma r^{\gamma} \, d w, \label{eq-ckls}$$ where $w$ is a Wiener process. Note that the linear drift is consistent with the physical measure formulation and choice of market price of risk in the original Vasicek model from [@vasicek] with $\gamma=0$ and the Cox-Ingersoll-Ross (CIR hereafter) model proposed in [@cir] with $\gamma=1/2$, see (\[eq:pdr:vas\]) and (\[eq:pdr:cir\]).
The price $P(\tau,r)$ of the discount bond, when the current level of the short rate is $r$ and time remaining to maturity is $\tau$, is then given by the solution to the partial differential equation $$-\partial_{\tau} P + \frac{1}{2} \sigma^2 r^{2 \gamma} \partial^2_{r} P + (\alpha + \beta r)\partial_{r} P - rP =0, \; r>0, \; \tau \in (0,T)
\label{PDE-1f}$$ satisfying the initial condition $P(0,r)=1$ for all $r>0$, see, e.g., [@kwok], [@brigo-mercurio]. Recall that in the case of Vasicek and CIR models the explicit solutions to bond pricing partial differential equations are known.
### Approximation formula due to Choi and Wirjanto
Consider the stochastic differential equation (\[eq-ckls\]) in the risk neutral measure for the evolution of the short rate $r$ and the corresponding partial differential equation (\[PDE-1f\]) for the bond price $P(\tau,r)$. The main result of the paper [@choi-wirjanto] by Choi and Wirjanto is the following approximation $P^{ap}$ for the exact solution $P^{ex}$:
[@choi-wirjanto Theorem 2] The approximate analytical solution $P^{ap}$ is given by $$\begin{aligned}
\label{1f-approximation-formula}
\ln P^{ap}(\tau,r) &=& -rB+\frac{\alpha}{\beta} (\tau-B)+ \left( r^{2 \gamma} +q \tau \right) \frac{\sigma^2}{4 \beta} \left[ B^2 + \frac{2}{\beta} (\tau-B) \right]
\nonumber \\
& & -q \frac{\sigma^2}{8 \beta^2} \left[B^2(2 \beta \tau-1) - 2B \left(2 \tau - \frac{3}{\beta} \right) + 2 \tau^2 - \frac{6 \tau}{\beta} \right]\end{aligned}$$ where $$q(r) = \gamma(2 \gamma -1)\sigma^2 r^{2(2 \gamma-1)} + 2 \gamma r^{2 \gamma-1} (\alpha+\beta r) \label{qr}$$ and $$B(\tau) = (e^{\beta \tau}-1)/\beta.$$
The derivation of the formula (\[1f-approximation-formula\]) is based on calculating the price as an expected value in the risk neutral measure. The tree property of conditional expectation was used and the integral appearing in the exact price was approximated to obtain a closed form approximation. The reader is referred to [@choi-wirjanto] for more details of the derivation of (\[1f-approximation-formula\]).
Authors furthermore showed that such an approximation coincides with the exact solution in the case of the Vasicek model [@vasicek]. Moreover, they compared the above approximation with the exact solution of the CIR model which is also known in a closed form. Graphical demonstration of relative mispricing, i.e., the relative error in the bond prices, has been also provided by the authors.
### Asymptotic analysis of the Choi and Wirjanto approximation formula
As it can be seen in the numerical examples given in [@choi-wirjanto], the error in bond prices is smaller in the case of $\tau$ small. Also, for $\tau=0$ the formula is exact. This suggests using $\tau$ as a small parameter in the asymptotic analysis.
Using the exact solution $P^{ex}_{CIR}$ in the case of $\gamma=1/2$ (i.e., the Cox-Ingersoll-Ross model), computing its expansion in $\tau$ around the point $\tau=0$ and comparing it with the expansion of the Choi and Wirjanto approximate formula $P^{ap}_{CIR}$ with $\gamma=1/2$ we obtain $$\ln P^{ap}_{CIR}(\tau,r) - \ln P^{ex}_{CIR}(\tau,r) = -\frac{1}{120} \sigma^2 \left[ \alpha \beta + r(\beta^2 - 4 \sigma^2) \right] \tau^5 + o(\tau^5)$$ as $\tau \rightarrow 0^+$. Considering logarithms of the bond prices enables us to estimate the relative error in the bond prices (the relative mispricing from the previous subsection) and the absolute error in the interest rates forming a term structure of interest rate.
The result of expanding the approximate and exact solutions in the case of the CIR model motivates finding a similar estimate also in the case of a general CKLS model, i.e., for arbitrary $\gamma$. In the paper [@stehlikova-sevcovic] we proved the following theorem:
[@stehlikova-sevcovic Theorem 3] \[theorem-accuracy-for-ln-wirjanto\] Let $P^{ap}$ be the approximative solution given by (\[1f-approximation-formula\]) and $P^{ex}$ be the exact bond price given as a unique complete solution to (\[PDE-1f\]). Then $$\ln P^{ap}(\tau,r) - \ln P^{ex}(\tau,r) = c_5(r) \tau^5 + o(\tau^5)$$ as $\tau \rightarrow 0^+$ where $$\begin{aligned}
\label{c5(r)wirjanto}
c_5(r) &=& -\frac{1}{120} \gamma r^{2(\gamma-2)} \sigma^2
\left[
2 \alpha^2 (-1+2 \gamma) r^2 + 4 \beta^2 \gamma r^4
- 8 r^{3+2\gamma} \sigma^2
\right.
\nonumber \\
& &
+ 2 \beta (1-5 \gamma + 6 \gamma^2) r^{2 (1+\gamma)} \sigma^2
+\sigma^4 r^{4 \gamma} (2\gamma-1)^2 (4 \gamma-3)
\\
& &
\left.
+ 2 \alpha r \left( \beta(-1+4 \gamma)r^2 + (2 \gamma -1)(3\gamma -2) r^{2 \gamma} \sigma^2 \right)
\right]. \nonumber\end{aligned}$$
Moreover, the method of the proof enabled to propose an approximation formula of a higher accuracy, as stated in the following theorem.
[@stehlikova-sevcovic Theorem 4] \[theorem-accuracy2-for-ln-wirjanto\] Let $P^{ex}$ be the exact bond price. Let us define an improved approximation $P^{ap2}$ by the formula $$\ln P^{ap2}(\tau,r) = \ln P^{ap}(\tau,r) - c_5(r) \tau^5 - c_6(r) \tau^6
\label{approx-higher-order}$$ where $\ln P^{ap}$ is given by (\[1f-approximation-formula\]), $c_5(\tau)$ is given by (\[c5(r)wirjanto\]) in Theorem \[theorem-accuracy-for-ln-wirjanto\] and $$c_6(r)=\frac{1}{6} \left( \frac{1}{2} \sigma^2 r^{2 \gamma} c_5''(r) + (\alpha+\beta r) c_5'(r) - k_5(r) \right)$$ where $c_5^\prime$ and $c_5^{\prime\prime}$ stand for the first and second derivative of $c_5(r)$ w. r. to $r$ and $k_5$ is defined by $$\begin{aligned}
\label{k5}
k_5(r)&=&
\frac{\gamma\sigma^2}{120}
r^{2 \left( -2 + {\gamma} \right) }
\left[ 6 {\alpha}^2 \beta \left( -1 + 2 {\gamma} \right) r^2 + 12 {\beta}^3 {\gamma} r^4
- 10 {\left( 1 - 2 {\gamma} \right) }^2 r^{1 + 4 {\gamma}} {\sigma}^4
\right. \nonumber \\
&&
+ 6 {\beta}^2 \sigma^2\left( 1 - 5 {\gamma} + 6 \gamma^2 \right) r^{2 \left( 1 + {\gamma} \right) }
\nonumber \\
&&
+ \beta r^{2 \gamma}\sigma^2 \left( -10 \left( 5 + 2 \gamma \right) r^3 +
3 {\left( 1 - 2 {\gamma} \right) }^2 \left( -3 + 4 {\gamma} \right) r^{2 {\gamma}} {\sigma}^2 \right) \nonumber \\
&&
+ 2 \alpha r \biggl( 3 {\beta}^2 \left( -1 + 4 {\gamma} \right) r^2 +
3 \beta \left( 2 - 7 {\gamma} + 6 {{\gamma}}^2 \right) r^{2 {\gamma}} {\sigma}^2
\nonumber \\
&&
\qquad - \left. 5 \left( -1 + 2 {\gamma} \right) r^{1 + 2 {\gamma}} {\sigma}^2 \biggr) \right]\,.\end{aligned}$$ Then the difference between the higher order approximation $\ln P^{ap2}$ given by (\[approx-higher-order\]) and the exact solution $\ln P^{ex}$ satisfies $$\ln P^{ap2}(\tau,r) - \ln P^{ex}(\tau,r) = o(\tau^6)$$ as $\tau \rightarrow 0^+$.
In Table \[irmodels-tab2\] we show $L_{\infty}$ and $L_2\,-\,$norms[^10] with respect to $r$ of the difference $\ln P^{ap} - \ln P^{ex}$ and $\ln P^{ap2} - \ln P^{ex}$ where we considered $r \in [0,0.15]$. We also compute the experimental order of convergence (EOC) in these norms. Recall that the experimental order of convergence gives an approximation of the exponent $\alpha$ of expected power law estimate for the error $\Vert\ln P^{ap}(\tau,.) - \ln P^{ex}(\tau,.)\Vert = O(\tau^{\alpha})$ as $\tau \rightarrow 0^+$. The $EOC_i$ is given by a ratio $$EOC_i = \frac{\ln (err_i/err_{i+1})}{\ln ( \tau_i/\tau_{i+1})},
\quad\hbox{where }\ \ err_i = \Vert\ln P^{ap}(\tau_i,.) - \ln P^{ex}(\tau_i,.)\Vert_p\,.$$
$\tau$ $\Vert\ln P^{ap} - \ln P^{ex}\Vert_{\infty}$ EOC $\Vert\ln P^{ap2} - \ln P^{ex}\Vert_{\infty}$ EOC
-------- ---------------------------------------------- ------- ----------------------------------------------- -------
1 $ 2.774 \times 10^{-7}$ 4.930 $4.682 \times 10^{-10}$ 7.039
0.75 $6.717 \times 10^{-8}$ 4.951 $ 6.181 \times 10^{-11}$ 7.029
0.5 $9.023 \times 10^{-9}$ 4.972 $3.576 \times 10^{-12}$ 7.004
0.25 $2.876 \times 10^{-10}$ – $2.786 \times 10^{-14}$ –
: The $L_{\infty}$ and $L_2\,-\,$errors for the original $\ln P_{CIR}^{ap}$ and improved $\ln P_{CIR}^{ap2}$ approximations. Parameters are set to be equal to $\alpha=0.00315$, $\beta=-0.0555$, $\sigma=0.0894$. Source: Stehlíková and Ševčovič [@stehlikova-sevcovic].
$\tau$ $\Vert\ln P^{ap} - \ln P^{ex}\Vert_{2}$ EOC $\Vert\ln P^{ap2} - \ln P^{ex}\Vert_{2}$ EOC
-------- ----------------------------------------- ------- ------------------------------------------ -------
1 $6.345 \times 10^{-8}$ 4.933 $9.828 \times 10^{-11}$ 7.042
0.75 1.535 $\times 10^{-8}$ 4.953 $1.296 \times 10^{-11}$ 7.031
0.5 2.061 $\times 10^{-9}$ 4.973 $7.492 \times 10^{-13}$ 7.012
0.25 6.563 $\times 10^{-11}$ – $5.805 \times 10^{-15}$ –
: The $L_{\infty}$ and $L_2\,-\,$errors for the original $\ln P_{CIR}^{ap}$ and improved $\ln P_{CIR}^{ap2}$ approximations. Parameters are set to be equal to $\alpha=0.00315$, $\beta=-0.0555$, $\sigma=0.0894$. Source: Stehlíková and Ševčovič [@stehlikova-sevcovic].
\[irmodels-tab2\]
In Table \[irmodels-tab2\] we show the $L_2\,-\,$error of the difference between the original and improved approximations for larger values of $\tau$. It turns out that the higher order approximation $P^{ap2}$ gives about twice better approximation of bond prices in the long time horizon up to 10 years.
### Approximation based on the Vasicek model
Our aim is to propose a formula which is as simple as possible, but still yields a good approximation to the exact bond prices. Using an approximation in calibration of the model requires many evaluations of its value for different sets of parameters, as well as times to maturity and the short rate levels. Therefore, its simple form can increase the efficiency of the calibration procedure. In particular, the approximation published by Stehlíková in [@stehlikova] presented in this section leads to a one-dimensional optimization problem.
Again, we consider the model (\[eq-ckls\]) in the risk neutral measure for the evolution of the short rate $r$ and the corresponding partial differential equation (\[PDE-1f\]) for the bond price $P(\tau,r)$.
Recall that in the case of Vasicek model, i.e., for $\gamma=0$, the solution $P_{vas}$ can be expressed in the closed form: $$\ln P_{vas}(\tau,r) = \left( \frac{\alpha}{\beta} + \frac{\sigma^2}{2 \beta^2} \right)
\left(\frac{1-e^{\beta \tau}}{\beta}+\tau \right)
+ \frac{\sigma^2}{4 \beta^3} (1-e^{\beta \tau})^2 + \frac{1-e^{\beta \tau}}{\beta} r.
\label{vasicek-price}$$ Now, let us consider a general model (\[eq-ckls\]) and the approximation of the bond price obtained by substituting the instantaneous volatility $\sigma r^{\gamma}$ for $\sigma$ in the Vasicek price (\[vasicek-price\]), i.e., $$\ln P^{ap}(\tau,r) = \left( \frac{\alpha}{\beta} + \frac{\sigma^2 r^{2 \gamma}}{2 \beta^2} \right)
\left(\frac{1-e^{\beta \tau}}{\beta}+\tau \right)
+ \frac{\sigma^2 r^{2 \gamma}}{4 \beta^3} (1-e^{\beta \tau})^2 + \frac{1-e^{\beta \tau}}{\beta} r.
\label{ckls-app-price}$$
[@stehlikova Theorem 1] \[theorem-accuracy-for-ln\] Let $P^{ap}$ be the approximate solution given by (\[ckls-app-price\]) and $P^{ex}$ be the exact bond price given as a solution to (\[PDE-1f\]). Then $$\ln P^{ap}(\tau,r) - \ln P^{ex}(\tau,r) = c_4(r) \tau^4 + o(\tau^4)$$ as $\tau \rightarrow 0^+$ where $$\begin{aligned}
\label{c5(r)}
c_4(r) &=& -\frac{1}{24} \gamma r^{2 \gamma -2} \sigma^2 [2 \alpha r + 2 \beta r^2 + (2 \gamma -1) r^{2 \gamma} \sigma^2].
\nonumber \end{aligned}$$
For the practical usage of the approximate formula, besides the order of accuracy also the absolute value of the error is significant.
Comparison of the approximation with the exact values in the case of CIR models and parameter values from [@choi-wirjanto] show (cf. [@stehlikova] for the exact figures) that for shorter maturities the differences are less than the accuracy to which the market data are quoted. Euribor, for example, is quoted in percentage points rounded to three decimal places. Moreover, Figure \[fig:approx-term-structures\] shows that even though the accuracy of this approximation is one order lower to that of the approximation from [@choi-wirjanto], it gives numerically comparable results for the real set of parameters.
Let us consider the calibration of the one-factor model based on the comparison of theoretical and market interest rates, where the parameters are chosen to minimize the function $$F=\frac{1}{mn} \sum_{i=1}^n \sum_{j=1}^m w_{ij} \left( R(\tau_j,r_i) - R_{ij} \right)^2, \label{ucelova-funkcia}$$ where $r_i$ ($i=1,\dots,n$) is the short rate observed on the $i$-th day, $\tau_j$ ($j=1,\dots,m$) is the $j$-th maturity of the interest rates in the data set, $R_{ij}$ is the interest rate with maturity $\tau_j$ observed on i-th day, $R(\tau,r)$ is the interest rate with maturity $\tau$ corresponding to the short rate $r$ computed from the model with the given parameters and $w_{ij}$ are the weights. In [@sevcovic-csajkova-1] and [@sevcovic-csajkova-2], this approach was used with $w_{ij}=\tau_j^2$ (i.e., giving more weight to fitting longer maturities) to calibrate Vasicek and CIR models using the explicit solutions for interest rates. To achieve the global minimum of the objective function, the authors used evolution strategies.
If we attempted to use this method to estimate a model with different $\gamma$ without analytical approximation, it would become computationally demanding, since each evaluation of the objective function would require numerical solutions of the PDE (\[PDE-1f\]). Note that the evaluation is needed for every member of the population in the evolution strategy (see [@sevcovic-csajkova-2] for details). Using an analytical approximation simplifies the computation of the objective function, but in general the dimension of the optimization problem is unchanged. We show that using the approximation proposed in this paper, we are able to reduce the calibration to a one-dimensional optimization problem which can be quickly solved using simple algorithms.
Hence we consider the criterion (\[ucelova-funkcia\]) with replacing $R(\tau,r)$ by its approximation $R^{ap}(\tau,r)$ calculated from (\[ckls-app-price\]). Note that the approximation formula $\ln P^{ap}$ is a linear function of parameters $\alpha$ and $\sigma^2$; it can be written as $$\ln P^{ap} (\tau,r)= c_0(\tau,r) + c_1(\tau,r) \alpha + c_2(\tau,r) \sigma^2,$$ where $$c_0 = \frac{1 - e^{\beta \tau}}{\beta} r, \; c_1 = \frac{1}{\beta} \left( \frac{1 - e^{\beta \tau}}{\beta} + \tau \right), \;
c_2 = \frac{r^{2 \gamma}}{2 \beta^2} \left( \frac{1- e^{\beta \tau}}{\beta} + \tau + \frac{ \left(1 - e^{\beta \tau} \right)^2}{2 \beta}
\right).$$ Hence taking the derivatives of (\[ucelova-funkcia\]) with respect to $\alpha$ and $\sigma^2$ and setting them equal to zero leads to a system of linear equations for these two parameters. It means that once we fix $\gamma$ and treat $\beta$ as parameter, we obtain the corresponding optimal values of $\alpha$ and $\sigma^2$ for each $\beta$. Substituting them into (\[ucelova-funkcia\]) then leads to a one-dimensional optimization problem. Doing this over a range of values of $\gamma$ allows us to find the optimal parameter $\gamma$ as well.
We show the proposed idea on simulated data. Once again, we consider the CIR model with parameters from [@choi-wirjanto] and simulate the daily term structures - interest rates with maturities of $1, 2, 3, \dots, 12$ months using the exact formula for CIR model - for a period of year. In the objective function (\[ucelova-funkcia\]) we use the weights $w_{ij}=\tau_j^2$ as in [@sevcovic-csajkova-1] and [@sevcovic-csajkova-2]. Afterwards we repeat the same procedure with maturities of 1, 2, 3, 4 and 5 years.
Results of the estimation for several values of $\gamma$ are presented in Table \[tab-calibration\], we show the estimated parameters and the optimal value of the objective function $F$.
$\gamma$ $\alpha$ $\beta$ $\sigma$ optimal value of $F$
---------- ---------- --------- ---------- -----------------------
0 0.00324 -0.0578 0.0176 1.1 $\times 10^{-12}$
0.25 0.00319 -0.0565 0.0403 2.9 $\times 10^{-13}$
0.5 0.00315 -0.0555 0.0896 1.1 $\times 10^{-15}$
0.75 0.00312 -0.0548 0.1912 6.3 $\times 10^{-13}$
1 0.00310 -0.0548 0.3813 2.5 $\times 10^{-12}$
: Estimation of the parameter $\gamma$ using the approximate formula for interest rates. The data were simulated using the exact formula with the parameters $\alpha=0.00315$, $\beta=-0.0555$, $\sigma=0.0894$, $\gamma=0.5$. Maturities used were $1,2,\dots,12$ months (above) and $1,2,\dots,5$ years (below). Source: Stehlíková, [@stehlikova]
$\gamma$ $\alpha$ $\beta$ $\sigma$ optimal value of $F$
---------- ---------- --------- ---------- -----------------------
0 0.00377 -0.0663 0.0214 1.0 $\times 10^{-8}$
0.25 0.00344 -0.0607 0.0432 2.4 $\times 10^{-9}$
0.5 0.00311 -0.0553 0.0860 2.2 $\times 10^{-10}$
0.75 0.00281 -0.0506 0.1688 6.7 $\times 10^{-9}$
1 0.00256 -0.0471 0.3238 2.7 $\times 10^{-8}$
: Estimation of the parameter $\gamma$ using the approximate formula for interest rates. The data were simulated using the exact formula with the parameters $\alpha=0.00315$, $\beta=-0.0555$, $\sigma=0.0894$, $\gamma=0.5$. Maturities used were $1,2,\dots,12$ months (above) and $1,2,\dots,5$ years (below). Source: Stehlíková, [@stehlikova]
\[tab-calibration\]
General one-factor models: power series expansions
--------------------------------------------------
The approximations considered in the previous sections share a common feature: their order of accuracy can be expressed in the form $$\ln P^{ap}(\tau,r) - \ln P (\tau,r) = c(r) \tau^{\omega} + o(\tau^{\omega}) \label{eq-order}$$ as $\tau \rightarrow 0^+$, where $P$ is the exact bond price and $P^{ap}$ is the proposed approximation. The relation (\[eq-order\]) asserts that the Taylor series of $\ln P^{ap}$ and $\ln P$ coincide up to the certain order. In particular, in [@stehlikova-sevcovic] it has been shown that for the formula for CKLS model from [@choi-wirjanto] the relation (\[eq-order\]) holds with $\omega=5$ and an improvement leading to $\omega=7$ has been derived. In [@stehlikova] a simple formula with $\omega=4$ has been proposed. Similar estimates hold in the case of multi-factor models. These results suggest that the Taylor expansion (either of the price itself and its logarithm) could be a good approximation too.
Let us consider a general one-factor model with constant coefficients $$d r = \mu(r) \, d t + \sigma(r) \, d w \label{eq:sde2}.$$ Recall that the price of the bond $P(\tau,r)$ is a solution to the partial differential equation $$-\partial_{\tau} P + \mu(r) \partial_{r} P + \frac{1}{2} \sigma^2(r) \partial^2_{r} P - rP=0 \label{eq:pde-tau}$$ for all $r>0$, $\tau \in (0,T)$ and the initial condition $P(0,r)=1$ for all $r>0$. Easy transformation of the PDE leads to the equation which is satisfied by the logarithm of the bond price, i.e., $f(\tau,r)= \log P(\tau,r)$: $$-\partial_{\tau} f = \frac{1}{2} \sigma^2(r) \left[ (\partial_{r} f)^2 + \partial^2_{rr} f \right] + \mu(r) \partial_{r} f - r =0
\label{eq-f}$$ for all $r>0$, $\tau \in (0,T)$ and the initial condition $f(0,r)=0$ for all $r>0$. Writing these functions in series expansions around $\tau=0$ in the form $$P(\tau,r) = \sum_{j=0}^{\infty} c_j(r) \tau^j, f(\tau,r) = \sum_{j=0}^{\infty} k_j(r) \tau^j \label{eq:p_f_series}$$ enables us to compute the parameters $c_j$ or $k_j$ recursively in the closed form. A practical usage of this approach is determined by the speed of convergence of these series for reasonable values of $\tau$ and $r$. Then, we can approximate the bond prices and their logarithms by terminating the infinite sums (\[eq:p\_f\_series\]) at a certain index $J$.
We show the results from [@stehlikova:taylor]. Firstly, the approximation is tested on CKLS model with the same parameters as in the previous chapter; the results suggest the possibility of practical usage of the proposed matter. As an another example, the Dothan model is considered. The Dothan model [@irmodels-dothan] assumes that the short rate in the risk neutral measure follows the stochastic differential equation $$dr = \mu r dt + \sigma r dw.$$ The zero-coupon bond in the Dothan model has an explicit solution, but it is computationally complicated (cf. [@brigo-mercurio]). Therefore, we use the Dothan bond prices computed in [@hansen-jorgensen] for which the error estimate is available. They are accurate to the given four decimal digits.
Setting $\mu(r)=\mu r$ and $\sigma(r)=\sigma r$ into the recursive formulae for coefficients results in the coefficients for the price and its logarithm. In the numerical experiments we use the values from [@hansen-jorgensen]. The authors price zero coupon bonds which pays 100 USD at maturity $T$ (hence its price is 100 times the value considered so far). Using their iterative algorithm, for $\tau=1,2,3,4,5,10$ they obtain the accuracy to four decimal digits for all combinations of parameters and in several cases also for higher maturities. Selected values from [@hansen-jorgensen] are used to test the approximation for a wider range of parameters and maturities, as shown in Table \[tab-dothan-2\].
parameters $\tau$ Taylor, J=3 Taylor, J=5 Taylor, J=7 exact [@hansen-jorgensen]
------------------------------ -------- ------------- ------------- ------------- --------------------------- --
$\mu=0.005$, $\sigma^2=0.01$ 1 96.5523 96.5523 96.5523 96.5523
2 93.2082 93.2082 93.2082 93.2082
3 89.9666 89.9663 89.9663 89.9663
4 86.8260 86.8251 86.8251 86.8251
5 83.7852 83.7830 83.7830 83.7830
10 70.0312 69.9977 69.9982 69.9982
$\mu=0.005$, $\sigma^2=0.02$ 1 96.5525 96.5525 96.5525 96.5525
2 93.2099 93.2098 93.2098 93.2098
3 89.9721 89.9715 89.9715 89.9715
4 86.8391 86.8370 86.8370 86.8370
5 83.8362 83.8056 83.8057 83.8057
10 70.4396 70.1530 70.1551 70.1551
$\mu=0.005$, $\sigma^2=0.03$ 1 96.5527 96.5527 96.5527 96.5527
2 93.2115 93.2113 93.2113 93.2113
3 89.9776 89.9767 89.9767 89.9767
4 86.8521 86.8491 86.8491 86.8491
5 83.8362 83.8287 83.8287 83.8287
10 70.4396 70.3112 70.3151 70.3151
: [Bond prices in the Dothan model with indicated parameters and maturities, and the initial value of the short rate $r_0=0.035$ - comparison of Taylor approximation with exact values. Source Stehlíková [@stehlikova:taylor]]{}
\[tab-dothan-2\]
The idea of the short time asymptotic expansion can be enhanced, by considering the so-called [exponent expansion]{} to derive a closed-form short-time approximation of the Arrow-Debrew prices, from which the prices of bonds or other derivatives can be obtained by a single integration. This technique, originally introduced in chemical physics by Makri and Miller [@makri], was introduced to finance by Capriotti [@ee]. In [@stehlikova-capriotti] by Stehlíková and Capriotti, it was employed to compute the bond prices in the Black-Karasinski model.
The exponent expansion is derived for the bond prices in short rate models with $r=r(x)$, where the auxiliary process has the form $$d x(t) = \mu(x) \,d t + \sigma \,d w,$$ where $\mu(x)$ is a drift function. Note that the process has a constant volatility $\sigma$, but in the general case it is possible to map to this case a general state dependent volatility function by means of an integral transformation. Note that this transformation is used also by Aït-Sahalia in [@ait-sahalia-transition] in his approximation of transition densities. The bond prices are not computed directly; instead, so-called Arrow-Debreu prices are approximated by a closed form formula and the bond prices are obtained by a single numerical integration. The Arrow-Debreu prices $\psi(x,T;x_0)$ are for each $x_0$ given as solutions to the partial differential equation (see [@shreve]) $$\label{fp}
\partial_{t} \psi = \Big( -r(x) - \partial_x \mu(x) + \frac{1}{2}\sigma^2 \partial_x^2 \Big) \psi,$$ with the initial condition $\psi(x, 0; x_0) = \delta(x-x_0)$. Looking for the solution in the form $$\psi(x, t; x_0) = \frac{1}{\sqrt{2\pi\sigma^2 t}}
\exp {\left[ -\frac{(x-x_0)^2}{2\sigma^2 t}-W(x,t;x_0)\right]},
\label{ansatz}$$ and inserting it into (\[fp\]) leads to a partial differential equation for $W(x, t; x_0)$. Writing it in the form $$W(x, t; x_0) = \sum_{n=0}^{\infty} W_n(x;x_0)\, t^n~,
\label{w}$$ allows a recursive computation of the functions $W_n(x;x_0)$ as solutions to first order linear ordinary differential equations.
This form of expansion for the bond prices results in a more rapid convergence especially for longer maturities, compared with the simple Taylor expansion described previously, see Table \[tab:ee-taylor\].
0.2cm
order Taylor exponent expansion Taylor exponent expansion
------- ---------- -------------------- ---------- --------------------
1 0.970000 0.969249 0.940000 0.937431
2 0.968045 0.968138 0.932179 0.933037
3 0.968123 0.968140 0.932807 0.933077
4 0.968141 0.968142 0.933097 0.933105
5 0.968142 0.968142 0.933118 0.933106
6 0.968142 0.968142 0.933110 0.933106
: [ Comparison of successive approximations of the bond price with six-months (left) and one-year (right) maturity in Black-Karasinski model with parameters $a=1$, $b=\ln 0.04$, $\sigma=0.85$, when the initial level of the short rate is $r=0.06$. ]{}
\[tab:ee-taylor\]
An important advantage that separates the exponential expansion is the possibility to systematically improve its accuracy over large time horizons by means of the convolution approach, cf. [@stehlikova-capriotti] for the algorithm. This allows to produce results accurate to more than 4 significant digits even for zero coupon bonds with maturities over 20 years. This is documented in Table \[tab:stehlikova-capriotti\] where the results are compared with Monte Carlo prices.
0.2cm
maturity convolution step: 5 convolution step: 2.5 convolution step: 1 MC
---------- --------------------- ----------------------- --------------------- --------
5 0.65949 0.65955 0.65966 0.6597
10 0.46139 0.46222 0.46229 0.4623
20 0.26812 0.26827 0.26831 0.2683
: [Bond prices computed with the 6th order Exponent Expansion and different convolution steps in Black-Karasinski model with parameters $a=1$, $b=\ln 0.04$, $\sigma=0.85$, when the initial level of the short rate is $r=0.06$, compared with the price obtained by Monte Carlo. Source: Stehlíková and Capriotti, [@stehlikova-capriotti].]{}
\[tab:stehlikova-capriotti\]
Fast time scale of volatility in stochastic volatility models
-------------------------------------------------------------
In the paper [@stehlikova-sevcovic-kybernetika] by Stehlíková and Ševčovič, studied a generalized CIR model with a stochastic volatility. The instantaneous interest rate (short rate) $r$ is modeled by the mean reverting process of the form (\[eq:cir\_sde\]) where the constant $\sigma$ appearing in the volatility function $\sigma \sqrt{r}$ is replaced by a square root of a stochastic dispersion $y$, i.e. $$d r = \kappa(\theta - r) \, d t + \sqrt{y} \sqrt{r} \, d w_r\,.
\label{CIR-short-rate-stoch}$$ The stochastic differential equation for the short rate is given by $$d y=\alpha(y) \, d t + v \sqrt{y} \, d w_y,
\label{slowly-osc}$$ with certain conditions given on the function $\alpha\!\!: [0,\infty) \rightarrow \mathbb{R}$ at zero and infinity, see [@stehlikova-sevcovic-kybernetika Assumption A] and a concrete example[^11] in [@stehlikova-sevcovic-kybernetika Lemma 1]. The differentials of the Wiener processes $d w_y$ and $d w_r$ are assumed to be uncorrelated.
The paper provides a tool for modeling the effects of rapidly oscillating stochastic volatility that can be observed in real markets, cf. [@sircar1], [@sircar2]. If the length of the time scale for dispersion $y$ is denoted by $\varepsilon$, the equation (\[slowly-osc\]) for the variable $y$ reads as follows: $$d y=\frac{\alpha(y)}{\varepsilon}\, d t + \frac{v \sqrt{y}}{\sqrt{\varepsilon}} \, d w_y.
\label{rapid-osc}$$ In what follows we will assume that $0< \varepsilon \ll 1$ is a small singular parameter. The density of the conditional distribution of the process is given by the solution to Fokker-Planck equation. The density $g(y)$ of its stationary distribution, which is widely used in the computations from [@stehlikova-sevcovic-kybernetika], is then given by the normalized solution to the stationary Fokker-Planck equation, which reads as $$\frac{v^2}{2} \partial^2_{y}(yg) - \partial_y(\alpha(y)g) =0 \label{eq:stacionarnyFP}$$ for the process (\[rapid-osc\]). Notice that the limiting density function $g$ is independent of the scaling parameter $\varepsilon>0$.
The market prices of risk functions are considered to be in the form $\tilde\lambda_1(t,r,y)=\lambda_1 \sqrt{r}\sqrt{y}$, $ \tilde\lambda_2(t,r,y)=\lambda_2 \sqrt{y}$, where $\lambda_1,\lambda_2\in \mathbb{R}$ are constants (note that this is a generalization of the original one-factor CIR model which assumes the market price of risk to be proportional to the square root of the short rate $r$). Then, we rewrite the partial differential equation for the bond price $P$ in the operator form: $$(\varepsilon^{-1} \mathcal{L}_0 + \varepsilon^{-1/2} \mathcal{L}_1 + \mathcal{L}_2 ) P^{\varepsilon} = 0,
\label{eq:bondprice2}$$ where the linear differential operators $\mathcal{L}_0, \mathcal{L}_1, \mathcal{L}_2$ are defined as follows: $$\mathcal{L}_0=\alpha(y) \partial_{y} + \frac {1}{2} v^2 y \partial^2_{y}, \:
\mathcal{L}_1= - \lambda_2 v y \partial_{y}, \:
\mathcal{L}_2= \partial_{t} + (\kappa (\theta-r) - \lambda_1 r y ) \partial_{r} + \frac{1}{2} r y \partial^2_{rr} - r.$$ Next we expand the solution $P^{\varepsilon}$ into Taylor power series: $$\label{expansion}
P^{\varepsilon}(t,r,y)= \sum_{j=0}^\infty \varepsilon^{\frac{j}{2}} P_j(t,r,y)$$ with the terminal conditions $P_0(T,r,y)=1, P_j(T,r,y)=0 \; \textrm{for } j \geq 1$ at expiry $t=T$. The main result of this paper is the examination the singular limiting behavior of a solution $P^\varepsilon$ as $\varepsilon\to 0^+$. More precisely, it determines the first three terms $P_0,P_1,P_2$ of the asymptotic expansion (\[expansion\]).
The main tool in the derivation is averaging with respect to the limiting distribution, whose density $g$ is given by (\[eq:stacionarnyFP\]), and is denoted by brackets $\langle \cdot \rangle$ in the following. In particular, the following two propositions are essential: Firstly, a function $\psi$, for which $\mathcal{L}_0\psi$ is bounded, satisfies $\langle \mathcal{L}_0 \psi \rangle =0$ (see [@stehlikova-sevcovic-kybernetika Lemma 3]). Secondly, [@stehlikova-sevcovic-kybernetika Lemma 4] gives an expression for $\psi_y$ and $\langle \mathcal{L}_1 \psi \rangle $, where $\psi$ is a solution of $\mathcal{L}_0 \psi = F$ with the right-hand side being a given function satisfying $\langle F \rangle =0$.
The solution $P^\varepsilon = P^\varepsilon(t,r,y)$ of the bond pricing equation (\[eq:bondprice2\]) can be approximated, for small values of the singular parameter $0<\varepsilon\ll 1$, by $$P^\varepsilon(t,r,y) \approx P_0(t,r) +\sqrt{\varepsilon} P_1(t,r) + \varepsilon P_2(t,r,y) + O(\varepsilon^\frac32)$$ and the main result of the paper lies in the derivation of the functions $P_0,P_1,P_2$. Note that the first two terms $P_0,P_1$ are independent of the $y$-variable representing unobserved stochastic volatility.
The first term $P_0$ is a solution to the averaged equation $\langle \mathcal{L}_2 \rangle P_0=0$, which is the partial differential equation for the bond price in one-factor CIR model with parameters set to the averaged values (with respect to the limiting distribution) from the model studied here. It has a form $$P_0(t,r) = A_0(t) e^{-B(t) r}, \label{eq:p0-form}$$ where the functions $A_0$ and $B$ are given by a system of ordinary differential equations which can be solved in a closed form. Neither the second term $P_1$ depends on the instantaneous level of the process $y$. The equation for the $P_1$ reads as $$\langle \mathcal{L}_2 P_1 \rangle = f(t) r e^{-B(t)r},$$ where the function $B$ comes from (\[eq:p0-form\]) and the function $f$ is obtained from the model parameters and the solution (\[eq:p0-form\]) in a closed form. The solution has the form $P_1(t,r) = (A_{10}(t) + A_{11}(t)r) e^{-B(t) r}$ with the function $B$ being the same as in (\[eq:p0-form\]) and the functions $A_{10}, A_{11}$ satisfying a system of linear ordinary differential equation. The next term in the expansion, $P_2$, non-trivially depends on the $y$-variable. It is decomposed into its expected value and zero-mean fluctuations as $$P_2(t,r,y)=\bar{P}_2(t,r) + \tilde{P}_2(t,r,y)$$ where $\langle \tilde{P}_2 \rangle =0$. The function $\tilde{P}_2$ can be computed by integration, using the results obtained so far. The function $\bar{P}_2$ satisfies the equation $$\langle \mathcal{L}_2 \bar{P}_2 \rangle = (a(t)+b(t)r+c(t)r^2)e^{-B(t)r},$$ where the functions $a,b,c$ are given, and therefore has the form $
\bar P_2(t,r) = (A_{20}(t) + A_{21}(t) r + A_{22}(t) r^2) e^{-B(t) r}$ where the function $B$ is the same as in (\[eq:p0-form\]) and the functions $A_{20}, A_{21}, A_{22}$ are solutions to a linear system of ODEs. More detailed computations can be found in [@stehlikova-sevcovic-kybernetika].
Recall the Fong-Vasicek model with stochastic volatility in which the short rate is given by the following pair of stochastic differential equation $$\begin{aligned}
d r &=& \kappa_1 (\theta_1 - r) \: d t + \sqrt{y} \: d w_1, \nonumber \\
d y &=& \kappa_2 (\theta_2 - y) \: d t + v \sqrt{y} \: d w_2. \label{eq:fvas}\end{aligned}$$ For a suitable choices of market prices of risk, computation of the bond prices can be reduced into solving ordinary differential equations. This computational simplicity makes it a suitable choice for assessing the quality of the approximation of the kind described above. Introducing fast time scale of volatility, the equation (\[eq:fvas\]) becomes (cf. equation (\[rapid-osc\])) $$d y = \frac{\kappa_2}{\varepsilon} (\theta_2 - y) \: d t + \frac{v}{\sqrt{\varepsilon}} \sqrt{y} \: d w_2. \label{eq:fvas2}$$ However, when estimating parameters using the real data, from the parameters of (\[eq:fvas2\]) we are able to obtain only $\theta_2$, $\tilde{\kappa}_2= \frac{\kappa_2}{\varepsilon}$ and $\tilde{v}=\frac{\kappa_2}{\sqrt{\varepsilon}}$. Hence, we are not able to reconstruct three parameters $\kappa_2, v, \varepsilon$ from two values $\tilde{\kappa}_2, \tilde{v}$.
Therefore, in the master thesis by Selečéniová [@dp-seleceniova], supervised by Stehlíková, another approach was used, following the parameterization used by Danilov and Mandal in [@danilov-mandal-1] and [@danilov-mandal-2]. Strong mean-reversion in the process for volatility can be characterized by a large value of $\kappa_y$. Hence we can define $\varepsilon=1/\kappa_y$ and expect it to be small enough to be used as a perturbation parameter. In [@dp-seleceniova], the derivation similar to that above has been made to compute the first two terms of the bond price expansion, leading to the approximation of the bond price of order zero $$P^\varepsilon(t,r,y) \approx P_0(t,r)$$ and of order one $$P^\varepsilon(t,r,y) \approx P_0(t,r) +\sqrt{\varepsilon} P_1(t,r).$$ Then, the resulting interest rates were compared with exact values. In Table \[tab:seleceniova\] we present sample results.
---------- ------------------------ ------------------------ ------------------------ --------- ---------
exact interest rate
maturity $y=1.6 \times 10^{-4}$ $y=2.4 \times 10^{-4}$ $y=3.2 \times 10^{-4}$ order 0 order 1
1 0.0424 0.0426 0.0429 0.0427 0.0432
2 0.0448 0.0451 0.0455 0.0451 0.0458
3 0.0470 0.0474 0.0478 0.0473 0.0482
4 0.0491 0.0495 0.0498 0.0493 0.0502
5 0.0510 0.0514 0.0517 0.0511 0.0521
6 0.0527 0.0531 0.0534 0.0528 0.0538
7 0.0543 0.0547 0.0550 0.0543 0.0553
8 0.0558 0.0561 0.0564 0.0557 0.0567
9 0.0572 0.0575 0.0578 0.0570 0.0580
10 0.0584 0.0587 0.0590 0.0582 0.0592
---------- ------------------------ ------------------------ ------------------------ --------- ---------
: [Interest rates from Fong-Vasicek model: comparison of order 0 and order 1 approximations with the exact values. Parameters are taken to be equal to $\kappa_1=0.109, \kappa_2 = 1.482, \theta_1 = 0.0652, \theta_2 - 0.000264, v= 0.01934, \lambda_1 = -11, \lambda_2 = -6, r=0.04$. Source: Selečéniová, [@dp-seleceniova].]{}
\[tab:seleceniova\]
Let us remark that even though the zero-order approximation of the bond price equals to the bond price from one-factor model with averaged coefficients, this is not the averaged bond price $\langle P(t,r,y) \rangle$. There is even a stronger result: The averaged bond price $\langle P(t,r,y) \rangle$, although it is a function of $t$ and $r$, does not equal to the bond price in any one-factor model, as it has been shown in [@miyazaki] which is also reprinted at the end of this thesis.
Convergence multiple-factor models
----------------------------------
The idea of approximating the bond prices in a model with general volatility by substituting the instantaneous volatility into a simple model of Vasicek type (i.e., with constant volatility) has been successfully applied also in multi-factor models:
Convergence models form a special class of two-factor models. A convergence model is used to model the entry of observed country into the monetary union (EMU). It describes the behavior of two short-term interest rates, the domestic one and the instantaneous short rate for EMU countries. European short rate is modeled using a one-factor model. It is assumed to have an influence on the evolution of the domestic short rate and hence it enters the SDE for its evolution. This kind of model was proposed for the first time in [@corzo-schwarz]. The model is based on Vasicek model, the volatilities of the short rates are constant. Analogical model of Cox-Ingersoll-Ross type, where the volatilities are proportional to the square root of the short rate, was considered in [@lacko-dp] and [@lacko-dphannover]. In the following sections we describe these two models and show how they price the bonds. Then we present a generalization with nonlinear volatility, which is analogous to the volatility in one-factor CKLS model.
Let us consider a model defined by the following system of SDEs: $$\begin{aligned}
\label{eq:2FactorModel}
d r &=& \mu _r (r,x,t)d t + \sigma_r (r,x,t)d w_1, \nonumber \\
d x &=& \mu _x (r,x,t)d t + \sigma_x (r,x,t)d w_2,\end{aligned}$$ where $\rho \in (-1,1)$ is the correlation between the increments of Wiener processes $W_1$ and $W_2$, i.e. $Cov(dW_1,dW_2)=\rho\, d t$. Process $x$ is a random process, which is connected with instantaneous rate. It can be a long-term interest rate, a short-term interest rate in another country, etc. Relations between real and risk-neutral parameters are analogous as in the one-factor case: $$\begin{aligned}
(\text{risk\--neutral drift function})_r = (\text{real drift function})_r - \lambda_r(r,x,t)\times(\text{volatility})_r, \\
(\text{risk\--neutral drift function})_x = (\text{real drift function})_x - \lambda_x(r,x,t)\times(\text{volatility})_x,\end{aligned}$$ where $\lambda_r$, $\lambda_x$ are market prices of risk of the short rate and the factor $x$ respectively.
If the short rate satisfies SDE (\[eq:2FactorModel\]) in the real measure and market prices of risk are $\lambda_r(r,x,t), \lambda_x(r,x,t)$, then the bond price $P$ satisfies the following PDE (assuming that the factor $x$ is positive): $$\begin{aligned}
\frac{\partial P}{\partial t}+(\mu_r(r,x,t)-\lambda_r(r,x,t)\sigma_r(r,x,t))\frac{\partial P}{\partial r}+(\mu_x(r,x,t)-\lambda_x(r,x,t)\sigma_x(r,x,t))\frac{\partial P}{\partial x}\\
+\frac{\sigma_r(r,x,t)^2}{2}\frac{\partial^2 P}{\partial r}
+ \frac{\sigma_x(r,x,t)^2}{2}\frac{\partial^2 P}{\partial x}+\rho \sigma_r(r,x,t) \sigma_x(r,x,t) \frac{\partial^2 P}{\partial r \partial x}-rP=0\\\end{aligned}$$ for $r, x > 0$, $t \in (0,T)$ and the terminal condition $P(r,x,T)=1$ for $r, x > 0$. The PDE is derived using It$\hat{\text{o}}$ lemma and construction of risk-less portfolio, see, e.g. [@kwok],[@brigo-mercurio].
### Convergence model of the CKLS type
The paper [@zikova-stehlikova] is focused a convergence model of the CKLS type. Recall that the exact bond prices are known in the case of Vasicek-type model and their computation can be simplified to numerical solution of ordinary differential equations in the case of CIR-type model with uncorrelated increments of the two Wiener processes. In [@zikova-stehlikova], the general CKLS model with uncorrelated Wiener processes (the effect of correlation can be seen only in higher order terms, when taking $\tau$ as a small parameter, numerical results presented in the paper show that the difference often occurs on decimal places which are not observable taking the precision of market quotes into account) is considered. Approximation formula from [@stehlikova] described in the previous section is used to compute European bond prices and in an analogous way, an approximation for domestic bond prices is proposed. It is tested numerically for CIR-type model and a general order of accuracy is derived. Then, a calibration procedure is suggested, tested on simulated data and applied to read data. The simple form of the approximation again allows relatively simple calibration procedure.
### A three-factor convergence model
A one-factor model is not always sufficient to model the European short rate in convergence model (as suggested by calibration results in [@zikova-stehlikova]), which affects also the appropriateness of the convergence model for the domestic currency. In paper [@stehlikova-zikova] by Stehlíková and Zíková, a three factor convergence model is suggested and provides first steps in the analysis of approximation formulae for domestic bond prices. The European short rate is modeled as a sum of two CKLS-type factor, as described in the previous point, and the domestic rate follows a process reverting to the European rate.
The fit of the convergence model from [@zikova-stehlikova] suggests looking for a more suitable approximation of the short rate. The paper [@halgasova-stehlikova-zikova] by Halgašová, Stehlíková and Zíková studies an estimation the short rate together with parameters of the model in Vasicek model. It is based on noting that for Vasicek model, the objective function (\[ucelova-funkcia\]) for the calibration is quadratic not only in parameters $\alpha$ and $\sigma^2$, but also in values of the short rates $r_1,\dots,r_n$.
Figure \[fig:halgasova-stehlikova-zikova\] shows a comparison of the estimated short rate from Euribor term structures with a market overnight rate. The choice of the time frame for the calibration was motivated by a possible use as an input for a convergence model: Slovakia adopted the Euro currency in 2009 and Estonia in 2011.
![Estimating the short rate from Euribor term structures and its comparison with overnight rate Eonia. Source: Halgašová, Stehlíková, Zíková, [@halgasova-stehlikova-zikova].[]{data-label="fig:halgasova-stehlikova-zikova"}](figures/habilitacia/tatra08 "fig:"){width="48.00000%"} ![Estimating the short rate from Euribor term structures and its comparison with overnight rate Eonia. Source: Halgašová, Stehlíková, Zíková, [@halgasova-stehlikova-zikova].[]{data-label="fig:halgasova-stehlikova-zikova"}](figures/habilitacia/tatra10 "fig:"){width="48.00000%"}
Using the approximation of the bond prices in the CKLS model, this algorithm can be modified for estimating the short rate also in the CKLS model. This has been done in the master thesis [@dp-mosny] by Mosný, supervised by Stehlíková. In the case of a general CKLS model, the objective function is not quadratic, but it is proposed to make a substitution $y_i = \sigma^2 r_i^{2 \gamma}$ in the objective function, which results in the new objective function which we minimize with respect to $\alpha, \beta, \sigma^2$ (model parameters), $r_1, \dots, r_n$ (short rates), $y_1, \dots, y_n$ (auxiliary variables treated as independent in the first step). In this way, for each $\beta$ a quadratic optimization problem is solved. For each $\beta$, there is therefore the optimal value of $\tilde{F}$ which is then used to find the optimal value of $\beta$. Note that the variables $r_i$ and $y_i$ are not independent, the ratio $y_i/r_i^{2 \gamma}$ is equal to $\sigma^2$. By treating them as independent variables, $y_i$ can be seen as approximations of $\sigma^2 r_i^{2 \gamma}$ when using real data. Hence the ratios $y_i/r_i^{2 \gamma}$ should provide a good approximation to $\sigma^2$. It is estimated as a median of these ratios.
### Convergence model of Vasicek type
The first convergence model was proposed in the paper [@corzo-schwarz] by Corzo and Schwartz in the real probability measure: $$\begin{aligned}
\label{eq:VasicekModel1}
d r_d &=& \left(a+b\left(r_e-r_d\right)\right)d t + \sigma_d d w_d,
\nonumber
\\
d r_e &=& \left(c\left(d-r_e\right)\right)d t + \sigma_e d w_e,\end{aligned}$$ where $Cov(dW_1,dW_2)=\rho d t$. They considered constant market prices of risk, i. e. $\lambda_d(r_d,r_e,\tau)=\lambda_d$ and $\lambda_e(r_d,r_e,\tau)=\lambda_e$. Hence for the European interest rate we have one-factor Vasicek model and we can easily price European bonds. Coefficient $b > 0$ expresses the power of attracting the domestic short rate to the European one with the possibility of deviation determined by the coefficient $a$. Rewriting the model into risk-neutral measure we obtain: $$\begin{aligned}
\label{eq:VasicekModel2}
d r_d &=& \left(a+b\left(r_e-r_d\right)- \lambda_d \sigma_d\right)d t + \sigma_d d w_d, \nonumber
\\
d r_e &=& \left(c\left(d-r_e\right) - \lambda_e \sigma_e\right)d t + \sigma_e d w_e,\end{aligned}$$ where $Cov[dW_d,dW_e] = \rho d t$. We consider a more general model in risk-neutral measure, in which the risk-neutral drift of the domestic short rate is given by a general linear function of variables $r_d$, $r_e$ and the risk-neutral drift of the European short rate is a general linear function of $r_e$. It means that the evolution of the domestic and the European short rates is given by: $$\begin{aligned}
\label{eq:VasicekModel3}
d r_d &=& \left(a_1+a_2r_d+a_3r_e\right)d t + \sigma_d d w_d, \\
d r_e &=& \left(b_1+b_2r_e\right)d t + \sigma_e d w_e,\end{aligned}$$ where $Cov[dW_d,dW_e] = \rho d t$. Note that the system (\[eq:VasicekModel3\]) corresponds to the system (\[eq:VasicekModel2\]) with $a_1=a-\lambda_d\sigma_d$, $a_2=-b$, $a_3=b$, $b_1=cd-\lambda_e\sigma_e$, $b_2=-c$. Price $P(r_d,r_e,\tau)$ of a bond with time to maturity $\tau=T-t$ then satisfies the PDE: $$\begin{aligned}
\label{eq:PDRvas}
-\frac{\partial P}{\partial \tau}+(a_1+a_2r_d+a_3r_e)\frac{\partial P}{\partial r_d}+(b_1+b_2r_e)\frac{\partial P}{\partial r_e}\nonumber
\\
+\frac{\sigma_d^2}{2}\frac{\partial^2 P}{\partial r_d^2}+\frac{\sigma_e^2}{2}\frac{\partial^2 P}{\partial r_e^2}+\rho \sigma_d \sigma_e \frac{\partial^2P}{\partial r_d \partial r_e} -r_dP&=0,\end{aligned}$$ for $r_d,r_e>0,$ $\tau \in (0,T)$ and the initial condition $P(r_d,r_e,0)=1$ for $r_d, r_e > 0.$ Its solution can be found in the same way as in the original paper [@corzo-schwarz]. Assuming the solution in the form $$\begin{aligned}
\label{eq:SolutionInSeparateForm}
P(r_d,r_e,\tau)=e^{A(\tau)-D(\tau)r_d-U(\tau)r_e}, \end{aligned}$$ and setting it into the equation (\[eq:PDRvas\]) we obtain the system of ordinary differential equations (ODEs): $$\begin{aligned}
\label{eq:EquationsVas}
\dot{D}(\tau)&=&1+a_2D(\tau),\nonumber \\
\dot{U}(\tau)&=&a_3D(\tau)+b_2U(\tau),\\
\dot{A}(\tau)&=&-a_1D(\tau)-b_1U(\tau)+\frac{\sigma_d^2 D^2(\tau)}{2}+\frac{\sigma_e^2 U^2(\tau)}{2}+\rho \sigma_d \sigma_e D(\tau) U(\tau)\nonumber \end{aligned}$$ with initial conditions $A(0)=D(0)=U(0)=0$. The solution of this system is given by: $$\begin{aligned}
\label{eq:SOLUTIONvas2}
D(\tau)&=&\frac{-1+e^{a_2\tau}}{a_2},\nonumber \\
U(\tau)&=& \frac{a_3\bigl(a_2-a_2 e^{b_2\tau}+b_2
\left( -1+e^{a_2\tau}\right) \bigl)}{a_2\left(a_2-b_2\right)b_2},\\
A(\tau)&=&\int_0^{\tau} -a_1D(s)-b_1U(s)+\frac{\sigma_d^2 D^2(s)}{2}+\frac{\sigma_e^2 U^2(s)}{2}+\rho \sigma_d \sigma_e D(s) U(s) \mbox{d}s.\nonumber \end{aligned}$$ Note that the function $A(\tau)$ can be easily written in the closed form without an integral. We leave it in this form for the sake of brevity. Furthermore, we consider only the case when $a_2 \neq b_2$. If $a_2 = b_2$, then $U(\tau)$ has another form, but it is a very special case and we will not consider it further.
### Convergence model of CIR type
Firstly we formulate the convergence model of CIR type (i.e. the volatilities are proportional to the square root of the short rates) in the real measure. $$\begin{aligned}
\label{eq:CIRmodelREAL}
d r_d &=& \left(a+b\left(r_e-r_d\right)\right)d t + \sigma_d \sqrt{r_d}d w_d, \nonumber\\
d r_e &=& \left(c\left(d-r_e\right)\right)d t + \sigma_e \sqrt{r_e}d w_e,\end{aligned}$$ where $Cov[dW_d,dW_e] = \rho d t$. If we assume the market prices of risk equal to $\lambda_e\sqrt{r_e}$, $\lambda_d\sqrt{r_d}$ we obtain risk neutral processes of the form: $$\begin{aligned}
\label{eq:CIRmodel1}
d r_d &=& \left(a_1+a_2r_d+a_3r_e\right)d t + \sigma_d \sqrt{r_d} d w_d, \nonumber \\
d r_e &=& \left(b_1+b_2r_e\right)d t + \sigma_e \sqrt{r_e}d w_e, \end{aligned}$$ where $Cov[dW_d,dW_e] = \rho d t$. In what follows we consider this general risk- neutral formulation (\[eq:CIRmodel1\]).
The European short rate is described by one-factor CIR model, so we are able to price European bonds using an explicit formula. Price of domestic bond $P(r_d,r_e,\tau)$ with maturity $\tau$ satisfies the PDE $$\begin{aligned}
\label{eq:PDRcir}
-\frac{\partial P}{\partial \tau}+(a_1+a_2r_d+a_3r_e)\frac{\partial P}{\partial r_d}+(b_1+b_2r_e)\frac{\partial P}{\partial r_e} \nonumber \\
+\frac{\sigma_d^2r_d^{2}}{2}\frac{\partial^2 P}{\partial r_d^2}+\frac{\sigma_e^2r_e^{2}}{2}\frac{\partial^2 P}{\partial r_e^2}+\rho \sigma_d \sqrt{r_d} \sigma_e \sqrt{r_e} \frac{\partial^2P}{\partial r_d \partial r_e} -r_dP&=0,\end{aligned}$$ for $r_d,r_e>0, \tau \in (0,T)$ with the initial condition $P(r_d,r_e,0)=1$ for $r_d, r_e > 0.$ It was shown in [@lacko-dp] (in a slightly different parametrization of the model) that solution in the form (\[eq:SolutionInSeparateForm\]) exists only when $\rho=0$. In this case we obtain system of ODEs $$\begin{aligned}
\label{eq:EquationsCIR}
\dot{D}(\tau)&=&1+a_2D(\tau)-\frac{\sigma_d^2 D^2(\tau)}{2},\nonumber \\
\dot{U}(\tau)&=&a_3D(\tau)+b_2U(\tau)-\frac{\sigma_e^2 U^2(\tau)}{2},\\
\dot{A}(\tau)&=&-a_1D(\tau)-b_1U(\tau),\nonumber\end{aligned}$$ with initial conditions $A(0)= D(0) = U(0) = 0,$ which can be solved numerically.
### Convergence model of CKLS type
We consider a model in which risk-neutral drift of the European short rate $r_e$ is a linear function of $r_e$, risk-neutral drift of the domestic short rate $r_d$ is a linear function of $r_d$ and $r_e$ and volatilities take the form $\sigma_er_e^{\gamma_e}$ and $\sigma_dr_d^{\gamma_d}$, i.e.
$$\begin{aligned}
\label{eq:CKLSmodel}
d r_d &=& (a_1+a_2 r_d + a_3 r_e)d t + \sigma_d r_d^{\gamma_d} dw_d, \nonumber \\
d r_e &=& (b_1+b_2r_e)d t + \sigma_e r_e^{\gamma_e} dw_e,\end{aligned}$$
where $Cov[dW_d,dW_e] = \rho d t$. Parameters $a_1, a_2, a_3, b_1, b_2 \in \mathbb{R}, \sigma_d, \sigma_e > 0, \gamma_d, \gamma_e \geq 0$ are given constants and $\rho \in (-1,1)$ is a constant correlation between the increments of Wiener processes $d W_d$ a $d W_e$. We will refer to this model as *two-factor convergence model of Chan-Karolyi-Longstaff-Sanders (CKLS) type*. The domestic bond price $P(r_d, r_e, \tau)$ with the maturity $\tau$ satisfies PDE: $$\begin{aligned}
\label{eq:PDRckls}
-\frac{\partial P}{\partial \tau}&+&(a_1+a_2r_d+a_3r_e)\frac{\partial P}{\partial r_d}+(b_1+b_2r_e)\frac{\partial P}{\partial r_e} \nonumber \\
&+&\frac{\sigma_d^2r_d^{2\gamma_d}}{2}\frac{\partial^2 P}{\partial r_d^2}+\frac{\sigma_e^2r_e^{2\gamma_e}}{2}\frac{\partial^2 P}{\partial r_e^2}+\rho \sigma_d r_d^{\gamma_d} \sigma_e r_e^{\gamma_e} \frac{\partial^2P}{\partial r_d \partial r_e} -r_dP=0,\end{aligned}$$ for $r_d,r_e>0, \tau \in (0,T),$ with initial condition $P(r_d,r_e,0)=1$ for $r_d, r_e > 0.$ Unlike for Vasicek and uncorrelated CIR model, in this case it is not possible to find solution in the separable form (\[eq:SolutionInSeparateForm\]). For this reason, we are looking for an approximative solution.
Approximation of the domestic bond price solution
-------------------------------------------------
The bond prices in the CKLS type convergence model are not known in a closed form. This is already the case for the European bonds, i.e. one-factor CKLS model. We use the approximation from [@stehlikova]. In this approximation we consider one-factor Vasicek model with the same risk-neutral drift and we set current volatility $\sigma r^{\gamma}$ instead of constant volatility into the closed form formula for the bond prices. We obtain $$\label{eq:Apr1factorCKLS}
\ln P^{ap}_{e}(\tau,r)=\left(\frac{b_1}{b_2}+\frac{\sigma^2r^{2\gamma}}{2b_2^2} \right) \left(\frac{1-e^{b_2\tau}}{b_2}+\tau \right)+\frac{\sigma^2r^{2\gamma}}{4b_2^3}\left(1-e^{b_2 \tau}\right)^2 + \frac{1-e^{b_2\tau}}{b_2}r.$$ We use this approach to propose an approximation for the domestic bond prices. We consider the domestic bond prices in Vasicek convergence model with the same risk-neutral drift and we set $\sigma_dr_d^{\gamma_d}$ instead of $\sigma_d$ and $\sigma_er_e^{\gamma_e}$ instead of $\sigma_e$ into (\[eq:SOLUTIONvas2\]). Hence, we have $$\label{eq:Apr2factorCKLS}
\ln P^{ap}=A-Dr_d-Ur_e$$ where $$\begin{aligned}
D(\tau)&=&\frac{-1+e^{a_2\tau}}{a_2},\nonumber \\
U(\tau)&=& \frac{a_3\bigl(a_2-a_2 e^{b_2\tau}+b_2
\left( -1+e^{a_2\tau}\right) \bigl)}{a_2\left(a_2-b_2\right)b_2},\nonumber \\
A(\tau)&=&\int_0^{\tau} -a_1D(s)-b_1U(s)+\frac{\sigma_d^2r_d^{2\gamma_d} D^2(s)}{2}+\frac{\sigma_e^2r_e^{2\gamma_e} U^2(s)}{2}\\
&+&\rho \sigma_dr_d^{\gamma_d} \sigma_er_e^{\gamma_e} D(s) U(s) \mbox{d}s.\nonumber \end{aligned}$$
In the CIR convergence model the domestic bond price $P^{CIR, \rho=0}$ has a separable form (\[eq:SolutionInSeparateForm\]) and functions $A, D, U$ are characterized by a system of ODEs (\[eq:EquationsCIR\]). This enables us to compute Taylor expansion of its logarithm around $\tau=0$. We can compare it with the expansion of proposed approximation $\ln P^{CIR, \rho=0, ap}$ (computed either using its closed form expression (\[eq:Apr2factorCKLS\]) or the system of ODEs (\[eq:SOLUTIONvas2\]) for Vasicek convergence model). More detailed computation can be found in [@ZIKOVA]. In this way we obtain the accuracy of the approximation for the CIR model with zero correlation: $$\label{eq:AccuracyCIR}
\text{ln}P^{CIR,\rho=0,ap}-\text{ln}P^{CIR,\rho=0}=\frac{1}{24}\left(-a_2\sigma_d^2r_d-a_1\sigma_d^2-a_3\sigma_d^2r_e \right)\tau^4 + o(\tau^4)\\$$ for $\tau \rightarrow 0^+$.
Let us consider real measure parameters: $a=0$, $b=2$, $\sigma_d=0.03$, $c=0.2$, $d=0.01$, $\sigma_e=0.01$ and market price of risk $\lambda_d=-0.25$, $\lambda_e=-0.1$. In the risk-neutral setting (\[eq:CIRmodel1\]) we have $a_1=a-\lambda_d\sigma_d=0.0075$, $a_2=-b=-2$, $a_3=b=2$, $b_1=cd-\lambda_e\sigma_e=0.003$, $b_2=-c=-0.2$, $\sigma_d=0.03$, $\sigma_e=0.01$. With the initial values for the short rates $r_d = 1.7 \%$ a $r_e = 1 \%$ we generate the evolution of domestic and European short rates using Euler-Maruyama discretization. In Table \[tab:rozdielCIR\] we compare the exact interest rate and the approximative interest rate given by (\[eq:Apr2factorCKLS\]). We observe very small differences. Note that the Euribor market data are quoted with the accuracy $10^{-3}$. Choosing other days, with other combination of $r_d$, $r_e$, leads to very similar results. The difference between exact and approximative interest rate remains nearly the same.
[c c]{}
Finally, we present a detailed derivation of the order of accuracy of the proposed approximation in the general case. We use analogous method as in [@stehlikova] and [@stehlikova-sevcovic] for one-factor models and in [@lacko-dp] to study the influence of correlation $\rho$ on bond prices in the convergence CIR model.
Let $f^{ex}=\ln P^{ex}$ be the logarithm of the exact price $P^{ex}$ of the domestic bond in two factor convergence model of CKLS type. It satisfies the PDE (\[eq:PDRckls\]). Let $f^{ap}=\ln P^{ap}$ be the logarithm of the approximative price $P^{ap}$ for the domestic bond price given by (\[eq:Apr2factorCKLS\]). By setting $f^{ap}$ to the left-hand side of (\[eq:PDRckls\]) we obtain non-zero right-hand side, which we denote as $h(r_d, r_e, \tau)$. We expand it into Taylor expansion and obtain that $$\label{h}
h(r_d,r_e,\tau)=k_3(r_d,r_e)\tau ^3 + k_4(r_d,r_e)\tau ^4 + o(\tau ^4),$$ for $\tau \rightarrow 0^+$, where $$\label{k3}
k_3(r_d,r_e)=\frac{1}{6}\sigma_d^{2}\gamma_d r_d^{2\gamma_d-2}\left( 2a_1r_d + 2a_2r_d^2 + 2a_3r_dr_e - r_d^{2\gamma_d}\sigma_d^2 + 2\gamma_dr_d^{2\gamma_d}\sigma_d^2 \right),$$ $$\begin{aligned}
\label{k4}
k_4(r_d,r_e)&=& \frac{1}{48} \frac{1}{r_e^2} r_d^{-2 +\gamma_d} \sigma_d
\Bigr(12 a_2^2 \gamma_d r_d^{2 + \gamma_d} r_e^2 \sigma_d - 16 \gamma_d r_d^{1 + 3 \gamma_d} r_e^2 \sigma_d^3 +
6 a_3 b_1 \gamma_e r_d^2 r_e^{1 + \gamma_e} \rho \sigma_e \\
&+&6 a_3 b_2 \gamma_e r_d^2 r_e^{2 + \gamma_e} \rho \sigma_e +
6 a_3^2 \gamma_d r_d r_e^{3 + \gamma_e} \rho \sigma_e- 3 a_3 \gamma_d r_d^{2 \gamma_d} r_e^{2 + \gamma_e} \rho \sigma_d^2 \sigma_e\\
&+& 3 a_3\gamma_d^2 r_d^{2\gamma_d}r_e^{2+\gamma_e}\rho\sigma_d^2\sigma_e +6 a_3\gamma_d\gamma_e r_d^{1+\gamma_d} r_e^{1+2 \gamma_e}\rho^2 \sigma_d \sigma_e^2-
3 a_3 \gamma_e r_d^2 r_e^{3 \gamma_e} \rho \sigma_e^3\\
&+& 3 a_3 \gamma_e^2 r_d^2 r_e^{3 \gamma_e} \rho \sigma_e^3 +
6 a_1 \gamma_d r_d r_e^2 \left(2 a_2 r_d^{\gamma_d} \sigma_d + a_3 r_e^{\gamma_e} \rho \sigma_e\right)\\
&+& 6 a_2 \gamma_d r_e^2 \bigl(\left(-1 + 2 \gamma_d\right) r_d^{3 \gamma_d} \sigma_d^3 +
a_3 r_d \left(2 r_d^{\gamma_d} r_e \sigma_d + r_d r_e^{\gamma_e} \rho \sigma_e\right)\bigl)\Bigr).\end{aligned}$$
We define function $g(\tau, r_d, r_e):=f^{ap}-f^{ex}=\ln P^{ap}-\ln P^{ex}$ as a difference between logarithm of the approximation and the exact price. Using the PDEs satisfied by $f^{ex}$ and $f^{ap}$ we obtain the following PDE for the function $g$: [$$\begin{aligned}
\label{PDRg}
-\frac{\partial g}{\partial \tau}&+&\left(a_1+a_2r_d+a_3r_e\right)\frac{\partial g}{\partial r_d}+\left(b_1+b_2r_e\right)\frac{\partial g}{\partial r_e} +
\frac{\sigma_d^2 r_d^{2\gamma_d}}{2} \left[\left( \frac{\partial g}{\partial r_d}\right)^2 + \frac{\partial^2 g}{\partial r_d^2}\right] \nonumber \\
&+& \frac{\sigma_e^2 r_e^{2\gamma_e}}{2} \left[\left( \frac{\partial g}{\partial r_e}\right)^2 + \frac{\partial^2 g}{\partial r_d^2}\right]
+\rho \sigma_d r_d^{\gamma_d} \sigma_e r_e^{\gamma_e} \left(\frac{\partial g}{\partial r_d} \frac{\partial g}{\partial r_e} +
\frac{\partial ^2 g}{\partial r_d \partial r_e} \right) \\
= h(r_d,r_e,\tau) &+& \frac{\sigma_d^2 r_d^{2\gamma_d}}{2} \left[ \left(\frac{\partial f^{ex}}{\partial r_d}\right)^2 -
\frac{\partial f^{ap}}{\partial r_d}\frac{\partial f^{ex}}{\partial r_d} \right]
+ \frac{\sigma_e^2 r_e^{2\gamma_e}}{2} \left[ \left(\frac{\partial f^{ex}}{\partial r_e}\right)^2 -
\frac{\partial f^{ap}}{\partial r_e}\frac{\partial f^{ex}}{\partial r_e} \right]\nonumber \\
&+& \rho \sigma_d r_d^{\gamma_d} \sigma_e r_e^{\gamma_e} \left[ 2 \frac{\partial f^{ex}}{\partial r_d}\frac{\partial f^{ex}}{\partial r_e}
-\frac{\partial f^{ap}}{\partial r_d}\frac{\partial f^{ex}}{\partial r_e}
-\frac{\partial f^{ex}}{\partial r_d}\frac{\partial f^{ap}}{\partial r_e} \right].\nonumber\end{aligned}$$ ]{}
Suppose that $g(r_d,r_e,\tau)=\sum_{k=\omega}^{\infty}c_{k}(r_d,r_e)\tau^{k}$. For $\tau=0$ is both the exact and approximative bond price equal to one, so $f^{ex}(r_d,r_e,0)=f^{ap}(r_d,r_e,0)=0$. It means that $\omega>0$ and on the left hand side of the equation (\[PDRg\]) the term with the lowest order is $c_{\omega}\omega\tau^{\omega-1}$. Now we investigate the order of the right hand side of the equation.
We know that $f^{ex}(r_d,r_e,0)=0$. It means that $f^{ex}=O(\tau)$ and also partial derivation $\frac{\partial f^{ex}}{\partial r_d}$ and $\frac{\partial f^{ex}}{\partial r_e}$ are of the order $O(\tau)$. From the approximation formula (\[eq:Apr2factorCKLS\]) we can see that $\frac{\partial f^{ap}}{\partial r_d}=O(\tau)$, $\frac{\partial f^{ap}}{\partial r_e}=O(\tau^2)$. Since $h(r_d, r_e, \tau)=O(\tau^3)$, the right hand side of the equation (\[PDRg\]) is at least of the order $\tau^2$. The left hand side of the equation (\[PDRg\]) is of the order $\tau^{\omega-1}$ and hence $\omega-1 \geq 2$, i.e. $\omega \geq 3$. It means that $$f^{ap}(r_d,r_e,\tau)-f^{ex}(r_d,r_e,\tau)=O(\tau^3).$$ Using this expression we can improve estimation of the derivative $\frac{\partial f^{ex}}{\partial r_e}$ as follows: $\frac{\partial f^{ex}}{\partial r_e}=\frac{\partial f^{ap}}{\partial r_e}+O(\tau^3)=O(\tau^2)+O(\tau^3)=O(\tau^2).$ We also estimate the terms on the right hand side in the equation (\[PDRg\]): $$\begin{aligned}
\label{odhad1}
\left(\frac{\partial f^{ex}}{\partial r_d}\right)^2 -
\frac{\partial f^{ap}}{\partial r_d}\frac{\partial f^{ex}}{\partial r_d} &=& \frac{\partial f^{ex}}{\partial r_d}
\left(\frac{\partial f^{ex}}{\partial r_d}-\frac{\partial f^{ap}}{\partial r_d}\right)=O(\tau).O(\tau^3)=O(\tau^4),\end{aligned}$$
$$\begin{aligned}
\label{odhad1.5}
\left(\frac{\partial f^{ex}}{\partial r_e}\right)^2 -
\frac{\partial f^{ap}}{\partial r_e}\frac{\partial f^{ex}}{\partial r_e}= \frac{\partial f^{ex}}{\partial r_e}
\left(\frac{\partial f^{ex}}{\partial r_e}-\frac{\partial f^{ap}}{\partial r_e}\right)=O(\tau^2).O(\tau^3)=O(\tau^5),\end{aligned}$$
$$\begin{aligned}
\label{odhad2}
2 \frac{\partial f^{ex}}{\partial r_d}\frac{\partial f^{ex}}{\partial r_e}
- \frac{\partial f^{ap}}{\partial r_d}\frac{\partial f^{ex}}{\partial r_e}
-\frac{\partial f^{ex}}{\partial r_d}\frac{\partial f^{ap}}{\partial r_e} =
\frac{\partial f^{ex}}{\partial r_d}
\left(\frac{\partial f^{ex}}{\partial r_e}-\frac{\partial f^{ap}}{\partial r_e}\right)\nonumber \\
+ \frac{\partial f^{ex}}{\partial r_e}
\left(\frac{\partial f^{ex}}{\partial r_d}-\frac{\partial f^{ap}}{\partial r_d}\right) = O(\tau).O(\tau^3)+O(\tau^2).O(\tau^3)=O(\tau^4)+O(\tau^5)=O(\tau^4).\end{aligned}$$
Since $h(r_d,r_e,\tau)=O(\tau^3)$, the right hand side of the equation (\[PDRg\]) is $O(\tau^3)$ and the coefficient at $\tau^3$ is the coefficient of the function $h(r_d,r_e,\tau)$ at $\tau^3$, i.e. $k_3(r_d,r_e)$. It means that $\omega=4$ and comparing the coefficients at $\tau^3$ on the left and right-hand side of (\[PDRg\]) we obtain $-4c_4(r_d,r_e)=k_3(r_d,r_e),$ i.e. $c_4(r_d,r_e)=-\frac{1}{4}k_3(r_d,r_e).$ Hence we have proved the following theorem.
\[VETApresnost\] Let $P^{ex}(r_d,r_e,\tau)$ be the price of the domestic bond in two-factor CKLS convergence model, i.e. satisfying equation (\[eq:PDRckls\]) and let $P^{ap}$ be the approximative solution defined by (\[eq:Apr2factorCKLS\]). Then $$\ln P^{ap}(r_d,r_e,\tau)-\ln P^{ex}(r_d,r_e,\tau)=c_4(r_d,r_e)\tau^4+ o(\tau^4)$$ for $\tau \rightarrow 0^+,$ where coefficient $c_4$ is given by $$\label{c4}
c_4(r_d,r_e)=-\frac{1}{24}\sigma_d^{2}\gamma_d r_d^{2\gamma_d-2}
\left( 2a_1r_d + 2a_2r_d^2 + 2a_3r_dr_e - r_d^{2\gamma_d}\sigma_d^2 + 2\gamma_dr_d^{2\gamma_d}\sigma_d^2 \right).$$
Note that if we substitute $\gamma_d=\frac{1}{2}$ and $\rho=0$ into Theorem \[VETApresnost\], we obtain the formula (\[eq:AccuracyCIR\]) for CIR model derived earlier in (\[eq:AccuracyCIR\]).
In some cases it is possible to improve an approximation by calculating more terms in Taylor expansion of the function $g=\ln P^{ap}-\ln P^{ex}$. It is so also in this case. Using that $f^{ap}-f^{ex}=O(\tau^4)$, we are able to improve estimates (\[odhad1\]) and (\[odhad2\]) and to deduce that also the coefficient at $\tau^4$ on the right hand side of equation (\[PDRg\]) comes only from the function $h$. Hence it is equal to $k_4(r_d,r_e)$, which is given by (\[k4\]). Comparing coefficients at $\tau^4$ on the left and right hand side of (\[PDRg\]) we obtain: $$\begin{aligned}
- 5c_5+(a_1+a_2r_d+a_3r_e)\frac{\partial c_4}{\partial r_d}+(b_1+b_2r_e)\frac{\partial c_4}{\partial r_e}\\
+\frac{\sigma_d^2r_d^{2\gamma_d}}{2}\frac{\partial^2 c_4}{\partial r_d^2}+\frac{\sigma_e^2r_e^{2\gamma_e}}{2}\frac{\partial^2 c_4}{\partial r_e^2}+4\rho\sigma_d r_d^{\gamma_d}\sigma_e r_e^{\gamma_e}\frac{\partial^2 c_4}{\partial r_d \partial r_e}=k_4,\end{aligned}$$ which enables us to express $c_5$ using already known quantities.
Let us define an approximation $\ln P^{ap2}$ by: $$\ln P^{ap2}(r_d,r_e,\tau)=\ln P^{ap}-c_4(r_d,r_e)\tau^4-c_5(r_d,r_e)\tau^5.$$ Then $\ln P^{ap2}- \ln P^{ex}=O(\tau^6)$ and therefore the new approximation $\ln P^{ap2}$ is of the order $O(\tau^6).$
Financial interpretation of the short rate factors and their evolution
----------------------------------------------------------------------
In the PhD thesis by Šesták [@phd-sestak], supervised by Ševčovič, the approximation formula from [@dp-halgasova] is used to estimate the model for European countries. The rate for each country is decomposed into a risk-free rate (common to all the countries) and a credit spread (specific for each country). The formula from [@dp-halgasova] is used to price bonds in this setting. The author suggests a calibration procedure which is computationally demanding since it involves a large data set - yields of all countries considered simultaneously (it is not possible to split this for each country, since the risk-free rate, which is one of the outputs, is shared by all the countries). Hence a simple approximate formula for the bond prices is crucial for a successful estimation.
Figure \[fig:sestak\] shows results of the estimation from [@phd-sestak]. Note how the very different evolution of the credit spread for Greece starts from a certain time, compared to the values obtained for the other counties.
![Estimating the risk-free rate and credit spread in the European countries. In the figure below, the values for Greece are shown in the right axis, for the other countries in the left axis. Source: Šesták, [@phd-sestak].[]{data-label="fig:sestak"}](figures/habilitacia/sestak-a "fig:"){width="70.00000%"}\
![Estimating the risk-free rate and credit spread in the European countries. In the figure below, the values for Greece are shown in the right axis, for the other countries in the left axis. Source: Šesták, [@phd-sestak].[]{data-label="fig:sestak"}](figures/habilitacia/sestak-b "fig:"){width="70.00000%"}
Conclusions
===========
In this survey we presented an overview of short rate models and presented some of the approaches to compute approximations of bond prices where the exact solutions are not available.
Firstly, we considered one-factor models. The simple models of Vasicek and Cox-Ingersoll-Ross admit closed form bond prices and therefore can serve as either basis for construction of analytical approximations or as testing cases for assessing numerical accuracy of different approximation formulae. Using partial differential approach to bond pricing enables us to derive their order of accuracy for small times remaining to maturity.
In the second part we dealt with multi-factor models – the process for the short rate written as a sum of two factors, second factor being the stochastic volatility or the European interest rate when modeling rates in a country before adoption of the Euro currency. In case of convergence model we provided also an example of a three-factor model, in which the European rate is modeled by a two-factor model. We studied similar analytic approximations for convergence models as in the case of one-factors models. Here we provided also a proof of accuracy of the proposed approximation; similar reasoning was applied also in the analyses of other models, where we only stated the results. Moreover, we studied the asymptotics of a fast time scale of volatility in stochastic volatility models.
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[^1]: Department of Applied Mathematics and Statistics, Comenius University, 842 48 Bratislava, Slovakia. Corresponding author: D. Ševčovič, sevcovic@fmph.uniba.sk The research has been supported by VEGA 1/0251/16 project and FP7-PEOPLE-2012-ITN project \#304617 - STRIKE. This survey chapter has been submitted to the book collection: Interest Rates: Global Trends, Macroeconomic Implications and Analysis, 2016 Nova Science Publishers, Inc., Hauppauge.
[^2]: for example, the rate on one-year loan today and next year
[^3]: interest rates on, for instance, one-year and ten-year loans are different
[^4]: It can be shown that it is possible if we assume an “idealized market” with no transaction costs, ability to buy or sell any desired amount of a security for its present price, to borrow/lend any amount of money for the short rate interest rate and operating in continuous time. This idealization of reality in the derivation of the equation for security prices might be another reason for being “satisfied” with a meaningful simple approximation of the short rate process, instead of requiring an extremely complex model for it.
[^5]: see *http://www.cboe.com/micro/VIX/vixintro.aspx*
[^6]: *www.cboe.com/VXTYN*
[^7]: *www.cboe.com/SRVX*
[^8]: Note that this model is a generalization of the one-factor CIR model and the choices for market prices of risk can be seen as generalizations of the said model too.
[^9]: Note that an explicit solution for the European bonds follows from the fact that we are using a classical Vasicek model for the European interest rates.
[^10]: $L_{p}$ and $L_{\infty}$ norms of a function $f$ defined on a grid with step $h$ are given by $\Vert f \Vert_p=\left( h \sum |f(x_i)|^p \right)^{1/p}$ and $\Vert f \Vert_{\infty}=\max |f(x_i)|$.
[^11]: The concrete example of a function $\alpha$ considered in the paper models a volatility clustering phenomenon where the dispersion can be observed in the vicinity of two local maxima of the density distribution. In particular, it uses a stochastic differential equation that leads to the limiting density of the volatility to be equal to a convex combination of two gamma densities, which has been proposed in [@iscam05]. However, the results are derived for a general process (\[slowly-osc\]), using the limiting distribution and its statistical moments.
|
---
abstract: 'Pulsed field emission from cold carbon-nanotube cathodes placed in a radiofrequency resonant cavity was observed. The cathodes were located on the backplate of a conventional $1+\frac{1}{2}$-cell resonant cavity operating at 1.3-GHz and resulted in the production of bunch train with maximum average current close to 0.7 Ampère. The measured Fowler-Nordheim characteristic, transverse emittance, and pulse duration are presented and, when possible, compared to numerical simulations. The implications of our results to high-average-current electron sources are briefly discussed.'
author:
- 'D. Mihalcea'
- 'L. Faillace'
- 'J. Hartzell'
- 'H. Panuganti'
- 'S. M. Boucher'
- 'A. Murokh'
- 'P. Piot'
- 'J. C. T. Thangaraj'
title: |
Ampère-Class Pulsed Field Emission from Carbon-Nanotube Cathodes\
in a Radiofrequency Resonator
---
Over the last decades field-emission (FE) $-$ the emission of electrons via tunneling effect $-$ has led to the development of compact electron sources that have been widely disseminated in microelectronics [@HP], electron microscopy [@SEM], and more recently as particle-accelerator sources [@brau]. FE enjoys a greater simplicity compared to other types of electron-emission mechanisms: it does not require auxiliary systems such as lasers employed in photoemission nor needs to be electrically heated by a filament as for thermionic cathode. A FE cathode operates with intense \[${\cal O}\mbox{(GV/m)}$\] electric fields applied at its surface. The field bends the potential barrier of the material and enhance the probability for electron tunneling. In practice, given the limited electric-field amplitude sustainable in common apparatus ($10-100$ MV/m), the generation of large field relies on field enhancement provided by sharp microscopic features present at the surface of the FE cathode. Such attribute results in the local field in the vicinity of the features $E_e=\beta_e E$ where $\beta_e$ is the enhancement factor and $E$ is the applied (macroscopic) field. Consequently the current density emitted from one of the sharp features is governed by the Fowler-Nordheim’s (FNÕs) law [@FN] $ {\pmb j}=a E_e^2\exp\left(-\frac{b}{E_e}\right) {\hat{\pmb n}}$, where $a\equiv \frac{1.42\times10^{-6}}{\Phi}\exp\left(\frac{10.4}{\Phi^{1/2}}\right) $ and $b\equiv -6.56 \times 10^9\Phi^{3/2}$ are constants that depend on the material work function $\Phi$ (in units of eV) [@minoux], ${\hat{\pmb n}}$ is the unitary vector normal to the local emitting surface. FE cathodes consisting of single field emitter have been proposed as source of ultra-bright electron bunches [@Hommelhoff]. Conversely, FE cathodes composed of a large number of field emitters are contemplated as high-current electron sources that could be deployed in a variety of contexts ranging from fundamental science along with medical, industrial, and security settings. Finally, the application of a time-dependent electric field results in the generation of electron bunches with finite duration [@IEEEpaper; @nature; @piotAPL; @UDCAPL]. Field emission from various materials have been extensively studied and most recently the use of carbon nanotube (CNT) has significantly increased [@minoux]. CNTs are allotropes of carbon with a cylindrical nanostructure and with exceptional electrical and mechanical properties. CNT’s are fibers with diameters ranging from 1 to 50 nm, coming in either single-wall or multi-wall strands and with aspect ratio of up to $\sim 1000$ [@zhu]. These properties yield substantial field enhancement factors and alongside with their low electrical resistance, high thermal stability and robustness to high temperatures, CNTs are excellent candidates for FE cathodes. CNTs can be synthesized as aligned field-emitter arrays (FEA) or configured as randomly oriented field emitters deposited on surfaces. Although FEAs are ideal for most applications, especially in vacuum microelectronics [@fursey], randomly oriented CNTs enjoy a simpler fabrication process and can be formed on shaped surfaces. The latter capability could enable the generation of transversely-tailored high-current electron beams such as needed for, e.g., electron-beam-based manipulations of high-intensity ion-beams accelerators [@iota].\
![Micrograph of CNT cathode surface (a) and photograph of the CNT cathode on its molybdenum substrate (b). \[fig:cntpic\]](cathode_newpics.png){width="45.00000%"}
In this Letter, we report on the experimental tests of pulsed emission from CNT cathodes. The cathodes were synthesized using an electrophoretic deposition (EPD) process and consisted of a layer of multiple allotropes of carbon including nanotubes, buckyballs, graphite, and amorphous carbon [@hartzell]; see Fig. \[fig:cntpic\](a). Emerging from this layer are a vast number of randomly-oriented nanotubes. Both the extremities and sides of the nanotubes can act as field emitters. The current formed by such cathodes can be obtained as the surface integration of the current density over the CNT deposition area and is written as $I= {\cal A} j$ for a flat surface where ${\cal A}$ is the effective emission area. Two cathodes were tested, a “large“ and a ”small" cathodes which respectively consisted of a 15-mm and 1.5-mm diameter CNT-coated circular area deposited on respectively a molybdenum and stainless steel substrate; see Fig. \[fig:cntpic\](b).
The experimental characterization of the cathodes was carried at the high-brightness electron beam source (HBESL) located at Fermilab [@carneiro]. The facility incorporates a radiofrequency (RF) gun followed by a beam line instrumented with various beam diagnostics depicted in Fig. \[fig:expsetup\]. The RF gun is a 1+1/2 cell resonant cavity operating on the TM$_{010,\pi}$ mode at $f_0=1.3$ GHz and is powered by a pulsed klystron capable of producing up to 2 MW of peak power. For the experiment reported in this Letter, the klystron was operated at 1 Hz with a pulse duration of 30 $\mu$s. The gun is nested in three magnetic lenses (referred to as ÒsolenoidsÓ) that are nominally used to control the beam divergence and transverse emittance. The two cathodes described above were mounted on a standard cathode-plug holder and inserted in the RF gun.
![Top-view schematics of the experimental setup of the HBESL facility. The “X’s” label indicate the location of diagnostics, “FC", “IG" and “GV" respectively stands for “faraday cup", “ion gauge", and vacuum “gate valve". Only beam-line elements pertinent to the experiment are shown. []{data-label="fig:expsetup"}](expsetup.png){width="45.00000%"}
The available diagnostics along the downstream accelerator beam line includes a Faraday cup, several transverse-density monitors consisting of remotely insertable Cerium-doped Yttrium Aluminum Garnet (YAG:Ce) scintillators, and a set of capacitive electromagnetic pick-ups that can detect the transient electromagnetic field produced by the passing electron bunches. Several dipole magnets are used to deflect the beam and infer its mean momentum. Finally, the beam transverse emittance $\varepsilon_u \equiv \frac{1}{m_e c}[{\mbox{$\langle{u^2}\rangle$}}{\mbox{$\langle{p_u^2}\rangle$}}-{\mbox{$\langle{up_u}\rangle$}}^2]^{1/2}$ can be measured using a multi-slit method [@lejeune] $-$ here $(u,p_u)$ refers to the position-momentum coordinate along the horizontal ($u=x$) or vertical ($u=y$) degree of freedom, ${\mbox{$\langle{.}\rangle$}}$ represents the statistical averaging over the beam phase space distribution, and $m_e$ and $c$ are the electron rest mass and the velocity of light.
To first characterize the field-emission process, the emitted current versus applied macroscopic field was measured for different conditions. The current inferred from the Faraday cup is time averaged and its functional dependence on the applied field is given by $\bar{I}= \frac{1}{\sqrt{2\pi}}{\cal A} a (\beta_e E)^{5/2} \exp\left(-\frac{b}{\beta_e E}\right)$ [@FNRF]. The $\bar{I}-E$ curves display the expected exponential dependence of the current; see Fig. \[fig:FNplots\]\[(a), and (c)\]. Furthermore when reported on a FN diagram $[1/E,\log(\bar{I} E^{-5/2})]$, the data appear as lines and a linear-regression analysis provides information on the averaged enhancement factor and effective emission area as summarized in Table \[tab:beta\] for four of the cases studied.
![Measured average current $\bar{I}$ as a function of applied microscopy field (a,b) and corresponding Fowler-Nordheim plots (c,d). The upper and lower rows respectively correspond to the large and small cathodes. The dashed lines in (c,d) represent linear polynomial fits. For plots (b,d) the blue and red symbols respectively correspond to data taken just after installation of the small cathode and 2 weeks later.[]{data-label="fig:FNplots"}](ie_iv_big_small_01122015.png){width="50.00000%"}
The settings of the three lenses were varied simultaneously and set to insure a zero magnetic field on the cathode surface and gave rise to a $\sim 10$% relative variation in produced beam current confirming a significant fraction of the current is actually transversely captured and transported up to the location of the Faraday cup. We found the values of the enhancement factors to be independent of the applied magnetic field \[Fig. \[fig:FNplots\](a,b)\] and to be qualitatively similar for the two cathodes used during our experiments; see Fig. \[fig:FNplots\](c,d). It should be noted that the effective area ${\cal A}$ is much smaller than expected for the small-area cathode.
configuration $\beta_e$ ${\cal A}\times 10^{16}$ (m$^2$)
------------------------------ -- ---------------------------------- -------------------------------------
small cath., B field off $468.1^{+73.4}_{-69.8} \pm 4.7$ $5.15^{+0.15}_{-0.22} \pm 0.10$
small cath., B field off[^1] $ 504.6^{+79.2}_{-75.2} \pm 5.1$ $ 1.57^{+0.05}_{-0.07} \pm 0.03 $
$\beta_e$ ${\cal A}\times 10^{7}$ (m$^2$)
large cath., B field off $ 395.3^{+62.0}_{-58.9} \pm 4.0$ $ 10.18^{+0.29}_{-0.44} \pm 37.62 $
large cath., B field on $ 468.1^{+73.4}_{-69.8} \pm 4.7$ $ 0.56^{+0.02}_{-0.02} \pm 2.81 $
: \[tab:beta\] Inferred enhancement factor $\beta_e$ and effective emission area ${\cal A}$ for the four operating cases considered in the text. The values are written as $A^{+u}_{-v}\pm w $, where $A$ is obtained for a nominal work function value $\Phi=4.9$ eV while the upper and lower uncertainties $u$ and $v$ are respectively evaluated for $\Phi=5.4$ and $4.5$ eV. The error bar $w$ is propagated from the uncertainty on the linear regression.
Assuming the same CNT density for both cathode we would anticipate the effective emission area associated to the small cathode to be $(1.5/15)^2=10^{-2}$ while a factor $\sim [10^{-10}-10^{-8}]$ is observed. A post-experiment inspection of the cathodes indicated some damages (dark spots) on the small cathode attributed to multipacting occurring due to the favorable secondary-emission yield of the stainless-steel substrate (strong multipacting emission was observed during operation of the small cathode). The small cathode also consistently degraded with time when operated at high field. The large cathode did not show any performance degradation despite being exposed to atmosphere for $\sim 4$ weeks between two subsequent tests.\
![Voltage detected from the electromagnetic pickup (a), corresponding bunching factor (b) and evolution of the bunching factor evaluated at $f=2.6$ GHz as function of $E_0$ (c). In (b) the data points are FFTs of different traces obtained for different values of $E_0$, the dash line represents a fit considering to a Gaussian bunch distribution (the shaded green area accounts for the uncertainties in the fit) and the solid thick line corresponds to the simulated bunch distribution using the [warp]{} program shown in (d). []{data-label="fig:BPM"}](figure_bunch.png){width="47.50000%"}
The beam temporal structure of the electron beam formed by the large cathode was characterized using an electromagnetic pick-up located 30 cm from the cathode. The transient voltage induced by the bunches was detected by a capacitive coupler and recorded on a 12-Gs oscilloscope; a typical trace is displayed in Fig. \[fig:BPM\](a). The detected signal can be factored as $V(t) = e(t)\sum_{n=1}^N \Lambda (t + nf_0^{-1})$ where $e(t)$ is the signal envelope, $\Lambda(t) \propto i(t)$ is the signal induced by one bunch. The amplitude of the fast-Fourier transform (FFT) of $V(t)$ provides the bunching factor $b(f)$ which is enhanced at harmonics of the bunch repetition frequency $f_0$ as shown in Fig. \[fig:BPM\](b). The relative amplitudes $b(f)/b(f_0)$ at harmonic frequencies $f=nf_0$ (with $n\ge 2$) provide an upper bound for the bunch duration. Analysis of the data presented in Fig. \[fig:BPM\](b), assuming the electron bunch follows a Gaussian temporal distribution, gives an rms bunch duration of $\sigma_t \simeq 67\pm 25$ ps at an applied field $E \in [5,12]$ MV/m. The electron-bunch duration is expected to scale as $\sigma_t \propto \sqrt{E}$ [@piotAPL] at the cathode. The bunch length increase with $E_0$ is supported by the observed increase of the $b(2f_0)/b(f_0)$ \[see Fig. \[fig:BPM\](c)\] but due to the limited resolution of our pulse-length-measurement technique and the complicated dynamics in the RF gun the functional dependence $\sigma_t(E)$ could not be characterized. In spite of these limitations, the measured pulse duration agrees reasonably well with particle-in-cell simulations performed with the [warp]{} framework [@vay] which includes a self-consistent field-emission model [@alex]; see Fig. \[fig:BPM\](b,d). The FE parameters used in the simulations are the one reported for the large cathode (solenoid off) in Table \[tab:beta\].
![Emittance measurement snapshots showing the beam transverse distribution at X3 (a), the transverse distribution of the beamlets transmitted through the multislit mask observed at X5 (b) with associated horizontal projections (red traces). Image (c) shows the reconstructed horizontal $(x,x'\equiv p_x/pz)$ trace space at the location of X3 from processing of images (a) and (b). These measurements were performed for the small cathode.[]{data-label="fig:emittance"}](figure_emit.png){width="49.50000%"}
An important figure of merit of the field-emitted beam is its transverse emittance. The horizontal emittance of the full bunch train was characterized for the small cathode. The multislit mask located a position X3 was inserted and the transmitted beamlets were observed at location X5. A measurement of the beamlet root-mean-square (rms) size at X5 provides information on the beam intrinsic divergence $\sigma'_u$ at X3. Together with a measurement of the rms transverse beam size $\sigma_u$ at X3, the divergence yield the value of the transverse normalized emittance as $\varepsilon_u = \beta\gamma \sigma'_u \sigma_u$ where $u \in[x,y]$ refers to one of the transverse degrees of freedom and $\beta\equiv (1-\gamma^{-2})^{1/2}$ where $\gamma$ is the relativistic Lorentz factor. An example of measurement with reconstructed phase space appears in Fig. \[fig:emittance\]. The measurement indicates a transverse horizontal emittance of $\varepsilon_x = 2.64 \pm 0.8$ $\mu$m for the small cathode. Measurements for the large cathode were compromised by the large energy spread.
Finally, the stability of a high-current electron source is crucial for some applications. We consequently tested the current evolution for a few hours and confirmed that the cathodes under test were able to sustain the production of high-average currents with very low jitter; see Fig. \[fig:stability\](a). A statistical analysis also indicates that typical relative rms fluctuation of $ \sigma_{\bar I} \equiv {\mbox{$\langle{\bar{I}^2}\rangle$}}^{1/2}/{\mbox{$\langle{\bar{I}}\rangle$}} \simeq 2$% was achieved over six-hour periods and independently of the mean operating current ${\mbox{$\langle{\bar{I}}\rangle$}}$; see Fig. \[fig:stability\](b,c).\
![Current evolution over a $>6$-hour period (a) for 100 (blue) , 300 (red), and 650 mA (green) with corresponding histograms (b) and rms fluctuations (c). []{data-label="fig:stability"}](figure_stab.png){width="50.00000%"}
In summary we have demonstrated the operation of a CNT cathode in the pulsed regime and produced bunch trains with operating average current up to $\bar{I}=0.65$ A and duration of $\sigma_t\simeq 70$ ps implying a charge per bunch $Q \simeq \bar{I}/f_0 \simeq 0.50 $ nC corresponding to a single-bunch peak current $\hat{I} = Q/(\sqrt{2\pi}\sigma_t)\simeq 3$ A. The explored cold-cathode technology coupled with a superconducting resonator could lead to the development of high-average current quasi-continuous-wave electron sources. The main challenge toward such an endeavor remains the temporal control of the emission process as electrons field-emitted at unfavorable times are most likely to hit the resonator wall. Such collisions could result in secondary electron emissions and possible multipacting (as observed in some of our experiments) or could ultimately result in a superconducting quench of the cavity. Therefore the development of gating schemes aimed at shortening the electron-bunch durations and preventing the back-propagation of electrons is crucial. A dual-frequency gun [@lewellen] supporting a fundamental and harmonic frequencies could effectively gate the emission of the CNT cathode to the proper phase of the accelerating RF wave. Since the CNT cathodes have a distinct threshold voltage, unlike thermionic cathodes, the bunch duration could be made much shorter, eliminating the need for a bunching structure before injection into a subsequent accelerator. In such a scenario, it should be possible to reach $\sim10$-ps bunch durations.\
We are grateful to D. P. Grote and J.-L. Vay for their help with [warp]{}, to B. Chase, P. Prieto, E. Lopez and R. Kellett for technical support and to E. Harms, S. Nagaitsev and V. Shiltsev for support. This work was funded via US Department of Energy (DOE) contract DE-SC0004459 with Radiabeam Technologies, LLC, and executed under CRADA agreement FRA-2013-0006 between Fermilab and Radiabeam Technology, LLC. Fermilab is operated by the Fermi Research Alliance, LLC. for the DOE under contract DE-AC02-07CH11359.
[99]{} I. Milne, K. B. K. Teo, E. Minoux, O. Groening, L. Gangloff, L. Hudanski, J.-P. Schnell, D. Dieumegard, F. Peauger, I. Y. Y. Bu, M. S. Bell, P. Legagneux, G. Hasko, and G. A. J. Amaratunga, J. Vac. Sci. Technol., B [**24**]{}, 345 (2006). doi: 10.1116/1.2161223 E. W. Müller, Z. Phys. [**131**]{}, 136 (1951); doi:10.1007/BF01329651 C. A. Brau, Nucl. Instr. Meth. A [**393**]{}, 426 (1997). doi:10.1016/S0168-9002(97)00538-X R.H. Fowler and L. Nordheim, Proc. Royal Soc. Of London. Series A, [**119**]{}, 173-181 (1928). E. Minoux , O. Groening , K. B. K. Teo , S. H. Dalal, L. Gangloff, J.-P. Schnell, L Hudanski, I. Y. Y. Bu, P. Vincent, P. Legagneux, G. A. J. Amaratunga, W. I. Milne, Nano Lett., [**5**]{} (11), 2135 (2005). doi: 10.1021/nl051397d P. Hommelhoff, Y. Sortais, A. Aghajani-Talesh, and M. A. Kasevich, Phys. Rev. Lett. [**96**]{}, 077401 (2006); doi 10.1103/PhysRevLett.96.077401 F. M. Charbonner, J. P. Barbour, L. F. Garett, W. P. Dyke, Proc. IEEE [**51**]{}, 991 (1963). 10.1109/PROC.1963.2379 K. B. K Teo, et al., Nature [**437**]{}, 968 (2005). P. Piot, C. A. Brau, B. K. Choi, B. Blomberg, W. E. Gabella, B. Ivanov, J. Jarvis, M. H. Mendenhall, D. Mihalcea, S. Panuganti, P. Prieto, J. Reid, Appl. Phys. Lett. [**104**]{}, 263504 (2014). S. V. Baryshev, S. Antipov. J. Shao, C. Jing, K. J. PŽrez Quintero, J. Qiu, W. Liu, W. Gai, A. D. Kanareykin, and A. V. Sumant, Appl. Phys. Lett. [**105**]{}, 203505 (2014). W. Zhu, C. Bower, O. Zhou, G. Kochanski, S. Jin, Appl. Phys. Lett. [**75**]{}, 873 (1999). doi: 10.1063/1.124541 G.N. Fursey, [*Field emission in vacuum micro-electronics*]{}, Kluwer Academic / Plenum Publishers, New York, 2005. G. Stancari, arXiv:1409.3615 \[physics.acc-ph\] (2014). J. Hartzell, R.B. Agustsson, S. Boucher, L. Faillace, A.Y. Murokh, A.V. Smirnov, W.A. Hubbard, C. Regan, in Proceedings of the North-American particle accelerator conference (NAPAC’13), Pasadena, CA USA, 394 (2013). J.-P. Carneiro, et al., Phys. Rev. ST Accel. Beams [**8**]{}, 040101 (2005). C. Lejeune and J. Aubert, “Emittance and Brightness: Definitions and Measurements", in Applied Charged Particle Optics, Part A, A. Septier, ed. (Academic Press, New York, 1980), p. 159. J. W. Wang and G. A. Loew, “Field Emission and RF breakdown in high-gradient room-temperature linac structure", report SLAC-PUB-7684 (unpublished, 1997). J. L. Vay, D. P. Grote, R. H. Cohen, and A. Friedman, Comput. Sci. Disc. [**5**]{} 014019 (2012). A. Seymour, D. Grote, D. Mihalcea, P. Piot, J.-L. Vay, “Beam dynamics simulations of optically-enhanced field emission from structured cathodes", Femilab preprint FERMILAB-CONF-14-363-APC, in proceedings of the Advanced Accelerator Conference (AAC14), San-Jose (in press, 2014). J. W. Lewellen and J. Noonan, Phys. Rev. ST Accel. Beams, [**8**]{}, 033502 (2005).
[^1]: data taken 2 weeks later than data on previous line.
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abstract: 'To date, the concept of topological order relies heavily on the properties of single-particle bands. Only recently it has been realized that interactions can have a dramatic impact on topological properties not only modifying the topology of the bands but also creating a topological order in an otherwise trivial system. Applying an extended version of the Bose-Hubbard model, we investigate a system which, being topologically trivial in the single-particle regime, harbors topologically nontrivial edge and interface states of repulsively bound photon pairs. Whereas binding of the photons in this model is captured by a standard local interaction term, an additional direct two-photon hopping renders the system topologically non-trivial. Besides their interaction-induced origin, predicted two-photon edge states exhibit a range of other unexpected features, including the robustness to collapse of the corresponding bulk band and the ability to coexist with the continuum of two-photon scattering states forming a bound state in the continuum. Performing rigorous calculation of the Zak phase for bound photon pairs, we prove the topological origin of the two-photon edge states.'
author:
- 'Andrei A. Stepanenko'
- 'Maxim A. Gorlach'
bibliography:
- 'TopologicalLib.bib'
title: 'Interaction-induced topological states of photon pairs'
---
Introduction {#sec:Intro}
============
Topological photonics offers a rich variety of remarkable functionalities including disorder-robust routing of light on a chip [@Lu2014; @Lu2016; @Khanikaev-review; @Ozawa_RMP; @Rider2019; @Smirnova-toporeview]. While topological states in classical optical systems form an established area of research [@Lu2014; @Lu2016; @Khanikaev-review; @Ozawa_RMP; @Rider2019; @Smirnova-toporeview], the emphasis is currently shifting towards topological states of quantum light [@Roushan2014; @Barik; @Tambasco; @Mittal-2018; @Blanco-Science; @Wang2019] with the potential of applications in topologically protected quantum information transfer, quantum computations and manipulation of entangled photons with quantum metasurfaces [@Sukhorukov-Science].
Just within one year, first realizations of single-photon topological states [@Barik; @Tambasco] and topologically protected sources of non-classical light [@Mittal-2018] have been reported. Moreover, previous theoretical analysis of entangled photons propagation in a topological system [@Rechtsman-Segev; @Mittal-Hafezi] has been followed by recent experiments [@Blanco-Science; @Wang2019]. In this context, it is especially important to investigate the implications of topological protection for more complex quantum states of light which can potentially uncover further exciting applications of topological photonics.
One of such intriguing states of quantum light is represented by doublons, which are bound photon pairs arising in discrete nonlinear arrays due to repulsive Kerr-type nonlinearity [@Mattis1986; @Winkler]. Quite counter-intuitive properties of doublon quasi-particles were analyzed in a series of theoretical papers in the context of bulk [@Valiente; @Valiente2009; @Menotti; @Bello-2017; @Wang-Liang] and edge doublon states [@Pinto; @Flach; @Zhang2012; @Zhang2013; @Longhi; @Gorlach-H-2017; @DiLiberto-EPJ] including more advanced concepts of doublons in two-dimensional geometries [@Salerno; @Salerno2019], Thouless pumping of doublons [@Angelakis2016; @YKe] and dissipatively bound photon pairs [@Lyubarov].
Driven by the ambitious goal to realize topological doublon edge states, we and several other groups have investigated a well-celebrated Su-Schrieffer-Heeger model (SSH) [@Su] in the two-photon regime with the effective on-site repulsive photon-photon interaction [@DiLiberto; @Gorlach-2017; @Marques2018]. However, since the analyzed model is topologically nontrivial even in the single-particle case, the emergence of two-photon edge states [@Gorlach-2017] it is not so surprising. Therefore, it is much more exciting to demonstrate topological states of doublons induced by interactions in an otherwise topologically trivial system.
Interestingly, such interaction-induced topological states are already known for classical systems characterized by the intensity-dependent coupling constants between some of the sites which give rise to the self-induced topological transitions [@Hadad; @Hadad-ACS; @Hadad-Nature].
To demonstrate interaction-induced topological states of photon pairs, we have recently proposed [@Olekhno] a one-dimensional system depicted schematically in Fig. \[fig:Sketch\](a) which, besides local photon-photon interaction also incorporates a direct two-photon hopping, which does not affect single-particle eigenstates and energies but becomes effective in the presence of two photons.
In this Article, we investigate and advance the concept of interaction-induced topological doublon states in the presence of the direct two-photon hopping, deriving the dispersion of bulk doublons and calculating the Zak phase for them. Our results prove the topological origin of the interaction-induced doublon states and provide valuable insights into the problem of topological characterization of few-body states.
![(a) The sketch of the system under study. Straight connecting lines represent single-photon tunneling amplitude $J$, whereas wavy lines illustrate effective two-photon hopping $P$, which enters the extended Bose-Hubbard Hamiltonian Eq. . (b) In the limit of strong interactions $U\gg J$ the dynamics of bound photon pair is governed by the Su-Schrieffer-Heeger model. []{data-label="fig:Sketch"}](Pic_1.pdf){width="1\linewidth"}
Quite importantly, topological states of doublons studied here should not be mixed with soliton-like nonlinear topological states of classical light arising in waveguide lattices with Kerr-type nonlinearity [@Lumer; @Leykam2016] or mean-field solutions of nonlinear Gross-Pitaevskii equation in the form of vortices [@Solnyshkov-NC] because of the few-body nature of topological states in our proposal.
The rest of the paper is organized as follows. In Sec. \[sec:Model\] we summarize our model and provide simple arguments to prove the existence of interaction-induced topological states. Section \[sec:Dispersion\] contains an in-depth analysis of the bulk properties of bound photon pairs including an analytical model for their dispersion and diagrams showing the evolution of doublon bands when the parameters of the model are varied. The properties of the edge and interface doublon states are examined in Sec. \[sec:Edge\], whereas our conclusions and outlook for future studies appear in Sec. \[sec:Concl\]. Technical details regarding the calculation of bulk doublons dispersion and Zak phase are summarized in Appendices A and B, respectively.
Summary of the model and doublon edge states {#sec:Model}
============================================
We search for the eigenstates of the system described by the extended version of Bose-Hubbard Hamiltonian: $$\begin{aligned}
\label{Hamiltonian}
\hat{H} &=& \omega_0\sum_{m} \hat{n}_m -J \sum_{m}({\hat{a}^{\dag}_{m}} {\hat{a}_{m+1}^{\vphantom{\dag}}}+{\hat{a}^{\dag}_{m+1}} {\hat{a}_{m}^{\vphantom{\dag}}})\nonumber\\
&&+ U \sum_{m} \hat{n}_m(\hat{n}_m-1)\nonumber\\
&& +\frac{P}{2}\sum_m ({\hat{a}^{\dag}_{2m}}{\hat{a}^{\dag}_{2m}}{\hat{a}_{2m+1}^{\vphantom{\dag}}}{\hat{a}_{2m+1}^{\vphantom{\dag}}} +\text{H.c.})\:,\end{aligned}$$ where we assume $\hbar=1$, ${\hat{a}^{\dag}_{m}}$ and ${\hat{a}_{m}^{\vphantom{\dag}}}$ are creation and annihilation operators for the photon in $m^{\rm{th}}$ cavity, $\hat{n}_m={\hat{a}^{\dag}_{m}}\,{\hat{a}_{m}^{\vphantom{\dag}}}$ is local photon number operator, $\omega_0$ is a cavity eigenfrequency and $J$ is photon tunneling amplitude. Term $\propto U$ is a standard term of Bose-Hubbard Hamiltonian describing local photon-photon interaction mediated by the nonlinearity of the medium, whereas an extra term $\propto P$ captures direct two-photon hopping. The latter two terms, obviously, do not come into play provided single-particle dynamics is studied, and hence no single-photon topological states are expected.
What is more remarkable, however, is the two-particle sector of this Hamiltonian. As it is straightforward to verify, the Hamiltonian Eq. conserves the number of particles and thus the two-photon wave function can be searched in the form $$\label{WaveFunc}
{\left|\psi\right>}=\frac{1}{\sqrt{2}}\,\sum_{m,n} \beta_{mn}\, {\hat{a}^{\dag}_{m}}\,{\hat{a}^{\dag}_{n}}\,{\left|0\right>}$$ with the usual normalization ${\left<\psi\left|\psi\right.\right>}=1$ and unknown superposition coefficients $\beta_{mn}$. As a consequence of bosonic symmetry, $\beta_{mn}=\beta_{nm}$ for any indices $m$ and $n$. Inserting Eqs. , into the Schr[ö]{}dinger equation $\hat{H}\,{\left|\psi\right>}=({\varepsilon}+2\,\omega_0)\,{\left|\psi\right>}$ with $2\,\omega_0$ used as an energy reference, we derive the linear system of equations: $$\begin{gathered}
({\varepsilon}-2U)\beta_{2m,2m}=-2J \beta_{2m+1,2m}\nonumber\\
-2J \beta_{2m,2m-1} + P \beta_{2m+1,2m+1}\:,\label{syst1}\\
({\varepsilon}-2U)\beta_{2m+1,2m+1} =-2J\beta_{2m+2,2m+1} \nonumber\\
-2J\beta_{2m+1,2m}+ P \beta_{2m,2m}\:,\label{syst2}\\
{\varepsilon}\beta_{m,n} = - J\beta_{m+1,n} - J \beta_{m-1,n}\notag\\
- J \beta_{m,n+1} - J \beta_{m,n-1}\mspace{12mu} (m\not=n)\:.\label{syst3}\end{gathered}$$ In the case of a finite array of length $N$ we additionally impose open boundary conditions $\beta_{00}=\beta_{m0}=0$ and $\beta_{N+1,N+1}=\beta_{m,N+1}=0$ with $m=1,2,\dots N$. As has been pointed out in Refs. [@Longhi; @DiLiberto; @Gorlach-2017], these equations can be reinterpreted as an eigenvalue problem for the single particle in a two-dimensional tight-binding lattice. In the latter model, photon-photon interactions $U$ are emulated by the detuning of resonance frequency for the diagonal cavities, whereas the two-photon hopping $P$ is represented as an additional coupling between the diagonal sites.
While the outlined one-dimensional two-particle model can be implemented with optical lattices [@Dutta] or with arrays of transmon qubits [@Roushan:2016NatPhys; @Roushan-Science; @Ye2019], the range of parameters attainable in both types of realization is quite limited, the constraints on the magnitude of the direct two-photon hopping being especially strict [@Dutta]. However, in view of the discussed 1D-2D mapping, the same physics can be emulated with two-dimensional classical arrays free of such limitations [@Platero2020], including, for instance, coupled waveguide lattices [@Mukherjee] or LC circuits [@Olekhno]. For that reason, we examine arbitrary ratios $U/J$ and $P/J$ revealing a full plethora of available effects. The only assumption that is made is the repulsive nature of nonlinearity $U>0$. The spectrum for $U<0$ is immediately recovered by calculating the two-photon states for the system with parameters $-U$ and $-P$ and by inverting the sign of the derived energy. Furthermore, to analyze both possible terminations of the array simultaneously, we focus our attention on the case of odd $N$ when the array starts and terminates with different tunneling links.
To grasp the main features of the proposed system, we start from a simplified model valid in the limit $U\gg J$. In such [*strong interaction limit*]{} the doublons are tightly bound, i.e. $\beta_{mm}$ coefficients are the dominant ones in the expansion Eq. . As such, we can rewrite the system Eqs. - in terms of $\beta_{mm}$ coefficients treating $\beta_{m+1,m}$ as perturbation and fully neglecting the coefficients $\beta_{m+p,m}$ for $p\geq 2$. This approach yields the problem: $$\begin{gathered}
({\varepsilon}-2U-2j)\beta_{2m,2m} =j \beta_{2m-1,2m-1}\nonumber\\
+ (j+P)\beta_{2m+1,2m+1}\:,\label{systeff}\\
({\varepsilon}-2U-2j)\beta_{2m+1,2m+1} =(j+P) \beta_{2m,2m} \nonumber\\
+ j \beta_{2m+2,2m+2}\:\label{systeff2}\end{gathered}$$ with the boundary conditions $$\begin{gathered}
({\varepsilon}-2U-j)\beta_{11} =j \beta_{22}\:,\label{boundeff}\\
({\varepsilon}-2U-j)\beta_{N,N} = (j+P)\beta_{N-1,N-1}\:,\label{boundeff2}\end{gathered}$$ where $j = J^2/U$ is the effective doublon hopping rate associated with two consecutive single-particle tunnelings to the neighboring cavity.
Equations , suggest that in the strong interaction limit the dynamics of a doublon is governed by the Su-Schrieffer-Heeger Hamiltonian [@Su] as illustrated by Fig. \[fig:Sketch\](b). This model is known to give rise to the two topologically nontrivial bands with the dispersion $$\label{effsol}
{\varepsilon}_{\pm}(k) = 2U + 2j \pm \sqrt{j^2 + (j+P)^2 +2j(j+P)\cos{2k}}\:.$$ Here Bloch wave number $k$ is defined such that the first Brillouin zone spans the range $[-\pi/2, \pi/2]$. According to Eq. , the bandgap closing occurs for $|j+P|=|j|$, since this condition renders two tunneling amplitudes equal. Specifically, for $P=0$ and $$\label{GapClosingSimple}
P=-2J^2/U$$ the bandgap closes at $k=\pm\pi/2$ and $k=0$, respectively. Furthermore, under a suitable parameter choice, doublon bands can be made dispersionless. As a condition for the flat band, we require that ${\varepsilon}_{+}(0)={\varepsilon}_{+}(\pi/2)$, i.e. $|2j+P|=|P|$, which can happen in two situations: (i) trivial case $P\gg j$ or $U\,P\gg J^2$, which implies both strong photon-photon interaction and strong two-photon hopping and (ii) nontrivial case when $j+P=0$ or $$\label{FlatBandSimple}
P=-J^2/U\:.$$ In the latter case half of tunneling links in the array vanishes, turning it into the collection of uncoupled dimers.
Besides the intuition about the properties of bulk doublon bands, the developed model also provides some insights into the properties of edge and interface states. While in the canonical SSH model the edge state arises at the center-of-bandgap frequency being localized at the weak link edge, this case is a bit different because of the interaction-induced detuning of the edge sites by $j$ as suggested by Eqs. , .
Solving Eq. together with Eqs. , for $N\gg 1$, we do not find any localized states near the site $(N,N)$. At the same time, similar analysis for $(1,1)$ site yields two states with degree of localization given by $$\label{LocLeftEdge}
z_{1,2}=e^{2ik}=\frac{j+P}{2\,j^3}\,\left[2j\,P+P^2\pm\sqrt{(2jP+P^2)^2+4\,j^4}\right]\:,$$ where localized states correspond to $|z|<1$. The energies of the edge states read: $$\label{EnLeftEdge}
{\varepsilon}_{1,2}=2U+j-\frac{1}{2j}\,\left[2jP+P^2\pm\sqrt{(2jP+P^2)^2+4j^4}\right]\:.$$ Equation shows that higher-energy state ${\varepsilon}_2$ is localized for any $P\not=0$, while the lower-energy state ${\varepsilon}_1$ is possible provided $$\label{LocCondition}
-\frac{2\,J^2}{U}<P<0\:.$$ Equation is equivalent to the condition $j>|j+P|$, which guarantees that $(1,1)$ site is the strong link edge. Hence, for parameter values given by Eq. , both of the edge states are Tamm-like. In the opposite case $P>0$ or $P<-2J^2/U$ site $(1,1)$ becomes a weak link edge and supports a single state with energy ${\varepsilon}_2$. Thus, for $P$ outside of the interval Eq. the state ${\varepsilon}_2$ is a topological one, transforming to the Tamm-like state when the condition Eq. is fulfilled. State ${\varepsilon}_1$ is a pure Tamm state. Note also that the boundaries of the interval in Eq. coincide with the points of closing and reopening of a bandgap between two doublon bands which illustrates the bulk-boundary correspondence for two-photon topological states in the strong interaction limit.
To further exemplify topological two-photon states, we analyze interface states localized at the boundary of two one-dimensional arrays with opposite dimerizations. If, for instance, $0^{\rm{th}}$ site is connected with the $1^{\rm{st}}$ and $-1^{\rm{st}}$ sites via the tunneling link $J$, the interface condition in the effective model takes the form: $$\label{IntCond}
\left({\varepsilon}-2U-2j\right)\,\beta_{00}=j\,\left(\beta_{11}+\beta_{-1,-1}\right)\:.$$ Hence, the interface site is not detuned with respect to the bulk ones and as a consequence the topological interface state is located exactly in the middle of bandgap ${\varepsilon}_{\rm{int}}=2U+2j$. If additionally $j>|j+P|$ (short-short defect case), the topological state is also accompanied by two trivial modes lying outside of doublon bandgap [@Blanco-PRL].
The developed model is only valid in the limit of $U\gg J$. In the next Sec. \[sec:Dispersion\] we derive a rigorous solution for the dispersion of bound photon pairs based on Bethe ansatz method and capture a range of intriguing phenomena beyond the canonical SSH model including the interaction of doublon bands with the continuum of scattering states.
Dispersion of bulk doublons {#sec:Dispersion}
===========================
To solve an infinite set of equations - and extract the dispersion of photon pairs, one needs some analytic expression for $\beta_{mn}$ coefficients. A powerful approach to this problem is provided by Bethe ansatz technique [@Essler; @Karbach]. The standard Bethe ansatz has the form: $$\label{eq:Bethe0}
\beta_{mn}=C\,\exp\left[i\frac{k}{2}\,(m+n)+i\frac{{\varkappa}}{2}(m-n)\right]$$ for $m\geq n$. In this expression, $k$ is Bloch wave number describing the motion of photon pair as a whole, whereas ${\varkappa}$ captures the relative motion of particles. Bound photon pairs are characterized by complex ${\varkappa}$, in which case the wave function decays with the increase of separation $(m-n)$ between the photons.
While such simple ansatz captures the properties of bound pairs in the limiting case $P=0$, it appears to be inconsistent with Eqs. - in the general case of $P\not=0$ and arbitrary $k$. To proceed with the analytical solution, we need to incorporate into the ansatz the presence of [*two*]{} sites in the unit cell. This extended unit cell shrinks the first Brillouin zone for doublons from $[-\pi, \pi]$ (as is the case for $P=0$) to $[-\pi/2,\pi/2]$ mixing the states with wave numbers $k$ and $k+\pi$. Therefore, we introduce the following modification of Bethe ansatz: $$\begin{gathered}
\beta_{mn}=C_{1}\,e^{ik(m+n)/2}\,e^{i{\varkappa}_1(m-n)/2}\nonumber\\
+C_{2}\,\,e^{i(k+\pi)(m+n)/2}\,e^{i{\varkappa}_2(m-n)/2}\label{eq:ansatz}\end{gathered}$$ with $m\geq n$ and $\text{Im}\,{\varkappa}_{1,2}>0$. The modified ansatz Eq. appears to be consistent with full system of equations Eqs. - and determines doublon dispersion as further detailed in Appendix A.
Omitting the details of the derivation, we would like to stress here several simple but illuminating results. The energies of doublon bands in the limiting case $k=\pm\pi/2$ can be found analytically: $$\label{EPi2}
{\varepsilon}_{\pm}={\mathop{\mathrm{sgn}}}{[2U\pm P]}\sqrt{(2U\pm P)^2+8J^2}\:.$$ Thus, bound photon pairs are always stable for wave numbers near the boundaries of the first Brillouin zone. In the strong interaction limit, the energies of the two bands scale as $(2U+P)$ and $(2U-P)$, which means that the effective photon-photon interaction $U$ defines the average energy of bound pair, whereas the two-photon hopping $P$ controls energy splitting between the two bands.
For $k=0$, energies of the doublon states read $$\begin{gathered}
{\varepsilon}'_+={\mathop{\mathrm{sgn}}}{[2U+P]}\sqrt{(2 U+P)^2+16J^2}\:,\\
{\varepsilon}'_{-}=2U-P\:,\end{gathered}$$ where ${\varepsilon}_-$ and ${\varepsilon}'_-$ (${\varepsilon}_+$ and ${\varepsilon}'_+$) can correspond to the same or to the different doublon bands. Note that the doublon band associated with ${\varepsilon}'_{-}$ can collapse intersecting with the continuum of two-photon scattering states for nonzero $k$ sufficiently far from the Brillouin zone boundaries. As we show in Appendix A, collapse of the doublon band occurs in the range of parameters $$-4J<2U-P<4J\:.$$
![Dispersion of two-photon excitations in the extended Bose-Hubbard model Eq. . (a,b) Doublon dispersion for the two representative cases: (a) strong interaction limit $U/J = 6$, $P/J = -0.5$; (b) moderate interactions $U/J = 1$, $P/J = -0.5$, when lower doublon band intersects with the continuum of scattering states and collapses. Two Tamm-like doublon edge states exist in the scattering continuum. In both cases, red solid lines correspond to bulk doublons, light green continuum shows energies of two-photon scattering states, horizontal black lines indicate energies of doublon edge states. (c) Evolution of doublon bands shown by red when two-photon hopping $P/J=-0.5$ is fixed and the interaction strength $U$ is varied. Black vertical lines indicate the parameter values used in panels (a) and (b). The green band shows the range of energies for the two-photon scattering states. (d,e) Evolution of doublon bands when photon-photon interaction $U/J$ is fixed and the two-photon hopping $P/J$ is varied. Vertical lines indicate the magnitude of $P$ corresponding to panels (a,b). (d) Strong interactions $U/J=6$. (e) Moderate interactions $U/J=1$. []{data-label="fig:DoublonZones"}](Pic_2.pdf){width="\linewidth"}
To illustrate the obtained solution further, we explore the dispersion of doublons in two characteristic situations with the same two-photon hopping $P/J=-0.5$ and different magnitude of the effective photon-photon interaction: sufficiently strong interactions $U/J=6$, Fig. \[fig:DoublonZones\](a) and moderate interactions $U/J=1$, Fig. \[fig:DoublonZones\](b). The former case, shown in Fig. \[fig:DoublonZones\](a), exhibits a remarkable agreement with the effective SSH model, both in terms of the boundaries of the bulk bands given by Eq. and spectral position of topological edge state. However, when the strength of interaction $U$ is decreased, one of the doublon bands intersects with the continuum of scattering states and collapses \[Fig. \[fig:DoublonZones\](b)\]. In agreement with our previous analysis, both doublon bands are stable near the edges of the Brillouin zone, while one of the bands becomes unstable in the vicinity of $k=0$. At the same time, the edge state near the first site persists coexisting with the continuum of scattering states.
It is also instructive to trace the evolution of doublon bands when the parameters of the model $U$ and $P$ are varied. Specifically, Fig. \[fig:DoublonZones\](c) illustrates the evolution of doublon bands with the interaction strength $U$ for the fixed value of two-photon tunneling $P$. In accordance with Eq. , we observe that the continuum of scattering states can be located right between two doublon bands provided $(2U+P)$ and $(2U-P)$ have different signs. The decreased spectral width of the lower band inside the continuum of scattering states for $0.3<U/J<1.75$ serves as an evidence of doublon collapse. In accordance with the simplified model developed in Sec. \[sec:Model\], doublon bands become dispersionless for $U/J=2$ \[cf. Eq. \] and the gap between them closes at $U/J=4$ \[cf. Eq. \]. In fact, such close agreement is not occasional, since the conditions for the flat band and for gap closing predicted by the simplified model coincide with those obtained from the rigorous solution as discussed in Appendix A.
In the strong interaction limit, two-photon hopping $P$ is the only parameter controlling the separation of two doublon bands. Figure \[fig:DoublonZones\](d) shows almost linear dependence of doublon energies on the magnitude of $P$, illustrating topological transitions due to closing and reopening of bandgap and the emergence of flat bands. The situation appears to be more complicated for moderate interactions $U/J=1$, when doublon bands interact with the scattering continuum collapsing and reviving \[Fig. \[fig:DoublonZones\](e)\]. The doublon state with energy $(2U-P)$ shown by the red dashed line in Fig. \[fig:DoublonZones\](e) appears to be especially robust crossing the entire scattering continuum.
Edge and interface topological doublon states {#sec:Edge}
=============================================
Closing and reopening of a bandgap between two doublon bands demonstrated in Sec. \[sec:Dispersion\] hints towards topological transitions happening in the system. While strong interaction limit $U\gg J$ is well-understandable in terms of the effective SSH-type model, the case of moderate interaction appears to be less intuitive. A characteristic example is presented in Fig. \[fig:DoublonZones\](b) when one of the doublon bands partially collapses and the edge state appears in the continuum of scattering states. These observations demonstrate two important features of our system.
. []{data-label="fig:EdgeState"}](Pic_3.pdf){width="\linewidth"}
First, the problem of bulk-boundary correspondence in two-particle topological models becomes more involved, since the corresponding doublon band can collapse leaving Berry connection undefined, whereas the topological state persists.
Second, the two-particle bound edge state can coexist with the continuum of scattering states as has been previously pointed out for a different two-particle model [@Zhang2012; @Zhang2013] thus providing a realization of the two-particle bound state in the continuum (BIC) [@Soljacic-bic].
To get further intuition about doublon BIC arising in this system, we analyze two geometries: (i) semi-infinite array with the edge state localized near the first site; (ii) domain wall between the two arrays with different dimerizations hosting the interface state. In the first scenario illustrated in Fig. \[fig:EdgeState\](a-d) we observe two edge states of bound photon pairs. The state with higher energy, shown in panels (a,b), still retains quite good localization close to exponential. At the same time, lower-energy state \[Fig. \[fig:EdgeState\](c,d)\] which lies deeper in the scattering continuum clearly exhibits non-exponential localization caused by the stronger interaction with the continuum. In turn, the interface state Fig. \[fig:EdgeState\](e) exhibits a symmetric profile with respect to the domain wall thus resembling the interface state at a long-long defect in the canonical SSH model [@Blanco-PRL]. However, in contrast to the SSH case, the localization of the interface state is non-exponential \[Fig. \[fig:EdgeState\](f)\] due to its hybridization with the two-photon scattering states.
Discussion and conclusions {#sec:Concl}
==========================
To summarize, we have investigated a system with interaction-induced topological order. Even though the single-particle model is topologically trivial, the two-particle bands feature a topological bandgap with doublon edge and interface states inside it. Quite interestingly, the observed two-particle topological states remain stable under the collapse of the bulk doublon band and, moreover, they can coexist with the scattering continuum thus providing a realization of the two-photon interaction-induced BIC.
![(a) Illustration of various two-photon Fock states ${\hat{a}^{\dag}_{m}}\,{\hat{a}^{\dag}_{n}}\,{\left|0\right>}$ comprising the two-photon wave function on a two-dimensional map. Boundaries of the inversion-symmetric unit cell used for the Zak phase calculation for doublon bands are shown by green dashed lines. (b) Phase diagram showing the magnitude of the Zak phase as a function of model parameters $U/J$ and $P/J$. Zak phase is equal to $\pi$ in the red shaded area of the diagram and equal to $0$ elsewhere. Areas shaded by light green and light purple indicate the parameter ranges corresponding to the collapse of lower and upper doublon bands, respectively. The black dashed line corresponds to the condition $P=2U$. []{data-label="fig:PhaseDiagram"}](Pic4.pdf){width="0.65\linewidth"}
To prove the topological origin of the observed doublon edge states, we perform the calculation of the Zak phase for bulk doublon bands. Our results shown in Fig. \[fig:PhaseDiagram\] and summarized in Appendix B suggest extremely rich physics associated with topological transitions, collapse and revival of doublon bands and edge states taking place as the parameters of the system are varied.
We believe that the physical realization of the proposed model can be based on cold atomic gases in optical lattices, arrays of coupled transmon qubits or other systems featuring a significant anharmonicity of on-site potential. At the same time, mapping of interacting one-dimensional two-body problem onto the two-dimensional classical setup possible for the class of Bose-Hubbard models opens another experimentally feasible route to emulate topological states in interacting systems.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge valuable discussions with Alexander Poddubny, Nikita Olekhno and Marco Di Liberto. This work was supported by the Russian Science Foundation (grant No. 16-19-10538). A.A.S. acknowledges partial support by Quantum Technologies Center, Faculty of Physics, Lomonosov Moscow State University. M.A.G. acknowledges partial support by the Foundation for the Advancement of Theoretical Physics and Mathematics “Basis".
Appendix A. Calculation of bulk doublon dispersion {#appendix-a.-calculation-of-bulk-doublon-dispersion .unnumbered}
==================================================
In this Appendix, we outline the rigorous solution for the dispersion of bulk doublons based on modified Bethe ansatz, Eq. . We seek the solution of equations $$\begin{gathered}
{\varepsilon}\beta_{m,n} = - J\beta_{m+1,n} - J \beta_{m-1,n}\nonumber\\
- J \beta_{m,n+1} - J \beta_{m,n-1}\mspace{12mu} (m\not=n),\label{sys1}\\
({\varepsilon}-2U)\beta_{2m,2m}=-2J \beta_{2m+1,2m}\nonumber\\
-2J \beta_{2m,2m-1} + P \beta_{2m+1,2m+1}\:,\label{sys2}\\
({\varepsilon}-2U)\beta_{2m+1,2m+1} =-2J\beta_{2m+2,2m+1} \nonumber\\
-2J\beta_{2m+1,2m}+ P \beta_{2m,2m}\label{sys3}\end{gathered}$$ assuming that $$\label{Bethe}
\begin{split}
\beta_{mn}=C_{1}\,e^{ik(m+n)/2}\,e^{i{\varkappa}_1(m-n)/2}\\
+C_{2}\,\,e^{i(k+\pi)(m+n)/2}\,e^{i{\varkappa}_2(m-n)/2}
\end{split}$$ for any $m\geq n$ and ${\mathop\mathrm{Im}\nolimits}{\varkappa}_{1,2}>0$. Inserting Eq. into the system Eqs. -, we get: $$\begin{gathered}
{\varepsilon}=-4J\,\cos\frac{k}{2}\,\cos\frac{{\varkappa}_1}{2}\:,\label{res1}\\
{\varepsilon}=4J\,\sin\frac{k}{2}\,\cos\frac{{\varkappa}_2}{2}\:,\label{res2}\\
C_1\,\left[{\varepsilon}-2U+4J\,\cos\frac{k}{2}\,e^{i{\varkappa}_1/2}-P\,e^{ik}\right]\notag\\
+C_2\,\left[{\varepsilon}-2U-4J\,\sin\frac{k}{2}\,e^{i{\varkappa}_2/2}+P\,e^{ik}\right]=0\:,\label{res3}\\
C_1\,\left[{\varepsilon}-2U+4J\,\cos\frac{k}{2}\,e^{i{\varkappa}_1/2}-P\,e^{-ik}\right]\notag\\
+C_2\,\left[{\varepsilon}-2U-4J\,\sin\frac{k}{2}\,e^{i{\varkappa}_2/2}+P\,e^{-ik}\right]=0\:.\label{res4}\end{gathered}$$ Here, Eqs. - are obtained from the single Eq. due to the linear independence of $e^{i{\varkappa}_1\,(m-n)/2}$ and $e^{i{\varkappa}_2\,(m-n)/2}$ for $m\not=n$. The system of four equations - defines four unknowns: ${\varepsilon}$, ${\varkappa}_1$, ${\varkappa}_2$ and the ratio $C_1/C_2$, while the absolute values of $C_1$ and $C_2$ are determined from the normalization of the two-photon wave function.
Excluding doublon energy from Eqs. - with Eqs. - and further rearranging them, we arrive to the linear system with respect to $C_1$ and $C_2$: $$\begin{aligned}
& C_1\,\left[4iJ\,\cos\frac{k}{2}\,\sin\frac{{\varkappa}_1}{2}-2U-P\,\cos k\right]\nonumber\\
& +iC_2\,P\,\sin k=0\:,\label{lin1}\\
&-iC_1\,P\,\sin k\nonumber\\
&+C_2\,\left[-4iJ\,\sin\frac{k}{2}\,\sin\frac{{\varkappa}_2}{2}-2U+P\,\cos k\right]=0\:.\label{lin2}\end{aligned}$$ Setting the determinant of this system to zero, we recover that $$\begin{gathered}
8J^2\,\sin k\,\sin\frac{{\varkappa}_1}{2}\,\sin\frac{{\varkappa}_2}{2}-4iJ\,\cos\frac{k}{2}\,\sin\frac{{\varkappa}_1}{2}\,(2U-P\,\cos k)\nonumber\\
+4iJ\,\sin\frac{k}{2}\,\sin\frac{{\varkappa}_2}{2}\,\left(2U+P\,\cos k\right)+4\,U^2-P^2=0\:.\label{LocCond}\end{gathered}$$ To provide an efficient numerical algorithm to calculate the dispersion of doublons, we introduce two auxiliary dimensionless variables: $$\begin{aligned}
x\equiv -i\cos\frac{k}{2}\,\sin\frac{{\varkappa}_1}{2}\:,\label{xdef}\\
y\equiv -i\sin\frac{k}{2}\,\sin\frac{{\varkappa}_2}{2}\:.\label{ydef}\end{aligned}$$ Making use of Eqs. , , , we get the following closed-form system of equations: $$\begin{aligned}
& f(x,y,k)\equiv x^2-y^2+\cos k=0\:,\label{num1}\\
& g(x,y,k)\equiv -16J^2\,x\,y+4Jx\,(2U-P\,\cos k)\nonumber\\
&-4Jy\,(2U+P\,\cos k)+4U^2-P^2=0\:.\label{num2}\end{aligned}$$ Separating real and imaginary parts for ${\varkappa}_1$ and ${\varkappa}_2$ and taking into account that ${\mathop\mathrm{Im}\nolimits}{\varkappa}_{1,2}>0$, we can show that the real parts of $x$, $y$ and ${\varepsilon}$ satisfy the following condition: $$\label{SignCond}
{\mathop{\mathrm{sgn}}}{\varepsilon}=-{\mathop{\mathrm{sgn}}}\,x={\mathop{\mathrm{sgn}}}\,y\:.$$ Hence, to find doublon dispersion one has to solve the system of algebraic equations - numerically, keeping only those pairs of $x$ and $y$ which satisfy an additional constraint Eq. . Doublon energy is then given by $${\varepsilon}=-4J\,{\mathop{\mathrm{sgn}}}x\,\sqrt{\cos^2\frac{k}{2}+x^2}\:.$$
Note that bound pairs coexist with the continuum of two-photon scattering states characterized by real $k$ and ${\varkappa}$. Bulk dispersion of such states is captured by the simple formula , where both parameters are real.
In some limiting cases, the expression for the doublon energy simplifies considerably. For instance, if $k=\pm\pi/2$, $x=-(2U\pm P)/(4J)$, whereas doublon energy $$\label{RootPi}
{\varepsilon}_{\pm}={\mathop{\mathrm{sgn}}}\left(2U\pm P\right)\,\sqrt{(2U\pm P)^2+8\,J^2}\:.$$ Hence, doublons with $k$ at the edge of Brillouin zone are always stable.
Another limiting case is $k=0$ when one of the doublon bands has $C_2=0$ and Bethe ansatz in the standard form Eq. can be applied. In such a case, the energy of this particular doublon band is given by $$\label{Root01}
{\varepsilon}'_{+}={\mathop{\mathrm{sgn}}}(2U+P)\,\sqrt{(2U+P)^2+16J^2}\:,$$ whereas the second doublon solution is localized purely on the diagonal and has the energy $$\label{Root02}
{\varepsilon}'_{-}=2U-P\:.$$ Note, however, that the latter doublon band is unstable for nonzero $k$ far enough from the Brillouin zone boundaries.
With the obtained rigorous solution, we can also find the condition for closing of the gap between two doublon bands. To this end, two roots for ${\varepsilon}(k=0)$ given by Eqs. and should coincide, i.e. $$\label{GapClosing}
UP=-2J^2\:.$$
In a similar manner we examine the condition for flat band. Inspecting Eqs. , , , we find out that ${\varepsilon}_{+}={\varepsilon}'_{-}$ and ${\varepsilon}_{-}={\varepsilon}'_{+}$ provided $$\label{FlatBand}
UP=-J^2\:.$$ Note that ${\varepsilon}_{+}$ and ${\varepsilon}'_{+}$ can correspond to the same or to the different doublon bands depending on parameter choice, and the subscript $\pm$ is used here just to label the solutions.
Quite interestingly, both of these results, Eqs. and coincide with those obtained from the simplified SSH-type effective model, Eqs. , . Additionally, we can find the conditions for collapse of the doublon state. Collapsing band is characterized by the decreased co-localization of photons, i.e. $x\rightarrow 0$. Solving Eqs. , for $k=0$ with $x=0$, we find the following condition: $$\label{CollapseCond}
2U-P=\pm 4J\:.$$ In a similar manner one can also examine the collapse of doublon with arbitrary wave number $k$.
Appendix B. Calculation of the Zak phase for bulk doublon bands {#appendix-b.-calculation-of-the-zak-phase-for-bulk-doublon-bands .unnumbered}
===============================================================
An obtained analytic solution for the dispersion of doublons allows us to evaluate the Zak phase for bulk doublon bands. To this end, we take the periodic part ${\left|u_k\right>}$ of the full doublon wave function in the following form:
$$\begin{split}
\label{WaveFuncCell}
{\left|u_k\right>} =\beta_{0,0}{\left|2_0\right>}+\beta_{1,1}{\left|2_1\right>}+\sqrt{2}\sum_{n=1}^{\infty} \beta_{n,-n} {\left|1_{n}1_{-n}\right>}\\+\sqrt{2}\sum_{n=2}^{\infty} \beta_{n,2-n} {\left|1_{n}1_{2-n}\right>}
+\dfrac{1}{\sqrt{2}}\sum_{n=1}^{\infty} \beta_{n-1,-n} {\left|1_{n-1}1_{-n}\right>}\\+\sqrt{2}\sum_{n=1}^{\infty} \beta_{n,1-n} {\left|1_{n}1_{1-n}\right>}+\dfrac{1}{\sqrt{2}}\sum_{n=1}^{\infty} \beta_{n+1,2-n} {\left|1_{n+1}1_{2-n}\right>}\:,
\end{split}$$
where ${\left|1_m 1_n\right>}\equiv {\hat{a}^{\dag}_{m}}\,{\hat{a}^{\dag}_{n}}\,{\left|0\right>}$ for $m\not=n$ and ${\left|2_m\right>}\equiv 2^{-1/2}\,{\hat{a}^{\dag}_{m}}\,{\left|0\right>}$. The corresponding choice of the unit cell is illustrated in Fig. \[fig:PhaseDiagram\](a). Importantly, such unit cell is inversion-symmetric which ensures quantization of the Zak phase in units of $\pi$ [@Zak].
In turn, the coefficients $\beta_{mn}$ are defined from Eq. and hence $$\begin{aligned}
{\left|u_k\right>}&=&C_1{\left|v_1\right>} +C_2{\left|v_2\right>}\:,\\
{\left|v_1\right>}&=& {\left|2_0\right>} +e^{ik}{\left|2_1\right>}+\sqrt{2}\sum_{n=1}^{\infty} e^{i{\varkappa}_1 n} {\left|1_{n}1_{-n}\right>}\nonumber\\&&
+\sqrt{2}\sum_{n=2}^{\infty} e^{i{\varkappa}_1(n-1)} e^{i k} {\left|1_{n}1_{2-n}\right>}\nonumber\\
&&+\dfrac{1}{\sqrt{2}}\sum_{n=1}^{\infty}e^{i{\varkappa}_1(n-\frac{1}{2})} \bigg{[} e^{-\frac{ik}{2}}{\left|1_{n-1}1_{-n}\right>}\nonumber\\&&+2e^{\frac{ik}{2}} {\left|1_{n}1_{1-n}\right>}+e^{\frac{3ik}{2}} {\left|1_{n+1}1_{2-n}\right>}\bigg{]}\:,\label{eq:v1}\end{aligned}$$ $$\begin{aligned}
{\left|v_2\right>}&=& {\left|2_0\right>} - e^{ik}{\left|2_1\right>}+\sqrt{2}\sum_{n=1}^{\infty} e^{i{\varkappa}_2 n} {\left|1_{n}1_{-n}\right>}\nonumber\\&&
- \sqrt{2}\sum_{n=2}^{\infty} e^{i{\varkappa}_2(n-1)}e^{ik} {\left|1_{n}1_{2-n}\right>}\nonumber\\
&&+\dfrac{1}{\sqrt{2}}\sum_{n=1}^{\infty}e^{i{\varkappa}_2(n-\frac{1}{2})} \bigg{[} -i e^{-\frac{ik}{2}}{\left|1_{n-1}1_{-n}\right>}\nonumber\\&&+2 i e^{\frac{ik}{2}} {\left|1_{n}1_{1-n}\right>} - i e^{\frac{3ik}{2}} {\left|1_{n+1}1_{2-n}\right>}\bigg{]}\:.\label{eq:v2}\end{aligned}$$
The Zak phase is defined in terms of Berry connection as $$\gamma=\int\limits_{-\pi/2}^{\pi/2}\,A(k)\,dk\:,$$ where Berry connection $A(k)$ $$\begin{aligned}
\label{eq:BerryConnection}
A(k)&=& i \bigg{\langle} u_k{\left|\frac{\partial u_k}{\partial k}\right>}
=i C_1^*\frac{\partial C_1}{\partial k}\langle v_1 {\left|v_1\right>}
+i C_1^*\frac{\partial C_2}{\partial k}\langle v_1 {\left|v_2\right>}\nonumber\\&&
+i C_1^*C_1 \bigg{\langle} v_1 {\left|\frac{\partial v_1}{\partial k}\right>}
+i C_1^*C_2 \bigg{\langle} v_1 {\left|\frac{\partial v_2}{\partial k}\right>} \nonumber\\
&&+i C_2^*\frac{\partial C_1}{\partial k}\langle v_2 {\left|v_1\right>}
+i C_2^*\frac{\partial C_2}{\partial k}\langle v_2 {\left|v_2\right>}\nonumber\\&&
+i C_2^*C_1 \bigg{\langle} v_2 {\left|\frac{\partial v_1}{\partial k}\right>}
+i C_2^*C_2 \bigg{\langle} v_2 {\left|\frac{\partial v_2}{\partial k}\right>} \:.\end{aligned}$$ Using Eqs. -, we calculate the scalar products: $$\begin{aligned}
\langle v_1{\left|v_1\right>}&=& \frac{2(1+x_{11})+3\sqrt{x_{11}}}{1-x_{11}}\:, \\
\langle v_1{\left|v_2\right>}&=&\langle v_2{\left|v_1\right>}^*= i\frac{\sqrt{x_{12}}}{1-x_{12}}\:,\\
\langle v_2{\left|v_2\right>}&=& \frac{2(1+x_{22})+3\sqrt{x_{22}}}{1-x_{22}}\:,\end{aligned}$$
$$\begin{aligned}
\bigg{\langle} v_1{\left|\frac{\partial v_1}{\partial k}\right>}&=&i{\varkappa}_1'\frac{8x_{11}+3\sqrt{x_{11}}(1+x_{11})}{2(1-x_{11})^2}\nonumber\\&&
+ i\frac{2(1+x_{11})+3\sqrt{x_{11}}}{2(1-x_{11})}\:, \\
\bigg{\langle} v_1{\left|\frac{\partial v_2}{\partial k}\right>}&=&-\frac{2i(1+x_{12})
+\sqrt{x_{12}}}{2(1-x_{12})}\nonumber\\&&
-{\varkappa}_2'\frac{\sqrt{x_{12}}(1+x_{12})}{2(1-x_{12})^2}\:,\\
\bigg{\langle} v_2{\left|\frac{\partial v_1}{\partial k}\right>}&=&-\frac{2i(1+x_{21}) -\sqrt{x_{12}}}{2(1-x_{21})}\nonumber\\&&
+{\varkappa}_1'\frac{\sqrt{x_{21}}(1+x_{21})}{2(1-x_{21})^2} \:,\\
\bigg{\langle} v_2{\left|\frac{\partial v_2}{\partial k}\right>}&=&i{\varkappa}_2'\frac{8x_{22}+3\sqrt{x_{22}}(1+x_{22})}{2(1-x_{22})^2}\nonumber\\&&
+i\frac{2(1+x_{22})+3\sqrt{x_{22}}}{2(1-x_{22})}\:,\end{aligned}$$
where $x_{\alpha\beta} = (e^{i{\varkappa}_\alpha})^*e^{i{\varkappa}_\beta}$.
The scalar products ${\left<\dfrac{\partial v_i}{\partial k}\right|}v_j\bigg{\rangle} $ are obtained by complex conjugation of $\bigg{\langle} v_j{\left|\dfrac{\partial v_i}{\partial k}\right>}$. Calculating Berry connection, we need to determine five quantities: $\dfrac{\partial C_1}{\partial k}$, $\dfrac{\partial C_2}{\partial k}$, $\dfrac{\partial {\varkappa}_1}{\partial k}$, $\dfrac{\partial {\varkappa}_2}{\partial k}$, $\dfrac{\partial {\varepsilon}}{\partial k}$. Differentiation of the identities Eqs. - with respect to $k$ yields only four equations. One more equation can be obtained differentiating the identity $$\label{NormCond}
\langle u_k{\left|u_k\right>} =1\:,$$ which is the normalization condition for the periodic part of the wave function. In these calculations, the gauge of the wave function should be fixed. To ensure smooth behavior of $C_1$ and $C_2$ coefficients with $k$, we choose their phases such that $(C_1+C_2) e^{i\phi}$ is real, where: $$\begin{aligned}
e^{i\phi} &=& \sqrt{\dfrac{-P e^{-ik}-2U+4iJ\sin{{\varkappa}_1/2}\cos{k/2}}{P e^{ik}+2U-4iJ\sin{{\varkappa}_1/2}\cos{k/2}}}\:.\end{aligned}$$ Such choice ensures, in particular, that the Berry connection is a smooth function of wave number $k$. The Zak phase calculation is then accomplished in several steps:
1\) Complex coefficients $C_{1,2}$ are calculated using normalization condition Eq. and Eqs. , .
2\) The derivatives $\dfrac{\partial C_1}{\partial k}$,$\dfrac{\partial C_2}{\partial k}$, $\dfrac{\partial {\varkappa}_1}{\partial k}$, $\dfrac{\partial {\varkappa}_2}{\partial k}$, $\dfrac{\partial {\varepsilon}}{\partial k}$ are evaluated from the differentiated identities Eqs. - and .
3\) The obtained quantities are inserted into Eq. and Berry connection is evaluated numerically in the entire Brillouin zone. The Zak phase is recovered by numerical integration.
Performing the calculation of the Zak phase for different values of model parameters, $U$ and $P$, we plot the phase diagram shown in Fig. \[fig:PhaseDiagram\](b) and analyze topological transitions happening in the system. For unit cell choice shown in Fig. \[fig:PhaseDiagram\](a), the link with the direct two-photon hopping is located inside the unit cell. For $U>1$, $\gamma=\pi$ is achieved exactly in the same range of parameters when $|j+P|<j$, i.e. the weak tunneling link appears [*inside*]{} the unit cell. This is consistent with the result expected from the effective SSH model with coupling constants equal to $j=J^2/U$ and $j+P$.
However, besides the analogy with the effective SSH model, our system also exhibits some distinctive properties. First, even if one of the doublon bands collapses, the Zak phase can still be defined using the wave functions for the remaining band. Furthermore, despite the collapse of the bulk doublon band, the doublon edge state can persist, which provides an interesting feature of two-photon topological states.
Even though this model is just a particular example, we believe that the present analysis provides valuable insights into the topological properties and bulk-boundary correspondence in nonlinear topological models.
|
---
abstract: 'The identification of a complete three-dimensional (3D) photonic band gap in real crystals always employs theoretical or numerical models that invoke idealized crystal structures. Thus, this approach is prone to false positives (gap wrongly assigned) or false negatives (gap missed). Therefore, we propose a purely experimental probe of the 3D photonic band gap that pertains to many different classes of photonic materials. We study position and polarization-resolved reflectivity spectra of 3D inverse woodpile structures that consist of two perpendicular nanopore arrays etched in silicon. We observe intense reflectivity peaks $(R > 90\%)$ typical of high-quality crystals with broad stopbands. We track the stopband width versus pore radius, which agrees much better with the predicted 3D photonic band gap than with a directional stop gap on account of the large numerical aperture used. A parametric plot of s-polarized versus p-polarized stopband width agrees very well with the 3D band gap and is model-free. This practical probe provides fast feedback on the advanced nanofabrication needed for 3D photonic crystals and stimulates practical applications of band gaps in 3D silicon nanophotonics and photonic integrated circuits, photovoltaics, cavity QED, and quantum information processing.'
author:
- Manashee Adhikary
- Ravitej Uppu
- 'Cornelis A.M. Harteveld'
- 'Diana A. Grishina'
- 'Willem L. Vos'
bibliography:
- '3D\_pbg\_probe.bib'
title: '**Experimental probe of a complete 3D photonic band gap**'
---
Introduction
============
Completely controlling the emission and the propagation of light simultaneously in all three dimensions (3D) remains a major outstanding target in the field of Nanophotonics [@Novotny2006book; @Joannopoulos2008PhotonicLight; @Lourtioz2008book; @Noginov2009book; @Ghulinyan2015book]. Particularly promising tools for this purpose are 3D photonic crystals with spatially periodic variations of the refractive index commensurate with optical wavelengths. The photon dispersion relations inside such crystals are organized in bands, analogous to electron bands in solids [@Ashcroft1976book; @Economou2010book], see for example Figure \[Fig:bandstruct\](a). When light waves inside a crystal are Bragg diffracted, directional energy gaps – known as stop gaps – arise for the relevant incident wavevector. When the stop gaps have a common overlap range for all wavevector and all polarizations, the 3D nanostructure has a photonic band gap. Within the band gap, no light modes are allowed in the crystal due to multiple Bragg interference [@vanDriel2000PRB; @Vos2000PLA; @Romanov2001PRE], hence the density of states (DOS) strictly vanishes. Since the local density of states also vanishes, the photonic band gap is a powerful tool to radically control spontaneous emission and cavity quantum electrodynamics (QED) of embedded quantum emitters [@Bykov1972JETP; @Yablonovitch1987PRL; @John1990PRL; @Vos2015CavityCrystals]. Applications of 3D photonic band gap crystals range from dielectric reflectors for antennae [@Smith1998MOTL] and for efficient photovoltaic cells [@Bermel2007OE; @Wehrspohn2012JO; @Koenderink2015science], via white light-emitting diodes [@David2012RPP], to elaborate 3D waveguides [@Li2003JOSA] for 3D photonic integrated circuits [@Tajiri2019Optica], and to thresholdless miniature lasers [@Tandaechanurat2011NP] and devices to control quantum noise for quantum measurement, amplification, and information processing [@Clerk2010RMP; @Vos2015CavityCrystals].
![(a) Band structure of an inverse woodpile photonic crystal calculated for a reduced pore radius $r/a$ = 0.22 and a relative permittivity $\varepsilon_{Si}=11.68$. The abscissa is the reduced wave vector in the $\Gamma Z$ high-symmetry direction. The stop gaps for $s$-polarized and $p$-polarized light are indicated by the red and blue bars, respectively, and $p$-polarized bands are shown in blue and $s$ bands in red [@Devashish2017PRB]. (b) The $\Gamma Z$ stop gap and photonic band gap as a function of the reduced pore radius $r/a$, with corresponding air volume fractions shown on the top abscissa. For $s$ and $p$ polarizations, the $\Gamma Z$ stop gap edges are shown as the blue and red dotted curves and the green and red dashed curves, respectively. The full black curves are the edges of the 3D band gap.[]{data-label="Fig:bandstruct"}](bandstruct_GammaZ_Rovera022.pdf "fig:"){width="70mm"} ![(a) Band structure of an inverse woodpile photonic crystal calculated for a reduced pore radius $r/a$ = 0.22 and a relative permittivity $\varepsilon_{Si}=11.68$. The abscissa is the reduced wave vector in the $\Gamma Z$ high-symmetry direction. The stop gaps for $s$-polarized and $p$-polarized light are indicated by the red and blue bars, respectively, and $p$-polarized bands are shown in blue and $s$ bands in red [@Devashish2017PRB]. (b) The $\Gamma Z$ stop gap and photonic band gap as a function of the reduced pore radius $r/a$, with corresponding air volume fractions shown on the top abscissa. For $s$ and $p$ polarizations, the $\Gamma Z$ stop gap edges are shown as the blue and red dotted curves and the green and red dashed curves, respectively. The full black curves are the edges of the 3D band gap.[]{data-label="Fig:bandstruct"}](gapmap_VF.pdf "fig:"){width="70mm"}
Thanks to extensive research efforts in nanotechnology, great strides have been made in the fabrication of 3D nanostructures that interact strongly with light such that they possess a 3D full and complete photonic band gap [@Lopez2003AM; @Benisty2006ProgrOptics; @Galisteo2011AM; @Vos2015CavityCrystals]. Remarkably, however, it remains a considerable challenge to decide firstly whether a 3D nanostructure has a *bona fide* photonic band gap functionality or not, and secondly to assess how broad such a band gap is, which is critical for the robustness of the functionality. It is natural to try to probe the photonic band gap via its influence on the DOS and LDOS by means of emission spectra or time-resolved emission dynamics of emitters embedded inside the photonic crystal [@Ogawa2004S; @Lodahl2004Nat; @Aoki2008NP; @Leistikow2011PRL]. However, such experiments are rather difficult for several practical reasons, that notably involve the emitter’s quantum efficiency [@Koenderink2002PRL], the choice of a suitable reference system [@Koenderink2003PSSA], and finite-size effects [@Hasan2018PRL].
Alternatively, the presence of a gap in the density of states may be probed by transmission or reflectivity [@Lin1998Nature; @Thijssen1999PRL; @Noda2000Science; @Blanco2000Nature; @Vlasov2001N; @Schilling2005APL; @Garcia-Santamaria2007AM; @Takahashi2009NM; @Staude2010OL; @Huisman2011PRB; @Frolich2013AM; @Marichy2016SR]. In such an experiment, a peak in reflectivity or a trough in transmission identifies a stopband in the real and finite crystal that is interpreted with a directional stop gap in the dispersion relations. By studying the 3D crystal over a sufficiently large solid angle, one expects to see a signature of a 3D photonic band gap. While reflectivity and transmission are readily measured, such probes suffer from two main limitations. One technical impediment is when a reflectivity or transmission experiment samples a too small angular range to safely assign a gap, whereas a broader range would reveal band overlap. A second class of impediment includes possible artifacts related to uncoupled modes [@Robertson1992PRL; @Sakoda2005book], fabrication imperfections, or unavoidable random disorder, all of which may lead either to erroneously assigned band gaps (‘false positive’) or to overlooked gaps (‘false negative’). To date, these issues are addressed by supplementing reflectivity or transmission experiments with theoretical or numerical results and deciding the presence of a band gap and its width from such results. Theory or numerical simulations, however, always require a model for the photonic crystal’s structure and the building blocks inside the unit cell. Such a model is necessarily an idealization of the real crystal structure and thus misses essential features. For instance, crystal models are often taken to be infinitely extended and then lack an interface that essentially determines reflectivity features [@Devashish2017PRB]. Or unavoidable disorder is not taken into account, while a certain degree of disorder may completely close a band gap [@Li2000PRB]. Or the crystal structure model lacks random stacking (occurring in self-organized structures) which affects the presence and width of a band gap [@Wang2003PRE]. Thus, in case that the ideal model differs from the real structure, the optical functionality of the crystal differs from the expected design for reasons that are far from trivial to identify [@Grishina2018Arxiv]. Therefore, the goal of this paper is to find a purely experimental identification of a photonic band gap, which is robust to artifacts as it avoids the need for modeling. To this end, we collect polarization and position-resolved reflectivity spectra with a large numerical aperture. By mapping the width of the observed stopband versus a characteristic structural feature (here: pore radii in inverse woodpile crystals) that tunes the average refractive index, and by parametrically plotting the width of the observed s-polarized stopband versus the p-polarized one, we arrive at an experimental probe to decide whether a photonic crystal has a band gap.
\[sec:Methods\]Samples and experimental
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Inverse woodpile crystals
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Here we study 3D photonic band gap crystals with the inverse woodpile crystal structure [@Ho1994SSC] made of silicon by CMOS-compatible means. The inverse woodpile structure is designed to consist of two identical two dimensional (2D) arrays of pores with radius $R$ running in the perpendicular $X$ and $Z$ directions. Each 2D array of pores has a centered-rectangular structure with lattice constants $a$ and $c$ in a ratio $a/c=\sqrt{2}$ for the crystal structure to be cubic with a diamond-like symmetry, as illustrated in a YouTube animation [@COPS2012youtube]. Inverse woodpile crystals have a broad 3D photonic band gap on account of their diamond-like structure [@Maldovan2004NM] with a maximum relative bandwidth of 25.4% for a reduced pore radius $r/a=0.245$ and a relative permittivity $\epsilon_{Si}=11.68$ typical of silicon backbone [@Hillebrand2003JAP; @Woldering2009JAP].
Figure \[Fig:bandstruct\](a) shows the band structure calculated for the $\Gamma Z$ high symmetry direction since in our experiments the axis of the incident light cone is along this direction. The stop gap is the frequency range where modes are forbidden in this high symmetry direction. The relative bandwidth of the stop gap, gauged as the gap width $\Delta \omega$ to mid-gap $\omega_c$ ratio, is wider for $s$-polarized light ($\Delta \omega/\omega_c = 36.5\%$) than for $p$-polarized light ($\Delta \omega/\omega_c = 27.6\%$), which is reasonable since in the former case the electric field is perpendicular to the first layer of pores so that light scatters more strongly from this layer. For the diamond-like inverse woodpile structure, the $\Gamma Z$ high-symmetry direction is equivalent to the $\Gamma X$ high-symmetry direction, and thus also their opposite counterparts *viz.* the $-\Gamma Z$ and $-\Gamma X$ high-symmetry directions [@Huisman2011PRB; @Devashish2017PRB]. Several bands have $s$ or $p$-polarized character following the assignment of Ref. [@Devashish2017PRB]. We refer to Bloch mode polarization to indicate their symmetry properties while being excited with either $s$ or $p$-polarized light incident from a high-symmetry direction (here the Z-direction).
Figure \[Fig:bandstruct\](b) shows the $\Gamma Z$ stop gaps for $s$ and $p$ polarization as a function of the reduced pore radius $r/a$, as well as the photonic band gap [@Huisman2011PRB]. The centers of all gaps shift to higher frequencies which makes sense, since a gap center frequency $\omega_c$ is equal to $\omega_c = \frac{c'}{n_{eff}}.k_{BZ}.G$ [@Vos1996PRB], with $c'$ the speed of light (not to be confused with the lattice parameter $c$), $n_{eff}$ the effective refractive index of the photonic crystal [@Datta1993PRB], and $G$ a structure-specific constant [@Vos1996PRB]. An increasing pore radius in Fig. \[Fig:bandstruct\](b) corresponds to an increasing air volume fraction and thus to a decreasing effective refractive index, hence to an increasing gap center frequency. As reported earlier, the 3D photonic band gap is the widest for $r/a=0.245$ and it is robust as it is open within the broad range $0.14 < r/a < 0.29$ [@Hillebrand2003JAP; @Woldering2009JAP]. When comparing the stop gaps and the 3D photonic band gap, we note that all lower edges nearly overlap, whereas the upper edges are all different. The overlap of the lower edges of the stop gaps and the band gap is robust as a function of pore radius $(r/a)$ and hence effective refractive index, which is a convenient feature that we will exploit.
![Scanning electron microscopy (SEM) image of the edge of silicon beam **A** with a cubic 3D inverse woodpile crystal in perspective view. The crystal consists of two sets of perpendicular pores along the $X$ and $Z$ directions with design radius $r_{d} = 160$ nm. The coordinate system used in the paper is shown with the origin at the lower right corner of the crystal. The crystal has lattice parameters $a = 680$ in the Y-direction, and $c$ in the X and Z-directions, with $c=a/\sqrt{2}$ [@VanDenBroek2012AFM; @Grishina20173DNanophotonics].[]{data-label="fig:SEM"}](SEM.png){width="66mm"}
The crystals are fabricated by etching pores into crystalline silicon using CMOS-compatible methods [@VanDenBroek2012AFM]. We employed deep reactive ion etching through an etch mask that was fabricated on the edge of a silicon beam [@Tjerkstra2011JVSTB; @Grishina2015Nanotech; @Grishina20173DNanophotonics]. Multiple crystals with different design pore radii $r_{d}$ and a constant lattice parameter $a = 680$ nm were fabricated on a silicon beam. One silicon beam, called **A**, contains eleven 3D crystals. We also present results obtained with another experimental setup on an older silicon beam **B** with five similar 3D crystals [@Grishina20173DNanophotonics]. Figure \[fig:SEM\] shows a scanning electron microscopy (SEM) image of one of our crystals with designed pore radius $r_d = 160$ nm ($r_d/a = 0.235$) on the edge of the silicon beam **A**. The dimensions of each crystal are typically $8 \times 10 \times 8 \mu$m$^3$. Figure \[fig:SEM\] shows that the sample geometry allows for good optical access to the $XY$ and $YZ$ crystal surfaces.
Near-infrared reflectivity microscope
-------------------------------------
![Setup to measure position-resolved microscopic broadband reflectivity. The Fianium SC is the broadband supercontinuum source, the long-pass glass filter F blocks the visible light at $\lambda < 850$ nm, the monochromator filters the light to a narrow band, HWP are half-wave plates, P are polarizers, and BS are beam splitters. Incident light is focused on the sample with a 100$\times$ objective that also collects the reflected light; the coordinate system is shown at top right. The NIR camera views the sample in reflection with a magnification of 250$\times$. The photodiodes PD1 and PD2 monitor the incident light power and measure signal from the crystal, respectively. []{data-label="fig:setup"}](Setup_reflectivity_manuscript3DPBG.pdf){width="80mm"}
We have developed a near-infrared microscope setup to collect position-resolved broadband reflectivity spectra of photonic nanostructures, as is shown in Figure \[fig:setup\]. The near-infrared range of operation is compatible with 3D silicon nanophotonics as it allows to avoid the intrinsic silicon absorption. The setup was developed with the option to collect in future light scattered perpendicular to the incident light. Furthermore, a spatial light modulator can be inserted to eventually perform wavefront shaping [@Vellekoop2007OptLt; @Mosk2012NP]. Therefore, we decided to use sequential scanning of wavelengths instead of measuring the spectrum at once with a spectrometer as in Refs. [@Ctistis2010PRB; @Huisman2011PRB].
In the optical setup shown in Figure \[fig:setup\], the silicon beam with the 3D crystals is mounted on an XYZ translation stage that has a step size of about $30$ nm. We use a broadband supercontinuum source (Fianium SC 400-4, 450 nm - 2400 nm) whose output is filtered by a long pass glass filter (Schott RG850) to block the unused visible range. The near infrared light is spectrally selected by a monochromator (Oriel MS257; 1200 lines/mm grating) with an output linewidth of about $\Delta \lambda = 1$ nm and a tuning precision better than 0.2 nm. The accessible range of wavelengths spans from 900 nm to 2120 nm (or wave numbers $\nu/c = 11000$ cm$^{-1}$ to $4700$ cm$^{-1}$) in the near infrared including the telecom bands. Using a combination of a linear polarizer and half wave plates, the linear polarization of the spectrally filtered light is selected and sent to an infrared apochromatic objective (Olympus MPlan Apo 100$\times$) to focus the light onto the sample’s $XY$ surface with a numerical aperture NA $=0.85$. The glass objective allows for access over the whole numerical aperture, instead of a blocked range around the axis as previously with a Schwarzschild reflecting objective [@Ctistis2010PRB; @Huisman2011PRB]. The NA corresponds to a collection solid angle of $0.95\pi$ sr. On account of the crystal symmetry mentioned above ($\Gamma Z$ equivalent with $\Gamma X$ and with the opposite counterparts), we effectively collect a solid angle of $3.8\pi$ sr.
![Image of the $XY$-surface of one of the 3D inverse woodpile crystals on beam **A** as seen with the IR camera in the setup. The bright spot is the focus of the incident light from the supercontinuum source filtered by the monochromator. The surface of the Si beam with the crystals is illuminated with a near infrared LED. []{data-label="fig:camera_image"}](camera_image.png){width="60mm"}
Light reflected by the sample is collected by the same objective as shown in Figure \[fig:setup\]. A beam splitter directs the reflected light towards the detection arm where the reflection from the sample is imaged onto an IR camera (Photonic Science InGaAs). In order to locate the focus of the input light on the surface, a near infrared LED is used to illuminate the sample surface. We use the XYZ translation stage to move the sample to focus the light on the desired location. An image as seen on the IR camera (see Fig. \[fig:camera\_image\]) reveals the $XY$ surface of the Si beam. The bright circular spot with a diameter of about 2 $\mu$m is the focus of light reflected from the crystal. The rectangular darker areas of about 8 $\mu$m $\times$10 $\mu$m are the XY surfaces of the 3D photonic crystals. They appear dark compared to the surrounding silicon since the LED illumination is outside the band gap of these crystals whose effective refractive index is less than that of silicon.
Once the input light beam is focused on the sample, the reflected light is sent to photodiode PD2 (Thorlabs InGaAs DET10D/M, 900 nm - 2600 nm) by flipping off the mirror in front of the camera. The photodiode records the reflected intensity $I_R$ as the monochromator scans the selected wavelength range. An analyzer in front of the detector selects the polarization of the reflected light. All reflectivity measurements are done for two orthogonal polarization states of the incident light, namely $s$ (electric field transverse to X-directed pores) and $p$ (electric field parallel to X-directed pores). A typical spectrum takes about 5 to 25 minutes to record depending on the chosen wavelength step size of typically 10 nm or 2 nm. Using the translation stage, the sample is moved in the Y-direction to select different crystals on the edge of the silicon beam.
To calibrate the reflectivity defined as $R \equiv I_R/I_0$, the spectral response $I_R$ of the crystals is referenced to the signal $I_0$ from a clean gold mirror that reflects $96\%$. Calibration also removes dispersive contributions from optical components in the setup. We ensure that the signal to noise ratio of the photodiode response is sufficient to detect signal in the desired range. Therefore, the detector photodiode is fed into a lock-in amplifier to amplify the signal with a suitable gain. Since a serial measurement mode holds the risk of possible temporal variations in the supercontinuum source, we simultaneously collect the output of the monochromator with photodiode PD1 in each reflectivity scan. This monitor spectrum is used to normalize out variations in the incident intensity $I_0$. Since it is tedious to dismount and realign the sample to take reference spectra during a position scan, we also take secondary reference measurements on bulk silicon outside the crystals which has a flat response $R = 31\%$ with respect to the gold mirror.
We also discuss data measured on similar silicon beam **B** and obtained with an older setup employing a Fourier transform spectrometer and a Schwarzschild reflecting objective [@Grishina20173DNanophotonics; @Huisman2011PRB; @Ctistis2010PRB]. The maximum reflectivities are lower than in the new setup (30% versus 90%) probably on account of a larger spot size in this setup (compared to Refs. [@Huisman2011PRB; @Ctistis2010PRB] we find that the focus diameter has over the years changed from $1$ to $5 \mu$m.) Nevertheless, the measured peak positions and bandwidths agree well with the newer ones.
Results
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\[sec:stopbands\]Stopbands
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![Measured reflectivity spectrum of a 3D crystal with designed pore radius $r_d = 130$ nm ($r_d/a=0.191$) on silicon beam **A** for $p$-polarized input light. The stopband is estimated as the full-width at half-maximum of the reflectivity peak, shown by the cyan area. The baseline reflectivity ($r_l$), maximum reflectivity ($r_m$) and half maximum are shown as the grey bars. The estimated error for both stopband edges is shown as the vertical dashed lines. []{data-label="fig:determine_stopband"}](show_stopband.pdf){width="80mm"}
Figure \[fig:determine\_stopband\] shows a reflectivity spectrum of a crystal with design pore radius $r_d = 130$ nm ($r_d/a=0.191$) recorded using our new setup. The broad and bright peak is the stopband that is associated with the main $\Gamma Z$ stop gap centered near $a/\lambda = 0.5$ in Figure \[Fig:bandstruct\](a). The stopband width is taken as the full width at half maximum (FWHM) of the reflectivity peak [@Vos2001NATO]. The baseline of the peak is taken as the minimum reflectivity in the long-wavelength limit at frequencies below the stopband, with the standard deviation in this frequency range as the error margin. Similarly, the maximum reflectivity is taken as the mean in a narrow range around the peak, with the standard deviation in this range taken as the error margin. The baseline, the maximum reflectivity, and the half maximum are shown in Figure \[fig:determine\_stopband\] as grey bars including their estimated errors. The errors are propagated into the estimates of the edges at half maximum of the peak.
![Reflectivity spectra of three different 3D photonic crystals on Si beam **A** with three designed pore radii $r_d = 130, 140$ and $160$ nm $(r_d/a = 0.191, 0.206, 0.235)$ (red circles, yellow diamonds and blue triangles, respectively). The stopbands appear at different frequency ranges. The gray squares represent reflectivity from bulk Si on the beam away from the crystals. []{data-label="fig:spectra_s9_s13_s18_si"}](reflectivity_3radii.pdf){width="80mm"}
Figure \[fig:spectra\_s9\_s13\_s18\_si\] shows reflectivity spectra measured on three 3D crystals on beam **A** with different designed pore radii $r_d = 130, 140, 160$ nm, as well as on the Si substrate. Here, a change in the ratio of pore radius to the lattice constant $r_d/a$, called as the reduced pore radius corresponds to a change in the pore radius only since the lattice constants in our crystals are kept constant at 680 nm. The constant reflectivity $R = 30.6 \pm {1.3} \%$ of the substrate agrees well with the Fresnel reflectivity of $31 \%$ expected for bulk silicon at normal incidence [@NSM]. Intense reflectivity peaks with maxima of $R_{m} = 96\%$ and $94\%$ are measured on the crystals with pore radii $r_{d} = 130$ nm and $140$ nm, respectively. Our observations are consistent with recent numerical results that perfect silicon inverse woodpile crystals with a thickness of only three unit cells reflect $99 \%$ of the incident light [@Devashish2017PRB]. The results are also consistent with $95 \%$ reflectivity on a direct silicon woodpile that was only one unit cell thick by Euser *et al.* [@Euser2008PRB]. We surmise that the current maximum reflectivities are higher than our previous results [@Huisman2011PRB; @Grishina2015Nanotech] due to improved nanofabrication and an improved optical setup. Figure \[fig:spectra\_s9\_s13\_s18\_si\] also shows that the center of the stopband shifts to higher frequencies with increasing pore radius. Such tuning of the stopband center with increasing pore radius qualitatively agrees with the behavior of the calculated stop gap and band gap shown in Figure \[Fig:bandstruct\](b).
The central question regarding reflectivity spectra as shown in Figure \[fig:spectra\_s9\_s13\_s18\_si\], is which feature of a measured reflectivity peak is representative of characteristic photonic crystal features, such as a (directional) photonic stop gap or a (omnidirectional) photonic band gap. In case of weakly interacting photonic crystals, Ref. [@Vos2001NATO] argued that the FWHM of a stopband collected with a low numerical aperture is a robust measure of a stop gap that is associated with one wave vector. Since such crystals weakly interact with light, there is a slim chance to find a photonic band gap. Using strongly interacting Si inverse opals, Palacios-Lid[ó]{}n *et al.* discussed that reflectivity collected over multiple high symmetry directions reveals a feature that is representative of the photonic band gap [@PalaciosLidon2002APL]. Huisman *et al.* proposed to combine measurements over several high-symmetry directions with a large numerical aperture since the band gap is associated with all wave vectors, hence $4 \pi$ sr solid angle [@Huisman2011PRB]. Here, we propose to extend these earlier probes by mapping stopbands for $s$ and $p$-polarized light as a function of a structural parameter, *viz.* the variation of the pore radii $r/a$, that entails the tuning of the effective refractive index.
Track pore radii from position-dependent stopband {#sec:position_dependent_stopbands}
-------------------------------------------------
To realize the mapping described above, we first identify a way to scan the pore radii. It is well-known from structural studies such as scanning electron microscopy on cleaved or milled crystals [@VanDenBroek2012AFM] and from non-destructive traceless X-ray tomography [@Grishina2018Arxiv]) that the radius of etched nanopores varies slightly around the designed value with position inside the crystal. By comparing the lower edge of the measured stopband with the calculated stop gap (*cf.* Figure \[Fig:bandstruct\](b)), we obtain an estimate of the local average pore radius $r$ at the position $\vec{\mathbf{r}}$ of the optical focus: $r (\vec{\mathbf{r}})$. In this comparison we take advantage of the feature in the band structures of inverse woodpile crystals that the lower edges of both the band gap and of the stop gap are nearly the same, see Figure \[Fig:bandstruct\](b), hence the determination is robust to the interpretation which gap is probed. For the three crystals in Figure \[fig:spectra\_s9\_s13\_s18\_si\], we derive the pore radii to be $r/a = 0.190\pm 0.001, 0.195\pm 0.001,~\textrm{and}~ 0.228\pm 0.002$, respectively, which agrees very well with the design ($r_d/a = 0.191,~0.206,~0.235$), where the small differencess are attributed to the depth-dependent pore radius discussed above. We note that since the probing direction is perpendicular to the $X$-directed pores in the crystals, the derived pore radii are effectively those of the pores that run in the $X$-direction.
. The grey areas at $Y < 0~\mu$m and $Y > 10~\mu$m indicate bulk silicon outside the crystal with a constant reflectivity near $31\%$. []{data-label="fig:Yscan"}](RmRl_bandedges_RoverA_S13Yscan_ppol){width="80mm"}
Next, we collect reflectivity spectra while scanning the focus across the crystal surface. Since we then effectively scan the pore radius $r$, we expect to scan the stopband in response. As an example, Figure \[fig:Yscan\] shows the results of a $Y$-scan across one of our crystals with design pore radius $r_d = 130$ nm ($r_d/a = 0.191$) on silicon beam **A**, measured with $p$-polarized light. While scanning the $Y$-position, a slight excursion occurred in the $X$-direction from $X = 2.8~\mu$m to $3.2~\mu$m due to imperfect alignment of the silicon beam axis with the vertical axis of the translation stage. From each collected spectrum, we derive the peak reflectivity $R_m$ and the minimum reflectivity below the stopband $R_l$ as shown in Figure \[fig:Yscan\](a). Inside the crystal there is substantial difference between $R_m$ (up to $R_m = 94.8 \%$) and $R_l$, hence the crystal’s reflectivity peaks are well-developed. Near the crystal edges ($Y = 0~\mu$m and $ 10~\mu$m) the difference between $R_m$ and $R_l$ rapidly decreases and both tend to about $31 \%$ since the focused light here is reflected by bulk silicon.
Figure \[fig:Yscan\](b) shows the edges of the measured stopband as a function of $Y$-position. Between $Y = 0~\mu$m and $ 10~\mu$m the lower edge shifts down from $5950$ to $5550$ cm$^{-1}$ and the upper edge shifts down from $7550$ to $6550$ cm$^{-1}$. In other words, both the center frequency of the stopband and its width decrease with increasing $Y$ as a result of the variation of the pore radii with position. The redshift of the stopband frequencies is mostly caused by the small excursion along $X$, since the radius of the $X$-directed pores decreases with increasing $X$.
By comparing the measured lower edges in Figure \[fig:Yscan\](b) with the theoretical gap maps shown in Fig. \[Fig:bandstruct\](b), we derive the local pore radius $r (\vec{\mathbf{r}})$ in the crystal that is plotted versus $Y$-position in Figure \[fig:Yscan\](c). The resulting $r (\vec{\mathbf{r}})/a $ is seen to vary from $0.197$ to $0.176$ about the design pore radius $r_d/a = 0.191$. Therefore, we can now combine all position-dependent data to make maps of stopband centers and stopband widths as a function of the pore radius.
Probing the 3D photonic band gap {#sec:band-gap}
--------------------------------
![Evolution of the stopband edges versus pore radius. The red and blue triangles represent upper edge of the stopband for $s$ and $p$-polarized light respectively. The red and blue circles represent the lower edge of the stopband for $s$ and $p$ polarized light. The stopband edges are inferred from the reflectivity peak measured on 11 crystals on the Si beam **A**. The solid lines indicate the edges of the photonic band gap. The upper edge of the $\Gamma Z$ stop gap for $s$ and $p$ polarized light are plotted as the red and blue dotted curves, respectively. For both polarizations, the experimental data agree well with the 3D photonic band gap edge. []{data-label="fig:bandedge_rad"}](bandedges_theory_expt_VF.pdf){width="80mm"}
We have applied the procedures described in sections \[sec:position\_dependent\_stopbands\] and \[sec:stopbands\] to reflectivity measured on many crystals on beam **A**. We also took multiple measurements along the $Y$-direction on two crystals to verify the consistency of all observations. From all collected reflectivity spectra, both $s$ and $p$ polarized, the lower and upper stopband edges are extracted, and are mapped as a function of $r/a$ in Figure \[fig:bandedge\_rad\]. The lower edge data form a continuous trace from reduced frequency $a/\lambda = 0.38$ at $r/a = 0.17$ to $a/\lambda = 0.50$ at $r/a = 0.245$. The data match well with the theory, which is obvious since we used the lower edge to estimate $r/a$ from the measured spectra. The upper edge data form a continuous trace from reduced frequency $a/\lambda = 0.42$ at $r/a = 0.17$ to $a/\lambda = 0.64$ at $r/a = 0.245$. It is remarkable that the upper edge data for both $s$ and $p$-polarized light mutually agree very well, especially for pores radii $r/a > 0.21$. This observation implies that the measured stopband is rather representative of the photonic band gap that is polarization insensitive, as opposed to a directional stop gap that is polarization sensitive. In comparison to theory, at pore radii $r/a < 0.21$, the upper edges are in between the theoretical upper edges of the band gap and the $p$-polarized edge of the directional stop gap. At larger radii $(r/a > 0.21)$ all measured upper edge data are near the theoretical upper band gap edge and differ from the stopband edges. This observation adds support to the notion that the structure-dependent stopbands represent the 3D photonic band gap, rather than a directional stop gap.
. (a) $s$-polarized data for beam **A** (red circles), and for beam **B** (yellow diamonds), (b) p-polarized data for beam **A** (blue circles). The cyan crosses, green asterisks, and magenta stars are numerical results for angle-averaged stopband, normal incidence, and band gap at $r/a=0.19$, respectively [@Devashish2017PRB]. The dashed red and dash dotted blue curves represent the width of the $\Gamma Z$ stop gap obtained from band structures for an infinite crystal for $s$ and $p$ polarized light, respectively. The magenta solid curve is the 3D photonic band gap from band structures for an infinite crystal. []{data-label="fig:rel-width_vs_radius"}](RelWidth_rovera_allpol_VF.pdf){width="80mm"}
To refine our reasoning, we plot in Figures \[fig:rel-width\_vs\_radius\](a,b) the relative stopband width (gap to mid-gap ratio) as a function of the reduced pore radius $r/a$ as derived from the lower edges. The large number of data in Figure \[fig:rel-width\_vs\_radius\](a) for Si beam **A** show that the width of the $s$-polarized stopband increases up to $r/a = 0.2$ before more or less saturating up to $r/a = 0.24$. The $s$-polarized data for Si beam **B** agree well with the data for beam **A** except for an outlier at $r/a = 0.24$. All data are close to the theoretical prediction for the width of the 3D photonic band gap and lie distinctly below the theoretical width of the stop gap. Figure \[fig:rel-width\_vs\_radius\](a) also shows results of $s$-polarized reflectivity simulated for a *finite* inverse woodpile crystal $r/a=0.19$ [@Devashish2017PRB], namely of a directional stopband, of an angle-averaged stopband (for a range of angles relevant for a reflecting objective with $NA = 0.65$), and of an omnidirectional band gap. With increasing aperture, the simulated stopband becomes narrower. From the comparison, it is apparent that our data match best with the width of the 3D photonic band gap. Figure \[fig:rel-width\_vs\_radius\](b) shows the $p$-polarized stopband widths versus pore radius. At pore radii $r/a < 0.21$, the stopband widths are in between the theoretical bandwidths of either the directional stopgap or the omnidirectional band gap. At larger radii $(r/a > 0.21)$, the measured stopband widths match better with the theoretical width of the band gap than with the stop gap width. From $p$-polarized finite-crystal simulations done at $r/a=0.19$ [@Devashish2017PRB], we learn that the bandwidths of the directional stop gap, of the angle-averaged stopgap, and of the band gap are near to each other, hence it is difficult given the variations in our data to discriminate between either feature. Considering the $s$ and $p$-polarized stopband widths jointly, we again find a much better agreement with the 3D photonic band gap than with the directional stop gap.
![Relative stopband width for p-polarization versus relative stopband width for s-polarization measured at the same position on crystals with a range of pore radii $r/a$ (blue circles). Black dashed-dotted line is the infinite-crystal theory result for the 3D photonic band gap, and red dashed curve the one for the $\Gamma Z$ stop gap. The cyan crosses and green asterisks are numerical results for angle-averaged stopband and normal incidence for $r/a=0.19$, respectively, and the magenta star is the band gap width simulated for a finite-thickness crystal with $r/a=0.19$ [@Devashish2017PRB], that are connected by the gray dotted line as a guide to the eye. []{data-label="fig:ppol-width_vs_spol-width"}](PvsS_bandwidths.pdf){width="80mm"}
The conclusions from Figures \[fig:bandedge\_rad\] and \[fig:rel-width\_vs\_radius\] are based on the agreement between measurements on one hand, and simulations and theory on the other hand. The latter invoke an idealized structural model, for instance, pores as infinite perfect cylinders which neglect pore tapering. Therefore, to find a criterion that is indeed free of theoretical or numerical modeling, we make a parametric plot of the width of the $p$-polarized stopband versus the width of the $s$-polarized stopband, as shown in Figure \[fig:ppol-width\_vs\_spol-width\]. In order to avoid systematic errors due to the position-dependence of the stopbands, we select data where both polarizations were measured on the same position on a crystal. For $s$-polarized stopband widths between $\Delta \omega/\omega_c = 17\%$ and $24\%$, the corresponding $p$-polarized stopband width increases linearly, and also from $17 \%$ to $24\%$. Such a linear increase is obviously expected for a 3D photonic band gap, even without detailed modeling, since a 3D band gap obviously entails a forbidden gap for both polarizations simultaneously [@Joannopoulos2008PhotonicLight]. In case of the alternative hypothesis that the stopbands correspond to directional $\Gamma Z$ stop gaps, the trend would be nonlinear and clearly different from the diagonal. Since this trend obviously does not match with our data, we can safely reject this hypothesis.
For comparison, the computer simulations on a finite-size crystal (with $r/a = 0.19$) in Ref. [@Devashish2017PRB] agree with the theory both for the omnidirectional photonic band gap and for the directional stopgap, where the former matches very well with our observations and the latter does not. The simulations have also been done for a numerical aperture comparable to a reflecting objective (as in Ref. [@Huisman2011PRB]), and this result is somewhat lower than our observations, which indicates that for a smaller NA than studied here the measured stopband is not representative of the band gap. Conversely, the numerical aperture $NA = 0.85$ used here and the correspondingly large overall solid angle of $3.8\pi$ sr is apparently sufficient to probe the omnidirectional photonic band gap.
Discussion {#sec:discussion}
==========
So far, we discussed the stopbands versus the radii of the pores that are specific to the inverse woodpile structure studied here [@Ho1994SSC]. In order to generalize our results to other classes of photonic band gap crystals, such as inverse opals, direct woodpiles, and even non-periodic ones [@Muller2017Optica], it is useful to realize that a varying pore size corresponds to the tuning of the filling fraction and thus the tuning of the effective refractive index [@Datta1993PRB], both of which pertain to all other classes of photonic band gap structures. As is shown in Figure \[fig:neff\_vs\_pore-radius\], the effective index of our crystals obtained from the band structures in the limit of zero frequency - is tuned from $3.5$ (silicon) to $1.0$ (air) by varying the pore size from $r/a = 0.0$ to a little over $0.3$. Both the filling fraction and the effective index are readily generalized to other 3D photonic band gap crystals. For instance, in inverse opals the filling fraction of the high-index backbone is known to vary with preparation conditions [@Wijnhoven2001CM], hence this can be used as a tuning knob. In direct woodpile crystals, the filling fraction is notably tuned by varying the width of the high-index nanorods [@Noda2000Science; @Tajiri2019Optica], and similarly in hyperuniform structures [@Muller2017Optica]. It is therefore that the top abscissae in Figures \[fig:rel-width\_vs\_radius\] and \[fig:ppol-width\_vs\_spol-width\] have been generalized to the effective refractive index. Therefore, the stopband width versus the effective index (as in Fig. \[fig:rel-width\_vs\_radius\]) or the $p$-polarized stopband width versus the $s$-polarized one also pertain as probes to other classes of band gap structures, and thus serve as experimental probes of the 3D photonic band gap in such other structures.
![Effective refractive index of inverse woodpile photonic crystals made of silicon as a function of the pore reduced pore radius $r/a$, obtained from the slope of the band structures in the limit of zero frequency.[]{data-label="fig:neff_vs_pore-radius"}](neff_porerad_inv_woodpile.pdf){width="80mm"}
It is generally agreed that the fabrication of 3D nanostructures necessary for photonic band gap physics is fairly challenging [@Wijnhoven1998Science; @Noda2000Science; @Blanco2000Nature; @Vlasov2001N]. Since the detailed 3D nanostructure critically determines the band gap functionality, it is important to have a non-destructive verification on the functionality. We propose that the practical probe methods presented here fill this gap by providing relatively fast feedback on a newly fabricated nanostructure. In a fundamentally holistic approach, one would not only verify the functionality but also the 3D structure since the latter usually serves to improve the understanding of the functionality, especially in ubiquitous situations where the function differs from the designed one. While studying the detailed 3D structure of a nanostructure is highly non-trivial, successful methods have been reported using X-ray techniques, notably small-angle X-ray scattering [@Wijnhoven2001CM], X-ray ptychography [@Furlan2018ApplMatToday], or traceless X-ray tomography [@Grishina2018Arxiv].
We expect that a practical probe of 3D photonic band gaps will boost their applications in several innovative fields. For instance, recent efforts by the Tokyo and Kyoto teams have demonstrated the use of 3D photonic band gap crystals as platforms for 3D photonic integrated circuits [@Tajiri2019Optica; @Ishizaki2013NatPhot]. In the field of photovoltaics that is of considerable societal interest, the use of 3D photonic band gap crystals is increasingly studied to enhance the collection efficiency by means of various kinds of photon management [@Bermel2007OE; @Wehrspohn2012JO; @Devashish2019PRB]. It is an essential feature of a 3D photonic band gap crystal to have a gap in the density of states, which in turn corresponds to the density of vacuum fluctuations. Therefore, quantum devices embedded inside a 3D band gap crystal are effectively shielded from quantum noise [@Clerk2010RMP], including quantum gates that manipulate qubits for quantum information processing.
Conclusion {#sec:conclusion}
==========
We present a purely experimental probe of the 3D band gap in real three-dimensional (3D) photonic crystals, without the need for theoretical or numerical modeling that invoke idealized and even infinite photonic crystals. As an exemplary structure, we study 3D inverse woodpile crystals made from silicon. We collected position and polarization-resolved reflectivity spectra of multiple crystals with different design parameters with a large numerical aperture and observed intense reflectivity peaks with maxima exceeding $90\%$ corresponding to the stopbands, typical of high-quality crystals. We track the stopband width versus pore radius, which agrees much better with the predicted 3D photonic band gap than with a directional stop gap. A parametric plot of s-polarized versus p-polarized stopband width is nearly a straight line, in agreement with the 3D band gap and at variance with the directional stop gap. Such a practical probe provides fast feedback on the advanced nanofabrication required for 3D photonic crystals and stimulates practical applications of band gaps in 3D silicon nanophotonics and photonic integrated circuits, photovoltaics, cavity QED, and quantum information processing.
Funding {#funding .unnumbered}
=======
NWO-TTW Perspectief Program ‘Free Form Scattering Optics’, NWO-FOM program ‘Stirring of Light!’, MESA+ section ‘Applied Nanophotonics (ANP)’.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Rajesh Nair, Simon Huisman, and Devashish for help with the photonic band structure calculations and Emre Yuce for early contributions in building the reflectivity setup.
|
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author:
- 'Hao Tang, James Glass'
bibliography:
- 'wordbias.bib'
date: |
Massachusetts Institute of Technology\
Computer Science and Artificial Intelligence Laboratory\
Cambridge MA, USA\
[`{haotang, glass}@mit.edu`]{}
title: 'On The Inductive Bias of Words in Acoustics-to-Word Models'
---
=1
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---
abstract: 'In this paper, we propose a structured linear parameterization of a feedback policy to solve the model-free stochastic optimal control problem. This parametrization is corroborated by a decoupling principle that is shown to be near-optimal under a small noise assumption, both in theory and by empirical analyses. Further, we incorporate a model-free version of the Iterative Linear Quadratic Regulator (ILQR) in a sample-efficient manner into our framework. Simulations on systems over a range of complexities reveal that the resulting algorithm is able to harness the superior second-order convergence properties of ILQR. As a result, it is fast and is scalable to a wide variety of higher dimensional systems. Comparisons are made with a state-of-the-art reinforcement learning algorithm, the Deep Deterministic Policy Gradient (DDPG) technique, in order to demonstrate the significant merits of our approach in terms of training-efficiency.'
author:
- 'Karthikeya S Parunandi$^1$, Aayushman Sharma$^2$, Suman Chakravorty$^3$ and Dileep Kalathil$^4$'
title: 'D2C 2.0: Decoupled Data-Based Approach for Learning to Control Stochastic Nonlinear Systems via Model-Free ILQR'
---
|
---
abstract: |
We consider the problem $$\begin{cases}
\triangle u+f(u)=0 & \text{in $\mathbb{R}^n$},\\
\displaystyle \lim_{\lvert x \rvert \to \infty} u(x) =0, \label{Intro2}
\end{cases}$$ where $$f(u)=-\omega u + u^p - u^q, \qquad \omega>0, ~~q>p>1.$$ It is known that a positive solution to exists if and only if $F(u):=\int_0^u f(s)ds >0$ for some $u>0$. Moreover, Ouyang and Shi in 1998 found that the solution is unique if $f$ satifies furthermore the condition that $\tilde{f}(u):= (uf'(u))'f(u)-uf'(u)^2 <0$ for any $u>0$. In the present paper we remark that this additional condition is unnecessary.
author:
- Shinji Kawano
title: Uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities
---
Introduction
============
We shall consider a boundary value problem $$\begin{cases}
u_{rr}+ \dfrac{n-1}{r}u_r+f(u)=0 & \text{for $r>0$}, \\
u_r(0)=0, \\
\displaystyle \lim_{r \to \infty} u(r) =0, \label{b}
\end{cases}$$ where $n \in \mathbb{N}$ and $$f(u)=-\omega u + u^p - u^q, \qquad \omega>0, ~~q>p>1.$$ The above problem arises in the study of $$\begin{cases}
\triangle u+f(u) =0 & \text{in $\mathbb{R}^n$},\\
\displaystyle \lim_{\lvert x \rvert \to \infty} u(x) =0. \label{a}
\end{cases}$$ Indeed, the classical work of Gidas, Ni and Nirenberg [@G1; @G2] tells us that any positive solution to is radially symmetric. On the other hand, for a solution $u(r)$ of , $v(x):=u(\lvert x \rvert)$ is a solution to .
The condition to assure the existence of positive solutions to (and so ) was given by Berestycki and Lions [@B1] and Berestycki, Lions and Peletier [@B2]:
A positive solution to exists if and only if $$F(u):=\int_0^u f(s)ds >0, \qquad \text{for some} \quad u>0. \label{existence}$$
Uniqueness of positive solutions to had long remained unknown. Finally in 1998 Ouyang and Shi [@OS] proved uniqueness for with $f$ satisfying the additional condition (See also Kwong and Zhang [@KZ]):
If $f$ satisfies furthermore the following condition, then the positive solution is unique; $$\tilde{f}(u):= (uf'(u))'f(u)-uf'(u)^2 <0, \qquad \text{for any} \quad u>0. \label{unique}$$
Following is the main result of the present paper:
If the nonlinearity $f$ satisfies the existence condition , then the uniqueness condition is automatically fulfilled.
\[thm\]
Proof of Theorem \[thm\].
=========================
The existence condition is equivalent to $$\omega < \omega_{p,q},$$ where $$\omega_{p,q}=\dfrac{2(q-p)}{(p+1)(q-1)} \left[ \dfrac{(p-1)(q+1)}{(p+1)(q-1)} \right] ^{\frac{p-1}{q-p}}.$$ (See Ouyang and Shi [@OS] and the appendix of Fukuizumi [@Fukuizumi].)
The uniqueness condition is equivalent to $$\omega < \eta_{p,q},$$ where $$\eta_{p,q}=\dfrac{q-p}{q-1}\left[ \dfrac{p-1}{q-1}\right]^{\frac{p-1}{q-p}}.$$
The proofs of these Lemmas are nothing but straightforward calculation and shall be omited.
It is apparent that $$0<\omega_{p,q}<\eta_{p,q},$$ which asserts the theorem.
[9]{} H. Berestycki and P. L. Lions, Nonlinear scalar field equation, I., *Arch. Rat. Math. Anal.* **82** (1983), 313-345. H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $\mathbb{R}^N$, *Indiana University Math. J.* **30** (1981), 141-157. R. Fukuizumi, Stability and instability of standing waves for nonlinear Schrödinger equations, *Tohoku Mathematical Publications*, **No.25**, 2003. B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximal principle, *Comm. Math. Phys.* **68** (1979), 209-243. B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, Mathematical analysis and applications, Part A, *Adv. in Math. Suppl. stud.* **7a**, Academic press, 1981, 369-402. S. Kawano, A remark on the uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities, preprint. M. K. Kwong and L. Zhang, Uniqueness of positive solutions of $\triangle u+f(u)=0$ in an annulus, *Differential Integral Equations* **4** (1991), 583-599. T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems, *J. Differential Equations* **146** (1998), 121-156. J. Wei and M. Winter, On a cubic-quintic Ginzburg-Landau equation with global coupling, *Proc. Amer. Math. Soc.* **133** (2005), 1787-1796.
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---
abstract: 'The paper deals with a theoretical model for interacting Tsallis holographic dark energy (THDE) whose infrared (IR) cut-off scale is set by the Hubble length. The interaction $Q$ between the dark sectors (dark energy and pressureless dark matter) of the universe has been assumed to be non-gravitational in nature. The functional form of $Q$ is chosen in such a way that it reproduces well known and most used interactions as special cases. We then study the nature of the THDE density parameter, the equation of state parameter, the deceleration parameter and the jerk parameter for this interacting THDE model. Our study shows that the universe undergoes a smooth transition from a decelerated to an accelerated phase of expansion in the recent past and also this transition occurs within the redshift interval \[0.637,0.962\]. This is well consistent with the present observations. It is shown the evolution of the the normalized Hubble parameter for our model and compared that with the latest Hubble parameter data. Finally, we also investigate both the stability and thermodynamic nature of this model in the present context.'
author:
- Abdulla Al Mamon
- Amir Hadi Ziaie
- Kazuharu Bamba
title: A Generalized Interacting Tsallis Holographic Dark Energy Model and its thermodynamic implications
---
Keywords: Tsallis holographic dark energy, interaction, thermodynamics.
Introduction
============
Many cosmological observations indicate that our Universe is now experiencing an accelerated expansion phase [@acc1; @acc2; @acc3; @acc4; @acc5]. A possible candidate to explain this cosmic acceleration is to consider some exotic matter, dubbed as dark energy (DE) which consists of approximately 68% of the total energy budget of our universe. However, the origin and nature of this DE are absolutely unknown. On the other hand, the second largest component of our universe is the dark matter (DM) which takes around 28% of the total energy density of the universe. Like the DE sector, DM sector is also not very well understood. Till now, a large number of theoretical models are taken into account to accommodate the present phase of acceleration and some excellent reviews on this topic can be found in [@de1; @de2; @de3]. However, the problem of the onset and nature of cosmic acceleration remains an open challenge of modern cosmology at present.\
In this context, holographic dark energy (HDE) is an interesting attempt to solve this problem (for details, see [@Hooft; @Hooft1995; @Cohen]) and some of its various scenarios can be found in [@hde1; @hde2; @hde3; @hde4; @hde5; @hde6; @hde7; @hde8; @hde9; @hde10; @hde11; @hde12; @hde13; @hde14; @hde15; @hde16; @hde17; @hde18; @hde19; @hde20]. In particular, a new HDE model has been proposed by using the holographic hypothesis and the Tsallis entropy [@tsahde], named Tsallis holographic dark energy (THDE) [@THDE; @tnote]. As a result, recently, several THDE models have been investigated and explored in different scenarios with an aim to search for the dynamics of the universe and one can look into [@thde1; @thde2; @thde3; @thde4; @thde5; @thde6; @thde7; @thde8; @thde9; @thde10; @thde11] for a comprehensive review.\
It is important to mention that observations admit an interaction between the dark sectors (DM and DE) of cosmos which can solve the coincidence problem and the tension in current observational values of the Hubble parameter [@obsint01; @obsint02; @obsint03; @gdo1; @gdo2; @ob1; @ob2; @ob3; @ob4; @ob5; @ob6; @ob7; @pav; @h1; @ig1; @ig2; @ig3; @ig4; @id1; @im1; @im2; @ie1; @iaamnc]. The scenario of interaction between DM and DE is one of such alternative models, which is the main subject interest of the present work. Recently, Zadeh et al. [@tnote] investigated the evolution of the THDE models with different IR cutoffs and studied their cosmological consequences under the assumption of a mutual interaction between the dark sectors of the universe. Following [@tnote], in this work, we are also interested in studying the dynamics of a flat FRW universe filled with a pressureless matter and THDE in an interacting scenario. In particular, we explore consequences of interacting THDE model in a more general scenario. In our setups, we study the evolution of our universe by considering an interaction between DM and THDE whose IR cutoff is the Hubble horizon. The nature of the THDE density parameter, the deceleration parameter, the jerk parameter and the THDE equation of state parameter have also been studied for the present model. Furthermore, we also investigate the stability and thermodynamic nature of this particular model in the present scenario. However, the present work is more general and also different from the similar work by Zhai et al. [@tnote] in different ways. Firstly, in this paper, the functional form of the interaction term is chosen in such a way that it can reproduce some well known and most used interactions (including [@tnote]) in the literature for some special cases (for details, see section \[sec2\]). Secondly, we study the evolution of jerk parameter for this general interaction term. We also plot the normalized Hubble parameter for our model and compared that with the latest Hubble parameter data. Additionally, we go one step further by investigating this scenario taking into account the thermodynamic considerations. In particular, we study the nature of the total entropy of the universe surrounded by a cosmological horizon.\
The paper is organized as follows. In the next section, we present a THDE model with Hubble scale as IR cutoff. Additionally, the results of considering a mutual interaction between the dark sectors of the universe are also investigated. In section \[sec-thermo\], we also explore the thermodynamical properties of the present model. Finally, in section \[conclusion\], we summarize the conclusions of this work.
Interacting THDE with Hubble Cutoff {#sec2}
===================================
The THDE model is based on the modified entropy-area relation [@tsahde] and the holographic dark energy hypothesis, was proposed in [@THDE] by introducing the following energy density $$\begin{aligned}
\label{Trho}
\rho_D=BL^{2\delta-4}\end{aligned}$$ where $B$ is an unknown parameter. By considering the Hubble horizon as the IR cutoff, i.e., $L=H^{-1}$, the energy density corresponding to THDE is obtained as $$\begin{aligned}
\label{Hrho}
\rho_D=BH^{-2\delta+4},\end{aligned}$$
In the large scale, our universe is homogeneous and isotropic and its geometry is best described by the spatially flat Friedmann-Robertson-Walker (FRW) metric $$\begin{aligned}
\label{frw}
ds^{2}=dt^{2}-a^{2}\left( t\right) \left[ dr^2
+r^{2}d\Omega^{2}\right],\end{aligned}$$ where $a(t)$ is the scale factor of the universe. Now, in such a spacetime, one can write down Friedmann equations as [@de1] $$\begin{aligned}
\label{frd}
H^{2}=\frac{1}{3m_{p}^{2}}\left(\rho_{m}+\rho_{D}\right)\end{aligned}$$ where, $m_p$ denotes the reduced Planck mass, $H=\frac{\dot{a}}{a}$ is the Hubble parameter and an overhead dot represents derivative with respect to the cosmic time $t$. Also, $\rho_m$ and $\rho_D$ represent the energy density of pressureless matter and the THDE density, respectively. The energy density parameter of THDE and pressureless matter can be expressed as $$\begin{aligned}
\label{3}
&&\Omega_{D}=\frac{\rho_{D}}{\rho_c}=\frac{B}{3m_{p}^{2}}H^{-2\delta+2}\\
&&\Omega_{m}=\frac{\rho_{m}}{\rho_c}\end{aligned}$$ where, $\rho_c=3m_{p}^{2}H^{2}$ denotes the critical energy density. Now, equation (\[frd\]) can be written as $$\begin{aligned}
\label{u}
\Omega_{m}+\Omega_{D}=1\end{aligned}$$ Also, the ratio of the energy densities is obtained as $$\label{r2}
r=\frac{\rho_m}{\rho_D}=\frac{\Omega_m}{\Omega_D}$$ Moreover, we assume that DM and DE interact with each other. Accordingly, the energy conservation equations become $$\begin{aligned}
\label{conm}
&&\dot{\rho}_m+3H\rho_m=Q\\
&&\dot{\rho}_D+3H(1+\omega_D)\rho_D=-Q\label{conD}\end{aligned}$$ where $\omega_D\equiv \frac{p_D}{\rho_D}$ denotes the [*equation of state*]{} (EoS) parameter of THDE, $p_{D}$ is the pressure of THDE and $Q$ indicates the rate of energy exchange between the dark sectors (DM and DE). Positive value of $Q$ indicates that there is an energy transfer from the THDE to the DM, while for $Q<0$, the reverse scenario happens. On the other hand, if $Q=0$ (i.e., non-intracting case), then the DM evolve as, $\rho_m \propto a^{-3}$. Hence, the interaction between the dark sectors is indeed a more general scenario to unveil the dynamics of the universe. In fact, there are many proposed interactions in the literature to study the dynamics of the universe (for details, one can look into [@ig1; @ig2; @ig3; @ig4; @id1; @im1; @im2; @ie1; @iaamnc; @HDESChange] and references therein), however, the exact functional form of $Q$ is still unknown to us. From the continuity equations (\[conm\]) and (\[conD\]), one can see that the interaction $Q$ could be any arbitrary function of the parameters $\rho_m$, $\rho_D$ and $H$. So, naturally, one can construct various interacting models to understand the dynamics of the universe in this framework. For mathematical simplicity, in the present work, we assume that the interaction is a linear combination of the dark sector densities given as $$\label{genQ}
Q=3H(b^2_{1}\rho_{m} + b^2_{2}\rho_{D}),$$ where, the parameters $b_{1}$ and $b_{2}$ are dimensionless constants. This type of functional forms of $Q$ has been studied recently by several authors [@ig1; @ig2; @ig3; @ig4; @id1] and the particular cases $b_{1}=0$, $b^2_{2}=\lambda$ in Ref. [@id1], $b_{2}=0$, $b^2_{1}=\alpha$ in Refs. [@im1; @im2], $b^2_{1}=\frac{\lambda_m}{3}$, $b^2_{2}=\frac{\lambda_D}{3}$ in Ref. [@ig1], $b_{1}=b_{2}=b$ in Ref. [@tnote] and $b_{2}=0$, $b_{1}=b$ in Ref. [@thde5]. Therefore, the general form of $Q$, given by equation (\[genQ\]), covers a wide range of other popular theoretical models for different choices of $b_{1}$ and $b_{2}$. The simplicity of the functional form of $Q$ (as given in equation (\[genQ\])), however, makes it very attractive and simple to study. Indeed, as DE and DM have not the same energy density (and hence contribution) within the universe dynamics and as we do not yet know their nature, it is reasonable to consider different contributions ($b_1\neq b_2$) for these dark components within the interaction term.\
Now, taking the time derivative of equation (\[frd\]), and by using equations (\[r2\]), (\[conm\]) and (\[conD\]), we get $$\begin{aligned}
\label{7}
\frac{\dot{H}}{H^{2}}=-\frac{3}{2}\Omega_{D}(1+\omega_{D}+r),\end{aligned}$$ Similarly, taking the time derivative of equation (\[Hrho\]) along with combining the result with equations (\[conD\]) and (\[7\]), we obtain $$\begin{aligned}
\label{w1}
\omega_{D}=\dfrac{1-\delta -\frac{b^2_{1}}{\Omega_D} + b^2_{1} - b^2_{2}}{1-(2-\delta)\Omega_{D}}.\end{aligned}$$ Taking the time derivative of equation (\[3\]) and by using equations (\[r2\]), (\[7\]) and (\[w1\]), we arrive at the following equation for THDE density parameter, as $$\begin{aligned}
\label{Omega}
\Omega_{D}^{\prime}=3(\delta-1)\Omega_{D}
\left[\dfrac{1-\Omega_{D} + {b^2_{1}\Omega_D} -b^2_{1}-{b^2_{2}\Omega_D}}{1-(2-\delta)\Omega_{D}}\right],\end{aligned}$$ where $\Omega_{D}^{\prime}=\frac{d\Omega_D}{d(\ln a)}$.\
Now, for simplicity, we re-expressed equation (\[3\]) as $$\begin{aligned}
H^{2} &=&\left(\frac{3m_{p}^{2}}{B}{\Omega_D}\right)^{\frac{1}{1-\delta}}, \nonumber \\
&=& H^2_{0} \left(\frac{\Omega_{D}}{\Omega^{0}_{D}}\right)^{\frac{1}{1-\delta}},\end{aligned}$$ which implies, $$\label{eqnh}
h=\frac{H}{H_0}=\left(\frac{\Omega_{D}}{\Omega^{0}_{D}}\right)^{\frac{1}{2(1-\delta)}},$$ where, $h$ is the normalized Hubble parameter, $\Omega^{0}_{D}$ is the present THDE density parameter and $H_{0}=\left(\frac{3m_{p}^{2}}{B}\Omega^{0}_{D}\right)^{\frac{1}{2(1-\delta)}}$, denotes the present value of $H$. Later, using equation (\[Omega\]) along with the above equation, we try to show the evolution of $h$ for this model and will compare it with that of observational Hubble parameter data.\
The deceleration parameter is defined as $$\begin{aligned}
\label{q}
q=-\frac{\ddot{a}}{aH^2}=-1-\frac{\dot{H}}{H^{2}},\end{aligned}$$ which is an important cosmological parameter to investigate the expansion history of the universe. In particular, $q<0$ indicates accelerated $({\ddot{a}}>0)$ expansion phase of our universe, whereas $q>0$ indicates a decelerated phase $({\ddot{a}}<0)$. In our model, $q$ evolves as $$\begin{aligned}
\label{q1}
q=\dfrac{(1-2\delta)\Omega_{D}+1-3{b^2_{1}}+3{b^2_{1}}\Omega_{D}-3{b^2_{2}}\Omega_{D}}{2[1-(2-\delta)\Omega_{D}]}.\end{aligned}$$ It is well known that the jerk parameter, a dimensionless third derivative of the scale factor with respect to cosmic time, provides a comparison between different DE models and the $\Lambda$CDM ($j=1$) model. It is given by [@jerk1; @jerk2; @jerk3] $$j=\frac{\frac{d^{3}a}{dt^{3}}}{aH^3}=q(2q + 1)+(1+z)\frac{dq}{dz}.$$ Finally, in order to estimate the stability of the model we consider the square of sound speed given as $$v^{2}_{s}=\frac{dp_{D}}{d\rho_{D}}=\omega_{D} + {\dot{\omega}}_{D} \frac{\rho_{D}}{{\dot{\rho}}_{D}}.$$ Using then equation (\[conD\]) along with equations (\[w1\]) and (\[Omega\]), the above equation can be re-expressed as $$\begin{aligned}
\label{vs2re}
v_s^2&=&\dfrac{b_1^2}{(\delta-2)\Omega_{D}(1+\Omega_{D}(\delta-2))^2}\nonumber\\&+&\dfrac{\left[1-b_2^2-\delta+b_1^2(1+\delta)+\Omega_D(\delta-1-b_1^2+b_2^2)\right]}{(1+\Omega_{D}(\delta-2))^2}.\nonumber\\\end{aligned}$$ The sign of $v^{2}_{s}$ is important to specify the stability of background evolution. $v^{2}_{s}>0$ ($v^{2}_{s}<0$) indicates a stable (unstable) model. It is important to note here that the expressions of $q$, $\omega_D$, $\Omega_D$ and $v^{2}_{s}$ are similar to the results of [@tnote] for the special choice, $b_{1}=b_{2}=b$. On the otherhand, if $b_{1}=b_{2}=0$, then the equations (\[w1\]), (\[q1\]), (\[Omega\]) and (\[vs2re\]) match to the relations derived in [@THDE]. As discussed earlier, thus the present work is more general in the literature.\
![The evolution of the THDE density parameter $\Omega_D$, as a function of $z$, is shown for the present model considering $\delta=1.4$, $\Omega^{0}_D=0.73$ and different values of $b^{2}_{1}$ and $b^{2}_{2}$, as indicated in panel.[]{data-label="figomegad"}](omegad.eps){width="8cm"}
![Evolution of $\omega_D$ as a function of $z$ is shown using $\delta=1.4$, $\Omega^{0}_D=0.73$ and different values of $b^{2}_{1}$ and $b^{2}_{2}$, as indicated in panel.[]{data-label="figeos"}](eos.eps){width="8cm"}
![The evolution of the deceleration parameter $q$ vs. $z$, is shown for $\delta=1.4$, $\Omega^{0}_D=0.73$ and different values of $b^{2}_{1}$ and $b^{2}_{2}$, as indicated in panel. Also, the horizontal line denotes $q(z)=0$.[]{data-label="figqz"}](q.eps){width="8cm"}
![The evolution of the cosmic jerk parameter $j(z)$ is shown for $\delta=1.4$, $\Omega^{0}_D=0.73$ and different values of $b^{2}_{1}$ and $b^{2}_{2}$, given in figure \[figomegad\].[]{data-label="figjz"}](jerk.eps){width="8cm"}
![The evolution of $h(z)$, as given in equation (\[eqnh\]), is shown by considering $\delta=1.4$ and $\Omega^{0}_D=0.73$. In this plot, the black dots correspond to the $H(z)$ data consisting 41 data points [@hub1; @hub2] with $1\sigma$ error bars and the corresponding error in $h(z)$ is given as [@zt4], $\sigma_{h}=h\sqrt{\frac{\sigma^{2}_{H_0}}{H^{2}_0} + \frac{\sigma^{2}_{H}}{H^{2}}}$. Here, $\sigma^{2}_{H}$, $\sigma^{2}_{H_0}$ are errors in $H$ and $H_0$ respectively. Also, the value of $H_0$ is taken from [@r16].[]{data-label="figh"}](h.eps){width="8cm"}
![Evolution of ${v}^{2}_{s}$ as a function of $z$ is shown for $\delta=1.4$, $\Omega^{0}_D=0.73$ and different values of $b^{2}_{1}$ and $b^{2}_{2}$, as indicated in panel.[]{data-label="figvs1"}](vs1.eps){width="8cm"}
![Evolution of ${v}^{2}_{s}$ as a function of $z$ is shown for same values of $b^2_{1}$, $b^2_{2}$ and $\Omega^{0}_D$ as given in figure \[figvs1\]. This plot is for $\delta=2.001$.[]{data-label="figvs2"}](vs2.eps){width="8cm"}
We plot the evolutionary trajectories for different cases of Tsallis parameter $\delta$ and interaction terms $b_{1}$ and $b_{2}$. For $\delta=1.4$ case and the initial condition $\Omega^{0}_D=0.73$, the evolutions of $\Omega_D$, $\omega_D$, $q$, $j$ and $h$, as a function of $z$, have been plotted in figures \[figomegad\], \[figeos\], \[figqz\], \[figjz\] and \[figh\], respectively. From figure \[figeos\], one can see that $\omega_D$ remains always in between $-1< \omega_D< -\frac{1}{3}$ at present, as expected. However, it crosses the phantom line ($\omega_{D}<-1$) in the near future as the value of the parameter pair ($b^2_{1},b^2_{2}$) increases.\
The evolution of $q(z)$, as a function of $z$, has been plotted in figure \[figqz\]. From this figure, it is clear that our model can describe the current accelerated universe, and the transition redshift $z_t$ (i.e., $q(z_{t})=0$) from the deceleration phase to an accelerated phase occurs within the interval \[0.637,0.962\], which are in good agreement with the results, $0.5<z_{t} <1$, as reported in [@jerk2; @jerk3; @zt1; @zt2; @zt3; @zt4; @zt5; @zt6; @zt7; @zt8]. The evolution of $j(z)$ has also been plotted in figure \[figjz\]. It is observed that $j$ stays positive and lies within (0.52-0.58) at late time, and further it tends to unity (or $\Lambda$CDM model) as $z\rightarrow -1$. This is an interesting result of the present analysis. In figure \[figh\], we have shown the evolution of $h$ (equation (\[eqnh\])) for the present model and compared it with the data points for $h(z)$ (within $1\sigma$ error bars) which have been obtained from the latest compilation of 41 data points of Hubble parameter measurements ( for details, see [@hub1; @hub2]). We observed from figure \[figh\] that the model reproduces the observed values of $h(z)$ quite well for each data point. Furthermore, we also checked that the nature of the evolution of $h(z)$ is hardly affected by a small change in the values of the parameters ($b^{2}_{1}$, $b^{2}_{2}$).\
For understanding the classical stability of our model, we also plot the square of sound speed in figures \[figvs1\] and \[figvs2\]. It has been found from figure \[figvs1\] that the model is unstable ($v^{2}_{s}<0$). However, the situation is completely different, i.e., $v^{2}_{s}>0$, for some higher value of $\delta$ (see figure \[figvs2\]). Thus, the stability of the interacting THDE model crucially depends on the choice of the parameter $\delta$.
Thermodynamics of interacting THDE {#sec-thermo}
==================================
In this section, we derive the rate of change of the total entropy and then examinethe validity of generalized second law of thermodynamics. It is well known that thermodynamical analysis of the gravity theory is an exciting research topic in the cosmological context and the thermodynamical properties which hold for a black hole are equally valid for a cosmological horizon [@Bekenstein; @Hawking; @gibbons; @jacobson; @paddy; @bak-rey; @horizon-temperature-1; @horizon-temperature-2; @horizon-temperature-3; @horizon-temperature-4; @horizon-temperature-5; @jamil]. In addition, the first law of thermodynamics which holds in a black hole horizon can also be derived from the first Friedmann equation in the FRW universe when the universe is bounded by an apparent horizon. This provides well motivation to select the apparent horizon as the cosmological horizon in order to examine the thermodynamic properties of any cosmological model. Motivated by the above arguments, here, we have considered the universe as a thermodynamic system that is bounded by the cosmological apparent horizon with the radius [@bak-rey] $$r_h= \left(H^2+ \frac{k}{a^2} \right)^{-1/2}.$$ For a spatially flat universe ($k= 0$), the above equation immediately give $$r_h = \frac{1}{H},$$ which is the Hubble horizon.\
If we consider $S_f$ and $S_h$ are the entropy of the fluid and the entropy of the horizon containing the fluid, then the total entropy ($S$) of the system can be expressed as $$S= S_f+ S_h.$$ According to the laws of thermodynamics, like any isolated macroscopic system, then $S$ should satisfy the following relations $$\dot{S}=\frac{dS}{dt}~\ge ~0~~~~~{\rm and}~~~~~\ddot{S}=\frac{d^{2}S}{dt^{2}}<0.$$ In this context, it is important to mention that the generalized second law (GSL) of thermodynamics and thermodynamic equilibrium (TE) refer to the inequalities $\dot{S}~\ge ~0$ and $\ddot{S}<0$ respectively. Furthermore, the GSL should be true throughout the evolution of the universe, while the TE should hold at least during the final phases of its evolution. We shall now examine the validity of GSL of thermodynamics and also the TE in the present context.\
Now, the entropy of the horizon $S_h$ can be derived as [@horizon-temperature-4], $$\begin{aligned}
\label{sp-thermo2}
S_h= \frac{A}{4G} = 8 \pi^2 r_h^2,\end{aligned}$$ where, $A= 4 \pi r_h^2$ and $r_h$ are the surface area and radius of the apparent horizon respectively. Also, the temperture of the apparent horizon is given by the relation [@horizon-temperature-4] $$T_h = \frac{1}{2 \pi r_h}.$$
As previously mentioned, we considered only the DE and DM as the components in the energy budget, so we can write $$S_f =S_d + S_m,$$ where, $S_d$ and $S_m$ represent the entropies of the DE and DM respectively, and $T$ is the temperture of the composite matter inside the horizon. Therefore, the first law of thermodynamics ($ TdS = dE + p dV$) can be written for the individual matter contents in the following form $$\begin{aligned}
T d S_d &= dE_d + p_d dV\label{sp-thermo5},\\
T dS_m &= dE_m + p_m dV = dE_m\label{sp-thermo4},\end{aligned}$$ where $V= \frac{4}{3}\pi r_h^3$, is the horizon volume. Also, $E_d= \frac{4}{3}\pi r_h^3 \rho_d$ and $E_m=\frac{4}{3} \pi r_h^3 \rho_m$ represent the internal energies of the DE and DM ($p_{m}=0$) respectively. Now, differentiating equations (\[sp-thermo2\]) and (\[sp-thermo4\]) and (\[sp-thermo5\]) with respect to time, we obtain
$$\label{eqsda}
\left(\dot{S}_d,\, \dot{S}_m,\, \dot{S}_h \right) = \left(\frac{4 \, \pi \, p_d \, r_h^2 \,\dot{r}_h + \dot{E}_d}{T},\frac{\dot{E}_m}{T},16 \pi^2 r_h \dot{r}_h \right).$$
Using equation (\[eqsda\]) along with the asumption that the fluid temperture $T$ should be equal to that of the horizon temperture $T_h$ [@id1], one can arrive at the expression $$\label{gslt}
\dot{S}= \dot{S}_d + \dot{S}_m + \dot{S}_h= 4 \pi^2 H r_h^6 \Bigl[\rho_m + (1+ w_d)\rho_d \Bigr]^2.$$ In fact, the above relation has already been established in the context of interacting DE, where DE, DM and radiation are inteacting with each other [@jamil]. Equation (\[gslt\]) by simple algebra takes the form $$\begin{aligned}
\label{gslt1.1}
S^\prime = \frac{16 \pi ^2}{H^4}\, \left( H^\prime \right)^2,\end{aligned}$$ which is always positive definite irrespective of the functional forms of $H$. Here, a ‘prime’ represents derivative with respect to $x=\ln(a)$. Differentiating equation (\[gslt1.1\]) once more, we obtain $$\label{eqspp}
S^{\prime \prime} = 2\,S^{\prime}\, \left( \frac{ H^{\prime \prime}}{H^{\prime}}- \frac{2\,H^{\prime}}{H} \right)=2\,S^{\prime}\, \phi,$$ where, we define $\phi= \left( \frac{ H^{\prime \prime}}{H^{\prime}}- \frac{2\,H^{\prime}}{H} \right)$.\
![Behavior of the rate of change of the total entropy $\dot{S}$ , defined in equation (\[gslt\]), is shown using the same values of the parameters $b^2_{1}$, $b^2_{2}$, $\Omega^{0}_D$ and $\delta$ as in Figure \[figomegad\].[]{data-label="figsdot"}](sdot.eps){width="8cm"}
![Behavior of the thermodynamic function $\phi$, defined in equation (\[eqspp\]), is shown using the same values of the parameters $b^2_{1}$, $b^2_{2}$, $\Omega^{0}_D$ and $\delta$ as in Figure \[figomegad\].[]{data-label="figspp"}](phi.eps){width="8cm"}
It deserves to mention here that for a thermodynamic equilibrium, $S^{\prime \prime}<0$, which further implies $\phi<0$. We now try to understand the behavior of $S^{\prime \prime}$ (or $\phi$) with the evolution of the universe for the interacting THDE model discussed in section \[sec2\]. For the present model, the evolution of the quantities $\dot{S}$ and $\phi$ are shown in figures \[figsdot\] and \[figspp\] respectively, for different values of the model parameters. Figure \[figsdot\] shows that the GSL is always satisfies in the present context. It is evident from figure \[figspp\] that $\phi$ undergoes a transition from positive to negative value in the past, where the THDE starts to dominate over the DM. It is also found $\phi$ to be negative at the current epoch and also remains negative at future, i.e, $z \rightarrow -1$. This indicates towards a TE of the universe.
Conclusions {#conclusion}
===========
In this paper, we have studied an accelerating cosmological model for the present universe which is filled with DM and THDE. The DM is assumed to be interact with the THDE whose IR cut-off scale is set by the Hubble length. As already discussed in section \[sec2\], the functional form of $Q$ is chosen in such a way that it reproduces well known and most used interactions in the literature for some specific values of the model parameters $b_{1}$ and $b_{2}$ [@tnote; @thde5; @ig1; @id1; @im1; @im2].\
In our setups, the behavior of various quantities, e.g., $\Omega_{D}$, $\omega_{D}$, $q$, $j$, $h$ and $v^2_{s}$ have been studied during the cosmic evolution. In figure \[figeos\], the plot of $\omega_{D}$ with $z$ shows $\omega_{D}<-\frac{1}{3}$ at the present epoch which indicates an accelerating phase of the universe. The evolution of $q$ shows that the universe is decelerating at early epoch and accelerating at present epoch. This explains both the observed growth of structures at the early times and the late time cosmic acceleration measurements. Also, the transition between the DM era and the THDE era takes place within the redshift interval \[0.637,0.962\], which are in good compatibility with several recent studies [@jerk2; @jerk3; @zt1; @zt2; @zt3; @zt4; @zt5; @zt6; @zt7; @zt8]. It is also observed that $j$ stays positive and approaches to the $\Lambda$CDM ($j=1$) model as $z\rightarrow -1$. Further, we studied the thermodynamic nature of the universe for this model. The basic motivation was to verify whether our model fulfills the thermodynamical requirements of the expanding universe. Our study shows that the GSL of thermodynamics is always satisfies and also indicates towards a TE of the universe.\
Furthermore, we noticed that the stability of our model crucially depends on the choice of the parameter $\delta$ (see figures \[figvs1\] & \[figvs2\]). Therefore, we conclude that for the deep understanding of behavior of interacting THDE, more investigations should be done. In a follow-up study, we would like to study the model by considering other IR cutoffs and some non-linear interaction between the dark sectors, which may modify the properties of THDE.
Acknowledgments
===============
The work of KB was supported in part by the JSPS KAKENHI Grant Number JP 25800136 and Competitive Research Funds for Fukushima University Faculty (19RI017).
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|
---
abstract: |
We find a counterpart of the classical fact that the regular representation $\mathfrak R(G)$ of a simple complex group $G$ is spanned by the matrix elements of all irreducible representations of $G$. Namely, the algebra of functions on the big cell $G_0 \subset G$ of the Bruhat decomposition is spanned by matrix elements of big projective modules from the category $\mathcal O$ of representations of the Lie algebra $\g$ of $G$, and has the structure of a $\ggbar$-module.
We extend both regular representations to the affine group $\hat G$, and show that the loop form of the Bruhat decomposition of $\hat G$ yields modified versions of $\mathfrak R(\hat G)$. They involve pairings of positive and negative level modules, with the total value of the central charge required for the existence of non-trivial semi-infinite cohomology. In this paper we consider in detail the case $G=SL(2,\C)$, the corresponding finite-dimensional and affine Lie algebras, and the closely related to them Virasoro algebra.
Using the Fock space realization, we show that both types of modified regular representations for the affine and Virasoro algebras become vertex operator algebras, whereas the ordinary regular representations have instead the structure of conformal field theories. We identify the inherited algebra structure on the semi-infinite cohomology when the central charge is generic. We conjecture that for the integral values of the central charge the semi-infinite cohomology coincides with the Verlinde algebra and its counterpart associated with the big projective modules.
address:
- |
Department of Mathematics, Yale University\
New Haven, CT 06520, USA
- |
Max-Planck-Institut für Mathematik\
D-53111 Bonn, Germany
author:
- 'Igor B. Frenkel'
- Konstantin Styrkas
title: |
Modified regular representations\
of affine and Virasoro algebras, VOA structure\
and semi-infinite cohomology.
---
Introduction.
=============
The study of the regular representation of a simple complex Lie group $G$ is at the foundation of representation theory of $G$. Realized as the space of regular functions on $G$, the regular representation $\mathfrak R(G)$ carries two compatible structures of a $G$-bimodule and of a commutative associative algebra. An algebro-geometric version of the Peter-Weyl theorem establishes the decomposition of $\mathfrak R(G)$ into a direct sum of subspaces, spanned by matrix elements of all irreducible finite-dimensional representations $V_\lambda$ of $G$, indexed by integral dominant highest weights $\lambda \in \mathbf P^+$. In other words, we have an isomorphism of $G$-bimodules $$\label{eq:Peter-Weyl classical}
\mathfrak R(G) = \bigoplus_{\lambda \in \mathbf P^+} V_\lambda \o V_\lambda^*.$$ where $V_\lambda^*$ is the dual representation of $G$. The multiplication in $\mathfrak R(G)$ can be described in representation-theoretic terms as a pairing of intertwining operators for the left and right $\g$-actions with appropriate structural coefficients. Thus the algebra structure on $\mathfrak R(G)$ encodes the information about the tensor category of finite-dimensional $\g$-modules.
The representations of $G$ can also be viewed as modules over the simple complex Lie algebra $\g$ associated with $G$. In the case of the Lie algebra $\g$ it is natural to consider a larger collection of modules - namely, the Bernstein-Gelfand-Gelfand category $\mathcal O$. Infinite-dimensional $\g$-modules from the category $\mathcal O$ are not integrable, and therefore their matrix elements cannot be regarded as functions on $G$. However, one can interpret them as functions on the open dense subset $G^o \subset G$, given by the Gauss decomposition $$\label{eq:Gauss decomposition}
G^o = N_- \cdot T \cdot N_+,$$ where $N_\pm$ is the upper and lower triangular unipotent subgroup of $G$, and $T$ is the diagonal maximal abelian subgroup. The space $\mathfrak R(G^o)$ of regular functions on $G^o$ does not have the structure of a representation of $G$. Nevertheless, the left and right infinitesimal actions of the Lie algebra $\g$ on this space are well-defined, and can be expressed in terms of explicit differential operators in the parameters of the Gauss decomposition . The enlarged regular representation $\mathfrak R(G^o)$ decomposes into the direct sum of bimodules spanned by the matrix coefficients of all “big” projective $\g$-modules $P_\lambda$, indexed by strictly anti-dominant highest weights $\lambda \in - \mathbf P^{++} = - (\mathbf P^+ + \rho)$, where $\rho$ is the half-sum of all positive roots of $\g$. Thus we obtain an isomorphism of $\g$-bimodules $$\label{eq:Peter-Weyl twisted}
\mathfrak R(G^o) \cong \bigoplus_{\lambda \in -\mathbf P^{++}} \( P_\lambda \o P_\lambda^* \) \biggr/ I_\lambda,$$ where $P_\lambda^*$ is the module dual to $P_\lambda$, and $I_\lambda$ is the sub-bimodule of the matrix coefficients which vanish identically on the universal enveloping algebra $\mathcal U(\g)$.
It is important to note that the dual modules $P_\lambda^*$ do not belong to the category $\mathcal O$, but to its “mirror image”, in which all highest weight modules are replaces by lowest weight modules. In order to stay in the category $\mathcal O$ we replace the open subset $G^o$ coming from the Gauss decomposition by the maximal cell in the Bruhat decomposition $$\label{eq:Bruhat decomposition}
G_0 = N_+ \cdot \mathbf{w_0} \cdot T \cdot N_+,$$ where $\mathbf {w_0}$ is the longest element of the Weyl group $W$. Then we obtain a version of the isomorphism , $$\label{eq:Peter-Weyl projective}
\mathfrak R(G_0) \cong \bigoplus_{\lambda \in -\mathbf P^{++}} \( P_\lambda \o P_\lambda^\star \) \biggr/ I_\lambda,$$ where the ’twisted’ duals $P_\lambda^\star$ differs from $P_\lambda^*$ by the automorphism $\omega$ of $\g$, which is induced by $\mathbf w_0$ and interchanges the positive and negative roots.
The theorems of Peter-Weyl type and the Gauss decomposition can be extended to the central extension of the loop group $\hat G$ associated to $G$, and to the corresponding affine Lie algebra $\ghat$ and its universal enveloping algebra $\mathcal U(\ghat)$. In this infinite-dimensional case the space $\mathfrak R(\hat G)$ of regular functions on $\hat G$ is decomposed into the direct sum of subspaces $\mathfrak R_k(\hat G)$, corresponding to the value $k \in \mathbb Z$ of the central charge. Using the version of the Gauss decomposition known as the Birkhoff decomposition, one can show (see [@PS]) that for any $k \in \mathbb Z$ there is an isomorphism $$\label{eq:Peter-Weyl affine}
\mathfrak R_k(\hat G) \cong \bigoplus_{\lambda \in \mathbf P^k_+} \hat V_{\lambda,k} \o \hat V_{\lambda,k}^*,$$ where $\lambda$ runs over the truncated alcove $\mathbf P^+_k \subset \mathbf P^+$, depending on $k$, and $\hat V_{\lambda,k}$ are the corresponding irreducible modules. Similarly, one can obtain decompositions of $\mathfrak R_k(\hat G^o)$ analogous to , where $\hat G^o$ is the maximal cell in the Birkhoff decomposition. Viewing the decomposition in terms of the Lie algebra $\ghat$ allows to extend it for all values of $k$, with $\mathbf P^+_k = \mathbf P^+$ for $k \notin \mathbb Q$.
To transform the dual module $\hat V_{\lambda,k}^*$ into a module from the category $\mathcal O$ for $\ghat$, one might apply again an automorphism of $\ghat$ which interchanges the positive and negative affine roots. However, it no longer belongs to the affine Weyl group, and the Bruhat decomposition for $\hat G$ does not have a maximal cell. To overcome this obstacle we consider instead an intermediate between the Birkhoff and the affine Bruhat decompositions - the loop version of the finite-dimensional Bruhat decomposition, and the corresponding big cell $$\label{eq:loop Bruhat decomposition}
\hat G_0 = LN_+ \cdot \mathbf{w_0} \cdot \widehat{LT} \cdot LN_+,$$ where $LN_\pm$ denote the loop groups with values in $\n_\pm$, and $\widehat{LT}$ is the central extension of the loop group with values in $T$. The decomposition is especially useful for explicit realizations of the left and right regular $\ghat$-actions in terms of differential operators. However, we are still in “semi-infinite” distance from the category $\mathcal O$, and need to further apply a well-known procedure of “changing the vacuum”, which has originated from the free field realizations of the Wakimoto modules and the irreducible representations $\hat V_{\lambda,k}$ (see [@FeFr; @BF]). As a result of this procedure we obtain a modified affine version of the extended regular representation , $$\label{eq:Peter-Weyl affine twisted}
\mathfrak R'_k(\hat G_0) \cong \bigoplus_{\lambda \in -\mathbf P^{++}}
\( \hat P_{\lambda,k-h^\vee} \o \hat P_{\lambda,-k-h^\vee}^\star \) \biggr/ \hat I_{\lambda,k},$$ where $\hat P_{\lambda,k-h^\vee}$ and $\hat P^\star_{\lambda,-k-h^\vee}$ are the projective $\ghat$-modules and their ’twisted’ duals, $\hat I_{\lambda,k}$ are appropriate sub-bimodules, and we assume $k \notin\mathbb Q$. The levels are shifted by the dual Coxeter number $h^\vee$, so that the diagonal $\ghat$-action has the level $-2h^\vee$.
Like the Wakimoto modules, the bimodule $\mathfrak R'_k(\hat G_0)$ is realized as a certain Fock space, with two commuting $\ghat$-actions described explicitly. This realization is similar to the standard realization of the Wakimoto modules, but the actions of $\ghat$ contain a crucial new ingredient - the vertex operators, directly related to the screening operators used to construct intertwining operators for the affine Lie algebra. We also establish that for $k \notin \mathbb Q$ the structure of the socle filtration of the non-semisimple bimodule $\mathfrak R'_k(\hat G_0)$ is the same as in the finite-dimensional case. In particular, $\mathfrak R'_k(\hat G_0)$ contains the distinguished sub-bimodule $$\label{eq:positive Peter-Weyl}
\mathfrak R'_k(\hat G) \cong \bigoplus_{\lambda \in \mathbf P^{+}} \hat V_{\lambda,k-h^\vee} \o \hat V_{\lambda,-k-h^\vee}^\star.$$ The shifts of the central charge by the dual Coxeter number no longer allow the interpretation of the bimodules in and as spaces of matrix elements of $\ghat$-modules. Nevertheless, the structures of these bimodules are completely analogous to those of bimodules $\mathfrak R(G_0)$ and $\mathfrak R(G)$!
The vacuum module $\hat V_{0,k}$ of the affine Lie algebra $\ghat$ carries an extremely rich additional structure of a vertex operator algebra (VOA); other $\ghat$-modules $\hat V_{\lambda,k}$ become its representations (see [@FZ]). We show in this paper that the bimodules $\mathfrak R'_k(\hat G)$ and $\mathfrak R'_k(\hat G_0)$ also admit a vertex operator structure, compatible with $\ghat$-actions. In the proof we use the explicit Fock space realization of these bimodules. As in the finite-dimensional case, the VOA structure of the modified regular representations $\mathfrak R'_k(\hat G)$ and $\mathfrak R'_k(\hat G_0)$ encodes the information about fusion rules of the corresponding tensor categories of $\ghat$-modules. Thus besides the vacuum modules there is a class of vertex operator algebras associated to affine Lie algebras with fixed central charges. It is also related to the algebras of chiral differential operators over the simple algebraic group $G$ recently studied in [@GMS] and [@AG]. A study of this relation might help to understand the geometric nature of the modified regular representations.
On the other hand, the original regular representation $\mathfrak R_k(\hat G)$ does not seem to have a VOA structure. Instead it has the structure of a two-dimensional conformal field theory, which is an object of a different nature despite having local properties similar to those of a VOA. The bimodule $\mathfrak R_k(\hat G_0)$ has a structure of a generalized (non-semisimple!) conformal field theory.
It is well-known that the representation theory of $\ghat$ is closely related to the representation theory of the corresponding $\mathcal W$-algebra via the quantum Drinfeld-Sokolov reduction. In particular, one expects to have an analogue of the Peter-Weyl theorem for the $\mathcal W$-algebras. In this paper we consider in detail the simplest case of $\g = \sl(2,\C)$, when the corresponding $\mathcal W$-algebra is the infinite-dimensional Virasoro algebra. We give explicit realizations of the Virasoro bimodules, analogous to and , and equip them with compatible VOA structures. The structures of the non-semisimple modified regular representations are quite parallel in all cases; we fully describe their socle filtrations. Generalizations of our constructions to higher rank Lie algebras are straightforward, but their $\mathcal W$-algebra versions require more technicalities; full details will be presented in a subsequent paper.
A remarkable feature of all the modified bimodules that appear in the decompositions of Peter-Weyl type is that the central charge of the diagonal subalgebra is always equal to the special values that appear in the semi-infinite cohomology theory [@Fe; @FGZ] - namely, $-2h^\vee$ for the affine Lie algebras and 26 for Virasoro. Moreover, thanks to a general result of [@LZ], the corresponding semi-infinite cohomology spaces inherit a VOA structure from the modified regular representations. In our case, they degenerate into commutative associative superalgebras, and for generic central charge we establish isomorphisms between cohomology groups with coefficients in the corresponding modified regular representations of the affine and Virasoro algebras and their finite-dimensional counterparts. In particular, we show that the 0th semi-infinite cohomologies of the affine and Virasoro algebras are isomorphic to the Grothendieck ring of finite-dimensional representations of $G$. We conjecture that for integral $k$ they lead to the Verlinde algebra and its projective counterpart.
This paper is organized as follows. In Section 1 we consider the Bruhat decomposition and the Peter-Weyl theorems in the finite-dimensional case with $G=SL(2,\C)$. We give a Fock space realization of the algebra $\mathfrak R(G_0)$, and obtain explicit formulas for the $\g$-actions and decomposition theorems, which will later be used as prototypes of the infinite-dimensional case. In the last subsection we compute the Lie algebra cohomology with coefficients in $\mathfrak R(G)$ and $\mathfrak R(G_0)$. In Section 2 we study the affine case, and use the loop version of the finite-dimensional Bruhat decomposition to obtain the modified Peter-Weyl theorems for the spaces $\mathfrak R'_k(\hat G)$ and $\mathfrak R'_k(\hat G_0)$. The Fock space realization of these spaces equips them with VOA structures compatible with the regular $\ghat$-actions. The semi-infinite cohomology of $\ghat$ with coefficients in the modified regular representations $\mathfrak R'_k(\hat G)$ and $\mathfrak R'_k(\hat G_0)$ for generic central charge is shown to be isomorphic to its finite-dimensional counterpart. In Section 3 we construct the analogues of the modified regular representations of the Virasoro algebra using the quantum Drinfeld-Sokolov reduction. We compute the corresponding semi-infinite cohomology groups using methods developed in string theory, and prove that they are isomorphic to their affine counterparts. Finally, in Section 4 we describe another class of vertex operator algebras obtained by the pairing of $\slhat$ and Virasoro modules. We also discuss generalizations of our results to Lie algebras of other types, and to the integral values of the central charge. We conclude with conjectures on relations of the semi-infinite cohomology of $\mathfrak R'_k(\hat G)$ and $\mathfrak R'_k(\hat G_0)$ for $k \in \Z_{>0}$ with the Verlinde algebra, its projective counterpart and twisted equivariant K-theory.
We wish to thank G. Zuckerman for sharing his expertise on semi-infinite cohomology, and S. Arkhipov, F. Malikov for valuable comments. I.B.F. is supported in part by NSF grant DMS-0070551.
Regular representation of $\sl(2,\C)$ on the big cell.
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Regular representations of $\sl(2,\C)$.
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Let $G = SL(2,\C)$. We define the left and right regular actions of $G$ on the space $\mathfrak R(G)$ of regular functions on $G$ by $$\label{eq:group regular actions}
(\pi_l(g)\psi)(h) = \psi(g^{-1} h), \qquad
(\pi_r(g)\psi)(h) = \psi(h \, g), \qquad
g,h \in G.$$ The multiplication in $\mathfrak R(G)$ intertwines both left and right regular actions.
Let $T, N_+$ denote the diagonal and unipotent upper-triangular subgroups of $G$. The group $W = \operatorname{Norm}(T)/T$ is called the Weyl group. The Bruhat decomposition $G = N_+ \cdot W \cdot T \cdot N_+$ implies that every $g \in G$ can be factored as $g = n \cdot w \cdot t \cdot n'$ for some $n,n' \in N_+, t \in T, w \in W$.
We denote by $G_0$ the big cell of the Bruhat decomposition, corresponding to the longest Weyl group element ${\mathbf w_0}$. Explicitly, $G_0$ is the dense open subset of $G$, consisting of $g \in G,$ $$\label{eq:explicit Bruhat decomposition}
g =
\begin{pmatrix} 1 && x \\ 0 && 1 \end{pmatrix}
\begin{pmatrix} 0 && -1 \\ 1 && 0 \end{pmatrix}
\begin{pmatrix} \zeta && 0 \\ 0 && \zeta^{-1} \end{pmatrix}
\begin{pmatrix} 1 && y \\ 0 && 1 \end{pmatrix}$$ for some $x,y \in \C$ and $\zeta \in \C^\times$. The variables $x,y,\zeta$ can be viewed as coordinates on $G_0$, and thus the algebra $\mathfrak R(G_0)$ of regular functions on $G_0$ is identified with the space $\C[x,y,\zeta^{\pm1}]$.
Let $\g = \sl(2,\C)$ be the Lie algebra of $G$, with the standard basis $$\mathbf e = \begin{pmatrix} 0 && 1 \\ 0 && 0 \end{pmatrix}, \qquad
\mathbf h = \begin{pmatrix} 1 && 0 \\ 0 && -1 \end{pmatrix}, \qquad
\mathbf f = \begin{pmatrix} 0 && 0 \\ 1 && 0 \end{pmatrix},$$ satisfying the commutation relations $[\mathbf h, \mathbf e] = 2 \mathbf e, \
[\mathbf h, \mathbf f] = -2 \mathbf f, \
[\mathbf e, \mathbf f] = \mathbf h.$ The nilpotent subalgebras $\n_{\pm}$ and the Cartan subalgebra $\h$ of $\g$ are defined by $\n_+ = \C \mathbf e, \, \h = \C \mathbf h, \, \n_- = \C \mathbf f.$ The element $\mathbf {w_0} \in W$ determines a Lie algebra involution $\omega$ of $\g$, such that $\omega(\n_\pm) = \n_\mp$ and $\omega(\h) = \h$, defined by $$\label{eq:Cartan automorphism}
\omega(\mathbf e) = -\mathbf f, \qquad
\omega(\mathbf h) = -\mathbf h, \qquad
\omega(\mathbf f) = -\mathbf e.$$
The infinitesimal regular actions of $\g$ on $\mathfrak R(G)$, corresponding to , are given by $$\label{eq:algebra regular actions}
(\pi_l(x) \psi)(g) = \frac d{dt} \psi(e^{-t \, x} g) \biggr|_{t=0}, \qquad
(\pi_r(x) \psi)(g) = \frac d{dt} \psi(g \, e^{t \, x} ) \biggr|_{t=0}, \qquad x \in \g, \, g \in G.$$
These formulas also define left and right infinitesimal actions of $\g$ on the space $\mathfrak R(G_0)$. (These actions cannot be lifted to the group $G$, because $G_0$ is not invariant under left and right shifts). Elementary calculations yield the following explicit description of the regular $\g$-actions (cf. [@FP]).
\[thm:classical action\] The regular $\g$-actions on $\mathfrak R(G_0)$ are given by $$\begin{split} \label {eq:classical left action}
\pi_l(\mathbf e) & = -\pd x,\\
\pi_l(\mathbf h) & = \zeta \pd \zeta - 2 x \pd x,\\
\pi_l(\mathbf f) & = - x \zeta \pd \zeta + x^2 \pd x + \zeta^{-2} \pd y.
\end{split}$$ $$\begin{split} \label {eq:classical right action}
\pi_r(\mathbf e) & = \pd y ,\\
\pi_r(\mathbf h) & = \zeta \pd \zeta - 2 y \pd y,\\
\pi_r(\mathbf f) & = y \zeta \pd \zeta - y^2 \pd y - \zeta^{-2} \pd x.
\end{split}$$
Bosonic realizations
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We now reformulate the constructions of the previous section in terms of Fock modules for certain Heisenberg algebras. These realizations admit generalizations to the affine and Virasoro cases, where the geometric approach to the regular representations becomes more subtle.
The operators $\beta = x, \ \gamma = -\pd x$ acting on polynomials in $y$ give a representation of the Heisenberg algebra with generators $\beta,\gamma$ and relation $[\beta,\gamma] = 1$. The polynomial space $\C[y]$ is then identified with its irreducible representation $F(\beta,\gamma)$, generated by a vector $\1$ satisfying $\gamma \, \1 = 0$.
The operators $\bar\beta = -y, \ \bar\gamma = \pd y$ generate a second Heisenberg algebra, acting irreducibly in the space $F(\bar\beta,\bar\gamma) \cong \C[x]$. Here and everywhere else in this paper the ’bar’ notation is used to denote the second copies of algebras and their generators; it does not denote the complex conjugation.
We identify $\h^* \cong \C$ so that $\mathbf P \cong \Z$. Whenever possible, we use the more invariant notation in order to avoid possible numeric coincidences. The operators $\1_\lambda =\zeta^\lambda$ and $a = \zeta \pd \zeta$ gives rise to the semi-direct product $\C[a] \ltimes \C[\mathbf P]$, with $\C[a]$ acting on $\C[\mathbf P]$ by derivations: $a \1_\lambda = \lambda \1_\lambda$.
Thus, we get a realization of the algebra $\mathfrak R(G_0)$ of regular functions on $G_0$, with the $\ggbar$-action described by the abstract versions of the formulas , .
\[thm:classical bimodule action\] The space $\mathbb F= F(\beta,\gamma) \otimes F(\bar\beta,\bar\gamma) \otimes \C[\mathbf P]$ gives a realization of the algebra $\mathfrak R(G_0)$. In particular,
1. The space $\mathbb F$ has a $\ggbar$-module structure, given by $$\label{eq:classical boson left}
\begin{split}
\mathbf e & = \gamma, \\
\mathbf h & = 2 \, \beta \gamma + a, \\
\mathbf f & = -\beta^2 \gamma - \beta \, a + \bar \gamma \, \1_{-2},
\end{split}$$ $$\label{eq:classical boson right}
\begin{split}
\bar {\mathbf e} & = \bar\gamma, \\
\bar {\mathbf h} & = 2 \, \bar\beta \bar\gamma + a, \\
\bar {\mathbf f} & = -\bar\beta^2 \bar\gamma - \bar\beta \, a + \gamma \, \1_{-2}.
\end{split}$$
2. The space $\mathbb F$ has a compatible commutative algebra structure (i.e. the multiplication in $\mathbb F$ intertwines the $\ggbar$-action).
By specializing the action to the subspace $\ker \bar\gamma \subset \mathbb F$, we get the following well-known realizations of $\g$-action in the spaces $F_\lambda = F(\beta,\gamma) \o \C\1_\lambda$: $$\begin{split} \label{eq:classical boson}
\mathbf e & = \gamma, \\
\mathbf h & = 2\beta \gamma + \lambda,\\
\mathbf f & = -\beta^2 \gamma - \lambda \, \beta.
\end{split}$$
Simultaneous rescaling of the extra terms in ,, involving the shift $\1_{-2}$, by any multiple $\epsilon$ would preserve all the $\ggbar$ commutation relations. For $\epsilon = 0$ such $\ggbar$-action degenerates into the product of two standard $\g$-actions . However, the multiplication in this naïve bimodule loses much of its rich structure, and no longer encodes the information about the fusion rules in the tensor category of finite-dimensional $\g$-modules.
$\ggbar$-module structure of the modified regular representation
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In this subsection we describe the socle filtration of the $\ggbar$-module $\mathbb F$.
For any $\lambda \in \h^*$, we denote by $V_\lambda$ the irreducible $\g$-module, generated by a highest weight vector $v_\lambda$ satisfying $\mathbf e \, v_\lambda = 0$ and $\mathbf h \, v_\lambda = \lambda \, v_\lambda$.
Recall that a $\g$-module $V$ is said to have a weight space decomposition, if $$V = \bigoplus_{\mu\in\h^*} V[\mu], \qquad V[\mu] = \left\{v \in V \, \bigr| \, \mathbf h \, v = \mu \, v \right\}.$$ The restricted dual space $V' = \bigoplus_{\mu\in\h^*} V[\mu]'$ can be equipped with a $\g$-action, defined by $$\<g \, v', v\> = - \< v', \, \omega(g) \, v\>, \qquad g \in \g, \, v \in V, \, v' \in V',$$ where $\omega$ is as in . We denote the resulting dual module $V^\star$.
We have an involution $\lambda \mapsto \lambda^\star$ of $\h^*$, determined by the condition $(V_\lambda)^\star \cong V_{\lambda^\star}$. This involution can also be defined by $\lambda^\star = - \mathbf{w_0} (\lambda)$, where $\mathbf {w_0}$ is the longest Weyl group element.
For $\g = \sl(2,\C)$, we have $\lambda^\star = \lambda$. However, we keep the notation $\lambda^\star$, to indicate how our constructions generalize to Lie algebras of higher rank, where the involution is nontrivial.
\[thm:classical bimodule structure\] There exists a filtration $$\label{eq:classical filtration}
0 \subset \mathbb F^{(0)} \subset \mathbb F^{(1)} \subset \mathbb F^{(2)} = \mathbb F$$ of $\ggbar$-submodules of $\mathbb F$, such that $$\begin{aligned}
\mathbb F^{(2)}/\mathbb F^{(1)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} V_{-\lambda-2} \o V^\star_{-\lambda-2} \label{eq:classical socle F2},\\
\mathbb F^{(1)}/\mathbb F^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \( V_\lambda \o V^\star_{-\lambda-2} \oplus V_{-\lambda-2} \o V^\star_\lambda \) \label{eq:classical socle F1},\\
\mathbb F^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P} V_\lambda \o V^\star_\lambda \label{eq:classical socle F0}.\end{aligned}$$
We introduce a filtration of $\ggbar$-submodules of $\mathbb F$ $$\label{eq:classical Fock filtration}
\dots \subset \mathbb F_{\le -2} \subset \mathbb F_{\le -1} \subset \mathbb F_{\le 0} \subset \mathbb F_{\le 1} \subset \mathbb F_{\le 2} \subset \dots,$$ satisfying $\bigcap_{\lambda\in\mathbf P} \ \mathbb F_{\le \lambda} = 0$ and $\bigcup_{\lambda\in\mathbf P} \ \mathbb F_{\le \lambda} = \mathbb F,$ where $$\mathbb F_{\le \lambda} = F(\beta,\gamma) \o F(\bar\beta,\bar\gamma) \o \bigoplus_{\mu \le \lambda} \C\1_\mu, \qquad \lambda \in \mathbf P.$$ It is clear that $\mathbb F_{\le \lambda} / \mathbb F_{<\lambda} \cong F_\lambda \o F_{\lambda^\star}$; moreover, for $\lambda <0$ we have $F_\lambda \cong V_\lambda$, and for $\lambda \ge 0$ there is a short exact sequence $0 \to V_\lambda\to F_\lambda \to V_{-\lambda-2}\to 0$. The linking principle for $\g$-modules implies that the successive quotients $\mathbb F_{\le \lambda} / \mathbb F_{<\lambda}$ and $\mathbb F_{\le \mu} / \mathbb F_{<\mu}$ of this filtration may be non-trivially linked only if $\mu = -\lambda-2$.
Thus we see that the $\ggbar$-module $\mathbb F$ splits into the direct sum of blocks $$\label{eq:classical double blocks}
\mathbb F = \mathbb F(-1) \oplus \bigoplus_{\lambda \in\mathbf P^+} \mathbb F(\lambda),$$ where $\mathbb F(-1) \cong V_{-1} \o V^\star_{-1}$, and $\mathbb F(\lambda)\cong
\(V_{-\lambda-2} \o V^\star_{-\lambda-2}\) + \(F_\lambda \o F_{\lambda^\star}\)$ for $\lambda \in \mathbf P^+$; another way to obtain the decomposition is by using the Casimir operator.
It remains to describe the structure of $\mathbb F(\lambda)$ for each $\lambda \in \mathbf P^+$. By construction, $\mathbb F(\lambda)$ can be included in a short exact sequence $0 \to V_{-\lambda-2} \o V^\star_{-\lambda-2} \to \mathbb F(\lambda) \to F_\lambda \o F_{\lambda^\star} \to 0.$ We conclude that there exists a filtration $0 \subset \mathbb F(\lambda)^{(0)} \subset \mathbb F(\lambda)^{(1)} \subset \mathbb F(\lambda)^{(2)} = \mathbb F(\lambda),$ such that $$\begin{aligned}
\mathbb F(\lambda)^{(2)}/\mathbb F(\lambda)^{(1)} &\cong
V_{-\lambda-2} \o V^\star_{-\lambda-2},\\
\mathbb F(\lambda)^{(1)}/\mathbb F(\lambda)^{(0)} &\cong
\( V_\lambda \o V^\star_{-\lambda-2} \) \oplus \( V_{-\lambda-2} \o V^\star_\lambda \) ,\\
\mathbb F(\lambda)^{(0)} &\cong
\( V_{-\lambda-2} \o V^\star_{-\lambda-2} \) + \( V_\lambda \o V^\star_\lambda \).\end{aligned}$$ In fact, the linking principle implies that the sum in $\mathbb F(\lambda)^{(0)}$ is direct: $$\mathbb F(\lambda)^{(0)} \cong
\( V_{-\lambda-2} \o V^\star_{-\lambda-2} \) \oplus \( V_\lambda \o V^\star_\lambda \).$$ Finally, we construct the filtration by setting $$\mathbb F^{(0)} = \mathbb F(-1) \oplus \bigoplus_{\lambda \in \mathbf P^+} \mathbb F(\lambda)^{(0)},
\qquad
\mathbb F^{(1)} = \mathbb F(-1) \oplus \bigoplus_{\lambda \in \mathbf P^+} \mathbb F(\lambda)^{(1)},$$ which obviously satisfies the required conditions , , .
For a Lie algebra $\g$ of higher rank, we will get a similar filtration of length $2\, l(\mathbf{w_0}) + 1$, and in addition to the regular blocks, corresponding to $\lambda \in \mathbf P^+$, and the most degenerate block $\mathbb F(-1)$, there will be all intermediate types.
The natural inclusion of algebras $\mathfrak R(G) \subset \mathfrak R(G_0)$ can be seen in the Fock space realizations.
\[thm:classical positive subalgebra\] There exists a subspace $\mathbf F \subset \mathbb F$ satisfying the following properties.
1. $\mathbf F$ is a subalgebra of $\mathbb F$, and is generated by the elements from the submodule $V_1 \o V_1^\star$, corresponding to the matrix elements of the canonical representation of $G$.
2. $\mathbf F$ is a $\ggbar$-submodule of $\mathbb F$, and is generated by the vectors $\{\1_\lambda\}_{\lambda \in \mathbf P^+}$. We have $$\label{eq:classical Peter-Weyl}
\mathbf F = \bigoplus_{\lambda\in\mathbf P^+} \mathbf F(\lambda) \cong \bigoplus_{\lambda\in \mathbf P^+} V_\lambda \o V^\star_\lambda.$$
3. The space $\mathbf F$ is a realization of the algebra $\mathfrak R(G)$.
In the polynomial realization, the generators of $\mathbf F$ from $V_1 \o V_1^\star$ are identified with functions $$\psi_{11} = \zeta, \qquad \psi_{12} = x \zeta, \qquad \psi_{21} = y \zeta, \qquad \psi_{22} = x y \zeta + \zeta^{-1},$$ which satisfy the relation $\psi_{11} \psi_{22} - \psi_{12} \psi_{21} = 1$. This establishes a very direct connection with the space of regular functions on the group $G = SL(2,\C)$.
The generalized Peter-Weyl theorem
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In this section we interpret the space $\mathfrak R(G_0)$ of regular functions on $G_0$ and its Fock space realization $\mathbb F$ as the algebra of matrix elements of all modules from the category $\mathcal O$.
Recall that the Bernstein-Gelfand-Gelfand category $\mathcal O$ consists of all finitely generated, locally $\n_+$-nilpotent $\g$-modules. In particular, $V_\lambda\in\mathcal O$ for any $\lambda$. If $V \in \mathcal O$, then $V^\star \in \mathcal O$.
For any $\g$-module $V$ we define $\mathbb M(V)$ to be the subspace of $\mathcal U(\g)'$, spanned by functionals $$\label{eq:matrix element}
\phi_{v,v'}(x) = \<v', x \, v\>, \qquad v \in V, \, v' \in V', \, x \in \mathcal U(\g),$$ where $\<\cdot,\cdot\>$ stands for the natural pairing between $V$ and $V'.$ The functionals are called matrix elements of the representation $V.$
\[thm:matrix elements\] Introduce a $\ggbar$-module structure on the restricted dual $\mathcal U(\g)'$ by $$\label{eq:dual enveloping bimodule}
(\pi_l(g) \phi) (x) = \phi(x g), \qquad\quad (\pi_r(g) \phi) (x) = - \phi(\omega(g) x)$$ for any $\phi \in \mathcal U(\g)', \, g\in \g, \, x \in \mathcal U(\g)$. Then
1. For any $\g$-module $V$, the space $\mathbb M(V)$ is a $\ggbar$-submodule of $\mathcal U(\g)'$.
2. For any $\varphi \in \mathcal U(\g)',$ there exists a $\g$-module $V$, such that $\varphi \in \mathbb M(V).$ Moreover, if $\varphi$ is $\n_+ \oplus \n_+$-nilpotent, then $V$ can be chosen from the category $\mathcal O.$
To show that $\mathbb M(V)$ is invariant under the left action of $\g$, we compute $$(\pi_l(g) \phi_{v,v'}) (x) = \phi_{v,v'}(x g) = \<v', x g \, v\> = \phi_{g v,v'}(x),$$ for any $x \in \mathcal U(\g),\, g \in \g,\, v \in V, \, v' \in V'.$ This shows that $y \phi_{v,v'}\in \mathbb M(V).$ The invariance under the right action follows from the computation $$(\pi_r(g) \phi_{v,v'}) (x) = - \phi_{v,v'}(\omega(g) x) = - \<v', \omega(g) x \, v\> = \<y v', x \, v\> = \phi_{v, y v'}(x).$$
For the second part, assume $\varphi \in \mathcal U(\g)'$. Denote by $V$ the subspace of $\mathcal U(\g)'$, generated from $\varphi$ by the left action of $\g$. Let $\varphi'$ be the restriction to $V$ of the unit $1 \in \mathcal U(\g) = \mathcal U(\g)''$. Equivalently, $\varphi'$ is the linear functional on $V$, determined by $\<\varphi', \psi\> = \psi(1),$ for any $\psi \in V \subset \mathcal U(\g)'$. We claim that $\varphi = \phi_{\varphi,\varphi'} \in \mathbb M(V)$. Indeed, for any $x \in \mathcal U(\g)$ we have $$\phi_{\varphi,\varphi'} (x) = \<\varphi', x \varphi\> = (x\varphi)(1) = \varphi(x).$$
Finally, if $\varphi$ is left-$\n_+$-nilpotent, then $V$ is locally $\n_+$-nilpotent. Since $V$ is generated by a single element $\varphi,$ it belongs to category $\mathcal O.$ The right-$\n_+$-nilpotency condition guarantees that $\varphi'$ belongs to the [*restricted*]{} dual space $V'$.
The elements of the universal enveloping algebra $\mathcal U(\g)$ may be regarded as the differential operators, acting on $\mathfrak R(G)$. This gives an interpretation of the regular functions on $G$ (or even on $G_0$) as linear functionals on $\mathcal U(\g)$, and thus to identifications of the spaces $\mathfrak R(G)$ and $\mathfrak R(G_0)$ with certain subspaces of $\mathcal U(\g)'$. In the explicit realizations $\mathbf F$ and $\mathbb F$ this correspondence is constructed using the algebraic analogue of the “co-unit” element of the Hopf algebra $\mathfrak R(G)$ - the linear functional $\<\cdot\>: \mathbb F \to \C$, defined by $$\<\beta^m \bar\beta^n \1_\lambda\> = \delta_{m,0} \delta _{n,0}.$$
The linear map $\vartheta: \mathbb F \to \mathcal U(\g)'$, defined by $v \mapsto \vartheta_v$, $$\label{eq:map iota}
\vartheta_v(x) = \<\pi_l(x) v\>, \qquad v \in \mathbb F, \ x \in \mathcal U(\g).$$ is an injective $\ggbar$-homomorphism.
In terms of the polynomial realization, $\<\cdot\>$ corresponds to evaluating a function $\psi(x,y,\zeta) \in \mathfrak R(G_0)$ at the element $\mathbf{w_0}$: $\<\psi\> = \psi(0,0,1)$. This implies that for any $v \in \mathbb F$ $$\label{eq:classical contravariant functional}
\<\mathbf e v\> = -\<\bar{\mathbf f} v\>, \qquad
\<\mathbf h v\> = -\<\bar{\mathbf h} v\>, \qquad
\<\mathbf f v\> = -\<\bar{\mathbf e} v\>.$$ Therefore, for any $g \in \g$ and $x \in \mathcal U(\g)$ we have $$\vartheta_{g v}(x) = \<x \, g v\> = \vartheta_v(x g) = (\pi_l(g)\vartheta_v)(x),$$ $$\vartheta_{\bar g v}(x) = \<x \, \bar g v\> = \<\bar g \, x v\> = - \<\omega(g) x v\> =
- \vartheta_v(\omega(g) x) = (\pi_r(g)\vartheta_v)(x).$$ We conclude that the map $\vartheta$ is a $\ggbar$-homomorphism. To prove that it is injective, we need to show that for any nonzero $v \in \mathbb F$ there exists an element $x \in \ggbar$ such that $\<x v\> \ne 0.$
Since $\mathbb F$ is locally $\n_+$-nilpotent, we can pick $k\ge0$ such that $\mathbf e^k v \ne 0,$ but $\mathbf e^{k+1} v = 0.$ Replacing $v$ by $\mathbf e^k v,$ we see that it suffices consider the case of $v \ne 0$ such that $\mathbf e v = 0.$ Similarly, we may assume that $\bar{\mathbf e} v = 0.$
A vector $v$ satisfying $\mathbf e v = 0 = \bar{\mathbf e} v$ must have the form $v = \sum_{\lambda\in\mathbf P} c_\lambda \1_\lambda$ with only finitely many $c_\lambda \ne 0.$ Using the formula for the Vandermonde determinant and the fact that $$\<\mathbf h^m v\> = \sum_{\lambda\in\mathbf P} c_\lambda \lambda^m, \qquad m \ge 0,$$ we conclude that $\<\mathbf h^k v\> = 0$ for all $k\ge0$ if and only if all $c_\lambda$ vanish. Thus, $\theta_v = 0$ is equivalent to $v = 0$, which means that $\vartheta$ is an injection.
The following statement is an algebraic version of the classical Peter-Weyl theorem.
The space $\mathfrak R(G)$ of regular functions on $G$ is spanned by the matrix elements of finite-dimensional irreducible $\g$-modules, $$\mathfrak R(G) \cong \bigoplus_{\lambda\in \mathbf P^+} \mathbb M(V_\lambda).$$ The decomposition of $\mathfrak R(G)$ as a $\ggbar$-module is given by $$\mathfrak R(G) \cong \bigoplus_{\lambda\in \mathbf P^+} V_\lambda \o V^\star_\lambda.$$
The subspace of $\mathcal U(\g)'$, corresponding to $\mathfrak R(G)$, is invariantly characterized as the restricted Hopf dual $\mathcal U(\g)'_{Hopf} \subset \mathcal U(\g)'$, defined by $$\mathcal U(\g)'_{Hopf} = \{ \phi\in \mathcal U(\g)' \bigr|
\exists \text{ two-sided ideal } J\subset \mathcal U(\g) \text { such that } \phi(J)=0 \text{ and }\operatorname{codim} J < \infty \}.$$
The extended space $\mathfrak R(G_0)$ corresponds to a larger subalgebra of $\mathcal U(\g)'$, spanned by the matrix elements of all modules in the category $\mathcal O$.
Recall that the category $\mathcal O$ has enough projectives; we denote by $P_\lambda$ the indecomposable projective cover of the irreducible module $V_\lambda$. It is known that every indecomposable module in the category $\mathcal O$ with integral weights is isomorphic to a subfactor of the projective module, corresponding to some anti-dominant integral weight $\lambda$. In particular, this means that it suffices to consider the matrix elements of the big projective modules $\{P_\lambda\}_{\lambda<0}$.
The following result can be regarded as a non-semisimple generalization of the Peter-Weyl theorem.
\[thm:generalized Peter-Weyl classical\] The space $\mathfrak R(G_0)$ of regular functions on $G_0$ is spanned by the matrix elements of all big projective modules in the category $\mathcal O$, $$\mathfrak R(G_0) \cong \bigoplus_{\lambda\in \mathbf P^+} \mathbb M(P_\lambda).$$ As a $\ggbar$-module, $\mathfrak R(G_0)$ is given by $$\mathfrak R(G_0) \cong \bigoplus_{\lambda\in-\mathbf P^{++}} \( P_{\lambda} \o P^\star_{\lambda} \) \biggr/ I_\lambda$$ where $I_\lambda$’s are the $\ggbar$-submodules of $P_{\lambda} \o P^\star_{\lambda}$, corresponding to identically vanishing matrix elements.
We use the realization of $\mathfrak R(G_0)$ in the Fock space $\mathbb F$. The inclusion provides the identification of $\mathbb F$ with a subspace of $\mathcal U(\g)'$. Since $\mathbb F$ is locally $\n_+ \oplus \n_+$-nilpotent, Proposition \[thm:matrix elements\] implies that for any $v \in \mathbb F$ there exists a $\g$-module $W \in \mathcal O$ such that $\vartheta_v \in \mathbb M(W).$
Let $W = W_1 \oplus W_2 \oplus \dots \oplus W_m$ be the decomposition of $W$ into a direct sum of indecomposable submodules. Each indecomposable component $W_i, \, i=1,\dots,m,$ is a subfactor of some big projective module $P_{\lambda_i}$. Then $\mathbb M(W_i) \subset \mathbb M(P_{\lambda_i})$, and therefore we have $$\mathbb M(W) = \mathbb M(W_1) + \mathbb M(W_2) + \dots + \mathbb M(W_m) \subset \bigoplus_{\lambda\in -\mathbf P^{++}} \mathbb M(P_\lambda),$$ which shows that $\vartheta(\mathbb F) \subset \bigoplus_{\lambda\in -\mathbf P^{++}} \mathbb M(P_\lambda).$ To prove that in fact $\vartheta(\mathbb F) = \bigoplus_{\lambda\in -\mathbf P^{++}} \mathbb M(P_\lambda)$, we compare the characters of the two spaces, and show that they have the same size.
For any $\lambda \in\mathbf P^+$ the $\ggbar$-module $\mathbb M(P_{-\lambda-2})$ is isomorphic to the quotient of the product $P_{-\lambda-2} \o P_{-\lambda-2}^\star$ by the kernel of the map $$\label{eq:matrix elements map}
\Theta_\lambda: P_{-\lambda-2} \o P_{-\lambda-2}^\star \to \mathcal U(\g)', \qquad \Theta_\lambda(v \o v') = \phi_{v,v'}.$$ Obviously, $I_\lambda = \ker \Theta_\lambda$ is a $\ggbar$-submodule of $P_{-\lambda-2} \o P_{-\lambda-2}^\star$; we describe it more explicitly. It is known that the module $P_{-\lambda-2}$ has a filtration $0 \subset P^{(0)} \subset P^{(1)} \subset P_{-\lambda-2}$ such that $$P^{(0)} \cong V_{-\lambda-2}, \qquad P^{(1)}/P^{(0)} \cong V_\lambda, \qquad P_{-\lambda-2}/P^{(1)} \cong V_{-\lambda-2},$$ and the dual filtration of the module $P_{-\lambda-2}^\star$ is given by $$0 \subset \Ann(P^{(1)}) \subset \Ann(P^{(0)}) \subset P_{-\lambda-2}^\star.$$ They determine a filtration of the tensor product $$\begin{split}
0 \subset P^{(0)} \o \Ann(P^{(1)}) \subset P^{(0)} \o \Ann(P^{(0)}) + P^{(1)} \o \Ann(P^{(1)}) \subset\\
\subset P^{(0)} \o P_{-\lambda-2}^\star + P^{(1)} \o \Ann(P^{(0)}) +
P_{-\lambda-2}\o \Ann(P^{(1)}) \subset\\
\subset P^{(1)} \o P_{-\lambda-2}^\star + P_{-\lambda-2} \o \Ann(P^{(0)})
\subset P_{-\lambda-2} \o P_{-\lambda-2}^\star.
\end{split}$$ If $v\in P^{(0)}$ and $v' \in \Ann(P^{(0)})$, then $\phi_{v,v'}$ is the zero functional, since for any $x \in \mathcal U(\g)$ we have $x\, v \in P^{(0)}$ and $\phi_{v,v'}(x) = \<v', x \, v\> = 0$. Hence the submodule $P^{(0)} \o \Ann(P^{(0)})$ lies in the kernel of the map $\Theta_\lambda$, and similarly does $P^{(1)} \o \Ann(P^{(1)})$. One can easily see that $$\Theta_\lambda \( P^{(1)} \o \Ann(P^{(0)}) \) = \mathbb M(V_\lambda),$$ $$\Theta_\lambda \( P^{(0)} \o P_{-\lambda-2}^\star \) = \Theta_\lambda \( P_{-\lambda-2}\o \Ann(P^{(1)})\) = \mathbb M(V_{-\lambda-2}).$$
It follows that the $\ggbar$-module $\mathbb M(P_{-\lambda-2})$ has a filtration $$0 \subset \mathbb M^{(0)} \subset \mathbb M^{(1)} \subset \mathbb M^{(2)} = \mathbb M(P_{-\lambda-2})$$ such that $$\begin{aligned}
\mathbb M^{(2)}/\mathbb M^{(1)} & \cong V_{-\lambda-2} \o V^\star_{-\lambda-2},\\
\mathbb M^{(1)}/\mathbb M^{(0)} & \cong ( V_\lambda \o V^\star_{-\lambda-2} ) \oplus ( V_{-\lambda-2} \o V^\star_\lambda ),\\
\mathbb M^{(0)} & \cong ( V_\lambda \o V^\star_\lambda ) \oplus ( V_{-\lambda-2} \o V^\star_{-\lambda-2} ).\end{aligned}$$
Thus, the block $\mathbb F(\lambda)$ of is identified with the subspace, spanned by the matrix elements of the big projective module $P_{-\lambda-2}$. Taking direct sums over all $\lambda \in \mathbf P^+$, adding the $\ggbar$-module $\mathbb M(P_{-1}) \cong V_{-1} \o V^\star_{-1}$, and comparing with Theorem \[thm:classical bimodule structure\], we see that $\bigoplus_{\lambda\in - \mathbf P^{++}} \mathbb M(P_\lambda)$ and $\mathbb F$ have the same characters. The statement of the theorem follows.
Cohomology of $\g$ with coefficients in regular representations
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The algebra $\mathfrak R(G)$ contains the subalgebra $\mathfrak R(G)^G$ of the conjugation-invariant functions on $G$, which is linearly spanned by the characters of the irreducible finite-dimensional representations. The subalgebra $\mathfrak R(G)^G$ is thus isomorphic to the Grothendieck ring of the finite-dimensional representations of $G$.
There is an isomorphism $\mathfrak R(G)^G \cong \C[\mathbf P]^W$, obtained by restricting the group characters to $\h$ and taking its Fourier expansion. Finally, the algebra $\mathfrak R(G)^G$ also admits a cohomological interpretation, which will be instrumental for further generalizations to the regular representations of the affine and Virasoro algebras. We briefly recall the definition of the cohomology of $\g$.
Let $\boldsymbol\Lambda = \bigwedge \g'$ be the exterior algebra of $\g'$ with unit $\1$. Then
1. The Clifford algebra, generated by $\{\iota(g), \, \eps(g')\}_{g \in \g, g' \in \g'}$ with relations $$\label{eq:Clifford relations}
\{\iota(x), \iota(y) \} = \{ \eps(x'), \eps(y') \} = 0, \qquad
\{\iota(x), \eps(y')\} = \<y',x\>,$$ acts irreducibly on $\boldsymbol\Lambda$, so that for any $\omega \in \boldsymbol\Lambda$ we have $$\iota(g) \1 = 0, \qquad \eps(g') \omega = g'\wedge\omega, \qquad g \in \g,\, g' \in \g', \, \omega \in \boldsymbol \Lambda.$$
2. $\boldsymbol\Lambda$ is a commutative superalgebra, $$\omega_1 \wedge \omega_2 = (-1)^{|\omega_1| \cdot |\omega_2|}\, \omega_2 \wedge \omega_1, \qquad \omega_1,\omega_2 \in \boldsymbol\Lambda,$$ where $|\cdot|$ is the natural grading on $\boldsymbol \Lambda$ satisfying $|\1| = 0, \ |\iota(g)| = -1, \ |\eps(g')| = 1.$
3. The $\g$-module structure on $\boldsymbol\Lambda$ is given by $$\pi_{\boldsymbol\Lambda}(x) = \sum_i \eps(g'_i) \iota([g_i,x]),$$ where $\{g_i\}$ is any basis of $\g$, and $\{g'_j\}$ is the corresponding dual basis of $\g'$.
The cohomology $H^\bullet(\g;V)$ of $\g$ with coefficients in a $\g$-module $V$ is the cohomology of the graded complex $C^\bullet(\g;V) = \boldsymbol\Lambda^\bullet \otimes V$, with the differential $$\label{eq:finite differential}
\mathbf d =\sum_i \eps(g'_i) \pi_V(g_i) - \frac12 \sum_{i,j} \eps(g'_i) \eps(g'_j) \iota([g_i,g_j]),$$ where $\{g_i\}$ is any basis of $\g$, and $\{g'_i\}$ is the dual basis of $\g'$.
The following is one of the fundamental results in Lie algebra cohomology, (see e.g. [@HS]).
\[thm:vanishing classical cohomology\] For any finite-dimensional $\g$-module $V$ we have $$\label{eq:de Rham}
H^\bullet(\g; V) \cong V^\g \o H_{DR}^\bullet(G),$$ where $H_{DR}^\bullet(G)$ denotes the holomorphic de Rham cohomology $H_{DR}^\bullet(G)$ of the Lie group $G$.
If $V$ is a commutative algebra with a compatible $\g$-action, then its cohomology inherits the multiplication from $V$ and $\boldsymbol\Lambda$, and $H^\bullet(\g;V)$ becomes itself a commutative superalgebra. Moreover, the isomorphism becomes an isomorphism of superalgebras, with respect to the cup product in $H_{DR}^\bullet(G)$.
The diagonal $\g$-action in $\mathbf F$ corresponds to the coadjoint action of $G$ in $\mathfrak R(G)$; thus, we get
There is an isomorphism of commutative superalgebras $$H^\bullet(\g;\mathbf F) = \C[\mathbf P]^W \o H_{DR}^\bullet(G).$$
Our next goal is to study the cohomology of $\g$ with coefficients in the extended regular representation $\mathbb F \cong \mathfrak R(G_0)$. For infinite-dimensional $\g$-modules Theorem \[thm:vanishing classical cohomology\] does not hold, and the cohomology $H^\bullet(\g;\mathbb F)$ does not reduce to $\mathbb F^\g \o H^\bullet_{DR}(G)$. We have instead
\[thm:classical cohomology\] There is an isomorphism of commutative superalgebras $$H^\bullet(\g;\mathbb F) \cong \C[\mathbf P]^W \o \sideset{}{^\bullet}\bigwedge \C^2.$$
It is easy to show using the results of [@W] that for $\lambda \ge -1$ $$H^n(\g;V_\lambda \o V^\star_{-\lambda-2}) = H^n(\g;V_{-\lambda-2} \o V^\star_\lambda) =
\begin{cases}
\C, & n=1,2\\
0, & \text{otherwise}
\end{cases}$$ and that for $\lambda \ge0$ we have $H^n(\g;V_{-\lambda-2} \o V^\star_{-\lambda-2}) = 0 $ for all $n$. The spectral sequence associated with the filtration of Theorem \[thm:classical bimodule structure\] can be used to show that $$\label{eq:classical big cohomology}
H^n(\g;\mathbb F(-1)) =
\begin{cases}
\C, & n=1,2\\
0, & \text{otherwise}
\end{cases},\qquad
H^n(\g;\mathbb F(\lambda)) =
\begin{cases}
\C, & n=0,2\\
\C^2, & n=1\\
0, & \text{otherwise}
\end{cases}, \quad \lambda \ge0.$$ Also, this spectral sequence shows that we have a natural isomorphism $H^0(\g;\mathbb F) \cong H^0(\g;\mathbf F).$
To explicitly get the generators of the commutative superalgebra $H^\bullet(\g;\mathbb F)$, we pick nonzero elements $$\chi \in H^0(\g;\mathbb F(1)), \qquad
\xi_{-1} \in H^1(\g;\mathbb F(-1)), \qquad
\eta_0 \in H^1(\g;\mathbb F(0)),$$ such that $\eta_0$ is not proportional to $\chi \, \xi_{-1}$. It is known that $H^0(\g;\mathbb F) \cong \C[\mathbf P]^W$ is isomorphic to the polynomial algebra $\C[\chi]$. It is also clear that $H^\bullet(\g;\mathbb F)$ is a free $\C[\chi]$-module. For each $\lambda\ge0$, the set $$B_{\le \lambda} = \{\xi_{-1}, \chi \xi_{-1}, \dots, \chi^{\lambda+1} \, \xi_{-1}\} \bigcup \{\eta_0,\chi \, \eta_0, \dots, \chi^\lambda \, \eta_0\}$$ consists of $2\lambda+3$ linearly independent elements, and in view of is a basis of $H^1(\g;\mathbb F_{\le \lambda})$. Finally, one can check that $\eta_0 \, \xi_{-1} \ne 0$, and thus the elements $\{\eta_0 \, \xi_{-1}, \chi \, \eta_0 \, \xi_{-1},\dots, \chi^{\lambda+1}\, \eta_0 \, \xi_{-1}\}$ give a basis of $H^2(\g;\mathbb F_{\le\lambda})$ for each $\lambda \ge -1$.
It follows that $H^\bullet(\g;\mathbb F) \cong \C[\chi] \otimes \bigwedge^\bullet[\xi_{-1},\eta_0]$, and the theorem is proven.
One of the ingredients in the exterior algebra part of the cohomology $H^\bullet(\g;\mathbb F)$ is the exterior algebra $\bigwedge^\bullet \h$, corresponding to $\bigwedge^\bullet[\eta_0]$ above. It would be interesting to obtain an invariant characterization of the remaining part of $H^\bullet(\g;\mathbb F)$ for arbitrary $\g$.
In each of the two-dimensional spaces $H^1(\g;\mathbb F(\lambda))$ there is a unique up to proportionality cohomology class $\xi_\lambda$ divisible by $\xi_{-1}$; the elements $\frac {\xi_\lambda}{\xi_{-1}}$ constitute a basis of $H^0(\g;\mathbb F) \cong \C[\mathbf P]^W$, associated with the characters of big projective modules (cf. [@La]).
Modified regular representations of the affine Lie algebra $\slhat$.
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Regular representations of $\slhat$
-----------------------------------
Let $\hat G$ be the central extension of the loop group $LG$, associated with $G=SL(2,\C)$ (see [@PS]), and let $\ghat$ be the corresponding Lie algebra. As we discussed in the introduction, there is no maximal cell in the affine Bruhat decomposition, and thus we will use the loop version of the finite-dimensional one. An additional advantage is that we get an explicit realization of the left and right regular $\ghat$-actions, analogous to the finite-dimensional case.
The standard basis of $\ghat$ consists of the elements $\{\mathbf e_n, \mathbf h_n, \mathbf f_n\}_{n \in \Z}$ and the central element $\mathbf k$, subject to the commutation relations $$\begin{gathered}
[\mathbf h_m, \mathbf e_n] = 2 \mathbf e_{m+n}, \qquad
[\mathbf h_m, \mathbf f_n] = -2 \mathbf f_{m+n}, \qquad
[\mathbf h_m, \mathbf h_n] = 2 m \delta_{m+n,0} \mathbf k,\\
[\mathbf e_m, \mathbf f_n] = \mathbf h_{m+n} + m \, \delta_{m+n,0} \mathbf k, \qquad
[\mathbf e_m, \mathbf e_n] = [\mathbf f_m, \mathbf f_n] = 0.\end{gathered}$$
The Lie algebra $\ghat$ has a $\Z$-grading $\ghat = \bigoplus_{n \in \Z} \ghat[n]$, determined by $$\deg \mathbf f_n = \deg \mathbf h_n = \deg \mathbf e_n = -n, \qquad \deg \mathbf k = 0,$$ We introduce subalgebras $\ghat_\pm = \bigoplus_{\pm n>0} \g[n]$; the finite-dimensional Lie algebra $\g$ is naturally identified with a subalgebra in $\ghat[0]$.
The element $\mathbf {w_0}$ of the classical Weyl group defines an involution $\hat\omega$ of $\ghat$, such that $$\label{eq:affine Cartan automorphism}
\hat\omega(\mathbf e_n) = - \mathbf f_n, \quad
\hat\omega(\mathbf h_n) = - \mathbf h_n, \quad
\hat\omega(\mathbf f_n) = - \mathbf e_n, \quad
\hat\omega(\mathbf k) = - \mathbf k.$$
We use the loop version of the finite-dimensional Bruhat decomposition , and factorize the central extension $\widehat{LT}$ into the product of loops that extend holomorphically inside and outside of the unit circle. The analogue of is the formal decomposition $$g = \exp \( \sum_{n\in\Z} x_n \mathbf e_n \) \ \mathbf{w_0}\, \tau^\mathbf k \
\exp \( \sum_{m < 0} \zeta_m \mathbf h_m \) \zeta^{\mathbf h_0} \exp \( \sum_{m > 0} \zeta_m \mathbf h_m \) \exp \( \sum_{n\in\Z} y_n \mathbf e_n \).$$ The polynomial algebra $\mathfrak R_0(\hat G_0) = \C[\{x_n\}, \{y_n\}, \{\zeta_{n\ne0}\},\zeta^{\pm1}]$ can be thought of as the algebra of regular functions on the big cell of the loop group, and for $\mathfrak R(\hat G_0)$ we get $$\mathfrak R(\hat G_0) = \mathfrak R_0(\hat G_0) \o \C[\tau^{\pm1}] = \bigoplus_{\varkappa \in \Z} \mathfrak R_\varkappa(\hat G_0), \qquad
\mathfrak R_\varkappa(\hat G_0) = \mathfrak R_0(\hat G_0) \o \C\tau^\varkappa$$
Note that for each $\varkappa$ the subspace $\mathfrak R_\varkappa(\hat G_0)$ is a $\ghatghat$-submodule of $\mathfrak R(\hat G_0)$, but it is not a subalgebra of $\mathfrak R(\hat G_0)$ when $\varkappa \ne 0$ ! It is easy to see that the infinitesimal regular $\ghat$-actions of the central element $\mathbf k$ on $\mathfrak R_\varkappa(\hat G_0)$ are given by $$\label{eq:unmodified central charges}
\pi_l(\mathbf k) = - \varkappa \cdot \Id, \qquad\qquad \pi_r(\mathbf k) = \varkappa \cdot \Id.$$ As vector spaces, all $\mathfrak R_\varkappa(\hat G_0)$ are identified with the same polynomial space, and one can compute the infinitesimal regular actions of $\ghat$ by treating $\varkappa$ as a complex parameter. In particular, the regular actions of $\ghat$ make sense for arbitrary $\varkappa \in \C$. Computations yield the following description, analogous to Proposition \[thm:classical action\].
\[thm:affine action\] The regular actions of $\ghat$ on $\mathfrak R_\varkappa(\hat G_0)$ are given by and $$\label{eq:left affine action}
\begin{split}
\pi_l(\mathbf e_n) &= -\pd{x_n},\\
\pi_l(\mathbf h_n) &= - 2\sum_{i\in\Z} x_i \pd{i_{n+n}} +
\begin{cases}
\pd{\zeta_n} + 2n \varkappa\, \zeta_{-n}, & n>0\\
\zeta \, \pd {\zeta}, & n=0\\
\pd{\zeta_n}, & n<0
\end{cases} ,\\
\pi_l(\mathbf f_n) &= \sum_{i,i' \in \Z} x_i x_{i'} \pd{x_{i+i'+n}} -
\sum_{j<0} x_{j-n}\pd{\zeta_j} - x_{-n} \zeta\, \pd {\zeta} - \sum_{j>0} x_{j-n} \(\pd{\zeta_j} + 2 j \varkappa\, \zeta_{-j} \) - \\
&- \varkappa \; n x_{-n} + \zeta^{-2} \sum_{j,j'>0} \mathrm s_{j'}(-2\zeta_1,-2\zeta_2,\dots) \, \mathrm s_j(-2\zeta_{-1},-2\zeta_{-2},\dots) \pd{y_{n-j+j'}},
\end{split}$$
$$\label{eq:right affine action}
\begin{split}
\pi_r(\mathbf e_n) &= \pd{y_n},\\
\pi_r(\mathbf h_n) &= -2\sum_{i\in\Z} y_i \pd{y_{i+n}} +
\begin{cases} \pd{\zeta_n} & n>0, \\ \zeta \, \pd {\zeta}, & n=0,\\ \pd{\zeta_n} - 2 n \varkappa\, \zeta_{-n}, & n<0. \end{cases} ,\\
\pi_r(\mathbf f_n) &= - \sum_{i,i' \in \Z} y_i y_{i'} \pd{y_{i+i'+n}} +
\sum_{j>0} y_{j-n}\pd{\zeta_j} + y_{-n} \zeta\, \pd {\zeta} + \sum_{j<0} y_{j-n} \(\pd{\zeta_j} - 2 j \varkappa\, \zeta_{-j} \) - \\
&- \varkappa \, n y_{-n} - \zeta^{-2} \sum_{j,j'>0} \mathrm s_{j'}(-2\zeta_{-1},-2\zeta_{-2},\dots) \, \mathrm s_j(-2\zeta_1,-2\zeta_2,\dots) \pd{x_{n+j-j'}},\\
\end{split}$$
where the Schur polynomials $\mathrm s_k(\a_1,\a_2,\dots)$ are defined by $$\mathrm s_m(\a_1,\a_2,\dots) = \sum_{\substack {l_1,l_2, \ldots \ge 0\\ l_1+2l_2 + \ldots = m}}
\frac {\a_1^{l_1} \a_2^{l_2} \dots}{l_1! l_2!\dots}.$$
The presence of the central extension requires the use of some elementary cases of the Campbell-Hausdorff formula in our computations; we use the identity $$\exp(B) \exp(tA) \equivt \exp \(t \sum_{j=0}^\infty \frac 1{j!} \underbrace{[B,\dots,[B,[B,A]]\dots]}_{j \text{ commutators}}\) \exp(B).$$ For example, to derive the last of , we use the formulas: $$\begin{split}
\exp \( \sum_{i\in\Z} y_i \mathbf e_i \) \exp \( t \mathbf f_n \) \equivt &
\exp \(t \mathbf f_n \)
\exp \(- t n y_{-n}\mathbf k + t \sum_{i\in\Z} y_i \mathbf h_{i+n} \) \times \\
&\times \exp \(-t \sum_{i,i'\in\Z} y_i y_{i'} \mathbf e_{i+i'+n} \)
\exp \( \sum_{i\in\Z} y_i \mathbf e_i \),\\
\exp \(\sum_{m>0} \zeta_m \mathbf h_m \) \exp \( t \mathbf f_n \) \equivt &
\exp \( t \sum_{j>0} \mathrm s_j(-2\zeta_1,-2\zeta_2,\dots) \mathbf f_{n+j} \)
\exp \( \sum_{m>0} \zeta_m \mathbf h_m \),\\
\zeta^{\mathbf h_0} \exp \( t \mathbf f_n \) \equivt &
\exp \( t \zeta^{-2}\, \mathbf f_n \) \zeta^{\mathbf h_0},\\
\exp \(\sum_{m<0} \zeta_m \mathbf h_m \) \exp \(t \mathbf f_n \) \equivt &
\exp \(t \sum_{j'>0} \mathrm s_{j'}(-2\zeta_{-1},-2\zeta_{-2},\dots) \mathbf f_{n-j'} \)
\exp \( \sum_{m<0} \zeta_m \mathbf h_m \),\\
\mathbf {w_0} \exp \( t \mathbf f_n \) \equivt &
\exp \(- t \, \mathbf e_n \) \mathbf {w_0}.
\end{split}$$ Combining these equations, we get the desired formulas. We leave the technical calculations to the reader.
Vertex operator algebras: review and useful examples
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We aim to endow $\mathfrak R_\varkappa(\hat G_0)$ (or its modification) with a structure similar to that of an associative commutative algebra. The relevant formalism is provided by the vertex algebra theory.
We recall the definitions of vertex and vertex operator algebras in the most convenient to us form. For more details and equivalent alternative definitions, we refer the reader to the books on the subject [@FLM; @BFr].
Let $\mathcal V$ be a vector space, equipped with a linear correspondence $$\label{eq:state field correspondence}
v \mapsto \mathcal Y(v,z) = \sum_{n \in \Z} v_{(n)} z^{-n-1}, \qquad v_{(n)} \in \End(\mathcal V).$$ We refer to such formal $\End(\mathcal V)$-valued generating functions as ’quantum fields’.
We say that $\mathcal V$ satisfies the locality property, if for any $a,b \in \mathcal V$ $$\label{eq:locality}
(z-w)^N [\mathcal Y(a,z),\mathcal Y(b,w)] = 0 \quad \text{ for } N \gg 0$$ in the ring of $\End(\mathcal V)$-valued formal Laurent series in two variables $z,w$.
A vector $\1 \in \mathcal V$ is called the vacuum vector, if it satisfies $$\label{eq:vacuum}
\mathcal Y(\1,z) = \Id_{\mathcal V}, \qquad \mathcal Y(v,z)\1 \bigr|_{z=0} = v.$$
An element $\mathcal D \in \End(\mathcal V)$, is called the infinitesimal translation operator, if it satisfies $$\label{eq:infinitesimal translation}
\qquad \mathcal D \, \1 = 0, \qquad\qquad [\mathcal D,\mathcal Y(v,z)] = \frac d{dz} \mathcal Y(v,z), \qquad \text{ for all } \ v \in \mathcal V.$$
The space $\mathcal V$ is called a vertex algebra, if it is equipped with a linear map , vacuum vector $\1$, and infinitesimal translation operator $\mathcal D$, satisfying the axioms , , above.
Vertex superalgebras are defined as usual by inserting $\pm$ signs according to parity. A vertex superalgebra $\mathcal V$ is called bi-graded, if it has $\Z$-gradings, $|\cdot|$ and $\deg$, $$\mathcal V = \bigoplus_{m,n\in\Z} \mathcal V^m[n], \qquad
\mathcal V^m[n] = \left\{ v \in\mathcal V \, \biggr| \, |v| = m \text{ and } \deg v = n \right\},$$ such that the parity in superalgebra is determined by $|\cdot|$, and for any homogeneous $v$ $$v \mapsto \mathcal Y(v,z) = \sum_{n\in\Z} v_{(n)} z^{-n-1}, \qquad \text { with } |v_{(n)}| = |v| \text{ and } \deg v_{(n)} = \deg v - n - 1 .$$ In particular, for the vacuum we must have $| \1 | = \deg \1 = 0$. Also, we write $|\mathcal Y(v,z)| = |v|$ and $\deg \mathcal Y(v,z) = \deg v$ for the quantum field $\mathcal Y(v,z)$, if the above conditions are satisfied.
A vertex algebra $\mathcal V$ is called a vertex operator algebra (VOA) of rank $c \in \C$, if there exists an element $\boldsymbol\omega \in \mathcal V$, usually called the Virasoro element, such that the operators $\{\mathcal L_n\}_{n \in \Z}$ defined by $$\mathcal Y(\boldsymbol\omega,z) = \sum_{n \in \Z} \mathcal L_n z^{-n-2},$$ satisfy $\mathcal L_{-1} = \mathcal D$, and the Virasoro commutation relations $$[\mathcal L_m,\mathcal L_n] = (m-n) \mathcal L_{m+n} + \delta_{m+n,0} \frac{m^3-m}{12}\, c.$$
We define the the normal ordered product of two quantum fields $X(z)$ and $Y(z)$ by $$:X(z)Y(w): = X_-(z)Y(w) + Y(w) X_+(z),$$ where $X_\pm(z)$ are the regular and principal parts of $X(z) = \sum_{n \in \Z} X_{(n)} z^{-n-1}$, $$X_+(z) = \sum_{n\ge0} X_{(n)} z^{-n-1}, \qquad X_-(z) = \sum_{n < 0} X_{(n)} z^{-n-1}.$$ For products of three or more quantum fields, the normal ordered product is defined inductively, starting from the left. In general, the normal ordered product is neither commutative nor associative.
The following ’reconstruction theorem’ is an effective tool for constructing vertex algebras.
\[thm:free field construction\] Let $\mathcal V$ be a vector space with a distinguished vector $\1$ and a family of pairwise local $\End(\mathcal V)$-valued quantum fields $\{X^\a(z)= \sum_{n \in \Z}X^\a_{(n)} z^{-n-1}\}_{\a \in \mathfrak I}.$ Suppose $\mathcal V$ is generated from $\1$ by the action of the Laurent coefficients of quantum fields $X^\a(w)$, and that the vectors $\{X^\a(z) \1 \bigr|_{z=0}\}_{\a \in \mathfrak I}$ are linearly independent in $\mathcal V$. Then the operators $$\mathcal Y\(X^{\a_1}_{(-n_1-1)}\dots X^{\a_k}_{(-n_k-1)} \1,z\) =
\ :X^{\a_1}(z)^{(n_1)} \dots X^{\a_k}(z)^{(n_k)}:,$$ where $X(z)^{(n)} = \frac 1{n!} \frac {d^n}{dz^n} X(z)$, satisfy and .
If a linear operator $\mathcal D \in \End(\mathcal V)$ satisfies $\mathcal D \1 = 0$ and $[\mathcal D,X^\a(z)] = \frac d{dz} X^\a(z)$ for every $\a \in \mathfrak I,$ then $[\mathcal D,\mathcal Y(v,z)] = \frac d{dz} \mathcal Y(v,z)$ for any $v \in \mathcal V$.
We say that a vertex algebra $\mathcal V$ has a PBW basis, associated with quantum fields $\{X^\a(z)\}_{\a\in\mathfrak I}$, if the index set $\mathfrak I$ is ordered, and we have a linear basis of $\mathcal V$, formed by the vectors $$\left\{X^{\a_1}_{(-n_1-1)}\dots X^{\a_k}_{(-n_k-1)} \1 \, \biggr| \,
n_1 \ge n_2 \ge \dots \ge n_k \ge 0, \text { and if } n_i=n_{i+1}, \text { then } \a_i\preceq\a_{i+1}\right\}.$$
For two mutually local quantum fields $X(z),Y(w)$ we introduce the operator product expansion (OPE) formalism, and write $$X(z)Y(w) \sim \sum_{j} \frac {C_j(w)}{(z-w)^j},$$ if for a finite collection of quantum fields $\{C_j(w)\}_{j=1,2,\dots}$ we have the equality $$X(z)Y(w) = \sum_{j} \frac {C_j(w)}{(z-w)^j} \; + :X(z)Y(w):$$ where $\frac 1{(z-w)^j}$ should be expanded into the Laurent series in non-negative powers of $\frac wz$. The importance of OPE lies in the fact that all commutators $[X_m,Y_n]$ of Laurent coefficients of quantum fields $X(z),Y(w)$ are completely encoded by the collection $\{C_j(w)\}$.
The remainder of this subsection presents some examples of vertex algebras, which will be used in this paper. All of these algebras are bi-graded and have a PBW basis associated with given quantum fields, for which we specify the OPEs.
We denote by $\hat F(\beta,\gamma)$ the vertex algebra generated by quantum fields $$\begin{aligned}
{5}
\beta(z) &= \sum_{n \in \Z} \beta_n z^{-n}, \qquad &|\beta(z)| &= 0, \qquad \deg \beta(z) &= 0, \\
\gamma(z) &= \sum_{n \in \Z} \gamma_n z^{-n-1}, \qquad &|\gamma(z)| &= 1, \qquad \deg \gamma(z) &= 0,\end{aligned}$$ with the operator product expansions $$\label{eq:beta-gamma OPE}
\beta(z)\gamma(w) \sim \frac 1{z-w}, \qquad \beta(z) \beta(w) \sim \gamma(z)\gamma(w) \sim 0.$$ The commutation relations for the underlying Heisenberg algebra are $$\label{eq:affine beta-gamma commutation}
[\beta_m,\gamma_n] = \delta_{m+n,0}, \qquad [\beta_m,\beta_n] = [\gamma_m,\gamma_n]=0.$$
We denote by $\hat\Lambda(\psi,\psi^*)$ the vertex superalgebra generated by quantum fields $$\begin{aligned}
{5}
\psi(z) &= \sum_{n\in\Z} \psi_n z^{-n-1}, \qquad & |\psi(z)| &= -1, \qquad &\deg \psi(z) &= 1,\\
\psi^*(z) &= \sum_{n\in\Z} \psi^*_n z^{-n}, \qquad & |\psi^*(z)| &= 1, \qquad &\deg \psi^*(z) &= 0, \end{aligned}$$ with the operator product expansions $$\psi(z)\psi(w) \sim \psi^*(z)\psi^*(w) \sim 0, \qquad \psi(z) \psi^*(w) \sim \frac 1{z-w}.$$ The (anti)-commutation relations for the underlying Clifford algebra are $$\label{eq:psi system relations}
\{\psi_m, \psi_n \} = \{ \psi^*_m, \psi^*_n \} = 0, \qquad
\{\psi_m, \psi^*_n\} = \delta_{m+n,0}.$$
We denote by $\ghat_k$ the vertex algebra generated by quantum fields $$X_n = \sum_{n \in \Z} X_n z^{-n-1}, \qquad |X(z)| = 0, \qquad \deg X(z) = 1, \qquad X \in \g,$$ with the operator product expansions $$X(z)Y(w) \sim \frac {[X,Y](w)}{z-w} + k \, \frac {\<X,Y\>}{(z-w)^2},\qquad k \in \C$$ where $\<\cdot,\cdot\>$ is the Killing form on $\g$. The number $k$ is called the level of $\ghat_k$.
We note that a module for the vertex algebra $\ghat_k$ is a $\Z$-graded $\ghat$-module $\hat V = \bigoplus_{n \ge n_0} \hat V[n]$, such that $\pi_{\hat V} (\mathbf k) = k \cdot \Id_{\hat V}$ and $\ghat[m] \hat V[n] \subset \hat V[m+n]$ for any $m,n \in \Z$.
We denote by $\vir_c$ the vertex algebra generated by the quantum field $$L(z) = \sum_{n\in\Z} L_n z^{-n-2}, \qquad |L(z)| = 0, \qquad \deg L(z) = 2,$$ with the operator product expansion $$L(z)L(w) \sim \frac {c/2}{(z-w)^4} + \frac {2 L(w)}{(z-w)^2} + \frac{L'(w)}{z-w}, \qquad c \in \C.$$ The number $c$ is called the central charge of $\vir_c$.
A module for the vertex algebra $\vir_c$ is a $\Z$-graded $\vir$-module $\tilde V = \bigoplus_{n \ge n_0} \tilde V[n]$, such that $\pi_{\tilde V} (\mathbf c) = c \cdot \Id_{\tilde V}$ and $L_{-m} \tilde V[n] \subset \tilde V[m+n]$ for any $m,n \in \Z$.
We denote by $\hat F_\varkappa(a)$ the vertex algebra generated by the quantum field $$a(z) = \sum_{n \in \Z} a_n z^{-n-1}, \qquad |a(z)| = 0, \qquad \deg a(z) = 1,$$ with the operator product expansion $$\label{eq:a OPE}
a(z)a(w) \sim \frac {2\varkappa}{(z-w)^2}, \qquad \varkappa \in \C.$$ The commutation relations for the underlying Heisenberg algebra $\mathcal H(a)$ are $$\label{eq:a commutation}
[a_m, a_n] = 2\varkappa \, m \, \delta_{m+n,0}.$$ Note that the operator $a_0$ is central and kills the vacuum.
Below we give the construction of a vertex algebra, which will be crucial for our future considerations. Let $\hat F_{-\varkappa}(\bar a)$ be defined similarly to $\hat F_\varkappa(a)$, so that $$\label{eq:bar a commutation}
[\bar a_m, \bar a_n] = -2\varkappa \, m \,\delta_{m+n,0}, \qquad
\bar a(z) \bar a(w) \sim - \frac {2\varkappa}{(z-w)^2}.$$
\[thm:dual Fock vertex structure\] Let $\varkappa \ne 0$. The space $\tilde{\mathbb F}_\varkappa = \hat F_\varkappa(a) \o \hat F_{-\varkappa}(\bar a) \o \C[\mathbf P]$ has a vertex algebra structure, extending those of $\hat F_\varkappa(a)$ and $\hat F_{-\varkappa}(\bar a)$, and such that $a_0 \1_\lambda = \bar a_0 \1_\lambda = \lambda \1_\lambda$.
Introduce the quantum fields $\{\mathbb Y(\mu,w)\}_{\mu\in\mathbf P}$ by $$\label{eq:double vertex operator}
\begin{split}
\mathbb Y(\mu,z) & = \exp\( \frac {\mu}{2\varkappa}\sum_{n<0} \frac {a_n}{-n} z^{-n}\)
\exp\(\frac {\mu}{2\varkappa}\sum_{n>0} \frac {a_n}{-n} z^{-n}\) \times\\
& \times
\exp\( -\frac {\mu}{2\varkappa}\sum_{n<0} \frac {\bar a_n}{-n} z^{-n}\)
\exp\( -\frac {\mu}{2\varkappa}\sum_{n>0} \frac {\bar a_n}{-n} z^{-n}\) \, \1_\mu .
\end{split}$$ Straightforward computations lead to the operator product expansions $$a(z) \bar a(w) \sim \bar a(z) a(w) \sim \mathbb Y(\mu,z) \mathbb Y(\nu,w) \sim 0,$$ $$a(z)\mathbb Y(\mu,w)\sim \bar a(z)\mathbb Y(\mu,w) \sim \frac {\mu \, \mathbb Y(\mu,w) }{z-w},$$ and establish mutual pairwise locality for the quantum fields $a(z),\bar a(z), \mathcal Y(\mu,z)$.
The vacuum is, of course, the vector $\1 \o \1 \o \1_0 \in \mathbb F_\varkappa$. We set $\mathcal Y(\1_\lambda,z) = \mathbb Y(\lambda,z)$ for any $\lambda \in \mathbf P$. The spanning and linear independence conditions of Proposition \[thm:free field construction\] are immediate. Finally, we set $\mathcal D \1_\lambda = \frac \lambda{2\varkappa} (a_{-1} - \bar a_{-1}) \1_\lambda$. The conditions on $\mathcal D$ amount to $$\label{eq:double vertex derivative}
\mathbb Y'(\lambda,z) = \frac \lambda{2\varkappa} \, \biggr(:a(z)\mathbb Y(\lambda,z): - :\bar a(z)\mathbb Y(\lambda,z):\biggr),$$ which is checked directly. Applying Proposition \[thm:free field construction\], we get the desired statement.
Theorem \[thm:dual Fock vertex structure\] should be compared with the construction of lattice vertex algebras. It is known that the space $\hat F_\varkappa(a) \o \C[\mathbf P]$ carries a vertex algebra structure only for special values of $\varkappa$, satisfying certain integrality conditions.
Bosonic realizations
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We now proceed to study the generalizations of the algebra $\mathfrak R(G_0)$. As in the finite-dimensional case, we study modules for the Lie algebra $\ghatghat$, which is equivalent to having two commuting actions of $\ghat$ on the same space.
As in the classical case, the regular $\ghat$-actions on $\mathfrak R_\varkappa(\hat G_0)$, described in Theorem \[thm:affine action\], can be reformulated in terms of representations of Heisenberg algebras. We note that the operators $$\begin{aligned}
\beta_n &= -y_{-n}\\
\gamma_n &= \pd {y_n}
\end{aligned}, \qquad\qquad
a_n = \begin{cases}
\pd {\zeta_n}, & n> 0 \\
\zeta \pd {\zeta}, & n = 0 \\
\pd {\zeta_n} - 2 n \varkappa \, \zeta_{-n}, & n < 0
\end{cases}$$ satisfy the commutation relations ,, and similarly for $$\begin{aligned}
\bar\beta_n &= x_{-n},\\
\bar\gamma_n &= - \pd {x_{n}}
\end{aligned}, \qquad\qquad
\bar a_n = \begin{cases}
\pd {\zeta_n} + 2n \varkappa \, \zeta_{-n} & n> 0 \\
\zeta \pd {\zeta}, & n = 0\\
\pd {\zeta_{n}} & n < 0\\
\end{cases}.$$ Note also that $\C[\zeta^{\pm1}] \cong \C[\mathbf P]$, and $a_0 = \bar a_0 = a$ act on $\C[\mathbf P]$ by derivations $a \1_\lambda = \lambda \1_\lambda$.
The formulas of Theorem \[thm:affine action\] are particularly simple, when written for the generating series $\mathbf e(z), \mathbf h(z), \mathbf f(z)$. For example, becomes
$$\label{eq:unordered bosons}
\begin{split}
\pi_r(\mathbf e(z)) &= \gamma(z),\\
\pi_r(\mathbf h(z)) &= 2 \beta(z)\gamma(z) + a(z),\\
\pi_r(\mathbf f(z)) &= -\beta(z)^2\gamma(z) - \beta(z)a(z) - \varkappa \beta'(z) +
\exp\(\frac 1\varkappa \sum_{n \ne 0} \frac{a_n- \bar a_n}{n} z^{-n} \) \bar \gamma(z) \1_{-2}.
\end{split}$$
Note that in this polynomial realization the constants are annihilated by $\{\mathbf e_n\}_{n \in \Z}$ and $\{\mathbf h_n\}_{n\ge0}$. The vertex algebra formalism requires a different choice of vacuum, and the introduction of normal ordering to make products of quantum fields well-defined. This procedure is well-known in the theory of Wakimoto modules (see [@BFr] and references therein), for which the $\ghat$-action is constructed by modifying the formulas originating from the semi-infinite flag variety. In particular, one expects the shifts of the levels of the representations by the dual Coxeter number $h^\vee = 2$.
The modifications of the formulas leads to the following result.
\[thm:affine bimodule action\] Let $\varkappa \ne 0,$ and let $k = \varkappa - h^\vee$ and $\bar k = -\varkappa - h^\vee$, and let $$\hat{\mathbb F}_\varkappa = \hat F(\beta,\gamma) \o \hat F(\bar\beta,\bar\gamma) \o \tilde{\mathbb F}_\varkappa.$$
1. The space $\hat{\mathbb F}_\varkappa$ has a $\ghatKK$-module structure, defined by $$\label{eq:affine boson left}
\begin{split}
\mathbf e(z) &= \gamma(z),\\
\mathbf h(z) &= 2:\beta(z) \gamma(z): + a(z),\\
\mathbf f(z) &= -:\beta(z)^2 \gamma(z): - \beta(z) a(z) - k \beta'(z) + \mathbb Y(-2,z) \bar\gamma(z),
\end{split}$$ $$\label{eq:affine boson right}
\begin{split}
\bar {\mathbf e}(z) &= \bar\gamma(z),\\
\bar {\mathbf h}(z) &= 2:\bar\beta(z) \bar\gamma(z): + \bar a(z),\\
\bar {\mathbf f}(z) &= -:\bar\beta(z)^2 \bar\gamma(z): - \bar\beta(z) \bar a(z) - \bar k \bar\beta'(z) + \mathbb Y(-2,z)\gamma(z).
\end{split}$$
2. The space $\hat {\mathbb F}_\varkappa$ has a compatible VOA structure with $\operatorname{rank}\hat {\mathbb F}_\varkappa = 6$. (Compatible means that the operators $\mathcal Y(v,z)$ are $\ghatKK$-intertwining operators in the VOA sense).
Similar formulas for the two commuting actions of $\ghat$ were suggested in [@FP], by analogy with the finite-dimensional Gauss decomposition of $G$. However, in order to get a meaningful VOA structure - and the corresponding semi-infinite cohomology theory! - one must incorporate the twist by $\mathbf {w_0}$, built into the Bruhat decomposition.
One can recover the original Wakimoto realization from by properly discarding the ’bar’ variables. We use superscripts ’W’ to distinguish the Wakimoto $\ghat_k$-action from .
The space $\hat W_{\lambda,k} = \hat F(\beta,\gamma)\o \hat F_\varkappa(a) \o \C \1_\lambda$ has the structure of a $\ghat_k$-module with $k = \varkappa - h^\vee$, defined by the formulas $$\begin{split} \label{eq:Wakimoto}
\mathbf e^W(z) &= \gamma(z),\\
\mathbf h^W(z) &= 2:\beta(z) \gamma(z): + a(z),\\
\mathbf f^W(z) &= -:\beta(z)^2 \gamma(z): - \beta(z) a(z) - k\, \beta'(z).
\end{split}$$ The $\ghat_k$-module $\hat W_{\lambda,k}$ is called the Wakimoto module.
It suffices to show that modifying the standard Wakimoto actions by the extra terms $$\begin{aligned}
\delta\mathbf f(z) & = \mathbf f(z) - \mathbf f^W(z) = \mathbb Y(-2,z) \bar\gamma(z), \\
\overline{\delta\mathbf f}(z) & = \bar{\mathbf f}(z) - \bar{\mathbf f}^W(z) = \mathbb Y(-2,z) \gamma(z),\end{aligned}$$ does not destroy the operator product expansions.
We begin by showing that the commutation relations for $\ghat_k$ hold. Only those involving the modified quantum field $\mathbf f(z)$ need to be considered. We have: $$\begin{aligned}
\mathbf e^W(z) \, \delta\mathbf f(w) &= \gamma(z) \mathbb Y(-2,w) \bar\gamma(w) \sim 0,\\
\mathbf h^W(z) \, \delta\mathbf f(w) &= \(2 :\beta(z) \gamma(z): + a(z)\) \mathbb Y(-2,w) \bar\gamma(w) \sim \\
& \sim a(z) \mathbb Y(-2,w) \bar \gamma(w) \sim -\frac {2 \mathbb Y(-2,w)}{z-w} \bar\gamma(w) = - \frac{2 \,\delta\mathbf f(w)}{z-w},\\
\mathbf f^W(z) \, \delta\mathbf f(w) &= -\beta(z) a(z)\gamma(z) \mathbb Y(-2,w) \bar\gamma(w) \sim \frac {2 \, \mathbb Y(-2,w)}{z-w} \beta(w) \bar\gamma(w),\\
\delta\mathbf f(z) \, \delta\mathbf f(w) &= \mathbb Y(-2,z) \bar\gamma(z) \mathbb Y(-2,w) \bar\gamma(w) \sim 0.\end{aligned}$$ Using the operator product expansions above we immediately check that $$\begin{aligned}
\mathbf e(z) \mathbf f(w) &= \mathbf e^W(z) \mathbf f^W(w) + \mathbf e^W(z) \, \delta\mathbf f(w) \sim
\( \frac k{(z-w)^2} + \frac {\mathbf h^W(w)}{z-w} \) + 0 = \frac k{(z-w)^2} + \frac {\mathbf h(w)}{z-w},\\
\mathbf h(z) \mathbf f(w) &= \mathbf h^W(z) \mathbf f^W(w) + \mathbf h^W(z) \, \delta\mathbf f(w) \sim
- \frac {2 \, \mathbf f^W(w)}{z-w} + 0 = - \frac {2 \, \mathbf f(w)}{z-w},\\
\mathbf f(z) \mathbf f(w) &= \mathbf f^W(z) \mathbf f^W(w) + \mathbf f^W(z) \, \delta\mathbf f(w) +
\delta\mathbf f(z) \, \mathbf f^W(w) + \delta\mathbf f(z) \, \delta\mathbf f(w) \sim \\
&\sim 0 + \frac {2 \, \mathbb Y(-2,w)}{z-w} \beta(w) \bar\gamma(w) + \frac {2 \, \mathbb Y(-2,z)}{w-z} \beta(z) \bar\gamma(z) + 0 \sim 0,\end{aligned}$$ and since the operator product expansions not involving $\mathbf f(z)$ are unchanged, we have proved the commutation relations for the (left) $\ghat_k$-action. Similarly, one verifies the commutation relations for the (right) $\ghat_{\bar k}$-action.
We now prove that the two actions of $\ghat_k$ and $\ghat_{\bar k}$ commute. We have $$\begin{aligned}
\bar {\mathbf e}^W(z) \, \delta\mathbf f(w) &= \bar \gamma(z) \mathbb Y(-2,w) \bar\gamma(w) \sim 0,\\
\bar {\mathbf h}^W(z) \, \delta\mathbf f(w) &= \(2 :\bar \beta(z) \bar \gamma(z): + \bar a(z) \) \mathbb Y(-2,w) \bar\gamma(w) \sim \\
& \sim 2 \frac{ \bar\gamma(z) }{z-w} \mathbb Y(-2,w) - \frac {2 \mathbb Y(-2,w)}{z-w} \bar\gamma(w) \sim 0,\end{aligned}$$ which implies that $\bar{\mathbf e}(z) \mathbf f(w) \sim \bar{\mathbf h}(z) \mathbf f(w) \sim 0.$ Finally, we compute $$\begin{aligned}
\bar{\mathbf f}^W(z) \, \delta\mathbf f(w) &= \biggr( -:\bar\beta(z)^2\bar\gamma(z): - \bar k \bar\beta'(z) -
\bar\beta(z) \bar a(z) \biggr) \biggr(\mathbb Y(-2,w) \bar\gamma(w) \biggr) \sim \\
& \sim -2 \, \frac{\bar\beta(z) \bar\gamma(z)}{z-w} \mathbb Y(-2,w) + \frac {\bar k}{(z-w)^2} \mathbb Y(-2,w) -\\
& - \( - \frac {2 \mathbb Y(-2,w)}{z-w} :\bar\beta(z) \bar\gamma(w): +
\frac { :\bar a(w) \mathbb Y(-2,w):}{z-w} - \frac {2 \mathbb Y(-2,w)}{(z-w)^2} \) \sim \\
& \sim \frac {(\bar k+2) \mathbb Y(-2,w)} {(z-w)^2} - \frac {:\bar a(w)\mathbb Y(-2,w):}{z-w} =
- \varkappa \frac {\mathbb Y(-2,w)} {(z-w)^2} - \frac {:\bar a(w)\mathbb Y(-2,w):}{z-w}.\end{aligned}$$ and similarly $$\overline{\delta\mathbf f}(z) \mathbf f^W(w) \sim
\varkappa \frac {\mathbb Y(-2,w)} {(z-w)^2} + \frac {:\bar a(w) \mathbb Y(-2,w):}{z-w} \sim
- \, \bar{\mathbf f}^W(z) \, \delta\mathbf f(w).$$ Therefore, $$\bar{\mathbf f}(z) \mathbf f(w) =
\bar{\mathbf f}^W(z) \mathbf f^W(w) + \bar{\mathbf f}^W(z) \, \delta\mathbf f(w) +
\overline{\delta\mathbf f}(z) \, \mathbf f^W(w) + \overline{\delta\mathbf f}(z) \, \delta\mathbf f(w) \sim 0,$$ and we have established the commutativity of the two $\ghat$-actions.
Proposition \[thm:free field construction\] implies that $\hat {\mathbb F}_\varkappa$ is a vertex algebra. The formulas can be written as $$\begin{split}
\mathbf e(z) &= \mathcal Y(\gamma_{-1}\1_0,z), \\
{\mathbf h}(z) &= \mathcal Y(2\beta_0\gamma_{-1}\1_0 + a_{-1}\1_0,z), \\
\mathbf f(z) &= \mathcal Y(-(\beta_0)^2\gamma_{-1}\1_0 - a_{-1}\beta_0\1_0 - k \beta_{-1}\1_0 - \bar\gamma_{-1}\1_{-2},z),
\end{split}$$ which means that the quantum fields are special cases of the operators $\mathcal Y(\cdot,z)$. The same is true for the quantum fields . Therefore, the vertex algebra structure is compatible (in the vertex algebra sense) with the $\ghatKK$-module structure on $\hat{\mathbb F}_\varkappa$.
To give $\hat{\mathbb F}_\varkappa$ a VOA structure we need to introduce the Virasoro element. The Sugawara construction for the affine algebra $\ghat_k$ gives a Virasoro quantum field with central charge $c = \frac {3k}{k+h^\vee} = 3 - \frac 6\varkappa$: $$\label{eq:Sugawara 1}
\begin{split}
L(z) & = \frac 1{2\varkappa} \( \frac{:\mathbf h^2(z):}2 + :\mathbf e(z)\mathbf f(z): + :\mathbf f(z)\mathbf e(z):\) = \\
& = \frac {:a(z)^2:}{4\varkappa} - \frac {a'(z)}{2\varkappa} \, - :\beta'(z) \gamma(z): +
\frac 1\varkappa \, \mathbb Y(-2,z) \gamma(z) \bar\gamma(z).
\end{split}$$ We also note that $$L(z) = L^W(z) - \frac 1\varkappa \, \mathbb Y(-2,z) \gamma(z) \bar\gamma(z),$$ where $L^W(z)$ is the Virasoro quantum field given by the Sugawara construction for the standard Wakimoto realization .
Similarly, the affine algebra $\ghat_{\bar k}$ produces another Virasoro quantum field with central charge $\bar c = \frac {3\bar k}{\bar k+h^\vee} = 3 + \frac 6\varkappa$: $$\label{eq:Sugawara 2}
\begin{split}
\bar L(z) & = - \frac 1\varkappa
\( \frac{:\bar {\mathbf h}^2(z):}2 + :\bar{\mathbf e}(z)\bar{\mathbf f}(z): + :\bar{\mathbf f}(z)\bar{\mathbf e}(z):\) = \\
& = - \frac {:\bar a(z)^2:}{4\varkappa} + \frac {\bar a'(z)}{2\varkappa} \, - :\bar \beta'(z) \bar \gamma(z): -
\frac 1\varkappa \, \mathbb Y(-2,z) \gamma(z) \bar\gamma(z).
\end{split}$$
We set $\mathcal L(z) = L(z) + \bar L(z) = L^W(z) + \bar L^W(z)$. To show that $\mathcal L_{-1} = \mathcal D$, we check that $$\label{eq:affine L_{-1}}
\mathcal Y(\mathcal L_{-1} v,z) = \frac d{dz} \mathcal Y(v,z), \quad v \in \hat{\mathbb F}_\varkappa,$$ which for all the generating quantum fields follows from straightforward computations.
Finally, the rank of the VOA $\hat{\mathbb F}_\varkappa$ is equal to $$\operatorname{rank}\hat{\mathbb F}_\varkappa = c + \bar c = \(3 - \frac 6\varkappa\) + \(3 + \frac 6\varkappa\) = 6.$$ This concludes the proof of the theorem.
$\ghatKK$-module structure of $\hat{\mathbb F}_\varkappa$ for generic $\varkappa$.
----------------------------------------------------------------------------------
We now prove the analogue of the Theorem \[thm:classical bimodule structure\], describing the structure of the $\ghatKK$-module $\hat {\mathbb F}_\varkappa$ for generic values of the parameter $\varkappa$.
For $\lambda \in \h^*, k \in \C$ we denote by $\hat V_{\lambda,k}$ the irreducible $\ghat_k$-module, generated by a vector $\hat v$ satisfying $\g_+ \hat v = \n_+ \hat v = 0$ and $\mathbf h \hat v = \lambda \, \hat v$.
For any $\ghat_k$-module $\hat V$, the restricted dual space $\hat V'$ can be equipped with a $\ghat_k$-action by $$\<g_n \, v', v\> = - \< v', \hat\omega(g_{-n}) v\>, \qquad v \in \hat V,\ v' \in \hat V', \ g \in \g,$$ where $\hat\omega$ is as in . We denote the resulting dual module by $\hat V^\star$.
An important source of $\ghat_k$-modules is the induced module construction. Any $\g$-module $V$ may be regarded as a module for the subalgebra $\mathfrak p = \bigoplus_{n\ge0} \ghat[n]$, with $\g[n]$ acting trivially for $n>0$ and $\mathbf k$ acting as the multiplication by a scalar $k \in \C.$ The induced $\ghat_k$-module $\hat V_k$ is defined as the space $$\label{eq:induced module}
\hat V_k = \mathcal U(\ghat) \o_{\mathcal U(\mathfrak p)} V,$$ with $\ghat_k$ acting by left multiplication.
For the remainder of this section, we will assume that complex numbers $\varkappa, k, \bar k$ satisfy $$\varkappa \notin \mathbb Q, \qquad k = \varkappa - h^\vee, \qquad \bar k = -\varkappa - h^\vee.$$
\[thm:affine bimodule structure\] There exists a filtration $$\label{eq:affine filtration}
0 \subset \hat{\mathbb F}_\varkappa^{(0)} \subset \hat{\mathbb F}_\varkappa^{(1)} \subset \hat{\mathbb F}_\varkappa^{(2)} = \hat{\mathbb F}_\varkappa$$ of $\ghatKK$-submodules of $\hat{\mathbb F}_\varkappa$ such that $$\begin{aligned}
\hat{\mathbb F}_\varkappa^{(2)}/\hat{\mathbb F}_\varkappa^{(1)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \hat V_{-\lambda-2,k} \o \hat V^\star_{-\lambda-2,\bar k}, \label{eq:affine socle F2}\\
\hat{\mathbb F}_\varkappa^{(1)}/\hat{\mathbb F}_\varkappa^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \( \hat V_{\lambda,k} \o \hat V^\star_{-\lambda-2,\bar k} \oplus
\hat V_{-\lambda-2,k} \o \hat V^\star_{\lambda,\bar k} \), \label{eq:affine socle F1}\\
\hat{\mathbb F}_\varkappa^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P} \hat V_{\lambda,k} \o \hat V^\star_{\lambda,\bar k}. \label{eq:affine socle F0}\end{aligned}$$
The operator $\mathcal L_0$ determines a $\Z$-grading $\deg$ of $\hat{\mathbb F}_\varkappa$, which is explicitly described by $$\label{eq:affine grading}
\deg \1_\lambda = 0, \qquad \deg X_n = -n \quad \text{ for } X = a,\bar a, \beta,\gamma, \bar\beta,\bar\gamma.$$ The lowest graded subspace $\hat {\mathbb F}_\varkappa[0] = F(\beta_0,\gamma_0) \o F(\bar\beta_0,\bar\gamma_0)\o \C[\mathbf P]$ of the vertex algebra $\hat {\mathbb F}_\varkappa$ is identified with the Fock space $\mathbb F$ for the finite-dimensional Lie algebra $\g$. Moreover, since $\varkappa$ is generic, $\hat{\mathbb F}_\varkappa$ can be constructed as the induced $\ghatKK$-module from the $\ggbar$-module $\mathbb F$: $$\hat {\mathbb F}_\varkappa = \mathcal U(\ghatghat) \o_{\mathcal U(\mathfrak p \oplus \mathfrak p)} \mathbb F.$$
We construct the filtration by inducing it from the finite-dimensional one : $$\hat {\mathbb F}_\varkappa^{(0)} = \mathcal U(\ghatghat) \o_{\mathcal U(\mathfrak p \oplus \mathfrak p)} \mathbb F^{(0)},
\qquad
\hat {\mathbb F}_\varkappa^{(1)} = \mathcal U(\ghatghat) \o_{\mathcal U(\mathfrak p \oplus \mathfrak p)} \mathbb F^{(1)}.$$
It is easy to check that ,, respectively imply ,,, which proves the theorem.
The analogue of the Corollary \[thm:classical positive subalgebra\], describing the realization of the subalgebra $\mathfrak R(G) \subset \mathfrak R(G_0)$, is given below.
\[thm:affine positive subalgebra\] There exists a subspace $\hat{\mathbf F}_\varkappa \subset \hat{\mathbb F}_\varkappa$, satisfying
1. $\hat{\mathbf F}_\varkappa$ is a vertex operator subalgebra of $\hat{\mathbb F}_\varkappa$, and is generated by the quantum fields , and $\mathbb Y(1,z)$. In particular, $\mathbf F$ is a $\ghatKK$-submodule of $\hat{\mathbb F}_\varkappa$.
2. As a $\ghatKK$-module, $\hat{\mathbf F}_\varkappa$ is generated by the vectors $\{\1_\lambda\}_{\lambda \in \mathbf P^+}$, and we have $$\label{eq:affine positive decomposition}
\hat{\mathbf F}_\varkappa \cong \bigoplus_{\lambda \in \mathbf P^+} \hat V_{\lambda,k} \o \hat V^\star_{\lambda, \bar k}.$$
As before, we identify the lowest graded subspace $\hat {\mathbb F}_\varkappa[0]\subset \hat {\mathbb F}_\varkappa$ with the Fock space $\mathbb F$ for the finite-dimensional Lie algebra $\g$. Recall from Corollary \[thm:classical positive subalgebra\] that the $\ggbar$-module $\mathbb F$ contains the distinguished submodule $\mathbf F$. We define the subspace $\hat {\mathbf F}_\varkappa$ as the $\ghatKK$-submodule of $\hat {\mathbb F}_\varkappa$, induced from $\mathbf F$: $$\hat {\mathbf F}_\varkappa = \mathcal U(\ghatghat) \o_{\mathcal U(\mathfrak p \oplus \mathfrak p)} \mathbf F.$$ It immediately follows from Corollary \[thm:classical positive subalgebra\] that $\hat{\mathbf F}_\varkappa$ is generated by the vectors $\{\1_\lambda\}_{\lambda \in \mathbf P^+}$, and has the decomposition .
Next, we need to show that $\hat {\mathbf F}_\varkappa$ is a vertex subalgebra. Let $\hat{\mathbf F}_\varkappa'$ denote the space, spanned by the Laurent coefficients of $\hat{\mathbb F}_\varkappa$-valued fields $\mathcal Y(a,z)b$ for all possible $a,b \in \hat {\mathbf F}_\varkappa$. We will establish that $\hat{\mathbf F}_\varkappa' = \hat{\mathbf F}_\varkappa$.
Indeed, $\hat{\mathbf F}_\varkappa'$ is a $\ghatKK$-submodule of $\hat{\mathbb F}_\varkappa$, and can be induced from its lowest graded component $\mathbf F' = \hat{\mathbf F}_\varkappa'[0]$, which is a $\ggbar$-submodule of $\mathbb F$. It suffices to prove that $\mathbf F' = \mathbf F$.
For any $a,b \in \mathbf F$, the lowest graded component of $\mathcal Y(a,z)b$ is equal to the product $a b$ in the algebra $\mathbb F$, and since $\mathbf F$ is a subalgebra, we have $a b \in \mathbf F$. (Note that any element $a \in \mathbf F$ can be obtained this way, for example, by taking $b = \1$). Using the commutation relations with the two copies of $\ghat$, we can prove that the lowest graded component of $\mathcal Y(a,z)b$ lies in $\mathbf F$ for any $a,b \in \hat {\mathbf F}_\varkappa$.
It follows that $\mathbf F' = \mathbf F$ and hence $\hat{\mathbf F}_\varkappa' = \hat{\mathbf F}_\varkappa$, which means that the restrictions of the operators $\mathcal Y(\cdot,z)$, corresponding to the subspace $\hat{\mathbf F}_\varkappa$, are well-defined. Thus $\hat{\mathbf F}_\varkappa$ is a vertex subalgebra of $\hat{\mathbb F}_\varkappa$. It is clear that as a vertex subalgebra $\hat{\mathbf F}_\varkappa$ is generated by the quantum fields , and $\{\mathbb Y(\lambda,z)\}_{\lambda \in \mathbf P^+}$, and the latter are generated by the single operator $\mathbb Y(1,z)$.
Finally, $\hat {\mathbf F}_\varkappa$ contains both $L^W(z)$ and $\bar L^W(z)$ - hence also $\mathcal L(z)$ - and therefore is a vertex operator subalgebra of $\hat {\mathbb F}_\varkappa$.
The vertex operator algebras $\hat{\mathbf F}_\varkappa$ and $\hat{\mathbb F}_\varkappa$ give explicit realizations of the modified regular representations $\mathfrak R'_\varkappa(\hat G)$ and $\mathfrak R'_\varkappa(\hat G_0)$ we discussed in the introduction. It would be interesting to construct them invariantly by using the correlation functions approach [@FZ], interpreting the rational functions $\<\1', \mathcal Y(v_1,z_1) \dots \mathcal Y(v_n,z_n) \1\>$ for $v_1,\dots,v_n \in \mathbb F \cong \hat{\mathbb F}_\varkappa[0]$ as solutions of differential equations similar to the Knizhnik-Zamolodchikov equations.
Semi-infinite cohomology of $\ghat$
-----------------------------------
The fact that the level of the diagonal action of $\ghat$ in the modified regular representations is equal to the special value $-2 h^\vee$ allows us to introduce the semi-infinite cohomology of $\ghat$ with coefficients in $\hat {\mathbb F}_\varkappa$ and in $\hat {\mathbf F}_\varkappa$. In this section we show that for generic values of $\varkappa$ these cohomologies lead to the same algebras of formal characters as in the finite-dimensional case.
We recall the definition of the semi-infinite cohomology [@Fe; @FGZ]. The main new ingredient is the “space of semi-infinite forms” $\boldsymbol{\hat\Lambda}^\semiinfty$, which replaces the finite-dimensional exterior algebra $\boldsymbol\Lambda$. We summarize its properties in the following
Let $\boldsymbol{\hat\Lambda}^\semiinfty = \bigwedge \ghat_- \o \bigwedge (\ghat'_+ \oplus \g')$. Then
1. The Clifford algebra, generated by $\{\iota(g_n), \eps(g'_n)\}_{g \in \g, g' \in \g',n \in \Z}$ with relations $$\label{eq:affine Clifford relations}
\{\iota(x_m), \iota(y_n) \} = \{ \eps(x'_m), \eps(y'_n) \} = 0, \qquad
\{\iota(x_m), \eps(y'_n)\} = \delta_{m,n}\, \<y',x\>.$$ acts irreducibly on $\boldsymbol{\hat\Lambda}^\semiinfty$, so that for any $\omega_- \in \bigwedge \ghat_-,\ \omega_+ \in \bigwedge (\ghat'_+ \oplus \g')$ we have $$\begin{aligned}
\iota(x_n) ( \omega_- \o 1 ) &=
\begin{cases} 0,& n \ge 0 \\ (x_n \wedge \omega_-) \o 1, & n<0 \end{cases}, \qquad
\eps(x'_n) (1 \o \omega_+) &=
\begin{cases} 1 \o (x'_n \wedge \omega_+), & n\ge0\\ 0,& n<0 \end{cases}.\end{aligned}$$
2. $\boldsymbol{\hat\Lambda}^{\semiinfty}$ is a bi-graded vertex superalgebra, with vacuum $\1 = 1 \o 1$, and generated by $$\begin{aligned}
{5}
\iota(x,z) &= \sum_{n\in\Z} \iota(x_n) z^{-n-1}, \qquad | \iota(x,z) | & = & -1, \quad &\deg \iota(x,z) &= 1, \quad & x &\in \g,\\
\eps(x',z) &= \sum_{n\in\Z} \eps(x'_{-n}) z^{-n}, \qquad | \eps(x',z) | & = & 1, \quad &\deg \eps(x',z) &= 0, \quad &x' &\in \g'.\end{aligned}$$
3. $\boldsymbol{\hat\Lambda}^\semiinfty$ has a $\ghat$-module structure on the level $\mathbf k = 2 h^\vee$, defined by $$\pi(x_n) = \sum_{m\in\Z} \sum_i :\eps((g'_i)_m)\iota([g_i,x]_{n+m}):,
\qquad x \in \g.$$
One can think of $\bigwedge \ghat_-$ as the space spanned by formal “semi-infinite” forms $$\omega = \xi'_{i_1} \wedge \xi'_{i_2} \wedge \xi'_{i_3} \wedge \dots,\qquad i_{n+1} = i_n + 1 \text{ for } n\gg0,$$ where $\{\xi_j\}_{j\in \mathbb N}$ is a homogeneous basis of $\ghat_-$. A monomial $\xi_{j_1} \wedge \dots \wedge \xi_{j_m} \in \bigwedge \ghat_-$ is identified with the semi-infinite form with the corresponding factors missing: $$\omega = \pm \ \xi'_1 \wedge \xi'_2 \wedge \dots \wedge \xi'_{j_1-1} \wedge \widehat{\xi'_{j_1}} \wedge \xi'_{j_1+1} \wedge \dots \wedge \xi_{j_m-1} \wedge \widehat{\xi_{j_m}} \wedge \xi_{j_m+1} \wedge\dots.$$ In other words, if $\xi_j \in \ghat$, then $\iota(\xi_j)$ operates as usual by eliminating the factor $\xi'_j$.
The BRST complex, associated with a $\ghat$-module $\hat V$ on the level $k = -2h^\vee$, is the complex $C^{\semiinfty+\bullet} (\ghat, \C\mathbf k;\hat V) = \boldsymbol{\hat\Lambda}^{\semiinfty+\bullet} \otimes \hat V$, with the differential $$\label{eq:affine differential}
\hat{\mathbf d} = \sum_{n\in\Z} \sum_i \eps((g'_i)_n) \pi_{\hat V}((g_i)_n) - \frac12 \sum_{m,n\in \Z} \sum_{i,j} :\eps((g_i)'_m) \eps((g_j)'_n) \iota([g_i,g_j]_{m+n}):,$$ where $\{g_i\}$ is any basis of $\g$, and $\{g'_i\}$ is the dual basis of $\g'$. The corresponding cohomology is denoted $H^{\semiinfty+\bullet}(\ghat,\C \mathbf k;\hat V)$.
The BRST complex above gives the relative (to the center) version of the semi-infinite cohomology. We don’t consider any other type of cohomology, and thus simply drop the word ’relative’ everywhere. The condition $k = -2h^\vee$ is equivalent to $\hat{\mathbf d}^2 = 0$.
If $\hat V$ is a vertex algebra, then its semi-infinite cohomology inherits a vertex superalgebra structure [@LZ].
The following theorem is similar to the reduction theorem of [@FGZ] (see also [@Li]), and relates the semi-infinite cohomology for generic values of $\varkappa$ with the classical cohomology of Lie algebras.
\[thm:reduction theorem\] Let $V$ be a $\ggbar$-module, and let $\varkappa \in \C$ be generic. Set $k = \varkappa - h^\vee$ and $\bar k = -\varkappa - h^\vee$, and denote $\hat V$ be the induced $\ghat_k \oplus \ghat_{\bar k}$-module. Then with respect to the diagonal $\g$-action $\hat V$ is a level $\mathbf k = -2 h^\vee$ module, and $$H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat V) \cong H^\bullet(\g,V).$$
As a vector space, the module $\hat V$ has a decomposition $$\hat V = \mathcal U(\ghat_-) \o V \o \mathcal U(\ghat_+)',$$ where we identified the factor $\mathcal U(\ghat_-)$, coming from the right induced action of $\ghat_{\bar k}$, with $\mathcal U(\ghat_+)'$ using the non-degenerate (since $\varkappa$ is generic!) contravariant pairing. Therefore, as vector spaces $$\label{eq:affine complex factorization}
C^\bullet(\ghat, \C\mathbf k; \hat V) = C_-^\bullet \o C_0^\bullet \o C_+^\bullet,$$ where $$C_-^\bullet = \bigwedge \ghat_- \o \mathcal U(\ghat_-), \qquad
C_0^\bullet = \bigwedge \g' \o V, \qquad
C_+^\bullet = \bigwedge \ghat'_+ \o \mathcal U(\ghat_+)^*$$
We write the differential $\hat{\mathbf d}$ as $$\hat{\mathbf d} = \mathbf d_- + \mathbf d_0 + \mathbf d_+ + \boldsymbol\delta,$$ where $\mathbf d_\pm$ are the BRST differentials for $\ghat_\pm$, $$\begin{aligned}
\mathbf d_- &= \sum_{n<0} \sum_i \eps((g'_i)_n) \pi_l((g_i)_n) - \frac 12 \sum_{m,n<0} \sum_{i,j} :\eps((g_i)'_m) \eps((g_j)'_n) \iota([g_i,g_j]_{m+n}):,\\
\mathbf d_+ &= \sum_{n>0} \sum_i \eps((g'_i)_n) \pi_r((g_i)_n) - \frac12 \sum_{m,n>0} \sum_{i,j} :\eps((g_i)'_m) \eps((g_j)'_n) \iota([g_i,g_j]_{m+n}): ,\end{aligned}$$ the differential $\mathbf d_0$ is defined as in with the $\g$-action $\pi_V$ replaced by $$\label{eq:spectral g-action}
\pi(x) = \pi_{\hat V}(x) + \sum_{n\ne0} :\eps((g_j)'_n) \iota([x,g_j]_n):, \qquad x \in \g,$$ and $\boldsymbol\delta$ includes all the remaining terms: $$\begin{aligned}
\boldsymbol \delta &= \sum_{n>0} \sum_i \eps((g'_i)_n) \pi_l((g_i)_n)
+ \sum_{n<0} \sum_i \eps((g'_i)_n) \pi_r((g_i)_n) - \\
&- \sum_{m>0,n<0} \sum_{i,j} :\eps((g_i)'_m) \eps((g_j)'_n) \iota([g_i,g_j]_{m+n}):.\end{aligned}$$
Following [@FGZ], we introduce the skewed degree $f\deg$ by $$f\deg (w_- \o w_0 \o w_+) = \deg w_+ - \deg w_-, \qquad w_\pm \in C_\pm,\ \ w_0 \in C_0,$$ where the ’$\deg$’ gradings in the complexes $C_\pm$ are inherited from $C^{\semiinfty}(\ghat,\mathbf k;\hat V)$. We set $$\mathfrak B^p = \left\{ v \in C^{\semiinfty}(\ghat,\mathbf k;\hat V) \ \biggr| \ f\deg v \ge p \right\}.$$ One can check that $\mathbf d_\pm$ and $\mathbf d_0$ preserve the filtered degree, and that $\boldsymbol \delta (\mathfrak B^p) \subset \mathfrak B^{p+1}$. Thus, $\{\mathfrak B^p\}_{p\in\Z}$ is a decreasing filtration of the complex $C^{\semiinfty}(\ghat,\mathbf k;\hat V)$, and the associated graded complex has the reduced differential $$\mathbf d_{red} = \mathbf d_- + \mathbf d_0 + \mathbf d_+.$$ We now compute the corresponding reduced cohomology, which will provide a bridge to $H^{\semiinfty+\bullet}(\ghat,\C\mathbf k;\hat V)$.
It is clear that $\mathbf d_\pm^2 = \{\mathbf d_+,\mathbf d_-\} = 0$, and that the differentials $\mathbf d_\pm: C_\pm^\bullet \to C_\pm^{\bullet+1}$ act in their respective factors of . One can also check that $(\mathbf d_0)^2 = \{\mathbf d_0,\mathbf d_\pm\} = 0$; moreover, $$\mathbf d_0 (C_-^\bullet \o C_0^\bullet \o C_+^\bullet) \subset (C_-^\bullet \o C_0^{\bullet+1} \o C_+^\bullet)$$ despite the fact that $\mathbf d_0$ does not act in $C_0^\bullet$.
It is a well-known fact in homological algebra that $$H^n(C_+,\mathbf d_+) = H^n(\ghat_+;\mathcal U(\ghat_+)') = \delta_{n,0} \, \C,$$ with $1 \o 1' \in C_+$ representing the non-trivial cohomology class. Similarly, one has $$H^n(C_-,\mathbf d_-) = H_{-n}(\ghat_-;\mathcal U(\ghat_-)) = \delta_{n,0} \, \C,$$ and $1 \o 1 \in C_-$ represents the non-trivial cohomology. Further, one can check that the subspace $1 \o C_0^\bullet \o 1 \subset C^{\semiinfty+\bullet}(\ghat,\C\mathbf k;\hat V)$ is stabilized by $\mathbf d_0$, and that the $\g$-action on that subspace reduces to the $\g$ action $1 \o \pi_V \o 1$. It follows that $$H^\bullet_{red}(\ghat, \C\mathbf k; \hat V) \cong H^\bullet(1 \o C_0 \o 1,\mathbf d_0) \cong H^\bullet(C_0,\mathbf d) \cong H^\bullet(\g,V).$$
We now return to the cohomology of $C^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat V)$. Since $\hat {\mathbf d}$ preserves the ’$\deg$’ grading, it can be computed separately for each subcomplex $C^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat V)[m], m \in \Z$. The filtration $\{\mathfrak B^p[m]\}_{p \in \Z}$ of this complex is finite for each $m$, and leads to a finitely converging spectral sequence with $E_1^{p,q}[m] = H^q_{red}(\mathfrak B^p[m]/ \mathfrak B^{p+1}[m])$.
For $m\ne 0$ we have $H^q_{red}(\mathfrak B^p[m]/ \mathfrak B^{p+1}[m])=0$ for all $p$, hence the spectral sequence is zero, and $H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat V)[m]=0$. For $m=0$ we note that $$\mathfrak B^0[0] = C^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat V)[0], \qquad
\mathfrak B^1[0] = 0,$$ which means that $E_1^{p,q}[0] = 0$ unless $p=0$, and the collapsing spectral sequence implies $$H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat V)[0] \cong H^\bullet_{red}(\mathfrak B^0[0]/ \mathfrak B^1[0]) \cong H^\bullet(\g;V).$$
This completes the proof of the theorem.
\[thm:affine cohomology\] The vertex superalgebras $H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat{\mathbf F}_\varkappa),
H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat{\mathbb F}_\varkappa)$ degenerate into commutative superalgebras. Moreover, we have commutative superalgebra isomorphisms $$\label{eq:affine semiinfinite isomorphisms}
H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat{\mathbf F}_\varkappa) \cong H^\bullet(\g; \mathbf F),\qquad
H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat{\mathbb F}_\varkappa) \cong H^\bullet(\g; \mathbb F).$$ In particular, $$H^{\semiinfty+0}(\ghat,\C \mathbf k;\hat{\mathbf F}_\varkappa) \cong
H^{\semiinfty+0}(\ghat,\C \mathbf k;\hat{\mathbb F}_\varkappa) \cong \C[\mathbf P]^W.$$
Theorem \[thm:reduction theorem\] gives us isomorphisms on the level of vector spaces. It is also clear from its proof that the semi-infinite cohomology is concentrated in the subspace of $\deg = 0$, and thus the operators $\mathcal Y(\cdot,z)$ on cohomology are reduced to their constant terms. In particular, they are independent of $z$, which means that the vertex superalgebra degenerates into a commutative superalgebra. The multiplication is easily traced back to the multiplications in $\mathbf F \cong \hat{\mathbf F}_\varkappa[0]$ and in the exterior algebra $\boldsymbol\Lambda = \bigwedge \g'$, which shows that are superalgebra isomorphisms.
Modified regular representations of the Virasoro algebra.
=========================================================
Virasoro algebra and the quantum Drinfeld-Sokolov reduction
-----------------------------------------------------------
In this section we present a construction of the regular representation of the Virasoro algebra, which goes in parallel with constructions in the previous sections. However, instead of beginning with a space of functions on the corresponding group (which is, strictly speaking, a semigroup in the complex case), we will use the quantum Drinfeld-Sokolov reduction [@FeFrDS] (see also [@BFr] and references therein), applied to the modified regular representations of $\ghat$ constructed in Section 1. As a result we obtain certain bimodules over the Virasoro algebra, which have the structure similar to their affine counterparts. The result of the quantum Drinfeld-Sokolov reduction applied to the actual regular representation of $\ghat$ should have a standard interpretation in terms of the space of functions on the Virasoro semigroup, but we will not need this fact for our purposes.
Recall that the Virasoro algebra $\vir$ is the infinite-dimensional complex Lie algebra, generated by $\{L_n\}_{n \in \Z}$ and a central element $\mathbf c,$ subject to the commutation relations $$[L_m, L_n] = (m-n) L_{m+n} + \frac {m^3-m}{12}\, \mathbf c.$$
The Virasoro algebra has a $\Z$-grading $\vir = \oplus_{n\in\Z} \vir[n]$, determined by $$\deg L_n = -n, \qquad \deg \mathbf c = 0.$$
There is a functorial correspondence between certain representations of affine Lie algebras and their $\mathcal W$-algebra counterparts, called the quantum Drinfeld-Sokolov reduction [@FeFrDS] (see also [@BFr] and references therein). We review this procedure for the case $\ghat = \slhat$, when the corresponding $\mathcal W$-algebra is identified with the Virasoro algebra.
For any $\ghat_k$-module $\hat V$, the complex $(C_{DS}(\hat V),\mathbf d_{DS})$, $$C_{DS}(\hat V) = \hat V \o \hat\Lambda(\psi,\psi^*), \qquad \mathbf d_{DS} = \sum_{n\in\Z} \psi^*_n \pi_{\hat V}(\mathbf e_n) + \psi^*_1,$$ is called the BRST complex of the quantum Drindeld-Sokolov reduction. The corresponding cohomology is denoted $H_{DS}(\hat V)$.
The BRST complex above is very similar to the semi-infinite cohomology complex for the nilpotent loop algebra $\nhat_+ = \bigoplus_{n\in\Z} \C \mathbf e_n$. Indeed, the corresponding space of semi-infinite forms $\boldsymbol{\Lambda}^\semiinfty(\nhat_+)$ is identified with $\hat\Lambda(\psi,\psi^*)$ by $\iota(\mathbf e_n) \equiv \psi_n, \ \eps(\mathbf e'_n) \equiv \psi^*_{-n}$, and the only modification is the additional term $\psi^*_1$ in the differential.
The BRST complex inherits the gradings $|\cdot|$ and $\deg$ from $\hat\Lambda(\psi,\psi^*)$ and $\hat V$. Since $|\mathbf d_{DS}| = 1$, the grading $|\cdot|$ descends to the cohomology $H_{DS}(\hat V)$. However, with respect to the other grading, the differential $\mathbf d_{DS}$ is not homogeneous. We introduce a modified grading $\deg'$ by $$\begin{gathered}
\deg' \mathbf e(z) = 0, \qquad
\deg' \mathbf h(z) = 1,\qquad
\deg' \mathbf f(z) = 2, \\
\deg' \psi(z) = 0,\qquad
\deg' \psi^*(z) = 1.\end{gathered}$$ The differential $\mathbf d_{DS}$ then satisfies $\deg' \mathbf d_{DS} = 0$, and the grading $\deg'$ descends to $H_{DS}(\hat V)$.
The cohomology $H^0_{DS}(\ghat_k)$ of the vacuum module inherits a vertex algebra structure. We have the following result (details of the proof can be found in [@BFr]).
\[thm:Drinfeld-Sokolov\] For $k \ne -h^\vee$ we have $H^0_{DS}(\ghat_k) \cong \vir_c$, where $c = 1 - \frac{6}{k+h^\vee} - 6k$.
For any $\ghat_k$-module $\hat V$, the vertex algebra $\vir_c \cong H^0_{DS}(\ghat_k)$ acts on $H^0_{DS}(\hat V)$. For $\varkappa \ne 0$, set $\tilde F_{\lambda,\varkappa} = H_{DS}^0(\hat W_{\lambda,\varkappa-h^\vee}).$ The following identifies the $\vir_c$-module structure on $\tilde F_{\lambda,\varkappa}$.
\[thm:Wakimoto to FF\] Let $\varkappa \ne 0$, and let $c = 13 - 6 \varkappa - \frac6\varkappa$. Then $\tilde F_{\lambda,\varkappa} \cong \hat F_\varkappa(a) \o \C \1_\lambda$ as a vector space, and the $\vir_c$-action is given by $$\label{eq:Feigin-Fuks}
L^F(z) = \frac 1{4\, \varkappa} :a(z)^2: + \frac{\varkappa-1}{2\, \varkappa} \, a'(z).$$
In the vector space factorization of $\hat W_{\lambda,\varkappa-h^\vee} = \hat F(\beta,\gamma) \o \hat F_\varkappa(a) \otimes \C \1_\lambda$, the differential $\mathbf d_{DS}$ acts only in the first component. Therefore, we must have $$\tilde F_{\lambda,\varkappa} = H_{DS}(\hat F(\beta,\gamma)) \o \hat F_\varkappa(a) \otimes \C \1_\lambda.$$ A spectral sequence reduces the cohomology $H_{DS}^0(\hat F(\beta,\gamma))$ to the cohomology of the semi-infinite Weil complex $\hat F(\beta,\gamma)\o\hat\Lambda(\psi,\psi^*)$. The latter splits into an infinite product of finite-dimensional Weil complexes, and thus has one-dimensional cohomology, concentrated in degree $0$.
The inclusion of vertex algebras $\ghat_{\varkappa-h^\vee} \hookrightarrow \hat W_{0,\varkappa-h^\vee}$ induces an inclusion $\vir_c \hookrightarrow \tilde F_{0,\varkappa}$, and the explicit formula for $L(z)$ in terms of $a(z)$ is a result of a direct computation.
The realization of Virasoro modules was known long before the quantum Drinfeld-Sokolov reduction, and is called the Feigin-Fuks construction in the literature. We use the superscript “F” to distinguish this standard action from the modified Virasoro actions, which we will be considering later.
Bosonic realization of the regular representation
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The Virasoro analogue of the Peter-Weyl theorem is more subtle than in the case of classical and affine Lie algebras. There is no clear way to calculate the two commuting $\vir$-actions in a way similar to Theorem \[thm:classical bimodule action\] and Theorem \[thm:affine action\]. However, there exists a Fock space realization analogous to Theorem \[thm:affine bimodule action\], which we will call the regular representation of the Virasoro algebra.
\[thm:Virasoro action\] Let $\varkappa \ne 0$, and let $c = 13 - 6 \varkappa - \frac 6\varkappa$ and $\bar c = 13 + 6 \varkappa + \frac 6\varkappa$.
1. The space $\tilde {\mathbb F}_\varkappa$ has a $\virCC$-module structure, defined by $$\begin{aligned}
L(z) &= \frac 1{4\varkappa} :a(z)^2: + \frac{\varkappa-1}{2\varkappa} a'(z)
- \frac 1\varkappa \, \mathbb Y(-2,z), \label{eq:Virasoro action 1}\\
\bar L(z) &= -\frac 1{4\varkappa} :\bar a(z)^2: + \frac{\varkappa+1}{2\varkappa} \bar a'(z)
+ \frac 1\varkappa \, \mathbb Y(-2,z). \label{eq:Virasoro action 2}\end{aligned}$$
2. The space $\tilde {\mathbb F}_\varkappa$ has a compatible VOA structure with $\operatorname{rank}\tilde {\mathbb F}_\varkappa = 26$.
The formulas , are nothing else but the result of the two-sided quantum Drinfeld-Sokolov reduction, which consists of two reductions applied separately to the two commuting $\ghat$-actions of Theorem \[thm:affine bimodule action\], cf. formulas , and Proposition \[thm:Wakimoto to FF\].
Rather than give detailed proof of this fact, we choose to verify the commutation relations directly. Introduce notation $$\begin{aligned}
\delta L(z) &= L(z) - L^F(z) = \frac 1\varkappa \, \mathbb Y(-2,z), \\
\overline{\delta L}(z) &= \bar L(z) - \bar L^F(z) = - \frac 1\varkappa \, \mathbb Y(-2,z).\end{aligned}$$ Without the additional terms $\delta L(z), \overline{\delta L}(z)$, both and give two commuting copies of the standard construction with the specified central charges. Therefore, it suffices to show that the presence of these extra terms does not violate the commutation relations for $\virCC$.
Straightforward computations immediately show that $$\delta L(z) \, \delta L(w) \sim \delta L(z) \, \overline{\delta L}(w) \sim \overline{\delta L}(z) \, \overline{\delta L}(w) \sim 0.$$
$$L^F(z) \mathbb Y(-2,w) \sim \frac {\mathbb Y(-2,w)}{(z-w)^2} - \frac 1{\varkappa} \frac{:a(w) \mathbb Y(-2,w):}{z-w},$$ $$\bar L^F(z) \mathbb Y(-2,w) \sim \frac {\mathbb Y(-2,w)}{(z-w)^2} + \frac 1{\varkappa} \frac{:\bar a(w) \mathbb Y(-2,w):}{z-w}.$$
We now prove the commutation relations for the action . We have $$\begin{aligned}
L(z)L(w) & - L^F(z) L^F(w) = L^F(z) \, \delta L(w) + \delta L(z) L^F(w) + \delta L(z) \, \delta L(w) \sim\\
& \sim \frac 1\varkappa \( \frac {\mathbb Y(-2,w)}{(z-w)^2}
- \frac 1{\varkappa} \frac{:a(w) \mathbb Y(-2,w):}{z-w}\)
+ \frac 1\varkappa \(\frac {\mathbb Y(-2,z)}{(z-w)^2}
+ \frac 1{\varkappa} \frac{:a(z) \mathbb Y(-2,w):}{z-w} \) \sim\\
& \sim \frac 1\varkappa \( \frac {2 \, \mathbb Y(-2,w)}{(z-w)^2} + \frac {\mathbb Y'(-2,w)}{z-w} \) \sim
\frac {2 \, \delta L(w)}{(z-w)^2} + \frac {(\delta L)'(w)}{z-w} ,\end{aligned}$$ and therefore $$\begin{aligned}
L(z)L(w) & \sim L^F(z) L^F(w) + \frac {2 \, \delta L(w)}{(z-w)^2} + \frac {(\delta L)'(w)}{z-w} \sim
\( \frac{c/2}{(z-w)^4} + \frac {2L^F(w)}{(z-w)^2} + \frac {(L^F)'(w)}{z-w} \) + \\
& + \frac {2 \, \delta L(w)}{(z-w)^2} + \frac {(\delta L)'(w)}{(z-w)^2} =
\frac{c/2}{(z-w)^4} + \frac {2L(w)}{(z-w)^2} + \frac {L'(w)}{z-w} .\end{aligned}$$ We have established that adding the extra term $\delta L(z)$ to the action preserves the commutation relations for $\vir_c.$ Similarly, the formula gives a representation of $\vir_{\bar c}.$
We now show that the two actions of $\vir_c$ and $\vir_{\bar c}$ commute. Using , we get $$\begin{split}
\delta L(z) \bar L^F(w) & \sim \frac 1\varkappa \( \frac {\mathbb Y(-2,z)}{(z-w)^2} - \frac 1{\varkappa} \frac{:\bar a(z) \mathbb Y(-2,z):}{z-w} \) \sim \\
& \sim \frac 1\varkappa \( \frac {\mathbb Y(-2,w)}{(z-w)^2} + \frac {\mathbb Y'(-2,w)}{z-w} -
\frac 1{\varkappa} \frac{:\bar a(w) \mathbb Y(-2,w):}{z-w} \) \sim \\
& \sim \frac 1\varkappa \( \frac {\mathbb Y(-2,w)}{(z-w)^2} - \frac 1{\varkappa} \frac{:a(w) \mathbb Y(-2,w):}{z-w} \) .
\end{split}$$ Note that implies $\delta L(z) \bar L^F(w) \sim - L^F(z) \, \overline{\delta L}(w)$, and thus $$L(z)\bar L(w) = L^F(z)\bar L^F(w) + L^F(z) \, \overline{\delta L}(w) + \delta L(z)\bar L^F(w) + \delta L(z)\overline{\delta L}(w) \sim 0,$$ which means that the two Virasoro actions commute.
It is easy to see that the formula can be written as $$L(z) = \mathcal Y\( \frac{(a_{-1})^2}{4\varkappa} \1_0 + \frac{\varkappa-1}{2\varkappa} a_{-2} \1_0 + \frac 1{\varkappa} \,\1_{-2},z\),$$ and similarly for , which means that the vertex algebra structure is compatible with $\virCC$-module structure on $\tilde {\mathbb F}_\varkappa$.
We introduce the VOA structure in $\tilde{\mathbb F}_\varkappa$ by setting $\mathcal L(z) = L(z) + \bar L(z) = L^F(z) + \bar L^F(z)$. One immediately checks that $\mathcal L(z)$ is a Virasoro quantum field with central charge $26,$ and satisfies $\mathcal L_{-1} \1_0 = 0.$ It suffices to check the remaining relation $$\label{eq:Virasoro L_{-1}}
[\mathcal L_{-1},\mathcal Y(v,z)] = \frac d{dz} \mathcal Y(v,z), \quad v \in \tilde{\mathbb F}_\varkappa,$$ for each of the generating quantum fields, which is done by direct computations.
$\virCC$-module structure of $\tilde{\mathbb F}_\varkappa$ for generic $\varkappa$.
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We now describe the socle filtration of the $\virCC$-module $\tilde {\mathbb F}_\varkappa$ for generic $\varkappa$, when it is completely analogous to the finite-dimensional and affine cases, given by Theorem \[thm:classical bimodule structure\] and Theorem \[thm:affine bimodule structure\]. In this subsection we assume that $$\varkappa \notin \mathbb Q, \qquad c = 13 - \frac 6\varkappa - 6\varkappa,\qquad \bar c = 13 + \frac 6\varkappa + 6\varkappa.$$
For a $\vir_c$-module $\tilde V$, the restricted dual space $\tilde V'$ can be equipped with a $\vir_c$-action by $$\<L_n \, v', v\> = \< v', L_{-n} v\>.$$ We denote the resulting dual module by $\tilde V^\star$.
We denote by $\tilde V_{\Delta,c}$ the irreducible $\vir_c$-module, generated by a highest weight vector $\tilde v$ satisfying $L_0 \, \tilde v = \Delta \, \tilde v$ and $L_n \tilde v = 0$ for $n>0$. For any $\lambda \in \h^*$, set $$\Delta(\lambda) = \frac {\lambda(\lambda+2)}{4\varkappa} - \frac \lambda2 , \qquad
\bar \Delta(\lambda) = -\frac {\lambda(\lambda+2)}{4\varkappa} - \frac \lambda2.$$
\[thm:Virasoro bimodule structure\] There exists a filtration $$\label{eq:Virasoro filtration}
0 \subset \tilde{\mathbb F}_\varkappa^{(0)} \subset \tilde{\mathbb F}_\varkappa^{(1)} \subset \tilde{\mathbb F}_\varkappa^{(2)} = \tilde{\mathbb F}_\varkappa$$ of $\virCC$-submodules of $\tilde{\mathbb F}_\varkappa$ such that $$\begin{aligned}
\tilde{\mathbb F}_\varkappa^{(2)}/\tilde{\mathbb F}_\varkappa^{(1)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \tilde V_{\Delta(-\lambda-2),c} \o \tilde V^\star_{\bar\Delta(-\lambda-2),\bar c} \label{eq:Virasoro socle F2},\\
\tilde{\mathbb F}_\varkappa^{(1)}/\tilde{\mathbb F}_\varkappa^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \( \tilde V_{\Delta(\lambda),c} \o \tilde V^\star_{\bar\Delta(-\lambda-2),\bar c} \oplus
\tilde V_{\Delta(-\lambda-2),c} \o \tilde V^\star_{\bar \Delta(\lambda),\bar c} \) \label{eq:Virasoro socle F1},\\
\tilde{\mathbb F}_\varkappa^{(0)} &\cong
\bigoplus_{\lambda\in \mathbf P} \tilde V_{\Delta(\lambda),c} \o \tilde V^\star_{\bar\Delta(\lambda),\bar c} \label{eq:Virasoro socle F0}.\end{aligned}$$
One can derive from Proposition \[thm:Wakimoto to FF\] that the for generic $\varkappa$ the reduction sends exact sequences of $\ghat_k$-modules to exact sequences of $\vir_c$-modules, which implies in particular that $$H_{DS}^n(\hat V_{\lambda,k}) = \begin{cases} \tilde V_{\Delta(\lambda),c}, & n=0 \\ 0, & n\ne 0 \end{cases},$$ It is then easy to check that the images $\tilde{\mathbb F}_\varkappa^{(0,1,2)}$ of the $\ghatKK$-submodules $\hat{\mathbb F}_\varkappa^{(0,1,2)}$ from Theorem \[thm:affine bimodule structure\] under the two-sided quantum Drinfeld-Sokolov reduction satisfy the required properties.
An alternative direct approach repeats the steps in the proof of Theorem \[thm:classical bimodule structure\]. In particular, we get a decomposition into blocks, $$\tilde{\mathbb F}_\varkappa = \tilde{\mathbb F}_\varkappa(-1) \oplus \bigoplus_{\lambda\in \mathbf P^+} \tilde{\mathbb F}_\varkappa(\lambda).$$
We also have the following Virasoro analogue of Corollary \[thm:classical positive subalgebra\] and Theorem \[thm:affine positive subalgebra\].
There exists a subspace $\tilde{\mathbf F}_\varkappa \subset \tilde{\mathbb F}_\varkappa$, satisfying
1. $\tilde{\mathbf F}_\varkappa$ is a vertex operator subalgebra of $\tilde{\mathbb F}_\varkappa$, and is generated by the quantum fields , and $\mathbb Y(1,z)$. In particular, $\tilde{\mathbf F}_\varkappa$ is a $\virCC$-submodule of $\tilde{\mathbb F}_\varkappa$.
2. As a $\virCC$-module, $\tilde{\mathbf F}_\varkappa$ is generated by the vectors $\{\1_\lambda\}_{\lambda \in \mathbf P^+}$, and we have $$\label{eq:Virasoro positive decomposition}
\tilde{\mathbf F}_\varkappa \cong \bigoplus_{\lambda \in \mathbf P^+} \tilde V_{\Delta(\lambda),c} \o \tilde V^\star_{\bar\Delta(\lambda), \bar c}.$$
The desired subspace $\tilde{\mathbf F}_\varkappa$ is the image of the vertex subalgebra $\hat{\mathbf F}_\varkappa$ under the two-sided quantum Drinfeld-Sokolov reduction. We leave technical details to the reader.
Semi-infinite cohomology of $\vir$
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The central charge for the diagonal action of $\vir$ in the modified regular representations is equal to the special value 26. In this section we study the semi-infinite cohomology of $\vir$ with coefficients in $\tilde {\mathbb F}_\varkappa$ and in $\tilde {\mathbf F}_\varkappa$.
The properties of the appropriate “space of semi-infinite forms” $\boldsymbol{\tilde\Lambda}^\semiinfty$ for the Virasoro algebra are summarized in the following
Set $\boldsymbol{\tilde\Lambda}^\semiinfty = \bigwedge \vir_- \o \bigwedge \vir'_+$, where $\vir_- = \bigoplus_{n\le-2} \C L_n$ and $\vir_+ = \bigoplus_{n\ge-1} \C L_n$. Then
1. The Clifford algebra, generated by $\{b_n, c_n \}_{n \in \Z}$ with relations $$\label{eq:bc system relations}
\{b_m, b_n \} = \{ c_m, c_n \} = 0, \qquad
\{b_m, c_n\} = \delta_{m+n,0}.$$ acts irreducibly on $\boldsymbol{\tilde\Lambda}^\semiinfty$, so that for any $\omega_- \in \bigwedge \vir_-, \, \omega_+ \in\bigwedge \vir'_+$ we have $$\begin{aligned}
b_n (1 \o \omega) &=
\begin{cases} 0,& n \ge -1 \\ 1 \o (L_n\wedge\omega), & n\le-2 \end{cases}, \qquad
c_n (\omega \o 1) &=
\begin{cases} (L'_{-n}\wedge\omega) \o 1, & n\le 1 \\ 0,& n\ge 2 \end{cases}.\end{aligned}$$
2. $\boldsymbol{\tilde\Lambda}^{\semiinfty}$ is a bi-graded vertex superalgebra, with vacuum $\1 = 1 \o 1$, generated by $$\begin{aligned}
{3}
b(z) &= \sum_{n\in\Z} b_n z^{-n-2}, \qquad | b(z) | = -1, \quad &\deg b(z) &= 2,\\
c(z) &= \sum_{n\in\Z} c_n z^{-n+1}, \qquad | c(z) | = 1, \quad &\deg c(z) &= -1.\end{aligned}$$
3. $\boldsymbol{\tilde\Lambda}^\semiinfty$ has a $\vir$-module structure with central charge $c = -26$, defined by $$\pi(L_n) = \sum_{m\in\Z} (m-n) :c_{-m} b_{n+m}:.$$
The BRST complex, associated with a $\vir$-module $\tilde V$ with central charge $c = 26$, is the complex $C^{\semiinfty+\bullet} (\vir, \C\mathbf c;\tilde V) = \boldsymbol{\tilde\Lambda}^{\semiinfty+\bullet} \otimes \tilde V$, with the differential $$\label{eq:Virasoro differential}
\tilde{\mathbf d} = \sum_{n\in\Z} c_{-n} \pi_{\tilde V}(L_n) - \frac 12 \sum_{m,n\in \Z} (m-n) :c_{-m} c_{-n} b_{m+n}:.$$ The corresponding cohomology is denoted $H^{\semiinfty+\bullet}(\vir,\C \mathbf c;\tilde V)$.
As in the affine case, the special value $c = 26$ of the central charge is required to ensure that $\tilde{\mathbf d}^2 = 0$.
The vertex superalgebras $H^{\semiinfty+\bullet}(\vir,\C \mathbf c;\tilde{\mathbf F}_\varkappa)$ and $H^{\semiinfty+\bullet}(\vir,\C \mathbf c;\tilde{\mathbb F}_\varkappa)$ degenerate into the commutative superalgebras, and we have commutative algebra isomorphisms $$\label{eq:Virasoro semi-infinite isomorphisms}
\begin{split}
H^{\semiinfty+\bullet}(\vir,\C \mathbf c;\tilde {\mathbf F}_\varkappa) \cong H^{\semiinfty+\bullet}(\ghat, \C\mathbf k;\hat{\mathbf F}_\varkappa), \qquad
H^{\semiinfty+\bullet}(\vir,\C \mathbf c;\tilde {\mathbb F}_\varkappa) \cong H^{\semiinfty+\bullet}(\ghat, \C\mathbf k;\hat{\mathbb F}_\varkappa).
\end{split}$$ In particular, $$H^{\semiinfty+0}(\vir,\C \mathbf c;\tilde {\mathbf F}_\varkappa) \cong
H^{\semiinfty+0}(\vir,\C \mathbf c;\tilde {\mathbb F}_\varkappa) \cong \C[\mathbf P]^W.$$
The problem of computing the semi-infinite cohomology of $\vir$, as well as its inherited algebra structure, has been extensively studied by mathematicians and physicists working in the string theory. We take advantage of these results, and construct our proof by combining entire blocks from previous papers.
We note that for both $\tilde{\mathbf F}_\varkappa$ and $\tilde{\mathbb F}_\varkappa$ the diagonal action of $\vir$ does not contain additional vertex operator shifts, and is equal to the sum of two standard Feigin-Fuks actions.
The comprehensive answer for the cohomology of tensor products of Feigin-Fuks and/or irreducible modules was given in [@LZ2] for the most difficult case of the central charge $c = c_{p,q}$, corresponding to $\varkappa = \frac pq \in \mathbb Q$. Simplified (for the case of generic $\varkappa$) version of their computations, and the spectral sequence associated with filtrations of Theorem \[thm:Virasoro bimodule structure\], yield
$$H^{\semiinfty+n}(\vir,\C\mathbf c; \tilde{\mathbf F}_\varkappa(\lambda)) =
\begin{cases}
\C, & n=0,3\\
0, & \text{otherwise}
\end{cases},$$ $$H^{\semiinfty+n}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa(-1)) =
\begin{cases}
\C, & n=1,2\\
0, & \text{otherwise}
\end{cases},\qquad
H^{\semiinfty+n}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa(\lambda)) =
\begin{cases}
\C, & n=0,2\\
\C^2, & n=1\\
0, & \text{otherwise}
\end{cases},$$ for each $\lambda \in \mathbf P^+$, as well as natural isomorphisms $$H^{\semiinfty+0}(\vir,\C\mathbf c; \tilde{\mathbf F}_\varkappa(\lambda)) \cong H^{\semiinfty+0}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa(\lambda)).$$ The algebra structure of $H^{\semiinfty+0}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa)$ is in fact independent of $\varkappa$, as can be seen from the change of variables $$p_n = \frac{a_n + \bar a_n}2, \qquad q_n = \frac {a_n - \bar a_n}{2\varkappa}.$$ Indeed, the new commutation relations become $[p_m,p_n] = [q_m,q_n] = 0$ and $[p_m,q_n] = \delta_{m+n,0}$, and the diagonal Virasoro action is given by $$\mathcal L(z) = :p(z)q(z): + p(z) - q(z).$$ For the special case $\varkappa = 1$, corresponding to the pairing of $c=1$ and $\bar c = 25$ modules, the cohomology of a bigger vertex algebra $\mathcal A_{2D} = \bigoplus_{\lambda,\mu \in \Z} \tilde F_{\lambda,1} \o \tilde F_{\mu,-1}$ was identified in [@WZ] with the polynomial algebra $\C[x,y]$ in two variables. The subalgebra $H^{\semiinfty+0}(\vir,\C c; \tilde{\mathbb F}_\varkappa)$ is therefore isomorphic to the polynomial algebra $\C[\chi]$, and we can take any nonzero cohomology class $\chi \in H^{\semiinfty+0}(\vir,\C c; \tilde{\mathbb F}_\varkappa(1))$ as the generator.
The vertex superalgebra structures on $H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbf F}_\varkappa)$ and $H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa)$ degenerate into commutative superalgebras. It is clear that both are free $\C[\chi]$-modules.
It follows immediately that $H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbf F}_\varkappa) \cong \C[\chi] \otimes \bigwedge^\bullet[\eta]$, where we can pick any non-zero element $\eta \in H^{\semiinfty+3}(\vir,\C\mathbf c; \tilde{\mathbf F}_\varkappa(0))$. This settles the case of $\tilde{\mathbf F}_\varkappa$.
To get the generators of $H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa)$, we pick non-zero representatives $$\xi_{-1} \in H^{\semiinfty+1}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa(-1)), \qquad
\eta_0 \in H^{\semiinfty+1}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa(0)),$$ such that $\eta_0$ is not proportional to $\chi \cdot \xi_{-1}$. One can check that $\eta_0 \xi_{-1} \ne 0$, and as in Theorem \[thm:classical cohomology\] it follows that $H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa) \cong \C[\chi] \otimes \bigwedge^\bullet[\xi_{-1},\eta_0]$. The statement now follows from Theorem \[thm:classical cohomology\] and Corollary \[thm:affine cohomology\].
It would be nice to get a direct proof of isomorphisms by using the techniques of the quantum Drinfeld-Sokolov reduction.
Extensions, generalizations, conjectures
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Heterogeneous vertex operator algebra
-------------------------------------
As we mentioned above, the vertex algebra construction for the Virasoro algebra can be obtained from their affine analogues by applying the two-sided quantum Drinfeld-Sokolov reduction to the left and right $\ghat$-action. One can consider a similar construction where the reduction is only applied to the affine action on one side, thus leading to a vertex operator algebra with two commuting actions of $\ghat_k$ and $\vir_{\bar c}$ with appropriate $k,\bar c$. Indeed, one can see that the following gives a direct realization of such a vertex algebra.
\[thm:dual pair action\] Let $\varkappa \ne 0$. Set $k = \varkappa - h^\vee,\ \bar c = 13 + 6\varkappa + \frac 6\varkappa$, and $\check {\mathbb F}_\varkappa = \hat F(\beta,\gamma) \o \tilde{\mathbb F}_\varkappa.$ Then
1. The space $\check{\mathbb F}_\varkappa$ has a $\ghat_k \oplus \vir_{\bar c}$-module structure, defined by $$\label{eq:heterogeneous affine current}
\begin{split}
\mathbf e(z) &= \gamma(z),\\
\mathbf h(z) &= 2:\beta(z) \gamma(z): + a(z),\\
\mathbf f(z) &= -:\beta(z)^2 \gamma(z): - \beta(z) a(z) - k \beta'(z) - \mathbb Y(-2,z),
\end{split}$$ $$\label{eq:heterogeneous Virasoro current}
\bar L(z) = -\frac {:\bar a(z)^2:}{4\varkappa} + \frac{\varkappa+1}{2\varkappa} \bar a'(z)
+ \frac 1\varkappa \, \mathbb Y(-2,z) \gamma(z).$$
2. The space $\check{\mathbb F}_\varkappa$ has a compatible VOA structure with $\operatorname{rank}\check{\mathbb F}_\varkappa = 28$.
The verification of commutation relations is straightforward. We define the Virasoro quantum field by $$\label{eq:heterogeneous total Virasoro}
\mathcal L(z) = \frac 1{2\varkappa} \(\frac {:\mathbf h(z)^2:}2 + :\mathbf e(z)\mathbf f(z): + :\mathbf f(z)\mathbf e(z):\)
+ \frac {\mathbf h'(z)}2 + \bar L(z)$$ The central charge for the Sugawara construction modified by $\frac {\mathbf h'(z)}2$ is equal to $\frac {3k}{k+h^\vee}-6k$, and we compute $$\operatorname{rank}\check{\mathbb F}_\varkappa = \( \frac{3(\varkappa-h^\vee)}\varkappa - 6(\varkappa-h^\vee) \) + \(13 + \frac 6\varkappa + 6\varkappa \) = 28.$$
We will call the vertex operator algebra of Theorem \[thm:dual pair action\] the heterogeneous VOA. Note (see [@Li] and references therein) that the central charge $c=28$ appears as the critical value in the study of 2D gravity in the light-cone gauge!
The structure of the bimodule $\check {\mathbb F}_\varkappa$ in the generic case is again quite similar to the non-semisimple bimodule $\mathfrak R(G_0)$. From now on we assume that
$$\varkappa \notin \mathbb Q, \qquad k = \varkappa - h^\vee, \qquad \bar c = 13 + \frac 6\varkappa + 6\varkappa.$$
\[thm:dual pair structure\] There exists a filtration $$0 \subset \check{\mathbb F}_\varkappa^{(0)} \subset \check{\mathbb F}_\varkappa^{(1)} \subset \check{\mathbb F}_\varkappa^{(2)} =
\check{\mathbb F}_\varkappa$$ of $\ghat_k \oplus \vir_{\bar c}$-submodules of $\check{\mathbb F}_\varkappa$, such that $$\begin{aligned}
\check{\mathbb F}_\varkappa^{(2)}/\check{\mathbb F}_\varkappa^{(1)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \hat V_{-\lambda-2,k} \o \tilde V^\star_{\bar\Delta(-\lambda-2),\bar c}, \\
\check{\mathbb F}_\varkappa^{(1)}/\check{\mathbb F}_\varkappa^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \( \hat V_{\lambda,k} \o \tilde V^\star_{\bar\Delta(-\lambda-2),\bar c} \oplus
\hat V_{-\lambda-2,k} \o \tilde V^\star_{\bar\Delta(\lambda),\bar c} \), \\
\check{\mathbb F}_\varkappa^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P} \hat V_{\lambda,k} \o \tilde V^\star_{\bar\Delta(\lambda),\bar c}.\end{aligned}$$
The heterogeneous VOA contains a vertex operator subalgebra, analogous to the classical Peter-Weyl subalgebra $\mathfrak R(G) \subset \mathfrak R(G_0)$.
There exists a subspace $\check{\mathbf F}_\varkappa \subset \check{\mathbb F}_\varkappa$, satisfying
1. $\check{\mathbf F}_\varkappa$ is a vertex operator subalgebra of $\check{\mathbb F}_\varkappa$, and is generated by the fields , , and $\mathbb Y(1,z)$. In particular, $\check{\mathbf F}_\varkappa$ is a $\ghat_k \oplus \vir_{\bar c}$-submodule of $\check{\mathbb F}_\varkappa$.
2. As a $\ghat_k \oplus \vir_{\bar c}$-module, $\check{\mathbf F}_\varkappa$ is generated by the vectors $\{\1_\lambda\}_{\lambda \in \mathbf P^+}$, and we have $$\check{\mathbf F}_\varkappa \cong \bigoplus_{\lambda \in \mathbf P^+} \hat V_{\lambda,k} \o \tilde V_{\bar\Delta(\lambda),\bar c}.$$
The proofs of the above theorems are obtained from their affine counterparts by applying the quantum Drinfeld-Sokolov reduction to (right) $\ghat_{\bar k}$-action, similarly to the Virasoro case.
The fact that the ranks of VOAs $\check{\mathbb F}_\varkappa$ and $\check{\mathbf F}_\varkappa$ are equal to 28 naturally leads to the consideration of the semi-infinite cohomology of these modules. We note that although the total Virasoro quantum field $\mathcal L(z)$ does not commute with $\ghat$, the spaces $\check{\mathbb F}_\varkappa$ and $\check{\mathbf F}_\varkappa$ can be regarded as modules over the semi-direct product $\virghat$, such that $$\begin{split}
[\mathcal L_m, \mathbf e_n] &= -(n+m+1)\, \mathbf e_{m+n},\\
[\mathcal L_m, \mathbf f_n] &= (m-n+1)\, \mathbf f_{m+n},\\
[\mathcal L_m, \mathbf h_n] &= -n\, \mathbf h_{m+n} + m(m+1)\delta_{m+n,0}\, k.
\end{split}$$
The semi-infinite cohomology is defined by the BRST complex of the subalgebra $\virnil$, where as before $\nhat_+ = \bigoplus_{n \in \Z} \C \mathbf e_n$ is the nilpotent loop subalgebra of $\ghat$. The condition $c=28$ ensures that the differential squares to zero.
The BRST complex, associated with a $\virnil$-module $\check V$ with Virasoro central charge $c=28$ is the complex $C^{\semiinfty+\bullet} (\vir, \C\mathbf c;\check V) = \boldsymbol{\tilde\Lambda}^{\semiinfty+\bullet} \o \hat\Lambda(\psi,\psi^*) \o \check V$, with the differential $$\label{eq:heterogeneous differential}
\begin{split}
\check{\mathbf d} &= \sum_{n\in\Z} c_{-n} \pi_{\check V}(\mathcal L_n) - \frac 12 \sum_{m,n\in \Z} (m-n) :c_{-m} c_{-n} b_{m+n}: + \\
&+ \sum_{n \in \Z} \psi^*_{-n} \pi_{\check V}(\mathbf e_n) - \sum_{m,n\in\Z} -(m+n+1) c_{-m} :\psi^*_{-n} \psi_{m+n}:
\end{split}$$ The corresponding cohomology is denoted $H^{\semiinfty+\bullet}(\virnil,\C \mathbf c;\check V)$.
The vertex algebras $H^{\semiinfty+\bullet}(\virnil,\C\mathbf c; \check{\mathbf F}_\varkappa)$ and $H^{\semiinfty+\bullet}(\virnil,\C\mathbb c; \check{\mathbf F}_\varkappa)$ degenerate into commutative algebra structures, and we have isomorphisms $$H^{\semiinfty+\bullet}(\virnil,\C\mathbf c; \check{\mathbf F}_\varkappa) \cong H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbf F}_\varkappa),$$ $$H^{\semiinfty+\bullet}(\virnil,\C\mathbf c; \check{\mathbb F}_\varkappa) \cong H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa).$$ In particular, $$H^{\semiinfty+0}(\virnil,\C\mathbf c; \check{\mathbf F}_\varkappa) \cong
H^{\semiinfty+0}(\virnil,\C\mathbf c; \check{\mathbb F}_\varkappa) \cong \C[\mathbf P]^W.$$
We use the technique from [@Li], where similar isomorphisms were established for relative cohomology spaces. Let $\mathbf d_{\n_+} = \sum_{n \in \Z} \psi^*_{-n} \pi(\mathbf e_n)$ be the BRST differential for $\n_+$; one can show that $\mathbf d_{\n_+}^2 = 0 = \{\mathbf d_{\n_+},\check{\mathbf d}\}$, which leads to the spectral sequence associated with decomposition $\check{\mathbf d} = \mathbf d_{\n_+} + (\check{\mathbf d} - \mathbf d_{\n_+})$. Computing the cohomology with respect to $\mathbf d_{\n_+}$ first, and using Proposition \[thm:Wakimoto to FF\], we get the desired statement. For full technical details (there is a slight difference between BRST reduction for $\n_+$ and quantum Drinfeld-Sokolov reduction, but it doesn’t affect the outcome) we refer the reader to [@Li].
General construction of vertex operator algebras and equivalence of categories
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The vertex operator algebras constructed in the previous sections can be built, like conformal field theories, by pairing the left and right modules from certain equivalent categories of representations of infinite-dimensional Lie algebras. The operators $\mathcal Y(\cdot,z)$ are then constructed by pairing the left and right intertwining operators. There is a unique choice of the structural coefficients for such pairing that would ensure the locality condition for the vertex operator algebras; these coefficients are determined by the tensor structure on the category of representations. Conversely, a natural VOA structure on a bimodule can be used to establish the equivalence of the left and right tensor categories.
The vertex operator algebra constructions in this paper deal with the pairings of different categories of modules. In the affine case, we pair the $\ghat$ modules on levels $k$ and $\bar k = -2h^\vee - k$, symmetric with respect to the critical level $-h^\vee$; the equivalence of the corresponding tensor categories was studied in [@Fi]. In the Virasoro case, we pair the modules with central charges $c$ and $\bar c = 26-c$.
The theorems of Peter-Weyl type can be extended to the quantum group $\mathcal U_q(\g)$, associated with $G$. On one hand, the modules from the category $\mathcal O$ can be $q$-deformed into modules over $\mathcal U_q(\g)$; on the other hand, one can define $q$-deformations $\mathfrak R_q(G)$ and $\mathfrak R_q(G_0)$ of the algebras of regular functions, which have especially simple description for $\g = \sl(2,\C)$. When $q$ is not a root of unity, we have the quantum analogues of isomorphisms , .
The Drinfeld-Kohno theorem establishes an isomorphism of tensor categories of representations of the quantum groups and affine Lie algebras when $q = \exp\(\frac{\pi i}{k+h^\vee}\)$. This equivalence, also extended to $\mathcal W$-algebras, was made explicit in [@S], where intertwining operators for $\mathcal U_q(\g)$ were directly identified with their VOA counterparts for $\ghat_k$ and $\mathcal W(\ghat_k)$; the key ingredients were the geometric results in [@V] on the homology of configuration spaces. The construction in [@S] built conformal field theories, associated to affine Lie algebras and $\mathcal W$-algebras based on their quantum group counterparts, and can be modified to produce the vertex algebras discussed in this paper.
The Drinfeld-Kohno equivalence also allows to couple categories of different types, producing in particular the heterogeneous VOA of the previous subsection. Another important case is the Frenkel-Kac construction, which corresponds to the pairing of modules for $\g$ and $\mathcal W(\ghat)$ with central charge $c=\dim \h$ (see [@F]). However, in general pairings between the quantum group and the affine Lie (or $\mathcal W$-) algebra lead to the generalized vertex algebra structures, satisfying a braided version of the commutativity axiom. An example of such structure was proposed in [@MR].
Integral central charge, semi-infinite cohomology and Verlinde algebras
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In this work we studied the structure of the generalized Peter-Weyl bimodules for $\ghat$ only for the generic values $k \notin \mathbb Q$ of the central charge. The structure of these bimodules when $k$ is integral is more complex and undoubtedly even more interesting. In the most special case when $k = \bar k = - h^\vee$, we get a regular representation of the affine Lie algebra $\ghat$ at the critical level, which can be viewed as the direct counterpart of the finite-dimensional case. This space admits a realization as a certain space of meromorphic functions on the affine Lie group $\hat G$, and subspaces of spherical functions with respect to conjugation give rise to solutions of the quantum elliptic Calogero-Sutherland system, generalizing the trigonometric analogue in the finite-dimensional case [@EFK].
Another special case is $k = - h^\vee+1, \bar k = -h^\vee - 1$, when the quantum group degenerates into its classical counterpart. In this case the left and right Fock spaces used in our construction each have separate vertex algebra structures, and the operators $\mathbb Y(-2,w)$, which play an important role in this paper, are factored into products of left and right vertex operators used in the basic representations of $\ghat$. The semi-infinite cohomology of the corresponding $\mathcal W$-algebras is fundamental to the string theory, and was studied in [@WZ] for the Virasoro algebra, and in [@BMP] for $\mathcal W_3$.
For positive integral $k$ one expects the existence of truncated versions $\hat{\mathbf F}_{k+h^\vee}$ and $\hat{\mathbb F}_{k+h^\vee}$ of our vertex operators algebras, similar to the truncation in the conformal field theory, where the positive dominant cone $\mathbf P^+$ is replaced by the alcove $\mathbf P^+_k \subset \mathbf P^+$. Then the relative semi-infinite cohomology of $\ghat$ with coefficients in $\hat{\mathbf F}_{k+h^\vee}$ and $\hat{\mathbb F}_{k+h^\vee}$ with respect to the center should be truncated correspondingly. The identification of the zero semi-infinite cohomology groups with the representation ring of $G$ in Corollary \[thm:affine cohomology\] leads to the following conjecture.
\[thm:conjecture\] For positive integral $k$, let $\mathbf V_k(\ghat)$ denote the Verlinde algebra associated with integrable level $k$ representations of $\ghat$, and let $\mathbb V_k(\ghat)$ denote its counterpart associated to the big projective modules (see [@La]). Then we have commutative algebra isomorphisms $$H^{\semiinfty+0}(\ghat,\C\mathbf k; \hat{\mathbf F}_{k+h^\vee}) \cong \mathbf V_k(\ghat), \qquad \qquad
H^{\semiinfty+0}(\ghat,\C\mathbf k; \hat{\mathbb F}_{k+h^\vee}) \cong \mathbb V_k(\ghat).$$
In other words, the most essential part of the VOA structure, embodied in the 0th cohomology, is equivalent to the structure of the fusion rules of the tensor category of $\ghat$-modules, encoded in the Verlinde algebra.
It was also realized recently (see [@FHT] and references therein) that the Verlinde algebra $\mathbf V_k(\ghat)$ admits an alternative realization in terms of twisted equivariant K-theory ${}^{k+h^\vee}K_G^{\dim G}(G)$ of a compact simple Lie group $G$ (which, in the notations of [@FHT], is the compact form of the complex Lie group which we denoted by $G$ in this paper).
Thanks to the results of [@FHT], the first isomorphism of our conjecture can be restated in a more invariant form.
We have a natural commutative algebra isomorphism $$H^{\semiinfty+0}(\ghat,\C\mathbf k; \mathfrak R'_{k+h^\vee}(\hat G)) \cong {}^{k+h^\vee}K_G^{\dim G}(G).$$
A conceivable direct geometric proof of the last isomorphism might combine the realization of the left hand side using the works [@GMS] and [@AG] with the interpretation of the right hand side given in the works [@FHT] and [@AS]. A similar K-theoretic interpretation of $H^{\semiinfty+0}(\ghat,\C\mathbf k; \hat{\mathbb F}_{k+h^\vee})$ in our second conjecture might add another twirl to the twisted equivariant K-theory.
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**From submodule categories to preprojective algebras.**
Claus Michael Ringel and Pu Zhang
Abstract: Let $\ssize S(n)$ be the category of invariant subspaces of nilpotent operators with nilpotency index at most $\ssize n$. Such submodule categories have been studied already in 1934 by Birkhoff, they have attracted a lot of attention in recent years, for example in connection with some weighted projective lines (Kussin, Lenzing, Meltzer). On the other hand, we consider the preprojective algebra $\ssize \Pi_n$ of type $\ssize \Bbb A_n$; the preprojective algebras were introduced by Gelfand and Ponomarev, they are now of great interest, for example they form an important tool to study quantum groups (Lusztig) or cluster algebras (Geiss, Leclerc, Schröer).
We are going to discuss the connection between the submodule category $\ssize \Cal S(n)$ and the module category $\ssize \mod \Pi_{n-1}$ of the preprojective algebra $\ssize \Pi_{n-1}$. Dense functors $\ssize \Cal S(n) \to \mod \Pi_{n-1}$ are known to exist: one has been constructed quite a long time ago by Auslander and Reiten, recently another one by Li and Zhang. We will show that these two functors are full, objective functors with index $\ssize 2n$, thus $\ssize \mod \Pi_{n-1}$ is obtained from $\ssize \Cal S(n)$ by factoring out an ideal which is generated by $\ssize 2n$ indecomposable objects.
As a byproduct we also obtain new examples of ideals in triangulated categories, namely ideals $\ssize \Cal I$ in a triangulated category $\ssize \Cal T$ which are generated by an idempotent such that the factor category $\ssize \Cal T/\Cal I$ is an abelian category.
[**1. Introduction.**]{} Let $k$ be a field. Let $S(n)$ be the category of invariant subspaces of nilpotent operators with nilpotency index at most $n$. For a detailed analysis of this category we refer to \[RS2\]. Let $k[x]$ be the polynomial ring in one variable $x$ with coefficients in $k$ and $\Lambda_n = k[x]/\langle x^n\rangle$. The objects of $\Cal S(n)$ are the pairs $(X,Y)$ where $Y$ is a $\Lambda_n$-module and $X$ is a submodule of $Y$ (or the corresponding inclusion maps $u\:X \to Y$). We denote the indecomposable $\Lambda_n$-module of length $i$ by $[i]$, and $[0]$ will denote the zero module.
Let $\Pi_n$ be the preprojective algebra of type $\Bbb A_n$ and $\mod \Pi_n$ the category of the $\Pi_n$-modules of finite length. The aim of this note is to show that $\mod \Pi_{n-1}$ is obtained from $\Cal S(n)$ by factoring out an ideal $\Cal I$ which is generated by $2n$ indecomposable objects — actually, we will exhibit two possible choices for $\Cal I$. Given an additive category $\Cal A$ and an ideal $\Cal I$ in $\Cal A$, we denote by $\Cal A/\Cal I$ the corresponding factor category (it has the same objects, and the homomorphisms in $\Cal A/\Cal I$ are the residue classes of the homomorphisms in $\Cal A$ modulo $\Cal I$). If $\Cal K$ is a class of objects of the category $\Cal A$, we denote by $\langle \Cal K\rangle$ the ideal of $\Cal A$ given by all maps which factor through a direct sum of objects in $\Cal K.$ Instead of writing $\Cal A/\langle \Cal K\rangle$, we just will write $\Cal A/\Cal K.$ If $\Cal A, \Cal B$ are Krull-Remak-Schmidt categories, then we say that a functor $F\:\Cal A \to \Cal B$ is [*objective,*]{} provided its kernel is generated by identity maps of objects (see the appendix). If $F$ is a dense, objective functor, then the number of isomorphism classes of indecomposable objects $A$ in $\Cal A$ with $F(A) = 0$ is called its index. Thus Theorem 1 asserts that there are full, dense, objective functors $\Cal S(n) \to \mod\Pi_{n-1}$ with index $2n.$ These functors are the main target of our considerations. Actually, the two functors $F,G\:\Cal S(n) \to \mod\Pi_{n-1}$ which we will deal with have been exhibited before: one of them has been constructed quite a long time ago by Auslander and Reiten \[AR2\], the other one recently by Li and Zhang \[LZ\], both are based on general considerations by Auslander \[A\], published already in 1965. What remains to be done is to show that the functors are full and objective. Both functors $F,G$ are given as compositions of functors which involve some further module categories and also these intermediate categories seem to be of interest.
First of all, let $T_2(\Lambda_n)$ be the ring of upper triangular $(2\times 2)$-matrices with coefficients in $\Lambda_n$. It is well-known (and easy to see) that the category $\mod T_2(\Lambda_n)$-modules can be identified with the category of maps between $\Lambda_n$-modules. Since the category $\Cal S(n)$ can be considered as the category of monomorphisms of $\Lambda_n$-modules, we see that $\Cal S(n)$ is a full subcategory of $\mod T_2(\Lambda_n)$, say with inclusion functor $\iota$. There is a second full embedding $\epsilon\: \Cal S(n) \to
\mod T_2(\Lambda_n)$, it sends the pair $(X,Y)$ in $\Cal S(n)$ to the canonical projection $Y \to Y/X$.
Next, given an algebra $\Lambda$ of finite representation type, we denote by $A(\Lambda)$ its (basic) Auslander algebra; it is defined as follows: let $E$ be a minimal Auslander generator for $\Lambda$, this is the direct sum of all indecomposable $\Lambda$-modules, one from each isomorphism class, and $A(\Lambda) = \End(E)^{\op}.$ Since the algebras $\Lambda_n$ are of finite representation type, we may consider $A_n = A(\Lambda_n)$. Of course, in this case $E = \bigoplus_{i=1}^n [i].$ In section 2, we study the Auslander algebra $A_n$ and the category $\Cal F(n)$ of the torsionless $A_n$-modules. Any Auslander algebra is quasi-hereditary, the Auslander algebras $A_n$ are quasi-hereditary in a unique way and the category $\Cal F(n)$ is just the category of $A_n$-modules with a $\Delta$-filtration.
The essential tool is the functor $$\alpha = \Cok\Hom_\Lambda(E,-) \: \mod T_2(\Lambda) \longrightarrow \mod A(\Lambda),$$ where $\Lambda$ is of finite representation type and $E$ is a minimal Auslander generator $\Lambda$; note that $\alpha$ sends a morphism $f$ in $\mod \Lambda$ (thus the object $f$ of the category $\mod T_2(\Lambda)$) to the cokernel of the induced map $\Hom_\Lambda(E,f)$; of course, $\Hom_\Lambda(E,f)$ is a map of $A(\Lambda)$-modules. This functor was considered already in 1965 by Auslander \[A\], and later by Auslander and Reiten in \[AR1\] and \[AR2\]. Section 3 and parts of section 6 are devoted to this functor.
Finally, let us note that $\Pi_{n-1}$ is a factor ring of $A_n$, namely $\Pi_{n-1} = A_n/\langle e\rangle$, where $e$ is an idempotent of $A_n$ such that $A_ne$ is indecomposable projective-injective. We will consider the functor $$\delta\:\mod A_n \to \mod \Pi_{n-1}$$ which sends any $A_n$-module $M$ to its largest factor module which is a $\Pi_{n-1}$-module, thus to $M/A_neM.$ Properties of this functor $\delta$ will be discussed in section 5. In particular, following \[DR\], we will look at the restriction $F_2$ of $\delta$ to the subcategory $\Cal F(n).$
Altogether, we will deal with the following functors $$%====================================
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\endpicture}$$ The upper composition $F = \delta\alpha\iota$ is the functor studied by Li and Zhang \[LZ\], the lower composition $G = \delta\alpha\epsilon$ is the one considered by Auslander and Reiten \[AR2\]. We will see that [*$F$ is a dense functor with kernel the ideal $\langle \Cal U\rangle,$ and $G$ is a dense functor with kernel the ideal $\langle \Cal V\rangle.$*]{} This yields the first part of Theorem 1.
The image of the functor $\alpha\iota$ is precisely the subcategory $\Cal F(n)$, thus we can write $F = F_2F_1$, where $F_1\:\Cal S(n) \to \Cal F(n)$ is the functor with $F_1(u) = \alpha(u)$, for $u\in \Cal S(n)$. It is known from the literature that the functors $F_1,F_2,G$ all are dense; our main concern is to show that they are full and objective, with index $n,n,2n$, respectively. This will be done in sections 4, 5, 6, respectively.
In order to compare the functors $F$ and $G$, we have to take into account the stable module category $\underline{\mod}\ \Pi_{n-1}$, it is obtained from the module category $\mod \Pi_{n-1}$ by factoring out the ideal generated by the identity maps of the projective modules. Since the algebra $\Pi_{n-1}$ is self-injective, the stable module category $\underline{\mod}\ \Pi_{n-1}$ is a triangulated category. We denote by $$\pi\:\mod \Pi_{n-1} \longrightarrow \underline{\mod}\ \Pi_{n-1}$$ the canonical projection. Note that this is a full, dense, objective functor of index $n-1$, its kernel is generated by the indecomposable projective $\Pi_{n-1}$-modules. We denote by $\Omega\:\underline{\mod}\ \Pi_{n-1} \to \underline{\mod}\ \Pi_{n-1}$ the syzygy functor, thus $\Omega(M)$ is the kernel of a projective cover of the $\Pi_{n-1}$-module $M$. $$\pi F = \Omega\pi G.$$ Section 8 draws the attention to the fact that in this setting we also obtain full, dense, objective functors $\Cal T \to \Cal A$ with finite index, such that $\Cal T$ is a triangulated category, $\Cal A$ an abelian category, thus with ideals in triangulated categories which are generated by an idempotent such that the corresponding factor categories are abelian.
In section 9 we provide illustrations concerning the change of the Auslander-Reiten quiver of $\Cal S(n)$ when we factor out the various ideals mentioned above. The authors are indebted to the referee for a careful reading of the paper and for many valuable comments concerning possible improvements of the paper, in particular, for suggesting to add section 9.
[**2. The Auslander algebra $A_n = A(\Lambda_n)$ and the subcategory $\Cal F(n)$ of $\mod A_n$.**]{} We recall that for $\Lambda$ a ring of finite representation type, $A(\Lambda)$ denotes the basic Auslander algebra of $\Lambda$, it is the opposite of the endomorphism ring of a minimal Auslander generator for $\Lambda$. We consider here the special case of $\Lambda = \Lambda_n =
k[x]/\langle x^n\rangle$ and its Auslander algebra $A_n = (\End E)^{\op}.$ Note that $E = \bigoplus_{i=1}^n[i]$, where $[i]$ is the indecomposable $\Lambda_n$-module of length $i$.
Let $P(i) = \Hom_\Lambda(E,[i]).$ This is an indecomposable projective $A_n$-module. The inclusions $[i] \to [i+1]$ in the category $\mod \Lambda_n$ yield a chain of inclusions $$P(1) \subset P(2) \subset \cdots \subset P(n-1) \subset P(n).$$ Let $\Delta(i) = P(i)/P(i-1)$ (with $P(0) = 0$). Note that $A_n$ is quasi-hereditary (for this ordering and only for this ordering) and the modules $\Delta(i)$ are the standard modules (but observe that the labeling of the simple $A_n$-modules exhibited here is the opposite of the labeling commonly used (see for example \[DR\]): in the present paper, it is the module $\Delta(1)$ which is projective and not $\Delta(n)$, and correspondingly, it is $P(n) = I(n)$ which is projective-injective and not $P(1)$). Let $T(i) = P(n)/P(i-1)$ for $1\le i \le n$. Note that $T(i)$ is also the largest submodule of $P(n-i+1)$ which is generated by $T(n)$. Let $T = \bigoplus_i T(i),$ this is the characteristic tilting module for the quasi-hereditary algebra $A_n$. We have avoided to refer to the labeling of the simple modules. If we use our labels, so that $P(n) = I(n)$ is the unique indecomposable module which is both projective and injective, then (v) can be reformulated as saying that (vi) [*the injective envelope $IM$ of $M$ is a direct sum of copies of $I(n)$,*]{} or also that (vii) [*the socle of $M$ is a direct sum of copies of $S(n)$.*]{} It has been shown in \[R\] that any Auslander algebra is left strongly quasi-hereditary. This means that any module with a $\Delta$-filtration has projective dimension at most $1$ (the implication (iii) $\implies$ (iv)). Note that most of the relevant properties of strongly quasi-hereditary algebras have been considered already in \[DR\] and Proposition 1 is just a reminder. We denote by $\Cal F(n)$ the full subcategory of $\mod A_n$ given by the $A_n$-modules which satisfy the equivalent conditions of Proposition 1. It follows directly from (i) or also (v) that $\Cal F(n)$ is closed under submodules. Proof of Proposition 1. (i) $\implies$ (ii). Let $M$ be torsionless. There is an embedding $M \to P$ with $P$ projective. Since the injective dimension of $T$ is $1$, the canonical map $\Ext^1(P,T) \to \Ext^1(M,T)$ is surjective. But $\Ext^1(P,T) = 0$, thus $\Ext^1(M,T) = 0.$
\(ii) $\implies$ (iii). We have to show that $\Ext^i(M,T) = 0$ for all $i\ge 1.$ Since $T$ has injective dimension $1$, we only have to look at $i = 1$, but this is assertion (ii).
\(iii) $\implies$ (iv). The modules $\Delta(i)$ have projective dimension at most 1.
\(iv) $\implies$ (v). Let $0 @>>> P_1 @>u>> P_0 @>>> M @>>> 0$ be a projective resolution. We have to show that $M$ embeds into a module which is both injective and projective. Consider the injective envelopes of $v_i\:P_i \to I(P_i)$ for $i=0,1$, this yields a commutative diagram with exact rows $$\CD
0 @>>> P_1 @>u>> P_0 @>>> M @>>> 0 \cr
@. @Vv_1 VV @VVv_0 V @VVf V \cr
0 @>>> I(P_1) @>u'>> I(P_0) @>>> M' @>>> 0 .
\endCD$$ Since $v_0$ is injective, the snake lemma yields an embedding of the kernel $K$ of $f$ into the cokernel of $v_1$. For any projective module $P$ with injective envelope $IP$, the cokernel $IP/P$ embeds into a projective-injective module (since the dominant dimension of $\Lambda_n$ is at least 2).
On the other hand, $u'$ is a monomorphism and $I(P_1)$ is injective, thus $u'$ is a split monomorphism. Thus $M'$ is a direct summand of $I(P_0)$ and this module is projective-injective. The exact sequence $0 \to K \to M \to M'$ shows that $M$ embeds into $IK\oplus M'$ which is projective-injective.
\(v) $\implies$ (i). If the injective envelope of $M$ is projective, then $M$ embeds into a projective module, thus $M$ is torsionless. $\square$ Proof: Write $M = P/U$ where $P$ is a direct sum of copies of $P(n)$ and $U$ is a submodule of $P$. Since the projective dimension of $M$ is at most 1, the submodule $U$ is projective. Now $P$ is injective, thus we may embed an injective envelope $IU$ of $U$ into $P$, this is a direct summand of $P$, say $P = IU\oplus C$ for some submodule $C$. Since $P$ is a direct sum of copies of $P(n)$, also $C$ is a direct sum of copies of $P(n) = T(1).$ On the other hand, the exact sequences $0 \to P(i) \to P(n) \to T(i+1) \to 0$ for $1\le i \le n$ (with $T(n+1) = 0$) show that for any indecomposable projective module $P'$, the module $I(P')/P'$ belongs to $\add T$, thus $IU/U$ belongs to $\add T.$ Altogether we see that $P/U = IU/U\oplus C$ belongs to $\add T.$ $\square$ The modules generated by $P(n) = T(1)$ are the modules with a $\nabla$-filtration (see \[DR\]), thus proposition 2 just asserts that modules which have both a $\Delta$-filtration and a $\nabla$-filtration belong to $\add T.$ [**3. The functor $\alpha\:\mod T_2(\Lambda) \to \mod A(\Lambda)$.**]{} Here we deal with an arbitrary representation-finite algebra $\Lambda$ with minimal Auslander generator $E$ and consider the corresponding Auslander algebra $A(\Lambda) = (\End E)^{\op}$. We have denoted $T_2(\Lambda)$ the ring of all upper triangular $(2\times 2)$-matrices with coefficients in $\Lambda$. The category of $T_2(\Lambda)$-modules may be seen as the category of all morphisms $X \to Y$ in $\mod\Lambda$. We consider the functor $$\alpha\:\mod T_2(\Lambda) \to \mod A(\Lambda)$$ defined by $\alpha(f) = \Cok\Hom_\Lambda(E,f)$ for a morphism $f$ in $\mod\Lambda$
Thus, if the number of isomorphism classes of indecomposable $\Lambda$-modules is $m$, [*then $\alpha\:\mod T_2(\Lambda) \to \mod A(\Lambda)$ is a full, dense, objective functor with index $2m$.*]{} For a slightly weaker statement we refer to Theorem 1.1 in \[AR2\]. Let us recall and complete the proof. Given a morphism $f\:X \to Y$ in $\mod\Lambda$, we obtain an exact sequence $$\Hom_\Lambda(E,X) @>\Hom_\Lambda(E,f)>> \Hom_\Lambda(E,Y) @>>> \alpha(f) @>>> 0.$$ Since the $A(\Lambda)$-modules $\Hom_\Lambda(E,X)$ and $\Hom_\Lambda(E,Y)$ are projective, we obtain in this way a projective presentation of $\alpha(f).$ Conversely, given an $A(\Lambda)$-module $M$, take a projective resolution $P_1 @>p>> P_0 \to M \to 0.$ Now, the category of projective $A(\Lambda)$-modules is equivalent to the category $\mod \Lambda$, thus we can assume that there are $\Lambda$-modules $X$ and $Y$ such that $P_1 = \Hom_\Lambda(E,X)$ and $P_0 = \Hom_\Lambda(E,Y)$ and a $\Lambda$-homomorphism $f\:X \to Y$ such that $\Hom_\Lambda(E,f) = p.$ In this way, we see that the functor is dense. Similarly, starting with an $A(\Lambda)$-homomorphism $M \to M'$, we can lift it to projective presentations of $M$ and $M'$ and using again the fact that the category of projective $A(\Lambda)$-modules is equivalent to the category $\mod \Lambda$, we see that $M \to M'$ is in the image of the functor $\alpha.$ Thus, it remains to calculate the kernel of $\alpha.$
Of course, under this functor $\alpha$ the two objects in $\Cal X$ are sent to zero. Thus the ideal $\langle \Cal X\rangle$ is contained in the kernel of $\alpha$. Conversely, assume that there is given a map $$(g_1,g_0)\:(f\:X_1\to X_0) \longrightarrow (f'\:X_1' \to X_0')$$ (thus $g_0f = f'g_1$), such that $\alpha(g_1,g_0) = 0.$ Thus the following diagram commutes and its rows are projective presentations: $$\CD
\Hom_\Lambda(E,X_1) @>\Hom_\Lambda(E,f)>> \Hom_\Lambda(E,X_0) @>e>> \alpha(f) @>>> 0\cr
@V\Hom_\Lambda(E,g_1)VV @VV\Hom_\Lambda(E,g_0)V @VV{\alpha(g_1,g_0)=0}V \cr
\Hom_\Lambda(E,X_1') @>\Hom_\Lambda(E,f')>> \Hom_\Lambda(E,X_0') @>e'>> \alpha(f') @>>> 0.
\endCD$$ Let us show that the map $(g_1,g_0)$ factors through $[1\ 0]\:X_0\oplus X_1 \to X_0.$ Now $e'\Hom_\Lambda(E,g_0) = 0$, thus there is a map $\widetilde h\:\Hom_\Lambda(E,X_0) \to
\Hom_\Lambda(E,X_1')$ such that $\Hom_\Lambda(E,f')\widetilde h = \Hom_\Lambda(E,g_0).$ Again using the equivalence of $\mod \Lambda_n$ and the category of projective $A_n$-modules, we get a map $h\:X_0 \to X_1'$ with $\widetilde h = \Hom_\Lambda(E,h)$ and $g_0 = f'h.$ Therefore $f'hf = g_0f = f'g_1$, and thus $f'(g_1-hf) = 0.$ As a consequence, the following diagram commutes $$\CD
X_1 @>f>> X_0 \cr
@V{\left[\smallmatrix f\cr 1\endsmallmatrix\right]}VV @VV1V \cr
X_0\oplus X_1 @>{\left[\smallmatrix 1&0\endsmallmatrix\right]}>> X_0 \cr
@V{\left[\smallmatrix h& g_1-hf\endsmallmatrix\right]}VV @VVg_0V \cr
X'_1 @>f'>> X'_0
\endCD$$ and the composition of the vertical maps is just $(g_1,g_0)$. Since $E$ is an Auslander generator, all $\Lambda_n$-modules belong to $\add E$, thus $[1\ 0]\:X_0\oplus X_1 \to X_0$ belongs to $\add \Cal X.$ This shows that the map $(g_1,g_0)$ belongs to $\langle\Cal X\rangle.$ $\square$ [**4. The functor $F_1:\Cal S(n) \to \Cal F(n)$.**]{} It may be appropriate to focus first the attention to the category $\Cal S(n)$ itself. An object $(X,Y)$ of $\Cal S(n)$ with inclusion map $u\:X \to Y$ will also be denoted by $u$. This stresses the fact that objects of $\Cal S(n)$ are given by maps in the category $\mod \Lambda_n$, thus we consider $\Cal S(n)$ as a full subcategory of the category of $T_2(\Lambda_n)$-modules. As we have mentioned, we denote by $\iota\:\Cal S(n) \to \mod T_2(\Lambda_n)$ the inclusion functor. Note that $\Cal S(n)$ turns out to be just the category of all Gorenstein-projective $T_2(\Lambda_n)$-modules, see for example \[Z\]. We consider the restriction $\alpha\iota$ of $\alpha$ to the full subcategory $\Cal S(n)$ of $\mod T_2(\Lambda_n)$, thus $$\alpha\iota(u) = \Cok \Hom_\Lambda(E,u).$$ for $(u\:X\to Y)$ in $\Cal S(n)$. Since $u$ is a monomorphism, also $\Hom_\Lambda(E,u)$ is a monomorphism, thus there is the following exact sequence $$0 \to \Hom_\Lambda(E,X) \to \Hom_\Lambda(E,Y) \to \alpha(u) \to 0.$$ Since both $\Hom_\Lambda(E,X)$ and $\Hom_\Lambda(E,Y)$ are projective $A_n$-modules, we see that the projective dimension of $\alpha(u)$ is at most $1$, thus $\alpha(u)$ belongs to $\Cal F(n).$ This shows that we can consider $\alpha\iota$ as a functor $\Cal S(n) \to \Cal F(n)$, we denote it by $F_1$ (thus $F_1(u) = \alpha(u)$, for $u\in \Cal S(n)$).
Under the functor $F_1$, we have $$\alignat 2
F_1([i],[i]) &= 0 &&1\le i \le n, \cr
F_1([i],[n]) &= P(n)/P(i) = T(i\!+\!1) &\qquad\text{for}\qquad &0\le i \le n\!-\!1, \cr
F_1([0],[j]) &= P(j) &&1\le j\le n\!-\!1. \cr
\endalignat$$ Proof. By definition, $F_1$ is the restriction of the functor $\alpha$ to the subcategory $\Cal S(n)$ of $\mod T_2(\Lambda_n)$. We have noted already that the image of $F_1$ consists of modules of projective dimension at most $1$. Also, conversely, if $M$ is a $A_n$-module of projective dimension at most $1$, say with a projective presentation $$0 @>>> P_1 @>p>> P_0 @>>> M @>>> 0,$$ then we can write $p = \Hom_\Lambda(E,f)$ for some map $f\:X_1 \to X_0$ in $\mod \Lambda_n$. Clearly, $f$ has to be a monomorphism, since $E$ is an Auslander generator. Thus we can assume that $f$ belongs to $S(n)$ and we have $F_1(f) = M.$
According to Proposition 3 the kernel of the functor $F_1$ is given by all morphisms in $\Cal S(n)$ which factor through a $T_2(\Lambda_n)$-module of the form $[1\ 0]\:V \oplus V' \to V.$ Thus assume we have the following commutative diagram $$ X @>u>> Y @VVV @VVg”V VV’ @>>> V @VVV @VVh”V X’ @>u’>> Y’. $*$ $$The commutativity of the lower square means that we have$$ \[h”,0\] = h”\[1,0\] = u’\[h,h’\] = \[u’h,u’h’\], $$ thus $u'h' = 0.$ Since $u'$ is a monomorphism, it follows that $h' = 0.$ But then also the diagram $$\CD
X @>u>> Y \cr
@Vg VV @VVg''V \cr
V @>1>> V \cr
@Vh VV @VVh''V \cr
X' @>u'>> Y'.
\endCD$$ commutes and the composition of the vertical maps is the same as the composition of the vertical maps in $(*)$. This shows that the kernel of the functor $F_1$ consists of all the morphisms in $S(n)$ which factor through objects of the form $(1\:V \to V)$, but this is just the ideal $\Cal U_1.$ $\square$ [**5. The functor $\delta\:\mod A_n \to \mod \Pi_{n-1}$ and its restriction $F_2$ to $\Cal F(n)$.**]{} Recall that the indecomposable projective $A_n$-modules are $P(i) = \Hom_\Lambda(E,[i])$ with $1\le i \le n$, if necessary, we will denote them by $P_A(i)$. The module $P(n)$ is also injective and we may choose an idempotent $e(n)$ in $A_n$ such that $P(n) = A_ne(n)$. Let $\Pi_n$ be the preprojective algebra of type $\Bbb A_n$; note that $A_n/\langle e(n)\rangle = \Pi_{n-1}$ (see \[DR\], Theorem 3, Theorem 4 and Chapter 7). We consider in $\mod A_n$ the following torsion pair: the torsion modules are the modules generated by $P(n)$, thus the modules with top being a direct sum of copies of $S(n)$, the torsionfree modules are the modules which do not have $S(n)$ as a composition factor (thus the torsionfree modules form a Serre subcategory). Note that the torsionfree modules are just the $A_n$-modules $M$ with $e(n)M = 0$, thus these are the $\Pi_{n-1}$-modules.
Given an $A_n$-module $M$, let $tM$ be its torsion submodule, and $$\delta M = M/tM.$$ The indecomposable projective $\Pi_{n-1}$-modules are factor modules of the modules $P_A(i)$ with $1\le i \le n\!-\!1$, we denote them by $P_\Pi(i) = P_A(i)/tP_A(i).$
Since we want to keep track of the selected objects $([i],[j])$ in $\Cal W$, let us repeat what happens under the functor $\delta$: $$\alignat 2
\delta(T(i)) &= 0 &\qquad\text{for}\qquad &1\le i \le n, \cr
\delta(P_A(j)) &= P_\Pi(j) &&1\le j\le n\!-\!1. \cr
\endalignat$$ Proof. First, let us show that the functor $\delta$ is dense. Thus, let $N$ be a $\Pi_{n-1}$-module, but consider it as an $A_n$-module. Actually, all the modules to be considered now are $A_n$-modules; in particular $I(i)$ will denote (as before) the indecomposable injective $A_n$-module corresponding to the vertex $i$.
Let $u\:N \to IN$ be an injective envelope of $N$ (as an $A_n$-module!) and $p\:PIN \to IN$ a projective cover of $IN$. Let $T'$ be the kernel of $p$. Now $IN$ is a direct sum of modules of the form $I(i)$ with $1\le i \le n-1$ and the exact sequence $0 \to T(i+1) \to P(n) \to I(i) \to 0$ shows that the kernel of the projective cover $P(n) \to I(i)$ is $T(i+1)$. Thus we see that $T'$ is a direct sum of copies of modules of the form $T(j)$ with $2\le j \le n.$
Forming the induced exact sequence with respect to $u$, we obtain the following commutative diagram with exact rows: $$\CD
0 @>>> T' @>v'>> \widetilde N @>p'>> N @>>> 0 \cr
@. @| @VVu'V @VVu V \cr
0 @>>> T' @>v>> PIN @>p>> IN @>>> 0
\endCD$$ The monomorphism $u'$ shows that $\widetilde N$ is torsionless, thus in $\Cal F(n)$. Clearly, $tT' = T'$ and $tN =0$, thus $t\widetilde N$ is the image of $v'$ and therefore $\delta(\widetilde N) = N.$ This shows that $\delta$ is dense. In the same way, we see that $\delta$ is also full. Namely, given a morphism $f\:N_1 \to N_2$ of $\Pi_{n-1}$-modules, we can extend $f$ to a morphism $f'\:I(N_1) \to I(N_2)$ and then lift it to a morphism $f''\:PI(N_1)
\to PI(N_2).$ Using the pullback property of the commutative square $$\CD
\widetilde N_2 @>p'>> N_2 \cr
@VVu'V @VVu V \cr
PI(N_2) @>p>> I(N_2)
\endCD$$ we finally obtain a map $\widetilde f\:\widetilde N_1 \to \widetilde N_2$ such that $\delta(\widetilde f) = f.$
It remains to determine the kernel of $\delta$. Since the modules $T(i)$ are generated by $T(1)$, they are torsion modules, thus $\delta(T(i)) = 0$ for all $1\le i \le n$. Conversely, let $M_1,M_2$ belong to $\Cal F(n)$ and let $f\:M_1 \to M_2$ be a homomorphism such that $\delta(f) = 0.$ This means that the image of $f$ is contained in the torsion submodule $tM_2.$ Now $tM_2$ is a submodule of $M_2 \in \Cal F(n)$ and $\Cal F(n)$ is closed under submodules, thus $tM_2$ has projective dimension at most $1$. On the other hand, $tM_2$ is generated by $P(n)$. Thus Proposition 2 asserts that $tM_2$ is in $\add T$. It follows that $f$ belongs to the ideal $\langle T\rangle.$
The last assertion has been mentioned already before. $\square$ In order to show the density of the functor $\delta$, the paper \[DR\] started with a $\Pi_{n-1}$-module $N$, considered it as an $A_n$-module and used a universal extension $$0 \to T' \to \widetilde N \to N \to 0$$ of $N$ by a module $T'$ in $\add T$. Of course, the pullback recipe given in the proof above provides such a universal extension. Looking at the exact sequence $$0 \to T' \to \widetilde N \to N \to 0$$ constructed in the proof of Proposition 5, one may decompose $T' = \bigoplus_i T(i)^{n(i)}$. Then $n(1) = 0$ and [*$n(i+1)$ is precisely the multiplicity of $S(i)$ in the socle of $N$.*]{} Combining Propositions 4 and 5 we obtain the first part of the following proposition: The last assertion relies on the fact that the set of projective objects of an abelian category is uniquely determined. Since $F([0],[j]) = P_{\Pi}(j)$, for $1\le j \le n-1$, we know that the objects $([0],[1]),\dots ,([0],[n-1])$ in $\Cal S(n)/\Cal U$ are the indecomposable projective objects in $\Cal S(n)/\Cal U$. Thus any equivalence between $\Cal S(n)/\Cal U$ and $\mod \Pi_{n-1}$ sends these objects to the indecomposable projective $\Pi_{n-1}$-modules. $\square$ [**6. The functor $G\:\Cal S(n) \to \mod\Pi_{n-1}$.**]{} In this section, we are going to analyze the functor $G\:\Cal S(n) \to \mod\Pi_{n-1}$. The essential observation is due to Auslander and Reiten, it is valid in the general setting of dealing with a representation-finite algebra $\Lambda$ as discussed in section 3. We may refer to Proposition 4.1 in \[AR1\], see also the formulation stated in \[AR2\] just after the proof of Theorem 1.1. But the reader should observe that for the proof one just has to use twice Yoneda isomorphisms. Namely, starting with the map $f\:X \to Y$ of $\Lambda$-modules, consider the map $\Hom_\Lambda(E,f)\:\Hom_\Lambda(E,X) \to \Hom_\Lambda(E,Y)$ of $A$-modules; by definition, $\alpha(f)$ is its cokernel, thus there is the exact sequence $$\CD
\Hom_\Lambda(E,X) @>\Hom_\Lambda(E,f)>> \Hom_\Lambda(E,Y) @>>> \alpha(f) @>>> 0.
\endCD$$ If we apply the functor $H = \Hom_A(\Hom_\Lambda(E,\Lambda),-)$, we obtain the sequence $$\CD
H(\Hom_\Lambda(E,X)) @>H(\Hom_\Lambda(E,f))>> H(\Hom_\Lambda(E,Y)) @>>> H(\alpha(f)) @>>> 0,
\endCD$$ it is exact, since $\Hom_\Lambda(E,\Lambda)$ is a projective $A$-module. In this way, we see that $H(\alpha(f)) = \Hom_A(\Hom_\Lambda(E,\Lambda),\alpha(f))$ is the cokernel of $H(\Hom_\Lambda(E,f)).$ There is the following commutative diagram $$\CD
H(\Hom_\Lambda(E,X)) @>H(\Hom_\Lambda(E,f))>> H(\Hom_\Lambda(E,Y)) \cr
@VVV @VVV \cr
\Hom_\Lambda(\Lambda,X) @>\Hom_\Lambda(\Lambda,f)>> \Hom_\Lambda(\Lambda,Y)
\endCD$$ where the vertical maps are Yoneda isomorphisms. The lower map may be identified with the map $f$ (again, this is due to Yoneda isomorphisms). Thus $f$ is an epimorphism if and only if $\Hom_\Lambda(\Lambda,f)$ is an epimorphism if and only if $\Hom_A(\Hom_\Lambda(E,\Lambda),\Hom_\Lambda(E,f))$ is an epimorphism, if and only of the cokernel of $\Hom_A(\Hom_\Lambda(E,\Lambda),\Hom_\Lambda(E,f))$ is zero, thus if and only if $\Hom_A(\Hom_\Lambda(E,\Lambda),\alpha(f)) = 0$. $\square$
Let us return to the special case of $\Lambda = \Lambda_n$ and $A = A_n$. Recall that we have chosen an idempotent $e(n)$ in $A$ such that $A_ne(n) = P(n) = \Hom_{\Lambda_n}(E,[n]) = \Hom_{\Lambda_n}(E,\Lambda_n)$ and we have identified $A_n/\langle e(n)\rangle = \Pi_{n-1}$. Thus, given an $A_n$-module $M$, the condition $\Hom_{A_n}(\Hom_ {\Lambda_n}(E,\Lambda_n),M) = 0$ can be rewritten as $\Hom_{A_n}(P(n),M) = 0,$ thus as $e(n)M = 0,$ and the modules $M$ with this property are just the $\Pi_{n-1}$ modules. Thus, in our case the Lemma asserts the following: [*Let $f$ be a morphism of $\Lambda_n$-modules. Then $f$ is an epimorphism if and only if $\alpha(f)$ is a $\Pi_{n-1}$-module.*]{} We consider now the functor $G = \delta\alpha\epsilon.$ If $u\:X \to Y$ in $\Cal S(n)$, then $\epsilon(u)$ is an epimorphism, thus $\alpha\epsilon(u)$ is a $\Pi_{n-1}$-module and therefore $$G(u) = \delta\alpha\epsilon(u) = \alpha\epsilon(u).$$ This shows that $G\:\Cal S(n) \to \mod\Pi_{n-1}$ is defined by $$G(X,Y) = \Cok\Hom_\Lambda(E,Y\to Y/X),$$ we form the cokernel map $q = \epsilon(u)\:Y \to Y/X$ of the inclusion map $u\:X\to Y$, apply the functor $\Hom_\Lambda(E,-)$ so that we obtain a map $$\Hom_\Lambda(E,q)\:\Hom_\Lambda(E,Y) \to \Hom_\Lambda(E,Y/X)$$ and take its cokernel $\Cok\Hom_\Lambda(E,q)$.
[**Remark.**]{} We may phrase the definition of $G$ also differently: Let $\Cal Q(n)$ be the category of all epimorphisms $(Y \to Z)$ in $\mod \Lambda_n$, this is again a subcategory of $\mod T_2(\Lambda_n)$ and we may look at the restriction of the functor $\alpha$ to $\Cal Q(n)$. Of course, there is an obvious categorical equivalence $\Cal S(n) \to \Cal Q(n)$, it sends an object $(u\:X \to Y)$ to the canonical map $(Y \to Y/X),$ this is just $\epsilon(u)$ and $G = \alpha\epsilon.$
Instead of looking at the categories $\Cal S(n)$ and $\Cal Q(n)$, we may consider the category $\Cal E(n)$ of all short exact sequences in the category $\mod \Lambda_n$. There are forgetful functors $\Cal E(n) \to \Cal S(n)$ and $\Cal E(n) \to \Cal Q(n)$ which send an exact sequence $0 @>>> X @>u>> Y @>q>> Z @>>> 0$ to $u$ or $q$, respectively. Obviously, both functors are categorical equivalences. Under the functor $G$, we have $$\alignat 2
G([i],[i]) &= 0 &&1\le i \le n, \cr
G([i],[n]) &= P_{\Pi}(n\!-\!i) &\qquad\text{for}\qquad &1\le i \le n\!-\!1, \cr
G([0],[j]) &= 0 &&1\le j\le n. \cr
\endalignat$$ (The second assertion is seen as follows: Let $q\:[n] \to [n-i]$ be a cokernel map for the inclusion map $u\:[i] \to [n]$. If we apply $\Hom_\Lambda(E,-)$ to the short exact sequence with maps $u$ and $q$, we obtain the exact sequence $$\CD
0 \to P(i) @>\Hom_\Lambda(E,u)>> P(n) @>\Hom_\Lambda(E,q)>> P(n-i).
\endCD$$ Now the embedding $\Hom_\Lambda(E,u)$ of $P(i)$ into $P(n)$ has as cokernel the module $T(i+1)$. In this way, $T(i+1)$ is embedded into $P(n-i)$; the image $U$ of this embedding is the largest submodule of $P(n-i)$ generated by $P(n)$. This shows that $P(n-i)/U$ is equal to $P_{\Pi}(n\!-\!i)$. On the other hand, $P(n-i)/U$ is just the cokernel of $\Hom_\Lambda(E,q)$, thus equal to $G([i],[n]).$) Proof. Proposition 3 asserts that $\alpha$ is a full functor from $\Cal Q(n)$ onto the category $\mod \Pi_{n-1}$ and that its kernel is the set of morphisms which factor through an object of the form $([1,0]\:V\oplus V' \to V).$ Under the equivalence $\Cal S(n) \to
\Cal Q(n)$, the objects of the form $([1,0]\:V\oplus V' \to V)$ in $\Cal Q(n)$ correspond to the objects of the form $(\left[\smallmatrix 0\cr 1
\endsmallmatrix\right]\:V' \to V\oplus V')$ in $\Cal S(n)$, but these are precisely the objects in $\add \Cal V.$ This shows that $G= \alpha\epsilon$ yields an equivalence $\Cal S(n)/\Cal V \to \mod\Pi_{n-1}$.
By proposition 3, we know that the functor $\alpha\:\mod T_2(\Lambda_n) \to \mod A_n$ is full and dense. Let us show that the restriction of $\alpha$ to $\Cal Q(n)$ is a dense functor $\Cal Q(n) \to \mod \Pi_{n-1}$. If $M$ is a $\Pi_{n-1}$-module, the density of $\alpha$ provides a map $f$ of $\Lambda_n$-modules such that $\alpha(f)$ is isomorphic to $M$. But since $\alpha(f)$ is a $\Pi_{n-1}$-module, we know that $f$ is an epimorphism (see the reformulation of the Lemma above), thus $f$ belongs to $\Cal Q(n)$.
The last assertion of Proposition 7 relies again on the fact that the set of projective objects of an abelian category is uniquely determined by the categorical structure. $\square$ This completes the proof of Theorem 1.
[**7. Comparison of the functors $F$ and $G$.**]{} We want to compare the functor $F = F_2F_1$ and $G$. In order to prove the equality $\pi F = \Omega\pi G,$ we start with an object $u\:X \to Y$ in $\Cal S(n)$, thus with an exact sequence $$0 @>>> X @>u>> Y @>q>> Z @>>> 0.$$ We apply the functor $\Hom_\Lambda(E,-)$ and obtain the exact sequence $$0 @>>> \Hom_\Lambda(E,X) @>\Hom_\Lambda(E,u)>> \Hom_\Lambda(E,Y) @>\Hom_\Lambda(E,q)>> \Hom_\Lambda(E,Z)$$ Now, the cokernel of $\Hom_\Lambda(E,q)$ is $G(u)$. The cokernel of $\Hom_\Lambda(E,u)$ and thus the image of $\Hom_\Lambda(E,q)$ is $F_1(u)$, thus there is the following exact sequence $$0 @>>> F_1(u) @>>> \Hom_\Lambda(E,Z) @>>> G(u) @>>> 0$$ and we can assume that the map $F_1(u) \to \Hom_\Lambda(E,Z)$ is an inclusion map. Since $G(u)$ is a $\Pi_{n-1}$-module, we have $tG(u) = 0$, therefore $t\Hom_\Lambda(E,Z) \subseteq F_1(u)$, and therefore $t\Hom_\Lambda(E,Z) =
tF_1(u).$ Thus we have the following exact sequence: $$0 @>>> F_1(u)/tF_1(u) @>>> \Hom_\Lambda(E,Z)/t\Hom_\Lambda(E,Z) @>>> G(u) @>>> 0.$$ Note that $F_1(u)/tF_1(u) = F(u)$ and that $\Hom_\Lambda(E,Z)/t\Hom_\Lambda(E,Z)$ is a projective $\Pi_{n-1}$-module. It follows that $F(u)$ coincides in $\underline \mod\ \Pi_{n-1}$ with $\Omega G(u).$ This completes the proof of Theorem 2. It should be stressed that for $n\ge 2$, there cannot exist an endofunctor $\phi$ of $\mod \Pi_{n-1}$ such that $F
= \phi G$ or $\phi F= G$. For example, if we would have $F = \phi G$, then the set of objects of $\Cal S(n)$ killed by $G$ would be contained in the set of objects killed by $F$. However $([0],[1])$ is killed by $G$, but not by $F$. Similarly, $([1],[n])$ is killed by $F$, but not by $G.$
[**8. Abelian factor categories of triangulated categories**]{} As a byproduct of our consideration, we see that here we deal with examples of triangulated categories $\Cal T$ with an ideal $\Cal I$ generated by an idempotent such that $\Cal T/\Cal I$ is abelian.
Namely, let $\Cal T = \underline{\Cal S}(n)$ be the stable category of $\Cal S(n)$, it is obtained from $\Cal S(n)$ by factoring out the ideal generated by the objects $([0],[n])$ and $([n],[n])$. Now $\Cal S(n)$ is in a natural way (see \[RS2\]) a Frobenius category such that $([0],[n])$ and $([n],[n])$ are the only indecomposable objects which are both projective and injective, thus the category $\Cal T$ is a triangulated category.
Let $\Cal I$ be the ideal in $\Cal T$ generated by the objects $([i],[j])$ with $1\le i \le n-1$ and either $i = j$ or $j = n$. Then, as additive categories, we have equivalences $$\Cal T/\Cal I \simeq \Cal S(n)/ \Cal U \simeq \mod \Pi_{n-1},$$ thus $\Cal T/\Cal I$ is an abelian category.
Similarly, let $\Cal J$ be the ideal in $\Cal T$ generated by the objects $([i],[j])$ with $1\le j \le n-1$ and either $i = j$ or $i = 0$. Then, as additive categories, we have equivalences $$\Cal T/\Cal J \simeq \Cal S(n)/ \Cal V \simeq \mod \Pi_{n-1},$$ thus $\Cal T/\Cal J$ is again an abelian category.
Here, the ideals $\Cal I, \Cal J$ of $\Cal T$ each are generated by $2n-2$ indecomposable objects, whereas the rank of the Grothendieck group $K_0(\Cal T/\Cal I)$ is $n-1$. One may compare this with examples which involve cluster categories and cluster tilted algebras, see \[BMR\] and \[K\]. Let $T$ be a cluster tilting object in a cluster category $\Cal T$ say of type $Q$, where $Q$ is a directed quiver with $n$ vertices. Then the factor category $\Cal T/T$ is an abelian category and the rank of the Grothendieck group $K_0(\Cal T/T)$ is equal to $n$, and this is also the number of isomorphism classes of indecomposable direct summands of $T$.
[**9. More objective functors $\Cal S(n) \to \mod \Pi_{n-1}$.**]{} The main part of the paper was devoted to a detailed study of two objective functors $\Cal S(n) \to \mod \Pi_{n-1}$, but there are also other ones.
As in the previous section, let us denote by $\Cal T$ the stable category of $\Cal S(n)$ by $\Cal T$. Let $\Cal I$ be again the ideal of $\Cal T$ generated by the objects $([i],[j])$ with $1\le i \le n-1$ and either $i=j$ or $j = n$, thus there is an equivalence $\zeta\:\Cal T/\Cal I \to \mod \Pi_{n-1}$.
Thus, [*let $\eta$ be any autoequivalence of $\Cal T$. Then the composition $$\Cal S(n) @>>> \Cal T @>\eta>> \Cal T @>>> \Cal T/\Cal I
@>\zeta>> \mod \Pi_{n-1}$$ (where the first and the third functor are the canonical projections) clearly is a full, dense, objective functor.*]{} We recall from \[RS1\] that the stable category $\Cal T$ has non-trivial autoequivalences, for example the endofunctor induced by the Auslander-Reiten translation $\tau$ is an autoequivalence of order $6$.
[**10. Auslander-Reiten orbits.**]{} Let $\tau$ be the Auslander-Reiten translation in $\Cal S(n)$. The paper \[RS1\] describes in detail how to obtain for the pair $(X,Y)$ in $\Cal S(n)$ the pair $\tau(X,Y)$.
We are interested in some of the objects of the form $([i],[j])$ (with $0\le i \le j \le n$). The objects $([n],[n])$ and $([0],[n])$ are projective-injective, thus they are sent to zero by $\tau.$ The following assertions for $1\le i \le n\!-\!1$ $$\align
\tau([0],[i]) &= ([i],[i])\cr
\tau([i],[i]) &= ([i],[n])\cr
\tau([i],[n]) &= ([0],[n\!-\!i])\cr
\endalign$$ are easily verified. Of course, $1\le i \le n\!-\!1$ implies that also $1\le n\!-\!i \le n\!-\!1$, thus we see that the set of objects of the form $([0],[i]),([i],[i]),([i],[n])$ with $1\le i\le n\!-\!1$ is closed under $\tau$. Let us present the corresponding parts of Auslander-Reiten components. First of all, for $2 \le i < \frac n2$, we deal with a $\tau$-orbit of length 6: $$%====================================
\hbox{\beginpicture
\setcoordinatesystem units <1cm,1cm>
\put{} at 0 .7
\multiput{} at -1 0 13 0 /
\arr{0.3 0.3}{0.7 0.7}
\arr{2.3 0.3}{2.7 0.7}
\arr{4.3 0.3}{4.7 0.7}
\arr{1.3 0.7}{1.7 0.3}
\arr{3.3 0.7}{3.7 0.3}
\arr{5.3 0.7}{5.7 0.3}
\setdots <1mm>
\plot .7 0 1.3 0 /
\plot 2.7 0 3.3 0 /
\plot 4.7 0 5.2 0 /
\setsolid
\arr{6.3 0.3}{6.7 0.7}
\arr{8.3 0.3}{8.7 0.7}
\arr{10.3 0.3}{10.7 0.7}
\arr{7.3 0.7}{7.7 0.3}
\arr{9.3 0.7}{9.7 0.3}
\arr{11.3 0.7}{11.7 0.3}
\setdots <1mm>
\plot 6.7 0 7.3 0 /
\plot 8.7 0 9.3 0 /
\plot 10.7 0 11.2 0 /
\put{$\ssize[i]\subseteq[i]$} at 4 0
\put{$\ssize[i]\subseteq[n]$} at 2 0
\multiput{$\ssize[0]\subseteq[n\!-\!i]$} at 0 0 12 0 /
\put{$\ssize[n\!-\!i]\subseteq[n\!-\!i]$} at 10 0
\put{$\ssize[n\!-\!i]\subseteq[n]$} at 8 0
\put{$\ssize[0]\subseteq[i]$} at 6 0
\put{} at 0 2.3
\setshadegrid span <.4mm>
\vshade 0 0.05 1.9 <,,,> 12 0.05 1.9 /
\setsolid
\arr{0.3 1.7}{0.7 1.3}
\arr{1.3 1.3}{1.7 1.7}
\arr{2.3 1.7}{2.7 1.3}
\arr{3.3 1.3}{3.7 1.7}
\arr{4.3 1.7}{4.7 1.3}
\arr{5.3 1.3}{5.7 1.7}
\arr{6.3 1.7}{6.7 1.3}
\arr{7.3 1.3}{7.7 1.7}
\arr{8.3 1.7}{8.7 1.3}
\arr{9.3 1.3}{9.7 1.7}
\arr{10.3 1.7}{10.7 1.3}
\arr{11.3 1.3}{11.7 1.7}
\setdashes <1mm>
% \plot 0 0.3 0 1.8 /
% \plot 12 0.3 12 1.8 /
\plot 0 0.3 0 2.3 /
\plot 12 0.3 12 2.3 /
\multiput{$\cdots$} at 1 2.3 6 2.3 11 2.3 /
\endpicture}$$ If $n\ge 4$ is even and $i =\frac n2$, there is the following orbit of length 3: $$%====================================
\hbox{\beginpicture
\setcoordinatesystem units <1cm,1cm>
\put{} at 0 .7
%\multiput{} at -1 0 13 0 /
\arr{0.3 0.3}{0.7 0.7}
\arr{2.3 0.3}{2.7 0.7}
\arr{4.3 0.3}{4.7 0.7}
\arr{1.3 0.7}{1.7 0.3}
\arr{3.3 0.7}{3.7 0.3}
\arr{5.3 0.7}{5.7 0.3}
\setdots <1mm>
\plot .7 0 1.3 0 /
\plot 2.7 0 3.3 0 /
\plot 4.7 0 5.2 0 /
\put{$\ssize[\frac n2]\subseteq[\frac n2]$} at 4 0
\put{$\ssize[\frac n2]\subseteq[n]$} at 2 0
\multiput{$\ssize[0]\subseteq[\frac n2]$} at 0 0 6 0 /
\put{} at 0 2.3
\setshadegrid span <.4mm>
\vshade 0 0.05 1.9 <,,,> 6 0.05 1.9 /
\setsolid
\arr{0.3 1.7}{0.7 1.3}
\arr{1.3 1.3}{1.7 1.7}
\arr{2.3 1.7}{2.7 1.3}
\arr{3.3 1.3}{3.7 1.7}
\arr{4.3 1.7}{4.7 1.3}
\arr{5.3 1.3}{5.7 1.7}
\setdashes <1mm>
\plot 0 0.3 0 2.3 /
\plot 6 0.3 6 2.3 /
\multiput{$\cdots$} at 1 2.3 5 2.3 /
\endpicture}$$ Finally, for $i=1$ we get: $$%====================================
\hbox{\beginpicture
\setcoordinatesystem units <1cm,1cm>
%%%%%%%%%%%%%%%%%%%%%%%%%%the case i = n-1
\put{} at 0 .7
\multiput{} at -1 0 13 0 /
\put{$\ssize[1]\subseteq[1]$} at 4 0
\put{$\ssize[1]\subseteq[n]$} at 2 0
\multiput{$\ssize[0]\subseteq[n\!-\!1]$} at 0 0 12 0 /
\put{$\ssize[0]\subseteq[n]$} at 1 -1
\arr{0.3 0.3}{0.7 0.7}
\arr{2.3 0.3}{2.7 0.7}
\arr{4.3 0.3}{4.7 0.7}
\arr{1.3 0.7}{1.7 0.3}
\arr{3.3 0.7}{3.7 0.3}
\arr{5.3 0.7}{5.7 0.3}
\arr{0.3 -.3}{0.7 -.7}
\arr{1.3 -.7}{1.7 -.3}
\setdots <1mm>
\plot .7 0 1.3 0 /
\plot 2.7 0 3.3 0 /
\plot 4.7 0 5.2 0 /
\setsolid
\put{$\ssize[n\!-\!1]\subseteq[n\!-\!1]$} at 10 0
\put{$\ssize[n\!-\!1]\subseteq[n]$} at 8 0
\put{$\ssize[0]\subseteq[1]$} at 6 0
\put{$\ssize[n]\subseteq[n]$} at 9 -1
\arr{6.3 0.3}{6.7 0.7}
\arr{8.3 0.3}{8.7 0.7}
\arr{10.3 0.3}{10.7 0.7}
\arr{7.3 0.7}{7.7 0.3}
\arr{9.3 0.7}{9.7 0.3}
\arr{11.3 0.7}{11.7 0.3}
\arr{8.3 -.3}{8.7 -.7}
\arr{9.3 -.7}{9.7 -.3}
\setdots <1mm>
\plot 6.7 0 7.3 0 /
\plot 8.7 0 9.3 0 /
\plot 10.7 0 11.2 0 /
\put{} at 0 2.3
\setshadegrid span <.4mm>
\vshade 0 0.05 1.9 <,z,,> 1 -1 1.9 <z,z,,> 2 0.05 1.9
<z,z,,> 8 0.05 1.9 <z,z,,> 9 -1 1.9
<z,z,,> 10 0.05 1.9 <z,,,> 12 0.05 1.9 /
\setsolid
\arr{0.3 1.7}{0.7 1.3}
\arr{1.3 1.3}{1.7 1.7}
\arr{2.3 1.7}{2.7 1.3}
\arr{3.3 1.3}{3.7 1.7}
\arr{4.3 1.7}{4.7 1.3}
\arr{5.3 1.3}{5.7 1.7}
\arr{6.3 1.7}{6.7 1.3}
\arr{7.3 1.3}{7.7 1.7}
\arr{8.3 1.7}{8.7 1.3}
\arr{9.3 1.3}{9.7 1.7}
\arr{10.3 1.7}{10.7 1.3}
\arr{11.3 1.3}{11.7 1.7}
\setdashes <1mm>
\plot 0 0.3 0 2.3 /
\plot 12 0.3 12 2.3 /
\multiput{$\cdots$} at 1 2.3 6 2.3 11 2.3 /
\endpicture}$$ [**The corresponding $\tau$-orbits in the category $\Cal F(n).$**]{} First, those for $2\le i < \frac n2$: $$ $$ Second, for $n\ge 4$ even and $i=\frac n2$: $$ $$ And finally, for $i=1$ we get: $$%====================================
\hbox{\beginpicture
\setcoordinatesystem units <1cm,1cm>
%%%%%%%%%%%%%%%%%%%%%%%%%%the case i = n-1
\put{} at 0 .7
\multiput{} at -1 0 13 0 /
\put{$\ssize P(n)/P(1)$} at 2 0
\multiput{$\ssize P(n\!-\!1)$} at 0 0 12 0 /
\put{$\ssize P(n)$} at 1 -1
\arr{0.3 0.3}{0.7 0.7}
\arr{2.3 0.3}{2.7 0.7}
\arr{1.3 0.7}{1.7 0.3}
\arr{5.3 0.7}{5.7 0.3}
\arr{0.3 -.3}{0.7 -.7}
\arr{1.3 -.7}{1.7 -.3}
\setdots <1mm>
\plot .7 0 1.3 0 /
\setsolid
\put{$\ssize P(n)/P(n-1)$} at 8 0
\put{$\ssize P(1)$} at 6 0
\arr{6.3 0.3}{6.7 0.7}
\arr{8.3 0.3}{8.7 0.7}
\arr{7.3 0.7}{7.7 0.3}
\arr{11.3 0.7}{11.7 0.3}
\setdots <1mm>
\plot 6.7 0 7.3 0 /
\plot 3.3 1 4.7 1 /
\plot 9.3 1 10.7 1 /
\put{} at 0 2.3
\setshadegrid span <.4mm>
\vshade 0 0.05 1.9 <,z,,> 1 -1 1.9 <z,z,,> 3 1 1.9
<z,z,,> 5 1 1.9 <z,z,,> 6 0.05 1.9 <z,z,,> 8 0.05 1.9 <z,z,,> 9 1 1.9
<z,z,,> 11 1 1.9 <z,,,> 12 0.05 1.9 /
\setsolid
\arr{0.3 1.7}{0.7 1.3}
\arr{1.3 1.3}{1.7 1.7}
\arr{2.3 1.7}{2.7 1.3}
\arr{3.3 1.3}{3.7 1.7}
\arr{4.3 1.7}{4.7 1.3}
\arr{5.3 1.3}{5.7 1.7}
\arr{6.3 1.7}{6.7 1.3}
\arr{7.3 1.3}{7.7 1.7}
\arr{8.3 1.7}{8.7 1.3}
\arr{9.3 1.3}{9.7 1.7}
\arr{10.3 1.7}{10.7 1.3}
\arr{11.3 1.3}{11.7 1.7}
\setdashes <1mm>
\plot 0 0.3 0 2.3 /
\plot 12 0.3 12 2.3 /
\multiput{$\cdots$} at 1 2.3 6 2.3 11 2.3 /
\endpicture}$$ The paper \[RS2\] describes in detail the Auslander-Reiten quivers of the categories $\Cal S(n)$ with $1\le n \le 6$. Similarly, in \[DR\] the Auslander-Reiten quivers of the categories $\Cal F(n)$ with $2\le n \le 5$ are presented. As the functor $F_1$ shows, the Auslander-Reiten quiver of $\Cal F(n)$ can be obtained from that of $\Cal S(n)$ by just deleting some vertices, thus it is easy to obtain the illustrations presented in \[DR\] from those in \[RS2\]. The deletion process explains also some of the features of the shape of the Auslander-Reiten quiver of $\Cal F(n)$: Of course, there is precisely one projective-injective vertex, namely the module $P(n) = I(n) = T(1)$. The remaining transjective orbits contain precisely two vertices, namely $T(n+1-i)$ and $P(i) = \tau_{\Cal F(n)} T(n+1-i)$, here $1\le i\le n-1$. As we now know, this concerns the $\tau$-orbit of $\Cal S(n)$ which contains the pairs $([i],[i])$ and $([n-i],[n-i])$: both pairs are killed by the functor $F_1$, but in-between the Auslander-Reiten sequence starting with $([0],[i])$ and ending in $([n-i],[n])$ is not touched and it yields under $F_1$ the Auslander-Reiten sequence starting with $P(i)$ and ending in $T(n+1-i).$ [**The corresponding $\tau$-orbits of the category $\Cal S(n)/\Cal U \simeq \mod \Pi_{n-1}$.**]{}
For $2\le i < \frac n2$: $$%====================================\Cal S(n)/\Cal U
\hbox{\beginpicture
\setcoordinatesystem units <1cm,1cm>
\put{} at 0 .7
\multiput{} at -1 0 13 0 /
\arr{0.3 0.3}{0.7 0.7}
\arr{5.3 0.7}{5.7 0.3}
\setdots <1mm>
\plot .7 0 1.3 0 /
\setsolid
\arr{6.3 0.3}{6.7 0.7}
\arr{11.3 0.7}{11.7 0.3}
\multiput{$\ssize P(n-i)$} at 0 0 12 0 /
\put{$\ssize P(i)$} at 6 0
\put{} at -1 0
\setdots <1mm>
\plot 3.3 1 4.7 1 /
\plot 9.3 1 10.7 1 /
\put{} at 0 2.3
\setshadegrid span <.4mm>
\vshade 0 0.05 1.9 <,z,,> 1 1 1.9
<z,z,,> 5 1 1.9 <z,z,,> 6 0 1.9 <z,z,,> 7 1 1.9
<z,z,,> 11 1 1.9 <z,,,> 12 0.05 1.9 /
\setsolid
\arr{0.3 1.7}{0.7 1.3}
\arr{1.3 1.3}{1.7 1.7}
\arr{2.3 1.7}{2.7 1.3}
\arr{3.3 1.3}{3.7 1.7}
\arr{4.3 1.7}{4.7 1.3}
\arr{5.3 1.3}{5.7 1.7}
\arr{6.3 1.7}{6.7 1.3}
\arr{7.3 1.3}{7.7 1.7}
\arr{8.3 1.7}{8.7 1.3}
\arr{9.3 1.3}{9.7 1.7}
\arr{10.3 1.7}{10.7 1.3}
\arr{11.3 1.3}{11.7 1.7}
\setdashes <1mm>
\plot 0 0.3 0 2.3 /
\plot 12 0.3 12 2.3 /
\multiput{$\cdots$} at 1 2.3 6 2.3 11 2.3 /
\endpicture}$$ For $n\ge 4$ even and $i=\frac n2$: $$%====================================\Cal S(n)/\Cal U
\hbox{\beginpicture
\setcoordinatesystem units <1cm,1cm>
\put{} at 0 .7
\arr{0.3 0.3}{0.7 0.7}
\arr{5.3 0.7}{5.7 0.3}
\setdots <1mm>
\plot .7 0 1.3 0 /
\multiput{$\ssize P(\frac n2)$} at 0 0 6 0 /
\put{$\ssize P(\frac n2)$} at 6 0
\plot 3.3 1 4.7 1 /
\put{} at 0 2.3
\setshadegrid span <.4mm>
\vshade 0 0.05 1.9 <,z,,> 1 1 1.9
<z,z,,> 5 1 1.9 <z,,,> 6 0 1.9 /
\setsolid
\arr{0.3 1.7}{0.7 1.3}
\arr{1.3 1.3}{1.7 1.7}
\arr{2.3 1.7}{2.7 1.3}
\arr{3.3 1.3}{3.7 1.7}
\arr{4.3 1.7}{4.7 1.3}
\arr{5.3 1.3}{5.7 1.7}
\setdashes <1mm>
\plot 0 0.3 0 2.3 /
\plot 6 0.3 6 2.3 /
\multiput{$\cdots$} at 1 2.3 5 2.3 /
\endpicture}$$ And finally, for $i=1$ we get: $$%====================================\Cal S(n)/\Cal U
\hbox{\beginpicture
\setcoordinatesystem units <1cm,1cm>
%%%%%%%%%%%%%%%%%%%%%%%%%%the case i = n-1
\put{} at 0 .7
\multiput{} at -1 0 13 0 /
\multiput{$\ssize P(n\!-\!1)$} at 0 0 12 0 /
\arr{0.3 0.3}{0.7 0.7}
\arr{5.3 0.7}{5.7 0.3}
\setdots <1mm>
\setsolid
\put{$\ssize P(1)$} at 6 0
\arr{6.3 0.3}{6.7 0.7}
\arr{11.3 0.7}{11.7 0.3}
\setdots <1mm>
\plot 3.3 1 4.7 1 /
\plot 9.3 1 10.7 1 /
\put{} at 0 2.3
\setshadegrid span <.4mm>
\vshade 0 0.05 1.9 <,z,,> 1 1 1.9
<z,z,,> 5 1 1.9 <z,z,,> 6 0 1.9 <z,z,,> 7 1 1.9
<z,z,,> 11 1 1.9 <z,,,> 12 0.05 1.9 /
\setsolid
\arr{0.3 1.7}{0.7 1.3}
\arr{1.3 1.3}{1.7 1.7}
\arr{2.3 1.7}{2.7 1.3}
\arr{3.3 1.3}{3.7 1.7}
\arr{4.3 1.7}{4.7 1.3}
\arr{5.3 1.3}{5.7 1.7}
\arr{6.3 1.7}{6.7 1.3}
\arr{7.3 1.3}{7.7 1.7}
\arr{8.3 1.7}{8.7 1.3}
\arr{9.3 1.3}{9.7 1.7}
\arr{10.3 1.7}{10.7 1.3}
\arr{11.3 1.3}{11.7 1.7}
\setdashes <1mm>
\plot 0 0.3 0 2.3 /
\plot 12 0.3 12 2.3 /
\multiput{$\cdots$} at 1 2.3 6 2.3 11 2.3 /
\endpicture}$$ Let us draw again the relevant components of $\Cal S(n)$ and use the shading in order to illustrate what remains when we delete the objects in $\Cal U$. Note that under the functor $F$, we have $$\alignat 2
F([i],[i]) &= 0 &&1\le i \le n, \cr
F([i],[n]) &= 0 &\qquad\text{for}\qquad &0\le i \le n\!-\!1, \cr
F([0],[j]) &= P_{\Pi}(j) &&1\le j\le n\!-\!1. \cr
\endalignat$$
First of all, for $2 \le i < \frac n2$, two objects of the $\tau_{\Cal S}$-orbit of $([i],[i])$ survive: $$%====================================
\hbox{\beginpicture
\setcoordinatesystem units <1cm,1cm>
\put{} at 0 .7
\multiput{} at -1 0 13 0 /
\arr{0.3 0.3}{0.7 0.7}
\arr{2.3 0.3}{2.7 0.7}
\arr{4.3 0.3}{4.7 0.7}
\arr{1.3 0.7}{1.7 0.3}
\arr{3.3 0.7}{3.7 0.3}
\arr{5.3 0.7}{5.7 0.3}
\setdots <1mm>
\plot .7 0 1.3 0 /
\plot 2.7 0 3.3 0 /
\plot 4.7 0 5.2 0 /
\setsolid
\arr{6.3 0.3}{6.7 0.7}
\arr{8.3 0.3}{8.7 0.7}
\arr{10.3 0.3}{10.7 0.7}
\arr{7.3 0.7}{7.7 0.3}
\arr{9.3 0.7}{9.7 0.3}
\arr{11.3 0.7}{11.7 0.3}
\setdots <1mm>
\plot 6.7 0 7.3 0 /
\plot 8.7 0 9.3 0 /
\plot 10.7 0 11.2 0 /
\put{$\ssize[i]\subseteq[i]$} at 4 0
\put{$\ssize[i]\subseteq[n]$} at 2 0
\multiput{$\ssize[0]\subseteq[n\!-\!i]$} at 0 0 12 0 /
\put{$\ssize[n\!-\!i]\subseteq[n\!-\!i]$} at 10 0
\put{$\ssize[n\!-\!i]\subseteq[n]$} at 8 0
\put{$\ssize[0]\subseteq[i]$} at 6 0
\put{} at 0 2.3
\setshadegrid span <.4mm>
\vshade 0 0.05 1.9 <,z,,> 1 1 1.9
<z,z,,> 5 1 1.9 <z,z,,> 6 0 1.9 <z,z,,> 7 1 1.9
<z,z,,> 11 1 1.9 <z,,,> 12 0.05 1.9 /
\setsolid
\arr{0.3 1.7}{0.7 1.3}
\arr{1.3 1.3}{1.7 1.7}
\arr{2.3 1.7}{2.7 1.3}
\arr{3.3 1.3}{3.7 1.7}
\arr{4.3 1.7}{4.7 1.3}
\arr{5.3 1.3}{5.7 1.7}
\arr{6.3 1.7}{6.7 1.3}
\arr{7.3 1.3}{7.7 1.7}
\arr{8.3 1.7}{8.7 1.3}
\arr{9.3 1.3}{9.7 1.7}
\arr{10.3 1.7}{10.7 1.3}
\arr{11.3 1.3}{11.7 1.7}
\setdashes <1mm>
\plot 0 0.3 0 2.3 /
\plot 12 0.3 12 2.3 /
\multiput{$\cdots$} at 1 2.3 6 2.3 11 2.3 /
\endpicture}$$ If $n\ge 4$ is even and $i =\frac n2$, the pair $([0],[i])$ is the only object in the $\tau_{\Cal S}$-orbit of $([i],[i])$ which survives: $$%====================================
\hbox{\beginpicture
\setcoordinatesystem units <1cm,1cm>
\put{} at 0 .7
%\multiput{} at -1 0 13 0 /
\arr{0.3 0.3}{0.7 0.7}
\arr{2.3 0.3}{2.7 0.7}
\arr{4.3 0.3}{4.7 0.7}
\arr{1.3 0.7}{1.7 0.3}
\arr{3.3 0.7}{3.7 0.3}
\arr{5.3 0.7}{5.7 0.3}
\setdots <1mm>
\plot .7 0 1.3 0 /
\plot 2.7 0 3.3 0 /
\plot 4.7 0 5.2 0 /
\put{$\ssize[\frac n2]\subseteq[\frac n2]$} at 4 0
\put{$\ssize[\frac n2]\subseteq[n]$} at 2 0
\multiput{$\ssize[0]\subseteq[\frac n2]$} at 0 0 6 0 /
\put{} at 0 2.3
\setshadegrid span <.4mm>
\vshade 0 0.05 1.9 <,z,,> 1 1 1.9
<z,z,,> 5 1 1.9 <z,,,> 6 0 1.9 /
\setsolid
\arr{0.3 1.7}{0.7 1.3}
\arr{1.3 1.3}{1.7 1.7}
\arr{2.3 1.7}{2.7 1.3}
\arr{3.3 1.3}{3.7 1.7}
\arr{4.3 1.7}{4.7 1.3}
\arr{5.3 1.3}{5.7 1.7}
\setdashes <1mm>
\plot 0 0.3 0 2.3 /
\plot 6 0.3 6 2.3 /
\multiput{$\cdots$} at 1 2.3 5 2.3 /
\endpicture}$$ Finally, for $i=1$, again two objects in the $\tau_{\Cal S}$-orbit of $([1],[1])$ survive: $$%====================================
\hbox{\beginpicture
\setcoordinatesystem units <1cm,1cm>
%%%%%%%%%%%%%%%%%%%%%%%%%%the case i = n-1
\put{} at 0 .7
\multiput{} at -1 0 13 0 /
\put{$\ssize[1]\subseteq[1]$} at 4 0
\put{$\ssize[1]\subseteq[n]$} at 2 0
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\put{$\ssize[0]\subseteq[n]$} at 1 -1
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\setdots <1mm>
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\put{$\ssize[n\!-\!1]\subseteq[n\!-\!1]$} at 10 0
\put{$\ssize[n\!-\!1]\subseteq[n]$} at 8 0
\put{$\ssize[0]\subseteq[1]$} at 6 0
\put{$\ssize[n]\subseteq[n]$} at 9 -1
\arr{6.3 0.3}{6.7 0.7}
\arr{8.3 0.3}{8.7 0.7}
\arr{10.3 0.3}{10.7 0.7}
\arr{7.3 0.7}{7.7 0.3}
\arr{9.3 0.7}{9.7 0.3}
\arr{11.3 0.7}{11.7 0.3}
\arr{8.3 -.3}{8.7 -.7}
\arr{9.3 -.7}{9.7 -.3}
\setdots <1mm>
\plot 6.7 0 7.3 0 /
\plot 8.7 0 9.3 0 /
\plot 10.7 0 11.2 0 /
\put{} at 0 2.3
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\vshade 0 0.05 1.9 <,z,,> 1 1 1.9
<z,z,,> 5 1 1.9 <z,z,,> 6 0 1.9 <z,z,,> 7 1 1.9
<z,z,,> 11 1 1.9 <z,,,> 12 0.05 1.9 /
\setsolid
\arr{0.3 1.7}{0.7 1.3}
\arr{1.3 1.3}{1.7 1.7}
\arr{2.3 1.7}{2.7 1.3}
\arr{3.3 1.3}{3.7 1.7}
\arr{4.3 1.7}{4.7 1.3}
\arr{5.3 1.3}{5.7 1.7}
\arr{6.3 1.7}{6.7 1.3}
\arr{7.3 1.3}{7.7 1.7}
\arr{8.3 1.7}{8.7 1.3}
\arr{9.3 1.3}{9.7 1.7}
\arr{10.3 1.7}{10.7 1.3}
\arr{11.3 1.3}{11.7 1.7}
\setdashes <1mm>
\plot 0 0.3 0 2.3 /
\plot 12 0.3 12 2.3 /
\multiput{$\cdots$} at 1 2.3 6 2.3 11 2.3 /
\endpicture}$$ [**The relevant components of $\Cal S(n)/\Cal V$**]{}. In the same way as we have presented components of $\Cal S(n)$ shading the parts which remain after deleting $\Cal U$, we now show what remains from these components when we remove $\Cal V$.
First of all, for $2 \le i < \frac n2$, two objects in the $\tau_{\Cal S}$-orbit of $([i],[i])$ survive: $$%====================================
\hbox{\beginpicture
\setcoordinatesystem units <1cm,1cm>
\put{} at 0 .7
\multiput{} at -1 0 13 0 /
\arr{0.3 0.3}{0.7 0.7}
\arr{2.3 0.3}{2.7 0.7}
\arr{4.3 0.3}{4.7 0.7}
\arr{1.3 0.7}{1.7 0.3}
\arr{3.3 0.7}{3.7 0.3}
\arr{5.3 0.7}{5.7 0.3}
\setdots <1mm>
\plot .7 0 1.3 0 /
\plot 2.7 0 3.3 0 /
\plot 4.7 0 5.2 0 /
\setsolid
\arr{6.3 0.3}{6.7 0.7}
\arr{8.3 0.3}{8.7 0.7}
\arr{10.3 0.3}{10.7 0.7}
\arr{7.3 0.7}{7.7 0.3}
\arr{9.3 0.7}{9.7 0.3}
\arr{11.3 0.7}{11.7 0.3}
\setdots <1mm>
\plot 6.7 0 7.3 0 /
\plot 8.7 0 9.3 0 /
\plot 10.7 0 11.2 0 /
\put{$\ssize[i]\subseteq[i]$} at 4 0
\put{$\ssize[i]\subseteq[n]$} at 2 0
\multiput{$\ssize[0]\subseteq[n\!-\!i]$} at 0 0 12 0 /
\put{$\ssize[n\!-\!i]\subseteq[n\!-\!i]$} at 10 0
\put{$\ssize[n\!-\!i]\subseteq[n]$} at 8 0
\put{$\ssize[0]\subseteq[i]$} at 6 0
\put{} at 0 2.3
\setshadegrid span <.4mm>
\vshade 0 1 1.9 <,z,,> 1 1 1.9 <z,z,,> 2 0 1.9 <z,z,,> 3 1 1.9 <z,z,,> 7 1 1.9
<z,z,,> 8 0 1.9 <z,z,,> 9 1 1.9 <z,,,> 12 1 1.9 /
\setsolid
\arr{0.3 1.7}{0.7 1.3}
\arr{1.3 1.3}{1.7 1.7}
\arr{2.3 1.7}{2.7 1.3}
\arr{3.3 1.3}{3.7 1.7}
\arr{4.3 1.7}{4.7 1.3}
\arr{5.3 1.3}{5.7 1.7}
\arr{6.3 1.7}{6.7 1.3}
\arr{7.3 1.3}{7.7 1.7}
\arr{8.3 1.7}{8.7 1.3}
\arr{9.3 1.3}{9.7 1.7}
\arr{10.3 1.7}{10.7 1.3}
\arr{11.3 1.3}{11.7 1.7}
\setdashes <1mm>
\plot 0 0.3 0 2.3 /
\plot 12 0.3 12 2.3 /
\multiput{$\cdots$} at 1 2.3 6 2.3 11 2.3 /
\endpicture}$$ Next, for $n\ge 4$ even and $i =\frac n2$, the pair $([i],[n])$ is the only object in the $\tau_{\Cal S}$-orbit of $([i],[i])$ which survives: $$%====================================
\hbox{\beginpicture
\setcoordinatesystem units <1cm,1cm>
\put{} at 0 .7
%\multiput{} at -1 0 13 0 /
\arr{0.3 0.3}{0.7 0.7}
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\arr{4.3 0.3}{4.7 0.7}
\arr{1.3 0.7}{1.7 0.3}
\arr{3.3 0.7}{3.7 0.3}
\arr{5.3 0.7}{5.7 0.3}
\setdots <1mm>
\plot .7 0 1.3 0 /
\plot 2.7 0 3.3 0 /
\plot 4.7 0 5.2 0 /
\put{$\ssize[\frac n2]\subseteq[\frac n2]$} at 4 0
\put{$\ssize[\frac n2]\subseteq[n]$} at 2 0
\multiput{$\ssize[0]\subseteq[\frac n2]$} at 0 0 6 0 /
\put{} at 0 2.3
\setshadegrid span <.4mm>
\vshade 0 1 1.9 <,z,,> 1 1 1.9 <z,z,,> 2 0 1.9 <z,z,,> 3 1 1.9 <z,,,> 6 1 1.9 /
\setsolid
\arr{0.3 1.7}{0.7 1.3}
\arr{1.3 1.3}{1.7 1.7}
\arr{2.3 1.7}{2.7 1.3}
\arr{3.3 1.3}{3.7 1.7}
\arr{4.3 1.7}{4.7 1.3}
\arr{5.3 1.3}{5.7 1.7}
\setdashes <1mm>
\plot 0 0.3 0 2.3 /
\plot 6 0.3 6 2.3 /
\multiput{$\cdots$} at 1 2.3 5 2.3 /
\endpicture}$$ Finally, for $i=1$, again two objects in the $\tau_{\Cal S}$-orbit of $([i],[i])$ survive: $$%====================================
\hbox{\beginpicture
\setcoordinatesystem units <1cm,1cm>
%%%%%%%%%%%%%%%%%%%%%%%%%%the case i = n-1
\put{} at 0 .7
\multiput{} at -1 0 13 0 /
\put{$\ssize[1]\subseteq[1]$} at 4 0
\put{$\ssize[1]\subseteq[n]$} at 2 0
\multiput{$\ssize[0]\subseteq[n\!-\!1]$} at 0 0 12 0 /
\put{$\ssize[0]\subseteq[n]$} at 1 -1
\arr{0.3 0.3}{0.7 0.7}
\arr{2.3 0.3}{2.7 0.7}
\arr{4.3 0.3}{4.7 0.7}
\arr{1.3 0.7}{1.7 0.3}
\arr{3.3 0.7}{3.7 0.3}
\arr{5.3 0.7}{5.7 0.3}
\arr{0.3 -.3}{0.7 -.7}
\arr{1.3 -.7}{1.7 -.3}
\setdots <1mm>
\plot .7 0 1.3 0 /
\plot 2.7 0 3.3 0 /
\plot 4.7 0 5.2 0 /
\setsolid
\put{$\ssize[n\!-\!1]\subseteq[n\!-\!1]$} at 10 0
\put{$\ssize[n\!-\!1]\subseteq[n]$} at 8 0
\put{$\ssize[0]\subseteq[1]$} at 6 0
\put{$\ssize[n]\subseteq[n]$} at 9 -1
\arr{6.3 0.3}{6.7 0.7}
\arr{8.3 0.3}{8.7 0.7}
\arr{10.3 0.3}{10.7 0.7}
\arr{7.3 0.7}{7.7 0.3}
\arr{9.3 0.7}{9.7 0.3}
\arr{11.3 0.7}{11.7 0.3}
\arr{8.3 -.3}{8.7 -.7}
\arr{9.3 -.7}{9.7 -.3}
\setdots <1mm>
\plot 6.7 0 7.3 0 /
\plot 8.7 0 9.3 0 /
\plot 10.7 0 11.2 0 /
\put{} at 0 2.3
\setshadegrid span <.4mm>
\vshade 0 1 1.9 <,z,,> 1 1 1.9 <z,z,,> 2 0 1.9 <z,z,,> 3 1 1.9 <z,z,,> 7 1 1.9
<z,z,,> 8 0 1.9 <z,z,,> 9 1 1.9 <z,,,> 12 1 1.9 /
\setsolid
\arr{0.3 1.7}{0.7 1.3}
\arr{1.3 1.3}{1.7 1.7}
\arr{2.3 1.7}{2.7 1.3}
\arr{3.3 1.3}{3.7 1.7}
\arr{4.3 1.7}{4.7 1.3}
\arr{5.3 1.3}{5.7 1.7}
\arr{6.3 1.7}{6.7 1.3}
\arr{7.3 1.3}{7.7 1.7}
\arr{8.3 1.7}{8.7 1.3}
\arr{9.3 1.3}{9.7 1.7}
\arr{10.3 1.7}{10.7 1.3}
\arr{11.3 1.3}{11.7 1.7}
\setdashes <1mm>
\plot 0 0.3 0 2.3 /
\plot 12 0.3 12 2.3 /
\multiput{$\cdots$} at 1 2.3 6 2.3 11 2.3 /
\endpicture}$$ Going from $\Cal S(n)$ to $\Cal F(n)$, or from $\Cal F(n)$ to $\mod\Pi_{n-1}$, the number of indecomposable objects decreases in both step by $n$. Here is the bookkeeping table. We denote the number of isomorphism classes of indecomposable objects in the category $\Cal C$ by $\#\ind\Cal C$. $$%====================================
\hbox{\beginpicture
\setcoordinatesystem units <3cm,.4cm>
\put{$n$} at 0 7.2
\put{$1$} at 0 6
\put{$2$} at 0 5
\put{$3$} at 0 4
\put{$4$} at 0 3
\put{$5$} at 0 2
\put{$6$} at 0 1
\put{$\#\ind\Cal S(n)$} at 1 7.2
\put{$2$} at 1 6
\put{$5$} at 1 5
\put{$10$} at 1 4
\put{$20$} at 1 3
\put{$50$} at 1 2
\put{$\infty$} at 1 1
\put{$\#\ind\Cal F(n)$} at 2 7.2
\put{$1$} at 2 6
\put{$3$} at 2 5
\put{$7$} at 2 4
\put{$16$} at 2 3
\put{$45$} at 2 2
\put{$\infty$} at 2 1
\put{$\#\ind\mod\Pi_{n-1}$} at 3 7.2
\put{$0$} at 3 6
\put{$1$} at 3 5
\put{$4$} at 3 4
\put{$12$} at 3 3
\put{$40$} at 3 2
\put{$\infty$} at 3 1
\plot -.3 6.7 3.5 6.7 /
\plot 0.4 8 0.4 0.5 /
\endpicture}$$ Going from $\underline{\Cal S}(n)$ to $\mod \Pi_{n-1}$ the number of indecomposables decreases by $2(n-1)$, going from $\mod \Pi_{n-1}$ to $\underline{\mod}\ \Pi_{n-1}$ the number decreases by $n-1$. Here are the actual numbers; for the triangulated categories $\underline{\Cal S}(n)$ and $\underline{\mod}\ \Pi_{n-1}$ we also list the tree type of the corresponding Auslander-Reiten quivers. $$%====================================
\hbox{\beginpicture
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\put{$\Bbb D_4$} at 1.3 4
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\plot 0.4 8 0.4 0.5 /
\endpicture}$$
[**11. Appendix: Objective functors.**]{} Let $\Cal A, \Cal B$ be additive categories and let $F\:\Cal A \to \Cal B$ be an (additive) functor. An object $A$ in $\Cal A$ will be called a [*kernel object*]{} for $F$ provided $F(A) = 0.$ The functor $F\:\Cal A \to \Cal B$ will be said to be [*objective*]{} provided any morphism $f\: A \to A'$ in $\Cal A$ with $F(f) = 0$ factors through a kernel object for $F$. If $F$ is an objective functor, then we will say that the [*kernel of $F$ is generated by*]{} $\Cal K,$ provided $\Cal K$ is a class of objects in $\Cal A$ such that $\add \Cal K$ is the class of all kernel objects for $F$.
Given an additive category $\Cal A$ and an ideal $\Cal I$ in $\Cal A$, we denote by $\Cal A/\Cal I$ the corresponding factor category (it has the same objects, and the homomorphisms in $\Cal A/\Cal I$ are the residue classes of the homomorphisms in $\Cal A$ modulo $\Cal I$). If $\Cal K$ is a class of objects of the category $\Cal A$, we denote by $\langle \Cal K\rangle$ the ideal of $\Cal A$ given by all maps which factor through a direct sum of objects in $\Cal K.$ Instead of writing $\Cal A/\langle \Cal K\rangle$, we just will write $\Cal A/\Cal K.$
If $F\:\Cal A \to \Cal B$ is a full, dense, objective functor and the kernel of $F$ is generated by $\Cal K$, then $F$ induces an equivalence between the category $\Cal A/\Cal K$ and $\Cal B$. We see that given a full, dense, objective functor $F\:\Cal A \to \Cal B$, the category $\Cal B$ is uniquely determined by $\Cal A$ and a class of indecomposable objects in $\Cal A$ (namely the class of indecomposable kernel objects for $F$); if $F$ is objective, but not necessarily full or dense, then $F$ induces an equivalence between the category $\Cal A/\Cal K$ and the image category of $F$. Here is an example: Let $\Cal B$ be the linearization of the chain of cardinality 3, thus $\Cal B$ has three objects $b_1,b_2,b_3$ with $\Hom(b_i,b_j) = k$ provided $i \le j$ and zero otherwise, such that the composition $\Hom(b_2,b_3)\otimes \Hom(b_1,b_2) \to \Hom(b_1,b_3)$ is the multiplication map. Let $\Cal A$ be the full subcategory of $\Cal B$ with objects $b_1,b_3$, thus $\Cal A$ is the linearization of a chain of cardinality 2. Let $\Cal K = \{b_2\}$ and $\Cal C = \Cal B/\Cal K$. The inclusion functor $F\:\Cal A \to \Cal B$ and the projection functor $G\:\Cal B \to \Cal C$ both are (full and) objective, however the composition $GF\:\Cal A \to \Cal B$ is not objective (none of the objects $b_1,b_3$ belongs to the kernel of $GF$, we have $\Hom_{\Cal A}(b_1,b_3) = k$ and any non-zero map $b_1 \to b_3$ is mapped to zero under $GF$). Note that the functor $F$ is not dense. (thus, the composition of full, dense, objective functors is full, dense, objective). Proof. Since $F, G$ both are full, also $GF$ is full. Let $a\:A_1\to A_2$ be a morphism with $GF(a) = 0.$ Since $G$ is objective, the morphism $F(a)$ factors through a kernel object $B$ for $G$, say $F(a) = b_2b_1$ where $b_1\:F(A_1) \to B$ and $b_2\:B \to F(A_2)$. By assumption, $F$ is dense, thus there is an isomorphism $b\:B \to F(A)$ for some object $A$ in $\Cal A$. Since $F$ is full, there is a map $a_1\:A_1 \to A$ such that $F(a_1) = bb_1$ and a map $a_2\:A \to A_2$ such that $F(a_2) = b_2b^{-1}$. We have $F(a) = b_2b^{-1}bb_1 = F(a_2)F(a_1) = F(a_2a_1),$ thus $F(a-a_2a_1) = 0.$ Since $F$ is objective, there is a kernel object $A'$ for $F$ such that $a-a_2a_1$ factors through $A'$, say $a-a_2a_1 = a_4a_3$, with $a_3\:A_1\to A', a_4\:A' \to A_2.$ It follows that $a = a_2a_1+a_4a_3 = \bmatrix a_2 &a_4\endbmatrix \bmatrix a_1 \cr a_3\endbmatrix,$ thus this map factors through $A\oplus A'$. But $GF(A\oplus A') = GF(A)\oplus GF(A').$ Now, $F(A)$ is isomorphic to $B$, thus $GF(A)$ is isomorphic to $G(B) = 0$. Also, $F(A') = 0,$ thus $GF(A') = 0$. This shows that $A\oplus A'$ is a kernel object for $GF.$ We recall that an additive category $\Cal A$ is said to be a Krull-Remak-Schmidt category, provided every object in $\Cal A$ is a (finite) direct sum of objects with local endomorphism rings. Assume now that $F\: \Cal A \to \Cal B$ is an objective functor between Krull-Remak-Schmidt categories $\Cal A$ and $\Cal B$. Then we are interested in the number $i_0(F)$ of isomorphism classes of indecomposable objects in $\Cal F$ which are kernel objects for $F$, as well as in the number $i_1(F)$ of isomorphism classes of indecomposable objects $B$ in $\Cal B$ such that $B$ is not isomorphic to an object of the form $F(A)$ where $A$ is an object in $\Cal A$. If at least one of the numbers $i_0(F), i_1(F)$ is finite, we call $i(F) = i_0(F)-i_1(F)$ the [*index*]{} of $F$.
The objective functors $F$ considered in the paper are also dense, in this case $i(F) = i_0(F)$ is the number of isomorphism classes of indecomposable kernel objects in $\Cal A$.
[**12. References.**]{}
[\[A\]]{} A. Auslander: Coherent functors. In: Proceedings of the Conference on Categorical Algebra. La Jolla 1965, Springer-Veriag, New York (1965), 189–231.
[\[AR1\]]{} M. Auslander, I. Reiten: Stable equivalence of dualizing $R$-varieties. Adv. Math. 12 (1974), 306–366.
[\[AR2\]]{} M. Auslander, I. Reiten: On the representation type of triangular matrix rings. J. London Math. Soc.(2), 12 (1976), 371–382.
[\[BMR\]]{} A. B. Buan, R. J. Marsh, I. Reiten: Cluster-tilted algebras. Trans. Amer. Math. Soc. 359 (2007), 323–332.
[\[DR\]]{} V. Dlab, C. M. Ringel: The module theoretical approach to quasi-hereditary algebras. In: Representations of Algebras and Related Topics (ed. H. Tachikawa and S. Brenner). London Math. Soc. Lecture Note Series 168. Cambridge University Press (1992), 200–224.
[\[K\]]{} B. Keller: On triangulated orbit categories. Doc. Math. 10 (2005), 551–581.
[\[LZ\]]{} Z.-W. Li, P. Zhang: A construction of Gorenstein-projective modules. J. Algebra 323 (2010), 1802–1812.
[\[R\]]{} C. M. Ringel: Iyama’s finiteness theorem via strongly quasi-hereditary algebras. Journal Pure Applied Algebra 214 (2010), 1687–1692.
[\[RS1\]]{} C. M. Ringel, M. Schmidmeier: The Auslander-Reiten translation in submodule categories. Trans. Amer. Math. Soc. 360 (2008), 691–716.
[\[RS2\]]{} C. M. Ringel, M. Schmidmeier: Invariant subspaces of nilpotent linear operators. I. J. reine angew. Math 614 (2008), 1–52.
[\[Z\]]{} P. Zhang, Gorenstein-projective modules and symmetric recollements J. Algebra 388 (2013), 65–80.
|
---
abstract: 'We establish a Penrose-like inequality for general (not necessarily time-symmetric) initial data sets of the Einstein-Maxwell equations, which satisfy the dominant energy condition. More precisely, it is shown that the ADM energy is bounded below by an expression which is proportional to the sum of the square root of the area of the outermost future (or past) apparent horizon and the square of the total charge. The proportionality constants depend on the solution to a linear elliptic equation which incorporates the charge. In addition, a corrected version of the Penrose-like inequality in [@Khuri] is presented.'
address: |
Department of Mathematics\
Stony Brook University\
Stony Brook, NY 11794
author:
- 'Marcus A. Khuri'
title: 'A Penrose-Like Inequality with Charge'
---
[^1]
Introduction {#sec1}
============
Consider an initial data set $(M, g, k, E)$ for the Einstein-Maxwell equations with vanishing magnetic field. Here $M$ is a Riemannian $3$-manifold with metric $g$, $k$ is a symmetric 2-tensor representing the second fundamental form of the embedding into spacetime, and $E$ denotes the electric field. It is assumed that the manifold has a boundary $\partial M$ consisting of an outermost apparent horizon. That is, if $H$ denotes mean curvature with respect to the normal pointing towards spatial infinity, then each boundary component $S\subset\partial M$ satisfies $\theta_{+}(S):=H_{S}+Tr_{S}k=0$ (future horizon) or $\theta_{-}(S):=H_{S}-Tr_{S}k=0$ (past horizon), and there are no other apparent horizons present. Moreover the data are taken to be asymptotically flat with one end, in that outside a compact set the manifold is diffeomorphic to the complement of a ball in $\mathbb{R}^{3}$, and in the coordinates given by this asymptotic diffeomorphism the following fall-off conditions hold $$\label{1}
|\partial^{m}(g_{ij}-\delta_{ij})|=O(|x|^{-m-1}),\text{ }\text{
}\text{ }|\partial^{m}k_{ij}|=O(|x|^{-m-2}),\text{ }\text{
}\text{ }|\partial^{m}E^{i}|=O(|x|^{-m-2}),\text{ }\text{ }\text{ }m=0,1,2,\text{ }\text{ }\text{as}\text{ }\text{
}|x|\rightarrow\infty.$$ With a vanishing magnetic field, the matter and current densities for the non-electromagnetic matter fields are given by $$\begin{aligned}
\label{2}
\begin{split}
2 \mu & = R + (Tr k)^2 - |k|_{g}^2 - 2|E|_{g}^2, \\
J & = div (k - (Tr k)g),
\end{split}\end{aligned}$$ where $R$ denotes the scalar curvature of $g$. The following inequality will be referred to as the dominant energy condition $$\label{3}
\mu \geq |J|_{g}.$$ Note that this dominant energy condition differs from the standard one, in that the energy density for the electric field is removed. Under these hypotheses and based on heuristic arguments of Penrose [@Penrose] which rely heavily on the cosmic censorship conjecture, the following inequality relating the ADM energy and the minimal area $\mathcal{A}$ required to enclose the boundary $\partial M$, has been conjectured to hold $$\label{4}
E_{ADM}\geq\sqrt{\frac{\mathcal{A}}{16\pi}}+\sqrt{\frac{\pi}{\mathcal{A}}}Q^{2},$$ where $Q=\lim_{r\rightarrow\infty}\frac{1}{4\pi}\int_{S_{r}}E^{i}\nu_{i}$ is the total electric charge, with $S_{r}$ coordinate spheres in the asymptotic end having unit outer normal $\nu$. Inequality has been proven by Jang [@Jang1] for time-symmetric initial data with a connected horizon, under the assumption that a smooth solution to the Inverse Mean Curvature Flow (IMCF) exists. Moreover in light of Huisken and Ilmanen’s work [@HuiskenIlmanen], the hypothesis of a smooth IMCF can be discarded. However without the assumption of a connected horizon, counterexamples [@WeinsteinYamada] are known to exist (these examples do not provide a contradiction to the cosmic censorship conjecture), although remains true [@Khuri3] if an auxiliary inequality holds between the area and charge. In the non-time-symmetric case this inequality has been proven under the additional hypothesis of spherically symmetric initial data [@Hayward]. In the general case, with a connected horizon, the validity of has been reduced to solving a coupled system of equations involving the generalized Jang equation and the IMCF [@DisconziKhuri]. In the case of equality, it is expected that the initial data arise from the Reissner-Nordström spacetime; this has been confirmed in the time-symmetric case [@DisconziKhuri].
In this paper we establish a Penrose-like inequality including charge, without any assumption on $k$ or on the connectedness of the boundary. The primary difficulty in the non-time-symmetric (and non-maximal) case is the lack of the following positive lower bound for the scalar curvature $$\label{5}
R\geq2|E|_{g}^{2}.$$ In order to circumvent this issue, we seek a deformation of the initial data to a new set $(\overline{M},\overline{g},\overline{E})$, where $\overline{M}$ is diffeomorphic to $M$, and the metric $\overline{g}$ and vector field $\overline{E}$ are related to $g$ and $E$ in a precise way described below. The purpose of the deformation is to obtain new initial data which satisfy in a weak sense, while preserving the relevant geometric and physical quantities, such as the charge density, total charge, ADM energy, and boundary area. The desired deformation is a generalization of a procedure introduced by Jang [@Jang] and studied extensively by Schoen and Yau [@SchoenYau]. More precisely, consider the product 4-manifold $(M \times \mathbb{R}, g + dt^2)$, and let $\overline{M}=\{t=f(x)\}$ be the graph of a function $f$ inside this setting. Then the induced metric on $\overline{M}$ is given by $\overline{g}=g+df^{2}$. In order to obtain the most desirable positivity property for the scalar curvature of the graph, the function $f$ should satisfy $$\label{6}
\left( g^{ij} - \frac{ f^i f^j}{1 + |\nabla f|_g^2 }\right)
\left( \frac{ \nabla_{ij}f}{ \sqrt{1 + |\nabla f|_g^2 }}
-k_{ij} \right) = 0,$$ where $\nabla$ denotes covariant differentiation with respect to the metric $g$, $f_{i}=\partial_{i}f$, and $f^{i}=g^{ij}f_{j}$. Equation is referred to as the Jang equation, and when it is satisfied $\overline{M}$ will be called the Jang surface. The scalar curvature of the Jang surface [@SchoenYau] is given by $$\label{7}
\overline{R}=2(\mu-J(w))+2|E|_{g}^{2}+
|h-k|_{\overline{g}}^{2}+2|q|_{\overline{g}}^{2}
-2\overline{div}(q),$$ here $\overline{div}$ is the divergence operator with respect to $\overline{g}$, $h$ is the second fundamental form of the graph $t=f(x)$ in the Lorentzian 4-manifold $(\overline{M} \times \mathbb{R}, \overline{g}-dt^2)$, and $w$ and $q$ are 1-forms given by $$\label{8}
h_{ij}=\frac{ \nabla_{ij}f}{ \sqrt{1 + |\nabla f|_g^2 }},\text{ }\text{ }\text{ }\text{ }
w_{i}=\frac{f_{i}}{\sqrt{1+|\nabla f|_{g}^{2}}},\text{
}\text{ }\text{ }\text{ }
q_{i}=\frac{f^{j}}{\sqrt{1+|\nabla f|_{g}^{2}}}(h_{ij}-k_{ij}).$$
The existence and regularity theory for equation is well-understood. In particular, it is shown in [@HanKhuri] and [@Metzger] that there exists a smooth solution on $M$ which blows-up in the form of a cylinder over the outermost apparent horizon, with $f(x)\rightarrow\infty$ ($-\infty$) at each component of $\partial M$ depending on whether it is a future (or past) apparent horizon. Let $\tau(x)=dist(x,\partial M)$, and denote the level sets of $\tau$ by $S_{\tau}$. If $|\theta_{\pm}(S_{\tau})|\sim\tau^{l}$ near a future (past) apparent horizon component of the boundary, then according to [@HanKhuri] the blow-up solution satisfies the following asymptotics near that boundary component $$\label{8.1}
\alpha^{-1}\tau^{-\frac{l-1}{2}}+\beta^{-1} \leq \pm f \leq \alpha\tau^{-\frac{l-1}{2}}+\beta,$$ for some positive constants $\alpha$ and $\beta$. Moreover the solution decays sufficiently fast at spatial infinity so that the ADM energies agree $E_{ADM}(\overline{g})=E_{ADM}(g)$.
When the dominant energy condition is satisfied, all terms appearing on the right-hand side of are nonnegative, except possibly the last term. Thus the scalar curvature is nonnegative modulo a divergence, so it may be described as weakly nonnegative. For the topic of interest here, a stronger condition than simple nonnegativity is required, more precisely we seek an inequality (holding in the weak sense) of the following form $$\label{9}
\overline{R}\geq 2|\overline{E}|_{\overline{g}}^{2},$$ where $\overline{E}$ is an auxiliary electric field defined on the Jang surface. This auxiliary electric field is required to satisfy three properties, namely $$\label{10}
|E|_{g}\geq|\overline{E}|_{\overline{g}},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\overline{div}\,\overline{E}=0,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\overline{Q}=Q,$$ where $\overline{Q}$ is the total charge defined with respect to $\overline{E}$. In particular, if the first inequality of is satisfied, then the dominant energy condition and the scalar curvature formula imply that holds weakly. It turns out that there is a very natural choice for this auxiliary electric field, namely $\overline{E}$ is the induced electric field on the Jang surface $\overline{M}$ arising from the field strength $F$ of the electromagnetic field on $(M \times \mathbb{R}, g + dt^2)$. More precisely $\overline{E}_i = F(N,X_i)$, where $N$ and $X_i$ are respectively the unit normal and canonical tangent vectors to $\overline{M}$ $$\label{11}
N=\frac{\partial_{t}-f^{i}\partial_{i}}{\sqrt{1+|\nabla f|_{g}^{2}}},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }X_{i}=\partial_{i}+f_{i}\partial_{t},$$ and $F = \frac{1}{2} F_{ab}dx^a\wedge dx^b$ is given by $F_{0i} = E_i$ and $F_{ij} = 0$ for $i=1,2,3$, with $x^i$, $i=1,2,3$ coordinates on $M$ and $x^{0}=t$. In matrix form $$\label{12}
F = \left(
\begin{array}{cccc}
0 & E_1 & E_2 & E_3 \\
- E_1 & 0 & 0 & 0 \\
- E_2 & 0 & 0 & 0 \\
- E_3 & 0 & 0 & 0
\end{array}
\right).$$ In [@DisconziKhuri] it is shown that $$\label{13}
\overline{E}_i = \frac{E_i + f_i f^j E_j}{\sqrt{1 + |\nabla f|^2_g}},$$ and that all the desired properties of hold. This auxiliary electric field was also used in [@Khuri2].
The fact that inequality holds in a weak sense, allows us to find (a proof is given in the next section) a unique positive solution to the prescribed scalar curvature equation $$\label{14}
\overline{\Delta}u-\frac{1}{8}\overline{R}u+\frac{1}{4}|\widehat{E}|_{\widehat{g}}^{2}u^{5}=0\text{ }\text{ }\text{ on }\text{ }\text{ }\overline{M}$$ with the following boundary conditions. Namely, $u$ vanishes asymptotically along the cylindrical ends of $\overline{M}$ or alternatively $u(x)\rightarrow 0$ as $x\rightarrow\partial M$, and $$\label{15}
u(x)=1+\frac{A}{|x|}+O\left(\frac{1}{|x|^{2}}\right)\text{ }\text{ }\text{ as }\text{ }\text{ }|x|\rightarrow\infty$$ for some constant $A$. Here $\overline{\Delta}$ is the Laplacian with respect to $\overline{g}$, $\widehat{g}=u^{4}\overline{g}$, and $\widehat{E}^{i}=u^{-4}\overline{E}^{i}$. Equation expresses the fact that the scalar curvature of $\widehat{g}$ is given by $$\label{16}
\widehat{R}=2|\widehat{E}|_{\widehat{g}}^{2}.$$ It follows that the conformally deformed initial data $(\overline{M},\widehat{g},\widehat{E})$ satisfies the desired version of the dominant energy condition. We point out that the process of conformally changing the Jang initial data in order to obtain favorable properties for the scalar curvature was first used by Schoen and Yau [@SchoenYau] in their proof of the positive mass theorem. In fact when $E=0$, the solution $u$ of coincides with the conformal factor used in [@SchoenYau].
We now state the main theorem. Recall that the Hawking mass of a surface $S\subset M$, with area $|S|$, is given by $$\label{16.1}
M_{H}(S)=\sqrt{\frac{|S|}{16\pi}}\left(1-\frac{1}{16\pi}\int_{S}H^{2}\right),$$ and that $S$ is said to be area outerminimizing if every surface which encloses it has area greater than or equal to $|S|$. If a connected surface $S$ encloses the boundary $\partial M$, the region between $S$ and spatial infinity will be denoted by $M_{S}$.
\[thm1\] Let $(M,g,k,E)$ be a smooth asymptotically flat initial data set for the Einstein-Maxwell equations with total charge $Q$, $div E=0$, and satisfying the dominant energy condition $\mu\geq|J|_{g}$. If the boundary consists of an outermost apparent horizon with components $\partial_{i}M$, $i=1,\ldots,n$, then $$\label{17}
E_{ADM}(g)\geq\frac{\sigma_{1}}{2(1+\sigma_{1})}\sum_{i=1}^{n}\sqrt{\frac{|\partial_{i}
M|_{g}}{\pi}} +\sigma_{2}\sqrt{\frac{\pi}{|\partial M|_{g}}}Q^{2}$$ *with* $$\label{18}
\sigma_{1}=\left(\sum_{i=1}^{n}\sqrt{4\pi|\partial_{i}M|_{g}}\right)^{-1}\parallel\overline{\nabla}
u\parallel^{2}_{L^{2}(\overline{M})},\text{ }\text{ }\text{ }\text{ }
\sigma_{2}=\sup_{S}\sqrt{\frac{|\partial M|_{g}}{|S|_{\widehat{g}}}}\min_{M_{S}}u^{4},$$ where the supremum is taken over all connected surfaces $S$ which enclose $\partial M$, are area outerminimizing, and have nonnegative Hawking mass all with respect to $\widehat{g}$.
Note that the constants $\sigma_{1}$, $\sigma_{2}$ are scale invariant making them independent of $|\partial M|_{g}$, and it is clear that $\sigma_{1}$ is strictly positive. It will be proven below that $\sigma_{2}$ is also strictly positive. Thus inequality has a similar structure to that of , and it applies in a more general setting without restriction on the number of boundary components. This theorem also applies without the assumption of time-symmetric or maximal data, whereas has so far only been confirmed with these added hypotheses. It turns out that the case of equality in cannot occur, which indicates that this inequality is not optimal. Lastly, similar Penrose-like inequalities have previously been discussed in [@Herzlich] and [@Khuri], without a contribution from the total charge. Issues with [@Herzlich] have been raised in [@BartnikChrusciel] and [@Malec] (and partially addressed in [@BartnikChrusciel]), while issues with [@Khuri] have been pointed out in [@Khuri1] and are resolved in the appendix of the present paper.
We remark that it is the special geometry of the Jang surface, namely that it blows-up as a cylinder over the horizon, which is responsible for a definite contribution of area from each boundary component to the right-hand side of . This will be examined in Section \[sec4\]. There it will also be shown that the constant $\sigma_{1}$ may be written as an infimum over all functions satisfying appropriate asymptotics.
The Conformal Factor {#sec2}
====================
In the work of Schoen and Yau [@SchoenYau] existence of a unique solution to the following boundary value problem was established: $$\label{19}
\overline{\Delta}z-\frac{1}{8}\overline{R}z=0\text{
}\text{ }\text{ on }\text{ }\text{ }\overline{M},$$ with $z(x)\rightarrow 0$ as $x\rightarrow\partial M$ and $z(x)\rightarrow 1$ as $|x|\rightarrow\infty$. The inequality (4.6) in [@SchoenYau] shows that the first eigenvalue, $\eta_{i}$, of the operator $\Delta-\frac{1}{8}K$ on $\partial_{i}M$, is strictly positive (here $K$ denotes Gaussian curvature). As observed by Schoen and Yau, $z\sim e^{\mp\sqrt{\eta_{i}}t}\zeta_{i}(y)$, that is the conformal factor $z$ is asymptotic to $e^{\mp\sqrt{\eta_{i}}t}\zeta_{i}(y)$ depending on whether the Jang surface blows up or down, where $\zeta_{i}$ is the corresponding first eigenfunction.
The same methods of [@SchoenYau] may also be used to establish the existence of a unique solution to the following boundary value problem: $$\label{19}
\overline{\Delta}u-\frac{1}{8}\overline{R}u+\frac{1}{4}|\overline{E}|^{2}_{\overline{g}}u=0\text{
}\text{ }\text{ on }\text{ }\text{ }\overline{M},$$ with $$\label{19.1}
u(x)\rightarrow 0\text{ }\text{ }\text{ as }\text{ }\text{ }x\rightarrow\partial M,\text{ }\text{ }
\text{ and }\text{ }\text{ }u(x)\rightarrow 1\text{ }\text{ }\text{ as }\text{ }\text{ }|x|\rightarrow\infty.$$ Note that equation is equivalent to equation . In light of and the dominant energy condition , a slightly modified version of (4.6) in [@SchoenYau] shows that the first eigenvalue, $\lambda_{i}$, of the operator $\Delta-\frac{1}{8}K+\frac{1}{4}(E\cdot n)^{2}$ on $\partial_{i}M$, is strictly positive (here $n$ denotes the unit normal to $\partial_{i}M$) if the initial data are slightly perturbed so that $\mu>|J|_{g}$ at $\partial M$. Moreover as in [@SchoenYau], $u\sim e^{\mp\sqrt{\lambda_{i}}t}\phi_{i}(y)$, where $\phi_{i}$ is the corresponding first eigenfunction.
For the purposes of the proof of Theorem \[thm1\], it will be convenient to consider auxiliary boundary value problems, for which the solutions $u_{T}$ will converge to $u$; this will also yield an alternate proof of existence for $u$. In order to describe the auxiliary problems, for each $T>0$ let $\overline{M}_{T}$ denote the portion of the Jang surface $\overline{M}$ which lies between the hyperplanes $t=\pm T$. Let $\chi_{T}(y)$ denote the one parameter family of functions defined on a given boundary component $\partial_{i}M$ as the restriction of $|q|_{\overline{g}}$ to $\partial_{i}\overline{M}_{T}$. According to the parametric estimates for the Jang equation [@SchoenYau], the sequence of functions $\chi_{T}$ is uniformly bounded and equicontinuous. Therefore after passing to a subsequence (still denoted by $\chi_{T}$ for convenience) we have that $\chi_{T}\rightarrow\chi$ as $T\rightarrow\infty$, for some continuous function $\chi$. There are two cases to consider, namely, case 1 when $\chi$ vanishes identically, and case 2 when $\chi$ does not vanish identically.
We will slightly perturb $u$ in order to prescribe appropriate boundary conditions on certain cylindrical ends. For large $T$ and $T_{0}$ ($T>T_{0}$), let $(\overline{M}_{T}-\overline{M}_{T_{0}})_{i}$ denote the component of $\overline{M}_{T}-\overline{M}_{T_{0}}$ associated with the boundary component $\partial_{i}M$. Let $i=1,\ldots,m$ index the boundary components which fall under case 1, and let $i=m+1,\ldots,n$ index the boundary components which fall under case 2. Set $\widehat{M}_{T}=\overline{M}-\bigcup_{i=1}^{m}(\overline{M}-\overline{M}_{T})_{i}$, that is, $\widehat{M}_{T}$ is the Jang surface after the cylindrical ends corresponding to case 1 have been removed. Consider the boundary value problem $$\label{20}
\overline{\Delta}u_{T}-\frac{1}{8}\overline{R}u_{T}+\frac{1}{4}|\overline{E}|^{2}_{\overline{g}}u_{T}=0\text{
}\text{ }\text{ on }\text{ }\text{ }\widehat{M}_{T},$$ $$\label{21}
\partial_{\overline{N}}u_{T}+\frac{1}{4}\overline{H}u_{T}=\frac{1}{4}\sqrt{\frac{16\pi}{|\partial_{i}\overline{M}_{T}|_{\widehat{g}_{T}}}}u_{T}^{3}
\text{ }\text{ }\text{ on }\text{ }\text{ }\partial_{i}\overline{M}_{T},\text{ }\text{ }i=1,\ldots,m,$$ $$\label{22}
u_{T}(x)\rightarrow 0\text{ }\text{ }\text{ as }\text{ }\text{ }x\rightarrow\partial_{i} M,\text{ }\text{ }i=m+1,\ldots,n,
\text{ }\text{ }
u_{T}(x)\rightarrow 1\text{ }\text{ }\text{ as }\text{ }\text{ }|x|\rightarrow\infty,$$ where the unit normal $\overline{N}$ (with respect to $\overline{g}$) points towards spatial infinity and $\widehat{g}_{T}=u_{T}^{4}\overline{g}$. Note that the boundary condition expresses the fact that the mean curvature of $\partial_{i}\overline{M}_{T}$, $i=1,\ldots,m$, with respect to $\widehat{g}_{T}$, is given by $\widehat{H}=\sqrt{\frac{16\pi}{|\partial_{i}\overline{M}_{T}|_{\widehat{g}_{T}}}}$. The solutions $u_{T}$ to this problem approximate the solution $u$ of for large $T$, as is shown in Theorem \[thm3\] below. Furthermore, as in [@SchoenYau] a separation of variables argument can be used to show that the solution $u_{T}$ possesses the same asymptotics as $u$ along the ends corresponding to $\partial_{i}M$, $i=m+1,\ldots,n$, namely $$\label{22.1}
u_{T}\sim e^{\mp\sqrt{\lambda_{i}}t}\phi_{i}(y).$$
\[thm2\] If $T$ is sufficiently large, then there exists a smooth positive solution to boundary value problem , , .
Consider the functional $$\begin{aligned}
\label{23}
\begin{split}
P(v)=&\frac{1}{2}\int_{\widehat{M}_{T}}\left(|\overline{\nabla}v|^{2}+\frac{1}{8}\overline{R}(1+v)^{2}-\frac{1}{4}|\overline{E}|^{2}_{\overline{g}}(1+v)^{2}\right)\\
&-\sum_{i=1}^{m}\frac{1}{8}\int_{\partial_{i}\overline{M}_{T}}\overline{H}(1+v)^{2}
+\sum_{i=1}^{m}\frac{\sqrt{\pi}}{2}\left(\int_{\partial_{i}\overline{M}_{T}}(1+v)^{4}\right)^{1/2}
\end{split}\end{aligned}$$ on the space of functions $$\label{24}
\mathcal{W}=\{v\in W_{loc}^{1,2}(\widehat{M}_{T})\mid |x|^{j-1}\overline{\nabla}^{j}v\in L^{2}(\widehat{M}_{T}),\text{ }j=0,1,\text{ }
1+v\in W^{1,2}_{0}(\widehat{M}_{T})\},$$ where $W^{1,2}_{0}(\widehat{M}_{T})$ is the closure, in the $W^{1,2}$-norm, of the space of smooth functions which have compact support when restricted to each cylindrical end indexed by $i=m+1,\ldots,n$. Here $W^{1,2}\subset L^{2}$ is the space of functions with square integrable first derivatives.
In order to establish the existence (as well as the asymptotic behavior) of a solution $v_{T}\in
\mathcal{W}\cap C^{\infty}(\widehat{M}_{T})$, it is enough, by the arguments of [@Herzlich], to show that for $T$ sufficiently large the functional $P$ is nonnegative. To see that this is the case, use formula and , and integrate the divergence term by parts to find that for any $v\in \mathcal{W}$, $$\begin{aligned}
\label{25}
\begin{split}
P(v)\geq&\int_{\widehat{M}_{T}}\left(\frac{3}{8}|\overline{\nabla}v|^{2}+\frac{1}{8}(\mu-|J|_{g})(1+v)^{2}\right)+
\sum_{i=1}^{m}\frac{\sqrt{\pi}}{2}\left(\int_{\partial_{i}\overline{M}_{T}}(1+v)^{4}\right)^{1/2}\\
&
-\sum_{i=1}^{m}\frac{1}{8}\int_{\partial_{i}\overline{M}_{T}}(\overline{H}-q(\overline{N}))(1+v)^{2}.
\end{split}\end{aligned}$$ According to [@BrayKhuri] (also [@BrayKhuri1]) $\overline{H}\rightarrow 0$ as $T\rightarrow\infty$, and since the boundary components $\partial_{i}\overline{M}_{T}$ belong to case 1 we have that $q(\overline{N})\rightarrow 0$ as $T\rightarrow\infty$. Moreover, the area of $\partial_{i}\overline{M}_{T}$ approximates the area of $\partial_{i}M$. It then follows from Jensen’s Inequality $$\label{26}
\left(\int_{\partial_{i}\overline{M}_{T}}(1+v)^{2}\right)^{2}\leq
|\partial_{i}\overline{M}_{T}|_{\overline{g}}\int_{\partial_{i}\overline{M}_{T}}(1+v)^{4},$$ that for $T$ sufficiently large $P$ is nonnegative.
It remains to show that $u_{T}=1+v_{T}$ is strictly positive. So suppose that $u_{T}$ is not positive and let $D_{-}$ be the domain on which $u_{T}<0$. Since $u_{T}\rightarrow 1$ as $|x|\rightarrow\infty$, the closure of $D_{-}\cap\overline{M}_{T}$ must be compact. Now multiply equation through by $u_{T}$ and integrate by parts to obtain $$\label{27}
\int_{D_{-}}|\overline{\nabla}u_{T}|^{2}\leq 0.$$ Note that if $D_{-}\cap\partial_{i}\overline{M}_{T}\neq\emptyset$, $i=1,\ldots,m$ then the same arguments used above to show that $P$ is nonnegative, must be employed. It follows that $u_{T}\geq 0$. To show that $u_{T}>0$, one need only apply Hopf’s maximum principle (the boundary condition of must be used to obtain this conclusion at $\partial_{i}\overline{M}_{T}$, $i=1,\ldots,m$).
Multiply equation by $u_{T}=1+v_{T}$ and integrate by parts to obtain $$\begin{aligned}
\label{28}
\begin{split}
\mathcal{P}(v_{T})&:=\lim_{r\rightarrow\infty}\frac{1}{2}\int_{|x|=r}u_{T}\partial_{\overline{N}}u_{T}\\
&\geq\int_{\overline{M}_{T}}\frac{1}{4}|\overline{\nabla}v_{T}|^{2}
+\left(\frac{1}{8}(\mu-|J|)+\frac{1}{16}|q|_{\overline{g}}^{2}\right)(1+v_{T})^{2}\\
& +\int_{\partial\overline{M}_{T}}\frac{1}{8}q(\overline{N})(1+v_{T})^{2}+\frac{1}{2}u_{T}\partial_{\overline{N}}u_{T}.
\end{split}\end{aligned}$$ A standard formula yields $$\label{29}
\partial_{\overline{N}}u_{T}=\frac{1}{4}\widehat{H}u_{T}^{3}-\frac{1}{4}\overline{H}u_{T},$$ where $\widehat{H}$ and $\overline{H}$ are the mean curvatures with respect to $\widehat{g}_{T}$ and $\overline{g}$, respectively. It follows that $$\begin{aligned}
\label{30}
\begin{split}
\mathcal{P}(v_{T})
&\geq\int_{\overline{M}_{T}}\frac{1}{4}|\overline{\nabla}v_{T}|^{2}
+\left(\frac{1}{8}(\mu-|J|)+\frac{1}{16}|q|_{\overline{g}}^{2}\right)(1+v_{T})^{2}\\
& +\int_{\partial\overline{M}_{T}}\frac{1}{8}(q(\overline{N})-\overline{H})(1+v_{T})^{2}+\frac{1}{8}\widehat{H}(1+v_{T})^{4}.
\end{split}\end{aligned}$$ The quantity $\mathcal{P}(v_{T})$ appears in the formula for the ADM energy of the metric $\widehat{g}_{T}$, more precisely $E_{ADM}(\widehat{g}_{T})=E_{ADM}(\overline{g})-\pi^{-1}\mathcal{P}(v_{T})$. Therefore it is important to estimate $\mathcal{P}(v_{T})$ from below.
\[lemma1\] Let $u_{T}=1+v_{T}$ be the function produced in Theorem \[thm2\], then $$\label{31}
\mathcal{P}(v_{T})\geq \int_{\overline{M}_{T}}\frac{1}{4}|\overline{\nabla}v_{T}|^{2}
+\left(\frac{1-\vartheta_{T}}{2}\right)
\sum_{i=1}^{n}\sqrt{\frac{\pi}{|\partial_{i}\overline{M}_{T}|_{\overline{g}}}}
\int_{\partial_{i}\overline{M}_{T}}(1+v_{T})^{2}$$ where $\vartheta_{T}\rightarrow 0$ as $T\rightarrow\infty$.
There are two cases to consider.
*Case 1: $\chi\equiv 0$.*
This case corresponds to the boundary components $\partial_{i}M$, $i=1,\ldots,m$. Here the methods of the proof of Theorem \[thm2\] apply to yield $$\begin{aligned}
\label{32}
\begin{split}
& \int_{\partial_{i}\overline{M}_{T}}\frac{1}{8}(q(\overline{N})-\overline{H})(1+v_{T})^{2}+\frac{1}{8}\widehat{H}(1+v_{T})^{4}\\
\geq &\left(\frac{1-\vartheta_{T}}{2}\right)\sqrt{\frac{\pi}{|\partial_{i}\overline{M}_{T}|_{\overline{g}}}}\int_{\partial_{i}\overline{M}_{T}}(1+v_{T})^{2},
\end{split}\end{aligned}$$ for some constants $\vartheta_{T}\rightarrow 0$ as $T\rightarrow\infty$.
*Case 2: $\chi$ does not vanish identically.*
This case corresponds to the boundary components $\partial_{i}M$, $i=m+1,\ldots,n$. For each such component there is a set of positive measure $\Omega_{i}\subset\partial_{i}M$ on which $\chi_{T_{j}}\geq 2\varepsilon>0$ for a subsequence of heights $T_{j}\rightarrow\infty$. Let $(\overline{M}_{T}-\overline{M}_{T'})\cap\Omega_{i}$ denote the portion of $(\overline{M}_{T}-\overline{M}_{T'})_{i}$ which, after projection onto the vertical cylinder over $\partial_{i}M$, corresponds with $\Omega_{i}\times(T',T)$. Similarly let $\partial_{i}\overline{M}_{T}\cap\Omega_{i}$ denote the portion of $\partial_{i}\overline{M}_{T}$ which, after projection onto the vertical cylinder over $\partial_{i}M$, corresponds with $\Omega_{i}\times \{T\}$. Since $|q|_{\overline{g}}$ is uniformly bounded in $C^{1}$, there is a $\delta>0$ independent of $j$, such that $|q|_{\overline{g}}\geq \varepsilon$ on $(\overline{M}_{T_{j}+\delta}-\overline{M}_{T_{j}-\delta})\cap\Omega_{i}$ for each $j$. If $\mathcal{N}(T-T_{0})$ denotes the number of $T_{j}$ in the interval $(T_{0},T)$, then using the asymptotics of $u_{T}$, it follows that for sufficiently large $T$ and $T_{0}$ we have $$\begin{aligned}
\label{33}
\begin{split}
\int_{(\overline{M}_{T}-\overline{M}_{T_{0}})_{i}}|q|_{\overline{g}}^{2}(1+v_{T})^{2}&\geq \int_{(\overline{M}_{T}-\overline{M}_{T_{0}})\cap\Omega_{i}}|q|_{\overline{g}}^{2}(1+v_{T})^{2}\\
&\geq\varepsilon^{2}\sum_{j=0}^{\mathcal{N}(T-T_{0})}\int_{(\overline{M}_{T_{j}+\delta}-\overline{M}_{T_{j}-\delta})\cap\Omega_{i}}(1+v_{T})^{2}\\
&\geq\delta\varepsilon^{2}\mathcal{N}(T-T_{0})\int_{\partial_{i}\overline{M}_{T}\cap\Omega_{i}}(1+v_{T})^{2}.
\end{split}\end{aligned}$$ Note that in the last step in the above sequence of inequalities, the factor $\delta$ can be pulled out in light of and the fact that the metric on $(\overline{M}_{T_{j}+\delta}-\overline{M}_{T_{j}-\delta})\cap\Omega_{i}$ approximates the product metric on $\Omega_{i}\times(T_{j}-\delta,T_{j}+\delta)$. Furthermore, using again yields $$\label{34}
\int_{\partial_{i}\overline{M}_{T}\cap\Omega_{i}}(1+v_{T})^{2}\geq C_{0}\int_{\partial_{i}\overline{M}_{T}}(1+v_{T})^{2},$$ for some positive constant $C_{0}$ independent of $T$. Hence $$\label{35}
\int_{(\overline{M}_{T}-\overline{M}_{T_{0}})_{i}}\frac{1}{16}|q|_{\overline{g}}^{2}(1+v_{T})^{2}\geq
C_{1}\mathcal{N}(T-T_{0})\sqrt{\frac{\pi}{|\partial_{i}\overline{M}_{T}|_{\overline{g}}}}\int_{\partial_{i}\overline{M}_{T}}(1+v_{T})^{2}.$$ From [@SchoenYau] (page 257) $$\label{36}
\widehat{g}_{T}\sim\phi_{i}^{4}(y)(d\rho^{2}+4\lambda_{i}\rho^{2}d\theta^{2})$$ for $\rho$ near zero, where $\rho=(2\sqrt{\lambda_{i}})^{-1}e^{\mp2\sqrt{\lambda_{i}}t}$ and $d\theta^{2}$ is the induced metric on $\partial_{i}M$. Therefore $$\label{37}
\widehat{H}\sim\phi_{i}^{-2}(y)\left(\frac{2}{\rho}\right)\sim 4\sqrt{\lambda_{i}}u_{T}^{-2}=4\sqrt{\lambda_{i}}(1+v_{T})^{-2}.$$ By applying , and using the fact that $q(\overline{N})-\overline{H}$ is uniformly bounded and $\mathcal{N}(T-T_{0})\rightarrow\infty$ as $T\rightarrow\infty$, we then have $$\begin{aligned}
\label{38}
\begin{split}
& \int_{(\overline{M}_{T}-\overline{M}_{T_{0}})_{i}}\frac{1}{16}|q|_{\overline{g}}^{2}(1+v_{T})^{2}
+\int_{\partial_{i}\overline{M}_{T}}\frac{1}{8}(q(\overline{N})-\overline{H})(1+v_{T})^{2}+\frac{1}{8}\widehat{H}(1+v_{T})^{4}\\
\geq & C_{1}\mathcal{N}(T-T_{0})\sqrt{\frac{\pi}{|\partial_{i}\overline{M}_{T}|_{\overline{g}}}}\int_{\partial_{i}\overline{M}_{T}}(1+v_{T})^{2}
-C_{2}\int_{\partial_{i}\overline{M}_{T}}(1+v_{T})^{2}\\
\geq &\left(1-C_{3}\mathcal{N}(T-T_{0})^{-1}\right)\sqrt{\frac{\pi}{|\partial_{i}\overline{M}_{T}|_{\overline{g}}}}\int_{\partial_{i}\overline{M}_{T}}(1+v_{T})^{2}
\end{split}\end{aligned}$$ for $T$ sufficiently large so that $C_{1}\mathcal{N}(T-T_{0})\geq 1$.
We may now combine , , and to obtain the desired result.
Proof of the Main Theorem {#sec3}
=========================
Consider the manifold $(\widehat{M}_{T},\widehat{g}_{T})$. Along the infinite cylindrical ends over $\partial_{i}M$, $i=m+1,\ldots,n$, the conformal factor $u_{T}$ decays exponentially fast. Therefore as in [@SchoenYau] these ends may be closed by adding a point at infinity. The remaining cylindrical ends, indexed by $i=1,\ldots,m$, correspond to the boundary components of $\widehat{M}_{T}$ which satisfy the hypotheses of Herzlich’s version of the positive mass theorem [@Herzlich]. Alternatively, these boundary components have zero Hawking mass $$\label{39}
M_{H}(\partial_{i}\widehat{M}_{T}):=\sqrt{\frac{|\partial_{i}\widehat{M}_{T}|_{\widehat{g}}}{16\pi}}\left(1-\frac{1}{16\pi}\int_{\partial_{i}\widehat{M}_{T}}\widehat{H}^{2}\right)= 0.$$ It follows that the ADM energy $E_{ADM}(\widehat{g}_{T})$ is nonnegative. Combining this with the lower bound for $\mathcal{P}(v_{T})$ yields a lower bound for the ADM energy of $g$, from the formula $$\label{40}
E_{ADM}(g)=E_{ADM}(\overline{g})=E_{ADM}(\widehat{g}_{T})+\pi^{-1}\mathcal{P}(v_{T}).$$
We now estimate the positive contributions from both terms on the right-hand side of . Let us begin with $E_{ADM}(\widehat{g}_{T})$. Consider a connected surface $S$ which encloses $\partial M$, is area outerminimizing, and has nonnegative Hawking mass all with respect to $\widehat{g}_{T}$. Let $\{S_{\varrho}\}_{\varrho=0}^{\infty}$ be a weak inverse mean curvature flow (see [@HuiskenIlmanen]) emanating from $S=S_{0}$. Then according to Geroch monotonicity [@HuiskenIlmanen] $$\label{41}
E_{ADM}(\widehat{g}_{T})\geq\int_{0}^{\infty}\left(\frac{|S_{\varrho}|_{\widehat{g}_{T}}^{1/2}}{(16\pi)^{3/2}}\int_{S_{\varrho}}\widehat{R}_{T}d\theta_{\widehat{g}_{T}}\right)d\varrho,$$ where $\widehat{R}_{T}$ is the scalar curvature of $\widehat{g}_{T}$. Let $\widehat{N}_{T}$ denote the unit normal to $S_{\varrho}$ with respect to $\widehat{g}_{T}$, and set $\widehat{E}_{T}^{i}=u_{T}^{-4}\overline{E}^{i}$. By $\widehat{R}_{T}=2|\widehat{E}_{T}|_{\widehat{g}_{T}}^{2}$, and with the help of Cauchy-Schwarz, , and Hölder’s inequality $$\begin{aligned}
\label{42}
\begin{split}
\int_{S_{\varrho}}|\widehat{E}_{T}|_{\widehat{g}_{T}}^{2}d\theta_{\widehat{g}_{T}}&\geq\int_{S_{\varrho}}\widehat{g}_{T}(\widehat{E}_{T},\widehat{N}_{T})^{2}d\theta_{\widehat{g}_{T}}\\
&=\int_{S_{\varrho}}\overline{g}(\overline{E},\overline{N})^{2}d\theta_{\overline{g}}\\
&\geq|S_{\varrho}|_{\overline{g}}^{-1}\left(\int_{S_{\varrho}}\overline{g}(\overline{E},\overline{N})d\theta_{\overline{g}}\right)^{2}\\
&=\frac{(4\pi\overline{Q})^{2}}{|S_{\varrho}|_{\overline{g}}}\\
&=\frac{|S_{\varrho}|_{\widehat{g}_{T}}}{|S_{\varrho}|_{\overline{g}}}\frac{(4\pi Q)^{2}}{|S_{\varrho}|_{\widehat{g}_{T}}}.
\end{split}\end{aligned}$$ A basic property of inverse mean curvature flow is that the area of the flow surfaces increases exponentially, in particular $|S_{\varrho}|_{\widehat{g}_{T}}=|S_{0}|_{\widehat{g}_{T}}e^{\varrho}$. Moreover, if $M_{S}$ denotes the region between spatial infinity and the surface $S$, then $$\label{43}
\frac{|S_{\varrho}|_{\widehat{g}_{T}}}{|S_{\varrho}|_{\overline{g}}}\geq\min_{M_{S}}u_{T}^{4}.$$ It follows that $$\label{44}
E_{ADM}(\widehat{g}_{T})\geq\left(\sqrt{\frac{|\partial M|_{g}}{|S|_{\widehat{g}_{T}}}}\min_{M_{S}}u_{T}^{4}\right)\sqrt{\frac{\pi}{|\partial M|_{g}}}Q^{2}.$$ Since this inequality is true for all surfaces $S$ which enclose $\partial M$, are area outerminimizing, and have nonnegative Hawking mass all with respect to $\widehat{g}_{T}$, we then have $$\label{45}
E_{ADM}(\widehat{g}_{T})\geq\sigma_{2,T}\sqrt{\frac{\pi}{|\partial M|_{g}}}Q^{2}$$ where $$\label{46}
\sigma_{2,T}=\sup_{S}\sqrt{\frac{|\partial M|_{g}}{|S|_{\widehat{g}_{T}}}}\min_{M_{S}}u_{T}^{4}.$$ Note that the set of surfaces $S$ which have the above desired properties is nonempty. To see this we may simply start an inverse mean curvature flow from one of the boundary components $\partial_{i}\widehat{M}_{T}$, $i=1,\ldots,m$, then for sufficiently large $\varrho$, each of the flow surfaces $S_{\varrho}$ encloses $\partial M$, is area outerminimizing, and has nonnegative Hawking mass. In particular, $\sigma_{2,T}$ is strictly positive.
The positive contribution from $\mathcal{P}(v_{T})$ will now be estimated. Suppose that $\mathcal{P}(v_{T})\leq\eta\sum_{i=1}^{n}\sqrt{\pi|\partial_{i}\overline{M}_{T}|_{\overline{g}}}$ for some positive constant $\eta$. Then by Lemma \[lemma1\] $$\label{47}
\int_{\overline{M}_{T}}\frac{1}{4}|\overline{\nabla}v_{T}|^{2}
+\left(\frac{1-\vartheta_{T}}{2}\right)
\sum_{i=1}^{n}\sqrt{\frac{\pi}{|\partial_{i}\overline{M}_{T}|_{\overline{g}}}}
\int_{\partial_{i}\overline{M}_{T}}(1+v_{T})^{2}
\leq\eta\sum_{i=1}^{n}\sqrt{\pi|\partial_{i}\overline{M}_{T}|_{\overline{g}}}.$$ However by Young’s inequality $$\label{48}
(1+v_{T})^{2}\geq 1-\frac{1}{\delta}+(1-\delta)v_{T}^{2}$$ for any $\delta>0$, and therefore $$\begin{aligned}
\label{49}
\begin{split}
&\int_{\overline{M}_{T}}\frac{1}{4}|\overline{\nabla}v_{T}|^{2}+(1-\delta)
\left(\frac{1-\vartheta_{T}}{2}\right)\sum_{i=1}^{n}\sqrt{\frac{\pi}
{|\partial_{i}\overline{M}_{T}|_{\overline{g}}}}\int_{\partial_{i}\overline{M}_{T}}v_{T}^{2}\\
\leq &(\eta-\frac{1}{2}(1-\delta^{-1})(1-\vartheta_{T}))
\sum_{i=1}^{n}\sqrt{\pi|\partial_{i}\overline{M}_{T}|_{\overline{g}}}.
\end{split}\end{aligned}$$ The left-hand side is nonnegative if $\delta-1\leq\sigma_{1,T}$ where $$\label{50}
\sigma_{1,T}=
\frac{\int_{\overline{M}_{T}}|\overline{\nabla}v_{T}|^{2}}{2(1-\vartheta_{T})
\sum_{i=1}^{n}\sqrt{\frac{\pi}{|\partial_{i}\overline{M}_{T}|_{\overline{g}}}}
\int_{\partial_{i}\overline{M}_{T}}v_{T}^{2}}.$$ It follows that $\eta\geq\delta^{-1}(\delta-1)(1-\vartheta_{T})/2$ for all such $\delta$. In particular by choosing $\delta=1+\sigma_{1,T}$ we conclude that $$\label{51}
\mathcal{P}(v_{T})\geq\frac{\sigma_{1,T}(1-\vartheta_{T})}{2(1+\sigma_{1,T})}\sum_{i=1}^{n}\sqrt{\pi
|\partial_{i}\overline{M}_{T}|_{\overline{g}}}.$$
The combination of , , and now produces $$\label{52}
E_{ADM}(g)\geq\frac{\sigma_{1,T}(1-\vartheta_{T})}{2(1+\sigma_{1,T})}\sum_{i=1}^{n}\sqrt{\frac{|\partial_{i}
\overline{M}_{T}|_{\overline{g}}}{\pi}} +\sigma_{2,T}\sqrt{\frac{\pi}{|\partial M|_{g}}}Q^{2}.$$
\[thm3\] After possibly passing to a subsequence, $u_{T}\rightarrow u$ in $C^{\infty}_{loc}(\overline{M})$ as $T\rightarrow\infty$, where $u$ is the unique solution of boundary value problem , .
Together , , and show that the sequence of functions $\{u_{T}\}$ is uniformly bounded in $W^{1,2}_{loc}(\overline{M})$. Thus with the help of elliptic estimates and Sobolev embeddings, a subsequence converges on compact subsets to a smooth uniformly bounded solution $u_{\infty}$ of $$\label{53}
\overline{\Delta}u_{\infty}-\frac{1}{8}\overline{R}u_{\infty}+\frac{1}{4}|\overline{E}|^{2}_{\overline{g}}u_{\infty}=0\text{
}\text{ }\text{ on }\text{ }\text{ }\overline{M},\text{ }\text{
}\text{ }\text{
}u_{\infty}=1+\frac{A_{\infty}}{|x|}+O(|x|^{-2})\text{ }\text{
}\text{ as }\text{ }\text{ }|x|\rightarrow\infty.$$ Moreover since $\overline{M}$ approximates a cylinder on regions where it blows-up, comparison with a bounded solution of the same equation on the cylinder (as is done in [@SchoenYau]) shows that $u_{\infty}(x)\rightarrow 0$ as $x\rightarrow\partial M$; in fact the decay rate is of exponential strength. Thus $u_{\infty}$ satisfies boundary value problem , , and therefore must coincide with the unique solution to this problem $u=u_{\infty}$.
Theorem \[thm3\] shows that after passing to a subsequence, $\sigma_{1,T}\rightarrow\sigma_{1}$ and $\sigma_{2,T}\rightarrow\sigma_{2}$ as $T\rightarrow\infty$. Theorem \[thm1\] now follows from .
Lastly we analyze what happens when equality occurs in Theorem \[thm1\]. By slightly modifying the arguments presented, we find that equality in implies that $$\label{54}
\int_{\overline{M}}|\overline{\nabla}u|^{2}=0,$$ and therefore $u$ must be constant. However this is impossible since $$\label{55}
u(x)\rightarrow\begin{cases}
1 & \text{as $|x|\rightarrow\infty$},\\
0 & \text{as $x\rightarrow\partial M$}.
\end{cases}$$ We conclude that the case of equality cannot occur.
Further Properties of the Constant $\sigma_{1}$ {#sec4}
===============================================
In this section it will be shown how the constant $\sigma_{1}$ of Theorem \[thm1\] may be redefined as an infimum over conformal factors which satisfy appropriate asymptotics. We also describe how the area of each boundary component $\partial_{i}M$, naturally arises and makes a definite contribution to the right-hand side of . First observe that a slight improvement of the estimate in Lemma \[lemma1\] is possible, by utilizing all of the terms in , namely $$\begin{aligned}
\label{56}
\begin{split}
\mathcal{P}(v_{T})&\geq \int_{\overline{M}_{T}}\frac{1}{4}|\overline{\nabla}v_{T}|^{2}+\left(\frac{1}{8}(\mu-|J|_{g})+\frac{1}{20}|q|_{\overline{g}}^{2}\right)(1+v_{T})^{2}\\
&+\left(\frac{1-\vartheta_{T}}{2}\right)
\sum_{i=1}^{n}\sqrt{\frac{\pi}{|\partial_{i}\overline{M}_{T}|_{\overline{g}}}}
\int_{\partial_{i}\overline{M}_{T}}(1+v_{T})^{2}.
\end{split}\end{aligned}$$ By following the arguments in Section \[sec3\], we obtain a lower bound of the form $$\label{57}
\mathcal{P}(v)\geq\frac{\overline{\sigma}_{1,T}(1-\vartheta_{T})}{2(1+\overline{\sigma}_{1,T})}\sum_{i=1}^{n}\sqrt{\pi
|\partial_{i}\overline{M}_{T}|_{\overline{g}}},$$ where $$\label{58}
\overline{\sigma}_{1,T}=
\frac{\int_{\overline{M}_{T}}|\overline{\nabla}v_{T}|^{2}+\left(\frac{1}{2}(\mu-|J|_{g})+\frac{1}{5}|q|_{\overline{g}}^{2}\right)(1+v_{T})^{2}}{2(1-\vartheta_{T})
\sum_{i=1}^{n}\sqrt{\frac{\pi}{|\partial_{i}\overline{M}_{T}|_{\overline{g}}}}
\int_{\partial_{i}\overline{M}_{T}}v_{T}^{2}}.$$ It follows that if $\overline{\sigma}_{1,T}\rightarrow\overline{\sigma}_{1}$ then $$\begin{aligned}
\label{59}
\begin{split}
\overline{\sigma}_{1}
&=\left(\sum_{i=1}^{n}\sqrt{4\pi|\partial_{i}M|_{g}}\right)^{-1}\int_{\overline{M}}|\overline{\nabla}u|^{2}+\left(\frac{1}{2}(\mu-|J|_{g})+\frac{1}{5}|q|_{\overline{g}}^{2}\right)u^{2}\\
&\geq\left(\sum_{i=1}^{n}\sqrt{4\pi|\partial_{i}M|_{g}}\right)^{-1}\inf_{w}\int_{\overline{M}}|\overline{\nabla}w|^{2}+\left(\frac{1}{2}(\mu-|J|_{g})+\frac{1}{5}|q|_{\overline{g}}^{2}\right)w^{2},
\end{split}\end{aligned}$$ where the infimum is taken over all smooth functions $w$ satisfying the following asymptotics $$\label{60}
w\sim e^{\mp\sqrt{\kappa_{i}}t}\text{ }\text{ }\text{ as }\text{ }\text{ }x\rightarrow\partial_{i}M,\text{ }\text{ }\text{ }\text{ }w\rightarrow 1\text{ }\text{ }\text{ as }\text{ }\text{ }
|x|\rightarrow\infty,$$ for some positive constants $\kappa_{i}\leq2\lambda_{i}$; here $\lambda_{i}$, as in Section \[sec2\], is the first eigenvalue of $\Delta-\frac{1}{8}K+\frac{1}{4}(E\cdot n)^{2}$ on $\partial_{i}M$. Therefore we may redefine the constant $\sigma_{1}$ appearing in Theorem \[thm1\] to be given by the infimum on the right-hand side of .
The infimum is realized by the unique (positive) solution of the equation $$\label{61}
\overline{\Delta}w-\left[\frac{1}{2}(\mu-|J|_{g})+\frac{1}{5}|q|_{\overline{g}}^{2}\right]w=0\text{ }\text{ }\text{ on }\text{ }\text{ }\overline{M},$$ with asymptotics $$\label{62}
e^{\pm\sqrt{\gamma_{i}}t}w\rightarrow\psi_{i}\text{ }\text{ }\text{ as }\text{ }\text{ }x\rightarrow\partial_{i}M,\text{ }\text{ }\text{ }\text{ }w\rightarrow 1\text{ }\text{ }\text{ as }\text{ }\text{ }
|x|\rightarrow\infty,$$ where $\gamma_{i}$ is the first eigenvalue and $\psi_{i}$ the corresponding eigenfunction for the operator $\Delta-\left[\frac{1}{2}(\mu-|J|_{g})+\frac{1}{5}\chi^{2}\right]$ on $\partial_{i}M$, with $\chi$ defined as in Section \[sec2\]. Note that this is consistent with the fact that $\gamma_{i}\leq2\lambda_{i}$. To see this, observe that for any $\xi\in C^{\infty}_{c}(\overline{M})$ $$\label{63}
\int_{\overline{M}}\left(-\overline{R}+2|\overline{E}|^{2}_{\overline{g}}+2(\mu-|J|_{g})+|q|_{\overline{g}}^{2}\right)\xi^{2}
\leq\int_{\overline{M}}(-|q|_{\overline{g}}^{2}+2\overline{div}(q))\xi^{2}
\leq\int_{\overline{M}}4|\overline{\nabla}\xi|^{2},$$ from which it follows that $$\label{64}
4\int_{\overline{M}}\left[|\overline{\nabla}\xi|^{2}+\left(\frac{1}{2}(\mu-|J|_{g})+\frac{1}{4}|q|_{\overline{g}}^{2}\right)\xi^{2}\right]
\leq8\int_{\overline{M}}\left[|\overline{\nabla}\xi|^{2}+\left(\frac{1}{8}\overline{R}-\frac{1}{4}|\overline{E}|^{2}_{\overline{g}}\right)\xi^{2}\right].$$ By translating the Jang surface in the $t$-direction as in [@SchoenYau] (page 254), we find $$\label{65}
\int_{\partial_{i}M}\left[|\nabla\varphi|^{2}+\left(\frac{1}{2}(\mu-|J|_{g})+\frac{1}{4}\chi^{2}\right)\varphi^{2}\right]
\leq2\int_{\partial_{i}M}\left[|\nabla\varphi|^{2}+\left(\frac{1}{8}K-\frac{1}{4}(E\cdot n)^{2}\right)\varphi^{2}\right]$$ for any $\varphi\in C^{\infty}(\partial_{i}M)$, so that $\gamma_{i}\leq2\lambda_{i}$.
Next we describe (heuristically) how the area of each boundary component $\partial_{i}M$, naturally arises and makes a definite contribution to the right-hand side of . This is primarily a consequence of the cylindrical geometry of the Jang surface near the horizon. Recall that as in , the goal is to obtain a lower bound for $\mathcal{P}(v_{T})$ and then let $T\rightarrow\infty$. Observe that since $(\overline{M}-\overline{M}_{t_{0}})_{i}$ approximates a cylinder for sufficiently large $t_{0}$, it follows that $$\label{66}
4\mathcal{P}(v)\geq\int_{\overline{M}}|\overline{\nabla}u|^{2}\geq\frac{1}{2}\sum_{i=1}^{n}\int_{(\overline{M}-\overline{M}_{t_{0}})_{i}}(\partial_{t}u)^{2}.$$ Furthermore, since $u\sim e^{\mp\sqrt{\lambda_{i}}(t-t_{0})}\phi_{i}$ where $\phi_{i}$ is the principal eigenfunction of the operator $\Delta-\frac{1}{8}K+\frac{1}{4}(E\cdot n)^{2}$ on $\partial_{i}M$, normalized so that $\parallel\phi_{i}\parallel_{L^{2}}^{2}=|\partial_{i}M|_{g}$, we find that $$\label{67}
\mathcal{P}(v)\geq\sum_{i=1}^{n}c_{i}\left(\int_{\partial_{i}M}\phi_{i}^{2}\right)\left(\int_{t_{0}}^{\infty}\lambda_{i}e^{-2\sqrt{\lambda_{i}}(t-t_{0})}\right)
=\sum_{i=1}\frac{c_{i}}{2}\sqrt{\lambda_{i}}|\partial_{i}M|_{g}$$ where the constants $c_{i}>0$ depend on $u$. As $\lambda_{i}$ is the principal eigenvalue for a self-adjoint elliptic operator on the 2-sphere, it should behave similarly to the principal eigenvalue of the Laplacian in that $\lambda_{i}\sim|\partial_{i}M|_{g}^{-1}$. Thus we find a natural contribution to the right-hand side of , from each boundary component, in the form of $\sqrt{|\partial_{i}M|_{g}}$. Another somewhat more vague approach to arrive at the same intuitive conclusion, is to realize that $\mathcal{P}(v)$ is related to the electrostatic capacity of $\partial M$, which in turn is related to $\sqrt{|\partial M|_{g}}$. There are several well-known results, and also conjectures (of Pólya and Szegö), concerning the relationship of capacity to the square root of boundary area in Euclidean space. It remains to be seen exactly how these generalize to a Riemannian manifold, although one result in this direction may be found in [@BrayMiao].
Appendix: The Uncharged Case {#sec5}
============================
In this section we make clear how the arguments above correct issues associated with the uncharged Penrose-like inequality discussed in [@Khuri]. Recall that two errors were pointed out in the erratum [@Khuri1]. The first concerns the constant $\sigma$ in the statement of Theorem 1.2 [@Khuri]; namely, this constant is in fact zero. The second error concerns Lemma 2.2 [@Khuri], in that the quantity $\overline{H}-q(\overline{N})$ may not necessarily approach zero as $r\rightarrow 0$. By setting the electric field $E=0$ in the results of the present paper, both problems are resolved; in particular, Lemma 2.2 [@Khuri] is not needed. However, for the convenience of the reader, we explicitly carry out the revised proofs below for the uncharged case.
Let $\chi_{T}(y)$ denote the one parameter family of functions defined on a given boundary component $\partial_{i}M$ as the restriction of $|q|_{\overline{g}}$ to $\partial_{i}\overline{M}_{T}$. According to the parametric estimates for the Jang equation [@SchoenYau], the sequence of functions $\chi_{T}$ is uniformly bounded and equicontinuous. Therefore after passing to a subsequence (still denoted by $\chi_{T}$ for convenience) we have that $\chi_{T}\rightarrow\chi$ as $T\rightarrow\infty$, for some continuous function $\chi$. There are two cases to consider, namely, case 1 when $\chi$ vanishes identically (in which case the conclusion of Lemma 2.2 holds), and case 2 when $\chi$ does not vanish identically.
Before considering both cases, we construct an appropriate conformal factor. In the work of Schoen and Yau \[2\] existence of a unique solution to the following boundary value problem was established: $$\label{a.0}
\overline{\Delta}u-\frac{1}{8}\overline{R}u=0\text{
}\text{ }\text{ on }\text{ }\text{ }\overline{M},$$ with $u(x)\rightarrow 0$ as $x\rightarrow\partial M$ and $u(x)\rightarrow 1$ as $|x|\rightarrow\infty$. A slightly modified version of (4.6) in [@SchoenYau] shows that the first eigenvalue, $\eta_{i}$, of the operator $\Delta-\frac{1}{8}K$ on $\partial_{i}M$, is strictly positive (here $K$ denotes Gaussian curvature). As observed by Schoen and Yau, $u\sim e^{\mp\sqrt{\eta_{i}}t}\zeta_{i}(y)$, that is the conformal factor $u$ is asymptotic to $e^{\mp\sqrt{\eta_{i}}t}\zeta_{i}(y)$ depending on whether the Jang surface blows up or down, where $\zeta_{1}$ is the corresponding first eigenfunction.
We will slightly perturb $u$ in order to prescribe appropriate boundary conditions on certain cylindrical ends. For large $T$ and $T_{0}$ ($T>T_{0}$), let $(\overline{M}_{T}-\overline{M}_{T_{0}})_{i}$ denote the component of $\overline{M}_{T}-\overline{M}_{T_{0}}$ associated with the boundary component $\partial_{i}M$. Let $i=1,\ldots,m$ index the boundary components which fall under case 1, and let $i=m+1,\ldots,n$ index the boundary components which fall under case 2. Set $\widehat{M}_{T}=\overline{M}-\bigcup_{i=1}^{m}(\overline{M}-\overline{M}_{T})_{i}$, that is, $\widehat{M}_{T}$ is the Jang surface after the cylindrical ends corresponding to case 1 have been removed. Consider the boundary value problem $$\label{a.1}
\overline{\Delta}u_{T}-\frac{1}{8}\overline{R}u_{T}=0\text{
}\text{ }\text{ on }\text{ }\text{ }\widehat{M}_{T},$$ $$\partial_{\overline{N}}u_{T}+\frac{1}{4}\overline{H}u_{T}=\frac{1}{4}\sqrt{\frac{16\pi}{|\partial_{i}\overline{M}_{T}|_{\widehat{g}_{T}}}}u_{T}^{3}
\text{ }\text{ }\text{ on }\text{ }\text{ }\partial_{i}\overline{M}_{T},\text{ }\text{ }i=1,\ldots,m,$$ $$u_{T}(x)\rightarrow 0\text{ }\text{ }\text{ as }\text{ }\text{ }x\rightarrow\partial_{i} M,\text{ }\text{ }i=m+1,\ldots,n,
\text{ }\text{ }
u_{T}(x)\rightarrow 1\text{ }\text{ }\text{ as }\text{ }\text{ }|x|\rightarrow\infty,$$ where the unit normal $\overline{N}$ points towards spatial infinity and $\widehat{g}_{T}=u_{T}^{4}\overline{g}$. The unique solution to this problem exists by Theorem \[thm2\] (with $E=0$), and it approximates the solution $u$ of for large $T$, as is shown in Theorem \[uncharged\_case\] below. Note that as in [@SchoenYau] a separation of variables argument can be used to show that this solution possesses the same asymptotics as $u$, namely $$\label{a.2}
u_{T}\sim e^{\mp\sqrt{\eta_{i}}t}\zeta_{i}(y)$$ along the ends corresponding to $\partial_{i}M$, $i=m+1,\ldots,n$.
Multiply equation by $u_{T}=1+v_{T}$ and integrate by parts to obtain $$\begin{aligned}
\begin{split}
Q(v_{T}):= &\lim_{r\rightarrow\infty}\frac{1}{2}\int_{|x|=r}u_{T}\partial_{\overline{N}}u_{T}\\
\geq &\int_{\overline{M}_{T}}\frac{1}{4}|\overline{\nabla}v_{T}|^{2}
+\left(\pi(\mu-|J|)+\frac{1}{16}|q|_{\overline{g}}^{2}\right)(1+v_{T})^{2}\\
&+\int_{\partial\overline{M}_{T}}\frac{1}{8}q(\overline{N})(1+v_{T})^{2}+\frac{1}{2}u_{T}\partial_{\overline{N}}u_{T}.
\end{split}\end{aligned}$$ A standard formula yields $$\partial_{\overline{N}}u_{T}=\frac{1}{4}\widehat{H}u_{T}^{3}-\frac{1}{4}\overline{H}u_{T},$$ where $\widehat{H}$ and $\overline{H}$ are the mean curvatures with respect to $\widehat{g}_{T}$ and $\overline{g}$, respectively. It follows that $$\begin{aligned}
\label{a.3}
\begin{split}
Q(v_{T})
\geq&\int_{\overline{M}_{T}}\frac{1}{4}|\overline{\nabla}v_{T}|^{2}
+\left(\pi(\mu-|J|)+\frac{1}{16}|q|_{\overline{g}}^{2}\right)(1+v_{T})^{2}\\
& +\int_{\partial\overline{M}_{T}}\frac{1}{8}(q(\overline{N})-\overline{H})(1+v_{T})^{2}+\frac{1}{8}\widehat{H}(1+v_{T})^{4},
\end{split}\end{aligned}$$ where $\mu$ and $J$ are given by with $E=0$.
*Case 1: $\chi\equiv 0$.*
This case corresponds to the boundary components $\partial_{i}M$, $i=1,\ldots,m$. Here the conclusion of Lemma 2.2 [@Khuri] is valid, and the boundary conditions of (3.1) in [@Khuri] hold. Thus the methods of [@Khuri] (as in the proof of Theorem 3.1) apply to yield $$\begin{aligned}
\label{a.4}
\begin{split}
& \int_{\partial_{i}\overline{M}_{T}}\frac{1}{8}(q(\overline{N})-\overline{H})(1+v_{T})^{2}+\frac{1}{8}\widehat{H}(1+v_{T})^{4}\\
&\geq \left(\frac{1-\vartheta_{T}}{2}\right)\sqrt{\frac{\pi}{|\partial_{i}\overline{M}_{T}|}}\int_{\partial_{i}\overline{M}_{T}}(1+v_{T})^{2},
\end{split}\end{aligned}$$ where the constants $\vartheta_{T}\rightarrow 0$ as $T\rightarrow\infty$. $\Box$
*Case 2: $\chi$ does not vanish identically.*
Here we may follow the proof in Lemma \[lemma1\] (of the present paper) directly to obtain $$\begin{aligned}
\label{a.6}
\begin{split}
& \int_{(\overline{M}_{T}-\overline{M}_{T_{0}})_{i}}\frac{1}{16}|q|_{\overline{g}}^{2}(1+v_{T})^{2}
+\int_{\partial_{i}\overline{M}_{T}}\frac{1}{8}(q(\overline{N})-\overline{H})(1+v_{T})^{2}+\frac{1}{8}\widehat{H}(1+v_{T})^{4}\\
\geq& \left(1-C\mathcal{N}(T-T_{0})^{-1}\right)\sqrt{\frac{\pi}{|\partial_{i}\overline{M}_{T}|}}\int_{\partial_{i}\overline{M}_{T}}(1+v_{T})^{2},
\end{split}\end{aligned}$$ for $T$ sufficiently large. $\Box$
By combining , , and we conclude that $$\label{a.7}
Q(v_{T})\geq \int_{\overline{M}_{T}}\frac{1}{4}|\overline{\nabla}v_{T}|^{2}
+\left(\frac{1-\vartheta_{T}}{2}\right)
\sum_{i=1}^{n}\sqrt{\frac{\pi}{|\partial_{i}\overline{M}_{T}|}}
\int_{\partial_{i}\overline{M}_{T}}(1+v_{T})^{2}.$$
\[uncharged\_case\] Let $(M,g,k)$ be an asymptotically flat initial data set for the Einstein equations satisfying the dominant energy condition. If the boundary consists of an outermost apparent horizon with components $\partial_{i}M$ having area $|\partial_{i}M|$, $i=1,\ldots,n$, then $$E_{ADM}(g)\geq\frac{\sigma}{2(1+\sigma)}\sum_{i=1}^{n}\sqrt{\frac{|\partial_{i}
M|}{\pi}}$$ where $$\sigma=\left(\sum_{i=1}^{n}\sqrt{4\pi|\partial_{i}M|}\right)^{-1}\parallel\overline{\nabla}
u\parallel^{2}_{L^{2}(\overline{M})}.$$
Consider the manifold $(\widehat{M}_{T},\widehat{g}_{T})$. Along the infinite cylindrical ends over $\partial_{i}M$, $i=m+1,\ldots,n$, the conformal factor $u_{T}$ decays exponentially fast. Therefore as in [@SchoenYau] these ends may be closed by adding a point at infinity. The remaining cylindrical ends, indexed by $i=1,\ldots,m$, correspond to the boundary components of $\widehat{M}_{T}$ which satisfy the hypotheses of Herzlich’s version of the positive mass theorem (Theorem 1.4 of [@Khuri]). It follows that $E_{ADM}(\widehat{g}_{T})$ is nonnegative. At this point, we can apply and follow the same procedure as in section $\S 4$ of [@Khuri] to obtain $$E_{ADM}(g)\geq\frac{\sigma_{T}(1-\vartheta_{T})}{2(1+\sigma_{T})}\sum_{i=1}^{n}\sqrt{\frac{|\partial_{i}\overline{M}_{T}|}{\pi}},$$ where $$\sigma_{T}=\frac{\int_{\overline{M}_{T}}|\overline{\nabla}v_{T}|^{2}}
{2(1-\vartheta_{T})\sum_{i=1}^{n}\sqrt{\frac{\pi}{|\partial_{i}\overline{M}_{T}|}}\int_{\partial_{i}\overline{M}_{T}}v_{T}^{2}}.$$ Moreover, the same arguments in the paragraph after (4.4) in [@Khuri] show that $u_{T}\rightarrow u_{\infty}$, where $u_{\infty}$ is a bounded solution of boundary value problem , and $\sigma_{T}\rightarrow\sigma_{\infty}$. Since there is a unique bounded solution of , it follows that $u_{\infty}=u$ and $\sigma_{\infty}=\sigma$, from which we obtain the desired result.
Note that in [@Khuri], in the last step, we replaced $\sigma_{\infty}$ with a different definition of $\sigma$ that employed an infimum. The purpose of this replacement was solely to give our result an expression akin to that in Herzlich’s Penrose-like inequality [@Herzlich]. Alas, this misguided sense of aesthetics resulted in the error mentioned at the beginning of the appendix.
Lastly, we mention that the definition of $\sigma$ in the current theorem is strictly positive, and $\sigma$ is dimensionless making it independent of the area of $\partial M$.
[99]{}
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[^1]: The author was partially supported by NSF Grants DMS-1007156 and DMS-1308753.
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---
author:
- |
Madhu Raka\
\
\
title: '**Good integers : A note on results of Jitman and Prugsapitak**'
---
Introduction.
==============
For fixed coprime nonzero integers $a$ and $b$, a positive integer $\ell$ is called a good integer with respect to $a$ and $b$ (see Moree [@Mor]), if there exists a positive integer $k$ such that $\ell|(a^k + b^k)$. Otherwise, $\ell$ is called a bad integer. Denote by $G_{(a,b)}$ the set of good integers defined with respect to $a$ and $b$. A positive integer $\ell$ is said to be oddly-good (with respect to $a$ and $b$) if $\ell|(a^k + b^k)$ for some odd integer $k \geq 1$, and evenly-good if $\ell|(a^k + b^k)$ for some even integer $k \geq 2$. Therefore, $\ell$ is good if it is oddly-good or evenly-good. Denote by $OG_{(a,b)}$ (resp., $EG_{(a,b)}$) the set of oddly-good (resp., evenly-good) integers. For a non-negative integer $\beta$, a positive integer $d$ is said to be $2^{\beta}$-good (with respect to $a$ and $b$) if $2^{\beta}d \in G_{(a,b)}$. Otherwise, $d$ is said to be $2^{\beta}$-bad. In the same fashion, $2^{\beta}$-oddly-good and $2^{\beta}$-evenly-good integers are defined. For an integer $\beta\geq 0$, denote by $G_{(a,b)}(\beta), OG_{(a,b)}(\beta)$ and $EG_{(a,b)}(\beta)$ the sets of $2^{\beta}$-good, $2^{\beta}$-oddly-good, and $2^{\beta}$-evenly-good integers, respectively.
In [@Jit], Jitman characterized good integers, oddly-good integers and considered their applications in coding theory. But there are some errors in this paper. In the proof of Proposition 2.1 of [@Jit], it is used that for odd integers $a$ and $b$, $$Ord_{2^{\beta}}(ab^{-1}) =2 ~~~ \Rightarrow ~~~ ab^{-1} \equiv -1 ( {\rm mod}~ 2^{\beta}) ~~~~ {\rm i. e.} ~~~2^{\beta}\mid a+b.$$ Again in the proof of Proposition 2.3 of [@Jit], it is used that $$Ord_{d}(ab^{-1}) =2k ~~~ \Rightarrow ~~~ (ab^{-1})^k \equiv -1 ( {\rm mod}~ d),$$ where $a$ and $b$ are coprime to $\ell=2^{\beta}d$, $\beta \geq 1$, $d$ is an odd positive integer and $b^{-1}$ denotes the multiplicative inverse of $b$ modulo $\ell$.
These are false statements as $$Ord_8(11)=2, ~~ 11^2 \equiv 1 ( {\rm mod}~ 8) ~~ {\rm but }~~ 11 \not\equiv -1( {\rm mod}~8).$$ $$Ord_{15}(11)=2, ~~ 11^2 \equiv 1 ( {\rm mod}~ 15) ~~ {\rm but } ~~ 11 \not\equiv -1( {\rm mod}~15).$$
Because of these errors, Proposition 2.1, Proposition 2.3, Theorem 2.1, Corollary 2.1, Theorem 3.1 and Corollary 3.3 of [@Jit] are no longer true. Proofs of some of otherwise correct results also need to be modified.
In a subsequent paper [@PJit], the authors Prugsapitak and Jitman tried to fix the second error (though not mentioning the error explicitly), but they still overlooked the first error. Because of this, Proposition 2.1, Proposition 2.2, Proposition 2.3, Corollary 2.1 and Corollary 2.2 of [@PJit] are again no longer true. As a consequence, applications of good integers in the study of self-dual negacyclic codes are also affected. In fact the statement that $$Ord_{d}(ab^{-1}) =2k ~~~ \Rightarrow ~~~ (ab^{-1})^k \equiv -1 ( {\rm mod}~ d)$$ holds only when $d$ is an odd prime power or $d=2$. The aim of this paper is to fix these errors and to rectify the above mentioned propositions and results.
Rectified results.
===================
Through out the paper let $a$, $b$ and $\ell=2^{\beta}d $, where $\beta \ge 0$ and $d$ an odd positive integer, be pairwise coprime non-zero integers. Let $b^{-1}$ denote the multiplicative inverse of $b$ modulo $\ell$ and $Ord_m(ab^{-1})$ denote the multiplicative order of $ab^{-1}$ modulo $m$ for a divisor $m$ of $\ell$. It is clear that $\ell \in G_{(a,b)}$ or $ d \in G_{(a,b)}(\beta)$ if and only if $(ab^{-1})^k \equiv -1 ( {\rm mod}~ \ell)$ for some positive integer $k$. Let $ x= ab^{-1}$. Denote by $2^{\gamma}||\ell$ if $\gamma $ is the largest integer such that $2^{\gamma}|\ell$. Note that if gcd$(a,b)=1$ and $\ell \in G_{(a,b)}$ then gcd$(a,\ell)=1$ and gcd$(b,\ell)=1$.
[**Lemma 1**]{}: Let $p$ be an odd prime and $r$ be a positive integer. Let $p^r$ be good and $s$ be the smallest positive integer such that $(ab^{-1})^s \equiv -1 ( {\rm mod}~ p^r)$. Then $Ord_{p^r}(ab^{-1})=2s$.
This is Proposition 2 of [@Mor].
The converse of Lemma 1 is also true. Let $ x= ab^{-1}$. If $Ord_{p^r}(x)=2s$, we have $p^r | (x^s-1)(x^s+1)$. It can not happen that $p^{i} | (x^s-1)$ and $p^j | (x^s+1)$ with $i+j=r, ~i \geq 1, j\ge 1$. Because then $p| (x^s-1)$ and $p | (x^s+1)$ which gives $p | 2$; not possible as $p$ is an odd prime. Hence either $p^r | (x^s-1)$ or $p^r | (x^s+1)$ but not both. If $p^r | (x^s-1)$, we get $Ord_{p^r}(x) \geq s$, not possible. Therefore $p^r$ must divide $(x^s+1)$.
In fact, we have a more general result.
[**Lemma 2**]{}: Let $a$, $b$ and $d$ be pairwise coprime odd integers. If $k$ is the smallest positive integer such that $(ab^{-1})^k \equiv -1 ( {\rm mod}~ d)$ then $Ord_{d}(ab^{-1})=2k.$
[**Proof**]{}: Let $k= 2^{\lambda} k'$, $\lambda \ge 0, ~k'$ odd. Let $x= ab^{-1}$. As $x^k \equiv -1 ( {\rm mod}~ d)$ we have $x^{2k} \equiv 1 ( {\rm mod}~ d)$. Therefore $Ord_{d}(x)\mid 2k.$ Let $Ord_{d}(x)= r = 2^{\mu} r'$, where $0\leq \mu \le \lambda+1, ~r'$ is odd and $ r'| k'$. Let $k'=r'r''$.
If $\mu \leq \lambda $, $x^{2^{\mu}r'} \equiv 1 ( {\rm mod}~ d)$, gives $x^{2^{\lambda}k'} \equiv (x^{2^{\mu}r'})^{2^{\lambda-\mu}r''} \equiv 1 ( {\rm mod}~ d)$, but $x^{2^{\lambda}k'} \equiv x^k \equiv -1 ( {\rm mod}~ d)$. Therefore $1\equiv -1 ( {\rm mod}~ d)$. This is not possible as $d$ is odd. Therefore we must have $\mu =\lambda + 1$.
If $d >1$, let $d = p_1^{e_1}p_2^{e_2}\cdots p_t^{e_t}$ where $p_i$ are odd primes and $e_i \geq 1$. As $x^r = x^{2^{\lambda +1}r'} \equiv 1 ( {\rm mod}~ p_i^{e_i})$ we have $p_i^{e_i} | (x^{2^{\lambda}r'}-1)(x^{2^{\lambda}r'}+1)$ for each $i$. As before, it can not happen that $p_i^{\alpha_i}| (x^{2^{\lambda}r'}-1)$ and $p_i^{\beta_i} | (x^{2^{\lambda}r'}+1)$ for some $\alpha_i \geq 1, \beta_i \ge 1$ with $\alpha_i+\beta_i=e_i$. Hence for each $i,~ 1 \le i \le t$, either $p_i^{e_i} |(x^{2^{\lambda}r'}-1)$ or $p_i^{e_i} | (x^{2^{\lambda}r'}+1)$ but not both. If for some $i$, $p_i^{e_i} | (x^{2^{\lambda}r'}-1)$, we get $x^k \equiv x^{2^{\lambda}k'} \equiv x^{2^{\lambda}r'r''} \equiv 1 ( {\rm mod}~ p_i^{e_i})$. Not possible as we are given that $ x^k \equiv -1 ( {\rm mod}~ p_i^{e_i})$ and $p_i$ is odd.
Hence $p_i^{e_i} \mid (x^{2^{\lambda}r'}+1)$ for all $i$. Therefore $d \mid (x^{2^{\lambda}r'}+1)$ i.e. $x^{r/2} \equiv -1 ( {\rm mod}~ d)$. Now the minimality of $k$ gives $r/2=k$. $\Box$\
The converse of Lemma 2 is not always true as illustrated in Section 1. Note that $$Ord_{2^{\beta}}(x)=\left \{ \begin{array}{ll} 1 & {\rm if} ~ \beta=1\\ 2 & {\rm if} ~ \beta \geq 2 ~{\rm and }~ x\equiv -1 ( {\rm mod}~ 2^{\beta}). \end{array}\right.$$ If $ \ell= 2^{\beta}p_1^{e_1}p_2^{e_2}\cdots p_t^{e_t}$ where $p_i$ are odd primes and $\beta \geq 0,~ e_i \geq 0$, we have $$Ord_{\ell}(x)= {\rm lcm}\big( Ord_ {2^{\beta}}(x), Ord_ {p_1^{e_1}}(x), Ord_ {p_2^{e_2}}(x), \cdots, Ord_ {p_t^{e_t}}(x) \big).$$
Following are some results of Moree [@Mor].
[**Lemma 3**]{} ([@Mor], Proposition 2): For an odd prime $p$, $Ord_{p^{e}}(x)= Ord_p(x)p^{\alpha}$ for some $\alpha\geq 0$.
[**Lemma 4**]{} ([@Mor], Theorem 1): Let $d>1$ be an odd integer. Then $ d \in G_{(a,b)}$ if and only if there exists an integer $s\geq 1$ such that $2^s|| Ord_p(x)$ for every prime $p$ dividing $d$.
[**Lemma 5**]{} ([@Jit], Proposition 2.2): Let $a,b,d>1$ be pairwise coprime odd integers. Then $ d \in G_{(a,b)}$ if and only if $ 2d \in G_{(a,b)}$.
The correct form of Proposition 2.1 of [@Jit] and Proposition 2.2 of [@PJit] is
[**Proposition 1**]{}: If $a$, $b$ are coprime odd integers and $\beta\geq 1$ is any integer, then the following are equivalent :
$ \begin{array}{ll} (1)& 2^{\beta}|a+b\\(2)&2^{\beta}\in G_{(a,b)}\\ (3)&\beta=1 ~{\rm or~} ab^{-1} \equiv -1 ( {\rm mod}~ 2^{\beta}). \end{array}$
The correct form of Proposition 2.3 of [@Jit] and Proposition 2.1 and Corollary 2.1 of [@PJit] is
[**Proposition 2**]{}: Let $a,b,d>1$ be pairwise coprime odd integers and $\beta \geq 2$ be any integer. Then $2^{\beta}d \in G_{(a,b)}$ if and only if $ab^{-1} \equiv -1 ( {\rm mod}~ 2^{\beta})$ and $2|| Ord_p(ab^{-1})$ for every prime $p$ dividing $d$.
[**Proof**]{}: Suppose $2^{\beta}d \in G_{(a,b)}$. Let $k$ be the smallest positive integer such that $2^{\beta}d|(a^k + b^k)$. This gives $2^{\beta}|(a^k + b^k)$. If $k$ is even $$a^k+b^k = (a^2)^{k/2}+(b^2)^{k/2}\equiv 1+1 \equiv 2 ( {\rm mod}~ 4),$$ as an odd square is always congruent $1$ modulo $4$. Therefore $k$ must be odd. But then $$a^k+b^k =(a+b)\big(a^{k-1}-a^{k-2}b+a^{k-3}b^2- \cdots + b^{k-1}\big).$$ The second factor on the right hand side is odd, it being a sum of odd terms taken odd number of times. Therefore $2^{\beta}|(a + b)$ which gives $ab^{-1} \equiv -1 ( {\rm mod}~ 2^{\beta})$. Also $k$ is smallest integer such that $d|(a^k + b^k)$, i. e., $x^k \equiv -1 ( {\rm mod}~ d)$. Then we have, by Lemma 2, $Ord_{d}(x)=2k,$ where $k$ is odd. Let $d = p_1^{e_1}p_2^{e_2}\cdots p_t^{e_t}$ where $p_i$ are odd primes and $e_i \geq 1$. Then, using Lemma 3, $$2k=Ord_d(x)= {\rm lcm}\big( Ord_{p_1}(x)p_1^{\alpha_1}, Ord_{p_2}(x) p_2^{\alpha_2}, \cdots, Ord_{p_t}(x)p_t^{\alpha_t} \big).$$ Also $x^k \equiv -1 ( {\rm mod}~ p_i)$ for all $i, 1\leq i \leq t$ and $k$ is odd. Therefore $Ord_{p_i}(x)$ is even and $2|| Ord_{p_i}(x)$ for each $i$.
Conversely let $2|| Ord_{p_i}(x)$ for each $p_i| d$. This gives $2|| Ord_{p_i^{e_i}}(x)$ for each $i$. Let $ Ord_{p_i^{e_i}}(x)=2r_i$, where $r_i$ is odd. Therefore $x^{r_i} \equiv -1 ( {\rm mod}~ p_i^{e_i})$ for all $i, 1\leq i \leq t$. Let $k= {\rm lcm}(r_1,r_2.\cdots,r_t)$, $k$ is odd and let $k=r_ir'_i$. Each of $r'_i$ is also odd. Then $x^k \equiv x^{r_i r'_i} \equiv (-1)^{r'_i} \equiv -1( {\rm mod}~ p_i^{e_i})$ for each $i$. Therefore $x^k \equiv -1 ( {\rm mod}~ d)$. Now $x \equiv -1 ( {\rm mod}~ 2^{\beta})$ implies $x^k \equiv -1 ( {\rm mod}~ 2^{\beta})$ as $k$ is odd. Hence $x^k \equiv -1 ( {\rm mod}~ 2^{\beta}d)$, i.e., $2^{\beta}d \in G_{(a,b)}$.
In view of the above results, Theorem 2.1, Corollary 2.1, Theorem 3.1 and Corollary 3.3 of [@Jit] should read as follows :
[**Theorem 1**]{}: Let $a$ and $b$ be pairwise coprime non-zero integers and let $\ell=2^{\beta}d$ be a positive integer such that $d$ is odd and $\beta \geq 0$.
1. If $ab$ is odd, then $\ell = 2^{\beta}d \in G_{(a,b)}$ if and only if one of the following statements hold\
$\begin{array}{ll} {\rm (a)}
& \beta \in \{0,1\} {\rm ~and ~} d=1\vspace{2mm}\\ {\rm (b)} & \beta \in \{0,1\},~ d \geq 3 {\rm ~and ~there~ exists} ~ s\geq 1 {\rm ~ such~ that~} 2^s|| Ord_p(ab^{-1}) {\rm ~ for} \\ & {\rm ~every~ prime~} p {\rm ~ dividing~} d.\vspace{2mm}\\{\rm(c)} & \beta \geq 2, d=1 {\rm ~ and~} ab^{-1} \equiv -1 ( {\rm mod}~ 2^{\beta}).\vspace{2mm}\\ {\rm(d)}& \beta \geq 2,~ d\geq 3,~ ab^{-1} \equiv -1 ( {\rm mod}~ 2^{\beta}) {\rm ~ and~} 2|| Ord_p(ab^{-1}) {\rm ~ for ~every~ prime} \\ & p {\rm ~ dividing~} d.\end{array}$
2. If $ab$ is even, then $\ell = 2^{\beta}d \in G_{(a,b)}$ if and only if one of the following statements hold\
$ \begin{array}{ll} {\rm (a)}& \beta =0 {\rm ~ and ~} d=1.\vspace{2mm}\\{\rm (b)}& \beta =0, ~ d \geq 3 {\rm ~ and ~there~ exists~} s\geq 1 {\rm ~ such ~ that~} 2^s|| Ord_p(ab^{-1}) {\rm ~ for ~every} \\ & {\rm ~prime~} p {\rm ~ dividing~} d. \end{array} $
[**Corollary 1**]{}: Let $a,~b$ and $\ell$ be pairwise coprime non-zero integers and let $\ell=2^{\beta}d$ be a positive integer such that $d$ is odd and $\beta \geq 0$. Let $\gamma \geq 0 $ be an integer such that $2^{\gamma}||a+b$. Then $\ell \in G_{(a,b)}$ if and only if one of the following statements hold.
1. $\ell =1,2$.
2. $d=1$ and $2\le \beta \leq \gamma$.
3. $d\geq 3$, $\beta \in \{0,1\}$ and $2^s|| Ord_p(ab^{-1})$ for some $s \geq 1$ and for every prime $p$ dividing $d$.
4. $d\geq 3$, $2\leq \beta \leq \gamma$ and $2|| Ord_p(ab^{-1})$ for every prime $p$ dividing $d$.
In 3 and 4, if $\ell \in G_{(a,b)}$ then $2^s|| Ord_{\ell}(ab^{-1})$ if and only if $2^s|| Ord_p(ab^{-1})$ for every prime $p$ dividing $d$.
[**Theorem 2**]{}: Let $a$ and $b$ be pairwise coprime non-zero integers and let $\ell=2^{\beta}d$ be a positive integer such that $d$ is odd and $\beta \geq 0$.
1. If $ab$ is odd, then $\ell = 2^{\beta}d \in OG_{(a,b)}$ if and only if one of the following statements hold\
$\begin{array}{ll} {\rm (a)}
& \beta \in \{0,1\} {\rm ~and ~} d=1\vspace{2mm}\\ {\rm (b)} & \beta \in \{0,1\},~ d \geq 3 {\rm ~and ~} 2|| Ord_p(ab^{-1}) {\rm ~ for} {\rm ~every~ prime~} p {\rm ~ dividing~} d.\vspace{2mm}\\{\rm(c)} & \beta \geq 2, d=1 {\rm ~ and~} ab^{-1} \equiv -1 ( {\rm mod}~ 2^{\beta}).\vspace{2mm}\\ {\rm(d)}& \beta \geq 2,~ d\geq 3,~ ab^{-1} \equiv -1 ( {\rm mod}~ 2^{\beta}) {\rm ~ and~} 2|| Ord_d(ab^{-1}) \end{array}$
2. If $ab$ is even, then $\ell = 2^{\beta}d \in OG_{(a,b)}$ if and only if one of the following statements hold\
$ \begin{array}{ll} {\rm (a)}& \beta =0 {\rm ~ and ~} d=1.\vspace{2mm}\\{\rm (b)}& \beta =0, ~ d \geq 3 {\rm ~ and ~} 2|| Ord_p(ab^{-1}) {\rm ~ for ~every} {\rm ~prime~} p {\rm ~ dividing~} d. \end{array} $
[**Corollary 2**]{}: Let $a,~b$ and $\ell$ be pairwise coprime non-zero integers and let $\ell=2^{\beta}d$ be a positive integer such that $d$ is odd and $\beta \geq 0$. Let $\gamma \geq 0 $ be an integer such that $2^{\gamma}||a+b$. Then $\ell \in OG_{(a,b)}$ if and only if one of the following statements hold.
1. $\ell =1,2$.
2. $d=1$ and $2\le \beta \leq \gamma$.
3. $d\geq 3$, $0\leq \beta \le \gamma$ and $2|| Ord_p(ab^{-1})$ for every prime $p$ dividing $d$. In that case $2|| Ord_{\ell}(ab^{-1})$.
[99]{}
S. Jitman, *Good integers and some applications in coding theory,* Cryptogr. Commun. DOI: 10.1007/s12095-017-0255-4 (2017), Volume 10, Issue 4, pp 685-704 (2018). P. Moree, *On the divisors of $a^k+b^k$,* Acta Arithmetica LXXX, 197-212 (1997). S. Prugsapitak and S. Jitman, Some generalizations of good integers and their applications in the study of self-dual negacyclic codes, arXiv:1801.04614v1 \[cs.IT\] 14 Jan 2018.
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[**Gap between the largest and smallest parts of partitions and Berkovich and Uncu’s conjectures** ]{}
[Wenston J.T. Zang]{}$^{1}$ and [Jiang Zeng]{}$^{2}$ $^{1}$Institute for Advanced Study in Mathematics\
Harbin Institute of Technology, Heilongjiang 150001, P.R. China\
$^{2}$Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille\
Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France\
$^1$zang@hit.edu.cn, $^2$zeng@math.univ-lyon1.fr
[**Abstract.**]{} We prove three main conjectures of Berkovich and Uncu (Ann. Comb. 23 (2019) 263–284) on the inequalities between the numbers of partitions of $n$ with bounded gap between largest and smallest parts for sufficiently large $n$. Actually our theorems are stronger than their original conjectures. The analytic version of our results shows that the coefficients of some partition $q$-series are eventually positive.
[**Keywords**]{}: Partition inequalities, Frobeinus coin problem, Non-negative $q$-series expansions, Injective maps.
[**AMS Classifications**]{}: 05A17, 05A20, 11P81.
Introduction
============
Let $n$ be a positive integer, a *partition* of $n$ is a nonincreasing finite sequence of positive integers $\lambda_1, \lambda_2, \ldots , \lambda_k$ whose sum is $n$. Each $\lambda_i$ is called a *part* of the partition. In [@Ber-Unc-19], Berkovich and Uncu proved various inequalities between the numbers of partitions with the bound on the largest part and some restrictions on occurrences of parts, and also make several conjectures. To be specific, we introduce the following definitions:
1. Let ${\mathcal C}_{L,s,2}(n)$ (resp. $c_{L,s,2}(n)$) be the set (resp. number) of partitions of $n$ with parts in the domain $\{s+1, \ldots, s+L\}$.
2. Let ${\mathcal F}_{L,s,k}(n)$ (resp. $f_{L,s,k}(n)$) denote the set (resp. number) of partitions of $n$ with the smallest part $s$, the largest part at most $L+s$, and no part equal to $k$.
In this paper, motivated by the open problems Conjecture 3.2, Conjecture 3.3 and Conjecture 7.1 in [@Ber-Unc-19], we shall prove the following two main theorems.
\[thm-main\] For integer $s\geq 1$, $L\geq 3$ and $s+L\geq k\geq \max\{s+1,L\}$, there exists an integer $M$ which only depends on $s$ such that for $n\ge M$, $$f_{L,s,k}(n)\geq c_{L,s,2}(n).$$
Berkovich and Uncu [@Ber-Unc-19 Theorems 1.1 and 3.1] proved Theorem \[thm-main\] for $s=1$ (resp. $s=2$), $k=L$ (resp. $k=L+1$) with $M=1$ (resp. $M=10$). They also conjectured the cases $k=s+L-1$ and $k=L$ of Theorem \[thm-main\] [@Ber-Unc-19 Conjectures 3.2 and 3.3].
By the elementary theory of partitions [@And-1998 Chapters 1–3] it is not difficult to see that the generating functions of $c_{L,s,2}(n)$’s and $f_{L,s,1}(n)$’s read as follows: $$\begin{aligned}
\sum_{n=0}^\infty c_{L,s,2}(n) q^n&=\frac{1}{(q^{s+1};q)_{L}},\label{gf1}\\
\sum_{n=1}^\infty f_{L,s,k}(n)q^n&=\frac{q^s(1-q^k)}{(q^s;q)_{L+1}}.\label{gf2}\end{aligned}$$ Here, we use the standard $q$-notation [@And-1998]: $$\begin{aligned}
(a;q)_n=(1-a)(1-aq)\ldots (1-aq^{n-1}).
$$
Recall that a series $\sum_{n\geq 0}a_nq^n\in {\mathbb R}[[q]]$ is called [*eventually positive*]{} if there exists an integer $M\geq 0$ such that $a_n>0$ for all $n>M$. For instance, Theorem \[thm-main\] and and imply that the $q$-series $$\frac{q^s(1-q^k)-(1-q^s)}{(q^s;q)_{L+1}}$$ is eventually positive. In general, we derive the following theorem.
\[thm-main0-2\] For integers $L\ge 3$, $s\geq 1$, $r\geq 0$ and $k_1>k_2\geq 1$, the series $$\begin{aligned}
\label{def:H}
H^*_{L,s,r,k_1,k_2}(q): =\frac{q^r(1-q^{k_1})-(1-q^{k_2})}{(q^s;q)_{L+1}}\end{aligned}$$ is eventually positive.
Set $r=s$, $k_1=k$ and $k_2=s$, we confirm the conjecture raised by Berkovich and Uncu [@Ber-Unc-19 Conjecture 7.1].
Let ${\mathcal A}=\{a_1,a_2,\ldots,a_m\}$ be a set of $m$ positive integers. Denote by $p_{\mathcal A}(n)$ the number of nonnegative integer solutions of the diophantine equation $a_1 x_1+\cdots +a_mx_m=n$, i.e., $$\sum_{n=0}^\infty p_{\mathcal A}(n)q^n=\frac{1}{(1-q^{a_1})\ldots (1-q^{a_m})}.$$ It should be noted that $p_{\mathcal A}(n)$ is closely related to the *Frobeinus coin problem*, see [@Alf-2000; @Sel-1977] or https://en.wikipedia.org/wiki/Coin\_problem for more details. We shall need the following result, see [@Be-Ge-Ko-01] or [@Wilf-1994 Theorem 3.15.2] for an elementary proof.
\[schur-thm\] If $\gcd(a_1,\ldots,a_m)=1$, then $$p_{\mathcal A}(n)\sim \frac{n^{m-1}}{(m-1)!a_1a_2\cdots a_m}.$$
This paper is organized as follows. In Section 2, we first give two weak forms of Theorem \[thm-main\], we then prove Theorem \[thm-main\] with the aid of these two weak forms. In Section 3, we give a proof of Theorem \[thm-main0-2\].
Proof of Theorem \[thm-main\]
=============================
In this section, we give a proof of Theorem \[thm-main\]. To this end, we first show the following two theorems, namely Theorems \[thm-asy\] and \[thm-main-inf\], which can be view as weak forms of Theorem\[thm-main\]. We then prove Theorem \[thm-main\] with the aid of Theorems \[thm-asy\] and \[thm-main-inf\].
\[thm-asy\] Given integer $s$ and $L\geq 3$, there exists $M_{L,s}$ depending on $L$ and $s$ such that for any $\max\{s+1, L\}\leq k\leq s+L$ and $n\geq M_{L,s}$, $$\label{equ-flskn-geq-cls3}
f_{L,s,k}(n)\geq c_{L,s,2}(n).$$
By definition, we see that $f_{L,s,k}(n)$ is the number of nonnegative integer solutions of the equation $sx_s+(s+1)x_{s+1}+\cdots+(s+L)x_{s+L}=n$, where $x_s\geq 1$ and $x_{k}=0$. Let $A:=\{s,s+1,\ldots,s+L\}$, from the definition of $p_A(n)$, we deduce that $$f_{L,s,k}(n)=p_{A\setminus \{k\}}(n-s).$$ Similarly, by the definition of $c_{L,s,2}(n)$, $$c_{L,s,2}(n)=p_{A\setminus\{s\}}(n).$$ As $L\geq 3$ both $A\setminus \{k\}$ and $A\setminus \{s\}$ contain two consecutive integers, thus $\gcd(A\setminus \{k\})=\gcd(A\setminus \{s\})=1$. Hence by Theorem \[schur-thm\], $$f_{L,s,k}(n)\sim \frac{k(n-s)^{L}}{L!s(s+1)\cdots (s+L)}$$ and $$c_{L,s,2}(n)\sim \frac{s(n-s)^{L}}{L!s(s+1)\cdots (s+L)}.$$ Therefore, $$\label{equ-sim-flsk}
f_{L,s,k}(n)-c_{L,s,2}(n)\sim \frac{(k-s)(n-s)^{L}}{L!s(s+1)\cdots (s+L)}.$$ From , we see that there exists $M_{L,s,k}$ such that for $n\geq M_{L,s,k}$, holds. When $L\ge s+1$, set $$M_{L,s}:=\max\{M_{L,s,L},M_{L,s,L+1},\ldots M_{L,s,s+L}\};$$ and when $L\le s$, set $$M_{L,s}:=\max\{M_{L,s,s+1},M_{L,s,s+2},\ldots M_{L,s,s+L}\}.$$ Clearly, for $n\ge M_{L,s}$, valid and $M_{L,s}$ only depends on $s$ and $L$. This completes the proof.
We next give another weak form of Theorem \[thm-main\].
\[thm-main-inf\] Let $L$, $s$ and $k$ be positive integers such that $L\geq 2s^3+5s^2+1$ and $L\leq k\leq s+L$. Then, for any $n\ge 2s^5+8s^4+s^3-14s^2+3s+1$, we have $$f_{L,s,k}(n)\geq c_{L,s,2}(n).$$
To prove Theorem \[thm-main-inf\], we shall build an injection $\phi: C_{L,s,2}(n)\to F_{L,s,k}(n)$. More specifically, we shall divide $\phi$ into five injections $\phi_i: C^i_{L,s,2}(n)\to F^i_{L,s,k}(n)$ for $1\leq i\leq 5$, where $\{C^1_{L,s,2}(n), \ldots, C^5_{L,s,2}(n)\}$ is a set partition of $ C_{L,s,2}(n)$, and $(F^1_{L,s,k}(n), \ldots, F^5_{L,s,k}(n))$ is a sequence of five disjoint subsets of $F_{L,s,k}(n)$. We denote each partition $\alpha\in C_{L,s,2}(n)$ by $\alpha=((s+1)^{f_{s+1}}\ldots (s+L)^{f_{s+L}})$, where $f_i$ is the number of occurrences of $i$ in $\alpha$. The five subsets $C^i_{L,s,2}(n)$ are defined as follows.
- $C^1_{L,s,2}(n)$ is the set of partitions in $C_{L,s,2}(n)$ such that $f_{k}=0$ and there exists $a\geq 2$ such that $f_{as}\geq 1$.
- $C^2_{L,s,2}(n)$ is the set of partitions in $C_{L,s,2}(n)$ such that $f_{k}=0$ and $f_{as}=0$ for all $a\geq 2$. Moreover, there exists an integer $j$ such that $2s^2+5s-1\ge j\ge s+1$ and $f_j\geq s$.
- $C^3_{L,s,2}(n)$ is the set of partitions in $C_{L,s,2}(n)$ such that $f_{k}=0$ and $f_{as}=0$ for all $a\geq 2$. Moreover, for any $j$ such that $2s^2+5s-1\ge j\ge s+1$ we have $f_j\leq s-1$.
- $C^4_{L,s,2}(n)$ is the set of partitions in $C_{L,s,2}(n)$ such that $f_{k}\geq 2$.
- $C^5_{L,s,2}(n)$ is the set of partitions in $C_{L,s,2}(n)$ such that $f_{k}=1$.
Similarly, we denote each partition $\beta\in F_{L,s,k}(n)$ by $\beta=(s^{g_s}(s+1)^{g_{s+1}}\ldots (s+L)^{g_{s+L}})$, where $g_i$ is the number of occurrences of $i$ in $\beta$. From the definition of $F_{L,s,k}(n)$, we see that $g_s\geq 1$ and $g_k=0$. Writing $k=rs+t$ with $0\leq t\leq s-1$, we define the five subsets $F^i_{L,s,2}(n)$ as follows.
- $F^1_{L,s,k}(n)$ is the set of partitions in $F_{L,s,k}(n)$ such that $r+1\geq g_s\geq 2$ and for any $2\leq i< g_s$, $g_{is}=0$.
- $F^2_{L,s,k}(n)$ is the set of partitions in $F_{L,s,k}(n)$ such that $g_{s}=1$ and there exists $i\geq 2$ such that $g_{is}=1$. Moreover, for any $j\neq 1,i$, we have $g_{js}=0$.
- $F^3_{L,s,k}(n)$ is the set of partitions in $F_{L,s,k}(n)$ such that $g_{s}=1$ and $g_{2s}+g_{3s}\geq 2$.
- $F^4_{L,s,k}(n)$ is the set of partitions in $F_{L,s,k}(n)$ such that $g_{s}\geq 2r-4$.
- $F^5_{L,s,k}(n)$ is the set of partitions in $F_{L,s,k}(n)$ such that $g_{s}=r-4$ and $g_{2s}\geq 1$.
Since $k\geq L\geq 2s^3+5s^2+1$, we derive that $r\geq 2s^2+5s\geq 7$. Therefore, $2r-4>r+1$, which implies that $F^1_{L,s,k}(n)\cap F^4_{L,s,k}(n)=\emptyset$; also $r-4\geq 3$, which implies that $F^1_{L,s,k}(n)\cap F^5_{L,s,k}(n)=\emptyset$, $F^2_{L,s,k}(n)\cap F^5_{L,s,k}(n)=\emptyset$ and $F^3_{L,s,k}(n)\cap F^5_{L,s,k}(n)=\emptyset$.
Now we proceed to construct the five injections explicitly.
\[lem-inj-1\] There is an explicit injection $\phi_1:C^1_{L,s,2}(n)\to F^1_{L,s,k}(n)$.
Let $\alpha=((s+1)^{f_{s+1}}\ldots (s+L)^{f_{s+L}})\in C^1_{L,s,2}(n)$ with $f_k=0$. Let $a\geq 2$ be the smallest integer such that $f_{as}\geq 1$. We define $$\label{equ-phi1}
\phi_1(\alpha):=(s^{g_s}\ldots (s+L)^{g_{s+L}})=(s^{a}\ldots (as)^{f_{as}-1}\ldots (s+L)^{f_{s+L}}).$$ Clearly, $|\phi_1(\alpha)|=|\alpha|=n$, $g_k=f_k=0$ and $g_s=a\geq 2$. Moreover from $L\leq k\leq s+L$ and $k=rs+t$, we deduce that $as\leq s+L\leq s+k=(r+1)s+t$. Thus $a\leq r+1$. Hence $r+1\geq a=g_s\geq 2$. From the choice of $a$, we see that for any $2\leq i< a$, we have $g_{is}=f_{is}=0$. From the above analysis, we derive that $\phi_1(\alpha)\in F^1_{L,s,k}(n)$.
It remains to show that $\phi_1$ is an injection. Let $$I^1_{L,s,k}(n)=\{\phi_1(\alpha)\colon \alpha\in C^1_{L,s,2}(n)\}$$ be the image set of $\phi_1$, which has been shown to be a subset of $F^1_{L,s,k}(n)$. We wish to construct a map $\psi_1\colon I^1_{L,s,k}(n)\rightarrow C^1_{L,s,2}(n)$ such that for any $\alpha\in C^1_{L,s,2}(n)$, $$\psi_1(\phi_1(\alpha))=\alpha.$$ Let $\beta=(s^{g_s}\ldots (s+L)^{g_{s+L}})\in I^1_{L,s,k}(n)$, that is, there exists $\alpha\in C^1_{L,s,2}(n)$ such that $\phi_1(\alpha)=\beta$. From the construction , we see that $sg_s\leq s+L$ and $sg_s\neq k$. Define $$\psi_1(\beta)=((s+1)^{g_{s+1}}\ldots (sg_s)^{g_{sg_s}+1}\ldots (s+L)^{g_{s+L}}).$$ It is easy to check that $\psi_1(\beta)\in C^1_{L,s,2}(n)$ and $\psi_1(\phi_1(\alpha))=\alpha$. This completes the proof.
For $s=3$, $L=110$, $k=112$ and $n=1205$, let $$\alpha:=(9^7,15^3,16^2,20^9,30^8,40^2,80,97^5).$$ It is trivial to check that $\alpha\in C^1_{110,3,2}(1205)$. Applying $\phi_1$ to $\alpha$, we see that $a=3$. Hence $$\phi_1(\alpha)=(3^3,9^6,15^3,16^2,20^9,30^8,40^2,80,97^5),$$ which is in $F^1_{110,3,112}(1205)$. Moreover, applying $\psi_1$ to $\phi_1(\alpha)$, we recover $\alpha$.
\[lem-inj-2\] There is an explicit injection $\phi_2: C^2_{L,s,2}(n)\to F^2_{L,s,k}(n)$.
For $\alpha=((s+1)^{f_{s+1}}\ldots (s+L)^{f_{s+L}})\in C^2_{L,s,2}(n)$, by definition we see $f_k=0$, for any $a\geq 2$ we have $f_{as}=0$. Moreover, there exists $s+1\leq j\leq 2s^2+5s-1$ such that $f_j\geq s$. We choose such $j$ to be minimum, that is, $j=\min\{i\colon f_{i}\geq s\}$. Define $$\label{equ-phi2}
\phi_2(\alpha)=(s^{g_s}\ldots (s+L)^{g_{s+L}})=(s^{1}\ldots (j)^{f_{j}-s}\ldots ((j-1)s)^1\ldots (s+L)^{f_{s+L}}).$$
From $k\geq L\geq 2s^3+5s^2+1$ and $j\leq 2s^2+5s-1$, we deduce that $k>s(2s^2+5s)>(j-1)s$. Thus $f_k=g_k=0$. Moreover, it is clear to see that $g_s=g_{(j-1)s}=1$ and for any $i\neq 1, j-1$, $g_{is}=f_{is}=0$. Furthermore, $|\phi_2(\alpha)|=|\alpha|-js+s+(j-1)s=n$. Hence we derive that $\phi_2(\alpha)\in F^2_{L,s,k}(n)$.
It remains to show that $\phi_2$ is an injection. Let $$I^2_{L,s,k}(n)=\{\phi_2(\alpha)\colon \alpha\in C^2_{L,s,2}(n)\}$$ be the image set of $\phi_2$, which has been shown to be a subset of $F^2_{L,s,k}(n)$. We wish to construct a map $\psi_2\colon I^2_{L,s,k}(n)\rightarrow C^2_{L,s,2}(n)$ such that for any $\alpha\in C^2_{L,s,2}(n)$, $$\psi_2(\phi_2(\alpha))=\alpha.$$ Let $\beta=(s^{g_s}\ldots (s+L)^{g_{s+L}})\in I^2_{L,s,k}(n)$, that is, there exists $\alpha\in C^2_{L,s,2}(n)$ such that $\phi_2(\alpha)=\beta$. From the definition of $F^2_{L,s,k}(n)$, we see that there exists a unique $i\ge 2$ such that $g_{is}=1$. By , we have $k\geq L>2s^2+5s-1\geq i+1\geq s+1$. Moreover, since $j$ is not a multiple of $s$, we see that $i+1$ is not a multiple of $s$. Hence we may define $$\psi_2(\beta)=((s+1)^{g_{s+1}}\ldots (i+1)^{g_{i+1}+s}\ldots (is)^0 \ldots(s+L)^{g_{s+L}}).$$ It is easy to check that $\psi_2(\beta)\in C^2_{L,s,2}$ and $\psi_2(\phi_2(\alpha))=\alpha$. This completes the proof.
Let $s=3$, $L=103$, $k=103$ and $n=1286$. Let $$\alpha=(10^1,11^3,20^7,28^2,31^7,46^9,52^3,65^4)$$ which is in $C^2_{103,3,2}(1286)$. It is clear that $j=11$. Applying $\phi_2$ to $\alpha$, we see that $$\phi_2(\alpha)=(3^1,10^1,20^7,28^2,30^1,31^7,46^9,52^3,65^4)$$ which is in $F^2_{103,3,103}(1286)$. Applying $\psi_2$ to $\phi_2(\alpha)$, we have $i=10$ and $\psi_2(\phi_2(\alpha))=\alpha$.
\[lem-inj-3\] There is an explicit injection $\phi_3: C^3_{L,s,2}(n)\to F^3_{L,s,k}(n)$.
Given $\alpha=((s+1)^{f_{s+1}}\ldots (s+L)^{f_{s+L}})\in C^3_{L,s,2}(n)$, by definition we see $f_k=0$, and for any $a\geq 2$ we have $f_{as}=0$. Moreover, $f_j\leq s-1$ for any $2s^2+5s-1\geq j\geq s+1$. We claim that there exists $i\geq 2s^2+5s+1$ such that $f_i\geq 1$. Otherwise, we see that $n=|\alpha|\leq (s-1)(s+1)+(s-1)(s+2)+\cdots+(s-1)(2s^2+5s-1)=2s^5+8s^4+s^3-14s^2+3s$, which is contradict to $n\ge 2s^5+8s^4+s^3-14s^2+3s+1$. Hence our claim has been verified.
From the above claim, we may set $j=\min\{i\colon i\geq 2s^2+5s+1\}$ and $j=cs+d$, where $1\leq d\leq s-1$. From $j\geq 2s^2+5s+1$ we see that $c\geq 2s+5$. Moreover, it is trivial to check that $$\label{equ-te-1-12}
j=cs+d=s+(s+1)(s+d)+s(c-s-d-2).$$ Notice that $c-s-d-2\geq 2s+5-s-(s-1)-2=4$. It is well known that $c-s-d-2$ can be uniquely written as $2x+3y$, where $0\leq y\leq 1$. Now we may define $\phi_3(\alpha)$ as follows. $$\begin{aligned}
\label{equ-phi3}
\phi_3(\alpha)&=&(s^{g_s}\ldots (s+L)^{g_{s+L}})\nonumber\\
&=&(s^{1}\ldots (s+d)^{f_{s+d}+s+1}\ldots (2s)^x \ldots (3s)^y\ldots j^{f_j-1}\ldots (s+L)^{f_{s+L}}).\end{aligned}$$ From $c-s-d-2\geq 4$ we see that $g_{2s}+g_{3s}=x+y\geq 2$. Moreover, $k\geq L> 3s$ yields that $g_k=f_k=0$. Furthermore, we may calculate $|\phi_3(\alpha)|$ as follows. $$\begin{aligned}
\label{equ-phi3-weight}
|\phi_3(\alpha)|&=&|\alpha|+s+(s+1)(s+d)+x\cdot 2s+y\cdot 3s -j\nonumber\\
&=&n+s+(s+1)(s+d)+s(2x+3y)-j\nonumber\\
&=&n+s+(s+1)(s+d)+s(c-s-d-2)-j\nonumber\\
&=&n.\end{aligned}$$ The last equation follows from . Hence $\phi_3(\alpha)\in F^3_{L,s,k}(n)$.
It remains to show that $\phi_3$ is an injection. Let $$I^3_{L,s,k}(n)=\{\phi_3(\alpha)\colon \alpha\in C^3_{L,s,2}(n)\}$$ be the image set of $\phi_3$, which has been shown to be a subset of $F^3_{L,s,k}(n)$. We wish to construct a map $\psi_3\colon I^3_{L,s,k}(n)\rightarrow C^3_{L,s,2}(n)$ such that for any $\alpha\in C^3_{L,s,2}(n)$, $$\psi_3(\phi_3(\alpha))=\alpha.$$ Let $\beta=(s^{g_s}\ldots (s+L)^{g_{s+L}})\in I^3_{L,s,k}(n)$, that is, there exists $\alpha\in C^3_{L,s,2}(n)$ such that $\phi_3(\alpha)=\beta$. From the definition of $F^3_{L,s,k}(n)$, we see that $g_{2s}+g_{3s}\geq 2$. By and the fact $f_i\leq s-1$ for all $s+1\leq i\leq 2s^2+5s-1$, we see that there exists a unique $s+1\leq i\leq 2s-1$ such that $g_i\geq s+1$. Moreover, by , we see that $$s+2sx+3sy+(s+d)(s+1)=j.$$ Thus $$s+2sg_{2s}+3sg_{3s}+i(s+1)=j\neq k.$$ Hence we may define $$\psi_3(\beta)=((s+1)^{g_{s+1}}\ldots i^{g_{i}-s-1}\ldots 2s^0\ldots 3s^0\ldots w^{g_w+1} \ldots(s+L)^{g_{s+L}}),$$ where $w=s+2sg_{2s}+3sg_{3s}+i(s+1)$. It is easy to check that $\psi_3(\beta)\in C^3_{L,s,2}(n)$ and $\psi_3(\phi_3(\alpha))=\alpha$. This completes the proof.
For example, let $s=3$, $L=105$, $k=105$ and $n=1057$. Let $$\alpha=(4^2,7^2,11^2,13^2,16^2,19^2,32^2,55,58^3,61^4,76^5)$$ be a partition in $C^3_{105,3,2}(1057)$. It is easy to see that $j=55$. Hence $c=18$ and $d=1$ and $c-s-d-2=12=2*6$. So $x=6$ and $y=0$. Applying $\phi_3$ on $\alpha$, $$\phi_3(\alpha)=(3,4^6,6^6,7^2,11^2,13^2,16^2,19^2,32^2,58^3,61^4,76^5).$$ It is trivial to check that $\phi_3(\alpha)\in F^3_{105,3,105}(1057)$. Applying $\psi_3$ to $\phi_3(\alpha)$ we recover $\alpha$.
\[lem-inj-4\] There is an explicit injection $\phi_4 : C^4_{L,s,2}(n)\to F^4_{L,s,2}(n)$.
Given $\alpha=((s+1)^{f_{s+1}}\ldots (s+L)^{f_{s+L}})\in C^4_{L,s,2}(n)$, by definition we see $f_k\geq 2$. Recall that $k=rs+t$, where $0\leq t\leq s-1$ and $r\geq 2s^2+5s\geq 7$. We may define $\phi_4(\alpha)$ as follows. $$\label{equ-phi4}
\phi_4(\alpha)=(s^{g_s}\ldots (s+L)^{g_{s+L}})
=(s^{f_k(r-2)}\ldots (2s+t)^{f_{2s+t}+f_k}\ldots k^0 \ldots (s+L)^{f_{s+L}}).$$ It is clear that $g_s=f_k(r-2)\geq 2(r-2)$. Moreover, $2s+t<7s+t\leq rs+t= k$ and $$|\phi_4(\alpha)|=|\alpha|+sf_k(r-2)+(2s+t)f_k-kf_k=n.$$ Hence $\phi_4(\alpha)\in F^4_{L,s,k}(n)$.
It remains to show that $\phi_4$ is an injection. Let $$I^4_{L,s,k}(n)=\{\phi_4(\alpha)\colon \alpha\in C^4_{L,s,2}(n)\}$$ be the image set of $\phi_4$, which has been shown to be a subset of $F^4_{L,s,k}(n)$. We wish to construct a map $\psi_4\colon I^4_{L,s,k}(n)\rightarrow C^4_{L,s,2}(n)$ such that for any $\alpha\in C^4_{L,s,2}(n)$, $$\psi_4(\phi_4(\alpha))=\alpha.$$ Let $\beta=(s^{g_s}\ldots (s+L)^{g_{s+L}})\in I^4_{L,s,k}(n)$, that is, there exists $\alpha\in C^4_{L,s,2}(n)$ such that $\phi_4(\alpha)=\beta$. From , the construction of $\phi_4$, we see that $g_s$ is a multiple of $(r-2)$. Moreover, $g_{2s+t}\geq g_s/(r-2)$. We may define $\psi_4$ as follows. $$\psi_4(\beta)=((s+1)^{g_{s+1}}\ldots (2s+t)^{g_{2s+t}-g_{s}/(r-2)}\ldots k^{g_s/(r-2)} \ldots(s+L)^{g_{s+L}}).$$ It is easy to check that $\psi_4(\beta)\in C^4_{L,s,2}(n)$ and $\psi_4(\phi_4(\alpha))=\alpha$. This completes the proof.
For $s=3$, $L=108$, $k=109$ and $n=1138$, set $$\alpha=(4^6,7^5,12^4,18^3,25^3,42^5,73^5,109^3).$$ Then $k=36*3+1$, so $r=36$ and $t=1$. Applying $\phi_4$ to $\alpha$, we derive that $$\phi_4(\alpha)=(3^{102},4^6,7^8,12^4,18^3,25^3,42^5,73^5).$$ It is trivial to check that $\phi_4(\alpha)\in F^4_{108,3,109}(1138)$. Applying $\psi_4$ to $\phi_4(\alpha)$ we recover $\alpha$.
\[lem-inj-5\] There is an explicit injection $\phi_5:C^5_{L,s,2}(n)\to F^5_{L,s,k}(n)$.
Given $\alpha=((s+1)^{f_{s+1}}\ldots (s+L)^{f_{s+L}})\in C^5_{L,s,2}(n)$, by definition we see $f_k=1$. Recall that $k=rs+t$, where $0\leq t\leq s-1$ and $r\geq 2s^2+5s\geq 7$. When $t\neq 0$, define $\phi_5(\alpha)$ as follows. $$\label{equ-phi5-1}
\phi_5(\alpha)=(s^{g_s}\ldots (s+L)^{g_{s+L}})
=(s^{r-4}\ldots (2s)^{f_{2s}+1}\ldots (2s+t)^{f_{2s+t}+1}\ldots k^0 \ldots (s+L)^{f_{s+L}}).$$ And when $t=0$, we set $\phi_5(\alpha)$ as given below. $$\label{equ-phi5-2}
\phi_5(\alpha)=(s^{g_s}\ldots (s+L)^{g_{s+L}})
=(s^{r-4}\ldots (2s)^{f_{2s}+2}\ldots k^0 \ldots (s+L)^{f_{s+L}}).$$ In either case, we see that $g_s=r-4$ and $g_{2s}\geq 1$. Moreover, it is trivial to check that $|\phi_5(\alpha)|=n$. This yields $\phi_5(\alpha)\in F^5_{L,s,k}(n)$.
It remains to show that $\phi_5$ is an injection. Let $$I^5_{L,s,k}(n)=\{\phi_5(\alpha)\colon \alpha\in C^5_{L,s,2}(n)\}$$ be the image set of $\phi_5$, which has been shown to be a subset of $F^5_{L,s,k}(n)$. We wish to construct a map $\psi_5\colon I^5_{L,s,k}(n)\rightarrow C^5_{L,s,2}(n)$ such that for any $\alpha\in C^5_{L,s,2}(n)$, $$\psi_5(\phi_5(\alpha))=\alpha.$$ Let $\beta=(s^{g_s}\ldots (s+L)^{g_{s+L}})\in I^5_{L,s,k}(n)$, that is, there exists $\alpha\in C^5_{L,s,2}(n)$ such that $\phi_5(\alpha)=\beta$. When $t\neq 0$, from we see $g_s=r-4$, $g_{2s}\ge 1$ and $g_{2s+t}\geq 1$. We may define $\psi_5$ as follows. $$\psi_5(\beta)=((s+1)^{g_{s+1}}\ldots (2s)^{g_{2s}-1}\ldots (2s+t)^{g_{2s+t}-1}\ldots k^{1} \ldots(s+L)^{g_{s+L}}).$$ When $t=0$, from , we have $g_s=r-4$ and $g_{2s}\ge 2$. Hence the map $\psi_5$ is defined as follows. $$\psi_5(\beta)=((s+1)^{g_{s+1}}\ldots (2s)^{g_{2s}-2}\ldots k^{1} \ldots(s+L)^{g_{s+L}}).$$ It is easy to check that in either case $\psi_5(\beta)\in C^5_{L,s,2}(n)$ and $\psi_5(\phi_5(\alpha))=\alpha$. This completes the proof.
For $s=3$, $L=103$, $k=105$ and $n=1217$, set $$\alpha=(6^2,9^5,12^8,17^4,35^6,42^5,73^5,105^1,106^1).$$ Then $k=35*3$, so $r=35$ and $t=0$. Applying $\phi_5$ to $\alpha$, we derive that $$\phi_5(\alpha)=(3^{31},6^4,9^5,12^8,17^4,35^6,42^5,73^5,106^1).$$ It is trivial to check that $\phi_5(\alpha)\in F^5_{103,3,105}(1217)$. Applying $\psi_5$ to $\phi_5(\alpha)$ we recover $\alpha$.
We are now in a position to prove Theorem \[thm-main-inf\].
[*Proof of Theorem \[thm-main-inf\].*]{} Given integer $s\geq 1$, $L\geq 2s^3+5s^2+1$, $s+L\ge k\ge L$ and $n\ge 2s^5+8s^4+s^3-14s^2+3s+1$, for any $\alpha\in C_{L,s,2}(n)$, we define $$\phi(\alpha)=\phi_i(\alpha)\;\; \text{if }\;\; \alpha\in C^i_{L,s,2}(n)\;\; \text{for}\;\; i=1,\ldots, 5.$$ From Lemmas \[lem-inj-1\]-\[lem-inj-5\], we deduce that $\phi(\alpha)\in F_{L,s,k}(n)$ and $\phi$ is an injection. This completes the proof.
We show that Theorem \[thm-main\] is a consequence of Theorems \[thm-asy\] and \[thm-main-inf\].
[*Proof of Theorem \[thm-main\].*]{} From Theorem \[thm-asy\], for any positive integer $s$ and $L$, there exists an integer $M_{L,s}$ such that for $n\geq M_{L,s}$, $$\label{eq-f-s-k-l}
f_{L,s,k}(n)\geq c_{L,s,2}(n).$$ Moreover, by Theorem \[thm-main-inf\], for $L\geq 2s^3+5s^2+1$ and $n\geq 2s^5+8s^4+s^3-14s^2+3s+1$, also holds. Hence, if we set $$M=\max\{M_{3,s},M_{4,s},\ldots,M_{2s^3+5s^2,s},2s^5+8s^4+s^3-14s^2+3s+1\},$$ then $M$ only depends on $s$, and holds for all $n\geq M$.
Proof of Theorem \[thm-main0-2\]
================================
Define the sequences $(a_n)$, $(b_n)$ and $(c_n)$ by $$\begin{aligned}
\sum_{n=0}^\infty a_nq^n&:=\frac{1-q}{(q^s;q)_{L+1}},\label{equ-def-an}\\
\sum_{n=0}^\infty b_nq^n&:=\frac{1}{(1-q^s)(q^{s+2};q)_{L-1}},\label{equ-def-bn}\\
\sum_{n=0}^\infty c_nq^n&:=\frac{1}{(q^{s+1};q)_L}.\label{equ-def-cn}\end{aligned}$$ We have the following result, which will play a curial role in the proof of Theorem \[thm-main0-2\].
\[lem-1-asy\] For any $s\geq 1$ and $L\geq 3$, $$a_n\sim \frac{n^{L-1}}{(L-1)!s(s+1)\cdots (s+L)}.$$
Writing $1-q=(1-q^{s+1})-(q-q^{s+1})$ we see that $$\begin{aligned}
\label{equ-sum-an}
\frac{1-q}{(q^s;q)_{L+1}}=\frac{1}{(1-q^s)(q^{s+2};q)_{L-1}}-\frac{q}{(q^{s+1};q)_L}.\end{aligned}$$ It follows from - that $$\label{equ-sum-an}
a_n=b_n-c_{n-1}\quad \textrm{for} \quad n\geq 1.
$$ For $L\geq 3$, as $\gcd(s,s+2,\ldots,s+L)=1$ and $\gcd(s+1,\ldots,s+L)=1$, by Theorem \[schur-thm\], we have $$\label{equ-sim-bn}
b_n\sim \frac{(s+1)n^{L-1}}{(L-1)!s(s+1)\cdots (s+L)},$$ and $$\label{equ-sim-cn}
c_n\sim \frac{sn^{L-1}}{(L-1)!s(s+1)\cdots (s+L)}.$$ Substituting and into , we deduce that $$a_n=b_n-c_{n-1}\sim \frac{n^{L-1}}{(L-1)!s(s+1)\cdots (s+L)}.$$ This completes the proof.
We proceed to show Theorem \[thm-main0-2\] with the aid of Lemma \[lem-1-asy\].
[*Proof of Theorem \[thm-main0-2\].*]{} For fixed integer $k$, define $$\begin{aligned}
\label{def:d}
\sum_{n=0}^\infty d_{k}(n)q^n=\frac{1-q^{k}}{(q^s;q)_{L+1}}.\end{aligned}$$ Comparing with we have $$\sum_{n=0}^\infty d_k(n)q^n
=(1+q+\cdots+q^{k-1})\sum_{n=0}^\infty a_nq^n.$$ Thus, for $n\geq k$, Lemma \[lem-1-asy\] implies that $$\begin{aligned}
d_k(n)&=a_n+a_{n-1}+\cdots+a_{n-k+1}\nonumber\\
&\sim \frac{n^{L-1}+\cdots+(n-k+1)^{L-1}}{(L-1)!s(s+1)\cdots (s+L)}\nonumber\\
&\sim \frac{kn^{L-1}}{(L-1)!s(s+1)\cdots (s+L)}.\label{equ-dkn-sim-in}\end{aligned}$$ For fixed integer $r\geq 0$, set $$\begin{aligned}
\label{def:e}
\sum_{n=0}^\infty e_{k,r}(n)q^n:=\frac{q^r(1-q^{k})}{(q^s;q)_{L+1}}.\end{aligned}$$ Then, for $n\ge r$, $$\label{equ-temp-e-d}
e_{k,r}(n)=d_k(n-r).$$ By , we deduce that $$\label{equ-dkn-sim-in-en}
e_{k,r}(n)\sim \frac{kn^{L-1}}{(L-1)!s(s+1)\cdots (s+L)}.$$ Therefore, it follows from , and that $$H^*_{L,s,r,k_1,k_2}(q)=\sum_{n=0}^\infty \left(e_{k_1,r}(n)-d_{k_2}(n)\right)q^n.$$ From and , we derive that $$e_{k_1,r}(n)-d_{k_2}(n)\sim \frac{(k_1-k_2)n^{L-1}}{(L-1)!s(s+1)\cdots (s+L)}.$$ Since $k_1>k_2$, there exists $M$ such that $e_{k_1,r}(n)-d_{k_2}(n)>0$ for $n>M$. Thus $H^*_{L,s,r,k_1,k_2}(q)$ is eventually positive. This completes the proof.
For some special cases it would be interesting to determine the smallest $M$ in Theorem \[thm-main\], see also Conjecture 5.3 in [@Ber-Unc-19].
[**Acknowledgments.**]{} This work was done during the second author’s visit to the Harbin Institute of Technology (HIT) in the summer of 2019. The first author was supported by the National Natural Science Foundation of China (No. 11801119). The second author would like to thank Institute for Advanced Study in Mathematics of HIT for the hospitality.
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|
---
abstract: 'Many real-world networks have properties of small-world networks, with clustered local neighborhoods and low average-shortest path (ASP). They may also show a scale-free degree distribution, which can be generated by growth and preferential attachment to highly connected nodes, or hubs. However, many real-world networks consist of multiple, inter-connected clusters not normally seen in systems grown by preferential attachment, and there also exist real-world networks with a scale-free degree distribution that do not contain highly connected hubs. We describe spatial growth mechanisms, not using preferential attachment, that address both aspects.'
author:
- Marcus Kaiser
- 'Claus C. Hilgetag'
title: 'Spatial Growth of Real-world Networks'
---
Introduction
============
Many real-world networks show [*small-world*]{} properties [@Watts1998]. Their average clustering coefficient, representing the proportion of direct links between the neighbors of a node, is higher than in same-size random networks, while they maintain a comparable average shortest path (ASP). The giant component of some of these networks has been shown to consist of several clusters, which contain strongly interlinked nodes and form only sporadic connections to other clusters. For instance, the cortical systems networks in macaque monkey and cat brains possess such a multi-cluster organization [@Hilgetag2000b]. Moreover, various complex linked systems have been described as [*scale-free*]{} networks [@Barabasi1999; @Huberman1999], in which the probability for a node possessing $k$ edges is $P(k)\propto k^{-\gamma}$. It has been suggested that this large class of networks may be generated by mechanisms of growth and preferential attachment, that is, the preferred linking of new nodes to already highly connected network nodes [@Barabasi1999]. An essential aspect of many real-world networks is, however, that they exist and develop in metric space. Therefore, questions arise how nodes are able to identify highly connected distant hubs and why they would attach to them, rather than to nearby nodes [@Caldarelli2002]. Moreover, long-range connections to hubs violate optimal wiring principles [@Cherniak1994]. For example, a city in New England would normally consider constructing a new highway to nearby Boston, rather than to faraway Los Angeles, even if Los Angeles represents a larger hub in the US highway system.
Previous spatial growth algorithms, in which the probability for edge formation decreased with node distance, predetermined the position of all nodes at the outset [@Waxman1988; @Yook2002]. Starting with the complete set of nodes, which was distributed randomly on a spatial grid, connections were established depending on distance [@Watts1999; @Kleinberg2000; @Eames2002]. Additionally, connected nodes could be drawn together by a [*posteriori*]{} pulling algorithm, which resulted in spatial clusters of connected nodes [@Segev2003]. Such mechanisms, however, appear unsuited as a general explanation for growing biological and artificial systems with newly forming nodes and connections.
Spatial Network Development Algorithm
=====================================
In an alternative approach, we employed a model of spatial growth in which the nodes, their positions and connections were established during development. Starting with one node at the central position (0.5; 0.5) of the square embedding space (edge length one), the following algorithm was used:\
1) A new node position was chosen randomly in two-dimensional space with coordinates in the interval \[0; 1\].\
2) Connections of the new node, $u$, with each existing node, $v$, were established with probability $$\label{exponential}
P(u,v) = \beta \ e^{-\alpha \ d(u, v)},$$ where $d(u, v)$ was the spatial (Euclidian) distance between the node positions, and $\alpha$ and $\beta$ were scaling coefficients shaping the connection probability [@Waxman1988].\
3) If the new node did not manage to establish connections, it was removed from the network. In that way, newly forming nodes could only be integrated within the vicinity of the existing network, making the survival of new nodes dependent on the spatial layout of the present nodes.\
4) The algorithm continued with the first step, until a desired number of nodes was reached.\
Parameter $\beta$ (“density”) served to adjust the general probability of edge formation and was chosen from the interval \[0; 1\]. The nonnegative coefficient $\alpha$ (“spatial range”) exponentially regulated the dependence of edge formation on the distance to existing nodes. Such spatial constraints are present during the development of many real networks. In biological systems, for instance, gradients of chemical concentrations, or molecule interactions decay exponentially with distance [@Murray1990].
The algorithm allowed some nodes to be established distant to the existing network, although with low probability. Subsequent nodes placed near to such ’pioneer’ nodes would establish connections to them and thereby generate new highly-connected regions away from the rest of the network. Through this mechanism multiple clusters were able to arise, resulting in networks in which nodes were clustered topologically as well as spatially.
In a slightly modified approach the growth model could employ a power-law to describe the dependence of edge formation on the spatial distance of nodes: $$\label{powerlaw}
P(u,v) = \sigma \ d(u,v)^{-\tau}.$$ By this mechanism the probability of establishing distant nodes would be increased even further. For example, simulating networks of similar size (50 networks; $n=100$; $density=0.04$; square embedding space edge length 100) for both types of distance dependencies, the power-law (Eq. \[powerlaw\], $\sigma=1$, $\tau=1$) resulted in higher total wiring length (6303) compared to networks generated by exponential edge probability (Eq. \[exponential\], $\alpha=0.35$, $\beta=1$, total wiring length 1077 units). In the following investigations, however, we concentrated on the exponential approach outlined above, since our simulations indicated that power-law edge probability was unable to yield small-world networks (tested parameter ranges $\sigma \in [0.004; 2]$ and $\tau \in [0.125; 64]$).
Another essential network feature investigated in the model was the presence or absence of hard spatial borders that limit network growth. Borders occur in many compartmentalized systems, be it mountains or water surrounding geographical regions, cellular membranes separating biochemical reaction spaces, or the skull limiting expansion of the brain. Depending on coefficient $\alpha$ and the network size, our simulated networks never reached a hard border (’virtually unlimited growth’), or quickly arrived at the spatial limits, so that new nodes could then only be established inside the existing networks. Naturally, virtually unlimited growth would eventually also arrive at the hard borders, after sufficiently sustained network growth. However, in the context of our simulations, growth could be considered virtually unlimited if for a chosen network size at the end of the algorithm all nodes were still far away from the borders (by at least 0.25 units).
In the following, we describe different types of spatially grown networks resulting from low or high settings for parameters $\alpha$ and $\beta$, and present examples of real-world networks corresponding to the generated types.
For the generated networks, two network properties are shown, which have been used previously to characterize complex networks [@Watts1998]. The average shortest path (ASP, similar, though not identical, to characteristic path length $L$ [@Watts1999]) of a network with $N$ nodes is the average number of edges that has to be crossed on the shortest path from any one node to another. $$\label{asp}
ASP = {1 \over N (N-1)} \sum_{i, j} d(i,j) \ \ \ \ \ with \ i\ne j,$$ where $d(i,j)$ is the length of the shortest path between nodes $i$ and $j$. The clustering coefficient of one node $v$ with $k_v$ neighbors is $$\label{clustercoef}
C_v = {|E(\Gamma_v)| \over {k_v \choose 2} },$$ where $|E(\Gamma_v)|$ is the number of edges in the neighborhood of $v$ and ${k_v \choose 2}$ is the number of possible edges [@Watts1999]. In the following analyses we use the term clustering coefficient as the average clustering coefficient for all nodes of a network.
Algorithms for network generation, calculation of network parameters and visualization were developed in Matlab (Release 12, MathWorks Inc., Natick) and also implemented in C for larger networks. For each parameter set and network size, 50 simulated networks were generated and analyzed (20 in the case of virtually unlimited growth, due to computational constraints).
Modeled Types of Networks
=========================
Sparse Networks (limited and virtually unlimited growth).
---------------------------------------------------------
For very small $\beta$ ($<0.01$), sparse networks were generated (Fig. \[sf\]a) in which only a small proportion of all possible edges was established. The resulting networks were highly linear, that is, exhibiting one-dimensional chains of nodes, independent of limited or virtually unlimited growth (parameter $\alpha$). The histograms of chain-lengths found in these networks, indicating the number of nodes in the chains, were similar to those of random networks with the same density. Unlike in random networks, however, the clustering coefficient was lower than the network density, and despite lacking clusters and hubs with large degree $k$, these networks possessed a power-law degree distribution, with high ASP (Fig. \[sf\]b). The power-law exponent was small, in the range of \[1.7; 2.1\]; and in the simulated networks of 100 nodes the cut-off for the maximum degree of the scale-free networks was 16. Given their low maximum degree, these networks with low clustering and long linear chains of nodes could be called linear scale-free.
![\[sf\] (a) Sparse network (density 0.42%) with 500 nodes obtained by limited growth ($\alpha=2$, $\beta=0.001$). (b) Cumulative degree probability $P(k)$ that a node possesses $k$ edges for the network shown in (a). A power-law of the degree distribution ($\gamma=2.43$) can be observed.](fig1.eps)
[**Example: German Highway System.**]{} We identified a linear scale-free organization in the German highway (“Autobahn”) system. The highway network of 1,168 nodes was compiled from data of the ’Autobahn-Informations-System’ [@auto]. The ratio of clustering coefficient and density of the highway system, which can be seen as a linearity coefficient, was 0.64. This system is also an example for a scale-free ($\gamma=2.8$), yet not small-world, network, as its ASP was twice as large as for comparable random networks.
A similar type of organization was also found for scale-free protein-protein interaction networks [@Jeong2001] ($k_{max}\approx 20$).
Dense Networks (limited and virtually unlimited growth).
--------------------------------------------------------
For higher edge probability ($\beta \rightarrow 1$), a noteworthy difference between limited and virtually unlimited growth became apparent. While it was impossible to generate high network density under virtually unlimited growth conditions, the introduction of spatial limits resulted in high density and clustering, as well as low ASP. This was due to the fact that, in the virtually unlimited case, new nodes at the borders of the existing network were surrounded by fewer nodes and therefore formed fewer edges than central nodes within the network. In the limited case, however, the network occupied the whole area of accessible positions. Therefore, new nodes could only be established within a region already dense with nodes and would form many connections.
Figure \[sw\] shows the relation between small-world graph properties and growth parameters $\alpha$ and $\beta$ for networks consisting of 100 nodes. The ratio of the clustering coefficient in spatial growth compared to random networks was larger than one (indicating small world graphs), if the values for $\alpha$ and $\beta$ were high (Fig. \[sw\]a). The ASP in the generated networks normalized by the ASP in random networks with similar density was similar for low values of $\alpha$ and high values of $\beta$. For these networks the likelihood of edge formation was high and — because of the low value of $\alpha$ — independent from spatial distance. Such networks resembled random growth, with the clustering coefficient possessing the same value as the density ($C/C_{random}\approx 1$).
In a small interval of intermediate values for $\alpha$ ($\alpha\approx 4$, $\beta=1$), networks exhibited properties of small-world networks (ASP and clustering coefficient shown in Fig. \[overview\]a). Here, the ASP was comparable to that in random networks of the same size ($ASP\approx ASP_{random}$), while the clustering coefficient was 39% higher than in random networks [@Watts1999 p. 114]. An overview of the parameter space and the resulting random, small-world, virtually unlimited or linear scale-free networks is given in Figure \[overview\]b.
[**Example: Cortical Connectivity.**]{} One biological example for small-world spatial networks with high clustering coefficient and high density are the well studied, clustered systems of long-range cortical connectivity in the cat and macaque monkey brains [@Scannell1999; @Young1993; @Hilgetag2000b]. We employed the model in order to generate networks with identical number of nodes and edges and comparable small-world properties. While small-world networks could be generated in the appropriate parameter range of the model (Fig. \[overview\]b), the biological networks featured even stronger clustering. We found, however, that such networks could be produced by extending the local range of high connection probability, so that $P=1$ for $distance_{cat}<0.18$, $distance_{macaque}<0.11$ and $P$ decaying exponentially as before for larger distances (this was implemented by setting $\alpha_{cat}=5$, $\alpha_{macaque}=8$ and for both networks $\beta=2.5$ and thresholding probabilities larger than one to one). The modified approach therefore combined specific features of the biological networks with the general model of limited spatial growth. This yielded networks with distributed, multiple clusters, and average densities of around 30% (for simulated cat brain connectivity) and 16% (monkey connectivity). Moreover, these networks had clustering coefficients of 50% and 40%, respectively, very similar to the biological brain networks [@Hilgetag2000b], as shown in Table \[brain\].
Comparison of the biological and simulated degree distributions, moreover, showed a significant correlation (Spearman’s rank correlation $\rho$ = 0.77 for the cat network, $P<3\times10^{-3}$; and $\rho$ = 0.9 for the macaque network, $P<2\times10^{-5}$). On the other hand, the BA-model [@Barabasi1999], using growth and preferential attachment, yielded similar densities and clustering coefficients, but was unable to generate multiple clusters as found in the real cortical networks.
[lllll]{} & $n$ & $d$ & $C_{cortical}$ & $C_{spatial \ growth}$\
\
cat & 55 & 0.30 & 0.55 & 0.5\
macaque & 73 & 0.16 & 0.46 & 0.4\
![\[sw\] Comparison of small-world properties of spatial and random networks for N=100 nodes. Each data point represents the average for 50 networks. (a) Ratio of the clustering coefficient C of the generated networks divided by the clustering coefficient for comparable random networks. A large ratio is one feature of small-world networks. (b) Ratio of the average-shortest paths, ASP, of spatial-growth and comparable random networks.](fig2.eps)
In contrast to limited growth, virtually unlimited growth simulations with high $\beta$ resulted in inhomogeneous networks with dense cores and sparser periphery. It is difficult to imagine realistic examples for strictly unlimited development, as all spatial networks eventually face internal or external constraints that confine growth, may it be geographical borders or limits of their energetic and material resources. However, virtually unlimited growth may be a good approximation for the early development of networks before reaching borders.
![\[overview\]Exploration of model parameter space. (a) For dense networks ($\beta=1$, $N=100$ nodes), an increased dependence of edge formation on distance (parameter $\alpha$) led to an increase of $ASP$ (diamonds) and a decrease in clustering coefficient $C$ (triangles). (b) Overview of network types for different spatial growth parameters ($N=100$ nodes). Low values of $\alpha$ made edge formation independent from distance and resulted in random networks (black). For large values of $\alpha$ only nodes near the existing network could establish connections, and the hard borders were not reached (virtually unlimited, green). The area labeled linear scale-free (blue) was a region in which sparse and highly linear networks showing a scale-free degree distribution occurred. Only a small part of the parameter space (red) showed properties of small-world networks. ](fig3.eps)
Classifying Types of Network Development
========================================
Different network growth types can be distinguished by assessing the evolution of network density and clustering coefficient. Growth with preferential attachment as well as spatial growth lead to clustering coefficients, $C(N)$, that depend on the current size of the network, that is, the number of nodes, $N$ (Fig. \[dev\]a). While $C(N)$ decreases with network size for networks generated by the BA-Model [@Barabasi1999], it remains constant for spatial-growth networks. Virtually unlimited or limited spatial growth can thus be distinguished, since density decreases with network size for unlimited growth, while remaining constant for limited growth (Fig. \[dev\]b).
[**Example: Evolution of metabolic networks.**]{} We applied this concept to classifying the development of real-world biological networks. The evolution of metabolic systems, for instance, can be seen as an incorporation of new substances and their metabolic interactions into an existing reaction network. Reviewing 43 metabolic networks in species of different organizational level [@Barabasi2000b], the clustering coefficient of these systems remained constant across the scale [@Ravasz2002], whereas their density (Fig. \[dev\]c) decreased with network size. This indicated features of virtually unlimited network growth. The relation between the number of links and nodes in these systems was linear (Fig. \[dev\]d), with a slope of 5.2, so that the number of interactions of a metabolite was not increasing with network size. Such linear growth may ensure that the metabolic systems remain connected (with the number of reactions larger than substances, as a necessary condition for connectedness), while not becoming too complex too quickly (as, for instance, with exponential addition of new reactions).
![\[dev\]Comparison of the dependence of clustering coefficient $C(N)$ and density on network size (number of nodes, $N$). (a) For the simulated networks the clustering coefficient remained constant for limited (triangles, $\alpha=5$, $\beta=1$) and virtually unlimited (boxes, $\alpha=200$, $\beta=1$) spatial growth, but decreased for growth with preferential attachment (diamonds). (b) Density was independent of network size only for limited spatial growth. (c) Density depending on network size ($N$) for the metabolic networks of 43 different organisms (15). (d) A critical measure for network development was the dependence of network size on the number of links. For metabolic networks, this relationship was strongly linear. ](fig4.eps)
Conclusions
===========
We have proposed a new kind of spatial growth mechanism, incorporating both limited and virtually unlimited growth, that can produce a variety of metric real-world networks. The metric is not limited to Euclidian space as in the discussed examples, but may also use measures of similarity to define the link probability (e.g., social relations, [@Watts2002]).
In contrast to previously studied spatial graphs [@Watts1999], networks generated by our model were always connected. Moreover, the approach was able to generate small-world graphs, which is thought not to be possible in the spatial graph model in which positions are chosen randomly [*before*]{} edge formation [@Watts1999]. Finally, the model was also able to produce scale-free networks with relatively low maximum degree, similar to, for example, the German highway system.
A systematic evaluation of model parameter space was carried out at the specific network size of 100 nodes, which was feasible computationally. It would be interesting to also evaluate larger or smaller network sizes and to investigate for them, if small-world networks can be generated in a larger range of parameters $\alpha$ and $\beta$.
Several algorithms have been proposed for the generation of different types of topological networks, in which links do not reflect physical distances, but merely the connectivity of the system [@Watts1998; @Barabasi1999; @Newman2001]. Examples for such networks include the World-Wide Web, financial transaction networks, and, to some extent, networks of airline transportation. The present model extends previous approaches to the development of spatial networks, such as cellular and brain connectivity networks, or food webs and many systems of social interactions. Spatial as well as temporal constraints shape network growth, and intrinsic or external spatial limits may determine essential features of the structural organization of linked systems, such as clustering and scaling properties. Borders, for instance, appear to have been critical for early chemical evolution, ensuring clustering of good replicators and preventing the spreading of short templates with limited replication function [@Szabo2002]. The same applies to cortical networks where elimination of growth limits results in a distorted network topology [@Kuida1998].
The specific spatio-temporal conditions for the development of different types of real-world networks warrant further investigation. They may be of additional interest, as local spatial growth mechanisms also imply global optimization of path lengths in connected systems [@Valverde2002].\
We thank N. Sachs, H. Jaeger, M. Zacharias and A. Birk for critical comments on the manuscript.
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|
---
abstract: |
We consider the weighted parabolic problem of the type $$\begin{split} \left\{\begin{array}{ll}
u_t-\dv(\omega_2(x)|\nabla u|^{p-2} \nabla u )= \lambda \omega_1(x) |u|^{p-2}u,& x\in\Omega,\\
u(x,0)=f(x),& x\in\Omega,\\
u(x,t)=0,& x\in\partial\Omega,\ t>0,\\
\end{array}\right.
\end{split}$$ for quite a general class of possibly unbounded weights $ \omega_1,\omega_2$ satisfying the Hardy-type inequality. We prove existence of a global weak solution in the weighted Sobolev spaces provided that $\lambda$ is smaller than the optimal constant in the inequality.
author:
- 'Iwona Skrzypczak[^1]'
- 'Anna Zatorska–Goldstein[^2]'
bibliography:
- 'AZG-IS-unbounded-arxiv.bib'
title: Existence to nonlinear parabolic problems with unbounded weights
---
Ø ø
\[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\]
1em
[ existence of solutions, Hardy inequalities, parabolic problems, weighted $p$–Laplacian, weighted Sobolev spaces]{}
[: 35K55, 35A01, 47J35. ]{}
Introduction
============
Our aim is to provide an existence result for a broad class of nonlinear parabolic equations $$\label{paraprob0}
u_t-\dv(\omega_2(x)|\nabla u|^{p-2}\nabla u)=\lambda \omega_1(x) |u|^{p-2}u,\qquad in\quad\Omega$$ where $p>2$, weight functions $\omega_1,\omega_2\ge 0$ are possibly unbounded, $\Omega\subseteq\rn$ is a bounded open set. We develop the previous results [@isazg1] by allowing $\omega_1$ to be unbounded, which entail challenges in functional analysis of the two-weighted Sobolev spaces $W_{(\omega_1,\omega_2)}^{1,p}(\Omega)$.
We impose the restrictions on the weights in order to control the structure of the two–weighted Sobolev spaces, as well as to ensure monotonicity of the leading part of the operator. Namely, we assume
- $\omega_1,\omega_2:\overline{\Omega}\to\R_+\cup\{0\}$ and $\omega_1,\omega_2\in L^{1}_{loc}(\Omega)$;
- $\omega_1^{-\frac{2}{p-2}}\in L^1(\Omega)$;
- for any $U\subset\subset \Omega$ there exists a constant $\omega_2(x)\geq c_U>0$ in $U$;
- $(\omega_1,\omega_2)$ is a pair of weights in Hardy inequality $$\label{eq:hardyintro}
K \int_\Omega \ |\xi|^p \omega_{1}(x) dx \leqslant \int_\Omega |\nabla \xi|^p \omega_{2}(x)dx;$$
Furthermore, assume that there exists $s>p$ such that
- for any $U\subset\subset \Omega$ we have a compact embedding $$W^{1,p}_{(\omega_1,\omega_2),0}(U)\subset\subset L^s_{\omega_1}(U);$$
- there exists $q\in\left(p,s\right)$, such that $$\omega_1^{-\frac{q}{s-q}}\in L^1_{loc}(\Omega)\quad\text{ and} \quad \omega_2^{\frac{q}{q-p}}\in L^1_{loc}(\Omega).$$
The assumptions are discussed in Subsetion \[ssec:comwei\].
The existence of solutions to problems $$u_t-\dv(a(x,t,u,\nabla u))= f,$$ where the involved operator is monotone and has $p$–growth, is very well understood, e.g. [@boc-ors1; @boc-ors2; @boccardo]. Nonetheless, this research concerns the autonomic case, i.e. when the right–hand side does not depend on the solution itself.
Various physical models (*combustion models*) involve semilinear parabolic problems of the form $$u_t-\Delta u= f(u).$$ Fujita’s Theory, developed since 1960s, analyses the possible singularities of solutions. There are known examples of problems, where solutions explode (blow-up) to infinity in finite time. More recent research in that directions was carried out by Giga and Kohn.
In [@vz] Vazquez and Zuazua, generalizing the seminal paper by Baras and Goldstein [@bargold], describe the asymptotic behaviour of the heat equation that reads $$u_t=\D u +V(x)u \quad\mathrm{and}\quad \D u +V(x)u+\mu u=0,$$ where $V(x)$ is an inverse–square potential. The key tool is an improved form of the Hardy–Poincaré inequality. The optimal constant in Hardy-type inequality indicates the critical $\lambda$ for blow-up or global existence, as well as the sharp decay rate of the solution.
This phenomenon is observed in wide range of parabolic problems, including semilinear equations, see e.g. [@AdChaRa; @anh; @bargold; @BV; @bh; @32; @gaap; @xiang; @vz]. In several papers, e.g. [@bbdg; @blanchet_09; @sharp], dealing with the rate of convergence of solutions to fast diffusion equations $u_t=\Delta u^m,$ the authors study the estimates for the constants in Hardy-Poincaré-type inequalities and their application. The weighted fast diffusion equation is getting attention [@weightmatteo1; @weightmatteo2].
In general, application of the general Hardy inequalities is expected to infer certain properties of solution to wide class of parabolic problems. The inspiration of our research was the paper of García Azorero and Peral Alonso [@gaap], who apply the Hardy inequality [[@gaap Lemma 2.1]]{} of the form $$\lambda_{N,p}\int_{\rN} \ |\xi|^p |x|^{-p} \ dx \le \int_{\rN} |\nabla \xi|^p\, dx ,$$ where $\lambda_{N,p}$ is optimal, to obtain the existence of weak solutions to the corresponding parabolic problem $$u_t -\Delta_p u=\frac{\lambda}{|x|^p}|u|^{p-2}u,\qquad 1<p <N.$$
We adapt some ideas of Anh, Ke [@anh], who consider the initial boundary value problem for a class of quasilinear parabolic equations involving weighted $p$-Laplace operator $$u_t-\dv(\sigma(x)|\nabla u|^{p-2} \nabla u )= \lambda |u|^{p-2}u-f(x,u),\qquad 2\leq p <N.$$
Our major difficulties are of technical nature and require more advanced setting than classical one in [@anh; @gaap]. We employ the two-weighted Sobolev spaces $W^{1,p}_{(\omega_1,\omega_2)}(\Omega)$, due to presence of general class of weights both in the leading part of the operator and on the right-hand side of . The key tool is truncation method of Boccardo, Murat [@boccardo].
Our main result is the following theorem.
\[theo:main\] Let $p> 2$, $\Omega\subseteq\rn$ be an open subset, $f\in L^2(\Omega)$. Assume that $\omega_1,\omega_2$ satisfy conditions (W1)–(W6).
There exist $\lambda_0=\lambda_0(p,N,\omega_1,\omega_2)$ such that for all $\lambda\in(0,\lambda_0)$, the parabolic problem $$\label{eq:main}\left\{\begin{array}{ll}
u_t-\dL u= \lambda \omega_1(x)|u|^{p-2}u & x\in\Omega,\\
u(x,0)=f(x)& x\in\Omega,\\
u(x,t)=0& x\in\partial\Omega,\ t>0,\\
\end{array}\right.$$ has a global weak solution $u \in L^p(0,T; W_{(\omega_1,\omega_2),0}^{1,p}(\Omega)),$ such that $
u_t \in L^{p'}(0,T; {W^{-1,p'}_{(\omega_1',\omega_2')}(\Omega)}),$ i.e. $$\int_{\Omega_T}\left( u_t\xi+\omega_2|\nabla u|^{p-2} \nabla u \nabla \xi -\lambda \omega_1(x) |u|^{p-2}u \xi\right)dx\,dt=0,$$ holds for each $\xi\in L^p(0,T; W_{(\omega_1,\omega_2),0}^{1,p}(\Omega))$. Moreover, $u\in L^\infty(0,T; L^2(\Omega_T))$.
In fact, the proof of the above theorem implies the existence to $$\left\{\begin{array}{ll}
u_t-\dL u= \lambda W(x)|u|^{p-2}u & x\in\Omega,\\
u(x,0)=f(x)& x\in\Omega,\\
u(x,t)=0& x\in\partial\Omega,\ t>0,\\
\end{array}\right.$$ with any $W(x)\leq\omega_1(x)$ without assumption $||W||_{L^\infty(\Omega_T)}<\infty.$
The conditions (W1)-(W6) are sastisfied by the following pairs of weights:
- $\omega_1(x)=|x|^{-p}$, $\omega_2(x)\equiv 1$. The compact embedding (W5) is given by [@fra-ser-ser Theorem 3.4] by Franchi, Serapioni, and Serra Cassano. This example retrieves the result of [@gaap];
- $\omega_1(x)=\left(dist(x,\partial\Omega)\right)^{\gamma-p}$, $\omega_2(x)=\left(dist(x,\partial\Omega)\right)^{\gamma}$, with some $\gamma<0$. The compact embedding (W5) is given by [@kuf-opic Example 18.15] by Kufner and Opic and holds for $\Omega$ with sufficiently regular boundary.
The paper is organised as follows. Section \[prelim\] provides disscusion on properties of the two-weighted Sobolev spaces and assumptions on the admissible weights. In Section \[sec:nonleig\] we stand the relation between the first eigenvalue of the elliptic operator and the optimal constant in the Hardy inequality. After the compactness results in Section \[sec:main\], the proof of Theorem \[theo:main\] is given.
Preliminaries {#prelim}
=============
Notation
--------
In the sequel we assume that $p> 2$, $\frac{1}{p}+\frac{1}{p'}=1$, $\O\subset \rn$ is an open subset not necessarily bounded. For $T>0$ we denote $\Omega_T=\Omega\times(0,T)$.
We denote $p$–Laplace operator by $$\Delta_p u = {\dv}( |\nabla u|^{p-2}\nabla u)$$ and $\omega$–$p$–Laplacian by $$\begin{aligned}
\Delta_p^{\omega } u&=& {\dv}(\omega|\nabla u|^{p-2}\nabla u),\label{Lpom2}\end{aligned}$$ with a certain weight function $\omega:\Omega\to\R$.
We use truncations $T_k(f)(x)$ defined as follows $$T_k(f)(x)=\left\{\begin{array}{ll}f & |f|\leq k;\\
k\frac{f}{|f|}& |f|\geq k.
\end{array}\right. .\label{Tk}$$
By $\langle f,g\rangle$ we denote the standard scalar product in $L^2(\Omega)$.
Let $B(r)\subset\rn$ denote the ball with the radius $r$, whose center shall be clear from the context. Then $|B(r)|$ is its Lebesgue’s measure, $\omega(B(r))$ its $\omega$-measure, i.e. $\omega(B(r))=\int_{B(r)}\omega(x)\,dx$.
Weighted Lebesgue and Sobolev spaces
------------------------------------
Suppose $\omega$ is a positive, Borel measurable, real function defined on an open set $\Omega\subset \rn$. Let $$\label{om'}
\omega'=\omega^{-1/(p-1)}.$$
We say that $\omega$ satisfies the $B_p$–condition on $\Omega$ ($\omega\in B_p(\Omega)$), if $$\label{Bp}
\omega'\in L^1_{{ loc}}(\Omega).$$
Note that any $\omega\in L^1_{loc}(\Omega)$, which is strictly positive inside $\Omega$ satisifes $B_p$ condition on $\Omega$.
When $1<p<\infty$ and $\omega\in B_p$, we have $\displaystyle L^{p}_{\omega,loc}(\Omega)\subseteq L^1_{{ loc}}(\Omega),$ see [@kuf-opic]. Moreover, for any $\omega\in L^1_{loc}(\Omega)$ and $s>p$ we have $$\label{emb:sp}
L^s_{\omega,loc}(\Omega)\subset L^p_{\omega,loc}(\Omega).$$
If $\nabla$ denotes distributional gradient , we denote $$\label{polnorma}
W^{1,p}_{(\omega_1,\omega_2)}(\Omega):= \left\{ f\in L^{p}_{\omega_1}(\Omega) : \nabla f\in (L_{\omega_2}^p (\Omega ))^N\right\}$$ with the norm $$\begin{gathered}
\| f\|_{W^{1,p}_{(\omega_1,\omega_2)}(\O)}\ :\,=\ \| f\|_{L^{p}_{\omega_1}(\O)} + \| \nabla f\|_{(L_{\omega_2}^p (\Omega ))^N}\\
=\left(\int_{\Omega}|f|^{p}{\omega_1(x)}dx \right)^\frac{1}{p} + \left(\int_{\Omega}\sum_{i=1}^N\left|\frac{\partial f}{\partial x_i} \right|^{p}{\omega_2(x)}dx \right)^\frac{1}{p}.\end{gathered}$$
\[factemb\] If $p>1$, $\Omega\subset \rn$ is an open set, $\omega_1,\omega_2$ satisfy $B_p$–condition , then
- $W^{1,p}_{(\omega_1,\omega_2)}(\Omega)$ defined by equipped with the norm $\| \cdot\|_{W_{(\omega_1,\omega_2)}^{1,p}(\Omega)}$ is a Banach space;
- $\displaystyle \overline{Lip_0(\Omega)} =\overline{C^\infty_0(\Omega)} =W^{1,p}_{(\omega_1,\omega_2),0}(\O),$ where the closure is in the norm $\| \cdot\|_{W_{(\omega_1,\omega_2)}^{1,p}(\Omega)}$;
- if $\omega_1,\omega_2$ are a pair in the Hardy-Poincaré inequality of the form , we may consider the Sobolev space $W^{1,p}_{(\omega_1,\omega_2),0}(\O)$ equipped with the norm $$\| f\|_{W^{1,p}_{(\omega_1,\omega_2), 0}(\O)} = \| \nabla f\|_{L^{p}_{\omega_2}(\O)}.$$
Operator $\dL$, given by , is hemicontinuous, i.e. for all $u,v,w\in W^{1,p}_{(\omega_1,\omega_2),0}(\O)$ the mapping $\lambda\mapsto\ll \dL(u+\lambda v),w\rr$ is continuous from $\r$ to $\r$.
We look for solutions in the space $L^p(0,T;W^{1,p}_{(\omega_1,\omega_2)}(\O))$, i.e. $$L^p(0,T;W^{1,p}_{(\omega_1,\omega_2)}(\O))=\left\{f\in L^p(0,T;L^{p}_{\omega_1}(\O)):\nabla f\in (L^p(0,T;L^{p}_{\omega_2}(\O)))^N\right\},$$ where $\nabla$ denotes distributional gradient with respect to the spacial variables, equipped with the norm $$\| f\|_{L^p(0,T;W^{1,p}_{(\omega_1,\omega_2)}(\O))}:= \left(\int_{0}^T\| f\|^p_{L^{p}_{\omega_1}(\O)}dt\right)^\frac{1}{p} + \left(\int_{0}^{T}\| \nabla f\|^p_{(L_{\omega_2}^p (\Omega ))^N}dt\right)^\frac{1}{p}.$$
### Dual spaces {#dual-spaces .unnumbered}
Let us stress that $$(L^p_{\omega }(\Omega))^*\neq L^{p'}_{\omega }(\Omega) ,\qquad \mathrm{but}\qquad (L^p_{\omega }(\Omega))^*= L^{p'}_{\omega '}(\Omega)$$ with $\omega'$ given by .
By $W^{-1,p'}_{(\omega_1',\omega_2')}(\O)$ we denote the dual space to $W^{1,p}_{(\omega_1,\omega_2),0}(\O)$ and the duality pairing is given by the standard scalar product. We note that $ L^{p'}(0,T;W^{-1,p'}_{(\omega_1',\omega_2')}(\O))$ is the dual space to $L^p(0,T;W^{1,p}_{(\omega_1,\omega_2),0}(\O))$.
Comments on admissible weights {#ssec:comwei}
------------------------------
We give here the reasons for which we assume the conditions (W1)-(W6).
- It is general assumption on the spaces: $L^{p}_{\omega_1}(\Omega)$ and $W^{1,p}_{(\omega_1,\omega_2)}(\Omega)$.
- To ensure that the weighted Sobolev space $W^{1,p}_{(\omega_1,\omega_2)}(\Omega)$ is a Banach space, we need to assume $\omega_1\in B_p(\Omega)$, cf. . It is necessary to assume a stronger condition $\omega_1^{-\frac{2}{p-2}}\in L_{loc}^1(\Omega)$, to obtain the embbedding $$L^p_{\omega_1,loc}(\Omega)\subset L^{p'}_{\omega_1',loc}(\Omega)$$and $\omega_1^{-\frac{2}{p-2}}\in L ^1(\Omega)$, to obtain $$W^{1,p}_{(\omega_1,\omega_2),0}(\Omega)\subset L^2(\Omega).$$
- It guarantees strict monotonicity of the operator.
Moreover, it implies that $\omega_2\in B_p(\Omega)$, cf. , which is necessary to ensure that $W^{1,p}_{(\omega_1,\omega_2)}(\Omega)$ is a Banach space.
- It is counterpart of the Poincaré inequality in $W^{1,p}_{(\omega_1,\omega_2),0}(\O)$.
We shall stress that there are multiple methods of deriving weights admissible in the Hardy inequalities having the form . In particular, the results of the first author [[@plap Theorem 4.1]]{} show that the weights may be generated by nonnegative solutions to the elliptic problem and the regularity conditions imposed on the weights are in fact expected regularity properties of the solutions.
- It is necessary for the compactness method of Boccardo and Murat [@boccardo].
To obtain (W5) the result by Franchi, Serapioni and Serra Cassano [@fra-ser-ser Theorem 3.4] can be applied. If one is equipped with another continuous embedding of the weighted Sobolev space into the weighted Lebesgue space, they may apply the results by Opic and Kufner [@OpKuf Sections 17 and 18] to obtain compact embedding on domains similar to John domains. For other ideas on compact embeddings in weighted Sobolev spaces we refer to [@anh Proposition 2.1] by Anh and Ke.
- Those are technical assumptions. Note that $\omega_1^{-\frac{q}{s-q}}\in L^1_{loc}(\Omega)$ may follow from (W2). It depends on the possible values of exponents $s$ and $q$.
As mentioned before the integrability assumption on $\omega_1^{-\frac{q}{s-q}}\in L^1_{loc}(\Omega)$ implies .
If $\Omega$ is bounded, $p\geq 2$ and $\omega_1,\omega_2$ satisfy (W1)-(W4), then$$W^{1,p}_{(\omega_1,\omega_2)}(\Omega)\subset
L^{p'}_{\omega_1'}(\Omega)=( L^p_{\omega_1}(\Omega))^*
\subset (W^{1,p}_{(\omega_1,\omega_2),0}(\Omega))^*=W^{-1,p'}_{(\omega_1',\omega_2')}(\Omega).$$ and $$L^p(0,T;
W^{1,p}_{(\omega_1,\omega_2)}(\Omega)) \subset L^p(0,T;L^{p'}_{\omega_1'}(\Omega))\subset L^{p'}(0,T;W^{-1,p'}_{(\omega_1',\omega_2')}(\Omega)).$$
If $\Omega$ is bounded, $2< p<s $ and $\omega_1,\omega_2$ satisfy (W1)-(W5), then $$\label{emb:chain2} W^{1,p}_{(\omega_1,\omega_2)}(\Omega)\subset \subset
L^s_{\omega_1}(\Omega)\subset L^{p'}_{\omega_1'}(\Omega)
\subset W^{-1,p'}_{(\omega_1',\omega_2')}(\Omega).$$Furthermore, $$W^{1,p}_{(\omega_1,\omega_2),0}(\Omega)\subset\subset L^2(\Omega)$$ and $$\label{eq:L2L2emb}
L^p(0,T;W^{1,p}_{(\omega_1,\omega_2),0}(\O))\subset L^2(0,T;L^2(\Omega))= L^2(\Omega_T).$$ Moreover, if additionally we have (W6), then $$\label{embLq}
L^s_{\omega_1,loc}(\Omega)\subset L^{q}_{loc} (\Omega)\quad \text{for $q\in(p,s)$}.$$
Auxiliary tools
---------------
For the sake of completeness we recall the general analytic tools necessary in our approach.
\[theo:VitConv\] Let $(X,\mu)$ be a positive measure space. If $\mu(X)<\infty $, $\{f_{n}\}$ is uniformly integrable, $f_{n}(x)\to f(x)$ a.e. and $|f(x)|<\infty $ a.e. in $X$, then $f\in {L}^1_\mu(X)$ and $f_{n}(x)\to f(x)$ in ${L}^1_\mu(X)$.
For the Aubin–Lions Lemmas we refer e.g. to [@simon].
\[AubinLionsLemmarefl\] Suppose $1<p<\infty$, $X,B,Y$ are the Banach spaces, $X\subset\subset B\subset Y$, $F$ is bounded in $L^p(0,T;X)$ and relatively compact in $L^p(0,T;Y)$ then $F$ is relatively compact in $L^p(0,T;B)$.
\[AubinLionsLemma\] Suppose $1\leq p<\infty$, $X,B,Y$ are the Banach spaces, $X\subset\subset B\subset Y$. If $F$ is bounded in $L^p(0,T;X)$ and $\frac{dF}{dt}$ is bounded in $L^r(0,T;Y)$, where $r>1$, then $F$ is relatively compact in $L^p(0,T;B)$.
For the Brezis Lieb Lemma we refer to [@brezlieb].
\[BrezisLiebLemma\] Suppose $\Omega\subset\rn$, $1\leq p<\infty$, and $\mu\geq 0$ is a Radon measure. If $f_n\to f$ a.e. in $\Omega$ and $(f_n)_n$ is bounded in ${L^p_\mu(\Omega)}$, then the following limit exists $$\lim_{n\to\infty}\left(\|f_n\|_{L^p_\mu(\Omega)}^p-\|f-f_n\|_{L^p_\mu(\Omega)}^p\right)= \|f \|_{L^p_\mu(\Omega)}^p$$ and the equality holds.
We have the following corollary of the above theorem.
\[coro:BrezisLiebLemma\] Suppose $\Omega\subset\rn$, $1\leq p<\infty$, and $\omega_1:\Omega\to\r\cup\{0\}$ is measurable. If $u_m\to u$ strongly in $L^p(0,T; L^p_{\omega_1}(\Omega))$, then $$\omega_1 |u_m|^{p-2}u_m\to\omega_1 |u|^{p-2}u\quad \mathrm{strongly\ in\ }L^{p'}(0,T; L^{p'}_{\omega_1'}(\Omega)).$$
If $u_m\to u$ strongly in $L^p(0,T; L^p_{\omega_1}(\Omega))$ and a.e. in $\Omega$, then Theorem \[BrezisLiebLemma\] yields that $$\int_{\Omega_T}\omega_1 |u_m|^p\,dx\,dt\to\int_{\Omega_T}\omega_1 |u |^p\,dx\,dt.$$ Equivalently, $$\int_{\Omega_T}\omega_1 \left||u_m|^{p-2}u_m\right|^\frac{p}{p-1}\,dx\,dt\to\int_{\Omega_T}\omega_1 \left||u |^{p-2}u \right|^\frac{p}{p-1}\,dx\,dt,$$ which, once again by Theorem \[BrezisLiebLemma\], implies $$|u_m|^{p-2}u_m\to |u|^{p-2}u\quad \mathrm{strongly\ in\ }L^{p'}(0,T; L^{p'}_{\omega_1 }(\Omega)).$$ When we observe that $$\begin{split}&\int_{\Omega_T}\omega_1 \left(|u_m|^{p-1}-|u|^{p-1} \right)^\frac{p}{p-1}\,dx\,dt\\&=\int_{\Omega_T}\omega_1' \left(\omega_1 |u_m |^{p-1}-\omega_1 |u |^{p-1} \right)^\frac{p}{p-1}\,dx\,dt,\end{split}$$ we conclude that $$\omega_1 |u_m|^{p-2}u_m\to\omega_1 |u|^{p-2}u\quad \mathrm{strongly\ in\ }L^{p'}(0,T; L^{p'}_{\omega_1'}(\Omega)).$$
The nonlinear eigenvalue problem {#sec:nonleig}
================================
The optimal constant in the Hardy–type inequality provides a spectral information for weighted problems. For the nonlinear eigenvalue problem $$-\mathrm{\dv}(|\nabla u|^{p-2} u \omega_2)=\lambda {|u|^{p-2}u}\omega_1,$$ we have the following variational characterisation of the first eigenvalue by the Rayleigh quotient $$\lambda_1=\inf\left\{\frac{\int_\Omega \omega_2|\nabla \phi |^{p}dx}{\int_\Omega \omega_1 |\phi |^{p} dx}:\phi\in W^{1,p}_{(\omega_1,\omega_2),0}(\Omega)\right\},$$ considered e.g. in [@AdChaRa; @anh; @dol-tos; @gaap; @vz].
Via the method of [@gaap], we obtain the following results.
Suppose $1<p<\infty$, $\Omega\subseteq\rn$. Assume further that $\omega_1,\omega_2:\Omega\to\R_+$ satisfy conditions (W1)-(W4) and $\lambda_{N,p}$ is the optimal left–hand side constant in the Hardy inequality .
Consider $\lambda_1 (m)$ — the first eigenvalue to the problem $$\left\{
\begin{array}{ll}
-\dv(\omega_2|\nabla \psi |^{p-2}\nabla \psi )=\lambda {W}_m |\psi |^{p-2}\psi & x\in \Omega\subset\rN,\\
\psi(x)=0 & x\in\partial\Omega,
\end{array}
\right.$$ where ${W}_m (x) = T_m(\omega_1(x))$ and $T_m$ is given by . Then $\lambda_1(m)\geqslant\lambda_{N,p}$ and moreover $\lim_{m\to\infty}\lambda_1(m)=\lambda_{N,p}$.
We define the first eigenvalues by the following Rayleigh quotients $$\begin{split}\lambda_1(m)=\inf\left\{\frac{\int_\Omega \omega_2|\nabla \phi |^{p}dx}{\int_\Omega {W}_m |\phi |^{p} dx}:\phi\in W^{1,p}_{(\omega_1,\omega_2),0}(\O)\right\},\\ \lambda_{N,p}=\inf\left\{\frac{\int_\Omega \omega_2|\nabla \phi |^{p}dx}{\int_\Omega \omega_1 |\phi |^{p} dx}:\phi\in W^{1,p}_{(\omega_1,\omega_2),0}(\O)\right\}.\end{split}$$
In particular, according to for each $\phi\in W^{1,p}_{(\omega_1,\omega_2),0}(\O)$ we have $$\lambda_{N,p}=\inf \frac{\int_\Omega \omega_2|\nabla \phi |^{p}dx}{\int_\Omega \omega_1 |\phi |^{p} dx}\leq \frac{\int_\Omega \omega_2|\nabla \phi |^{p}dx}{\int_\Omega {W}_m |\phi |^{p} dx}.$$ Then $\lambda_{N,p}\leq \lambda_1(m)$, $(\lambda_1(m))_{m\in\n}$ is a nonincreasing sequence, and $\lim_{m\to\infty}\lambda_1(m)$ exists.
We prove that $\lim_{m\to\infty}\lambda_1(m)=\lambda_{N,p}$ by contradiction. Suppose $\lim_{m\to\infty}\lambda_1(m)=\lambda_{N,p}+2{\varepsilon}$ with a certain ${\varepsilon}>0$. Let us take $\phi_0\in W^{1,p}_{(\omega_1,\omega_2),0}(\O)$ such that $$\frac{\int_\Omega \omega_2|\nabla \phi_0 |^{p}dx}{\int_\Omega \omega_1 |\phi_0 |^{p} dx}<\lambda_{N,p}+{\varepsilon}.$$ On the other hand, due to the Lebesgue Monotone Convergence Theorem we notice that $$\frac{\int_\Omega \omega_2|\nabla \phi_0 |^{p}dx}{\int_\Omega {W}_m |\phi_0 |^{p} dx}\xrightarrow[{m\to\infty}]{}\frac{\int_\Omega \omega_2|\nabla \phi_0 |^{p}dx}{\int_\Omega \omega_1 |\phi_0 |^{p} dx},$$ thus there exists $m_0$, such that $$\lambda_{N,p}\leq\frac{\int_\Omega \omega_2|\nabla \phi_0 |^{p}dx}{\int_\Omega {W}_{m_0} |\phi_0 |^{p} dx}\leq \lambda_{N,p} + \frac{3}{2}{\varepsilon}.$$ and $$\lambda_{N,p}+2{\varepsilon}\leq \lambda_1(m_0)\leq \lambda_{N,p}+\frac{3}{2}{\varepsilon}.$$
Suppose $1<p<\infty$, $\Omega\subseteq\rn$. Assume further that $\omega_1,\omega_2:\Omega\to\R_+$ satisfy conditions (W1)-(W4), $\lambda_{N,p}$ is the optimal left–hand side constant in the Hardy inequality and the nonlinear operator ${\cal L}_\lambda$ in $W^{1,p}_{(\omega_1,\omega_2),0}(\O)$ is given by$${\cal L}_\lambda u =
-\dv(\omega_2|\nabla u|^{p-2}\nabla u)- \lambda\omega_1(x)|u|^{p-2}u.$$ For $\lambda\leq\lambda_{N,p}$, we have positivity of the operator. Moreover, for sufficiently big $\lambda$, the operator is unbounded from below.
The result for small $\lambda$s results from the Hardy inequality . If $\lambda$ is bigger than the optimal constant in the already mentioned inequality, we reach the goal as an easy consequence of a density argument and the existence of $\phi\in C_0^\infty (\Omega)$, such that $\langle {\cal L}_\lambda \phi,\phi\rangle <0$. We can assume that $||\phi||_p=1$. We define $u_\mu(x)=\mu^\frac{N}{p}\phi(\mu x)$ and we have $||u_\mu||_p =1$. Due to the homogeneity of the operator we conclude that $\langle {\cal L}_\lambda u_\mu,u_\mu\rangle =\mu^p
\langle {\cal L}_\lambda \phi,\phi\rangle<0.$
Existence {#sec:main}
=========
This section is divided into two subsections. The first one concerns necessary compactness properties, while the second one provides the proof of the main result.
Compactness results
-------------------
Before we start the proof of the main theorem we need to adjust [@boccardo Lemma 4.2] in the following way.
\[theo:boc-mur-appl\] Suppose $p>2$ and $\omega_1,\omega_2$ satisfy (W1)-(W5). Assume further that $$\label{eq:lem:boc-mur}
(u_m)_t=h_m\qquad in\ {\cal D}'(\Omega),$$ where $h_m$ — bounded in $L^{p'}(0,T; W^{-1,p'}_{(\omega_1,\omega_2)}(U))$ and $u_m\xrightharpoonup[m\to\infty]{} u$ in $L^{p}(0,T; W^{1,p}_{(\omega_1,\omega_2),0}(\Omega)).$
Then
- $u_m\xrightarrow[m\to\infty]{} u$ strongly in $L^{p}(0,T; L^{s}_{\omega_1}(U));$
- $u_m\xrightarrow[m\to\infty]{} u$ a.e. in $\Omega_T$ (up to a subsequence).
Let us consider a function $\phi(x,t)=\psi(x)\eta(t)$, where $\psi\in {\cal D}(\Omega)$ and $\eta\in {\cal D}(0,T)$, and set $v_m=\phi u_m$. For any bounded open subset $U$, such that $\mathrm{supp}\phi\subset U\subset\Omega, $ we have $$(v_m)_t=(\phi u_m)_t=\phi(u_m)_t+\phi_tu_m=\phi h_m+\phi_tu_m.$$
Then $v_m$ is bounded in $L^{p}(0,T; W^{1,p}_{(\omega_1,\omega_2)}(U))$ and, due to , $(v_m)_t$ is bounded in $L^{p'}(0,T; W^{-1,p'}_{(\omega_1',\omega_2')}(\Omega)).$ We are going to apply the Aubin–Lions Lemma (Theorem \[AubinLionsLemma\]). Let us note that if $p>2$, then (W5) and gives $$W^{1,p}_{(\omega_1,\omega_2),0}(U)\subset\subset L^s_{\omega_1}(U) \subset W^{-1,p'}_{(\omega_1',\omega_2')}(\Omega).$$ Therefore $v_m$ is relatively compact in $L^p(0,T;L^s_{\omega_1}(U)).$
Moreover, since we know , strong convergence in Lebesgue’s space implies convergence almost everywhere.
For the convenience of the reader, we provide the following extension of [@bocmurpuel Lemma 5] with the proof.
\[thm:bmpuel\] Let $U$ be a bounded open subset in $\rn$, $U_T:=U\times(0,T)$, $2<p<\infty$ and $\omega_1,\omega_2$ satisfy (W1)-(W4). Assume that $\nu_m\rightharpoonup \nu$ weakly in $L^{p}(0,T; W^{1,p}_{(\omega_1,\omega_2),0}(U))$ and a.e. in $U_T$, and $$\int_{U_T} \omega_2 \left[ |\nabla \nu_m|^{p-2}\nabla \nu_m- |\nabla \nu |^{p-2}\nabla \nu \right]\nabla(\nu_m-\nu)\,dx\,dt\to 0.\label{eq:limzero}$$ Then $\nabla\nu_m\to \nabla\nu$ strongly in $L^{p}(0,T; (L^{p}_{\omega_2}(U))^N)$, when $m\to\infty$.
We adapt the proof of [@bocmurpuel Lemma 5] to the weighted setting. Let $D_m$ be defined by $$D_m(x)= \left[ |\nabla \nu_m|^{p-2}\nabla \nu_m- |\nabla \nu |^{p-2}\nabla \nu \right]\nabla(\nu_m-\nu).$$ By the monotonicity of $\Delta_p^{\omega_2}$ we note that $\omega_2(x)D_m\geq 0$. Since , observe that $D_m\to 0$ in $L^1(0,T;L^1_{\omega_2}(U))$ strongly. Thus, up to a subsequence $D_m\to 0$ a.e. in $U_T$. Recall $U_T$ is bounded. Suppose $X\subset U$ is a maximal set of full Lebesgue’s measure (and therefore of full $\omega_2$-measure), where for each $x\in X$ we have $$|\nu(x)|<\infty,\quad |\nabla \nu(x)|<\infty,\quad \nu_m(x)\to \nu(x),\quad D_m(x)\to 0.$$ Clearly $\omega_2|\nabla \nu_m|^p\geq 0$ and $0\leq D_m(x)$. Moreover, $$\begin{gathered}
D_m(x)=|\nabla \nu_m|^p+|\nabla \nu|^p-|\nabla \nu_m|^{p-2}\nabla \nu_m \nabla \nu-|\nabla \nu |^p \nabla \nu \nabla \nu_m\geq \\
\geq |\nabla \nu_m|^p -c(x)\left(|\nabla \nu_m|^{p-1}+|\nabla \nu_m|\right),
\end{gathered}$$ with $c(x)$ dependent on $X$, but not on $m$. As $D_m(x)\to 0$, we infer that $|\nabla \nu_m|$ is uniformly bounded on $X$.
Let us take arbitrary $x_0\in X$ and denote $$\zeta_m=\nabla \nu_m(x_0),\qquad \zeta =\nabla \nu (x_0).$$ Observe that $\omega_2(x_0)>0$ and $(\zeta_m)$ is a bounded sequence. Set $\zeta_*$ as one of its cluster points. Recall $D_m(x_0)\to 0$ and note that $$D_m(x_0)\to (|\zeta_*|^{p-2}\zeta_*-|\zeta |^{p-2}\zeta )(\zeta_*-\zeta).$$ Thus, $\zeta=\zeta_*$ is a unique cluster point of whole the sequence and $\nabla \nu_m(x_0)\to\nabla \nu (x_0)$ for arbitrary $x_0\in X$. Then $$\omega_2|\nabla \nu_m|^p\to \omega_2|\nabla \nu |^p\qquad \mathrm{in}\quad X.$$ It implies uniform integrability of the sequence $ |\nabla u_m|^{p } $ in $L^1_{\omega_2}(X)$, which implies uniform integrability in $L^1_{\omega_2}(U)$.
Therefore, Vitali’s Convergence Theorem (Theorem \[theo:VitConv\]) yields that $$\int_U \omega_2 \left( |\nabla \nu_m|^{p } - |\nabla \nu |^{p } \right)dx\to 0\quad \text{for}\quad m\to\infty$$and the claim follows.
Next we use the following modification of [@boccardo Theorem 4.1].
\[theo:boc-mur\]Assume $p>2$, $\omega_1,\omega_2$ satisfy (W1)-(W6). Suppose $$\label{eq:prob:boc-mur}
(u_m)_t-\dL (u_m)=g_m\qquad in\ {\cal D}'(\Omega),$$ moreover $g_m\xrightarrow[m\to\infty]{} g$ in $L^{p'}(0,T; W^{-1,p'}_{(\omega_1',\omega_2')}(\Omega))$ and $u_m\xrightharpoonup[m\to\infty]{} u$ in $L^{p}(0,T; W^{1,p}_{(\omega_1,\omega_2),0}(\Omega)).$
Then, for any fixed $k>0$, we have a strong convergence the gradients $$\nabla T_k(u_m)\xrightarrow[m\to\infty]{} \nabla T_k(u)\qquad in\quad L^{p}\left(0,T; (L^{p}_{\omega_2}(U))^N\right).$$
We define $S_k(s)=\int_0^s T_k(r)\,dr$, where $T_k$ is given by . Then for any $\phi\in{\cal D}(\Omega_T)$ and any $\zeta\in L^{p}(0,T;W^{1,p}_{(\omega_1,\omega_2)}(\Omega))$ such that $\zeta_t\in L^{p'}(0,T;W^{-1,p'}_{(\omega_1,\omega_2)}(\Omega))$ we have $$\int_{\Omega_T} \zeta_t\phi T_k(\zeta)dxdt=-\int_{\Omega_T} \phi_t\,S_k(\zeta)dxdt.$$
We fix compact sets $K\subset \Omega_T$ and $U\subset \Omega$, such that $K\subset(0,T)\times U \subset \Omega_T$. We take an arbitrary function $\phi_K\in{\cal D}(\Omega_T)$ with $\mathrm{supp}\,\phi_K\subset K\subset\subset \Omega_T,$ such that $0\leq \phi_K\leq 1$ with $\phi_K=1$ on $K$. Then we test by $$w_m=(T_k(u_m)-T_k(u))\phi_K$$ getting $$\begin{gathered}
0= \int_{\Omega_T} (u_m)_t\phi_K \left[T_k(u_m)-T_k(u)\right]dxdt\\+\int_{\Omega_T} \phi_K \omega_2 |\nabla u_m|^{p-2}\nabla u_m [\nabla T_k(u_m)-\nabla T_k(u)]dxdt\\
+\int_{\Omega_T}\omega_2 |\nabla u_m|^{p-2}\nabla u_m[T_k(u_m)-T_k(u)]\nabla\phi_K dxdt\\-\int_{\Omega_T} g_m[T_k(u_m)-T_k(u)]\phi_K dxdt\\
=J_m^1+J_m^2+J_m^3+J_m^4.\end{gathered}$$
We deal with $J_m^1$ and $J_m^4$ in the similar way. We note that either sequence $((u_m)_t)_m$ or $(g_m)_m$ are bounded sequences in $ L^{p'}(0,T; W^{-1,p'}_{(\omega_1',\omega_2')}(\Omega))$. Therefore, Theorem \[theo:boc-mur-appl\] implies that up to a subsequence $T_k(u_m)-T_k(u)\xrightarrow[m\to\infty]{} 0$ strongly in $L^{p}(0,T; L^{p}_{\omega_1,loc}(\Omega))$, as we have (W5) and . Then $J^1_m,J_m^4\to 0$ as $m\to \infty$. As for $J_m^3$, we apply the Hölder inequality, to get $$\begin{split} J_m^3&= \int_{\Omega_T}\omega_2 |\nabla u_m|^{p-2}\nabla u_m[T_k(u_m)-T_k(u)]\nabla\phi_K dxdt \\
&= \int_{U_T}\omega_2 |\nabla u_m|^{p-2}\nabla u_m[T_k(u_m)-T_k(u)]\nabla\phi_K dxdt \\
&\leq const\left[ \int_0^T\left(\int_{U}[T_k(u_m)-T_k(u)]^q dx \right)^\frac{p}{q}dt\right]^\frac{1}{p} \cdot\\&\qquad\qquad\qquad\cdot\left[ \int_0^T \int_{U}\omega_2 |\nabla u_m|^{p} dx\,dt\right]^\frac{p-1}{p}\left( \int_{U}\omega_2^{\frac{q}{q-p}}dx\right)^\frac{q-p}{qp},
\end{split}$$ where $c_H>0$, $U\subset\subset\Omega$ such that $\mathrm{supp}\phi_K\subset(0,T)\times U$, and $q$ comes from (W6). Then $J_m^3$ tends to zero. Indeed,
- by Theorem \[theo:boc-mur-appl\] we obtain $T_k(u_m)-T_k(u)\to 0$ strongly in $L^p(0,T;L^s_{\omega_1}(U))$. Notice that (W6) ensures that there exists $q$ such that $$L^p(0,T;L^s_{\omega_1}(U))\subset L^p(0,T;L^q(U)).$$
- weak convergence of $(u_m)$ in $L^{p}(0,T;W^{1,p}_{(\omega_1,\omega_2),0}(\Omega))$ implies its uniform boundedness in this space (up to a subsequence), thus $\int_{U_T}\omega_2 |\nabla u_m|^{p} dxdt<C$, with a constant $C$ independent of $m$;
- $\int_{U }\omega_2^\frac{q}{q-p}dx<\infty$ due to (W6).
As $J_m^1+J_m^2+J_m^3+J_m^4=0$ and $\lim_{m\to\infty}(
J_m^1+J_m^3+J_m^4)=0$, then also $\lim_{m\to\infty}J_m^2=0$, i.e. $$\label{eq:dlaEm}
\int_{\Omega_T} \phi_K \omega_2 |\nabla u_m|^{p-2}\nabla u_m [\nabla T_k(u_m)-\nabla T_k(u)]dxdt\xrightarrow[m\to\infty]{}0.$$
Let us observe that if $m\to\infty$, then $E_m$, given by$$E_m=\int_{\Omega_T} \phi_K \omega_2\left[|\nabla T_k(u_m)|^{p-2}\nabla T_k(u_m)-|\nabla T_k(u)|^{p-2}\nabla T_k(u)\right] [\nabla T_k(u_m)-\nabla T_k(u)]dxdt,$$ tends to $0$. Indeed, $$\begin{split}\label{Emsplit}E_m=\int_{K} \phi_K\omega_2 |\nabla u_m |^{p-2}\nabla u_m [\nabla T_k(u_m)-\nabla T_k(u)]dxdt+\\-\int_{\Omega_T} \phi_K\omega_2 |\nabla u_m |^{p-2}\nabla u_m [\nabla T_k(u_m)-\nabla T_k(u)]\chi_{\{u_m>k\}}dxdt+\\
-\int_{\Omega_T} \phi_K \omega_2 |\nabla T_k(u)|^{p-2}\nabla T_k(u) [\nabla T_k(u_m)-\nabla T_k(u)]dxdt,\end{split}$$ where the first term converges to zero because of . Since $\nabla T_k(u_m) \chi_{\{u_m>k\}}=0$, the second term reads $$\int_{\Omega_T} \phi_K\omega_2 |\nabla u_m |^{p-2}\nabla u_m \nabla T_k(u) \chi_{\{u_m>k\}}dxdt,$$ where $|\nabla u_m |$ is bounded in $L^{p }(0,T;L^{p }_{\omega_2}(\Omega))^N) $ and for $m\to\infty$ we have $\nabla T_k(u) \chi_{\{u_m>k\}}\to \nabla T_k(u)\chi_{\{u >k\}}$ strongly in $L^{p}(0,T;L^{p}_{\omega_2}(U))^N).$ Then the Monotone Convergence Theorem and fact that $u_m$ is nondecreasing give the point. The third term in converges to zero, because $$T_k(u_m)-T_k(u)\xrightharpoonup[m\to\infty]{} 0\quad \mathrm{weakly\ in\ }L^{p}(0,T; W^{1,p}_{(\omega_1,\omega_2)}(\Omega)).$$
We have proven that for $m\to\infty$ we have $E_m\to 0.$ Recall weak convergence $u_m\xrightharpoonup[m\to\infty]{} u$ in $L^{p}(0,T; W^{1,p}_{(\omega_1,\omega_2),0}(\Omega))$ and a.e. in $\Omega_T$. Therefore, Theorem \[thm:bmpuel\] for $\nu=T_k(u)$ and $\nu_m=T_k(u_m)$ yields $$\nabla u_m\xrightarrow[m\to\infty]{} \nabla u \quad \mathrm{strongly\ in\ }L^{p}(0,T; (L^{p}_{\omega_2}(U))^N).$$
Proof of the main result
------------------------
In order to construct a weak solution to we first consider a truncated problem $$(u_m)_t-\dL u_m = \lambda T_m(\omega_1) |u_m|^{p-2}u_m ,$$ where $T_m$ is given by . Existence of the solution to the truncated problem is a consequence of the following result.
\[thm:extrun\] Let $p> 2$, $\Omega\subseteq\rn$ be an open subset, $f\in L^2(\Omega)$ and $\omega_1,\omega_2$ satisfy (W1)-(W5).
There exists $\lambda_0=\lambda_0(p,N,\omega_1,\omega_2)$, such that for all $\lambda\in(0,\lambda_0)$ the parabolic problem $$\left\{\begin{array}{ll}
u_t-\dL u= \lambda W(x)|u|^{p-2}u & x\in\Omega,\\
u(x,0)=f(x)& x\in\Omega,\\
u(x,t)=0& x\in\partial\Omega,\ t>0,\\
\end{array}\right.$$ where $W:\Omega\to\R_{+}$ is such that $$W(x)\leq \min\{m,\omega_1(x)\}$$ with a certain $m\in\R_+$, has a global weak solution $u \in L^p(0,T; W_{(\omega_1,\omega_2),0}^{1,p}(\Omega)),$ such that $
u_t \in L^{p'}(0,T; {W^{-1,p'}_{(\omega_1',\omega_2')}(\Omega)}),$ i.e. $$\int_{\Omega_T}\left( u_t\xi+\omega_2|\nabla u|^{p-2} \nabla u \nabla \xi +\lambda W(x) |u|^{p-2}u \xi\right)dx\,dt=0,$$ holds for each $\xi\in L^p(0,T; W_{(\omega_1,\omega_2),0}^{1,p}(\Omega))$. Moreover, $u\in L^\infty(0,T; L^2(\Omega_T))$.
In our previous paper [@isazg1] another embedding result was used, namely [@anh Proposition 2.1]. The theorem holds true as well, when we assume (W5) instead of that one.
Let us present the proof of the main result.
We consider $u_m$ — the solution to the truncated problem $$\label{eq:mtrunc}\left\{\begin{array}{ll}
w_t-\dL w= \lambda T_m(\omega_1) |w|^{p-2}w & x\in\Omega\\
w(x,0)=f(x)& x\in\Omega\\
w(x,t)=0& x\in\partial\Omega,\ t>0,\\
\end{array}\right.$$ where $T_m$ is given by . Due to Theorem \[thm:extrun\] there exists a solution $u_m$ to the problem such that $$u_m \in L^p(0,T; W^{1,p}_{(\omega_1,\omega_2),0}(\O))\cap L^\infty(0,T; L^2(\Omega )),\quad (u_m)_t \in L^{p'}( 0,T; W^{-1,p'}_{(\omega_1,\omega_2)}(\Omega )).$$
We are going to let $m\to\infty$. To obtain a priori estimate we test the problem by $u_m$ getting $$\begin{gathered}
\frac{1}{2}\frac{d}{dt}\|u_m\|^2_{L^2(\Omega)}+\int_{\Omega } \omega_2|\nabla u_m|^pdx=\lambda\int_{\Omega } T_m(\omega_1)|u_m|^p\,dx\leq\\\leq \lambda\int_{\Omega } \omega_1 |u_m|^p\,dx
\leq \frac{\lambda}{K}\int_{\Omega } \omega_2|\nabla u_m|^pdx,\end{gathered}$$ where the last inequality is allowed due to the Hardy inequality . Note that the density of Lipschitz and compactly supported functions in $W^{1,p}_{(\omega_1,\omega_2),0}(\O))$ is given by Fact \[factemb\]. Therefore, $$\frac{1}{2}\frac{d}{dt}\|u_m\|^2_{L^2(\Omega)}+\left(1-\frac{\lambda}{K}\right)\int_{\Omega } \omega_2|\nabla u_m|^pdx\leq 0.$$
Note that $$\int_0^T\frac{d}{dt}\|u_m\|^2_{L^2(\Omega)}dt =\|u_m (\cdot,T)\|_{L^2(\Omega)}^2- \|f\|_{L^2(\Omega)}^2.$$
Summing up, we obtain $$\begin{gathered}
\frac{1}{2}\| u_m (\cdot,T)\|_{L^2(\Omega)}^2+ \left(1-\frac{\lambda}{K}\right)\int_0^T\|\nabla u_m (\cdot,t)\|^p_{L_{\omega_2}^p(\Omega )}dt \leq \frac{1}{2}\|f \|_{L^2(\Omega)}^2. \end{gathered}$$ In particular, this implies
- $(u_m)_{m\in\n}$ is bounded in $ L^\infty(0,T; L^2(\Omega ))$;
- $(u_m)_{m\in\n}$ is bounded in $ L^p( 0,T; W_{(\omega_1,\omega_2),0}^{1,p}(\Omega)).$
Thus, there exists a function $u \in L^p(0,T; W_{(\omega_1,\omega_2),0}^{1,p}(\Omega ))\cap L^\infty(0,T; L^2(\Omega ))$ with $u_t \in L^{p'}( 0,T; W_{(\omega_1,\omega_2)}^{-1,p'}(\Omega))$, such that and up to a subsequence, we have $$\begin{aligned}
u_m\xrightharpoonup[m\to\infty]{\ \ *\ \ } u& \mathrm{in}& L^\infty(0,T; L^2(\Omega )),\label{eq:ulimit}\\
u_m\xrightharpoonup[m\to\infty]{\ \ \ \ \ } u& \mathrm{in}& L^p( 0,T; W_{(\omega_1,\omega_2),0}^{1,p}(\Omega )).\nonumber\end{aligned}$$
We know that for each $\xi\in L^p(0,T; W_{(\omega_1,\omega_2),0}^{1,p}(\Omega))$ the following equality holds $$\label{eq:weakum}
\int_{\Omega_T}\left( (u_m)_t\xi+\omega_2|\nabla u_m|^{p-2} \nabla u_m \nabla \xi +\lambda T_m(\omega_1) |u_m|^{p-2}u_m \xi\right)dx\,dt=0.$$ We have to show that the limit function $u$ from is the weak solution to , i.e. $$\int_{\Omega_T}\left( u_t\xi+\omega_2|\nabla u|^{p-2} \nabla u \nabla \xi +\lambda \omega_1 |u|^{p-2}u \xi\right)dx\,dt=0$$ holds for each $\xi\in L^p(0,T; W_{(\omega_1,\omega_2),0}^{1,p}(\Omega))$. Let us note that the integral above is well–defined on this class, in particular $ L^p(0,T; W_{(\omega_1,\omega_2),0}^{1,p}(\Omega))\subset L^2(\Omega_T)$. The weak convergence of gradients is not enough to pass to the limit with $\int_{Q_T}\omega_2|\nabla u_m|^{p-2} \nabla u_m \nabla \xi$. Thus, the first step is to get strong convergence of gradients. We follow the spirit of Boccardo and Murat to obtain a strong convergence of the gradients of trucations and apply it in splitted into $$\begin{gathered}
0=\int_{\Omega_T} (u_m)_t\xi dx\,dt+\int_{\Omega_T\cap \{|u_m|\leq k\}} \omega_2|\nabla u_m|^{p-2} \nabla u_m \nabla \xi dx\,dt+\\
+\int_{\Omega_T\cap \{|u_m|> k\}} \omega_2|\nabla u_m|^{p-2} \nabla u_m \nabla \xi dx\,dt+\int_{\Omega_T} \lambda T_m(\omega_1) |u_m|^{p-2}u_m \xi dx\,dt\\=A_1^m+A_2^m+A_3^m+A_4^m.\end{gathered}$$
The convergence of $A_1^m$ to $\int_{\Omega_T} u_t\xi\,dxdt$ can be obtained by integrating by parts and by the Lebesgue’s Monotone Convergence Theorem since $(u_m)_m$ is a nondecreasing sequence.
In the case of $A_2^m$ we first remark that over this set we can write $$A_2^m=\int_{\Omega_T} \omega_2|\nabla T_k(u_m)|^{p-2} \nabla T_k(u_m) \nabla \xi dx\,dt.$$ Now we engage the method of Boccardo and Murat via Theorem \[theo:boc-mur\] to get $$\nabla T_k(u_m)\xrightarrow[m\to\infty]{} \nabla T_k(u)\quad \mathrm{in}\quad L^{p}\left(0,T; (L^{p}_{\omega_2}(U))^N\right).$$ The assumptions of Theorem \[theo:boc-mur\] are satisfied, because besides the weak convergence of functions, we have $$\label{gmconv}g_m=\lambda\omega_1 |u_m|^{p-2}u_m\xrightarrow[m\to\infty]{} \lambda\omega_1 |u|^{p-2}u=g$$ in $L^{p'}(0,T; W^{-1,p'}_{(\omega_1',\omega_2')}(\Omega))$. To justify this we apply the Aubin–Lions Lemma (Theorem \[AubinLionsLemmarefl\]). Since we assume (W5) and we know , we have $$W^{1,p}_{(\omega_1,\omega_2),0}(U)\subset\subset L^{p}_{ \omega_1}(U)\subset W^{-1,p'}_{(\omega_1',\omega_2')}(\Omega).$$ Then we infer that $u_m\to u$ strongly in $L^p(0,T;L^p_{\omega_1}(U))$. Strongly convergent sequence has a subsequence convergent almost everywhere. If it is necessary, we pass to such subsequence, but we do not change the notation. Note that $$\begin{gathered}
\|g_m\|^{p'}_{ L^{p'}( 0,T; L_{ \omega_1' }^{ p'}(\Omega ))}=\lambda\int_{\Omega_T}\omega_1'\left|\omega_1|u_m|^{p-1}\right|^\frac{p}{p-1}dxdt=\\
=\lambda\int_{\Omega_T}\omega_1^{-\frac{1}{p-1}} \omega_1^{ \frac{p}{p-1}}|u_m|^{p } dxdt=\lambda\int_{\Omega_T}\omega_1 |u_m|^{p } dxdt<\infty \end{gathered}$$ and thus$$g_m\in L^{p'}( 0,T; L_{ \omega_1' }^{ p'}(U))\subset L^{p'}( 0,T; W_{(\omega_1',\omega_2')}^{-1,p'}(\Omega )).$$ According to the Brezis–Lieb Lemma (Corollary \[coro:BrezisLiebLemma\]) the strong convergence of $u_m\to u$ in $L^p(0,T;L^p_{\omega_1}(U))$ implies the strong convergence $$\omega_1 |u_m|^{p-2}u_m\to \omega_1|u|^{p-2}u_m\quad\mathrm{ in}\ L^{p'}(0,T;L^{p'}_{\omega_1'}(U)),$$ which entails strong convergence $g_m \to g$ in $L^{p'}(0,T;L^{p'}(0,T;L^{p'}_{\omega_1'}(U))$ and in turn also . This finishes the case of $A_2^m$.
We easily show that the Hölder inequality implies that $A_3^m\leq s(k),$ with a certain constant $s$ depending on $k$. Indeed, $$\begin{gathered}
\int_{\Omega_T\cap \{|u_m|> k\}} \omega_2|\nabla u_m|^{p-2} \nabla u_m \nabla \xi dx\,dt\leq \\\leq\left(\int_{\Omega_T\cap \{|u_m|> k\}} \omega_2|\nabla u_m|^{p}dx\,dt\right)^\frac{p-1}{p}\left(\int_{\Omega_T\cap \{|u_m|> k\}} \omega_2| \nabla \xi|^p dx\,dt\right)^\frac{1}{p}\\
\leq const \left(\int_{\Omega_T\cap \{|u|> k\}} \omega_2| \nabla \xi|^p dx\,dt\right)^\frac{1}{p}=s(k).\end{gathered}$$ Note that the integral on the right–hand side above is finite even for $k=0$ and that the sequence $(u_m)$ is nondecreasing (and thus $\{|u_m|> k\}\subset\{|u |> k\}$).
It suffices to show that $A_4^m- \int_{\Omega_T} \lambda \omega_1 |u |^{p-2}u \xi dx\,dt\to 0$, when $m\to\infty$. We show that both expressions below tend to zero $$\begin{gathered}
\int_{\Omega_T} T_m(\omega_1) |u_m|^{p-2}u_m \xi dx\,dt-\int_{\Omega_T} \omega_1 |u |^{p-2}u \xi dx\,dt=\\
=\int_{\Omega_T} ( T_m(\omega_1) -\omega_1)|u_m|^{p-2}u_m \xi dx\,dt
+ \int_{\Omega_T} (|u_m|^{p-2}u_m-|u|^{p-2}u)\omega_1 \xi dx\,dt=\\=B_1^m+B_2^m.\end{gathered}$$
To deal with $B_1^m$ we recall that $(| u_m|^{p-2} u_m)_m$ is bounded in $ L^{p'}( 0,T; W_{(\omega_1',\omega_2')}^{-1,p'}(\Omega ))$ (cf. the case of $A_2^m$), while $T_m(\omega_1)\nearrow\omega_1$, so the Lebesgue Monotone Convergence Theorem implies $B_1^m\to 0$ as $m\to\infty$.
Let us concentrate on $B_2^m$. We have $$\begin{gathered}
|B_2^m|\leq \left(\int_{\Omega_T} \left||u_m|^{p-2}u_m-|u|^{p-2}u\right|^\frac{p}{p-1}\omega_1 dxdt\right)^\frac{p-1}{p} \left(\int_{\Omega_T} \omega_1 |\xi|^p dx\,dt\right)^\frac{1}{p}.\end{gathered}$$ We employ the Brezis–Lieb Lemma (Corollary \[coro:BrezisLiebLemma\]) to get $$\omega_1 |u_m|^{p-2}u_m\to \omega_1|u|^{p-2}u_m\quad\mathrm{ in}\ L^{p'}(0,T;L^{p'} (U)),$$ leading to $$|u_m|^{p-2}u_m\to |u|^{p-2}u_m\quad\mathrm{ in}\ L^{p'}(0,T;L^{p'}_{\omega_1}(U)).$$ which implies that $B_2^m\to 0$ as $m\to\infty$.
We have shown that for every $k\in\n$ $$\int_{\Omega_T}\left( u_t\xi+\omega_2|\nabla u|^{p-2} \nabla u \nabla \xi +\lambda \omega_1 |u|^{p-2}u \xi\right)dx\,dt=s(k).$$ Since $s(k)\to 0$ when $k\to\infty$, we conclude that $u$ is the desired solution.
[^1]: email address: iskrzypczak@mimuw.edu.pl
[^2]: email address: azator@mimuw.edu.pl\
The research of A.Z.-G. has been supported by the Foundation for Polish Science grant no. POMOST BIS/2012-6/3 and by the NCN grant no. 2012/05/E/ST1/03232 (years 2013 - 2017).
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abstract: 'Graph embedding techniques, which learn low-dimensional representations of a graph, are achieving state-of-the-art performance in many graph mining tasks. Most existing embedding algorithms assign a single vector to each node, implicitly assuming that a single representation is enough to capture all characteristics of the node. However, across many domains, it is common to observe pervasively overlapping community structure, where most nodes belong to multiple communities, playing different roles depending on the contexts. Here, we propose [`persona2vec`]{}, a graph embedding framework that efficiently learns multiple representations of nodes based on their structural contexts. Using link prediction-based evaluation, we show that our framework is significantly faster than the existing state-of-the-art model while achieving better performance.'
author:
- |
Jisung Yoon[^1]\
Department of Industrial and Management Engineering\
Pohang University of Science and Technology\
Pohang, Gyeongbuk, Korea 37673\
`jisung.yoon@postech.ac.kr`\
Kai-Cheng Yang\
Luddy School of Informatics, Computing, and Engineering\
Indiana University\
Bloomington, Indiana, USA 47408\
`yangkc@iu.edu`\
Woo-Sung Jung\
Department of Industrial and Management Engineering\
Pohang University of Science and Technology\
Pohang, Gyeongbuk, Korea 37673\
`wsjung@postech.ac.kr`\
Yong-Yeol Ahn\
Luddy School of Informatics, Computing, and Engineering\
Indiana University\
Bloomington, Indiana, USA 47408\
`yyahn@iu.edu`\
bibliography:
- 'ref.bib'
title: 'Persona2vec: A Flexible Multi-role Representations Learning Framework for Graphs'
---
Introduction
============
Graph embedding maps the nodes in a graph to continuous and dense vectors that capture relations among the nodes [@Perozzi2014deepwalk; @grover2016node2vec; @Tang2015line]. Resulting node representations allow direct applications of algebraic operations and common algorithms, facilitating graph mining tasks such as node classification [@sen2008collective; @Perozzi2014deepwalk], community detection [@fortunato2010community; @yang2016modularity], link prediction [@grover2016node2vec] and visualization [@Tang2015line]. Most methods map each node to a single vector, implicitly assuming that a single representation is sufficient to capture the full characteristics of a node.
However, nodes often play multiple roles. For instance, people have multiple roles, or “personas”, across contexts (e.g. professor, employee, and so on) [@ahn2010link; @coscia2014uncovering; @leskovec2009community; @leskovec2010empirical]. Similarly, proteins and other biological elements play multiple functionalities [@palla2005uncovering; @gavin2006proteome; @ahn2010link]. Another example is the polysemy of words when their relations are modeled with graphs; many words possess multiple meanings differentiated by the contexts [@chen2014unified; @li2015multi; @iacobacci2015sensembed]. Explicit modeling of such multiplicity and overlapping clusters has been fruitful not only for community detection [@rosvall2014memory; @coscia2014uncovering; @epasto2017ego], but also for improving the quality of embedding [@li2015multi; @epasto2019single]. Yet, with the scarcity of embedding methods embracing this idea, the full potential of this approach has not been properly explored.
In this paper, we propose [`persona2vec`]{}, a scalable framework that builds on the idea of ego-splitting [@epasto2017ego], the process of identifying local structural contexts of a node via performing local community detection on the node’s ego-network. For each detected local community (role), we transform each node into multiple personas if there are multiple local communities to which the node belongs. After the split, the original node is replaced by the new persona nodes that inherit the connection from each local community, producing a new persona graph. Instead of separating a node’s persona nodes from each other completely, we add directed, weighted edges between personas to capture their origin. In doing so, we allow the direct application of the existing graph embedding methods. In addition, we take an approach of considering persona-based learning as fine-tuning of the base graph embedding, achieving both efficiency and balance between information from the original graph and the persona graph. Compared with the previous approach [@epasto2019single], our framework is conceptually simpler to understand and practically easier to implement. Furthermore, it achieves state-of-the-art performance in the link prediction tasks while being much faster. Our implementation of [`persona2vec`]{} is publicly available at <https://github.com/jisungyoon/persona2vec>.
Proposed method: [`persona2vec`]{}
==================================
[`persona2vec`]{} creates a *persona graph*, where some nodes are split into multiple personas. We then apply a graph embedding algorithm to the persona graph to learn the embeddings of the personas (see Fig. \[fig:ego\_split\]). Let us explain the method formally. Let $G = (V, E)$ be a graph with a set of nodes $V$ and a set of edges $E$. $|V|$ and $|E|$ denote the number of nodes and edges respectively. Let $f\colon v \rightarrow \mathbb{R}^d$ be the embedding function that maps a node $v$ to a $d$-dimensional vector space ($d \ll |V|$).
Refined Ego-splitting
---------------------
{width="80.00000%"}
We adopt and refine the ego-splitting method [@epasto2017ego; @epasto2019single]. For each node in the original graph, we first extract its ego graph, remove the ego, and identify the local clusters. Every cluster in the ego graph leads to a new persona node in the persona graph (see Fig. \[fig:ego\_split\]a, c). For example, if we consider each connected component as a local community with a connected component algorithm, node $C$ in the original graph belongs to two non-overlapping clusters $\{A,B\}$ and $\{D,E,F\}$ in its ego-network. Given these two clusters, in the persona graph, $C$ is split into $C_1$ and $C_2$ to represent the two roles in respective clusters. $C_1$ and $C_2$ inherit the connections of $C$ from both clusters separately (see Fig. \[fig:ego\_split\]c). On the other hand, node $A$ only belongs to one ego cluster $\{B,C\}$, so it does not split into multiple personas.
Original graph $G(V,E)$; weight parameter $\lambda$; non-overlapping local clustering algorithm $\mathcal{C}$ Persona graph $G_P(V_P, E_P)$; node to personas mapping $V2P$; persona to local cluster mapping $P2C$
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$P_{v_o} \leftarrow \mathcal{C}(v_o)$ Create $v_p$ Add $v_p$ to $G_P$,$V2P(v_o)$ $P2C(v_p) \leftarrow p$ $w \leftarrow $ weight of edge Add original edges $(v_p, v'_p, w), (v'_p, v_p, w)$ to $E_P$ $k^o \leftarrow \text{out-degree sequence after adding original edges}$ Add persona edges $(v_i, v_j, k^o_i \times \lambda), (v_j, v_i, k^o_j \times \lambda)$ to $E_P$
Any graph clustering algorithm can be employed for splitting a node into personas. The simplest algorithm is considering each connected component in the ego-network (sans the ego) as a cluster. This approach is fast and works well on sparse graphs. However, in dense graphs, ego-networks are more likely to form fewer connected component, thus other algorithms such as the Louvain method [@blondel2008fast], Infomap [@rosvall2008maps], and label propagation [@raghavan2007near] would be more appropriate.
In previous studies, the personas get disconnected without retaining the information about their origin, creating isolated components in the splitting process [@epasto2017ego; @epasto2019single]. Because of this disconnectedness, common embedding methods could not be directly applied to the ego-split graph. A previous study attempted to address this issue by imposing a regularization term in the cost function to penalize separation of persona nodes originating from the same node [@epasto2019single].
Here, instead of adopting the regularization strategy, we add weighted *persona edges* between the personas, maintaining the connectedness between them after the splitting (see Fig. \[fig:ego\_split\]c). Because the persona graph stays connected, classical graph algorithms and graph embedding methods can now be readily applied without any modification. As we will show later, our strategy achieves both better scalability and better performance.
In persona graph, we set the weights of the unweighted original edges as $1$ and tune the strength of the connections among personas with $\lambda$. Persona edges are directed and weighted, with weight $\lambda k^{\text{o}}_i$, where $k^{\text{o}}_i$ is the out-degree of the persona node after splitting (see Fig. \[fig:ego\_split\]c). Assigning weight proportional to $k^{\text{o}}_i$ helps the random walker explores both the local neighbors and other parts of the graph connected to the other personas regardless of its out-degree $k^{\text{o}}_i$.
Imagine node $u$, which is split into $n_p$ personas. Consider one of the personas $i$ with out-degree $k^{\text{o}}_i$ and persona edges with weight $w_i$. Then the probability $p_i$ that an unbiased random walker at $i$ visits neighbors connected with the original edge at the next step is $ \frac{k^{\text{o}}_i}{k^{\text{o}}_i + n_pw_i}$. If we set constant weight $w_i=\lambda$, then $p_i = \frac{k^{\text{o}}_i}{k^{\text{o}}_i + n_p\lambda} = \frac{1}{1 + \frac{n_p}{k^{\text{o}}_i}\lambda}$, which depends on $k^{\text{o}}_i$. A random-walker would not explore its local neighborhood if $n_p \gg k^{\text{o}}_i$, while the opposite happens when $n_p \ll k^{\text{o}}_i$. Instead, assigning the weight proportional to $k^{\text{o}}_i$, namely $w_i=\lambda k^{\text{o}}_i$, removes such bias because $p_i=\frac{k^{\text{o}}_i}{k^{\text{o}}_i + n_p\lambda k^{\text{o}}_i} = \frac{1}{1 + n_p\lambda}$, which is independent of $k^{\text{o}}_i$. Our experiments also show that using the out-degree yields better performance than assigning the identical weight to each persona edge. Our algorithm for refined ego-splitting is described in Algorithm \[alg:ego\_splitting\]. Note that it can be generalized to the directed graphs.
Persona graph embedding
-----------------------
As explained above, any graph embedding algorithm that recognizes edge direction and weight can be readily applied to the persona graph. Although we use `node2vec` as the embedding method here, other embedding methods can also be employed. We initialize the persona vectors with the vectors from the original graph before ego-splitting (see Fig. \[fig:ego\_split\]b) to leverage the information from the original graph structure. Persona nodes that belong to the same node in the original graph are thus initialized with the same vector. We then execute the embedding algorithm for a small number of epochs to fine-tune the embedding vectors with the information from the persona graph (see Fig. \[fig:ego\_split\]). Experiments show that usually only one epoch of training is enough. Our full algorithm is described in Algorithm \[alg:persona2vec\].
Original graph $G(V,E)$; embedding dimension $d$; number of walks per node for base embedding $\mathcal{\gamma}_b$; random walk length for base embedding $t_b$; window size for base embedding $w_b$; number of walks per node for persona embedding $\mathcal{\gamma}_p$; random walk length for persona embedding $t_p$; window size for persona embedding $w_p$; learning rate $\alpha$; refined ego-splitting method <span style="font-variant:small-caps;">RefEgoSplit</span>; node to personas mapping $V2P$; a graph embedding method (e.g. `DeepWalk`, `Node2vec`) <span style="font-variant:small-caps;">EmbeddingFunc</span> $\Phi_{G_P}$, A $N_P \times d$ matrix with $d$-dimensional vector representations for all $N_P$ persona nodes
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$\Phi_{G_P}(v_p) = \Phi_{G}(v_o) $
Complexity
----------
The persona graph is usually larger than the original graph, but not too large. Node $u$ with degree $k_u$ may be split into at most $k_u$ personas. In the worst case, the number of nodes in the persona graph can reach $O(|E|)$. But, in practice, only a subset of nodes split into personas, and the number of personas rarely reaches the upper bound. If we look at the persona edges, for a node $u$ with degree $k_u$, at most $O(k_u^2)$ new persona edges may be added. Thus, the whole persona graph has at most $O(|V| \times k_{\text{max}}^2)$ or $O(|V|^3$) ($\because k_{\text{max}} \le |V|$) extra persona edges. If graph’s degree distribution follows a power-law distribution $P(k) \sim k^{-\gamma} $, then $k_{\text{max}} \sim |V|^{1 / \gamma - 1}$. Hence, it could be $O(|V|^{\gamma + 1/\gamma - 1})$ and it is between $O(|V|^2)$ and $O(|V|^3)$ ($\because 2 \le \gamma \le 3$ in general). However, real graph tends to be sparse and $k_i\ll|V|$. If we further assume $k_i < \sqrt{|E|}$ holds for every node, then $\sum^{|V|}_{n=1} k_n^2 \leq \sum^{|V|}_{n=1} k_n \sqrt{|E|} = 2|E|\sqrt{|E|}$. Under this assumption, the upper bound becomes $O(|E|^{3/2})$. Similarly, with the scale-free condition, the upper bound could be $O(|E||V|^{1 / \gamma - 1})$, which is between $O(|E||V|^{1/2})$ and $O(|E||V|)$. Again, in practice, the number of persona edges is much smaller than this upper bound. To illustrate, we list the number of nodes and persona edges in the persona graph for the graphs we use in this paper in Table \[tab:statistics\]. All considered, the extra nodes and edges do not bring too much space complexity burden in practice.
Assessing the time complexity requires consideration of the two steps: ego-splitting and embedding. The ego-splitting algorithm has complexity of $O(|E|^{3/2} + \sqrt{|E|} T(|E|))$ in the worst case, where $|E|$ is the number of edges in the original graph and $T(|E|)$ is the complexity of detecting the ego clusters in the graph with $|E|$ edges [@epasto2017ego]. The embedding on the persona graph, which dominates the whole embedding procedure, has complexity $O(|V_p|\gamma t w d(1 + \log(|V_p|)))$ where $|V_p|$ is the number of nodes, $\gamma$ is the number of random walkers, $d$ is the embedding dimension, and $w$ is the window size [@chen2018harp].
[r]{}[0.3]{} {width="30.00000%"}
The final complexity is $O(|E|^{3/2} + \sqrt{|E|} T(|E|)) + O(|V|\gamma t w d(1 + \log(|V|)))$. Removing the constant factors and assuming close-to-linear local community detection algorithm, the whole process has time complexity of $O(|E|^{3/2})$ with space complexity of $O(|E|^{3/2}$) if $k_i < \sqrt{|E|}$ holds. Complexity can be increased depending on the clustering algorithms on the ego-network.
To test the validity of our assumptions, we sample 1,000 graphs from a public network repository [@nr]. We apply the refined ego-splitting with connected component algorithms on these samples and report the actual number of persona edges $|E_p|$ with respect to the practical upper bound $|E|^{3/2}$ in Fig. \[fig:complexity\], which shown that the actual number of persona edges $|E_p|$ rarely exceeds the tighter upper bound that we proposed and is usually orders of the magnitude smaller.
Dataset Type $|V|$ $|E|$ |$V_p$| |$V_p|/|V|$ |$E_p$| $|E_p|/|E^{3/2}|$
-------------- ------------ -------- --------- --------- ------------- ----------- ------------------- --
PPI Undirected 3,863 38,705 16,734 4.34 132,932 0.0175
ca-HepTh Undirected 9,877 25,998 16,071 1.86 33,524 0.0800
ca-AstroPh Undirected 17,903 197,301 25,706 1.44 29,012 0.0003
wiki-vote Directed 7,066 103,633 21,476 3.04 118,020 0.0035
soc-epinions Directed 75,877 508,836 220,332 2.90 3,550,594 0.0098
: **Descriptive statistics in the graphs used in the evaluation.** We report the number of nodes $|V|$, number of edges $|E|$, number of nodes in the persona graph $|V_p|$, the ratio of $|V_p|$ over $|V|$, number of persona edges $|E_p|$ added in ego-splitting, and the ratio of $|E_p|$ over $|E^{3/2}|$ which is the upper bound of space complexity.
\[tab:statistics\]
Case Study
==========
![ **Illustrative examples.** (a) The Zachary’s Karate club network with the force-atlas layout [@zachary1977information]. Nodes are colored by communities detected by the Louvain modularity method [@blondel2008fast]. (b) The persona graph. Nodes are colored by k-means clusters [@macqueen1967some] from the embedding vectors. Coordinates of the persona nodes come from the 2-D projection of the embedding with t-SNE [@van2008visualizing]. Light grey lines represent the persona edges. (c) The word association network, clusters around the word “Newton”. Coordinates of the words come from the 2-D projection of the embedding vectors with UMAP [@mcinnes2018umap]. Word colors correspond to the clusters obtained by k-means clustering [@macqueen1967some] on the embedding vectors. []{data-label="fig:persona2vec_example"}](figures/example_merged_vertical.pdf){width="1\columnwidth"}
Before diving into systematic evaluations, we provide two illustrative examples: Zachary’s Karate club network and a word association network.
#### Case Study: Zachary’s Karate club network
We use the Zachary’s Karate club network [@zachary1977information], a well-known example for the community detection. Nodes represent members of the Karate club, and edges represent ties among the members (see Fig. \[fig:persona2vec\_example\]a). Although it is often considered to have two large disjoint communities, smaller overlapping communities can also be seen, highlighted by nodes such as `1`, `3`, `28`, and `32`. In Fig. \[fig:persona2vec\_example\]b, we present the persona graph of the network. [`persona2vec`]{} successfully recognizes these bridge nodes and place their personas in reasonable places. Take node `1` for example. It splits into four persona nodes, which then end up in two different communities. The orange and green communities are clearly separated as a result.
#### Case Study: word association network
Word association network captures how people associate words together (free association task). The dataset was originally assembled from nearly 750,000 responses [@nelson2004university]. In Fig. \[fig:persona2vec\_example\]c, we shows the [`persona2vec`]{} clusters around the word “Newton”. We use the Louvain method [@blondel2008fast] to split the personas of each word. [`persona2vec`]{} successfully captures multiple contexts of the word “Newton”. For instance, the red persona is associated with “scientists” and “philosopher”, grey one is linked to the physics, and yellow one is associated with “apple” (note that there is a cookie called “(Fig) Newton” in the U.S.). Furthermore, [`persona2vec`]{} also captures different nuances of the word “law” that are related to the crime (brown cluster) and the legal concepts (orange cluster).
Numerical Experiment
====================
Link Prediction Task
--------------------
To systematically evaluate the performance and scalability of the [`persona2vec`]{} framework, we perform a link prediction task using real-world graphs [@grover2016node2vec; @abu2017learning]. Link prediction aims to predict missing edges in a graph with partial information, which is useful for many tasks such as suggesting new friends on social networks or recommending products. It has been employed as a primary task to evaluate the performance of unsupervised graph embedding methods [@abu2017learning; @zhang2018arbitrary].
We follow the task setup from the literature [@grover2016node2vec; @abu2017learning]. First, the edge set of an input graph is divided equally and randomly into $E_{\text{train}}$ and $E_{\text{test}}$. We then refine $E_{\text{test}}$ using a rejection sampling based on the criterion that, even when we remove all edges in $E_{\text{test}}$, the graph should be connected as a single component. $E_{\text{train}}$ is used to train the models, and $E_{\text{test}}$ is used as positive examples for the prediction task. Second, a negative edge set $E_{(-)}$ of non-existent random edges with the same size of $E_{\text{test}}$ are generated as negative examples for testing. The performance of a model is measured by its ability to correctly distinguish $E_{\text{test}}$ and $E_{(-)}$ after being trained on $E_{\text{train}}$. We then report ROC-AUC.
Datsets
-------
To facilitate the comparison with the state-of-the-art baseline, we use five graph datasets that are publicly available and previously used [@epasto2019single]. We summarize them as follows.
*PPI* is a protein-protein interaction graph of *Homo sapiens* [@stark2006ppiorigin]. Nodes represent proteins and edges represent physical interactions between the proteins. *ca-HepTh* is a scientific collaboration graph. It represents the co-authorship among researchers from the Theoretical High Energy Physics field, derived from papers on arXiv. *ca-AstropPh* is also scientific collaboration graph, but from Astrophysics. *wiki-vote* is a voting network, each node is a Wikipedia user and a directed edge from node $i$ to node $j$ represents that user $i$ voted for user $j$ to become an administrator. *soc-epinions* is a voting graph from a general consumer review site `Epinions.com`, each node is a member, and a directed edge from node $i$ to node $j$ means that member $i$ trusted member $j$.
For *PPI*, we use the prepossessed version from the `node2vec` project web page [@grover2016node2vec], while other graphs are downloaded from the SNAP library homepage [@snapnets]. We use the largest component of the undirected graphs and the largest weakly connected component of the directed ones. The statistics of all the graphs are reported in Table \[tab:statistics\].
Methods
-------
The state-of-the-art method in this link prediction task is `SPLITTER`[@epasto2019single], which also models multiple roles. As reported in the paper, it outperforms various exiting algorithms ranging across non-embedding methods like `Jaccard Coefficient`, `Common Neighbors,` and `Adamic-Adar` as well as embedding methods like Laplacian `EigenMaps` [@belkin2002laplacian], `node2vec` [@grover2016node2vec], `DNGR` [@cao2016deep], `Asymmetric` [@abu2017learning] and `M-NMF` [@wang2017community].
Given the state-of-the-art performance of `SPLITTER`, for simplicity, we compare our framework with `SPLITTER` using the identical task setup and datasets. In addition, because our method can be considered as an augmentation of a single-role embedding method, and because we use `Node2vec` as the base embedding method, we also employ `Node2vec`. We run the link prediction task using the original authors’ implementation of `Node2vec` and `SPLITTER`. The parameters are also kept consistent with the original paper.
[`persona2vec`]{} and `SPLITTER` have multiple representations on each node, which leads to non-unique similarity estimations between two nodes. Hence, we define the similarity score of a pair of nodes on [`persona2vec`]{} as the maximum dot-product of embedding vectors between any pair of their personas. We found that, among experiment with three aggregation functions *min*, *max*, *mean*, the highest performance is achieved with *max*, same with `SPLITTER` [@epasto2019single]. For `SPLITTER`, we use maximum cosine similarity, following the author’s note in their implementation.
#### `Node2vec` (baseline method)
For `Node2vec`, we set random walk length $t=40$, the number of walks per node $\gamma=10$, random walk parameters $p = q = 1$, the window size $w=5$, and the initial learning rate $\alpha=0.025$. In the original paper, they learn an additional logistic regression classifier over the Hadamard product of the embedding of two nodes for the link prediction. In general, the logistic regression classifier improves the performance. Here, we report results on `Node2vec` with both dot products and the logistic regression classifier.
#### `SPLITTER` (baseline method)
For `SPLITTER`, we use the same parameters in their paper [@epasto2019single] and `Node2vec` baseline. We use `node2vec` with random walk parameters $p = q = 1$.
#### [`persona2vec`]{} (our proposed method)
We set the hyper-parameters of the original graph embedding with $t_{b}=40$, $\gamma_{b}=10$, $w_{b}=5$. For the persona embedding, we set $t_{p}=80$, $\gamma_{p}=5$, $w_{p}=2$ to better capture the micro-structure of the persona graph. The size of the total trajectories is determined by random walk length $t_{*}$ times number of walks per node $\gamma_{*}$, so we keep $t_{*} \gamma_{*}$ constant to roughly preserve the amount of information used in the embedding. For both embedding stages, we use the $\alpha=0.025$, and `node2vec` with the random walk parameters $(p = q = 1)$ as the graph embedding function.
Experiment Results
------------------
{width="\textwidth"}
Fig. \[fig:performance\_result\] shows the link prediction performance of [`persona2vec`]{} in comparison with the baselines. Overall, [`persona2vec`]{} yields superior performance across graphs and across a range of hyperparameter choice. We show that augmenting `Node2vec` by considering personas significantly improves the link prediction performance, evinced by the significant performance gain (see Table \[tab:performance\_summary\]).
Method PPI ca-HepTh ca-AstroPh wiki-vote soc-epinions
------------------------- ------------------- ----------- ------------ ------------------- -----------------------
`Node2vec` 0.585 0.825 0.901 0.694 0.547 $\pm$ 0.007
`Node2vec`\* 0.662 $\pm$ 0.001 0.848 0.914 0.705 $\pm$ 0.001 0.767 $\pm$ 0.002
`SPLITTER` 0.856 0.903 0.982 0.931 **0.961 $\pm$ 0.001**
`SPLITTER`\* 0.853 0.898 0.984 0.931 0.954 $\pm$ 0.001
**[`persona2vec`]{}\*** **0.879** **0.927** **0.985** **0.936** **0.961**
**Performance gain** 0.294 0.102 0.084 0.242 0.414$\pm$ 0.007
: **Performance of [`persona2vec`]{} with $\lambda=0.5$.** All methods use $d=128$. `Node2vec`\* refers `Node2vec` with the logistic regression classifier, `SPLITTER`\* refers `SPLITTER` with one epoch, and [`persona2vec`]{}\* refers [`persona2vec`]{} with $\lambda=0.5$, our suggested default. Performance gain is performance difference between `Node2vec` and [`persona2vec`]{}\*. We omit the standard error which is smaller than $10^{-3}$.
\[tab:performance\_summary\]
As expected, larger dimensions lead to better performance, although [`persona2vec`]{} achieves reasonable results even with tiny embedding dimensions like 8 or 16. We also show how the performance of [`persona2vec`]{} varies with $\lambda$. For undirected graphs, larger $\lambda$ is beneficial but the trend saturates quickly. For directed graphs, however, optimal performance is achieved with smaller values of $\lambda$. In practice, we suggest starting with $\lambda=0.5$ as a default parameter because the overall variation brought by $\lambda$ is not substantial and even when the performance increases with $\lambda$, near-optimal performance can be achieved at $\lambda = 0.5$. When compared with the `SPLITTER` baseline, [`persona2vec`]{} shows on par or better performances given the same embedding dimensions across a wide range of $\lambda$. We also report the performance summary for [`persona2vec`]{} with $\lambda=0.5$ (our suggested default) compared with the best baselines in Table \[tab:performance\_summary\], which show that [`persona2vec`]{} outperforms the baseline consistently.
[r]{}[0.3]{} {width="0.3\columnwidth"}
In addition to the performance of the link prediction task, we also report the execution time of [`persona2vec`]{} and `SPLITTER` to compare their scalabilities in practice (see Fig. \[fig:speed\_result\]). Note that the reported execution time is on the link-prediction task, with half of the edges removed from the original graph. `SPLITTER` runs the embedding procedures for 10 epochs by default in the original implementation, whereas [`persona2vec`]{} only runs for one epoch. For a fair comparison, we also report the results of `SPLITTER` with one epoch of training. When being limited to only one epoch, `SPLITTER`’s performance slightly suffers on three graphs while it goes up or stays stable for the other two.
Nevertheless, [`persona2vec`]{} is more efficient—39 to 58 times faster than `SPLITTER` with $10$ epochs and five to eight times faster than `SPLITTER` with one epoch, while consistently outperforming both. The main reason behind the drastic difference is the overhead from the extra regularization term in the cost function of `SPLITTER`, which [`persona2vec`]{} gets rid of. In sum, [`persona2vec`]{} outperforms the previous state-of-the-art method both in terms of scalability and link prediction performance.\
\
Related Work
============
In addition to graph embedding, our work is closely related to the research of identifying overlapping communities in graphs. Various non-embedding methods such as link clustering [@ahn2010link; @PhysRevE.80.016105], clique percolation [@palla2005uncovering], and mixed membership stochastic blockmodel [@airoldi2008mixed] have been proposed. Another thread of works focuses on using local graph structure to extract community information [@coscia2014uncovering; @epasto2015ego; @epasto2017ego]. Specifically, Epasto et al. introduce the persona graph method for detecting overlapping communities in graphs [@epasto2017ego], leveraging ego-network partition. The combination of ego-network analysis and graph embedding methods is still rare. An example is `SPLITTER` [@epasto2019single], which we use as the baseline in this paper. Instead of constraining the relations between personas with a regularization term, we propose a simpler and more efficient way of adding persona edges to the graph.
Our work is also related to the word disambiguation problem in word embedding. Recently, word embedding techniques [@mikolov2013efficient; @mikolov2013distributed; @pennington2014glove] have been extensively applied to various NLP tasks as the vectorized word representations can effectively capture syntactic and semantic information. Although some words have multiple senses depending on the context, the original word embedding methods only assign one vector to each word. Li *et al*. shows that embedding that is aware of multiple word senses and provides vectors for each specific sense does improve the performance for some NLP tasks [@li2015multi]. For this issue, some utilize the local context information and clustering for identifying word sense [@reisinger2010multi; @wu2015sense; @neelakantan2015efficient], some resort to external lexical database for disambiguation [@rothe2015autoextend; @iacobacci2015sensembed; @camacho2016nasari; @chen2014unified; @jauhar2015ontologically; @pelevina2017making], while some combine topic modeling methods with embedding [@liu2015topical; @liu2015learning; @cheng2015contextual; @zhang2016improving]. We adopt the idea of assigning multiple vectors to each node in the graph to represent different roles as well as exploiting local graph structure for the purpose.
Conclusions
===========
We present [`persona2vec`]{}, a framework for learning multiple node representations based on the node’s local structural contexts. [`persona2vec`]{} first performs ego-splitting, where nodes with multiple non-overlapping local communities in their ego-networks are replaced with corresponding persona nodes. The persona nodes inherit the edges from the original graph and remain connected by newly added persona edges, forming the persona graph. Initialized by the embedding of the original graph, the embedding algorithm applied to the persona graph yields the final representations. Instead of assigning only one vector to every node with multiple roles, [`persona2vec`]{} learns vectors for each of the personas.
With extensive link prediction evaluations, we demonstrate that [`persona2vec`]{} achieves the state-of-the-art performance while being able to scale better. Moreover, our method is easy to comprehend and implement without losing any flexibility for incorporating other embedding algorithms, presenting great potential for applications. The possible combination with various algorithms provides vast space for further exploration.
Broader Impact {#broader-impact .unnumbered}
==============
The graph (relational) structure is ubiquitous across many complex systems, including physical, social, economic, biological, neural, and information systems, and thus fundamental graph algorithms have far-reaching impacts across many areas of sciences. Graph embedding, in particular, removes the barrier of translating methods to the special graph data structure, opening up a powerful way to transfer existing algorithms to the graphs and relational data. Therefore, graph embedding methods are being actively adopted in many fields and may continue to have strong broader impacts across all areas of sciences that deal with graph structure. Furthermore, given that it is natural to assume overlapping clusters in most real networks, multi-role embedding methods may find numerous applications in physical, biological, and social sciences.
At the same time, the advancement of graph algorithms can adversely impact our society because more and more data about our everyday life is getting captured by smart devices and our online activities, and in turn harnessed by companies and governments. The ability to predict relationships between entities, people, and traits, as well as human behaviors, has been called out as a threat to personal privacy and even our social systems, sometimes referred to as “surveillance capitalism”. Therefore, we believe that it is also crucial to study the risk brought by strong prediction algorithms on social graphs and ways to protect privacy while providing utility.
[^1]: Co-affiliate with Luddy School of Informatics, Computing, and Engineering, Indiana University
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abstract: |
The Mallows model on $S_n$ is a probability distribution on permutations, $q^{d(\pi,e)}/P_n(q)$, where $d(\pi,e)$ is the distance between $\pi$ and the identity element, relative to the Coxeter generators. Equivalently, it is the number of inversions: pairs $(i,j)$ where $1\leq i<j\leq n$, but $\pi_i>\pi_j$. Analyzing the normalization $P_n(q)$, Diaconis and Ram calculated the mean and variance of $d(\pi,e)$ in the Mallows model, which suggests the appropriate $n \to \infty$ limit has $q_n$ scaling as $1-\beta/n$. We calculate the distribution of the empirical measure in this limit, $u(x,y)\, dx\, dy = \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} \delta_{(i,\pi_i)}$. Treating it as a mean-field problem, analogous to the Curie-Weiss model, the self-consistent mean-field equations are $\frac{\partial^2}{\partial x \partial y} \ln u(x,y) = 2 \beta u(x,y)$, which is an integrable PDE, known as the hyperbolic Liouville equation. The explicit solution also gives a new proof of formulas for the blocking measures in the weakly asymmetric exclusion process, and the ground state of the $\mathcal{U}_q(\mathfrak{sl}_2)$-symmetric XXZ ferromagnet.
[**Keywords:**]{} Mallows model, random permutation, Liouville equation, ASEP, XXZ model. .2 cm [**MCS numbers: 82B05, 82B10, 60B15**]{}
address: |
Department of Mathematics\
Hylan Building\
University of Rochester\
Rochester, NY 14627
author:
- Shannon Starr
date: 'March 2, 2009'
title: 'Thermodynamic Limit for the Mallows Model on $S_n$'
---
Introduction and Main Results
=============================
The Coxeter generators of the symmetric group $S_n$ are the transpositions $(1,2)$, $(2,3)$, …, $(n-1,n)$. The height of a permutation is defined distance to the identity element $e$, $$d(\pi,e)\, =\, \min\{k \geq 0\, :\, \exists\, \tau_1,\dots,\tau_k \in \{(1,2),\dots,(n-1,n)\}\ \text{ such that }\ \pi = \tau_1 \cdots \tau_k\}\, .$$ More generally, $d(\pi_1,\pi_2) = d(\pi_2^{-1} \pi_1,e)$. It is easy to see that $d(\pi,e) = d(\pi^{-1},e)$. In fact, another formula is $$d(\pi,e)\, =\, |\{(i,j) \in {\mathbb{Z}}^2\, :\, 1\leq i<j\leq n\ \text{ and }\ \pi_i>\pi_j\}|\, .$$ The pairs $(i,j)$ are called inversions of $\pi$. In [@DiaconisRam], Diaconis and Ram studied the Mallows measure, which is a probability measure on $S_n$ given by $$\mathbbm{P}_{n}^{q}(\pi)\, =\, \frac{q^{d(\pi,e)}}{P_n(q)}\, ,$$ with $P_n(q)$ being a normalization constant. Actually, Diaconis and Ram studied a Markov chain on $S_n$ for which the Mallows model gives the limiting distribution. This was followed up by another paper on a related topic by Benjamini, Berger, Hoffman and Mossel (BBHM) [@BBHM] who related the biased shuffle and the Mallows model to the asymmetric exclusion process and the “blocking” measures (of Liggett, see [@Liggett], Chapter VIII, especially Example 2.8 and the end of Section 3). They did this using Wilson’s height functions [@Wilson]. We will discuss this more in Section \[sec:Applications\]. For now, let it suffice that Diaconis and Ram identified the explicit formula for the normalization which they remarked is the “Poincaré polynomial”: $$P_n(q)\, =\, \prod_{i=1}^{n} \left(\frac{q^{i}-1}{q-1}\right)\, =\, [n]_q!\, =\, [n]_q \cdots [1]_q\, ,\quad \text{where}\quad
[n]_q\, =\, \frac{q^n-1}{q-1}\, .$$ Further references for the statistical applications[^1] of the Mallows model can be found in their paper.
Note that, physically speaking, one would define a Hamiltonian energy function $H_n : S_n \to {\mathbb{R}}$ as $$H_n(\pi)\, =\, \frac{1}{n-1}\, \sum_{1\leq i<j\leq n} \mathbbm{1}_{(0,\infty)}(\pi_i-\pi_j)\, .$$ In this case, one thinks of $\pi = (\pi_1,\dots,\pi_n)$ as some type of constrained spin system, where each of the components $\pi_1,\dots,\pi_n$ are spins in $\{1,\dots,n\}$ as in a Potts model. The choice of the normalization of the Hamiltonian is then standard for mean-field models. One would be most interested in the free energy $$f_n(\beta)\, =\, -\frac{1}{\beta n}\, \ln \sum_{\pi \in S_n} e^{-\beta H_n(\pi)}\, .$$ For our purposes, we prefer to consider the mathematically simpler “pressure” $$p_n(\beta)\, =\, \frac{1}{n}\, \ln \left(\frac{1}{n!}\, \sum_{\pi \in S_n} e^{-\beta H_n(\pi)}\right)\, .$$ (Note that, contrary to the usual conventions of statistical physics, we have divided the partition function, which is $Z_n(\beta) = \sum_{\pi in S_n} e^{-\beta H_n(\pi)}$, by the infinite-temperature partition function $Z_n(0)=n!$, which is equivalent to starting with a normalized [*a priori*]{} measure rather than counting measure on $S_n$.) It is trivial to see that this is given precisely by the Poincaré polynomial described by Diaconis and Ram: $$p_n(\beta)\, =\, \frac{1}{n} \ln\frac{P_n(e^{-\beta/(n-1)})}{n!}\, =\, \frac{1}{n} \ln \frac{[n]_{e^{-\beta/(n-1)}}!}{n!}\, .$$ With this scaling, it is also easy to calculate the limit: $$p(\beta)\, =\, \lim_{n \to \infty} p_n(\beta)\, =\, \int_0^1 \ln\left(\frac{1-e^{-\beta x}}{\beta x}\right)\, dx\, ,$$ (which can be solved explicitly using the polylogarithm function). From this one can calculate the mean and variance in the limiting Gibbs measure. For instance, one can calculate ${\mathbb{E}}(d(\pi,e) - n(n-1)/4)^2 \sim n^3/72$ in the uniform measure on $S_n$.
To go beyond the statistics of $d(\pi,e)$ it seems worthwhile to study the empirical measure of $\pi$: $$\frac{1}{n} \sum_{i=1}^{n} \delta_{(i,\pi_i)}\, ,$$ which is a normalized measure on $\{1,\dots,n\} \times \{1,\dots,n\}$. More specifically, this is a random measure. Rescaling the discrete cube $\{1,\dots,n\} \times \{1,\dots,n\}$ to $[0,1] \times [0,1]$, it is easy to see that the random empirical measure converges, in probability, to the non-random Lebesgue measure, when $\beta=0$ (the uniform case). Our main theorem generalizes this result.
\[thm:main\] For any $\beta \in {\mathbb{R}}$, $$\lim_{\epsilon \downarrow 0} \lim_{n \to \infty}
\mathbbm{P}_n^{1-\beta/n}\left\{\left|\frac{1}{n} \sum_{i=1}^{n} f(i/n,\pi_i/n) - \int_{[0,1] \times [0,1]} f(x,y) u(x,y)\, dx\, dy\right| > \epsilon\right\}\, =\, 0\, ,$$ for every continuous function $f : [0,1] \times [0,1] \to {\mathbb{R}}$, where $$u(x,y)\, =\, \frac{(\beta/2) \sinh(\beta/2)}{\big(e^{\beta/4} \cosh(\beta[x-y]/2) - e^{-\beta/4} \cosh(\beta[x+y-1]/2)\big)^2}\, .$$
Note that (one can show) the limit $\beta \to 0$ gives $1$. The proof of Theorem \[thm:main\] uses a rigorous version of mean-field theory, as in the solution of the Curie-Weiss model. An interesting feature is that the self-consistent mean-field equation leads us to the characterization of $u$ as the solution of an integrable PDE $$\frac{\partial^2}{\partial x \partial y} \ln u(x,y)\, =\, 2 \beta u(x,y)\, .$$ It is not unusual for mean-field problems to lead to integrable PDE’s. We demonstrate this briefly in the next section with the ubiquitous toy model, the Curie-Weiss ferromagnet.
Toy Model: The Curie-Weiss Ferromagnet
======================================
We include this section merely to point out that mean-field problems often do lead to integrable PDE. However the issue is serious: in fact there is a recent paper by Genovese and Barra which we recommend for more details [@GenoveseBarra]. Our approach merely summarizes their results (in our own words) as well as the earlier paper by Barra, himself [@Barra]. The configuration space of the CW model is $\Omega_N = \{+1,-1\}^N = \{\sigma = (\sigma_1,\dots,\sigma_n)\, :\, {\sigma}_1,\dots,{\sigma}_n = \pm 1\}$. For technical reasons, we choose the Hamiltonian as $$H_N({\sigma},t,x)\, =\, - \frac{t}{2N}\, \sum_{i,j=1}^{N} {\sigma}_i {\sigma}_j - x \sum_{i=1}^{N} {\sigma}_i\, ,$$ we assume $t\geq 0$ and $x \in {\mathbb{R}}$. Defining $m_N({\sigma}) = N^{-1} \sum_{i=1}^{N} {\sigma}_i$, which takes values in $[-1,1]$, we see that $$H_N\, =\, - N \left(\frac{t m_N^2}{2} + x m_N\right)\, .$$ Therefore, defining $$p_N(t,x)\, =\, \frac{1}{N} \ln \sum_{{\sigma}\in \Omega_N} e^{-H_N({\sigma},t,x)}\, ,$$ we easily see that $$\frac{\partial}{\partial t} p_N(t,x)\, =\, \frac{1}{2} \langle m_N^2 \rangle_{N,t,x}\, ,$$ and $$\frac{\partial^2}{\partial x^2} p_N(t,x)\, =\, N \left( \langle m_N^2 \rangle - \langle m_N\rangle^2\right)\, ,$$ where $$\langle f \rangle\, =\, \langle f\rangle_{N,t,x}\, =\, \frac{\sum_{{\sigma}\in \Omega_N} f(\sigma) e^{-H_N({\sigma},t,x)}}{\sum_{{\sigma}\in \Omega_N} e^{-H_N({\sigma},t,x)}}\, .$$ Actually it is easier to consider the “order parameter,” $$u_N(t,x)\, =\, \langle m_N \rangle_{N,t,x}\, =\, \frac{\partial}{\partial x} p_N(t,x)\, ,$$ from which $p_N(t,x)$ can be calculated by solving the ODE: $$\begin{cases}
\frac{\partial}{\partial x} p_N(t,x)\, =\, u_N(t,x) & \text{ for $x \in {\mathbb{R}}$,}\\
p_N(t,x) - |x| \to \frac{1}{2} t^2 & \text{ as $x \to \pm \infty$.}
\end{cases}$$ Then we see that $u_N(t,x)$ satisfies the viscous Burgers equation (with velocity equal to the negative amplitude): $$\begin{cases}
\frac{\partial}{\partial t}\, u_N(t,x)\, =\, u_N(t,x)\, \frac{\partial}{\partial x}\, u_N(t,x) + \frac{1}{2N} \cdot \frac{\partial^2}{\partial x^2} u_N(t,x)
& \text{ for $t>0$ and $x \in {\mathbb{R}}$,}\\
u_N(0,x)\, =\, \tanh(x)
& \text{ for $x \in {\mathbb{R}}$.}
\end{cases}$$ This is an integrable PDE, using the Cole-Hopf transform. See, for instance, Chapter 4 of Whitham, [@Whitham]. Actually, this leads to a solution in terms of Gaussian integrals. The analogous transform in spin-configuration notation is the Hubbard-Stratonovich transform: $$e^{N t m^2/2}\, =\, \int_{-\infty}^{\infty} \frac{e^{Nt (mx - x^2/2)}}{\sqrt{2\pi/t}}\, dx\, ,$$ which “linearizes” the dependence of the Hamiltonian on $m_N$, in the exponential. This trick is used to solve the Curie-Weiss model. See, for example, Thompson [@Thompson].
Note that in the $N \to \infty$ limit, one obtains $u(t,x) = \lim_{N \to \infty} u_N(t,x)$ being the vanishing-viscosity solution of the inviscid Burgers equation. Shocks correspond to phase transitions. The Lax-Oleinik variational formula for solutions of hyperbolic conservation laws applies. See for example, Section 3.4.2 of Evans [@Evans]. In this context we claim that this is equivalent to the Gibbs variational formula, in the mean-field limit. We review this next.
Gibbs Variational Formula {#sec:SCMFE}
=========================
Let us begin by considering a general problem in classical statistical mechanics. Suppose that $\mathcal{X}$ is a compact metric space, and suppose that there is a two-body interaction $$h : \mathcal{X} \times \mathcal{X} \to {\mathbb{R}}\cup \{+\infty\}\, .$$ We assume that $h$ is bounded below. Then for each $N \geq 2$, one can consider the mean-field Hamiltonian $H_N : \mathcal{X}^N \to {\mathbb{R}}\cup \{+\infty\}$ $$H_N(x_1,\dots,x_N) = \frac{1}{N-1}\, \sum_{1\leq i<j\leq N} h(x_i,x_j)\, .$$ Suppose that there is also an [*a priori*]{} measure $\mu_0$ on $\mathcal{X}$, which we assume is normalized so that $$\int_{\mathcal{X}} d\mu_0(x)\, =\, 1\, .$$ Then the thermodynamic quantities are the partition function, $$Z_N(\beta)\,
=\, \int_{\mathcal{X}^N} e^{-\beta H_N(x_1,\dots,x_N)}\, d\mu_0(x_1) \cdots d\mu_0(x_N)\, ,$$ the pressure, $$p_N(\beta)\,
=\, \frac{1}{N}\, \ln Z_N(\beta)\, ,$$ and the Boltzmann-Gibbs measure $$d\mu_N^{\beta}(x_1,\dots,x_N)\,
=\, \frac{e^{-\beta H_N(x_1,\dots,x_N)}}{Z_N(\beta)}\, d\mu_0(x_1) \cdots d\mu_0(x_N)\, .$$ Physically, it is more correct to consider the free energy rather than the pressure, $f_N(\beta) = - \frac{1}{\beta} p_N(\beta)$. But we will consider $p_N(\beta)$, which seems slightly easier to handle, mathematically.
We will write $\mu_0^N$ for the measure $d\mu_0^N(x_1,\dots,x_N) = d\mu_0(x_1) \cdots d\mu_0(x_N)$. Also, if $f$ is a function, then we use the short-hand $\mu(f)$ for $\int f d\mu$. Then, according to the Gibbs variational principle, we have $$\label{eq:GibbsVariational}
p_N(\beta)\, =\, \max_{\mu_N \in \mathcal{M}_{+,1}(\mathcal{X}^N)} \frac{1}{N} \left[S_N(\mu_N, \mu_0^{\otimes N}) - \beta \mu_N(H_N)\right]\, ,$$ where $S_N(\mu_N, \mu_0^{\otimes N})$ is the relative entropy (and $\mathcal{M}_{+,1}(\mathcal{X}^N)$ denotes all Borel probability measures on $\mathcal{X}^N$) $$S_N(\mu_N, \mu_0^{\otimes N})\, =\, \begin{cases} \mu_0^N(\phi(d\mu_N/d\mu_0^N)) & \text{ if $\mu_N$ is absolutely continuous with respect to $\mu_0^N$,}\\
-\infty & \text{ otherwise,}
\end{cases}$$ and $\phi(x) = - x \ln(x)$, which is $0$ if $x=0$. Also, the unique $\mu_N$ maximizing the Gibbs variational formula (the “arg-max”) is the Boltzmann-Gibbs measure $\mu_N^{\beta}$.
A natural ansatz for the optimizing measure is $\mu_N = \mu^N$, for some measure $\mu \in \mathcal{M}_{+,1}(\mathcal{X})$. Probabilistically, this means that all the $x_1,\dots,x_N$ are independent and identically distributed. Technically, this cannot usually be exact for finite $N$. But it leads to a simpler formula because $$S_N(\mu^N,\mu_0^N)\, =\, N S_1(\mu,\mu_0)\quad \text{ and }\quad
\mu^N(H_N)\, =\, \frac{N}{2}\, \mu^2(h)\, ,$$ and one hopes that the formula may become exact in the thermodynamic limit. Mark Fannes, Herbert Spohn and Andre Verbeure proved that this approach is rigorous in the $N \to \infty$ limit [@FSV]:
\[prop:FSV\] The limiting pressure exists, $p(\beta) = \lim_{N \to \infty} p_N(\beta)$, and solves the variational problem $$p(\beta)\, =\, \max_{\mu \in \mathcal{M}_{+,1}(\mathcal{X})} [S_1(\mu,\mu_0) - \frac{\beta}{2} \mu^2(h)]\, .$$ Moreover, any subsequential limit of the sequence $(\mu_N^\beta)$ is a mixture of infinite product measures $\mu^{\infty}$, for $\mu$’s maximizing the right-hand-side of the formula above.
Note that the Gibbs variational principle (\[eq:GibbsVariational\]) is true in general for all Hamiltonians whether they are mean-field or not. (See, for instance, Lemma II.3.1 from Israel’s monograph [@Israel], or any other textbook on mathematical statistical mechanics, for a rigorous proof which also applies directly in the thermodynamic limit.) But the product ansatz which seems to yield the formula from the proposition is not generally valid, since there are nontrivial correlations in the true Boltzmann-Gibbs state. Nevertheless Fannes, Spohn and Verbeure proved the mean-field limit in the $N\to\infty$ limit, using de Finetti’s theorem (which states that all infinitely exchangeable measures are mixtures of product states) and properties of the relative entropy.
Because one has $\mu^2 = \mu \times \mu$, one replaces the linear form $\mu_N(H_N)$ by the nonlinear one $\mu^2(h)$. Fannes, Spohn and Verbeure actually proved their theorem more generally for quantum statistical mechanics models, such as the Dicke maser, but it also applies to classical models. For the quantum models, one replaces de Finetti’s theorem by the non-commutative analogue, Störmer’s theorem. (See [@Aldous] and references therein for a detailed survey of de Finetti’s theorem, and refer to Fannes, Spohn and Verbeure’s paper and references therein for the noncommutative analogue, which we will not need.) With Eugene Kritchevski, we tried to find a simpler proof of the specialization of Proposition \[prop:FSV\] to the classical case. But there were several errors in our proof, which have been brought to my attention by Alex Opaku, to whom I am grateful. Fortunately, Fannes, Spohn and Verbeure’s original paper definitely does also apply to classical models.
Application to the Mallows model {#sec:Mallows,}
================================
\[sec:Liouville\]
We take for $\mathcal{X}$ the unit square $[0,1] \times [0,1]$. Suppose that $f,g : [0,1] \to {\mathbb{R}}$ are probability densities: $f,g\geq 0$ and $\int_0^1 f(x)\, dx = \int_0^1 g(y)\, dy = 1$. For simplicity, later on, we also assume that there are constants $0<c<C<\infty$ such that $c \leq f,g\leq C$. Then we take the [*a priori*]{} measure to be $$d\mu_0(x,y)\, =\, f(x) g(y)\, dx\, dy\, .$$ We take the interaction to be $$h((x_1,y_1),(x_2,y_2))\, =\, \theta(x_1-x_2) \theta(y_2-y_1)
+ \theta(x_2-x_1) \theta(y_1-y_2)\, ,$$ where $\theta : {\mathbb{R}}\to {\mathbb{R}}$ is the Heaviside function, $$\theta(x)\, =\, \begin{cases} 1 & \text{ if $x>0$,}\\
0 & \text{ if $x<0$.}
\end{cases}$$ Since $\mu_0$ is absolutely continuous with respect to Lebesgue measure, all $x_1,\dots,x_N$ and $y_1,\dots,y_N$ are distinct, with probability 1. (This is why we do not bother to specify $\theta$ at the discontinuity point $0$.)
Let $X_1<\dots<X_N$ and $Y_1<\dots<Y_N$ be any points. Then for any $\sigma, \tau \in S_N$, the symmetric group, we have $$\frac{d\mu_N^{\beta}}{d\mu_0^N}((X_{\sigma_1},Y_{\tau_1}),\cdots,(X_{\sigma_N},Y_{\tau_N}))\,
=\, \mathbbm{P}_N^{\exp(-\beta/(N-1))}(\sigma^{-1} \tau)\, ,$$ where $\mathbbm{P}_N^q$ is the Mallows measure on $S_N$. So studying the limit of the $\mu_N^{\beta}$’s gives us direct information on the limit of $\mathbbm{P}^{1-\beta/N}_N$. For any fixed $\sigma \in S_N$, the permutation $\sigma^{-1} \tau$ is uniform on $S_N$, if $\tau$ is. Because of this, we have the following result for the marginal of $\mu_N^{\beta}$ on $(x_1,\dots,x_N)$, $$\int_{\mathcal{X}^N} U(x_1,\dots,x_N)\, d\mu_N^{\beta}((x_1,y_1),\dots,(x_N,y_N))\,
=\, \int_{[0,1]^N} U(x_1,\dots,x_N) f(x_1)\cdots f(x_N)\, dx_1\cdots dx_N\, ,$$ and the marginal on $(y_1,\dots,y_N)$, $$\int_{\mathcal{X}^N} U(y_1,\dots,y_N)\, d\mu_N^{\beta}((x_1,y_1),\dots,(x_N,y_N))\,
=\, \int_{[0,1]^N} U(y_1,\dots,y_N) g(y_1)\cdots g(y_N)\, dy_1\cdots dy_N\, ,$$ for all bounded, continuous functions $U : (0,1)^N \to {\mathbb{R}}$. Enforcing these conditions on the marginals, Proposition \[prop:FSV\] yields the following: $$\label{eq:FSV}
p(\beta)\, =\, \max_{\mu \in \mathcal{M}_{+,1}(f,g)} [S_1(\mu,\mu_0) - \beta \mu^2(h)]\, ,$$ where $\mathcal{M}_{+,1}(f,g)$ is the set of all probability measures $\mu \in \mathcal{M}_{+,1}(\mathcal{X})$ such that $d\mu(x,y)$ has marginals $f(x) dx$ and $g(y) dy$.
Suppose that $\mu$ is any arg-max of the right-hand-side of (\[eq:FSV\]). Since we have chosen $\mu_0$ to be absolutely continuous with respect to Lebesgue measure on $\mathcal{X}$, the same must be true of $\mu$. Otherwise the relative entropy would be $-\infty$. So we can write $$d\mu(x,y)\, =\, u(x,y)\, dx\, dy\, .$$ Then it is easy to see that the Euler-Lagrange equations for (\[eq:FSV\]) are $$\label{eq:EulerLagrange}
\ln u(x,y)\, =\, \ln f(x) + \ln g(y) + C - \beta \int_{\mathcal{X}} u(x',y') [\theta(x-x') \theta(y'-y) + \theta(x'-x) \theta(y-y')]\, dx' dy'\, ,$$ for some constant, $C<\infty$. Therefore, $u$ solves the equation $$\label{eq:EL}
\begin{cases}
u(x,y)\, =\, \frac{1}{\mathcal{Z}} f(x) g(y) e^{- \beta \int_{\mathcal{X}} h((x,y),(x',y')) u(x',y')\, dx'\, dy'}
& \text{ for $(x,y) \in \mathcal{X}$,}\\
\int_0^1 u(x,y)\, dy\, =\, f(x) & \text{ for $x \in [0,1]$,}\\
\int_0^1 u(x,y)\, dx\, =\, g(y) & \text{ for $y \in [0,1]$,}
\end{cases}$$ where $\mathcal{Z}$ is a normalization constant.
Since $u(x,y)$ solves an integral equation it can be differentiated both with respect to $x$ and $y$. Doing so yields the partial differential equation $$\label{eq:LiouvillePDE}
\frac{\partial^2}{\partial x\partial y} \ln u(x,y)\, =\, 2 \beta u(x,y)\, ,$$ known as the hyperbolic Liouville equation. This equation arises naturally in differential geometry, related to the problem of choosing a metric on a given manifold. I am very grateful to S.G. Rajeev for important information regarding this PDE. One of the facts he imparted is the symmetry of the differential equation under the following general transformation: $$\label{eq:symmetry}
\begin{split}
v(x,y)\, =\, F'(x) G'(y) u(F(x),G(y)) \qquad&\\
\Rightarrow
\frac{\partial^2}{\partial x \partial y} \ln v(x,y)\,
&=\, \frac{\partial^2}{\partial x \partial y} u(F(x),G(y)) + \frac{\partial}{\partial y}\left(\frac{F''(x)}{F'(x)}\right)
+ \frac{\partial}{\partial x} \left(\frac{G''(y)}{G'(y)}\right)\\
&=\, \frac{\partial^2}{\partial x \partial y} u(F(x),G(y))\\
&=\, 2 \beta F'(x) G'(y) u(F(x),G(y))\\
&=\, 2 \beta v(x,y)\, .
\end{split}$$ So, if $\frac{\partial^2}{\partial x\partial y} \ln u = \beta u$ then the same is true for $v(x,y) = F'(x) G'(y) u(F(x),G(y))$.
Our real goal is to solve the Euler-Lagrange equation (\[eq:EL\]). But as a first step, we want to consider the Cauchy problem for (\[eq:LiouvillePDE\]). In other words, we want to consider the problem $$\label{eq:Liouville}
\begin{cases}
\frac{\partial^2}{\partial x \partial y} \ln u(x,y)\, =\, 2 \beta u(x,y)
& \text{ for $(x,y) \in [0,L_1] \times [0,L_2]$,}\\
u(x,0)\, =\, \phi(x) & \text{ for $x \in [0,L_1]$,}\\
u(0,y)\, =\, \psi(y) & \text{ for $y \in [0,L_2]$,}
\end{cases}$$ for some $L_1,L_2 > 0$ and $\phi : [0,L_1] \to {\mathbb{R}}$, $\psi : [0,L_2] \to {\mathbb{R}}$ both positive and continuous.
Note that $\frac{\partial^2}{\partial x \partial y}$ is a wave operator, with characteristics directed along $x$ and $y$. Specifically, defining $\xi = (x+y)/\sqrt{2}$ and $\zeta = (x-y)/\sqrt{2}$, we have $\frac{\partial^2}{\partial x \partial y} = \frac{1}{2} (\frac{\partial^2}{\partial \xi^2} - \frac{\partial^2}{\partial \zeta^2})$, the usual wave operator. Therefore, D’Alembert’s formula for solutions of the wave equation allow us to reformulate (\[eq:Liouville\]) as an integral equation, $$\label{eq:CauchyIntegral}
\ln u(x,y)\, =\, \ln \phi(x) + \ln \psi(y) - \ln \alpha + 2 \beta \int_{[0,x] \times [0,y]} u(x',y')\, dx'\, dy'\, ,$$ which we prefer. This equation is supposed to be solved for all $(x,y) \in [0,L_1] \times [0,L_2]$. We have introduced the number $\alpha = \phi(0)$, which we also assumed equals $\psi(0)$, for consistency since both are supposed to give $u(0,0)$. (Note that the initial surface, $([0,L_1] \times \{0\}) \cup (\{0\} \times [0,L_2])$, is [*not*]{} a non-characteristic surface. This is the reason that our Cauchy problem does not require initial data for the tangential derivative of $u$ even though the wave equation is second order.) We refer to Evans textbook for PDE’s, (especially Section 2.4 on the wave equation and Section 4.6 on the Cauchy-Kovalevskaya theorem).
As we will see, the symmetry (\[eq:symmetry\]) is the key to solving both the Euler-Lagrange equation (\[eq:EL\]) and the Cauchy problem (\[eq:CauchyIntegral\]).
The Cauchy Problem
==================
We start with uniqueness for the Cauchy problem.
\[lem:IVP\] For any $L_1,L_2>0$, the Cauchy problem (\[eq:CauchyIntegral\]) having $\phi=\psi=\alpha=1$ has at most one solution in the class of nonnegative integrable functions.
Since $\phi=\psi=\alpha=1$, equation (\[eq:CauchyIntegral\]) simplifies to $$\ln u(x,y)\, =\, 2 \beta \int_{[0,x] \times [0,y]} u(x',y')\, dx'\, dy'\, .$$ Assuming that $u$ is nonnegative and integrable, this implies that $\ln u$ is bounded and continuous. Then, using these properties in the right-hand-side of the equation again (similarly as one does to prove elliptic regularity) we deduce that $\ln u$ is continuously differentiable and globally Lipschitz. In particular, it is continuous up to the boundary.
Now suppose that there are two solutions $u$ and $v$. Letting $z = \ln u - \ln v$, we have $$z(x,y)\, =\, 2 \beta \int_0^x \int_0^y [1-e^{-z(x',y')}] u(x',y')\, dx'\, dy'\, .$$ Since both $\ln u$ and $\ln v$ are bounded, we see that $z$ is as well. Therefore, there exists a constant $K<\infty$ such that $|1 - e^{-z}| \leq K |z|$ for all values of $z$ in the range. So we have $$|z(x,y)|\, \leq\, \beta K \|u\|_{\infty} \int_0^x \int_0^y |z(x',y')|\, dx'\, dy'\, .$$ A version of Gronwall’s lemma then implies that $z \equiv 0$. We outline this now, although our argument can probably be improved.
Let $Z(t) = \sup\{ |z(x,y)|\, :\, (x,y) \in (0,L_1) \times (0,L_2)\, ,\ xy\leq t\}$. Then we obtain, after making the change of variables $(x,y) \mapsto (x,t)$ where $t= xy$, and using Fubini-Tonelli to integrate over $x$ first, $$Z(t)\, \leq\,\beta K \|u\|_{\infty} \int_0^t \ln(t/t') Z(t')\, dt'\, .$$ We rewrite this as $$Z(t)\, \leq\, \beta K \|u\|_{\infty} \int_0^t [\ln(t/L_1L_2) - \ln(t'/L_1L_2)] Z(t')\, dt'\, .$$ Since $\ln(t/L_1L_2)\leq 0$ for $t\leq L_1L_2$, and since $Z\geq 0$, we can drop the term $\ln(t/L_1L_2) Z(t')$ in the integrand to obtain $$Z(t)\, \leq\, \beta K \|u\|_{\infty} \int_0^t |\ln(t'/L_1L_2)| Z(t')\, dt'\, .$$ Finally, setting $\zeta(t) = \int_0^t |\ln(t'/L_1L_2)| Z(t')\, dt'$, this leads to $$\zeta'(t)\, \leq\, \beta K \|u\|_{\infty} |\ln(t/L_1L_2)| \zeta(t)\, .$$ By Gronwall’s inequality (see for example Appendix B of Evans [@Evans]), we obtain $$\zeta(t)\, =\, e^{\beta K \|u\|_{\infty} \int_0^t |\ln(t'/L_1L_2)|\, dt'} \zeta(0)\, =\, e^{\beta K \|u\|_{\infty} (1+|\ln(t/L_1L_2)|) t/L_1L_2} \zeta(0)\, .$$ But $\zeta(0) = 0$. Hence $\zeta(t)=0$ for all $t$. This implies $Z(t)=0$ for all $t$ which implies $z(x,y) = 0$ for all $x,y$.
Next we derive the explicit solution of (\[eq:CauchyIntegral\]), for the case $\phi=\psi=\alpha=1$.
\[cor:Cauchy\] Suppose that $L_1,L_2>0$ and either $\beta\leq 0$ or $L_1L_2 < 1/\beta$. Then the unique solution of the Cauchy problem (\[eq:CauchyIntegral\]) with $\phi=\psi=\alpha=1$ is $$u(x,y)\, =\, (1 - \beta xy)^{-2}\, .$$
Uniqueness was proved in Lemma \[lem:IVP\], and it is trivial to check that this solves the PDE (\[eq:LiouvillePDE\]). Therefore, assuming that $u$ is integrable on $[0,L_1] \times [0,L_2]$, we may derive D’Alembert’s formula by standard calculus: $$\begin{aligned}
\ln u(x,y)\,
&=\, \int_0^x \frac{\partial}{\partial x} \ln u(x',y)\, dx' + \ln \psi(y)\\
&=\, \int_{(0,x) \times (0,y)} \frac{\partial^2}{\partial x \partial y} \ln u(x',y')\, dx'\, dy'
+ \int_0^x \frac{\partial}{\partial x} \ln \phi(x')\, dx' + \ln \psi(y)\\
&=\, 2 \beta \int_{(0,x) \times (0,y)} u(x',y')\, dx'\, dy' + \ln \psi(y) + \ln \phi(x) - \ln \alpha\, .\end{aligned}$$ The only issue is to check integrability, which amounts to checking $\inf_{(x,y) \in [0,L_1] \times [0,L_2]} 1-\beta xy>0$. This holds if and only if $\beta\leq 0$ or $L_1 L_2 < 1/\beta$.
Let us briefly explain one approach to deriving this formula. For nonlinear PDE’s one always first guesses a scaling solution, in hopes of finding an explicit formula. Because of the hyperbolic nature it makes sense to look for a solution $u(x,y) = U(xy)$ for some $U(z)$. This leads to the ODE $$\frac{d}{dz} \ln U(z) + z \frac{d^2}{dz^2} \ln U(z)\, =\, 2 \beta U(z)\, ,$$ which can also be expressed as $$\frac{d}{dz} \left(z\, \frac{d}{dz} \ln U(z)\right)\, =\, 2 \beta U(z)\, .$$ The idea of using a power law solution is natural because the derivative of the logarithm results in a power law, itself. Trying $U(z) = (1+cz)^p$ leads to $$\ln U(z)\, =\, p \ln(1+cz)\quad \Rightarrow\quad
z \frac{d}{dz} \ln \phi(z)\, =\, \frac{cpz}{1+cz}\, =\, p - \frac{p}{1+cz}\quad
\Rightarrow \quad
\frac{d}{dz} \left(z\, \frac{d}{dz} \ln U(z)\right)\, =\, \frac{cp}{(1+cz)^2}\, .$$ So, taking $p=-2$ and $c=-\beta$, this solves the equation, and gives $U(z) = (1-\beta z)^{-2} \Rightarrow u(x,y) = (1-\beta x y)^{-2}$. Finally, we are led to the solution of the general Cauchy problem.
\[cor:GenCauchy\] Suppose that $\phi, \psi : [0,1] \to {\mathbb{R}}$ are continuous and satisfy $c\leq \phi,\psi\leq C$, for some constants $0<c<C<1$. Also suppose that $\phi(0) = \psi(0) = \alpha$ for some $\alpha$. Then the Cauchy problem (\[eq:CauchyIntegral\]) has a solution if and only if $\beta \leq 0$ or $\int_0^{1} \phi(x)\, dx \int_0^{1} \psi(y)\, dy < \alpha/\beta$. In case a solution exists, it is unique and equals $$\label{eq:GenSol}
u(x,y)\, =\, \frac{\alpha \phi(x) \psi(y)}{(\alpha-\beta \Phi(x) \Psi(y))^{2}}\, ,$$ where $\Phi(x) = \int_0^x \phi(x')\, dx'$ and $\Psi(y) = \int_0^y \psi(y')\, dy'$.
Suppose that $u$ is any solution of (\[eq:CauchyIntegral\]). Let $v$ be given by $$v(x,y)\, =\, \frac{u(\Phi^{-1}(\alpha^{1/2} x),\Psi^{-1}(\alpha^{1/2} y))}{\alpha \Phi'(\Phi^{-1}(\alpha^{1/2} x))\Psi'(\Psi^{-1}(\alpha^{1/2} y))}\, .$$ Then, by the symmetry (\[eq:symmetry\]), $v$ is a solution of Liouville’s PDE (\[eq:LiouvillePDE\]) on the domain $[0,\alpha^{-1/2} \Phi(1)]\times [0,\alpha^{-1/2} \Psi(1)]$. But $v(x,0) = v(0,y) = 1$ because $\Phi' = \phi$ and $\Psi'=\psi$. So uniqueness, the conditions for existence, and the formula for the solution all follow from Lemma \[lem:IVP\] and Corollary \[cor:Cauchy\].
Solving the Euler-Lagrange Equation
===================================
By general principles, we know that a solution of (\[eq:EL\]) always exists: specifically, the optimizer in Proposition \[prop:FSV\]. Next we calculate it, and prove uniqueness.
\[lem:marginal\] If $f=g=1$, then the unique solution of (\[eq:EL\]) is given by (\[eq:GenSol\]) for $$\phi(z)\, =\, \psi(z)\, =\, \frac{\beta e^{-\beta z}}{1-e^{-\beta}}\, ,\quad
\Phi(z)\, =\, \Psi(z)\, =\, \frac{1-e^{-\beta z}}{1-e^{-\beta}}\quad \text{and}\quad
\alpha\, =\, \frac{\beta}{1-e^{-\beta}}\, .$$
Suppose $u$ solves (\[eq:EL\]). Note that $\lim_{x \to 0} h((x,y),(x',y')) = \theta(y-y')$ for all $(x',y') \in \mathcal{X}$. By the dominated convergence theorem, this implies $$\lim_{x \to 0} \int_{\mathcal{X}} h((x,y),(x',y')) u(x',y')\, dx'\, dy'\,
=\, \int_0^1 \int_0^1 \theta(y-y') u(x',y')\, dx'\, dy'\,
=\, \int_0^1 \theta(y-y')\, dy'\, =\, y\, ,$$ where we used the fact that $\int_0^1 u(x',y')\, dx' = 1$ for all $y'$. So $$\psi(y)\, =\, \lim_{x \to 0} u(x,y)\, =\, \frac{1}{\mathcal{Z}} e^{-\beta y}\, .$$ Similar arguments lead to $\phi(x)\, =\, \lim_{y \to 0} u(x,y)\, =\, \frac{1}{\mathcal{Z}} e^{-\beta x}$. Since $\int_0^1 u(x,y)\, dy = 1$ for all $x \in [0,1]$, it again follows from the dominated converge theorem, taking the limit $x \to 0$, that $\int_0^1 \psi(y)\, dy$ must also be $1$. So $\mathcal{Z} = (1-e^{-\beta})/\beta$. Checking, the reader will easily see that this gives the stated value for $\phi$, $\psi$ and $\alpha$. Integrating, it also leads to $\Phi$ and $\Psi$.
Uniqueness follows from uniqueness of the Cauchy problem, Corollary \[cor:GenCauchy\]. Since this is the only possible solution, and since a solution exists, this must be it.
Substituting in, and simplifying leads to the formula $$\label{eq:solution}
u(x,y)\, =\, \frac{(\beta/2) \sinh(\beta/2)}{\big(e^{\beta/4} \cosh(\beta[x-y]/2) - e^{-\beta/4} \cosh(\beta[x+y-1]/2)\big)^2}\, .$$ Therefore, we arrive at the final formula.
As long as $c\leq f,g\leq C$ for some $0<c<C<1$, and $\int_0^1 f(x)\, dx = \int_0^1 g(y)\, dy = 1$, the unique solution of (\[eq:EL\]) is $$u(x,y)\, =\, \frac{(\beta/2) \sinh(\beta/2) f(x) g(y)}{\big(e^{\beta/4} \cosh(\beta[F(x)-G(y)]/2) - e^{-\beta/4} \cosh(\beta[F(x)+G(y)-1]/2)\big)^2}\, ,$$ where $F(x) = \int_0^x f(x')\, dx'$ and $G(y) = \int_0^y g(y')\, dy'$.
Suppose that $u$ is a solution of (\[eq:EL\]) under the conditions stated. Define $$v(x,y)\, =\, \frac{u(F^{-1}(x),G^{-1}(y))}{f(F^{-1}(x)) g(G^{-1}(y))}\, ,$$ analogously to the proof of Corollary \[cor:GenCauchy\]. Note that $F$ and $G$ are continuously, strictly increasing bijections of $[0,1]$. Using (\[eq:EL\]), we see that $$\begin{aligned}
\ln v(x,y)\,
&=\, \ln u(F^{-1}(x),G^{-1}(y)) - \ln f(F^{-1}(x)) - \ln g(G^{-1}(y))\\
&=\, -\ln \mathcal{Z} - \beta \int_{\mathcal{X}} h((F^{-1}(x),G^{-1}(y)),(x',y')) u(x',y')\, dx'\, dy'\, .\end{aligned}$$ Making the change-of-variables $x'' = F(x')$ and $y'' = F(y')$, we see that $dx' = dx''/f(F^{-1}(x''))$ and $dy' = dy''/g(G^{-1}(y''))$. So we have $$\ln v(x,y)\, =\, - \ln \mathcal{Z} - \beta \int_{\mathcal{X}} h((F^{-1}(x),G^{-1}(y)),(F^{-1}(x''),G^{-1}(y''))) v(x'',y'')\, dx''\, dy''\, .$$ But the Heaviside function satisfies $\theta(F(x)-F(x')) = \theta(x-x')$ for any continuous, strictly increasing function $F$. For this reason, $$h((F^{-1}(x),G^{-1}(y)),(F^{-1}(x''),G^{-1}(y'')))\,
=\, h((x,y),(x'',y''))\, .$$ In other words, $v$ also solves (\[eq:EL\]), except that $$\int_0^1 v(x,y)\, dy\, =\, \int_0^1 v(x,y)\, dx\, =\, 1\, ,$$ using the change-of-variables formula, again. So uniqueness and the formula follows from Lemma \[lem:marginal\].
Proof of Main Result
====================
We now explain the minor details needed to go from Proposition \[prop:FSV\] to a proof of Theorem \[thm:main\]. According to Fannes, Spohn and Verbeure’s result, $\mu_{N}^{\beta}$ must converge weakly to a mixture of i.i.d., product measures, each of whose 1-particle marginal optimizes $S_1(\mu,\mu_0) - \frac{\beta}{2} \mu^2(h)$. But $\mu_N^{\beta}$ has marginals on $(x_1,\dots,x_N)$ and $(y_1,\dots,y_N)$ equal to the product measures of $f(x)\, dx$ and $g(y)\, dy$, respectively. Therefore, according to the weak law of large numbers (WLLN), we know that all the $\mu$’s in the support of the directing measure for the limit of $\mu_N^{\beta}$, must have $x$ marginal equal to $f(x)\, dx$ and $y$ maginal $g(y)\, dy$. Hence, this constraint can be imposed when looking for an optimizer. This is actually a relevant comment because all optimizers, for all choices of [*a priori*]{} measure $\mu_0$, have the same value/pressure: that due to the Mallows measure on $S_n$. For concreteness, we will now take $f=g=1$.
Now suppose that $\mu$ optimizes the Gibbs formula. It must be absolutely continuous with respect to $\mu_0$ in order to not have the relative entropy equal to $-\infty$. So we can write $$d\mu(x,y)\, =\, u(x,y)\, dx\, dy\, ,$$ where $u(x,y)$ is absolutely continuous. Choosing any continuous function $\phi : [0,1] \times [0,1] \to {\mathbb{R}}$, with $$\int_{\mathcal{X}} u(x,y) \phi(x,y)\, dx\, dy\, =\, 0\, ,$$ we can take $$u_{\epsilon}(x,y)\, =\, (1+\epsilon \phi(x,y)) u(x,y)\, .$$ For $|\epsilon| < 1/\|\phi\|_{\infty}$, we have that $u_{\epsilon}$ is a probability measure. It is easy to see that $$S_1(\mu_{\epsilon},\mu_0)\, =\, S_1(\mu,\mu_0) - \epsilon \int \phi u \ln u
- \int [1+\epsilon \phi] \ln[1+\epsilon \phi] u\, .$$ Since $1+\epsilon \phi$ is bounded away from $0$ (and infinity) for the $\epsilon$ we are considering, it is clear that both integrals above are well-defined. Moreover, it is clear that $$\int [1+\epsilon \phi] \ln[1+\epsilon \phi] u\, =\, o(\epsilon)\, ,$$ because $\ln[1+\epsilon \phi] = \epsilon \phi + o(\epsilon)$ and $\int \phi u = 0$. A similar calculation also shows that $$\mu_{\epsilon}^2(h)\, =\, \mu^2(h) + \epsilon \mu^2([\phi(x,y)+\phi(x',y')] h) + O(\epsilon^2)\, .$$ Since $\mu$ is supposed to be the optimizer, the terms linear in $\epsilon$ must vanish: $$\int \phi u \ln u\, =\, \frac{\beta}{2}\, \mu^2([\phi(x,y) + \phi(x',y')] h)\, .$$ Since $h$ is symmetric, by varying over all $\phi$ orthogonal to $u$, we deduce that $$u(x,y) \ln u(x,y)\, =\, \beta u(x,y) \int_{\mathcal{X}} h((x,y),(x',y')) u(x,y)\, dx'\, dy' + C u(x,y)\, ,$$ for some constant $C$. (The reason we cannot assume $C=0$ is because we left out one direction for $\phi$, namely the direction parallel to $u$, so that there is an indeterminacy in this direction, as seen using the Riesz representation theorem.) In other words, we have just deduced equation (\[eq:EulerLagrange\]). On the other hand, we have also proved that this equation has a unique solution given by (\[eq:solution\]). Therefore, $\mu_N^{\beta}$ does converge weakly to the i.i.d., product measure of $\mu$, where $d\mu(x,y) = u(x,y)\, dx\, dy$.
Because of all this, if we take the empirical measure with respect to $\mu_N^{\beta}$, $$\frac{1}{N}\, \sum_{i=1}^{N} \delta_{(x_i,y_i)}\, ,$$ then this does satisfy just the type of convergence claimed in Theorem \[thm:main\]. But, taking the order statistics $X_1<\dots<X_N$ and $Y_1<\dots<Y_N$, we do have $(x_i,y_i) = (X_{\sigma_i},Y_{\tau_i})$ for some permutations $\sigma,\tau \in S_N$. Moreover (by commutativity of addition) $$\frac{1}{N}\, \sum_{i=1}^{N} f(x_i,y_i)\, =\,
\frac{1}{N}\, \sum_{i=1}^{N} f(X_i,Y_{\pi_i})\, ,$$ where $\pi = \tau \sigma^{-1}$. As noted before, $(X_1,\dots,X_N)$ and $(Y_1,\dots,Y_N)$ are distributed as the order statistics coming from Lebesgue measure, the effect of the Hamiltonian is only present in the Mallow model $\mathbbm{P}_N^{\exp(-\beta/(N-1))}$-measure of $\pi$. By the WLLN for the order statistics, we see that, defining $$g_N(x,y)\, =\, \sum_{i,j=1}^{N} f(X_i,Y_j) \mathbbm{1}_{((i-1)/N,i/N]}(x) \mathbbm{1}_{((j-1)/N,j/N]}(y)\, ,$$ we have that the random function $g_N$ converges in probability to $f$, everywhere in $(0,1]\times (0,1]$. Therefore, since $$\frac{1}{N}\, \sum_{i=1}^{N} f(x_i,y_i)\, =\, \frac{1}{N}\, \sum_{i=1}^{N} g_N(i/N,\pi_i/N)\, ,$$ we do deduce the theorem from the corresponding result for $\mu_N^{\beta}$. Finally note that taking $\exp(-\beta/(n-1))$ versus $1-\beta/n$ in the theorem does not matter, since the probability measures are continuous with respect to $\beta$, and $\exp(-\beta/(n-1)) = 1 - \beta(1+o(1))/n$.
Applications {#sec:Applications}
============
The ground state of the $\mathcal{U}_q(\mathfrak{sl}_2)$-symmetric XXZ quantum spin system, and the invariant measures of the asymmetric exclusion process on an interval can be obtained from $\mathbbm{P}_N^{q}$. See Koma and Nachtergaele’s paper [@KomaNachtergaele] and Gottstein and Werner’s paper [@GottsteinWerner] for information about the XXZ model. For information about the blocking measures and the asymmetric exclusion process, we find it convenient to refer to Benjamini, Berger, Hoffman and Mossel (BBHM), [@BBHM]. The reader can easily deduce information for the XXZ model, since there is a perfect dictionary between these two. An excellent reference for this is Caputo’s review [@Caputo].
An interesting perspective on the ground state of the quantum XXZ ferromagnet was discovered by Bolina, Contucci and Nachtergaele in [@BCN]. They viewed the ground state of the quantum spin system as a thermal Boltzmann-Gibbs state for a classical model at inverse temperature $\beta = \ln(q^{-2})$. The state space they considered was the set of all up-right paths from $(0,0)$ to $(m,n) \in \mathbb{Z}^2$ (with $m,n\geq 0$). The Hamiltonian energy function for such a path is the energy under the path, and above the $x$-axis. Note that the Hamiltonian for the Mallows model also has a graphical representations as the number of “crossings” of the permutation. Using their representation, they explained some symmetries of the ground state of the XXZ model, and obtained estimates which were later useful in their follow-up paper, [@BCN2]. The two models are related, but only the Mallows model is manifestly a mean-field model.
We consider the (nearest neighbor) asymmetric exclusion process on $\{1,\dots,N\}$, with hopping rate to the left $p$ and hopping rate to the right $1-p$, and $q = (1-p)/p$. We no longer use $p$ or $p_N$ for the pressure, instead we use it for the hopping rate as expressed above. As BBHM explain, the invariant measure of the ASEP is a push-forward of $\mathbbm{P}_N^q$. Given a permutation $\pi \in S_N$ and a particle configuration $\eta = (\eta_1,\dots,\eta_N) \in \{0,1\}^N$, let $\pi \eta = (\eta_{\pi_1},\dots,\eta_{\pi_N})$. Let $(1^k,0^{N-k}) = (1,\dots,1,0,\dots,0)$ with $k$ $1$’s and $N-k$ $0$’s. Then, taking a random permutation $\pi$, distributed according to $\mathbbm{P}_N^q$, and letting $$\eta^{(k,N-k)} = \pi (1^k,0^{N-k})\, ,$$ the law of $\eta^{(k,N-k)}$ is the invariant measure for the ASEP, with $k$ particles and $N-k$ holes. As BBHM explain, this is an instance of Wilson’s general height function approach to tiling and shuffling [@Wilson][^2].
The question we can answer is the non-random limiting density of $\eta^{(k,N-k)}$ in the scaling limit, $N \to \infty$, $p_N = \frac{1}{2} + \beta/4N$, $k_N = \lfloor{y N}\rfloor$. (Note that this corresponds to $q_N = 1- \beta(1+o(1))/N$.) Namely, for a continuous function $f : [0,1] \to {\mathbb{R}}$, we have $$\lim_{\epsilon \downarrow 0} \lim_{N \to \infty} \mathbbm{P}^{1-\beta/N}_{N}\left\{
\left| \frac{1}{N}\, \sum_{i=1}^N f(i/N) \eta^{(\lfloor{y N}\rfloor,\lceil{(1-y)N}\rceil)}_i - \int_0^1 f(x) \rho(x;y)\, dx\right| > \epsilon\right\}\, =\, 0\, ,$$ for all $y \in [0,1]$, where $$\rho(x;y)\, =\, \int_0^y u(x,y')\, dy'\, .$$ The scaling $p_N = \frac{1}{2} + \beta/4N$ is the regime typically called “weakly asymmetric.” See, for example, Enaud and Derrida’s paper [@EnaudDerrida], following the matrix method used, for example by Derrida, Lebowitz and Speer [@DerridaLebowitzSpeer]. Note that while they considered the nonequilibrium case, we consider the particle conserving, equilibrium case. On the other hand, we are sure that the formula above is known.
The integral for $\rho(x;y)$ is readily evaluated. Setting $\phi$, $\psi$, $\Phi$, $\Psi$ and $\alpha$ as in Lemma \[lem:marginal\], $$\begin{aligned}
\rho(x;y)\,
&=\, \int_0^{y} \frac{\alpha \phi(x) \psi(y')}{(\alpha - \beta \Phi(x) \Psi(y'))^2}\, dy'\\
&=\, \frac{\alpha \phi(x)}{\beta \Phi(x) (\alpha - \beta \Phi(x) \Psi(y'))} \bigg|_0^y\\
&=\, \frac{\phi(x) \Psi(y)}{\alpha - \beta \Phi(x) \Psi(y)}\, .\end{aligned}$$ Substituting in, and doing minor algebraic simplifications, we obtain $$\rho(x;y)\, =\, \frac{(1-e^{-\beta y}) e^{-\beta x}}{(1-e^{-\beta}) - (1-e^{-\beta x})(1-e^{-\beta y})}\, .$$ From this formula it is obvious that the $\beta \to 0$ limit recovers $\rho(x;y) \equiv y$, as it should (for the symmetric case). Also, after further “simplifications,” we obtain $$\rho(x;y)\, =\, \frac{e^{\beta(\frac{1}{2}-x)/2} \sinh(\beta y/2)}{e^{\beta/4} \cosh(\beta [x-y]/2) - e^{-\beta/4} \cosh(\beta[x+y-1]/2)}\, .$$ In particular, one can observe that the particle-hole/reflection symmetry is manifest in this formula due to the invariance under the transformation $(\beta,x) \mapsto (-\beta,1-x)$.
Finally, we note that we can partially undo the scaling limit by taking $\beta \to \infty$ with $x=y+t/\beta$ (assuming $0<y<1$). Approximating $\sinh(\beta y/2) \approx \frac{1}{2} e^{\beta y/2}$ and noting that $e^{-\beta/2} \cosh(\beta [x+y-1]/2) \to 0$ since $|x+y-1|<1$, we obtain $$\rho(x;y) \to \frac{1}{1+e^t}\, .$$ This is not correctly normalized due to the fact that $dx = dt/\beta$, and $\beta \to \infty$. On the other hand, this does recover the actual lattice scaling limit for the density (modulo a reflection), as has been previously calculated for the XXZ model by Dijkgraaf, Orlando and Reffert in Appendix A of [@DijkgraafOrlandoReffert].
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported in part by a U.S. National Science Foundation grant, DMS-0706927. I am very grateful to the following people for useful discussions and suggestions: S. G. Rajeev, Alex Opaku, Carl Mueller, Bruno Nachtergaele, Wolfgang Spitzer and Pierluigi Contucci. I also thank the anonymous referees for their useful suggestions for improvement.
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[^1]: Independently, a similar $q$-deformed combinatorial formula was explained for a problem in quantum statistical mechanics, the ground state of the ferromagnetic $\mathcal{U}_q(\mathfrak{sl}_2)$-symmetric XXZ quantum spin chain, by Bolina, Contucci and Nachtergaele [@BCN]. We will comment more on this in Section 8.
[^2]: Because of this, let us note that the ground state of the XXZ model is also a projection, or marginal, of the Mallows model for permutations (using the correspondence between the ASEP and the XXZ model [@Caputo]). This raises an interesting point for further consideration: are other integrable models projections of mean-field models?
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abstract: |
Recently, Krukier et al. \[Generalized skew-Hermitian triangular splitting iteration methods for saddle-point linear systems, Numer. Linear Algebra Appl. 21 (2014) 152-170\] proposed an efficient *generalized skew-Hermitian triangular splitting* (GSTS) iteration method for nonsingular saddle-point linear systems with strong skew-Hermitian parts. In this work, we further use the GSTS method to solve *singular* saddle-point problems. The semi-convergence properties of GSTS method are analyzed by using singular value decomposition and Moore-Penrose inverse, under suitable restrictions on the involved iteration parameters. Numerical results are presented to demonstrate the feasibility and efficiency of the GSTS iteration methods, both used as solvers and preconditioners for GMRES method.
*MSC:* 65F08; 65F10; 65F20
address: 'School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, PR China'
author:
- Yan Dou
- 'Ai-Li Yang'
- 'Yu-Jiang Wu'
bibliography:
- 'References.bib'
title: 'On semi-convergence of generalized skew-Hermitian triangular splitting iteration methods for singular saddle-point problems'
---
singular saddle-point problems; skew-Hermitian triangular splitting; iteration method; semi-convergence; Moore-Penrose inverse; singular value decomposition
Introduction
============
Consider the following saddle-point linear system: $$\label{01}
\mathcal{A}\,u\equiv\left(
\begin{array}{cc}
M & E \\
-E^{*} & 0 \\
\end{array}
\right)\left(
\begin{array}{c}
u_{1} \\
u_{2} \\
\end{array}
\right)=\left(
\begin{array}{c}
f_{1} \\
f_{2} \\
\end{array}
\right)\equiv f,$$ where $M\in \mathbb{C}^{p\times p}$ is a Hermitian positive definite matrix, $E\in \mathbb{C}^{p\times q}$ is a rectangular matrix satisfying $q\leq p$, and $f\in \mathbb{C}^{p+q}$ is a given vector in the range of $\mathcal{A}\in \mathbb{C}^{(p+q)\times (p+q)}$, with $f_{1}\in \mathbb{C}^{p}$ and $f_{2}\in \mathbb{C}^{q}$. This kind of linear systems arise in a variety of scientific and engineering applications, such as computational fluid dynamics, constrained optimization, optimal control, weighted least-squares problems, electronic networks, computer graphic etc, and typically result from mixed or hybrid finite element approximation of second-order elliptic problems or the Stokes equations; see [@BPW20051; @SSY19981; @WSY2004139; @ZBY2009808; @Brezzi1991].
When matrix $E$ is of full column rank, the saddle-point matrix $\mathcal{A}$ is nonsingular. A number of effective iteration methods, such as matrix splitting iteration methods, Minimum residual methods, Krylov subspace iteration methods etc, have been proposed in the literature to approximate the unique solution of the nonsingular saddle-point problems ; see [@BGP20041; @BGL20051; @EG19941645; @FRSW1998527; @KGW20001300; @B2009447; @BW20082900] and the references therein. Recently, Krukier et al. [@Krukier2013] proposed a generalized skew-Hermitian triangular splitting (GSTS) iteration method for solving the linear systems with strong skew-Hermitian parts. When used for approximating the solution of the nonsingular saddle-point problem , the GSTS method can be described as follows.
\[met:1\](The GSTS iteration method) Given initial guesses $u^{(0)}_{1}\in \mathbb{C}^{p}$ and $u^{(0)}_{2}\in \mathbb{C}^{q}$, for $k=0,1,2\ldots$, until $u^{(k)}=[u_1^{(k)};u_2^{(k)}]$ convergence
1. compute $u_2^{(k+1)}$ from $$\label{02}
u_2^{(k+1)}=u_2^{(k)}+\tau B^{-1}\left[\omega_1E^*M^{-1}\left(f_1-Eu_2^{(k)}\right)+(1-\omega_1)E^*u_1^{(k)}+f_2\right];$$
2. compute $u_2^{(k+1)}$ from $$\label{19}
u_1^{(k+1)}=(1-\tau) u_1^{(k)}+ M^{-1}\left[E\left((\omega_2-\tau)u_2^{(k)}-\omega_2u_2^{(k+1)}\right)+\tau f_1\right],$$
where $\omega_{1}$ and $\omega_{2}$ are two nonnegative acceleration parameters with at least one of them being nonzero, $\tau$ is a positive parameter, $B\in\mathbb{C}^{q\times q}$ is a Hermitian positive definite matrix, which is chosen as an approximation of the Shur complement $S_M:=E^{*}M^{-1}E$.
Theoretical analysis and numerical experiments in [@Krukier2013] have shown that the GSTS iteration method is convergent under suitable restrictions on iteration parameters. Moreover, no matter as a solver or as a preconditioner for GMRES method, the GSTS method is robust and effective for solving the large sparse nonsingular saddle-point linear systems. However, matrix $E$ in saddle-point matrix $\mathcal{A}$ is rank deficient in many real world applications, such as the discretization of incompressible steady state Stokes problem with suitable boundary conditions; see [@WSY2004139; @ZhangWei2010139]. In this case, the saddle-point linear systems are always singular and consistent. The Uzawa algorithm and its variants [@ZBY2009808; @ZLW2014334], Hermitian and skew-Hermitian splitting iteration method [@Bai2010171; @BGN2002603; @Li20122338], general stationary linear iteration method [@Cao20081382; @ZhangWei2010139], Krylov subspace methods (preconditioned by block-diagonal, block-tridiagonal or constraint preconditioners) [@WSY2004139; @BW199737; @ZhangShen2013116] etc, can be used to approximate a solution of the singular and consistent saddle-point linear system.
In this work, owing to the high efficiency of the GSTS iteration method used for solving the nonsingular saddle-point linear systems, we will further analyze the feasibility and efficiency of the GSTS iteration method when it is used for solving the singular saddle-point problems with Hermitian positive definite matrix $M\in \mathbb{C}^{p\times p}$ and rank deficient matrix $E\in \mathbb{C}^{p\times q}$. Since matrix $E$ is rank deficient, the Shur complement $S_M=E^{*}M^{-1}E$ is Hermitian positive semi-definite. As the approximation of Shur complement, it may be better if we choose matrix $B$ being a Hermitian positive semi-definite matrix and having the same null space with Shur complement $S_M$. In this way, matrix $B$ is singular, we replace iteration scheme in Method \[met:1\] by the scheme of the form $$\label{14}
u_2^{(k+1)}=u_2^{(k)}+\tau B^{\dag}\left[\omega_1E^*M^{-1}\left(f_1-Eu_2^{(k)}\right)+(1-\omega_1)E^*u_1^{(k)}+f_2\right],$$ where $B^{\dag}$ is the Moore-Penrose inverse [@BermanPlemmons1994; @Kucera2011] of the singular matrix $B$, which satisfies $$B=BB^{\dag}B,\quad B^{\dag}=B^{\dag}BB^{\dag},\quad BB^{\dag}=(BB^{\dag})^{*}, \quad B^{\dag}B=(B^{\dag}B)^{*}.$$ The convergence properties of the GSTS iteration methods, with Hermitian positive definite and singular Hermitian positive semi-definite matrices $B$, will be carefully analyzed. Moreover, the feasibility and efficiency of the GSTS iteration methods for singular and consistent saddle-point problems will also be numerically verified.
The remainder part of this work is organized as follows. In Section \[sec2\] we give the semi-convergence concepts of the GSTS iteration methods with different choices of matrix $B$, i.e., $B$ is Hermitian positive definite and singular Hermitian positive semi-definite. When $B$ is Hermitian positive definite, the semi-convergence properties of the GSTS iteration method are analyzed in Section \[sec3\]. In Section \[sec4\], we give the semi-convergence properties of the GSTS method with $B$ being singular Hermitian positive semi-definite. In Section \[sec5\], numerical results are presented to show the feasibility and effectiveness of the GSTS iteration methods for solving the singular saddle-point linear systems. Finally, in Section \[sec6\], we end this work with a brief conclusion.
Basic concepts and lemmas {#sec2}
=========================
We split matrix $\mathcal{A}$ into its Hermitian and skew-Hermitian parts, i.e., $\mathcal{A}=\mathcal{A}_{H}+\mathcal{A}_{S}$, where $$\label{32}
\mathcal{A}_{H}=\frac{1}{2}(\mathcal{A}+\mathcal{A}^{*})=\left(
\begin{array}{cc}
M & 0 \\
0 & 0 \\
\end{array}
\right),\quad
\mathcal{A}_{S}=\frac{1}{2}(\mathcal{A}-\mathcal{A}^{*})=\left(
\begin{array}{cc}
0 & E \\
-E^{*} & 0 \\
\end{array}
\right).$$ Let $\mathcal{K}_{L}$ and $\mathcal{K}_{U}$ be, respectively, the strictly lower-triangular and the strictly upper-triangular parts of $\mathcal{A}_{S}$ satisfying $$\label{08}
\mathcal{A}_{S}=\mathcal{K}_{L}+\mathcal{K}_{U}=\left(
\begin{array}{cc}
0 & 0 \\
-E^{*} & 0 \\
\end{array}
\right)+\left(
\begin{array}{cc}
0 & E \\
0 & 0 \\
\end{array}
\right),$$ and denote $$\label{21}
\mathcal{B}_{c}=\left(
\begin{array}{cc}
M & 0 \\
0 & B \\
\end{array}
\right).$$
In the following two subsections, we give some basic concepts and useful lemmas for the analysis of the semi-convergence properties of the GSTS iteration methods according to the choices of matrix $B$.
Matrix $B$ is Hermitian positive definite
-----------------------------------------
Firstly, we consider the case that matrix $B$ used in Method \[met:1\] is Hermitian positive definite. Combining iteration schemes and , the GSTS iteration method can be rewritten as $$\label{20}
u^{(k+1)}=u^{(k)}-\tau\mathcal{B}(\omega_{1},\omega_{2})^{-1}(\mathcal{A}u^{(k)}-f),$$ where $$\label{25}
\mathcal{B}(\omega_{1},\omega_{2})=(\mathcal{B}_{c}+\omega_{1}\mathcal{K}_{L})\mathcal{B}_{c}^{-1}
(\mathcal{B}_{c}+\omega_{2}\mathcal{K}_{U}).$$ The iteration matrix is $$\label{13}
\mathcal{G}(\omega_{1},\omega_{2},\tau)=I-\tau\mathcal{B}(\omega_{1},\omega_{2})^{-1}\mathcal{A}.$$
Iteration scheme can be induced from the splitting $$\mathcal{A}=\mathcal{M}(\omega_{1},\omega_{2},\tau)-\mathcal{N}(\omega_{1},\omega_{2},\tau),$$ where $$\mathcal{M}(\omega_{1},\omega_{2},\tau)=(1/\tau)\mathcal{B}(\omega_{1},\omega_{2}), \quad\mathcal{N}(\omega_{1},\omega_{2},\tau)=(1/\tau) \left(\mathcal{B}(\omega_{1},\omega_{2})-\tau\mathcal{A}\right).$$ Hence, matrix $\mathcal{M}(\omega_{1},\omega_{2},\tau)$, or $\mathcal{B}(\omega_{1},\omega_{2})$, can be viewed as a preconditioner for the saddle-point linear system , which may be used to accelerate the convergence rate of Krylov subspace methods, such as the generalized minimum residual (GMRES) method and the quasi-minimal residual (QMR) method.
For the semi-convergence of iteration scheme , we give the following useful lemma.
[@BermanPlemmons1994]\[lem:6\] The iterative scheme $$u^{(k+1)}=u^{(k)}-\mathcal{M}^{-1}(\mathcal{A}u^{(k)}-f)$$ is semi-convergent, if and only if its iteration matrix $\mathcal{G}=I-\mathcal{M}^{-1}\mathcal{A}$ satisfies
1. The pseudo-spectral radius of matrix $\mathcal{G}$ is less than $1$, i.e., $$\gamma(\mathcal{G}):=\max \{|\lambda|:\lambda\in\sigma(\mathcal{G})\setminus{1}\}<1,$$ where $\sigma(\mathcal{G})$ is the set of eigenvalues of matrix $\mathcal{G}$;
2. $\text{index}(I-\mathcal{G})=1$, or equivalently, $\text{rank}(I-\mathcal{G})=\text{rank}((I-\mathcal{G})^{2})$.
Matrix $B$ is singular and Hermitian positive semi-definite
-----------------------------------------------------------
When matrix $E$ in is rank deficient, the Shur complement $S_M=E^{*}M^{-1}E$ is singular and Hermitian positive semi-definite. As the approximation of $S_M$, matrix $B$ is chosen as $E^{*}P^{-1}E$, where $P$ is an approximation of $M$ and is Hermitian positive definite. Hence, matrix $B$ is singular and Hermitian positive semi-definite, and has the same null space with Shur complement $S_M$.
Owing to the singularity of matrix $B$, we replace iteration scheme by and obtain a more generalized GSTS iteration method. Based on and , the GSTS iteration method can be rewritten as $$\label{22}
u^{(k+1)}=u^{(k)}-\tau\mathcal{B}(\omega_{1},\omega_{2})^{\dag}(\mathcal{A}u^{(k)}-f),$$ where $$\mathcal{B}(\omega_{1},\omega_{2})=(\mathcal{B}_{c}+\omega_{1}\mathcal{K}_{L})\mathcal{B}_{c}^{\dag}
(\mathcal{B}_{c}+\omega_{2}\mathcal{K}_{U}).$$ Iteration matrix is $$\label{23}
\mathcal{G}(\omega_{1},\omega_{2},\tau)=I-\tau\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}.$$ Here, matrix $\mathcal{B}(\omega_{1},\omega_{2})$ can also be viewed as a preconditioner for singular saddle-point linear system . The difference is that the preconditioner $\mathcal{B}(\omega_{1},\omega_{2})$ introduced in this subsection is singular.
Comparing with iteration scheme , we need one more condition to keep the semi-convergence of iteration scheme since matrix $B$ is singular.
\[lem:1\][@Cao20081382] The iterative scheme $$u^{(k+1)}=u^{(k)}-\mathcal{M}^{\dag}(\mathcal{A}u^{(k)}-f)$$ is semi-convergent if and only if the following three conditions are fulfilled:
1. The pseudo-spectral radius of matrix $\mathcal{G}$ is less than $1$, i.e., $\gamma(\mathcal{G})<1$, where $\mathcal{G}\equiv I-\mathcal{M}^{\dag}\mathcal{A}$ is the iteration matrix;
2. $(\mathcal{M}^{\dag}\mathcal{A})$=$(\mathcal{A})$;
3. index$(I-\mathcal{G})$=1, or equivalently, rank$(I-\mathcal{G})$=rank$((I-\mathcal{G})^{2})$.
The semi-convergence of GSTS method with $B$ being Hermitian positive definite {#sec3}
==============================================================================
Since matrix $B$ is nonsingular, it is easy to see that matrix $\mathcal{B}(\omega_{1},\omega_{2})$ is invertible. The inverse matrix of $\mathcal{B}(\omega_{1},\omega_{2})$ has the following explicit form $$\mathcal{B}(\omega_{1},\omega_{2})^{-1}=\left(
\begin{array}{cc}
M^{-1}-\omega_{1}\omega_{2}M^{-1}EB^{-1}E^{*}M^{-1} & -\omega_{2}M^{-1}EB^{-1} \\
\omega_{1}B^{-1}E^{*}M^{-1} & B^{-1} \\
\end{array}
\right).$$ The iteration matrix $\mathcal{G}(\omega_{1},\omega_{2},\tau)$ can be written as $$\mathcal{G}(\omega_{1},\omega_{2},\tau)=\left(
\begin{array}{cc}
(1-\tau)I_{p}-\tau\omega_{2}(1-\omega_{1})M^{-1}EB^{-1}E^{*} & -\tau M^{-1}E(I_q-\omega_{1}\omega_{2}B^{-1}E^{*}M^{-1}E) \\
\tau(1-\omega_{1})B^{-1}E^{*} & I_{q}-\tau\omega_{1}B^{-1}E^{*}M^{-1}E \\
\end{array}
\right).$$
In the following, we further study the semi-convergence properties of GSTS iteration method in which matrix $B$ is Hermitian positive definite. In fact, we only need to verify the two conditions presented in Lemma \[lem:6\].
The conditions for index$(I-\mathcal{G}(\omega_{1},\omega_{2},\tau))=1$
-----------------------------------------------------------------------
Let matrices $\mathcal{A}$ and $\mathcal{B}(\omega_{1},\omega_{2})$ be defined by and , respectively. Then, we have index$(I-\mathcal{G}(\omega_{1},\omega_{2},\tau))=1$, or equivalently, $$\label{31}
\text{rank}(I-\mathcal{G}(\omega_{1},\omega_{2},\tau))=\text{rank}((I-\mathcal{G}(\omega_{1},\omega_{2},\tau))^{2}).$$
Inasmuch as $\mathcal{G}(\omega_{1},\omega_{2},\tau)=I-\tau\mathcal{B}(\omega_{1},\omega_{2})^{-1}\mathcal{A}$, equality holds if $$\text{null}((\mathcal{B}(\omega_{1},\omega_{2})^{-1}\mathcal{A})^{2})=\text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{-1}\mathcal{A}).$$ It is obvious that $\text{null}((\mathcal{B}(\omega_{1},\omega_{2})^{-1}\mathcal{A})^{2})\supseteq \text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{-1}\mathcal{A})$, we only need to prove $$\label{27}
\text{null}((\mathcal{B}(\omega_{1},\omega_{2})^{-1}\mathcal{A})^{2})\subseteq \text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{-1}\mathcal{A}).$$
Let $x=(x_{1}^{*},x_{2}^{*})^{*}\in\mathbb{C}^{p+q}$ satisfy $(\mathcal{B}(\omega_{1},\omega_{2})^{-1}\mathcal{A})^{2}x=0$. Denote $y=\mathcal{B}(\omega_{1},\omega_{2})^{-1}\mathcal{A}x$, then simple calculation gives $$\label{26}
y=\left(
\begin{array}{c}
y_{1} \\
y_{2} \\
\end{array}
\right)=\left(
\begin{array}{cc}
(I_{p}+\omega_{2}(1-\omega_{2})M^{-1}EB^{-1}E^{*})x_{1}+M^{-1}E(I_q-\omega_{1}\omega_{2}B^{-1}E^{*}M^{-1}E)x_{2} \\
B^{-1}E^{*}\left((\omega_{1}-1)x_{1}+\omega_{1}M^{-1}Ex_{2}\right)
\end{array}\right).$$ In the following, we only need to prove $y=0$. From $\mathcal{B}(\omega_{1},\omega_{2})^{-1}\mathcal{A}y=(\mathcal{B}(\omega_{1},\omega_{2})^{-1}\mathcal{A})^{2}x=0$, we have $\mathcal{A}y=0$, i.e., $$\label{24}
My_{1}+Ey_{2}=0 \quad\text{and}\quad -E^{*}y_{1}=0.$$ Note that $M$ is nonsingular, solving $y_{1}$ from the first equality of and taking it into the second equality, it follows that $E^{*}M^{-1}Ey_{2}=0$. Hence, $$(Ey_{2})^{*}M^{-1}(Ey_{2})=y_{2}(E^{*}M^{-1}Ey_{2})=0.$$ Owing to the Hermitian positive definiteness of matrix $M^{-1}$, we can obtain that $Ey_{2}=0$. Taking it into the first equality of gives $y_{1}=0$. Furthermore, using $Ey_{2}=0$ and , we have $$Ey_{2}=EB^{-1}E^{*}\left((\omega_{1}-1)x_{1}+\omega_{1}M^{-1}Ex_{2}\right)=0.$$ Since matrix $B$ is Hermitian positive definite, we can derive, with similar technique, that $$E^{*}\left((\omega_{1}-1)x_{1}+\omega_{1}M^{-1}Ex_{2}\right)=0,$$ which means $$y_2=B^{-1}E^{*}\left((\omega_{1}-1)x_{1}+\omega_{1}M^{-1}Ex_{2}\right)=0.$$ Thus, $\mathcal{B}(\omega_{1},\omega_{2})^{-1}\mathcal{A}x=y=0$, i.e., the inclusion relation holds.
The conditions for $\gamma(\mathcal{G}(\omega_{1},\omega_{2},\tau))<1$
----------------------------------------------------------------------
Assume that the column rank of $E$ is $r$, i.e., $r=\text{rank}(E)$. Let $$\label{10}
E=U(E_{r},0)V^{*}$$ be the singular value decomposition of $E$, where $U\in\mathbb{C}^{p\times p}$ and $V\in\mathbb{C}^{q\times q}$ are two unitary matrices, $E_{r}=(\Sigma_{r},0)^*\in\mathbb{C}^{p\times r}$ and $\Sigma_{r}=\text{diag}(\sigma_{1}, \sigma_{2}, \cdots, \sigma_{r})$, with $\sigma_{i}$ being the singular value of matrix $E$.
We partition matrix $V$ as $V=(V_{1},V_{2})$ with $V_{1}\in\mathbb{C}^{q\times r}$, $V_{2}\in\mathbb{C}^{q\times (q-r)}$ and define $$\label{17}
\mathcal{P}=\left(
\begin{array}{cc}
U & 0 \\
0 & V \\
\end{array}
\right).$$ It is obvious that $\mathcal{P}$ is a $(p+q)\times(p+q)$ unitary matrix, and the iteration matrix $\mathcal{G}(\omega_{1},\omega_{2},\tau)$ is unitarily similar to the matrix $\hat{\mathcal{G}}(\omega_{1},\omega_{2},\tau)=\mathcal{P}^{*}\mathcal{G}(\omega_{1},\omega_{2},\tau)\mathcal{P}$. Hence, the pseudo-spectral radii of matrices $\hat{\mathcal{G}}(\omega_{1},\omega_{2},\tau)$ and $\mathcal{G}(\omega_{1},\omega_{2},\tau)$ are same, we in the following only need to analyze the pseudo-spectral radius of matrix $\hat{\mathcal{G}}(\omega_{1},\omega_{2},\tau)$.
Denoting $\hat{M}=U^{*}MU$ and $\hat{B}=V^{*}BV$, we have $$\label{28}
\hat{B}^{-1}=\left(\begin{array}{cc}
\hat{B}^{-1}_{11} & \hat{B}^{-1}_{12} \\
\hat{B}^{-1}_{21} & \hat{B}^{-1}_{22} \\\end{array}
\right)=\left(\begin{array}{cc}
V^{*}_{1}B^{-1}V_{1} & V^{*}_{1}B^{-1}V_{2} \\
V^{*}_{2}B^{-1}V_{1} & V^{*}_{2}B^{-1}V_{2} \\
\end{array}\right).$$ Furthermore, we can derive that $$\hat{\mathcal{G}}(\omega_{1},\omega_{2},\tau)=\left(
\begin{array}{cc}
\hat{\mathcal{G}}_{1}(\omega_{1},\omega_{2},\tau) & 0 \\
\hat{\mathcal{L}}(\omega_{1},\omega_{2},\tau) & I_{q-r} \\
\end{array}
\right),$$ where $$\hat{\mathcal{G}}_{1}(\omega_{1},\omega_{2},\tau)=\left(
\begin{array}{cc}
(1-\tau)I_{p}-\tau\omega_{2}(1-\omega_{1})\hat{M}^{-1}E_{r}\hat{B}^{-1}_{11}E^{*}_{r} & -\tau \hat{M}^{-1}E_{r}(I_q-\omega_{1}\omega_{2}\hat{B}^{-1}_{11}E^{*}_{r}\hat{M}^{-1}E_{r}) \\
\tau(1-\omega_{1})\hat{B}^{-1}_{11}E^{*}_{r} & I_{q}-\tau\omega_{1}\hat{B}^{-1}_{11}E^{*}_{r}\hat{M}^{-1}E_{r} \\
\end{array}
\right)$$ and $$\hat{\mathcal{L}}(\omega_{1},\omega_{2},\tau)=\left(
\begin{array}{cc}
\tau(1-\omega_{1})\hat{B}^{-1}_{21}E^{*}_{r} & -\tau\omega_{1}\hat{B}^{-1}_{21}E^{*}_{r}\hat{M}^{-1}E_{r} \\
\end{array}
\right).$$ Then, $\gamma(\hat{\mathcal{G}}(\omega_{1},\omega_{2},\tau))<1$ holds if we have $\rho(\hat{\mathcal{G}}_{1}(\omega_{1},\omega_{2},\tau))<1$.
Note that $\hat{\mathcal{G}}_{1}(\omega_{1},\omega_{2},\tau)$ is the iteration matrix of GSTS iteration method applied to the nonsingular saddle-point problem $$\label{15}
\hat{\mathcal{A}}\hat{u}:=\left(
\begin{array}{cc}
\hat{M} & E_{r} \\
-E^{*}_{r} & 0 \\
\end{array}
\right)\left(
\begin{array}{c}
\hat{u}_{1} \\
\hat{u}_{2} \\
\end{array}
\right)=\left(
\begin{array}{c}
\hat{f}_{1} \\
\hat{f}_{2} \\
\end{array}
\right)=:\hat{f}.$$ Moreover, in the iteration process, we have $$\label{16}
\hat{\mathcal{B}}(\omega_{1},\omega_{2})=(\hat{\mathcal{B}}_{c}+\omega_{1}\hat{\mathcal{K}}_{L})\hat{\mathcal{B}}_{c}^{-1}(\hat{\mathcal{B}}_{c}+\omega_{2}\hat{\mathcal{K}}_{U})\quad \text{and}\quad\hat{\mathcal{B}}_{c}=\left(
\begin{array}{cc}
\hat{M} & 0 \\
0 & \hat{B}_{11} \\
\end{array}
\right),$$ where $\hat{B}_{11}\in \mathbb{C}^{r\times r}$ defined in is Hermitian positive definite, and $\hat{\mathcal{K}}_{L}$ and $\hat{\mathcal{K}}_{U}$ are the strictly lower-triangular and the strictly upper-triangular parts of $\hat{\mathcal{A}}_{S}=(1/2)(\hat{\mathcal{A}}-\hat{\mathcal{A}}^{*})$, respectively.
For convenience, we denote by $$\alpha:=\frac{z^{*}E_{r}^{*}\hat{M}^{-1}E_{r}z}{z^{*}z}\quad \text{and} \quad \beta_1:=\frac{z^{*}\hat{B}_{11}z}{z^{*}z}.$$ By making use of Theorem 3.3 in [@Krukier2013], we derive the following result.
\[lem:7\] Denote $\tilde{\omega}=(\omega_{1}-1)(\omega_{2}-1)$. Let matrices $\mathcal{A}$ and $\mathcal{B}(\omega_{1},\omega_{2})$ be defined by and , respectively. Then $\gamma(\mathcal{G}(\omega_{1},\omega_{2},\tau))<1$ holds, provided that the parameters $\omega_{1}$, $\omega_2$ satisfy $$\tilde{\omega}<\frac{\alpha+\beta_1}{\alpha},$$ and the parameter $\tau$ satisfies
1. if $[\beta_1+(1-\tilde{\omega})\alpha]^{2}-4\alpha\beta_1\leq 0$, then $$0<\tau<\frac{\beta_1+(1-\tilde{\omega})\alpha}{\alpha};$$
2. if $[\beta_1+(1-\tilde{\omega})\alpha]^{2}-4\alpha\beta_1> 0$, then $$0<\tau<\frac{\beta_1+(1-\tilde{\omega})\alpha-\sqrt{[\beta_1+(1-\tilde{\omega})\alpha]^{2}
-4\alpha\beta_1}}{\alpha}.$$
Using Lemma \[lem:6\] and combining the above analyses, we finally obtain the following semi-convergence properties of GSTS iteration method.
Let parameters $\omega_1$, $\omega_2$ and $\tau$ satisfy the conditions of Lemma \[lem:7\] and matrix $B$, as an approximation of Shur complement $S_M$, be Hermitian positive definite. Then, the GSTS iteration method used for solving singular saddle-point linear system is semi-convergent.
The semi-convergence of GSTS method with $B$ being singular and Hermitian positive semi-definite {#sec4}
================================================================================================
In this section, we particularly choose matrix $B$ as $B=E^{*}P^{-1}E$, where $P$, as an approximation of $M$, is Hermitian positive definite. Hence, matrix $B$ is singular and has the same null space with Shur complement $S_M$.
In this case, matrix $\mathcal{B}_{c}$ defined in is singular. We can write matrix $\mathcal{B}(\omega_{1},\omega_{2})$ as $$\label{11}
\begin{split}
\mathcal{B}(\omega_{1},\omega_{2})
&=\left(
\begin{array}{cc}
M & 0 \\
-\omega_{1}E^{*} & B \\
\end{array}
\right) \left(
\begin{array}{cc}
M^{-1} & 0 \\
0 & B^{\dag}
\end{array}
\right) \left(
\begin{array}{cc}
M & \omega_{2}E \\
0 & B \\
\end{array}
\right) \\
&=\left(
\begin{array}{cc}
M & \omega_{2}E \\
-\omega_{1}E^{*} & B-\omega_{1}\omega_{2}E^{*}M^{-1}E \\
\end{array}
\right),
\end{split}$$ where $B^{\dag}$ is the Moore-Penrose inverse of $B$. Since $BB^{\dag}E^{*}=B^{\dag}BE^{*}=E^{*}$ [@ZhangShen2013116], the Moore-Penrose inverse of singular matrix $\mathcal{B}(\omega_{1},\omega_{2})$ has the form of $$\label{12}
\mathcal{B}(\omega_{1},\omega_{2})^{\dag}=\left(
\begin{array}{cc}
M^{-1}-\omega_{1}\omega_{2}M^{-1}EB^{\dag}E^{*}M^{-1} & -\omega_{2}M^{-1}EB^{\dag} \\
\omega_{1}B^{\dag}E^{*}M^{-1} & B^{\dag} \\
\end{array}
\right).$$
In the following subsections, we analyze the semi-convergence properties of GSTS iteration method according to Lemma \[lem:1\].
The conditions for $\gamma(\mathcal{G}(\omega_{1},\omega_{2},\tau))<1$
----------------------------------------------------------------------
Based on the singular value decomposition of $E$ defined in , we have $$\begin{aligned}
B=E^{*}P^{-1}E&=&(V_{1},V_{2})\left(
\begin{array}{cc}
\Sigma_{r} & 0 \\
0 & 0 \\
\end{array}
\right)\left(
\begin{array}{c}
U_{1}^{*} \\
U_{2}^{*} \\
\end{array}
\right)P^{-1}(U_{1},U_{2})\left(
\begin{array}{cc}
\Sigma_{r} & 0 \\
0 & 0 \\
\end{array}
\right)\left(
\begin{array}{c}
V_{1}^{*} \\
V_{2}^{*} \\
\end{array}
\right)\\
&=&(V_{1},V_{2})\left(
\begin{array}{cc}
\Sigma_{r}\hat{P}\Sigma_{r} & 0 \\
0 & 0 \\
\end{array}
\right)\left(
\begin{array}{c}
V_{1}^{*} \\
V_{2}^{*} \\
\end{array}
\right),\end{aligned}$$ where $\hat{P}=U_{1}^{*}P^{-1}U_{1}$. The Moore-Penrose inverse of $B$ can be written as $$B^{\dag}=(V_{1},V_{2})\left(
\begin{array}{cc}
(\Sigma_{r}\hat{P}\Sigma_{r})^{-1} & 0 \\
0 & 0 \\
\end{array}
\right)\left(
\begin{array}{c}
V_{1}^{*} \\
V_{2}^{*} \\
\end{array}
\right).$$ Using the unitary matrix $\mathcal{P}$ defined in , iteration matrix $\mathcal{G}(\omega_{1},\omega_{2},\tau)$ is unitarily similar to the matrix $\hat{\mathcal{G}}(\omega_{1},\omega_{2},\tau)=\mathcal{P}^{*}\mathcal{G}(\omega_{1},\omega_{2},\tau)\mathcal{P}$. Hence, we in this subsection only need to analyze $\gamma(\hat{\mathcal{G}}(\omega_{1},\omega_{2},\tau))<1$.
Define matrices $\hat{M}=U^{*}MU$ and $\hat{S}_P=\Sigma_{r}\hat{P}\Sigma_{r}$, then $$\hat{\mathcal{G}}(\omega_{1},\omega_{2},\tau)=\mathcal{P}^{*}\mathcal{G}(\omega_{1},\omega_{2},\tau)\mathcal{P}
=\left(
\begin{array}{cc}
\hat{\mathcal{G}}_{1}(\omega_{1},\omega_{2},\tau) & 0 \\
\hat{\mathcal{L}}(\omega_{1},\omega_{2},\tau) & I_{q-r} \\
\end{array}
\right),$$ where $$\hat{\mathcal{G}}_{1}(\omega_{1},\omega_{2},\tau)=\left(
\begin{array}{cc}
(1-\tau)I_{p}-\tau\omega_{2}(1-\omega_{1})\hat{M}^{-1}E_{r}\hat{S}_P^{-1}E^{*}_{r} & -\tau \hat{M}^{-1}E_{r}+\tau\omega_{1}\omega_{2}\hat{M}^{-1}E_{r}\hat{S}_P^{-1}E^{*}_{r}\hat{M}^{-1}E_{r} \\
\tau(1-\omega_{1})\hat{S}_P^{-1}E^{*}_{r} & I_{q}-\tau\omega_{1}\hat{S}_P^{-1}E^{*}_{r}\hat{M}^{-1}E_{r} \\
\end{array}
\right)$$ and $$\hat{\mathcal{L}}(\omega_{1},\omega_{2},\tau)=\left(
\begin{array}{cc}
\tau(1-\omega_{1})V_{2}^{*}V_{1}\hat{S}_P^{-1}E^{*}_{r} & -\tau\omega_{1}V_{2}^{*}V_{1}\hat{S}_P^{-1}E^{*}_{r}\hat{M}^{-1}E_{r} \\
\end{array}
\right).$$ As $E_{r}$ is of full column rank and $\hat{S}_P^{-1}$ is nonsingular, then $\hat{\mathcal{L}}(\omega_{1},\omega_{2},\tau)\neq 0$, so $\gamma(\hat{\mathcal{G}}(\omega_{1},\omega_{2},\tau))<1$ if and only if $\rho(\hat{\mathcal{G}}_{1}(\omega_{1},\omega_{2},\tau))<1$.
Analogously, $\hat{\mathcal{G}}_{1}(\omega_{1},\omega_{2},\tau)$ is the iteration matrix of the GSTS iteration method applied for the nonsingular saddle-point problem $$\label{03}
\hat{\mathcal{A}}\hat{u}=\left(
\begin{array}{cc}
\hat{M} & E_{r} \\
-E^{*}_{r} & 0 \\
\end{array}
\right)\left(
\begin{array}{c}
\hat{u}_{1} \\
\hat{u}_{2} \\
\end{array}
\right)=\left(
\begin{array}{c}
\hat{f}_{1} \\
\hat{f}_{2} \\
\end{array}
\right)=\hat{f},$$ where $\hat{M}$ is a Hermitian positive definite matrix, $E_{r}$ is of full column rank and $$\label{04}
\hat{\mathcal{B}}(\omega_{1},\omega_{2})=(\hat{\mathcal{B}}_{c}+\omega_{1}\hat{\mathcal{K}}_{L})\hat{\mathcal{B}}_{c}^{-1}(\hat{\mathcal{B}}_{c}
+\omega_{2}\hat{\mathcal{K}}_{U}),$$ with $$\hat{\mathcal{B}}_{c}=\left(
\begin{array}{cc}
\hat{M} & 0 \\
0 & \hat{S}_P \\
\end{array}
\right)$$ being Hermitian positive definite since $\hat{S}_P\in \mathbb{C}^{r\times r}$ is Hermitian positive definite.
Under this situation, we denote by $$\alpha:=\frac{z^{*}E_{r}^{*}\hat{M}^{-1}E_{r}z}{z^{*}z}\quad \text{and} \quad \beta_{2}:=\frac{z^{*}\hat{S}_Pz}{z^{*}z},$$ By making use of Theorem 3.3 in [@Krukier2013], we derive the following result.
\[lem:2\] Denote $\tilde{\omega}=(\omega_{1}-1)(\omega_{2}-1)$. Let matrices $\mathcal{A}$ and $\mathcal{B}(\omega_{1},\omega_{2})$ be defined by and , respectively, and $B$=$E^{*}P^{-1}E$. Then $\gamma(\mathcal{G})<1$ holds, provided that the parameters $\omega_{1}$, $\omega_{2}$ satisfy $$\tilde{\omega}<\frac{\alpha+\beta_{2}}{\alpha},$$ and the parameter $\tau$ satisfies
1. if $[\beta_{2}+(1-\tilde{\omega})\alpha]^{2}-4\alpha\beta_{2}\leq 0$, then $$0<\tau<\frac{\beta_{2}+(1-\tilde{\omega})\alpha}{\alpha};$$
2. if $[\beta_{2}+(1-\tilde{\omega})\alpha]^{2}-4\alpha\beta_{2}> 0$, then $$0<\tau<\frac{\beta_{2}+(1-\tilde{\omega})\alpha-\sqrt{[\beta_{2}+(1-\tilde{\omega})\alpha]^{2}-4\alpha\beta_{2}}}{\alpha}.$$
The conditions for $\text{null}(\mathcal{M}^{\dag}\mathcal{A})=\text{null}(\mathcal{A})$
----------------------------------------------------------------------------------------
From iteration scheme , we have $\mathcal{M}=\tau\mathcal{B}(\omega_{1},\omega_{2})$, which means $$\text{null}(\mathcal{M}^{\dag}\mathcal{A})=\text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}).$$ In the following, we only need to verify $\text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})=\text{null}(\mathcal{A})$.
\[lem:3\] Let matrices $\mathcal{A}$ and $\mathcal{B}(\omega_{1},\omega_{2})$ be defined by and , respectively, and $B$=$E^{*}P^{-1}E$ with $P$ being Hermitian positive definite. Then $\text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})=\text{null}(\mathcal{A})$.
Let $x\in\mathbb{C}^{p+q}$ satisfy $\mathcal{B}(\omega_{1},\omega_{2})\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}x=0$, then $$\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}x=\mathcal{B}(\omega_{1},\omega_{2})^{\dag}
(\mathcal{B}(\omega_{1},\omega_{2})\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}x)=0.$$ So, we have $$\text{null}(\mathcal{B}(\omega_{1},\omega_{2})\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})\subseteq \text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}).$$ Note that $\text{null}(\mathcal{B}(\omega_{1},\omega_{2})\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})\supseteq \text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})$ is obvious, we get $$\label{05}
\text{null}(\mathcal{B}(\omega_{1},\omega_{2})\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})= \text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}).$$ Simple calculation gives $$\label{06}
\mathcal{B}(\omega_{1},\omega_{2})\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}=\left(
\begin{array}{cc}
I_{p} & 0 \\
0 & B^{\dag}B \\
\end{array}
\right)\left(
\begin{array}{cc}
M & E \\
-E^{*} & 0 \\
\end{array}
\right)=\left(
\begin{array}{cc}
M & E \\
-E^{*} & 0 \\
\end{array}
\right)=\mathcal{A}.$$ Hence, using and , we finally obtain that $\text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})=\text{null}(\mathcal{A})$.
the conditions for index$(I-\mathcal{G}(\omega_{1},\omega_{2},\tau))=1$
-----------------------------------------------------------------------
\[lem:4\] Let matrices $\mathcal{A}$ and $\mathcal{B}(\omega_{1},\omega_{2})$ be defined by and , respectively, and $B$=$E^{*}P^{-1}E$, with $P$ being Hermitian positive definite. Then $\text{index}(I-\mathcal{G}(\omega_{1},\omega_{2},\tau))=1$ or equivalently, $$\label{29}
\text{rank}(I-\mathcal{G}(\omega_{1},\omega_{2},\tau))=\text{rank}((I-\mathcal{G}(\omega_{1},\omega_{2},\tau))^{2}).$$
Since $I-\mathcal{G}(\omega_{1},\omega_{2},\tau)=\tau\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}$, the equality holds if $$\text{null}((\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})^{2})=\text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}).$$ Since $\text{null}((\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})^{2})\supseteq \text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})$ is obvious, we only need to prove $$\text{null}((\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})^{2})\subseteq \text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}).$$
Suppose that $x=(x_{1}^{*},x_{2}^{*})^{*}\in\mathbb{C}^{p+q}$ satisfies $(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})^{2}x=0$, we have $$\begin{aligned}
\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}x&=&\left(
\begin{array}{cc}
I_{p}+\omega_{2}(1-\omega_{2})M^{-1}EB^{\dag}E^{*} & M^{-1}E-\omega_{1}\omega_{2}M^{-1}EB^{\dag}E^{*}M^{-1}E \\
(\omega_{1}-1)B^{\dag}E^{*} & \omega_{1}B^{\dag}E^{*}M^{-1}E \\
\end{array}
\right)\left(
\begin{array}{c}
x_{1} \\
x_{2} \\
\end{array}
\right)\\
&=&\left(
\begin{array}{cc}
(I_{p}+\omega_{2}(1-\omega_{2})M^{-1}EB^{\dag}E^{*})x_{1}+(M^{-1}E-\omega_{1}\omega_{2}M^{-1}EB^{\dag}E^{*}M^{-1}E)x_{2} \\
(\omega_{1}-1)B^{\dag}E^{*}x_{1}+(\omega_{1}B^{\dag}E^{*}M^{-1}E)x_{2} \\
\end{array}
\right)\\
&=&\left(
\begin{array}{c}
y_{1} \\
y_{2} \\
\end{array}
\right)\equiv y.\end{aligned}$$ In the following, we only need to prove $\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}x=y=0$. Owing to $\text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})=\text{null}(\mathcal{A})$ and $$\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A}y=(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})^{2}x=0,$$ we have $\mathcal{A}y=0$, i.e., $$\label{07}
My_{1}+Ey_{2}=0 \quad \text{and} \quad -E^{*}y_{1}=0.$$ Since $M$ is nonsingular, solving $y_{1}$ from the first equality of and taking into the second equality, we have $E^{*}M^{-1}Ey_{2}=0$, which means $$(Ey_{2})^{*}M^{-1}(Ey_{2})=y_{2}(E^{*}M^{-1}Ey_{2})=0.$$ Owing to the positive definiteness of matrix $M^{-1}$, we can obtain that $Ey_{2}=0$. Hence, using the first equality of gives $y_{1}=0$.
Using $Ey_{2}=0$ and $B^{\dag}BE^{*}=E^{*}$[@ZhangShen2013116], we have $$\begin{aligned}
y_{2}&=&(\omega_{1}-1)B^{\dag}E^{*}x_{1}+(\omega_{1}B^{\dag}E^{*}M^{-1}E)x_{2}\\
&=&B^{\dag}E^{*}P^{-1}[(\omega_{1}-1)EB^{\dag}E^{*}x_{1}+(\omega_{1}EB^{\dag}E^{*}M^{-1}E)x_{2}]\\
&=&B^{\dag}E^{*}P^{-1}(Ey_{2})=0\end{aligned}$$ Finally, we obtain $y=(y_{1}^{*},y_{2}^{*})^{*}=0$, so $\text{null}((\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})^{2})=\text{null}(\mathcal{B}(\omega_{1},\omega_{2})^{\dag}\mathcal{A})$.
Using Lemmas \[lem:1\] and \[lem:2\]-\[lem:4\], we obtain the semi-convergence property of GSTS iteration method for singular saddle-point linear systems.
Let parameters $\omega_1$, $\omega_2$ and $\tau$ satisfy the conditions of Lemma \[lem:2\] and matrix $B=E^{*}P^{-1}E$ with $P$ being a Hermitian positive definite approximation of matrix $M$. Then, the GSTS iteration method used for solving singular saddle-point linear system is semi-convergent.
In GSTS iteration method, we can particularly choose $P=M$ since $P$ is an approximation of $M$. In this case, we have $B=E^{*}M^{-1}E=S_M$. Simple calculation gives $$\alpha=\frac{z^{*}E_{r}^{*}\hat{M}^{-1}E_{r}z}{z^{*}z}
=\frac{z^{*}(\Sigma_{r},0)U^{*}M^{-1}U\left(
\begin{array}{c}
\Sigma_{r} \\
0 \\
\end{array}
\right)z}{z^{*}z}=\frac{z^{*}\Sigma_{r}U_{1}^{*}M^{-1}U_{1}\Sigma_{r}z}{z^{*}z},$$ and $$\beta_{2}=\frac{z^{*}\hat{S}_Pz}{z^{*}z}=\frac{z^{*}\Sigma_{r}\hat{P}^{-1}\Sigma_{r}z}{z^{*}z}=\frac{z^{*}\Sigma_{r}U_{1}^{*}P^{-1}U_{1}\Sigma_{r}z}{z^{*}z}
=\frac{z^{*}\Sigma_{r}U_{1}^{*}M^{-1}U_{1}\Sigma_{r}z}{z^{*}z}.$$ Obviously, under this assumption, we have $\alpha=\beta_{2}$. The convergence property of the particular GSTS method becomes
Denote $\tilde{\omega}=(\omega_{1}-1)(\omega_{2}-1)$. Let matrices $\mathcal{A}$ and $\mathcal{B}(\omega_{1},\omega_{2})$ be defined by and , respectively, and $B$=$E^{*}M^{-1}E$. Then, the GSTS iteration method is semi-convergent, provided that the parameters $\omega_{1}$, $\omega_{2}$ satisfy $\tilde{\omega}<2$ and the parameter $\tau$ satisfies
1. if $0\leq\tilde{\omega}<2$, then $0<\tau<2-\tilde{\omega}$;
2. if $\tilde{\omega}<0$, then $0<\tau<2-\tilde{\omega}-\sqrt{\tilde{\omega}(\tilde{\omega}-4)}$.
Numerical results {#sec5}
=================
In this section, we assess the feasibility and robustness of the GSTS iteration methods for solving the singular saddle-point problems . In addition, the preconditioning effects of the GSTS preconditioners for GMRES(10) and QMR methods will also be tested.
Consider the Stokes equations of the following form $$\label{34}
\left\{
\begin{split}
&-\nu \Delta \textbf{u}+\nabla p = \textbf{f},\quad \text{in }\Omega,\\
&-\nabla\cdot \textbf{u}=0,\quad \text{in }\Omega,
\end{split}\right.$$ where $\Omega$ is an open bounded domain in $\mathbb{R}^2$, vector $\textbf{u}$ represents the velocity in $\Omega$, function $p$ represents pressure, and the scalar $\nu>0$ is the viscosity constant. The boundary conditions are $\textbf{u} = (0,0)^T$ on the three fixed walls $(x = 0, y = 0, x = 1)$, and $\textbf{u} = (1, 0)^T$ on the moving wall $(y = 1)$.
Dividing $\Omega$ into a uniform $l\times l$ grid with mesh size $h=1/l$ and discretizing by the “marker and cell” (MAC) finite difference scheme [@HW19652182; @Elman19991299], the singular saddle-point system is obtained, where $$M=\nu\left( \begin{array}{cc} A_1 & 0 \\ 0 & A_2 \end{array} \right)\in\mathbb{R}^{2l(l-1)\times 2l(l-1)},\quad E^*=(E_1,E_2)\in\mathbb{R}^{l^2\times 2l(l-1)}.$$ The coefficient matrix $\mathcal{A}$ of has the following properties: $M$ is symmetric and positive definite, $\text{rank}(E) = l^2-1$, thus $\mathcal{A}$ is singular.
Based on the different choices of matrix $P$ and parameters $\omega_1$, $\omega_2$ and $\tau$, we test five cases of the GSTS iteration method listed in Table \[tab:01\]. In order to reduce the complexity for finding the experimental optimal values of $\omega_1$, $\omega_2$ and $\tau$, we particularly choose $\omega_{1}=\tau$. The last two cases of GSTS iteration method reduce to the generalized successive overrelaxation (GSOR) methods discussed in [@ZBY2009808]. Here, $\omega_{\exp}$ and $\tau_{\exp}$ in GSTS methods denote the experimental optimal values of the iteration parameters $\omega_{1}$ and $\tau$, respectively, while $\omega_{\mathrm{opt}}$ and $\nu_{\mathrm{opt}}$ in GSOR denote the theoretical optimal values; see Theorem 4.1 in [@ZBY2009808].
[\*[3]{}[l]{}]{} Methods& Preconditioning matrix $B$ &Parameters\
GSTS I &$E^{*}M^{-1}E$ & $\omega_1=\tau=\tau_{\exp}$, $\omega_2=\omega_{\exp}$\
GSTS II &$I+E^{*}P^{-1}E$, with $P=\text{diag}(M)$ & $\omega_1=\tau=\tau_{\exp}$, $\omega_2=\omega_{\exp}$\
GSTS III &$I+E^{*}P^{-1}E$, with $P=\text{tridiag}(M)$ & $\omega_1=\tau=\tau_{\exp}$, $\omega_2=\omega_{\exp}$\
GSOR I &$\frac{\tau}{\nu}(I+E^{*}P^{-1}E)$, with $P=\text{diag}(M)$ & $\omega_1=\tau=\omega_{\mathrm{opt}}$, $\omega_2=0$, $\nu=\tau_{\mathrm{opt}}$\
GSOR II &$\frac{\tau}{\nu}(I+E^{*}P^{-1}E)$, with $P=\text{tridiag}(M)$ & $\omega_1=\tau=\omega_{\mathrm{opt}}$, $\omega_2=0$, $\nu=\tau_{\mathrm{opt}}$\
In actual computations, we choose $l=25$ and the right-hand-side vector $f\in \mathbb{R}^{3l^{2}-2l}$ such that the exact solution of is $u^{*}=(1,2,\cdots,3l^{2}-2l)^{T}\in \mathbb{R}^{3l^{2}-2l}$. The iteration methods are started from zero vector and terminated once the current iterate $x^{(k)}$ satisfies $$\label{40}
\text{RES}=\sqrt{\frac{\|f_{1}-Mu^{(k)}_{1}-Eu^{(k)}_{2}\|^{2}_{2}+\|f_{2}+E^*u^{(k)}_{1}\|^{2}_{2}}{\|f_{1}\|^{2}_{2}+\|f_{2}\|^{2}_{2}}}< 10^{-6}.$$ In addition, all codes were run in MATLAB \[version 7.10.0.499 (R2010a)\] in double precision and all experiments were performed on a personal computer with 3.10GHz central processing unit \[Intel(R) Core(TM) Duo i5-2400\] and 3.16G memory.
In Table \[tab:02\], we present the numerical results including iteration steps (denoted as IT), elapsed CPU time in seconds (denoted as CPU) and relative residuals (denoted as RES) of the GSTS and GSOR iteration methods listed in Table \[tab:01\] and GMRES method. From the numerical results we see that all the testing methods can converge to the approximate solutions. The five cases of GSTS method perform better than GMRES method in iteration steps and CPU times. During the five cases of GSTS method, the second and third cases, i.e., GSTS II and GSTS III, always outperform the fourth and fifth cases, i.e., GSOR I and GSOR II methods, respectively, especially for the elapsed CPU time. In GSTS I, we choose $B$ being a singular matrix. Comparing with GMRES and other four cases of GSTS method, GSTS I uses the least iteration number and CPU time to achieve the stop criterion.
[rl\*[5]{}[c]{}]{} &Method&$\omega_{\exp}(\omega_{\mathrm{opt}})$&$\tau_{\exp}(\tau_{\mathrm{opt}})$&IT&CPU&RES\
$\nu=1$ & GSTS I & 0.98 & 1.01 & 3 & 0.0156 & 8.3841e-7\
& GSTS II & 0.99 & 1.03 & 12 & 0.0312 & 9.7573e-7\
& GSTS III & 0.97 & 1.02 & 13 & 0.0468 & 9.6413e-7\
& GSOR I & 0.89 & 2.07 & 14 & 0.1248 & 4.0305e-7\
& GSOR II & 0.89 & 2.11 & 14 & 0.1872 & 4.0232e-7\
& GMRES & $-$ & $-$ & 182 & 1.3104 & 9.9927e-7\
$\nu=0.01$ & GSTS I & 0.99 & 1.00 & 3 & 0.0468 & 6.6282e-7\
& GSTS II & 0.01 & 0.30 & 73 & 0.1872 & 9.7501e-7\
& GSTS III & 0.02 & 0.34 & 63 & 0.1872 & 9.6153e-7\
& GSOR I & 0.38 & 0.24 & 68 & 0.4524 & 8.1165e-7\
& GSOR II & 0.43 & 0.28 & 58 & 0.5304 & 8.5245e-7\
& GMRES & $-$ & $-$ & 404 & 6.5988 & 9.7660e-7\
$\nu=0.0001$ & GSTS I & 0.98 & 1.00 & 3 & 0.0312 & 1.4299e-12\
& GSTS II & 0.01 & 0.17 & 115 & 0.3901 & 9.5478e-7\
& GSTS III & 0.01 & 0.24 & 110 & 0.3276 & 9.4855e-7\
& GSOR I & 0.24 & 0.14 & 164 & 0.7176 & 9.1062e-7\
& GSOR II & 0.32 & 0.20 & 118 & 0.8580 & 9.8351e-7\
& GMRES & $-$ & $-$ & 637 & 15.6004 & 9.7598e-7\
In addition to using GSTS as an iteration solver, we also use it to precondition GMRES method. The preconditioning effects of the five cases of GSTS method are compared with those of the Hermitian and skew-Hermitian splitting (HSS) preconditioner [@BGL20051; @BGN2002603; @BG200520] and the constraint preconditioner [@KGW20001300; @ZhangWei2010139; @Cao20081382; @ZhangShen2013116]. Here, the non-singular constraint preconditioner (CP) is of the form $$\mathcal{P}_c=\left( \begin{array}{cc} P & E \\ -E^* & I \end{array} \right),$$ where $P$ is an approximate matrix of $M$ and $I$ is an identity matrix. We name the constraint preconditioners $\mathcal{P}_c$ with $P=\text{diag}(M)$ and $P=\text{tridiag}(M)$, respectively, as CP I and CP II preconditioners. The HSS preconditioner is of the form $$\mathcal{P}_{h}=(\alpha I+\mathcal{A}_{H})(\alpha I+\mathcal{A}_{S}),$$ where $\alpha>0$ is a constant, $\mathcal{A}_{H}$ and $\mathcal{A}_{S}$ are defined in . In the implementation, $\alpha$ is chosen to be the experimental optimal value.
[l\*[9]{}[c]{}]{} &&&&&&\
Method&&IT&CPU&&IT&CPU&&IT&CPU\
GSTS I && 14 & 0.0780 && 11 & 0.1560 && 3 & 0.1248\
GSTS II && 17 & 0.0624 && 18 & 0.1872 && 4 & 0.1248\
GSTS III && 17 & 0.0757 && 16 & 0.2028 && 3 & 0.1404\
GSOR I && 26 & 0.0780 && 27 & 0.6084 && 8 & 0.1560\
GSOR II && 26 & 0.0936 && 22 & 0.7332 && 6 & 0.1716\
CP I && 34 & 0.0793 && 29 & 0.3225 && 41 & 0.4209\
CP II && 34 & 0.0880 && 28 & 0.3573 && 41 & 0.4370\
HSS && 21 & 0.0816 && 22 & 0.4212 && 19 & 0.4056\
In Table \[tab:03\], we list the iteration numbers and CPU times of the preconditioned GMRES methods used for solving singular saddle-point linear system . From the numerical results, we see that the superiorities of GSTS preconditioners, comparing with the constraint and HSS preconditioners, become more and more evident with the decrease of parameter $\nu$. This may be because the smaller of the parameter $\nu$, the stronger of the skew-Hermitian part of the saddle-point matrix. In addition, we can also find that the preconditioning effect of singular GSTS I preconditioner is the best one during the eight preconditioners listed in Table \[tab:03\].
Thus, we can conclude that the GSTS iteration methods, no matter used as solvers or as preconditioners for GMRES method, are always feasible and effective for solving singular saddle-point linear systems.
Conclusion {#sec6}
==========
In this work, we used the GSTS iteration methods to solve *singular* saddle-point linear system . For each of the two choices of preconditioning matrix $B$, the semi-convergence conditions of GSTS iteration method were derived. Numerical results verified the effectiveness of the GSTS method both used as a solver and as a preconditioner for the GMRES method.
However, the GSTS method involves three iteration parameters $\omega_1$, $\omega_2$ and $\tau$. The choices of these parameters were not discussed in this work since it is a very difficult and complicated task. Considering that the efficiency of GSTS method largely depends on the values of these parameters, how to determine efficient and easy calculated parameters should be a direction of future research.
|
---
abstract: 'We report on the coherence properties of single photons from chromium-based colour centres in diamond. We use field-correlation and spectral lineshape measurements to reveal the interplay between slow spectral wandering and fast dephasing mechanisms as a function of temperature. We show that the zero-phonon transition frequency and its linewidth follow a power-law dependence on temperature indicating that the dominant fast dephasing mechanisms for these centres are direct electron-phonon coupling and phonon-modulated Coulomb coupling to nearby impurities. Further, the observed reduction in the quantum yield for photon emission as a function of temperature is consistent with the opening of additional nonradiative channels through thermal activation to higher energy states predominantly and indicates a near-unity quantum efficiency at 4 K.'
author:
- 'T. Müller$^{1}$, I. Aharonovich$^{2}$, Z. Wang$^{3}$, X. Yuan$^{3}$, S. Castelletto$^{4}$, S. Prawer$^{5}$, M. Atatüre$^{1}$'
title: 'Phonon-induced dephasing of chromium colour centres in diamond'
---
Diamond plays a key role in a wide range of applications in both electronics [@electronics] and photonics [@Greentree] today. Due to its wide bandgap it hosts a variety of optically active centres in the visible and the near infrared part of the spectrum [@Zaitsev]. The nitrogen-vacancy (NV) centre has attracted great attention due to its remarkable spin properties as a model system for quantum technologies [@Gruber; @Wrachtrup; @Degen]. Other optically active centres such as silicon-vacancy [@Neu], nickel [@Nadolinny], xenon [@Xenon] and chromium [@Igor1] have also shown desirable photonic properties such as short lifetimes, predominant emission into the zero-phonon line (ZPL), and spectral tuning via DC Stark effect [@Tamarat; @Muller; @Bassett]. However, the ZPL spectrum of diamond colour centres are typically broader than the radiative linewidth evidencing the influence of dephasing mechanisms on the optical transitions [@footnote1]. These dephasing mechanisms can range from slow spectral wandering of the transition due to charge fluctuations in the environment to fast dephasing processes. The former can be remedied by direct feedback control on the transition, which has been demonstrated for NV centres [@Acosta]. The latter is irreversible and presents a fundamental limit to the single-photon coherence even with feedback. Therefore, it is essential to identify the source for these mechanisms, as well as the extent of their contributions to the spectral broadening of these centres. Here, we investigate the dephasing mechanisms influencing the chromium colour centres and report that the predominant sources for fast dephasing are direct electron-phonon coupling and phonon-modulated Coulomb coupling to nearby impurities, as manifested by the power-law temperature dependence of the transition frequency and linewidth. Further, the temperature dependence of the quantum yield reveals the existence of a thermal excitation mechanism to higher energy states similar to the neutral silicon-vacancy centres in diamond [@Warwick].
![(Color online)(a) Schematic of the confocal microscope used for the experiments. BS is a beamsplitter and LP is a 740 nm longpass filter to remove residual laser reflection from luminescence by the chromium emitters. (b) Lifetime measured for the 773.3 nm transition line in figure 1a) inset. A value of 1.7 ns is extracted from the single exponential fit. (c) Autocorrelation measurement for the 773.3 nm line with g(2)(0) = 0.28. This value reflects the finite system response time and background emission from the crystal. The respective fit again reveals an excited state lifetime of 1.7 ns. []{data-label="fig1"}](Figure1_b.jpg "fig:"){width="47.00000%"}\
Chromium centres can be formed either by chemical vapour deposition (CVD) growth, where the etched sapphire substrate acts as the source for chromium atoms [@Igor2], or by implantation into bulk diamond [@Castelletto]. While the centres in bulk diamond exhibit narrower emission linewidth than those grown in nano- or microcrystals, they suffer from low photon collection efficiency due total internal reflection, as well as reduced photostability [@Muller]. We therefore concentrate on centres in microcrystals (size [$\sim$]{}2 microns) for the investigation of dephasing mechanisms. The experimental setup used for the optical measurements is illustrated in Fig. 1(a). A temperature-controlled cryostat and a homemade confocal optical microscope are used to access the centres optically. The chromium centres are excited by a continuous wave laser at 700 nm and the emission ($\sim 780$ nm) is collected into a single-mode fiber. A time-correlated photon counting setup is used for excited-state lifetime measurements, as well as intensity-correlation measurements. The excited-state lifetimes of chromium centres in bulk, micro- and nano-crystal diamonds can vary significantly [@Castelletto; @Siyushev]. Figures 1(b) and 1(c) display exemplary results for these measurements performed at 4 K on a single chromium centre in a microcyrstal, and the value of 1.7 ns extracted from the measurements is consistent with the range of values reported to-date. Based on this value, transform-limited photon emission is expected to result in a coherence time of 3.4 ns. A calibrated and scanable Michelson interferometer and a spectrometer with 8-GHz resolution are used in parallel to quantify the degree of dephasing via the first-order coherence of the photon emission \[see Fig. 1(a)\].
![(Color online) First-order field autocorrelation measurement, $g^{(1)}$, performed at liquid helium temperature on centres A and B, as well as a chromium centre implanted in bulk diamond. (a) For centre A the visibility extracted from interference fringes at different time delays between the two arms of a Michelson interferometer (filled black circles) exhibits a Gaussian decay pattern with a $\tau_{1/e} = 62$ ps (solid red curve). An exponential function is shown for comparison (dotted black curve). The measured $g^{(1)}$ for a chromium centre located in bulk diamond (filled blue circles) display a coherence time longer than 700 ps (solid red curve). (b) For centre B the extracted visibility (filled black circles) follows an exponential decay profile (solid red curve) with a coherence time of 57 ps. Again, a Gaussian profile (dotted black curve) is shown for comparison along with the $g^{(1)}$ of the chromium centre in bulk diamond. []{data-label="fig:fig2"}](Figure2.jpg "fig:"){width="45.00000%"}\
Figures 2(a) and 2(b) display the measured interference visibility $V = (I_{max}-I_{min})/(I_{max}+I_{min})$ as a function of relative time delay due to path-length difference for photons from two colour centres situated in two different microcrystals (labelled A and B, with ZPL at 756 nm and 770 nm, respectively). Filled black circles are the experimental data and the solid red curves are the theoretical fits. Common to both colour centres is the coherence timescale approximately 30 times lower than the predicted transform limit. The coherence functions for these centres are, however, fundamentally different from each other. Centre A shown in Fig. 2(a) displays a predominantly Gaussian decay of coherence (the solid red curve is a Gaussian fit with $\tau_{1/e} = 62$ ps), whereas for centre B shown in Fig. 2(b) the coherence decays exponentially (the solid red curve displays an exponential fit with $\tau_{1/e} = 57$ ps). The exponential form of the coherence function for centre B indicates a dominant contribution from the irreversible dephasing mechanisms to the emission spectrum, while the Gaussian-like coherence function of centre A limits an exponential contribution to a timescale not shorter than 210 ps. This corresponds to 750-MHz upper bound to the irreversible broadening for the linewidth for this centre. The relatively short coherence times exhibited by both emitters as well as the different profiles in spectra and coherence functions indicate that the immediate environment of the emitters has a varying impact on their photonic properties. In order to emphasize the effect of the environment on photonic coherence, similar measurements performed on a chromium centre implanted directly in bulk diamond with lower density of impurities is shown in both panels as filled blue circles. This centre shows a ZPL at 790 nm. A coherence time longer than 700 ps is extracted for a Gaussian coherence function (solid red curve) - an order of magnitude longer than the coherence time measured for nanodiamonds.\
{width="90.00000%"}\
The degree of optical dephasing can be revealed in first-order coherence as well as spectral measurements. If the line broadening is small compared to the natural linewidth, the former is more accurate, whereas for strong dephasing, the latter provides a better means of determining emission linewidths. Figure 3(a) displays the characteristic broadening and red shift of the emission spectrum of another chromium centre (labeled centre C) as a function of lattice temperature between 9 K and 126 K. An analysis of the spectrum reveals that the Gaussian contribution remains constant for all temperatures, while the Lorentzian contribution is temperature dependent. This leads to a switch over from a predominantly Gaussian-like lineshape at low temperatures to a Lorentzian lineshape at higher temperatures for the 10 emitters investigated. This can be seen in the linear-log plots of the lineshape at two different temperatures displayed in the two insets of Fig. 3(a). The diamond bandgap itself will be modified as a function of temperature, following the Varshni law [@Varshni; @Varshni_footnote], and shallow centres will inherit this behaviour. Figure 3(b) displays the fractional change of the transition frequency ($\Delta\omega / \omega$) as a function of temperature, following a power law of the form $\Delta\omega / \omega \propto T^{\alpha}$ with $\alpha$=3.3. This frequency shift is distinctly stronger than the Varshni temperature dependence of bandgap energy for diamond, as is evident in Fig. \[fig:fig3\](b) (black dotted curve). This reveals that all optically relevant states are sufficiently far in energy from the band edges, as expected for deep defects in diamond. Fig. 3(c) shows the temperature dependence of the linewidth $\gamma$ of the Lorentzian contribution to the emission spectrum of centre C, following a power law of the form $\gamma \propto T^{\beta}$, with $\beta$=3.7. These measurements were repeated for 10 separate centres, and the average values of $\alpha$ and $\beta$ are $3.5\pm0.3$ and $3.4\pm0.3$, respectively. In the low temperature regime data suggests a temperature-independent dephasing mechanism is present for this centre limiting the linewidth to 4.1 GHz. This residual linewidth varies from centre to centre. We note however that a value of 4.1 GHz is within the resolution of the spectrometer ($\sim$ 8 GHz).
A strong candidate for such temperature-dependent broadening mechanism is interaction with lattice phonons. The simplest model for electron-phonon coupling considers quasi-elastic scattering of phonons with a Debye density of states [@Maradudin] from a non-degenerate transition of a centre. This model predicts a $T^7$-dependence for dephasing and a $T^4$-dependence for the fractional change of the transition frequency. The strong deviation of the $\beta$ values for chromium centres from the prediction by this model suggests the presence of an additional mechanism for temperatures below the Debye temperature $T_{\textrm{D} }$. An example of this is already seen in the NV linewidth, which exhibits a $T^5$ behaviour in ultrapure diamond due to Jahn-Teller effect in the excited states [@Fu]. Further, experimentally determined values of $\alpha = 3.1$ and $\beta = 3$ have been reported for the N3 colour centre related to nitrogen aggregates in diamond [@Halperin]. The particular temperature dependence was attributed to a pronounced deviation of the actual phonon density of states in diamond from the Debye model [@Smith], which was confirmed experimentally for the N3 colour centre using the vibrational structure in the N3 absorption spectrum. A numerical evaluation of linewidth and transition frequency shift using the modified phonon density of states in bulk diamond as a function of temperature according to Ref. [@Maradudin] results in $a = 3.1$ and $b = 3.7$, which is in reasonable agreement with the values reported here. This model is plausible for chromium centres as well, although an experimental mapping of the phonon density of states is not feasible due to the small Huang-Rhys parameter [@Huang] of these centres, which is less than 0.05 [@Siyushev]. That said, the variation of $\alpha$ and $\beta$ from centre to centre suggests that a mechanism governed more by local rather than global properties of the material may be playing a central role in the photon dephasing.
An alternative model takes into account explicitly the impurity-rich nature of a material and incorporates the effect of phonons to the coupling of the centres (e.g. Coulombic and spin-dipolar) to nearby lattice impurities and other centres [@Hizhnyakov]. Consequently, a time-varying potential at the location of a centre leads to a $T^3$ dependence of the dephasing rate, valid for the temperature range $T_0 < T < T_D$, with $T_D$ the Debye temperature and $T_0 = c^{3/8}T_D$. The coefficient $c$ is the impurity concentration in the host crystal. Applying this general principle to the chromium emitters, an impurity concentration as high as 1 part-per-million gives $T_0 \sim 10$ K and $T_D = 1850$ K for diamond, so the temperature regime over which the chromium centres are investigated lies within this range. The presence of charges in the immediate environment of the emitter is already evident from the slow-wandering type decay of coherence exhibited by some of the centres (Gaussian coherence profile in Fig \[fig:fig2\] (a)). This confirms the plausibility of such a dephasing mechanism. Further investigation of the temperature dependent dephasing as a function of concentration $c$ of defects is necessary for an unambiguous identification of the dephasing type.
{width="95.00000%"}\
Coupling of a colour centre to a vibronic mode also leads to reduced ZPL intensity with increasing temperature [@Huang; @Davies; @Warwick]: $$\begin{split}
I_0(T)= &I_0(0) \exp{\left[-S \coth\left(\frac{\hbar\omega}{2k_{\textrm{B}}T}\right)\right]}\\
&\times J_0\left[S \, \textrm{csch}\left(\frac{\hbar\omega}{2k_BT}\right)\right]
\label{eq:zplintensity}
\end{split}$$
where $I_0$ is the fraction of emission intensity decaying into the ZPL, $S$ is the Huang-Rhys factor, $\omega$ is the vibronic mode frequency, $\hbar$ is the reduced Planck constant, $k_{\textrm{B}}$ is the Boltzmann constant, and $J_0$ is the zeroeth order Bessel function of the first kind. For chromium centres $S < 0.05$ [@Siyushev] and $\hbar\omega$ is on the order of about 10 m*e*V, similar to other diamond centres [@Zaitsev; @Igor1; @Warwick]. Equation (1) applied to chromium centres yields 95% of the ZPL intensity to be sustained within the temperature range we study here. In strong contrast, Fig. 3(a) displays the significantly more rapid reduction of the ZPL intensity observed as a function of temperature. Such a strong reduction in intensity is uncharacteristic for $S<0.05$ and suggests an additional mechanism taking part.
The quantum efficiency, $\eta$, of a transition is determined by the presence of nonradiative channels connecting the excited and the ground states of a centre and is defined as $\eta=P_r/(P_r+P_{nr})$ where $P_r$ ($P_{nr}$) is the probability for a radiative (nonradiative) transition. Nonradiative channels contributing to $P_{nr}$ can become available as a function of temperature via thermal excitation from the excited state of the centre to higher energy nonradiative states. These states can either belong to the centre, or they could be formed by the impurity density in the vicinity of the emitters. Assuming that the selection rules do not affect the thermalization process, the quantum efficiency becomes temperature dependent: $$\eta(T)=\frac{1}{1+ge^{-\frac{E_{\textrm{th}}}{k_{\textrm{B}}T}}}.
\label{eq:quantumyield}$$
The nonradiative decay process is assumed here to have a barrier with activation energy $E_{\textrm{th}}$, and the constant $g=g_{\textrm{th}}/g_{\textrm{ex}}$ is the ratio of degeneracies between the thermally activated level and the excited state [@Warwick]. The presence of a thermal excitation mechanism to additional states is further evidenced by the bunching behaviour in intensity autocorrelation measurements [@Igor1], which appears predominantly for room-temperature measurements and is suppressed at low temperatures. Including this process, the overall intensity change of the ZPL as a function of temperature is given by [@Warwick; @Feng; @Collins]: $$\begin{split}
I_0(T)=&\left(\frac{I(0)}{1+ge^{\left(-\frac{E_{\textrm{th}}}{k_{\textrm{B}}T}\right)}}\right)\exp{\left[-S \coth\left({\frac{\hbar\omega}{2k_{\textrm{B}}T}}\right)\right]}\\&\times J_0\left[S \, \textrm{csch}\left(\frac{\hbar\omega}{2k_{\textrm{B}}T}\right)\right].
\end{split}$$ Fig. 4 (a) presents the calculated intensity of the ZPL according to equation 3 as a function of the number degeneracy of the state and the temperature (for a fixed value of the energy separation, $E_{th}$ = 19.6). Similarly, Fig. 4 (b) presents the intensity as a function of $E_{th}$ and temperature for a fixed value of the degeneracy, $g = 19.0$. Clearly, the intensity is sustained until the temperature is high enough to allow thermalization to be significant and the number of degeneracy determines how rapidly the intensity is reduced beyond this temperature. Figures 4c and 4d present experimental results (blue circles) for two separate chromium centres superimposed on the Eq. 3 (solid black curves) using $E_{\textrm{th}}$ and $g$ as fitting parameters. The corresponding values for best fits are given in Table 1 along with those reported for the neutral silicon-vacancy centre [@Warwick] for comparison. The values for both $g$ and $E_{\textrm{th}}$ for centre C displayed in Fig. 4(c) are higher than those for centre D displayed in Fig. 4(d). The difference in energy separation and degeneracy of the thermally occupied states is not surprising given that the chromium emitters are known to be in an environment comprising a high density of impurity atoms such as additional nitrogen, oxygen and silicon [@Siyushev; @Igor1]. While the model assumes thermal coupling to only one additional state, we interpret from the high values of $g$ and $E_{\textrm{th}}$ that the chromium emitters couple to an ensemble of states with an *effective* energy barrier $E_{\textrm{th}}$ and an overall degeneracy ratio $g$. Intuitively, the number of states or the degeneracy in the ensemble is expected to increase with larger energy barriers $E_{\textrm{th}}$, since the number of allowed energy states introduced by impurities around the chromium emitters should be higher closer to the valence band edge. This situation is depicted in panels (c) and (d) of Fig. 4. In principle, this model can be extended to include a quasi-continuum band with density of states $g(E)$ formed by the impurities at a mean energy barrier $E_{\textrm{th}}$. Straightforwardly, $P_{nr}$ in Eq. (2) then becomes $\int{g(E)\exp(-\frac{E}{k_{\textrm{B}}T})dE}$. We note that using a Gaussian density of states of finite width around a mean energy barrier yields the same functional dependence, however independent measurements would be necessary to reveal the location of the energy barrier per emitter, supported by a density-functional theory calculations of the delocalized electronic states due to the impurities in diamond. The temperature-dependent model we use results in the values of 11% and 23% for the room temperature quantum efficiency for centre C and D, respectively. These efficiencies are consistent with previous reports on chromium centres at room temperature [@Castelletto] and indicates that the strong suppression of thermal excitation allows quantum efficiencies at low temperatures, e.g. 4 K, of order unity.
$g$ $E_{\textrm{th}}$ $\hbar \omega$ $S$
---------- ------------- ------------------- ---------------- ------
Centre C 19.0 $\pm$8 19.6 $\pm$3.5 meV 28.9 meV 0.05
Centre D 4.3 $\pm$2 4.4 $\pm$1.7 meV 28.9 meV 0.05
SiV$^0$ 1-3 5 meV 28 meV 1.5
\[1ex\]
: Fit parameters for the two chromium centres shown in figure and SiV centres for comparison.
\[table:parameters\]
In summary, the temperature dependence of photon emission from single chromium centres in diamond revealed the interplay between slow and fast dephasing mechanisms and the nature of coupling to the impurities in the vicinity of the centres. Due to these coupling mechanisms, the photon coherence times remain about 3.5 times in bulk diamond and 25 times in microcrystals below the theoretical value for transform-limit photon emission. Nonradiative losses due to thermal excitation to additional states are dominant at elevated temperatures, but the quantum efficiency of chromium centres approaches unity at low temperature operation. Deterministic implantation of these centres in ultrapure diamond, or their controlled incoporation in high purity CVD microdiamonds, is therefore needed to make these systems available as good-quality single photon sources.
[**Acknowledgments**]{}
We gratefully acknowledge financial support by the University of Cambridge and the European Research Council (FP7/2007-2013)/ERC Grant agreement no. 209636. Z. Wang and Y. Xin acknowledge the USTC-Cambridge exchange programme for future physicists. We thank C.-Y. Lu for technical assistance, E. Neu for helpful discussions, and D. McCutcheon and A. Nazir for theoretical support.
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---
abstract: 'The multi-GPU open-source package QCDGPU for lattice Monte Carlo simulations of pure SU(N) gluodynamics in external magnetic field at finite temperature and O(N) model is developed. The code is implemented in OpenCL, tested on AMD and NVIDIA GPUs, AMD and Intel CPUs and may run on other OpenCL-compatible devices. The package contains minimal external library dependencies and is OS platform-independent. It is optimized for heterogeneous computing due to the possibility of dividing the lattice into non-equivalent parts to hide the difference in performances of the devices used. QCDGPU has client-server part for distributed simulations. The package is designed to produce lattice gauge configurations as well as to analyze previously generated ones. QCDGPU may be executed in fault-tolerant mode. Monte Carlo procedure core is based on PRNGCL library for pseudo-random numbers generation on OpenCL-compatible devices, which contains several most popular pseudo-random number generators.'
author:
- |
Vadim Demchik[^1], Natalia Kolomoyets[^2] \
\
[*[Dnipropetrovsk National University, 49010 Dnipropetrovsk, Ukraine]{}*]{}
title: |
QCDGPU: open-source package for Monte Carlo lattice simulations\
on OpenCL-compatible multi-GPU systems
---
[*Keywords:*]{} lattice gauge theory, Monte Carlo simulations, GPGPU, OpenCL
Introduction
============
Nowadays graphics processing units (GPU) play a rather important role in high-performance computing (HPC). The proportion of computing systems equipped with the special accelerators (GPUs, MIC, etc.) is growing among supercomputers, which is reflected in the well-known TOP500 list of the most powerful supercomputing systems of the world [@top500:2013]. The most popular programming languages for general-purpose computing on GPU (GPGPU) are CUDA (for NVIDIA GPUs only) and Open Computing Language (OpenCL) [@OpenCL:2011]. Currently more than 80% of all scientific researches are using GPU accelerators performed with CUDA [@hgpustat:2013].
One of the main approaches to study high-energy physics (HEP) phenomena is the lattice Monte Carlo simulations. In 1974 Kenneth G. Wilson proposed a formulation of quantum chromodynamics (QCD) on a space-time lattice [@Wilson:1974sk] – a lattice gauge theory (LGT), which allows to calculate infinite-dimensional path integral with the procedure of computation of finite sums. LGT has many important features, in particular, LGT makes it possible to study low-energy limit of QCD, which is not achievable by analytic methods. In the limit of an infinite number of lattice sites and zero lattice spacing, LGT becomes an ordinary quantum field theory (QFT). Numerical results, obtained by means of lattice approximation, depend on number of lattice sites, the using of lattices with large size is preferable. Moreover, some phenomena could be studied on big lattices only, because small lattices are not sensitive to such effects. The use of large lattices puts special demands on computer systems on which the investigation is performed. Thus, the need for high computing performance in addition to the existence of well parallelized algorithms makes the application of GPUs for lattice simulations particularly important. Now HEP is one of the main consumers of supercomputing facilities.
Major HEP research collaborations develop software environments to achieve their scientific goals (see below). The standard practice is to incorporate into packages special utilities for data exchange among collaborations. While software development programmers optimize their code according to the hardware available for collaboration. So, due to severe competition among hardware manufacturers it is necessary to take into consideration the cross-platform principles while constructing a new HEP package.
![Currently $\sim 20\%$ of all scientific researches using GPU accelerators are performed with OpenCL [@hgpustat:2013][]{data-label="fig:openclstat"}](papers-cuda-opencl-percents.pdf){width="60.00000%"}
Obviously, each software package has limited scope of scientific tasks, which could be solved with it. Current HEP development leads to emerging the problems beyond this scope. The spontaneous vacuum magnetization at high temperature [@Demchik:2012vf], phase transition behavior in external fields [@Cea:2001pc], dependence of the phase transition order in O(N) models on coupling constant value [@Bordag:2012nh] and so on are some of such tasks. In 2008 we created the IDS package [@Demchik:2009ni], which allows to research quantum effects in external chromomagnetic field. It was written in ATI IL for AMD/ATI GPUs and provided fast derivation of a huge statistic, desired for solving applied tasks. After the deprecation to maintain ATI IL by AMD, the demand to port the package to a modern GPGPU language has been appeared. Currently $\sim 20\%$ of all scientific researches using GPU accelerators are performed with OpenCL [@hgpustat:2013] (see figure \[fig:openclstat\]). So, OpenCL was chosen to provide a multi-platform usage.
In this paper we present [[QCDGPU]{}]{} package and describe its program architecture and possibilities. The general aim of the package is production of lattice gauge field configurations for pre described models with following statistical processing of different measurable quantities. The current version of [[QCDGPU]{}]{} allows to study SU(2), SU(3) gauge theories as well as O(N) models. The extension of available groups in the package may be made by linking the file with appropriate algebra and core of the Monte Carlo procedure for particular groups. The number of space-time dimensions is one of the run-time parameters in [[QCDGPU]{}]{}. $3+1$ dimensional space-time is assumed by default. The list of measurable quantities may be changed accordingly to the problem investigated. All lattice simulations as well as measurements can be performed whether with single or double precisions. The package is easily-scalable to the number of available devices.
The paper is organized as follows. The overview of existing packages is made in the Sect.\[sect:relworks\]. The structure of the [[QCDGPU]{}]{} package is shown in the Sect.\[sect:architecture\]. The description of multi-GPU mode and the capability of distributed simulations are provided in the Sect.\[sect:multigpu\]. Some performance results are shown in Sect.\[sect:performance\]. The last Sect.\[sect:discussion\] is devoted to discussion of the scope of the package and summarizes the results.
Related Works {#sect:relworks}
=============
While a great amount of new experimental data appeared, modern high-energy physics requires extremely resource-intensive computations, in particular Monte Carlo lattice simulations. Therefore, only big scientific collaborations, which have enough computation time on supercomputers can run such simulations. As usual such collaborations (UKQCD, USQCD, TWQCD, PTQCD, etc.) have own software packages for simulations. The most well-known among them are:
- [**FermiQCD**]{}: an open C++ library for development of parallel Lattice Quantum Field Theory computations [@DiPierro2005qx],
- [**MILC**]{}: an open code of high performance research software written in C for doing $SU(3)$ lattice gauge theory simulations on several different (MIMD) parallel computers [@MILC],
- [**QDP++/Chroma**]{}: package supporting data-parallel programming constructs for lattice field theory and in particular lattice QCD [@Edwards:2004sx].
The first paper relating application of GPUs in HEP lattice simulations was published in 2007 [@Egri:2006zm]. The authors of this work used OpenGL as programming language and for the first time denoted the need to store lattice data in the form of four-component vectors. Shortly after of the publication, NVIDIA unveiled a new architecture CUDA and OpenGL ceased to be used as a GPGPU-computation language in further works.
Recently some open-source software packages have been developed targeted to use GPUs:
- [**QUDA**]{}: a library for performing calculations in lattice QCD on CUDA-ready GPUs [@Clark:2009wm],
- [**PTQCD**]{}: a collection of lattice SU(N) gauge production programs on CUDA-ready GPUs [@Cardoso:2011xu],
- [**cuLGT**]{}: code for gauge fixing in lattice gauge field theories with CUDA-ready GPUs [@Schrock:2013nta],
and several other packages with closed-access source codes ([@Bach:2012iw], [@Chakrabarty:2012rv], [@Chiu:2011bm], [@Bonati:2011dv], [@Alexandru:2011sc], [@Kim:2010br]). Some HEP collaborations link special GPU-libraries for their projects to engage GPGPU computing possibility without code refactoring. In particular, MILC collaboration uses QUDA package [@Shi:2010aq], but only in single-device mode now. USQCD collaboration also has powered its QDP++/Chroma software with CUDA [@Winter:2011an].
Most of the GPU-targeted packages are based on NVIDIA CUDA environment. However, the development of HPC market leads to increasing of cross-platform heterogeneous computing importance. In spite of unchallenged leadership of CUDA the software packages adapted for CUDA-ready devices could not be executed on other (even hardware-compatible) accelerators because of its closed standard.
QCDGPU Package Architecture {#sect:architecture}
===========================
The [[QCDGPU]{}]{} package architecture is based on the full platform-independence principle. All modules of the package are written in C++ and OpenCL with minimal external libraries dependence. [[QCDGPU]{}]{} can run equally well both on Windows operating system (OS), and on Linux without any code modifications. The OS-independence is implemented by the inclusion into the package of special file [[platform.h]{}]{}, which adapts the OS-dependent commands by means of precompiler directives.
The package is tested on different OSs, OpenCL SDK of different vendors and on several devices:
- [**OS:**]{} Windows XP, Windows 7, OpenSUSE 11.4, OpenSUSE 12.2;
- [**SDK:**]{} NVIDIA CUDA SDK 5.5, AMD APP SDK 2.8.1, Intel SDK for Applications OpenCL 2013;
- [**devices:**]{} AMD Radeon HD7970, HD6970, HD5870, HD5850, HD4870, HD4850 (single-precision mode), NVIDIA GeForce GTX 560 Ti, NVIDIA GeForce GTX 560 M, Intel Core i7-2600, Intel Core i7-2630QM, AMD Phenom II X6.
The package can be executed on all versions of OpenCL (1.0, 1.1, 1.2) without any code changes.
Package Structure
-----------------
![Structure of [[QCDGPU]{}]{} package.[]{data-label="fig:structure"}](program-structure.pdf){width="60.00000%"}
The schematic diagram of the [[QCDGPU]{}]{} is shown in figure \[fig:structure\]. The package core compounds of the block [[CLinterface]{}]{}, which provides the interaction of host code with computing devices. It performs all services for the preparation of devices, run of kernels and release the host memory and devices. Next important block is the block with physical model description [[Model description]{}]{} ([[SUNCL]{}]{} or [[ONCL]{}]{} depending on the model under investigation). Memory organization on the host in accordance with the physical model (gauge group, space-time dimension, etc.) is made in this block, as well as preparation and configuration of kernels in accordance with simulation conditions. Algorithms based on pseudo-random number generators are the base of Monte Carlo procedure. Library [[PRNGCL]{}]{} performs the function of generating pseudo-random numbers with required generator selection. The block [[Big lattice]{}]{} is designed to provide the possibility to produce simulations on large lattices, and to use multiple devices on a single host. The package performs all the necessary calculations on computing devices and provides the final simulation results to the host. Statistical analysis of the results over the run is performed by the block [[Data analysis]{}]{}. Validation of the data is performed by the block [[CPU Results checking]{}]{}, which produces control measurements of required quantities on the last gauge configuration by CPU means. The block is basically designed for debugging (for example, if a device has non-ECC memory), it can independently produce correspondent lattice gauge configurations on the same pseudo-random numbers as the device. The block can be turned off to save host resources. The interaction of several copies of the main program on different hosts is performed by a separate program [[QCDGPU-dispatcher]{}]{}. The program realizes task scheduling for the available copies of the main program according to the parameter space of the problem under investigation.
The results of simulations are written in separate text files that are sent for further processing by external means. Each file contains the startup parameters needed for reproduction of the run, as well as run average values and average configurations quantities table. Also, for the possibility of resuming the interrupted simulation, the possibility of regular saving of computation package state is realized, it allows to interrupt the simulation at any time, and provides basic fault tolerance (power or hardware failure). Saving frequency is set at the beginning of the simulations by the corresponding parameter. Saved file with the computation state is portable (the calculation can be continued on another device).
Detailed description of the package blocks is below.
CLinteface Module
-----------------
Block [[CLinterface]{}]{} is designed to hide all preparatory work for OpenCL-kernels startup from the main program. At the same time fine adjustment of all programming units is available. Every memory object and kernel obtain its own ID number and further kernel execution, memory object binding to the kernel, results output and so on are carried out by this ID number. This principle is initially used in the OpenCL standard [@OpenCL:2011], but the numbering of objects is produced mainly for the purposes of memory usage monitoring, users are not granted with the possibility to use these internal ID numbers.
As far as lattice simulation implies multiply startup of the same kernels, block [[CLinterface]{}]{} allows to adjust parameters of each kernel startup by default. Meanwhile, there is only kernel’s ID number in startup command arguments, which makes the program code shorter. At the same time there is a separate control of workgroupsize in case of Intel OpenCL SDK usage, as it returns overestimated values workgroupsize for some devices.
The unit also controls compute device errors. All errors are recorded in .log-file. Noncritical errors don’t cause stop of the program.
The block also performs caching of previously compiled programs to reduce the startup time of their launch. Built-in compute cache is realized only in the NVIDIA CUDA SDK. AMD APP SDK stores only the last compiled program, while Intel SDK for applications OpenCL makes re-compilation while each startup. In spite of compute cache in the NVIDIA CUDA SDK, there is a problem with the re-compilation necessity of dependent files (included in the device-side OpenCL-program by directive [[\#include]{}]{}) – SDK up to 5.5 version does not monitor such file changes. All mentioned above necessitates to create own compute cache. This compute cache is realized by creating .bin-files with compiled code for a particular device with distinct compilation parameters. .inf files are created along with these files, in which compilation-specific and additional parameters (program number, platform, device, program options, MD5-hash of program source, compilation timestamp) are indicated. For each source code MD5-hash is calculated, which is de facto ID number of the source code version. New .bin and .inf are created if startup parameters change. While changing MD5-hash of the source code, old .bin and .inf files are overwritten. Such internal compute cache can be turned off in startup parameters.
The unit can perform a run-time profiling of kernels and memory objects. It allows to optimize new kernels during designing of them. By default profiling is turned off.
Startup parameters are passed to kernels by 3 means:
1. by determined parameters of internal precompiler;
2. by constant buffers;
3. directly by binding values.
Undoubtedly, from the performance point of view, the most preferred way of passing parameters to kernels is the first mean. But it necessitates its re-compilation. That’s why only rarely changed parameters are passed by this mean (lattice geometry, gauge group, precision and so on). Often changed parameters are passed mainly by the second mean (coupling constant values, magnetic field flux and so on), which are common for all kernels. Specific parameters for the particular kernel are passed by the third mean (reducing size, memory offsets, etc.). Such division allows to use computational time efficiently, which is formed from both program execution time, and its compilation time.
SUNCL and ONCL Modules
----------------------
The package block of [[QCDGPU]{}]{}, which is responsible for the physical model description, is modules [[SUNCL]{}]{} and [[ONCL]{}]{}, which provide simulation SU(N) gluodynamics and O(N) models correspondingly. As it is well known, the gauge fields in the group SU(N) are presented with $N \times N$ complex matrices. These matrices are associated with lattice links. In case of O(N) models the fields are set with $N$-vectors, which are associated with lattice sites. That’s why it is possible to unify storage of lattice data in the memory by the following means. The fastest index – number of lattice site. Next index – spatial direction of lattice link. In case of O(N)-model, this index is not used. The slowest index is connected with gauge group. At the same time group matrices are represented as structures of defined quantity 4-vectors, each of which contains a piece of information about corresponding matrix. This is due to the architecture of GPU-devices memory.
In order to carry out lattice update, lattice is traditionally divided into even and odd sites (checkerboard scheme), and, if necessary, into separate parts, which makes it possible to study big lattices (see below). (Pseudo)heat-bath algorithm is used for SU(N) model update. Improved Metropolis algorithm [@Bordag:2012nh] is used for O(N) update.
Due to the equalizing of array lengths, as well as offsets in accordance with workgroup size, it achieves coalesced memory access, which has a positive effect on overall performance.
Pseudo-Random Number Generators
-------------------------------
All pseudo-random numbers (PRN) needed for kernel operation are produced by own library [[PRNGCL]{}]{}. The library is a porting and development of the library [[PRNGCAL]{}]{}, written for AMD/ATI GPUs on ATI CAL [@Demchik:2010fd]. The most popular pseudo-random number generators (PRNG), which are used in HEP lattice simulations (RANMAR, RANECU, XOR128, XOR7, MRG32k3a, RANLUX with different luxury levels), are realized in it. Realization of Park-Miller PRNG and “Constant generator”, which produces a given constant, is included for testing and debugging purposes. The selection of the generator used is made by one external parameter, which gives the possibility to check obtained results stability in relation to PRNG used.
By default, the package generates PRNs by the number of threads equal to the parameter value CL\_DEVICE\_IMAGE3D\_MAX\_WIDTH (maximal width of 3D image), which the device returns. In practice this parameter connects to GPU device architecture and does not depend on the vendor. Our observations show, that using this parameter as launched threads quantity allows to achieve the best performance. Users can choose the quantity of threads for PRNs producing manually.
As soon as almost all realized generators do not have a general scheme of multi-stream execution (apart from MRG32k3a) PRNG parallelization is made by various initializations of seed tables. Every PRNG thread keeps own seed table with own initial values.
The generator initialization is made by one value of the external parameter ([[RANDSERIES]{}]{}), on which base PRNG seed tables are initialized. At the same time, if this parameter equals zero, then system time is chosen as its value. The value of parameter [[RANDSERIES]{}]{} definitely reproduces simulation results, so the value along with the name of PRNG used is presented in output files.
Multi-GPU Mode and Distributed Simulations {#sect:multigpu}
==========================================
One of the main features of package [[QCDGPU]{}]{} is its possibility to launch in multi-GPU mode. At this time several devices on the host system can be used for one simulation. It is evident that it is necessary to divide lattice into parts. In the general case dividing lattice is made into unequal parts, which partly allows to hide the difference in devices performances.
Since the package is mainly designed to research finite-temperature effects, dividing lattice into parts is made in first spatial coordinate direction $L_1$. This is due to the fact that the temperature is associated with the time direction $L_t$ and the case high temperature corresponds to $L_t<L_1$.
Divided lattice simulation differs from a full lattice simulation only in necessity to carry out the exchanging values of boundary sites. To divide the lattice into parts so-called second-level checkerboard scheme is used. It is carried out in the alternate update of odd and even parts of the lattice. In this case, the border information exchange is performed in asynchronous mode – while even sites update is being made, fulfil the information at border sites is taken place and vise verse. So while dividing the lattice, it is preferable to use an even quantity of parts. Dividing the lattice is also used in cases when compute device memory is not enough for the simulation.
The package also allows to make simulations on several hosts at the same time. Each copy of the main computational program launches on the corresponding host, while parameters passing and launch control are carried out by external program [[QCDGPU-dispatcher]{}]{}. A very simple scheme is used for interaction of the control and computational programs: in case of computational program launch in this mode, the program waits for the special file [[finish.txt]{}]{} deleting before simulation begins. If there is [[QCDGPU]{}]{} and [[QCDGPU-dispatcher]{}]{} in general folder, it means “the previous run is completed – you can collect the results, waiting for the next run”. After collecting files with simulation results, the control program creates a special file [[init.dat]{}]{}, in which parameters of next run are written, and file [[finish.txt]{}]{} is deleted. As soon as the computational program does not find the file [[finish.txt]{}]{}, it reads new parameters of the launch from the file [[init.dat]{}]{}. The cycle repeats. At the same time in case of using several copies of computational program [[QCDGPU-dispatcher]{}]{} sequentially looks through catalogues and set a new task for the first free host. Names of files with results contain simulation finish time and unique prefix of computational program copy, which makes each file unique.
Performance Results {#sect:performance}
===================
In order to demonstrate some benchmarks we performed at several MC simulations for O(4), SU(2), SU(3) models on various lattices on the following GPUs: NVIDIA GeForce GTX 560 M (Windows 7), NVIDIA GeForce GTX 560 Ti, AMD Radeon HD 7970 (OpenSUSE 12.2), HD 6970, HD 5870 (OpenSUSE 11.4). For all MC simulations the “hot” lattice initialization and RANLUX pseudo-random number generator with luxury level 3 were used. For O(4) model $NHIT=100$ tries were used to update each lattice site (this provides acceptance rate up to 50%). For SU(2) and SU(3) models $NHIT=10$ and reunitarization were used. One bulk sweep was performed to decorrelate configurations to be measured. There are two types of sweeps: thermalization and working sweeps. During thermalization sweep each lattice element (sites or links) is updated. In working sweep the lattice is updated and some quantities are measured. So, working sweeps are some longer than thermalization sweeps. Here we present the timings for working sweeps only.
[**Model**]{} [**Device**]{} [**Parts**]{} [**Lattice**]{} [**Single**]{} [**Double**]{}
--------------- ------------------ --------------- ----------------- ---------------- ----------------
O(4) GTX560M 1 $16^4$ 0.08 0.12
$24^4$ 0.42 0.6
$32^4$ 1.31 1.87
4 $48^4$ 7.79 11.11
GTX560Ti 1 $32^4$ 0.72 0.99
4 $48^4$ 4.64 6.10
HD 5870 2 $32^4$ 0.96 1.56
HD 6970 2 $32^4$ 0.73 1.3
HD 7970 2 $32^4$ 0.66 0.95
12 $72^4$ 20.35 36.2
HD 7970+GTX560Ti 4 $32^4$ 0.72 0.96
8 $48^4$ 2.97 4.1
SU(2) GTX560Ti 1 $44^4$ 0.95 1.14
HD 5870 1 $30^4$ 0.22 0.24
HD 6970 1 $30^4$ 0.20 0.22
HD 7970 1 $48^4$ 4.64 6.10
SU(3) GTX560Ti 1 $32^4$ 0.56 1.17
$36^4$ 0.90 1.86
HD 6970 1 $24^4$ 0.24 0.23
HD 7970 1 $28^4$ 0.13 0.22
: The timings of one working sweep for O(4), SU(2) and SU(3) models in single- and double-precision mode (in seconds).
\[tab:tab1\]
The performance results are collected in the Table 1. The gauge model and computing devices used in MC simulations are shown in first and second columns, correspondingly. Due to the memory limit of particular computing device whole lattice should be divided into several parts to perform MC simulations. The number of parts is presented in the third column. Obviously, the bigger number of lattice parts means the bigger number of boundary elements transmission between host and device (or between devices in multi-GPU mode), which reduces the overall performance. The last two columns contain the timings for single- and double-precision mode simulations, correspondingly.
Among all the data in the table the case of cooperative simulation of one lattice on two different OpenCL platforms is shown (AMD Radeon HD 7970 on AMD APP SDK 2.8 and NVIDIA GeForce GTX 560 Ti on NVIDIA CUDA 5.5). In this case the timings are better only on 10-25% than simulation on a single best device. Nevertheless, simultaneous multi-platform simulations might be interesting for very big lattices.
In [@Cardoso:2011xu] Cardoso and Bicudo reported the timings $6\times 10^{-4}$ ($7\times 10^{-4}$ with double precision) and $2\times 10^{-3}$ ($4\times 10^{-3}$ with DP) seconds per single sweep on lattice $8^4$ for SU(2) and SU(3) models, respectively. The authors used NVIDIA GeForce GTX 580 and NVIDIA CUDA. We performed the same performance measurements with [[QCDGPU]{}]{} (lattice sweep with reunitarization, without any measurements) and obtain the following values in seconds: $6\times 10^{-4}$ ($10^{-3}$ with DP) for SU(2) and $1.3\times 10^{-3}$ ($4.9\times 10^{-3}$ with DP) for SU(3) on NVIDIA GeForce GTX 560 Ti and $2.0\times 10^{-3}$ ($2.6\times 10^{-3}$ with DP) for SU(2) and $3.4\times 10^{-3}$ ($5.1\times 10^{-3}$ with DP) for SU(3) on AMD Radeon HD 7970.
In actual MC simulations the trivial parallelization scheme (each computing device receives from dispatcher unique parameters set for simulation of whole lattice) we often use. The best performance results are obtained in this case. Undoubtedly, due to many tuning parameters (such as number of lattice parts, size of parts for different computing devices, part sequence, workgroup sizes, etc.) the [[QCDGPU]{}]{} package performance of multi-GPU mode is the subject for special research.
Discussion {#sect:discussion}
==========
In the present work a new package [[QCDGPU]{}]{} is introduced. This package is designed for Monte Carlo simulations of SU(N) gluodynamics in external field and O(N) model on OpenCL-compatible devices. The package allows to carry out the simulations for very big lattices in single- or multi-GPU mode. Simulations can be run with single or double precision. The package claims low demands to the host CPU and practically does not load it, which provides the possibility to use the package along with other traditional computational programs.
The [[QCDGPU]{}]{} package allows to investigate most popular LGT models in N-dimensional space-time. The architecture of the [[QCDGPU]{}]{} provides a possibility to easily add new LGT model into the package.
If the size of the lattice under investigation allows its location in device memory, all necessary operations are carried out on device memory, the result returns to the host program after finishing the simulation. If the lattice is too big to locate it in device memory, it is divided into parts and work on separate parts is made by all computing devices available at the host. [[QCDGPU]{}]{} package allows the simultaneous run of several instances of the computational program on all tested OSs.
The current version of the program uses trivial parallelization scheme to distribute computing – every computing node gets separate task for simulation. At the same time OS type on every used host is not important. The main requirement is to create a folder with shared access at the host. Now it is made within the local network framework and by the means of virtual private network (VPN) organization for remote nodes. Using small pieces of text information between task scheduling module [[QCDGPU-dispatcher]{}]{} and the host does not impose load on the network.
Built-in mechanism for saving the computational state while achieving some conditions (every N sweeps or every M seconds) allows to interrupt long calculations without threat of data loss and to continue them on another available device. It is also very useful while often power failures.
At present we are working on including to the package the following possibilities:
- using the base of the built-in profiling mechanism, as well as additional micro-benchmarks to make automatic adjustments of package start-up parameters;
- realization of RHMC algorithm for including fermionic fields on a lattice;
- running of one simulation on several hosts at the same time on the base of MPI, which is very important in case of accounting of fermionic fields. Using mixed-precision possibility will be realized to reduce amount of exchanging information.
In this work we did not bring attention to the details of realized algorithms, as well as to detailed description of the package parameters. The package is being constantly developed. In the first place methods and algorithms needed for actual research in the Quantum Chromoplasma Laboratory of Dnepropetrovsk National University is realized. Obtained physical results are published in particular in the works: [@Demchik:2012vf], [@Bordag:2012nh], etc.
Source codes of the package [[QCDGPU]{}]{} and some examples of result files are in the open access at\
<https://github.com/vadimdi/QCDGPU>.
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[^1]: E-mail: vadimdi@yahoo.com
[^2]: E-mail: rknv7@mail.ru
|
---
abstract: 'For a finite dimensional simple complex Lie algebra $\mathfrak{g}$, Lie bialgebra structures on $\mathfrak{g}[[u]]$ and $\mathfrak{g}[u]$ were classified by Montaner, Stolin and Zelmanov. In our paper, we provide an explicit algorithm to produce $r$-matrices which correspond to Lie bialgebra structures over polynomials.'
address: 'Department of Mathematical Sciences, University of Gothenburg, Sweden. Email: iulia@chalmers.se; md1jm@chalmers.se'
author:
- IULIA POP
- 'JULIA YERMOLOVA–MAGNUSSON'
title: 'New $r$-Matrices for Lie Bialgebra Structures over Polynomials'
---
Introduction
============
In the recent paper [@SZ], Montaner, Stolin and Zelmanov discussed the classification of Lie bialgebras and corresponding quantum groups over Taylor series and polynomials. We devote the introduction to giving an overview of the main results in [@SZ].
Lie bialgebra structures over Taylor series
-------------------------------------------
Let $\mathfrak{g}$ denote a finite dimensional simple complex Lie algebra. Consider $\mathfrak{g}[[u]]$ the algebra of Taylor series over $\mathfrak{g}$. In order to classify Lie bialgebra structures on $L:=\mathfrak{g}[[u]]$, one needs some information on the corresponding Drinfeld double. As it is known (see [@D]), any Lie bialgebra structure $\mu$ on $L$ induces a Lie algebra structure on the space of restricted functionals $L^*$ on $L$. Moreover, the vector space $D_{\mu}(L):=L\oplus L^*$ can be equipped with a unique Lie algebra structure, which extends the brackets on $L$ and $L^*$ respectively, and is such that the natural nondegenerate form on $D_{\mu}(L)$ is invariant. The Lie algebra $D_{\mu}(L)$ is called the Drinfeld double corresponding to $\mu$.
In [@SZ] it was proved that $D_{\mu}(L)$ is either a trivial extension of $L$ or has one of the following forms:
I. The Lie algebra $\mathfrak{g}((u))$ of Laurent polynomials over $\mathfrak{g}$ with the nondegenerate symmetric bilinear form $Q_1$ defined by $$Q_1(xf_1(u),yf_2(u))=K(x,y)\cdot T_1(f_1(u)f_2(u)),$$ for any $x, y\in \mathfrak{g}$, $f_1(u),f_2(u)\in\mathbb{C}((u))$, where $K$ denotes the Killing form on $\mathfrak{g}$ and $T_1$ is given by $T_1(u^k)=0$ for all $k\geq 0$, $T_1(u^{-1})=1$ and $T_1(u^{-k-1})=a_k$, for all $k\geq 1$.
II\. The Lie algebra $\mathfrak{g}((u))\oplus \mathfrak{g}$ with the nondegenerate symmetric bilinear form $Q_2$ defined by $$Q_2(x_1f_1(u)+x_2,y_1f_2(u)+y_2)=K(x_1,y_1)\cdot T_2(f_1(u)f_2(u))-K(x_2,y_2),$$ for any $x_1, y_1, x_2, y_2 \in \mathfrak{g}$, $f_1(u),f_2(u)\in\mathbb{C}((u))$, and $T_2$ is given by $T_2(u^k)=0$ for all $k\geq 1$, $T_2(1)=1$ and $T_2(u^{-k})=a_k$, for all $k\geq 1$.
III\. The Lie algebra $\mathfrak{g}((u))\oplus (\mathfrak{g}+\varepsilon\mathfrak{g})$, where $\varepsilon^2=0$, with the nondegenerate symmetric bilinear form $Q_2$ defined by $$Q_3(x_1f_1(u)+x_2+\varepsilon x_3,y_1f_2(u)+y_2+\varepsilon y_3)=$$ $$K(x_1,y_1)\cdot T_3(f_1(u)f_2(u))-K(x_3,y_2)-K(x_2,y_3)-a\cdot K(x_2,y_2),$$ for any $x_1, y_1, x_2, y_2,x_3, y_3\in \mathfrak{g}$, $f_1(u),f_2(u)\in\mathbb{C}((u))$, where $T_3(u^k)=0$ for all $k\geq 2$, $T_3(u)=1$, $T_3(1)=a$ and $T_3(u^{-k})=a_{k+1}$, for all $k\geq 1$.
In each of the above cases, consider
$$a(u):=1+\sum_{k=1}^{\infty}a_ku^k.$$
Let $\mathrm{Aut}_0(\mathbb{C}[[u]])$ be the group of infinite series $\gamma=x+\gamma_2x^2+\gamma_3x^3+...$, with respect to substitution. In [@SZ] it was shown that up to an automorphism $\gamma\in\mathrm{Aut}_0(\mathbb{C}[[u]])$, one may suppose that $a(u)=1$.
[@SZ] Let $\mu$ be any Lie bialgebra structure on $\mathfrak{g}[[u]]$. Then there exists $\gamma\in\mathrm{Aut}_0(\mathbb{C}[[u]])$ such that $D_{\mu}(\mathfrak{g}[[u]])$ is isomorphic, via $\gamma$, to one of the Lie algebras in cases I–III with $a(u)=1$.
Lie bialgebra structures over polynomials
-----------------------------------------
Let us now focus on the classification of Lie bialgebra structures on $\mathfrak{g}[u]$. Denote by $\delta$ any such structure. It was shown in [@SZ] that $\delta$ can be naturally extended to a Lie bialgebra structure $\bar{\delta}$ on $\mathfrak{g}[[u]]$, by letting $$\bar{\delta}(\sum_{n=0}^{\infty}x_nu^n)=\sum_{n=0}^{\infty}\delta(x_nu^n).$$ The corresponding Drinfeld double $D_{\bar{\delta}}(\mathfrak{g}[[u]])$ is either a trivial extension of $\mathfrak{g}[[u]]$ or is isomorphic to one of the Lie algebras in cases I–III. We note that $a(u)$ is not necessarily constant. An automorphism of $\mathbb{C}[[u]]$ is not necessarily an automorphism of $\mathbb{C}[u]$. Moreover, there are certain restrictions that have to be imposed on $a(u)$, depending on the case.
**Case I.** Let us suppose that $D_{\bar{\delta}}(L)=\mathfrak{g}((u))$. This implies that there exists a Lagrangian subalgebra $W$ of $\mathfrak{g}((u))$, with bracket induced by $\bar{\delta}$, such that $W\oplus L=\mathfrak{g}((u))$. It was shown that the following properties hold:
\(i) $W$ is bounded, i.e., there exists a natural number $n$ such that $W\subseteq u^n\mathfrak{g}[u^{-1}]$.
\(ii) $W\cdot\mathbb{C}[[u^{-1}]]$ is an order in $\mathfrak{g}((u^{-1}))$. Here we recall that an order in $\mathfrak{g}((u^{-1}))$ is a Lie subalgebra $V$ for which there exist natural numbers $k$ and $n$ such that $u^{-n}\mathfrak{g}[[u^{-1}]]\subseteq V \subseteq u^k\mathfrak{g}[[u^{-1}]]$.
\(iii) Using the general theory of orders [@S2], the authors show that, by means of a gauge transformation $\sigma(u)\in \mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$, one can embed $W\cdot\mathbb{C}[[u^{-1}]]$ into a special order denoted by $\mathbb{O}_\alpha$, constructed in the following way:
Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$ with the corresponding set of roots $R$ and a choice of simple roots $\Gamma$. Denote by $\mathfrak{g}_{\alpha}$ the root space corresponding to a root $\alpha$. Let $\mathfrak{h}(\mathbb{R})$ be the set of all $h\in\mathfrak{h}$ such that $\alpha(h)\in\mathbb{R}$ for all $\alpha\in R$. Consider the valuation on $\mathbb{C}((u^{-1}))$ defined by $v(\sum_{k\geq n}a_{k}u^{-k})=n$. For any root $\alpha$ and any $h\in\mathfrak{h}(\mathbb{R})$, set $M_{\alpha}(h)$:=$\{f\in\mathbb{C}((u^{-1})):v(f)\geq \alpha(h)\}$. Consider $$\mathbb{O}_{h}:=\mathfrak{h}[[u^{-1}]]\oplus(\oplus_{\alpha\in R}M_{\alpha}(h)\otimes\mathfrak{g}_{\alpha}).$$
Vertices of the above simplex correspond to vertices of the extended Dynkin diagram of $\mathfrak{g}$, the correspondence being given by the following rule: $$0\leftrightarrow\ -\alpha_{\max}$$$$h_{i}\leftrightarrow\alpha_{i}$$ where $\alpha_{i}(h_{j})=\delta_{ij}/k_{j}$ and $k_{j}$ are given by the relation $\sum
k_{j}\alpha_{j}=\alpha_{\max}$. One writes $\mathbb{O}_{\alpha}$ instead of $\mathbb{O}_{h}$ if $\alpha$ is the root which corresponds to the vertex $h$ and $\mathbb{O}_{-\alpha_{\max}}$ instead of $\mathbb{O}_0$.
Then there exists $\sigma(u)\in \mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$ such that $\sigma(u)(W\cdot\mathbb{C}[[u^{-1}]])\subseteq \mathbb{O}_{\alpha}$, where $\alpha$ is either a simple root or $-\alpha_{\max}$. Here one makes the remark that for any $\sigma\in\mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$, there exists a natural embedding $$\mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])\hookrightarrow\mathrm{Aut}_{\mathbb{C}((u^{-1}))}(\mathfrak{g}((u^{-1}))),$$ defined by the formula $\sigma(u^{-k}x)=u^{-k}\sigma(x)$, for any $x\in\mathfrak{g}[u]$; hence one can regard $\sigma(u)$ as acting on $\mathfrak{g}((u^{-1}))$.
[@SZ]\[condCaseI\] Let $\alpha_{\max}=\sum k_j\alpha_j$, $\alpha_j\in\Gamma$ and $\sigma\in\mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$.
\(1) Assume that $\sigma(u)(W\cdot\mathbb{C}[[u^{-1}]])\subseteq \mathbb{O}_{-\alpha_{\max}}$. Then $\frac{1}{a(u)}$ is a polynomial of degree at most 2.
\(2) Assume that $\sigma(u)(W\cdot\mathbb{C}[[u^{-1}]])\subseteq \mathbb{O}_{\alpha_i}$, for some $i$. Then $\frac{1}{a(u)}$ is a polynomial of degree at most 2 if $k_i=1$, and $\frac{1}{a(u)}$ is a polynomial of degree at most 1 if $k_i>1$.
**Case II.** Let us suppose $\bar{\delta}$ satisfies the condition $D_{\bar{\delta}}(L)=\mathfrak{g}((u))\oplus \mathfrak{g}$ and $W$ is the corresponding Lagrangian subalgebra in the double transversal to $L$. We first note that any $\sigma(u)\in \mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$ induces an automorphism $\tilde{\sigma}(u)$ of $\mathfrak{g}((u))\oplus \mathfrak{g}$ defined by $\tilde{\sigma}(u)=\sigma(u)\oplus \sigma(0)$. Then, similarly to Case I, one can show that there exists $\sigma(u)\in \mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$ satisfying $\tilde{\sigma}(u)(W\cdot\mathbb{C}[[u^{-1}]])\subseteq \mathbb{O}_{\alpha}\oplus\mathfrak{g}$, where $\alpha$ is either a simple root or $-\alpha_{\max}$. Moreover the following result holds:
[@SZ]\[condCaseII\] Let $\alpha_{\max}=\sum k_j\alpha_j$, $\alpha_j\in\Gamma$ and $\sigma\in\mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$.
\(1) Assume that $\tilde{\sigma}(u)(W\cdot\mathbb{C}[[u^{-1}]])\subseteq \mathbb{O}_{-\alpha_{\max}}\oplus\mathfrak{g}$. Then $\frac{1}{a(u)}$ is a polynomial of degree at most 1.
\(2) Assume that $\tilde{\sigma}(u)(W\cdot\mathbb{C}[[u^{-1}]])\subseteq \mathbb{O}_{\alpha_i}\oplus\mathfrak{g}$, for some $i$. Then $\frac{1}{a(u)}$ is a polynomial of degree at most 1, if $k_i=1$ and $\frac{1}{a(u)}$ is a constant if $k_i>1$.
**Case III.** Let us suppose $\bar{\delta}$ satisfies the condition $D_{\bar{\delta}}(L)=\mathfrak{g}((u))\oplus (\mathfrak{g}+\varepsilon\mathfrak{g})$, where $\varepsilon^2=0$, and $W$ is the corresponding Lagrangian subalgebra in the double transversal to $L$.
Any $\sigma(u)\in \mathrm{Ad}(\mathfrak{g}[u])$ induces an automorphism $\sigma(0)\in\mathrm{Ad}(\mathfrak{g})$, which in turn gives an automorphism $\bar{\sigma}(0)$ of $\mathfrak{g}+\varepsilon\mathfrak{g}$ defined by $\bar{\sigma}(0)(x+\varepsilon y)=\sigma(0)(x)+\varepsilon\sigma(0)(y)$. Then $\tilde{\sigma}(u)=\sigma(u)\oplus \bar{\sigma}(0)$ is an automorphism of $\mathfrak{g}((u))\oplus (\mathfrak{g}+\varepsilon\mathfrak{g})$. One can prove that there exists $\sigma(u)\in \mathrm{Ad}(\mathfrak{g}[u])$ such that $\tilde{\sigma}(u)(W\cdot\mathbb{C}[[u^{-1}]])\subseteq \mathbb{O}_{\alpha}\oplus (\mathfrak{g}+\varepsilon \mathfrak{g})$, where $\alpha$ is either a simple root or $-\alpha_{\max}$. The following result is similar to Theorem \[condCaseI\] and \[condCaseII\]:
[@SZ]\[condCaseIII\] Let $\alpha_{\max}=\sum k_j\alpha_j$, $\alpha_j\in\Gamma$ and $\sigma\in\mathrm{Ad}(\mathfrak{g}[u])$.
\(1) Assume that $\tilde{\sigma}(u)(W\cdot\mathbb{C}[[u^{-1}]])\subseteq \mathbb{O}_{-\alpha_{\max}}\oplus(\mathfrak{g}+\varepsilon \mathfrak{g})$. Then $\frac{1}{a(u)}$ is a constant.
\(2) Assume that $\tilde{\sigma}(u)(W\cdot\mathbb{C}[[u^{-1}]])\subseteq \mathbb{O}_{\alpha_i}\oplus\mathfrak{g}$, for some $i$. Then $\frac{1}{a(u)}$ is a constant if $k_i=1$, and the above inclusion is impossible if $k_i>1$.
In [@SZ] it was noticed that, by means of a change of variable in $\mathbb{C}[u]$ and rescaling the nondegenerate bilinear form $Q$, one may assume that $a(u)$ has one of the following forms:
1. $a(u)=1/(1-c_1u)(1-c_2u)$, for non-zero constants $c_1\neq c_2$
2. $a(u)=1/(1-u)^2$
3. $a(u)=1/1-u$
4. $a(u)=1$.
Lie bialgebra structures on $\mathfrak{g}[u]$ in Case I
=======================================================
In this section we will focus on Case I. We first note that the nondegenerate bilinear form on $\mathfrak{g}((u))$ is given by the formula $$Q_{a(u)}(f_1(u),f_2(u))=\mathrm{Res}_{u=0}(K(f_1(u),f_2(u))\cdot a(u)),$$ where $K$ is the Killing form of the Lie algebra $\mathfrak{g}((u))$ over $\mathbb{C}((u))$.
In [@SZ] the following result was proved:
[@SZ] There exists a one-to-one correspondence between Lie bialgebra structures $\delta$ on $\mathfrak{g}[u]$ satisfying $D_{\bar{\delta}}(\mathfrak{g}[[u]])=\mathfrak{g}((u))$ and bounded Lagrangian subalgebras $W$ of $\mathfrak{g}((u))$, with respect to the nondegenerate bilinear form $Q_{a(u)}$, and transversal to $\mathfrak{g}[[u]]$.
If $W$ is a bounded Lagrangian subalgebra of $\mathfrak{g}((u))$ transversal to $\mathfrak{g}[[u]]$, then $W\oplus \mathfrak{g}[u]=\mathfrak{g}[u,u^{-1}]$. The equality implies that one can choose dual bases in $W$ and $\mathfrak{g}[u]$ with respect to $Q_{a(u)}$. The corresponding Lie bialgebra structure $\delta$ can be reconstructed starting from $W$ in the following way: let us choose a system of Chevalley-Weyl generators $e_{\alpha}$, $e_{-\alpha}$, $h_{\alpha}$, for all positive roots $\alpha$, such that $K(e_{\alpha},e_{-\alpha})=1$ and $h_{\alpha}=[e_{\alpha},e_{-\alpha}]$. The canonical basis of $\mathfrak{g}[u]$ is formed by $e_{\alpha}u^k$, $e_{-\alpha}u^k$, $h_{\alpha}u^k$, for all positive roots $\alpha$ and all natural $k$. Denote these elements by $e_{\alpha,k}$. Let $w_{\alpha,k}$ be a dual basis in $W$ with respect to the the nondegenerate bilinear form $Q_{a(u)}$ and consider the $r$-matrix $$r(u,v)=\sum_{\alpha,k} e_{\alpha,k}\otimes w_{\alpha,k}.$$ Then $$\delta(f(u))=[f(u)\otimes 1+1\otimes f(v),r(u,v)],$$ for all $f(u)\in \mathfrak{g}[u]$.
\[max\_ord\] Suppose that $W$ is a bounded Lagrangian subalgebra of $\mathfrak{g}((u))$, with respect to $Q_{a(u)}$ and transversal to $\mathfrak{g}[[u]]$. Then there exists $\sigma\in\mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$ such that $\sigma(u)(W)\subseteq \mathbb{O}_{\alpha}\cap \mathfrak{g}[u,u^{-1}]$, where $\alpha$ is either a simple root or $-\alpha_{\max}$.
Since $W$ is bounded, we have $W\subseteq \mathfrak{g}[u,u^{-1}]$. From [@SZ] we also know that there exists $\sigma(u)\in \mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$ such that $\sigma(u)(W\cdot\mathbb{C}[[u^{-1}]])\subseteq \mathbb{O}_{\alpha}$, where $\alpha$ is either a simple root or $-\alpha_{\max}$.
On the other hand, $\sigma(u)(\mathfrak{g}[u,u^{-1}])=\mathfrak{g}[u,u^{-1}]$. One obtains the inclusions: $\sigma(u)(W)\subseteq\sigma(u)(W\cdot\mathbb{C}[[u^{-1}]])\cap \sigma(u)(\mathfrak{g}[u,u^{-1}])\subseteq \mathbb{O}_{\alpha}\cap \mathfrak{g}[u,u^{-1}]$.
In what follows we will restrict ourselves to the case $\alpha=-\alpha_{\max}$. Let us make the remark that $\mathbb{O}_{-\alpha_{\max}}\cap\mathfrak{g}[u,u^{-1}]=\mathfrak{g}[u^{-1}]$. Consider also the Lie algebra $\mathfrak{g}\oplus\mathfrak{g}$, together with the nondegenerate bilinear form $$\bar{Q}((x_1,y_1),(x_2,y_2))=K(x_1,x_2)-K(y_1,y_2).$$
Let $a(u)=1/(1-c_1u)(1-c_2u)$, for non-zero constants $c_1\neq c_2$. There exists a one-to-one correspondence between Lagrangian subalgebras $W$ of $\mathfrak{g}((u))$, with respect to $Q_{a(u)}$, which are transversal to $\mathfrak{g}[[u]]$ and satisfy $W \subseteq \mathfrak{g}[u^{-1}]$, and Lagrangian subalgebras in $\mathfrak{g}\oplus\mathfrak{g}$, with respect to $\bar{Q}$, transversal to $\mathrm{diag}(\mathfrak{g})$.
Assume $W$ is a Lagrangian subalgebra of $\mathfrak{g}((u))$ which is transversal to $\mathfrak{g}[[u]]$ and such that $W \subseteq \mathfrak{g}[u^{-1}]$. This implies that $W \supseteq \mathfrak{g}[u^{-1}]^{\perp}=(1-c_1u)(1-c_2u)u^{-2}\mathfrak{g}[u^{-1}]=
(u^{-1}-c_1)(u^{-1}-c_2)\mathfrak{g}[u^{-1}]$.
The quotient $\frac{W}{(u^{-1}-c_1)(u^{-1}-c_2)\mathfrak{g}[u^{-1}]}$ is a subalgebra of $\frac{\mathfrak{g}[u^{-1}]}{(u^{-1}-c_1)(u^{-1}-c_2)\mathfrak{g}[u^{-1}]}$ which can be identified with $\mathfrak{g}\oplus\mathfrak{g}$. Indeed, let us consider the epimorphism $\psi: \mathfrak{g}[u^{-1}]\longrightarrow \mathfrak{g}\oplus\mathfrak{g}$ defined by $\psi(x)=(x,x)$, for any $x\in \mathfrak{g}$ and $\psi(xu^{-1})=(xc_1,xc_2)$. Then $\mathrm{Ker}(\psi)=(u^{-1}-c_1)(u^{-1}-c_2)\mathfrak{g}[u^{-1}]$, which implies that $\frac{\mathfrak{g}[u^{-1}]}{(u^{-1}-c_1)(u^{-1}-c_2)\mathfrak{g}[u^{-1}]}$ is isomorphic to $\mathfrak{g}\oplus\mathfrak{g}$ via an isomorphism $\hat{\psi}$ induced by $\psi$.
Let $\bar{W}$ be the image of $W$ in $\mathfrak{g}\oplus\mathfrak{g}$. Since $W$ is a Lagrangian subalgebra of $\mathfrak{g}((u))$, $\bar{W}$ is a Lagrangian subalgebra of $\mathfrak{g}\oplus\mathfrak{g}$. Moreover, $\mathfrak{g}[[u]]$ projects onto $\mathrm{diag}(\mathfrak{g})$. Then $\bar{W}$ is transversal to $\mathrm{diag}(\mathfrak{g})$ since $W$ is transversal to $\mathfrak{g}[[u]]$. One can easily check that the correspondence $W\mapsto\bar{W}$ is bijective. This ends the proof.
\[sol\_I\] Let $c_1$, $c_2$ be different and non-zero complex constants, and $$r_{c_1,c_2}=\sum_{\alpha>0}(c_1e_{-\alpha}\otimes e_{\alpha}+c_2e_{\alpha}\otimes e_{-\alpha}+\frac{c_1+c_2}{4}h_{\alpha}\otimes h_{\alpha}),$$
$$r_0(u,v)= \frac{1-(c_1+c_2)u+c_1c_2uv}{v-u}\Omega-r_{c_1,c_2}.$$ Then $r_0(u,v)$ provides a Lie bialgebra structure on $\mathfrak{g}[u]$.
Let $\bar{W_0}$ be the Lie subalgebra of $\mathfrak{g}\oplus\mathfrak{g}$ spanned by the pairs $(e_{-\alpha},0)$, $(0,e_{\alpha})$, $(h_{\alpha},-h_{\alpha})$, for all positive roots $\alpha$. This is obviously a Lagrangian subalgebra of $\mathfrak{g}\oplus\mathfrak{g}$ complementary to the diagonal. Then the corresponding Lagrangian subalgebra $W_0$ of $\mathfrak{g}((u))$ is spanned by $\psi^{-1}(e_{-\alpha},0)$, $\psi^{-1}(0,e_{\alpha})$, $\psi^{-1}(h_{\alpha},-h_{\alpha})$ and $(u^{-1}-c_1)(u^{-1}-c_2)\mathfrak{g}[u^{-1}]$. We have: $\psi^{-1}(e_{-\alpha},0)=\frac{(u^{-1}-c_2)e_{-\alpha}}{c_1-c_2}$, $\psi^{-1}(0,e_{\alpha})=\frac{(u^{-1}-c_1)e_{\alpha}}{c_2-c_1}$, $\psi^{-1}(h_{\alpha},-h_{\alpha})=\frac{(2u^{-1}-c_1-c_2)h_{\alpha}}{c_1-c_2}$.
Let us choose the following basis in $W_0$: $v^{-k}(v^{-1}-c_1)(v^{-1}-c_2)e_{-\alpha}$, $v^{-k}(v^{-1}-c_1)(v^{-1}-c_2)e_{\alpha}$, $\frac{1}{2}v^{-k}(v^{-1}-c_1)(v^{-1}-c_2)h_{\alpha}$, for all $k\geq 0$, $e_{\alpha}(v^{-1}-c_1)$, $e_{-\alpha}(v^{-1}-c_2)$, $\frac{1}{2}h_{\alpha}(v^{-1}-\frac{c_1+c_2}{2})$. The corresponding dual elements in $\mathfrak{g}[u]$ are respectively $e_{\alpha}u^{k+1}$, $e_{-\alpha}u^{k+1}$, $h_{\alpha}u^{k+1}$, $e_{-\alpha}$, $e_{\alpha}$, $h_{\alpha}$.
One can check by a straightforward computation that the $r$-matrix constructed from these dual bases is precisely $r_0(u,v)$.
Let us note that $r_0(u,v)$ can be rewritten in the following form: $$r_0(u,v)=\frac{1-c_1v-c_2u+c_1c_2uv}{v-u}\Omega+(c_1-c_2)r_{\mathrm{DJ}},$$ where $r_{\mathrm{DJ}}=\frac{1}{2}(\sum_{\alpha>0}e_{\alpha}\wedge e_{-\alpha}+\Omega)$.
Let $a(u)=1/(1-c_1u)(1-c_2u)$, where $c_1$ and $c_2$ are different non-zero complex numbers. Let $r(u,v)$ be an $r$-matrix which corresponds to a Lagrangian subalgebra $W$ of $\mathfrak{g}((u))$, with respect to the form $Q_{a(u)}$, such that $W \subseteq \mathfrak{g}[u^{-1}]$. Then $$r(u,v)=\frac{1-c_1v-c_2u+c_1c_2uv}{v-u}\Omega+(c_1-c_2)r,$$ where $r\in \mathfrak{g}\otimes\mathfrak{g}$ verifies: $r+r^{21}=\Omega$ and $\mathrm{CYB}(r)=0$.
Since $W$ is a Lagrangian subalgebra of $\mathfrak{g}((u))$ which is transversal to $\mathfrak{g}[[u]]$ and $W \subseteq \mathfrak{g}[u^{-1}]$, it is uniquely defined by a Lagrangian subalgebra $\bar{W}$ of $\mathfrak{g}\oplus \mathfrak{g}$ transversal to $\mathrm{diag}(\mathfrak{g})$. On the other hand, Lagrangian subalgebras with this property are in a one-to-one correspondence with solutions of the modified classical Yang–Baxter equation, i.e., $r+r^{21}=\Omega$ and $\mathrm{CYB}(r)=0$ (see [@RS; @S1]). If $r(u,v)$ corresponds to $W$, then it is uniquely defined by a constant $r$-matrix and the non-constant part of $r(u,v)$ is given by the same formula as $r_0(u,v)$. This ends the proof.
Consider the Lie algebra $\mathfrak{g}+\varepsilon\mathfrak{g}$ endowed with the following invariant form: $\bar{Q}_{\varepsilon}(x_1+\varepsilon x_2,y_1+\varepsilon y_2)=K(x_1,y_2)+K(x_2,y_1)$.
Let $a(u)=1/(1-u)^2$. There exists a one-to-one correspondence between Lagrangian subalgebras $W$ of $\mathfrak{g}((u))$, with respect to $Q_{a(u)}$, which are transversal to $\mathfrak{g}[[u]]$ and satisfy $W \subseteq \mathfrak{g}[u^{-1}]$, and Lagrangian subalgebras in $\mathfrak{g}+\varepsilon\mathfrak{g}$, with respect to $\bar{Q}_{\varepsilon}$, transversal to $\mathfrak{g}$.
Assume $W$ is a Lagrangian subalgebra of $\mathfrak{g}((u))$ which is transversal to $\mathfrak{g}[[u]]$ and such that $W \subseteq \mathfrak{g}[u^{-1}]$. This implies that $W \supseteq \mathfrak{g}[u^{-1}]^{\perp}=(1-u)^2u^{-2}\mathfrak{g}[u^{-1}]=
(u^{-1}-1)^2\mathfrak{g}[u^{-1}]$.
The quotient $\frac{W}{(u^{-1}-1)^2\mathfrak{g}[u^{-1}]}$ is therefore a subalgebra of the Lie algebra $\frac{\mathfrak{g}[u^{-1}]}{(u^{-1}-1)^2\mathfrak{g}[u^{-1}]}$.
On the other hand the Lie algebra $\frac{\mathfrak{g}[u^{-1}]}{(u^{-1}-1)^2\mathfrak{g}[u^{-1}]}$ can be identified with $\mathfrak{g}+\varepsilon\mathfrak{g}$. Indeed let $\psi:\mathfrak{g}[u^{-1}]\longrightarrow \mathfrak{g}+\varepsilon\mathfrak{g}$ be given by $\psi(x)=x$, $\psi(xu^{-1})=x(1+\varepsilon)$, for all $x\in\mathfrak{g}$. Then $\psi$ is an epimorphism whose kernel equals $(u^{-1}-1)^2\mathfrak{g}[u^{-1}]$.
The image $\bar{W}$ of $W$ in $\mathfrak{g}+\varepsilon\mathfrak{g}$ is obviously a Lagrangian subalgebra transversal to $\mathfrak{g}$ (we also note that $\mathfrak{g}[[u]]\cap \mathfrak{g}[u^{-1}]=\mathfrak{g}$). One can check that the correspondence which associates $\bar{W}$ to $W$ is bijective.
Let $$r_0(u,v)=\frac{(u-1)(v-1)}{v-u} \Omega.$$ Then $r_0(u,v)$ provides a Lie bialgebra structure on $\mathfrak{g}[u]$.
Take $\bar{W_0}=\varepsilon\mathfrak{g}$ with canonical basis $\varepsilon e_{\alpha}$, $\varepsilon e_{-\alpha}$, $\varepsilon h_{\alpha}$, for all positive roots $\alpha$.
Let $W_0$ be the Lagrangian subalgebra of $\mathfrak{g}((u))$ which corresponds to $\bar{W_0}$. Then $\frac{W_0}{(u^{-1}-1)^2\mathfrak{g}[u^{-1}]}$ is spanned by $(u^{-1}-1)e_{\alpha}$, $(u^{-1}-1)e_{-\alpha}$ and $(u^{-1}-1)h_{\alpha}$. Therefore the Lie algebra $W_0$ is spanned by these elements together with $(u^{-1}-1)^2u^{-k}e_{\alpha}$, $(u^{-1}-1)^2u^{-k}e_{-\alpha}$, $(u^{-1}-1)^2u^{-k}h_{\alpha}$, for all positive roots $\alpha$ and all natural $k$.
The basis in $W_0$ which is dual to the canonical basis in $\mathfrak{g}[u]$ is $(u^{-1}-1)e_{\alpha}$, $(u^{-1}-1)e_{-\alpha}$, $(u^{-1}-1)h_{\alpha}$, $(u^{-1}-1)^2u^{-k-2}e_{\alpha}$, $(u^{-1}-1)^2u^{-k-2}e_{-\alpha}$, $(u^{-1}-1)^2u^{-k-2}h_{\alpha}$. An easy computation shows that the $r$-matrix constructed from the dual bases has the form $r_0(u,v)$.
Let $a(u)=1/(1-u)^2$ and $r(u,v)$ be an $r$-matrix which corresponds to a Lagrangian subalgebra $W$ of $\mathfrak{g}((u))$, with respect to the form $Q_{a(u)}$, such that $W \subseteq \mathfrak{g}[u^{-1}]$. Then $$r(u,v)= \frac{(u-1)(v-1)}{v-u} \Omega+r,$$ where $r\in\mathfrak{g}\wedge\mathfrak{g}$ verifies the classical Yang–Baxter equation $\mathrm{CYB}(r)=0$.
Since $W$ is a Lagrangian subalgebra of $\mathfrak{g}((u))$ which is transversal to $\mathfrak{g}[[u]]$ and $W \subseteq \mathfrak{g}[u^{-1}]$, it is uniquely defined by a Lagrangian subalgebra $\bar{W}$ of $\mathfrak{g}+\varepsilon\mathfrak{g}$ transversal to $\mathfrak{g}$. On the other hand, Lagrangian subalgebras $\bar{W}$ with this property are in a one-to-one correspondence with skew-symmetric solutions of the classical Yang–Baxter equation (see [@S1]).
Let $a(u)=1/1-u$. There exists a one-to-one correspondence between Lagrangian subalgebras $W$ of $\mathfrak{g}((u))$, with respect to $Q_{a(u)}$, which are transversal to $\mathfrak{g}[[u]]$ and satisfy $W \subseteq \mathfrak{g}[u^{-1}]$, and Lagrangian subalgebras in $\mathfrak{g}\oplus\mathfrak{g}$ transversal to $\mathrm{diag}(\mathfrak{g})$.
Since $W$ is Lagrangian and contained in $\mathfrak{g}[u^{-1}]$, we have $W \supseteq \mathfrak{g}[u^{-1}]^{\perp}=u^{-2}(1-u)\mathfrak{g}[u^{-1}]=u^{-1}(1-u^{-1})\mathfrak{g}[u^{-1}]$.
Then $\frac{W}{u^{-1}(1-u^{-1})\mathfrak{g}[u^{-1}]}$ is a subalgebra of $\frac{\mathfrak{g}[u^{-1}]}{u^{-1}(1-u^{-1})\mathfrak{g}[u^{-1}]}$.
On the other hand, $\frac{\mathfrak{g}[u^{-1}]}{u^{-1}(1-u^{-1})\mathfrak{g}[u^{-1}]}$ is isomorphic to $\mathfrak{g}\oplus\mathfrak{g}$ via a morphism $\psi(x)=(x,x)$, $\psi(xu^{-1})=(0,x)$, for all $x\in\mathfrak{g}$. The projection of $W$ onto $\mathfrak{g}\oplus\mathfrak{g}$ becomes a Lagrangian subalgebra which is complementary to the diagonal.
Let $$r_0(u,v)=\frac{1-u}{v-u} \Omega-r_{DJ},$$ where $r_{\mathrm{DJ}}=\frac{1}{2}(\sum_{\alpha>0}e_{\alpha}\wedge e_{-\alpha}+\Omega)$. Then $r_0(u,v)$ provides a Lie bialgebra structure on $\mathfrak{g}[u]$.
Let $\bar{W_0}$ be the Lie subalgebra of $\mathfrak{g}\oplus\mathfrak{g}$ spanned by the pairs $(e_{-\alpha},0)$, $(0,e_{\alpha})$, $(h_{\alpha},-h_{\alpha})$, for all positive roots $\alpha$. Then the corresponding Lagrangian subalgebra $W_0$ of $\mathfrak{g}((u))$ is spanned by the elements $\psi^{-1}(e_{-\alpha},0)=(1-u^{-1})e_{-\alpha}$, $\psi^{-1}(0,e_{\alpha})=u^{-1}e_{\alpha}$, $\psi^{-1}(h_{\alpha},-h_{\alpha})=(1-2u^{-1})h_{\alpha}$ and contains $u^{-1}(1-u^{-1})\mathfrak{g}[u^{-1}]$.
The basis in $W_0$ which is dual to the canonical basis of $\mathfrak{g}[u]$ is the following: $(1-u)u^{-k-1}e_{\alpha}$, $(1-u)u^{-k-1}e_{-\alpha}$, $(1-u)u^{-k-1}h_{\alpha}$, $(u^{-1}-1)e_{-\alpha}$, $u^{-1}e_{\alpha}$, $-\frac{1}{4}(1-2u^{-1})h_{\alpha}$. The corresponding $r$-matrix is $r_0(u,v)$.
Consequently, the following result holds:
Let $a(u)=1/1-u$ and $r(u,v)$ be an $r$-matrix which corresponds to a Lagrangian subalgebra $W$ of $\mathfrak{g}((u))$, with respect to the form $Q_{a(u)}$, such that $W \subseteq \mathfrak{g}[u^{-1}]$. Then $$r(u,v)= \frac{1-u}{v-u} \Omega-r,$$ where $r\in\mathfrak{g}\otimes\mathfrak{g}$ verifies $r+r^{21}=\Omega$ and $\mathrm{CYB}(r)=0$.
The case $a(u)=1$ can be treated in a similar manner and the following results can be easily proved:
Let $a(u)=1$. There exists a one-to-one correspondence between Lagrangian subalgebras $W$ of $\mathfrak{g}((u))$, with respect to $Q_{a(u)}$, which are transversal to $\mathfrak{g}[[u]]$ and satisfy $W \subseteq \mathfrak{g}[u^{-1}]$, and Lagrangian subalgebras in $\mathfrak{g}+\varepsilon\mathfrak{g}$ transversal to $\mathfrak{g}$.
Let $a(u)=1$ and $r(u,v)$ be an $r$-matrix which corresponds to a Lagrangian subalgebra $W$ of $\mathfrak{g}((u))$, with respect to the form $Q_{a(u)}$, such that $W \subseteq \mathfrak{g}[u^{-1}]$. Then $$r(u,v)= \frac{\Omega}{v-u}+r,$$ where $r\in\mathfrak{g}\wedge\mathfrak{g}$ verifies the classical Yang–Baxter equation $\mathrm{CYB}(r)=0$.
Lie bialgebra structures on $\mathfrak{g}[u]$ in Case II
========================================================
We will analyse in more detail the case $D_{\bar{\delta}}(\mathfrak{g}[[u]])=\mathfrak{g}((u))\oplus\mathfrak{g}$. We note that the double is endowed with the following nondegenerate bilinear form: $$Q_{a(u)}(f_1(u)+x_1,f_2(u)+x_2)=\mathrm{Res}_{u=0}(u^{-1}a(u)K(f_1(u),f_2(u)))-K(x_1,x_2),$$ for all $f_1(u), f_2(u)\in\mathfrak{g}((u))$ and $x_1,x_2\in\mathfrak{g}$.
According to [@SZ], the following statement holds:
There exists a one-to-one correspondence between Lie bialgebra structures $\delta$ on $\mathfrak{g}[u]$ satisfying $D_{\bar{\delta}}(\mathfrak{g}[[u]])=\mathfrak{g}((u))\oplus\mathfrak{g}$ and bounded Lagrangian subalgebras $W$ of $\mathfrak{g}((u))$, with respect to the nondegenerate bilinear form $Q_{a(u)}$, and transversal to $\mathfrak{g}[[u]]$.
If $W$ is a bounded Lagrangian subalgebra of $\mathfrak{g}((u))\oplus\mathfrak{g}$ transversal to $\mathfrak{g}[[u]]$, then $W\oplus\mathfrak{g}[u]=\mathfrak{g}[u,u^{-1}]\oplus\mathfrak{g}$. Then the corresponding $r$-matrix can be constructed by choosing dual bases in $W$ and $\mathfrak{g}[u]$ with respect to $Q_{a(u)}$ and projecting onto $\mathfrak{g}[u,u^{-1}]$.
For any $\sigma\in\mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$, denote by $\tilde{\sigma}(u)=\sigma(u)\oplus\sigma(0)$, regarded as an automorphism of $\mathfrak{g}((u))\oplus\mathfrak{g}$. Then the following result holds (we omit the proof which is similar to that of Proposition \[max\_ord\]):
Suppose that $W$ is a bounded Lagrangian subalgebra of $\mathfrak{g}((u))\oplus\mathfrak{g}$, with respect to $Q_{a(u)}$ and transversal to $\mathfrak{g}[[u]]$. Then there exists $\sigma\in\mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$ such that $\tilde{\sigma}(u)(W)\subseteq (\mathbb{O}_{\alpha}\cap \mathfrak{g}[u,u^{-1}])\oplus\mathfrak{g}$, where $\alpha$ is either a simple root or $-\alpha_{\max}$.
From now on we will restrict ourselves to the case $-\alpha_{\max}$, so that we will study Lagrangian subalgebras $W\subseteq\mathfrak{g}[u^{-1}]\oplus\mathfrak{g}$ transversal to $\mathfrak{g}[[u]]$ . Let us recall that such subalgebras exist if only if $1/a(u)$ has degree at most 1. By a change of variable or rescaling the form $Q$, we have two situations: $a(u)=1/1-u$ or $a(u)=1$.
Let $a(u)=1/1-u$. There exists a one-to-one correspondence between Lagrangian subalgebras $W$ of $\mathfrak{g}((u))\oplus\mathfrak{g}$, with respect to $Q_{a(u)}$, which are transversal to $\mathfrak{g}[[u]]$ and satisfy $W \subseteq \mathfrak{g}[u^{-1}]\oplus\mathfrak{g}$, and Lagrangian subalgebras in $\mathfrak{g}\oplus\mathfrak{g}$ transversal to $\mathrm{diag}(\mathfrak{g})$.
We immediately note that $W$ must contain $(\mathfrak{g}[u^{-1}]\oplus\mathfrak{g})^{\perp}=(1-u)u^{-1}\mathfrak{g}[u^{-1}]=(u^{-1}-1)\mathfrak{g}[u^{-1}]$. The quotient $\frac{W}{(u^{-1}-1)\mathfrak{g}[u^{-1}]}$ is a subalgebra of $\frac{\mathfrak{g}[u^{-1}]}{(u^{-1}-1)\mathfrak{g}[u^{-1}]}\oplus\mathfrak{g}$, obviously identified with $\mathfrak{g}\oplus\mathfrak{g}$. The conclusion follows by arguments similar to those in Section 2.
Consequently, one obtains the following result (whose proof we omit, being similar to those in Section 2):
\[sol\_II\] Let $a(u)=1/1-u$ and $r(u,v)$ be an $r$-matrix which corresponds to a Lagrangian subalgebra $W$ of $\mathfrak{g}((u))\oplus\mathfrak{g}$,with respect to the form $Q_{a(u)}$, such that $W \subseteq \mathfrak{g}[u^{-1}]\oplus\mathfrak{g}$. Then $$r(u,v)= \frac{u(1-v)}{v-u} \Omega+r,$$ where $r\in\mathfrak{g}\otimes\mathfrak{g}$ verifies $r+r^{21}=\Omega$ and $\mathrm{CYB}(r)=0$.
The remaining case $a(u)=1$ can be treated analogously.
Let $a(u)=1$. There exists a one-to-one correspondence between Lagrangian subalgebras $W$ of $\mathfrak{g}((u))\oplus\mathfrak{g}$, with respect to $Q_{a(u)}$, which are transversal to $\mathfrak{g}[[u]]$ and satisfy $W \subseteq \mathfrak{g}[u^{-1}]\oplus\mathfrak{g}$, and Lagrangian subalgebras in $\mathfrak{g}\oplus\mathfrak{g}$ transversal to $\mathrm{diag}(\mathfrak{g})$.
We note that $W$ must contain $u^{-1}\mathfrak{g}[u^{-1}]$ and $\frac{W}{u^{-1}\mathfrak{g}[u^{-1}]}$ is a subalgebra of $\frac{\mathfrak{g}[u^{-1}]}{u^{-1}\mathfrak{g}[u^{-1}]}\oplus\mathfrak{g}$, which is isomorphic to $\mathfrak{g}\oplus\mathfrak{g}$. Conclusion follows easily.
Let $a(u)=1$ and $r(u,v)$ be an $r$-matrix which corresponds to a Lagrangian subalgebra $W$ of $\mathfrak{g}((u))\oplus\mathfrak{g}$,with respect to the form $Q_{a(u)}$, such that $W \subseteq \mathfrak{g}[u^{-1}]\oplus\mathfrak{g}$. Then $$r(u,v)= \frac{v}{v-u} \Omega+r,$$ where $r\in\mathfrak{g}\otimes\mathfrak{g}$ verifies $r+r^{21}=\Omega$ and $\mathrm{CYB}(r)=0$.
$r$-matrices of the form $\frac{v}{v-u} \Omega+p(u,v)$, where $p(u,v)\in\mathfrak{g}[u,v]$, are called *quasi-trigonometric* and were classified in [@KPSST; @PS].
Lie bialgebra structures on $\mathfrak{g}[u]$ in Case III
=========================================================
Finally, let us treat the case $D_{\bar{\delta}}(\mathfrak{g}[[u]])=\mathfrak{g}((u))\oplus(\mathfrak{g}+\varepsilon\mathfrak{g})$. with the nondegenerate bilinear form on the double given by the formula $$Q_{a(u)}(f_1(u)+x_2+\varepsilon x_3,f_2(u)+y_2+\varepsilon y_3)=\mathrm{Res}_{u=0}(u^{-2}a(u)K(f_1(u),f_2(u))-$$ $$-K(x_3,y_2)-K(x_2,y_3),$$ for any $f_1(u),f_2(u)\in\mathfrak{g}((u))$ and $x_2,x_3,y_2,y_3\in\mathfrak{g}$.
According to [@SZ], the following result holds:
There exists a one-to-one correspondence between Lie bialgebra structures $\delta$ on $\mathfrak{g}[u]$ satisfying $D_{\bar{\delta}}(\mathfrak{g}[[u]])=\mathfrak{g}((u))\oplus(\mathfrak{g}+\varepsilon\mathfrak{g})$ and bounded Lagrangian subalgebras $W$ of $\mathfrak{g}((u))\oplus(\mathfrak{g}+\varepsilon\mathfrak{g})$, with respect to the nondegenerate bilinear form $Q_{a(u)}$, and transversal to $\mathfrak{g}[[u]]$.
If $W$ is Lagrangian in $\mathfrak{g}((u))\oplus(\mathfrak{g}+\varepsilon\mathfrak{g})$ and transversal to $\mathfrak{g}[[u]]$, then $W\oplus\mathfrak{g}[u]=\mathfrak{g}[u,u^{-1}]\oplus((\mathfrak{g}+\varepsilon\mathfrak{g})$. The corresponding $r$-matrix can be found by choosing dual bases in $W$ and $\mathfrak{g}[u]$ respectively and projecting onto $\mathfrak{g}[u,u^{-1}]$.
Recall that any $\sigma(u)\in \mathrm{Ad}(\mathfrak{g}[u])$ induces an automorphism $\sigma(0)\in\mathrm{Ad}(\mathfrak{g})$, which in turn gives an well-defined automorphism $\bar{\sigma}(0)$ of $\mathfrak{g}+\varepsilon\mathfrak{g}$ via $\bar{\sigma}(0)(x+\varepsilon y)=\sigma(0)(x)+\varepsilon\sigma(0)(y)$. Then $\tilde{\sigma}(u)=\sigma(u)\oplus \bar{\sigma}(0)$ is an automorphism of $\mathfrak{g}((u))\oplus (\mathfrak{g}+\varepsilon\mathfrak{g})$. The following proposition can be proved similarly to Proposition \[max\_ord\]:
Suppose that $W$ is a bounded Lagrangian subalgebra of $\mathfrak{g}((u))\oplus(\mathfrak{g}+\varepsilon\mathfrak{g})$, with respect to $Q_{a(u)}$ and transversal to $\mathfrak{g}[[u]]$. Then there exists $\sigma\in\mathrm{Ad}_{\mathbb{C}[u]}(\mathfrak{g}[u])$ such that $\tilde{\sigma}(u)(W)\subseteq (\mathbb{O}_{\alpha}\cap \mathfrak{g}[u,u^{-1}])\oplus(\mathfrak{g}+\varepsilon\mathfrak{g})$, where $\alpha$ is either a simple root or $-\alpha_{\max}$.
Such subalgebras exist only when $\alpha$ has coefficient one in the decomposition of $\alpha_{\max}$ or is $-\alpha_{\max}$. Moreover $a(u)$ should be a constant. Without loss of generality, one may assume that $a(u)=1$.
Let $a(u)=1$. There exists a one-to-one correspondence between Lagrangian subalgebras $W$ of $\mathfrak{g}((u))\oplus(\mathfrak{g}+\varepsilon\mathfrak{g})$, with respect to $Q_{a(u)}$, which are transversal to $\mathfrak{g}[[u]]$ and satisfy $W \subseteq \mathfrak{g}[u^{-1}]\oplus(\mathfrak{g}+\varepsilon\mathfrak{g})$, and Lagrangian subalgebras in $\mathfrak{g}+\varepsilon\mathfrak{g}$ transversal to $\mathfrak{g}$.
Being isotropic, $W$ must include $(\mathfrak{g}+\varepsilon\mathfrak{g})^{\perp}=\mathfrak{g}[u^{-1}]$. The quotient $W/\mathfrak{g}[u^{-1}]$ is a Lagrangian subalgebra in $\mathfrak{g}+\varepsilon\mathfrak{g}$ complementary to $\mathfrak{g}$.
Consequently, we obtain the description of the corresponding $r$-matrices:
Let $a(u)=1$ and $r(u,v)$ be an $r$-matrix which corresponds to a Lagrangian subalgebra $W$ of $\mathfrak{g}((u))\oplus(\mathfrak{g}+\varepsilon\mathfrak{g})$, with respect to the form $Q_{a(u)}$, such that $W \subseteq \mathfrak{g}[u^{-1}]\oplus(\mathfrak{g}+\varepsilon\mathfrak{g})$. Then $$r(u,v)= \frac{uv}{v-u} \Omega+r,$$ where $r\in\mathfrak{g}\wedge\mathfrak{g}$ verifies $\mathrm{CYB}(r)=0$.
$r$-matrices of the form $\frac{uv}{v-u} \Omega+p(u,v)$, where $p(u,v)\in \mathfrak{g}[u,v]$, are called *quasi-rational* and were studied in [@SY].
Quasi-twist equivalence between Lie bialgebra structures
========================================================
We first remind the reader the notion of twist equivalence between Lie bialgebra structures, according to [@KS]:
Two Lie bialgebra structures $\delta_1$ and $\delta_2$ on a Lie algebra $L$ are called *twist-equivalent* is there exists a Lie algebra isomorphism $f: D_{\delta_1}(L)\longrightarrow D_{\delta_2}(L)$ satisfying the following properties:
\(1) $Q_1(x,y)=Q_2(f(x),f(y))$, for any $x, y\in D_{\delta_1}(L)$, where $Q_i$ denotes the canonical form on $D_{\delta_i}(L)$, $i=1,2$.
\(2) $f\circ j_1=j_2$, where $j_i$ is the canonical embedding of $L$ into $D_{\delta_i}(L)$.
Secondly, let us also recall the notion of quantum twisting for Hopf algebras (see [@ES]). Let $A:=A(m,\Delta,\epsilon,S$) be a Hopf algebra with multiplication $m:A\otimes A\rightarrow A$, coproduct $\Delta:A \rightarrow A\otimes A$, counit $\epsilon:A\rightarrow\mathbb C$, and antipode $S:A\to A$.
An invertible element $F\in A\otimes A$, $F=\sum_i f^{(1)}_i\otimes f^{(2)}_i$ is called a *quantum twist* if it satisfies the cocycle equation $$F^{12}(\Delta\otimes{\rm id})(F)=F^{23}({\rm id}\otimes\Delta)(F)\,,$$ and the “unital” normalization condition $$(\epsilon \otimes{\rm id})(F)=({\rm id}\otimes\epsilon )(F)=1\,.$$
One can now define a twisted Hopf algebra $A^{(F)}:=A^{(F)}(m,\Delta^{(F)},\epsilon,S^{(F)}$) which has the same multiplication $m$ and the counit mapping $\epsilon$ but the twisted coproduct and antipode $$\Delta^{(F)}(a)=F\Delta(a)F^{-1},\quad\;S^{(F)}(a)=u\,S(a)u^{-1}, \quad\;u= \sum_i
f^{(1)}_{i}S(f^{(2)}_i).$$
Regarding the quantization of twist-equivalent Lie bialgebra structures, the following result was proved by Halbout in [@H]:
Suppose $\delta_1$ and $\delta_2$ are twist-equivalent and let $(A_1,\Delta_1)$ be a quantization of $(L, \delta_1)$. Then there exists a quantization $(A_2,\Delta_2)$ of $(L, \delta_2)$ such that $A_2$ is obtained from $A_1$ via a quantum twist.
Let $\delta_1$ and $\delta_2$ be two Lie bialgebra structures on $L$. Assume that there exists an automorphism $\sigma$ of $L$ such that, for any $a\in L$, $\delta_2(a)=(\sigma\otimes \sigma)(\delta_1(\sigma^{-1}(a)))$. Then $\sigma$ is not necessarily extendable to an isomorphism $\bar{\sigma}: D_{\delta_1}(L)\longrightarrow D_{\delta_2}(L)$. In fact, it might happen that the doubles are not isomorphic even as Lie algebras. Let us consider an example. Recall that the $r$-matrix
$$r(u,v)=\frac{1-uv}{v-u}\Omega+\sum_{\alpha>0} e_{\alpha}\wedge e_{-\alpha}$$ induces a Lie bialgebra structure on $\mathfrak{g}[u]$ for which the classical double is $(\mathfrak{g}[u,u^{-1}],Q_{1/(1-u^2)})$.
Let us make the change of variable $2u_1=u+1$, $2v_1=v+1$, which is an automorphism of $\mathfrak{g}[u]$. Then $$r(u,v)=2(\frac{u_1(1-v_1)}{v_1-u_1}\Omega+r_{DJ}),$$ where $r_{DJ}=(r_0+\Omega)/2$. One can notice that $r(u,v)$ is proportional to the solution obtained in Theorem \[sol\_II\]. However, this solution corresponds to a Lie bialgebra structure for which the double is $\mathfrak{g}[u,u^{-1}]\oplus\mathfrak{g}$. Obviously the Lie algebras $\mathfrak{g}[u,u^{-1}]$ and $\mathfrak{g}[u,u^{-1}]\oplus\mathfrak{g}$ are not isomorphic.
$\sigma:\mathfrak{g}[u]\longrightarrow\mathfrak{g}[u]$ given by $\sigma(u)=pu+q$ cannot be extended to $\mathfrak{g}[u,u^{-1}]$.
Two Lie bialgebra structures $\delta_1$ and $\delta_2$ on $\mathfrak{g}[u]$ are called *quasi-twist equivalent* if $\delta_2(a)=(\sigma\otimes \sigma)(\delta_1(\sigma^{-1}(a)))$, for any $a\in \mathfrak{g}[u]$. Equivalently, the corresponding $r$-matrices satisfy the relation $r_2(u,v)=C\cdot r_1(\sigma(u),\sigma(v))$, for some constant $C$.
Recall that in Section 2 we studied the Lie bialgebra structures on $\mathfrak{g}[u]$ whose double is $\mathfrak{g}[u,u^{-1}]$ endowed with the form $Q_{a(u)}$, where $a(u)=1/(1-c_1u)(1-c_2u)$, for any non-zero constants $c_1\neq c_2$.
By Corollary \[sol\_I\], the double $(\mathfrak{g}[u,u^{-1}],Q_{1/(1-c_1u)(1-c_2u)})$ leads to the following $r$-matrix:
$$r_{c_1,c_2}(u,v)= \frac{1-(c_1+c_2)u+c_1c_2uv}{v-u}\Omega-r_{c_1,c_2},$$ where $$r_{c_1,c_2}=\sum_{\alpha>0}(c_1e_{-\alpha}\otimes e_{\alpha}+c_2e_{\alpha}\otimes e_{-\alpha}+\frac{c_1+c_2}{4}h_{\alpha}\otimes h_{\alpha}).$$
For any non-zero complex constants $c_1\neq c_2$, $d_1\neq d_2$, the Lie bialgebra structures with corresponding $r$-matrices $r_{c_1,c_2}(u,v)$ and $r_{d_1,d_2}(u,v)$, are quasi-twist equivalent.
Let us first notice that there exist unique $p$, $q$ such that $d_1=\frac{c_1p}{1-c_1q}$ and $d_2=\frac{c_2p}{1-c_2q}$. Since we also have
$$r_{d_1,d_2}(u,v)= \frac{1-(d_1+d_2)u+d_1d_2uv}{v-u}\Omega-r_{d_1,d_2},$$ a straightforward computation gives
$$r_{d_1,d_2}(u,v)(1-c_1q)(1-c_2q)=\frac{1-(c_1+c_2)(pu+q)+c_1c_2(pu+q)(pv+q)}{v-u}\Omega$$$$-p\cdot r_{c_1,c_2}.$$
Thus $$r_{d_1,d_2}(u,v)=\frac{p}{(1-c_1q)(1-c_2q)}\cdot r_{c_1,c_2}(pu+q,pv+q).$$
We have obtained that the Lie bialgebra structures on $\mathfrak{g}[u]$ corresponding to $r_{c_1,c_2}$ and $r_{d_1,d_2}$ are quasi-twist equivalent.
Concerning the quantization of the quasi-twist equivalent structures in the example above, the following conjecture was proposed by A. Stolin:
**Conjecture.** Quantizations of Lie bialgebra structures on $\mathfrak{g}[u]$ defined by $r_{c_1,c_2}(u,v)$ can be chosen isomorphic as quasi-Hopf algebras.
Here we recall that two Hopf algebras $(A,\Delta_1)$ and $(A,\Delta_2)$ are isomorphic as quasi-Hopf algebras if there is an invertible element $F\in A\otimes A$ and some $A$-invariant element $\Phi$ of $A^{\otimes3}$ such that $F^{12}(\Delta\otimes{\rm id})(F)=F^{23}({\rm id}\otimes\Delta)(F)\cdot\Phi$ and $\Delta_2=F\Delta_1F^{-1}$.
**Acknowledgment.** The authors are thankful to Professor A. Stolin for fruitful discussions.
[8]{} Drinfeld, V.: Quantum groups. Proc. ICM (Berkeley 1986) **1**, 798–820. American Mathematical Society (1987) Etingof, P., Schiffmann, O.: Lectures on Quantum Groups. Somerville, MA: International Press (1998) Halbout, G.: Formality theorem for Lie algebras and quantization of twists and coboundary $r$-matrices. Adv. Math. **207**, 617–633 (2006) Karolinsky, E., Stolin, A.: Classical dynamical $r$-matrices, Poisson homogeneous spaces and Lagrangian subalgebras. Lett. Math. Phys. **60**, 257–274 (2002) Khoroshkin, S., Pop, I., Samsonov, M., Stolin, A., Tolstoy, V.: On some Lie bialgebra structures on polynomial algebras and their quantization. Commun. Math. Phys. **282**(3), 625–662 (2008) Pop, I., Stolin, A.: Lagrangian subalgebras and quasi-trigonometric $r$-matrices. Lett. Math. Phys. **85**(2–3), 249–262 (2008) Reshetikhin, N., Semenov-Tian-Shansky, M.: Quantum R-matrices and factorization problems. J. Geom. Phys. **5**(4), 533–550 (1988) Montaner, F., Stolin, A., Zelmanov, E.: Classification of Lie bialgebras and quantum groups over polynomials and Taylor series. To appear. Stolin, A.: Some remarks on Lie bialgebra structures on simple complex Lie algebras. Commun. Alg. **27**(9), 4289–4303 (1999) Stolin, A.: A geometrical approach to rational solutions of the classical Yang-Baxter equation. Part I. Symposia Gaussiana, Conf. A: Mathematics and Theoretical Physics (Munich 1993), 347–357. Walter de Gruyter (1995) Stolin, A., Yermolova–Magnusson, J.: The 4th structure. Czech. J. Phys. **56**(10/11), 1293–1297 (2006)
|
---
author:
- 'F. Rodler'
- 'R. Deshpande'
- 'M. R. Zapatero Osorio'
- 'E. L. Martín'
- 'M.M. Montgomery'
- 'C. del Burgo'
- 'O. L. Creevey'
date: 'Received ?; accepted ?'
title: 'Search for radial velocity variations in eight M-dwarfs with NIRSPEC/Keck II'
---
[Radial velocity (RV) measurements from near-infrared spectra have become a potentially powerful tool to search for planets around cool stars and sub-stellar objects. As part of a large survey to characterize M-dwarfs using NIRSPEC at Keck II, we obtained spectra of eight late M-dwarfs (spectral types M5.0-M8.0) during two or more observing epochs per target. These spectra were taken with intermediate spectral resolving powers ($R\sim20,000$) in the $J$-band. ]{} [We search for relative radial velocity variability in these late M-dwarfs and test the NIRSPEC capability of detecting short period brown dwarf and massive planetary companions around low-mass stars in the $J$-band ($\approx 1.25~\mu$m). Additionally, we reanalyzed the data of the M8-type star vB10 (one of our targets) presented in Zapatero Osorio et al. (2009), which were obtained with the same instrumentation as our data.]{} [To achieve a precise RV measurement stability, the NIRSPEC spectra are self-calibrated by making use of the telluric absorption lines, which are present in the observed spectra and used as a long-term stable reference. In the modeling process a multi-parameter $\chi^2$-optimization is employed to generate an accurate description of the observation. The telluric lines allow us to model the instrumental profile of the spectrograph and the determination of the Doppler shift of the stellar absorption lines. ]{} [For the entire M-dwarf sample, we do not find any evidence of relative RV variations induced by a short period brown dwarf or massive planetary companion. The typical RV precision of the measurements is between 180 and 300 m s$^{-1}$, which is sufficient to detect hot Neptunes around M-dwarfs. Also, we find that the spurious RV shift in Zapatero et al. (2009) of the star VB10 was caused by asymmetries in the instrumental profile between different observing epochs, which were not taken into account in their analysis.]{}
Introduction
============
The search for extrasolar planets has led to more than 700 confirmed discoveries[^1] by using all detection techniques. Up to now, most of them have been detected by means of the radial velocity (RV) technique using high-resolution spectrographs ($R=\lambda / \Delta \lambda \ge 40,000$) at optical wavelengths. Most discoveries are giant gaseous planets (typically hot Neptunes and Jupiters) of short periods (of a few days) around stars of spectral types F, G and K.
As potential hosts to rocky planetary companions, M-dwarfs have become increasingly popular as targets for RV searches (e.g. Endl et al. 2006, Charbonneau et al. 2009, Mayor et al. 2009, Zechmeister et al. 2009). Very cool stars such as M-dwarfs are the most abundant type ($\sim70 \%$) of stars in the solar neighborhood and the Milky Way in general (Henry at al. 1997). The effective temperatures and masses of M-dwarfs, respectively, are in the range 3700 to 2200 K and 0.5 to 0.07 solar masses for the M0 to M9.5 spectral types. They exhibit prominent absorption features corresponding to strong neutral atoms, H$_2$0, FeH, VO, CO, and TiO. Owing to the low masses of these objects, the reflex motion of the host star due to the gravitational pull of the extrasolar planet is higher and more easily detectable than for more massive host stars. Since M-dwarfs are very cool stars in comparison with solar-type stars, short period planets would more likely be situated in the habitable zone.
M-dwarfs emit most of their energy around $1.1-1.3~\mu {\rm m}$, in the near-infrared (NIR), while they appear very faint at optical wavelengths. First attempts to measure RV variations among very cool M-dwarfs at NIR wavelengths were done by Martín et al. (2006). They achieved a RV precision of around 300 m s$^{-1}$ for the M9.5-dwarf LP944-20 by using the spectrograph NIRSPEC, which is mounted on the Keck II telescope in Hawaii (McLean et al. 1998). Recently, several research groups have reported high-precision RV measurements taken in the NIR with CRIRES (Käufl et al. 2004), mounted at the UT1/VLT in the Paranal Observatory of ESO in Chile. Bean et al. (2010a) conducted high-resolution spectroscopic data of over 60 M-dwarfs (spectral types M4-M9) and used a NH$_3$ gas cell spectrum as a stable reference, and report an RV precision of better than $5~{\rm m~s^{-1}}$. Figueira et al. (2010) took observations of the planetary candidate TW Hya and achieved a RV precision better than $10~{\rm m~s^{-1}}$ by adopting telluric lines as a stable reference. Blake et al. (2010) report RV measurements of 59 M- and L-dwarfs using the Keck/NIRSPEC spectrograph, with the aim to detect low-mass companions. They made use of strong CO absorption features around 2.3 $\mu$m in M- and L-dwarfs and achieved RV precisions between 50 and 200 m s$^{-1}$. Tanner et al. (2010) report preliminary results of a late M-dwarf survey by using Keck/NIRSPEC with RV precisions between 150-300 m s$^{-1}$.
In 2009, Pravdo & Shaklan (2009) announced a massive planet around the M-dwarf vB10 discovered by means of astrometrical data. Zapatero Osorio et al. (2009; hereafter ZO09) made use of our NIRSPEC data set and found evidence for RV variations, which supported the planet hypothesis. They achieved a RV precision of about 300 m s$^{-1}$. However, this planet was later refuted by different groups: Bean et al. (2010b), who took high-resolution spectra ($R=\lambda/\Delta\lambda\sim100,000$) with CRIRES and who achieved a RV precision of $\sim 10$ m s$^{-1}$, and by Anglada-Escudé et al. (2010). Additionally, Lazorenko et al. (2011) carried out an astrometric survey using the FORS2 camera of the ESO/VLT on Cerro Paranal, Chile, but found no evidence for the existence of a massive planet orbiting vB10. As part of this work, we aimed at finding out what had caused the spurious RV variations in the data analysis of ZO09.
Here, we report relative RV measurements of 8 late M-dwarfs with NIRSPEC, and we support the capability of this instrument to detect giant planetary companions with short orbital periods. In Section 2 we describe our M-dwarf sample, our observations and data reduction. In Section 3 we outline the details of the data analysis, followed by the results and discussion (Section 4).
Observations and Data Reduction
===============================
As part of our M-dwarf survey (Deshpande et al., in prep.), we observed 8 M-dwarfs (2M2331, GJ1156, GJ406, GJ905, LHS1363, RXJ2208.2, and vB10) at two or more epochs (Table \[xoxo:T1\]) using the NIRSPEC instrument, mounted on the Keck II telescope on the summit of Mauna Kea in Hawaii (McLean et al. 1998). We aimed at conducting RV precision tests, and searching for RV drifts which could be interpreted as massive planets orbiting those M-dwarfs. Our sample comprised dwarfs with spectral types of M5.0-M8.0 and masses between 0.14 and 0.075 M$_\odot$. Table \[xoxo:T4\] provides a list of the spectral types, $J$-band magnitude and the projected stellar rotational broadening $v \sin i$ of the targets.
NIRSPEC is a cross-dispersed, cryogenic echelle spectrometer employing a $1024\times
1024$ ALADDIN InSb array detector. In the echelle mode, we selected the NIRSPEC-3 ($J$-band) filter and an entrance slit width of $0.432\arcsec$ (i.e. 3 pixels along the dispersion direction of the detector), except for the 2001 June observations of vB10, where we used an entrance slit width of $0.576\arcsec$. The corresponding spectral resolving powers were $R= \lambda / \Delta \lambda \approx 22,700$ and $R \approx 17,800$, respectively for the $0.432\arcsec$ slit and the $0.576\arcsec$ slit. The length of both slits was $12\arcsec$. All observations were carried out at an echelle angle of $\sim 63^\circ$. This instrumental setup provided a wavelength coverage from 1.148 to $1.346~\mu$m split into 10 different echelle orders, a nominal dispersion ranging from 0.164 (blue) to 0.191$~{\rm \AA~pix^{-1}}$ (red wavelengths). Weather conditions (seeing and atmospheric transparency) were fine during the observations, except for the 2008 epoch, which was hampered by cirrus and strong wind. Table \[xoxo:T1\] lists the individual exposure times and the signal-to-noise ratios (SNRs) on average per spectral pixel in the stellar continua for each observing epoch.
For each target, the spectra were collected at two different positions along the entrance slit. This nodding allowed later the removal of the OH sky emission lines. For the identification of atmospheric telluric absorption, near-infrared featureless stars of spectral types A0-A2 were observed close in time (on average 3 min before or after the target observations) and position to our targets.
Raw data were reduced using the echelle package within [IRAF]{}[^2]. Nodded images were subtracted to remove sky background and dark current. White light spectra obtained with the same instrumental configuration and for each target observation were used to flat-field the data. By adopting the [apall]{} task, we first identified and optimally centered the echelle orders in the two individual nodding frames for each target and traced these orders by adopting a second-order Legendre polynomial along the dispersion axis. In the next step, we extracted the one-dimensional spectra for each echelle order adopting the same aperture/trace parameters for both, the target as well as an arc-lamp exposure of Ar, Kr, and Xe, which was always acquired after observing the target and before pointing the telescope at the next star. The air wavelengths of the arc lines were identified using the NIST[^3] database, and we produced preliminary wavelength calibration fits using a third-order Legendre polynomial along the dispersion axis and a second-order one perpendicular to it. The mean rms of the fits was $0.03~{\rm \AA}$, or 0.7 km s$^{-1}$.
[lcccc]{} Obs. date & UT & Slit & Exp. (s) & SNR$^{a}$\
\
2007-Jun-24 & 14:33 & $0.432 \times 12$ & $2\times200$ & $\sim50$\
2007-Jun-25 & 14:12 & $0.432 \times 12$& $2\times200$ & $\sim60$\
\
2007-Apr-30 & 7:03 & $0.432 \times 12$ & $2\times120$ & $\sim50$\
2007-Dec-23 & 15:32 & $0.432 \times 12$ & $2\times~30$ & $\sim50$\
\
2007-Jun-25 & 14:42 & $0.432 \times 12$ & $2\times~20$ & $\sim110$\
2007-Oct-27 & 10:51 & $0.432 \times 12$ & $2\times120$ & $\sim280$\
\
2007-Jun-24 & 7:24 & $0.432 \times 12$ & $2\times120$ & $\sim140$\
2007-Jun-25 & 15:55& $0.432 \times 12$ & $4\times300$ & $\sim240$\
\
2007-Oct-26 & 12:09 & $0.432 \times 12$ & $2\times300$ & $\sim110$\
2007-Oct-27 & 12:13 & $0.432 \times 12$ & $2\times300$ & $\sim110$\
\
2007-Oct-26 & 12:44 & $0.432 \times 12$ & $2\times300$ & $\sim60$\
2007-Oct-27 & 12:56 & $0.432 \times 12$ & $2\times300$ & $\sim50$\
\
2007-Jun-24 & 13:49 & $0.432 \times 12$ & $2\times120$ & $\sim70$\
2007-Jun-25 & 13:40 & $0.432 \times 12$ & $2\times120$ & $\sim70$\
\
2001-Jun-15 & 14:06 & $0.576 \times 12$ & $2\times100$ & $\sim60$\
2001-Nov- 2 & 4:43 & $0.432 \times 12$ & $2\times120$ & $\sim60$\
2001-Nov- 2 & 5:39 & $0.432 \times 12$ & $2\times120$ & $\sim60$\
2007-Jun-25 & 13:22 & $0.432 \times 12$ & $2\times120$ & $\sim70$\
2008-Jul-28 & 6:07 & $0.432 \times 12$ & $2\times120$ & $\sim20$\
$^{a}$ SNR on average in the pseudo stellar continuum per spectral pixel around 1.265 $\mu$m.\
[lcccccl]{} Name & Sp. type & $J$ & $v \sin i$ & Ref.\
& & & (km s$^{-1}$)\
2MJ2331-2749 & M7.0 & 11.65 & $<12$ & Des11\
GJ406 & M5.5 & 7.09 & $\sim3$ &Rei10\
GJ905 & M5.0 & 6.88 & $<3$ & Rei10\
GJ1156 & M5.0 & 8.52 & $17.2\pm2.9$ & Des11\
LHS1363 & M6.5 & 10.48 & $<12$ & Des11\
LP412-31 & M6.5 & 10.48 & $17.6\pm3.2$ & Des11\
RXJ2208.2 & M5.0 & 10.60 & $18.6\pm2.3$ & Des11\
vB10 & M8.0 & 9.91 & 6.5 & Moh03\
\
\
\
\
Relative Radial Velocity Method
===============================
We measured the RV of the stars relatively to the telluric lines present in the spectra as well as to a selected epoch of the star, employing a self-calibration approach. The radial velocity of the telluric lines is constant in all wavelengths down to a level of 10 m s$^{-1}$ (e.g., Figueira et al. 2010, Seifahrt & Käufl 2008), which is about a magnitude smaller than the velocity precision we can achieve with NIRSPEC. The basics of the self-calibration method have been extensively described (e.g. Valenti et al. 1995, Endl et al. 2000, Bean et al. 2010a), so that we just give a concise description of the method here and point out the important aspects of its implementation.
Briefly, the main idea is to model the observations and thereby determine the relative RV shift, and perform a fine tuning of the wavelength solution at the same time. Basically, the model spectrum is the product of a high-resolution telluric spectrum with a Doppler-shifted version of a high-resolution reference spectrum of the star. This product of those two spectra is then subjected to a convolution with the instrumental profile (IP) of the spectrograph, and finally binned to the sampling of the observed data. By variation of the free parameters of the model (Table \[xoxo:T2\]), the best fit model is evaluated by $\chi^2$ statistics. The input Doppler-shift which yields the best fit represents the measured RV.
Since our method requires the presence of telluric lines in the spectra, we restricted the analysis to the echelle orders 66, 60, 58, and 57, which were heavily contaminated mainly by absorption lines of water vapor. These four orders correspond to the wavelength ranges of $\lambda \sim 1.147$ to 1.163 $\mu$m, 1.261 to 1.279 $\mu$m, 1.304 to 1.323 $\mu$m, and 1.327 to 1.346 $\mu$m, respectively. For the echelle order numbering we refer to McLean et al. (2007). Subsequently, we subdivided each spectral order into 5 equidistant pixel chunks of 200 pixel each (i.e. for all four orders together we have 20 chunks). This step was done to simplify the process of improving the model, to speed up the calculations and to account for variations of the IP throughout each spectral order. Each of the following steps was carried out on each chunk individually, and the SNR was determined by $$\label{xoxoequ:50}
{\rm SNR} = S_{\star} / \sqrt{S_{\star} + S_{\rm BG} + {S_{\rm BG2} + \rm RON}^2\times 2n} ,$$ where $S_{\star}$ denotes the signal level from a star in electrons, integrated over an aperture of $n$ pixels, $S_{\rm BG}$ is the signal level of the sky background, $S_{\rm BG2}$ is the signal level from the sky background of the frame taken at the other nodding position, and RON denotes the read-out-noise level per pixel in rms electrons (for NIRSPEC, RON=65e$^{-1}$). The noise errors were propagated in the following data analysis steps.
Step 1: Telluric template spectrum and determination of the instrumental profile
--------------------------------------------------------------------------------
For the calculation of the atmospheric transmission spectrum, we used the Line-By-Line Radiative Transfer Model (LBLRTM) code, which is based on the FASCODE algorithm (Clough et al. 1992). LBLRTM is available as fortran source code [^4] and runs on various platforms. As molecular database we adopted HITRAN (Rothman et al. 2005), which contains the 42 most prominent molecules and isotopes present in the atmosphere of the Earth. Following the approach presented by Seifahrt et al. (2010), we created a high-resolution theoretical telluric spectrum for each observed spectrum by accounting for the air mass of the star as well as the weather conditions (water vapour density column, temperature and pressure profiles) during the observations. We retrieved the weather information from the Global Data Assimilation System (GDAS). GDAS models are available in 3 hours intervals for any location around the globe[^5].
![Comparison between the observed telluric spectrum (points), and the theoretical model (line). The observed telluric spectrum was taken by using a featureless A-star (HD181414), which was observed with NIRSPEC in the $J$-band on 2007-06-25. The rms of the telluric model fit to the observed data is about 1%. \[xoxo:F1A\]](xoxo-fg123.ps)
To calculate a first version of the instrumental profile (IP) of the spectrograph, we made use of the A-star observations next to our targets. First of all, we normalized the spectrum in such a way that the flux in the telluric continuum was at one. Next, we refined the wavelength solution of the observed telluric spectrum with the appropriate high-resolution theoretical telluric spectrum by adopting a second order polynomial. We then determined a preliminary version of the IP as the sum of 7 Gaussian profiles in a similar way as described in Valenti et al. (1995): Around a central Gaussian we grouped 3 Gaussians on each side of it, which allow to account for asymmetries in the IP. Free parameters were the height and width of the central Gaussian, plus the heights of the six satellite Gaussians (c.f. Table \[xoxo:T2\]). To reduce the number of free parameters per chunk, and to ensure that the method works robust, the positions and the widths of these satellites were fixed and set [*a priori*]{} in such a way that their half-widths overlapped.
Next, we convolved the high-resolution theoretical spectrum with the determined preliminary IP and compared the resulting spectrum with the observed A-star spectrum. For a few telluric lines, we realized that the theoretical spectrum under- or overestimated the line-depths. To produce a better match between theory and observation, we iteratively carried out a fine tuning of the line-depths in the high-resolution theoretical telluric spectrum, then again convolved the modified telluric spectrum with the IP and evaluated the result with the observation by means of $\chi^2$-statistics. The iterations were carried out until the reduced $\chi^2$ reached 1. Fig. \[xoxo:F1A\] shows a comparison between an A-star spectrum (HD 181414) and the fit of the refined theoretical telluric spectrum as well as the residuals of the model fit to the A-star spectrum. The rms of the telluric model fit to the observed data is about 1% on average. The refined high-resolution telluric model spectrum from now on served the purpose of the telluric template spectrum. In the final step, we refined the wavelength solution of the observed spectrum, and then re-calculated the IP by adopting this new telluric template spectrum.
Step 2: Stellar template spectrum
---------------------------------
Due to lack of appropriate theoretical model spectra which fit the stellar absorption features in the $J$-band, we created the stellar template spectrum for one selected reference epoch by calculating an IP-free and telluric-free version of the target spectrum. Concerning the reference epoch, we selected that epoch in which the stellar spectrum showed the highest SNR. To produce the stellar template, we first applied the refined wavelength solution of the A-star spectrum (which was taken - on average - 3 minutes before or after the target observations) to the observed target spectrum of the same epoch. Since the telluric lines were present in the target spectrum, we needed to remove them from the spectrum. In preparation for this, we convolved the appropriate theoretical telluric spectrum with the IP. Then, we divided the target spectrum by the convolved theoretical telluric spectrum (Fig. \[xoxo:F1A\]). Similar to Bean et al. (2010a) and Blake et al. (2010), we found that this approach led to smaller uncertainties than when the usual method of the telluric lines removal was carried out, where the target spectrum is simply divided by the appropriate A-star spectrum.
To create the final stellar template spectrum, we deconvolved the telluric-free target observation by the IP by employing the maximum-entropy method (MEM) with 5 times oversampling of the output spectrum. In the final step, we applied the refined wavelength solution that we had obtained for the A-star spectrum to the 5-times over-sampled IP-free stellar spectrum, which from there served the purpose of the stellar template.
![Example model components and fit for the radial velocity measurements. The components are given in the two top panels: the spectrum of the high-resolution theoretical telluric spectrum (top), and the deconvolved and RV-shifted version of the telluric free stellar spectrum (bottom). We note that the scale of the flux is different in each panel for better visibility. In the lower panel, we show the observed spectrum (points) and the best-fit model (line).\[xoxo:F3A\]](xoxo-fg3.ps)
------------------------------------------------ -----------
Parameter degree of
freedom
Stellar absorption line depth 1
Linear stellar continuum trend 2
Doppler shift of stellar template 1
Telluric absorption line depth 1
Amplitude and width of main Gaussian 2
Amplitudes of satellite Gaussians 6
${\rm2^{nd}}$ order wavelength solution vector 3
------------------------------------------------ -----------
: Free parameters in the model per chunk.[]{data-label="xoxo:T2"}
Step 3: Fitting the observed data
---------------------------------
For each target, we first determined the barycentric velocity differences $\Delta v_{{\rm bc},t}$ for all observation epochs $t$ with respect to that one of the stellar template epoch. This correction was calculated by use of the JPL ephemeris DE200 (Standish 1990).
We constructed the model of the observation by multiplying the telluric template with the stellar template, where its Doppler shift is one of the free parameters. The resulting combination spectrum was subjected to a convolution with the IP that was determined in Step 1, and a new wavelength solution was calculated. Subsequently, all the free parameters (i.e. line-depths in the models, IP, ...; See Table \[xoxo:T2\]) were refined by employing Brent’s optimization algorithm, and the fit to the observed data was evaluated by using $\chi^2$ statistics. The search range for the Doppler shift was $\Delta v_{{\rm bc},t}\pm15$ km s$^{-1}$ with a step width of 10 m s$^{-1}$. We note that such a large interval would also allow us to detect large relative RV variations due to unseen massive companions such as low-mass stars and brown dwarfs. We calculated the $\chi^2$ values for each Doppler shift and then determined the exact $\chi^2$-minimum by using a Gaussian fit. That Doppler shift which led to the overall best fit model ($\chi^2$-minimum) constituted the measured RV of the star in the chunk, relatively to the stellar template.
To determine the global (i.e. all chunks together) RV measurement, we combined all the RV measurements in all chunks into one by considering the following restrictions: Each chunks was given a specific weight which was determined from the average SNR in the stellar continuum, plus the number of telluric lines and stellar absorption lines which were present in that chunk, plus the depths of the stellar lines. Furthermore, we rejected chunks in which the RV-measurement constituted clearly an outlier ($3 \sigma$ above / below average of all RV measurements) by adopting sigma-clipping. No chunks were rejected for the stars GJ905, GJ1156, LHS1363, RXJ2208.2, and vB10. For 2MJ2331-2749 and LP412-31, one chunk was rejected each, while for GJ406 two chunks were rejected. All those rejected chunks were located in noisy areas with SNR levels lower than 40 on average. We attribute these spurious RV shifts to improper stellar templates which contained artifacts coming from the deconvolution of low-SNR data.
The global RV measurement was then determined as the arithmetic weighted mean of the un-rejected chunks. The error of the global RV measurement was determined as the weighted standard deviation of the un-rejected RV measurements in the chunks.
Results and Discussion
======================
We analyzed the data sets with our relative radial velocity measurement approach and determined the relative RV measurements with respect to the selected reference epoch. For any of the eight M-dwarfs in our sample, we have not found significant evidence of relative RV variations at the level of 3$\sigma$ (Table \[xoxo:T3\]), where $\sigma$ stands for the measurements uncertainty. The RV precisions are in the order of 180-300 m s$^{-1}$, except for the observations in July 2008, which were taken at low SNR.
We investigated the period and mass range of companions which could be detected with such RV precisions. We determined the minimum mass of the planet by employing a Monte-Carlo analysis, thereby probing planetary orbits with different parameters and investigating how many of these orbits could be recovered for the five measurements of vB10. We considered only the case of a circular orbit and the mass of vB10, which is $m_\star=0.078~{\rm M}_\odot$. Fig. \[xoxo:F1\] shows the 3$\sigma$ detection limit. We find that for companions with only a few days period, even planets with minimum masses of $m_{\rm{p}}\sin i \ge 0.3~M_{\rm Jup}$ can be detected with a RV precision of $\sim220$ m s$^{-1}$.
![Monte-Carlo analysis for the five vB10 measurements. For the mass of vB10, we adopted 0.078 M$_\odot$ from Pravdo & Shaklan (2009). It is shown that with a RV precision of $\sim220$ m s$^{-1}$ even hot Jupiters with minimum masses $m_{\rm p} \sin i> 0.3~$M$_{\rm Jup}$ could be detected around late M-dwarfs with 3$\sigma$ confidence. We note that for a larger number of measurements the number of aliasing peaks can be significantly decreased.\[xoxo:F1\]](out.dat.ps)
In Fig. 4 we show our relative RVs of vB10 and the measurements by ZO09. We note that for a proper comparison, we adopted the same reference epoch as in ZO09. The agreement between ZO09 and our measurements is within 1$\sigma$ of the quoted uncertainties for all epochs except for the 2001 epoch (BJD = 2452076). We provide next an explanation for the discrepancy of this one measurement. Similar to our data analysis, ZO09 used the telluric lines present in the target spectra as a stable reference, but contrary to our analysis they did not account for any IP variations in their analysis, but calculated the RVs by cross correlation. In our analysis, we do not see any RV shift exceeding the RV-precision for any measurement. We get evidence that the different instrumental setting used on 2001-Jun-15 (0.576“ slit instead of the standard setting of 0.432”) produced an asymmetric instrumental profile (Fig. \[xoxo:F2\]), which led to a significant RV shift when a simple cross correlation is adopted for the RV determination. Our results clearly demonstrate the importance of modeling the IP especially when observations are carried out with different instrumental settings.
![Our RV measurements of vB10 (crosses) vs. the measurements of ZO09 (open circles). The RV precision given by our analysis is about $~220$ m s$^{-1}$, except for the last epoch in 2008, which was hampered by bad weather. We prove that the RV shift in the work of ZO09 in the first 2001 epoch (BJD$=245~2076$) originates from unaccounted asymmetries in the IP rather than from a planetary companion. \[xoxo:F2A\]](xoxo-fg2.ps)
![Instrumental profiles (IPs) of NIRSPEC for two different observing epochs of vB10. On 2001-06-15, a broader slit was used ($0.576"$; solid line) than for the reference epoch ($0.432"$; 3007-06-25; dashed line). To visualize the asymmetries between both IPs, we calculated the ratio between both IPs and scaled the resulting function for better visibility (dotted line). ZO09 did not account for these asymmetries between both IPs, which led to a spurious RV shift of about 1 km s$^{-1}$ for the 2001-06-15 measurement in their data analysis.\[xoxo:F2\]](new-ip-nogrid6.ps)
[lcccc]{} Obs. Date & BJD & rel. RV\
& 245 0000+ & (m s$^{-1}$)\
\
2007-Jun-24 & 4276.10874 & $194\pm224$\
2007-Jun-25 & 4277.09424 & [*ref. epoch*]{}\
\
2007-Apr-30 & 4220.79751 & [*reference epoch*]{}\
2007-Dec-23 & 4458.14962 & $-77\pm 238$\
\
2007-Jun-25 & 4277.11249 & $-233\pm 201$\
2007-Oct-27 & 4400.95657 & [*reference epoch*]{}\
\
2007-Apr-30 & 4220.81513 & $141\pm185$\
2007-Dec-22 & 4457.16844 & [*reference epoch*]{}\
\
2007-Oct-26 & 4400.01234 & $-27\pm196$\
2007-Oct-27 & 4401.01511 & [*reference epoch*]{}\
\
2007-Oct-26 & 4400.03654 & [*reference epoch*]{}\
2007-Oct-27 & 4401.04491 & $298\pm260$\
\
2007-Jun-24 & 4276.07879 & $-189\pm 307$\
2007-Jun-25 & 4277.07266 & [*reference epoch*]{}\
\
2001-Jun-15 & 2076.08951 & $19\pm230$\
2001-Nov-02 & 2215.70560 & $69\pm223$\
2001-Nov-02 & 2215.74225 & $-3\pm217$\
2007-Jun-25 & 4277.05865 & [*reference epoch*]{}\
2008-Jul-28 & 4675.75669 & $131\pm497$\
We compare our results to the work of Blake et al. (2010), who searched for companions to M- and L-dwarf wby using NIRSPEC at a spectral resolving power of $\sim25,000$ in the $K$-band. They adopted one spectral order covering the wavelength range from 2.285 to 2.318 $\mu$m to measure the dense and strong CO-absorption line pattern present in those dwarfs. As a stable wavelength reference, they made use of the CH$_4$ telluric absorption lines present in the observations. Similarly to us, they employed a self calibrating approach, with the difference that they adopted theoretical models for M- and L-dwarfs, which well-described the observations.
Blake et al. obtained measurements with SNRs in the range of 50 to 100 in the pseudo stellar continua, and they report RV precisions of 100-300 m/s for slowly rotating late-M and L dwarfs. The uncertainty of Blake et al. in the K-band is in agreement with our derivation of 180-300 m/s in view of our SNRs. However, our wavelength coverage is about twice that in Blake et al. According to the relative RV precision formulae, we should have obtained better velocity precision in terms of wavelength coverage, which is not the case. We conclude that both the larger number of deep lines (more than 30 lines with a line depth larger than 50%) in the CO-band region as compared to the J-band (only a few lines with a depth larger than 25%) as well as the use of theoretical template spectra instead of deconvolved stellar spectra appear to compensate for the shorter wavelength coverage in a similar factor (c.f. Equation 6 in Butler et al. 1996).
We note that Reiners et al. (2010) and Rodler et al. (2011) carried out theoretical RV precision studies of M- and L-dwarfs, by adopting theoretical models of M-dwarfs (e.g. del Burgo et al. 2009). As result, they find that the highest RV precision for M-dwarfs is attained in the $Y$ band around $1~\mu$m, rather than in the $J-$ , $H-$ or $K$-band. For L-dwarfs, however, Rodler et al. (2011) reported that the highest RV precision is attained in the $J$-band.
We conclude that for an accurate relative RV determination with NIRSPEC, a self-calibrating approach, which accounts for changes in the instrumental setting, produces the best measurements in terms of RV precision. Although with our RV precision we would be able to detect massive hot Neptunes around late M-dwarfs, we have not found any brown dwarf or massive planetary companion in our survey. Additionally, the re-analysis of the data of the M8-dwarf vB10 presented in ZO09 now clearly confirms the non-existence of a massive planet orbiting that dwarf and agrees with the results by other research groups (e.g. Anglada-Escudé et al. 2010; Bean et al. 2010b; Lazorenko et al. 2011).
We thank to those of the Hawaiian ancestry on whose sacred mountain we are privileged to be guests. We are grateful to H. Bouy, N. Dello-Russo, P.-B. Ngoc, R. Tata, and R. Vervack for helping to obtain the 2007 and 2008 NIRSPEC spectra. FR thanks to A. Seifahrt for his help with LBLRTM, and to M. Zechmeister and M. Endl for discussions on the self-calibrating approach. This work has been supported by the Spanish Ministerio de Eduación y Ciencia through grant AYA2007-67458. Partial support for this research was provided by RoPACS, a Marie Curie Initial Training Network funded by the European Commission’s Seventh Framework Programme. The Center for Exoplanets and Habitable Worlds is supported by the Pennsylvania State University, the Eberly College of Science and the Pennsylvania Space Grant Consortium. This work was partly funded by the Fundação para a Ciência e a Tecnologia (FCT)-Portugal through the project PEst-OE/EEI/UI0066/201. We would furthermore like to thank the anonymous referee for valueable comments to improve the article.
Anglada-Escudé, G., Shkolnik, E., Weinberger, A., Thompson, I., Osip, D., Debes, J.2010, ApJ, 711L, 24A Bean, J. L., Seifahrt, A., Hartman, H., et al., 2010a, ApJ, 713, 410 Bean, J. L., Seifahrt, A., Hartman, H., et al., 2010b, ApJ, 711, 19 Blake, C. H., Charbonneau, D., White, R. J., 2010, ApJ, 723, 684 Butler R. P., Marcy G. W., Williams E., et al., 1996, PASP, 108, 500 Charbonneau, D., Berta, Z. K., Irwin, J. et al., 2009, Nature, 462, 891 Clough S. A., Iacono M. J., & Moncet J.-L-, 1992, J. Geophys. Res., 97, 15761 del Burgo C., Martín E. L., Zapatero Osorio M. R., Hauschildt P., 2009, A&A, 501, 1059 Endl M., Kürster M., Els S., 2000, A&A, 362, 585 Endl, M., Cochran, W. D., Kürster, M., et al., 2006, ApJ, 649, 436 Figueira, P., Pepe, F., Melo, C. H. F., et al., 2010, A&A, 511, 55 Henry, T., Ianna, P., Kirkpatrick, D., et al., 1997, ApJ, 114, 388 Mart[í]{}n, E. L., Guenther, E., Zapatero Osorio, M. R., et al., 2006, ApJ, 644, 75 Mayor M., Bonfils, X., Forveille, T., et al., 2009, A&A, 507, 487 McLean, I. S., Becklin, E. E., Bendiksen O., et al., 2998, Proc. SPIE, 3354, 566 McLean, I. S., Prato, L., McGovern, M. R., et al. 2007, ApJ, 658, 1217 Mohanty, S., & Basri, G., 2003, ApJ, 583, 451 Käufl, H.-U., Ballester, P., Biereichel, P., el al., 2004, SPIE, 5492, 1218
Lazorenko, P. F., Sahlmann, J., Ségransan, D., et al., 2011, A&A, 527, 25 Pravdo, S. H.,& Shaklan, S. B., 2009, ApJ, 700, 623 Reiners, A., Basri, G., 2010, ApJ, 710, 924 Reiners, A., Bean, J. L., Huber, K. F, et al., 2010, ApJ, 710, 432 Rodler, F., del Burgo, C., Martín E. L., et al., 2011, A&A, in press. (arXiv:1105.2287) Rothman L. S., Jacquemart D., et al., 2005, JQSRT, 96, 139 Seifahrt, A., & Käufl, H. U., 2008, A&A, 491, 929 Seifahrt, A., Käufl, H. U., Zängl, G., et al., 2010, A&A, 524, 11 Standish, E. M., Jr., 1990, A&A, 233, 252 Tanner A., White R., Bailey J., et al., 2010, arXiv:1012.4882 Valenti, J. A., Butler, R. P., Marcy, G. W., 1995, PASP, 107, 966 Zapatero Osorio, M. R., Martín, E. L., del Burgo, C., et al., 2009, A&A, 505, 5 Zechmeister, M., Kürster, M., Endl, M., 2009, A&A, 505, 859
[^1]: The Extrasolar Planets Encyclopedia; http://www.exoplanet.eu; 2011-Nov-15
[^2]: [IRAF]{} is distributed by the National Optical Astronomy Observatory (NOAO), which is operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation in the USA.
[^3]: http://physics.nist.gov/PhysRefData/ASD/lines\_form.html
[^4]: Source code and manuals are available under [http://rtweb.aer.com/lblrtm\_description.html]{}
[^5]: GDAS webpage: [http://ready.arl.noaa.gov/READYamet.php]{}
|
---
abstract: 'Using general identities for difference operators, as well as a technique of symbolic computation and tools from probability theory, we derive very general $k$th order ($k\geq 2$) convolution identities for Bernoulli and Euler polynomials. This is achieved by use of an elementary result on uniformly distributed random variables. These identities depend on $k$ positive real parameters, and as special cases we obtain numerous known and new identities for these polynomials. In particular we show that the well-known identities of Miki and Matiyasevich for Bernoulli numbers are special cases of the same general formula.'
address:
- |
Department of Mathematics and Statistics\
Dalhousie University\
Halifax, Nova Scotia, B3H 4R2, Canada
- |
Department of Mathematics\
Tulane University\
New Orleans, LA 70118
author:
- Karl Dilcher
- Christophe Vignat
title: General convolution identities for Bernoulli and Euler polynomials
---
[^1]
Introduction
============
The Bernoulli and Euler numbers and polynomials have been studied extensively over the last two centuries, both for their numerous important applications in number theory, combinatorics, numerical analysis and other areas of pure and applied mathematics, and for their rich structures as interesting objects in their own right. The Bernoulli numbers $B_n$, $n=0,1,2,\ldots$, can be defined by the exponential generating function $$\label{1.1}
\frac{z}{e^z-1} = \sum_{n=0}^\infty B_n\frac{z^n}{n!}\qquad (|z|< 2\pi).$$ They are rational numbers, the first few being 1, $-\frac{1}{2}$, $\frac{1}{6}$, 0, $-\frac{1}{30}$, 0, $\frac{1}{42},\ldots$, with $B_{2k+1}=0$ for $k\geq 1$. For the most important properties see, for instance, [@AS Ch. 23] or its successor [@DLMF Ch. 24]. Other good references are [@GKP], [@Jo], or [@No]. For a general bibliography, see [@DSS].
Numerous linear and nonlinear recurrence relations for these numbers are known, and such relations also exist for the Bernoulli [*polynomials*]{} and for Euler numbers and polynomials which will be defined later. This paper deals with [*nonlinear*]{} recurrence relations, the prototype of which is Euler’s well-known identity $$\label{1.2}
\sum_{j=0}^n \binom{n}{j}B_jB_{n-j} = -nB_{n-1} - (n-1)B_n\qquad (n\geq 1).$$ This can also be seen as a convolution identity. Two different types of convolution identities were discovered more recently, namely $$\label{1.3}
\sum_{j=2}^{n-2}\frac{B_jB_{n-j}}{j(n-j)}
-\sum_{j=2}^{n-2}\binom{n}{j}\frac{B_jB_{n-j}}{j(n-j)}
= 2H_n\frac{B_n}{n} \qquad (n\geq 4)$$ by Miki [@Mi], where $H_n=1+\frac{1}{2}+\dots+\frac{1}{n}$ is the $n$th harmonic number, and $$\label{1.4}
(n+2)\sum_{j=2}^{n-2}B_jB_{n-j}
-2\sum_{j=2}^{n-2}\binom{n+2}{j}B_jB_{n-j} = n(n+1)B_n \qquad (n\geq 4)$$ by Matiyasevich [@Ma]; see also [@Ag] and the references therein. These two identities, which are remarkable in that they combine two different types of convolutions, were later extended to Bernoulli polynomials by Gessel [@Ge2] and by Pan and Sun [@PS], respectively. Gessel [@Ge2] also extended to third-order convolutions, i.e., sums of products of three Bernoulli numbers. Later Agoh [@Ag] found different and simpler proofs of the polynomial analogues of and and proved numerous other similar identities involving Bernoulli, Euler, and Genocchi numbers and polynomials. Subsequently Agoh and the first author [@AD] extended the polynomial analogue of to convolution identities of arbitrary order, and did the same for Euler polynomials. Meanwhile, following different lines of investigation, Dunne and Schubert [@DS] derived an identity that has both and as special cases, and Chu [@Ch] obtained a large number of convolution identities, some of them extending and .
It is the purpose of this paper to contribute to the recent work summarized above and to further extend the identities and of Miki and Matiyasevich. In Section 2 we state a general result concerning second-order convolutions, and derive some consequences. In Section 3 we introduce a symbolic notation with a related calculus, and use it to state and prove a very general identity for Bernoulli polynomials. This is then used in Section 4, along with some methods from probability theory, to prove a general higher-order convolution identity which gives the main result of Section 2 as a special case. In Section 5 we apply most of the methods from Sections 3 and 4 to Euler polynomials and again obtain general higher-order convolution identities. Finally, in Section 6, we state and prove several further consequences of each of our main theorems. We conclude this paper with some further remarks in Section 7.
Identities for Bernoulli polynomials
====================================
The [*Bernoulli polynomials*]{} can be defined by $$\label{2.1}
B_n(x):=\sum_{j=0}^n\binom{n}{j}B_jx^{n-j},$$ or equivalently by the generating function $$\label{2.2}
\frac{ze^{xz}}{e^z-1}=\sum_{n=0}^\infty B_n(x)\frac{z^n}{n!}\qquad (|z|< 2\pi).$$ For the first few Bernoulli polynomials, see Table 1 in Section 5. They have the special values $$\label{2.3}
B_n(0) = B_n,\qquad B_n(\tfrac{1}{2})=(2^{1-n}-1)B_n,\qquad B_n(1)=(-1)^nB_n,$$ ($n=0, 1, 2,\ldots$), where the first identity is immediate from comparing with , and the other two follow from easy manipulations of the generating function . We also require the Pochhammer symbol (or rising factorial) $(z)_k$, defined for $z\in\mathbb C$ and integers $k\geq 0$ by $$\label{2.4}
(z)_k = \frac{\Gamma(z+k)}{\Gamma(z)} = z(z+1)\dots (z+k-1),$$ where the right-hand product is valid for $k\geq 1$.
We are now ready to state our first main result, which will be proved later.
\[thm:Thm1\] For integers $n\geq 1$ and real numbers $a, b> 0$ we have $$\begin{aligned}
\sum_{l=0}^{n}\binom{n}{l}\frac{(a)_{l}(b)_{n-l}}{(a+b)_{n}}B_{l}(x)B_{n-l}(x)
&= \sum_{l=0}^{n}\binom{n}{l}\frac{a(b)_{l}+b(a)_{l}}{(a+b)_{l+1}}B_lB_{n-l}(x)
\label{2.5}\\
&\qquad+ \frac{ab}{(a+b+1)(a+b)}nB_{n-1}(x).\nonumber \end{aligned}$$
The remainder of this section will be devoted to deriving a number of consequences of this general identity. First, it is clear by that we get an analogous identity for Bernoulli [*numbers*]{} by simply deleting the variable $x$. The most immediate special case is obtained by setting $a=b=1$. With $(1)_n=n!$ and $(2)_n=(n+1)!$, some straightforward manipulations involving the binomial coefficients in lead to the following identity, which was earlier obtained in [@AD].
For all $n\geq 1$ we have $$\label{2.6}
(n+2)\sum_{l=0}^n B_l(x)B_{n-l}(x)
=2\sum_{l=0}^n\binom{n+2}{l+2}B_lB_{n-l}(x)+\binom{n+2}{3}B_{n-1}(x).$$
When $x=0$, this identity becomes trivial for odd $n$ since one of $B_l, B_{n-l}$ will be zero except in the cases $l=1$ and $l=n-1$. For even $n\geq 4$, however, we have the following identity.
For all even $n\geq 2$ we have $$\label{2.7}
(n+2)\sum_{l=0}^n B_lB_{n-l}=2\sum_{l=0}^n\binom{n+2}{l+2}B_lB_{n-l}.$$
This identity, although different in appearance, is equivalent to . For our next corollary we need the [*shifted harmonic numbers*]{} which for real $a>0$ and integers $n\geq 1$ are defined by $$\label{2.8}
H_{a,n} := \sum_{j=0}^{n-1}\frac{1}{j+a}.$$ Obviously, $H_{1,n}=H_n$. As we shall see, the following result can be considered as an infinite class of generalizations of Miki’s identity .
For real $a>0$ and integers $n\geq 1$ we have $$\begin{aligned}
\sum_{l=0}^{n-1}\binom{n}{l}\frac{(a)_{l}(n-l-1)}{(a)_{n}}B_{l}(x)B_{n-l}(x)
&= \sum_{l=1}^{n}\binom{n}{l}\frac{a(l-1)!+(a)_{l}}{(a)_{l+1}}B_lB_{n-l}(x)
\label{2.9}\\
&\qquad+ \frac{n}{a+1}B_{n-1}(x)+H_{a,n}B_n(x).\nonumber\end{aligned}$$
The idea of proof is to divide both sides of by $b$ and then take the limit as $b\rightarrow 0$. On the left-hand side we have for $0\leq l\leq n-1$, $$\frac{1}{b}\cdot\frac{(a)_l(b)_{n-l}}{(a+b)_n}
=\frac{(a)_l(b+1)\dots(b+n-l-1)}{(a+b)_n}
\rightarrow\frac{(a)_l(n-l-1)!}{(a)_n}$$ as $b\rightarrow 0$, and on the right-hand side, for $1\leq l\leq n$, $$\frac{1}{b}\cdot\frac{a(b)_l+b(a)_l}{(a+b)_{l+1}}
=\frac{a(b+1)\dots(b+l-1)+(a)_l}{(a+b)_{l+1}}
\rightarrow\frac{a(l-1)!+(a)_l}{(a)_{l+1}}$$ as $b\rightarrow 0$. To take care of the terms that were left out in the limits above, we note that the term for $l=0$ in the right-hand sum and the term for $l=n$ in the left-hand sum of combine to give $$\binom{n}{0}\frac{a+b}{a+b}B_n(x)-\frac{(a)_n}{(a+b)_n}B_n(x)
=\frac{(a+b)_n-(a)_n}{(a+b)_n}B_n(x),$$ and we have $$\lim_{b\rightarrow 0}\frac{(a+b)_n-(a)_n}{b(a+b)_n}
=\frac{1}{(a)_n}\left.\frac{d}{dx}(x)_n\right|_{x=a} = H_{a,n},$$ where the second equation follows directly from applying the product rule repeatedly to the right-hand side of . Putting everything together, we get .
As illustrations of Corollary 3 we state the cases $a=1$ and $a=2$ separately.
For integers $n\geq 1$ we have $$\begin{aligned}
\frac{n}{2}\sum_{l=1}^{n-1}\frac{B_l(x)}{l}\frac{B_{n-l}(x)}{n-l}
&= \sum_{l=1}^{n}\binom{n}{l}\frac{B_l}{l}B_{n-l}(x)
+\frac{n}{2}B_{n-1}(x)+H_{n-1}B_n(x),\label{2.10} \\
(n+2)\sum_{l=0}^{n-1}(l+1)B_{l}(x)&\frac{B_{n-l}(x)}{n-l}
=\sum_{l=1}^{n}\binom{n+2}{l+2}(l^2+l+1)\frac{B_l}{l}B_{n-l}(x)\label{2.11}\\
&\qquad+(n+1)(n+2)\left(\frac{n}{3}B_{n-1}(x)+H_{2,n}B_n(x)\right).\nonumber\end{aligned}$$
After some easy manipulations, with $a=1$ gives $$\label{2.12}
\sum_{l=0}^{n-1}B_l(x)\frac{B_{n-l}(x)}{n-l}
=\sum_{l=1}^{n}\binom{n}{l}\frac{B_l}{l}B_{n-l}(x)
+\frac{n}{2}B_{n-1}(x)+H_nB_n(x),$$ and with $a=2$, gives . We now exploit the symmetry on the left-hand side of and rewrite the sum as $$\frac{B_n(x)}{n}+\frac{1}{2}\sum_{l=1}^{n-1}
\left(\frac{1}{n-l}+\frac{1}{l}\right)B_l(x)B_{n-l}(x) = \frac{B_n(x)}{n}
+\frac{n}{2}\sum_{l=1}^{n-1}\frac{B_l(x)}{l}\frac{B_{n-l}(x)}{n-l}.$$ Finally we subtract $\frac{1}{n}B_n(x)$ from both sides of and note that $H_nB_n(x)$ then becomes $H_{n-1}B_n(x)$.
Using a technique that involves generating functions for Stirling numbers and Nörlund polynomials, Gessel [@Ge2] obtained as a polynomial analogue of Miki’s identity . When $x=0$, then we also have symmetry in the sum on the right-hand side of ; we can therefore use again the identity $1/(n-l)+1/l=n/l(n-l)$, upon which we easily recover Miki’s identity.
Some further consequences of Theorem 1 and Corollary 3 will be derived in the final section of this paper.
Symbolic notation and general identities
========================================
[**1.**]{} The use of symbolic notation in dealing with Bernoulli numbers and polynomials goes back to J. Blissard in the 1860s. Subsequently it was used by many other authors, among them É. Lucas in the 1870s and 1880s. Later it was put on a firm theoretical foundation as part of “the classical umbral calculus"; see, e.g., [@Ge1] or [@RT].
Here we propose and use a system of symbolic notation that is in some respects similar to the classical umbral calculus, but is different in that it is related to probability theory. Also, this system of notation is more specific to Bernoulli numbers and polynomials and (later in this paper) Euler numbers and polynomials.
The basis for our symbolic notation for Bernoulli numbers and polynomials are two symbols, $\mathcal{B}$ and $\mathcal{U}$, which are complementary to each other or, as we shall see, annihilate each other. First, we define the [*Bernoulli symbol*]{} $\mathcal{B}$ by $$\label{3.1}
\mathcal{B}^{n}=B_{n}\qquad (n=0, 1,\ldots)$$ so that, for instance, can be rewritten as $$\label{3.1a}
B_n(x) = (x+\mathcal{B})^n.$$ Furthermore, with we have $$\label{3.2}
\exp\left(\mathcal{B}z\right)=\sum_{n=0}^\infty\mathcal{B}^n\frac{z^n}{n!}
=\frac{z}{e^z-1}.$$ We also require several independent Bernoulli symbols $\mathcal{B}_{1},\dots,\mathcal{B}_{k}$. Independence means that if we have any two Bernoulli symbols, say $\mathcal{B}_1$ and $\mathcal{B}_2$, then $$\label{3.2a}
\mathcal{B}_1^{k}\mathcal{B}_2^{\ell}=B_kB_\ell.$$ Second, the [*uniform symbol*]{} $\mathcal{U}$ is defined by $$\label{3.3}
f(x+\mathcal{U})=\int_0^1f(x+u)du.$$ Here and elsewhere we assume that $f$ is an analytic function for which the objects in question exist. From we immediately obtain, in analogy to , $$\label{3.4}
\mathcal{U}^{n}=\frac{1}{n+1}\qquad(n=0, 1,\ldots),$$ and using this, we get $$\label{3.5}
\exp\left(\mathcal{U}z\right)=\sum_{n=0}^\infty\mathcal{U}^n\frac{z^n}{n!}
=\frac{e^z-1}{z}.$$ From and we now deduce $$\exp\left(z\left(\mathcal{B}+\mathcal{U}\right)\right)
=\sum_{n=0}^\infty\left(\mathcal{B}+\mathcal{U}\right)^n\frac{z^n}{n!} = 1,$$ which means that $\mathcal{B}$ and $\mathcal{U}$ annihilate each other, i.e., $(\mathcal{B}+\mathcal{U})^n=0$ for all $n\neq 0$, in the sense that $$\label{3.6}
f(x+\mathcal{B}+\mathcal{U})=f(x),$$ or in other words, we have the equivalence $$\label{3.7}
f(x)=g(x+\mathcal{U})\quad\Leftrightarrow\quad g(x)=f(x+\mathcal{B}).$$ Finally, we note that immediately gives, for any $u\in\mathbb R$, $$\label{3.8}
uf'(x+u\mathcal{U}) = f(x+u)-f(x),$$ a difference equation that will be used repeatedly.
It is well known that the Bernoulli polynomials are closely related to the calculus of finite differences; see, e.g., the classic books [@Jo] or [@No]. It is therefore not surprising that methods from difference calculus turn out to be useful in the proofs of our main results. Let $\Delta_u$ be the [*forward difference operator*]{} defined by $$\label{3.9}
\Delta_uf(x)=f(x+u)-f(x).$$ With two (in general) distinct differences $u_1, u_2$ we compute $$\begin{aligned}
\Delta_{u_1}\Delta_{u_2}f(x)&=\left(f(x+u_2+u_1)-f(x+u_1)\right)
-\left(f(x+u_2)-f(x)\right) \\
&=\left(f(x+u_1+u_2)-f(x)\right)-\left(f(x+u_1)-f(x)\right)
-\left(f(x+u_2)-f(x)\right)\\
&=\Delta_{u_1+u_2}f(x)-\Delta_{u_1}f(x)-\Delta_{u_2}f(x),\end{aligned}$$ which gives the operator identity $$\label{3.10}
\Delta_{u_1+u_2}=\Delta_{u_1}\Delta_{u_2}+\Delta_{u_1}+\Delta_{u_2}.$$ Similarly, one obtains $$\begin{aligned}
\Delta_{u_1+u_2+u_3} &= \Delta_{u_1}\Delta_{u_2}\Delta_{u_3}
+\Delta_{u_1}\Delta_{u_2}+\Delta_{u_1}\Delta_{u_3}+\Delta_{u_2}\Delta_{u_3}\label{3.11} \\
&\quad +\Delta_{u_1}+\Delta_{u_2}+\Delta_{u_3}.\nonumber\end{aligned}$$ To generalize these identities, we use the following notation: For a fixed integer $k\geq 1$ and for any subset $J\subseteq\{1,\ldots,k\}$, we denote $$\label{3.12}
\Delta_J:=\prod_{j\in J}\Delta_{u_j},$$ and we let $|J|$ be the cardinality of $J$. We can now state and prove the following simple but important lemma.
For any $k\geq 1$ and for real numbers $u_1,\ldots,u_k$ we have $$\label{3.13}
\Delta_{u_1+\dots+u_k}=\sum_{j=1}^k\sum_{|J|=j}\Delta_J.$$
The case $k=1$ is trivial, and we immediately see that $k=2$ and $k=3$ give the identities and , respectively.
This result can be proved by induction on $k$ in a straightforward way. Alternatively, and more formally, we can use the shift operator $$f(x+u) = e^{u\partial}f(x),$$ with the differential operator $\partial=\frac{d}{dx}$. Then we have $\Delta_u=e^{u\partial}-1$, and $$\Delta_{u_1+\dots+u_k}=e^{(u_{1}+\dots+u_{k})\partial}-1
=\sum_{j=1}^k\sum_{|J|=j}\prod_{\ell\in J}\left(e^{u_{\ell}\partial}-1\right),$$ and the result follows.
[**3.**]{} We now apply results from the first two parts of this section to obtain a general identity for Bernoulli symbols, and thus for Bernoulli numbers and polynomials. In what follows, we assume that for a fixed integer $k\geq 1$, $u_1,\ldots,u_k$ are real parameters. To simplify notations, we write, for a subset $J\subseteq\{1,\ldots,k\}$, $$\label{3.14}
u_{J} := \prod_{j\in J}u_{j},\qquad
\left(u\mathcal{B}\right)_{J} := \sum_{j\in J}u_{j}\mathcal{B}_{j},\qquad
\overline{J}=\{1,\dots,k\}\setminus J.$$ The following is, in fact, a restatement of an intermediate result in [@AD].
Let $u_1+\dots+u_k=1$. Then we have $$\label{3.15}
\frac{1}{n!}\left(x+u_1\mathcal{B}_1+\dots+u_k\mathcal{B}_k\right)^n
=\sum_{j=1}^k\sum_{|J|=j}\frac{u_J}{(n+1-j)!}
\left(x+\mathcal{B}_0+(u\mathcal{B})_{\overline{J}}\right)^{n-j+1},$$ where $\mathcal{B}_0,\dots,\mathcal{B}_k$ are independent Bernoulli symbols.
We apply the operator identity to the function $$f(x):=\frac{1}{(n+1)!}
\left(x+\mathcal{B}_0+u_1\mathcal{B}_1+\dots+u_k\mathcal{B}_k\right)^{n+1}.$$ Then the left-hand side of gives, with , $$\begin{aligned}
\Delta_{u_1+\dots+u_k}f(x) &= \Delta_1f(x)=f(x+1)-f(x)\label{3.16}\\
&= f'(x+\mathcal{U})
=\frac{1}{n!}\left(x+u_1\mathcal{B}_1+\dots+u_k\mathcal{B}_k\right)^n,\nonumber\end{aligned}$$ where in the last step we used , i.e., $\mathcal{B}_0$ is annihilated by $\mathcal{U}$. Similarly, we have for any $i=1,\ldots, k$, again using , $$\begin{aligned}
\Delta_{u_i}f(x) &= f(x+u_i)-f(x) = u_if'(x+u_i\mathcal{U}) \label{3.17}\\
&= u_i\frac{1}{n!}\left(x+\mathcal{B}_0
+(u\mathcal{B})_{\{1,\ldots,k\}\setminus\{i\}}\right)^n,\nonumber\end{aligned}$$ having used the fact that the uniform symbol $\mathcal{U}$ annihilated the Bernoulli symbol $\mathcal{B}_i$; note that the coefficients $u_i$ have to match for the annihilation (i.e., identity ) to apply. Using the definition and successively applying , we get $$\Delta_Jf(x)=\frac{u_J}{(n-j+1)!}
\left(x+\mathcal{B}_{0}+(u\mathcal{B})_{\overline{J}}\right)^{n+1-j}.$$ Finally, applying to this and to , we immediately get .
While the case $k=1$ is trivial, for $k=2$ and $k=3$ we get the following identities.
[**Examples.**]{} For $u_1+u_2=1$, we have $$\begin{aligned}
\left(x+u_1\mathcal{B}_1+u_2\mathcal{B}_2\right)^n
&= u_1\left(x+\mathcal{B}_0+u_2\mathcal{B}_2\right)^n
+u_2\left(x+\mathcal{B}_0+u_1\mathcal{B}_1\right)^n\\
&\quad +u_1u_2n\left(x+\mathcal{B}_0\right)^{n-1},\end{aligned}$$ and for $u_1+u_2+u_2=1$, $$\begin{aligned}
&\left(x+u_1\mathcal{B}_1+u_2\mathcal{B}_2+u_3\mathcal{B}_3\right)^n
= \left[u_1\left(x+\mathcal{B}_0+u_2\mathcal{B}_2+u_3\mathcal{B}_3\right)^n+o.t.\right]\\
&\qquad+\left[nu_1u_2\left(x+\mathcal{B}_0+u_3\mathcal{B}_3\right)^{n-1}+o.t.\right]
+n\left(n-1\right)u_1u_2u_3\left(x+\mathcal{B}_0\right)^{n-2},\end{aligned}$$ where “o.t." in each of the first and second rows stands for the “other terms" obtained by cyclically permuting the subscripts $\{1,2,3\}$.
[**Remarks.**]{} (1) The left-hand side of , and in fact also the terms on the right-hand side, could be written as Bernoulli polynomials of higher order, as defined in identity (30) in [@EMOT p. 39]. We will not pursue this further.
\(2) It is clear from the proof of Lemma 2 that, more generally, for any analytic function $f$ and $u_1+\dots+u_k=1$ we have $$f(x+u_1\mathcal{B}_1+\dots+u_k\mathcal{B}_k)
=\sum_{j=1}^n\sum_{|J|=j}u_J
f^{(j-1)}\left(x+\mathcal{B}_0+(u\mathcal{B})_{\overline{J}}\right),$$ and in particular, for $k=2$ and $u_1+u_2=1$, $$\begin{aligned}
f(x+u_1\mathcal{B}_1+u_2\mathcal{B}_2)
&=u_2f(x+\mathcal{B}_0+u_1\mathcal{B}_1)+u_1f(x+\mathcal{B}_0+u_2\mathcal{B}_2)\\
&\quad +u_1u_2f'(x+\mathcal{B}_0).\end{aligned}$$
The main results of this paper are based on Lemma 2 and an analogue for Euler polynomials, and will be obtained by considering the expectation when $u_1,\ldots,u_k$ are taken to be certain random variables.
Generalization and proof of Theorem 1
=====================================
[**1.**]{} In this section we prove a higher-order analogue of Theorem 1, of which the latter is an immediate consequence. The proof uses some probabilistic methods which will be summarized in a brief subsection.
For integers $k\geq 2$ and $n\geq 0$ and for positive real parameters $a_1,\ldots,a_k$ we have $$\begin{aligned}
&\sum_{l_1+\dots+l_k=n}\binom{n}{l_1,\ldots,l_k}
\frac{(a_1)_{l_1}\dots(a_k)_{l_k}}{(a_1+\dots+a_k)_n}B_{l_1}(x)\dots B_{l_k}(x)
=\sum_{j=1}^k\sum_{|J|=j}\frac{a_J n!}{(n+1-j)!}\label{4.0} \\
&\qquad\times\sum_{\substack{l_0+l_1+\dots+l_{k-j}\\=n+1-j}}\binom{n+1-j}{l_0,l_1,\ldots,l_{k-j}}
\frac{(a_{i_{j+1}})_{l_1}\dots(a_{i_k})_{l_{k-j}}}{(a_1+\dots+a_k)_{n+1-l_0}}
B_{l_0}(x)B_{l_1}\dots B_{l_{k-j}}.\nonumber\end{aligned}$$
When $k=2$, we immediately get Theorem 1. For $k=3$ and $a_1=a_2=a_3=1$ we get, after some easy transformations and renaming the summation indices, $$\begin{gathered}
(n+3)\sum_{i+j+l=n}B_i(x)B_j(x)B_l(x)
= 3\sum_{i+j+l=n}\binom{n+3}{i}B_i(x)B_jB_l\label{4.0a}\\
\qquad+3\sum_{i+j=n-1}\binom{n+3}{i}B_i(x)B_j
+\binom{n+3}{5}B_{n-2}(x),\nonumber\end{gathered}$$ valid for $n\geq 3$; this is Corollary 1 in [@AD]. Other special cases with $k=3$ will be considered later, in Section 6. For arbitrary $k\geq 2$, with $a_1=\dots=a_k=1$, we recover Theorem 1 in [@AD], which for $x=0$ gives a $k$th order analogue of Matiyasevich’s identity , namely $$\sum_{l_1+\dots+l_k=n}B_{l_1}\dots B_{l_k}
=\frac{1}{n+k}\sum_{j=1}^k\binom{k}{j}
\sum_{\substack{l_0+l_1+\dots+l_{k-j}\\=n+1-j}}\binom{n+k}{l_0}
B_{l_0}B_{l_1}\dots B_{l_{k-j}}.$$
[**2.**]{} We now summarize some facts from probability theory that will be used in the proofs that follow. For the basics we refer the reader to any introductory text in probability theory, e.g., [@Du] or [@Wa]. For the interplay between probability theory and umbral calculus, see [@SW].
We assume that $X$ is a continuous random variable with probability density function $f_X(x)$, i.e., $$\label{4.1}
{\rm Pr}(X\leq x) = \int_{-\infty}^x f_X(y)dy.$$ Given a measurable function $g:{\mathbb R}\rightarrow {\mathbb R}$ such that the image random variable $g(X)$ is absolutely integrable, its expectation can be expressed as $$\label{4.2}
{\mathbb E}g(X) = \int_{-\infty}^\infty g(y)f_X(y)dy.$$ The main tool in this section is the use of random variables with a gamma distribution of “scale parameter" 1. We write such a random variable as $X \sim\Gamma_a$, with “shape parameter" $a>0$, defined by the density function $$\label{4.3}
f_X(x;a) = \begin{cases}
\tfrac{1}{\Gamma(a)}x^{a-1}e^{-x} &\hbox{for}\; x\geq 0,\\
0 &\hbox{otherwise}.
\end{cases}$$ Then from the definition of the gamma function, $$\Gamma(s) = \int_0^\infty x^{s-1}e^{-x}dx,$$ and with and we immediately get, for an integer $n\geq 1$, $$\label{4.4}
{\mathbb E}(\Gamma_a^n) = \int_0^\infty y^n\tfrac{1}{\Gamma(a)}y^{a-1}e^{-y}dy
=\frac{\Gamma(a+n)}{\Gamma(a)} = (a)_n.$$ An essential property of the gamma distribution is additivity, i.e., if $\Gamma_{a_1},\ldots,\Gamma_{a_k}$ are independent gamma distributed random variables, then $$\label{4.5}
\Gamma_{a_1}+\dots+\Gamma_{a_k} \sim \Gamma_{a_1+\dots+a_k},$$ where the symbol $\sim$ indicates that the random variables on both sides have the same distribution. The relation follows from the fact that the density probability function for the sum of two independent random variables is the convolution of the individual ones; see, e.g., [@Wa p. 107].
The next important tool is the choice of random coefficients $u_1,\ldots, u_k$ such that $(u_1,\ldots, u_k)$ follows a Dirichlet distribution with parameters $(a_1,\ldots, a_k)$. This is equivalent to choosing $k$ independent gamma random variables $\Gamma_{a_i}$, each having shape parameter $a_i$, and to define $$\label{4.6}
u_i=\frac{\Gamma_{a_i}}{\Gamma_{a_1}+\dots+\Gamma_{a_k}},\qquad 1\leq i\leq k;$$ note that $u_1+\dots+u_k=1$. For Dirichlet distributions in general, see, e.g., [@JK p. 231]. We now need an important property of gamma random variables, namely that $\Gamma_a+\Gamma_b$ and $\Gamma_a/(\Gamma_a+\Gamma_b)$ are independent when $\Gamma_a$ and $\Gamma_b$ are. In fact, this characterizes gamma random variables; see [@Lu]. This is easily extended to the statement that $$\label{4.7}
\Gamma_{a_1}+\dots+\Gamma_{a_k}\quad\hbox{and}\quad
\frac{\Gamma_{a_i}}{\Gamma_{a_1}+\dots+\Gamma_{a_k}},\qquad 1\leq i\leq k,$$ are independent. The importance of this lies in the fact that ${\mathbb E}(XY)={\mathbb E}(X){\mathbb E}(Y)$ for independent random variables $X$ and $Y$.
Combining all of the above, we first note that for any positive integers $l_1,\ldots, l_k$ we have by , $$\begin{aligned}
{\mathbb E}[(\Gamma_{a_1}+\dots+\Gamma_{a_k})^{l_1+\dots+l_k}
(u_1^{l_1}\dots u_k^{l_k})]
&={\mathbb E}(\Gamma_{a_1}^{l_1}\dots\Gamma_{a_k}^{l_k})\label{4.8}\\
&={\mathbb E}(\Gamma_{a_1}^{l_1})\dots
{\mathbb E}(\Gamma_{a_k}^{l_k}).\nonumber\end{aligned}$$ On the other hand, using the independence of the terms in , we see that the left-hand side of is equal to $$\label{4.9}
{\mathbb E}(\Gamma_{a_1+\dots+a_k}^{l_1+\dots+l_k})
{\mathbb E}(u_1^{l_1}\dots u_k^{l_k}),$$ having also used . Finally, applying to the right-hand side of and to , we get $$\label{4.10}
{\mathbb E}(u_1^{l_1}\dots u_k^{l_k})
=\frac{(a_1)_{l_1}\dots (a_k)_{l_k}}{(a_1+\dots+a_k)_{l_1+\dots+l_k}}.$$ This identity will be used repeatedly in what follows.
We are now ready to prove Theorem 2; as we shall see, much of the work was already done in obtaining the identities and .
We choose $u_1,\ldots, u_k$ as in . Since $u_1+\dots+u_k=1$, we can rewrite the $n$th power term on the left-hand side of as follows, and then apply a multinomial expansion, using : $$\begin{aligned}
&\left(u_1(x+\mathcal{B}_1)+\dots+u_k(x+\mathcal{B}_k)\right)^n\label{4.11}\\
&\qquad\qquad=\sum_{l_1+\dots+l_k=n}\binom{n}{l_1,\ldots,l_k}
u_1^{l_1}\dots u_k^{l_k}B_{l_1}(x)\dots B_{l_k}(x).\nonumber\end{aligned}$$ Similarly, we use multinomial expansions for the powers on the right of , this time combining the terms $x+\mathcal{B}_0$ for the sake of applying : $$\begin{aligned}
&\left(x+\mathcal{B}_0+(u\mathcal{B})_{\overline{J}}\right)^{n-j+1}\label{4.12}\\
&\qquad=\sum_{\substack{l_0+l_1+\dots+l_{k-j}\\=n+1-j}}\binom{n+1-j}{l_0,l_1,\ldots,l_{k-j}}
B_{l_0}(x)\left(u_{i_{j+1}}\mathcal{B}_{i_{j+1}}\right)^{l_1}\dots
\left(u_{i_k}\mathcal{B}_{i_k}\right)^{l_{k-j}}\nonumber\\
&\qquad=\sum_{\substack{l_0+l_1+\dots+l_{k-j}\\=n+1-j}}\binom{n+1-j}{l_0,l_1,\ldots,l_{k-j}}
u_{i_{j+1}}^{l_1}\dots u_{i_k}^{l_{k-j}}B_{l_0}(x)B_{l_1}\dots B_{l_{k-j}},\nonumber\end{aligned}$$ where we have also used . All that remains to be done now is to compute the expectation on both sides of , which mainly involves applying to the right-hand sides of and . In particular, keeping the first notation in in mind, we have $$\begin{aligned}
{\mathbb E}(u_Ju_{i_{j+1}}^{l_1}\dots u_{i_k}^{l_{k-j}})
&=\frac{(a_{i_1})_1\dots(a_{i_j})_1(a_{i_{j+1}})_{l_1}\dots(a_{i_k})_{l_{k-j}}}
{(a_1+\dots+a_k)_{n+1-l_0}} \\
&=a_J\frac{(a_{i_{j+1}})_{l_1}\dots(a_{i_k})_{l_{k-j}}}{(a_1+\dots+a_k)_{n+1-l_0}},\end{aligned}$$ where we have used the fact that $(a)_1=a$ and, in the denominator, that $1+\dots +l_1+\dots+l_{k-j}=j+(n-j+1)-l_0=n+1-l_0$. This completes the proof.
Euler numbers and polynomials
=============================
The Euler numbers and polynomials are often considered in parallel with their Bernoulli analogues. Indeed, they are similar in various respects, including their importance in the classical calculus of finite differences (see, e.g., [@Jo] or [@No]). In this section we follow the outlines of the previous sections to derive analogous results for Euler polynomials and, to a lesser extent, Euler and Genocchi numbers.
The [*Euler numbers*]{} $E_n$, $n=0, 1, 2,\ldots$, can be defined by $$\label{5.1}
\frac{2}{e^z+e^{-z}} = \sum_{n=0}^\infty E_n\frac{z^n}{n!}\qquad
(|z|< \tfrac{\pi}{2}).$$ The Euler numbers are all integers with $E_n=0$ when $n$ is odd; the first few values are listed in Table 1. The [*Euler polynomials*]{} can be defined by $$\label{5.2}
E_n(x):=\sum_{j=0}^n\binom{n}{j}\frac{E_j}{2^j}(x-\tfrac{1}{2})^{n-j},$$ or equivalently by the generating function $$\label{5.3}
\frac{2e^{xz}}{e^z+1}=\sum_{n=0}^\infty E_n(x)\frac{z^n}{n!}\qquad (|z|< \pi).$$ A key consequence of is the functional equation $$\label{5.4}
E_n(x)+E_n(x+1)=2x^n,\qquad n=0,1,2,\ldots,$$ which gives rise to numerous applications. One important difference to the Bernoulli case is the fact that $E_n(0)$ is [*not*]{} the $n$th Euler number. The [*Genocchi numbers*]{} $G_n$, are often used instead; they are closely related to the Bernoulli numbers via $$\label{5.5}
G_n := 2(1-2^n)B_n\qquad (n=0, 1, 2,\ldots).$$ These numbers are all integers; the first few values are also listed in Table 1.
$n$ $B_n$ $E_n$ $G_n$ $B_n(x)$ $E_n(x)$
----- --------- ------- ------- ---------------------------------------------------------- ----------------------------------------------------
0 1 1 0 1 1
1 $-1/2$ 0 1 $x-\tfrac{1}{2}$ $x-\tfrac{1}{2}$
2 $1/6$ $-1$ $-1$ $x^2-x+\tfrac{1}{6}$ $x^2-x$
3 0 0 0 $x^3-\tfrac{3}{2}x^2+\tfrac{1}{2}x$ $x^3-\tfrac{3}{2}x^2+\tfrac{1}{4}$
4 $-1/30$ 5 1 $x^4-2x^3+x^2-\tfrac{1}{30}$ $x^4-2x^3+x$
5 0 0 0 $x^5-\tfrac{5}{2}x^4+\tfrac{5}{3}x^3-\tfrac{1}{6}x$ $x^5-\tfrac{5}{2}x^4+\tfrac{5}{2}x^2-\tfrac{1}{2}$
6 $1/42$ $-61$ $-3$ $x^6-3x^5+\tfrac{5}{2}x^4-\tfrac{1}{2}x^2+\tfrac{1}{42}$ $x^6-3x^5+5x^3-3x$
: $B_n, E_n, G_n, B_n(x)$ and $E_n(x)$ for $0\leq k\leq 6$.
By elementary manipulations of the relevant generating functions, we get $$\label{5.6}
E_n(0)=\frac{1}{n+1}G_{n+1},\qquad E_n(\tfrac{1}{2})=2^{-n}E_n \qquad
(n=0, 1, 2,\ldots).$$ The Euler polynomial analogue of Theorem 1 can now be stated as follows.
For integers $n\geq 1$ and real numbers $a, b> 0$ we have $$\begin{aligned}
\sum_{l=0}^{n}\binom{n}{l}&\frac{(a)_{l}(b)_{n-l}}{(a+b)_{n}}E_{l}(x)E_{n-l}(x)
= \frac{4}{n+1}B_{n+1}(x) \label{5.6a}\\
&-\frac{2}{n+1}\sum_{l=0}^{n+1}\binom{n+1}{l}
\frac{(a)_{l}+(b)_{l}}{(a+b)_l}E_l(0)B_{n+1-l}(x)\nonumber\end{aligned}$$
As in the case of Theorem 1, this result follows from a higher-order convolution identity that will be proved later. As a special case of , for $a=b=1$, we get the following Euler polynomial analogue of Matiyasevich’s identity: $$(n+2)\sum_{l=0}^{n}\binom{n}{l}E_{l}(x)E_{n-l}(x)
=4(n+2)B_{n+1}(x)-4\sum_{l=0}^{n+1}\binom{n+2}{l}B_l(x)E_{n+1-l}(0).$$ This identity was earlier obtained as Corollary 2 in [@AD].
As we develop a formalism parallel to that involving the Bernoulli symbol, we note that the analogue of $B_n$ is $E_n(0)$. Thus, we define the [*Euler symbol*]{} $\mathcal E$ by $$\label{5.7}
{\mathcal E}^n = E_n(0),\qquad n=0,1,2,\ldots,$$ and elementary manipulation of the generating function gives $$\label{5.8}
E_n(x) = (x+{\mathcal E})^n,\qquad n=0,1,2,\ldots.$$ The analogue to the uniform symbol $\mathcal U$ defined in Section 3 is the uniform discrete symbol $\mathcal V$ with generating function $$\label{5.9}
e^{z{\mathcal V}} = \tfrac{1}{2}+\tfrac{1}{2}e^z,$$ or equivalently defined by $$f(x+\mathcal{V}) = \frac{f(x)+f(x+1)}{2}$$ for an analytic function $f$; this is a discrete analogue of . With a change of variable we have for any real $u$, $$\label{5.10}
f(x+u{\mathcal V}) = \frac{1}{2}f(x)+\frac{1}{2}f(x+u),$$ which is analogous to , and which will be just as useful. Next, by multiplying , setting $x=0$, with , we see that in analogy with and we have ${\mathcal E}+{\mathcal V}=0$ in the sense that $$\label{5.11}
f(x+{\mathcal E}+{\mathcal V}) = f(x),$$ or in other words, $$\label{5.12}
f(x)=g(x+\mathcal{V})\quad\Leftrightarrow\quad g(x)=f(x+\mathcal{E}).$$
[**3.**]{} The functional equations and give rise to the definition of the [*discrete forward difference operator*]{} $\delta_u$ defined by $$\label{5.13}
\delta_uf(x) = \frac{f(x)+f(x+u)}{2}.$$ Thus, in particular, we have $\delta_1E_n(x)=x^n$ and by , $$\label{5.14}
\delta_uf(x) = f(x+u{\mathcal V}).$$ In analogy to we now compute $$\begin{aligned}
2\delta_{u_1}\delta_{u_2}f(x)
&=\frac{f(x)+f(x+u_2)}{2}+\frac{f(x+u_1)+f(x+u_2+u_1)}{2} \\
&=\frac{f(x+)+f(x+u_1+u_2)}{2}+\frac{f(x)+f(x+u_1)}{2}\\
&\qquad+\frac{f(x)+f(x+u_2)}{2}-f(x), \end{aligned}$$ which gives the operator identity $$\label{5.15}
\delta_{u_1+u_2}=2\delta_{u_1}\delta_{u_2}-\delta_{u_1}-\delta_{u_2}+1.$$ Similarly, one obtains $$\begin{aligned}
\delta_{u_1+u_2+u_3} &= 4\delta_{u_1}\delta_{u_2}\delta_{u_3}
-2\delta_{u_1}\delta_{u_2}-2\delta_{u_1}\delta_{u_3}-2\delta_{u_2}\delta_{u_3}\label{5.16} \\
&\qquad +\delta_{u_1}+\delta_{u_2}+\delta_{u_3}.\nonumber\end{aligned}$$ In general, using the notation $\delta_J$, with the same meaning as in , where again $J\subseteq\{1,\ldots,k\}$, we have the following result.
For even $k\geq 2$ we have $$\label{5.17}
\delta_{u_1+\dots+u_k} = 1-\sum_{j=1}^k\sum_{|J|=j}(-2)^{j-1}\delta_J,$$ and for odd $k\geq 1$, $$\label{5.18}
\delta_{u_1+\dots+u_k} = \sum_{j=1}^k\sum_{|J|=j}(-2)^{j-1}\delta_J.$$
These identities can be proved by straightforward induction, with as induction beginning. Using notation from , we now obtain the following result.
Let $u_1+\dots+u_k=1$. Then for even $k\geq 2$ we have $$\label{5.19}
(n+1)\left(x+u_1\mathcal{E}_1+\dots+u_k\mathcal{E}_k\right)^n
=\sum_{j=1}^k(-2)^j\sum_{|J|=j}
\left(x+\mathcal{B}+(u\mathcal{E})_{\overline{J}}\right)^{n+1},$$ and for odd $k\geq 1$, $$\label{5.20}
\left(x+u_1\mathcal{E}_1+\dots+u_k\mathcal{E}_k\right)^n
=\sum_{j=1}^k(-2)^{j-1}\sum_{|J|=j}
\left(x+\mathcal{E}_0+(u\mathcal{E})_{\overline{J}}\right)^n,$$ where $\mathcal{E}_0,\dots,\mathcal{E}_k$ are independent Euler symbols.
Comparing with , we get the operator identity $\Delta_u = 2\delta_u-2$, and thus for even $k\geq 2$ we have with , $$\label{5.22}
\Delta_{u_1+\dots+u_k} = \sum_{j=1}^k(-2)^j\sum_{|J|=j}\delta_J.$$ Since $u_1+\dots u_k=1$, we have by , $$\label{5.23}
\Delta_{u_1+\dots+u_k}f(x) = f'(x+\mathcal{U}).$$ Now let $$f(x):=\left(x+\mathcal{B}+u_1\mathcal{E}_1+\dots+u_k\mathcal{E}_k\right)^{n+1},$$ and apply to this function. On the left-hand side, using , the symbols $\mathcal{U}$ and $\mathcal{B}$ cancel each other, and we get the left-hand side of . To obtain the right-hand side, we first note that for any $i=1,\ldots,k$ we have by , $$\label{5.24}
\delta_{u_i}f(x)
=\left(x+\mathcal{B}+(u\mathcal{E})_{\{1,\ldots,k\}\setminus\{i\}}\right)^{n+1},$$ having used the notation in and the fact that the symbols $\mathcal{V}$ and $\mathcal{E}_i$ cancel each other. As in , the coefficients $u_i$ have to match for this cancellation to apply. Successively applying and using the notation , we get $$\delta_Jf(x) = \left(x+\mathcal{B}+(u\mathcal{E})_{\overline{J}}\right)^{n+1}.$$ This, combined with , completes the proof of .
The proof of is very similar: Instead of use and apply it to $$f(x):=\left(x+\mathcal{E}_0+u_1\mathcal{E}_1+\dots+u_k\mathcal{E}_k\right)^n.$$ While the right-hand side is evaluated as before, for the left-hand side we use with $u=1$.
[**4.**]{} We are now ready to state and prove the main result of this section.
Let $n\geq 0$ and $k\geq $ be integers and $a_1,\ldots,a_k$ positive parameters. Then for even $k\geq 2$ we have $$\begin{gathered}
\sum_{l_1+\dots+l_k=n}\binom{n}{l_1,\ldots,l_k}
\frac{(a_1)_{l_1}\dots(a_k)_{l_k}}{(a_1+\dots+a_k)_n}E_{l_1}(x)\dots E_{l_k}(x)
=\sum_{j=1}^k\frac{(-2)^j}{n+1}\label{5.25} \\
\quad\times\sum_{|J|=j}\sum_{\substack{l_0+l_1+\dots\\+l_{k-j}=n+1}}\binom{n+1}{l_0,\ldots,l_{k-j}}
\frac{(a_{i_{j+1}})_{l_1}\dots(a_{i_k})_{l_{k-j}}}{(a_1+\dots+a_k)_{n+1-l_0}}
B_{l_0}(x)E_{l_1}(0)\dots E_{l_{k-j}}(0),\nonumber\end{gathered}$$ and for odd $k\geq 1$, $$\begin{gathered}
\sum_{l_1+\dots+l_k=n}\binom{n}{l_1,\ldots,l_k}
\frac{(a_1)_{l_1}\dots(a_k)_{l_k}}{(a_1+\dots+a_k)_n}E_{l_1}(x)\dots E_{l_k}(x)
=\sum_{j=1}^k(-2)^{j-1}\label{5.26} \\
\quad\times\sum_{|J|=j}\sum_{\substack{l_0+l_1+\dots\\+l_{k-j}=n}}\binom{n}{l_0,\ldots,l_{k-j}}
\frac{(a_{i_{j+1}})_{l_1}\dots(a_{i_k})_{l_{k-j}}}{(a_1+\dots+a_k)_{n-l_0}}
E_{l_0}(x)E_{l_1}(0)\dots E_{l_{k-j}}(0).\nonumber\end{gathered}$$
For $k=2$, the identity reduces to Theorem 3. In the special case $a_1=\dots=a_k=1$, the identities and reduce to Theorems 2 and 3, respectively, in [@AD]. Other special cases can be found in Section 6.
The proof is almost identical to that of Theorem 2: We expand the powers on both sides of and using the multinomial theorem, and then compute the expectation on both sides by way of , having chosen $u_1,\ldots,u_k$ as in .
The left-hand sides of and are obtained just as in the expansion , with Euler instead of Bernoulli symbols and polynomials, and having used in place of . The right-hand sides are expanded as in , with the appropriate exponent and with “Bernoulli" replaced by “Euler" where appropriate. Applying then completes the proofs of both identities.
Some further identities
=======================
In this final section we state and prove some further consequences of our main results from Sections 2, 4 and 5, respectively.
Consequences of Theorem 1
-------------------------
In addition to the two identities in Corollary 4, we can obtain one more consequence of Corollary 3 by multiplying both sides of by $a$ and then taking the limit as $a\rightarrow\infty$. Then all the terms in the sum on the left disappear, with the exception of the $l=n-1$ term. On the right, the fraction in the sum tends to 1 for $l\geq 2$, and to 2 for $l=1$. Putting everything together, we get the following consequence.
For integers $n\geq 1$ we have $$\label{2.13}
\sum_{l=0}^n\binom{n}{l}B_lB_{n-l}(x) = n(x-1)B_{n-1}(x)-(n-1)B_n(x).$$
For $x=0$ this is Euler’s identity , but it is also a special case of identity (5.11.2) in [@Ha].
We can obtain even more consequences from Theorem 1 by setting $a=b=\varepsilon$ and then taking the limit as $\varepsilon\rightarrow\infty$, or by considering the terms in as power series in $\varepsilon$. We begin with the first case.
For integers $n\geq 1$ we have $$\label{2.14}
\sum_{l=0}^n\binom{n}{l}\frac{1}{2^l}B_lB_{n-l}(x)
= \frac{n}{2^n}(2x-1)B_{n-1}(2x)-\frac{n-1}{2^n}B_n(2x)-\frac{n}{4}B_{n-1}(x).$$
With $a=b=\varepsilon$, the following limits are obvious: $$\lim_{\varepsilon\rightarrow\infty}\frac{(\varepsilon)_l(\varepsilon)_{n-l}}{(2\varepsilon)_n}=\frac{1}{2^n},\qquad
\lim_{\varepsilon\rightarrow\infty}\frac{2\varepsilon(\varepsilon)_l}{(2\varepsilon)_{l+1}}=\frac{1}{2^l},\qquad
\lim_{\varepsilon\rightarrow\infty}\frac{\varepsilon^2}{(2\varepsilon+1)(2\varepsilon)}=\frac{1}{4}.$$ Hence we have $$\label{2.15}
\frac{1}{2^n}\sum_{l=0}^n\binom{n}{l}B_l(x)B_{n-l}(x)
=\sum_{l=0}^n\binom{n}{l}\frac{1}{2^l}B_lB_{n-l}(x)+\frac{n}{4}B_{n-1}(x).$$ The sum on the left has a well-known evaluation (see (50.11.2) in [@Ha]): $$\sum_{l=0}^n\binom{n}{l}B_l(x)B_{n-l}(x)=n(2x-1)B_{n-1}(2x)-(n-1)B_n(2x).$$ This, with , immediately gives .
We note that can also be obtained as a special case of identity (6.1) in [@AD]. For the next statement, again using $a=b=\varepsilon$, we need the [*second-order harmonic numbers*]{}, defined by $H_0^{(2)}:=0$ and $$H_n^{(2)}:=\sum_{j=1}^n\frac{1}{j^2}\qquad(n\geq 1).$$ We can now prove the following result.
For integers $n\geq 1$ we have $$\begin{aligned}
\sum_{l=1}^{n-1}\left(H_{n-1}-H_{l-1}\right)
\frac{B_l(x)}{l}&\frac{B_{n-l}(x)}{n-l}
=\sum_{l=1}^{n}\binom{n}{l}\left(H_l+\frac{1}{l}\right)\frac{B_l}{l}B_{n-l}(x)\label{2.16} \\
&+nB_{n-1}(x)+\frac{1}{2}\left(H_{n-1}^2+3H_{n-1}^{(2)}\right)B_n(x).\nonumber\end{aligned}$$
Setting $a=b=\varepsilon$ in and dividing both sides by $\varepsilon$, we have to expand the following terms. First, for $1\leq l\leq n-1$ we have $$\begin{aligned}
\frac{1}{\varepsilon}\frac{(a)_{l}(b)_{n-l}}{(a+b)_{n}}
&=\frac{(\varepsilon+1)\dots(\varepsilon+l-1)(\varepsilon+1)\dots(\varepsilon+n-l-1)}{2(2\varepsilon+1)\dots(2\varepsilon+n-1)}\label{2.17}\\
&=\frac{(l-1)!(n-l-1)!}{2(n-1)!}
\prod_{j=1}^{l-1}\left(1+\frac{\varepsilon}{j}\right)
\prod_{j=1}^{n-l-1}\left(1+\frac{\varepsilon}{j}\right)
\prod_{j=1}^{n-1}\left(1+\frac{2\varepsilon}{j}\right)^{-1} \nonumber\\
&=\frac{(l-1)!(n-l-1)!}{2(n-1)!}\left(1+\left(H_{l-1}+H_{n-l-1}-2H_{n-1}\right)\varepsilon+O(\varepsilon^2)\right).\nonumber\end{aligned}$$ Next, for $l\geq 1$ we get $$\begin{aligned}
\frac{1}{\varepsilon}\frac{a(b)_{l}+b(a)_{l}}{(a+b)_{l+1}}
&=\frac{(\varepsilon+1)\dots(\varepsilon+l-1)}{(2\varepsilon+1)\dots(2\varepsilon+l)}
=\frac{1}{l}\left(1+\frac{2\varepsilon}{l}\right)^{-1}
\prod_{j=1}^{l-1}\frac{1+\frac{\varepsilon}{j}}{1+\frac{2\varepsilon}{j}}\label{2.18}\\
&=\frac{1}{l}\left(1-\frac{2\varepsilon}{l}+\dots\right)
\prod_{j=1}^{l-1}\left(1-\frac{\varepsilon}{j}+\dots\right) \nonumber \\
&=\frac{1}{l}\left(1-\left(H_l+\frac{1}{l}\right)\varepsilon+O(\varepsilon^2)\right),\nonumber\end{aligned}$$ where we have used the fact that $H_{l-1}+2/l=H_l+1/l$. Next, we have $$\label{2.19}
\frac{1}{\varepsilon}\frac{ab}{(a+b+1)(a+b)}=\frac{1}{2(1+2\varepsilon)}
=\frac{1}{2}\left(1-2\varepsilon+O(\varepsilon^2)\right).$$ Finally, we collect on the right the terms left out in the two sums, namely $$\frac{1}{\varepsilon}\left(1-2\frac{(a)_n}{(a+b)_n}\right)B_n(x).$$ We expand $$\begin{aligned}
2\frac{(a)_n}{(a+b)_n}&=\frac{2(\varepsilon)_n}{(2\varepsilon)_n}
=\prod_{j=1}^{n-1}\frac{\varepsilon+j}{2\varepsilon+j} \\
&=\prod_{j=1}^{n-1}\left(1+\frac{\varepsilon}{j}\right)\left(1+\frac{2\varepsilon}{j}\right)^{-1}
=\prod_{j=1}^{n-1}\left(1-\frac{\varepsilon}{j}+\frac{2\varepsilon^2}{j^2}+\dots\right) \\
&=1-H_{n-1}\varepsilon+\left(2H_{n-1}^{(2)}+\sum_{1\leq j<k\leq n-1}\frac{1}{jk}\right)\varepsilon^2+O(\varepsilon^3).\end{aligned}$$ Now the double sum in the last term can clearly be written as $(H_{n-1}^2-H_{n-1}^{(2)})/2$, and thus $$\label{2.20}
\frac{1}{\varepsilon}\left(1-2\frac{(a)_n}{(a+b)_n}\right)
=H_{n-1}-\left(\frac{3}{2}H_{n-1}^{(2)}+\frac{1}{2}H_{n-1}^2\right)\varepsilon+O(\varepsilon^2).$$ If we substitute – into and let $\varepsilon\rightarrow 0$, we recover the polynomial analogue of Miki’s identity. Finally, if we equate the coefficients of $\varepsilon$, we immediately get after multiplying both sides by $-1$ and exploiting symmetry in .
Consequences of Theorem 2
-------------------------
We restrict our attention to the case $k=3$ and $a_1=a_2=a_3=\varepsilon$. Furthermore, to avoid double indices, we set $i=l_1$, $j=l_2$, $l=l_3$ on the left, and $i=l_0$, $j=l_1$, $l=l_2$ on the right of . Then we get, after dividing by $n!$, $$\begin{gathered}
\sum_{i+j+l=n}\frac{(\varepsilon)_i(\varepsilon)_j(\varepsilon)_l}
{(3\varepsilon)_n}\frac{B_i(x)}{i!}\frac{B_j(x)}{j!}\frac{B_l(x)}{l!}
=3\sum_{i+j+l=n}\frac{\varepsilon(\varepsilon)_j(\varepsilon)_l}
{(3\varepsilon)_{j+l+1}}\frac{B_i(x)}{i!}\frac{B_j}{j!}\frac{B_l}{l!}\label{6.9}\\
+3\sum_{i+j=n-1}\frac{\varepsilon^2(\varepsilon)_j}{(3\varepsilon)_{j+2}}
\frac{B_i(x)}{i!}\frac{B_j}{j!}
+\frac{\varepsilon^3}{(3\varepsilon)_3}\frac{B_{n-2}(x)}{(n-2)!},\nonumber\end{gathered}$$ valid for all $n\geq 2$ and $\varepsilon>0$. This will be the basis for the various results in this subsection, and also immediately gives .
For a first easy consequence we let $\varepsilon\rightarrow\infty$ on both sides of . Then the limit of the four fractions involving $\varepsilon$ are easily seen to be $3^{-n}$, $3^{-j-l-1}$, $3^{-j-2}$, and $3^{-3}$, respectively. Thus, after multiplying both sides by $3^nn!$, we immediately get the following result.
For integers $n\geq 2$ we have $$\begin{gathered}
\sum_{i+j+l=n}\binom{n}{i,j,l}B_i(x)B_j(x)B_l(x)
=\sum_{i+j+l=n}\binom{n}{i,j,l}3^iB_i(x)B_jB_l\label{6.10} \\
+n\sum_{i=0}^{n-1}\binom{n-1}{i}3^iB_i(x)B_{n-1-i}+n(n-1)3^{n-3}B_{n-2}(x).\nonumber\end{gathered}$$
For the next consequence of we set $x=0$, for greater simplicity of the statement. The proof is tedious, and we leave the details to the interested reader.
For integers $n\geq 2$ we have $$\begin{aligned}
&\frac{1}{3}\sum_{\substack{i+j+l=n\\i,j,l\geq 1}}
\frac{B_i}{i}\frac{B_j}{j}\frac{B_l}{l}
=\sum_{\substack{i+j+l=n\\i,j,l\geq 1}}\binom{n-1}{i-1}
\frac{B_i}{i}\frac{B_j}{j}\frac{B_l}{l}
+\sum_{l=1}^{n-2}\binom{n-1}{l+1}\frac{B_l}{l}\frac{B_{n-l-1}}{n-l-1}\\
&+\sum_{l=1}^{n-1}\left(3H_{n-1}-2H_{l-1}+\tfrac{1}{n}\right)\frac{B_l}{l}\frac{B_{n-l}}{n-l}
-2\sum_{l=1}^{n-1}\binom{n-1}{l-1}\left(2H_l+\tfrac{1}{l}\right)\frac{B_l}{l}\frac{B_{n-l}}{n-l} \\
&+\frac{n-1}{6}B_{n-2}+\left(\frac{1}{(n-1)n}-3\right)B_{n-1}
-2\left(\frac{2}{n}H_{n-1}+H_{n-1}^2+2H_{n-1}^{(2)}+\frac{3}{n^2}\right)\frac{B_n}{n}.\end{aligned}$$
This can be seen as a third-order analogue of Miki’s identity. Note the difference in complexity between this result and the third-order analogue of Matiyasevich’s identity given in . See also [@AD (6.5)] for a third-order “Miki analogue" for Bernoulli polynomials. To prove Corollary 9, one can use a similar method as in the proof of Corollary 7, and proceed as follows:
– Collect the “edge" and “corner" terms in the sums of .
– Divide both sides of by $\varepsilon^2$.
– Expand the various fractions involving $\varepsilon$ in a similar way as in and .
– Equate the constant terms (i.e., let $\varepsilon\rightarrow 0$) to obtain a first identity.
– Equate the coefficients of $\varepsilon$ to obtain a second identity.
Interestingly, in this case the first identity turns out to be equivalent to , Miki’s original identity. The second one is Corollary 9.
Consequences of Theorems 3 and 4
--------------------------------
Given the similarities between Theorems 1 and 3, it is clear that Euler analogues of Corollaries 2–7 could easily be derived; recall that an analogue of Corollary 1 is already stated following Theorem 3. Here we restrict ourselves to only a few more consequences; we also skip the proofs which are again similar to the proof of Corollary 7.
For integers $n\geq 2$ we have $$\begin{aligned}
\sum_{l=1}^{n-2}\frac{E_l(x)}{l}\frac{E_{n-l-1}(x)}{n-l-1}
&=4\sum_{l=1}^{n-1}\binom{n-2}{l-1}H_{l-1}\frac{B_{n-l}(x)}{n-l}\frac{E_l(0)}{l}\\
&\qquad\qquad+2H_{n-2}\frac{E_{n-1}(x)}{n-1}+4\frac{H_{n-1}}{n-1}\frac{E_n(0)}{n},\end{aligned}$$ and for $n\geq 1$, $$\begin{aligned}
&\sum_{l=1}^{n-1}\left(H_{n-1}-H_{l-1}\right)
\frac{E_l(x)}{l}\frac{E_{n-l}(x)}{n-l}
=\frac{1}{2}\left(H_{n-1}^2+3H_{n-1}^{(2)}\right)\frac{E_n(x)}{n} \\
&\quad+\sum_{l=1}^n\binom{n-1}{l-1}\left(H_{l-1}^2+3H_{l-1}^{(2)}\right)
\frac{B_{n+1-l}(x)}{n+1-l}\frac{E_l(0)}{l}
+\frac{1}{n}\left(H_n^2+3H_n^{(2)}\right)\frac{E_{n+1}(0)}{n+1}.\end{aligned}$$
Finally, to obtain third-order analogues of Miki’s identity, we start with for $k=3$ and follow the outline described after Corollary 9. This leads to the following identities.
For integers $n\geq 2$ we have $$\begin{aligned}
\sum_{l=1}^{n-1}&\left(\frac{E_l(x)}{l}\frac{E_{n-l}(x)}{n-l}
-\frac{E_l(0)}{l}\frac{E_{n-l}(0)}{n-l}\right) \\
&\qquad\qquad=\sum_{\substack{i+j+l=n\\i,j,l\geq 1}}\binom{n-1}{i}
E_i(x)\frac{E_j(0)}{j}\frac{E_l(0)}{l}+2H_{n-1}\frac{E_n(x)}{n},\end{aligned}$$ and $$\begin{aligned}
\frac{1}{3}&\sum_{\substack{i+j+l=n\\i,j,l\geq 1}}
\frac{E_i(x)}{i}\frac{E_j(x)}{j}\frac{E_l(x)}{l}
=-2\left(H_{n-1}^2+2H_{n-1}^{(2)}\right)\frac{E_n(x)}{n} \\
&+\sum_{\substack{i+j+l=n\\i,j,l\geq 1}}\binom{n-1}{i}
\left(H_{j-1}+H_{l-1}-3H_{j+l-1}\right)E_i(x)\frac{E_j(0)}{j}\frac{E_l(0)}{l}\\
&+\sum_{l=1}^{n-1}\left(3H_{n-1}-H_{l-1}-H_{n-l-1}\right)
\left(\frac{E_l(x)}{l}\frac{E_{n-l}(x)}{n-l}
-\frac{E_l(0)}{l}\frac{E_{n-l}(0)}{n-l}\right).\end{aligned}$$
Using the special values , numerous identities involving Genocchi and/or Euler numbers could also be obtained.
Final remarks
=============
[**1.**]{} By choosing different values of the parameters in Theorems 1–4, many more identities for Bernoulli and Euler numbers and polynomials could be obtained, some relatively simple, and others of increasing complexity. We have shown in this paper that the original identities of Miki and Matiyasevich and their various extensions are special cases of very general class of identities.
This is not the first common extension of the identities of Miki and Matiyasevich. In fact, using generating functions, Dunne and Schubert [@DS] recently proved the following result.
For any integer $n\geq 2$ and real $p\geq 0$ we have $$\begin{aligned}
\sum_{l=1}^{n-1}(2l)_p(2n-2l)_p&\frac{B_{2l}}{2l}\frac{B_{2n-2l}}{2n-2l}
= 2B_{2n}\frac{\Gamma(2n+2p)}{(2n)!}\sum_{l=1}^{2n-1}
\frac{\Gamma(p+l)\Gamma(p+1)}{\Gamma(2p+l+1)}\label{7.1} \\
&+\frac{2\Gamma(p+1)}{(2n)!}\sum_{l=1}^n\binom{2n}{2l}
\frac{\Gamma(p+2l)\Gamma(2p+2n)}{\Gamma(2p+2l+1)}B_{2l}B_{2n-2l}.\nonumber\end{aligned}$$
The case $p=0$ gives Miki’s identity, while for $p=1$ we get $$\sum_{l=1}^nB_{2l}B_{2n-2l} = \frac{1}{n+1}
\sum_{l=1}^n\binom{2n+2}{2l+2}B_{2l}B_{2n-2l} + 2nB_{2n},$$ which is equivalent to Matiyasevich’s identity.
The identity actually follows from Theorem 1 if we take $a=b=p$ and $x=0$, then replace $n$ by $2n$ and extract the end terms in the sums. After some manipulations we then get $$\begin{aligned}
\sum_{l=1}^{n-1}(2l)_p(2n-2l)_p&\frac{B_{2l}}{2l}\frac{B_{2n-2l}}{2n-2l}
= \frac{\Gamma^2(p)}{(2n)!}\left((2p)_{2n}-2(p)_{2n}\right) \label{7.2} \\
&+\frac{2\Gamma(p+1)}{(2n)!}\sum_{l=1}^n\binom{2n}{2l}
\frac{\Gamma(p+2l)\Gamma(2p+2n)}{\Gamma(2p+2l+1)}B_{2l}B_{2n-2l}.\nonumber\end{aligned}$$ Comparing with shows that after some simplification we have $$\sum_{l=1}^{2n-1}\frac{\Gamma(p+l)}{\Gamma(2p+l+1)}
=\frac{\Gamma(p)}{\Gamma(2p+1)} - \frac{\Gamma(p+2n)}{p\Gamma(2p+2n)}.$$ This last identity can be proved independently, for instance by manipulating the integral representation of Euler’s beta function.
[25]{}
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[^1]: The first author was supported in part by the Natural Sciences and Engineering Research Council of Canada
|
---
abstract: 'Quantum spin dephasing is caused by inhomogeneous coupling to the environment, with resulting limits to the measurement time and precision of spin-based sensors. The effects of spin dephasing can be especially pernicious for dense ensembles of electronic spins in the solid-state, such as nitrogen-vacancy (NV) color centers in diamond. We report the use of two complementary techniques, spin bath driving, and double quantum coherence magnetometry, to enhance the inhomogeneous spin dephasing time ($T_2^*$) for NV ensembles by more than an order of magnitude. In combination, these quantum control techniques (i) eliminate the effects of the dominant NV spin ensemble dephasing mechanisms, including crystal strain gradients and dipolar interactions with paramagnetic bath spins, and (ii) increase the effective NV gyromagnetic ratio by a factor of two. Applied independently, spin bath driving and double quantum coherence magnetometry elucidate the sources of spin ensemble dephasing over a wide range of NV and bath spin concentrations. These results demonstrate the longest reported $T_2^*$ in a solid-state electronic spin ensemble at room temperature, and outline a path towards NV-diamond DC magnetometers with broadband femtotesla sensitivity.'
author:
- Erik Bauch
- 'Connor A. Hart'
- 'Jennifer M. Schloss'
- 'Matthew J. Turner'
- 'John F. Barry'
- Pauli Kehayias
- Swati Singh
- 'Ronald L. Walsworth'
bibliography:
- 'references.bib'
title: 'Ultralong dephasing times in solid-state spin ensembles via quantum control'
---
[^1]
[^2]
Introduction {#introduction .unnumbered}
============
Solid-state electronic spins, including defects in silicon carbide [@Klimov2015; @Widmann2015; @Heremans2016; @Koehl2017; @Tarasenko2017], phosphorus spins in silicon [@Abe2010; @Tyryshkin2012], and silicon-vacancy [@Hepp2014; @Heremans2016; @Rose2017] and nitrogen-vacancy (NV) centers [@Doherty2013] in diamond, have garnered increasing relevance for quantum science and sensing experiments. In particular, NV centers in diamond have been extensively studied and deployed in diverse applications facilitated by long NV spin coherence times [@Balasubramanian2009; @Stanwix2010] at ambient temperature, as well as optical preparation and readout of NV spin states [@Doherty2013]. Many applications utilize dense NV spin ensembles for high-sensitivity DC magnetic field sensing [@Barry2016; @Bucher2017] and wide-field DC magnetic imaging [@LeSage2013; @Glenn2015; @Shao2016; @Tetienne2016; @Glenn2017], including measurements of single-neuron action potentials [@Barry2016], paleomagnetism [@Fu2017; @Glenn2017], and current flow in graphene [@Tetienne2016]. For NV ensembles, the DC magnetic field sensitivity is typically limited by dephasing of the NV sensor spins. In such instances, spin interactions with an inhomogeneous environment (see Fig.\[fig:fig1\]a) limit the experimental sensing time to the spin dephasing time $T_2^{*} \, \lesssim\,1\,\upmu$s [@Acosta2010; @Kubo2011; @Grezes2015; @Choi2017a]. Hahn echo and dynamical decoupling protocols can restore the NV ensemble phase coherence by isolating the NV sensor spins from environmental noise and, in principle, permit sensing times approaching the spin lattice relaxation ($T_1\sim\,$ms) [@DeLange2010; @Pham2012; @Bar-Gill2013]. However, these protocols restrict sensing to AC signals within a narrow bandwidth. For this reason, the development of high sensitivity, broadband magnetometers requires new approaches to extend $T_2^*$ for NV ensembles while retaining the ability to measure DC signals. [ To date, spin dephasing mechanisms for NV ensembles have not been systematically studied, as spatially inhomogeneous effects do not lead to single NV spin dephasing, which has traditionally been the focus of the NV-diamond literature [@Balasubramanian2009; @Maurer2012; @Fang2013; @Mamin2014]. Here, we characterize and control the dominant NV spin ensemble dephasing mechanisms by combining two quantum control techniques, double quantum (DQ) coherence magnetometry [@Fang2013; @Mamin2014] and spin bath driving [@DeLange2012; @Knowles2014]. We apply these techniques to three isotopically engineered $^{12}$C samples with widely varying nitrogen and NV concentrations. In combination, we show that these quantum control techniques can extend the NV spin ensemble $T_2^*$ by more than an order of magnitude. ]{} [ Several inhomogeneous spectral broadening mechanisms can]{} contribute to NV spin ensemble dephasing in bulk diamond. [ First,]{} the formation of negatively-charged NV$^-$ centers [ (with electronic spin $S=1$)]{} requires the incorporation of nitrogen into the diamond lattice. As a result, paramagnetic substitutional nitrogen impurities (P1 centers, $S=1/2$) [@Smith1959; @Cook1966; @Loubser1978] typically persist at densities similar to or exceeding the NV concentration, leading to a ‘spin bath’ that couples to the NV spins via incoherent dipolar interactions[, with a magnitude that can vary significantly across the NV ensemble. Second, $^{13}$C nuclei ($I=1/2$) can be a considerable source of NV spin dephasing in diamonds with natural isotopic abundance ($1.07\,\%$), with the magnitude of this effect varying spatially due to the random location of $^{13}$C within the diamond lattice [@Mizuochi2009; @Dreau2012]. Such NV spin ensemble dephasing, however, can be greatly reduced through isotope engineering of the host diamond material [@Balasubramanian2009]. Third, ]{} strain is well-known to affect the diamond crystal and the zero-magnetic-field splitting between NV spin states [@Jamonneau2015; @Trusheim2016]. [ The exact contribution of strain gradients to NV spin ensemble dephasing has not been quantified rigorously because strain varies throughout and between samples, and is in part dependent upon the substrate used for diamond growth[@Gaukroger2008; @Hoa2014].]{} Furthermore, the interrogation of spatially large NV ensembles requires the design of uniform magnetic bias fields to minimize magnetic field gradients across the detection volume.
[ We assume that the relevant NV spin ensemble dephasing mechanisms are independent and can be summarized by Eqn.\[eqn:rate\], $$\label{eqn:rate}
\begin{split}
\frac{1}{T_2^*} \approx & \frac{1}{T_2^*\{\text{NV-}^{13}\mathrm C \}} + \frac{1}{T_2^*\{\text{NV-N}\}} + \frac{1}{T_2^*\{\text{other spins}\}} + \\ & \frac{1}{T_2^*\{\text{strain grad.}\}} +
\frac{1}{T_2^*\{\text{B-field grad.}\}} + \\ & \frac{1}{T_2^*\{\text{temp. fluctuations}\}} + ...,
\end{split}$$ where $T_2^*\{\cdot \}$ describes the $T_2^*$-limit imposed by a particular dephasing mechanism, and the “$\approx$”-symbol indicates that individual dephasing rates add approximately linearly. ]{}
DQ magnetometry employs the $\{-1,+1\}$ sub-basis of the NV spin$-1$ system for quantum sensing. In this basis, noise sources that shift the $|\pm 1\rangle$ states in common mode (e.g., strain inhomogeneities [ and spectrum drifts due to temperature fluctuations of the host diamond; fourth and sixth term in Eqn.\[eqn:rate\], respectively]{}) are suppressed by probing the energy difference between the $|+1\rangle$ and $|-1\rangle$ spin states. In addition, the NV DQ spin coherence accumulates phase due to an external magnetic field at twice the rate of traditional single quantum (SQ) coherence magnetometry, for which the $|0\rangle$ and $|+1\rangle$ (or $|-1\rangle$) spin states are probed. DQ magnetometry provides enhanced susceptibility to target magnetic field signals while also making the spin coherence twice as sensitive to magnetic noise, including interactions with the paramagnetic spin bath. We therefore use resonant radiofrequency control to decouple the bath spins from the NV sensors [ (second and third term in Eqn.\[eqn:rate\])]{}. By employing both DQ magnetometry and spin bath driving [ with isotopically enriched samples]{}, we elucidate and effectively eliminate the dominant sources of NV spin ensemble dephasing, realizing up to a $16\times$ extension of the ensemble $T_2^*$ in diamond. These techniques are also compatible with Ramsey-based DC sensing, and we find up to an $8\times$ improvement in DC magnetic field sensitivity. Our results should enable broadband DC sensing using NV spin ensembles with spin interrogation times approaching those used in AC sensing; and may aid in the fabrication of optimized samples for a wide range of solid-state sensor species.
![[NV ensemble spectroscopy of diamond spin bath. (a)]{} The inhomogeneously broadened electron spin resonance (ESR) linewidth of nitrogen-vacancy (NV) ensembles is a complex function of the local environment within the diamond sample, which includes a diverse bath of electronic and nuclear spins. Inset: Schematics of NV ensemble ESR spectra in the single quantum and double quantum bases, and for double quantum with spin-bath drive. [(b)]{} Spin-1 ground state of the NV center. [(c)]{} Imaging of the longitudinal strain component $M_z$ of one NV orientation class across a $1$-$\,$mm$^2$ field of view for Sample B. An optical microscope image of the diamond surface (left) is included for reference with a red box outlining the field of view shown in the NV strain image. [(d)]{} NV double electron-electron resonance (DEER) spectrum of Sample B, showing six nitrogen groups ($1-6$) attributed to $^{14}$N electronic spins with an external field $B_0 = 8.5\,$mT aligned along a \[111\]-crystallographic axis (see main text). Linewidths are Fourier-broadened. The peaks labeled $i$ and $ii$ correspond to dipole-forbidden transitions of the $^{14}$N electronic spins ($\Delta m_I \neq 0$, see Suppl.XI). The simulated spectrum using the full nitrogen Hamiltonian is shown in red, with linewidth and amplitudes chosen to reflect the experimental data.[]{data-label="fig:fig1"}](Figure_1_v7.pdf)
Double Quantum Magnetometry {#double-quantum-magnetometry .unnumbered}
---------------------------
The enhanced sensitivity to magnetic fields and insensitivity to common-mode noise sources in this DQ basis can be understood by considering the full ground-state Hamiltonian for NV centers, given by (neglecting the hyperfine interaction) [@Doherty2013], $$\label{eqn:NVHamiltonian}
\begin{split}
H/h = & D\,\mathbf S_z^2 + \frac{\gamma_{NV}}{2\pi} \mathbf B \cdot \mathbf S + M_z \mathbf S_z^2 + \\
& M_x \mathbf (\mathbf S_y^2 - \mathbf S_x^2) + M_y (\mathbf S_x \mathbf S_y + \mathbf S_y \mathbf S_x),
\end{split}$$ where $D\approx 2.87\,$GHz is the zero-field spin-state splitting, $\mathbf S = \{\mathbf S_x, \mathbf S_y, \mathbf S_z\}$ are the dimensionless spin-1 operators, $\mathbf B = \{B_x, B_y, B_z\}$ are the local magnetic field components, $\gamma_{NV}/2\pi \approx 28\,$GHz/T is the NV gyromagnetic ratio, and $\{M_x, M_y, M_z\}$ describe the strain and electric field contributions to $H$ [@Barson2017]. Ignoring terms $\propto \mathbf S_x$, $\mathbf S_y$ due to the large zero-field splitting $D$ and a small applied bias $B_z \gtrsim 10\,$mT along $z$, the transition frequencies $f_{\pm1}$ (see Fig.\[fig:fig1\]b) are $$\label{eqn:nvenergies}
f_{\pm1} \approx D + M_z \pm \frac{\gamma_{NV}}{2\pi} B_z.$$ On-axis strain contributions ($\propto\,M_z$) as well as temperature fluctuations ($\frac{\partial D}{\partial T} \approx -74$kHz/K) [@Acosta2010; @Toyli2013] shift the $f_{\pm1}$ transitions linearly. Thus, when performing DQ magnetometry where the difference $\Delta f = f_{+1} -f_{-1}$ is probed, their effects are to first order suppressed. In addition, a pertubative analysis of the complete Hamiltonian in Eqn.\[eqn:NVHamiltonian\] (see Suppl.VII) shows that the effects of off-axis strain contributions ($\propto M_x, M_y$) on DQ magnetometry are reduced by a factor $\sqrt{M_x^2 + M_y^2}/(\gamma_{NV} B_z/\pi)$, proportional to the bias magnetic field $B_z$. Similarly, the effects of off-axis magnetic fields ($\propto B_x, B_y$) on DQ magnetometry are suppressed due to the large zero-field splitting $D$, and are also largely common-mode. Working in the DQ basis at moderate bias fields can therefore lead to an enhancement in $T_2^*$ for NV ensembles if strain inhomogeneities, small off-axis magnetic field gradients ($B_x, B_y \ll D$), or temperature fluctuations are significant mechanisms of inhomogeneous spin dephasing. This result should be contrasted with single NV measurements in which $T_2^*$ and $T_2$ in the DQ basis were found to be approximately half the values in the SQ basis, i.e., $\tau^\text{coh}_\text{DQ} \approx \tau^\text{coh}_\text{SQ}/2$ [@Fang2013; @Mamin2014]. Since spatial inhomogeneities are not relevant for single centers, the reduced decay times are attributed to an increased sensitivity to magnetic noise in the DQ basis due to the paramagnetic spin bath. For example, using vector magnetic microscopy (VMM) [@Glenn2017], we mapped the on-axis strain component $M_z$ in a $1\,$mm$^2$-region for one of the three NV ensemble diamond samples studied in this work ($[\text N]=0.75\,$ppm, Sample B) to quantify the length-scale and magnitude of strain inhomogeneity (Fig. \[fig:fig1\]c). From this analysis, we estimate an average strain gradient $M_z/L \approx 2.8\,$kHz/$\upmu$m, which, as we show below, is in good agreement with the observed SQ $T_2^*$ in our samples.
Spin Bath Driving {#spin-bath-driving .unnumbered}
-----------------
To mitigate NV spin dephasing due to the spin bath, we drive the bath electronic spins [@DeLange2012; @Knowles2014] using resonant radiofrequency (RF) radiation. In Fig.\[fig:fig1\]d, we display the spin resonance spectrum of a nitrogen-rich diamond sample ($[\text N]=0.75\,$ppm, Sample B), recorded via the NV double electron-electron resonance (DEER) technique [@Slichter1990] in the frequency range 100 - 500MHz (see Suppl.IX). The data reveal 6 distinct spectral peaks attributed to $^{14}$N substitutional defects in the diamond lattice. The resonance peaks have an approximate amplitude ratio of 1:3:1:3:3:1 resulting from the four crystallographic Jahn-Teller orientations of the nitrogen defects at two possible angles with respect to an applied bias magnetic field ($B_z= 8.5\,$mT, aligned along the \[111\]-axis), as well as 3 hyperfine states [@Ammerlaan1981; @Davies1979; @Davies1981] (see Suppl.IX for details). [Additional smaller peaks $i$ and $ii$ are attributed to dipole-forbidden nitrogen spin transitions and other electronic dark spins]{} [@Yamamoto2013]. In pulsed spin bath driving [@DeLange2012], a multi-frequency RF $\pi$-pulse is applied to each of the bath spin resonances midway through the NV Ramsey sequence, decoupling the bath from the NV sensor spins in analogy to a refocusing $\pi$-pulse in a spin echo sequence [@DeLange2010]. Alternatively, the bath spins can be driven with continuous wave (CW) [@DeLange2012; @Knowles2014]. In this case, the Rabi drive strength $\Omega_\text{Bath}$ at each bath spin resonance frequency must significantly exceed the characteristic coupling strength $\gamma$ between the bath spins and NV centers, i.e., $\Omega_{Bath}/\gamma \gg 1$, to achieve effective decoupling. Under this condition, the baths spins undergo many Rabi oscillations during the characteristic dipolar interaction time $1/\gamma$. [As a result, the dipolar interaction with the bath is incoherently averaged and the NV spin dephasing time increases.]{}
Results {#results .unnumbered}
=======
We studied three diamond samples with increasing nitrogen concentrations that are summarized in Table\[tab:tab1\]. Samples A ($[\text N] \lesssim 0.05\,$ppm) and B ($[\text N] = 0.75\,$ppm) each consist of a $^{14}$N-doped, $\approx 100\,\upmu$m-thick chemical-vapor-deposition (CVD) layer (99.99$\%~^{12}$C) deposited on top of a diamond substrate. Sample C ($[\text N] = 10\, $ppm) possesses a $ 40\,\upmu$m-thick, $^{15}$N-doped CVD layer (99.95$\%~^{12}$C) on a diamond substrate. For all three samples, the nitrogen-limited NV dephasing times can be estimated from the average dipolar interaction strength between electronic spins giving $T_{2,\text{NV-N}}^* \approx 350\,\upmu$s, $23\,\upmu$s, and $2\,\upmu$s for Samples A, B, and C, respectively. Analysis and measurements suggest that the $^{13}$C nuclear spin bath limit to $T_2^*$ is $\approx 100\,\upmu$s for Samples A and B, and $\approx 20\,\upmu$s for Sample C (for details, see Suppl.V). All samples are unirradiated and the N-to-NV conversion efficiency is $\lesssim 1\%$. Contributions from NV-NV dipolar interactions to $T_2^*$ can therefore be neglected.
The parameter regime covered by Samples A, B, and C was chosen to best illustrate the efficacy of DQ coherence magnetometry and spin bath driving.
[ccccccccccc]{}\
Sample & \[N\] & $^{13}$C & \[NV\] & $T_2^{\text{meas}}$ & $T_{2,\text{SQ}}^{*,\text{meas}}$ & $T_{2,\text{DQ}}^{*,\text{meas}}$ & $T_{2,\text{NV-N}}^{*,\text{est}}$ & $T_{2,\mathrm{NV-^{13}C}}^{*,\text{est}}$ & $T_{2,\mathrm{NV-(^{13}C+N)}}^{*,\text{est}}$ & $dM_z^{\text{meas}}/dL$\
& (ppm) & (%) & (cm$^{-3})$ & ($\upmu$s) & ($\upmu$s) & ($\upmu$s) & ($\upmu$s) & ($\upmu$s) & ($\upmu$s) & (MHz/$\upmu$m)\
\
A &$\lesssim 0.05$ & 0.01 & $\sim 3 \times 10^{12}$ & $\gtrsim 630$ & $5-12$ & 34(2) & 350 & 100 & 78 & n/a\
B & 0.75 & 0.01 & $\sim 10^{14}$ & $250-300$ & $1-10$ & 6.9(5) & 23 & 100 & 19 & 0.0028\
C & 10 & 0.05 & $\sim 6 \times 10^{15}$ & $15-18$ & $0.3-1.2$ & $0.60(2)$ & 2 & 20 & 2 & n/a\
We measured $T_2^*$ values in the SQ and DQ bases, denoted $T_{2,\text{SQ}}^*$ and $T_{2,\text{DQ}}^*$ from here on, by performing a single- or two-tone $\pi/2 - \tau - \pi/2$ Ramsey sequence, respectively (see inset Fig.\[fig:fig2\]). In both instances, the observed Ramsey signal exhibits a characteristic stretched exponential decay envelope that is modulated by the frequency detunings of the applied NV drive(s) from the NV hyperfine transitions. We fit the data to the expression $C_0 \exp\left[-(\tau / T_2^*)^p\right] \sum_{i} \cos(2 \pi f_i (\tau-\tau_{0,i}))$, where the free parameters in the fit are the maximal contrast $C_0$ at $\tau=0$, dephasing time $T_2^*$, stretched exponential parameter $p$, time-offsets $\tau_{0,i}$, and (up to) three frequencies $f_{i}$ from the NV hyperfine splittings. The $p$ value provides a phenomenological description of the decay envelope, which depends on the specific noise sources in the spin bath as well as the distribution of individual resonance lines within the NV ensemble. For a purely magnetic-noise-limited spin bath, the NV ensemble decay envelope exhibits simple exponential decay ($p=1$) [@Abragam1983; @Dobrovitski2008]; whereas a non-integer p-value ($p \neq 1$) suggests magnetic and/or strain gradient-limited NV spin ensemble dephasing.
Strain-dominated dephasing (Sample A: low nitrogen density regime) {#strain-dominated-dephasing-sample-a-low-nitrogen-density-regime .unnumbered}
------------------------------------------------------------------
Experiments on Sample A ($[\text N]\lesssim 0.05\, $ppm, $^{14}$N) probed the low nitrogen density regime. In different regions of this diamond, the measured SQ Ramsey dephasing time varies between $T_{2,\text{SQ}}^* \simeq 5 - 12\,\upmu$s, with $1<p<2$. Strikingly, even the longest measured $T_{2,\text{SQ}}^*$ is $\sim 30\times$ shorter than the calculated $T_{2, \text{NV-N}}^*$ given by the nitrogen concentration of the sample ($\gtrsim 350\,\upmu$s, see Table\[tab:tab1\]) and is approximately $10\times$ smaller than the expected SQ limit due to $0.01\%$ $^{13}$C spins ($\simeq 100\,\upmu$s). This discrepancy indicates that dipolar broadening due to paramagnetic spins is not the dominant NV dephasing mechanism. Indeed, the spatial variation in $T_{2,\text{SQ}}^*$ and low concentration of nitrogen and $^{13}$C spins suggests that crystal lattice strain inhomogeneity is the main source of NV spin ensemble dephasing in this sample. For the measured NV ensemble volume ($\sim 10^4\,\upmu\mathrm{m}^3$) and the reference strain gradient (Fig.\[fig:fig1\]c), we estimate a strain gradient limited dephasing time of $\sim 6\,\upmu$s, in reasonable agreement with the observed $T_{2,\text{SQ}}^*$.
![ [NV Ramsey measurements for low nitrogen density sample (Sample A, ${[\text N] \lesssim 0.05\,}$ ppm) at an applied bias magnetic field of ${B_0 = 2.2\,}$mT.]{} Comparison of time-domain data and resulting fit values for the NV spin ensemble $T_2^*$ for the [single quantum (SQ) coherence, $\{0,+1\}$ (blue, upper); and the double quantum (DQ) coherence, $\{+1, -1\}$ (black, lower).]{} Upper inset: Illustration of DQ Ramsey protocol with two-tone microwave (MW) pulses, where $\hat{U}_{S=1}(\pi/2)$ is the spin-1 unitary evolution operator [@Mamin2014]. For SQ measurements, a single-tone MW pulse is applied instead to generate the pseudo-spin-1/2 unitary evolution operator $\hat{U}_{S=1/2}(\pi/2)$. Lower inset: Discrete Fourier transform of the SQ (solid blue) and DQ (dashed black) Ramsey measurements with a MW drive detuned 0.4MHz from the $\{0,\pm 1\}$ transitions. NV sensor spins accumulate phase twice as quickly in the DQ basis as in the SQ basis.[]{data-label="fig:fig2"}](Figure_2.pdf)
Measurements in the DQ basis at moderate bias magnetic fields are to first order strain-insensitive, and therefore provide a means to eliminate the dominant contribution of strain to NV spin ensemble dephasing. Fig.\[fig:fig2\] shows data for $T_{2}^*$ in both the SQ and DQ bases for an example region of Sample A with SQ dephasing time $T_{2,\text{SQ}}^* = 5.8(2) \,\upmu$s and $p = 1.7(2)$. For these measurements, we applied a small $2.2\,$mT bias field parallel to one NV axis (misalignment angle $<3\degree$) to lift the $|\pm 1\rangle$ degeneracy, and optimized the magnet geometry to reduce magnetic field gradients over the sensing volume (see Suppl.VI). In the DQ basis, we find $T_{2,\text{DQ}}^* = 34(2) \upmu$s with $p=1.0(1)$, which is a $ \sim 6\times$ improvement over the measured $T_{2}^*$ in the SQ basis. We observed similar $T_{2}^*$ improvements in the DQ basis in other regions of this diamond. Our results suggest that in the low nitrogen density regime, dipolar interactions with the $^{13}$C nuclear spin bath are the primary decoherence mechanism when DQ basis measurements are employed to remove strain and temperature effects. Specifically, the measured $T_{2,\text{DQ}}^*$ and $p$ values in Sample A are consistent with the combined effect of NV dipolar interactions with (i) the $0.01\,\%$ concentration of $^{13}$C nuclear spins ($T_{2,\text{N-$^{13}$C}}^*/2 \simeq 50\,\upmu$s) and (ii) residual nitrogen spins $[\text N]\sim 0.05\,$ppm; with an estimated net effect of $T_{2,\text{DQ}}^* \simeq 39\,\upmu$s. Diamond samples with greater isotopic purity $(^{12}$C$ > 99.99\%)$ would likely yield further enhancements in $T_{2,\text{DQ}}^*$.
Strain- and dipolar-dominated dephasing (Sample B: intermediate nitrogen density regime) {#strain--and-dipolar-dominated-dephasing-sample-b-intermediate-nitrogen-density-regime .unnumbered}
----------------------------------------------------------------------------------------
Although Sample B ($[\text N]=0.75\,$ppm, $^{14}$N) contains more than an order of magnitude higher nitrogen spin concentration than Sample A ($[\text N]\lesssim 0.05\, $ppm), we observed SQ Ramsey dephasing times $T_{2,\text{SQ}}^{*} \simeq 1 - 10\, \upmu$s in different regions of Sample B, which are similar to the results from Sample A. We conclude that strain inhomogeneities are also a significant contributor to NV spin ensemble dephasing in Sample B . Comparative measurements of $T_{2}^*$ in both the SQ and DQ bases yield a more moderate increase in $T_{2,\text{DQ}}^*$ for Sample B than for Sample A. Example Ramsey measurements of Sample B are displayed in Fig.\[fig:fig3\], showing $T_{2,\text{SQ}}^*$ = $1.80(6)\,\upmu$s in the SQ basis increasing to $T_{2,\text{DQ}}^*$ = $6.9(5)\,\upmu$s in the DQ basis, a $\sim 4\times$ extension. The observed $T_{2,\text{DQ}}^*$ in Sample B approaches the expected limit set by dipolar coupling of NV spins to residual nitrogen spins in the diamond ($T_{2,\text{N-NV}}^*/2 \simeq 12\,\upmu$s), but is still well below the expected DQ limit due to $0.01\,\%$ $^{13}$C nuclear spins ($\simeq 50\,\upmu$s).
![ [NV Ramsey measurements for intermediate nitrogen density sample (Sample B, (${[\text N]=0.75\,}$ ppm) at an applied bias magnetic field of ${B_0 = 8.5\,}$mT.]{} Comparison of time-domain data and resulting fit values for the NV spin ensemble $T_2^*$ for the [single quantum (SQ) coherence $\{0,+1\}$ (blue, 1^st^ from top); the SQ coherence with spin-bath drive (blue, 2^nd^ from top); the DQ coherence with no drive (black, 3^rd^ from top); and the DQ coherence with spin-bath drive (black, 4^th^ from top). There is a $16.2\times$ improvement of $T_2^*$ with spin-bath drive when the DQ coherence is used for sensing compared to SQ with no drive. Inset: Two-tone NV Ramsey protocol with applied spin-bath bath drive resonant with nitrogen spins.]{} []{data-label="fig:fig3"}](Figure_3.pdf)
Measuring NV Ramsey decay in both the SQ and DQ bases while driving the nitrogen spins, either via application of CW or pulsed RF fields [@DeLange2012; @Knowles2014], is effective in revealing the electronic spin bath contribution to NV ensemble dephasing. With continuous drive fields of Rabi frequency $\Omega_N = 2\,$MHz applied to nitrogen spin resonances $1-6$, $i$, and $ii$ (see Fig. \[fig:fig1\]d), we find that $T_{2,\text{SQ+Drive}}^* = 1.94(6)\,\upmu$s, which only marginally exceeds $T_{2,\text{SQ}}^* = 1.80(6)\,\upmu$s. This result is consistent with NV ensemble SQ dephasing being dominated by strain gradients in Sample B, rendering spin bath driving ineffective in the SQ basis. In contrast, DQ Ramsey measurements exhibit a significant additional increase in $T_2^*$ when the bath drive is applied, improving from $T_{2,\text{DQ}}^* = 6.9(5) \,\upmu$s to $T_{2,\text{DQ+Drive}}^* = 29.2(7)\,\upmu$s. This $\sim 16\times$ improvement over $T_{2,\text{SQ}}^*$ confirms that, for Sample B without spin bath drive, dipolar interactions with the nitrogen spin bath are the dominant mechanism of NV spin ensemble dephasing in the DQ basis. Note that the NV dephasing time for Sample B with DQ plus spin bath drive is only slightly below that for Sample A with DQ alone ($\approx 34\,\upmu$s). We attribute this $T_2^*$ limit in Sample B primarily to NV dipolar interactions with $0.01\%$ $^{13}$C nuclear spins. There is also an additional small contribution from magnetic field gradients over the detection volume ($\sim 10^4\,\upmu\mathrm{m}^3$) due to the four times larger applied bias field ($B_0$ = $8.5\,$mT), relative to Sample A, which was used in Sample B to resolve the nitrogen ESR spectral features (see Suppl. TableS3 and S4). We obtained similar extensions of $T_2^*$ using pulsed driving of the nitrogen bath spins (see Supp.X). We also characterized the efficacy of CW spin bath driving for increasing $T_2^*$ in both the SQ and DQ bases (see Fig.\[fig:fig4\]a). While $T_{2,\text{SQ}}^*$ remains approximately constant with varying Rabi drive frequency $\Omega_{N}$, $T_{2,\text{DQ}}^*$ exhibits an initial rapid increase and saturates at $T_{2,\text{DQ}}^* \approx 27\,\upmu$s for $\Omega_{N} \gtrsim 1\,$MHz (only resonances $1-6$ are driven here). To explain the observed trend, we introduce a model that distinguishes between (i) NV spin ensemble dephasing due to nitrogen bath spins, which depends upon bath drive strength $\Omega_\text{N}$, and (ii) dephasing from drive-independent sources (including strain and $^{13}$C spins), $$\label{eqn:t2stardrive}
1/T_{2}^* = 1/T_{2,\text{NV-N}}^*(\Omega_\text{N}) + 1/T_{2,\text{other}}^*.$$ Taking the coherent dynamics of the bath drive into account (see Suppl.VIII), the data is well described by the functional form $$\label{eqn:t2stardrive2}
1/T_{2,\text{NV-N}}^*(\Omega_\text{N}) = \Delta m \times \gamma_\text{NV-N} \frac{\delta_\text{N}^2}{\delta_\text{N}^2 + \Omega_\text{N}^2},$$ where $\Delta m = 1 (2)$ is the change in spin quantum number in the SQ (DQ) basis and $\delta_\text{N} = \gamma_\text{N}/2\pi$ is the Lorentzian linewidth (half width at half max) of the nitrogen spin resonances measured through DEER ESR (Fig.\[fig:fig1\]d). Although we find that NV and nitrogen spins have comparable $T_2^*$ ($\gamma_\text{NV-N} \approx \gamma_\text{N}$, see Suppl.XI), the effective linewidth $\delta_\text{N}$ relevant for bath driving is increased due to imperfect overlap of the nitrogen spin resonances caused by a small misalignment angle of the applied bias magnetic field.
Using the NV-N dipolar estimate for Sample B, $\gamma_\text{NV-N} \approx 2\pi \times 7\,$kHz, $\delta_\text{N} \approx 80\,$kHz extracted from DEER measurements (Suppl.XI), and a saturation value of $T_{2,\text{other}}^* \approx 27\,\upmu$s, we combine Eqns.\[eqn:t2stardrive\] and\[eqn:t2stardrive2\] and plot the calculated $T_{2}^*$ as a function of $\Omega_\text{N}$ in Fig.\[fig:fig4\]a (black, dashed line). The good agreement between the model and our data in the DQ basis suggests that Eqns.\[eqn:t2stardrive\] and\[eqn:t2stardrive2\] capture the dependence of $T_{2}^*$ on drive field magnitude (i.e., Rabi frequency). Alternatively, we fit the model to the DQ data (red, solid line) and extract $\gamma_\text{NV-N}^{fit} = 2\pi \times 9.3(2)$kHz and $\delta_\text{N}^{fit} = 60(3)\,$kHz, in reasonable agreement with our estimated parameters. In summary, the results from Sample B show that the combination of spin bath driving and sensing in the DQ basis suppresses inhomogeneous NV ensemble dephasing due to both interactions with the nitrogen spin bath and strain-gradients. Similar to Sample A, further enhancement in $T_{2}^*$ could be achieved with improved isotopic purity, as well as reduced magnetic-gradients due to the applied magnetic bias field.
![ Application of quantum control techniques to extend NV spin ensemble dephasing time (${T_2^*}$) and increase DC magnetic field sensitivity. (a) Ramsey measurements of $T_2^*$ in the single quantum (SQ, blue) and double quantum (DQ, black) bases for different spin-bath drive strengths (Rabi frequencies) for Sample B ($\mathrm [\text N] = 0.75\,$ppm) at $B_0 = 8.5\,$mT. Black dashed-dotted line is calculated from a model of NV spins that are dipolar-coupled to a multi-component spin bath (Eqn. \[eqn:t2stardrive\]). The red solid line is a fit of the model to the $T_2^*$ data (see main text for details). [(b)]{} Same as (a) but for Sample C ($[\text N]=10\,$ppm) and $B_0 = 10.3\,$mT. [(c)]{} Measured $T_{2,\text{N-NV}}^* \equiv 2\times T_{2,\text{DQ}}^*$ as a function of nitrogen concentration for Samples B, C, D, E. [Samples were selected to have a predominately electronic nitrogen (P1) spin bath using DEER ESR measurements.]{} The black dashed-dotted line is the dipolar-interaction-estimated dependence of $T_2^*$ on nitrogen concentration (Suppl.V). We fit the data using an orthogonal-distance-regression routine to account for the uncertainties in \[N\] and $T_2^*$. A fit to the form $1/T_2^* = A_\text{NV-N} [\text N]$ yields $A_\text{N-NV} = 2\pi \times 16.6(2.6)\,$kHz/ppm \[$1/A_\text{NV-N} = 9.6(1.8)\,\upmu \mathrm{s}\,\cdot\,$ppm\]. The red shaded region indicates the 95% standard error of the fit value for $A_\text{N-NV}$. The black dashed line is the expected scaling extracted from numerical simulations using a second-moment analysis of the NV ensemble ESR linewidth (see text for details). [(d)]{} Measured Ramsey DC magnetometry signal $S \propto C \sin(\phi(\tau))$ for Sample B, in the SQ and DQ bases, as well as the DQ [sub-basis]{} with spin-bath drive (see main text for details). There is a $36\times$ faster oscillation in the DQ [sub-basis]{} with spin-bath drive compared to SQ with no drive. This greatly enhanced DC magnetic field sensitivity is a direct result of the extended $T_2^*$, with the sensitivity enhancement given by $2\times \sqrt{\tau_\text{DQ+Drive}/\tau_\text{SQ}}$ at equal contrast. The slight decrease in observed contrast in the DQ + drive case for $|B_{DC}| > 0.05\,$mT is a result of changes in the Zeeman resonance frequencies of the nitrogen spins due to the applied test field $B_{DC}$, which was not corrected for in these measurements. []{data-label="fig:fig4"}](Figure_4_big.pdf)
Dipolar-dominated dephasing (Sample C: high nitrogen density regime) {#dipolar-dominated-dephasing-sample-c-high-nitrogen-density-regime .unnumbered}
--------------------------------------------------------------------
Spin bath driving results for Sample C ($[\text N]=10\,$ppm, $^{15}$N) are shown in Fig.\[fig:fig4\]b. At this high nitrogen density, interactions with the nitrogen bath dominate NV spin ensemble dephasing, and $T_{2,\text{SQ}}^*$ and $T_{2,\text{DQ}}^*$ both exhibit a clear dependence on spin bath drive strength $\Omega_\text{N}$. With no drive ($\Omega_\text{N}=0$), we measured $T_{2,\text{DQ}}^* \approx T_{2,\text{SQ}}^*/2$, in agreement with dephasing dominated by a paramagnetic spin environment and the twice higher precession rate in the DQ basis [@Fang2013; @Mamin2014; @MacQuarrie2015]. Note that this result is in contrast to the observed DQ basis enhancement of $T_{2}^*$ at lower nitrogen density for Samples A and B (Figs.\[fig:fig2\] and\[fig:fig3\]). We also find that $T_2^*$ in Sample C increases more rapidly as a function of spin bath drive amplitude in the DQ basis than in the SQ basis, such that $T_{2,\text{DQ}}^*$ surpasses $T_{2,\text{SQ}}^*$ with sufficient spin bath drive strength. We attribute the $T_2^*$-limit in the SQ basis ($\simeq 1.8\,\upmu$s) to strain inhomogeneities in this sample, whereas the longest observed $T_2^*$ in the DQ basis ($\simeq 3.4\,\upmu$s) is in agreement with dephasing due to the $0.05\%\,^{13}$C and 0.5ppm residual $^{14}$N spin impurities. The latter were incorporated during growth of this $^{15}$N sample (see Suppl.TableS5). In Fig.\[fig:fig4\]c we plot $T_{2,\text{NV-N}}^* \equiv 2\times T_{2,\text{DQ}}^*$ versus sample nitrogen concentration $[\text N]$ [to account for the twice faster dephasing of the DQ coherence.]{} To improve the range of $[\text N]$ coverage, we include DQ data for additional diamonds, Samples D ($[\text N] = 3\,$ppm) and E ($[\text N]= 48\,$ppm). To our knowledge, the dependence of the NV spin ensemble dephasing time on $[\text N]$ has not previously been experimentally reported. Fitting the data to the function $1/T_{2,\text{NV-N}}^* = A_\text{NV-N}\cdot[\text N]$ (red shaded region), we find the characteristic NV-N interaction strength for NV ensembles to be $A_\text{NV-N} = 2\pi \times 16.6(2.6)\,$kHz/ppm \[$1/A_\text{N-NV} = 9.6(1.8)\,\upmu\mathrm{s}\cdot\mathrm{ppm}$\] [in the SQ sub-basis]{}. This value is about $1.8\times$ larger than the dipolar-estimate $\gamma_{\text{e-e}} = 2\pi \times 9.1\,$kHz/ppm (black dashed-dotted line), which is used above in estimates of NV dephasing due to the nitrogen spin bath. We also performed numerical spin bath simulations for the NV-N spin system and determine the second moment of the dipolar-broadened single NV ESR linewidth [@Abragam1983 Ch. III and IV]. By simulating $10^4$ random spin bath configurations, we extract the ensemble-averaged dephasing time from the distribution of the single NV linewidths [@Dobrovitski2008]. The results of this simulation (black dashed line) are in excellent agreement with the experiment and confirm the validity of our obtained scaling for $T_{2,\text{NV-N}}^*(\mathrm N)$. Additional details of the simulation are provided in Ref. [@Bauch2017].
Ramsey DC Magnetic Field Sensing {#ramsey-dc-magnetic-field-sensing .unnumbered}
--------------------------------
We demonstrated that combining the two quantum control techniques can greatly improve the sensitivity of Ramsey DC magnetometry. Fig.\[fig:fig4\]d compares the accumulated phase for SQ, DQ, and DQ plus spin bath drive measurements of a tunable static magnetic field of amplitude $B_{DC}$, for Sample B. Sweeping $B_{DC}$ leads to a characteristic observed oscillation of the Ramsey signal $S \propto C \sin(\phi)$, where $C = C_0 \exp[-\left(\tau/T_2^*\right)^p]$ is the measurement contrast and $\phi = \Delta m \times \gamma_{NV} B_{DC} \tau$ is the accumulated phase during the free precession interval $\tau \approx T_2^*$. Choosing $\tau_{\text{SQ}}=1.308\,\upmu$s and $\tau_\text{DQ+Drive}=23.99\,\upmu$s (see Suppl.XII), we find a $36.3(1.9)\times$ faster oscillation period (at equal measurement contrast) when DQ and spin bath driving are both employed, compared to a SQ measurement. This enhancement in phase accumulation, and hence DC magnetic field sensitivity, agrees well with the expected improvement ($2 \times \tau_\text{DQ+Drive}/\tau_\text{{SQ}}$ = 36.7).
Discussion {#discussion .unnumbered}
==========
Our results (i) characterize the dominant spin dephasing mechanisms for NV ensembles in bulk diamond (strain and interactions with the paramagnetic spin bath); and (ii) demonstrate that the combination of DQ magnetometry and spin bath driving can greatly extend the NV spin ensemble $T_2^*$. For example, in Sample B we find that these quantum control techniques, when combined, provide a $16.2\times$ improvement in $T_2^*$. Operation in the DQ basis protects against common-mode inhomogeneities and enables an extension of $T_2^*$ for samples with $[\text N] \lesssim 1$ppm. In such samples, strain inhomogeneities are found to be the main causes of NV spin ensemble dephasing. In samples with higher N concentration ($[\text N] \gtrsim 1$ppm), spin bath driving in combination with DQ sensing provides an increase of the NV ensemble $T_2^*$ by decoupling paramagnetic nitrogen and other electronic dark spins from the NV spins. Our results suggest that quantum control techniques may allow the NV ensemble $T_{2}^*$ to approach the bare Hahn echo coherence time $T_2$. Note that spin bath driving may also be used to enhance the NV ensemble $T_2$ in Hahn echo, dynamical decoupling [@DeLange2010; @Pham2012], and spectral decomposition experimental protocols [@Bar-Gill2012]. Furthermore, we showed that the combination of DQ magnetometry and spin bath driving allows improved DC Ramsey magnetic field sensing. The relative enhancement in photon-shot-noise-limited sensitivity (neglecting experimental overhead time) is quantified by $2 \times\sqrt{\zeta}$, where the factor of two accounts for the enhanced gyromagnetic ratio in the DQ basis and $\zeta \equiv T_{2,\text{DQ}}^*/T_{2,\text{SQ}}^*$ is the ratio of maximally achieved $T_2^*$ in the DQ basis (with spin bath drive when advantageous) and non-driven $T_2^*$ in the SQ basis. For Samples A, B, and C, we calculate $2 \times \sqrt{\zeta} = 5.2\times$, $8.1\times$, and $3.9\times$, respectively, using our experimental values. In practice, increasing $T_{2}^*$ also decreases the fractional overhead time associated with NV optical initialization and readout, resulting in even greater DC magnetic field sensitivity improvements and an approximately linear sensitivity enhancement with $\zeta$ (see Suppl.XII). We expect that these quantum control techniques will remain effective when integrated with other approaches to optimize NV ensemble magnetic field sensitivity, such as high laser power and good N-to-NV conversion efficiency. In particular, conversion efficiencies of $1-30\,\%$ have been reported for NV ensemble measurements [@Acosta2010; @Wolf2015; @Grezes2015; @Barry2016], such that the nitrogen spin bath continues to be a relevant spin dephasing mechanism. There are multiple avenues for further improvement in NV ensemble $T_2^*$ and DC magnetic field sensitivity, beyond the gains demonstrated in this work. First, the $^{13}$C limitation to $T_2^*$, observed for all samples, can be mitigated via improved isotopic purity (\[$^{12}$C\] $>99.99\,\%$); or possibly through driving of the nuclear spin bath [@London2013]. Second, more efficient RF delivery will enable faster spin bath driving (higher Rabi drive frequency $\Omega_\text{N}$), which will be critical for decoupling denser nitrogen baths and thereby extending $T_2^* \propto \Omega_\text{N}^2/\delta_{\text N}^2 \propto \Omega_\text{N}^2/[\text N]^2$ (see Eqn. \[eqn:t2stardrive2\]). Third, short NV ensemble $T_2^*$ times have so far prevented effective utilization of more exotic readout techniques, e.g., involving quantum logic [@Jiang2009; @Neumann2010; @Lovchinsky2016] or spin-to-charge-conversion [@Shields2015; @Jaskula2017]. Such methods offer greatly improved NV spin-state readout fidelity but introduce substantial overhead time, typically requiring tens to hundreds of microseconds per readout operation. The NV spin ensemble dephasing times demonstrated in this work ($T_2^*\gtrsim 20\,\upmu$s) may allow effective application of these readout schemes, which only offer sensitivity improvements when the sequence sensing time (set by $T_2^*$ for DC sensing) is comparable to the added overhead time. We note that the NV ensemble $T_2^*$ values obtained in this work are the longest for any electronic solid-state spin system at room temperature (see comparison Fig.S2) suggesting that state-of-the-art DC magnetic field sensitivity [@Barry2016; @Chatzidrosos2017] may be increased to $\sim 100\,$fT/$\sqrt{\text{Hz}}$ for optimized NV ensembles in a diamond sensing volume $\sim (100\, \upmu$m)$^3$ (see discussion on NV ensemble DC magnetic field sensitivity optimization in Barry et al. [@Barry2016]). In conclusion, DQ magnetometry in combination with spin bath driving allows for order-of-magnitude increase in the NV ensemble $T_2^*$ in diamond, providing a clear path to ultra-high sensitivity DC magnetometry with NV ensemble coherence times approaching $T_2$.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank David Le Sage for his initial contributions to this project. We thank Joonhee Choi, Soonwon Choi, and Renate Landig for fruitful discussions. This material is based upon work supported by, or in part by, the United States Army Research Laboratory and the United States Army Research Office under Grant No. W911NF1510548; the National Science Foundation Electronics, Photonics and Magnetic Devices (EPMD), Physics of Living Systems (PoLS), and Integrated NSF Support Promoting Interdisciplinary Research and Education (INSPIRE) programs under Grants No. ECCS-1408075, PHY-1504610, and EAR-1647504, respectively; and Lockheed Martin under award A32198. This work was performed in part at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the National Science Foundation under NSF award no. 1541959. CNS is part of Harvard University. P. K. acknowledges support from the Intelligence Community Postdoctoral Research Fellowship Program. J. M. S. was supported by a Fannie and John Hertz Foundation Graduate Fellowship and a National Science Foundation (NSF) Graduate Research Fellowship under Grant 1122374.
Author contributions statement {#author-contributions-statement .unnumbered}
==============================
E. B., C. A. H., J. M. S., M. J. T., J. F. B., and R. L. W. conceived the experiments, C. A. H. and E. B. conducted the experiments and analyzed the results. P. K. provided the strain analysis. E. B. and S. S. provided the spin bath simulation. All authors contributed to and reviewed the manuscript. R. L. W. supervised the work.
[^1]: these authors contributed equally to this work
[^2]: these authors contributed equally to this work
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abstract: 'We explore the postulates of string no-scale supergravity in the context of free-fermionic string models. The requirements of vanishing vacuum energy, flat directions of the scalar potential, and stable no-scale mechanism impose strong restrictions on possible string no-scale models, which must possess only two or three moduli, and a constrained massless spectrum. The soft-supersymmetry-breaking parameters involving all twisted and untwisted fields are given explicitly. This class of models contain no free parameters, , in principle all supersymmetric particle masses and interactions are completely determined. A computerized search for free-fermionic models with the desired properties yields a candidate $SU(5)\times U(1)$ model containing extra (,) matter representations that allow gauge coupling unification at the string scale. Our candidate model possesses a bening non-universal assignment of supersymmetry breaking scalar masses, which may have interesting low-energy experimental consequences.'
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0.5cm [*$^1$Center for Theoretical Physics, Department of Physics, Texas A&M University\
*]{} [*College Station, TX 77843–4242, USA\
*]{} [*$^2$Astroparticle Physics Group, Houston Advanced Research Center (HARC)\
*]{} [*The Mitchell Campus, The Woodlands, TX 77381, USA\
*]{} [*$^3$CERN Theory Division, 1211 Geneva 23, Switzerland\
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[CTP-TAMU-60/94]{}\
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February 1995
Introduction {#Introduction}
============
Experiments at LEP and the Tevatron have given the strongest to-date support to the Standard Model of the strong and electroweak interactions. Yet despite all this experimental evidence, physicists believe that this model is incomplete. One possible completion of the Standard Model is embedded in the physics of supersymmetry. In fact, the same experimental evidence confirming the Standard Model, can also be used to support its supersymmetric extension. This can be seen through the unification of the gauge couplings at very high energies in supersymmetric models, but perhaps more pervasively from the fact that supersymmetric models have built-in mechanisms that make them look almost identical to the Standard Model at presently available facilities. This extreme similarity with the Standard Model occurs at the tree-level for energies below the threshold for production of supersymmetric particles, and also at one-loop if the supersymmetric mass scales exceed the electroweak scale. This similarity is not totally devoid of predictivity, since in supersymmetric models the lightest Higgs boson is expected to be light ($m_h\sim M_Z$). In view of these facts, attention has turned strongly towards supersymmetric models, in particular those that can be understood as low energy limits of more fundamental theories, such as grand unification, supergravity, and superstrings. These models are highly predictive, with all supersymmetric physics typically dependent on four or less parameters.
Four-parameter supersymmetric models are obtained as low-energy effective supergravity models with assumed universal soft-supersymmetry-breaking terms: the scalar mass ($m_0$), the gaugino mass ($m_{1/2}$), the trilinear scalar coupling ($A$), and (at low energies) the ratio of Higgs vacuum expectation values ($\tan\beta$). Most phenomenological analyses are content with exploring such a four-dimensional parameter space. However, theoretically speaking it is not clear that all possible combinations of these parameters are consistent, since in a specific supergravity model, , one specified by the Kähler function $G$ and the gauge kinetic function $f$, the parameters $m_0,m_{1/2},A$ can be explicitly calculated in terms of the gravitino mass ($m_{3/2}$). Guidance in this matter has come from string-inspired choices for $G$ and $f$, which lead to simple values for the ratios $m_0/m_{1/2}$ and $A/m_{1/2}$, and thus to two-parameter supersymmetric models.[^1]
Despite this great reduction in the model parameters, several questions remain unanswered: (i) can one construct an explicit string-derived model where the various ratios of soft supersymmetry breaking terms are calculated, and at the same time the usual low-energy phenomenology is explained? (ii) does this model possess a sufficiently suppressed cosmological constant? (iii) how is the scale of supersymmetry breaking ($m_{3/2}$) determined in such model? This model would be a “no-parameter" model.
No-scale supergravity [@LN] provides satisfactory answers to the latter two questions, , vanishing cosmological constant (at the tree level) and dynamical determination of $m_{3/2}$ via the no-scale mechanism. [*String no-scale supergravity*]{} is postulated to be the subset of string models which can provide satisfactory answers to all three questions. This subset is rather restricted, since in practice it is seen that most known string models [*do not*]{} obey the postulates of string no-scale supergravity.
Our purpose in the present paper is to explore the postulates of string no-scale supergravity in the context of fermionic string models. Our investigations lead to a set of constraints on the spectrum and interactions in realistic string models, as well as to novel predictions for the soft-supersymmetry-breaking parameters. We are also able to provide an existence proof that realistic string no-scale supergravity models do exist. A search for models of this type with the gauge group $SU(5)\times U(1)$ turns up a rather interesting phenomenon: among the class of models which we have explored, a necessary condition to satisfy the postulates of no-scale supergravity appears to be the existence in the spectrum of extra (,) matter representations that allow gauge coupling unification at the string scale.
This paper is organized as follows. In Sec. \[No-scale\] we summarize the postulates of no-scale supergravity and the no-scale mechanism. In Sec. \[Fermionic\] we discuss the Kähler potential and the superpotential in free-fermionic string models, and compute the vacuum energy, and the quantity ${\rm Str}\,{\cal M}^2$. We also discuss the conditions under which these quantities would vanish, as required in no-scale models. In Sec. \[Soft\] we compute the soft-supersymmetry-breaking parameters, including the Goldstino composition, the gaugino and scalar masses and the $A$ terms. We also discuss the origin of the $\mu$ term and the associated parameter $B$. In Sec. \[Field\] we discuss the normalization of the fields and how these affect the observable Yukawa couplings. In Sec. \[Possible\] we perform a search for realistic string no-scale free-fermionic models, and present evidence for the conjecture mentioned in the previous paragraph. We also study the supersymmetry-breaking parameters in the candidate model found. Finally, in Sec. \[Conclusions\] we summarize our conclusions, in Appendix \[Redefinitions\] we collect some details about the transformation between the string and supergravity bases, and in Appendix \[AppB\] we give details of the calculation of the twisted sector Kähler potential in a specific model.
No-scale supergravity {#No-scale}
=====================
A supergravity theory is specified by two functions, the Kähler function $$G=K+\ln|W|^2\ ,
\label{Gdef}$$ where $K$ is the Kähler potential and $W$ the superpotential, and the gauge kinetic function $f$. All masses and interactions are explicitly calculable from these inputs. In particular, the (tree-level) scalar potential is given by $$V=e^G(G^IG_I-3)+V_D\ ,
\label{Vdef}$$ where the sum is over all scalar fields in the spectrum, $G_I=\partial_I G$, $G^I=G^{I\bar J}G_{\bar J}$, and $G^{I\bar J}$ is the inverse Kähler metric (, the transpose of the inverse of $G_{I\bar J}=K_{I\bar J}$). The second term in Eq. (\[Vdef\]) is the contribution from the $D$-terms; we will assume in what follows that this vanishes at the minimum of the potential. The scalar potential is used to determine the vacuum, its energy, and any flat directions it may have. Also, small deviations around it determine the supersymmetry-breaking masses and couplings of the scalar fields. Finally, derivatives of the Kähler function determine the supersymmetry-breaking masses of the (non-chiral) fermions, and derivatives of the gauge kinetic function determine the supersymmetry-breaking gaugino masses.
Spontaneous breakdown of supergravity induces a mass for the gravitino $$m_{3/2}=\VEV{e^{G/2}}=\VEV{e^{K/2}\,|W|}\ .
\label{m3/2def}$$ This relation shows that supersymmetry breaking can only occur if $\VEV{W}\not=0$. All soft-supersymmetry-breaking parameters are proportional to $m_{3/2}$, with typically ${\cal O}(1)$ coefficients of proportionality. Therefore, $m_{3/2}$ values are expected to be not much higher than the electroweak scale. Moreover, restoring the dimensions in Eq. (\[m3/2def\]) one sees that the right-hand-side has units of $10^{18}\GeV$ and therefore a strong suppression of $e^{K/2}$ or $\VEV{W}$ is typically necessary. Two scenarios for $\VEV{W}\not=0$ have received the most attention in the literature: gaugino condensation in the hidden sector [@GauginoConden], giving $W\sim e^{-b\pi^2/g^2}$; and string tree-level breaking via coordinate-dependent compactifications [@Coordinate], giving $W\sim1$.
No-scale supergravity is defined by three constraints on a supergravity model:
- The vacuum energy vanishes ($V_0=\VEV{V}=0$) by suitable choice of the Kähler function ($G$) [@Cremmer].
- At the minimum of the potential there are flat directions (“moduli") which leave the value of $m_{3/2}$ undetermined [@EKNI+II].
- The quantity ${\rm Str}\,{\cal M}^2$ should vanish at the minimum. This constraint protects the potential from large one-loop corrections which would otherwise force $m_{3/2}=0$ or $m_{3/2}=M_{Pl}$ [@EKNI+II].
These three constraints impose severe restrictions on the possible $G$ and $f$ functions. Particularly non-trivial is the last one, , ${\rm Str}\,{\cal
M}^2\equiv 2Qm^2_{3/2}$, with [@EKNI+II; @FKZ] $$Q=N-1-G^I(R_{I\bar J}-H_{I\bar J})G^{\bar J}\ ,
\label{Qdef}$$ where $N$ is the total number of chiral superfields, and $$\begin{aligned}
R_{I\bar J}&=&\partial_I\partial_{\bar J}\ln\det G_{M\bar N}\ ,\label{Rdef}\\
H_{I\bar J}&=&\partial_I\partial_{\bar J}\ln\det {\rm Re\,}(f_{ab})\
.\label{Hdef}\end{aligned}$$
If the above three conditions are satisfied, the low-energy theory, obtained by renormalization-group evolution from the Planck scale down to the electroweak scale, will be undetermined to the extent that $m_{3/2}$ is undetermined, as it depends on the undetermined moduli vacuum expectation values (VEVs). The low-energy one-loop effective potential ($V_{\rm eff}$) then depends on the usual Higgs fields, as well as the moduli fields. The [*no-scale mechanism*]{} [@Lahanas] consists of minimizing this potential with respect to all these fields, thus determining the Higgs [*and*]{} moduli vacuum expectation values. The thusly determined moduli VEVs then determine $m_{3/2}$, and therefore all of the supersymmetry breaking masses.
The no-scale mechanism has an additional unsuspected consequence: it may solve the strong CP problem [@thetaQCD]. Indeed, the equally undetermined imaginary parts of the moduli fields leave the $\theta_{\rm QCD}$ parameter undetermined, , the potential in the imaginary directions is also flat.[^2] According to the usual argument, non-perturbative QCD dynamics at low energies determines $\theta_{\rm QCD}=0$, which in our language corresponds to lifting the imaginary flat directions, giving zero VEVs to the corresponding fields.
In practice, this procedure is subtle and complicated by the existence of a remnant vacuum energy term at high energies. This field-independent term ($Cm_{3/2}^4$) needs to be added to the low-energy effective potential to ensure its renormalization-scale independence [@aspects; @KPZ; @No-scale], , $$V_{\rm eff}=V_{\rm tree}+{1\over64\pi^2}{\rm Str}\,{\cal M}^4\left(\ln{{\cal
M}^2\over Q^2}-{3\over2}\right)-Cm_{3/2}^4\ ,
\label{V1def}$$ where $V_{\rm tree}$ is the tree-level Higgs potential. It is worth mentioning that just as the usual radiative breaking mechanism (, minimization of $V_{\rm eff}$ with respect to the Higgs fields) does not always work, the no-scale mechanism may also not work. This happens when the effective potential does not have a good minimum in the moduli directions. In the case of a single modulus field (as we discuss below), it can be shown that a necessary condition for a good minimum is ${\rm Str}\,{\cal M}^4>0$ [@No-scale]. If the explicit $m_{3/2}$-dependent contribution to ${\rm Str}\,{\cal M}^4$ is negative, then [@No-scale] $${\rm Str}\,{\cal M}^4>0\ \Longrightarrow\ {m_{3/2}\over m_{\tilde q}}\lsim{\cal
O}(1)\ ,
\label{Str>0}$$ which imposes restrictions on the allowed low-energy parameter space.
Fermionic string models {#Fermionic}
=======================
The discussion in the previous section can be applied to any supergravity or superstring model. However, the requirement of ${\rm Str}\,{\cal M}^2=0$ can only be investigated if the full spectrum of the model is known, as in string models. We are interested in exploring the three postulates of string no-scale supergravity in the context of string models built within the free-fermionic formulation [@FFF]. Our motivation for such choice is that fairly realistic models already exist in this construction [@revamped; @others; @search; @Moscow], and we would like to know whether this class of models satisfies the postulates, or what constraints may need to be imposed on model-building so that these postulates are satisfied.
Generalities {#Generalities}
------------
All of the level-one free-fermionic models built to date have been based on the simplest supersymmetry-generating basis vector $S$. This choice is not unique, but should suffice for our present purposes of investigating the viability of no-scale supergravity in free-fermionic models. In this class of models, all states in the spectrum fall into three [*sets*]{}, depending on the quantum numbers they carry with respect to some internal symmetries of the two-dimensional world-sheet theory [@KLN]. The spectrum further divides itself into two sectors: untwisted and twisted. More specifically, all matter fields carry charges under three world-sheet $U(1)$ currents. The sum of these three currents provides the additional current which extends the manifest $N=1$ world-sheet supersymmetry to $N=2$ world-sheet supersymmetry [@KLN], as required for $N=1$ space-time supersymmetry to exist. The scalar components of untwisted (or Neveu-Schwarz) matter superfields carry one of three possible types of charges, whereas the twisted matter superfields carry two of them, , $$\begin{tabular}{lcc}
&Untwisted&Twisted\\
First set&$\{1,0,0\}$&$\{0,\half,\half\}$\\
Second set&$\{0,1,0\}$&$\{\half,0,\half\}$\\
Third set&$\{0,0,1\}$&$\{\half,\half,0\}$
\end{tabular}
\label{charges}$$ The charges of their fermionic partners are obtained by a uniform shift of $\{-\half,-\half,-\half\}$. From these charge assignments it follows immediately what kind of cubic superpotential couplings are allowed in a model. Indeed, the sum of the charges of the three fields in question must be $\{1,1,1\}$; charge conservation follows from the two required shifts by $\{-\half,-\half,-\half\}$, since a cubic coupling contains two fermionic fields and one scalar field. Thus, one gets five types of cubic couplings $$\begin{tabular}{ccc}
$U^{(1)}\,U^{(2)}\,U^{(3)}$&$U^{(1)}\,T^{(1)}\,T^{(1)}$&
$T^{(1)}\,T^{(2)}\,T^{(3)}$\\
&$U^{(2)}\,T^{(2)}\,T^{(2)}$&\\
&$U^{(3)}\,T^{(3)}\,T^{(3)}$&\\
\end{tabular}
\label{couplings}$$ where $U^{(I)}$ \[$T^{(I)}$\] denotes a generic untwisted (twisted) field belonging to the $I$-th set.
Several features of the Kähler function for the untwisted sector of such models had been known for some time [@oldUT], and have been recently further clarified, extended, and applied to specific models in Ref. [@LNY94]. The corresponding contributions from the twisted sector have been known for some time in simple models [@oldTS], and have been only recently calculated in realistic models [@LNY95]. Abstracting all that is known about free-fermionic models, we write $$G=-\ln(S+\bar S)+\sum_{I=1,2,3} K_{(I)}+K_{\rm TS}+\ln|W|^2\ ,
\label{GFFF}$$ with $$K_{(I)}=-\ln\left[1-\sum_i^{n_I}
\alpha_i\bar\alpha_i+\coeff{1}{4}\left(\sum_i^{n_I}\alpha_i^2\right)
\left(\sum_i^{n_I}\bar\alpha_i^2\right)
\right]\ ,
\label{Kdef}$$ where $n_I$ represents the number of untwisted fields in set $I$, and set-indices $I=1,2,3$ on the $\alpha_i$ (, $\alpha^{(I)}_i$) are understood. (The twisted sector contribution $K_{\rm TS}$ will be addressed below.) The Kähler function in Eqs. (\[GFFF\]),(\[Kdef\]) is written in the “string basis". For purposes of low-energy effective supergravity analyses, it is more convenient to make suitable field redefinitions of the $\alpha_i$ to exhibit the moduli fields that may be present in the spectrum, , to go from the “string basis" to the “supergravity basis". In the class of fermionic models which we consider here, three possibilities for the untwisted moduli space of any of the sets were identified in Ref. [@LNY94]:
\(i) $[SU(1,1)/U(1)]^2$, with two moduli fields denoted by $\tau_1,\tau_2$ (a “$\tau_1,\tau_2$ set"). [^3]
\(ii) $SU(1,1)/U(1)$ with one modulus field denoted by $\tau$ (a “$\tau$ set").
\(iii) No moduli at all (an “$\alpha$ set").
Whichever of the three possibilities may be realized for a given set depends on the choice of basis and GSO projections of the fermionic model. In what follows, if a set has any modular symmetry at all, we perform a field redefinition of the fields in that set which leaves the Kähler function ($G$) unchanged. (Details of these manipulations are given in Appendix A.) The redefined $K_{(I)}$ are given by:
- $\tau_1,\tau_2$ set: $$%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
K=-\ln\left[(\tau_1+\bar\tau_1)(\tau_2+\bar\tau_2)-\sum_i^{n_\phi}(\phi_i+\bar\phi_i)^2\right]\ ,
\label{TUdef}$$ where $n_\phi=n_I-2$, since two of the $\alpha_i$ are transformed into the moduli $\tau_1,\tau_2$. The scalar fields parametrize the Kähler manifold $SO(2,2+n_\phi)/SO(2)\times SO(2+n_\phi)$, which has as a subspace the moduli space $SO(2,2)/SO(2)\times SO(2)\approx[SU(1,1)/U(1)]^2$.
- $\tau$ set: $$K=-\ln\left[(\tau+\bar \tau)^2-\sum_i^{n_\psi}(\psi_i+\bar\psi_i)^2\right]\ ,
\label{taudef}$$ where $n_\psi=n_I-1$, since one of the $\alpha_i$ is transformed into the modulus $\tau$. The scalar fields parametrize the Kähler manifold $SO(2,1+n_\psi)/SO(2)\times SO(1+n_\psi)$,[^4] which has as a subspace the moduli space $SO(2,1)/SO(2)\approx SU(1,1)/U(1)$.
The modular symmetries exhibited above can be extended from the Kähler potential ($K$) to the whole Kähler function ($G=K+\ln|W|^2$) if the matter fields have suitable transformation properties under the modular symmetries [@LNY94]. In fact, the superpotential should transform as a modular form of weight $-1$. For instance, if the modulus field in question belongs to the first untwisted set, then the modular weights of the matter fields are the negative of the first world-sheet $U(1)$ charge in Eq. (\[charges\]), , $$\begin{tabular}{crccr}
$U^{(1)}$& $-1$ &\qquad\qquad &$T^{(1)}$& $0$\\
$U^{(2)}$& $0$ & &$T^{(2)}$& $-\coeff{1}{2}$\\
$U^{(3)}$& $0$ & &$T^{(3)}$& $-\coeff{1}{2}$\\
\end{tabular}
\label{ModularWeights}$$ With this modular weight assignment one can verify that all cubic superpotential couplings in Eq. (\[couplings\]) have modular weight $-1$. A similar argument holds for moduli in the other sets. When quartic or higher-order superpotential couplings are considered, the resulting modular weight imbalance has to be compensated by the insertion of suitable powers of the Dedekind eta function [@modinv].
It is important to reiterate that modular symmetries inferred from the Kähler potential must be respected by the whole Kähler function. If presumed moduli fields appear in the cubic superpotential, then the symmetry is explicitly broken and the presumed moduli are to be discarded. This phenomenon is quite common in free-fermionic models and has the beneficial effect of reducing the number of moduli in the model. Therefore, in what follows, when discussing the number and type of moduli and their impact on the vacuum energy and other properties of the models, we refer to fields which have been indentified as untwisted moduli [*and*]{} that have no superpotential couplings.
The twisted sector contribution to the Kähler potential ($K_{\rm TS}$) has been obtained some time ago for simple models with a specific type and number of twisted sectors [@oldTS]. It turns out that the result found in Ref. [@oldTS] in fact applies to realistic free-fermionic models with a large number of twisted sectors [@LNY95] (see Appendix \[AppB\] for details). The result is $$K_{\rm TS}=
\sum_i^{n_{T1}} \beta^{(1)}_i\bar\beta^{(1)}_i\ e^{{1\over2}[K_{(2)}+K_{(3)}]}
+\sum_i^{n_{T2}} \beta^{(2)}_i\bar\beta^{(2)}_i\ e^{{1\over2}[K_{(1)}+K_{(3)}]}
+\sum_i^{n_{T3}} \beta^{(3)}_i\bar\beta^{(3)}_i\ e^{{1\over2}[K_{(1)}+K_{(2)}]}
\label{KTS}$$ where the $\beta^{(I)}_i$ are twisted sector fields that belong to the $I$-th set, $n_{T1,T2,T3}$ are the numbers of these fields, and $K_{(1,2,3)}$ are given in Eq. (\[Kdef\]). It is important to realize that this result is only valid to lowest order in the twisted matter fields.
Computation of $V_0$
--------------------
The computation of the scalar potential in Eq. (\[Vdef\]) requires the knowledge of $G^{I\bar J}$, , the transpose of the inverse of the matrix of second derivatives of the Kähler potential. The computation is simplified by that fact that the matrix $G_{I\bar J}=K_{I\bar J}$ possesses a block-diagonal form once the twisted sector matter fields are set at their zero vacuum expectation values. Indeed, schematically we have $$K_{I\bar J}\sim
\bordermatrix{&U^{(1)}&U^{(2)}&U^{(3)}&T^{(1)}&T^{(2)}&T^{(3)}\cr
U^{(1)}&X_{U^{(1)}}&[T^{(3)}]^2&[T^{(2)}]^2&0&T^{(2)}&T^{(3)}\cr
U^{(2)}&[T^{(3)}]^2&X_{U^{(2)}}&[T^{(1)}]^2&T^{(1)}&0&T^{(3)}\cr
U^{(3)}&[T^{(2)}]^2&[T^{(1)}]^2&X_{U^{(3)}}&T^{(1)}&T^{(2)}&0\cr
T^{(1)}&0&T^{(1)}&T^{(1)}&X_{T^{(1)}}&0&0\cr
T^{(2)}&T^{(2)}&0&T^{(2)}&0&X_{T^{(2)}}&0\cr
T^{(3)}&T^{(3)}&T^{(3)}&0&0&0&X_{T^{(3)}}\cr}\ .
\label{schematic}$$ At this point of the calculation all derivatives have been taken and we can set $\VEV{T^{(1,2,3)}}=0$, which reveals the block-diagonal structure with $$K^{I\bar J}={\rm diag}\,\{
X^{-1}_{U^{(1)}},X^{-1}_{U^{(2)}},X^{-1}_{U^{(3)}},
X^{-1}_{T^{(1)}},X^{-1}_{T^{(2)}},X^{-1}_{T^{(3)}}\}\ .
\label{inv-schematic}$$ (There is an additional contribution to $K^{I\bar J}$ from the dilaton field.) For each of these blocks one can compute $G^IG_I=G^{I\bar J}G_{\bar J}G_I$. This calculation depends on the superpotential since $G_I=K_I+\partial_I \ln W$, , $$G^IG_I=K^IK_I+K^I\partial_I\ln W+K^{\bar I}\partial_{\bar I}\ln\overline W
+K^{I\bar J}\partial_I\ln W\partial_{\bar J}\ln\overline W\ .
\label{GIGI}$$ With the help of Mathematica [@Mathematica] we obtain for the untwisted fields $$\begin{aligned}
&{\rm Dilaton:}\qquad &\left[K^IK_I\right]^{(S)}=1\ .\label{KKS}\\
&\tau_1,\tau_2\ {\rm set:}\qquad
&\left[K^IK_I\right]^{(\tau_1,\tau_2)}=2\quad(\forall\ n_\phi)\
.\label{KKTU}\\
&{\rm \tau\ set:}\qquad &\left[K^IK_I\right]^{(\tau)}=2\quad(\forall\ n_\psi)\
.\label{KKtau}\\
&{\rm \alpha\ set:}\qquad
&\left[K^IK_I\right]^{(\alpha)}=\sum_i^{n_I}\alpha_i\bar\alpha_i\ .
\label{KKalpha}\end{aligned}$$ We do not show the corresponding results for the twisted fields since $\VEV{K^IK_I}=0$ in this case (, $\VEV{K_\beta}\propto\VEV{\bar\beta}=0$). The scalar potential then becomes $$V=e^G\left[ \left\{\begin{array}{cc}1,&\lambda_f=1\\ (S+\bar
S)^2|G_S|^2,&\lambda_f=0\end{array}\right\}+2n_{\rm\tau_1\tau_2}
+2n_\tau+n_\alpha\sum_i^{n_I}\alpha_i\bar\alpha_i-3+F(\beta,\partial\ln W)
\right]\ ,
\label{V}$$ where the sum of the three kinds of sets $n_{\rm\tau_1\tau_2}+n_\tau+n_\alpha=3$ is fixed. The term $n_\alpha\sum_i\alpha_i\bar\alpha_i$ is meant to represent however many $\alpha$-set contributions may exist in a given model. Also, $\lambda_f=1$ indicates that $W$ does not depend on $S$, whereas $\lambda_f=0$ indicates that $W$ does depend on $S$ in which case $G_S=-1/(S+\bar
S)+(\partial_S W)/W$. The contributions which depend on $\partial_I\ln W$, $\partial_{\bar I}\ln\overline W$, or the twisted fields are collectively denoted by $F(\beta,\partial\ln W)$. The minimum of the potential is given by $$V_0=e^G\,[\lambda_f+2n_{\rm \tau_1\tau_2}+2n_\tau-3]\ ,
\label{V0}$$ with $\VEV{\alpha}=\VEV{\beta}=0$, $\VEV{\partial_I\ln W}=0$,[^5] and [*if*]{} $\lambda_f=0$ also
In view of Eq. (\[V0\]), there are only two choices for the untwisted modular symmetry which are consistent with $V_0=0$, namely $$\begin{tabular}{cccc}
$\lambda_f$&$n_{\rm \tau_1\tau_2}$&$n_\tau$&$n_\alpha$\\
1&1&0&2\\
1&0&1&2
\end{tabular}\qquad
\begin{tabular}{c}
Moduli\\
$S,\tau_1,\tau_2$\\
$S,\tau$
\end{tabular}
\label{V0=0}$$ Note that the dilaton is required to be a modulus field, and that only one set contributes moduli fields. If these requirements are satisfied, we would obtain a model with zero vacuum energy and flat directions, thus satisfying the first two no-scale supergravity postulates. Moreover, the gravitino mass is given by (see Eq. (\[m3/2def\])) $$m^2_{3/2}=\left\{\begin{array}{c}
{\VEV{|W|^2}\over
\VEV{(S+\bar
S)\,[(\tau_1+\bar\tau_1)(\tau_2+\bar\tau_2)-\sum_i(\phi_i+\bar\phi_i)^2]}}\\
{\VEV{|W|^2}\over
\VEV{(S+\bar S)\,[(\tau+\bar\tau)^2-\sum_i(\psi_i+\bar\psi_i)^2]}}
\end{array}\right.
\label{m3/2}$$ for each of the two cases in Eq. (\[V0=0\]), and is undetermined as anticipated. In these equations the values of $\VEV{\phi_i,\psi_i}$ are determined by the flatness conditions $\VEV{\partial_I\ln W}=0$ (typically $\VEV{\phi_i}=\VEV{\psi_i}=0$).
Computation of $Q$ {#computationofQ}
------------------
We now compute $Q$ using the formula in Eq. (\[Qdef\]). The basic quantity to be computed is the determinant of $G_{M\bar N}=K_{M\bar N}$. Since we know the untwisted sector contribution to $K$ exactly, whereas we only have a first-order approximation to the twisted sector contribution (see Eq. (\[KTS\])), we address the untwisted sector first. As in the computation of $G^IG_I$ above, the $K_{M\bar N}$ matrix is block-diagonal (three untwisted sets and the dilaton) and (with the help of Mathematica) we obtain $$\begin{aligned}
&{\rm Dilaton:}\qquad &\left[\det G_{M\bar N}\right]^{(S)}=(S+\bar S)^{-2}
\ .\label{detS}\\
&\tau_1,\tau_2\ {\rm set:}\qquad &\left[\det G_{M\bar
N}\right]^{(\tau_1,\tau_2)}=2^{n_\phi}
\left[(\tau_1+\bar\tau_1)(\tau_2+\bar\tau_2)-
\sum_i^{n_\phi}(\phi_i+\bar\phi_i)^2\right]^{-n_\phi-2}\label{detTU}\\
&{\rm \tau\ set:}\qquad &\left[\det G_{M\bar N}\right]^{(\tau)}=
2^{n_\psi+1}\left[(\tau+\bar \tau)^2-
\sum_i^{n_\psi}(\psi_i+\bar\psi_i)^2\right]^{-n_\psi-1}\label{dettau}\\
&{\rm \alpha\ set:}\qquad &\left[\det G_{M\bar N}\right]^{(\alpha)}=
\left[1-\sum_i^{n_I}
\alpha_i\bar\alpha_i+\coeff{1}{4}\left(\sum_i^{n_I}\alpha_i^2\right)
\left(\sum_i^{n_I}\bar\alpha_i^2\right)\right]^{-n_I}\label{detalpha}\end{aligned}$$ From these results and Eqs. (\[GFFF\],\[Kdef\],\[TUdef\],\[taudef\]) we see that $R_{I\bar J}=\partial_I\partial_{\bar J}\ln\det G_{M\bar N}$ for each block is just a multiple of $G_{I\bar J}$ for that block, , $$\begin{aligned}
&{\rm Dilaton:}\qquad &\left[R_{I\bar J}\right]^{(S)}=2 G_{S\bar S}^{(S)}
\ .\label{RS}\\
&\tau_1,\tau_2\ {\rm set:}\qquad &\left[R_{I\bar
J}\right]^{(\tau_1,\tau_2)}=(n_\phi+2)G_{I\bar
J}^{(\tau_1,\tau_2)}\label{RTU}\\
&{\rm \tau\ set:}\qquad &\left[R_{I\bar J}\right]^{(\tau)}=(n_\psi+1)G_{I\bar
J}^{(\tau)}\label{Rtau}\\
&{\rm \alpha\ set:}\qquad &\left[R_{I\bar J}\right]^{(\alpha)}=n_I G_{I\bar
J}^{(\alpha)}\label{Ralpha}\end{aligned}$$ With the above observation, the quantity that appears in $Q$ can be readily obtained: for each block $G^IR_{I\bar J}G^{\bar J}\propto G^IG_{I\bar J}G^{\bar
J}=G^IG_I$, and these quantities have been given in Eqs. (\[KKS\])–(\[KKalpha\]) (at the minimum $\VEV{G^IG_I}=\VEV{K^IK_I}$).
The computation of $Q$ also involves evaluating $H_{I\bar
J}=\partial_I\partial_{\bar J}\ln\det {\rm Re\,}(f_{ab})$. In string models, $f_{ab}$ receives tree-level and one-loop contributions only [@f1loop]. Writing $f_{ab}=\delta_{ab}S+f^{\rm 1-loop}$ (suitable for level-one Kac-Moody constructions), and neglecting the one-loop contribution, we obtain $\det {\rm Re\,}(f_{ab})=[{1\over2}(S+\bar S)]^{d_f}$, where $d_f$ is the dimension of the gauge group ($d_f\gg1$). Also, $H_{S\bar S}=-d_f G_{S\bar S}$ and the contribution to $Q$ is $G^IH_{I\bar J}G^{\bar J}=-d_f G^SG_S=-\lambda_f
d_f$ (at the minimum).
The total contribution to $Q$ is then $$\begin{aligned}
Q&=&\left[1+(2+n_\phi)n_{\rm \tau_1\tau_2}+(1+n_\psi)n_\tau+n_I n_\alpha\right]
- 1\nonumber\\
&&-\left[\{2\lambda_f+2(n_\phi+2)n_{\rm \tau_1\tau_2}
+2(n_\psi+1)n_\tau\}+\lambda_f d_f\right]\nonumber\\
&&+Q_{\rm TS}\ ,
\label{Qresult}\end{aligned}$$ where the terms are displayed in correspondence with those in Eq. (\[Qdef\]), and $Q_{\rm TS}$ is the twisted sector contribution to $Q$. For the two cases in Eq. (\[V0=0\]), which give $V_0=0$, we obtain $$Q=\left\{\begin{array}{c} 2n_I-n_\phi-4-d_f+Q_{\rm TS}\\
2n_I-n_\psi-3-d_f+Q_{\rm TS}\end{array}\right.\ ,
\label{QV0=0}$$ where “$2n_I$" is meant to represent the sum of the untwisted fields in the two sets which do not contain moduli.
Now let us address the twisted sector contribution to $Q$ (, $Q_{\rm TS}$). Of the two cases in Eq. (\[V0=0\]), consistent with $V_0=0$, the second case corresponds to a specific string model which will be discussed in Sec. \[Possible\] below; we focus on this case in what follows. The complete Kähler potential in this case is (see Appendix \[AppB\] for details) $$\begin{aligned}
K&=&-\ln(S+\bar S)
-\ln\left[(\tau+\bar \tau)^2-\sum_i^{n_\psi}(\psi_i+\bar\psi_i)^2\right]
+\sum_i^{n_{U2}}\alpha^{(2)}_i\bar\alpha^{(2)}_i
+\sum_i^{n_{U3}}\alpha^{(3)}_i\bar\alpha^{(3)}_i\nonumber\\
+&&\!\!\!\!\!\!\sum_i^{n_{T1}}\beta^{(1)}_i\bar\beta^{(1)}_i
+{1\over\left[(\tau+\bar
\tau)^2-\sum_i^{n_\psi}(\psi_i+\bar\psi_i)^2\right]^{1/2}}
\left(\sum_i^{n_{T2}}\beta^{(2)}_i\bar\beta^{(2)}_i
+\sum_i^{n_{T3}}\beta^{(3)}_i\bar\beta^{(3)}_i\right)
\label{completeK}\end{aligned}$$ which is valid to first order in the twisted fields $\beta_i^{(1,2,3)}$, and the untwisted fields $\alpha_i^{(2,3)}$, and corresponds to $2n_I=n_{U2}+n_{U3}$ in Eq. (\[QV0=0\]). The contribution to $Q_{\rm TS}$ from the first twisted set comes only from the $n_{T1}$ contribution to the total number of chiral superfields ($N$) in Eq. (\[Qdef\]), since $K_{\beta^{(1)}\bar\beta^{(1)}}={\bf1}$ has unit determinant and thus $R_{\beta^{(1)}\bar\beta^{(1)}}=0$. The second and third twisted sets contribute to $N$ ($n_{T2},n_{T3}$), and in principle get tangled up with the fields in the first untwisted set ($\tau,\psi_i$). However, since $\VEV{G_{\beta^{(2,3)}}}=0$, we only need to worry about their possible additional contributions to $[R_{I\bar J}]^{(\tau)}$ in Eq. (\[Rtau\]). This means that in this enlarged determinant (involving $\tau,\psi_i,\beta^{(2)}_i,\beta^{(3)}_i$) we can set $\beta^{(2,3)}_i=0$ after calculating the derivatives but before calculating the determinant. The enlarged determinant is then the one given in Eq. (\[dettau\]) times the factor $\left[(\tau+\bar
\tau)^2-\sum_i^{n_\psi}(\psi_i+\bar\psi_i)^2\right]^{-(n_{T2}+n_{T3})/2}$. That is, the exponent in Eq. (\[dettau\]) and the coefficient in Eq. (\[Rtau\]) receive a further contribution of $-(n_{T2}+n_{T3})/2$. The result for $Q$ is then $$\begin{aligned}
Q&=&[1+(1+n_\psi)+n_{U2}+n_{U3}+n_{T1}+n_{T2}+n_{T3}]-1\nonumber\\
&&-[\{2+2(n_\psi+1+(n_{T2}+n_{T3})/2)\}+d_f]\nonumber\\
&=&n_{U2}+n_{U3}+n_{T1}-n_\psi-d_f-3\ .
\label{Qformula}\end{aligned}$$
What are the prospects for obtaining $Q=0$? In typical models one observes that $n_{U2}\sim n_{U3}\sim n_\psi\sim{\cal O}(10)\ll n_{T1}$. On the other hand $d_f>d_{\rm SM}=12$, although in realistic models we expect a number significantly exceeding this lower bound since, , the hidden sector gauge group needs to be large enough for supersymmetry breaking via gaugino condensation to occur at a sufficiently high scale; a typical value would be $d_f\sim{\cal O}(50-100)$. Thus, it is not inconceivable that models can be found where the various contributions to $Q$ cancel each other out. In Sec. \[Possible\] we exhibit a model where this cancellation is almost perfect.
Soft-supersymmetry-breaking parameters {#Soft}
======================================
With the knowledge of the Kähler function, the scalar potential, and the gauge kinetic function one can compute the usual soft-supersymmetry-breaking parameters on which the low-energy model predictions depend so crucially.
Goldstino composition
---------------------
Before we proceed with these calculations, it is instructive to determine the field dependence of the goldstino field, which has received considerable attention in “model-independent" approaches to this problem. The goldstino, which is eaten by the gravitino upon spontaneous breaking of supergravity, is given by $$\widetilde\eta=\VEV{e^{G/2}G_I}\chi^I\ ,
\label{etadef}$$ where $\chi^I$ are the fermionic partners of the scalar fields which appear in the scalar potential (\[Vdef\]). At the minimum of the scalar potential we have $\VEV{G_{\alpha,\beta}}=0$. In computing the $G_I$ derivatives, for present purposes it suffices to approximate the Kähler function in Eqs. (\[TUdef\]),(\[taudef\]) in the limit where the $\psi_i,\phi_i$ fields have vevs much smaller than the moduli vevs. This gives $$\begin{aligned}
&{\rm Dilaton:}\qquad &\VEV{G_I}^{(S)}=-\lambda_f(S+\bar S)^{-1}\label{G_IS}\\
&\tau_1,\tau_2\ {\rm set:}\qquad
&\VEV{G_I}^{(\tau_1,\tau_2)}\approx\{-(\tau_1+\bar\tau_1)^{-1},-(\tau_2
+\bar\tau_2)^{-1},\nonumber\\
&&\qquad\qquad\qquad\qquad 2(\phi_i+\bar\phi_i)[(\tau_1+\bar\tau_1)(\tau_2
+\bar\tau_2)]^{-1}\}\label{G_ITU}\\
&{\rm \tau\ set:}\qquad &\VEV{G_I}^{(\tau)}\approx\{-2(\tau+\bar\tau)^{-1},
2(\psi_i+\bar\psi_i)(\tau+\bar\tau)^{-2}\}\label{G_Itau}\end{aligned}$$ These results can be substituted back into Eq. (\[etadef\]) to obtain $\widetilde\eta$. The final step is to express the $\chi^I$ fields in terms of the properly normalized $\widehat\chi^I$ fields. This operation entails a rescaling of the fields which will be discussed in detail in Sec. \[Field\]. Let us just quote the results: $$\begin{aligned}
\widehat S&=&{S\over\VEV{S+\bar S}}\ ,\quad \widehat
\tau_1={\tau_1\over\VEV{\tau_1+\bar\tau_1}}\
,\quad
\widehat \tau_2={\tau_2\over\VEV{\tau_2+\bar\tau_2}}\ ,\quad
\widehat\tau={\sqrt{2}\,\tau\over\VEV{\tau+\bar\tau}}\ ,
\label{redefinitions1}
\\
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\widehat\phi_i&=&{\sqrt{2}\,\phi_i\over\VEV{(\tau_1+\bar\tau_1)(\tau_2+\bar\tau_2)}^{1/2}}\
,\qquad
\widehat\psi_i={\sqrt{2}\,\psi_i\over\VEV{\tau+\bar\tau}}\ .
\label{redefinitions2}\end{aligned}$$ The goldstino fields corresponding to the two cases in Eq. (\[V0=0\]) are then $$\widetilde\eta\propto\left\{
\begin{array}{l}
\widehat S+\widehat\tau_1+\widehat\tau_2
+\sqrt{2}\,\sum_i^{n_\phi}{\VEV{\phi_i+\bar\phi_i}\over\VEV{(\tau_1
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
+\bar\tau_1)(\tau_2+\bar\tau_2)}^{1/2}}\,(\widehat\phi_i+\widehat{\bar\phi_i})\\\widehat S+\sqrt{2}\,\widehat\tau+
\sqrt{2}\,\sum_i^{n_\psi}{\VEV{\psi_i+\bar\psi_i}\over\VEV{\tau+\bar\tau}}\,
(\widehat\psi_i+\widehat{\bar\psi_i})
\end{array}\right.
\longrightarrow
\widetilde\eta=\left\{
\begin{array}{l}
{1\over\sqrt{3}}(\widehat S+\widehat\tau_1+\widehat\tau_2)\\
{1\over\sqrt{3}}(\widehat S+\sqrt{2}\,\widehat\tau)
\end{array}\right.
\label{goldstino}$$ where the second form holds in the limit $\VEV{\psi_i,\phi_i}\approx0$. Thus, we get a goldstino field which contains substantial components of both “dilaton" and “moduli". Note that, in principle light matter fields also appear, although their contribution is highly suppressed, by a factor $\VEV{\phi}/\VEV{\rm moduli}\sim{\cal O}(10^2/10^{18})$.
Gaugino masses
--------------
The properly normalized gaugino masses are obtained from the expression $$M_a={e^{G/2}\over 2{\rm Re}\, f_a}\sum_I \partial_I f_a\ G^I,
\label{Madef}$$ where the sum over $I$ runs over all matter fields which $f_a$ depends on. For $f_a$ we use the following one-loop (although correct to all orders) expression [@f1loop] $$f_a=k_a S
-\coeff{1}{16\pi^2}B^{(\tau_1,\tau_2)}_a\ln|\eta(\tau_1)\eta(\tau_2)|^4\cdot
n_{\rm \tau_1\tau_2}
-\coeff{1}{16\pi^2}B^{(\tau)}_a\ln|\eta(\tau)|^4\cdot n_\tau\ ,
\label{fadef}$$ where the level of the Kac-Moody algebra is one ($k_a=1$), and $\eta$ is the Dedekind eta function. Also, $B_a$ is a quantity which depends on the massless sector of the theory and their modular weights, as well as on the coefficient $\delta_{\rm GS}$ which arises in the Green-Schwarz modular-anomaly cancellation. For our present purposes, a detailed specification of $B_a$ is not required. The derivatives $\partial_I f_a$ in the expression for $M_a$ are non-zero only for $S,\tau_1,\tau_2,\tau$. One obtains, , $\partial_{\tau_1} f_a={1\over16\pi^3}B^{(\tau_1,\tau_2)}_a \widehat
G_2(\tau_1)$, where $\widehat G_2(\tau_1)=G_2(\tau_1)-2\pi/(\tau_1+\bar\tau_1)$, and the Eisenstein function $G_2$ is related to the Dedekind function via $G_2(\tau_1)=-4\pi\partial_{\tau_1}\ln\eta(\tau_1)$ [@Cvetic]. We also note that $\widehat G_2(\tau_1)$ has zeroes at $\tau_1=1,e^{i\pi/6}$.
The other ingredient in the expression for $M_a$ is $G^I=G^{I\bar J}G_{\bar
J}=K^{I\bar J}(K_{\bar J}+\partial_{\bar J}\ln\overline W)=
K^{I\bar J}K_{\bar J}=K^I$ at the minimum. This expression can be evaluated, with the result $$\begin{aligned}
&{\rm Dilaton:}\qquad &[K^I]^{(S)}=-(S+\bar S)\ ,\label{K^IS}\\
&\tau_1,\tau_2\ {\rm set:}\qquad
&[K^I]^{(\tau_1,\tau_2)}=\{-(\tau_1+\bar\tau_1),-(\tau_2+\bar
\tau_2),-(\phi_i+\bar\phi_i)\}\ ,\label{K^ITU}\\
&{\rm \tau\ set:}\qquad &[K^I]^{(\tau)}=\{-(\tau+\bar\tau),
-(\psi_i+\bar\psi_i)\}\ ,\label{K^Itau}\\
&{\rm \alpha\ set:}\qquad &[K^I]^{(\alpha)}=\{\alpha_i+{\cal O}(\alpha^2_i)\}\
.\label{K^Ialpha}\end{aligned}$$ With these results we finally obtain (only $I=S,\tau_1,\tau_2,\tau$ are relevant) $$\begin{aligned}
M_a={m_{3/2}\over2{\rm Re}\, f_a}&&\hspace{-0.5cm}\Bigl\{-(S+\bar
S)\lambda_f\nonumber\\
&&-\coeff{1}{16\pi^2}B^{(\tau_1,\tau_2)}_a[(\tau_1+\bar\tau_1)\widehat
G_2(\tau_1)+(\tau_2+\bar\tau_2)\widehat
G_2(\tau_2)]\cdot n_{\rm \tau_1\tau_2}\nonumber\\
&&-\coeff{1}{16\pi^2}B^{(\tau)}_a(\tau+\bar\tau)\widehat G_2(\tau)\cdot
n_\tau\Bigr\}\ .
\label{Ma}\end{aligned}$$ For the two cases in Eq. (\[V0=0\]) $\lambda_f=1$ and thus the tree-level contribution to $M_a$ is non-zero and therefore dominant, giving nearly (up to small one-loop corrections) [*universal*]{} gaugino masses, , $$M_a=m_{1/2}=m_{3/2}\ .
\label{m1/2}$$
Scalar masses
-------------
The scalar masses are obtained by taking second derivatives of the scalar potential.[^6] These masses have two sources: “supersymmetric" masses from the superpotential, and supersymmetry-breaking masses from the Kähler potential. For the untwisted fields, the latter can be deduced from the expression for $V$ given in Eq. (\[V\]) (only the $K^IK_I$ term in Eq. (\[GIGI\]) matters). We see that neither the $\phi_i$ nor the $\psi_i$ fields appear, thus $$\widetilde m_{\widehat\phi_i}=\widetilde m_{\widehat\psi_i}=0\ ,
\label{mphipsi}$$ where $\widetilde m_f$ represents the supersymmetry-breaking contribution to the mass of the scalar field $f$. On the other hand, the $\alpha_i$ do appear in $V$ and their mass is given by $$\widetilde m_{\widehat\alpha_i}=m_{3/2}\ .
\label{malpha}$$ In other words, untwisted sector fields in sets with moduli receive no (tree-level) supersymmetry-breaking masses, whereas those in sets with no moduli receive a universal mass equal to the gravitino mass. Obviously, it also follows that the moduli are massless (including the dilaton since $\lambda_f=1$ is required).
Turning to the twisted field scalar masses, let us again consider the second case in Eqs. (\[V0=0\],\[QV0=0\]), with the Kähler potential given in Eq. (\[completeK\]). For the fields in the first untwisted set \[$\beta^{(1)}_i$\] it is clear that their (Kähler potential) masses are equal to those of the second \[$\alpha^{(2)}_i$\] and third \[$\alpha^{(3)}_i$\] untwisted set fields. Indeed, in this case $K_{\beta^{(1)}\bar\beta^{(1)}}={\bf1}$ and $K^IK_I=\sum_i^{n_{T1}}\beta^{(1)}_i\bar\beta^{(1)}_i$, and as expected $$\widetilde m_{\widehat\beta^{(1)}_i}=m_{3/2}\ .
\label{mbeta1}$$ The scalar masses of the second and third twisted set fields can be obtained by considering the following portion of the Kähler potential in Eq. (\[completeK\]) $$K_0(\tau,\bar\tau)+
K_1(\tau,\bar\tau)\sum_i^{n_\psi}(\psi_i+\bar\psi_i)^2
+K_2(\tau,\bar\tau)\left(\sum_i^{n_{T2}}\beta^{(2)}_i\bar\beta^{(2)}_i
+\sum_i^{n_{T3}}\beta^{(3)}_i\bar\beta^{(3)}_i\right)\ ,
\label{ApproxK}$$ with $$K_0=-\ln(\tau+\bar \tau)^2\,,\qquad K_1={1\over(\tau+\bar\tau)^2}\,,\qquad
K_2={1\over\tau+\bar \tau}\ ,
\label{K0K1K2}$$ where we have performed an expansion to first order in both $\psi_i$ and $\beta_i^{(2,3)}$. The approximate expression in Eq. (\[ApproxK\]) is sufficient to compute the scalar masses. After some algebra we obtain for this subset of the fields (, $\tau,\psi_i,\beta^{(2)}_i,\beta^{(3)}_i$) $$\begin{aligned}
K^IK_I&=&{K_{0\tau}K_{0\bar\tau}\over K_{0\tau\bar\tau}}\nonumber\\
&&+2K_1\left[1
-{K_{0\tau}K_{1\bar\tau}+K_{0\bar\tau}K_{1\tau}\over 2K_1K_{0\tau\bar\tau}}
-{K_{0\tau}K_{0\bar\tau}\over K_{0\tau\bar\tau}}
{K_1K_{1\tau\bar\tau}-2K_{1\tau}K_{1\bar\tau}\over 2(K_1)^2K_{0\tau\bar\tau}}
\right]\sum_i^{n_\psi}(\psi_i+\bar\psi_i)^2\nonumber\\
&&+K_2\left[1-{K_{0\tau}K_{0\bar\tau}\over K_{0\tau\bar\tau}}\,{(\ln
K_2)_{\tau\bar\tau}\over K_{0\tau\bar\tau}}\right]
\left(\sum_i^{n_{T2}}\beta^{(2)}_i\bar\beta^{(2)}_i
+\sum_i^{n_{T3}}\beta^{(3)}_i\bar\beta^{(3)}_i\right)\ .
\label{KIKIapprox}\end{aligned}$$ That is $$\begin{aligned}
\widetilde m^2_{\widehat\psi_i}&=&1
-{K_{0\tau}K_{1\bar\tau}+K_{0\bar\tau}K_{1\tau}\over 2K_1K_{0\tau\bar\tau}}
-{K_{0\tau}K_{0\bar\tau}\over K_{0\tau\bar\tau}}
{K_1K_{1\tau\bar\tau}-2K_{1\tau}K_{1\bar\tau}\over 2(K_1)^2K_{0\tau\bar\tau}}
\ ,
\label{psimass}
\\
\widetilde m^2_{\widehat\beta^{(2)}_i}&=&\widetilde m^2_{\widehat\beta^{(3)}_i}
=1-{K_{0\tau}K_{0\bar\tau}\over K_{0\tau\bar\tau}}\,{(\ln
K_2)_{\tau\bar\tau}\over K_{0\tau\bar\tau}}
\ ,
\label{betamass}\end{aligned}$$ where we have properly normalized the $\psi_i$ and $\beta^{(2,3)}_i$ fields by absorbing the overall factors (see Eqs. (\[normalizations1\],\[normalizations3\])). For the choices of $K_0,K_1,K_2$ in Eq. (\[K0K1K2\]), the first term in Eq. (\[KIKIapprox\]) is $K_{0\tau}K_{0\bar\tau}/K_{0\tau\bar\tau}=2$, as expected from the vacuum energy calculation above. It also follows that $\widetilde m^2_{\widehat\psi_i}=0$ (confirming the result in Eq. (\[mphipsi\])) and the new result $$\widetilde m_{\widehat\beta^{(2)}_i}=\widetilde m_{\widehat\beta^{(3)}_i}=0\ .
\label{mbeta2beta3}$$ The expression for the $\beta^{(2,3)}_i$ masses in Eq. (\[betamass\]) agrees with that given in Ref. [@BIM]; the expression for the $\psi_i$ masses is new.
To summarize, for the model with Kähler potential given in Eq. (\[completeK\]), the scalar masses of all fields are the following multiples of $m_{3/2}$ $$\begin{tabular}{ccrcccr}
$U^{(1)}\,$:&$\psi_i$& $0$ &\qquad\qquad &$T^{(1)}\,$:&$\beta^{(1)}_i$& $1$\\
$U^{(2)}\,$:&$\alpha^{(2)}_i$& $1$ & &$T^{(2)}\,$:&$\beta^{(2)}_i$& $0$\\
$U^{(3)}\,$:&$\alpha^{(2)}_i$& $1$ & &$T^{(3)}\,$:&$\beta^{(3)}_i$& $0$\\
\end{tabular}
\label{ScalarMasses}$$ Note the close correlation between the scalar masses and the corresponding modular weights of the matter fields given in Eq. (\[ModularWeights\]) (where the same choice of moduli fields was made). The scalar mass spectrum in Eq. (\[ScalarMasses\]) is non-universal. This situation is likely to be an important model-building constraint, given what we know about needed near-degeneracies in certain low-energy squark and slepton masses. For example, data on $K^0-\bar K^0$ mixing and leptonic flavor-changing decays like $\mu\to e\gamma$ strongly constrain the mass differences for squarks and sleptons of the first two generations with the same electric charge but of different flavor [@EN]. The scenario which appears to emerge in string no-scale supergravity seems to explain this phenomenological requirement naturally: since all light chiral matter fields usually arise from the twisted sector, one would assign the first two generations to the second and third sets (with vanishing scalar masses); the third generation could be assigned to any of the sets.
Fermion masses
--------------
Supersymmetry breaking can also induce masses for (non-chiral) fermions in real representations of the gauge group. These (unnormalized) masses are given by the following expression [@FKZ] $$\left(M_f\right)_{IJ}=m_{3/2}\left(G_{IJ}-G_{IJ\bar K}G^{\bar
K}+\coeff{1}{3}G_IG_J\right)\ .
\label{FermionMasses}$$ As above, we focus on the model with Kähler potential given in Eq. (\[completeK\]). For fermions in the second and third untwisted sets \[$\alpha^{(2,3)}_i$\] and in all of the twisted sets \[$\beta^{(1,2,3)}_i$\] one has $G_{IJ}\equiv0$ and $\VEV{G_I}=0$, and thus $$m_{\widehat\alpha^{(2)}_i}=m_{\widehat\alpha^{(3)}_i}=0\ ,\qquad
m_{\widehat\beta^{(1)}_i}=m_{\widehat\beta^{(2)}_i}=
m_{\widehat\beta^{(3)}_i}=0\ .
\label{mfermions0}$$ The remaining fields are $S,\tau,\psi_i$. If we make the simplifying assumption $\VEV{\psi_i}=0$ (, $\VEV{G_{\psi_i}}=0$) one can show that the normalized fermion mass matrix reduces to $$\left(M_f\right)_{IJ}=m_{3/2}\ \bordermatrix{
&\widehat S&\widehat\tau&\widehat\psi_j\cr
\widehat S&2/3&-\sqrt{2}/3&0\cr
\widehat\tau&-\sqrt{2}/3&1/3&0\cr
\widehat\psi_i&0&0&\delta_{ij}\cr}\ .
\label{FermionMassMatrix}$$ To obtain this result we have made use of the various normalization factors given in Eqs. (\[redefinitions1\],\[redefinitions2\]). This matrix has zero determinant, indicating the presence of a massless eigenstate, namely the goldstino ($\widetilde\eta$). Indeed, from Eq. (\[FermionMassMatrix\]) this eigenstate is $\widetilde\eta=(\widehat S+\sqrt{2}\widehat\tau)/\sqrt{3}$, in agreement with our previous result in Eq. (\[goldstino\]) (for $\VEV{\psi_i}=0$). From Eq. (\[FermionMassMatrix\]) it also follows that the orthogonal linear combination $\widetilde\eta_\perp=(\sqrt{2}\widehat
S-\widehat\tau)/\sqrt{3}$, and all of the $\widehat\psi_i$ get masses equal to the gravitino mass, , $$m_{\widetilde\eta_\perp}=m_{3/2}\ ,\qquad m_{\widehat\psi_i}=m_{3/2}\ .
\label{mfermions1}$$
A consistency check
-------------------
In the previous three subsections we have computed all of the supersymmetry breaking masses, in particular for the model with Kähler potential given in Eq. (\[completeK\]). One can then perform a consistency check of result for $Q$ given in Eq. (\[Qformula\]), since we can calculate directly ${\rm
Str}\,{\cal M}^2=\sum_j(-1)^{2j}(2j+1){\cal M}^2_j=
2Qm^2_{3/2}$. The masses of the complex scalars ($j=0$) are given in Eq. (\[ScalarMasses\]) and contribute to the supertrace (in units of $m^2_{3/2}$) in the amount of $2(n_{U2}+n_{U3}+n_{T1})$. The masses of the Majorana fermions ($j=1/2$) are given in Eqs. (\[mfermions0\],\[mfermions1\]) and contribute $-2(1+n_\psi)$, whereas the Majorana gaugino masses (given in Eq. (\[m1/2\])) contribute $-2d_f$. Finally the gravitino contributes $-4$. Putting it all together gives $Q=n_{U2}+n_{U3}+n_{T1}-n_\psi-d_f-3$, which is the result found in Eq. (\[Qformula\]) by less direct means.
$A$ terms
---------
The supersymmetry-breaking cubic scalar couplings (or $A$ terms) are contained in the term $e^G\,K^I\partial_I\ln W$ (plus hermitian conjugate) of the scalar potential in Eq. (\[GIGI\]). The main input required to evaluate these couplings is the value of $K^I$ for each of the types of untwisted and twisted states. For the untwisted states these inputs are given in Eqs. (\[K\^Itau\],\[K\^Ialpha\]), , $$K^{\psi_i}=-\psi_i\ ,\quad
K^{\alpha^{(2)}_i}\approx\alpha^{(2)}_i\ ,\quad
K^{\alpha^{(3)}_i}\approx\alpha^{(3)}_i\ .
\label{K^Iu}$$ For the twisted states in the model with Kähler potential given in Eq. (\[completeK\]), the first twisted set fields \[$\beta^{(1)}_i$\] have the same functional dependence as the second and third untwisted set fields \[$\alpha^{(2,3)}_i$\], and therefore the result is as above: $K^{\beta^{(1)}_i}\approx\beta^{(1)}_i$. For the second and third twisted set fields, an intermediate step in the calculation that yields the result in Eq. (\[KIKIapprox\]) gives $$K^{\beta^{(2,3)}_i}\approx
{K_{0\tau\bar\tau}K_2-K_{0\bar\tau}K_{2\tau}\over K_{0\tau\bar\tau}K_2}
\,\beta^{(2,3)}_i\ .
\label{K^Ibeta}$$ Inserting the values for $K_0,K_2$ (Eq. (\[K0K1K2\])) one finds a zero result to first order, , $$K^{\beta^{(1)}_i}\approx\beta^{(1)}_i\ ,\quad
K^{\beta^{(2)}_i}\approx0\ ,\quad
K^{\beta^{(3)}_i}\approx0\ .
\label{K^It}$$
With the above results one can proceed to compute the $A$ terms for all the types of cubic couplings given in Eq. (\[couplings\]), with the field identifications given in Eq. (\[ScalarMasses\]). The expression to manipulate is $e^GK^I\partial_I\ln W=e^{G/2}e^{K/2}K^I\partial_I
W=m_{3/2}e^{K/2}K^I\partial_I W$. Since the cubic superpotential does not depend on $S$ or $\tau$, all we need to do is take derivatives with respect to the untwisted and twisted matter fields. Each time one such field is removed from a cubic coupling by the $\partial_I W$ operation, the corresponding $K^I$ factor puts it back in restoring the original coupling, although a coefficient ($0,1,-1$) is picked up in this process. After summing over all fields in a given cubic coupling, and over all cubic couplings one ends up with $m_{3/2}\,c\,e^{K/2} W=m_{3/2}\,c\,\widehat W$ where $\widehat
W(\widehat\phi)=e^{K/2}\,W(\phi)$ is the superpotential written in terms of the properly normalized fields, as discussed in Sec. \[Field\]. The constant $c$ is common to all of the types of cubic couplings in Eq. (\[couplings\]), and in fact $c=1$, , $$\begin{tabular}{rr}
$\psi\alpha^{(2)}\alpha^{(3)}$ : &$-1+1+1=1$\\
$\psi\beta^{(1)}\beta^{(1)}$ : &$-1+1+1=1$\\
$\alpha^{(2)}\beta^{(2)}\beta^{(2)}$ : &$1+0+0=1$\\
$\alpha^{(3)}\beta^{(3)}\beta^{(3)}$ : &$1+0+0=1$\\
$\beta^{(1)}\beta^{(2)}\beta^{(3)}$ : &$1+0+0=1$\\
\end{tabular}
\label{Aterms}$$ Thus we conclude that for all cubic couplings $$A=m_{3/2}\ .
\label{A}$$ (In passing we note that in the old no-scale models, $K^I=\{-[T+\bar T-C_i\bar
C_i],\vec0\}$, and therefore $A\equiv0$.)
$\mu$ and $B$
-------------
The possible origin of the low-energy Higgs mixing parameter $\mu$ (and its associated supersymmetry-breaking bilinear coupling $B$) has been discussed in the literature for some time. It is well-known that this term ($\mu h_1h_2$) must be present in the superpotential, and have a magnitude comparable to all other dimensional parameters of the low-energy theory. In the framework of string theory, where explicit mass parameters are not present in the superpotential, the nature of the $\mu$ term is particularly intriguing. Three scenarios have been put forward:
- The low-energy theory possesses an additional singlet field ($N$) which couples to the two Higgs doublets ($\lambda Nh_1h_2$) and gets a vacuum expectation value which effectively produces $\mu=\lambda\VEV{N}$ [@singlet]. Even though such couplings proliferate in fermionic string models, in all known instances the singlet fields are heavy and decouple from the low-energy spectrum.
- The quadratic $\mu$ term arises as an effective non-renormalizable fourth- (or higher) order term in the superpotential, , ${1\over
M}\lambda_4H\bar Hh_1h_2$ where $M\sim10^{18}\GeV$ is the string scale [@nonren]. In this case $\mu={1\over M}\lambda_4\VEV{H\bar H}$; for $\mu\sim1\TeV$, one requires $\VEV{H\bar H}^{1/2}\sim10^{11}\GeV$ which is typical of hidden sector matter condensates in string models.
- The quadratic $\mu$ term is built into the theory through the Kähler potential, and becomes non-zero and of ${\cal O}(m_{3/2})$ upon supersymmetry breaking [@oldUT; @kl; @FKZ].
Let us first address the third scenario. From the calculation of the fermion masses in Eq. (\[mfermions1\]) ($m_{\widehat\psi_i}=m_{3/2}$) one could think that these may come from a superpotential $\mu$ term, , $\mu\widehat\psi\widehat\psi$ with $\mu={1\over2}m_{3/2}$. However, Eq. (\[mphipsi\]) shows that the corresponding scalar masses vanish ($\widetilde m_{\widehat\psi_i}=0$), a result apparently inconsistent with the possible presence of a superpotential $\mu$ term. It turns out that things are more intricate and the interpretation of a $\mu$ term is not inconsistent. What happens is that one has to split the Kähler function into two pieces, one which is absorbed into the superpotential to provide the $\mu$ term as suggested in Refs. [@oldUT; @kl; @FKZ], and another one which remains as part of the Kähler function. The resolution to the puzzle comes from realizing that these two pieces give equal and opposite contributions to the squared scalar masses (but not to the fermion masses). Thus, such a Kähler-induced $\mu$-term does break supersymmetry, even though it can be incorporated into the superpotential.
Let us illuminate this result by studying a simple example in detail. Consider the Kähler potential $K=-\ln[(\tau+\bar\tau)^2-(\psi+\bar\psi)^2]$, which gives $\widetilde m_{\widehat\psi}=0$ and $m_{\widehat\psi}=m_{3/2}$. Let us expand the Kähler function to first order in the $\psi$ field $$K\approx-\ln(\tau+\bar\tau)^2+{2\over(\tau+\bar\tau)^2}\,\psi\bar\psi
+{1\over(\tau+\bar\tau)^2}\,(\psi\psi+\bar\psi\bar\psi)\ .
\label{Example}$$ Ignoring the last term in this expression (to be absorbed into $W$) the scalar mass can be obtained from Eq. (\[betamass\]) with $K_0=-\ln(\tau+\bar\tau)^2$ and $K_2=2/(\tau+\bar\tau)^2$. The result is $\widetilde m^2_{\widehat\psi}=-m^2_{3/2}$. The last term in Eq. (\[Example\]) can be lumped with the superpotential $W\to
We^{K_1\psi\psi}\approx W+WK_1\psi\psi$ with $K_1=1/(\tau+\bar\tau)^2$. Proper normalization entails multiplying W times $e^{K/2}$, thus giving the new superpotential term $e^{K/2}WK_1\psi\psi={1\over2}m_{3/2}\widehat\psi\widehat\psi$, where we have also properly normalized the $\psi$ field.[^7] We therefore get $\mu={1\over2}m_{3/2}$, which leads to a superpotential fermion mass $m_{\widehat\psi}=2\mu=m_{3/2}$. This also entails a superpotential scalar mass-squared $\widetilde
m^2_{\widehat\psi}=4\mu^2=m^2_{3/2}$, which when added to the Kähler potential mass-squared found above gives the expected vanishing result.[^8]
If this mechanism for generation of the $\mu$ term is present in realistic string models, one has to be careful with its embedding into the traditional supergravity-induced soft-supersymmetry-breaking parameters. For the squared scalar masses of the Higgs doublets one normally writes $(m_1^2+\mu^2)|H_1|^2
+(m^2_2+\mu^2)|H_2|^2$. The Kähler-induced $\mu$ term has the property that $m^2_1=m^2_2=-\mu^2$, which is what would need to be used as initial conditions in the corresponding renormalization group equations. (In practice we do not find this kind of $\mu$ term present in realistic string models, at least involving the two Higgs doublets which are to remain in the light spectrum.)
Now let us consider the second mechanism for generating a $\mu$ term, namely via a non-renormalizable coupling in the superpotential. Given such coupling we would like to know what is the associated supersymmetry breaking $B$ term. Essentially this term arises in a very similar manner as the $A$ terms discussed above, that is, from the $e^GK^I\partial_I\ln W$ term in the scalar potential. There is one crucial difference: the non-renormalizable couplings may need to be multiplied by powers of the Dedekind eta function to restore modular invariance of the superpotential, as discussed after Eq. (\[ModularWeights\]). This possible new dependence on the moduli must be taken into account when taken the derivatives of $W$. Let us first list the types of quartic and quintic superpotential couplings in free-fermionic models. In the same spirit as the cubic couplings given in Eq. (\[couplings\]), the types of higher-order couplings can be deduced from Ref. [@KLN], $$\begin{tabular}{cccc}
Quartic couplings&\qquad\qquad&\multicolumn{2}{c}{Quintic couplings}\\
$ \eta^0\,[T^{(1)}]^2\,[T^{(2)}]^2$&
&$\eta^0\,[T^{(1)}]^2\,[T^{(2)}]^2\,U^{(3)}$
&$\eta^0\,[T^{(1)}]^3\,T^{(2)}\,T^{(3)}$\\
$\eta^0\,[T^{(1)}]^2\,[T^{(3)}]^2$&
&$\eta^0\,[T^{(1)}]^2\,[T^{(3)}]^2\,U^{(2)}$&
$\eta^2\,[T^{(2)}]^3\,T^{(1)}\,T^{(3)}$\\
$\eta^2\,[T^{(2)}]^2\,[T^{(3)}]^2$&
&$\eta^4\,[T^{(2)}]^2\,[T^{(3)}]^2\,U^{(1)}$
&$\eta^2\,[T^{(3)}]^3\,T^{(1)}\,T^{(2)}$\\
\end{tabular}
\label{NRTs}$$ Next we need to determine if there is a modular weight imbalance, but this can only be done in a specific class of models, such as the ones with Kähler potential given in Eq. (\[completeK\]). The modular weights are then given in Eq. (\[ModularWeights\]). For every unit of modular weight imbalance we multiply the non-renormalizable coupling by $\eta^2(\tau)$, as indicated in Eq. (\[NRTs\]). Note that in some instances there is no modular weight imbalance. The value of the corresponding $B$ parameters can then be determined in analogy with the procedure followed for the $A$ terms, except when a power of $\eta^p$ is present. In this case one adds to the result the quantity $$-(\tau+\bar\tau)(\partial_\tau\eta^p)/\eta^p
=p/2+(\tau+\bar\tau)(p/4\pi)\widehat G_2(\tau)\ .
\label{AddOn}$$ With the use of Eqs. (\[K\^Iu\],\[K\^It\]) one can determine the $B$ parameters for each of the types of quartic or quintic couplings, as follows (in units of $m_{3/2}$) $$\begin{tabular}{cc}
Quartic couplings&$B$\\
$[\beta^{(1)}]^2\,[\beta^{(2)}]^2$
&$2$\\
$[\beta^{(1)}]^2\,[\beta^{(3)}]^2$
&$2$\\
$[\beta^{(2)}]^2\,[\beta^{(3)}]^2$
&$1+{\tau+\bar\tau\over2\pi}\,\widehat G_2(\tau)$\\
\end{tabular}
\label{QuarticBs}$$ $$\begin{tabular}{lccc}
Quintic couplings&$B$&Quintic couplings&$B$\\
$[\beta^{(1)}]^2\,[\beta^{(2)}]^2\,\alpha^{(3)}$
&$3$
&$[\beta^{(1)}]^3\,\beta^{(2)}\,\beta^{(3)}$
&$3$\\
$[\beta^{(1)}]^2\,[\beta^{(3)}]^2\,\alpha^{(2)}$
&$3$
&$[\beta^{(2)}]^3\,\beta^{(1)}\,\beta^{(3)}$
&$2+{\tau+\bar\tau\over2\pi}\,\widehat G_2(\tau)$\\
$[\beta^{(2)}]^2\,[\beta^{(3)}]^2\,\psi$
&$1+{\tau+\bar\tau\over\pi}\,\widehat G_2(\tau)$
&$[\beta^{(3)}]^3\,\beta^{(1)}\,\beta^{(2)}$
&$2+{\tau+\bar\tau\over2\pi}\,\widehat G_2(\tau)$\\
\end{tabular}
\label{QuinticBs}$$ Concerning the calculated values of the $B$ terms, we should note that $\widehat
G_2(\tau)\approx{\pi^2\over3}(1-24e^{-2\pi\tau})-2\pi/(\tau+\bar\tau)$ with $\widehat G_2(\tau=1)=0$ [@Cvetic]. Also, for values of $\tau$ in the fundamental domain (, $\tau\ge1$ if $\tau$ is real) $\widehat
G_2(\tau)\ge0$. Therefore, the integer values of $B$ shown in Eqs. (\[QuarticBs\],\[QuinticBs\]) are actually the minimum possible values.
Field normalizations and Yukawa couplings {#Field}
=========================================
The superpotential couplings in free-fermionic models are easily calculable in the string basis. Moreover, in specific models the fermion Yukawa couplings have led to structures which bear close resemblance to the observed hierarchical fermion mass spectrum. On the other hand, results about the Yukawa couplings in the string model are not necessarily directly related to those which would be observed at low energies. The possible snag lies in the normalization of the fields in the supergravity lagrangian. For the scalar fields the relevant term is $$K_{ij}\partial_\mu\phi_i\partial^\mu\bar\phi_j\ .
\label{kinetic-terms}$$ If the Kähler potential ($K$) is non-trivial, then the scalar fields would need to be normalized appropriately. If this effect propagates to the Yukawa couplings, the physical ones may differ from those naively expected.
We seek a matrix $A$ such that $$\phi=A\widehat\phi\ ,\qquad \bar\phi=\bar A\widehat{\bar\phi}\ ,
\label{Adef}$$ where $\widehat\phi,\widehat{\bar\phi}$ are the properly normalized fields. From the condition $\partial_\mu\phi^T
K\partial^\mu\bar\phi=\partial_\mu\widehat\phi^TA^TK\bar
A\partial^\mu\widehat{\bar\phi}=\partial_\mu\widehat\phi^T
\partial^\mu\widehat{\bar\phi}$ we obtain $$A=(K^{-1/2})^T\ ,\qquad \bar A=K^{-1/2}\ .
\label{AA}$$ Here $K^{-1/2}$ is the matrix obtained by taking the square root of $K^{M\bar N}$, where $K^{M\bar N}$ has been used above in calculating, , $K^I=K^{I\bar J}K_{\bar J}$. If $K^{-1/2}$ is obtained, one can determine the physical Yukawa couplings (which couple properly normalized fields) from the expression [@BIM; @FKZ] $$\widehat\lambda_{ijk}=e^{K/2}\,(K^{-1/2})_{ii'}\,(K^{-1/2})_{jj'}\,
(K^{-1/2})_{kk'}\,\lambda_{i'j'k'}\ ,
\label{yuks}$$ where $\lambda_{ijk}$ are the couplings which appear in the superpotential of the string model. That is, $\widehat W(\widehat\phi)=e^{K/2}\,W(\phi)$.
Calculating the square root of a matrix can be a complicated task. Before we attempt this, it is quite illuminating to consider the limit of small matter field vevs. To leading order we only keep the diagonal contributions to $K_{M\bar N}$, and obtain for the untwisted matter fields $$\begin{aligned}
&\tau_1,\tau_2\ {\rm set:}\qquad &[K^{-1/2}]^{(\tau_1,\tau_2)}
\approx{\rm diag}\,[(\tau_1+\bar\tau_1),(\tau_2+\bar\tau_2),\nonumber\\
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&&\qquad\qquad\qquad\qquad\qquad\coeff{1}{\sqrt{2}}(\tau_1+\bar\tau_1)^{1/2}(\tau_2+\bar
\tau_2)^{1/2}{\bf1}_{n_\phi}]\ ,\label{K-1/2TU}\\
&\tau\ {\rm set:}\qquad &[K^{-1/2}]^{(\tau)}
\approx{\rm diag}\,[\coeff{1}{\sqrt{2}}(\tau+\bar\tau),
\coeff{1}{\sqrt{2}}(\tau+\bar\tau){\bf1}_{n_\psi}]
\ ,\label{K-1/2tau}\\
&{\rm \alpha\ set:}\qquad &[K^{-1/2}]^{(\alpha)}\approx{\rm
diag}\,[{\bf1}_{n_I}]\ ,\label{K-1/2alpha}\end{aligned}$$ where ${\bf1}_n$ is a vector with $n$ unit entries. In the case of the model with Kähler potential given in Eq. (\[completeK\]), and in this same approximation, we obtain the explicit normalized fields as follows $$\begin{aligned}
\widehat S&=&{S\over\VEV{S+\bar S}}\ ,\quad
\widehat\tau={\sqrt{2}\,\tau\over\VEV{\tau+\bar\tau}}\ ,\quad
\widehat\psi_i={\sqrt{2}\,\psi_i\over\VEV{\tau+\bar\tau}}\ ;
\label{normalizations1}
\\
\widehat\alpha^{(2)}_i&=&\alpha^{(2)}_i\ ,\quad
\widehat\alpha^{(3)}_i=\alpha^{(3)}_i\ ,\quad
\widehat\beta^{(1)}_i=\beta^{(1)}_i\ ;
\label{normalizations2}
\\
\widehat\beta^{(2)}_i&=&{\beta^{(2)}_i\over\VEV{\tau+\bar\tau}^{1/2}}\ ,
\qquad
\widehat\beta^{(3)}_i={\beta^{(3)}_i\over\VEV{\tau+\bar\tau}^{1/2}}\ .
\label{normalizations3}\end{aligned}$$
Now let us determine the properly normalized cubic Yukawa couplings. From Eq. (\[completeK\]) we have $\VEV{e^{K/2}}=1/[(S+\bar
S)^{1/2}(\tau+\bar\tau)]$. For the various types of cubic couplings in Eq. (\[couplings\]) we write generically $\widehat\lambda=f\lambda$, with the normalization factor $f$ given by $$\begin{tabular}{rr}
$\psi\alpha^{(2)}\alpha^{(3)}$ :
&$e^{K/2}\,\coeff{1}{\sqrt{2}}(\tau+\bar\tau)\,(1)\,(1)=\coeff{1}{2}g$\\
$\psi\beta^{(1)}\beta^{(1)}$ :
&$e^{K/2}\,\coeff{1}{\sqrt{2}}(\tau+\bar\tau)\,(1)\,(1)=\coeff{1}{2}g$\\
$\alpha^{(2)}\beta^{(2)}\beta^{(2)}$ :
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&$e^{K/2}\,(1)\,(\tau+\bar\tau)^{1/2}\,(\tau+\bar\tau)^{1/2}=\coeff{1}{\sqrt{2}}g$\\
$\alpha^{(3)}\beta^{(3)}\beta^{(3)}$ :
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&$e^{K/2}\,(1)\,(\tau+\bar\tau)^{1/2}\,(\tau+\bar\tau)^{1/2}=\coeff{1}{\sqrt{2}}g$\\
$\beta^{(1)}\beta^{(2)}\beta^{(3)}$ :
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&$e^{K/2}\,(1)\,(\tau+\bar\tau)^{1/2}\,(\tau+\bar\tau)^{1/2}=\coeff{1}{\sqrt{2}}g$
\end{tabular}
\label{NormYuks}$$ where we have used the result $g^2=1/{\rm Re}\, S$. We conclude that all the normalized cubic Yukawa couplings are independent of the moduli,[^9] but depend on the dilaton (or the gauge coupling).
Our assumption that the $K^{-1/2}$ matrices are nearly diagonal can be justified by studying some simple cases where the calculation can be done exactly. For instance, let us take a $\tau$ set with $n_\psi=1$, which amounts to a $2\times2$ matrix. The two matrices of relevance are $$K_{M\bar N}=\coeff{2}{X^2}\left(
\begin{array}{cc}a^2+b^2&-4ab\\-4ab&a^2+b^2\end{array}\right)\ ,
\qquad
K^{-1}=\coeff{1}{2}\left(
\begin{array}{cc}a^2+b^2&2ab\\2ab&a^2+b^2\end{array}\right)\ ,
\label{2x2}$$ where $a=\tau+\bar\tau$, $b=\psi+\bar\psi$, and $X=a^2-b^2$. From $K^{-1}$ we can compute the square root, $$K^{-1/2}=\coeff{1}{\sqrt{2}}\left(
\begin{array}{cc}a&b\\b&a\end{array}\right)\ ,
\label{K-1/2}$$ and therefore $$\left(\begin{array}{c}\tau\\ \psi\end{array}\right)=
\coeff{1}{\sqrt{2}}\left(
\begin{array}{cc}\tau+\bar\tau&\psi+\bar\psi\\ \psi+\bar\psi&\tau+\bar\tau
\end{array}\right)\left(\begin{array}{c}\widehat\tau\\
\widehat\psi\end{array}\right)\ ,
\label{normalized}$$ which agrees with Eq. (\[K-1/2tau\]) in the limit $\VEV{\psi+\bar\psi}/\VEV{\tau+\bar\tau}\to0$. However, the exact expression allows us to study the possibility of [*moduli-matter*]{} mixing. For the all-untwisted superpotential coupling $\lambda_{\psi\alpha\alpha}\psi\alpha\alpha$, we find in an obvious notation $$\begin{aligned}
\widehat\lambda_{\widehat\psi\widehat\alpha\widehat\alpha}&=&
{\coeff{1}{\sqrt{2}}(\tau+\bar\tau)\,\lambda_{\psi\alpha\alpha}
\over (S+\bar S)^{1/2}[(\tau+\bar\tau)^2-(\psi+\bar\psi)^2]^{1/2}}\ ,\\
\label{paa}
\widehat\lambda_{\widehat\tau\widehat\alpha\widehat\alpha}&=&
{\coeff{1}{\sqrt{2}}(\psi+\bar\psi)\,\lambda_{\psi\alpha\alpha}
\over (S+\bar S)^{1/2}[(\tau+\bar\tau)^2-(\psi+\bar\psi)^2]^{1/2}}\ .
\label{taa}\end{aligned}$$ The novelty here is a new (although small) Yukawa coupling between matter ($\widehat\alpha$) and moduli ($\widehat\tau$) fields, of order $\widehat\lambda_{\widehat\tau\widehat\alpha\widehat\alpha}
={\cal O}[(\psi+\bar\psi)/(\tau+\bar\tau)]$. Otherwise, the results derived above in the diagonal approximation are quite accurate.
The above exact calculation for $n_\psi=1$ does not allow quantification of possible [*matter-matter*]{} mixing through field normalizations. One can repeat the exercise for $n_\psi=2$ to study the magnitude of such mixings. The square root of such $3\times3$ matrix can be readily obtained by the use of Sylvester’s formula [@Sylvester] $$P(U)=\sum_{r=1}^n P(\lambda_r)\prod_{j\not=r}{\lambda_j
I-U\over\lambda_j-\lambda_r}\ ,
\label{Sylvester}$$ where $U$ is the given matrix, $P$ is the required operation (square root in our case), and $\lambda_r$ are the eigenvalues of $U$. Mathematica can be programmed to calculate $K^{-1/2}$ using Sylvester’s formula, but the result is messy. In the limit of interest we find, for example $$\widehat\lambda_{\widehat\psi_1\widehat\alpha\widehat\alpha}\approx
{\coeff{1}{\sqrt{2}}\lambda_{\psi_1\alpha\alpha}\over(S+\bar S)^{1/2}}
+{\coeff{1}{\sqrt{2}}\lambda_{\psi_2\alpha\alpha}\over(S+\bar S)^{1/2}}
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\left[{\coeff{1}{\sqrt{2}}(\psi_1+\bar\psi_1)\coeff{1}{\sqrt{2}}(\psi_2+\bar\psi_2)\over(\tau+\bar\tau)^2}\right]^2\ .
\label{mm}$$ The possible matter-matter mixing is therefore very small ${\cal O}[(\psi+\bar\psi)/(\tau+\bar\tau)]^4$. Matter-matter mixing through the Kähler potential (as in the case of the $\mu$ term) or through the superpotential are therefore the realistic possible sources of mixing. For $n_\psi=2$ the matter-moduli mixing is also highly suppressed ${\cal
O}[(\psi+\bar\psi)/(\tau+\bar\tau)]^2$.
Possible realistic models {#Possible}
=========================
In the previous sections we have explored what consequences would string no-scale supegravity models have regarding the soft-supersymmetry-breaking terms and the low-energy Yukawa couplings. However, it remains to be shown that such models actually exist, , that the first postulate of string no-scale supergravity is satisfied. Here we describe a search for such models, and the properties of any that may be found.
The search for models
---------------------
We have performed a computerized search of free-fermionic string models with the desired properties. The two cases in Eq. (\[V0=0\]) indicate that we should look for models which [*effectively*]{} possess one untwisted set with moduli (a $\tau_1,\tau_2$ set or a $\tau$ set) and the other two untwisted sets with all moduli projected out. As discussed in Sec. \[Generalities\], this determination should be done after the cubic superpotential of the model is calculated, since we would discard “moduli" which have superpotential couplings.[^10]
Let us consider the kind of truncations of the moduli space which could occur if moduli have superpotential couplings. If we start off with a $\tau$ set, all that can happen is that the modulus field appears in $W$, and therefore the set would effectively become an $\alpha$ set. In the case of a $\tau_1,\tau_2$ set, if both fields appear in $W$ we have an $\alpha$ set, whereas if only one appears we have a $\tau$ set. In the latter case ($\tau_1,\tau_2\,{\rm set}\to\tau\,{\rm set}$) we would perform the field redefinition in Eq. (\[taudef\]), instead of that in Eq. (\[TUdef\]).
With the restriction that we obey Eqs. (\[V0=0\]) after moduli present in $W$ are discarded, we have performed a search for free-fermionic models following the methods described in Ref. [@search]. A free-fermionic model is specified by a basis of $n$ basis vectors of boundary conditions, plus an $n\times n$ matrix of GSO projections (the “$k$-matrix"). Our search is based on the reasonable assumption that the basis vectors of the fermionic model contain five “standard" vectors which have appeared in all known models of this kind, these are denoted by ${\bf1},S,b_1,b_2,b_3$. Since we are interested in models with $SU(5)\times U(1)$ observable gauge symmetry, we also assume the presence of two other vectors ($b_4,b_5$) which have been used in the two $SU(5)\times U(1)$ string models in the literature [@revamped; @search]. The eighth and last vector, called $\alpha$, is decisive. In Ref. [@search] this vector was allowed to take all possible values consistent with the free-fermionic model-building rules. In addition, the $k$-matrix was varied at random. This search was specifically focused on finding models which allow unification of the low-energy gauge couplings at the string scale. As such, the model had to contain five ’s and two ’s of $SU(5)$ (a “5/2 model"), as opposed to the original “revamped" model which is a “4/1 model". In fact, one such 5/2 model was found, which we will call the “search" model. The “search" model is in fact an example of a class of 5/2 models, with slightly varying properties.
Our purpose here is somewhat different, since the search is more constrained. We seek either 4/1 or 5/2 models with a single [*effective*]{} untwisted $\tau_1,\tau_2$ or $\tau$ set (and two effective untwisted $\alpha$ sets). Our search procedure consists of picking representative $\alpha$ vectors and varying the $k$-matrix at random.
- $\alpha=\alpha_{\rm search}$. In 10,000 $k$-matrices we find $\sim6\%$ $N=1$ supersymmetric models, of which 19 are 5/2 models and none are 4/1 models. All these models possess two $\tau_1,\tau_2$ sets and one $\alpha$ set, since untwisted (Neveu-Schwarz) states like the moduli depend only on the choice of basis vectors, and not on the $k$-matrix. Calculation of the cubic superpotential reveals that the 5/2 models divide into two “Yukawa sets": 1/3/2 and 2/3/2,[^11] with the 2/3/2 case preferred phenomenologically [@search]. Moreover, we discover that [*all*]{} models with the 1/3/2 Yukawa set have their $\tau_1,\tau_2$ sets broken down to $\tau$ sets, whereas [*all*]{} the models with the preferred 2/3/2 Yukawa set have one $\tau_1,\tau_2$ set broken to an $\alpha$ set and the other $\tau_1,\tau_2$ set broken to a $\tau$ set. Therefore, the 2/3/2 models (like the “search" model) possess precisely the desired moduli content. (The 1/3/2 models give $V_0>0$.) Schematically $$\begin{array}{c}
{\rm 5/2\ models}\\ \alpha=\alpha_{\rm search}\end{array}
\qquad\left\{
\begin{tabular}{ccc}
Yukawa sets& Moduli&$V_0$\\
1/3/2& $\left\{\begin{array}{l}\tau_1\tau_2\to\tau\\ \tau_1\tau_2\to\tau\\
\alpha\to\alpha\end{array}\right.$&$V_0>0$\\
&&\\
2/3/2& $\left\{\begin{array}{l}\tau_1\tau_2\to\tau\\ \tau_1\tau_2\to\alpha\\
\alpha\to\alpha\end{array}\right.$&$V_0=0$\\
\end{tabular}
\right.
\label{5/2}$$
- $\alpha=\alpha_{\rm revamped}$. In 5,000 $k$-matrices we find 41 4/1 models and no 5/2 models. All these models contain one $\tau_1,\tau_2$ set and two $\tau$ sets. The 4/1 models come with two possible Yukawa sets (1/3/3 and 2/3/3, the latter is preferred), and in [*all*]{} instances we find that the $\tau$ sets are unbroken, whereas the $\tau_1,\tau_2$ set is broken to an $\alpha$ set, , $$\begin{array}{c}
{\rm 4/1\ models}\\ \alpha=\alpha_{\rm revamped}\end{array}
\qquad\left\{
\begin{tabular}{ccc}
Yukawa sets& Moduli&$V_0$\\
$\begin{array}{c}1/3/3\\ 2/3/3\end{array}$&
$\left\{\begin{array}{l}\tau_1\tau_2\to\alpha\\ \tau\to\tau\\
\tau\to\tau\end{array}\right.$&$V_0>0$\\
\end{tabular}
\right.
\label{4/1}$$
- $\alpha=\alpha_{\rm 3a}$. We choose two other $\alpha$ vectors which belong to “class 3a" in the notation of Ref. [@search]. Models with $\alpha$ vectors in this class are expected to be 4/1 models ($\alpha_{\rm
revamped}$ belongs to this class). We obtain the same result as in Eq. (\[4/1\]).
- $\alpha=\alpha_{\rm price}$. Unlike our previous choices for $\alpha$, $\alpha_{\rm price}$ (introduced in Ref. [@price]) produces both 4/1 and 5/2 models. In this case the three sets are $\tau$ sets and the three moduli appear in the superpotential, , $$\begin{array}{c}
{\rm 5/2,\,4/1\ models}\\ \alpha=\alpha_{\rm price}\end{array}
\qquad\left\{
\begin{tabular}{cc}
Moduli&$V_0$\\
$\begin{array}{l}\tau\to\alpha\\\tau\to\alpha\\\tau\to\alpha\end{array}$
&$V_0<0$\\
\end{tabular}
\right.
\label{price}$$ These models are not realistic, but we consider them since we want to establish a connection between the value of $V_0$ and the 4/1 or 5/2 nature of a model.
- [*Change $b_4,b_5$*]{}. We finally allow for changes in the core basis, in addition to varying $\alpha$. In this case 4/1 and 5/2 models are found, although very unappealing ones (, with no Yukawa couplings!). Nonetheless, a sample case yields $$\begin{array}{c}
{\rm 5/2,\,4/1\ models}\\ {\rm change\ b_4,b_5}\end{array}
\qquad\left\{
\begin{tabular}{cc}
Moduli&$V_0$\\
$\begin{array}{l}\tau\to\tau\\\tau\to\tau\\\tau\to\alpha\end{array}$
&$V_0>0$\\
\end{tabular}
\right.
\label{changeb4b5}$$
The above search for models, although limited in extent, provides support for the following $${\rm Conjecture:}\qquad {\rm 4/1\ models\ always\ give}\ V_0\not=0\ .
\label{conjecture}$$ This would imply that a [*necessary*]{} condition for $SU(5)\times U(1)$ string no-scale supergravity models is a 5/2 field content. This condition is consistent with the string-theory nature of the model which requires unification at the string scale, which can be accomplished in a 5/2 model. Moreover, a realistic model which satisfies the postulates of string no-scale supergravity already exists, namely the “search" model of Ref. [@search].
A realistic example
-------------------
As we just saw, the “search" model of Ref. [@search] is a good candidate for a string no-scale supergravity model, with a single effective untwisted $\tau$ set and a Kähler potential of the general form given in Eq. (\[completeK\]). With this information it should be possible to get a good idea of what the spectrum of sparticles may look like. First of all, we note that any model of this kind would be a [*no-parameter model*]{} [@LNZprep]. That is, the whole supersymmetric spectrum would be unambigously determined. Indeed, with the ability to compute all soft-supersymmetry-breaking parameters (including $\mu$ and $B$) in terms of $m_{3/2}$, the high-energy theory would be determined up to the value of $m_{3/2}$. At low energies one new parameter arises, namely $\tan\beta$, but one also has two radiative electroweak symmetry breaking conditions which therefore allow one to determine $m_{3/2}$ and $\tan\beta$. The no-scale mechanism can then be used to compute the quantity $C$ in Eq. (\[V1def\]). Thus, if the radiative breaking conditions can be solved, we would have a complete determination of the sparticle spectrum. In this exercise the top-quark mass is not an independent parameter: the top-quark Yukawa coupling at the string scale is a hallmark prediction of string models [@t-paper],[^12] and the value of $\tan\beta$ would be self-consistently determined by the radiative breaking conditions.
Let us first address the question of the value of $Q$ in the “search" model of Ref. [@search]. The formula in Eq. (\[Qformula\]) is $Q=n_{U2}+n_{U3}+n_{T1}-n_\psi-d_f-3$. The gauge group is $SU(5)\times
SO(10)\times SU(4)\times U(1)^6$ which gives $d_f=90$. The number of untwisted and twisted fields is: $n_\psi=13,n_{U2}=14,n_{U3}=16$ and $n_{T1}=80,n_{T2}=80,n_{T3}=68$, where we count $p$-dimensional representations of the gauge group as $p$. Putting it all together gives $$Q=14+16+80-13-90-3=110-106=4\ ,
\label{Q}$$ which is remarkably close to the desired zero result ([*c.f.*]{}, if all terms were to be added in magnitude, we are off by 2%).[^13] Pragmatically speaking $Q\not=0$ and a destabilizing one-loop correction to the scalar potential is expected. However, given our incomplete knowledge of string dynamics (, the role played by the anomalous $U_A(1)$) and of additional contributions to $Q$ from massive string states [@FKZ] and string loop corrections, we are not ready to discard this model hastily. For instance, if the one-loop corrections to the gaugino masses (see Eq. (\[Ma\])) increased them by $4.4\%$, we would obtain $Q=0$. Such small string one-loop shifts on the scalar and gaugino masses are expected and quantify our statement that $Q=4$ is a “small" number. We will therefore proceed exploring the manifold observable implications of this model, carrying the $Q=4$ result as a warning flag.
Considering the spectrum of the “search" model of Ref. [@search], the various relevant observable fields (and the sets they belong to) are as follows $$\begin{tabular}{lccc}
Set&Untwisted fields&Twisted fields\\
First&$\Phi_0,\Phi_1;\,h_1,\bar h_1$&$F_0,F_1,F_4,\bar F_4$\\
Second&$h_2,\bar h_2$&$F_2,\bar f_2,l^c_2;\,\bar F_5,\bar
f_5,l^c_5$\\
Third&$\Phi_3,\Phi_5;\,h_3,\bar h_3$&$F_3,\bar f_3,l^c_3;\, h_{45},\bar h_{45}$
\end{tabular}
\label{set-states}$$ The possible moduli are $\Phi_{0,1,3,5}$, of which all but $\Phi_1$ appear in the cubic superpotential (given in Eq. (6.3a) in Ref. [@search]). Therefore, the first set is a $\tau$ set, whereas the other two are $\alpha$ sets. In Ref. [@search] it was argued that $F_4=\{Q_4,d^c_4,\nu^c_4\}$ should contain the third generation squarks, whereas $F_0,F_1,\bar F_4$ contain either Higgs particles or intermediate scale particles, therefore the first and second generation squarks and sleptons belong to the second and third twisted sets. Moreover, the light Higgs boson doublets are located inside $h_1$ and $\bar h_{45}$, which in the usual notation correspond to $H_1,H_2$ respectively. Contrasting Eq. (\[set-states\]) with the general result for the scalar masses in this class of models in Eq. (\[ScalarMasses\]) we obtain the following spectrum of supersymmetry-breaking scalar masses: $$\begin{aligned}
&{\rm First\ generation:}&m^2_{Q_1,U_1^c,D_1^c,L_1,E_1^c}=0\
,\label{first}\\
&{\rm Second\ generation:}&m^2_{Q_2,U_2^c,D_2^c,L_2,E_2^c}=0\
,\label{second}\\
&{\rm Third\ generation:}&\left\{
\begin{array}{l}
m^2_{Q_3,D^c_3}=m_{3/2}\\
m^2_{U^c_3,L_3,E^c_3}=0
\end{array}
\right.\ ,\label{third}\\
&{\rm Higgs\ masses:}&m^2_{H_1}=0,\quad m^2_{H_2}=0\ . \label{higgses}\end{aligned}$$ We also know that the gaugino masses are degenerate $m_{1/2}=m_{3/2}$ (see Eq. (\[m1/2\])), and that the $A$ parameter is universal $A=m_{3/2}$ (see Eq. (\[A\])). The $\mu$ parameter is expected to arise at the quintic level in the superpotential (no suitable terms exist at the quartic level). Since one of the light Higgs doublets belongs to the first untwisted set ($h_1$) and the other one to the third twisted set ($\bar h_{45}$), Eq. (\[NRTs\]) singles out only one possible type of quintic term: $[\beta^{(2)}]^2\,[\beta^{(3)}]^2\,\psi$. Moreover, from Eq. (\[QuinticBs\]) we get $B=[1+{\tau+\bar\tau\over\pi}\,\widehat G_2(\tau)]m_{3/2}$, which has a minimum at $B=m_{3/2}$. In sum, without identifying the specific quintic term giving rise to $\mu$, we can nonetheless predict the corresponding $B$ parameter. Therefore, our no-parameter model reduces in practice to a one-parameter model until we compute such a quintic term.
An important ingredient in the viability of our candidate model is that the extra (,) matter representations have suitable masses to allow gauge coupling unification at the string scale. In fact, the $Q$ and $D^c$ components of these representations should acquire different masses ($\sim10^{12}\GeV$ and $\sim10^6\GeV$ respectively [@LNZI]), but this is allowed since $SU(5)\times U(1)$ is broken at the string scale. Morever, the $Q$ mass scale is determined by an effective superpotential mass term, whereas the $D^c$ mass scale is obtained by working out the eigenvalues of the extended Higgs triplet mass matrix [@search].
To summarize, the above candidate model has several notable properties: (i) a $Q$ parameter tantalizingly close to zero, (ii) a benign non-universal spectrum of supersymmetry-breaking scalar masses which bears close resemblance to the old “no-scale" result, (iii) a universal trilinear term $A=m_{3/2}$, (iv) a $\mu$ parameter at the quintic level of superpotential interactions associated with $B\ge m_{3/2}$, and (v) extra , representations dynamically required and likely to possess the desired mass spectrum. The low-energy consequences of this model will be explored in detail elsewhere [@LNZprep].
Conclusions {#Conclusions}
===========
In this paper we have explored the postulates of string no-scale supergravity in the context of free-fermionic string models. These postulates are not trivially satisfied, and in fact impose important new restrictions on string model building. In particular the moduli sector should be rather minimal, and the massless matter spectrum and gauge group should be correlated in a way such that the parameter $Q$ vanishes. We have given plausibility arguments indicating that this condition may be possible to satisfy in specific models, and in fact presented a model where $Q$ is very close to zero. For all (untwisted and twisted) matter fields we have computed explicitly the associated supersymmetry breaking parameters. Models of this kind are in fact “no-parameter" models, with all needed masses and couplings completely determined. A search for free-fermionic models which satisfy the minimal necessary conditions yielded one candidate model with the $SU(5)\times U(1)$ observable gauge group, calculated novel supersymmetry breaking parameter space, and several desirable properties regarding the stability of the no-scale mechanism. This search also appears to imply that viable $SU(5)\times U(1)$ models always contain additional matter representations that allow unification at the string scale.
It is interesting to remark that the models studied in this paper possess a goldstino composition with significant admixtures of both “dilaton" and “moduli". This hybrid scenario borrows desirable features from the two extremes, where either one or the other dominates the goldstino field. From the dilaton admixture we get a tree-level contribution to the gaugino mass, and from the moduli admixture (or more properly old no-scale) we get universality of scalar masses in the subset of fields where it is desired.
We should also note that the no-scale supergravity realized in free-fermionic models \[$SO(2,n)/SO(2)\times SO(n)$\] differs in structure from that in the old no-scale models \[$SU(n,1)/U(1)\times SU(n)$\]. The effects of the different structures is most evident in the computation of the vacuum energy and in the computation of the supersymmetry breaking parameters.
Throughout our discussion we have said little about the supersymmetry breaking mechanism which creates $\VEV{W}\not=0$, thus implicitly assuming that its precise nature would not affect our results. In gaugino condensation models, the non-perturbative superpotential depends explicity on $S$ and fixes its value, but this $S$-dependence is likely to also affect the calculation of $V_0$. Nonetheless, it may be possible to retain all the desirable no-scale supergravity properties [@FKZ]. In coordinate-dependent compactifications we expect our results to hold since $W$=constant, although here the question is: what determines $S$? The no-scale mechanism extended to the $S$ field would appear to answer this question. A third possibility to obtain $\VEV{W}\not=0$ appears possible in models with an anomalous $U_A(1)$. In this case various singlet fields would acquire vacuum expectations values $\vev{\phi}/M\sim1/10$ and a cubic term in the superpotential would give $\VEV{W}=(\vev{\phi}/M)^3\sim10^{-3}$, if the flatness conditions can be simultaneously satisfied. The crucial question in the phenomenologically viability of these supersymmetry breaking mechanisms is the whether the calculated value of $m_{3/2}$ is of electroweak scale size, since all supersymmetry-breaking parameters are proportional to $m_{3/2}$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Costas Kounnas for useful and encouraging discussions and for reading the manuscript. We would also like to thank Kajia Yuan for useful discussions at the earlier stages of this project, for reading the manuscript, and for consultations regarding the twisted sector Kähler potential. This work has been supported in part by DOE grant DE-FG05-91-ER-40633.
Field redefinitions {#Redefinitions}
===================
As discussed in Sec. \[Fermionic\], the modular properties of the theory are evident in the supergravity basis, while the direct string calculations are in the string basis. Here we discuss how one goes from one basis to the other, or more specifically, how Eqs. (\[TUdef\]) and (\[taudef\]) are obtained from Eq. (\[Kdef\]). Our purpose is to identify a transformation which leaves the Kähler function ($G$) invariant. In this way all results which follow from it will not depend on the transformation. Some discussion of the relevant transformation has been given in Ref. [@FKPZII].
The main observation is that the two forms of the untwisted Kähler potential are simply two different parametrizations of the metric for the same coset space: $SO(2,n)/SO(2)\times SO(n)$. Following Gilmore [@Gilmore], we introduce complex variables $t_j=x_j+iy_j$ with $1\le j\le n+2$, which describe the coset space of dimension $2n$ (two of the variables are auxiliary). The first parametrization is in terms of the variables $\alpha_j$ $$\alpha_j={t_j\over t_{n+1}-it_{n+2}}\ ,\qquad 1\le j\le n\ ,
\label{alphas}$$ which have the following properties [@Gilmore] $$\begin{aligned}
\left|\sum_{j=1}^n\alpha^2_j\right|&<&1\ , \label{prop1}\\
Y(\alpha)=
1-2\sum_{j=1}^n\alpha_j\bar\alpha_j+\left|\sum_{j=1}^n\alpha^2_j\right|^2&=&
{4\over|t_{n+1}-it_{n+2}|^2}>0\ .\label{prop2}\end{aligned}$$ This parametrization corresponds to that given in Ref. [@LNY94] and used in Eq. (\[Kdef\]). In fact, $Y(\alpha)$ is the argument of the logarithm in the Kähler potential. The second parametrization is in terms of the variables $\beta_k$ $$\beta_k={t_k\over t_1-t_{n+2}}\ ,\qquad 2\le k\le n+1\ ,
\label{betas}$$ which have the property $$Y(\beta)=\sum_{k=2}^n(\beta_k-\bar\beta_k)^2-(\beta_{n+1}-\bar\beta_{n+1})^2=
{4\over|t_{n+2}-t_1|^2}>0\ .
\label{prop}$$ Performing the phase transformation $\beta_k\to i\beta_k$, we get $$Y(\beta)\to
Y(\beta)=(\beta_{n+1}+\bar\beta_{n+1})^2-\sum_{k=2}^n(\beta_k+\bar\beta_k)^2\ ,
\label{Ytau}$$ which reproduces Eq. (\[taudef\]) with the identifications $\beta_{n+1}\to\tau$, $\beta_k\to\psi_k$. Moreover, we can rewrite Eq. (\[Ytau\]) as follows $$\begin{aligned}
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Y(\beta)&=&(\beta_{n+1}+\bar\beta_{n+1})^2-(\beta_{n}+\bar\beta_{n})^2-\sum_{k=2}^{n-1}(\beta_k+\bar\beta_k)^2\nonumber\\
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&=&(\tau_1+\bar\tau_1)(\tau_2+\bar\tau_2)-\sum_{i=1}^{n_\phi}(\phi_i+\bar\phi_i)^2
\label{YTU}\end{aligned}$$ where the second equality follows from the identifications $\beta_{n+1}+\beta_n\to \tau_1$ and $\beta_{n+1}-\beta_n\to \tau_2$. This result reproduces Eq. (\[TUdef\]).
We now show that the transformation $\alpha\leftrightarrow
t\leftrightarrow\beta$ leaves $G$ unchanged. Let us write $e^G$ as follows $$e^G={|W|^2\over Y_0Y_1Y_2Y_3}={\lambda_{ijk}\lambda^*_{i'j'k'}\over Y_0}\,
{\alpha^{(1)}_i\bar\alpha^{(1)}_{i'}\over Y_1(\alpha)}\,
{\alpha^{(2)}_j\bar\alpha^{(2)}_{j'}\over Y_2(\alpha)}\,
{\alpha^{(3)}_k\bar\alpha^{(3)}_{k'}\over Y_3(\alpha)}\ ,
\label{e^G}$$ where $Y_0=(S+\bar S)$, $Y_{1,2,3}=e^{-K_{(1,2,3)}}$ with the $K$’s in Eq. (\[Kdef\]), and the superpotential is $W=\lambda_{ijk}\alpha^{(1)}_i\alpha^{(2)}_j\alpha^{(3)}_k$. Now we note that from Eqs. (\[betas\]),(\[prop\]) we can write $${\beta_i\bar\beta_{i'}\over Y(\beta)}={t_it_{i'}\over|t_1-t_{n+2}|^2}\cdot
{|t_{n+2}-t_1|^2\over4}=\coeff{1}{4}t_i\bar t_{i'}\ ,
\label{bb}$$ whereas from Eqs. (\[alphas\]),(\[prop2\]) we can write $${\alpha_i\bar\alpha_{i'}\over
Y(\alpha)}={t_it_{i'}\over|t_{n+1}-it_{n+2}|^2}\cdot
{|t_{n+1}-it_{n+2}|^2\over4}=\coeff{1}{4}t_i\bar t_{i'}\ .
\label{aa}$$ Therefore we conclude that $\beta_i\bar\beta_{i'}/Y(\beta)=\alpha_i\bar\alpha_{i'}/Y(\alpha)$, and thus Eq. (\[e\^G\]) shows that $e^G$ remains invariant when written in terms of the $\beta$ variables. Note that the transformation involves the Kähler potential [*and*]{} the superpotential, and that the superpotential has the same couplings when written in terms of the $\beta$ variables. It appears unnecessary to relate the $\alpha$ to the $\beta$ variables directly (, eliminating $t$), although this has apparently been done in Ref. [@FKPZII].
Example of twisted sector Kähler potential {#AppB}
==========================================
The twisted sector contribution to the Kähler potential in free-fermionic models was calculated in Ref. [@oldTS] for a simple model with $N=1$ spacetime supersymmetry and fermionic basis ${\cal
B}=\{{\bf1},S,b_1,b_2,b_3\}$. This model has only three (massless) twisted sectors: $b_1,b_2,b_3$. The result, obtained to lowest order in the twisted fields, is $$K_{\rm TS}=
\sum_i^{n_{T1}} \beta^{(1)}_i\bar\beta^{(1)}_i\ e^{{1\over2}[K_{(2)}+K_{(3)}]}
+\sum_i^{n_{T2}} \beta^{(2)}_i\bar\beta^{(2)}_i\ e^{{1\over2}[K_{(1)}+K_{(3)}]}
+\sum_i^{n_{T3}} \beta^{(3)}_i\bar\beta^{(3)}_i\ e^{{1\over2}[K_{(1)}+K_{(2)}]}
\label{oldKTS}$$ where $\beta^{(1,2,3)}$ are the twisted fields (numbering $n_{T1,T2,T3}$) in the $b_{1,2,3}$ sectors, and $K_{(1,2,3)}$ are the contributions to the Kähler potential from the untwisted fields as given in Eq. (\[Kdef\]). Realistic free-fermionic models contain more basis vectors and a great deal more massless twisted sectors. For instance, the “search" model discussed above has basis ${\cal B}=\{{\bf1},S,b_1,b_2,b_3,b_4,b_5,\alpha\}$ and 22 massless twisted sectors. Despite this enlargement of the model, we conjecture that the structure of the twisted sector Kähler potential remains as simple as in Eq. (\[oldKTS\]) once we generalize the concept of twisted sector to “twisted set" with the meaning given in Sec. \[Generalities\]. In what follows we prove this conjecture by explicit calculation in the context of the “search" model.[^14]
We start by listing all of the massless states of the model divided into untwisted and twisted fields and by the set they belong to $$\begin{tabular}{lcc}
Set&Untwisted fields&Twisted fields\\
&$\Phi_0,\Phi_1$&$F_0,F_1$\quad {\small$[b_1]$}\\
First&$\Phi_{23},\bar\Phi_{23}$&$F_4,\bar F_4$\\
&$h_1,\bar h_1$&$\F_3,\Fb_{1,2,4},D_{1,2,5,6}$\\
&&\\
&$\eta_1,\bar\eta_1$&$F_2,\bar f_2,l^c_2$\quad {\small$[b_2]$}\\
Second&$\Phi_{31},\bar\Phi_{31}$&$\bar F_5,\bar f_5,l^c_5$\\
&$h_2,\bar h_2$&$\F_{1,5,6},\Fb_3,D_{3,7},T_{1,3}$\\
&&\\
&$\Phi_3,\Phi_5$&$F_3,\bar f_3,l^c_3$\quad {\small$[b_3]$}\\
Third&$\eta_2,\bar\eta_2$&$h_{45},\bar h_{45}$\\
&$\Phi_{12},\bar\Phi_{12}$
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&$\phi_{45},\bar\phi_{45},\phi^+,\bar\phi^+,\phi^-,\bar\phi^-,\phi_{3,4},\bar\phi_{3,4}$\\
&$h_3,\bar h_3$&$\F_{2,4},\Fb_{5,6},D_4,T_2$
\end{tabular}
\label{AllStates}$$ (The $[b_{1,2,3}]$ that appear next to some states indicates that these states belong to that particular twisted sector.) With the exception of $\Phi_{0,1,3,5}$, the above fields carry charges under $SU(5)\times SU(4)\times SO(10)$ and various $U(1)$ symmetries: $\Phi_{23,31,12},\bar\Phi_{12,31,12},\eta_{1,2},\bar\eta_{1,2}$ and $\phi_{45},\bar\phi_{45}$, $\phi^+,\bar\phi^+$, $\phi^-,\bar\phi^-$, $\phi_{3,4},\bar\phi_{3,4}$ are gauge singlets; $F_{0,1,2,5}$ ($\bar F_{4,5}$) are () under $SU(5)$, $\bar f_{2,3,5}$ ($l^c_{2,5,3}$) are () under $SU(5)$; $\F_{1,2,3,4,5,6}$ ($\Fb_{1,2,3,4,5,6}$) are () under $SU(4)$, and $D_{1,2,3,4,5,6,7}$ are under $SU(4)$; $T_{1,2,3}$ are under $SO(10)$.
To verify our conjecture (at least to lowest order) we work in the string basis and expand the exponentials in Eq. (\[oldKTS\]) to first order in the untwisted states: $K_{(I)}\approx\sum_{i}^{n_I}\alpha^{(I)}_i\bar\alpha^{(I)}_i$. We end up with generic terms of the form $$\beta^{(I)}_i\bar\beta^{(I)}_i
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+\coeff{1}{2}\beta^{(I)}_i\bar\beta^{(I)}_i\sum_j\left[\alpha^{(J)}_j\bar\alpha^{(J)}_j+\alpha^{(K)}_j\bar\alpha^{(K)}_j\right]\ ,
\label{generic}$$ where $J,K\not=I$. To verify the presence of the quartic term with the $1\over2$ coefficient we need to compute string scattering amplitudes of the type $\VEV{\beta^{(I)}_i\bar\beta^{(I)}_i\alpha^{(J)}_j\bar\alpha^{(J)}_j}$, which should exhibit a term proportional to $s$ (in fact $(g^4/4)\,(s/2)$). This type of calculations have been performed in detail in Ref. [@LNY94]. Here we give the new results of interest and point out subtleties that need to be dealt with in the process.
To verify the old result in Ref. [@oldTS], we need to pick $\beta$’s from the twisted sectors $b_{1,2,3}$ (see Eq. (\[AllStates\])). For instance, for $\alpha^{(1)}=\Phi_{23}$ and $\beta^{(1)}=F_{0,1}$ we find $$\VEV{F_{0,1}\Phi_{23}\Phi_{23}^\dagger F_{0,1}^\dagger}={g^2\over4}\,{su\over
t} \ ,
\label{Ex1}$$ which exhibits no term $\propto s$ since both $\alpha$ and $\beta$ belong to the same set. The $su/t$ term is just the expected graviton exchange contribution. There are no further contributions since $F_{0,1}$ and $\Phi_{23}$ do not have any $U(1)$ charges in common (, no “D terms") or appear in the same superpotential coupling (, no “F terms").[^15] Now let us consider $\alpha^{(2)}=\Phi_{31},\bar\Phi_{31}$ and $\beta^{(1)}=F_{0,1}$, $$\begin{aligned}
\VEV{F_{0,1}\Phi_{31}\Phi_{31}^\dagger F_{0,1}^\dagger}&=&
{g^2\over4}\left[{s\over2}+{su\over t}+2\ln2 s\right]
-{g^2\over2}\left({s-u\over t}-1\right)\ ,\label{Ex2a}\\
\VEV{F_{0,1}\bar\Phi_{31}\bar\Phi_{31}^\dagger F_{0,1}^\dagger}&=&
{g^2\over4}\left[{s\over2}+{su\over t}-2\ln2 s\right]
+{g^2\over2}\left({s-u\over t}-1\right)\ .
\label{Ex2b}\end{aligned}$$ In these expressions we see the expected graviton exchange term ($\propto
su/t$) and also the contact term $(g^2/4)(s/2)$, signaling the non-trivial quartic coupling in the Kähler function (see Eq. (\[generic\])). The last term is just a “D term": under a common $U(1)$ $F_{0,1}$ carry $-1/2$ charge, whereas $\Phi_{31}\,(\bar\Phi_{31})$ carries $+1\,(-1)$ charge, which give $D=-{1\over2}|F_{0,1}|^2+|\Phi_{31}|^2-|\bar\Phi_{31}|^2$. Therefore, we expect gauge boson exchange \[$\propto(s-u)/t$\] and a contact term from $-{g^2\over2}D^2$, all evidently present in Eqs. (\[Ex2a\],\[Ex2b\]). We do not expect an “F term" since there is no superpotential coupling involving these fields. The last matter concerns the disturbing term $\propto 2\ln2 s$. In Ref. [@LNY94] it was shown that the proper coordinates in which to write the untwisted sector Kähler function may be linear combinations of the string coordinates, in this case $\chi_1={1\over\sqrt{2}}(\Phi_{31}+\bar\Phi_{31})$ and $\chi_2={-i\over\sqrt{2}}(\Phi_{31}-\bar\Phi_{31})$. The amplitudes to consider are then $\VEV{F_{0,1}\chi_{1,2}\chi^\dagger_{1,2}F^\dagger_{0,1}}=
{1\over2}\VEV{F_{0,1}\Phi_{31}\Phi_{31}^\dagger F_{0,1}^\dagger}
+{1\over2}\VEV{F_{0,1}\bar\Phi_{31}\bar\Phi_{31}^\dagger F_{0,1}^\dagger}
={g^2\over4}[s/2+su/t]$, which is the expected result.
Continuing in this fashion one can calculate all of the terms of the form (\[generic\]), where the $\beta$’s come exclusively from the $b_{1,2,3}$ sectors. In each instance one can account for all pieces of the amplitudes, thus confirming the old result in Ref. [@oldTS]. We have conjectured that this result can be extended to all states in the spectrum. This conjecture has been verified explicitly for all twisted sectors in this model. For instance, consider $\alpha^{(1)}=\Phi_0$ and $\beta^{(1)}=F_4$,[^16] $$\VEV{F_4\Phi_0\Phi_0^\dagger F_4^\dagger}={g^2\over4}\,{su\over t}-{g^2\over2}
\ .
\label{Ex3}$$ According to our conjecture, in this case we do expect a term $\propto s$ since both fields belong to the first set. We also get the usual graviton exchange term, and a contact term $-g^2/2$. This latter is an “F term" originating from the superpotential coupling ${1\over2}g\sqrt{2}F_4\bar F_4\Phi_0$. Another example would be to take $\alpha^{(2)}=\Phi_{31},\bar\Phi_{31}$ and $\beta^{(1)}=F_4$. The result in this case is identical to that in Eqs. (\[Ex2a\],\[Ex2b\]); repeating the subsequent discussion shows the appearance of the term $\propto s$, as conjectured. A lot more of these brute force calculations shows that the conjecture holds for all states in the massless spectrum. That is, in Eq. (\[oldKTS\]) one is to intepret $\beta^{(I)}$ as fields belonging to the $I$-th set as defined in Eq. (\[charges\]).
We should add that we have also verified the results of Ref. [@LNY94] for the untwisted sector of the “search" model. The novelty here is that some of the untwisted singlet fields (some of the presumed moduli) have superpotential couplings (in fact $\Phi_{0,3,5}$), a feature that does not alter the structure of the Kähler function derived in Ref. [@LNY94]. We have also repeated the above exercise (and therefore proven our conjecture) for the “revamped" model of Ref. [@revamped]. For remarks regarding the generality of these results see Ref. [@LNY95].
In the main text we have used the twisted sector contribution to the Kähler potential to write down Eq. (\[completeK\]). In this case we have one untwisted set which is a $\tau$ set, and the other two are $\alpha$ sets. In Eq. (\[completeK\]) the two $\alpha$ sets have been expanded to first order in $\alpha^{(2,3)}_i$. The same expansion has been carried out in the exponents in Eq. (\[oldKTS\]), reducing it to unity for the first twisted set, and to the square-root of the argument of the $\ln$ in the first untwisted set Kähler potential. Equation (\[completeK\]) then follows immediately. We should remark that these various approximations in calculating the Kähler function are immaterial as far as the observable quantities which we have calculated are concerned.
[99]{} For reviews see A. B. Lahanas and D. V. Nanopoulos, Phys. Rep. [**145**]{} (1987) 1;\
D. V. Nanopoulos, “The march towards no-scale supergravity", CERN-TH.7423/94 (hep-th/9411281). J.-P. Derendinger, L.E. Ibáñez and H.P. Nilles, ; M. Dine, R. Rohm, N. Seiberg and E. Witten, . C. Kounnas and M. Porrati, ; M. Porrati and F. Zwirner, ; C. Kounnas and B. Rostand, ; I. Antoniadis, . E. Cremmer, S. Ferrara, C. Kounnas, and , . J. Ellis, C. Kounnas, and , and . S. Ferrara, C. Kounnas, and F. Zwirner, . J. Ellis, A. Lahanas, , and K. Tamvakis, . J. Ellis, K. Enqvist, and , and . S. Kalara, , and , . S. Kelley, , , H. Pois, and K. Yuan, . C. Kounnas, I. Pavel, and F. Zwirner, . S. Kelley, , , and , (hep-ph/9409223). I. Antoniadis, C. Bachas, and C. Kounnas, Nucl. Phys. B [**289**]{} (1987) 87; I. Antoniadis and C. Bachas, Nucl. Phys. B [**298**]{} (1988) 586; H. Kawai, D.C. Lewellen, and S.H.-H. Tye, Phys. Rev. Lett. [**57**]{} (1986) 1832; Phys. Rev. D [**34**]{} (1986) 3794; Nucl. Phys. B [**288**]{} (1987) 1; R. Bluhm, L. Dolan, and P. Goddard, Nucl. Phys. B [**309**]{} (1988) 330; H. Dreiner, J. L. Lopez, D. V. Nanopoulos, and D. Reiss, Nucl. Phys. B [**320**]{} (1989) 401. I. Antoniadis, J. Ellis, J. Hagelin and D.V. Nanopoulos, Phys. Lett. B [**231**]{} (1989) 65. A. Faraggi, , and K. Yuan, ; I. Antoniadis, G. Leontaris, and J. Rizos, ; A. Faraggi, , , . , , and K. Yuan, . For a recent review see, , (hep-th/9405278) (to appear in Surveys in High Energy Physics). S. Kalara, , and , . I. Antoniadis, J. Ellis, E. Floratos, , and T. Tomaras, ; S. Ferrara, L. Girardello, C. Kounnas, and M. Porrati, . , , and K. Yuan, . S. Ferrara, L. Girardello, C. Kounnas, and M. Porrati, . , , and K. Yuan, to appear. S. Wolfram, [*Mathematica, a system for doing mathematics by computer*]{} (Addison-Wesley, Redwood City, 1991). L. Dixon, V. Kaplunovsky, and J. Louis, ; I. Antoniadis, K. Narain, and T. Taylor, ; J. Derendinger, S. Ferrara, C. Kounnas, and F. Zwirner, . M. Cvetic, , . J. Ellis and , . J. E. Kim and H. P. Nilles, and ; E. J. Chun, J. E. Kim, and H. P Nilles, . and , ; J. Casas and C. Muñoz, . G. Giudice and A. Masiero, ; V. Kaplunovsky and J. Louis, . A. Brignole, L. Ibáñez, and C. Muñoz, . U. Ellwanger, LPTHE Orsay 94-106 (hep-ph/9501227). P. Binetruy and E. Dudas, and LPTHE Orsay 94/73 (hep-ph/9411413). See , R. Frazer, W. Duncan, and A. Collar, [*Elementary Matrices*]{} (Macmillan, New York, 1946). I. Antoniadis, J. Ellis, S. Kelley, and , . , , and A. Zichichi, in preparation. , , and , . , , and A. Zichichi, . S. Ferrara, C. Kounnas, M. Porrati, and F. Zwirner, . R. Gilmore, [*Lie Group, Lie Algebras, and Some of Their Applications*]{} (John Wiley & Sons, New York, 1974), Ch. 9, section V.
[^1]: Note however that such parametrization may be inadequate since non-universal scalar masses are not uncommon in string-derived supergravities.
[^2]: This is certainly the case at tree-level in the Kähler potential and for moduli-independent Yukawa couplings. In free-fermionic models, moduli dependence of the superpotential does not arise until the quartic order [@modinv].
[^3]: In Ref. [@LNY94], $\tau_1,\tau_2$ were denoted by $T,U$. Such notation would cause confusion here.
[^4]: We note that in the old no-scale models [@EKNI+II], the Kähler potential was assumed to be of the form $K\propto \ln(T+\bar T - \sum_i C_i \bar C_i)$, which yields the metric of the space $SU(1,n_C+1)/U(1)\times SU(n_C+1)$. The old no-scale ansatz and the string free-fermionic $\tau$-set result agree only for $n_\psi=n_C=0$.
[^5]: Since $F(\beta,\partial\ln W)$ is generally a complicated expression, it may be possible to find special minima for particular non-zero values of $\VEV{\partial_I \ln W}$. In the case of the old no-scale models, $F(\beta,\partial\ln W)=\sum_i |\partial_{C_i}\ln W|^2>0$ [@EKNI+II], and therefore $\VEV{\partial_{C_i}\ln W}=0$ is required.
[^6]: One has to properly normalize the fields (, $\psi\to\widehat\psi$) to obtain the physical masses. The normalization factors are given in Eqs. (\[redefinitions1\],\[redefinitions2\],\[normalizations1\], \[normalizations2\],\[normalizations3\]), and can be trivial (, $1$) in many instances.
[^7]: This procedure can be easily generalized to what would be the case of interest with $\psi\psi\to\psi_1\psi_2$, as discussed in Ref. [@BIM]. In $SU(5)\times
U(1)$ free-fermionic models one finds $\psi_1={1\over\sqrt{2}}(h_1+h_2)$, $\psi_2={-i\over\sqrt{2}}(h_1-h_2)$ with $h_1\,(h_2)$ a () of $SU(5)$ [@LNY94]. It follows that $(\psi_1+\bar\psi_1)^2+(\psi_2+\bar\psi_2)^2=2h_1h_2+2(h_1h_2)^*+2h_1h^*_1
+2h_2h^*_2$ and $K_1=K_2=2/(\tau+\bar\tau)^2$. The new superpotential term is $\mu\hat h_1\hat h_2$ with $\mu=m_{3/2}$, which gives $\widetilde
m_{\widehat h_1}=\widetilde m_{\widehat h_2}=0$ and $m_{\widehat h_1\widehat
h_2}=m_{3/2}$.
[^8]: Note that in principle the coefficients $K_1,K_2$ could be related in a different manner, with even the $K_2$ piece leading to a vanishing scalar mass (if $K_2\propto1/(\tau+\bar\tau)$) and to the so-called “super-soft" supersymmetry breaking terms [@Ellwanger] which do not break supersymmetry.
[^9]: This property does not appear to allow the dynamical determination of Yukawa couplings via the no-scale mechanism, as recently advocated [@KPZ; @Dudas].
[^10]: Note that even if a field appears in the superpotential, it may still be a flat direction if the fields coupled to it conspire in the appropriate way. In what follows we disregard such exceptional possibilities.
[^11]: An “$m/n/p$ Yukawa set" includes $m$ potential up-quark like Yukawa couplings, $n$ potential down-quark like Yukawa couplings, and $p$ potential charged-lepton Yukawa couplings [@search].
[^12]: When considering a Yukawa coupling at the string scale, care must be taken to include any normalization factors that may arise, as discussed in Sec. \[Field\] especially in Eq. (\[NormYuks\]).
[^13]: In contrast, the second paper in Ref. [@Coordinate] presents a model with zero vacuum energy where $Q=-272$. Also, in the “revamped" model [@revamped], for the value of $V_0$ closest to zero (it cannot be exactly zero), we get $Q=-83$.
[^14]: For a more general discussion of twisted sector Kähler potentials in fermionic models see Ref. [@LNY95].
[^15]: A listing of all $U(1)$ charges associated with the fields in Eq. (\[AllStates\]) is given in Table 4 in Ref. [@search]; the cubic and quartic superpotential is given in Eq. (6.3) of this same reference.
[^16]: This amplitude involves the Ising model correlator $\VEV{\sigma f f \sigma}={1\over2}|z_\infty|^{-1/4}[4(1-z)^{-2}+z^{-1}]^{1/2}$, which has been calculated using the methods of Ref. [@KLN].
|
---
author:
- '[^1]'
- '[^2]'
title: Pulsating White Dwarfs
---
Introduction {#sec:intro}
============
White dwarf stars are the final evolutionary state of stars with initial masses up to 8.5–10.6 M$_\odot$[@Woosley15], corresponding to 95 – 97 % of all stars. The fraction depends on the stellar metallicity, which affects both the Initial-Mass-Function and the Initial-to-Final-Mass Relation. For single stars, the minimum mass of a present day white dwarf is around 0.30–0.45 M$_\odot$[@Kilic07], because progenitors that would become lower mass white dwarfs have main sequence evolution time larger than the age of the Universe. Such masses correspond, considering the mass-radius relation of white dwarfs, to a minimal $\log g\simeq 6.5$. Evolutionary models e.g. by Ref. [@Romero15] indicate that the maximum surface gravity for main sequence A stars, which have similar optical spectra to DA white dwarfs, corresponds to $\log g \leq 4.75$, including very low metallicity. There is therefore a gap between low mass white dwarfs and main sequence stars, $4.75~\leq~\log~g~\leq~6.5$.
Most white dwarfs do not generate energy from nuclear fusion, but radiate due to residual gravitational contraction. Because of the degenerate equation of state, contraction is accompanied by a loss of thermal energy instead of increase as in the case of ideal gases; the evolution of white dwarfs is therefore often simply described as cooling. The radius of an average white dwarf star is of the same order of the Earth’s radius, which implies that they have small surface area, resulting in very large cooling times; it takes approximately $10^{10}$ years for the effective temperature of a $\sim 0.6 M_{odot}$ white dwarf to decrease from $100\,000$ K to near $5\,000$ K. Consequently, the cool $\sim 0.6 M_{odot}$ ones are still visible and among the oldest objects in the Galaxy[@GarciaBerro16]. Therefore, studying white dwarfs is extremely important to comprehend the processes of stellar formation and evolution in the Milky Way[@Winget87; @Campos16].
The progenitors of white dwarfs lose most of their envelope in the giant phases, where mass loss depends on metallicity. If the remainder H mass were above $\simeq 10^{-4} M*$, or the He mass above $\simeq 10^{-2} M*$ there would be observable nuclear burning in the white dwarf phase. The limits depend on the mass of the white dwarf. Most white dwarfs have atmospheres dominated by H, and the remainder by He. All other elements are only small traces, much less abundant than in the Sun, due to separation in the strong gravitational field[@Schatzman48]. The lightest elements float to the surface once the white dwarf cools below The He-core white dwarf stars in the mass range $0.2-0.45~M_\odot$, referred to as low-mass white dwarfs, are usually found in close binaries, often double degenerate systems[@Marsh95], being most likely a product of interacting binary star evolution. More than 70% of those studied by Ref. [@Kilic11] with masses below $0.45~M_\odot$ and all but a few with masses below $0.3~M_\odot$ show radial velocity variations[@Brown13; @Gianninas14]. Ref. [@Kilic07] suggests single low-mass white dwarfs result from the evolution of old metal-rich stars that truncate evolution before the helium flash due to severe mass loss. They also conclude all white dwarfs with masses below $\simeq 0.3~M_\odot$ must be a product of binary star evolution involving interaction between the components, otherwise the lifetime of the progenitor on the main sequence would be larger than the age of the Universe.
In Fig. \[single\] we show the results of our effective temperature and surface gravity determinations for all candidates from SDSS. We calculated the single star evolutionary models shown in the figure with the MESA[@MESA] evolutionary code, including diffusion. In Fig. \[double\] the evolutionary models are those with rotation and diffusion of Ref. [@Istrate16].
{width="90.00000%"}
Even though the low resolution hydrogen lines observed in SDSS spectra are poor surface gravity indicators below $T_\mathrm{eff} \simeq 10\,000$ K, considering the SDSS spectra are concentrated mainly outside the Galaxy disk, we were surprised that several thousand stars were classified by the SDSS pipeline as A and B stars. Considering their lifetimes on the main sequence smaller than 1 Gyr, and their distance modulus $(m-M)\geq 14.5$ at the SDSS bright saturation, if these stars were main sequence stars, there would be a considerable population of young stars very far from the galactic disk. Using their measured radial velocities, and proper motions if available, [@Pelisoli16] estimated their U, V, W velocities and show there would be a large number of hypervelocity A stars, not detected up to date. If these stars are in fact low mass counterparts of interacting binary evolution, similar to the models of [@Althaus13; @Istrate16], they are mainly concentrated in the galactic disk. Considering we do not know their metallicities, and that low ionization potential metals contribute significantly to the electron pressure, we estimated their surface gravities with two sets of models, a pure hydrogen model and a solar composition model. The estimated surface gravities with solar metallicity models were on average $\Delta \log g \simeq 0.5$ dex smaller, but not systematically. Our plotted values are the solar metallicity ones.
{width="90.00000%"}
Interacting Binaries
--------------------
Ref. [@Pietrzynski12] found an RR Lyrae with 0.26 $M_\odot$, and Ref. [@Latour16] found a 0.23 $M_\odot$ pulsating subdwarf (sdBV). Ref. [@Istrate16] interacting binary with mass exchange models show that during a hydrogen shell burning pulse, an extremely low mass white dwarf can cross the main sequence, horizontal branch and even giant instability strip. Ref. [@Karczmarek17] estimate that up to 5% of stars that cross the RR Lyrae and Cepheid instability strip are binaries.
DA white dwarf stars with masses $M\leq 0.45~M_\odot$ and $T_\mathrm{eff} < 20\,000$ K are Low Mass and Extremely Low Mass (ELM) as found by Refs. [@Brown10], [@Kilic11], [@Brown12], [@Brown13], [@Gianninas14], [@Gianninas15] and [@Brown16]. Refs. [@Hermes12] – [@Bell16a] found pulsations in eight of these ELMs, similar to the pulsations seen in DAVs (ZZ Ceti stars), as described in Ref. [@VanGrootel13]. Ref. [@Maxted14] found 17 pre-ELMs, i.e., helium–core white dwarf precursors, and Ref. [@Maxted14a; @Gianninas16] report pulsations in six of them. Pulsations are an important tool to study the stellar interior, and Refs. [@Corsico14] – [@Istrate16a] report on theoretical models and pulsations of ELMs. Refs. [@Kepler16a] and [@Kepler16b] show there are thousands of stars, photometrically classified as blue horizontal branch stars by Ref. [@Xue08; @Xue11; @Carollo16], that have spectroscopic estimated surface gravities much higher than main sequence stars ($\log g \geq 4.75$) and therefore must have radii smaller than the Sun, classifying them as sdAs, in line with the hot subdwarfs reviewed by Ref. [@Heber16]. Ref. [@Pelisoli16] discuss they are possibly Extremely Low Mass white dwarf stars. Refs. [@Kepler16a; @Fusillo15] show that photometrically selected white dwarfs have a contamination around 40%. Even the ones selected also from proper motion by Ref. [@Munn17] show significant contamination by non-white dwarf objects, when spectra are available.
Most stars that produced white dwarfs are born in binaries or multiple systems. Ref. [@Lada06] demonstrates that while around 70% of stars more massive than the Sun are in binaries, two-thirds of the most common stars, M type dwarf stars, are single. More than 10% of the spectroscopically identified white dwarfs in SDSS have red companions[@Kepler16a; @Rebassa16]. Refs. [@Farihi10; @Nebot11] show that nearly 25% of all main sequence binaries are close enough that mass transfer interactions occur when the more massive star becomes a red giant or an asymptotic giant star. If mass transfer exceeds the Eddington limit, the secondary star is not able to accrete the transferred material and the system evolves through a common envelope phase, i.e., the core of the giant and the main sequence companion orbit within the outer layers of the giant star, leading to the shrinkage of the orbit and the release of orbital energy. The orbital energy deposited into the envelope eventually ejects it. Therefore a close binary is formed by the core of the giant star and a main sequence companion, later a close white dwarf-main sequence binary. An ELM will be formed if the envelope is ejected before the helium-flash, which would happen if the star has a low initial mass, i.e., $M\lesssim 2 M_\odot$, to reach conditions to fuse helium in the core before it becomes degenerate.
Mass Distribution
=================
We estimated the masses of all DA white dwarfs found by Ref. [@Kleinman13], [@Kepler15] and [@Kepler16a]. There were no new optical stellar spectra in SDSS Data Release 13. For the DA mass distribution, we only consider spectra with S/N$\geq 15$ to have reliable mass determinations. From $T_\mathrm{eff}$ and $\log g$ values obtained from our fits, after correcting to 3D convection following Ref. [@Tremblay13a], we use the mass–radius relations of Refs. [@Althaus05], [@Renedo10] and [@Romero15] to calculate the stellar mass.
Considering that white dwarfs with larger mass have smaller radius, and therefore can only be seen to smaller distances in a magnitude limited survey as SDSS, we calculated the density by correcting the visible volume with the $1/V_\mathrm{max}$ method of Ref. [@Schmidt68], up to a maximum g=19 magnitude. For DAs with $T_\mathrm{eff} \geq 10000$ K, N=4054, we estimate $\langle M \rangle=0.647\pm 0.002~M_\odot$, $T_\mathrm{eff} \geq 13000$ K, N=3637, $\langle M \rangle=0.646\pm 0.002~M_\odot$, $T_\mathrm{eff} \geq 16000$ K, N=3012, $\langle M \rangle=0.641\pm 0.002~M_\odot$, $T_\mathrm{eff} \geq 25000$ K, N=1121, $\langle M \rangle=0.613\pm 0.003~M_\odot$. The distribution shows that the DA and DB distributions have very different shapes. The DA’s has a tail to larger masses, while the DB’s is extended to lower masses. This is probably reflecting some limitation in the progenitors that can undergo very-late thermal pulses and become DBs.
Pulsations
==========
During the cooling of the white dwarf star, partial ionization zones of C, O, He and H develop at subsequently lower $T_\mathrm{eff}$. Such partial ionization zones increase the opacity and cause pulsations. For C/O, the stars are called pulsating PG 1159 stars, DOVs or GW VIr stars, and occurs at $140\,000~\mathrm{K} \lesssim T_\mathrm{eff} \lesssim 75\,000$ K. For He, $32\,000~\mathrm{K} \lesssim T_\mathrm{eff} \lesssim 22\,000$ K and are called DBVs. For H, $13\,000~\mathrm{K} \lesssim T_\mathrm{eff} \lesssim 10\,500$ K and are called DAVs or ZZ Ceti stars. Recent additions are the DQVs, with $22\,000~\mathrm{K} \lesssim T_\mathrm{eff} \lesssim 18\,000$ K, and with $10\,000~\mathrm{K} \lesssim T_\mathrm{eff} \lesssim 8\,000$ K, the ELMVs and the pre-ELMVs or EL CVn stars.
Class Number
------------ --------
DAVs 181
DBVs 23
DOVs$^{1}$ 22
ELMVs 11
pre-ELMVs 5
DQVs 3
: Number of known pulsating white dwarfs.
\[tab:tab-1\]\
[$^{1}$ Pulsating PG 1159 stars.]{}
Magnetic Fields
===============
Ref. [@GarciaBerro16a] presents a review on magnetic fields in white dwarf stars. When examining each candidate SDSS spectrum by eye, [@Kleinman13; @Kepler15; @Kepler16a] found 822 stars with Zeeman splittings indicating magnetic fields above 2 MG — the limit where the line splitting becomes too small to be identified at the SDSS spectral resolution [@Kepler13]. The mean fields, estimated following [@Kulebi09], range from 2 MG to 700 MG. We caution that stars with large fields are difficult to identify because fields above around 30 MG intermixes subcomponents between different hydrogen series components so much that, depending on effective temperature and signal-to-noise, it becomes difficult to identify the star as containing hydrogen at all, and affecting even the colors significantly. Both the low field limit and the high field limit are totally dominated by systematic effects, not the real limits. The effect of the magnetic field on pulsations has been estimated by [e.g. @Jones89; @Tremblay15].
Rotation
========
In general the measured rotation period for single white dwarfs ranges from 1 h to 18 d, with a median around 1 d[@Kawaler15]. The fastest single white dwarf rotator from asteroseismological measurements (Table \[rot\]) is the $0.79~M_\odot$ DAV SDSS J161218.08+083028.1 discovered by Ref. [@Castanheira13], assuming the two observed periods at 115.0 s and 117.0 s are two components of a rotation triplet.
-------------------------- --------------------- ---------------------- --------- ---------------------
Star $P_{\rm rot}$ \[h\] $T_\mathrm{eff}$[^3] Type $M$ \[$M_{\odot}$\]
RX J2117.1+3412 28 170000 GW Vir 0.72
PG 1159-035 33 140000 GW Vir 0.54
NGC 1501 28 134000 \[WCE\] 0.56
PG 2131+066 5 95000 GW Vir 0.55
PG 1707+427 16 85000 GW Vir 0.53
PG 0122+200 37 80000 GW Vir 0.53
PG 0112+104 10.17 31040 DBV 0.58
KIC 8626021 43 29700 DBV 0.56
EC 20058-5234 2 25500 DBV 0.65
GD 358 29 23740 DBV 0.54
SDSS J083702.16+185613.4 1.13 13590 ZZ Ceti 0.88
G 226-29 9 12510 ZZ Ceti 0.83
G 185-32 15 12470 ZZ Ceti 0.67
SDSS J113655.17+040952.6 2.6 12330 ZZ Ceti 0.55
SDSS J161218.08+083028.1 [**0.93**]{} 12330 ZZ Ceti 0.79
Ross 548 37 12300 ZZ Ceti 0.63
GD 165 50 12220 ZZ Ceti 0.68
LP 133-144 41.8 12150 ZZ Ceti 0.59
KIC 11911480 86.4 12160 ZZ Ceti 0.58
L 19-2 13 12070 ZZ Ceti 0.69
HS 0507+0435 41 12010 ZZ Ceti 0.73
EC 14012-1446 14.4 12020 ZZ Ceti 0.72
KUV 11370+4222 5.56 11940 ZZ Ceti 0.72
G 29-38 32 11910 ZZ Ceti 0.72
KUV 02464+3239 90.7 11620 ZZ Ceti 0.70
HL Tau 76 53 11470 ZZ Ceti 0.55
SDSS J171113.01+654158.3 16.4 11130 ZZ Ceti 0.90
GD 154 50.4 11120 ZZ Ceti 0.65
KIC 4552982 15.0 10860 ZZ Ceti 0.71
SDSS J094000.27+005207.1 11.8 10590 ZZ Ceti 0.82
-------------------------- --------------------- ---------------------- --------- ---------------------
: Rotation periods of white dwarfs as determined via asteroseismology.
\[rot\]
Differential rotation in white dwarfs was studied by Refs. [@Charpinet09] – [@Hermes16], using the change in rotation splitting of non-radial pulsations.
Axions and Dark Mass
====================
Axions are the best candidates for dark mass[@Ringwald16]. Refs.[@Isern03; @Isern10; @Corsico12; @Corsico12a; @Corsico16; @Battich16] show white dwarf pulsations and luminosity function are consistent with extra cooling caused by axions of masses around $17\pm 4$ meV.
Fitting Models
==============
The pulsation spectra exhibited by ZZ Ceti stars strongly depends on their inner chemical profile. There are several processes affecting the chemical profiles that are still not accurately determined. See [@Geronimo17] for a study of the impact of the current uncertainties in stellar evolution on the expected pulsation properties of ZZ Ceti stars.
Each single period is determined by the integral of the pulsation kernel (or work function) over the whole star. It cannot distinguish among different distributions, as demonstrated for example by [@Montgomery03]. If different modes are not independent, i.e., they sample the same regions of the distributions, they carry less information than independent modes. [@Giammichele16] propose one could determine the whole chemical distribution profile from pulsations. [@Giammichele17] performed a test using ten periods, namely the modes l=1, k=2,3,4,5,6 and l=2, k=3,4,5,6,7. Note that in this tests, a sequence of consecutive modes k=2,3,4,5,6 is needed to sample the structure, sequence that is usually not seen real stars. ZZ Ceti stars shows different pulsation spectra: hot stars show few short modes that sample the inner parts of the star while the cool stars show many long modes, which sample the outer parts of the stars. These characteristics must be taken into account when asteroseismology is applied to white dwarfs. A good example of a hot ZZ Ceti pulsator is G 117-B15A, with three modes. Were it not for the convection description problem, that introduces an uncertainty of $\Delta T_{\rm eff} \simeq 500$ K, and the problem with line broadening that gives different $\log g$ from different spectral lines, we could use the 3 modes of G 117-B15A to get 3 structural parameters, plus $dP/dt$ to estimate the core mean molecular weight.
Kepler satellite
================
Observations of pulsating white dwarfs with the Kepler satellite are limited to the brightest objects due to the relatively small size of the telescope, but its long observations has allowed not only an exquisite precision in the pulsation spectra but also the discovery of outbursts lasting hours ([@Bell15; @Hermes15; @Bell16]). These outbursts resemble the [*forte*]{} episode observed in 1996 for the DBV GD 358 by [@Nitta98; @Kepler03].
0.2cm [*Acknowledgments*]{}: S.O.K. and A.D.R. are financed by Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brasil. This research has made use of NASA’s Astrophysics Data System and of the cross-match service provided by CDS, Strasbourg. Funding for the Sloan Digital Sky Survey has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. The SDSS web site is www.sdss.org. Part of the work has been based on observations obtained at the Southern Astrophysical Research (SOAR) telescope, which is a joint project of the Ministério da Ciência, Tecnologia, Inovações e Comunicações (MCTIC) da República Federativa do Brasil, the U.S. National Optical Astronomy Observatory (NOAO), the University of North Carolina at Chapel Hill (UNC), and Michigan State University (MSU).
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[^1]: <kepler@if.ufrgs.br>
[^2]: <alejandra.romero@ufrgs.br>
[^3]: The effective temperatures and masses are corrected to 3D convection[@Tremblay13].
|
---
abstract: 'Let $ \mathfrak{f} $ run over the space $ H_{4k} $ of primitive cusp forms of level one and weight $ 4k $, $ k \in {\mathbf{N}}$. We prove an explicit formula for the mixed moment of the Hecke $ L $-function $ L(\mathfrak{f}, 1/2) $ and the symmetric square $L$-function $ L({\operatorname{sym}}^2\mathfrak{f}, 1/2)$, relating it to the dual mixed moment of the double Dirichlet series and the Riemann zeta function weighted by the ${}_3F_{2}$ hypergeometric function. Analysing the corresponding special functions by the means of the Liouville-Green approximation followed by the saddle point method, we prove that the initial mixed moment is bounded by $\log^3k$.'
address:
- 'Department of Mathematical Sciences, University of Gothenburg and Chalmers University of Technology, SE-412 96 Göteborg, Sweden'
- 'Laboratoire Painlevé LABEX-CEMPI, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France'
- 'Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina st., Moscow, 119991, Russia'
- 'Laboratoire Painlevé LABEX-CEMPI, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France'
author:
- Olga Balkanova
- Gautami Bhowmik
- Dmitry Frolenkov
- Nicole Raulf
nocite: '[@*]'
title: 'Mixed moment of $GL(2)$ and $GL(3)$ $L$-functions'
---
Introduction
============
Let $H_{2k}$ be the normalised Hecke basis for the space of holomorphic cusp forms of even weight $2k \geq 2$ with respect to the full modular group. Every function $ \mathfrak{f} \in H_{2k} $ has a Fourier expansion of the form $$\mathfrak{f}(z)=\sum_{n\geq 1}\lambda_{\mathfrak{f}}(n)n^{k-1/2}\exp(2\pi inz),
\quad \lambda_{\mathfrak{f}}(1)=1.$$
Consider the mixed moment at the critical point: $$\label{mixmoment}
{\mathcal{M}}(0,0):=\sum_{\mathfrak{f}\in H_{4k}} \omega(\mathfrak{f}) L(\mathfrak{f},1/2)L({\operatorname{sym}}^2\mathfrak{f}, 1/2), \quad
\omega(\mathfrak{f}) := \frac{12\zeta(2)}{(4k-1)L({\operatorname{sym}}^2\mathfrak{f},1)},$$ where the corresponding $L$-functions are defined for $\Re{s}>1$ as $$\label{L def}
L(\mathfrak{f},s):=\sum_{n=1}^{\infty}\frac{\lambda_\mathfrak{f}(n)}{n^s}, \quad L({\operatorname{sym}}^2\mathfrak{f},s):=\zeta(2s)\sum_{n=1}^{\infty}\frac{\lambda_\mathfrak{f}(n^2)}{n^s},$$ and admit the analytic continuation to the whole complex plane. Note that we consider only $k$ divisible by $4$ in because otherwise $ L(\mathfrak{f},1/2)$ is identically zero.
The mixed moment was studied in [@BBFR] by combining an explicit formula for the first moment of symmetric square $L$-functions and an approximate functional equation for the Hecke $L$-function. This approach along with the Liouville-Green method appeared to be quite effective, producing an asymptotic formula with an arbitrary power saving error term.
However, the same problem [*without extra smooth averaging*]{} is much more difficult in view of analysis of non-diagonal terms. For this reason, we modify the methodology of [@BBFR], relying entirely on the method of analytic continuation. More precisely, we prove an explicit formula for the mixed moment , which contains the diagonal main term of size $\log{k}$, the non-diagonal main term of size $k^{-1/2}$ and dual mixed moments weighted by ${}_3F_{2}$ hypergeometric functions. In order to state explicitly non-diagonal terms in , it is required to introduce the generalised Dirichlet $L$-function $$\label{Lgeneralised}
\mathscr{L}_n(s) := \frac{\zeta(2s)}{\zeta(s)} \sum_{q=1}^{\infty}
\frac{1}{q^s} \left(\sum_{1\leq t \leq 2q;t^2 \equiv n {\ (\textup{mod}\ 4q)}}1\right),$$ and the associated double Dirichlet series $$L^{-}_{f}(s) :=
\frac{\Gamma(3/4)}{2\sqrt{\pi}}\sum_{n<0}\frac{\mathscr{L}_{n}(1/2)}{|n|^{s+1/2}}, \quad L^{-}_{g}(s):=\frac{\Gamma(3/4)}{4\sqrt{\pi}}\sum_{n<0}\frac{\mathscr{L}_{4n}(1/2)}{|n|^{s+1/2}}.$$
\[main thm\] For any $\epsilon>0$ the following formula holds $$\label{M(0,0) exact formula1}
{\mathcal{M}}(0,0)=2{\mathcal{M}}^D(0,0)+2{\mathcal{M}}^{ND}(0,0)+\frac{1}{2\pi i}\int_{(0)}G_{2k}(0,s)ds+O\left(\frac{k^{\epsilon}}{k}\right).$$ Here the main diagonal term is given by $$\begin{gathered}
{\mathcal{M}}^D(0,0)=
\frac{\zeta(3/2)}{2}
\Biggl(\frac{\pi}{2}-3\log{2\pi}+3\gamma+
2\frac{\zeta'(3/2)}{\zeta(3/2)}
+\frac{\Gamma'}{\Gamma}(2k-1/4)+\frac{\Gamma'}{\Gamma}(2k+1/4)\Biggr).\end{gathered}$$ The main non-diagonal term is smaller in size and depends on the special value of the double Dirichlet series: $${\mathcal{M}}^{ND}(0,0)=
\frac{2^{3/2}\pi}{\Gamma(3/4)}\frac{\Gamma(2k-1/4)}{\Gamma(2k+1/4)}L^{-}_{g}(1/4).$$ Finally, the last term involves the product of the Riemann zeta function with the double Dirichlet series weighted by ${}_3F_{2}$ hypergeometric function: $$\begin{gathered}
G_{2k}(0,s)=
\frac{2^{5/2}}{\Gamma^2(3/4)}\Gamma(2k-1/4)\Gamma(3/4-2k)
\Gamma(1/2+s)\Gamma(1/4-s)\zeta(1/2-2s)
\\\times
\Biggl(
\left(1-2^{2s-1/2}\right)L^{-}_{f}(s)-
\frac{\left(1-2^{2s+1/2}\right)}{2^{2s}}L^{-}_{g}(s)
\Biggr){}_3F_{2}\left(2k-\frac{1}{4},\frac{3}{4}-2k,\frac{1}{4}-s;\frac{1}{2},\frac{3}{4}; 1\right).\end{gathered}$$
The formula contains the error term $O\left(k^{-1+\epsilon}\right)$ with the goal to shorten the statement of the main result. This error term can be replaced by a completely explicit expression. See , Lemma \[lem:second type\] and Lemma \[lem:et1at0v\] for details.
The first and the second terms in can be estimated as follows: $${\mathcal{M}}^D(0,0)\ll \log{k}, \quad {\mathcal{M}}^{ND}(0,0)\ll k^{-1/2}.$$ The analysis of the third term given by the integral of $G_{2k}(0,s)$ is the core part of this paper. Using the representation for $G_{2k}(0,s)$ in terms of the Mellin transform $\hat{g}_{2k}(u,v;s)$ studied in Lemma \[lem:gMelTrans\], it is required to estimate the weighted mixed moment of the double Dirichlet series and the Riemann zeta function: $$\label{eq:dualmoment}
\int_{-\infty}^{\infty}L^{-}_{f,g}(ir)\zeta(1/2-2ir)\hat{g}_{2k}(0,0;ir)dr.$$
The contribution of $|r|>3k$ is negligibly small because in this range the function $\hat{g}_{2k}(0,0;ir)$ is of rapid decay by Lemma \[Lem gMellin big r\]. In the remaining range, we use the representation for $\hat{g}_{2k}(0,0;ir)$ as an integral of the function $\Phi_{2k}(u;x)$ defined by . Applying the Liouville-Green approximation for $\Phi_{2k}(u;x)$ in terms of $Y_0$ and $J_0$ Bessel functions, we prove that for ${\mathbf{k}}:=4k-1$ we have (see Lemma \[lem:LGapprox\]) $$\begin{gathered}
\label{eq:LGgooir}
\hat{g}_{2k}(0,0;ir)=-2^{3/2}\pi^{1/2}
\int_0^{\pi/2}\frac{(\tan x)^{2ir}}{(\sin(2x))^{1/2}}
Y_0({\mathbf{k}}x)x^{1/2}dx-\\
-2^{3/2}\pi^{1/2}
\int_0^{\pi/2}\frac{(\tan x)^{2ir}}{(\sin(2x))^{1/2}}
J_0({\mathbf{k}}x)x^{1/2}dx+O(k^{-3/2}).\end{gathered}$$ We remark that taking the absolute values to estimate the integrals in and using standard estimates for the Bessel functions yields $\hat{g}_{2k}(0,0;ir)\ll k^{-1/2}$, and consequently ${\mathcal{M}}(0,0) \ll k^{1/2+\epsilon}$. In order to improve these bounds, we analyse the integrals in further by making the partition of unity and replacing the Bessel functions with their asymptotic formulas. Consequently, it is required to study the oscillating integral (see Lemma \[int:saddle\]) $$\label{int:saddle2}
\int_0^{\pi/2}\frac{\beta(x)\exp(i{\mathbf{k}}h(x))}{(\sin(2x))^{1/2}}
dx,$$ where $\beta=\beta(x)$ is a smooth characteristic function vanishing at the end points, and $$h(x)=-x+\frac{2r}{{\mathbf{k}}}\log(\tan x).$$ A possible approach to estimate the integral is the saddle point method. However, as $4r\rightarrow k$ we encounter the problem of two coalescing saddle points. It is known that in this case the considered integral has different behaviour in three different ranges: $r$ is small, $r$ is near $k/4$, $r$ is large. As the standard saddle point method cannot be applied in such situation, we follow instead [@BlHan Section 9.2], which describes the method that was originally developed by Chester, Friedman and Ursell [@CFU], with some additional ideas due to Bleistein [@Bl]. As a result, we obtain a uniform expansion of in terms of the Airy function (see ), which yields the following result.
\[Lem gMellin small r\] Let $\delta$ be some fixed constant such that $0<\delta<1/4$. For ${\mathbf{k}}^{1/2-\delta}<r\le{\mathbf{k}}$ we have $$\label{gMellin small r est}
\hat{g}_{2k}(0,0;ir)\ll
\frac{1}{{\mathbf{k}}^{5/6}}\min\left(1,\frac{{\mathbf{k}}^{1/12}}{|{\mathbf{k}}-4r|^{1/4}}\right)
+\frac{k^{-1/4-3\delta}+k^{-1/2}}{r}.$$
This lemma is the key ingredient for the proof of the second main theorem.
\[maincor\] The following upper bound holds $$\label{eq:mainformula}
{\mathcal{M}}(0,0) \ll \log^{3} k.$$
The estimate of Theorem \[maincor\] is at the edge of current technology. A similar result in the $q$-aspect was proved by Petrow [@Pet]. More precisely, [@Pet Theorem 1] refines the estimate of Conrey and Iwaniec [@CI] for the cibic moment of central $L$-values of level $q$ cusp forms twisted by quadratic characters of conductor $q$. See also [@PY; @PY2; @Y] for related results. The role of the dual moment in this case is played by the weighted fourth moment of Dirichlet $L$-functions, as shown in [@Pet Theorem 2].
Another related problem is the cubic moment of cusp form $L$-functions of level one and large individual weight. In this direction, Peng [@Peng] proved that $$\label{eq:peng}
\sum_{\mathfrak{f} \in H_{4k}}\omega(\mathfrak{f})L^3(\mathfrak{f},1/2)\ll k^{\epsilon}.$$ The method of this paper allows replacing $k^{\epsilon}$ in by $\log^5{k}$. Consequently, it is possible to obtain a slightly improved subconvexity estimate $$L(\mathfrak{f},1/2)\ll k^{1/3}\log^{2}{k}.$$ The dual moment in this case is the fourth moment of the Riemann zeta function weighted by the ${}_3F_{2}$ hypergeometric function.
The common characteristic of all mentioned moments is that in order to derive asymptotic formulas, it is required to evaluate the dual moments very carefully taking into consideration the oscillatory behaviour of the corresponding ${}_3F_{2}$ hypergeometric functions. Our results yield a uniform approximation of ${}_3F_{2}$ in terms of simpler functions, which may be useful for further study of these problems.
The main difference between the mixed moment and the above mentioned moments is the appearence of the double Dirichlet series. This turns out to be a specific characteristic of symmetric square $L$-functions. Similar phenomenon in the level aspect was discovered by Iwaniec-Michel [@IM] and Blomer [@Blo]. Furthermore, it is expected that refining the asymptotic formula of Munshi-Sengupta [@MS] for the mixed moment in the level aspect, it should be possible to obtain a second main term of size $q^{-1/2}$ which involves special values of a certain double Dirichlet series.
The paper is organised as follows. Section \[sec:2\] is devoted to the generalised Dirichlet $L$-functions and the associated double Dirichlet series. In Section \[sec:3\] we recall the explicit formula for the twisted first moment of symmetric square $L$-functions. Section \[sec:4\] is the core part of the paper, containing all required estimates for special functions. Finally, in Section \[sec:5\] we prove Theorem \[main thm\] and Theorem \[maincor\].
Generalised Dirichlet $L$-functions {#sec:2}
===================================
As stated in Theorem \[main thm\], the non-diagonal terms of the mixed moment can be expressed in terms of the double Dirichlet series associated to the generalised Dirichlet $L$-function . In this section, we gather various results related to $\mathscr{L}_n(s)$ that are required for evaluation of the non-diagonal terms.
According to [@Byk Section 1], the function $ \mathscr{L}_n(s) $ considered as a function of $ s $ does not vanish only if $ n \equiv 0,1 {\ (\textup{mod}\ 4)} $. The completed $L$-function $$\mathscr{L}_{n}^{*}(s) =
(\pi/|n|)^{-s/2} \Gamma(s/2+1/4-{\operatorname{sgn}}{n}/4) \mathscr{L}_n(s)$$ satisfies the functional equation (see [@Z Proposition 3, p. 130]) $$\mathscr{L}_{n}^{*}(s)=\mathscr{L}_{n}^{*}(1-s).$$ For any $ \epsilon>0 $ we have (see [@BF1 Lemma 4.2]) $$\label{eq:subcL1}
\mathscr{L}_{n}(1/2)\ll |n|^{\theta+\epsilon},$$ where $ \theta=1/6 $ is the best known subconvexity exponent for Dirichlet $ L $-functions obtained by Conrey and Iwaniec in [@CI]. It follows from and the Phragmen-Lindelöf principle that for any $ \epsilon>0 $ and $\Re{u}>0$ the following upper bound holds $$\label{eq:subcL}
\mathscr{L}_{n}(1/2+u)\ll |n|^{\max(\theta(1-2\Re{u}),0)+\epsilon}.$$
Similarly to [@BBFR Section 2.3], we define $f$ as the combination of the Maa[ß]{}-Eisenstein series of weight $ 1/2 $ and level $ 4 $ at the cusps $ \infty $ and $ 0 $, namely $$f=f(z;s):= \zeta(4s-1)
\left(E_{\infty}(z;s;1/2) + \frac{1+i}{4^s} E_{0}(z;s;1/2)\right).$$ Furthermore, we define $g$ as follows $$g=g(z;s):=\frac{1}{2}\left( f\left(\frac{z}{4};s\right)+f\left(\frac{z+2}{4};s\right)\right).$$ Using [@BBFR (2.26), (2.27)], we associate to the functions $f$ and $g$ the double Dirichlet series $$\label{eq:lf}
L^{+}_{f}(s) :=
\frac{\Gamma(1/4)}{2\sqrt{\pi}}\sum_{n>0}\frac{\mathscr{L}_{n}(1/2)}{n^{s+1/2}},
\quad
L^{-}_{f}(s) :=
\frac{\Gamma(3/4)}{2\sqrt{\pi}}\sum_{n<0}\frac{\mathscr{L}_{n}(1/2)}{|n|^{s+1/2}},$$ $$\label{eq:lg}
L^{+}_{g}(s):=\frac{\Gamma(1/4)}{4\sqrt{\pi}}\sum_{n>0}\frac{\mathscr{L}_{4n}(1/2)}{n^{s+1/2}},
\quad
L^{-}_{g}(s):=\frac{\Gamma(3/4)}{4\sqrt{\pi}}\sum_{n<0}\frac{\mathscr{L}_{4n}(1/2)}{|n|^{s+1/2}}.$$
\[thm:lfuncteq\] The functions $L^{\pm}_{f}(s)$ and $L^{\pm}_{g}(s)$ have a meromorphic continuation to the whole complex plane and satisfy the functional equations $$\begin{gathered}
L^{+}_{g}(s) =
\frac{-\pi^{2s+2}}{\sqrt{2}\Gamma^2(1/2+s)\sin^{2}{\pi s}}
\left(\frac{\sin \pi(-s-1/4)}{\pi}L^{+}_{f}(-s)
- \frac{L^{-}_{f}(-s)}{\Gamma^2(3/4)} \right),\end{gathered}$$ $$\begin{gathered}
\label{eq:lgs}
L^{-}_{g}(s) = \frac{\pi^{2s+2}}{\sqrt{2}\Gamma^2(1/2+s)\sin^{2}{\pi s}}
\left(-\frac{\sin \pi(-s+1/4)}{\pi} L^{-}_{f}(-s)
+ \frac{L^{+}_{f}(-s)}{\Gamma^2(1/4)}\right).\end{gathered}$$ Furthermore, $L^{\pm}_{f}(s)$ and $L^{\pm}_{g}(s)$ are holomorphic in ${\mathbf{C}}$ except for a double pole at $s=1/2$.
See [@BBFR Theorem 2.4].
\[cor Lfg coeff relation\] Assume that we have the Laurent series $$\label{Laurent series Lfg}
L^{\pm}_{f,g}(s+1/2)=\frac{c_{f,g}^{\pm}(-2)}{s^2}+\frac{c_{f,g}^{\pm}(-1)}{s}+O(1).$$ Then the coefficients $c_f^{\pm}$, $c_g^{\pm}$ satisfy the following identites: $$\label{eq:firstidentityc}
\frac{c^{+}_{f}(-2)}{\Gamma(1/4)}-
\frac{c^{-}_{f}(-2)}{\Gamma(3/4)}=0$$ $$\frac{c^{+}_{f}(-1)}{\Gamma(1/4)}-
\frac{c^{-}_{f}(-1)}{\Gamma(3/4)}+
\frac{c^{-}_{f}(-2)\pi}{\Gamma(3/4)}=0$$ $$\label{relation for MT 1}
\frac{16\sqrt{\pi}}{\Gamma(3/4)}\left(
c_f^{-}(-2)(1-\sqrt{2})+
c_g^{-}(-2)(\sqrt{2}-1/2)
\right)=1.$$
To evaluate the Laurent expansion for $L^{+}_{f}( s) $ we apply [@BBFR (2.7), (2.9), (2.10)] and [@GR (9.137.11)], getting $$\begin{gathered}
L^{+}_{f}( s+1/2) =
\frac{C \delta \widehat{b}_{\infty,0}}{s^2}
\frac{(2\pi)^{s+1/2}}{2} \frac{2^{3/4}}{\Gamma(s+3/4)}
{}_2F_{1}\left(-1/4,3/4; s+3/4; 1/2\right) \\
+
\frac{1}{s} \frac{(2 \pi)^{s+1/2}}{2} \Bigg(\frac{2^{3/4}}{\Gamma(s+3/4)}
{}_2F_{1}\left(-1/4,3/4; s+3/4; 1/2\right) C \delta (\widehat{a}_{\infty,0}
+ 2 \log{\delta} \, \widehat{b}_{\infty,0}) \\
- \frac{2^{-1/4}}{\Gamma(s+7/4)}
{}_2F_{1}\left(3/4,3/4;s+7/4;1/2\right) C \delta \widehat{b}_{\infty,0}\Bigg)
+ O(1). \end{gathered}$$
Note that in our case (see [@BBFR (2.26), (2.27)]) $C=\sqrt{2},$ $\delta=1/2$ and $$\label{coeff:ab}
\alpha_{\infty,0}=\gamma-\log{4\pi}, \quad \widehat{a}_{\infty,0}=(\gamma-\log{8\pi})/2,\quad b_{\infty,0}=1/2, \quad\widehat{b}_{\infty,0}=1/4.$$
Applying [@GR (9.131.1)] we infer $$\begin{split}
{}_2F_{1} & \left(-1/4,3/4; s+3/4; 1/2\right)
=
\left(\frac{1}{2}\right)^{s+1/4}
\hyp\left(s+1, s; s+3/4; 1/2\right) \\
&=
\left(\frac{1}{2}\right)^{s+1/4}
\left(1 +
\frac{\Gamma(s+3/4)}{\Gamma(s+1) \Gamma(s)}
\sum_{n=0}^{\infty}
\frac{\Gamma(s+2+n) \Gamma(s+1+n)}{(n+1)! \Gamma(s+7/4+n)}
\left(\frac{1}{2}\right)^{n+1}\right),
\end{split}$$ and therefore, $$\begin{split}
\frac{1}{s^2} &
\frac{(2\pi)^{s+1/2}}{2^{1/4} \Gamma(s+3/4)}
{}_2F_{1}\left(-1/4,3/4; s+3/4; 1/2\right)=
\frac{1}{s^2}
\frac{\pi^{s+1/2}}{\Gamma(s+3/4)} \\
& \qquad
+
\frac{1}{s}
\frac{\pi^{s+1/2}}{2 \Gamma^2(s+1)} \sum_{n=0}^{\infty}
\frac{\Gamma(s+2+n) \Gamma(s+1+n)}{(n+1)! \Gamma(s+7/4+n)}
\left(\frac{1}{2}\right)^{n}.
\end{split}$$ Since $$\frac{\pi^{s+1/2}}{\Gamma(s+3/4)} =
\frac{\pi^{1/2}}{\Gamma(3/4)} + \frac{\pi^{1/2}}{\Gamma(3/4)}
\left(\log \pi - \psi(3/4)\right) s + O\left(s^2\right)$$ and $$\begin{gathered}
\lim_{s \rightarrow 0} \frac{\pi^{s+1/2}}{2 \Gamma^2(s+1)}
\sum_{n=0}^{\infty}
\frac{\Gamma(s+2+n) \Gamma(s+1+n)}{(n+1)! \Gamma(s+7/4+n)}
\left(\frac{1}{2}\right)^{n} \\
=
\frac{(2\pi)^{1/2}}{2^{3/2}} \sum_{n=0}^{\infty}
\frac{\Gamma(1+n) \Gamma(1+n)}{n! \Gamma(7/4+n)}
\left(\frac{1}{2}\right)^{n} \\
=
\frac{(2\pi)^{1/2}}{2^{3/2} \Gamma(7/4)}
\hyp\left(1, 1; 7/4; 1/2\right)
=
\frac{(2\pi)^{1/2}}{2^{3/2} \Gamma(3/4)}
\left(\psi(7/8) - \psi(3/8)\right),\end{gathered}$$ we obtain $$\begin{split}
L^{+}_{f}( s+1/2)
&=
\frac{\pi^{1/2} C \delta \widehat{b}_{\infty,0}}{\Gamma(3/4)}
\frac{1}{s^2}
+
\frac{\pi^{1/2} C \delta}{\Gamma(3/4)}
\Big(\left(\log \pi - \psi(3/4)\right) \widehat{b}_{\infty,0} \\
& \quad
+
\widehat{a}_{\infty,0} + 2 \log{\delta} \,
\widehat{b}_{\infty,0}\Big) \frac{1}{s} + O(1).
\end{split}$$
Similarly, using [@GR (9.137.11)] we have $$\begin{split}
L^{-}_{f}(s+1/2)
&=
\frac{(2 \pi)^{s+1/2} C \delta \widehat{b}_{\infty,0}}{2^{11/4}
\Gamma(s+5/4) s^2} {}_2F_{1}\left(1/4,5/4;s+5/4;1/2\right) \\
& \qquad
+
\frac{(2 \pi)^{s+1/2} C \delta (\widehat{a}_{\infty,0}
+ 2 \log{\delta} \, \widehat{b}_{\infty,0})}{2^{11/4} \Gamma(s+5/4) s}
{}_2F_{1}\left(1/4,5/4;s+5/4;1/2\right) \\
& \qquad
+
\frac{(2 \pi)^{s+1/2} C \delta \widehat{b}_{\infty,0}}{2s}
\frac{2^{1/4}}{\Gamma(s+5/4)} {}_2F_{1}\left(1/4,1/4;s+5/4;1/2\right)
+ O(1).
\end{split}$$ In this case [@GR (9.131.1)] yields $$\begin{split}
& {}_2F_{1}\left(1/4,5/4;s+5/4;1/2\right)
= \\
& \qquad
\left(\frac{1}{2}\right)^{s-1/4}
+ \left(\frac{1}{2}\right)^{s-1/4}
\frac{\Gamma(s+5/4) s}{2 \Gamma^2(s+1)} \sum_{n=0}^{\infty}
\frac{\Gamma(s+2+n) \Gamma(s+1+n)}{(n+1)! \Gamma(s+9/4+n)}
\left(\frac{1}{2}\right)^n
\end{split}$$ so that $$\begin{split}
L^{-}_{f}(s+1/2) &=
\frac{\pi^{1/2} C \delta \widehat{b}_{\infty,0}}{\Gamma(1/4)}
\frac{1}{s^2}
+
\frac{\pi^{1/2} C \delta}{\Gamma(1/4)}
\Bigg(\left(\log \pi - \psi(5/4)\right) \widehat{b}_{\infty,0} \\
& \quad
+
4 \widehat{b}_{\infty,0}
+
\widehat{a}_{\infty,0} + 2 \log{\delta} \, \widehat{b}_{\infty,0}\Bigg)
\frac{1}{s} + O(1).
\end{split}$$
Now let us look at $ L^{+}_{g}( s) $. Since $$\begin{split}
L^{+}_{g}( s+1/2) &=
\frac{(2 \pi)^{s+1/2}}{2} \frac{\delta^{2s+1}}{C}
\Bigg(
\frac{1}{s^2} \frac{2^{3/4} b_{\infty,0}}{\Gamma(s+3/4)}
\hyp(-1/4, 3/4; s+3/4;1/2) \\
& \qquad
+
\frac{1}{s}
\bigg(\frac{2^{3/4} a_{\infty,0}}{\Gamma(s+3/4)} \hyp(-1/4, 3/4; s+3/4;1/2) \\
& \qquad
- \frac{2^{-1/4} b_{\infty, 0}}{\Gamma(s+7/4)} \hyp(3/4, 3/4; s+7/4; 1/2)
\bigg)
\Bigg)
+ O(1)
\end{split}$$ we obtain, using the same transformations for the hypergeometric functions as before, $$\begin{split}
L^{+}_{g}( s+1/2) &=
\frac{1}{s^2}
\frac{(\pi \delta^2)^{1/2} b_{\infty,0}}{\Gamma(3/4) C} \\
& \quad
+
\frac{1}{s} \frac{(\pi \delta^2)^{1/2}}{\Gamma(3/4) C}
\Big(\big(\log(\pi \delta^2) - \psi(3/4)\big) b_{\infty,0}
+ a_{\infty,0}\Big)
+ O(1). \\
\end{split}$$ Lastly, we show that $$\begin{split}
L^{-}_{g}( s+1/2)
&=
\frac{1}{s^2} \frac{\delta}{C} \frac{\pi^{1/2}}{\Gamma(1/4)}
b_{\infty, 0} \\
& \quad
+
\frac{1}{s} \frac{\delta}{C} \frac{\pi^{1/2}}{\Gamma(1/4)}
\left(\left(\log(\pi \delta^2) - \psi\left(1/4\right)\right)
b_{\infty, 0}
+ a_{\infty, 0}\right)
+ O(1). \\
\end{split}$$
The identity follows immediately from the expansion for $ L^{+}_{f}( s) $ and $ L^{-}_{f}( s) $. Furthermore, $$\begin{gathered}
\frac{c_f^+(-1)}{\Gamma(1/4)} - \frac{c_f^+(-1)}{\Gamma(3/4)}
+ \frac{c_f^-(-2) \pi}{\Gamma(3/4)} =
\frac{\pi^{1/2} C \delta}{\Gamma(1/4) \Gamma(3/4)}
\Bigg(\left(\log \pi - \psi(3/4)\right) \widehat{b}_{\infty,0}
+
\widehat{a}_{\infty,0} \\+ 2 \log{\delta} \,
\widehat{b}_{\infty,0}
-
\Big(\left(\log \pi - \psi(5/4)\right) \widehat{b}_{\infty,0}
+
4 \widehat{b}_{\infty,0}
+
\widehat{a}_{\infty,0} + 2 \log{\delta} \, \widehat{b}_{\infty,0}\Big)
+
\pi \widehat{b}_{\infty,0}\Bigg) \\
=
\frac{\pi^{1/2} C \delta \widehat{b}_{\infty,0}}{\Gamma(1/4) \Gamma(3/4)}
\Big(- \psi(3/4) + \psi(1/4) + \pi\Bigg)
=
0 \end{gathered}$$ by [@GR (8.365.10), p. 905]. Finally, $$\begin{split}
\frac{16 \sqrt{\pi}}{\Gamma(3/4)}
&
\left(c_f^-(-2) \left(1 - \sqrt{2}\right)
+ c_g^{-}(-2) \left(\sqrt{2} - 1/2\right)\right) \\
&=
\frac{16 \sqrt{\pi}}{\Gamma(3/4)}
\left(\frac{\pi^{1/2} C \delta \widehat{b}_{\infty,0}}{\Gamma(1/4)}
\left(1 - \sqrt{2}\right)
+
\frac{\pi^{1/2} \delta b_{\infty, 0}}{\Gamma(1/4) C}
\left(\sqrt{2} - 1/2\right)\right) \\
&=
8 \sqrt{2} \delta
\left(C \widehat{b}_{\infty,0} \left(1 - \sqrt{2}\right)
+
\frac{b_{\infty, 0}}{C} \left(\sqrt{2} - 1/2\right)\right) =
1
\end{split}$$ since according to we have $ C = \sqrt{2} $, $ \delta = 1/2 $, $ \widehat{b}_{\infty,0} = 1/4 $ and $ b_{\infty, 0} = 1/2 $.
\[thm second mom Lfg\] The following estimates hold $$\int_0^T|L^{\pm}_{g}(it)|^2dt\ll T(\log T)^4,\quad
\int_0^T|L^{\pm}_{f}(it)|^2dt\ll T(\log T)^4.$$
This is a direct consequence of [@Mu Theorem 5.1 (iv)]. Note that the Fourier-Whittaker expansion of the functions $f(z;s)$ and $g(z;s)$ can be found in [@BBFR (2.26), (2.27)].
Explicit formula for the twisted first moment of symmetric square $L$-functions {#sec:3}
===============================================================================
For $0<x<1$ and $0\le\Re{u}<2k-3/2$ let $$\begin{gathered}
\label{defpsi}
\Psi_k(u;x) :=
x^k \frac{\Gamma(k-1/4-u/2) \Gamma(k+1/4-u/2)}{\Gamma(2k)}\\ \times
{}_2F_{1}\left(k-\frac{1}{4}-\frac{u}{2}, k+\frac{1}{4}-\frac{u}{2}; 2k; x\right),\end{gathered}$$ $$\begin{gathered}
\label{defphi}
\Phi_k(u;x) :=
\frac{\Gamma(k-1/4-u/2) \Gamma(3/4-k-u/2)}{\Gamma(1/2)}\\ \times
{}_2F_{1}\left(k-\frac{1}{4}-\frac{u}{2}, \frac{3}{4}-k-\frac{u}{2}; 1/2; x\right),\end{gathered}$$ where $ {}_2F_{1}(a,b;c;x) $ is the Gauss hypergeometric function. For simplicity, let us introduce the following notation $$\label{defpsiphi0}
\Psi_k(x) :=\Psi_k(0;x),\quad
\Phi_k(x) :=\Phi_k(0;x).$$ For $1-2k<\Delta<1/2-\Re{u}$ let $$\begin{gathered}
\label{eq:integralI}
I_k(u;x):=\frac{1}{2\pi i}\int_{(\Delta)}\frac{\Gamma(k-1/2+w/2)}{\Gamma(k+1/2-w/2)}\Gamma(\frac{1}{2}-u-w)\\ \times \sin\left( \pi \frac{1/2+u+w}{2}\right)x^wdw.\end{gathered}$$
According to [@BF1 (5.3)], for $x>2$ we have $$\label{eq:integralI x>2}
I_k(u;x)=(-1)^k\frac{\cos(\pi(1/4+u/2))}{2^{1/2+u}\pi^{1/2}}x\Psi_k\left(u;\frac{4}{x^2}\right).$$
According to [@BF1 (5.5)], for $0<x<2$ we have $$\label{eq:integralI x<2}
I_k(u;x)=(-1)^k\frac{\sin(\pi(1/4+u/2))}{\pi^{1/2}}x^{1/2-u}\Phi_k\left(u;\frac{x^2}{4}\right).$$
Now we are ready to state the explicit formula for the twisted first moment of symmetric square $L$-functions.
\[lem:EF\] For $0\le\Re{u}<4k-3/2$ we have $$\begin{gathered}
\sum_{\mathfrak{f} \in H_{4k}} \omega(\mathfrak{f}) \lambda_{\mathfrak{f}}(l) L({\operatorname{sym}}^2\mathfrak{f}, 1/2+u)
=\\
M^{D}(u,l) \delta_{l=\Box} + M^{ND}(u,l) + ET_1(u,l) + ET_2(u,l),\end{gathered}$$ where $$\delta_{l=\Box} =
\begin{cases}
1 & \text{if } l \text{ is a full square,} \\
0 & \text{otherwise},
\end{cases}$$ $$\begin{gathered}
\label{eq:MT}
M^D(u,l^2)
=
\frac{\zeta(1+2u)}{l^{1/2+u}} + \sqrt{2}(2\pi)^{3u}\cos{\pi(1/4+u/2)}
\times \\
\frac{\zeta(1-2u)}{l^{1/2-u}} \frac{\Gamma(2k-1/4-u/2) \Gamma(2k+1/4-u/2)
\Gamma(1-2u)}{\Gamma(2k+1/4+u/2) \Gamma(2k-1/4+u/2)\Gamma(1-u)},\end{gathered}$$ $$\label{eq:MNDT}
M^{ND}(u,l)
=
\frac{(2\pi)^{1/2+u}}{2l^{1/4-u/2}} \frac{\Gamma(2k-1/4-u/2)}{\Gamma(2k+1/4+u/2)}
\mathscr{L}_{-4l}(1/2+u),$$ $$\label{eq:ET1}
ET_1(u,l) =
(2\pi)^{1/2+u}\sum_{1\leq n<2\sqrt{l}}\frac{\mathscr{L}_{n^2-4l}(1/2+u)}{n^{1/2-u}}I_{2k}\left(u;\frac{n}{l^{1/2}}\right),$$ $$\label{eq:ET2}
ET_2(u,l) =
(2\pi)^{1/2+u}\sum_{n>2\sqrt{l}}\frac{\mathscr{L}_{n^2-4l}(1/2+u)}{n^{1/2-u}}I_{2k}\left(u;\frac{n}{l^{1/2}}\right).$$
See [@BF1 (2.9), (5.6)].
The role of the shift $u$ is to guarantee the absolute convergence of the integral .
Special functions {#sec:4}
=================
For a function $h(x)$, we denote its Mellin transform by $$\label{Mellin def}
\hat{h}(s)=\int_0^{\infty}h(x)x^{s-1}dx.$$
Let us define for $0<x<1$ $$\label{def f}
f_{2k}(u,v;x):=\frac{x^{1/2+v}}{(1-x)^{1/2+v}}I_{2k}\left(u;\frac{2}{(1-x)^{1/2}}\right),$$ and $f_{2k}(u,v;x):=0$ for $x>1$.
For $0<x<\infty$ let $$\label{def g}
g_{2k}(u,v;x):=\frac{x^{1/2+v}}{(1+x)^{1/2+v}}I_{2k}\left(u;\frac{2}{(1+x)^{1/2}}\right).$$
In this section we analyse the Mellin transforms of the functions $f_{2k}(u,v;x)$ and $g_{2k}(u,v;x)$.
Mellin transform of $f_{2k}$
----------------------------
For $\Re{s}>-1/2-\Re{v}$ and $\Re{v}<2k$, $0\le\Re{u}<4k-1$, the Mellin transform of the function can be written in three different ways: $$\label{fMellin1}
\hat{f}_{2k}(u,v;s)=
\frac{2\cos(\pi(1/4+u/2))}{2^{1/2+u}\pi^{1/2}}
\int_0^1\frac{(1-x)^{s+v-1/2}}{x^{1+v}}
\Psi_{2k}\left(u;x\right)dx,$$ $$\begin{gathered}
\label{fMellin2}
\hat{f}_{2k}(u,v;s)=\Gamma(1/2+s+v)
\frac{1}{2\pi i}\int_{(\Delta)}\frac{\Gamma(2k-1/2+w/2)}{\Gamma(2k+1/2-w/2)}\\ \times \Gamma(\frac{1}{2}-u-w)
\sin\left( \pi \frac{1/2+u+w}{2}\right)\frac{\Gamma(1/2-v-w/2)}{\Gamma(1+s-w/2)}2^wdw,\end{gathered}$$ where $1-4k<\Delta<\min(1-2\Re{v},1/2-\Re{u})$, and $$\begin{gathered}
\label{fMellin3}
\hat{f}_{2k}(u,v;s)=
\frac{2^{1/2-u}\sin(\pi(3/4+u/2))}{\pi^{1/2}}\Gamma(1/2+s+v)\\\times
\frac{\Gamma(2k-1/4-u/2)\Gamma(2k+1/4-u/2)\Gamma(2k-v)}{\Gamma(4k)\Gamma(2k+1/2+s)}\\\times
{}_3F_{2}\left(2k-\frac{1}{4}-\frac{u}{2}, 2k+\frac{1}{4}-\frac{u}{2},2k-v; 4k, 2k+1/2+s; 1\right).\end{gathered}$$
It follows from the definition of the Mellin transform that $$\label{fMellin0}
\hat{f}_{2k}(u,v;s)=
\int_0^1\frac{x^{s+v-1/2}}{(1-x)^{1/2+v}}I_{2k}\left(u;\frac{2}{(1-x)^{1/2}}\right)dx.$$ Substituting into we obtain .
Assuming first that $\Re{u}>0$, we substitute to . For $\Re{u}>0,$ $\Re{w}<1-2\Re{v},$ $\Re{s}>-1/2-\Re{v}$, the resulting double integral converges absolutely. Changing the order of integration and using [@HMF (5.12.1)], namely $$\int_0^1\frac{x^{s+v-1/2}}{(1-x)^{1/2+w/2+v}}dx=
\Gamma(1/2+s+v)\frac{\Gamma(1/2-v-w/2)}{\Gamma(1+s-w/2)},$$ we obtain . Note that the integral on the right-hand side of converges absolutely provided that $\Re{s}>-1/2-\Re{u}-\Re{v}.$
Moving the line of integration in to the left and crossing the poles at $w=1-4k-2j$, we finally prove .
For $-1/4<\Re{v}<2k$ we have $$\label{fMellin -1/4}
\hat{f}_{2k}(0,v;-1/4)=
\frac{\Gamma^2(1/4+v)}{\pi^{1/2}}
\frac{\Gamma(2k-1/4)\Gamma(2k-v)}{\Gamma(2k+1/4)\Gamma(2k+v)}.$$
Rewriting for $u=0$, we obtain $$\begin{gathered}
\hat{f}_{2k}(0,v;-1/4)=
\frac{\Gamma(1/4+v)}{\pi^{1/2}}
\frac{\Gamma(2k-1/4)\Gamma(2k-v)}{\Gamma(4k)}
{}_2F_{1}\left(2k-\frac{1}{4}, 2k-v; 4k; 1\right).\end{gathered}$$ Then follows by applying [@HMF (15.4.20)].
\[Lem fMellin 1/2\] The following estimates hold $$\label{fMellin 1/2}
\hat{f}_{2k}(0,0;1/2), \quad \frac{\partial}{\partial s}\hat{f}_{2k}(0,0;s)\Bigg|_{s=1/2}\quad
\ll\frac{k^{\epsilon}}{k^2}.$$
To prove we apply together with the Liouville-Green approximation of the function $\Psi_{2k}\left(x\right)$ obtained in [@BF1]. More precisely, using [@BF1 (6.58), (6.62), (6.64), (6.68)], we have that $$\begin{gathered}
\label{psik LG}
\Psi_{2k}\left(\frac{1}{\cosh^2{\sqrt{\xi}/2}} \right)\left( \xi\sinh^2{\sqrt{\xi}}\right)^{1/4}=\\
C_K\left(
\sqrt{\xi}K_0(u\sqrt{\xi})-\frac{\xi}{u}K_1(u\sqrt{\xi})B_K(0;\xi)\right)+
O\left(\frac{\sqrt{\xi}K_0(u\sqrt{\xi})}{u^{3}}\min\left(\sqrt{\xi}, \frac{1}{\xi}\right)\right),\end{gathered}$$ where $u=2k-1/2,$ $C_K=2+O(k^{-1})$ and $$B(0;\xi)=\frac{1}{16}\left( \frac{\coth{\sqrt{\xi/4}}}{\sqrt{\xi}}-\frac{2}{\xi}\right).$$ Note that there is a typo in the formula [@BF1 (6.58)] for $B(0;\xi).$ Instead of $\coth{\sqrt{\xi}}$ there should be $\coth{\sqrt{\xi/4}}.$ It follows from and the standard bounds on the $K$-Bessel functions [@HMF (10.25.3), (10.30.2), (10.30.3)] that $$\label{psik LG2}
\Psi_{2k}\left(\frac{1}{\cosh^2{\sqrt{\xi}/2}} \right)\left( \xi\sinh^2{\sqrt{\xi}}\right)^{1/4}\ll
\sqrt{\xi}K_0(u\sqrt{\xi}).$$ Applying and making the change of variable $x=\cosh^{-2}{\sqrt{\xi}/2}$, we obtain $$\begin{gathered}
\hat{f}_{2k}(0,0;1/2)\ll
\int_0^1x^{-1}\Psi_{2k}\left(x\right)dx\ll
\int_0^{\infty}\Psi_{2k}\left(\frac{1}{\cosh^2{\sqrt{\xi}/2}} \right)\frac{\sinh{\sqrt{\xi}/2}}{\cosh{\sqrt{\xi}/2}}
\frac{d\xi}{\xi^{1/2}}.\end{gathered}$$ Then according to we have $$\label{fMellin 1/2 2}
\hat{f}_{2k}(0,0;1/2)\ll
\int_0^{\infty}|K_0(u\sqrt{\xi})|
\frac{\tanh{\sqrt{\xi}/2}}{\sinh^{1/2}{\sqrt{\xi}}}
\frac{d\xi}{\xi^{1/4}}.$$ Estimating the $K$-Bessel function by the means of [@HMF (10.25.3), (10.30.3)] completes the proof of the first estimate in . The derivative of $\hat{f}_{2k}(0,0;s)$ can be estimated similarly since it follows from that $$\frac{\partial}{\partial s}\hat{f}_{2k}(0,0;s)\Bigg|_{s=1/2}\ll
\int_0^1\frac{\log(1-x)}{x}\Psi_{2k}\left(x\right)dx.$$
The following estimate holds $$\label{fMellin ir}
\hat{f}_{2k}(0,0;ir)\ll\frac{k^{\epsilon}}{k(1+|r|)^2}.$$
For $|r|\ll1$ we estimate trivially: $$\hat{f}_{2k}(0,0;ir)\ll
\int_0^1\frac{(1-x)^{-1/2}}{x}\Psi_{2k}\left(x\right)dx.$$ Repeating the arguments of Lemma \[Lem fMellin 1/2\], we obtain $$\hat{f}_{2k}(0,0;ir)\ll
\int_0^{\infty}|K_0(u\sqrt{\xi})|
\frac{d\xi}{\xi^{1/4}\sinh^{1/2}{\sqrt{\xi}}}.$$ Using [@HMF (10.25.3), (10.30.2)] we prove .
Now let us consider the case $|r|\gg 1$. Introducing the notation $$\label{Y def}
T_{2k}(x):=(1-x)^{1/2}\Psi_{2k}(x),$$ we have $$\label{fMellinY}
\hat{f}_{2k}(0,0;ir)=
\frac{1}{\pi^{1/2}}\int_0^1\frac{(1-x)^{ir-1}}{x}T_{2k}(x)dx.$$ Integrating by parts three times, we obtain $$\label{fMellinY2}
\hat{f}_{2k}(0,0;ir)\ll\frac{1}{(1+|r|)^3}
\int_0^1(1-x)^{ir+2}\left(T_{2k}(x)x^{-1}\right)'''dx.$$
According to [@BF1 (6.47)], the function $T_{2k}(x)$ satisfies the differential equation $$\label{eq:diffurf21}
T_{2k}''(x)-(u^2\alpha(x)+\beta(x))T_{2k}(x)=0,$$ where $u=2k-1/2$ and $$\label{def fandg}
\alpha(x):=\frac{1}{x^2(1-x)}, \quad \beta(x):=-\frac{1}{4x^2(1-x)^2}+\frac{3}{16x(1-x)}.$$ Differentiating yields $$T_{2k}'''(x)=(u^2\alpha'(x)+\beta'(x))T_{2k}(x)+(u^2\alpha(x)+\beta(x))T'_{2k}(x).$$ Consequently, $$\begin{gathered}
\label{3rd derivative}
\left(T_{2k}(x)x^{-1}\right)'''=\left(\frac{u^2\alpha(x)+\beta(x)}{x}+\frac{6}{x^3}\right)T'_{2k}(x)\\+
\left(\frac{u^2\alpha'(x)+\beta'(x)}{x}-3\frac{u^2\alpha(x)+\beta(x)}{x^2}-\frac{6}{x^4}\right)T_{2k}(x).\end{gathered}$$ Substituting into , we have $$\begin{gathered}
\label{fMellinY3}
\hat{f}_{2k}(0,0;ir)\ll\frac{1}{(1+|r|)^3}\\\times
\int_0^1(1-x)^{2}\left(\frac{u^2|\alpha'(x)|+|\beta'(x)|}{x}+\frac{u^2|\alpha(x)|+|\beta(x)|}{x^2}+\frac{1}{x^4}\right)|T_{2k}(x)|dx\\+
\frac{1}{(1+|r|)^3}\left|\int_0^1(1-x)^{2+ir}
\left(\frac{u^2\alpha(x)+\beta(x)}{x}+\frac{6}{x^3}\right)T'_{2k}(x)dx\right|.\end{gathered}$$ To estimate the second integral in , we integrate it by parts, getting $$\begin{gathered}
\label{fMellinY4}
\frac{1}{(1+|r|)^3}\int_0^1(1-x)^{2+ir}
\left(\frac{u^2\alpha(x)+\beta(x)}{x}+\frac{6}{x^3}\right)T'_{2k}(x)dx\\\ll
\frac{1}{(1+|r|)^2}\int_0^1(1-x)
\left(\frac{u^2|\alpha(x)|+|\beta(x)|}{x}+\frac{1}{x^3}\right)|T_{2k}(x)|dx+\\
\int_0^1 \frac{(1-x)^2}{(1+|r|)^3}
\left(\frac{u^2|\alpha'(x)|+|\beta'(x)|}{x}+\frac{u^2|\alpha(x)|+|\beta(x)|}{x^2}+\frac{1}{x^4}\right)|T_{2k}(x)|dx.\end{gathered}$$ Note that various constants are omitted since we are using the $\ll $ sign. Substituting into , we obtain $$\begin{gathered}
\label{fMellinY5}
\hat{f}_{2k}(0,0;ir)\ll\frac{1}{(1+|r|)^3}\\\times
\int_0^1(1-x)^{2}\left(\frac{u^2|\alpha'(x)|+|\beta'(x)|}{x}+\frac{u^2|\alpha(x)|+|\beta(x)|}{x^2}+\frac{1}{x^4}\right)|T_{2k}(x)|dx\\+
\frac{1}{(1+|r|)^2}\int_0^1(1-x)
\left(\frac{u^2|\alpha(x)|+|\beta(x)|}{x}+\frac{1}{x^3}\right)|T_{2k}(x)|dx.\end{gathered}$$ Consider the second integral in . Using , and making the change of variable $x=\cosh^{-2}{\sqrt{\xi}/2}$, we show that $$\begin{gathered}
\label{fMellinY6}
\frac{1}{(1+|r|)^2}\int_0^1(1-x)
\left(\frac{u^2|\alpha(x)|+|\beta(x)|}{x}+\frac{1}{x^3}\right)|T_{2k}(x)|dx\\\ll
\frac{1}{(1+|r|)^2}\int_0^1(1-x)^{1/2}
\left(\frac{u^2}{x^3}+\frac{1}{x^3(1-x)}\right)|\Psi_{2k}(x)|dx\\ \ll
\frac{1}{(1+|r|)^2}
\int_0^{\infty}\Big|\Psi_{2k}\left(\frac{1}{\cosh^2{\sqrt{\xi}/2}} \right)\Big|
\left(u^2\cosh^{2}\frac{\sqrt{\xi}}{2}\sinh^{2}\frac{\sqrt{\xi}}{2}
+\cosh^{4}\frac{\sqrt{\xi}}{2}\right)\frac{d\xi}{\xi^{1/2}}.\end{gathered}$$ Applying and estimating the $K$-Bessel function using [@HMF (10.25.3), (10.30.3)], we obtain $$\begin{gathered}
\label{fMellinY7}
\frac{1}{(1+|r|)^2}\int_0^1(1-x)
\left(\frac{u^2|\alpha(x)|+|\beta(x)|}{x}+\frac{1}{x^3}\right)|T_{2k}(x)|dx\\\ll
\frac{1}{(1+|r|)^2}
\int_0^{\infty}\frac{|K_0(u\sqrt{\xi})|}{\sinh^{1/2}{\sqrt{\xi}}}
\left(u^2\cosh^{2}\frac{\sqrt{\xi}}{2}\sinh^{2}\frac{\sqrt{\xi}}{2}
+\cosh^{4}\frac{\sqrt{\xi}}{2}\right)\frac{d\xi}{\xi^{1/4}}\\
\ll\frac{k^{\epsilon}}{k(1+|r|)^2}.\end{gathered}$$ Consider the first integral in . Using , we have for $0<x<1$ $$\label{fandg derivative}
\alpha'(x)\ll\frac{1}{x^3(1-x)}+\frac{1}{x^2(1-x)^2}, \quad
\beta'(x)\ll-\frac{1}{x^3(1-x)^2}+\frac{1}{x^2(1-x)^3}.$$ Using , , and making the change of variable $x=\cosh^{-2}{\sqrt{\xi}/2}$, we obtain $$\begin{gathered}
\label{fMellinY8}
\frac{1}{(1+|r|)^3}\int_0^1(1-x)^{2}\left(\frac{u^2|\alpha'(x)|+|\beta'(x)|}{x}+\frac{u^2|\alpha(x)|+|\beta(x)|}{x^2}+\frac{1}{x^4}\right)|T_{2k}(x)|dx\\\ll
\frac{1}{(1+|r|)^3}\int_0^1(1-x)^{1/2}
\left(\frac{u^2(1-x)}{x^4}+\frac{u^2}{x^3}+\frac{1}{x^4}+\frac{1}{x^3(1-x)}\right)\left|\Psi_{2k}(x)\right|dx\\\ll
\frac{1}{(1+|r|)^3}
\int_0^{\infty}\left|\Psi_{2k}\left(\frac{1}{\cosh^2{\sqrt{\xi}/2}} \right)\right|
\Biggl(u^2\cosh^{2}\frac{\sqrt{\xi}}{2}\sinh^{4}\frac{\sqrt{\xi}}{2}+\\
+u^2\cosh^{2}\frac{\sqrt{\xi}}{2}\sinh^{2}\frac{\sqrt{\xi}}{2}
+\cosh^{4}\frac{\sqrt{\xi}}{2}\sinh^{2}\frac{\sqrt{\xi}}{2}
+\cosh^{4}\frac{\sqrt{\xi}}{2}\Biggr)\frac{d\xi}{\xi^{1/2}}.\end{gathered}$$ Applying and the standard bounds on the $K$-Bessel function [@HMF (10.25.3), (10.30.3)], we have $$\begin{gathered}
\label{fMellinY9}
\frac{1}{(1+|r|)^3}\int_0^1(1-x)^{2}\left(\frac{u^2|\alpha'(x)|+|\beta'(x)|}{x}+\frac{u^2|\alpha(x)|+|\beta(x)|}{x^2}+\frac{1}{x^4}\right)\\ \times |T_{2k}(x)|dx
\ll\frac{k^{\epsilon}}{k(1+|r|)^2}.\end{gathered}$$ Substituting and into , we complete the proof of .
Mellin transform of $g_{2k}$
----------------------------
\[lem:gMelTrans\] Assume that $-1/2-\Re{v}<\Re{s}<1/4-\Re{u}/2$ and $0\le\Re{u}<4k-1/2$. Then the Mellin transform of the function $g_{2k}(u,v;x)$ can be written as follows: $$\label{gMellin1}
\hat{g}_{2k}(u,v;s)=
\frac{2^{1/2-u}\sin(\pi(1/4+u/2))}{\pi^{1/2}}
\int_0^1\frac{(1-x)^{s+v-1/2}}{x^{s+u+3/4}}
\Phi_{2k}\left(u;x\right)dx,$$ $$\begin{gathered}
\label{gMellin2}
\hat{g}_{2k}(u,v;s)=\Gamma(1/2+s+v)
\frac{1}{2\pi i}\int_{(\Delta)}\frac{\Gamma(2k-1/2+w/2)}{\Gamma(2k+1/2-w/2)}\Gamma(\frac{1}{2}-u-w)\\\times
\sin\left( \pi \frac{1/2+u+w}{2}\right)\frac{\Gamma(w/2-s)}{\Gamma(1/2+v+w/2)}2^wdw,\end{gathered}$$ where $\max(1-4k,2\Re{s})<\Delta<1/2-\Re{u}$, $$\begin{gathered}
\label{gMellin3}
\hat{g}_{2k}(u,v;s)=
\frac{2^{1/2-u}\sin(\pi(1/4+u/2))}{\pi^{1/2}}\Gamma(1/2+s+v)\\\times
\frac{\Gamma(2k-1/4-u/2)\Gamma(3/4-2k-u/2)\Gamma(1/4-u/2-s)}{\Gamma(1/2)\Gamma(3/4+v-u/2)}\\\times
{}_3F_{2}\left(2k-\frac{1}{4}-\frac{u}{2},\frac{3}{4}-2k-\frac{u}{2},\frac{1}{4}-\frac{u}{2}-s;\frac{1}{2},\frac{3}{4}+v-\frac{u}{2}; 1\right).\end{gathered}$$
It follows from and that $$\label{gMellin0}
\hat{g}_{2k}(u,v;s)=
\int_0^{\infty}\frac{x^{s+v-1/2}}{(1+x)^{1/2+v}}I_{2k}\left(u;\frac{2}{(1+x)^{1/2}}\right)dx.$$ Applying to evaluate , we obtain . Assuming that $\Re{u>0}$, we substitute to . For $\Re{u>0},$ $\Re{w}>2\Re{s},$ $\Re{s}>-1/2-\Re{v}$, the resulting double integral converges absolutely. Changing the order of integration and applying [@HMF (5.12.3)], namely $$\int_0^{\infty}\frac{x^{s+v-1/2}}{(1+x)^{1/2+w/2+v}}dx=
\Gamma(1/2+s+v)\frac{\Gamma(w/2-s)}{\Gamma(1+v+w/2)},$$ we prove . Note that the integral on the right-hand side of converges absolutely provided that $\Re{s}>-1/2-\Re{u}-\Re{v}.$ Moving the line of integration in to the right and crossing the poles at $w=1/2-u+j$, we obtain .
For $\Re{v}>-1/4$ the following equality holds $$\label{gMellin -1/4}
\hat{g}_{2k}(0,v;-1/4)=-\sqrt{2}\sin(\pi v)
\frac{\Gamma^2(1/4+v)}{\pi^{1/2}}
\frac{\Gamma(2k-1/4)\Gamma(2k-v)}{\Gamma(2k+1/4)\Gamma(2k+v)}.$$ In particular, $$\label{gMellin -1/4 v=0}
\hat{g}_{2k}(0,0;-1/4)=0.$$
According to we have $$\begin{gathered}
\hat{g}_{2k}(0,v;-1/4)=
\frac{\Gamma(1/4+v)\Gamma(2k-1/4)\Gamma(3/4-2k)}{\Gamma(3/4+v)\pi^{1/2}}
{}_2F_{1}\left(2k-\frac{1}{4},\frac{3}{4}-2k;\frac{3}{4}+v; 1\right).\end{gathered}$$ Applying [@HMF (15.4.20)], this expression simplifies to $$\hat{g}_{2k}(0,v;-1/4)=
\frac{\Gamma^2(1/4+v)}{\pi^{1/2}}
\frac{\Gamma(2k-1/4)\Gamma(3/4-2k)}{\Gamma(1+v-2k)\Gamma(2k+v)}.$$ Finally, using [@HMF (5.5.3)] we obtain .
\[Lem gMellin v 1/2-v\] For $v\to 1/4$ the following asymptotic formulas hold $$\label{gMellin v 1/2-v}
\hat{g}_{2k}(0,v;1/2-v)=\frac{2^{3/2}}{2v-1/2}\frac{\Gamma(2k-1/4)}{\Gamma(2k+1/4)}+O(1),$$ $$\label{gMellin dif v 1/2-v}
\frac{\partial}{\partial s}\hat{g}_{2k}(0,v;s)\Bigg|_{s=1/2-v}=
\frac{2^{5/2}}{(2v-1/2)^2}\frac{\Gamma(2k-1/4)}{\Gamma(2k+1/4)}+
O(1).$$ Furthermore, $$\label{gMellin 1/2}
\hat{g}_{2k}(0,0;1/2)=2^{3/2}\Gamma(-1/2)+O(k^{-1+\epsilon}),$$ $$\begin{gathered}
\label{gMellin dif 1/2}
\frac{\partial}{\partial s}\hat{g}_{2k}(0,0;s)\Bigg|_{s=1/2}=
-2^{5/2}\Gamma(-1/2)\frac{\Gamma'}{\Gamma}(-1/2)\\+
2^{3/2}\Gamma(-1/2)\left(2\frac{\Gamma'}{\Gamma}(2k)+2\log2-\pi\right)+O(k^{-1+\epsilon}).\end{gathered}$$
For $u=0$, $0\le\Re{v}\le1/2$ and $0<\Re{s}<1/4$, we move the line of integration in to $-2+2\Re{s}<\Delta<2\Re{s}$ crossing the pole at $w=2s$. Hence $$\begin{gathered}
\label{gMellin 0vs}
\hat{g}_{2k}(0,v;s)=
2^{2s+1}\frac{\Gamma(2k-1/2+s)}{\Gamma(2k+1/2-s)}\Gamma(\frac{1}{2}-2s)\sin\left( \pi \frac{1/2+2s}{2}\right)\\+
\Gamma(1/2+s+v)\frac{1}{2\pi i}\int_{(\Delta)}\frac{\Gamma(2k-1/2+w/2)}{\Gamma(2k+1/2-w/2)}\Gamma(\frac{1}{2}-u-w)\\ \times
\sin\left( \pi \frac{1/2+u+w}{2}\right)\frac{\Gamma(w/2-s)}{\Gamma(1/2+v+w/2)}2^wdw,\end{gathered}$$ where $-2+2\Re{s}<\Delta<\min(2\Re{s},1/2).$ Therefore, is now valid for $\Re{s}<5/4.$ Choosing $\Delta=0$, we obtain $$\begin{gathered}
\label{gMellin 0v1/2-v 2}
\hat{g}_{2k}(0,v;1/2-v)=
2^{2-2v}\frac{\Gamma(2k-v)}{\Gamma(2k+v)}\Gamma(2v-\frac{1}{2})\sin(3\pi/4-\pi v)+\\+
\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\Gamma(2k-1/2+ir)}{\Gamma(2k+1/2-ir)}\Gamma(\frac{1}{2}-2ir)
\frac{\sin(\pi/4+\pi ir)2^{2ir}dr}{(ir-1/2+v)}.\end{gathered}$$ Estimating the integral above trivially using Stirling’s formula, we conclude the proof of . Another direct consequence of the representation is . Finally, the formulas and can also be derived from by taking the derivative with respect to $s$.
\[Lem gMellin big r\] For $|r|>3k$ and any $A>0$ we have $$\label{gMellin big r}
\hat{g}_{2k}(0,0;ir)\ll\frac{1}{|r|^A}.$$
Using the representation , we obtain $$\hat{g}_{2k}(0,0;ir)=
\frac{\Gamma(1/2+ir)}{\pi^{1/2}}
\sum_{j=0}^{\infty}\frac{(-1)^j}{j!}
\frac{\Gamma(2k-1/4+j)\Gamma(3/4-2k+j)\Gamma(1/4-ir+j)}{\Gamma(1/2+j)\Gamma(3/4+j)}.$$ It follows from [@HMF (5.4.4), (5.6.6)] that $$|\Gamma(1/2+ir)|\ll\exp(-\pi|r|/2)\text{ and }|\Gamma(1/4-ir+j)|\ll\Gamma(1/4+j).$$ Consequently, $$\hat{g}_{2k}(0,0;ir)\ll
\exp(-\pi|r|/2)
\sum_{j=0}^{\infty}
\frac{|\Gamma(3/4-2k+j)|\Gamma(2k-1/4+j)\Gamma(1/4+j)}{\Gamma(1+j)\Gamma(1/2+j)\Gamma(3/4+j)}.$$ According to [@HMF (5.5.3)] we have $|\Gamma(3/4-2k+j)|=|\Gamma(2k+1/4-j)|^{-1}.$ Furthermore, $$\frac{\Gamma(1/4+j)}{\Gamma(1+j)\Gamma(1/2+j)\Gamma(3/4+j)}\ll\frac{1}{\Gamma^2(1+j)}.$$ As a result, $$\begin{gathered}
\label{gMellin big r1}
\hat{g}_{2k}(0,0;ir)\ll\exp(-\pi|r|/2)
\sum_{j=0}^{2k-1}
\frac{\Gamma(2k-1/4+j)}{\Gamma(2k+1/4-j)\Gamma^2(1+j)}+\\
\exp(-\pi|r|/2)\sum_{j=2k}^{\infty}
\frac{\Gamma(2k-1/4+j)\Gamma(j+3/4-2k)}{\Gamma^2(1+j)}.\end{gathered}$$ Using Stirling’s formula [@HMF (5.11.3)] we obtain that for $0<j<2k-1$ $$\begin{gathered}
\frac{\Gamma(2k-1/4+j)}{\Gamma(2k+1/4-j)\Gamma^2(1+j)}\asymp
\frac{(2k+j)^{-1/4}}{(2k-j)^{1/4}j^2}\frac{\Gamma(2k+j)}{\Gamma(2k-j)\Gamma^2(j)}\asymp\\
\frac{(2k+j)^{-3/4}}{(2k-j)^{-1/4}j}
\frac{(2k+j)^{2k+j}}{(2k-j)^{2k-j}j^{2j}}=
\frac{(2k-j)^{1/4}}{(2k+j)^{3/4}j}\exp(s_1(k,j)),\end{gathered}$$ where $$s_1(k,j):=(2k+j)\log(2k+j)-(2k-j)\log(2k-j)-2j\log j.$$ The function $s_1(k,j)$ attains its maximum at the point $j=k\sqrt{2}$, and therefore, $$\begin{gathered}
\label{gMellin big r2}
\exp(-\pi|r|/2)\sum_{j=0}^{2k-1}
\frac{\Gamma(2k-1/4+j)}{\Gamma(2k+1/4-j)\Gamma^2(1+j)}\ll\\
\exp(-\pi|r|/2)\sum_{j=1}^{2k-1}\frac{(2k-j)^{1/4}}{(2k+j)^{3/4}j}\exp(s_1(k,j))\ll\\
\exp(-\pi|r|/2+s_1(k,k\sqrt{2}))k^{1/2}\ll\frac{1}{|r|^A}\end{gathered}$$ for $|r|>3k.$ In the same way we show that for $j>2k$ $$\frac{\Gamma(2k-1/4+j)\Gamma(j+3/4-2k)}{\Gamma^2(1+j)}\asymp
\frac{(j-2k)^{1/4}}{(2k+j)^{3/4}j}\exp(s_2(k,j)),$$ where $$s_2(k,j):=(2k+j)\log(2k+j)-(j-2k)\log(j-2k)-2j\log j.$$ The function $s_2(k,j)$ is decreasing. For $j=2k$ it follows from [@HMF (5.5.5)] that $$\frac{\Gamma(2k-1/4+j)\Gamma(j+3/4-2k)}{\Gamma^2(1+j)}\asymp
\frac{\Gamma(4k-1/4)}{\Gamma^2(2k+1)}\ll\frac{2^{4k}}{k^{7/4}}.$$ Finally, we obtain $$\begin{gathered}
\label{gMellin big r3}
\exp(-\pi|r|/2)\sum_{j=2k}^{\infty}
\frac{\Gamma(2k-1/4+j)\Gamma(j+3/4-2k)}{\Gamma^2(1+j)}\ll\\
\exp(-\pi|r|/2)\sum_{j=2k}^{\infty}\frac{2^{4k}}{j^{3/2}}\ll\frac{1}{|r|^A}\end{gathered}$$ for $|r|>3k.$ Substituting and into , we prove the lemma.
To investigate the behavior of the function $\hat{g}_{2k}(0,0;ir)$ for $|r|\le3k$ we use the formula .
\[lem:LGapprox\] Let ${\mathbf{k}}:=4k-1.$ We have $$\begin{gathered}
\label{gMellin BessInt}
\hat{g}_{2k}(0,0;ir)=-2^{3/2}\pi^{1/2}
\int_0^{\pi/2}\frac{(\tan x)^{2ir}}{(\sin(2x))^{1/2}}
Y_0({\mathbf{k}}x)x^{1/2}dx-\\
-2^{3/2}\pi^{1/2}
\int_0^{\pi/2}\frac{(\tan x)^{2ir}}{(\sin(2x))^{1/2}}
J_0({\mathbf{k}}x)x^{1/2}dx+O(k^{-3/2}),\end{gathered}$$ $$\label{gMellin triv.est.}
\hat{g}_{2k}(0,0;ir)\ll\frac{1}{k^{1/2}}.$$
It follows from that $$\label{gMellin1 at 00}
\hat{g}_{2k}(0,0;ir)=
\frac{1}{\pi^{1/2}}
\int_0^{\pi^2/4}\frac{(\tan\sqrt{\xi})^{2ir}}{(\cos\sqrt{\xi})^{1/2}}
\Phi_{2k}\left(\cos^2\sqrt{\xi}\right)\frac{d\xi}{\sqrt{\xi}}.$$ In order to prove , we apply the following approximation of the function $\Phi_{2k}$ (see [@BF1 Theorems 6.5 and 6.10, Corollary 6.9] for details): $$\begin{gathered}
\label{eq:Phi2k}
\Phi_{2k}(\cos^2\sqrt{\xi}) =
\frac{-\pi}{\xi^{1/4}(\sin{\sqrt{\xi}})^{1/2}}
\Biggl[\sqrt{\xi}Y_0((4k-1)\sqrt{\xi}) + \\
\sqrt{\xi}J_0((4k-1)\sqrt{\xi})
+ O\left(\frac{1}{k}\left|\sqrt{\xi}Y_0((4k-1)\sqrt{\xi})\right|\right)\Biggr].\end{gathered}$$ Substituting into and estimating the error term using standard estimates on the $Y$-Bessel function [@HMF (10.7.1), (10.7.8)], we obtain . Applying [@HMF (10.7.1), (10.7.8)] to estimate the integrals in we prove .
The estimate is sufficiently good for our purposes only if $r\ll {\mathbf{k}}^{1/2-\delta}$. For $r\gg{\mathbf{k}}^{1/2-\delta}$, it is required to analyse more carefully. We consider further only the first integral in , as the second integral can be treated similarly. The idea is to replace the $Y$-Bessel function in by its asymptotic formula [@HMF (10.7.8)]. To this end, we first make the following partition of unity: $$\label{partition of unity}
\alpha_1(x)+\beta(x)+\alpha_2(x)=1 \quad\hbox{for}\quad0\le x\le\frac{\pi}{2},$$ where $\alpha_{1,2}(x)$, $\beta(x)$ are smooth infinitely differentiable functions such that for some small $\varepsilon>0$ (to be chosen later), we have $$\label{partition of unity1}
\alpha_1(x)=1 \text{ for } 0\le x\le \varepsilon,\quad
\alpha_1(x)=0 \text{ for } x\ge 2\varepsilon,$$ $$\label{partition of unity2}
\alpha_2(x)=1 \text{ for } \frac{\pi}{2}-\varepsilon\le x\le\frac{\pi}{2},\quad
\alpha_2(x)=0 \text{ for } 0\le x\le \frac{\pi}{2}-2\varepsilon,$$ $$\label{partition of unity3}
\beta(x)=1 \text{ for }2\varepsilon\le x\le \frac{\pi}{2}-2\varepsilon,\quad
\beta(x)=0 \text{ for } 0\le x\le \varepsilon, \frac{\pi}{2}-\varepsilon\le x\le \frac{\pi}{2},$$ and $\alpha_{1,2}^{(j)}(x)\ll\varepsilon^{-j}$, $\beta^{(j)}(x)\ll\varepsilon^{-j}.$
For $|r|>1$ the following holds $$\begin{gathered}
\label{gMellin bettaBessInt}
\hat{g}_{2k}(0,0;ir)\ll
\left|\int_0^{\pi/2}\frac{\beta(x)(\tan x)^{2ir}}{(\sin(2x))^{1/2}}
Y_0({\mathbf{k}}x)x^{1/2}dx\right|+
\\
\left|
\int_0^{\pi/2}\frac{\beta(x)(\tan x)^{2ir}}{(\sin(2x))^{1/2}}
J_0({\mathbf{k}}x)x^{1/2}dx\right|
+O\left(\frac{k^{-1+\epsilon}+k^{1/2}\varepsilon^{3/2}}{r}\right).\end{gathered}$$
As the first step, we use the partition of unity to rewrite the integrals in . Then to prove the lemma, it is required to estimate the contribution of integrals with $\alpha_{1,2}(x).$ All these integrals can be analysed similarly. Therefore, we consider only $$\label{Y alpha 1}
I_1:=\int_0^{\pi/2}\frac{\alpha_1(x)Y_0({\mathbf{k}}x)x^{1/2}}{(\sin(2x))^{1/2}}
(\tan x)^{2ir}dx.$$ Integrating by parts we obtain $$\label{Y alpha 2}
I_1\ll\frac{1}{r}\int_0^{\pi/2}\frac{\partial}{\partial x}\left[
\alpha_1(x)Y_0({\mathbf{k}}x)x^{1/2}(\sin(2x))^{1/2}\right](\tan x)^{2ir}dx.$$ Evaluating the derivative and estimating the integral trivially with the use of [@HMF (10.7.1), (10.7.8)], we complete the proof of .
For simplicity, let us assume further that $r>0$. The case $r<0$ can be treated in the same way.
\[int:saddle\] Let $\delta$ be some fixed constant such that $0<\delta<1/4$. Then for $r>1$ and $\varepsilon=k^{-1/2-2\delta}$ we have $$\label{gMellin bettaInt}
\hat{g}_{2k}(0,0;ir)\ll\frac{1}{k^{1/2}}
\left|\int_0^{\pi/2}\frac{\beta(x)\exp(i{\mathbf{k}}h(x))}{(\sin(2x))^{1/2}}
dx\right|+\frac{k^{-1/4-3\delta}+k^{-1/2}}{r}+\frac{1}{k^{5/4-\delta}},$$ where $$\label{h def}
h(x)=-x+\frac{2r}{{\mathbf{k}}}\log(\tan x).$$
We substitute the asymptotic formulas for Bessel functions [@GR (8.451.1), (8.451.2)] into and estimate the error terms by its absolute value, obtaining $$\hat{g}_{2k}(0,0;ir)\ll\frac{1}{k^{1/2}}
\sum_{\pm}\left|\int_0^{\pi/2}\frac{\beta(x)\exp(ih_{\pm}(x))}{(\sin(2x))^{1/2}}
dx\right|+\frac{k^{-1/4-3\delta}}{r}+\frac{1}{k^{5/4-\delta}},$$ where $$\label{h_pm def}
h_{\pm}(x)=\pm{\mathbf{k}}x+2r\log(\tan x).$$ The integral with $h_{+}(x)$ can be estimated using [@T Lemma 4.3]. More precisely, $$\frac{1}{k^{1/2}}
\left|\int_0^{\pi/2}\frac{\beta(x)\exp(ih_{+}(x))}{(\sin(2x))^{1/2}}
dx\right|\ll\frac{1}{k^{1/2}}\max_{0<x<\pi/2}\frac{\beta(x)(\sin(2x))^{1/2}}{{\mathbf{k}}\sin(2x)+4r}\ll\frac{1}{rk^{1/2}}.$$
The classical approach to estimate the integral on the right-hand side of is the saddle point method (also called the method of steepest descent). Another possibility is the stationary phase method, which is in some sense (see discussion in [@BlHan pp. 276–279]) an analogue of the saddle point method for Fourier-type integrals. The first step in all these methods is to determine the so-called saddle points of the function $h(x)$ defined as zeros of $h'(x)=0.$ Using we find that $$\label{h derivative}
h'(x)=-1+\frac{4r}{{\mathbf{k}}}\frac{1}{\sin(2x)}.$$ It is convenient to introduce two new parametrs $\vartheta$ and $\mu$ such that: $$\label{varteta def}
\sin(2\vartheta)=\frac{4r}{{\mathbf{k}}},\quad0<\vartheta<\frac{\pi}{4}\quad\hbox{if}\quad 4r\le{\mathbf{k}},$$ $$\label{mu def}
\cosh(2\mu)=\frac{4r}{{\mathbf{k}}},\quad \mu>0\quad\hbox{if}\quad 4r>{\mathbf{k}}.$$ Then the saddle points of the function $h(x)$ are $$\label{sadpoints def1}
x_1=\vartheta, \quad x_2=\frac{\pi}{2}-\vartheta\quad\hbox{if}\quad 4r\le{\mathbf{k}},$$ $$\label{sadpoints def2}
x_3=\frac{\pi}{4}-i\mu, \quad x_4=\frac{\pi}{4}+i\mu\quad\hbox{if}\quad 4r>{\mathbf{k}}.$$ We consider only the case $4r\le{\mathbf{k}}$ since the second case can be analysed similarly. Note that the condition $r\gg{\mathbf{k}}^{1/2-\delta}$ implies that $$\label{teta conditions}
\vartheta\gg{\mathbf{k}}^{-1/2-\delta}>k^{-1/2-2\delta}=\varepsilon.$$ Thus both saddle points belong to the interval of integration.
An important observation is that as $4r\rightarrow{\mathbf{k}}$ [*the saddle points coalesce*]{}. It is known that in this case the behaviour of the integral changes. Therefore, it is required to analyse three different ranges:
- $r$ is small,
- $r$ is near ${\mathbf{k}}/4$,
- $r$ is large.
The case of coalescing saddle points is usually described in books, see [@BlHan Section 9.2] and [@Wong Section 7.4]. It is well known that the standard saddle point method does not work in this situation and a more refined analysis is required. Therefore, we mainly follow [@BlHan Section 9.2]. This approach was originally developed by Chester, Friedman and Ursell [@CFU], with some additional ideas due to Bleistein [@Bl].
The main idea of the method is to change the variable of integration such that the integral can be written in terms of the Airy function, which has for real $x$ the following representation: $$\label{Airy def}
Ai(ax^{2/3})=\frac{x^{1/3}}{2\pi}\int_{-\infty}^{\infty}\exp\left(ix\left(\frac{y^3}{3}+ay\right)\right)dy.$$ For simplicity, let us denote $$\label{g def}
g(x):=\frac{\beta(x)}{(\sin(2x))^{1/2}}.$$ Our goal is to estimate the integral (see ) $$\label{I Int}
I=\int_0^{\pi/2}g(x)\exp(i{\mathbf{k}}h(x))dx, \quad h(x)=-x+\frac{\sin\vartheta}{2}\log(\tan x).$$ To this end, following [@BlHan (9.2.6)] we define a new variable $t$ such that: $$\label{t variable def}
h(x)=\frac{t^3}{3}-\gamma^2 t+\rho,$$ where the constants $\gamma$ and $\rho$ are chosen such that the point $x=x_1$ corresponds to $t=-\gamma$ and the point $x=x_2$ corresponds to $t=\gamma.$ These conditions yield $$h(x_1)=\frac{2\gamma^3}{3}+\rho,\quad h(x_2)=-\frac{2\gamma^3}{3}+\rho,$$ and therefore, $$\label{gamma rho expression1}
\frac{4\gamma^3}{3}=h(x_1)-h(x_2),\quad 2\rho=h(x_1)+h(x_2).$$ Evaluating $h(x_{1,2})$ we find that $\rho=-\pi/4$ and $$\label{gamma rho expression2}
\frac{4\gamma^3}{3}=\frac{\pi}{2}-2\vartheta+\sin(2\vartheta)\log(\tan\vartheta).$$ Note that for $0<\vartheta<\pi/4$ the right-hand side of is positive, and that for $\vartheta=\pi/4$ we obtain $\gamma=0$. Changing the variable $x$ in the integral by $t$ defined by , we obtain an analogue of [@BlHan (9.2.18), (9.2.19)], namely $$\label{I Int2}
I=\exp(i{\mathbf{k}}\rho)\int_{-\infty}^{\infty}G_0(t,\vartheta)\exp\left(i{\mathbf{k}}(t^3/3-\gamma^2 t)\right)dt,$$ where $$\label{G0 def}
G_0(t,\vartheta)=g(x(t))\frac{dx}{dt}.$$ As in [@BlHan (9.2.20)] we write $$\label{H0 def}
G_0(t,\vartheta)=a_0+a_1t+(t^2-\gamma^2)H_0(t,\vartheta),$$ where (see [@BlHan (9.2.21),(9.2.22)]) $$\label{a0a1 def}
a_0=\frac{G_0(\gamma,\vartheta)+G_0(-\gamma,\vartheta)}{2},\quad
a_1=\frac{G_0(\gamma,\vartheta)-G_0(-\gamma,\vartheta)}{2\gamma}.$$ Note that $a_0$ and $a_1$ are chosen such that the function $H_0(t,\vartheta)$ is regular at points $t=\pm\gamma.$
To evaluate $a_{0,1}$, as well as to analyse the properties of $H_0(t,\vartheta)$, we need some preliminary results.
\[lem derivatives\] For $\vartheta<\pi/4$ we have $$\label{dx/dt at gamma}
\frac{dx}{dt}\Biggr|_{t=\pm\gamma}=\sqrt{\gamma\tan(2\vartheta)},$$ $$\label{d2x/dt2 at gamma}
\frac{d^2x}{dt^2}\Biggr|_{t=\pm\gamma}=\mp\frac{1}{3}\left(
4\gamma+2\gamma\tan^2(2\vartheta)-\sqrt{\frac{\tan(2\vartheta)}{\gamma}}
\right).$$ For $\vartheta=\pi/4$ we have $$\label{dx/dt at 0}
\frac{dx}{dt}\Biggr|_{t=0}=2^{-1/3},\quad
\frac{d^2x}{dt^2}\Biggr|_{t=0}=0,\quad
\frac{d^3x}{dt^3}\Biggr|_{t=0}=-1.$$
First, consider the case $\vartheta<\pi/4$, $t=-\gamma$. We can write $$\label{taylor series 1}
x-\vartheta=\sum_{n=0}^{\infty}b_n(t+\gamma)^n,\quad
h'(x)=\sum_{n=0}^{\infty}c_n(t+\gamma)^n.$$ Let us compute $b_i$, $c_i$ for $i=0,1,2.$ Note that $b_0=0$ since the point $x=\vartheta$ corresponds to $t=-\gamma$. We have $$\label{taylor series 2}
h'(x)=-1+\frac{\sin(2\vartheta)}{\sin(2x)}=-\frac{2(x-\vartheta)}{\tan(2\vartheta)}+
(x-\vartheta)^2\left(\frac{4}{\tan^2(2\vartheta)}+2\right)+O((x-\vartheta)^3).$$ Substituting the expansion for $(x-\vartheta)$ from into , we show that $$\label{c coeff}
c_0=0,\quad
c_1=-\frac{2b_1}{\tan(2\vartheta)},\quad
c_2=-\frac{2b_2}{\tan(2\vartheta)}+b_1^2\left(\frac{4}{\tan^2(2\vartheta)}+2\right).$$ It follows from that $$\label{t variable def conseq}
h'(x)\frac{dx}{dt}=t^2-\gamma^2=-2\gamma(t+\gamma)+(t+\gamma)^2.$$ Substituting into yields $$\label{b,c coeff relation}
c_1b_1=-2\gamma,\quad
c_2b_1+2c_1b_2=1.$$ Using and , we obtain $$b_1=\sqrt{\gamma\tan(2\vartheta)},\quad
b_2=\frac{1}{6}\left(
4\gamma+2\gamma\tan^2(2\vartheta)-\sqrt{\frac{\tan(2\vartheta)}{\gamma}}
\right).$$ This proves and for $t=-\gamma$. The case $t=\gamma$ is similar.
Second, consider $\vartheta=\pi/4$. In that case $\gamma=0.$ We can write $$\label{taylor series 3}
x-\frac{\pi}{4}=\sum_{n=0}^{\infty}d_nt^n,\quad
h'(x)=\sum_{n=0}^{\infty}e_nt^n.$$ We proceed to compute $d_i$, $e_i$ for $i=0,1,2,3.$ Note that $d_0=0$. Furthermore, we have $$\label{taylor series 4}
h'(x)=-1+\frac{1}{\sin(2x)}=2\left(x-\frac{\pi}{4}\right)^2+\frac{10}{3}\left(x-\frac{\pi}{4}\right)^4
+O((x-\pi/4)^6).$$ Substituting the expansion for $(x-\pi/4)$ from into , we show that $$\label{e coeff}
e_0=e_1=0,\quad
e_2=2d_1^2,\quad
e_3=4d_1d_2,\quad
e_4=2d^2_2+4d_1d_3+\frac{10}{3}d_1^4.$$ Substituting into gives $$\label{d,e coeff relation}
e_2d_1=1,\quad
2e_2d_2+e_3d_1=0,\quad
3e_2d_3+2e_3d_2+e_4d_1=0.$$ Using and we finally show that $$d_1=2^{-1/3},\quad
d_2=0,\quad
d_3=-\frac{1}{6}.$$ This completes the proof of .
\[lem a0a1\] For $\vartheta<\pi/4$ we have $$\label{a0a1 value}
a_0=\sqrt{\frac{\gamma}{\cos(2\vartheta)}},\quad
a_1=0,$$ and for $\vartheta=\pi/4$ we have $$\label{a0a1 pi/4value}
a_0=2^{-1/3},\quad a_1=0.$$
Consider the case $\vartheta<\pi/4$. It follows from , , and that $$a_0=\frac{\beta(\vartheta)+\beta(\pi/2-\vartheta)}{2}\sqrt{\frac{\gamma}{\cos(2\vartheta)}},\quad
a_1=\frac{\beta(\pi/2-\vartheta)-\beta(\vartheta)}{2\gamma}\sqrt{\frac{\gamma}{\cos(2\vartheta)}}.$$ As a consequence of we obtain .
Consider the case $\vartheta=\pi/4$. It follows from , , and that $a_0=2^{-1/3}$ and $$a_1=\frac{d}{dt}G_0\left(t,\frac{\pi}{4}\right)\Bigr|_{t=0}=
g'(\pi/4)\left(\frac{dx}{dt}\Bigr|_{t=0}\right)^2+g(\pi/4)\frac{d^2x}{dt^2}\Bigr|_{t=0}=
\frac{d^2x}{dt^2}\Bigr|_{t=0}=0.$$ This proves .
Substituting into and using Lemma \[lem a0a1\], we obtain the following representation for our integral $$\begin{gathered}
I=\exp(i{\mathbf{k}}\rho)a_0\int_{-\infty}^{\infty}\exp\left(i{\mathbf{k}}(t^3/3-\gamma^2 t)\right)dt+\\
\frac{\exp(i{\mathbf{k}}\rho)}{i{\mathbf{k}}}\int_{-\infty}^{\infty}H_0(t,\vartheta)d\exp\left(i{\mathbf{k}}(t^3/3-\gamma^2 t)\right).\end{gathered}$$ Using and integrating by parts yields $$I=\exp(i{\mathbf{k}}\rho)\frac{2a_0\pi}{{\mathbf{k}}^{1/3}}Ai(-\gamma^2{\mathbf{k}}^{2/3})
-\frac{\exp(i{\mathbf{k}}\rho)}{i{\mathbf{k}}}\int_{-\infty}^{\infty}\frac{d}{dt}\left(H_0(t,\vartheta)\right)\exp\left(i{\mathbf{k}}(t^3/3-\gamma^2 t)\right)dt.$$ Since the function $H_0(t,\vartheta)$ has a finite number of intervals of monotonicity, we can estimate the integral in the formula above by its absolute value, getting $$\label{I Int4}
I=\exp(i{\mathbf{k}}\rho)\frac{2a_0\pi}{{\mathbf{k}}^{1/3}}Ai(-\gamma^2{\mathbf{k}}^{2/3})+
O\left(\frac{1}{{\mathbf{k}}}\max_{t}|H_0(t,\vartheta)|\right).$$ The final step is to estimate the function $H_0(t,\vartheta)$.
For $\vartheta\le\pi/4$ satisfying we have $$\label{H0 estimate}
\max_{t}|H_0(t,\vartheta)|\ll\frac{1}{\sqrt{\vartheta}}.$$
It is required to consider separately three different cases: $\vartheta\rightarrow\frac{\pi}{4},$ $\vartheta\rightarrow0$ and the third case when $\vartheta$ is some fixed number.
First, let us assume that $\vartheta$ is some fixed number. Then $\gamma$ (see ) is also some fixed number. We start by estimating $H_0(t,\vartheta)$ near the points $t=\pm\gamma.$ According to [@BlHan (9.2.24)], we have $$\label{H0 at pm gamma}
\lim_{t\rightarrow\pm\gamma}H_0(t,\vartheta)=\pm\frac{1}{2\gamma}\frac{d}{dt}G_0(t,\vartheta)\Bigr|_{t=\pm\gamma}.$$ Using , and Lemma \[lem derivatives\], we prove that $H_0(t,\vartheta)$ is bounded near the points $t=\pm\gamma.$ Other critical points of $H_0(t,\vartheta)$ are $t\rightarrow\pm\infty.$ In this case we use , and , getting $$\label{H0 expression}
H_0(t,\vartheta)=\frac{g(x(t))}{h'(x(t))}-\frac{a_0}{t^2-\gamma^2}=
\frac{\beta(x(t))\sqrt{\sin(2x(t))}}{\sin(2\vartheta)-\sin(2x(t))}-\frac{a_0}{t^2-\gamma^2}.$$ Consequently, for all $t$ that do not belong to a neighbourhood of $\pm\gamma$, the function $H_0(t,\vartheta)$ is trivially bounded by a constant.
Second, consider the case $\vartheta\rightarrow\frac{\pi}{4}.$ It is enough to prove for $\vartheta=\pi/4.$ In this case we have $\gamma=0$ and (see ) $$\label{H0 at 0}
\lim_{t\rightarrow0}H_0(t,\pi/4)=\frac{1}{2}\frac{d^2}{dt^2}G_0(t,\pi/4)\Bigr|_{t=0}.$$ It follows from that $$\label{G0 second der}
\frac{d^2}{dt^2}G_0(t,\pi/4)=
g''(x(t))\left(\frac{dx}{dt}\right)^3+3g'(x(t))\frac{dx}{dt}\frac{d^2x}{dt^2}+
g(x(t))\frac{d^3x}{dt^3}.$$ Using the fact that all derivatives of $g(x)$ at $x=\pi/4$ are finite and applying Lemma \[lem derivatives\], we conclude that the limit in is finite. For $t$ outside of a neighbourhood of $0$, the function $H_0(t,\pi/4)$ is trivially bounded by a constant using .
Third, consider the case $\vartheta\rightarrow0$. Let us estimate the right-hand side of . We remark that $\gamma$ is a constant for small $\vartheta$ . Therefore, it is only required to estimate the derivative of $G_0(t,\vartheta).$ It follows from that $$\frac{d}{dt}G_0(t,\vartheta)=g'(x(t))\left(\frac{dx}{dt}\right)^2+g(x(t))\frac{d^2x}{dt^2}.$$ Using and Lemma \[lem derivatives\] we obtain the estimate $$\label{G0 derivative est}
\frac{d}{dt}G_0(t,\vartheta)\Bigr|_{t=\pm\gamma}\ll\frac{\tan(2\vartheta)}{\sin^{3/2}(2\vartheta)}+\frac{1}{\sin^{1/2}(2\vartheta)}
\ll\frac{1}{\vartheta^{1/2}},$$ which completes the proof of .
Substituting into we obtain for ${\mathbf{k}}^{1/2-\delta}<r\le{\mathbf{k}}/4$ (see ) that $$\label{I Int5}
I=\exp(i{\mathbf{k}}\rho)\frac{2a_0\pi}{{\mathbf{k}}^{1/3}}Ai(-\gamma^2{\mathbf{k}}^{2/3})+
O\left(\frac{1}{(r{\mathbf{k}})^{1/2}}\right).$$ Since $Ai(-x)\ll\min(1,x^{-1/4})$ we have $$\label{I Int6}
I\ll\frac{1}{{\mathbf{k}}^{1/3}}\min\left(1,\frac{1}{\gamma^{1/2}{\mathbf{k}}^{1/6}}\right)+
\frac{1}{(r{\mathbf{k}})^{1/2}}.$$ It follows from and that $$\begin{gathered}
\gamma=2^{1/3}\left(\frac{\pi}{4}-\vartheta\right)+O\left((\pi/4-\vartheta)^5\right)=
2^{-2/3}\arccos\frac{4r}{{\mathbf{k}}}+O\left(\arccos^5\frac{4r}{{\mathbf{k}}}\right)=\\=
2^{-1/6}\left(1-\frac{4r}{{\mathbf{k}}}\right)^{1/2}+O\left((1-4r/{\mathbf{k}})^{3/2}\right).\end{gathered}$$ Consequently, $$\label{I Int7}
I\ll\frac{1}{{\mathbf{k}}^{1/3}}\min\left(1,\frac{1}{(1-4r/{\mathbf{k}})^{1/4}{\mathbf{k}}^{1/6}}\right)+
\frac{1}{(r{\mathbf{k}})^{1/2}}.$$
Using to estimate the integral in , we finally prove Lemma \[Lem gMellin small r\].
Explicit formula for the mixed moment {#sec:5}
=====================================
This section is devoted to proving an explicit formula for the mixed moment $$\sum_{\mathfrak{f}\in H_{4k}} \omega(\mathfrak{f}) L(\mathfrak{f},1/2)L({\operatorname{sym}}^2\mathfrak{f}, 1/2).$$ To this end, we introduce two complex variables $u$, $v$ with sufficiently large real parts and consider the shifted moment $${\mathcal{M}}(u,v)=\sum_{\mathfrak{f}\in H_{4k}} \omega(\mathfrak{f}) L(\mathfrak{f},1/2+v)L({\operatorname{sym}}^2\mathfrak{f}, 1/2+u).$$ This enables us to use the technique of analytic continuation.
Let us assume for simplicity that $0<\Re{u}<1$ and $\Re{v}>3/4+\Re{u}/2.$ Using and Lemma \[lem:EF\] we obtain $$\label{M(u,v) 1}
{\mathcal{M}}(u,v)={\mathcal{M}}^D(u,v)+{\mathcal{M}}^{ND}(u,v)+{\mathcal{ET}}_1(u,v)+{\mathcal{ET}}_2(u,v),$$ where $$\label{MD MND def}
{\mathcal{M}}^D(u,v)=\sum_{l=1}^{\infty}\frac{M^{D}(u,l^2)}{l^{1+2v}},\quad
{\mathcal{M}}^{ND}(u,v)=\sum_{l=1}^{\infty}\frac{M^{ND}(u,l)}{l^{1/2+v}},$$ $$\label{ET1 ET2 def}
{\mathcal{ET}}_1(u,v)=\sum_{l=1}^{\infty}\frac{ET_1(u,l)}{l^{1/2+v}},\quad
{\mathcal{ET}}_2(u,v)=\sum_{l=1}^{\infty}\frac{ET_2(u,l)}{l^{1/2+v}}.$$ As a consequence of and we obtain $$\begin{gathered}
\label{MD(u,v)}
{\mathcal{M}}^D(u,v)=
\zeta(3/2+2v+u)\zeta(1+2u) \\ + \zeta(3/2+2v-u)\zeta(1-2u)\sqrt{2}
(2\pi)^{3u}\cos{\pi(\frac{1}{4}+\frac{u}{2})}\\ \times
\frac{\Gamma(2k-1/4-u/2) \Gamma(2k+1/4-u/2)
\Gamma(1-2u)}{\Gamma(2k+1/4+u/2) \Gamma(2k-1/4+u/2)\Gamma(1-u)},\end{gathered}$$ $$\begin{gathered}
\label{MD(0,v)}
{\mathcal{M}}^D(0,v)=
\frac{\zeta(3/2+2v)}{2}
\Biggl(-3\log{2\pi}+\frac{\pi}{2}+3\gamma\\+
2\frac{\zeta'(3/2+2v)}{\zeta(3/2+2v)}
+\frac{\Gamma'}{\Gamma}(2k-1/4)+\frac{\Gamma'}{\Gamma}(2k+1/4)\Biggr).\end{gathered}$$ Similarly, it follows from , and that $$\label{MND(u,v)}
{\mathcal{M}}^{ND}(u,v)=
\frac{(2\pi)^{1/2+u}}{2}\frac{\Gamma(2k-1/4-u/2)}{\Gamma(2k+1/4+u/2)}
\sum_{l=1}^{\infty}\frac{\mathscr{L}_{-4l}(1/2+u)}{l^{3/4+v-u/2}},$$ $$\label{MND(0,v)}
{\mathcal{M}}^{ND}(0,v)=
\frac{2^{3/2}\pi}{\Gamma(3/4)}\frac{\Gamma(2k-1/4)}{\Gamma(2k+1/4)}L^{-}_{g}(1/4+v).$$
We remark that a part of the main term is also contained in ${\mathcal{ET}}_1(u,v)$ and ${\mathcal{ET}}_2(u,v)$, which we analyse in detail in the next two subsections.
Analysis of ${\mathcal{ET}}_2(u,v)$
-----------------------------------
For $0<\Re{u}<1,$ $\Re{v}>3/4+\Re{u}/2$ we have $$\begin{gathered}
\label{ET2 0}
{\mathcal{ET}}_2(u,v)=
(2\pi)^{1/2+u}2^{1+2v}\frac{1}{2\pi i}\\ \times \int_{(\sigma)}
\left(
\sum_{\substack{n=1\\ n\equiv0(2)}}^{\infty}\sum_{\substack{m=1\\ m\equiv0(4)}}^{\infty}+
\sum_{\substack{n=1\\ n\equiv1(2)}}^{\infty}\sum_{\substack{m=1\\ m\equiv1(4)}}^{\infty}
\right)
\frac{\mathscr{L}_{m}(1/2+u)}{m^{1/2+v+s}n^{1/2-u-2s}}\hat{f}_{2k}(u,v;s)ds,\end{gathered}$$ where $1/2-\Re{v}<\sigma<-1/4-\Re{u}/2.$
Substituting into we obtain $$\label{ET2 1}
{\mathcal{ET}}_2(u,v)=
\sum_{l=1}^{\infty}\frac{(2\pi)^{1/2+u}}{l^{1/2+v}}
\sum_{n>2\sqrt{l}} \frac{\mathscr{L}_{n^2-4l}(1/2+u)}{n^{1/2-u}}I_{2k}\left(u;\frac{n}{l^{1/2}}\right).$$ It follows from that $I_{2k}\left(u;x\right)\sim x^{1-4k}$ as $x\rightarrow\infty.$ Thus $$\sum_{l=1}^{\infty}\frac{1}{l^{1/2+v}}
\sum_{n>2\sqrt{l}} \frac{\mathscr{L}_{n^2-4l}(1/2+u)}{n^{1/2-u}}I_{2k}\left(u;\frac{n}{l^{1/2}}\right)\ll
\sum_{l=1}^{\infty}\frac{l^{1/4+\Re{u}/2}}{l^{1/2+\Re{v}}}.$$ And we see that the double series on the right-hand side of converges absolutely provided that $\Re{v}>3/4+\Re{u}/2.$ Changing the order of summation in and making the change of variables $m=n^2-4l$, we obtain $${\mathcal{ET}}_2(u,v)=(2\pi)^{1/2+u}
\sum_{n=1}^{\infty}
\sum_{\substack{0<m<n^2\\ m\equiv n^2(4)}}\frac{\mathscr{L}_{m}(1/2+u)2^{1+2v}}{(n^2-m)^{1/2+v}n^{1/2-u}}
I_{2k}\left(u;\frac{2n}{(n^2-m)^{1/2}}\right).$$ Rewriting this using yields $$\begin{gathered}
\label{ET2 2}
{\mathcal{ET}}_2(u,v)=(2\pi)^{1/2+u}2^{1+2v}
\left(
\sum_{\substack{n=1\\ n\equiv0(2)}}^{\infty}\sum_{\substack{m=1\\ m\equiv0(4)}}^{\infty}+
\sum_{\substack{n=1\\ n\equiv1(2)}}^{\infty}\sum_{\substack{m=1\\ m\equiv1(4)}}^{\infty}
\right)\\\times
\frac{\mathscr{L}_{m}(1/2+u)}{m^{1/2+v}m^{1/2-u}}
f_{2k}\left(u,v;\frac{m}{n^2}\right).\end{gathered}$$ Applying the Mellin inversion formula for $f_{2k}\left(u,v;m/n^2\right)$ completes the proof.
For $\Re{v}>3/4$ we have $$\label{ET2(0,v)}
{\mathcal{ET}}_2(0,v)=\frac{1}{2\pi i}\int_{(\sigma)}F_{2k}(v,s)ds,$$ where $1/2-\Re{v}<\sigma<-1/4$ and $$\begin{gathered}
\label{F def}
F_{2k}(v,s)=
(2\pi)^{1/2}2^{1+2v}
\Biggl(
\left(1-2^{2s-1/2}\right)\frac{2\sqrt{\pi}}{\Gamma(1/4)}L^{+}_{f}(s+v)-\\
-\left(1-2^{2s+1/2}\right)\frac{4\pi^{1/2}}{2^{1+2v+2s}\Gamma(1/4)}L^{+}_{g}(s+v)
\Biggr)
\zeta(1/2-2s)\hat{f}_{2k}(0,v;s).\end{gathered}$$
We first let $u=0$ in . Then for $\Re{v}>3/4$ the following formula holds $$\begin{gathered}
\label{ET2(0,v)1}
{\mathcal{ET}}_2(0,v)=
(2\pi)^{1/2}2^{1+2v}\frac{1}{2\pi i}\int_{(\sigma)}
\left(
\sum_{\substack{n=1\\ n\equiv0(2)}}^{\infty}\sum_{\substack{m=1\\ m\equiv0(4)}}^{\infty}+
\sum_{\substack{n=1\\ n\equiv1(2)}}^{\infty}\sum_{\substack{m=1\\ m\equiv1(4)}}^{\infty}
\right)\\\times
\frac{\mathscr{L}_{m}(1/2)}{m^{1/2+v+s}n^{1/2-2s}}\hat{f}_{2k}(0,v;s)ds,\end{gathered}$$ where $1/2-\Re{v}<\sigma<-1/4.$ It follows from that $$\label{Lg 1}
\sum_{\substack{m=1\\ m\equiv0(4)}}^{\infty}\frac{\mathscr{L}_{m}(1/2)}{m^{1/2+v+s}}=
\frac{4\pi^{1/2}}{4^{1/2+v+s}\Gamma(1/4)}L^{+}_{g}(s+v).$$ Since $ \mathscr{L}_n(s) $ vanishes if $n \equiv 2,3 {\ (\textup{mod}\ 4)} $, we obtain using and that $$\begin{gathered}
\label{Lg 2}
\sum_{\substack{m=1\\ m\equiv1(4)}}^{\infty}\frac{\mathscr{L}_{m}(1/2)}{m^{1/2+v+s}}=
\sum_{m=1}^{\infty}\frac{\mathscr{L}_{m}(1/2)}{m^{1/2+v+s}}-
\sum_{\substack{m=1\\ m\equiv0(4)}}^{\infty}\frac{\mathscr{L}_{m}(1/2)}{m^{1/2+v+s}}\\=
\frac{2\sqrt{\pi}}{\Gamma(1/4)}L^{+}_{f}(s+v)-\frac{4\pi^{1/2}}{4^{1/2+v+s}\Gamma(1/4)}L^{+}_{g}(s+v).\end{gathered}$$ Furthermore, $$\label{zeta21}
\sum_{\substack{n=1\\ n\equiv0(2)}}^{\infty}\frac{1}{n^{1/2-2s}}=\frac{\zeta(1/2-2s)}{2^{1/2-2s}},$$ $$\label{zeta22}
\sum_{\substack{n=1\\ n\equiv1(2)}}^{\infty}\frac{1}{n^{1/2-2s}}=\left(1-\frac{1}{2^{1/2-2s}}\right)\zeta(1/2-2s).$$ Substituting , , and into we prove .
For $1/2<\Re{v}<3/4$ we have $$\begin{gathered}
\label{ET2(0,v)1/2 3/4}
{\mathcal{ET}}_2(0,v)=
\frac{1}{2\pi i}\int_{(0)}F_{2k}(v,s)ds+\\ \sqrt{2\pi}2^{2v}\frac{\Gamma^2(1/4+v)}{\Gamma(1/4)}\frac{\Gamma(2k-1/4)\Gamma(2k-v)}{\Gamma(2k+1/4)\Gamma(2k+v)}
L^{+}_{f}(v-1/4),\end{gathered}$$ where $F_{2k}(v,s)$ is defined by .
The function $F_{2k}(v,s)$ has a simple pole at $s=-1/4$ coming from $\zeta(1/2-2s).$ Moving the line of integration in to $\sigma=0$ we cross this pole, obtaining $$\label{ET2(0,v) 3}
{\mathcal{ET}}_2(0,v)=-\operatorname{res}_{s=-1/4}F_{2k}(v,s)+
\frac{1}{2\pi i}\int_{(0)}F_{2k}(v,s)ds.$$ The right-hand side of shows that ${\mathcal{ET}}_2(0,v)$ can be continued to the region $\Re{v}>1/2.$ To prove it remains to evaluate the residue. Using we have $$\operatorname{res}_{s=-1/4}F_{2k}(v,s)=\frac{2^{1/2+2v}\pi}{\Gamma(1/4)}\hat{f}_{2k}(0,v;-1/4)L^{+}_{f}(v-1/4).$$ Applying we prove the lemma.
For $0\le\Re{v}<1/2$ we have $$\begin{gathered}
\label{ET2(0,v)0 1/2}
{\mathcal{ET}}_2(0,v)=\operatorname{res}_{s=1/2-v}F_{2k}(v,s)+
\frac{1}{2\pi i}\int_{(0)}F_{2k}(v,s)ds\\
+\sqrt{2\pi}2^{2v}\frac{\Gamma^2(1/4+v)}{\Gamma(1/4)}\frac{\Gamma(2k-1/4)\Gamma(2k-v)}{\Gamma(2k+1/4)\Gamma(2k+v)}
L^{+}_{f}(v-1/4),\end{gathered}$$ where $F_{2k}(v,s)$ is defined by .
The function $F_{2k}(v,s)$ has a double pole at $s=1/2-v$ coming from $L^{+}_{f,g}(s+v).$ To prove the analytic continuation of ${\mathcal{ET}}_2(0,v)$ to the region $\Re{v}<1/2$, we apply [@CMR Corollary 2.4.2, p. 55], which yields for $\Re{v}<1/2$ that $$\begin{gathered}
{\mathcal{ET}}_2(0,v)=-\operatorname{res}_{s=-1/4}F_{2k}(v,s)+\operatorname{res}_{s=1/2-v}F_{2k}(v,s)+
\frac{1}{2\pi i}\int_{(0)}F_{2k}(v,s)ds.\end{gathered}$$
\[lem:second type\] The following formula holds $$\label{ET2(0,0)expression1}
{\mathcal{ET}}_2(0,0)={\mathcal{M}}^{ND}(0,0)+\operatorname{res}_{s=1/2-v}F_{2k}(v,s)+
\frac{1}{2\pi i}\int_{(0)}F_{2k}(v,s)ds.$$
Comparing and , we find that in order to prove , it is required to show that $$\sqrt{2\pi}\Gamma(1/4)L^{+}_{f}(-1/4)=
\frac{2^{3/2}\pi}{\Gamma(3/4)}L^{-}_{g}(1/4).$$ Since $\Gamma(1/4)\Gamma(3/4)=\pi\sqrt{2}$ we need to verify that $$\sqrt{\pi}L^{+}_{f}(-1/4)=2^{1/2}L^{-}_{g}(1/4),$$ and this follows from .
For any $\epsilon>0$ we have $$\label{ET2(0,0)expression2}
{\mathcal{ET}}_2(0,0)={\mathcal{M}}^{ND}(0,0)+O\left(\frac{k^{\epsilon}}{k}\right),$$ $$\label{ET2(0,0) estimate}
{\mathcal{ET}}_2(0,0)\ll k^{-1/2}.$$
This follows immediately from , , Lemma \[Lem fMellin 1/2\], Theorem \[thm second mom Lfg\] and the estimate .
Analysis of ${\mathcal{ET}}_1(u,v)$
-----------------------------------
Assume that $0<\Re{u}<1$ and $\Re{v}>3/4+\Re{u}/2+\max(\theta(1-2\Re{u}),0)$. Then the following formula holds $$\begin{gathered}
\label{ET1 0}
{\mathcal{ET}}_1(u,v)=
(2\pi)^{1/2+u}2^{1+2v} \frac{1}{2\pi i}\times \\ \int_{(\sigma)}
\left(
\sum_{\substack{n=1\\ n\equiv0(2)}}^{\infty}\sum_{\substack{m=1\\ m\equiv0(4)}}^{\infty}+
\sum_{\substack{n=1\\ n\equiv1(2)}}^{\infty}\sum_{\substack{m=-1\\ m\equiv1(4)}}^{\infty}
\right)
\frac{\mathscr{L}_{-m}(1/2+u)}{m^{1/2+v+s}n^{1/2-u-2s}}\hat{g}_{2k}(u,v;s)ds,\end{gathered}$$ where $1/2-\Re{v}<\sigma<-1/4-\Re{u}/2.$
Substituting into we show that $$\label{ET1 1}
{\mathcal{ET}}_1(u,v)=
\sum_{l=1}^{\infty}\frac{(2\pi)^{1/2+u}}{l^{1/2+v}}
\sum_{0<n<2\sqrt{l}}\frac{\mathscr{L}_{n^2-4l}(1/2+u)}{n^{1/2-u}}I_{2k}\left(u;\frac{n}{l^{1/2}}\right).$$ It follows from that $I_{2k}\left(u;x\right)\sim x^{1/2-\Re{u}}$ as $x\rightarrow0.$ Using this fact and applying , we obtain $$\begin{gathered}
\sum_{l=1}^{\infty}\frac{1}{l^{1/2+v}}
\sum_{0<n<2\sqrt{l}}\frac{\mathscr{L}_{n^2-4l}(1/2+u)}{n^{1/2-u}}I_{2k}\left(u;\frac{n}{l^{1/2}}\right)\ll
\sum_{l=1}^{\infty}\frac{1}{l^{1/2+\Re{v}}}\frac{l^{1/2+\max(\theta(1-2\Re{u}),0)+\epsilon}}{l^{1/4-\Re{u}/2}}.\end{gathered}$$ Therefore, the double series on the right-hand side of converges absolutely provided that $\Re{v}>3/4+\Re{u}/2+\max(\theta(1-2\Re{u}),0)$. Changing the order of summation in and making the change of variables $-m=n^2-4l$, we have $${\mathcal{ET}}_1(u,v)=(2\pi)^{1/2+u}
\sum_{n=1}^{\infty}
\sum_{\substack{m=1\\ m+n^2\equiv0(4)}}\frac{\mathscr{L}_{-m}(1/2+u)2^{1+2v}}{(n^2+m)^{1/2+v}n^{1/2-u}}
I_{2k}\left(u;\frac{2n}{(n^2+m)^{1/2}}\right).$$ Applying , we obtain $$\begin{gathered}
\label{ET1 2}
{\mathcal{ET}}_1(u,v)=(2\pi)^{1/2+u}2^{1+2v}
\left(
\sum_{\substack{n=1\\ n\equiv0(2)}}^{\infty}\sum_{\substack{m=1\\ m\equiv0(4)}}^{\infty}+
\sum_{\substack{n=1\\ n\equiv1(2)}}^{\infty}\sum_{\substack{m=1\\ m\equiv-1(4)}}^{\infty}
\right)\\\times
\frac{\mathscr{L}_{-m}(1/2+u)}{m^{1/2+v}m^{1/2-u}}
g_{2k}\left(u,v;\frac{m}{n^2}\right).\end{gathered}$$ Using the Mellin inversion formula for $g_{2k}\left(u,v;m/n^2\right)$ we prove the lemma.
For $\Re{v}>3/4$ the following representation takes place $$\label{ET1(0,v)3/4}
{\mathcal{ET}}_1(0,v)=\frac{1}{2\pi i}\int_{(\sigma)}G_{2k}(v,s)ds,$$ where $1/2-\Re{v}<\sigma<-1/4$ and $$\begin{gathered}
\label{G def}
G_{2k}(v,s)=
(2\pi)^{1/2}2^{1+2v}
\Biggl(
\left(1-2^{2s-1/2}\right)\frac{2\sqrt{\pi}}{\Gamma(3/4)}L^{-}_{f}(s+v)-\\-
\left(1-2^{2s+1/2}\right)\frac{4\pi^{1/2}}{2^{1+2v+2s}\Gamma(3/4)}L^{-}_{g}(s+v)
\Biggr)
\zeta(1/2-2s)\hat{g}_{2k}(0,v;s).\end{gathered}$$
Letting $u=0$ in , we obtain for $\Re{v}>3/4+\theta$ $$\begin{gathered}
\label{ET1(0,v)1}
{\mathcal{ET}}_1(0,v)=
(2\pi)^{1/2}2^{1+2v}\frac{1}{2\pi i}\times \\
\int_{(\sigma)}
\left(
\sum_{\substack{n=1\\ n\equiv0(2)}}^{\infty}\sum_{\substack{m=1\\ m\equiv0(4)}}^{\infty}+
\sum_{\substack{n=1\\ n\equiv1(2)}}^{\infty}\sum_{\substack{m=1\\ m\equiv-1(4)}}^{\infty}
\right)
\frac{\mathscr{L}_{-m}(1/2)}{m^{1/2+v+s}n^{1/2-2s}}\hat{g}_{2k}(0,v;s)ds,\end{gathered}$$ where $1/2-\Re{v}<\sigma<-1/4.$ It follows from that $$\label{Lg 3}
\sum_{\substack{m=1\\ m\equiv0(4)}}^{\infty}\frac{\mathscr{L}_{-m}(1/2)}{m^{1/2+v+s}}=
\frac{4\pi^{1/2}}{4^{1/2+v+s}\Gamma(3/4)}L^{-}_{g}(s+v).$$ Recall that $ \mathscr{L}_n(s) $ vanishes for $n \equiv 2,3 {\ (\textup{mod}\ 4)} $. Consequently, using and we show that $$\begin{gathered}
\label{Lg 4}
\sum_{\substack{m=1\\ m\equiv-1(4)}}^{\infty}\frac{\mathscr{L}_{-m}(1/2)}{m^{1/2+v+s}}=
\sum_{m=1}^{\infty}\frac{\mathscr{L}_{-m}(1/2)}{m^{1/2+v+s}}-
\sum_{\substack{m=1\\ m\equiv0(4)}}^{\infty}\frac{\mathscr{L}_{-m}(1/2)}{m^{1/2+v+s}}\\=
\frac{2\sqrt{\pi}}{\Gamma(3/4)}L^{-}_{f}(s+v)-\frac{4\pi^{1/2}}{4^{1/2+v+s}\Gamma(3/4)}L^{-}_{g}(s+v).\end{gathered}$$ Applying , , and to evaluate , we prove .
For $1/2<\Re{v}<3/4$ the following formula holds $$\begin{gathered}
\label{ET1(0,v)1/2 3/4}
{\mathcal{ET}}_1(0,v)=
\frac{1}{2\pi i}\int_{(0)}G_{2k}(v,s)ds
\\-\sqrt{\pi}2^{1+2v}\sin(\pi v)
\frac{\Gamma^2(1/4+v)}{\Gamma(3/4)}\frac{\Gamma(2k-1/4)\Gamma(2k-v)}{\Gamma(2k+1/4)\Gamma(2k+v)}
L^{-}_{f}(v-1/4),\end{gathered}$$ where $G_{2k}(v,s)$ is defined by .
The function $G_{2k}(v,s)$ has a simple pole at $s=-1/4$ from $\zeta(1/2-2s).$ Moving the line of integration in to $\sigma=0$ we cross this pole, getting $$\label{ET1(0,v) 3}
{\mathcal{ET}}_1(0,v)=-\operatorname{res}_{s=-1/4}G_{2k}(v,s)+
\frac{1}{2\pi i}\int_{(0)}G_{2k}(v,s)ds.$$ The right-hand side of proves the analytic continuation of ${\mathcal{ET}}_1(0,v)$ to the region $\Re{v}>1/2.$ Then to complete the proof of , it remains to evaluate the residue. Using we have $$\operatorname{res}_{s=-1/4}G_{2k}(v,s)=\frac{2^{1/2+2v}\pi}{\Gamma(1/4)}\hat{g}_{2k}(0,v;-1/4)L^{-}_{f}(v-1/4).$$ The lemma follows by applying .
\[lem:et1at0v\] For $0\le\Re{v}<1/2$ we have $$\begin{gathered}
\label{ET1(0,v)0 1/2}
{\mathcal{ET}}_1(0,v)=\operatorname{res}_{s=1/2-v}G_{2k}(v,s)+\frac{1}{2\pi i}\int_{(0)}G_{2k}(v,s)ds
\\-\sqrt{\pi}2^{1+2v}\sin(\pi v)
\frac{\Gamma^2(1/4+v)}{\Gamma(1/4)}\frac{\Gamma(2k-1/4)\Gamma(2k-v)}{\Gamma(2k+1/4)\Gamma(2k+v)}
L^{+}_{f}(v-1/4),\end{gathered}$$ where $G_{2k}(v,s)$ is defined by and $$\label{res G 1/2-v}
\operatorname{res}_{s=1/2-v}G_{2k}(v,s)=M_1(v)+M_2(v),$$ $$\begin{gathered}
\label{res G 1/2-v M1}
M_1(v)=
\frac{2^{5/2+2v}\pi}{\Gamma(3/4)}
\Biggl[
c_f^{-}(-1)\zeta(2v-1/2)\hat{g}_{2k}(0,v;1/2-v)\left(1-2^{1/2-2v}\right)\\+
c_f^{-}(-2)
\Biggl(
\zeta(2v-1/2)\left(1-2^{1/2-2v}\right)\frac{\partial}{\partial s}\hat{g}_{2k}(0,v;s)\Bigg|_{s=1/2-v}\\-
2\zeta'(2v-1/2)\hat{g}_{2k}(0,v;1/2-v)\left(1-2^{1/2-2v}\right)\\ -
\zeta(2v-1/2)\hat{g}_{2k}(0,v;1/2-v)2^{3/2-2v}\log2
\Biggr)\Biggr],\end{gathered}$$ $$\begin{gathered}
\label{res G 1/2-v M2}
M_2(v)=
\frac{8\pi}{\Gamma(3/4)}
\Biggl[
c_g^{-}(-1)\zeta(2v-1/2)\hat{g}_{2k}(0,v;1/2-v)\left(1-2^{2v-3/2}\right)\\+
c_g^{-}(-2)
\Biggl(
\zeta(2v-1/2)\left(1-2^{2v-3/2}\right)\frac{\partial}{\partial s}\hat{g}_{2k}(0,v;s)\Bigg|_{s=1/2-v}\\-
2\zeta'(2v-1/2)\hat{g}_{2k}(0,v;1/2-v)\left(1-2^{2v-3/2}\right)\\+
\zeta(2v-1/2)\hat{g}_{2k}(0,v;1/2-v)2^{2v-1/2}\log2
\Biggr)
\Biggr].\end{gathered}$$
The function $G_{2k}(v,s)$ has a double pole at $s=1/2-v$ from $L^{-}_{f,g}(s+v).$ To prove the analytic continuation of ${\mathcal{ET}}_1(0,v)$ to the region $\Re{v}<1/2$ we apply [@CMR Corollary 2.4.2, p. 55]. Consequently, for $\Re{v}<1/2$ we have $$\begin{gathered}
\label{ET1(0,v) 4}
{\mathcal{ET}}_1(0,v)=\operatorname{res}_{s=1/2-v}G_{2k}(v,s)-\operatorname{res}_{s=-1/4}G_{2k}(v,s)+\\
\frac{1}{2\pi i}\int_{(0)}G_{2k}(v,s)ds.\end{gathered}$$ Then it follows from that $$\begin{gathered}
\label{res G 1/2-v 0}
\operatorname{res}_{s=1/2-v}G_{2k}(v,s)=\\
\frac{2^{5/2+2v}\pi}{\Gamma(3/4)}
\operatorname{res}_{s=1/2-v}\Biggl(
\zeta(1/2-2s)\hat{g}_{2k}(0,v;s)\left(1-2^{2s-1/2}\right)L^{-}_{f}(s+v)
\Biggr)\\+
\frac{8\pi}{\Gamma(3/4)}
\operatorname{res}_{s=1/2-v}\Biggl(
\zeta(1/2-2s)\hat{g}_{2k}(0,v;s)\left(1-2^{-2s-1/2}\right)L^{-}_{g}(s+v)
\Biggr).\end{gathered}$$ Let $H(s)$ be an arbitrary function that is holomorphic at $s=1/2-v$. Using , we obtain the Laurent series $$\begin{gathered}
\label{Laurent series1}
H(s)L^{-}_{f,g}(s+v)=\frac{c_{f,g}^{-}(-2)H(1/2-v)}{(s+v-1/2)^2}\\+\frac{c_{f,g}^{-}(-1)H(1/2-v)+c_{f,g}^{-}(-2)H'(1/2-v)}{s+v-1/2}+O(1).\end{gathered}$$ Applying to evaluate , we prove .
Analytic continuation
---------------------
Finally, we obtain the following decomposition for the mixed moment.
For $\Re{v}\ge0$ we have $$\label{M(u,v) exact formula}
{\mathcal{M}}(0,v)={\mathcal{M}}^D(0,v)+{\mathcal{M}}^{ND}(0,v)+{\mathcal{ET}}_1(0,v)+{\mathcal{ET}}_2(0,v),$$ where ${\mathcal{M}}^D(0,v)$ is defined by and ${\mathcal{M}}^{ND}(0,v)$ by . Furthermore, the terms ${\mathcal{ET}}_1(0,v)$ and ${\mathcal{ET}}_2(0,v)$ are given by and for $\Re{v}>3/4$, by and for $1/2<\Re{v}\le3/4$ and by and for $0\le\Re{v}<1/2.$
In order to prove the theorem, it remains to show that the right-hand side of is holomorphic for $\Re{v}\ge0.$ More precisely, we need to consider points $v=3/4$ and $v=1/4.$ The only summands on the right-hand side of that are not holomorphic at $v=3/4$ come from and , namely: $$\begin{gathered}
\label{res at 3/4}
\sqrt{2\pi}2^{2v}\frac{\Gamma^2(1/4+v)}{\Gamma(1/4)}\frac{\Gamma(2k-1/4)\Gamma(2k-v)}{\Gamma(2k+1/4)\Gamma(2k+v)}
L^{+}_{f}(v-1/4)-\\
-\sqrt{\pi}2^{1+2v}\sin(\pi v)
\frac{\Gamma^2(1/4+v)}{\Gamma(3/4)}\frac{\Gamma(2k-1/4)\Gamma(2k-v)}{\Gamma(2k+1/4)\Gamma(2k+v)}
L^{-}_{f}(v-1/4)=\\=
\sqrt{\pi}2^{2v}\Gamma^2(1/4+v)\frac{\Gamma(2k-1/4)\Gamma(2k-v)}{\Gamma(2k+1/4)\Gamma(2k+v)}\times \\
\left(\frac{\sqrt{2}}{\Gamma(1/4)}L^{+}_{f}(v-1/4)-
\frac{2\sin(\pi v)}{\Gamma(3/4)}L^{-}_{f}(v-1/4)
\right).\end{gathered}$$ Therefore, to prove that the right-hand side of is holomorphic at $v=3/4$, it is sufficient to show that $$\frac{\sqrt{2}}{\Gamma(1/4)}L^{+}_{f}(v-1/4)-
\frac{2\sin(\pi v)}{\Gamma(3/4)}L^{-}_{f}(v-1/4)$$ is holomorphic at $v=3/4.$ Using Theorem \[cor Lfg coeff relation\] and the asymptotic formula $$\sin(\pi v)=\frac{1}{\sqrt{2}}-\frac{\pi(v-3/4)}{\sqrt{2}}+O((v-3/4)^2),$$ we obtain $$\begin{gathered}
\frac{\sqrt{2}}{\Gamma(1/4)}L^{+}_{f}(v-1/4)-
\frac{2\sin(\pi v)}{\Gamma(3/4)}L^{-}_{f}(v-1/4)=\\=
\frac{1}{(v-3/4)^2}\left(
\frac{c^{+}_{f}(-2)\sqrt{2}}{\Gamma(1/4)}-
\frac{c^{-}_{f}(-2)\sqrt{2}}{\Gamma(3/4)}
\right)+\\+
\frac{1}{v-3/4}\left(
\frac{c^{+}_{f}(-1)\sqrt{2}}{\Gamma(1/4)}-
\frac{c^{-}_{f}(-1)\sqrt{2}}{\Gamma(3/4)}+
\frac{c^{-}_{f}(-2)\pi\sqrt{2}}{\Gamma(3/4)}
\right)+O(1)=O(1).\end{gathered}$$ Thus the right-hand side of is holomorphic at $v=3/4$.
The only summands on the right-hand side of that are not holomorphic at $v=1/4$ come from and , namely $${\mathcal{M}}^{ND}(0,v)+\operatorname{res}_{s=1/2-v}G_{2k}(v,s).$$
Let us consider the $M_1(v)$ part of $\operatorname{res}_{s=1/2-v}G_{2k}(v,s)$ given by . Using and , we obtain a Laurent series for $M_1(v)$ at the point $v=1/4$: $$\begin{gathered}
\label{M1 1/4 1}
M_1(v)=
\frac{2^{5/2}\pi}{\Gamma(3/4)}
c_f^{-}(-2)
\Biggl(
\zeta(2v-1/2)\left(2^{2v}-2^{1/2}\right)\frac{\partial}{\partial s}\hat{g}_{2k}(0,v;s)\Bigg|_{s=1/2-v}\\-
\frac{2^{3}\zeta(0)\log2}{2v-1/2}\frac{\Gamma(2k-1/4)}{\Gamma(2k+1/4)}
\Biggr)+O(1).\end{gathered}$$ Using and the fact that $$\zeta(2v-1/2)\left(2^{2v}-2^{1/2}\right)=(2v-1/2)\zeta(0)2^{1/2}\log2+O((2v-1/2)^2),$$ we show that $M_1(v)=O(1)$ as $v\to 1/4.$
Let us consider the $M_2(v)$ part of $\operatorname{res}_{s=1/2-v}G_{2k}(v,s)$ given by . Applying , we obtain $$\begin{gathered}
\label{M2 Laurent 1/4 1}
M_2(v)=\frac{8\pi}{\Gamma(3/4)}c_g^{-}(-2)
\zeta(2v-1/2)\left(1-2^{2v-3/2}\right)\frac{\partial}{\partial s}\hat{g}_{2k}(0,v;s)\Bigg|_{s=1/2-v}\\+
\frac{1}{2v-1/2}\frac{8\pi}{\Gamma(3/4)}\frac{\Gamma(2k-1/4)}{\Gamma(2k+1/4)}
\Biggl(
c_g^{-}(-1)\zeta(0)2^{1/2}\\+
c_g^{-}(-2)
\Biggl(
-2^{3/2}\zeta'(0)+
\zeta(0)2^{3/2}\log2
\Biggr)
\Biggr).\end{gathered}$$ In order to evaluate a Laurent series for the remaining term we use together with the following formula $$\zeta(2v-1/2)\left(1-2^{2v-3/2}\right)=\frac{\zeta(0)}{2}+
\frac{\zeta'(0)-\zeta(0)\log2}{2}(2v-1/2)+O((2v-1/2)^2).$$ Consequently, $$\begin{gathered}
\label{M2 Laurent 1/4 2}
M_2(v)=\frac{8\pi}{\Gamma(3/4)}\frac{\Gamma(2k-1/4)}{\Gamma(2k+1/4)}
\frac{c_g^{-}(-2)2^{3/2}\zeta(0)}{(2v-1/2)^2}
+\\+
\frac{8\pi}{\Gamma(3/4)}\frac{\Gamma(2k-1/4)}{\Gamma(2k+1/4)}
\frac{c_g^{-}(-1)2^{1/2}\zeta(0)}{2v-1/2}+O(1).\end{gathered}$$
It follows from and that $$\begin{gathered}
\label{MND Laurent 1/4}
{\mathcal{M}}^{ND}(0,v)=\frac{2^{7/2}\pi}{\Gamma(3/4)}\frac{\Gamma(2k-1/4)}{\Gamma(2k+1/4)}
\frac{c_g^{-}(-2)}{(2v-1/2)^2}
+\\+
\frac{2^{5/2}\pi}{\Gamma(3/4)}\frac{\Gamma(2k-1/4)}{\Gamma(2k+1/4)}
\frac{c_g^{-}(-1)}{2v-1/2}
+O(1).\end{gathered}$$ Since $M_1(v)=O(1),$ applying and we conclude that the sum $${\mathcal{M}}^{ND}(0,v)+\operatorname{res}_{s=1/2-v}G_{2k}(v,s),$$ and consequently the right-hand side of , are holomorphic at $v=1/4.$
The following asymptotic formula holds $$\label{resG=MD}
\operatorname{res}_{s=1/2}G_{2k}(0,s)={\mathcal{M}}^{D}(0,0)+O(k^{-2}).$$
We compare the leading terms. It follows from [@HMF (5.11.2)] that $$\label{psi relation}
2\frac{\Gamma'}{\Gamma}(2k)=\frac{\Gamma'}{\Gamma}(2k-1/4)+\frac{\Gamma'}{\Gamma}(2k+1/4)+O(k^{-2}).$$ Therefore, implies that the leading term of ${\mathcal{M}}^{D}(0,0)$ is equal to $$\label{MD(0,0)leading term}
\zeta(3/2)\frac{\Gamma'}{\Gamma}(2k).$$ Let us compute the leading term of $\operatorname{res}_{s=1/2}G_{2k}(0,s).$ It follows from and that the leading term is $$\begin{gathered}
\label{resG leading term1}
2\frac{\Gamma'}{\Gamma}(2k)\frac{16\pi}{\Gamma(3/4)}\Biggl(
c_f^{-}(-2)(1-\sqrt{2})\zeta(-1/2)\Gamma(-1/2)+\\+
c_g^{-}(-2)(\sqrt{2}-1/2)\zeta(-1/2)\Gamma(-1/2)
\Biggr).\end{gathered}$$ Using the functional equation for the Riemann zeta function [@HMF (25.4.1)] we have $$\zeta(-1/2)\Gamma(-1/2)=\frac{\zeta(3/2)}{2\sqrt{\pi}}.$$ Consequently, the leading term of $\operatorname{res}_{s=1/2}G_{2k}(0,s)$ is as follows: $$\label{resG leading term2}
\frac{\Gamma'}{\Gamma}(2k)\zeta(3/2)\frac{16\sqrt{\pi}}{\Gamma(3/4)}\left(
c_f^{-}(-2)(1-\sqrt{2})+
c_g^{-}(-2)(\sqrt{2}-1/2)
\right).$$ Finally, applying , we find that is equal to .
Proof of main theorems
----------------------
Asymptotic formula is a direct consequence of for $v=0$. More precisely, we replace ${\mathcal{ET}}_2(0,v)$ by , ${\mathcal{ET}}_1(0,v)$ by , and apply .
Consider and note that all summands except the integral can be trivially bounded by $\log k$. The final step is to show that $$\label{integral of G estimate}
\frac{1}{2\pi i}\int_{(0)}G_{2k}(0,s)ds\ll\log^{3} k,$$ where $G_{2k}(0,s)$ is given by . In view of , in order to establish , we need to prove that $$\label{integral of G estimate2}
\frac{1}{2\pi i}\int_{(0)}L^{-}_{f,g}(s)\zeta(1/2-2s)\hat{g}_{2k}(0,0;s)ds\ll\log^{3} k.$$ Let $r=\Im{s}$. By Lemma \[Lem gMellin big r\] the contribution of $|r|>3k$ is negligible. As a result, we are left to show that $$\label{integral of G estimate3}
\int_{-3k}^{3k}L^{-}_{f,g}(ir)\zeta(1/2-2ir)\hat{g}_{2k}(0,0;ir)dr\ll\log^{3} k.$$ Let $\delta$ be some fixed constant such that $0<\delta<1/4$. For $|r|<{\mathbf{k}}^{1/2-\delta}$ we apply the trivial bound . Then it is required to establish that $$\label{integral of G estimate4}
\int_{{\mathbf{k}}^{1/2-\delta}}^{3k}L^{-}_{f,g}(ir)\zeta(1/2-2ir)\hat{g}_{2k}(0,0;ir)dr\ll\log^{3} k.$$ With this goal, we apply Lemma \[Lem gMellin small r\], Theorem \[thm second mom Lfg\], the following estimate for the second moment of the Riemann zeta function $$\label{zeta 2moment}
\int_{T}^{T+H}|\zeta(1/2+ir)|^2dr\ll H\log T$$ over short intervals $H\gg T^{1/3}$, and the Cauchy-Schwarz inequality. Consequently, we show that the contribution of the second summand on the right-hand side of to is negligibly small. So our problem got reduced to proving that $$\label{integral of G estimate5}
\int_{{\mathbf{k}}^{1/2-\delta}}^{3k}|L^{-}_{f,g}(ir)\zeta(1/2-2ir)|\frac{1}{{\mathbf{k}}^{5/6}}\min\left(1,\frac{{\mathbf{k}}^{1/12}}{|{\mathbf{k}}-4r|^{1/4}}\right)dr\ll\log^{3} k.$$ Opening the minimum we obtain three integrals: $$\begin{gathered}
\label{integral of G estimate6}
\int_{r_1}^{r_2}\frac{|L^{-}_{f,g}(ir)\zeta(1/2-2ir)|}{{\mathbf{k}}^{3/4}|{\mathbf{k}}-4r|^{1/4}}dr+
\int_{r_2}^{r_3}|L^{-}_{f,g}(ir)\zeta(1/2-2ir)|\frac{dr}{{\mathbf{k}}^{5/6}}+\\
\int_{r_3}^{r_4}\frac{|L^{-}_{f,g}(ir)\zeta(1/2-2ir)|}{{\mathbf{k}}^{3/4}|{\mathbf{k}}-4r|^{1/4}}dr
\ll\log^{3} k,\end{gathered}$$ where $r_1={\mathbf{k}}^{1/2-\delta}$, $r_2={\mathbf{k}}/4-{\mathbf{k}}^{1/3}$, $r_3={\mathbf{k}}/4+{\mathbf{k}}^{1/3},r_4=3k.$ To estimate the second integral we apply the Cauchy-Schwarz inequality, Theorem \[thm second mom Lfg\] and , getting $$\label{integral of G estimate7}
\int_{r_2}^{r_3}|L^{-}_{f,g}(ir)\zeta(1/2-2ir)|\frac{dr}{{\mathbf{k}}^{5/6}}
\ll\frac{\log^{5/2} k}{k^{1/6}}.$$ Let us now consider the first integral in . Applying the Cauchy-Schwarz inequality and Theorem \[thm second mom Lfg\], we obtain $$\label{integral of G estimate8}
\int_{r_1}^{r_2}\frac{|L^{-}_{f,g}(ir)\zeta(1/2-2ir)|}{{\mathbf{k}}^{3/4}|{\mathbf{k}}-4r|^{1/4}}dr
\ll\frac{\log^{2} k}{{\mathbf{k}}^{1/4}}\left(\int_{r_1}^{r_2}\frac{|\zeta(1/2-2ir)|^2}{|{\mathbf{k}}-4r|^{1/2}}dr\right)^{1/2}.$$ Making the change of variable ${\mathbf{k}}-4r=x$, and then performing a dyadic partition of unity, we prove using that $$\label{integral of G estimate9}
\int_{r_1}^{r_2}\frac{|\zeta(1/2-2ir)|^2}{|{\mathbf{k}}-4r|^{1/2}}dr\ll k^{1/2}\log^2k,$$ where one of the logarithms comes from the partition of unity. Substituting into we obtain $$\label{integral of G estimate10}
\int_{r_1}^{r_2}\frac{|L^{-}_{f,g}(ir)\zeta(1/2-2ir)|}{{\mathbf{k}}^{3/4}|{\mathbf{k}}-4r|^{1/4}}dr
\ll\log^{3} k.$$ Finally, the third integral in can be estimated in the same way as the first one. This completes the proof.
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---
abstract: 'In the model of randomly perturbed graphs we consider the union of a deterministic graph ${{\mathcal{G}}_\alpha}$ with minimum degree $\alpha n$ and the binomial random graph $\gnp$. This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac’s theorem and the results by Posá and Koršunov on the threshold in $\gnp$. In this note we extend this result in ${{\mathcal{G}}_\alpha}\cup \gnp$ to sparser graphs with $\alpha=o(1)$. More precisely, for any $\varepsilon>0$ and $\alpha \colon \mathbb{N} \mapsto (0,1)$ we show that a.a.s. ${{\mathcal{G}}_\alpha}\cup \gnp{n}{\beta /n}$ is Hamiltonian, where $\beta = -(6 + {\varepsilon}) \log(\alpha)$. If $\alpha>0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $\alpha=O(1/n)$ the random part $\gnp$ is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into $\gnp$.'
address:
- 'Max Hahn-Klimroth, [hahnklim@math.uni-frankfurt.de]{}, Goethe University, Mathematics Institute, 10 Robert Mayer St, Frankfurt 60325, Germany.'
- 'Giulia Satiko Maesaka, [giulia.maesaka@uni-hamburg.de]{}, Universität Hamburg, Fachbereich Mathematik, 55 Bundesstr., Hamburg 20146, Germany.'
- 'Yannick Mogge, [yannick.mogge@tuhh.de]{}, Hamburg University of Technology, Mathematics Institute, 3 Am Schwarzenberg-Campus, Hamburg 21073, Germany.'
- 'Samuel Mohr, [samuel.mohr@tu-ilmenau.de]{}, Ilmenau University of Technology, Mathematics Institute, 25 Weimarer St, Ilmenau 98693, Germany.'
- 'Olaf Parczyk, [o.parczyk@lse.ac.uk]{}, London School of Economics, Department of Mathematics, Houghton St, London, WC2A 2AE, UK.'
author:
- 'Max Hahn-Klimroth, Giulia S. Maesaka, Yannick Mogge, Samuel Mohr, and Olaf Parczyk'
bibliography:
- 'literature.bib'
title: Random perturbation of sparse graphs
---
[^1]
Introduction and results {#Sec_intro}
========================
For $\alpha\in(0,1)$ we let ${{\mathcal{G}}_\alpha}$ be an $n$-vertex graph with minimum degree $\delta({{\mathcal{G}}_\alpha}) \ge \alpha n$. A famous result by Dirac [@Dirac] says that if $\alpha \ge 1/2$ and $n \ge 3$, then ${{\mathcal{G}}_\alpha}$ contains a Hamilton cycle, i.e. a spanning cycle through all vertices of ${{\mathcal{G}}_\alpha}$. This motivated the more general questions of determining the smallest $\alpha$ such that ${{\mathcal{G}}_\alpha}$ contains a given spanning structure. For example, there are results for trees [@komlos2001spanning], factors [@hajnal1970proof], powers of Hamilton cycles [@komlos1998posa; @komlos1998proof], and general bounded degree graphs [@bottcher2009proof]. This is a problem for deterministic graphs that belongs to the area of extremal graph theory.
We can consider similar questions for random graphs, in particular, for the binomial random graph model $\gnp{n}{p}$, which is the probability space over $n$-vertex graphs with each edge being present with probability $p$ independent of all the others. Analogous to the smallest $\alpha$ we are looking for a function $\hat{p}=\hat{p}(n) \colon \mathbb{N} \mapsto (0,1)$ such that if $p=\omega(\hat{p})$ the probability that $\gnp{n}{p}$ contains some spanning subgraph tends to $1$ as $n$ tends to infinity and for $p=o(\hat{p})$ it tends to $0$. We call this $\hat{p}$ the threshold function for the respective property (an easy sufficient criteria for its existence can be found in [@bollobas1987threshold]) and if the first/second statement holds we say that $\gnp{n}{p}$ has/does not have this property asymptotically almost surely (a.a.s.). One often says that $\gnp{n}{p}$ undergoes a *phase transition* at $\hat p$. For the Hamilton cycle problem Posá [@posa1976hamiltonian] and Koršunov [@korshunov1976solution] proved independently that $\hat{p}=\log n/n$ gives the threshold. Similar as above there was a tremendous amount of research on determining the thresholds for various spanning structures, e.g. for matchings [@erdHos1966existence], trees [@krivelevich2010embedding; @montgomery2019spanning], factors [@johansson2008factors], powers of Hamilton cycles [@kuhn2012posa; @nenadov2019powers], and general bounded degree graphs [@alon1992spanning; @ferber2017embedding; @ferber2018spanning; @riordan2000spanning]. An extensive survey by Böttcher can be found in [@bottcher2017large].
Motivated by the smoothed analysis of algorithms [@spielman2004smoothed], both these worlds were combined by Bohman, Frieze, and Martin [@BFM03]. For any fixed $\alpha >0$, they defined the model of randomly perturbed graphs as the union ${{\mathcal{G}}_\alpha}\cup \gnp{n}{p}$. They showed that $1/n$ is the threshold for a Hamilton cycle, meaning that there is a graph ${{\mathcal{G}}_\alpha}$ such that with $p=o(1/n)$ there a.a.s. is no Hamilton cycle in ${{\mathcal{G}}_\alpha}\cup \gnp{n}{p}$ and for any ${{\mathcal{G}}_\alpha}$ and $p=\omega(1/n)$ there a.a.s. is a Hamilton cycle in ${{\mathcal{G}}_\alpha}\cup \gnp{n}{p}$. It is important to note that in $\gnp$, $p = 1/n$ is also the threshold for an almost spanning cycle, this is for any $\varepsilon>0$ a cycle on at least $(1-\varepsilon)n$ vertices. It should be further remarked that if $p=o(\log n/n)$ there are a.a.s. isolated vertices in $\gnp{n}{p}$ and the purpose of ${{\mathcal{G}}_\alpha}$ is to compensate for this and to help in turning the almost spanning cycle into a Hamilton cycle.
This first result on randomly perturbed graphs [@BFM03] sparked a lot of subsequent research on the thresholds of spanning structures in this randomly perturbed graphs model, e.g. trees [@bottcher2019universality; @joos2018spanning; @krivelevich2017bounded], factors [@balogh2019tilings], powers of Hamilton cycles [@bedenknecht2018powers; @BMPPM18], and general bounded degree graphs [@BMPPM18]. As for a Hamilton cycle there is often a $\log$-factor difference to the thresholds in $\gnp{n}{p}$ alone, which is there for local reasons similar to isolated vertices. In most of these cases a ${{\mathcal{G}}_\alpha}$, that is responsible for the lower bound, is the complete imbalanced bipartite graph $K_{\alpha n,(1-\alpha)n}$. In this model there are also results with lower bounds on $\alpha$ [@bennett2017adding; @dudek2018powers; @han2019tilings; @nenadov2018sprinkling] and for Ramsey-type problems [@das2019vertex; @das2019ramsey].
Hamiltonicity in randomly perturbed sparse graphs {#Sec_results}
-------------------------------------------------
The aim of this note is to investigate a new direction. Instead of fixing an $\alpha \in (0,1)$ in advance we allow $\alpha$ to tend to zero with $n$. This extends the range of ${{\mathcal{G}}_\alpha}$ to sparse graphs and we want to determine the threshold probability in ${{\mathcal{G}}_\alpha}\cup \gnp$. For example, with $\alpha=1/\log n$ we have a sparse deterministic graph ${{\mathcal{G}}_\alpha}$ with minimum degree $n/\log n$. Then $p=\omega(1/n)$ does not suffice in general, but it is sufficient to take ${{\mathcal{G}}_\alpha}\cup \gnp{n}{\Theta(\log \log n)/n}$ to [a.a.s.]{} guarantee a Hamilton cycle. More generally, we can prove the following.
\[Thm\_hamiltonicity\] Let $\alpha = \alpha(n) : \mathbb{N} \mapsto (0,1)$ and $\beta = \beta(\alpha) = -(6+o(1)) \log(\alpha)$. Then a.a.s. ${{\mathcal{G}}_\alpha}\cup \gnp{n}{\beta /n}$ is Hamiltonian.
This extends the result of Bohman, Frieze, and Martin [@BFM03] for constant $\alpha>0$. For even $n$ a direct consequence of this theorem is the existence of a perfect matching in the same graph. To prove Theorem \[Thm\_hamiltonicity\] we use a result by Frieze [@Frieze86] to find a very long path in $\gnp$ alone and then use the switching technique developed in [@BMPPM18] to turn this into a Hamilton cycle. As it turns out, our method allows to prove the existence of a perfect matching with a slightly lower edge probability.
\[Thm\_PM\] Let $\alpha = \alpha(n) : \mathbb{N} \mapsto (0,1)$ and $\beta = \beta(\alpha) = -(4+o(1)) \log(\alpha)$. Then a.a.s. ${{\mathcal{G}}_\alpha}\cup \gnp{n}{\beta /n}$ contains a perfect matching.
To see that in both theorems $\beta$ is optimal up to the constant factor, consider ${{\mathcal{G}}_\alpha}=K_{\alpha n,(1-\alpha)n}$ and note that there cannot be a perfect matching, if we have more than $\alpha n$ isolated vertices on the $(1-\alpha)n$ side. The number of isolated vertices in $\gnp{n}{\beta /n}$ roughly is $n (1-\beta /n)^{n-1} \cong n \exp(-\beta)$, which is larger than $\alpha n$ if $\beta = o(-\log(\alpha))$.
For proving results in the randomly perturbed graphs model good almost spanning results are essential. Typically, by almost spanning one means that for any ${\varepsilon}>0$ we can embed the respective structure on at least $(1-{\varepsilon})n$ vertices. For paths and cycles in $\gnp{n}{C/n}$ this can, for example, be done using expansion properties and the DFS-algorithm [@krivelevich2016long]. These almost spanning results are much easier than the spanning counterpart, because there is always a linear size set of available vertices. But for the proof of Theorem \[Thm\_hamiltonicity\] this is not sufficient, because if $\alpha=o(1)$ we will not be able to take care of a linear sized leftover. Instead we exploit that we have $\gnp{n}{\beta/n}$ and use the following result showing that we can find a long cycle consisting of all but sublinearly many vertices.
\[Lemma\_almost\_hamiltonian\] Let $0 < \beta=\beta(n) \leq \log n$. Then $\gnp{n}{\beta/n}$ a.a.s. contains a cycle of length at least $${\left({1 - {\left({1-o(1)}\right)} \beta \exp{\left({-\beta}\right)}}\right)}n.$$
This is optimal, because this is asymptotically the size of the $2$-core (maximal subgraph with minimum degree $2$) of $\gnp$ [@frieze2016introduction Lemma 2.16]. A similar result holds for large matchings.
\[Lemma\_almost\_pm\] Let $0 < \beta=\beta(n) \leq \log n$. Then $\gnp{n}{\beta/n}$ a.a.s. contains a matching consisting of at least ${\left({1 - {\left({1-o(1)}\right)} \exp{\left({-\beta}\right)}}\right)}n$ vertices.
Again this is optimal, because the number of isolated vertices is a.a.s. $(1+o(1))e^{-\beta}n$ [@frieze2016introduction Theorem 3.1]. Observe, that also a bipartite variant of this lemma holds, which can be proved by removing small degree vertices and employing Halls theorem.
\[Lemma\_almost\_pm\_bip\] Let $0 < \beta=\beta(n) \leq \log n$. Then the bipartite binomial random graph $\gnp{n,n}{\beta/n}$ a.a.s. contains a matching consisting of at least ${\left({1 - {\left({1-o(1)}\right)} \exp{\left({-\beta}\right)}}\right)}n$ edges.
Bounded degree trees in randomly perturbed sparse graphs
--------------------------------------------------------
After Hamilton cycles and perfect matchings, the next natural candidates are $n$-vertex trees with maximum degree bounded by a constant $\Delta$. In $\gnp$ the threshold $\log n/n$ was determined in a breakthrough result by Montgomery [@montgomery2019spanning], in ${{\mathcal{G}}_\alpha}$ it is enough to have a fixed $\alpha>1/2$ [@komlos1995proof], and in ${{\mathcal{G}}_\alpha}\cup \gnp$ with constant $\alpha>0$ the threshold is $1/n$ [@krivelevich2017bounded]. To obtain a result similar to Theorem \[Thm\_hamiltonicity\] for bounded degree trees using our approach we need an almost spanning result similar to Lemma \[Lemma\_almost\_hamiltonian\]. With a similar approach as for Theorem \[Thm\_hamiltonicity\] and \[Thm\_PM\] we obtain the following modular statement.
\[Thm\_trees\] Let $\Delta \ge 2$ be an integer and suppose that $\alpha,\beta,{\varepsilon}\colon \mathbb{N} \mapsto [0,1]$ are such that $4 (\Delta+1) {\varepsilon}< \alpha^{\Delta + 1}$ and a.a.s. $\gnp{n}{\beta/n}$ contains a given tree with maximum degree $\Delta$ on $(1 - {\varepsilon})n$ vertices. Then any tree with maximum degree $\Delta$ on $n$ vertices is a.a.s. contained in the union ${{\mathcal{G}}_\alpha}\cup \gnp{n}{\beta/n}$.
Next we discuss the almost spanning results that we can obtain in the relevant regime. Improving on a result of Alon, Krivelevich, and Sudakov [@alon2007embedding], Balogh, Csaba, Pei, and Samotij [@balogh2010large] proved that for $\Delta \ge 2$ there exists a $C>0$ such that for ${\varepsilon}>0$ a.a.s. $\gnp{n}{\beta/n}$ contains any tree with maximum degree $\Delta$ on at most $(1-{\varepsilon})n$ vertices provided that $\beta \ge \tfrac{C}{{\varepsilon}} \log \tfrac{1}{{\varepsilon}} $. For the proof they only require that the graph satisfies certain expander properties. This can be extended to the range where ${\varepsilon}\to 0$ and $\omega(1)=\beta \le \log n$ and following along the lines of their argument we get the following.
\[lem:almost\_tree\] For $\Delta \ge 2$ there exists a $C>0$ such that for any $0 < \beta=\beta(n) \leq \log n$ and ${\varepsilon}={\varepsilon}(n)>0$ with $\beta \ge \tfrac{C}{{\varepsilon}} \log \tfrac{1}{{\varepsilon}}$ the following holds. $\gnp{n}{\beta/n}$ a.a.s. contains any bounded degree tree on at most ${\left({1 - {\varepsilon}}\right)}n$ vertices.
Then together with Theorem \[Thm\_trees\] we obtain the following.
\[cor:trees\] For $\Delta \ge 2$ there exists a $C>0$ such that for $\alpha = \alpha(n) : \mathbb{N} \mapsto (0,1)$ and $\beta = \beta(\alpha) = C \alpha^{-(\Delta+1)} \log \tfrac{1}{\alpha}$ the following holds. Any $n$-vertex tree $T$ with maximum degree $\Delta$ is a.a.s. contained in ${{\mathcal{G}}_\alpha}\cup \gnp{n}{\beta /n}$.
The proof for the dense case in [@krivelevich2017bounded] uses regularity and it is unlikely to give anything better in the sparse regime. As remarked in [@alon2007embedding] the condition on the almost spanning embedding in $\gnp{n}{\beta/n}$ could possibly be improved to $\beta > \log \tfrac{C}{{\varepsilon}}$, then covering almost all non-isolated vertices. More precisely this asks for the following.
\[que:tree\] For every integer $\Delta$ there exists $C>0$ such that with $0<\beta=\beta(n)\le \log n$ the following holds. Is any given tree with maximum degree $\Delta$ on $$(1-C \exp(-\beta))n$$ vertices a.a.s. contained in $\gnp{n}{\beta/n}$?
With Theorem \[Thm\_trees\] this would then give that already $\beta=-(\Delta+1)\log (C \alpha)$ suffices, which would be optimal up to the constant factors. We want to briefly argue why it is possible to answer this question for large families of trees and what the difficulties are. For simplicity we only discuss the case $\beta=\log \log n$ and note that by Lemma \[lem:almost\_tree\] above we can embed trees on roughly $(1-1/\log \log n)n$ vertices. A very helpful result for handling trees by Krivelevich [@krivelevich2010embedding] states that for any integer $n,k>2$, a tree on $n$ vertices either has at least $n/4k$ leaves or a collection of at least $n/4k$ bare paths (internal vertices of the path have degree $2$ in the tree) of length $k$. If there are at least $n /(4\log \log n)$ leaves, we can embed the tree obtained after removing the leaves. Then we can use a fresh random graph and Lemma \[Lemma\_almost\_pm\_bip\] to find a matching for all the leaves, completing the embedding of the tree.
On the other hand, if there are at least $n \log \log n/(4 \log n)$ bare paths of length $\log n/\log \log n$, it is possible to embed all but $n /\log n$ of these paths, which are all but $n /\log \log n$ vertices. Then one has to connect the remaining paths, again using ideas from [@montgomery2019spanning]. In between both cases it is not clear what should be done, because we might have $n / \log n$ leaves and $n/(4\log \log n)$ bare paths of length $\log \log n$. The length of the paths are too short to connect them and the leaves are too few for the above argument. Answering this questions and thereby improving the result of Alon, Krivelevich, and Sudakov [@alon2007embedding] is a challenging open problem.
Other spanning structures
-------------------------
As mentioned above, embeddings of spanning structures in ${{\mathcal{G}}_\alpha}$, $\gnp{n}{p}$, and ${{\mathcal{G}}_\alpha}\cup \gnp{n}{p}$ for fixed $\alpha>0$ have also been studied for other graphs such as powers of Hamilton cycles, factors, and general bounded degree graphs. In most of these cases almost spanning embeddings (e.g. Ferber, Luh, and Nguyen [@ferber2017embedding]) can be generalised such that previous proofs can be extended to the regime $\alpha=o(1)$ with $\beta=\alpha^{-1/C}$, similar to what we do in Corollary \[cor:trees\]. Further improvements seem to be hard, because better almost spanning results are similar in difficulty to spanning results in $\gnp{n}{p}$ alone. We want to discuss this on one basic example, the triangle factor, which is the disjoint union of $n/3$ triangles.
In ${{\mathcal{G}}_\alpha}$ we need $\alpha \ge 2/3$, in $\gnp$ the threshold is $n^{-2/3}\log^{1/3}n $, and in ${{\mathcal{G}}_\alpha}\cup \gnp$ with a fixed $\alpha>0$ it is $n^{-2/3}$. Note that the $\log$-term in $\gnp$ is needed to ensure that every vertex is contained in a triangle, which is essential for a triangle factor. Using Janson’s inequality [@frieze2016introduction Theorem 21.12] it is not hard to prove the almost spanning result for a triangle factor on at least $(1-\varepsilon)n$ vertices with $p=\omega(n^{-2/3})$. This can be generalised to $\gnp{n}{\beta n^{-2/3}}$ giving a.a.s. a triangle factor on at least $(1-C/\beta)n$ vertices. Again, this can only give something with $\beta=\alpha^{-1/C}$ in ${{\mathcal{G}}_\alpha}\cup \gnp{n}{\beta n^{-2/3}}$ and to improve this we ask the following.
Let $0<\beta=\beta(n)\le \log^{1/3} n$. Does $\gnp{n}{\beta n^{-2/3}}$ a.a.s. contain a triangle factor on at least $$\left(1- (1-o(1))\exp(-\beta^3)\right)n$$ vertices?
Observe, that this is a.a.s. the number of vertices of $\gnp{n}{\beta n^{-2/3}}$ that are not contained in a triangle. Similar questions for other factors or more general structures would be of interest. It took a long time until Johannson, Kahn, and Vu [@johansson2008factors] determined the threshold for the triangle factor. This conjecture seems to be of similar difficulty, whereas for our purposes it would already be great to obtain a triangle factor on at least $(1-C \exp(-\beta^3))n$ vertices for some $C>1$.
For the remainder of this note we prove Theorem \[Thm\_hamiltonicity\] and \[Thm\_trees\] in Section \[Sec\_hamil\_proof\] and \[Sec\_trees\_proof\] respectively.
Hamiltonicity {#Sec_hamil_proof}
=============
We will prove the following proposition that will be sufficient to prove the theorem together with known results on Hamilton cycles in $\gnp$.
\[Prop\_hamiltonicity\_large\] Let $\alpha = \alpha(n) : \mathbb{N} \mapsto (0,1)$ such that $\alpha = \omega(n^{-1/6})$, and let $\beta = \beta(\alpha) = -(6+o(1)) \log(\alpha)$. Then a.a.s. ${{\mathcal{G}}_\alpha}\cup \gnp{n}{\beta /n}$ is Hamiltonian.
Proof of Theorem \[Thm\_hamiltonicity\] {#proof-of-theoremthm_hamiltonicity .unnumbered}
---------------------------------------
Let $\alpha, \beta>0$ such that $\beta = -(6+o(1)) \log(\alpha)$. If $\alpha = O(n^{-1/6})$, we have $\beta \ge (1+o(1)) \log n$ and we can infer that a.a.s. there is a Hamilton cycle in $G(n,\beta/n)$ (this follows from an improvement on the result concerning the threshold for Hamiltonicity [@komlos1983limit]). On the other hand, if $\alpha = \omega(n^{-1/6})$, then we apply Proposition \[Prop\_hamiltonicity\_large\] to a.a.s. get the Hamilton cycle.
Proof of [Proposition]{} \[Prop\_hamiltonicity\_large\] {#proof-of-propositionprop_hamiltonicity_large .unnumbered}
-------------------------------------------------------
To prove the proposition we apply the following strategy. We first find a long path in $\gnp$ alone. Then, by considering the union with ${{\mathcal{G}}_\alpha}$, we obtain a reservoir structure for each vertex that allows us to extend the length of the path iteratively. Finally, we will also be able to close this path to a cycle on all vertices. W.l.o.g. we can assume that $\alpha<1/10$.
Finding a long path {#finding-a-long-path .unnumbered}
-------------------
Let $P=p_1, \dots, p_\ell$ be the longest path that we can find in ${\mathcal{G}}_1=\gnp{n}{(\beta-1)/n}$ and let $V' = {\left\{{ v_1,\dots,v_k }\right\}} = V({\mathcal{G}}_1) \setminus {\left\{{ p_1, ..., p_\ell }\right\}}$ be the left-over. Then, by Lemma \[Lemma\_almost\_hamiltonian\], we get a.a.s. that $$\begin{aligned}
k={\left|{V'}\right|}=n-\ell \le {\left({1-o(1)}\right)} \beta \exp{\left({1-\beta}\right)}n. \label{Eq_SizeLeftover}
\end{aligned}$$ Next, let $P'$ be a collection of vertices of $P$, where we take every other vertex, excluding the last, that is $$\begin{aligned}
\label{Def_Pprime} P' = {\left\{{ p_i : i \equiv 0 \pmod 2 }\right\}} \setminus {\left\{{ p_\ell}\right\}}
\end{aligned}$$ In the following, we will work on $P'$ instead of all of $P$, ensuring that certain absorbing structures do not overlap.
Absorbing the left-over {#absorbing-the-left-over .unnumbered}
-----------------------
We now consider the union ${{\mathcal{G}}_\alpha}\cup {\mathcal{G}}_1$. The following absorbing structure is the key to the argument.
\[Def\_absorber\] For any vertices $u, v \in V({{\mathcal{G}}_\alpha}\cup {\mathcal{G}}_1)$ let $$\begin{aligned}
\label{Def_Buv} {{\boldsymbol{B}}(u, v)} = {\left\{{ x \in N_{{{\mathcal{G}}_\alpha}}(u) \cap P' \mid N_P(x) \subseteq N_{{{\mathcal{G}}_\alpha}}(v) }\right\}}.
\end{aligned}$$
\(1) at (0,0) [$p_1$]{}; (2) at (1.5, 0) [$p_2$]{}; (3) at (3, 0) [$p_3$]{}; (4) at (4.5, 0) […]{}; (5) at (6,0) [$p_{j-1}$]{}; (6) at (7.5, 0) [$p_j$]{}; (7) at (9, 0) [$p_{j+1}$]{}; (8) at (10.5, 0) […]{}; (9) at (12, 0) [$p_\ell$]{}; (10) at (7.5,2) [$v$]{};
\(1) – (2) – (3) – (4) – (5) – (6) – (7) – (8) – (9); (10) edge \[orange\] (5); (10) edge \[orange\] (7); (9) edge \[orange, bend left=30\] (6);
(anchor) at (14.1,0) ;
\(1) at (0,0) [$p_1$]{}; (2) at (1.5, 0) [$p_2$]{}; (3) at (3, 0) [$p_3$]{}; (4) at (4.5, 0) […]{}; (5) at (6,0) [$p_{j-1}$]{}; (6) at (13.5, 0) [$p_j$]{}; (7) at (9, 0) [$p_{j+1}$]{}; (8) at (10.5, 0) […]{}; (9) at (12, 0) [$p_\ell$]{}; (10) at (7.5,0) [$v$]{};
\(1) – (2) – (3) – (4) – (5); (7) – (8) – (9); (10) edge \[orange\] (5); (10) edge \[orange\] (7); (9) edge \[orange\] (6);
(anchor) at (14.1,0) ;
If for some $v \in V'$ there is an $p_j \in {{\boldsymbol{B}}(p_\ell, v)}$ we can proceed as follows (see Figure \[Fig\_AbsorberBuv\]). By definition we have $p_{j-1}, p_{j+1} \in N_{{{\mathcal{G}}_\alpha}(v)}$ and $p_{j} \in N_{{{\mathcal{G}}_\alpha}}(p_\ell) \cap P$. Then $p_j$ can be replaced by $v$ in the path $P$ and can now be appended to the path $P$ at $p_\ell$. So we get the path $\tilde{P} = p_1, \dots, p_{j-1},v, p_{j+1}, \dots, p_\ell, p_j$, where $\tilde{P} \subset P \cup {{\mathcal{G}}_\alpha}$.
To iterate this argument we show that a.a.s. for any pair of vertices $u$ and $v$, the set ${{\boldsymbol{B}}(u, v)}$ is large enough.
\[Lemma\_SizeOfBuv\] We have a.a.s. ${\left|{{{\boldsymbol{B}}(u, v)}}\right|} \geq \alpha^3 n / 4$ for any $u,v \in V({{\mathcal{G}}_\alpha}\cup {\mathcal{G}}_1)$.
Let $u, v$ be arbitrary vertices in $V=V({{\mathcal{G}}_\alpha}\cup {\mathcal{G}}_1)$. The set ${{\boldsymbol{B}}(u, v)}$ is uniformly distributed over $P'$, because $\gnp{n}{(\beta-1)/n}$ is sampled independently of the deterministic graph ${{\mathcal{G}}_\alpha}$. Then by definition $$\begin{aligned}
{\mathbb{E}}{\left\lbrack{{\left|{{{\boldsymbol{B}}(u, v)}}\right|}}\right\rbrack} \ge \frac{9}{10} \alpha^3 {\left|{P'}\right|} \geq \frac{2}{5} \alpha^3 {\left({1 - {\left({1-o(1)}\right)} \beta \exp{\left({1-\beta}\right)}}\right)} n \geq \alpha^3 n / 3. \label{Eq_ErwBuv}
\end{aligned}$$ An immediate consequence of ${{\boldsymbol{B}}(u, v)}$ being uniformly settled over $\gnp{n}{(\beta-1)/n}$ is that ${\left|{{{\boldsymbol{B}}(u, v)}}\right|} \sim {{\rm Bin}}({\left|{P'}\right|}, \alpha^3)$. It follows from and the Chernoff bound that there is a sufficiently small, but constant, $\delta > 0$ s.t. $$\begin{aligned}
{{\mathbb{P}}}{\left({{\left|{{{\boldsymbol{B}}(u, v)}}\right|} < \alpha^3 n/4 }\right)} \leq {{\mathbb{P}}}{\left({{\left|{{{\boldsymbol{B}}(u, v)}}\right|} < (1- \delta){\mathbb{E}}{\left\lbrack{{\left|{{{\boldsymbol{B}}(u, v)}}\right|}}\right\rbrack} }\right)} \leq \exp {\left({ - \delta^2 / 8 \alpha^3 n }\right)} < \exp {\left({- \sqrt{n} }\right)}. \label{Eq_ConcBuv}
\end{aligned}$$ The lemma follows from a union bound over all $\binom{n}{2}$ choices for $u,v$ and .
We now have everything at hand to absorb all but two of the left-over vertices $v \in V'$ onto a path of length $n-2$. We do this inductively using Algorithm \[Algo\_IncreaseP\].
Define $\ell_1 = \ell$, $P_1 = P$ with $P_1 = u^1_1 \dots u^1_{\ell_1}$ Define for any $u, v$ the set $B_1(u,v)={{\boldsymbol{B}}(u, v)}$ Define $V_1' = V'$ $\tilde{P} = P_{k}$
Let $\tilde{P}, B_i(\cdot, \cdot)$ be defined as in Algorithm \[Algo\_IncreaseP\]. In order to see that the algorithm terminates with $\tilde{P} = P_{k}$ it suffices to prove, that $B_i(u,v)$ is not empty for any $u,v \in V$ and $i=1\dots k$. By definition of $P'$ in we have $|{{\boldsymbol{B}}(u, v)} \setminus B_i(u,v)| \le i$ and using Claim \[Lemma\_SizeOfBuv\] and we get $$\begin{aligned}
\label{Eq_LeftOverVertices}
|B_i(u,v)| \ge \alpha^3 n/8,
\end{aligned}$$ whenever $\beta \exp {\left({1- \beta}\right)} < \alpha^3 / 8$. As this holds by definition of $\beta = -(6+o(1)) \log(\alpha)$ and with $\alpha < 1/10$, we get that holds for all $u,v \in V$ and any $i=1,\dots,k$.
Closing the cycle {#closing-the-cycle .unnumbered}
-----------------
We have found a path $\tilde{P} = p_1, \dots, p_{n-2}$ and we are left with two vertices $v_{k-1}, v_{k}$ that are not on the path. It is possible to close the Hamilton cycle by absorbing $v_{k-1}$ and $v_{k}$ if there is an edge between $A := B_k(p_1,v_{k-1})$ and $B := B_k(p_{n-2},v_k)$. Indeed, we then have w.l.o.g. $i<j$ such that $p_i \in A$, $p_j \in B$, and there is an edge $p_ip_j$. By definition of $A$ and $B$ we can then obtain the Hamilton cycle $$p_i,p_1,\dots,p_{i-1},v_{k-1},p_{i+1},\dots,p_{j-1},v_k,p_{j+1},\dots,p_{n-2},p_j.$$
It remains to prove that we have an edge between $A$ and $B$. For this we reveal ${\mathcal{G}}_2 = \gnp{n}{1/n}$. As $|A|,|B| \ge \alpha^3 n/8$ by we get $$\begin{aligned}
{\mathbb{E}}{\left\lbrack{ e_{{\mathcal{G}}_2} {\left({A, B}\right)} }\right\rbrack} \geq \frac{1}{n} \cdot {\left({\frac{\alpha^3 n}{16}}\right)}^2 = \omega(1),
\end{aligned}$$ as $\alpha = \omega (n^{-1/6})$. Together with Chernoff’s inequality this implies that a.a.s $e_{{\mathcal{G}}_2} {\left({A, B}\right)}>0$. As the union of ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ can be coupled as a subgraph of $\gnp{n}{\beta/n}$ this implies that a.a.s. there is a Hamilton cycle in ${{\mathcal{G}}_\alpha}\cup \gnp$ and finishes the proof of [Proposition]{} \[Prop\_hamiltonicity\_large\].
Observe, that when running the same proof for Theorem \[Thm\_PM\] we can obtain the better constant by adapting the definition of the ${{\boldsymbol{B}}(u, v)}$ to the setup of perfect matchings and then proving that a.a.s. $|{{\boldsymbol{B}}(u, v)}| \ge \alpha^2n/4$. We spare the details here.
Bounded degree trees {#Sec_trees_proof}
====================
[Theorem]{} \[Thm\_trees\] is modular, which turns almost spanning embeddings in the random graph into spanning embeddings in the union ${{\mathcal{G}}_\alpha}\cup \gnp{n}{\beta/n}$. The proof is very similar to the proof for Hamilton cycles and we will spare some details.
Proof of [Theorem]{} \[Thm\_trees\] {#proof-of-theoremthm_trees .unnumbered}
-----------------------------------
Let ${{\mathcal{G}}_\alpha}$ be given and ${\mathcal{G}}= \gnp{n}{\beta/n}$. Let ${\mathcal{T}}$ be an arbitrary tree on $n$ vertices with maximum degree $\Delta$. Denote by ${\mathcal{T}}_{{\varepsilon}}$ the tree obtained from ${\mathcal{T}}$ by the following construction.
1. Set ${\mathcal{T}}_0 = {\mathcal{T}}$.
2. In every step $i$, check whether ${\mathcal{T}}_i$ has at most $(1 - {\varepsilon})n$ vertices.
- If this is the case, set ${\mathcal{T}}_{\varepsilon}= {\mathcal{T}}_i$ and finish the process.
- Otherwise, create ${\mathcal{T}}_{i+1}$ by deleting one leaf of ${\mathcal{T}}_i$.
We denote by $L$ the left-over, that are the vertices removed during construction of ${\mathcal{T}}_{{\varepsilon}}$. Then $$\begin{aligned}
{\left|{V({\mathcal{T}}_{{\varepsilon}})}\right|} \leq (1 - {\varepsilon})n, \qquad {\left|{L}\right|} \leq {\varepsilon}n+1, \qquad \text{and} \qquad V({\mathcal{T}}) = V({\mathcal{T}}_{{\varepsilon}}) \cup L.\end{aligned}$$ Next we let $T$ be an independent subset of the vertices of ${\mathcal{T}}_{\varepsilon}$ such that the vertices in $T$ do not have neighbours outside of ${\mathcal{T}}_{\varepsilon}$ with respect to ${\mathcal{T}}$. Observe, that there exists such a $T$ such that $|T| \ge \frac{(1-\Delta {\varepsilon})n}{\Delta+1}$.
By assumption we a.a.s. have an embedding ${\mathcal{T}}_{\varepsilon}'$ of ${\mathcal{T}}_{\varepsilon}$ into ${\mathcal{G}}$ and we denote by $T'$ the image of $T$ under this embedding. We adapt Definition \[Def\_absorber\] and define for any two vertices $u,v$ $${{\boldsymbol{B}}}(u,v) = {\left\{{ x \in N_{{{\mathcal{G}}_\alpha}}(u) \cap T' \mid N_{{\mathcal{T}}_{\varepsilon}'}(x) \subset N_{{{\mathcal{G}}_\alpha}(v)}}\right\}}.$$ As before, if we want to embed a vertex $w$ that is a neighbour of an already embedded vertex $u$ in ${\mathcal{T}}_{\varepsilon}$ and $v$ is an available vertex we can do it if ${{\boldsymbol{B}}}(u,v)$ is non-empty. More precisely, with $x \in {{\boldsymbol{B}}}(u,v)$, we can embed the vertex embedded onto $x$ to $v$, embed $w$ to $x$, and obtain a valid embedding of ${\mathcal{T}}_{\varepsilon}$ with an additional neighbour of $u$. Analogous to Claim \[Lemma\_SizeOfBuv\] we get the following.
\[Claim\_Buv\] We have a.a.s. ${\left|{{{\boldsymbol{B}}(u, v)}}\right|} \geq \frac{\alpha^{\Delta+1} n}{4(\Delta+1)}$ for any $u,v \in V({{\mathcal{G}}_\alpha}\cup {\mathcal{G}})$.
Therefore, similar to Algorithm \[Algo\_IncreaseP\], we can iteratively append leaves to ${\mathcal{T}}_{\varepsilon}$ to obtain an embedding of ${\mathcal{T}}$ into ${{\mathcal{G}}_\alpha}\cup {\mathcal{G}}$. As in every step we lose at most one vertex from each ${{\boldsymbol{B}}}(u,v)$ this works as long as $${\left|{L}\right|} \leq {\varepsilon}n+1 < {\left|{{{\boldsymbol{B}}}(u,v)}\right|},$$ which holds by Claim \[Claim\_Buv\] and the assumption on ${\varepsilon}$ and $\alpha$.
[^1]: The research on this project was initiated during a workshop in Cuxhaven. We would like to thank the Hamburg University of Technology for their support. OP was supported by Technische Universität Ilmenau, the Carls Zeiss Foundation, and DFG Grant PA 3513/1-1. MHK was supported by Stiftung Polytechnische Gesellschaft. SM was supported by DFG Grant 327533333. GSM is supported by the European Research Council (Consolidator Grant PEPCo 724903).
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abstract: 'We find a sufficient condition to establish that certain abelian groups are not CI-groups with respect to ternary relational structures, and then show that the groups $\Z_3\times\Z_2^2$, $\Z_7\times\Z_2^3$, and $\Z_5\times\Z_2^4$ satisfy this condition. Then we completely determine which groups $\Z_2^3\times\Z_p$, $p$ a prime, are CI-groups with respect to binary and ternary relational structures. Finally, we show that $\Z_2^5$ is not a CI-group with respect to ternary relational structures.'
title: 'CI-groups with respect to ternary relational structures: new examples'
---
---------------------------------------------- -------------------------------------------------
[Edward Dobson]{} [Pablo Spiga]{}
[Department of Mathematics and Statistics]{} [Dipartimento di Matematica Pura e Applicata]{}
[Mississipi State University]{} [Università degli Studi di Milano-Bicocca]{}
[PO Drawer MA]{} [Via Cozzi 53]{}
[Mississipi State, MS 39762 USA]{} [20126 Milano, Italy]{}
[dobson@math.msstate.edu]{} [pablo.spiga@unimib.it]{}
---------------------------------------------- -------------------------------------------------
Introduction {#intro}
============
In recent years, there has been considerable interest in which groups $G$ have the property that any two Cayley graphs of $G$ are isomorphic if and only if they are isomorphic by a group automorphism of $G$. Such a group is a called a CI-group with respect to graphs, and this problem is often referred to as the Cayley isomorphism problem. The interested reader is referred to [@Li2002] for a survey on CI-groups with respect to graphs. Of course, the Cayley isomorphism problem can and has been considered for other types of combinatorial objects. Perhaps the most significant such result is a well-known theorem of Pálfy [@Palfy1987] which states that a group $G$ of order $n$ is a CI-group with respect to every class of combinatorial objects if and only if $n = 4$ or $\gcd(n,\varphi(n)) = 1$, where $\varphi$ is the Euler phi function. In fact, in proving this result, Pálfy showed that if a group $G$ is not a CI-group with respect to some class of combinatorial objects, then $G$ is not a CI-group with respect to quaternary relational structures. As much work has been done on the case of binary relational structures (i.e., digraphs), until recently there was a “gap" in our knowledge of the Cayley isomorphism problem for $k$-ary relational structures with $k = 3$. As additional motivation to study this problem, we remark that a group $G$ that is a CI-group with respect to ternary relational structures is necessarily a CI-group with respect to binary relational structures.
Although Babai [@Babai1977] showed in $1977$ that the dihedral group of order $2p$ is a CI-group with respect to ternary relational structures, no additional work was done on this problem until the first author considered the problem in $2003$ [@Dobson2003]. Indeed, in [@Dobson2003] a relatively short list of groups is given and it is proved that every CI-group with respect to ternary relational structures lies in this list (although not every group in this list is necessarily a CI-group with respect to ternary relational structures). Additionally, several groups in the list were shown to be CI-groups with respect to ternary relational structures. Recently, the second author [@Spiga2008] has shown that two groups given in [@Dobson2003] are not CI-groups with respect to ternary relational structures, namely $\Z_3\ltimes Q_8$ and $\Z_3\times Q_8$. In this paper, we give a sufficient condition to ensure that certain abelian groups are not CI-groups with respect to ternary relational structures (Theorem \[main\]), and then show that $\Z_3\times\Z_2^2$, $\Z_7\times\Z_2^3$, and $\Z_5\times\Z_2^4$ satisfy this condition in Corollary \[coro1\] (and so are not CI-groups with respect to ternary relational structures). We then show that $\Z_5\times \Z_2^3$ is a CI-group with respect to ternary relational structures. As the first author has shown [@Dobson2010a] that $\Z_2^3\times\Z_p$ is a CI-group with respect to ternary relational structures provided that $p\ge 11$, we then have a complete determination of which groups $\Z_2^3\times\Z_p$, $p$ a prime, are CI-groups with respect to ternary relational structures.
The group $\Z_2^3\times\Z_p$ is a CI-group with respect to color ternary relational structures if and only if $p\not = 3$ and $7$.
We will show that both $\Z_2^3\times\Z_3$ and $\Z_2^3\times\Z_7$ are CI-groups with respect to binary relational structures. As it is already known that $\Z_2^4$ is a CI-group with respect to binary relational structures [@Li2002], we have the following result.
The group $\Z_2^3\times\Z_p$ is a CI-group with respect to color binary relational structures for all primes $p$.
We are then left in the situation of knowing whether or not any subgroup of $\Z_2^3\times\Z_p$ is a CI-group with respect to binary or ternary relational structures, with the exception of $\Z_2^2\times\Z_7$ with respect to ternary relational structures (as $\Z_2^2\times\Z_7$ is a CI-group with respect to binary relational structures [@KovacsM2009]). We show that $\Z_2^2\times\Z_7$ is a CI-group with respect to ternary relational structures (which generalizes a special case of the main result of [@KovacsM2009]) and we prove the following.
The group $\Z_2^2\times\Z_p$ is a CI-group with respect to color ternary relational structures if and only if $p\neq 3$.
Finally, using Magma [@Magma] and GAP [@GAP], we show that $\Z_2^5$ is not a CI-group with respect to ternary relational structures.
We conclude this introductory section with the formal definition of the objects we are interested in.
[A [*$k$-ary relational structure*]{} is an ordered pair $X =
(V,E)$, with $V$ a set and $E$ a subset of $V^k$. Furthermore, a [*color $k$-ary relational structure*]{} is an ordered pair $X =
(V,(E_1,\ldots,E_c))$, with $V$ a set and $E_1,\ldots,E_c$ pairwise disjoint subsets of $V^k$. If $k = 2,3$, or $4$, we simply say that $X$ is a (color) binary, ternary, or quaternary relational structure.]{}
The following two definitions are due to Babai [@Babai1977].
\[def:2\][For a group $G$, define $g_L:G\rightarrow G$ by $g_L(h)
= gh$, and let $G_L = \{g_L:g\in G\}$. Then $G_L$ is a permutation group on $G$, called the [*left regular representation of $G$*]{}. We will say that a (color) $k$-ary relational structure $X$ is a [ *Cayley (color) $k$-ary relational structure of $G$*]{} if $G_L\le\Aut(X)$ (note that this implies $V = G$). In general, a combinatorial object $X$ will be called a [*Cayley object of $G$*]{} if $G_L\le\Aut(X)$.]{}
\[def:3\][For a class ${\cal C}$ of Cayley objects of $G$, we say that $G$ is a [*CI-group with respect to ${\cal C}$*]{} if whenever $X,Y\in{\cal
C}$, then $X$ and $Y$ are isomorphic if and only if they are isomorphic by a group automorphism of $G$.]{}
It is clear that if $G$ is a CI-group with respect to [*color*]{} $k$-ary relational structures, then $G$ is a CI-group with respect to $k$-ary relational structures.
[For $g,h$ in $G$, we denote the commutator $g^{-1}h^{-1}gh$ of $g$ and $h$ by $[g,h]$.]{}
The main ingredient and Theorem A {#thm}
=================================
We start by proving the main ingredient for our proof of Theorem A.
\[main\] Let $G$ be an abelian group and $p$ an odd prime. Assume that there exists an automorphism $\alpha$ of $G$ of order $p$ fixing only the zero element of $G$. Then $\Z_p\times G$ is not a $\mathrm{CI}$-group with respect to color ternary relational structures. Moreover, if there exists a ternary relational structure on $G$ with automorphism group $\la G_L,\alpha\ra$, then $\Z_p\times G$ is not a $\mathrm{CI}$-group with respect to ternary relational structures.
Since $\alpha$ fixes only the zero element of $G$, we have $\vert G\vert \equiv 1\ (\mod p)$ and so $\gcd (p,\vert G\vert)=1$.
For each $g\in G$, define $\hat{g}:\Z_p\times G\to\Z_p\times G$ by $\hat{g}(i,j) = (i,j+g)$. Additionally, define $\tau,\gamma,\bar{\alpha}:\Z_p\times G\to\Z_p\times G$ by $\tau(i,j) = (i + 1,j)$, $\gamma(i,j) = (i,\alpha^i(j))$, and $\bar{\alpha}(i,j) = (i,\alpha(j))$. Then $(\Z_p\times G)_L =
\la\tau,\hat{g}:g\in G\ra$.
Clearly, $\la G_L,\alpha\ra =G_L\rtimes \la \alpha\ra$ is a subgroup of ${\mathop{\mathrm{Sym}}}(G)$ (where $G_L$ acts on $G$ by left multiplication and $\alpha$ acts as an automorphism). Note that the stabilizer of $0$ in $\la G_L,\alpha\ra$ is $\la\alpha\ra$. As $\alpha$ fixes only $0$, we conclude that for every $g\in G$ with $g\neq 0$, the point-wise stabilizer of $0$ and $g$ in $\langle G_L,\alpha\rangle $ is $1$. Therefore, by [@Wielandt1969 Theorem 5.12], there exists a color Cayley ternary relational structure $Z$ of $G$ such that $\Aut(Z) = \la G_L,\alpha\ra$. If there exists also a ternary relational structure with automorphism group $\la G_L,\alpha\ra$, then we let $Z$ be one such ternary relational structure.
Let $$U = \{((0_{\Z_p},g),(0_{\Z_p},h)):(0_G,g,h)\in
E(Z)\}\quad \mathrm{and}\quad S =
\{([\hat{g},\gamma](1,0_G),[\hat{g},\gamma](2,0_G)):g\in G\}\cup U$$ and define a (color) ternary relational structure $X$ by $$V(X) = \Z_p\times G\quad
\mathrm{and}\quad E(X) = \{k(0_{\Z_p\times G},s_1,s_2):(s_1,s_2)\in S,k\in(\Z_p\times G)_L\}.$$
If $Z$ is a color ternary relational structure, then we assign to the edge $k(0_{\Z_p\times G},s_1,s_2)$ the color of the edge $(0_G,g,h)$ in $Z$ if $(s_1,s_2)\in U$ and $(s_1,s_2) = ((0_{\Z_p},g),(0_{\Z_p},h))$, and otherwise we assign a fixed color distinct from those used in $Z$. By definition of $X$ we have $(\Z_p\times G)_L\le\Aut(X)$ and so $X$ is a (color) Cayley ternary relational structure of $\Z_p\times G$.
We claim that $\bar{\alpha}\in \Aut(X)$. As $\bar{\alpha}$ is an automorphism of $\Z_p\times G$, we have that $\bar{\alpha}\in\Aut(X)$ if and only if $\bar{\alpha}(S) = S$ and $\bar{\alpha}$ preserves colors (if $X$ is a color ternary relational structure). By definition of $Z$ and $U$, we have $\bar{\alpha}(U) = U$ and $\bar{\alpha}$ preserves colors (if $X$ is a color ternary relational structure). So, it suffices to consider the case $s\in S-U$, i.e., $s=
([\hat{g},\gamma](1,0),[\hat{g},\gamma](2,0))$ for some $g\in G$. Note that now we need not consider colors as all the edges in $S-U$ are of the same color. Then $\bar{\alpha}\hat{g}(i,j) = (i,\alpha(j) +
\alpha(g)) = \widehat{\alpha(g)}\bar{\alpha}(i,j)$. Thus $\bar{\alpha}\hat{g} = \widehat{\alpha(g)}\bar{\alpha}$. Similarly, $\bar{\alpha}\hat{g}^{-1} = \widehat{\alpha(g)}^{-1}\bar{\alpha}$. Clearly $\bar{\alpha}$ commutes with $\gamma$, and so $\bar{\alpha}[\hat{g},\gamma] = [\widehat{\alpha(g)},\gamma]\bar{\alpha}$. As $\bar{\alpha}$ fixes $(1,0)$ and $(2,0)$, we see that
$$\begin{aligned}
\bar{\alpha}(s)=\bar{\alpha}([\hat{g},\gamma](1,0),[\hat{g},\gamma](2,0)) & = &
(\bar{\alpha}[\hat{g},\gamma](1,0),\bar{\alpha}[\hat{g},\gamma](2,0))\\
& =&
([\widehat{\alpha(g)},\gamma]\bar{\alpha}(1,0),
[\widehat{\alpha(g)},\gamma]\bar{\alpha}(2,0))\\
& = &
([\widehat{\alpha(g)},\gamma](1,0),[\widehat{\alpha(g)},\gamma](2,0))\in
(S-U).\end{aligned}$$
Thus $\bar{\alpha}(S) = S$, $\bar{\alpha}$ preserves colors (if $X$ is a color ternary relational structure) and $\bar{\alpha}\in\Aut(X)$.
We claim that $\gamma^{-1}(\mathbb{Z}_p\times G)_L\gamma$ is a subgroup of $\Aut(X)$. We set $\tau'=\gamma^{-1}\tau\gamma$ and $g'=\gamma^{-1} \hat{g}\gamma$, for $g\in G$. Note that $\tau'=\tau \bar{\alpha}^{-1}$. As $\bar{\alpha}\in\Aut(X)$, we have that $\tau'\in\Aut(X)$. Therefore it remains to prove that $\langle g' : g\in G\rangle$ is a subgroup of $\Aut(X)$. Let $e\in
E(X)$ and $g\in G$. Then $e=k((0,0),s)$, where $s\in S$ and $k=\tau^a\widehat{l}$, for some $a\in\mathbb{Z}_p$, $l\in G$. We have to prove that $g'(e)\in E(X)$ and has the same color of $e$ (if $X$ is a color ternary relational structure).
Assume that $s\in U$. As $g'(i,j)=(i,j+\alpha^{-i}(g))$, by definition of $U$, we have $g'[k((0,0),s)]\in E(X)$ and has the same color of $e$ (if $X$ is a color ternary relational structure). So, it remains to consider the case $s\in S - U$, i.e., $s=([\widehat{x},\gamma](1,0),[\widehat{x},\gamma](2,0))$ for some $x\in G$. As before, we need not concern ourselves with colors because all the edges in $S-U$ are of the same color.
Set $m=k\widehat{\alpha^{-a}(g)}$. Since $\bar{\alpha}\widehat{g}=\widehat{\alpha(g)}\bar{\alpha}$ and $\bar{\alpha},\gamma$ commute, we get $\bar{\alpha}g'=(\alpha(g))'\bar{\alpha}$. Also observe that as $G$ is abelian, $g'$ commutes with $\widehat{h}$ for every $g,h\in G$. Hence
$$\begin{aligned}
g'k&=&\gamma^{-1}\widehat{g}\gamma\tau^a\widehat{l}=\gamma^{-1}\widehat{g}\tau^a\gamma\bar{\alpha}^a\widehat{l}=\gamma^{-1}\tau^a\widehat{g}\bar{\alpha}^a\gamma\widehat{l}\\
&=&\tau^a\gamma^{-1}\bar{\alpha}^{-a}\widehat{g}\bar{\alpha}^a\gamma\widehat{l}=\tau^a(\alpha^{-a}(g))'\widehat{l}=\tau^a
\widehat{l}(\alpha^{-a}(g))'\\
&=&k\widehat{\alpha^{-a}(g)}\widehat{\alpha^{-a}(g)}^{-1}\gamma^{-1}\widehat{\alpha^{-a}(g)}\gamma=m[\widehat{\alpha^{-a}(g)},\gamma] \end{aligned}$$
and $$\begin{aligned}
g'[k((0,0),s)]&=&g'k((0,0),[\widehat{x},\gamma](1,0),[\widehat{x},\gamma](2,0))\\
&=&m[\widehat{\alpha^{-a}(g)},\gamma]((0,0),[\widehat{x},\gamma](1,0),[\widehat{x},\gamma](2,0))\\
&=&m((0,0),[\widehat{\alpha^{-a}(g)},\gamma][\widehat{x},\gamma](1,0),[\widehat{\alpha^{-a}(g)},\gamma][\widehat{x},\gamma](2,0))\\
&=&m((0,0),[\widehat{\alpha^{-a}(g)x},\gamma](1,0),[\widehat{\alpha^{-a}(g)x},\gamma](2,0))\in E(X).\end{aligned}$$
This proves that $g'\in \Aut(X)$. Since $g$ is an arbitrary element of $G$, we have $\gamma^{-1}G_L\gamma\subseteq \Aut(X)$. As claimed, $\gamma^{-1}(\mathbb{Z}_p\times G)_L\gamma$ is a regular subgroup of $\Aut(X)$ conjugate in ${\mathop{\mathrm{Sym}}}(\mathbb{Z}_p\times G)$ to $(\mathbb{Z}_p\times G)_L$.
We now have that $Y = \gamma(X)$ is a Cayley (color) ternary relational structure of $\Z_p\times G$ as $\Aut(Y) =
\gamma\Aut(X)\gamma^{-1}$. We will next show that $Y\not = X$. Assume by way of contradiction that $Y=X$. As $\gamma(0,g)=(0,g)$ for every $g\in G$, the permutation $\gamma$ must map edges of $U$ to themselves, so that $\gamma(S-U) = S - U$. We will show that $\gamma(S-U)\not = S - U$. Note that we need not concern ourselves with colors because as all the edges derived from $S - U$ via translations of $(\Z_p\times G)_L$ have the same color. Observing that
$$\begin{aligned}
([\hat{g},\gamma](1,0),[\hat{g},\gamma](2,0)) & = &
(\hat{g}^{-1}\gamma^{-1}\hat{g}\gamma(1,0),\hat{g}^{-1}\gamma^{-1}\hat{g}\gamma(2,0)) =
(\hat{g}^{-1}\gamma^{-1}\hat{g}(1,0),\hat{g}^{-1}\gamma^{-1}\hat{g}(2,0))\\
& = & (\hat{g}^{-1}\gamma^{-1}(1,g),\hat{g}^{-1}\gamma^{-1}(2,g))
=
(\hat{g}^{-1}(1,\alpha^{-1}(g),\hat{g}^{-1}(2,\alpha^{-2}(g))\\
& = & ((1,\alpha^{-1}(g) - g),(2,\alpha^{-2}(g) - g)),\end{aligned}$$
we see that $\gamma(S - U) = \{((1,g - \alpha(g)),(2,g -
\alpha^2(g))):g\in G\}$. Moreover, as $S-U=\{(1,\alpha^{-1}(g)-g),(2,\alpha^{-2}(g)-g): g \in G\}$, we conclude that for each $g\in G$, there exists $h_g\in G$ such that
$$g - \alpha(g) = \alpha^{-1}(h_g) - h_g{\rm\ \ \ \ \ and\ \ \ \ \ }g
- \alpha^2(g) = \alpha^{-2}(h_g) - h_g.$$
Setting $\iota:G\to G$ to be the identity permutation, we may rewrite the above equations as
$$(\iota - \alpha)(g) = (\alpha^{-1} - \iota)(h_g){\rm \ \ \ \ \ and \ \
\ \ \ }(\iota - \alpha^2)(g) = (\alpha^{-2} - \iota)(h_g).$$
Computing in the endomorphism ring of the abelian group $G$, we see that $(\alpha^{-2} - \iota) =
(\alpha^{-1}+\iota)(\alpha^{-1} - \iota)$. Applying the endomorphism $(\alpha^{-1} + \iota)$ to the first equation above, we then have that
$$(\alpha^{-1} + \iota)(\iota - \alpha)(g) = (\alpha^{-1} +
\iota)(\alpha^{-1} - \iota)(h_g)
= (\alpha^{-2} - \iota)(h_g)
= (\iota - \alpha^2)(g).$$
Hence $(\alpha^{-1} + \iota)(\iota - \alpha) = \iota -
\alpha^2$, and so $$0=(\alpha^{-1}+\iota)(\iota-\alpha)-(\iota-\alpha^2)=((\alpha^{-1}+\iota)-(\iota+\alpha))(\iota-\alpha)=(\alpha^{-1}-\alpha)(\iota-\alpha),$$ (here $0$ is the endomorphism of $G$ that maps each element of $G$ to $0$). As $\alpha$ fixes only $0$, the endomorphism $\iota-\alpha$ is invertible, and so we see that $\alpha^{-1} -\alpha= 0$, and $\alpha = \alpha^{-1}$. However, this implies that $p=\vert\alpha\vert = 2$, a contradiction. Thus $\gamma(S - U)\not = S - U$ and so $Y\not = X$.
We set $T = \gamma(S)$, so that $((0,0),t)\in E(Y)$ for every $t\in T$, where if $X$ is a color ternary relational structure we assume that $\gamma$ preserves colors. Now suppose that there exists $\beta\in\Aut(\Z_p\times G)$ such that $\beta(X) = Y$. Since $\gcd(p,\vert G\vert)=1$, we obtain that $\Z_p\times 1_G$ and $1_{\Z_p}\times G$ are characteristic subgroups of $\Z_p\times G$. Therefore $\beta(i,j) = (\beta_1(i),\beta_2(j))$, where $\beta_1\in\Aut(\Z_p)$ and $\beta_2\in\Aut(G)$. As $\beta$ fixes $(0,0)$, we must have that $\beta(S) = T$. As there is no element of $T$ of the form $((2,x_1),(1,y_1))$, we conclude that $\beta_1 = 1$ as $\beta_1(i) = i$ or $2i$. As $\bar{\alpha}\in\Aut(X)$ and $X\neq Y$, we have that $\beta_2\not\in\la\alpha\ra$. Now observe that $\beta(U) = U$. Thus $\beta_2\in\Aut(Z) = \la G_L,\alpha\ra$. We conclude that $\beta_2\in\la\alpha\ra$, a contradiction. Thus $X,Y$ are not isomorphic by a group automorphism of $\Z_p\times G$, and the result follows.
The following two lemmas, which in our opinion are of independent interest, will be used (together with Theorem \[main\]) in the proof of Corollary \[coro1\].
\[Stabkary\] Let $G$ be a transitive permutation group on $\Omega$. If $x\in\Omega$ and $\Stab_G(x)$ in its action on $\Omega - \{x\}$ is the automorphism group of a $k$-ary relational structure with vertex set $\Omega - \{x\}$, then $G$ is the automorphism group of a $(k+1)$-ary relational structure.
Let $Y$ be a $k$-ary relational structure with vertex set $\Omega - \{x\}$ and automorphism group $\Stab_G(x)$ in its action on $\Omega - \{x\}$. Let $W = \{(x,v_1,\ldots,v_k):(v_1,\ldots,v_k)\in E(Y)\}$, and define a $(k+1)$-ary relational structure $X$ by $V(X) = \Omega$ and $E(X) = \{g(w):w\in W \,\mathrm{ and }\, g\in G \}$. We claim that $\Aut(X) = G$. First, observe that $\Stab_G(x)$ maps $W$ to $W$. Also, if $e\in E(X)$ and $e = (x,v_1,\ldots,v_k)$ for some $v_1,\ldots,v_k\in \Omega$, then there exists $ (x,u_1,\ldots,u_k)\in W$ and $g\in G$ with $g(x,u_1,\ldots,u_k) = (x,v_1,\ldots,v_k)$. We conclude that $g(x) = x$ and $g(u_1,\ldots,u_k) = (v_1,\ldots,v_k)$. Hence $g\in\Stab_G(x)$ and $(v_1,\ldots,v_k)\in E(Y)$. Then $W$ is the set of all edges of $X$ with first coordinate $x$.
By construction, $G\le\Aut(X)$. For the reverse inclusion, let $h\in\Aut(X)$. As $G$ is transitive, there exists $g\in G$ such that $g^{-1}h\in\Stab_{\Aut(X)}(x)$. Note that as $g\in G$, the element $g^{-1}h\in G$ if and only if $h\in G$. We may thus assume without loss of generality that $h(x) = x$. Then $h$ stabilizes set-wise the set of all edges of $X$ with first coordinate $x$, and so $h(W) = W$ and $h$ induces an automorphism of $Y$. As $\Aut(Y) = \Stab_G(x)\leq G$, the result follows.
\[autosemiregular\] Let $m\ge 2$ be an integer and $\rho\in {\mathop{\mathrm{Sym}}}(\Z_{ms})$ be a semiregular element of order $m$ with $s$ orbits. Then there exists a digraph with vertex set $\Z_{ms}$ and with automorphism group $\la\rho\ra$.
For each $i\in \Z_s$, set $$\rho_i = (0,1,\ldots,m-1)\cdots(im,im + 1,\ldots,im+m-1)\quad \textrm{and}\quad V_i = \{im + j:j\in\Z_m\}.$$ We inductively define a sequence of graphs $\Gamma_0,\ldots,\Gamma_{s-1} = \Gamma$ such that the subgraph of $\Gamma$ induced by $\Z_{(i+1)m}$ is $\Gamma_i$, the indegree in $\Gamma$ of a vertex in $V_i$ is $i+1$, and $\Aut(\Gamma_i) = \la\rho_i\ra$, for each $i\in\Z_s$.
We set $\Gamma_0$ to be the directed cycle of length $m$ with edges $\{(j,j+1): j \in \Z_m\}$ and with automorphism group $\la\rho_0\ra$. Inductively assume that $\Gamma_{s-2}$, with the above properties, has been constructed. We construct $\Gamma_{s-1}$ as follows. First, the subgraph of $\Gamma_{s-1}$ induced by $\Z_{(s-1)m}$ is $\Gamma_{s-2}$. Then we place the directed $m$ cycle $\{((s-1)m+j,(s-1)m+j+1):j \in \Z_m\}$ whose automorphism group is $\la((s-1)m,(s-1)m + 1,\ldots,(s-1)m+m-1)\ra$ on the vertices in $V_{s-1}$. Additionally, we declare the vertex $(s-1)m$ to be outadjacent to $(s-2)m$ and to every vertex that $(s-2)m$ is outadjacent to that is not contained in $V_{s-2}$. Finally, we add to $\Gamma_{s-1}$ every image of one of these edges under an element of $\langle\rho_{s-1}\rangle$.
By construction, $\rho_{s-1}$ is an automorphism of $\Gamma_{s-1}$ and the subgraph of $\Gamma_{s-1}$ induced by $\Z_{(s-1)m}$ is $\Gamma_{s-2}$. Then each vertex in $\Gamma_{s-1}\cap V_i$ has indegree $i+1$ for $0\le i\le s-2$, while it is easy to see that each vertex of $V_{s-1}$ has indegree $s$. Finally, if $\delta\in \Aut(\Gamma_{s-1})$, then $\delta$ maps vertices of indegree $i+1$ to vertices of indegree $i+1$, and so $\delta$ fixes set-wise $V_i$, for every $i\in\Z_s$. Additionally, the action induced by $\langle\delta\rangle$ on $V_{s-1}$ is necessarily $\la((s-1)m,(s-1)m + 1,\ldots,(s-1)m+m-1)\ra$ as this is the automorphism group of the subgraph of $\Gamma_{s-1}$ induced by $V_{s-1}$. Moreover, arguing by induction, we may assume that the action induced by $\delta$ on $V(\Gamma_{s-1}) - V_{s-1}$ is given by an element of $\langle\rho_{s-2}\rangle$. If $\delta\not\in\la\rho_{s-1}\ra$, then $\Aut(\Gamma_{s-1})$ has order at least $m^2$, and there is some element of $\Aut(\Gamma_{s-1})$ that is the identity on $V(\Gamma_{s-2})$ but not on $V_{s-1}$ and vice versa. This however is not possible as each vertex of $V_{s-2}$ is outadjacent to exactly one vertex of $V_{s-1}$. Then $\Aut(\Gamma_{s-1}) = \la\rho_{s-1}\ra$ and the result follows.
\[coro1\] None of the groups $\Z_3\times\Z_2^2$, $\Z_7\times\Z_2^3$, or $\Z_5\times\Z_2^4$ are CI-groups with respect to ternary relational structures.
Observe that $\Z_2^2$ has an automorphism $\alpha_3$ of order $3$ that fixes $0$ and acts regularly on the remaining $3$ elements, and similarly, $\Z_2^3$ has an automorphism $\alpha_7$ of order $7$ that fixes $0$ and acts regularly on the remaining $7$ elements. As a regular cyclic group is the automorphism group of a directed cycle, we see that $\la(\Z_3\times\Z_2^2)_L,\alpha_3\ra$ and $\la(\Z_7\times\Z_2^3)_L,\alpha_7\ra$ are the automorphism groups of ternary relational structures by Lemma \[Stabkary\]. The result then follows by Theorem \[main\].
Now $\Z_2^4$ has an automorphism $\alpha_5$ of order $5$ that fixes $0$ and acts semiregularly on the remaining $15$ points. Then $\la\alpha_5\ra$ in its action on $\Z_2^4 - \{0\}$ is the automorphism group of a binary relational structure by Lemma \[autosemiregular\]. By Lemma \[Stabkary\], there exists a ternary relational structure with automorphism group $\la(\Z_5\times\Z_2^4)_L,\alpha_5\ra$. The result then follows by Theorem \[main\].
Before proceeding, we will need terms and notation concerning complete block systems.
Let $G\le {\mathop{\mathrm{Sym}}}(n)$ be a transitive permutation group (acting on $\Z_n$, say). A subset $B\subseteq\Z_n$ is a [*block for $G$*]{} if $g(B) = B$ or $g(B)\cap B = \emptyset$ for every $g\in G$. Clearly $\Z_n$ and its singleton subsets are always blocks for $G$, and are called [*trivial blocks*]{}. If $B$ is a block, then $g(B)$ is a block for every $g\in G$, and the set ${\cal B} = \{g(B):g\in G\}$ is called a [*complete block system for $G$*]{}, and we say that $G$ [*admits*]{} ${\cal B}$. A complete block system is [*nontrivial*]{} if its blocks are nontrivial. Observe that a complete block system is a partition of $\Z_n$, and any two blocks have the same size. If $G$ admits ${\cal B}$ as a complete block system, then each $g\in G$ induces a permutation of ${\cal B}$, which we denote by $g/{\cal B}$. We set $G/{\cal B} = \{g/{\cal B}:g\in G\}$. The kernel of the action of $G$ on ${\cal B}$, denoted by $\fix_G({\cal B})$, is then the subgroup of $G$ which fixes each block of ${\cal B}$ set-wise. That is, $\fix_G({\cal B}) = \{g\in G:g(B) = B{\rm\ for\ all\ }B\in{\cal B}\}$. For fixed $B\in{\cal B}$, we denote the set-wise stabilizer of $B$ in $G$ by $\Stab_G(B)$. That is $\Stab_G({\cal B}) = \{g\in G:g(B) = B\}$. Note that $\fix_G({\cal B}) = \cap_{B\in{\cal B}}\Stab_G(B)$. Finally, for $g\in\Stab_G(B)$, we denote by $g\vert_B$ the action induced by $g$ on $B\in{\cal B}$.
Note that Corollary \[coro1\], together with the fact that $\Z_2^3\times\Z_p$, $p\ge 11$, is a CI-group with respect to color ternary relational structures [@Dobson2010a], settles the question of which groups $\Z_2^3\times\Z_p$ are CI-groups with respect to color ternary relational structures except for $p = 5$. Our next goal is to show that $\Z_2^3\times\Z_5$ is a CI-group with respect to color ternary relational structures. From a computational point of view, the number of points is too large to enable a computer to determine the answer without some additional information. Lemma $6.1$ in [@Dobson2010a] is the only result that uses the hypothesis $p\geq 11$. For convenience, we report [@Dobson2010a Lemma $6.1$].
\[refine\] Let $p\geq 11$ be a prime and write $H=\Z_2^3\times \Z_p$. For every $\phi\in {\mathop{\mathrm{Sym}}}(H)$, there exists $\delta\in \la H_L,\phi^{-1}H_L\phi\ra$ such that $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra$ admits a complete block system consisting of $8$ blocks of size $p$.
In particular, to prove that $\Z_2^3\times \Z_5$ is a CI-group with respect to color ternary relational structures, it suffices to prove that Lemma \[refine\] holds true also for the prime $p=5$. We begin with some intermediate results which accidentally will also help us to prove that $\Z_2^3\times\Z_7$ is a CI-group with respect to color binary relational structures. (Here we denote by $\Alt(X)$ the alternating group on the set $X$ and by $\Alt(n)$ the alternating group on $\{1,\ldots,n\}$.)
\[dtlemma\] Let $P_1$ and $P_2$ be partitions of $\Z_n$ where each block in $P_1$ and $P_2$ has order $p\geq 2$. Then there exists $\phi\in \Alt(\Z_n)$ such that $\phi(P_1) = P_2$.
Let $P_1 = \{\Delta_1,\ldots,\Delta_{n/p}\}$ and $P_2 = \{\Omega_1,\ldots,\Omega_{n/p}\}$. As $\Alt(n)$ is $(n-2)$-transitive, there exists $\phi\in \Alt(n)$ such that $\phi(\Delta_i) = \Omega_i$, for $i\in\{1,\ldots, n/p - 1\}$. As both $P_1$ and $P_2$ are partitions, we see that $\phi(\Delta_{n/p}) = \Omega_{n/p}$ as well.
\[altlem\] Let $n = 8p$, $G = (\Z_2^3\times\Z_p)_L$ and $\delta\in {\mathop{\mathrm{Sym}}}(n)$. Suppose that $\la G,\delta^{-1}G\delta\ra$ admits a complete block system ${\cal C}$ with $p$ blocks of size $8$ such that $\Alt(C)\le\Stab_{\la G,\delta^{-1}G\delta\ra}(C)\vert_C$, where $C\in{\cal C}$. Then there exists $\gamma\in\la G,\delta^{-1}G\delta\ra$ such that $\la G,\gamma^{-1}\delta^{-1}G\delta\gamma\ra$ admits a complete block system ${\cal B}$ with $4p$ blocks of size $2$.
Clearly both $G$ and $\delta^{-1}G\delta$ are regular, and so both $\fix_G({\cal C})$ and $\fix_{\delta^{-1}G\delta}({\cal C})$ are semiregular of order $8$. As $\Alt(8)$ is simple and as $\fix_{\la G,\delta^{-1}G\delta\ra}({\cal C})\vert_C\tl\Stab_{\la G,\delta^{-1}G\delta\ra}({\cal C})\vert_C$, we have that $\Alt(C)\le\fix_{\la G,\delta^{-1}G\delta\ra}({\cal C})\vert_C$, for every $C\in{\cal C}$. Let $J\le\fix_G({\cal C})$ and $K\le\fix_{\delta^{-1}G\delta}({\cal C})$ be both of order $2$. Fix $C_0\in{\cal C}$, and let ${\cal O}_1,\ldots,{\cal O}_4$ be the orbits of $J\vert_{C_0}$, and ${\cal O}_1',\ldots,{\cal O}_4'$ be the orbits of $K\vert_{C_0}$. By Lemma \[dtlemma\], there exists $\gamma_0\in\fix_{\la G,\delta^{-1}G\delta\ra}({\cal C})$ such that $\gamma_0^{-1}({\cal O}_i') = {\cal O}_i$, for each $i\in\{1,\ldots,4\}$. Hence the orbits of $J\vert_{C_0}$ and $(\gamma_0^{-1}K\gamma_0)\vert_{C_0}$ are identical.
Recall that two transitive actions are equivalent if and only if the stabilizer of a point in one action is the same as the stabilizer of a point in the other [@DixonM1996 Lemma 1.6B]. Suppose now that the action of $\fix_{\la G,\delta^{-1}G\delta\ra}({\cal
C})$ on $C_0$ is equivalent to the action of $\fix_{\la G,\delta^{-1}G\delta\ra}({\cal C})$ on $C\in{\cal C}$. Let $\omega_J$ generate $J$ and let $\omega_K$ generate $K$. As the orbits of $J\vert_{C_0}$ and $(\gamma_0^{-1}K\gamma_0)\vert_{C_0}$ are identical and $|\omega_J|=|\omega_K|=2$, we see that $\omega_J\vert_{C_0} = (\gamma_0^{-1}\omega_K\gamma_0)\vert_{C_0}$. Hence $(\omega_J\gamma_0^{-1}\omega_K\gamma_0)\vert_{C_0} = 1$ and so $(\omega_J\gamma_0^{-1}\omega_K\gamma_0)\vert_{C} = 1$. Therefore the orbits of $J\vert_{C}$ and $(\gamma_0^{-1}K\gamma_0)\vert_{C}$ are identical.
Define an equivalence relation $\equiv$ on ${\cal C}$ by $C\equiv C'$ if and only if the action of $\fix_{\la G,\delta^{-1}G\delta\ra}({\cal C})$ on $C$ is equivalent to the action of $\fix_{\la G,\delta^{-1}G\delta\ra}({\cal C})$ on $C'$. Since $\Alt(8)$ has only one permutation representation of degree $8$ [@Cameron1981 Theorem 5.3], we obtain that $C\not\equiv C'$ if and only if the action of $\fix_{\la G,\delta^{-1}G\delta\ra}({\cal C})\vert_{C\cup C'}$ on $C'$ is not faithful. Thus $C\not \equiv C'$ if and only if there exists $\alpha\in\fix_{\la G,\delta^{-1}G\delta\ra}({\cal C})$ such that $\alpha\vert_C =
1$ but $\alpha\vert_{C'}\not = 1$.
Let $E_0$ be the $\equiv$-equivalence class containing $C_0$ and set $$L_1 = \{\alpha\in\fix_{\la G,\delta^{-1}G\delta\ra}({\cal C}):\alpha\vert_{C} = 1 \textrm{ for every }C\in E_0\}.$$ Let $C_1$ be in ${\cal C}$ with $C_1\not\equiv C_0$ and let $E_1$ be the $\equiv$-equivalence class containing $C_1$. Then there exists $\omega\in\fix_{\la G,\delta^{-1}G\delta\ra}({\cal C})$ with $\omega\vert_{C_0} = 1$ and $\omega\vert_{C_1}\not = 1$. From the definition of $\equiv$, we see that $\omega\vert_C = 1$, for every $C\in E_0$, that is, $\omega\in L_1$ and $L_1\not = 1$. As $L_1\tl\fix_{\la G,\delta^{-1}G\delta\ra}({\cal C})$ and $\Alt(8)$ is simple, we conclude that $\Alt(C_1)\le L_1\vert_{C_1}$.
As both $J$ and $K$ are semiregular of order $2$, the groups $J\vert_{C_1}$ and $(\gamma_0^{-1}K\gamma_0)\vert_{C_1}$ are generated by even permutations. So $J\vert_{C_1}\le L_1\vert_{C_1}$ and $(\gamma_0^{-1} K\gamma_0)\vert_{C_1}\le L_1\vert_{C_1}$. By Lemma \[dtlemma\], there exists $\gamma_1\in L_1$ such that the orbits of $J\vert_{C_1}$ and $(\gamma_1^{-1}\gamma_0^{-1}K\gamma_0\gamma_1)\vert_{C_1}$ are identical. In particular, the orbits of $J\vert_{C}$ and $(\gamma_1^{-1}\gamma_0^{-1}K\gamma_0\gamma_1)\vert_{C}$ are identical, for every $C\in E_1$. Furthermore, as $L_1\vert_C = 1$ for every $C\in E_0$, we have that the orbits $J\vert_C$ and $(\gamma_1^{-1}\gamma_0^{-1}K\gamma_0\gamma_1)\vert_C$ are identical for every $C\in E_0\cup E_1$.
Applying inductively the previous two paragraphs to the various $\equiv$-equivalence classes, we find $\gamma\in\la G,\delta^{-1}G\delta\ra$ such that the orbits of $J$ and $(\gamma^{-1}\delta^{-1}K\delta\gamma)$ are identical. Since $|J|=2$, we get $J=\gamma^{-1}\delta^{-1}K\delta\gamma$. As $J\tl G$ and $\gamma^{-1}\delta^{-1}K\delta\gamma\tl\gamma^{-1}\delta^{-1}G\delta\gamma$, we obtain $J\tl\la G,\gamma^{-1}\delta^{-1}G\delta\gamma\ra$ and the orbits of $J$ form a complete block system for $\la G,\gamma^{-1}\delta^{-1}G\delta\gamma\ra$ of $4p$ blocks of size $2$.
The proof of the following result is analogous to the proof of [@Dobson2010a Lemma 6.1].
\[pbig\] Let $H$ be an abelian group of order $\ell p$, where $\ell < p$ and $p$ is prime. Let $\phi\in {\mathop{\mathrm{Sym}}}(H)$. Then there exists $\delta\in \la H_L,\phi^{-1}H_L\phi\ra$ such that $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra$ admits a complete block system with blocks of size $p$.
\[57tool\] Let $p \ge 5$, $H = \Z_2^3\times\Z_p$, and $\phi\in {\mathop{\mathrm{Sym}}}(H)$. Then either there exists $\delta\in \la H_L,\phi^{-1}H_L\phi\ra$ such that $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra$ admits a complete block system with blocks of size $p$ or $ \la H_L,\phi^{-1}H_L\phi\ra$ admits a complete block system ${\cal B}$ with blocks of size $8$ and $\fix_K({\cal B})\vert_B$ is isomorphic to a primitive subgroup of $\AGL(3,2)$, for $B\in {\cal B}$.
Set $K=\la H_L,\phi^{-1}H_L\phi\ra$. As $H$ has a cyclic Sylow $p$-subgroup, we have by [@DixonM1996 Theorem 3.5A] that $K$ is doubly-transitive or imprimitive. If $K$ is doubly-transitive, then by [@Li2003 Theorem 1.1] we have that $\Alt(H)\le K$. Now Lemma \[dtlemma\] reduces this case to the imprimitive case. Thus we may assume that $K$ is imprimitive with a complete block system ${\cal C}$.
Suppose that the blocks of ${\cal C}$ have size $\ell p$, where $\ell = 2$ or $4$. Notice that $p > \ell$. As $H$ is abelian, $\fix_{H_L}({\cal C})$ is a semiregular group of order $\ell p$ and $\fix_{\phi^{-1}H_L\phi}({\cal C})$ is also a semiregular group of order $\ell p$. Then, for $C\in{\cal C}$, both $\fix_{H_L}({\cal C})\vert_C$ and $\fix_{\phi^{-1}H_L\phi}({\cal C})\vert_C$ are regular groups of order $\ell p$. Let $C\in{\cal C}$. By Lemma \[pbig\], there exists $\delta\in \la\fix_{H_L}({\cal C}),\fix_{\phi^{-1}H_L\phi}({\cal C})\ra$ such that $\la\fix_{H_L}({\cal C}),\fix_{\delta^{-1}\phi^{-1}H_L\phi\delta}({\cal C})\ra\vert_{C}$ admits a complete block system ${\cal B}_{C}$ consisting of blocks of size $p$. Let $C'\in{\cal C}$ with $C'\not = C$. Arguing as above, there exists $\delta'\in \la\fix_{H_L}({\cal C}),\fix_{\delta^{-1}\phi^{-1}H_L\phi\delta}({\cal C})\ra$ such that $\la\fix_{H_L}({\cal C}),\fix_{\delta'^{-1}\delta^{-1}\phi^{-1}H_L\phi\delta\delta'}({\cal C})\ra\vert_{C'}$ admits a complete block system ${\cal B}_{C'}$ consisting of blocks of size $p$. Note that $\delta'\vert_{C}\in\la\fix_{H_L}({\cal C}),\fix_{\delta^{-1}\phi^{-1}H_L\phi\delta}({\cal C})\ra\vert_{C}$ and so $\la\fix_{H_L}({\cal C}),\fix_{\delta'^{-1}\delta^{-1}\phi^{-1}H_L\phi\delta\delta'}({\cal C})\ra\vert_{C}$ admits ${\cal B}_{C'}$ as a complete block system. Repeating this argument for every block in ${\cal C}$, we find $\delta\in \la\fix_{H_L}({\cal C}),\fix_{\phi^{-1}H_L\phi}({\cal C})\ra$ such that $\la\fix_{H_L}({\cal C}),\fix_{\delta^{-1}\phi^{-1}H_L\phi\delta}({\cal C})\ra\vert_{C}$ admits a complete block system ${\cal B}_C$ consisting of blocks of size $p$. Let ${\cal B} = \cup_{C}{\cal B}_C$. We claim that ${\cal B}$ is a complete block system for $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra$, which will complete the argument in this case.
Let $\rho\in H_L$ be of order $p$. By construction, $\rho\in\fix_{H_L}({\cal B})$. As $H$ is abelian, $\fix_{H_L}({\cal C})\vert_{C}$ is abelian, for every $C\in{\cal C}$. Then ${\cal B}_C$ is formed by the orbits of some subgroup of $\fix_{H_L}({\cal C})\vert_{C}$ of order $p$, and as $\la\rho\ra\vert_{C}$ is the unique subgroup of $\fix_{H_L}({\cal C})\vert_{C}$ of order $p$, we obtain that ${\cal B}_C$ is formed by the orbits of $\la\rho\ra\vert_{C}$. Then ${\cal B}$ is formed by the orbits of $\la\rho\ra\tl H_L$ and ${\cal B}$ is a complete block system for $H_L$. An analogous argument for $\delta^{-1}\phi^{-1}\la\rho\ra\phi\delta$ gives that ${\cal B}$ is a complete block system for $\delta^{-1}\phi^{-1}H_L\phi\delta$. Then ${\cal B}$ is a complete block system for $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra$ with blocks of size $p$, as required.
Suppose that the blocks of ${\cal C}$ have size $8$. Now $H_L/{\cal C}$ and $\phi^{-1}H_L\phi/{\cal C}$ are cyclic of order $p$, and as $\Z_p$ is a CI-group [@Babai1977 Theorem 2.3], replacing $\phi^{-1}H_L\phi$ by a suitable conjugate, we may assume that $\la H_L,\phi^{-1}H_L\phi\ra/{\cal C} = H_L/{\cal C}$. Then $K/{\cal C}$ is regular and $\Stab_K(C) = \fix_K({\cal C})$, for every $C\in{\cal C}$.
Suppose that $\Stab_K({\cal C})\vert_C$ is imprimitive, for $C\in{\cal C}$. By [@DixonM1996 Exercise 1.5.10], the group $K$ admits a complete block system ${\cal D}$ with blocks of size $2$ or $4$. Then $K/{\cal D}$ has degree $2p$ or $4p$ and, by Lemma \[pbig\], there exists $\delta\in K$ such that $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra/{\cal D}$ admits a complete block system ${\cal B}'$ with blocks of size $p$. In particular, ${\cal B}'$ induces a complete block system ${\cal B}''$ for $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra$ with blocks of size $2p$ or $4p$, and we conclude by the case previously considered applied with $\mathcal{C}={\cal B}''$. Suppose that $\Stab_K({\cal C})\vert_C$ is primitive, for $C\in{\cal C}$. If $\Stab_K({\cal C})\vert_C\ge \Alt(C)$, then the result follows by Lemma \[altlem\], and so we may assume this is not the case. By [@Li2003 Theorem 1.1], we see that $\Stab_K({\cal C})\vert_C\le\AGL(3,2)$. The result now follows with ${\cal B} = {\cal C}$.
\[5tool\] Let $H = \Z_2^3\times\Z_5$ and $\phi\in {\mathop{\mathrm{Sym}}}(H)$. Then there exists $\delta\in \la H_L,\phi^{-1}H_L\phi\ra$ such that $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra$ admits a complete block system with blocks of size $5$.
Set $K=\la H_L,\phi^{-1}H_L\phi\ra$. By Lemma \[57tool\], we may assume that $K$ admits a complete block system ${\cal B}$ with blocks of size $8$ and with $\Stab_K({\cal B})\vert_B\le\AGL(3,2)$, for $B\in {\cal B}$. As $\vert\AGL(3,2)\vert = 8\cdot 7\cdot 6\cdot 4$, we see that a Sylow $5$-subgroup of $K$ has order $5$. Let $\la\rho\ra$ be the subgroup of $H_L$ of order $5$. So $\la\rho\ra$ is a Sylow $5$-subgroup of $K$. Then $\phi^{-1}\la\rho\ra\phi$ is also a Sylow $5$-subgroup of $K$, and by a Sylow theorem there exists $\delta\in K$ such that $\delta^{-1}\phi^{-1}\la\rho\ra\phi\delta = \la\rho\ra$. We then have that $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra$ has a unique Sylow $5$-subgroup, whose orbits form the required complete block system ${\cal B}$.
We are finally ready to prove Theorem A.
If $p$ is odd, then the paragraph following the proof of Corollary \[coro1\] shows that it suffices to prove that Lemma \[refine\] holds for the prime $p=5$. This is done in Corollary \[5tool\]. If $p=2$, then the result can be verified using GAP or Magma.
Proof of Corollaries A and B
============================
Before proceeding to our next result we will need the following definitions.
\[defin1\][Let $G$ be a permutation group on $\Omega$ and $k\geq 1$. A permutation $\sigma\in {\mathop{\mathrm{Sym}}}(\Omega)$ lies in the $k$-closure $G^{(k)}$ of $G$ if for every $k$-tuple $t\in\Omega^k$ there exists $g_t\in G$ (depending on $t$) such that $\sigma(t) = g_t(t)$. We say that $G$ is $k$-closed if the permutations lying in the $k$-closure of $G$ are the elements of $G$, that is, $G^{(k)}=G$. The group $G$ is $k$-closed if and only if there exists a color $k$-ary relational structure $X$ on $\Omega$ with $G=\Aut(X)$, see [@Wielandt1969].]{}
[For color digraphs $\Gamma_1$ and $\Gamma_2$, we define the [*wreath product of $\Gamma_1$ and $\Gamma_2$*]{}, denoted $\Gamma_1\wr\Gamma_2$, to be the color digraph with vertex set $V(\Gamma_1)\times V(\Gamma_2)$ and edge set $E_1\cup E_2$, where $E_1 =\{((x_1,y_1),(x_1,y_2)):x_1\in V(\Gamma_1), (y_1,y_2)\in E(\Gamma_2)\}$ and the edge $((x_1,y_1),(x_1,y_2))\in E_1$ is colored with the same color as $(y_1,y_2)$ in $\Gamma_2$, and $E_2 = \{((x_1,y_1),(x_2,y_2)):(x_1,x_2)\in E(\Gamma_1), y_1,y_2\in V(\Gamma_2)\}$ and the edge $((x_1,y_1),(x_2,y_2))
\in E_2$ is colored with the same color as $(x_1,x_2)$ in $\Gamma_1$. ]{}
[For permutation groups $G\le {\mathop{\mathrm{Sym}}}(X)$ and $H\le {\mathop{\mathrm{Sym}}}(Y)$, we define the [*wreath product of $G$ and $H$*]{}, denoted $G\wr H$, to be the permutation group $G\wr H\le {\mathop{\mathrm{Sym}}}(X\times Y)$ consisting of all permutations of the form $(x,y)\mapsto(g(x),h_x(y))$, $g\in G$, $h_x\in H$.]{}
The following very useful result (see [@Babai1977 Lemma 3.1]) characterizes CI-groups with respect to a class of combinatorial objects.
\[CItool\] Let $H$ be a group and let ${\cal K}$ be a class of combinatorial objects. The following are equivalent.
1. $H$ is a CI-group with respect to ${\cal K}$,
2. whenever $X$ is a Cayley object of $H$ in ${\cal K}$ and $\phi\in {\mathop{\mathrm{Sym}}}(H)$ such that $\phi^{-1}H_L\phi\le\Aut(X)$, then $H_L$ and $\phi^{-1}H_L\phi$ are conjugate in $\Aut(X)$.
From Theorem A, it suffices to show that $\Z_2^3\times\Z_3$ and $\Z_2^3\times\Z_7$ are CI-groups with respect to color binary relational structures. As the transitive permutation groups of degree $24$ are readily available in GAP or Magma, it can be shown using a computer that $\Z_2^3\times \Z_3$ is a CI-group with respect to color binary relational structures. It remains to consider $H=\Z_2^3\times \Z_7$.
Fix $\phi\in {\mathop{\mathrm{Sym}}}(H)$ and set $K=\la H_L,\phi^{-1}H_L\phi\ra$. Assume that there exists $\delta\in K$ such that $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra$ admits a complete block system with blocks of size $7$. Now, it follows by [@Dobson2010a] (see the two paragraphs following the proof of Corollary \[coro1\]) that $H_L$ and $\delta^{-1}\phi^{-1}H_L\phi\delta$ are conjugate in $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra^{(3)}$. Since $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra^{(3)}\le \la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra^{(2)}$, the corollary follows from Lemma \[CItool\] (and from Definition \[defin1\]).
Assume that there exists no $\delta\in K$ such that $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra$ admits a complete block system with blocks of size $7$. By Lemma \[57tool\], the group $K$ admits a complete block system ${\cal C}$ with blocks of size $8$ and $\fix_K({\cal C})\vert_C$ is isomorphic to a primitive subgroup of $\AGL(3,2)$, for $C\in {\cal C}$. Suppose that $7$ and $\vert\fix_K({\cal C})\vert$ are relatively prime. So, a Sylow $7$-subgroup of $K$ has order $7$. We are now in the position to apply the argument in the proof of Corollary \[5tool\]. Let $\la\rho\ra$ be the subgroup of $H_L$ of order $7$. Then $\phi^{-1}\la\rho\ra\phi$ is a Sylow $7$-subgroup of $K$, and by a Sylow theorem there exists $\delta\in K$ such that $\delta^{-1}\phi^{-1}\la\rho\ra\phi\delta = \la\rho\ra$. We then have that $\la H_L,\delta^{-1}\phi^{-1}H_L\phi\delta\ra$ has a unique Sylow $7$-subgroup, whose orbits form a complete block system with blocks of size $7$, contradicting our hypothesis on $K$. We thus assume that $7$ divides $|\fix_K({\cal C})|$ and so $\fix_K({\cal C})$ acts doubly-transitively on $C$, for $C\in {\cal C}$.
Fix $C\in {\cal C}$ and let $L$ be the point-wise stabilizer of $C$ in $\fix_K({\cal C})$. Assume that $L\not = 1$. Now, we compute $K^{(2)}$ and we deduce that $H_L$ and $\phi^{-1}H_L\phi$ are conjugate in $K^{(2)}$, from which the corollary will follow from Lemma \[CItool\]. As $L\tl \fix_K({\cal C})$, we have $L\vert_{C'}\tl \fix_K({\cal C})\vert_{C'}$, for every $C'\in{\cal C}$. As a nontrivial normal subgroup of a primitive group is transitive [@Wielandt1964 Theorem 8.8], either $L\vert_{C'}$ is transitive or $L\vert_{C'} = 1$. Let $\Gamma$ be a Cayley color digraph on $H$ with $K^{(2)}=\Aut(\Gamma)$. Let ${\cal C} = \{C_i:i\in\Z_7\}$ where $C_i = \{(x_1,x_2,x_3,i):x_1,x_2,x_3\in\Z_2\}$, and assume without loss of generality that $C = C_0$. Suppose that there is an edge of color $\kappa$ from some vertex of $C_i$ to some vertex of $C_j$, where $i\not = j$. Then there is an edge of color $\kappa$ from some vertex of $C_0$ to some vertex of $C_{j-i}$. Additionally, $j - i$ generates $\Z_7$, so there is a smallest integer $s$ such that $L\vert_{C_{s(j-i)}} = 1$ while $L\vert_{C_{(s+1)(j-i)}}$ is transitive. As there is an edge of color $\kappa$ from some vertex of $C_{s(j-i)}$ to some vertex of $C_{(s+1)(j-i)}$, we conclude that there is an edge of color $\kappa$ from every vertex of $C_{s(j-i)}$ to every vertex of $C_{(s+1)(j-i)}$. This implies that there is an edge of color $\kappa$ from every vertex of $C_i$ to every vertex of $C_j$, and then $\Gamma$ is the wreath product of a Cayley color digraph $\Gamma_1$ on $\Z_7$ and a Cayley color digraph $\Gamma_2$ on $\Z_2^3$. Since $\fix_K({\cal C})$ is doubly-transitive on $C$, we have $\Aut(\Gamma_2)\cong {\mathop{\mathrm{Sym}}}(8)$. Therefore $K^{(2)}=\Aut(\Gamma_1)\wr\Aut(\Gamma_2)\cong \Aut(\Gamma_1)\wr{\mathop{\mathrm{Sym}}}(8)$. By [@DobsonM2009 Corollary 6.8] and Lemma \[CItool\] $H_L$ and $\phi^{-1}H_L\phi$ are conjugate in $K^{(2)}$. We henceforth assume that $L = 1$, that is, $\fix_K({\cal C})$ acts faithfully on $C$, for each $C\in {\cal C}$.
Define an equivalence relation on $H$ by $h\equiv k$ if and only if $\Stab_{\fix_K({\cal C})}(h) = \Stab_{\fix_K({\cal C})}(k)$. The equivalence classes of $\equiv$ form a complete block system ${\cal D}$ for $K$. As $\fix_K({\cal C})\vert_C$ is primitive and not regular, each equivalence class of $\equiv$ contains at most one element from each block of ${\cal C}$. We conclude that ${\cal D}$ either consists of $8$ blocks of size $7$ or each block is a singleton. Since we are assuming that $K$ has no block system with blocks of size $7$, we have that each block of ${\cal D}$ is a singleton.
Fix $C$ and $D$ in ${\cal C}$ with $C\neq D$ and $h\in C$. Now, $\Stab_{\fix_K({\cal C})}(h)$ is isomorphic to a subgroup of $\GL(3,2)$ and acts with no fixed points on $D$. From [@DixonM1996 Appendix B]), we see that $\AGL(3,2)$ is the only doubly-transitive permutation group of degree $8$ whose point stabilizer admits a fixed-point-free action of degree $8$. Therefore $\fix_K({\cal C})\cong \AGL(3,2)$. Additionally, $\Stab_{\fix_K({\cal C})}(h)\vert_D$ is transitive on $D$.
Suppose that $\Gamma$ is a color digraph with $K^{(2)} = \Aut(\Gamma)$ and suppose that there is an edge of color $\kappa$ from $h$ to $\ell\in E$, with $E\in{\cal C}$ and $E\neq D$. Then $\Stab_{\fix_K({\cal C})}(h)\vert_E$ is transitive, and so there is an edge of color $\kappa$ from $h$ to every vertex of $E$. As $\fix_K({\cal C})$ is transitive on both $C$ and $E$, we see that there is an edge of color $\kappa$ from every vertex of $C$ to every vertex of $D$. We conclude that $\Gamma$ is a wreath product of two color digraphs $\Gamma_1$ and $\Gamma_2$, where $\Gamma_1$ is a Cayley color digraph on $\Z_7$ and $\Gamma_2$ is either complete or the complement of a complete graph, and $K^{(2)} = \Aut(\Gamma_1)\wr {\mathop{\mathrm{Sym}}}(8)$. The result then follows by the same arguments as above.
From Corollary \[coro1\] and Theorem A, it suffices to show that $\Z_2^2\times \Z_7$ is a CI-group with respect to color ternary relational structures. As the transitive permutation groups of degree $28$ are readily available in GAP or Magma, it can be shown using a computer that $\Z_2^2\times \Z_7$ is a CI-group with respect to color ternary relational structures. (We note that a detailed analysis similar to the proof of Corollary A for the group $\Z_2^3\times \Z_7$ also gives a proof of this theorem.)
Concluding remarks
==================
In the rest of this paper, we discuss the relevance of Theorem A to the study of CI-groups with respect to ternary relational structures. Using the software packages [@Magma] and [@GAP], we have determined that $\Z_2^5$ is not a CI-group with respect to ternary relational structures. Here we report an example witnessing this fact: the group $G$ has order $2048$, $V$ and $W$ are two [*nonconjugate*]{} elementary abelian regular subgroups of $G$, and $X=(\{1,\ldots,32\},E)$ is a ternary relational structure with $G=\Aut(X)$. $$\begin{aligned}
V&=&
\langle\mbox{\footnotesize{(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)}},\\
&&\langle\mbox{\footnotesize{(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,
15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)}},\\
&&\langle\mbox{\footnotesize{ (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)}},\\
&&\langle\mbox{\footnotesize{ (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)}},\\
&&\langle\mbox{\footnotesize{ (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)}}\rangle,\\
W&=&\langle\mbox{\footnotesize{(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)}},\\
&&\mbox{\footnotesize{ (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,
15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)}},\\
&&\mbox{\footnotesize{ (1,5)(2,6)(3,7)(4,8)(9,14)(10,13)(11,16)(12,15)(17,22)(18,21)(19,24)(20,
23)(25,29)(26,30)(27,31)(28,32)}},\\
&&\mbox{\footnotesize{ (1,9)(2,10)(3,11)(4,12)(5,14)(6,13)(7,16)(8,15)(17,27)(18,28)(19,25)(20,26)(21,32)(22,31)(23,30)(24,29)}},\\
&&\mbox{\footnotesize{ (1,17)(2,18)(3,20)(4,19)(5,22)(6,21)(7,23)(8,24)(9,27)(10,28)(11,26)(12,25)(13,32)(14,31)(15,29)(16,30)}}\rangle,\\
G&=&\langle V,W,\mbox{\footnotesize{(25,26)(27,28)(29,30)(31,32),(1,11)(2,12)(3,9)(4,10)(5,13)(6,
14)(7,15)(8,16)(17,19)(18,20)(25,27)(26,28)}}\rangle,\\
E&=&\{g((1,3,9)),g((1,5,25)): g\in G\}.\end{aligned}$$
[For a cyclic group $M=\langle g\rangle$ of order $m$ and a cyclic group $\langle z\rangle$ of order $2^d$, $d\geq 1$, we denote by $D(m,2^d)$ the group $\langle z\rangle\ltimes M$ with $g^z=g^{-1}$.]{}
Combining Theorem A with [@Dobson2003 Theorem 9], [@Dobson2003 Lemma 6], the construction given in [@Spiga2008] and the previous paragraph, we have the following result which lists every group that can be a CI-group with respect to ternary relational structures (although not every group on the list needs to be a CI-group with respect to ternary relational structures).
\[new\] If $G$ is a CI-group with respect to ternary relational structures, then all Sylow subgroups of $G$ are of prime order or isomorphic to $\Z_4$, $\Z_2^d$, $1\le d\le 4$, or $Q_8$. Moreover, $G = U\times
V$, where $\gcd(\vert U\vert, \vert V\vert) = 1$, $U$ is cyclic of order $n$, with $\gcd(n,\varphi(n)) = 1$, and $V$ is one of the following:
1. $\Z_2^d$, $1\le d\le 4$, $D(m,2)$, or $D(m,4)$, where $m$ is odd and $\gcd(nm,\varphi(nm)) = 1$,
2. $\Z_4$, $Q_8$.
Furthermore,
1. if $V = \Z_4$, $Q_8$, or $D(m,4)$ and $p\mid
n$ is prime, then $4\mathrel{\not|}(p - 1)$,
2. if $V = \Z_2^d$, $d\ge 2$, or $Q_8$, then $3\not\vert\ n$,
3. if $V = \Z_2^d$, $d\ge 3$, then $7\not\vert\ n$,
4. if $V = \Z_2^4$, then $5\not\vert\ n$.
[10]{}
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|
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abstract: 'In this paper we study existence of ground state solution to the following problem $$(- \Delta)^{\alpha}u = g(u) \ \ \mbox{in} \ \ \mathbb{R}^{N}, \ \ u \in H^{\alpha}(\mathbb R^N)$$ where $(-\Delta)^{\alpha}$ is the fractional Laplacian, $\alpha\in (0,1)$. We treat both cases $N\geq2$ and $N=1$ with $\alpha=1/2$. The function $g$ is a general nonlinearity of Berestycki-Lions type which is allowed to have critical growth: polynomial in case $N\geq2$, exponential if $N=1$.'
address:
- 'Departamento de Matemática Universidade Federal de Campina Grande, 58429-970, Campina Grande - PB - Brazil'
- 'Faculdade de Matemática Universidade Federal do Pará 66075-110, Belém - PA, Brazil'
- 'Departamento de Matemática Instituto de Matemática e Estatística Universidade de São Paulo Rua do Matão 1010, 05508-090 São Paulo, SP, Brazil '
author:
- 'Claudianor O. Alves'
- 'Giovany M. Figueiredo'
- Gaetano Siciliano
title: Ground state solutions for fractional scalar field equations under a general critical nonlinearity
---
10000
[^1]
Introduction
============
In the present paper, we are interesting in the existence of ground state solution for a class of nonlocal problem of the following type $$\tag{P}\label{eq:P}
(- \Delta)^{\alpha}u = g(u), \quad \mbox{in} \quad \mathbb{R}^N$$ where $N \geq 1$, $\alpha \in (0,1),$ $(- \Delta)^{\alpha}$ denotes the fractional Laplacian operator and $g$ is a $C^{1}-$function verifying some conditions which will be mentioned later on.
The main motivation for this paper comes from the papers Berestycki and Lions [@BL] and Berestycki, Gallouet and Kavian [@BGK] which have studied the existence of solution for in the local case $\alpha =1$, that is, for a class of elliptic equations like $$\label{LE1*}
- \Delta u = g(u), \quad \mbox{in} \quad \mathbb{R}^N,$$ where $N \geq 2$, $\Delta$ denotes the Laplacian operator and $g$ is a continuous function verifying some conditions. In [@BL], Berestycki and Lions have assumed $N \geq 3$ and the following conditions on $g$: $$- \infty < \liminf_{s \to 0^+}\frac{g(s)}{s} \leq \limsup_{s \to 0^+}\frac{g(s)}{s}\leq -m<0, $$ $$\limsup_{s \to 0^+}\frac{g(s)}{s^{2^{*}-1}}\leq 0, $$ $$\mbox{there is} \quad \xi>0 \, \, \mbox{such that} \,\, G(\xi)>0, $$ where $G(s)=\int_{0}^{s}g(t)\,dt$.
In [@BGK], Berestycki, Gallouet and Kavian have studied the case where $N=2$ and the nonlinearity $g$ possesses an exponential growth of the type $$\limsup_{s \to 0^+}\frac{g(s)}{e^{\beta s^2}}=0, \quad \forall \beta >0.$$
In the two papers above mentioned, the authors have used the variational method to prove the existence of solution for . The main idea is to solve the minimization problem $$\min \left\{\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^{2} \,dx \,:\, \int_{\mathbb{R}^N}G(u)\,dx=1 \right\}$$ and $$\min \left\{\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^{2}\,dx \,:\, \int_{\mathbb{R}^N}G(u)\,dx=0 \right\}$$ for $N \geq 3$ and $N=2$ respectively. After that, the authors showed that the minimizer functions of the above problem are in fact ground state solutions of (\[LE1\*\]). By a ground state solution, we mean a solution $u \in H^{1}(\mathbb{R}^N)$ which satisfies $$E(u) \leq E(v) \quad \mbox{for all nontrival solution} \ v \ \text{of} \ (\ref{LE1*}),$$ where $E:H^{1}(\mathbb{R}^N) \to \mathbb{R}$ is the energy functional associated to given by $$E(u)=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^{2}\,dx - \int_{\mathbb{R}^N}G(u)\,dx.$$
After, Jeanjean and Tanaka in [@JJTan] showed that the mountain pass level of $E$ is a critical level and it is indeed the lowest critical level.
In the above mentioned papers, the nonlinearity does not have critical growth. Motivated by this fact, Alves, Montenegro and Souto in [@AlvesSoutoMontenegro] have studied the existence of ground state solution for by supposing that $g(s)=f(s)-s$ and that $f$ may have critical growth, more precisely, the following condition were considered: $$\lim_{s \to 0}\frac{f(s)}{s}=0$$ $$\limsup_{s \to +\infty}\frac{f(s)}{s^{2^{*}-1}}\leq 1, \,\, \mbox{if} \,\, N \geq 3$$ $$\lim_{s \to +\infty}\frac{f(s)}{e^{\beta s^{2}}}=0 \,\, (\beta < \beta_0) \,\, \mbox{if} \,\, \beta > \beta_0 \,\, ( \beta < \beta_0) \,\, \mbox{when} \,\, N=2$$ $$H(s)=f(s)s-2F(s) \geq 0 \quad \forall s > 0 \quad \mbox{where} \quad F(s)=\int_{0}^{s}f(t)\,dt,$$ there is $\tau >0$ and $q \in (2,2^{*})$ if $N \geq 3$ and $q \in (2, +\infty)$ if $N=2$ such that $$f(s) \geq \tau s^{q-1}, \quad \forall s \geq 0.$$
By using the variational method, the authors in [@AlvesSoutoMontenegro] give a unified approach in order to deal with subcritical and critical case. However, we would like to point out that the Concentration Compactness Principle of Lions [@lionsI] was crucial for the case $N \geq 3$. For the case $N=2$, as in the previous references, a Trudinger-Moser inequality due to Cao [@Cao] was the main tool used. A similar study was made for the critical case and $N \geq 3$ in Zhang and Zou [@ZZ].
After a review bibliographic, we have observed that there is no a version of the paper [@AlvesSoutoMontenegro] for the fractional Laplacian operator. Motivated by this fact, we have decide to study this class of problem. However, we would like point out that some estimates made in [@AlvesSoutoMontenegro] are not immediate for fractional Laplacian operator. For example, there is some restriction to use Concentration Compactness Principle of Lions [@lionsI] as mentioned in Palatucci and Pisante [@PalatucciPisante Theorem 1.5] for the dimension $ N\geq 2$ and $\alpha \in (0,1)$. To overcome this difficulty, we use a new approach which do not use the Concentration Compactness Principle of [@PalatucciPisante]. For the dimension $N=1$ and $\alpha =1/2$, we use a Trudinger-Moser inequality due to Ozawa [@Ozawa] which also permits to apply variational methods in this case. Here, it is very important to mention that Zhan, do Ó and Squassina [@ZOS Theorem 4.1] studied the existence of ground state solution for for $N \geq 2$, by supposing that $g$ satisfies $$\lim_{s \to +\infty}\frac{g(s)}{s^{2^{*}_\alpha -1}}=b>0.$$ These condition is not assumed in our paper, and so, our results complete the studied made in that paper.
Before stating our main results, we must fix some notations. We will look for weak solutions of hence the natural setting involves the fractional Sobolev spaces $H^\alpha(\mathbb R^N)$ defined as $$H^\alpha(\mathbb R^N)=\big\{u\in L^2(\mathbb R^N):\,
(-\Delta)^{\alpha/2}u \in L^2(\mathbb R^N)\big\}$$ endowed with scalar product and (squared) norm given by $$(u,v)= \int_{\mathbb R^{N}} (-\Delta)^{\alpha/2}u (-\Delta)^{\alpha/2}v \, dx+ \int_{\mathbb R^{N}} uv \, dx,
\qquad
\|u\|^2=|(-\Delta)^{\alpha/2}u |_2^2+ |u|_2^2.$$ It is well known that $H^\alpha(\mathbb R^N)$ is a Hilbert space with the above scalar product. We are denoting with $|u|_{p} = (\int_{\mathbb R^{N}}|u|^{p}dx)^{1/p}$ the $L^{p}-$norm of $u$, and by $(-\Delta)^\alpha$ the fractional Laplacian, which is the pseudodifferential operator defined via the Fourier transform of the following way $$\mathcal F((-\Delta)^{\alpha}u)=|\cdot|^{2\alpha}\mathcal Fu.$$ It is known that $H^{\alpha}(\mathbb R^{N})$ has continuos embedding into $L^{q}(\mathbb R^{N})$ for suitable $q$ depending on $N$: we will denote by $C_{q}>0$ the embedding constant.
It is useful to introduce also the homogeneous fractional Sobolev space $$\mathcal{D}^{\alpha, 2}({\mathbb R}^N) = \Big\{u \in L^{2^{*}_{\alpha}}({\mathbb R}^N): \ \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u |^{2} \, dx < \infty \Big\}$$ where hereafter $2^{*}_{\alpha}=\frac{2N}{N-2 \alpha}$ for $N \geq 2$. It is well known that the following inequality holds $$\label{definicaoS}
S \, \left( \int_{{\mathbb R}^N} |u|^{2^{*}_{\alpha}} \, dx \right)^{2/2^{*}_{\alpha}} \leq \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u |^{2} \, dx \quad\hbox{for all $u \in \mathcal{D}^{\alpha, 2}({\mathbb R}^N)$}$$ for some positive $S>0$. For these facts and the relation between the fractional Laplacian and the fractional Sobolev space $H^{\alpha}(\mathbb R^{N})$, we refer the reader to classical books on Sobolev space, and to the monograph [@DPV].
We will study by variational methods: its solutions will be found as critical points of a $C^{1}$ functional $I:H^{\alpha}(\mathbb R^{N}) \to \mathbb R$. Actually our results concern the existence of [*ground state solutions*]{}, that is a solution $u \in H^\alpha({\mathbb R}^N)$ such that $I(u) \leq I(v)$ for every nontrivial solution $v \in H^\alpha({\mathbb R}^N)$ of . In view of this, we make the following assumptions on the nonlinearity $f$. More precisely we assume that $f:\mathbb R\to \mathbb R$ is a $C^{1}$-function satisfying
1. \[f\_[1]{}\] $\displaystyle\lim_{s \rightarrow 0^+} f(s)/ s =0$;
2. \[f\_[2]{}\] $\displaystyle \limsup_{s \rightarrow + \infty} f(s) / s^{2^{*}_\alpha -1} \leq 1$;
3. \[f\_[3]{}\] $ f(s) s - 2F(s) \geq 0$ for $s > 0$, where $F(s)= \int_0^s f( t) \, dt$;
4. \[f\_[4]{}\] $f(s) \geq \tau s^{q-1}$, $s \in {\mathbb R}$ with $s \geq 0$, where
- If $N\geq2$, we assume $q \in (2, 2^{*}_{\alpha})$ and $$\tau>\tau^{*}:=\left[ 2^{(2\alpha-N)/2\alpha} S^{-N/2\alpha} \frac N\alpha \left(\frac{2N}{N-2\alpha}\right)^{(N-2\alpha)/2\alpha}\right]^{(q-2)/2}
\left( \frac{q-2}{2q} \right)^{(q-2)/2} C_q^{q/2},$$
- If $N=1$, we assume $q>2$ and $$\tau>\tau^* = \left( \frac{q-2}{q} \right)^{(q-2)/2} C_q^{q/2};$$
5. \[f\_[5]{}\] there exist $\omega \in (0,\pi)$ and $\beta_0 \in (0, \omega],$ such that $$\lim_{s\rightarrow +\infty}\frac{f(s)}{e^{\beta s^2}}=0, \ \forall \beta > \beta_0,\quad \mbox{and} \quad
\lim_{s \rightarrow +\infty}\frac{f(s)}{e^{\beta s^2}}=+\infty, \ \forall \beta < \beta_0.$$
As we can see, a critical growth for the function $f$ is allowed. Note also that a weaker condition than the usual Ambrosetti-Rabinowitz condition is imposed on $f$, see condition .
Our main results are the following one.
\[groundN3\] Suppose that $N \geq 2$ and $f$ satisfies -. Then problem admits a ground state solution which is non-negative, radially symmetric and decreasing.
\[groundN1\] Suppose that $N = 1$ and $f$ satisfies , and . Then problem admits a ground state solution a ground state solution which is non-negative, radially symmetric and decreasing.
The plan of the paper is the following: In Section \[tools\] we study the case $N\geq2$. We first introduce the variational framework, then give some preliminaries results and Lemmas which will be useful to prove Theorem \[groundN3\]. In Section \[tools1\] we consider the case $N=1$ and $\alpha=1/2$, where again, after some preliminaries, the proof of Theorem \[groundN1\] is given.
Before concluding this introduction, we would like to cite some papers involving the fractional Laplacian operator where the problem is related to the problem in some sense, see for example, Ambrosio [@Am], Barrios, Colorado, de Pablo and Sánchez [@barrios], Frank and Lenzmann [@FL], Felmer, Quass and Tan [@FQT], Iannizzotto and Squassina [@IS], Zhang, do Ó and Squassina [@ZOS] and their references.
[**Notations**]{} As a matter of notations, we will use in all the paper the letter $C, \overline C, C', \ldots$ to denote suitable positive constants whose exact value is insignificant for our purpose.
The case $N\geq2$ {#tools}
=================
The variational framework
-------------------------
The energy functional $I: H^\alpha({\mathbb R}^N) \rightarrow {\mathbb R}$ associated to equation is defined as follows $$I(u) = \frac 12 \| u \|^2 - \int_{{\mathbb R}^N} F(u) \, dx.$$ Under assumptions and , $I \in C^1(H^{\alpha}({\mathbb R}^N), {\mathbb R})$ with Frechét derivative given by $$I'(u)[v] = \displaystyle\int_{\mathbb{R}^{N}}(-\Delta)^{\alpha/2}u(-\Delta)^{\alpha/2}v dx+ \displaystyle\int_{\mathbb{R}^{N}} uv dx -
\int_{{\mathbb R}^N} f(u) v dx, \quad \forall u,v \in H^\alpha({\mathbb R}^N).$$ Hence the critical points are easily seen to be weak solutions to .
We remark two inequalities which will be frequently used in the sequel. From and , for any $\varepsilon > 0$ there exists $C_{\varepsilon} > 0$ such that $$\label{ESTIMATIVA1}
|f(s) | \leq \varepsilon |s|+ C_{\varepsilon} |s|^{2^{*}_{\alpha} -1} \quad \hbox{for all $s >0$}$$ and, then by integration, $$\label{ESTIMATIVA2}
|F(s) | \leq \frac{\varepsilon}{2} s^{2} + C_{\varepsilon} |s|^{2^{*}_{\alpha} } \quad \hbox{for all $s >0$}.$$
Once we intend to find nonnegative solution, we will assume that $f(s) =0$ for every $s \leq 0$. Let us consider the set of non-zero critical points of $I$, that is non trivial solution of , $$\Sigma = \{ u \in H^\alpha({\mathbb R}^N) \setminus \{0 \}: I'(u) =0 \},$$ and define $$m = \inf_{u \in \Sigma} I(u)$$ the so called [*ground state level*]{}.
Now, denoting with $G(u) = F(u) -\displaystyle\frac{u^2}{2}$, the primitive of $g(u) = f(u) -u$, let us introduce the set $$\label{eq:M}
\mathcal{M}= \left\{ u \in H^\alpha({\mathbb R}^N) \setminus \{0 \}: \int_{{\mathbb R}^N} G(u) \, dx =1 \right \} $$ and $$\label{eq:D}
T(u) =\frac 12 \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u|^2 dx,\qquad D = \inf_{u\in \mathcal M} T(u).
$$ In particular $$2 D = \inf_{u \in \mathcal{M}} \Big\{ \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u|^2 dx\Big\}.
$$ It is worth to point out that if we define the $C^{1}$ functional $$\label{eq:J}
J(u) := \int_{\mathbb R^{N}} G(u)dx -1,$$ it holds from : $$\label{eq:positividade}
u\in \mathcal M \Longrightarrow J'(u)[u]=\int_{\mathbb R^{N}}(f(u)u - u^{2}) dx =
\int_{\mathbb R^{N}} (f(u) u - 2F(u))dx +2\int_{\mathbb R^{N}}G(u)\geq 2.$$ The last information will be used later on.
In addition, we define the min-max level associated to the functional $I$ $$\label{eq:b}
b = \inf_{\gamma \in \Gamma} \max_{t \in [0, 1]} I(\gamma(t))$$ where $$\Gamma = \{\gamma \in C\left([0, 1], H^{\alpha}({\mathbb R}^N) \right): \gamma(0) =0 \ \hbox{ and } \ I(\gamma(1)) < 0\}$$ which is not empty since $I$ has a Mountain Pass Geometry.
Let us define also the set, usually called [*Pohozaev manifold*]{}, $$\mathcal{P} = \left\{u \in H^\alpha({\mathbb R}^N, {\mathbb R}) \setminus \{ 0 \}: \frac{N-2\alpha}{2} \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u|^2 \, dx = N \int_{{\mathbb R}^N} G(u) \, dx \right\}.$$ which, according to [@ChangWang Proposition 4.1], contains any weak solution of . If we denote by $$p = \inf_{u \in \mathcal{P}} I(u),$$ from [@LilianeRquelSquassina Lemma 2.4] it holds that $$\label{eq:valordep}
p = \frac {\alpha}{N} \left(\frac{N-2\alpha}{2N} \right)^{(N-2\alpha)/2\alpha} (2D)^{N/2\alpha}.$$
Some preliminary stuff
----------------------
At this point we establish some preliminary results which will be useful in order to prove Theorem \[groundN3\].
\[blowbound\] It holds $$\frac {\alpha}{N} \left(\frac{N-2\alpha}{2N} \right)^{(N-2\alpha)/2\alpha} (2D)^{N/2\alpha} \leq b$$ where $b$ is the min-max level of $I$ defined in .
Indeed, from [@LilianeRquelSquassina Lemma 2.3], for each $\overline \gamma \in \Gamma$ with $$\Gamma = \{\overline{\gamma} \in C\left([0, 1], H^\alpha({\mathbb R}^N) \right): \overline{\gamma}(0) =0 \ \hbox{and} \ I(\overline{\gamma}(1)) < 0 \}$$ it results $\overline{\gamma}([0, 1]) \cap \mathcal{P} \neq \emptyset$. Then, there exists $t_0 \in [0, 1]$ such that $\overline{\gamma}(t_0) \in \mathcal{P}$. So $$p \leq I(\overline{\gamma}(t_0)) \leq \max_{t \in [0, 1]} I(\overline{\gamma}(t))$$ from where it follows that $p \leq b$ and the result follows from .
The next result is standard. We recall the proof for the reader’s convenience.
\[Mmanif\] The set $\mathcal{M}$ defined in is not empty and a $C^1$ manifold.
Observe that, fixed $0\not\equiv\varphi\in C^{\infty}_{0}(\mathbb R^{N}), \varphi\geq0$ the function $h(t)=\int_{\mathbb R^{N}}G(t\varphi)dx $ is strictly negative for small $t$ and $h'(t)>0$ for $t$ large; this implies that there exists some $\bar t>0$ such that $\bar t\varphi\in \mathcal M.$ Moreover $\mathcal M$ is a $C^{1}$ manifold in virtue of .
The next steps consists in proving the boundedness of the minimizing sequences in $H^{\alpha}(\mathbb{R}^N)$ for the problem $$\label{MINIMIZA1}
\min \left\{\frac{1}{2}\int_{\mathbb{R}^N}|(- \Delta u)^{\frac{\alpha}{2}} \,dx \,:\, \int_{\mathbb{R}^N}G(u)\,dx=1 \right\}.$$
\[bdedminseq\] Any minimizing sequence $\{ u_n \}\subset \mathcal M$ for $T$ is bounded in $H^\alpha({\mathbb R}^N)$.
Let $\{ u_n \}\subset \mathcal M $ be a minimizing sequence for $T$, then $$T(u_{n}) =\frac 12 \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u_n|^2 dx \longrightarrow D \quad \hbox{as $n \rightarrow + \infty$}$$ and $$\int_{{\mathbb R}^N} G(u_n) dx = 1, \quad \hbox{that is}, \quad \int_{{\mathbb R}^N} \left( F(u_n) - \frac 12 u_n^2 \right) dx =1.$$ Then $$\label{boundgradA}
\frac 12\int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u_{n}|^2 \, dx \leq C \quad \hbox{for all $n \in {\mathbb N}$ and for some constant $C > 0$ }$$ and $$\int_{{\mathbb R}^N} F(u_n) \,dx = 1 +\frac 12 \int_{{\mathbb R}^N} u_n ^2 \, dx.$$ By using (\[ESTIMATIVA1\]) with $\varepsilon = 1/4$, we get $$1 +\frac 12 \int_{{\mathbb R}^N} u_n^2 \, dx \leq \frac 14 \int_{{\mathbb R}^N} u_n ^2 \, dx + C_{1/4} \int_{{\mathbb R}^N} |u_n|^{2^{*}_{\alpha}} \, dx.$$ Then, for every $n \in {\mathbb N}$, by using , it follows $$\frac 12 \int_{{\mathbb R}^N} u_n^2 dx \leq C_{1/4} \, \int_{{\mathbb R}^N} |u_n|^{2^{*}_{\alpha}} dx
\leq C_{1/4} C \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u_{n}|^2 \, dx \leq \overline C.$$ Consequently $\{ u_n \} $ is bounded also in $L^2({\mathbb R}^N)$ and this ensures its boundedness in $H^\alpha({\mathbb R}^N)$.
By the Ekeland Variational Principle we can assume that the minimizing sequence $\{u_{n}\}$ is also a [*Palais-Smale sequence*]{}, that is, there exists a sequence of Lagrange multipliers $\{ \lambda_n \} \subset \mathbb R$ such that $$\label{eq:minimizing}
\frac 12 \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u_{n}|^2 \, dx \longrightarrow D \qquad \hbox{as $n \rightarrow + \infty$}$$ and $$\label{LAMBDA1}
T'(u_n) - \lambda_n J'(u_n) \longrightarrow 0 \ \hbox{in } (H^\alpha({\mathbb R}^N))^{-1}\qquad \text{ as } n \rightarrow + \infty.$$ In the remaining part of this section, $\{\lambda_{n}\}$ will be the associated sequence of Lagrange multipliers. At this point it is useful to establish some properties of the levels $D$ and $b$.
\[Dposit\] The number $D$ given by is positive, namely, $ D > 0$.
Clearly by definition $D \geq 0$. Suppose, by contradiction, that $D =0$. If $\{ u_n \} $ is a minimizing sequence for $D =0$, then $$\frac 12 \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u_n|^2 dx \to 0 \quad \hbox{as} \quad n \to + \infty$$ and $$1 = \int_{{\mathbb R}^N} G(u_n) \, dx = \int_{{\mathbb R}^N} \left( F(u_n) - \frac 12 u_n^2 \right) dx.$$ Then, for any $\varepsilon>0$, see (\[ESTIMATIVA2\]), $$1 + \frac 12\int_{{\mathbb R}^N} u_n^2 \, dx = \int_{{\mathbb R}^N} F(u_n) \, dx \leq \frac{\varepsilon}{2} \int_{{\mathbb R}^N} u_n^2 \, dx
+ \frac{C_{\varepsilon}}{2^{*}_{\alpha}} \int_{{\mathbb R}^N} |u_n|^{2^{*}_{\alpha}} \, dx$$ so that $$1+ \frac 12(1 - \varepsilon) \int_{{\mathbb R}^N} u_n^2 \, dx \leq C_{\varepsilon} \int_{{\mathbb R}^N} |u_n|^{2^{*}_{\alpha}} \, dx \leq C_{\varepsilon} C \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u_{n}|^2 \, dx.$$ By choosing $\varepsilon = 1/2$, we obtain $$1 \leq C_{ 1/2} C \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u_n|^2 \, dx \longrightarrow 0 \qquad \hbox{ as $n \longrightarrow + \infty$}.$$ This contradiction concludes the proof.
\[lambdanD\] The sequence of Lagrange multipliers $\{ \lambda_n \}$ associated to the minimizing sequence $\{u_{n}\}$ is bounded. More precisely, we have that $$0<\liminf_{n \to +\infty}\lambda_n \leq \limsup_{n \to + \infty} \lambda_n \leq D.$$ Hence, for some subsequence, still denoted by $\{\lambda_n\}$, we can assume that $\lambda_n \to \lambda^{*}$, for some $\lambda^{*} \in (0, D]$.
By , $$\label{eq:T'J'}
2T(u_{n}) =T'(u_n)[u_n] = \lambda_n J'(u_n)[u_n] +o_{n}(1).$$ Then, from $$2T(u_n) \geq 2\lambda_{n} +o_{n}(1)$$ which implies, taking into account , $$\limsup_{n \rightarrow + \infty} \lambda_n \leq \frac 12 \limsup_{n \rightarrow + \infty} \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u_{n}|^2 dx =D.$$ Since $\{u_{n}\}$ is a bounded minimizing sequence, it is easy to see that $|J'(u_{n})[u_{n}] |= |\int_{\mathbb R^{N}} g(u_{n}) u_{n}| \leq C$, and then by and the fact that $2T(u_{n})\to 2D>0$, we infer that $$\liminf_{n \to +\infty}\lambda_n >0.$$ The proof is thereby completed.
In the sequel, we will show that a minimizing sequence for $(\ref{MINIMIZA1})$ can be choose nonnegative and radially symmetric around the origin. Note that for our proof we do not need to consider the “odd extension” of the nonlinearity, as it is usually done in the literature to show that the minimizing sequence can be replaced by the sequence of the absolute values. In fact we will prove that the minimizing sequence can be replaced, roughly speaking, with the sequence of the positive parts.
\[positividade\] Any minimizing sequence $\{u_n\}$ for $(\ref{MINIMIZA1})$ can be assumed radially symmetric around the origin and nonnegative.
To begin with, we recall that $F(s)=0$ for all $s \leq 0$. Thus, $F(u_n)=F(u_n^{+})$ for all $n \in \mathbb{N}$ with $u_n^{+}=\max\{0,u_n\}$. From this, the equality $$\int_{{\mathbb R}^N}G(u_n)\,dx=1, \quad \forall n \in \mathbb{N}$$ leads to $$\int_{{\mathbb R}^N}G(u_n^+)\,dx \geq 1, \quad \forall n \in \mathbb{N}.$$ Defining the function $h_n:[0,1] \to {\mathbb R}$ by $$h_n(t)=\int_{{\mathbb R}^N}G(t u_n^+)\,dx$$ the conditions on $f$ yield that $h$ is continuous with $h_n(1) \geq 1$. Once $u_n^+ \not=0$ for all $n \in \mathbb{N}$, the condition ensures that $h_n(t)<0$ for $t$ close to 0. Thus there is $t_n \in (0,1]$ such that $h_n(t_n)=1$, that is, $$\int_{{\mathbb R}^N}G(t_n u_n^+)\,dx = 1, \quad \forall n \in \mathbb{N},$$ implying that $t_n u_n^+ \in \mathcal{M}$. On the other hand, we also know that $$\int_{{\mathbb R}^N}|(-\Delta)^{\alpha/2}u_n^+|^{2}\,dx \leq \int_{{\mathbb R}^N}|(-\Delta)^{\alpha/2}u_n|^{2}\,dx.$$ Once $t_n \in (0,1]$, the last inequality gives $$D \leq T(t_n u_n^+)\leq T(u_n)=D+o_n(1)$$ that is, $$t_n u_n^{+} \in \mathcal{M} \quad \mbox{and} \quad T(t_n u_n^+) \to D,$$ showing that $\{t_n u_n^+\}$ is a minimizing sequence for $T$. Thereby, without lost of generality, we can assume that $\{u_n\}$ is a nonnegative sequence.
Moreover, by noticing that $$\int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u_n^{*}|^2 \, dx \leq \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u_n|^2 \, dx, \quad \forall n \in \mathbb{N}$$ and $$\int_{\mathbb{R}^N}G(u_{n}^{*})\, dx= \int_{\mathbb{R}^N}G(u_{n})\, dx, \quad \forall n \in \mathbb{N}$$ where $u_n^{*}$ is the Schwartz symmetrization of $u_n$, any minimizing sequence can be assumed radially symmetric, non-negative and decreasing in $r=|x|$.
In what follow, we will use that the embedding $$\label{imersao}
H_{rad}^{\alpha}({\mathbb R}^N) \hookrightarrow L^{p}({\mathbb R}^N)$$ is compact for all $p \in (2,2_{\alpha}^{*})$, see Lions [@Lions] for more details.
Due to the boundedness in $H^{\alpha}(\mathbb R^{N})$ of the (non-negative and radial symmetric) minimizing sequence $\{u_{n}\}$ (see Lemma \[bdedminseq\]) we can assume that $\{u_{n}\}$ has a weak limit in $H^{\alpha}(\mathbb R^{N})$ denoted hereafter with $u$. Observe also that, by the boundedness in $L^{2}(\mathbb R^{N})$ we have the uniform decay $|u_{n}(x)| \leq C |x|^{-N/2}$, see [@Am Lemma 1]. Therefore, passing to a subsequence, if necessary, we deduce that the weak limit $u$ is non-negative, radially symmetric and decreasing.
It turns out that the weak limit $u$ is a solution of the minimizing problem we were looking for. Before to see this some preliminary lemmas are in order to recover some compactness.
\[nuSD\] Assume that $v_n:=u_n-u\rightharpoonup 0$ in $H^{\alpha}(\mathbb R^{N})$ and $\displaystyle\int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2} v_n|^{2} \, dx \to L>0$. Then $$D \geq 2^{-2 \alpha/N}S.$$
First of all, we recall the limit $T'(u_n) - \lambda_n J'(u_n) \rightarrow 0$ as $n \rightarrow + \infty$ gives $$T'(u_n)[u_n]-\lambda_{n} J'(u_n)[u_n]=o_n(1).$$ Using standard arguments, it is possible to prove that $$T'(u_n)[u_n]-\lambda J'(u_n)[u_n]=T'(v_n)[v_n]-\lambda_{n} J'(v_n)[v_n]+T'(u)[u]-\lambda^{*} J'(u)[u] + o_n(1)$$ and $$T'(u)-\lambda^{*} J'(u)=0 \quad \mbox{in} \quad (H^{\alpha}(\mathbb{R}^N))^{-1}.$$ Then $T'(v_n)[v_n]-\lambda_{n} J'(v_n)[v_n]=o_n(1)$, or equivalently, $$\int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2} v_n|^{2} \, dx =
\lambda_n \, \int_{{\mathbb R}^N} f(v_n) v_n \, dx - \lambda_n \int_{{\mathbb R}^N} v_n^2 \, dx + o_n(1).$$ Using the growth conditions on $f$, fixed $q \in (2,2^{*}_{\alpha})$ and given $\varepsilon >0$, there exists $C=C(\varepsilon, q)>0$ such that $$f(t)t \leq \varepsilon t^{2}+C|t|^{q}+(1+\varepsilon)|t|^{2^{*}_\alpha}, \quad \forall t \in \mathbb{R}.$$ From this, $$\int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}v_n|^2 \, dx \leq
\lambda_n \left(\varepsilon \int_{{\mathbb R}^N} v_n^2\, dx + C \int_{{\mathbb R}^N} |v_n|^q dx +(1+\varepsilon) \int_{{\mathbb R}^N} |v_n|^{2^{*}_{\alpha}} \, dx\right) + o_n(1).$$ Now, using the definition of $S$, see (\[definicaoS\]), we get $$\begin{gathered}
\label{eq:LCS}
\int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}v_n|^2\, dx \\ \leq
\lambda_n \left(\varepsilon \int_{{\mathbb R}^N} v_n^2 dx + C \int_{{\mathbb R}^N} |v_n|^q dx +(1+\varepsilon) \left( \frac{1}{S} \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2} v_n|^{2} \, dx\right)^{2^{*}_{\alpha}/ 2} \right)+ o_n(1).\end{gathered}$$ Passing to the limit in , recalling that $\{v_n\}$ is bounded, $$\label{eq:L}
\displaystyle\int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}v_n|^2 \, dx \longrightarrow L$$ and that $u_n \to 0$ in $L^{q}({\mathbb R}^N)$ (see (\[imersao\])), we find $$L\leq \lambda^*\left( \varepsilon C_1 + (1+\varepsilon)\left(\frac{L}{S}\right)^{2^{*}_{\alpha}/2} \right).$$ By the arbitrariety of $\varepsilon$, we derive $L\leq D \left(L/S \right)^{2^{*}_{\alpha}/2}$, or equivalently, $$\label{eq:SDL}
S^{2^{*}_{\alpha}/2}\leq D L^{2 \alpha /(N-2 \alpha)}.$$ On the other hand implies that $L = 2D -\displaystyle\int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2}u|^2 dx \leq 2D$. Hence becomes $$S^{2^{*}_{\alpha}/2}\leq 2^{2 \alpha /(N-2 \alpha)} D^{2^{*}_{\alpha}/2},
\quad \text{ i.e. } \quad D \geq 2^{-2 \alpha/N}S$$ and the proof is finished.
In the next result the condition $\tau>\tau^{*}$ given in plays a crucial role.
\[lambdabbound\] It holds $$b < \frac {\alpha}{ N} \left(\frac{N-2\alpha}{2N} \right)^{(N-2\alpha)/2\alpha} 2^{(N-2\alpha)/2\alpha} S^{N/2\alpha}.$$
Take $\varphi \in H^\alpha({\mathbb R}^N)$ such that $\| \varphi \| =1$ and $| \varphi |_q^2 = C_q^{-1}$. From definition of $ b = \inf_{\gamma \in \Gamma} \max_{t \in [0, 1]} I(\gamma(t))$ and $$\begin{aligned}
b \leq \max_{ t \geq 0} I(t \varphi) & \leq & \max_{t \geq 0} \left\{ \frac{t^2}{2} - \tau \frac{t^q}{q} \int_{{\mathbb R}^N} |\varphi|^q \, dx \right\} \\
& = & \max_{t \geq 0} \left\{ \frac{t^2}{2} - \tau \frac{t^q}{q} C_q^{-q/2} \right\} \\
& = & \frac{q-2}{2q} \, \frac{C_q^{q/(q-2)}}{\tau^{2/(q-2)}}.\end{aligned}$$ This gives (by the definition of $\tau^{*}$) exactly the conclusion.
\[weaklim\] If $u_n \rightharpoonup u$ in $H^{\alpha}({\mathbb R})$, then $u_n \to u$ in $\mathcal D^{\alpha,2}(\mathbb{R}^N)$. In particular, $u_n \to u$ in $L^{2^{*}_{\alpha}}(\mathbb{R}^N)$.
Of course $v_n=u_n -u \rightharpoonup 0$ in $H^{\alpha}({\mathbb R})$. Suppose by contradiction that $u_n \not\to u $ in $\mathcal D^{\alpha,2}(\mathbb{R}^N)$. Thereby, $\displaystyle\int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2} v_n|^{2} \, dx \to L>0$ for some subsequence.Then, by Lemma \[nuSD\], $$\label{eq:DS}
D \geq 2^{- 2\alpha/N} S.$$ On the other hand, from Lemma \[blowbound\] $$\frac {\alpha}{N} \left(\frac{N-2\alpha}{2N} \right)^{(N-2\alpha)/2\alpha} (2D)^{N/2\alpha} \leq b,$$ from which, using , it follows that $$\frac {\alpha}{N} \left(\frac{N-2\alpha}{2N} \right)^{(N-2\alpha)/2\alpha} 2^{(N-2\alpha )/2\alpha} S^{N/2\alpha} \leq b.$$ This contradicts Lemma \[lambdabbound\] and finishes the proof.
Proof of Theorem \[groundN3\] {#subsec:th1}
-----------------------------
At this point we wish to show that $D$ is attained by $u$, where $u$ is the weak limit of $\{ u_n \}$. First of all, we know that $$\label{TDN}
T(u) = \frac 12\int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2} u|^2 \, dx \leq \liminf_{n \rightarrow + \infty} \frac 12 \int_{{\mathbb R}^N}
|(-\Delta)^{\alpha/2} u_n|^2 \, dx =D$$ so we just need to prove that $u\in \mathcal M$.
By [@Am Lemma 1], there is $R>0$ such that $$\frac{1}{2}u_n^{2}-F(u_n) \geq 0 \quad \forall n \in \mathbb{N} \quad \mbox{in } \mathbb R^{N} \setminus B_{R},
$$ $B_{R}$ being the ball of radius $R$ centered in $0.$ Since $$\int_{B_R}F(u_n)\,dx=\frac{1}{2}\int_{B_R}u_n^{2}\,dx+\int_{\mathbb{R}^N \setminus B_R}\left(\frac{1}{2}u_n^{2}-F(u_n)\right)\,dx +1$$ and $u_n \to u $ in $L^{2^{*}_{\alpha}}(B_R)$, the above information together with the Fatous’ Lemma gives $$\displaystyle\int_{B_R}F(u)\, dx \geq
\frac{1}{2}\int_{B_R}u^{2}\,dx+\int_{\mathbb{R}^N \setminus B_R}\left(\frac{1}{2}u^{2}-F(u)\right)\,dx +1$$ which leads to $$\displaystyle\int_{{\mathbb R}^N} G(u) \, dx \geq 1.$$ Suppose by contradiction that $$\int_{{\mathbb R}^N} G(u) \, dx > 1$$ and define $h: [0, 1] \rightarrow {\mathbb R}$ by $h(t) = \int_{{\mathbb R}^N} G(tu) \, dx$. The growth conditions on $f$ ensure that $h(t) < 0$ for $t$ close to $0$ and $h(1) = \int_{{\mathbb R}^N} G(u) \, dx > 1$. Then, by the continuity of $h$, there exists $t_0 \in (0, 1)$ such that $h(t_0)=1$. Then, $$\int_{{\mathbb R}^N} G(t_0 u) \, dx = 1 \Longleftrightarrow t_0 u \in \mathcal{M}.$$ Consequently, by $$D \leq T(t_0 u) = \frac{t_0^2}{2} \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2} u|^2 \, dx = t_0^2 \, T(u) \leq t_0^2 \, D < D$$ which is absurd. Thus $ \int_{{\mathbb R}^N} G(u) \, dx =1 $, i.e. $ u\in \mathcal M$. The fact that the solution $u$ of the minimizing problem gives rise to a ground state solution, follows by standard arguments; indeed, since $ u$ is a solution of the minimizing problem , i.e. $D =T(u)= \inf_{w\in \mathcal M}T(w)$, then there exists an associated Lagrange multiplier $\lambda$ such that, in a weak sense, $$(-\Delta)^{\alpha} u = \lambda g( u).$$ Now by testing the previous equation on the same minimizer $u$, we deduce that $$2T(u)=\lambda\int_{\mathbb R^{N}}g(u)u dx = \lambda J'(u)[u] \geq 2\lambda$$ so that it has to be, by Lemma \[Dposit\], $T(u)\geq\lambda>0$. Setting $u_{\sigma}(x):=u(\sigma x)$ for $\sigma>0$, we easily see that $$(-\Delta)^{\alpha} u_{\sigma} = \lambda \sigma^{2\alpha}g(u_{\sigma}).$$ Choosing $\sigma=\lambda^{1/2\alpha}$ we obtain a solution of . Arguing as in [@BL Theorem 3], $u_\sigma$ is a ground state solution.
The case $N=1$ and $\alpha= 1/2$ {#tools1}
================================
The variational framework
-------------------------
As for the previous case, let us consider the set of nontrivial solutions of , namely $$\Sigma = \{ u \in H^{1/2}({\mathbb R}) \setminus \{0 \}: I'(u) =0 \},$$ and let $$m = \inf_{u \in \Sigma} I(u).$$
Denoting with $G(u) = F(u) -\displaystyle\frac{u^2}{2}$, the primitive of $g(u) = f(u) -u$, we introduce the set $$\label{eq:MN=1}
\mathcal{M}= \left\{ u \in H^{1/2}({\mathbb R}) \setminus \{0 \}: \int_{{\mathbb R}} G(u) \, dx =0 \right \} ,$$ and $$\label{DefD}
T(u) =\frac 12 \int_{{\mathbb R}} |(-\Delta)^{1/4}u|^2 dx,\qquad D = \inf_{u\in \mathcal M} T(u).
$$ it is, again as before, $$2 D = \inf_{u \in \mathcal{M}} \Big\{ \int_{{\mathbb R}} |(-\Delta)^{1/4}u|^2 dx\Big\}.
$$ We point out here that, since we will deal with minimizing sequences $\{u_{n}\}$ for the minimization problem , as in the previous Section we suppose that $u_{n}$ is non-negative and radially symmetric. Moreover, we again define the min-max level associated to the functional $I$ $$\label{eq:bN=1}
b = \inf_{\gamma \in \Gamma} \max_{t \in [0, 1]} I(\gamma(t))$$ where $$\Gamma = \{\gamma \in C\left([0, 1], H^{\alpha}({\mathbb R}) \right): \gamma(0) =0 \ \hbox{ and } \ I(\gamma(1)) < 0\}.$$
Some preliminary stuff
----------------------
Let us start with the following important result due to T. Ozawa [@Ozawa]
\[th:Ozawa\] There exists $0 < \omega \leq \pi$ such that, for all $r \in (0, \omega)$, there exists $H_{r}>0$ satisfying $$\label{I1}
\int_{\mathbb{R}}( e^{r u^2 }-1) \,d x \leq H_{r}|u|^{2}_{2},$$ for all $u\in H^{1/2}(\mathbb{R})$ with $|(-\Delta)^{1/4} u|^2_{2}\leq 1$.
At this point we establish some preliminary results which will be useful in order to prove Theorem \[groundN1\].
\[Mmanif1\] The set $\mathcal{M}$ defined in is not empty and a $C^1$ manifold.
Consider $w \in C^{\infty}_{0}({\mathbb R})$ with $w(x)>0$ and define a function $$h(t)=\displaystyle\int_{{\mathbb R}}G(tw)dx=\displaystyle\int_{{\mathbb R}}F(tw) dx-\frac {t^{2}}{2} \displaystyle\int_{{\mathbb R}}w^{2}dx.$$ From for $t>0$ small we have $$h(t)\leq \frac{\varepsilon -1}{2}t^{2}\displaystyle\int_{{\mathbb R}}w^{2}dx.$$ For $\varepsilon <1$, we get $h(t)<0$ for $t>0$ small.
Now using we obtain $$h(t) \geq \tau \frac{t^{q}}{q}\displaystyle\int_{{\mathbb R}}w^{q}dx-\frac {t^{2}}{2} \displaystyle\int_{{\mathbb R}}w^{2}dx
\quad {and }\quad h'(t) \geq \lambda t^{q-1}\displaystyle\int_{{\mathbb R}}w^{q}dx- t \displaystyle\int_{{\mathbb R}}w^{2}dx.$$ Then, $h(t)>0$ for $t>0$ large and $h'(t)>0$ for $t>0$ large. Then there is a $\overline{t}>0$ such that $$\displaystyle\int_{{\mathbb R}}G(\overline{t}w)\,dx=h(\overline{t})=0.$$
Now we prove that $\mathcal{M}$ is a manifold. Indeed, if $w\in \mathcal{M}$, then $w \not=0$. Then, from and the fact that $\lim_{|x|\to \infty}w(x)=0 $, there exists $ x_0 \in {\mathbb R}$ such that $g(w(x_0))<0$. Thereby, by continuity, there is an open interval $B_\delta(x_0)$ such that $$g(w(x))<0, \quad \forall x \in B_\delta(x_0).$$ As a consequence we can always find a $\phi \in C_0^{\infty}({\mathbb R}) \subset H^{1/2}({\mathbb R})$ such that $J'(w)[\phi]=\int_{{\mathbb R}}g(w)\phi \,dx < 0$, showing that $J'(w) \not=0$.
\[ConvergenceofF\] Assume that $f$ satisfies , and . Let $\{v_n\}\subset H^{1}(\mathbb R)$ be a sequence of radial functions such that $$v_n \rightharpoonup v \ \ \mbox{in} \ \ H^{1/2}({\mathbb R})$$ and $$\displaystyle\sup_{n}|(-\Delta)^{1/4} u|^{2}_{2}=\rho <1 \ \ \mbox{and} \ \ \displaystyle\sup_{n}|v_n|^{2}_{2}=M <\infty.$$ Then, $$\displaystyle\int_{{\mathbb R}}F(v_n)dx\longrightarrow \displaystyle\int_{{\mathbb R}}F(v)dx.$$
Without loss of generality, we can assume that there is $v \in H^{1/2}({\mathbb R})$, radial, such that $$v_n \rightharpoonup v \ \ \mbox{in} \ \ H^{1/2}({\mathbb R}), \ \ v_n(x) \to v(x) \ \ \mbox{a.e in} \ \ {\mathbb R}\ \ \mbox{and} \ \ \displaystyle\lim_{|x|\to +\infty} v_n(x)=0, \ \ \mbox{uniformly in} \ \ n.$$ Using the Theorem \[th:Ozawa\], we know that for each $m\in(0,1)$ and $M>0$, there exists $C(m,M)>0$ such that $$\displaystyle\sup_{u\in \texttt{B}}\int_{{\mathbb R}}(e^{w u^2}-1) \, d x \leq C(m,M),$$ where $$\texttt{B}=\biggl\{ u \in H^{1/2}({\mathbb R}) : \ |(-\Delta)^{1/4} u|^{2}_{2}\leq m \ \ \mbox{and} \ \ |u|^{2}_{2}\leq M \biggl\}.$$ Now, choose $\varepsilon>0$ small enough such that $m=\frac{\rho}{(1-\epsilon)^{2}} \in (0,1)$ and set $t=\frac{\omega}{(1-\epsilon)^{2}} > w \geq \beta_0$. Then, $$\displaystyle\int_{{\mathbb R}}(e^{t v_{n}^2}-1) dx=\displaystyle\int_{{\mathbb R}}(e^{t(1-\epsilon)^{2} (\frac{v_{n}}{1-\epsilon})^2}-1)dx
= \displaystyle\int_{{\mathbb R}}(e^{\omega (\frac{v_{n}}{1-\epsilon})^2}-1)dx.$$ Since $v_{n}\in \texttt{B}$ we have $$\displaystyle\int_{{\mathbb R}}(e^{t v_{n}^2}-1)dx \leq \displaystyle\sup_{u\in \texttt{B}}\int_{{\mathbb R}}(e^{\omega u^2}-1) \,d x \leq C(m,M).$$ Now, setting $P(s)=F(s)$ and $Q(s)=e^{ts^{2}}-1$, from , and the last inequality, we get $$\displaystyle\lim_{s\to 0}\frac{P(s)}{Q(s)}=\displaystyle\lim_{s\to+ \infty}\frac{P(s)}{Q(s)}=0,$$ $$\displaystyle\sup_{n\to+ \infty}\int_{{\mathbb R}}Q(v_n) dx<\infty$$ and $$P(v_{n}(x))\longrightarrow P(v(x)) \ \ \mbox{a.e in} \ \ {\mathbb R}.$$ Consequently the hypotheses of the Compactness Lemma of Strauss [@BL Theorem A.I] are fulfilled. Hence $P(v_{n})$ converges to $P(v)$ in $L^{1}({\mathbb R})$, and then $$\displaystyle\int_{{\mathbb R}}F(v_n)dx\longrightarrow \displaystyle\int_{{\mathbb R}}F(v)dx$$ concluding the proof.
The relation between the ground state level and the minimax level defined in is given in the following
\[Dandb\] The numbers $D$ and $b$ satisfy the inequality $D\leq b$.
Arguing as in Lemma \[Mmanif1\], given $v\in H^{1/2}({\mathbb R})$ with $v^{+}=\max\{v,0\}\neq 0$, there is $t_{0}>0$ such that $t_0 v^{+} \in \cal{M}$. Then, $$D\leq \frac{t_{0}^{2}}{ 2} \int_{{\mathbb R}} |(-\Delta)^{1/4} v^{+}|^2 \, dx = I(t_{0}v^{+})\leq \displaystyle\max_{t\geq 0}I(tv^{+}).$$ On the other hand, since $f(s)=0$ for $s\leq 0$, if $v\in H^{1/2}({\mathbb R})$, $v\neq 0$ with $v^{+}=0$, then $\max_{t\geq 0}I(tv)=\infty$. Hence in any case $D\leq b$.
\[Dposit\] The number $D$ given by is positive, namely, $ D > 0$.
By definition $D\geq 0$. Assume by contradiction that $D=0$ an let $\{u_n\}$ be a (non-negative and radial) minimizing sequence in $H^{1/2}({\mathbb R})$ for $T$, that is, $$\displaystyle\int_{{\mathbb R}} |(-\Delta)^{1/4}u_n|^2 dx \to 0 \ \ \mbox{and} \ \ \displaystyle\int_{{\mathbb R}} G(u_n) \, dx =0.$$ For each $\mu_n>0$, the function $v_n(x):=u_n( x / \mu_n)$ satisfies $$\displaystyle\int_{{\mathbb R}} |(-\Delta)^{1/4}v_n|^2dx=
\displaystyle\int_{{\mathbb R}} |(-\Delta)^{1/4}u_n|^2dx \ \ \mbox{and} \ \ \displaystyle\int_{{\mathbb R}} G(v_n) \, dx =0.$$ Since $$\displaystyle\int_{{\mathbb R}} v_n^{2}dx= \mu_{n}^{2}\displaystyle\int_{{\mathbb R}} u_n^{2}dx,$$ we choose $\mu_{n}^{2}=|u_{n}|_{2}^{-2}
$ to obtain $$\displaystyle\int_{{\mathbb R}} |(-\Delta)^{1/4}v_n|^2 dx\to 0, \ \ \displaystyle\int_{{\mathbb R}} v_n^{2}dx=1 \ \ \mbox{and} \ \ \displaystyle\int_{{\mathbb R}} G(v_n) \, dx =0.$$ and we can assume that there exists $ v \in H^{1/2}({\mathbb R})$, radial, such that $v_n \rightharpoonup v$ in $H^{1/2}({\mathbb R})$. From Lemma \[ConvergenceofF\] we get $$\displaystyle\int_{{\mathbb R}}F(v_n)dx\longrightarrow \displaystyle\int_{{\mathbb R}}F(v)dx.$$ Note that $\int_{{\mathbb R}} G(v_n) \, dx =0$ implies $\int_{{\mathbb R}} F(v_n) \, dx =\frac 12$ and $\int_{{\mathbb R}} F(v) \, dx =\frac 12$. Then $v \neq 0$. But $$\displaystyle\int_{{\mathbb R}} |(-\Delta)^{1/4}v|^2dx \leq \liminf_{n\to+\infty} \displaystyle\int_{{\mathbb R}} |(-\Delta)^{1/4}v_n|^2dx
\longrightarrow 0,$$ implies $v=0$ which is an absurd.
\[lambdabbound1\] We have $ b < 1/2.$
It is sufficient to repeat the same argument of the Lemma \[lambdabbound\], recalling that now by it is $$\tau^* = \left( \frac{q-2}{q} \right)^{(q-2)/2} C_q^{q/2}$$ concluding the proof.
Proof of Theorem \[groundN1\] {#sectmainres1}
-----------------------------
At this point we will show that $D$ is attained by $u$, where $u$ is the weak limit of $\{ u_n \}$. Indeed, since $u_n \rightharpoonup u$ in $H^{1/2}({\mathbb R})$ we have $$\label{TD}
T(u) = \frac 12\int_{{\mathbb R}} |(-\Delta)^{1/4} u|^2 \, dx \leq \liminf_{n \rightarrow + \infty} \frac 12 \int_{{\mathbb R}}
|(-\Delta)^{1/4} u_n|^2 \, dx =D.$$ Moreover, by Lemma \[ConvergenceofF\] we have $$\displaystyle\int_{\mathbb{R}}F(u) dx= \displaystyle\liminf_{n\to\infty}\displaystyle\int_{\mathbb{R}}F(u_n) dx\geq \frac 12 \displaystyle\int_{\mathbb{R}}u^{2} dx$$ leading to $$\displaystyle\int_{{\mathbb R}} G(u) \, dx \geq 0.$$ As in the previous case $N\geq2$, we just need to prove that $u\in \mathcal M$, i.e. $ \int_{{\mathbb R}^N} G(u) \, dx = 0$.
We again argue by contradiction by supposing that $$\int_{{\mathbb R}} G(u) \, dx > 0.$$ As in the previous section, we set $h: [0, 1] \rightarrow {\mathbb R}$ by $h(t) = \int_{{\mathbb R}} G(tu) \, dx$. Using the growth condition of $f$ we have $h(t) < 0$ for $t$ close to $0$ and $h(1) = \int_{{\mathbb R}} G(u) \, dx > 0$. Then, by the continuity of $h$, there exists $t_0 \in (0, 1)$ such that $h(t_0)=0$, that is $t_{0}u\in \mathcal M$. Consequently, by $$D \leq T(t_0 u) = \frac{t_0^2}{2} \int_{{\mathbb R}^N} |(-\Delta)^{\alpha/2} u|^2 \, dx = t_0^2 \, T(u) \leq t_0^2 \, D < D$$ which is absurd.
As for the case $N\geq2$, one show that the minimizer $u$ of gives rise to a ground state solution of .
[1]{}
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[^1]: Claudianor Alves was partially supported by CNPq/Brazil Proc. 304036/2013-7 ; Giovany M. Figueiredo was partially supported by CNPq, Brazil; Gaetano Siciliano was partially supported by Fapesp and CNPq, Brazil.
|
---
address: |
Vrije Universiteit Brussel - Interuniversity Institute for High Energies\
Pleinlaan 2, 1050 Brussel, Belgium
author:
- '<span style="font-variant:small-caps;">S. Blyweert</span>, on behalf of the ATLAS and CMS Collaborations'
title: 'TOP-QUARK MASS MEASUREMENTS AT THE LHC'
---
Introduction
============
The top quark, which was discovered in 1995 at the Tevatron, is the heaviest currently known fundamental particle. Its mass is an important parameter of the standard model, since it is an important input for the global Electro-Weak fits. These fits can be used to constrain the mass of the SM Brout-Englert-Higgs boson, and are also a consistency check of the standard model. $m_{\rm t}$ has been measured already by the CDF and D0 collaborations with great precision, resulting in $m_{\rm t} = 173.2 \pm 0.9$ GeV as the current world average [@TevatronCombi; @TevTopMassTalk]. A precise measurement of $m_{\rm t}$ by the ATLAS and CMS collaborations at the Large Hadron Collider (LHC) would provide an independent cross-check of this value and would help to further reduce the total uncertainty on the world average of $m_{\rm t}$.
In proton-proton collisions at $\sqrt{s} = 7$ TeV top quarks are dominantly produced in pairs. In 2011 the LHC delivered more than 5 fb$^{-1}$ to both the ATLAS and CMS experiments, corresponding to about $8 \cdot 10^5$ ${\rm t\bar{t}}$ pairs per experiment. This gave both ATLAS and CMS the opportunity to perform precise measurements of $m_{\rm t}$ in the different decay channels. Since top quarks decay most of the time into a b quark and a W boson, ${\rm t}\bar{\rm t}$ events can be categorized, according to the decay of the W bosons, in dileptonic (${\rm t\bar{t} \to b\bar{b}\ell\nu_{\ell}\ell'\nu_{\ell'}}$), all-jets (${\rm t\bar{t} \to b\bar{b}qq'q''q'''}$) and $\ell$+jets (${\rm t\bar{t} \to b\bar{b}\ell\nu_{\ell}qq'}$) events. More information about the production and decay of ${\rm t\bar{t}}$ events can be found in Ref. [@TTbarXS].
Direct measurements of the top-quark mass
=========================================
Measurement of the top-quark mass in the dilepton channel
---------------------------------------------------------
The $m_{\rm t}$ measurement from CMS in the dilepton channel is performed with 2.3 fb$^{-1}$ of data [@DileptonCMS]. Events in this channel are selected by requiring exactly two leptons (electrons (e) or muons ($\mu$)) with $p_{\rm T} > 20$ GeV and $|\eta| < 2.4$, at least 2 jets with $p_{\rm T} > 30$ GeV and $|\eta| < 2.4$, missing transverse energy $\not\!\!\!E_{\rm T} > 30$ GeV, and at least one b-tagged jet. In the ee and $\mu\mu$ channels events with $76 {\rm~GeV} < m_{\ell\ell} < 106$ GeV are rejected. The event-by-event top-quark mass $m_{\rm KINb}$ is reconstructed with the KINb algorithm [@KINb]. For each of the possible jet-quark assignments the kinematic equations are solved multiple times per event, each time varying the reconstructed kinematics within their resolutions. The jet-quark assignment with the largest number of solutions is selected. Finally, $m_{\rm KINb}$ is extracted by taking the mean of a Gaussian fit to the distribution of the reconstructed top quark mass for all the different solutions of the kinematic equations, for the chosen jet-quark assignment.
The extraction of $m_{\rm t}$ is then performed by applying the template method. Templates which are sensitive to $m_{\rm t}$ are constructed for different top-quark mass hypotheses. The value of $m_{\rm t}$ is then extracted by doing a maximum likelihood fit of these templates to the distribution of $m_{\rm KINb}$ observed in data. By applying this technique, CMS measures $m_{\rm t} = 173.3 \pm 1.2 {\rm (stat)} ^{+2.5}_{-2.6} {\rm (syst)}$ GeV, where the systematic uncertainty is dominated by the global Jet Energy Scale (JES) uncertainty and the uncertainty on the flavour-dependent JES. This is the most precise measurement of $m_{\rm t}$ in the dilepton channel, with similar precision as the most recent result from D0 [@DileptonD0].
Measurement of the top-quark mass in the all-jets channel
---------------------------------------------------------
ATLAS performed the first measurement at the LHC of $m_{\rm t}$ in the all-jets channel, using 2.04 fb$^{-1}$ of data [@AllHadrATLAS]. Events are selected by asking $\geq 5$ jets with $p_{\rm T} > 55$ GeV and $|\eta|<4.5$, and a 6$^{\rm th}$ jet with $p_{\rm T} > 30$ GeV and $|\eta|<4.5$. Two of these jets, with $p_{\rm T} > 55$ GeV and inside the inner detector acceptance ($|\eta|<2.5$), must be b-tagged. A cut on the $\not\!\!E_{\rm T}$ significance is also applied: $\not\!\!E_{\rm T} / \sqrt{H_{\rm T}} < 3$, where $H_{\rm T}$ is the scalar sum of the $p_{\rm T}$ of all the jets in the events. The ${\rm t \bar{t}}$ event topology is reconstructed using a ’mass $\chi^2$’: $$\chi^2 = \frac{(m_{j_1,j_2} - m_{\rm W})^2}{\sigma^2_{\rm W}} + \frac{(m_{j_1,j_2,b_1} - m_{\rm t})^2}{\sigma^2_{\rm t}} + \frac{(m_{j_3,j_4} - m_{\rm W})^2}{\sigma^2_{\rm W}} + \frac{(m_{j_3,j_4,b_2} - m_{\rm t})^2}{\sigma^2_{\rm t}},$$ where $\sigma_{\rm W} = 10.2$ GeV and $\sigma_{\rm t} = 17.4$ GeV are the respective mass resolutions from simulation. For every event, the jet-quark assignment with the lowest $\chi^2$ is taken, requiring $50 {\rm~GeV} < m_{j_1,j_2} < 110$ GeV and $50 {\rm~GeV} < m_{j_3,j_4} < 110$ GeV. This $\chi^2$ is minimized as a function of $m_{\rm W}$ and $m_{\rm t}$. Only events with $\chi^2_{\rm min} < 8$ are considered in the extraction of $m_{\rm t}$.
The template method is applied to the distribution of $m_{jjb}$ values from the selected jet-quark assignment, as observed in data. Each selected event contributes two values to this distribution. ATLAS measures $m_{\rm t} = 174.9 \pm 2.1 ({\rm stat}) \pm 3.8 ({\rm syst})$ GeV. The systematic uncertainty is dominated by the uncertainty on Initial and Final State Radiation (ISR/FSR), the uncertainty on the data-driven multijet background and the JES uncertainty.
Measurement of the top-quark mass in the $\ell$+jets channel
------------------------------------------------------------
Both ATLAS and CMS have measured $m_{\rm t}$ in the $\ell$+jets channel [@LeptonJetsATLAS; @LeptonJetsCMS]. ATLAS selects events by asking exactly 1 isolated electron ($E_{\rm T} > 25$ GeV) or muon ($p_{\rm T} > 20$ GeV). The events need to have $\geq 4$ jets with $p_{\rm T} > 25$ GeV and $|\eta| < 2.5$, of which at least one is b-tagged. Multijet events are rejected by asking $\not\!\!\!E_{\rm T} > 35$ GeV and $m^{\rm T}_{\rm W} > 25$ GeV (e+jets), or $\not\!\!\!E_{\rm T} > 20$ GeV and $\not\!\!\!E_{\rm T} + m^{\rm T}_{\rm W} > 60$ GeV ($\mu$+jets). ATLAS is using two different approaches to measure $m_{\rm t}$, designed to reduce the JES uncertainty. Both analyses are based on the template method.
In the 1d-analysis, $R_{32} \equiv \frac{m_{\rm t}^{\rm reco}}{m_{\rm W}^{\rm reco}}$ is calculated for every event, where $m_{\rm t}^{\rm reco}$ and $m_{\rm W}^{\rm reco}$ are the reconstructed invariant masses of the hadronically decaying top quark and W boson, respectively. A kinematic fit is used to select a jet-quark assignment. The likelihood $L$ of the kinematic fit of the selected jet-quark assignment has to pass: $\ln{L} > -50$. The jets assigned to the $\rm t \to bqq'$ decay need to have $p_{\rm T} > 40$ GeV, and $m_{\rm W}^{\rm reco}$ has to fulfill $60 {\rm ~GeV} < m_{\rm W}^{\rm reco} < 100$ GeV.
The 2d-analysis, a combined measurement of $m_{\rm t}$ and a global Jet energy Scale Factor (JSF), is performed by a template fit to $m_{\rm t}^{\rm reco}$ and $m_{\rm W}^{\rm reco}$. In this analysis, the jet triplet assigned to the hadronic top decay is chosen as the one with maximum $p_{\rm T}$, considering only triplets with $50 {\rm ~GeV} < m_{\rm W}^{\rm reco} < 110$ GeV. Finally a kinematic fit is performed to the chosen jet triplet.
Both analyses are applied on 1.04 fb$^{-1}$ of data. The 2d-analysis has a slightly smaller uncertainty: $m_{\rm t} = 174.5 \pm 0.6 ({\rm stat}) \pm 2.3 ({\rm syst})$ GeV. The systematic uncertainty is dominated by the uncertainty on ISR/FSR and the uncertainty on the b-jet energy scale.
The CMS measurement of $m_{\rm t}$ uses 4.7 fb$^{-1}$ of data in the $\mu$+jets channel [@LeptonJetsCMS]. Events are selected by asking exactly 1 isolated muon with $p_{\rm T} > 30$ GeV and $|\eta| < 2.1$, $\geq 4$ jets with $p_{\rm T} > 30$ GeV and $|\eta| < 2.4$, of which at least 2 are b-tagged. For every possible jet-quark assignment, a kinematic fit is applied with 3 mass constraints: an equal-mass constraint of $m_{\rm t}^{\rm lept}$ and $m_{\rm t}^{\rm hadr}$, and 2 $m_{\rm W}$ constraints. The kinematic fit returns the fitted top-quark mass $m_{{\rm t},i}^{\rm fit}$ and the fit probability $P_{\rm fit}^i$. Wrong jet-quark assignments are rejected by asking $P_{\rm fit}^i > 0.2$.
To extract $m_{\rm t}$ from the data, the Ideogram method is used in a combined measurement of $m_{\rm t}$ and the Jet Energy Scale (JES). In this method, a likelihood is calculated for every event: $$\mathcal{L} ({\rm event} \mid m_{\text{t}},\text{JES}) = \left( \sum_{i=1}^{n} P_{\rm fit}^i \cdot P \left( m_{{\rm t}, i}^{\rm fit}, m_{{\rm W},i}^{\rm reco} \mid m_{\rm t}, {\rm JES} \right) \right)^{\sum_{i=1}^n P_{\rm fit}^i}.$$ The distributions $P \left( m_{{\rm t}, i}^{\rm fit}, m_{{\rm W},i}^{\rm reco} \mid m_{\rm t}, {\rm JES} \right)$ for all possible jet-quark assignments (correct assignments, wrong assignments and unmatched assignments) are taken from simulation. The individual event likelihoods are combined in a global likelihood, from which the measured $m_{\rm t}$ and JES values can be extracted. With this method, CMS measures $m_{\rm t} = 172.6 \pm 0.6 ({\rm stat}) \pm 1.2 ({\rm syst})$ GeV. The systematic uncertainty is dominated by the b-jet energy scale uncertainty and the uncertainty on the factorization scale. The systematic uncertainty does not include the uncertainty on color reconnection and on the underlying event.
Measurement of the top-quark mass from the $\rm t \bar{t}$ cross section at $\sqrt{s} = 7$ TeV
==============================================================================================
One of the problems of the direct measurements of $m_{\rm t}$ is that they use the mass definition from Monte Carlo generators, which is not related to $m_{\rm t}$ in a well-defined renormalization scheme ($m_{\rm t}^{\rm pole}$ or $m_{\rm t}^{\rm \overline{MS}}$) in a straightforward way. These masses can however be extracted from the measured cross section of top quark pair production $\sigma_{\rm t \bar{t}}$, since the theoretical dependency of $\sigma_{\rm t\bar{t}}$ on $m_{\rm t}^{\rm pole}$ or $m_{\rm t}^{\rm \overline{MS}}$ is known. ATLAS used $\sigma_{\rm t \bar{t}}$ measured from 35 pb$^{-1}$ of data in the $\ell$+jets channel, and obtains $m_{\rm t}^{\rm pole} = 166.4 ^{+7.8}_{-7.3}$ GeV for the pole mass [@MassFromXSATLAS]. CMS used $\sigma_{\rm t\bar{t}}$ measured in the dilepton final state from 1.14 fb$^{-1}$ of data, and measures $m_{\rm t}^{\rm pole} = 170.3 ^{+7.3}_{-6.7}$ GeV and $m_{\rm t}^{\rm \overline{MS}} = 163.1 ^{+6.8}_{-6.1}$ GeV for the pole mass and the $\rm \overline{MS}$ mass, respectively [@MassFromXSCMS]. Both the ATLAS and CMS results are in agreement with previous measurement performed by CDF and D0 [@TevTopMassTalk].
Measurement of the mass difference between top and antitop quarks
=================================================================
One of the fundamental symmetries in the standard model, the invariance under CPT transformations, can be tested by measuring the difference in mass between a particle and the corresponding antiparticle. Since the top quark is the only quark that decays before hadronization can take place, this difference can be measured directly. CMS performed a measurement using 1.09 fb$^{-1}$ of data in the $\mu$+jets channel [@TopMassDiff]. The events were splitted in two distinct samples according to the charge of the lepton. In each of these two samples, the mass of the hadronically decaying top quarks was measured and finally both masses were subtracted from eachother. This resulted in $\Delta m_{\rm t} = -1.20 \pm 1.21 ({\rm stat}) \pm 0.47 ({\rm syst})$ GeV. The smallness of the systematic uncertainty, when compared to $m_{\rm t}$ measurements, can be explained by the cancellation of most systematics by taking the difference.
Conclusion and outlook
======================
Currently the most precise measurements of $m_{\rm t}$ are performed by the Tevatron experiments, but the ATLAS and CMS results are getting more and more precise. An overview of all these results can be found in Fig. \[fig:OverviewMtop\].
The next step is to combine all the CMS and ATLAS measurements, on which work has already started, and finally to combine these also with the CDF and D0 results. CMS already performed a combination of its measurements, resulting in $m_{\rm t} = 172.6 \pm 0.4 (\text{stat}) \pm 1.2(\text{syst})$ GeV.
With the huge amount of data which will be recorded by ATLAS and CMS, the precision on $m_{\rm t}$ is expected to increase. The uncertainties are currently dominated by the jet energy scale systematic uncertainty, which can be reduced once more data is available to be analyzed.
References {#references .unnumbered}
==========
[99]{} CDF and D0 Collaboration, “Combination of CDF and DO results on the mass of the top quark using up to 5.8 fb$^{-1}$ of data”, (2011). `arXiv:1107.5255`.
O. Brandt, “Measurements of the top quark mass at the Tevatron”, these proceedings. `arXiv:1204.0919`.
I. Aracena, “Top Pair Production at $\rm E_{cm} = 7$ TeV”, these proceedings.
CMS Collaboration, “Measurement of the top-quark mass in the dilepton channel in pp collisions at $\sqrt{s} = 7$ TeV”, [*CMS Physics Analysis Summary*]{} [**CMS-PAS-TOP-11-016**]{} (2012).
CMS Collaboration, “Measurement of the ${\rm t \bar{t}}$ production and the top-quark mass in the dilepton channel in pp collisions at $\sqrt{s} = 7$ TeV”, [*JHEP*]{} [**07**]{} (2011) 049. `arXiv:1105.5661`.
D0 Collaboration, “Measurement of the top quark mass in ${\rm p \bar{p}}$ collisions using events with two leptons”, submitted to [*Phys. Rev. Lett.*]{} `arXiv:1201.5172`.
ATLAS Collaboration, “Determination of the Top Quark Mass with a Template Method in the All-Hadronic Decay channel using 2.04 fb$^{-1}$ of ATLAS data”, [*ATLAS Note*]{} [**ATLAS-CONF-2012-030**]{} (2012).
ATLAS Collaboration, “Measurement of the top quark mass with the template method in the top antitop $\to$ lepton + jets channel using ATLAS data”, submitted to [*Eur. Phys. J.*]{} [**C**]{}, `arXiv:1203.5755`.
CMS Collaboration, “Measurement of the top quark mass in the muon+jets channel”, [*CMS Physics Analysis Summary*]{} [**CMS-PAS-TOP-11-015**]{} (2012).
ATLAS Collaboration, “Determination of the Top-Quark Mass from the $\rm t\bar{t}$ Cross Section Measurement in pp Collisions at $\sqrt{s} = 7$ TeV with the ATLAS detector”, [*ATLAS Note*]{} [**ATLAS-CONF-2011-054**]{} (2011).
CMS Collaboration, “Determination of the Top Quark Mass from the $\rm t \bar{t}$ Cross Section at $\sqrt{s} = 7$ TeV”, [*CMS Physics Analysis Summary*]{} [**CMS-PAS-TOP-11-008**]{} (2011).
CMS Collaboration, “Measurement of the mass difference between top and antitop quarks”, [*CMS Physics Analysis Summary*]{} [**CMS-PAS-TOP-11-019**]{} (2011).
|
---
abstract: 'Let $T$ be a tile in $\mathbb{Z}^n$, meaning a finite subset of $\mathbb{Z}^n$. It may or may not tile $\mathbb{Z}^n$, in the sense of $\mathbb{Z}^n$ having a partition into copies of $T$. However, we prove that $T$ does tile $\mathbb{Z}^d$ for some $d$. This resolves a conjecture of Chalcraft.'
author:
- 'Vytautas Gruslys[^1]'
- Imre Leader
- 'Ta Sheng Tan[^2]'
bibliography:
- 'tilings.bib'
nocite: '[@*]'
title: Tiling with arbitrary tiles
---
Introduction
============
Let $T$ be a *tile*, by which we mean a finite non-empty subset of ${\mathbb{Z}}^n$ for some $n$. It is natural to ask if ${\mathbb{Z}}^n$ can be partitioned into copies of $T$, that is, into subsets each of which is isometric to $T$. If such a partition exists, we say that $T$ *tiles* ${\mathbb{Z}}^n$.
For instance, consider the following tiling of ${\mathbb{Z}}^2$ by translates of the $S$-shaped tetromino.
![ The $S$-shaped tetromino tiles ${\mathbb{Z}}^2$. []{data-label="fig:tetromino-s"}](\diagrams{diag-17})
As another example, the one-dimensional tile $\mathtt{X.X}$ (to be understood as $\{1, 3\}$) tiles ${\mathbb{Z}}$, and so does $\mathtt{XX.X}$. On the other hand, $\mathtt{XX.XX}$ is a one-dimensional tile that does not tile ${\mathbb{Z}}$. Does it tile some space of higher dimension? The following diagram shows that $\mathtt{XX.XX}$ does tile ${\mathbb{Z}}^2$.
![ This pattern is formed from disjoint copies of $\mathtt{XX.XX}$; copies of the pattern may be stacked vertically to tile ${\mathbb{Z}}^2$. []{data-label="fig:xx.xx"}](\diagrams{diag-19})
A similar pattern works for $\mathtt{XXX.XX}$ in ${\mathbb{Z}}^2$. However, one can check by hand that $\mathtt{XXX.XXX}$ does not tile ${\mathbb{Z}}^2$. Does it tile ${\mathbb{Z}}^3$, or ${\mathbb{Z}}^d$ for some $d$? What about more complicated one-dimensional tiles?
Let us now consider a couple of two-dimensional examples. Let $T$ denote the $3 \times 3$ square with the central point removed. Clearly $T$ does not tile ${\mathbb{Z}}^2$, since the hole in a copy of $T$ cannot be filled. However, in ${\mathbb{Z}}^3$ there is enough space for one copy of $T$ to fill the hole of another. (Of course, this in no way implies that $T$ does tile ${\mathbb{Z}}^3$.)
For a ‘worse’ example, consider the $5 \times 5$ square with the central point removed. Two copies of such tile cannot be interlinked in ${\mathbb{Z}}^3$. However, there is, of course, enough space in ${\mathbb{Z}}^4$ to fill the hole, as demonstrated in the following diagram.
![ The diagram on the right is four-dimensional and shows a $5 \times 5 \times 5 \times 5$ region of ${\mathbb{Z}}^4$. Let $x_1, x_2, x_3, x_4$ be the directions of ${\mathbb{Z}}^4$. Each of the five $5 \times 5 \times 5$ cubes corresponds to a fixed value of $x_1$. Increasing the value of $x_1$ by $1$ means jumping from a cube to the cube on its right. This four-dimensional diagram contains two copies of the two-dimensional tile depicted on the left side. One copy is horizontal and can be found in the top left part of the diagram. The second copy is formed by the vertical columns. []{data-label="fig:coveringholes"}](\diagrams{diag-20})
Chalcraft [@mathforum; @mathoverflow] made the remarkable conjecture that every tile $T \subset {\mathbb{Z}}$, or even $T \subset {\mathbb{Z}}^n$, does tile ${\mathbb{Z}}^d$ for some $d$.
Let $T \subset {\mathbb{Z}}^n$ be a tile. Then $T$ tiles ${\mathbb{Z}}^d$ for some $d$.
It is not important if reflections are allowed when forming copies of a tile. Indeed, any reflection of an $n$-dimensional tile can be obtained by rotating it in $n+1$ dimensions. It is also not important if only connected tiles are considered, as it is an easy exercise to show that any disconnected tile in ${\mathbb{Z}}^n$ tiles a connected tile in ${\mathbb{Z}}^{2n}$.
In this paper we prove Chalcraft’s conjecture.
[thm]{}[main]{} \[thm:main\] Let $T \subset {\mathbb{Z}}^n$ be a tile. Then $T$ tiles ${\mathbb{Z}}^d$ for some $d$.
Interestingly, the problem is not any easier for tiles $T \subset {\mathbb{Z}}$. Indeed, the proof for one-dimensional tiles seems to us to be as hard as the general problem.
The plan of the paper is as follows. In Section \[sec:simplecase\] we prove a special case of the theorem, namely when $T$ is an interval in ${\mathbb{Z}}$ with one point removed. The aim of this section is to demonstrate some of the key ideas in a simple setting. The proof of the general case builds on these ideas and on several additional ingredients. We give a proof of Theorem \[thm:main\] in Section \[sec:generalcase\]. Finally, in Section \[sec:conclusion\] we give some open problems.
We end the section with some general background. A lot of work has been done about tiling ${\mathbb{Z}}^2$ by polyominoes (a polyomino being a connected tile in ${\mathbb{Z}}^2$). Golomb [@golomb66] proved that every polyomino of size at most $6$ tiles ${\mathbb{Z}}^2$. In [@golomb70] he also proved that there is no algorithm which decides, given a finite set of polyominoes, if ${\mathbb{Z}}^2$ can be tiled with their copies – this is based on the work of Berger [@berger66], who showed a similar undecidability result for Wang tiles (which are certain coloured squares). However, it is not known if such an algorithm exists for single polyominoes. A related unsolved problem is to determine whether there is a polyomino which tiles ${\mathbb{Z}}^2$ but such that every tiling is non-periodic. On the other hand, Wijshoff and van Leeuwen [@wijshoff84] found an algorithm which determines if disjoint translates (rather than translates, rotations and reflections) of a single given polyomino tile ${\mathbb{Z}}^2$. A vast number of results and questions regarding tilings of ${\mathbb{Z}}^2$ by polyominoes and other shapes are compiled in Grünbaum and Shephard [@grunbaum13].
One may also wish to know if a given polyomino tiles some finite region of ${\mathbb{Z}}^2$, say a rectangle. This class of questions has also received significant attention, producing many beautiful techniques and invariants – see, for example, [@conway90; @golomb89; @klarner69]. In the context of this paper, we observe that there are tiles which cannot tile any (finite) cuboid of any dimension. For example, consider the plus-shaped tile of size $5$ in ${\mathbb{Z}}^2$: this tile cannot cover the corners of any cuboid. In fact, there are one-dimensional such tiles. For example, it turns out that any symmetric tile $T \subset {\mathbb{Z}}$ whose associated polynomial $p(x) = \sum_{t \in T} x^t$ does not have all of its roots on the unit circle cannot tile a cuboid (see [@mathoverflow]).
Tiling ${\mathbb{Z}}^d$ by an interval minus a single point {#sec:simplecase}
===========================================================
Overview {#subsec:overview}
--------
Before starting the proof of Theorem \[thm:main\], we demonstrate some of the key ideas in a simple setting, where the tile is a one-dimensional interval with one point removed. We give a self-contained proof of the general case in Section \[sec:generalcase\], but it will build on the ideas in this section.
We write $[k] = \{1, \dotsc, k\}$.
[thm]{}[simpletheorem]{} \[thm:simplecase\] Fix integers $k \ge 3$ and $i \in \{2, \dotsc, k-1\}$ and let $T$ be the tile $[k] {\setminus}\{i\}$. Then $T$ tiles ${\mathbb{Z}}^d$ for some $d$.
The tile $T = [k] {\setminus}\{i\}$ will remain fixed throughout this section.
The proof is driven by two key ideas. A first natural idea is to use *strings*, where a string is a one-dimensional infinite line in ${\mathbb{Z}}^d$ with every $k$-th point removed. Note that any string is a union of disjoint copies of $T$. An obvious way to use strings would be to partition ${\mathbb{Z}}^d$ into them. Although this is an attractive idea, it is not possible, for the following simple reason: if we consider just the fixed cuboid $[k]^d \subset {\mathbb{Z}}^d$, then every string intersects it in exactly $0$ or $k-1$ points, but the order of $[k]^d$ is not divisible by $k-1$.
This suggests a refinement of the idea. We will try to use strings parallel to $d-1$ of the $d$ directions, while the remaining direction will be special and copies of $T$ parallel to it will be used even without forming strings. In other words, we will view ${\mathbb{Z}}^d$ as ${\mathbb{Z}}\times {\mathbb{Z}}^{d-1}$, that is, as being partitioned into $(d-1)$-dimensional *slices* according to the value of the first coordinate. We will first put down some tiles parallel to the first direction (each such tile intersects multiple slices), and then complete the tilings in each slice separately by strings.
To do this we need another idea. What subsets of ${\mathbb{Z}}^{d-1}$ can be tiled by strings? Note that a partial tiling of ${\mathbb{Z}}^{d-1}$ by strings can be identified with a partial tiling of the discrete torus ${{\mathbb{Z}}_{k}}^{d-1}$ (where ${{\mathbb{Z}}_{k}}$ denotes the integers modulo $k$), where a tile in ${{\mathbb{Z}}_{k}}^{d-1}$ means any line with one point removed. The size of ${{\mathbb{Z}}_{k}}^{d-1}$ is $k^{d-1}
\equiv 1 \pmod{k-1}$, so any such partial tiling of ${{\mathbb{Z}}_{k}}^{d-1}$ must leave out $1 \pmod{k-1}$ points. Of course, it is far from true that any subset of ${{\mathbb{Z}}_{k}}^{d-1}$ of size a multiple of $k-1$ may be partitioned into tiles. However, our plan is to find a large supply of sets that do have this property. In particular, it turns out that a key idea will be to find a large set $C \subset {{\mathbb{Z}}_{k}}^{d-1}$ such that for any choice of distinct elements $x_1, \dotsc, x_m \in C$ with $m \equiv 1 \pmod{k-1}$, $T$ does tile ${{\mathbb{Z}}_{k}}^{d-1} {\setminus}\{x_1, \dotsc, x_m\}$.
![ A partial tiling of ${\mathbb{Z}}_k^2$ (here $k=5$) corresponds to a partial tiling of ${\mathbb{Z}}^2$ by strings. []{data-label="fig:torustoperiodic"}](\diagrams{diag-1})
These ideas work together as follows (see Figure \[fig:specialdimension\]). First, in ${\mathbb{Z}}\times
{{\mathbb{Z}}_{k}}^{d-1}$ (for large $d$) we find a subset $X$ which is a disjoint union of translates of $T \times \{0\}^{d-1}$ and has the property that for any $n \in {\mathbb{Z}}$ the set $\{ x \in {{\mathbb{Z}}_{k}}^{d-1} : (n, x) \in
X\}$ is a subset of $C$ of size congruent to $1$ modulo $k-1$. Then $T$ tiles $(\{n\} \times {{\mathbb{Z}}_{k}}^{d-1}) {\setminus}X$. This holds for all $n \in {\mathbb{Z}}$, so in fact $T$ tiles ${\mathbb{Z}}\times {{\mathbb{Z}}_{k}}^{d-1}$, and hence it tiles ${\mathbb{Z}}^d$, establishing Theorem \[thm:simplecase\].
![ The aim is to put down tiles parallel to one of the directions so that the remainder of each slice could be tiled by strings. This diagram only symbolically visualises this principle. In particular, the slices here are two-dimensional, while in the proof they can have much higher dimension. []{data-label="fig:specialdimension"}](\diagrams{diag-2})
The rest of this section is organised as follows. In Section \[subsec:removedcorners-simple\] we consider partial tilings by strings. In Section \[subsec:specialdimension-simple\] we consider the special direction. Both ideas are combined in Section \[subsec:formalities-simple\], where a full proof of Theorem \[thm:simplecase\] is given.
Tiling ${{\mathbb{Z}}_{k}}^d$ with some elements removed {#subsec:removedcorners-simple}
--------------------------------------------------------
For any $1 \le j \le d$, define the *$j$-th corner* of ${{\mathbb{Z}}_{k}}^d$ to be $c_{j,d}$ where $$c_{j,d} =
(
\underbrace{ 0,\dotsc,0,
\overset{ \overset{ \mathclap{j\text{-th coordinate}}
}{ \downarrow
}
}{ k-1
},
0,\dotsc,0
}_{d \text{ coordinates}}
)
\in {{\mathbb{Z}}_{k}}^d.$$ Write $C_d = \{c_{j,d} : j = 1, \dotsc, d\}$ for the set of corners.
![ The set of corners $C_4$ when $k = 6$. In this diagram the space ${{\mathbb{Z}}_{6}}^4 = \{(x_1, x_2, x_3, x_4) : x_i \in
\{0, \dots, 5\}\}$ is split from left to right, according to the value of $x_4$, into $6$ three-dimensional slices. []{data-label="fig:corners"}](\diagrams{diag-3})
Looking ahead, our aim later will be to provide some copies of $T$ in the $x_1$-direction in ${\mathbb{Z}}\times {{\mathbb{Z}}_{k}}^d$, at heights corresponding to points of $C_d$, and in such a way that what remains in each ${{\mathbb{Z}}_{k}}^d$ can be partitioned into lines with one point removed. But first we need to create a useful supply of such subsets of ${{\mathbb{Z}}_{k}}^d$.
Recall that $|{{\mathbb{Z}}_{k}}^d| \equiv 1\pmod{k-1}$, so if $T$ tiles some set $X \subset {{\mathbb{Z}}_{k}}^d$ (here and in the remainder of this section $T$ is identified with its image under the projection ${\mathbb{Z}}\to {{\mathbb{Z}}_{k}}$, so its copies in ${{\mathbb{Z}}_{k}}^d$ are lines with one point removed), then $|{{\mathbb{Z}}_{k}}^d {\setminus}X|
\equiv 1\pmod{k-1}$. In this section we will prove Lemma \[lem:removedcorners\], which is an approximate converse of this statement.
\[lem:removedcorners\] Let $d \ge 1$ and suppose that $S \subset C_d$ is such that $|S| \equiv 1~(\text{mod } k-1)$ and $|S| \le
d - \log_k{d}$. Then $T$ tiles ${{\mathbb{Z}}_{k}}^d {\setminus}S$.
In fact, this lemma holds even without the assumption that $|S| \le d - \log_k{d}$, but we keep it for the sake of simpler presentation.
We will prove Lemma \[lem:removedcorners\] at the end of this section. Meanwhile, we collect the tools needed for the proof. In fact, there are several ways to prove Lemma \[lem:removedcorners\]. The method outlined here is quite general, and we will build on it in Section \[sec:generalcase\].
We start with a simple proposition.
\[prop:coverbutone\] Let $d \ge 1$ and $x \in {{\mathbb{Z}}_{k}}^d$. Then $T$ tiles ${{\mathbb{Z}}_{k}}^d {\setminus}\{x\}$.
Use induction on $d$. If $d = 1$, then ${{\mathbb{Z}}_{k}}{\setminus}\{x\}$ is itself a translate of $T$. Now suppose that $d \ge 2$ and write $x = (x_1, \dotsc, x_d)$, $\hat{x} = (x_1, \dotsc, x_{d-1})$. By the induction hypothesis, for each $j \in {{\mathbb{Z}}_{k}}$, $({{\mathbb{Z}}_{k}}^{d-1}{\setminus}\{\hat{x}\}) \times \{j\}$ can be tiled with copies of $T$. It remains to tile $\{\hat{x}\} \times ({{\mathbb{Z}}_{k}}
{\setminus}\{x_d\})$, but this is itself a copy of $T$.
![ The induction step in the proof of Proposition \[prop:coverbutone\]. The grey cube represents $x$. The vertical column in which $x$ lies, without $x$ itself, is a copy of $T$. Each horizontal slice minus the point in this column can be tiled by the induction hypothesis. []{data-label="fig:onemissing"}](\diagrams{diag-4})
Let $X \subset {{\mathbb{Z}}_{k}}^d$ (for any $d \ge 1$) be such that $T$ tiles ${{\mathbb{Z}}_{k}}^d {\setminus}X$. We will say that such $X$ is a *hole* in ${{\mathbb{Z}}_{k}}^d$. The intuition for $X$ is that it is a set that remains uncovered after an attempt to tile ${{\mathbb{Z}}_{k}}^d$ by copies of $T$.
We can identify $X$ with a higher-dimensional set ${{X}\ifthenelse{\isempty{}}{'}{'^{()}}} = X \times \{0\} \subset {{\mathbb{Z}}_{k}}^{d+1}$. One can easily verify that ${{X}\ifthenelse{\isempty{}}{'}{'^{()}}}$ is a hole in ${{\mathbb{Z}}_{k}}^{d+1}$. More importantly, we will show in the following proposition that a single additional point of $X'$ can be covered in exchange for leaving the $(d+1)$-st corner of ${{\mathbb{Z}}_{k}}^{d+1}$ uncovered (see Figure \[fig:movepoint\]). This is why, for any $S \subset {{\mathbb{Z}}_{k}}^d$, we define $${{S}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}} = (S \times \{0\}) \cup \{ c_{d+1, d+1} \}
\subset {{\mathbb{Z}}_{k}}^{d+1}.$$ Note that the definition of ${{S}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}}$ and the definition of $S$ being a hole depend not only on $S$, but also on the dimension of the underlying discrete torus ${{\mathbb{Z}}_{k}}^d$. For $m \ge 1$, we will use the shorthand ${{S}^{\ifthenelse{\isempty{m}}{\dagger}{\dagger(m)}}}$ to denote the result of $m$ consecutive applications of the ${{}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}}$ operation to $S$, that is, $${{S}^{\ifthenelse{\isempty{m}}{\dagger}{\dagger(m)}}} = S{\underbrace{^{\dagger\dotsc\dagger}}_{m}}
\subset {{\mathbb{Z}}_{k}}^{d+m}.$$
![ Suppose $S$ is the subset of ${{\mathbb{Z}}_{k}}^2$ given in the diagram on the left (here $k=6$). The diagram in the middle depicts ${{S}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}}$, and the diagram on the right depicts ${{S}^{\ifthenelse{\isempty{2}}{\dagger}{\dagger(2)}}}$. Observe that $S$ is a hole in ${{\mathbb{Z}}_{k}}^2$, but ${{S}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}}$ and ${{S}^{\ifthenelse{\isempty{2}}{\dagger}{\dagger(2)}}}$ are not holes in ${{\mathbb{Z}}_{k}}^3$ and ${{\mathbb{Z}}_{k}}^4$, respectively. []{data-label="fig:dagger"}](\diagrams{diag-5})
\[prop:movepoint\] Let $d \ge 1$ and let $X \subset {{\mathbb{Z}}_{k}}^d$ be a hole. Then for each $x \in X$ the set ${{(X {\setminus}\{x\})}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}}$ is a hole in ${{\mathbb{Z}}_{k}}^{d+1}$.
![ An illustration of the statement of Proposition \[prop:movepoint\]. The aim is to show that $T$ tiles ${{\mathbb{Z}}_{k}}^{d+1} \setminus (X \setminus \{x\})^{\dagger}$. []{data-label="fig:movepoint"}](\diagrams{diag-6})
Use a tiling of ${{\mathbb{Z}}_{k}}^d {\setminus}X$ for $({{\mathbb{Z}}_{k}}^d {\setminus}X) \times \{0\}$, and one copy of $T$ to cover $\{x\}\times({{\mathbb{Z}}_{k}}{\setminus}\{k-1\})$. By Proposition \[prop:coverbutone\], $({{\mathbb{Z}}_{k}}^d{\setminus}\{(0,\dotsc,0)\})\times
\{k-1\}$ and $({{\mathbb{Z}}_{k}}^d{\setminus}\{x\})\times\{i\}$, $i \in
{{\mathbb{Z}}_{k}}{\setminus}\{0,k-1\}$, can each be tiled with copies of $T$.
![ An illustration of the proof of Proposition \[prop:movepoint\]. The bottom horizontal piece is tilable because $X$ is a hole, and the other horizontal pieces are tilable by Proposition \[prop:coverbutone\]. The remaining vertical column is a copy of $T$. []{data-label="fig:movepointproof"}](\diagrams{diag-7})
We will apply Proposition \[prop:movepoint\] inductively, that is, in the form of the following corollary.
\[cor:exchangeall-simple\] Let $d \ge 1$ and let $X \subset {{\mathbb{Z}}_{k}}^d$ be a hole. Then for any distinct elements $x_1, \dotsc x_m \in X$, the set ${{(X {\setminus}\{x_1, \dotsc, x_m\})}^{\ifthenelse{\isempty{m}}{\dagger}{\dagger(m)}}}$ is a hole in ${{\mathbb{Z}}_{k}}^{d+m}$.
We are now ready to prove Lemma \[lem:removedcorners\].
Write $|S| = m$ and $r = d-m$. By symmetry, we can assume that $$S =
\left\{ c_{j,d} : j = r+1, \dotsc, d \right\}.$$
Our aim is to prove that $S$ is a hole in ${{\mathbb{Z}}_{k}}^d$. Note that $S = {{{\emptyset}}^{\ifthenelse{\isempty{m}}{\dagger}{\dagger(m)}}}$, where the empty set ${\emptyset}$ is considered as a subset of ${{\mathbb{Z}}_{k}}^r$. Therefore by Corollary \[cor:exchangeall-simple\] it suffices to find a hole $X \subset {{\mathbb{Z}}_{k}}^r$ with $|X| = m$.
This can be done by partitioning ${{\mathbb{Z}}_{k}}^r$ into a singleton $\{x\}$ and copies of $T$ (this can be done by Proposition \[prop:coverbutone\]), and letting $X$ be the union of $\{x\}$ and the appropriate number of copies of $T$. By assumption, $m \equiv 1\pmod{k-1}$ so the only potential problem with this construction of $X$ is if $|{{\mathbb{Z}}_{k}}^r| < m$. However, this is ruled out by the assumption that $m \le d - \log_k{d}$.
Using one special direction to get $T$-tilable slices {#subsec:specialdimension-simple}
-----------------------------------------------------
The purpose of this section is to demonstrate that tiles in the first direction in ${\mathbb{Z}}\times {{\mathbb{Z}}_{k}}^{d-1}$ (that is, translates of $T \times \{0\}^{d-1}$) can be combined in such a way that the uncovered part of each slice can be tiled by copies of $T$ using Lemma \[lem:removedcorners\]. The exact claim is as follows.
\[lem:specialdimension\] There exists a number $\ell \ge 1$ such that for any set $C$ of order $|C| \ge \ell$ there is a set $X \subset {\mathbb{Z}}\times C$, satisfying:
1. $X$ is a union of disjoint sets of the form $(T + n) \times \{c\}$ with $n \in {\mathbb{Z}}$ and $c
\in C$;
2. $|(\{n\} \times C) \cap X| \equiv 1~ (\text{mod } k-1)$ for every $n \in {\mathbb{Z}}$.
![ A possible construction of $X$. In this example the aim is to have $1$ modulo $6$ elements covered in each column. []{data-label="fig:specialdimensionin2d"}](\diagrams{diag-8})
We start with the following trivial proposition.
\[prop:deconvolution-simple\] There is a function $f : {\mathbb{Z}}\to \{0, \dots, k-2\}$ such that for each $x \in {\mathbb{Z}}$ $$\sum_{y \in T} f(x - y) \equiv 1~(\text{mod }k-1).$$
Start by defining $f(n) = 0$ for $-k+1 \le n \le -1$. Now define $f(n)$ for $n \ge 0$ as follows. Suppose that for some $n \ge 0$ the values of $f(j)$ are already defined for all $j$ such that $-k+1 \le j \le n-1$. Then the value of $f(n)$ is uniquely defined by $$f(n) \hspace{5pt} \equiv \hspace{5pt}
1 \hspace{5pt} - \sum_{y \in T {\setminus}\{1\}} f(n+1 - y)
\hspace{5pt} (\text{mod }k-1).$$
Define $f(n)$ for all $n \le -k$ in a similar way.
Now Lemma \[lem:specialdimension\] can be proved quickly.
Write $\ell = 2k(k-2)$ and suppose that $|C| \ge \ell$. Let $f : {\mathbb{Z}}\to \{0, \dots, k-2\}$ be as given by Proposition \[prop:deconvolution-simple\]. The aim is to choose subsets $S_n \subset C$ for every $n \in {\mathbb{Z}}$, with orders satisfying $|S_n| = f(n)$, and such that $S_m \cap S_n = {\emptyset}$ whenever $m \neq n$ and $(T+m)\cap(T+n) \neq {\emptyset}$. Then $X$ can be taken to be $\bigcup_{n\in{\mathbb{Z}}} (T+n) \times S_n$.
Fix any enumeration of ${\mathbb{Z}}$, and define the sets $S_n$ one by one in that order. When defining $S_n$, there can be at most $2k - 1$ choices of $m$ with $S_m$ already defined and $m-n \in T-T$. Moreover, $|S_m| \le k-2$ for each $m$. Therefore to be able to find $S_n$ it is enough to have $|C| -
(2k - 1)(k-2) \ge f(n)$. Finally, this condition is ensured by the choice of $\ell$, completing the proof.
Completing the proof of Theorem \[thm:simplecase\] {#subsec:formalities-simple}
--------------------------------------------------
It was noted in Section \[subsec:overview\] that Lemmas \[lem:removedcorners\] and \[lem:specialdimension\] together imply that for some $d \ge 1$ $$\begin{aligned}
&T \text{ tiles } {\mathbb{Z}}\times{{\mathbb{Z}}_{k}}^{d-1},
\label{eq:zzkdtilable}\\
\shortintertext{and therefore}
&T \text{ tiles } {\mathbb{Z}}^d,
\label{eq:zdtilable}
\end{aligned}$$ implying Theorem \[thm:simplecase\]. However, some abuse of notation is already present in the statement of (\[eq:zzkdtilable\]). In this section we will carefully explain what is meant by (\[eq:zzkdtilable\]), why it follows from the two lemmas and how it implies (\[eq:zdtilable\]). In doing so, we will complete the proof of Theorem \[thm:simplecase\].
To avoid confusion, within this section we use quite precise language. Although this might seem pedantic here, for later it will be very important to have precise notation available. We denote the elements of ${{\mathbb{Z}}_{k}}$ by ${\overline{x}}$ for $x \in {\mathbb{Z}}$ (instead of identifying them with $x$, which was our preferred notation in the rest of the section), and we will denote the image of $T$ under the natural projection $\pi : {\mathbb{Z}}\to {{\mathbb{Z}}_{k}}$ by $\pi(T)$ rather than simply by $T$.
Fix a large $d$ (more precisely, first let $\ell$ be as given by Lemma \[lem:specialdimension\] and then fix $d$ such that $d-1 - \log_k (d-1) \ge \ell$).
Denote the projection map ${\mathbb{Z}}\to{{\mathbb{Z}}_{k}}$ by $\pi$, and consider the following subsets of ${\mathbb{Z}}\times {{\mathbb{Z}}_{k}}^{d-1}$:
$$\begin{aligned}
{5}
\mathsf{T}_1 &=
\hspace{6pt} T && \times \,{\,\left\{\,{\overline{0}}\,\right\}\,}&& \times {\,\left\{\,{\overline{0}}\,\right\}\,}&& \times \dotsb && \times {\,\left\{\,{\overline{0}}\,\right\}\,}, \\
\mathsf{T}_2 &=
\big\{\,0\,\big\} && \times \pi(T) && \times {\,\left\{\,{\overline{0}}\,\right\}\,}&& \times \dotsb && \times {\,\left\{\,{\overline{0}}\,\right\}\,}, \\
&\hspace{6pt}\vdots && && && && \\
\mathsf{T}_d &=
\big\{\,0\,\big\} && \times \,{\,\left\{\,{\overline{0}}\,\right\}\,}&& \times {\,\left\{\,{\overline{0}}\,\right\}\,}&& \times \dotsb && \times \pi(T).
\end{aligned}$$
Recall from Section \[subsec:removedcorners-simple\] the definition of $$C_{d-1} =
\left\{ (\,
\underbrace{ {\overline{0}}\,,\,\dotsc\,,\,{\overline{0}}\,,
\overset{ \overset{ \mathclap{j\text{-th coordinate}}
}{ \downarrow
}
}{ \,{\overline{k-1}}\,
},
\,{\overline{0}}\,,\,\dotsc\,,\,{\overline{0}}
}_{d-1 \text{ coordinates}}
\,) : j = 1,\dotsc,d-1
\right\}
\subset {{\mathbb{Z}}_{k}}^{d-1}.$$ By Lemma \[lem:specialdimension\], there is a set $X \subset {\mathbb{Z}}\times C_{d-1}$, which is a union of disjoint translates of $\mathsf{T}_1$ and for each $n \in {\mathbb{Z}}$ satisfies $|(\{n\} \times C_{d-1}) \cap X| \le d-1 - \log_k (d-1)$ and $|(\{n\} \times C_{d-1}) \cap X| \equiv 1\pmod{k-1}$. Hence, by Lemma \[lem:removedcorners\], $(\{n\}\times{{\mathbb{Z}}_{k}}^{d-1})
{\setminus}X$ is a union of disjoint translates of $\mathsf{T}_2, \dotsc, \mathsf{T}_d$ for each $n \in {\mathbb{Z}}$. Therefore ${\mathbb{Z}}\times {{\mathbb{Z}}_{k}}^{d-1}$ is a union of disjoint translates of $\mathsf{T}_1, \dotsc,
\mathsf{T}_d$ (this is exactly what is meant by (\[eq:zzkdtilable\])).
More explicitly, there are integers $1 \le {t(\alpha)}
\le d$ and ${x_1(\alpha)}, \dotsc, {x_d(\alpha)} \in {\mathbb{Z}}$, indexed by $\alpha\in A$, such that ${\mathbb{Z}}\times{{\mathbb{Z}}_{k}}^{d-1}$ is the disjoint union $${\mathbb{Z}}\times{{\mathbb{Z}}_{k}}^{d-1} = \bigsqcup_{\alpha\in A}
\Big[
\mathsf{T}_{{t(\alpha)}}+
\big(
\,{x_1(\alpha)},
\,{\overline{{x_2(\alpha)}}}\,,
\,\dotsc\,,
\,{\overline{{x_d(\alpha)}}}\,
\big)
\Big].$$
From this it follows that, in fact, ${\mathbb{Z}}^d$ is $T$-tilable. Indeed, consider the following subset of ${\mathbb{Z}}^d$: $$\begin{aligned}
{5}
\mathsf{T}_1' &=
\hspace{4pt}T && \times \{0\} && \times \{0\}
&& \times \dotsb && \times \{0\}, \\
\mathsf{T}_2' &=
\{0\} && \times \hspace{4pt}T && \times \{0\}
&& \times \dotsb && \times \{0\}, \\
&\hspace{6pt}\vdots && && && && \\
\mathsf{T}_d' &=
\{0\} && \times \{0\} && \times \{0\}
&& \times \dotsb && \times \hspace{4pt}T.
\end{aligned}$$ Then we can express ${\mathbb{Z}}^d$ as the disjoint union $${\mathbb{Z}}^d =
\bigsqcup_{
\substack{
\alpha\in A \\
\mathclap{c_2,\dotsc,c_d\in{\mathbb{Z}}}
}
}
\Big[
\mathsf{T}_{{t(\alpha)}}'+
\big(
{x_1(\alpha)},
\,{x_2(\alpha)}+kc_2\,,
\,\dotsc\,,
\,{x_d(\alpha)}+kc_d\,
\big)
\Big]. \qedhere$$
The general case {#sec:generalcase}
================
Recall the statement of the main theorem.
In this section we prove the main theorem by generalising the approach demonstrated in Section \[sec:simplecase\]. We have to account for two ways in which Theorem \[thm:simplecase\] is a special case: firstly, the tile can be multidimensional; secondly, even in the one-dimensional case the tile can have more complicated structure than in Section \[sec:simplecase\].
It turns out that dealing with the first issue does not add significant extra difficulty to the proof, provided that the right setting is chosen. Namely, most of the intermediate results will be stated in terms of abelian groups rather than integer lattices. This way a multidimensional tile $T \subset {\mathbb{Z}}^b$ can be considered as being one-dimensional, if ${\mathbb{Z}}^b$ (rather than ${\mathbb{Z}}$) is chosen as the underlying abelian group. Moreover, this point of view is vital for comparing periodic tilings of an integer lattice with tilings of a discrete torus, already an important idea in the proof of the special case.
On the other hand, dealing with the second issue requires significant effort. It involves finding the right way to generalise the two key ideas from Section \[sec:simplecase\], as well as introducing a new ingredient that allows the argument to be applied iteratively.
We now introduce some definitions. Given an abelian group $G$, we call any non-empty subset $T \subset G$ a *tile* in $G$. Given abelian groups $G_1,\dotsc,G_d$ and corresponding tiles $T_i\subset G_i$, consider the following subsets of $G_1 \times \dotsb \times G_d$: $$\begin{aligned}
{4}
\mathsf{T}_1 &= \hspace{3pt} T_1 &&\times \{0\}
&&\times \dotsb &&\times \{0\}, \\
\mathsf{T}_2 &= \{0\} &&\times \hspace{3pt} T_2
&&\times \dotsb &&\times \{0\}, \\
&\hspace{6pt}\vdots && && && \\
\mathsf{T}_d &= \{0\} &&\times \{0\}
&&\times \dotsb &&\times \hspace{3pt} T_d.
\end{aligned}$$ Any translate of such $\mathsf{T}_i$ (that is, a set of the form $\mathsf{T}_i + x$ for $x \in G_1\times\dotsb\times G_d$) is called a *copy of $T_i$*. We say that a subset $X\subset G_1\times\dotsb\times G_d$ is *$(T_1,\dotsc,T_d)$-tilable* if $X$ is a disjoint union of copies of $T_1,\dotsc,T_d$.
It will often be the case that $(G_1, T_1) = \dotsb =
(G_d, T_d) = (G, T)$. Then we will use the term *$T$-tilable* as a shorthand for $(T,\dotsc,T)$- tilable.
More generally, we may consider subsets of $G_1^{d_1}
\times \dotsb \times G_m^{d_m}$ where $G_1, \dotsc,
G_m$ are abelian groups with tiles $T_i \subset G_i$. In this setting we would say that a subset is $({d_1\cdotT_1}, \dotsc, {d_m\cdotT_m})$-tilable. In other words, each ${d_i\cdotT_i}$ replaces $$\underbrace{T_i, \dotsc, T_i}_{d_i}.$$ However, we suppress “${1\cdot}$” in the notation. So, for example, we could say that a subset of $G_1^7 \times G_2 \times G_3^{10}$ is $({7\cdotT_1},
T_2, {10\cdotT_3})$-tilable.
A summary of the proof {#subsec:summary}
----------------------
Let $T$ be a fixed finite tile in ${\mathbb{Z}}^b$. Without loss of generality assume that $T \subset [k]^b$ for some $k \ge 1$. Then, writing $\pi:{\mathbb{Z}}^b\to{{\mathbb{Z}}_{k}}^b$ for the projection map, $\pi(T)$ is a tile in $G = {{\mathbb{Z}}_{k}}^b$.
In the light of the argument from Section \[sec:simplecase\], one might hope to find a positive integer $d$ and a large family $\mathcal{F}$ of disjoint subsets of $G^d$ with the property that whenever a subfamily $\mathcal{S} \subset \mathcal{F}$ with $|\mathcal{S}| \equiv 1 \pmod{|T|}$ is chosen, the set $G^d {\setminus}(\bigcup_{S\in\mathcal{S}}S)$ is $\pi(T)$-tilable. However, this seems to be achievable only in the case when $\pi(T)$ is in a certain sense a ‘dense’ subset of $G$.
If $\pi(T)$ is sparse, we achieve a weaker aim. Namely, we find a certain set $X \subset G^d$ which has sufficiently nice structure and is a denser subset of $G^d$ than $\pi(T)$ is of $G$. Also, we find a large family $\mathcal{F}$ of disjoint subsets of $X$ such that for any $\mathcal{S} \subset \mathcal{F}$ of appropriate size $X {\setminus}(\bigcup_{S \in \mathcal{S}} S)$ is $\pi(T)$-tilable. Taking copies of $T$ in the special direction, we can now tile ${\mathbb{Z}}^b \times X$.
Repeating this process, we can use copies of $T$ and ${\mathbb{Z}}^b \times X$ to tile ${\mathbb{Z}}^p \times Y$ for an even denser subset $Y \subset
G^l$. After finitely many iterations of this procedure we tile the whole of ${\mathbb{Z}}^q \times G^m$ for some possibly large $q$ and $m$. From this it follows that ${\mathbb{Z}}^{q + bm}$ is $T$-tilable.
The rest of this section is organised as follows. In Section \[subsec:removedcorners\] we show how any tile in a (finite) abelian group $H$ can be used to *almost* tile a sufficiently nice denser subset of $H^d$ for some $d$. This is the most complicated part of the proof, but it shares a similar structure with the simpler argument in Section \[subsec:removedcorners- simple\].
In Section \[subsec:specialdimension\] we show how one special dimension can be used to cover the gaps in every slice. The argument is almost identical to the one in Section \[subsec:specialdimension-simple\].
In Section \[subsec:general\] we observe some simple transitivity properties of tilings. They enable the iterative application of the process. The ideas in this section are fairly straightforward.
Finally, in Section \[subsec:formalproof\] we compile the tools together and complete the proof of Theorem \[thm:main\].
Almost tiling denser multidimensional sets {#subsec:removedcorners}
------------------------------------------
Our goal is to prove the following lemma.
[lem]{}[constructdenser]{} \[lem:constructdenser\]
Let $T \subsetneq G$ be a tile in a finite abelian group $G$. Then there is a set $A \subset G$, with $T \subsetneq A$, having the following property. Given any $d_0 \ge 1$, there is some $d \ge d_0$ and a family $\mathcal{F}$ consisting of at least $d_0$ pairwise disjoint subsets of $A^d$ such that $$G \times
\left(
A^d {\setminus}\bigcup_{S \in \mathcal{S}} S
\right)
\subset
G^{d+1}$$ is $T$-tilable whenever $\mathcal{S} \subset
\mathcal{F}$ satisfies $|\mathcal{S}| \equiv 1~ (\text{mod }|T|)$.
Before presenting the proof, we make a few definitions that will hold throughout this section. First, let $G$ and $T$ be fixed as in the statement of Lemma \[lem:constructdenser\]. Since $T \neq G$, we can fix an $x \in G$ such that $T + x \neq T$. Define $$\begin{aligned}
&{T^\text{up}}= T+x,\\
&{C^\text{up}}= {T^\text{up}}{\setminus}T,\\
&{C_\text{down}}= T{\setminus}{T^\text{up}},\\
&A = T\cup{T^\text{up}}.
\end{aligned}$$
![ A four-dimensional diagram that shows $A^4$. The sets ${C_\text{down}}$ and ${C^\text{up}}$ are marked on two of the axes. In this example $|A| = 5$ and $|{C_\text{down}}| = |{C^\text{up}}| = 2$. This and the following four-dimensional diagrams in this section should be understood more generally as depicting $A^d$ for any $d$, the three-dimensional slices representing copies of $A^{d-1}$. []{data-label="fig:fourdimensions"}](\diagrams{diag-10})
We will use $A$ from this definition in the proof of Lemma \[lem:constructdenser\]. For the family $\mathcal{F}$ we will take all sets of the following form. For any integers $1\le i\le d$, write $$C_{i,d}= \underbrace{ {C_\text{down}}\times\dotsb\times{C_\text{down}}\times
\overset{ \overset{ \mathclap{i\text{-th component}}
}{ \downarrow
}
}{ {C^\text{up}}}\times
{C_\text{down}}\times\dotsb\times{C_\text{down}}}_{d \text{ components}} \subset A^d.$$ Also write $C_{0,d} = ({C_\text{down}})^d$. Note that if $i \neq j$, then $C_{i,d} \cap C_{j,d} = {\emptyset}$.
Finally, as $T$ is fixed, we can simply say *tilable* instead of $T$-tilable.
![ A four-dimensional diagram, which extends the previous diagram. Note that $C_{3,4}$ and $C_{4,4}$ both intersect two three-dimensional slices, because in this example $|{C_\text{down}}| = |{C^\text{up}}| = 2$. []{data-label="fig:cornersinfourdimensions"}](\diagrams{diag-11})
One of the reasons why these definitions are useful is that they allow the following analogue of Proposition \[prop:coverbutone\].
\[prop:removedcset\] For any integers $d \ge 1$ and $0 \le i \le d$, the set $A^d {\setminus}C_{i,d}$ is tilable.
Use induction on $d$. If $d=1$, observe that $A = {C^\text{up}}\sqcup T
= {C_\text{down}}\sqcup {T^\text{up}}$, and so $A{\setminus}C_{i,1}$ ($=
A{\setminus}{C^\text{up}}\text{ or } A{\setminus}{C_\text{down}}$) is a translate of $T$.
Now suppose that $d\ge 2$ and without loss of generality assume that $i \neq d$. By the induction hypothesis, for each $g \in A$, the slice $(A^{d-1}{\setminus}C_{i,d-1})
\times \{g\}$ can be $T$-tiled. It remains to tile the set $C_{i,d-1}\times(A{\setminus}{C_\text{down}})=
C_{i,d-1}\times{T^\text{up}}$, but this is obviously a union of disjoint copies of $T$.
![ The induction step in the proof of Proposition \[prop:removedcset\]. The set $C_{i,d-1} \times {T^\text{up}}$ is a union of copies of ${T^\text{up}}$. In each slice it remains to tile a copy of $A^{d-1} {\setminus}C_{i,d-1}$. This can be done by the induction hypothesis. []{data-label="fig:removedcset"}](\diagrams{diag-12})
We now make a series of definitions that are useful for lifting subsets of lower-dimensional spaces to higher-dimensional spaces.
A *basic set* is a set of the form $A^d,\,
G \times A^d$ or $\{g\} \times A^d$ for some $g \in G$, with $d$ any positive integer. Let $X$ be a subset of a basic set $\Omega$ and write $\Omega = W \times A^d$ (so $W = A^0,\,G$ or $\{g\}$ for some $g \in G$). We define $${{X}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}} = (X \times {C_\text{down}}) \cup (W \times C_{d+1,d+1})
\subset W \times A^{d+1}.$$
![ An illustration of the definition of ${{X}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}}$, building on Figure \[fig:fourdimensions\]. The diagram on the left is four-dimensional and represents a generic set $X \subset W \times A^d$. The diagram on the right is five-dimensional and represents the corresponding ${{X}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}}$. We stress that this is an abstract illustration. In particular, here $|A| = 5$ and $|W| = 3$, while in fact we always have either $|W| = 1$ or $|W| = |G| \ge |A|$. []{data-label="fig:dagger-general"}](\diagrams{diag-13})
Moreover, for any $m \ge 1$ we use the shorthand ${{X}^{\ifthenelse{\isempty{m}}{\dagger}{\dagger(m)}}}$ to denote the result of $m$ consecutive applications of the ${{}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}}$ operation to $X$, that is, $$\begin{aligned}
{{X}^{\ifthenelse{\isempty{m}}{\dagger}{\dagger(m)}}} &= X{\underbrace{^{\dagger\dotsc\dagger}}_{m}} \\
&= \big(
X \times C_{0,m}
\big)
\cup
\left(
W \times C_{d+1,d+m}
\right)
\cup
\dotsb
\cup
\left(
W \times C_{d+m,d+m}
\right) \\
&\subset W \times A^{d+m}.
\end{aligned}$$
For the final definition, we say that $X$ is a *hole* in $\Omega$ if $\Omega {\setminus}X$ is tilable. Note that these definitions depend not only on $X$, but also on the underlying basic set $\Omega$. Therefore we will only use them when the underlying set is explicitly stated or clear from the context.
\[prop:usecset\] Let $d \ge 1$ and let $X$ be a hole in $A^d$. Suppose that $C_{i,d} \subset X$ for some $0 \le i \le d$. Then ${{(X {\setminus}C_{i,d})}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}}$ is a hole in $A^{d+1}$.
Partition $A^{d+1} {\setminus}{{(X {\setminus}C_{i,d})}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}}$ into four sets
1. $C_{i,d} \times (A {\setminus}{C^\text{up}})$ — tilable, because $A {\setminus}{C^\text{up}}= T$;
2. $(A^d {\setminus}X) \times {C_\text{down}}$ — tilable, because $A^d {\setminus}X$ is tilable;
3. $(A^d {\setminus}C_{i,d}) \times (A {\setminus}({C^\text{up}}\cup {C_\text{down}}))$ — tilable by Proposition \[prop:removedcset\];
4. $(A^d {\setminus}C_{0,d}) \times {C^\text{up}}$ — tilable by Proposition \[prop:removedcset\].
![ An illustration of the proof of Proposition \[prop:usecset\]. The three-dimensional diagram on the left represents a hole $X \subset A^d$ which contains $C_{i,d}$. The four-dimensional diagram on the right represents ${{(X {\setminus}C_{i,d})}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}}$ and demonstrates why it is a hole in $A^{d+1}$. []{data-label="fig:usecset"}](\diagrams{diag-14})
This proposition is the most useful for us in the form of the following corollary.
\[cor:mcsets\] Let $d \ge 1$ and suppose that $0 \le i_1, \dotsc, i_m
\le d$ are distinct integers. Then $${{ \big(
A^d {\setminus}(C_{i_1, d} \cup \dotsb \cup C_{i_m, d})
\big)
}^{\ifthenelse{\isempty{m}}{\dagger}{\dagger(m)}}}$$ is a hole in $A^{d+m}$.
Use induction on $m$. The base case $m=1$ is a special case of Proposition \[prop:usecset\], so suppose that $m \ge 2$. Note that $$\begin{aligned}
&{{ \big(
A^d {\setminus}(C_{i_1,d} \cup \dotsb \cup C_{i_m,d})
\big)}^{\ifthenelse{\isempty{m}}{\dagger}{\dagger(m)}}} \\
=&
{{
\left(
{{ \big(
A^d {\setminus}(C_{i_1,d} \cup \dotsb \cup C_{i_{m-1},d})
\big)
}^{\ifthenelse{\isempty{m-1}}{\dagger}{\dagger(m-1)}}}
{\setminus}C_{i_m,d+m-1}
\right)
}^{\ifthenelse{\isempty{}}{\dagger}{\dagger()}}}
\end{aligned}$$ so it is a hole in $A^{d+m-1}$ by the induction hypothesis and Proposition \[prop:usecset\].
Now we have the tools needed for the proof of Lemma \[lem:constructdenser\].
Fix any $d \ge (1 + |G|/|T|)d_0$ and write $\mathcal{F} = \{C_{i,d} : i = 1, \dotsc, d\}$. By symmetry, it is enough to find a tiling for the set $$M_m =
G \times
\left(
A^d {\setminus}(
C_{d-m+1,d} \cup \dotsb \cup C_{d,d}
)
\right)$$ for every choice of $m \le d_0$ with $m \equiv 1\pmod{|T|}$. Fix one such value of $m$, and let $M = M_m$ be the corresponding set that we have to tile.
Define $r = d - m$ and $\Omega = G \times A^r$. We will construct a partition $\mathcal{B}$ of the set $\Omega$, satisfying:
- $\mathcal{B}$ consists of the set $Y_0 =
G \times C_{0,r}$ and copies of the tile $T$;
- for each $1 \le i \le r$, there is some $y_i \in G$ such that the set $Y_i =
(T+y_i)\times C_{i,r}$ is exactly the union of some copies of $T$ in $\mathcal{B}$;
- each $y \in G$ appears at least $t= (m-1) / |T|$ times in the list $y_1, \dotsc, y_r$.
![ By constructing the partition $\mathcal{B}$ we show that the set $\bigcup_{i=0}^r Y_i$ (grey in this diagram) is a hole in $\Omega = G \times A^r$. In fact, $\bigcup_{i\in I\cup\{0\}} Y_i$ is a hole for any $I \subset [r]$. []{data-label="fig:blueprint"}](\diagrams{diag-15}){width="\textwidth"}
We start the construction by fixing any list $y_1,\dotsc, y_r$ such that each member of $G$ appears exactly $t$ times in $y_1, \dotsc, y_{t|G|}$ (in particular, this list satisfies the final condition displayed above). Note that such a list exists since $r \ge t |G|$.
Now we use induction to construct, for each $0 \le j \le r$, a partition $\mathcal{B}_j$ of $G \times A^j$ such that the first two conditions are satisfied when $\mathcal{B}$ and $r$ are replaced by $\mathcal{B}_j$ and $j$.
Let $\mathcal{B}_0 = \{G\}$. Having defined $\mathcal{B}_{j-1}$, let $\mathcal{B}_j$ consist of the following sets (see Figure \[fig:blueprintconstruction\]):
1. $G\times C_{0,j}$,
2. $X \times \{b\}$ for each $X \in
\mathcal{B}_{j-1}$ that is a copy of $T$ and each $b \in {C_\text{down}}$,
3. $\{g\} \times \{a\} \times {T^\text{up}}$ for each $g \in G {\setminus}(T + y_j)$ and each $a \in A^{j-1}$,
4. $(T+y_j) \times \{a\} \times \{b\}$ for each $a \in A^{j-1}$ and each $b \in {T^\text{up}}= A{\setminus}{C_\text{down}}$.
One can easily check that $\mathcal{B}_j$ is a partition of $G \times A^j$ with the required properties. In particular, the sets of the first two types cover $G \times A^{j-1} \times {C_\text{down}}$, and the remaining sets cover $G \times A^{j-1} \times (A {\setminus}{C_\text{down}})$.
This concludes the construction of $\mathcal{B}$.
![ The induction step in the construction of the partition $\mathcal{B}$. []{data-label="fig:blueprintconstruction"}](\diagrams{diag-16}){width=".9\textwidth"}
Define (recalling that $Y_0 = G \times C_{0,r}$ and $Y_i = (T+y_i) \times C_{i,r}$ for $1 \le i
\le t|G|$) $$S =
\Omega {\setminus}\left(
\bigcup_{i = 0}^{t|G|} Y_i
\right).$$
The point is that $S$ is tilable by the restriction of $\mathcal{B}$, and hence $S \times C_{0,m}$ is also tilable. Therefore it only remains to prove that $M {\setminus}\left(S \times C_{0,m} \right)$ is tilable, because this would imply that $M$ is tilable. Observe that $M {\setminus}\left(S \times C_{0,m} \right) = (G \times A^d) {\setminus}{{S}^{\ifthenelse{\isempty{m}}{\dagger}{\dagger(m)}}}$, so it remains to prove that ${{S}^{\ifthenelse{\isempty{m}}{\dagger}{\dagger(m)}}}$ is a hole.
To prove this, fix any $g \in G$ and write $\Omega_g$ = $\{g\} \times A^r$. Then $\Omega_g$ intersects $Y_0$ and exactly $t|T| = m-1$ of the $Y_1, \dotsc, Y_{t|G|}$. In other words, $$\Omega_g \cap S =
\{g\} \times
\left(
A^r {\setminus}\bigcup_{k=1}^m C_{j_k,r}
\right)$$ for some $0 = j_1 < j_2 < \dotsb < j_m \le r$. By Corollary \[cor:mcsets\], ${{(\Omega_g \cap S)}^{\ifthenelse{\isempty{m}}{\dagger}{\dagger(m)}}}$ is a hole in $\{g\} \times A^d$. This holds for any $g \in G$, so in fact ${{S}^{\ifthenelse{\isempty{m}}{\dagger}{\dagger(m)}}} = \bigcup_{g \in G}
{{(\Omega_g \cap S)}^{\ifthenelse{\isempty{m}}{\dagger}{\dagger(m)}}}$ is a hole in $G \times A^d$, completing the proof.
Using one special dimension to cover certain subsets in slices {#subsec:specialdimension}
--------------------------------------------------------------
In this section we show how one special dimension can be used to lay foundations for a tiling so that the tiling can be completed in each slice separately using Lemma \[lem:constructdenser\].
Here is the main result of this section. Its statement and proof are very similar to Lemma \[lem:specialdimension\] from Section \[sec:simplecase\].
[lem]{}[coverholes]{} \[lem:coverholes\] Let $t, b \ge 1$ be integers and $T$ a finite tile in ${\mathbb{Z}}^b$. Further, let $S$ be a set and let $\mathcal{F}$ be a family consisting of at least $(t-1)|T|^2$ pairwise disjoint subsets of $S$. Then there is a set $X\subset{\mathbb{Z}}^b \times S$, satisfying:
- $X$ is a union of disjoint sets of the form $(T+x)\times A$ with $x\in{\mathbb{Z}}^b$ and $A\in\mathcal{F}$, and
- for each $x\in{\mathbb{Z}}^b$ there is some $m \equiv
1~(\text{mod }t)$ such that $
\{y \in S : (x, y) \in X\}
$ is a union of $m$ distinct members of $\mathcal{F}$.
We will deduce Lemma \[lem:coverholes\] from the following simple deconvolution type statement.
Let $t \ge 1$ and $b \ge 0$ be integers, $T$ a finite tile in ${\mathbb{Z}}^b$, and $f : {\mathbb{Z}}^b \to {\mathbb{Z}}$ a function. Then there is a function $g : {\mathbb{Z}}^b \to \{0, \dotsc, t-1\}$ such that for each $x \in {\mathbb{Z}}^b$ $$\sum_{y \in T} g(x-y) \equiv f(x)\,(\text{mod } t).$$
Use induction on $b$. The base case $b = 0$ is trivial (${\mathbb{Z}}^0$ being the trivial group), so suppose that $b \ge 1$. For any $n \in {\mathbb{Z}}$, write $$T_n = \{x \in {\mathbb{Z}}^{b-1} :
(x, n) \in T\}.$$
Without loss of generality, assume that $T_0 \neq {\emptyset}$ and $T_n = {\emptyset}$ for all $n < 0$. Write $k$ for the greatest integer such that $T_k \neq {\emptyset}$. In other words, $[0, k]$ is the minimal interval containing the projection of $T$ in the last coordinate.
Set $g(x, n)=0$ for all $x \in {\mathbb{Z}}^{b-1}$ and all $n \in {\mathbb{Z}}$ such that $-k \le n \le -1$. The next step is to define $g(x, n)$ for all $n \ge 0$ and $x \in {\mathbb{Z}}^{b-1}$. Consider $N = 0, 1, \dotsc$ in turn, at each step having defined $g(x, n)$ whenever $-k \le n \le N - 1$ and $x \in {\mathbb{Z}}^{b-1}$. By the induction hypothesis, we can define $g(x, N)$ so that for all $x \in {\mathbb{Z}}^{b-1}$ $$\sum_{y \in T_0} g(x - y, N) \equiv f(x, N) -
\sum_{
\mathclap{
\substack{
1 \le j \le k \\
z \in T_j
}
}
}
g(x-z, N-j) \hspace{10pt}
(\text{mod } t).$$
The final step is to define $g(x, n)$ when $n \le -k-1$. The argument is similar. Consider $N = -k-1, -k-2, \dotsc$ in turn, at each step having defined $f(x, n)$ whenever $n \ge N+1$. By the induction hypothesis, we can define $g(x, N)$ so that for each $x \in {\mathbb{Z}}^{b-1}$ $$\sum_{y \in T_k} g(x - y, N) \equiv f(x, N+k) -
\sum_{
\mathclap{
\substack{
0 \le j \le k-1 \\
z \in T_j
}
}
}
g(x-z, N+k-j) \hspace{10pt}
(\text{mod } t).$$
These three steps together define $g$ completely, and it is easy to see that $g$ satisfies the required condition.
We get Lemma \[lem:coverholes\] as a quick corollary. In its proof we write ${\mathbb{N}}$ for $\{1, 2, \dotsc\}$.
Let $g : {\mathbb{Z}}^b \to \{0,\dotsc,t-1\}$ be such that for each $x \in {\mathbb{Z}}^b$ $$\sum_{y \in T} g(x - y) \equiv 1 \pmod{t}.$$
Let $z_1, z_2, \dotsc$ be any enumeration of the elements of ${\mathbb{Z}}^b$. We will define sets $F_1, F_2, \dotsc \subset
\mathcal{F}$ such that $|F_n| = g(z_n)$ for any $n \in {\mathbb{N}}$, and $F_m \cap F_n = {\emptyset}$ for any distinct $m, n \in {\mathbb{N}}$ with $(T+z_m) \cap (T+z_n)
\neq {\emptyset}$. Then we will be done by taking $$X = \bigcup_{
\substack{
n \in {\mathbb{N}}\\
A \in F_n
}
}
\big[
(T + z_n) \times A
\big].$$
The $F_n$ can be defined inductively. Indeed, suppose that for some $N \in {\mathbb{N}}$ the sets $F_1, \dotsc, F_{N-1}$ are already defined. Then we can define $F_N$ to consist of exactly $g(z_N)$ elements of $\mathcal{F}$ that are not contained in $F_n$ for any $n \le N-1$ with $(T + z_n) \cap (T + z_N) \neq {\emptyset}$. This is possible, because we have at most $|T|^2 - 1$ choices for such $n$, and $$g(z_N) + (|T|^2 - 1) \max_{n \le N-1} |F_n|
\le (t-1) |T|^2 \le |\mathcal{F}|.
\qedhere$$
General properties of tilings {#subsec:general}
-----------------------------
In this section we prove some transitivity results for tilings. The underlying theme is, expressed very roughly, ‘if $B$ is $A$-tilable with the help of $k$ extra dimensions, and $C$ is $B$-tilable with the help of $\ell$ extra dimensions, then $C$ is $A$-tilable with the help of $k+\ell$ extra dimensions’.
To avoid making the notation, which is already somewhat cumbersome, even more complicated we allow ourselves to abuse it in places where this is unlikely to create ambiguity. For example, given a tiling $X = \bigsqcup X_\alpha$ we may refer to the sets $X_\alpha$ as tiles (technically, they are not tiles, but copies of tiles). Otherwise, the proofs in this section are fairly straightforward.
\[prop:tilabletransitive\] Let $G,\;G_1,\dotsc,G_m$ and $H_1,\dotsc,H_n$ be abelian groups with tiles $A\subset B\subset C
\subset G,\; T_i\subset B_i\subset G_i$ and $U_i\subset C_i \subset H_i$. Suppose that $$\begin{aligned}
{2}
B_1 \times \dotsb \times B_m \times B
&{\;\text{ is }\;}&&
(T_1,\dotsc,T_m, A)
\text{-tilable}
\label{cond:b-is-a-tilable}
\shortintertext{and that}
C_1 \times \dotsb \times C_n \times C^d
&{\;\text{ is }\;}&&
(U_1, \dotsc, U_n, {d\cdotB})
\text{-tilable.}
\label{cond:c-is-b-tilable}
\shortintertext{Then}
B_1 \times \dotsb \times B_m
\times C_1 \times \dotsb \times C_n \times C^d
&{\;\text{ is }\;}&&
(
T_1, \dotsc, T_m,
U_1, \dotsc, U_n, {d\cdotA}
)
\text{-tilable}.
\notag
\end{aligned}$$
Let us unravel the statement of this proposition. Intuitively, condition (\[cond:b-is-a-tilable\]) asserts that ‘$B$ is almost $A$-tilable’ – the extra dimensions $B_1, \dotsc, B_m$ are used to fill the gaps. Similarly, condition (\[cond:c-is-b-tilable\]) asserts that ‘$C^d$ is almost $B$-tilable’ – here we use the extra dimensions $C_1, \dotsc, C_n$. Finally, the conclusion states that ‘$C^d$ is almost $A$-tilable’ – we use all the extra dimensions, $B_1, \dotsc, B_m$ and $C_1, \dotsc, C_n$, to complete this tiling.
For each tile $X$ in the $(U_1, \dotsc, U_n,{d\cdotB})$ -tiling of $C_1 \times \dotsb \times C_n \times
C^d$, partition the set $B_1 \times \dotsb \times
B_m \times X$ in one of the two following ways:
- if $X$ is a copy of $B$, then partition $B_1 \times \dotsc \times B_m \times X$ into its $(T_1, \dotsc, T_m,A)$-tiling;
- otherwise (that is, if $X$ is a copy of one of the $U_1, \dotsc, U_n$), partition the set into copies of $X$, namely $\{b\} \times X$ for each $b \in
B_1 \times \dotsc \times B_n$.
This produces a $(T_1, \dotsc, T_m, U_1,
\dotsc, U_n, {d\cdotA})$-tiling of $B_1 \times \dotsb
\times B_m \times C_1 \times \dotsb \times C_n
\times C^d$.
In the proof of the main theorem, we will apply this result in the following more compact form.
\[cor:tilingtransitive\] Let $G$ and $H$ be abelian groups with tiles $T \subset G$ and $A \subset B \subset H$. Suppose that $$\begin{aligned}
{2}
\label{eq:ghbtilable}
G^k \times H^{\ell} \times B^d
&{\;\text{ is }\;}&&
({k\cdotT}, {(\ell+d)\cdotA})
\text{-tilable}
\shortintertext{and that}
\label{eq:ghtilable}
G^u \times H^v
&{\;\text{ is }\;}&&
({u\cdotT}, {v\cdotB})
\text{-tilable.}
\shortintertext{Then}
G^{du+k} \times H^{dv+\ell}
&{\;\text{ is }\;}&&
({(du+k)\cdotT}, {(dv+l)\cdotA})
\text{-tilable.}
\notag
\end{aligned}$$
Use induction on $d$. The base case $d = 0$ is trivial, so suppose that $d \ge 1$.
Rewrite ($\ref{eq:ghbtilable}$) to state that $$G^k \times H^l \times B^{d-1} \times B
\text{ is }
({k\cdotT}, {(\ell+d-1)\cdotA}, A)
\text{-tilable.}$$ Now Proposition \[prop:tilabletransitive\] applied to this and (\[eq:ghtilable\]) implies that $$G^k \times H^l \times B^{d-1}
\times G^u \times H^v
\text{ is }
({k\cdotT}, {(\ell+d-1)\cdotA},
{u\cdotT}, {v\cdotA})
\text{-tilable,}$$ which after reordering and combining terms becomes the statement that $$G^{u+k} \times H^{v+l} \times B^{d-1}
\text{ is }
({(u+k)\cdotT}, {(v+l+d-1)\cdotA})
\text{-tilable.}$$ Finally, apply the induction hypothesis to this and $(\ref{eq:ghtilable})$ to conclude the proof.
The following straightforward proposition allows tilings to be lifted via surjective homomorphisms.
\[prop:lifting\] Let $G, H$ and $G_1, \dotsc, G_n$ be abelian groups with tiles $T \subset G$ and $U_i \subset G_i$, and let $\rho : G \to H$ be a surjective homomorphism that is injective on $T$. If $G_1 \times \dotsb \times G_n \times H$ is $(U_1, \dotsc, U_n, \rho(T))$-tilable, then $G_1 \times \dotsb \times G_n \times G$ is $(U_1, \dotsc, U_n, T)$-tilable.
For any tile $X$ in the $(U_1, \dots, U_n,
\rho(T)$)-tiling of $G_1 \times \dotsb \times G_n
\times H$, let $\hat{X}$ denote the set $$\hat{X} =
\{
(x_1, \dotsc, x_n, x) \in
G_1 \times \dotsb \times G_n \times G :
(x_1, \dotsc, x_n, \rho(x)) \in X
\}.$$
For every $X$, partition $\hat{X}$ in one of the two following ways:
- if $X$ is a copy of $U_i$ for some $1 \le i \le n$, then partition $\hat{X}$ into copies of $U_i$ in the obvious way;
- if $X$ is a copy of $\rho(T)$, then $X = \{(x_1, \dotsc, x_n)\} \times \rho(T+x)$ for some $x_i \in G_i$ and $x \in G$. Hence $\hat{X} = \{(x_1, \dotsc, x_n)\} \times
(T + x + \ker(\rho))$, and as $\rho$ is injective on $T$, this can be partitioned into copies of $T$.
Since the sets $\hat{X}$ partition $G_1 \times
\dotsb \times G_n \times G$, this produces a $(U_1, \dotsc, U_n, T)$-tiling for it.
An inductive application of this proposition gives the following result, which we will use in the proof of the main theorem.
\[cor:completelifting\] Let $G$ and $H$ be abelian groups, and let $T \subset G$ be a tile. Moreover, suppose that a surjective homomorphism $\rho : G \to H$ is injective on $T$. If $G^k \times H^\ell$ is $({k\cdotT}, {\ell\cdot\rho(T)})$-tilable, then $G^{k+\ell}$ is $T$-tilable.
Proof of the main theorem {#subsec:formalproof}
-------------------------
The tools needed for the proof Theorem \[thm:main\] are now available.
Without loss of generality assume that $T
\subset [k]^b$, where $k \in {\mathbb{N}}$. Write $G =
{{\mathbb{Z}}_{k}}^b$ and let $\pi : {\mathbb{Z}}^b \to G$ be the projection map. In particular, $\pi(T)$ is a tile in $G$.
Suppose that $A \subset G$ is a tile. Then there exist integers $p \ge 0$ and $q \ge 1$ such that $({\mathbb{Z}}^b)^p \times G^q$ is $({p\cdotT},
{q\cdotA})$-tilable.
Use reverse induction on $|A|$. If $|A| = |G|$ then in fact $A = G$, and the claim holds with $p = 0$, $q = 1$. So suppose that $|A| \le |G|-1$.
Applying Lemma \[lem:constructdenser\] to the tile $A$ with fixed large $d_0$ produces a number $d_1 \ge d_0$, a set $B$ such that $A \subsetneq B
\subset G$ and a family $\mathcal{F}$ ($|\mathcal{F}| \ge d_0$) of pairwise disjoint subsets of $B^{d_1}$ with the property that for any subfamily $\mathcal{S} \subset \mathcal{F}$ of size satisfying $|\mathcal{S}| \equiv 1\pmod{|A|}$, the set $$G \times
\left(
B^{d_1} {\setminus}\bigcup_{S\in\mathcal{S}}
S
\right)$$ is $A$-tilable.
Since $d_0$ is large, Lemma \[lem:coverholes\] gives a set $X \subset {\mathbb{Z}}^b \times G \times B^{d_1}$ that is a disjoint union of copies of $T$, and such that for every $x \in {\mathbb{Z}}^b$ the slice $\{y \in G\times
B^{d_1} : (x, y) \in X\}$ is a hole in $G \times B^{d_1}$. Therefore ${\mathbb{Z}}^b \times G \times B^{d_1}$ is $(T, {(d_1+1)\cdotA})$-tilable.
By the induction hypothesis, there exist $u \ge 0$ and $v \ge 1$ such that $({\mathbb{Z}}^b)^{u} \times
G^{v}$ is ${({u\cdotT}, {v\cdotB})}$-tilable. Now apply Corollary \[cor:tilingtransitive\] to conclude that the claim holds with $p = d_{1}u+1$ and $q = d_{1}v+1$. This proves the claim.
To finish the proof of the theorem, apply the claim to the tile $\pi(T)$. This gives $p \ge 0$ and $q \ge 1$ such that $({\mathbb{Z}}^b)^p \times G^q$ is $({p\cdotT},
{q\cdot\pi(T)})$-tilable. Hence, by Corollary \[cor:completelifting\], $({\mathbb{Z}}^b)^{p+q}$ is $T$-tilable.
Concluding remarks and open problems {#sec:conclusion}
====================================
We mention in passing that all our tilings are (or can be made to be) periodic. Also, our copies of $T$ arise only from translations and permutations of the coordinates – in particular, ‘positive directions stay positive’.
We have made no attempt to optimise the dimension $d$ in Theorem \[thm:main\]. What can be read out of the proof is the following.
Let $T \subset {\mathbb{Z}}^n$ be a tile and suppose that $T \subset [k]^n$. Then $T$ tiles ${\mathbb{Z}}^d$, where $d = \lceil \exp(100 (n \log k)^2)
\rceil$.
Thus our upper bound on $d$ is superpolynomial in the variable $k^n$. We believe that there should be an upper bound on $d$ in terms only of the size and dimension of $T$. Even in the case $n=1$ this seems to be a highly non-trivial question.
For any positive integer $t$ there is a number $d$ such that any tile $T \subset {\mathbb{Z}}$ with $|T| \le t$ tiles ${\mathbb{Z}}^d$.
On the other hand, it is easy to see that there cannot be a bound just in terms of the dimension of the tile. Indeed, given any $d$ it is possible to find a one-dimensional tile that does not tile ${\mathbb{Z}}^d$. Such a tile $T$ can be constructed by fixing an integer $k$ and taking two intervals of length $k$, distance $k^2 - 1$ apart, where in between the intervals only every $k$-th point is present in the tile. For example, if $k = 4$ then the resulting tile would be $$\mathtt{XXXX...X...X...X...XXXX}$$
Suppose that $T$ tiles ${\mathbb{Z}}^d$. Choose a large integer $N$ and consider the cuboid $[N]^d$. Fix one of the $d$ directions and only consider the copies of $T$ in this direction that intersect $[N]^d$. There can be at most $N^{d-1} (N/(k^2 + 2k - 1) + 2) = O(N^d / k^2)$ such tiles and they cover at most $O(N^d / k)$ elements of $[N]^d$. Since there are $d$ directions, at most $O(dN^d / k)$ elements of the cuboid can be covered with tiles, but this number is less than $N^d$ for large $k$. Therefore, if $k$ is large enough, then $T$ does not tile ${\mathbb{Z}}^d$.
Finally, we do not know how to find reasonable lower bounds, even for seemingly simple tiles. In particular, we do not know the smallest dimension that can be tiled by a given interval with the central point removed. The largest tile of this shape for which we know the answer is $\mathtt{XXX.XXX}$: it does not tile ${\mathbb{Z}}^2$ (by case analysis), but it tiles ${\mathbb{Z}}^3$. (One way to achieve this is to adapt the argument of Section \[sec:simplecase\], to make it work in the case of this tile for $d=3$.)
Let $T$ be the tile $
\underbrace{\mathtt{XXXXX}}_k
.
\underbrace{\mathtt{XXXXX}}_k
$ . What is the least $d$ for which $T$ tiles ${\mathbb{Z}}^d$?
Just getting rough bounds on $d$ would be very interesting. We do not even know if $d \to \infty$ as $k \to \infty$.
[^1]: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, CB30WB Cambridge, United Kingdom; e-mail: .
[^2]: Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia; e-mail: `tstan@um.edu.my`. This author acknowledges support received from the Ministry of Education of Malaysia via FRGS grant FP048-2014B.
|
---
abstract: 'We propose a system of highly efficient photoemitters comprising metal-molecule multilayered structures. In the proposed structure, the absorption in the molecular layer is greatly enhanced through quantum interference between the split modes arising from the coupling of the layered excitons and the plasmons sustained by the metal layer. Furthermore, the large interaction volume between surface plasmons and excitons causes exciton superradiance, which results in the extremely efficient photoemission. This finding indicates the possibility of designing highly efficient photoemitters based on simple layered structures.'
author:
- Takuya Matsuda
- Hajime Ishihara
title: 'Proposal of highly efficient photoemitter with strong photon-harvesting capability and exciton superradiance'
---
PS.
For the realization of high-efficiency solar cells, photo-emitting devices, and so forth, smart designs of the light-matter coupling are necessary [@tang87; @baluschev06; @riesen16; @peng06; @reinke06; @nowy08; @hayashi10]. In particular, the scheme of high-efficiency photon harvesting is crucial for the development of energy-saving technologies. For example, a high concentration of harvested light energy in the photoactive parts of devices is important for realizing efficient photonic functions. One approach to achieving the high concentration of harvested light energy is to utilize nanometallic structures sustaining surface plasmon (SP) resonance [@wiederrecht04; @kuhn06; @ueno08; @ishi11; @yano13; @osaka14]. It has been proposed that the quantum interference between the split coupled modes comprising SPs and molecular excitons leads to strong energy concentration in molecules [@ishi11; @yano13; @osaka14], which has been discussed as a type of Fano resonance (FR) appearing through the interference between the sharp discrete resonance and much broader resonance [@luk10]. However, such effects usually require highly sophisticated designs and the fabrication of nanometallic structures to ensure strict control of the system parameters. Thus, it is desirable to realize systems for efficient photon-harvesting with simpler designs such as those with layered structures. In studies on the control of the light-matter interaction, it was reported that metal-molecular composite layered structures easily induce large Rabi splitting (RS) due to the strong interaction between excitons and SPs [@bellessa04; @bonnand06; @hakala09; @cade09]. Also, Hayashi [*et al*]{}. proposed highly sensitive sensors based on the interference between the SP mode and waveguide mode in multilayered structures [@hayashi16]. However, such structures have received little interest for application to photon-harvesting through the mode interference effect (Fano resonance).
On the other hand, to realize highly efficient photoemitters, not only an efficient photon-harvesting scheme but also an efficient photoemission scheme must be realized. It is known that highly efficient photoemission occurs through exciton superradiance. As the interaction volume (or coherence volume) between excitons and the radiation field increases, the radiative decay rate increases (so-called exciton superradiance) [@feldmann87; @nakamura89; @itoh90; @knoester92; @bjork95; @agranovich97]. For example, for very high quality CuCl thin films, the radiative decay time was observed to reach 100 fs order [@ichimiya09], where excitons with a very high coherence volume over the entire sample are strongly coupled with the radiation field.
If we can design simple layered structures in which energy concentration and exciton superradiance are simultaneously realized, it will greatly contribute to the realization of photoemitters with high efficiency. In this paper, we demonstrate a condition under which strong energy concentration occurs in metal-molecule composite multilayered structures, where SP-induced exciton superradiance occurs simultaneously.
![Sketch of the assumed composite layered structure and $p$-polarized incident light is assumed to excite the SP modes.[]{data-label="fig1"}](fig1.eps){width="60mm"}
We consider a Kretschmann configuration consisting of a prism, a metal layer, a nonactive layer, a molecular layer, and a vacuum as illustrated in Fig. \[fig1\]. Unless otherwise noted, we use the same parameters as in Ref. . We treat a prism as a semi-infinite layer. The nonactive layer is TiO$_{2}$ and its thickness $\thick_{\rm nonac}$ is $2~\nm$. We assume that the molecular layer with thickness $\thick_{\rm m}=26~\nm$ is a PVA-TDBC one. This structure enables SP modes to propagate along the vacuum-side surface of the molecular layer. The dielectric function of the metal layer is modeled by the three critical point pole pairs (CP3) model with the parameters of Ag [@lu10].
We consider the following Hamiltonian: $\oH=\oHex+\oHI$, where $\oHex=\hbar\wex\oexd\oex$ represents the resonance energy of a bare exciton in the molecular layer and $\hbar\wex=2.1~\eV$. The bosonic operator $\oex~(\oexd)$ stands for the annihilation (creation) operator of an exciton. $\oHI= -\int\dd\vr~\ovPex(\vr)\cdot\vE(\vr,t)$ represents the interaction between excitons and the electric field. The excitonic polarization is represented as $\ovPex(\vr)=\vdimP(\vr)\oex+\Hc$, where the expansion coefficient $\vdimP(\vr)$ is expressed as $\vdimP(\vr)=\vP~\ee^{\ii\vkp\cdot\vrp}\big\{{\it \Theta}(z-z_{j-1})-{\it \Theta}(z-z_{j})\big\}$, where $\vP$ is the transition dipole moment density, and $\vkp$ and $\vrp$ are a wavevector and position vector parallel to the film surface, respectively. ${\it \Theta}(z)$ is the Heaviside step function. Here, we consider only the excitonic polarization as the coordinate phase mode, and the relative motion of excitons is treated as that in the bulk. We express the electric field satisfying the Maxwell equation in integral form as $$\begin{aligned}
\label{eq:Maxwell}
\vE(\vr,\w)
=
\vEz(\vr,\w)+\int\dd\vr^{\prime}~\bar{\mG}(\vr,\vr^{\prime},\w)\cdot\vdimP(\vr^{\prime})\braket{\oex(\w)}.\end{aligned}$$ The dyadic Green’s function $\bar{\mG}(\vr,\vr^{\prime},\w)$ satisfies the equation $\big[\rot\rot-\eps(\vr,\w)\w^{2}/c^{2}\big]\bar{\mG}(\vr,\vr^{\prime},\w)=\w^{2}/(\epsz c^{2})\delta(\vr-\vr^{\prime})~\bar{\munit}$ [@chew95], where $\eps(\vr,\w)$ reflects the sample geometry determined by the background dielectric constants of the different layers. In this expression, $c$ is the speed of light and $\epsz$ is the vacuum permittivity. Solving the equation of motion for excitons and the Maxwell equation simultaneously, we obtain the self-consistent equation set as
$$\begin{aligned}
\label{eq:bex}
&\big[\hbar(\wex-\w-\ii\dampex/2)+\mathcal{A}(\w)\big]
\braket{\oex(\w)}
\nonumber\\
&=
\int\dd\vr~\vdimP^{\ast}(\vr)\cdot\vEz(\vr,\w),
\\
\label{eq:rad}
\mathcal{A}(\w)
&\equiv
-\int\dd\vr\int\dd\vr^{\prime}~
\vdimP^{\ast}(\vr)\cdot\bar{\mG}(\vr,\vr^{\prime},\w)\cdot\vdimP(\vr^{\prime}),\end{aligned}$$
where $\vEz(\vr,\w)$ is the incident electric field. In this expression, we phenomenologically introduce nonradiative damping $\dampex$ whose value is $49~\meV$. Equation describes the energy correction of excitons including the effect of SPs through $\bar{\mG}(\vr,\vr^{\prime},\w)$. We can now determine the self-consistent field by substituting $\braket{\oex(\w)}$ from Eq. into Eq. .
First, we investigate the absorptivity spectrum in the molecular layer while varying the thicknesses of the metal layer. We evaluate the absorptivity spectrum in the $j$th layer, $A_{j}(\w)~(j \in\mathbb{N})$, using the following expressions [@kim12]: $A_{j}(\w)
\equiv
\frac{1}{S_{0}(\w)}
\int_{z_{j-1}}^{z_{j}}\dd z~
Q_{j}(z,\w)$, where $Q_{j}(z,\w)$ is the optical power dissipation of the $j$th layer in the $z$ direction. $S_0(\w)=(1/2)\Re\{E_{x,0}H_{y,0}^\ast\}$ is the magnitude of the input time-averaged Poynting vector, where $E_{x,0}$ is the $x$ component of the electric field in the input region and $H_{y,0}$ is the $y$ component of the magnetic field in the input region. If the absorptivity at a particular layer is 1, it means that an incident light power is totally concentrated into this layer without reflectance and transmittance.
![Absorptivity in the molecular layer plotted as a function of the incident angle and incident photon energy for different thicknesses of the metal layer: (a) $\thick_{\rm me}=66~\nm$, (b) $\thick_{\rm me}=50~\nm$, (c) $\thick_{\rm me}=34~\nm$, (d) $\thick_{\rm me}=22~\nm$, and (e) $\thick_{\rm me}=12~\nm$.[]{data-label="fig2"}](fig2.eps){width="\linewidth"}
Here, we assume that the $p$-polarized incident light is used to excite the SP modes. In Fig. \[fig2\], we plot the absorptivity in the molecular layer as a function of the incident angle and incident photon energy. In Figs. \[fig2\](a)-(e), we confirm the crossover behaviour from RS to FR with decreasing thickness of the metal layer. As shown in Figs. \[fig2\](a,b), in the RS regime the absorption by the two modes in the molecular layer is enhanced at particular incident angles. On the other hand, in the FR regime, the absorption of only one of the modes in the molecular layer is greatly enhanced over a wide range of the incident angle as shown in Figs. \[fig2\](d,e). Within the thickness regime in our system, the thicker metal film generates a larger plasmonic dipole moment that leads to the stronger coupling with the molecular excitons. On the other hand, with the decrease in the metal film thickness, the coupling strength between the plasmons and excitons rapidly decreases. Also, note that, in FR regime, we do not need to carefully choose the incident angle to enhance the absorption in the molecular layer.
To find suitable conditions to enhance the absorption in the molecular layer, we examine the following two cases: (i) the thickness of the metal layer $\thick_{\rm me}$ is $50~\nm$ and the incident angle $\theta$ is $49^{\circ}$ in the RS regime, and (ii) $\thick_{\rm me}=22~\nm$ and the incident angle $\theta=50^{\circ}$ in the FR regime.
{width="\linewidth"}
\[fig3\]
In case (i) (RS regime), as shown in Fig. \[fig3\](a), the enhancement of absorption in the metal layer cannot be avoided at the peak energy of the absorption in the molecular layer because the Rabi-split modes contain both plasmon and exciton components owing to their strong coupling. Figure \[fig3\](b) shows the electric field intensity as a function of position $z$ and incident photon energy. As shown in Fig. \[fig3\](b), the electric field is enhanced at the peak positions of absorption in Fig. \[fig3\](a). Thus, the RS regime is not suitable for efficient photon-harvesting because the incident photon energy cannot be concentrated into only the molecular layer.
.[]{data-label="fig4"}](fig4.eps){width="\linewidth"}
In case (ii) (FR regime), we find the condition of strong suppression of the absorption in the metal layer ($2\%$) as shown by the red solid line in Fig. \[fig4\](a), and, at the same frequency, marked enhancement of the absorption in the molecular layer ($96\%$) shown by the blue dashed line in Fig. \[fig4\](a). This value is considerably superior to that for the system in Ref. . On the other hand, in the single molecular layer \[see the green dotted line in Fig. \[fig4\](a)\], the absorptivity does not reach $40\%$ even under the optimum condition, the incident angle of $s$-polarized light of $\theta=80^{\circ}$ for the same thickness. Figure \[fig4\](b) shows the electric field profile plotted similarly to that in Fig. \[fig3\](b). This profile indicates that excitons are not very strongly coupled with SPs, and the energy concentration occurs in the molecular layer [@ishi11] because, in the FR regime, the splitting between exciton-plasmon coupled modes is not larger than the spectral width of the respective modes, and at the excitonic resonance energy, these two peaks are overlapped, where the excitonic components are constructively superposed, while the plasmonic components destructively superposed. This quantum interference clearly appears in Fig 4(a). Namely, the plasmonic absorption is strongly suppressed and the excitonic resonance is remarkably enhanced. (Note that the electric field intensity at around 2.1 eV is suppressed at the molecular layer in spite of the greatly enhanced absorptivity because of the interference between the incident field and the strong radiated field from the induced polarization in layered structures.) In this way, strong energy concentration into the molecular layer alone is possible in the FR regime, which is advantageous for realizing efficient photon-harvesting with simple layered structures.
Next, we examine the possibility of exciton superradiance in our proposed structure. To evaluate the radiative decay time of excitons, we find the roots $\{\tilde{\w}\}$ of ${\rm det}[\hbar(\wex-\tilde{\w})+\mathcal{A}(\tilde{\w})]=0$ in the LHS of Eq. . Here, we set the nonradiative damping included in the dielectric function of Ag to zero for the technical reason to extract the pure radiative width of excitons. In fact, the superradiance is hardly affected by the the nonradiative damping of Ag because the plasmonic excitation is strongly suppressed in the considered condition. The roots $\{\tilde{\w}\}$ provide the complex eigenfrequencies $\{\tilde{\w}\}$ of the coupled mode, whose real parts $\Re\{\tilde{\w}\}$ give the eigenfrequencies including the radiative shift and whose imaginary parts $\Im\{\tilde{\w}\}$ correspond to the radiative decay rate. Thus, we can describe the radiative decay time as $\tau_{\rm R}=1/(-2\Im\{\tilde{\w}\})$.
By evaluating $\tau_{\rm R}$, we find that the radiative decay time of excitons in our proposed structure ($\tau_{\rm R}$=0.012 PS. ) is much shorter than that in the single layered structure with the optimum condition, the incident angle of $s$-polarized light of $\theta=80^{\circ}$ ($\tau_{\rm R}$=0.046 PS. ), because of the large interaction volume between coherently extended wavefunction of excitons and SPs. These results indicate that our proposed structure has the major advantage of highly efficient photoemitters owing to the compatibility of the strong absorption and large radiative width.
Finally, we calculate excitonic population spectrum to discuss the amount of luminescence in the molecular layer within the linear response regime. The excitonic population spectrum can be expressed as [@ishikawa13] $$\begin{aligned}
N_\text{ex}(\w)
&=
|\braket{\oex(\w)}|^2
\nonumber\\
&=
\Big|
\frac{\int\dd\vr~\vdimP^{\ast}(\vr)\cdot\vEz(\vr,\w)}{\hbar(\wex-\w-\ii\dampex/2)+\mathcal{A}(\w)}
\Big|^2.\end{aligned}$$ In Fig. \[fig5\], we plot the excitonic population spectrum versus the incident photon energy. By calculating the integrated intensity of each excitonic population spectrum, we find that the intensity in the present structure is about 24 times larger than that of the single molecular layer under the optimum condition, the incident angle of $s$-polarized light of $\theta=80^{\circ}$. From these results, we can expect marked strengthening of the luminescence utilizing our proposed structure.
![(Color online) Excitonic population spectrum versus incident photon energy. Both spectra are normalized by the peak of the red solid line.[]{data-label="fig5"}](fig5.eps){width="\linewidth"}
To conclude, we find that high-efficiency energy concentration in the photoactive part of a device can be realized with a simple metal-molecule multilayered structure, where the absorption at the metal layer is greatly suppressed by the Fano resonance effect even for this simple structure. We propose a high-efficiency photoemitter using this mechanism, where strong light-harvesting by the above mechanism and strong photoemission by exciton superradiance occur simultaneously. To demonstrate the potential of our proposed structure, we numerically calculate the absorption, radiative decay rate, and population of a layered exciton under particular conditions. As a result, we find a condition under which every aspect commonly shows excellent performance. We hope that our findings will enable the design of high-efficiency photoemitters based on simple layered structures.
We thank T. Yano and N. Yokoshi for helpful discussions. This work was partially supported by Grant-in-Aid for JSPS Fellows No.16J11326 from MEXT, Japan, and by JSPS KAKENHI Grant No.JP16H06504 in Scientific Research on Innovative Areas “Nano-Material Optical Manipulation”.
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---
abstract: 'Inelastic neutron scattering (INS) from polycrystalline antiferromagnetic LaMnAsO, LaMnSbO, and BaMnAsF are analyzed using a $J_1-J_2-J_c$ Heisenberg model in the framework of the linear spin-wave theory. All three systems show clear evidence that the nearest- and next-nearest-neighbor interactions within the Mn square lattice layer ($J_1$ and $J_2$) are both antiferromagnetic (AFM). However, for all compounds studied the competing interactions have a ratio of $2J_2/J_1 < 1$, which favors the square lattice checkerboard AFM structure over the stripe AFM structure. The inter-plane coupling $J_c$ in all three systems is on the order of $\sim 3\times10^{-4}J_1$, rendering the magnetic properties of these systems with quasi-two-dimensional character. The substitution of Sb for As significantly lowers the in-plane exchange coupling, which is also reflected in the decrease of the N[é]{}el temperature, $T_{\rm N}$. Although BaMnAsF shares the MnAs sheets as LaMnAsO, their $J_1$ and $J_2$ values are substantially different. Using density functional theory, we calculate exchange parameters $J_{ij}$ to rationalize the differences among these systems.'
author:
- Farhan Islam
- Elijah Gordon
- Pinaki Das
- Yong Liu
- Liqin Ke
- 'Douglas L. Abernathy'
- 'Robert J. McQueeney'
- David Vaknin
- 'Rrobert J. McQueeney'
- David Vaknin
bibliography:
- 'paper1111-v3.bib'
title: 'Spin Dynamics in the Antiferromagnetic Oxy- and Fluoro- Pnictides: LaMnAsO, LaMnSbO, and BaMnAsF'
---
=1
Introduction
============
Manganese (Mn) based pnictide compounds with Mn$Pn$ ($Pn =$ P, As, Sb, and Bi) layers have been in the spotlight by virtue of their intriguing magnetic properties, most notably the recently discovered Dirac semimetals $A$Mn$Pn_2$ ($A=$ Ca, Sr, and Ba) [@Park2011; @Farhan2014; @Huang2017]. The quasi two-dimensional (2D) $A$Mn$Pn_2$ have been recognized as the three-dimensional (3D) analogs of the 2D graphene with linearly dispersing bands that cross at the Fermi energy [@Farhan2014]. Generally, the Mn atoms are arranged in a square lattice or in a slightly distorted orthorhombic lattice and undergo antiferromagnetic (AFM) ordering. Of particular interest is the coupling of magnetism to Dirac fermions, which in ideal cases can deliver a Weyl semimetal with unique bulk magnetotransport and optical properties. A few compounds exhibit uniform canting with a finite ferromagnetic (FM) moment that can further remove the degeneracy of the Dirac bands to furnish Weyl states [@Park2011; @Wang2011; @Sanchez-Barriga2016; @Liu2017; @Rahn2017; @Liu2019].
$A$Mn$_2Pn_2$ ($A=$ Ca, Sr, Ba) is another class of AFM semiconductors with similar Mn$Pn$ layers and with partially localized Mn moments. These compounds are not known to possess Dirac-like bands, but can become metallic with doping [@An2009; @Pandey2012a; @Lamsal2013]. It has been reported that the substitution of K for Ba in (Ba$_{1-x}$K$_x$)Mn$_2$As$_2$ shows a novel magnetic ground state below $T_{\rm C} \simeq 100$ K, in which itinerant ferromagnetism associated with the As bands coexists with a collinear local-moment AFM ordering associated with the Mn atoms with $T_{\rm N}\simeq 480$ K (for $x =0.2$) [@Pandey2013; @Ueland2015]. We note that other reports associate the FM in this system to simple canting of the AFM magnetic moments that gives rise to the observed weak-FM signal [@Glasbrenner2014].
It is clear that systematic studies of the evolution of the magnetism among all of these square-lattice Mn pnictides are necessary. The magnetism in these systems is dominated by the Mn$Pn$ square layer where Mn-$Pn$-Mn superexchange couplings via Mn-$Pn$-Mn between the nearest- and next-nearest-neighbor (NN and NNN, $J_1-J_2$) Mn spins lead to a checkerboard-type AFM order. Easy axis anisotropy results in Mn moments that point normal to the square layers. The AFM Mn$Pn$ planes are coupled via intervening layers by a much weaker AFM or FM exchange coupling $J_c$. Analysis of magnetic excitations obtained by inelastic neutron scattering (INS) of polycrystalline (Ba$_{1-x}$K$_x$)Mn$_2$As$_2$ determined competing AFM $J_1-J_2$ exchange interactions and the much weaker interplane coupling $J_c$ of a Heisenberg model [@Ramazanoglu2017]. In fact, INS studies of other layered Mn$Pn$ compounds, such as BaMn$_2$Bi$_2$ [@Calder2014], and the topological semimentals $A{\mathrm{MnBi}}_{2}$ ($A=\mathrm{Sr}, \mathrm{Ca}$)[@Rahn2017] and ${\mathrm{YbMnBi}}_{2}$.[@Soh2019], have also established the presence of competing AFM $J_1-J_2$ exchange couplings.
![(Color online) Crystal and magnetic structure of LaMnSbO and LaMnAsO (C-type) on the right and BaMnAsF (G-type) on the left.[]{data-label="Fig:Crystal"}](structure_final_v2.pdf){width="1.5in"}
Another class of AFM compounds that shares similar Mn$Pn$ planes is $R$Mn$PnB$ ($R=$ La, Ce, Pr, Ba, ...and $B =$ O and F, referred to here as Mn-1111 compounds) [@Emery2010; @Kimber2010; @Tsukamoto2011; @Emery2011; @Lee2012; @Zhang2015; @McGuire2016; @Saparov2013; @Zhang2016]. Recently it has been suggested that one such compound, LaMnAsO, can be hole doped by substitution of Sr for La and undergo insulator-to-metal transition and exhibit thermoelectric properties [@Sun2012; @Hanna2013].
In this manuscript, we report on the measurements and analysis of INS data from polycrystalline samples of Mn-based 1111 pnictides — LaMnSbO, LaMnAsO, and BaMnAsF. They all belong to the $P4/nmm$ space group, and both LaMnSbO and LaMnAsO have C-type AFM order, whereas BaMnAsF has G-type AFM order, as depicted in Fig. \[Fig:Crystal\]. By analyzing the spin-waves in the framework of a Heisenberg model, we determine the exchange interactions $J_1$, $J_2$, and $J_c$ and show that all compounds demonstrate a significant competitive AFM NNN interaction ($J_2$), that places these systems close to a magnetic instability between checkerboard and stripe AFM order. The very weak inter-plane interaction ($J_c$) renders the spectra with quasi-2D characteristics. Density functional theory (DFT) calculations confirm the magnetic ground states, the average magnetic moment, and the exchange parameters determined experimentally. Confirmation of these energy scales provides theoretical grounds for predicting by design new materials for potentially novel ground states in Mn$Pn$ systems, such as spin liquids or magnetic topological materials. applications.
Experimental Details
====================
*Sample Preparation*: Polycrystalline samples of LaMnAsO, LaMnSbO, and BaMnAsF were synthesized by a solid-state reaction method. The stoichiometric chemicals of La and Ba pieces, Mn, As, Sb, MnO and BaF$_2$ powder were weighed and mixed in a glovebox under argon atmosphere. The mixtures were pressed into pellets under a pressure of 12 MPa. The pellets were loaded into alumina crucibles and sealed in quartz tubes. The quartz ampoules were slowly heated up to 500 $^\circ$C at a ramping rate of 100 $^\circ$C/h. After a dwell time of 6 hours, the ampoules were heated up to 780 $^\circ$C/h at the same rate and held at that temperature for 6 hours. These prereacted samples were crushed and ground in the glovebox. The powder was pressed into pellets and sintered at 1100 1 $^\circ$C/h for 12 hours in an evacuated quartz tube. After sintering, the furnace was cooled down to room temperature at a rate of 200 $^\circ$C/h. To improve the homogeneity and get rid of impurity phases, the final step was repeated once.
![INS intensities $S(Q,E)$ for $E_i =50$ and 150 meV for (a) and (d) LaMnSbO, (b) and (e) LaMnAsO, and (c) and (f) BaMnAsF, respectively as indicated. Columns of scattering emanating at $\sim 1.6 $ and at 3.5 [Å]{}$^{-1}$ are due to magnons. []{data-label="Fig:rawdata"}](rawdata_allsamples_v2.pdf){width="3.4"}
Powder x-ray diffraction (XRD) measurements were performed on a PANalytical MPD diffractometer using Co K$\alpha$ radiation. Magnetization measurements were performed by using Physical Property Measurement System (PPMS, Quantum Design) equipped with Vibrating Sample Magnetometer (VSM). All three 1111 compounds crystallize in a tetragonal $P4/nmm $ crystal symmetry, with the lattice parameters listed in Table \[tab:params\] , with no change in crystal symmetry down to base temperature ($T = 12$ K). LaMnSbO and LaMnAsO adopt a C-type AFM ground state and BaMnAsF into a G-type with varying [[$T_{\textrm N}$]{}]{} as listed in Table \[tab:params\] (based on Refs. \[ \]).
*Inelastic Neutron Scattering*: INS measurements were carried out on the ARCS spectrometer at the Spallation Neutron Source at Oak Ridge National Laboratory. Each polycrystalline sample was placed in a cylindrical aluminum sample can and mounted on the cold tip of a closed-cycle helium cryostat. Measurements were performed at $T = 10$ K with incident energies, $E_i = 50, 150, 300,$ and $500$ meV with an energy resolution of $3-5$ % of $E_i$. The data were corrected for both aluminum (sample holder) and hydrogen scattering (due to surface adsorption of water by exposure of the polycrystalline sample to air). Incoherent nuclear scattering from a vanadium standard was used to correct for the variation of the detector efficiency. The dynamical structure factor $S(Q,E)$, where $Q$ is the momentum transfer and $E$ is the energy transfer were used to get $Q$- and $E$-cuts for refined fitting.
*Modeling with $\textsc{SpinW}$*: We use $\textsc{SpinW}$, a <span style="font-variant:small-caps;">matlab</span> library, to model the magnetic excitations and fit the INS data [@Toth2015]. We set up the crystal properties for each compound using documented lattice constants, space-group, atomic position of magnetic atoms, neutron scattering form factor, and magnetic structure. We specify the Heisenberg interactions between $ab$-plane nearest neighbor ($J_1$) and next nearest neighbor ($J_2$), c-axis nearest neighbor ($J_c$), and single-ion anisotropy ($D$). The powder-averaged spin wave spectrum are calculated by averaging over a large number of momentum transfer vectors on the surface of a sphere of radius $Q$. The Heisenberg spin Hamiltonian for the $J_1$-$J_2$-$J_c$-$D$ model can be written as:
$$\begin{split}
H = &J_{1}\sum_{i\neq j\in ab}{{{\bf S}}_i\cdot{{\bf S}}_j} + J_{2}\sum_{i\neq k\in ab}{{{\bf S}}_i\cdot{{\bf S}}_k} \\
& + J_{c}\sum_{i\neq l\in c}{{{\bf S}}_i\cdot{{\bf S}}_l} + D\sum_{i}{(S_i^z)^2}.
\label{eq:Heisenberg}
\end{split}$$
We compare the exchange parameters and single-ion anisotropy values extracted from experiments with those obtained from DFT.
*DFT calculational details:* Spin-polarized DFT+$U$ calculations, within the Dudarev scheme [@Dudarev1998], were carried out in the Vienna Ab-initio Simulation Package (<span style="font-variant:small-caps;">vasp</span>) [@Kresse1999; @Kresse1996] by employing the projected-augmented wave method [@Blochl1994]. The exchange-correlation functional used is the generalized gradient approximation of Perdew, Burke, and Ernzerhof [@Perdew1996]. The difference between the effective on-site Coulomb and exchange parameters, denoted as $U$ (0–5 eV), was used to simulate additional Mn $d$-orbital on-site electron-electron correlations. Plane wave cutoff energy was set at 500 eV and the energy threshold for calculation was set at $10^{-6}$ eV. Exchange parameters are calculated using an energy-mapping analysis [@Xiang2013]. The total energies of four different collinear spin configurations are calculated and mapped to [Eq. (\[eq:Heisenberg\])]{} to extract the three exchange parameters, $J_1$, $J_2$, and $J_c$ (computational details can be found in the Supporting Information).
To determine the single-ion anisotropy term, $SD$ in [Eq. (\[eq:Heisenberg\])]{}, we calculate the magnetocrystalline anisotropy energy (MAE) of each compound. MAE originates from the spin-orbit coupling (SOC) [@ke2015prb; @ke2019prb]. We include SOC using the second-variation method [@koelling1977jpcs; @Li1990; @Shick1997] in our calculations. Starting from the experimental spin configuration of each compound, we calculate $SD$=$E_{a}{-}E_{c}$, where $E_{a}$ and $E_{c}$ are the total energies (per Mn) of the system with spins aligned along the $a$ or $c$ axis, respectively, and $S$ is the magnitude of Mn spin.
Results and Discussion
======================
LaMnSbO LaMnAsO BaMnAsF
------------------ ----------- ------------ ----------
$a$ ([Å]{}) 4.236 4.111 4.26
$c$ ([Å]{}) 9.545 9.026 9.559
$z_A$ 0.619 0.633 0.661
$z_{P}$ 0.181 0.168 0.154
$T_{\rm N}$ (K) 255 360 338
$SJ_{1}$ (meV) 40(4) 48(4) 35(4)
$SJ_{2}$ (meV) 17(2) 18(3) 10(2)
$SJ_{c}$ (meV) -0.01$^*$ -0.01$^*$ 0.01$^*$
$SD_{c}$ (meV) -0.07(2) -0.045(30) -0.06(4)
$J_2/J_1$ 0.42(6) 0.38(7) 0.29(6)
Energy Gap (meV) 8(2) 9(2) 7(2)
\[tab:params\]
: Lattice parameters $a$ and $c$ of LaMnAsO, LaMnSbO, and BaMnAsF in space group $P4/nmm$. The atomic positions of La and Ba, are at ($\frac{1}{4},\frac{1}{4},z_A$) and As and Sb at ($\frac{1}{4},\frac{1}{4},z_P$). $J_1$, $J_2$, $J_c$, and $D$ are the exchange couplings between intralayer NN, NNN, interlayer NN, and the single-ion anisotropy, respectively as obtained from our modeling of INS data.
$^*$ \* The value for $J_c$ is the upper limit modeling is not sensitive to values in the range of 0.01 to $10^{-4}$ meV. Numbers in bracket are the uncertainty in the last digit of a value.
Measured and Simulated Spin-waves Spectra
-----------------------------------------
Figure \[Fig:rawdata\] shows INS intensity maps, proportional to $S(Q,E)$ for polycrystalline LaMnSbO, LaMnAsO, and BaMnAsF at $T=10$ K for two incident energies $E_i = 50$ and 150 meV. Each $S(Q,E)$ map has a major contribution in the elastic region near $E=0$ due to elastic Bragg reflections and incoherent scattering (neutron energy loss is positive). The $S(Q,E)$ data also includes strong intensities that grow as $Q^2$ due to INS from phonons. The magnetic INS in our samples form steep columns, that emanate from (1 0 0) and (1 2 0) magnetic Bragg reflections for LaMnSbO and LaMnAsO, and from the (1 0 $\frac{1}{2}$) and (1 2 $\frac{1}{2}$) reflections for BaMnAsF that slightly open into cones at high energies. Due to the fast-falling off of the magnetic form factor of Mn$^{2+}$, magnon scattering intensity practically vanishes for $Q \geq 4.5$ [Å]{}$^{-1}$. To analyze the magnetic spectra, we focus our analysis to $Q \leq 4.5$ [Å]{}$^{-1}$. In this region, the intensity due to magnetic scattering is still contaminated by phonon scattering and other background contributions that can complicate the modeling. We cleaned up the phonon signal by fitting phonon peaks in the high-$Q$ region $Q \geq 4.5$ [Å]{}$^{-1}$ with a Gaussian function and estimating its intensity in the low-Q region by interpolation from the high-Q region.
![(Left column) measured inelastic neutron scattering data at $E_i$ = 150 meV for (a) LaMnSbO, (b) LaMnAsO, and (c) BaMnAsF as indicated. (Right column,) (d-f) Corresponding calculated spectra using the best fit parameters given in Table \[tab:params\]. The shaded areas in the calculated panels are kinematically inaccessible regions for neutrons at the specified energy and set up.[]{data-label="Fig:ins150mev"}](ins150mev.pdf){width="3.4in"}
To model the magnetic spectra we follow a procedure similar to that provided in Ref. \[\] and using the Heisenberg Hamiltonian in Eq. (\[eq:Heisenberg\]). In the linear approximation, spin operators in Eq. are transformed into bosonic operators with the Holstein-Primakoff approximation, leading to spin wave dispersion relations $$\begin{aligned}
\bigg[ \frac{\hbar \omega ( \bf{q})}{2S} \bigg]^2 =
&\bigg [ 2J_1 - J_2(2-\cos{q_xa}-\cos{q_ya}) \\
&- J_c(1 - \cos{q_zc}) + D \bigg]^2 \\
&-\bigg [ J_1 \{ \cos{\frac{(q_x + q_y)a}{2}} + \cos{\frac{(q_x - q_y)a}{2}} \}\bigg]^2
\end{aligned}
\label{eq:3}$$ for C-type structure, and $$\begin{aligned}
\bigg[ \frac{\hbar \omega ( \bf{q})}{2S} \bigg]^2 =
&\bigg [ 2J_1 - J_2(2-\cos{q_xa}-\cos{q_ya}) + J_c + D \bigg]^2 \\
&-\bigg [ J_1 \{ \cos{\frac{(q_x + q_y)a}{2}} + \cos{\frac{(q_x - q_y)a}{2}} \} \\
& +J_c\cos{(\frac{q_zc}{2})}\bigg]^2
\end{aligned}
\label{eq:4}$$ for G-type structure where $\textbf{q}$ is the wave vector measured relative to a $\Gamma$-point at a magnetic Bragg peak, and $a$ and $c$ are the lattice parameters for the tetragonal $P4/nmm $ unit cell.
We first make a rough estimate of $J_1$ and $J_2$ and subsequently refine $D$ and $J_c$ by fitting to the low energy portion of the magnetic spectrum. After refining $D$ and $J_c$, we perform fits to the full magnetic spectrum by fixing $D$ and $J_c$ and varying $J_1$ and $J_2$. This process is repeated until good convergence is achieved, although additional constraints, described below, were necessary to optimize $J_1$ and $J_2$.
Using $\textsc{SpinW}$ we calculate magnon dispersion and the powder-averaged intensities $S(Q,E)$ by Monte Carlo sampling of 50000 $Q$-vectors for a given magnitude of $Q$, from 0.1 - 4.2 [Å]{}$^{-1}$ as shown in Figs. \[Fig:ins150mev\] and \[Fig:ins50mevgap\]. Different $E-$ and $Q$-cuts were fit by using the non-linear least-squares process to capture major features of the INS spectra.
[*Spin gap ($\Delta$) and single ion anisotropy $D$:*]{} To estimate $D$ we focus on Fig. \[Fig:ins50mevgap\] with spectra obtained at $E_i = 50$ meV, where it can be seen that there is a gap in the spin-wave spectrum of $\approx 6$ meV for each compound. Figure \[Fig:gapcut\] shows energy averaged over a limited range of $Q$ and centered at $Q_{(100)} \pm0.2$ Å$^{-1}$ for LaMnAsO and LaMnSbO and at the $Q_{(10\frac{1}{2})} \pm0.2$ Å$^{-1}$ for BaMnAsF, obtained from data shown in Fig. \[Fig:ins50mevgap\]. The solid lines are best fit to the experimental data using the parameters listed in Table \[tab:params\]. For LaMnAsO we identify significant magnetic INS contribution from MnO that is present as an impurity phase (for details on the magnetic INS contribution of MnO polycrystalline see Ref. \[\]).
![ (Left column) measured inelastic neutron scattering data at $E_i$ = 50 meV for (a) LaMnSbO, (b) LaMnAsO, and (c) BaMnAsF as indicated. (Right column) (d-f) Corresponding calculated spectra using the best fit parameters given in Table \[tab:params\]. The shaded areas in the calculated panels are kinematically inaccessible regions neutrons at the specified energy and set up. Notice the weak but detectable minimum at the M-point for LaMnSbO that also shows in the calculations.[]{data-label="Fig:ins50mevgap"}](ins50mevgap_v2.pdf){width="3.4"}
[*Two-dimensionality of the systems:*]{} The $J_c$ term in Eqs. (\[eq:3\]) and (\[eq:4\]) determines the interlayer correlations. For all three samples, we find that the value of $J_c$ is negligibly small. Although we kept the value of $|J_c|$ fixed at 0.01 meV, this serves as an upper bound as modeling the data using $|J_c|$ as small as $10^{-4}$ meV yields similar results. As $|J_c|$ increases above 0.01 meV, we visually notice that columns of excitations emanate from (10$L$) magnetic Bragg peaks in our models, which is not observed experimentally. To get more insight into $J_c$, we make $Q$ cuts near roughly $E\simeq 25$ meV where the INS data is relatively cleaner and free from phonon and multiple scattering signals as shown in Fig. \[Fig:jccut\]. The $Q$-cuts in Fig. \[Fig:jccut\] all show characteristic quasi-2D features with a tail that extends to large $Q$ values due. This is similar to a Warren lineshape which corresponds to the powder averaging of rod of scattering. This behavior can be contrasted with similar cuts in the INS of the more 3D-like BaMn$_2$As$_2$ for which the scattering is modulated with peaks that are near ($H$0$L$) reflections [@Ramazanoglu2017].
![Q-cut around low energy region ($E\approx 25 \pm 5$ meV) for (a) LaMnSbO, (b) LaMnAsO, and (c) BaMnAsF showing the 2D nature of the spin excitations. The region is chosen on the $E_i=150 $ meV data with relatively cleaner region where signal from phonons is absent. These cuts are used to estimate an upper limit for $J_c$. []{data-label="Fig:jccut"}](jccut_allsamples_v2.pdf){width="2.5in"}
[*Relation between $J_1$ and $J_2$ and their determination*]{}: Fixing $J_c$ and $D$, we proceed by systematically varying $J_1$ and $J_2$ to model energy-cuts as shown in Fig. \[Fig:vanhovecut\]. We calculate $\chi^2$ values for numerous combinations of $J_1$ and $J_2$ to search for its minimum to obtain the best fit to the data. A 3D plot of $\chi^{2}(J_1,J_2)$ is shown in Fig. \[Fig:chi2fig2\] and the optimal values are listed in Table \[tab:params\] (note that in Fig. \[Fig:chi2fig2\] we present 1/$\chi^{2}(J_1,J_2)$ for color enhancement purposes).
![Energy-cuts for the full $S(Q,E)$ spectrum for (a) LaMnSbO, (b) LaMnAsO, and (c) BaMnAsF. Additional phonons and other backgrounds were subtracted by similar methods described in [@Ramazanoglu2017]. Circular symbols on the left and right side of the vertical dashed line denote the data extracted from $E_i$ = 50 meV and 150 meV respectively. (b) We determined the presence of MnO in LaMnAsO sample and the magnon signal from AFM MnO is shown in the shaded region. (c) Obvious phonon signal was detected near 20 meV of BaMnAsF spectrum which could not be subtracted in a systematic manner. Hence we decided to omit those points.[]{data-label="Fig:vanhovecut"}](vanhovecut_allsamples_v3.pdf){width="3in"}
![3D plot of 1/$\chi^{2}(J_1,J_2)$ showing the relation between $J_1$ and $J_2$, obtained from best fits to energy cuts. The dashed line is extracted by determining the range of energy scale for the X-point and mapping it to corresponding $SJ_1$ and $SJ_2$ values. Solid red line is $SJ_1 = 2SJ_2$ above which the stripe structure is favored.[]{data-label="Fig:chi2fig2"}](chi2_figures_v4.pdf){width="3in"}
![Calculated spin-waves dispersions along principal directions of single crystals using the best fit parameters obtained in this study.[]{data-label="Fig:singlecrystal"}](singlecrystal_allsamples_lineonly.pdf){width="3.3in"}
Figure \[Fig:chi2fig2\] shows that the minima in $\chi^2$ form a shallow valley which does not allow for a precise determination of $J_1$ and $J_2$. We can improve this situation by exploiting the extrema (van Hove singularities) in the spin wave dispersion to further constrain the values. For example, Fig. \[Fig:singlecrystal\] shows the spin wave dispersion obtained for each compound using the parameters from Table I. The maximum between the M- and Z-point gives rise to a van-Hove singularity that results in a peak in the magnetic spectra. Whereas the M-point energy is evident in the measured and calculated spectra for LaMnSbO, see Fig. \[Fig:ins50mevgap\] (a) and (d), for LaMnAsO and BaMnAsF we can only estimate this point from $E_i=150$ meV with larger uncertainty. Our best estimates of the minimum at the M-point is at 23, 50, and 58 meV with a standard deviation of 2, 5, and 5 meV for LaMnSbO, LaMnAsO, and BaMnAsF, respectively. Similarly, looking at $E_{i} = 150$ and $E_i=300$ meV data, (see, Fig. \[Fig:ins300mev\] in SI) we estimate that the spin-wave bandwidth (corresponding to the X-point) of LaMnSbO, LaMnAsO, and BaMnAsF to be at $90\pm3$, $120\pm3$, and $95\pm5$ meV. It is worth noting that the kinematic constraint of neutron does not allow us to get a good handle on the X-point, which would have significantly narrowed the uncertainties of $J_1$ and $J_2$.
First-Principles Calculations
-----------------------------
The magnetic ground states of LaMnSbO, LaMnAsO, and BaMnAsF, independent of $U$, are correctly predicted using first-principles calculations. Extracted $SJ_1$, $SJ_2$, $SJ_c$, and $SD$ for various $U$ values are shown in Fig. \[Fig:fpcalc\]. As $U$ increases the localization of Mn d-states, $SJ_1$, $SJ_2$, and $SD$ values decrease in magnitude while $SJ_c$ experiences little change. Quantitative agreement of theoretical $SJ_1$ and $SJ_2$ values are most consistent with INS experiments at $U\simeq 0$ eV for LaMnSbO, $U \simeq 1$ eV for LaMnAsO, and $U \simeq 2$ eV for BaMnAsF. The C-type magnetic structure,found in LaMnSbO and LaMnAsO is readily explained by AFM intralayer and FM interlayer couplings, while the G-type magnetic structure of BaMnAsF arises from AFM interlayer coupling.
In agreement with experiment, DFT+$U$ calculations also find competing AF NN and NNN interactions within the square lattice layer. For all compounds, experiments confirm that $2J_2/J_1<1$ which is a necessary condition for the observed intra-layer checkerboard AFM order. However, it is somewhat surprising that this frustration is rather large. For example, we find $2J_2/J_1=0.84$ for LaMnSbO which is responsible for low-lying M-point spin waves. These results suggest two interesting possibilities. The first is that the square lattice Mn pnictides may adopt stripe AFM order for relatively larger $J_2$ values. Even more interesting is the possibility that such materials can be tuned into quantum disordered regime with $2J_2/J_1 \approx 1$ hosting a spin liquid.
Calculated Mn moments, as shown in Fig. \[Fig:fpcalc\], range from 3.49 to 4.33 $\mu_B/\text{Mn}$, increasing with $U$, showing greater localization as a function of increasing $U$, as expected. Additional electron-electron correlation, required to more accurately describe the INS data, slightly overestimates the on-site Mn moments found in these systems, i.e., 3.45 $\mu_{B,exp}$ $vs$ 3.60 $\mu_B$ for LaMnSbO at $U$ = 0 eV, 3.34 $\mu_{B,exp}$ $vs$ 3.74 $\mu_B$ for LaMnAsO at $U$ $\simeq 1$ eV, and 3.65 $\mu_{B,exp}$ $vs$ 4.02 $\mu_B$ for BaMnAsF at $U$ $\simeq 2$ eV [@McGuire2016; @Zhang2016; @Saparov2013]. Comparison of moment sizes and absolute values of $J_1$ and $J_2$ suggest small effective $U$ and indicate a degree of delocalization of Mn $d-$electrons in all three systems.
![(a) Magnetic moments localized on Mn for the AF1 state as a function of $U$ obtained from the DFT calculations. First principle calculations of (b-d) $SJ_1$, $SJ_2$, (e-g) $SJ_c$, and $SD$ vs experimentally determined values for LaMnSbO, LaMnAsO, and BaMnAsF respectively. Dashed lines are values obtained from the spin-waves analysis as listed in Table \[tab:params\]. The shaded regions refer to experimental error found in $SJ_1$, $SJ_2$, $SJ_c$, and $D$. Best agreement with theory and experiment occur at $U \simeq 0, 1$, and 2 eV for LaMnSbO, LaMnAsO, and BaMnAsF, respectively.[]{data-label="Fig:fpcalc"}](fpcalc.pdf){width="3.5in"}
Conclusions
===========
We have extracted the magnetic excitations of polycrystalline antiferromagnetic LaMnAsO, LaMnSbO, and BaMnAsF from inelastic neutron scattering data by removing signals from the sample holder, some phonons, and other background features. We analyzed the magnetic spectra in the framework of inplane $J_1-J_2$ and out-of plane $J_c$ exchange coupling of a Heisenberg model using $\textsc{SpinW}$. We also provide theoretical results using spin-polarized DFT + $U$ calculations, that to a large extent agree with the experimental results. Our analysis shows that for all three samples $J_1$ and $J_2$ are antiferromagnetic with a ratio $2J_2/J_1 < 1$ consistent with square lattice checkerboard order, but with $J_2$ large enough to consider effects of magnetic frustration. We note that the largest $2J_2/J_1$ ratio is obtained for LaMnSbO which may explain the relatively lower $T_N$ compared to the the other compounds[@Calder2014; @Rahn2017; @Soh2019; @Ramazanoglu2017]. The inter-plane coupling $J_c$ in all three systems is on order of $\sim 3\times10^{-4}J_1$ rendering these systems’ quasi-two-dimensional magnetic properties. Such a weak $J_c$ is due to the intervening rocksalt LaO and BaF layers, which effectively reduce the interlayer coupling compared to Mn-122 and Mn-112 square lattice antiferromagnets.
With regard to the intralayer exchange couplings, these are controlled by both steric effects from the rocksalt layers and the $Pn$ ligands (lattice parameters) and the hybridization of Mn with the specific $Pn$ ligand. For example, in LaMn$Pn$O series, $J_1$ and $J_2$ are progressively reduced for heavier $Pn$ atoms due primarily to an increase in the Mn-Mn nearest-neighbor distance. The larger distance between intralayer Mn atoms results in much weaker hybridization and exchange. On the other hand, we can also compare LaMnSbO and BaMnAsF, which have nearly the same Mn-Mn distance but different $Pn$ ligands. In this case, $SJ_1$ and $SJ_2$ are larger for the heavier Sb ligand, likely due to the increased hybridization from the extended p-orbitals of the Sb atom as compared to As.
Acknowledgments
---------------
This research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358.
[——–]{}
Supporting Information
----------------------
[ ]{}
All isotropic spin exchange calculations used a $7 \times 7 \times 3$ $k$-point mesh within a $2a \times 2b\times 2c$ supercell (with respect to the conventional unit cell), MAE calculations for LaMnSbO and LaMnAsO used a $18 \times18\times 8$ k-point mesh within the conventional unit cell, and MAE calculations for BaMnAsF used a $18 \times 18 \times 4$ k-point mesh within a $1a\times 1b \times 2c$ supercell.
The total spin exchange energies, per supercell (16 f.u.), of the ordered spin states are given as $$E_x=(n_1J_1+n_2J_2+n_3J_c)(S_\text{Mn})^2,$$ where $E_x$ is the energy of different ordered spin state ($x = {\rm AF1} - {\rm AF4}$) relative to the ground state magnetic structure (AF1), and $S_\text{Mn}$ is the total spin, 5/2, localized on Mn$^{2+}$. The ordered spin states, values for , and the relative energies of the ordered spin states are shown in Fig. \[Fig:AFstructure\], and Tables \[tab:sx1\] and \[tab:sx2\].
![$2a \times 2b \times 2c$ supercell with respect to the conventional unit cell used to construct construct states AF1 - AF4. These ordered spin states were used to extract $J_1$, $SJ_2$, and $SJ_c$. Blue (filled) spheres represent down-spin Mn and white (open) spheres represent up-spin Mn. Thick and thin bonds correspond to $J_1$ and $J_c$ interactions, respectively.[]{data-label="Fig:AFstructure"}](OrderedSpinStates.pdf){width="2.8in"}
Ordered Spin States $n_1$ $n_2$ $n_3$
--------------------- ------- ------- -------
AF1 -32 32 16
AF2 -32 32 -16
AF3 0 -32 16
AF4 -24 24 12
: Ordered spin states $n_1$-$n_2$ values used to extract $J_1$-$J_c$ using energy mapping analysis.
\[tab:sx1\]
-- --- ----------- ------------ --------- --------
AF1 AF2 AF3 AF4
0 [ 0.00]{} 2.00 1679.98 633.81
1 [ 0.00]{} 2.11 1381.14 545.47
2 [ 0.00]{} 2.03 1154.85 465.91
3 [ 0.00]{} 1.87 991.27 400.91
4 [ 0.00]{} 1.67 875.01 348.91
5 [ 0.00]{} 1.48 792.71 307.27
0 [ 0.00]{} 1.17 2249.12 873.94
1 [ 0.00]{} 1.48 1934.91 761.54
2 [ 0.00]{} 1.60 1670.39 653.50
3 [ 0.00]{} 1.60 1462.89 562.08
4 [ 0.00]{} 1.53 1302.24 486.94
5 [ 0.00]{} 1.42 1177.00 425.39
0 0.00 [ -1.44]{} 2046.78 814.73
1 0.00 [ -1.24]{} 1762.11 687.49
2 0.00 [ -1.05]{} 1531.27 581.03
3 0.00 [ -0.87]{} 1349.55 495.11
4 0.00 [ -0.72]{} 1205.99 425.94
5 0.00 [ -0.60]{} 1090.78 369.89
-- --- ----------- ------------ --------- --------
: Relative energies (meV/16 f.u.) of the four ordered spin states used to calculate $SJ_1$-$SJ_c$ . Red values signify the lowest energy state.
\[tab:sx2\]
![(Left column) measured inelastic neutron scattering data at $E_i$ = 300 meV for (a) LaMnSbO, (b) LaMnAsO, and (c) BaMnAsF as indicated. (right column) (d-f) Corresponding calculated spectra using the best fit parameters given in Table \[tab:params\]. The shaded areas in the calculated panels are kinematically inaccessible regions for neutrons at the specified energy and set up.[]{data-label="Fig:ins300mev"}](ins300mev.pdf){width="3.1in"}
|
---
abstract: 'We have extended the ’t Hooft-Nobbenhuis complex transformations to include mass. Under these new transformations, Schrodinger, Dirac, Klein-Gordon and Einstein general relativity equations are invariant. The non invariance of the cosmological constant in Einstein field equations dictates it to vanish thus solving the longstanding cosmological constant problem.'
author:
- 'Arbab I. Arbab[^1] and Hisham M. Widatallah[^2]'
title: 'A mass-extended ’t Hooft-Nobbenhuis complex transformations and their consequences'
---
Introduction
============
Recently ’t Hooft and Nobbenhuis have introduced complex space-time transformations under which the invariance of the ground - state would associate the vacuum state to a zero cosmological constant. These transformations are such that the space-time coordinates $x^\mu\rightarrow ix^\mu$. Under these transformations, the Hamiltonian of a nonrelativistic particle $H\rightarrow -H$ and the boundary conditions of the physical states do not become invariant. This is because while the real part of the states goes to zero as $x\rightarrow \infty$, the imaginary part doesn’t. Consequently, all states except the ground state ($\psi (x)=\rm const.$) will break this symmetry. In the context of quantum theory, hermiticity, normalization and boundary conditions will not transform as in the usual symmetry transformations for a non-relativistic particle.
In as much as in quantum mechanics energy and momentum are described by $p=-i\hbar \vec{\nabla}$ and $E=i\hbar\frac{\partial}{\partial t}$, any coordinates transformations will inevitably transform $p$ and $E$. Moreover, in relativity theory mass and energy are related. Hence, the coordinate transformations will necessarily require the mass transformations too. This latter transformations have been overlooked by ’t Hooft and Nobbenhuis in their complex transformations.
We argue that the inclusion of mass transformation will resolve the shortcomings of the theory pertaining to the hermiticity, normalization and boundary conditions. The resulting complex transformations, referred to the ’t Hooft-Nobbenhuis mass-extended transformations, lead to interesting physics when applied to Schrodinger, Dirac, Einstein general relativity and Maxwell equations. Under these transformations, the hermitian operators are transformed into antihermitian operators. Moreover, the wavefunctions that are periodic in real spaces are also periodic in imaginary space. These pretty transformations urge us to formulate our laws of nature in a complex space instead of the present real space.
The mass-extended ’t Hooft-Nobbenhuis transformations
=====================================================
’t Hooft and Nobbenhuis have recently introduced a complex space-time transformations, and identified it as a symmetry of laws of nature. In quantum mechanics, the space-time transformations will transform momentum ($p$) and energy ($E$) since the latter are expressed by $$\vec{p}=-i\hbar\vec{\nabla}\,,\qquad\qquad E=i\hbar\frac{\partial}{\partial t}\,.$$ From the theory of relativity, one knows that mass ($m$) and energy are related by $$E=mc^2\,,\qquad\qquad p=mv\,.$$ Hence, Eq.(1) and (2) will yield $$E'= -iE\,,\qquad \vec{p}\,'=-i\vec{p}\,,\qquad m'= -im\,.$$ Therefore, the full complex transformations will become $$\vec{r}\,'= -i\vec{r}\,,\qquad t'= -it\,,\qquad m'= -im\,.$$ We refer to these transformations as ’t Hooft-Nobbenhis mass-extended transformations. These transformations do not alter the Einstein mass-energy equation, $E=\sqrt{c^2p^2+m_0^2c^4}$ and the Lorentz transformations. According to Eq.(4), one notices that $$E't'= Et\,,\qquad \vec{p}\,'\cdot \vec{r}\,'= \vec{p}\cdot
\vec{r}\,,\qquad \hbar\,'=\hbar\,,\qquad c\,'=c$$ so that the wave phase does not change. This guarantees the normalization and boundary conditions of all physical states. Hence, Eq.(4) is a symmetry of laws of nature.
Schrodinger equation
====================
Applying the transformations in Eq.(4) to Schrodinger equation $$i\hbar\frac{\partial \psi}{\partial t}=-\frac {\hbar^2}{2m}\nabla^2\psi+V\psi$$ shows that Schrodinger equation is invariant provided $V(ix)=-iV(x)$. The probability and current densities in Schrodinger formalism are defined by $$\rho=\psi^*\psi\,,\qquad\vec{J}=\frac{\hbar}{2mi}\left(\psi^*\vec{\nabla} \psi-\psi\vec{\nabla} \psi^*\right)$$ so that the continuity equation reads $$\vec{\nabla}\cdot\vec{J}+\frac{\partial \rho}{\partial t}=0\,.$$ Applying the transformations (4) in Eq.(7) yields $$\vec{J}\,'=i\vec{J}\,,\qquad\qquad \rho\,'=i\rho\,.$$ This implies that the continuity equation, Eq.(8), is invariant under the transformation in Eq.(4) too.
Non relativistic particle motion
================================
Following ’t Hooft and Nobbenhuis , we discuss here a one dimensional motion of a non-relativistic particle. Consider the Hamiltonian $$H=\frac{p^2}{2m}+V(x)\,.$$ Under the transformations in Eq.(4), the above equation yields $$H'=\frac{-p^2}{-2mi}+V(ix)=-i\left(\frac{p^2}{2m}+V(x)\right)=-iH$$ where, $V(ix)=-iV(x)$. This can be realized for certain potentials.
If we now consider a harmonic oscillator where the Hamiltonian is given by $$H=\hbar\,\omega\,(a^+a+\frac{1}{2})$$ where[^3] $$a=\sqrt{\frac{m\omega}{2\hbar}}\left(x+\frac{ip}{m\omega}\right)\,,\qquad a^+=\sqrt{\frac{m\omega}{2\hbar}}\left(x-\frac{ip}{m\omega}\right).$$ Under the transformations in Eq.(4), one has $$a'= -a\,,\qquad a'^+=- a^+\,,\qquad H'= -iH\,,\qquad m'\omega'=-m\omega\,.$$ The state wavefunction, $\psi_n(x)$, which is given by $$\psi_n(x)=\frac{1}{\sqrt{2^nn!}}\left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \exp(-\frac{m\omega}{2\hbar} x^2)\,H_n(\sqrt{\frac{m\omega}{\hbar}}\,x)$$ where $$H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}\left(e^{-x^2}\right)\,,\qquad n=0, 1, 2, \ldots$$ transforms as, $\psi\,'_n(x)= (-1)^n \sqrt{i}\,\psi_n(x)$. This is also consistent with Eqs.(7) and (9). Hence, the boundary and normalization conditions are satisfied for all states. This is unlike the original ’t Hooft-Nobbenhuis transformations, where these two properties of the wavefunction are lost. Moreover, the transformed energy and momentum operators are antihermitian.
Classical scalar field
======================
Consider a real scalar field $\Phi(x)$ defined by the Lagrangian density $${\cal L}=-\frac{1}{2}(\partial_\mu\Phi)^2-V(\Phi)\,,\qquad V(\Phi)=\frac{1}{2}m^2\Phi^2+\lambda \Phi^4$$ and the Hamiltonian density $${\cal H}=\frac{1}{2}\,\Pi^2+\frac{1}{2}(\nabla\Phi)^2+V(\Phi)\,,\qquad \Pi(x)=\partial_0\Phi$$ where $\lambda$ is a constant. Now if $\Phi(x)$ transforms as $$\Phi'(x)\equiv\Phi(ix)=-i\Phi(x)\qquad\Rightarrow\qquad \Pi'(ix)=-\Pi(x)$$ the above Lagrangian and Hamiltonian will be invariant, i.e., $\cal L'=\cal L$ and $\cal H'=H$. Moreover, the action $S=\int {\cal L}\, d^4x=S\,'$.
Maxwell equations
=================
Defining the electromagnetic tensor $F_{\mu\nu}$ as \[3\] $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$$ Maxwell equations read $$\partial_\mu F^{\mu\nu}=\mu_0 J^\nu\,,\qquad \partial_\mu F_{\nu\lambda}+\partial_\nu F_{\lambda\mu}+\partial_\lambda F_{\mu\nu}=0\,.$$ Under the transformations in Eq.(4), the charge ($q$), current ($I$) and vector potential $A_\mu$ transform as $$q'= q\,,\qquad I'=-iI\,,\qquad A'_{\mu}= -iA_\mu\,.$$ Therefore, the electromagnetic tensor, the charge and current densities transform as $$F\,'_{\mu\nu}=-F_{\mu\nu}\,,\qquad \vec{J}\,'=i\vec{J}\,,\qquad \rho'= i \rho\,.$$ Hence, Maxwell equations are invariant under Eq.(4). If we extend our analysis to Yang-Mills field, the Lagrangian will involve quadratic, cubic and quartic couplings of the filed $A_\mu$. In this case, one has $F_{\mu\nu}\rightarrow F^a_{\mu\nu}$, where $$F^a_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu+if^{abc}A^b_\mu A^c_\nu$$ where $f^{abc}$ are the structure constants, we will get an invariant Lagrangian, since $$F^a_{\mu\nu}\,'=-F^a_{\mu\nu}\,.$$
Quantum electrodynamics-QED
===========================
The QED Lagrangian density for a free particle with rest mass $m_0$ is given by \[2\] $${\cal L}=i\hbar \,\bar{\psi}\,\gamma^\mu D_\mu\psi-m_0c\,\bar{\psi}\,\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$ where $D_\mu=\partial_\mu-i\frac{e}{\hbar}A_\mu\,.$ This Lagrangian is invariant under Eq.(4) and (17), viz., $\cal L'=\cal
L$, provided that $\bar{\psi}\,'\,\psi'=i\,\bar{\psi}\,\psi$, where we have assumed $\gamma\,'^\mu=\gamma^\mu$. This can be satisfied if $\psi\,'=\sqrt{i}\,\psi$ and $\bar{\psi}\,'=\sqrt{i}\,\bar{\psi}$. This ushers in the direction that the probability and current density transform as $ \rho\,'=i\,\rho$ and $\vec{J}\,'=i\,\vec{J}$. This is in agreement with the transformations in Eqs. (9) and (17).
General theory of relativity
============================
Einstein equations for general relativity are \[4\] $$R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}=8\pi\,G T_{\mu\nu}\,.$$ The particle equation of motion is given by the geodesic equation $$\frac{d^2x^\mu}{d\tau^2}+\Gamma^\mu_{\nu\lambda}u^\nu
u^\lambda=0$$ where $\Gamma^\mu_{\nu\lambda}$ are the Christoffel symbols, $\tau$ and $u^\mu$ are the proper time and velocity, respectively. Under the transformations (4), Eq.(28) yields $$\Gamma^\mu_{\nu\lambda}\,'=-i\,\Gamma\,^\mu_{\nu\lambda}\,.$$ Using the transformations in Eq.(4), one finds that[^4] $$\rho'_m=\rho_m\,,\qquad G'=-G\,,\,\qquad R'_{\mu\nu}=-R_{\mu\nu}\,,\qquad T'_{\mu\nu}=T_{\mu\nu}$$ where $\rho_m$ is the matter density and $G$ is Newton constant. Hence, Einstein general relativity equations are invariant under the transformations in Eq.(4). However, the existence of the cosmological in the Einstein field equations will violate the invariance. The way out of this is that the cosmological constant must be zero. Thus, the vanishing of the cosmological constant is thus because its existence violates the symmetry defined in Eq.(4). But if the cosmological constant has to be present in the Einstein field equations, it must change sign under the transformations in Eq.(4). In this case, the Einstein field equations are invariant. This global invariance is a quite interesting merit that the original ’t Hooft- Nobbenhuis transformations had wished for. Hence, the reason for the vanishing of the cosmological constant is now understood.
Concluding remarks
==================
We have extended in this work the complex space-time transformation postulated by ’t Hooft and Nobbenhuis to include mass. This extended transformation remedied the problems of the original transformations. These transformations can be considered as special case of scale transformation. We have found n in this work that all physical laws are invariant under the complex space-time and mass transformations. Hence, the extended complex transformations are the symmetry of laws of nature.
References {#references .unnumbered}
==========
Gerard ’t Hooft, Stefan Nobbenhuis, *Class. Quantum Grav*. 23, 3819 (2006).\
Bjorken, J. D., and Drell, S. D., *Relativistic Quantum Mechanics*, McGraw-Hill (1964).\
Jackson, D., *Classical Electrodynamics*, John Wiley & Sons Inc. (1962).\
Weinber, S., *Gravitation and cosmology*, John Wiley & Sons Inc. (1972).\
[^1]: Email: aiarbab@uofk.edu
[^2]: hisham@ictp.it
[^3]: $[\omega]=T\,^{-1}.$
[^4]: $[G]=M^{-1}L^3T^{-2}$ and \[$\rho_m]=ML^{-3}$.
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The provocative proposal that the mechanism of high temperature superconductivity in layered cuprates may be related to the exotic properties of low dimensional quantum spin systems, such as the RVB (Resonanting Valence Bond) state [@Anderson], has been a major driving force behind the rapid advance of [*quantum magnetism*]{}. Naturally, the initial emphasis was placed on understanding materials involving Cu$^{2+}$ ions with the 3d$^{9}$ configuration ($S=\frac{1}{2}$) [@Hase; @Azuma]. More recently, $S=\frac{1}{2}$ quantum magnets involving V$^{4+}$ [@Ueda1; @Ueda2; @Ueda3] and Ti$^{3+}$ [@Tokura; @Keimer; @Ueda4; @Ueda5; @Beynon; @Seidel] ions with 3d$^{1}$ configurations have been attracting strong attention. Potential interests include the realization of doped metallic states in 3d$^{1}$ [*Mott insulators*]{}, and ultimately, superconductivity[@Ueda3]. In addition, the near degeneracy of the $t_{2g}$-orbitals often gives rise to the orbital and/or charge ordering [@Nagaosa]. In the orbital ordered state, the configuration of the occupied 3d$_{xy,yz,zx}$-orbitals exhibits a long range order. On the other hand, the valence of ions differs site by site in charge ordered states. The additional orbital and charge degrees of freedom make the underlying physics of 3d$^{1}$ quantum spin systems more intriguing, yet more complicated. A fascinating example is a mixed valence system NaV$_{2}$O$_{5}$ [@Ueda2; @Ohama1] (the average valence of V-ions is +4.5). NaV$_{2}$O$_{5}$ undergoes orbital and charge ordering at $T_{c}=34K$, where a large energy gap $E_{g}=98K$ opens [@Ohama1; @Fagot; @Ohama2; @Yoshihama]. The ratio $2E_{g}/k_{B}T_{c}=6$ is much larger than the BCS value 3.5 expected for conventional spin-Peierls transitions with lattice-dimerization. Intensive theoretical and experimental efforts have been underway to understand the exotic phase transition.
An equally exciting new avenue to investigate $S=\frac{1}{2}$ quantum spin systems with 3d$^{1}$ electrons is the titanates involving Ti$^{3+}$ ions. Starting from the metal-insulator transition in Sr$_{1-x}$La$_{x}$TiO$_{3}$[@Tokura], growing efforts are under way in search for a new form of quantum magnetism in titanates: strong orbital fluctuations in LaTiO$_{3}$[@Keimer]; a quasi 1D $S=\frac{1}{2}$ chain system (Na,Li)TiSi$_{2}$O$_{6}$ [@Ueda4]; 3D Pyrochlore system MgTi$_{2}$O$_{4}$[@Ueda5]; and a quasi-2D layered system TiOCl[@Beynon; @Seidel].
Among these titanates, the TiOCl system has particularly unique characteristics. First, Ti$^{3+}$O$^{2-}$ form bi-layers separated by Cl$^{-}$ bi-layers. The quasi-2D layered structure may be considered a close analogue to those realized in high $T_{c}$ cuprates. In fact, Beynon and Wilson [@Beynon] reported the very little temperature dependence in uniform susceptibility $\chi(T)$, and discussed the possible realization of a RVB ground state in the Mott-insulator. They also noted that $\chi(T)$ is very sensitive to impurities, and demonstrated that a Sc$^{3+}$ ion ($S=0$) substituted into a Ti$^{3+}$ site ($S=\frac{1}{2}$) gives rise to a localized spin $S=\frac{1}{2}$, in analogy with creation of a free spin by Zn$^{2+}$ ($S=0$) substituted into Cu$^{2+}$ ($S=\frac{1}{2}$) sites in cuprates. Second, in a very recent report, Seidel et al. observed a sharp, nearly isotropic drop of $\chi(T)$ below $\sim100K$ in defect-free samples, signaling the emergence of an energy gap[@Seidel]. The drop in $\chi(T)$ is even more pronounced in high quality single crystals. Since $\chi(T)$ above $\sim200K$ can be fitted with a $S=\frac{1}{2}$ Heisenberg chain model with the nearest neighbor exchange interaction $J=660K$, Seidel et al. proposed, based on LDA+U calculations, that the effective dimension of the TiO layers are reduced from 2D to 1D by an orbital order at Ti$^{3+}$ sites along the a- or b-axis. The gapped behavior of $\chi(T)$ was attributed to a spin-Peierls transition [@Seidel]. Third, but related to the second point, the near degeneracy of the energy levels of different 3d$^{1}$ orbital configurations [*without mixed-valence nature*]{} might make TiOCl an ideal model system to investigate spin-Peierls-[*like*]{} transitions with additional orbital degrees of freedom but probably without charge order.
In this Letter, we report the first NMR investigation of TiOCl. While the spin-Peierls transition in CuGeO$_{3}$ ([*without*]{} orbital and charge degrees of freedom) and the exotic order in NaV$_{2}$O$_{5}$ ([*with*]{} orbital [*and*]{} charge degrees of freedom) have seen many detailed microscopic investigations, to the best of our knowledge this is the first successful microscopic experiments reported for TiOCl. We demonstrate that TiOCl reveals a unique spin-gap behavior accompanied by lattice instabilities, and undergoes successive phase transitions at $T_{c1}=94\pm2K$ and $T_{c2}=66\pm2 K$. Unlike CuGeO$_{3}$ and NaV$_{2}$O$_{5}$, the fluctuation effects in TiOCl are so strong above $T_{c1}$ that a pseudo spin-gap manifests itself as high as $T^{*}=135\pm10K$. Moreover, our observation of a broad continuum in the NMR lineshape data indicates that the intermediate phase between $T_{c1}$ and $T_{c2}$ is [*not*]{} a simple, dimerized state in 1D. Below $T_{c2}$, TiOCl undergoes a first order phase transition to a fully gapped state (rather than to the gapless RVB state) with an extraordinarily large energy gap $E_{g}=430\pm60K$. The unusually large energy gap ($2E_{g}/k_{B}T_{c1,c2}=10\sim 15$) as well as the presence of the intermediate state between $T_{c1}$ and $T_{c2}$ suggest that the observed phase transitions are not conventional spin-Peierls transitions, and point towards the significance of the roles played by the additional orbital degrees of freedom over the entire temperature range.
Our TiOCl single crystals were synthesized by standard vapor-transport techniques from TiO$_{2}$ and TiCl$_{3}$[@Seidel]. A large number of very thin, flaky single crystals with typical dimensions of 2mm$\times$2mm were assembled on a Macor sample holder. NMR measurements were conducted by applying an external magnetic field in parallel with the aligned crystal c-axis. Random alignment within the TiO-plane prevented us from conducting measurements along the a- and b-directions. We emphasize that, unlike $^{63,65}$Cu and $^{51}$V NMR, the sensitivity of $^{47,49}$Ti NMR is notoriously low [@sensitivity]. Very short transverse relaxation times $T_{2}$ at Ti sites and the small volume of the available crystals made the NMR measurements even more difficult. We needed to average spin-echo signals up to $\sim10^{6}$ scans to obtain a reasonable signal to noise ratio. In general, NMR signal intensities increase as $1/T$ in proportion to the Boltzman factor. However, the exponentially growing spin-lattice relaxation time $T_{1}$ below $T_{c1}$ slows down the NMR pulse-sequence, hence the measurements at lower temperatures were equally difficult and time consuming. Despite our intensive efforts, we have been able to find $^{47,49}$Ti NMR signals only for the central transition from the $I_{z}=+\frac{1}{2}$ to $-\frac{1}{2}$ state. The detection of $^{35}$Cl NMR signal was somewhat easier, and we did manage to find the $I_{z}=\pm\frac{3}{2}$ to $\pm\frac{1}{2}$ transitions above $T_{c1}$ [@nuq]. All the nuclear spin-lattice relaxation data were deduced by fitting the nuclear spin recovery after an inversion $\pi$-pulse to the standard rate equations. We confirmed that $^{47}$Ti (nuclear spin $I=5/2$) and $^{49}$Ti (nuclear spin $I=7/2$) NMR gives identical $1/T_{1}$. Given that the magnetic recovery process of nuclear magnetization $M(t)$ is dominated by different terms for $^{47}$Ti ($M(t)\sim exp[-15t/T_{1}]$) and $^{49}$Ti ($M(t)\sim exp[-28t/T_{1}]$), we conclude that $1/T_{1}$ is dominated entirely by magnetic fluctuations at all temperatures. The difference in the recovery characteristic of $M(t)$ also helped us identify the pairs of $^{47}$Ti and $^{49}$Ti lines below $T_{c2}$, where the NMR lines split into doublets.
The experimental information on the spin degrees of freedom is summarized in Fig. 1. Quite generally, the $^{47,49}$Ti nuclear spin-lattice relaxation rate, caused by low frequency magnetic fluctuations may be expressed as[@Moriya], $$\frac{1}{T_{1}} = T\frac{ k_{B}\gamma_{n}^{2}}{\mu_{B}^{2} \hbar}
\sum_{{\bf q}} | F({\bf q}) |^{2} \frac{\chi''({\bf
q},\omega_{n})}{\omega_{n}},
\label{T1}$$ where $F({\bf q})$ is the form factor of the electron-nucleus hyperfine couplings, and $\chi''({\bf q},\omega_{n})$ is the imaginary part of the dynamical susceptibility at the observed NMR frequency $\omega_{n}$. $1/T_{1}$ appears to asymptote to a constant value near $300K$, $1/T_{1}\sim 1400$ sec$^{-1}$. This implies that $\chi''({\bf q},\omega_{n})$ asymptotes to a Curie law at higher temperatures. This is consistent with the behavior expected for typical $S=\frac{1}{2}$ 1D Heisenberg chains such as Sr$_{2}$CuO$_{3}$ [@Takigawa]. However, we caution that $1/T_{1}\sim const.$ may also be expected for $S=\frac{1}{2}$ 2D Heisenberg model[@Imai93]. Certainly $\chi(T)$ of TiOCl also fits nicely to the 1D Heisenberg chain model between $200K$ and $800K$ [@Seidel], but we recall that NaV$_{2}$O$_{5}$ turns out to be a 2D spin-charge-orbital hybrid system [@Gaulin] despite the similarly nice fit of $\chi(T)$ to the 1D model.
In general, the growth of short-range order enhances low frequency spin fluctuations (hence $1/T_{1}T$) with decreasing temperature in the absence of a gap in the spin excitation spectrum. The most striking feature in Fig.1 is that $1/T_{1}T$ begins to [*decrease*]{} below $T^{*}\sim135K$. This implies that low frequency spin fluctuations are suppressed below $T^{*}$, by almost two orders of magnitude between $135K$ and $65K$. Interestingly, the observed behavior of $1/T_{1}T$ is qualitatively similar to the pseudo-gap phase in underdoped high $T_{c}$ cuprates. Needless to say, it does not necessarily imply that the underlying mechanism is identical.
We can also probe the lattice degrees of freedom by observing the EFG (Electric Field Gradient) reflected on NMR lineshapes. In Fig.2, we present the $^{47,49}$Ti and $^{35}$Cl NMR lineshapes for the central transition. We observed a single NMR line for both Ti and Cl down to $T_{c1}=94\pm2 K$. For $^{35}$Cl, we also managed to detect $I_{z}=\pm\frac{3}{2}$ to $I_{z}=\pm\frac{1}{2}$ satellite transitions, but again there was only one kind of signal [@nuq]. These results indicate that there is only one kind of Ti and Cl site in TiOCl within our experimental resolution, and rule out any potential orbital order configurations above $T_{c1}$ that would lead to more than one inequivalent sites. Below $T_{c1}$, both $^{47,49}$Ti and $^{35}$Cl NMR central lines begin to broaden, signaling a second order phase transition. We confirmed that the magnitude of the $^{35}$Cl NMR line splitting is inversely proportional to the external magnetic field. Therefore the line splitting is caused by the second-order nuclear quadrupole interaction with the EFG. We also found that the drop of $1/T_{1}T$ is accelerated below $T_{c1}$. In addition, close inspection of the $\chi(T)$ data reported in [@Seidel] reveals a kink at $T_{c1}$, followed by a rapid decrease with temperature with positive curvatures. These results suggest that a clear gap structure emerges in the spin excitation spectrum at $T_{c1}$, accompanied by a [*static*]{} distortion of the lattice. The NMR lineshape exhibits a broad continuum below $T_{c1}$ down to $T_{c2}$ as summarized in Fig.3(b). The broad continuous distribution of the local EFG environment implies the presence of numerous inequivalent Ti and Cl sites in the TiOCl lattice. The most plausible scenario is that the emergence of the spin-gap at $T_{c1}$ is accompanied by an orbital-order (possibly incommensurate). That is, the phase transition at $T_{c1}$ is [*not*]{} a simple spin-Peierls transition due to lattice dimerization, even though the well-developed short-rage spin order below $400 K$ and the gapped behavior of $\chi(T)$ below $\sim100K$ [@Seidel] suggest the contrary.
We found additional evidence for the involvement of the lattice in the phase transition at $T_{c1}$ by $^{35}$Cl nuclear relaxation data $^{35}1/T_{1}$ as shown in Fig.3(a). $^{35}1/T_{1}$ is two orders of magnitude slower than $1/T_{1}$ at $^{47,49}$Ti-sites. This implies that the magnetic hyperfine coupling of $^{35}$Cl nuclear spins with Ti 3d$^{1}$ electron spins is at least by an order of magnitude smaller. The qualitatively different temperature dependence between $^{35}1/T_{1}$ \[see Fig.3(a)\] and $1/T_{1}$ \[see Fig.1(a)\] suggests that slowing of EFG fluctuations below $T^{*}$ toward the phase transition at $T_{c1}$ causes the cusp of $^{35}1/T_{1}$ at $T_{c1}$[@quadrupole]. Even more straightforward evidence is that the $^{35}$Cl spin-echo decay (transverse $T_{2}$ relaxation) in Fig.3(c) and (d) shows typical motional narrowing effects below $T^{*}$ towards $T_{c1}$ due to lattice softening. The slow lattice dynamics at or below the $^{35}$Cl NMR frequency (37.6 MHz) transforms the spin echo-decay to Lorentzian-like as $T_{c1}$ is approached. Taken together with the precursive suppression observed for $1/T_{1}T$ below $T^{*}$ in Fig.1(a), we conclude that the drastic softening of the lattice below $T^{*}$ drives the observed pseudo spin-gap behavior prior to the actual second-order phase transition at $T_{c1}=94K$. The high onset temperature $\sim200K$ of the gradual decrease of $1/T_{1}$ suggests that phonon softening at higher frequency scales begins at much higher temperature.
At $T_{c2}$, the continuum of the $^{35}$Cl and $^{47,49}$Ti NMR lineshape suddenly collapses into doublets, whose line positions correspond to the two extrema observed above $T_{c2}$. This strongly suggests that the unit cell of TiOCl has two and only two inequivalent Ti and Cl sites below $T_{c2}$ [@tensor]. The low-frequency Ti 3d$^{1}$ spin fluctuations exhibit an activation behavior below $T_{c2}$, as shown in Fig.1(b). From the fit to an exponential form, $1/T_{1}T\sim
exp(-E_{g}/k_{B}T)$, we deduce the energy gap $E_{g}=430\pm60K$. These results are consistent with the formation of a singlet ground state at $T_{c2}$ by lattice dimerization. Our results are also consistent with the recent observation of doubling of the unit cell along the b-axis at $66\pm1K$ by Y.S. Lee et al. based on x-ray scattering measurements[@x-ray]. On the other hand, there are many experimental signatures which are at odds with conventional second order spin-Peierls transitions. $1/T_{1}$ \[Fig.1\], the NMR lineshapes \[Fig.2\], and $^{35}1/T_{1}$ \[Fig.3(a)\] change discontinuously at $T_{c2}$. $^{35}$Cl spin-echo decay curve also changes suddenly to a slow Gaussian type at $T_{c2}$ \[Fig.3(d)\]. This indicates that the soft vibration of the lattice disappears suddenly below $T_{c2}$. All these NMR results as well as the discontinuous change in $\chi(T)$ [@Seidel] and the x-ray data [@x-ray] at $T_{c2}$ indicate that the phase transition at $T_{c2}$ is first order. Furthermore, the observed energy gap is unusually large, $2E_{g}/k_{B}T_{c1,c2}=10\sim 15$. In the case of conventional, mean-field spin-Peierls gap, one would expect the ratio to be close to the BCS value, $2E_{g}/k_{B}T_{c}=3.5$, as observed for CuGeO$_{3}$ ($2E_{g}/k_{B}T_{c}=3.3$[@Kikuchi; @Regneau]). The observed ratio in TiOCl is even larger than that for NaV$_{2}$O$_{5}$, $2E_{g}/k_{B}T_{c}\sim 6$ [@Ohama1; @Yoshihama]. It is important to realize that $E_{g}$ is comparable to the apparent exchange energy $J=660 K$ estimated by the 1D fit of $\chi(T)$ [@Seidel]. Such a large magnitude of the energy gap $E_{g}$ strongly suggests that the spin excitations from the singlet ground state is dressed by other electronic degrees of freedom, most likely of the orbital origin.
To summarize, TiOCl is a rather unique layered $S=\frac{1}{2}$ quantum-material with a pre-existing pseudo spin-gap above $T_{c1}$, the unconventional intermediate spin and lattice (probably orbital) states between $T_{c1}$ and $T_{c2}$, and a first order phase transition into a singlet ground state with an unusually large energy gap. At first sight, the strong fluctuation effects evidenced in the pseudo spin-gap behavior suggest a genuinely 1D nature of the TiOCl along the a- or b-axis, achieved by an orbital order well above $T_{c1}$. We recall that the pre-existing orbital order above $T_{c1}$ may reduce the effective dimensions of the TiO layer from 2D to 1D, as suggested by Seidel et al. based on LDA+U calculations[@Seidel]. On the other hand, the additional orbital degrees of freedom along the two orthogonal directions within the 2D TiO-layers may cause strong orbital fluctuations, which could effectively suppress the tendency towards a spin-Peierls transition at finite temperature. In such a scenario, the manifestation of the pseudo spin-gap above $T_{c1}$ may be the consequence of the suppression of the spin-Peierls transition, and the successive phase transitions at $T_{c1}$ and $T_{c2}$ may arise from a competition between different orbital orders. In addition, if the second nearest-neighbor exchange interaction $J_{a}'$ in the zig-zag chain structure along the a-axis (see Fig.1(c) in [@Seidel]) exceeds $0.241J_{a}$ (where $J_{a}$ being the nearest-neighbor exchange interaction along the a-axis), a spontaneous spin dimerization along the a-axis may be favoured [@second] over the unit-cell doubling along the b-axis observed below $T_{c2}$, introducing frustration. In any case, the unprecedented behavior of TiOCl points towards the crucial roles played by the orbital degrees of freedom and their fluctuations below $\sim200K$. Recalling that TiOCl is not a mixed-valence system such as NaV$_{2}$O$_{5}$, TiOCl may be an ideal model spin-Peierls system with additional orbital degrees of freedom, and we call for futher microscopic studies.
We thank P.A. Lee, Y.S. Lee, G. Sawatzky, V. Kiryukhin, and B.D. Gaulin for helpful discussions, and I. Affleck for calling our attention to Ref. [@second]. The work at M.I.T. was supported by NSF DMR 98-08941.
P.W. Anderson, Science [**253**]{}, 1196 (1987). M. Hase et al., Phys. Rev. Lett. [**70**]{}, 3651 (1993). M. Azuma et al., Phys. Rev. Lett. [**73**]{}, 3463 (1994). H. Iwase et al., J. Phys. Soc. Jpn. [**65**]{}, 2397 (1996). M. Isobe and Y. Ueda, J. Phys. Soc. Jpn. [**65**]{},1178 (1996). T. Yamauchi et al., Phys. Rev. Lett. [**89**]{}, 057002 (2002). Y. Tokura et al., Phys. Rev. Lett [**70**]{}, 2126 (1993). B. Keimer et al., Phys. Rev. Lett [**85**]{}, 3946 (2000). M. Isobe et al., J. Phys. Soc. Jpn. [**71**]{}, 1423 (2002). M. Isobe and Y. Ueda, J. Phys. Soc. Jpn. [**71**]{}, 1848 (2002). R.J. Beynon and J. A. Wilson, J. Phys. Cond. Matt. [ **5**]{}, 1983 (1993). A. Seidel et al., cond-matt/0206374 (To appear in Phys. Rev. B). Y. Tokura and N. Nagaosa, Science [**288**]{}, 462 (2000). T. Ohama et al., J. Phys. Soc. Jpn. [**66**]{}, 545 (1997). Y. Fagot-Revurat et al., Phys. Rev. Lett [**84**]{}, 4176 (2000). T. Ohama et al., Phys. Rev. [**B 59**]{}, 3299 (1999). T. Yoshihama et al., J. Phys. Soc. Jpn. [**67**]{}, 744 (1998). The natural abundance of $^{47,49}$Ti is low (7.3% and 5.5%, respectively). The nuclear gyromagnetic ratio is also rather small ($^{47,49}\gamma_{n}/2\pi=2.4000$ and $2.4005$ MHz/Tesla, respectively). The $^{35}$Cl quadrupole tensor along the c-axis was determined to be $^{35}\nu_{Q}^{c}=1.641\pm 0.005 MHz$ at 130K. T. Moriya, J. Phys. Soc. Jpn. [**18**]{}, 516 (1963). M. Takigawa et al., Phys. Rev. Lett [**76**]{}, 4612 (1996). T. Imai et al., Phys. Rev. Lett [**70**]{}, 1002 (1993). B.D. Gaulin et al., Phys. Rev. Lett [**84**]{}, 3446 (2000). Strictly speaking, fitting the $^{35}$Cl nuclear spin recovery $M(t)$ to a purely magnetic form may not be valid near $T_{c1}$. However, it is sufficient for the present purpose. Our preliminary attempts to deduce more detailed information on the lattice distortion at $T_{c1}$ and $T_{c2}$ have been unsuccessful, as NMR signals are too weak to detect satellite transitions below $T_{c1}$. We defer further efforts until much larger single crystals become available. Y.S. Lee et al., private communications. J. Kikuchi et al., J. Phys. Soc. Jpn. [**63**]{}, 872 (1994). L. P. Regnault et al., Phys. Rev. [**B 53**]{}, 5579 (1996). F.D.M. Haldane, Phys. Rev. [**B25**]{}, 4925 (1982). Also see S. Eggert, Phys. Rev. [**B54**]{}, R9612 (1996).
|
---
author:
- 'Jasmine A. Nirody'
- 'Richard M. Berry'
- George Oster
bibliography:
- 'ref.bib'
title: The Limiting Speed of the Bacterial Flagellar Motor
---
The bacterial flagellar motor (BFM) drives swimming in a wide variety of bacterial species, making it crucial for several fundamental processes including chemotaxis and community formation [@Berg2003; @korobkova2006hidden; @Bai2010; @sourjik2012responding]. Accordingly, gaining a mechanistic understanding of this motor’s function has been a fundamental challenge in biophysics. The relative ease with which the output of a single motor can be measured in real time, by observing with light microscopy rotation of a large label attached to the motor, has made it one of the best studied of all large biological molecular machines.
![The bacterial flagellar motor consists of a series of large concentric rings that attach to a flagellar filament via a flexible hook. An active motor can have between 1 and 11 torque-generating stator units. Stators interact with protein ‘spokes’ (FliG) along the rotor’s edge to drive motor rotation.[]{data-label="schematic"}](Figure0.pdf){width="40.00000%"}
However, because of its complexity and localization to the membrane, atomic structures of the entire motor are not yet available. Still, relatively detailed models have been developed using a combination of partial crystal structures [@Lloyd1999; @Brown2002; @Lee2010], cross-linking and mutagenesis [@Zhou1998; @braun2004arrangement; @lowder2005flig], and electron microscopic and cryo-electron tomography images [@khan1992cytoplasmic; @suzuki2004structure] (Fig \[schematic\]).
Arguably the most important physical probe into the *dynamics* of a molecular motor is its torque-speed relationship. For the BFM, this curve was shown to have two distinct regimes (Fig. \[intro\]). This characteristic feature of the BFM was long held as the first ‘checkpoint’ for any theoretical model of the motor. However, recent experiments showed that the number of torque-generating units (*stators*) in the motor is load-dependent—that is, published torque-speed curves most likely contain measurements from motors with different numbers of engaged stators [@Lele2013; @tipping2013load]. Specifically, at high loads (low speeds) a motor can have up to 11 engaged stators, while at low loads (high speeds) motors typically operate with only one stator.
This finding shed doubt on several fundamental results about the dynamics of the BFM, including, importantly, its behavior at low loads. A seminal set of experiments, termed ‘resurrection’ experiments, studied the dependence of motor speed on the number of stators at various external loads [@block1984successive; @Reid2006; @Yuan2008]. In these experiments, paralyzed cells were allowed to begin rotating slowly, and discrete increases in speed were interpreted as the addition of torque-generating units. Surprisingly, while up to 11 increases of near-equal size were observed at high loads, only a single such ‘jump’ was observed at low loads.
These results quickly led to a series of reworked theoretical models, all of which required that the limiting speed of the motor be independent of the number of torque-generators [@meacci2009dynamics; @bai2009model; @meacci2011dynamics]. However, it is likely that low-load measurements were never performed on motors with more than one stator, leaving open the question of how the BFM behaves in the zero-torque (high-speed) limit.
Here, we predict that the limiting speed of the BFM increases with the number of active stators. This prediction is due to our assumption that the stator is not in contact with the rotor in between steps, or ‘power strokes’ (i.e., the *duty ratio* of the motor is less than 1). We recently presented a model for torque-generation in flagellar motors with a single stator [@mandadapu2015mechanics]. Here, we extend this model to motors with multiple stators. Our model is a specific example of such a mechanism; however, most models involving a conformational change in stator structure will share this property. This is because such mechanisms likely require stators to ‘reset’ between steps.
We argue that these mechanisms have a significant effect on the motor’s duty ratio only at low loads. In this way, our model, and others in this category, remain compatible with current evidence that the BFM must have a high duty ratio to be processive at high loads. Experiments testing this hypothesis, if successful, would be the first to explicitly quantify the behavior of a multi-stator motor in the low-load regime.
**Overview.** We have implicated a steric interaction between the stator and rotor in torque generation [@mandadapu2015mechanics]. Here, we briefly describe our proposed mechanism. Further details, including explicit forms of the Langevin equations used in simulation, can be found in [@mandadapu2015mechanics], as well as in the supplementary material.
![Recent experiments have shown that the number of torque-generators (stators) is not constant across applied loads. Therefore, it is likely that previously measured torque-speed curves (here, data from [@fung1995powering]) were generated using motors with varying numbers of stators: points in the high-load regime correspond to motors with up to 11 stators and points at low loads to motors with only one. Red dashed line separates the high-load, mechanically-limited and low-load, kinetically-limited regimes of the curve; the latter is the focus of this article.[]{data-label="intro"}](Figure1.pdf){width="49.00000%"}
Stators drive the rotation of the motor by stepping along protein ‘spokes’ around the periphery of the *rotor*, which is a large ring that connects to the flagellar filament via a flexible hook. This interaction is analogous to parents pushing on the handles of a merry-go-round on the playground for their children’s amusement.
Individual steps are initiated by the arrival of protons at ion-binding sites within the stator complex. During the power stroke, conformational changes in the stator apply a steric force onto the spokes of the rotor wheel, rotating it a discrete step-length $\ell$. Stators apply no productive torque to the rotor between power strokes. Because the BFM lives at low Reynolds number, the rotor also exhibits no productive movement in between steps.
We assumed that there are 26 spokes along the edge of the rotor ([@sowa2005direct], although see, e.g., [@Lee2010; @paul2011architecture]). A ‘perfect’ power stroke is defined as a step of length $\ell = \frac{2\pi}{26}$ rad, leaving the stator in contact with the neighboring spoke. These steps are observed through the rotation of a small bead (the ‘load’) attached to a truncated flagellar hook. When the connection between the rotor and the bead is soft, discrete motor steps ‘blur’ into a seemingly continuous trajectory. Experimentally, steps have been directly observed by slowing the motor down to a speed of approximately 10 Hz [@sowa2005direct]. Simulation trajectories showing steps for high-speed (near-zero load) motors with one and seven engaged stators are shown in Fig. \[preds\](a)-(b).
{width="90.00000%"}
**Motor speed at low loads increases with number of stators.** From simulations, we predict that the maximum speed of the motor is *not* ‘universal’ as currently assumed, but dependent on the number of engaged torque-generators (Fig. \[preds\](c), open red markers). In their recent paper, Lo *et al.* computed torque-speed curves for a chimeric sodium-driven motor [@lo2013mechanism]. Low-load measurements on these motors were performed using a 100 nm-diameter gold bead (inset, Fig. \[preds\](c)).
This data was collected from motors with between 1 and 5 active stators, with results from motors with higher stator numbers corresponding to faster peaks (Fig. \[preds\](c), blue markers). The authors chose to focus on the dynamics of single-stator motors, leaving open the implications of this data for how the zero-torque speed depends on stator number. The existence of multiple discrete peaks at low load strongly supports the idea that the maximum speed is dependent on the number of stators, at least in chimeric motors. While experimental results characterizing how the zero-torque speed varies with the number of stators have yet to be published on the wild-type, our predictions should hold for both Na$^+$ and H$^+$ motors.
Previously, Ryu and coauthors reported a set of general conditions that must be met in order for the limiting speed to be independent of the number of engaged stators [@ryu2000torque]. First, the rate at which steps are initiated must be independent of the relative position of the rotor and the stator. This position is dependent on both the external load and the actions of any other engaged stators. Therefore, the ‘decision’ of a stator to step should be ignorant of both these factors. Second, stators must engage the rotor for the majority of their cycle (that is, the BFM’s duty ratio $DR \approx 1$). Using reasoning based on dynamics at high load, the authors concluded that the duty ratio of the stators was indeed very high. Experiments reporting that the speed at low loads was independent of stator number soon followed [@Yuan2008], which seemed to lend strong support to both of the proposed requirements [@ryu2000torque].
We assume that the stators are disengaged from the rotor for a large part of their cycle at low loads, resulting in a violation of the second condition. Unlike most recently proposed mechanisms (but see [@boschert2015loose]), we assume motor rotation and ion flow can be *loosely coupled*: an ion passage may not always result in appreciable rotation of the rotor. The *stator’s* motion, however, is tightly coupled to ion flow—that is, an ion passage is both necessary and sufficient for the initiation of a stator’s power stroke. Therefore, loose coupling in our model does not arise from some form of ion leakage, as may be expected [@boschert2015loose; @oosawa1982mechanism; @oosawa1983coupling; @oosawa1986loose; @Berry1993]. Instead, it is due to the fact that stator steps are rarely ‘perfect’ in multiple-stator motors: if stator steps overlap, a portion of the second stroke is ‘wasted’ because the rotor is pushed out of the later-firing stator’s reach.
These properties seem contrary to present assumptions that stators in the BFM must have a high duty ratio. However, arguments in support of $DR \approx 1$ are largely based on motor dynamics at high load. We show that our prediction that $DR < 1$ at low loads arises from fundamental differences in motor dynamics between the two regimes. In this way, we argue that our proposed mechanism is compatible with experimental evidence for a high duty ratio at high loads.
**Kinetically-limited stators have low duty ratios.** A stator initiates a step when protons arrive at a specified binding site within the complex. The mechanochemical cycle of the stator then has two phases: moving and waiting, characterized by timescales $T_m$ and $T_w$, respectively [@meacci2009dynamics]. If $T_S$ is the time that a stator engages the rotor during a complete cycle ($T_m + T_w$), a single-stator motor has duty ratio $DR = T_S/(T_m + T_w)$.
The waiting time between power strokes $T_w$ depends on the rate of proton arrivals at the binding site on a stator unit. These arrivals are Poissonian with rate $k_{\text{on}} = k_0\exp \left[\lambda\Delta G_{\text{ij}}/k_BT\right]$. Here, $\Delta G_{\text{ij}}$ is the thermodynamic contribution of the ion motive force and $k_BT$ is Boltzmann’s constant multiplied by temperature [@xing2006torque; @bai2009model]. For simplicity, we choose $\lambda=0.5$ as done in previous studies [@xing2006torque]. The parameter $k_0$ is a function of the pH of the external periplasm; lower pH corresponds to higher proton concentration and thus a speedier arrival at the site. At room temperature and pH 7.0, $\langle T_w \rangle = 1/k_\text{on} = 0.2$ ms for single-stator motors.
The average moving time is estimated through the relation $\omega \approx \ell/\left(\langle T_m \rangle + \langle T_w \rangle\right)$ [@meacci2009dynamics]. The average motor speed $\omega$ is also related to the load drag coefficient $\zeta_L$ by $\zeta_L\omega \approx \tau$, where $\tau$ is the motor torque [@Berg2003; @inoue2008torque]. In our simulations, the motor is limited by proton arrivals at very low loads ($\langle T_m \rangle \approx 0.01$ ms), while at high loads, $\langle T_m \rangle \approx 10$ ms surpasses $\langle T_w \rangle$. These values are consistent with previous studies [@meacci2009dynamics; @meacci2011dynamics].
Because we predict that motor rotation is driven by steric forces, a stator must be in contact with the rotor for a large part of a productive power stroke ($T_S/T_m \approx 1$). Previous models of torque-generation have similarly considered the mechanochemical cycle of the BFM to consist of moving and waiting phases [@meacci2009dynamics; @meacci2011dynamics]. However, our model is unique in assuming that stators disengage from the rotor between subsequent power strokes. This results in $DR < 1$ for single-stator motors at low loads, as the waiting time is no longer negligible compared to the moving time in this regime (Fig. \[preds\](d)). The waiting time may even surpass $\langle T_m \rangle$, as shown in Fig. \[preds\](a)-(b).
The waiting time until a proton binds to any one of $N$ independently-stepping stators is exponentially distributed with rate $N\times k_{\text{on}}$. Therefore, $\langle T_w \rangle$ is shortened as additional stators are recruited. The subsequent increase in duty ratio (Fig. \[preds\](d)) results in an increase in limiting speed with the number of stators.
**High duty ratios at high loads.** Here, we address two arguments which have been used to assert that the duty ratio of the BFM must be very high: (i) the observation that the number of steps per revolution $n_{\text{steps}}$ increases as additional torque-generating units were recruited [@samuel1995fluctuation; @samuel1996torque], and (ii) a calculation determining that a motor with a low duty ratio cannot be processive due to ‘unwinding’ of the tether connection between the rotor and load [@Berg2003]. Though these arguments are based on observations at high load, they were taken as support for a zero-torque speed independent of stator number. This extrapolation was possible largely due to the absence of a proposed physical mechanism for rotation of the BFM. Such a mechanism is now provided in our model [@mandadapu2015mechanics]. To this end, we show that these arguments can be consolidated with our proposed mechanism, as well as with the corresponding prediction that $DR < 1$ at low loads.
Samuel and Berg used fluctuation analysis to determine that the number of steps per revolution was proportional to stator number [@samuel1995fluctuation; @samuel1996torque]. In the absence of a specific physical mechanism, this result was interpreted to mean that a motor decreases its elementary step size as it recruits torque generators. This in turn implied a motor with a high duty ratio, in which each unit acts with the $N-1$ others to rotate a fixed distance $d$ [@ryu2000torque].
This observation holds in the high-load (low-speed) regime, which is where these measurements were made. Even though stators disengage between subsequent strokes, the duty ratio of the motor remains very high because the time spent within a power stroke is far greater than the pauses between subsequent strokes ($DR = T_s/(T_m+T_w) \approx T_s/T_m \approx 1$). Furthermore, the rotor is likely always in contact with at least one stator as the steps of individual stators almost certainly overlap. This accounts for the observed proportional increase in $n_{\text{steps}}$ with the number of active stators.
Stator steps still may overlap at low loads (high speeds), though they are less likely to do so because $T_m$ is shorter than at high loads. Our simulations predict that similar analyses in this regime will detect a sublinear increase in $n_\text{steps}$ with stator number (Fig. \[preds\](e)).
A second argument for a high duty ratio in the BFM was posed by Howard Berg, who determined that if the BFM did not have a duty ratio of close to unity, it could not be processive [@Berg2003]. The reasoning behind this is as follows. Consider an experiment where a cell is tethered to a surface by the hook of its flagella and is spun about by the rotation of the motor at its base. The cell body is large in comparison to the flagellar motor, and accordingly the viscous drag on it is much larger than that on the BFM’s rotor. Therefore, if there are no stators to prevent it, the wound tether between the motor and the cell will unwind exponentially: $\theta = \theta_0\exp(-\alpha t)$, where $\theta_0$ is the initial twist and $\alpha$ is the torsional spring constant divided by the rotational drag coefficient of the rotor. A simple calculation showed that unless a motor had a duty ratio of very close to unity, this tether would unwind too quickly for the stator units to keep up.
We note that concrete evidence is still lacking that slowly-rotating tethered motors do not ‘lose’ steps to the tether connection unwinding. Support for tightly-coupled mechanisms came from reports that the number of ions per revolution was directly proportional to motor speed [@Meister1987]. However, it was later shown that a loosely-coupled mechanism also produced a linear relationship with the same slope, but non-zero intercept [@Berry1993]. Regardless, our model construction and parameter choice is such that the unwinding of the tether does not overwhelm the stator in our simulations (see supplementary information) [@mandadapu2015mechanics]. A final resolution awaits experiments measuring how the ion flux at stall (zero speed) differs between single- and multi-stator motors.
In contrast to the high-load regime, the relative drags of the bead and the rotor are comparable at low loads. As we approach the zero-torque limit, the rotor drag may surpass that of the load [@meacci2009dynamics; @meacci2011dynamics]. For example, we estimated the drag coefficient for the low-load measurement in [@lo2013mechanism] to be $\zeta_L \approx 0.005 $ pN-nm-s-rad$^{-1}$, which is lower than $\zeta_R \approx 0.02$ pN-nm-s-rad$^{-1}$ [@Berg2003]. In this case, the bead will move forward as the tether connection unwinds.
More generally, the characteristic timescale of the load’s motion is given by its frictional drag coefficient divided by the spring constant: $t_L = \zeta_L/\kappa$. A single-stator motor should have a power stroke of comparable length. Note that this is not necessary for a multi-stator motor: steps from different stators may overlap, extending the period during which at least one unit is engaged.
To illustrate, we consider the second-smallest bead used by Lo *et al* [@lo2013mechanism]. Estimating $\zeta_L = 0.04 $ pN-nm-s-rad$^{-1}$ and choosing a conservative spring constant $\kappa = 150$ pN-nm-rad$^{-1}$ (at the lower edge of the measured range [@block1989compliance]), the characteristic timescale of the load is $t_L = \zeta_L/\kappa \approx 0.27$ ms. A single-stator motor with this load rotated at $\approx$ 110 Hz [@lo2013mechanism]. Recall that motor speed $\omega \approx d/(\langle T_m \rangle + \langle T_w \rangle)$, where the step size $\ell = \frac{1}{26}$ rev and $ \langle T_w \rangle \approx 0.2$ ms. Then $\langle T_m\rangle~\approx \left(\frac{1}{26}\right)/110 - 2$e-4 $\approx 0.15$ ms, which is enough time for the load to (at least partially) ‘catch up’ to the rotor.
**Conclusions.** The dynamics of the BFM across applied loads have been of great interest since a two-regime torque-speed curve was proposed several decades ago. Recent experiments reporting that the number of stators in a motor varies across loads have opened some interesting questions, and reopened several more.
For instance, the zero-torque speed has been assumed to be independent of the number of engaged stators based on the results of early ‘resurrection’ experiments [@block1984successive; @Reid2006; @Yuan2008]. Theoretical models after these results were reported have all been constructed to reproduce this behavior at low loads. However, recent experiments strongly suggest that these experiments were never performed on motors with more than a single stator [@Lele2013].
In opposition to current assumptions, our simulations predict that the limiting (zero-torque) speed of the BFM increases with stator number. This relationship arises from our assumption that stators detach from the motor when they pause between steps. This assumption is common to most models in which a conformational change in the stator drives motor rotation. This results in a low duty ratio for motors at low load, where the waiting time between steps is at least on the order of the time spent in a power stroke. Because the power stroke duration is much longer at high loads, the duty ratio in this regime is not affected by this unbound state. In this way, our mechanism is consistent with evidence that processive motors at high load must have a high duty ratio.
Recently, Lo *et al.* presented evidence of increasing zero-torque-speed with stator number in chimeric, sodium-driven motors [@lo2013mechanism]. However, this result was not fully explored as the authors focused on understanding single-stator motor dynamics. Further experiments, especially on wild-type motors, would directly test the hypothesis presented here, and be the first to explicitly characterize the low-load behavior of the flagellar motor.
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---
abstract: 'Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and first-order Reed-Muller codes have been an object of intensive study by many coding theorists. In this paper, we investigate some binary shortened code of two families of linear codes from APN functions and some $p$-ary shortened codes from PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$-ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal in the sense that they meet some bound on linear codes. These results show high potential for shortening to be used in designing good codes.'
author:
- 'Can Xiang, [^1]Chunming Tang [^2] and Cunsheng Ding [^3]'
title: 'Shortened linear codes from APN and PN functions [^4] '
---
Linear code, shortened code, PN function,APN function, $t$-design
Introduction {#Sec-introduct}
============
Let ${{\mathrm{GF}}}(q)$ denote the finite field with $q=p^m$ elements, where $p$ is a prime and $m$ is a positive integer. An $[v,\, k,\,d]$ linear code ${{\mathcal{C}}}$ over ${{\mathrm{GF}}}(q)$ is a $k$-dimensional subspace of ${{\mathrm{GF}}}(q)^v$ with minimum (Hamming) distance $d$. Let $A_i$ denote the number of codewords with Hamming weight $i$ in a code ${{\mathcal{C}}}$ of length $v$. The weight enumerator of ${{\mathcal{C}}}$ is defined by $
1+A_1z+A_2z^2+ \cdots + A_v z^v.
$ The sequence $(1,A_1,\ldots,A_v)$ is called the weight distribution of ${{\mathcal{C}}}$ and it is an important research topic in coding theory, as it contains crucial information to estimate the error correcting capability. Thus the study of the weight distribution attracts much attention in coding theory and much work focuses on the determination of the weight distributions of linear codes (see, for examples, [@sihem2020; @sihem2017; @ding2018; @Ding16; @DingDing2; @Ding15; @TXF2017; @zhou20132; @zhou20131; @Tangit2016]). Denote by ${{\mathcal{C}}}^\bot$ and $(A_0^{\perp}, A_1^{\perp}, \dots, A_\nu^{\perp})$ the dual code of a linear code ${{\mathcal{C}}}$ and its weight distribution, respectively. It is known that the *Pless power moments* [@HP10] $$\begin{aligned}
\label{eq:PPM}
\sum_{i=0}^\nu i^t A_i= \sum_{i=0}^t (-1)^i A_i^{\perp} \left [ \sum_{j=i}^t j ! S(t,j) q^{k-j} (q-1)^{j-i} \binom{\nu-i}{\nu -j} \right ],
\end{aligned}$$ play an important role in calculating the weight distributions of linear codes, where $A_0=1$, $0\le t \le \nu$ and $S(t,j)=\frac{1}{j!} \sum_{i=0}^j (-1)^{j-i} \binom{j}{i} i^t$. A code ${{\mathcal{C}}}$ is said to be a $t$-weight code if the number of nonzero $A_i$ in the sequence $(A_1, A_2, \cdots, A_v)$ is equal to $t$. We call a $[v,k,d]$ code *distance-optimal* if no $[v,k,d+1]$ code exists and *dimension-optimal* if no $[v,k+1,d]$ code exists. An $[v,k,d]$ code is said to be length-optimal if there is no $[v',k,d]$ code exists with $v' < v$. A code is said to be optimal if it is distance-optimal, dimension-optimal and length-optimal.
Let ${{\mathcal{C}}}$ be a $[\nu,k,d]$ linear code over ${{\mathrm{GF}}}(q)$ and $T$ a set of $t$ coordinate positions in ${{\mathcal{C}}}$. We use $\mathcal C^T$ to denote the code obtained by puncturing $\mathcal C$ on $T$, which is called the *punctured code* of $\mathcal C$ on $T$. Let $\mathcal C(T)$ be the set of codewords which are $\mathbf{0}$ on $T$. We now puncture $\mathcal C(T)$ on $T$, and obtain a linear code $\mathcal C_{T}$, which is called the *shortened code* of $\mathcal C$ on $T$. The following property plays an important role in determining the parameters of the punctured and shortened codes of $\mathcal{C}$ in [@HP10 Theorem 1.5.7].
[@HP10]\[lem:C-S-P\] Let ${{\mathcal{C}}}$ be a $[\nu,k,d]$ linear code over ${{\mathrm{GF}}}(q)$ and $d^{\perp}$ the minimum distance of $\mathcal C^{\perp}$. Let $T$ be any set of $t$ coordinate positions. Then
- $\left ( \mathcal C_{T} \right )^{\perp} = \left ( \mathcal C^{\perp} \right)^T$ and $\left ( \mathcal C^{T} \right )^{\perp} = \left ( \mathcal C^{\perp} \right)_T$.
- If $t<\min \{d, d^{\perp} \}$, then the codes $\mathcal C_{T}$ and $\mathcal C^T$ have dimension $k-t$ and $k$, respectively.
It is worth noting that the shortening and puncturing technologies are two important approaches to constructing new linear codes. Very recently, Tang et al. obtained some ternary linear codes with few weights by shortening and puncturing a class of ternary codes in [@Tangdcc2019]. Afterwards, they presented a general theory for punctured and shortened codes of linear codes supporting t-designs and generalized Assmus-Mattson theorem in [@Tangit2019]. Some linear codes and $t$-designs can be obtained and their parameters can be also derived. However, till now few results on constructing punctured and shortened codes have been done and it is in general hard to determine their weight distributions. Motivated by this fact, we obtain some shortened codes of linear codes from almost perfect nonlinear (APN for short) and perfect nonlinear (PN for short) functions, and determine their parameters. Some of these codes are optimal or almost optimal.
The rest of this paper is arranged as follows. Section \[sec-pre\] introduces some notation and results related to group characters, gauss sums, t-designs and linear codes from APN and PN functions. Section \[sec-general\] gives some general results on shortened codes. Section \[sec-APN\] investigates some shortened codes of binary linear codes from APN functions. Section \[sec-PN\] studies some shortened codes of two classes of special linear codes from PN functions. Section \[sec-summary\] concludes this paper and makes concluding remarks.
Preliminaries {#sec-pre}
=============
In this section, we briefly recall some results on group characters, Gauss sums, t-designs, and linear codes from APN and PN functions. These results will be used later in this paper. We begin this section by fixing some notations throughout this paper.
- $p^*=(-1)^{(p-1)/2}p$.
- $\zeta_p=e^{\frac{2\pi \sqrt{-1}}{p}}$ is the primitive $p$-th root of unity.
- ${{\mathrm{GF}}}(q)^*={{\mathrm{GF}}}(q)\setminus \{0\}$.
- ${{\mathrm{Tr}}}_{q/p}$ is the trace function from ${{\mathrm{GF}}}(q)$ to ${{\mathrm{GF}}}(p)$.
- $\textup{SQ}$ and $\textup{N\textup{SQ}}$ denote the set of all squares and nonsquares in ${{\mathrm{GF}}}(p)^{*}$, respectively.
- $\eta$ and $\bar{\eta}$ are the quadratic characters of ${{\mathrm{GF}}}(q)^{*}$ and ${{\mathrm{GF}}}(p)^{*}$, repsectively. We extend these quadratic characters by letting $\eta(0)=0$ and $\bar{\eta}(0)=0$.
Group characters and Gauss sums
-------------------------------
An additive character of ${{\mathrm{GF}}}(q)$ is a nonzero function $\chi$ from ${{\mathrm{GF}}}(q)$ to the set of nonzero complex numbers such that $\chi(x+y)=\chi(x) \chi(y)$ for any pair $(x, y) \in {{\mathrm{GF}}}(q)^2$. For each $b\in {{\mathrm{GF}}}(q)$, the function $$\begin{aligned}
\label{dfn-add}
\chi_b(c)=\zeta_p^{{{\mathrm{Tr}}}(bc)} \ \ \mbox{ for all }
c\in{{\mathrm{GF}}}(q)\end{aligned}$$ defines an additive character of ${{\mathrm{GF}}}(q)$. When $b=0$, $\chi_0(c)=1 \mbox{ for all } c\in{{\mathrm{GF}}}(q),
$ and $\chi_0$ is called the [*trivial additive character*]{} of ${{\mathrm{GF}}}(q)$. The character $\chi_1$ in (\[dfn-add\]) is called the [*canonical additive character*]{} of ${{\mathrm{GF}}}(q)$. It is well known that every additive character of ${{\mathrm{GF}}}(q)$ can be written as $\chi_b(x)=\chi_1(bx)$ [@LN Theorem 5.7]. The orthogonality relation of additive characters is given by $$\sum_{x\in {{\mathrm{GF}}}(q)}\chi_1(ax)=\left\{
\begin{array}{rl}
q & \mbox{ for }a=0,\\
0 & \mbox{ for }a\in {{\mathrm{GF}}}(q)^*.
\end{array} \right.$$
The Gauss sum $G(\eta, \chi_1)$ over ${{\mathrm{GF}}}(q)$ is defined by $$\begin{aligned}
G(\eta, \chi_1)=\sum_{c \in {{\mathrm{GF}}}(q)^*} \eta(c) \chi_1(c) = \sum_{c \in {{\mathrm{GF}}}(q)} \eta(c) \chi_1(c)\end{aligned}$$ and the Gauss sum $G(\bar{\eta}, \bar{\chi}_1)$ over ${{\mathrm{GF}}}(p)$ is defined by $$\begin{aligned}
G(\bar{\eta}, \bar{\chi}_1)=\sum_{c \in {{\mathrm{GF}}}(p)^*} \bar{\eta}(c) \bar{\chi}_1(c)
= \sum_{c \in {{\mathrm{GF}}}(p)} \bar{\eta}(c) \bar{\chi}_1(c),\end{aligned}$$ where $\bar{\chi}_1$ is the canonical additive character of ${{\mathrm{GF}}}(p)$.
The following four lemmas are proved in [@LN Theorem 5.15, Theorem 5.33, Corollary 5.35] and [@DingDing2 Lemma 7], respectively.
[@LN] \[lem-32A1\] Let $q=p^m$ and $p$ be an odd prime. Then $$\begin{aligned}
G(\eta,\chi_1)&=&(-1)^{m-1}(\sqrt{-1})^{(\frac{p-1}{2})^2m}\sqrt{q}\\
&=&\left\{
\begin{array}{lll}
(-1)^{m-1}\sqrt{q} & \mbox{ for }p\equiv 1\pmod{4},\\
(-1)^{m-1}(\sqrt{-1})^{m}\sqrt{q} & \mbox{ for }p\equiv 3\pmod{4}.
\end{array} \right.\end{aligned}$$ and $$G(\bar{\eta}, \bar{\chi}_1)= \sqrt{-1}^{(\frac{p-1}{2})^2 } \sqrt{p}=\sqrt{p*}.$$
[@LN] \[lem-32A2\] Let $\chi$ be a nontrivial additive character of ${{\mathrm{GF}}}(q)$ with $q$ odd, and let $f(x)=a_2x^2+a_1x+a_0 \in {{\mathrm{GF}}}(q)[x]$ with $a_2 \ne 0$. Then $$\sum_{c \in {{\mathrm{GF}}}(q)} \chi(f(c)) = \chi(a_0-a_1^2(4a_2)^{-1}) \eta(a_2) G(\eta, \chi).$$
[@LN] \[lem-charactersum-evenq\] Let $\chi_b$ be a nontrivial additive character of ${{\mathrm{GF}}}(q)$ with $q$ even and $f(x)=a_2x^2+a_1x+a_0\in {{\mathrm{GF}}}(q)[x]$, where $b\in {{\mathrm{GF}}}(q)^*$. Then $$\sum_{c\in {{\mathrm{GF}}}(q)}\chi_b(f(c))=\left\{\begin{array}{ll}
\chi_b(a_0)q & \mbox{ if }a_2=ba_{1}^{2},\\
0 & \mbox{ otherwise. }
\end{array} \right.$$
[@DingDing2] \[lem-bothcharac\] If $m \ge 2$ is even, then $\eta(y)=1$ for each $y \in {{\mathrm{GF}}}(p)^*$. If $m \ge 1$ is odd, then $\eta(y)=\bar{\eta}(y)$ for each $y \in {{\mathrm{GF}}}(p)$.
Let $e$ be a positive integer and $(a, b)\in {{\mathrm{GF}}}(q)^2$, define the exponential sum $$\label{def S}
S_e (a, b)=\sum\limits_{x\in {{\mathrm{GF}}}(q)}\chi_1\left(a
x^{p^{e}+1}+b x\right).$$ Then we have the following results in [@Coult] , [@Cao2016], [@coulter2002] and [@YCD2006].
[@Coult]\[expsum\] Let $e$ be a positive integer and $m$ be even with $\gcd(m,e)=1$. Let $p=2$, $q=2^m$ and $a\in {{\mathrm{GF}}}(q)^*$. Then $$\begin{aligned}
S_e(a,0)=\left\{
\begin{array}{l}
(-1)^{\frac{m}{2}} 2^{\frac{m}{2}} ~~~~~~\mbox{ if $a \ne \alpha^{3t}$ for any $t$,} \\
-(-1)^{\frac{m}{2}} 2^{\frac{m}{2}+1} ~\mbox{ if $a = \alpha^{3t}$ for some $t$,}
\end{array}
\right.\end{aligned}$$ where $\alpha$ is a generator of ${{\mathrm{GF}}}(q)^*$.
[@Cao2016]\[expsum1\] Let $e,h$ be positive integers and $m$ be even with $\gcd(m,e)=1$. Let $p=2$, $q=2^m$ and $a\in {{\mathrm{GF}}}(q)^*$. Then $$\begin{aligned}
\sum_{b\in {{\mathrm{GF}}}(q)^*} \left( S_e(a,b)\right)^h =\left\{
\begin{array}{ll}
(2^m-1) 2^{\frac{m}{2}\cdot h} & \mbox{ if $h$ is even and $a \ne \alpha^{3t}$ for any $t$,} \\
(2^{m-2}-1) 2^{(\frac{m}{2}+1) \cdot h} & \mbox{ if $h$ is even and $a = \alpha^{3t}$ for some $t$,}
\end{array}
\right.\end{aligned}$$ where $\alpha$ is a generator of ${{\mathrm{GF}}}(q)^*$.
[@coulter2002] \[lem-psum\] Let $p$ be an odd prime, $q=p^m$, and $e$ be any positive integer such that $m/\gcd(m,e)$ is odd. Suppose $a\in {{\mathrm{GF}}}(q)^* $ and $b \in {{\mathrm{GF}}}(q)^*$. Let $x_{a,b}$ be the unique solution of the equation $$a^{p^e}x^{p^{2e}}+a x+ b^{p^e}=0.$$ Then $$\begin{aligned}
S_e (a, b)=
\left\{
\begin{array}{ll}
(-1)^{m-1} \sqrt{q} \eta(-a) \chi_1(-a x_{a,b}^{p^e+1} ), & \mbox{if}~ p\equiv 1 ~mod~4, \\
(-1)^{m-1} i^{3m} \sqrt{q} \eta(-a) \chi_1(-a x_{a,b}^{p^e+1} ), & \mbox{if}~ p\equiv 3 ~mod~4. \\
\end{array}
\right.\end{aligned}$$
[@YCD2006]\[lem-sumxp\] Let $p$ be an odd prime, $q=p^m$, and $e$ be any positive integer such that $m/\gcd(m,e)$ is odd. Suppose $a\in {{\mathrm{GF}}}(q)^* $ and $b \in {{\mathrm{GF}}}(q)^*$. Let $x_{a,b}$ be the unique solution of the equation $a^{p^e}x^{p^{2e}}+a x+ b^{p^e}=0$ and $\Delta=\sum_{c\in {{\mathrm{GF}}}(p)^*}S_e (\ ac,\ bc )$. Then we have the following results.
- If $m$ is odd, then $$\begin{aligned}
\Delta=
\left\{
\begin{array}{ll}
0, & \mbox{if ${{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1})=0$}, \\
\eta(a)\eta( {{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1}) )\sqrt{q} \sqrt{p^*}, & \mbox{if $p\equiv 1 ~mod~4$ and ${{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1})\neq 0$ }. \\
\eta(a)\eta( {{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1}) )i^{3m}\sqrt{q} \sqrt{p^*}, & \mbox{if $p\equiv 3 ~mod~4$ and ${{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1})\neq 0$ }.
\end{array}
\right.\end{aligned}$$
- If $m$ is even, then $$\begin{aligned}
\Delta=
\left\{
\begin{array}{ll}
-(p-1)\eta(a)\sqrt{q}, & \mbox{if $p\equiv 1 ~mod~4$ and ${{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1})=0$}, \\
\eta(a)\sqrt{q}, & \mbox{if $p\equiv 1 ~mod~4$ and ${{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1})\neq 0$}, \\
-i^m (p-1)\eta(a)\sqrt{q}, & \mbox{if $p\equiv 3 ~mod~4$ and ${{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1})=0$}, \\
i^m \eta(a)\sqrt{q}, & \mbox{if $p\equiv 3 ~mod~4$ and ${{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1})\neq 0$}, \\
\end{array}
\right.\end{aligned}$$
[@YCD2006]\[lem-NN0\] Let $p$ be an odd prime, $m$ and $e$ be positive integers such that $m/\gcd(m,e)$ is odd. Let $q=p^m$. Define $$\begin{aligned}
\hat{N}_0(a,b)=\sharp\{x\in {{\mathrm{GF}}}(q):{{\mathrm{Tr}}}_{q/p}(ax^{p^e+1}+bx)=0\}.\end{aligned}$$ Then we have the following results.
- If $a=0$ and $b=0$, then $\hat{N}_0(a,b)=q$.
- If $a=0$ and $b\neq 0$, then $\hat{N}_0(a,b)=p^{m-1}$.
- If $a\neq 0$ and $b=0$, then $$\begin{aligned}
\hat{N}_0(a,b)=
\left\{
\begin{array}{ll}
p^{m-1} & \mbox{if $m$ is odd}, \\
\frac{1}{p} \left ( q-(p-1)\eta(a)\sqrt{q} \right), & \mbox{if $m$ is even and $p\equiv 1 ~mod~4$}, \\
\frac{1}{p}\left( q-i^m (p-1)\eta(a)\sqrt{q} \right), & \mbox{if $m$ is even and $p\equiv 3 ~mod~4$.}
\end{array}
\right.\end{aligned}$$
- If $a\neq 0$ and $b\neq 0$, then $\hat{N}_0(a,b)=\frac{1}{p}(q+\Delta)$, where $R$ was given in Lemma \[lem-sumxp\].
$t$-designs and related results
-------------------------------
Let $k$, $t$ and $v$ be positive integers with $1\leq t \leq k \leq v$. Let ${{\mathcal{P}}}$ be a set of $v \ge 1$ elements, and let ${{\mathcal{B}}}$ be a set of $k$-subsets of ${{\mathcal{P}}}$. The *incidence structure* ${{\mathbb{D}}}= ({{\mathcal{P}}}, {{\mathcal{B}}})$ is said to be a $t$-$(v, k, \lambda)$ [*design*]{} if every $t$-subset of ${{\mathcal{P}}}$ is contained in exactly $\lambda$ elements of ${{\mathcal{B}}}$. The elements of ${{\mathcal{P}}}$ are called points, and those of ${{\mathcal{B}}}$ are referred to as blocks. We usually use $b$ to denote the number of blocks in ${{\mathcal{B}}}$. A $t$-design is called [*simple*]{} if ${{\mathcal{B}}}$ has no repeated blocks. A $t$-design is called symmetric if $v = b$ and trivial if $k = t$ or $k = v$. When $t \geq 2$ and $\lambda=1$, a $t$-design is called a [*Steiner system*]{} and traditionally denoted by $S(t,k, v)$.
Linear codes and $t$-designs are companions. A $t$-design $\mathbb D=(\mathcal P, \mathcal B)$ induces a linear code over GF($p$) for any prime $p$. Let $\mathcal P=\{p_1, \dots, p_{\nu}\}$. For any block $B\in \mathcal B$, the *characteristic vector* of $B$ is defined by the vector $\mathbf{c}_{B}
=(c_1, \dots, c_{\nu})\in \{0,1\}^{\nu}$, where $$\begin{aligned}
c_i=
\left\{
\begin{array}{ll}
1, & \text{if}~ p_i \in B, \\
0, & \text{if}~ p_i \not \in B.
\end{array}
\right.\end{aligned}$$ For a prime $p$, a *linear code* $\mathsf{C}_{p}(\mathbb D)$ over the prime field $\mathrm{GF}(p)$ from the design $\mathbb D$ is spanned by the characteristic vectors of blocks of $\mathbb B$, which is the subspace $\mathrm{Span}\{\mathbf{v}_{B}: B\in \mathcal B\}$ of the vector space $\mathrm{GF}(p)^{\nu}$. Linear codes $\mathsf{C}_{p}(\mathbb D)$ from designs $\mathbb D$ have been studied and documented in the literature (see, for examples, [@AK92; @Ding15; @Ton98; @Ton07]).
On the other hand, a linear code ${{\mathcal{C}}}$ may induce a $t$-design under certain conditions, which is formed by supports of codewords of a fixed Hamming weight in ${{\mathcal{C}}}$. Let $\mathcal P(\mathcal C)$ be the set of the coordinate positions of ${{\mathcal{C}}}$, where $\# \mathcal P(\mathcal C)=v$ is the length of $\mathcal C$. For a codeword $\mathbf c =(c_i)_{i\in \mathcal P(\mathcal C)}$ in ${{\mathcal{C}}}$, the *support* of $\mathbf c$ is defined by $$\begin{aligned}
\mathrm{Supp}(\mathbf c) = \{i: c_i \neq 0, i \in \mathcal P(\mathcal C)\}.\end{aligned}$$ Let $\mathcal B_{w}(\mathcal C)
=\{\mathrm{Supp}(\mathbf c): wt(\mathbf{c})=w
~\text{and}~\mathbf{c}\in \mathcal{C}\}$. For some special $\mathcal C$, $\left (\mathcal P(\mathcal C), \mathcal B_{w}(\mathcal C) \right)$ is a $t$-design. In this way, many $t$-designs are derived from linear codes ( see, for examples, [@AK92; @Ding18dcc; @Ding18jcd; @DLX17; @HKM04; @HMT05; @KM00; @MT04; @Ton98; @Ton07]). A major approach to constructing $t$-designs from linear codes is the use of linear codes with $t$-homogeneous or $t$-transitive automorphism groups(see [@ding2018 Theorem 4.18]). Another major approach to constructing $t$-designs from codes is the use of the Assmus-Mattson Theorem [@AM74; @HP10]. The following Assmus-Mattson Theorem for constructing simple $t$-designs is given in [@AM69].
\[thm-AMTheorem\] Let $\mathcal C$ be a linear code over $\mathrm{GF}(q)$ with length $\nu$ and minimum weight $d$. Let $\mathcal C^{\perp}$ with minimum weight $d^{\perp}$ denote the dual code of $\mathcal C$. Let $t~(1\le t <\min \{d, d^{\perp}\})$ be an integer such that there are at most $d^{\perp}-t$ weights of $\mathcal C$ in the range $\{1,2, \ldots, \nu-t\}$. Then the following holds:
- $(\mathcal P(\mathcal C), \mathcal B_{k}(\mathcal C))$ is a simple $t$-design provided that $A_k \neq 0$ and $d \leq k \leq w$, where $w$ is defined to be the largest integer satisfying $w \leq \nu$ and $$w-\left\lfloor \frac{w+q-2}{q-1} \right\rfloor <d.$$
- $(\mathcal P(\mathcal C^{\perp}), \mathcal B_{k}(\mathcal C^{\perp}))$ is a simple $t$-design provided that $A_k^\perp \neq 0$ and $d^\perp \leq k \leq w^\perp$, where $w^\perp$ is defined to be the largest integer satisfying $w^\perp \leq \nu$ and $$w^\perp-\left\lfloor \frac{w^\perp+q-2}{q-1} \right\rfloor <d^\perp.$$
We will need the following results on the punctured and shortened codes of ${{\mathcal{C}}}$ in [@Tangit2019 Lemma 3.1,Theorem 3.2] .
[@Tangit2019]\[lem:P:k:k+t\] Let $\mathcal C$ be a linear code of length $\nu$ and minimum distance $d$ over $\mathrm{GF}(q)$ and $d^{\perp}$ the minimum distance of $\mathcal C^{\perp}$. Let $t$ and $k$ be two positive integers with $0< t <\min \{d, d^{\perp}\}$ and $1\le k\le \nu-t$. Let $T$ be a set of $t$ coordinate positions in $\mathcal C$. Suppose that $\left ( \mathcal P(\mathcal C) , \mathcal B_i(\mathcal C) \right )$ is a $t$-design for all $i$ with $k\le i \le k+t$. Then $$A_k(\mathcal C^T) =\sum_{i=0}^t \frac{\binom{\nu-t}{k} \binom{k+i}{t} \binom{t}{i} }{\binom{\nu-t}{k-t+i} \binom{\nu}{t}} A_{k+i}(\mathcal C).$$
[@Tangit2019] \[thm:sct-code\] Let $\mathcal C$ be a $[\nu, \bar{k}, d]$ linear code over $\mathrm{GF}(q)$ and $d^{\perp}$ be the minimum distance of $\mathcal C^{\perp}$. Let $t$ be a positive integer with $0< t <\min \{d, d^{\perp}\}$. Let $T$ be a set of $t$ coordinate positions in $\mathcal C$. Suppose that $\left ( \mathcal P(\mathcal C) , \mathcal B_i(\mathcal C) \right )$ is a $t$-design for any $i$ with $d \le i \le \nu-t$. Then the shortened code $\mathcal C_T$ is a linear code of length $\nu-t$ and dimension $\bar{k}-t$. The weight distribution $\left ( A_k(\mathcal C_T) \right )_{k=0}^{\nu-t}$ of $\mathcal C_T$ is independent of the specific choice of the elements in $T$. Specifically, $$A_k(\mathcal C_T) =\frac{ \binom{k}{t} \binom{\nu-t}{k}}{ \binom{\nu }{t} \binom{\nu-t}{k-t}}A_k(\mathcal C).$$
Linear codes from APN and PN functions
--------------------------------------
Let $m, \tilde{m}$ be two positive integers with $m \geq \tilde{m}$ and $F$ be a mapping from ${{\mathrm{GF}}}(p^{m})$ to ${{\mathrm{GF}}}(p^{\tilde{m}})$. Define $$\delta_F=\max\{\delta_F(a,b): a \in {{\mathrm{GF}}}(p^{m})^*, b \in {{\mathrm{GF}}}(p^{\tilde{m}})\},$$ where $\delta_F(a,b)=\#\{x \in {{\mathrm{GF}}}(p^{m}) : F(x+a)-F(x)=b\}$, $a\in{{\mathrm{GF}}}(p^{m})$ and $b\in {{\mathrm{GF}}}(p^{\tilde{m}}) $. The function $F(x)$ is called PN function if $\delta_F=p^{m-\tilde{m}}$, and it is called APN function if $m=\tilde{m}$ and $\delta_F=2$. From the above definition one immediately has that $F(x)$ is PN if and only if $F(x+a)-F(x)$ is balanced for each $a\in {{\mathrm{GF}}}(p^{m})^{*}$. Currently, all known PN and APN functions over ${{\mathrm{GF}}}(p^{m})$ can be summarized in [@ding2018; @Budaghyan-Helleseth; @Coulter-Henderson-Hu; @Coulter-Matthews; @Dembowski-Ostrom; @Ding-Yuan; @Zha-Kyureghyan-Wang; @car2015; @car2016]. It is known that PN and APN functions are very important functions to constructing linear codes with good parameters (see, for examples,[@YCD2006; @YCD2005; @sihem2019; @WLZ2020]).
Let $q=p^m$ and ${{\mathcal{C}}}$ denote the linear code of length $q$ as follows: $$\begin{aligned}
\label{eq:cf}
{{\mathcal{C}}}=\left\{\left(({{\mathrm{Tr}}}_{q/p}(af(x)+bx+c)\right)_{x \in {{\mathrm{GF}}}(q)}:a,b,c\in {{\mathrm{GF}}}(q) \right\},\end{aligned}$$ where $f(x)$ is a polynomial over ${{\mathrm{GF}}}(q)$. Then we can regard ${{\mathrm{GF}}}(q)$ as the set of the coordinate positions $\mathcal{P}(\mathcal C)$ of $\mathcal C$. It is known that ${{\mathcal{C}}}$ has dimension $2m+1$ and the weight distribution in Table \[tab-cf\] when $p=2$, $m\geq 5$ is odd and $f(x)=x^{s}$ is an APN function, where $s$ takes the following values [@ding2018].
- $s=2^e+1$, where $\gcd(e,m) = 1$ and $e$ is a positive integer.
- $s=2^{2e}-2^e+1$, where $e$ is a positive integer and $\gcd(e,m) = 1$.
- $s=2^{(m-1)/2}+3$.
- $s=2^{(m-1)/2}+2^{(m-1)/4}-1$, where $m \equiv 1 ~(~mod ~4~)$.
- $s=2^{(m-1)/2}+2^{(3m-1)/4}-1$, where $m \equiv 3 ~(~mod~4~)$.
When $f(x)=x^{2^e+1}$, $p=2$ and $m\geq 4$ is even with $\gcd(e,m) = 1$, the code ${{\mathcal{C}}}$ defined in (\[eq:cf\]) has dimension $2m+1$ and the weight distribution in Table \[tab-cf1\] [@ding2018].
Weight Multiplicity
----------------------- -----------------------------
$0$ $1$
$2^{m-1}-2^{(m-1)/2}$ $ (2^m-1)2^{m-1}$
$2^{m-1}$ $ (2^m-1)(2^{m+1}-2^{m}+2)$
$2^{m-1}+2^{(m-1)/2}$ $ (2^m-1)2^{m-1}$
$2^{m}$ $1$
: The weight distribution of ${{\mathcal{C}}}$ for $m$ odd []{data-label="tab-cf"}
Weight Multiplicity
----------------------- ------------------------
$0$ $1$
$2^{m-1}-2^{m/2}$ $ (2^m-1)2^{m-2}/3$
$2^{m-1}-2^{(m-2)/2}$ $ (2^m-1)2^{m+1}/3$
$2^{m-1}$ $ 2(2^m-1)(2^{m-2}+1)$
$2^{m-1}+2^{(m-2)/2}$ $ (2^m-1)2^{m+1}/3$
$2^{m-1}+2^{m/2}$ $ (2^m-1)2^{m-2}/3$
$2^{m}$ $1$
: The weight distribution of ${{\mathcal{C}}}$ for $m$ even []{data-label="tab-cf1"}
It is known that the code ${{\mathcal{C}}}$ defined in (\[eq:cf\]) has dimension $2m+1$ and a few weights when $p$ is an odd prime and $f(x)=x^{s}$ is a PN function. If $s$ takes the following values [@ding2018; @LQ2009]
- $s=2$,
- $s=p^e+1$, where $m/\gcd(m,e)$ is odd.
- $s=(3^e+1)/2$, where $p=3$, $e$ is odd and $\gcd(m,e)=1$,
then $f(x)=x^s$ is a PN and also planar function, $\mathrm{Tr}_{q/p}(\beta f(x)))$ is a weakly regular bent function [@GHHK2012; @HHKWX2009] for any $\beta\in
{{\mathrm{GF}}}(q)^*$ , and the code ${{\mathcal{C}}}$ defined in (\[eq:cf\]) has four or six weights [@LQ2009].
Let $f(x)$ be a function from ${{\mathrm{GF}}}(q)$ to ${{\mathrm{GF}}}(p)$, the Walsh transform of $f$ at a point $\beta \in {{\mathrm{GF}}}(q)$ is defined by $$\mathcal{W}_{f}(\beta)=
\sum_{x\in{{\mathrm{GF}}}(q)}\zeta_p^{f(x)
-{{\mathrm{Tr}}}_{q/p}(\beta x)}.$$ The function $f(x)$ is said to be a $p$-ary bent function, if $|\mathcal{W}_f(\beta)|=p^{\frac{m}{2}}$ for any $\beta\in \mathbb{F}_q$. A bent function $f(x)$ is weakly regular if there exists a complex $u$ with unit magnitude satisfying $\mathcal{W}_f(\beta)=up^{\frac{m}{2}}\zeta_p^{f^*(\beta)}$ for some function $f^*(x)$. Such function $f^*(x)$ is called the dual of $f(x)$. A weakly regular bent function $f(x)$ satisfies $$\mathcal{W}_f(\beta)=\varepsilon \sqrt{p^*}^{m} \zeta_p^{f^*(\beta)},$$ where $\varepsilon =\pm 1$ is called the sign of the Walsh Transform of $f(x)$. Let $\mathcal{RF}$ be the set of $p$-ary weakly regular bent functions with the following two properties:
- $f(0)=0$; and
- $f(ax)=a^hf(x)$ for any $a\in {{\mathrm{GF}}}(p)^*$ and $x\in {{\mathrm{GF}}}(q)$, where $h$ is a positive even integer with $\gcd(h-1,p-1)=1$.
We will need the following results on $p$-ary weakly regular bent functions in [@Tangit2016].
[@Tangit2016] \[2lem11\] Let $\beta\in {{\mathrm{GF}}}(q)^*$ and $f(x)\in \mathcal{RF}$ with $\mathcal{W}_f(0)=\varepsilon\sqrt{p^*}^{m}$. Define $$N_{f,\beta}=\#\{x\in {{\mathrm{GF}}}(q):
f(x)=0 ~\textrm{and}~ {{\mathrm{Tr}}}_{q/p}(\beta x)=0\}.$$ If $f^*(\beta)=0$, then $$N_{f,\beta}=
\left\{
\begin{array}{ll}
p^{m-2}+\varepsilon \bar{\eta}^{m/2}(-1)(p-1)p^{(m-2)/2}, & \hbox{if $m$ is even;} \\
p^{m-2}, & \hbox{if $m$ is odd.}
\end{array}
\right.$$
[@Tangit2016] \[2lem14\] Let $\beta\in {{\mathrm{GF}}}(q)^*$ and $f(x)\in \mathcal{RF}$ with $\mathcal{W}_f(0)=\varepsilon\sqrt{p^*}^{m}$. Let $$N_{sq,\beta}=\#\{x\in {{\mathrm{GF}}}(q):
f(x)\in \textrm{SQ} ~\textrm{and}~ {{\mathrm{Tr}}}_{q/p}(
\beta x)=0 \},$$ and $$N_{nsq,\beta}=\#\{x\in {{\mathrm{GF}}}(q):
f(x)\in \textrm{N\textrm{SQ}} ~\textrm{and}~ {{\mathrm{Tr}}}_{q/p}(
\beta x)=0 \}.$$ We have the following results.
- If $m$ is even and $f^*(\beta)=0$, then $$N_{sq,\beta}=N_{nsq,\beta}=\frac{p-1}{2}\left(p^{m-2}-\varepsilon \bar{\eta}^{m/2}(-1)p^{(m-2)/2}\right).$$
- If $m$ is odd and $f^*(\beta)=0$, then $$N_{sq,\beta}=\frac{p-1}{2}\left( p^{m-2}+\varepsilon \sqrt{p*}^{m-1} \right)$$ and $$N_{nsq,\beta}= \frac{p-1}{2}\left( p^{m-2}-\varepsilon \sqrt{p*}^{m-1} \right).$$
Shortened binary linear codes with special weight distributions {#sec-general}
===============================================================
In this section, we give some general results on the shortened codes of linear codes with the weight distributions in Tables \[tab-cf\] and \[tab-cf1\].
Let $T$ be a set of $t$ coordinate positions in ${{\mathcal{C}}}$ (i.e., $T$ is a $t$-subset of $\mathcal P({{\mathcal{C}}})$ ). Define $$\Lambda _{T,w}({{\mathcal{C}}})=\{\mathrm{Supp}(\mathbf c):~wt(\mathbf{c})=w, ~\mathbf{c}\in {{\mathcal{C}}}~and~T\subseteq \mathrm{Supp}(\mathbf c) \}.$$ and $\lambda _{T,w}({{\mathcal{C}}}) = \# \Lambda _{T,w}({{\mathcal{C}}})$.
Shortened linear codes holding $t$-designs
------------------------------------------
Let $p=2$ and $q=2^m$. Notice that if a binary code ${{\mathcal{C}}}$ has length $2^m$ and the weight distribution in Table \[tab-cf\] (resp. Table \[tab-cf1\]), then the code ${{\mathcal{C}}}$ holds $3$-design (resp. $2$-design) in [@ding2018; @DT2020ccds]. The following two theorems are easily derived from Theorem \[thm:sct-code\], Tables \[tab-cf\] and \[tab-cf1\], and we omit their proofs.
\[main-design1\] Let $m\geq 5$ be odd, and ${{\mathcal{C}}}$ be a binary linear code with length $2^m$ and the weight distribution in Table \[tab-cf\]. Let $T$ be a $t$-subset of $\mathcal P({{\mathcal{C}}})$. We have the following results.
- If $t=1$, then the shortened code ${{\mathcal{C}}}_{T}$ is a $[2^{m}-1, 2m, 2^{m-1} - 2^{(m -1)/2}]$ binary linear code with the weight distribution in Table \[tab-des11\].
- If $t=2$, then the shortened code ${{\mathcal{C}}}_{T}$ is a $[2^{m}-2, 2m-1, 2^{m-1} - 2^{(m -1)/2}]$ binary linear code with the weight distribution in Table \[tab-des12\].
- If $t=3$, then the shortened code ${{\mathcal{C}}}_{T}$ is a $[2^{m}-3, 2m-2, 2^{m-1} - 2^{(m -1)/2}]$ binary linear code with the weight distribution in Table \[tab-des13\].
Weight Multiplicity
------------------------------- ---------------------------------------------
$0$ $1$
\[2mm\] $2^{m-1}-2^{(m-1)/2}$ $ 2^{(m-5)/2} (2^m-1) (2 + 2^{(1 + m)/2})$
\[2mm\] $2^{m-1}$ $-1 + 2^{m-1} + 2^{2m-1}$
\[2mm\] $2^{m-1}+2^{(m-1)/2}$ $ 2^{(m-5)/2} (2^m-1) (-2 + 2^{(1 + m)/2})$
\[2mm\]
: The weight distribution of ${{\mathcal{C}}}_T$ for $m$ odd and $t =1$ []{data-label="tab-des11"}
Weight Multiplicity
------------------------------- ---------------------------------------------------
$0$ $1$
\[2mm\] $2^{m-1}-2^{(m-1)/2}$ $ 2^{(m-7)/2} (-4 + 2^{2 + m} + 2^{(1 + 3 m)/2})$
\[2mm\] $2^{m-1}$ $-1 +2^{2m-2} $
\[2mm\] $2^{m-1}+2^{(m-1)/2}$ $ 2^{(m-7)/2} (4 - 2^{2 + m} + 2^{(1 + 3 m)/2})$
\[2mm\]
: The weight distribution of ${{\mathcal{C}}}_T$ for $m$ odd and $t =2$ []{data-label="tab-des12"}
Weight Multiplicity
------------------------------- -------------------------------------------------------------
$0$ $1$
\[2mm\] $2^{m-1}-2^{(m-1)/2}$ $ -2^{( m-3)/2} + 3\cdot 2^{(3m-7)/2} + 2^{m-3} + 2^{2m-4}$
\[2mm\] $2^{m-1}$ $(-1 + 2^{m-2}) (1+2^{ m-1})$
\[2mm\] $2^{m-1}+2^{(m-1)/2}$ $ 2^{( m-3)/2} - 3\cdot 2^{(3m-7)/2} + 2^{m-3} + 2^{2m-4}$
\[2mm\]
: The weight distribution of ${{\mathcal{C}}}_T$ for $m$ odd and $t =3$[]{data-label="tab-des13"}
\[exa-des11\] Let $m=5$ and $T$ be a $1$-subset of $\mathcal P({{\mathcal{C}}})$. Then the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-design1\] is a $[31,10,12]$ binary linear code with the weight enumerator $1+310z^{12}+527z^{16}+186z^{20}$. The code ${{\mathcal{C}}}_{T}$ is optimal. The dual code of ${{\mathcal{C}}}_{T}$ has parameters $[31,21,5]$ and is optimal according to the tables of best known codes maintained at http://www.codetables.de.
\[exa-des12\] Let $m=5$ and $T$ be a $2$-subset of $\mathcal P({{\mathcal{C}}})$. Then the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-design1\] is a $[30,9,12]$ linear code with the weight enumerator $1+190z^{12}+255z^{16}+66z^{20}$. The code ${{\mathcal{C}}}_{T}$ is optimal. The dual code of ${{\mathcal{C}}}_{T}$ has parameters $[30,21,4]$ and is optimal according to the tables of best known codes maintained at http://www.codetables.de.
\[exa-des13\] Let $m=5$ and $T$ be a $3$-subset of $\mathcal P({{\mathcal{C}}})$. Then the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-design1\] is a $[29,8,12]$ binary linear code with the weight enumerator $1+114z^{12}+119z^{16}+22z^{20}$. The code ${{\mathcal{C}}}_{T}$ is optimal. The dual code of ${{\mathcal{C}}}_{T}$ has parameters $[29,21,3]$ and is almost optimal according to the tables of best known codes maintained at http://www.codetables.de.
\[main-design2\] Let $m\geq 4 $ be even, and ${{\mathcal{C}}}$ be a binary linear code with length $2^m$ and the weight distribution in Table \[tab-cf1\]. Let $T$ be a $t$-subset of $\mathcal P({{\mathcal{C}}})$. We have the following results.
- If $t=1$, then the shortened code ${{\mathcal{C}}}_{T}$ is a $[2^{m}-1, 2m, 2^{m-1} - 2^{m/2}]$ binary linear code with the weight distribution in Table \[tab-des21\].
- If $t=2$, then the shortened code ${{\mathcal{C}}}_{T}$ is a $[2^{m}-2, 2m-1, 2^{m-1} - 2^{m/2}]$ binary linear code with the weight distribution in Table \[tab-des22\].
Weight Multiplicity
------------------------------- ---------------------------------------------------------
$0$ $1$
\[2mm\] $2^{m-1}-2^{m/2}$ $ 1/3 \cdot 2^{-3 + m/2} (2 + 2^{m/2}) (-1 + 2^m) $
\[2mm\] $2^{m-1}-2^{(m-2)/2}$ $ 1/3 \cdot 2^{m/2} (-1 + 2^{m/2}) (1 + 2^{m/2})^2 $
\[2mm\] $2^{m-1}$ $ (2^m-1) (1+2^{ m-2})$
\[2mm\] $2^{m-1}+2^{(m-2)/2}$ $ 1/3 \cdot 2^{m/2} (-1 + 2^{m/2})^2 (1 + 2^{m/2}) $
\[2mm\] $2^{m-1}+2^{m/2}$ $ 1/3 \cdot 2^{-3 + m/2} (-2 + 2^{m/2}) (-1 + 2^m) $
\[2mm\]
: The weight distribution of ${{\mathcal{C}}}_T$ for $m$ even and $t=1$ []{data-label="tab-des21"}
Weight Multiplicity
------------------------------- ------------------------------------------------------------------
$0$ $1$
\[2mm\] $2^{m-1}-2^{m/2}$ $ 1/3 \cdot 2^{m/2-4} (2 + 2^{m/2}) (2^m +2^{1+m/2}-2) $
\[2mm\] $2^{m-1}-2^{(m-2)/2}$ $ 1/3 \cdot 2^{m/2-1} (1 + 2^{m/2}) (2^m +2^{m/2}-2) $
\[2mm\] $2^{m-1}$ $ (2^{m-1}-1)(1+2^{ m-2})$
\[2mm\] $2^{m-1}+2^{(m-2)/2}$ $ 1/3 \cdot 2^{m/2-1} (-1 + 2^{m/2}) (2^m -2^{m/2}-2) $
\[2mm\] $2^{m-1}+2^{m/2}$ $ 1/3 \cdot 2^{m/2-4} (4 + 2^{1 + m/2} + 2^{3 m/2} - 2^{2 + m})$
\[2mm\]
: The weight distribution of ${{\mathcal{C}}}_T$ for $t=2$ and $m$ even []{data-label="tab-des22"}
\[exa-des21\] Let $m=4$ and $T$ be a $1$-subset of $\mathcal P({{\mathcal{C}}})$. Then the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-design2\] is a $[15,8,4]$ linear code with the weight enumerator $1+15z^{4}+100z^{6}+75 z^{8}+60 z^{10}+5 z^{12}$. This code ${{\mathcal{C}}}_{T}$ is optimal. Its dual ${{\mathcal{C}}}_{T}^\perp$ has parameters $[15,7,5]$ and is optimal according to the tables of best known codes maintained at http://www.codetables.de.
\[exa-des22\] Let $m=4$ and $T$ be a $2$-subset of $\mathcal P({{\mathcal{C}}})$. Then the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-design2\] is a $[14,7,4]$ binary linear code with the weight enumerator $1+11z^{4}+60z^{6}+35z^{8}+20 z^{10}+z^{12}$. This code ${{\mathcal{C}}}_{T}$ is optimal. Its dual ${{\mathcal{C}}}_{T}^\perp$ has parameters $[14,7,4]$ and is optimal according to the tables of best known codes maintained at http://www.codetables.de.
Several general results on shortened codes
-------------------------------------------
\[lem-cf\] Let $m\geq 5$ be odd (resp. $m\geq 4$ be even), and ${{\mathcal{C}}}$ be a binary linear code with the length $2^m$ and the weight distribution in Table \[tab-cf\] (resp. Table \[tab-cf1\]). Then the dual code ${{\mathcal{C}}}^\bot$ of ${{\mathcal{C}}}$ has parameters $[2^m,2^m-2m-1,6]$.
The weight distribution in Table \[tab-cf\] (or \[tab-cf1\]) means that the dimension of ${{\mathcal{C}}}$ is $2m+1$. Thus, the dual code ${{\mathcal{C}}}^\bot$ of ${{\mathcal{C}}}$ has dimension $2^m-2m-1$. Since the code length of ${{\mathcal{C}}}$ is $2^m$, from the weight distribution in Table \[tab-cf\] (or \[tab-cf1\]) and the first seven Pless power moments in (\[eq:PPM\]), it is easily obtain that $A_6({{\mathcal{C}}}^\perp)> 0$ and $A_i({{\mathcal{C}}}^\perp)=0$ for any $i\in \{1,2,3,4,5\}$. The desired conclusions then follow .
\[main-31\] Let $m\geq 5$ be odd (resp. $m\geq 4$ be even), and ${{\mathcal{C}}}$ be a binary linear code with length $2^m$ and the weight distribution in Table \[tab-cf\] (resp. Table \[tab-cf1\]). Let $T$ be a $4$-subset of $\mathcal P({{\mathcal{C}}})$ and $\lambda _{T,6}({{\mathcal{C}}}^\bot)=\lambda$, then $\lambda =0$ or $1$. Furthermore, we have the following results.
- If $m\geq 5$ is odd and $\lambda =0$, then the shortened code ${{\mathcal{C}}}_{T}$ is a $[2^{m}-4, 2m-3, 2^{m-1} - 2^{(m -1)/2}]$ binary linear code with the weight distribution in Table \[tab-31\].
- If $m\geq 5$ is odd and $\lambda =1$, then the shortened code ${{\mathcal{C}}}_{T}$ is a $[2^{m}-4, 2m-3, 2^{m-1} - 2^{(m -1)/2}]$ binary linear code with the weight distribution in Table \[tab-32\].
Weight Multiplicity
------------------------------- --------------------------------------------------------
$0$ $1$
\[2mm\] $2^{m-1}-2^{(m-1)/2}$ $ -2^{( m-3)/2} + 2^{m-3} + 2^{2 m-5} + 2^{(3 m-5)/2}$
\[2mm\] $2^{m-1}$ $-1 - 2^{m-2} + 4^{ m-2}$
\[2mm\] $2^{m-1}+2^{(m-1)/2}$ $ 2^{( m-3)/2} + 2^{m-3} + 2^{2 m-5} - 2^{(3 m-5)/2}$
\[2mm\]
: The weight distribution of ${{\mathcal{C}}}_T$ for $\lambda =0$ []{data-label="tab-31"}
Weight Multiplicity
------------------------------- -----------------------------------------------------------------------
$0$ $1$
\[2mm\] $2^{m-1}-2^{(m-1)/2}$ $ 3\times 2^{m-4} - 2^{(-3 + m)/2} + 2^{-5 + 2 m} + 2^{(-5 + 3 m)/2}$
\[2mm\] $2^{m-1}$ $ 2^{-4} (-8 + 2^m) (2 + 2^m)$
\[2mm\] $2^{m-1}+2^{(m-1)/2}$ $ 3\times 2^{m-4} + 2^{(-3 + m)/2} + 2^{-5 + 2 m} - 2^{(-5 + 3 m)/2}$
\[2mm\]
: The weight distribution of ${{\mathcal{C}}}_T$ for $\lambda =1$ []{data-label="tab-32"}
By definition, we have $$\lambda=\lambda _{T,6}({{\mathcal{C}}}^\perp )=\# \Lambda _{T,6}({{\mathcal{C}}}^\perp)=\# \{\mathrm{Supp}(\mathbf c):~wt(\mathbf{c})=6, ~\mathbf{c}\in {{\mathcal{C}}}^\perp~and~T\subseteq \mathrm{Supp}(\mathbf c) \}.$$ Suppose that $\lambda \geq2$. There exist $\mathrm{Supp}(\mathbf c_1), \mathrm{Supp}(\mathbf c_2)\in \Lambda _{T,6}({{\mathcal{C}}}^\perp)$. Then $\mathbf c_1+\mathbf c_2 \in {{\mathcal{C}}}^\bot$ and the weight $wt(\mathbf c_1+\mathbf c_2)\leq 4$. This is a contradiction to the minimal distance $6$ of ${{\mathcal{C}}}^\bot$ in Lemma \[lem-cf\]. Thus, $\lambda =0$ or $1$.
We now prove the two cases as follows.
() The case that $\lambda=0$ and $m$ is odd.
By Lemma \[lem-cf\], the minimal distance of ${{\mathcal{C}}}^\bot$ is 6. Thus, $$\begin{aligned}
\label{eq-A}
A_1\left ( \left (\mathcal C^{\perp} \right )^{T} \right )=A_2\left ( \left (\mathcal C^{\perp} \right )^{T} \right )=0,~
A_1\left ( \left (\mathcal C_{T} \right )^{\perp} \right )=A_2\left ( \left ({{\mathcal{C}}}_{T} \right )^{\perp} \right )=0\end{aligned}$$ and the shortened code ${{\mathcal{C}}}_{T}$ has length $n=2^m-4$ and dimension $k=2m-3$ from $\lambda _{T,6}({{\mathcal{C}}}^\perp )=0$ and Lemma \[lem:C-S-P\]. By the definition and Lemma \[lem-cf\], we have $A_i\left ({{\mathcal{C}}}_T \right )=0$ for $i\not \in \{0, i_1, i_2, i_3\} $, where $i_1=2^{m-1}-2^{(m-1)/2}$, $i_2=2^{m-1}$ and $i_3=2^{m-1}+2^{(m-1)/2}$. Therefore, from (\[eq-A\]) and (\[eq:PPM\]), the first three Pless power moments $$\begin{aligned}
\left\{
\begin{array}{l}
A_{i_1} + A_{i_2} +A_{i_3} = 2^{2m-3}-1, \\
i_1 A_{i_1} + i_2 A_{i_2} + i_3 A_{i_3} = 2^{2m-3-1}(2^m-4), \\
i_1^2 A_{i_1} + i_2^2 A_{i_2} + i_3^2 A_{i_3} = 2^{2m-3-2}(2^m-4)(2^m-4+1).
\end{array}
\right.\end{aligned}$$ yield the weight distribution in Table \[tab-31\]. This completes the proof of ().
() The case that $\lambda =1$ and $m$ is odd.
The proof is similar to that of (). Since $\lambda _{T,6}({{\mathcal{C}}}^\perp )=1$ and the minimal distance of ${{\mathcal{C}}}^\bot$ is 6, from Lemma \[lem:C-S-P\] we have $$\begin{aligned}
\label{eq-AA}
& A_1\left ( \left (\mathcal C^{\perp} \right )^{T} \right )=0,~A_2\left ( \left (\mathcal C^{\perp} \right )^{T} \right )=1,~ \nonumber \\
& A_1\left ( \left (\mathcal C_{T} \right )^{\perp} \right )=0,~~A_2\left ( \left ({{\mathcal{C}}}_{T} \right )^{\perp} \right )=1.\end{aligned}$$ Then the desired conclusions follow from (\[eq-AA\]), the definitions and the first three Pless power moments of (\[eq:PPM\]). This completes the proof.
\[le-even31\] Let $m\geq 4$ be even, and ${{\mathcal{C}}}$ be a binary linear code with length $2^m$ and the weight distribution in Table \[tab-cf1\]. Let $T$ be a $3$-subset of $\mathcal P({{\mathcal{C}}})$. Suppose $\lambda _{T,6}({{\mathcal{C}}}^\bot)=\lambda$, then $ A_1\left ( \left (\mathcal C^{\perp} \right )^{T} \right )= A_2\left ( \left (\mathcal C^{\perp} \right )^{T} \right )=0$, $A_3\left ( \left (\mathcal C^{\perp} \right )^{T} \right )=\lambda$ and $ A_4\left ( \left (\mathcal C^{\perp} \right )^{T} \right )=2\cdot (2^{m-2}-1)^2- 3\lambda $.
By Lemma \[lem-cf\], the minimal distance of ${{\mathcal{C}}}^\bot$ is 6. Thus, from $\#T=3$ and the definitions, we have $
A_1\left ( \left (\mathcal C^{\perp} \right )^{T} \right )=A_2\left ( \left (\mathcal C^{\perp} \right )^{T} \right )=0
$ and $A_3\left ( \left (\mathcal C^{\perp} \right )^{T} \right )=\lambda$. Note that the code ${{\mathcal{C}}}$ has length $2^m$ and dimension $2m+1$. By Lemma \[lem-cf\], Table \[tab-cf1\] and the first seven Pless power moments of (\[eq:PPM\]), we have $$A_6({{\mathcal{C}}}^{\perp})= \frac{1}{45}\cdot 2^{m-4} (2^m-4)^2 (2^m-1).$$ Further, from Theorem \[thm-AMTheorem\], Lemmas \[lem-cf\] and \[lem:P:k:k+t\], we have $(\mathcal P(\mathcal C^{\perp}), \mathcal B_{6}(\mathcal C^{\perp}))$ is a $2$-design and $$A_4\left ( \left (\mathcal C^{\perp} \right )^{\{t_1,t_2\}} \right )=\frac{\binom{ 6}{ 2 }}{\binom{ q }{ 2 }}\cdot A_6({{\mathcal{C}}}^{\perp}) =\frac{2}{3}\cdot (2^{m-2}-1)^2$$ for any $\{t_1,t_2\} \subseteq \mathcal P({{\mathcal{C}}})$. Since $\#T=3$ and the minimal distance of ${{\mathcal{C}}}^\bot$ is 6, $$A_4\left ( \left (\mathcal C^{\perp} \right )^{T} \right )= \binom{ 3}{ 2 }\left ( A_4\left ( \left (\mathcal C^{\perp} \right )^{\{t_1,t_2\}} \right )- \lambda \right ).$$ Then the desired conclusions follow.
\[main-even32\] Let $m\geq 4$ be even, and ${{\mathcal{C}}}$ be a binary linear code with length $2^m$ and the weight distribution in Table \[tab-cf1\]. Let $T$ be a $3$-subset of $\mathcal P({{\mathcal{C}}})$. Suppose $\lambda _{T,6}({{\mathcal{C}}}^\bot)=\lambda$, then the shortened code ${{\mathcal{C}}}_{T}$ is a $[2^{m}-3, 2m-2, 2^{m-1}-2^{m/2}]$ binary linear code with the weight distribution in Table \[tab-even32\].
The proof is similar to that of Theorem \[main-31\]. From Lemma \[lem:C-S-P\] and $\#T=3$, the shortened code ${{\mathcal{C}}}_{T}$ has length $n=2^m-3$ and dimension $k=2m-2$. By the definitions and the weight distribution in Table \[tab-cf1\], then $A_i\left ({{\mathcal{C}}}_T \right )=0$ for $i\not \in \{0, i_1, i_2, i_3,i_4,i_5\} $, where $i_1=2^{m-1}-2^{m/2}$, $i_2=2^{m-1}-2^{(m-2)/2}$ , $i_3=2^{m-1}$, $i_4=2^{m-1}+2^{(m-2)/2}$ and $i_5=2^{m-1}+2^{m/2}$. Moreover, from Lemmas \[lem:C-S-P\] and \[le-even31\] we have $ A_1\left ( \left (\mathcal C_{T} \right )^{\perp} \right )= A_2\left ( \left (\mathcal C_{T} \right )^{\perp} \right )=0$, $A_3\left ( \left (\mathcal C_{T} \right )^{\perp} \right )=\lambda$ and $ A_4\left ( \left (\mathcal C_{T} \right )^{\perp} \right )=2\cdot (2^{m-2}-1)^2- 3\lambda $. Therefore, the first five Pless power moments of (\[eq:PPM\]) yield the weight distribution in Table \[tab-even32\]. This completes the proof.
Weight Multiplicity
------------------------------- -----------------------------------------------------------------------------------------
$0$ $1$
\[2mm\] $2^{m-1}-2^{m/2}$ $ 1/3 \cdot 2^{ m/2-5} (8 + 2^{3 + m/2} + 2^{3 m/2} + 2^{2 + m} + 12 \lambda)$
\[2mm\] $2^{m-1}-2^{(m-2)/2}$ $ 1/3 \cdot 2^{m/2-3} ((2 + 2^{m/2}) (-8 + 3 \cdot 2^{m/2} + 2^{1 + m}) - 6 \lambda)$
\[2mm\] $2^{m-1}$ $ -1 + 4^{ m-2}$
\[2mm\] $2^{m-1}+2^{(m-2)/2}$ $ 1/3 \cdot 2^{m/2-3} ((-2 + 2^{m/2}) (-8 - 3 \cdot 2^{m/2} + 2^{1 + m}) + 6 \lambda) $
\[2mm\] $2^{m-1}+2^{m/2}$ $ 1/3 \cdot 2^{ m/2-5} (-8 + 2^{3 + m/2} + 2^{3 m/2} - 2^{2 + m} - 12 \lambda)$
\[2mm\]
: The weight distribution of ${{\mathcal{C}}}_T$ for $\lambda _{T,6}({{\mathcal{C}}}^\bot)=\lambda$ []{data-label="tab-even32"}
Shortened linear codes from APN functions {#sec-APN}
=========================================
Let $p=2$ and $q=2^m$. In this section, we study some shortened codes ${{\mathcal{C}}}_{T}$ of linear codes ${{\mathcal{C}}}$ defined by (\[eq:cf\]) and determine their parameters, where $f(x)=x^{2^e+1}$ is a special APN function, $e$ is a positive integer, and $\gcd(m,e)=1$. It is known that ${{\mathcal{C}}}$ has the weight distribution in Tables \[tab-cf\] (resp. Tables \[tab-cf1\]) when $m$ is odd (resp. $m$ is even).
Let $T$ be a $t$-subset of $P(\mathcal C)$. We will consider some shortened code ${{\mathcal{C}}}_{T}$ of ${{\mathcal{C}}}$ for the case $m\geq 4$ and $t=3$ (or $t=4$).
Some shortened codes for the case $t=4$ and $m$ odd
---------------------------------------------------
We notice that it is difficult to determine the conditions satisfied by the four coordinate positions $x_1, x_2, x_3, x_4\in T$ such that $ \lambda _{T,6}({{\mathcal{C}}}^\perp )=0 $ (or 1) in Theorem \[main-31\]. We start with the special code satisfying Theorem \[main-31\] and give the necessary and sufficient conditions in Theorem \[main-32\]. In order to prove Theorem \[main-32\], we need the result in the following lemma.
\[lem-solu1\] Let $e$ and $m\geq 4$ be positive integers with $\gcd(m,e)=1$. Let $q=2^m$ and $ \{ x_1, x_2, x_3, x_4\}$ $\subseteq {{\mathrm{GF}}}(q) $. Denote $S_i=x_1^i+x_2^i+x_3^i+x_4^i$. Let $N$ be the number of solutions $(x,y)\in ({{\mathrm{GF}}}(q))^2$ of the system of equations $$\begin{aligned}
\label{eqx3}
\left\{
\begin{array}{l}
x_1+x_2+x_3+x_4+x+y=0, \\ [2mm]
x_1^{2^e+1}+x_2^{2^e+1}+x_3^{2^e+1}+x_4 ^{2^e+1}+x^{2^e+1}+y^{2^e+1}=0 \\
\end{array}
\right.\end{aligned}$$ where $x_1,x_2,x_3,x_4,x,y$ are pairwise distinct. Then $N =4$ if $S_1 \neq 0$ and ${{\mathrm{Tr}}}_{q/2}(\frac{S_{2^e+1}}{S_1^{2^e+1}}+1)=0$ , and $N=0$ otherwise.
By definition, the system (\[eqx3\]) can be written as follows : $$\begin{aligned}
\label{eq-con1}
\left\{
\begin{array}{l}
x+y=S_1 , \\ [2mm]
x^{2^e+1}+y^{2^e+1}+S_{2^e+1}=0 \\
\end{array}
\right.\end{aligned}$$ Substituting $y = x + S_1$, the second equation of (\[eq-con1\]) leads to $$\begin{aligned}
\label{eq-con2}
& x^{2^e+1}+(x+S_1)^{2^e+1}+S_{2^e+1} \nonumber \\
& = x^{2^e+1}+(x^{2^e}+S_1^{2^e})(x+S_1)+S_{2^e+1} \nonumber \\
&=S_1 x ^{2^e}+S_1^{2^e} \cdot x+ S_1^{2^e+1}+S_{2^e+1} =0.\end{aligned}$$ Further, Equation (\[eq-con2\]) with $S_1\neq 0$ is equivalent to $$\begin{aligned}
\label{eq-con3}
\frac{S_1 x ^{2^e}+S_1^{2^e} \cdot x+ S_1^{2^e+1}+S_{2^e+1}}{S_1^{2^e+1}}=(\frac{x}{S_1})^{2^e}+\frac{x}{S_1}+1+\frac{S_{2^e+1}}{S_1^{2^e+1}}=0.\end{aligned}$$ This means that Equation (\[eq-con2\]) with $S_1\neq 0$ has only two different solutions $x=u,v\in {{\mathrm{GF}}}(q)$ if and only if ${{\mathrm{Tr}}}_{q/2}(\frac{S_{2^e+1}}{S_1^{2^e+1}}+1)=0$, and no solution otherwise. Note that if $(x,y)$ is the solution of (\[eqx3\]), then so is $(y,x)$. Thus, $N=4$ when Equation (\[eq-con2\]) with $S_1\neq 0$ has solutions. Note that if (\[eqx3\]) has nontrivial solution (i.e., not all being zero or pairwise equal), all elements $x_1,x_2,x_3,$ $x_4, x,y$ have to be distinct since the cyclic code with zero set $\{1, 2^e+1\}$ has minimum distance $5$, where $\gcd(m,e)=1$. Thus, $x_1,x_2,x_3,$ $x_4, u,v$ are pairwise distinct. Then the desired conclusions follow.
\[main-32\] Let $e$ and $m\geq 4$ be positive integers with $\gcd(m,e)=1$. Let $q=2^m$, $f(x)=x^{2^e+1}$ and ${{\mathcal{C}}}$ be defined in (\[eq:cf\]). Let $\alpha$ be a primitive element of ${{\mathrm{GF}}}(q)$ and $T=\{x_1,x_2,x_3,x_4\}$ be a $4$-subset of $\mathcal P({{\mathcal{C}}})$ . Then $\lambda _{T,6}({{\mathcal{C}}}^\bot)=1$ iff $\sum_{i=1}^4 x_i \neq 0$ and ${{\mathrm{Tr}}}_{q/2} \left(\sum_{i=1}^4 x_i^{2^e+1}/(\sum_{i=1}^4 x_i^{2^e+1}+1 \right)=0$, and $\lambda _{T,6}({{\mathcal{C}}}^\bot)=0$ otherwise.
By definitions, the dual code ${{\mathcal{C}}}^\bot$ of ${{\mathcal{C}}}$ has minimum distance 6. For any $\{ x_1, x_2, x_3, x_4\} \subseteq {{\mathrm{GF}}}(q)$, there exists a unique set $\{x_1,x_2,x_3,x_4,x,y\}$ satisfying Equation (\[eqx3\]) and $0 \in \{x_1,x_2,x_3,x_4,x,y\}$, since the code with zero set $\{1, 2^e+1\}$ has minimum distance $5$, where $\gcd(m,e)=1$. From the definitions we have $\lambda _{T,6}({{\mathcal{C}}}^\bot)=\frac{N}{2!}-1$, where $N$ was defined in Lemma \[lem-solu1\]. Then the desired conclusions follow from Lemma \[lem-solu1\].
By Theorems \[main-32\] and \[main-31\], we have the following theorem, which is one of the main results in this paper.
\[main-odd4\] Let $m\geq 5$ be odd and $e$ be a positive integer with $\gcd(m,e)=1$. Let $q=2^m$, $f(x)=x^{2^e+1}$ and ${{\mathcal{C}}}$ be defined in (\[eq:cf\]). Let $\alpha$ be a primitive element of ${{\mathrm{GF}}}(q)$ and $T=\{x_1,x_2,x_3,x_4\}$ be a $4$-subset of $\mathcal P({{\mathcal{C}}})$ . Then $\lambda _{T,6}({{\mathcal{C}}}^\bot)=1$ if $\sum_{i=1}^4 x_i \neq 0$ and ${{\mathrm{Tr}}}_{q/2}\left(\sum_{i=1}^4 x_i^{2^e+1}/(\sum_{i=1}^4 x_i^{2^e+1}\right)=1$, and $\lambda _{T,6}({{\mathcal{C}}}^\bot)=0$ otherwise. The parameters of the shortened code ${{\mathcal{C}}}_{T}$ were given in Theorem \[main-31\].
\[exa-31odd\] Let $m=5$,$e=1$ and $T=\{\alpha^1,\alpha^2,\alpha^4,\alpha^5\}$. Then $\gamma =\alpha^{17}$, ${{\mathrm{Tr}}}_{q/2} \left(\frac{\gamma}{\gamma^3} \right)=0$ and $\lambda _{T,6}({{\mathcal{C}}}^\bot)=0$, where $\gamma = \alpha^1+\alpha^2+\alpha^4+\alpha^5$. The shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-odd4\] is a $[28,7,12]$ binary linear code with the weight enumerator $1+66z^{12}+55z^{16}+6z^{20}$. The code ${{\mathcal{C}}}_{T}$ is optimal according to the tables of best known codes maintained at http://www.codetables.de.
\[exa-32odd\] Let $m=5$,$e=1$ and $T=\{\alpha^1,\alpha^2,\alpha^3,\alpha^4\}$. Then $ \gamma=\alpha^{24}$, ${{\mathrm{Tr}}}_{q/2} \left(\frac{\gamma}{\gamma^3} \right)=1$, and $\lambda _{T,6}({{\mathcal{C}}}^\bot)=1$, where $\gamma = \alpha^1+\alpha^2+\alpha^3+\alpha^4$. The shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-odd4\] is a $[28,7,12]$ binary linear code with the weight enumerator $1+68z^{12}+51z^{16}+8z^{20}$. The code ${{\mathcal{C}}}_{T}$ is optimal according to the tables of best known codes maintained at http://www.codetables.de.
Some shortened codes for the case $t=3$ and $m$ even
----------------------------------------------------
For any $T=\{x_1,x_2,x_3\}\subseteq \mathcal P({{\mathcal{C}}})$, we notice that it is difficult to determine the value of $\lambda$ in Theorem \[main-even32\]. We will study a class of special linear codes ${{\mathcal{C}}}$ defined by (\[eq:cf\]), where $f(x)=x^{2^e+1}$ is an APN function, $m\geq 4$ is even and $\gcd(e,m) = 1$. Note that the code ${{\mathcal{C}}}$ satisfies Theorem \[main-even32\]. In the following, we will determine the value of $\lambda$ in Theorem \[main-even33\] and the parameters of the shortened code ${{\mathcal{C}}}_{T}$. We need the result in the following lemma.
\[lem-solu2\] Let $e$ be a positive integer, $m$ be even with $\gcd(m,e)=1$, and $q=2^m$. Let $\hat{N}$ be the number of solutions $(x,y,z)\in ({{\mathrm{GF}}}(q))^3$ of the system of equations $$\begin{aligned}
\label{eqx4}
\left\{
\begin{array}{l}
x+y+z=a, \\ [2mm]
x^{2^e+1}+y^{2^e+1}+z ^{2^e+1}=b \\
\end{array}
\right.\end{aligned}$$ where $a,b\in {{\mathrm{GF}}}(q)$ and $a^{2^e+1}\neq b$. Then $\hat{N} =2^m+(-2)^{m/2}-2$ if $a^{2^e+1}+b$ is not a cubic residue, and $\hat{N}=2^m+(-2)^{m/2+1}-2$ if $a^{2^e+1}+b$ is a cubic residue.
Replacing $x$ with $x + a$, $y$ with $y + a$ and $z$ with $z + a$, we have $$\begin{aligned}
\label{eqx5}
\left\{
\begin{array}{l}
x+y+z=0, \\ [2mm]
x^{2^e+1}+y^{2^e+1}+z ^{2^e+1}=a^{2^e+1}+b \\
\end{array}
\right.\end{aligned}$$ By $x +
y + z = 0$, we have the following equation $$\begin{aligned}
\label{eqx6}
x^{2^e}y+y^{2^e}x=a^{2^e+1}+b.\end{aligned}$$ Thus, $\hat{N}$ equals the number of solutions $(x,y)\in ({{\mathrm{GF}}}(q))^2$ to Equation (\[eqx6\]).
Replacing $y$ with $xy$ in Equation (\[eqx6\]), we get $$\begin{aligned}
\label{eqx7}
x^{2^e}y+y^{2^e}x= x^{2^e+1}y+y^{2^e}x^{2^e+1}=x^{2^e+1}(y+y^{2^e}) =a^{2^e+1}+b.\end{aligned}$$ Since $a^{2^e+1}+b \neq 0$, a rearrangement of Equation (\[eqx7\]) yields $$y+y^{2^e} =(a^{2^e+1}+b)x^{-(2^e+1)}.$$ Then $$\begin{aligned}
\label{eqx8}
{{\mathrm{Tr}}}((a^{2^e+1}+b)x^{-(2^e+1)})=0.\end{aligned}$$ Further, from Lemma \[expsum\] we get $$\begin{aligned}
\label{eqx9}
&\# \{x\in {{\mathrm{GF}}}(q)^*: {{\mathrm{Tr}}}((a^{2^e+1}+b)x^{-(2^e+1)})=0\} \nonumber \\
&= \# \{x\in {{\mathrm{GF}}}(q)^*: {{\mathrm{Tr}}}((a^{2^e+1}+b)x^{(2^e+1)})=0\} \nonumber \\
&=\frac{1}{2}\cdot \sum_{u\in {{\mathrm{GF}}}(2)}\sum_{x\in {{\mathrm{GF}}}(q)^*}(-1)^{{{\mathrm{Tr}}}(u(a^{2^e+1}+b)x^{(2^e+1)})} \nonumber \\
&=\frac{1}{2}\cdot(q-2+\sum_{x\in {{\mathrm{GF}}}(q)}(-1)^{{{\mathrm{Tr}}}((a^{2^e+1}+b)x^{(2^e+1)})}) \nonumber \\
&=\left\{
\begin{array}{l}
\frac{1}{2}\cdot(q-2+(-1)^{\frac{m}{2}} 2^{\frac{m}{2}}) ~~~~~~~~\mbox{ if $a^{2^e+1}+b$ is not a cubic residue,} \\
\frac{1}{2}\cdot(q-2-(-1)^{\frac{m}{2}} 2^{\frac{m}{2}+1} ) ~~~~\mbox{ if $a^{2^e+1}+b$ is a cubic residue.}
\end{array}
\right.\end{aligned}$$ Note that if $(x,y)$ is the solution of Equation (\[eqx6\]), then so is $(y,x)$. This means that $ \hat{N}$ equals twice the number of solutions $x\in {{\mathrm{GF}}}(q)^*$ to Equation (\[eqx8\]). Then the desired conclusions follow from Equation (\[eqx9\]).
\[main-even33\] Let $e$ be a positive integer and $m\geq 4$ be even with $\gcd(m,e)=1$. Let $q=2^m$, $f(x)=x^{2^e+1}$ and ${{\mathcal{C}}}$ be defined in (\[eq:cf\]). Let $\alpha$ be a primitive element of ${{\mathrm{GF}}}(q)$ and $T=\{x_1,x_2,x_3\}$ be a $3$-subset of $\mathcal P({{\mathcal{C}}})$. Then $$\begin{aligned}
\lambda=\lambda _{T,6}({{\mathcal{C}}}^\bot) &=\left\{
\begin{array}{l}
\frac{1}{6}\cdot(q-2+(-1)^{\frac{m}{2}} 2^{\frac{m}{2}})-1 ~~~~~~~~\mbox{ if $a^{2^e+1}+b$ is not a cubic residue} \\
\frac{1}{6}\cdot(q-2-(-1)^{\frac{m}{2}} 2^{\frac{m}{2}+1} )-1 ~~~~\mbox{ if $a^{2^e+1}+b$ is a cubic residue,}
\end{array}
\right.\end{aligned}$$ where $a=\sum_{i=1}^3 x_i$ and $b=\sum_{i=1}^3 x_i^{2^e+1}$. Moreover, the shortened code ${{\mathcal{C}}}_{T}$ is a $[2^{m}-3, 2m-2, 2^{m-1}-2^{m/2}]$ binary linear code with the weight distribution in Table \[tab-even32\].
By definitions, the code ${{\mathcal{C}}}$ has parameters $[2^m,2m+1,2^{m-1}-2^{m/2}]$ and the weight distribution in Table \[tab-cf1\]. By Lemma \[lem-cf\], the minimal distance of ${{\mathcal{C}}}^\bot$ is 6. Consider the system of equations given by $$\begin{aligned}
\label{eqxexp}
\left\{
\begin{array}{l}
x_1+x_2+x_3+x+y+z=0, \\ [2mm]
x_1^{2^e+1}+x_2^{2^e+1}+x_3^{2^e+1}+x^{2^e+1}+y^{2^e+1}+z^{2^e+1}=0. \\
\end{array}
\right.\end{aligned}$$ There exists only one set $\{x_1,x_2,x_3,x,y,z\}$ satisfying $0 \in \{x_1,x_2,x_3,x,y,z\}$ and Equation (\[eqxexp\]), since the code with zero set $\{1, 2^e+1\}$ has minimum distance 5. Therefore, from the definitions we have $\lambda _{T,6}({{\mathcal{C}}}^\bot)=\frac{\hat{N}}{3!}-1$, where $\hat{N}$ was defined in Lemma \[lem-solu2\]. Then the desired conclusions follow from Lemma \[lem-solu2\] and Theorem \[main-even32\].
\[exa-31even\] Let $m=4$, $e=1$ and $T=\{\alpha^1,\alpha^2,\alpha^4\}$. Then $\lambda=0$ and the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-even32\] is a $[13,6,4]$ binary linear code with the weight enumerator $1+7z^{4}+36z^{6}+15z^{8}+4 z^{10}+z^{12}$. This code ${{\mathcal{C}}}_{T}$ is optimal. Its dual ${{\mathcal{C}}}_{T}^\perp$ has parameters $[13,7,4]$ and is optimal according to the tables of best known codes maintained at http://www.codetables.de.
\[exa-32even\] Let $m=4$,$e=1$ and $T=\{\alpha^2,\alpha^5,\alpha^7\}$. Then $\lambda=2$ and the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-even32\] is a $[13,6,4]$ binary linear code with the weight enumerator $1+8z^{4}+34z^{6}+15z^{8}+6 z^{10}$. This code ${{\mathcal{C}}}_{T}$ is optimal. Its dual ${{\mathcal{C}}}_{T}^\perp$ has parameters $[13,7,3]$ and is almost optimal according to the tables of best known codes maintained at http://www.codetables.de.
Some shortened codes for the case $t=4$ and $m$ even
----------------------------------------------------
For any $T=\{x_1,x_2,x_3,x_4\}\subseteq \mathcal P({{\mathcal{C}}})$, Magma programs show that the weight distribution of ${{\mathcal{C}}}_{T}$ for some code ${{\mathcal{C}}}$ is very complex. Thus, it is difficult to determine their parameters in general. In this subsection, we will study a class of special linear codes ${{\mathcal{C}}}$ with the weight distribution in Table \[tab-cf1\] and determine parameters of ${{\mathcal{C}}}_{T}$ with the special set $T$ and $t=\#T=4$ in Theorem \[main-even41\].
In order to determine the parameters of ${{\mathcal{C}}}_{T}$, we need the next two lemmas.
\[lem-re\] Let $m\geq 4$ be even and $q=2^m$. Define $$\begin{aligned}
\label{eqx-re}
& R_{(3,i)}=\# \{x \in {{\mathrm{GF}}}(q)^*: {{\mathrm{Tr}}}_{q/2}(x)=i~and~ \mbox{x is a cubic residue} \} \nonumber \\
& \bar{R}_{(3,i)}=\# \{x \in {{\mathrm{GF}}}(q)^*: {{\mathrm{Tr}}}_{q/2}(x)=i~ and~ \mbox{x is not a cubic residue} \},\end{aligned}$$ where $i=0$ or $i=1$. Then $$\begin{aligned}
& R_{(3,0)}=\frac{1}{6}(2^m-2+(-2)^{m/2 +1}) \\
& R_{(3,1)}=\frac{2^m-1}{3}-\frac{1}{6}(2^m-2+(-2)^{m/2 +1}) \\
& \bar{R}_{(3,0)}= (2^{m-1}-1)-\frac{1}{6}(2^m-2+(-2)^{m/2 +1}) \\
& \bar{R}_{(3,1)}= \frac{2^m}{3}-\frac{(-2)^{m/2}}{3}.\end{aligned}$$
By definitions, we get $$\begin{aligned}
R_{(3,0)}&=\frac{1}{3} \cdot \# \{x \in {{\mathrm{GF}}}(q)^*: {{\mathrm{Tr}}}_{q/2}(x^3)=0\} \\
& =\frac{1}{6}\sum_{z\in {{\mathrm{GF}}}(2)}\sum_{x\in {{\mathrm{GF}}}(q)^*}(-1)^{z{{\mathrm{Tr}}}_{q/2}(x^3)} \\
& =\frac{1}{6} \left (q-2+\sum_{x\in {{\mathrm{GF}}}(q)}(-1)^{{{\mathrm{Tr}}}_{q/2}(x^3)} \right).\end{aligned}$$ Then the value of $R_{(3,0)}$ follows from Lemma \[expsum\]. Note that it has $\frac{q-1}{3}$ cubic residues in ${{\mathrm{GF}}}(q)^*$ and $$\# \{x \in {{\mathrm{GF}}}(q)^*: {{\mathrm{Tr}}}_{q/2}(x)=0 \}=\frac{q}{2}-1.$$ Then the desired conclusions follow from $$\begin{aligned}
\left\{ \begin{array}{l}
R_{(3,0)}+\bar{R}_{(3,0)}=\frac{q}{2}-1, \\
R_{(3,0)}+R_{(3,1)}=\frac{q-1}{3}, \\
\bar{R}_{(3,0)}+\bar{R}_{(3,0)}=\frac{2(q-1)}{3}
\end{array}
\right.\end{aligned}$$ This completes the proof.
\[lem-even41\] Let $m\geq 4$ be even, $e$ be positive integer with $\gcd(m,e)=1$, and $q=2^m$. Let $N_{(0,1)}$ be the number of solutions $(x,y,z,u)\in ({{\mathrm{GF}}}(q))^4$ of the system of equations $$\begin{aligned}
\label{eqx-even1}
\left\{
\begin{array}{l}
x+y+z+u=0, \\ [2mm]
x^{2^e+1}+y^{2^e+1}+z^{2^e+1}+u ^{2^e+1}=1 \\
\end{array}
\right.\end{aligned}$$ where $x,y,z,u$ are pairwise distinct. Then $N_{(0,1)} =q(q-2-(-1)^{m/2}2^{m/2+1}).$
By definitions, we have $$\begin{aligned}
N_{(0,1)}
&=\frac{1}{q^2} \sum_{x,y,z,u \in {{\mathrm{GF}}}(q)}\sum_{a,b\in {{\mathrm{GF}}}(q)}\chi_1(b(x+y+z+u))\chi_1(a(x^{2^e+1}+y^{2^e+1}+z^{2^e+1}+u ^{2^e+1}-1)) \\
&=\frac{1}{q^2} \sum_{a,b\in {{\mathrm{GF}}}(q)}\chi_1(-a) \left(\sum_{x\in {{\mathrm{GF}}}(q)}\chi_1(ax^{2^e+1}+bx) \right)^4 \\
&=\frac{1}{q^2} (\sum_{a\in {{\mathrm{GF}}}(q)}\chi_1(-a) (\sum_{x\in {{\mathrm{GF}}}(q)}\chi_1(ax^{2^e+1}))^4 + \\
&~~~~~~~~~~\sum_{a\in {{\mathrm{GF}}}(q)^*}\sum_{b\in {{\mathrm{GF}}}(q)^*}\chi_1(-a) ( \sum_{x\in {{\mathrm{GF}}}(q)}\chi_1(ax^{2^e+1}+bx) )^4 ) \\
&=\frac{1}{q^2} (\sum_{a\in {{\mathrm{GF}}}(q)}(-1)^{{{\mathrm{Tr}}}(-a)} (\sum_{x\in {{\mathrm{GF}}}(q)}\chi_1(ax^{2^e+1}))^4 + \\
&~~~~~~~~~~\sum_{a\in {{\mathrm{GF}}}(q)^*}(-1)^{{{\mathrm{Tr}}}(-a)} \sum_{b\in {{\mathrm{GF}}}(q)^*}( \sum_{x\in {{\mathrm{GF}}}(q)}\chi_1(ax^{2^e+1}+bx) )^4 ) \\
&=\frac{1}{q^2}\cdot (q^4+ (R_{(3,0)}-R_{(3,1)})\cdot 2^{m+4}+(\bar{R}_{(3,0)}-\bar{R}_{(3,1)})\cdot 2^{m} + \\
&~~~~~~~~~~~~ (R_{(3,0)}-R_{(3,1)})\cdot(2^{3m+2}-2^{2m+4})+ (\bar{R}_{(3,0)}-\bar{R}_{(3,1)})\cdot (2^m-1)2^{2m} ),\end{aligned}$$ where $R_{(3,0)},R_{(3,1)},\bar{R}_{(3,0)}$ and $\bar{R}_{(3,1)}$ were defined in (\[eqx-re\]) and the last equality holds from Lemmas \[expsum\] and \[expsum1\]. Then the desired conclusions follow from Lemma \[lem-re\].
\[main-even41\] Let $m\geq 4$ be even and $e$ be a positive integer with $\gcd(m,e)=1$. Let $q=2^m$, $f(x)=x^{2^e+1}$ and ${{\mathcal{C}}}$ be defined in (\[eq:cf\]). Let $T={{\mathrm{GF}}}(4)=\{ 0, 1, w,w^2\} \subseteq {{\mathrm{GF}}}(q)$, where $w$ is a generator of ${{\mathrm{GF}}}(4)^*$. Then the shortened code ${{\mathcal{C}}}_{T}$ is a $[2^{m}-4, 2m-3, 2^{m-1}-2^{m/2}]$ binary linear code with the weight distribution in Table \[tab-even41\].
Weight Multiplicity
------------------------------- ------------------------------------------------------------------------------------------------
$0$ $1$
\[2mm\] $2^{m-1}-2^{m/2}$ $ 1/3 \cdot 2^{ m/2-6} (-16 + 2^{3m/2} - 2^{m+1}(-4+(-1)^{m/2}) - 2^{4 + m/2}(-1+(-1)^{m/2}))$
\[2mm\] $2^{m-1}-2^{(m-2)/2}$ $ 1/24 \cdot (2^{m/2+2}+2^m )(2^m+(-1)^{m/2}2^{m/2}-2)$
\[2mm\] $2^{m-1}$ $ -1 + 2^{2m-5}-(-1)^{m/2} 2^{ 3m/2 -4}$
\[2mm\] $2^{m-1}+2^{(m-2)/2}$ $ 1/24 \cdot (-2^{m/2+2}+2^m )(2^m+(-1)^{m/2}2^{m/2}-2) $
\[2mm\] $2^{m-1}+2^{m/2}$ $ 1/3 \cdot 2^{ m/2-6} (16 + 2^{3m/2} - 2^{m+1}(4+(-1)^{m/2}) + 2^{4 + m/2}(1+(-1)^{m/2}))$
\[2mm\]
: The weight distribution of ${{\mathcal{C}}}_T$ in Theorem \[main-even41\] []{data-label="tab-even41"}
By definitions and Lemma \[lem-cf\], ${{\mathcal{C}}}$ is a $[2^{m}, 2m+1, 2^{m-1}-2^{m/2}]$ with the weight distribution in Table \[tab-cf1\] and the minimal distance of ${{\mathcal{C}}}^\bot$ is 6. Note that $\lambda _{T,6}({{\mathcal{C}}}^\bot)=0$ from Theorem \[main-32\] and the definition of $T$. Thus, the minimal distance of $\left (\mathcal C^{\perp} \right )^{T}$ is at least 3. This means that $$\begin{aligned}
\label{eq-4A12}
A_1\left ( \left (\mathcal C^{\perp} \right )^{T} \right )=A_2\left ( \left (\mathcal C^{\perp} \right )^{T} \right )=0.~
$$ Further, from Theorem \[main-even33\] and the definition of $T$, we have $$\begin{aligned}
\label{eq:A3}
\lambda _{\hat{T},6}({{\mathcal{C}}}^\bot)=\frac{1}{6}\cdot(q-2-(-1)^{\frac{m}{2}} 2^{\frac{m}{2}+1} )-1\end{aligned}$$ for any $\hat{T}=\{\hat{x}_1, \hat{x}_2, \hat{x}_3\} \subseteq T$. Therefore, $$\begin{aligned}
\label{eq-4A3}
A_3\left ( \left (\mathcal C^{\perp} \right )^{T} \right )= \binom{4 }{ 3 }\cdot \left(\frac{1}{6}\cdot(q-2-(-1)^{\frac{m}{2}} 2^{\frac{m}{2}+1} )-1 \right).~
$$ Note that the solutions of the system (\[eqx-even1\]) have symmetrical property and $(x,y,z,u)=(0,1,w,w^2)$ is a solution of the system (\[eqx-even1\]). Thus, from the definitions and Lemma \[lem-even41\] we get $$\begin{aligned}
\label{eq-84}
\lambda _{T,8}({{\mathcal{C}}}^\bot)=\frac{N_{(0,1)}}{4!}-1-\binom{4 }{ 3 }\cdot \lambda _{\hat{T},6}({{\mathcal{C}}}^\bot),\end{aligned}$$ where $N_{(0,1)}$ was defined in Lemma \[lem-even41\] and $\lambda _{\hat{T},6}({{\mathcal{C}}}^\bot)$ was defined in (\[eq:A3\]). By the proof of Lemma \[le-even31\], we obtain $$\begin{aligned}
\label{eq-62}
A_4\left ( \left (\mathcal C^{\perp} \right )^{\{\bar{x}_1,\bar{x}_2\}} \right ) =\frac{2}{3}\cdot (2^{m-2}-1)^2\end{aligned}$$ for any $\{\bar{x}_1,\bar{x}_2\} \subseteq T $. It is obvious that there exists only two 3-subsets $\bar{T}$ of $T$ with $\{\bar{x}_1,\bar{x}_2\} \subseteq \bar{T} \subseteq T$ for any $\{\bar{x}_1,\bar{x}_2\} \subseteq T $. Thus, from the definitions we have $$\begin{aligned}
\label{eq-4A4}
A_4\left ( \left (\mathcal C^{\perp} \right )^{T} \right )= \binom{4 }{ 2 }\cdot \left( A_4\left ( \left (\mathcal C^{\perp} \right )^{\{\bar{t}_1,\bar{t}_2\}} \right )-2 \cdot \lambda _{\hat{T},6}({{\mathcal{C}}}^\bot) \right)+
\lambda _{T,8}({{\mathcal{C}}}^\bot).\end{aligned}$$ Combining Equations (\[eq:A3\]), (\[eq-84\]) and (\[eq-62\]), Equation (\[eq-4A4\]) yields $$\begin{aligned}
\label{eq-even4A4}
A_4\left ( \left (\mathcal C^{\perp} \right )^{T} \right )= 4\cdot \left(2^{m-2}-1 \right)^2-\frac{8}{3} \cdot \left( 2^m-2-(-1)^{\frac{m}{2}} 2^{\frac{m}{2}+1} \right)+\frac{N_{(0,1)}}{24}+15.\end{aligned}$$
Note that the shortened code ${{\mathcal{C}}}_{T}$ has length $2^m-4$ and dimension $2m-3$, since $\#T=4$ and Lemma \[lem:C-S-P\]. By the definitions and the weight distribution in Table \[tab-cf1\], then $A_i\left ({{\mathcal{C}}}_T \right )=0$ for $i\not \in \{0, i_1, i_2, i_3,i_4,i_5\} $, where $i_1=2^{m-1}-2^{m/2}$, $i_2=2^{m-1}-2^{(m-2)/2}$ , $i_3=2^{m-1}$, $i_4=2^{m-1}+2^{(m-2)/2}$ and $i_5=2^{m-1}+2^{m/2}$. Hence, from Lemma \[lem:C-S-P\] and Equations (\[eq-4A12\]), (\[eq-4A3\]) and (\[eq-even4A4\]), the first five Pless power moments of (\[eq:PPM\]) yield the weight distribution in Table \[tab-even41\]. This completes the proof.
\[exa-even41\] Let $m=4$ and $e=1~(or~3)$. Then the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-even41\] is a $[12,5,4]$ binary linear code with the weight enumerator $1+3z^{4}+24z^{6}+3z^{8}+z^{12}$. This code ${{\mathcal{C}}}_{T}$ is optimal. Its dual ${{\mathcal{C}}}_{T}^\perp$ has parameters $[12,7,4]$ and is optimal according to the tables of best known codes maintained at http://www.codetables.de.
Shortened linear codes from PN functions {#sec-PN}
========================================
In this section, we study some shortened codes of linear codes ${{\mathcal{C}}}$ from the special PN functions and determine their parameters.
Let $p$ be odd prime and $q=p^m$. Let $f(x)=x^2$ and ${{\mathcal{C}}}$ defined by (\[eq:cf\]). It is known that the code ${{\mathcal{C}}}$ is a $[q,2m+1]$ linear code with the the weight distribution in [@LQ2009]. Note that the code ${{\mathcal{C}}}$ is affine invariant, and thus holds $2$-designs. Then the following theorem is easily derived from the parameters of the code ${{\mathcal{C}}}$ in [@LQ2009] and Theorem \[thm:sct-code\], and we omit its proof.
\[main-pdesign2\] Let $p$ be an odd prime, $m$ and $t$ be positive integers. Let $q=p^m$, $f(x)=x^2$ and ${{\mathcal{C}}}$ be defined in (\[eq:cf\]). Suppose $T$ is a $t$-subset of $\mathcal P({{\mathcal{C}}})$. We have the following results.
- It $t=1$, then the shortened code ${{\mathcal{C}}}_{T}$ is a $[2^{m}-1, 2m]$ linear code with the weight distribution in Table \[tab-px2odd1\] (resp. Table \[tab-px2even1\]) when $m$ is odd (resp. even).
- It $t=2$, then the shortened code ${{\mathcal{C}}}_{T}$ is a $[2^{m}-2, 2m-1]$ linear code with the weight distribution in Table \[tab-px2odd2\] (resp. Table \[tab-px2even2\]) when $m$ is odd (resp. even).
Weight Multiplicity
------------------------------------------ -------------------------------------------------------------------
$0$ $1$
\[2mm\] $p^{m-1}(p-1)$ $ (p^m-1) (1 + p^{m-1}) $
\[2mm\] $p^{m-1}(p-1)-p^{\frac{m-1}{2}}$ $ 1/2 \cdot (p-1) p^{(m-3)/2} (p^m-1) (p + p^{(1 + m)/2}) $
\[2mm\] $p^{m-1}(p-1)+p^{\frac{m-1}{2}}$ $ 1/2 \cdot (p-1) p^{(m-3)/2} (p^m-1) (-p + p^{(1 + m)/2}) $
\[2mm\]
: The weight distribution of ${{\mathcal{C}}}_T$ for $m$ odd and $t=1$ []{data-label="tab-px2odd1"}
Weight Multiplicity
------------------------------------------ ------------------------------------------------------------
$0$ $1$
\[2mm\] $p^{m-1}(p-1)$ $ p^{2 m-2} -1 $
\[2mm\] $p^{m-1}(p-1)-p^{\frac{m-1}{2}}$ $ ( p-1) (-p^{(m-1)/2} + p^{2m-2} + 2 p^{(3m-3)/2}) /2 $
\[2mm\] $p^{m-1}(p-1)+p^{\frac{m-1}{2}}$ $ ( p-1) (p^{(m-1)/2} + p^{2m-2} - 2 p^{(3m-3)/2} )/2 $
\[2mm\]
: The weight distribution of ${{\mathcal{C}}}_T$ for $m$ odd and $t=2$ []{data-label="tab-px2odd2"}
Weight Multiplicity
------------------------------------- --------------------------------------------------------
$0$ $1$
$(p-1)(p^{m-1}-p^{\frac{m-2}{2}}) $ $ p^{ m/2-1} (p + p^{m/2}-1) (p^m-1)/2 $
$(p-1) p^{m-1}-p^{\frac{m-2}{2}} $ $ ( p-1) p^{m/2-1} (p^{m/2}-1) (1 + p^{m/2})^2 /2 $
$(p-1) p^{m-1}$ $ p^{m}-1$
$(p-1)p^{m-1}+p^{\frac{m-2}{2}}$ $ ( p-1) p^{m/2-1} (p^{m/2}-1)^2 (1 + p^{m/2}) /2 $
$(p-1)(p^{m-1}+p^{\frac{m-2}{2}})$ $ p^{ m/2-1} (-p + p^{m/2}+1) (p^m-1)/2 $
: The weight distribution of ${{\mathcal{C}}}_T$ for $m$ even and $t=1$ []{data-label="tab-px2even1"}
Weight Multiplicity
------------------------------------- --------------------------------------------------------------
$0$ $1$
$(p-1)(p^{m-1}-p^{\frac{m-2}{2}}) $ $p^{m/2-2} (p^{m/2}-1) (-1 + p + p^{m/2}) (p + p^{m/2})/2 $
$(p-1) p^{m-1}-p^{\frac{m-2}{2}} $ $ (p-1) p^{m/2-2} (1 + p^{m/2}) (-p + p^{m/2} + p^m)/2 $
$(p-1) p^{m-1}$ $ p^{m-1}-1$
$(p-1)p^{m-1}+p^{\frac{m-2}{2}}$ $ (p-1) p^{m/2-2} (1 + p^{m/2}) (-p - p^{m/2} + p^m)/2 $
$(p-1)(p^{m-1}+p^{\frac{m-2}{2}})$ $p^{m/2-2} (p^{m/2}+1) (1 - p + p^{m/2}) (p^{m/2}-p)/2 $
: The weight distribution of ${{\mathcal{C}}}_T$ for $m$ even and $t=2$ []{data-label="tab-px2even2"}
\[exa-p1\] Let $m=3$, $p=3$ and $T$ be a $1$-subset of $\mathcal P({{\mathcal{C}}})$. Then the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-pdesign2\] is a $[26, 6, 15]$ linear code with the weight enumerator $1+312 z^{15}+260 z^{18}+ 156 z^{21}$. This code ${{\mathcal{C}}}_{T}$ is optimal. Its dual ${{\mathcal{C}}}_{T}^\perp$ has parameters $[26, 20, 4]$ and is optimal according to the tables of best known codes maintained at http://www.codetables.de.
\[exa-p5\] Let $m=4$, $p=3$ and $T$ be a $1$-subset of $\mathcal P({{\mathcal{C}}})$. Then the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-pdesign2\] is a $[80, 8, 48]$ linear code with the weight enumerator $1+1320 z^{48}+2400 z^{51}+ 80 z^{54}+1920 z^{57}+840 z^{60}$. This code ${{\mathcal{C}}}_{T}$ is optimal. Its dual ${{\mathcal{C}}}_{T}^\perp$ has parameters $[80, 72, 4]$ and is optimal according to the tables of best known codes maintained at http://www.codetables.de.
\[exa-p3\] Let $m=5$, $p=3$ and $T$ be a $2$-subset of $\mathcal P({{\mathcal{C}}})$. Then the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-pdesign2\] is a $[241, 9, 153]$ linear code with the weight enumerator $1+8010 z^{153}+6560 z^{162}+ 5112 z^{171}$. This code ${{\mathcal{C}}}_{T}$ is optimal. Its dual ${{\mathcal{C}}}_{T}^\perp$ has parameters $[241, 232, 3]$ and is optimal according to the tables of best known codes maintained at http://www.codetables.de.
\[exa-p4\] Let $m=4$, $p=3$ and $T$ be a $2$-subset of $\mathcal P({{\mathcal{C}}})$. Then the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-pdesign2\] is a $[79, 7, 48]$ linear code with the weight enumerator $1+528 z^{48}+870 z^{51}+ 26 z^{54}+552 z^{57}+210 z^{60}$. This code ${{\mathcal{C}}}_{T}$ is almost optimal. Its dual ${{\mathcal{C}}}_{T}^\perp$ has parameters $[79,72,3]$ and is almost optimal according to the tables of best known codes maintained at http://www.codetables.de.
In the following, we will consider the shortened code ${{\mathcal{C}}}_{T}$ of ${{\mathcal{C}}}$ for the case $\#T=p$. In order to determine the parameters of ${{\mathcal{C}}}_{T}$, we need the next lemmas.
[@HD2019]\[lem-equalities\] Let $q=p^m$ with $p$ an odd prime. Then $$\begin{aligned}
\lefteqn{\sharp \{a\in {{\mathrm{GF}}}(q)^*:\eta(a)=1\mbox{ and }{{\mathrm{Tr}}}_{q/p}(a)=0\} } \\
&=&\left\{
\begin{array}{ll}
\frac{p^{m-1}-1-(p-1)p^{\frac{m-2}{2}}(\sqrt{-1})^{\frac{(p-1)m}{2}}}{2} & \mbox{ for even $m$,} \\
\frac{p^{m-1}-1}{2} & \mbox{ for odd $m$.}
\end{array}\right.\end{aligned}$$ and $$\begin{aligned}
\lefteqn{ \sharp \{a\in {{\mathrm{GF}}}(q)^*:\eta(a)=-1\mbox{ and }{{\mathrm{Tr}}}_{q/p}(a)=0\} } \\
&=&\left\{
\begin{array}{ll}
\frac{p^{m-1}-1+(p-1)p^{\frac{m-2}{2}}(\sqrt{-1})^{\frac{(p-1)m}{2}}}{2} & \mbox{ for even $m$,}\\
\frac{p^{m-1}-1}{2} & \mbox{ for odd $m$.}
\end{array}\right.\end{aligned}$$
[@HD2019] \[lem-equalities1\] Let $q=p^m$ with $p$ an odd prime and $(a,b)\in {{\mathrm{GF}}}(q)^2$. Denote
$$\begin{aligned}
\label{eqn-50}
N_0(a,b)=\sharp\{x\in {{\mathrm{GF}}}(q):{{\mathrm{Tr}}}_{q/p}(ax^2+bx)=0\}.\end{aligned}$$
Then the following results follow.
- If $m$ is odd, then $$\begin{aligned}
\label{eqn-5}
\lefteqn{ N_0(a,b)=} \nonumber \\
&
\left\{\begin{array}{ll}
p^m & \mbox{ if }(a,b)=(0,0),\\
p^{m-1} & \mbox{ if }a=0,\ b\neq 0,\mbox{ or }a\neq 0,\ 0={{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}),\\
p^{m-1}+p^{\frac{m-1}{2}}(-1)^{\frac{(p-1)(m+1)}{4}} &
\mbox{ if }a\neq 0,\ 0\neq{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}),\ \eta(a)\eta'\left(-{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a})\right)=1,\\
p^{m-1}+p^{\frac{m-1}{2}}(-1)^{\frac{(p-1)(m+1)+4}{4}} &
\mbox{ if }a\neq 0,\ 0\neq{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}),\ \eta(a)\eta'\left(-{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a})\right)=-1,
\end{array} \right.\end{aligned}$$
- If $m$ is even, then $$\begin{aligned}
\label{eqn-6}
N_0(a,b)=
\left\{\begin{array}{ll}
p^m & \mbox{ if }(a,b)=(0,0),\\
p^{m-1} & \mbox{ if }a=0,\ b\neq 0,\\
p^{m-1}+(p-1)p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)+4}{4}} & \mbox{ if }a\neq 0,\ 0={{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}),\ \eta(a)=1,\\
p^{m-1}+(p-1)p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)}{4}} & \mbox{ if }a\neq 0,\ 0={{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}),\ \eta(a)=-1,\\
p^{m-1}+p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)}{4}} &
\mbox{ if }a\neq 0,\ 0\neq{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}),\ \eta(a)=1,\\
p^{m-1}+p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)+4}{4}} &
\mbox{ if }a\neq 0,\ 0\neq{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}),\ \eta(a)=-1,
\end{array} \right.\end{aligned}$$
\[lem-sign\] Let $q=p^m$ with $p$ an odd prime and $a\in {{\mathrm{GF}}}(q)^*$. Define $f(x)={{\mathrm{Tr}}}_{q/p}(-\frac{1}{4a}x^2)$. Then the dual of $f(x)$ is $f^*(x)={{\mathrm{Tr}}}_{q/p}(ax^2)$ and the sign of the Walsh transform of $f(x)$ is $$\varepsilon=\eta(-\frac{1}{4a})(-1)^{\frac{m(p-1)(p-3)}{8}+m-1}=\eta(a)(-1)^{\frac{m(p-1)(p-3)}{8}+m-1+\frac{q-1}{2}}.$$
Note that $f(x)$ is a weakly regular bent function. Then the desired conclusions follow from the definitions and Lemmas \[lem-32A1\] and \[lem-32A2\].
\[lem-psign\] Let $q=p^m$ with $p$ an odd prime, $a\in {{\mathrm{GF}}}(q)^*$ and $e$ be any positive integer such that $m/\gcd(m,e)$ is odd. Define $f(x)={{\mathrm{Tr}}}_{q/p}(ax^{p^e+1})$. Then the dual of $f(x)$ is $f^*(\beta )={{\mathrm{Tr}}}_{q/p}(-a (x_{a,-\beta}) ^{p^e+1})$ for any $\beta \in {{\mathrm{GF}}}(q)$ and the sign of the Walsh transform of $f(x)$ is $$\varepsilon=\eta(a)(-1)^{\frac{m(p-1)(p-3)}{8}+m-1+\frac{q-1}{2}},$$ where $x_{a,-\beta}$ is the unique solution of the equation $$a^{p^e}x^{p^{2e}}+a x+ (-\beta)^{p^e}=0.$$
Note that $f(x)$ is a weakly regular bent function. By definitions, we have $$f^*(\beta )={{\mathrm{Tr}}}_{q/p}(-a (x_{a,-\beta}) ^{p^e+1}).$$ From the definitions and Lemma \[lem-psum\], we have $$\begin{aligned}
\varepsilon =
\left\{
\begin{array}{ll}
(-1)^{m-1+\frac{q-1}{2}+\frac{m(p-1)}{4}} \eta(a) , & \mbox{if}~ p\equiv 1 ~mod~4, \\
(-1)^{m-1+\frac{q-1}{2}+\frac{m(p-3)}{4}} \eta(a) , & \mbox{if}~ p\equiv 3 ~mod~4. \\
\end{array}
\right.\end{aligned}$$ Then the desired conclusions follow.
\[lem-psign1\] Let $q=p^m$ with $p$ an odd prime, $a\in {{\mathrm{GF}}}(q)^*$, $\gamma \in {{\mathrm{GF}}}(p)$ and $e$ be any positive integer such that $m/\gcd(m,e)$ is odd. Define $$\tilde{N}_b=\# \{b\in {{\mathrm{GF}}}(q): {{\mathrm{Tr}}}_{q/p}(a(x_{a,b}) ^{p^e+1})=\gamma ~and~{{\mathrm{Tr}}}_{q/p}(b)=0\}$$ where $x_{a,b}$ is the unique solution of the equation $
a^{p^e}x^{p^{2e}}+a x+ (b)^{p^e}=0.
$ If ${{\mathrm{Tr}}}_{q/p}(a)=0$, then $$\tilde{N}_b =
\left\{
\begin{array}{ll}
p^{m-2}+\varepsilon \bar{\eta}^{m/2}(-1)(p-1)p^{(m-2)/2}, & \hbox{if $\gamma=0$ and $m$ is even;} \\
p^{m-2}, & \hbox{if $\gamma=0$ and $m$ is odd.} \\
\frac{p-1}{2}\left(p^{m-2}-\varepsilon \bar{\eta}^{m/2}(-1)p^{(m-2)/2}\right) & \hbox{if $\gamma \neq 0$ and $m$ is even;} \\
\frac{p-1}{2}\left( p^{m-2}+\varepsilon \sqrt{p*}^{m-1} \right) & \hbox{if $\gamma \in SQ$ and $m$ is odd;} \\
\frac{p-1}{2}\left( p^{m-2}-\varepsilon \sqrt{p*}^{m-1} \right). & \hbox{if $\gamma \in NSQ$ and $m$ is odd;} \\
\end{array}
\right.$$ where $\varepsilon$ is given in Lemma \[lem-psign\].
By definitions, we have $$a^{p^e}(x_{a,b})^{p^{2e}}+a x_{a,b}+ (b)^{p^e}=(a x_{a,b}^{p^e})^{p^e}+a x_{a,b}+ (b)^{p^e}=0,$$ which deduces $${{\mathrm{Tr}}}_{q/p}(a x_{a,b}^{p^e} +a x_{a,b}+ (b)^{p^e})={{\mathrm{Tr}}}_{q/p}(a x_{a,b}^{p^e} +a x_{a,b})={{\mathrm{Tr}}}_{q/p}\left((a^{p^{-e}}+a) x_{a,b} \right)=0.$$ When ${{\mathrm{Tr}}}_{q/p}(b)=0$, we have $$\begin{aligned}
\label{eqn-61}
\tilde{N}_b=\# \{x\in {{\mathrm{GF}}}(q): {{\mathrm{Tr}}}_{q/p}(a x ^{p^e+1})=\gamma ~and~{{\mathrm{Tr}}}_{q/p}\left((a^{p^{-e}}+a) x \right)=0\}.\end{aligned}$$ Note that $x=1$ is the unique solution of the equation $$a^{p^e}(x)^{p^{2e}}+a x+ (-a^{p^{-e}}-a)^{p^e}=0 .$$ By Lemma \[lem-psign\], we get $$\begin{aligned}
\label{eqn-62}
f^*(a^{p^{-e}}+a)={{\mathrm{Tr}}}_{q/p}(-a)=0\end{aligned}$$ when ${{\mathrm{Tr}}}_{q/p}(a)=0$, where $f^*$ is the dual of ${{\mathrm{Tr}}}_{q/p}(ax^{p^e+1})$. By Equations (\[eqn-61\]) and (\[eqn-62\]), the desired conclusions follow from Lemmas \[2lem11\], \[2lem14\] and \[lem-psign\].
\[main-51\] Let $p$ be an odd prime and $m$ be a positive integer. Let $q=p^m$, $f(x)=x^2$ and ${{\mathcal{C}}}$ be defined in (\[eq:cf\]). Let $T={{\mathrm{GF}}}(p)$. Then the shortened code ${{\mathcal{C}}}_{T}$ is a $[2^{m}-p, 2m-2]$ linear code. If $m$ is odd, the weight distribution of ${{\mathcal{C}}}_{T}$ is given in Table \[tab-51\]; if $m\geq 2$ is even, the weight distribution of ${{\mathcal{C}}}_{T}$ is given in Table \[tab-52\], where $B=(-1)^{\frac{m(p-1)(p-3)}{8}+m-1+\frac{q-1}{2}} \cdot \sqrt{p^*}^{m-1}$, $B_1=\frac{p^{m-1}-1-(p-1)p^{\frac{m-2}{2}}(\sqrt{-1})^{\frac{(p-1)m}{2}}}{2}$ and $B_2= (-1)^{\frac{m(p-1)(p-3)}{8}+m-1+\frac{q-1}{2}} \bar{\eta}^{m/2}(-1) \cdot (p-1)p^{(m-2)/2}$.
Weight Multiplicity
--------------------------------------------------------------------- ----------------------------------------------
$0$ $1$
\[2mm\] $p^{m-1}(p-1)$ $ (p^{m-1}-1)(p^{m-2}+1)$
\[2mm\] $p^{m-1}(p-1)-p^{\frac{m-1}{2}}(-1)^{\frac{(p-1)(m+1)}{4}}$ $ \frac{p^{m-1}-1}{2}\cdot (p-1)(p^{m-2}+B)$
\[2mm\] $p^{m-1}(p-1)+p^{\frac{m-1}{2}}(-1)^{\frac{(p-1)(m+1)}{4}}$ $ \frac{p^{m-1}-1}{2}\cdot (p-1)(p^{m-2}-B)$
\[2mm\]
: The weight distribution of ${{\mathcal{C}}}_T$ for $m$ odd []{data-label="tab-51"}
Weight Multiplicity
---------------------------------------------------------------- -------------------------------------------------
$0$ $1$
$p^{m-1}(p-1)$ $ p^{m-1}-1$
$p^{m-1}(p-1)-(p-1)p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)+4}{4}}$ $ B_1\cdot (p^{m-2}+B_2) $
$p^{m-1}(p-1)-(p-1)p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)}{4}}$ $ (p^{m-1}-1-B_1)\cdot (p^{m-2}-B_2)$
$p^{m-1}(p-1)-p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)}{4}} $ $ B_1\cdot (p^{m-1}-p^{m-2}-B_2) $
$p^{m-1}(p-1)-p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)+4}{4}}$ $ (p^{m-1}-1-B_1)\cdot (p^{m-1}-p^{m-2}+B_2) $
: The weight distribution of ${{\mathcal{C}}}_T$ for $m$ even []{data-label="tab-52"}
By definitions and Lemma \[lem:C-S-P\], the shortened code ${{\mathcal{C}}}_{T}$ has length $2^m-p$ and dimension $2m-2$. Since $T={{\mathrm{GF}}}(p)$, the weight distribution of ${{\mathcal{C}}}_{T}$ is the same as the code $$\begin{aligned}
\label{eq:cf1}
\hat{{{\mathcal{C}}}}=\left\{\left({{\mathrm{Tr}}}_{q/p}(ax^2+bx)\right)_{x \in {{\mathrm{GF}}}(q)}:a,b\in {{\mathrm{GF}}}(q)~and~{{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0 \right\}.\end{aligned}$$
For each $(a,b) \in {{\mathrm{GF}}}(q) \times {{\mathrm{GF}}}(q) $ with ${{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0$, define the corresponding codeword $$\begin{aligned}
\label{eqn-mcodeword}
{{\mathbf{c}}}(a,b)=\left(({{\mathrm{Tr}}}_{q/p}(ax^2+bx)_{x\in {{\mathrm{GF}}}(q)}\right) \in \hat{{{\mathcal{C}}}}.\end{aligned}$$ Then the Hamming weight of ${{\mathbf{c}}}(a,b)$ is $$\begin{aligned}
{{\mathtt{wt}}}({{\mathbf{c}}}(a,b))&=& q-N_0(a,b)\end{aligned}$$ with ${{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0$, where $N_0(a,b)$ was defined in Equation (\[eqn-50\]). We discuss the value of ${{\mathtt{wt}}}({{\mathbf{c}}}(a,b))$ in the following two cases.
() The case that $m$ is odd.
From Equation (\[eqn-5\]), we get $$\begin{aligned}
& & {{\mathtt{wt}}}({{\mathbf{c}}}(a,b))=q-N_0(a,b) \nonumber \\
& & =
\left\{\begin{array}{ll}
0 & \mbox{ if }(a,b)=(0,0), {{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0,\\
p^{m-1}(p-1) & \mbox{ if }a=0,\ b\neq 0,{{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0, \\
~ & \mbox{ or }a\neq 0,\ {{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a})=0, {{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0\\
p^{m-1}(p-1)-p^{\frac{m-1}{2}}(-1)^{\frac{(p-1)(m+1)}{4}} & \mbox{ if }a\neq 0, {{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0,\\
~ & ~{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a})\neq 0,\ \eta(a)\bar{\eta}\left(-{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a})\right)=1,\\
p^{m-1}(p-1)+p^{\frac{m-1}{2}}(-1)^{\frac{(p-1)(m+1)}{4}} & \mbox{ if }a\neq 0, {{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0,\\
~ & ~{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a})\neq 0,\ \eta(a)\bar{\eta} \left(-{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a})\right)=-1.
\end{array} \right. \\ [2mm]
& & = \left\{\begin{array}{ll}
0 & \mbox{ with $1$ time},\\
p^{m-1}(p-1) & \mbox{ with $(p^{m-1}-1)(p^{m-2}+1)$ time},\\
p^{m-1}(p-1)-p^{\frac{m-1}{2}}(-1)^{\frac{(p-1)(m+1)}{4}} & \mbox{ with $ \frac{p^{m-1}-1}{2}\cdot (p-1)(p^{m-2}+B)$ time},\\
p^{m-1}(p-1)+p^{\frac{m-1}{2}}(-1)^{\frac{(p-1)(m+1)}{4}} & \mbox{ with $\frac{p^{m-1}-1}{2}\cdot (p-1)(p^{m-2}-B)$ time}, \\
\end{array} \right.\end{aligned}$$ when $(a,b)$ runs through ${{\mathrm{GF}}}(q) \times {{\mathrm{GF}}}(q)$, where $B=(-1)^{\frac{m(p-1)(p-3)}{8}+m-1+\frac{q-1}{2}} \cdot \sqrt{p^*}^{m-1}$ and the frequency is obtained based on Lemmas \[lem-equalities\], \[lem-sign\], \[2lem11\] and \[2lem14\]. We first compute the frequency $A_w$ of the nonzero weight $w$, where $$w=p^{m-1}(p-1)-p^{\frac{m-1}{2}}(-1)^{\frac{(p-1)(m+1)}{4}},$$ and $$A_w=\sharp \{(a,b):a\neq 0,\ {{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}) \neq 0 ,\ \eta(a)\bar{\eta} \left(-{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a})\right)=1,\ {{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0\}.$$ Clearly, the number of $a \in {{\mathrm{GF}}}(q)^*$ such that ${{\mathrm{Tr}}}_{q/p}(a)=0$ and $\eta(a)=1$, is $\bar{n}_a=\frac{p^{m-1}-1}{2}$ by Lemma \[lem-equalities\]; if we fix $a$ with ${{\mathrm{Tr}}}_{q/p}(a)=0$ and $\eta(a)=1$, the number of $b$ such that ${{\mathrm{Tr}}}_{q/p}(b)=0$ and $ \bar{\eta} (-{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}))=1$, is $\bar{n}_b=\frac{p-1}{2}(p^{m-2}+ (-1)^{\frac{m(p-1)(p-3)}{8}+m-1+\frac{q-1}{2}} \cdot \sqrt{p^*}^{m-1})$ by Lemmas \[lem-sign\] and \[2lem14\]. Meanwhile, the number of $a \in {{\mathrm{GF}}}(q)^*$ such that ${{\mathrm{Tr}}}_{q/p}(a)=0$ and $\eta(a)=-1$, is $\hat{n}_a=\frac{p^{m-1}-1}{2}$ by Lemma \[lem-equalities\]; if we fix $a$ with ${{\mathrm{Tr}}}_{q/p}(a)=0$ and $\eta(a)=-1$, the number of $b$ such that ${{\mathrm{Tr}}}_{q/p}(b)=0$ and $ \bar{\eta} (-{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}))=-1$, is $\hat{n}_b=\frac{p-1}{2}(p^{m-2}+ (-1)^{\frac{m(p-1)(p-3)}{8}+m-1+\frac{q-1}{2}} \cdot \sqrt{p^*}^{m-1})$ by Lemmas \[lem-sign\] and \[2lem14\]. Hence, $$A_w=\bar{n}_a \bar{n}_b+\hat{n}_a \hat{n}_b= \frac{p^{m-1}-1}{2}\cdot (p-1)(p^{m-2}+ (-1)^{\frac{m(p-1)(p-3)}{8}+m-1+\frac{q-1}{2}} \cdot \sqrt{p^*}^{m-1}).$$ The frequencies of other nonzero weights can be similarly derived.
() The case that $m$ is even.
From Equation (\[eqn-6\]), we get $$\begin{aligned}
& & {{\mathtt{wt}}}({{\mathbf{c}}}(a,b))=q-N_0(a,b) \nonumber \\
& & =
\left\{\begin{array}{ll}
0 & \mbox{ if }(a,b)=(0,0),{{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0,\\ [2mm]
p^{m-1}(p-1) & \mbox{ if }a=0,\ b\neq 0, {{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0,\\ [2mm]
p^{m-1}(p-1)-(p-1)p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)+4}{4}} & \mbox{ if }a\neq 0,\ 0={{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}),\ \eta(a)=1,\\
~ & ~{{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0, \\ [2mm]
p^{m-1}(p-1)-(p-1)p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)}{4}} & \mbox{ if }a\neq 0,\ 0={{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}),\ \eta(a)=-1,\\
~ & ~{{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0, \\ [2mm]
p^{m-1}(p-1)-p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)}{4}} & \mbox{ if }a\neq 0,\ 0\neq{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}),\ \eta(a)=1,\\
~ & ~{{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0, \\ [2mm]
p^{m-1}(p-1)-p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)+4}{4}} & \mbox{ if }a\neq 0,\ 0\neq{{\mathrm{Tr}}}_{q/p}(\frac{b^2}{4a}),\ \eta(a)=-1,\\
~ & ~{{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0.
\end{array} \right. \\ [2mm]
& & =
\left\{\begin{array}{ll}
0 & \mbox{ with $ 1 $ time},\\
p^{m-1}(p-1) & \mbox{ with $ p^{m-1}-1 $ time},\\
p^{m-1}(p-1)+(p-1)p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)}{4}} & \mbox{ with $ B_1\cdot (p^{m-2}+B_2) $ time}, \\
p^{m-1}(p-1)-(p-1)p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)}{4}} & \mbox{ with $ (p^{m-1}-1-B_1)\cdot (p^{m-2}-B_2) $ time}, \\
p^{m-1}(p-1)-p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)}{4}} & \mbox{ with $ B_1\cdot (p^{m-1}-p^{m-2}-B_2) $ time}, \\
p^{m-1}(p-1)+p^{\frac{m-2}{2}}(-1)^{\frac{m(p-1)}{4}} & \mbox{ with $ (p^{m-1}-1-B_1)\cdot (p^{m-1}-p^{m-2}+B_2) $ time}, \\
\end{array} \right.\end{aligned}$$ where $B_1=\frac{p^{m-1}-1-(p-1)p^{\frac{m-2}{2}}(\sqrt{-1})^{\frac{(p-1)m}{2}}}{2}$, $B_2= (-1)^{\frac{m(p-1)(p-3)}{8}+m-1+\frac{q-1}{2}} \bar{\eta}^{m/2}(-1) \cdot (p-1)p^{(m-2)/2}$, and the frequency is easy to obtain based on Lemmas \[lem-equalities\], \[lem-sign\], \[2lem11\] and \[2lem14\].
By the above two cases, the weight distributions in Tables \[tab-51\] and \[tab-52\] follow. This completes the proof.
\[exa-51\] Let $m=3$ and $p=3$. Then the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-51\] is a $[24,4,15]$ linear code with the weight enumerator $1+48z^{15}+32z^{18}$. This code ${{\mathcal{C}}}_{T}$ is optimal. Its dual ${{\mathcal{C}}}_{T}^\perp$ has parameters $[24,20,3]$ and is optimal according to the tables of best known codes maintained at http://www.codetables.de.
\[exa-53\] Let $m=4$ and $p=3$. Then the shortened code ${{\mathcal{C}}}_{T}$ in Theorem \[main-51\] is a $[78,6,48]$ linear code with the weight enumerator $1+240z^{48}+240z^{51}+26z^{54}+192 z^{57}+30 z^{60}$. This code ${{\mathcal{C}}}_{T}$ is almost optimal. Its dual ${{\mathcal{C}}}_{T}^\perp$ has parameters $[78,72,2]$ and is almost optimal according to the tables of best known codes maintained at http://www.codetables.de.
\[main-52\] Let $p$ be an odd prime, $m$ and $e$ be positive integers such that $m/\gcd(m,e)$ is odd. Let $q=p^m$, $f(x)=x^{p^e+1}$ and ${{\mathcal{C}}}$ be defined in (\[eq:cf\]). Let $T={{\mathrm{GF}}}(p)$. Then the parameters of the shortened code ${{\mathcal{C}}}_{T}$ are the same as that of ${{\mathcal{C}}}_{T}$ in Theorem \[main-51\].
The proof is similar to that of Theorem \[main-51\]. Recall that the code ${{\mathcal{C}}}$ has length $q$ and dimension $2m+1$. By Lemma \[lem:C-S-P\], the shortened code ${{\mathcal{C}}}_{T}$ has length $2^m-p$ and dimension $2m-2$. Since $T={{\mathrm{GF}}}(p)$, the weight distribution of ${{\mathcal{C}}}_{T}$ is the same as the code $$\begin{aligned}
\label{eq:cf2}
\hat{{{\mathcal{C}}}}=\left\{\left({{\mathrm{Tr}}}_{q/p}(ax^{p^e+1}+bx)\right)_{x \in {{\mathrm{GF}}}(q)}:a,b\in {{\mathrm{GF}}}(q)~and~{{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0 \right\}.\end{aligned}$$
For each $(a,b) \in {{\mathrm{GF}}}(q) \times {{\mathrm{GF}}}(q) $ with ${{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0$, define the corresponding codeword $$\begin{aligned}
{{\mathbf{c}}}(a,b)=\left(({{\mathrm{Tr}}}_{q/p}(ax^{p^e+1}+bx)_{x\in {{\mathrm{GF}}}(q)}\right) \in \hat{{{\mathcal{C}}}}.\end{aligned}$$ Then the Hamming weight of ${{\mathbf{c}}}(a,b)$ is $$\begin{aligned}
\label{eq-word}
{{\mathtt{wt}}}({{\mathbf{c}}}(a,b))=q-\hat{N}_0(a,b),\end{aligned}$$ where ${{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0$ and $\hat{N}_0(a,b)$ was defined in Lemma \[lem-NN0\].
We need prove the value of ${{\mathtt{wt}}}({{\mathbf{c}}}(a,b))$ and its frequencies in the following four cases.
- () $m$ is odd and $p\equiv 1~mod~4$.
- () $m$ is odd and $p\equiv 3~mod~4$.
- () $m$ is even and $p\equiv 1~mod~4$.
- () $m$ is even and $p\equiv 3~mod~4$.
Next we only give the proof for the case () and omit the proofs for the other there cases whose proofs are similar.
Suppose that $m$ is odd and $p\equiv 1~mod~4$. From Equation (\[eq-word\]) and Lemma \[lem-NN0\], we get $$\begin{aligned}
& & {{\mathtt{wt}}}({{\mathbf{c}}}(a,b))=q-\hat{N}_0(a,b) \nonumber \\
& & =
\left\{\begin{array}{ll}
0 & \mbox{ if }(a,b)=(0,0), {{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0,\\
p^{m-1}(p-1) & \mbox{ if }a=0,\ b\neq 0,{{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0, \\
~ & \mbox{ or }a\neq 0,\ b=0, {{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0,\\
~ & \mbox{ or }a\neq 0,\ b \neq 0, {{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1})= 0, {{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0,\\
p^{m-1}(p-1)-p^{m/2-1} \sqrt{p^*} & \mbox{ if }a\neq 0, b\neq 0, {{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1})\neq 0, {{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0,\\
~~ & ~ \eta(a)\eta( {{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1}) )=1. \\
p^{m-1}(p-1)+p^{m/2-1} \sqrt{p^*} & \mbox{ if }a\neq 0, b\neq 0, {{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1})\neq 0, {{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0,\\
~~ & ~\eta(a)\eta( {{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1}) )=-1.
\end{array} \right. \\ [2mm]
& & = \left\{\begin{array}{ll}
0 & \mbox{ with $1$ time},\\
p^{m-1}(p-1) & \mbox{ with $(p^{m-1}-1)(p^{m-2}+1)$ time},\\
p^{m-1}(p-1)-p^{\frac{m-1}{2}} & \mbox{ with $ \frac{p^{m-1}-1}{2}\cdot (p-1)(p^{m-2}+B)$ time},\\
p^{m-1}(p-1)+p^{\frac{m-1}{2}} & \mbox{ with $\frac{p^{m-1}-1}{2}\cdot (p-1)(p^{m-2}-B)$ time}, \\
\end{array} \right.\end{aligned}$$ when $(a,b)$ runs through ${{\mathrm{GF}}}(q) \times {{\mathrm{GF}}}(q)$, where $B=(-1)^{\frac{m(p-1)(p-3)}{8}+m-1+\frac{q-1}{2}} \cdot \sqrt{p^*}^{m-1}$ and the frequency is obtained by Lemmas \[lem-psign\] and \[lem-equalities\]. As an example, we just compute the frequency $A_w$ of the nonzero weight $w$, where $$w=p^{m-1}(p-1)-p^{\frac{m-1}{2}}$$ and $$A_w=\sharp \{(a,b):a\neq 0, b\neq 0, {{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1})\neq 0, {{\mathrm{Tr}}}_{q/p}(a)={{\mathrm{Tr}}}_{q/p}(b)=0, \eta(a)\eta( {{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1}) )=1 \}.$$ Clearly, the number of $a \in {{\mathrm{GF}}}(q)^*$ such that ${{\mathrm{Tr}}}_{q/p}(a)=0$ and $\eta(a)=1$, is $\bar{n}_a=\frac{p^{m-1}-1}{2}$ by Lemma \[lem-equalities\]; if we fix $a$ with ${{\mathrm{Tr}}}_{q/p}(a)=0$ and $\eta(a)=1$, the number of $b$ such that ${{\mathrm{Tr}}}_{q/p}(b)=0$ and $\eta( {{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1}) )=1$ , is $\bar{n}_b=\frac{p-1}{2}\left( p^{m-2}+(-1)^{\frac{m(p-1)(p-3)}{8}+m-1+\frac{q-1}{2}}\cdot \sqrt{p*}^{m-1} \right)$ by Lemma \[lem-psign\]. Meanwhile, the number of $a \in {{\mathrm{GF}}}(q)^*$ such that ${{\mathrm{Tr}}}_{q/p}(a)=0$ and $\eta(a)=-1$, is $\hat{n}_a=\frac{p^{m-1}-1}{2}$ by Lemma \[lem-equalities\]; if we fix $a$ with ${{\mathrm{Tr}}}_{q/p}(a)=0$ and $\eta(a)=-1$, the number of $b$ such that ${{\mathrm{Tr}}}_{q/p}(b)=0$ with $b \neq 0$ and $\eta( {{\mathrm{Tr}}}_{q/p}(a(x_{a,b})^{p^e+1}) )=-1$ , is $\frac{p-1}{2}\left( p^{m-2}+(-1)^{\frac{m(p-1)(p-3)}{8}+m-1+\frac{q-1}{2}} \sqrt{p*}^{m-1} \right)$ by Lemma \[lem-psign\]. Hence, $$A_w=\bar{n}_a \bar{n}_b+\hat{n}_a \hat{n}_b= \frac{p^{m-1}-1}{2}\cdot (p-1)(p^{m-2}+ (-1)^{\frac{m(p-1)(p-3)}{8}+m-1+\frac{q-1}{2}} \cdot \sqrt{p^*}^{m-1}).$$ The frequencies of other nonzero weights can be similarly derived. This completes the proof of the weight distribution of Table in \[tab-51\] for the case $m$ odd and $p\equiv 1~mod~4$.
The proofs of the other three cases are similar. The desired conclusions follow from Equation (\[eq-word\]), Lemmas \[lem-NN0\], \[lem-equalities\] and \[lem-psign\]. This completes the proof.
Concluding remarks {#sec-summary}
==================
In this paper, we mainly investigated some shortened codes of linear codes from PN and APN functions and determined their parameters. The obtained codes have a few weights and some of these codes are optimal or almost optimal. Specifically, the main results are summarized as follows:
- For any binary linear code ${{\mathcal{C}}}$ with length $q=2^m$ and the weight distribution in Table \[tab-cf\], we gave a general result on their shortened code ${{\mathcal{C}}}_{T}$ with $\#T=4$ in Theorem \[main-31\]. Meanwhile, when $m$ is odd, the parameters of the shortened code ${{\mathcal{C}}}_{T}$ of a class of binary linear codes from APN functions was determined in Theorem \[main-32\].
- For any binary linear code ${{\mathcal{C}}}$ with length $q=2^m$ and the weight distribution in Table \[tab-cf1\], we gave a general result on their shortened code ${{\mathcal{C}}}_{T}$ with $\#T=3$ in Theorem \[main-even32\]. Further, the parameters of the shortened code ${{\mathcal{C}}}_{T}$ of a special class of linear codes from APN functions were determined for $\#T=3$ and $\#T=4$ in Theorems \[main-even33\] and \[main-even41\].
- Two classes of $p$-ary shortened code ${{\mathcal{C}}}_{T}$ from PN functions were presented and their parameters were also determined in Theorems \[main-51\] and \[main-52\], where $p$ is an odd prime.
Some of these shortened codes in this paper are optimal or almost optimal. Many codes with good parameters can be produced by using shortening and puncturing technologies. It seems hard to determine their weight distributions. The reader is cordially invited to settle these problems.
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[^1]: C. Xiang is with the College of Mathematics and Informatics, South China Agricultural University, Guangzhou, Guangdong 510642, China (email:cxiangcxiang@hotmail.com).
[^2]: C. Tang is with School of Mathematics and Information, China West Normal University, Nanchong, Sichuan 637002, China, and also with the Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China (email: tangchunmingmath@163.com).
[^3]: C. Ding is with the Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China (email: cding@ust.hk).
[^4]: The research of C. Xiang was supported by the National Natural Science Foundation of China under grant number 11701187. The research of C. Tang was supported by National Natural Science Foundation of China under grant number 11871058 and China West Normal University (14E013, CXTD2014-4 and the Meritocracy Research Funds). The research of C. Ding was supported by the Hong Kong Research Grants Council, Project No. 16301020.
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abstract: 'We analyze the impact of a proposed tidal instability coupling $p$-modes and $g$-modes within neutron stars on GW170817. This non-resonant instability transfers energy from the orbit of the binary to internal modes of the stars, accelerating the gravitational-wave driven inspiral. We model the impact of this instability on the phasing of the gravitational wave signal using three parameters per star: an overall amplitude, a saturation frequency, and a spectral index. Incorporating these additional parameters, we compute the Bayes Factor ([$\ln B^{pg}_{!pg}$]{}) comparing our [$p\text{-}g$]{} model to a standard one. We find that the observed signal is consistent with waveform models that neglect [$p\text{-}g$]{} effects, with ${\ensuremath{\ln B^{pg}_{!pg}}}= \lnBAbstract$. By injecting simulated signals that do not include [$p\text{-}g$]{} effects and recovering them with the [$p\text{-}g$]{} model, we show that there is a $\simeq \FAPlogU$ probability of obtaining similar [$\ln B^{pg}_{!pg}$]{} even when [$p\text{-}g$]{} effects are absent. We find that the [$p\text{-}g$]{} amplitude for 1.4 $M_\odot$ neutron stars is constrained to $\lesssim {{\textcolor{black}{\ensuremath{\mathrm{few}\times10^{-7}}}}}$, with maxima a posteriori near $\sim {{\textcolor{black}{\ensuremath{10^{-7}}}}}$ and [$p\text{-}g$]{} saturation frequency $\sim {{\textcolor{black}{\ensuremath{70\, \mathrm{Hz}}}}}$. This suggests that there are less than a few hundred excited modes, assuming they all saturate by wave breaking. For comparison, theoretical upper bounds suggest a [$p\text{-}g$]{} amplitude $\lesssim 10^{-6}$ and $\lesssim 10^{3}$ modes saturating by wave breaking. Thus, the measured constraints only rule out extreme values of the [$p\text{-}g$]{} parameters. They also imply that the instability dissipates $\lesssim {{\textcolor{black}{\ensuremath{10^{51}\, \mathrm{ergs}}}}}$ over the entire inspiral, i.e., less than ${{\textcolor{black}{\ensuremath{\text{a few percent}}}}}$ of the energy radiated as gravitational waves.'
bibliography:
- '../refs.bib'
title: ' Constraining the $p$-mode–$g$-mode tidal instability with GW170817 '
---
Introduction {#section:introduction}
============
Detailed analysis of the gravitational-wave (GW) signal received from the first binary neutron star (NS) coalescence event (GW170817 [@GW170817]) constrains the tidal deformability of NSs and thus the equation of state (EOS) above nuclear saturation density [@GW170817SourceProperties; @EOS; @De2018]. Studies of NS tidal deformation typically focus on the linear, quasi-static tidal bulge induced in each NS by its companion. Such deformations modify the system’s binding energy and GW luminosity and thereby alter its orbital dynamics. The degree of deformation is often expressed in terms of the tidal deformability $\Lambda_i \propto (R_i/m_i)^5$ of each component [@Flanagan2008], or a particular mass-weighted average thereof ($\tilde{\Lambda}$) [@GW170817SourceProperties]. The strong dependence on compactness $R/m$ means that a stiffer EOS, which has larger $R$ for the same $m$, imprints a larger tidal signals than a softer EOS. Current analyses of GW data from the LIGO [@LIGO] and Virgo [@Virgo] detectors favor a soft EOS [@EOS; @GW170817GRB]. Specifically, [@GW170817SourceProperties] finds $\tilde{\Lambda} \lesssim {{\textcolor{black}{\ensuremath{730}}}}$ at the 90% credible level for all waveform models considered, allowing for the components to spin rapidly. The pressure at twice nuclear saturation density is also constrained to $P = {{\textcolor{black}{\ensuremath{3.5^{+2.7}_{-1.7}\times10^{34}\ \mathrm{dyn}/\mathrm{cm}^2}}}}$ (median and 90% credible region) [@EOS] assuming small component spins. In addition to GW phasing, the EOS-dependence of $\tilde{\Lambda}$ should correlate with post-merger signals [@gw170817postmerger], possible tidal disruptions, and kilonova observations [@gw170817kilonova]. Observed light-curves for the kilonova suggest a lower bound of $ \tilde{\Lambda}\gtrsim {{\textcolor{black}{\ensuremath{200}}}}$ [@Radice2018; @Coughlin2018].
Although some dynamical tidal effects are incorporated in these analyses (see, e.g., [@Hinderer2016; @GW170817SourceProperties]), the impact of several types of dynamical tidal effects are neglected because they are assumed to be small or have large theoretical uncertainties. These effects arise because tidal fields, in addition to raising a quasi-static bulge, excite stellar normal modes. Three such excitation mechanisms are (i) resonant linear excitation, (ii) resonant nonlinear excitation, and (iii) non-resonant nonlinear excitation (see, e.g., [@Andersson2018]). The first occurs when the GW frequency (the oscillation frequency of the tidal field) sweeps through a mode’s natural frequency (see, e.g., [@Lai1994; @Reisenegger:94; @Ho1999; @Lai:06; @Flanagan:07; @Yu2017a; @Yu2017b; @Xu2017]). However, since the GW frequency increases rapidly during the late inspiral, the time spent near resonance is too short to excite modes to large amplitudes. As a result, for modes with natural frequencies within the sensitive bands of ground-based GW detectors, the change in orbital phasing is expected to be small ($\Delta \Psi \lesssim 10^{-2}\textrm{ rad}$) unless the stars are rapidly rotating [@Ho1999; @Lai:06; @Flanagan:07]. The impact of resonant nonlinear mode excitation (i.e., the parametric subharmonic instability) is likewise limited by the swiftness of the inspiral [@Weinberg2013].
The proposed [$p\text{-}g$]{} tidal instability is a non-resonant, nonlinear instability in which the tidal bulge excites a low-frequency buoyancy-supported $g$-mode and a high-frequency pressure-supported $p$-mode [@Weinberg2013; @Venumadhav2014; @Weinberg2016; @Zhou2017]. It occurs in the inner core of the NS, where the stratification is weak and the shear due to the tidal bulge is especially susceptible to instability. Unlike resonantly excited modes, an unstable [$p\text{-}g$]{} pair continuously drains energy from the orbit once excited, even after the orbital frequency changes significantly. There are many potentially unstable [$p\text{-}g$]{} pairs, each becoming unstable at a different frequency and growing at a different rate. Although there is considerable uncertainty about the number of unstable pairs, their exact growth rates, and how they saturate, estimates suggest that the impact could be measurable with current detectors [@Essick2016].
In this letter, we investigate the possible impact of the [$p\text{-}g$]{} instability on GW170817 using the phenomenological model developed in [@Essick2016]. The model describes the energy dissipated by the instability within each NS, indexed by $i$, in terms of three parameters: (i) an overall amplitude $A_i$, which is related to the number of modes participating in the instability, their growth rates, and their saturation energies, (ii) a frequency $f_i$ corresponding to when the instability saturates, and (iii) a spectral index $n_i$ describing how the saturation energy evolves with frequency. In Section \[section:phenomenologlical model\], we describe our models in detail. In Section \[section:model selection\], we compare the statistical evidence for models that include the [$p\text{-}g$]{} instability relative to those that do not. In Section \[section:parameter exclusion\], we investigate the constraints on the [$p\text{-}g$]{} parameters from GW170817, and in Section \[section:discussion\] we conclude.
Phenomenological Model {#section:phenomenologlical model}
======================
Following [@Essick2016], we extend a post-Newtonian (PN) waveform by including a parametrized model of the [$p\text{-}g$]{} instability. For the initial PN model, we use the `TaylorF2` frequency-domain approximant (see, e.g., [@Buonanno2009]) terminated at the inner-most stable circular orbit, which includes the effects of linear tides ($\tilde{\Lambda}$) and spins aligned with the orbital angular momentum (the impact of mis-aligned spins on [$p\text{-}g$]{} effects is not known). Waveform systematics between different existing approximants may be important for small [$p\text{-}g$]{} effects. However, by comparing the waveform mismatches between several other models (`TaylorF2`, `SEOBNRT`, `PhenomDNRT`, and `PhenomPNRT`, see [@GW170817SourceProperties]), we find these systematic become important for [$p\text{-}g$]{} effects roughly an order of magnitude smaller than the upper limits set by our analysis (see Section \[section:parameter exclusion\]). We expect `TaylorF2` to be reasonably accurate and defer a complete analysis of waveform systematics to future work.
Assuming the [$p\text{-}g$]{} effects are a perturbation to `TaylorF2`, we find that they modify the phase in the frequency-domain by
$$\Delta \Psi(f) = - \frac{2C_1}{3B^2(3-n_1)(4-n_1)}\left[ \Theta_1 \left(\frac{f}{f_{\rm ref}}\right)^{n_1-3} + (1-\Theta_1) \left(\frac{f_1}{f_{\rm ref}}\right)^{n_1-3}\left( \left(4-n_1\right) - \left(3-n_1\right)\left(\frac{f}{f_1}\right)\right)\right] + (1 \leftrightarrow 2),
\label{eq:dPsi}$$
where $f_i$ is the saturation frequency, $f_\mathrm{ref} \equiv 100\, \mathrm{Hz}$ is a reference frequency with no intrinsic significance, $C_i = [2m_i/(m_1+m_2)]^{2/3} A_i$, $B = (32/5)(G\mathcal{M}\pi f_\mathrm{ref}/c^3)^{5/3}$, $\mathcal{M}=(m_1 m_2)^{3/5}/(m_1+m_2)^{1/5}$, and $\Theta_i = \Theta(f-f_i)$ where $\Theta$ is the Heaviside function. This approximant is slightly different than that of [@Essick2016] because they incorrectly applied the saddle-point approximation to obtain the frequency-domain waveform from time-domain phasing [@Cutler1994]. This correction renders the [$p\text{-}g$]{} instability slightly more difficult to measure than predicted in [@Essick2016], although the observed behavior is qualitatively similar. Specifically, we find that in order to achieve the same $|\Delta \Psi|$, $A_i$ needs to be larger than [@Essick2016] found by a factor of $\sim (4-n_i)$, although the precise factor also depends on the other [$p\text{-}g$]{} parameters.
The $\Delta \Psi$ expression contains three types of terms: a constant term, a linear term $\propto (1-\Theta_i)f$, and a power-law term $\propto\Theta_i f^{n_i-3}$. The constant term corresponds to an overall phase offset and is degenerate with the orbital phase at coalescence. The linear term corresponds to a change in the time of coalescence; because the [$p\text{-}g$]{} instability transfers energy from the orbit to stellar normal modes, the binary inspirals faster than it would if the effect was absent. The power-law term accounts for the competition between the rate of [$p\text{-}g$]{} energy dissipation and the rate of inspiral, both of which increase as $f$ increases. As argued in [@Essick2016], we expect $n_i<3$, which implies that the phase shift accumulates primarily at frequencies just above the “turn-on" (saturation) frequency $f\gtrsim f_i$.
When $n_i < 3$, [$p\text{-}g$]{} effects are most important at lower frequencies whereas linear tides ($\tilde{\Lambda}$) and spins ($\chi_i = c S_i/G m_i^2$, where $S_i$ is the spin-angular momentum of each component) have their largest impact at higher frequencies (see, e.g., [@Agathos2014]). The priors placed on the latter quantities can, however, affect our inference of [$p\text{-}g$]{} parameters.
In order to account for a possible dependence on the component masses ($m_i$), we parametrize our model using a Taylor expansion in the [$p\text{-}g$]{} parameters around $m_i=1.4M_\odot$ and sample from the posterior using the first two coefficients. Our model computes $A_i$ as $$A_i(m_i) = A_0 + \left(\left.\frac{dA}{dm}\right|_{1.4M_\odot}\right)\left(m_i - 1.4M_\odot\right),$$ and uses $A_0$ and $dA/dm$ instead of $A_1$ and $A_2$. The model uses similar representations for $f_i$ and $n_i$ in terms of the parameters $f_0$, $df/dm$, $n_0$, and $dn/dm$. We assume a uniform prior on $\log_{10} A_0$ within $10^{-10} \leq A_0 \leq 10^{-5.5}$, a uniform prior in $f_0$ within $10\,\mathrm{Hz} \leq f_0 \leq 100\,\mathrm{Hz}$, and a uniform prior in $n_0$ within $-1 \leq n_0 \leq 3$. The priors on the first-order terms ($dA/dm, df/dm, dn/dm$) are the same as those in [@Essick2016]; when $m_1 \sim m_2$, they imply $A_1 \sim A_2$, etc.
We investigate GW170817 using data from several different frequency bands and with different spin priors, but unless otherwise noted we focus on results for data above 30 Hz with $|\chi_i|\leq 0.89$. Throughout this letter, results from GW170817 were obtained using the same data conditioning as [@GW170817SourceProperties], including the removal of a short-duration noise artifact from the Livingston data ([@glitch_mitigation] and discussion in [@GW170817]) along with other independently measured noise sources (see, e.g., [@P1700260; @iLIGO_SeisCleaning; @Meadors2014; @Tiwari2015]), calibration [@Cahillane2017; @Viets2018], marginalization over calibration uncertainties, and whitening procedures [@Cornish2015; @Littenberg2015].
Model Selection {#section:model selection}
===============
Using GW data from GW170817, we perform Bayesian model selection. We compare a model that includes linear tides, spin components alinged with the orbital angular momentum, and PN phasing effects up to 3.5 PN phase terms ([$\mathcal{H}_{!pg}$]{}) to an extension of this model that also includes [$p\text{-}g$]{} effects ([$\mathcal{H}_{pg}$]{}). Since we have nested models ([$\mathcal{H}_{!pg}$]{} is obtained from [$\mathcal{H}_{pg}$]{} as $A_i\rightarrow0$)[^1], we use the Savage-Dickey Density Ratio (see, e.g., [@Dickey1970; @Verdinelli1995; @Wagenmakers2010]) to estimate the Bayes Factor ($B^{pg}_{!pg}=p(D|{\ensuremath{\mathcal{H}_{pg}}})/p(D|{\ensuremath{\mathcal{H}_{!pg}}})$, where $D$ refers to the observed data). Specifically, we sample from the model’s posterior distribution [@LALInference] and calculate
$$\begin{aligned}
\label{equation:sddr derivation}
\lim\limits_{A_i\rightarrow0}\left[\frac{p(A_i|D, {\ensuremath{\mathcal{H}_{pg}}})}{p(A_i|{\ensuremath{\mathcal{H}_{pg}}})}\right] & = \lim\limits_{A_i\rightarrow 0}\left[\frac{1}{p(D|{\ensuremath{\mathcal{H}_{pg}}})} \int d \theta df_i \, dn_i \, p(D|\theta, A_i, f_i, n_i; {\ensuremath{\mathcal{H}_{pg}}}) \, p(\theta|{\ensuremath{\mathcal{H}_{pg}}}) \, p(f_i, n_i|A_i, {\ensuremath{\mathcal{H}_{pg}}})\right] \nonumber \\
& = \frac{1}{p(D|{\ensuremath{\mathcal{H}_{pg}}})} \int d \theta \, p(D|\theta; {\ensuremath{\mathcal{H}_{!pg}}}) \, p(\theta|{\ensuremath{\mathcal{H}_{!pg}}}) \left[\frac{p(\theta|{\ensuremath{\mathcal{H}_{pg}}})}{p(\theta|{\ensuremath{\mathcal{H}_{!pg}}})}\right] \int df_i \, dn_i \, p(f_i, n_i|A_i, {\ensuremath{\mathcal{H}_{pg}}}) \nonumber \\
& = \frac{p(D|{\ensuremath{\mathcal{H}_{!pg}}})}{p(D|{\ensuremath{\mathcal{H}_{pg}}})} \left< \frac{p(\theta|{\ensuremath{\mathcal{H}_{pg}}})}{p(\theta|{\ensuremath{\mathcal{H}_{!pg}}})} \right>_{p(\theta|D, {\ensuremath{\mathcal{H}_{!pg}}})},\end{aligned}$$
where $\theta$ refers to all parameters besides the [$p\text{-}g$]{} phenomenological parameters, we note that $\int df dn\, p(f_i,n_i|A_i, {\ensuremath{\mathcal{H}_{pg}}})=1\, \forall\, A_i$, and $\left<x\right>_p$ denotes the average of $x$ with respect to the measure defined by $p$. Assuming that $p(\theta|{\ensuremath{\mathcal{H}_{pg}}})=p(\theta|{\ensuremath{\mathcal{H}_{!pg}}})$, we determine [$\ln B^{pg}_{!pg}$]{} from the ratio, as $A_i \rightarrow 0$, of the marginal distribution of $A_i$ *a priori* to the distribution *a posteriori* $$\label{equation:sddr}
{\ensuremath{\ln B^{pg}_{!pg}}}= \lim\limits_{A_i\rightarrow 0} \left[\ln p(A_i|{\ensuremath{\mathcal{H}_{pg}}}) - \ln p(A_i|D, {\ensuremath{\mathcal{H}_{pg}}})\right].$$ We confirmed that this estimate agrees with estimates from both nested sampling [@skilling2006] and thermodynamic integration [@Nicolas2006].
![ Distributions of [$\ln B^{pg}_{!pg}$]{} due to sampling uncertainty for different values of $f_\mathrm{low}$. The solid red curves assume high-spin priors ($|\chi_i|\leq0.89$) and the dashed blue curves assume low-spin priors ($|\chi_i|\leq0.05$). []{data-label="figure:lnB"}](prl-lnB-logU.pdf){width="\columnwidth"}
Figure \[figure:lnB\] shows ${\ensuremath{\ln B^{pg}_{!pg}}}$ as a function of $f_\mathrm{low}$, the minimum GW frequency considered. At a given $f_{\rm low}$, we show the distribution of ${\ensuremath{\ln B^{pg}_{!pg}}}$ due to the sampling uncertainty from the finite length of our MCMC chains. The solid and dashed curves correspond to the high-spin ($|\chi_i| \leq 0.89$) and low-spin ($|\chi_i|\leq0.05$) priors discussed in [@GW170817; @GW170817SourceProperties; @EOS].
For certain combinations of $f_\mathrm{low}$ and $|\chi_i|$, we find ${\ensuremath{\ln B^{pg}_{!pg}}}>0$, suggesting [$\mathcal{H}_{pg}$]{} is more likely than [$\mathcal{H}_{!pg}$]{}. In order to assess how likely such values are, we calculate [$\ln B^{pg}_{!pg}$]{} for a large number of simulated, high-spin signals with $A_i=0$ and distinct realizations of detector noise from times near GW170817. We find that simulated signals without [$p\text{-}g$]{} effects can readily produce ${\ensuremath{\ln B^{pg}_{!pg}}}$ at least as large as the ones we measured from GW170817. In particular, [$\ln B^{pg}_{!pg}$]{} for the 30 Hz, high-spin data corresponds to a False Alarm Probability (FAP) $\approx \FAPlogU$. We focus on the 30 Hz, high-spin data because it corresponds to the largest bandwidth investigated and the largest signal-to-noise ratio. The high-spin prior is the most inclusive prior considered, and therefore allows the most model freedom when fitting [$p\text{-}g$]{} effects.
In our model of the instability, the phase shift $\Delta \Psi$ accumulates primarily at frequencies just above the saturation frequency $f\gtrsim f_i$. Therefore, if it is present, its impact should become more apparent as we decrease the minimum GW frequency considered from $f_\mathrm{low} \gg f_i$ to $f_{\rm low}\lesssim f_i$. We do see some indication of this behavior in Fig. \[figure:lnB\]. However, we note that if our phenomenological model breaks down at $f<f_i$ due to poor modeling of the pre-saturation behavior (e.g., if our step-function turn-on at $f_i$ is not a good approximation to the instability’s induced phase shift), we might expect [$\ln B^{pg}_{!pg}$]{} to decrease as we lower $f_\mathrm{low}$ below $f_i$.
Parameter Inference {#section:parameter exclusion}
===================
{width="\textwidth"}\
{width="\textwidth"}\
We now investigate the constraints obtained from GW170817. Figure \[figure:corner\] shows the joint posterior distributions for both [$\mathcal{H}_{pg}$]{} and [$\mathcal{H}_{!pg}$]{}. We find that [$\mathcal{H}_{pg}$]{} and [$\mathcal{H}_{!pg}$]{} yield similar posterior distributions for all non-[$p\text{-}g$]{} parameters, including both extrinsic and intrinsic parameters. The constraints on the chirp mass ($\mathcal{M}$), effective spin $\chi_\mathrm{eff} = (m_1\chi_1 + m_2\chi_2)/(m_1+m_2)$, and $\tilde{\Lambda}$ are slightly weaker in [$\mathcal{H}_{pg}$]{} than [$\mathcal{H}_{!pg}$]{}. This is because [$\mathcal{H}_{pg}$]{} provides extra freedom to the signal’s duration in the time-domain.
Regarding the [$p\text{-}g$]{} parameters, we find a noticeable peak near $A_0\sim10^{-7}$ with a flat tail to small $A_0$. We find $A_0 \leq {{\textcolor{black}{\ensuremath{3.3\times10^{-7}}}}}$ assuming a uniform-in-$\log_{10} A_0$ prior and $A_0 \leq {{\textcolor{black}{\ensuremath{6.8\times10^{-7}}}}}$ assuming a uniform-in-$A_0$ prior, both at 90% confidence.[^2] We also find a peak at $f_0\sim{{\textcolor{black}{\ensuremath{70\, \mathrm{Hz}}}}}$. The peaks persist when we analyze the data from each interferometer separately, with reasonably consistent locations and shapes (Fig. \[figure:corner\]). However, we find that the simulated signals with $A_i=0$ can produce similar peaks, suggesting they may be due to noise alone. Similar to [@Essick2016], we find that $n_i$ is not strongly constrained and the gradient terms in the Taylor expansions are not measurable.
Theoretical arguments suggest an upper bound of $A_0 \lesssim 10^{-6}$ [@Essick2016]. Therefore, our $A_0$ constraint only rules out the most extreme values of the [$p\text{-}g$]{} parameters.
Discussion {#section:discussion}
==========
While GW170817 is consistent with models that neglect [$p\text{-}g$]{} effects, it is also consistent with a broad range of [$p\text{-}g$]{} parameters. The constraints from GW170817 imply that there are $\lesssim{{\textcolor{black}{200}}}$ excited modes at $f=100\,\mathrm{Hz}$, assuming all modes grow as rapidly as possible and saturate at their breaking amplitudes ($\lambda=\beta=1$ in Eq. (7) of [@Essick2016]) and that the frequency at which modes become unstable is well approximated by $f_0$. For comparison, theoretical arguments suggest an upper bound of $\sim 10^{3}$ modes saturating by wave breaking [@Essick2016]. More modes may be excited if they grow more slowly or saturate below their wave breaking energy.
We can also use the measured constraints to place upper limits on the amount of energy dissipated by the [$p\text{-}g$]{} instability. As Fig. \[figure:energy\] shows, [$p\text{-}g$]{} effects dissipate $\lesssim {{\textcolor{black}{\ensuremath{2.7\times10^{51}\, \mathrm{ergs}}}}}$ throughout the entire inspiral at 90% confidence. In comparison, GWs carry away $\gtrsim {{\textcolor{black}{\ensuremath{10^{53}\, \mathrm{ergs}}}}}$. This implies time-domain phase shifts $|\Delta \phi| \lesssim {{\textcolor{black}{\ensuremath{7.6\, \mathrm{rad}}}}}$ ( orbits) at $100\textrm{ Hz}$ and $|\Delta \phi| \lesssim {{\textcolor{black}{\ensuremath{32\, \mathrm{rad}}}}}$ ( orbits) at $1000\textrm{ Hz}$ after accounting for the joint uncertainty in component masses, spins, linear tides, and [$p\text{-}g$]{} effects.
A $g$-mode with natural frequency $f_g$ is predicted to become unstable at a frequency $f_{\rm crit}\simeq 45\, \mathrm{Hz} (f_g/10^{-4}\lambda f_{\rm dyn})^{1/2}$, where $f_{\rm dyn}$ is the dynamical frequency of the NS and $\lambda$ is a slowly varying function typically between $0.1-1$ [@Weinberg2016; @Essick2016]. Since the modes grow quickly, the frequency at which the instability saturates is likely close to the frequency at which the modes become unstable ($f_0\simeq f_{\rm crit}$). If we assume that the observed peak near $f_0\sim{{\textcolor{black}{\ensuremath{70\, \mathrm{Hz}}}}}$ is not due to noise alone, then the maximum a posteriori estimate for $f_0$ along with approximate values for the masses (1.4 $M_\odot$) and radii (11 km) of the components [@EOS] imply $f_g \simeq {{\textcolor{black}{0.5\, \mathrm{Hz}}}}$.
![ Upper limits on the cumulative enegy dissipated by the [$p\text{-}g$]{} instability as a function of frequency. We note the relatively strong constraints at lower frequencies, where [$p\text{-}g$]{} effects are more pronounced. []{data-label="figure:energy"}](prl-intenergy-logU.pdf){width="\columnwidth"}
With several more signals comparable to GW170817, it should be possible to improve the amplitude constraint to $A_0 \lesssim 10^{-7}$. Obtaining even tighter constraints will likely require many more detections, especially since most events will have smaller SNR. Future measurements will also benefit from a better understanding of how the instability saturates. To date, there have only been detailed theoretical studies of the instability’s threshold and growth rate [@Weinberg2013; @Venumadhav2014; @Weinberg2016; @Zhou2017], not its saturation. As a result, we cannot be certain of the fidelity of our phenomenological model.
While this letter was in final internal review, related work was posted [@Reyes2018] with, in particular, the conclusion that the [$\mathcal{H}_{!pg}$]{} model is strongly favored over the [$\mathcal{H}_{pg}$]{} model by a factor of at least $10^4$. We are investigating possible reasons for the differences between our conclusions.
The authors gratefully acknowledge the support of the United States National Science Foundation (NSF) for the construction and operation of the LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS) and the Foundation for Fundamental Research on Matter supported by the Netherlands Organisation for Scientific Research, for the construction and operation of the Virgo detector and the creation and support of the EGO consortium. The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, the Department of Science and Technology, India, the Science & Engineering Research Board (SERB), India, the Ministry of Human Resource Development, India, the Spanish Agencia Estatal de Investigación, the Vicepresidència i Conselleria d’Innovació, Recerca i Turisme and the Conselleria d’Educació i Universitat del Govern de les Illes Balears, the Conselleria d’Educació, Investigació, Cultura i Esport de la Generalitat Valenciana, the National Science Centre of Poland, the Swiss National Science Foundation (SNSF), the Russian Foundation for Basic Research, the Russian Science Foundation, the European Commission, the European Regional Development Funds (ERDF), the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the Lyon Institute of Origins (LIO), the Paris Île-de-France Region, the National Research, Development and Innovation Office Hungary (NKFI), the National Research Foundation of Korea, Industry Canada and the Province of Ontario through the Ministry of Economic Development and Innovation, the Natural Science and Engineering Research Council Canada, the Canadian Institute for Advanced Research, the Brazilian Ministry of Science, Technology, Innovations, and Communications, the International Center for Theoretical Physics South American Institute for Fundamental Research (ICTP-SAIFR), the Research Grants Council of Hong Kong, the National Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research Corporation, the Ministry of Science and Technology (MOST), Taiwan and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, MPS, INFN, CNRS and the State of Niedersachsen/Germany for provision of computational resources. N. Weinberg was supported in part by NASA grant NNX14AB40G.
[^1]: Since we use a uniform-in-$\log_{10} A_0$ prior, [$\mathcal{H}_{pg}$]{} does not formally include $A_i=0$. Nonetheless, our lower limit on $A_i$ is sufficiently small that [$\mathcal{H}_{!pg}$]{} is effectively nested in [$\mathcal{H}_{pg}$]{}.
[^2]: The upper limit with a uniform-in-$A_0$ prior is larger only because we weight larger values of $A_0$ more *a priori* than with a uniform-in-$\log_{10} A_0$ prior.
|
---
abstract: 'Cold-start is a very common and still open problem in the Recommender Systems literature. Since cold start items do not have any interaction, collaborative algorithms are not applicable. One of the main strategies is to use pure or hybrid content-based approaches, which usually yield to lower recommendation quality than collaborative ones. Some techniques to optimize performance of this type of approaches have been studied in recent past. One of them is called feature weighting, which assigns to every feature a real value, called weight, that estimates its importance. Statistical techniques for feature weighting commonly used in Information Retrieval, like TF-IDF, have been adapted for Recommender Systems, but they often do not provide sufficient quality improvements. More recent approaches[@fbsm2015; @lfw] estimate weights by leveraging collaborative information via machine learning, in order to learn the importance of a feature based on other users opinions. This type of models have shown promising results compared to classic statistical analyzes cited previously. We propose a novel graph, feature-based machine learning model to face the cold-start item scenario, learning the relevance of features from probabilities of item-based collaborative filtering algorithms.'
author:
- Cesare Bernardis
- Maurizio Ferrari Dacrema
- Paolo Cremonesi
bibliography:
- 'bibliography.bib'
title: 'A novel graph-based model for hybrid recommendations in cold-start scenarios'
---
|
---
abstract: 'Studying exoplanets with their parent stars is crucial to understand their population, formation and history. We review some of the key questions regarding their evolution with particular emphasis on giant gaseous exoplanets orbiting close to solar-type stars. For masses above that of Saturn, transiting exoplanets have large radii indicative of the presence of a massive hydrogen-helium envelope. Theoretical models show that this envelope progressively cools and contracts with a rate of energy loss inversely proportional to the planetary age. The combined measurement of planetary mass, radius and a constraint on the (stellar) age enables a global determination of the amount of heavy elements present in the planet interior. The comparison with stellar metallicity shows a correlation between the two, indicating that accretion played a crucial role in the formation of planets. The dynamical evolution of exoplanets also depends on the properties of the central star. We show that the lack of massive giant planets and brown dwarfs in close orbit around G-dwarfs and their presence around F-dwarfs are probably tied to the different properties of dissipation in the stellar interiors. Both the evolution and the composition of stars and planets are intimately linked.'
address:
- 'Laboratoire Lagrange, UMR 7293, Université de Nice-Sophia Antipolis, CNRS, Observatoire de la Côte d’Azur, 06304 Nice Cedex 04, France'
- 'UCO/Lick Observatory, University of California, Santa Cruz, CA 95064, USA'
author:
- Tristan Guillot
- 'Douglas N.C. Lin'
- Pierre Morel$^1$
- Mathieu Havel$^1$
- Vivien Parmentier$^1$
bibliography:
- 'roscoff\_guillot+2014.bib'
title: Evolution of exoplanets and their parent stars
---
Introduction
============
The discovery of 51Pegb [@Mayor+Queloz1995] heralded the birth of a new field: exoplanetology. The study of exoplanets is important in itself, but they also offer us a new window to study their parent stars, the interactions between stars and planets and the processes that led to their formation.
This paper stems from a presentation given in October 2013 in Roscoff, France, at the school “The ages of stars”. We present a few key elements based on the authors’ work to understand the physical and dynamical evolution of exoplanets and their parent stars. This is by no means a proper review of this extremely rich field.
We first examine the physical mechanisms that govern the evolution of fluid planets, brown dwarfs and stars. In Section 3, we show that the global composition of exoplanets may be derived from evolution models and linked to the composition of their parent star. In Section 4, we discuss how star-planet interactions shape the population of close-in exoplanets and brown dwarfs, leading some to be swallowed by their star.
Thermal evolution of exoplanets
===============================
The wide range of exoplanets discovered include objects from a fraction of an Earth-mass to giant planets many times the mass of our Jupiter. It also includes objects orbiting only a few stellar radii away from their parent stars to objects at orbital distances beyond 100AU. We focus on objects for which both a mass and a radius may be determined, corresponding to the ones discovered transiting in front of their parent star. We also choose to restrict ourselves to giant planets and brown dwarfs, i.e. fluid objects mostly made of hydrogen and helium. Because of their large masses (more than about 100 times the mass of the Earth), these objects inherit a large amount of internal energy from their gravitational contraction. Because of a non-negligible thermal expansion coefficient, the progressive loss of this initial energy results in both a cooling and a contraction of the planets. It can indeed be shown that the energy loss per unit time (i.e. the [*intrinsic luminosity*]{}) follows a modified Kelvin-Helmholtz relation: $$L\approx \eta {GM^2\over R\tau},
\label{eq:lapprox}$$ where $\tau$ is the age, and $\eta$ is a factor that hides most of the complex physics. In the approximation that Coulomb and exchange terms can be neglected, $\eta\approx\theta/(\theta +1)$ where $\theta$ is the electron degeneracy factor. The poor compressibility of giant planets in their mature evolution stages imply that $\eta\ll 1$ ($\eta\sim 0.03$ for Jupiter): the luminosity is not obtained from the entire gravitational potential, but from the much more limited reservoir constituted by the thermal internal energy [@Guillot2005]. The evolution of giant planets and brown dwarfs is thus akin to the pre-main sequence evolution of stars [see also @Burrows+97; @Chabrier+Baraffe2000].
Giant planets thus gradually cool and contract but with a timescale that depends also on the amount of irradiation from their parent star [e.g. @Hubbard1977]. Giant planets and brown dwarfs are thought to be mostly convective but planets very close to their star (such as 51 Peg b) develop an outer radiative zone that progressively extends to the deeper levels [@Guillot+96; @GS02]. This radiative zone and the atmosphere to which it is linked [see @Parmentier+Guillot2014] govern the rate of cooling and contraction.
Another factor affecting the sizes of exoplanets is whether they contain more or less heavy elements[^1]: Everything else being equal, planets with a higher fraction of heavy elements tend to be smaller.
![Theoretical and observed mass-radius relations. The black line is applicable to the evolution of solar composition planets, brown dwarfs and stars, when isolated or nearly isolated (as Jupiter, Saturn, Uranus and Neptune, defined by diamonds and their respective symbols), after 5 Ga of evolution. The dotted line shows the effect of a $15\mea$ core on the mass-radius relation. Orange and yellow curves represent the mass-radius relations for heavily irradiated planets with equilibrium temperatures of 1000 and 2000K, respectively, and assuming that 0.5% of the incoming stellar luminosity is dissipated at the center. For each irradiation level, two cases are considered: a solar-composition planet with no core (top curve), and one with a $100\mea$ central core (bottom curve). Circles with error bars correspond to known planets, brown dwarfs and low-mass stars, color-coded as a function of their equilibrium temperature(below 750, 1500, 2250K and above 2250K, respectively, from darkest to lightest). \[From [@Guillot+Gautier2014]\][]{data-label="fig:exo_M_R"}](figs/geophys_guillot_f8-eps-converted-to.pdf){width="\hsize"}
Figure \[fig:exo\_M\_R\] compares a mass-radius diagram of known exoplanets, brown dwarfs and low-mass stars against some theoretical relations. After about 5 Ga of evolution, the mass-radius curve for non-irradiated hydrogen-helium planets has a maximum for a mass of $4\,\rm M_{Jup}$ and a radius slightly above that of Jupiter. For planets of larger masses, degeneracy effects in the equation of state begin to dominate, thermal effects become less important and cannot oppose as efficiently the increased compression: the radius decreases with mass until in the stellar domain nuclear reactions set in. However, for heavily-irradiated hydrogen-helium planets (i.e. for “hot Jupiters”), the radius tends to decrease with mass: for smaller masses the warm atmosphere is less bounded to the planet because of the weaker gravity [see @Guillot2005]. The presence of a massive core removes that effect. The variety of irradiation level and compositions is responsible for an ensemble of known giant exoplanet that is “trumpet-shaped” in that diagram.
A link between stellar and exoplanetary compositions
====================================================
The possibility to measure planetary masses, radii, irradiation levels and to constrain the age through the stellar age thus offers the possibility to determine the global compositions of giant exoplanets. However, a problem remains: a significant fraction of hot Jupiters is found to have radii above that theoretically predicted for hydrogen-helium planets [@Bodenheimer+2001; @GS02]. A number of explanations have been proposed [e.g. @Laughlin+11]. For energetic considerations, the most plausible class of models appear to be those which invoke the dissipation of a small fraction (of order 1% or less) of the stellar flux relatively deep in the planetary interior as a mean to slow the planetary cooling and contraction [@GS02; @Laine+2008; @BS10; @ArrasSocrates10]. A definitive test of these models would be an accurate measurement of stellar ages coupled to the determination of precise planetary masses and radii for a statistically significant number of objects.
Under the assumption that this class of models prevails, it is relatively straightforward to parametrize this ’missing physics’ through the dissipation of a fixed fraction of the irradiation energy into the planetary interior. This fraction may be chosen so that model radii for hydrogen-helium planets are always (or almost always) above the observational constraint. The global amount of heavy elements present in the planet may then be inferred from evolution models [@Guillot+06].
![Mass fraction of heavy elements in the planets as a function of the metallicity of their parent star. [*Top panel*]{}: Result for hot Jupiters with masses smaller than $2\,\rm M_{Jup}$. The evolution model assumes that 0.5% of the incoming irradiation flux is dissipated at the planet’s center. Circles which are labeled 1 to 23 correspond to the CoRoT giant planets. Gray symbols correspond to a subset of known transiting systems [@Guillot08; @Laughlin+11]. Unphysical negative values for $M_Z$ correspond to insufficient heat sources leading to a radius that is larger than observed. \[From [@Moutou+2013].\] [*Bottom panel*]{}: Same plot for weakly irradiated planets. \[From [@MF11].\] In both panels, a curve shows a linear relation between $\log M_Z$ and \[Fe/H\] fitted to the results for weakly irradiated planets.[]{data-label="fig:correlation"}](figs/FeH_vs_Mz.pdf){width="10cm"}
Figure \[fig:correlation\] shows the result of such an exercice when applied to hot Jupiters with known masses and radii. In that plot, the mass of heavy elements is plotted against the metallicity of the parent star. For some objects, the chosen (small) value of the dissipation parameter imply an unphysical negative mass of heavy elements. This is for example the case of the still unexplained large radius of CoRoT-2b [see @Guillot+Havel2011]. However, two robust results are the requirement of very large masses in heavy elements (above $100\,\rm M_\oplus$) for some objects, and the correlation between the mass of heavy elements inferred in the planets and that in the parent star [@Guillot+06; @Burrows+07; @Guillot08; @Laughlin+11; @Moutou+2013]. Interestingly, these two results are also observed when one uses a subset of the ensemble of planets that is weakly irradiated and does not require any extra heat source to explain their radius [@MF11].
The large amount of heavy elements shows that solids were efficiently stored and delivered into the planets. This must have occurred in the circumstellar disk at the time of the formation of the parent star and its planets. Furthermore, the correlation between planet and star ’metallicity’ shows that the storage/accretion/delivery of solids was strongly favored by an increased metallicity. This favor scenarios in which giant planets are formed by the accretion of a central core followed by the capture of a hydrogen-helium envelope which may be polluted in variable amounts by dust and planetesimals.
Dynamical interactions between stars and close-in exoplanets
============================================================
Another link between the properties of stars and of their close-in exoplanets is a direct consequence of tidal interactions between them. Figure \[fig:exo\_M\_Teff\] shows the masses of known exoplanets as a function of the effective temperature of their parent star. The symbol sizes are inversely proportional to the planets’ orbital periods which effectively helps focusing on short period planets. Obvious observational biases are indicated in the figure: stars with low effective temperature are faint which makes it difficult to observe planets around them with either radial velocimetry or photometry in the visible. Similarly, at high effective temperatures two effects combine against the discovery of small planets: the fast rotation of F-dwarfs makes the spectral lines broader which reduces the sensitivity of radial velocity surveys and the stars become larger which also acts against the discovery of transiting planets by photometry. A paucity of close-in companions is obvious for masses above about $5\,\rm M_{Jup}$ and around stars with effective temperatures between about 5000 and 6000 K. This cannot be explained by an observational bias: if present, massive companions around G-dwarfs would be at least as easily detected as massive companions to F-dwarfs.
![Mass (times the sine of the orbital the inclination) of known exoplanets and brown dwarfs as a function of effective temperature of the parent stars. Some of the key systems are labelled. The size of the symbols is inversely proportional to their orbital period. (Large symbols correspond to important tidal interactions between the star and the companion.) \[Figure adapted from [@Bouchy+2011].\][]{data-label="fig:exo_M_Teff"}](figs/exo_M_Teff.png){width="\hsize"}
Given the small different in mass between F- and G-dwarfs, the different planet populations must be due to a different dynamical evolution caused by different dissipation regimes. This may be shown by defining the inward migration timescale [see @Barker+Ogilvie2009]: $$\begin{split}
\tau_{\rm mig}=12\,{\rm Ma}&\ \left(Q_*\over 10^6\right)\left(\mp\over 1\,\mjup\right)^{-1}\left(M_*\over 1\,\msol\right)^{8/3}\left(R_*\over 1\,\rsol\right)^{-5}\\
&\times\left(P_{\rm orb}\over 1{\,\rm day}\right)^{13/3}\left(1-{P_{\rm orb}\over P_{\rm spin}}\right)^{-1}
\end{split}
\label{eq:tau_mig}$$ where $M_*$ and $R_*$ are the stellar mass and radius, respectively, $\mp$ is the companion’s mass, $P_{\rm orb}$ its orbital period and $P_{\rm spin}$ the stellar spin period. $Q'_*\equiv Q_*/k_2$ is the equivalent stellar tidal dissipation factor defined as the ratio between the tidal dissipation factor and the second Love number. The value of $\tau_{\rm mig}$ for known brown dwarfs and exoplanetary companions for an assumed fixed $Q'_*=10^6$ ranges from $10^6$yrs and more for G-dwarfs and only about $10^5$yrs and more for F-dwarfs (Guillot, Lin & Morel, in preparation).
We interpret this difference as arising from the engulfment of massive planets and brown dwarfs around G-dwarfs and their preservation around F-dwarfs. This has two reasons: (i) F-dwarfs are known to be rapid rotators which have a weaker magnetic braking due to their small outer convective zone. They are hence less efficient at extracting angular momentum. On the other hand in systems in which a G-dwarf is spun-up by a close-in companion, stellar winds and magnetic fields yield a rapid loss of angular momentum from the system and a fast runaway migration of the companion onto the central star. (ii) G-dwarfs have a radiative center while F-dwarfs have a central convective zone. This implies that gravity waves propagating in the inner radiative zone may break and dissipate their energy in G-dwarfs but not in F-dwarfs [see @Barker+Ogilvie2010].
Figure \[fig:tides\] shows the result of a dynamical model that includes tidal interactions between stars and their companion [@Barker+Ogilvie2009], stellar evolution [@Morel+Lebreton2008], the magnetic braking of stars [@Bouvier+1997], and a consistent calculation of tidal dissipation ($Q'_*$) by gravity waves [@Barker+Ogilvie2010]. The latter is calculated consistently by including the evolution of the stellar interior which enter the calculation of $Q'_*$ [see also @Barker2011]. The figure shows for stars of 0.8 to 1.4$\,\rm M_\odot$ and companions of 0.2 to 200$\rm M_{Jup}$ on an initial 3-day orbital period the fraction of the star’s main-sequence lifetime on which the companion is able to survive.
![Contour plot showing the ratio of the lifetime age to the main sequence lifetime of a companion of mass $M_{\rm pla}$ orbiting around a star of mass $M_{\rm star}$ initially on a 3 day orbit and assuming \[Fe/H\]=0.0. The model used is the full internal gravity wave model \[see text\]. The points correspond to the observed low-metallicity population, with $-0.15 \le\rm [Fe/H] < 0.15$. \[Figure from Guillot, Lin & Morel (in preparation).\][]{data-label="fig:tides"}](figs/mpla_vs_mstar_tides.png){width="\hsize"}
We find that massive companions around G-dwarfs generate tides that break at the star center, leading to a strong dissipation, their inward migration and eventual demise. The process is much less efficient for small-mass companions essentially because the waves that they exert have a too small amplitude to break except at late ages when the star stars leaving the main sequence. For massive companions, the inward migration is slow because of the large initial angular momentum. Around F-dwarfs, very little migration occurs both because internal gravity waves cannot reach the center and because of their weak magnetic braking.
Massive close-in planets and brown dwarfs are therefore engulfed preferentially around G-dwarfs in qualitative agreement with the observations. Actually, while this goes in the right direction, it may be argued that the observations show an even stronger deficit of massive companions around G-dwarfs than found by the models. This clearly requires further work.
Conclusion
==========
A proper understanding of the properties, formation and evolution of planetary systems requires combining observations and theoretical studies of both the planets and their parent stars. This clearly demonstrated by the two examples that we discussed: (i) the determination of the global composition of giant exoplanets which appears to be directly linked to the metallicity of the parent star; (ii) the fate of close-in massive exoplanets and brown dwarfs which depends both on the planetary and stellar properties and the magnitude of their tidal interactions. This is of course not restricted to these two examples. The prospect of discovering thousands of transiting planets and measuring at the same time both the stellar and planetary properties very accurately thanks to a number of ground-based and space-based projects (including CHEOPS, TESS, PLATO) is extremely promising.
[^1]: Here, anything other than hydrogen and helium is dubbed “heavy element”.
|
---
author:
- 'Sofiane M. Boucenna'
- 'Florian K[ü]{}hnel'
- Tommy Ohlsson
- Luca Visinelli
date: ', '
title: |
Novel Constraints on Mixed Dark-Matter Scenarios of\
Primordial Black Holes and WIMPs
---
Introduction {#sec:Introduction}
============
Despite the considerable evidence in favor for the existence of non-baryonic dark matter (DM) in the Universe, the nature of the DM continues to remain unknown and its identification is a major challenge in modern physics. To date, all experimental efforts have yielded null or inconclusive results. From the point of view of particle physics, many theories and paradigms have been proposed to model the DM. A stable weakly interacting massive particle (WIMP) is among the best motivated candidates for describing the presence of DM. WIMPs emerge in various extensions of the Standard Model (SM), which aim to solve other theoretical issues. The natural appearance of WIMPs in such models provides a non-empirical support in their favor.
For masses around the GeV-TeV scale, the annihilation cross-section of WIMPs into lighter particles, computed from a thermal production scenario, is typically of the order of what is obtained from the mediation of the weak interaction. Thus, the relic abundance of WIMPs obtained naturally accounts for the present abundance of the DM. Indeed, after the WIMPs have *chemically* decoupled from the primordial plasma at the freeze-out temperature, their abundance is approximately \_[WIMP]{}h\^[2]{} 0.1 , \[eq:omega\] where $\svz_{\rm f.o.}$ is the thermal average of the velocity and the annihilation cross-section of the DM into lighter particles at freeze-out and $\Omega_{\rm WIMP} \equiv \rW / \rho_{\rm c}$ is a density parameter with $\rW$ being the energy density of WIMPs and $\rho_{\rm c}$ the critical energy density of the Universe.[^1] Although the number of WIMPs is fixed at the moment of chemical decoupling, an efficient kinetic equilibrium is still maintained for a certain time through the exchange of momentum between WIMPs and lighter particles. Eventually, the *kinetic* decoupling of WIMPs occurs at a certain temperature $\Tk$ and the WIMP velocity distribution is fixed.
The non-trivial interplay between the known weak-scale physics and DM phenomenology allows for concrete model-building possibilities, relating WIMPs to various aspects of physics lying beyond the SM. This leads to links with, e.g., supersymmetry [@Jungman:1995df], universal extra-dimensions [@Hooper:2007qk], and various baryogenesis [@Boucenna:2013wba; @Cui:2015eba] and neutrino mass models [@Boucenna:2014zba; @Restrepo:2013aga]. Perhaps more importantly, the WIMP paradigm is testable, since it provides many direct and indirect detection prospects through recoil off nuclei and annihilation into detectable SM particles. The ever-growing level of sensitivity reached by WIMP detection experiments offers a great hope that WIMPs will be unambiguously discovered in the near future.
One of the main search strategies for WIMPs is the indirect detection of the products of their annihilation in and beyond our Galaxy. The WIMP annihilation rate is proportional to the square of their number density, implying that denser regions offer much higher detection prospects. Such regions include, e.g., the Galactic center and nearby dwarf spheroidal galaxies. Clearly, any physical mechanism, which increases the density of DM particles, is advantageous from the point of view of indirect detection. A compelling example of such physics is offered by primordial black holes (PBHs), namely black holes that have been produced in the very early Universe.
PBHs have received considerable attention, since they were first postulated [@1967SvA....10..602Z; @Carr:1974nx]. The interest in them constituting (parts of) the DM [@1975Natur.253..251C] has recently been revived [@Jedamzik:1996mr; @Niemeyer:1997mt; @Jedamzik:2000ap; @Frampton:2010sw; @Capela:2012jz; @Griest:2013aaa; @Belotsky:2014kca; @Young:2015kda; @Frampton:2015xza; @Bird:2016dcv; @Kawasaki:2016pql; @Carr:2016drx; @Kashlinsky:2016sdv; @Clesse:2016vqa; @Green:2016xgy; @Kuhnel:2017pwq; @Akrami:2016vrq; @Nakama:2017xvq; @Deng:2017uwc; @Bernal:2017vvn; @Green:2017qoa; @Kuhnel:2017bvu; @Kannike:2017bxn; @Kuhnel:2017ofn; @Guo:2017njn; @Nieuwenhuizen:2017pto] through the discovery of black-hole binary mergers [@Abbott:2016blz; @Abbott:2016nmj]. The possible PBH formation mechanisms are very diverse and there is a plethora of scenarios that lead to their formation. All of these have in common that they require a mechanism to generate large overdensities. Often these overdensities are of inflationary origin [@Hodges:1990bf; @Carr:1993aq; @Ivanov:1994pa]. When re-entering the cosmological horizon, these overdensities collapse if they are larger than a certain medium- and shape-dependent threshold. Here, the case of radiation domination is the most often considered in the literature. Other scenarios for PBH formation exist, such as those where the source of the inhomogeneities are first-order phase transitions [@Jedamzik:1996mr], Higgs fluctuations during inflation [@Burda:2016mou; @Espinosa:2017sgp], bubble collisions [@Crawford:1982yz; @Hawking:1982ga; @Deng:2017uwc], collapse of cosmic strings [@Hogan:1984zb; @Hawking:1987bn], necklaces [@Matsuda:2005ez] or domain walls [@Berezin:1982ur]. We refer to App. \[sec:Primordial-Black–Hole-Formation\] for more details. In some regions of the parameter space, the energy density of PBHs $\rho_{\rm BH}$ can describe the observed DM abundance.
In this paper, we are not assuming any specific formation mechanism of PBHs, since we only require that PBHs are formed prior to the kinetic decoupling of WIMPs from the primordial plasma. We assume a scenario in which the DM is mixed of PBHs and WIMPs, i.e., the energy density of DM is given by $\rDM = \rW + \rho_{\rm BH}$. Thus, the fraction of PBHs $f$ is defined as f , \[eq:Fraction\] so that the corresponding fraction of WIMPs is $\rW = (1 - f )\.\rDM$. Either PBHs or WIMPs could, in principle, account for the total DM abundance in the Universe. However, even though their existence is well motivated for different reasons, there is no evidence that one of them must account for the total DM abundance. Indeed, a mixed DM scenario, consisting of both PBHs and WIMPs, seems more likely. Such a scenario allows the early-formed PBHs to accrete WIMPs around them and seed the formation of a compact dark clump [@Dokuchaev:2002ts; @Ricotti:2009bs; @Mack:2006gz; @Lacki:2010zf; @Saito:2010ts; @Eroshenko:2016yve]. This would enhance the annihilation of WIMPs in bound orbits around PBHs even for values of $f$ that are many orders of magnitude below unity.
Assuming a certain simplified density profile of the DM in the central region of the clump, various groups studied the signal produced by DM annihilation in these overdensities [@Lacki:2010zf; @Saito:2010ts; @Eroshenko:2016yve; @Zhang:2010cj; @Sandick:2010qu; @Sandick:2010yd; @Sandick:2011zs]. Here, following Ref. [@Eroshenko:2016yve], we aim to derive the DM density profile from first principles. Furthermore, our goal is to derive reliable bounds on the parameter space of the mixed PBH-WIMP scenario, using as little assumptions as possible.
The rest of this paper is organized as follows. In Sec. \[sec:Dark–Matter-Density-Spikes-from-Primordial-Black-Holes\], we describe the generation of DM density spikes in the presence of PBHs. Next, in Sec. \[sec:Annihilation-Signal-from-Dark–Matter-Density-Spikes\], we investigate the associated enhancement of the annihilation signal. Then, in Sec. \[sec:Results\], we present, discuss, and summarize our results on constraints on the PBH DM fraction. In Sec. \[sec:Conclusions\], we provide our conclusions. Finally, in the appendices, we give details on the PBH formation mechanism of PBHs (App. \[sec:Primordial-Black–Hole-Formation\]), the kinetic decoupling (App. \[sec:Kinetic-Decoupling\]), and a derivation of the WIMP density (App. \[sec:Derivation-of-the-WIMP-density\]).
Dark-Matter Density Spikes from Primordial Black Holes {#sec:Dark--Matter-Density-Spikes-from-Primordial-Black-Holes}
======================================================
Once WIMPs have kinetically decoupled from the primordial plasma, they are gravitationally bound to PBHs and form density spikes, where WIMP annihilation might be boosted even to present days. Comparing the expected annihilation signal with data from the [Fermi]{} telescope, we can constrain the PBH-WIMP parameter space. In this section, we revise and extend the derivation of the density of WIMPs around a PBH, following Ref. [@Eroshenko:2016yve]. We assume the presence of a sufficient amount of WIMPs.
In a radiation-dominated Universe, the Hubble rate depends on the cosmic time $t$ and the temperature of the plasma $T$ as H(T) = = = , \[eq:H(T)\] where $\rho$ is the energy density of radiation and we introduce the quantity \[eq:Alpha\] in terms of the effective number of relativistic degrees of freedom $g_{*}(T)$. For practical purposes, we set $g_{*}(\Tk) = 61.75$. At temperature $\Tk$, the scattering of WIMPs off radiation becomes inefficient in exchanging momentum and WIMPs kinetically decouple from the plasma. From the moment of decoupling, the WIMP momentum $p$ decreases with the scale factor $a(t)$ according to $p \propto 1/a(t)$, which leads to a WIMP temperature: T\_[WIMP]{} = , . \[eq:TWIMP\] Note that the behavior of $T_{\rm WIMP}$ differs from the evolution of $T$, which scales as $a^{-2}(t)$. We refer to App. \[sec:Kinetic-Decoupling\] for additional details on the derivation of $\Tk$.
Suppose that, at time $t_{i}$ prior the kinetic decoupling $\tk$, a PBH of mass $\MB$ forms in a radiation-dominated Universe with the energy density $\rho$. Using Eq. , we have $\rho = 3 / ( 32 \pi\.G\.t^{2} )$. In order for particles to be gravitationally affected by the PBH, the energy density within a sphere of radius $\ri( \tk )$ must equal $\rho$: = . \[eq:rho\] This yields ( ) = = r\_[g]{} ( )\^[2 / 3]{}, \[eq:ri(t)\] with the Schwarzschild radius $r_{\rm g} \equiv 2\.G\MB$. For $r > \ri( \tk )$, the kinematic properties of WIMPs are the same as what is obtained for the standard radiation-cosmological history of the Universe, while for $r < \ri( \tk )$, the gravitational attraction of the PBH governs the WIMPs orbiting.
Now, we focus on a WIMP at position $r_{i}$ and with velocity $\bv$ when the PBH forms at time $t_{i}$. If $\tau_{\rm orb}$ is the period of the WIMP’s orbital motion around the PBH, it would spend only a fraction $ 2\,\d t/\tau_{\rm orb}$ at distances between $r$ and $r+\d r$,[^2] where $\d t$ is the time it takes for the WIMP to move from $r$ to $r+\d r$. Then, at any later time $t>t_{i}$, we have the mass relation \_[bound]{}( r ) 4 r\^2r = 4 r\^\_[i]{}r\_[i]{}\^[2]{}\^\_[i]{}( r\^\_[i]{} ) \^[3]{}f\_( ) , \[eq:DensityR\] which implies that the density of WIMPs in bound orbits around the PBH is given by \_[bound]{}( r ) = r\^\_[i]{}r\_[i]{}\^[2]{}\^\_[i]{}( r\^\_[i]{} ) \^[3]{}f\_( ) , \[eq:DensityR\] where the WIMP velocity distribution $f_{\Brm}( \bv )$, the radial velocity $\d r / \d t$, and the orbital period $\tau_{\rm orb}$ are given in App. \[sec:Derivation-of-the-WIMP-density\] in Eqs. , , and , respectively. Reference [@Eroshenko:2016yve] lacks details on the subtleties in the computation of Eq. , which deals with the conditions that the WIMP orbit is bound to the PBH. We derive these details in App. \[sec:Derivation-of-the-WIMP-density\].[^3] Accounting for the fact that the present DM density has decreased at least to the value [@Bertone:2005xz] \_[max]{} = , \[eq:MaxRho\] where $t_{0}$ is the age of the Universe, we estimate the present WIMP density profile around a PBH to be ( r ) = . \[eq:rho( r )\] Figure \[fig:PlotProfile\] shows the WIMP density profile bound to a PBH as a function of the rescaled radius $x \equiv r / r_{\rm g}$, obtained from Eq. with different values of $\MB$, namely $\MB = 10\.M_{\odot}$, $\MB = 10^{-2}\.M_{\odot}$, $\MB = 10^{-5}\.M_{\odot}$, and $\MB = 10^{-12}\.M_{\odot}$, where $M_{\odot}$ is the solar mass. The horizontal line represents the value of $\rho_{\rm max}$ in Eq. for $m_{\chi} = 100\,{\rm GeV}$ and $\svz = 3 \times 10^{-26}\,{\rm cm}^{3} / {\rm s}$, and thus, we find $\rho_{\rm max} \simeq 1.4 \times 10^{-14}\,{\rm g / cm}^{3}$.
The WIMP density profile for the smallest value of $\MB$ constitutes an envelope to the profiles for the other values and is defined as \_[0]{}( r ) \_[0]{} ( r ) . We have numerically confirmed that profiles with even smaller values of $\MB$ than $\MB = 10^{-12}\,M_\odot$ converge towards $\rho_{0}( r )$. Increasing the value of $\MB$ from $10^{-12}\.M_{\odot}$ to $10\.M_{\odot}$ and even further, the different profiles decouple from $\rho_{0}( r )$ and cross $\rho_{\rm max}$ at decreasing values of $r$, as is indicated in Fig. \[fig:PlotProfile\]. Furthermore, we denote by $\bar{x}$ the “critical” value of $x$ at which $\rho_{0}( \bar{x} ) = \rho_{\rm max}$. This corresponds to the “critical” PBH mass $\Mb$. Since the WIMP density profile for values below $\rho_{\rm max}$ rapidly goes to zero as a function of $x$, we model the DM density profile around the PBH as \_[max]{}( |[x]{} - x ) . \[eq:RhoApprox\] For PBHs with masses $\MB \lesssim \Mb$, the decoupling of the corresponding profiles from $\rho_{0}( r )$ occurs at values of the density below $\rho_{\rm max}$, so that these PBHs share the same constant value of $\bar{x}$, whereas for $\MB \gtrsim \Mb$, $\bar{x}$ is proportional to a power of $\MB$.
Annihilation Signal from Dark-Matter Density Spikes {#sec:Annihilation-Signal-from-Dark--Matter-Density-Spikes}
===================================================
In Sec. \[sec:Dark–Matter-Density-Spikes-from-Primordial-Black-Holes\], we have derived the expression for the WIMP density profile around PBHs. Now, we proceed to calculate the expected signal from these overdense regions. The number of WIMP annihilations in the vicinity of a PBH per unit time is given by = \^[3]{} rn\^[2]{}( r ) = r\_[g]{}\^[3]{}xx\^[2]{}\^[2]{}( x ) , \[eq:Gamma\] where again $x \equiv r / r_{\rm g}$. Based on what we displayed in Fig. \[fig:PlotProfile\] and on the parametrization of Eq. , the decay rate in Eq. is $$\begin{aligned}
\GB
&\simeq
\frac{4\pi\.\svz}{m_{\chi}^{2}}\,r_{\rm g}^{3}\,
\int_{0}^{\bar{x}} \d x\;x^{2}\rho_{\rm max}^{2}
\notag
\\
&=
\frac{4\pi\.\svz}{3 m_{\chi}^{2}}\,\(2G\MB\)^{3}\,
\bar{x}^{3} \rho_{\rm max}^{2}
\; .
\label{eq:GammaApprox}\end{aligned}$$ Since $\bar{x}$ is constant for $\MB \lesssim \Mb$, we obtain $\GB \propto \MB^{3}$, while the dependence of $\bar{x}$ on $\MB$ for $\MB \gtrsim \Mb$ yields milder relations.
In Fig. \[fig:PlotGamma\], we show $\GB$ as a function of $\MB$ for different values of $m_{\chi}$, which are chosen as $m_{\chi} = 10\,{\rm GeV}$, $m_{\chi} = 100\,{\rm GeV}$, $m_{\chi} = 1\,{\rm TeV}$, and $m_{\chi} = 10\,{\rm TeV}$. Nevertheless, for small values of $\MB$ (according to Fig. \[fig:PlotGamma\]), we have $\GB \propto \MB^{3}$. In general, for the different regimes of $\MB$, the complete power law behavior of $\GB$ is manifest in Fig. \[fig:PlotGamma\]. The subscript ‘BH’ reminds us that $\GB$ depends on $\MB$.
The byproducts of WIMPs annihilating around the PBHs contribute to the isotropic flux of gamma rays coming from our Galaxy (“gal”) and the extragalactic (“ex”) components. In fact, the spectrum of gamma rays contains various components like individual sources, the diffuse Galactic emission, and the residual isotropic diffuse gamma-ray background, which comprises the unresolved extragalactic emissions as well as the residual Galactic foregrounds. Therefore, the total expected differential flux of gamma rays is given by [@Cirelli:2012ut] = 4.|\_[gal]{} + |\_[ex]{} . \[eq:dPhi/dE/dOmega\] The Galactic component is not isotropic, and hence, the dependence on the differential solid angle $\d\Omega$. However, its minimum value with respect to the direction contributes an irreducible background to the isotropic flux. In the following, we assume that the energy density in the PBHs tracks the DM density profile in both the Galactic and the extragalactic environments as in Eq. , i.e., $\rho_{\rm BH} = f\.\rDM$.
For a PBH of mass $\MB$ and for a specific annihilation channel, the expected Galactic component of the gamma-ray flux per solid angle is [@Chen:2009uq; @Cirelli:2009dv; @Cirelli:2012ut] |\_[gal]{} = \_[l.o.s.]{}s\_[H]{}(r) , \[eq:DiffFluxGalactic\] where the integral is taken along the line of sight (l.o.s.) $s$, $\d N_{\gamma} / \d E$ is the number of photons $N_\gamma$ produced by the WIMP annihilation channel considered per unit energy $E$, $\rho_{\rm H}(r)$ is the DM distribution in the halo, and we account for the fact that WIMPs contribute a fraction $1 - f$ of the total dark-matter energy density. The coordinate $r$ appearing in $\rho_{\rm H}(r)$ is defined as r = r(s, ) , where $r_{\odot} = 8.33\,{\rm kpc}$ is the distance from the solar system to the Galactic center, $s$ parametrizes the distance from the solar system along the l.o.s., and $\psi$ is the angle between the direction of observation in the sky and the Galactic center. Similarly, the extragalactic component is [@Chen:2009uq; @Cirelli:2009dv; @Cirelli:2012ut] |\_[ex]{} = \_[0]{}\^z. , \[eq:DiffFluxExtra\] where the optical depth $\tau_{\rm opt}(z)$ accounts for the attenuation of high-energy gamma rays due to their scattering with UV extragalactic photons as a function of the redshift $z$, and $H( z )$ is the Hubble rate as a function $z$.
Results on Constraints on the Primordial Black-Hole Dark-Matter Fraction {#sec:Results}
========================================================================
Equations and are analogous to what is obtained when considering (a single component) *decaying* DM of mass $m_{\rm DM}$ and decay rate $\Gamma_{\rm DM}$ if we perform the substitution = . \[eq:analogy\] Therefore, we can translate the experimental bounds on decaying DM to our parameter spaces.
First, we constrain the PBH physics in the parameter space given by $\MB$ and $f$. We use the results presented in Ref. [@Ando:2015qda], where the extragalactic gamma-ray background measured by [Fermi]{} is used to set limits on the DM decay rate $\Gamma_{\rm DM}$ for different decay channels and DM particle masses. Thus, we convert the bounds obtained in Ref. [@Ando:2015qda] to bounds on $f$, given $\GB$ in Eq. and the analogy in Eq. . To this end, we focus on the bounds obtained from the $b\bar{b}$ channel, corresponding to the results in Fig. 3.(f) of Ref. [@Ando:2015qda].
In Fig. \[fig:PlotBounds\], we present the bounds on $f$ as a function of $\MB$ obtained using the analysis on decaying dark matter in Ref. [@Ando:2015qda]. We have used $\svz = 3 \times 10^{-26}\,$cm$^{3}$/s and presented two results, corresponding to $m_{\chi} = 100\,{\rm GeV}$ (red dashed curve) and $m_{\chi} = 1\,{\rm TeV}$ (green dot-dashed curve), respectively. The gray-shaded areas stem from other techniques to constrain the PBH parameter space, see the caption of Fig. \[fig:PlotBounds\]. For $m_{\chi} = 100\,{\rm GeV}$, we find that the corresponding bound is, in some regions, several orders of magnitude stronger than *all* currently existing bounds for PBHs with masses $10^{-12}\,M_{\odot} \lesssim \MB \lesssim 10^{4}\,M_{\odot}$.
Second, we convert the bounds on decaying DM to bounds on $f$ in the WIMP parameter space spanned by $m_{\chi}$ and $\svz$ in order to be as model-independent as possible from the point of view of WIMPs. In this case, we fix the PBH parameters to some representative benchmark values, namely $\MB = 10^{-12}\,M_{\odot}$, $\MB = 10^{-5}\,M_{\odot}$, $\MB = 10^{-2}\,M_{\odot}$, and $\MB = 10\,M_{\odot}$.
In Fig. \[fig:PlotBounds-sigmav-mchi\], we display density plots for the PBH fraction $f$ as a function of $m_{\chi}$ and $\svz$ for the four above-mentioned values of $\MB$. The colored regions of these plots represent $f$ with the color scale indicating the value of $\log_{10}f$. White regions show areas in which the value of $f>1$ and are therefore excluded. The hatched regions mark the areas of the WIMP parameter space that are excluded by the search of gamma rays from DM annihilation in dwarf satellite galaxies coming from the combined analysis of the [Fermi]{} and [MAGIC]{} telescopes [@Ahnen:2016qkx] for the $b\bar{b}$ channel. This bound assumes that WIMPs account for the total DM of the Universe and is therefore only valid for $f \ll 1$, otherwise it should be properly rescaled for the considered value of $f$. We show it to illustrate the interplay between the WIMP indirect-detection bounds and $f$. Figure \[fig:PlotBounds-sigmav-mchi\] provides the maximal value of $f$ for each WIMP parameter pair $m_{\chi}$ and $\svz$.
Conclusions {#sec:Conclusions}
===========
In this work, we have derived constraints on mixed DM scenarios consisting of PBHs and WIMPs. Here, the PBHs efficiently accrete the WIMPs, leading to spikes in their DM density profile. We have precisely calculated this profile (see Fig. \[fig:PlotProfile\]) as well as the associated enhanced WIMP annihilation signal (see Fig. \[fig:PlotGamma\]). These are shown to potentially provide strong constraints on the allowed PBH DM fraction $f$. In particular, we have demonstrated that experimental knowledge on WIMP DM may be used to significantly constrain both the PBH and WIMP parameter spaces. Our results are summarized in Figs. \[fig:PlotBounds\] and \[fig:PlotBounds-sigmav-mchi\].
We should stress that the exclusion limits derived in this work are for the mixed scenario of PBH and WIMP DM. In particular, these constraints by no means restrict the abundance of PBHs for DM particles of much larger masses and/or smaller annihilation cross-sections.
We are grateful to M. Taoso for useful discussions on [Fermi]{} bounds. S.B. thanks the “Roland Gustafssons Stiftelse f[ö]{}r teoretisk fysik” for financial support. F.K. and L.V. acknowledge support by the Swedish Research Council (Vetenskapsrdet) through contract No. 638-2013-8993 and the Oskar Klein Centre for Cosmoparticle Physics. T.O. acknowledges support by the Swedish Research Council (Vetenskapsrdet) through contract No. 2017-03934 and the KTH Royal Institute of Technology for a sabbatical period at the University of Iceland.
Primordial Black-Hole Formation {#sec:Primordial-Black--Hole-Formation}
===============================
When re-entering the cosmological horizon, a density perturbation collapses to a PBH if it is larger than a certain medium- and shape-dependent threshold. Here, the best studied (and highly idealized) case is that of a collapse of spherical Gaussian overdensities within the epoch of radiation domination. The vast majority of the literature assumes rapid dynamics, leading to a black hole with a mass proportional to the horizon mass, and hence, a monochromatic mass spectrum. However, it can be shown that even if the initial density spectrum was monochromatic, the phenomenon of critical collapse [@Niemeyer:1997mt] will inevitably lead to a PBH mass spectrum which is spread out, shifted towards lower masses and lowered, leading to potentially large effects ( Ref. [@Kuhnel:2015vtw]). Under the assumption of spherical symmetry, it has been shown [@Choptuik:1992jv; @Koike:1995jm; @Niemeyer:1999ak; @Gundlach:1999cu; @Gundlach:2002sx] that the functional dependence of the PBH mass $\MB$ on the density contrast $\delta$ follows the critical scaling relation = kM\_[H]{}. ( - \_ )\^[\_]{} , \[eq:M-delta-scaling\] where $M_{\rm H}$ is the mass contained in a Hubble patch at a given time $t$, namely M\_[H]{} \~ 10\^[15]{} ( ) . \[eq:Moft\] In Eq. , the constant $k$, the threshold $\delta_{\crm}$, and the critical exponent $\gamma_{\crm}$ all depend on the nature of the fluid containing the overdensity $\delta$ at horizon-crossing [@Musco:2012au]. In radiation-dominated models, which are the focus of this paper, repeated studies have shown that $\gamma_{\crm} \simeq 0.36$ [@Koike:1995jm; @Niemeyer:1999ak; @Musco:2004ak; @Musco:2008hv; @Musco:2012au] and $\delta_{\rm c} \simeq 0.45$ [@Musco:2004ak; @Musco:2008hv; @Musco:2012au]. In accordance with Ref. [@Niemeyer:1997mt], we set $k = 3.3$. Precise numerical computations [@Musco:2004ak; @Musco:2008hv; @Musco:2012au] have confirmed the above scaling law, which has been shown to apply over more than ten orders of magnitude in density contrast [@Musco:2008hv]. Applying the Press–Schechter formalism [@1974ApJ...187..425P] for spherical collapse and assuming a Gaussian perturbation profile, one can express the ratio $\beta$ of the PBH energy density to the total energy density at the time of PBH *formation* as k.\^[2\_]{} [erfc]{} ( ) , \[eq:BetaNormalisation\] which holds for $\sigma \ll \delta_{\crm}$ with $\sigma$ being the variance of the primordial power spectrum of density perturbations generated by the model of inflation. From the above specified $\beta$, we can express the PBH DM fraction $f$ via f = 2.4.\^[eq]{} , \[eq:Fraction-2\] where $\Omega_{\rm DM}^{\rm eq} \approx 0.42$ and $\beta^{\rm eq}$ are the DM density and the PBH mass fraction at matter-radiation equality, respectively.[^4]
In the main part of this work, we derive the flux $\Phi_{\gamma} |_{\MB}$ of gamma rays coming from WIMP annihilation in halos around the PBHs assuming a certain, fixed $\MB$. However, for a given extended mass distribution $f( \MB )$, specifying the number of PBHs per unit mass, the final constraint can be easily derived from these results. This utilizes the following weighting: Let $I_{\rm bin} \in \Nbb$ denote the number of mass bins $\{ [ M_{{\rm BH}, i},\.M_{{\rm BH}, i + 1} ],\, 1 \leq i \leq I_{\rm bin}\}$. Then, we have \_ |\_[total]{} \_[i = 1]{}\^[I\_[bin]{}]{}.\_ |\_\^[(i)]{} \_[M\_[[BH]{}, i]{}]{}\^[M\_[[BH]{}, i + 1]{}]{} , \[eq:ConstraintNSCapture\] where $\Phi_{\gamma} \big|_{\MB}^{(i)}$ is evaluated for a mass within each bin $i$ and the bin number $I_{\rm bin}$ should be chosen such that, to the precision sought, it does not matter where exactly in each bin $i$ the quantity $\Phi_{\gamma} \big|_{\MB}^{(i)}$ is evaluated ( Ref. [@Kuhnel:2017pwq]).
Kinetic Decoupling {#sec:Kinetic-Decoupling}
==================
We consider the single-particle Boltzmann equation, neglecting other effects related to the medium and restricting to Boltzmann statistics. After the chemical decoupling has occurred at temperature $T_{\rm chem} \sim m_{\chi} / 20$, the relevant Boltzmann collision equation for the phase space density $f_{\Brm}$ reads [@Bernstein:1985; @Bernstein:1988] E ( - H\_[ p]{} ) f\_( ) = C\_[el]{}\[ f\_ \] . \[eq:BoltzmannF\] Here, the elastic collision rate is given by [@Bringmann:2006mu; @Bringmann:2009vf] C\_[el]{}\[ f\_ \] = 2Ė( T ) ( m\_T.\_[p]{}\^[2]{} + \_[p]{} + 3 ) f\_( ) , \[eq:Cel\[f\]\] where $\gamma(T)$ is the momentum relaxation rate. Introducing the number density and the kinetic temperature of $\chi$ as n\_ = f\_( ) , T\_ = p\^[2]{}f\_( ) , \[eq:NChiTChi\] respectively, the expression for $T_{\chi}$ is obtained by integrating Eq. by $\int\mspace{-2mu}\d^{3}\bp\;p^{2} / [ (2\pi)^{3} E ]$, obtaining \_ + 2HT\_ - ( T ) ( T - T\_ ) = 0 . \[eq:BoltzmannT\] The temperature of kinetic decoupling is defined as [@Bringmann:2006mu; @Bringmann:2009vf] = |\_[T.0]{} = ( )\^[ 1 / 4]{} , \[eq:Tk\] where $\Gamma(\cdot)$ is the gamma function. This expression of $\Tk$ coincides, up to a numerical factor, with other definitions available in the literature [@Visinelli:2015eka].
Derivation of the WIMP Density {#sec:Derivation-of-the-WIMP-density}
==============================
In deriving the WIMP density, our starting point is [@Eroshenko:2016yve] ( r ) = r\_[i]{} r\_[i]{}\^[2]{}\_[i]{}( r\_[i]{} ) \^[3]{} f\_( ) , \[eq:DensityRa\] which is exactly the same equation as Eq. . The specific expressions for $\tau_{\rm orb}$, $\d t / \d r$, and $f_{\Brm}( \bv )$ will be specified below in Eqs. , , and , respectively. The energy of a particle of mass $m_{\chi}$ at position $r_{i}$ and with velocity $v_{i}$ is given by E = .\^[2]{} - = - ( - \_[i]{}\^[2]{} ) , \[eq:E\] where in the last equality we have expressed the velocity in terms of $\beta_{i} \equiv v_{i} / c$ and the radius $r_{i}$ in terms of $r_{\rm g}$. We have then set $c = r_{\rm g} = 1$. Requiring that the orbit is bound gives $E \leq 0$ or $r_{i} < 1 / \beta_{i}^{2}$. The orbital period and the inverse radial velocity are \_[orb]{} = 2Ġ ( )\^[3 / 2]{} = ( - \_[i]{}\^[2]{} )\^[-3 / 2]{} \[eq:period\] and
$$\begin{aligned}
\frac{\d t}{\d r}
&=
\left[
\frac{2}{m_{\chi}}
\big[
E - U( r )
\big]
-
\left(
\frac{l}{m_{\chi} r}
\right)^{\mspace{-4mu} 2}\.
\right]^{-1 / 2}
\notag
\\
&=
\left[
\beta_{i}^{2} - \frac{1}{r_{i}} + \frac{1}{r}
-
\left(
\frac{r_{i}\.\beta_{i}}{r}
\right)^{\mspace{-4mu} 2}\!
\left(
1 - y^{2}
\right)
\right]^{-1 / 2}
\; ,
\label{eq:radialspeed}\end{aligned}$$
respectively. Above, $y = \cos\theta$, the angular momentum $l$ is given by $l = m_{\chi}\.r_{i}\.v_{i}\.\sin\theta$, and we have expressed all radial quantities in units of $r_{\rm g}$. We rewrite this expression as $$\begin{aligned}
\frac{\d t}{\d r}
&=
\frac{r}{r_{i}\.\beta_{i}}
\left[
\left(
\frac{r}{r_{i}\.\beta_{i}}
\right)^{\!2}\!
\left(
\beta_{i}^{2} - \frac{1}{r_{i}} + \frac{1}{r}
\right)
-
\left(
1 - y^{2}
\right)
\right]^{-1 / 2}
\notag
\\
&=
\frac{r}{r_{i}\.\beta_{i}}
\frac{1}{\sqrt{y^{2} - y_{m}^{2}\,}}
\; ,
\label{eq:radialspeed1}\end{aligned}$$
where y\_[m]{} . \[eq:ym\] The condition that the orbit is confined between the aphelion and the perihelion implies that $y^{2} > y_{m}^{2}$ [@Eroshenko:2016yve]. Inserting Eqs. and into Eq. and writing $\d^{3}\bv = 2\pi\.\beta_{i}^{2}\.\d \beta_{i}\,\d y$, the expression for the WIMP density reduces to $$\begin{aligned}
\rho( r )
&=
\frac{8}{r}\int_{0}^{\infty}\!\d\beta_{i}\;\beta_{i}\.f_{\Brm}( \beta_{i} )
\,\int_{0}^{\infty}\!\d r_{i}\;r_{i}\,\rho_{i}( r_{i} )\;
\notag
\\
&\phantom{=\;} \times
\left(
\frac{1}{r_{i}}
-
\beta_{i}^{2}
\right)^{\!3 / 2}
\int_{y_{m}}^{1}\!\frac{\d y}{\sqrt{y^{2} - y_{m}^{2}\,}}
\; .
\label{eq:DensityRSimplified}\end{aligned}$$ Here, we explore the parameter space further in order to perform the integration. Since $0 \leq y_{m}^{2} \leq 1$, we obtain 0 1 + ( )\^[2]{} ( - \_[i]{}\^[2]{} - ) 1 . \[eq:Condition\] The upper bound leads to r\_[i]{} , \[eq:UpperCondition\] whereas the lower bound gives 1 + ( )\^[2]{} ( - \_[i]{}\^[2]{} - ) 0 , \[eq:LowerCondition\] and the equality has the two positive roots $r_{i, 1}= r$ and $r_{i, 2} = r\,\big\{ [ 1 + 4 / ( r\.\beta_{i}^{2} ) ]^{1 / 2} - 1 \big\} / 2$. Defining r\_[+]{} ( r\_[i, 1]{}, r\_[i, 2]{} ) , r\_[-]{} ( r\_[i, 1]{}, r\_[i, 2]{} ) , \[eq:RPlusAndRMinus\] the condition in Eq. is satisfied for $r_{i} \leq r_{-}$ or $r_{i} \geq r_{+}$. Finally, setting $x_{m} \equiv \sqrt{1 - y_{m}^{2}\,}$, Eq. reads $$\begin{aligned}
\rho( r )
&=
\frac{8}{r}\int_{0}^{\infty}\!\d\beta_{i}\;\beta_{i} \,f_{\Brm}( \beta_{i} )
\int_{0}^{\infty}\!\d r_{i}\;r_{i}\,\rho_{i}( r_{i} )
\notag
\\
&\phantom{=\;}\times
\left(
\frac{1}{r_{i}}
-
\beta_{i}^{2}
\right)^{\!3 / 2}
\ln\frac{1 + x_{m}}{y_{m}}\.\Theta
\; ,
\label{eq:DensityR1}\end{aligned}$$ where $\Theta$ is defined in terms of the Heaviside function $\theta( \cdot )$ as $$\begin{aligned}
\Theta
&\equiv
\theta\!
\left(
\frac{1}{\beta_{i}^{2}} - r_{i}
\right)
\theta\!
\left(
r_{i} - \frac{r}{1 + r \beta_{i}^{2}}
\right)
\notag
\\
&\phantom{=\;\.} \times
\big[\.
\theta\mspace{-1mu}
\left(
r_{-} - r_{i}
\right)
+
\theta\mspace{-1mu}
\left(
r_{i} - r_{+}
\right)
\big]
\; .
\label{eq:Theta}\end{aligned}$$ Here, we use the Maxwell–Boltzmann distribution f\_( \_[i]{} ) = . ( - ) , \[eq:BoltzmannDistribution\] where $\bar{\sigma} \equiv \sqrt{T / m_{\chi}\,}$. Furthermore, the initial energy density and the temperature at radius $r_{i}$ are given by (cf. the discussion in Sec. 3 in Ref. [@Eroshenko:2016yve])
$$\begin{aligned}
\rho_{i}( r_{i} )
&=
\rk\,\theta\!
\left(
\ri - r_{i}
\right)
\notag
\\
&\phantom{=\;}
+
\rk
\left(
\frac{\ri}{r_{i}}
\right)^{\mspace{-4mu}9 / 4}\,
\theta\!
\left(
r_{i} - \ri
\right)
\. ,
\label{eq:rhoi(ri)}
\\
T( r_{i} )
&=
\Tk\,\theta\!
\left(
\ri - r_{i}
\right)
\notag
\\
&\phantom{=\;}
+
\Tk
\left(
\frac{\ri}{r_{i}}
\right)^{\mspace{-4mu}3 / 2}\,
\theta\!
\left(
r_{i} - \ri
\right)
\. ,
\label{eq:T(ri)}\end{aligned}$$
respectively.
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[^1]: Note that $\rho_{\rm c} = 3\.H_{0}^{2} / 8 \pi G$, where $G$ is the gravitational constant, $H_{0}$ is the present value of the Hubble rate, and $h \equiv H_{0} / (100\,{\rm km} / {\rm s} / {\rm Mpc})$. In the following, the Planck mass $\MP = \sqrt{1/ G\,} \simeq 1.221 \times 10^{19}\,$GeV is also used.
[^2]: The factor of 2 comes from the fact that the WIMP passes twice by the same radius given the symmetry of the orbit [@Eroshenko:2016yve].
[^3]: It should be noted that our derivation does not take into account general relativistic effects and extreme eccentricities of the WIMP orbits such as those that intersect the PBH. We also work under the assumption that the WIMPs are non-relativistic. Nevertheless, all of these approximations are justified by the fact that the temperature of kinetic decoupling, at which the physics of interest takes place, is much smaller than the WIMP masses considered.
[^4]: When using critical collapse instead of the horizon-mass approximation, at each instance of time $t$ of horizon re-entry of a given mode, we have a to deal with an extended PBH mass spectrum. Hence, one looses the one-to-one correspondence between $M_{\rm BH}$ and $t$. In order to evaluate $\beta$ (or $f$) at a given time, one needs to take into account the scaling of the PBH population, behaving as matter in radiation domination, leading to different amplifications for different modes (see Ref. [@Kuhnel:2015vtw] for details).
|
---
abstract: |
In this paper, by using a Tsallis-Pareto-type function and the multisource thermal model, the elliptic flow coefficients of particles $\pi ^{\pm }$, $K^{\pm }$, $p+\overline{p}$, $\Lambda +\overline{%
\Lambda }$, and $K_{S}^{0}$ produced in Pb-Pb collisions at the center-of-mass energy of $\sqrt{s_{NN}}=5.02$ TeV are investigated. In the process of collisional evolution, because of geometric structure, pressure gradient, and thermal diffusion effects, deformation and translation occurred in the isotropic emission source, leading to anisotropy in the azimuth distribution of the final-state particles. Based on these dynamic factors, the dependence of elliptic flow on transverse momentum is described as well.
PACS: 14.65.Bt, 13.85.Hd, 24.10.Pa
author:
- 'Er-Qin Wang'
- 'Yin-Qun Ma'
- 'Li-Na Gao'
- 'San-Hong Fan'
title: |
Elliptic flow of identified particles in Pb-Pb collisions at $\sqrt{%
s_{NN}}=5.02$ TeV
---
Introduction
============
As collision energy has gradually increased in recent years, high-energy physics has developed rapidly. On the one hand, the energy range of nucleus-nucleus collisions has been broadened [@1; @2; @3; @4]. On the other hand, the kinds of final-state particles measured by detectors have become more explicit [@5; @6; @7]. This creates better conditions for obtaining a deep understanding of the collision mechanism. The distribution of high-energy final-state particles is important to understand the evolutionary mechanism of fluid dynamics, where as the flow effect of final-state particles is meaningful for the new material form, quark-gluon plasma (QGP) [@8; @9; @10]. The formation of QGP requires an extremely high-temperature, high-density environment. It is a state of released quarks and gluons that is similar to an ideal fluid. From an anisotropic azimuth analysis of final-state particles measured at the Relativistic Heavy Ion Collider (RHIC) [@11] and the Large Hadron Collider (LHC) [@12], it can be seen that the generated material unaffected by gravity is QGP under the condition of strong coupling. The quarks and gluons in the high-temperature, high-density state are affected by multiple factors. By means of the pressure gradient, the heterogeneity of energy density and the asymmetry of the geometric structure at the early stage of collisions are converted to the anisotropy of final-state particle momentum and finally manifest as the flow effect [@121; @122].
In the evolutionary process of high-energy collisions, there are two main stages, chemical freeze-out and dynamic freeze-out. The former occurs in the formation stage of different kinds of particles, and the decay and generation of particles remain in dynamic balance. This is an inelastic collision process. The second process occurs later, in the diffusion stage. Momentum and energy are maintained in a thermal equilibrium state in an elastic collision process. After the two stages, as the temperature drops, the final-state particles are ejected from the action system. Various physical properties of the final-state particles are then measured by detectors, such as the longitudinal momentum spectrum [@13; @14], the rapidity (pseudorapidity) distribution [@15; @16], the multiplicity distribution [@17; @18], and the flow effect [19,20,101,102,103,104,105,106,107,108]{}. By analysis of the final-state distribution using various theoretical models, the dynamic evolutionary mechanism, phase graph information, and particle attribution of quantum chromodynamics were deduced.
In non-central nucleus-nucleus collisions, the main coefficient of the flow effect is the second-order harmonic, which is called elliptic flow ($v_{2}$). The value is used to represent collective motion in the system. Collective motion is one of the characteristics formed in collisions of QGP. The flow effect that is caused by the asymmetry of the initial geometric structure and the heterogeneous energy of the action system includes direct flow, elliptic flow, and triangular flow. All the harmonics are quantified by the coefficient ($v_{n}$) of Fourier decomposition [@21; @22]:
$$\frac{dN}{d\varphi }\varpropto 1+2\sum v_{n}\cos \left[ n(\varphi -\Psi _{n})\right] , \label{1}$$
Similar long-range ridge structures and positive coefficients $v_{2}$ have been observed in experiments [@19]. In theory, it is assumed that these are based on the collective effect caused by hydrodynamic evolution of colliding particles.
Previous studies [@231; @232; @233] have presented a description of elliptic flow over a smaller range. Moreover, the isotropic hypothesis on the transverse plane and the translation and expansion effects of the emission source are used. In this paper, based on the multisource thermal model, using the distribution of the Tsallis-Pareto-type function, and at the center-of-mass energy of $\sqrt{s_{NN}}=$5.02 TeV, the dependence of the elliptic flow of the identified particles ($\pi ^{\pm }$, $K^{\pm }$, $p+\overline{p}$, $\Lambda +\overline{\Lambda }$, and $K_{S}^{0}$) in different centrality intervals in Pb-Pb collisions on transverse momentum is described [@24]. The multisource thermal model is a statistical model that is based on the one-dimensional string model [@301] and the fireball model [@302] and was developed from the thermalized cylinder model [303,304]{}. According to the multisource thermal model, many local emission sources are formed along the incident direction in high-energy collisions, and the final-state particles and jets are generated by these emission sources. In the rest frame of an emission source, the source is isotropic, that is, the final particles produced by the emission source are assumed to emit isotropically. Due to differences in impact parameters, centralities, position in space, or energy density, the emission source’s temperature, excitation degree, and particle yield ratio may vary. In comparison with previous work [@231; @232; @233] by the multisource thermal model, not only is the range of transverse momentum larger, but also the identification of the final-state particles is more accurate.
Model and formulation
=====================
In this paper, using the multisource thermal model [@25; @26; @27; @28; @29] and a Tsallis-Pareto-type function [@30; @31; @32; @33], the elliptic flow of identified particles in Pb-Pb collisions is analyzed. For each source in the multisource model, the Tsallis-Pareto-type function shows excellent reproducibility of the spectral measurement of many particles; the form is: $$\frac{d^{2}N}{dydp_{T}}=\frac{dN}{dy}Kp_{T}\left[ 1+\frac{m_{T}-m_{0}}{nC}\right] ^{-n}, \label{2}$$where $$\begin{aligned}
K &=&\frac{(n-1)(n-2)}{nC\left[ nC+(n-2)m_{0}\right] }, \label{3} \\
m_{T} &=&\sqrt{m_{0}^{2}+p_{T}^{2}},\end{aligned}$$where $m_{0}$ is the rest mass, $y$ is the rapidity, and $N$ is the number of particles. According to some non-extensive thermodynamic particle models, the free parameter $C$, which is related to the average particle energy, represents the mean effective temperature in the interacting system, $dN/dy$ is the particle output at different rapidity intervals, and $n$ indicate the non-extensivity of the process, which is the departure of the spectra from the Boltzmann distribution. After integrating for rapidity, the distribution density function of the transverse momentum is:
$$f(p_{T})=\frac{dN}{dp_{T}}=N_{0}Kp_{T}\left[ 1+\frac{m_{T}-m_{0}}{nC}\right]
^{-n}, \label{5}$$
where $N_{0}$ denotes the normalization constant, which depends on the free parameters $n$ and $C$. Hence, it is natural that $\int\nolimits_{0}^{\infty
}f(p_{T})dp_{T}=1$.
Related work [@34] has shown that the transverse momentum distribution of the final-state particles formed in nucleus-nucleus collisions satisfies the Tsallis-Pareto-type function. In accordance with the Monte Carlo method, by Eq. (5), the transverse momentum $p_{T}$ can be extracted. In this expression, $R_{0}$ represents random numbers uniformly distributed on \[0,1\], and $p_{T}$ can be given as:
$$\int_{0}^{p_{T}}f(p_{T})dp_{T}<R_{0}<\int_{0}^{p_{T}+dp_{T}}f(p_{T})dp_{T}.
\label{6}$$
Under the assumption of an isotropic emission source, the azimuth distribution of final-state particles is even, and the distribution function is:
$$f_{\varphi }\left( \varphi \right) =\frac{1}{2\pi }.$$
By the Monte Carlo method, the random number of the azimuth can be obtained as:
$$\varphi =2\pi R,$$
where $R$ represents a random number distributed on \[0, 1\]. Let the beam direction be the $Oz$ axis, and let the reaction plane be the $xOz$ plane. Therefore, the momentum components are
$$\begin{aligned}
p_{x} &=&p_{T}\cos \varphi , \\
p_{y} &=&p_{T}\sin \varphi .\end{aligned}$$
Due to the geometric structure of the participant, the pressure gradient, and interaction with the medium, the emission source deforms and translates in its rest frame. Hence, an anisotropic emission source is introduced in the multisource thermal model. To quantify the deformation and translation of the emission source, $a_{x}$ ($a_{y}$) and $b_{x}$ ($b_{y}$) express the deformation and translation of the emission source along the $Ox$ ($Oy$) axis, $a_{x}>1$ ($<1$) represents expansion (compression), and $b_{x}>0$ ($<0$) represents translation along the positive (negative) axis. Generally, for particles with different centrality intervals and transverse momentum, different $a_{x}$ ($a_{y}$) or $b_{x}$ ($b_{y}$) are obtained. As a first approximation, the empirical relationship can be expressed as:
$$a_{x}=1+k_{1}\exp (-\frac{p_{T}}{\lambda _{1}})+k_{2}p_{T},$$
where $k_{1}$, $\lambda _{1}$, $k_{2\text{ }}$are free parameters. For simplicity, the default is $a_{y}=1$ and $b_{x,y}=0$. Because of deformation, the above $p_{x}$ is revised to become:$$p_{x}^{\prime }=a_{x}p_{x}+b_{x}.$$Then the converted transverse momentum is: $$p_{T}^{\prime }=\sqrt{p_{x}^{\prime 2}+p_{y}^{2}}.$$Finally, the elliptic flow of final-state particles can be represented as: $$v_{2}=\left\langle \frac{p_{x}^{\prime 2}-p_{y}^{2}}{p_{x}^{\prime
2}+p_{y}^{2}}\right\rangle . \label{17}$$
Comparisons with experimental data
==================================
Using the multisource thermal model, the anisotropic spectrum data of various particles generated in Pb-Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV [@24] are studied and analyzed. The particles $\pi ^{\pm }$, $K^{\pm }$, $p+\overline{p}$, $\Lambda +\overline{\Lambda }$, and $K_{S}^{0}$ are located in different centrality intervals within 0–70% and depend on $v_{2}
$ of the transverse momentum $p_{T}$. The rapidity is in the range $\left\vert y\right\vert <0.5$. For particles $\pi ^{\pm }$, $K^{\pm }$, and $p+\overline{p}$, the measurements in hypercenter collisions (0–1%) are also shown.
Figure 1 shows the elliptic flow $v_{2}(p_{T})$ of meson $\pi ^{\pm }$ generated in a Pb-Pb collision at energy $\sqrt{s_{NN}}=5.02$ TeV in different centrality intervals. The data measured by the ALICE Collaboration in different centrality intervals are represented by different solid symbols, and the statistical and systematic errors are both considered in the error bar [@24]. The curves are fitted to results generated by the Tsallis-Pareto-type function in the framework of the multisource thermal model. Table 1 shows the fitted free parameters ($C$, $n$, $k_{1}$, $\lambda
_{1}$, and $k_{2}$), $\chi ^{2}$ and the degrees of freedom (dof). Clearly the model results are consistent with the experimental data. In the calculation, the data fitting indicates that the effective temperature $C$ increases as the centrality percentage decreases, but that the value of $n$remains unchanged and is assumed to be 9. It is obvious that $v_{2}$ increases with $p_{T}$ in the low $p_{T}$ region, and then decreases slowly in the high $p_{T}$ region. The transverse momentum corresponding to the maximum value increases with increasing particle mass. This trend is reflected in the values of $k_{1}$, $\lambda _{1}$, and $k_{2}$. Moreover, it is not hard to find that the parameter $k_{1}$ first increases rapidly with the centrality percentage and then slowly decreases. Finally, the values of $\chi ^{2}/dof$ are in a reasonable range, which is not only affected by experimental errors, but is also related to the inaccuracy of the theoretical calculation results.
Figure 2 shows that $v_{2}(p_{T})$ of $K^{\pm }$ in the given centrality interval. Similarly to Fig. 1, the solid symbols also represent the experimental data recorded by the ALICE Collaboration, and the error bar includes the statistical and systematic errors. The curves are the results of fitting using the Tsallis-Pareto-type function. The fitting parameters, $\chi ^{2}$ and dof are also listed in Table 1. It is apparent that the experimental data are well fitted by the model results. In the calculation, the values of effective temperature $C$ decrease from the central to peripheral collisions and are systematically larger than those for particles $\pi ^{\pm }$. As the centrality percentage increases, the values of $k_{1}$ first increase rapidly, then slowly decrease, as shown in Fig. 1.
Figure 3 shows the $v_{2}$ of $p+\overline{p}$, which depends on the transverse momentum. Figures 4 and 5 show the relationship between the elliptic flow and the transverse momentum spectrum of $\Lambda +\overline{\Lambda }$ and $K_{S}^{0}$ respectively. The solid symbols are the data points, and the curves show the model results. The fitted parameter values, dof and $\chi ^{2}$, are included in Table 1. It is evident that the fits are in good agreement with the experimental data. However, as shown in Fig. 4, in the given centrality interval of 60–70%, there is a datum point located at $p_{T}=9$ Gev/c that deviates seriously from the fitted value. The physical mechanism underlying this deviation is not yet understood. Similarly, when moving from central to peripheral collisions, $C$ increases, and $k_{1}$ increases rapidly, then decreases slowly. Overall, the model fits the spectrum $v_{2}(p_{T})$ of identified particles measured in different centrality intervals by ALICE in Pb+Pb collisions at approximately $\sqrt{s_{NN}}=5.02$ TeV.
Based on the fitted results shown in Figs. 1–5, Figure 6 shows the dependency relationship between the expansion factor $a_{x}$ and the transverse momentum $p_{T}$ in the given centrality interval for different particles $\pi ^{\pm }$, $K^{\pm }$, $p+\overline{p}$, $\Lambda +\overline{\Lambda }$, and $K_{S}^{0}$. For a certain particle, $a_{x}(p_{T})$ are different in different centrality intervals. The curves with maximum and minimum dependency relationship were chosen based on Eq.(11) and are represented by solid and dashed lines respectively. The variation trends are similar, but the ranges are slightly different. Furthermore, as the particle mass increases, the range also increases. Figure 7 shows the fitting parameter $C$, which depends on the variation of centrality. When moving from central to peripheral collisions, the effective temperature $C$gradually declines.
Discussion and conclusions
==========================
According to the fitted results from the above comparisons, the fitted free parameter $C$ is actually not the real temperature (the kinetic freeze-out temperature) of the emission source, but the effective temperature. As is well known, the interacting system at kinetic freeze-out (the last stage of collision) is influenced not only by thermal motion, but also by the flow effect. The real temperature of the emission source should reflect the thermal motion of the particles, and therefore the real temperature of the source is the kinetic freeze-out temperature. The effective temperature extracted from the elliptic flow spectrum includes thermal motion and the flow effect of the particles. By dissecting the effective temperature, it is possible to obtain the real temperature of the interacting system. The relationships between effective temperature, real temperature, and flow velocity are not totally clear. Therefore, the value of effective temperature obtained in this work is higher than the kinetic freeze-out temperature.
Table 1 shows that the parameter $k_{1}$ first increases rapidly with centrality percentage and then decreases slowly. It reaches a maximum as the centrality percentage reaches about 30%. In addition, Fig. 6 shows that $a_{x}$ decreases with increasing transverse momentum $p_{T}$. However, Fig. 7 shows that the parameter $C$ declines gradually from central to peripheral collisions. As for the dependency relationship, it can be readily understood.
From the participant-spectator geometric structure, it can be seen that as centrality percentage increases, the extent of the overlapping parts decreases, whereas the asymmetry rises. There is an approximate linear relationship between the elliptic flow and the eccentricity ratio of the participant. Hence, with increasing centrality percentage, the elliptic flow also grows. However, $v_{2}$ of particles in peripheral collisions is slightly smaller than in central collisions. This may be due to shorter system life under peripheral collisions, resulting in small $v_{2}$. Hence, $k_{1}$ first increases rapidly with the centrality percentage and then decreases slowly.
However, as the centrality percentage rises, the effective temperature $C$ declines gradually. In accordance with the geometric structure of collisions, as the centrality percentage decreases, the number of involved nucleons increases, and the overlapping parts also increase, leading to higher energy density and strength of interaction, which manifests as higher temperature. The effective temperature $C$ obtained in this study was higher than the true temperature. The reason for this was that the effective temperature incorporates the true temperature and the flow effect. The value excluding the flow effect should be equal to the true temperature. Fig. 7 shows that for particles with considerable mass, the low variation ranges of effective temperature are similar.
In short, based on the multisource model, by introducing a Tsallis-Pareto-type function, the elliptic flow of identified particles generated in Pb-Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV was correctly analyzed. Therefore, in the collision process, the asymmetry, expansion, and translation effects of geometric structure affect the dynamics of the final-state particles.
**Data Availability**
The data used to support the findings of this study are included within the article.
**Ethical Approval**
The authors declare that they are in compliance with ethical standards regarding the content of this paper.
**Conflicts of Interest**
The authors declare that they have no conflicts of interest regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China Grant Nos. 11447137 and 11575103 and the Doctoral Scientific Research Foundation of Taiyuan Normal University under Grant No. I170108.
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$n$$\lambda $
Figure particles centrality $C(GeV)$ $n$ $k_{1}$ $\lambda _{1}$ $k_{2}$ $\chi ^{2}/dof$
-------- --------------- ------------ ---------- ----- --------- ---------------- --------- -----------------
Fig.1 $\pi ^{\pm }$ 0–1% 1.00 9 0.17 2.35 0.001 4/17
Fig.1 $\pi ^{\pm }$ 0–5% 1.10 9 0.27 2.35 0.001 6/17
Fig.1 $\pi ^{\pm }$ 5–10% 1.10 9 0.49 2.35 0.004 2/17
Fig.1 $\pi ^{\pm }$ 10–20% 0.80 9 0.60 2.35 0.004 3/17
Fig.1 $\pi ^{\pm }$ 20–30% 0.60 9 0.65 2.35 0.004 2/17
Fig.1 $\pi ^{\pm }$ 30–40% 0.50 9 0.64 2.35 0.004 2/17
Fig.1 $\pi ^{\pm }$ 40–50% 0.40 9 0.59 2.35 0.004 7/17
Fig.1 $\pi ^{\pm }$ 50–60% 0.40 9 0.54 2.35 0.006 1/17
Fig.1 $\pi ^{\pm }$ 60–70% 0.40 9 0.48 2.40 0.005 11/17
Fig.2 $K^{\pm }$ 0–1% 2.60 9 0.28 2.35 0.000 5/12
Fig.2 $K^{\pm }$ 0–5% 2.20 9 0.40 2.35 0.000 12/12
Fig.2 $K^{\pm }$ 5–10% 1.70 9 0.66 2.35 0.002 8/12
Fig.2 $K^{\pm }$ 10–20% 1.25 9 0.86 2.25 0.002 4/12
Fig.2 $K^{\pm }$ 20–30% 1.00 9 0.98 2.15 0.003 2/12
Fig.2 $K^{\pm }$ 30–40% 0.72 9 0.83 2.35 0.002 6/12
Fig.2 $K^{\pm }$ 40–50% 0.68 9 0.84 2.20 0.002 2/12
Fig.2 $K^{\pm }$ 50–60% 0.55 9 0.64 2.35 0.003 1/12
Fig.2 $K^{\pm }$ 60–70% 0.40 9 0.44 2.40 0.005 1/12
Fig.3 $p+\bar{p}$ 0–1% 3.50 9 0.40 2.40 0.002 10/15
Fig.3 $p+\bar{p}$ 0–5% 4.40 9 0.70 2.40 0.002 33/15
Fig.3 $p+\bar{p}$ 5–10% 2.80 9 1.05 2.35 0.002 25/15
Fig.3 $p+\bar{p}$ 10–20% 1.70 9 1.25 2.35 0.006 18/15
Fig.3 $p+\bar{p}$ 20–30% 1.30 9 1.25 2.35 0.007 23/15
Fig.3 $p+\bar{p}$ 30–40% 1.10 9 1.25 2.35 0.007 12/15
Fig.3 $p+\bar{p}$ 40–50% 0.95 9 1.10 2.35 0.006 8/15
Fig.3 $p+\bar{p}$ 50–60% 0.75 9 0.97 2.35 0.006 2/15
Fig.3 $p+\bar{p}$ 60–70% 0.75 9 0.77 2.35 0.006 1/15
Figure particles centrality $C(GeV)$ $n$ $k_{1}$ $\lambda _{1}$ $k_{2}$ $\chi ^{2}/dof$
-------- -------------------------- ------------ ---------- ----- --------- ---------------- --------- -----------------
Fig.4 $\Lambda +\bar{\Lambda}$ 0–5% 4.20 9 0.58 3.00 0.005 12/7
Fig.4 $\Lambda +\bar{\Lambda}$ 5–10% 3.00 9 1.20 2.55 0.007 3/7
Fig.4 $\Lambda +\bar{\Lambda}$ 10–20% 2.10 9 1.57 2.30 0.009 2/7
Fig.4 $\Lambda +\bar{\Lambda}$ 20–30% 1.40 9 1.60 2.30 0.009 1/7
Fig.4 $\Lambda +\bar{\Lambda}$ 30–40% 1.10 9 1.42 2.40 0.009 1/7
Fig.4 $\Lambda +\bar{\Lambda}$ 40–50% 0.90 9 1.26 2.55 0.005 1/7
Fig.4 $\Lambda +\bar{\Lambda}$ 50–60% 0.80 9 1.07 2.50 0.009 1/7
Fig.4 $\Lambda +\bar{\Lambda}$ 60–70% 0.60 9 0.70 2.50 0.005 4/7
Fig.5 $K_{s}^{0}$ 0–5% 2.10 9 0.38 2.20 0.002 3/8
Fig.5 $K_{s}^{0}$ 5–10% 1.70 9 0.65 2.20 0.002 2/8
Fig.5 $K_{s}^{0}$ 10–20% 1.20 9 0.78 2.20 0.006 1/8
Fig.5 $K_{s}^{0}$ 20–30% 0.90 9 0.83 2.20 0.005 1/8
Fig.5 $K_{s}^{0}$ 30–40% 0.70 9 0.79 2.20 0.008 1/8
Fig.5 $K_{s}^{0}$ 40–50% 0.60 9 0.73 2.20 0.006 1/8
Fig.5 $K_{s}^{0}$ 50–60% 0.55 9 0.63 2.40 0.003 1/8
Fig.5 $K_{s}^{0}$ 60–70% 0.40 9 0.44 2.45 0.005 1/8
![$v_{2}(p_{T})$ of $\protect\pi ^{\pm }$ in a given centrality interval arranged into panels of various centrality classes [@24]. The data, which were measured by the ALICE Collaboration in various centrality classes, are represented in the figure by different symbols. Statistical and systematic uncertainties are shown as bars. The curves are the results of this study fitted using the Tsallis-Pareto-Type function and the multisource ideal gas model.](fig1.ps){width="12cm"}
![As for Fig. 1, but showing $v_{2}(p_{T})$ of $K ^{\pm }$ for a given centrality [@24].](fig2.ps){width="12cm"}
![As for Fig. 1, but showing $v_{2}(p_{T})$ of $p+\bar{p}$ for a given centrality [@24].](fig3.ps){width="12cm"}
![As for Fig. 1, but showing $v_{2}(p_{T})$ of $\Lambda+\bar{\Lambda}$ for a given centrality. [@24]](fig4.ps){width="12cm"}
![As for Fig. 1, but showing $v_{2}(p_{T})$ of $K_{s}^{0}$ for a given centrality. [@24]](fig5.ps){width="12cm"}
{width="12cm"}
{width="12cm"}
|
---
abstract: |
The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. In this paper, we consider the intuitionistic fuzzification of the concept of sub-hyperquasigroups in a hyperquasigroup and investigate some properties of such sub-hyperquasigroups. In particular, we investigate some natural equivalence relations on the set of all intuitionistic fuzzy sub-hyperquasigroups of a hyperquasigroup.\
[*2000 Mathematics Subject Classification:*]{} 20N20, 20N25.\
[*Keywords:*]{} hyperquasigroup, fuzzy sub-hyperquasigroup, intuitionistic fuzzy sub-hyperquasigroup, quasigroup.
author:
- |
**Wies[ł]{}aw A. Dudek$^{\rm a}$, Bijan Davvaz$^{\rm b},$ Young Bae Jun$^{\rm c,*}$\
[*$^{\rm a}$ Institute of Mathematics, Technical University,*]{}\
[*Wybrzeże Wyspiańskiego 27, 50-370 Wroc[ł]{}aw, Poland*]{}\
[*$^{\rm b}$ Department of Mathematics, Yazd University, Yazd, Iran*]{}\
[*$^{\rm c}$ Department of Mathematics Educations, Gyeongsang National University,*]{}\
[*Chinju 660-701, Korea*]{}**
title: '**On intuitionistic fuzzy sub-hyperquasigroups of hyperquasigroups**'
---
[^1]
Introduction and preliminaries
==============================
The theory of hyperstructures which is a generalization of the concept of algebraic structures first was introduced by Marty [@14] and then many researchers have been worked on this new field of modern algebra and developed it. A short review of the theory of hyperstructures appear in [@4] and [@16]. A recent book [@3] contains a wealth of applications. There are applications to the following subjects: geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, combinatorics, codes, artificial intelligence, and probabilities. The theory of fuzzy sets proposed by Zadeh [@17] has achieved a great success in various fields. Out of several higher order fuzzy sets, intuitionistic fuzzy sets introduced by Atanassov [@1; @2; @2a] have been found to be highly useful to deal with vagueness. Gau and Buehrer [@IEEE23-610] presented the concept of vague sets. But, Burillo and Bustince [@FSS79-403] showed that the notion of vague sets coincides with that of intuitionistic fuzzy sets. Szmidt and Kacprzyk [@FSS118-467] proposed a non-probabilistic-type entropy measure for intuitionistic fuzzy sets. De et al. [@FSS117-209] studied the Sanchez’s approach for medical diagnosis and extended this concept with the notion of intuitionistic fuzzy set theory. Dengfeng and Chuntian [@PRL23-221] introduced the concept of the degree of similarity between intuitionistic fuzzy sets, presented several new similarity measures for measuring the degree of similarity between intuitionistic fuzzy sets, which may be finite or continuous, and gave corresponding proofs of these similariry measures and discussed applications of the similarity measures between intuitionistic fuzzy sets to pattern recofnition problems. The notion of join space has been introduced by Prenowitz and used by him and afterwards together Jantosciak to build again several branches of geometry. A join space is a hypergroup with additional conditions. A generalization of join spaces for the point of view of independence, dimension etc., is that of cambiste hypergroups studied by Freni. Noticing that a hypergroup is a hyperquasigroup with the associative hyperoperation, the results of this paper will make a contribution to discuss a generalization of join spaces, to deal with several notions in geometries since there are deep relations between geometries and hypergroups (or, to say multigroups), and to develop the intuitionistic fuzzy theory in several algebraic structures.
A [*hypergroupoid*]{} $(G,\circ )$ is a non-empty set $G$ with a [*hyperoperation*]{} $\circ$ defined on $G$, i.e., a mapping of $G\times G$ into the family of non-empty subsets of $G$. If $(x,y)
\in G\times G,$ its image under $\circ$ is denoted by $x\circ y$. If $A,B \subseteq G$ then $A\circ B$ is given by $A\circ B=\bigcup
\{x\circ y \ | \ x\in A, \ y\in B\}$. $x\circ A$ is used for $\{
x\}\circ A$ and $A\circ x$ for $A\circ \{ x\}$.
\[def11\] A hypergroupoid $(G,\circ )$ is called a [*hypergroup*]{} if for all $x,y,z\in G$ the following two conditions hold:
- $x\circ (y\circ z)=(x\circ y)\circ z$,
- $x\circ G = G\circ x=G$.
The second condition, called the [*reproduciblity condition*]{}, means that for any $x,y\in G$ there exist $u,v\in G$ such that $y\in x\circ u$ and $y\in v\circ x$.
A hypergroupoid satisfying this condition is called a [*hyperquasigroup*]{}. Thus a hypergroup is a hyperquasigroup with the associative hyperoperation.
A non-empty subset $K$ of a hyperquasigroup $(G,\circ )$ is called a [*sub-hyperquasigroup*]{} if $(K,\circ)$ is a hyperquasigroup.
The concept of fuzzy sets was introduced by Zadeh [@17] in 1965. A mapping $\mu :X \to [0,1]$, where $X$ is an arbitrary non-empty set, is called a [*fuzzy set*]{} in $X$. The [*complement*]{} of $\mu$, denoted by $\mu^c$, is the fuzzy set in $X$ given by $\mu^c(x)=1-\mu (x)$ for all $x \in X$.
For any fuzzy set $\mu$ in $X$ and any $t\in [0,1]$ we define two sets $$U(\mu;t)=\{x\in X\ |\ \mu(x)\geq t\} \ \ \ \ {\rm and } \ \ \ \
L(\mu;t)=\{x\in X\ |\ \mu(x)\leq t\},$$ which are called an [*upper* ]{} and [*lower $t$-level cut*]{} of $\mu$ and can be used to the characterization of $\mu$.
In 1971, Rosenfeld [@15] applied the concept of fuzzy sets to the theory of groups and studied fuzzy subgroups of a group. Davvaz applied in [@6] fuzzy sets to the theory of algebraic hyperstructures and studied their fundamental properties. Further investigations are contained in [@5], [@7] and [@8].
(cf. [@6]) Let $(G,\circ )$ be a hypergroup (resp. hyperquasigroup) and let $\mu$ be a fuzzy set in $G$. Then $\mu$ is said to be a [*fuzzy sub-hypergroup*]{} (resp. [*fuzzy sub-hyperquasigroup*]{}) of $G$ if the following axioms hold:
- $\min\{\mu(x),\mu(y)\} \leq\inf\{\mu(z)\ |\ z\in x\circ
y\}$ for all $x,y\in G$,
- for all $x,a\in G$ there exists $y\in G$ such that $x\in a\circ y$ and $$\min\{\mu(a),\mu(x)\}\leq\mu(y),$$
- for all $x,a\in G$ there exists $z\in G$ such that $x\in z\circ a$ and $$\min\{\mu(a),\mu(x)\}\leq\mu(z).$$
As an important generalization of the notion of fuzzy sets in $X$, Atanassov [@1] introduced the concept of [*intuitionistic fuzzy sets*]{} defined on a non-empty set $X$ as objects having the form $$A=\{(x,\mu_A(x), \lambda_A(x)) \ | \ x\in X\},$$ where the functions $\mu_A:X\to [0,1]$ and $\lambda_A:X\to
[0,1]$ denote the [*degree of membership*]{} (namely $\mu_A(x)$) and the [*degree of nonmembership*]{} (namely $\lambda_A(x)$) of each element $x\in X$ to the set $A$ respectively, and $0\leq\mu_A(x)+\lambda_A(x) \leq 1$ for all $x \in X$.
Such defined objects are studied by many authors (see for example two journals: 1. [*Fuzzy Sets and Systems*]{} and 2. [*Notes on Intuitionistic Fuzzy Sets*]{}) and have many interesting applications not only in mathematics (see Chapter 5 in the book [@2a]). In particular, Kim, Dudek and Jun in [@11] introduced the notion of an intuitionistic fuzzy subquasigroup of a quasigroup. Also in [@12], Kim and Jun introduced the concept of intuitionistic fuzzy ideals of semigroups.
For every two intuitionistic fuzzy sets $A$ and $B$ in $X$ we define (cf. [@2]):
- $A\subseteq B$ iff $\mu_A(x)\leq\mu_B(x)$ and $\lambda_A(x)\geq\lambda_B(x)$ for all $x\in X$,
- $A^c=\{(x, \lambda_A(x),\mu_A (x)) \ | \ x\in X\}$,
- $A\cap B=\{(x,\min\{\mu_A(x),\mu_B(x)\},
\max\{\lambda_A(x),\lambda_B(x)\}) \ | \ x\in X\}$,
- $A\cup B=\{(x,\max\{\mu_A(x),\mu_B(x)\},
\min\{\lambda_A(x),\lambda_B(x)\}) \ | \ x\in X\}$,
- $\Box A=\{(x,\mu_A(x),\mu^c_A(x)) \ | \ x\in X\}$,
- $\Diamond A=\{(x,\lambda^c_A(x),\lambda_A(x)) \ | \ x\in X\}$.
Intuitionistic fuzzy sub-hyperquasigroups
=========================================
For the sake of simplicity, we shall use the symbol $A=(\mu_A,
\lambda_A)$ for the intuitionistic fuzzy set $A=\{(x,\mu_A(x),\lambda_A(x) \ | \ x\in X\}$.
In what follows, let $G$ denote a hyperquasigroup, and we start by defining the notion of intuitionistic fuzzy sub-hyperquasigroups.
Based on [@11], we can extend the concept of the intuitionistic fuzzy subquasigroup to the concept of intuitionistic fuzzy sub-hyperquasigroups in the following way:
\[def21\] An intuitionistic fuzzy set $A=(\mu_A,
\lambda_A)$ in $G$ is called an [*intuitionistic fuzzy sub-hyperquasigroup*]{} of $G$ ($IFSH$ of $G$ for short) if
- $\min\{\mu_A(x),\mu_A(y)\}\leq\inf\{\mu_A(z)\ |\
z\in x\circ y\}$ for all $x,y\in G$,
- for all $x,a\in G$ there exist $y,z\in G$ such that $x\in (a\circ y)\cap (z\circ a)$ and $$\min\{\mu_A(a),\mu_A(x)\}\leq\min\{\mu_A(y),\mu_A(z)\},$$
- $\sup\{\lambda_A(z)\ |\ z\in x\circ y\}\leq\max\{\lambda_A(x),\lambda_A(y)\}$ for all $x,y\in G$,
- for all $x,a\in G$ there exist $y,z\in G$ such that $x\in (a\circ y)\cap (z\circ a)$ and $$\max\{\lambda_A(y),\lambda_A(z)\}\leq\max\{\lambda_A(a),\lambda_A(x)\}.$$
\[lem22\] If $A=(\mu_A, \lambda_A)$ is an $IFSH$ of $G$, then so is $\Box A=(\mu_A, \mu^c_A)$.
It is sufficient to show that $\mu_A^c$ satisfies the third and fourth conditions of Definition \[def21\]. For $x,y\in G$ we have $$\min\{\mu_A(x),\mu_A(y)\}\leq\inf\{\mu_A(z)\ |\ z\in x\circ y\}$$ and so $$\min\{1-\mu_A^c (x), 1-\mu_A^c (y)\} \leq\inf\{1-\mu_A^c (z)\ |\ z\in x\circ y\}.$$ Hence $$\min\{1-\mu_A^c(x),1-\mu_A^c(y)\}\leq 1-\sup\{\mu_A^c(z)\ |\ z\in
x\circ y\}$$ which implies $$\sup\{\mu_A^c (z)\ |\ z\in x\circ y\}\leq 1-\min\{1-\mu_A^c(x),
1-\mu_A^c(y)\}.$$ Therefore $$\sup\{\mu_A^c(z)\ | \ z\in x\circ y\}\leq\max\{\mu_A^c(x),\mu_A^c(y)\}.$$ Hence the third condition of Definition \[def21\] is verified.
Now, let $a,x\in G.$ Then there exist $y,z\in G$ such that $x\in
a\circ y, \ x\in z\circ a$ and $$\min\{\mu_A (a),\mu_A (x)\} \leq \min\{\mu_A (y),\mu_A (z)\}.$$ So $$\min\{1-\mu_A^c(a),1-\mu_A^c(x)\}\leq\min\{1-\mu_A^c(y),1-\mu_A^c(z)\}.$$ Hence $$\max\{\mu_A^c(y),\mu_A^c(z)\}\leq\max\{\mu_A^c(a),\mu_A^c(x)\},$$ and the fourth condition of Definition \[def21\] is satisfied.
\[lem23\] If $A=(\mu_A, \lambda_A)$ is an $IFSH$ of $G$, then so is $\Diamond A=(\lambda^c_A,\lambda_A)$.
The proof is similar to the proof of Lemma \[lem22\].
Combining the above two lemmas it is not difficult to see that the following theorem is valid.
\[th24\] $A=(\mu_A, \lambda_A)$ is an $IFSH$ of $\,G$ if and only if $\Box A$ and $\Diamond A$ are $IFSHs$ of $\,G$. $\Box$
\[cor25\] $A=(\mu_A, \lambda_A)$ is an $IFSH$ of $\,G$ if and only if $\mu_A$ and $\lambda^c_A$ are fuzzy sub-hyperquasigroups of $\,G$. $\Box$
\[th26\] If $\,A=(\mu_A, \lambda_A)$ is an $IFSH$ of $G$ then the upper $t$-level cut $U(\mu_A;t)$ of $\mu_A$ and the lower $t$-level cut $L(\lambda_A;t)$ of $\lambda_A$ are sub-hyperquasigroups of $G$ for every $t\in Im (\mu_A)\cap
Im(\lambda_A)$.
Let $t\in Im (\mu_A)\cap Im(\lambda_A) \subseteq [0,1]$ and let $x,y\in U(\mu_A; t)$. Then $\mu_A(x)\geq t$ and $\mu_A(y)\geq t$ and so $\min\{\mu_A (x),\mu_A(y)\}\geq t$. It follows from the first condition of Definition \[def21\] that $\inf\{\mu_A(z)\ |
\ z\in x\circ y\}\geq t$. Therefore for all $z\in x\circ y$ we have $z\in U(\mu_A;t)$, so $x\circ y\subseteq U(\mu_A;t)$. Hence for all $a\in U(\mu_A;t)$ we have $a\circ U(\mu_A;t)\subseteq
U(\mu_A;t)$ and $U(\mu_A;t)\circ a\subseteq U(\mu_A;t)$. Now, let $x\in U(\mu_A;t)$ then there exist $y,z\in G$ such that $x\in
a\circ y$, $x\in z\circ a$ and $\min\{\mu_A(x),\mu_A(a)\}\leq
\min\{\mu(y),\mu(z)\}$. Since $x,a\in U(\mu_A; t)$, we have $t\leq\min\{\mu_A(x),\mu_A(a)\}$ and so $t\leq\min\{\mu_A(y),
\mu_A(z)\}$ which implies $y\in U(\mu_A;t)$, $z\in U(\mu_A;t)$ and these prove that $U(\mu_A;t)\subseteq a\circ U(\mu_A;t)$ and $U(\mu_A;t)\subseteq U(\mu_A;t)\circ a$. Hence $a\circ U(\mu_A;t)=
U(\mu_A;t)= U(\mu_A;t)\circ a$.
Now let $x,y\in L(\lambda_A;t)$. Then $\lambda_A(x)\leq t$, $\lambda_A (y)\leq t$ and, consequently, $\max \{\lambda_A
(x),\lambda_A(y)\} \leq t$. It follows from the third condition of Definition \[def21\] that $\sup\{\lambda_A(z)\ |\ z\in x\circ
y\}\leq t$. Therefore for all $z\in x\circ y$ we have $z\in
L(\lambda_A;t)$, so $x\circ y \subseteq L(\lambda_A;t)$. Hence for all $a\in L(\lambda_A;t)$ we have $a\circ L(\lambda_A;t)\subseteq
L(\lambda_A;t)$ and $L(\lambda_A;t)\circ a \subseteq
L(\lambda_A;t)$. Now, let $x\in L(\lambda_A;t)$. Then there exist $y,z\in G$ such that $x\in a\circ y$, $x\in z\circ a $ and $\max\{\lambda_A(y),\lambda_A(z)\}\leq
\max\{\lambda(a),\lambda(x)\}$. Since $x,a\in L(\lambda_A;t)$, we have $\max\{\lambda_A(a), \lambda_A(x)\}\leq t$ and so $\max\{\lambda_A(y),\lambda_A(z)\}\leq t$ which implies $y\in
L(\lambda_A;t)$, $\,z\in L(\lambda_A;t)$ and these prove that $L(\lambda_A;t)\subseteq a\circ L(\lambda_A;t)$ and $L(\lambda_A;t)\subseteq L(\lambda_A;t)\circ a$. Thus $a\circ
L(\lambda_A;t) = L(\lambda_A;t) = L(\lambda_A;t)\circ a$.
\[th27\] If $A=(\mu_A, \lambda_A)$ is an intuitionistic fuzzy set in $G$ such that the non-empty sets $U(\mu_A;t)$ and $L(\lambda_A;t)$ are sub-hyperquasigroups of $G$ for all $t\in [0,1],$ then $A=(\mu_A, \lambda_A)$ is an $IFSH$ of $G$.
For $t\in [0,1]$, assume that $U(\mu_A;t)\neq
\emptyset$ and $L(\lambda_A;t)\neq\emptyset$ are sub-hyperquasigroups of $G$. We must show that $A=(\mu_A,
\lambda_A)$ satisfies the all conditions in Definition \[def21\]. Let $x,y\in G$, we put $t_0=\min\{\mu_A
(x),\mu_A(y)\}$ and $t_1=\max\{\lambda_A(x),\lambda_A(y)\}$. Then $x,y\in U(\mu_A;t_0)$ and $x,y\in L(\lambda_A;t_1)$. So $x\circ y
\subseteq U(\mu_A;t_0)$ and $x\circ y \subseteq L(\lambda_A;t_1)$. Therefore for all $z\in x\circ y$ we have $\mu_A(z)\geq t_0$ and $\lambda_A(z)\leq t_1$ which imply $$\inf\{\mu_A(z) \ | \ z\in x\circ y\}\geq\min\{\mu_A(x),\mu_A(y)\}$$ and $$\sup\{\lambda_A (z) \ | \ z\in x\circ y\}\leq\max\{\lambda_A(x),
\lambda_A(y)\}$$ The conditions $(1)$ and $(3)$ of Definition \[def21\] are verified.
Now, let $x,a\in G$. If $t_2=\min\{\mu_A(a),\mu_A(x)\}$, then $a,x\in U(\mu_A;t_2)$. So there exist $y_1,z_1\in U(\mu_A;t_2)$ such that $x\in a\circ y_1$ and $x\in z_1\circ a$. Also we have $t_2\leq\min\{\mu_A(y_1),\mu_A(z_1)\}$. Therefore the condition $(2)$ of Definition \[def21\] is verified. If we put $t_3=\max\{\lambda_A(a),\lambda_A(x)\}$, then $a,x\in
L(\lambda_A;t_3)$. So there exist $y_2,z_2\in L(\lambda_A; t_3)$ such that $x\in a\circ y_2$ and $x\in z_2\circ a$ and we have $\max\{\lambda_A(y_2),\lambda_A(y_2)\}\leq t_3$, and so the condition $(4)$ of Definition \[def21\] is verified. This completes the proof.
\[cor28\] Let $K$ be a sub-hyperquasigroup of a hyperquasigroup $(G,\circ)$. If fuzzy sets $\mu$ and $\lambda$ are defined on $G$ by $$\mu(x)=\left\{\begin{array}{cl}\alpha_0 &\text{if \, $x\in K$},\\
\alpha_1 &\text{if \, $x\in G\setminus K$,}\end{array}
\right. \qquad\lambda(x)=\left\{\begin{array}{cl}\beta_0
&\text{if \, $x\in K$},\\
\beta_1 &\text{if \, $x\in G\setminus K$,}\end{array}
\right.$$ where $\,0\leq\alpha_1< \alpha_0$, $0\leq\beta_0<\beta_1\,$ and $\,\alpha_i+\beta_i\leq 1$ for $i=0,1,$ then $A=(\mu,\lambda )$ is an $IFSH$ of $\,G$ and $\,U(\mu;\alpha_0 )=K=L(\lambda;\beta_0)$. $\Box$
\[cor29\] Let $\chi_{_K}$ be the characteristic function of a sub-hyperquasigroup $K$ of $\,(G,\circ )$. Then $K=(\chi_{_K},\chi^c_{_K})$ is an $IFSH$ of $\,G$. $\Box$
\[th210\] If $\,A=(\mu_{A},\lambda_A)\,$ is an $IFSH$ of $\,G$, then for all $\,x\in G$ we have\
$\mu_{A}(x)=\sup\{\alpha \in [0,1]\ |\ x\in
U(\mu_{A};\alpha ) \}$
\
and\
$ \lambda_A(x)=\inf\{\alpha \in [0,1]\ |\ x\in
L(\lambda_A ;\alpha)\}.$
Let $\,\delta=\sup\{\alpha\in [0,1]\ |\ x\in
U(\mu_{A};\alpha )\}\,$ and let $\,\varepsilon >0\,$ be given. Then $\delta -\varepsilon <\alpha$ for some $\,\alpha \in [0,1]$ such that $\,x\in U(\mu_{A};\alpha)$. This means that $\,\delta
-\varepsilon <\mu_{A}(x)\,$ so that $\,\delta \leq\mu_{A}(x)\,$ since $\,\varepsilon\,$ is arbitrary.
We now show that $\,\mu_{A}(x)\leq\delta.$ If $\,\mu_{A}(x)=\beta$, then $\,x\in U(\mu_{A};\beta)\,$ and so $$\beta\in\{\alpha\in [0,1]\ |\ x\in U(\mu_{A};\alpha )\}.$$ Hence $$\mu_{A}(x)=\beta\leq\sup\{\alpha\in [0,1]\ |\ x\in
U(\mu_{A};\alpha)\}=\delta .$$ Therefore $$\mu_{A}(x)=\delta=\sup\{\alpha\in [0,1]\ |\ x\in
U(\mu_{A};\alpha)\}.$$ Now let $\,\eta =\inf\{\alpha \in [0,1]\ |\ x\in
L(\lambda_A;\alpha)\}$. Then $$\inf\{\alpha\in [0,1]\ |\ x\in L(\lambda_A;\alpha)\}<\eta
+\varepsilon$$ for any $\,\varepsilon >0,\,$ and so $\,\alpha <\eta
+\varepsilon\,$ for some $\,\alpha\in [0,1]\,$ with $\,x\in
L(\lambda_A ;\alpha)$. Since $\lambda_A(x)\leq\alpha\,$ and $\,\varepsilon\,$ is arbitrary, it follows that $\,\lambda_A(x)\leq\eta$.
To prove $\,\lambda_A(x)\geq\eta$, let $\,\lambda_A(x)=\zeta$. Then $\,x\in L(\lambda_A ;\zeta)\,$ and thus $\,\zeta\in\{\alpha\in [0,1]\ |\ x\in L(\lambda_A ;\alpha)\}$. Hence $$\inf\{\alpha\in [0,1]\ |\ x\in L(\lambda_A ;\alpha)\}\leq\zeta,$$ i.e. $\;\eta\leq\zeta =\lambda_A(x).$ Consequently $$\lambda_A(x)=\eta=\inf\{\alpha\in [0,1]\ |\ x\in L(\lambda_A
;\alpha)\},$$ which completes the proof.
\[th211\] Let $\Omega$ be a non-empty finite subset of $\,[0,1]$. If $\,\{K_{\alpha}\ |\ \alpha\in\Omega \}$ is a collection of sub-hyperquasigroups of $\,G\,$ such that
- $G= \bigcup\limits_{\alpha\in\Omega} K_{\alpha}$,
- $\alpha >\beta\,\Longleftrightarrow\,
K_{\alpha}\subset K_{\beta}$ for all $\alpha,\beta\in\Omega$,
then an intuitionistic fuzzy set $A=(\mu_A,\lambda_A)$ defined on $G$ by
$\mu_{A}(x)=\sup\{\alpha\in\Omega\ |\ x\in
K_{\alpha}\}$ and $\lambda_A(x)=\inf\{\alpha\in\Omega\
|\ x\in K_{\alpha}\}$
is an $IFSH$ of $G$.
According to Theorem \[th27\], it is sufficient to show that the non-empty sets $U(\mu_{A};\alpha)$ and $L(\lambda_A;\beta)$ are sub-hyperquasigroups of $G$. We show that $U(\mu_{A};\alpha)=K_{\alpha}$. This holds, since $$\begin{array}{ll}
x \in U(\mu_{A};\alpha) & \Longleftrightarrow \mu_A(x)\geq \alpha \\
& \Longleftrightarrow \sup \{\gamma \in \Omega \ | \
x \in K_{\gamma} \} \geq \alpha \\
& \Longleftrightarrow \exists \gamma_0 \in \Omega , \
x \in K_{\gamma_0}, \ \gamma_0 \geq \alpha \\
& \Longleftrightarrow x \in K_{\alpha} \ \ \ ({\rm since} \
K_{\gamma_0} \subseteq K_{\alpha} ).
\end{array}$$ Now, we prove that $L(\lambda ; \beta )\not = \emptyset$ is a sub-hyperquasigroup of $G$. We have $$\begin{array}{ll}
x \in L(\lambda_{A};\beta) & \Longleftrightarrow \lambda_A(x)\leq \beta \\
& \Longleftrightarrow \inf \{\gamma \in \Omega \ | \
x \in K_{\gamma} \} \leq \beta \\
& \Longleftrightarrow \exists \gamma_0 \in \Omega , \
x \in K_{\gamma_0}, \ \gamma_0 \leq \beta \\
& \Longleftrightarrow x \in \displaystyle \bigcup_{\gamma \leq \beta} K_{\gamma}
\end{array}$$ and hence $L(\lambda_A ; \beta )= \displaystyle \bigcup_{\gamma \leq \beta} K_{\gamma}$. It is not difficult to see that the union of any family of increasing sub-hyperquasigroups of a given hyperquasigroup is a sub-hyperquasigroup. This completes the proof.
Relations
=========
Let $\alpha\in [0,1]$ be fixed and let $IFSH(G)$ be the family of all intuitionistic fuzzy sub-hyperquasigroups of a hyperquasigroup $G$. For any $A=(\mu_A,\lambda_A)$ and $B=(\mu_B,\lambda_B)$ from $IFSH(G)$ we define two binary relations $\frak U^{\alpha}$ and $\frak L^{\alpha}$ on $IFSH(G)$ as follows: $$(A,B)\in\frak U^{\alpha}\Longleftrightarrow U(\mu_{A};\alpha) =
U(\mu_B;\alpha)$$ and $$(A,B)\in\frak L^{\alpha}\Longleftrightarrow L(\lambda_A
;\alpha)=L(\lambda_B;\alpha)\, .$$ These two relations $\frak U^{\alpha}$ and $\frak L^{\alpha}$ are equivalence relations. Hence $IFSH(G)$ can be divided into the equivalence classes of $\frak U^{\alpha}$ and $\frak L^{\alpha}$, denoted by $[A]_{\frak U^{\alpha}}$ and $[A]_{\frak L^{\alpha}}$ for any $A=(\mu_A,\lambda_A)\in IFSH(G)$, respectively. The corresponding quotient sets will be denoted by $IFSH(G)/\frak
U^{\alpha}$ and $IFSH(G)/\frak L^{\alpha}$, respectively.
For the family $S(G)$ of all sub-hyperquasigroups of $G$ we define two maps $U_{\alpha}$ and $L_{\alpha}$ from $IFSH(G)$ to $S(G)\cup\{\emptyset\}$ by putting $$U_{\alpha}(A)=U(\mu_{A};\alpha) \ \ \text{ and } \ \
L_{\alpha}(A)=L(\lambda_A;\alpha)$$ for each $A=(\mu_A,\lambda_A)\in IFSH(G)$.
It is not difficult to see that these maps are well-defined.
\[lem31\] For any $\alpha\in
(0,1)$ the maps $U_{\alpha}$ and $L_{\alpha}$ are surjective.
Let $\bold 0$ and $\bold 1$ be fuzzy sets in $G$ defined by $\bold 0(x)=0$ and $\bold 1(x)=1$ for all $x\in G$. Then $\bold 0_{\sim}=(\bold 0,\bold 1)\in IFSH(G)$ and $U_{\alpha}(\bold 0_{\sim})=L_{\alpha}(\bold 0_{\sim})=
\emptyset$ for any $\alpha\in (0,1)$. Moreover for any $K\in S(G)$ we have $K_{\sim} = (\chi_{_K},\chi^c_{_K})\in IFSH(G)$, $U_{\alpha}(K_{\sim}) =U(\chi_{_K};\alpha)=K$ and $L_{\alpha}(K_{\sim})=L(\chi^c_{_K};\alpha)=K$. Hence $U_{\alpha}$ and $L_{\alpha}$ are surjective.
\[th31\] For any $\alpha\in
(0,1)$ the sets $IFSH(G)/\frak U^{\alpha}$ and $IFSH(G)/\frak
L^{\alpha}$ are equipotent to $S(G)\cup\{\emptyset\}$.
Let $\alpha\in (0,1)$. Putting $U^*_{\alpha}([A]_{\frak U^{\alpha}}) = U_{\alpha}(A)$ and $L^*_{\alpha}([A]_{\frak L^{\alpha}})=L_{\alpha}(A)$ for any $A=(\mu_A,\lambda_A)\in IFSH(G)$, we obtain two maps $$U^*_{\alpha}:IFSH(G)/\frak U^{\alpha}\to S(G)\cup \{\emptyset\} \
\ {\rm and } \ \ L^*_{\alpha}:IFSH(G)/\frak L^{\alpha}\to
S(G)\cup\{\emptyset\}.$$ If $U(\mu_{A};\alpha)=U(\mu_B;\alpha)$ and $L(\lambda_A;\alpha)=L(\lambda_B;\alpha)$ for some $A=(\mu_A,\lambda_A)$ and $B=(\mu_B,\lambda_B)$ from $IFSH(G)$, then $(A,B)\in\frak U^{\alpha}$ and $(A,B)\in\frak L^{\alpha}$, whence $[A]_{\frak U^{\alpha}}=[B]_{\frak U^{\alpha}}$ and $[A]_{\frak L^{\alpha}}=[B]_{\frak L^{\alpha}}$, which means that $U*_{\alpha}$ and $L^*_{\alpha}$ are injective.
To show that the maps $U^*_{\alpha}$ and $L_{\alpha}$ are surjective, let $K\in S(G)$. Then for $K_{\sim}=(\chi_{_K},\chi^c_{_K})\in IFSH(G)$ we have $U^*_{\alpha}([K_{\sim}]_{\frak U^{\alpha}}) =
U(\chi_{_K};\alpha)=K$ and $L^*_{\alpha}([K_{\sim}]_{\frak
L^{\alpha}})= L(\chi^c_{_K};\alpha)= K$. Also $\bold
0_{\sim}=(\bold 0,\bold 1)\in IFSH(G)$. Moreover $U^*_{\alpha}([\bold 0_{\sim}]_{\frak U^{\alpha}})=U(\bold
0;\alpha)=\emptyset$ and $L^*_{\alpha}([\bold 0_{\sim}]_{\frak
L^{\alpha}}) = L(\bold 1;\alpha)=\emptyset .$ Hence $U^*_{\alpha}$ and $L^*_{\alpha}$ are surjective.
Now for any $\alpha \in [0,1]$ we define a new relation $\frak
R^{\alpha}$ on $IFSH(G)$ by putting: $$(A,B)\in\frak R^{\alpha}\Longleftrightarrow U(\mu_{A};\alpha)\cap
L(\lambda_A;\alpha) =U(\mu_B;\alpha)\cap L(\lambda_B;\alpha),$$ where $A=(\mu_A,\lambda_A)$ and $B=(\mu_B,\lambda_B)$. Obviously $\frak R^{\alpha}$ is an equivalence relation.
\[lem33\] The map $\;I_{\alpha}: IFSH(G)\to S(G)\cup\{\emptyset\}$ defined by $$I_{\alpha}(A)=U(\mu_{A};\alpha)\cap L(\lambda_A;\alpha),$$ where $A=(\mu_{A},\lambda_A)$, is surjective for any $\alpha\in
(0,1)$.
If $\alpha\in (0,1)$ is fixed, then for $\bold 0_{\sim}=(\bold 0,\bold 1)\in IFSH(G)$ we have $$I_{\alpha}(\bold 0_{\sim})=U(\bold 0;\alpha)\cap L(\bold 1;\alpha)
= \emptyset\, ,$$ and for any $K\in S(G)$ there exists $K_{\sim}=(\chi_{_K},\chi^c_{_K})\in IFSH(G)$ such that $I_{\alpha}(K_{\sim})=U(\chi_{_K};\alpha)\cap
L(\chi^c_{_K};\alpha) = K$.
\[th34\] For any $\alpha\in (0,1)$ the quotient set $IFSH(G)/\frak R^{\alpha}$ is equipotent to $S(G)\cup\{\emptyset\}$.
Let $I^*_{\alpha} : IFSH(G)/\frak R^{\alpha}\to
S(G)\cup\{\emptyset\}$, where $\alpha\in (0,1)$, be defined by the formula: $$I^*_{\alpha}([A]_{\frak R^{\alpha}}) = I_{\alpha}(A) \ \ \text{
for each } \ \ [A]_{\frak R^{\alpha}}\in IFSH(G)/\frak R^{\alpha}.$$ If $I^*_{\alpha}([A]_{\frak R^{\alpha}})=I^*_{\alpha}([B]_{\frak
R^{\alpha}})$ for some $[A]_{\frak R^{\alpha}},\, [B]_{\frak
R^{\alpha}}\in IFSH(G)/\frak R^{\alpha}$, then $$U(\mu_{A};\alpha)\cap L(\lambda_A;\alpha)=U(\mu_B;\alpha)\cap
L(\lambda_B;\alpha) ,$$ which implies $(A,B)\in\frak R^{\alpha}$ and, in the consequence, $[A]_{\frak R^{\alpha}}=[B]_{\frak R^{\alpha}} .$ Thus $I^*_{\alpha}$ is injective.
It is also onto because $I^*_{\alpha}(\bold
0_{\sim})=I_{\alpha}(\bold 0_{\sim})=\emptyset$ for $\bold
0_{\sim}=(\bold 0,\bold 1)\in IFSH(G)$, and $I^*_{\alpha}(K_{\sim}) = I_{\alpha}(K) = K$ for $K\in S(G)$ and $K_{\sim}=(\chi_{_K},\chi^c_{_K})\in IFSH(G)$.
Connections with binary quasigroups
===================================
A groupoid $(Q,\cdot)$ is called a (*binary*) *quasigroup* if each of the equations $ax=b$ and $ya=b$ has a unique solution for any $a,b\in Q$. Since a non-empty subset of $Q$ closed with respect to this operation is not in general a quasigroup we must use the another equivalent definition of a quasigroup. A quasigroup $(Q,\cdot)$ can be defined (cf. [@olga]) as an algebra $(Q,\cdot,\setminus, \, / )$ with three binary operation such that $(Q,\cdot )$ is a quasigroup in the above sense and $$x\setminus y=z\Leftrightarrow xz=y \ \ \text{ and } \ \ x/ y=z
\Leftrightarrow zy=x$$ for all $x,y,z\in Q$. In this case a non-empty subset of $Q$ is a subquasigroup of $(Q,\cdot)$ (and $(Q,\cdot ,\setminus,\, /)$) if and only if it is closed with respect to these three operations. This gives the possibility to the introduction of a good definition of intuitionistic fuzzy subquasigroups of binary quasigroups [@11].
\[def35\] Let $(Q,\cdot)$ be a quasigroup. An intuitionistic fuzzy set $A=(\mu_A, \lambda_A)$ in $Q$ is called an *intuitionistic fuzzy subquasigroup* of $Q$ if
- $\min \{ \mu_A(x), \mu_A(y) \} \leq \mu_A(x*y)$
- $\lambda_A(x*y)\leq \max \{ \lambda_A(x), \lambda_A(y)
\}$
hold for all $x,y\in Q$ and $*\in\{\cdot,\setminus,\, /\}.$
In this case an intuitionistic fuzzy set $A=(\mu_A, \lambda_A)$ is an intuitionistic fuzzy subquasigroup of $(Q,\cdot,\setminus, \, /
)$ if and only if all non-empty $U(\mu;t)$ and $L(\mu;t)$ are subquasigroups of $(Q,\cdot,\setminus, \, / )$ (cf. [@11]).
A hyperquasigroup $(G, \circ )$ is called *regular* if $$x \in y\circ z \ \ {\rm implies } \ \ y\in x \circ z \ \ {\rm and
} \ \ z \in y \circ x$$ for all $x,y,z \in G$. Let $(G,\circ )$ be a regular hyperquasigroup. The relation $\beta^*$ is the smallest equivalence relation on $G$ such that the quotient $G/\beta^*$, the set of all equivalence classes, is a quasigroup. $\beta^*$ is called the fundamental equivalence relation on $G$ and $G/\beta^*$ is called the fundamental quasigroup. The equivalence relation $\beta^*$ was introduced by Koskas [@13] and studied mainly by Corsini [@4] and Freni [@9], [@10] concerning hypergroups and Vougiouklis [@16] concerning $H_v$-groups.
Let us denote by ${\cal U}$ the set of all finite products of elements of $G$ as follows: $$x \beta y \ {\rm if \ and \ only \ if } \ \{x,y \} \subseteq u \
{\rm for \ some} \ u \in {\cal U}.$$ The fundamental relation $\beta^*$ is the transitive closure of the relation $\beta$ (see Theorem 1.2.2 in [@16]). Suppose $\beta^*(a)$ is the equivalence class containing $a \in G$. Then the product “$\cdot$” on $G/\beta^*$ is defined as follows: $$\beta^*(a)\cdot\beta^*(b)=\beta^*(c)\ \ {\rm for \ all} \ \
c\in\beta^*(a)\circ\beta^*(b).$$ In this case, each of the equations $\beta^*(a)\cdot
\beta^*(x)=\beta^*(b)$ and $\beta^*(y)\cdot \beta^*(a)=\beta^*(b)$ has a unique solution for any $\beta^*(a), \beta^*(b) \in
G/\beta^*$. The quasigroup $(G/\beta^*, \cdot , \setminus , / )$ corresponds to quasigroup $(G/\beta^* , \cdot )$, where $$\begin{array}{lll}
\beta^*(x)\setminus\beta^*(y)=\beta^*(z) & \Longleftrightarrow &
\beta^*(x) \cdot \beta^*(z)=\beta^*(y),\\
\beta^*(x) / \beta^*(y)=\beta^*(z) & \Longleftrightarrow &
\beta^*(z) \cdot \beta^*(y)=\beta^*(x).
\end{array}$$ Let $\mu$ be a fuzzy set in $G$. The fuzzy set $\mu_{\beta^*}$ in $G/\beta^*$ is defined as follows: $$\mu_{\beta^*}:G/\beta^*\to [0,1], \quad
\beta^*(x)\mapsto \sup\{\mu(a)\ | \ a\in\beta^*(x)\}.$$ Now, we have
\[th36\] Let $G$ be a regular hyperquasigroup and $A=(\mu_A, \lambda_A)$ an intuitionistic fuzzy sub-hyperquasigroup of $G$. Then $A/\beta^*=(\mu_{\beta^*}, \lambda_{\beta^*})$ is an intuitionistic fuzzy subquasigroup of the fundamental quasigroup $G/\beta^*$.
[**Acknowledgements.**]{} The authors are highly grateful to the referees for their valuable comments and suggestions for improving the paper.
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[^1]: [*E-mail address:*]{} dudekim.pwr.wroc.pl (W. A. Dudek), davvazyazduni.ac.ir (B. Davvaz), ybjunnongae.gsnu.ac.kr (Y. B. Jun) ()
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abstract: 'Radiation hydrodynamics simulations based on the one-fluid two-temperature model may violate the law of energy conservation because the governing equations are expressed in a nonconservative formulation. Here, we maintain the important physical requirements by employing a strategy based on the key concept that the mathematical structures associated with the conservative and nonconservative equations are preserved, even at the discrete level. To this end, we discretize the conservation laws and transform them via exact algebraic operations. The proposed scheme maintains the global conservation errors within the round-off level. In addition, a numerical experiment concerning the shock tube problem suggests that the proposed scheme well agrees with the jump conditions at the discontinuities regulated by the Rankine–Hugoniot relationship. The generalized derivation allows us to employ arbitrary central difference, artificial dissipation, and Runge–Kutta methods.'
address: 'Department of Aerospace Engineering, Tohoku University, 6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai, Miyagi 980-8579, Japan'
author:
- Takashi Shiroto
- Soshi Kawai
- Naofumi Ohnishi
title: 'Structure-preserving operators for thermal-nonequilibrium hydrodynamics'
---
Radiation hydrodynamics ,Nonequilibrium hydrodynamics ,Conservative scheme ,Structure-preserving scheme
Introduction {#sec:1}
============
Radiation hydrodynamics (RHD)[@MihalasMihalas] is one of the major techniques employed in laser-plasma simulations. In RHD, a neutral charge is assumed for a fluid composed of ions and electrons. This assumption allows large-scale plasma simulations to be performed, as the grid interval is not limited by the Debye length, unlike the conventional particle-in-cell (PIC) method. This type of simulation has been employed in laser-plasma simulations that model, for example, the implosion dynamics of inertial confinement fusion (ICF)[@Nuckolls1972].
One of the most simplified RHD simulations combines radiative transfer and one-temperature hydrodynamics, which are employed by the FastRad3D code[@Bates2016]. However, thermal nonequilibrium typically exists between the ions and electrons of a laser plasma. The incident laser is absorbed by the inverse bremsstrahlung process, and the energy is deposited on the electrons. Thus, the one-fluid two-temperature (1F2T) model has been employed in order to investigate plasma hydrodynamics more accurately [@Fujioka2004-2; @Dangelo2013; @Tanaka2015]. The governing equations of the 1F2T model excluding viscous and heat conduction effects are expressed in the form $$\begin{aligned}
\frac{\partial \rho}{\partial t}+\nabla\cdot(\rho \mathbf{u})=0,\label{eq:1.1}\\
\frac{\partial \rho \mathbf{u}}{\partial t}+\nabla\cdot(\rho \mathbf{uu}+p_\mathrm{i}+p_\mathrm{e})=0,\label{eq:1.2}\\
\frac{\partial \rho e_\mathrm{i}}{\partial t}+\nabla\cdot(\rho e_\mathrm{i}\mathbf{u})+p_\mathrm{i}\nabla\cdot\mathbf{u}=0,\label{eq:1.3}\\
\frac{\partial \rho e_\mathrm{e}}{\partial t}+\nabla\cdot(\rho e_\mathrm{e}\mathbf{u})+p_\mathrm{e}\nabla\cdot\mathbf{u}=0,\label{eq:1.4}\end{aligned}$$ where $\rho$ is the mass density, $\mathbf{u}=^\mathrm{T}[u,v,w]$ is the flow velocity, $e$ is the specific internal energy, and $p$ is the pressure. The subscripts “$\mathrm{i}$” and “$\mathrm{e}$” denote the ions and electrons, respectively. Note that nonhydrodynamic energy transports such as radiative transfer are neglected. These nonconservative equations have been used in many RHD simulations [@Colombier2005; @Cao2015].
Three main methods of solving the 1F2T model exist. The first (hereafter referred to as “nonconservative method") discretizes the governing equations directly with the Lagrangian or arbitrary Lagrangian–Eulerian (ALE) grids[@Hirt1974]. This method has been adopted for use in the RHD codes known as HYDRA[@Marinak2001], DRACO[@Keller1999; @Radha2005], and PINOCO[@Nagatomo2007], which are specialized for ICF implosions. However, because the nonconservative equations are discretized directly using the artificial dissipation, the law of energy conservation is not satisfied, especially near the discontinuities such as a shock wave. For example, it has been reported that Helios-CR[@MacFarlane2006], a one-dimensional RHD code, has an error of several percent with regard to the law of energy conservation in the typical cases. The second discretization technique is based on approximate Riemann solvers and methods of this kind are employed by CRASH[@vanderHolst2011], FLASH[@Fryxell2000], RAGE[@Gittings2008], and RAICHO[@Ohnishi2012], which are primarily implemented in astrophysical simulations. Note that some of these codes incorporate adaptive mesh refinement (AMR) for the performance of high-resolution simulations, and cross-code comparisons are also conducted[@Joggerst2014]. These codes are designed to solve the Euler equation and the hydrodynamic heating is artificially divided between the ions and electrons in proportion to the pressure ratio. Therefore, the results are not guaranteed to converge to the exact solutions, because the governing equations are not discretized directly. However, the energy conservation is satisfied even in the discrete form.
The most recently developed method (hereafter, “conservative method") is implemented in the ASTER[@Igmenshchev2016] code developed at the Laboratory of Laser Energetics, University of Rochester. This code solves the conservation laws of mass, momentum, and total energy, along with the energy equation given in Eq. (\[eq:1.4\]). Thus, the conservation law of total energy is satisfied automatically and the numerical solutions converge to the exact solutions, provided the solutions do not possess any discontinuities. However, relatively high-intensity lasers ($\sim$10$^{15}\mathrm{\ W/cm^2}$) are usually employed in typical laser-plasma investigations[@Smalyuk2010; @Hu2012; @Haines2016; @Shiroto2016], which yield strong shock waves. The internal energies of the ions and electrons may not be accurately determined near these shock waves. This problem has a critical impact on RHD simulations, because the emission and absorption coefficients of radiative transfer are strongly dependent on the electron internal energy.
In this paper, we propose a structure-preserving scheme that can overcome the above problems. The proposed scheme should achieve the following requirements: (i) solve the partially nonconservative equations (\[eq:1.1\])–(\[eq:1.4\]), (ii) maintain the error of energy conservation at the round-off level, and (iii) get a good agreement with exact solutions including discontinuities. In Section \[sec:2\], we derive the conservative scheme through a discussion using the discrete mathematics. The mathematical keys for the exact derivation are the product and quotient rules in the discrete calculus; therefore, these tools are also introduced. The numerical methods employed in this study are described in Section \[sec:3\]. Furthermore, in Section \[sec:4\], the results of a numerical experiment on an extended shock-tube problem are discussed, and the superiority of the proposed scheme over the existing nonconservative and conservative methods is demonstrated. The accuracy of the proposed scheme is verified in Section \[sec:5\] via a linear advection problem concerning entropy waves. Section \[sec:6\] presents the conclusion of this article.
Mathematical background of the proposed approach {#sec:2}
================================================
Causes of energy conservation law violation {#sec:2.0}
-------------------------------------------
Equations (\[eq:1.1\])–(\[eq:1.4\]) can be mathematically transformed into the law of energy conservation by using the product and quotient rules, $$\begin{aligned}
\frac{\partial \left(\rho e_\mathrm{i}+\rho e_\mathrm{e}+\frac12\rho |\mathbf{u}|^2\right)}{\partial t}+
\nabla\cdot\left(\rho e_\mathrm{i}\mathbf{u}+\rho e_\mathrm{e}\mathbf{u}+\frac12\rho |\mathbf{u}|^2\mathbf{u}
+p_\mathrm{i}\mathbf{u}+p_\mathrm{e}\mathbf{u}\right)=0.\label{eq:1.5}\end{aligned}$$ Physically, the law of energy conservation should be maintained by solving Eqs. (\[eq:1.1\])–(\[eq:1.4\]). However, when these equations are solved in discrete form, the energy conservation is often violated. Sometimes, this error can be fatal to the simulations. The conservation law is violated because the energy equations (\[eq:1.3\]) and (\[eq:1.4\]) are expressed in nonconservative formulation. To explain why the conservation laws are violated in the discrete form, we convert Eq. (\[eq:1.1\]) into nonconservative form using the product rule, as an example. Thus, Eq. (\[eq:1.1\]) becomes $$\begin{aligned}
\frac{\partial \rho}{\partial t}+\rho\nabla\cdot\mathbf{u}+\mathbf{u}\cdot\nabla\rho=0.\label{eq:1.6}\end{aligned}$$ The forward-time central-space method is then applied to Eqs. (\[eq:1.1\]) and (\[eq:1.6\]) to obtain the discretized equations in the finite difference method (FDM). Thus, $$\begin{aligned}
\frac{\rho^{n+1}_j-\rho^n_j}{\Delta t}+\frac{\langle\rho u\rangle^n_{j^+}
-\langle \rho u\rangle^n_{j^-}}{\Delta x}=0,\label{eq:1.7}\\
\frac{\rho^{n+1}_j-\rho^n_j}{\Delta t}+\rho^n_j\frac{\langle u\rangle^n_{j^+}-\langle u\rangle^n_{j^-}}{\Delta x}
+u^n_j\frac{\langle \rho\rangle^n_{j^+}-\langle \rho\rangle^n_{j^-}}{\Delta x}=0,\label{eq:1.8}\end{aligned}$$ where “$\langle \rangle$” denotes an arbitrary interpolation operator at the half points ($j^+=j+1/2, j^-=j-1/2$), and $n$ and $j$ are the indices of time and space, respectively. Note that the spatial derivatives in the $y$ and $z$ directions are omitted for simplicity. Of course, Eqs. (\[eq:1.7\]) and (\[eq:1.8\]) correspond to Eqs. (\[eq:1.1\]) and (\[eq:1.6\]), respectively. We take the summation of Eqs. (\[eq:1.7\]) and (\[eq:1.8\]) over the computational domain in order to discuss the global conservation in the discrete system; thus, we obtain $$\begin{aligned}
\sum_{j=1}^N\frac{\rho^{n+1}_j-\rho^n_j}{\Delta t}+\sum_{j=1}^N\frac{\langle \rho u\rangle^n_{j^+}
-\langle\rho u\rangle^n_{j^-}}{\Delta x}=0,\label{eq:1.9}\\
\sum_{j=1}^N\frac{\rho^{n+1}_j-\rho^n_j}{\Delta t}+\sum_{j=1}^N\rho^n_j\frac{\langle u\rangle^n_{j^+}-\langle u\rangle^n_{j^-}}{\Delta x}
+\sum_{j=1}^Nu^n_j\frac{\langle\rho\rangle^n_{j^+}-\langle\rho\rangle^n_{j^-}}{\Delta x}=0,\label{eq:1.10}\end{aligned}$$ where $N$ is the number of grids. The advection terms of Eq. (\[eq:1.9\]) are cancelled, yielding $$\begin{aligned}
\sum_{j=1}^N\frac{\rho^{n+1}_j-\rho^n_j}{\Delta t}+\frac{\langle \rho u\rangle^n_{N+1/2}
-\langle\rho u\rangle^n_{1/2}}{\Delta x}=0.\label{eq:1.11}\end{aligned}$$ Therefore, the total mass of the discrete system does not change if the periodic \[$\langle \rho u\rangle_{1/2}=\langle\rho u\rangle_{N+1/2}$\] or Neumann \[$\langle\rho u\rangle_{1/2}=\langle\rho u\rangle_{N+1/2}=0$\] boundaries are applied. In contrast, Eq. (\[eq:1.10\]) violates the conservation laws, as the advection terms are not generally cancelled.
The above example is the most simple explanation of why the conservation laws are violated in the discrete form, and of why the mathematical structure of the product rule should be preserved, even at the discrete level. Below, we discuss our idea of how to construct a structure-preserving scheme.
Product and quotient rules in discrete form {#sec:2.1}
-------------------------------------------
The nonconservative energy equation is obtained from the conservation laws of mass, momentum, and energy. Although the derivation is performed using the product and quotient rules, these formulae may violate the global conservation in the discrete form. Therefore, we introduce the discrete product and quotient rules.
First, we introduce the product rule in discrete form: $$\begin{aligned}
\frac{f^{n+1}g^{n+1}-f^ng^n}{\Delta t}&=\frac{f^{n+1}g^{n+1}-f^{n+1}g^n+f^{n+1}g^n-f^ng^n}{\Delta t},\notag\\
&=f^{n+1}\frac{g^{n+1}-g^n}{\Delta t}+\frac{f^{n+1}-f^n}{\Delta t}g^n,\label{eq:2.2}\end{aligned}$$ where $f$ and $g$ are arbitrary functions depending on $t$. Algebraic operations performed on the numerator, which add and subtract $f^{n+1}g^n$, correspond to the same approach as that used to prove the original formula in the differential form. The other forms of the product rule are obtained in a similar manner, where $$\begin{aligned}
\frac{f^{n+1}g^{n+1}-f^ng^n}{\Delta t}=\frac{f^{n+1}+f^n}{2}\frac{g^{n+1}-g^n}{\Delta t}+
\frac{f^{n+1}-f^n}{\Delta t}\frac{g^{n+1}+g^n}{2}.\label{eq:2.4}\end{aligned}$$
The quotient rule in discrete form is also obtained via the same strategy, yielding $$\begin{aligned}
\frac{f^{n+1}/g^{n+1}-f^n/g^n}{\Delta t}&=\frac{g^{n+1}+g^n}{2g^{n+1}g^n}\frac{f^{n+1}-f^n}{\Delta t}-
\frac{f^{n+1}+f^n}{2g^{n+1}g^n}\frac{g^{n+1}-g^n}{\Delta t}.\label{eq:2.10}\end{aligned}$$ The quotient rule of Eq. (\[eq:2.10\]) is mathematically equivalent to the product rule of Eq. (\[eq:2.4\]); however, we must expand the kinetic energy term as a function of the mass and momentum. The quotient rule is useful when a conservative variable appears in the denominator of a function.
Derivation of structure-preserving scheme {#sec:2.2}
-----------------------------------------
In this subsection, we derive a scheme which exactly satisfies the law of energy conservation with solving the internal energy of ions and electrons. It is the structure-preserving scheme that the mathematical structure associating the conservative formulation with the nonconservative one is strictly maintained. The Euler equation is discretized using the FDM approach, such that $$\begin{aligned}
\frac{\rho^{n+1}_j-\rho^n_j}{\Delta t}+\frac{\langle\rho u\rangle^n_{j^+}
-\langle\rho u\rangle^n_{j^-}}{\Delta x}=0,\label{eq:2.11}\\
\frac{(\rho u)^{n+1}_j-(\rho u)^n_j}{\Delta t}+
\frac{\langle\rho u^2+p\rangle^n_{j^+}-
\langle\rho u^2+p\rangle^n_{j^-}}{\Delta x}=0,\label{eq:2.12}\\
\frac{(\rho e+\frac12\rho u^2)^{n+1}_j-(\rho e+\frac12\rho u^2)^n_j}{\Delta t}
+\frac{\langle\rho eu+\frac12\rho u^3+p u\rangle^n_{j^+}-
\langle\rho eu+\frac12\rho u^3+pu\rangle^n_{j^-}}{\Delta x}=0,\label{eq:2.13}\end{aligned}$$ where $p=p(\rho, e)$ is the total pressure. As shown in Section \[sec:2.0\], the conservation laws of mass, momentum, and energy are strictly satisfied in the system. Furthermore, although we employ the Euler explicit method here, the scheme can be extended to the Runge–Kutta (RK) methods. Proof of this is given in \[sec:a\]. The multidimensional description is given in \[sec:b\].
The structure-preserving scheme is obtained by expanding Eq. (\[eq:2.13\]). The time derivative of the kinetic energy is expanded in the form $$\begin{aligned}
\frac{(\rho u^2)^{n+1}_j-(\rho u^2)^n_j}{\Delta t}&=
\frac{1}{\Delta t}\left[ \left\{\frac{(\rho u)^2}{\rho}\right\}^{n+1}_j - \left\{\frac{(\rho u)^2}{\rho}\right\}^n_j\right],\notag\\
&=\frac{\rho^{n+1}_j+\rho^n_j}{2\rho^{n+1}_j\rho^n_j}\frac{\{(\rho u)^2\}^{n+1}_j-\{(\rho u)^2\}^n_j}{\Delta t}-
\frac{\{(\rho u)^2\}^{n+1}_j+\{(\rho u)^2\}^n_j}{2\rho^{n+1}_j\rho^n_j}\frac{\rho^{n+1}_j-\rho^n_j}{\Delta t},\notag\\
&=\frac{\rho^{n+1}_j+\rho^n_j}{2\rho^{n+1}_j\rho^n_j}\left\{(\rho u)^{n+1}_j+(\rho u)^n_j\right\}
\frac{(\rho u)^{n+1}_j-(\rho u)^n_j}{\Delta t}
-\frac{\{(\rho u)^2\}^{n+1}_j+\{(\rho u)^2\}^n_j}{2\rho^{n+1}_j\rho^n_j}\frac{\rho^{n+1}_j-\rho^n_j}{\Delta t},\notag\\
&=\frac{\rho^{n+1}_j+\rho^n_j}{2\rho^{n+1}_j\rho^n_j}\left\{(\rho u)^{n+1}_j+(\rho u)^n_j\right\}
\left(-\frac{\langle \rho u^2+p\rangle^n_{j^+}-
\langle\rho u^2+p\rangle^n_{j^-}}{\Delta x}\right)\notag\\
&+\frac{\{(\rho u)^2\}^{n+1}_j+\{(\rho u)^2\}^n_j}{2\rho^{n+1}_j\rho^n_j}
\frac{\langle\rho u\rangle^n_{j^+}-\langle\rho u\rangle^n_{j^-}}{\Delta x}.\label{eq:2.18}\end{aligned}$$ Here, the quotient and product rules of Eqs. (\[eq:2.10\]) and (\[eq:2.4\]), respectively, are used in the derivation. Some readers may feel that this expansion is unnecessary, as both $\rho^{n+1}_j$ and $u^{n+1}_j$ have already been obtained in Eqs. (\[eq:2.11\]) and (\[eq:2.12\]). However, this operation is necessary as it clarifies the contributions of the ion and electron pressures on the right hand side (RHS) of Eq. (\[eq:2.18\]). This clarification is required in order to prove that the proposed scheme qualifies as the discretized equations of Eqs. (\[eq:1.3\]) and (\[eq:1.4\]). In order to obtain the nonconservative formulation, Eq. (\[eq:2.18\]) is substituted into Eq. (\[eq:2.13\]), yielding $$\begin{aligned}
\frac{(\rho e)^{n+1}_j-(\rho e)^n_j}{\Delta t}+
\frac{\langle\rho eu\rangle^n_{j^+}-\langle\rho eu\rangle^n_{j^-}}{\Delta x}+
\frac{\langle pu\rangle^n_{j^+}-\langle pu\rangle^n_{j^-}}{\Delta x}
-\frac{\rho^{n+1}_j+\rho^n_j}{4\rho^{n+1}_j \rho^n_j}\{(\rho u)^{n+1}_j+(\rho u)^n_j\}
\frac{\langle p\rangle^n_{j^+}-\langle p\rangle^n_{j^-}}{\Delta x}=\notag\\
-\frac{(\rho^2 u^2)^{n+1}_j+(\rho^2 u^2)^n_j}{4\rho^{n+1}_j\rho^n_j}
\frac{\langle \rho u\rangle^n_{j^+}-\langle \rho u\rangle^n_{j^-}}{\Delta x}
+\frac{\rho^{n+1}_j+\rho^n_j}{4\rho^{n+1}_j \rho^n_j}\{(\rho u)^{n+1}_j+(\rho u)^n_j\}
\frac{\langle \rho u^2\rangle^n_{j^+}-\langle \rho u^2\rangle^n_{j^-}}{\Delta x}\notag\\
-\frac12\frac{\langle \rho u^3\rangle^n_{j^+}-\langle \rho u^3\rangle^n_{j^-}}{\Delta x}.\label{eq:2.19}\end{aligned}$$ This equation provides important information regarding the energy conservation in the 1F2T model. The left hand side (LHS) of Eq. (\[eq:2.19\]) is a discretized formulation of the nonconservative energy equation $$\begin{aligned}
\frac{\partial (\rho e)}{\partial t}+
\frac{\partial (\rho e u)}{\partial x}+
p\frac{\partial u}{\partial x}=0
\qquad \mathrm{or} \qquad
\frac{\partial (\rho e)}{\partial t}+
\frac{\partial(\rho e u)}{\partial x}+
\frac{\partial (p u)}{\partial x}-u\frac{\partial p}{\partial x}=0.\label{eq:2.21}\end{aligned}$$ The rest of the terms on the RHS are error terms that are cancelled out when $\Delta t \to 0$ and $\Delta x \to 0$. The information on this error is lost in Eq. (\[eq:2.21\]); therefore, it is difficult to reconstruct the error terms using intuitive discretizations. Furthermore, note that Eq. (\[eq:2.21\]) is the sum of Eqs. (\[eq:1.3\]) and (\[eq:1.4\]), which correspond to the first law of thermodynamics for ions and electrons. This relationship must be satisfied even at the discrete level; therefore, Eq. (\[eq:2.19\]) provides a constraint condition that regulates the law of energy conservation in the discrete 1F2T model.
Finally, the structure-preserving scheme for the 1F2T model can be written in the form $$\begin{aligned}
\frac{(\rho e_\mathrm{s})^{n+1}_j-(\rho e_\mathrm{s})^n_j}{\Delta t}+
\frac{\langle\rho e_\mathrm{s}u\rangle^n_{j^+}-\langle\rho e_\mathrm{s}u\rangle^n_{j^-}}{\Delta x}+
\frac{\langle p_\mathrm{s}u\rangle^n_{j^+}-\langle p_\mathrm{s}u\rangle^n_{j^-}}{\Delta x}
-\frac{\rho^{n+1}_j+\rho^n_j}{4\rho^{n+1}_j \rho^n_j}\{(\rho u)^{n+1}_j+(\rho u)^n_j\}
\frac{\langle p_\mathrm{s}\rangle^n_{j^+}-\langle p_\mathrm{s}\rangle^n_{j^-}}{\Delta x}\notag\\
=
-\frac{(\rho^2 u^2)^{n+1}_j+(\rho^2 u^2)^n_j}{8\rho^{n+1}_j\rho^n_j}
\frac{\langle \rho u\rangle^n_{j^+}-\langle \rho u\rangle^n_{j^-}}{\Delta x}
+\frac{\rho^{n+1}_j+\rho^n_j}{8\rho^{n+1}_j \rho^n_j}\{(\rho u)^{n+1}_j+(\rho u)^n_j\}
\frac{\langle \rho u^2\rangle^n_{j^+}-\langle \rho u^2\rangle^n_{j^-}}{\Delta x}\notag\\
-\frac14\frac{\langle \rho u^3\rangle^n_{j^+}-\langle \rho u^3\rangle^n_{j^-}}{\Delta x}.\label{eq:2.22}\end{aligned}$$ The terms incorporating the internal energy ($e$) or pressure ($p$) in Eq. (\[eq:2.19\]) can be easily separated because of the physical requirements. However, the other terms have no explicit restrictions for partition. Equation (\[eq:2.22\]) distributes the error terms equally among the ions and electrons; hence, the symmetry of Eqs. (\[eq:1.3\]) and (\[eq:1.4\]) is retained. The Rankine–Hugoniot relationship and the law of equipartition can only be reproduced using this approach. This will be mentioned in the later verification. Note that the extension to the multitemperature model is straightforward, because of the law of equipartition.
Shock capturing method {#sec:2.2.1}
----------------------
When solving the hydrodynamic field with the discontinuities, the shock capturing method is required to obtain the entropy solutions. Here, we derive the structure-preserving scheme including artificial dissipation terms. The discretized Euler equation Eqs. (\[eq:2.11\])–(\[eq:2.13\]) is modified as $$\begin{aligned}
\frac{\rho^{n+1}_j-\rho^n_j}{\Delta t}+\frac{\langle\rho u\rangle^n_{j^+}
-\langle\rho u\rangle^n_{j^-}}{\Delta x}=0,\label{eq:2.23}\\
\frac{(\rho u)^{n+1}_j-(\rho u)^n_j}{\Delta t}+
\frac{\langle\rho u^2+p\rangle^n_{j^+}-
\langle\rho u^2+p\rangle^n_{j^-}}{\Delta x}=
\frac{\langle A\rangle^n_{j^+}-\langle A\rangle^n_{j^-}}{\Delta x},\label{eq:2.24}\\
\frac{(\rho e+\frac12\rho u^2)^{n+1}_j-(\rho e+\frac12\rho u^2)^n_j}{\Delta t}
+\frac{\langle\rho eu+\frac12\rho u^3+p u\rangle^n_{j^+}-
\langle\rho eu+\frac12\rho u^3+pu\rangle^n_{j^-}}{\Delta x}=
\frac{\langle B\rangle^n_{j^+}-\langle B\rangle^n_{j^-}}{\Delta x},\label{eq:2.25}\end{aligned}$$ where $A$ and $B$ are the artificial dissipations required to capture the discontinuities. The nonconservative equations about the internal energy of ions and electrons are derived by the same way in Sec. \[sec:2.2\]: $$\begin{aligned}
\frac{(\rho e_\mathrm{s})^{n+1}_j-(\rho e_\mathrm{s})^n_j}{\Delta t}+
\frac{\langle\rho e_\mathrm{s}u\rangle^n_{j^+}-\langle\rho e_\mathrm{s}u\rangle^n_{j^-}}{\Delta x}+
\frac{\langle p_\mathrm{s}u\rangle^n_{j^+}-\langle p_\mathrm{s}u\rangle^n_{j^-}}{\Delta x}
-\frac{\rho^{n+1}_j+\rho^n_j}{4\rho^{n+1}_j \rho^n_j}\{(\rho u)^{n+1}_j+(\rho u)^n_j\}
\frac{\langle p_\mathrm{s}\rangle^n_{j^+}-\langle p_\mathrm{s}\rangle^n_{j^-}}{\Delta x}\notag\\
=-\frac{\rho^{n+1}_j+\rho^n_j}{8\rho^{n+1}_j \rho^n_j}\{(\rho u)^{n+1}_j+(\rho u)^n_j\}
\frac{\langle A\rangle^n_{j^+}-\langle A\rangle^n_{j^-}}{\Delta x}
+\frac{\langle B_\mathrm{s}\rangle^n_{j^+}-\langle B_\mathrm{s}\rangle^n_{j^-}}{\Delta x}
-\frac{(\rho^2 u^2)^{n+1}_j+(\rho^2 u^2)^n_j}{8\rho^{n+1}_j\rho^n_j}
\frac{\langle \rho u\rangle^n_{j^+}-\langle \rho u\rangle^n_{j^-}}{\Delta x}\notag\\
+\frac{\rho^{n+1}_j+\rho^n_j}{8\rho^{n+1}_j \rho^n_j}\{(\rho u)^{n+1}_j+(\rho u)^n_j\}
\frac{\langle \rho u^2\rangle^n_{j^+}-\langle \rho u^2\rangle^n_{j^-}}{\Delta x}
-\frac14\frac{\langle \rho u^3\rangle^n_{j^+}-\langle \rho u^3\rangle^n_{j^-}}{\Delta x}.\label{eq:2.26}\end{aligned}$$ Note that the first term on the RHS about $A$, which is associated with the viscosity, is equally separated between ions and electrons since the viscosity transfers the momentum whose contributions of ions and electrons are inseparable.
Summary of the proposed approach {#sec:2.2.2}
--------------------------------
The summary of the proposed approach toward construction of the structure-preserving scheme is as follows:
1. Discretize the conservation laws of mass, momentum, and total energy Eqs. (\[eq:2.27\])–(\[eq:2.29\]) so that the conservation laws are automatically satisfied even in the discrete level.
2. Expand the time derivative of kinetic energy in Eq. (\[eq:2.29\]) using the discrete product and quotient rules to obtain the nonconservative energy equation about $\rho (e_\mathrm{i}+e_\mathrm{e})$ with error terms.
3. Separate the contributions of ions and electrons in the discretized nonconservative energy equation. The terms with the pressure $p_\mathrm{s}$ and the specific internal energy $e_\mathrm{s}$ are separated by the physically accurate way. The rest terms only including the density $\rho$ and the velocity $u$ are mathematically equally shared by assuming the law of equipartition.
4. Solve the discretized equations about the density $\rho$, the momentum $\rho u$, the internal energy of ions $\rho e_\mathrm{i}$, and that of electrons $\rho e_\mathrm{e}$.
$$\begin{aligned}
\frac{\partial \rho}{\partial t}+\frac{\partial (\rho u)}{\partial x}=0,\label{eq:2.27}\\
\frac{\partial (\rho u)}{\partial t}+\frac{\partial (\rho u^2+p_\mathrm{i}+p_\mathrm{e})}{\partial x}=0,\label{eq:2.28}\\
\frac{\partial (\rho e_\mathrm{i}+\rho e_\mathrm{e}+\frac12\rho u^2)}{\partial t}+
\frac{\partial (\rho e_\mathrm{i}u+\rho e_\mathrm{e}u+\frac12\rho u^3+p_\mathrm{i}u+p_\mathrm{e}u)}{\partial x}=0.\label{eq:2.29}\end{aligned}$$
The above methodology is also available if the artificial dissipation terms are included for the purpose of the shock capturing method.
Numerical implementations {#sec:3}
=========================
In this study, the fourth-order Padé-type interpolation[@Kobayashi1999] is utilized, with $$\begin{aligned}
\frac14 \langle f\rangle_{j-1/2}+\langle f\rangle_{j+1/2}+
\frac14 \langle f\rangle_{j+3/2}=\frac32(f_j+f_{j+1}),\label{eq:3.1}\end{aligned}$$ where $f$ is the flux of an arbitrary conservative equation. This interpolation operator has the linearity as follows: $$\begin{aligned}
\langle f_1+f_2 \rangle = \langle f_1 \rangle + \langle f_2 \rangle,\label{eq:3.15}\\
\langle \alpha f_1 \rangle = \alpha \langle f_1 \rangle ,\label{eq:3.16}\end{aligned}$$ where $\alpha$ is an arbitrary constant. Time integration is performed using the third-order total variation diminishing (TVD) RK method[@shu1988].
Artificial dissipations are modeled using the bulk viscosity $\beta$ and the thermal conductivity $\kappa_\mathrm{s}$ for ions and electrons, where $$\begin{aligned}
A=\beta\frac{\partial u}{\partial x},\label{eq:3.4}\\
B_\mathrm{s}=\frac12u\beta\frac{\partial u}{\partial x}+
\kappa_\mathrm{s}\frac{\partial e_\mathrm{s}}{\partial x}.\label{eq:3.5}\end{aligned}$$ These transport coefficients are modeled in a similar manner to the localized artificial diffusivity (LAD) scheme[@Kawai2008; @Kawai2010a] $$\begin{aligned}
\beta=C_\beta \rho f_\mathrm{sw}
\left|\frac{\partial^r}{\partial x^r}\left(\frac{\partial u}{\partial x}\right)\right|\Delta x^{r+2},\label{eq:3.7}\\
\kappa_\mathrm{s}=C_\kappa \frac{\rho a}{e_\mathrm{i}+e_\mathrm{e}}
\left|\frac{\partial^r e_\mathrm{s}}{\partial x^r}\right|\Delta x^{r+1},\label{eq:3.8}\\
f_\mathrm{sw}=H\left(-\frac{\partial u}{\partial x}\right),\label{eq:3.9}\end{aligned}$$ where $H$ is the Heaviside function, $a$ is the sound speed, and $r=4$, according to the typical LAD usage. $C_\beta$ and $C_\kappa$ are the nondimensional parameters of the LAD scheme and fixed to $5$ and $2$, respectively. High-order derivatives obtained from these compact schemes have noisy profiles; thus, the obtained $\beta$ and $\kappa_\mathrm{s}$ should be smeared using the appropriate truncated Gaussian blur[@Cook2004]. The first and fourth derivatives are derived using the fourth-order compact schemes: $$\begin{aligned}
\frac14\left(\frac{\partial f}{\partial x}\right)_{j-1}+\left(\frac{\partial f}{\partial x}\right)_{j}+
\frac14\left(\frac{\partial f}{\partial x}\right)_{j+1}=\frac32\frac{f_{j-1}-f_{j+1}}{2\Delta x},\label{eq:3.10}\\
\frac14\left(\frac{\partial^4 f}{\partial x^4}\right)_{j-1}+\left(\frac{\partial^4 f}{\partial x^4}\right)_{j}+
\frac14\left(\frac{\partial^4 f}{\partial x^4}\right)_{j+1}=\frac32\frac{f_{j-2}-
4f_{j-1}+6f_{j}-4f_{j+1}+f_{j+2}}{\Delta x^4}.\label{eq:3.11}\end{aligned}$$ Note that tridiagonal matrices appearing in the compact schemes are solved using the Thomas algorithm.
Hydrodynamic simulations with compact differences are often coupled with low-pass filters[@Lele1992; @Gaitonde2000], owing to the stabilization of numerical dispersion especially at high wavenumbers. Low-pass filters are usually applied to the conservative variables. The 1F2T model only has three conservation laws of mass, momentum, and energy although the number of governing equations Eqs. (\[eq:1.1\])–(\[eq:1.4\]) is four. Therefore, the filtering scheme should be applied to the 1F2T model carefully. We derive a constraint condition that should be preserved through the filtering operations: $$\begin{aligned}
\rho^\star=\lceil\rho\rfloor,\label{eq:3.12}\\
u^\star=\frac{\lceil\rho u\rfloor}{\lceil\rho\rfloor},\label{eq:3.13}\\
e^\star_\mathrm{i}=\frac{\lceil\rho e_\mathrm{i}\rfloor}{\lceil\rho\rfloor}+
\frac14\left(\frac{\lceil\rho u^2\rfloor}{\lceil\rho\rfloor}-
\frac{\lceil\rho u\rfloor^2}{\lceil\rho\rfloor^2}\right),\label{eq:3.14}\\
e^\star_\mathrm{e}=\frac{\lceil\rho e_\mathrm{e}\rfloor}{\lceil\rho\rfloor}+
\frac14\left(\frac{\lceil\rho u^2\rfloor}{\lceil\rho\rfloor}-
\frac{\lceil\rho u\rfloor^2}{\lceil\rho\rfloor^2}\right),\label{eq:3.15}\end{aligned}$$ where the superscript “$\star$” denotes the filtered quantities and “$\lceil \ \rfloor$” is the filtering operator. The momentum and total energy composed of the filtered primitive variables are $$\begin{aligned}
\rho^\star u^\star=\lceil\rho u\rfloor,\\
\rho^\star e^\star_\mathrm{i}+\rho^\star e^\star_\mathrm{e}+\frac12 \rho^\star (u^\star)^2=
\lceil \rho e_\mathrm{i}+\rho e_\mathrm{e}+\frac12\rho u^2 \rfloor,\end{aligned}$$ where we assume that the filtering operator also has the linearity. Therefore, the conservation laws are maintained when the filtering operators are used like Eqs. (\[eq:3.12\])–(\[eq:3.15\]). In this study, an eighth-order compact filter[@Shiroto2017a] is used in the final step of the RK integration to stabilize the numerical dispersion, such that $$\begin{aligned}
\alpha_\mathrm{f} g_{j-1/2}+ g_{j+1/2}+ \alpha_\mathrm{f} g_{j+3/2}=
\sum_{n=0}^3 b_n(1-2\alpha_\mathrm{f})(f_{j+1+n}+f_{j-n}),\label{eq:3.2}\\
\lceil f \rfloor_j = f_j - (g_{j+\frac12}-g_{j-\frac12}),\label{eq:3.3}\end{aligned}$$ where $b_0=-\frac{35}{256}$, $b_1=\frac{21}{256}$, $b_2=-\frac{7}{256}$, and $b_3=\frac{1}{256}$. The filtering parameter $\alpha_\mathrm{f}$ is set to 0.495.
Verification via shock tube problem {#sec:4}
===================================
The shock tube problem is employed to verify the effects of the conservative and nonconservative schemes, as this problem includes discontinuous solutions, i.e., the shock wave and contact discontinuity[@Sod1978]. Here, we extend the shock tube problem to the 1F2T model, so that the proposed scheme can be verified using the exact solution.
We assume that a diaphragm separating the high- and low-pressure sections of a tube bursts at $t=0$ and $x=0.5$. The computational domain is $0 \le x \le 1$ and the number of grids is 201. The initial conditions are almost identical to the original Sod’s problem: $\rho_\mathrm{L}=1$, $\rho_\mathrm{R}=0.125$, $u_\mathrm{L}=u_\mathrm{R}=0$, $e_{\mathrm{i,L}}=1.5$, and $e_{\mathrm{i,R}}=e_{\mathrm{e,L}}=e_{\mathrm{e,L}}=1$, where the subscripts “$\mathrm{L}$” and “$\mathrm{R}$” denote the areas on the left and right sides of the diaphragm, respectively. The quantity $e_\mathrm{i}+e_\mathrm{e}$ is identical to the original Sod’s problem. Therefore, these conditions simply add the temperature nonequilibrium between the ions and electrons to the original problem. Note that $e$ is associated with $p$ by the thermally and calorically ideal equations of state (EOS), with the ratio of specific heat $\gamma=1.4$. The boundaries satisfy the Neumann condition such that the summations of the mass and energy are fixed to the initial values.
Here we show the exact solution of the 1F2T shock tube problem. The governing equations can be described by the following quasi-linear form: $$\begin{aligned}
\begin{bmatrix}
\dfrac{\partial \rho}{\partial t} \\[2ex]
\dfrac{\partial u}{\partial t} \\[2ex]
\dfrac{\partial e_\mathrm{i}}{\partial t} \\[2ex]
\dfrac{\partial e_\mathrm{e}}{\partial t}
\end{bmatrix}
+
\begin{bmatrix}
u & \rho & 0 & 0 \\[2ex]
\dfrac{(\gamma-1)(e_\mathrm{i}+e_\mathrm{e})}{\rho} & u & \gamma-1 & \gamma-1 \\[2ex]
0 & (\gamma-1)e_\mathrm{i} & u & 0 \\[2ex]
0 & (\gamma-1)e_\mathrm{e} & 0 & u
\end{bmatrix}
\begin{bmatrix}
\dfrac{\partial \rho}{\partial x} \\[2ex]
\dfrac{\partial u}{\partial x} \\[2ex]
\dfrac{\partial e_\mathrm{i}}{\partial x} \\[2ex]
\dfrac{\partial e_\mathrm{e}}{\partial x}
\end{bmatrix}
=
\begin{bmatrix}
0 \\[2ex] 0 \\[2ex] 0 \\[2ex] 0
\end{bmatrix}.\label{eq:4.1}\end{aligned}$$ This is a hyperbolic system which has the eigenvectors $\mathbf{k}$ and the corresponding eigenvalues $\lambda$: $$\begin{aligned}
\mathbf{k}_1=\begin{bmatrix}
\rho \\ -a \\ (\gamma-1)e_\mathrm{i} \\ (\gamma-1)e_\mathrm{e}
\end{bmatrix},\quad
\mathbf{k}_2=\begin{bmatrix}
\rho \\ 0 \\ -e_\mathrm{i} \\ 0
\end{bmatrix},\quad
\mathbf{k}_3=\begin{bmatrix}
\rho \\ 0 \\ 0 \\ -e_\mathrm{e}
\end{bmatrix},\quad
\mathbf{k}_4=\begin{bmatrix}
\rho \\ a \\ (\gamma-1)e_\mathrm{i} \\ (\gamma-1)e_\mathrm{e}
\end{bmatrix},\label{eq:4.2}\\
\lambda_1 = u-a,\quad \lambda_2=\lambda_3=u,\quad \lambda_4=u+a,\label{eq:4.3}\end{aligned}$$ where $a=\sqrt{\gamma(p_\mathrm{i}+p_\mathrm{e})/\rho}$ is the sound speed. The first and fourth eigenmodes denote the pressure waves while the second and third ones are the entropy waves for ions and electrons, respectively. According to the eigenstructure, the ratio of internal energy $e_\mathrm{s}/(e_\mathrm{i}+e_\mathrm{e})$ only changes at the contact surface. The exact solutions for each internal energies are easily derived from the original Sod’s solution with this fact.
Figure \[fig:4.2\] shows a time history of the errors with respect to the global conservation of energy. As mentioned in Section \[sec:1\], the nonconservative method obviously violates the law of energy conservation, because this method discretizes the nonconservative equations. On the other hand, the conservative and proposed methods keep the errors within the machine zero level of the double-precision floating-point numbers ($\sim$2$~\times 10^{-16}$). This is because the law of energy conservation is discretized directly using these techniques. The behaviors of these errors differ from each other; however, this discrepancy has no physical meaning, as these are the round-off rather than truncation errors. Therefore, the conservative and proposed schemes are proven to be conservative. Moreover, the round-off errors of these schemes do not linearly accumulate in the numerical experiments. This is primarily because we employ the finite-volume-method (FVM) like interpolation and filtering schemes[@Kobayashi1999; @Shiroto2017a]. Note that this is the key to maintain the global conservation errors at the round-off level.
Figures \[fig:4.3\] and \[fig:4.4\] are spatial profiles of $e_\mathrm{i}$ and $e_\mathrm{e}$, respectively. The proposed scheme well agrees with the exact solutions including the discontinuities. However, profiles obtained by the existing nonconservative and conservative methods clearly deviate from the exact solutions between the shock wave and contact surface. The exact solutions are associated with the Rankine–Hugoniot relationship; thus, they are influenced by the local principles such as the conservation laws. Therefore, Figures \[fig:4.3\] and \[fig:4.4\] indicate that the nonconservative and conservative schemes violate some of these principles. For the nonconservative scheme, $e_\mathrm{i}$ and $e_\mathrm{e}$ remain identical between the discontinuities, while $e_\mathrm{i}+e_\mathrm{e}$ is lower than the exact solution. In other words, the nonconservative scheme maintains the law of equipartition but violates the law of energy conservation at the discrete level. In contrast, the conservative scheme reproduces the spatial profile of $e_\mathrm{i}+e_\mathrm{e}$ but cannot maintain the equilibrium of the ions and electrons between the discontinuities. This finding suggests that the conservative scheme maintains the law of energy conservation, but the law of equipartition is violated near the discontinuities. Note that the law of equipartition is strongly related to the symmetry of Eqs. (\[eq:1.3\]) and (\[eq:1.4\]). Thus, this symmetry should be preserved at the discrete level. Both the nonconservative scheme, which directly discretizes the nonconservative equations, and the proposed scheme determine $e_\mathrm{i}$ and $e_\mathrm{e}$ with the symmetric formulation. In contrast, although the energy equation of Eq. (\[eq:1.4\]) is discretized directly by the conservative scheme, $e_\mathrm{i}$ is indirectly obtained from the other discretized equations; this explains why the existing conservative scheme violates the law of equipartition. The derivation of the proposed scheme is unique; the mathematical structures of the governing equations must be preserved in order to maintain the important physical principles.
![\[fig:4.2\] (Color online) Time histories of errors with respect to law of energy conservation for examined schemes. ](global){width="80.00000%"}
![\[fig:4.3\] (Color online) Internal energy profiles for ions $e_\mathrm{i}$ at $t=0.2$, for examined schemes and exact solution. ](ei){width="80.00000%"}
![\[fig:4.4\] (Color online) Internal energy profiles for electrons $e_\mathrm{e}$ at $t=0.2$, for examined schemes and exact solution. ](ee){width="80.00000%"}
Accuracy verification {#sec:5}
=====================
The spatial accuracy is assessed to check whether the formal accuracy is reproduced by the proposed scheme. The 1F2T hydrodynamic equations are decomposed into an eigenstructure comprised of two pressure waves and the entropy waves of the ions and electrons. We introduce this test problem to examine the spatial accuracy of the 1F2T model: $$\begin{aligned}
\rho = 1.1+0.1\cos(2\pi x),\label{eq:5.1}\\
u = 1,\label{eq:5.2}\\
p_\mathrm{i} = 1.1+0.1\cos(4\pi x), \label{eq:5.3}\\
p_\mathrm{e} = 1.1-0.1\cos(4\pi x), \label{eq:5.4}\\
\gamma=1.4, \label{eq:5.5}\end{aligned}$$ where $x\in[0,1]$. The initial profiles are depicted in Fig. \[fig:5.1\]. Again, the EOS is thermally and calorically ideal. These conditions eliminate the pressure waves which correspond to the nonlinear field. Therefore, this is a linearized problem involving ion and electron entropy waves, whose solutions are easily obtained via the hyperbolic solvers. A periodic condition is applied to the boundaries, so that the initial conditions are recovered at $t=1$. The time interval $\Delta t=10^{-5}$ is sufficiently small to make the temporal error negligible.
Figure \[fig:5.3\] is a log-log graph to illustrate the accuracy verification. The results show that $\rho$, $e_\mathrm{i}$, and $e_\mathrm{e}$ have the fourth-order formal accuracy of the compact scheme (Eq. (\[eq:3.1\])). Note that $e_\mathrm{i}$ and $e_\mathrm{e}$ are not discretized separately; the discretizations are only applied to the conservation laws of mass, momentum, and energy. On the other hand, the spatial accuracy of $u$ cannot be observed via this numerical experiment, because $u$ and the static pressure $p_\mathrm{i}+p_\mathrm{e}$ are initially constant in the entire region. Equation (\[eq:2.12\]) is simplified by assuming $u^n_j=U=\mathrm{const.}$ and $p^n_j=\mathrm{const.}$, such that $$\begin{aligned}
\rho^{n+1}\frac{u^{n+1}_j-u^n_j}{\Delta t}+U\frac{\rho^{n+1}_j-\rho^n_j}{\Delta t}
+U\frac{\langle \rho u\rangle^n_{j^+}-\langle \rho u\rangle^n_{j^-}}{\Delta x}=0.\label{eq:5.6}\end{aligned}$$ Here, the relationship $\langle \rho u^2\rangle=U\langle \rho u\rangle$ is valid at the discrete level, because of the linearity of the interpolation operators. Hence, $u^{n+1}_j=u^n_j$ is obtained by substituting Eq. (\[eq:2.11\]) into Eq. (\[eq:5.6\]). This is why the $u$ errors are independent of the grid interval, so that the error-norm always maintains the round-off level.
Note that the discretizations are not based on the FVM but, rather, on the FDM, although FVM-like schemes of interpolation and filtering are utilized in this investigation.
![\[fig:5.1\] (Color online) Initial conditions of linear advection problem. ](entropy_wave){width="80.00000%"}
![\[fig:5.3\] (Color online) Accuracy verification for all primitive variables. ](log_primitive){width="80.00000%"}
Conclusions {#sec:6}
===========
In this article, a structure-preserving scheme for the 1F2T hydrodynamic equations is proposed, with the aim of improving the reliability of compressible RHD simulations. The proposed scheme exactly satisfies the conservation laws and thus the Rankine–Hugoniot relationship.
The key of constructing a physically accurate scheme that satisfies the important physical principles is to maintain the mathematical structure of the governing equations, even in the discrete form. Specifically, the product rule and the symmetry of the energy equations of each species must be maintained. Therefore, the proposed approach does not discretize the energy equations of the ions and electrons directly but, rather, discretizes the energy conservation law using the FDM approach. The ion and electron energy equations in the discrete form are derived using the product rule; this is the same strategy as that used to derive the nonconservative equations in differential form. This derivation yields error terms in the energy equations, which should be separated equally in accordance with the law of equipartition.
Verification via the shock tube problem demonstrates that the proposed scheme maintains the global conservation error to within the round-off level and well agrees with the Rankine–Hugoniot relationship of the 1F2T model. In other words, the proposed scheme strictly preserves the conservation laws of mass, momentum, and energy, and the law of equipartition in the discrete form. Moreover, accuracy verification based on the linear advection of entropy waves reveals that the proposed scheme yields the formal accuracy.
Although the proposed scheme possesses the prefer features explained above, some issues remain toward practical RHD simulations, such as those considering the ICF implosion. For example, the scheme must be generalized to curvilinear coordinates for spherical[@Kidder1974]/cylindrical[@Piriz2002] implosions, non-ideal EOS, and magnetohydrodynamics (MHD) for magnetized fast ignition [@Fujioka2013; @Nagatomo2015]. This problem may be solved by our strategy using a previous work about MHD scheme [@Kawai2013]. Furthermore, our approach may be used in hypersonic hydrodynamics of re-entries. It is modeled by multitemperature hydrodynamics regarding the translational, rotational, vibrational and excitation modes [@Park1989; @Sakai2001].
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by a Grant-in-Aid from the Japan Society for the Promotion of Science (JSPS) Fellows, No. 15J02622. T.S. wishes to thank Dr. Atsushi Sunahara (Purdue University) for valuable discussions on the physical background of the 1F2T model.
Usage of Runge–Kutta method in proposed scheme {#sec:a}
==============================================
High-order time integration is important for the performance of high-resolution simulations. Here, the implementations of the RK methods in the proposed scheme are presented.
First-order RK method {#sec:a1}
---------------------
The first-order RK method is identical to the Euler explicit method. The terms in Eq. (\[eq:2.22\]) can be classified into the following three components: $$\begin{aligned}
ADV^{n+1,n,n}_{\mathrm{s},j}=DFS^{n+1,n,n}_{\mathrm{s},j}+ERR^{n+1,n,n}_{\mathrm{s},j},\label{eq:a.1}\\
%
ADV^{k,l,m}_{\mathrm{s},j}\equiv \frac{(\rho e_\mathrm{s})^k_j-(\rho e_\mathrm{s})^l_j}{\Delta t}+
\frac{\langle \rho e_\mathrm{s}u \rangle^m_{j^+}-\langle \rho e_\mathrm{s} u\rangle^m_{j^-}}{\Delta x}+
\frac{\langle p_\mathrm{s}u \rangle^m_{j^+}-\langle p_\mathrm{s} u\rangle^m_{j^-}}{\Delta x}\notag\\
-\frac{\rho^k_j+\rho^l_j}{4\rho^k_j \rho^l_j}\{(\rho u)^k_j+(\rho u)^l_j \}
\frac{\langle p_\mathrm{s} \rangle^m_{j^+}-\langle p_\mathrm{s} \rangle^m_{j^-}}{\Delta x},\label{eq:a.2}\\
%
DFS^{k,l,m}_{\mathrm{s},j}\equiv -\frac{\rho^k_j+\rho^l_j}{8\rho^k_j \rho^l_j}\{(\rho u)^k_j+(\rho u)^l_j \}
\frac{\langle A \rangle^m_{j^+}-\langle A \rangle^m_{j^-}}{\Delta x}+
\frac{\langle B_\mathrm{s} \rangle^m_{j^+}-\langle B_\mathrm{s} \rangle^m_{j^-}}{\Delta x},\label{eq:a.3}\\
%
ERR^{k,l,m}_{\mathrm{s},j}\equiv -\frac{(\rho^2 u^2)^k_j+(\rho^2 u^2)^l_j}{8\rho^k_j\rho^l_j}
\frac{\langle \rho u \rangle^m_{j^+}-\langle \rho u \rangle^m_{j^-}}{\Delta x}+
\frac{\rho^k_j+\rho^l_j}{8\rho^k_j \rho^l_j}\{(\rho u)^k_j+(\rho u)^l_j \}
\frac{\langle \rho u^2 \rangle^m_{j^+}-\langle \rho u^2 \rangle^m_{j^-}}{\Delta x}\notag\\
-\frac14\frac{\langle \rho u^3\rangle^m_{j^+}-\langle \rho u^3\rangle^m_{j^-}}{\Delta x},\label{eq:a.4}\end{aligned}$$ where $ADV$, $DFS$, and $ERR$ represent the advection, artificial dissipation, and error components, respectively.
Second-order RK method {#sec:a2}
----------------------
Here, the proposed scheme is extended using the second-order RK method. The conservation laws of mass, momentum, and energy are discretized as follows: $$\begin{aligned}
\frac{\rho^{*}_j-\rho^n_j}{\Delta t}+\frac{\langle\rho u\rangle^n_{j^+}
-\langle\rho u\rangle^n_{j^-}}{\Delta x}=0,\label{eq:a.5}\\
%
\frac{(\rho u)^{*}_j-(\rho u)^n_j}{\Delta t}+
\frac{\langle\rho u^2+p\rangle^n_{j^+}-
\langle\rho u^2+p\rangle^n_{j^-}}{\Delta x}=
\frac{\langle A\rangle^n_{j^+}-\langle A\rangle^n_{j^-}}{\Delta x},\label{eq:a.6}\\
%
\frac{(\rho e+\frac12\rho u^2)^{*}_j-(\rho e+\frac12\rho u^2)^n_j}{\Delta t}
+\frac{\langle\rho eu+\frac12\rho u^3+p u\rangle^n_{j^+}-
\langle\rho eu+\frac12\rho u^3+pu\rangle^n_{j^-}}{\Delta x}=
\frac{\langle B\rangle^n_{j^+}-\langle B\rangle^n_{j^-}}{\Delta x},\label{eq:a.7}\\
%
\frac{\rho^{n+1}_j-\rho^n_j}{\Delta t}
+\frac{\langle\rho u\rangle^n_{j^+}-\langle\rho u\rangle^n_{j^-}}{2\Delta x}
+\frac{\langle\rho u\rangle^*_{j^+}-\langle\rho u\rangle^*_{j^-}}{2\Delta x}=0,\label{eq:a.8}\\
%
\frac{(\rho u)^{n+1}_j-(\rho u)^n_j}{\Delta t}
+\frac{\langle\rho u^2+p\rangle^n_{j^+}-\langle\rho u^2+p\rangle^n_{j^-}}{2\Delta x}
+\frac{\langle\rho u^2+p\rangle^*_{j^+}-\langle\rho u^2+p\rangle^*_{j^-}}{2\Delta x}\notag\\
=\frac{\langle A\rangle^n_{j^+}-\langle A\rangle^n_{j^-}}{2\Delta x}
+\frac{\langle A\rangle^*_{j^+}-\langle A\rangle^*_{j^-}}{2\Delta x},\label{eq:a.9}\\
%
\frac{(\rho e+\frac12\rho u^2)^{n+1}_j-(\rho e+\frac12\rho u^2)^n_j}{\Delta t}
+\frac{\langle\rho eu+\frac12\rho u^3+p u\rangle^n_{j^+}-\langle\rho eu+\frac12\rho u^3+pu\rangle^n_{j^-}}{2\Delta x}\notag\\
+\frac{\langle\rho eu+\frac12\rho u^3+p u\rangle^*_{j^+}-\langle\rho eu+\frac12\rho u^3+pu\rangle^*_{j^-}}{2\Delta x}
=\frac{\langle B\rangle^n_{j^+}-\langle B\rangle^n_{j^-}}{2\Delta x}
+\frac{\langle B\rangle^*_{j^+}-\langle B\rangle^*_{j^-}}{2\Delta x},\label{eq:a.10}
%\end{aligned}$$ where “$*$” denotes the internal time-step of the second-order RK method. Equations (\[eq:a.5\])–(\[eq:a.7\]) and (\[eq:a.8\])–(\[eq:a.10\]) correspond to the primary and secondary RK steps, respectively. Obviously, the following energy equations are obtained from Eq. (\[eq:a.5\])–(\[eq:a.7\]): $$\begin{aligned}
ADV^{*,n,n}_{\mathrm{s},j}=DFS^{*,n,n}_{\mathrm{s},j}+ERR^{*,n,n}_{\mathrm{s},j}.\label{eq:a.11}\end{aligned}$$ In addition, the spatial-difference terms of Eq. (\[eq:a.8\])–(\[eq:a.10\]) can be interpreted as arithmetic averages of “$n$” and “$*$” steps. Thus, the energy equations in the second step are expressed as $$\begin{aligned}
\frac{ADV^{n+1,n,n}_{\mathrm{s},j}+ADV^{n+1,n,*}_{\mathrm{s},j}}{2}=
\frac{DFS^{n+1,n,n}_{\mathrm{s},j}+DFS^{n+1,n,*}_{\mathrm{s},j}}{2}+
\frac{ERR^{n+1,n,n}_{\mathrm{s},j}+ERR^{n+1,n,*}_{\mathrm{s},j}}{2}.\label{eq:a.12}\end{aligned}$$
Third-order TVD RK method
-------------------------
The implementation of the third-order TVD RK method is derived using a similar method to the second-order case. The first step is expressed as $$\begin{aligned}
ADV^{\dagger,n,n}_{\mathrm{s},j}=DFS^{\dagger,n,n}_{\mathrm{s},j}+ERR^{\dagger,n,n}_{\mathrm{s},j},\label{eq:a.13}\end{aligned}$$ where “$\dagger$” indicates the primary internal time-step of the third-order TVD RK. The second step is given as $$\begin{aligned}
\frac{ADV^{\dagger\dagger,\dagger,\dagger}_{\mathrm{s},j}-3ADV^{\dagger\dagger,\dagger,n}_{\mathrm{s},j}}{4}=
\frac{DFS^{\dagger\dagger,\dagger,\dagger}_{\mathrm{s},j}-3DFS^{\dagger\dagger,\dagger,n}_{\mathrm{s},j}}{4}+
\frac{ERR^{\dagger\dagger,\dagger,\dagger}_{\mathrm{s},j}-3ERR^{\dagger\dagger,\dagger,n}_{\mathrm{s},j}}{4},\label{eq:a.14}\end{aligned}$$ where “$\dagger\dagger$” indicates the secondary internal time-step of the third-order TVD RK. The final step is obtained by weighting these internal steps, such that $$\begin{aligned}
\frac{ADV^{n+1,n,n}_{\mathrm{s},j}+ADV^{n+1,n,\dagger}_{\mathrm{s},j}+4ADV^{n+1,n,\dagger\dagger}_{\mathrm{s},j}}{6}=\notag\\
\frac{DFS^{n+1,n,n}_{\mathrm{s},j}+DFS^{n+1,n,\dagger}_{\mathrm{s},j}+4DFS^{n+1,n,\dagger\dagger}_{\mathrm{s},j}}{6}+
\frac{ERR^{n+1,n,n}_{\mathrm{s},j}+ERR^{n+1,n,\dagger}_{\mathrm{s},j}+4ERR^{n+1,n,\dagger\dagger}_{\mathrm{s},j}}{6}.\label{eq:a.15}\end{aligned}$$
Multidimensional description {#sec:b}
============================
Multidimensional scheme is required for the numerical simulations of ICF implosions because hydrodynamic instabilities such as Rayleigh–Taylor instability [@Takabe1985; @Betti1998] are one of the fundamental physics to determine the fusion gain. Here, we introduce a multidimensional description of the proposed approach. Before the derivation of multidimensional scheme, we define two vector differential operators in the discrete form: $$\begin{aligned}
\mathrm{Grad\ } \phi|^n_{i,j,k}=\begin{bmatrix}
\dfrac{\langle \phi \rangle^n_{i+\frac12,j,k}-\langle \phi \rangle^n_{i-\frac12,j,k}}{\Delta x} \\
\dfrac{\langle \phi \rangle^n_{i,j+\frac12,k}-\langle \phi \rangle^n_{i,j-\frac12,k}}{\Delta y} \\
\dfrac{\langle \phi \rangle^n_{i,j,k+\frac12}-\langle \phi \rangle^n_{i,j,k-\frac12}}{\Delta z}
\end{bmatrix},\label{eq:b.1}\\
\mathrm{Div\ } \mathbf{\Psi}|^n_{i,j,k}=
\dfrac{\langle \psi_{x} \rangle^n_{i+\frac12,j,k}-\langle \psi_{x}\rangle^n_{i-\frac12,j,k}}{\Delta x}+
\dfrac{\langle \psi_{y} \rangle^n_{i,j+\frac12,k}-\langle \psi_{y}\rangle^n_{i,j-\frac12,k}}{\Delta y}+
\dfrac{\langle \psi_{z} \rangle^n_{i,j,k+\frac12}-\langle \psi_{z}\rangle^n_{i,j,k-\frac12}}{\Delta z}\end{aligned}$$ where $\phi$ is an arbitrary scalar function, $\mathbf{\Psi}=^\mathrm{T}[\psi_x,\psi_y,\psi_z]$ is an arbitrary vector function, and $(i, j, k)$ are the spatial indices over $(x, y, z)$ directions, respectively. The three-dimensional Euler equation is discretized by using these operators as follows: $$\begin{aligned}
\frac{\rho^{n+1}_{i,j,k}-\rho^n_{i,j,k}}{\Delta t}+\mathrm{Div\ }(\rho \mathbf{u})|^n_{i,j,k}=0,\label{eq:b.3}\\
\frac{(\rho \mathbf{u})^{n+1}_{i,j,k}-(\rho \mathbf{u})^n_{i,j,k}}{\Delta t}+\mathrm{Div\ }(\rho \mathbf{uu})|^n_{i,j,k}
+\mathrm{Grad\ } p|^n_{i,j,k} =\mathbf{0},\label{eq:b.4}\\
\frac{(\rho e+\frac12\rho |\mathbf{u}|^2)^{n+1}_{i,j,k}-(\rho e+\frac12\rho |\mathbf{u}|^2)^n_{i,j,k}}{\Delta t}
+\mathrm{Div\ }\left.\left( \rho e \mathbf{u} + \frac12\rho|\mathbf{u}|^2\mathbf{u} + p\mathbf{u}\right)\right|^n_{i,j,k}=
0,\label{eq:b.5}\end{aligned}$$ Note that the key of our approach in Sec. \[sec:2.2\] is the expansion of the time derivative. Hence, the three-dimensional scheme is also obtained by the same way. The discretized equations corresponding to Eqs. (\[eq:1.3\]) and (\[eq:1.4\]) are $$\begin{aligned}
\frac{(\rho e_\mathrm{s})^{n+1}_{i,j,k}-(\rho e_\mathrm{s})^n_{i,j,k}}{\Delta t}+
\mathrm{Div\ }(\rho e_\mathrm{s} \mathbf{u}+p_\mathrm{s} \mathbf{u})|^n_{i,j,k}-
\frac{\rho^{n+1}_{i,j,k}+\rho^n_{i,j,k}}{4\rho^{n+1}_{i,j,k}\rho^n_{i,j,k}}
\{(\rho \mathbf{u})^{n+1}_{i,j,k}+(\rho \mathbf{u})^n_{i,j,k}\}\cdot
\left(\mathrm{Grad\ }p_\mathrm{s}|^n_{i,j,k}\right)=\notag\\
-\frac{(|\rho \mathbf{u}|^2)^{n+1}_{i,j,k}+(|\rho \mathbf{u}|^2)^n_{i,j,k}}{8\rho^{n+1}_{i,j,k}\rho^n_{i,j,k}}
\mathrm{Div\ }(\rho \mathbf{u})|^n_{i,j,k}
+\frac{\rho^{n+1}_{i,j,k}+\rho^n_{i,j,k}}{8\rho^{n+1}_{i,j,k}\rho^n_{i,j,k}}
\{(\rho \mathbf{u})^{n+1}_{i,j,k}+(\rho \mathbf{u})^n_{i,j,k}\}\cdot\mathrm{Div\ }(\rho \mathbf{uu})|^n_{i,j,k}\notag\\
-\mathrm{Div\ }\left.\left(\frac14\rho|\mathbf{u}|^2\mathbf{u}\right)\right|^n_{i,j,k}.\end{aligned}$$ Extension to the shock capturing scheme is self-evident.
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---
abstract: 'Using the geometry of quadrics of a projective plane $\PG(2,q)$ a family of $(6,q^3(q^2-1)(q-1)/3+(q^2+1)(q^2+q+1),4;3)_q$ constant dimension subspace codes is constructed.'
author:
- |
Antonio Cossidente\
Dipartimento di Matematica Informatica ed Economia\
Università della Basilicata\
Contrada Macchia Romana\
I-85100 Potenza\
Italy\
antonio.cossidente@unibas.it\
Francesco Pavese\
Dipartimento di Matematica Informatica ed Economia\
Università della Basilicata\
Contrada Macchia Romana\
I-85100 Potenza\
Italy\
francesco.pavese@unibas.it
title: Veronese subspace codes
---
[**Proposed Running Head:**]{} Veronese subspace codes
[**Corresponding Author:**]{}\
Antonio Cossidente\
Dipartimento di Matematica Informatica ed Economia\
Università della Basilicata\
Contrada Macchia Romana\
I-85100 Potenza\
Italy\
antonio.cossidente@unibas.it
\[section\] \[theorem\][Lemma]{} \[theorem\][Conjecture]{} \[theorem\][Remark]{} \[theorem\][Corollary]{} \[theorem\][Proposition]{} \[theorem\][Definition]{} \[theorem\][Result]{}
PS. @headings[ oddhead evenhead oddfoot evenfoot[oddfoot]{} ]{}
[**KEYWORDS:**]{} projective bundle; constant dimension subspace code; Singer cyclic group; Veronese map;
[**AMS MSC:**]{} 51E15, 05B25
Introduction
============
Let $V$ be an $n$–dimensional vector space over $\GF(q)$, $q$ any prime power. The set $S(V)$ of all subspaces of $V$, or subspaces of the projective space $\PG(V)$, forms a metric space with respect to the [*subspace distance*]{} defined by $d_s(U,U')=\dim (U+U')- \dim(U\cap U')$. In the context of subspace codes, the main problem is to determine the largest possible size of codes in the space $(S(V),d_s)$ with a given minimum distance, and to classify the corresponding optimal codes. The interest in these codes is a consequence of the fact that codes in the projective space and codes in the Grassmannian over a finite field referred to as subspace codes and constant–dimension codes, respectively, have been proposed for error control in random linear network coding. An $(n,M,d;k)_q$ constant–dimension subspace code (CDC) is a set $\cal C$ of $k$–subspaces of $V$ with $\vert{\cal C}\vert =M$ and minimum subspace distance $d_s({\cal C})=\min\{d_s(U,U') \vert U,U'\in {\cal C}, U\ne U' \}=d$. The smallest open constant–dimension case occurs when $n = 6$ and $k = 3$. From a projective geometry point of view it translates in the determination of the maximum number of planes in $\PG(5,q)$ mutually intersecting in at most one point. In [@HKK], the authors show that the maximum size of a binary subspace code of packet length $n=6$, minimum subspace distance $d=4$ and constant dimension $k=3$ is $M=77$. Therefore the maximum number of planes in $\PG(5,2)$ mutually intersecting in at most one point is $77$. In the same paper, the authors, with the aid of a computer, classify all $(6,77,4;3)_2$ subspace codes into $5$ isomorphism types [@HKK Table 6] and a computer–free construction of one isomorphism type [@HKK Table 6, A] is provided. This last isomorphism type is then generalized to any $q$ providing a family of $(6,q^6+2q^2+2q+1,4;3)_q$ subspace codes [@HKK Lemma 12]. In [@CP1] the authors provided a construction of families of $(6,q^6+2q^2+2q+1,4;3)_q$ subspace codes potentially including the infinite family constructed in [@HKK].
In this paper we construct a family of $(6,q^3(q^2-1)(q-1)/3+(q^2+1)(q^2+q+1),4;3)_q$ CDC. Our approach is purely geometric and the construction relies on the geometry of quadrics of a projective plane $\PG(2,q)$. More precisely, we use the correspondence between quadrics of $\PG(2,q)$ and points of $\PG(5,q)$. In this setting, we show that a special net of conics (circumscribed bundle) yields a $(6,q^3(q^2-1)(q-1)/3,4;3)_q$ CDC admitting the linear group $\PGL(3,q)$ as an automorphism group. Although the size of such a code asymptotically reaches the theoretical upper bound of a $(6,M,4;3)_q$ CDC [@HKK], it turns out that it can be enlarged. This is done in the second part of the paper where we are able to find a set of further $(q^2+1)(q^2+q+1)$ planes of $\PG(5,q)$ mutually intersecting in at most one point and extending the previous code. The $(6,q^3(q^2-1)(q-1)/3+(q^2+1)(q^2+q+1),4;3)_q$ CDC so obtained admits the normalizer of a Singer cyclic group of $\PGL(3,q)$ as an automorphism group.
The Veronese embedding
======================
Let $\PG(2,q)$ the Desarguesian projective plane of order $q$. A [*quadric*]{} of $\PG(2,q)$ is the locus of zeros of a quadratic polynomial, say $a_{11}X_1^2+a_{22}X_2^2+a_{33}X_3^2+a_{12}X_1X_2+a_{13}X_1X_3+a_{23}X_2X_3$. There are six parameters associated to such a curve and hence the set of quadrics of $\PG(2,q)$ forms a $5$–dimensional projective space. There exist four kinds of quadrics in $\PG(2,q)$, three of which are [*degenerate*]{} (splitting into lines, which could be in the plane $\PG(2,q^2)$) and one of which is [*non–degenerate*]{} [@JWPH1].
The [*Veronese map v*]{} defined by $$a_{11}X_1^2+a_{22}X_2^2+a_{33}X_3^2+a_{12}X_1X_2+a_{13}X_1X_3+a_{23}X_2X_3\mapsto (a_{11},a_{22},a_{33},a_{12},a_{13},a_{23}),$$ is the correspondence between plane quadrics and the points of $\PG(5,q)$,
The quadrics in $\PG(2,q)$ are:
- $q^2+q+1$ repeated lines;
- $(q^2+q+1)(q+1)q/2$ quadrics consisting of two distinct lines of $\PG(2,q)$ (bi–lines);
- $(q^2+q+1)(q-1)q/2$ quadrics consisting of two distinct conjugate lines of $\PG(2,q^2)$ (imaginary bi–lines).
- $q^5-q^2$ non–degenerate quadrics (conics).
We will say that a bi–line or an imaginary bi–line is [*centered*]{} at $A$ if its lines meet in the point $A$.
When $q$ is even, all tangent lines to a conic $\cal C$ pass through a point of $\PG(2,q)$ called the [*nucleus*]{} of $\cal C$.
It is not difficult to see that a quadric of $\PG(2,q)$ is degenerate if and only if its parameters satisfy the polynomial $$P_1:=X_4X_5X_6+X_1X_6^2+X_2X_5^2+X_3X_4^2,$$ when $q$ is even and $$P_2:=X_1X_2X_3+2X_4X_5X_6+X_1X_6^2+X_2X_5^2+X_3X_4^2,$$ when $q$ is odd.
Notice that from [@HT Theorem 25.1.3] the image of the Veronese map [*v*]{} is the dual of the image of the map $\zeta$ defined in [@HT p. 146].
The group $G:=\PGL(3,q)$ acts on $\PG(2,q)$ and so it also acts naturally on the plane quadrics, and hence also on $\PG(5,q)$. The four sets of quadrics described above are $G$–orbits. With a slight abuse of notation we will denote by $G$ the group $\PGL(3,q)$ acting on $\PG(5,q)$. We will denote by ${\cal O}_i$, $i=1,2,3,4$ the images under [*v*]{} in $\PG(5,q)$ of the four types of quadrics, respectively. It turns out that ${\cal O}_1$ is the [*Veronese surface*]{} when $q$ is odd and a plane (called [*degenerate Veronese surface*]{}) when $q$ is even. The orbits ${\cal O}_i$, $i=1,2,3$, partition the cubic hypersurface ${\cal S}$ of $\PG(5,q)$ with equation $P_1=0$ when $q$ is even and with equation $P_2=0$ when $q$ is odd.
It should be noted that under the map [*v*]{} a $k$–dimensional linear system of quadrics of $\PG(2,q)$ corresponds to a $(k-1)$–dimensional projective subspace of $\PG(5,q)$. This means that pencils, nets and webs of quadrics, are represented by lines, planes and solids of $\PG(5,q)$, respectively.
Let us fix a point $A$ of $\PG(2,q)$. The $q+1$ lines passing through $A$ considered as repeated lines, the $q(q+1)/2$ bi–lines centered at $A$ and the $q(q-1)/2$ imaginary bi–lines centered at $A$ form a net that under the Veronese map [*v*]{} corresponds to a plane $\pi_A$ contained in $\cal S$ and meeting ${\cal O}_1$ at $q+1$ points forming either a conic ($q$ odd) or a line ($q$ even). Hence, there is a set $\cal N$ of $q^2+q+1$ such planes. Also, through a point $P\in {\cal S}\setminus {\cal O}_1$ there passes exactly one plane of $\cal N$ whereas through a point of ${\cal O}_1$ there pass $q+1$ planes of $\cal N$ . It follows that two distinct planes in $\cal N$ meet in a point of $\cO_1$.
Let us fix two distinct points of $\PG(2,q)$, say $A$ and $B$. Let $\ell$ be the line joining $A$ and $B$. There are $q^2+q$ bi–lines of $\PG(2,q)$ containing $\ell$. These bi–lines together $\ell$ (considered as a repeated line) form a net that under the Veronese map [*v*]{} corresponds to a plane $\pi_{\ell}$ contained in $\cal S$ and tangent to ${\cal O}_1$. Hence, there is a set $\cal T$ of $q^2+q+1$ such planes. Also, through a point $P\in{\cal O}_2$ there pass exactly two planes of $\cal T$ whereas through a point $P\in{\cal O}_1$ there passes exactly one plane of $\cal T$. It follows that two distinct planes in $\cal T$ meet in a point of $\cO_2$.
Circumscribed bundles
=====================
There exists a collection of $q^2+q+1$ conics in $\PG(2,q)$ that mutually intersect in exactly one point, and hence serve as the lines of another projective plane on the points of $\PG(2,q)$. Such a collection of conics is called a [*projective bundle*]{} of $\PG(2,q)$. For more details on projective bundles, see [@BBEF].
Let us embed $\PG(2,q)$ into $\PG(2,q^3)$, and let $\sigma$ be the period $3$ collineation of $\PG(2,q^3)$ fixing $\PG(2,q)$. Let us fix a triangle $T$ of vertices $P$, $P^\sigma$, $P^{\sigma^2}$ in $\PG(2,q^3)$. Up to date, the known types of projective bundles are as follows [@DGG], [@DGG1]:
- [*circumscribed bundle*]{} consisting of all conics of $\PG(2,q)$ containing the vertices of $T$. This exists for all $q$;
- [*inscribed bundle*]{} consisting of all conics of $\PG(2,q)$ that are tangent to the three sides of $T$. This exists for all odd $q$;
- [*self–polar bundle*]{} consisting of all conics of $\PG(2,q)$ with respect to which $T$ is self–polar. This exists for all odd $q$.
From [@BBEF] the conics of a circumscribed bundle form a net.
A cyclic group of $G$ permuting points (lines) of $\PG(2,q)$ in a single orbit is called a [*Singer cyclic group*]{} of $G$. A generator of a Singer cyclic group is called a [*Singer cycle*]{}.
A Singer cyclic group of $G$ has order $q^2+q+1$ and its normalizer in $G$ turns out to be a metacyclic group of order $3(q^2+q+1)$. For more details, see [@huppert].
[All these projective bundles are invariant under the normalizer of a Singer cyclic group of $G$.]{}
Let ${\cal B}$ be a circumscribed bundle of $\PG(2,q)$. We will need the following result, which extends [@CP1 Lemma 3.2].
\[bundle\] Consider two distinct conics $C_0,C_\infty$ of a circumscribed bundle $\cal B$. If $q$ is even, their nuclei are distinct. If $q$ is odd, for a point $P\in\PG(2,q)$ the polar lines of $P$ with respect to $C_0$ and $C_\infty$ are distinct.
If $q$ is even or if $q$ is odd and $P=C_0\cap C_\infty$ then the result follows from [@CP1 Lemma 3.2]. Assume that $q$ is odd and $P\ne C_0\cap C_\infty$. Let $r_0$ and $r_\infty$ be the polar lines of $P$ with respect to $C_0$ and $C_\infty$, respectively. By way of contradiction let $r_0=r_\infty$. If $P\in C_0$ then $P\in r_0$ but $P\not\in r_\infty$, a contradiction.
Let $A_0,A_{\infty}$, be the symmetric $3\times 3$ matrices associated to $C_0$ and $C_{\infty}$, respectively. Let ${\cal F}=\{C_\lambda,\lambda\in\GF(q)\cup\{\infty\} \}$ be the pencil generated by $C_0$ and $C_\infty$. We have that the quadrics of $\cal F$ are the conics (non degenerate quadrics) of $\cal B$ through $C_0\cap C_\infty$ and they cover all points of $\PG(2,q)$ . Let $\perp_\lambda$ denote the polarity associated with the conic in $C_\lambda\in {\cal F}$. The product $\perp_0\perp_\infty$ is then a projectivity of $\PG(2,q)$ fixing $P$ whose associated matrix is $A_0^{-t}A_\infty$, where $t$ denotes transposition. In other terms $(A_0^{-t}A_ {\infty})(P^t)=\rho P^t$, for some $\rho\in\GF(q)\setminus\{0\}$. Analogously, $\perp_0\perp_{\lambda}$ is a projectivity of $\PG(2,q)$ whose associated matrix is $A_0^{-t}(A_0+\lambda A_{\infty})$ and fixing $P$. Indeed, $(A_0^{-t}(A_0+\lambda A_{\infty}))(P^t)=(I+\lambda A_0^{-t}A_{\infty})(P^t)=P^t+\lambda\rho P^t=(1+\lambda\rho)(P^t)$. It turns out that $(P^{\perp_{\lambda}})^{\perp_0}=P$ if and only if $P^{\perp_{\lambda}}=r_0$ if and only if $r_0^{\perp_{\lambda}}=P$ for every $\lambda\in\GF(q)\setminus\{0\}$. Let $\lambda_0\in\GF(q)\setminus\{0\}$ such that $P\in C_{\lambda_0}$. Then $P\in P^{\perp_{\lambda_0}}=r_0$ and hence $P\in r_0$, a contradiction.
[Notice that if $q$ is odd then the projectivity obtained as the product of two polarities associated to distinct conics of a circumscribed bundle is fixed point free.]{}
Since ${\cal B}$ is stabilized by the normalizer $N$ of a Singer cyclic group $S$ of $G$ that is maximal in $G$ [@BHR] we get that ${\cal B}^G$ has size $q^3(q^2-1)(q-1)/3$.
From [@RF] the group $G$ has three orbits on points and lines of $\PG(2,q^3)$. The orbits on points are the $q^2+q+1$ points of $\PG(2,q)$, the $(q^3-q)(q^2+q+1)$ points on lines of $\PG(2,q^3)$ that are not in $\PG(2,q)$ and the remaining set $E$ of $q^3(q^2-1)(q-1)$ points of $\PG(2,q^3)$. The $G$–orbits on lines are the $q^2+q+1$ lines of $\PG(2,q)$, the $(q^3-q)(q^2+q+1)$ lines meeting $\PG(2,q)$ in a point and the set $L$ of $q^3(q^2-1)(q-1)$ lines external to $\PG(2,q)$. The group $N$ fixes a triangle $T$ whose vertices are points of $E$ and whose edges are lines of $L$.
We will need the following lemma.
\[5arc\] Let $T^G$ be the orbit of $T$ under $G$. If $T_1$ and $T_2$ are distinct elements of $T^G$ then the union of their vertices always contains a $5$–arc, i.e., $5$ points of $\pi$ no three of which are collinear.
The stabilizer of $T$ in $G$ contains $N$ that is maximal in $G$ and hence $\vert T^G\vert =q^3(q^2-1)(q-1)/3$. Let us consider the incidence structure whose points are the points of $E$ and whose blocks are the vertex sets of the triangles of $T^G$. The incidence relation is containment. It turns out that through a point of $E$ there pass exactly one triangle of $T^G$. Analogously, let us consider the incidence structure whose points are the lines of $L$ and whose blocks are the edge sets of the triangles of $T^G$. The incidence relation is containment. It turns out that through a line of $L$ there exists exactly one triangle of $T^G$ having that line as an edge. As a consequence, the union of two triangles of $T^G$ always contains a $5$–arc of $\pi$.
Two distinct circumscribed bundles of ${\cal B}^G$ share at most one conic.
Let ${\cal B}_1$ and ${\cal B}_2$ be two distinct circumscribed bundles in ${\cal B}^G$. Let $T_i$ be the triangle associated to ${\cal B}_i$, $i=1,2$. By way of contradiction assume that $C_1$ and $C_2$ are distinct conics in ${\cal B}_1\cap{\cal B}_2$. Then $C_1$ and $C_2$, considered as conics of $\pi$, contain the vertices of both triangles $T_1$ and $T_2$. From Lemma \[5arc\] and from [@JWPH1 Corollary 7.5] we get a contradiction.
Under the Veronese map [*v*]{} the circumscribed bundles in ${\cal B}^G$ correspond to a set $\cal C$ of $q^3(q^2-1)(q-1)/3$ planes of $\PG(5,q)$ mutually intersecting in at most one point. Since no quadric in a bundle is degenerate, a plane of $\cal C$ is always disjoint from ${\cal S}$.
Two special webs of quadrics
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Firstly, we recall some basic properties of three–dimensional non–degenerate quadrics.
A [*hyperbolic quadric*]{} ${\cal Q}^+(3,q)$ of $\PG(3,q)$ consists of $(q+1)^2$ points of $\PG(3,q)$ and $2(q+1)$ lines that are the union of two reguli. A [*regulus* ]{} is the set of lines intersecting three skew lines and has size $q+1$. Through a point of ${\cal Q}^+(3,q)$ there pass two lines belonging to different reguli. A plane of $\PG(3,q)$ is either secant to ${\cal Q}^+(3,q)$ and meets ${\cal Q}^+(3,q)$ in a conic or it is tangent to ${\cal Q}^+(3,q)$ and meets ${\cal Q}^+(3,q)$ in a bi–line.
An [*elliptic quadric*]{} ${\cal Q}^-(3,q)$ of $\PG(3,q)$ consists of $q^2+1$ points of $\PG(3,q)$ such that no three of them are collinear. A plane of $\PG(3,q)$ is either secant to ${\cal Q}^-(3,q)$ and meets ${\cal Q}^-(3,q)$ in a conic or it is tangent to ${\cal Q}^-(3,q)$ and meets ${\cal Q}^-(3,q)$ in a point. For more details on hyperbolic and elliptic quadrics in a three–dimensional projective space we refer to [@JWPH].
Let $P_1,P_2$ be two distinct points of $\PG(2,q)$. Since $G$ is $2$–transitive on points of $\PG(2,q)$ we can always assume that $P_1=(1,0,0)$ and $P_2=(0,1,0)$. The set of quadrics of $\PG(2,q)$ passing through $P_1$ and $P_2$ are those having the coefficients $a_{11}=a_{22}=0$ and forms a web $W$. Under the Veronese map [*v*]{}, $W$ corresponds to the solid [*v*]{}$(W)$ with equations $X_1=X_2=0$. The solid [*v*]{}$(W)$ intersects ${\cal S}$ into the set of points satisfying the equations $X_4(2X_5X_6-X_3X_4)=0$ and $X_4(X_5X_6-X_3X_4)=0$ accordingly as $q$ is odd or even, respectively. In both cases, this set consists of a hyperbolic quadric $Q$ and a plane tangent $\pi$ to $Q$ at the point $R=(0,0,1,0,0,0)$. In particular, $\pi$ meets $Q$ at $2q+1$ points forming a bi–line centered at $R$. The point $R$ corresponds to the repeated line $P_1P_2$ and the remaining $2q$ points correspond to the bi–lines of $W$ centered at $P_1$ and $P_2$. It is easily seen that the number of such solids (hyperbolic solids) is $q(q+1)(q^2+q+1)/2$.
Assume that $P_1,P_2$ are points of $\PG(2,q^2)\setminus \PG(2,q)$ conjugate over $\GF(q)$. Since $G$ is transitive on points of $\PG(2,q^2)\setminus\PG(2,q)$ we can assume that $P_1=(1,\alpha,0)$ and so $P_2=(1,\alpha^q,0)$, where $\alpha$ is a primitive element of $\GF(q^2)$ over $\GF(q)$. Again, the set of quadrics of $\PG(2,q^2)$ passing through $P_1$ and $P_2$ are those whose coefficients satisfy $a_{11}=\alpha^{q+1}a_{22}$ and $a_{12}=-(\alpha+\alpha^q)a_{22}$ and forms a web $U$ . Under the Veronese map [*v*]{}, $U$ corresponds to the solid [*v*]{}$(U)$ with equations $X_1=\alpha^{q+1}X_2$ and $X_4=-(\alpha+\alpha^q)X_2$. The solid [*v*]{}$(U)$ intersects ${\cal S}$ into the set of points satisfying the equations $X_2(X_6^2+\alpha^{q+1}X_5^2+(\alpha+\alpha^q)X_5X_6+((\alpha+\alpha^q)^2-\alpha^{q+1})X_2X_3)=0$ and $X_2((\alpha+\alpha^q)^2X_2X_3+\alpha^{q+1}X_6^2+(\alpha+\alpha^q)X_5X_6+X_5^2)=0$ accordingly as $q$ is odd or even, respectively. Notice that the polynomial $X^2+(\alpha+\alpha^q)X+\alpha^{q+1}$ is irreducible over $\GF(q)$ and that, if $q$ is odd, $\alpha^{q+1}$ is a nonsquare element of $\GF(q)$. Therefore, in both cases, this set consists of an elliptic quadric $Q'$ and a plane $\pi$ tangent to $Q'$ at the point $R=(0,0,1,0,0,0)$. In this case, the number of such solids (elliptic solids) is $q(q-1)(q^2+q+1)/2$.
The plane $\pi$ is contained in $\cal S$, belongs to $\cal T$ and meets ${\cal O}_1$ at the point $R$.
In the sequel a hyperbolic or elliptic solid will be denoted by $\Sigma=(\tau,Q)$ where $\tau\in{\cal T}$ is contained in $\Sigma$ and $Q$ is the three–dimensional hyperbolic or elliptic quadric contained in $\Sigma\cap{\cal S}$. We will denote by $\cal H$ and ${\cal E}$ the set of hyperbolic solids and elliptic solids, respectively.
Now, we do investigate how two solids (elliptic or hyperbolic) can intersect.
\[mcsh\] Let $\Sigma_1=(\pi_1,Q_1)$, $\Sigma_2=(\pi_2,Q_2)$ be two distinct hyperbolic solids. Then, one of the following cases occur:
- $\Sigma_1 \cap \Sigma_2$ is a plane, $\pi_1 = \pi_2$ and $\vert Q_1 \cap Q_2 \vert = q+1$;
- $\Sigma_1 \cap \Sigma_2$ is a plane, $\pi_1 = \pi_2$ and $\vert Q_1 \cap Q_2 \vert = 1$;
- $\Sigma_1 \cap \Sigma_2$ is a plane, $\vert \pi_1 \cap \pi_2 \vert = 1$ and $\vert Q_1 \cap Q_2 \vert = q+2$;
- $\Sigma_1 \cap \Sigma_2$ is a line, $\vert \pi_1 \cap \pi_2 \vert = 1$ and $\vert Q_1 \cap Q_2 \vert = 2$;
Let us assume that $\Sigma_i$ corresponds to the web defined by the points $A_i,B_i$, $i=1,2$. Let $\ell_i$ be the line $A_iB_i$, $i=1,2$.
- The pairs $A_1,B_1$ and $A_2,B_2$ share a point and $\ell_1=\ell_2$. Then we can assume that $A_2=B_1$. In this case it is clear that $\pi_1=\pi_2$ and the $q+1$ points of $Q_1\cap Q_2$ correspond to the bi–lines centered at $A_2=B_1$ of the relevant webs together with $\ell_1=\ell_2$ considered as a repeated line.
- The pairs $A_1,B_1$ and $A_2,B_2$ share no point and $\ell_1=\ell_2$. In this case it is clear that $\pi_1=\pi_2$ and the point of $Q_1\cap Q_2$ corresponds to $\ell_1=\ell_2$ considered as a repeated line.
- The pairs $A_1,B_1$ and $A_2,B_2$ share the point $A_2=B_1=\ell_1\cap \ell_2$. In this case the planes $\pi_1$ and $\pi_2$ share only the point corresponding to the bi–line $\ell_1\ell_2$. On the other hand, $Q_1\cap Q_2$ contains the $q+1$ points corresponding to the bi–lines centered at a point of the line $A_1B_2$ and containing the line $A_1B_2$ and the line through $A_2$. Also, $Q_1\cap Q_2$ contains the point corresponding to the bi–line $\ell_1\ell_2$.
- The pairs $A_1,B_1$ and $A_2,B_2$ share no point and $\ell_1\ne \ell_2$.
- $\ell_1\cap\ell_2=A_2$.
In this case $\pi_1$ and $\pi_2$ share only the point corresponding to the bi–line $\ell_1\ell_2$. Here, $Q_1\cap Q_2$ consists of the two points corresponding to the bi–line $\ell_1A_1B_2$ centered in $A_1$ and the bi–line $\ell_1B_1B_2$ centered at $B_2$. The line joining the points of $Q_1\cap Q_2$ lies on $\pi_1$.
- The points $A_1,B_1,A_2,B_2$ form a $4$–arc in $\PG(2,q)$. In this case $\pi_1$ and $\pi_2$ share only the point corresponding to the bi–line $\ell_1\ell_2$. Here, $Q_1\cap Q_2$ consists of the two points corresponding to the bi–line containing the lines$A_1A_2$ and $B_1B_2$ and the bi–line containing $A_1B_2$ and $A_2B_1$.
- $\ell_1\cap\ell_2=B_1$. In this case by switching $\ell_1$ and $\ell_2$ we are again in the case $4.1$.
\[mcse\] Let $\Sigma_1=(\pi_1,Q_1)$, $\Sigma_2=(\pi_2,Q_2)$ be two distinct elliptic solids. One of the following cases occur:
- $\Sigma_1\cap\Sigma_2$ is a plane, $\pi_1=\pi_2$ and $\vert Q_1\cap Q_2\vert =1$;
- $\Sigma_1\cap\Sigma_2$ is a line, $\vert\pi_1\cap\pi_2\vert =1$ and $\vert Q_1\cap Q_2\vert =2$;
Let us assume that $\Sigma_i$ corresponds to the web defined by the points $A_i,A_i^q$, $i=1,2$. Let $\ell_i$ be the line $A_iA_i^q$, $i=1,2$.
- Assume that $\ell_1=\ell_2$. In this case it is clear that $\pi_1=\pi_2$ and that the unique intersection point between $Q_1$ and $Q_2$ corresponds to the repeated line $\ell_1=\ell_2$.
- Assume that $\ell_1\ne \ell_2$. In this case $\pi_1$ and $\pi_2$ share a unique point corresponding to the bi–line $\ell_1\ell_2$. Here $Q_1\cap Q_2$ consists of the two points corresponding to the two imaginary bi–lines containing the lines $A_1A_2$, $A_1^qA_2^q$ and $A_1A_2^q$, $A_1^qA_2$, respectively.
Two special nets of quadrics {#nets}
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As already observed, the Singer cyclic group $S$ permutes the points (lines) of $\PG(2,q)$ in a single orbit. Under the action of $S$, the set of $q(q+1)(q^2+q+1)/2$ bi–lines of $\PG(2,q)$ is partitioned into $q(q+1)/2$ orbits of size $q^2+q+1$. Let us fix one of the $q(q+1)/2$ orbits of bi–lines, say $b$, and let us consider the incidence structure whose points are the lines of $\PG(2,q)$ and whose blocks are the bi–lines of $b$. It turns out that a line $\ell$ is contained in exactly two bi–lines, say $b_1$ and $b_2$, of $b$ centered at two distinguished points of $\ell$, say $A_1$ and $A_2$, respectively. Let $s$ be the unique element of $S$ such that $A_1^s = A_2$. Then $b_1^s = b_2$.
Let ${\cal P}_{A_1}$ and ${\cal P}_{A_2}$ be the pencils of lines with vertices $A_1$ and $A_2$. Clearly, $s$ is a projectivity sending ${\cal P}_{A_1}$ to ${\cal P}_{A_2}$ that does not map the line $\ell$ onto itself. In [@steiner] it is proved that the set of points of intersection of corresponding lines under $s$ is a conic $C$ passing through $A_1$ and $A_2$ (Steiner’s argument). The projectivity $s$ maps the tangent line to $C$ at $A_1$ onto the line $\ell$ and the line $\ell$ onto the tangent line to $C$ at $A_2$. Moreover, for any two distinct points $A$ and $B$ of a conic there exists a projectivity $\psi\in S$ sending $A$ to $B$ and such that $C$ is the set of points of intersection of corresponding lines under $\psi$. Assume that $b_i=\ell\ell_i$, $i=1,2$. Since $s$ sends $\ell_1$ to $\ell$ and $\ell$ to $\ell_2$, it follows that $\ell_i$ is tangent to $C$ at $A_i$, $i=1,2$. Embed $\PG(2,q)$ into $\PG(2,q^3)$. We have denoted by $T$ be the unique triangle of $\PG(2,q^3)$ fixed by $S$. Considering ${\cal P}_{A_1}$ and ${\cal P}_{A_2}$ as pencils in $\PG(2,q^3)$ and repeating the previous argument, a conic $\bar C$ of $\PG(2,q^3)$ passing through the vertices of $T$ and containing $C$ arises. It follows that $C$ is a member of the circumscribed bundle $\cal B$ of $\PG(2,q)$ left invariant by $S$. Let $A_3\in C\setminus\{A_1,A_2\}$ and let $b_3$ the bi–line of $b$ centered at $A_3$. Let $s'$ be the unique element of $S$ sending $A_1$ to $A_3$. Then $b_1^{s'}=b_3$. Steiner’s argument with $s$ replaced by $s'$, applied to the pencils ${\cal P}_{A_1}$ and ${\cal P}_{A_3}$, gives rise to a conic $C'$ that necessarily belongs to ${\cal B}$. Furthermore, being unique the conic of $\cal B$ through two distinct points of $\PG(2,q)$ it follows that $C=C'$. Since $A_1^{s'}=A_3$ and $A_1\in\ell_1$ then $A_1^{s'}=A_3\in\ell_1^{s'}$. On the other hand, the point $\ell_1^{s'}\cap\ell_1$ lies on $C$ and of course it lies on $\ell_1$. Since the line $\ell_1$ is tangent to $C$ at $A_1$ we have that $\ell_1^{s'}\cap\ell_1$ is the point $A_1$ or, in other words, $\ell_1^{s'}$ is the line $A_1A_3$. Analogously, the point $\ell^{s'}\cap\ell$ lies on $C$ and of course it lies on $\ell$. Therefore $\ell^{s'}$ is either the line $A_1A_2$ or the line $A_1A_3$. Since $\ell_1\ne\ell$ it follows that $\ell^{s'}=A_2A_3$. We have showed that $b_3$ is the bi–line containing the lines $A_1A_3$ and $A_2A_3$.
We have proved the following Proposition.
\[conics\] For any conic $C$ of ${\cal B}$ there exists two distinguished points $P_1$ and $P_2$ of $C$ such that the elements of $b$ centered at a point of $C$ are as follows: $t_{P_1}r$, $t_{P_2}r$, $r_1r_2$, where $t_{P_i}$ is the tangent line to $C$ at $P_i$, $i=1,2$, $r$ is the line $P_1P_2$, $r_i$ is the line $PP_i$, $i=1,2$, and $P$ ranges over $C\setminus\{P_1,P_2\}$.
\[corr\] [Notice that there exists a one to one correspondence between the orbits of $S$ on bi–lines and secant lines to $C$.]{}
\[conf\] [With the notation introduced in Proposition \[conics\], notice that, from [@JWPH1 Table 7.7] the pencil generated by the bi–lines $r_1r_2$ and $ru$, where $t_{P1}\cap t_{P_2}\in u$ and $P_1,P_2,P_3\not\in u$, contains exactly a further bi–line. Moreover, this bi–line is centered at a point of $C$. Indeed, let $C$ be the conic with equation $X_1X_3-X_2^2=0$. The stabilizer of $C$ in $G$ is isomorphic to $\PGL(2,q)$ and acts $3$–transitively on points of $C$. Hence, without loss of generality, we can assume that $P_1=(1,0,0)$, $P_2=(0,0,1)$ and $P_3=(1,1,1)$. Let $v_i$ be the line joining the point $P_3$ and the point $P_i$, $i=1,2$. Then$b_3=v_1v_2$. Notice that $U=t_{P_1}\cap t_{P_2}=(0,1,0)$. Let $u$ be a line passing through $U$ and containing none of the points $P_i$, $i=1,2,3$. Let $b_4$ be the bi–line $uP_1P_2$. Then $b_3\cap b_4$ consists of the four points $P_1,P_2, (1,1,t), (1,t,t)$, with $t \ne 0,1$. It turns out that the bi–line consisting of the lines $(1,t,t)P_2$ and $(1,1,t)P_1$ is centered at the point $(1,t,t^2)\in C$.]{}
The following Proposition could be of some interest.
\[pp\] The incidence structure whose points are the elements of $b$ and whose lines are the conics of the circumscribed bundle $\cal B$, where a bi–line is incident with a conic if it is centered at one of its points, forms a projective plane.
We have that $\vert b \vert = \vert {\cal B} \vert = q^2+q+1$. Since through a point of $\PG(2,q)$ there pass $q+1$ conics of $\cal B$, we have that a bi–line of $b$ is incident with $q+1$ conics of $\cal B$. On the other hand a conic is incident with $q+1$ bi–lines of $b$. In particular we have seen that to a conic $C$ of $\cal B$ are associated two distinguished points $P, P^s \in C$, where $s \in S$ and all the bi–lines of $b$ incident with $C$ contain both $P, P^s$. Let us consider now a conic of ${\cal B} \setminus \{ C \}$. Then it is necessarily of the form $C^\mu$, for some non–trivial element $\mu \in S$. Then two possibilities occur according as one of the points $P^\mu, P^{s \mu}$ does belong to $C$ or does not. If the first case occurs then, assuming that $P^\mu$ is the point belonging to $C$, we have that $C \cap C^{\mu} = {P^\mu}$. If $t_{P}$ denotes the tangent line to $C$ at the point $P$, it turns out that $t_{P}^{\mu}$ is the tangent line to $C^{\mu}$ at the point $P^{\mu}$ and $t_{P}^{\mu} = PP^{\mu}$. Therefore the unique bi–line of $b$ incident with both $C$ and $C^{\mu}$ is centered at $P^{\mu}$. If the latter case occurs, then, by construction (a la Steiner) , $P P^{\mu} \cap P^{s} P^{s \mu} = P P^{\mu} \cap P^{s} P^{\mu s} = P P^{\mu} \cap (P P^{\mu})^{s}$ is the unique point in common between $C$ and $C^{\mu}$. Therefore the unique bi–line of $b$ incident with both $C$ and $C^{\mu}$ is centered at $C \cap C^{\mu}$.
With the notation introduced in Proposition \[conics\] let us consider the three bi–lines $t_{P_1}r$, $t_{P_2}r$ and $r_1r_2$, where $r_1\cap r_2=P\in C\setminus\{P_1,P_2\}$. Under the map [*v*]{} they correspond to three points $R_1,R_2,R_3$ of ${\cal O}_2$, respectively. From the classification of pencils of quadrics of $\PG(2,q)$ in [@JWPH1 Table 7.7 ] the line joining $R_1$ and $R_2$ corresponds to the unique pencil $\cal P$ whose members are all bi–lines and having a base consisting of $q+2$ points. Hence the line $R_1R_2$ is completely contained in ${\cal O}_2$. In particular, the bi–lines of $\cal P$ are those containing the line $r$ and the line $t_{P_1}\cap t_{P_2}A$, where $A$ ranges over $r$. It follows that the bi–line corresponding to $R_3$ cannot belong to $\cal P$. Let [*v*]{}$(b)$ be the image of $b$ under [*v*]{}. Of course [*v*]{}$(b)$ contains $R_i$, $i=1,2,3$. Let $\pi_{e}$ be the plane of $\PG(5,q)$ generated by $R_1,R_2,R_3$ and let $\Pi_e$ denote the set of planes obtained in this way.
The plane $\pi_{e}$ meets ${\cal O}_2$ at $2q$ points consisting of the line $R_1R_2$ and of further $q-1$ points. Also, the plane $\pi_e$ meets $v(b)$ in $q+1$ points containing $R_1,R_2,R_3$. Indeed, from Remark \[conf\], through the point $R_3$ there are $q-2$ lines intersecting ${\cal O}_2$ in three points and $v(b)$ in two points.
It follows that the points of $\pi_{e}\cap{\cal O}_2$ correspond to the bi–lines of $b$ centered at points of $C$. We have that $\vert \Pi_e\vert =q(q+1)(q^2+q+1)/2$
On the other hand, the plane $\pi_{e}$ is contained in the hyperbolic solid defined by the points $P_1,P_2$ and then $\pi_{e}\cap {\cal O}_2$ consists of a conic and a line secant to it.
Since the number of hyperbolic solids equals $\vert \Pi_e\vert$ and each plane of $\Pi_e$ is contained in at least a hyperbolic solid, it follows that there exists a one to one correspondence between planes of $\Pi_e$ and hyperbolic solids.
Now, let $S'$ be the unique Singer cyclic group of $\PGL(3,q^2)$ containing $S$. It is clear that the circumscribed bundle ${\cal B}'$ of $\PG(2,q^2)$ fixed by $S'$ induces the circumscribed bundle ${\cal B}$ of $\PG(2,q)$ fixed by $S$.
Let $b_1'$ be the imaginary bi–line containing the lines $r,r^q$ and centered at the point $P\in\PG(2,q)$. Let $C$ be a conic of ${\cal B}$ through $P$ and let $\bar C$ be the unique conic of ${\cal B}'$ containing $C$.
Let $\bar b$ be the orbit of $b_1'$ under $S'$. As already observed above there exist two points, say $P_1,P_2$ on $\bar C$ such that all elements of $\bar b$ centered at a point of $\bar C$ pass through $P_1$ and $P_2$. Also, since $P_1\cup P_2\in r\cup r^q$ and the tangent line to $C$ at $P$ is a line of $\PG(2,q)$, it follows that $P_1,P_2\not\in\PG(2,q)$. Under the action of $S$, $\bar b$ is partitioned into $q^2-q+1$ orbits of size $q^2+q+1$. Among these, we denote by $b'$ the unique $S$–orbit consisting of imaginary bi–lines. It turns out that a member of $b'$ consists of the lines $z=RP_1$ and $z^q=RP_2$ for some $R\in C$. Let $R_1,R_2\in C$, $R_1\ne R_2$. Let $r_i=R_iP_1$ and $r_i^q=R_iP_2$, $i=1,2$. Since $r_1\cap r_2=P_1$ it follows that $r_1^q\cap r_2^q=P_1^q$ and then $P_2=P_1^q$.
Notice that the line $P_1P_2$ arises from a line $a$ of $\PG(2,q)$ that is external to $C$. Let $A=t_{P_1}\cap t_{P_2}$, where $t_{P_1}$ and $t_{P_2}$ are the tangent lines to $\bar C$ at $P_1$ and $P_2$, respectively. Then $A\in\PG(2,q)$. Indeed, when $q$ is odd, $A$ is the conjugate of $a$ with respect to $C$. When $q$ is even, $A$ is the nucleus of both $C$ and $\bar C$.
\[conics1\] For any conic $C$ of ${\cal B}$ there exists two distinguished points $P$ and $P^q$ of $\bar C$ not on $C$ such that the elements of $b'$ centered at a point of $C$ are of the form $XP,XP^q$, where $X$ ranges over $C$.
\[corr1\] [Notice that there exists a one to one correspondence between the orbits of $S$ on imaginary bi–lines and lines external to $C$]{}.
Similar arguments used in Proposition \[pp\] give the following result.
\[pp1\] The incidence structure whose points are the elements of $b'$ and whose lines are the conics of the circumscribed bundle $\cal B$, where an imaginary bi–line is incident with a conic if it is centered at one of its points, forms a projective plane.
Let $d_i$ be the bi–line consisting of the lines $a$ and $D_iA$, $i=1,2$, where $D_1,D_2$ are distinct points of $a$.
With the notation introduced in Proposition \[conics1\], let us consider two bi–lines of the form $a,D_iA$, $i=1,2$ and the imaginary bi–line $XP, XP^q$, for some $X\in C$. Under the map [*v*]{} they correspond to three points $R_1,R_2,R_3$, respectively. The points $R_1$ and $R_2$ are in $\cO_2$, whereas $R_3 \in \cO_3$. From the classification of pencils of quadrics of $\PG(2,q)$ in [@JWPH1 Table 7.7 ] the line joining $R_1$ and $R_2$ corresponds to the unique pencil $\cal P$ whose members are all bi–lines and having a base consisting of $q+2$ points. Hence the line $R_1 R_2$ is completely contained in ${\cal O}_2$. In particular, the bi–lines of $\cal P$ are those containing the line $a$ and the line $D A$, where $D$ ranges over $a$. Of course the imaginary bi–line corresponding to $R_3$ cannot belong to $\cal P$. Let [*v*]{}$(b')$ be the image of $b'$ under [*v*]{}. Of course [*v*]{}$(b')$ contains $R_i$, $i=1,2,3$. Let $\pi_{i}$ be the plane of $\PG(5,q)$ generated by $R_1,R_2,R_3$ and let $\Pi_i$ denote the set of planes obtained in this way.
The plane $\pi_{i}$ meets ${\cal O}_2$ in the line $R_1R_2$ and $\cO_3$ in further $q+1$ points. Indeed, from the classification of pencils of quadrics of $\PG(2,q)$ in [@JWPH1 Table 7.7 ], through the point $R_3$ there are $q$ lines intersecting ${\cal O}_3$ in two points and $\cO_2$ in one point. Each of these lines corresponds to the unique pencil consisting of $q-2$ conics, a bi–line and two imaginary bi–lines. Also, there exists a unique line through the point $R_3$ intersecting both $\cO_2$, $\cO_3$ in one point. Such a line corresponds to the unique pencil consisting of $q-1$ conics one bi–line and one imaginary bi–line. It follows that the points of $\pi_{i}\cap{\cal O}_3$ correspond to the imaginary bi–lines of $b'$ centered at points of $C$. We have that $\vert \Pi_i\vert =q(q-1)(q^2+q+1)/2$
On the other hand, the plane $\pi_{i}$ is contained in the elliptic solid defined by the points $P,P^q$ and then $\pi_{i}\cap {\cal O}_3$ consists of a conic and $\pi_{i} \cap \cO_2$ of a line external to it.
Since the number of elliptic solids equals $\vert \Pi_i\vert$ and each plane of $\Pi_i$ is contained in at least an elliptic solid, it follows that there exists a one to one correspondence between planes of $\Pi_i$ and elliptic solids.
Lifting Singer cycles
=====================
Here, we assume that $q$ is odd. From [@huppert], we may assume that $S$ is given by $$\left(
\begin{array}{ccc}
\omega & 0 & 0\\
0 &\omega^q & 0\\
0 & 0 & \omega^{q^2}
\end{array}
\right),$$ where $\omega$ is a primitive element of $\GF(q^3)$ over $\GF(q)$. It follows that the lifting of $S$ to a collineation of $\PG(5,q)$ fixing the Veronese surface ${\cal O}_1$ has the following canonical form $A=diag(S^2,S^{q+1})$ [@BBCE].
The group $\langle A\rangle$ has order $q^2+q+1$. Geometrically, $\langle A\rangle$ fixes two planes of $\PG(5,q)$ , say $\rho_1$, $\rho_2$, and partition the remaining points of $\PG(5,q)$ into Veronese surfaces, [@BBCE Corollary 5]. In particular, the planes $\rho_1$ and $\rho_2$ are both full orbits of $\langle
A\rangle$ and disjoint from the cubic hypersurface ${\cal S}$ [@BCS].
From [@BBCE] the cubic hypersurface ${\cal S}$ is partitioned under $\langle A\rangle$ into Veronese surfaces. The hypersurface ${\cal S}$ has $(q^2+1)(q^2+q+1)$ points and hence it consists of $q^2+1$ Veronese surfaces.
The construction of subspace codes
==================================
In this Section we prove our main result.
Two distinct planes of $\Pi_e$ can meet in at most one point.
Let $\sigma_1,\sigma_2$ be two distinct planes of $\Pi_e$. From Section \[nets\] there exist uniquely determined hyperbolic solids $\Sigma_1=(\pi_1,Q_1)$ and $\Sigma_2=(\pi_2,Q_2)$ of $\cal H$ containing $\sigma_1$ and $\sigma_2$, respectively. Let $c_i=\sigma_i\cap Q_i$ be the conic in $\Sigma_i$, $i=1,2$.
Assume first that $c_1,c_2$ belong to the same $S$–orbit. Then, from Proposition \[pp\], $c_1$ and $c_2$ share exactly one point. Since $S$ permutes the planes of $\cal T$ in a single orbit we have that $\pi_1\ne\pi_2$. Therefore, from Proposition \[mcsh\], $Q_1\cap Q_2$ consists of either $2$ or $q+2$ points ($q+1$ points on a line together with a further point $Y$). Assume that $y=\sigma_1\cap\sigma_2$ is a line. If $\vert Q_1 \cap Q_2 \vert = 2$, then the conics $c_1$ and $c_2$ should share two points, a contradiction. If $\vert Q_1 \cap Q_2 \vert = q+2$, then it turns out that $c_1 \cap c_2 = \{Y\}$. On the other hand, since $y \subseteq \Sigma_1\cap\Sigma_2$, the line $y$ contains $Y$ and must be secant to both $Q_1$ and $Q_2$. Hence, again, the conics $c_1$ and $c_2$ should share two points, a contradiction.
Assume that $c_1,c_2$ do not belong to the same $S$–orbit. Then $c_1$ and $c_2$ have no point in common. Assume that $y=\sigma_1\cap\sigma_2$ is a line. If $Q_1 \cap Q_2$ consists of either $2$ or $q+1$ or $q+2$ points, then since $y \subseteq \Sigma_1\cap\Sigma_2$, from Proposition \[mcsh\], the conics $c_1$ and $c_2$ should share at least one point, a contradiction. If $\vert Q_1 \cap Q_2 \vert = 1$, since $y \subseteq \Sigma_1\cap\Sigma_2$, then the line $y$ either contains the point $Q_1 \cap Q_2$ and, again, the conics $c_1$ and $c_2$ should share one point, a contradiction, or the line $y$ is secant to both $c_1$, $c_2$ and $y \cap (c_1 \cup c_2)$ consists of four distinct points. If this last case occurs, then, under the inverse of the map [*v*]{}, these four points correspond to four distinct bi–lines having in common $q+1$ points of a line $z$ and a further point $Z \notin z$. In particular, let $c_i'$ denote the conic of the circumscribed bundle $\cal B$ locus of centers of the bi–lines corresponding to points of $c_i$, $i=1,2$. It turns out that $c_1' \ne c_2'$ (see Remark \[corr\]) and $z$ is the polar line of the point $Z$ with respect to both $c_1'$ and $c_2'$, when $q$ is odd or $Z$ is the nucleus of both $c_1'$ and $c_2'$, when $q$ is even. But this contradicts Lemma \[bundle\].
Two distinct planes of $\Pi_i$ can meet in at most one point.
Let $\sigma_1,\sigma_2$ be two planes of $\Pi_i$. From Section \[nets\] there exist uniquely determined elliptic solids $\Sigma_1=(\pi_1,Q_1)$ and $\Sigma_2=(\pi_2,Q_2)$ of $\cal E$ containing $\sigma_1$ and $\sigma_2$, respectively. Let $c_i=\sigma_i\cap Q_i$ be the conic in $\Sigma_i$, $i=1,2$.
Assume first that $c_1,c_2$ belong to the same $S$–orbit. Then, from Proposition \[pp1\], $c_1$ and $c_2$ share exactly one point. Since $S$ permutes the planes of $\cal T$ in a single orbit we have that $\pi_1\ne\pi_2$. Therefore, from Proposition \[mcse\], $Q_1\cap Q_2$ consists of $2$ points. Assume that $y=\sigma_1\cap\sigma_2$ is a line, then the conics $c_1$ and $c_2$ should share two points, a contradiction.
Assume that $c_1,c_2$ does not belong to the same $S$–orbit. Then $c_1$ and $c_2$ have no point in common. Assume that $y=\sigma_1\cap\sigma_2$ is a line. If $Q_1 \cap Q_2$ consists of $2$ points, then, since $y \subseteq \Sigma_1\cap\Sigma_2$, from Proposition \[mcse\], the conics $c_1$ and $c_2$ should share at least one point, a contradiction. If $\vert Q_1 \cap Q_2 \vert = 1$, since $y \subseteq \Sigma_1\cap\Sigma_2$, then the line $y$ either contains the point $Q_1 \cap Q_2$ and, again, the conics $c_1$ and $c_2$ should share one point, a contradiction, or the line $y\subset \cO_2$ is external to both $c_1$, $c_2$. If this last case occurs, then, under the inverse of the map [*v*]{}, the points of $y$ correspond to bi–lines having in common $q+1$ points of a line $z$ and a further point $Z \notin z$. In particular let $c_i'$ denote the conic of the circumscribed bundle $\cal B$ locus of centers of the imaginary bi–lines corresponding to points of $c_i$, $i=1,2$. It turns out that $c_1' \ne c_2'$ (see Remark \[corr1\]) and $z$ is the polar line of the point $Z$ with respect to both $c_1'$ and $c_2'$, when $q$ is odd or $Z$ is the nucleus of both conics $c_1'$ and $c_2'$, when $q$ is even. But this, again, contradicts Lemma \[bundle\].
The set ${\cal C}\cup\Pi_i\cup\Pi_e\cup{\cal N}$ consists of $q^3(q^2-1)(q-1)/3+(q^2+1)(q^2+q+1)$ planes mutually intersecting in at most one point.
- Assume that $\sigma_1\in {\cal C}$ and $\sigma_2\in \Pi_i\cup\Pi_e\cup{\cal N}$. In this case $\sigma_1\subset {\cal O}_4$ and $\sigma_2$ always contains a line in ${\cal O}_2\cup {\cal O}_3$ and hence if $\sigma_1\cap\sigma_2$ was a line then $\sigma_1$ should contain a point of ${\cal O}_2\cup{\cal O}_3$.
- Assume that $\sigma_1\in \Pi_i\cup\Pi_e$ and $\sigma_2\in{\cal N}$. In this case $\sigma_2\subset {\cal S}$ whereas $\sigma_1$ meets $\cal S$ in the union of a conic and a line $r$. Hence if $\sigma_1\cap\sigma_2$ was a line, such a line should be $r$. From [@JWPH1 Table 7.7 ] the line $r$ corresponds to the unique pencil of quadrics containing only bi–lines. On the other hand, a line of $\sigma_2$ corresponds to a pencil of quadrics always containing imaginary bi–lines or at most $q$ bi–lines.
- Assume that $\sigma_1 \in \Pi_e$ and $\sigma_2 \in {\Pi_i}$.
Let $\Sigma_1=(\pi_1,Q_1)$ the unique hyperbolic solid of $\cal H$ containing $\sigma_1$ and let $\Sigma_2=(\pi_2,Q_2)$ the unique elliptic solid of $\cal E$ containing $\sigma_2$. Notice that $Q_1\setminus \pi_1$ is always disjoint from $Q_2\setminus \pi_2$. Let $c_i=\sigma_i\cap Q_i$ be the conic in $\Sigma_i$, $i=1,2$. Assume that $\sigma_1\cap\sigma_2$ is a line $r$ Since $r\subset\Sigma_1\cap\Sigma_2$ it follows that $r\subset\pi_1\cap\pi_2$. Under the inverse of the map [*v*]{}, the points of $r$ correspond to bi–lines having in common $q+1$ points of a line $z$ and a further point $Z \notin z$. In particular, let $c_i'$ denote the conic of the circumscribed bundle $\cal B$ locus of centers of the (imaginary) bi–lines corresponding to points of $c_1$ ($c_2$). It turns out that $z$ is secant to $c_1'$ and external to $c_2'$. In particular, $z$ is the polar line of the point $Z$ with respect to both $c_1'$ and $c_2'$, when $q$ is odd or $Z$ is the nucleus of both conics $c_1'$ and $c_2'$, when $q$ is even. But this, again, contradicts Lemma \[bundle\].
There exists a constant dimension subspace code $\cal K$ with parameters $(6,q^3(q^2-1)(q-1)/3+(q^2+1)(q^2+q+1),4;3)_q$.
The code $\cal K$ admits a group of order $3(q^2+q+1)$ as an automorphism group. It is the normalizer of a Singer cyclic group of $\PGL(3,q)$.
[We say that a constant dimension subspace code is [*complete*]{} if it is maximal with respect to set–theoretic inclusion. Some computer tests performed with MAGMA [@magma] yield that our code is not complete when $q=3$. Indeed there exist other $39$ planes that can be added to our set. However, when $q=4,5$ our code is complete. We conjecture that our code is complete whenever $q \ge 4$.]{}
[10]{}
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abstract: 'Babson and Steingrímsson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted $stat$. Given a permutation $\pi$, let $des(\pi)$ denote the descent number of $\pi$ and $maj(\pi)$ denote the major index of $\pi$. Babson and Steingrímsson conjectured that $(des,stat)$ and $(des,maj)$ are equidistributed on $S_n$. Foata and Zeilberger settled this conjecture using q-enumeration, generating functions and Maple packages ROTA and PERCY. Later, Burstein provided a bijective proof of a refinement of this conjecture. In this paper, we give a new bijective proof of this conjecture.'
---
[**A new bijective proof of Babson and Steingrímsson’s conjecture**]{}
Joanna N. Chen$^1$, Shouxiao Li$^2$
$^{1}$College of Science\
Tianjin University of Technology\
Tianjin 300384, P.R. China
$^2$ College of Computer and Information Engineering\
Tianjin Agricultural University\
Tianjin 300384, P.R. China
$^1$joannachen@tjut.edu.cn, $^2$shouxiao09009@163.com.
[**Keywords**]{}: Euler-Mahonian, bijection, involution
[**AMS Subject Classifications**]{}: 05A05, 05A15
Introduction
============
In this paper, we give a new bijective proof of a conjecture of Babson and Steingrímsson [@Babson] on Euler-Mahonian statistics.
Let $S_n$ denote the set of all the permutations of $[n]=\{1,2,\cdots,n\}$. Given a permutation $\pi=\pi_1 \pi_2 \cdots \pi_n \in S_n$, a descent of $\pi$ is a position $i \in [n-1]$ such that $\pi_i > \pi_{i+1}$, where $\pi_i$ and $\pi_{i+1}$ are called a descent top and a descent bottom, respectively. An ascent of $\pi$ is a position $i \in [n-1]$ such that $\pi_i < \pi_{i+1}$, where $\pi_i$ is called an ascent bottom and $\pi_{i+1}$ is called an ascent top. The descent set and the ascent set of $\pi$ are given by $$Des(\pi)=\{i\colon \pi_i>\pi_{i+1}\},$$ $$Asc(\pi)=\{i\colon \pi_i<\pi_{i+1}\}.$$ The set of the inversions of $\pi$ is $$Inv(\pi)=\{(i,j)\colon 1\leq i< j \leq n, \pi_i > \pi_j\}.$$ Let $des(\pi)$, $asc(\pi)$ and $inv(\pi)$ be the descent number, the ascent number and the inversion number of $\pi$, which are defined by $des(\pi)=|Des(\pi)|$, $asc(\pi)=|Asc(\pi)|$ and $inv(\pi)=|Inv(\pi)|$, respectively. The major index of $\pi$, denoted $maj(\pi)$, is given by $$maj(\pi)=\sum_{i \in Des(\pi)} i.$$
Suppose that $st_1$ is a statistic on the object $Obj_1$ and $st_2$ is a statistic on the object $Obj_2$. If $$\sum_{\sigma \in Obj_1} q^{st_1(\sigma)}=\sum_{\sigma \in Obj_2} q^{st_2(\sigma)},$$ we say that the statistic $st_1$ over $Obj_1$ is equidistributed with the statistic $st_2$ over $Obj_2$.
A statistic on $S_n$ is said to be Eulerian if it is equidistributed with the statistic $des$ on $S_n$. While a statistic on $S_n$ is said to be Mahonian if it is equidistributed with the statistic $inv$ on $S_n$. It is well-known that $$\sum_{\pi \in S_n} q^{inv(\pi)}=\sum_{\pi \in S_n} q^{maj(\pi)}=[n]_q !,$$ where $[n]_q=1+q+\cdots +q^{n-1}$ and $[n]_q!=[n]_q [n-1]_q \cdots [1]_q$. Thus, the major index $maj$ is a Mahonian statistic. A pair of statistics on $S_n$ is said to be Euler-Mahonian if it is equidistributed with the joint distribution of the descent number and the major index.
In [@Babson], Babson and Steingrímsson introduced generalized permutation patterns, where they allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Let $\mathcal{A}$ be the alphabet $\{a,b,c,\ldots\}$ with the usual ordering. We write patterns as words in $\mathcal{A}$, where two adjacent letters may or may not be separated by a dash. Two adjacent letters without a dash in a pattern indicates that the corresponding letters in the permutation must be adjacent. Given a generalized pattern $\tau$ and a permutation $\pi$, we say a subsequence of $\pi$ is an occurrence (or instance) of $\tau$ in $\pi$ if it is order-isomorphic to $\tau$ and satisfies the above dash conditions. Let $(\tau)\pi$ denote the number of occurrences of $\tau$ in $\pi$. Here, we see $(\tau)$ as a generalized pattern function. For example, an occurrence of the generalized pattern $b$-$ca$ in a permutation $\pi=\pi_1\pi_2 \cdots \pi_n$ is a subsequence $\pi_i\pi_j\pi_{j+1}$ such that $i<j$ and $\pi_{j+1}<\pi_i<\pi_j$. For $\pi=4753162$, we have $(b$-$ca)\pi=4$.
Further, Babson and Steingrímsson [@Babson] showed that almost all of the Mahonian permutation statistics in the literature can be written as linear combinations of generalized patterns. We list some of them below. $$maj=(a-cb)+(b-ca)+(c-ba)+(ba),$$ $$stat=(ac-b)+(ba-c)+(cb-a)+(ba).$$
They conjectured that the statistic $(des,stat)$ is Euler-Mahonian.
\[stat\] The distribution of the bistatistic $(des,stat)$ is equal to that of $(des,maj)$.
In 2001, D. Foata and D. Zeilberger [@Foata] gave a proof of this conjecture using q-enumeration and generating functions and an almost completely automated proof via Maple packages ROTA and PERCY.
Given a permutation $\pi=\pi_1 \pi_2 \cdots \pi_n$, let $F(\pi)=\pi_1$ be the first letter of $\pi$ and $$adj(\pi)=|\{i \colon 1 \leq i \leq n \text{~and~} \pi_i- \pi_{i+1}=1\}|,$$ where $\pi_{n+1}=0$. Burstein [@Burstein] provided a bijective proof of the following refinement of Conjecture \[stat\] as follows.
Statistics $(adj, des, F, maj, stat)$ and $(adj, des, F, stat, maj)$ are equidistributed over $S_n$ for all $n$.
In this paper, we will give a new bijective proof of Conjecture \[stat\], which does not preserve the statistic $adj$.
A new bijective proof of Conjecture \[stat\]
============================================
In this section, we recall a particular bijection $\varphi$ on $S_n$ that maps the inversion number to the major index, which is due to Carlitz [@Carlitz] and stated more clearly in [@remmel] and [@skandera]. Based on this, we give an analogous bijection which proves Conjecture \[stat\].
To give a description of $\varphi$, we first recall two labeling schemes for permutations. It involves accounting for the effects of inserting a new largest element into a permutation, and so it is known as the insertion method.
Given a permutation $\sigma=\sigma_1 \sigma_2 \cdots \sigma_{n-1}$ in $S_{n-1}$. To obtain a permutation $\pi \in S_n$, we can insert $n$ in $n$ spaces, namely, immediately before $\sigma_1$ or immediately after $\sigma_i$ for $1 \leq i \leq n$. In order to keep track of how the insertion of $n$ affects the inversion number and the major index, we may define two labelings of the $n$ inserting spaces.
The inv-labeling of $\sigma$ is given by numbering the spaces from right to left with $0,1,\ldots,n-1$. The maj-labeling of $\sigma$ is obtained by labeling the space after $\sigma_{n-1}$ with $0$, labeling the descents from right to left with $1,2,\ldots,\rm{des(\sigma)}$ and labeling the remaining spaces from left to right with $des(\sigma)+1,\ldots,n$. As an example, let $\sigma=13287546$, the inv-labeling of $\sigma$ is $$_8 1_7 3_62_5 8_4 7_3 5_2 4_1 6_0,$$ while the maj-labeling of $\sigma$ is given by $$_5 1_6 3_4 2_7 8_3 7_2 5_1 4_8 6_0.$$
For $n \geq 2$, we define the map $$\phi_{inv,n}\colon \{0,1,\ldots,n-1\}\times S_{n-1} \to S_n$$ by setting $\phi_{inv,n}(i,\sigma)$ to be the permutation obtained by inserting $n$ in the space labeled $i$ in the inv-labeling of $\sigma$. By changing the inv-labeling to maj-labeling, we obtain the map $\phi_{maj,n}$. As an example, $$\phi_{inv,9}(3,13287546)=132879546 ~~\text{and}~~ \phi_{maj,9}(3,13287546)=132897546.$$
For maps $\phi_{inv,n}$ and $\phi_{maj,n}$, we have the following two lemmas.
\[inv\] For $\sigma \in S_{n-1}$ and $i \in \{0,1, \ldots, n-1 \}$, we have $$inv(\phi_{inv,n}(i,\sigma))=inv(\sigma)+i.$$
\[maj\] For $\sigma \in S_{n-1}$ and $i \in \{0,1, \ldots, n-1 \}$, we have $$maj(\phi_{maj,n}(i,\sigma))=maj(\sigma)+i.$$
Lemma \[inv\] is easy to verified. For a detailed proof of Lemma \[maj\], see [@Haglund].
Given a permutation $\pi \in S_n$, let $\pi^{(i)}$ be the restriction of $\pi$ to the letters $1,2,\ldots,i$ for $1 \leq i \leq n$ . For $2 \leq i \leq n$, let $$(c_i,\pi^{(i-1)})= \phi_{inv,i}^{-1}(\pi^{(i)}),$$ $$(m_i,\pi^{(i-1)})= \phi_{maj,i}^{-1}(\pi^{(i)}).$$ By setting $c_1=0$ and $m_1=0$, we obtain two sequences $c_1c_2 \cdots c_n$ and $m_1m_2 \cdots m_n$ in $E_n$, where $$E_n=\{w=w_1 \cdots w_n| w_i\in [0,i-1], 1 \leq i \leq n\}.$$ Define $\gamma(\pi)=c_1c_2 \cdots c_n $ and $\mu(\pi)=m_1m_2 \cdots m_n $. It is not hard to check that both $\gamma$ and $\mu$ are bijections. The sequence $c_1c_2 \cdots c_n$ is called the inversion table of $\pi$, while $m_1m_2 \cdots m_n$ is called the major index table of $\pi$. Moreover, we have $\sum_{i=1}^{n}c_i=inv(\pi)$ and $\sum_{i=1}^{n}m_i=maj(\pi).$ As an example, let $\pi=13287546$, then $\pi^{(8)}=\pi$ and $\mu(\pi)=00204056$ as computed in Table \[majtable\].
$i$ $\pi^{(i-1)}$ $m_i$
----- ----------------------------------------------- -------
$8$ $_4 1_5 3_3 2_6 7_2 5_1 4_7 6_0$ $6$
$7$ $_3 1_4 3_2 2_5 5_1 4_6 6_0$ $5$
$6$ $_3 1_4 3_2 2_5 5_1 4_0$ $0$
$5$ $_2 1_3 3_1 2_4 4_0 $ $4$
$4$ $_2 1_3 3_1 2_0 $ $0$
$3$ $_1 1_2 2_0$ $2$
$2$ $_1 1_0$ $0$
: The computation of the major index table of $13287546$.[]{data-label="majtable"}
Now, we can define the bijection $\varphi$ that maps the inversion number to the major index by letting $\varphi=\mu^{-1}\gamma$. Clearly, $\varphi$ is a bijection which proves that statistics $inv$ and $maj$ are equidistributed over $S_n$. In the following of this section, we will construct a bijection $\rho$ to prove Conjecture \[stat\], which is, to some extent, an analogue of the above bijection $\varphi$.
First, we define a stat-labeling of $\sigma \in S_n$. Label the descents of $\sigma$ and the space after $\sigma_n$ by $0,1,\ldots,des(\sigma)$ from left to right. Label the space before $\sigma_0$ by $des(\sigma)+1$. The ascents of $\sigma$ are labeled from right to left by $des(\sigma)+2, \ldots, n$. As an example, for $\sigma=13287546$, we have the stat-labeling of $\sigma$ as follows $$_5 1_8 3_0 2_7 8_1 7_2 5_3 4_6 6_4.$$ Based on the stat-labeling, we define the map $\phi_{stat,n}$ for $n \geq 2$. For a permutation $\sigma=\sigma_1 \sigma_2 \ldots \sigma_{n-1}$ and $0 \leq i \leq n-1$, define $\phi_{stat,n}(i,\sigma)$ to be the permutation obtained by inserting $n$ in the position labeled $i$ in the stat-labeling of $\sigma$. For instance, $\phi_{stat,9}(7,13287546)=132987546$.
It should be noted that unlike the properties of $\phi_{inv,n}$ and $\phi_{maj,n}$ stated in Lemma \[inv\] and Lemma \[maj\], we deduce that $\phi_{stat,n}(i,\sigma)) \neq stat
(\sigma)+i$ for $i=des(\sigma)+1$. For the map $\phi_{stat,n}$, we have the following property.
\[statlem\] For $\sigma \in S_{n-1}$ and $i \in \{0,1,\ldots, des(\sigma),des(\sigma)+2, \ldots, n-1 \}$, we have $$stat(\phi_{stat,n}(i,\sigma))=stat
(\sigma)+i.$$
First, we recall that $stat=(ac$-$b)+(ba$-$c)+(cb$-$a)+(ba)$. To prove this lemma, we have to consider the changes of the statistic $stat$ brought by inserting $n$ into $\sigma$. Assume that $\sigma=\sigma_1 \sigma_2\cdots \sigma_{n-1}$, there are three cases for us to consider.
- Case 1 $\colon$ $n$ is inserted into the space after $\sigma_{n-1}$.
Write $\pi=\sigma n$. Clearly, the insertion of $n$ does not bring new $ac$-$b$, $cb$-$a$ and $ba$ patterns. While $n$ can form new $ba$-$c$ patterns of $\pi$ with the $ba$ patterns of $\sigma$. It follows that $stat(\pi)-stat(\sigma)=(ba)\sigma=des(\sigma)$. Notice that the label of the space after $\sigma_{n-1}$ is $des(\sigma)$. Hence, in this case the lemma holds.
- Case 2 $\colon$ $n$ is inserted into a descent.
Suppose that $\tau$ is the permutation obtained by inserting $n$ to the position $i$ and $\sigma_i> \sigma_{i+1}$. Moreover, let this descent be the $k$-th descent from left to right. We claim that $stat(\tau) -stat(\sigma)=k-1$.
The insertion of $n$ forms some new $ac$-$b$ patterns, and the number of these new patterns is $|\{j, ~j>i ~\text{and}~ \sigma_j>\sigma_i\}|$. Moreover, $n$ forms $k-1$ new $ba$-$c$ patterns with the former $k-1$ descents, while it destroys the $ba$-$c$ patterns of $\sigma$ by the number $|\{j, ~j>i ~\text{and}~ \sigma_j>\sigma_i\}|$. It is easy to verify the functions $(cb$-$a)$ and $(ba)$ do not change. Hence, we conclude that $stat(\tau) -stat(\sigma)=k-1$. The claim is verified. Notice that the label of the $k$-th descent from left to right is also $k-1$. It follows that the lemma holds for this case.
- Case 3 $\colon$ $n$ is inserted into an ascent.
Suppose that $p$ is the permutation obtained by inserting $n$ into the position $i$, where $\sigma_i < \sigma_{i+1}$. Moreover, we assume that there are $k$ descents to the left of position $i$. We claim that $stat(p)-stat(\sigma)=k+n-i+1$.
Now we proceed to prove this claim. The insertion of $n$ to $\sigma$ brings new $ac$-$b$ patterns, the number of which is $|\{j, ~j \geq i+1 ~\text{and}~ \sigma_j \geq \sigma_{i+1}\}|$. Moreover, the insertion of $n$ brings $k$ new $ba$-$c$ patterns with the former $k$ descents. Also, $n\sigma_{i+1}\sigma_l$, where $l>i+1$ and $\sigma_l<\sigma_{i+1}$, forms a new $cb$-$a$ pattern of $p$. Notice that $(ba)p-(ba)\sigma=1$. By combining all above, we see that $stat(\tau) -stat(\sigma)=k+n-i+1$. The claim is verified. By the stat-labeling of $\sigma$, we see that the label of this position is also $k+n-i+1$. Hence, in this case the lemma holds.
By combining the three cases above, we compete the proof.
Based on the stat-labeling, we define the stat table of a permutation. Given $\pi \in S_n$, for $2 \leq i \leq n$, let $$(s_i,\pi^{(i-1)})= \phi_{stat,i}^{-1}(\pi^{(i)}).$$ Set $s_1=0$ and $\nu(\pi)=s_1 s_2 \cdots s_n$. It is easily checked that $\nu$ is a bijection from $S_n$ to $E_n$. As an example, $\nu(52718346)=01112216$, which is computed in Table \[stattable\].
$i$ $\pi^{(i-1)}$ $s_i$
----- ----------------------------------------------- -------
$8$ $_3 5_0 2_7 7_1 1_6 3_5 4_4 6_2 $ $6$
$7$ $_3 5_0 2_1 1_6 3_5 4_4 6_2 $ $1$
$6$ $_3 5_0 2_1 1_5 3_4 4_2 $ $2$
$5$ $_2 2_0 1_4 3_3 4_1$ $2$
$4$ $_2 2_0 1_3 3_1 $ $1$
$3$ $_2 2_0 1_1 $ $1$
$2$ $_1 1_0$ $1$
: The computation of the stat table of $52718346$.[]{data-label="stattable"}
Now we can give the definition of the map $\rho$ which proves Conjecture \[stat\]. Given $\pi \in S_n$, let $\sigma= \rho(\pi)$, where $\sigma$ can be constructed as follows. Assume that $F(\pi)=k$ and $\pi^{(k)}=k p_2 \cdots p_k $. Then, let $\sigma^{(k)}=k (k-p_k) (k-p_{k-1}) \cdots (k-p_2)$. Assume that $s=s_1 s_2 \cdots s_n=\nu(\pi)$. For $k+1 \leq i \leq n$, let $\sigma^{(i)}=\phi_{maj,i}(s_i,\sigma^{(i-1)})$. Clearly, $\sigma=\sigma^{(n)}$ can be constructed by the procedure above. As an example, we compute $\rho(\pi)$, where $\pi=52718346$. It is straightforward to see that $k=5$ and $\pi^{(5)}=52134$.
$i$ $s_i$ $\sigma^{(i)}$
----- ------- ----------------------------------
$5$ $2$ $_3 5_2 1_4 2_5 4_1 3_0 $
$6$ $2$ $_3 5_4 6_2 1_5 2_6 4_1 3_0$
$7$ $1$ $_3 5_4 6_2 1_5 2_6 4_7 7_1 3_0$
$8$ $6$ $56128473$
: The computation of $\rho(52718346)$.[]{data-label="rho"}
By the computation in Table \[rho\], we see that $\rho(52718346)=56128473$. Notice that in this example, the descent number and the first letter of both of the preimage and image of $\rho$ are the same. In fact, these properties always holds.
\[F\] For $\pi \in S_n$, we have $F(\pi)=F(\rho(\pi))$ and $des(\pi)=des(\rho(\pi))$.
Suppose that $\sigma=\rho(\pi)$ and $k=F(\pi)$. We proceed to show that $F(\sigma^{(i)})=F(\pi^{(i)})=k$ and $des(\sigma^{(i)})=des(\pi^{(i)})$ for $k \leq i \leq n$ by induction. Let $\pi^{(k)}=k p_2 \cdots p_k $, then we have $\sigma^{(k)}=k (k-p_k) (k-p_{k-1}) \cdots (k-p_2)$. It is routine to check that $F(\sigma^{(k)})=F(\pi^{(k)})=k$ and $des(\sigma^{(k)})=des(\pi^{(k)})$, hence, we omit the details here.
Now assume that $des(\sigma^{(l)})=des(\pi^{(l)})=d$ and $F(\sigma^{(l)})=F(\pi^{(l)})=k$ for $k \leq l \leq n-1 $. We proceed to show that $des(\sigma^{(l+1)})=des(\pi^{(l+1)})$ and $F(\sigma^{(l+1)})=F(\pi^{(l+1)})=k$.
Let $s=s_1 s_2 \cdots s_n=\nu(\pi)$. By the constructions of $\nu$ and $\rho$, we may see that $\pi^{(l+1)}=\phi_{stat,l+1}(s_{l+1},\pi^{(l)})$ and $\sigma^{(l+1)}=\phi_{maj,l+1}(s_{l+1},\sigma^{(l)})$. Notice that both in the maj-labeling of $\sigma^{(l)}$ and the stat-labeling of $\pi^{(l)}$, the descents and the space after the last letter are labeled by $\{0,1,\ldots,d\}$, the space before the first element is labeled by $d+1$, and the ascents are labeled by $\{d+2, \ldots, l\}$. Since $F(\pi^{(l)})=k=F(\pi)$, it is easy to see that $s_{l+1} \neq d+1$. It follows that $F(\sigma^{(l+1)})=F(\pi^{(l+1)})=k$.
If $s_{l+1}<d+1$, then $l+1$ is inserted into the descents or the space after the last element of $\pi^{(l)}$ and $\sigma^{(l)}$. Hence, we deduce that $des(\sigma^{(l+1)})=des(\pi^{(l+1)})=d$. If $s_{l+1}> d+1$, then $l+1$ is inserted into the ascents of $\pi^{(l)}$ and $\sigma^{(l)}$. Hence, we deduce that $des(\sigma^{(l+1)})=des(\pi^{(l+1)})=d+1$. Combining the two cases above, we have $des(\sigma^{(l+1)})=des(\pi^{(l+1)})$. Notice that $\sigma^{(n)}=\sigma$ and $\pi^{(n)}=\pi$, we complete the proof.
Base on the construction of $\rho$ and Lemma \[F\], we have the following theorem.
\[bijection\] The map $\rho$ is an involution on $S_n$.
Given a permutation $\pi \in S_n$, it suffices for us to show that $\rho^2(\pi)=\pi$. That is, writing $\sigma=\rho(\pi)$, we need to show that $\rho(\sigma)=\pi$.
Let $F(\pi)=k$, then by Lemma \[F\], we know that $k=F(\pi)=F(\sigma)$. Write $\pi^{(k)}=k p_2 \cdots p_k$, then we have $\sigma^{(k)}=k (k-p_k) \cdots (k-p_2) $. Assume that $\nu(\pi)=s=s_1 s_2 \cdots s_n$ and $\mu(\sigma)=m_1 m_2 \cdots m_n$. By the construction of $\rho$, we have $\sigma^{(i)}=\phi_{maj,i}(s_i,\sigma^{(i-1)})$ for $k+1 \leq i \leq n$. It follows that $m_i=s_i$ for $k+1 \leq i \leq n$.
Suppose that $\alpha=\rho(\sigma)$, in the following, we proceed to show that $\alpha^{(i)}=\pi^{(i)}$ for $k \leq i \leq n $ by induction. By the definition of $\rho$, we have $\alpha^{(k)}=kp_2\cdots p_k=\pi^{(k)}$. Assume that $\alpha^{(i)}=\pi^{(i)}$ holds for $k \leq i \leq n-1$, we aim to show that $\alpha^{(i+1)}=\pi^{(i+1)}$. To achieve this, we need to mention the following property of the maj-labeling and the stat-labeling of a single permutation.
For a permutation $p\in S_n$, assume the maj-labeling of $p$ is $f_0f_1 \cdots f_n$, while the stat-labeling of $p$ is $h_0h_1 \cdots h_n$. Then it is easily checked that
$$\label{label}
f_i+h_i = \left\{
\begin{array}{ll}
des(p), & \mbox{if $i$ is a descent of $p$ or $i=n$,}\\[3pt]
n+des(p)+2, & \mbox{if $i$ is an ascent of $p$,}\\[6pt]
2des(p)+2, & \mbox{if $i=0$.}
\end{array} \right.$$
Let $\nu(\sigma)=l_1 l_2 \cdots l_n$ and $d=des(\alpha^{(i)})$. Then, we have $$\alpha^{(i+1)} =\phi_{maj,i+1}(l_{i+1},\alpha^{(i)}) = \left\{
\begin{array}{ll}
\phi_{stat,i+1}(d-l_{i+1},\alpha^{(i)}), & \mbox{if $0 \leq l_{i+1} \leq d $,}\\[3pt]
\phi_{stat,i+1}(n+d+2-l_{i+1},\alpha^{(i)}), & \mbox{if $d+2 \leq l_{i+1} \leq n $.}
\end{array} \right.$$ By the proof of Lemma \[F\], we see that $des(\sigma^{(i)})=des(\alpha^{(i)})=d$. Recall that $\nu(\sigma)=l_1 l_2 \cdots l_n$ and $\mu(\sigma)=m_1 m_2 \cdots m_n$. Hence, it follows from (\[label\]) that $$\begin{aligned}
\alpha^{(i+1)}=& ~\phi_{stat,i+1}(m_{i+1},\alpha^{(i)})\\[3pt]
=& ~\phi_{stat,i+1}(s_{i+1},\pi^{(i)})\\[3pt]
=& ~\pi^{(i+1)}.\end{aligned}$$ Notice that $\alpha=\alpha^{(n)}$ and $\pi=\pi^{(n)}$. Hence, we have $\pi=\rho(\sigma)$, namely, $\rho^2(\pi)=\pi$. This completes the proof.
Indeed, the involution $\rho$ also preserves some addtional statistics, which is stated in the following proposition.
\[stat=maj\] For any $\pi=\pi_1 \pi_2 \cdots \pi_n \in S_n$, we have $maj(\rho(\pi))=stat(\pi)$ and $stat(\rho(\pi))=maj(\pi)$.
Write $\sigma=\rho(\pi)$. Let $k=F(\pi)$ and $\pi^{(k)}=k p_2 \cdots p_k$. Then we have $\sigma^{(k)}=
k (k-p_k) \cdots (k-p_2)$. In the following, we will show that $maj(\sigma^{(i)})=stat(\pi^{(i)})$ for $k \leq i \leq n$ by induction.
First, we show that $maj(\sigma^{(k)})=stat(\pi^{(k)})$. By the proof of Lemma \[F\], we have $des(\pi^{(k)})=des(\sigma^{(k)}).$ Hence $(ba)\pi^{(k)}=(ba)\sigma^{(k)}$. It suffices for us to show that $$\label{**}
(ac-b)\pi^{(k)}+(ba-c)\pi^{(k)}+(cb-a)\pi^{(k)}=
(a-cb)\sigma^{(k)}+(b-ca)\sigma^{(k)}+(c-ba)\sigma^{(k)}$$ Given a pattern $p=p_1 p_2 \cdots p_k$, by putting a line under $p_1$(resp. $p_k$), we mean that an instance of $p$ must begin(resp. end) with the leftmost(resp. rightmost) letter of the permutation. By putting a dot under $p_1$, we mean that an instance of $p$ must not begin with the leftmost letter. For instance, $(\b{c}b$-$a)$ is a function which maps a permutation, say $\tau=\tau_1 \tau_2 \cdots \tau_n$, to $|\{i, \tau_1 \tau_2\tau_{i} ~\text{forms a 321 pattern of}~\tau\}|.$
By the definition of $\rho$, we know that $$\begin{aligned}
&(ac-b)\pi^{(k)}+(ba-c)\pi^{(k)}+(cb-a)\pi^{(k)}\\
=&(ac-b)\pi^{(k)}+(ba-c)\pi^{(k)}+(\b{c}b-a)\pi^{(k)}+(\d{c}b-a)\pi^{(k)}\\
=&(b-ac)\sigma^{(k)} + (a-cb)\sigma^{(k)} + (\b{c}-b-\b{a})\sigma^{(k)} + (\d{c}-ba)\sigma^{(k)}
\end{aligned}$$ Hence, to prove (\[\*\*\]), we need to prove that for any permutation $p \in S_k$ with $p_1=k$, $$\label{***}
(\b{c}-ba)p+(b-ca)p=(\b{c}-b-\b{a})p+(b-ac)p.$$ We define sets $A(p),C(p)$ and multisets $ B(p),D(p)$ as follows. $$\begin{aligned}
A(p)&=\{p_i \colon p_1p_i p_{i+1} \text{ forms a $321$ pattern of } p\},\\
B(p)&=\{p_i \colon p_ip_j p_{j+1} \text{ forms a $231$ pattern of } p\},\\
C(p)&=\{p_i \colon p_1p_i p_{m} \text{ forms a $321$ pattern of } p\},\\
D(p)&=\{p_i \colon p_ip_j p_{j+1} \text{ forms a $213$ pattern of } p\}.
\end{aligned}$$ To prove (\[\*\*\*\]), it is enough to show that $A \cup B =C\cup D$, where the union operator is a multiset union. First, we show that $C\cup D \subseteq A \cup B$.
Let $p_k=a$, then we know that $C(p)=\{a+1, a+2, \ldots, k-1\}$. If $p_i \in C(p)$ is a descent top, it is easy to see that $p_i \in A(p)$. If $p_i \in C(p)$ is an ascent bottom, we claim that $p_i \in B(p)$. This claim will be proved together with case 4 in the following.
Given an element $p_i$ in $D(p)$, if the multiplicity of $p_i$ is $x$, there exists a set $$\{j_1,j_1+1, j_2, j_2+1,\ldots,j_x,j_x+1\},$$ which is ordered by increasing order, satisfying that $$p_ip_{j_1}p_{j_1+1},~ p_ip_{j_2}p_{j_2+1},~\ldots,~
p_ip_{j_x}p_{j_x+1}$$ are instances of $213$ pattern. We claim that there exists $j_1+1 \leq r_1 < j_{2}$ such that $p_i p_{r_1}p_{r_1+1}$ forms a $231$ pattern.
Choose the smallest $g_1$ such that $j_1+1 \leq g_1< j_2$ and $g_1$ is a descent. If $p_{g_1+1}<p_i$, then $p_i p_{g_1}p_{g_1+1}$ forms a $231$ pattern, the claim is verified. Otherwise, we seek the smallest $g_1+1 \leq g_2< j_2$ such that $g_2$ is a descent. If $p_{g_2+1}<p_i$, the claim is verified. If not, we repeat the above process. Since $p_i > p_{j_2}$, the process must be terminated. Hence, the claim is verified. By a similar means, we deduce that there exists $j_l+1 \leq r_l < j_{l+1}$ where $1 \leq l \leq x-1$ such that $p_i p_{r_l}p_{r_l+1}$ forms a $231$ pattern. The claim is verified.
To analyze the element $p_i$ in $D(p)$, we consider four cases.
- $p_i$ is a descent top and $1 < p_i \leq a-1$.
Suppose that the multiplicity of $p_i$ of this type in $D(p)$ is $x$. By the above statement, we see that $p_i p_{r_l}p_{r_l+1}$ forms a $231$ pattern, where $j_l+1 \leq r_l < j_{l+1}$ and $1 \leq l \leq x-1$. Thus, we deduce that there are $x-1$ $p_i$s in $B(p)$. Notice that there is one $p_i$ left in $B(p)$. Clearly, we can set this $p_i$ to be the element of $A(p)$ consisting of $p_1$, $p_i$ and $p_{i+1}$.
- $p_i$ is an ascent bottom and $1 \leq p_i \leq a-1$.
Suppose that the multiplicity of $p_i$ of this type in $D(p)$ is $x$. Similarly, we know that $p_i p_{r_l}p_{r_l+1}$ forms a $231$ pattern, where $j_l+1 \leq r_l < j_{l+1}$ and $1 \leq l \leq x-1$. Since $p_i < p_{i+1}$, we have $j_1>i+1$. Based on this, it can be easily seen that there exists $r_0$ such that $i+1 \leq r_0 < j_1$ and $p_i p_{r_0}p_{r_0+1}$ forms a $231$ pattern. Hence, we deduce that in this case there are $x$ $p_i$s in $B(p)$.
- $p_i$ is a descent top and $a+1 \leq p_i \leq k-1$.
Suppose that the multiplicity of $p_i$ of this type in $D(p)$ is $x$. Similarly, we deduce that $p_i p_{r_l}p_{r_l+1}$ forms a $231$ pattern, where $j_l+1 \leq r_l < j_{l+1}$ and $1 \leq l \leq x-1$. What’s more, it follows from $p_i>a$ that there exists $j_x \leq r_x<m$ such that $p_i p_{r_{x}}a$ forms a $231$ pattern. Hence, we deduce that $p_i$ in $B(p)$ and its multiplicity is $x$.
- $p_i$ is an ascent bottom and $a+1 \leq p_i \leq k-1$.
Suppose that the multiplicity of $p_i$ of this case in $D(p)$ is $x$. Notice that $p_i$ is also an element of $C(p)$ with multiplicity equals $1$. Hence, in this case, we have to prove that there are $x+1$ $p_i$s in $B(p)$.
Similarly with the above cases, we deduce that $p_i p_{r_l}p_{r_l+1}$ forms a $231$ pattern, where $j_l+1 \leq r_l < j_{l+1}$ and $1 \leq l \leq x-1$. Since $p_i>a$, there exists $j_x \leq r_x<m$ such that $p_i p_{r_{x}}a$ forms a $231$ pattern. By $p_i < p_{i+1}$, we have $j_1>i+1$. Based on this, it can be easy seen that there exists $r_0$ such that $i+1 \leq r_0 < j_1$ and $p_i p_{r_0}p_{r_0+1}$ forms a $231$ pattern. Hence, we deduce that $p_i$ in $B(p)$ and its multiplicity is $x+1$.
Combining all above, we deduce that $C\cup D \subseteq A \cup B$. By a similar analysis, we can prove that $A \cup B \subseteq C\cup D$. We omit it here. As an example, if $p=978452613$, we have $A(p)=\{5,6,8\}$, $B(p)=\{2,4,4,5,7\}$, $C(p)=\{4,5,6,7,8\}$ and $D(p)=\{2,4,5\}$. It can be verified that $A \cup B =C\cup D$. This proves that $maj(\sigma^{(k)})=stat(\pi^{(k)})$.
Now assume that $maj(\sigma^{(i)})=stat(\pi^{(i)})$ for $k \leq i \leq n-1$, we proceed to show that $maj(\sigma^{(i+1)})=stat(\pi^{(i+1)})$. Write $\nu(\pi)=s_1 s_2 \cdots s_n$. Then, by Lemma \[maj\], Lemma \[statlem\] and the construction of $\rho$, we have $$\begin{aligned}
maj(\sigma^{(i+1)})=&~ s_{i+1}+maj(\sigma^{(i)}) \\[3pt]
=& ~s_{i+1}+stat(\pi^{(i)}) \\[3pt]
=& ~ stat(\pi^{(i+1)}).\end{aligned}$$ This proves $maj(\sigma^{(i)})=stat(\pi^{(i)})$ for $k \leq i \leq n$. Notice that $\pi=\pi^{(n)}$ and $\sigma=\sigma^{(n)}$. We deduce that $maj(\sigma)=maj(\rho(\pi))=stat(\pi)$. By Theorem \[bijection\], we see that $\rho$ is an involution. This implies that $stat(\rho(\pi))=maj(\pi)$. This completes the proof.
Combining Lemma \[F\], Theorem \[bijection\] and Proposition \[stat=maj\], we give a proof of Conjecture \[stat\].
It should be mentioned that Burstein [@Burstein] provided a direct bijective proof of a refinement of Conjecture \[stat\]. The bijection $\chi$ is given as follows. Given a permutation $\pi \in S_n$ with $F(\pi)=k$. Let $\pi'=\chi(\pi)$ with $\pi'(1)=k$ and $$\label{label}
\pi'(i) = \left\{
\begin{array}{ll}
k-\pi(n+2-i), & \mbox{if $\pi(n+2-i)<k$,}\\[3pt]
n+k+1-\pi(n+2-i), & \mbox{if if $\pi(n+2-i)>k$.}
\end{array} \right.$$ In addition to preserving the statistics $des$ and $F$, the bijection $\chi$ also preserves the statistic $adj$, while our bijection does not. As an example, set $\pi=543617982$, then $\rho(\pi)=\sigma=539784621$ and $\chi(\pi)=\pi'=537684921$. It can be checked that $adj(\pi) = adj(\pi')$, while $adj(\pi) \neq adj(\sigma)$. Moreover, it is easily seen that for $\pi \in S_n$ with $F(\pi)=n$, we have $\rho(\pi)=\chi(\pi)=\sigma$. We note that in this case both Burstein and us have to prove $maj(\sigma)=stat(\pi)$. Different form our proof in Proposition \[stat=maj\], Burstein gave the following two relations, which implies that $maj(\sigma)=stat(\pi)$. $$maj(\pi)+stat(\pi)=(n+1)des(\pi)-(F(\pi)-1),$$ $$maj(\pi)+maj(\sigma)=(n+1)des(\pi)-(F(\pi)-1).$$
[**Acknowledgments.**]{} We wish to thank the anonymous referees for their valuable comments and suggestions.
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L. Carlitz, A combinatorial property of q-Eulerian numbers, Amer. Math. Monthly, 82 (1975), 51-54.
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abstract: 'Given an ample line bundle $L$ on a $K3$ surface $S$, we study the slope stability with respect to $L$ of rank-$3$ Lazarsfeld-Mukai bundles associated with complete, base point free nets of type $g^2_d$ on curves $C$ in the linear system $\vert L\vert$. When $d$ is large enough and $C$ is general, we obtain a dimensional statement for the variety $W^2_d(C)$. If the Brill-Noether number is negative, we prove that any $g^2_d$ on any smooth, irreducible curve in $\vert L\vert$ is contained in a $g^r_e$ which is induced from a line bundle on $S$, thus answering a conjecture of Donagi and Morrison. Applications towards transversality of Brill-Noether loci and higher rank Brill-Noether theory are then discussed.'
address: 'Humboldt Universität zu Berlin, Institut für Mathematik, 10099 Berlin'
author:
- 'Margherita Lelli–Chiesa'
---
Introduction and statement of the results
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Many results of Brill-Noether theory regarding a general point in the moduli space $M_g$, which parametrizes isomorphism classes of smooth, irreducible curves of genus $g$, have been proved by studying curves lying on $K3$ surfaces. One of the advantages of considering an irreducible curve $C\subset S$, where $S$ is a smooth $K3$ surface, is that some interesting properties, such as the Clifford index, do not change while moving $C$ in its linear system (cf. [@green]). Moreover, Brill-Noether theory on $C$ is strictly connected with the geometry of some moduli spaces of vector bundles on the $K3$ surface. Indeed, given a complete, base point free linear series $A$ on $C$, one associates with the pair $(C,A)$ a vector bundle on $S$, the so-called Lazarsfeld-Mukai bundle, denoted by $E_{C,A}$.
Lazarsfeld-Mukai bundles were first used by Lazarsfeld, in order to show that, given a $K3$ surface $S$ such that ${\mathrm{Pic}}(S)={\mathbb{Z}}\cdot L$, a general curve $C\in\vert L\vert$ satisfies the Gieseker-Petri Theorem, that is, for any line bundle $A\in {\mathrm{Pic}}(C)$ the Petri map $$\mu_{0,A}:H^0(C,A)\otimes H^0(C,\omega_C\otimes A^\vee)\to H^0(C,\omega_C)$$ is injective (cf. [@lazarsfeld], [@pareschi], or [@lazarsfeld1] for a more geometric argument).
It is natural to investigate what happens if the Picard number of $S$ is greater than $1$. In order to do this, having denoted by $\vert L\vert_s$ the locus of smooth, connected curves in the linear system $\vert L\vert$ and chosen two positive integers $r,d$, one studies the natural projection $\pi:{\mathcal W}^r_d(\vert L\vert)\to \vert L\vert_s$, whose fibre over $C$ coincides with the Brill-Noether variety $W^r_d(C)$. We set $g:=1+L^2/2$; this coincides with the genus of curves in $\vert L\vert_s$.
At first we look at the cases where $\rho(g,r,d)<0$. Following [@donagi], we say that a line bundle $M$ is [*adapted*]{} to $\vert L\vert$ whenever
1. \[pe2\] $h^0(S,M)\ge2$, $h^0(S,L\otimes M^\vee)\geq 2$,
2. \[pe4\] $h^0(C,M\otimes{\mathcal O}_{C})$ is independent of the curve $C\in\vert L\vert _s$.
Conditions (i) and (ii) ensure that $M\otimes {\mathcal O}_C$ contributes to the Clifford index of $C$ and ${\mathrm{Cliff}}(M\otimes {\mathcal O}_C)$ is the same for any $C\in\vert L\vert_s$.
Donagi and Morrison ([@donagi] Theorem (5.1’)) proved that, if $A$ is a complete, base point free pencil $g^1_d$ on a nonhyperelliptic curve $C\in\vert L\vert_s$ and $\rho(g,1,d)<0$, then $\vert A\vert$ is contained in the restriction to $C$ of a line bundle $M\in{\mathrm{Pic}}(S)$ which is adapted to $\vert L\vert$ and such that ${\mathrm{Cliff}}(M\otimes {\mathcal O}_C)\leq {\mathrm{Cliff}}(A)$. The same is expected to hold true for any linear series of type $g^r_d$ with $\rho(g,r,d)<0$ (compare with [@donagi] Conjecture (1.2)). We prove this conjecture for $r=2$ under some mild hypotheses on $L$.
\[thm:magari\] Let $S$ be a $K3$ surface and $L\in{\mathrm{Pic}}(S)$ be an ample line bundle such that a general curve in $\vert L\vert$ has genus $g$, Clifford dimension $1$ and maximal gonality $k=\left\lfloor \frac{g+3}{2}\right\rfloor$. Let $A$ be a complete, base point free $g^2_d$ on a curve $C\in\vert L\vert_s$ such that $\rho(g,2,d)<0$.
Then, there exists $M\in{\mathrm{Pic}}(S)$ adapted to $\vert L\vert$ such that the linear system $\vert A\vert$ is contained in $\vert M\otimes {\mathcal O}_C\vert$ and ${\mathrm{Cliff}}(M\otimes {\mathcal O}_C)\leq {\mathrm{Cliff}}(A)$. Moreover, one has $c_1(M)\cdot C\leq (4g-4)/3$.
We recall that the assertion that $\vert A\vert$ is contained in $\vert M\otimes {\mathcal O}_C\vert$ is equivalent to the requirement $h^0(C, A^\vee\otimes M\otimes {\mathcal O}_C)>0$. The assumption on the gonality $k$ is used for computational reasons; however, the methods of our proof might be adapted in order to treat the cases where $k$ is not maximal. It was proved by Ciliberto and Pareschi (cf. [@ciliberto] Proposition 3.3) that the ampleness of $L={\mathcal O}_S(C)$ forces $C$ to have Clifford dimension $1$ with only one exception occurring for $g=10$.
The case of pencils is very particular, since it involves vector bundles of rank $2$. Donagi and Morrison used the fact that any non-simple, indecomposable Lazarsfeld-Mukai bundle of rank $2$ can be expressed as an extension of the image and the kernel of a nilpotent endomorphism, which both have rank $1$. Their proof cannot be adapted to linear series with $r>1$, corresponding to Lazarsfeld-Mukai bundles of rank at least $3$. Our techniques consist of showing that, under the hypotheses of Theorem \[thm:magari\], the rank-$3$ Lazarsfeld-Mukai bundle $E=E_{C,A}$ is given by an extension $$0\to N\to E\to E/N\to 0,$$ where $N\in{\mathrm{Pic}}(S)$ and $E/N$ is a $\mu_L$-stable, torsion free sheaf of rank $2$. When $E$ is $\mu_L$-unstable, the line bundle $N$ coincides with its maximal destabilizing sheaf and the determinant of $E/N$ plays the role of the line bundle $M$ in the statement. Something similar happens if $E$ is properly $\mu_L$-semistable.
This suggests that the notion of stability might play a fundamental role in a general proof of the Donagi-Morrison Conjecture.
Now, we turn our attention to the cases where $\rho(g,r,d)\geq 0$. In the course of their proof of Green’s Conjecture for curves on arbitrary $K3$ surfaces, Aprodu and Farkas (cf. [@aprodu]) showed that, if $L$ is an ample line bundle on a $K3$ surface such that a general curve $C\in\vert L\vert$ has Clifford dimension $1$ and gonality $k$, given $d>g-k+2$, any dominating component of ${\mathcal W}^1_d(\vert L\vert)$ corresponds to simple Lazarsfeld-Mukai bundles. In particular, when the gonality is maximal this ensures that, if $C$ is general in its linear system and the Brill-Noether number $\rho(g,1,d)$ is positive, the variety $W^1_d(C)$ is reduced and of the expected dimension. In the case $\rho(g,1,d)=0$, one finds that $W^1_d(C)$ is $0$-dimensional, even though not necessarily reduced.
It is natural to wonder to what extent such a result can be expected to hold for linear series of type $g^r_d$ with $r>1$. We prove the following theorem.
\[thm:principale\] Let $S$ be a $K3$ surface and $L\in{\mathrm{Pic}}(S)$ be an ample line bundle such that a general curve in $\vert L\vert$ has genus $g$, Clifford dimension $1$ and maximal gonality $k=\left\lfloor \frac{g+3}{2}\right\rfloor$. Fix a positive integer $d$ such that $\rho(g,2,d)\geq 0$ and assume $(g,d)\not\in\{(2,4),(4,5),(6,6),(10,9)\}$. Then, the following hold:
1. \[aa\] If $d> \frac{3}{4}g+2$, no dominating component of $\mathcal{W}^2_d(\vert L\vert)$ corresponds to rank-$3$ Lazarsfeld-Mukai bundles which are not $\mu_L$-stable.
2. \[bb\] If $d\leq\frac{3}{4}g+2$, let ${\mathcal W}$ be a dominating component of ${\mathcal W}^2_d(\vert L\vert)$ that corresponds to Lazarsfeld-Mukai bundles which are not $\mu_L$-stable. Then, there exists $M\in{\mathrm{Pic}}(S)$ adapted to $\vert L\vert$ such that, for a general $(C,A)\in{\mathcal W}$, the linear system $\vert A\vert$ is contained in $\vert M\otimes {\mathcal O}_C\vert$ and ${\mathrm{Cliff}}(M\otimes {\mathcal O}_C)\leq {\mathrm{Cliff}}(A)$. Moreover, $c_1(M)\cdot C\leq (4g-4)/3$.
Unlike case (\[aa\]), case (\[bb\]) does not exclude the existence of dominating components of ${\mathcal W}^2_d(\vert L\vert)$ which correspond to either $\mu_L$-stable or properly $\mu_L$-semistable Lazarsfeld-Mukai bundles. However, general points of such a component ${\mathcal W}$ give nets $g^2_d$, which are all contained in the restriction of the same line bundle $M\in {\mathrm{Pic}}(S)$ to curves in $\vert L\vert$. Furthermore, the Clifford index of $M\otimes {\mathcal O}_C$ is the same for any $C\in\vert L\vert_s$ and does not exceed $d-4$.
For a curve $C\in\vert L\vert_s$ and for a fixed value of $d$, we define the variety $$\widetilde{W}^2_d(C):=\{A\in W^2_d(C)\,\,\vert\,\, A\textrm{ is base point free}\},$$ which is an open subscheme of $W^2_d(C)$, not necessarily dense. The following result is a direct consequence of Theorem \[thm:principale\].
Under the same hypotheses of Theorem \[thm:principale\], for a general $C\in\vert L\vert_s$ the following hold.
1. If $d> \frac{3}{4}g+2$, the variety $\widetilde{W}^2_d(C)$ is reduced of the expected dimension $\rho(g,2,d)$.
2. If $d\leq\frac{3}{4}g+2$, let $W$ be an irreducible component of $\widetilde{W}^2_d(C)$ which either is non-reduced or has dimension greater than $\rho(g,2,d)$. Then, there exists an effective divisor $D\subset S$ such that ${\mathcal O}_S(D)$ is adapted to $\vert L\vert$ and, for a general $A\in W$, the linear system $\vert A\vert$ is contained in $\vert {\mathcal O}_C(D)\vert$ and ${\mathrm{Cliff}}({\mathcal O}_C(D))\leq {\mathrm{Cliff}}(A)$.
Aprodu and Farkas’ result follows from a parameter count for spaces of Donagi-Morrison extensions corresponding to non-simple Lazarsfeld-Mukai bundles of rank $2$. The strategy used to prove Theorem \[thm:principale\] consists, instead, of counting the number of moduli of $\mu_L$-unstable and properly $\mu_L$-semistable Lazarsfeld-Mukai bundles of rank $3$; this involves Artin stacks that parametrize the corresponding Harder-Narasimhan and Jordan-Hölder filtrations.
The plan of the paper is as follows. Sections \[background1\] and \[background2\] give background information on Lazarsfeld-Mukai bundles and stability of sheaves on $K3$ surfaces.
In Section \[bistrot\] we present a different proof of Aprodu and Farkas’ result and show that, if $\rho(g,1,d)>0$, the Lazarsfeld-Mukai bundles corresponding to general points of any dominating component of ${\mathcal W}^1_d(\vert L\vert)$ are not only simple, but even $\mu_L$-stable (Theorem \[thm:stable\]). We introduce stacks of filtrations, studied for instance by Bridgeland in [@bridgeland] and Yoshioka in [@yoshioka], and explain our parameter count in an easier case. The space of Lazarsfeld-Mukai bundles $E$, such that the bundles appearing in the Harder-Narasimhan filtration of $E$ have prescribed Mukai vectors, turns out to be an Artin stack, whose dimension can be computed by using some well known facts regarding morphisms between semistable sheaves.
In Section \[tempo\] we look at the different types of possible Harder-Narashiman and Jordan-Hölder filtrations of a rank-$3$ Lazarsfeld-Mukai bundle $E$ with $\det(E)=L$ and $c_2(E)=d$. If the determinants of both the subbundles $E_i$ and the quotient sheaves $E^j$, given by the filtration of $E$, have at least $2$ global sections, their restriction to a general curve $C\in\vert L\vert$ contributes to the Clifford index. This is used in order to bound from below the intersection products between the first Chern classes of the sheaves $E_i$ and $E^j$.
In Sections \[mare\], \[section:cola\], \[spiaggia\] we estimate the number of moduli of pairs $(C,A)$ corresponding to rank-$3$ Lazarsfeld-Mukai bunldes which are not $\mu_L$-stable. The subdivision in three sections reflects the different methods necessary to treat various types of filtrations, depending on their length and on the rank of the sheaves $E_i$ and $E^j$. At the end of Section \[spiaggia\] the proofs of both Theorem \[thm:magari\] and Theorem \[thm:principale\] are given.
In Section \[bus\], an application towards transversality of Brill-Noether loci and Gieseker-Petri loci is presented. Recall that the Gieseker-Petri locus $GP_g$ consists, by definition, of curves inside $M_g$ that violate the Gieseker-Petri Theorem. For values of $r,d$ such that $\rho(g,r,d)\geq 0$, one defines the component of $GP_g$ of type $(r,d)$ as $$GP^r_{g,d}:=\{[C]\in M_g\,\vert\,\exists\,(A,V)\in G^r_d(C)\textrm{ with }\ker\mu_{0,V}\neq 0\},$$ where $\mu_{0,V}$ is the Petri map. The subscheme $$\widetilde{GP}^r_{g,d}:=\{[C]\in M_g\,\vert\,\exists\,A\in W^r_d(C)\setminus W^{r+1}_d(C)\textrm{ with }\ker\mu_{0,A}\neq 0\}$$ is open in $GP^r_{g,d}$ but not necessarily dense. We prove the following:
\[thm:tra\] Let $r\geq 3$, $g\geq 0$, $d\leq g-1$ be positive integers such that $\rho(g,r,d)<0$ and $d-2r+2\geq \lfloor(g+3)/2\rfloor$. If $r\geq 4$, assume $d^2>4(r-1)(g+r-2)$. For $r=3$, let $d^2>8g+1$. If $-1$ is not represented by the quadratic form $$Q(m,n)=(r-1)m^2+mnd+(g-1)n^2,\,\,\,m,n\in\mathbb{Z},$$ then:
1. $M^r_{g,d}\not\subset M^1_{g,f}$ for $f<(g+2)/2$.
2. $M^r_{g,d}\not\subset \widetilde{GP}^1_{g,f}$ for $f\geq (g+2)/2$.
3. $M^r_{g,d}\not\subset M^2_{g,e}$ if $e<d-2r+5$ and $\rho(g,2,e)<0$.
4. $M^r_{g,d}\not\subset \widetilde{GP}^2_{g,e}$ if $e<\min\left\{\frac{17}{24}g+\frac{23}{12}, d-2r+5\right\}$ and $\rho(g,2,e)\geq 0$.
The assumption on the quadratic form $Q$ is a mild hypothesis. For instance, it is automatically satisfied when $r$ and $g$ are odd and $d$ is even.
In the last section we exhibit an application of our methods to higher rank Brill-Noether Theory. We give a negative answer to Question 4.2 in [@new], which asks whether the second Clifford index ${\mathrm{Cliff}}_2(C)$, associated with rank-$2$ vector bundles on a curve $C$, equals ${\mathrm{Cliff}}(C)$ whenever $C$ is a Petri curve. We analyze what happens in genus $11$ and look at the Noether-Lefschetz divisor ${\mathcal{NL}}^4_{11,13}$, which consists of curves that lie on a $K3$ surface $S\subset \mathbb{P}^4$ with Picard number at least $2$; this coincides with the locus of curves $[C]\in M_{11}$ such that ${\mathrm{Cliff}}_2(C)<{\mathrm{Cliff}}(C)$ (cf. [@gan]). We prove the following:
\[thm:chisa\] A general curve $[C]\in {\mathcal{NL}}^4_{11,13}$ satisfies the Gieseker-Petri Theorem.
In other words, the Gieseker-Petri divisor $GP_{11}$ and the Noether-Lefschetz divisor ${\mathcal{NL}}^4_{11,13}$ are transversal.\
**Acknowledgements:** This paper is part of my Ph.D. thesis and I am grateful to my advisor Gavril Farkas for discussions. I would like to thank Marian Aprodu for an inspiring conversation had last February in Berlin. A special thank goes to Peter Newstead for giving me the opportunity of spending a productive period at the Newton Institute in Cambridge and suggesting to me the genus-$11$ problem and further applications to higher rank Brill-Noether theory.
Lazarsfeld-Mukai bundles {#background1}
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In this section we briefly recall the definition and the main properties of Lazarsfeld-Mukai bundles (LM bundles in the sequel) associated with complete, base point free linear series on curves lying on $K3$ surfaces. We refer to [@lazarsfeld], [@lazarsfeld1], [@pareschi] for the proofs. Let $S$ be a $K3$ surface and $C\subset S$ a smooth connected curve of genus $g$. Any base point free linear series $A\in W^r_d(C)\setminus W^{r+1}_d(C)$ can be considered as a globally generated sheaf on $S$; therefore, the evaluation map $\mathrm{ev}_{A,S}:H^0(C,A)\otimes{\mathcal O}_S\to A$ is surjective and one defines the bundle $F_{C,A}$ to be its kernel, i.e., $$\label{equation:prima}
0\to F_{C,A}\to H^0(C,A)\otimes{\mathcal O}_S\to A\to 0.$$ The LM bundle associated with the pair $(C,A)$ is, by definition, $E_{C,A}:=F_{C,A}^\vee$. By dualizing (\[equation:prima\]), one finds that $E_{C,A}$ sits in the following short exact sequence: $$\label{equation:seconda}
0\to H^0(C,A)^\vee\otimes{\mathcal O}_S\to E_{C,A}\to \omega_C\otimes A^\vee\to 0;$$ in particular, $E_{C,A}$ is equipped with a ($r+1$)-dimensional subspace of sections. The following proposition summarizes the most important properties of $E_{C,A}$:
\[prop:basic\] If $E_{C,A}$ is the LM bundle corresponding to a base point free linear series $A\in W^r_d(C)\setminus W^{r+1}_d(C)$, then:
- ${\mathrm{rk}}\, E_{C,A}=r+1$.
- $\det E_{C,A}=L$, where $C\in\vert L\vert$.
- $c_2( E_{C,A})=d$.
- The bundle $E_{C,A}$ is globally generated off the base locus of $\omega_C\otimes A^\vee$.
- $h^0(S, E_{C,A})=h^0(C,A)+h^0(C,\omega_C\otimes A^\vee)=r+1+g-d+r$,\
$h^1(S, E_{C,A})=h^2(S, E_{C,A})=0$.
- $\chi(S, E_{C,A}\otimes F_{C,A})=2(1-\rho(g,r,d))$.
In particular, if $\rho(g,r,d)<0$, the LM bundle $E_{C,A}$ is non-simple.
Being a LM bundle is an open condition. Indeed, a vector bundle $E$ of rank $r+1$ is a LM bundle whenever $h^1(S,E)=h^2(S,E)=0$ and there exists $\Lambda\in G(r+1,H^0(S,E))$ such that the degeneracy locus of the evaluation map $ev_\Lambda:\Lambda\otimes{\mathcal O}_S\to E$ is a smooth connected curve.
Analogously, given $(C,A)$ as above, one defines a rank-$r$ vector bundle $M_A$ on $C$ as the kernel of $\mathrm{ev}_{A,C}:H^0(C,A)\otimes{\mathcal O}_C\to A$. It turns out that $$H^0(C,M_A\otimes \omega_C\otimes A^\vee)=\ker\mu_{0,A}.$$ Similarly, by tensoring (\[equation:seconda\]) by $F_{C,A}$ and taking cohomology, one shows that $$H^0(S,E_{C,A}\otimes F_{C,A})\simeq H^0(C,F_{C,A}\otimes \omega_C\otimes A^\vee).$$ Moreover, there is the following short exact sequence: $$\label{nuvole}
0\to {\mathcal O}_C\to F_{C,A}\otimes \omega_C\otimes A^\vee\to M_A\otimes \omega_C\otimes A^\vee\to 0.$$ Having denoted by $\pi: {\mathcal W}^r_d(\vert L\vert)\to \vert L\vert_s$ the natural projection and by $\mu_{1,A,S}$ the composition of the Gaussian map $\mu_{1,A}:\ker\mu_{0,A}\to H^0(C,\omega_C^2)$ with the transpose of the Kodaira-Spencer map $\delta_{C,S}^\vee:H^0(C,\omega_C^2)\to (T_C\vert L\vert)^\vee=H^1(C,{\mathcal O}_C)$, one has that ${\mathrm{Im}}(d\pi_{(C,A)})\subset \mathrm{Ann}({\mathrm{Im}}(\mu_{1,A,S}))$. Sard’s Lemma applied to the projection $\pi$ implies that, if ${\mathcal W}\subset {\mathcal W}^r_d(\vert L\vert)$ is a dominating component and $C$ is general in its linear system, the sequence (\[nuvole\]) is exact on the global sections for any $(C,A)\in(\pi\vert_{\mathcal W})^{-1}(C)$ such that $A$ is base point free and $h^0(C,A)=r+1$; indeed, the coboundary map $H^0(C, M_A\otimes \omega_C\otimes A^\vee)\to H^1(C,\omega_C)$ coincides, up to a scalar factor, with $\mu_{1,A,S}$. As a consequence, the simplicity of $E_{C,A}$ is equivalent to the injectivity of $\mu_{0,A}$. In particular, if general points of ${\mathcal W}$ are complete, base point free linear series corresponding to simple LM bundles, the fiber $(\pi\vert_{\mathcal W})^{-1}(C)$ over a general $C\in\vert L\vert_s$ is reduced of the expected dimension. Standard Brill-Noether theory implies that no component of ${\mathcal W}^r_d(\vert L\vert)$ is entirely contained in ${\mathcal W}^{r+1}_d(\vert L\vert)$. Therefore, the variety $W^r_d(C)$ is reduced of the expected dimension for a general $C\in\vert L\vert_s$ if no dominating component ${\mathcal W}$ of ${\mathcal W}^r_d(\vert L\vert)$ is of one of the following types:
1. For $(C,A)\in{\mathcal W}$ general, $A$ is complete, base point free and $E_{C,A}$ is non-simple.
2. \[referendum\] For $(C,A)\in{\mathcal W}$ general, $A$ is not base point free and $\ker\mu_{0,A}\neq 0$.
In order to exclude (\[referendum\]), one can proceed by induction on $d$ because, if $B$ denotes the base locus of $A$ and $\ker\mu_{0,A}\neq 0$, then $\mu_{0,A(-B)}\neq 0$, too.
Mumford stability for sheaves on $K3$ surfaces {#background2}
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For later use, we recall some facts about coherent sheaves on smooth projective surfaces referring to [@lehn] and [@shatz] for most of the proofs. Let $S$ be a smooth, projective surface over $\mathbb{C}$ and $H$ an ample line bundle on it. Given a torsion free sheaf $E$ on $S$ of rank $r$, the $H$-slope of $E$ is defined as $$\mu_H(E)=\frac{c_1(E)\cdot c_1(H)}{r},$$ and $E$ is called $\mu_H$-semistable (resp. $\mu_H$-stable) in the sense of Mumford-Takemoto if for any subsheaf $0\neq F\subset E$ with ${\mathrm{rk}}\, F<{\mathrm{rk}}\, E$, one has $\mu_H(F)\leq \mu_H(E)$ (resp. $\mu_H(F)< \mu_H(E)$). The Harder-Narasimhan filtration of $E$ (HN filtration in the sequel) is the unique filtration $$0=E_0\subset E_1\subset\ldots\subset E_s=E,$$ such that $E^i:=E_i/E_{i-1}$ is a torsion free, $\mu_H$-semistable sheaf for $1\leq i\leq s$, and $\mu_H(E_{i+1}/E_i)<\mu_H(E_i/E_{i-1})$ for $1\leq i\leq s-1$. Such a filtration always exists. It can be easily checked that, if $E$ is a vector bundle, the sheaves $E_i$ are locally free; moreover, $$\mu_H(E_1)>\mu_H(E_2)>\ldots >\mu_H(E).$$ The sheaf $E_1$ is called the maximal destabilizing sheaf of $E$; the number $\mu_H(E_1)$ is the maximal slope of a proper subsheaf of $E$ and, among the subsheaves of $E$ of slope equal to $\mu_H(E_1)$, the sheaf $E_1$ has maximal rank. In particular, $E_1$ is $\mu_H$-semistable.
Now, we assume $E$ is $\mu_H$-semistable. A Jordan-Hölder filtration of $E$ (later on, JH filtration) is a filtration $$0=JH_0(E)\subset JH_1(E)\subset\ldots\subset JH_s(E)=E,$$ such that all the factors $\mathrm{gr}_i(E):=JH_i(E)/JH_{i-1}(E)$ are torsion free, $\mu_H$-stable sheaves of slope equal to $\mu_H(E)$. This implies that $\mu_H(JH_i(E))=\mu_H(E)$ for $1\leq i\leq s$. The Jordan-Hölder filtration always exists but is not uniquely determined, while the graded object $\mathrm{gr}(E):=\oplus_i \mathrm{gr}_i(E)$ is.
The following result concerns morphisms between $\mu_H$-semistable and $\mu_H$-stable sheaves on $S$ (cf. [@shatz], [@friedman]).
\[prop:morfismi\] Given two torsion free sheaves $E$ and $F$ on $S$, the following holds:
1. \[utile\] If $E$ and $F$ are $\mu_H$-semistable and $\mu_H(E)>\mu_H(F)$, then ${\mathrm{Hom}}(E,F)=0$.
2. If $E$ and $F$ are $\mu_H$-stable, $\mu_H(E)=\mu_H(F)$ and there exists $0\neq\varphi\in {\mathrm{Hom}}(E,F)$, then ${\mathrm{rk}}\, E={\mathrm{rk}}\, F$ and $\varphi$ is an isomorphism in codim $\leq 1$ (in particular it is injective).
In the case where $S$ is a $K3$ surface, by Serre duality $H^2(S,E)\simeq{\mathrm{Hom}}(E,{\mathcal O}_S)^\vee$; hence (\[utile\]) implies that, if $E$ is $\mu_H$-semistable and $\mu_H(E)>0$, then $h^2(S,E)=0$.
From now on, we assume $S$ to be a $K3$ surface. Throughout the paper we will often use the following fact:
\[lem:marghe\] Let $E,Q\in\mathrm{Coh}(S)$ be torsion free and ${\mathrm{rk}}\, E\geq 2$. If $E$ is globally generated off a finite number of points, $h^2(S,E)=0$ and there exists a surjective morphism $\varphi:E\to Q$, then $h^0(S,Q^{\vee\vee})\geq 2$.
Being a quotient of $E$ , the sheaf $Q$ is globally generated off a finite set. If ${\mathrm{rk}}\, Q\geq 2$, this trivially implies $h^0(S,Q^{\vee\vee})\geq h^0(S,Q)\geq 2$. On the other hand, if $Q$ has rank $1$, then $Q=N\otimes I$, where $N\in{\mathrm{Pic}}(S)$ and $I$ is the ideal sheaf associated with a $0$-dimensional subscheme of $S$. Since $N$ is a quotient of $E$ off a finite number of points, it has no fixed components, thus it is base point free (cf. [@donat]). The statement follows by remarking that $N=Q^{\vee\vee}$ cannot be trivial because $h^2(S,E)=0$.
Another useful result is the following one (cf. Lemma 3.1 in [@green]):
\[lem:evvai\] Let $E$ be a vector bundle of rank $r$ on $S$ which is globally generated off a finite number of points. If $h^2(S,E)=0$, then $h^0(S,\det E)\geq 2$.
Since the natural map $\wedge^rH^0(S,E)\otimes{\mathcal O}_S\to\wedge^rE=\det E$ is surjective off a finite number of points, the line bundle $\det E$ is base point free. Therefore, it is enough to show that $\det E$ is non-trivial. This follows by remarking that, given a general $V\in G(r,H^0(S,E))$, the natural map $ev_V:V\otimes {\mathcal O}_S\to E$ is injective but is not an isomorphism since $h^2(S,E)=0$. Therefore, $\det ev_V$ gives a section of $\det E$ vanishing on a non-zero effective divisor.
Last but not least, we recall some notation and results from [@tata]. The Mukai vector of a sheaf $E\in\mathrm{Coh}(S)$ is defined as: $$v(E):=\mathrm{ch}(E)(1+\omega)=\mathrm{rk}(E)+c_1(E)+(\chi(E)-\mathrm{rk}(E))\omega\in H^*(S,{\mathbb{Z}})=H^{2*}(S,\mathbb{Z}),$$ where $H^4(S,{\mathbb{Z}})$ is identified with ${\mathbb{Z}}$ by means of the fundamental cocycle $\omega$. The Mukai lattice is the pair $(H^*(S,{\mathbb{Z}}),\langle,\rangle)$, with $\langle,\rangle$ being the symmetric bilinear form on $H^*(S,{\mathbb{Z}})$ whose definition is the following: $$\langle v,w\rangle:=-\int_Sv^*\wedge w,$$ where, if $v=v^0+v^1+v^2$ with $v^i\in H^{2i}(S,{\mathbb{Z}})$, we set $v^*:=v^0-v^1+v^2$. Given $E,F\in\mathrm{Coh}(S)$, we define the Euler characteristic of the pair $(E,F)$ as $$\chi(E,F):=\sum_{i=0}^2(-1)^i \dim{\mathrm{Ext}}^i(E,F),$$ and it turns out that $\chi(E,F)=-\langle v(E),v(F)\rangle$.
Given a Mukai vector $v\in H^*(S,{\mathbb{Z}})$, let ${{\mathcal M}}(v)$ be the moduli stack of coherent sheaves on $S$ of Mukai vector $v$. If $H\in{\mathrm{Pic}}(S)$ is ample, we denote by ${{\mathcal M}}_H(v)^{\mu ss}$ (resp. ${{\mathcal M}}_H(v)^{\mu s}$) the moduli stack parametrizing isomorphism classes of $\mu_H$-semistable (resp. $\mu_H$-stable) sheaves on $S$ with Mukai vector $v$. Recall that any $\mu_H$-stable sheaf is simple and that any irreducible component of ${{\mathcal M}}_H(v)^{\mu s}$ has dimension equal to $\langle v,v\rangle +1$. Moreover, if $\gcd(v^0,v^1. H)=1$, then $\mu_H$-semistability and $\mu_H$-stability coincide.
Stability of Lazarsfeld-Mukai bundles of rank $2$ {#bistrot}
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Let $S$ be a smooth, projective $K3$ surface and consider a line bundle $L\in\mathrm{Ample}(S)$ such that a general curve $C\in\vert L\vert_s$ has genus $g$, Clifford dimension $1$ and maximal gonality $k=\left\lfloor \frac{g+3}{2}\right\rfloor$. In this section we prove that, if $C$ is general in its linear system and $\rho(g,1,d)>0$, the LM bundle associated with a general complete, base point free $g^1_d$ on $C$ is $\mu_L$-stable.
Fix a rank-$2$ LM bundle $E=E_{C,A}$ corresponding to a complete, base point free pencil $A\in W^1_d(C)$ with $C\in\vert L\vert_s$; Proposition \[prop:basic\] implies that $$v(E)=2+c_1(L)+(g-d+1)\omega.$$ We assume $E$ is not $\mu_L$-stable. In the case where $E$ is $\mu_L$-unstable (resp. properly $\mu_L$-semistable) we consider its HN filtration (resp. JH filtration) $0\subset M\subset E$, which gives a short exact sequence $$\label{domenica}
0\to M\to E\to N\otimes I_\xi\to 0,$$ where $M$ and $N$ are two line bundles such that $\mu_L(M)>\mu_L(E)=g-1>\mu_L(N)$ (resp. $\mu_L(M)=\mu_L(E)=\mu_L(N)$) and $I_\xi$ is the ideal sheaf of a $0$-dimensional subscheme $\xi\subset S$ of length $l=d-c_1(N)\cdot c_1(M)$. By Lemma \[lem:marghe\], we know that $h^0(S,N)\geq 2$. First of all, we prove the following:
\[lem:uova\] In the situation above, if general curves in $\vert L\vert_s$ have Clifford dimension $1$ and (constant) gonality $k$, one has $c_1(M)\cdot c_1(N)\geq k$.
We remark that $h^2(S,M)=0$ since $\mu_L(M)>0$. Therefore, if $$2>h^0(S,M)\geq\chi(M)=2+c_1(M)^2/2,$$ then $c_1(M)^2<0$ and the inequality $\mu_L(M)\geq g-1$ implies $c_1(M)\cdot c_1(N)\geq g+1\geq k$.\
From now on, we assume $h^0(S,M)\geq 2$. Since $\omega_C\otimes N^\vee\vert_C=M\otimes{\mathcal O}_C$, the line bundle $N\vert_C$ contributes to ${\mathrm{Cliff}}(C)$. The short exact sequence $$0\to M^\vee\to N\to N\otimes{\mathcal O}_C\to 0$$ gives $h^0(C,N\otimes{\mathcal O}_C)\geq h^0(S,N)$. It follows that $$\begin{aligned}
{\mathrm{Cliff}}(N\otimes{\mathcal O}_C)&=&c_1(N)\cdot (c_1(N)+c_1(M))-2h^0(C,N\otimes{\mathcal O}_C)+2\\
&\leq& c_1(N)^2+c_1(N)\cdot c_1(M)-2\chi(N)-2h^1(S,N)+2\\
&=&-2+c_1(N)\cdot c_1(M)-2h^1(S,N).\end{aligned}$$ Since ${\mathrm{Cliff}}(N\otimes{\mathcal O}_C)\geq k-2$, then $c_1(M)\cdot c_1(N)\geq k+2h^1(S,N)\geq k$.
Our goal is to count the number of moduli of $\mu_L$-unstable and properly $\mu_L$-semistable LM bundles of rank $2$.
Fix a nonnegative integer $l$ and a non-trivial, globally generated line bundle $N$ on $S$ such that, having defined $M:=L\otimes N^\vee$, either $\mu_L(M)=\mu_L(N)=g-1$ or $\mu_L(M)>g-1>\mu_L(N)$. We consider the moduli stack $\mathcal{E}_{N,l}$ parametrizing filtrations $0\subset M\subset E$ with $[M]\in{{\mathcal M}}(v(M))(\mathbb{C})$ and $[E/M]\in{{\mathcal M}}(v(N\otimes I_\xi))(\mathbb{C})$, where $l(\xi)=l$. Note that, since both $N$ and $M$ are line bundles, the stack ${{\mathcal M}}(v(M))$ has a unique $\mathbb{C}$-point endowed with an automorphism group of dimension $1$, while ${{\mathcal M}}(v(N\otimes I_\xi))$ is corepresented by the Hilbert scheme $S^{[l]}$ parametrizing $0$-dimensional subschemes of $S$ of length $l$. Two filtrations $0\subset M\subset E$ and $0\subset M'\subset E'$ are equivalent whenever there exists a commutative diagram $$\xymatrix{
M\ar[r]\ar[d]_{\varphi_1}&E\ar[d]^{\varphi_2}\\
M'\ar[r]&E',\\
}$$ where $\varphi_1$ and $\varphi_2$ are two isomorphisms (cf. [@bridgeland] for the proof that $\mathcal{E}_{N,l}$ is algebraic). The stack $\mathcal{E}_{N,l}$ can be alternatively described as the moduli stack of extensions of type (\[domenica\]). Let $p:{{\mathcal E}}_{N,l}\to {{\mathcal M}}(v(M))\times {{\mathcal M}}(v(N\otimes I_\xi))$ be the natural morphism of stacks mapping the short exact sequence (\[domenica\]) to $(M,N\otimes I_\xi)$. The fiber of $p$ over the $\mathbb{C}$-point $(M,N\otimes I_\xi)$ of ${{\mathcal M}}(v(M))\times {{\mathcal M}}(v(N\otimes I_\xi))$ is the quotient stack $$[{\mathrm{Ext}}^1(N\otimes I_\xi,M)/{\mathrm{Hom}}(N\otimes I_\xi,M)],$$ where the action of ${\mathrm{Hom}}(N\otimes I_\xi,M)$ over ${\mathrm{Ext}}^1(N\otimes I_\xi,M)$ is the trivial one (cf. [@bridgeland]); it follows that in general $p$ is not representable.
We define $\tilde{P}_{N,l}$ to be the closure of the image of ${{\mathcal E}}_{N,l}$ under the natural projection $q:{{\mathcal E}}_{N,L}\to{{\mathcal M}}(v(E))$, which maps the point of ${{\mathcal E}}_{N,L}$ given by (\[domenica\]) to $[E]$. The morphism $q$ is representable (cf. proof of Lemma (4.1) in [@bridgeland]) and the fiber of $q$ over a $\mathbb{C}$-point of $\tilde{P}_{N,l}$ corresponding to $E$ is the Quot-scheme $\mathrm{Quot}_S(E,P)$, where $\mathrm{P}$ is the Hilbert polynomial of $N\otimes I_\xi$. We denote by $P_{N,l}$ the open substack of $\tilde{P}_{N,l}$ whose $\mathbb{C}$-points correspond to vector bundles $E$ satisfying $h^1(S,E)=h^2(S,E)=0$.
Let ${{\mathcal G}}_{N,l}\to P_{N,l}$ be the Grassmann bundle with fiber over a point $[E]\in P_{N,l}(\mathbb{C})$ equal to $G(2,H^0(S,E))$. A $\mathbb{C}$-point of ${{\mathcal G}}_{N,l}$ is a pair $(E,\Lambda)$ and comes endowed with an automorphism group equal to $\mathrm{Aut}(E)$. We consider the rational map $$h_{N,l}:{{\mathcal G}}_{N,l}\dashrightarrow {\mathcal W}^1_d(\vert L\vert ),$$ mapping a general point $(E,\Lambda)\in{{\mathcal G}}_{N,l}(\mathbb{C})$ to the pair $(C_\Lambda,A_\Lambda)$, where $C_\Lambda$ is the degeneracy locus of the evaluation map $ev_\Lambda:\Lambda\otimes{\mathcal O}_S\to E$, which is injective for a general $\Lambda\in G(2,H^0(S,E))$, and $\omega_{C_\Lambda}\otimes A_\Lambda^\vee$ is the cokernel of $ev_\Lambda$. Notice that $d:=c_1(N)\cdot c_1(M)+l$. Since while mapping to ${\mathcal W}^1_d(\vert L\vert )$ we forget the automorphisms, the fiber of $h_{N,l}$ over $(C,A)$ is the quotient stack $$[\mathbb{P}({\mathrm{Hom}}(E_{C,A},\omega_C\otimes A^\vee)^\circ)/\mathrm{Aut}(E_{C,A})],$$ where ${\mathrm{Hom}}(E_{C,A},\omega_C\otimes A^\vee)^\circ$ denotes the open subgroup of ${\mathrm{Hom}}(E_{C,A},\omega_C\otimes A^\vee)$ consisting of those morphisms whose kernel is isomorphic to ${\mathcal O}_S^{\oplus 2}$, and $\mathrm{Aut}(E_{C,A})$ acts on $\mathbb{P}({\mathrm{Hom}}(E_{C,A},\omega_C\otimes A^\vee)^\circ)$ by composition. In particular, $h_{N,l}$ is not representable. As remarked in Section \[background1\], one has $${\mathrm{Hom}}(E_{C,A},\omega_C\otimes A^\vee)\simeq H^0(S,E_{C,A}\otimes E_{C,A}^\vee);$$ it is trivial to check that $${\mathrm{Hom}}(E_{C,A},\omega_C\otimes A^\vee)^\circ\simeq \mathrm{Aut}(E_{C,A}).$$ Therefore, the action of $\mathrm{Aut}(E_{C,A})$ on $\mathbb{P}({\mathrm{Hom}}(E_{C,A},\omega_C\otimes A^\vee)^\circ)$ is transitive and the stabilizer of any point is the subgroup generated by $\mathrm{Id}_{E_{C,A}}$; as a consequence, any fiber of $h_{N,l}$ has dimension $-1$ (cf. [@gomez] for the definition of the dimension of a locally Noetherian algebraic stack). We denote by ${\mathcal W}_{N,l}$ the closure of the image of $h_{N,l}$. The following holds:
\[prop:no\] Assume that $P_{N,l}$ be non-empty and let ${\mathcal W}$ be an irreducible component of ${\mathcal W}_{N,l}$. Then $$\dim{\mathcal W}\leq g+d-k,$$ where $k$ is the gonality of any curve in $\vert L\vert_s$.
Proposition \[prop:morfismi\], together with the fact that $h^0(S,I_\xi)= 0$ if $l>0$, implies that $$\dim{\mathrm{Hom}}(M,N\otimes I_\xi)=\left\{\begin{array}{ll}1&\textrm{if }M\simeq N,\,\xi=\emptyset\\0&\textrm{otherwise}\end{array}\right..$$ It follows that the dimension of the fibers of $p$ is constant and equals $-\chi(M,N\otimes I_\xi)$, unless $M\simeq N$ and $l=0$, in which case it is $-\chi(M,N\otimes I_\xi)+1$.
Regarding the fibers of $q$, it is well known (cf. [@lehn] Proposition 2.2.8) that, given $[\varphi:E\to N\otimes I_\xi]\in\mathrm{Quot}_S(E,P)$, the following holds: $$\label{quot}\begin{array}{lll}
\dim{\mathrm{Hom}}(K,N\otimes I_\xi)-\dim{\mathrm{Ext}}^1(K,N\otimes I_\xi)&\leq& \dim_{[\xi]}\mathrm{Quot}_S(E,P)\\&\leq&\dim{\mathrm{Hom}}(K,N\otimes I_\xi),
\end{array}$$ where $K=\ker\varphi$; moreover, if ${\mathrm{Ext}}^1(K,N\otimes I_\xi)=0$, then $\mathrm{Quot}_S(E,P)$ is smooth in $[\varphi]$ of dimension equal to $\dim{\mathrm{Hom}}(K,N\otimes I_\xi)$. Since $K\simeq M$, if $M\simeq N$ and $l=0$, the fibers of $q$ are smooth of dimension $1$; indeed, ${\mathrm{Ext}}^1(N,N)\simeq H^1(S,{\mathcal O}_S)=0$. Otherwise, the fibers of $q$ are $0$-dimensional. It follows that, if $P_{N,l}$ is non-empty, then: $$\begin{aligned}
\dim {{\mathcal G}}_{N,l}&=&\dim P_{N,l}+2(g-d+1)\\
&=&\dim{{\mathcal M}}(v(M))+\dim{{\mathcal M}}(v(N\otimes I_\xi))+\langle v(M),v(N\otimes I_\xi)\rangle+2(g-d+1)\\
&=&2l-2+c_1(M)\cdot c_1(N)-\frac{c_1(M)^2}{2}-\frac{c_1(N)^2}{2}-2+l+2(g-d+1)\\
&=&3l+2g-2d-2-(g-1)+2c_1(M)\cdot c_1(N)\\
&=&g+d-1-c_1(N)\cdot c_1(M)\\
&\leq&g+d-1-k,\end{aligned}$$ where we have used that $c_1(M)+c_1(N)=c_1(L)$ and $d=c_1(M)\cdot c_1(N)+l$, and the last inequality follows from Lemma \[lem:uova\]. The statement is a consequence of the fact that the fibers of $h_{N,l}$ are quotient stacks of dimension equal to $-1$.
We can finally prove the following result:
\[thm:stable\] Assume that general curves in $\vert L\vert_s$ have Clifford dimension $1$ and maximal gonality $k=\left\lfloor \frac{g+3}{2}\right\rfloor$.
- If $\rho(g,1,d)>0$, any dominating component of ${\mathcal W}^1_d(\vert L\vert)$ corresponds to $\mu_L$-stable LM bundles. In particular, if $C\in\vert L\vert_s$ is general, the variety $W^1_d(C)$ is reduced and has the expected dimension $\rho(g,1,d)$.
- If $\rho(g,1,k)=0$ and $C\in\vert L\vert_s$ is general, then $W^1_k(C)$ has dimension $0$.
When $\rho(g,1,d)>0$, we show that no component ${\mathcal W}$ of ${\mathcal W}^1_d(\vert L\vert_s)$ corresponding to either $\mu_L$-unstable or properly $\mu_L$-semistable LM bundles dominates $\vert L\vert$. Proposition \[prop:no\] gives: $$\dim{\mathcal W}\leq g+d-k\leq g+d-\frac{g+2}{2}.$$ Our claim follows by remarking that any dominating component of ${\mathcal W}^1_d(\vert L\vert)$ has dimension at least $g+\rho(g,1,d)$ and that $\rho(g,1,d)>d-\frac{g+2}{2}$ whenever $d>\frac{g+2}{2}$.
If $k=\frac{g+2}{2}$, that is, $\rho(g,1,k)=0$, our parameter count shows that any dominating component of ${\mathcal W}^1_k(\vert L\vert)$ has dimension $g$; hence, if $C\in\vert L\vert$ is general, $W^1_k(C)$ is $0$-dimensional, even though not necessarily reduced. By induction on $d$, one excludes the existence of components of ${\mathcal W}^1_d(\vert L\vert)$ whose general points correspond to linear series which are not base point free.
Lazarsfeld-Mukai bundles of rank $3$ which are not $\mu_L$-stable {#tempo}
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We fix a LM bundle $E=E_{C,A}$ associated with a complete, base point free $g^2_d$ on a smooth connected curve $C\in \vert L\vert_s$ with $L\in\mathrm{Ample}(S)$. By Proposition \[prop:basic\], we have $$v(E)=3+c_1(L)+(2+g-d)\omega,$$ where $g=g(C)$. We assume that $E$ is not $\mu_L$-stable and, in the case where it is $\mu_L$-unstable, we look at its HN filtration: $$0=E_0\subset E_1\subset\ldots\subset E_s=E.$$ On the other hand, if $E$ is properly $\mu_L$-semistable, we consider its JH filtration: $$0=JH_0(E)\subset JH_1(E)\subset\ldots\subset JH_s(E)=E.$$
We first consider the cases where either $E$ is properly $\mu_L$-semistable and $JH_1(E)$ has rank $2$, or $E$ is $\mu_L$-unstable, $\mathrm{rk}\,E_1=2$ and $E_1$ is $\mu_L$-stable. Under these hypotheses, $E$ sits in the following short exact sequence: $$\label{rango2}
0\to M\to E\to N\otimes I_\xi\to 0,$$ where $M=JH_1(E)$ (resp. $M=E_1$) is a $\mu_L$-stable vector bundle of rank $2$, $N$ is a line bundle and $I_\xi$ is the ideal sheaf of a $0$-dimensional subscheme $\xi\subset S$. Moreover, $$\label{bank}
\mu_L(M)\geq\mu_L(E)=\frac{2g-2}{3}\geq\mu_L(N\otimes I_\xi)=\mu_L(N),$$ with the former inequality being strict whenever the latter one is. We have that $c_1(L)=c_1(E)=c_1(M)+c_1(N)$ and $d=c_2(E)=c_1(N)\cdot c_1(M)+l(\xi)+c_2(M)$, where $l(\xi)$ denotes the length of $\xi$. We prove the following:
\[lem:det\] Assume a general curve $C\in\vert L\vert_s$ has Clifford dimension $1$ and gonality $k$. In the above situation, one has $c_1(N)\cdot c_1(M)\geq k$ and $$\label{nene}
d\geq \frac{3}{4}k+\frac{7}{6}+\frac{g}{3}.$$
As $E$ is globally generated off a finite number of points, $N$ is base point free and non-trivial, thus $h^0(S,N)\geq 2$ and $\mu_L(N)>0$. The inequality $\mu_L(M)>0$ implies that $h^2(S,M)=0$ and, since $\mu_L(\det M)=2\mu_L(M)$, we have that $h^2(S,\det M)=0$, too. Therefore, $h^0(S,\det M)\geq\chi(\det M)=2+c_1(M)^2/2$ and, if $h^0(S,\det M)< 2$, then $c_1(M)^2\leq-2$ and $c_1(N)\cdot c_1(M)\geq(4g+2)/3>k$ by the first inequality in (\[bank\]), which gives $$c_1(M)^2+c_1(N)\cdot c_1(M)\geq\frac{4g-4}{3}.$$ On the other hand, if $h^0(S,\det M)\geq 2$, then $N\vert_C$ contributes to ${\mathrm{Cliff}}(C)$ and one shows, as in the proof of Lemma \[lem:uova\], that $c_1(N)\cdot c_1(M)\geq k+2h^1(S,N)\geq k$.
The $\mu_L$-stability of $M$ implies that $$-2\leq\langle v(M),v(M)\rangle=c_1(M)^2-4\chi(M)+8=4c_2(M)-c_1(M)^2-8.$$ Therefore, we have $$d= c_1(N)\cdot c_1(M)+c_2(M)+l(\xi)\geq c_1(N)\cdot c_1(M)+\frac{c_1(M)^2}{4}+\frac{6}{4}\geq \frac{3}{4}k+\frac{7}{6}+\frac{g}{3};$$ this concludes the proof.
Now, we assume that either $E$ is $\mu_L$-unstable, $\mathrm{rk}\,E_1=1$ and $E/E_1$ is $\mu_L$-stable, or $E$ is properly $\mu_L$-semistable and its JH filtration is of type $0\subset JH_1(E)\subset E$ with $\mathrm{rk}\,JH_1(E)=1$. Denoting by $N$ the line bundle $E_1$ (resp. $JH_1(E)$), one has a short exact sequence: $$\label{rango1}
0\to N\to E\to E/N\to 0,$$ where $E/N$ is a rank-$2$, $\mu_L$-stable, torsion free sheaf on $S$ such that $$\mu_L(N)\geq\mu_L(E)\geq\mu_L(E/N),$$ and either both inequalities are strict, or none is. We prove the following:
\[lem:endo1\] In the above situation, if a general curve $C\in\vert L\vert_s$ has Clifford dimension $1$ and gonality $k$, then $c_1(N)\cdot c_1(E/N)\geq k$.
As in the proof of Lemma \[lem:marghe\] one shows that $h^0(S,E/N)\geq 2$. Since $E/N$ is stable, then $\mu_L(E/N)>0$ and $h^2(S,E/N)=0$. Moreover, the vector bundle $(E/N)^{\vee\vee}$ is globally generated off a finite number of points and $h^0(S,\det (E/N))\geq 2$ by Lemma \[lem:evvai\] because $\det (E/N):=\det (E/N)^{\vee\vee}$.
Since $\mu_L(N)=c_1(N)\cdot (c_1(N)+c_1(E/N))\geq (2g-2)/3>0$, we have $h^2(S,N)=0$.\
Hence, if $h^0(S,N)<2$, then $c_1(N)^2<0$ and $c_1(N)\cdot c_1(E/N)\geq (2g+4)/3> k$. Otherwise, $N\otimes{\mathcal O}_C$ contributes to the Clifford index and this implies $c_1(N)\cdot c_1(E/N)\geq k$, too.
The cases still to be considered are the following ones:
1. \[one\] $E$ is $\mu_L$-unstable with HN filtration $0\subset E_1\subset E_2\subset E$.
2. \[two\] $E$ is properly $\mu_L$-semistable with JH filtration $0\subset JH_1(E)\subset JH_2(E)\subset E$.
3. \[three\] $E$ is $\mu_L$-unstable with HN filtration $0\subset E_1\subset E$ and $E_1$ is a properly $\mu_L$- semistable vector bundle of rank $2$.
4. \[four\] $E$ is $\mu_L$-unstable with HN filtration $0\subset E_1\subset E$ and $E_1$ is a line bundle such that $E/E_1$ is a properly $\mu_L$- semistable torsion free sheaf of rank $2$.
In all these cases one has four short exact sequences:
$$\label{pizza}0\to N\to E\to E/N\to 0$$
$$\label{arr}0\to M\to E\to N_1\otimes I_{\xi_1}\to 0,$$
$$\label{arr1}0\to N\to M\to N_2\otimes I_{\xi_2}\to 0,$$
$$\label{rango1b}0\to N_2\otimes I_{\xi_2}\to E/N\to N_1\otimes I_{\xi_1}\to 0,$$
where $N$, $N_1$, $N_2$ are line bundles, $I_{\xi_1}$ and $I_{\xi_2}$ denote the ideal sheaves of two $0$-dimensional subschemes $\xi_1,\xi_2\subset S$, the sheaf $E/N$ has rank-$2$ and no torsion, while $M$ is a vector bundle of rank $2$. Moreover, the following inequalities hold: $$\label{lun}
\mu_L(N)\geq\mu_L(N_2)\geq\mu_L(N_1),$$ $$\label{mar}
\mu_L(N)\geq\frac{2g-2}{3}\geq\mu_L(N_1);$$ in particular, $\mu_L(N)=\mu_L(N_2)$ (resp. $\mu_L(N_1)=\mu_L(N_2)$) whenever $M$ (resp. $E/N$) is properly $\mu_L$-semistable, that is, in cases (\[two\]) and (\[three\]) (resp. in cases (\[two\]) and (\[four\])). Analogously, equalities in (\[mar\]) force $E$ to be properly $\mu_L$-semistable with JH-filtration $0\subset N\subset M\subset E$, that is, one is in case (\[two\]).
\[lem:caso3\] In the above situation, $N_1\otimes{\mathcal O}_C$ always contributes to the Clifford index of $C\in\vert L\vert_s$. Moreover, one of the following occurs:
1. \[uno\] Both $N\otimes{\mathcal O}_C$ and $N_2\otimes{\mathcal O}_C$ contribute to the Clifford index of $C\in\vert L\vert_s$.
2. \[due\] The inequality $c_1(N)\cdot (c_1(N_1)+c_1(N_2))\geq \frac{2g+4}{3}$ holds and either $N_2\otimes{\mathcal O}_C$ contributes to the Clifford index of $C$ or $c_1(N_2)\cdot (c_1(N)+c_1(N_1))\geq g$.
3. \[tre\] The linear series $N\otimes{\mathcal O}_C$ contributes to the Clifford index of $C\in\vert L\vert_s$ and one has the inequality $c_1(N_2)\cdot c_1(N)>\frac{1}{2}c_1(N)\cdot (c_1(N_1)+c_1(N_2))$.
4. \[quattro\] The inequality $c_1(N)\cdot c_1(N_2)\geq \frac{g+5}{3}$ holds.
In particular, if a general $C\in\vert L\vert_s$ has Clifford dimension $1$ and gonality $k$, then $$\label{sese}
d\geq c_1(N)\cdot c_1(N_1)+c_1(N)\cdot c_1(N_2)+c_1(N_1)\cdot c_1(N_2)\geq\frac{3}{2}k.$$
Being a quotient of $E$ off a finite set, $N_1$ is base point free and non-trivial, thus $h^0(S,N_1)\geq 2$ and $\mu_L(N_1)>0$. By the “Strong Bertini’ s Theorem” (cf. [@donat]), $N_1$ is nef. Proposition \[prop:morfismi\] implies $h^2(S,N)=h^2(S,N_2)=0$ because of (\[lun\]). Analogously, $\mu_L(N_2\otimes N)=\mu_L(N_2)+\mu_L(N)>0$ and $h^2(S,N_2\otimes N)=0$. Moreover, the following holds: $$\begin{aligned}
c_1(N_2\otimes N)^2&=&c_1(N_2)^2+c_1(N)^2+2c_1(N_2)\cdot c_1(N)\\
&\geq&c_1(N)^2+c_1(N_2)\cdot c_1(N)+c_1(N_1)\cdot c_1(N)+c_1(N_1)^2\\
&=&\mu_L(N)+c_1(N_1)^2>0,\end{aligned}$$ where we have used that, since $\mu_L(N_2)\geq\mu_L(N_1)$, then $$\label{banana}
c_1(N_2)^2+c_1(N_2)\cdot c_1(N)\geq c_1(N_1)^2+c_1(N_1)\cdot c_1(N),$$ and that $c_1(N_1)^2\geq 0$ because $N_1$ is nef. We obtain that $$h^0(S, N_2\otimes N)\geq \chi(N_2\otimes N)=2+\frac{1}{2}c_1(N_2\otimes N)^2> 2,$$ thus $N_1\otimes{\mathcal O}_C$ always contributes to the Clifford index of $C\in\vert L\vert_s$.
If both $h^0(S,N_2)\geq 2$ and $h^0(S,N)\geq 2$, we are in case (\[uno\]).
If $h^0(S,N_2)\geq 2$ and $h^0(S,N)<2$, we show that (\[due\]) occurs. Since $\chi(N)<2$, one has $c_1(N)^2<0$ and $c_1(N)\cdot (c_1(N_1)+c_1(N_2))\geq\mu_L(E)+2=(2g+4)/3$ by the first inequality in (\[mar\]). Since $\mu_L(N\otimes N_1)>0$, then $h^2(S,N\otimes N_1)=0$. Moreover, one can show that $$c_1(N\otimes N_1)^2\geq \mu_L(N_1)+c_1(N_2)^2>c_1(N_2)^2.$$ It follows that, if $c_1(N\otimes N_1)^2<0$, then $c_1(N_2)^2<0$ and $$2g-2<2c_1(N)\cdot c_1(N_2)+2c_1(N_1)\cdot c_1(N_2),$$ that is, $c_1(N_2)\cdot (c_1(N)+c_1(N_1))\geq g$. On the other hand, if $c_1(N\otimes N_1)^2\geq0$, then $h^0(S,N\otimes N_1)\geq 2$ and $N_2\otimes{\mathcal O}_C$ contributes to the Clifford index.
From now on, assume $h^0(S,N_2)<2$, hence $c_1(N_2)^2<0$. Since $\det E/N\simeq N_1\otimes N_2$, Lemma \[lem:evvai\] implies $h^0(S,N_1\otimes N_2)\geq 2$. Thus, if $h^0(S,N)\geq 2$, the linear series $N\otimes{\mathcal O}_C$ contributes to the Clifford index of $C\in \vert L\vert_s$. Furthermore, inequality (\[banana\]), together with the fact that $c_1(N_2)^2<0\leq c_1(N_1)^2$, implies that $c_1(N_2)\cdot c_1(N)>c_1(N_1)\cdot c_1(N)$. We obtain $$c_1(N_2)\cdot c_1(N)>\frac{1}{2}c_1(N)\cdot (c_1(N_1)+c_1(N_2)),$$ and we are in case (\[tre\])
It remains to treat the case where both $h^0(S,N_2)<2$ and $h^0(S,N)<2$. Under these hypotheses, $c_1(N_2)^2<0$ and $c_1(N)^2<0$ and we obtain $$\begin{aligned}
2g-2&\leq&c_1(N_1)^2+2c_1(N_1)\cdot c_1(N)+2c_1(N_1)\cdot c_1(N_2)+2c_1(N)\cdot c_1(N_2)-4\\
&=&2c_1(N)\cdot c_1(N_2)+2\mu_L(N_1)-c_1(N_1)^2-4\\
&\leq& 2c_1(N)\cdot c_1(N_2)+\frac{4g-4}{3}-4.\end{aligned}$$ As a consequence, $c_1(N)\cdot c_1(N_2)\geq\frac{g+5}{3}$ and we are in case (\[quattro\]).
Now, we assume that $C$ has Clifford dimension $1$ and gonality $k$ and prove inequality (\[sese\]). One shows, as in Lemma \[lem:uova\], that $$\label{pilvia}
c_1(N_1)\cdot (c_1(N)+c_1(N_2))\geq k,$$ because $N_1\otimes {\mathcal O}_C$ always contributes to the Clifford index of $C\in\vert L\vert_s$. Analogously, if $N\otimes {\mathcal O}_C$ (resp. $N_2\otimes {\mathcal O}_C$) contributes to ${\mathrm{Cliff}}(C)$, then $c_1(N)\cdot (c_1(N_1)+c_1(N_2))\geq k$ (resp. $c_1(N_2)\cdot (c_1(N)+c_1(N_1))\geq k$); therefore, the last part of the statement is proved if either (\[uno\]) or (\[due\]) occurs (use that $(2g+4)/3\geq k$).
In case (\[tre\]), one arrives at the same conclusion by adding inequality (\[pilvia\]) and $$\label{fra}c_1(N)\cdot c_1(N_2)>\frac{1}{2}c_1(N)\cdot (c_1(N_1)+c_1(N_2))\geq \frac{k}{2}.$$ Similarly, in case (\[quattro\]), one uses that $c_1(N)\cdot c_1(N_2)\geq(g+5)/3\geq k/2$.
\[cor:lasagna\] Assume $C\in\vert L\vert_s$ has Clifford dimension $1$ and maximal gonality $k=\left\lfloor \frac{g+3}{2}\right\rfloor$ and let $E$ be the Lazarsfeld-Mukai bundle associated with a complete, base point free net $A\in W^2_d(C)$. If $E$ is not $\mu_L$-stable, $d<\frac{3}{4}k+\frac{7}{6}+\frac{g}{3}$ and $(g,d)\neq(6,6)$, then $E$ is given by an extension of type (\[rango1\]), with $N\in{\mathrm{Pic}}(S)$ and $E/N$ a $\mu_L$-stable, torsion free sheaf of rank $2$ such that $\mu_L(N)\geq (2g-2)/3\geq\mu_L(E/N)$.
Apply Lemma \[lem:det\] and Lemma \[lem:caso3\] and remark that $\left\lceil\frac{3}{4}k+\frac{7}{6}+\frac{g}{3}\right\rceil\leq\left\lceil\frac{3}{2}k\right\rceil$ unless $g=6$.
Cases with a $\mu_L$-stable subbundle of rank $2$ and $L$-slope $\geq\mu_L(E)$ {#mare}
===============================================================================
We assume that a general curve in $\vert L\vert$ has Clifford dimension $1$ and maximal gonality. In this section we show that, if $C\in\vert L\vert_s$ is general, the LM bundle $E$ corresponding to a general, complete, base point free $g^2_d$ on $C$ is neither properly $\mu_L$-semistable with JH filtration $0\subset JH_1(E)\subset E$ and $\mathrm{rk}\,JH_1(E)=2$, nor $\mu_L$-unstable with a $\mu_L$-stable, rank-$2$ vector bundle $E_1$ as maximal destabilizing sheaf .
Fix a positive integer $d$. Choose $l\in\mathbb{N}$ and a non-trivial, globally generated line bundle $N$ such that $$\label{ale}
\mu_L(N)\leq\frac{2g-2}{3}\leq \frac{(c_1(L)-c_1(N))\cdot c_1(L)}{2},$$ and impose that these are either two equalities or two strict inequalities. Set $$\begin{aligned}
c_1&:=&c_1(L)-c_1(N),\\
c_2&:=&d-c_1.c_1(N)-l,\\
\chi&:=&g-d+5-\chi(N)+l,\end{aligned}$$ and define the vector $v:=2+c_1+(\chi-2)\omega\in H^*(S,{\mathbb{Z}})$. The following construction is analogous to that of Section \[bistrot\].
Let $\mathcal{E}_{N,l}$ be the moduli stack of filtrations $0\subset M\subset E$, where $[M]\in {{\mathcal M}}_L(v)^{\mu s}(\mathbb{C})$ and $[E/M]\in {{\mathcal M}}(v(N\otimes I_\xi))(\mathbb{C})$ with $l(\xi)=l$. This is alternatively described as the moduli stack of extensions $$\label{imp}
0\to M\to E\to N\otimes I_\xi\to 0,$$ with $M$ and $\xi$ as above.
If $p:{{\mathcal E}}_{N,l}\to {{\mathcal M}}_L(v)^{\mu s}\times {{\mathcal M}}(v(N\otimes I_\xi))$ denotes the morphism of Artin stacks mapping the short exact sequence (\[imp\]) to $(M,N\otimes I_\xi)$, the fiber of $p$ over the point of $M_L(v)^{\mu s}\times {{\mathcal M}}(v(N\otimes I_\xi))$ corresponding to the pair $(M,N\otimes I_\xi)$ is the quotient stack $$[{\mathrm{Ext}}^1(N\otimes I_\xi,M)/{\mathrm{Hom}}(N\otimes I_\xi,M)].$$ Define $\tilde{P}_{N,l}$ to be the closure of the image of ${{\mathcal E}}_{N,L}$ under the natural projection $$q:{{\mathcal E}}_{N,L}\to{{\mathcal M}}(v(E)),$$ which sends the isomorphism class of extension (\[imp\]) to $[E]\in{{\mathcal M}}(v(E))(\mathbb{C})$. The morphism $q$ is representable and the fiber of $q$ over the point of $\tilde{P}_{N,l}$ corresponding to $[E]$ is the Quot-scheme $\mathrm{Quot}_S(E,P)$, where by $P$ we denote the Hilbert polynomial of $N\otimes I_\xi$. We consider the open substack $P_{N,l}\subset\tilde{P}_{N,l}$, whose $\mathbb{C}$-points are isomorphism classes of vector bundles $E$ such that $h^1(S,E)=h^2(S,E)=0$.
\[lem:dim\] The stack $P_{N,l}$, if nonempty, has dimension $$\dim P_{N,l}=2l+\langle v,v\rangle+\langle v(N\otimes I_\xi),v\rangle.$$
We claim that the dimension of the fibers of $p$ is constant. Indeed, Serre duality and Proposition \[prop:morfismi\] imply that $\dim {\mathrm{Ext}}^2(N\otimes I_\xi,M)=\dim{\mathrm{Hom}}(M,N\otimes I_{\xi})=0$ for any $[M]\in {{\mathcal M}}_L(v)^{\mu s}(\mathbb{C})$ and $\xi\in S^{[l]}$. This shows that ${{\mathcal E}}_{N,l}$, if nonempty, has dimension equal to $$\dim (M_L(v)^{\mu s}\times {{\mathcal M}}(v(N\otimes I_\xi)))-\chi(N\otimes I_\xi,M)=2l-1+1+\langle v,v\rangle+\langle v(N\otimes I_\xi),v\rangle;$$ note that this coincides with the dimension computed by Yoshioka (cf. Lemma 5.2 in [@yoshioka]). The statement follows by remarking that, if $P_{N,l}$ is nonempty, then $\dim P_{N,l}=\dim\tilde{P}_{N,l}=\dim{{\mathcal E}}_{N,l}$ because the Quot-schemes corresponding to the fibers of $q$ are $0$-dimensional (use inequalities analogous to (\[quot\])).
We consider the Grassmann bundle ${{\mathcal G}}_{N,l}\to P_{N,l}$, whose fiber over $[E]\in P_{N,l}(\mathbb{C})$ is $G(3,H^0(S,E))$, and the rational map $h_{N,l}:{{\mathcal G}}_{N,l}\dashrightarrow {\mathcal W}^2_d(\vert L\vert )$. The fiber of $h_{N,l}$ over a pair $(C,A)$ is the quotient stack $$[\mathbb{P}({\mathrm{Hom}}(E_{C,A},\omega_C\otimes A^\vee)^\circ)/\mathrm{Aut}(E_{C,A})],$$ where ${\mathrm{Hom}}(E_{C,A},\omega_C\otimes A^\vee)^\circ\subset {\mathrm{Hom}}(E_{C,A},\omega_C\otimes A^\vee)$ consists, by definition, of morphisms with kernel isomorphic to ${\mathcal O}_S^{\oplus 3}$. This quotient stack has dimension equal to $-1$, as in Section \[bistrot\]. Our goal is to estimate the dimension of the closure of the image of $h_{N,l}$, which is denoted by ${\mathcal W}_{N,l}$. We first prove the following:
\[lem:occhio\] If ${{\mathcal G}}_{N,l}$ is nonempty, then $$\dim{{\mathcal G}}_{N,l}= g+\rho(g,2,d)+\chi(M,N\otimes I_\xi).$$ Moreover, $\chi(M,N\otimes I_\xi)\leq\frac{4}{3}g+\frac{8}{3}-d-\frac{3}{2}c_1(N)\cdot c_1$.
We use that $$\begin{aligned}
2(\rho(g,2,d)-1)=\langle v(E),v(E)\rangle&=&\langle v(N\otimes I_\xi),v(N\otimes I_\xi)\rangle+\langle v,v\rangle+2\langle v(N\otimes I_\xi),v\rangle\\
&=&2l-2+\langle v,v\rangle+2\langle v(N\otimes I_\xi),v\rangle;\end{aligned}$$ this implies that $$\begin{aligned}
\dim {{\mathcal G}}_{N,l}=\dim P_{N,l} +3(h^0(S,E)-3)&=&2\rho(g,2,d)-\langle v(N\otimes I_\xi),v\rangle+3(g-d+2)\\&=&g+\rho(g,2,d)+\chi(M,N\otimes I_\xi),\end{aligned}$$ as soon as ${{\mathcal G}}_{N,l}$ is nonempty.
Since $\chi(M,N\otimes I_\xi)=-\langle v(N\otimes I_\xi),v\rangle=2\chi(N\otimes I_\xi)+\chi-4-c_1(N)\cdot c_1$, the last part of the statement follows by remembering that $\chi(E)=\chi+\chi(N\otimes I_\xi)=g-d+5$ and that $$\frac{c_1(N)^2}{2}\leq\frac{g-1}{3}-\frac{c_1(N)\cdot c_1}{2}$$ because $\mu_L(E)\geq\mu_L(N\otimes I_\xi)$.
In conclusion, we prove the following:
\[prop:cambridge\] Assume that a general curve in $\vert L\vert_s$ has Clifford dimension $1$ and maximal gonality $k=\left\lfloor \frac{g+3}{2}\right\rfloor$. Let ${\mathcal W}\subset {\mathcal W}_{N,l}$ be an irreducible component of ${\mathcal W}^2_d(\vert L\vert)$; then, $\rho(g,2,d)>0$ and ${\mathcal W}$ does not dominate the linear system $\vert L\vert$.
Lemma \[lem:det\] gives $c_1(N)\cdot c_1\geq k\geq (g+2)/2$ and $d\geq \frac{3}{4}k+\frac{7}{6}+\frac{g}{3}\geq\frac{17}{24}g+\frac{23}{12}$; in particular, $\rho(g,2,d)> 0$. By Lemma \[lem:occhio\], we have $$\begin{aligned}
\dim{{\mathcal G}}_{N,l}&\leq& g+\rho(g,2,d)+\frac{4}{3}g+\frac{8}{3}-d-\frac{3}{2}k\\
&\leq& g+\rho(g,2,d)+\frac{4}{3}g+\frac{8}{3}-d-\frac{3}{4}g-\frac{3}{2}\\&=&g+\rho(g,2,d)+\frac{7}{12}g+\frac{7}{6}-d.\end{aligned}$$ Since any fiber of $h_{N,l}$ is an algebraic stack of dimension $-1$, then $$\dim{\mathcal W}\leq g+\rho(g,2,d)+\frac{7}{12}g+\frac{13}{6}-d.$$ The right hand side is strictly smaller than $g+\rho(g,2,d)$ because $d> \frac{7g+26}{12}$. It follows that ${\mathcal W}$ cannot dominate $\vert L\vert$.
Cases with a $\mu_L$-stable quotient sheaf of rank $2$ and $L$-slope $\leq\mu_L(E)$ {#section:cola}
===================================================================================
In this section we count the number of moduli of rank-$3$ LM bundles $E$, which are either properly $\mu_L$-semistable with JH filtration $0\subset JH_1(E)\subset E$ where $JH_1(E)$ is a line bundle, or $\mu_L$-unstable with maximal destabilizing sheaf $E_1$ such that $E/E_1$ is a $\mu_L$-stable, torsion free sheaf of rank $2$.
Fix an integer $d\geq 4$. Choose $N\in{\mathrm{Pic}}(S)$ such that $$\label{zia}
\mu_L(N)\geq\frac{2g-2}{3}\geq\frac{(c_1(L)-c_1(N))\cdot c_1(L)}{2},$$ with equality holding either everywhere or nowhere.
As before, we set $c_1':=c_1(L)-c_1(N)$, $c_2':=d-c_1'\cdot c_1(N)$, $\chi':=g-d+5-\chi(N)$, $v':=2+c_1'+(\chi'-2)\omega\in H^*(S,{\mathbb{Z}})$.
We denote by ${{\mathcal F}}_{N}$ the algebraic stack of extensions $$\label{sole}
0\to N\to E\to E/N\to 0,$$ where $E/N$ defines a point of ${{\mathcal M}}_L^{\mu s}(v')$. Equivalently, ${{\mathcal F}}_{N}$ is the moduli stack of filtrations $0\subset N\subset E$ such that $[E/N]\in{{\mathcal M}}_L^{\mu s}(v')(\mathbb{C})$. Consider the two projections $p:{{\mathcal F}}_N\to {{\mathcal M}}_L^{\mu s}(v')\times {{\mathcal M}}(v(N))$ and $q:{{\mathcal F}}_N\to {{\mathcal M}}(v(E))$ and define $\tilde{R}_{N}$ to be the closure of the image of $q$. The open substack $R_{N}\subset\tilde{R}_{N}$ consists, by definition, of points corresponding to bundles $E$ such that $h^1(S,E)=h^2(S,E)=0$. We look at the Grassmann bundle ${{\mathcal G}}_N\to R_N$ with fiber over $[E]\in R_N(\mathbb{C})$ equal to $G(3,H^0(S,E))$. The closure of the image of ${{\mathcal G}}_N$ under the rational map $h_N:{{\mathcal G}}_N\dashrightarrow {\mathcal W}^2_d(\vert L\vert)$ is denoted by ${\mathcal W}_N$. As before, the fibers of $h_N$ are quotient stacks of dimension $-1$.
\[lem:fontana\] The stack ${{\mathcal G}}_N$, if nonempty, has dimension $$\dim{{\mathcal G}}_N= g+\rho(g,2,d)+\chi(E/N,N).$$
The fiber of $p$ over a point of ${{\mathcal M}}_L^{\mu s}(v')\times {{\mathcal M}}(v(N))$ corresponding to $(E/N,N)$ is the quotient stack $[{\mathrm{Ext}}^1(E/N,N)/{\mathrm{Hom}}(E/N,N)]$. Since $\mu_L(N)\geq\mu_L(E/N)$ and $E/N$ is $\mu_L$-stable, Serre duality and Proposition \[prop:morfismi\] imply that ${\mathrm{Ext}}^2(E/N,N)=0$; hence, the dimension of the fibers of $p$ is constantly equal to $-\chi(E/N,N)=\langle v(N),v'\rangle$. The morphism $q$ is representable and, as in the previous sections, one shows that its fibers are Quot-schemes of dimension $0$. Therefore, if $R_N$ is nonempty, one has: $$\dim R_N=\dim\tilde{R}_N=\dim{{\mathcal F}}_N=\langle v',v'\rangle+\langle v(N),v'\rangle.$$ The statement follows by proceeding as in the proof of Lemma \[lem:occhio\].
The next Lemma gives an upper bound for $\chi(E/N,N)$.
\[lem:burraco\] Assume that a general curve $C\in\vert L\vert_s$ has Clifford dimension $1$ and maximal gonality $k=\left\lfloor \frac{g+3}{2}\right\rfloor$. If $R_N$ is nonempty, then $\chi(E/N,N)\leq \frac{3}{2}g-2d+3$ for any $E/N$ corresponding to a point of ${{\mathcal M}}_L^{\mu s}(v')$.
Consider the extension (\[sole\]), where $[E]\in R_N(\mathbb{C})$. Since $\mu_L(N)>0$, one has $h^1(S,E/N)=h^2(S,N)=0$. As in Lemma \[lem:marghe\] one obtains $\chi(E/N)=h^0(S,E/N)\geq 2$, hence $\chi(N)=\chi(E)-\chi(E/N)\leq g-d+3$. As a consequence: $$\begin{aligned}
\chi(E/N,N)&=&2\chi(N)+\chi'-4-c_1(N)\cdot c_1'\\
&=&g-d+1+\chi(N)-c_1(N)\cdot c_1'\\
&\leq &2g-2d+4-c_1(N)\cdot c_1'\\
&\leq&\frac{3}{2}g-2d+3,\end{aligned}$$ where the last inequality follows from Lemma \[lem:endo1\].
Finally, we prove the following:
\[prop:fiducia\] We assume that a general curve in $\vert L\vert$ has Clifford dimension $1$ and maximal gonality $k=\left\lfloor \frac{g+3}{2}\right\rfloor$. If $d>\frac{3}{4}g+2$, no irreducible component ${\mathcal W}$ of ${\mathcal W}^2_d(\vert L\vert)$ which is contained in ${\mathcal W}_N$ dominates the linear system $\vert L\vert$.
Let ${\mathcal W}\subset{\mathcal W}_N$ be an irreducible component of ${\mathcal W}^2_d(\vert L\vert)$. Since any fiber of $h_N$ is an Artin stack of dimension equal to $-1$, Lemma \[lem:fontana\] and Lemma \[lem:burraco\] imply that $$\dim{\mathcal W}\leq g+\rho(g,2,d)+\frac{3}{2}g-2d+4.$$
If $\rho(g,2,d)\geq 0$, the condition $d>\frac{3}{4}g+2$ prevents the map ${\mathcal W}\to \vert L\vert$ from being dominant.
Now we show that, if $d$ is small enough and $C\in\vert L\vert_s$, any complete base point free $g^2_d$ on $C$, whose LM bundle is given by an extension of type (\[sole\]), is contained in a linear series which is induced from a line bundle on $S$.
\[prop:vai\] Let $S$ and $L$ be as in the hypotheses of Proposition \[prop:fiducia\] and $A$ be a complete, base point free $g^2_d$ on a curve $C\in\vert L\vert_s$. If $d<(5g+13)/6$ and the LM bundle $[E_{C,A}]\in R_N(\mathbb{C})$ for some $N\in{\mathrm{Pic}}(S)$, the linear system $\vert A\vert$ is contained in the restriction to $C$ of the linear system $\vert L\otimes N^\vee\vert$ on $S$. Moreover, $L\otimes N^\vee$ is adapted to $\vert L\vert$ and ${\mathrm{Cliff}}( L\otimes N^\vee\otimes{\mathcal O}_C)\leq {\mathrm{Cliff}}(A)=d-4$.
By hypothesis, $E=E_{C,A}$ sits in a short exact sequence like (\[sole\]), where $E/N$ is $\mu_L$-stable and $\mu_L(N)\geq (2g-2)/3\geq \mu_L(E/N)$. Since $\mu_L(N)>0$, then $h^2(S,N)=0$.
The $\mu_L$-stability of $E/N$ implies $$-2\leq\langle v',v'\rangle=4c_2'-(c_1')^2-8,$$ thus $c_2'\geq 3/2+(c_1')^2/4$.
If $h^0(S,N)<2$, then $c_1(N)^2\leq -2$, which implies $(c_1')^2+2c_1(N)\cdot c_1'\geq 2g$ and $c_1'\cdot c_1(N)\geq (2g+4)/3$. In particular, $$d=c_1'\cdot c_1(N)+c_2'\geq c_1'\cdot c_1(N)+\frac{3}{2}+\frac{(c_1')^2}{4}\geq \frac{g}{2}+\frac{3}{2}+\frac{g+2}{3}=\frac{5g+13}{6},$$ thus a contradiction. Therefore, one has both $h^0(S,N)\geq 2$ and $h^0(S,\det E/N)\geq 2$.
Remark that $(E/N)^{\vee\vee}$ is globally generated off a finite set and $$h^i(S,(E/N)^{\vee\vee})=h^i(S,E/N)=0\textrm{ for }i=1,2.$$ Since $\det E/N=\det (E/N)^{\vee\vee}$ is base point free and non trivial, if $h^1(S,\det E/N)\neq0$, then $(c_1')^2=0$ and Proposition (1.1) in [@green] implies the existence of a smooth elliptic curve $\Sigma\subset S$ such that $$(E/N)^{\vee\vee}={\mathcal O}_S(\Sigma)\oplus{\mathcal O}_S(\Sigma).$$ Such equality would contradict the stability of $E/N$, thus we conclude that $(c_1')^2\geq 2$ (and $c_2'\geq 2$) and $$\label{treno}
h^1(S,\det E/N)=0.$$ This ensures that $h^0(C,\det E/N\otimes {\mathcal O}_C)$ does not depend on the curve $C\in \vert L\vert_s$ (cf. [@donagi] Lemma (5.2)). Hence, the line bundle $\det E/N=L\otimes N^\vee$ is adapted to $\vert L\vert$.
We obtain: $$\begin{aligned}
{\mathrm{Cliff}}(\det E/N\otimes{\mathcal O}_C)&=&c_1(E/N)^2+c_1(N)\cdot c_1(E/N)-2h^0(C,\det E/N\otimes{\mathcal O}_C)+2\\
&\leq&c_1(E/N)^2+c_1(N)\cdot c_1(E/N)-2h^0(S,\det E/N)+2\\
&=&c_1(N)\cdot c_1(E/N)-2-2h^1(S,\det E/N)\\
&=&d-c_2(E/N)-2\\
&\leq&d-4.\end{aligned}$$
It remains only to prove that $h^0(C,\det E/N\otimes {\mathcal O}_C\otimes A^\vee)>0$. Consider the following diagram: $$\xymatrix{
0\ar[r]&H^0(C,A)^\vee\otimes{\mathcal O}_S\ar[r]&E\ar[r]^{\alpha}&\omega_C\otimes A^\vee\ar[r]&0.\\
&&N\ar@{^{(}->}[u]_{\gamma}&&
}$$ Since $h^2(S,N)=0$, the composition $\alpha\circ\gamma\neq 0$. This implies ${\mathrm{Hom}}(N,\omega_C\otimes A^\vee)\neq 0$ and we have finished because $N^\vee\otimes\omega_C\otimes A^\vee\simeq\det E/N\otimes{\mathcal O}_C\otimes A^\vee$.
Remaining cases {#spiaggia}
===============
In this section we consider rank-$3$ LM bundles $E$ of type (\[one\]), (\[two\]), (\[three\]), (\[four\]) on page 12, such that $\det E=L$ and $c_2(E)=d$ is fixed.
Choose $l_2\in\mathbb{N}$ and two line bundles $N,N_2\in{\mathrm{Pic}}(S)$ such that $N_1:=L\otimes(N\otimes N_2)^\vee$ is globally generated and non-trivial, and the following holds: $$\begin{aligned}
\label{mami}\mu_L(N)\geq&\mu_L(N_2)\geq&\mu_L(N_1),\\
\label{pa}\mu_L(N)\geq&\frac{2g-2}{3}\geq&\mu_L(N_1),\end{aligned}$$ where in (\[pa\]) either both the inequalities are strict, or none is.
Set $v:=v(N)$, $v_1:=v(N_1\otimes I_{\xi_1})$ and $v_2:=v(N_2\otimes I_{\xi_2})$, with $l(\xi_2)=l_2$ and $$l(\xi_1)=l_1:=d-l_2-c_1(N)\cdot c_1(N_1)-c_1(N)\cdot c_1(N_2)-c_1(N_1)\cdot c_1(N_2).$$
Define ${{\mathcal F}}_{N,N_2,l_2}$ to be the moduli stack of extensions $$0\to N_2\otimes I_{\xi_2}\to E/N\to N_1\otimes I_{\xi_1}\to 0,$$ where $\xi_i\subset S$ is a $0$-dimensional subscheme of length $l_i$ for $i=1,2$. We consider the projections $p_2:{{\mathcal F}}_{N,N_2,l_2}\to {{\mathcal M}}(v_2)\times {{\mathcal M}}(v_1)$ and $q_2:{{\mathcal F}}_{N,N_2,l_2}\to {{\mathcal M}}(v(E/N))$, and we denote by $Q_{N,N_2,l_2}$ the closure of the image of $q_2$.
If ${{\mathcal E}}_{N,N_2,l_2}$ is the moduli stack of extensions $$0\to N\to E\to E/N\to 0,$$ where $[E/N]\in Q_{N,N_2,l_2}(\mathbb{C})$, consider the morphisms $p_1:{{\mathcal E}}_{N,N_2,l_2}\to {{\mathcal M}}(v) \times Q_{N,N_2,l_2}$ and $q_1:{{\mathcal E}}_{N,N_2,l_2}\to {{\mathcal M}}(v(E))$. The closure of the image of $q_1$ is denoted by $\tilde{P}_{N,N_2,l_2}$ and its open substack, consisting of points which correspond to vector bundles $E$ such that $h^1(S,E)=h^2(S,E)=0$, by $P_{N,N_2,l_2}$.
Remark that, if $E$ is a LM bundle of type (\[one\]), (\[two\]), (\[three\]) or (\[four\]), there exist $N,N_2$ and $l_2$ such that $[E]$ defines a point of $P_{N,N_2,l_2}$. In order to count the number of moduli of such bundles, we start by proving the following:
\[lem:leo\] The stack $Q_{N,N_2,l_2}$, if nonempty, has dimension $$\dim Q_{N,N_2,l_2}=2l_1+2l_2-2+\langle v_1,v_2\rangle,$$ unless $N_1\simeq N_2$, $l_2\neq 0$ and $l_1=0$. In this case, for any component $Q\subset Q_{N,N_2,l_2}$, the following inequality holds: $$\dim Q\leq 2l_1+2l_2-1+\langle v_1,v_2\rangle.$$
The fiber of $p_2$ over the point of ${{\mathcal M}}_L(v_2)\times{{\mathcal M}}_L(v_1)$ given by $(N_2\otimes I_{\xi_2},N_1\otimes I_{\xi_1})$ is the quotient stack $$[{\mathrm{Ext}}^1(N_1\otimes I_{\xi_1},N_2\otimes I_{\xi_2})/{\mathrm{Hom}}(N_1\otimes I_{\xi_1},N_2\otimes I_{\xi_2})].$$ Since $\mu_L(N_2)\geq\mu_L(N_1)$, if either $N_1\not\simeq N_2$ or $N_1\simeq N_2$, $l_1\neq0$ and $l_2= 0$, one finds that $${\mathrm{Hom}}(N_2\otimes I_{\xi_2}, N_1\otimes I_{\xi_1})=0.$$ In these cases, the fibers of $p_2$ have constant dimension equal to $-\chi(N_1\otimes I_{\xi_1}, N_2\otimes I_{\xi_2})$ while the fibers of $q_2$ are $0$-dimensional Quot-schemes, hence the statement follows.
If $N_1\simeq N_2$ and $l_1=l_2= 0$, the conclusion is the same because the fibers of $p_2$ have constant dimension equal to $-\chi(N_1\otimes I_{\xi_1}, N_2\otimes I_{\xi_2})+1$ and the fibers of $q_2$ are smooth Quot-schemes of dimension $1$. Indeed, ${\mathrm{Hom}}(N,N)=1$ and ${\mathrm{Ext}}^1(N,N)=0$.
On the other hand, if $N_1\simeq N_2$ and $l_2\neq 0$, the fibers of $p_2$ do not necessarily have constant dimension; indeed, $\dim{\mathrm{Hom}}(N_1\otimes I_{\xi_2}, N_1\otimes I_{\xi_1})$ depends on the reciprocal position of $\xi_1$ and $\xi_2$. Since $\mathcal{H}om(I_{\xi_2},{\mathcal O}_S)\simeq\mathcal{H}om(I_{\xi_2},I_{\xi_2})\simeq {\mathcal O}_S$ (cf. [@okonek]), one shows that $$\mathcal{H}om(I_{\xi_2},I_{\xi_1})\simeq\{f\in{\mathcal O}_S\,\vert\, f\cdot I_{\xi_2}\subseteq I_{\xi_1}\}=:(I_{\xi_1}:I_{\xi_2})=I_{\xi_1\setminus (\xi_1\cap \xi_2)};$$ hence, one finds that $$\dim{\mathrm{Hom}}(N_1\otimes I_{\xi_2}, N_1\otimes I_{\xi_1})=H^0(S,\mathcal{H}om(I_{\xi_2}, I_{\xi_1}))=\left\{\begin{array}{ll}1&\textrm{ if }\xi_1\subseteq \xi_2\\0&\textrm{ otherwise }\end{array}\right..$$
As in [@yoshioka], let ${{\mathcal N}}^0_{N,N_2,l_2}$ (resp. ${{\mathcal N}}^1_{N,N_2,l_2}$) be the substack of ${{\mathcal M}}(v_2)\times{{\mathcal M}}(v_1)$ whose points correspond to pairs $(N_1\otimes I_{\xi_2},N_1\otimes I_{\xi_1})$ such that $\xi_1\not\subseteq\xi_2$ (resp. $\xi_1\subseteq\xi_2$), that is, $\dim{\mathrm{Hom}}(N_1\otimes I_{\xi_2}, N_1\otimes I_{\xi_1})=0$ (resp. $\dim{\mathrm{Hom}}(N_1\otimes I_{\xi_2}, N_1\otimes I_{\xi_1})=1$). Remark that ${{\mathcal N}}^0_{N,N_2,l_2}$ and ${{\mathcal N}}^1_{N,N_2,l_2}$ are complementary and that, being open, ${{\mathcal N}}^0_{N,N_2,l_2}$ is dense in ${{\mathcal M}}(v_2)\times{{\mathcal M}}(v_1)$ provided $l_1\neq0$.
We define ${{\mathcal F}}_{N,N_2,l_2}^{0}:=(p_2)^{-1}({{\mathcal N}}^0_{N,N_2,l_2})$ and ${{\mathcal F}}_{N,N_2,l_2}^{1}:=(p_2)^{-1}({{\mathcal N}}^1_{N,N_2,l_2})$ and we denote by $Q_{N,N,l_2}^{0}$ and $Q_{N,N,l_2}^{1}$ the closures of the images under $q_2$ of ${{\mathcal F}}_{N,N_2,l_2}^{0}$ and ${{\mathcal F}}_{N,N_2,l_2}^{1}$ respectively. Since the fibers of $q_2$ are Quot-schemes, we obtain that: $$\begin{aligned}
\dim Q_{N,N_2,l_2}^{0}&=&\dim {{\mathcal F}}_{N,N_2,l_2}^{0}=\dim{{\mathcal N}}^0_{N,N_2,l_2}+\langle v_1,v_2\rangle\leq2l_1+2l_2-2+\langle v_1,v_2\rangle,\\
\dim Q_{N,N_2,l_2}^{1}&\leq&\dim {{\mathcal F}}_{N,N_2,l_2}^{1}=\dim{{\mathcal N}}^1_{N,N_2,l_2}+\langle v_1,v_2\rangle+1\leq2l_1+2l_2-1+\langle v_1,v_2\rangle,\end{aligned}$$ where the last inequality in the second row is strict, unless the stack ${{\mathcal N}}^1_{N,N_2,l_2}$ is dense in ${{\mathcal M}}(v_2)\times{{\mathcal M}}(v_1)$, that is, $l_1=0$.
The statement follows because every component of $Q_{N,N,l_2}$ is contained either in $Q_{N,N,l_2}^{0}$ or in $Q_{N,N,l_2}^{1}$.
By proceeding as in Lemma \[lem:leo\], one proves the following:
\[prop:sedici\] Let $Z$ be a nonempty irreducible component of $P_{N,N_2,l_2}$. We have that $$\begin{aligned}
\label{maggio}\dim Z&=& 2l_1+2l_2+\langle v_2,v\rangle+\langle v_1,v\rangle+\langle v_1,v_2\rangle-\alpha,\end{aligned}$$ where $\alpha$ satisfies:
1. \[unob\] If $N$, $N_1$, $N_2$ are all non-isomorphic, then $\alpha=3$.
2. \[dueb\] Assume $N\simeq N_1\simeq N_2$. If $l_2\neq 0$ and $l_1=0$, then $\alpha\in\{1,2,3\}$. If $l_1\neq 0$ and $l_2=0$, one has $\alpha\in\{2,3\}$. In all the other cases, $\alpha=3$. If $N\simeq N_1\not\simeq N_2$, one has $\alpha=3$ unless $l_1=0$, in which case $\alpha\in\{2,3\}$.
3. \[treb\] If $N\simeq N_2\not\simeq N_1$, then $\alpha=3$ unless $l_2=0$, in which case $\alpha\in\{2,3\}$.
4. \[quattrob\] Assume $N_1\simeq N_2\not\simeq N$. Then $\alpha=3$ except when $l_2\neq0$ and $l_1= 0$; in this case $\alpha\in\{2,3\}$.
Note that LM bundles of type (\[one\]) lie in some $P_{N,N_2,l_2}$ with $N,N_1$, $N_2$ as in case (\[unob\]). Analogously, if $E$ is a LM bundle of type (\[three\]) (resp. of type (\[four\])), there exist $N,N_2,N_1=L\otimes(N\otimes N_2)^\vee$ as in (\[unob\]) or (\[treb\]) (resp. as in (\[unob\]) or (\[quattrob\])) and $l_2\in\mathbb{N}$ such that $[E]\in P_{N,N_2,l_2}(\mathbb{C})$. On the other hand, if a bundle of type (\[two\]) defines a point of $P_{N,N_2,l}$, then $\mu_L(N)=\mu_L(N_2)=\mu_L(N_1)$ and any case of the previous proposition may occur.
Now, we consider the Grassmann bundle $\psi:{{\mathcal G}}_{N,N_2,l_2}\to P_{N,N_2,l_2}$ with fiber over a point of $P_{N,N_2,l_2}$ corresponding to a bundle $E$ equal to $G(3,H^0(S,E))$ and denote by ${\mathcal W}_{N,N_2,l_2}$ the closure of the image of the rational map $h_{N,N_2,l_2}:{{\mathcal G}}_{N,N_2,l_2}\dashrightarrow{\mathcal W}^2_d(\vert L\vert)$.
\[lem:marta\] Assume that general curves in $\vert L\vert$ have Clifford dimension $1$ and maximal gonality $k=\left\lfloor \frac{g+3}{2}\right\rfloor$. Then, for any irreducible component ${\mathcal W}$ of ${\mathcal W}_{N,N_2,l_2}$, one has $$\begin{aligned}
\dim{\mathcal W}&\leq& \frac{1}{4}g+d+\frac{3}{2}-\alpha,\end{aligned}$$ where $\alpha$ is as in Proposition \[prop:sedici\].
Let ${{\mathcal G}}$ be an irreducible component of ${{\mathcal G}}_{N,N_2,l_2}$ such that ${\mathcal W}=\overline{h_{N,N_2,l_2}({{\mathcal G}})}$. Since ${{\mathcal G}}=\psi^{-1}(Z)$ for some irreducible component $Z$ of $P_{N,N_2,l_2}$, Proposition \[prop:sedici\] implies that: $$\begin{aligned}
\dim{{\mathcal G}}&=&3(g-d+2)+\dim Z\\
&=&3(g-d)+12-\alpha-2\chi(E)+2l_1+2l_2+\\
& &c_1(N)\cdot c_1(N_1)+c_1(N)\cdot c_1(N_2)+c_1(N_1)\cdot c_1(N_2)\\
&=&g-d+2-\alpha+2(l_1+l_2)+c_1(N)\cdot (c_1(N_1)+c_1(N_2))+c_1(N_1)\cdot c_1(N_2)\\
&=&g+d+2-\alpha-c_1(N)\cdot c_1(N_1)-c_1(N)\cdot c_1(N_2)-c_1(N_1)\cdot c_1(N_2)\\
&\leq&g+d+2-\alpha-\frac{3}{2}k\\
&\leq&\frac{1}{4}g+d+\frac{1}{2}-\alpha,\end{aligned}$$ where we have used Lemma \[lem:caso3\] and the fact that $k\geq (g+2)/2$. The statement follows since the fibers of $h_{N,N_2,l_2}$ are quotient stacks of dimension $-1$.
Finally, we prove the following:
\[prop:nutella\] Assume that general curves in $\vert L\vert$ have Clifford dimension $1$ and maximal gonality $k=\left\lfloor \frac{g+3}{2}\right\rfloor$. Fix a positive integer $d$ such that $(g,d)\not\in\{(2,4),(4,5),(6,6),(10,9)\}$. Let ${\mathcal W}\subset {\mathcal W}_{N,N_2,l_2}$ be an irreducible component of ${\mathcal W}^2_d(\vert L\vert)$. Then $\rho(g,2,d)\geq 0$ and ${\mathcal W}$ does not dominate $\vert L\vert$.
Lemma \[lem:caso3\] implies $d\geq\frac{3}{2}k$, hence $\rho(g,2,d)\geq0$. Lemma \[lem:marta\] gives: $$\dim{\mathcal W}\leq \frac{1}{4}g+d+\frac{3}{2}-\alpha.$$ Therefore, ${\mathcal W}$ cannot dominate $\vert L\vert$ if $$\frac{1}{4}g+d+\frac{3}{2}-\alpha< g+\rho(g,2,d)=-g+3d-6,$$ that is, $d>\frac{5}{8}g+\frac{15}{4}-\frac{\alpha}{2}$. In particular, as $\alpha\geq1$, it is enough to require $d>\frac{5}{8}g+\frac{13}{4}=:h$. Such inequality is satisfied always except for $$(g,d)\in\{(2,4),(3,5),(4,5),(5,6),(6,6),(6,7),(8,8),(10,9),(14,12)\}.$$ If $(g,d)=(6,6)$, the linear system $\vert L\vert$ can be dominated by ${\mathcal W}$. In all the other cases $d=\lfloor h\rfloor$ and we check whether $\alpha>2h-2\left\lfloor h\right\rfloor+1$, which would prevent ${\mathcal W}$ from being dominant. This holds true if $(g,d)\not\in\{(2,4),(4,5),(10,9)\}$ (use that the case $\alpha=1$ may occur only when parametrizing LM bundles of type (\[two\]) and that, if $\gcd(2g-2,3)=1$, there do not exist properly $\mu_L$-semistable bundles of Mukai vector $v(E)$).
The four cases which are not covered by Proposition \[prop:nutella\] might be treated by “ad hoc” arguments but this is not the purpose of the paper.
Proofs of Theorem \[thm:magari\] and Theorem \[thm:principale\] are now straightforward.
Being non-simple, the LM bundle $E_{C,A}$ is not $\mu_L$-stable. Since $d<\frac{2}{3}g+2$, Corollary \[cor:lasagna\] implies the existence of a line bundle $N\in{\mathrm{Pic}}(S)$ such that $E_{C,A}\in R_N(\mathbb{C})$. The statement thus follows directly from Proposition \[prop:vai\].
Case (\[aa\]) trivially follows from Proposition \[prop:cambridge\], Proposition \[prop:fiducia\] and Proposition \[prop:nutella\].
Now, let $\frac{2}{3}g+2\leq d\leq \frac{3}{4}g+2$. Given ${\mathcal W}$ an irreducible component of ${\mathcal W}^2_d(\vert L\vert)$ which dominates $\vert L\vert$ and whose general point corresponds to a LM bundle that is not $\mu_L$-stable, Proposition \[prop:cambridge\] and Proposition \[prop:nutella\] imply the existence of a line bundle $N\in{\mathrm{Pic}}(S)$ such that ${\mathcal W}\subset {\mathcal W}_N$. The statement follows from Proposition \[prop:vai\].
Transversality of some Brill-Noether loci {#bus}
=========================================
We apply our results in order to prove Theorem \[thm:tra\] in the introduction.
\[thm:gon\] Let $r\geq 3$, $g\geq 0$, $d\leq g-1$ be positive integers such that $\rho(g,r,d)<0$ and $d-2r+2\geq \lfloor(g+3)/2\rfloor$. If $r\geq 4$, assume $d^2>4(r-1)(g+r-2)$. For $r=3$, let $d^2>8g+1$. If $-1$ is not represented by the quadratic form $$Q(m,n)=(r-1)m^2+mnd+(g-1)n^2\,\,\,m,n\in\mathbb{Z},$$ there exists a smooth curve $C\subset \mathbb{P}^r$ of genus $g$, degree $d$ and maximal gonality $\left\lfloor \frac{g+3}{2}\right\rfloor$. Moreover, one can choose $C$ such that for any complete, base point free $g^1_{e}$ on $C$ with $\rho(g,1,e)\geq 0$ the Petri map is injective.
Notice that the inequalities $d\leq g-1$ and $d^2>4(r-1)(g-1)$ trivially imply $d>4(r-1)$.
In order to prove the first part of the statement, we proceed as in [@gabi1] Theorem 3 paying special attention to our slightly different hypotheses. Rathmann’s Theorem implies the existence of a $2r-2$-degree $K3$ surface $S\subset \mathbb{P}^r$ and a smooth curve $C\subset S$ of degree $d$ and genus $g$ such that ${\mathrm{Pic}}(S)=\mathbb{Z}H\oplus\mathbb{Z}C$, where $H$ is the hyperplane section of $S$. Our assumption on $Q$ implies that $S$ does not contain $(-2)$-curves. As in [@gabi1], one shows that the line bundle $L:={\mathcal O}_S(C)$ is ample by Nakai-Moishezon criterion (if $D\subset S$ is an effective divisor, use that $D^2\geq 0$ and $D\cdot H>2$, in order to show that $C\cdot D>0$). Hence, $C$ has Clifford dimension $1$ (cf. [@ciliberto] Proposition 3.3).
Assume that $C$ has gonality $k<\left\lfloor \frac{g+3}{2}\right\rfloor$. We reach a contradiction by showing that $k\geq d-2r+2$. If $A$ is a complete, base point free pencil $g^1_k$ on $C$, by [@donagi] Theorem (4.2) there exists an effective divisor $D\equiv mH+nC$ on $S$, such that $\vert A\vert$ is contained in the linear system $\vert {\mathcal O}_C(D)\vert$ and the following conditions are satisfied: $$h^0(S,{\mathcal O}_S(D))\geq 2,\,\,h^0(S,{\mathcal O}_S(C-D))\geq 2,\,\,C\cdot D\leq g-1,\,\, {\mathrm{Cliff}}(D\vert_C)= {\mathrm{Cliff}}(A).$$ In particular, as remarked in [@donagi] page 60, the last equality implies that $$h^1(S,{\mathcal O}_S(D))=h^1(S,{\mathcal O}_S(C-D))=0,$$ thus $c_1(D)^2>0$ and $c_1(C-D)^2>0$. Moreover, one has $$k=2+{\mathrm{Cliff}}(D\vert_C)=D\cdot (C-D).$$ We show that $$f(m,n)=D\cdot C-D^2=-(2r-2)m^2+d(1-2n)m+(n-n^2)(2g-2)\geq d-2r+2,$$ for values of $m$ and $n$ satisfying the following inequalities:
1. \[nuovo1\] $(r-1)m^2+mnd+n^2(g-1)>0$,
2. \[nein\] $(r-1)m^2+(mn-m)d+(1-n)^2(g-1)>0$,
3. \[nuovo2\] $2<(2r-2)m+nd<d-2$,
4. \[nuovo3\] $md+(2n-1)(g-1)\leq 0$.
Assume first that $n=1$, and set $a = -m$. Then (\[nuovo2\]) implies $0 < a < (d-2)/(2r-2)$. Inequality (\[nuovo1\]) is equivalent to $(r-1)a^2-ad+g-2\geq 0$, whence $$a\leq\frac{d-\sqrt{d^2-4(r-1)(g-2)}}{2r-2}.$$ We have $f(-a,1)\geq d-2r+2$ whenever $1\leq a\leq d/(2r-2)-1$. For either $r\geq 4$ or $r=3$ and $d^2-8g\geq8$, this holds true because $d^2-4(r-1)(g-2)\geq4r(r-1)>4(r-1)^2$. If $r=3$ and $d^2-8g<8$, then $d^2-8g=4$ and $d\equiv2\mod4$. Hence, (\[nuovo2\]) implies that $1\leq a< (d-4)/4$. Remark that $f(-a,1)=d-2r+2$ whenever $a=1$, that is, $C\equiv C-H$. The case $n=0$ can be treated similarly by using (\[nein\]) instead of (\[nuovo1\]), and one obtains that $f(m,0)\geq d-2r+2$ with equality holding only for $m=1$, that is, $D\equiv H$.
If $n<0$, inequalities (\[nuovo1\]), (\[nuovo2\]) and (\[nuovo3\]) imply that $-\alpha n<m\leq(g-1)(1-2n)/d$, where $$\alpha=\frac{d+\sqrt{d^2-4(r-1)(g-1)}}{2r-2}.$$ Therefore, one has $$f(m,n)\geq \min\left\{f(-\alpha n,n),f\left(\frac{(g-1)(1-2n)}{d},n\right)\right\}.$$ Analogously, if $n\geq 2$, then $\max\{-\beta n, (2-nd)/(2r-2)\}<m\leq(g-1)(1-2n)/d$, where $$\beta=\frac{d-\sqrt{d^2-4(r-1)(g-1)}}{2r-2};$$ this gives $$f(m,n)\geq \min\left\{f\left(\frac{(g-1)(1-2n)}{d},n\right), \max\left\{f(-\beta n,n),f\left(\frac{2-nd}{2r-2},n\right)\right\}\right\}.$$ Computations in [@gabi1] give $\max\left\{f(-\beta n,n),f\left((2-nd)/(2r-2),n\right)\right\}>d-2r+2$ if $n\geq 2$, and $f(-\alpha n,n)>d-2r+2$ when $n<0$, unless $r=3$, $n=-1$ and $d^2-8g=4$. In this case, $d\equiv 2\mod 4$ and $m\geq (d+4)/4$ by (\[nuovo2\]); one uses that $f((d+4)/4,-1)>d-4$. In order to conclude the proof that $C$ has maximal gonality, it is enough to remark that the function $$h(n):=f\left(\frac{(g-1)(1-2n)}{d},n\right)=\frac{g-1}{2}\left[\frac{(2n-1)^2(d^2-4(r-1)(g-1))}{d^2}+1\right]$$ reaches its minimum for $n=1/2$ and $h(0)\geq d-2r+2$ by direct computation.
Concerning the last part of the statement, assume $C$ is general in its linear system and let $A$ be a complete, base point free pencil $g^1_{e}$ on $C$ such that $\rho(g,1,e)\geq 0$ and $\ker\mu_{0,A}\neq 0$. The bundle $E=E_{C,A}$ is non-simple, hence it cannot be $\mu_L$-stable. As a consequence, there exists a short exact sequence $$\label{fu}
0\to M\to E\to N\otimes I_{\xi}\to 0,$$ where $M,N$ are line bundles, $I_{\xi}$ is the ideal sheaf of a $0$-dimensional subscheme $\xi\subset S$ and $c_1(M)\cdot C\geq \mu_L(E)=g-1\geq c_1(N)\cdot C$. If sequence (\[fu\]) does not split, then $$h^0(S,E\otimes E^\vee)\leq 1+\dim{\mathrm{Hom}}(M,N\otimes I_\xi)+\dim{\mathrm{Hom}}(N\otimes I_\xi,M).$$ Since $\mu_L(M)\geq \mu_L(N)$, if ${\mathrm{Hom}}(M,N\otimes I_\xi)\neq 0$ then $M\simeq N$ and $C=2c_1(M)$, which is absurd. It follows that $N^\vee\otimes M$ is non-trivial and effective. Since $S$ does not contain $(-2)$-curves, one has $$c_1(N^\vee\otimes M)^2=C^2-4c_1(N)\cdot c_1(M)=2g-2-4c_1(N)\cdot c_1(M)\geq 0;$$ this contradicts Lemma \[lem:uova\], which states that $c_1(N)\cdot c_1(M)\geq k\geq(g+2)/2$. Thus, $\xi=\emptyset$ and sequence (\[fu\]) splits. We have to show that, if $E=N\oplus M$ is a splitting LM bundle, the rational map $\chi:G(2,H^0(S,E))\dashrightarrow \vert L\vert$ cannot be dominant. Remark that $\chi$ factors through the rational map $h_E:G(2,H^0(S,E))\dashrightarrow {\mathcal W}^1_e(\vert L\vert)$, whose fiber over a point $(C,A)\in{\mathrm{Im}}\, h_E$ is at least $1$-dimensional since it is isomorphic to $\mathbb{P}({\mathrm{Hom}}(E_{C,A},\omega_C\otimes A^\vee)^\circ)$, where ${\mathrm{Hom}}(E_{C,A},\omega_C\otimes A^\vee)^\circ$ is an open subgroup of ${\mathrm{Hom}}(E_{C,A},\omega_C\otimes A^\vee)\simeq H^0(S,E_{C,A}\otimes E_{C,A}^\vee)$. This is enough to conclude because $\rho(g,1,e)\geq 0$, hence $\dim G(2,H^0(S,E))=2(g-e+1)\leq g$.
\[thm:pappa\] Let $g,r,d$ satisfy the hypotheses of Theorem \[thm:gon\]. The curve $C$ can be chosen such that, if $$e< \min\left\{d-2r+5,\frac{17}{24}g+\frac{23}{12}\right\},$$ then $C$ does not have any complete, base point free net $g^2_{e}$ for which the Petri map is non-injective.
Let $S\subset \mathbb{P}^r$ be as in the proof of Theorem \[thm:gon\] and $C$ be general in its linear system. Let $A$ be a complete, base point free net on $C$ of degree $d_A< \frac{17}{24}g+\frac{23}{12}$; if $\rho(g,2,d_A)\geq 0$, assume moreover that $\ker\mu_{0,A}\neq 0$. Corollary \[cor:lasagna\] and Proposition \[prop:vai\] imply that $\vert A\vert$ is contained in the linear system $\vert{\mathcal O}_C(D)\vert$ for some effective divisor $D\equiv mH+nC$ on $S$ such that: $$h^0(S,{\mathcal O}_S(D))\geq 2,\,\,h^0(S,{\mathcal O}_S(C-D))\geq 2,\,\,C\cdot D\leq \frac{4g-4}{3},\,\, {\mathrm{Cliff}}(D\vert_C)\leq {\mathrm{Cliff}}(A).$$ In fact, the Lazarsfeld-Mukai bundle $E:=E_{C,A}$ is given by an extension: $$0\to N\to E\to E/N\to 0,$$ where $N:={\mathcal O}_S(C-D)$ and $E/N$ is a $\mu_L$-stable torsion free sheaf of rank $2$ on $S$. As in the proof of Proposition \[prop:vai\], one shows that $D^2>0$, hence $h^1(S,{\mathcal O}_S(D))=0$. Moreover, one obtains that $h^1(S,N)=0$ because the equality $(C-D)^2=0$ would imply $d\geq (5g+4)/6$, which is absurd. As a consequence, one has $$\label{cara}
d_A-4 ={\mathrm{Cliff}}(A)\geq {\mathrm{Cliff}}(D\vert_C)=D\cdot C-2h^0(S,{\mathcal O}_C(D\vert_C))+2=D\cdot (C-D)-2,$$ and equality holds only if $D^2=2$ and $c_2(E/N)=2$ (cf. proof of Proposition \[prop:vai\]); in particular, for $D\equiv H$, the inequality is strict. We show that $$\label{semi}
f(m,n):=D\cdot (C-D)\geq d-2r+2,$$ and, if equality holds, then either $D\equiv H$ or $D\equiv C-H$. Computations are similar to those in Theorem \[thm:gon\], but now, instead of having $D\cdot C\leq g-1$, we only know that $D\cdot C\leq(4g-4)/3$. Therefore, inequality (\[nuovo3\]) must be replaced with
- \[bla\] $md+(2n-\frac{4}{3})(g-1)\leq 0$.
The cases $n\in\{0,1\}$ can be treated exactly as before. For $n<0$, we have $$f(m,n)\geq \min\left\{f(-\alpha n,n),f\left(\frac{(g-1)(\frac{4}{3}-2n)}{d},n\right)\right\}.$$ If $n\geq 2$, then $$f(m,n)\geq \min\left\{f\left(\frac{(g-1)(\frac{4}{3}-2n)}{d},n\right), \max\left\{f(-\beta n,n),f\left(\frac{2-nd}{2r-2},n\right)\right\}\right\}.$$ Therefore, it is enough to show that $$g(n):=f\left(\frac{(g-1)(\frac{4}{3}-2n)}{d},n\right)-d+2r-2> 0\textrm{ for }n< 0 \textrm{ or }n\geq 2.$$ One can write $g(n)=an^2+bn+c$, with $$\begin{aligned}
a&=&-4(2r-2)\left(\frac{g-1}{d}\right)^2+2d\left(\frac{g-1}{d}\right),\\
b&=&\frac{16}{3}(2r-2)\left(\frac{g-1}{d}\right)^2-\frac{8}{3}d\left(\frac{g-1}{d}\right),\\
c&=&-\frac{16}{9}(2r-2)\left(\frac{g-1}{d}\right)^2+\frac{4}{3}d\left(\frac{g-1}{d}\right)-d+2r-2.\end{aligned}$$ Since $a>0$ and $0<-b/(2a)<1$, our claim follows if $g(0)=c>0$, or equivalently, if $$\frac{3}{4}<\frac{g-1}{d}<\frac{3}{8}\left(\frac{d-2(r-1)}{r-1}\right).$$ The left inequality is trivial since $d\leq g-1$. The right inequality is equivalent to the condition $8(g-1)(r-1)<3d^2-6d(r-1)$, which is satisfied as well (if $r\geq 4$, use that $8(g-1)(r-1)<2d^2-8(r-1)^2$ and $d> 4(r-1)$; if $r=3$, use that $d^2>8g+1$ and either $(g,d)=(12,11)$ or $d\geq 12$ by manipulation of the hypotheses).
We conclude that $d_A\geq d-2r+4$ and the inequality is strict unless equalities hold both in (\[cara\]) and (\[semi\]), thus $D\equiv C-H$ and $(C-H)^2=2$. This case can be excluded since it would imply $d=g+r-3\geq g$.
Remark that the condition $e<\frac{17}{24}g+\frac{23}{12}$ is automatically satisfied if $\rho(g,2,e)<0$.
The proof of Theorem \[thm:tra\] is now trivial: apply Theorem \[thm:gon\] and Theorem \[thm:pappa\] and proceed by induction on $f$ and $e$ in order to deal with pencils $g^1_f$ and nets $g^2_e$ which have a nonempty base locus.
Noether-Lefschetz divisor and Gieseker-Petri divisor in genus 11
================================================================
The Clifford index ${\mathrm{Cliff}}(C)$ is one of the most important invariants of an algebraic curve $C$. In [@lange] Lange and Newstead defined the analogue of the Clifford index for higher rank vector bundles in the following way. If ${\mathcal U}_C(n,d)$ denotes the moduli space of semistable rank-$n$ vector bundles of degree $d$ on a genus-$g$ curve $C$, given $E\in {\mathcal U}_C(n,d)$, the Clifford index of $E$ is $$\gamma(E):=\mu(E)-\frac{2}{n}h^0(C,E)+2\geq 0,$$ where $\mu(E)$ denotes the slope of $E$. For any positive integer $n$, one defines the higher Clifford index of $C$ $${\mathrm{Cliff}}_n(C):=\mathrm{min}\{\gamma(E)\,\vert\, E\in {\mathcal U}_C(n,d),\,h^0(C,E)\geq 2n,\,\mu(E)\leq g-1\}.$$ A natural question is whether higher Clifford indices are new invariants, different from the ones arising in classical Brill-Noether theory. In [@lange] Lange and Newstead reformulated a conjecture of Mercat (cf. [@mercat]) in a slightly weaker form predicting: $$\label{rugby}
{\mathrm{Cliff}}_n(C)={\mathrm{Cliff}}(C);$$ remark that trivially ${\mathrm{Cliff}}_n(C)\leq {\mathrm{Cliff}}(C)$, while the opposite inequality is largely non-trivial. When $n=2$, the conjecture has been proved for a general curve in $M_g$ if $g\leq 16$ by Farkas and Ortega (cf. [@angela]) and the same is expected to hold true in any genus. However, if $g\geq 11$, there are examples of curves with maximal Clifford index ${\mathrm{Cliff}}(C)=\left\lfloor\frac{g-1}{2}\right\rfloor$ that violate (\[rugby\]) for $n=2$. These have been constructed in [@angela], [@gan], [@lange], [@peter], [@new] as sections of $K3$ surfaces with Picard number at least $2$. We recall that the $K3$ locus $${\mathcal K}_g:=\{[C]\in M_g\,\vert\, C\subset S,\,S\textrm{ is a }K3\textrm{ surface}\}$$ is irreducible of dimension $19+g$ if either $g=11$ or $g\geq 13$ (cf. [@cil]). In particular, ${\mathcal K}_{11}=M_{11}$ and a general curve $[C]\in M_{11}$ lies on a unique $K3$ surface with Picard number $1$ (cf [@muk]). Given two positive integers $r,d$ such that $d^2>4(r-1)g$ and $d$ does not divide $2r-2$, one defines the Noether-Lefschetz divisor inside ${\mathcal K}_g$ as $${\mathcal{NL}}^r_{g,d} := \left\{ [C] \in {\mathcal K}_g \, \left\vert\begin{array}{l} C \subset S, \, S\textrm{ is a } K3 \textrm{ surface}, \,{\mathrm{Pic}}(S) \supset {\mathbb{Z}}C \oplus {\mathbb{Z}}H, \\
H \textrm{ nef }, \, H^2 = 2r - 2, \, C^2 = 2g - 2, \, C\cdot H = d \end{array}\right.\right\}.$$ In [@gan] it is proved that a curve $C$ of genus $11$ violates Mercat’s conjecture for $n=2$ whenever $[C]\in{\mathcal{NL}}^4_{11,13}$.
Since some of the curves exhibited in [@lange], [@peter], [@new] do not satisfy the Gieseker-Petri Theorem, Lange and Newstead asked whether ${\mathrm{Cliff}}_2(C)={\mathrm{Cliff}}(C)$ whenever $C$ is a Petri curve (Question 4.2 in [@new]). We prove Theorem \[thm:chisa\], which gives a negative answer to this question.
Let $S\subset \mathbb{P}^4$ be a $K3$ surface such that ${\mathrm{Pic}}(S)={\mathbb{Z}}C\oplus{\mathbb{Z}}H$, where $H$ is the hyperplane section, $H^2=6$, $C^2=20$ and $C\cdot H=13$. Denote by $L$ the line bundle ${\mathcal O}_S(C)$. We show that, if $C\in\vert L\vert$ is general, then $[C]$ does not lie in the Gieseker-Petri locus $GP_{11}$. Recall that $GP_{11}$ has pure codimension $1$ in $M_{11}$ (cf. [@marghem]) and decomposes in the following way: $$GP_{11}=M^2_{11,9}\cup GP^2_{11,10}\cup\bigcup_{d=7}^{10} GP^1_{11, d},$$ where $M^2_{11,9}$ is a Brill-Noether divisor. Therefore, proving the transversality of ${\mathcal{NL}}^4_{11,13}$ and $GP_{11}$ is equivalent to showing that in the above situation, if $C\in\vert L\vert$ is general, then $C$ has no $g^2_9$ and the varieties $G^2_{10}(C)$ and $G^1_d(C)$ for $7\leq d\leq 10$ are smooth of the expected dimension.
We proceed as in the previous section; since the hypotheses of Theorem \[thm:gon\] are not satisfied, explicit computations must be performed. Direct calculations imply that $S$ does not contain any $(-2)$-curve. Moreover, $C$ is an ample line bundle on $S$ by Proposition 2.1 in [@new]. As a consequence, $C$ has Clifford dimension $1$ (cf. Proposition 3.3 in [@ciliberto]) and ${\mathrm{Cliff}}(C)=5$ (cf. Proposition 3.3 in [@gan]). In particular, $C$ has maximal gonality $k=7$ and has no $g^2_d$ for $d\leq 8$. Hence, in order to prove that $G^2_9(C)= \emptyset$, it is enough to exclude the existence of complete, base point free $g^2_{9}$ on $C$. Similarly, the condition $[C]\not\in GP^2_{11,10}$ is equivalent to the requirement for $G^2_{10}(C)$ to be smooth of the expected dimension $\rho(11,2,10)$ in the points corresponding to complete, base point free linear series. Analogously, by induction on $d$, if the Petri map associated with any complete, base point free pencil of degree $7\leq d\leq 10$ is injective, then $[C]\not\in \cup_{d=7}^{10}GP^1_{11,d}$.
For any $A\in G^2_9(C)$, the Petri map $\mu_{0,A}$ is non-injective for dimension reasons and the bundle $E=E_{C,A}$ is non-simple, hence it cannot be $\mu_L$-stable. Since $$\gcd(rk\, E, c_1(E)^2)=\gcd(3,20)=1,$$ there are no properly semistable sheaves of Mukai vector $v(E)=(3,C,4)$; hence, $E$ is $\mu_L$-unstable. By Corollary \[cor:lasagna\], $E$ sits in the short exact sequence $$\label{panda}
0\to N\to E\to E/N\to 0,$$ where $N\in{\mathrm{Pic}}(S)$ is its maximal destabilizing sheaf and the quotient $E/N$ is a $\mu_L$-stable torsion free sheaf of rank $2$. Having denoted by $D$ the first Chern class of $E/N$, Proposition \[prop:vai\] implies that the line bundle ${\mathcal O}_C(D)$ contributes to ${\mathrm{Cliff}}(C)$. Moreover, as in the proof of the aforementioned proposition, one shows that $$\begin{aligned}
\label{dis1}D^2&\geq& 2,\\
\label{dis2}c_2(E/N)&\geq &\frac{3}{2}+\frac{1}{4}D^2.\end{aligned}$$ Furthermore, Lemma \[lem:endo1\] gives $$\label{dis3}
c_1(N)\cdot c_1(E/N)=(C-D)\cdot D\geq k=7.$$ Since $$9=c_2(E)=c_2(E/N)+(C-D)\cdot D\geq \frac{3}{2}+\frac{1}{4}D^2+(C-D)\cdot D,$$ the divisor $D\equiv mH+nC$ must satisfy $$\left\{\begin{array}{l}C\cdot D=13m+20n=9\\D^2=6m^2+20n^2+26mn=2(m+n)(3m+10n)=2\end{array}\right..$$ One shows that this system admits no integral solution. As a consequence, a general curve in $\vert L\vert$ has no linear series of type $g^2_9$.
Analogously, given a complete, base point free $A\in G^2_{10}(C)$ with $\ker\mu_{0,A}\neq 0$, the LM bundle $E=E_{C,A}$ is $\mu_L$-unstable and its maximal destabilizing sheaf is a line bundle $N$ such that $E/N$ is $\mu_L$-stable by Corollary \[cor:lasagna\]. With the same notation as above, inequalities (\[dis1\]), (\[dis2\]), (\[dis3\]) still hold true and the following cases must be considered: $$\begin{aligned}
(a)\,\,\left\{\begin{array}{l}C\cdot D=10\\D^2=2\\(c_2(E/N)=2)\end{array}\right.\,\,\,\,\,\,\,\,&
(b)\,\,\left\{\begin{array}{l}C\cdot D=9\\D^2=2\\ (c_2(E/N)=3)\end{array}\right.\\
(c)\,\,\left\{\begin{array}{l}C\cdot D=11\\D^2=4\\ (c_2(E/N)=3)\end{array}\right.\,\,\,\,\,\,\,\,&
(d)\,\,\left\{\begin{array}{l}C\cdot D=13\\D^2=6\\ (c_2(E/N)=3)\end{array}\right..\\
\end{aligned}$$ These systems have no integral solutions except for (d), which is satisfied by $$(m,n)=(1,0).$$ Therefore, $N={\mathcal O}_S(C-H)$ and $v(E/N)=(2,H,2)$. Since $\langle v(E/N),v(E/N)\rangle=-2$, the sheaf $E/N$ is uniquely determined.
By applying first ${\mathrm{Hom}}(E,-)$ and then ${\mathrm{Hom}}(-,N)$ and ${\mathrm{Hom}}(-,E/N)$ to the short exact sequence (\[panda\]), one shows that $$h^0(S,E\otimes E^\vee)\leq 2+\dim{\mathrm{Hom}}(N, E/N)+\dim{\mathrm{Hom}}(E/N,N)$$ and the inequality is strict if the sequence does not split. Since $\mu_L(N)>\mu_L(E/N)$, Proposition \[prop:morfismi\] implies that ${\mathrm{Hom}}(N,E/N)=0$. Let $0\neq \alpha\in{\mathrm{Hom}}(E/N,N)$. Since both ${\mathrm{Im}}\,\alpha$ and $\ker\alpha$ are torsion free sheaves of rank $1$, there exists an effective divisor $D_1$ on $S$ and two $0$-dimensional subschemes $\xi_1,\xi_2\subset S$ such that $E/N$ is given by an extension $$0\to{\mathcal O}_S(2H-C+D_1)\otimes I_{\xi_1}\to E/N\to {\mathcal O}_S(C-H-D_1)\otimes I_{\xi_2}\to 0.$$ The $\mu_L$-stability of $E/N$ implies that $$13/2=\mu_L(E/N)<(C-H-D_1)\cdot C=-D_1\cdot C+7;$$ since $C$ has positive intersection with any non-trivial effective divisor, $D_1=0$. It follows that $$3=c_2(E/N)=(2H-C)\cdot (C-H)+l(\xi_1)+l(\xi_2)\geq 7,$$ which is absurd. Hence, ${\mathrm{Hom}}(E/N,N)=0$ and (\[panda\]) splits. As a consequence, the bundle $E=N\oplus E/N$ is uniquely determined.
We look at the rational map $\chi:G(3,H^0(S,E))\dashrightarrow \vert L\vert$; this cannot be dominant since $\dim G(3,H^0(S,E))=9$. Therefore, a general curve $C\in\vert L\vert$ does not lie in $GP^2_{11,10}$.
It remains to show that, if $C\in\vert L\vert$ is general, then $[C]\not\in \cup_{d=7}^{10}GP^1_{11,d}$. It is enough to prove that for any complete, base point free $g^1_d$ on $C$ the Petri map is injective. One can proceed exactly as in the last part of the proof of Theorem \[thm:gon\] since $S$ does not contain $(-2)$-curves.
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|
---
author:
- 'Tomohiro <span style="font-variant:small-caps;">Isobe</span> and Shogo <span style="font-variant:small-caps;">Tanimura</span>[^1]'
title: |
A method for systematic construction of Bell-like inequalities\
and a proposal of a new type of test
---
Introduction
============
From the early time in the history of quantum mechanics, doubts about validity or completeness of quantum mechanics have been raised repeatedly. For instance, Einstein [@EPR1935] and Bohr [@Bohr1935] devoted themselves to enthusiastic debates. De Broglie proposed the pilot wave theory as an alternative to quantum mechanics. Bohm [@Bohm1952] and other people developed de Broglie’s idea as the hidden variable theory. Bell [@Bell1964] formulated his famous inequality in the context of the hidden variable theory and proposed a possible test to compare prediction of quantum mechanics with predictions of other alternative theories. Since his proposal, many experiments have been performed [@CH1978; @Aspect1981] and they support validity of quantum mechanics with increasing accuracy [@Sakai2006]. Many people [@Froissart1981; @Peres1999; @Collins2004] have proposed generalizations of the Bell inequality. So, today there seems no room for putting a doubt on validity of quantum mechanics.
However, in this paper we attempt to propose a new test for checking validity of quantum mechanics from a different viewpoint. At least, we give a new explanation on the Bell inequality. This explanation gives an insight for understanding implication of quantum mechanics and suggests a way for producing various kinds of paradoxical inequalities which are not equivalent to the original Bell inequality.
The hidden variable theory consists of the following assumptions: (1) The state of a physical system is specified not only by a quantum state $ \psi $, but also by a variable $ \lambda $ or a set of variables $ \lambda = ( \lambda_1, \lambda_2, \cdots , \lambda_n ) $ which we do not know. (2) Any physical observable $ A $ is a function of $ \psi $ and $ \lambda $. Once the state $ \psi $ and the value of the hidden variable $ \lambda $ are specified, the value $ A ( \psi, \lambda ) \in {{\mathbb R}}$ of the observable is uniquely determined. (3) The variable $ \lambda $ obeys a probability distribution $ P( \psi, \lambda ) $, which is nonnegative, $ P \ge 0 $, and normalized, $ \int P d \lambda = 1 $. (4) Additionally, in the local hidden variable theory, the value or the distribution of the hidden variable cannot be influenced by superluminous signals.
In the scheme of the hidden variable theory, the expectation value of the observable $ A $ is calculated as $${\langle}A {\rangle}= \int A ( \psi, \lambda ) P( \psi, \lambda ) \, d \lambda
\label{hidden variable}$$ with the probability distribution $ P( \psi, \lambda ) $ of the hidden variable $ \lambda $. So, a question arises; can the hidden variable theory mimic the quantum theory completely? In other words, does the probability distribution of the hidden variable which reproduces the predictions of the quantum theory for any observables mathematically exist?
Bell proved an inequality which bounds the expectation value of a certain observable in the scheme of the local hidden variable theory. Clauser, Horne, Shimony and Holt [@CHSH1969; @CH1974] reformulated Bell’s inequality in a form more suitable for experimental tests. It tells that the expectation value of a quantity $ S = A_1 B_1 + A_1 B_2 + A_2 B_1 - A_2 B_2 $ must be in the range $$-2 \: \le \: {\langle}S {\rangle}\: \le \: 2
\qquad
\mbox{: hidden variable theory}
\label{BCHSH inequality},$$ if the local hidden variable theory is correct. We call (\[BCHSH inequality\]) the BCHSH inequality abbreviating the names of the authors; Bell, Clauser, Horne, Shimony and Holt. On the other hand, the quantum theory predicts that $$-2 \sqrt{2} \: \le \: {\langle}S {\rangle}\: \le \: 2 \sqrt{2}
\qquad
\mbox{: quantum theory}.
\label{quantum inequality}$$ If experiments yield the value of $ {\langle}S {\rangle}$ in the range $ -2 \sqrt{2} \le {\langle}S {\rangle}< -2 $ or in $ 2 < {\langle}S {\rangle}\le 2 \sqrt{2} $, the BCHSH inequality is violated and we can conclude that the hidden variable theory is wrong and the quantum theory is correct. After proposal of the BCHSH inequality, many experiments have been performed [@CH1978; @Aspect1981; @Aspect1982; @Sakai2006] and they revealed violation of the BCHSH inequality. So, there is no doubt of failure of the hidden variable theory.
However, the meaning of the quantity $ S = A_1 B_1 + A_1 B_2 + A_2 B_1 - A_2 B_2 $ seems obscure. A lot of generalizations of the BCHSH inequality have been proposed by other researchers [@Froissart1981; @Peres1999; @Collins2004] and a systematic method for generalization also has been given by Avis, Moriyama, and Owari [@Avis2009], who used methods of the operations research. But it is still desirable to construct Bell-like inequalities with understanding of the physical principle which enables the construction. In addition, it is noted that the BCHSH inequality is not a fair test for the hidden variable theory in a sense explained below.
We classify types of inequalities which examine validity of the hidden variable theory and the quantum theory. In general, for a physical quantity $ T $, each theory predicts that the expectation value of $ T $ falls in some range as $$\begin{aligned}
&& a \: \le \: {\langle}T {\rangle}\: \le \: b
\qquad
\mbox{: hidden variable theory},
\label{general BCHSH inequality}
\\
&& c \: \le \: {\langle}T {\rangle}\: \le \: d
\qquad
\mbox{: quantum theory}.
\label{general quantum inequality}\end{aligned}$$ Then by measuring the experimental value we can judge validity of the two theory. We classify tests into four types: $$\begin{aligned}
&& \mbox{type 1: } \,
c < a < b < d
\nonumber \\
&& \mbox{type 2: } \,
c < a < d < b, \quad \mbox{or} \quad a < c < b < d
\nonumber \\
&& \mbox{type 3: } \,
a < c < d < b
\nonumber \\
&& \mbox{type 4: } \,
c < d < a < b, \quad \mbox{or} \quad a < b < c < d.
\label{classification}\end{aligned}$$
According to this scheme, the BCHSH inequality belongs to the type 1, where the range of prediction of the hidden variable theory is included in the range of the quantum theory. So, in any experiment, it cannot happen that only the hidden variable theory is correct and the quantum theory is wrong. However, if we have a test of the type 2 with $ c < a < d < b $ and if we get the experimental value in $ d < {\langle}T {\rangle}\le b $, we should conclude that the hidden variable theory is correct and the quantum theory is wrong. On the other hand, the Kochen-Specker theorem [@Kochen1967] and the Greenberger-Horne-Shimony-Zeilinger test [@Mermin1990; @GHSZ1990; @Mermin1993] belong to the type 4, where the range of the hidden variable theory and the range of the quantum theory are completely disjoint. Of course, we do not expect that any experiments invalidate the quantum theory in the real world. But it is desirable for strengthening validity of the quantum theory to have a test which can reveal even failure of the quantum theory and success of the hidden variable theory. Passing such a severe test like type 2 or type 4, the quantum theory will become more persuasive and reliable.
In this paper we explain several mathematical reasons of violation of the BCHSH inequality. We also give a method for making systematic generalizations of the BCHSH inequality; this method is a main result of this paper. As a product of the main result, we invent a quantity $$\begin{aligned}
T
&=&
A_1 B_3
+ A_1 B_6
+ A_2 B_3
- A_2 B_6
\nonumber \\ &&
+ A_3 B_2
+ A_3 B_5
+ A_1 B_2
- A_1 B_5
\nonumber \\ &&
+ A_2 B_1
+ A_2 B_4
+ A_3 B_1
- A_3 B_4,
\label{T intro}\end{aligned}$$ where the observables $ A_i $ $(i=1,2,3) $ and $ B_j $ $(j=1,\cdots,6) $ take their values in $ \{ 1, -1 \} $ and $ A_i $ commutes with $ B_j $. We will show that the two theory predict the range of the expectation value as $$\begin{aligned}
&& -6 \; \le \; {\langle}T {\rangle}\; \le \; 6
\qquad \qquad
\mbox{: hidden variable theory},
\nonumber \\
&& -6 \sqrt{2} \; \le \; {\langle}T {\rangle}\; \le \; 2 \sqrt{2}
\quad
\mbox{: quantum theory}.\end{aligned}$$ Hence this set of inequalities belongs to the type 2, which offers a severer and fairer test for comparing the quantum theory and the hidden variable theory than the conventional BCHSH inequality.
Bell-Clauser-Horne-Shimony-Holt inequality
==========================================
In this section we present a brief review of the BCHSH inequality. The constituents of the BCHSH inequality are four observables $ A_1 $, $ A_2 $, $ B_1 $ and $ B_2 $. Each observable takes $ +1 $ or $ -1 $ as its value. It is assumed that $ A_i $ and $ B_j $ $ (i,j=1,2) $ are simultaneously measurable. However, $ A_1 $ and $ A_2 $ are not necessarily simultaneously measurable. $ B_1 $ and $ B_2 $ are not either. The quantity $ S $ is defined as $$\begin{aligned}
S
&=& A_1 B_1 + A_1 B_2 + A_2 B_1 - A_2 B_2
\nonumber \\
&=& A_1 ( B_1 + B_2 ) + A_2 ( B_1 - B_2 ).
\label{S}\end{aligned}$$
The above formulation is interpreted as follows. Suppose we have a pair of spin-half particles or a pair of photons. The two particles are labeled with $ A $ and $ B $, respectively. The observable $ A_i $ is interpreted as a spin component of the spin-half particle or a polarization of the photon. The index $ i =1,2 $ specifies the direction of the polarization detector. The observable $ B_j $ is interpreted as a spin component of the other particle or a polarization of the other photon. Two detectors acting on the two particles $ A $ and $ B $ are spatially separated, and hence, an event observed at one detector cannot make influence on an event observed at the other detector. This separation justifies defining the value of $ A_i B_j $ by a product of observed values of $ A_i $ and $ B_j $. So, by varying the directions of the detectors and by generating pairs of particles repeatedly, we accumulate data for the combined observables, $ (A_1, B_1) $, $ (A_1, B_2) $, $ (A_2, B_1) $ and $ (A_2, B_2) $. By making product of the measured values and by taking their average and adding them up, we get the average of $ S $, $$\begin{aligned}
{\langle}S {\rangle}&=&
{\langle}A_1 B_1 {\rangle}+ {\langle}A_1 B_2 {\rangle}+ {\langle}A_2 B_1 {\rangle}- {\langle}A_2 B_2 {\rangle}.
\label{average of S}\end{aligned}$$ The hidden variable theory and the quantum theory give different predictions on the range of $ {\langle}S {\rangle}$ as seen below.
In the context of the hidden variable theory, the values $ A_i ( \psi, \lambda ) $, $ B_j ( \psi, \lambda )
\in \{ +1, -1 \} $ are determined depending on the quantum state $ \psi $ and the hidden variable $ \lambda $ of the system. When $ B_1 + B_2 = \pm 1 \pm 1= \pm 2 $, we have $ B_1 - B_2 = 0 $. When $ B_1 + B_2 = \pm 1 \mp 1=0 $, we have $ B_1 - B_2 = \pm 2 $. So, one of $ (B_1+B_2) $ or $(B_1-B_2) $ is always 0 and the other is $ \pm 2 $. The coefficients $ A_1, A_2 $ are also $ +1 $ or $ -1 $. Hence, possible values of the quantity $ S = A_1 ( B_1 + B_2 ) + A_2 ( B_1 - B_2 ) $ are $ \pm 2 $. Since the probability distribution is assumed to be nonnegative and normalized, the average $$\begin{aligned}
{\langle}S {\rangle}&=&
\int
( A_1 B_1 + A_1 B_2 + A_2 B_1 - A_2 B_2 )
\, P ( \psi, \lambda ) \, d \lambda
\label{local calculation}\end{aligned}$$ is in $ -2 \le {\langle}S {\rangle}\le 2 $. This proves the BCHSH inequality (\[BCHSH inequality\]).
Let us turn to the quantum theory. In the quantum theory, $ A_i $ and $ B_j $ are not functions but operators or matrices. A typical choice for them is $$\begin{aligned}
A_1 &=& \sigma_z \otimes I,
\label{A_1}
\\
A_2 &=& (\sigma_z \cos 2 \theta + \sigma_x \sin 2 \theta) \otimes I,
\\
B_1 &=& I \otimes (\sigma_z \cos \theta + \sigma_x \sin \theta),
\\
B_2 &=& I \otimes (\sigma_z \cos \theta - \sigma_x \sin \theta).
\label{B_2}\end{aligned}$$ These are operators acting on the two-qubit Hilbert space $ {{\mathbb C}}^2 \otimes {{\mathbb C}}^2 $ and $ I $ is the two-dimensional identity matrix. The parameter $ \theta $ is adjustable for specifying directions of the detectors. By substituting the Pauli matrices, we get the matrix representation for $ S $, $$\begin{aligned}
S_\theta
& = & A_1 B_1 + A_1 B_2 + A_2 B_1 - A_2 B_2
\nonumber \\
& = &
2 \cos \theta
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\nonumber \\
&&
+ 2 \cos 2 \theta \sin \theta
\begin{pmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & -1 & 0
\end{pmatrix}
+ 2 \sin 2 \theta \sin \theta
\begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0
\end{pmatrix}.
\label{S theta}\end{aligned}$$ The eigenvalues of $ S_\theta $ are $$\begin{aligned}
&& \{ s_1 (\theta), s_2 (\theta), s_3 (\theta), s_4 (\theta) \}
\nonumber \\
&&
=
\Big\{
2 \cos 2 \theta, \,
-2 \cos 2 \theta, \,
2 \sqrt{1 + \sin^2 2 \theta}, \,
-2 \sqrt{1 + \sin^2 2 \theta} \,
\Big\}.\end{aligned}$$
For a general state we get an expectation value $ {\langle}S_\theta {\rangle}= \sum_{i=1}^4 w_i s_i (\theta) $, which is a convex combination of $ \{ s_i (\theta) \}_{i=1,\cdots,4} $. Then we introduce sets of expectation values as $$\begin{aligned}
Q_S ( \theta )
& :=&
\Big\{
\sum_{i=1}^4 w_i s_i (\theta) \, \Big| \,
w_i \in {{\mathbb R}}, \, 0 \le w_i \le 1, \, \sum_{i=1}^4 w_i = 1 \,
\Big\}
\nonumber \\
&=&
\Big[ -2 \sqrt{1 + \sin^2 2 \theta},
\; 2 \sqrt{1 + \sin^2 2 \theta} \, \Big] \subset {{\mathbb R}},
\\
Q_S
& := & \bigcup_{ 0 \le \theta \le 2 \pi} Q_S ( \theta )
= \big[ -2 \sqrt{2}, \; 2 \sqrt{2} \, \big]
\subset {{\mathbb R}},
\\
B_S
& := &
\big\{
( \theta, s ) \, \big| \,
0 \le \theta \le 2 \pi, \, s \in Q_S (\theta)
\big\}
\subset {{\mathbb R}}^2.
\label{B_S}\end{aligned}$$ Hence the range $ Q_S $ implies that $ -2 \sqrt{2} \le {\langle}S {\rangle}\le 2 \sqrt{2} $ in the quantum theory. This proves the inequality (\[quantum inequality\]).
We proved that the range of the expectation value allowed by the hidden variable theory is $$\begin{aligned}
H_S := [ -2, \, 2 \, ] \subset {{\mathbb R}}.\end{aligned}$$ Then it holds that $ H_S \subset Q_S (\theta) $ for any $ \theta $. So, the quantity $ S $ provides only the type 1 test. The band $ B_S $ formed by values allowed by the quantum theory is shown in the figure \[FIG2\] as the painted domain.
In particular, if we take the spin singlet state $$\begin{aligned}
\psi
&=&
\psi_1 + \psi_2
=
| \! \uparrow \downarrow {\rangle}+ | \! \downarrow \uparrow {\rangle}\nonumber \\
&=&
\frac{1}{\sqrt{2}}
\begin{pmatrix} 1 \\ 0 \end{pmatrix}
\otimes
\begin{pmatrix} 0 \\ 1 \end{pmatrix}
-
\frac{1}{\sqrt{2}}
\begin{pmatrix} 0 \\ 1 \end{pmatrix}
\otimes
\begin{pmatrix} 1 \\ 0 \end{pmatrix}
=
\frac{1}{\sqrt{2}}
\begin{pmatrix} 0 \\ 1 \\ -1 \\ 0 \end{pmatrix},
\label{singlet}\end{aligned}$$ the expectation value becomes $$\begin{aligned}
{\langle}\psi | S_\theta | \psi {\rangle}& = &
- 2 \cos \theta
- 2 \sin 2 \theta \sin \theta.
\label{singlet value S}\end{aligned}$$ At $ \theta = \pm \frac{\pi}{4} $, we get $ {\langle}\psi | S_\theta | \psi {\rangle}= -2 \sqrt{2} $. At $ \theta = \pm \frac{3}{4} \pi $, we get $ {\langle}\psi | S_\theta | \psi {\rangle}= 2 \sqrt{2} $. Thus the maximum violation of the BCHSH inequality is attained at these angles.
Here we explain implication of locality. In the context of the hidden variable theory, locality or nonlocality is formulated as follows. Locality requests that the probability distribution is independent of the directions of the detectors, which are placed far from each other and from the source of the particle pair. Namely, in the local theory we calculate the average with the formula $${\langle}A_i B_j {\rangle}= \int A_i ( \psi, \lambda ) B_j ( \psi, \lambda )
P ( \psi, \lambda ) \, d \lambda
\quad
\mbox{: local hidden variable theory}.
\label{local}$$ On the other hand, the nonlocal hidden variable theory allows that the probability distribution depends on the directions of the far separated detectors. Hence, the above formula is replaced by $${\langle}A_i B_j {\rangle}= \int A_i ( \psi, \lambda ) B_j ( \psi, \lambda )
P_{ij} ( \psi, \lambda ) \, d \lambda
\quad
\mbox{: nonlocal hidden variable theory}.
\label{nonlocal}$$ The calculation (\[local calculation\]) is based on the local hidden variable theory, not on the nonlocal theory. It is to be emphasized that the BCHSH inequality is proved in the context of the local hidden variable theory.
In the context of the quantum theory, we adopt the conventional interpretation which tells that locality implies commutativity of observables separated by a spacelike distance. The operators $ A_i \, (i=1,2) $ act on the Hilbert space of the particle $ A $ while the operators $ B_j \, (j=1,2) $ act on the Hilbert space of the particle $ B $. So, they are acting on different spaces, and hence they commute.
It should be mentioned that commutativity is not a necessary condition for locality. Deutsch [@Deutsch2004] constructed a model in which spacelike-separated observables do not commute and showed that no contradiction arises in his model. Hence, identification of locality and commutativity should not be accepted without question.
Why is the BCHSH inequality violated?
=====================================
In this section we give several explanations on violation of the BCHSH inequality. Here we consider in the context of the quantum theory.
One of the interpretations of the violation is interference effect. If the system is in a mixed state described by the density matrix $$\begin{aligned}
\rho
&=&
| \psi_1 {\rangle}{\langle}\psi_1 | +
| \psi_2 {\rangle}{\langle}\psi_2 |
\nonumber \\
&=&
\frac{1}{2}
\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}
\otimes
\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
+
\frac{1}{2}
\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
\otimes
\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}
=
\frac{1}{2}
\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix},\end{aligned}$$ the expectation value becomes $$\begin{aligned}
{\mbox{Tr} \,}( S_\theta \, \rho )
= {\langle}\psi_1 | S_\theta | \psi_1 {\rangle}+ {\langle}\psi_2 | S_\theta | \psi_2 {\rangle}=
- 2 \cos \theta\end{aligned}$$ and it stays in the range $ -2 \le {\mbox{Tr} \,}( S_\theta \, \rho ) \le 2 $ being consistent with the BCSHS inequality. Instead of the mixed state, if we substitute the pure state $ \rho = | \psi {\rangle}{\langle}\psi | $ with (\[singlet\]), the off-diagonal elements of $ S_\theta $ in (\[S theta\]) contribute to the expectation value as $$\begin{aligned}
{\langle}\psi | S_\theta | \psi {\rangle}&=&
{\langle}\psi_1 | S_\theta | \psi_1 {\rangle}+ {\langle}\psi_2 | S_\theta | \psi_2 {\rangle}+ {\langle}\psi_1 | S_\theta | \psi_2 {\rangle}+ {\langle}\psi_2 | S_\theta | \psi_1 {\rangle}\nonumber \\
&=&
- 2 \cos \theta
- 2 \sin 2 \theta \sin \theta.\end{aligned}$$ The cross terms $ {\langle}\psi_1 | S_\theta | \psi_2 {\rangle}+ {\langle}\psi_2 | S_\theta | \psi_1 {\rangle}$ represent interference effect, which causes the violation of the BCHSH inequality. The interference of the two terms in the superposed state $ \psi = | \!\! \uparrow \downarrow {\rangle}+ | \!\! \downarrow \uparrow {\rangle}$ is also called entanglement effect. This kind of explanation for the violation of the BCHSH inequality can be found in recent textbooks [@Shimizu2003].
Method for systematic construction of Bell-like inequalities
============================================================
Here we give another explanation for the violation of the BCHSH inequality, which gives a hint for generalizing the inequality. This explanation is based on noncommutativity of quantum observables. The eigenvalues of $ B_1 = \sigma_z \cos \theta + \sigma_x \sin \theta $ are $ \{ 1, -1 \} $. The eigenvalues of $ B_2 = \sigma_z \cos \theta - \sigma_x \sin \theta $ are $ \{ 1, -1 \} $, too. However, the eigenvalues of $ B_1 + B_2 $ are not $ \{ 2, 0, -2 \} $. Actually, the eigenvalues of $ B_1 + B_2 = \sigma_z \, 2 \cos \theta $ are $ \{ 2 \cos \theta, -2 \cos \theta \} $, which become $ \{ \sqrt{2}, - \sqrt{2} \} $ at $ \theta = \pm \frac{\pi}{4} $ or $ \pm \frac{3}{4} \pi $ particularly. This trick is written in the typical form as $$\begin{aligned}
\frac{1}{\sqrt{2}} ( \sigma_z + \sigma_x ) +
\frac{1}{\sqrt{2}} ( \sigma_z - \sigma_x ) =
\sqrt{2} \, \sigma_z.
\label{trick}\end{aligned}$$ While each term $ \frac{1}{\sqrt{2}} ( \sigma_z \pm \sigma_x ) $ in the left hand side has the spectrum $ \{ 1, -1 \} $, the right term $ \sqrt{2} \, \sigma_z $ has the spectrum $ \{ \sqrt{2}, - \sqrt{2} \} $. Symbolically, we can say that [*1 + 1 is not 2 but $ \sqrt{2} $ in the quantum theory.*]{} This example tells that [*the eigenvalues of a sum of noncommutative operators are not equal to the sum of eigenvalues of the respective operators in general*]{}. This is an elementary fact of linear algebra. The sum or the product of eigenvalues of operators $ B_1 $ and $ B_2 $ is equal to eigenvalues of $ B_1 + B_2 $ or $ B_1 B_2 $ only in their simultaneous eigenvector. Namely, the proposition $$\begin{aligned}
B_1 \phi_1 = b_1 \phi_1, \:
B_2 \phi_2 = b_2 \phi_2
\, \Rightarrow \,
( B_1 + B_2 ) \phi = ( b_1 + b_2 ) \phi, \:
( B_1 B_2 ) \phi = ( b_1 b_2 ) \phi
\quad
\label{naive}\end{aligned}$$ holds only when the state vectors $ \phi_1 $, $ \phi_2 $ and $ \phi $ belong to the common eigenspace of $ B_1 $ and $ B_2 $. On the other hand, if $ B_1 $ and $ B_2 $ are noncommutative, there is a state vector which is not decomposable into the common eigenspaces of $ B_1 $ and $ B_2 $. Then the naive calculation like (\[naive\]) does not hold.
The hidden variable theory assumes that the observables $ A_i $ and $ B_j $ have some values $ A_i (\lambda) $ and $ B_j (\lambda) $ at any time even when the observables are not measured[^2]. It also assumes that the values obey the ordinary arithmetic rule. The reasoning based on these assumptions leads to the BCHSH inequality (\[BCHSH inequality\]). But, in the quantum theory, values cannot be assigned to the noncommuting observables simultaneously and the naive arithmetic rule is not applicable to their values. Hence the BCHSH inequality is violated.
This kind of reasoning reveals the trick for making the BCHSH quantity $ S $. We begin with $$\begin{aligned}
S
=
\sqrt{2} \,
( \sigma_z \otimes \sigma_z
+ \sigma_x \otimes \sigma_x ).
\label{begin}\end{aligned}$$ Note that the spectra of $ \sigma_z \otimes \sigma_z $ and $ \sigma_x \otimes \sigma_x $ are both $ \{ 1, -1 \} $. Since $ \sigma_z \otimes \sigma_z $ and $ \sigma_x \otimes \sigma_x $ commute, the naive arithmetic rule is applicable to them and the spectrum of $ S $ should be a subset of $ \{ 2 \sqrt{2}, \, 0, \, -2 \sqrt{2} \} $; actually the spectrum of $ S $ is $ \{ 2 \sqrt{2}, \, -2 \sqrt{2} \} $. Then by applying the trick (\[trick\]) for rewriting $ S $ we get $$\begin{aligned}
S
&=&
\sqrt{2} \,
( \sigma_z \otimes \sigma_z
+ \sigma_x \otimes \sigma_x )
\nonumber \\
&=&
\sqrt{2} \Big[
\sigma_z \otimes \frac{1}{2} \{ ( \sigma_z + \sigma_x ) + ( \sigma_z - \sigma_x ) \}
+ \sigma_x \otimes \frac{1}{2} \{ ( \sigma_z + \sigma_x ) - ( \sigma_z - \sigma_x ) \}
\Big]
\nonumber \\
&=&
\sigma_z \otimes \frac{1}{\sqrt{2}} ( \sigma_z + \sigma_x )
+ \sigma_z \otimes \frac{1}{\sqrt{2}} ( \sigma_z - \sigma_x )
\nonumber \\ &&
+ \sigma_x \otimes \frac{1}{\sqrt{2}} ( \sigma_z + \sigma_x )
- \sigma_x \otimes \frac{1}{\sqrt{2}} ( \sigma_z - \sigma_x )
\nonumber \\
&=& A_1 B_1 + A_1 B_2 + A_2 B_1 - A_2 B_2,
\label{how to make it}\end{aligned}$$ which is the quantity maximally violating the BCHSH inequality. By introducing the adjustable parameter $ \theta $ we get the quantity $ S_\theta $ expressed in terms of the observables (\[A\_1\])-(\[B\_2\])
Actually, Bell [@Bell1966; @Mermin1993] himself noticed that values of noncommuting observables do not obey the naive additive law (\[naive\]) but he did not utilize this property to derive his inequality. Seevinck and Uffink [@Seevinck2007] argued that noncommutativity is related to violation of the original BCHSH inequality but they did not consider a method for generating variations of the BCHSH inequality.
New inequality
==============
Here we build a new quantity $ T $ which satisfies a new type of the Bell-like inequality. We begin with the quantity $$T = 2 \sqrt{2} \, (
\sigma_x \otimes \sigma_x +
\sigma_y \otimes \sigma_y +
\sigma_z \otimes \sigma_z ).
\label{starting T}$$ The three terms $ \sigma_a \otimes \sigma_a $ $ ( a = x,y,z ) $ are mutually commutative and their spectra are $ \{ 1, -1 \} $. Therefore, the spectrum of $ T $ should be a subset of $ \{ 6 \sqrt{2}, \, 2 \sqrt{2}, \, -2 \sqrt{2}, \, -6 \sqrt{2} \} $. The matrix representation of $ T $ is calculated as $$\begin{aligned}
\frac{1}{2 \sqrt{2}} \, T
&=&
\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
\otimes
\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
+
\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}
\otimes
\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}
+
\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
\otimes
\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0
\end{pmatrix}
+
\begin{pmatrix}
0 & 0 & 0 &-1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
-1& 0 & 0 & 0
\end{pmatrix}
+
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 &-1 & 0 & 0 \\
0 & 0 &-1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 &-1 & 2 & 0 \\
0 & 2 &-1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}.
\label{starting T matrix}\end{aligned}$$ It is easily seen that the eigenvalues of $ T $ are $ 2 \sqrt{2} $ (three-fold degeneracy) and $ -6 \sqrt{2} $ (no degeneracy); the corresponding eigenvectors are $$\begin{aligned}
2 \sqrt{2} : \:
a
\begin{pmatrix}
1 \\ 0 \\ 0 \\ 0
\end{pmatrix}
+b
\begin{pmatrix}
0 \\ 1 \\ 1 \\ 0
\end{pmatrix}
+c
\begin{pmatrix}
0 \\ 0 \\ 0 \\ 1
\end{pmatrix},
\qquad
-6 \sqrt{2} :
\begin{pmatrix}
0 \\ 1 \\ -1 \\ 0
\end{pmatrix}.\end{aligned}$$ In other words, the states with the eigenvalue $ T = 2 \sqrt{2} $ are the triplet spin state, while the state with the eigenvalue $ T = -6 \sqrt{2} $ is the singlet spin state. Hence the quantum theory predicts the bound $$-6 \sqrt{2} \; \le \; {\langle}T {\rangle}\; \le \; 2 \sqrt{2}
\qquad \mbox{: quantum theory}
\label{T quantum}$$ of the expectation value for any state.
Next, by applying the trick (\[trick\]) several times, we rewrite $ T $ as $$\begin{aligned}
T
&=&
2 \sqrt{2} \, (
\sigma_x \otimes \sigma_x +
\sigma_y \otimes \sigma_y +
\sigma_z \otimes \sigma_z )
\nonumber \\
&=&
\frac{2 \sqrt{2}}{4} \Big[
\sigma_x \otimes
\{ (\sigma_x + \sigma_y) + (\sigma_x - \sigma_y)
+ (\sigma_z + \sigma_x) - (\sigma_z - \sigma_x) \}
\nonumber \\ && \qquad
+ \sigma_y \otimes
\{ (\sigma_y + \sigma_z) + (\sigma_y - \sigma_z)
+ (\sigma_x + \sigma_y) - (\sigma_x - \sigma_y) \}
\nonumber \\ && \qquad
+ \sigma_z \otimes
\{ (\sigma_z + \sigma_x) + (\sigma_z - \sigma_x)
+ (\sigma_y + \sigma_z) - (\sigma_y - \sigma_z) \}
\Big].\end{aligned}$$ By introducing $$\begin{aligned}
&& A_1 = \sigma_x \otimes I, \qquad \qquad
A_2 = \sigma_y \otimes I, \qquad \qquad
A_3 = \sigma_z \otimes I,
\nonumber \\
&& B_1 = \frac{1}{\sqrt{2}} \, I \otimes (\sigma_y + \sigma_z), \quad
B_2 = \frac{1}{\sqrt{2}} \, I \otimes (\sigma_z + \sigma_x), \quad
B_3 = \frac{1}{\sqrt{2}} \, I \otimes (\sigma_x + \sigma_y),
\nonumber \\
&& B_4 = \frac{1}{\sqrt{2}} \, I \otimes (\sigma_y - \sigma_z), \quad
B_5 = \frac{1}{\sqrt{2}} \, I \otimes (\sigma_z - \sigma_x), \quad
B_6 = \frac{1}{\sqrt{2}} \, I \otimes (\sigma_x - \sigma_y), \quad\end{aligned}$$ we reach the expression $$\begin{aligned}
T
&=&
A_1 ( B_3 + B_6 + B_2 - B_5 ) +
A_2 ( B_1 + B_4 + B_3 - B_6 ) +
A_3 ( B_2 + B_5 + B_1 - B_4 )
\nonumber \\
&=&
A_1 ( B_3 + B_6 )
+ A_2 ( B_3 - B_6 )
\nonumber \\ &&
+ A_3 ( B_2 + B_5 )
+ A_1 ( B_2 - B_5 )
\nonumber \\ &&
+ A_2 ( B_1 + B_4 )
+ A_3 ( B_1 - B_4 ),
\label{T}\end{aligned}$$ which was shown at (\[T intro\]) in Introduction.
The hidden variable theory is applicable to $ T $ in this form. In the context of the hidden variable theory, $ A_i $ and $ B_j $ are functions of the hidden variable $ \lambda $ and the values of $ A_i ( \lambda ) $ and $ B_j ( \lambda ) $ are in $ \{ 1, -1 \} $. Then it is easily seen from the expression (\[T\]) that the possible values of $ T ( \lambda ) $ are in $ \{ 6, 2, -2, -6 \} $. Hence, the average $ {\langle}T {\rangle}= \int T ( \lambda ) P ( \lambda ) d \lambda $ is in the range $$-6 \; \le \; {\langle}T {\rangle}\; \le \; 6
\qquad \mbox{: hidden variable theory}.
\label{T hidden}$$ The range (\[T quantum\]) of the prediction of the quantum theory and the range (\[T hidden\]) of the hidden variable theory have the overlap $ [ -6, 2 \sqrt{2} ] $ where both the theories hold, and the region $ [ -6 \sqrt{2}, -6 ) \cup ( 2 \sqrt{2}, 6 ] $ where only one of the two theories holds. Thus, $ T $ is a quantity which realizes the type 2 test.
Moreover, we introduce a parameter $ \theta $ and define $ T_\theta $ as the polynomial (\[T\]) of $$\begin{aligned}
&& A_1 = ( \sigma_z \cos 2 \theta + \sigma_x \sin 2 \theta ) \otimes I,
\label{A1} \\
&& A_2 = ( \sigma_z \cos 2 \theta + \sigma_y \sin 2 \theta ) \otimes I, \\
&& A_3 = \sigma_z \otimes I, \\
&& B_1 = I \otimes ( \sigma_y \cos \theta + \sigma_z \sin \theta ), \\
&& B_2 = I \otimes ( \sigma_z \cos \theta + \sigma_x \sin \theta ), \\
&& B_3 = I \otimes ( \sigma_x \cos \theta + \sigma_y \sin \theta ), \\
&& B_4 = I \otimes ( \sigma_y \cos \theta - \sigma_z \sin \theta ), \\
&& B_5 = I \otimes ( \sigma_z \cos \theta - \sigma_x \sin \theta ), \\
&& B_6 = I \otimes ( \sigma_x \cos \theta - \sigma_y \sin \theta ).
\label{B6}\end{aligned}$$ The parameter $ \theta $ is interpreted as an angle which specifies the directions of the detectors. The matrix representation of $ T_\theta $ is $$T_\theta
=
2 (\cos \theta + \sin \theta )
\begin{pmatrix}
1 & (1-i) \cos 2 \theta & 0 & 0 \\
(1+i) \cos 2 \theta & -1 & 2 \sin 2 \theta & 0 \\
0 & 2 \sin 2 \theta & -1 & -(1-i) \cos 2 \theta \\
0 & 0 & -(1+i) \cos2 \theta & 1
\end{pmatrix}.$$
When $ \theta = \frac{\pi}{4} $, $ T_\theta $ is reduced to the original form (\[starting T matrix\]). The eigenvalues of $ T_\theta $ are $$\begin{aligned}
&& \{ t_1 (\theta),
t_2 (\theta),
t_3 (\theta),
t_4 (\theta) \}
\nonumber \\
&& =
\Big\{
2 ( \cos \theta + \sin \theta )
\big(
- \sin 2 \theta
\pm \sqrt{ \cos^2 2 \theta + 2 + 2 \sin 2 \theta}
\big),
\nonumber \\ && \quad \quad
2 ( \cos \theta + \sin \theta )
\big(
\sin 2\theta
\pm \sqrt{ \cos^2 2 \theta + 2 - 2 \sin 2 \theta}
\big)
\Big\}.\end{aligned}$$ The sets of values allowed by the quantum theory are denoted as $$\begin{aligned}
Q_T ( \theta )
& := &
\Big\{
\sum_{i=1}^4 w_i t_i (\theta) \, \Big| \,
w_i \in {{\mathbb R}}, \, 0 \le w_i \le 1, \, \sum_{i=1}^4 w_i = 1 \,
\Big\}
\subset {{\mathbb R}},
\\
Q_T
& := & \bigcup_{ 0 \le \theta \le 2 \pi} Q_T ( \theta )
= [ -6 \sqrt{2}, \; 6 \sqrt{2} \, ]
\: \subset {{\mathbb R}},
\\
B_T
& := & \{ ( \theta, t ) \, | \,
0 \le \theta \le 2 \pi, \, t \in Q_T (\theta) \}
\subset {{\mathbb R}}^2.
\label{B_T}\end{aligned}$$ And the range allowed by the hidden variable theory is denoted as $$\begin{aligned}
H_T := [ -6, \, 6 \, ] \: \subset {{\mathbb R}}.\end{aligned}$$ In the figure \[FIG3\] the band $ B_T $ is shown as the painted region. At any fixed value of $ \theta $, the predictions of the two theories have some overlap, namely, we have $ Q_T ( \theta ) \cap H_T \ne \varnothing $. It also happens that $ Q_T ( \theta ) \subset H_T $ for some $ \theta $. But it never happens that $ Q_T ( \theta ) \supset H_T $ at any $ \theta $. In this sense, the tests of type 2 and type 3 are realized. It is also to be noted that $ Q_T \supset H_T $, namely, the whole set of quantum predictions is wider than predictions of the hidden variable theory.
In particular, the singlet state $ \psi $ of (\[singlet\]) yields the expectation value $$\begin{aligned}
f ( \theta )
= {\langle}\psi | T_\theta | \psi {\rangle}= -2 ( \cos \theta + \sin \theta )
( 1 + 2 \sin 2 \theta ).
\label{singlet value for T}\end{aligned}$$ The range $ F
:= \{ f ( \theta ) \, | \, 0 \le \theta \le 2 \pi \}
= [ -6 \sqrt{2}, \, 6 \sqrt{2} ] $ covers $ H_T $ completely. This means that violation of the bound of the hidden variable theory is observed with the state $ \psi $.
If we take another entangled state $$\begin{aligned}
\chi
=
| \! \uparrow \uparrow {\rangle}+ | \! \downarrow \downarrow {\rangle}=
\frac{1}{\sqrt{2}}
\begin{pmatrix} 1 \\ 0 \end{pmatrix}
\otimes
\begin{pmatrix} 1 \\ 0 \end{pmatrix}
+
\frac{1}{\sqrt{2}}
\begin{pmatrix} 0 \\ 1 \end{pmatrix}
\otimes
\begin{pmatrix} 0 \\ 1 \end{pmatrix}
=
\frac{1}{\sqrt{2}}
\begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}
\label{entangled}\end{aligned}$$ instead of the singlet state $ \psi $, it yields $$\begin{aligned}
g ( \theta )
= {\langle}\chi | T_\theta | \chi {\rangle}= 2 ( \cos \theta + \sin \theta ).\end{aligned}$$ The range $ G
:= \{ g ( \theta ) \, | \, 0 \le \theta \le 2 \pi \}
= [ -2 \sqrt{2}, \, 2 \sqrt{2} ] $ is included in $ H_T $ completely. This means that violation of the hidden variable theory will be never observed in measurement of $ T $ with the state $ \chi $.
Discussions
===========
Here we give a summary of this study. In this paper we pointed out that violation of the BCHSH inequality can be understood as a result of noncommutativity of quantum observables. For noncommuting operators, an eigenvalue of a sum of operators does not coincide with a sum of eigenvalues of the respective operators. Using this property, we invented a method to build systematically Bell-like observables and showed that the conventional BCHSH inequality is reconstructed by this method. This diminished the ad hoc nature of the BCHSH observable.
We classified possible tests of the hidden variable theory and the quantum theory. We pointed out that there was no chance in the conventional BCHSH test to reveal invalidity of the quantum theory with validity of the hidden variable theory. By applying our method, we constructed the new observable $ T $ and calculated the range of its average in the contexts of the hidden variable theory and the quantum theory, respectively. It was shown that there is a chance that the new test with $ T $ reveals invalidity of the quantum theory with validity of the hidden variable theory.
Of course, we do not aim to deny validity of the quantum theory, but we aim to support it by passing the new severer test. It is also our purpose to clarify the implication of violation of various types of Bell-like inequalities.
There are several remaining problems concerning the generalized inequality. First one is the existence problem in a mathematical sense. The Bell-like inequality is a necessary condition for existence of the probability distribution which satisfies (\[local\]). For the case of the conventional BCHSH inequality, Fine [@Fine1982] proved that the set of inequalities $$\begin{aligned}
&& -2 \: \le \:
{\langle}A_1 B_1 {\rangle}+ {\langle}A_1 B_2 {\rangle}+ {\langle}A_2 B_1 {\rangle}- {\langle}A_2 B_2 {\rangle}\: \le \: 2,
\\
&& -2 \: \le \:
{\langle}A_1 B_2 {\rangle}+ {\langle}A_2 B_1 {\rangle}+ {\langle}A_2 B_2 {\rangle}- {\langle}A_1 B_1 {\rangle}\: \le \: 2,
\\
&& -2 \: \le \:
{\langle}A_2 B_1 {\rangle}+ {\langle}A_2 B_2 {\rangle}+ {\langle}A_1 B_1 {\rangle}- {\langle}A_1 B_2 {\rangle}\: \le \: 2,
\\
&& -2 \: \le \:
{\langle}A_2 B_2 {\rangle}+ {\langle}A_1 B_1 {\rangle}+ {\langle}A_1 B_2 {\rangle}- {\langle}A_2 B_1 {\rangle}\: \le \: 2\end{aligned}$$ is a necessary and sufficient condition for existence of the probability distribution of the hidden variable. Although there are studies on tightness of some variations of the BCHSH inequalities [@Garg1984; @Laskowski2004; @Pal2009], finding the necessary and sufficient condition for existence of the probability distribution of the hidden variable for our observable $ T $ is left as an open problem.
The second one is a practical problem. In principle, the test which we proposed can be implemented in experiment using pairs of photons or spin-half particles. But our choice involves nine observables (\[A1\])-(\[B6\]) with variable angle. Hence, our scheme is still too cumbersome for practical use. It is more desirable to reduce the number of observables to make experiments easier.
The third one is extensibility of our scheme. The proposed observable $ T $ is defined in the two-qubit Hilbert space. It is possible to extend our scheme to multi-qubit systems. It is also desirable to construct a Bell-like quantity with less number of observables.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to express our sincere thanks to Prof. T. Iwai and Prof. Y. Y. Yamaguchi for their valuable comments on our study. We thank Prof. Valerio Scarani, who taught us the work by A. Fine and the work by D. Collins and N. Gisin; both are related to tightness of the BCHSH inequality. The referee also gave us comments useful for improving our manuscript. We thank I. Tsutsui, T. Ichikawa, M. Ozawa, A. Hosoya, M. Kitano, H. Kobayashi, and S. Ogawa for their interests in our study and for their helpful comments. This work was supported by the Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science, Grant No. 22540410.
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[^1]: E-mail: tanimura@i.kyoto-u.ac.jp
[^2]: In the following, we do not write dependence on $ \psi $ of $ A_i (\psi, \lambda) $ and $ P (\psi, \lambda) $ explicitly.
|
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\#1>[\#1]{} \#1> [.16667em]{} &[..1em]{} \#1[[*\#1*]{}]{} \#1
\[section\] \[thm\][Proposition]{}
@b@ld \#1[@b@ld-.01em@b@ld-@b@ld .02em@b@ld-@b@ld-.012em.02em@b@ld]{}
addtoreset[thm]{}[section]{} addtoreset[equation]{}[section]{}
1 [**On solutions of the $q$-hypergeometric equation .3> with $q^{N}=1$** ]{} 2> [Yoshihiro Takeyama$^{\,\diamond}$]{} 1.5> [*Research Institute for Mathematical Sciences, Kyoto University, Kyoto 6068502, Japan*]{}
1.75>
[We consider the $q$-hypergeometric equation with $q^{N}=1$ and $\alpha, \beta, \gamma \in {\Bbb Z}$. We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic function. We prove that the subspace of solutions is two-dimensional over the field of quasi-constants. We get a basis for this space explicitly. In terms of this basis, we represent the $q$-hypergeometric function of the Barnes type constructed by Nishizawa and Ueno. Then we see that this function has logarithmic singularity at the origin. This is a difference between the $q$-hypergeometric functions with $0<|q|<1$ and at $|q|=1$. 1.4>]{} 0>
Introduction
============
Consider the $q$-hypergeometric equation { (1-D\_[q]{})(1-q\^[-1]{}D\_[q]{})-t(1-q\^D\_[q]{})(1-q\^D\_[q]{}) } (t)=0, \[uqhyp\] where $D_{q}$ is the $q$-difference operator defined by $(D_{q}\varphi)(t):=\varphi(qt)$. In this paper, we solve (\[uqhyp\]) with $q^{N}=1$ and $\alpha, \beta, \gamma \in {\Bbb Z}$ explicitly on a special space of functions and represent the $q$-hypergeometric function of the Barnes type at $|q|=1$ in terms of our solutions.
Let us recall some results about solutions to (\[uqhyp\]). In the case of $0<|q|<1$, one of the solutions to (\[uqhyp\]) is the basic hypergeometric function $\varphi(\alpha, \beta, \gamma ; t)$ [@GR] defined by (, , ; t):= \_[k=0]{}\^ t\^[k]{}, \[defbhyp\] where $(x)_{n}:=\prod_{j=0}^{n-1}(1-q^{j}x).$ We can get this solution by setting (t)=\_[k=0]{}\^c\_[k]{}t\^[k]{}, c\_[k]{} \[powerseries\] and solving a recursion relation for $\left\{ c_{k} \right\}$. In the case of $|q|=1$, we can not get solutions in this manner because the coefficient in (\[defbhyp\]) does not converge. However, some solutions are constructed in terms of a contour integral by Nishizawa and Ueno [@NU]. In this paper, we compare the solutions of these two types under the condition that $q^{N}=1$ and $\alpha, \beta, \gamma \in {\Bbb Z}$.
First we try to find solutions in a similar way to the case of $0<|q|<1$. Then we consider solutions of more general form than (\[powerseries\]) because of the following reason. In the limit as $q \to 1$, the equation (\[uqhyp\]) goes to the hypergeometric differential equation: { t(1-t)+(-(++1)t)-}F(t)=0. It is known that if $\gamma \not\in {\Bbb Z}$ there exist two independent solutions at $t=0$ of the form F(t)=\_[k=0]{}\^a\_[k]{}t\^[k+]{}. \[class\] However, if $\gamma \in {\Bbb Z}$ one of the two independent solutions is represented as F\_[1]{}(t)+F\_[2]{}(t) \[logclass\] where $F_{1}(t)$ and $F_{2}(t)$ are functions of the from (\[class\]) (see [@AAR], for example).
Now let us return to the equation (\[uqhyp\]) with $0<|q|<1$. When we consider solutions of the form (\[class\]), we get two independent solutions if $\gamma \not \in {\Bbb Z}$. One of them is the basic hypergeometric function (\[defbhyp\]) and the other is given by t\^[1-]{}(-+1, -+1, 2-; t) =\_[k=0]{}\^ t\^[k+1-]{}. Here we note that if $\gamma \in {\Bbb Z}$ one of these solutions does not make sense. Then we can construct another solution in the form (\[logclass\]) (see [@E] for details).
From the consideration above, we formulate our problem as follows. We set $t=q^{x}$ and rewrite (\[uqhyp\]) as a difference equation for a function of $x$, see (\[qhyp\]). We try to find solutions of the following form: \_[j=0]{}\^[n]{}\_[k=0]{}\^c\_[jk]{}x\^[j]{}q\^[kx]{}, c\_[jk]{} . \[first\] Here the part of $j>0$ corresponds to the term in (\[logclass\]) with logarithmic singularity. Now we note that the function $q^{Nx}$ is invariant under the shift $x \mapsto x+1$. Hence we can rewrite (\[first\]) as \_[j=0]{}\^[n]{}\_[k=0]{}\^[N-1]{}f\_[jk]{}(x)x\^[j]{}q\^[kx]{}, \[second\] where $f_{jk}(x)$ is a periodic function with a period 1, that is a quasi-constant for the difference equation (\[qhyp\]). We solve the $q$-hypergeometric equation with $q^{N}=1$ and $\alpha, \beta, \gamma \in {\Bbb Z}$ on the space of functions of the form (\[second\]).
The result is as follows (Theorem \[th2\]). The space of solutions is two-dimensional over the field of quasi-constants, and all the solutions are represented as (\[logclass\]), that is $n=0$ or $1$ in (\[second\]).
Next we consider solutions to (\[uqhyp\]) with $|q|=1$. One of the solutions is the $q$-hypergeometric function of the Barnes type at $|q|=1$, see (\[defhyp\]). We can deal with the integral representation of this function in the framework of the $q$-twisted cohomology at $|q|=1$ [@T]. It is shown that, if $q$ is not a root of unity and the parameters $\alpha, \beta$ and $\gamma$ are generic, we can construct two independent solutions to (\[uqhyp\]) in terms of the integral of the Barnes type by taking two independent homologies. However, in the case that $q^{N}=1$ and $\alpha, \beta, \gamma \in {\Bbb Z}$, the function (\[defhyp\]) is a unique solution of this form. In Theorem \[th3\], we write down an explicit formula for this function in terms of the basis constructed in Theorem \[th2\]. Then we see that if the parameters $\alpha, \beta$ and $\gamma $ satisfy some condition, the $q$-hypergeometric function at $|q|=1$ has logarithmic singularity. On the other hand, the $q$-hypergeometric function with $0<|q|<1$ defined by (\[defbhyp\]) has no logarithmic singularity. This is a difference between the cases of $0<|q|<1$ and $|q|=1$.
Acknoledgement {#acknoledgement .unnumbered}
==============
The author thanks Masaki Kashiwara and Tetsuji Miwa for valuable remarks.
A basis for the space of solutions
==================================
Let $N$ be a integer with $N \ge 2$. The $q$-hypergeometric difference equation is defined by L(x)=0, L:=(1-D)(1-q\^[-1]{}D)-q\^[x]{}(1-q\^D)(1-q\^D), \[qhyp\] where $D$ is the difference operator defined by $D\Psi(x):=\Psi(x+1)$. In this paper, we consider the equation (\[qhyp\]) with q:=e\^, , , { 1, , N}, . \[condition\] Note that the equation (\[qhyp\]) is symmetric with respect to $\alpha$ and $\beta$, and hence we can assume $\beta \le \alpha$ without loss of generality.
We denote by ${\cal C}$ the field of periodic meromorphic functions of $x$ with a period 1. This field is a space of quasi-constants for (\[qhyp\]) in the sense that if $\Psi$ is a solution to (\[qhyp\]) then $f\Psi$ is also a solution for any $f \in {\cal C}$.
Let us find a solution $\Psi$ of the following form: (x)=\_[j=0]{}\^[n]{}\_[k=0]{}\^[N-1]{} f\_[jk]{}x\^[j]{}q\^[kx]{}, f\_[jk]{} . \[assume\] It is easy to see the following.
\[unique\] The expression (\[assume\]) is unique, that is \_[j=0]{}\^[n]{}\_[k=0]{}\^[N-1]{} f\_[jk]{}x\^[j]{}q\^[kx]{}=0 f\_[jk]{}=0, where $f_{jk} \in {\cal C}$.
Let ${\cal S}$ be the space of functions of the form (\[assume\]): :={ \_[j=0]{}\^[n]{}\_[k=0]{}\^[N-1]{} f\_[jk]{}x\^[j]{}q\^[kx]{} | n 0, f\_[jk]{} }. Set :={ \_[k=0]{}\^[N-1]{} z\_[k]{}q\^[kx]{} | z\_[k]{} }. Note that $L{\cal P} \subset {\cal P}$ and $L{\cal S} \subset {\cal S}$ because $q^{Nx} \in {\cal C}$.
The following result holds.
\[th2\] The subspace of solutions to (\[qhyp\]) in ${\cal S}$ is two-dimensional over ${\cal C}$. A basis $\{\Psi_{1}, \Psi_{2}\}$ of this space is given as follows:
1\. $ \gamma \le \beta \le \alpha$ case. \_[1]{}(x)&=&\_[k=0]{}\^[N-]{}q\^[kx]{},\
\_[2]{}(x)&=&x\_[1]{}(x)+ \_[k=1]{}\^[N-]{} \_[j=1]{}\^[k]{}( - )q\^[kx]{}\
&& -(1-q\^[-]{}) \_[k=N-+1]{}\^[N-]{} q\^[kx]{}\
&& -(1-q\^[-1]{})\_[k=N-+1]{}\^[N-1]{} q\^[kx]{}. \[base1\]
2\. $ \beta < \gamma \le \alpha$ case. \_[1]{}(x)=\_[k=0]{}\^[N-]{}q\^[kx]{}, \_[2]{}(x)=\_[k=N-+1]{}\^[N-]{} q\^[kx]{}. \[base2\]
3\. $ \beta \le \alpha < \gamma$ case. \_[1]{}(x)&=&\_[k=N-+1]{}\^[N-]{} q\^[kx]{},\
\_[2]{}(x)&=&x\_[1]{}(x)- (1-q\^[N-+1]{})\_[k=0]{}\^[N-]{} q\^[kx]{}\
&& +\_[k=N-+2]{}\^[N-]{} \_[j=N-+2]{}\^[k]{}( - )q\^[kx]{}\
&& -(1-q\^[-]{}) \_[k=N-+1]{}\^[N-]{} q\^[kx]{}. \[base3\] Here we set a\_[k]{}:=(1-q\^[k]{})(1-q\^[-1+k]{}), b\_[k]{}:=(1-q\^[+k]{})(1-q\^[+k]{}).
Here we prove the theorem in the first case. The proof for the other case is similar.
Set =\_[j=0]{}\^[n]{}x\^[j]{}P\_[j]{}, P\_[j]{}:=\_[k=0]{}\^[N-1]{}f\_[jk]{}q\^[kx]{} . From Proposition \[unique\], we see that $L\Psi=0$ is equivalent to the following: LP\_[j]{}+\_[t=j+1]{}\^[n]{}( t )L\_[t-j]{}P\_[t]{}=0, (j=n, n-1, , 0), \[cond\] where L\_[k]{}:=2\^[k]{}(1-q\^[x]{})D\^[2]{}-((1+q\^[-1]{})-q\^[x]{}(q\^+q\^))D.
It is easy to solve (\[cond\]) for $j=n$ and $n-1$. The solution is given by P\_[n]{}=f\_[n]{}\_[1]{}, P\_[n-1]{}=f\_[n-1]{}\_[1]{}+nf\_[n]{}(\_[2]{}-x\_[1]{}), \[sol\] where $f_{n}, f_{n-1} \in {\cal C}$.
Now we prove that $n<2$. If $n \ge 2$, there is a solution $P_{n-2}$ to LP\_[n-2]{}+(n-1)(L\_[1]{}P\_[n-1]{}+L\_[2]{}P\_[n]{})=0, where $P_{n-1}$ and $P_{n}$ are given in (\[sol\]). Especially, we have Q\_[n]{}:=(n-1)(L\_[1]{}P\_[n-1]{}+L\_[2]{}P\_[n]{}) L[P]{}. \[mustrel\]
On the other hand, we can see that L[P]{} \_[k=0]{}\^[N-1]{} z\_[k]{} q\^[kx]{}, (z\_[k]{} ) {
[l]{} \_[j=N-+1]{}\^[N-]{} z\_[j]{}+z\_[N-+1]{}=0, (=1),\
\_[j=N-+1]{}\^[N-1]{} z\_[j]{}+q\^[-Nx]{}z\_[0]{}=0, (=1).
. \[contracond\] However, it can be checked that $Q_{n}$ does not satisfy (\[contracond\]) if $n>2$ and $f_{n}\not=0$. This contradicts (\[mustrel\]).
Hence $n=0$ or $1$, and $\{\Psi_{1}, \Psi_{2}\}$ is a basis.
The $q$-hypergeometric function at $q^{N}=1$
============================================
We recall the definition of the $q$-hypergeometric function at $|q|=1$.
Set $q=e^{2\pi i\omega}, \omega >0$. The $q$-hypergeometric function of the Barnes type is given as follows [@NU; @T]: (, , ; x):= \_[C]{}q\^[xz]{} dz. \[defhyp\] Here the function $\langle z \rangle$ is defined by z := S\_[2]{}(z | 1, ), where $S_{2}(z)$ is the double sine function. We refer the reader to [@JM] for the double sine function. The contour $C$ is the imaginary axis $(-i \infty, i\infty)$ except that the poles at -+[Z]{}\_[0]{}+\_[0]{}, -+[Z]{}\_[0]{}+\_[0]{} are on the left of $C$ and the poles at \_[0]{}+\_[0]{}, -+[Z]{}\_[1]{}+\_[1]{} are on the right of $C$. The integral (\[defhyp\]) is absolutely convergent if 0< [Re]{} x < 1++[Re]{} -[Re]{} -[Re]{} . \[conv\] Then the function $\Psi(x)$ satisfies the equation (\[qhyp\]) with $q=e^{2\pi i\omega}$.
Now we consider $\Psi(x)$ under the condition (\[condition\]). We also assume that +N-. \[condition2\] Then the integral (\[defhyp\]) converges if $0<{\rm Re}x<1$ because $\omega=1/N$ and (\[conv\]).
\[th3\] Under the conditions (\[condition\]) and (\[condition2\]), the $q$-hypergeometric function $\Psi$ satisfies $\Psi \in {\cal S}$ and is represented explicitly as follows:
1\. $\gamma \le \beta \le \alpha$ case. =\_[1]{}. \[rel1\]
2\. $\beta < \gamma \le \alpha$ case. = { \_[1]{}+ \_[2]{} }. \[rel2\]
3\. $\beta \le \alpha < \gamma$ case. && = ( C\_[, ]{}\^\_[1]{}-\_[2]{} ), \[rel3\]\
&& C\_[, ]{}\^:= 1-(\_[j=1]{}\^[N-+1]{}+\_[j=N-++1]{}\^[N-1]{}+\_[j=N-++1]{}\^[N-1]{}) . \[defC\] Here $\left\{\Psi_{1}, \Psi_{2}\right\}$ in (\[rel1\]), (\[rel2\]) and (\[rel3\]) is the basis of the space of solutions given in (\[base1\]), (\[base2\]) and (\[base3\]), respectively.
Let us calculate the integral in (\[defhyp\]). We denote by $\Phi(z)$ the integrand of (\[defhyp\]): (z):=q\^[xz]{} . By using = , we see the following under the condition (\[condition\]): (z+N)=q\^[Nx]{}(z). Hence, we have (1-q\^[Nx]{})\_[C]{}(z)dz= ( \_[C]{}-\_[C+N]{} )(z)dz \[shift\] From (\[condition\]), we can take the line $-\frac{1}{2}+i{\Bbb R}$ as the contour $C$. Then the right hand side of (\[shift\]) is given by the sum of residues at $z=0, \cdots , N-1$. Therefore, we get (x)= \_[k=0]{}\^[N-1]{}[res]{}\_[z=k]{}(z).
By using = , we can represent the function $\Phi(z)$ in terms of $q^{z}$ and calculate residues of this function explicitly.
It is easy to see that all the poles at $z=0, \cdots N-1$ are simple if $\beta \le \alpha$ and $\gamma \le \alpha$. Then we find the formulae (\[rel1\]) and (\[rel2\]) easily by using (1-q)(1-q\^[2]{})(1-q\^[N-1]{})=N. \[basicrel\]
Let us consider the case of $\beta \le \alpha < \gamma$. The poles at $z=0, \cdots , N-\gamma$ and $z=N-\alpha+1, \cdots , \beta$ are simple, and it is easy to calculate residues at these poles. The result is as follows: \_[z=k]{}= - (1-q\^[N-+1]{}) , (k=0, , N-) and \_[z=k]{}&=& -\
&& (1-q\^[-]{}) , (k=N-+1, , ). Here we used (\[basicrel\]).
Next we calculate residues at $z=N-\gamma+1, \cdots , N-\alpha$. Note that these points are double poles. For $k=N-\gamma+1, \cdots , N-\alpha$, we have && [res]{}\_[z=k]{}(z)=\
&& =q\^[kx]{}[res]{}\_[z=0]{} ( \_[j=1]{}\^[k]{} \_[j=+k]{}\^[N-1]{} \_[j=1]{}\^[k+-1-N]{} \_[j=+k]{}\^[N-1]{}). \[res\] By substituting && q\^[zx]{}=1+xz+o(z),\
&& = {
[l]{} -z\^[-1]{}++o(1), (j0 [mod]{} N),\
+z+o(z), (j 0 [mod]{} N),
z 0, . we find && (\[res\])=q\^[kx]{}[res]{}\_[z=0]{} ( ()\^[2]{}D\_[k]{}z\^[-2]{}+ D\_[k]{}(x-E\_[k]{})z\^[-1]{}+o(1) )\
&& =D\_[k]{}(x-E\_[k]{})q\^[kx]{}, where && D\_[k]{}= \_[j=1]{}\^[k]{} \_[j=+k]{}\^[N-1]{} \_[j=1]{}\^[k+-1-N]{} \_[j=+k]{}\^[N-1]{},\
&& E\_[k]{}=1- ( \_[j=1]{}\^[k]{}+\_[j=+k]{}\^[N-1]{}+\_[j=1]{}\^[k+-1-N]{}+\_[j=+k]{}\^[N-1]{} ) . By using (\[basicrel\]), we see D\_[k]{}= . Moreover, we have && E\_[N-+1]{}=C\_[, ]{}\^,\
&& E\_[k]{}-E\_[k-1]{}=-, where $C_{\alpha, \beta}^{\gamma}$ is given by (\[defC\]). Hence we find E\_[k]{}=C\_[, ]{}\^+ \_[j=N-+2]{}\^[k]{}( - ).
From this calculation, we get the relation (\[rel3\]).
Theorem \[th3\] implies that the $q$-hypergeometric function of the Barnes type has logarithmic singularity in the case that $q^{N}=1$ and $\beta \le \alpha < \gamma$.
[II]{} .05
Andrews, G. E., Askey, R. and Roy, R&, , Encyclopedia of Mathematics and its Applications 71. Cambridge Univ. Press., 1999.
Exton, H&, , Ellis Horwood Limited., 1983.
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Takeyama, Y&, [*The $q$-twisted cohomology and the $q$-hypergeometric function at $|q|=1$*]{}, Publ. RIMS, Kyoto Univ., [**37**]{} (2001), 71-89.
|
---
abstract: |
A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, *optimal* $1$-planar graphs, i.e. those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized, this has not been the case for $k \geq 2$. For $k=2,3$ and $4$, upper bounds on the edge density have been developed for the case of simple graphs by Pach and Tóth, Pach et al. and Ackerman, which have been used to improve the well-known “Crossing Lemma”. Recently, we proved that these bounds also apply to non-simple $2$- and $3$-planar graphs without homotopic parallel edges and self-loops.
In this paper, we completely characterize optimal $2$- and $3$-planar graphs, i.e., those that achieve the aforementioned upper bounds. We prove that they have a remarkably simple regular structure, although they might be non-simple. The new characterization allows us to develop notable insights concerning new inclusion relationships with other graph classes.
author:
- |
Michael A. Bekos$^1$, Michael Kaufmann$^1$, Chrysanthi N. Raftopoulou$^2$\
\
$^1$Wilhelm-Schickhard-Institut für Informatik, Universität Tübingen, Germany\
`{bekos,mk}@informatik.uni-tuebingen.de`\
$^2$School of Applied Mathematical & Physical Sciences, NTUA, Greece\
`crisraft@mail.ntua.gr`
bibliography:
- 'references.bib'
title: 'On Optimal 2- and 3-Planar Graphs'
---
Introduction {#sec:introduction}
============
Topological graphs, i.e. graphs that usually come with a representation of the edges as Jordan arcs between corresponding vertex points in the plane, form a well-established subject in the field of geometric graph theory. Besides the classical problems on crossing numbers and crossing configurations [@DBLP:journals/dcg/AlonE89; @DBLP:journals/dcg/LovaszPS97; @DBLP:journals/jgt/Turan77], the well-known ”Crossing Lemma” [@ACNS82; @Lei83] stands out as a prominent result. Researchers on graph drawing have followed a slightly different research direction, based on extensions of planar graphs that allow crossings in some restricted local configurations [@DBLP:journals/tcs/BinucciGDMPST15; @DBLP:journals/algorithmica/CheongHKK15; @DBLP:journals/tcs/DidimoEL11; @DBLP:journals/cj/GiacomoDLMW15; @KU14]. The main focus has been on *1-planar graphs*, where each edge can be crossed at most once, with early results dating back to Ringel [@Ringel65] and Bodendiek et al. [@BSW84]. Extensive work on generation [@DBLP:journals/siamdm/Suzuki10], characterization [@DBLP:conf/cocoon/HongELP12], recognition [@DBLP:journals/corr/Brandenburg16a], coloring [@Borodin95], page number [@BB0R15], etc. has led to a very good understanding of structural properties of 1-planar graphs.
Pach and Tóth [@PachT97], Pach et al. [@PachRTT06] and Ackerman [@DBLP:journals/corr/Ackerman15] bridged the two research directions by considering the more general class of *$k$-planar graphs*, where each edge is allowed to be crossed at most $k$ times. In particular, Pach and Tóth provided significant progress, as they developed techniques for upper bounds on the number of edges of simple $k$-planar graphs, which subsequently led to upper bounds of $5n -10$ [@PachT97], $5.5n - 11$ [@PachRTT06] and $6n-12$ [@DBLP:journals/corr/Ackerman15] for simple $2$-, $3$- and $4$-planar graphs, respectively. An interesting consequence was the improvement of the leading constant in the ”Crossing Lemma”. Note that for general $k$, the current best bound on the number of edges is $4.1 \sqrt k n$ [@PachT97].
Recently, we generalized the result and the bound of Pach et al. [@PachRTT06] to non-simple graphs, where non-homotopic parallel edges as well as non-homotopic self-loops are allowed [@BKR16]. Note that this non-simplicity extension is quite natural and not new, as for planar graphs, the density bound of $3n-6$ still holds for such non-simple graphs.
In this paper, we now completely characterize optimal non-simple $2$- and $3$-planar graphs, i.e. those that achieve the bounds of $5n-10$ and $5.5n-11$ on the number of edges, respectively; refer to Theorems \[thm:2-characterization\] and \[thm:3-characterization\]. In particular, we prove that the commonly known $2$-planar graphs achieving the upper bound of $5n-10$ edges, are in fact, the only optimal $2$-planar graphs. Such graphs consist of a crossing-free subgraph where all not necessarily simple faces have size $5$. At each face there are $5$ more edges crossing in its interior. We correspondingly show that the optimal $3$-planar graphs have a similar simple and regular structure where each planar face has size $6$ and contains $8$ additional crossing edges.
The remainder of this paper is structured as follows: In Section \[sec:preliminaries\] we introduce preliminary notions and notation. In Section \[sec:properties\] we present several structural properties of optimal $2$- and $3$-planar graphs that we use in Sections \[sec:2planar\] and \[sec:3planar\] in order to give their characterizations. We conclude in Section \[sec:discussion\] with further notable insights and research directions.
Preliminaries {#sec:preliminaries}
=============
Let $G$ be a (not necessarily simple) *topological graph*, i.e. $G$ is a graph drawn on the plane, so that the vertices of $G$ are distinct points in the plane, its edges are Jordan curves joining the corresponding pairs of points, and:
no edge passes through a vertex different from its endpoints,
no edge crosses itself and
no two edges meet tangentially.
Let $\Gamma(G)$ be such a drawing of $G$. The *crossing graph* $\mathcal{X}(G)$ of $G$ has a vertex for each edge of $G$ and two vertices of $\mathcal{X}(G)$ are connected by an edge if and only if the corresponding edges of $G$ cross in $\Gamma(G)$. A connected component of $\mathcal{X}(G)$ is called *crossing component*. Note that the set of crossing components of $\mathcal{X}(G)$ defines a partition of the edges of $G$. For an edge $e$ of $G$ we denote by $\mathcal{X}(e)$ the crossing component of $\mathcal{X}(G)$ which contains $e$.
An edge $e$ in $\Gamma(G)$ is called a *topological edge* (or simply *edge*, if this is clear in the context). Edge $e$ is called *true-planar*, if it is not crossed by any other edge in $\Gamma(G)$. The set of all true-planar edges of $\Gamma(G)$ forms the so-called *true-planar skeleton* of $\Gamma(G)$, which we denote by $\Pi(G)$. Since $G$ is not necessarily simple, we will assume that $\Gamma(G)$ contains neither *homotopic parallel edges* nor *homotopic self-loops*, that is, both the interior and the exterior regions defined by any self-loop or by any pair of parallel edges contain at least one vertex. For a positive integer $s$, a cycle of length $s$ is called *true-planar $s$-cycle* if it consists of true-planar edges of $\Gamma(G)$. If $e$ is a true-planar edge, then $\mathcal{X}(e)=\{e\}$, while for a chord $e$ of a true-planar $s$-cycle that has no vertices in its interior, it follows that all edges of $\mathcal{X}(e)$ are also chords of this $s$-cycle. Let $\mathcal{F}_s=\{v_1,v_2,\ldots,v_s\}$ be a facial $s$-cycle of $\Pi(G)$ with length $s \geq 3$. The order of the vertices (and subsequently the order of the edges) of $\mathcal{F}_s$ is determined by a walk around the boundary of $\mathcal{F}_s$ in clockwise direction. Since $\mathcal{F}_s$ is not necessarily simple, a vertex or an edge may appear more than once in this order; see Figure \[fig:non\_simple\_face\]. More in general, a *region* in $\Gamma(G)$ is defined as a closed walk along non-intersecting segments of Jordan curves that are adjacent either at vertices or at crossing points of $\Gamma(G)$. The *interior* and the *exterior* of a connected region are defined as the topological regions to the right and to the left of the walk.
![ (a) A non-simple face $\{v_1,\ldots,v_7\}$, where $v_6$ is identified with $v_4$. Different configurations used in (b–d) Lemma \[lem:crossing\_twice\], and (e–f) Lemma \[lem:crossing\_adjacent\].[]{data-label="fig:2_planar_polygon_conf"}](images/pre_non_simple_face "fig:"){width="\textwidth"} \[fig:non\_simple\_face\]
![ (a) A non-simple face $\{v_1,\ldots,v_7\}$, where $v_6$ is identified with $v_4$. Different configurations used in (b–d) Lemma \[lem:crossing\_twice\], and (e–f) Lemma \[lem:crossing\_adjacent\].[]{data-label="fig:2_planar_polygon_conf"}](images/pre_cross_twice "fig:"){width="\textwidth"} \[fig:crossing\_twice\_reverse\]
![ (a) A non-simple face $\{v_1,\ldots,v_7\}$, where $v_6$ is identified with $v_4$. Different configurations used in (b–d) Lemma \[lem:crossing\_twice\], and (e–f) Lemma \[lem:crossing\_adjacent\].[]{data-label="fig:2_planar_polygon_conf"}](images/pre_cross_twice "fig:"){width="\textwidth"} \[fig:crossing\_twice\]
![ (a) A non-simple face $\{v_1,\ldots,v_7\}$, where $v_6$ is identified with $v_4$. Different configurations used in (b–d) Lemma \[lem:crossing\_twice\], and (e–f) Lemma \[lem:crossing\_adjacent\].[]{data-label="fig:2_planar_polygon_conf"}](images/pre_cross_twice "fig:"){width="\textwidth"} \[fig:crossing\_twice\_2\]
![ (a) A non-simple face $\{v_1,\ldots,v_7\}$, where $v_6$ is identified with $v_4$. Different configurations used in (b–d) Lemma \[lem:crossing\_twice\], and (e–f) Lemma \[lem:crossing\_adjacent\].[]{data-label="fig:2_planar_polygon_conf"}](images/pre_cross_adjacent "fig:"){width="\textwidth"} \[fig:crossing\_adjacent\_2\]
![ (a) A non-simple face $\{v_1,\ldots,v_7\}$, where $v_6$ is identified with $v_4$. Different configurations used in (b–d) Lemma \[lem:crossing\_twice\], and (e–f) Lemma \[lem:crossing\_adjacent\].[]{data-label="fig:2_planar_polygon_conf"}](images/pre_cross_adjacent "fig:"){width="\textwidth"} \[fig:crossing\_adjacent\]
Drawing $\Gamma(G)$ is called *$k$-planar* if every edge in $\Gamma(G)$ is crossed at most $k$ times. Accordingly, a graph is called *$k$-planar* if it admits a $k$-planar drawing. An *optimal $k$-planar* graph is a $k$-planar graph with the maximum number of edges. In particular, we consider optimal $2$- and $3$-planar graphs achieving the best-known upper bounds of $5n-10$ and $5.5n-11$ edges. For an optimal $k$-planar graph $G$ on $n$ vertices, a $k$-planar drawing $\Gamma(G)$ of $G$ is called *planar-maximal crossing-minimal* or simply PMCM-drawing, if and only if $\Gamma(G)$ has the maximum number of true-planar edges among all $k$-planar drawings of $G$ and, subject to this restriction, $\Gamma(G)$ has also the minimum number of crossings.
Consider two edges $(u,v)$ and $(u',v')$ that cross at least twice in $\Gamma(G)$. Let $c$ and $c'$ be two crossing points of $(u,v)$ and $(u',v')$ that appear consecutively along $(u,v)$ in this order from $u$ to $v$ (i.e., there is no other crossing point of $(u,v)$ and $(u',v')$ between $c$ and $c'$). W.l.o.g. we can assume that $c$ and $c'$ appear in this order along $(u',v')$ from $u'$ to $v'$ as well. In Figures \[fig:crossing\_twice\_reverse\] and \[fig:crossing\_twice\] we have drawn two possible crossing configurations. First we drew edge $(u,v)$ as an arc with $u$ above $v$ and the edge-segment of $(u',v')$ between $u$ and $c$ to the right of $(u,v)$. The edge-segment of $(u',v')$ between $c$ and $c'$, starts at $c$ and ends at $c'$ either from the right (Figure \[fig:crossing\_twice\_reverse\]) or from the left (Figure \[fig:crossing\_twice\]) of $(u,v)$, yielding the two different crossing configurations.
For $k \in \{2,3\}$, let $\Gamma(G)$ be a PMCM-drawing of an optimal $k$-planar graph $G$ in which two edges $(u,v)$ and $(u',v')$ cross more than once. Let $c$ and $c'$ be two consecutive crossings of $(u,v)$ and $(u',v')$ along $(u,v)$, and let $R_{c,c'}$ be the region defined by the walk along the edge segment of $(u,v)$ from $c$ to $c'$ and the one of $(u',v')$ from $c'$ to $c$. Then, $R_{c,c'}$ has at least one vertex in its interior and one in its exterior. \[lem:crossing\_twice\]
Consider first the crossing configuration of Figure \[fig:crossing\_twice\_reverse\]. Since $c$ and $c'$ are consecutive along $(u,v)$ and $(u',v')$ does not cross itself, vertex $u'$ lies in the exterior of $R_{c,c'}$, while vertex $v'$ in the interior of $R_{c,c'}$. Hence, the lemma holds. Consider now the crossing configuration of Figure \[fig:crossing\_twice\]. Since $c$ and $c'$ are consecutive along $(u,v)$, vertices $u'$ and $v'$ are in the exterior of $R_{c,c'}$. Assume now, to the contrary, that $R_{c,c'}$ contains no vertices in its interior. W.l.o.g. we further assume that $(u,v)$ and $(u',v')$ is a *minimal crossing pair* in the sense that, $R_{c,c'}$ cannot contain another region $R_{p,p'}$ defined by any other pair of edges that cross twice; for a counterexample see Figure \[fig:crossing\_twice\_2\]. Let $nc(u,v)$ and $nc(u',v')$ be the number of crossings along $(u,v)$ and $(u',v')$ that are between $c$ and $c'$, respectively (red in Figure \[fig:crossing\_twice\]). Observe that by the “minimality” criterion of $(u,v)$ and $(u',v')$ we have $nc(u,v) = nc(u',v')$. We redraw edges $(u,v)$ and $(u',v')$ by exchanging their segments between $c$ and $c'$ and eliminate both crossings $c$ and $c'$ without affecting the $k$-planarity of $G$; see the dotted edges of Figure \[fig:crossing\_twice\]. This contradicts the crossing minimality of $\Gamma(G)$.
For $k \in \{2,3\}$, let $\Gamma(G)$ be a PMCM-drawing of an optimal $k$-planar graph $G$ in which two edges $(u,v)$ and $(u,v')$ incident to a common vertex $u$ cross. Let $c$ be the first crossing of them starting from $u$ and let $R_{c}$ be the region defined by the walk along the edge segment of $(u,v)$ from $u$ to $c$ and the one of $(u,v')$ from $c$ to $u$. Then, $R_{c}$ has at least one vertex in its interior and one in its exterior. \[lem:crossing\_adjacent\]
Since $c$ is the first crossing point of $(u,v)$ and $(u,v')$ along $(u,v)$ from $u$, vertex $v'$ is not in the interior of $R_{c}$. If $u \neq v'$, then $v'$ is indeed in the exterior of $R_{c}$. Otherwise, if $u=v'$ and there is no other vertex in the exterior of $R_{c}$, then $(u,v')$ is a homotopic self-loop; a contradiction. Assume now, to the contrary, that $R_{c}$ contains no vertices in its interior. W.l.o.g. we further assume that $(u,v)$ and $(u,v')$ is a *minimal crossing pair* in the sense that, $R_{c}$ cannot include another region $R_{p}$ defined any other pair of crossing edges incident to a common vertex; for an example see Figure \[fig:crossing\_adjacent\_2\]. Denote by $nc(u,v)$ and $nc(u,v')$ the number of crossings along $(u,v)$ and $(u,v')$ that are between $u$ and $c$, respectively (red drawn in Figure \[fig:crossing\_adjacent\]). First assume that $nc(u,v) = nc(u,v')$. We proceed by eliminating crossing $c$ without affecting the $k$-planarity of $G$; see the dotted-drawn edges of Figure \[fig:crossing\_adjacent\]. This contradicts the crossing minimality of $\Gamma(G)$. It remains to consider the case where $nc(u,v) \neq nc(u,v')$. Assume w.l.o.g. that $nc(u,v) > nc(u,v')$. By the “minimality”assumption there is an edge $(u'',v'')$ that crosses at least twice edge $(u,v)$. By Lemma \[lem:crossing\_twice\], $R_{c}$ is not an empty region; a contradiction.
In our proofs by contradiction we usually deploy a strategy in which starting from an optimal $2$- or $3$-planar graph $G$, we modify $G$ and its drawing $\Gamma(G)$ by adding and removing elements (vertices or edges) without affecting its $2$- or $3$-planarity. Then, the number of edges in the derived graph forces $G$ to have either fewer or more edges than the ones required by optimality (contradicting the optimality or the $3$-planarity of $G$, resp.). To deploy the strategy, we must ensure that we do not introduce homotopic parallel edges or self-loops, and that we do not violate basic properties of $\Gamma(G)$ (e.g., introduce a self-crossing edge). We next show how to select and draw the newly inserted elements.
A Jordan curve $[u,v]$ connecting vertex $u$ to $v$ of $G$ is called a *[potential edge]{}* in drawing $\Gamma(G)$ if and only if $[u,v]$ does not cross itself and is not a homotopic self-loop in $\Gamma(G)$, that is, either $u \neq v$ or $u=v$ and there is at least one vertex in the interior and the exterior of $[u,v]$. Note that $u$ and $v$ are not necessarily adjacent in $G$. However, since each topological edge $(u,v) \in E$ of $G$ is represented by a Jordan curve in $\Gamma(G)$, it follows that edge $(u,v)$ is by definition a [potential edge]{}of $\Gamma(G)$ among other [potential edges]{}that possibly exist. Furthermore, we say that vertices $v_1,v_2,\dots,v_s$ define a *[potential empty cycle]{}* $\mathcal{C}_s$ in $\Gamma(G)$, if there exist [potential edges]{}$[v_i,v_{i+1}]$, for $i=1,\dots, s-1$ and [potential edge]{}$[v_1,v_s]$ of $\Gamma(G)$, which
do not cross with each other and
the walk along the curves between $v_1,v_2,\dots,v_s,v_1$ defines a region in $\Gamma(G)$ that has no vertices in its interior.
Note that $\mathcal{C}_s$ is not necessarily simple.
For $k \in \{2,3\}$, let $\Gamma(G)$ be a PMCM-drawing of a $k$-planar graph $G$. Let also $\mathcal{C}_s$ be a [potential empty cycle]{}of length $s$ in $\Gamma(G)$ and assume that $\kappa$ edges of $\Gamma(G)$ are drawn completely in the interior of $\mathcal{C}_s$, while $\lambda$ edges of $\Gamma(G)$ are crossing[^1] the boundary of $\mathcal{C}_s$. Also, assume that if one focuses on $\mathcal{C}_s$ of $\Gamma(G)$, then $\mu$ pairwise non-homotopic edges can be drawn as chords completely in the interior of $\mathcal{C}_s$ without deviating $k$-planarity.
1. \[prp:nonoptim\] If $\mu > \kappa + \lambda$, then $G$ is not optimal.
2. \[prp:boundary\] If $G$ is optimal and $\mu = \kappa + \lambda$, then all boundary edges of $\mathcal{C}_s$ exist[^2] in $\Gamma(G)$.
\[lem:exchange\]
(\[prp:nonoptim\]) If we could replace the $\kappa + \lambda$ edges of $\Gamma(G)$ that are either drawn completely in the interior of $\mathcal{C}_s$ or cross the boundary of $\mathcal{C}_s$ with the $\mu$ ones that one can draw exclusively in the interior of $\mathcal{C}_s$, then the lemma would trivially follow. However, to do so we need to ensure that this operation introduces neither homotopic parallel edges nor homotopic self-loops. Since the edges that we introduce are [potential edges]{}, it follows that no homotopic self-loops are introduced. We claim that homotopic parallel edges are not introduced either. In fact, if $e$ and $e'$ are two homotopic parallel edges, then both must be drawn completely in the interior of $\mathcal{C}_s$, which implies that $e$ and $e'$ are both newly-introduced edges; a contradiction, since we introduce $\mu$ pairwise non-homotopic edges. (\[prp:boundary\]) In the exchanging scheme that we just described, we drew $\mu$ edges as chords exclusively in the interior of $\mathcal{C}_s$. Of course, one can also draw the boundary edges of $\mathcal{C}_s$, as long as they do not already exist in $\Gamma(G)$. Since $G$ is optimal, these edges must exist in $\Gamma(G)$.
![ (a–c) A [potential empty cycle]{}$\mathcal{C}_s$ with (a) $s=5$ and five chords with two crossings each, (b) $s=6$ and six chords with at most two crossings each, and (c) $s=6$ and eight chords with at most three crossings each. (d) Configuration used in the proof of Property \[prp:2planar\_triangle\].[]{data-label="fig:chord_conf"}](images/pre_chords "fig:"){width="\textwidth"} \[fig:2\_planar\_5gon\]
![ (a–c) A [potential empty cycle]{}$\mathcal{C}_s$ with (a) $s=5$ and five chords with two crossings each, (b) $s=6$ and six chords with at most two crossings each, and (c) $s=6$ and eight chords with at most three crossings each. (d) Configuration used in the proof of Property \[prp:2planar\_triangle\].[]{data-label="fig:chord_conf"}](images/pre_chords "fig:"){width="\textwidth"} \[fig:2\_planar\_6gon\]
![ (a–c) A [potential empty cycle]{}$\mathcal{C}_s$ with (a) $s=5$ and five chords with two crossings each, (b) $s=6$ and six chords with at most two crossings each, and (c) $s=6$ and eight chords with at most three crossings each. (d) Configuration used in the proof of Property \[prp:2planar\_triangle\].[]{data-label="fig:chord_conf"}](images/pre_chords "fig:"){width="\textwidth"} \[fig:3\_planar\_6gon\]
![ (a–c) A [potential empty cycle]{}$\mathcal{C}_s$ with (a) $s=5$ and five chords with two crossings each, (b) $s=6$ and six chords with at most two crossings each, and (c) $s=6$ and eight chords with at most three crossings each. (d) Configuration used in the proof of Property \[prp:2planar\_triangle\].[]{data-label="fig:chord_conf"}](images/2planar_triangle "fig:"){width="\textwidth"} \[fig:2\_planar\_triangle\]
Note that in Lemma \[lem:exchange\] the $\kappa$ edges that are drawn completely in the interior of the [potential empty cycle]{}$\mathcal{C}_s$ and the $\lambda$ edges that cross its boundary, are the only edges that have at least one edge-segment within $\mathcal{C}_s$. This means that we can compute $\kappa+\lambda$ by counting the edges that have at least one edge-segment within $\mathcal{C}_s$. In the following sections, there will be some standard cases where we apply Lemma \[lem:exchange\]. In most of them, a [potential empty cycle]{}$\mathcal{C}_s$ on five or six vertices is involved, that is, $5 \leq s \leq 6$. If $s=5$, then one can draw five chords in the interior of $\mathcal{C}_s$ without affecting its $2$- or $3$-planarity; see Figure \[fig:2\_planar\_5gon\]. If $s=6$, then one can draw either six or eight chords in the interior of $\mathcal{C}_s$ without affecting its $2$- or $3$-planarity, respectively; see Figures \[fig:2\_planar\_6gon\] and \[fig:3\_planar\_6gon\].
Properties of optimal 2- and 3-planar graphs {#sec:properties}
============================================
In this section, we investigate properties of optimal $2$- and $3$-planar graphs.We prove that a PMCM-drawing $\Gamma(G)$ of an optimal $2$- or $3$-planar graph $G$ can contain neither true-planar cycles of a certain length nor a pair of edges that cross twice. We use these properties to show that $\Gamma(G)$ is *quasi-planar*, i.e. it contains no $3$ pairwise crossing edges. First, we give the following definition. Let $R$ be a simple closed region that contains at least one vertex of $G$ in its interior and one in its exterior. Let $H_1$ ($H_2$) be the subgraph of $G$ whose vertices and edges are drawn entirely in the interior (exterior) of $R$. Note that $H_1$ ($H_2$) is not necessarily an induced subgraph of $G$, since there could be edges that exit and enter $R$. We refer to $H_1$ and $H_2$ as the *compact subgraphs* of $\Gamma(G)$ defined by $R$. The following lemma, used in the proofs for several properties of optimal $2$- and $3$-planar graphs, bounds the number of edges in any compact subgraph of $\Gamma(G)$.
Let $\Gamma(G)$ be a drawing of an optimal $2$- or $3$-planar graph $G$ and let $H$ be a compact subgraph of $\Gamma(G)$ on $n'$ vertices that is defined by a closed region $R$. If $n'\geq 2$, $H$ has at most $5n'-6$ edges if $G$ is optimal $2$-planar, and at most $5.5n' - 6.5$ edges if $G$ is optimal $3$-planar. Furthermore, there exists at least one edge of $G$ crossing the boundary of $R$ in $\Gamma(G)$. \[prp:connected\]
We prove this property for the class of $3$-planar graphs; the proof for the class of $2$-planar graphs is analogous. So, let $\Gamma(G)$ be a drawing of an optimal $3$-planar graph $G=(V,E)$ with $n$ vertices and $m$ edges. Let $H_1$ and $H_2$ be two compact subgraphs of $\Gamma(G)$ defined by a closed region $R$. For $i=1,2$ let $n_i$ and $m_i$ be the number of vertices and edges of $H_i$. Suppose that $n_1\geq 2$. In the absence of $\Gamma(H_2)$, drawing $\Gamma(H_1)$ might contain homotopic parallel edges or self-loops. To overcome this problem, we subdivide an edge-segment of the unbounded region of $\Gamma(H_1)$ by adding one vertex.[^3] The derived graph, say $H'_1$, has $n'_1=n_1+1$ vertices and $m'_1=m_1+1$ edges. Since $H'_1$ has no homotopic parallel edges or self-loops and $n'_1 \geq 3$, it follows that $m'_1 \leq 5.5n'_1-11$, which gives $m_1\leq 5.5n_1-6.5$.
For the second part, assume for the sake of contradiction that no edge of $G$ crosses the boundary of $R$. This implies that $m=m_1+m_2$. We consider first the case where $n_1,n_2 \geq 2$. By the above we have that $m_1\leq 5.5n_1-6.5$and $m_2 \leq 5.5n_2 - 6.5$. Since $n=n_1+n_2$ and $m=m_1+m_2$, it follows that $m \leq 5.5n -13$; a contradiction to the optimality of $G$. Since a graph consisting only of two non-adjacent vertices cannot be optimal, it remains to consider the case where either $n_1=1$ or $n_2=1$. W.l.o.g. assume that $n_1=1$. Since $n_2 \geq 2$, it follows that $m_2 \leq 5.5n_2 - 6.5$, which implies $m \leq 5.5n - 12$; a contradiction to the optimality of $G$.
For two compact subgraphs $H_1$ and $H_2$ defined by a closed region $R$, Property \[prp:connected\] implies that the drawings of $H_1$ and $H_2$ cannot be “separable”. In other words, either there exists an edge connecting a vertex of $H_1$ with a vertex of $H_2$, or there exists a pair of edges, one connecting vertices of $H_1$ and the other vertices of $H_2$, that cross in the drawing $\Gamma(G)$.
In a PMCM-drawing $\Gamma(G)$ of an optimal $2$-planar graph $G$ there is no empty true-planar cycle of length three. \[prp:2planar\_triangle\]
Assume to the contrary that there exists an empty true-planar $3$-cycle $\mathcal{C}$ in $\Gamma(G)$ on vertices $u$, $v$ and $w$. Since $G$ is connected and since all edges of $\mathcal{C}$ are true-planar, there is neither a vertex nor an edge-segment in $\mathcal{C}$, i.e., $\mathcal{C}$ is a chordless facial cycle of $\Pi(G)$. This allows us to add a vertex $x$ in its interior and connect $x$ to vertex $u$ by a true-planar edge. Now vertices $u$, $x$, $u$, $w$ and $v$ define a [potential empty cycle]{}of length five, and we can draw five chords in its interior without violating $2$-planarity and without introducing homotopic parallel edges or self-loops; refer to Figure \[fig:2\_planar\_triangle\]. The derived graph $G'$ has one more vertex than $G$ and six more edges. Hence, if $n$ and $m$ are the number of vertices and edges of $G$ respectively, then $G'$ has $n'=n+1$ vertices and $m'=m+6$ edges. Then $m'=5n'-9$, which implies that $G'$ has more edges than allowed; a contradiction.
The number of vertices of an optimal $3$-planar graph $G$ is even. \[prp:3planar\_even\_order\]
Follows directly from the density bound of $5.5n-11$ of $G$.
A PMCM-drawing $\Gamma(G)$ of an optimal $3$-planar graph $G$ has no true-planar cycle of odd length. \[prp:3planar\_odd\_cycle\]
Let $s \geq 1$ be an odd number and assume to the contrary that there exists a true-planar $s$-cycle $\mathcal{C}$ in $\Gamma(G)$. Denote by $G_1$ ($G_2$, respectively) the subgraph of $G$ induced by the vertices of $\mathcal{C}$ and the vertices of $G$ that are in the interior (exterior, respectively) of $\mathcal{C}$ in $\Gamma(G)$ without the chords of $\mathcal{C}$ that are in the exterior (interior, respectively) of $\mathcal{C}$ in $\Gamma(G)$. For $i=1,2$, observe that $G_i$ contains a copy of $\mathcal{C}$. Let $n_i$ and $m_i$ be the number of vertices and edges of $G_i$ that do not belong to $\mathcal{C}$. Based on graph $G_i$, we construct graph $G_i'$ by employing two copies of $G_i$ that share cycle $\mathcal{C}$. Observe that $G_i'$ is $3$-planar, because one copy of $G_i$ can be embedded in the interior of $\mathcal{C}$, while the other one in its exterior. Hence, in this embedding, there exist neither homotopic self-loops nor homotopic parallel edges. Let $n_i'$ and $m_i'$ be the number of vertices and edges of $G_i'$ that do not belong to $\mathcal{C}$. If $G$ has $n$ vertices and $m$ edges, then by construction the following equalities hold:
\[il:1\] $n_i'= 2n_i+s$,
\[il:2\] $m_i'= 2m_i+s$,
\[il:3\] $n = n_1+n_2+s$, and
\[il:4\] $m = m_1+m_2+s$.
We now claim that $n_i' \geq 3$. When $s \geq 3$ the claim clearly holds. Otherwise (i.e., $s=1$), cycle $\mathcal{C}$ is degenerated to a self-loop which must contain at least one vertex in its interior and its exterior. Hence, the claim follows. Property \[prp:3planar\_even\_order\] in conjunction with Eq.(\[il:1\]) implies that $G_i'$ is not optimal, that is, $m_i'<5.5n_i'-11$. Hence, by Eq.(\[il:2\]) it follows that $2m_i+s < 5.5(2n_i+s)-11$. Summing up over $i$, we obtain that $2(m_1+m_2+s) < 5.5 (2n_1+2n_2+2s)-22$. Finally, from Eq.(\[il:3\]) and Eq.(\[il:4\]) we conclude that $m<5.5n-11$; a contradiction to the optimality of $G$.
In a PMCM-drawing $\Gamma(G)$ of an optimal $2$-planar graph $G$ there is no pair of edges that cross twice with each other. \[prp:2\_planar\_cross\_twice\]
Assume to the contrary that $(u,u')$ and $(v,v')$ cross twice in $\Gamma(G)$ at points $c$ and $c'$. By $2$-planarity no other edge of $\Gamma(G)$ crosses $(u,u')$ and $(v,v')$. Let $R_{c,c'}$ be the region defined by the walk along the edge segment of $(u,u')$ between $c$ and $c'$ and the edge segment of $(v,v')$ between $c'$ and $c$. As mentioned in the proof of Lemma \[lem:crossing\_twice\], there exist two crossing configurations for $(u,u')$ and $(v,v')$; see Figures \[fig:crossing\_twice\_reverse\] and \[fig:crossing\_twice\]. In the crossing configuration of Figure \[fig:crossing\_twice\_reverse\], vertices $v$ and $v'$ are in the interior of $R_{c,c'}$, while vertices $u$ and $u'$ in its exterior. Hence, $u\neq v$ and $u'\neq v'$ hold. We redraw $(u,u')$ and $(v,v')$ by exchanging the middle segments between $c$ and $c'$ and eliminate both crossings $c$ and $c'$ without affecting $2$-planarity; see the dotted edges of Figure \[fig:crossing\_twice\_reverse\]. Note that since $u\neq v$ and $u'\neq v'$ the two edges cannot be homotopic self-loops. Also, no homotopic parallel edges are introduced, since this would imply that at least one of the two edges already exists in $\Gamma(G)$ violating $2$-planarity. Now consider the crossing configuration of Figure \[fig:crossing\_twice\]. By Lemma \[lem:crossing\_twice\], $R_{c,c'}$ has at least one vertex in its interior. By $2$-planarity we have that no edge of $G$ crosses the boundary of $R_{c,c'}$; a contradiction to Property \[prp:connected\].
In a PMCM-drawing $\Gamma(G)$ of an optimal $3$-planar graph $G$ there is no pair of edges that cross more than once with each other. \[prp:3\_planar\_cross\_twice\]
We have already noted that a pair of edges cannot cross more than twice in $\Gamma(G)$. Assume to the contrary that two edges $(u,v)$ and $(u',v')$ of $G$ cross (exactly) twice in $\Gamma(G)$. Figures \[fig:3\_planar\_cross\_twice\_general\_reverse\] and \[fig:3\_planar\_cross\_twice\_general\] illustrate the two possible different crossing configurations. Let $c$ and $c'$ be their crossing points. By Lemma \[lem:crossing\_twice\] it follows that the region $R_{c,c'}$ that is defined by the walk along the the edge segment of $(u,v)$ between $c$ and $c'$ and the edge segment of $(u',v')$ between $c'$ and $c$ has at least one vertex in its interior. Let $G_{c,c'}$ be the subgraph of $G$ that is drawn completely in the interior of $R_{c,c'}$ in $\Gamma(G)$. By $3$-planarity, there exist at most two edges $e$ and $e'$ that cross $(u,v)$ and $(u',v')$ respectively.
In both crossing configurations we proceed to define two Jordan curves $[u,u']_1$ and $[u,u']_2$ in $\Gamma(G)$ with endpoints $u$ and $u'$, so that their union contains only in its interior the vertices of $G_{c,c'}$; see Figures \[fig:3\_planar\_cross\_twice\_general\_reverse\] and \[fig:3\_planar\_cross\_twice\_general\]. Curve $[u,u']_1$ emanates from vertex $u$, follows edge $(u,v)$ up to point $c$ and ends at vertex $u'$ by following edge $(u',v')$. Curve $[u,u']_2$ emanates from vertex $u'$, follows edge $(u',v')$ up to point $c$, follows edge $(u,v)$ up to point $c'$, follows edge $(u',v')$ up to point $c$ and ends at vertex $u$ by following edge $(u,v)$.
We now claim that both curves $[u,u']_1$ and $[u,u']_2$ are [potential edges]{}. By definition, our claim holds when $u \neq u'$. Assume now that $u=u'$. Let $R_{c}$ be the region defined by the walk along the edge-segment of $(u',v')$ from $u'$ to $c$ and the edge-segment of $(u,v)$ from $c$ to $u$ (where $u=u'$). By Lemma \[lem:crossing\_adjacent\] $R_{c}$ has at least one vertex in its interior and at least one vertex in its exterior. This implies that the first of our curves, i.e. $[u,u']_1$, which encloses region $R_c$ is a [potential edge]{}.
Now, assume to the contrary that $[u,u']_2$ is not a [potential edge]{}. Then $u=u'$. Let $R_{c'}$ be the region defined by the walk along the edge-segment of $(u',v')$ from $u'$ to $c$, the edge-segment of $(u,v)$ from $c$ to $c'$, the edge-segment of $(u',v')$ from $c'$ to $c$ and the edge-segment of $(u,v)$ from $c$ to $u$ (where $u=u'$). Since $G_{c,c'}$ lies in the interior of $R_{c'}$ and $[u,u']_2$ is not a [potential edge]{}, region $R_{c'}$ has no vertices in its exterior; refer to Figure \[fig:3\_planar\_cross\_twice\_special\_reverse\]. Note that in Figure \[fig:3\_planar\_cross\_twice\_special\] we illustrate the same case assuming $u=u'=v=v'$. By Property \[prp:3planar\_odd\_cycle\] [potential edge]{}$[u,u']_1$ must be crossed (as otherwise it is a true-planar self-loop in $\Gamma(G)$). This implies that there exists at least one edge that crosses $[u,u']_1$. This edge must also cross $(u,v)$ or $(u',v')$ and is therefore either edge $e$ or edge $e'$. Suppose w.l.o.g. that $[u,u']_1$ is edge $e$; see Figure \[fig:3\_planar\_cross\_twice\_special\_reverse\]. Let $p$ be the crossing point of $e$ and $(u,v)$. Now edge $(u,v)$ has exactly three crossings. We redraw $(u,v)$ and $(u',v')$ by exchanging their edge-segments between their common endpoint $u$ and their first crossing $c$, so as to eliminate $c$. Let $[u,v]$ and $[u',v']$ be the new curves in $\Gamma(G)$. Since $G$ is crossing minimal, it follows that at least one of $[u,v]$ or $[u',v']$ must be homotopic parallel to an existing edge in $\Gamma(G)$. Since $(u,v)$ has already three crossings in $\Gamma(G)$, [potential edge]{}$[u',v']$ cannot exist in $\Gamma(G)$, as otherwise it would introduce a fourth crossing on $(u,v)$. Hence, [potential edge]{}$[u,v]$ must exist in $\Gamma(G)$ and this is edge $e'$. Now we focus on edge $e$. Edge $e$ has an endpoint in the interior of $R_c$ and crosses $[u,u']_2$. However, since $R_{c'}$ has no vertices in its exterior, and edges $(u,v)$ and $(u',v')$ have already three crossings, edge $e$ must end at vertex $u=u'$. In this case, edges $e$ and $(u,v)$ have $u$ as a common endpoint and cross at point $p$. Hence, region $R_p$ defined by the walk along the edge segment of $(u,v)$ from $u$ to $p$ and the edge segment of $e$ from $p$ to $u$ contains at least one vertex in its interior. However, $R_p$ is contained in the exterior of $R_{c'}$, and therefore there exists at least one vertex in the exterior of $R_{c'}$, which is a contradiction. Hence, $[u,u']_2$ is a [potential edge]{}.
![ Configurations used in Property \[prp:3\_planar\_cross\_twice\]. \[fig:3\_planar\_one\_crossing\_2\]](images/prop_2planar_cross_twice "fig:"){width="\textwidth"} \[fig:3\_planar\_cross\_twice\_general\_reverse\]
![ Configurations used in Property \[prp:3\_planar\_cross\_twice\]. \[fig:3\_planar\_one\_crossing\_2\]](images/prop_3planar_cross_twice "fig:"){width="\textwidth"} \[fig:3\_planar\_cross\_twice\_general\]
![ Configurations used in Property \[prp:3\_planar\_cross\_twice\]. \[fig:3\_planar\_one\_crossing\_2\]](images/prop_2planar_cross_twice "fig:"){width="\textwidth"} \[fig:3\_planar\_cross\_twice\_special\_reverse\]
![ Configurations used in Property \[prp:3\_planar\_cross\_twice\]. \[fig:3\_planar\_one\_crossing\_2\]](images/prop_3planar_cross_twice "fig:"){width="\textwidth"} \[fig:3\_planar\_cross\_twice\_special\]
We proceed by removing from $\Gamma(G)$ all vertices and edges of $G_{c,c'}$, edges $e$, $(u,v)$, $(u',v')$ as well as the edge that crosses $(u',v')$, if any. Then, the cycle formed by [potential edges]{}$[u,u']_1$ and $[u,u']_2$ becomes empty and this allows us to follow an approach similar to the one described in the proof of Lemma \[lem:exchange\]. More precisely, we add in the interior of this [potential empty cycle]{}two vertices $x$ and $y$, such that $u$, $x$ and $y$ form a path (in this order) that is completely drawn in its interior. The union of this path with $[u,u']_1$ and $[u,u']_2$ defines in the derived drawing a new (non-simple) [potential empty cycle]{}of length six. In its interior one can embed $8$ additional edges as in Figure \[fig:3\_planar\_6gon\]. Summarizing, if $G_{c,c'}$ has $n_{c,c'}$ vertices and $m_{c,c'}$ edges, we removed from $G$ exactly $n_{c,c'}$ vertices and at most $m_{c,c'}+4$ edges and this allowed us to introduce two new vertices and $10$ edges without affecting $3$-planarity. Let $G'$ be the derived $3$-planar graph. The fact that $G'$ contains neither homotopic parallel edges nor homotopic self-loops can be argued as in the proof of Lemma \[lem:exchange\].(\[prp:nonoptim\]). If $G$ has $n$ vertices and $m$ edges, then $G'$ has $n' = n -n_{c,c'}+2$ vertices and $m'$ edges, where $m' \geq m -m_{c,c'}+ 6$ edges. We distinguish two cases depending on whether $G_{c,c'}$ has one or more vertices. If $n_{c,c'}=1$, then $m_{c,c'}=0$. Also, $G'$ has exactly one more vertex than $G$. Since $G$ is optimal, by Property \[prp:3planar\_even\_order\] it follows that $G'$ cannot be optimal. Hence, $m' < 5.5n' - 11$, which implies that $m < 5.5n - 11.5$; a contradiction to the density of $G$. On the other hand if $n_{c,c'}\geq 2$, by Property \[prp:connected\] we have that $m_{c,c'}\leq 5.5n_{c,c'}-6.5$, as $G_{c,c'}$ is a compact subgraph of $\Gamma(G)$ defined by $R_{c,c'}$. This gives $m'\geq 5.5n'-9.5$, that is $G'$ has more edges than allowed; a clear contradiction.
Now assume that $\Gamma(G)$ contains three mutually crossing edges $(u,v)$, $(u',v')$ and $(u'',v'')$. In Figures \[fig:quasi\_1\]–\[fig:quasi\_4\] we have drawn four possible crossing configurations. First, we drew $(u,v)$ and $(u',v')$ w.l.o.g. as vertical and horizontal line-segments that cross at point $c$. Then, we placed vertex $u''$ and drew the first segment of its edge crossing w.l.o.g. the edge-segment of $(u',v')$ between $u'$ and $c$ at point $c'$ from above. So the middle segment of $(u'',v'')$ starts at $c'$ and has to end at edge $(u,v)$, either from left or right, and either in the lower or in the upper segment. This gives rise to the four configurations demonstrated in Figures \[fig:quasi\_1\]–\[fig:quasi\_4\], which we examine in more details in the following. Note that the endpoints of the three edges are not necessarily distinct (e.g., in Figure \[fig:quasi\_1\_1\] we illustrate the case where $u=u''$ and $v'=v''$ for the crossing configuration of Figure \[fig:quasi\_1\]). For each crossing configuration, one can draw curves connecting the endpoints of $(u,v)$, $(u',v')$ and $(u'',v'')$ (red colored in Figures \[fig:quasi\_1\]–\[fig:quasi\_4\]), which define a region that has no vertices in its interior. This region fully surrounds $(u,v)$ and $(u',v')$ and the two segments of $(u'',v'')$ that are incident to vertices $u''$ and $v''$.
![ Crossing configurations for three mutually crossing edges. Potential edges are drawn solid red. Jordan curves that can either be potential edges or homotopic self-loops are drawn dotted red.[]{data-label="fig:prp_quasi"}](images/prop_quasi_planar "fig:"){width="\textwidth"} \[fig:quasi\_1\]
![ Crossing configurations for three mutually crossing edges. Potential edges are drawn solid red. Jordan curves that can either be potential edges or homotopic self-loops are drawn dotted red.[]{data-label="fig:prp_quasi"}](images/prop_quasi_planar "fig:"){width="\textwidth"} \[fig:quasi\_2\]
![ Crossing configurations for three mutually crossing edges. Potential edges are drawn solid red. Jordan curves that can either be potential edges or homotopic self-loops are drawn dotted red.[]{data-label="fig:prp_quasi"}](images/prop_quasi_planar "fig:"){width="\textwidth"} \[fig:quasi\_3\]
![ Crossing configurations for three mutually crossing edges. Potential edges are drawn solid red. Jordan curves that can either be potential edges or homotopic self-loops are drawn dotted red.[]{data-label="fig:prp_quasi"}](images/prop_quasi_planar "fig:"){width="\textwidth"} \[fig:quasi\_4\]
![ Crossing configurations for three mutually crossing edges. Potential edges are drawn solid red. Jordan curves that can either be potential edges or homotopic self-loops are drawn dotted red.[]{data-label="fig:prp_quasi"}](images/prop_quasi_planar "fig:"){width="\textwidth"} \[fig:quasi\_1\_1\]
Each of the crossing configurations of Figures \[fig:quasi\_2\]-\[fig:quasi\_4\] induces at least $5$ [potential edges]{}. \[clm:1\]
Observe that all solid-drawn red curves of Figures \[fig:quasi\_2\]–\[fig:quasi\_4\] are indeed [potential edges]{}: If for example $[u',u'']$ of Figure \[fig:quasi\_2\] is not a [potential edge]{}, then $u'=u''$ and $[u',u'']$ is a self-loop with no vertex either in its interior or in its exterior; a contradiction to Lemma \[lem:crossing\_adjacent\].
The crossing configuration of Figure \[fig:quasi\_1\] induces at least four [potential edges]{}. \[clm:2\]
As in Claim \[clm:1\] we can prove that $[u',u'']$, $[u,v']$, $[u',v]$ and $[v,v'']$ are [potential edges]{}.
The configuration of Figure \[fig:quasi\_1\] induces a [potential empty cycle]{} $\mathcal{C}$ of length $\geq 4$. Each of the configurations of Figures \[fig:quasi\_2\]–\[fig:quasi\_4\] induces a [potential empty cycle]{} $\mathcal{C}$ of length $\geq 5$. \[crl:crl\]
In the case where the crossing configuration of Figure \[fig:quasi\_1\] induces exactly four [potential edges]{}, there exists at least one vertex in the interior of region $\mathcal{T}$ defined by the walk along the edge segment of $(u,v)$ between $c$ and $c''$, the edge segment of $(u'',v'')$ between $c''$ and $c'$ and the edge segment of $(u',v')$ between $c'$ and $c$. \[clm:3\]
By Claim \[clm:2\], $[u,u'']$, and $[v',v'']$ must be homotopic self-loops; see Figure \[fig:quasi\_1\_1\]. In this case, edges $(u,v)$ and $(u'',v'')$ are incident to a common vertex, namely $u=u''$ and cross. By Lemma \[lem:crossing\_adjacent\] region $R_{c''}$ (red-shaded in Figure \[fig:quasi\_1\_1\]) has at least one vertex in its interior. Since $R_{c''}$ is the union of the interior of $\mathcal{T}$ and the homotopic self-loop $[u,u'']$, $\mathcal{T}$ contains at least one vertex in its interior.
A PMCM-drawing $\Gamma(G)$ of an optimal $2$-planar graph $G$ is quasi-planar. \[prp:2\_planar\_quasi\]
Assume to the contrary that there exist three mutually crossing edges $(u,v)$, $(u',v')$ and $(u'',v'')$ in $\Gamma(G)$; see Figure \[fig:prp\_quasi\]. By Corollary \[crl:crl\], there is a [potential empty cycle]{}$\mathcal{C}$ of length at least $4$. By $2$-planarity, there is no other edge crossing $(u,v)$, $(u',v')$ or $(u'',v'')$. Hence, the only edges that are drawn in the interior of $\mathcal{C}$ are $(u,v)$ and $(u',v')$, while $(u'',v'')$ is the only edge that crosses the boundary of $\mathcal{C}$.
First, consider the case where $\mathcal{C}$ is of length $\geq 5$. Since we can draw at least five chords completely in the interior of $\mathcal{C}$ as in Figure \[fig:2\_planar\_5gon\] or \[fig:2\_planar\_6gon\] without violating its $2$-planarity, it follows by Lemma \[lem:exchange\].(\[prp:nonoptim\]) (for $\kappa+\lambda=3$ and $\mu \geq 5$) that $G$ is not optimal; a contradiction. Finally, consider the case where $\mathcal{C}$ is of length four. In this case, we have the crossing configuration of Figure \[fig:quasi\_1\]. By Claim \[clm:3\] there is at least one vertex in the interior of region $\mathcal{T}$. More in general, let $G_{\mathcal{T}}$ be the compact subgraph of $G$ that is completely drawn in the interior of region $\mathcal{T}$. Since edges $(u,v)$, $(u',v')$ and $(u'',v'')$ have already two crossings, it follows that no edge of $G$ crosses the boundary of $\mathcal{T}$; a contradiction to Property \[prp:connected\].
A PMCM-drawing $\Gamma(G)$ of an optimal $3$-planar graph $G$ is quasi-planar. \[prp:3\_planar\_quasi\]
As in the case of $2$-planar optimal graphs, assume that there exist three mutually crossing edges $(u,v)$, $(u',v')$ and $(u'',v'')$ in $\Gamma(G)$. By Corollary \[crl:crl\], there is always a [potential empty cycle]{}$\mathcal{C}$ of length at least $4$. Since $(u,v)$, $(u',v')$ and $(u'',v'')$ have already two crossings each, there exist at most three other edges that cross $(u,v)$, $(u',v')$ or $(u'',v'')$. Hence, the only edges that are drawn in the interior of $\mathcal{C}$ are $(u,v)$ and $(u',v')$, while $(u'',v'')$ and at most three other edges of $\Gamma(G)$ cross the boundary of $\mathcal{C}$. We distinguish three cases depending on whether $\mathcal{C}$ has length $6$, $5$ or $4$.
Consider first the case where $\mathcal{C}$ has length six. Since we can draw eight chords completely in the interior of $\mathcal{C}$ as in Figure \[fig:3\_planar\_6gon\] without deviating $3$-planarity, it follows by Lemma \[lem:exchange\].(\[prp:nonoptim\]) (for $\kappa+\lambda=6$ and $\mu=8$) that $G$ is not optimal; a contradiction.
Consider now the case where $\mathcal{C}$ has length five. We claim that at least one boundary edge of $\mathcal{C}$ does not exist in $\Gamma(G)$. In order to prove the claim, we consider the four crossing configurations of Figure \[fig:prp\_app\_quasi\] separately. In Figure \[fig:quasi\_app\_1\], if [potential edge]{}$[u',v]$ is an edge in $\Gamma(G)$, then it crosses twice $(u'',v'')$, contradicting Property \[prp:3\_planar\_cross\_twice\]. For Figures \[fig:quasi\_app\_2\]–\[fig:quasi\_app\_4\], if all red drawn curves belong to $\Gamma(G)$, then $(u'',v'')$ crosses $(u,v)$, $(u',v')$ and at least two of the boundary edges of $\mathcal{C}$, violating $3$-planarity. Hence, our claim follows. We proceed by removing edges $(u,v)$, $(u',v')$ and $(u'',v'')$ and any other edge crossing the boundary of $\mathcal{C}$ from $\Gamma(G)$, and we add five chords in the interior of $\mathcal{C}$, along with one “missing” boundary edge of $\mathcal{C}$. Let $G'$ be the derived graph. Note that, we removed at most six edges and added at least six. This implies that $G'$ is also optimal. However, $\mathcal{C}$ is a true-planar $5$-cycle in the drawing of $G'$, contradicting Property \[prp:3planar\_odd\_cycle\].
![ Crossing configurations for three mutually crossing edges. Potential edges are drawn solid red. Jordan curves that can either be potential edges or homotopic self-loops are drawn dotted red.[]{data-label="fig:prp_app_quasi"}](images/prop_quasi_planar "fig:"){width="\textwidth"} \[fig:quasi\_app\_1\]
![ Crossing configurations for three mutually crossing edges. Potential edges are drawn solid red. Jordan curves that can either be potential edges or homotopic self-loops are drawn dotted red.[]{data-label="fig:prp_app_quasi"}](images/prop_quasi_planar "fig:"){width="\textwidth"} \[fig:quasi\_app\_2\]
![ Crossing configurations for three mutually crossing edges. Potential edges are drawn solid red. Jordan curves that can either be potential edges or homotopic self-loops are drawn dotted red.[]{data-label="fig:prp_app_quasi"}](images/prop_quasi_planar "fig:"){width="\textwidth"} \[fig:quasi\_app\_3\]
![ Crossing configurations for three mutually crossing edges. Potential edges are drawn solid red. Jordan curves that can either be potential edges or homotopic self-loops are drawn dotted red.[]{data-label="fig:prp_app_quasi"}](images/prop_quasi_planar "fig:"){width="\textwidth"} \[fig:quasi\_app\_4\]
![ Crossing configurations for three mutually crossing edges. Potential edges are drawn solid red. Jordan curves that can either be potential edges or homotopic self-loops are drawn dotted red.[]{data-label="fig:prp_app_quasi"}](images/prop_quasi_planar "fig:"){width="\textwidth"} \[fig:quasi\_app\_1\_1\]
![ Crossing configurations for three mutually crossing edges. Potential edges are drawn solid red. Jordan curves that can either be potential edges or homotopic self-loops are drawn dotted red.[]{data-label="fig:prp_app_quasi"}](images/prop_quasi_planar "fig:"){width="\textwidth"} \[fig:3\_planar\_quasi\]
It remains to consider the case where $\mathcal{C}$ is of length four. By Claim \[clm:3\] there is at least one vertex in the interior of region $\mathcal{T}$. As in the proof of Property \[prp:2\_planar\_quasi\], we denote by $G_{\mathcal{T}}$ the subgraph of $G$ completely drawn in region $\mathcal{T}$. $G_{\mathcal{T}}$ is a compact subgraph of $\Gamma(G)$ and by Property \[prp:connected\], it follows that if $G_{\mathcal{T}}$ has $n_{\mathcal{T}}\geq 2$ vertices, then it has $m_{\mathcal{T}}\leq 5.5n_{\mathcal{T}}-6.5$ edges (note that if $n_{\mathcal{T}}=1$, then $m_{\mathcal{T}}=0$). We replace $G_{\mathcal{T}}$ with one vertex, say $x$, we keep edges $(u,v)$, $(u',v')$ and $(u'',v'')$ and remove any edge crossing $(u,v)$, $(u',v')$ or $(u'',v'')$ in $\Gamma(G)$. We redraw the edge-segment of $(u,v)$ incident to $v$ so as to be incident to $u'$ (without introducing new crossings). Finally, we add edges $(x,u)$, $(x,u')$ and $(x,v')$; see Figure \[fig:3\_planar\_quasi\]. The derived graph $G'$ has $n'=n-n_{\mathcal{T}}+1$ vertices and at least $m'\geq m-m_{\mathcal{T}}$ edges, where $n$ and $m$ are the number of vertices and edges of $G$. For $n_{\mathcal{T}}\geq 2$, we have that $m'\geq 5.5n'-10$, i.e., $G'$ has more edges than allowed. In the case where $n_{\mathcal{T}}=1$ and $m_{\mathcal{T}}=0$, it follows that $G'$ has the same number of edges as $G$ and is therefore optimal. However, [potential edges]{}$[u,v']$, $[u',u'']$ and $[u',v'']$ can be added in $\Gamma(G')$ (if not present) forming thus a true-planar $3$-cycle; a contradiction to Property \[prp:3planar\_odd\_cycle\].
We next present a refinement of the notion of [potential edges]{}. In particular, we focus on two main categories of [potential edges]{}that we will heavily use in Sections \[sec:2planar\] and \[sec:3planar\]. Consider a pair of vertices $u$ and $v$ of $G$ that are not necessarily distinct. We say that $u$ and $v$ form a *corner pair* if and only if an edge $(u,u')$ crosses an edge $(v,v')$ for some $u'$ and $v'$ in $\Gamma(G)$; see Figure \[fig:corner\_pair\]. Let $c$ be the crossing point of $(u,u')$ and $(v,v')$. Then, any Jordan curve $[u,v]$ joining vertices $u$ and $v$ induces a region $R_{u,v}$ that is defined by the walk along the edge-segment of $(u,u')$ from $u$ to $c$, the edge segment of $(v,v')$ from $c$ to $v$ and the curve $[u,v]$ from $v$ to $u$. We call $[u,v]$ *corner edge* with respect to $(u,u')$ and $(v,v')$ if and only if $R_{u,v}$ has no vertices of $\Gamma(G)$ in its interior.
![ (a-b) vertices $u$ and $v$ form a corner pair; (c-d) vertices $u$ and $v$ form a [side]{}pair; (e) at least one of the two potential [side-edges]{}exists.[]{data-label="fig:crossing_confs"}](images/prop_corner "fig:"){width="\textwidth"} \[fig:corner\_pair\]
![ (a-b) vertices $u$ and $v$ form a corner pair; (c-d) vertices $u$ and $v$ form a [side]{}pair; (e) at least one of the two potential [side-edges]{}exists.[]{data-label="fig:crossing_confs"}](images/prop_corner "fig:"){width="\textwidth"} \[fig:corner\_pair\_same\]
![ (a-b) vertices $u$ and $v$ form a corner pair; (c-d) vertices $u$ and $v$ form a [side]{}pair; (e) at least one of the two potential [side-edges]{}exists.[]{data-label="fig:crossing_confs"}](images/prop_parallel "fig:"){width="\textwidth"} \[fig:parallel\_pair\]
![ (a-b) vertices $u$ and $v$ form a corner pair; (c-d) vertices $u$ and $v$ form a [side]{}pair; (e) at least one of the two potential [side-edges]{}exists.[]{data-label="fig:crossing_confs"}](images/prop_parallel "fig:"){width="\textwidth"} \[fig:parallel\_pair\_same\]
![ (a-b) vertices $u$ and $v$ form a corner pair; (c-d) vertices $u$ and $v$ form a [side]{}pair; (e) at least one of the two potential [side-edges]{}exists.[]{data-label="fig:crossing_confs"}](images/prop_parallel "fig:"){width="\textwidth"} \[fig:parallel\_pair\_homotopic\]
In a PMCM-drawing $\Gamma(G)$ of an optimal $k$-planar graph $G$ any corner edge $[u,v]$ is a potential edge. \[prp:corner\]
By the definition of potential edges, the property holds when $u \neq v$. Consider now the case where $u=v$. In this case $[u,v]$ is a self-loop; see Figure \[fig:corner\_pair\_same\]. If the property does not hold, then it follows that $[u,v]$ is a self-loop with no vertices either in its interior or in its exterior. However, this contradicts Lemma \[lem:crossing\_adjacent\], and the property holds.
We say that vertices $u$ and $v$ form a *[side]{}pair* if and only if there exist edges $(u,u')$ and $(v,v')$ for some $u'$ and $v'$ such that they both cross a third edge $(w,w')$ in $\Gamma(G)$ and additionally $(u,u') \neq (v,v')$; see Figure \[fig:parallel\_pair\] or \[fig:parallel\_pair\_same\]. Let $c$ and $c'$ be the crossing points of $(u,u')$ and $(v,v')$ with $(w,w')$, respectively. Assume w.l.o.g. that $c$ and $c'$ appear in this order along $(w,w')$ from vertex $w$ to vertex $w'$. Also assume that the edge-segment of $(u,u')$ between $u$ and $c$ is on the same side of edge $(w,w')$ as the edge-segment of $(v,v')$ between $v$ and $c'$; refer to Figure \[fig:parallel\_pair\]. Then, any Jordan curve $[u,v]$ joining vertices $u$ and $v$ induces a region $R_{u,v}$ that is defined by the walk along the edge-segment of $(u,u')$ from $u$ to $c$, the edge segment of $(w,w')$ from $c$ to $c'$, the edge segment of $(v,v')$ from $c'$ to $v$ and the curve $[u,v]$ from $v$ to $u$. We call $[u,v]$ *[side-edge]{}* w.r.t. $(u,u')$ and $(v,v')$ if and only if $R_{u,v}$ has no vertices of $\Gamma(G)$ in its interior. Since by Properties \[prp:2\_planar\_quasi\] and \[prp:3\_planar\_quasi\] edges $(u,u')$ and $(v,v')$ cannot cross with each other (as they both cross $(w,w')$), it follows that region $R_{u,v}$ is well-defined. Symmetrically we define region $R_{u',v'}$ and [side-edge]{}$[u',v']$ with respect to $(u,u')$ and $(v,v')$.
In a PMCM-drawing $\Gamma(G)$ of an optimal $k$-planar graph $G$ with $k \in \{2,3\}$ at least one of the [side-edges]{}$[u,v]$, $[u',v']$ is a [potential edge]{}. \[prp:parallel\]
Before giving the proof, note that since edges $(u,u')$, $(v,v')$ and $(w,w')$ do not mutually cross, curves $[u,v]$ and $[u',v']$ cannot cross themselves. Now, for a proof by contradiction, assume that neither $[u,v]$ nor $[u',v']$ are potential edges. This implies that $u=v$, $u'=v'$ and both $[u,v]$ and $[u',v']$ are self-loops that have no vertices in their interiors or their exteriors. Figure \[fig:parallel\_pair\_homotopic\] illustrates the case where both $[u,v]$ and $[u',v']$ are self-loops with no vertices in their interiors; the other cases are similar. It is not hard to see that $(u,u')$ and $(v,v')$ are homotopic [side-edges]{}; a contradiction.
We say that $(u,u')$ and $(v,v')$ are *[side-apart]{}* if and only if both [side-edges]{}$[u,v]$ and $[u',v']$ are [potential edges]{}.
Characterization of optimal 2-planar graphs {#sec:2planar}
===========================================
By using the properties we proved in Section \[sec:properties\], in this section we examine some more structural properties of optimal $2$-planar graphs in order to derive their characterization (see Theorem \[thm:2-characterization\]).
Let $\Gamma(G)$ be a PMCM-drawing of an optimal $2$-planar graph $G$. Any edge that is crossed twice in $\Gamma(G)$ is a chord of a true-planar $5$-cycle in $\Gamma(G)$. \[lem:2\_planar\_small\_faces\]
Let $(u,v)$ be an edge of $G$ that is crossed twice in $\Gamma(G)$ by edges $(u',v')$ and $(u'',v'')$ at points $c$ and $c'$, respectively. Note that, by Property \[prp:2\_planar\_cross\_twice\] edges $(u',v')$ and $(u'',v'')$ are not identical. We assume w.l.o.g. that $c$ and $c'$ appear in this order along $(u,v)$ from vertex $u$ to vertex $v$. We also assume that the edge-segment of $(u',v')$ between $u'$ and $c$ is on the same side of edge $(u,v)$ as the edge-segment of $(u'',v'')$ between $u''$ and $c'$; refer to Figure \[fig:2\_general\]. By Property \[prp:corner\] corner edges $[u,u']$, $[u,v']$, $[v,u'']$ and $[v,v'']$ are [potential edges]{}. By Property \[prp:parallel\] at least one of [side-edges]{}$[u',u'']$ and $[v',v'']$ is a [potential edge]{}. Assume w.l.o.g. that $[v',v'']$ is a [potential edge]{}.
![ Different configurations used in Lemma \[lem:2\_planar\_small\_faces\].[]{data-label="fig:2_planar_potential_parallel"}](images/2planar_2_crossing "fig:"){width="\textwidth"} \[fig:2\_general\]
![ Different configurations used in Lemma \[lem:2\_planar\_small\_faces\].[]{data-label="fig:2_planar_potential_parallel"}](images/2planar_2_crossing "fig:"){width="\textwidth"} \[fig:2\_6gon\]
![ Different configurations used in Lemma \[lem:2\_planar\_small\_faces\].[]{data-label="fig:2_planar_potential_parallel"}](images/2planar_2_crossing "fig:"){width="\textwidth"} \[fig:2\_5gon\]
![ Different configurations used in Lemma \[lem:2\_planar\_small\_faces\].[]{data-label="fig:2_planar_potential_parallel"}](images/2planar_2_crossing "fig:"){width="\textwidth"} \[fig:2\_5gon\_extend\]
![ Different configurations used in Lemma \[lem:2\_planar\_small\_faces\].[]{data-label="fig:2_planar_potential_parallel"}](images/2planar_2_crossing "fig:"){width="\textwidth"} \[fig:2\_5gon\_final\]
First consider the case that $[u',u'']$ is also a [potential edge]{}; see Figure \[fig:2\_6gon\]. In this case, vertices $u$, $v'$, $v''$, $v$, $u''$ and $u'$ define a [potential empty cycle]{}$\mathcal{C}$ on six vertices (shaded in gray in Figure \[fig:2\_6gon\]). Edges $(u,v)$, $(u',v')$ and $(u'',v'')$ are drawn in the interior of $\mathcal{C}$, and there exist at most two other edges that cross $(u',v')$ or $(u'',v'')$. In total there exist at most five edges that have an edge-segment within $\mathcal{C}$. However, in the interior of $\mathcal{C}$ one can draw six chords as in Figure \[fig:2\_planar\_6gon\] without deviating $2$-planarity. By Lemma \[lem:exchange\].(\[prp:nonoptim\]) for $\kappa+\lambda\leq5$ and $\mu=6$, it follows that $G$ is not optimal; a contradiction.
To complete the proof, it remains to consider the cases where $[u',u'']$ is not a [potential edge]{}; see Figure \[fig:2\_5gon\]. In this case, $[u',u'']$ is a homotopic self-loop (hence, the red-shaded region of Figure \[fig:2\_5gon\] contains no vertices in its interior). Vertices $u$, $v'$, $v''$, $v$ and $u'$ define a [potential empty cycle]{}$\mathcal{C}$ on five vertices (shaded in gray in Figure \[fig:2\_5gon\]). However, in the interior of $\mathcal{C}$ one can draw five chords as in Figure \[fig:2\_planar\_5gon\] without deviating $2$-planarity. By Lemma \[lem:exchange\].(\[prp:boundary\]), for $\kappa+\lambda\leq 5$ and $\mu=5$, it follows that all boundary edges of $\mathcal{C}$ exist in $\Gamma(G)$. Furthermore, $\kappa+\lambda=5$ must hold, which implies that there exist two edges (other than $(u,v)$), say $e$ and $e'$, that cross $(u',v')$ and $(u'',v'')$ respectively.
If $\mathcal{C}$ is a true-planar $5$-cycle in $\Gamma(G)$ the lemma holds. If it is not, then at least one of edges $e$ or $e'$ crosses a boundary edge of $\mathcal{C}$. Suppose w.l.o.g. that edge $e$ crosses $(v',v'')$ of $\mathcal{C}$ at point $p$ and let $w$ and $w'$ be the endpoints of $e$ (other cases are similar). Observe that $e$ already has two crossings in $\Gamma(G)$. By $2$-planarity, either the edge-segment of $(w,w')$ between $w$ and $p$ or the one between $w'$ and $p$ is drawn completely in the exterior of $\mathcal{C}$. Suppose w.l.o.g. that this edge-segment is the one between $w$ and $p$. Then vertices $v'$, $w$ and $v''$ define a [potential empty cycle]{}$\mathcal{C}'$ on three vertices; see Figure \[fig:2\_5gon\_extend\]. We proceed as follows: We remove edges $(u,v)$, $(u',v')$, $(u'',v'')$, $e$ and $e'$ and replace them with five chords drawn in the interior of $\mathcal{C}$ (as in Figure \[fig:2\_5gon\_final\]). The derived graph $G'$ has the same number of edges as $G$. However, $\mathcal{C'}$ becomes a true-planar $3$-cycle in $G'$, contradicting Property \[prp:2planar\_triangle\].
By Lemma \[lem:2\_planar\_small\_faces\], any edge of $G$ that is crossed twice in $\Gamma(G)$ is a chord of a true-planar $5$-cycle. So, it remains to consider edges of $G$ that have only one crossing in $\Gamma(G)$. In fact, the following lemma states that there are no such edges in $\Gamma(G)$.
Let $\Gamma(G)$ be a PMCM-drawing of an optimal $2$-planar graph $G$. Then, every edge of $\Gamma(G)$ is either true-planar or has exactly two crossings. \[lem:2\_planar\_one\_crossing\]
As shown in the proof of Lemma \[lem:2\_planar\_small\_faces\], for any edge $e$ of $G$ that is crossed twice in $\Gamma(G)$, both edges that cross $e$ also have two crossings in $\Gamma(G)$. So, the crossing component $\mathcal{X}(e)$ consists exclusively of edges with two pairwise crossings. This implies that if edges $(u,v)$ and $(u',v')$ cross in $\Gamma(G)$ and $(u,v)$ has only one crossing, then the same holds for $(u',v')$; see Figure \[fig:2\_planar\_one\_crossing\_before\]. Vertices $u$, $v'$, $v$ and $u'$ define a [potential empty cycle]{}$\mathcal{C}$ on four vertices (gray-shaded in Figure \[fig:2\_planar\_one\_crossing\_before\]). Since edges $(u,v)$ and $(u',v')$ have only one crossing each, the boundary of $\mathcal{C}$ exists in $\Gamma(G)$ and are true-planar edges. We proceed by removing edge $(u',v')$. Now $\mathcal{C}$ is split into two true-planar $3$-cycles; see Figure \[fig:2\_planar\_one\_crossing\_after\]. In both of them, we plug the $2$-planar pattern of Figure \[fig:2\_planar\_triangle\]. In total, we removed one edge and added two vertices and a total of $12$ edges, without creating any homotopic parallel edges or self-loops. Hence, if $G$ has $n$ vertices and $m$ edges, the derived graph $G'$ is $2$-planar and has $n'=n+2$ vertices and $m'=m+11$ edges. Hence $m'=5n'-9$, i.e. $G'$ has more edges than allowed; a contradiction.
![ Different configurations used in Lemma \[lem:2\_planar\_one\_crossing\].[]{data-label="fig:2_planar_one_crossing"}](images/2planar_one_crossing "fig:"){width="\textwidth"} \[fig:2\_planar\_one\_crossing\_before\]
![ Different configurations used in Lemma \[lem:2\_planar\_one\_crossing\].[]{data-label="fig:2_planar_one_crossing"}](images/2planar_one_crossing "fig:"){width="\textwidth"} \[fig:2\_planar\_one\_crossing\_after\]
The true-planar skeleton $\Pi(G)$ of a PMCM-drawing $\Gamma(G)$ of an optimal $2$-planar graph is connected. \[prp:2\_planar\_skeleton\_connected\]
Assume to the contrary that $\Pi(G)$ is not connected and let $H$ be a connected component of $\Pi(G)$. By Property \[prp:connected\] either there exists an edge $(u,v)$ with $u \in H$ and $v \in G \setminus H$, or two crossing edges $e_1 \in H$ and $e_2 \in G \setminus H$. In the first case, $(u,v)$ is not a true-planar edge. By Lemma \[lem:2\_planar\_small\_faces\], there exists a true-planar $5$-cycle with chord $(u,v)$ connecting $u$ to $v$ in $\Pi(G)$; a contradiction. In the second case, edges $e_1$ and $e_2$ belong to the same crossing component and by Lemma \[lem:2\_planar\_small\_faces\], there exists a true-planar $5$-cycle with $e_1$ and $e_2$ as chords, therefore connecting their endpoints in $\Pi(G)$; a contradiction.
The true-planar skeleton $\Pi(G)$ of a PMCM-drawing $\Gamma(G)$ of an optimal $2$-planar graph $G$ contains only faces of length $5$, each of which contains $5$ crossing edges in $\Gamma(G)$. \[lem:2\_planar\_faces\]
Since $\Pi(G)$ is connected (by Lemma \[prp:2\_planar\_skeleton\_connected\]), all faces of $\Pi(G)$ are also connected. By Lemmas \[lem:2\_planar\_small\_faces\] and \[lem:2\_planar\_one\_crossing\], all crossing edges are chords of true-planar $5$-cycles. We claim that $\Pi(G)$ has no chordless faces. First, $\Pi(G)$ cannot contain a chordless face of size $\geq 4$, as otherwise we could draw in its interior a chord, contradicting the optimality of $G$. Property \[prp:2planar\_triangle\] ensures that $\Pi(G)$ contains no faces of size $3$. Finally, faces of size $1$ or $2$ correspond to homotopic self-loops and parallel edges.
We are now ready to state the main theorem of this section.
A graph $G$ is optimal $2$-planar if and only if $G$ admits a drawing $\Gamma(G)$ without homotopic parallel edges and self-loops, such that the true-planar skeleton $\Pi(G)$ of $\Gamma(G)$ spans all vertices of $G$, it contains only faces of length $5$ (that are not necessarily simple), and each face of $\Pi(G)$ has $5$ crossing edges in its interior in $\Gamma(G)$. \[thm:2-characterization\]
For the forward direction, consider an optimal $2$-planar graph $G$. By Lemma \[lem:2\_planar\_faces\], the true-planar skeleton $\Pi(G)$ of its $2$-planar PMCM-drawing $\Gamma(G)$ contains only faces of length $5$ and each face of $\Pi(G)$ has $5$ crossing edges in its interior in $\Gamma(G)$. Since the endpoints of two crossing edges are within a true-planar $5$-cycle (by Lemmas \[lem:2\_planar\_small\_faces\] and \[lem:2\_planar\_one\_crossing\]) and since $\Pi(G)$ is connected (by Lemma \[prp:2\_planar\_skeleton\_connected\]), $\Pi(G)$ spans all vertices of $G$. This completes the proof of this direction.
For the reverse direction, denote by $n$, $m$ and $f$ the number of vertices, edges and faces of $\Pi(G)$. Since $\Pi(G)$ spans all vertices of $G$, it suffices to prove that $G$ has exactly $5n-10$ edges. The fact that $\Pi(G)$ contains only faces of length $5$ implies that $5f=2m$. By Euler’s formula for planar graphs, $m=5(n-2)/3$ and $f=2(n-2)/3$ follows. Since each face of $\Pi(G)$ contains exactly $5$ crossing edges, the total number of edges of $G$ equals $m+5f=5n-10$.
Characterization of optimal 3-planar graphs {#sec:3planar}
===========================================
In this section we explore several structural properties of optimal $3$-planar graphs to derive their characterizations (see Theorem \[thm:3-characterization\]).
Let $\Gamma(G)$ be a PMCM-drawing of an optimal $3$-planar graph $G$, and suppose that there exists a [potential empty cycle]{}$\mathcal{C}$ of $6$ vertices in $\Gamma(G)$, such that the potential boundary edges of $\mathcal{C}$ exist in $\Gamma(G)$. Let $E_{\mathcal{C}}$ be the set of edge-segments within $\mathcal{C}$. If the conditions C.\[cnd:1\] and C.\[cnd:2\] hold, then $\mathcal{C}$ is an empty true-planar $6$-cycle in $\Gamma(G)$ and all edges with edge-segments in $E_{\mathcal{C}}$ are drawn as chords in its interior.
1. \[cnd:1\] $|E_{\mathcal{C}}|\leq 8$, and,
2. \[cnd:2\] every edge-segment of $E_{\mathcal{C}}$ has at least one crossing in the interior of $\mathcal{C}$.
\[lem:size9\]
We start with the following observation: If $e$ is an edge of $G$, then due to $3$-planarity at most one edge-segment of $e$ belongs to $E_{\mathcal{C}}$. More precisely, if $E_{\mathcal{C}}$ contains at least two edge-segments of $e$, then we claim that $e$ has at least four crossings. By Condition C.\[cnd:2\] each of the two edge-segments of $e$ contributes one crossing to $e$. Since $\mathcal{C}$ is empty and contains two edge-segments of $e$, it follows that $e$ exists and enters $\mathcal{C}$. Hence, $e$ has two more crossings, summing up to a total of at least four crossings.
Let $v_1,\dots,v_6$ be the vertices of $\mathcal{C}$. If all edges with edge-segments in $E_{\mathcal{C}}$ completely lie in $\mathcal{C}$, then $\mathcal{C}$ is a true-planar $6$-cycle and the lemma trivially holds. Otherwise, there is at least one edge $e$ with an edge-segment in $E_{\mathcal{C}}$, that crosses a boundary edge of $\mathcal{C}$. W.l.o.g. we can assume that $e$ crosses $(v_1,v_6)$ of $\mathcal{C}$ at point $c$ (refer to Figure \[fig:cs1\]). If $w$ and $w'$ are the two endpoints of $e$, then by the observation we made at the beginning of the proof it follows that either the edge-segment of $(w,w')$ between $w$ and $c$ or the one between $c$ and $w'$ is drawn completely in the exterior of $\mathcal{C}$ (as otherwise $e$ would have at least two edge-segments in $E_{\mathcal{C}}$). W.l.o.g. assume that this is the edge-segment between $w$ and $c$. Then, corner edges $[v_1,w]$ and $[w,v_6]$ are [potential edges]{}(by Property \[prp:corner\]).
![ Different configurations used in (a)-(d) Lemma \[lem:size9\], (e) Lemma \[lem:3\_planar\_independent\].[]{data-label="fig:replacements_2"}](images/3planar_6gon_bound "fig:"){width="\textwidth"} \[fig:cs1\]
![ Different configurations used in (a)-(d) Lemma \[lem:size9\], (e) Lemma \[lem:3\_planar\_independent\].[]{data-label="fig:replacements_2"}](images/3planar_6gon_bound "fig:"){width="\textwidth"} \[fig:cs2\]
![ Different configurations used in (a)-(d) Lemma \[lem:size9\], (e) Lemma \[lem:3\_planar\_independent\].[]{data-label="fig:replacements_2"}](images/3planar_6gon_bound "fig:"){width="\textwidth"} \[fig:7\_stick\]
![ Different configurations used in (a)-(d) Lemma \[lem:size9\], (e) Lemma \[lem:3\_planar\_independent\].[]{data-label="fig:replacements_2"}](images/3planar_6gon_bound "fig:"){width="\textwidth"} \[fig:cp\]
![ Different configurations used in (a)-(d) Lemma \[lem:size9\], (e) Lemma \[lem:3\_planar\_independent\].[]{data-label="fig:replacements_2"}](images/3planar_6gon_bound "fig:"){width="\textwidth"} \[fig:cs\_final\]
![ Different configurations used in (a)-(d) Lemma \[lem:size9\], (e) Lemma \[lem:3\_planar\_independent\].[]{data-label="fig:replacements_2"}](images/3planar_independent "fig:"){width="\textwidth"} \[fig:independent\]
Recall that $e$ has one crossing in the interior of $\mathcal{C}$ (by Condition C.\[cnd:2\] of the lemma) and one more crossing with edge $(v_1,v_6)$. By $3$-planarity, it follows that edge $e$ may have at most one more crossing, say with edge $e'$. Note that $e'$ may or may not have an edge-segment in $E_{\mathcal{C}}$. Vertices $w$, $v_1$, $\dots$, $v_6$ define a [potential empty cycle]{}$\mathcal{C}'$ on $7$ vertices (see Figure \[fig:cs2\]). The set $E_{\mathcal{C}'}$ of edge-segments within $\mathcal{C}'$ contains all edge-segments of $E_{\mathcal{C}}$ (that is, $E_{\mathcal{C}} \subseteq E_{\mathcal{C}'}$) plus at most two additional edge-segments: the one defined by edge $(v_1,v_6)$, and possibly an edge-segment of $e'$. Hence $|E_{\mathcal{C}'}| \leq 10$. In the following we make some observations in the form of claims.
All edges with an edge-segment in $E_{\mathcal{C}'}$ have at least one crossing in the interior of $\mathcal{C}'$. \[nclm:1\]
The claim clearly holds for all edge-segments of $E_{\mathcal{C}}$ (recall that $E_{\mathcal{C}}\subset E_{\mathcal{C}'}$). Since $(v_1,v_{6})$ and $e'$ (if it exists) both cross $e$ in the interior of $\mathcal{C}'$, the remaining edge-segments within $\mathcal{C}'$ (i.e., the ones defined by edges $(v_1,v_{6})$ and $e'$) have at least one crossing in the interior of $\mathcal{C}'$.
At least one edge with an edge-segment in $E_{\mathcal{C}'}$ crosses one edge of $\mathcal{C}'$. \[nclm:2\]
If all edges with an edge-segment in $E_{\mathcal{C}'}$ do not cross $\mathcal{C}'$, then all edges with an edge-segment in $E_{\mathcal{C}'}$ can be drawn completely in the interior of $\mathcal{C}'$. Hence, all [potential edges]{}of $\mathcal{C}'$ can be added in $\Gamma(G)$ (if they are not present already). Then, $\mathcal{C}'$ is a true-planar $7$-cycle contradicting Property \[prp:3planar\_odd\_cycle\].
By Claim \[nclm:2\], there is an edge $g$ that crosses a boundary edge, say $[w,v_1]$, of $\mathcal{C}'$ at point $c'$; see Figure \[fig:7\_stick\].
All boundary edges of $\mathcal{C}'$ exist in $\Gamma(G)$ and $g$ has one crossing in the interior of $\mathcal{C}'$. \[nclm:3\]
To prove this claim, we remove all edges with an edge-segment in $E_{\mathcal{C}'}$ (recall that $|E_{\mathcal{C}'}| \leq 10$) and replace them with the $10$ edges of the $3$-planar crossing pattern of Figure \[fig:cp\], i.e., we redraw the segment of $g$ in the interior of $\mathcal{C}'$ so that:
$g$ emanates from vertex $v_6$ of $\mathcal{C}'$,
$g$ crosses only [potential edge]{}$[w,v_1]$ at point $c'$, and
$g$ has no other crossings in the interior of $\mathcal{C}'$.
This allows us to add all boundary edges of $\mathcal{C}'$ in $\Gamma(G)$ (if they are not present). Hence, $3$-planarity is preserved and the derived graph has at least as many edges as $G$. Since $G$ is optimal, it follows that all boundary edges of $\mathcal{C}'$ must exist in $\Gamma(G)$, which completes the proof of the claim.
We follow an analogous approach to the one we used for expanding $\mathcal{C}$ (that has $6$ vertices) to $\mathcal{C}'$ (that has $7$ vertices). We can find an endpoint of $g$, say $z$, such that $w$, $z$, $v_1$, $v_2$, $\dots$, $v_6$ define a [potential empty cycle]{}$\mathcal{C}''$ on $8$ vertices. Furthermore, the set $E_{\mathcal{C}''}$ of edge-segments within $\mathcal{C}''$ has at most $12$ elements (at most two more than $E_{\mathcal{C}'}$). We proceed by removing all edges with an edge-segment in $E_{\mathcal{C}''}$ and split $\mathcal{C}''$ into two true-planar cycles of length $6$ and $4$, by adding true-planar chord $(v_1,v_6)$; see Figure \[fig:cs\_final\]. In the interior of the $6$-cycle, we add $8$ crossing edges as in Figure \[fig:3\_planar\_6gon\]. In the interior of the $4$-cycle, we add a vertex $x$ with a true planar edge $(v_1,x)$. Vertices $v_1$, $x$, $v_1$, $v_6$, $w$ and $z$ define a new [potential empty cycle]{}on $6$ vertices, allowing us to add $8$ more crossing edges. In total, we removed at most $12$ edges, added a vertex and $18$ edges. If $n$ and $m$ are the number of vertices and edges of $G$, then the derived graph $G'$ has $n'=n+1$ vertices and $m'\geq m+6$ edges. The last equation gives $m'\geq 5.5n'-10.5$, i.e. $G'$ has more edges than allowed; a contradiction.
Let $(u,v)$ be an edge of $G$ that is crossed by two edges $(u_1,v_1)$ and $(u_2,v_2)$ in $\Gamma(G)$ at points $c_1$ and $c_2$. By Property \[prp:3\_planar\_cross\_twice\] edges $(u_1,v_1)$ and $(u_2,v_2)$ are not identical. We assume w.l.o.g. that $c_1$ and $c_2$ appear in this order along $(u,v)$ from $u$ to $v$. We also assume that the edge-segment of $(u_1,v_1)$ between $u_1$ and $c$ is on the same side of edge $(u,v)$ as the edge-segment of $(u_2,v_2)$ between $u_2$ and $c_2$; refer to Figure \[fig:independent\]. Vertices $u_1$, $u_2$ and $v_1$, $v_2$ define two [side]{}pairs. By Property \[prp:parallel\], at least one of [side-edges]{}$[u_1,u_2]$ and $[v_1,v_2]$ is a [potential edge]{}of $\Gamma(G)$. Recall that if both [side-edges]{}$[u_1,u_2]$ and $[v_1,v_2]$ are [potential edges]{}of $\Gamma(G)$, then edges $(u_1,v_1)$ and $(u_2,v_2)$ are called [side-apart]{}.
Let $\Gamma(G)$ be a PMCM-drawing of an optimal $3$-planar graph $G$. If $(u,v)$ is crossed by [side-apart]{}edges $(u_1,v_1)$ and $(u_2,v_2)$ in $\Gamma(G)$, then it is a chord of an empty true-planar $6$-cycle. \[lem:3\_planar\_independent\]
Refer to Figure \[fig:independent\]. Since $(u_1,v_1)$ and $(u_2,v_2)$ are [side-apart]{}, [side-edges]{}$[u_1,u_2]$ and $[v_1,v_2]$ are [potential edges]{}. By Property \[prp:corner\], corner edges $[u, u_1]$, $[u,v_1]$, $[u,u_2]$ and $[v,v_2]$ are [potential edges]{}. Hence, vertices $u$, $v_1$, $v_2$, $v$, $u_2$ and $u_1$ define a [potential empty cycle]{}$\mathcal{C}$ on six vertices (gray-shaded in Figure \[fig:independent\]). Edges $(u,v)$, $(u_1,v_1)$ and $(u_2,v_2)$ are drawn completely in the interior of $\mathcal{C}$ and there exist at most five other edges either drawn in the interior of $\mathcal{C}$ or crossing its boundary: at most one that crosses $(u,v)$, and at most four others that cross $(u_1,v_1)$ and $(u_2,v_2)$. Since we can draw eight chords in the interior of $\mathcal{C}$ as in Figure \[fig:3\_planar\_6gon\], by Lemma \[lem:exchange\].(\[prp:boundary\]), for $\kappa+\lambda\leq 8$ and $\mu=8$, all boundary edges of $\mathcal{C}$ exist in $\Gamma(G)$. Furthermore $\kappa+\lambda=8$ must hold. Note that the set $E_{\mathcal{C}}$ of edge-segments within $\mathcal{C}$ contains only edge-segments of these $\kappa+\lambda$ edges. Also, these $8$ edges have exactly one edge-segment within $\mathcal{C}$ that is crossed in the interior of $\mathcal{C}$. Hence, conditions C.\[cnd:1\] and C.\[cnd:2\] of Lemma \[lem:size9\] are satisfied and there exists an empty true-planar $6$-cycle that has $(u,v)$ as chord.
Let $\Gamma(G)$ be a PMCM-drawing of an optimal $3$-planar graph $G$. If $e$ is crossed by two [side-apart]{}edges in $\Gamma(G)$, then all edges of $\mathcal{X}(e)$ are chords of an empty true-planar $6$-cycle. \[lem:3\_planar\_xing\_comp\]
The lemma follows by the observation that since $e$ is a chord of an empty true-planar $6$-cycle (by Lemma \[lem:3\_planar\_independent\]), all edges of $\mathcal{X}(e)$ are also chords of this $6$-cycle.
Let $\Gamma(G)$ be a PMCM-drawing of an optimal $3$-planar graph $G$. Any edge that is crossed three times in $\Gamma(G)$ is a chord of an empty true-planar $6$-cycle in $\Gamma(G)$. \[lem:3\_planar\_small\_faces\]
Our proof is based on a case analysis and in order to lighten the presentation we will use intermediate observations in the form of claims. Let $(u,v)$ be an edge of $G$ that crosses edges $(u_i,v_i)$ in $\Gamma(G)$, for $i=1,2,3$. Let also $c_1$, $c_2$ and $c_3$ be the corresponding crossing points as they appear along $(u,v)$ from vertex $u$ to vertex $v$; see Figure \[fig:3\_planar\_three\_crossing\]. We assume w.l.o.g. that the edge-segment of $(u_i,v_i)$ between $u_i$ and $c_i$ is on the same side of edge $(u,v)$ as the edge-segment of $(u_j,v_j)$ between $u_j$ and $c_j$, for $1\leq i<j\leq 3$. Consider the crossing component $\mathcal{X}((u,v))$. We distinguish two cases depending on whether there exists an edge in $\mathcal{X}((u,v))$ that crosses two [side-apart]{}edges or not. Assume that there is an edge of $\mathcal{X}((u,v))$ that crosses two [side-apart]{}edges. Then, by Lemma \[lem:3\_planar\_xing\_comp\] all edges of $\mathcal{X}((u,v))$, including $(u,v)$, are chords of an empty true-planar $6$-cycle and the lemma follows. Assume now that there exists no edge in $\mathcal{X}((u,v))$ that crosses two [side-apart]{}edges. Hence, for edge $(u,v)$, that crosses edges $(u_1,v_1)$, $(u_2,v_2)$ and $(u_3,v_3)$, we have that any two edges $(u_i,v_i)$ and $(u_j,v_j)$ with $1\leq i<j\leq 3$, are not [side-apart]{}. Observe that by definition, exactly one of [side-edges]{}$[u_i,u_j]$ or $[v_i,v_j]$ is not a [potential edge]{}. In the following claim, we refine this observation.
Either [side-edges]{}$[u_1,u_2]$, $[u_1,u_3]$ and $[u_2,u_3]$ are not [potential edges]{}or [side-edges]{}$[v_1,v_2]$, $[v_1,v_3]$ and $[v_2,v_3]$ are not [potential edges]{}. \[clm:fan\_planar\]
Consider [side-edges]{}$[u_1,u_2]$ and $[u_2,u_3]$ and assume that both are not [potential edges]{}. It follows that $[u_1,u_2]$ and $[u_2,u_3]$ are both homotopic self-loops. Hence, $u_1=u_2=u_3$. We will prove that [side-edge]{}$[u_1,u_3]$ is not a [potential edge]{}either. Let $R_{1,2}$ be the region defined by the edge-segment of $(u_1,v_1)$ between $u_1$ and crossing $c_1$, the edge-segment of $(u,v)$ between crossings $c_1$ and $c_2$ and the edge-segment of $(u_2,v_2)$ between $c_2$ and $u_2$ (recall that $u_1=u_2$; see Figure \[fig:3\_planar\_small\_faces\_example\]). Similarly, we define regions $R_{2,3}$ and $R_{1,3}$. Observe that $R_{1,2}\cup R_{2,3} = R_{1,3}$. Since $[u_1,u_2]$ and $[u_2,u_3]$ are homotopic self-loops, $R_{1,2}$ and $R_{2,3}$ do not contain any vertex in their interiors. Hence, $R_{1,3}$ does not contain any vertex in its interior either. More in general, since $R_{1,2} \cup R_{2,3} = R_{1,3}$ we can prove that whenever any two of $[u_1,u_2]$, $[u_2,u_3]$ and $[u_1,u_3]$ are not [potential edges]{}, then the third one is not a [potential edge]{}either. Similarly, we can prove that whenever any two of $[v_1,v_2]$, $[v_2,v_3]$ and $[v_1,v_3]$ are not [potential edges]{}, then the third one is not a [potential edge]{}either.
![ Different configurations used in Lemma \[lem:3\_planar\_small\_faces\].](images/3planar_three_crossing "fig:"){width="\textwidth"} \[fig:3\_planar\_three\_crossing\]
![ Different configurations used in Lemma \[lem:3\_planar\_small\_faces\].](images/3planar_fan_crossing "fig:"){width="\textwidth"} \[fig:3\_planar\_small\_faces\_example\]
![ Different configurations used in Lemma \[lem:3\_planar\_small\_faces\].](images/3planar_fan_crossing "fig:"){width="\textwidth"} \[fig:3fan\]
![ Different configurations used in Lemma \[lem:3\_planar\_small\_faces\].](images/3planar_fan_crossing "fig:"){width="\textwidth"} \[fig:3fan\_middle\_a\]
![ Different configurations used in Lemma \[lem:3\_planar\_small\_faces\].](images/3planar_fan_crossing "fig:"){width="\textwidth"} \[fig:3fan\_middle\_b\]
Finally, we show that at least two of $[u_1,u_2]$, $[u_1,u_3]$ and $[u_2,u_3]$ or at least two of $[v_1,v_2]$, $[v_1,v_3]$ and $[v_2,v_3]$ are not [potential edges]{}, which, by our previous arguments, implies that the third [side-edge]{}is not a [potential edge]{}either. If for example $[u_1,u_2]$ and $[u_1,u_3]$ are [potential edges]{}, then neither $[v_1,v_2]$ nor $[v_1,v_3]$ is a [potential edge]{}, as otherwise, either $(u_1,v_1)$ and $(u_2,v_2)$ are [side-apart]{}or $(u_1,v_1)$ and $(u_3,v_3)$ are [side-apart]{}, contradicting our previous observation.
By Claim \[clm:fan\_planar\] we can assume w.l.o.g. that [side-edges]{}$[u_1,u_2]$, $[u_1,u_3]$ and $[u_2,u_3]$ are not [potential edges]{}in $\Gamma(G)$. This implies that regions $R_{1,2}$, $R_{2,3}$ and $R_{1,3}$ do not contain any vertex in their interiors (and also $u_1=u_2=u_3$). Hence, each edge of $\mathcal{X}(e)$ which is crossed by three edges in $\Gamma(G)$ complies with the crossing pattern of Figure \[fig:3fan\], where the red-shaded region has no vertices in its interior. Now, vertices $u$, $v_1$, $v_2$, $v$, $u_2$ and $u_1$ define a [potential empty cycle]{}$\mathcal{C}$ on six vertices. Our goal is to use Lemma \[lem:size9\], whose precondition C.\[cnd:1\] requires at most $8$ edge-segments within $\mathcal{C}$. Note that since $(u,v)$ has three crossings and since each of $(u_1,v_1)$, $(u_2,v_2)$ and $(u_3,v_3)$ has one crossing, there may exist at most $10$ with at least one edge-segment within $\mathcal{C}$; see also Figure \[fig:3\_planar\_three\_crossing\]. In the following, we prove that this is not the case.
Any edge crossing $(u_2,v_2)$ in the interior of $\mathcal{C}$ must also cross $(u_1,v_1)$ or $(u_3,v_3)$. \[clm:shared\_crossing\]
Suppose that edge $(u',v')$ crosses $(u_2,v_2)$ at point $c_2'$ in the interior of $\mathcal{C}$. Recall that $c_2$ denotes the crossing point between $(u_2,v_2)$ and $(u,v)$. Since $(u_2,v_2) \in \mathcal{X}((u,v))$, edge $(u_2,v_2)$ is not crossed by [side-apart]{}edges. So, edges $(u',v')$ and $(u,v)$ are not [side-apart]{}, and exactly one of [side-edges]{}$[u,u']$ or $[v,v']$ is not a [potential edge]{}. Assume w.l.o.g. that [side-edge]{}$[u,u']$ is not a [potential edge]{}; see Figure \[fig:3fan\_middle\_a\]. This implies that $u=u'$ and that the region $R_{u,u'}$ defined by the edge-segment of $(u,v)$ between $u$ and $c_2$, the edge-segment of $(u_2,v_2)$ between $c_2$ and $c_2'$ and the edge-segment of $(u',v')$ between $c_2'$ and $u'$ has no vertices in its interior (red-shaded in Figure \[fig:3fan\_middle\_a\]). Then, edge $(u',v')$ must cross $(u_1,v_1)$, as otherwise vertex $v_1$ would be in the interior of $R_{u,u'}$; see Figure \[fig:3fan\_middle\_b\]. This completes the proof of this claim.
Recall that our goal is to use Lemma \[lem:size9\]. Claim \[clm:shared\_crossing\] implies that there exist at most four other edges that cross edges $(u_1,v_1)$, $(u_2,v_2)$ or $(u_3,v_3)$, i.e. we have at most $8$ edges that are either drawn in the interior of $\mathcal{C}$ or cross its boundary. Since one can draw eight chords in the interior of $\mathcal{C}$ as in Figure \[fig:3\_planar\_6gon\], by Lemma \[lem:exchange\].(\[prp:boundary\]), for $\kappa+\lambda\leq8$ and $\mu=8$, it follows that the boundary edges of $\mathcal{C}$ exist in $\Gamma(G)$. Furthermore $\kappa+\lambda=8$ must hold. Note that the set $E_{\mathcal{C}}$ of edge-segments within $\mathcal{C}$ contains only edge-segments of these $\kappa+\lambda$ edges. Also, these $8$ edges have exactly one edge-segment within $\mathcal{C}$ that is crossed in the interior of $\mathcal{C}$. Hence conditions C.\[cnd:1\] and C.\[cnd:2\] of Lemma \[lem:size9\] are satisfied, and therefore we conclude that $(u,v)$ is a chord of a true planar $6$-cycle.
By Lemma \[lem:3\_planar\_small\_faces\], any edge of $G$ that is crossed three times in $\Gamma(G)$ is a chord of an empty true-planar $6$-cycle. In the following, we consider edges of $G$ that have two or fewer crossings in $\Gamma(G)$. Hence, their crossing components contain edges with at most two crossings. Our approach is slightly different than the one we followed in the proof of Lemma \[lem:2\_planar\_small\_faces\] for the optimal $2$-planar graphs.
Let $\Gamma(G)$ be a PMCM-drawing of an optimal $3$-planar graph $G$ and let $\mathcal{X}$ be a crossing component of $\Gamma(G)$. Then, there is at least one edge in $\mathcal{X}$ that has three crossings. \[lem:3\_planar\_small\_faces\_2\]
Assume to the contrary that there exists a crossing component $\mathcal{X}$ where all edges have at most two crossings. We distinguish two cases depending on whether $\mathcal{X}$ contains an edge with two crossings or not. Assume first that $\mathcal{X}$ does not contain an edge with two crossings. Then, $|\mathcal{X}|=2$. W.l.o.g. assume that $\mathcal{X}=\{e,e'\}$. The four endpoints of edges $e$ and $e'$ define a [potential empty cycle]{}$\mathcal{C}$ on $4$ vertices; see Figure \[fig:3\_planar\_one\_crossing\_before\]. Since $e$ and $e'$ have only one crossing each, the [potential edges]{}of the boundary of $\mathcal{C}$ exist in $\Gamma(G)$ and are true-planar edges. Note that there are no other edges passing through the interior of $\mathcal{C}$. We proceed by removing edges $e$ and $e'$ and replace them with the $3$-planar pattern of Figure \[fig:3\_planar\_one\_crossing\_after\]. In particular we add a vertex $x$ in the interior of $\mathcal{C}$ and true-planar edge $(v',x)$. Vertices $u$, $v'$, $x$, $v'$, $v$ and $u'$ define a [potential empty cycle]{}on six vertices, and we can add $8$ crossing edges in its interior as in Figure \[fig:3\_planar\_6gon\]. If $G$ has $n$ vertices and $m$ edges, the derived graph $G'$ has $n'=n+1$ vertices and $m'=m-2+8$ edges Then, $G'$ is $3$-planar and has $m'=5.5n'-10.5$ edges, that is, $G'$ has more edges than allowed by $3$-planarity; a contradiction.
To complete the proof, assume that there exists an edge $(u,v)\in \mathcal{X}$ which has two crossings, say with $(u_1,v_1)$ and $(u_2,v_2)$. By Lemma \[lem:3\_planar\_independent\], $(u_1,v_1)$ and $(u_2,v_2)$ are not [side-apart]{}. Since all edges in $\mathcal{X}$ have at most two crossings, adopting the proof of Lemma \[lem:2\_planar\_small\_faces\] we can prove that the endpoints of $(u,v)$, $(u',v')$ and $(u'',v'')$ define a [potential empty cycle]{}$\mathcal{C}$ on five vertices, with at most five edges passing through its interior. We proceed by redrawing these five edges as chords of $\mathcal{C}$ (as in Figure \[fig:2\_planar\_5gon\]). All its boundary edges are true-planar in the new drawing. The derived graph is optimal, as it has at least as many edges as $G$. Observe, however, that $\mathcal{C}$ becomes a true-planar $5$-cycle in the new drawing; a contradiction to Property \[prp:3planar\_even\_order\].
![ Configurations used in Lemma \[lem:3\_planar\_small\_faces\_2\].](images/3planar_one_crossing "fig:"){width="\textwidth"} \[fig:3\_planar\_one\_crossing\_before\]
![ Configurations used in Lemma \[lem:3\_planar\_small\_faces\_2\].](images/3planar_one_crossing "fig:"){width="\textwidth"} \[fig:3\_planar\_one\_crossing\_after\]
. \[fig:3\_planar\_one\_crossing\_1\]
The proof of Lemma \[prp:3\_planar\_skeleton\_connected\] is similar to the one of Lemma \[prp:2\_planar\_skeleton\_connected\] and is therefore omitted.
The true planar skeleton $\Pi(G)$ of a PMCM-drawing $\Gamma(G)$ of an optimal $3$-planar graph is connected. \[prp:3\_planar\_skeleton\_connected\]
The true-planar skeleton $\Pi(G)$ of a PMCM-drawing $\Gamma(G)$ of an optimal $3$-planar graph $G$ contains only faces of length $6$, each of which contains $8$ crossing edges in $\Gamma(G)$. \[lem:3\_planar\_faces\]
Since by Lemma \[prp:3\_planar\_skeleton\_connected\] $\Pi(G)$ is connected, all faces of $\Pi(G)$ are connected as well. By Lemma \[lem:3\_planar\_small\_faces\], any edge that is crossed three times in $\Gamma(G)$ is a chord of an empty true-planar $6$-cycle in $\Gamma(G)$. By Lemma \[lem:3\_planar\_small\_faces\_2\], an edge $e$ that is crossed fewer than three times belongs to a crossing component $\mathcal{X}(e)$ containing an edge that is crossed three times. This last edge defines an empty true-planar $6$-cycle in $\Gamma(G)$ and by the observation we made in the proof of Lemma \[lem:3\_planar\_xing\_comp\] all edges of $\mathcal{X}(e)$, including $e$, are also chords of this cycle. So, every crossing edge is a chord of a true-planar $6$-cycle. Note that one cannot embed nine edges in the interior of a true-planar $6$-cycle without deviating $3$-planarity but at most eight. We claim that $\Pi(G)$ has no chordless faces. First, we observe that $\Pi(G)$ cannot contain a chordless face of size $\geq 4$, as otherwise we could draw in its interior at least one chord, which would contradict the optimality of $G$. Also, by Property \[prp:3planar\_odd\_cycle\] $\Pi(G)$ contains no faces of length $3$. Finally, observe that $\Pi(G)$ cannot contain faces of length $1$ or $2$, as those would correspond to homotopic self-loops and parallel edges. This completes the proof.
We say that a chord of a cycle of length $2s$ is a *middle chord* if the two paths along the cycle connecting its endpoints both have length $s$. Next we state the main theorem of this section.
A graph $G$ is optimal $3$-planar if and only if $G$ admits a drawing $\Gamma(G)$ without homotopic parallel edges and self-loops, such that the true-planar skeleton $\Pi(G)$ of $\Gamma(G)$ spans all vertices of $G$, it contains only faces of length $6$ (that are not necessarily simple), and each face of $\Pi(G)$ has $8$ crossing edges in its interior in $\Gamma(G)$ such that one of the middle chords is missing. \[thm:3-characterization\]
For the forward direction, consider an optimal $3$-planar graph $G$. By Lemma \[lem:3\_planar\_faces\], the true-planar skeleton $\Pi(G)$ of its $3$-planar PMCM-drawing $\Gamma(G)$ contains only faces of length $6$ and each face of $\Pi(G)$ has $8$ crossing edges in its interior in $\Gamma(G)$. By Property \[prp:3\_planar\_quasi\], one of the three middle chords of each face of $\Pi(G)$ cannot be present. Since the endpoints of two crossing edges are within a true-planar $6$-cycle (by Lemmas \[lem:3\_planar\_small\_faces\] and \[lem:3\_planar\_small\_faces\_2\]) and since $\Pi(G)$ is connected (by Lemma \[prp:3\_planar\_skeleton\_connected\]), $\Pi(G)$ spans all vertices of $G$. This completes the proof of this direction.
For the reverse direction, denote by $n$, $m$ and $f$ the number of vertices, edges and faces of $\Pi(G)$. Since $\Pi(G)$ spans all vertices of $G$, it suffices to prove that $G$ has exactly $5.5n-11$ edges. The fact that $\Pi(G)$ contains only faces of length $6$ implies that $6f=2m$. By Euler’s formula for planar graphs, $m=3(n-2)/2$ and $f=(n-2)/2$ follows. Since each face of $\Pi(G)$ contains exactly $8$ crossing edges, the total number of edge of $G$ equals to $m+8f=5.5n-11$.
Further Insights From Our Work {#sec:discussion}
==============================
In this section, we give new insights which follow from the new characterization of optimal $2$- and $3$-planar graphs. For simple optimal $3$-planar graphs we can note the following. Since the planar skeleton of an optimal $3$-planar graph consists exclusively of faces of length $6$, it cannot be simple. Hence, simple $3$-planar graphs do not reach the bound of $5.5n -11$ edges. Note that the best-known lower bound for simple optimal $3$-planar graph is $5.5n-15$ [@PachT97].
Simple $3$-planar graphs have at most $5.5n - 11.5$ edges. \[rec:simple3-planar\]
A *bar-visibility representation* of a graph is a representation where vertices are represented as horizontal bars, and edges as vertical segments, called *visibilities*, between corresponding bars. In the traditional bar-visibility model, a visibility edge is not allowed to cross any other bar except for the two bars at its endpoints. A central result here is due to Tamassia and Tollis [@DBLP:journals/dcg/TamassiaT86] who showed that any biconnected planar graph admits a bar-visibility representation, which can be computed in linear time. The variant of *bar 1-visibility* allows each visibility edge to cross at most one vertex bar. This model allows to represent also non-planar graphs in a limited way, e.g., the number of edges of a bar 1-visible graph on $n$ vertices can be at most $6n-20$ [@DBLP:journals/jgaa/DeanEGLST07]. Notable is a result by Brandenburg [@DBLP:journals/jgaa/Brandenburg14] who showed that $1$-planar graphs admit bar 1-visibility representations; see also [@DBLP:journals/jgaa/Evans0LMW14].
We follow a similar technique to the one of Brandenburg [@DBLP:journals/jgaa/Brandenburg14] to prove that simple optimal $2$-planar graphs are bar 1-visible. Since the faces defined by the true-planar skeleton $\Pi(G)$ of a simple optimal $2$-planar graph $G$ have size $5$, we can construct a bar-visibility representation $\mathcal{L}(G)$ of $\Pi(G)$ based on an $s$-$t$ ordering of $\Pi(G)$ [@DBLP:journals/dcg/TamassiaT86]. In the $s$-$t$ ordering each face is oriented such that it consists of a source and a target vertex joined by two chains of vertices (one on the left and one on the right). Since $\Pi(G)$ consists of faces of length $5$, the two chains have either $1$ and $2$ vertices each, or, $0$ and $3$ vertices each. In $\mathcal{L}(G)$, the source and target bars of a face $f$ see each other through a vertical visibility edge $b_f$ and the bars of the two chains are arranged to the left and to the right of $b_f$. Now it is straightforward to extend the bars of the two chains towards $b_f$, such that the bars of the two chains are vertically overlapping, and all five crossing edges of that face are realized. We conclude this observation in the following corollary.
Simple optimal $2$-planar graphs admit bar $1$-visibility representations. \[rec:2-planar-bar1\]
In a *fan-planar drawing* of a graph an edge can cross only edges with a common endpoint. Graphs that admit fan-planar drawings are called *fan-planar*. Fan-planar graphs have been introduced by Kaufmann and Ueckerdt [@KU14], who proved that every simple $n$-vertex fan-planar drawing has at most $5n-10$ edges, and that this bound is tight for $n \geq 20$. This density result immediately implies that optimal $3$-planar graphs are not fan-planar. On the other hand, the density bound of $2$-planar graphs is the same as the one for fan-planar graphs. Binucci et al. [@DBLP:journals/tcs/BinucciGDMPST15] already investigated the relationship between these two classes and in particular they proved that there exist $2$-planar graphs that are not fan-planar. Our characterization for simple optimal $2$-planar graphs, however, implies that all optimal $2$-planar graphs are fan-planar, as their PMCM-drawings are in fact fan-planar. We conclude this observation in the following corollary.
Simple optimal $2$-planar graphs are optimal fan-planar. \[rec:2-planar-fan\]
Our characterizations naturally lead to many open questions. In the following we name a few.
- What is the complexity of the recognition problem for optimal $2$- and $3$-planar graphs?
- What is the exact upper bound on the number of edges of simple optimal $3$-planar graphs? We conjecture that they do not have more than $5.5n-15$ edges.
- Theorems \[thm:2-characterization\] and \[thm:3-characterization\] imply that optimal $2$- and $3$-planar graphs have a fully triangulated planar subgraph. Can this property be proved for optimal $4$-planar or more in general for optimal $k$-planar graphs? Proving this property would be useful to derive better density bounds for $k\geq4$.
- By Properties \[prp:2\_planar\_quasi\] and \[prp:3\_planar\_quasi\], optimal $2$- and $3$-planar graphs are quasi-planar. Angelini et al. [@ABBL17] proved that every simple $k$-planar graph is $(k+1)$-quasi planar for $k \geq 3$ (i.e., it can be drawn with no $k+1$ pairwise crossing edges). Our results about optimal $2$-planar and even more about optimal $3$-planar graphs give indications that the result by Angelini et al. [@ABBL17] may hold also for $k=2$.
- We have found a RAC drawing (i.e., a drawing in which all crossing edges form right angles) with at most one bend per edge for the optimal $2$-planar graph obtained from the dodecahedron as its true-planar structure. Is this generalizable to all simple optimal $2$-planar graphs?
[^1]: Note that the boundary edges of $\mathcal{C}_s$ are not necessarily present in $\Gamma(G)$.
[^2]: We say that a Jordan curve $[u,v]$ *exists* in $\Gamma(G)$ if and only if $[u,v]$ is homotopic to an edge in $\Gamma(G)$.
[^3]: One can view this process as replacing $\Gamma(H_2)$ with a single vertex; thus no homotopic parallel edges exist in $\Gamma(H_1)$. Then we move this vertex towards the edge-segment we want to subdivide until it touches it.
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The chemical potential of the electron gas on a two dimensional lattice
V.Celebonovic
Institute of Physics,Pregrevica 118,11080 Zemun- Beograd,Yugoslavia
e-mail: vladan@phy.bg.ac.yu
Abstract: The chemical potential of the electron gas on a two-dimensional rectangular lattice is determined.An approximate expression for $\exp(-\mu/T)$ is obtained,and its second order approximation is discussed in some detail.
Introduction
Investigations of correlated electron systems of various dimensionality are one of the priorities of contemporary condensed matter physics (for example \[1\]).The driving force behind the scientific interest in such systems is the fact that they exist in high temperature superconductors and organic conductors.Although the existence of correlated electrons in these materials is regarded as an “established fact”,it is far from being completely clear how does one theoretically describe them \[2\].The question is if it is possible to use the Fermi liquid model,or the Luttinger liquid model has to be used.For a recent introduction to the Luttinger liquid theory see for example \[3\].
The aim of the present paper is to calculate the chemical potential of the electron gas on a two dimensional rectangular lattice.Theoretically,this is an interesting problem in itself,and the experimental motivation is amply discussed in the literature (for example \[4\]). The calculation will be performed within the standard Fermi liquid (FL) model.This presumption could be criticized on the grounds that there are recent results indicating the breakdown of the FL model in cuprate superconductors and heavy-fermion systems(such as \[5\]).The idea of the present calculation is to contribute to this result,in a different way: by calculating the chemical potential,then separately the electrical conductivity and finally drawing conclusions by comparison with experimental data.
Calculations
The lattice axes are designated by x and y.It can be shown that the number density of a 2D Fermi liquid is given by \[6\]: $$\label{(1)}\ n=\frac{<N>}{S}= \int\frac{exp(\beta(\mu-\epsilon))}{1+exp(\beta(\mu-
\epsilon))} 2 \frac{d^{2}p}{(2\pi\hbar)^{2}}$$ where $n$ denotes the number density and all the other symbols have their usual meanings and $p=\hbar k$. It then follows that $dp=\hbar\frac{\partial k}{\partial\epsilon}d\epsilon$,where $\epsilon$ denotes the energy.Accordingly, $$\label{(2)}\ d^{2}p = dp_{x}dp_{y}=\hbar^{2}\frac{\partial
k_{x}}{\partial\epsilon_{x}}\frac{\partial
k_{y}}{\partial\epsilon_{y}}d\epsilon_{x}d\epsilon_{y}$$
Assuming that the energy bands along each of the lattice axes have the form $\epsilon=-2tcos(ks)$,where $t$ denotes the hopping integral and $s$ the lattice constant along the axis,it follows that $$\label{(3)} \frac{\partial k_{x}}{\partial\epsilon_{x}}=\frac{1}{2s_{x}t_{x}}\frac{d\epsilon_{x}}{\surd(1-(\epsilon_{x}/(2t_{x}))^{2})}$$ An analogous relation is valid along the y axis.This implies that $$\label{(4)}\ d^{2}p = \frac{\hbar^{2}}{4
s_{x}t_{x}s_{y}t_{y}}\frac{d\epsilon_{x}d\epsilon_{y}}{\surd(1-
(\epsilon_{x}/(2t_{x}))^{2}\surd 1-(\epsilon_{y}/(2t_{y}))^{2}}$$ Using the expressions for the particle energy along the lattice axes,the exponential factor in eq.(1) can be expressed as $$\label{(5)} \exp(\beta(\mu-\epsilon))=\exp(\beta(\mu-(\epsilon_{x}+\epsilon_{y})))$$ Inserting eqs.(4) and (5) into eq.(1),after some algebra leads to $$\begin{aligned}
n = \nonumber
\frac{1}{8\pi^{2}s_{x}t_{x}s_{y}t_{y}}\int\int\frac{d\epsilon_{x}d\epsilon_{y}}{1+exp(\beta(\epsilon_{x}+\epsilon_{y}-\mu))}
\nonumber \\
\frac{1}{\surd(1-(\epsilon_{x}/(2t_{x})^{2})}
\frac{1}{\surd(1-(\epsilon_{y}/(2t_{y})^{2}))}\end{aligned}$$
which is the required equation linking various measurable parameters of the electron gas and the lattice with the chemical potential of the electron gas.The limits of integration are $\pm2t$ along each of the lattice axes,and the problem is how to solve this integral,which can not be done in closed analytical form.Looking at things in a purely mathematical way,the easiest way to solve this integral is to develop the sub-integral function into series. Reasoning in this way,the Fermi distribution function can be re-expressed as follows,under the condition that $\epsilon_{x}+\epsilon_{y}\prec\mu$:
$$\label{(7)}
\frac{1}{1+exp(\beta(\epsilon_{x}+\epsilon_{y}-\mu))}=\sum_{l=0}^{l=\infty}(-1)^lexp(\beta
l(\epsilon_{x}+\epsilon_{y}-\mu))$$
Introducing this development into eq.(6),and slightly grouping the terms in it,one gets the following expression:
$$\begin{aligned}
n=\nonumber
\frac{1}{8\pi^{2}s_{x}t_{x}s_{y}t_{y}}\sum_{l=0}^{l=\infty}(-1)^{l}\int\int
exp(\beta l (\epsilon_{x}+\epsilon_{y}-\mu))\nonumber\\
\frac{d\epsilon_{x}d\epsilon_{y}}{\surd
(1-(\epsilon_{x}/2t_{x})^{2})\surd(1-(\epsilon_{y}/2t_{y})^{2})}\end{aligned}$$
Integrating this expression,within the limits of $\pm2t$ along both axes, and performing some algebra,leads to the following final result for the filling factor of a Fermi gas on a two dimensional lattice: $$n=\nonumber \frac{1}{2s_{x}s_{y}}
\sum_{l=0}^{\infty}(-1)^{l}exp(-\beta\mu
l)I_{0}(2lt_{x}\beta)I_{0}(2lt_{y}\beta)$$ where the symbol $I_{n}(x)$ denotes the modified Bessel function of the first kind of the order $n$ and $\beta$ is the inverse temperature.The functions $I_{n}(x)$ are defined by: $$I_{n}(x)=\sum_{k=0}^{k=\infty}\frac{(x/2)^{n+2k}}{k!\Gamma(n+k+1)}$$
Discussion
Equation (9) represents the general result of this paper,expressed in its most compact form.The obvious question is the applicability of this result to real physical systems. Limiting the summation in eq$(9)$ to terms with $l\le2$ gives $$\begin{aligned}
n=\frac{1}{2s_{x}s_{y}}[1-\exp(-\mu/T)I_{0}(\frac{2t_{x}}{T}))I_{0}\frac{2t_{y}}{T}+\nonumber\\ \exp(-2 \mu/T)I_{0}(\frac{4t_{x}}{T})I_{0}(\frac{4t_{y}}{T})]\end{aligned}$$
Solving this equation for $\exp(-\mu/T)$ leads to
$$\begin{aligned}
exp(-\mu/T)= (I_{0}(\frac{2t_{x}}{T})I_{0}(\frac{2t_{y}}{T})\pm\nonumber\\\sqrt{I_{0}^{2}(\frac{2t_{x}}{T})I_{0}^{2}(\frac{2t_{y}}{T})-4(1-2ns_{x}s_{y})I_{0}(\frac{4t_{x}}{T})I_{0}(\frac{4t_{y}}{T})})/(2I_{0}(\frac{4t_{x}}{T})I_{0}(\frac{4t_{y}}{T})) \end{aligned}$$
Developing this equation into series up to second order terms in $t_{x}$ and $t_{y}$ as small parameters,gives the following expression for $exp(-\mu/T)$:
$$\begin{aligned}
exp(-\mu/T)\cong\frac{1}{2}(1-\sqrt{8ns_{x}s_{y}-3})+\frac{1}{2}[
(t_{x}/T)^{2}+(t_{y}/T)^{2}]\nonumber\\(\frac{16ns_{x}s_{y}-5}{\sqrt{8ns_{x}s_{y}-3}}-3)+\frac{1}{2}(\frac{t_{x}t_{y}}{T^{2}})^{2}(9+\frac{16ns_{x}s_{y}(9-16ns_{x}s_{y})-17}{(8ns_{x}s_{y}-3)^{3/2}})+.\end{aligned}$$
Compare this result with \[7\] where an expression for the chemical potential of a 1D electron gas has been derived.It follows from eq.(13) that there exists a lower bound on the values of $n$ for which it is applicable.The condition for validity of eq.(13) is $8ns_{x}s_{y}-3\succ0$ which implies $n\succ\frac{3}{8s_{x}s_{y}}$.The implication of this limitation is that eq.(13) can not be used for extremely small values of $n$. Inspection of this expression shows that it represents the dependence of the chemical potential of a 2D electron gas on various measurable parameters of the system.As the lattice constants along both lattice axes are taken into account,it follows that this equation gives the possibility of analyzing the influence of high external pressure on the chemical potential. This possibility could turn out as being useful in calculations of the pressure dependence of the electrical conductivity of 2D organic conductors.
An interesting problem is the value of $n$ (the filling factor) for which the chemical potential becomes equal to zero.It follows after some algebra that $\mu=0$ for $n=n_{0}$,where
$$\begin{aligned}
n_{0}=\frac{7T^{4}-T^{2}[1+16(t_{x}^{2}+t_{y}^{2})]+32(t_{x}^{2}+t_{y}^{2})^{2}}{16s_{x}s_{y}(T^{2}-2(t_{x}^{2}+t_{y}^{2}))^{2}}\nonumber\\\sqrt{T^{4}+10T^{2}(t_{x}^{2}+t_{y}^{2})+(t_{x}^{2}+t_{y}^{2})^{2}}\end{aligned}$$
The outright implication of this result is that the Lieb-Wu theorem \[8\] is not generally applicable to 2D systems. It can be applied only for certain values of various parameters of the system which enter into eq.(14).It also follows from the last equation that $n_{0}$ is a function of the external pressure to which the specimen is subdued.
Conclusions
In this paper we have determined the chemical potential of the electron gas on a 2D rectangular lattice.A general expression linking the filling factor,the chemical potential and various measurable parameters of this system was obtained.Starting from this general expression,an approximate expression for the chemical potential has been obtained. It turns out that this expression is much more complicated than the corresponding expression for a 1D electron gas.This expression can be applied in analyzing real experimental data,and in calculations of the electrical conductivity of Q2D organic metals,which will be done in future work.
Acknowledgement
This work was performed as a part of the project 1231 financed by the MNTRS in Beograd.The purchase of one of the references (J.Low Temp.Phys) was made possible by a donation of the Royal Dutch Embassy in Yugoslavia.
References
Springer Verlag,Heidelberg,(1998).
T.Götzfried,A.Weber,K.Heuser et. al.,J.Low Temp.Phys.,[**127**]{},51 (2002).
W.A.Benjamin Inc.,London (1976).
and preprint cond-mat/0207529 (2002).
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---
abstract: 'Time-periodic (Floquet) drive is a powerful method to engineer quantum phases of matter, including fundamentally non-equilibrium states that are impossible in static Hamiltonian systems. One characteristic example is the anomalous Floquet insulator, which exhibits topologically quantized chiral edge states similar to a Chern insulator, yet is amenable to bulk localization. We study the response of this topological system to time-dependent noise, which breaks the topologically protecting Floquet symmetry. Surprisingly, we find that the quantized response, given by partially filling the fermionic system and measuring charge pumped per cycle, remains quantized up to finite noise amplitude. We trace this robust topology to an interplay between diffusion and Pauli blocking of edge state decay, which we postulate should be robust against interactions. We comment on quantization of other topological responses in the absence of Floquet symmetry and potential experimental realizations.'
author:
- 'Christopher I. Timms, Michael H. Kolodrubetz'
bibliography:
- 'dissipative\_afai.bib'
title: Quantized Floquet topology with temporal noise
---
*Introduction* – Periodic Floquet drive is an indispensable tool in engineered quantum systems [@Aidelsburger_2013; @Wang_2013; @Miyake_2013; @Roushan_2016; @Boyers_2019; @Oka_2019]. Recently, Floquet drive has enabled the realization of fundamentally non-equilibrium phases of matter, such as Floquet time crystals [@Keyserlingk2016; @Else2016; @Choi_2017; @Zhang_2017; @rovny2018observation; @autti2018observation; @else2019discrete; @khemani2019brief] and Floquet symmetry-protected topological states (SPTs) [@chandran2014many; @Nathan_2015; @Roy2016; @Keyserlingk2016a; @Else2016a; @potter2016classification; @Roy2017; @po2016chiral; @Potter2017; @Harper2017; @Roy2017a; @Po2017; @Potter2018; @Reiss2018]. A quintessential example of Floquet SPT is the anomalous Floquet-Anderson insulator (AFAI), which has topologically protected chiral edge states similar to a Chern insulator but with a fully localizable bulk [@rudner2013anomalous; @titumphysical; @leykam2016anomalous; @maczewsky2017observation]. Topologically protected transport in the AFAI can be measured via current flowing through the system [@titumphysical; @kundu2020quantized], magnetization density in a fully-filled patch within the bulk [@nathan2019anomalous], or quantized transport of quantum information at the edge [@po2016chiral; @PhysRevB.98.054309; @fidkowski2019interacting].
![Illustration of quantized non-adiabatic pumping in the presence of noise. (a) The 2D system is placed on a cylinder with the top half filled and bottom half empty. Current $Q$ around the cylinder per Floquet cycle is quantized without noise. Noise is added by disordering the timings of the 5-step drive ([Fig. \[Figure.Model\]]{}). (b) For weak noise, $Q$ goes to a topological plateau before decaying when the edge states start to depopulate at times of order the Thouless time.[]{data-label="Figure.Intro"}](fig1.pdf){width="\columnwidth"}
All of these non-equilibrium states are protected by discrete time-translation symmetry of the Floquet Hamiltonian, $H(t) = H(t+T)$, where $T=2\pi/\Omega$ is the driving period. In this Letter, we ask what happens to the AFAI upon breaking time-translation symmetry via time-dependent random noise. A similar question has been studied in the case of a Floquet SPT protected by chiral symmetry [@sieberer2018statistical; @rieder2018localization], where the authors found that edge states decay at a slow but finite rate set by diffusion. In this work, we instead find that for the two most realistic experimental protocols, namely bulk magnetization or current measurements in partially-filled samples, the topological response remains *fully protected* over a time scale that diverges in the thermodynamic limit. We trace this topological protection back to Pauli blocking, which prevents diffusive loss of the topological edge state pumping up to approximately the Thouless time. We argue that the results should hold for many-body localization as well as Anderson localization, and comment on the potential for experimental realization.
*Model* – Throughout this paper, we study a single-particle model of the anomalous Floquet-Anderson insulator (AFAI) with time-dependent noise. We start from the original AFAI model [@titumphysical], which involves a 5-step Floquet drive. The first four steps involve hopping between sites of the two sublattices. Specifically, for step $\ell\in \{1,2,3,4\}$, the Hamiltonian is $H_\ell=-J\sum_{<ij>_\ell} c_i^\dagger c_j$, where $<ij>_\ell$ indicates the bonds that are “turned on” during step $\ell$, as illustrated in [Fig. \[Figure.Model\]]{}a. During step 5, a sublattice-dependent potential of strength $\Delta$ is applied: $H_5=\Delta \sum_{j} \eta_j c_j^\dagger c_j$, where $\eta_j=+1$ ($-1$) on the A (B) sublattice. Each Hamiltonian $H_\ell$ is present for time $T_\ell$, which in the absence of temporal order is just $T_\ell=T/5$, where $T=2\pi/\Omega$ is the driving period. The hopping Hamiltonians $H_{1-4}$ are chosen such that, for the fine-tuned value $J=J_0\equiv 5 \Omega /4$, bulk electrons undergo a “cyclotron” orbit during each Floquet cycle and return to their original site, as illustrated in [Fig. \[Figure.Model\]]{}. A static chemical potential disorder is added throughout the cycle with Hamiltonian $H_\mathrm{dis}=\sum{\mu_j c_j^\dagger c_j}$, where each $\mu_j$ is uniformly sampled from the interval $[-W,W]$. Units are set by $\Omega=\hbar=1$, and we choose $\Delta = 0.4\Omega$ and $J=J_0=5\Omega/4$ throughout.
![Noisy AFAI model. (a) First 4 steps of drive protocol. Hopping occurs on bonds labeled $1$ for $0<t<T_1$, on bonds labeled $2$ for $T_1 < t< T_1+T_2$, etc. (b) Noise is added by randomly changing the time over which the Hamiltonians are present, $T_\ell=T(1+\delta_\ell)/5$. The random noise $\delta_\ell\in[-W_T,W_T]$ is different for each “Floquet” cycle. (c) Charge pumped per Floquet cycle for 1D spatial disorder with $W=0.2$ and $L=100$. Dashed lines show non-quantized plateau value.[]{data-label="Figure.Model"}](fig2.pdf){width="\columnwidth"}
In this work, we modify the Hamiltonian by adding temporal disorder (noise). Explicitly, noise is introduced via random modification of the Floquet timing: $T_\ell = T(1+\delta_\ell)/5$, where $\delta_\ell \in [-W_T,W_T]$ is sampled independently during each Floquet cycle [^1]. Naively, one expects that noise will immediately destroy the Floquet topological phase, as it breaks the time periodicity [@sieberer2018statistical]. Yet, as we will show, the topological response remains robust against weak noise due to special properties of the AFAI’s topological response.
In our numerics, we measure topologically protected non-adiabatic charge pumping for a cylinder of $L_x=2L$ and $L_y=L$ lattice sites [@titumphysical]. As shown in [Fig. \[Figure.Intro\]]{}a, the system is initialized with one half of the cylindrical crystal filled with particles and the other half left empty. We measure the charge pumped per cycle, $$\label{ChargePumpedEquation}
Q = \int_{0}^{\tilde T} dt \langle\psi(t)|J_x|\psi(t)\rangle,$$ where $J_x$ is the current in the $x$-direction and $\tilde T = \sum_\ell T_\ell$ is the “Floquet” period appropriately modified by noise. In the absence of temporal disorder, Titum et. al [@titumphysical] demonstrated quantization of $Q$ in the presence of spatial disorder. One may think of this quantization as coming from the single filled edge state, which pumps $Q=1$ per cycle in the topological phase, while the localized bulk states do not carry current. In the presence of temporal disorder, the bulk states no longer remain localized; we now demonstrate how this affects $Q$.
*One-dimensional disorder* – Large two-dimensional (2D) lattices without translation symmetry are computationally challenging to simulate. Therefore, as a warmup problem in which we can address large system sizes, we begin by implementing one-dimensional (1D) spatial disorder in the y-direction, meaning that for site $j=(x,y)$, $\mu_j$ only depends on the $y$ position. Such 1D disorder is an equivalent to the Floquet-Thouless energy pump [@kolodrubetz2018topological], where the pump parameter $\lambda$ plays the role of the conserved momentum $k_x$. With 1D spatial disorder, large system sizes and long times may be readily simulated.
Some characteristic traces of $Q$ vs. $t$ are shown in [Fig. \[Figure.Model\]]{}c. For weak temporal and spatial disorder, the charge approaches a plateau value and remains nearly perfectly quantized up to more than 2000 drive cycles. As $W_T$ is increased, the plateau value of the pumped charge is no longer quantized and the pumped charge begins to decay at late times. To quantify this behavior, we define two quantities: the plateau value of pumped charge, $Q^{*}$, and the decay time scale, $\tau$.
![Finite size effects for 1D disorder. (a) System size dependence of $Q$ for $W=0.5$ and $W_T=0.6$. The dashed lines show times $\tau \sim L^2$, illustrating that the pumped charge begins to decay on of order the Thouless time, which is set by diffusion. (b) Comparison of plateau value $Q^\ast$ for actual time disorder and “Floquet time disorder,” in which the same random pattern of $\delta_\ell$ is repeated indefinitely. Finite size effects have been removed by extrapolating to $L\to\infty$ using a linear fit to $Q^\ast$ versus $1/L$ at large $L$.[]{data-label="Figure.finite_size_and_Qstar_1d_disorder"}](fig3.pdf){width="\columnwidth"}
{width="80.00000%"}
The key to understanding these quantities is their dependence on system size $L$, shown in [Fig. \[Figure.finite\_size\_and\_Qstar\_1d\_disorder\]]{}. We see that the plateau value $Q^{*}$ does not depend on system size, while $\tau$ increases sharply with system size. We have confirmed that this finite size dependence of $\tau$ reflects the fact that temporal disorder causes the particles of the system to undergo a diffusive random walk [@rieder2018localization]. The consequence of this diffusion is that the sharp density edge separating the top and bottom half of the system spreads diffusively into a smooth density dependence, until eventually the edge state starts to depopulate on a time scale of order the Thouless time, $t_\mathrm{D} =L^2/D$ with diffusion constant $D$. Since non-adiabatic charge pumping comes entirely from the edge state, the loss of edge state occupation corresponds to a loss of the signal in $Q$, and thus $\tau$ will be proportional to the Thouless time.
We can now draw two important conclusions about the system with one-dimensional disorder. First, the pumped charge reaches a plateau that eventually decays on a time scale $\tau \sim L^2$. Importantly, this implies that the plateau will be infinitely long-lived in the thermodynamic limit, where the topological phase is defined. Second, we learned that the plateau value $Q^\ast$ loses quantization as either spatial or temporal disorder are added, similar to the smooth “topological crossover” of the 1D Floquet-Thouless energy pump that has added spatial disorder [@kolodrubetz2018topological]. We postulate that this physics is, in fact, exactly captured by that of the Floquet-Thouless energy pump. Specifically, we consider the behavior of a related Floquet system created by randomly sampling the times $T_1$, $T_2$, $\ldots$, $T_5$ as before, but then repeating this random sequence for each Floquet cycle. Such a Floquet system will achieve a plateau value $Q^\ast$ and then stay there [@kolodrubetz2018topological], as there is no diffusion to prevent localization. We refer to this construction as “Floquet time disorder.”
We compare the results of actual time disorder and Floquet time disorder in [Fig. \[Figure.finite\_size\_and\_Qstar\_1d\_disorder\]]{}, showing that they match within error bars after extrapolation to the thermodynamic limit. Importantly, each realization of the Floquet time disorder can be analyzed in the language of the Floquet-Thouless energy pump, meaning that our non-topological response with time disorder is obtained by averaging over the non-quantized responses from the Floquet-Thouless energy pump. This explains why the response is not quantized, and provides a valuable method for defining (average) topology in this temporally disordered system.
*Two-dimensional disorder* – Having understood one-dimensional disorder, we can now make predictions for full two-dimensional disorder, in which $\mu_j$ is chosen independently for each site. The dependence of $\tau$ on system size should be the same with 2D disorder, since diffusion still applies. This means that the plateau value $Q^\ast$ should again be infinitely long-lived in the thermodynamic limit. However, a more interesting fact comes out of thinking about this plateau value. Unlike the case of 1D disorder, 2D disorder has a non-trivial topological phase (the AFAI) that survives to finite disorder, with a sharp transition from $Q=1$ to $Q=0$ at finite $W_c$ [@titumphysical]. Therefore, our analysis of 1D disorder implies that the non-trivial topological phase will also survive for weak Floquet time disorder, since this is a perturbative deformation of the original AFAI model. Given the equivalence of pumping for time disorder and Floquet time disorder, we thus predict that the AFAI is *stable* to weak temporal noise.
This intuition is confirmed numerically in [Fig. \[Figure.2d\_disorder\]]{} using the same technique as Titum et. al [@titumphysical]. Specifically, for a given realization of Floquet time disorder, we calculate the Floquet quasienergies $\epsilon^F_n$ and determine the statistics of their nearest-neighbor level spacings: $\Delta_n \equiv \epsilon^F_{n+1}-\epsilon^F_{n}$. We calculate the $r$-statistic [@PhysRevB.75.155111]: $$r_n = \mathrm{min}\left[\Delta_n,\Delta_{n+1}\right]/\mathrm{max}\left[\Delta_n,\Delta_{n+1}\right],$$ whose average value is a useful indicator of level repulsion. $\langle r \rangle$ converges to the Poisson value, $r_P \approx 0.39$, for localized systems that do not display level repulsion and to the circular unitary ensemble (CUE) value, $r_C \approx 0.6$, for delocalized systems. In the present case of non-interacting particles, both the topologically non-trivial phase at low $W$ and the topologically trivial phase at high $W$ are localized, giving $r_P$. Right at the phase transition, the system delocalizes, creating a sharp peak with CUE level statistics. This peak was shown to be a sensitive indicator of the phase transition for the Floquet model [@titumphysical], and we see this holds with Floquet time disorder as well (Figure \[Figure.2d\_disorder\]a) [^2]. Therefore, for a given realization of Floquet time disorder, we can obtain the critical disorder value $W_c$ by finding this peak.
There is one notable effect of Floquet time disorder, namely that different realizations of time disorder yield different values for this critical $W_c$, as seen in Figure \[Figure.2d\_disorder\]b. In other words, Floquet time disorder does not self average. This means that there is not a sharp transition from topologically non-trivial to trivial, but rather a topologically non-trivial phase for $W < W_{c,min}$, a topologically trivial phase for $W > W_{c,max}$, and a crossover region in between where the response is not quantized. The full phase diagram showing these three regions is plotted in [Fig. \[Figure.2d\_disorder\]]{}c, with best estimates for the phase transition lines $W_{c,min/max}$. The topological phase survives up to a relatively large finite value $W_T \approx 0.6$.
*Discussion* – We have shown that the two-dimensional anomalous Floquet-Anderson insulator is stable to weak temporal noise. The argument involves constructing a related Floquet system for a given noise realization and then arguing that if each such realization is topological, then their noise-average is topological as well. This argument should hold for other types of environmental noise, and therefore we postulate that the AFAI is stable to a wide class of weak dissipative couplings. Correlated noise would kill this argument, hence we leave generic non-Markovian baths for future work.
While we numerically studied the topological response via charge pumping in a half-filled system, our arguments indicate that a similar story should hold for other proposed experimental measurements of the anomalous Floquet insulator [@nathan2017quantized; @nathan2019anomalous; @PhysRevB.98.054309]. For instance, topologically quantized magnetization for a filled region of linear size $\ell$ [@nathan2017quantized] should hold up to time $\tau \sim \ell^2$ and remain measurable by the same protocols. This fact will be important in practical experimental realizations, as there are always finite noise sources – such as laser fluctuations or spontaneous emission into lattice lasers – that break the Floquet symmetry of the problem.
It has recently been argued that the AFAI is stable to interactions [@nathan2019anomalous], and we suspect the same will be true in the presence of noise. An interesting question is how noise affects other topological invariants that have been identified in the AFAI [@nathan2019hierarchy], which are also theoretically measurable. Finally, we speculate that similar ideas may be used to demonstrate stability in other Floquet topological phases, such as the Floquet topological superconductor, with possible implications for robust quantum information and computation [@PhysRevLett.106.220402; @PhysRevB.100.041102].
*Acknowledgments* – We would like to acknowledge useful discussions with L. Sieberer, P. Titum, and F. Nathan. This work was performed with support from the National Science Foundation through award number DMR-1945529 and the Welch Foundation through award number AT-2036-20200401. We used the computational resources of the Lonestar 5 cluster operated by the Texas Advanced Computing Center at the University of Texas at Austin and the Ganymede and Topo clusters operated by the University of Texas at Dallas’ Cyberinfrastructure & Research Services Department.
[^1]: Note that one can equivalently keep each step fixed at $T/5$ and rescale the Hamiltonian by a factor $1+\delta_\ell$.
[^2]: Note that, for both $W_T=0$ and finite $W_T$, the plateau value $Q^\ast$ is not a sensitive indicator of the phase transition, requiring inaccesibly large system size in order to see a sharp transition
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abstract: 'Learning to optimize has emerged as a powerful framework for various optimization and machine learning tasks. Current such “meta-optimizers” often learn in the space of continuous optimization algorithms that are point-based and uncertainty-unaware. To overcome the limitations, we propose a meta-optimizer that learns in the algorithmic space of both point-based and population-based optimization algorithms. The meta-optimizer targets at a meta-loss function consisting of both cumulative regret and entropy. Specifically, we learn and interpret the update formula through a population of LSTMs embedded with sample- and feature-level attentions. Meanwhile, we estimate the posterior directly over the global optimum and use an uncertainty measure to help guide the learning process. Empirical results over non-convex test functions and the protein-docking application demonstrate that this new meta-optimizer outperforms existing competitors. The codes and the supplement information are publicly available at: <https://github.com/Shen-Lab/LOIS>.'
author:
- |
Yue Cao, Tianlong Chen, Zhangyang Wang, Yang Shen\
Departments of Electrical and Computer Engineering & Computer Science and Engineering\
Texas A&M University, College Station, TX 77840\
`{cyppsp,wiwjp619,atlaswang,yshen}@tamu.edu`\
bibliography:
- 'Ref.bib'
title: Learning to Optimize in Swarms
---
Introduction
============
Optimization provides a mathematical foundation for solving quantitative problems in many fields, along with numerical challenges. The *no free lunch* theorem indicates the non-existence of a universally best optimization algorithm for all objectives. To manually design an effective optimization algorithm for a given problem, many efforts have been spent on tuning and validating pipelines, architectures, and hyperparameters. For instance, in deep learning, there is a gallery of gradient-based algorithms specific to high-dimensional, non-convex objective functions, such as Stochastic Gradient Descent [@robbins1951stochastic], RmsDrop [@tieleman2012lecture], and Adam [@kingma2014adam]. Another example is in *ab initio* protein docking whose energy functions as objectives have extremely rugged landscapes and are expensive to evaluate. Gradient-free algorithms are thus popular there, including Markov chain Monte Carlo (MCMC) [@gray_proteinprotein_2003] and Particle Swarm Optimization (PSO) [@moal_swarmdock_2010].
To overcome the laborious manual design, an emerging approach of meta-learning (learning to learn) takes advantage of the knowledge learned from related tasks. In meta-learning, the goal is to learn a meta-*learner* that could solve a set of problems, where each sample in the training or test set is a particular problem. As in classical machine learning, the fundamental assumption of meta-learning is the generalizability from solving the training problems to solving the test ones. For optimization problems, a key to meta-learning is how to efficiently utilize the information in the objective function and explore the space of optimization algorithms.
In this study, we introduce a novel framework in meta-learning, where we train a meta-optimizer that learns in the space of both point-based and population-based optimization algorithms for continuous optimization. To that end, we design a novel architecture where a population of RNNs (specifically, LSTMs) jointly learn iterative update formula for a population of samples (or a swarm of particles). To balance exploration and exploitation in search, we directly estimate the posterior over the optimum and include in the meta-loss function the differential entropy of the posterior. Furthermore, we embed feature- and sample-level attentions in our meta-optimizer to interpret the learned optimization strategies. Our numerical experiments, including global optimization for nonconvex test functions and an application of protein docking, endorse the superiority of the proposed meta-optimizer.
Related work
============
Meta-learning originated from the field of psychology [@ward1937reminiscence; @harlow1949formation]. [@bengio1995search; @bengio1991learning; @bengio1992optimization] optimized a learning rule in a parameterized learning rule space. [@zoph2016neural] used RNN to automatically design a neural network architecture. More recently, learning to learn has also been applied to sparse coding [@gregor2010learning; @wang2016learning; @chen2018theoretical; @liu2018alista], plug-and-play optimization [@ryu2019plug], and so on.
In the field of learning to optimize, [@andrychowicz2016learning] proposed the first framework where gradients and function values were used as the features for RNN. A coordinate-wise structure of RNN relieved the burden from the enormous number of parameters, so that the same update formula was used independently for each coordinate. [@li2016learning] used the history of gradients and objective values as states and step vectors as actions in reinforcement learning. [@chen2017learning] also used RNN to train a meta-*learner* to optimize black-box functions, including Gaussian process bandits, simple control objectives, and hyper-parameter tuning tasks. Lately, [@wichrowska2017learned] introduced a hierarchical RNN architecture, augmented with additional architectural features that mirror the known structure of optimization tasks.
The target applications of previous methods are mainly focused on training deep neural networks, except [@chen2017learning] focusing on optimizing black-box functions. There are three **limitations** of these methods. First, they learn in a limited algorithmic space, namely point-based optimization algorithms that use gradients or not (including SGD and Adam). So far there is no method in learning to learn that reflects population-based algorithms (such as evolutionary and swarm algorithms) proven powerful in many optimization tasks. Second, their learning is guided by a limited meta loss, often the cumulative regret in sampling history that primarily drives exploitation. One exception is the expected improvement (EI) used by [@chen2017learning] under Gaussian processes. Last but not the least, these methods do not interpret the process of learning update formula, despite the previous usage of attention mechanisms in [@wichrowska2017learned].
To overcome aforementioned limitations of current learning-to-optimize methods, we present a new meta-optimizer with the following **contributions**:
- (*Where to learn*): We learn in an extended space of both point-based and population-based optimization algorithms;
- (*How to learn*): We incorporate the posterior into meta-loss to guide the search in the algorithmic space and balance the exploitation-exploration trade-off.
- (*What more to learn*): We design a novel architecture where a population of LSTMs jointly learn iterative update formula for a population of samples and embedded sample- and feature-level attentions to explain the formula.
Method
======
Notations and background
------------------------
We use the following convention for notations throughout the paper. Scalars, vectors (column vectors unless stated otherwise), and matrices are denoted in lowercase, bold lowercase, and bold uppercase, respectively. Superscript $^{\prime}$ indicates vector transpose.
Our goal is to solve the following optimization problem: $$\bm{x}^*= \arg\min_{\bm{x} \in \mathbb{R}^n} f(\bm{x}).$$ Iterative optimization algorithms, either point-based or population-based, have the same generic update formula: $$\bm{x}^{t+1} = \bm{x}^t + \delta \bm{x}^t,$$ where $\bm{x}^t$ and $\delta \bm{x}^t$ are the sample (or a single sample called “particle" in swarm algorithms) and the update (a.k.a. step vector) at iteration $t$, respectively. The update is often a function $g(\cdot)$ of historic sample values, objective values, and gradients. For instance, in point-based gradient descent, $$\delta \bm{x}^t= g(\{\bm{x}^\tau, f(\bm{x}^\tau), \nabla f(\bm{x}^\tau) \}_{\tau=1}^t) = -\alpha \nabla f(\bm{x}^t),$$ where $\alpha$ is the learning rate. In particle swarm optimization (PSO), assuming that there are $k$ samples (particles), then for particle $j$, the update is determined by the entire population: $$\delta \bm{x}_j^t = g(\{\{\bm{x}_j^\tau, f(\bm{x}_j^\tau), \nabla f(\bm{x}_j^\tau) \}_{j=1}^k\}_{\tau=1}^{t}) = w\delta \bm{x}_j^{t-1} + r_1(\bm{x}^t_j-\bm{x}_j^{t*}) + r_2(\bm{x}^t_j-\bm{x}^{t*}),$$ where $\bm{x}_j^{t*}$ and $\bm{x}^{t*}$ are the best position (with the smallest objective value) of particle $j$ and among all particles, respectively, during the first $t$ iterations; and $w, r_1, r_2$ are the hyper-parameters often randomly sampled from a fixed distribution (e.g. standard Gaussian distribution) during each iteration.
In most of the modern optimization algorithms, the update formula $g(\cdot)$ is analytically determined and fixed during the whole process. Unfortunately, similar to what the *No Free Lunch Theorem* suggests in machine learning, there is no single best algorithm for all kinds of optimization tasks. Every state-of-art algorithm has its own best-performing problem set or domain. Therefore, it makes sense to learn the optimal update formula $g(\cdot)$ from the data in the specific problem domain, which is called “learning to optimize”. For instance, in [@andrychowicz2016learning], the function $g(\cdot)$ is parameterized by a recurrent neural network (RNN) with input $\nabla f(\bm{x}^t)$ and the hidden state from the last iteration: $g(\cdot)=\text{RNN}(\nabla f(\bm{x}^t), \bm{h}^{t-1})$. In [@chen2017learning], the inputs of RNN are $\bm{x}^t, f(\bm{x}^t)$ and the hidden state from the last iteration: $g(\cdot)=\text{RNN}(\bm{x}^t,f(\bm{x}^t), \bm{h}^{t-1})$.
Population-based learning to optimize with posterior estimation
---------------------------------------------------------------
We describe the details of our population-based meta-optimizer in this section. Compared to previous meta-optimizers, we employ $k$ samples whose update formulae are learned from the population history and are individually customized, using attention mechanisms. Specifically, our update rule for particle $i$ could be written as: $$g_i(\cdot) = \text{RNN}_i\left(\alpha_i^{\text{inter}}(\{\alpha_j^{\text{intra}}(\{\bm{S}_j^\tau\}_{\tau=1}^t)\}_{j=1}^k), \bm{h}^{t-1}_i\right)$$ where $\bm{S}_j^t=(\bm{s}^t_{j1}, \bm{s}^t_{j2}, \bm{s}^t_{j3}, \bm{s}^t_{j4})$ is a $n \times 4$ feature matrix for particle $j$ at iteration $t$, $\alpha_j^{\text{intra}}(\cdot)$ is the intra-particle attention function for particle $j$, and $\alpha_i^{\text{inter}}(\cdot)$ is the $i$-th output of the inter-particle attention function. $\bm{h}^{t-1}_i$ is the hidden state of the $i$th LSTM at iteration $t-1$.
For typical population-based algorithms, the same update formula is adopted by all particles. We follow the convention to set $g_1(\cdot)=g_2(\cdot)=...=g_k(\cdot)$, which suggests $\text{RNN}_i=\text{RNN}$ and $\alpha_j^{\text{intra}}(\cdot) = \alpha^{\text{intra}}(\cdot) $.
We will first introduce the feature matrix $\bm{S}_j^\tau$ and then describe the intra- and inter- attention modules.
### Features from different types of algorithms
Considering the expressiveness and the searchability of the algorithmic space, we consider the update formulae of both point- and population-based algorithms and choose the following four features for particle $i$ at iteration $t$:
- gradient: $\nabla f(\bm{x}_i^t)$
- momentum: $\bm{m}_i^t = \sum_{\tau=1}^t (1-\beta)\beta^{t-1}\nabla f(\bm{x}_i^\tau) $
- velocity: $\bm{v}_i^t = \bm{x}_i^t - \bm{x}_i^{t*}$
- attraction: $\frac{\sum_{j}\left(\exp({-\alpha d_{ij}^2})(\bm{x}_i^t-\bm{x}_j^t)\right)}{\sum_{j}\exp({-\alpha d_{ij}^2})}$, for all $j$ that $f(\bm{x}_j^t) < f(\bm{x}_i^t)$. $\alpha$ is a hyperparameter and $d_{ij}=||\bm{x}_i^t - \bm{x}_j^t||_2$.
These four features include two from point-based algorithms using gradients and the other two from population-based algorithms. Specifically, the first two are used in gradient descent and Adam. The third feature, velocity, comes from PSO, where $\bm{x}_i^{t*}$ is the best position (with the lowest objective value) of particle $i$ in the first $t$ iterations. The last feature, attraction, is from the Firefly algorithm [@Yang:2009:FAM:1814087.1814105]. The attraction toward particle $i$ is the weighted average of $\bm{x}_i^t - \bm{x}_j^t$ over all $j$ such that $f(\bm{x}_j^t) < f(\bm{x}_i^t)$; and the weight of particle $j$ is the Gaussian similarity between particle $i$ and $j$. For the particle of the smallest $f(\bm{x}_i^t)$, we simply set this feature vector to be zero. In this paper, we use $\beta=0.9$ and $\alpha=1$.
It is noteworthy that each feature vector is of dimension $n\times 1$, where $n$ is the dimension of the search space. Besides, the update formula in each base-algorithm is linear w.r.t. its corresponding feature. To learn a better update formula, we will incorporate those features into our model of deep neural networks, which is described next.
### Overall model architecture
{width="\textwidth"}
Fig. \[figure:1\]a depicts the overall architecture of our proposed model. We use a population of LSTMs and design two attention modules here: feature-level (“intra-particle”) and sample-level (“inter-particle”) attentions. For particle $i$ at iteration $t$, the intra-particle attention module is to reweight each feature based on the context vector $\bm{h}^{t-1}_i$, which is the hidden state from the $i$-th LSTM in the last iteration. The reweight features of all particles are fed into an inter-particle attention module, together with a $k \times k$ distance similarity matrix. The inter-attention module is to learn the information from the rest $k-1$ particles and affect the update of particle $i$. The outputs of inter-particle attention module will be sent into $k$ identical LSTMs for individual updates.
### Attention mechanisms
For the intra-particle attention module, we use the idea from [@bahdanau2014neural; @bansal2018can; @chen2019abd]. As shown in Fig. \[figure:1\]b, given that the $j$th input feature of the $i$th particle at iteration $t$ is $\bm{s}^t_{ij}$, we have:
$$\begin{aligned}
b^t_{ij} = \bm{v}_a^T\tanh{(\bm{W}_a\bm{s}^t_{ij} + \bm{U}_a\bm{h}^t_{ij} )}, \quad p^t_{ij} = \frac{\exp(b^t_{ij})}{\sum_{r=1}^4 \exp(b^t_{ir})},
\end{aligned}$$
where $\bm{v}_a \in \mathbb{R}^n$, $\bm{W}_a \in \mathbb{R}^{n \times n}$ and $\bm{U}_a \in \mathbb{R}^{n \times n}$ are the weight matrices, $\bm{h}^{t-1}_i \in \mathbb{R}^n$ is the hidden state from the $i$th LSTM in iteration $t-1$, $b^t_{ij}$ is the output of the fully-connected (FC) layer and $p^t_{ij}$ is the output after the softmax layer. We then use $p^t_{ij}$ to reweight our input features: $$\bm{c}^t_i = \sum_{r=1}^4 p^t_{ir}\bm{s}^t_{ir},$$
where $ \bm{c}^t_i \in \mathbb{R}^{n}$ is the output of the intra-particle attention module for the $i$th particle at iteration $t$.
For inter-particle attention, we model $\delta \bm{x}^t_i$ for each particle $i$ under the impacts of the rest $k-1$ particles. Specific considerations are as follows.
- The closer two particles are, the more they impact each other’s update. Therefore, we construct a kernelized pairwise similarity matrix $\bm{Q}^t \in \mathbb{R}^{k \times k}$ (column-normalized) as the weight matrix. Its element is $ q^t_{ij}= \frac{exp(-\frac{||\bm{x}^t_i - \bm{x}^t_j||^2}{2l})}{\sum_{r=1}^k exp(-\frac{||\bm{x}^t_r - \bm{x}^t_j||^2}{2l})}$.
- The similar two particles are in their intra-particle attention outputs ($ \bm{c}^t_i$, local suggestions for updates), the more they impact each other’s update. Therefore, we introduce another weight matrix $\bm{M}^t \in \mathbb{R}^{k \times k}$ whose element is $m_{ij} = \frac{\exp\left((\bm{c}^t_i)^\prime \bm{c}^t_j \right)}{\sum_{r=1}^k \exp\left((\bm{c}^t_r)^\prime \bm{c}^t_j\right)}$ (normalized after column-wise softmax).
As shown in Fig. \[figure:1\]b, the output of the inter-particle module for the $j$th particle will be: $$\bm{e}^t_j = \gamma\sum_{r=1}^k {m^t_{rj} q^t_{rj} \bm{c}^t_r} + \bm{c}^t_j,$$ where $\gamma$ is a hyperparameter which controls the ratio of contribution of rest $k$-1 particles to the $j$th particle. In this paper, $\gamma$ is set to be 1 without further optimization.
### Loss function, posterior estimation, and model training
Cumulative regret is a common meta loss function: $L(\bm{\phi})=\sum_{t=1}^T \sum_{j=1}^{k}f(\bm{x}^t_j)$. However, this loss function has two main drawbacks. First, the loss function does not reflect any exploration. If the search algorithm used for training the optimizer does not employ exploration, it can be easily trapped in the vicinity of a local minimum. Second, for population-based methods, this loss function tends to drag all the particles to quickly converge to the same point.
To balance the exploration-exploitation tradeoff, we bring the work from [@cao2019bayesian] — it built a Bayesian posterior distribution over the global optimum $\bm{x}^*$ as $p(\bm{x}^*|\bigcup_{t=1}^T D_t)$, where $D_t$ denotes the samples at iteration $t$: $D_t=\left\{\left(\bm{x}_j^t, f(\bm{x}_j^t)\right)\right\}_{j=1}^k$. We claim that, in order to reduce the uncertainty about the whereabouts of the global minimum, the best next sample can be chosen to minimize the entropy of the posterior, $h\left(p(\bm{x}^*|\bigcup_{t=1}^T D_t)\right)$. Therefore, we propose a loss function for function $f(\cdot)$ as: $$\ell_f(\bm{\phi})=\sum_{t=1}^T \sum_{j=1}^{k}{f(\bm{x}^t_j)} + \lambda h\left(p(\bm{x}^*|\bigcup_{t=1}^T D_t)\right),$$ where $\lambda$ controls the balance between exploration and exploitation and $\bm{\bm{\bm{\phi}}}$ is a vector of model parameters.
Following [@cao2019bayesian], the posterior over the global optimum is modeled as a Boltzmann distribution: $$p(\bm{x}^*|\bigcup_{t=1}^T D_t) \propto \exp(-\rho \hat{f}(\bm{x})),$$ where $\hat{f}(\bm{x})$ is a function estimator and $\rho$ is the annealing constant. In the original work of [@cao2019bayesian], both $\hat{f}(\bm{x})$ and $\rho$ are updated over iteration $t$ for active sampling. In our work, they are fixed since the complete training sample paths are available at once.
Specifically, for a function estimator based on samples in $\bigcup_{t=1}^T D_t$, we use a Kriging regressor [@chiles_geostatistics:_2012] which is known to be the best unbiased linear estimator (BLUE): $$\hat{f}(\bm{x})=f_0(\bm{x}) + \left(\bm{\kappa}(\bm{x})\right)^\prime(\bm{K}+\epsilon^2\bm{I})^{-1}(\bm{y}-\bm{f_0}),$$ where $f_0(\bm{x})$ is the prior for $\mathrm{E}[f(\bm{x})]$ (we use $f_0(\bm{x})=0$ in this study); $\bm{\kappa}(\bm{x})$ is the kernel vector with the $i$th element being the kernel, a measure of similarity, between $\bm{x}$ and $\bm{x}_i$; $\bm{K}$ is the kernel matrix with the $(i,j)$-th element being the kernel between $\bm{x}_i$ and $\bm{x}_j$; $\bm{y}$ and $\bm{f_0}$ are the vector consisting of $y_1,\ldots,y_{n_t}$ and $f_0(\bm{x}_1),\ldots,f_0(\bm{x}_{n_t})$, respectively; and $\epsilon$ reflects the noise in the observation and is often estimated to be the average training error (set at 2.1 in this study).
For $\rho$, we follow the annealing schedule in [@cao2019bayesian] with one-step update: $$\rho=\rho_0\cdot\exp\left((h_0)^{-1}\left|\bigcup_{t=1}^T D_t\right|^\frac{1}{n}\right),$$ where $\rho_0$ is the initial parameter of $\rho$ ($\rho_0=1$ without further optimization here); $h_0$ is the initial entropy of the posterior with $\rho=\rho_0$; and $n$ is the dimensionality of the search space.
In total, our meta loss for $m$ functions $f_q(\cdot)$ ($q=1,\ldots,m$) (analogous to $m$ training examples) with L2 regularization is $$L(\bm{\phi}) = \frac{1}{m}\sum_{q=1}^{m} \ell_{f_q}(\bm{\phi}) + C||\bm{\phi}||_2^2.$$
To train our model we use the optimizer Adam which requires gradients. The first-order gradients are calculated numerically through TensorFlow following [@andrychowicz2016learning]. We use coordinate-wise LSTM to reduce the number of parameters. In our implementation the length of LSTM is set to be 20. For all experiments, the optimizer is trained for 10,000 epochs with 100 iterations in each epoch.
Experiments
===========
We test our meta-optimizer through convex quadratic functions, non-convex test functions and an optimization-based application with extremely noisy and rugged landscapes: protein docking.
Learn to optimize convex quadratic functions
--------------------------------------------
In this case, we are trying to minimize a convex quadratic function: $$f(\bm{x}) = ||\bm{A}_q\bm{x}-\bm{b}_q||_2^2,$$ where $\bm{A}_q \in \mathbb{R}^{n \times n}$ and $\bm{b}_q \in \mathbb{R}^{n \times 1}$ are parameters, whose elements are sampled from i.i.d. normal distributions for the training set. We compare our algorithm with SGD, Adam, PSO and DeepMind’s LSTM (DM\_LSTM) [@andrychowicz2016learning]. Since different algorithms have different population sizes, for fair comparison we fix the total number of objective function evaluations (sample updates) to be 1,000 for all methods. The population size $k$ of our meta-optimizer and PSO is set to be 4, 10 and 10 in the 2D, 10D and 20D cases, respectively. During the testing stage, we sample another 128 pairs of $\bm{A}_q$ and $\bm{b}_q$ and evaluate the current best function value at each step averaged over 128 functions. We repeat the procedure 100 times in order to obtain statistically significant results.
As seen in Fig. \[fig:qua\_performance\], our meta-optimizer performs better than DM\_LSTM in the 2D, 10D, and 20D cases. Both meta-optimizers perform significantly better than the three baseline algorithms (except that PSO had similar convergence in 2D).
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We also compare our meta-optimizer’s performances with and without the guiding posterior in meta loss. As shown in the supplemental Fig. S1, including the posterior improves optimization performances especially in higher dimensions. Meanwhile, posterior estimation in higher dimensions presents more challenges. The impact of posteriors will be further assessed in ablation study.
Learn to optimize non-convex Rastrigin functions
------------------------------------------------
We then test the performance on a non-convex test function called Rastrigin function: $$f(\bm{x}) = \sum_{i=1}^{n}x_i^2 - \sum_{i=1}^{n} \alpha\cos{(2\pi x_i)} + \alpha n,$$ where $\alpha=10$. We consider a broad family of similar functions $f_q(\bm{x})$ as the training set: $$\label{equ:ras_train}
f_q(\bm{x}) = ||\bm{A}_q\bm{x}-\bm{b}_q||_2^2 - \alpha\bm{c}_q\cos(2\pi\bm{x}) + \alpha n,$$ where $\bm{A}_q \in \mathbb{R}^{n \times n}$, $\bm{b}_q \in \mathbb{R}^{n \times 1}$ and $\bm{c}_q \in \mathbb{R}^{n \times 1}$ are parameters whose elements are sampled from i.i.d. normal distributions. It is obvious that Rastrigin is a special case in this family with $\bm{A}=\bm{I}, \bm{b}=\{0,0,\ldots,0\}^\prime , \bm{c}=\{1,1,\ldots,1\}^\prime $.
During the testing stage, 100 i.i.d. trajectories are generated in order to reach statistically significant conclusions. The population size $k$ of our meta-optimizer and PSO is set to be 4, 10 and 10 for 2D, 10D and 20D, respectively. The results are shown in Fig. \[fig:ras\_performance\]. In the 2D case, our meta-optimizer and PSO perform fairly the same while DM\_LSTM performs much worse. In the 10D and 20D cases, our meta-optimizer outperforms all other algorithms. It is interesting that PSO is the second best among all algorithms, which indicates that population-based algorithms have unique advantages here.
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Transferability: Learning to optimize non-convex functions from convex optimization
-----------------------------------------------------------------------------------
We also examine the transferability from convex to non-convex optimization. The hyperparameter $\alpha$ in Rastrigin family controls the level of ruggedness for training functions: $\alpha=0$ corresponds to a convex quadratic function and $\alpha=10$ does the rugged Rastrigin function. Therefore, we choose three different values of $\alpha$ (0, 5 and 10) to build training sets and test the resulting three trained models on the 10D Rastrigin function. From the results in the supplemental Fig. S2, our meta-optimizer’s performances improve when it is trained with increasing $\alpha$. The meta-optimizer trained with $\alpha=0$ had limited progress over iterations, which indicates the difficulty to learn from convex functions to optimize non-convex rugged functions. The one trained with $\alpha=5$ has seen significant improvement.
Interpretation of learned update formula
----------------------------------------
In an effort to rationalize the learned update formula, we choose the 2D Rastrigin test function to illustrate the interpretation analysis. We plot sample paths of our algorithm, PSO and Gradient Descent (GD) in Fig \[fig:attention\]a. Our algorithm finally reaches the funnel (or valley) containing the global optimum ($\bm{x}=\bm{0}$), while PSO finally reaches a suboptimal funnel. At the beginning, samples of our meta-optimizer are more diverse due to the entropy control in the loss function. In contrast, GD is stuck in a local minimum which is close to its starting point after 80 samples.
{width="80.00000%"}
To further show which factor contributes the most to each update, we plot the feature weight distribution over the first 20 iterations. Since for particle $i$ at iteration $t$, the output of its intra-attention module is a weighted sum of its 4 features: $\bm{c}^t_i = \sum_{r=1}^4 p^t_{ir}\bm{s}^t_{ir}$, we hereby sum $p^t_{ir}$ for the $r$-th feature over all particles $i$. The final weight distribution (normalized) over 4 features reflecting the contribution of each feature at iteration $t$ is shown in Fig. \[fig:attention\]b. In the first 6 iterations, the population-based features contribute to the update most. Point-based features start to play an important role later.
Finally, we examine in the inter-particle attention module the level of particles working collaboratively or independently. In order to show this, we plot the percentage of the diagonal part of $\gamma \bm{Q}^t \odot \bm{M}^t + \bm{I}$: $\frac{\mathrm{tr}(\gamma \bm{Q}^t \odot \bm{M}^t+ \bm{I})}{\sum{\gamma \bm{Q}^t \odot \bm{M}^t} + \bm{I}}$ ($\odot$ denotes element-wise product), as shown in Fig. \[fig:attention\]c. It can be seen that, at the beginning, particles are working more collaboratively. With more iterations, particles become more independent. However, we note that the trace (reflecting self impacts) contributes 67%-69% over iterations and the off-diagonals (impacts from other particles) do above 30%, which demonstrates the importance of collaboration, a unique advantage of population-based algorithms.
Ablation study
--------------
How and why our algorithm outperforms DM\_LSTM is both interesting and important to unveil the underlying mechanism of the algorithm. In order to deeply understand each part of our algorithms, we performed an ablation study to progressively show each part’s contribution. Starting from the DM\_LSTM baseline (**B**$_0$), we incrementally crafted four algorithms: running DM\_LSTM for $k$ times under different initializations and choosing the best solution (**B**$_1$); using $k$ independent particles, each with the two point-based features, the intra-particle attention module, and the hidden state (**B**$_2$); adding the two population-based features and the inter-particle attention module to **B**$_2$ so as to convert $k$ independent particles into a swarm (**B**$_3$); and eventually, adding an entropy term in meta loss to **B**$_3$, resulting in our **Proposed** model.
We tested the five algorithms (**B**$_0$–**B**$_3$ and the **Proposed**) on 10D and 20D Rastrigin functions with the same settings as in Section 4.2. We compare the function minimum values returned by these algorithms in the table below (reported are means $\pm$ standard deviations over 100 runs, each using 1,000 function evaluations).
\[hbt!\]
Dimension **B**$_0$ **B**$_1$ **B**$_2$ **B**$_3$ **Proposed**
----------- ---------------- ---------------- ---------------- -------------- ---------------
10 55.4$\pm$13.5 48.4$\pm$10.5 40.1$\pm$9.4 20.4$\pm$6.6 12.3$\pm$5.4
20 140.4$\pm$10.2 137.4$\pm$12.7 108.4$\pm$13.4 48.5$\pm$7.1 43.0 $\pm$9.2
Our key observations are as follows. i) **B**$_1$ v.s. **B**$_0$: their performance gap is marginal, which proves that our performance gain is not simply due to having $k$ independent runs; ii) **B**$_2$ v.s. **B**$_1$ and **B**$_3$ v.s. **B**$_2$: Whereas including intra-particle attention in **B**$_2$ already notably improves the performance compared to **B**$_1$, including population-based features and inter-particle attention in **B**$_3$ results in the largest performance boost. This confirms that our algorithm majorly benefits from the attention mechanisms; iii) **Proposed** v.s. **B**$_3$: adding entropy from the posterior gains further, thanks to its balancing exploration and exploitation during search.
Application to protein docking
------------------------------
We bring our meta-*optimizer* into a challenging real-world application. In computational biology, the structural knowledge about how proteins interact each other is critical but remains relatively scarce [@mosca2013interactome3d]. Protein docking helps close such a gap by computationally predicting the 3D structures of protein-protein complexes given individual proteins’ 3D structures or 1D sequences [@smith2002prediction]. *Ab initio* protein docking represents a major challenge of optimizing a noisy and costly function in a high-dimensional conformational space [@cao2019bayesian].
Mathematically, the problem of *ab initio* protein docking can be formulated as optimizing an extremely rugged energy function: $f(\bm{x}) = \Delta G(\bm{x})$, the Gibbs binding free energy for conformation $\bm{x}$. We calculate the energy function in a CHARMM 19 force field as in [@moal_swarmdock_2010] and shift it so that $f(\bm{x})=0$ at the origin of the search space. And we parameterize the search space as $\mathbb{R}^{12}$ as in [@cao2019bayesian]. The resulting $f(\bm{x})$ is fully differentiable in the search space. For computational concern and batch training, we only consider 100 interface atoms. We choose a training set of 25 protein-protein complexes from the protein docking benchmark set 4.0 [@hwang_protein-protein_2010] (see Supp. Table S1 for the list), each of which has 5 starting points (top-5 models from ZDOCK [@ZDOCKserver2014]). In total, our training set includes 125 instances. During testing, we choose 3 complexes (with 1 starting model each) of different levels of docking difficulty. For comparison, we also use the training set from Eq. \[equ:ras\_train\] ($n=12$). All methods including PSO and both versions of our meta-optimizer have $k=10$ particles and 40 iterations in the testing stage.
As seen in Fig. \[fig:dock\], both meta-optimizers achieve lower-energy predictions than PSO and the performance gains increase as the docking difficulty level increases. The meta-optimizer trained on other protein-docking cases performs similarly as that trained on the Rastrigin family in the easy case and outperforms the latter in the difficult case.
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Conclusion
==========
Designing a well-behaved optimization algorithm for a specific problem is a laborious task. In this paper, we extend point-based meta-*optimizer* into population-based meta-*optimizer*, where update formulae for a sample population are jointly learned in the space of both point- and population-based algorithms. In order to balance exploitation and exploration, we introduce the entropy of the posterior over the global optimum into the meta loss, together with the cumulative regret, to guide the search of the meta-*optimizer*. We further embed intra- and inter- particle attention modules to interpret each update. We apply our meta-optimizer to quadratic functions, Rastrigin functions and a real-world challenge – protein docking. The empirical results demonstrate that our meta-optimizer outperforms competing algorithms. Ablation study shows that the performance improvement is directly attributable to our algorithmic innovations, namely population-based features, intra- and inter-particle attentions, and posterior-guided meta loss.
### Acknowledgments {#acknowledgments .unnumbered}
This work is in part supported by the National Institutes of Health (R35GM124952 to YS). Part of the computing time is provided by the Texas A&M High Performance Research.
|
---
abstract: |
We establish symmetrization results for the solutions of the linear fractional diffusion equation $\partial_t u +(-\Delta)^{\sigma/2}u=f$ and its elliptic counterpart $h v +(-\Delta)^{\sigma/2}v=f$, $h>0$, using the concept of comparison of concentrations. The results extend to the nonlinear version, $\partial_t u
+(-\Delta)^{\sigma/2}A(u)=f$, but only when $A:{{\mathbb R}}_+\to{{\mathbb R}}_+$ is a concave function. In the elliptic case, complete symmetrization results are proved for $\,B(v)
+(-\Delta)^{\sigma/2}v=f$ when $B(v)$ is a convex nonnegative function for $v>0$ with $B(0)=0$, and partial results when $B$ is concave. Remarkable counterexamples are constructed for the parabolic equation when $A$ is convex, resp. for the elliptic equation when $B$ is concave. Such counterexamples do not exist in the standard diffusion case $\sigma=2$.
author:
- 'Juan Luis Vázquez [^1] and Bruno Volzone [^2]'
title: |
**Symmetrization for Linear and Nonlinear\
Fractional Parabolic Equations\
of Porous Medium Type**\
---
Introduction {#sec.intro}
============
The techniques of symmetrization are a very popular tool of obtaining a priori estimates for the solutions of different partial differential equations, notably those of elliptic and parabolic type. Symmetrization techniques appear in classical works like [@MR0046395; @PS1951]. The application of Schwarz symmetrization to obtaining a priori estimates for elliptic problems is described by Weinberger in [@Wein62], see also [@Maz]. The standard elliptic result refers to the solutions of an equation of the form $$Lu=f, \qquad Lu=-\sum_{i,j} \partial_i(a_{ij}\partial_j u)\,,$$ posed in a bounded domain $\Omega\subseteq {{\mathbb R}^N}$; the coefficients $\{a_{ij}\}$ are assumed to be bounded, measurable and satisfy the usual ellipticity condition; finally, we take zero Dirichlet boundary conditions on the boundary $\partial\Omega$. The now classical analysis, introduced by Talenti [@Talenti1], leads to pointwise comparison between the symmetrized version of the actual solution of the problem $u(x)$ and the radially symmetric solution $v(|x|)$ of some radially symmetric model problem which is posed in a ball with the same volume as $\Omega$. Sharp a priori estimates for the solutions are then derived. We refer to the papers [@Talenti4; @ATL90] in the framework of linear operators, and [@Talenti3; @FerMess] where comparison results are obtained when dealing with nonlinear elliptic operators of divergence type. There is a large literature in this topic with many interesting developments.
When this technique is applied to parabolic equations, the general program of comparison with a model problem of radial type still works, but the result of pointwise comparison need not hold and has to be replaced by comparison of $L^p$ norms at every time $t>0$. Actually, a more basic result, called [*comparison of concentrations*]{} is true, cf. Bandle [@Bandle; @Band2] where linear parabolic equations with smooth coefficients are discussed. Such results have been extended in works like [@Mossino], [@AlvVolpVolz1], [@VolpVolz] for weak solutions of linear parabolic problems with discontinuous coefficients. Relevant definitions about symmetrization, rearrangements and concentration are recalled in Section \[sec.prelim\].
[Elliptic approach to nonlinear parabolic problems.]{} An extension of the symmetrization results to nonlinear parabolic equations of possibly degenerate type, more precisely of the porous medium type, was done by the first author in [@Vsym82]. The paper considers the evolution problem $$\label{evol.pbm}
\partial_t u=\Delta A(u), \quad u(0)=u_0,$$ where $A$ a monotone increasing real function[^3] and $u_0$ is a suitably given initial datum which is assumed to be integrable. For simplicity the problem was posed for $x\in {{\mathbb R}^N}$, but bounded open sets can be used as spatial domains. The novel idea of the paper was to use the famous Crandall-Liggett Implicit Discretization theorem [@CL71] to reduce the evolution problem to a sequence of nonlinear elliptic problems of the iterative form $$- h\,\Delta A(u(t_k))+u(t_k)=u(t_{k-1}),\quad k=1,2, \cdots,$$ where $t_k=kh$, and $h>0$ is the time step per iteration. Writing $A(u)=v$, the resulting chain of elliptic problems can be written in the common form $$\label{ell.eq}
h \,L v + B(v)= f\,, \quad B=A^{-1}.$$ General theory of these equations, cf. [@BBC1975], ensures that the solution map: $$T:f\mapsto u=B(v)$$ is a contraction in some Banach space, which happens to be $L^1(\Omega)$. Note that the constant $h>0$ is not essential, it can be put to 1 by scaling. In that context, the symmetrization result can be split into two results:
\(i) the first one applies to rearranged right-hand sides and solutions. It says that if two r.h.s. functions $f_1, f_2$, are rearranged and satisfy a concentration comparison of the form $f_1 \prec f_2$, then the same applies to the solutions, in the form $B(v_1)\prec B(v_2)$.[^4]
\(ii) The second result aims at comparing the solution $v$ of equation with a non-rearranged function $f$ with the solution $\tilde v$ corresponding to $f^\#$, the radially decreasing rearrangement of $f$. We obtain that $\tilde v$ is a rearranged function and $ B(v^\#)\prec
B(\tilde v)$, i.e., $B(v)$ is less concentrated than $B(\tilde v)$.
This precise pair of comparison results can be combined to obtain similar results along the whole chain of iterations $u(t_k)$ of the evolution process, if discretized as indicated above. This allows in turn to conclude the symmetrization theorems (concentration comparison and comparison of $L^p$ norms) for the evolution problem . This approach has had a large expansion in the past decades, cf. [@VANS05] and references. There is no difficulty in considering equations with a right-hand side, like $u_t=\Delta A(u)+f$, as long as $f\in L^1(Q_T)$, $Q_T={{\mathbb R}^N}\times(0,T)$. We also mention how time discretization and symmetrization tools can be combined together to get interesting comparison results for some types of parabolic equations with double nonlinearity, such as $b(u)_t=\Delta A(u)+f$, with special assumptions on $b$, see [@Diaz1] and [@AlvVolpVolz2], and to equations with weights [@Reyes]. The technique also applies for $p$-Laplacian operators, cf. [@Vport].
[Equations with fractional operators.]{} The study of elliptic and parabolic equations involving nonlocal operators, usually of fractional type, is currently the subject of great attention. In that sense, it is quite natural to investigate how to apply symmetrization techniques to the elliptic equations like $$(-\Delta)^{\sigma/2}v=f,$$ where the standard Laplacian operator $\Delta$ is replaced by one of the fractional Laplacian operators $(-\Delta)^{\sigma/2}$, $0<\sigma<2$, as defined in [@Landkof72; @Stein70]. The study of this question has been successfully implemented by the second author and Di Blasio in a recent paper [@BV] by an interesting technique of analysis of Steiner symmetrization of an equivalent extended problem, based on the extension technique used by Caffarelli and Silvestre for the definition of $\sigma$-Laplacian operator, [@CaffS]. For previous uses of Steiner symmetrization in standard elliptic problems see [@ATLD]. The results of [@BV] include a comparison of concentrations, in the form $v^\#\prec \tilde v $, that parallels the result that holds in the standard Laplacian case; note however that no pointwise comparison is obtained, so the result looks a bit like the parabolic results of the standard theory as mentioned above.
In the present paper we are interested in considering the application of such symmetrization techniques to linear or nonlinear parabolic equations with similar fractional Laplacian operators. To be specific, we will focus on the equations of the form $$\label{nolin.parab}
\partial_t u +(-\Delta)^{\sigma/2}A(u)=f, \qquad 0<\sigma<2\,.$$ Following the theory for the standard Laplacian just sketched, we want to consider as nonlinearity $A$ an increasing real function such that $A(0)=0$, and we may accept some other regularity conditions as needed, like $A$ smooth with $A'(u)>0$ for all $u>0$. The problem is posed in the whole space ${{\mathbb R}^N}$. We want to pay special attention to the form $A(u)=u^m$ with $m>0$; the equation is then called the Fractional Heat Equation (FHE) when $m=1$, the Fractional Porous Medium Equation (FPME) if $m>1$, and the Fractional Fast Diffusion Equation (FFDE) if $m<1$. Let us recall that the linear equation $\partial_t u +(-\Delta)^{\sigma/2}u=0$ is a model of so-called anomalous diffusion, a much studied topic in physics, see for instance [@AT; @JKOlla; @MMM; @VIKH; @WZ] and the references therein. The interest in these operators has a long history in Probability since the fractional Laplacian operators of the form $(-\Delta)^{\sigma/2}$ are infinitesimal generators of stable Lévy processes, see [@Applebaum; @Bertoin; @Valdinoc].
For $A(u)=u^m$ and general $m>0$ we obtain a nonlinear diffusion model; the theory of existence of weak solutions for the initial value problem has been addressed by the first author and collaborators in [@pqrv; @pqrv2; @vazBaren], and the main properties have been obtained. In particular, if we take initial data in $L^1$, then an $L^1$-contraction semigroup is generated, and the Crandall-Liggett discretization theorem applies. Extension of this results to general smooth $A$ is done in [@pqrv4].
The application of the method of implicit time discretization leads to the nonlinear equation of elliptic type $$\label{nolin.ell}
h\,(-\Delta)^{\sigma/2}v+B(v) =f$$ posed again in the whole space ${{\mathbb R}^N}$ or in an open subdomain $\Omega\subset {{\mathbb R}^N}$ with zero Dirichlet boundary conditions; $h>0$ is a non-important constant, and the nonlinearity $B$ is the inverse function to the monotone function $A$ that appears in the parabolic equation.
[Organization of the paper and main results.]{} Section \[sec.prelim\] contains the preliminaries about symmetrization and mass concentration that we will need.
$ \bullet$ As a first step of our analysis, we address in Section \[sect.ell\] the issue of comparison of concentrations for rearranged functions, more precisely how to compare the rearrangement of the solution of an elliptic problem with data $f$ with the solution of a radial problem with data $f^{\#}$, rearrangement of $f$. The technique used in [@BV] does not work for the modified equation with lower order term . We supply in this paper the proof of elliptic concentration comparison in the two forms that are needed to try pass to the parabolic problem via discretization in time. However, the results are complete only in the case where $B$ is a convex function and $\Omega={{\mathbb R}^N}$. Though the elliptic results are used here as a step towards the parabolic theory, they have an interest in themselves as an improvement on the symmetrization result developed in [@BV].
$ \bullet$ Complementing this analysis, we prove in Section \[sec.ell.count\] that one the elliptic comparison results that is needed to build a good parabolic theory is false in the case of a concave $B$ of the form $B(v)=v^m$, $0<m<1$.
$ \bullet$ The main issue of symmetrization for linear or nonlinear fractional parabolic equations is addressed in Section \[sec.par\]. After the iteration steps described above, the elliptic results allow to conclude similar comparison results for the mild solutions of the evolution problem when $A$ is concave, i.e., in the range of exponents $0<m<1$ of the Fractional Fast Diffusion Equation (FFDE), and also in the most popular case, the Fractional Heat Equation (FHE) $$\label{ell.eq2}
\partial_t u +(-\Delta)^{\sigma/2}u=f, \qquad u(0)=u_0.$$
A number of consequences are derived from the symmetrization result in the form of a priori estimates, much in the manner these consequences are derived in the case of equation involving the standard Laplacian. In particular, we can use the Barenblatt solutions of the FHE and FFDE constructed in [@vazBaren] as a worst-case to obtain a priori estimates for the solutions in the $L^p$ spaces, a result that was one of the main corollaries of paper [@Vsym82]. Such consequences are important in developing the general theory of such equations, which is one of the aims of the symmetrization techniques. In order to keep a reasonable length for this paper, we have decided to explain such consequences in a companion paper, [@VazVol2].
$ \bullet$ Returning to the presentation of the main results, an important gap was therefore left in the analysis, namely to examine what happens with this approach when applied to the FPME with $m>1$, or more generally to when $A$ is not concave. To our surprise, the result of comparison of concentrations is false for the evolution problem, i.e., for the FPME with $m>1$. As a consequence, we cannot use the Barenblatt solutions of the FPME as a worst case to obtain a priori estimates for the solutions in the $L^p$ spaces. The surprising negative results about mass comparison for parabolic equations are described in Section \[sec.neg.par\]. This is to be seen in parallel to the counterexample found for the elliptic equation ; by the way, this one was found later and is less intuitive.
We conclude by a short section containing comments, extensions and open problems.
Preliminaries on symmetrization {#sec.prelim}
===============================
A measurable real function $f$ defined on ${{\mathbb R}}^{N}$ is called *radially symmetric* (*radial*, for short) if there is a function $\widetilde{f}:[0,\infty)\rightarrow {{\mathbb R}}$ such that $f(x)=\widetilde{f}(|x|)$ for all $x\in {{\mathbb R}}^{N}$. We will often write $f(x)=f(r)$, $r=|x|\ge0$ for such functions by abuse of notation. We say that $f$ is *rearranged* if it is radial, nonnegative and $\widetilde{f}$ is a right-continuous, non-increasing function of $r>0$. A similar definition can be applied for real functions defined on a ball $B_{R}(0)=\left\{x\in{{\mathbb R}}^{N}:|x|<R\right\}$.
Now, let $\Omega$ be an open set of $\mathbb{R}
^{N}$ and $f$ be a real measurable function on $\Omega$. We will denote by $\left\vert
\cdot\right\vert $ the $N$-dimensional Lebesgue measure. We define the *distribution function* $\mu_{f}$ of $f$ as$$\mu_{f}\left( k\right) =\left\vert \left\{ x\in\Omega:\left\vert f\left(
x\right) \right\vert >k\right\} \right\vert \text{ , }k\geq0,$$ and the *decreasing rearrangement* of $f$ as$$f^{\ast}\left( s\right) =\sup\left\{ k\geq0:\mu_{f}\left( k\right)
>s\right\} \text{ , }s\in\left( 0,\left\vert \Omega\right\vert \right).$$ We may also think of extending $f^{\ast}$ as the zero function in $[|\Omega|,\infty)$ if $\Omega$ is bounded. From this definition it turns out that $\mu_{f^{\ast}}=\mu_{f}$ (*i.e.,* $f$, and $f^{\ast}$ are equi-distributed) and $f^{\ast}$ is exactly the *generalized inverse* of $\mu_{f}$. Furthermore, if $\omega_{N\text{ }}$ is the measure of the unit ball in $\mathbb{R}
^{N}$ and $\Omega^{\#}$ is the ball of $\mathbb{R}
^{N}$ centered at the origin having the same Lebesgue measure as $\Omega,$ we define the function $$f^{\#}\left( x\right) =f^{\ast}(\omega_{N}\left\vert x\right\vert
^{N})\text{ \ , }x\in\Omega^{\#},$$ that will be called *spherical decreasing rearrangement* of $f$. From this definition it follows that $f$ is rearranged if and only if $f=f^{\#}$.
For an exhaustive treatment of rearrangements we refer to [@Bandle], [@Kawohl], or the appendix in [@Talenti2]. Here, we just recall the conservation of the $L^{p}$ norms (coming from the definition of rearrangements and the classical *Cavalieri principle*): for all $p\in[1,\infty]$ $$\|f\|_{L^{p}(\Omega)}=\|f^{\ast}\|_{L^{p}(|0,\Omega|)}=\|f^{\#}\|_{L^{p}(\Omega^{\#})}\,,$$ as well as the classical Hardy-Littlewood inequality (see [@MR0046395])$$\int_{\Omega}\left\vert f\left( x\right) g\left(
x\right) \right\vert dx\leq\int_{0}^{\left\vert \Omega\right\vert }f^{\ast
}\left( s\right) g^{\ast}\left( s\right) ds=\int_{\Omega^{\#}}f^{\#}(x)\,g^{\#}(x)\,dx\,,
\label{HardyLit}$$ where $f,g$ are measurable functions on $\Omega$.
$\bullet$ We will often deal with two-variable functions of the type $$\label{f}f:\left( x,y\right) \in\mathcal{C}_{\Omega}\rightarrow f\left( x,y\right)
\in{\mathbb{R}}$$ defined on the cylinder $\mathcal{C}_{\Omega}:=\Omega\times\left( 0,+\infty\right) $, and measurable with respect to $x.$ In that case, it will be convenient to define the so-called [*Steiner symmetrization*]{} of $\mathcal{C}_{\Omega}$ with respect to the variable $x$, namely the set Furthermore, we will denote by $\mu
_{f}\left( k,y\right) $ and $f^{\ast}\left( s,y\right) $ the distribution function and the decreasing rearrangements of (\[f\]), with respect to $x$ for $y$ fixed, and we will also define the function$$f^{\#}\left( x,y\right) =f^{\ast}(\omega_{N}|x|^{N},y)$$ which is called the *Steiner symmetrization* of $f$, with respect to the line $x=0.$ Clearly, $f^{\#}$ is a spherically symmetric and decreasing function with respect to $x$, for any fixed $y$.
$\bullet$ There are some interesting differentiation formulas which turn out to be very useful in our approach. Typically, they are used when one wants to get sharp estimates satisfied by the rearrangement $u^{\ast}$ of a solution $u$ to a certain parabolic problem, for in that context it becomes crucial to differentiate with respect to the extra variable $y$ under the integral symbol, in the form $$\int_{\{u(x,y)>u^{*}(s,y)\}}\frac{\partial u}{\partial y}(x,y)\,dx\,.$$ For the sake of completeness, we recall here two formulas, of first and second order, available in literature. The following proposition can be found in [@Mossino], and is a generalization of a well-known result by Bandle (see [@Bandle]).
\[BANDLE\] Suppose that $f\in H^{1}(0,T;L^{2}(\Omega))$ for some $T>0$. Then $$f^{*}\in H^{1}(0,T;L^{2}(0,|\Omega|))$$ and if $|\left\{f(x,t)=f^{*}(s,t)\right\}|=0$ for a.e. $(s,t)\in(0,|\Omega|)\times(0,T)$, the following differentiation formula holds: $$\int_{f(x,y)>f^{*}(s,y)}\frac{\partial f}{\partial y}(x,y)\,dx=\int_{0}^{s}\frac{\partial f^{*}}{\partial y}(\tau,y)\,d\tau.\label{Rakotoson}$$
Moreover, the following second order differentiation formula (which was also proved in [@ATLD] in a more regular framework) is due to Mercaldo and Ferone (see [@MR1649548]):
\[Ferone-Mercaldo\] Let $f\in W^{2,\infty}\left( \mathcal{C}_{\Omega}\right) $. Then for almost every $y\in(0,+\infty)$ the following differentiation formula holds: $$\begin{aligned}
\int_{f\left( x,y\right) >f^{\ast}\left( s,y\right) }\frac{\partial^{2}f}{\partial y^{2}}\left( x,y\right) dx & =\frac{\partial^{2}}{\partial
y^{2}}\int_{0}^{s}f^{\ast}\left( \tau,y\right) d\tau-\int_{f\left(
x,y\right) =f^{\ast}\left( s,y\right) }\frac{\left( \frac{\partial
f}{\partial y}\left( x,y\right) \right) ^{2}}{\left\vert \nabla
_{x}f\right\vert }\,d\mathcal{H}^{N-1}\left( x\right) \\
& \!\!\!+\left( \int_{f\left( x,y\right) =f^{\ast}\left( s,y\right)
}\!\frac{\frac{\partial f}{\partial y}\left( x,y\right) }{\left\vert
\nabla_{x}f\right\vert }\,d\mathcal{H}^{N-1}\left( x\right) \!\right)
^{2}\!\left( \!\int_{f\left( x,y\right) =f^{\ast}\left( s,y\right)
}\!\frac{1}{\left\vert \nabla_{x}f\right\vert }\,d\mathcal{H}^{N-1}\left(
x\right) \!\right) ^{-1}\!.\end{aligned}$$
Mass concentration
------------------
We will provide estimates of the solutions of our elliptic and parabolic problems in terms of their integrals. For that purpose, the following definition, taken from [@Vsym82], is remarkably useful.
Let $f,g\in L^{1}_{loc}({{\mathbb R}}^{N})$ be two radially symmetric functions on ${{\mathbb R}}^{N}$. We say that $f$ is less concentrated than $g$, and we write $f\prec
g$ if for all $R>0$ we get $$\int_{B_{R}(0)}f(x)dx\leq \int_{B_{R}(0)}g(x)dx.$$
The partial order relationship $\prec$ is called *comparison of mass concentrations*. Of course, this definition can be suitably adapted if $f,g$ are radially symmetric and locally integrable functions on a ball $B_{R}$. Besides, if $f$ and $g$ are locally integrable on a general open set $\Omega$, we say that $f$ is less concentrated than $g$ and we write again $f\prec g$ simply if $f^{\#}\prec g^{\#}$, but this extended definition has no use if $g$ is not rearranged.
The comparison of mass concentrations enjoys a nice equivalent formulation if $f$ and $g$ are rearranged, whose proof we refer to [@MR0046395], [@Chong], [@VANS05]:
\[lemma1\] Let $f,g\in L^{1}(\Omega)$ be two rearranged functions on a ball $\Omega=B_{R}(0)$. Then $f\prec g$ if and only if for every convex nondecreasing function $\Phi:[0,\infty)\rightarrow [0,\infty)$ with $\Phi(0)=0$ we have $$\int_{\Omega}\Phi(f(x))\,dx\leq \int_{\Omega}\Phi(g(x))\,dx.$$ This result still holds if $R=\infty$ and $f,g\in L^{1}_{loc}({{\mathbb R}}^{N})$ with $g\rightarrow0$ as $|x|\rightarrow\infty$.
From this Lemma it easily follows that if $f\prec g$ and $f,g$ are rearranged, then $$\|f\|_{L^{p}(\Omega)}\leq \|g\|_{L^{p}(\Omega)}\quad \forall p\in[1,\infty].$$
Elliptic Problems with lower order term {#sect.ell}
=======================================
Recall of existence, uniqueness and main properties
---------------------------------------------------
As explained in the Introduction, the implicit time discretization scheme directly connects the analysis of the evolution equation to solving the elliptic equation . Therefore, we start our analysis by the following nonlocal Dirichlet problem with homogeneous boundary condition: $$\label{eq.1}
\left\{
\begin{array}
[c]{lll}\left( -\Delta\right)^{\sigma/2}v+ B(v)=f\left( x\right) & & in\text{ }\Omega,\\[6pt]
v=0 & & on\text{ }\partial\Omega,
\end{array}
\right. $$ where $\Omega$ is an open bounded set of ${\mathbb{R}}^{N}$, $\sigma\in(0,2)$ and $f$ is an integrable function defined in $\Omega$ (we will also take $\Omega={\mathbb{R}}^{N}$, and then no boundary condition is assumed, see below). We assume that the nonlinearity is given by a function $B:{{\mathbb R}}_{+}\rightarrow{{\mathbb R}}_{+}$ which is smooth and monotone increasing with $B(0)=0$ and $B'(v) >0$. It is not essential to consider negative values for our main results, but the general theory can be done in that greater generality, just by assuming that $B$ is extended to a function $B:{{\mathbb R}}_{-}\rightarrow{{\mathbb R}}_{-}$ by symmetry, $B(-v)=-B(v)$. The fractional-Laplacian operator $(-\Delta)^{\sigma/2}$ acts on functions $u$ in $\Omega$ and is defined through the spectral decomposition of $u$, in terms of eigenvalues and eigenfunctions of the Laplacian $-\Delta$ with homogeneous boundary conditions. Note that we have changed a bit the notation with respect to equation in the introduction, by eliminating the constant $h>0$, but the change is inessential for the comparison results.
As explained in [@CT] and [@Colorado], when working in a bounded domain the fractional Laplacian $(-\Delta)^{\sigma/2}$ can still be defined as a Dirichlet-to-Neumann map (in the same flavor of the construction in [@CaffS] for $\Omega={{\mathbb R}^N}$), and this allows to connect nonlocal problems involving $(-\Delta)^{\sigma/2}$ to suitable degenerate-singular, local problems defined in one more space dimension. In our case, a solution to problem (\[eq.1\]) is defined as the trace of a properly defined Dirichlet-Neumann problem as follows. If $w$ is a weak solution to the local problem $$\left\{
\begin{array}
[c]{lll}-\operatorname{div}_{x,y}\left( y^{1-\sigma}\nabla w\right) =0 & & in\text{
}\mathcal{C}_{\Omega},\\[6pt]
\ w=0 & & on\text{ }\partial_{L}\mathcal{C}_{\Omega},\\[6pt]
\displaystyle{-\frac{1}{\kappa_{\sigma}}\lim_{y\rightarrow0^{+}}y^{1-\sigma}\,\dfrac{\partial w}{\partial y}(x,y)}+\,B(w(x,0))=f\left( x\right) &
&
in\text{ }\Omega,
\end{array}
\right. \label{eq.3}$$ where $\mathcal{C}_{\Omega}:=\Omega\times\left( 0,+\infty\right) $ is the cylinder of basis $\Omega$, $\partial_{L}\mathcal{C}_{\Omega}:=\partial\Omega\times\lbrack0,+\infty)$ is its lateral boundary, and $\kappa_{\sigma}$ is the constant (see [@CaffS]) $$\kappa_{\sigma}:=\frac{2^{1-\sigma}\,\Gamma(1-\frac{\sigma}{2})}{\Gamma(\frac{\sigma}{2})},$$ then the trace of $w$ over $\Omega$, $\text{Tr}_{\Omega}(w)=w(\cdot,0)=:v$ is said a solution to problem (\[eq.1\]). To make this more precise, we introduce the concept of weak solution to problem . It is convenient to define the weighted energy space $$X_{0}^{\sigma/2}(\mathcal{C}_{\Omega})=\left\{ w\in H^{1}(\mathcal{C}_{\Omega
}),\,w=0\,\text{ on }\partial_{L}\mathcal{C}_{\Omega}\,\,:\int_{\mathcal{C}_{\Omega}}y^{1-\sigma}|\nabla_{x,y} w(x,y)|^{2}\,dxdy<\infty\right\}\,,$$ equipped with the norm $$\Vert w\Vert_{X_{0}^{\sigma/2}}:=\left( \int_{\mathcal{C}_{\Omega}}y^{1-\sigma}\,|\nabla w(x,y)|^{2}\,dxdy\right) ^{1/2}.\label{norm}$$ Then, following [@pqrv], [@pqrv2] we provide the following definition
\[definition2\] Let $\Omega$ be an open bounded set of ${{\mathbb R}^N}$ and $f\in L^{1}(\Omega)$. We say that $w\in X_{0}^{\sigma/2}(\mathcal{C}_{\Omega})$ is a weak solution to if $Tr_{\Omega}(B(w))=:B(w(x,0))\in L^{1}(\Omega)$ and $$\int_{\mathcal{C}_{\Omega}}y^{1-\sigma}\nabla_{x,y} w\cdot \nabla_{x,y}\varphi\,dx\,dy+\int_{\Omega} B(w(x,0))
\,\varphi(x,0)dx=\kappa_{\sigma}\int_{\Omega}f(x)\,\varphi(x,0)dx\label{weakfor}$$ for all the test functions $\varphi\in C^{1}(\overline{\mathcal{C}_{\Omega}})$ vanishing on the lateral boundary $\partial_{L}\mathcal{C}_{\Omega}$.
If $w$ is a solution to the “extended problem” , then the trace function $v=\text{Tr}_{\Omega}w$ will be called a weak solution to problem . Concerning existence of solutions, their smoothness and $L^{1}$ contraction properties, we excerpt some known results from [@pqrv], [@pqrv2], which can be extended for our more general nonlinearity $B$.
\[th.exist\] For any $f\in L^{\infty}(\Omega)$ there exists a unique weak solution $w\in X_{0}^{\sigma/2}(\mathcal{C}_{\Omega})$ to problem , such that $\text{Tr}_{\Omega}(B(w))\in L^{\infty}(\Omega)$. Moreover,
[(i)]{} Regularity: we have $w\in C^{\alpha}(\mathcal{C}_{\Omega})$ for every $\alpha<\sigma$ if $\sigma\le 1$ (resp. $w\in C^{1,\alpha}(\mathcal{C}_{\Omega})$ for every $\alpha<\sigma-1$ if $\sigma> 1$). Arguing as in [[@CT]]{}, higher regularity of $w$ depends easily on higher regularity of $f$ and $B$.
[(ii)]{} $L^{1}$ contraction: if $w,\widetilde{w}$ are the solutions to corresponding to data $f,\widetilde{f}$, the following $L^{1}$ contraction property holds: $$\int_{\Omega}\left[B(w(x,0))-B(\widetilde{w}(x,0))\right]_{+}dx\leq\int_{\Omega}[f(x)-\widetilde{f}(x)]_{+}dx.\label{contraction}$$ In particular, we have that $w\geq0$ in $\overline{\mathcal{C}}_{\Omega}$ whenever $f\geq0$ on $\Omega$. Furthermore, if we put $u:=B(w(\cdot,0))$, then for all $p\in [1, \infty]$ we have $$\|u\|_{L^{p}(\Omega)}\leq \|f\|_{L^{p}(\Omega)}.$$
[(iii)]{} For data $f\in L^1(\Omega)$ the weak solution is obtained as the limit of the solutions of approximate problems with $f_n\in L^1(\Omega)\cap L^\infty(\Omega)$, $f_n\to f$ in $L^1$, since then the sequence $\{B(w_n(x,0))\}_n$ also converges in $L^1$ to some $B(w(x,0)$, and $\|B(w(x,0))\|_1\le \|f\|_1$, hence $v_n$ in uniformly bounded in $L^p$ for all small $p$. Property (ii) holds for such limit solutions.
We give a short account of the proof of these results for the reader’s convenience. See more details on this issue in the forthcoming work [@pqrv4]. In order to get the existence of a weak solution, we first define the integral function of $B$ $$G(t)=\int_{0}^{t}B(\xi)d\xi\,,$$ then we minimize the functional $$\mathcal{J}(w)=\frac{1}{2\kappa_{\sigma}}\int_{\mathcal{C}_{\Omega}}y^{1-\sigma}\,|\nabla
w|^{2}dx\,dy+\int_{\Omega}G(|w(x,0)|)dx-\int_{\Omega}f(x)\,w(x,0)dx$$ over the space $X_{0}^{\sigma/2}(\mathcal{C}_{\Omega})$. where $A:=B^{-1}$ is the inverse of $B$. Arguing as in [@pqrv2], and using the Hölder, trace, and Young inequalities we find that the functional $\mathcal{J}$ is coercive on $\mathcal{X}$. In order to prove that $\mathcal{J}$ is weak lower semi-continuous, let $\left\{w\right\}_{n}$ be a sequence in $X_{0}^{\sigma/2}(\mathcal{C}_{\Omega})$ converging weakly to $w$. By the trace embedding theorem, we have (up to subsequences) $$w_{n}(\cdot,0)\rightarrow w(\cdot,0)\quad\text{strong in }L^{q}(\Omega)\,\,\,\forall q\in[1,2N/(N-\sigma)),$$ then $$w_{n}(\cdot,0)\rightarrow w(\cdot,0)\quad a.e.\,in\,\Omega.$$ Now Fatou’s lemma implies that $\mathcal{J}$ is weakly lower semicontinuous on $X_{0}^{\sigma/2}(\mathcal{C}_{\Omega})$. Then there exists a minimizer $w\in X_{0}^{\sigma/2}(\mathcal{C}_{\Omega})$ of $\mathcal{J}$. Furthermore, a truncation argument shows that we can suppose $w(\cdot,0)\in L^{\infty}(\Omega)$ and $$\|w(\cdot, 0)\|_{L^{\infty}(\Omega)}\leq A(\|f\|_{L^{\infty}(\Omega)}).$$ Finally, computing the first variation of $\mathcal{J}$ in the direction of any $\varphi\in X_{0}^{\sigma/2}(\mathcal{C}_{\Omega})$ we obtain that $w$ is a weak solution to in the sense of definition . The contraction property in Theorem \[th.exist\] follows by the arguments of [@pqrv]-[@pqrv2].
$\bullet$ Let us now consider problem in the whole ${{\mathbb R}^N}$, where the fractional Laplacian is defined by a singular integral. The problem is $$\label{whole}
\left\{
\begin{array}
[c]{lll}\left( -\Delta\right) ^{\sigma/2}v+ B(v)=f\left( x\right) & & in\text{ }{{\mathbb R}}^{N}\\
& & \\
v(x)\rightarrow0 & & as\text{ }|x|\rightarrow\infty,
\end{array}
\right. $$ where $f\in L^{1}({{\mathbb R}^N})\cap L^{\infty}({{\mathbb R}^N})$, and we can define again a suitable meaning of weak solution, making use of a proper extension problem. Indeed, if we denote by $X^{\sigma/2}(\mathcal{C}_{{{\mathbb R}}^{N}})$, being $\mathcal{C}_{{{\mathbb R}}^{N}}:={{\mathbb R}}_{+}^{N+1} $ the upper half-space, the completion of $C^{\infty}(\overline{\mathcal{C}_{{{\mathbb R}}^{N}}})$ with respect to the norm with $\Omega$ replaced by ${{\mathbb R}}^{N}$, then a solution $v$ to is the trace on ${{\mathbb R}^N}$ of a weak solution $w\in
X^{\sigma/2}(\mathcal{C}_{{{\mathbb R}}^{N}})$ to the problem $$\left\{
\begin{array}
[c]{lll}-\operatorname{div}_{x,y}\left( y^{1-\sigma}\nabla w\right) =0 & & in\text{
}{{\mathbb R}}^{N}\times(0,+\infty)\\[6pt]
\displaystyle{-\frac{1}{\kappa_{\sigma}}\lim_{y\rightarrow0^{+}}y^{1-\sigma}\,\dfrac{\partial w}{\partial y}(x,y)}+\,B(w(x,0))=f\left( x\right) &
&
x\in{{\mathbb R}}^{N}.
\end{array}
\right. \label{eq.6}$$ Of course, we mean that $w\in X^{\sigma/2}(\mathcal{C}_{{{\mathbb R}}^{N}})$ is a weak (energy) solution to if equality holds, with $\Omega$ replaced by ${{\mathbb R}^N}$. In order to obtain the existence and uniqueness of solution to problem , we can ague as in [@pqrv2]. For any $R>0$, we consider the solution $w_{R}$ to corresponding to the data $f_{R}=f\chi_{B_{R}(0)}$, where $\Omega$ is the ball $B_{R}(0)$ centered at the origin. If the data $f$ is nonnegative (the case of changing sign data can be treated as in [@pqrv]), we obtain an increasing sequence of non-negative solutions $\left\{w_{R}\right\}$ converging to a weak solution $w$ to the problem in the upper half-space. Then the contraction property holds in ${{\mathbb R}^N}$, from which uniqueness and preserving sign property follow. Moreover, if $u=B(w(\cdot,0))$ then we have $$\|u\|_{L^{1}({{\mathbb R}^N})}\leq \|f\|_{L^{1}({{\mathbb R}^N})},\quad
\|u\|_{L^{\infty}({{\mathbb R}^N})}\leq \|f\|_{L^{\infty}({{\mathbb R}^N})}.$$
[**Remarks.**]{} The approximation method we have used to prove the comparison theorem in the whole ${{\mathbb R}}^{N}$ actually says that we can approximate the solution $v$ to problem with the fractional Laplacian on ${{\mathbb R}}^{N}$ by a sequence of solutions of Dirichlet problems of the type with the fractional laplacian defined on balls, with homogeneous boundary data.
We point out that Theorem \[th.exist\] and the related considerations of existence of solutions on ${{\mathbb R}^N}$ still hold if $B:{{\mathbb R}}\rightarrow{{\mathbb R}}$ is assumed to be increasing and $B(0)>0$ (and this remark will enter in Subsection 3.5). If we want to extend $B$ to the whole real axis it suffices to set $B(-v)=2B(0)-B(v)$ for all $v\geq0$.
From now on, we will always assume that the right-hand side $f$ is nonnegative.
The extended problem and concentration comparison
-------------------------------------------------
Let us address the comparison issue. Our goal here is to compare the solution $v$ to with the solution $V$ to the problem $$\left\{
\begin{array}
[c]{lll}\left( -\Delta\right) ^{\sigma/2}V+\, B(V)=f^{\#}\left( x\right) & & in\text{ }\Omega^{\#}\\[6pt]
V=0 & & on\text{ }\partial\Omega^{\#}.
\end{array}
\right. \label{eq.4}$$ A reasonable way to do that is to compare the solution $w$ to with the solution $\psi$ to the problem $$\left\{
\begin{array}
[c]{lll}-\operatorname{div}_{x,y}\left( y^{1-\sigma}\nabla \psi\right) =0 & & in\text{
}\mathcal{C}_{\Omega^{\#}}\\[6pt]
\psi=0 & & on\text{ }\partial_{L}\mathcal{C}_{\Omega^{\#}}\\[6pt]
\displaystyle{-\frac{1}{\kappa_{\sigma}}\lim_{y\rightarrow0^{+}}y^{1-\sigma}\,\dfrac{\partial \psi}{\partial y}(x,y)}+\,B(\psi(x,0))=f^{\#}\left(
x\right)
& & in\text{ }\Omega^{\#},
\end{array}
\right. \label{eq.5}$$ where $\psi(x,0)=V(x)$. According to [@BV], using the change of variables $$z=\left( \frac{y}{\sigma}\right) ^{\sigma},$$ problems and become respectively
$$\left\{
\begin{array}
[c]{lll}-z^{\nu}\dfrac{\partial^{2}w}{\partial z^{2}}-\Delta_{x}w=0 & & in\text{
}\mathcal{C}_{\Omega}\\
& & \\
w=0 & & on\text{ }\partial_{L}\mathcal{C}_{\Omega}\\
& & \\
-\dfrac{\partial w}{\partial z}\left( x,0\right) =\,\sigma^{\sigma-1}\kappa_{\sigma}\left(f\left( x\right)-B(w(x,0))\right) & & in\text{ }\Omega,
\end{array}
\right. \label{eq.23}$$
and $$\left\{
\begin{array}
[c]{lll}-z^{\nu}\dfrac{\partial^{2}\psi}{\partial z^{2}}-\Delta_{x}\psi=0 & & in\text{
}\mathcal{C}_{\Omega}^{\#}\\
& & \\
\psi=0 & & on\text{ }\partial_{L}\mathcal{C}_{\Omega}^{\#}\\
& & \\
-\dfrac{\partial \psi}{\partial z}\left( x,0\right) =\,\sigma^{\sigma-1}\kappa_{\sigma}\left(f^{\#}\left( x\right)-B(\psi(x,0))\right) & & in\text{ }\Omega^{\#}.
\end{array}
\right. \label{eq.24}$$ where $\nu:=2\left( \sigma-1\right) /\sigma.$ Then, the problem reduces to prove the concentration comparison between the solutions $w(x,z)$ and $\psi(x,z)$ to - respectively. Following [@BV], using standard symmetrization tools (among which the differentiation formulas in Propositions - are essential), if we introduce the function $${Z}(s,z)=\int_{0}^{s}(w^{\ast}(\tau,z)-\psi^{\ast}(\tau,z))d\tau\,,$$ then we get the inequality $$-z^{\nu}{Z}_{zz}-p\left( s\right)
{Z}_{ss}\leq0\label{symineq}$$ for a.e. $(s,z)\in D:=\left( 0,|\Omega| \right)
\times\left( 0,+\infty\right) $ . Obviously, we have $$Z(0,y)={Z}_{s}(|\Omega|,y)=0\label{boundcond}.$$ A crucial point in our arguments below is played by the derivative of $Z$ with respect to $z$. Due to the boundary conditions contained in -, we have $$\label{Z_yboundary.formula}
{Z}_{z}(s,0)\geq \theta_{\sigma}\int_{0}^{s} (B(w^*(\tau,0))-B(\psi^{\ast}(\tau,0))\, d\tau$$ where $$\theta_{\sigma}:=\sigma^{\sigma-1}\kappa_{\sigma}.$$ Now observe that the function $$Y(s,0)=\int_{0}^{s}B(w^*(\tau,0))-B(\psi^*(\tau,0))\, d\tau$$ has the same points of maximum or minimum and the same regions of monotonicity than ${Z}(s,0)$.
Comparison result for concave $B$
---------------------------------
\[thm.ell.concave\] Let $v$ be the nonnegative solution of problem $\eqref{eq.1}$ posed in a bounded domain with zero Dirichlet boundary condition, nonnegative data $f\in L^1(\Omega)$ and nonlinearity $B(v)$ given by a concave function with $B(0)=0$ and $B'(v)>0$ for all $v>0$. If $V$ is the solution of the corresponding symmetrized problem, we have $$v^\#(x)\prec V(x).$$ The same is true if $\Omega={{\mathbb R}^N}$.
[*Proof.*]{} In this case we pose the problem first in a bounded domain $\Omega$ of ${{\mathbb R}^N}$ with smooth boundary. We also assume that $f$ is smooth, bounded and compactly supported, since the comparison result for general data can be obtained later by approximation using the $L^1$ dependence of the map $f\mapsto B(v)$.
\(i) We want to prove that $Z(s,0)\le 0$ for all $s\in [0,|\Omega|]$. It is easy to prove that a positive maximum of $Z(s,z)$ cannot happen at the lateral boundary $s=|\Omega|$ for $z>0$ by the stated boundary conditions and the Hopf’s boundary principle.
In order to study the possible positive maximum at the line $z=0$ we proceed as follows. The concavity of $B$ implies that for $a,b\ge 0$ we have $B(a)-B(b)\ge B'(a)(a-b)$. Using this and , it follows that $$\begin{aligned}
& Z_{z}(s,0)\geq \theta_{\sigma}Y(s,0)\nonumber
\geq
\theta_{\sigma}\int_{0}^{s}B^{\prime}(v^{\ast}(\tau))[w^{\ast}(\tau,0)-\psi^{\ast}(\tau,0)]d\tau\nonumber
\\&=\theta_{\sigma}\int_{0}^{s}B^{\prime}(v^{\ast}(\tau))Z_{s}(\tau,0)d\tau.\label{dery}\end{aligned}$$ If we set $g(s):=B^{\prime}(v^{\ast}(s))$, since $B^{\prime}$ is decreasing, we notice that $g$ is an increasing function bounded from below by $g(0)=B^{\prime}(\|v\|_{\infty})$. If the positive maximum of $Z(s,0)$ happens for $s=s_0$ then, using an integration by parts in we can write $$\begin{aligned}
&Z_{z}(s_{0},0)\geq\theta_{\sigma}\int_{0}^{s_0}B^{\prime}(v^{\ast}(s))Z_{s}(s,0)ds\nonumber\\
&=\theta_{\sigma}\left[g(0)Z(s_{0},0)+\int_{0}^{s_0}[Z(s_{0},0)-Z(s,0)]dg(s)\right]>0\label{relmaxZY}\end{aligned}$$ which is impossible because $Z_{z}(s_{0},0)\leq0$. Then $Z(s_0,0)\leq0$, that is $Z\leq0$, namely $$\int_{0}^{s}w^{\ast}(\tau,z)\,d\tau\leq\int_{0}^{s}\psi^{\ast}(\tau,z)\,d\tau.$$
Another remark is that either $Z\equiv 0$ or $$Z<0 \text{ in } (0,|\Omega|)\times[0,\infty)\label{negativity}:$$ indeed, if $Z\not\equiv0$ for the previous arguments it cannot be $Z=0$ in some points of $(0,|\Omega|)\times(0,\infty)$ (otherwise it would reach the maximum in this domain, hence it would be constantly 0 by the maximum principle). On the other hand, if $Z(s_{0},0)=0$ for some point $s_{0}\in
(0,|\Omega|)$, by the Hopf boundary maximum principle we have $Z_{z}(s_{0},0)<0$, but by we have $Z_{z}(s_{0},0)\geq0$.
\(ii) Here is a simpler proof in the important special case of the linear fractional diffusion, i.e., when $B(v)=\,v$. Indeed, from we have the inequality $$Z_{z}(s,0)\geq\theta_{\sigma} Z(s,0).\nonumber\\$$ Now we simply observe that can be rewritten as $$-p(s)^{-1}Z_{zz}-z^{-\nu}
Z_{ss}\leq0$$ therefore, multiplying both sides by $Z_{+}$ and integrating by parts over the strip $[0,|\Omega|]\times(0,+\infty)$, the boundary conditions and the fact that $Z(s,z)\rightarrow0$ as $z\rightarrow\infty$ imply $$\begin{aligned}
&\int_{0}^{|\Omega|}p(s)^{-1}Z_{z}(s,0)Z_{+}(s,0)ds+\int_{0}^{\infty}\int_{0}^{|\Omega|}z^{-\nu}|\left(Z_{+}\right)_{s}|^{2}ds\,dz\\
&+\int_{0}^{\infty}\int_{0}^{|\Omega|}
p(s)^{-1}|\left(Z_{+}\right)_{z}|^{2}ds\,dz\leq0\end{aligned}$$ namely $$\int_{0}^{\infty}\int_{0}^{|\Omega|}z^{-\nu}|\left(Z_{+}\right)_{s}|^{2}ds\,dz
+\int_{0}^{\infty}\int_{0}^{|\Omega|}
p(s)^{-1}|\left(Z_{+}\right)_{z}|^{2}ds\,dz\leq0.$$ hence $Z_{+}\equiv0$.
$\bullet$ [*Problem in the whole space.*]{} The previous arguments still apply if the problem is posed in the whole space ${{\mathbb R}}^{N}$, namely if $v$ solves . Indeed, in this case we may use the boundary condition $Z_{s}(s,y)\rightarrow0$ as $s\rightarrow\infty$. Alternatively, according to what remarked in Section 3, we may approximate the solution $w$ to the elliptic problem in the upper half-space , with nonnegative $f\in L^{1}({{\mathbb R}^N})\cap L^{\infty}({{\mathbb R}^N})$, with the family $w_R$ of solutions to problems of the type $$\left\{
\begin{array}
[c]{lll}-\operatorname{div}_{x,y}\left( y^{1-\sigma}\nabla w_{R}\right) =0 & & in\text{
}\mathcal{C}_{B_{R}},\\[6pt]
w_{R}=0 & & on\text{ }\partial_{L}\mathcal{C}_{B_{R}},\\[6pt]
\displaystyle{-\frac{1}{\kappa_{\sigma}}\lim_{y\rightarrow0^{+}}y^{1-\sigma}\,\dfrac{\partial w_R}{\partial y}(x,y)}+\,B(w_{R}(x,0))=f_{R}\left(
x\right)
&
& in\text{ }B_{R},
\end{array}
\right. \label{eq.7}$$ where $B_{R}$ is a ball of radius $R$ at the origin. According to Theorem \[thm.ell.concave\], we obtain $$\int_{0}^{s}w_{R}^{\ast}(\tau,y)d\tau\leq\int_{0}^{s}\psi_{R}^{\ast}(\tau,y)d\tau\label{compball}$$ for all $s\in[0,|B_{R}|]$ and $y\geq0$ where $\psi_{R}$ is the solution to $$\left\{
\begin{array}
[c]{lll}-\operatorname{div}_{x,y}\left( y^{1-\sigma}\nabla \psi_{R}\right) =0 & & in\text{
}\mathcal{C}_{B_{R}}\\[6pt]
\psi_{R}=0 & & on\text{ }\partial_{L}\mathcal{C}_{B_{R}}\\[6pt]
\displaystyle{-\frac{1}{\kappa_{\sigma}}\lim_{y\rightarrow0^{+}}y^{1-\sigma}\,\dfrac{\partial \psi_R}{\partial
y}(x,y)}+\,B(\psi_{R}(x,0))=f_{R}^{\#}\left(
x\right) & & in\text{ }B_{R}.
\end{array}
\right. \label{eq.8}$$ Then we get (see Theorem 7.3 in [@pqrv2]) $w_{R}\rightarrow w$ and $\psi_{R}\rightarrow \psi$ as $R\rightarrow\infty$, where $\psi$ solves $$\left\{
\begin{array}
[c]{lll}-\operatorname{div}_{x,y}\left( y^{1-\sigma}\nabla \psi\right) =0 & & in\text{
}{{\mathbb R}}^{N}\times(0,+\infty)\\[6pt]
\displaystyle{-\frac{1}{\kappa_{\sigma}}\lim_{y\rightarrow0^{+}}y^{1-\sigma}\,\dfrac{\partial \psi}{\partial y}(x,y)}+\,B(\psi(x,0))=f^{\#}\left(
x\right)
& & x\in{{\mathbb R}}^{N}.
\end{array}
\right. \label{eq.10}$$ Therefore, letting $R\rightarrow\infty$ in we find $$\int_{0}^{s}w^{\ast}(\tau,y)d\tau\leq\int_{0}^{s}\psi^{\ast}(\tau,y)d\tau\label{eq.11}$$ for all $s\geq0$ and $y\geq0$.
[**Remark.**]{} We also wanted to prove that $Y(s,0)\le 0$, i.e., $$\int_{0}^{s}B(v^{\ast}(\tau))d\tau \leq\int_{0}^{s}B(V^{\ast}(\tau))d\tau.\label{comparison}$$ but it did not work. See next section.
Comparison of concentrations for radial problems
------------------------------------------------
This second result is a variation and extension of the previous comparison result. We consider the same assumptions on $B$ and $\Omega$.
\[thm.ell.concave.rad\] Let $v_1, v_2$ be two nonnegative solutions of problem $\eqref{eq.1}$ posed in a ball $B_{R}(0)$, with $R\in(0,+\infty]$ with zero Dirichlet boundary conditions if $R<+\infty$, nonnegative radially symmetric decreasing data $f_1, f_2\in L^1(B_{R}(0))$ and nonlinearity $B(v)$ given by a concave function for $v\ge0$, with $B(0)=0$ and $B'(v)>0$ for all $v>0$. Then $v_1$ and $v_2$ are rearranged, and $$f_1\prec f_2 \quad \mbox{implies} \quad v_1\prec v_2\,.$$
[*Proof.*]{} As in the proof of Theorem \[thm.ell.concave\], we arrive at the inequality $$-z^{\nu}Z_{zz}^{1,2}-p\left( s\right)
Z_{ss}^{1,2}\leq0\label{symineq2}$$ satisfied a.e. in the strip $(0,|B_{R}(0)|)\times(0,+\infty)$ by the function $$Z^{1,2}(s,z)=\int_{0}^{s}(w_{1}^{\ast}-w_{2}^{\ast})d\tau$$ where $w_1$ and $w_2$ are the solutions of the extensions problems associated to $v_1$ and $v_{2}$ respectively. Concerning the boundary conditions, since we have $f_1\prec f_2$ we get $$\begin{aligned}
&Z_{y}^{1,2}(s,0)\geq\theta_{\sigma}\int_{0}^{\tau} (B(w_{1}^*(\tau,0))-B(w_{2}^{\ast}(\tau,0))\,
d\tau+\int_{0}^{s}\left(f^{\ast}_{2}-f^{\ast}_{1}\right)d\tau\\
&\geq \int_{0}^{s} (B(w_{1}^*(\tau,0))-B(w_{2}^{\ast}(\tau,0))d\tau.\end{aligned}$$ Then we conclude as in the proof of Theorem \[thm.ell.concave\].
Comparison results for convex $B$
---------------------------------
In the case of a convex nonlinearity we prove a stronger result in the whole space. For the sake of clarity, we first prove the result when $B$ is a *superlinear* nonlinearity in the sense that is made precise next:
\[thm.ell.convex\] Let $v$ be the nonnegative solution of problem $\eqref{whole}$ posed in $\Omega={{\mathbb R}^N}$, nonnegative data $f\in
L^1({{\mathbb R}^N})$ and nonlinearity given by a convex function $B:{{\mathbb R}}_{+}\rightarrow{{\mathbb R}}_{+}$ which is smooth, and superlinear: $B(v)\geq \varepsilon v$ for some $\varepsilon>0$ and all $v\geq0$. If $V$ is the solution of the corresponding symmetrized problem, we have $$v^\#\prec V, \qquad B(v^\#) \prec B(V).$$
[**Remark.**]{} The simplest example of superlinear nonlinearity is of course the linear case, $B(v)=cv$. A nontrivial example from the literature would be the remarkable nonlinearity $A(t)=\log(1+t)$ in the model of logarithmic diffusion, [@pqrv3]. Then, $B(s)=A^{-1}(s)=e^{s}-1$, $s\geq0$. We will relax the restriction $B(v)\geq \varepsilon v$ below by approximation.
[*Proof.*]{} In order to prove that $Z(s,0)\le 0$ for all $s\in [0,\infty)$ we argue as follows. We have $ B(w^*(\tau,0))-B(\psi^*(\tau,0)=B'(\xi)(w^*(\tau,0))-\psi^*(\tau,0))$, where $\xi $ is an intermediate value between $w^*(\tau,0)$ and $\psi^*(\tau,0)$. Since $B$ is convex, $B'$ is an increasing real function and $$B(w^*(\tau,0))-B(\psi^*(\tau,0))\le B'(w^*(\tau,0)))(w^*(\tau,0))-\psi^*(\tau,0))$$ Due to the maximum principle and the boundary conditions , unless $Z$ is constant, the maximum of $Z$ can be achieved either on the half-line $\left\{(0,z):z\geq0\right\}$ or on the segment line $\left\{(s,0):s\in[0,\infty)\right\}$. Suppose this second circumstance occurs, and let $(s_{0},0)$ be a maximum point. Assume $s_{0}>0$. We also have $Z_z(s_0,0)<0$ by Hopf’s maximum principle, and by , this leads to $Y(s_0,0)<0$. Then for $s>s_0$ $$\begin{array}{l}
\displaystyle Y(s,0)-Y(s_0,0)=\int_{s_0}^s [B(v^*(\tau))- B(V^{\ast}(\tau))]\, d\tau\\
[6pt]
\displaystyle \le \int_{s_0}^s
B'(w^*(\tau,0))(v^*(\tau)-V^{\ast}(\tau))\, d\tau.
\end{array}$$ After integration by parts $$\begin{array}{l}
\displaystyle Y(s,0)-Y(s_0,0) \le \left[B'(v^*(\tau))(Z(\tau,0)-Z(s_0,0))\right]_{s_0}^{s} -\\
[6pt]
\displaystyle \int_{s_0}^s B''(v^*(\tau))v^*_s(\tau)(Z(\tau,0)-Z(s_0,0))d\tau.
\end{array}$$ Since $Z$ has a maximum at $s_0$ and $B'$ is positive, the first term in the RHS is nonpositive. As for the second, we have: $B''>0$, $v^*_s<0$, and $Z(s,0)-Z(s_0,0)\le 0$, hence the last term is also nonpositive. We conclude that $Y(s,0)\le Y(s_0,0)<0$ for all $s>s_0$. This is a contradiction, because by the conservation of mass property (see proposition \[prop.7\] below) we have $Y(\infty,0)=0$. Then $s_{0}=0$ and $Z\leq0$.
\(ii) Once we have $Z(s,z)\le 0$ we also want to prove that $Y(s,0)\le 0$. We use the fact that $s=0$ is a point of maximum of $Z$ and write $$Y(s,0) \le \left[B'(v^*(\tau))Z(\tau,0)\right]_{0}^s -
\int_{0}^s B''(v^*(\tau))v^*_s(\tau)Z(\tau,0))\,d\tau\le 0.$$ Also, we obtain the same result by using Lemma \[lemma1\], taking advantage of the convexity of $B$ and choosing any convex, increasing function $\Phi:[0,\infty)\rightarrow[0,\infty)$. This ends the proof of the concentration comparison theorem in this case.
$\bullet$ The only remaining question here is then to prove that $\|B(w(x,0))\|_{L^1}=\|B(\psi(x,0))\|_{L^1}$. Under the additional assumption $B(s)\ge \varepsilon s$ for all $s>0$, this will be a consequence of the following mass conservation result for the solutions of the elliptic equation.
\[prop.7\] Let $v$ be the weak solution of $ (-\Delta)^{\sigma/2}v+ B(v)=f$ with $f\in L^1({{\mathbb R}^N})$ nonnegative and let $u=B(v)$, with $B$ satisfying the same assumptions as in Theorem [\[thm.ell.convex\]]{}. Then we have $$\int_{{{\mathbb R}^N}} u(x)\,dx=\int_{{{\mathbb R}^N}} f(x)\,dx.$$
[*Proof.*]{} Using a nonnegative nonincreasing cutoff function $\zeta(s)$ such that $\zeta(s)=1$ for $0\leq s\leq 1$ and $\zeta(s)=0$ for $s\geq2$, we rescale such function to $\zeta_R(x)=\zeta(|x|/R)$. Then we have $$\displaystyle \int_{{{\mathbb R}^N}} f(x)\,\zeta_R(x)\,dx-\int_{{{\mathbb R}^N}} u(x)\,\zeta_R(x)\,dx=\int_{{{\mathbb R}^N}} v\,((-\Delta)^{\sigma/2}\zeta_R)\,dx\label{conservation}$$ Due to the superlinearity assumption, if $u(x)=B(v(x))$ we get $$\left|\int_{{{\mathbb R}^N}} v\,(-\Delta)^{\sigma/2}\zeta_R\,dx\right| \le \frac{1}{\varepsilon}\int_{{{\mathbb R}^N}} |u(x)(-\Delta)^{\sigma/2}\zeta_R|\,dx \le
\frac{c}{R^{\sigma}} \int_{{{\mathbb R}^N}} |u(x)|\,dx$$ which in the limit $R\to \infty$ tends to zero.
$\bullet$ The property of mass conservation for solutions in the whole space is also true for some convex $B$ that are not superlinear, like $B(v)=v^p$ with some $p>1$ but near 1, but it is not true for all $p>1$. However, the comparison result we are looking for (which will extend Theorem \[thm.ell.convex\] for such kind of nonlinearity) will be true and the proof proceeds by an approximation process, approximating $B(v)$ by $B_{\varepsilon}(v)=B(v)+{\varepsilon}v$, solving the approximate problem, deriving the comparison result and passing to the limit. The details are as follows.
\[prop.8\] Suppose $f\in L^{1}({{\mathbb R}^N})$ and $B:{{\mathbb R}}_{+}\rightarrow{{\mathbb R}}_{+}$ is smooth, convex, $B(0)=0$ and $B^{\prime}(v)>0$ for all $v>0$. Let $v_{{\varepsilon}}$ be the solution of problem , with nonlinearity given by $B_{{\varepsilon}}(v)=B(v)+{\varepsilon}v$. Then $v_{{\varepsilon}}\rightarrow v$ as ${\varepsilon}\rightarrow0$ pointwise and in $L^{1}({{\mathbb R}^N})$.
We first prove the result on a bounded domain $\Omega$. Suppose that $f\in L^{\infty}(\Omega)$, let $v_{{\varepsilon}}$ be the solution to , and let $w_{{\varepsilon}}$ be its $\alpha-$ harmonic extension to the cylinder $\mathcal{C}_{\Omega}$. We have that $w_{{\varepsilon}}$ is obtained as minimizer of the functional $$\mathcal{J}_{{\varepsilon}}(w)=\frac{1}{2\kappa_{\sigma}}\int_{\mathcal{C}_{\Omega}}y^{1-\sigma}\,|\nabla
w|^{2}dx\,dy+\int_{\Omega}G_{{\varepsilon}}(|w(x,0)|)dx-\int_{\Omega}f\,w(x,0)dx$$ with $$G_{{\varepsilon}}(t)=\int_{0}^{t}\left[B(\xi)+{\varepsilon}\xi\right]d\xi$$ over the space $X_{0}^{\sigma/2}(\mathcal{C}_{\Omega})$. Moreover the trace $v_{\varepsilon}$ over $\Omega$ of $w_{\varepsilon}$ is bounded and $$\|w_{\varepsilon}(\cdot, 0)\|_{L^{\infty}(\Omega)}\leq A(\|f\|_{L^{\infty}(\Omega)}).\label{limitw}$$ Taking $w_{{\varepsilon}}$ as a test function in the weak formulation of problem , namely in the formula $$\int_{\mathcal{C}_{\Omega}}y^{1-\sigma}\,\nabla w_{{\varepsilon}}\cdot \nabla\varphi\,dx\,dy+\int_{\Omega} B(w_{{\varepsilon}}(x,0))
\,\varphi(x,0)dx+{\varepsilon}\int_{\Omega}w_{{\varepsilon}}(x,0)
\,\varphi(x,0)dx=\kappa_{\sigma}\int_{\Omega}f(x)\,\varphi(x,0)dx\label{weakforve}$$ for $\varphi\in X_{0}^{\sigma/2}(\mathcal{C}_{\Omega})$, the Young and trace inequalities imply that $\left\{w_{{\varepsilon}}\right\}$ is bounded in $X_{0}^{\sigma/2}(\mathcal{C}_{\Omega})$. Then we can extract a subsequence $\left\{w_{{\varepsilon}}\right\}$ (we used the same labeling for simplicity) such that $$w_{{\varepsilon}}\rightharpoonup w\quad \text{weak\,in }\,X_{0}^{\sigma/2}(\mathcal{C}_{\Omega}).$$ Then the compactness of the trace embedding inequality gives $$w_{{\varepsilon}}(\cdot,0)\rightarrow w(\cdot,0)\quad\text{as}\,{\varepsilon}\rightarrow0\,\text{ strong in }L^{q}(\Omega)\,\,\,\forall q\in[1,2N/(N-\sigma)).$$ Using , Lebesgue’s dominated convergence implies $$\int_{\Omega}B(w_{{\varepsilon}}(\cdot,0))\varphi(\cdot,0)\,dx\rightarrow \int_{\Omega}B(w(\cdot,0))\varphi(\cdot,0)\,dx$$ for all $\varphi\in X_{0}^{\sigma/2}(\mathcal{C}_{\Omega})$. This is enough to pass to the limit in and obtain that $w$ is the weak solution to , that is $v=w(\cdot,0)$ solves
If $f\in L^{\infty}({{\mathbb R}^N})$, let $v$ be the solution to . We know that the solution $v_{{\varepsilon}}$ to with nonlinearity $B_{{\varepsilon}}$ is the trace on ${{\mathbb R}^N}$ of the solution $w_{{\varepsilon}}$ to the problem , with the nonlinearity $B_{{\varepsilon}}$. By the arguments we recalled in Subsection 3.1, we have that the sequence of solutions $\left\{w_{{\varepsilon}}^{R}\right\}_{R>0}$ to problem , defined on the cylinder $\mathcal{C}_{B_{R}(0)}$, with nonlinearity $B_{{\varepsilon}}$ and data $f_{R}=f\chi_{B_{R}(0)}$, converges pointwise to $w_{{\varepsilon}}$ as $R\rightarrow\infty$, that is $$w_{{\varepsilon}}^{R}\rightarrow w_{{\varepsilon}}\quad\text{pointwise as }R\rightarrow\infty.\label{conve1}$$ Moreover, if $v_{{\varepsilon}}^{R}=w_{{\varepsilon}}^{R}(\cdot,0)$, the arguments explained above show that $$v_{{\varepsilon}}^{R}\rightarrow v^{R}\quad\text{as }{\varepsilon}\rightarrow0 \text{ strong in }L^{q}(B_{R}(0))\,\,\forall q\in[1,2N/(N-\sigma))\label{conve2}$$ where $v^{R}$ is the solution to the Dirichlet problem , posed on the ball $B_{R}(0)$. In addition, we have $$v^{R}\rightarrow v\quad\text{pointwise as } R\rightarrow\infty.\label{conve3}$$ Then ,, implies that $v_{{\varepsilon}}$ converges to $v$ pointwise and in $L^{1}({{\mathbb R}^N})$.
Then we are able to prove the comparison result of Theorem \[thm.ell.convex\] also for power nonlinearities like $B(v)=v^p$ for large $p>1$ (which means that we can include the fast diffusion range when we pass to the parabolic setting in Section \[sec.par\])
\[genconvex\] Let $v$ be the nonnegative solution of problem $\eqref{whole}$, nonnegative data $f\in L^1({{\mathbb R}^N})$ and the nonlinearity given by a convex function $B:{{\mathbb R}}_{+}\rightarrow{{\mathbb R}}_{+}$, with $B(0)=0$ and $B^{\prime}(v)>0$ for all $v>0$. If $V$ is the solution of the corresponding symmetrized problem, the conclusion of Theorem [\[thm.ell.convex\]]{} still holds.
By virtue of Theorem \[thm.ell.convex\] we have $$v_{{\varepsilon}}^\#\prec V_{{\varepsilon}}, \qquad B(v_{{\varepsilon}}^\#) \prec B(V_{{\varepsilon}})\label{compeps}$$ for all ${\varepsilon}>0$, being $v_{{\varepsilon}}$, $V_{{\varepsilon}}$ the solution of problem and its symmetrized with the nonlinearity $B_{{\varepsilon}}$. By Proposition \[prop.8\] we have that for all $s>0$ $$\int_{0}^{s}v_{{\varepsilon}}^{\ast}\,d\tau\rightarrow\int_{0}^{s}v^{\ast}\,d\tau,\quad\int_{0}^{s}V_{{\varepsilon}}^{\ast}\,d\tau\rightarrow\int_{0}^{s}V^{\ast}\,d\tau$$ as ${\varepsilon}\rightarrow0$. Passing to the limit in we find the desired result.
Second comparison result for convex $B$
---------------------------------------
Here is the second result, about comparison of concentrations for radial problems. We leave the proof to the reader.
\[thm.ell.convex2\] Let $v_1, v_2$ be two nonnegative solutions of problem $\eqref{eq.1}$ posed in $\Omega={{\mathbb R}^N}$, with nonnegative radially symmetric decreasing data $f_1, f_2\in L^1(\Omega)$ and nonlinearity $B(v)$ given by a convex function with $B(0)=0$ and $B'(v)>0$ for all $v>0$. Then, $v_1$ and $v_2$ are rearranged, and for $f_1\prec f_2$ we have $$v_1(x)\prec v_2(x), \qquad B(v_1)\prec B(v_2)\,.$$
[**Remark.**]{} These results are in perfect agreement with the results of [@Vsym82] for the standard Laplacian case.
Counterexample for elliptic concentration comparison with convex powers {#sec.ell.count}
=======================================================================
If we compare the results of the preceding section for convex $B$ and concave $B$ we realize that the conclusion is weaker in the latter case. This seemed to us a possible defect in the technique since in the case of the standard diffusion $\sigma=2$ the results are identical.But it turned out that in the fractional equation the concave case has an essential difficulty.
Our aim is now to prove that actually *the general concentration comparison does not hold* in the whole ${{\mathbb R}}^{N}$, for an equation of the form $$\label{eqell.h}
h \, L_{\sigma}(u^m)+u =f, \qquad L_{\sigma}=(-\Delta)^{\sigma/2},\,h>0,$$ with $m>1$. The precise result is stated in Theorem \[counter.ell\]. The reduction to power-like nonlinearity simplifies the calculations and is the most important case in the applications. The solutions satisfy $ u\rightarrow0 \text{ as } |x|\rightarrow\infty.$ Comparing this equation to , we notice that here we denote by $u=B(v)=v^{1/m}$ the unknown function. Equation is posed in ${{\mathbb R}^N}$ with nonnegative and integrable data $f(x)$. We want to describe the asymptotic behaviour of the solution as $|x|\to\infty$, more precisely its rate of decay. We will focus on the dependence of the behaviour on the constant $h>0$. This constant is important since it represents the time increment when discretizing the evolution problem. We stress the dependence by often denoting the solution as $u(x;h)$. Our main result says that roughly speaking $$u(x;h) \sim \,h\,|x|^{-(N+\sigma)}$$ when $|x|$ is large and $h$ small. We assume $m\ge 1$. Note that the parameter $h$ can be changed, or fixed to 1, by using the scaling $$\widetilde u(x)= a\,u(bx), \qquad \widetilde f(x)= a\,f(bx)\,.$$ If $a^{m-1}b^{\sigma}\widetilde h=h$, then $\widetilde u$ satisfies: $\widetilde h L_{\sigma}\, (\widetilde u^m)+\widetilde u =\widetilde f$. In other words, $u(x;\widetilde h)=a u(bx;h)$. This will be of great use in deriving the negative implication for the concentration analysis in Section \[sec.neg\].
Elliptic positivity estimate via subsolutions
---------------------------------------------
The first step in our asymptotic positivity analysis of solutions of is to ensure that solutions with positive data remain positive in some region. We only need a special case that we establish next.
\[lemma.uniflower\] Let $u(x;h)$ be the solution of with RHS $f(x)\ge0$ such that $f(x)\ge 1$ for $|x|\le 1$. We assume that $m>0$. Then, for every $R<1$ there are constants $A_1, h_1>0$ (depending on $R$) such that $$u(x;h)\ge A_1 \quad \mbox{ for } \quad |x|\le R, \ 0<h<h_1.$$
[*Proof.*]{} $\bullet$ First, we construct a subsolution for a related problem that has an explicit form and compact support. Let $g(x)$ be the explicit function, $$g(x)=\frac12 (1-r^2)_+^{\sigma/2}, \quad r=|x|\ge 0\,.$$ Getoor [@Getoor], Theorem 5.2, proves that $L_{\sigma} g(x)=c_0>0$ on the ball of radius 1 where $g$ is positive, while $L_{\sigma/2}g<0$ for $r>1$, with an explicit formula that goes to minus infinity as $r\to 1$ and behaves as $\sim r^{-(N+\sigma)}$ when $r\to\infty$. Next, we consider the following combination $$f_1(x):=h L_{\sigma}g + g^{1/m}.$$ This can be seen as follows: the solution $u$ of equation corresponding to RHS $f_1$ is $u_1=g^{1/m}$.
Let us now try to estimate $f_1$: for $r>1$ we have $f_1<0$. For $r\le 1$ we have $f_1= c_o h+ g^{1/m}>0$, besides $f_1\le (1/2)^{1/m}+hc_0<1$ if $h<
(1-2^{-m})/c_0$. Under these restrictions on $h$, $u_1=g^{1/m}$, the solution for RHS $f_1$, serves as a subsolution for the RHS $f(x)=\chi_{1}(0)$, the characteristic function of the ball of radius $1$. This means that the solution $u$ corresponding to such $f$ is equal or larger than $u_1=g^{1/m}$. Since $u_1(x)$ is uniformly positive in the ball of radius $1/2$, $u(x)$ is uniformly positive in the ball of radius $1/2$ when $0<h<h_1$.
$\bullet$ By means of scalings to put the dimensions in $x$, $u$ and $h$ as in the statement.
We now proceed with the asymptotic estimate from below.
\[lower.ell.est\] Let $u(x;h)$ be the solution of with RHS $f(x)\ge0$ such that $f(x)\ge 1$ in the ball $B_1(0)$. We assume that $m\ge 1$. Then there are constants $ C_-,R_1, h_1 >0$ such that $$u(x;h)\ge C_-\,h\,|x|^{-(N+\sigma)}$$ if $|x|\ge R_1$ and $0<h<h_1$.
[*Proof.*]{} We will use the standard comparison theorem to reduce the case where $f$ is a smooth version of the characteristic function of the ball $B_1(0)$. Then known theory says that $u\le 1$ everywhere and is continuous, radially symmetric and decreasing in $r=|x|$. In fact, a bootstrap argument shows that $u\in C^{\infty}$. Since $\|u\|_1\le \|f\|_1=\omega_N$ we also have a first decay for $u$ near infinity of the form $$u(r)\le C\,r^{-N}.$$ Of course, this first estimate is not sharp, in view of our next results.
$\bullet$ We want to construct a subsolution of the form $$U^m(x;h)=G(|x|)+ h^m\,F^m(|x|)\,,$$ which will be valid for $0<h<h_1$. Here the functions $G, F\ge 0$ and the constant $h_1>0$ have to chosen carefully, as explained below.
We take $G(r)=0$ for $r=|x|\ge 1/2$ so that $U(x;h)=hF(x)$ there. If $G$ is also smooth we have $L_{\sigma}\,G$ bounded and we can also choose $G$ so that $$L_{\sigma}G\le -C_1r^{-(N+\sigma)} \quad \mbox{for} \quad r>1/2.$$ We may choose as $G$ a smoothed version of the previous Getoor function, using convolution.
We also need $F$ positive, smooth and $F(r)\sim C_2r^{-(N+\sigma)}$ as $r\to\infty$ to get the desired conclusion after the comparison argument: $u(x;h)\ge
U(x;h)\ge C\,h\,r^{-(N+\sigma)}$ (if $r$ is large and $h\sim 0$, see below)
To check the property of subsolution we proceed as follows. We have $$L_{\sigma}U^m=L_{\sigma} G(x)+ h^m L_{\sigma}F^m(x)$$ As we have pointed out, our choice of $G$ leads to the above estimate for $L_{\sigma}G$ with negative sign. We also have $F\le C_2r^{-(N+\sigma)}$ for $r>1/2$, by Lemma 2.1 in [@BV2012] we have that since $F^m=O(r^{-(N+\sigma)m})$ and $(N+\sigma)m>N$, we can choose $F$ so that $ |L_{\sigma}F^m|\leq
C_{3}r^{-(N+\sigma)}$ for some positive constant $C_3$ and $r>1/2$. Then we will have $$U+ h\,L_{\sigma}U^m\le h\left(F + L_{\sigma}G+h^mL_{\sigma}F^m\right) \le h( C_2r^{-(N+1)} -C_1r^{-(N+1)}+ h^m L_{\sigma}F^m),$$ which will be negative for all $r>1/2$ if $$C_2+h^m C_3 < C_1\,.$$ In order to make sure that such constants can be obtained, we fix first $G$ and this determines $C_1$. We then use a tentative $F_0$ for the function $F$ and multiply it by a small constant so that $F$ and $L_{1/2}F^m$ are smaller than $C_1/2$. Indeed, we can take $C_{2}<C_{1}/2$ and $h<(C_{1}/2C_{3})^{1/m}=:h_1$. Finally, we may take $h_1=1$.
$\bullet$ The next step is to use the viscosity method to compare $u$ and $U$ in the $Q=\{|x|\ge 1/2\}$, and this will prove that $U(x;h)\le u(x;h)$ in $Q$ if $h<h_1$.
The following inequality establishes a suitable comparison of the boundary conditions at $|x|=1/2$: $$U(x;h)=hF(1/2)< A_1\le u(x;h).$$ Here we use Lemma with the choice $R=1/2$, which gives $u(x;h)\geq A_{1}$ for some constant $A_{1}$, $|x|\leq1/2$ and $h$ sufficiently small. Now $$hF(1/2)\leq 2^{N+1}\,h_{1}C_{2}<A_{1}$$ up to choose $C_{2}$ properly and $h$ under a further bound. Once this is justified, we argue at the first point where $u$ and $U$ touch. Actually, we must use and approximation $u_{\varepsilon}$ instead of $u$.
[**Remark.**]{} The only restriction on $m$ is $m(N+\sigma)>N$, which means $m>m_1=N/(N+\sigma)$.
Upper bound estimate
--------------------
\[up.ell.est\] Let $u(x;h)$ be the solution of with RHS $f$ such that $0\le f(x)\le 1$ in the ball $B_1(0)$ and $f(x)=0$ for $|x|>1$. We assume that $m\ge 1$. Then there are constants $ C_+,R_2, h_2 >0$ such that $$u(x;h)\le C_+\,h\,|x|^{-(N+\sigma)}$$ if $|x|\ge R_2$ and $0<h<h_2$.
[*Proof.*]{} We will construct a super-solution of the form $$U^m(x;h)= G(x) + b^mh^m\,F(x)^m\,,$$ where $F\ge 0$ is chosen as before and $b>0$. We will use the fact that $F(r)\sim C_1r^{-(N+\sigma)}$ as $r\to\infty$. It follows that there is a large constant $k>0$ such that $$kF+L_{\sigma}\,F^m\ge 0 \quad \mbox{everywhere in } \ {{\mathbb R}^N}.$$ Next we choose $G\ge 0$ compactly supported in a ball of radius $R_1>1$, and such that $L_{\sigma}G=c_0>0$ on the support. As $r\to\infty$, we get the usual $L_{\sigma
}G\sim -C\,r^{-(N+\sigma)}$. We also need $G(1)> 1$. Note that for $G=0$ we have $U=bhF$. In any case $U$ is nonnegative, $U\ge 0$. We also have $$L_{\sigma}U^m=L_{\sigma}G+ b^mh^m L_{\sigma}F^m(x)$$ We perform an analysis by regions. Thus, when $G=0$ we have $$U+ hL_{\sigma}U^m = h(b F + b^mh^mL_{\sigma}F^m + L_{\sigma}G)\ge 0$$ The final inequality is obtained as follows: we first put $b>b_0$ so that $$(b/2)F + L_{\sigma}G\ge 0$$ (recall that $L_{\sigma}G=O(r^{-(N+\sigma)})$). Then we put $b>2k(bh)^m$ to have $$(b/2)F + (bh)^mL_{\sigma}F^m\ge 0,$$ i.e. if $b^{m-1}h^mk<1/2$; this imposes an upper bound on $h$.
On the other hand, where $G>0$ we have $$U+ h L_{\sigma}U^m \ge h(c_0+(bh)^mL_{\sigma}F^m)\ge 0$$ if $C_3(bh)^m\le c_0$ (we use the fact that $L_{\sigma}F^m$ is bounded). Both conditions are fulfilled if $0<h<h_2$.
$\bullet$ Now the viscosity method works in the region $Q=\{|x|\ge 1\}$, and this will prove that $U(x,h)\ge
u(x,h)$ in $Q$. Indeed, the boundary condition at $r=1$ is $$U(1)\ge G(1)\ge 1\geq u(x;h).$$ by the maximum principle. This ends the proof.
[**Remark.**]{} The only restriction on $m$ is $m(N+\sigma)>N$, which means $m>m_1=N/(N+\sigma)$.
Scaled data. Negative concentration result {#sec.neg}
------------------------------------------
We have done the argument for a solution with data of height 1 supported in the ball of radius $R=1$. If we want to change the radius to $R\ne 1$ and the height to $A$ we can use the scaling $$\label{scal.frm}
\widetilde u(x;h)= A u(x/R; A^{m-1}R^{-\sigma} h)$$ Using this formula and the result of Theorem \[lower.ell.est\] applied to $u$, we see that the comparison result holds in an $h$-interval of the form $$0<h<h_1(R)=h_1\,R^{\sigma}A^{-(m-1)},$$ and the new result is $$\widetilde u(x; h)\ge C_-\frac{hA^{m}R^N}{|x|^{N+\sigma}},$$ valid for large $x$ and $h$ suitably small. We are interested in conservation of mass, i.e., $A=R^{-N}$. In that case, denoting the new solution by $u_R(x;h)$ we have $$u_R(x; h)\ge \frac{C_-}{R^{N(m-1)}}\frac{h}{|x|^{N+\sigma}}.$$
$\bullet$ In the same way, the scaling formula applies in combination with the result of Theorem \[up.ell.est\] in the $h$-interval: $0<h<h_2(R)=h_2\,RA^{-(m-1)},$ and the new result is $$u(x; h)\le C_+\frac{hA^mR^N}{|x|^{N+\sigma}}$$ Under conservation of mass, $A=R^{-N}$, denoting the solution by $u_R(x;h)$ we have $$u_R(x; h)\le \frac{C_+}{R^{N(m-1)}}\frac{h}{|x|^{N+\sigma}}.\label{C_+}$$
We are ready to arrive at a contradiction in the comparison of concentrations.
\[counter.ell\] If $m>1$ there exist two nonnegative, compactly supported, bounded, radially symmetric and rearranged functions, $f$ and $f_R$, such that $f_R
\prec f$, and nevertheless the corresponding solutions $u(x;h)$ and $u_R(x;h)$ do not obey the same relation.
[*Proof.*]{} Let us choose $f=\chi_{1}$, then rescaled function $ f_R (x)=R^{-N}f(x/R)=R^{-N}\chi_{R}$ is compactly supported in the ball $B_{R}$, has height $R^{-N}$ and it less concentrated than $f$ if $R>1$. Let us consider the solutions $u$ and $u_R$ that they produce, with the same coefficient $h$. If we apply Theorem \[lower.ell.est\] and inequality to $u$ and $u_{R}$ respectively, we have that $u_R(x)<u(x)$ if $C_+<C_-R^{N(m-1)}$, on the condition that $x$ is large enough, and $h$ is small enough: $$|x|\ge R_-, \quad |x|\ge R_+R; \qquad h<h_1, \quad h< h_2R^{N(m-1)+\sigma}.$$ Since $u_R$ and $u$ have the same mass (because they are solutions to equation with the same $h$ and data having the same mass, see the remark below), this means that $u_R$ cannot be less concentrated than $u$ for such small values of $h$.
Once we have the contradiction for the equation with some $h$ we may put $h=1$ by scaling.
[**Remark.**]{} Here we see that the contradiction is obtained only for $m>1$. For $m\le 1$, $C_+$ will always be larger than $C_-R^{N(m-1)}$, and there is contradiction, just as predicted by the theory, cf. Theorem \[thm.ell.convex2\].
[**Remark.**]{} In order to prove the conservation of the mass for the nonlinearity $A(u)=u^{m}$ with $m>1$, we can argue as in Proposition \[prop.7\]. Indeed, suppose that $v$ be the weak solution of $ (-\Delta)^{\sigma/2}v+ B(v)=f$ with $f\in L^1({{\mathbb R}^N})$ nonnegative and let $u= B(v)$, with $B:{{\mathbb R}}_{+}\rightarrow{{\mathbb R}}_{+}$ be a concave function, strictly increasing, such that $B(0)=0$. Suppose first that $f\in L^{\infty}({{\mathbb R}^N})$ and $|f|\leq
K$. By Theorem \[th.exist\], we have that $u=B(v)\leq K$. By the convexity of $A=B^{-1}$, we have that the function $t\in{{\mathbb R}}_{+}\rightarrow
A(t)/t\in{{\mathbb R}}_{+}$ is increasing, then $$\frac{v}{u}=\frac{A(B(v))}{B(v)}\leq \frac{A(K)}{K}.$$ Now if $\zeta$ is the usual cutoff function and $\zeta_R(x)=\zeta(Rx)$ is its rescaled version, we still find equation . We also have $$\left|\int_{{{\mathbb R}^N}} v\,(-\Delta)^{\sigma/2}\zeta_R\,dx\right| \le \frac{A(K)}{K}\int_{{{\mathbb R}^N}} |u(x)(-\Delta)^{\sigma/2}\zeta_R|\,dx \le
\frac{c}{R^{\sigma}} \int_{{{\mathbb R}^N}} |u(x)|\,dx$$ which in the limit $R\to \infty$ tends to zero. Then by we conclude $$\int_{{{\mathbb R}^N}} u(x)\,dx=\int_{{{\mathbb R}^N}} f(x)\,dx.$$ If $f$ is in $L^{1}({{\mathbb R}^N})$ we proceed by approximation.
[**Last Remark.**]{} We want to point out the comparison performed in this section looks too contrived. Actually, this is partly due to the fact that it is the translation into the elliptic framework of the more natural parabolic counterexample, constructed in Section \[sec.neg.par\].
Symmetrization for the parabolic problem {#sec.par}
========================================
For simplicity of exposition, we start with the case $f=0$. We briefly remind the concept of *weak solution* to the Cauchy problem associated to a fractional parabolic equation : $$\label{eqcauchy}
\left\{
\begin{array}
[c]{lll}u_t+(-\Delta)^{\sigma/2}A(u)=0 & & x\in{{\mathbb R}}^{N}\,,t>0\\[6pt]
u(x,0)=u_{0}(x) & & x\in{{\mathbb R}}^{N}.
\end{array}
\right. $$ Here, $u_{0}$ is an integrable function on ${{\mathbb R}}^{N}$, nonnegative in our applications), the nonlinearity $A(u)$ is a nonnegative concave function with $A(0)=0$ and $A'(u)>0$ for all $u>0$ (extended antisymmetrically in the general two-signed theory). Set $B:=A^{-1}$. As in the elliptic case we rewrite problem as the following quasi-stationary problem $$\label{eqcauchy1}
\left\{
\begin{array}
[c]{lll}-\operatorname{div}_{x,y}\left( y^{1-\sigma}\nabla w\right) =0 & & (x,y)\in{{\mathbb R}}^{N}\times(0,\infty)\,,t>0
\\[6pt]
\dfrac{1}{\kappa_{\sigma}}\displaystyle{\lim_{y\rightarrow0^+}y^{1-\sigma}\,\dfrac{\partial w}{\partial y}}-\dfrac{\partial B(w)}{\partial t}=0 & &
x\in{{\mathbb R}}^{N},\,y=0,t>0,\\
[8pt]
w(x,0,0)=A(u_{0}(x))& & x\in{{\mathbb R}}^{N}.
\end{array}
\right. $$ Then we have the following definition
\[def.weak.par\] We say that $w$ is a weak energy solution to problem if $w\in L^{2}_{loc}((0,\infty);X^{\sigma/2}(\mathcal{C}_{{{\mathbb R}^N}}))$, the function $u(x,t):=(B(w(x,0,t))$ is in the space $C([0,\infty);L^{1}({{\mathbb R}^N}))$ and the following identity holds $$\int_{0}^{\infty}\int_{{{\mathbb R}}^{N}}u\frac{\partial
\varphi}{\partial
t}\,dx\,dt-\frac{1}{\kappa_{\sigma}}\int_{0}^{\infty}\int_{\mathcal{C}_{{{\mathbb R}^N}}}y^{1-\sigma}\,\nabla_{x,y}\,w\cdot\nabla_{x,y}\,\varphi\,dx\,dy\,dt=0$$ for all test functions $\varphi\in C_{0}^{1}(\overline{{{\mathbb R}}_{+}^{N+1}}\times[0,\infty))$. Finally, the initial data are taken in the sense that $$\lim_{t\rightarrow0}u(\cdot,t)=u_{0}(x)\quad \in\,L^{1}({{\mathbb R}}^{N}).$$
If $w$ is a solution to , we sill say that $u(x,t):=(B(w(x,0,t))$ is a *weak solution* to the Cauchy problem . We refer to [@pqrv], [@pqrv2] for questions related to existence and uniqueness of weak solutions to problem .
The theorems we are going to prove, Theorems \[thm.par.convex\] and \[thm.par.convex.f\], will come from the combination of two ingredients: the existence of a *mild solution* to problem that is reduced to solving some elliptic problems by applying the Crandall-Liggett theory for $m$-accretive operators, and the comparison theorems \[genconvex\], \[thm.ell.convex2\], already proved for elliptic problems. Therefore, we will devote a subsection to review this material for the reader’s convenience.
Abstract evolution equations and accretive operators. The semigroup approach {#Appendix}
----------------------------------------------------------------------------
Let $X$ be a Banach space and $\mathcal{A}:D(\mathcal{A})\subset X\rightarrow X$ a nonlinear operator defined on a suitable subset of $X$. Let us consider the problem $$\label{eqcauchyabstract.3}
\left\{
\begin{array}
[c]{lll}u^{\prime}(t)+\mathcal{A}(u)=f, & & t>0,\\[4pt]
u(0)=u_{0}\,, & &
\end{array}
\right.$$ where $u_{0}\in X$ and $f\in L^{1}(I;X)$ for some interval $I$ of the real axis. For a wide class of operators, in particular the ones considered in this paper, a very efficient way to approach such problem is to use an implicit time discretization scheme that we describe next. Suppose to be specific that $I=[0,T]$ (but this can be replaced by any interval $[a,b]$ and the procedure is similar). The method consists in taking first a partition of the interval, say, $t_k=kh$ for $k=0,1,\ldots n$ and $h=T/n$, and then solving the system of difference relations $$\frac{u_{h,k}-u_{h,k-1}}{h}+\mathcal{A}(u_{h,k})=f_{k}^{(h)}\label{discrellprob}$$ for $k=0,1,\ldots n$, where we pose $u_{h,0}=u_{0}$. The data set $\left\{f_{k}^{(h)}:k=1,\ldots,n\right\}$ is supposed to be a suitable discretization of the source term $f$, corresponding to the time discretization we choose. This process is called *implicit time discretization scheme* (ITD for short) of the equation $u^{\prime}(t)+\mathcal{A}(u)=f$. It can be rephrased in the form $$u_{h,k}=J_{h}(u_{h,k-1}+hf_{k}^{(h)})$$ where the operator $$J_{\lambda}=(I+\lambda \mathcal{A})^{-1},\,\lambda>0$$ is called the *resolvent operator*, being $I$ the identity operator. Therefore, the application of the method needs the operator $\mathcal{A}$ to have a well-defined family of resolvents with good properties. When the ITD is solved, we construct a *discrete approximate solution* $\left\{u_{h,k}\right\}_{k}$. By piecing together the values $u_{h,k}$ we form a piecewise constant function, $u_h(t)$, typically defined through $$u_{h}(t)=u_{h,k}\quad\text{if }t\in[(k-1)h,kh]\label{interpol}$$ (or some other interpolation rule, like linear interpolation). Then the main question consists in verifying if such function $u_{h}$ converges somehow as $h\rightarrow0$ to a solution $u$ (which we hope to be a classical, strong, weak, or other type of solution) to problem . To this regard, we first choose a suitable discretization $\left\{f_{k}^{(h)}\right\}$ in time of the source term $f$, such that the piecewise constant interpolation of this sequence produces a function $f^{(h)}(t)$ (defined by means of ) verifies the property $$\|f^{(h)}-f\|_{L^{1}(0,T;X)}\rightarrow0\quad\text{as }h\rightarrow0.$$ By means of these discrete approximate solutions we introduce the following notion of *mild solution*:
We say that $u\in C((0,T);X)$ is a mild solution to if it is obtained as uniform limit of the approximate solutions $u_{h}$, as $h\rightarrow0$. The initial data are taken in the sense that $u(t)$ is continuous in $t=0$ and $u(t)\rightarrow u_{0}$ as $t\rightarrow0$. Besides, we say that $u\in C((0,\infty);X)$ is a mild solution to in $[0,\infty)$ if $u$ is a mild solution to the same problem in any compact subinterval $I\subset [0,\infty)$.
In order to state a positive existence result, we need to restrict the class of operators according to the following definitions.
\[AcRank\]Let $\mathcal{A}:D(\mathcal{A})\subset X\rightarrow X$ be a nonlinear, possibly unbounded operator. Let $R_{\lambda}(\mathcal{A})$ be the range of $I+\lambda \mathcal{A}$, a subset of $X$.
[(i)]{} The operator $\mathcal{A}$ is said accretive if for all $\lambda>0$ the map $I+\lambda \mathcal{A}$ is one-to-one onto $R_{\lambda}(\mathcal{A})\subset X$, and the resolvent operator $J_{\lambda}:R_{\lambda}(\mathcal{A})\rightarrow X$ is a (non-strict) contraction in the $X$-norm (i.e., a Lipschitz map with Lipschitz norm 1).
[(ii)]{} We say that $\mathcal{A}$ satisfies the rank condition if $R_{\lambda}(\mathcal{A})\supset\overline{D(\mathcal{A})}$ for all $\lambda>0$. In particular, the rank condition is satisfied if $R_{\lambda}(\mathcal{A})=X$ for all $\lambda>0$; in this case, if $\mathcal{A}$ is accretive, we say that $\mathcal{A}$ is $m$-accretive.
We are now ready to state the desired semigroup generation result, that generalizes the classical result of Hille-Yosida (valid in Hilbert spaces and for linear $\mathcal{A}$) and the variant by Lumer and Phillips (valid in Banach spaces, still for linear $\mathcal{A}$), and provides the existence and uniqueness of mild solutions for problems of the type in the case $f\equiv0$:
\[CrLigg\] Suppose that $\mathcal{A}$ is an accretive operator satisfying the rank condition. Then for all data $u_{0}\in \overline{D(\mathcal{A})}$ the limit $$S_{t}(\mathcal{A})u_{0}=\lim_{n\rightarrow\infty}(J_{t/n}(\mathcal{A}))^{n}u_{0}.\label{CrLig}$$ exists uniformly with respect to $t$, on compact subset of $[0,\infty)$, and $u(t)=S_{t}(\mathcal{A})u_{0}\in C([0,\infty): X)$. Moreover, the family of operators $\left\{S_{t}(\mathcal{A})\right\}_{t>0}$ is a strongly continuous semigroup of contractions on $\overline{D(\mathcal{A})}\subset X$.
Using a popular notation in the linear framework, we could write $S_{t}(\mathcal{A})u_{0}=e^{-t\mathcal{A}}u_{0}$, and because of this analogy formula is called the *Crandall-Liggett exponential formula for the nonlinear semigroup generated by* $-\mathcal{A}$. The problem with this very general and useful result is that the $X$-valued function $u(t)=S_t(\mathcal{A})u_0$ solves the equation only in a mild sense, that is not necessarily a strong solution or a weak solution. Though it is known that strong solutions are automatically mild, the correspondence between mild and weak solutions is not always clear. For the FPME this issue has been discussed in detail in [@pqrv; @pqrv2].
In addition, the Crandall-Liggett Theorem result can be extended when we consider nontrivial source term $f$, according to the following result
\[existmildsol\] Suppose that $\mathcal{A}$ is $m$-accretive. If $f\in L^{1}(0,\infty;X)$ and $u_{0}\in\overline{D(\mathcal{A})}$. Then the abstract problem has a unique mild solution $u$, obtained as limit of the discrete approximate solution $u_{h}$ by ITD scheme described above, as $h\rightarrow0$: $$u(t):=\lim_{h\rightarrow0}u_{h}(t)\,,$$ and the limit is uniform in compact subsets of $[0,\infty)$. Moreover, $u\in C([0,\infty);X)$ and for any couple of solutions $u_{1}$, $u_{2}$ corresponding to source terms $f_1$, $f_2$ we have $$\|u_{1}(t)-u_{2}(t)\|_{X}\leq\|u_{1}(s)-u_{2}(s)\|_{X}+\int_{s}^{t}\|f_{1}(\tau)-f_{2}(\tau)\|_{X}d\tau$$ for all $0\leq s<t$.
There is a wide literature on these topics, starting with the seminal paper by Crandall and Liggett [@CL71], see also [@Cr86] and the general reference [@Barbu]. These notes are based on Chapter 10 of the book [@vazquezPME], cf. the references therein. The last formula we have mentioned introduces the correct concept of uniqueness for the constructed class of solutions. Characterizing the uniqueness of different concepts of solution is a difficult topic already discussed by Bénilan in his thesis [@BeTh].
Parabolic Symmetrization
------------------------
In order to apply this theory we have to check that the operator associated to our evolution problem $\mathcal{\mathcal{A}}$ is $m$-accretive or that it is accretive and the rank condition holds, in the sense of definition \[AcRank\]. A main question in this approach to nonlinear evolution is the corrected identification of the operator. This has been done in [@pqrv] as follows.
If $u_{0}\in L^{1}({{\mathbb R}^N})\cap L^{\infty}({{\mathbb R}^N})$, we introduce the nonlinear operator $\mathcal{A}_0: D(\mathcal{A}_0)\subset L^{1}({{\mathbb R}^N})\rightarrow L^{1}({{\mathbb R}^N})$, defined by $$\mathcal{A}_0(u):=(-\Delta)^{\sigma/2}A(u)\,,$$ with domain $$D(\mathcal{A}_0 ):=\left\{v\in L^1({{\mathbb R}^N})\cap L^{\infty}({{\mathbb R}^N}): \mathcal{A}_0(v)\in L^1({{\mathbb R}^N})\cap L^{\infty}({{\mathbb R}^N}) \right\}.$$ Returning to the results of Subsection \[sect.ell\].1, we see that the contractive property implies that this operator is accretive in the space $X=L^1({{\mathbb R}^N})$. On the other hand, Theorem \[th.exist\] and its extension on ${{\mathbb R}^N}$ gives the rank condition in $L^1\cap L^\infty$. By closing this operator with respect to the norm $\|.\|_1$ we find an operator $\mathcal{A}$ that is $m$-accretive in $L^1({{\mathbb R}^N})$.
Therefore, we can use Theorem \[existmildsol\] that implies that there is a unique mild solution to , obtained as a limit of discrete approximate solutions by the ITD scheme. In the case $A(u)=u^m$ the extra regularity of these solutions is discussed in detail in the papers [@pqrv; @pqrv2]. For general $A$ see [@pqrv4].
We can now use this method to prove a symmetrization result for Fractional Fast Diffusion Equations, including in particular the well-known linear fractional heat equation, $$u_t+(-\Delta)^{\sigma/2}u=0.$$
\[thm.par.convex\] Let $u$ be the mild nonnegative solution of the FPME with $0<\sigma<2$, posed in $\Omega={{\mathbb R}^N}$, with initial data $u_0\in L^1({{\mathbb R}^N})\ge 0$ and nonlinearity $A(u)$ given by a concave function with $A(0)=0$ and $A'(u)>0$ for all $u>0$. Let $v$ be the solution of the corresponding symmetrized problem $$\label{eqcauchysymm}
\left\{
\begin{array}
[c]{lll}v_t+(-\Delta)^{\sigma/2}A(v)=0 & & x\in{{\mathbb R}}^{N}\,, \ t>0,\\[6pt]
v(x,0)=u_{0}^{\#}(x) & & x\in{{\mathbb R}}^{N}\,.
\end{array}
\right. $$ Then we have for all $t>0$ $$u^\#(|x|,t)\prec v(|x|,t).\label{concentrationcomp}$$ In particular, we have $\|u(\cdot,t)\|_p \le\|v(\cdot,t)\|_p$ for every $t>0$ and every $p\in [1,\infty]$.
[*Proof.*]{} According to what explained before, we use the implicit time discretization scheme. For each time $T>0$, we divide the time interval $[0,T]$ in $n$ subintervals $(t_{k-1},t_{k}]$, where $t_{k}=kh$ and $h=T/n$. We construct then the function $u_{h}$ which is piecewise constant in each interval $(t_{k-1},t_{k}]$, by $$u_{h}(x,t)=
\left\{
\begin{array}
[c]{lll}u_{h,1}(x) & & if\,\,t\in[0,t_{1}]\\[6pt]
u_{h,2}(x) & & if\,\,t\in(t_{1},t_{2}]
\\
[6pt]
\cdots
\\
[6pt]
u_{h,n}(x) & & if\,\,t\in(t_{n-1},t_{n}]
\end{array}
\right. $$ where $u_{h,k}$ solves the equation $$h(-\Delta)^{\sigma/2}A(u_{h,k})+u_{h,k}=u_{h,k-1}\label{eq.18}$$ with the initial value $u_{h,0}=u_{0}$. Similarly, concerning the symmetrized problem , we define the piecewise constant function $v_{h}$ by $$v_{h}(x,t)=
\left\{
\begin{array}
[c]{lll}v_{h,1}(x) & & if\,t\in[0,t_{1}]\\
[6pt]
v_{h,2}(x) & & if\,t\in(t_{1},t_{2}]
\\
[6pt]
\cdots
\\
[6pt]
v_{h,n}(x) & & if\,t\in(t_{n-1},t_{n}]
\end{array}
\right. $$ where $v_{h,k}(x)$ solves the equation $$h(-\Delta)^{\sigma/2}A(v_{h,k})+v_{h,k}=v_{h,k-1}\label{eq.20}$$ with the initial value $v_{h,0}=u_{0}^{\#}$. Our aim is now to compare the solution $u_{h,k}$ to with the solution . We proceed by induction. Using Theorem \[genconvex\], we get $$A(u^{\#}_{h,1})\prec A(v_{h,1}).$$ If we suppose by induction that $u^{\#}_{h,k-1}\prec v_{h,k-1}$ and call $\widetilde{u}_{h,k}$ the (radially decreasing) solution to the equation $$h(-\Delta)^{\sigma/2}A(\widetilde{u}_{h,k})+\widetilde{u}_{h,k}=u^{\#}_{h,k-1},$$ Theorem \[genconvex\] and Theorem \[thm.ell.convex2\] imply $$\label{eq.21}
A(u^{\#}_{h,k})\prec A(\widetilde{u}_{h,k})\prec A(v_{h,k})\,,$$ hence holds for all $k=1,\ldots,n$. Therefore, by the definition of $u_{h}$ and $v_{h}$, we find $$A(u_{h}(\cdot,t)^{\#})\prec A(v_{h}(\cdot,t))\label{eq.22}$$ for all times $t$. Using Lemma \[lemma1\] with the choice $\Phi=F\circ B$, where $F\ge 0$ is convex and $F(0)=0$, we obtain $$\int_{{{\mathbb R}^N}} F(u_{h}^{\#}(x,t))dx\leq\int_{{{\mathbb R}^N}} F(v_{h}(x,t))dx\,,$$ which in turn yields $$u_{h}^{\#}(\cdot,t)\prec v_{h}(\cdot, t).\label{conccomph}$$ Now Crandall-Liggett Theorem implies $$u_{h}\rightarrow u,\quad v_{h}\rightarrow v\,\, \text{uniformly}.$$ Then passing to the limit in we get the result.
Symmetrization for the equation with a left-hand side
-----------------------------------------------------
We now consider the case $f\in L^1(Q)$, $Q={{\mathbb R}^N}\times(0,\infty)$ and $f\not\equiv 0$. In that case the semigroup generation Theorem \[existmildsol\] can still be applied to obtain the so-called [*unique mild solution* ]{} of the evolution problem $$\label{eqcauchy.f}
\left\{
\begin{array}
[c]{lll}u_t+(-\Delta)^{\sigma/2}A(u)=f & & x\in{{\mathbb R}}^{N}, \ t>0\,,\\[6pt]
u(x,0)=u_{0}(x) & & x\in{{\mathbb R}}^{N}.
\end{array}
\right. $$ As explained in Subsection \[Appendix\], we need to perform a discretization of $f$ adapted to the time mesh $t_k=kh$ that we have used above, let us call it $\{f_k^{(h)}\}$, so that the piecewise constant (or linear in time) interpolation of this sequence produces a function $f^{(h)}(x,t)$ such that $\|f-f^{(h)}\|_1\to 0$ as $h\to 0$. Then we use the previous implicit discretization scheme, now in the form $$\label{disc.ev.eqn}
\frac{1}{h}(u_{h,k}-u_{h,k-1})+(-\Delta)^{\sigma/2}A(u_{h,k})=f_k^{(h)}\,,$$ to produce the semi-discrete function $\{u_{h}(x,t_k)=u_{h,k}(x): k=0,1,\cdots\}$, which after interpolation in time serves as $h$-approximation to the mild solution $u(x,t)$. According to we have to solve the elliptic problems $$\label{eq.18b}
h(-\Delta)^{\sigma/2}A(u_{h,k})+u_{h,k}=u_{h,k-1}+ h\,f_k^{(h)}\,,$$ and we can use the theory developed in Section \[sect.ell\]. Then we have the following result
\[thm.par.convex.f\] Let $u$ be the nonnegative mild solution of the FPME with $0<\sigma<2$, posed in $\Omega={{\mathbb R}^N}$, with initial data $u_0\in L^1({{\mathbb R}^N})$, $u_0\ge 0$, right-hand side $f\in L^1(Q)$, $f\ge 0$, and nonlinearity $A(u)$ given by a concave function with $A(0)=0$ and $A'(u)>0$ for all $u>0$. Let $v$ be the solution of the symmetrized problem $$\label{eqcauchysymm.f}
\left\{
\begin{array}
[c]{lll}v_t+(-\Delta)^{\sigma/2}A(v)=f^{\#}(|x|,t) & & x\in{{\mathbb R}}^{N}\,, \ t>0,\\[6pt]
v(x,0)=u_{0}^{\#}(x) & & x\in{{\mathbb R}}^{N},
\end{array}
\right. $$ where $f^{\#}(|x|,t)$ means symmetrization of $f(x,t)$ w.r. to $x$ for a.e. time $t>0$. Then, for all $t>0$ we have $$u^\#(|x|,t)\prec v(|x|,t).$$ In particular, we have $\|u(\cdot,t)\|_p \le\|v(\cdot,t)\|_p$ for every $t>0$ and every $p\in [1,\infty]$.
The proof follows the lines of Theorem \[thm.par.convex\], so we leave the details to the reader.
[**Remark.**]{} As an easy extension, we can have also a result about comparison of concentrations for the radial solutions of two evolution problems, if we assume that the initial data satisfy the condition $u_{0,1}\prec u_{0,2}$ and the right-hand sides satisfy $f_1(\cdot,t)\prec f_{2}(\cdot,t)$ for almost all $t>0$. The conclusion is that $u_1(\cdot,t)\prec u_2(\cdot,t)$ for all $t>0$. Let us remind the reader that the result holds only if $A$ is linear or concave, as assumed above.
Negative result about concentration comparison for the Fractional PME {#sec.neg.par}
=====================================================================
As in the elliptic case, it came to us as a surprise that the comparison result could not be proved for general nonlinearities $A$ without the assumption of concavity. It turns out that for convex powers it does not hold. Here we will state and prove the negative result about concentration comparison for solutions of the Fractional Diffusion Equation in the range of exponents $m>1$, usually known as Slow Diffusion. We first argue in a formal way, since we give later the justification of some details.
$\bullet$ Let us consider the Fractional PME: $u_t+(-\Delta)^{\sigma/2}u^m=0$ in ${{\mathbb R}^N}$ and consider first the Barenblatt solution that was studied in [@vazBaren]. Here we suppose that $$m>(N-\sigma)/N=:m_c.$$ Let us fix the mass to 1 for simplicity. The Barenblatt solution has the form $$U_{1}(x,t)=t^{-\alpha}F_{m,1}(\xi), \qquad \xi=|x|t^{-\beta}$$ with $\alpha=N\beta$ and $\beta=1/(N(m-1)+\sigma)$ and $F_{m,1}$ is the Barenblatt profile of mass 1. It is also known that as $\xi\to\infty$ we have $$F_{m,1}(\xi)\sim C\,\xi^{-(N+\sigma)}.\label{asymptobeh}$$ This means that for large $x$ and $t\sim 0$ (so that $\xi\sim \infty$) we get $$U_{1}(x,t)\sim C\,t^{\lambda}|x|^{-(N+1)}, \qquad \lambda=\beta\sigma.$$ This approximation holds uniformly for all $|x|\ge C$ large and all $0<t<\tau$ if $\tau$ is small enough, that is the error is higher order small $$U_{1}(x,t)= C\,t^{\lambda}|x|^{-(N+\sigma)}(1+\varepsilon).$$ Now, let us choose the initial data to associate to the equation. Let $$u_0(x)=\phi(|x|)\ge 0\,,$$ where $\phi$ is a smooth and compactly supported function in the ball of radius one, having mass 1. Suppose also that $\phi$ is rearranged. Let us call $u$ the solution to the equation with such choice of the data. We have that $(-\Delta)^{\sigma/2}u_0^m$ is a bounded function that decreases at infinity like $C_1\,|x|^{-(N+\sigma)}$. By virtue of the equation we have $$u_t(x,0)=-(-\Delta)^{\sigma/2}u_0^m$$ so that for small $t$ we have approximately $$u(x,t)\sim - t\,(-\Delta)^{1/2}u_0^m$$ and this behaves as $t\to\infty$ like $Ct|x|^{-(N+\sigma)}$. A rigorous proof of this behaviour will be given in the next two subsections.
The conclusion is that the concentration comparison result is not true. The reason is that the exponent $$\lambda=
\sigma\beta=\frac{\sigma}{\sigma+N(m-1)}$$ is less than 1 (precisely for $m>1$). Clearly, $U_0=\delta(x)$ is more concentrated than $u_0$, but $u(x,t)$ is larger than $U(x,t)$ for very large $x$ if $t$ is quite small. This is incompatible with satisfying the concentration comparison and having the same mass.
In the graphics of Figure 1 we show the relative evolution of two solutions in time. Initially the concentrations are ordered, later they are not. The parameter $m$ is 2.
\[figure\_Rates\] {width="80.00000%" height="43.00000%"} {width="80.00000%"}
Supersolution. First tail estimate
----------------------------------
Let us now give a rigorous derivation of the tail behaviour. Fist, we have a preparatory step.
\[upper.tail.par\] Let $u(x,t)$ be a classical solution of the FPME with initial data $u_0(x)\ge0$ such that $u_0(x)\le 1$ in the ball $B_1(0)$ and $u_0(x)\le
\,|x|^{-(N+\sigma)}$ for $|x|>1$. Then there is a time $t_1>0$ such that $$\label{rate1.tail}
u(x,t)\le 2\, |x|^{-(N+\sigma)}$$ if $|x|\ge R$ and $R$ is large enough and $0<t<t_1$.
[*Proof.*]{} We consider the FPME for $m>1$ and initial data $u_0$ is 1 in the ball of radius 2. Then for all times the solution will be bounded by 1.
We want to construct a super-solution of the form $$U(x,t)= (1+bt)\,F(|x|),$$ where $F\ge 0$ has to chosen. I will need $F(r)\sim C_1r^{-(N+\sigma)}$ as $r=|x|\to\infty$ to get the desired conclusion after the comparison argument in the following way: $$u(x,t)\le U(x,t)\le 2F(r)\le 2C_1\,r^{-(N+\sigma)}$$ if $r$ is large and $t\le t_1/b \sim 0$.
To establish such comparison we first note that $U_t=b F(x).$ Using the notation $$L_{\sigma}=(-\Delta)^{\sigma/2}$$, we also have $$L_{\sigma}U^m=(1+bt) L_{\sigma}F^m(x)$$ As we have pointed out, $F(r)\sim C_1r^{-(N+\sigma)}$ as $r\to\infty$ so that $ L_{\sigma}F^m=O(r^{-(N+\sigma)})$ for $r>1$ (cf. Lemma 2.1 of [@BV2012]). It follows that there is a constant $k>0$ such that $$kF+L_{\sigma}\,F^m\ge 0, \quad \mbox{everywhere in } \ {{\mathbb R}^N}.$$ Therefore, we will have $$U_t+ L_{\sigma}U^m = bF + (1+bt)^mL_{\sigma}F^m\ge 0$$ if $b>k(1+bt)$, i.e. if $b>k$ and $t<(b-k)/kb$, for instance for $b=2k$ and $t<1/b=1/2k$.
Under such assumptions, the viscosity method will work in the exterior region $Q=\{(x,t): |x|\ge 1, 0<t<t_1\}$, and this will prove that $U(x,t)\le u(x,t)$ in $Q$ as desired.
We finally check the application of the viscosity method. Indeed, the boundary condition at $r=1$ is $$U(1,t)\ge F(1)\ge 1.$$ so $U(x,t)\ge u(x,t)$ on the lateral boundary of $Q$ located at $r=1$. Same comparison is trivial for $t=0$. We only need to argue by contradiction at the first point where the classical solution $u$ touches $U$ from below to conclude that $u(x,t)$ is strictly less than $U(x,t)$ in $Q$. The contradiction at the point of contact is explained in [@BV2012]. The construction of classical solutions is done in [@pqrv4].
[**Remarks.**]{} (i) We have done the argument for $R=1$. If we want to change the radius to $R>1$ we may use the scaling of the equation.
\(ii) Lower estimates that match the tail behaviour are derived and used in [@StanV2013].
Supersolution. Sharp tail estimate
----------------------------------
Let $u(x,t)$ be a classical solution of the FPME with initial data $u_0(x)\ge0$ such that $u_0(x)\le 1$ in the ball $B_1(0)$ and $u_0(x)=0$ for $|x|>1$. Then there is a time $t_1>0$ and constants $C^*$ and $R$ such that $$\label{rate2.tail}
u(x,t)\le C^* t\, |x|^{-(N+\sigma)}$$ if $|x|\ge R$ and $R$ is large enough and $0<t<t_1$.
[*Proof.*]{} We proof is a delicate variation of the preceding one. We still consider the FPME for $m>1$ and initial data $u_0$ is 1 in the ball of radius 2. For all times the solution will be bounded by 1. We consider a supersolution of the form $$U^m(x,t)= G(x) + b^mt^m\,F(x)^m,$$ where $F\ge 0$ is chosen as before. I take $F(r)\sim C_1r^{-(N+\sigma)}$ as $r\to\infty$. Again, it follows that there is a constant $k>0$ such that $$kF+L_{\sigma}\,F^m\ge 0, \quad \mbox{everywhere in } \ {{\mathbb R}^N}.$$ Next we choose $G\ge 0$ compactly supported and such that $L_{\sigma}G=c_0>0$ on the support. As $r\to\infty$, we get the usual $L_{\sigma}G\sim
-cr^{-(d+2s)}$.
We get the formula $$U_t=b(G(x)+ b^mt^m\,F^m(x))^{(1/m)-1}t^{m-1}F^m(x)\,,$$ which reduces to $U_t=bF$ when $G=0$. In any case it is nonnegative, $U_t\ge 0$. We also have $$L_{\sigma}U^m=L_{\sigma}G+ b^mt^m L_{\sigma}F^m(x).$$ Then when $G=0$ we will have $$U_t+ L_{\sigma}U^m = bF + (bt)^mL_{\sigma}F^m + L_{\sigma}G\ge 0$$ if $b>b_0$, $b>2k(bt)^m$, i.e. if $b^{m-1}t^mk<1/2$, which imposes a condition above on $t$.
On the other hand for $G>0$ we have $$U_t+ L_{\sigma}U^m \ge c_0+(bt)^mL_{\sigma}F^m\ge 0$$ if $C_2(bt)^m\le c_0$. Both conditions are fulfilled if $0<t<t_1$.
Is this is the case the viscosity method will work in the region $Q=\{|x|\ge 1, 0<t<t_1\}$. and this will prove that $U(x,t)\le u(x,t)$ in $Q$.
Indeed, the boundary condition at $r=1$ is $$U(1,t)\ge G(1)\ge 1.$$
[**Remarks.**]{} The rate of decay of the tail of such solutions at infinity is optimal as a consequence of the construction of suitable sub-solutions with the same exponents in the $x$ and $t$ dependence, which is done in [@StanV2013].
The counterexample is heavily technical. Surprisingly, the situation becomes much clearer when we let $m\to\infty$. This is studied in [@VazMesa].
Comments, extensions and open problems
======================================
-In a companion paper [@VazVol2] we will use symmetrization results of this paper to obtain sharp a priori estimates with best constants for some functional embeddings involving the solutions of the linear fractional heat equation or its fast diffusion relative.
-As an extension of the above results, we could consider equations that involve a more general version of the fractional Laplacian operator, in the same way that the standard symmetrization applies to elliptic equations with coefficients.
-The elliptic and parabolic counterexamples have been constructed for the problems posed on the whole space. They could also be constructed for solutions defined on a bounded domain, say a ball, with zero Dirichlet boundary conditions. The argument is as follows: we consider the problems posed in a sequence of balls $B_R$ expanding so that $R\to\infty$ with same data of compact support. According to [@pqrv; @pqrv2] the solutions $u_R$ converge to the solutions of the limit problem in the whole space. Now, for the limit equation there is a counterexample. We deduce that there is a counterexample before the limit. We leave the details to the reader.
-Another interesting problem would be obtaining a priori estimates for solutions of elliptic and parabolic problems of this type with Neumann boundary conditions using symmetrization techniques. A good indication is that conservation of mass is true for both elliptic and parabolic problems.
Let us now list some open problems that have arisen in the course of the work:
-We do not know how to deal with concave nonlinearities $A$ in bounded domains.
-We do not know how to do the elliptic or parabolic comparison in the case of more general function $A$, if it is neither concave or convex.
-Finally, we wonder if there is a partial or alternative theory that replaces the failure of the concentration comparison result for the fractional porous medium equation, i.e. the equation $$\partial_t u +(-\Delta)^{\sigma/2}u^m=0$$ with $m>1$.
[**Acknowledgments**]{}
Both authors partially supported by the Spanish project MTM2011-24696. We thank Felix del Teso for the computations supporting Figure 1.
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[^1]: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain. E-mail: [juanluis.vazquez@uam.es]{}
[^2]: Dipartimento per le Tecnologie, Facoltà di Ingegneria, Università degli Studi di Napoli “Parthenope”, 80143 Italia. E-mail: [bruno.volzone@uniparthenope.it]{}
[^3]: More generally, $A$ can be a maximal monotone graph, but that generality is of no concern for us here.
[^4]: See the definitions of the order relation $\prec$ in Section \[sec.prelim\].
|
---
abstract: 'We report a Fe K$\beta$ x-ray emission spectroscopy study of local magnetic moments in the rare-earth doped iron pnictide Ca$_{1-x}RE_x$Fe$_2$As$_2$ ($RE$=La, Pr, and Nd). In all samples studied the size of the Fe local moment is found to decrease significantly with temperature and goes from $\sim \!\! 0.9 \mu_B$ at T = 300 K to $\sim \!\! 0.45 \mu_B$ at T = 70 K. In the collapsed tetragonal (cT) phase of Nd- and Pr-doped samples (T$<$70K) the local moment is quenched, while the moment remains unchanged for the La-doped sample, which does not show lattice collapse. Our results show that Ca$_{1-x}RE_x$Fe$_2$As$_2$ ($RE$= Pr and Nd) exhibits a spin-state transition and provide direct evidence for a non-magnetic Fe$^{2+}$ ion in the cT-phase, as predicted by Yildirim. We argue that the gradual change of the the spin-state over a wide temperature range reveals the importance of multiorbital physics, in particular the competition between the crystal field split Fe 3$d$ orbitals and the Hund’s rule coupling.'
author:
- 'H. Gretarsson'
- 'S. R. Saha'
- 'T. Drye'
- 'J. Paglione'
- Jungho Kim
- 'D. Casa'
- 'T. Gog'
- 'W. Wu'
- 'S. R. Julian'
- 'Young-June Kim'
title: 'Spin-state transition in the Fe-pnictides'
---
The interesting orbital physics found in many 3$d$ and 4$d$ transition metal compounds, such as manganites [@Nagaosa2000; @Hotta2006] and ruthenates [@Hotta2006], seems to play an important role in the iron based superconductors as well [@Kruger2009; @Lv2009; @Lv2010; @Lee2009; @Lee2010; @Chan2009; @Chan2010]. In the iron pnictides, many low-energy probes such as transport [@Fisher2010], scanning tunnelling microscopy [@Davis2010], inelastic neutron scattering [@Pengcheng2009], angle-resolved photoemission spectroscopy [@Shen2011; @Shimojima2010], and most recently magnetic torque measurements [@Kasahara2012] have reported a strong in-plane anisotropy of electronic properties. These results have spurred a great deal of interest in the orbital physics of the iron pnictides, in particular the possibility of orbital order [@Kruger2009; @Lv2009; @Lv2010; @Lee2009; @Lee2010; @Chan2009; @Chan2010].
An important aspect of the orbital physics is the competition between the Hund’s rule coupling constant ${\emph J_H}$ and the crystal field splitting, $\Delta_{{\rm CF}}$. In the case of LaCoO$_3$, the energy scales of $\Delta_{{\rm CF}}$ and ${\emph J_H}$ are similar, resulting in spin-state transition; Co$^{3+}$ ions take on a low-spin state (S=0) at low temperature, but go into thermally excited high/intermediate-spin (S=2 or S=1) states at elevated temperature [@Blasse1965; @Sawatzky1996]. Applied pressure, either chemical or hydrostatic, can alter $\Delta_{{\rm CF}}$ through lattice changes and affect the balance between ${\emph J_H}$ and $\Delta_{{\rm CF}}$. A high-spin state was for instance observed in the case of La$_{0.82}$Sr$_{0.18}$CoO$_3$ in which Sr dopants exert negative chemical pressure [@Lengsdorf2007], while a low-spin state was observed under hydrostatic pressure [@VankoPRB2006; @Lengsdorf2007].
Among the iron based superconductors, CaFe$_2$As$_2$ offers perhaps the best system to investigate the competition between $\Delta_{{\rm CF}}$ [@CF] and ${\emph J_H}$, and its effect on the spin-state. Like many iron pnictides, CaFe$_2$As$_2$ goes from a high temperature tetragonal phase (T-phase) to an orthorhombic and antiferromagnetically (AFM) ordered phase, below $T_N \approx$ 170 K [@Canfield2008]. More importantly, CaFe$_2$As$_2$ takes on yet another structural phase at low temperatures through application of a modest pressure of 0.35 GPa [@Kreyssig2008] or chemical doping, with rare-earths [@Saha2012] or phosphorus [@Kasahara2011]. Upon entering this phase, known as the collapsed tetragonal phase (cT-phase), the lattice undergoes a $\sim\!\! 10 \%$ reduction along the [*c*]{}-axis and an $\sim\!\! 2 \%$ increase along the [*a*]{}-axis. This is accompanied by a disappearance of the AFM order [@Kreyssig2008], supression of spin fluctuations [@Pratt2009], and recovery of Fermi liquid behavior [@Kasahara2011]. It is thus clear that an unusually dramatic lattice instability exists in CaFe$_2$As$_2$ and its doped counterparts, and that the magnetic and the electronic structure are strongly influenced by this instability. In particular, the change in $\Delta_{{\rm CF}}$, due to the distortion of the FeAs$_4$ tetrahedra in the cT-phase [@Saha2012], can result in a very different spin-state of the Fe$^{2+}$ ion.
-2cm -1cm {width="1.8\columnwidth"}
In this Letter, we use x-ray emission spectroscopy (XES), which is a very sensitive probe of local instantaneous spin moment, to investigate the spin-state of the Fe$^{2+}$ ion in Ca$_{0.92}$Nd$_{0.08}$Fe$_2$As$_2$ (Nd-doped), Ca$_{0.85}$Pr$_{0.15}$Fe$_2$As$_2$ (Pr-doped), and Ca$_{0.78}$La$_{0.22}$Fe$_2$As$_2$ (La-doped) as a function of temperature. We find that local moments disappear in the cT-phase of the Nd- and Pr-doped samples, indicating that the Fe$^{2+}$ ions go through a spin-state transition by taking on the low-spin state; this confirms earlier calculations by Yildirim [@Yildirim2009]. X-ray absorption near edge spectra (XANES) taken at the Fe $K$-edge reveal large changes in the electronic structure due to the collapse of the lattice. Surprisingly for all the samples studied, the Fe local moment is found to decrease significantly with decreasing temperature in the T-phase, from $\sim \!\! 0.9 \mu_B$ at T = 300 K to $\sim \!\! 0.45 \mu_B$ below T$\sim$70 K. This behavior could be described as a thermally induced spin-state crossover which arises due to the competition between $\Delta_{{\rm CF}}$ and ${\emph J_H}$. Our findings seem to suggest that spin-state degeneracy could be important for the understanding of iron pnictides, as recently proposed by Chaloupka and Khaliullin [@Khaliullin2012].
The XES measurement was performed at the Advanced Photon Source on the undulator beamline 9ID-B using the identical setup as in Ref. [@emission2011]. The XANES spectra in the partial fluorescence yield mode (PFY-XANES) was measured by monitoring the Fe K$\beta$ emission line across the Fe $K$-edge. X-ray diffraction measurements were performed using a Cu tube source with a graphite (002) monochromator, and a four-circle diffractometer. For all temperature dependence studies closed-cycle refrigerators were used. Details of the growths and characterization of the single-crystal samples have been reported in earlier publications [@Wu2011; @Saha2012].
The local moment sensitivity of the Fe K$\beta$ emission line $(3p
\rightarrow 1s)$ originates from a large overlap between the $3p$ and $3d$ orbitals. This interaction is mainly driven by the presence of a net magnetic moment $(\mu)$ in the $3d$ valence shell [@Tsutsumi1976; @Peng1994] and causes the K$\beta$ emission line to split into a main peak K$\beta_{1,3}$ and a low-energy satellite K$\beta^\prime$. A schematic diagram of the Fe K$\beta$ emission process is shown in Fig. \[fig01\] (a) inset for both non-magnetic (red) and magnetic (blue) Fe$^{2+}$ in the atomic limit. Filled and empty circles represent electrons and holes, respectively, and $\Delta E$ represents the splitting of K$\beta_{1,3}$ and K$\beta^\prime$. Information on the local moment of Fe can be extracted using the overall shape of the Fe K$\beta$ emission spectra by applying the integrated absolute difference (IAD) analysis [@Vanko2006]. In Fig. \[fig01\] (a) we demonstrate how this method works by showing Fe K$\beta$ XES data for the Nd-doped sample taken at T = 300 K along with a non-magnetic FeCrAs reference spectrum [@emission2011; @Wu2009; @Ishida1996]. Relative to the main line in FeCrAs, we see that the Nd-doped K$\beta_{1,3}$ peak shifts towards higher energy, while the intensity and the width of this peak also change; a contribution from K$\beta^\prime$ on the lower energy side becomes visible now. These changes are all attributed to the existence of a local moment. To follow the IAD procedure from Ref. , the area underneath each spectrum was normalized to unity. The reference spectrum was then subtracted from the sample spectrum, and the resulting difference plotted. For display purpose, the difference was magnified by a factor of 4. The IAD value can be extracted by integrating the absolute value of the difference spectrum. This quantity is found to be linearly proportional to the local spin magnetic moment of the Fe atom [@Vanko2006]. This method has recently been applied to study various iron-based superconductors [@emission2011; @Chen2011; @Simonelli2012].
Fe K$\beta$ emission lines obtained at different temperatures are shown in Fig. \[fig01\] (b) - (e) for both the Pr- and La-doped samples. At T = 300 K the samples show the same characteristics as the Nd-doped. However, at T = 45 K significant changes can be observed, the K$\beta_{1,3}$ shifts towards lower energy and the contribution from K$\beta^\prime$ is supressed. This is well captured in the difference spectra and provides evidence for a decreased local moment. The change is much larger for the Pr- than for the La-doped sample; in fact a complete supression of the difference spectra is observed for the Pr-doped sample. It should be noted that such a strong thermally induced change is surprising given that neither the presence of long-range order nor carrier doping had any affect on local magnetic moment in other iron based superconductors [@emission2011].
![(Color online) The temperature dependence of (a) c-axis lattice parameters, and (b) the IAD values derived from the XES spectra. Same symbols are used in both panels. On the right hand side of panel (b), the local magnetic moment scale ($\mu$) as described in Ref. are shown. Thick dashed lines are guides to the eyes. The solid red line and thin dashed black lines are fits to 3-state and 2-state models, respectively. The spin and orbital configuration of the 3-state model used for the fitting is shown in the inset. The grey area represents the detection limit of the IAD method. []{data-label="fig02"}](fig02){width="\columnwidth"}
In order to extract quantitative information about the evolution of the local moment in these samples we have studied detailed temperature dependence of the IAD values. The results are plotted in Fig. \[fig02\] (b), in which the right-hand side of the figure is the local moment scale determined from the IAD value of $\rm K_2Fe_4Se_5$ [@emission2011]. Dashed lines are given as guides to the eye. At room temperature all three samples have local moments around $\sim\!\! 0.9{\rm \mu_B}$. With decreasing temperature this local moment decreases and at T = 70 K the local moment has already been reduced by a factor of 2, to a value of $\sim\!\! 0.45{\rm \mu_B}$. Below T = 70 K a noticable difference among the three samples can be observed. Both Nd- and Pr-doped samples show an abrupt drop in the IAD value down to an undetectable level, while the La-doped sample shows no change in the same temperature range. The detection limit of the IAD technique is shown as a grey area and indicates that at low temperature the local moment for Nd- and Pr-doped samples is $<\!\!0.2{\rm \mu_B}$. The evolution of the c-axis lattice constant, as determined from the position of the (008) Bragg peak, is shown in Fig. \[fig02\] (a). Data were obtained on cooling and shows the T- to cT-phase transition clearly for the Nd- and Pr-doped samples. One should note that the lattice constant change is unusually large even well above the cT-T transition temperature. The linear thermal expansion coefficient of the La-doped sample is almost a factor of 10 larger than that of Ba(Fe,Co)$_2$As$_2$ [@Luz2009].
The change of the local moment from T= 300 K to 70 K is puzzling. In LaCoO$_3$, a continuous increase in the local spin moment as a function of increased temperature was observed [@VankoPRB2006]. Using a three spin-state model, this behavior was explained as arising from spin-state transition, in which the low-spin Co$^{3+}$ ions ($S$=0) are thermally excited into a magnetic state ($S$=1 or 2), resulting in a larger local moment. To determine whether changes in the spin-state can account for the observed change in local moment, we follow the analysis on LaCoO$_3$ [@VankoPRB2006] and analyze the data in terms of thermally excited localized three spin-states model [@Saitoh1997]. The effective moment can then be expressed as:
$$\overline{\mu}(T)=A\sum_{i=0}^{2} g\mu_B\sqrt{S_i(S_i+1)}\nu_ie^{-\Delta_i/k_BT}/Z$$
where $i$ indexes the states with spin $S_i$ and energy $\Delta_i$ relative to the non-magnetic spin state ($S_0$=0), $\nu_i$ is the degeneracy factor and $g$=2, while $Z=\sum_{i=0}^{2}\nu_ie^{-\Delta_i/k_BT}$. We assume that the partially itinerant nature of the iron pnictides reduces the local moment size by an overall scaling factor $A$ [@emission2011]. We include both intermediate ($S_1$=1) and high spin-state ($S_2$=2) of the Fe$^{2+}$ ($d^6$) ion. In the ionic picture the FeAs$_4$ tetrahedral crystal field splits the 3$d$ orbitals into lower $e$ and upper $t_{2}$; a small tetragonal crystal field further splits the $t_{2}$ states into upper $d_{xz}/d_{yz}$ and lower $d_{xy}$ [@Kruger2009]. This gives us an estimate for the orbital degeneracy and results in an overall degeneracy (including spin) of $\nu_0$=1, $\nu_1$=6 and $\nu_2=10$ (see inset of Fig. \[fig02\] (b)). This spin-state crossover model describes the temperature dependence of the local moments above 70 K very well, as shown as the solid line in Fig. \[fig02\] (b), which was obtained using $\Delta_1 = 8$ meV, $\Delta_2 = 45$ meV and an overall scaling factor of $A = 0.3$. The model also reveals that the continuous drop in local moment comes from high-spin Fe$^{2+}$ ions going into a low-spin state, i.e. the ratio of $S_0$:$S_1$:$S_2$ spin-states goes from $\sim$0.15:0.60:0.25 at 300 K to $\sim$0.40:0.60:0 at 70 K. We also tried to fit the data using just two spin states (S=0 and S=2). The fit result plotted in Fig. 2(b) quickly reaches a plateau, underestimating the local moment at higher temperatures. We note that $\Delta_i$ is a phenomenological energy splitting between spin-states, and does not correspond to real energy scales such as $\Delta_{{\rm CF}}$ or ${\emph J_H}$. The behavior of the La-doped sample at lower temperatures (T$<$70K) deviates from the model calculation, which could originate from our assumption that $\Delta_i$ and $\nu_i$ are constants, when in principle they can vary with temperature.
Our analysis suggests that the Fe$^{2+}$ ions consists of a mixture of different spin-states; this provides an important clue as to the energy scale of $\Delta_{{\rm CF}}$ and ${\emph J_H}$. That is, the energy scale of $\Delta_{{\rm CF}}$ and ${\emph J_H}$ in Ca$_{1-x}RE_x$Fe$_2$As$_2$ must be comparable in order for the spin-state crossover to occur. Since a high-spin Fe$^{2+}$ ion has a larger ionic radius than its low-spin counterpart [@Shannon1976] the lattice is forced to expand as the local moment increases. This could help explain the large linear thermal expansion coefficient observed in all our samples [@Radaelli2002]. We emphasize that Ca$_{1-x}$RE$_x$Fe$_2$As$_2$ might be a rare example where a spin-state crossover can be observed. We speculate that the small ionic radius of Ca, compared to Sr or Ba, exerts negative chemical pressure in Ca$_{1-x}RE_x$Fe$_2$As$_2$ and splits the spin states. In (Ba,Sr)Fe$_2$As$_2$ these spin states might be degenerate, resulting in a temperature independent local moment. In a similar experiment on SrFe$_2$As$_2$, indeed no sign of spin-state crossover was observed [@Bondino2012]. The idea of degenerate Fe$^{2+}$ spin-states has recently been proposed in order to understand the magnetic behavior of the iron pnictides [@Khaliullin2012].
The sudden collapse of the Fe$^{2+}$ moments below T = 70 K for both the Pr- and Nd-doped samples is too sharp to be explained by the crossover model in Eq. (1). Large structural changes can often be associated with changes in the electronic structure and thus affect the magnetism directly [@Chen2011]. To investigate whether that is the case in Fig. \[fig03\] we show the XANES spectra obtained above and below the cT-transition temperature. The absorption spectra consist of a prominent pre-edge peak A and three higher energy features B, C, and D. Similar features have been observed in BaFe$_2$As$_2$ [@Bittar2011] and were found to be independent of doping. Now both Nd- and Pr-doped samples show a difference going from T = 75 K to T = 45 K. In particular C shifts towards higher energy and B becomes more pronounced, while A stays more or less unchanged. The La-doped sample showed no noticable difference going from T = 100 K to T =45 K. In order to see whether the T- to cT-phase transition could explain this difference we simulated XANES spectra using the FDMNES code [@FDMNES] with a radius of 8${\rm \AA}$. The reported tetragonal crystal structure of Ca$_{0.91}$Nd$_{0.09}$Fe$_2$As$_2$ at T = 105 K and T = 80K [@Saha2012] was used for T- and cT-phase, respectively. The simulated spectra capture fairly well the changes we observed for the Nd- and Pr-doped samples.
![(Color online) Fe K-edge x-ray absorption near edge spectra taken in the partial fluorescence yield mode by monitoring the Fe K$\beta$ emission line. Spectra for Nd- and Pr-doped samples were taken at T = 75 K and 45 K, and at T = 100 K and 45 K for La-doped. The simulated spectrum was calculated for a Nd-doped crystal in the T and cT phase. It has been shifted in energy to match the pre-edge peak A for the experimental scans. Spectra were offset for clarity.[]{data-label="fig03"}](fig03){width="\columnwidth"}
Theoretical calculations show that the T- to cT-phase transition is accompanied by the formation of interlayer As-As dimers [@Yildirim2009] which are believed to form below the critical interlayer As-As distance of 3${\rm \AA}$ and cause the lattice to suddenly collapse [@Saha2012]. This collapse changes both the As-Fe-As angle and the Fe-As bond length, distorting the FeAs$_4$ tetrahedra. In previous XANES studies of Fe-pnictides, features similar to C and D have been assigned to hybridized Fe/As 4p states [@Bittar2011; @Ignace2010]. These states are also known to vary strongly due to local bonding and symmetry effects [@Ignace2010]. The loss of local magnetic moment can thus be understood as coming from the distortion of the FeAs$_4$ tetrahedra, which changes the electronic structure and forces the Fe$^{2+}$ ion to the low spin-state. In a recent NMR experiment (much slower probe) [@Long2012] evidence for a large suppression of the local Fe moment in the cT-phase have been reported, supporting our findings.
In summary, we have studied the spin-state of the Fe$^{2+}$ ion in Ca$_{1-x}$RE$_x$Fe$_2$As$_2$ (RE=La,Pr, and Nd) as a function of temperature. The continuous decrease of the local moment in the T-phase of all the samples is explained through spin-state crossover which cause the local moment to decrease by a factor of two. The ionic radius of the Fe$^{2+}$ ion decreases with decreasing moment size, allowing the $c$-axis to contract, which can explain the large linear thermal expansion coefficient in these compounds. In the Nd- and Pr-doped samples this reduction causes the interlayer As-As distance to cross the critical value of 3${\rm \AA}$, resulting in the formation of As-As dimers and transition into the collapsed tetragonal phase, which is accompanied by a total loss of local moment as predicted by Yildirim [@Yildirim2009].
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abstract: 'Quantum dot photonic crystal membrane lasers were fabricated and the large signal modulation characteristics were studied. We find that the modulation characteristics of quantum dot lasers can be significantly improved using cavities with large spontaneous emission coupling factor. Our experiments show, and simulations confirm, that the modulation rate is limited by the rate of carrier capture into the dots to around 30GHz in our present system.'
author:
- Bryan Ellis
- Ilya Fushman
- Dirk Englund
- Bingyang Zhang
- Yoshihisa Yamamoto
- Jelena Vučković
title: Dynamics of Quantum Dot Photonic Crystal Lasers
---
Recent advances in microfabrication technology have enabled researchers to control the electromagnetic environment and, consequently, the spontaneous emission rate of an emitter by coupling it to a semiconductor microcavity [@ref:Gerard_micropillar; @ref:Englund_enhancement]. In a microcavity where the spontaneous emission rate is significantly enhanced, a large fraction of spontaneously emitted photons can be coupled into a single optical mode. This unique property has been used to demonstrate microcavity semiconductor lasers with thresholds dramatically reduced relative to conventional lasers [@ref:Loncar_PClaser]. Thresholds can be further reduced by using quantum dots as the gain medium because of reduced active area and nonradiative recombination. Recently, quantum dot lasers exhibiting high spontaneous emission coupling factors and very few quantum dots as a gain medium were demonstrated [@ref:Strauf_PClaser]. In addition, microcavity lasers can be designed to have very broad modulation bandwidth because the relaxation oscillation can be shifted beyond the cavity cutoff frequency [@ref:Yamamoto_microcavity]. A recent experiment demonstrated direct modulation rates far exceeding 100GHz in quantum well photonic crystal (PC) lasers [@ref:Altug_modulation]. Lasers for applications such as high-speed optical telecommunications or on-chip optical interconnects could benefit from a combination of reduced thresholds and increased modulation bandwidth. In this letter we examine the factors limiting the modulation rate of quantum dot photonic crystal lasers and demonstrate large-signal direct modulation rates approaching 30GHz in quantum dot structures.
In a quantum dot laser the maximum modulation bandwidth is limited by either the frequency of the relaxation oscillations or the rate of carrier capture into the quantum dots depending on which is smaller. In conventional quantum dot lasers at low pump powers, the relaxation oscillation frequency is significantly smaller than the rate of carrier capture into the dots. This frequency increases with input power, so the modulation bandwidth can be enhanced by increasing pump power. This technique was used to demonstrate small-signal modulation rates of several tens of GHz [@ref:Fathpour_tunnel], but relatively large pump powers were necessary making these lasers impractical for low power applications. In addition the large signal modulation rates are considerably slower because the turn-on delay times are on the order of a nanosecond [@ref:Grundmann_turnon; @ref:Bhattacharya_turnon]. This could be resolved by using a microcavity laser with enhanced spontaneous emission coupling factor, which increases the relaxation oscillation frequency (as demonstrated for quantum well lasers [@ref:Altug_modulation]). In that case, the maximum modulation rate would be limited by the rate of carrier capture into the dots at practically achievable pump powers. To demonstrate the utility of this approach we fabricated quantum dot photonic crystal lasers and investigated their large-signal modulation characteristics. We find that the maximum modulation rate is limited by the rate of carrier capture into the dots, as predicted.
The laser cavities employed in our experiment are high-Q linear 3-hole defect PC cavities in a GaAs membrane (Fig. 1a). Finite-Difference Time Domain simulations were used to design the cavities to have a high Q resonance near the center of the quantum dot gain spectrum (Fig. 1b). The membrane is approximately 135nm thick and contains one layer of high density (600/$\mu m^{2}$) InAs quantum dots. Electron beam lithography is used to define the photonic crystal pattern in PMMA. A Chlorine-based reactive ion etch is used to create the holes. Finally an Al$_{0.9}$Ga$_{0.1}$As sacrificial layer is first oxidized, then removed in a KOH solution to release the membrane. After fabrication the structures are placed inside a He-flow cryostat and cooled to 5K (necessary for operation of the InAs/GaAs quantum dots with shallow quantum confinement). The lasers are optically pumped using a Ti-Sapphire laser, and the emission is detected using a liquid nitrogen cooled spectrometer or streak camera.
To determine the cavity photon lifetime $\tau_p$, the quality factor of the cavities was measured well below threshold using continuous wave pumping. Fits to a Lorentzian lineshape indicate that the cold-cavity quality factors are around 3000, corresponding to a cavity photon lifetime of about 1.5ps. To investigate the dynamics of the structures, we pumped them with 3ps pulses at an 80MHz repetition rate using the Ti-Sapphire laser. Streak camera measurements of the rise time of photoluminescence from quantum dots in bulk GaAs indicate that the carrier capture time is around 10ps for a wide range of pump powers. Because the carrier capture time is longer than the cavity photon lifetime, it will ultimately determine the maximum modulation bandwidth of the lasers. Figure 1c shows an L-L curve taken under pulsed pumping conditions and a fit to the rate equations for one of the PC lasers. The L-L curve exhibits a threshold of around 1$\mu W$ confirming that the structures are lasing. In the best structures (where the cavity mode is near the center of the gain spectrum) threshold values are around 250nW average power, while in other structures with more absorption and less gain, threshold values were measured at several $\mu W$ average power. From fits to the light in-light out curve we estimate that in our structures the spontaneous emission coupling factor $\beta$ is around 0.2. To confirm that the spontaneous emission rate in our cavities is significantly enhanced we used a streak camera to compare the decay time of photoluminescence from quantum dots in bulk GaAs and cavity coupled dots in nonlasing devices. The measurements show that the dot lifetime is significantly reduced from the bulk value of 2.5ns to 300ps when coupled to the PC cavity.
To investigate the large-signal modulation response, emission from the lasers above threshold was collected by the streak camera. One of the main advantages of cavity-QED enhanced lasers is the decreased rise time because spontaneous emission rapidly builds up the photon number in the laser mode [@ref:Altug_modulation]. Experimentally we find that as pump power is increased the rise time is reduced to 13.5ps when the laser is pumped at about 5 times threshold (Figure 2b). Experiments performed at 10 and 15 times threshold indicate that the rise time is pinned at about 12ps even at very high pump powers. From simulations we conclude that the rise time of quantum dot lasers is limited by the carrier capture time. However, in high-$\beta$ lasers this limit is practically achievable because it is approached at lower pump powers relative to threshold (as opposed to quantum dot lasers not employing stong cavity effects where higher power pumping is needed).
Above threshold, higher pump powers lead to faster decay times due to increased stimulated emission rates. Small-mode volume PC cavities can be used to achieve large photon densities and speed up this process. Figure 3a shows the laser response at various pump powers, demonstrating the reduction in decay time with increasing pump power. We observed a minimum decay time of 8.5ps at pump powers around 5 times threshold (Figure 2a). For higher pump powers the laser response appears largely unchanged. We attribute this to large carrier densities causing the gain to saturate preventing further decrease of the decay time, but more work is necessary to characterize saturation effects in our quantum dots. Based on the measured laser response and the results of simulations, we predict our laser can be modulated at large-signal modulation rates approaching 30GHz.
To accurately model the photonic crystal laser modulation characteristics, the usual rate equation model must be adapted to include the finite relaxation time from the wetting layer into the quantum dots. We employed a three-level rate equation model adapted from [@ref:Coldren_Corzine; @ref:OBrien_rateeq] for the photon density P, the quantum dot ground state carrier density N$_g$ and the wetting layer carrier density N$_w$: $$\frac{dN_w}{dt}=R_p-\frac{N_w}{\tau_w}-\frac{N_w}{\tau_c}$$ $$\frac{dN_g}{dt}=\frac{N_w}{\tau_c}-\frac{N_g}{\tau_{sp}}-\frac{N_g}{\tau_{nr}}-GP$$ $$\frac{dP}{dt}=\Gamma G P+\Gamma \beta \frac{N_g}{\tau_{sp}}-\frac{P}{\tau_p}$$ Here $\tau_{sp}$ and $\tau_{nr}$ are the spontaneous emission lifetime and nonradiative lifetime of the dots, $\tau_w$ is the lifetime of carriers in the wetting layer including spontaneous and nonradiative recombination, $\tau_p$ the photon lifetime, $\tau_c$ the carrier capture time into the dots, $\Gamma$ the confinement factor, and $R_p$ the pump rate. Streak camera measurements of the wetting layer response indicate that for our samples $1/\tau_w \approx 1/100ps$. From estimates of the overlap of the mode volume with the gain medium we estimate that $\Gamma$=0.028 in our lasers. We have assumed a linear gain model $$G=G_o*(N_g-N_{tr})$$ where $G_o$ is the linear gain coefficient and $N_{tr}$ is the transparency carrier density. From fits to the L-L curve we estimate that in our system $G_o = 8.13*10^{-6} s^{-1}$ and $N_{tr}=3.22*10^{17} cm^{-3}$. The linear gain model was chosen to give good quantitative agreement when the lasers are operated around threshold, but the model will overestimate the gain well above threshold. A more sophisticated gain model is necessary to model the modulation response of the lasers when gain saturation effects become significant. Figure 4b shows the simulated laser response at various pump powers demonstrating good agreement between theory and experiment.
For practical applications, room temperature operation of the quantum dot microcavity lasers will be necessary. Quantum dot lasers have been shown to have a strongly temperature-dependent modulation response. Recent work on tunnel injection quantum dots has demonstrated that fast relaxation rates ($\sim 1.7ps$) at room temperature are achievable. These dots have been used to demonstrate 25GHz small signal modulation bandwidth [@ref:Fathpour_tunnel]. We believe this bandwidth can be significantly improved using a PC laser cavity.
In summary, we have investigated the large-signal modulation characteristics of quantum dot PC lasers. We demonstrated that cavity-QED effects can be used to combine ultra-low threshold operation and improved bandwidth. Because of their low-power consumption and high-speed operation, quantum dot microcavity structures have the potential to significantly improve optical interconnect and photonic integrated circuit technology.
Financial support for this work was provided by the MARCO Interconnect Focus Center, NSF grants No. ECS-0424080 and No. ECS-0421483, the Stanford Graduate Fellowship, and the NDSEG fellowship. Financial assistance for B.Y. Zhang was provided in part by JST/SORST.
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abstract: |
We present constraints on the cosmological and reionization parameters based on the cumulative mass function of the Ly-$\alpha$ systems. We evaluate the formation rate of bound objects and their cumulative mass function for a class of flat cosmological models with cold dark matter plus cosmological constant or dark energy with constant equation of state encompassing different reionization scenarios and compare it with the cumulative mass function obtained from the Ly-$\alpha$ transmitted flux power spectrum.\
We find that the analysis of the cumulative mass function of the Ly-$\alpha$ systems indicates a reionization redshift $z_r=24.2\pm4$ (68%CL) in agreement with the value found on the basis of the WMAP anisotropy measurements, setting constraints on the amplitude of the density contrast, $\sigma_8=0.91\pm0.04$ (68%CL), similar to those derived from the X-ray cluster temperature function.\
Our joint analysis of Ly-$\alpha$ cumulative mass function and WMAP anisotropy measurements shows that the possible current identification of a running of the slope, $dn_s/d{\rm ln}k\ne 0$, at $k_p$=0.05Mpc$^{-1}$ (multipole ${\it l} \approx 700$) is mainly an effect of the existing degeneracy in the amplitude-slope plane at this scale, the result being consistent with the absence of running, the other constraints based on WMAP data remaining substantially unchanged.
Finally, we evaluate the progress on the determination the considered parameters achievable by using the final temperature anisotropy data from WMAP and from the forthcoming [Planck]{} satellite that will significantly improve the sensitivity and reliability of these results.
This work has been done in the framework of the [Planck]{} LFI activities.
address: |
IASF/CNR, Istituto di Astrofisica Spaziale e Fisica Cosmica, Sezione di Bologna,\
Consiglio Nazionale delle Ricerche, Via Gobetti 101, I-40129 Bologna, Italy\
$^{\rm b}$ Institute of Space Sciences, Bucharest-Magurele, R-76900, Romania
author:
- 'L.A. Popa $^{\rm a,b}$, C. Burigana $^{\rm a}$, and N. Mandolesi'
title: 'Non-linear evolution of the cosmological background density field as diagnostic of the cosmological reionization'
---
0.5truecm
[^1]
Cosmology: cosmic microwave background – large scale structure – dark matter
Introduction
============
The cosmological density background is assumed to have been seeded at some early epoch in the evolution of the Universe, inflation being the most popular of the current theories for the origin of the cosmological structures (see e.g. Kolb & Turner 1990, Linde 1990).\
Measurements of the Cosmic Microwave Background (CMB) anisotropies present at the epoch of the recombination at large scales ($R {\,\lower2truept\hbox{${>\atop\hbox{\raise4truept\hbox{$\sim$}}}$}\,}8h^{-1}$Mpc), always outside the horizon during the radiation-dominated era, carry precious information on the power spectrum of density field. Moving to intermediate and small scales the background density field encodes information related to the non-linear evolution of the gravitational clustering, the time evolution of the galaxy bias relative to the underlying mass distribution (see e.g. Hoekstra et al. 2002, Verde et al. 2002), and the magnitude of the peculiar motions and bulk flows in the redshift space (Kaiser 1987). At these scales the spectrum of the background density field can be constrained by complementing CMB measurements with other astronomical data (see Kashlinsky (1998) for the evaluation of the spectrum of the density field from a variety of astronomical data on scales 1$h^{-1}$Mpc $\le$ R $\le$ 100$h^{-1}$Mpc ).
In particular, on scales less then 5 $h^{-1}$Mpc the spectrum of the density field can be constrained by using observations of the spatial distribution of high-redshift collapsed objects as quasars and galaxies and information on their macroscopic properties (Efstathiou & Ress 1988, Kashlinsky & Jones 1991, Kashlinsky 1993). The detection of high-z quasars by the Sloan Digital Sky Survey (Becker et al. 2001, White et al. 2003, Fan et al. 2003) and by the Keck telescope (Vogt et al. 1994, Songaila & Cowie 2002) as well as the detection of high-redshift galaxies (see Kashlinsky (1998) and the references therein) are indications about the existence of such early collapsed objects at redshifts between 2 and 6, on mass scales of $\sim$ 10$^{10}$M$_{\odot}$.\
The [*rms*]{} mass fluctuations over a sphere of radius of 8$h^{-1}$Mpc, $\sigma_8$, fixes the amplitude of the density field power spectrum and then determines the redshift of collapse, $z_c$, of an object with a given mass-scale. Consequently, the observation of such objects can set significant constraints on different cosmological theories.\
Recently, the WMAP [^2] team (Spergel et al. 2003, Verde et al. 2003) highlighted the relevance of complementing the CMB and LSS measurements with the Ly-$\alpha$ forest data (the absorptions observed in quasar spectra by the neutral hydrogen in the intergalactic medium) in constraining the cosmological parameters (Croft et al. 1998; Zaldarriaga et al. 2003 and references therein) and the shape and amplitude of the primordial density field at small scales.
In this paper we investigate the dynamical effects of the non-linear evolution of the density field implied by the high-z collapsed objects as diagnostic of the reionization history of the Universe. The reionization is assumed to be caused by the ionizing photons produced in star-forming galaxies and quasars when the cosmological gas falls into the potential wells caused by the cold dark matter halos. In this picture the reionization history of the Universe is a complex process that depends on the evolution of the background density field and of the gas properties in the intergalactic medium (IGM) and on their feedback relation. The evolution of the background density field, that determines the formation rate of the bound objects, is a function of the grow rate of the density perturbations and depends on the assumed underlying cosmological model. The evolution of the gas in the IGM is a complex function of the gas density distribution and of the gas density-temperature relation. The latter is related to the spectrum (amplitude and shape) of the ionizing radiation, to the reionization history parametrized by some reionization parameters (the reionization redshift, $z_r$, and the reionization temperature, $T_r$), and to the assumed cosmological parameters.\
As shown by hydrodynamical simulations (Cen et al. 1994; Zhang et al. 1995; Hernquist et al. 1996; Theuns et al. 1998), the gas in the IGM is highly inhomogeneous, leading to the non-linear collapse of the structures. In this process the gas is heated to its virial temperature. The photoionization heating and the expansion cooling cause the gas density and temperature to be tightly related. Finally, the temperature-mass relation for the gas in the IGM at the time of virilization determines the connection between the gas density and the matter density at the corresponding scales.
Taking advantage of the results of a number of hydrodynamical simulations (Cen et al. 1994; Miralda-Escudé et al. 2000; Chiu, Fan & Ostriker 2003) and semi-analytical models (Gnedin & Hui 1998; Miralda-Escudé, Haehnelt & Rees 2000, Chiu & Ostriker 2000), we re-assess in this paper the possibility to use the mass function of Ly-$\alpha$ systems to place constraints on spatially flat cosmological models with cold dark matter plus cosmological constant or dark energy, encompassing different reionization scenarios. We use the Press-Schechter theory (Press & Schechter 1974) to compute the comoving number density of Ly-$\alpha$ systems per unit redshift interval from the Ly-$\alpha$ transmitted flux power spectrum (Croft et al. 2002) and examine the constraints on the cosmological parameters. In our analysis we take into account the connection between the non-linear dynamics of the gravitational collapse and the properties of the gas in the IGM through the virial temperature-mass relation. We address the question of the consistency of the WMAP and Ly-$\alpha$ constraints on the cosmological parameters, paying a particular attention to the degeneracy between the running of the effective spectral index and the power spectrum amplitude. Finally, we discuss the improvements achievable with the final WMAP temperature anisotropy data and, in particular, the impact of the next CMB temperature anisotropy measurements on-board the ESA [Planck]{} [^3] satellite.
Early object formation mass function
====================================
Formation rates
---------------
The most accurate way used to assess the formation rates of the high-z collapsed objects is based on numerical simulations. A valid alternative is offered by the Press-Schechter theory (Press & Schechter 1974, Bond et al. 1991) extensively tested by numerical simulations for both open and flat cosmologies (Lacey & Cole 1994, Eke, Cole & Frenk 1996, Viana & Liddle 1996).
According to the Press-Schechter theory, the fraction of the mass residing in gravitationally bounded objects is given by: $$f_{{\rm coll}}(z) \approx \sqrt{\frac{2}{\pi}} \frac{1} {\sigma(R_f,z)}
\int_{\delta_c(z)}^{\infty} \exp{\left[-\frac{\delta_c^2(z)}
{2 \sigma(R_f,z)}\right]}
d\,\delta_c(z) \,.$$ Here $\delta_c(z)$ is the redshift dependent density threshold required for the collapse; $R_f$ is the filtering scale associated with the mass scale $M=4 \pi R_f^3 \rho_b/3$, $\rho_b$ being the comoving background density; $\sigma(R_f,z)$ is the [*rms*]{} mass fluctuation within the radius $R_f$: $$\begin{aligned}
\sigma^2(R_f,z)=\int_0^{\infty}\frac{d \,k}{k}\Delta^2(k,z)W^2(kR_f)\, ,\end{aligned}$$ where W(x) is the window function chosen to filter the density field. For a top-hat filtering $W(x)=3({\rm sin}x-x{\rm cos}x)/x^3$ while for a Gaussian filtering $W(x)=\exp(-x^2)$. As we do not find significant differences between the results derived by adopting the two considered window functions for some representative cases, we present in this work the results obtained by using the Gaussian smoothing. In the above equation $\Delta^2(k,z)$ is referred as the power variance and is related to the matter power spectrum $P(k,z)$ through: $$\Delta^2(k,z)=\frac{1}{2 \pi^2}k^3P(k,z)\,.$$ Motivated in the framework of the spherical collapse model and calibrated by N-body numerical simulations, the linear density threshold of the collapse $\delta_c(z)$ was found to vary at most by $\simeq 5$ % with the background cosmology (see e.g. Lilje 1992, Lacey & Cole 1993, Eke, Cole & Frenk 1996). However, the choice of $\delta_c(z)$ depends on the type of collapse. For the spherical collapse the standard choice is $\delta_c(0)=1.7 \pm 0.1$ while for pancake formation or filament formation its value is significantly smaller (Monaco 1995). For the purpose of this work we assume that the collapse have occurred spherically and use $\delta_c(0)=1.686$ that is the conventional choice for the case of the flat cosmological models (Eke, Cole & Frenk 1996). The time evolution of $\delta_c$ depends on the background cosmology: $$\delta_c(z)=\delta_c(0)\frac{D(z)}{D(0)} \, ,$$ where $D(z)$ is the linear growth function of the density perturbation, given in general form by (Heath 1977, Carroll, Press & Turner 1992, Hamilton 2001): $$D(a)=\frac{5\Omega_m}{2 a f(a)} \int_0^a f^3(a) d \,a \, .$$ Here $a=(1+z)^{-1}$ is the cosmological scale factor normalized to unity at the present time ($a_0=1$), $\Omega_m$ is the matter density energy parameter at the present time and $f(a)$ specifies the time evolution of the scale factor for a given cosmological model: $$\begin{aligned}
\frac{d\,a}{d \,t}=\frac{H_0}{f(a)}, \hspace{0.4cm}
f(a)=\left[ 1+\Omega_m\left(\frac{1}{a}-1\right) +
\Omega_{de}\left( \frac{1}{a^{1+3w}}-1\right) \right]^{-1/2}.\end{aligned}$$ In the above equation $H_0$ is the present value of the Hubble paramenter, $\Omega_{de}$ is the present value of the energy density parameter of the dark energy, $w=p/\rho_{de} \sim -1$ defines the dark energy equation of state, and $\Omega_m$ is the matter energy density parameter. Equation (6) reduces to that of a $\Lambda$CDM model for $w=-1$.
The comoving number density of the gravitationally collapsed objects within the mass interval $ d\,M$ about $M$ at a redshift $z$ is given by (Viana & Liddle 1996): $$n(M,z)d\,M=-\sqrt{\frac{2}{\pi}}\frac{\rho_b}{M}\frac{\delta_c}{\sigma^2(R_f,z)}
\frac{d\,\sigma(R_f,z)}{d\,M}
\exp{\left[-\frac{\delta_c^2}{2 \sigma^2(R_f,z)}\right]}d\,M.$$ We are interested in the formation rate of the high-z collapsed objects at a given redshift. According to Sasaki method (Sasaki 1994), the comoving number density of bounded objects with the mass in the range $dM$ about M, which virilized in the redshift interval $dz$ about $z$ and survived until the redshift $z_f$ without merging with other systems is given by: $${\rm N }(M,z)d\,M d\,z=\left[
-\frac{\delta_c^2}{\sigma^2(R_f,z)}
\frac{n(M,z)} {\sigma(R_f,z)}
\frac{d\, \sigma(R_f,z)}{d\,z}\right]
\frac{\sigma(R_f,z)}{\sigma(R_f,z_f)}
d\,M \,d\,z\, ,$$ where: $$\sigma(R_f,z)=\sigma(R_f,0)\frac{D(z)}{D(0)}\frac{1}{1+z}.$$ The total comoving number density of the bounded objects per unit redshift interval with the mass exceeding $M$ (the cumulative mass function) is given by: $${\bf N}(>M)=\int^{\infty}_M {\rm N}(M,z)dM.$$
Mass-temperature relation for the virilized gas in the IGM
----------------------------------------------------------
The fraction of the mass of the gas in collapsed virilized halos can be calculated if the probability distribution function (PDF) for the gas overdensity is known: $$\begin{aligned}
f_{{\rm coll}}(z)=\int_{\Delta_c(z)}^\infty {\bf \Delta} P_V({\bf \Delta})
d\,{\bf \Delta}\,.\end{aligned}$$ Here $P_V({\bf \Delta})$ is the volume-weighted PDF for the gas overdensity ${\bf \Delta}=\rho_g/\rho_{bar}$, where $\rho_g$ is the gas density and $\rho_{bar}$ is the mean density of baryons. In the above equation $\Delta_c(z)$ is the halo density contrast at virilization. Based on hydrodynamical simulations, Miralda-Escudé et al. (2000) found for the volume-weighted probability distribution, $P_V({\bf \Delta})$, the following fitting formula: $$P_V({\bf \Delta)}d\, {\bf \Delta}=A \exp{\left[-\frac{({\bf \Delta}^{-2/3}-C_0)^2}
{2(2\delta_0/3)^2}\right]}{\bf \Delta}^{-\beta} d \, {\bf \Delta}.$$ In this equation $\delta_0$ is the linear [*rms*]{} gas density fluctuation and $\beta$ is a parameter that describe the gas density profile ( $\rho_g \sim r^{-\beta}$ for an isotermal gas). As shown by the numerical simulations (see Table 1 from Chiu, Fan & Ostriker 2003) the redshift evolution of $\delta_0$ depends on the underlying cosmological model. Assuming the same fraction of baryons and dark matter in collapsed objects, we compute the redshift dependence of $\delta_0$ on the cosmological parameters by using an iterative procedure (Chiu, Fan & Ostriker 2003) requiring the equality of the equations (1) and (10) representing the collapsed mass fraction. The parameters $A$ and $C_0$ were obtained by requiring the normalization to unity of the total volume and mass. For the redshift dependence of $\beta$ in the considered redshift range we take the values $\beta \approx {\rm min}[2.5, \, 3.2-4.73/(1+z)]$ obtained by Chiu, Fan & Ostriker (2003) through a fit to their hydrodinamical simulations.
As we are interested to apply equation (9) to the virilized gas, we need to know the temperature-density and the mass-temperature relations for the virilized gas in the IGM.\
The temperature-density relation is determined by the reionization scenario and the underlying cosmological model. Hydrodynamical simulations can predict this relation accurately, but the limited computer resources restrict the number of cosmological models and reionization histories that can be studied. For this reason we evaluate the temperature-density relation at the redshifts of interest by using the semi-analytical model developed by Hui & Gnedin (1997) that permits to study the reionization models by varying the amplitude, spectrum, the epoch of reionization and the underlying cosmological model. According to this model, for the case of uniform reionization models, the mean temperature-density relation is well approximated by a power-law equation of state that can be written as: $$T=T_0(1+{\bf \Delta})^{\gamma-1},$$ where $T_0$ and $\gamma$ are analytically computed as functions of the reionization temperature $T_r$, the reionization redshift $z_r$, the matter and baryon energy density parameters $\Omega_m$ and $\Omega_{bar}$, and the Hubble parameter $H_0$.
According to the virial theorem (Lahav et al. 1991, Lilje et al. 1992), the virial mass-temperature relation at any redshift $z$ can be written as (Eke, Cole & Frenk 1996; Viana & Liddle 1996; Kitayama & Suto 1997; Wang & Steinhardt 1988): $$\frac{M_{vir}}{10^{15} h^{-1} M_{\odot}}=\left(\frac{k_BT_{vir}/{\it f}_{\beta}}{0.944{\rm keV}}\right)^{3/2}
[(1+z)^3 \Omega_0 \Delta_c]^{-1/2}
\left[1-\frac{2 \Omega_{de}(z)}{\Delta_c \Omega_m(z)}\right]^{-3/2} ,$$ where $k_B$ is the Boltzmann constant, $T_{vir}$ is the temperature of the virilized gas, $\Delta_c$ is the density contrast at virilization and ${\it f}_{\beta}= {\it f}_u \mu/\beta$, where ${\it f}_u$ is the fudge factor (of order of unity) that allows for deviations from the simplistic spherical model and $\mu$ is the proton molecular weight. Different analyses adopting similar mass-temperature relations disagree on the value of $f_{\beta}$ because of the uncertainties in the numerical simulations. We adopt here $f_{\beta}=1$, as indicated by the most extensive simulation results obtained by Eke, Cole & Frenk (1996).
Throughout this paper we consider that the virilization takes place at the collapse time, $t(z_c)$, that is half of the turn-around time: $t(z_c)=t(z_{ta})/2$, $z_{ta}$ being the redshift at which ${\dot R}(z_{ta})=0$ ($R$ is the radius of a spherical overdensity).\
For $\Lambda$CDM models the density contrast at virilization is a function of $\Omega_m$ only. For quintessence models the density contrast at virilization becomes a function of $\Omega_m$ and $w$ and can be written as (Wang & Steinhardt 1988): $$\Delta_c(z=z_c)=\frac{\rho_{clust}(z_c)}{\rho_b(z_c)}=
\zeta\left(\frac{R_{ta}}{R_{vir}}\right)^3
\left(\frac{1+z_{ta}}{1+z_c}\right)^3,$$ where: $\zeta(z_{z_{ta}})=
\rho_{clust}(z_{ta})/\rho_b(z_{ta})$; $R_{ta}$ and $R_{vir}$ are the radius at $z_{ta}$ and $z_c$ respectively: $$\zeta(z_{ta})=(3 \pi/4)^2\Omega_m(z_{ta})
^{-0.79+0.26 \Omega_m(z_{ta})-0.06w},$$ $$\frac{R_{vir}}{R_{ta}}=\frac{1-\eta_v/2}{2+\eta_t-3\eta_v/2},$$ where $\eta_t=2 \zeta^{-1}\Omega_{de}(z_{ta})/\Omega_m(z_{ta})$ and $\eta_v=2 \zeta^{-1}[(1+z_c)/(1+z_{ta})]^3
\Omega_{de}(z_c)/\Omega_m(z_c)$.\
The energy density parameters and the Hubble parameter evolve with the scale factor according to: $$\begin{aligned}
\Omega_m(a)=\frac{\Omega_0f^2(a)}{a}, \hspace{0.3cm}
\Omega_{de}(a)=\frac{\Omega_{de}f^2(a)}{a^{1+3w}},
\hspace{0.3cm} H(a)=\frac{H_0}{a f(a)},\end{aligned}$$ where $f(a)$ is given by the equation (6).\
The scale factors for collapse, $a_c$, and turn-around, $a_{ta}$, was computed from the spherical collapse model (Lahav et al. 1991, Eke, Cole & Frenk 1996).\
For any region inside the radius $R$ that was overdense by $\Delta_i$ with respect to the background at some initial time $t_i$ corresponding to the redshift $z_i$, we solve the set of equations: $$\begin{aligned}
\int^{a_{ta}}_0 f(a)d\,a= \frac{H_0} {H_i} \int_0^{s_{ta}}g(s) d\,s,
\hspace{0.3cm}
\int^{a_{c}}_0 f(a)d\,a= 2\frac{H_0} {H_i} \int_0^{s_{ta}}g(s) d\,s,\end{aligned}$$ where: $$\begin{aligned}
\frac{ds}{dt}=\frac{H_i}{g(s)} \hspace{0.2cm} {\rm and} \hspace{0.2cm}
g(s)=\left[1+\Omega_i(1+\Delta_i)\left( \frac{1}{s}-1\right)+\Omega_{de,i}
(s^2-1)\right]^{-1/2}.\end{aligned}$$ In the above equations, solved by using an iterative procedure, $s=R/R_i$ is the scale factor of a spherical perturbation with the initial radius $R_i$ and $s_{ta}=R_{ta}/R_i$ is its scale factor at the turn-around (see Appendix A in Eke, Cole & Frenk 1996). The average density perturbation inside the radius $R$ is given by: $$\Delta(R,z)=\frac{3}{R^3}\int_0^R R^2 \delta(R,z) d\,R,$$ where $\delta(R,z)$ is related to the power spectrum of the density field $P(k,z)=|\delta_k(z)|^2$ through: $$\delta(R,z)=\frac{1}{(2 \pi)^3}\int\delta_k(z) e^{-ikR}d^3\,k.$$ As initial conditions for the spherical infall we choose the epoch given by $z_i=1100$ when the growth of perturbations is fully determined by the linear theory. The initial values of the parameters $H_i=H(z_i)$, $\Omega_i=\Omega(z_i)$ and $\Omega_{de,i}=\Omega_{de}(z_i)$ are given by the equation (17) and for the normalization of the density field at $z_i$, $\sigma_8(z_i)$, we take: $$\sigma_8(z_i)=\sigma_8(0)\frac{D(z_i)}{D(0)}\frac{1}{1+z_i}.$$ We adopt for $\sigma_8$ at the present time the value obtained from the analysis of the local X-ray temperature function for flat cosmological models with a mixture of cold dark matter and cosmological constant or dark energy with constant equation of state (Wang & Steinhardt 1998): $$\begin{aligned}
\sigma_8=(0.50-0.1\Theta)\Omega_m^{-\gamma(\Omega_m,\Theta)}\, ,\end{aligned}$$ where: $$\begin{aligned}
\gamma(\Omega_m,\Theta)=0.21 -0.22w+0.33\Omega_m+0.25\Theta \nonumber\end{aligned}$$ and $$\begin{aligned}
\Theta=(n_s-1)+(h-0.65). \nonumber\end{aligned}$$
Results
=======
Cosmological constraints from Ly-$\alpha$ observations
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The study Ly-$\alpha$ transmitted flux power spectrum has become increasingly important for cosmology as it is probing the absorptions produced by the low density gas in voids or mildly overdense regions. This gas represents an accurate tracer of the distribution of the dark matter at the early stages of the structure formation. One of the most important application is to recover the linear matter power spectrum $P_L(k)$ from the flux power spectrum $P_F(k)$ and inferring the cosmological parameters of the underlying cosmological model.\
On the observational side there are recent analyses by McDonald et al. (2000) and Croft et al. (2002) that obtain results for the transmitted flux power spectrum $P_F(k)$ in agreement with each other within the error bars. Two different methods have been proposed to constrain the cosmological parameters: McDonald et al. (2000) and Zaldarriaga et al. (2001) directly compare $P_F(k)$ with the predictions of the cosmological models, while Croft et al. (2002) and Gnedin & Hamilton (2002) use an analytical fitting function to recover the matter power spectrum, $P_L(k)$, from the flux power spectrum $P_F(k)$. The WMAP team (Verde et al. 2003) used the analytical fitting function obtained by Gnedin & Hamilton (2002) to convert $P_F(k)$ into $P_L(k)$.\
In a recent work, Seljak, McDonald & Makarov (2003) investigate the cosmological implications of the conversion between the measured flux power spectrum and the matter power spectrum, pointing out several issues that lead to the expansion of the errors on the inferred cosmological parameters.
We compute the total comoving number density, $N_{Ly-\alpha}$, of Ly-$\alpha$ systems per unit redshift interval that survived until ${z}=2.72$ without merging with other systems from the flux transmission power spectrum, $P_F(k)$, obtained by Croft at al. (2002) for their fiducial sample with mean absorption redshift ${\bar z}=2.72$. We then compare $N_{Ly-\alpha}$ with the theoretical predictions for the same function, $N_{th}$, obtained for a class of cosmological models encompassing the dark energy contribution with constant equation of state and different reionization scenarios.\
Our fiducial background cosmology is described by a flat cosmological model, $\Omega_0=\Omega_m +\Omega_{de}=1$, with the following parameters at the present time: $\Omega_{bar}h^2=0.024\pm 0.001$, $\Omega_mh^2=0.14\pm0.02$, $h=0.72\pm0.05$, as indicated by the best fit of the power law $\Lambda$CDM model of WMAP data (Spergel et al. 2003). In our analysis we allow to vary the primordial scalar spectral index $n_s$, the parameter $w$ describing the equation of state for the dark energy, the reionization redshift $z_r$, and the reionization temperature $T_r$. We assume adiabatic initial conditions and neglect the contribution of the tensorial modes.
{width="16cm"}
\[\]
Our parameter vector ${\bf p}=(n_s,w,z_r,T_r)$ has four dimensions. We create a grid of model predictions for the each choice of the parameters in the grid:
- $n_s=(0.7, 0.8, 0.85, 0.99, 1.1, 1.15, 1.2, 1.3)$
- $w=(-0.4, -0.5, -0.6, -0.65, -0.7, -0.75, -0.8, -0.85, -0.9, -0.95, -1)$
- $z_r=(5, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 35, 40)$
- $T_{r,4}=(1.6, 1.8, 1.9, 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.8, 3)$
Here $T_{r,4}$ is the reionization temperature in units of $10^4$K.\
The density perturbations $\delta_k(z_i)$ at the initial redshift $z_i=1100$ and the matter transfer function at $z=2.72$ was computed for each set of parameters in the grid by using the CMBFAST code version 4.2 (Seljak & Zaldarriaga 1996). Then we evaluate the linear matter power spectrum, $P_L(k)$, with the appropriate normalization.\
The Ly-$\alpha$ transmission power spectrum for the fiducial sample at ${\bar z}=2.72$ proves linear scales in the range $R = (11.4 - 1750)$ km/s (see Table 2 from Croft et al. 2002). For the purpose of this work we consider scales up to 791 km/s, which are non-linear today. We compute the averaged density perturbation $\Delta_i$ inside each scale $R$ at the initial redshift $z_i$ and the corresponding redshift of collapse, $z_c$, as described in the previous section. Then, assuming that the virilization takes place at the collapse time, we evaluate the temperature-density relation at $z_c$ and compute the appropriate virial mass at $z=2.72$ by using the mass-temperature relation. The virial mass obtained in this way is related to the filtering scale through $M_{vir}=(4 \pi / 3) R_f^3 \rho_b({\bar z})$ with $ \rho_b({\bar z})=(3H_0^2 / 8 \pi G) \Omega_0(1+{\bar z})^3$. We found that the filtering scale obtained in this way, $R_f \approx R_J/2$, is proportional to the Jeans scale, $R_J$, corresponding to the the Jeans mass $M_J= 1.5 \, {\rm T}_4 (1+z_c)^{-3/2}\Omega_m^{-1/2} 10^{10}{\rm h}^{-1}
{\rm M}_{\odot}\,$ ($T_4$ being the gas temperature in units of $10^4$ K), but the exact relation depends on the cosmological model and reionization parameters.\
Figure 1 presents few dependences of the gas properties on the background cosmology and reionization parameters. Panels a) and b) show the dependence of the gas temperature $T_0$ and of the parameter $\gamma$ on the redshift of collapse $z_c$. Panel c) and d) present the dependence of the virilized mass on the linear scale $R$ and on the mass scale $M(R)$, as an indication on the fraction of the virilized mass at the given scale. In panel e) we show the dependence of the clumping factor ${\it C}=<\rho^2_g>/< \rho_{bar}>^2$ on the virial mass. Panel f) presents the dependence of the filtering scale $R_f$ on the linear scale $R$.\
We note that the filtering scale obtained in this way depends on our parameter vector: $R_f=R_f(k,{\bf p})$, where $k$ is the wave number corresponding to the linear scale $R$.\
For each choice of the parameters in our simulation grid we compute $N_{Ly-\alpha}$ according to the equations (7) – (9) by filtering the transmission power spectrum $P_F(k)$ at each wave number $k$ with the corresponding filtering scale $R_f(k,{\bf p})$ and apply the same procedure to compute $N_{th}$ from the linear power spectrum $P_L(k)$. We compare $ N_{Ly-\alpha}$ and $N_{th}$ by computing a Gaussian approximation of the likelihood function: $${\it L}(N_{Ly-\alpha};{\bf p}) \propto \prod_{i=1}^{17}=\exp\left[-\frac{1}{2}
\left( \frac{N^i_{Ly-\alpha}-N^i_{th}}{\sigma_i}\right)^2\right] \, ,$$ where $i$ runs over different points \[we are using first 17 bins from the $P_F(k)$\] and $\sigma_i$ is the error bar associated to $N_{Ly-\alpha}$ on each point. The full chi-squared goodness of fit is $\chi^2 \approx -2 $ln$ {\it L}$ with 12 degrees of freedom (ndf). We define the confidence level (CL) as the upper tail probability of the chi-squared distribution and calculate $\chi^2({\rm CL},{\rm ndf})$ for a given CL by inverting the chi-squared distribution.\
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Figure 2 presents $N_{Ly-\alpha}$ and $N_{th}$ functions obtained for few choices of the parameters in the simulation grid. In each of the panels a), b), and c) we indicate in the top the values of the parameters in common to the two curves reported in each of the panels a), b), and c) are indicated in the top while the values of the parameters associated only to one of the two curves of each of the panels a), b), and c) are indicated in the bottom together with the corresponding confidence level obtained from our analysis. Panel d) shows an example of two quasi-degenerated models and their predictions for $N_{Ly-\alpha}$.
Each simulated $P_L(k)$ at $z=2.72$ was parametrized by its effective slope (Peacock & Dodds 1996), the running of the slope and the power variance at the pivot point $k^*_p$=0.03 s/km: $$n^*_{{\rm eff}}(k^*_p)=\frac{d {\rm ln}P_L(k)}{d{\rm ln}k} (k=k^*_p/2) \, ,$$ $$\frac{d n^*_{\rm eff}}{d {\rm ln} k}(k=k^*_p/2) \, ,$$ $$\Delta^2_*(k^*_p)=2 \pi^2 k^3 P_L(k)|_{k^*_p} \, .$$
Assuming full ionization (ionization fraction $x_e$=1) for an easier comparison with the results of the WMAP team, we compute for each model in our grid the optical depth to the last scattering, $\tau$, and the normalization of the matter power spectrum at the present time in terms of $\sigma_{8}$.\
Once we have computed the value of the $\chi^2$ for every choice of the parameters of our simulation grid we marginalize along one direction at a time to get the four-dimensional constraints on the considered parameters.
In Table 1 we report the constraints [^4] we have obtained at $68\%$ CL and $95\%$ CL. Figure 3 presents our $68\%$ CL contour in $n^*_{{\rm eff}}$ - $\Delta^2_*$ plane, indicating the best fit values and in Figure 4 we compare the best fit linear power spectrum, $P_L(k)$, at $z=2.72$ ($68\%$ CL) with the transmission power spectrum $P_F(k)$. The error bars include the statistical error and a systematic error due to the undeterminations in $\Omega_m$, $\Omega_{de}$ and $H_0$.\
Our values for $n^*_{{\rm eff}}$ and $\Delta^2_*$ are in a good agreement with those obtained by Seljak, McDonald & Makarov (2003) indicating the same degeneracy direction in $n^*_{{\rm eff}}$ - $\Delta^2_*$ plane, but are only marginally consistent with the similar values obtained by Croft et al. (2002).\
We find that the analysis of the cumulative mass function of the Ly-$\alpha$ systems indicates a reionization redshift in agreement, within the error bars, with the value found on the basis of the WMAP anisotropy measurements, setting constraints on the amplitude of the density contrast similar to those derived from the X-ray cluster temperature function.
Ly-$\alpha$
------------------------------------------- -------------------- --------------------
Parameter $68\%$ CL $95\%$ CL
$n_s$ 1.001$\pm$0.034 0.975$\pm$0.047
$T_{r,4}$ 2.317$\pm$0.205 2.244$\pm$0.227
$z_r$ 24.195$\pm$3.976 22.313$\pm$4.814
$w$ $-$0.689$\pm$0.141 $-$0.729$\pm$0.144
$\sigma_8$ 0.911$\pm$0.038 0.919$\pm$0.039
$\tau^a$ 0.148$\pm$0.035 0.132$\pm$0.041
$\sigma_8 e^{-\tau}$ 0.786$\pm$0.041 0.805$\pm$0.047
$n^*_{{\rm eff}}$ $-$2.550$\pm$0.034 $-$2.576$\pm$0.047
$d n^*_{{\rm eff}}/d {\rm ln}k$ $-$0.017$\pm$0.004 $-$0.018$\pm$0.005
$\Delta^2_*$ 0.666$\pm$0.113 0.618$\pm$0.131
$^a$Assumes ionization fraction, $x_e$=1.
: Cosmological parameter constraints from Ly-$\alpha$ cumulative mass function. The values of $n^*_{{\rm eff}}$, $d n^*_{{\rm eff}}/d {\rm ln}k$ and $\Delta^2_*$ are evaluated at the pivot point $k^*_p$=0.03 s/km and $z=2.72$; $T_{r,4}$ is the reionization temperature in units of $10^4$K.
{width="13cm"}
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\[\]
Combined CMB and Ly-$\alpha$ analysis
-------------------------------------
By jointly considering the WMAP anisotropy data and the Ly-$\alpha$ observations, we investigate the cosmological constraints obtained on the basis of the analysis of the Ly-$\alpha$ cumulative mass function. To this purpose, we use here only the accurate WMAP temperature anisotropy (TT) power spectrum. In fact, although crucial to probe the cosmological reionization, the inclusion of the polarization (ET) power spectrum derived from WMAP \[see also the DASI detection/upper limit on (E and B) polarization power spectrum; Kovac et al. 2002\] does not change significantly our quantitative results, as we have verified for a representative set of cases. We will take into account the polarization information in future works. We ran the CMBFAST code v4.2 with the COBE normalization option to generate the CMB temperature anisotropy power spectra for the considered grid of parameters and then renormalized each computed power spectra, $C_\ell$, of the grid to minimize the $\chi^2$ when compared to the WMAP data (we find that this renormalization does not change appreciably the final best fit and error bar results). We vary $n_s$, $w$ and $z_r$ in the same range as in the previous analysis, but for this case we also allow to vary the effective running of the slope $dn_{s}/d{\rm ln}k|_{k_p}$ at $k_p=0.05$Mpc$^{-1}$, the same pivot wavenumber used in the analysis by the WMAP team (Spergel et al. 2003).\
Our parameter vector has also four dimensions: ${\bf p}=(n_s, w, z_r, dn_{s}/d{\rm ln}k$), where $dn_{s}/d{\rm ln}k$ was free to vary in the range $[-0.1 , 0.1]$ with a step of 0.01. We compute the $\chi^2$ for each choice of the parameters in the grid, comparing the simulated CMB temperature anisotropy power spectra with the WMAP anisotropy power spectrum, following the same procedure as in the previous analysis. In addition, for each choice of parameters in the grid we generate the matter transfer function and evaluate the linear matter power spectrum $P_L(k)$ at the present time with the normalization given by the equation (22). For each $P_L(k)$ we compute the effective slope $n_{{\rm eff}}(k_p)$ and the running of the slope $d n_{\rm eff}/d {\rm ln}k$ at the pivot wavenumber $k_p=0.05$Mpc$^{-1}$, as given by equations (24)–(26) by only replacing $k^*_p$ with $k_p$. Note the difference between the two definitions of the running of the slope: $d n_s/d {\rm ln}k$ is the matter power law free parameter (see Spergel et al. 2003 and the CMBFAST code v4.2) while $d n_{\rm eff}/d {\rm ln}k$ is obtained from the matter transfer function shape.\
WMAP+Ly-$\alpha$
------------------------------------------- -------------------- --------------------
Parameter $68\%$ CL $95\%$ CL
$n_s$ 1.034$\pm$0.043 1.011$\pm$0.059
$T_{r,4}$ 2.320$\pm$0.194 2.228$\pm$0.229
$z_r$ 26.131$\pm$3.105 25.495$\pm$3.728
$w$ $-$0.804$\pm$0.122 $-$0.831$\pm$0.115
$\sigma_8$ 0.945$\pm$0.035 0.949$\pm$0.032
$\tau^a$ 0.167$\pm$0.028 0.159$\pm$0.032
$\sigma_8 e^{-\tau}$ 0.800$\pm$0.037 0.810$\pm$0.040
$n_{{\rm eff}}$ $-$2.151$\pm$0.045 $-$2.154$\pm$0.047
$dn_{\rm eff}/d\,{\rm ln}k$ 0.018$\pm$0.004 0.017$\pm$0.005
$dn_{s}/d\,{\rm ln}k$ 0.023$\pm$0.022 0.024$\pm$0.023
$^a$Assumes ionization fraction, $x_e$=1.
: Cosmological constraints from the joint WMAP and Ly-$\alpha$ analysis. The values of $n_{{\rm eff}}$, $dn_{s}/d\,{\rm ln}k$ and $dn_{\rm eff}/d\,{\rm ln}k$ are evaluated at the pivot wavenumber $k_p$=0.05Mpc$^{-1}$; $T_{r,4}$ is the reionization temperature in units of $10^4$K.
{width="15cm"}
\[\]
We generate Monte Carlo Markov Chains (MCMC) using the combined $\chi^2$ obtained from the Ly-$\alpha $ cumulative mass function and WMAP anisotropy analysis (the number of model elements is of about $3 \times 10^6$). In this way we sample the chi-squared probability distribution in the combined parameter space. We run the MCMC with $dn_s/d{\rm ln}k=0$ and varying $dn_s/d{\rm ln}k$ in the range mentioned above.\
Before of discussing the results obtained by using the pivot wavenumber $k_p$=0.05Mpc$^{-1}$ we briefly report on the results we derived by using the pivot point $k^*_p$=0.03 s/km as in Sect. 3.1 but by exploiting only the WMAP (TT) power spectrum or WMAP combined to the Ly-$\alpha$ information. They are shown again in Figure 3: note how for this pivot point choice the poor sensitivity of WMAP at the small scales accessible to Ly-$\alpha$ observations does not improve but slightly worses the parameter recovery, the two kinds of observations having similar degeneracy directions in the $n^*_{\rm eff} - \Delta^2_*$ plane. On the contrary, the situation improves by considering a pivot wavenumber (namely at $k_p$=0.05Mpc$^{-1}$) at larger scales. We report in Table 2 the results obtained from the joint WMAP and Ly-$\alpha$ analysis from the MCMC with $dn_s/d{\rm ln}k \ne 0$ that can be compared with the WMAP results (Spergel et al. 2003; Peiris et al. 2003).\
Panel a) in Figure 5 presents the constraints in $n_{\rm eff}$ - $dn_s/d{\rm ln}k$ plane obtained from the analysis of the WMAP anisotropy measurements and the joint WMAP and Ly-$\alpha$ analysis from MCMC with $dn_s/d{\rm ln}k \ne 0$. We found that both analyses favour a positive running of the slope $dn_s/d{\rm ln}k \approx 0.023$ and an effective spectral index $n_{eff}\approx-2.2$ at $k_p$=0.05 Mpc$^{-1}$.\
Panels b) and c) present the constraints in the $n_{{\rm eff}}$ – $\sigma_8e^{-\tau}$ plane and $\sigma_8e^{-\tau}$ – $dn_{\rm eff}/d{\rm ln}k$ plane obtained from MCMC with $dn_s/d{\rm ln}k =0$ [^5]. From the Monte Carlo Markov chain with $dn_s/d{\rm ln}k =0$ we found $n_{{\rm eff}}\approx -2.19$ and $dn_{{\rm eff}}/d{\rm ln}k \approx 0.01$.
Our results differ from the value of the effective running of the slope $dn_s/d{ln}k \approx -0.03$ found by the WMAP team (Spergel et al. 2003; Peiris et al. 2003) at the same pivot wavenumber and indicate that a possible identification of a running of the slope, $dn_s/d{\rm ln}k\ne 0$, at $k_p$=0.05Mpc$^{-1}$ (multipole ${\it l} \approx 700$) with the current data is mainly an effect of the existing degeneracy in the amplitude-slope plane at this scale, the result being clearly consistent with the absence of running.
In Figure 6 we compare the cosmological parameter constraints at $68\%$ CL from the Ly-$\alpha$ analysis and the joint WMAP and Ly-$\alpha$ analysis with the cosmological parameter “simulated” constraints on cosmological parameters achievable by WMAP after 4 years of observations and by the combination of the three “cosmological” channels of [Planck]{} (Mandolesi et al. 1998, Puget et al. 1998, Tauber 2000) at 70, 100, and 143 GHz considering only the multipoles $\ell \le 1500$ and neglecting the Galactic and extragalactic foreground contamination. We assume Gaussian symmetric beams with the nominal resolution, a sky coverage of $80\%$, the cosmic variance and nominal noise sensitivity as sources of error, and neglect for simplicity possible systematic effects. Only the information from the temperature (TT) power spectrum is again considered. In the case of the “simulated” data we assume exactly the current WMAP data and error bars at $\ell {\,\lower2truept\hbox{${<\atop\hbox{\raise4truept\hbox{$\sim$}}}$}\,}300$, being the uncertainty in that multipole range dominated by the cosmic variance. We consider as fiducial model the best fit power law $\Lambda$CDM model to the WMAP data with $dn_s/d\,{\rm ln}k=0$ (Table 1 from Spergel at al. 2003).
Note the improvement on power spectrum and reionization parameters achievable by using the final WMAP data (improvement of about a factor of two) and that (of about a further factor of two) achievable with [Planck]{} by using only the temperature anisotropy data. Note also the role of [Planck]{} in reducing the error bar for the parameter $w$ defining the equation of state of the dark energy component parameter.
We find that adding the current Ly-$\alpha$ information to the simulated WMAP 4-yr data only slightly reduces the error bars (of course, the relative improvement is significantly smaller by adding them to the simulated [Planck]{} data).
![Cosmological parameter constraints at $68\%$ CL from the Ly-$\alpha$ and WMAP+Ly-$\alpha$ analysis compared with the cosmological parameter constraints achievable by WMAP after 4 years of observations and by [Planck]{}. For all panels the meaning of the symbols is the same as in panel a). See also the text.](cosmo.eps){width="15cm"}
\[\]
Discussion and conclusions
==========================
The recent detection of high values of the electron optical depth to the last scattering (Kogut et al. 2003, Spergel et al. 2003) by the WMAP satellite (Bennett et al. 2003) implies the existence of an early epoch of reionization of the Universe at $z_r\sim 20$, fundamentally important for understanding the formation and evolution of the structures in the Universe.\
As the reionization is assumed to be caused by the ionizing photons produced during the early stages of star-forming galaxies and quasars, we evaluate this effect by computing the cumulative mass function of the high-z bound objects for a class of flat cosmological models with cold dark matter plus cosmological constant or dark energy with constant equation of state, encompassing different reionization scenarios. Our fiducial cosmological model has $\Omega_bh^2=0.024\pm 0.001$, $\Omega_mh^2=0.14\pm0.02$, $h=0.72\pm0.05$ as indicated by the best fit power law $\Lambda$CDM model of WMAP data (Spergel et al. 2003).
Assuming that the virilization takes place at the collapse time and a constant baryon/dark matter ratio in collapsed objects, we compute the fraction of the mass residing in gravitationally bounded systems as a function of the redshift of collapse at each linear scale and of the virial mass-temperature relation. We evaluate the formation rate of bound objects at $z=2.72$ and their cumulative mass function was compared with the cumulative mass function obtained from the Ly-$\alpha$ transmission power spectrum (Croft et al. 2002).\
Our method allows to study reionization models by varying the amplitude, spectrum, and epoch of the reionization and the cosmological parameters. In the same time, as the high-z bound objects are rare fluctuations of the overdensity field, the tail of the cumulative mass function is sensitive to the [*rms*]{} mass fluctuations within the filtering scale $\sigma(R_f,z)$.\
We find that the analysis of the cumulative mass function of the Ly-$\alpha$ systems indicates a reionization redshift in agreement with the value found on the basis of the WMAP anisotropy measurements, setting constraints on the amplitude of the power spectrum, $\sigma_8$, similar to those derived from the X-ray cluster temperature function.
Our joint analysis of Ly-$\alpha$ cumulative mass function and WMAP anisotropy measurements shows that a possible identification of a running of the slope, $dn_s/d{\rm ln}k\ne 0$, at $k_p$=0.05Mpc$^{-1}$ (multipole ${\it l} \approx 700$) is mainly an effect of the existing degeneracy in the amplitude-slope plane at this scale, the result being clearly consistent with the absence of running, the other constraints based on WMAP remaining substantially unchanged.
We also shown that, for the set of cosmological models studied in this work, the error bars on the considered parameters can be reduced by about a factor of two by using the final WMAP data. The temperature anisotropy data from the forthcoming [Planck]{} satellite will further improve the sensitivity on these parameters, by another factor of two, and also the reliability of these results thanks to the better foreground subtraction achievable with the wider frequency coverage and the improved sensitivity, resolution and systematic effect control. This information jointed with the great improvement on the study of the Ly-$\alpha$ forest trasmission power spectrum (Seljak et al. 2002) achievable by the increase of the number of quasar spectrum measures expected from the Sloan Digital Sky Survey will allow to significantly better constrain the properties of the primordial density field at small scales.
Acknowledgements
================
We acknowledge the use of the computing system at [Planck]{}-LFI Data Processing Center in Trieste and the staff working there. LAP acknowledge the financial support from the European Space Agency. It is a pleasure to thank to U. Seljak and M. Zaldarriaga for the use of the CMBFAST code v4.2 employed in the computation of the CMB power spectra and the matter transfer functions.
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[^1]: The address to which the proofs have to be sent is:\
Carlo Burigana\
IASF/CNR, Istituto di Astrofisica Spaziale e Fisica Cosmica, Sezione di Bologna, Consiglio Nazionale delle Ricerche, Via Gobetti 101, I-40129 Bologna, Italy\
fax: +39-051-6398724\
e-mail: burigana@bo.iasf.cnr.it
[^2]: http://lambda.gsfc.nasa.gov
[^3]: http://astro.estec.esa.nl/Planck/
[^4]: We also verified that the final results do not change significantly by using the transmission power spectra from McDonald & Miralda-Escudè (1999) at $z = 2.4, 3$ and 3.9.
[^5]: We have verified that the value of $dn_{\rm eff}/d{\rm ln}k$ may depend quite critically on the choice of the step in $k$ used to numerically compute the (central) derivatives of the power spectrum. For the considered models we find in practice quite stable results for steps in $k$ less than $\simeq 5$ % of $k$, while using larger steps significantly affects the final results.
|
---
abstract: 'A second-order accurate modular algorithm is presented for a standard BDF2 code for the Navier-Stokes equations (NSE). The algorithm exhibits resistance to solver breakdown and increased computational efficiency for increasing values of grad-div parameters. We provide a complete theoretical analysis of the algorithms stability and convergency. Computational tests are performed and illustrate the theory and advantages over monolithic grad-div stabilizations.'
author:
- 'Y. Rong[^1]'
- 'J. A. Fiordilino[^2]'
title: 'Numerical analysis of a bdf2 modular grad-div Stabilization method for the Navier-Stokes equations'
---
Introduction
============
A common, powerful tool for improving solution quality for fluid flow problems is grad-div stabilization [@Jenkins; @Linke2; @LeBorne; @Olshanskii2; @Olshanskii4]. This technique typically involves adding $\gamma \nabla \nabla \cdot u_{h}$, nonzero for most finite element velocity-pressure pairs, which penalizes mass conservation and improves solution accuracy. It was first introduced in [@Franca] and has been widely studied since, both analytically and computationally [@Decaria; @Jenkins; @Layton3; @Linke2; @Lube; @Olshanskii; @Olshanskii2; @Olshanskii4].
Unfortunately, grad-div stabilization also exhibits increased coupling in the linear system’s matrix, efficiency loss and solver breakdown, and classical Poisson locking [@Bowers; @Guermond; @Heister; @Linke; @LeBorne; @Olshanskii; @Olshanskii2]. In particular, since the matrix arising from grad-div term is singular, large grad-div parameter $\gamma$ values can cause solver breakdown [@Glowinski]. This difficulty cannot always be circumvented since recommended parameter choices vary greatly, e.g., from $\mathcal{O}(h^{2})$ to $\mathcal{O}(10^{4})$ for different applications, finite elements, and meshes [@Decaria; @Jenkins; @John2; @Olshanskii4; @Roos]. An alternate realization of grad-div stabilization with greater computational efficiency was introduced in [@Fiordilino] for the backward Euler time discretization. Herein, we show how to implement modular grad-div stabilization for any multistep time discretization and perform analysis and testing for the BDF2 case.
To begin, consider the incompressible time-dependent NSE: Find the fluid velocity $u:\Omega\times[0,T]\rightarrow\mathbb{R}^{d}$ and pressure $p:\Omega\times(0,T]\rightarrow\mathbb{R}$ satisfying: $$\label{NSE}
\begin{aligned}
&u_{t}+u\cdot\nabla u-\nu\Delta u+\nabla p
= f,\; \mathrm{and} \;
\nabla\cdot u=0\;
\mathrm{in} \;\Omega,\\
&u=0\;
\mathrm{on} \;\partial\Omega,\; \mathrm{and} \;
\int_{\Omega} p\,dx
= 0,\\
&u(x,0)=u^{0}(x)\;
\mathrm{in} \;\Omega.\\
\end{aligned}$$ Here, the domain $\Omega\subset \mathbb{R}^{d}$($\mathrm{d}$=2,3) is a bounded polyhedron, $f$ is the body force and $\nu$ is the fluid viscosity. Suppressing the spacial discretization for the moment, we consider the following two step method that uncouples the grad-div solve.\
$\emph{Step 1}:$ Given $u^{n-1}, u^{n}$, find $\hat{u}^{n+1}$ and $p^{n+1}$ satisfying: $$\begin{aligned}
&\frac{3\hat{u}^{n+1} - 4u^{n} + u^{n-1}}{2\Delta t}
+ (2u^{n}-u^{n-1})\cdot\nabla\hat{u}^{n+1}
- \nu \Delta\hat{u}^{n+1}
+ \nabla p^{n+1}
= f^{n+1}, \label{step01.1} \\
&\nabla \cdot \hat{u}^{n+1}=0. \label{step01.2}\end{aligned}$$ $\emph{Step 2}:$ Given $\hat{u}^{n+1}$, find $u^{n+1}$ satisfying: $$\begin{aligned}
&\frac{3u^{n+1}-3\hat{u}^{n+1}}{2\Delta t}
- \beta \nabla \nabla \cdot \frac{3u^{n+1} - 4u^{n} + u^{n-1}}{2\Delta t}
- \gamma \nabla \nabla \cdot u^{n+1} = 0. \label{step02}\end{aligned}$$ In the above, $\beta \geq 0$ and $ \gamma \geq 0$ are application-dependent grad-div stabilization parameters. The combined effect of $\emph{Step 1}$ and $\emph{Step 2}$ is a consistent BDF2 time discretization of the following model: $$\label{grad-div NSE}
\begin{aligned}
&u_{t} - \beta \nabla \nabla \cdot u_{t} - \gamma \nabla \nabla \cdot u
+ u\cdot\nabla u - \nu\Delta u+\nabla p
= f.
\end{aligned}$$ In [@Fiordilino], two minimally intrusive, modular algorithms were developed for backward Euler, which implemented grad-div stabilization. These algorithms effectively treated issues resulting from increased coupling and solver breakdown. Although the second steps of each of these algorithms can be used here when $\beta \equiv 0$, they cannot be used when $\beta > 0$; that is, the dispersive term [@Decaria2; @Layton4; @Prohl], associated with $\beta$ demands special attention. In the case $\beta > 0$, the time-discretizations in both steps must be consistent with one another. In particular, for the BDFk family of methods:\
$\emph{Step 1}:$ Find $\hat{u}^{n+1}$ and $p^{n+1}$ satisfying: $$\begin{aligned}
&\frac{1}{\Delta t}\big(a_{0}\hat{u}^{n+1} + \sum^{S}_{s=1} a_{s} u^{n+1-s}\big)
+ U\cdot\nabla\hat{u}^{n+1}
- \nu \Delta\hat{u}^{n+1}
+ \nabla p^{n+1}
= f^{n+1}, \\
&\nabla \cdot \hat{u}^{n+1}=0. \end{aligned}$$ $\emph{Step 2}:$ Find $u^{n+1}$ satisfying: $$\begin{aligned}
&\frac{a_{0}}{\Delta t}\big(u^{n+1}-\hat{u}^{n+1}\big)
- \beta \nabla \nabla \cdot \frac{\sum^{S}_{s=0} a_{s} u^{n+1-s}}{\Delta t}
- \gamma \nabla \nabla \cdot u^{n+1} = 0, \end{aligned}$$ where $U$ denotes either $\hat{u}^{n+1}$ or a consistent extrapolation. A similar generalization can be made for general linear multistep methods.
This paper is arranged as follows. Section $\ref{Preliminaries}$ introduces notation, lemmas, and necessary preliminaries. In Section $\ref{Algorithm and Stability}$, a fully-discrete modular grad-div stabilization algorithm (*BDF2-mgd*) and its unconditional, nonlinear, energy stability are presented. A complete error analysis is given in Section $\ref{Error Analysis}$ where second-order convergence is proven for the modular method. Numerical experiments are provided to confirm the effectiveness of *BDF2-mgd* in Section $\ref{Numerical Tests}$. In particular, the algorithm maintains the positive impact of grad-div stabilization while resisting debilitating slow down for $0\leq \gamma \leq 20,000$ or $0 \leq \beta \leq 8,000$. Conclusions follow in Section $\ref{Conclusion}$.
Preliminaries {#Preliminaries}
=============
We use the standard notations $H^{k}(\Omega), H^{k}_{0}(\Omega)$, and $L^{p}(\Omega)$ to denote Sobolev spaces and $L^{p}$ spaces; see, e.g., [@Ad]. The $L^{2}(\Omega)$ inner product and its induced norm are denoted by $(\cdot,\cdot)$ and $\|\cdot\|$, respectively. Let $\|\cdot\|_{L^{p}}$ and $\|\cdot\|_{k}$ denote the $L^{p}(\Omega)$ ($p\neq2$) norm and $H^{k}(\Omega)$ norm. The space $H^{-k}(\Omega)$ denotes the dual space of $H^{k}_{0}(\Omega)$ and its norm is denoted by $\|\cdot\|_{-k}$. Throughout the paper, we use $C$ to denote a generic positive constant varying in different places but never depending on mesh size, time step, and grad-div parameters. For functions $v(x,t)$, we define the following norms: $$\begin{aligned}
\begin{aligned}
& \|v\|_{\infty,k}:=ess\sup_{[0,T]}\|v(\cdot,t)\|_{k}\, ,\quad
\|v\|_{p,k}:=(\int^{T}_{0}\|v(\cdot,t)\|^{p}_{k}dt)^{\frac{1}{p}}\,
\end{aligned}\end{aligned}$$ for $1\leq p<\infty$. The velocity space $X$, pressure space $Q$, and divergence free space $V$ are defined as follows. $$\begin{aligned}
\begin{aligned}
& X:=H^{1}_{0}(\Omega)^{d}=\{v\in H^{1}(\Omega)^{d}:v|_{\partial\Omega}=0\},\\
& Q:=L^{2}_{0}(\Omega)=\{q\in L^{2}(\Omega):\int_{\Omega}q\,dx=0\},\\
& V:=\{v\in X:(\nabla\cdot v,q)=0\quad\forall q\in Q\}.
\end{aligned}\end{aligned}$$ Define the skew-symmetric trilinear form $$\begin{aligned}
b(u,v,w):=
\frac{1}{2}(u\cdot\nabla v,w)-\frac{1}{2}(u\cdot\nabla w,v)
\quad \forall\;u,v,w \in X.\end{aligned}$$ Then, we have the following estimates for $b$ (see, e.g., Lemma 2.2 in [@Layton2]): $$\begin{aligned}
\label{tri}
& b(u,v,w)\leq C\|\nabla u\|\|\nabla v\|\|\nabla w\|,\label{tri1} \\
& b(u,v,w)\leq C\|u\|^{\frac{1}{2}}\|\nabla u\|^{\frac{1}{2}}\|\nabla v\|\|\nabla w\|,\label{tri2}\\
& b(u,v,w)\leq C\|u\|\|v\|_{2}\|\nabla w\|.\label{tri3}\end{aligned}$$ Divide the simulation time $T$ into $N$ smaller time intervals with $[0,T]=\bigcup\limits_{n=0}^{N-1}[t^{n},t^{n+1}]$, where $t^{n}=n\Delta t, \, T=N\Delta t$. We may define the following discrete norms: $$\begin{aligned}
\begin{aligned}
& |\|v\||_{\infty,k}:=\max_{0\leq n\leq N}\|v(\cdot,t^{n})\|_{k}\, ,\quad
|\|v\||_{p,k}:=(\Delta t\sum^{N}_{n=0}\|v(\cdot,t^{n})\|^{p}_{k})^{\frac{1}{p}}\,.
\end{aligned}\end{aligned}$$
Let $\Omega_{h}$ be a quasi-uniform mesh of $\Omega$ with $\overline{\Omega}=\bigcup\limits_{K\in\Omega_{h}}K$. Denote $h=\sup\limits_{K\in\Omega_{h}}diam(K)$. Let $X_{h}\subset X$ and $Q_{h}\subset Q$ be the finite element spaces. Assume that $X_{h}$ and $Q_{h}$ satisfy approximation properties of piecewise continuous polynomials on quasi-uniform meshes of local degrees $k$ and $m$, respectively: $$\begin{aligned}
& \inf\limits_{v_{h}\in X_{h}}\|u-v_{h}\|
\leq Ch^{k+1}|u|_{k+1}\qquad u\in X\cap H^{k+1}(\Omega)^{d},\\
& \inf\limits_{v_{h}\in X_{h}}\|u-v_{h}\|_{1}
\leq Ch^{k}|u|_{k+1}\qquad\;\;u\in X\cap H^{k+1}(\Omega)^{d},\\
& \inf\limits_{q_{h}\in Q_{h}}\|p-q_{h}\|
\leq Ch^{m+1}|p|_{m+1}\qquad\;\;\; p\in Q\cap H^{m+1}(\Omega).\end{aligned}$$ Furthermore, we assume that $X_{h}$ and $Q_{h}$ satisfy the usual discrete inf-sup condition: $$\begin{aligned}
\label{inf-sup}
\begin{aligned}
\inf\limits_{q\in Q_{h}}\sup\limits_{v\in X_{h}}\frac{(q,\nabla\cdot v)}{\|\nabla v\|\|q\|}\geq C_{0}>0.
\end{aligned}\end{aligned}$$ The discrete divergence-free space $V_{h}$ is defined by $$\begin{aligned}
\begin{aligned}
V_{h}:=\{v_{h}\in X_{h}:(\nabla\cdot v_{h},q_{h})=0\quad\forall q_{h}\in Q_{h}\}.
\end{aligned}\end{aligned}$$ Note that the well-known Taylor-Hood mixed finite element is one such example satisfying the above assumptions with $k=2, m=1$.
The following lemmas will be useful in later analyses. For their proofs, see Theorem 1.1 on p. 59 of [@Girault] for Lemma \[lemma0\], Lemma 2 of [@RY2] for Lemma \[lemma1\], and Lemma 5.1 on p. 369 of [@Heywood] for Lemma \[gronwall2\].
\[lemma0\] Suppose that the finite element spaces satisfy ($\ref{inf-sup}$). Then, for any $u\in V$, we have $$\begin{aligned}
\begin{aligned}
\inf\limits_{v_{h}\in V_{h}}\|\nabla(u-v_{h})\|
\leq C\inf\limits_{v_{h}\in X_{h}}\|\nabla(u-v_{h})\|.
\end{aligned}\end{aligned}$$
\[lemma1\] If $g_{t}, g_{tt}, g_{ttt}\in L^{2}(0,T;H^{r}(\Omega))$, then we have $$\label{lemma1.1}
\begin{aligned}
\|g^{n+1}-2g^{n}+g^{n-1}\|^{2}_{r}\leq C\Delta t^{3}\int^{t^{n+1}}_{t^{n-1}}\|g_{tt}\|^{2}_{r}dt,
\end{aligned}$$ $$\label{lemma1.2}
\begin{aligned}
\|3g^{n+1}-4g^{n}+g^{n-1}\|^{2}_{r}\leq C\Delta t\int^{t^{n+1}}_{t^{n-1}}\|g_{t}\|^{2}_{r}dt,
\end{aligned}$$ $$\label{lemma1.3}
\begin{aligned}
\|\frac{3g^{n+1}-4g^{n}+g^{n-1}}{2\Delta t}-g_{t}(t^{n+1})\|^{2}_{r}
\leq C\Delta t^{3}\int^{t^{n+1}}_{t^{n-1}}\|g_{ttt}\|^{2}_{r}dt.
\end{aligned}$$
[(The discrete Gronwall’s lemma,without $\Delta t$-restriction)]{}\[gronwall2\] Suppose that $n$ and $N$ are nonnegative integers,$n\leq N$. The real numbers $a_{n},b_{n},c_{n},\kappa_{n},\Delta t,C$ are nonnegative and satisfy $$\begin{aligned}
\begin{aligned}
& a_{N}+\Delta t\sum\limits_{n=0}^{N}b_{n}\leq\Delta t\sum\limits_{n=0}^{N-1}\kappa_{n}a_{n}+\Delta t\sum\limits_{n=0}^{N}c_{n}+C.
\end{aligned}\end{aligned}$$ Then, $$\begin{aligned}
\begin{aligned}
& a_{N}+\Delta t\sum\limits_{n=0}^{N}b_{n}\leq\exp(\Delta t\sum\limits_{n=0}^{N-1}\kappa_{n})(\Delta t\sum\limits_{n=0}^{N}c_{n}+C).
\end{aligned}\end{aligned}$$
The BDF2 modular grad-div stabilization algorithm and its stability {#Algorithm and Stability}
===================================================================
We propose the following fully-discrete modular grad-div stabilization algorithm for approximating solutions of ($\ref{NSE}$).\
*BDF2-mgd*:\
$\emph{Step 1}:$ Given $u^{n-1}_{h}, u^{n}_{h} \in X_{h}$, find $(\hat{u}^{n+1}_{h}, p^{n+1}_{h}) \in (X_{h},Q_{h})$ satisfying: $$\begin{aligned}
&(\frac{3\hat{u}^{n+1}_{h} - 4u^{n}_{h} + u^{n-1}_{h}}{2\Delta t},v_{h})
+ b(2u^{n}_{h}-u^{n-1}_{h},\hat{u}^{n+1}_{h},v_{h})
+ \nu (\nabla\hat{u}^{n+1}_{h}, \nabla v_{h}) \nonumber \\
& - (p^{n+1}_{h},\nabla \cdot v_{h})
= (f^{n+1},v_{h}) \quad \forall v_{h} \in X_{h}, \label{step1.1} \\
&(\nabla \cdot \hat{u}^{n+1}_{h}, q_{h})=0 \quad \forall q_{h} \in Q_{h}. \label{step1.2}\end{aligned}$$ $\emph{Step 2}:$ Given $\hat{u}^{n+1}_{h} \in X_{h}$, find $u^{n+1}_{h} \in X_{h}$ satisfying: $$\begin{aligned}
&(\frac{3u^{n+1}_{h}-3\hat{u}^{n+1}_{h}}{2\Delta t},v_{h})
+ \beta (\nabla \cdot \frac{3u^{n+1}_{h} - 4u^{n}_{h}
+ u^{n-1}_{h}}{2\Delta t}, \nabla \cdot v_{h}) \nonumber \\
&+ \gamma (\nabla \cdot u^{n+1}_{h}, \nabla \cdot v_{h}) = 0 \quad \forall v_{h} \in X_{h}. \label{step2}\end{aligned}$$
When $\beta = 0$, Step 2 is equivalent to Step 2 appearing in [@Fiordilino] with $\gamma \leftarrow \frac{2}{3}\gamma$.
Step 2 of *BDF2-mgd* appears to be overdetermined since both the tangential and normal components of the solution are prescribed on the boundary. However, due to the zeroth-order term, it is not; a unique solution always exists, Theorem \[existence\], and converges to the true NSE solution, Theorems \[error\] and \[error0\].
\[existence\] Suppose $f^{n+1}\in H^{-1}(\Omega)^{d}$ and $u^{n-1}_{h}, u^{n}_{h} \in X_{h}$. Then, there exists unique solutions $\hat{u}^{n+1}_{h}, u^{n+1}_{h} \in X_{h}$ to *BDF2-mgd*.
The proof follows by similar arguments as in Theorem 5 of [@Fiordilino].
Next, we analyze the stability of *BDF2-mgd*. We first prove an important lemma for the stability analysis. Unconditional, nonlinear, energy stability is then proven in Theorem $\ref{stability}$.
\[stability\_lemma\] Consider *BDF2-mgd*, then the following identities hold for Step 2 (\[step2\]): $$\begin{aligned}
\label{stability1}
\begin{aligned}
\|\hat{u}^{n+1}_{h}\|^{2}
=
&\|u^{n+1}_{h}\|^{2} + \|\hat{u}^{n+1}_{h} - u^{n+1}_{h}\|^{2}
+ \frac{4}{3}\gamma \Delta t \|\nabla \cdot u^{n+1}_{h}\|^{2}
\\
&+ \frac{\beta}{3} \Big(\|\nabla \cdot u^{n+1}_{h}\|^{2} - \|\nabla \cdot u^{n}_{h}\|^{2}
+ \|\nabla \cdot (2u^{n+1}_{h}-u^{n}_{h})\|^{2}
\\
&
- \|\nabla \cdot (2u^{n}_{h}-u^{n-1}_{h})\|^{2}
+\| \nabla \cdot (u^{n+1}_{h} - 2u^{n}_{h} + u^{n-1}_{h}) \|^{2} \Big),
\end{aligned}\end{aligned}$$ and $$\begin{aligned}
\label{stability1.0}
\begin{aligned}
&(\frac{3u^{n+1}_{h} - 4u^{n}_{h} + u^{n-1}_{h}}{2\Delta t},\hat{u}^{n+1}_{h}-u^{n+1}_{h})
\\
&=
\frac{\beta}{6\Delta t}\|\nabla\cdot(3u^{n+1}_{h} - 4u^{n}_{h} + u^{n-1}_{h})\|^{2}
+
\frac{\gamma}{3}(\nabla\cdot u^{n+1}_{h},\nabla\cdot(3u^{n+1}_{h} - 4u^{n}_{h} + u^{n-1}_{h})).
\end{aligned}\end{aligned}$$
Selecting $v_{h}=\frac{4\Delta t}{3}u^{n+1}_{h}$ in ($\ref{step2}$), we have $$\begin{aligned}
\label{pf1.1}
\begin{aligned}
&2\|u^{n+1}_{h}\|^{2} - 2(\hat{u}^{n+1}_{h},u^{n+1}_{h})
+ \frac{4}{3}\gamma \Delta t \|\nabla \cdot u^{n+1}_{h}\|^{2}
+ \frac{\beta}{3} \Big(\|\nabla \cdot u^{n+1}_{h}\|^{2}
- \|\nabla \cdot u^{n}_{h}\|^{2}
\\
&+ \|\nabla \cdot (2u^{n+1}_{h}-u^{n}_{h})\|^{2}
- \|\nabla \cdot (2u^{n}_{h}-u^{n-1}_{h})\|^{2}
+\| \nabla \cdot (u^{n+1}_{h} - 2u^{n}_{h} + u^{n-1}_{h}) \|^{2} \Big)
=0,
\end{aligned}\end{aligned}$$ where we have used the identity $2(3a-4b+c)a=a^{2}-b^{2}+(2a-b)^{2}-(2b-c)^{2}+(a-2b+c)^{2}$ on the third term. For the second term in (\[pf1.1\]), using the following polarization identity $$\begin{aligned}
\label{pf1.2}
\begin{aligned}
(\hat{u}^{n+1}_{h},u^{n+1}_{h})
&=\frac{1}{2}\|\hat{u}^{n+1}_{h}\|^{2}
+\frac{1}{2}\|u^{n+1}_{h}\|^{2}
-\frac{1}{2}\|\hat{u}^{n+1}_{h}-u^{n+1}_{h}\|^{2}
\end{aligned}\end{aligned}$$ yields the first identity (\[stability1\]). The second follows by setting $v_{h}=\frac{3u^{n+1}_{h} - 4u^{n}_{h} + u^{n-1}_{h}}{3}$ in ($\ref{step2}$).
We are now in a position to prove unconditional stability.
\[stability\] Suppose $f\in L^{2}(0,T;H^{-1}(\Omega)^{d})$, then the following holds for all $N\geq 1$.
$$\begin{aligned}
\label{stability2}
\begin{aligned}
&\|u^{N}_{h}\|^{2}
+ \|2u^{N}_{h}-u^{N-1}_{h}\|^{2}
+ (\frac{2\gamma\Delta t}{3} + \beta)\|\nabla\cdot u^{N}_{h}\|^{2}
+ (\frac{2\gamma\Delta t}{3} + \beta)\|\nabla\cdot(2u^{N}_{h}-u^{N-1}_{h})\|^{2}
\\&\quad
+ 4\gamma \Delta t \sum^{N-1}_{n=1}\|\nabla \cdot u^{n+1}_{h}\|^{2}
+ 2\nu \Delta t\sum^{N-1}_{n=1}\|\nabla \hat{u}^{n+1}_{h}\|^{2}
\\
&
\leq
\frac{2\Delta t}{\nu}\sum^{N-1}_{n=1}\|f^{n+1}\|^{2}_{-1}
+ \|u^{1}_{h}\|^{2}
+ \|2u^{1}_{h}-u^{0}_{h}\|^{2}
\\
&\quad
+ (\frac{2\gamma\Delta t}{3} + \beta) \|\nabla \cdot u^{1}_{h}\|^{2}
+ (\frac{2\gamma\Delta t}{3} + \beta) \|\nabla \cdot (2u^{1}_{h}-u^{0}_{h})\|^{2}.
\end{aligned}\end{aligned}$$
Set $v_{h}=\hat{u}^{n+1}_{h}$ in ($\ref{step1.1}$) and $q_{h}=p^{n+1}_{h}$ in ($\ref{step1.2}$). Adding these two equations and rearranging the discrete time derivative yields $$\begin{aligned}
\label{pf2.1}
\begin{aligned}
&(\frac{3u^{n+1}_{h} - 4u^{n}_{h} + u^{n-1}_{h}}{2\Delta t},u^{n+1}_{h})
+(\frac{3u^{n+1}_{h} - 4u^{n}_{h} + u^{n-1}_{h}}{2\Delta t},\hat{u}^{n+1}_{h}-u^{n+1}_{h})
\\
& +(\frac{3\hat{u}^{n+1}_{h} - 3u^{n+1}_{h}}{2\Delta t},\hat{u}^{n+1}_{h})
+ \nu\|\nabla \hat{u}^{n+1}_{h}\|^{2}
=
(f^{n+1},\hat{u}^{n+1}_{h}).
\end{aligned}\end{aligned}$$ Consider the resulting time derivative terms. Use the identity $2(3a-4b+c)a=a^{2}-b^{2}+(2a-b)^{2}-(2b-c)^{2}+(a-2b+c)^{2}$ on the first term and both (\[stability1.0\]) of Lemma \[stability\_lemma\] and the identity on the second term. Apply the polarization identity to the third term. Then, $$\begin{aligned}
\label{pf2.3}
\begin{aligned}
&\frac{1}{4\Delta t}
\Big(\|u^{n+1}_{h}\|^{2} - \|u^{n}_{h}\|^{2}
+ \|2u^{n+1}_{h}-u^{n}_{h}\|^{2} - \|2u^{n}_{h}-u^{n-1}_{h}\|^{2}
+ \|u^{n+1}_{h}-2u^{n}_{h}+u^{n-1}_{h}\|^{2}\Big)
\\&\quad
+ \frac{\gamma}{6}
\Big(\|\nabla\cdot u^{n+1}_{h}\|^{2} - \|\nabla\cdot u^{n}_{h}\|^{2}
+ \|\nabla\cdot(2u^{n+1}_{h}-u^{n}_{h})\|^{2} - \|\nabla\cdot(2u^{n}_{h}-u^{n-1}_{h})\|^{2}
\\&\quad
+ \|\nabla\cdot(u^{n+1}_{h}-2u^{n}_{h}+u^{n-1}_{h})\|^{2}\Big)
+ \frac{\beta}{6\Delta t}\|\nabla\cdot(3u^{n+1}_{h} - 4u^{n}_{h} + u^{n-1}_{h})\|^{2}
\\&\quad
+ \frac{3}{4\Delta t}
\Big(\|\hat{u}^{n+1}_{h}\|^{2} - \|u^{n+1}_{h}\|^{2}
+ \|\hat{u}^{n+1}_{h}-u^{n+1}_{h}\|^{2}\Big)
+ \nu\|\nabla \hat{u}^{n+1}_{h}\|^{2}
\\&
=
(f^{n+1},\hat{u}^{n+1}_{h}).
\end{aligned}\end{aligned}$$ Multiply ($\ref{pf2.3}$) by $4\Delta t$ and use ($\ref{stability1}$) of Lemma \[stability\_lemma\]. Then $$\begin{aligned}
\label{pf2.4}
\begin{aligned}
&\|u^{n+1}_{h}\|^{2} - \|u^{n}_{h}\|^{2}
+ \|2u^{n+1}_{h}-u^{n}_{h}\|^{2} - \|2u^{n}_{h}-u^{n-1}_{h}\|^{2}
+ \|u^{n+1}_{h}-2u^{n}_{h}+u^{n-1}_{h}\|^{2}
\\&\quad
+ \frac{2\gamma\Delta t}{3}
\Big(\|\nabla\cdot u^{n+1}_{h}\|^{2} - \|\nabla\cdot u^{n}_{h}\|^{2}
+ \|\nabla\cdot(2u^{n+1}_{h}-u^{n}_{h})\|^{2} - \|\nabla\cdot(2u^{n}_{h}-u^{n-1}_{h})\|^{2}
\\&\quad
+ \|\nabla\cdot(u^{n+1}_{h}-2u^{n}_{h}+u^{n-1}_{h})\|^{2}\Big)
+ \frac{2\beta}{3}\|\nabla\cdot(3u^{n+1}_{h} - 4u^{n}_{h} + u^{n-1}_{h})\|^{2}
\\&\quad
+ \beta \Big(\|\nabla \cdot u^{n+1}_{h}\|^{2} - \|\nabla \cdot u^{n}_{h}\|^{2}
+ \|\nabla \cdot (2u^{n+1}_{h}-u^{n}_{h})\|^{2}
- \|\nabla \cdot (2u^{n}_{h}-u^{n-1}_{h})\|^{2}
\\&\quad
+\| \nabla \cdot (u^{n+1}_{h} - 2u^{n}_{h} + u^{n-1}_{h}) \|^{2} \Big)
+ 6\|\hat{u}^{n+1}_{h} - u^{n+1}_{h}\|^{2}
\\&\quad
+ 4\gamma\Delta t\|\nabla \cdot u^{n+1}_{h}\|^{2}
+ 4\nu\Delta t\|\nabla \hat{u}^{n+1}_{h}\|^{2}
\\&
=
4\Delta t(f^{n+1},\hat{u}^{n+1}_{h}).
\end{aligned}\end{aligned}$$ Summing ($\ref{pf2.4}$) from $n=1$ to $N-1$ yields $$\begin{aligned}
\label{pf2.5}
\begin{aligned}
&\|u^{N}_{h}\|^{2}
+ \|2u^{N}_{h}-u^{N-1}_{h}\|^{2}
+ \frac{2\gamma\Delta t}{3}\|\nabla\cdot u^{N}_{h}\|^{2}
+ \frac{2\gamma\Delta t}{3}\|\nabla\cdot(2u^{N}_{h}-u^{N-1}_{h})\|^{2}
+ \beta \|\nabla \cdot u^{N}_{h}\|^{2}
\\&\quad
+ \beta \|\nabla \cdot (2u^{N}_{h}-u^{N-1}_{h})\|^{2}
+ 4\gamma \Delta t \sum^{N-1}_{n=1}\|\nabla \cdot u^{n+1}_{h}\|^{2}
+ 4\nu \Delta t\sum^{N-1}_{n=1}\|\nabla \hat{u}^{n+1}_{h}\|^{2}
\\
&
\leq
4\Delta t\sum^{N-1}_{n=1}(f^{n+1},\hat{u}^{n+1}_{h})
+ \|u^{1}_{h}\|^{2}
+ \|2u^{1}_{h}-u^{0}_{h}\|^{2}
\\
&\quad
+ \frac{2\gamma\Delta t}{3}\|\nabla\cdot u^{1}_{h}\|^{2}
+ \frac{2\gamma\Delta t}{3}\|\nabla\cdot(2u^{1}_{h}-u^{0}_{h})\|^{2}
+ \beta \|\nabla \cdot u^{1}_{h}\|^{2}
+ \beta \|\nabla \cdot (2u^{1}_{h}-u^{0}_{h})\|^{2}.
\end{aligned}\end{aligned}$$ Finally, using the Cauchy-Schwarz-Young inequality on the first term on the right hand side completes the proof.
Lemma \[stability\_lemma\] and Theorem \[stability\] imply stability of $ \hat{u}_{h} $ with respect to $ |\|\cdot\||_{\infty, 0} $.
Error Analysis {#Error Analysis}
==============
In this section, we provide $\acute{a}$ priori error estimates for *BDF2-mgd*. In particular, we show that *BDF2-mgd* is second-order convergent. Denote $u^{n}=u(t^{n})$ for $n=0,1,\cdots,N$ (and similarly for all other variables). The errors are denoted by $$\begin{aligned}
\begin{aligned}
e_{u}^{n}=u^{n}-u^{n}_{h},\quad
e_{\hat{u}}^{n}=u^{n}-\hat{u}^{n}_{h},\quad
e_{p}^{n}=p^{n}-p^{n+1}_{h}.
\end{aligned}\end{aligned}$$ Decompose the velocity errors $$\begin{aligned}
\begin{aligned}
&e_{u}^{n}=\eta^{n} - \phi^{n}_{h},\quad
\eta^{n}:=u^{n} - \tilde{u}^{n},\quad
\phi^{n}_{h}:=u^{n}_{h} - \tilde{u}^{n},
\\
&e_{\hat{u}}^{n}=\eta^{n} - \psi^{n}_{h},\quad
\psi^{n}_{h}:=\hat{u}^{n}_{h} - \tilde{u}^{n},
\end{aligned}\end{aligned}$$ where $\tilde{u}^{n}$ denotes an interpolant of $u^{n}$ in $V_{h}$.
Define the following consistency errors. For all $v_{h}\in V_{h}$, $$\begin{aligned}
\label{c-errors}
\begin{aligned}
\tau^{n+1}(v_{h}):=
(\frac{3u^{n+1} - 4u^{n} + u^{n-1}}{2\Delta t} - u_{t}^{n+1},v_{h})
- b(u^{n+1}-2u^{n}+u^{n-1},u^{n+1},v_{h}).
\end{aligned}\end{aligned}$$
\[c-errors-bound\] Assume the true solution $u$ satisfies the following, $$\begin{aligned}
\label{regularity1}
\begin{aligned}
u\in L^{\infty}(0,T;H^{1}(\Omega)^{d}), \;
u_{tt}\in L^{2}(0,T;H^{1}(\Omega)^{d}), \;
u_{ttt}\in L^{2}(0,T;H^{-1}(\Omega)^{d}).
\end{aligned}\end{aligned}$$ Then, $\forall \sigma>0$, we have $$\begin{aligned}
\begin{aligned}
|\tau^{n+1}(v_{h})|
\leq
\frac{C}{2\sigma}\Delta t^{3}\Big(\int^{t^{n+1}}_{t^{n-1}}\|u_{ttt}\|_{-1}^{2}dt
+ \int^{t^{n+1}}_{t^{n-1}}\|\nabla u_{tt}\|^{2}dt\Big)
+ \sigma\|\nabla v_{h}\|^{2}.
\end{aligned}\end{aligned}$$
For an arbitrary $\sigma>0$, $$\begin{aligned}
\label{pf3.1}
\begin{aligned}
&|\tau^{n+1}(v_{h})|
\\&\leq
\|\frac{3u^{n+1} - 4u^{n} + u^{n-1}}{2\Delta t} - u_{t}^{n+1}\|_{-1}\|\nabla v_{h}\|
+ C\|\nabla(u^{n+1}-2u^{n}+u^{n-1})\|\|\nabla u^{n+1}\|\|\nabla v_{h}\|
\\&\leq
\frac{1}{2\sigma}\|\frac{3u^{n+1} - 4u^{n} + u^{n-1}}{2\Delta t} - u_{t}^{n+1}\|_{-1}^{2}
\\&\quad
+ \frac{C}{2\sigma}\|\nabla u^{n+1}\|^{2}\|\nabla(u^{n+1}-2u^{n}+u^{n-1})\|^{2}
+ \sigma\|\nabla v_{h}\|^{2}
\\&\leq
\frac{C}{2\sigma}\Delta t^{3} \Big(\int^{t^{n+1}}_{t^{n-1}}\|u_{ttt}\|_{-1}^{2}dt
+ \int^{t^{n+1}}_{t^{n-1}}\|\nabla u_{tt}\|^{2}dt \Big)
+ \sigma\|\nabla v_{h}\|^{2},
\end{aligned}\end{aligned}$$ where we use the Cauchy-Schwarz-Young inequality and Lemma $\ref{lemma1}$.
Once again, we require a key lemma, regarding Step 2, to prove convergence.
The following inequality holds. $$\begin{aligned}
\label{erroranalysis1}
\begin{aligned}
\|\psi^{n+1}_{h}\|^{2}
&\geq
\|\phi^{n+1}_{h}\|^{2} + \|\phi^{n+1}_{h}-\psi^{n+1}_{h}\|^{2}
+ \frac{\beta}{3} \Big( \|\nabla\cdot\phi^{n+1}_{h}\|^{2} - \|\nabla\cdot\phi^{n}_{h}\|^{2}
+ \|\nabla\cdot(2\phi^{n+1}_{h}-\phi^{n}_{h})\|^{2}
\\&\quad
- \|\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h})\|^{2}
+ \frac{1}{2}\|\nabla\cdot(\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h})\|^{2} \Big)
+ \frac{2\gamma\Delta t}{3}\|\nabla\cdot\phi^{n+1}_{h}\|^{2}
\\&\quad
- \frac{C\beta d(1+2\Delta t)}{3}\int^{t^{n+1}}_{t^{n-1}}\|\nabla\eta_{t}\|^{2}dt
- \frac{\beta\Delta t}{3}\|\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h})\|
- \frac{2\gamma d\Delta t}{3}\|\nabla\eta^{n+1}\|^{2}.
\end{aligned}\end{aligned}$$
At time $t^{n+1}$, for all $v_{h} \in X_{h}$, the true solution $u$ satisfies $$\begin{aligned}
\label{pf4.1}
\begin{aligned}
&(\frac{3u^{n+1}-3u^{n+1}}{2\Delta t},v_{h})
+ \beta (\nabla \cdot \frac{3u^{n+1} - 4u^{n}
+ u^{n-1}}{2\Delta t}, \nabla \cdot v_{h})
+ \gamma (\nabla \cdot u^{n+1}, \nabla \cdot v_{h}) = 0.
\end{aligned}\end{aligned}$$ Subtracting ($\ref{pf4.1}$) from ($\ref{step2}$), we have $$\begin{aligned}
\label{pf4.2}
\begin{aligned}
&(\frac{3e_{u}^{n+1} - 3e_{\hat{u}}^{n+1}}{2\Delta t},v_{h})
+ \beta (\nabla \cdot \frac{3e_{u}^{n+1} - 4e_{u}^{n}
+ e_{u}^{n-1}}{2\Delta t}, \nabla \cdot v_{h})
+ \gamma (\nabla \cdot e_{u}^{n+1}, \nabla \cdot v_{h}) = 0.
\end{aligned}\end{aligned}$$ Setting $v_{h}=\phi^{n+1}_{h}$ in ($\ref{pf4.2}$), using similar identities as in Theorem \[stability\_lemma\], and rearranging terms yields $$\begin{aligned}
\label{pf4.3}
\begin{aligned}
\|\psi^{n+1}_{h}\|^{2}
&=
\|\phi^{n+1}_{h}\|^{2} + \|\phi^{n+1}_{h}-\psi^{n+1}_{h}\|^{2}
+ \frac{\beta}{3} \Big( \|\nabla\cdot\phi^{n+1}_{h}\|^{2} - \|\nabla\cdot\phi^{n}_{h}\|^{2}
+ \|\nabla\cdot(2\phi^{n+1}_{h}-\phi^{n}_{h})\|^{2}
\\&\quad
- \|\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h})\|^{2}
+ \|\nabla\cdot(\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h})\|^{2} \Big)
+ \frac{4\gamma\Delta t}{3}\|\nabla\cdot\phi^{n+1}_{h}\|^{2}
\\&\quad
- \frac{2\beta}{3}(\nabla\cdot(3\eta^{n+1}-4\eta^{n}+\eta^{n-1}),\nabla\cdot\phi^{n+1}_{h})
- \frac{4\gamma\Delta t}{3}(\nabla\cdot\eta^{n+1},\nabla\cdot\phi^{n+1}_{h}).
\end{aligned}\end{aligned}$$ Split $-\frac{2\beta}{3}(\nabla\cdot(3\eta^{n+1}-4\eta^{n}+\eta^{n-1}),\nabla\cdot\phi^{n+1}_{h})$ into $ - \frac{2\beta}{3}(\nabla\cdot(3\eta^{n+1}-4\eta^{n}+\eta^{n-1}),\nabla \cdot (\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h})) -\frac{2\beta}{3}(\nabla\cdot(3\eta^{n+1}-4\eta^{n}+\eta^{n-1}),\nabla \cdot (2\phi^{n}_{h}-\phi^{n-1}_{h}))$. Using the Cauchy-Schwarz-Young inequality and Lemma $\ref{lemma1}$. Then, the following three inequalities hold, $$\begin{aligned}
\label{pf4.4}
\begin{aligned}
&|\frac{2\beta}{3}(\nabla\cdot(3\eta^{n+1}-4\eta^{n}+\eta^{n-1}),\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h}))|
\\&\leq
\frac{2\beta\sqrt{d}}{3}\|\nabla(3\eta^{n+1}-4\eta^{n}+\eta^{n-1})\|\|\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h})\|
\\&\leq
\frac{C\beta d}{3}\int^{t^{n+1}}_{t^{n-1}}\|\nabla\eta_{t}\|^{2}dt
+ \frac{\beta\Delta t}{3}\|\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h})\|^{2}
\\&\leq
\frac{C\beta d}{3}\int^{t^{n+1}}_{t^{n-1}}\|\nabla\eta_{t}\|^{2}dt
+ \frac{\beta\Delta t}{3}\big(\|\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h})\|^{2} + \|\nabla \cdot \phi^{n}_{h}\|^{2}\big),
\end{aligned}\end{aligned}$$ $$\begin{aligned}
\label{pf4.5}
\begin{aligned}
&|\frac{2\beta}{3}(\nabla\cdot(3\eta^{n+1}-4\eta^{n}+\eta^{n-1}),\nabla\cdot(\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h}))|
\\&\leq
\frac{2\beta\sqrt{d}}{3}\|\nabla(3\eta^{n+1}-4\eta^{n}+\eta^{n-1})\|\|\nabla\cdot(\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h})\|
\\&\leq
\frac{2C\beta d\Delta t}{3}\int^{t^{n+1}}_{t^{n-1}}\|\nabla\eta_{t}\|^{2}dt
+ \frac{\beta}{6}\|\nabla\cdot(\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h})\|^{2},
\end{aligned}\end{aligned}$$ and $$\begin{aligned}
\label{pf4.6}
\begin{aligned}
|\frac{4\gamma\Delta t}{3}(\nabla\cdot\eta^{n+1},\nabla\cdot\phi^{n+1}_{h})|
&\leq
\frac{4\gamma\sqrt{d}\Delta t}{3}\|\nabla\eta^{n+1}\|\|\nabla\cdot\phi^{n+1}_{h}\|
\\&\leq
\frac{2\gamma d\Delta t}{3}\|\nabla\eta^{n+1}\|^{2}
+ \frac{2\gamma\Delta t}{3}\|\nabla\cdot\phi^{n+1}_{h}\|.
\end{aligned}\end{aligned}$$ Combining ($\ref{pf4.3}$) - ($\ref{pf4.6}$) completes the proof.
Next, we give the main error result for *BDF2-mgd* when $ \beta>0 $.
\[error\] Assume the true solution $u, p$ satisfy ($\ref{regularity1}$) and the following regularity $$\begin{aligned}
\label{regularity2}
\begin{aligned}
&u\in L^{\infty}(0,T;H^{k+1}(\Omega)^{d})\cap L^{2}(0,T;H^{k+1}(\Omega)^{d}), \\
&u_{t}\in L^{2}(0,T;H^{k+1}(\Omega)^{d}), \quad
p\in L^{2}(0,T;H^{m+1}(\Omega)).
\end{aligned}\end{aligned}$$ Then, we have the following estimates for *BDF2-mgd*. $$\begin{aligned}
\label{erroranalysis2}
\begin{aligned}
&\|e_{u}^{N}\|^{2} + \|2e_{u}^{N}-e_{u}^{N-1}\|^{2}
+ (\frac{2\gamma\Delta t}{3} + \beta)
\Big( \|\nabla\cdot e_{u}^{N}\|^{2} + \|\nabla\cdot(2e_{u}^{N}-e_{u}^{N-1})\|^{2} \Big)
\\&\quad
+ 2\nu \Delta t \sum^{N-1}_{n=1}\|\nabla e_{\hat{u}}^{n+1}\|^{2}
+ 2\gamma\Delta t \sum^{N-1}_{n=1}\|\nabla\cdot e_{u}^{n+1}\|^{2}
\\&\leq
C\exp(C^{*}T)\bigg\{\inf\limits_{v_{h}\in X_{h}}
\Big( \beta (1+\Delta t) \|\nabla(u - v_{h})_{t}\|_{2,0}^{2}
+ \frac{1}{\nu}\|(u - v_{h})_{t}\|_{2,0}^{2}
\\&\quad
+ (\frac{\gamma^{2}\Delta t}{\beta} + \gamma + \nu + \frac{1}{\nu}) |\|\nabla(u - v_{h})\||_{2,0}^{2}
+ (\frac{2\gamma\Delta t}{3} + \beta + \frac{1}{\nu^{2}}) \|\nabla(u - v_{h})\|_{\infty,0}^{2}
\\&\quad
+ \|u - v_{h}\|_{\infty,0}^{2} \Big)
+ \frac{1}{\nu} \inf\limits_{q_{h}\in Q_{h}} |\|p - q_{h}\||_{2,0}^{2}
+ \frac{1}{\nu}\Delta t^{4}
\\&\quad
+ \|e_{u}^{1}\|^{2} + \|2e_{u}^{1}-e_{u}^{0}\|^{2}
+ (\frac{2\gamma\Delta t}{3} + \beta)
\Big( \|\nabla\cdot e_{u}^{1}\|^{2}
+ \|\nabla\cdot(2e_{u}^{1}-e_{u}^{0})\|^{2} \Big)\bigg\}.
\end{aligned}\end{aligned}$$
At time $t^{n+1}$, the true solution $u, p$ satisfies $$\begin{aligned}
&(\frac{3u^{n+1} - 4u^{n} + u^{n-1}}{2\Delta t},v_{h})
+ b(2u^{n}-u^{n-1},u^{n+1},v_{h})
+ \nu (\nabla u^{n+1}, \nabla v_{h}) \nonumber \\
& - (p^{n+1},\nabla \cdot v_{h})
= (f^{n+1},v_{h}) + \tau^{n+1}(v_{h}) \quad \forall v_{h} \in X_{h}, \label{pf5.1} \\
&(\nabla \cdot u^{n+1}, q_{h})=0 \quad \forall q_{h} \in Q_{h}. \label{pf5.2}\end{aligned}$$ Subtracting ($\ref{step1.1}$) and ($\ref{step1.2}$) from ($\ref{pf5.1}$) and ($\ref{pf5.2}$), respectively, we have $$\begin{aligned}
&(\frac{3e_{\hat{u}}^{n+1} - 4e_{u}^{n} + e_{u}^{n-1}}{2\Delta t},v_{h})
+ b(2u^{n}-u^{n-1},u^{n+1},v_{h})
- b(2u^{n}_{h}-u^{n-1}_{h},\hat{u}^{n+1}_{h},v_{h}) \nonumber \\
& + \nu (\nabla e_{\hat{u}}^{n+1}, \nabla v_{h})
- (e_{p}^{n+1},\nabla \cdot v_{h})
= \tau^{n+1}(v_{h}) \quad \forall v_{h} \in X_{h}, \label{pf5.3} \\
&(\nabla \cdot e_{\hat{u}}^{n+1}, q_{h})=0 \quad \forall q_{h} \in Q_{h}. \label{pf5.4}\end{aligned}$$ Set $v_{h}=\psi^{n+1}_{h} \in V_{h}$ in equation ($\ref{pf5.3}$), then $$\begin{aligned}
\label{pf5.5}
\begin{aligned}
&(\frac{3\eta^{n+1} - 4\eta^{n} + \eta^{n-1}}{2\Delta t},\psi^{n+1}_{h})
- (\frac{3\phi^{n+1}_{h} - 4\phi^{n}_{h} + \phi^{n-1}_{h}}{2\Delta t},\phi^{n+1}_{h})
\\&
- (\frac{3\phi^{n+1}_{h} - 4\phi^{n}_{h} + \phi^{n-1}_{h}}{2\Delta t},\psi^{n+1}_{h}-\phi^{n+1}_{h})
- (\frac{3\psi^{n+1}_{h} - 3\phi^{n+1}_{h}}{2\Delta t},\psi^{n+1}_{h})
\\&
+ b(2u^{n}-u^{n-1},u^{n+1},\psi^{n+1}_{h})
- b(2u^{n}_{h}-u^{n-1}_{h},\hat{u}^{n+1}_{h},\psi^{n+1}_{h})
\\&
+ \nu (\nabla\eta^{n+1}, \nabla\psi^{n+1}_{h})
- \nu \|\nabla\psi^{n+1}_{h}\|^{2}
- (p^{n+1} - q_{h},\nabla \cdot \psi^{n+1}_{h})
= \tau^{n+1}(\psi^{n+1}_{h}).
\end{aligned}\end{aligned}$$ Here, $q_{h}\in Q_{h}$ is arbitrary. Furthermore, setting $v_{h}=\frac{3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}}{3} \in V_{h}$ in ($\ref{pf4.2}$) and rearranging terms yields $$\begin{aligned}
\label{pf5.6}
\begin{aligned}
&(\frac{3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}}{2\Delta t},\psi^{n+1}_{h}-\phi^{n+1}_{h})
\\&
= \frac{\gamma}{3}(\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}),\nabla\cdot\phi^{n+1}_{h})
+ \frac{\beta}{6\Delta t}\|\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1})\|^{2}
\\&\quad
- \frac{\gamma}{3}(\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}),\nabla\cdot\eta^{n+1})
\\&\quad
- \frac{\beta}{6\Delta t}(\nabla\cdot(3\eta^{n+1} - 4\eta^{n} + \eta^{n-1}),\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1})).
\end{aligned}\end{aligned}$$ Combine ($\ref{pf5.5}$) and ($\ref{pf5.6}$) and rearrange. Then, $$\begin{aligned}
\label{pf5.7}
\begin{aligned}
&(\frac{3\phi^{n+1}_{h} - 4\phi^{n}_{h} + \phi^{n-1}_{h}}{2\Delta t},\phi^{n+1}_{h})
+ \frac{\gamma}{3}(\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}),\nabla\cdot\phi^{n+1})
\\&\quad
+ \frac{\beta}{6\Delta t}\|\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1})\|^{2}
+ \nu \|\nabla\psi^{n+1}_{h}\|^{2}
\\&\quad
+ \frac{3}{4\Delta t}(\|\psi^{n+1}_{h}\|^{2} - \|\phi^{n+1}_{h}\|^{2} + \|\psi^{n+1}_{h}-\phi^{n+1}_{h}\|^{2})
\\&=
(\frac{3\eta^{n+1} - 4\eta^{n} + \eta^{n-1}}{2\Delta t},\psi^{n+1}_{h})
+ \frac{\gamma}{3}(\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}),\nabla\cdot\eta^{n+1})
\\&\quad
+ \frac{\beta}{6\Delta t}(\nabla\cdot(3\eta^{n+1} - 4\eta^{n} + \eta^{n-1}),\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}))
\\&\quad
+ b(2u^{n}-u^{n-1},u^{n+1},\psi^{n+1}_{h})
- b(2u^{n}_{h}-u^{n-1}_{h},\hat{u}^{n+1}_{h},\psi^{n+1}_{h})
\\&\quad
+ \nu (\nabla\eta^{n+1}, \nabla\psi^{n+1}_{h})
- (p^{n+1} - q_{h},\nabla \cdot \psi^{n+1}_{h})
- \tau^{n+1}(\psi^{n+1}_{h}).
\end{aligned}\end{aligned}$$ Multiplying ($\ref{pf5.7}$) by $4\Delta t$ and use ($\ref{erroranalysis1}$). Then, $$\begin{aligned}
\label{pf5.8}
\begin{aligned}
&\|\phi^{n+1}_{h}\|^{2} - \|\phi^{n}_{h}\|^{2}
+ \|2\phi^{n+1}_{h}-\phi^{n}_{h}\|^{2}
- \|2\phi^{n}_{h}-\phi^{n-1}_{h}\|^{2}
+ \|\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h}\|^{2}
\\&\quad
+ (\frac{2\gamma\Delta t}{3} + \beta) \Big( \|\nabla\cdot\phi^{n+1}_{h}\|^{2} - \|\nabla\cdot\phi^{n}_{h}\|^{2}
+ \|\nabla\cdot(2\phi^{n+1}_{h}-\phi^{n}_{h})\|^{2}
\\&\quad
- \|\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h})\|^{2}
+ \|\nabla\cdot(\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h})\|^{2} \Big)
\\&\quad
+ \frac{2\beta}{3}\|\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1})\|^{2}
+ 6 \|\psi^{n+1}_{h}-\phi^{n+1}_{h}\|^{2}
\\&\quad
+ 4\nu \Delta t \|\nabla\psi^{n+1}_{h}\|^{2}
+ 2\gamma\Delta t\|\nabla\cdot\phi^{n+1}_{h}\|^{2}
\\&\leq
2(3\eta^{n+1} - 4\eta^{n} + \eta^{n-1},\psi^{n+1}_{h})
+ \frac{4\gamma\Delta t}{3}(\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}),\nabla\cdot\eta^{n+1})
\\&\quad
+ \frac{2\beta}{3}(\nabla\cdot(3\eta^{n+1} - 4\eta^{n} + \eta^{n-1}),\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}))
\\&\quad
+ 4\Delta t b(2u^{n}-u^{n-1},u^{n+1},\psi^{n+1}_{h})
- 4\Delta t b(2u^{n}_{h}-u^{n-1}_{h},\hat{u}^{n+1}_{h},\psi^{n+1}_{h})
\\&\quad
+ 4\nu\Delta t (\nabla\eta^{n+1}, \nabla\psi^{n+1}_{h})
- 4\Delta t(p^{n+1} - q_{h},\nabla \cdot \psi^{n+1}_{h})
- 4\Delta t\tau^{n+1}(\psi^{n+1}_{h})
\\&\quad
+ C\beta d(1+2 \Delta t)\int^{t^{n+1}}_{t^{n-1}}\|\nabla\eta_{t}\|^{2}dt
+ 2\gamma d\Delta t\|\nabla\eta^{n+1}\|^{2}
\\&\quad
+ \frac{\beta}{2} \|\nabla\cdot(\phi^{n+1}_{h}-2\phi^{n+1}_{h}+\phi^{n}_{h})\|^{2}
+ \beta \Delta t \big( \|\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h})\|^{2}
+ \|\nabla\cdot\phi^{n}_{h}\|^{2} \big).
\end{aligned}\end{aligned}$$
Next, we need to bound the terms on the right hand side of ($\ref{pf5.8}$). Applying Lemma $\ref{lemma1}$, the Poincaré-Friedrichs inequality, and the Cauchy-Schwarz-Young inequality, for an arbitrary $\delta>0$, we have $$\begin{aligned}
\label{pf5.9}
\begin{aligned}
2(3\eta^{n+1} - 4\eta^{n} + \eta^{n-1},\psi^{n+1}_{h})
&\leq
C\|3\eta^{n+1} - 4\eta^{n} + \eta^{n-1}\|\|\nabla\psi^{n+1}_{h}\|
\\&\leq
\frac{C}{\delta\nu}\int^{t^{n+1}}_{t^{n-1}}\|\eta_{t}\|^{2}dt + \delta\nu \Delta t \|\nabla\psi^{n+1}_{h}\|^{2}.
\end{aligned}\end{aligned}$$ $$\begin{aligned}
\label{pf5.10}
\begin{aligned}
&\frac{4\gamma\Delta t}{3}(\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}),\nabla\cdot\eta^{n+1})
\\&\leq
\frac{4\gamma\sqrt{d}\Delta t}{3}\|\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1})\|\|\nabla\eta^{n+1}\|
\\&\leq
\frac{\beta}{3}\|\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1})\|^{2}
+ \frac{4 d\gamma^{2}\Delta t^{2}}{3\beta}\|\nabla\eta^{n+1}\|^{2}.
\end{aligned}\end{aligned}$$ $$\begin{aligned}
\label{pf5.11}
\begin{aligned}
&\frac{2\beta}{3}(\nabla\cdot(3\eta^{n+1} - 4\eta^{n} + \eta^{n-1}),\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}))
\\&\leq
\frac{2 \beta\sqrt{d}}{3}
\|\nabla(3\eta^{n+1} - 4\eta^{n} + \eta^{n-1})\|\|\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1})\|
\\&\leq
\frac{\beta}{3}\|\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1})\|^{2}
+ \frac{C\beta d \Delta t}{3}\int^{t^{n+1}}_{t^{n-1}}\|\nabla\eta_{t}\|^{2}dt.
\end{aligned}\end{aligned}$$ Furthermore, $$\begin{aligned}
\label{pf5.12}
\begin{aligned}
&4\nu\Delta t (\nabla\eta^{n+1}, \nabla\psi^{n+1}_{h})
\leq
\frac{4\nu\Delta t}{\delta}\|\nabla\eta^{n+1}\|^{2} + \delta\nu \Delta t \|\nabla\psi^{n+1}_{h}\|^{2}.
\end{aligned}\end{aligned}$$ $$\begin{aligned}
\label{pf5.13}
\begin{aligned}
&- 4\Delta t(p^{n+1} - q_{h},\nabla \cdot \psi^{n+1}_{h})
\leq
\frac{4d\Delta t}{\delta\nu}\|p^{n+1} - q_{h}\|^{2} + \delta\nu \Delta t \|\nabla\psi^{n+1}_{h}\|^{2}.
\end{aligned}\end{aligned}$$ Applying Lemma $\ref{c-errors-bound}$ yields $$\begin{aligned}
\label{pf5.14}
\begin{aligned}
&- 4\Delta t\tau^{n+1}(\psi^{n+1}_{h})
\\&\leq
\frac{C\Delta t^{4}}{\delta\nu}\int^{t^{n+1}}_{t^{n-1}}\|u_{ttt}\|_{-1}^{2}dt
+ \frac{C\Delta t^{4}}{\delta\nu}\int^{t^{n+1}}_{t^{n-1}}\|\nabla u_{tt}\|^{2}dt
+ \delta\nu\Delta t\|\nabla\psi^{n+1}_{h}\|^{2}.
\end{aligned}\end{aligned}$$ For the nonlinear terms, we treat them as follows. Adding and subtracting $4\Delta t b(2u^{n}_{h}-u^{n-1}_{h},u^{n+1},\psi^{n+1}_{h})$ yields $$\begin{aligned}
\label{pf5.15}
\begin{aligned}
&4\Delta t b(2u^{n}-u^{n-1},u^{n+1},\psi^{n+1}_{h})
- 4\Delta t b(2u^{n}_{h}-u^{n-1}_{h},\hat{u}^{n+1}_{h},\psi^{n+1}_{h})
\\&=
4\Delta t \Big( b(2\eta^{n}-\eta^{n-1},u^{n+1},\psi^{n+1}_{h})
- b(2\phi^{n}_{h}-\phi^{n-1}_{h},u^{n+1},\psi^{n+1}_{h})
\\&\quad
+ b(2\hat{u}_{h}^{n}-\hat{u}_{h}^{n-1},\eta^{n+1},\psi^{n+1}_{h}) \Big).
\end{aligned}\end{aligned}$$ Then, $$\begin{aligned}
\label{pf5.16}
\begin{aligned}
&4\Delta t b(2\eta^{n}-\eta^{n-1},u^{n+1},\psi^{n+1}_{h})
\leq
4C\Delta t \|\nabla(2\eta^{n}-\eta^{n-1})\|\|\nabla u^{n+1}\|\|\nabla\psi^{n+1}_{h}\|
\\&\leq
\frac{4C\Delta t}{\delta\nu} \|\nabla(2\eta^{n}-\eta^{n-1})\|^{2}\|\nabla u^{n+1}\|^{2}
+ \delta\nu\Delta t\|\nabla\psi^{n+1}_{h}\|^{2}
\\&\leq
\frac{16C\Delta t}{\delta\nu} ( \|\nabla\eta^{n}\|^{2} + \|\nabla\eta^{n-1}\|^{2} ) \|\nabla u\|_{\infty,0}^{2}
+ \delta\nu\Delta t\|\nabla\psi^{n+1}_{h}\|^{2},
\end{aligned}\end{aligned}$$ $$\begin{aligned}
\label{pf5.17}
\begin{aligned}
&-4\Delta t b(2\phi^{n}_{h}-\phi^{n-1}_{h},u^{n+1},\psi^{n+1}_{h})
\leq
4C\Delta t \|2\phi^{n}_{h}-\phi^{n-1}_{h}\|\|u^{n+1}\|_{2}\|\nabla\psi^{n+1}_{h}\|
\\&\leq
\frac{4C\Delta t}{\delta\nu} \|2\phi^{n}_{h}-\phi^{n-1}_{h}\|^{2}\|u^{n+1}\|_{2}^{2}
+ \delta\nu\Delta t\|\nabla\psi^{n+1}_{h}\|^{2}
\\&\leq
\frac{8C\Delta t}{\delta\nu} ( \|2\phi^{n}_{h}-\phi^{n-1}_{h}\|^{2} + \|\phi^{n}_{h}\|^{2}) \|u^{n+1}\|_{2}^{2}
+ \delta\nu\Delta t\|\nabla\psi^{n+1}_{h}\|^{2},
\end{aligned}\end{aligned}$$ $$\begin{aligned}
\label{pf5.18}
\begin{aligned}
&4\Delta t b(2\hat{u}_{h}^{n}-\hat{u}_{h}^{n-1},\eta^{n+1},\psi^{n+1}_{h})
\leq
4C\Delta t \|\nabla(2\hat{u}_{h}^{n}-\hat{u}_{h}^{n-1})\|\|\nabla\eta^{n+1}\|\|\nabla\psi^{n+1}_{h}\|
\\&\leq
\frac{4C\Delta t}{\delta\nu} \|\nabla(2\hat{u}_{h}^{n}-\hat{u}_{h}^{n-1})\|^{2}\|\nabla\eta^{n+1}\|^{2}
+ \delta\nu\Delta t\|\nabla\psi^{n+1}_{h}\|^{2}
\\&\leq
\frac{16C\Delta t}{\delta\nu} ( \|\nabla\hat{u}_{h}^{n}\|^{2} + \|\nabla\hat{u}_{h}^{n-1}\|^{2} ) \|\nabla\eta\|_{\infty,0}^{2}
+ \delta\nu\Delta t\|\nabla\psi^{n+1}_{h}\|^{2}.
\end{aligned}\end{aligned}$$ Setting $\delta= \frac{2}{7}$ and using the estimates ($\ref{pf5.9}$)-($\ref{pf5.18}$) in ($\ref{pf5.8}$) yields $$\begin{aligned}
\label{pf5.19}
\begin{aligned}
&\|\phi^{n+1}_{h}\|^{2} - \|\phi^{n}_{h}\|^{2}
+ \|2\phi^{n+1}_{h}-\phi^{n}_{h}\|^{2}
- \|2\phi^{n}_{h}-\phi^{n-1}_{h}\|^{2}
+ \|\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h}\|^{2}
\\&\quad
+ (\frac{2\gamma\Delta t}{3} + \beta) \Big( \|\nabla\cdot\phi^{n+1}_{h}\|^{2} - \|\nabla\cdot\phi^{n}_{h}\|^{2}
+ \|\nabla\cdot(2\phi^{n+1}_{h}-\phi^{n}_{h})\|^{2}
- \|\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h})\|^{2}
\\&\quad
+ \frac{1}{2}\|\nabla\cdot(\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h})\|^{2} \Big)
+ 6 \|\psi^{n+1}_{h}-\phi^{n+1}_{h}\|^{2}
+ 2\nu \Delta t \|\nabla\psi^{n+1}_{h}\|^{2}
+ 2\gamma\Delta t\|\nabla\cdot\phi^{n+1}_{h}\|^{2}
\\&\leq
\frac{C\Delta t}{\nu} \|u^{n+1}\|_{2}^{2} ( \|2\phi^{n}_{h}-\phi^{n-1}_{h}\|^{2} + \|\phi^{n}_{h}\|^{2}) + \beta \Delta t (\|\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h})\|^{2} + \|\nabla\cdot\phi^{n}_{h}\|^{2})
\\&\quad
+ {C\beta d \Delta t}\int^{t^{n+1}}_{t^{n-1}}\|\nabla\eta_{t}\|^{2}dt
+ \frac{C}{\nu}\int^{t^{n+1}}_{t^{n-1}}\|\eta_{t}\|^{2}dt
+ C(\frac{d\gamma^{2}\Delta t}{\beta} + \nu)\Delta t\|\nabla\eta^{n+1}\|^{2}
\\&\quad
+ \frac{Cd\Delta t}{\nu}\|p^{n+1} - q_{h}\|^{2}
+ \frac{C\Delta t^{4}}{\nu}\int^{t^{n+1}}_{t^{n-1}}\|u_{ttt}\|_{-1}^{2}dt
+ \frac{C\Delta t^{4}}{\nu}\int^{t^{n+1}}_{t^{n-1}}\|\nabla u_{tt}\|^{2}dt
\\&\quad
+ \frac{C\Delta t}{\nu} ( \|\nabla\eta^{n}\|^{2} + \|\nabla\eta^{n-1}\|^{2} )
+ \frac{C\Delta t}{\nu} ( \|\nabla\hat{u}_{h}^{n}\|^{2} + \|\nabla\hat{u}_{h}^{n-1}\|^{2} ) \|\nabla\eta\|_{\infty,0}^{2}.
\end{aligned}\end{aligned}$$ Sum ($\ref{pf5.19}$) from $n=1$ to $N-1$ to get $$\begin{aligned}
\label{pf5.20}
\begin{aligned}
&\|\phi^{N}_{h}\|^{2} + \|2\phi^{N}_{h}-\phi^{N-1}_{h}\|^{2}
+ (\frac{2\gamma\Delta t}{3} + \beta)
\Big( \|\nabla\cdot\phi^{N}_{h}\|^{2} + \|\nabla\cdot(2\phi^{N}_{h}-\phi^{N-1}_{h})\|^{2} \Big)
\\&\quad
+ \frac{1}{2}\sum^{N-1}_{n=1}\|\nabla\cdot(\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h})\|^{2}
+ 2\nu \Delta t \sum^{N-1}_{n=1}\|\nabla\psi^{n+1}_{h}\|^{2}
+ 2\gamma\Delta t \sum^{N-1}_{n=1}\|\nabla\cdot\phi^{n+1}_{h}\|^{2}
\\&\leq
\Delta t\sum^{N-1}_{n=1} \Big(C\nu^{-1}\|u^{n+1}\|_{2}^{2} ( \|\phi^{n}_{h}\|^{2} + \|2\phi^{n}_{h}-\phi^{n-1}_{h}\|^{2} ) + \beta (\|\nabla\cdot\phi^{n}_{h}\|^{2} + \|\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h})\|^{2}) \Big)
\\&\quad
+ \frac{C\beta d \Delta t}{3}\|\nabla\eta_{t}\|_{2,0}^{2}
+ \frac{C}{\nu}\|\eta_{t}\|_{2,0}^{2}
+ C(\frac{d\gamma^{2}\Delta t}{\beta} + \nu + \frac{1}{\nu}) |\|\nabla\eta\||_{2,0}^{2}
\\&\quad
+ \frac{Cd}{\nu} |\|p - q_{h}\||_{2,0}^{2}
+ \frac{C\Delta t^{4}}{\nu}\|u_{ttt}\|_{2,-1}^{2}
+ \frac{C\Delta t^{4}}{\nu}\|\nabla u_{tt}\|_{2,0}^{2}
+ \frac{C}{\nu^{2}}( \nu\Delta t\sum^{N-1}_{n=0} \|\nabla\hat{u}_{h}^{n}\|^{2} ) \|\nabla\eta\|_{\infty,0}^{2}
\\&\quad
+ \|\phi^{1}_{h}\|^{2} + \|2\phi^{1}_{h}-\phi^{0}_{h}\|^{2}
+ (\frac{2\gamma\Delta t}{3} + \beta)
\Big( \|\nabla\cdot\phi^{1}_{h}\|^{2} + \|\nabla\cdot(2\phi^{1}_{h}-\phi^{0}_{h})\|^{2} \Big).
\end{aligned}\end{aligned}$$ Denote $C^{*}=\max\{\frac{C}{\nu}|\|u\||_{2,2}^{2},1\}$. Then, Lemma \[gronwall2\], the boundedness of $\nu\Delta t \sum\limits^{N-1}_{n=1} \|\nabla\hat{u}_{h}^{n+1}\|^{2}$ (Theorem $\ref{stability}$), and taking infimums over $V_{h}$ and $Q_{h}$ yield $$\begin{aligned}
\label{pf5.21}
\begin{aligned}
&\|\phi^{N}_{h}\|^{2} + \|2\phi^{N}_{h}-\phi^{N-1}_{h}\|^{2}
+ (\frac{2\gamma\Delta t}{3} + \beta)
\Big( \|\nabla\cdot\phi^{N}_{h}\|^{2} + \|\nabla\cdot(2\phi^{N}_{h}-\phi^{N-1}_{h})\|^{2} \Big)
\\&\quad
+ \frac{1}{2}\sum^{N-1}_{n=1}\|\nabla\cdot(\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h})\|^{2}
+ 2\nu \Delta t \sum^{N-1}_{n=1}\|\nabla\psi^{n+1}_{h}\|^{2}
+ 2\gamma\Delta t \sum^{N-1}_{n=1}\|\nabla\cdot\phi^{n+1}_{h}\|^{2}
\\&\leq
C\exp(C^{*}T)\bigg\{\inf\limits_{v_{h}\in V_{h}}
\Big( \beta(1+\Delta t) \|\nabla(u - v_{h})_{t}\|_{2,0}^{2}
+ \frac{1}{\nu}\|(u - v_{h})_{t}\|_{2,0}^{2}
\\&\quad
+ (\frac{\gamma^{2}\Delta t}{\beta} + \nu + \frac{1}{\nu}) |\|\nabla(u - v_{h})\||_{2,0}^{2}
+ \frac{1}{\nu^{2}} \|\nabla(u - v_{h})\|_{\infty,0}^{2} \Big)
\\&\quad
+ \frac{1}{\nu} \inf\limits_{q_{h}\in Q_{h}} |\|p - q_{h}\||_{2,0}^{2}
+ \frac{1}{\nu}\Delta t^{4}
\\&\quad
+ \|\phi^{1}_{h}\|^{2} + \|2\phi^{1}_{h}-\phi^{0}_{h}\|^{2}
+ (\frac{2\gamma\Delta t}{3} + \beta)
\Big( \|\nabla\cdot\phi^{1}_{h}\|^{2} + \|\nabla\cdot(2\phi^{1}_{h}-\phi^{0}_{h})\|^{2} \Big)\bigg\}.
\end{aligned}\end{aligned}$$ Then, using Lemma \[lemma0\] and the triangle inequality completes the proof.
The above result has dependence on $\beta^{-1}$. Consequently, we consider the convergency of *BDF2-mgd* when $ \beta=0 $ separately.
\[error0\] Assume the true solution $u, p$ satisfy ($\ref{regularity1}$) and ($\ref{regularity2}$). Then, when $ \beta=0 $, we have the following estimates for *BDF2-mgd*. $$\begin{aligned}
\label{erroranalysis0}
\begin{aligned}
&\|e_{u}^{N}\|^{2} + \|2e_{u}^{N}-e_{u}^{N-1}\|^{2}
+ \frac{2\gamma\Delta t}{3}
\Big( \|\nabla\cdot e_{u}^{N}\|^{2} + \|\nabla\cdot(2e_{u}^{N}-e_{u}^{N-1})\|^{2} \Big)
\\&\quad
+ 2\nu \Delta t \sum^{N-1}_{n=1}\|\nabla e_{\hat{u}}^{n+1}\|^{2}
+ 2\gamma\Delta t \sum^{N-1}_{n=1}\|\nabla\cdot e_{u}^{n+1}\|^{2}
\\&\leq
C\exp(C^{**}T)
\bigg\{\inf\limits_{v_{h}\in X_{h}}
\Big( \frac{1}{\nu}\|(u - v_{h})_{t}\|_{2,0}^{2}
+ \|u - v_{h}\|_{\infty,0}^{2}
+ ( \gamma + \nu + \frac{1}{\nu}) |\|\nabla(u - v_{h})\||_{2,0}^{2}
\\&\quad
+ (\frac{2\gamma\Delta t}{3} + \frac{1}{\nu^{2}}) \|\nabla(u - v_{h})\|_{\infty,0}^{2} \Big)
+ \frac{1}{\nu} \inf\limits_{q_{h}\in Q_{h}} |\|p - q_{h}\||_{2,0}^{2}
+ \frac{1}{\nu}\Delta t^{4}
\\&\quad
+ \|e_{u}^{1}\|^{2} + \|2e_{u}^{1}-e_{u}^{0}\|^{2}
+ \gamma\Delta t
\|\nabla\cdot e_{u}^{1}\|^{2}
+ \gamma\Delta t \|\nabla\cdot(2e_{u}^{1}-e_{u}^{0})\|^{2} \bigg\}.
\end{aligned}
\end{aligned}$$
Similar to ($ \ref{pf5.8} $), we have $$\begin{aligned}
\label{pf6.1}
\begin{aligned}
&\|\phi^{n+1}_{h}\|^{2} - \|\phi^{n}_{h}\|^{2}
+ \|2\phi^{n+1}_{h}-\phi^{n}_{h}\|^{2}
- \|2\phi^{n}_{h}-\phi^{n-1}_{h}\|^{2}
+ \|\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h}\|^{2}
\\&\quad
+ \frac{2\gamma\Delta t}{3} \Big( \|\nabla\cdot\phi^{n+1}_{h}\|^{2} - \|\nabla\cdot\phi^{n}_{h}\|^{2}
+ \|\nabla\cdot(2\phi^{n+1}_{h}-\phi^{n}_{h})\|^{2}
\\&\quad
- \|\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h})\|^{2}
+ \|\nabla\cdot(\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h})\|^{2} \Big)
\\&\quad
+ 6 \|\psi^{n+1}_{h}-\phi^{n+1}_{h}\|^{2}
+ 4\nu \Delta t \|\nabla\psi^{n+1}_{h}\|^{2}
+ 2\gamma\Delta t\|\nabla\cdot\phi^{n+1}_{h}\|^{2}
\\&\leq
2(3\eta^{n+1} - 4\eta^{n} + \eta^{n-1},\psi^{n+1}_{h})
+ \frac{4\gamma\Delta t}{3}(\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}),\nabla\cdot\eta^{n+1})
\\&\quad
+ 4\Delta t b(2u^{n}-u^{n-1},u^{n+1},\psi^{n+1}_{h})
- 4\Delta t b(2u^{n}_{h}-u^{n-1}_{h},\hat{u}^{n+1}_{h},\psi^{n+1}_{h})
\\&\quad
+ 4\nu\Delta t (\nabla\eta^{n+1}, \nabla\psi^{n+1}_{h})
- 4\Delta t(p^{n+1} - q_{h},\nabla \cdot \psi^{n+1}_{h})
- 4\Delta t\tau^{n+1}(\psi^{n+1}_{h})
\\&\quad
+ \frac{4\gamma d\Delta t}{3}\|\nabla\eta^{n+1}\|^{2}.
\end{aligned}\end{aligned}$$ Since $ \beta=0 $, we estimate $\frac{4\gamma\Delta t}{3}(\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}),\nabla\cdot\eta^{n+1}) $ as follows, $$\begin{aligned}
\label{pf6.2}
\begin{aligned}
&\frac{4\gamma\Delta t}{3}(\nabla\cdot(3\phi_{h}^{n+1} - 4\phi_{h}^{n} + \phi_{h}^{n-1}),\nabla\cdot\eta^{n+1})
\\&\leq
\frac{4\gamma\sqrt{d}\Delta t}{3}
\Big( 2\|\nabla\cdot \phi_{h}^{n+1} \|
+ 2\|\nabla\cdot \phi_{h}^{n} \|
+ \|\nabla\cdot(\phi_{h}^{n+1} - 2\phi_{h}^{n} + \phi_{h}^{n-1})\|\Big)
\|\nabla\eta^{n+1}\|
\\&\leq
\frac{2\gamma\Delta t}{3}\|\nabla\cdot(\phi_{h}^{n+1} - 2\phi_{h}^{n} + \phi_{h}^{n-1})\|^{2}
%\\&\quad
+ \frac{\gamma\Delta t}{2}\|\nabla\cdot \phi_{h}^{n+1}\|^{2}
+ \frac{\gamma\Delta t}{2}\|\nabla\cdot \phi_{h}^{n}\|^{2}
+ \frac{70\gamma d \Delta t}{9} \|\nabla\eta^{n+1}\|^{2}.
\end{aligned}\end{aligned}$$ Then we have $$\begin{aligned}
\label{pf6.3}
\begin{aligned}
&\|\phi^{n+1}_{h}\|^{2} - \|\phi^{n}_{h}\|^{2}
+ \|2\phi^{n+1}_{h}-\phi^{n}_{h}\|^{2}
- \|2\phi^{n}_{h}-\phi^{n-1}_{h}\|^{2}
+ \|\phi^{n+1}_{h}-2\phi^{n}_{h}+\phi^{n-1}_{h}\|^{2}
\\&\quad
+ \frac{2\gamma\Delta t}{3}
\Big( \|\nabla\cdot\phi^{n+1}_{h}\|^{2} - \|\nabla\cdot\phi^{n}_{h}\|^{2}
+ \|\nabla\cdot(2\phi^{n+1}_{h}-\phi^{n}_{h})\|^{2}
- \|\nabla\cdot(2\phi^{n}_{h}-\phi^{n-1}_{h})\|^{2}
\Big)
\\&\quad
+ \frac{\gamma\Delta t}{2} (
\|\nabla\cdot\phi^{n+1}_{h}\|^{2}
- \|\nabla\cdot\phi^{n}_{h}\|^{2}
)
% \\&\quad
+ 6 \|\psi^{n+1}_{h}-\phi^{n+1}_{h}\|^{2}
+ 2\nu \Delta t \|\nabla\psi^{n+1}_{h}\|^{2}
+ \gamma\Delta t\|\nabla\cdot\phi^{n+1}_{h}\|^{2}
\\&\leq
\frac{C\Delta t}{\nu} \|u^{n+1}\|_{2}^{2} ( \|2\phi^{n}_{h}-\phi^{n-1}_{h}\|^{2} + \|\phi^{n}_{h}\|^{2} + \|\phi^{n-1}_{h}\|^{2} )
\\&\quad
+ \frac{C}{\nu}\int^{t^{n+1}}_{t^{n-1}}\|\eta_{t}\|^{2}dt
+ C (\nu + \gamma d)\Delta t\|\nabla\eta^{n+1}\|^{2}
\\&\quad
+ \frac{Cd\Delta t}{\nu}\|p^{n+1} - q_{h}\|^{2}
+ \frac{C\Delta t^{4}}{\nu}\int^{t^{n+1}}_{t^{n-1}}\|u_{ttt}\|_{-1}^{2}dt
+ \frac{C\Delta t^{4}}{\nu}\int^{t^{n+1}}_{t^{n-1}}\|\nabla u_{tt}\|^{2}dt
\\&\quad
+ \frac{C\Delta t}{\nu} ( \|\nabla\eta^{n}\|^{2} + \|\nabla\eta^{n-1}\|^{2} )
+ \frac{C\Delta t}{\nu} ( \|\nabla\hat{u}_{h}^{n}\|^{2} + \|\nabla\hat{u}_{h}^{n-1}\|^{2} ) \|\nabla\eta\|_{\infty,0}^{2}.
\end{aligned}\end{aligned}$$ Sum ($\ref{pf6.3}$) from $n=1$ to $N-1$ to get $$\begin{aligned}
\label{pf6.4}
\begin{aligned}
&\|\phi^{N}_{h}\|^{2} + \|2\phi^{N}_{h}-\phi^{N-1}_{h}\|^{2}
+ \frac{2\gamma\Delta t}{3}
\Big( \|\nabla\cdot\phi^{N}_{h}\|^{2} + \|\nabla\cdot(2\phi^{N}_{h}-\phi^{N-1}_{h})\|^{2} \Big)
\\&\quad
+ 2\nu \Delta t \sum^{N-1}_{n=1}\|\nabla\psi^{n+1}_{h}\|^{2}
+ \gamma\Delta t \sum^{N-1}_{n=1}\|\nabla\cdot\phi^{n+1}_{h}\|^{2}
\\&\leq
\frac{C\Delta t}{\nu} \sum^{N-1}_{n=1} \|u^{n+1}\|_{2}^{2} ( \|\phi^{n}_{h}\|^{2} + \|2\phi^{n}_{h}-\phi^{n-1}_{h}\|^{2} )
+ C(\gamma d + \nu + \frac{1}{\nu}) |\|\nabla\eta\||_{2,0}^{2}
\\&\quad
+ \frac{C}{\nu}\|\eta_{t}\|_{2,0}^{2}
+ \frac{Cd}{\nu} |\|p - q_{h}\||_{2,0}^{2}
+ \frac{C\Delta t^{4}}{\nu}\|u_{ttt}\|_{2,-1}^{2}
+ \frac{C\Delta t^{4}}{\nu}\|\nabla u_{tt}\|_{2,0}^{2}
+ \frac{C}{\nu^{2}} \|\nabla\eta\|_{\infty,0}^{2}
\\&\quad
+ \|\phi^{1}_{h}\|^{2} + \|2\phi^{1}_{h}-\phi^{0}_{h}\|^{2}
+ \frac{7\gamma\Delta t}{6} \|\nabla\cdot\phi^{1}_{h}\|^{2}
+ \frac{2\gamma\Delta t}{3} \|\nabla\cdot(2\phi^{1}_{h}-\phi^{0}_{h})\|^{2} \Big).
\end{aligned}\end{aligned}$$ Denote $C^{**}=\frac{C}{\nu}|\|u\||_{2,2}^{2}$. The result then follows by similar arguments as in Theorem \[error\].
Under the assumptions of Theorem $\ref{error}$, suppose that $(X_{h},Q_{h})$ is given by P2-P1 Taylor-Hood approximation elements ($k=2, m=1$). Then, the following estimate holds for *BDF2-mgd*. $$\begin{aligned}
\label{taylor-hood-error}
\begin{aligned}
&\|e_{u}^{N}\|^{2} + \|2e_{u}^{N}-e_{u}^{N-1}\|^{2}
+ (\frac{2\gamma\Delta t}{3} + \beta)
\Big( \|\nabla\cdot e_{u}^{N}\|^{2} + \frac{1}{2}\|\nabla\cdot(2e_{u}^{N}-e_{u}^{N-1})\|^{2} \Big)
\\&\quad
+ 2\nu \Delta t \sum^{N-1}_{n=1}\|\nabla e_{\hat{u}}^{n+1}\|^{2}
+ 2\gamma\Delta t \sum^{N-1}_{n=1}\|\nabla\cdot e_{u}^{n+1}\|^{2}
\\&\leq
C \Big( h^{6} + h^{4} + \Delta t \,h^{4} + \Delta t^{4}
\\&\quad
+ \|e_{u}^{1}\|^{2} + \|2e_{u}^{1}-e_{u}^{0}\|^{2}
+ ( \gamma\Delta t + \beta)
(\|\nabla\cdot e_{u}^{1}\|^{2}
+ \|\nabla\cdot(2e_{u}^{1}-e_{u}^{0})\|^{2}) \Big).
\end{aligned}
\end{aligned}$$
Numerical Tests {#Numerical Tests}
===============
In this section, we consider three test problems to illustrate the stability, convergence, and effectiveness of *BDF2-mgd*. First, we consider the Taylor-Green benchmark problem to compute convergence rates and test both computational efficiency and pressure-robustness. We follow with 2D channel flow over a step, where the effect of *BDF2-mgd* on reducing the divergence error is illustrated. Moreover, it is shown how $ \gamma $ and $ \beta $ influence this effect. Finally, we simulate flow past a cylinder to further present the effectiveness of *BDF2-mgd*. For all tests, we compare *BDF2-mgd* with BDF2 (Non-Stabilized) and BDF2 with standard grad-div stabilization (Standard Stabilized). All tests are implemented using FreeFem++ [@Hecht].
Test of Convergence and Pressure Robustness {#Taylor-Green}
-------------------------------------------
The Taylor-Green benchmark problem is commonly used to test convergence rates of new algorithms. As such, we first illustrate convergence rates. The domain is $ [0,1]\times [0,1] $ and final time is $ T=1 $. Finite element meshes are generated via Delaunay-Vornoi triangulations with $ m $ points on each side of the boundary. The true solution is given by $$\begin{aligned}
\begin{aligned}
&u(x,y,t) = ( -\cos(\omega\pi x)\sin(\omega\pi y), \sin(\omega\pi x)\cos(\omega\pi y) ) \exp( -2\omega^{2}\pi^{2}t/\tau ),\\
&p(x,y,t) = -\frac{1}{4} \Big( \cos(2\omega\pi x) + \cos(2\omega\pi y) \Big) \exp( -4\omega^{2}\pi^{2}t/\tau ).
\end{aligned}\end{aligned}$$ Here, $ \omega=1 $, $ \tau=100$, and $ Re=\frac{1}{\nu}=100 $. The body force $ f $, initial condition, and boundary condition are determined by the true solution. The grad-div parameters are set to $ \gamma=1 $, $ \beta=0.2 $. The time step is $ \Delta t = 1/m$ where we vary $ m=16, 24, 32, 40 $, and 48 to calculate convergence rates. Table \[taylorgreen\_rate\] presents the results which are consistent with our theoretical analysis.
To test computational efficiency, we set $ m=32 $ and vary $ \gamma$ and $\beta $. We compare computational times of Standard Stabilized and *BDF2-mgd*; for $ \gamma = \beta = 0 $, Standard Stabilized is equivalent to Non-Stabilized. For Standard Stabilized and $ Step\; 1 $ of *BDF2-mgd*, we use a standard GMRES solver. If GMRES fails to converge at a single iterate, we denote the result with an “F". For $ Step\; 2 $ of *BDF2-mgd*, since it leads to an SPD system with same sparse coefficient matrix, at each timestep, we use UMFPACK. The results are presented in Table \[taylorgreen\_time\]. The computing time of Standard Stabilized generally increases as $ \gamma $ and $ \beta $ increase. However, computing time of *BDF2-mgd* is unaffected and therefore increasingly more efficient than Standard Stabilized. Interestingly, GMRES fails to converge when $ \gamma \simeq 20$ and $ \beta \simeq 0.8 $, which are not very large values.
Lastly, we consider the issue of pressure-robustness. An advantage of grad-div stabilization is that appropriate selection of the grad-div parameter $ \gamma $ can reduce the effect of the pressure error on the velocity error. Generally, for non-stabilized methods, velocity error estimates result in $ \nu^{-1}\inf\limits_{q_{h}\in Q_{h}}|\|p-p_{h}\||^{2}_{2,0} $ on the right hand side; see, e.g., Theorem 24 on p. 168 of [@Layton]. This same term appears for *BDF2-mgd* in Theorems \[error\] and \[error0\]. However, for standard grad-div stabilized methods, this term is replaced by $ \gamma^{-1}\inf\limits_{q_{h}\in Q_{h}}|\|p-p_{h}\||^{2}_{2,0} $ in theoretical analyses. To investigate the sharpness of our results, we vary $ Re $ while fixing $ \Delta t = 1/m = 1/32$ and $ \gamma=1 $, $ \beta=0.2 $. We compare the velocity and pressure errors of Non-Stabilized, Standard Stabilized, and *BDF2-mgd*. Results are presented in Table \[taylorgreen\_Re\]. It is clear that *velocity errors of Non-Stabilized*, especially for the divergence and gradient, *grow as Re increases*; this is consistent with the corresponding theoretical result. Alternatively, as $ Re $ is increased, velocity errors of Standard Stabilized and *BDF2-mgd* are consistent with one another and maintain good approximations. This suggests that the effect of $ Re $ appearing in our analysis is not sharp. This is an open problem, Section \[Conclusion\].
$m$ $|\| e_{u} \||_{\infty,0}$ Rate $|\| \nabla \cdot e_{u} \||_{\infty,0}$ Rate $|\|\nabla \cdot e_{u} \||_{2,0}$ Rate $|\| e_{p} \||_{2,0}$ Rate
----- ---------------------------- ------ ----------------------------------------- ------ ----------------------------------- ------ ----------------------- ------
16 2.47E-04 - 3.33E-03 - 2.82E-03 - 1.26E-04 -
24 8.07E-05 2.76 1.37E-03 2.19 1.18E-03 2.15 6.28E-05 1.72
32 3.54E-05 2.86 7.21E-04 2.24 6.24E-04 2.21 2.87E-05 2.73
40 1.90E-05 2.79 5.00E-04 1.64 4.34E-04 1.63 2.08E-05 1.43
48 1.12E-05 2.88 3.58E-04 1.83 3.11E-04 1.82 1.32E-05 2.50
: Errors and rates of velocity and pressure for *BDF2-mgd* using the Taylor-Hood element.[]{data-label="taylorgreen_rate"}
--------- ---------- --------------------- ------------
$\beta$ $\gamma$ Standard Stabilized *BDF2-mgd*
0 0 **17.77** **25.09**
0 0.2 **30.91** **20.10**
0 2 **55.29** **20.45**
0 20 F (**339.01**) **27.99**
0 200 F (**507.41**) **23.88**
0 2,000 F (**421.66**) **17.34**
0 20,000 F (**27.44**) **20.04**
0.01 0.2 27.79 22.28
0.02 0.2 32.89 22.33
0.04 0.2 64.37 21.68
0.08 0.2 69.31 23.97
0.8 0.2 F 25.87
8 0.2 F 19.53
80 0.2 F 21.20
800 0.2 F 17.43
8,000 0.2 F 18.64
--------- ---------- --------------------- ------------
: Computational time and solver breakdown for Standard and BDF2-mgd with increasing grad-div parameters.[]{data-label="taylorgreen_time"}
[max width=]{}
----- ---------- ------------ ---------- ---------- ------------ ---------- ---------- ------------ ---------- ---------- --
Standard Standard Standard
Stabilized Stabilized Stabilized
1 1.26E-03 1.26E-03 1.26E-03 3.61E-05 2.43E-05 5.33E-05 4.05E-03 4.05E-03 4.09E-03 5.79E-07
1e1 2.45E-05 2.42E-05 2.70E-05 4.71E-04 1.39E-04 2.01E-04 5.48E-04 3.67E-04 2.07E-03 1.09E-05
1e2 9.95E-05 1.80E-05 3.57E-05 1.20E-02 5.40E-04 6.44E-04 1.27E-02 2.80E-03 6.65E-03 2.99E-05
1e3 1.05E-03 5.85E-05 8.90E-05 1.03E-01 7.20E-04 7.51E-04 1.31E-01 9.04E-03 1.15E-02 3.57E-05
1e4 1.30E-01 2.23E-04 2.62E-04 6.27 7.61E-04 7.78E-04 7.81 2.40E-02 3.01E-02 3.64E-05
1e5 3.57E-01 3.99E-04 3.50E-04 16.36 7.76E-04 7.84E-04 25.23 3.61E-02 3.95E-02 3.64E-05
1e6 4.33E-01 4.32E-04 3.63E-04 20.34 7.78E-04 7.85E-04 32.44 3.84E-02 4.09E-02 3.64E-05
----- ---------- ------------ ---------- ---------- ------------ ---------- ---------- ------------ ---------- ---------- --
: Comparison of velocity and pressure errors with increasing $ Re $.[]{data-label="taylorgreen_Re"}
2D Channel Flow Over a Step
---------------------------
We now illustrate the effect of $ Step\; 2 $ of *BDF2-mgd* by comparing Non-Stabilized, Standard Stabilized, and *BDF2-mgd* simulations of 2D channel flow over a step [@Fragos; @John3]. The channel considered here is $ [0,40]\times [0,10] $ with a $ 1\times 1 $ step on the bottom for $x \in [5,6] $. A flow with $ \nu=1/600$ passes though this channel from left to right. For boundary conditions, the left inlet and right outlet are given by $$\begin{aligned}
\begin{aligned}
&u(0,y,t) = u(40,y,t) = y(10-y)/25,\\
&v(0,y,t) = v(40,y,t) = 0.
\end{aligned}\end{aligned}$$ No-slip, $ u=0 $, boundary conditions are imposed elsewhere. Taylor-Hood elements are used, comprising a mesh with $ 31,089 $ degrees of freedom. The body force $ f = 0 $, final time $ T = 40 $, and time step $ \Delta t = 0.01$. The selected grad-div parameters are $ \gamma = 0.1, 0.2, 1 $ and $ \beta = 0, 0.1, 0.2, 1 $. $ \|\nabla\cdot u(t^{n})\| $ is computed and plotted in Figure \[figure-step1\]. Also, plots of flow speed and divergence contours, at the final time, with $ \gamma=1, \beta=0 $, are presented in Figure \[figure-step2\].
![$ \|\nabla\cdot u\| $ vs time for Non-Stabilized and *BDF2-mgd*. []{data-label="figure-step1"}](step_divu.pdf){width="70.00000%"}
![Flow speed and divergence contours at time $ t=40 $ for Non-Stabilized (top), Standard Stabilized (middle) and *BDF2-mgd* (down) with $ \gamma=1, \beta=0 $. []{data-label="figure-step2"}](step_speed.png "fig:"){width="46.00000%"} ![Flow speed and divergence contours at time $ t=40 $ for Non-Stabilized (top), Standard Stabilized (middle) and *BDF2-mgd* (down) with $ \gamma=1, \beta=0 $. []{data-label="figure-step2"}](step_DivContour.png "fig:"){width="50.00000%"}
As shown in Figure \[figure-step1\], $ Step\; 2 $ of *BDF2-mgd* greatly reduces the divergence error $\|\nabla\cdot u\|$ compared with Non-Stabilized. Observing the curves of different $ \gamma $ and $ \beta $, it’s interesting to find that the value of $ \beta $ determines the minimum divergence error that can be reached in the beginning and the value of $ \gamma $ determines the long-time divergence error. This is consistent with [@Fiordilino]. In Figure \[figure-step2\], we see that results for $ Step\; 2 $ of *BDF2-mgd* are consistent with Standard Stablilzed; both reduce divergence error, especially around the step.
2D Channel Flow Past a Cylinder
-------------------------------
In order to further test the effectiveness of *BDF2-mgd*, we consider channel flow past a cylinder [@Schafer]. Like the Taylor-Green benchmark, this is a common test problem for new algorithms. The channel domain is $ [0,2.2]\times [0,0.41] $ with a cylinder of diameter $ 0.1 $ within. The center of the cylinder is $ (0.2, 0.2) $. A flow with $ \nu=0.001, \rho=1 $ passes though this channel from left to right. No body forces are present, $ f = 0 $. Left in-flow and right out-flow boundaries are given by $$\begin{aligned}
\begin{aligned}
&u(0,y,t) = u(2.2,y,t) = \frac{6y(0.41-y)}{0.41^{2}}\sin(\frac{\pi t}{8}),\\
&v(0,y,t) = v(2.2,y,t) = 0.
\end{aligned}\end{aligned}$$ The no-slip boundary condition is prescribed elsewhere.
We use Taylor-Hood elements on a mesh with $ 41,042 $ degree of freedom and final time $ T = 8 $. The time step is $ \Delta t = 0.001$. The grad-div parameters are set to $ \gamma = 5\nu $ and $ \beta = 0$. Drag $ c _{d}(t) $ and lift $ c_{l}(t) $ coefficients are calculated; maximum values are presented in Table \[table=cylinder\]. The pressure difference between the front and back of the cylinder ($ \Delta p(t)=p(0.15,0.2,t)-p(0.25,0.2,t) $) and both the $L^2(0,T;L^{2}(\Omega))$ and $L^{\infty}(0,T;L^{2}(\Omega))$ norms of the velocity divergence are also tabulated in Table \[table=cylinder\]. Furthermore, Figure \[figure-cylinder\] shows velocity speed and vectors for *BDF2-mgd* at times $ t=4, 6, 7, 8$, which are consistent with that in [@Bowers; @Fiordilino; @John4; @Linke].
In Table \[table=cylinder\], we see that grad-div stabilization effectively reduces the divergence error, as expected. This results in improved accuracy of Standard Stabilized and *BDF2-mgd* over the Non-Stabilized solution. In particular, both stabilized algorithms produce accurate lift coefficients and smaller divergence errors.
Method $c^{max}_{d}$ $c^{max}_{l}$ $\Delta p^{N}_{h}$ $|\|\nabla \cdot u_{h}|\|_{2,0}$ $\| \nabla \cdot u^{N}_{h}\|$
--------------------- --------------- --------------- -------------------- ---------------------------------- -------------------------------
Non-Stabilized 2.950 **0.441** -0.1084 **1.967** **0.186**
Standard Stabilized 2.950 0.477 -0.1115 0.859 0.072
*BDF2-mgd* 2.950 0.475 -0.1115 0.906 0.074
: Maximum lift, drag coefficients, pressure drop, and divergence quantities for flow past a cylinder.[]{data-label="table=cylinder"}
![Flow speed and vectors for flow past a cylinder at times t=4, 6, 7, and 8. []{data-label="figure-cylinder"}](cylinder_speed.png "fig:"){width="48.00000%"} ![Flow speed and vectors for flow past a cylinder at times t=4, 6, 7, and 8. []{data-label="figure-cylinder"}](cylinder_vectors.png "fig:"){width="48.00000%"}
Conclusion {#Conclusion}
==========
We developed a BDF2 time-discrete, modular grad-div stabilization algorithm (*BDF2-mgd*) for the time dependent Navier-Stokes equations. Compared with methods implementing standard grad-div stabilization, our algorithm produces consistent numerical approximations while avoiding solver breakdown for large grad-div parameters. We prove that this algorithm is unconditionally, nonlinearly, energy stable and second-order accurate in time. Numerical tests illustrate the theoretical results and computational efficiency.
To impose discrete versions of $-\beta \nabla \nabla \cdot u_t - \gamma \nabla \nabla \cdot u$, modular grad-div requires a solve of the form $\big(\frac{1}{\Delta t}I + (\frac{\beta}{\delta t} + \gamma) G\big)u = RHS$, where $G$ is the symmetric positive semi-definite grad-div matrix. For constant $\Delta t$, efficiency increases can exploit the fact that the matrix is fixed. For variable timestep and $\beta =0$, the matrix is a variable shift of $G$ and efficient algorithms exist exploiting this structure. Important next steps include investigating, analytically, the $\nu$ dependence of $ \nu^{-1}\inf\limits_{q_{h}\in Q_{h}}|\|p-p_{h}\||^{2}_{2,0} $ in Theorems \[error\] and \[error0\], extending these results to alternative numerical methods, and including sparse, effective variants of grad-div stabilization.
[00]{}
[^1]: Xi’an Jiaotong University, School of Mathematics and Statistics, Xi’an, Shaanxi 710049, China. Support from NSFC grants 11171269 and 11571274 and China Scholarship Council grant 201606280154.
[^2]: University of Pittsburgh, Department of Mathematics, Pittsburgh, PA 15260. The research presented herein was partially supported by NSF grants CBET 1609120 and DMS 1522267. J.A.F. is supported by the DoD SMART Scholarship.
|
---
abstract: 'We consider a model for substrate-depletion oscillations in genetic systems, based on a stochastic differential equation with a slowly evolving external signal. We show the existence of critical transitions in the system. We apply two methods to numerically test the synthetic time series generated by the system for early indicators of critical transitions: a detrended fluctuation analysis method, and a novel method based on topological data analysis (persistence diagrams).'
title: Critical transitions in a model of a genetic regulatory system
---
<span style="font-variant:small-caps;">Jesse Berwald</span>
<span style="font-variant:small-caps;">Marian Gidea</span>
(Communicated by the associate editor name)
Introduction {#section:introduction}
============
Gene expression is the process by which the genetic code is used to synthesize functional gene products (proteins, functional RNAs). The timing and the level of gene production is specified by a wide range of mechanisms, termed gene regulation. It is believed that a significant number of genes express cyclically, with about $10-15$% of genes directly regulated by the circadian molecular clock. Such gene expression oscillations allow for rapid adaptation to changes in intracellular and environmental conditions. In general, the phase and amplitude of gene expression depend on the function of the gene and internal and external stimuli. It has been postulated that gene expression oscillation is a basic property of all genes, not necessarily connected with any specific gene function [@Ptitsyn2007].
Gene regulatory systems are intrinsically stochastic. Stochasticity originates in the statistical uncertainty of the chemical reactions between molecules, and is inversely proportional to the square root of the number of molecules. Thus, lower numbers of interacting molecules yield increasingly significant statistical fluctuations. Besides intrinsic stochasticity, gene expression is also subjected to extrinsic stochasticity, due to environmental effects. In general, stochastic fluctuations are seen as a source of robustness and stability, but sometimes can adversely affect a cell function [@Leier2006].
A motivation for the modeling and simulation of gene regulatory systems comes from synthetic biology, an emerging field of research devoted to the design and construction of biological systems, based on engineering principles. An interesting analogy in [@Tyson2003] compares a molecular network to an electrical circuit, where, instead of resistors, capacitors and transistors connected via circuits one has genes, proteins, and metabolites connected via chemical reactions and molecular pathways. Some milestone achievements in this direction have been reported, e.g., in [@Bulter2004; @Elowitz; @Gardner]. Relatedly, one would also like to achieve a clear understanding of the functionality of the molecular circuits and networks through measurements of various product outputs, in the same way one understands an electrical circuit through measurements of current, voltage, and resistance. In particular, one would like to discern possible ways in which systems subjected to varying parameters and noise may switch between different stable regimes, or more generally, between potential attractors.
It is the concept of suddenly shifts amongst stable regimes which we explore in this paper. We investigate the occurrence of these [ *critical transitions*]{} in a model of a genetic regulatory system. By a critical transition we mean a sudden change of a system from one stable regime (fixed point, limit cycle) to an unstable regime, possibly followed by some other stable regime. We consider a simple genetic circuit that exhibits an oscillatory regime, and we study the behavior of the system under noise. Explicitly, we consider a two-gene model whose oscillations depend on several parameters. We show that the system undergoes a critical transition under slow parameter drift. We accomplish this by recording the time series generated by this model and analyzing the critical transitions. We utilize numerical methods to identify early warning signals indicative of critical transitions in the synthetic data.
First, we apply a statistical method, based on detrended fluctuation analysis, to analyze these time series data. The method is described in detail in Subsection \[subsection:DFA\]. The tests are performed using a windowed analysis of the data and reveal that the autocorrelation of the time series increases towards $1$, and the variance of the time series distribution grows steadily prior to a critical transition. These signs are consistent with early warning indicators of critical transitions described by others; for instance, see results in Scheffer, [*et al*]{} [@Scheffer2009].
Secondly, we apply a method from topological data analysis, based on [*persistence diagrams*]{}, which we describe in more detail in Subsection \[subsection:persistence\]. Again, we consider windows of data from the time series. In this case, we consider these windows as strings of data points to which we associate filtrations of Rips complexes and for which we generate associated persistence diagrams to analyze the topology of the data at different resolutions. The persistence diagrams reveal qualitative changes in the topology of the strings of data points prior to the critical transitions: the distribution of data points becomes more widespread and/or asymmetric. While the detrended fluctuation analysis has been previously used for detection of critical transitions, the application of persistence diagrams, a method from topological data analysis, is novel.
We also make a comparison between the two methods. While the detrended fluctuation analysis introduces artificial choices and possible bias (see also [@Bryce2012]), the proposed topological method is inherently robust. Note that both methods can be applied to detect or predict critical transitions in experimental data as well. As such, the results from model testing may serve as benchmarks for testing data measured from real world sources.
Background
==========
In this section we briefly describe critical transitions in the context of fast-slow systems. We consider both deterministic and stochastic systems. Then we present a simple genetic regulatory model of the substrate-depletion oscillator type.
Critical transitions {#sub:critical}
--------------------
A fast-slow system of ordinary differential equations is a system of the type $$\begin{aligned}
\label{eqn:f} x' &=& f(x,y), \\
\label{eqn:s}{\varepsilon}y' &=&g(x,y),\end{aligned}$$ where $(x,y)\in\mathbb{R}^m\times\mathbb{R}^n$, $f:\mathbb{R}^{m+n}\to\mathbb{R}^m$, $g:\mathbb{R}^{m+n}\to\mathbb{R}^n$ are $C^{r}$-functions with $r\geq 3$, ${\varepsilon}>0$ is a small parameter, and ${}'=\frac{d}{dt}$. One can regard $y$ as a fast variable. Rescaling the time $t={\varepsilon}\tau$ yields $$\begin{aligned}
\label{eqn:ftau} \dot x &=&{\varepsilon}f(x,y), \\
\label{eqn:stau} \dot y &=& g(x,y),\end{aligned}$$ where $\,\dot{}= \frac{d}{d\tau}$. The singular limit of , when ${\varepsilon}\to 0$ gives the slow subsystem, and the singular limit of , gives the fast subsystem. The critical set is defined as $$C_0=\{(x,y)\,:\,g(x,y)=0\}$$ and consists of equilibrium points for the fast subsystem. If the Jacobian $\frac{\partial g}{\partial y}$ is nonsingular on $C_0$, then $C_0$ is an $m$-dimensional manifold, and is the graph of a smooth function $y=h_0(x)$. The slow subsystem is determined by $x'=f(x,h_0(x))$ and restricts to $C_0$. In the case that all eigenvalues of $\frac{\partial g}{\partial y}$ at a point have non-zero real parts, then the point is normally hyperbolic. The set of normally hyperbolic points forms a normally hyperbolic invariant manifold (NHIM). In particular, it has stable and unstable manifolds. The NHIM can have attractive components, where all the eigenvalues of $\frac{\partial g}{\partial y}$ have negative real part, and repelling components, where at least one eigenvalue has positive real part. An attractive component has an $n$-dimensional stable manifold, while a repelling component has a non-trivial unstable manifold.
Fenichel’s Theorem [@Fenichel79] implies that, for all sufficiently small ${\varepsilon}$, every compact submanifold (with boundary) $S_0$ of $C_0$ can be continued to a NHIM $S_{\varepsilon}$ (not uniquely defined) for the flow of , , which is the graph of a smooth function $y=h_{\varepsilon}(x)$. The stable and unstable manifolds of $S_0$ continue to stable and unstable manifolds of $S_{\varepsilon}$. The flow on $S_{\varepsilon}$ converges to the slow flow as ${\varepsilon}\to 0$. Such a manifold $S_{\varepsilon}$ is referred to as a slow manifold.
We define a critical transition for this type of system following [@Kuehn2011]. We assume that the critical set $C_0$ can be decomposed as $C_0=S^a_0\cup S^r_0\cup S^b_0$, where $S^a_0$ is an attractive NHIM, $S^r_0$ a repelling NHIM, and $S^b_0$ is the part of $C_0$ that is not normally hyperbolic (corresponding to bifurcation points). By definition, a point $p_0=(x_0,y_0)$ on $C_0$ that is not normally hyperbolic is a critical transition if there exists a concatenation of trajectories $\gamma_0$, $\gamma_1$, where $\gamma_0:[t_{-1},t_0]\to \mathbb{R}^{m+n}$, $\gamma_1:[t_0,t_1]\to \mathbb{R}^{m+n}$ satisfy the following properties:
- $\gamma_0(t_{-1},t_0)$ is a trajectory of the slow subsystem, oriented from $\gamma_0(t_{-1})$ to $\gamma_0(t_0)$, contained in the attracting NHIM $S^a_0$;
- $\gamma_0(t_0)=\gamma_1(t_0)=p_0\in S^b_0$ is a point that is not normally hyperbolic;
- $\gamma_1(t_0,t_1)$ is a trajectory of the fast subsystem, oriented from $\gamma_1(t_0)$ to $\gamma_1(t_1)$.
When we consider the dynamics of the system for ${\varepsilon}>0$ sufficiently small, a trajectory starting near $\gamma_0$ follows closely the slow dynamics around $\gamma_0$ for some time, after which it transitions to follow the fast dynamics near $\gamma_1$ for a period of time. In [@Kuehn2011; @Kuehn2012] several types of bifurcations are examined to determine whether or not they exhibit the characteristics of critical transitions, under some suitable conditions on the smoothness, compactness, and non-degeneracy on the system. We summarize the findings below:
- for $m=n=1$, saddle-node (fold) bifurcation points determine critical transitions;
- for $m=n=1$, subcritical pitchfork bifurcation points determine critical transitions;
- for $m=n=1$, transcritical bifurcation points determine critical transitions;
- for $m=2$ and $n=1$, subcritical non-degenerate Hopf bifurcations determine critical transitions.
Incorporating noise {#sec:noise}
-------------------
It is often important for the understanding of a physical system to incorporate stochastic effects. We consider the Langevin form of , $$\begin{aligned}
\label{eqn:fst} dx &=& f(x,y)+\sigma_1dW_1, \\
\label{eqn:sst} dy&=&\frac{1}{{\varepsilon}}g(x,y)+\frac{1}{\sqrt{{\varepsilon}}}\sigma_2dW_2,\end{aligned}$$ where $\sigma_1$, $\sigma_2$ represent noise levels (depending on ${\varepsilon}$), and $W_1,W_2$ are one-dimensional Wiener processes (Brownian motions). Assuming $\sigma_1,\sigma_2$ are sufficiently small, the sample paths of the system , stay near $S^a_0$ with high probability, up to a neighborhood of the critical transition, after which they exit the neighborhood. In [@Kuehn2011; @Kuehn2012], it is argued that the following behaviors are typical of a system prior to a critical transition:
- The system recovery from small perturbations is ‘critically’ slows down;
- The variance in the time series increases steadily;
- The autocorrelation of the time series increases towards $1$;
- The distribution of the time series becomes more asymmetric.
We note that (iv) from above depends on whether or not the underlying bifurcation has symmetry. For example, a saddle-node bifurcation as in (a) from above will typically yield asymmetric fluctuations, while a pitchfork bifurcation as in (b) from above will typically yield symmetric fluctuations. A complementary characteristic to (iv) is that the distribution of the time series loses its normality, for example it changes from uni-modal to multi-modal.
Such system response characteristics can be monitored numerically and serve as ‘early warnings’ of critical transitions in real-world systems. Examples include Earth’s climate, ecological systems, global finance, asthma attacks or epileptic seizures; see, e.g., [@Ditlevsen2010; @Scheffer2009; @Scheffer2012; @Thompson2010], and the references therein.
A simple genetic circuit
------------------------
We briefly describe a model of simple genetic circuit which generates oscillations of varying amplitude. The model consists of two genes, one producing the protein $R(t)$, and the other producing the protein $X(t)$. The protein $X(t)$ is a substrate for the activator protein $R(t)$ that is produced in an autocatalytic process. As $R(t)$ accumulates, the production of $R(t)$ accelerates until there is an explosive conversion of the whole of $X(t)$ into $R(t)$. This rapid change corresponds to a critical transition in the underlying system. With the substrate $X(t)$ depleted, the autocatalytic reaction terminates, and the activator $R(t)$ degrades in time. This allows the level of $X(t)$ to grow again, leading to another cycle of explosive growth in $R(t)$. This process is know as a substrate-depletion oscillator.
An example of this mechanism is the oscillation of the M-phase-promoting factor (MPF) activator in the frog egg, where the substrate is the phosphorylated form of the B-cyclin-dependent kinase (note that the true mechanism involves several other proteins and reactions) [@Novak1993]. A mathematical model for the substrate-depletion oscillator is given by the following system: $$\begin{aligned}
\label{eqn:X} X'(t) &=& k_1S -[k_0'+k_0E_P(R(t))]X(t), \\
\label{eqn:R} R'(t) &=&[k_0'+k_0E_P(R(t))]X(t)-k_2R(t).\end{aligned}$$ Here $E_P(R(t))$ represents the level of the phosphorylated version of the protein $R(t)$ – involved with $R(t)$ in a mutual activation process – given by $E_P(R)=G(k_3R,k_4,J,K)$, where $G$, the Goldbeter-Koshland function, is defined by $$G(u,v, J,K)=
\frac{2uK}{v-u+vJ+uK+\sqrt{(v-u+vJ+uK)^2-4(v-u)uK}}.$$ The Goldbeter-Koshland function represents the equilibrium concentration of the phosphorylated form of a protein, for a phosphorylation-dephosphorylation reaction governed by Michelis-Menten kinetics. The Goldbeter-Koshland function is responsible for creating a switch-like signal-response in the evolution of the protein $R(t)$. The quantities $k_0,k'_0,k_1,k_2,k_3,k_4,J,K,S$ are parameters. The parameter $S$ is the strength of a signal, representing the rate of synthesis of the substrate $X$, which we regard as an external input to the system.
This system presents both positive and negative feedback. The positive feedback loop creates a bistable system and the negative-feedback loop drives the system back and forth between two stable steady states. In what follows, we will modify the simple genetic circuit in and by considering the external input $S$ as a slowly varying parameter, in addition to including a stochastic term. We will study critical transitions in the resulting system.
Model {#section:model}
=====
We consider a new model of a genetic regulatory network with a slowly dependent signal, given by , , with $S$ being now a slowly evolving parameter, i.e. $$\begin{aligned}
\label{eqn:XS0} X'(t) &=& k_1S(t) -[k_0'+k_0E_P(R(t))]X(t), \\
\label{eqn:RS0} R'(t) &=&[k_0'+k_0E_P(R(t))]X(t)-k_2R(t),\\
\label{eqn:S0} S'(t)&=&{\varepsilon},\end{aligned}$$ where ${\varepsilon}>0$ is small. We will fix the parameter as in [@Tyson2003], $k_0=0.04,k'_0=0.01, k_1=k_2=k_3=1,k_4=0.3,J=K=0.05$.
The fast subsystem is obtained by letting ${\varepsilon}\to 0$ yielding $$\begin{aligned}
\label{eqn:Xf} X'(t) &=& k_1S(t) -[k_0'+k_0E_P(R(t))]X(t), \\
\label{eqn:Rf} R'(t) &=&[k_0'+k_0E_P(R(t))]X(t)-k_2R(t),\\
\label{eqn:Sf} S'(t)&=&0.\end{aligned}$$
We rescale time $\tau={\varepsilon}t$, and we rewrite the corresponding system relative to the rescaled time $$\begin{aligned}
\label{eqn:Xr} {\varepsilon}\dot X &=& k_1S -[k_0'+k_0E_P(R)]X, \\
\label{eqn:Rr} {\varepsilon}\dot R &=&[k_0'+k_0E_P(R )]X -k_2R ,\\
\label{eqn:Sr} \dot S &=&1.\end{aligned}$$ Sample trajectories for this system are plotted in Fig. \[sdevol\]. From the above equations the slow subsystem is obtained by letting ${\varepsilon}\to 0$ yielding $$\begin{aligned}
\label{eqn:Xs} 0 &=& k_1S -[k_0'+k_0E_P(R)]X, \\
\label{eqn:Rs} 0 &=&[k_0'+k_0E_P(R )]X -k_2R ,\\
\label{eqn:Ss} \dot S &=&1.\end{aligned}$$ From this we see that the critical submanifold is given by $$C_0=\left\{(X,R,S)\,,\, X=\frac{k_1S}{k_0'+k_0E_P({\frac{k_1}{k_2}S)}}, R=\frac{k_1}{k_2}S\right\}.$$
The critical submanifold consists of equilibrium points of the fast subsystem. The stability of the $C_0$ is determined by the eigenvalues of the Jacobi matrix evaluated at the equilibrium points $$J=\left(
\begin{array}{cc}
-[k'_0+k_0E_P(R)] & -[k_0\partial_RE_P(R)X] \\
k'_0+k_0E_P(R) & k_0\partial_RE_P(R)X-k_2\\
\end{array}
\right).$$
For the values of the parameters chosen above, we find that the stability at an equilibrium point changes at $S_{crit1}=0.13326703$ and $S_{crit2}=0.34680193$, respectively. The corresponding equilibria are $X_{crit1}=5.41285587$, $R_{crit1}=0.13326703$, and $X_{crit2}=1.07370450$, $R_{crit2}=0.34680193$.
![A sample trajectory, and the $X$-time series (higher value) and $R$-time series (lower values).[]{data-label="sdevol"}](sdevol00-eps-converted-to.pdf){width="100.00000%"}
Precisely, for $S<S_{crit1}$ and $S>S_{crit2}$ the equilibrium point is stable. For $S\in(S_{crit1}, S_{crit2})$ the equilibrium point is unstable, and there exists a periodic orbit that is asymptotically stable, whose existence can be established numerically, as observed in Fig. \[sdevol\]. The values $S=S_{crit1}$, $S=S_{crit2}$ yield subcritical Hopf bifurcations, where an unstable equilibrium point is turned into a stable one and a small unstable periodic orbit is born (or vice versa). In addition, one expects canard-type solutions in some exponentially small neighborhoods of $S_{crit1}$, $S_{crit2}$, relative to ${\varepsilon}$ (see, e.g., [@KrupaS2001]). In Fig. \[nullclines\] we plot the nullclines of the system for the critical points $S_{crit1}$ and $S_{crit2}$.
$$\begin{array}{cc}
\includegraphics[width=0.5\textwidth, clip, keepaspectratio]
{nullclines01_v2-eps-converted-to.pdf} &
\includegraphics[width=0.5\textwidth, clip, keepaspectratio]
{nullclines02_v2-eps-converted-to.pdf}
\end{array}$$
The specific stochastic differential equation (SDE) system associated to , , can be written $$\begin{aligned}
\label{eqn:Xr_stoch} \dot X &=&\frac{1}{{\varepsilon}}(k_1S -[k_0'+k_0E_P(R)]X)+\frac{\sigma_1}{\sqrt{{\varepsilon}}}dW_1, \\
\label{eqn:Rr_stoch} \dot R &=&\frac{1}{{\varepsilon}}([k_0'+k_0E_P(R )]X -k_2R)+\frac{\sigma_2}{\sqrt{{\varepsilon}}}dW_2,\\
\label{eqn:Sr_stoch} \dot S &=&1,\end{aligned}$$ where $W_1,W_2$ represent Brownian motions, and $\sigma_1,\sigma_2$ are noise levels. Since the parameter values $S_{crit1}$ and $S_{crit2}$ yield subcritical Hopf bifurcations, the theory from Subsection \[sub:critical\] allows us to conclude that the corresponding points $(X_{crit1}, R_{crit1})$, $(X_{crit2},
R_{crit2})$ determine critical transitions. A typical trajectory of the stochastic system in the phase space and its corresponding $R$-time series are shown in Fig. \[substratedepletion\]. Examining the $R$-time series, one can see that a critical transition occurs near time $t \approx 1000$. We note that a similar analysis for activator-inhibitor oscillations has been performed in [@Kuehn2012].
As mentioned in Section \[section:introduction\], noise in the form of random fluctuations arises naturally in gene regulatory networks. One typically distinguishes between intrinsic noise, inherent in the biochemical reactions, and extrinsic noise, originating in the random variation of the externally set control parameters. Both types of noise can be model by augmenting the governing rate equations with additive or multiplicative stochastic terms. We refer the interested reader to [@Hasty2000].
![Phase space of the model described by , , , and a corresponding $R$-time series.[]{data-label="substratedepletion"}](substratedepletion00-eps-converted-to.pdf){width="100.00000%"}
In our model we only consider the effects of additive noise, which can be thought of as a randomly varying external field acting on the biochemical reactions. The field enters into the governing rate equations as an additive stochastic term in the Langevin equation. We choose to focus on additive extrinsic noise as this could be used as a switch and/or amplifier for gene expression, which has potential applications to gene therapy [@Hasty2000]. Switching mechanisms are exactly the type of phenomena that we would like to capture via the critical transitions approach.
Methods {#sec:methods}
=======
We use the model proposed in Section \[section:model\] to generate a synthetic time series given by successive reading of one of the variables. We investigate the synthetic time series for early warning signs of critical transitions. Below we describe two such detection methods: a well-known detrended fluctuation analysis method, and a novel method inspired by topological data analysis.
Detrended fluctuation analysis {#subsection:DFA}
------------------------------
Detrended fluctuation analysis (DFA) is a technique introduced by Hurst half a century ago to analyze fluctuations in time series. The DFA procedure has been widely used for early detection of critical transitions [@Livnia2007; @Scheffer2009; @Thompson2010]. We outline the algorithm below.
### Algorithmic description of DFA
The DFA procedure takes as input a time series $(s_k,z_k),\, k=1,\ldots, N$, where $s_k$ is the instant of time of the $k$-th measurement (not necessarily equally spaced), and $z_k$ is the $k$-th measurement of some observable. To detect whether the system undergoes a critical transition, the DFA algorithms proceeds as follows:
Interpolation
: Choose an optimal step size $\Delta t$, and interpolate the given time series such that it is evenly spaced in time. Denote the new series $(t_k, x_k)$, with $t_{k}=k\Delta t$.
Detrending
: One way to remove a general trend from statistical data is by subtracting a moving average. For example, using a Gaussian kernel $$G_k(t)=\frac{1}{\sqrt{2\pi}d}\exp\left(-\frac{(t-k\Delta t)^2}{2d^2}\right)$$ of bandwidth $d$, one may compute the weighted average of $x_k$, $$X(k\Delta t)=\frac{\sum_{i=1}^{N}G_k(i\Delta t)x_i }{\sum_{i=1}^{N}G_k(i\Delta t)}.$$ Subtracting the weighted average from the time series yields the detrended series, $$y_k=x_k-X(k\Delta t)$$.
Instead of Gaussian kernel detrending, alternative detrending methods can be used, such as linear, cubic spline, or Fourier interpolation, depending on the nature of the data. We also explore these additional methods in Section \[sec:results\]
Lag-1 autocorrelation
: The final step involves fitting a first-order autoregressive (AR(1)) process $$y_{k+1}=c_ky_k+\sigma \xi_k,$$ to the detrended time series $y_k$, where $(\sigma
\xi_k)_{k\in\mathbb{N}}$ is white noise of intensity $\sigma$. In order to compute $c_k$, choose a sliding window of size $w$ and determine the least-squares fit $$y_{j+1} \approx c_ky_j,\quad \textrm{ for } j=k,\ldots, k+w-1.$$ Hence, for each window we extract the value of $AR(1)$ $c_k$. Recall that the lag-1 autocorrelation $AR(1)$ is $0$ for white noise and close to $1$ for red (autocorrelated) noise.
### Detection of critical transitions via DFA {#sec:dfa-detection}
The following criterion has been proposed for the detection of critical transition [@Scheffer2009]:
- Given a time series measured from a system approaching a critical transition, the DFA outputs a time series $(t_k,y_k)$ for which
1. the autocorrelation has a general trend which increases towards $1$;
2. the variance has a positive trend.
In certain special cases, this criterion has a rigorous justification [@Kuehn2011; @Kuehn2012].
The DFA method is an effective tool in detecting early signs of critical transitions in noisy data. However, the method comes with several significant drawbacks, such as its sensitivity to the procedures and parameters used in processing the data. For instance, the sample frequency, detrending method (e.g., the bandwidth of the Gaussian detrending), or the size of the sliding window all have a strong effect on the conclusions drawn from the algorithm in subsection \[subsection:DFA\] and hence on the power of the DFA method to serve as a prediction tool. One concern is that the measurement of the $AR(1)$ values as well as the variance are strongly influenced by the fit of the detrending method, with a poor fit being likely to signal ‘false positives’ for critical transitions (see [@Bryce2012]).
Persistence diagrams {#subsection:persistence}
--------------------
As mentioned in Section \[section:introduction\], we propose to use tools from the field of topological data analysis as a new method to detect critical transitions in dynamical systems. In particular, we leverage the stability properties of persistence diagrams to detect critical transitions. Topological persistence is a relatively recent development that forms the core of topological data analysis and has been widely used to extract relevant information from noisy data (see [@Edelsbrunner2008] for background in persistence topology in general). There are numerous applications, including computer vision, cluster analysis, biological networks, cancer survival analysis, and granular material (see [@Chazal2011; @Edelsbrunner2008; @Gameiro2012; @Kondic2012; @Nicolau2011] and the references listed there).
In this section we describe the way in which we adapt this method to observe changes in time series from systems approaching or undergoing critical transitions. The key idea is to extract from the time series consecutive strings of data points of a fixed length, which we regard as individual point cloud data sets. To each such point cloud we assign a topological invariant, namely its persistence diagram. Roughly speaking, the persistence diagram is a representation of the data set in an abstract metric space which encodes information about topological features of the data.
The highlight of this method is that when the system undergoes a critical transition, the topological features associated to the point cloud data sets also change significantly. The fact that the corresponding persistence diagrams and distances between them can be computed algorithmically enables us to describe these changes quantitatively.
### Description of persistence diagrams
We describe the concept of a persistence diagram associated with point cloud data starting with an informal description. From a high-level perspective, the data analysis pipeline works as follows: $$\text{Data }
\implies \text{ Filtration }
\implies \text{ Persistence Module }
\implies \text{ Persistence Diagram}$$ We focus on the first two and the fourth parts of this pipeline, and only briefly detail the algebraic aspects of the third component below. Suppose that one is provided with a point cloud data set, $X_0$, that is an approximation of some geometric shape. One would like to infer from the data the topological information on that shape. However, a finite collection of points has only trivial topology. One way to convert the collection of points into a non-trivial topological space is to replace the points of the set by balls of a certain radius $\epsilon$. One then computes the topological features of the resulting set, $X_{\epsilon}$. Typical invariants resulting from this computation, which serve to classify the set, include the number connected components along with the number of ‘tunnels’ and number of ‘cavities’ (known as Betti numbers).
Of course, the topology of $X_{\epsilon}$ depends on the choice of the radius of the balls in this construction. Instead of fixing a certain radius, topological persistence considers all possible radii, from some sufficiently small value, up to a sufficiently large radius. This growth of the radius yields the filtration step above. As the radius is gradually increased, new topological features will be ‘born’ and certain existing ones will ‘die’. A schematic representation of this process is depicted in Fig. \[persistence\]. The birth and death of each topological feature at a given dimension is recorded by a persistence diagram. This is a collection (multiset actually) of (birth,death) times in $\mathbb{R}^2$. The 0- and 1-dimensional diagrams for the associated filtration are shown in the bottom row of Fig. \[persistence\].
The lifespan of a feature is easily computed by calculating (death time) - (birth time). A topological feature with a long lifespan, measured by the range of radii over which it ‘persists’, is likely to capture an essential topological feature of the underlying space from which the data was sampled. On the other hand, short lived features are likely to result from ‘noise’ in the data. However, rather than discriminating between what is an essential feature of the topology and what is not, the persistence diagram method provides a summary of topological features that appear and disappear throughout the variation of the radii of the balls, as well as a ranking of the significance of these features, expressed in terms of the lifespans.
![A collection of points in the plane, resembling a ‘noisy’ circle, is given. At each instant of time $t=1,\ldots,4$, around each point we constructs disks of radii $\delta_1/2,\ldots,
\delta_4/2$, respectively, with $\delta_1<\delta_2<\delta_3<\delta_4$. The corresponding Rips complexes are also constructed at each instant. The topological features of the Rips complexes change as time increases. At time $t=1$ there are 8 connected components and no 1-dimensional hole. At time $t=2$ the connected components coalesce into a single component (thus in the 0-dimensional diagram there are actually 7 deaths represented at $(1,2)$), and a 1-dimensional hole is born. Both the single connected component and the 1-dimensional hole survive to $t=3$. At time $t=4$ the 1-dimensional hole dies as it fills in (the death is represented at $(2,4)$), while the single connected component continues living (in fact, it has infinite lifespan). The $\Diamond$ at $(1,4)$ represents this fact. []{data-label="persistence"}](noisycircle0-eps-converted-to.pdf){width="70.00000%"}
We now continue with a concise, formal description of persistence diagrams. For an introduction to algebraic topology and homology, see [@Hatcher2002]; surveys of persistent topology can be found in [@Edelsbrunner2008; @Zomorodian2005] Given point cloud data $X\subset \mathbb{R}^d$, i.e., a collection of points in $\mathbb{R}^d$, and $\delta>0$, we associate to them the Rips complex $\mathcal{R}_\delta$. This is, by definition, the abstract simplicial complex whose $0$-simplices are points $x_\alpha$, and whose $k$-simplices are given by unordered $(k+1)$-tuples of points $\{x_{\alpha_j}\}_{j=0,\ldots,k}$ which are pairwise within a distance $\delta$. Figure \[persistence\] provides an example for $X \subset \mathbb{R}^2$. For all $0<a<b$ we have $\mathcal{R}_a\subset\mathcal{R}_b$. That is, the family $\{\mathcal{R}_\delta\}_{\delta>0}$ forms a filtration.
Denote by $H_p(\mathcal{R}_a)$ the $p$-homology of $\mathcal{R}_a$ with $\mathbb{Z}_2$ coefficients. Heuristically, the homology of a simplicial complex provides information about the topological features of the complex, e.g., the number of connected components, tunnels and cavities in that complex. The inclusion $\mathcal{R}_a\hookrightarrow
\mathcal{R}_b$ induces the homomorphisms $f^{a,b}_*:H_*(\mathcal{R}_a) \to H_*(\mathcal{R}_b)$ in all dimensions. Note that the image $F^{a-\rho,b}$ of $f^{a-\delta,a}_*$ in $H_*(\mathcal{R}_b)$ is independent of $\rho$ for all $\rho>0$ sufficiently small. We denote this image by $F^{a-,b}_*$.
A real value $c>0$ is called a homological critical value if there exists $q$ such that the homomorphism $f^{c-\rho,c}_q:H_q(\mathcal{R}_{c-\rho}) \to H_q(\mathcal{R}_{c})$ is not an isomorphism for all sufficiently small $\rho>0$. The image $F_q^{c-,c}$ of $f^{c-\rho,c}_q$ in $H_q(\mathcal{R}_{c})$ is independent of $\rho$, if this is small enough. The quotient group $B_q^c=H_q(\mathcal{R}_c)/F_q^{c-,c}$ is called the $q$-th birth group at $\mathcal{R}_c$, and it captures the homology classes that did not exist in $\mathcal{R}_{c-\rho}$. A homology class $\alpha \in H_q(\mathcal{R}_c)$ is born in $\mathcal{R}_c$ if it represents a non-trivial element in $B_q^c$, that is, the canonical projection of $\alpha$ is non-zero.
Now consider the homomorphism $g^{a,b}_q:B_q^a\to H_q(\mathcal{R}_b)/F^{a-,b}$, where $g^{a,b}_q([\alpha])=[f^{a,b}_q(\alpha)]$, for $\alpha \in H_q(\mathcal{R}_b)$, where the notation $[\cdot]$ stands for equivalence class. We set $g_q^{a,b}=0$ for all $b>0$ sufficiently large. The kernel $D_q^{a,b}$ of the map $g^{a,b}_q$ is called the death subgroup of $B_q^a$ at $\mathcal{R}_q^b$. A homology class $\alpha\in H_q(\mathcal{R}_a)$ dies entering $\mathcal{R}_b$ if $[\alpha]\in D_q^{a,b}$ but $[\alpha]\not\in D_q^{a,b-\rho}$, for $\rho>0$ sufficiently small. The degree $r$ of the death value $b$ of $B^a_q$ is defined by $r=\textrm{rank}D_q^{a,b}-\textrm{rank}D_q^{a,b-}$. The sum of the degrees of all death value of the birth group $B_q^a$ is clearly equal to $\textrm{rank}(B^a_q)$. The birth time of a homology class $\alpha$ is the value $a>0$ where the $\alpha$ is born in $\mathcal{R}_a$, and the death time is the value $b>0$ where $\alpha$ dies in $\mathcal{R}_b$.
The $q$-persistence diagram of the filtration $(\mathcal{R}_\delta)_{\delta>0}$ is defined as a multiset $\mathcal{P}_q$ in $\mathbb{R}^2$ consisting of points of the type $z_i=(a,b_i)$, where $a$ is a birth value corresponding to a non-trivial group $B_p^a$, and $b_i$ is a death value of $B_p^a$; the point $z_i$ appears in the diagram with multiplicity equal to the degree $r_i$ of the death value $b_i$. Since deaths occur after births, all points $(a,b_i)$ lie above the diagonal set of $\mathbb{R}^2$. By default, the diagonal set of $\mathbb{R}^2$ is part of the persistence diagram, representing all trivial homology generators that are born and die at every level. Each point on the diagonal has infinite multiplicity. The axes of the persistence diagram are birth values on the horizontal axis and death values on the vertical axis. Again, see Figure \[persistence\] for a schematic representation of the construction of a persistence diagram.
It is convenient to define a metric on the space of persistence diagrams. A number of options exist. A fairly standard metric is the $p$-Wasserstein metric. On the set of the $q$-persistence diagrams consider the $p$-Wasserstein metric, $1\leq p\leq\infty$, defined by $$d_p (\mathcal{P}^1_q, \mathcal{P}^2_q) = \left(\inf_{\phi} \sum_{z \in \mathcal{P}^1_q} \| z - \phi(z) \|_{\infty}^p\right)^{1/p},$$ where $\mathcal{P}^1_q, \mathcal{P}^2_q$ are two $q$-persistence diagrams, and the sum is taken over all bijections $\phi : \mathcal{P}_q^1 \rightarrow \mathcal{P}^2_q$. The set of bijections, $\{ \phi:\mathcal{P}^1_q \to \mathcal{P}^2_q\}$, is nonempty owing to the fact that each diagram includes the diagonal set, allowing one to match off-diagonal elements in one diagram with diagonal elements in another when their numbers differ.
The space of $q$-persistence diagrams together with the $p$-Wasserstein metric forms a metric space, which is complete and separable. The Wasserstein distance takes the ‘best’ matching; that is, it minimizes the distance, relative to the $L_p$ norm, that one has to shift generators in $\mathcal{P}^1_q$ to match them with those in $\mathcal{P}^2_q$. In probability theory, and in particular cases with continuous or weighted distributions, the Wasserstein metric is sometimes termed the ‘earth mover distance’, which refers to the operation of transforming one distribution into another with the minimal change in mass. In what follows we set $p=2$ and drop the reference to $p$. For details on the Wasserstein metric, see [@Cohen-Steiner2010].
One of the remarkable properties of persistence diagrams is their stability, meaning that small changes in the initial point cloud data produce persistence diagrams that are close to one another relative to Wasserstein metric. The stability results are very general for the ‘bottleneck distance’, when $p=\infty$, and more restrictive for the Wasserstein metric with $p<\infty$. The essence of the stability results, as shown in [@Chazal2009; @Cohen-Steiner2010; @EdelsbrunnerM2012], is that the persistence diagrams depend Lipschitz-continuously on point cloud data.
In applications, the stability result ensures the robustness of the data analysis performed via persistence diagrams, which makes them a powerful alternative to statistical methods. This is particularly useful in context of data from stochastic systems, since persistence diagrams turn out to be quite versatile in distinguishing between small but relevant features in a data set and noise.
### Detection of critical transitions via persistence diagrams
We now describe how to apply this method to detect critical transitions in time series. Consider a time series $(t_j,x_j)$, $j=1,\ldots, J$, with $x_j\in\mathbb{R}^d$. (In the case of a time series obtained from the model discussed in Section \[section:model\], we will chose $d=1$). Assume that the time series $(t_j,x_j)$ is obtained as a time discretization of a process $(t,x_t)$ which is Lipschitz continuous. (This is indeed the case when a time series is obtained from a Langevin equation, as in Subsection \[section:model\].) To each $t_i$ we associate a string of $N$ consecutive data points points from the time series, with $N$ sufficiently large, which we denote $$\begin{aligned}
\label{eq:data-string}
t_i\mapsto X_i=(x_i,x_{i+1}, \ldots, x_{i+N-1}).\end{aligned}$$ We regard each $X_i$ as a point cloud set in $\mathbb{R}^d$. We compute the persistence diagrams $\mathcal{P}_*(X_i)$ of $X_i$, in all dimensions, and follow the evolution of the persistence diagrams in time.
Diagrams corresponding to nearby times will be close to one another, due to the Lipschitz continuity of the process underlying the time series and to the robustness of persistence diagrams. Within this context, persistence diagrams that are near to each other in time, but relatively far from one another in the Wasserstein metric, indicate a sudden change in the time series. Therefore, we propose the following empirical criterion for detection of critical transitions in slow-fast systems:
- Persistence diagrams undergo significant changes, measured using the Wasserstein metric, prior to a critical transition.
This criterion follows from the following heuristic argument. If the noise level in the Langevin equation is small, then far from a critical transition the time series follows closely, with high probability, a trajectory of the slow subsystem. A point cloud associated to a data string displays significant topological features similar to those of the slow manifold, plus less significant topological features due to noise. In addition, the corresponding persistence diagrams at nearby times are close to one another relative to the Wasserstein metric.
When the system undergoes a critical transition, the time series ceases to follow the slow manifold, as the dynamics enters a transient regime. The topological features associated to the slow manifold are destroyed, and new topological features appear in the point cloud structure. Furthermore, if the system moves to a different stable regime after a finite time, the point cloud will reflect the topological features associated to that regime. Critically, for a point cloud data [*near*]{} a critical transition the corresponding persistence diagrams shift away from those diagrams corresponding to data far from the critical transition. Consequently, successive distances between diagrams in this region exhibit a large jump prior to a critical transition.
Results {#sec:results}
=======
We numerically solve the SDE defined in – using the Euler-Maruyama procedure, with stepsize $0.01$ and noise level $\sigma_1=\sigma_2=0.02$. We fix the rate of change of the parameter $S$ to be ${\varepsilon}=10^{-4}$. As output, we choose the time series given by the $R$-component; a particular realization of this time series is shown in the righthand panel of Fig. \[substratedepletion\]. Since the solution values are dense in time, we subsample the time series by taking every $10$-th data point. The time series follows a slowly varying attractive equilibrium point, until it reaches a critical transition, at which point it enters an oscillatory mode. To test for early signs of the critical transition, we truncate the time series before it enters the oscillatory regime. This truncated region is shown in Fig. \[cutoff\].
![$R$-time series for the gene regulatory network model from Section \[section:model\] cut off before the critical transition.[]{data-label="cutoff"}](beforecritical0-eps-converted-to.pdf){width="70.00000%"}
DFA analysis {#subsection:DFAanalysis}
------------
We conduct three experiments to detect critical transitions using the DFA[^1] methodology on the time series generated by our model. The first experiment uses a Gaussian kernel to detrend the time series; in the second experiment we use a cubic spline interpolation; and in the third experiment, we use Fourier interpolation. For each of the detrended time series we compute $AR(1)$ and the variance for a windowed time series. The results are summarized in Fig. \[substratedepletionDFA\]. All three experiments show $AR(1)$ increases to $1$ as the system approaches the transition, while the variance also grows steadily, both behaviors being consistent with a critical transition.
![DFA analysis of the time series using (top to bottom) Gaussian kernel, cubic spline, and Fourier interpolation detrending.[]{data-label="substratedepletionDFA"}](substratedepletionDFA-0-eps-converted-to.pdf "fig:"){width="100.00000%"}\
![DFA analysis of the time series using (top to bottom) Gaussian kernel, cubic spline, and Fourier interpolation detrending.[]{data-label="substratedepletionDFA"}](substratedepletionDFAcubic-0-eps-converted-to.pdf "fig:"){width="100.00000%"}![DFA analysis of the time series using (top to bottom) Gaussian kernel, cubic spline, and Fourier interpolation detrending.[]{data-label="substratedepletionDFA"}](substratedepletionDFAfourier-0-eps-converted-to.pdf "fig:"){width="100.00000%"}
Persistence diagram analysis {#subsection:persistenceanalysis}
----------------------------
We compute the $0$-dimensional persistence diagrams for strings of $N$ data points $X_i$ (see Eq. ) as the $t_i$ approaches a critical transition, as for the example time series in Fig. \[cutoff\]. We construct Rips complexes, with an initial radius of $\delta=10^{-4}$ around each data point, and then grow the radii of the balls $\delta,2\delta,\ldots,
n\delta$, where $n$ is chosen large enough so that the final complex in the resulting filtration has a single connected component. From this filtration we compute the $0$-persistence diagrams. We choose the size of the data sets $N=300$; larger sizes make little difference in the qualitative behavior of the diagrams.
The $0$-dimensional persistence diagrams are easy to interpret: they track of the births and deaths of connected components in the Rips complex, as the radii of the balls increase. Note that all births occur at the same time, when the radius of balls is zero and we have $N$ disjoint points. After this initial stage, a large number of connected components die as the radii of the balls increase, as connected components merge with one another.
$\begin{array}{cc}
\includegraphics[width=0.45\textwidth, clip, keepaspectratio]{sd4070-0-eps-converted-to.pdf} & \includegraphics[width=0.45\textwidth, clip, keepaspectratio]{sd9680-0-eps-converted-to.pdf}
\end{array}$
Consider Fig. \[fig:persistence\], where the persistence diagrams for a data string far from the critical transition (left panel) and a data string close to the critical transition (right panel) are shown. For data far from the critical transition, points cluster near the attractive equilibrium. Thus, when balls are constructed in the Rips filtration around these points, they will quickly yield a robust connected component around the attractive equilibrium , plus a small number of scattered connected components corresponding to points that escape for brief periods time from the equilibrium point due to stochastic effects. The implication is that, in the corresponding persistence diagram, the vertical spread of the death times is relatively small, and consists of a small numbers of points away from the diagonal (accounting for the robust connected component and a few outliers), plus many short-lived points close to the diagonal (accounting for noise).
Conversely, when a data string originates from close to a critical transition, the points tend to spread further away from the attractive equilibrium point, due to changes in the potential field. Heuristically, the equilibrium loses its attractiveness. This causes the distribution of the data points from the time series to grow. The implication is that, in the corresponding persistence diagram, the vertical spread of the death times is much larger, with a tendency to form multiple small clusters.
The visual inspection of persistence diagrams provides intuition, but is not a precise way to indicate the approach to a critical transition. To quantify the above assessment, we study the behavior of the Wasserstein distances between consecutive diagrams which are summarized in Fig. \[fig:sd\]. In the figure, time increases from left to right. The figures in the top row represent persistence diagrams for data sampled far from the critical transition, and those in the middle represent persistence diagrams for data sampled close to the critical transition. The five time frames captured in each column spread over a time interval of size $\Delta t=0.5$. We then compute the Wasserstein distances between consecutive persistence diagrams. These changes in the persistence diagrams are quantified in the bottom row of Fig. \[fig:sd\]. The solid curve, corresponding to data near the critical transition, shows a significant increase in the distances between consecutive diagrams as the point cloud anayzed near the critical transition. The dotted curve, corresponding to distances between diagrams far from the critical transition, shows only small variations in the consecutive distances. The computed distance are indicated by the symbols on each curve and are placed between the diagrams from which they were computed.[^2]
![The $0$-persistence diagrams on the top row correspond to consecutive strings of data sampled far from the critical transition; the diagrams in the middle row correspond to consecutive strings of data sampled near the critical transition. The bottom curves describe the distances between consecutive diagrams. The plot labels are positioned between two successive diagrams with values on the $y$-axis indicating the 2-Wasserstein distances between them.[]{data-label="fig:sd"}](persdia_distances_samplefig2-0-eps-converted-to.pdf){width="105.00000%"}
Note that the variance of the lifespans is related to the variance in the time series. An increase in the vertical spread while approaching a critical transition is consistent with the findings by the DFA method in Subsection \[subsection:DFAanalysis\]. Also, the change observed in the clustering of the diagram coordinates is related to asymmetric or multimodal properties of the data, as mentioned in Subsection \[subsection:persistence\].
Conclusions
===========
The substrate-depletion oscillator that we analyze in this paper is a realistic model for certain types of molecular regulation circuits studied experimentally. The methods for detecting critical transitions that we propose are suitable for the analysis of real data as well. Indeed, most of the experimental data obtained about gene regulatory networks (e.g., data obtained from microarrays, or reverse transcriptase polymerase chain reaction) is limited by background noise, and both the DFA and persistence diagram methods are robust to noise in data, as long as the noise does not overwhelm the signal. Also, in comparison with the DFA method, which necessitates a number of ‘ad-hoc’ choices of statistical parameters and procedures, the persistence diagram method appears more robust and objective.
In current and future work, we are developing a theoretical framework for the empirical criterion proposed in this paper. Namely, we plan to establish rigorously that bifurcation-induced critical transitions determine large changes in the persistence diagrams, and, conversely, large changes in the persistence diagrams imply the existence of bifurcations.
Acknowledgement {#acknowledgement .unnumbered}
===============
A portion of this work has been done while M.G. was a member of the IAS, to which he is very grateful. Also, we thank Konstantin Mischaikow, Miroslav Kramar, Vidit Nanda, and Rebecca M. Jones for many useful discussions on this subject.
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[^1]: The DFA procedure that we use here was implemented in Matlab by Rebecca M. Jones [@Jones]
[^2]: For the computation of persistence diagrams we use the Perseus software developed by Vidit Nanda [@Nanda]. The Wasserstein distances were computed using software written by Miroslav Kramar [@Miro2013].
|
---
abstract: 'The Next-To-Minimal-Supersymmetric extension of the Standard Model (NMSSM) has been in the focus of extensive studies in the past two decades. In anticipation of the LHC era, the interest in automatized tools that can calculate collider signatures has grown. We present the implementation of the NMSSM into the event generator WHIZARD. In addition to a brief review of the implementation, we discuss the testing and validation procedure. Phenomenological studies will not be presented here.'
author:
- Jürgen Reuter
- Felix Braam
bibliography:
- 'sample.bib'
title: The NMSSM implementation in WHIZARD
---
[ address=[University of Freiburg, Institute of Physics, Hermann-Herder-Str. 3, 79104 Freiburg, Germany]{} ]{}
[ address=[University of Freiburg, Institute of Physics, Hermann-Herder-Str. 3, 79104 Freiburg, Germany]{} ]{}
The Multi-Purpose Event Generator WHIZARD
=========================================
WHIZARD [@Kilian:2007gr] is a multi-purpose Monte-Carlo event generator for the Standard Model (SM) and beyond (BSM). It uses a multi-channel adaptive phase space integration based on the VAMP algorithm [@Ohl:1998jn], which is called by a highly efficient phase space grid decomposition relying on heuristics of the underlying resonance structure of field theoretic scattering amplitudes. The matrix element are delivered by the generator O’Mega [@Moretti:2001zz] which avoids in an optimal way all redundancies of tree-level scattering amplitudes and combines gauge-invariant substructures. WHIZARD was the first generator for full matrix elements for the MSSM [@whiz_susy] and has been validated by many groups and projects both for supersymetric models and alternative BSM models [@whiz_bsm]. Among the available models are half a dozen variants of Little Higgs models, extra-dimensional models, several (extended) supersymmetric models, general anomalous gauge, top and Higgs couplings, general unitarization models for electroweak scattering processes, noncommutative versions of the Standard Model etc.
WHIZARD can be accessed via the Hepforge web site: <http://projects.hepforge.org/whizard>. The alpha version of release 2.0.0 has been finished just before the SUSY conference, including the first implementation of the Next-to-Minimal Standard Model (NMSSM) in a multi-particle event generator, which is the main topic here. We briefly want to review the new features of version 2.0.0, which brings a major improvement over the older version 1. WHIZARD has been completely restructured in a fully object-oriented way. It allows for an arbitrary setup of structure functions, comprises a fully flexible script language for defining scales, cuts, analyses variables as arbitrary functions of kinematic variables and particle lists, it contains a rudimentary own parton shower (which will be further developed during the development of WHIZARD 2 and connected with a module for simulating multiple interactions which is right now being developed). WHIZARD is able to write out standard event formats like LHA, LHEF, STDHEP, HepMC etc. It comes with its own graphics analysis tool, while an interface to ROOT will also be available in WHIZARD 2. Also, an interface to DELPHES [@Ovyn:2009tx] for a fast detector simulation is under way.
Although, WHIZARD is able to deliver matrix elements for $2\to 16$ processes (or higher multiplicities) and integrate processes with at least up to ten particles in the final states, this might not be sufficient for BSM models with a discrete parity symmetry as suggested by the existence of dark matter. Hence, WHIZARD 2 is also able to integrate and simulate long cascade decay chains with full spin correlations. A further improvement is a full-fledged interface to FeynRules [@feynrules] for a more automatized inclusion of new physics models for both WHIZARD versions 1 and 2.
The NMSSM
=========
The Next-to-Minimal Supersymmetric Standard Model (NMSSM) is a viable extension of the MSSM in the Higgs sector of the model (see e.g. the references in [@Accomando:2006ga]). The main motivation for the NMSSM is a possible solution to the so-called $\mu$ problem, the fact, that the supersymmetric parameter $\mu$ in the superpotential is dimensionful and should in principle be of the order of the SUSY breaking scale. A working electroweak symmetry breaking demands this parameter, however, in the ball park of a few hundred GeV. The NMSSM tackles this problem by enlarging the particle spectrum by a single left-chiral superfield $S$, being a singlet under the SM gauge group. Their are two general options to generate a quartic potential term for the scalar part of $S$: either a $D$ term of an additional $U(1)$ guage symmetry (like e.g. in the PSSSM [@Kilian:2006hh]) or by a $F$ term from a cubic superpotential term (conventional NMSSM). For the conventions, we follow those of the SUSY Les Houches Accord 2 [@slhaetc] (for a recent review on NMSSM conventions cf. also [@Accomando:2006ga]). This is a quite general approach, but neglecting possible CP, $R$-parity, or flavor violation. The suporpotential of the NMSSM is given by $$\label{eq:nmssmsup}
W_{NMSSM} = W_{MSSM} - \epsilon_{ab}\lambda {S} {H}^a_1 {H}^b_2 +
\frac{1}{3}
\kappa {S}^3 + \mu' S^2 +\xi_F S \ ,$$ where we neglect an explicit $\mu'$ term and the Fayet-Iliopoulos term for the singlet superfield. The most general soft-breaking terms for the NMSSM $$\label{eq:nmssmsoft}
V_\mathrm{soft} = V_{2,MSSM} + V_{3,MSSM} + m_\mathrm{S}^2 | S |^2 +
(-\epsilon_{ab}\lambda A_\lambda {S} {H}^a_1 {H}^b_2 +
\frac{1}{3} \kappa A_\kappa {S}^3
+ m_{S}'^2 S^2 +\xi_S S
+ \mathrm{h.c.}) \ ,$$ using the conventions from [@slhaetc].
The field content of the NMSSM is almost the same as for the MSSM, except for an additional scalar and pseudoscalar Higgs boson, denoted by $H_3^0$ and $A_2^0$, as well as a fifth neutralino, $\tilde{\chi}_5^0$ coming from the additional singlino component.
Implementation and Validation
=============================
As mentioned in the section about the model before, we are sticking to the SLHA 2 conventions. This fixes the uncertainties in signs and phases. As for the MSSM, the WHIZARD implementation uses explicitly positive masses for charginos and neutralinos, and puts signs and complex phases instead in the corresponding mixing matrices. In contrast to the MSSM, the NMSSM implementation is more flexible than the MSSM one (although now thanks to Björn Herrmann there is a completely general MSSM implementation in WHIZARD), as it allows not only for a full CKM matrix but also for a left-/right mixing for all generations, but no inter-generational mixing. The generalization to the CP-non-conserving case is easily possible.
The NMSSM is a whole is an incredibly complicated model with order 6,700 couplings (including quartic and Goldstone couplings, most of each are fortunately not of phenomenological importance). It is mandatory to check an implementation as far as possible. For this task, we followed the strategy given in [@Hagiwara:2005wg]: unitarity checks for $2\to 2$ and $2\to 3$ scattering amplitudes have been performed, and we tested Ward- and Slavnov-Taylor identities for both gauge and supersymmetry. Another check was to reproduce the correct MSSM limit for the NMSSM, namely letting the trilinear singlet Higgs coupling and the cubic singlet coupling approach zero, sending the singlet vev to infinity while keeping the combination of the latter two fixed to $\mu$: $\lambda \to 0$, $\kappa \to 0$, $\left< S
\right> \to \infty$, $\left< S \right> \lambda \to \mu$.
![\[tab1\] Comparison between WHIZARD and the FeynRules implementation within Madgraph. The green color shows agreement, at least up to the Monte Carlo integration error. We used $\sqrt{s} = 3$ TeV to be above all thresholds.](tautau3000_1 "fig:"){height=".4\textheight"} ![\[tab1\] Comparison between WHIZARD and the FeynRules implementation within Madgraph. The green color shows agreement, at least up to the Monte Carlo integration error. We used $\sqrt{s} = 3$ TeV to be above all thresholds.](tautau3000_2 "fig:"){height=".4\textheight"}
What is by far one of the most stringent tests is the comparison with an independent implementation. For that purpose we are using a FeynRules generated NMSSM model file for MadEvent/Madgraph [@mgme] as well as CalcHEP [@Pukhov:2004ca]. Table \[tab1\] shows the comparison for $\tau^+\tau^-$ initial state processes as an example. The corresponding SLHA 2 input file is left out here due to reasons of limited space. Our comparison contains far more than 600 processes, showing full agreement between the two implementations.
Conclusions
===========
We presented the complete implementation of the NMSSM implementation into the multi-purpose event generator WHIZARD. A vast sample of consistency checks and comparisons with independent implementations show the correctness of the code. WHIZARD is available as a modern high-level tool for NMSSM signal and background simulations for the LHC era. A detailed version of the comparison and validation of the NMSSM implementations will appear soon [@bfr].
The authors are supported by the Ministerium für Bildung und Kultur of the state Baden-Württemberg by the program ZO IV and the German Research Society (DFG) under grant no. Re 2850/1-1. JR would like to thank to Aspen Center for Physics for their hospitality. We are also grateful to Ulrich Ellwanger for valuable remarks and discussions, as well as Benjamin Fuks, Neil Christensen, Claude Duhr and Christian Speckner for the FeynRules collaboration.
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---
abstract: 'To describe non-equilibrium transport processes in a quantum device with infinite baths, we propose to formulate the problems as a reduced-order problem. Starting with the Liouville-von Neumann equation for the density-matrix, the reduced-order technique yields a finite system with open boundary conditions. We show that with appropriate choices of subspaces, the reduced model can be obtained systematically from the Petrov-Galerkin projection. The self-energy associated with the bath emerges naturally. The results from the numerical experiments indicate that the reduced models are able to capture both the transient and steady states.'
author:
- Weiqi Chu
- Xiantao Li
bibliography:
- 'JCTC.bib'
title: 'A reduced-order modeling approach for electron transport in molecular junctions'
---
Introduction
============
In the past decades, there has been significant progress in the investigation of molecular electronics and quantum mechanical transport [@aviram1989molecular; @reed1999molecular; @joachim2000electronics], one emerging issue among which is the modeling of interfaces or junctions between molecular entities [@aradhya2013single; @cahen2005energetics; @cahen2003electron; @hwang2009energetics]. The junctions encompass two sections: (i) a molecular core at the nanometer scale that bridges two metallic devices; (ii) the surrounding areas from contacting materials. Notable examples include quantum dots, quantum wires, and molecule-lead conjunctions. The junctions play an essential role in determining the functionality and properties of the entire device and structure, such as photovoltaic cells [@brabec2004organic; @bredas2009molecular], intramolecular vibrational relaxation [@poulsen2001path; @potter2000transport; @everitt1998vibrational; @nibbering1991femtosecond], infrared chromophore spectroscopy, and photochemistry [@pshenichnikov1995time; @joo1995ultrafast; @becker1989femtosecond; @lang1999aqueous]. At such a small spatial and temporal scale, modeling the transport properties and processes demands a quantum theory that directly targets the electronic structures.
Such problems have been traditionally treated with the Landauer-Büttiker formalism [@landauer1970electrical; @buttiker1985generalized; @buttiker1986four], which aims at computing the steady-state of a system interacting with two or more macroscopic electrodes, and the non-equilibrium Green’s function (NEGF) approach, which, often based on the tight-binding (TB) representation, can naturally incorporate the external potential and predict the steady-state current [@cini1980time]. This approach was later extended to the first-principle level [@lang1995resistance; @taylor2001ab; @brandbyge2002density] using the density-functional theory (DFT) [@hohenberg1964inhomogeneous; @Kohn1965].
Due to the dynamic nature and the involvement of electron excitations, one natural computational framework for transport problems is the time-dependent density-functional theory (TDDFT) [@kurth2005time; @runge1984density; @stefanucci2004time; @burke2005density; @cheng2006simulating], which extends the DFT to model electron dynamics. This effort was initiated by Stefanucci and Almbladh [@stefanucci2004time-prb; @stefanucci2004time], and Kurth et al. [@kurth2005time], where the wave functions are projected into the center and bath regions. An algorithm was developed to propagate the wave functions confined to the center region so that the influence from the bath is taken into account. This is later treated by using the complex absorbing potential (CAP) method [@baer2004ab] by Varga [@varga2011time]. One computational challenge from this framework is the computation of the initial eigenstates. Kurth et al. [@kurth2005time] addressed this issue by diagonalizing the Green’s function. However, the normalization is still nontrivial, since the wave functions also have components in the bath regions. Another issue is that the CAP method is usually developed for constant external potentials. For time-dependent scalar potentials, a gauge transformation is usually needed to express the absorbing boundary condition [@antoine2003unconditionally], and it is not yet clear how this can be implemented within CAP.
Another framework is based on the Liouville-von Neumann (LvN) equation [@sanchez2006molecular; @subotnik2009nonequilibrium] to compute the density-matrix operator directly. One advantage of the LvN approach is that the initial density-matrix can be obtained quite easily from the Green’s function. Therefore diagonalization and normalization are not needed. To incorporate the influence of the bath, the LvN equation has been modified by adding a driving term at the contact regions according to the potential bias. This approach was later extended by Zelovich and coworkers [@zelovich2014state; @zelovich2015molecule], which is again motivated by the CAP method. Despite the heuristic derivation [@zelovich2014state], these methods are still empirical in modeling the electron transport problem. In particular, the steady state and transient predicted by the driven LvN equation have not been compared with those from the full model.
This paper follows the density-matrix-based framework. Rather than using the approach by Sánchez et al. [@sanchez2006molecular], we derive the open quantum system using the reduced-order techniques that have been widely successful in many engineering applications [@bai2002krylov; @freund2000krylov; @villemagne1987model]. We first formulate the full quantum system as a large-dimensional dynamical system with low-dimensional input and output. This motivates a subspace projection approach, which has been the most robust method in reduced-order modeling [@bai2002krylov; @freund2000krylov]. In particular, we employ the Petrov-Galerkin projection, a standard tool in numerical computations, [[*e.g.*]{}]{}, linear systems, eigenvalue problems, matrix equations, and partial differential equations (PDEs)[@smith1985numerical; @johnson2012numerical; @lambert1973computational; @morton2005numerical]. With appropriate choices of the subspaces, we obtain a reduced LvN equation, modeling an open quantum system where the computational domain only consists of the center and contact regions. We illustrate the procedure for a one-dimensional model system, as a first step to treat more realistic systems. The numerical results have shown that the reduced LvN equations can capture both the transient and the steady state solutions.
The rest of the paper is organized as follows. In Sec. \[sec: 2\], we provide a detailed account of our methodology, including the mathematical framework and the derivation of the reduced models. In Sec. \[sec: results\], we present results from some numerical experiments to examine the effectiveness of the derived models. Sec. \[sec: summary\] summarizes the methodology and provides an outlook of future works.
Methods and algorithms {#sec: 2}
======================
The density-matrix formulation
------------------------------
Following the conventions from existing literature [@brandbyge2002density; @kurth2005time; @zelovich2014state; @zelovich2015molecule], we consider a molecular junction, where a molecule is connected to two semi-infinite leads. More specifically, the physical domain for the entire system is denoted by $\Omega$, divided into three parts, $\Omega_{L}, \Omega_{C}$, and $\Omega_{R}$ representing respectively the left lead, the center region, and the right lead, as illustrated in Figure \[fig: schematic1\].
![(Color online) Schematic representation of a two semi-infinite lead junction model consisting of two semi-infinite leads: left lead (L), right lead (R), and an extended molecule (C) in the center.[]{data-label="fig: schematic1"}](schematic1.eps){width="80.00000%"}
We will start with the LvN equation, which for molecular conduction problems, has been proposed and implemented in a series of papers. [@sanchez2006molecular; @subotnik2009nonequilibrium; @zelovich2014state; @zelovich2015molecule]. The LvN equation governs the dynamics of the density-matrix operator $\hat{\rho}$, which can be connected to the wave functions ([[*e.g.*]{}]{}, the Kohn-Sham orbitals) as follows, $$\hat{\rho}(\bm r,\bm r',t) = \sum_{j} n_{j} \hat{\psi}_{j}(\bm r,t)\hat{\psi}_{j}(\bm r',t)^{*},$$ with $n_{j}$ being the occupation numbers. The equation can be derived from a time-dependent Schrödinger equation (TDSE), and for the entire system $\Omega$, it can be written as, $$\label{eq: LvN}
i {\partial_t} \hat{\rho}(t) = \hat{H}(t)\hat{\rho}(t) - \hat{\rho}(t)\hat{H}(t)=[\hat{H}(t),\hat{\rho}(t)].$$
Here the bracket is the usual quantum commutator, which we will generalize as follows, $$\label{eq: comm}
[\hat{A},\hat{B}] := \hat{A}^{*}\hat{B} - \hat{B}^{*}\hat{A}.$$ Here $A^*$ denotes the conjugate transpose (or Hermitian transpose of $A$). Notice that with this generalization, $A$ or $B$ can be non-Hermitian.
Our goal is to derive an [*open*]{} quantum system for the density-matrix at the center region $\Omega_C$, where the influence from the leads is implicitly incorporated. For convenience, we first assume that the entire system has been appropriately discretized in $\Omega$ so that $\rho(\bm r,\bm r',t)$ is a matrix defined at certain grid points, here denoted by $\Omega_{\Delta}$ with $\Delta$ indicating the grid size. Namely, $\rho(\bm r, \bm r', t)$ is the density-matrix with $\bm r, \bm r' \in \Omega_{\Delta}$. This can be obtained by using a finite-difference scheme, especially in real-space methods [@beck2000real]. As a result, one arrives at a matrix-valued infinite-dimensional system, and hence we will drop the $\hat{\quad}$ notation from now on. A similar system can also be obtained using the TB approximation, where the wave functions are projected to atomic-centered orbitals, in which case, the LvN equation would contain the overlap matrix on the left hand side when the basis functions are not orthogonal. [@zelovich2015molecule; @sankey1989ab] However, it would not affect our following reduction method.
Following the setup by Cini [@cini1980time], we treat the problem as an initial value problem (IVP), starting with an initial density $\rho_{0}=f_{eq}(\mu-H_{0})$ as an equilibrium density at $t=0$. Such setup is particularly amenable for numerical computations. While it is challenging to compute the wave function in a subdomain, which in general requires solving nonlinear eigenvalue problems and normalization [@Inglesfield1981], efficient algorithms are available to calculate the density-matrix in a sub-domain [@KellyCar92; @lin2011selinv; @williams1982green]. These algorithms take advantage of the relation between the density-matrix and the Green’s function, $$\label{eq: g-to-rho}
\rho = \frac{1}{2\pi i} \oint_C G(z) dz, \quad G(z)= (zI - H)^{-1},$$ where the contour encloses all the occupied states. The restrictions of the density-matrix to a finite subdomain can be obtained by $E^* \rho E$, where the operator $E$, with proper arrangement, can be written simply as $E^*=[I, \quad 0],$ with the identity operator $I$ corresponding to the subdomain and the zero matrix corresponding to the exterior (bath). This observation, together with , reduces the problem to the computation of the following expression that we have slightly generalized the linear algebraic system to, $$\label{eq: reduced-G}
[ \times \quad 0] (zI - H)^{-1} \left[
\begin{array}{c}
\times \\
0 \end{array}\right],$$ where the left and right vectors have finite supports. Although this amounts to solving an infinite-dimensional linear system, a finite number of unknowns are needed due to the multiplication by the sparse vector on the left and right. For one-dimensional (or quasi one-dimensional) systems, an iterative scheme can be used [@godfrin1991method; @pecchia2008non] to invert the block tri-diagonal matrix. For multi-dimensional problems, a discrete boundary element method [@Li2012] can be used [@li2016pexsi]. We will refer to these algorithms in general as [*selective inversion* ]{}[@lin2011selinv].
Although our model works with the density-matrix, our primary interest is in the electric current induced by a time-dependent external potential that is switched on at $t=0_+$. Similar to the theory of linear response [@gross1985local; @dobson1997time; @gunnarsson1976exchange], we consider $H(t)$ as a deviation from its initial value $H_{0}$ and write $H(t)=H_{0}+{{\delta H}}(t)$ with ${{\delta H}}(t)$ being the applied potential from the leads. The response of the system due to the external potential could be represented in terms of the perturbed density, $${{\delta\rho}}(t) :=\rho(t) - \rho_{0}, \quad {{\delta\rho}}(0)=0,$$ which satisfies a response equation, $$\label{eq: drhoeq}
i\frac{d}{dt}{{\delta\rho}}(t)= [H(t),{{\delta\rho}}(t)] + \Theta(t).$$ Here $\Theta(t) = [{{\delta H}}(t),\rho_{0}]$ is a non-homogeneous term that incorporates the influence from the external potential.
As is customary [@kurth2005time; @li2019absorbing; @sanchez2006molecular; @zelovich2014state], we neglect the direct coupling between the two leads and partition the density-matrix and the Hamiltonian operator in accordance with the partition of the domain indicated in Figure \[fig: schematic1\]. In this case, Eq translates to $$\label{eq: blockLvN}
i\frac{d}{dt}
\left(
\begin{array}{ccc}
{{\delta\rho}}_{LL} & {{\delta\rho}}_{LC} & {{\delta\rho}}_{LR} \\
{{\delta\rho}}_{CL} & {{\delta\rho}}_{CC} & {{\delta\rho}}_{CR} \\
{{\delta\rho}}_{RL} & {{\delta\rho}}_{RC} & {{\delta\rho}}_{RR} \\
\end{array}
\right)
=
\left[
\left(
\begin{array}{ccc}
H_{LL} & H_{LC} & 0 \\
H_{CL} & H_{CC} & H_{CR} \\
0 & H_{RC} & H_{RR} \\
\end{array}
\right),
\left(
\begin{array}{ccc}
{{\delta\rho}}_{LL} & {{\delta\rho}}_{LC} & {{\delta\rho}}_{LR} \\
{{\delta\rho}}_{CL} & {{\delta\rho}}_{CC} & {{\delta\rho}}_{CR} \\
{{\delta\rho}}_{RL} & {{\delta\rho}}_{RC} & {{\delta\rho}}_{RR} \\
\end{array}
\right)
\right] + \Theta.$$
We are interested in the case when $\delta H$ corresponds to scalar potentials in the leads, given by $ U_L(t)$ and $ U_R(t).$ Then the matrix function $\Theta(t)$ can be written as, $$\label{eq: Theta(t)}
\Theta(t) = \left[\begin{array}{ccc} 0 & U_L(t) \rho_{LC}(0) & ( U_L(t) - U_R(t)) \rho_{LR}(0) \\
- U_L(t) \rho_{CL}(0) & 0 & - U_R(t) \rho_{CR}(0) \\
- ( U_L(t) - U_R(t)) \rho_{RL}(0) & U_R(t) \rho_{RC}(0) & 0\end{array}\right].$$
In practice, to mimic the infinite leads, one has to pick much larger regions $\Omega_{L/R}$ to prevent the finite size effect, [[*e.g.*]{}]{}, a recurrence. This makes a direct implementation using Eq impractical and requires model reduction tools to reduce the complexity of the full problem.
There are six unknown blocks in the density-matrix $\delta\rho:$ the blocks $\delta\rho_{LL}$ (and $\delta\rho_{RR}$) are semi-infinite, and this is where an appropriate reduction is needed. It suffices to illustrate the reduction of the degrees of freedom in the left bath. A direct computation yields $$\label{eq: leftLvN}
i\frac{d}{dt} \delta \rho_{LL}(t) = [H_{LL}(t),\delta\rho_{LL}(t)] + F_{L}(t),$$ where $H_{LL}(t)=H_{LL}(0)+{{\delta H}}_{LL}(t)$ and ${{\delta H}}_{LL}(t)$ is the external potential imposed on the left lead. $F_{L}(t)$ represents the influence from the interior and can be extracted from , $$\label{eq: UL}
F_{L}(t)=H_{LC}{{\delta\rho}}_{CL}(t) - {{\delta\rho}}_{LC}(t)H_{CL}+{\Theta_{LL}(t)}.$$
Now our key observation is that Eqs and constitute an infinite-dimensional control problem with control variables $\delta \rho_{CL}$ and output ${{\delta\rho}}_{LL}$. In practice, only the entries in ${{\delta\rho}}_{LL}$ near the interface (between $\Omega_L$ and $\Omega_C$) are needed. Such a large-dimensional dynamical system with low-dimensional input and output can be effectively treated by using the reduced-order techniques [@bai2002krylov; @freund2000krylov; @gugercin2013model; @lucia2004reduced; @nayfeh2005reduced].
General Petrov-Galerkin projection methods
------------------------------------------
Motivated by the development of reduced-order modeling techniques [@gugercin2013model; @lucia2004reduced; @decoster1976comparative] that have been widely used in control problems [@villemagne1987model], circuit simulation [@freund2000krylov], and microelectromechanical systems [@nayfeh2005reduced], etc., we propose a Petrov-Galerkin projection approach to derive a reduced model from the infinite-dimensional LvN Eq . The objective is to provide a reduced dynamics for the device region that captures both the transient and the steady state.
The first ingredient is to pick an appropriate subspace where the approximate solution is sought. To start with, we pick an $n$-dimensional subspace $\mathcal{V}_{L}$ spanned by a group of basis functions $\{\varphi_{i}\}_{i=1}^{n}$. The subspace can be expressed in a matrix form as $V_{L}= [\varphi_1, \varphi_2, \cdots, \varphi_n]$: $\mathcal{V}_{L}=\text{Range}(V_L).$ Throughout this paper, we will not distinguish a subspace $\mathcal{V}_{L}$ and its matrix representation $V_L$.
In practice, the basis functions can be standard hat functions centered at certain grid points, as shown in Figure \[fig: spaceV\], or Gaussian-like functions that mimic atomic orbitals.
(-3,0) – (13,0); iin [1,6,9,10,11,12]{} [ (i-1,0) – (i,4); (i+1,0) – (i,4); ]{} iin [-2,...,12]{} (i,0) – (i,0.3);
iin [4.5]{} [ (12, i) node [$\varphi_1$]{}; (11, i) node [$\varphi_2$]{}; (10, i) node [$\varphi_3$]{}; (9, i) node [$\varphi_4$]{}; (6, i) node [$\varphi_5$]{}; (3.5, i) node [$\cdots$]{}; (1, i) node [$\varphi_n$]{}; ]{} (5,-0.8) node [$\Omega_{L}$]{};
With the subspace set up, one can seek a low-rank approximation of ${{\delta\rho}}_{LL}(t)$ as ${{\delta\widetilde{\rho}}}_{LL}$ in the following form, $$\label{eq: g1}
{{{\delta\widetilde{\rho}}}}_{LL}(t) := V_{L} D_{LL}(t) V_{L}^{*},$$ where the $n\times n$ matrix $D_{LL}(t)$ represents the nodal values. This representation automatically guarantees that the resulting density-matrix is Hermitian and semi positive-definite, as long as $D_{LL}$ has those properties. The residual error from this approximation can be directly deduced from the LvN equation by subtraction, $${\mathcal{E}}(D_{LL},t) = iV_L \frac{d}{dt} D_{LL}(t) V_L^* - [H(t), V_{L} D_{LL}(t) V_{L}^{*}] -F_L(t).$$ The second ingredient to determine $D_{LL}$ is by projecting the residual error to the orthogonal complement of a test subspace, $\mathcal{W}_L$, spanned by the columns of $W_{L}$, that is $$\label{eq: g3}
W^{*}_{L}\mathcal{E}(D_{LL})W_{L}=0.$$
This yields a finite-dimensional system, and the reduction procedure described above is known in general as the Petrov-Galerkin projection, which has been a classical numerical method in the solutions of differential equations [@larsson2008partial], order-reduction problems [@bai2002krylov; @freund2000krylov], and matrix equations [@jaimoukha1994krylov; @jbilou2006projection].
The reduced equation from the Petrov-Galerkin projection Eqs to can be written as, $$\label{eq: reducedleftLvN}
i\frac{d}{dt} D_{LL}(t) = [{{\widetilde{H}}}_{LL}M_{L}, D_{LL}] - \widetilde{F}_{L}(t),$$ where the matrices are given by $$\begin{aligned}
M_{L}&=\left(V_{L}^{*}W_{L}\right)^{-1},\\
{{\widetilde{H}}}_{LL}(t)&=V_{L}^{*}H_{LL}(t)W_{L},\\
\widetilde{F}_{L}(t)&=M_{L}^{*}W_{L}^{*}F_{L}(t)W_{L}M_{L}.
\end{aligned}$$ Notice that in we have used the generalized notation of commutators . At this point, we will keep the subspaces spanned by $V_L$ and $W_L$ at the abstract level, and the specific choices will be discussed in the next section.
The same model reduction procedure can be applied to the right lead and it yields a similar finite-dimensional equation, $$\label{eq: reducedrightLvN}
i\frac{d}{dt} D_{RR}(t) = [ {{\widetilde{H}}}_{RR}M_{R}, D_{RR}] - \widetilde{F}_{R}(t).$$ Eqs and are related by the non-homogeneous terms $\widetilde{F}_{\alpha}(t), \alpha={L,R}$ that involve the evolution of ${{\delta\rho}}_{C\alpha}$ and their Hermitian transpose.
In the center region, no reduction is needed and we will retain this part of Eq . Therefore, we can construct a Petrov-Galerkin projection for the [*entire*]{} system, by [*gluing* ]{} the subspaces as follows,
$$\label{eq: VW}
V = \left[
\begin{array}{ccc}
V_{L} & 0 & 0 \\
0 & I_{n_{C}} & 0 \\
0 & 0 & V_{R} \\
\end{array}
\right], \quad
W = \left[
\begin{array}{ccc}
W_{L} & 0 & 0 \\
0 & I_{n_{C}} & 0 \\
0 & 0 & W_{R} \\
\end{array}
\right].$$
We seek an approximate solution $$\label{eq: delrho}
{{\delta\rho}}(t) \approx {{\delta\widetilde{\rho}}}(t) := VD(t)V^{*},$$ for the projected dynamics of Eq , such that, $$i\frac{d}{dt} W^{*}{{\delta\widetilde{\rho}}}(t)W = W^{*} \big( [H(t),{{\delta\widetilde{\rho}}}(t)] + \Theta(t) \big) W.$$
Direct computations yield, $$\label{eq: Dequation}
i\frac{d}{dt} D(t) = [{H}_\text{eff},D] + \widetilde{\Theta}(t),$$ where ${H}_\text{eff}$ is the reduced [Hamiltonian]{}, $$\label{eq: tildeH}
{H}_\text{eff} =
\left[
\begin{array}{ccc}
V_{L}^{*}H_{LL}W_{L}\left( V^{*}_{L}W_{L}\right)^{-1} &V^{*}_{L}H_{LC} & 0 \\
H_{CL}W_{L}\left( V^{*}_{L}W_{L}\right)^{-1} & H_{CC} & H_{CR}W_{R}\left( V^{*}_{R}W_{R} \right)^{-1} \\
0 & V^{*}_{R}H_{RC} & V^{*}_{R}H_{RR}W_{R}\left( V^{*}_{R}W_{R} \right)^{-1} \\
\end{array}
\right],$$ and $\widetilde{\Theta}(t)$ is given by $$\widetilde{\Theta}(t) = M^{*}W^{*} \Theta(t) WM = M^{*}W^{*}[{{\delta H}}(t), \rho_{0}]WM.$$ Here the matrix $M$ is block-diagonal, $$\label{eq: M}
M = \left[
\begin{array}{ccc}
\left( V^{*}_{L}W_{L}\right)^{-1} & 0 & 0 \\
0 & I_{n_{C}} & 0 \\
0 & 0 & \left( V^{*}_{R}W_{R}\right)^{-1} \\
\end{array}
\right].$$
It is worthwhile to point out that the subspaces can also be time-dependent. This offers the flexibility to pick subspaces that evolve in time. It should also be emphasized that our discussions regarding the Petrov-Galerkin projection is suitable for general cases and not limited to one-dimensional junction models, [[*i.e.*]{}]{}, the typical lead-molecule-lead structures. With appropriate domain decomposition, it can be applied to high-dimensional systems with more general device structures.
The selection of the subspaces
------------------------------
In this section, we discuss specific choices of the subspaces in the Galerkin-Petrov projection. Without loss of generality, we again start by considering the left lead $\Omega_{L}$. Let $\Omega_{\Gamma_{\!L}} \subset \Omega_{L}$ be a subdomain in the left lead that is adjacent to the center region, as shown in Figure \[fig: schematic2\]. $\Omega_{\Gamma_{\!L}}$ and $\Omega_{\Gamma_{\!R}}$ are often referred to as contact regions that have direct coupling with the interior [@williams1982green; @do2014non]. In our case, we pick $\Omega_{\Gamma_{\!R}}$ in such a way that the remaining component in the lead has no coupling with the center region, [[*i.e.*]{}]{}, $H_{i,j}=0$ for $i \in \Omega_C$ amd $j \in \Omega_{R}-\Omega_{\Gamma_{\!R}}$. This imposes a lower bound on the size of the contact region.
![(Color online) A schematic representation of junction model with contact regions in green.[]{data-label="fig: schematic2"}](schematic2){width="70.00000%"}
In reduced-order modeling problems, the subspaces are often chosen based on how the input/control variables enter the large-dimensional system, [[*e.g.*]{}]{}, see the review papers [@bai2002krylov; @freund2000krylov]. In our setting, we consider the dynamics in the left lead, given the density-matrix in the contact region. So we pick the basis $V_L$ so that $V^{*}_{L}$ acts as a restriction operator from $\Omega_{L}$ to $\Omega_{\Gamma_{\!L}}$, $$\label{eq: VL}
V^{*}_{L} = [0, I_{n_{\Gamma,L}}],$$ where $I_{n_{\Gamma,L}}$ is an identity matrix with the dimension $n_{\Gamma,L}$ being the number of grid points in $\Omega_{\Gamma_{\!L}}$.
The same procedure can be applied to the other lead region. When the subspaces are combined (cf. Eq ), we have, $$\label{eq: V'}
V = \left[
\begin{array}{ccc}
0 & 0 & 0,\\
I_{n_{\Gamma,L}} & 0 & 0 \\
0 & I_{n_{C}} & 0 \\
0 & 0 & I_{n_{\Gamma,R}} \\
0 & 0 & 0
\end{array}
\right].$$ The entire density-matrix is approximated as in . It is now clear that $V$ is a restriction operator to an extended center domain, $ \Omega_{\widetilde{C}} = \Omega_{\Gamma_{\!L}}\cup \Omega_{C} \cup \Omega_{\Gamma_{\!R}}$. Consequently, $D$ in Eq becomes the density-matrix in $\widetilde{C}$, $$D(t) = {{\delta\widetilde{\rho}}}(t)|_{\Omega_{\widetilde{C}}\times\Omega_{\widetilde{C}}}.$$
It remains to choose the subspaces $W_{L/R}$. Motivated by the Green’s function approach for quantum transport [@kadanoff1962quantum; @caroli1971direct; @datta2005quantum], we consider the test space, $$\label{eq: WL}
W_{L}(\varepsilon) = (\varepsilon I - H_{LL})^{-1}V_{L},$$ where $\varepsilon \in \mathbb{C}$ is in the resolvent space of the Hamiltonian $H_{LL}$. We require that $\text{Im} \big(\varepsilon\big) <0 $ to ensure the stability of the reduced models. In this case, it corresponds to the advanced Green’s function as the imaginary part of $\varepsilon$ goes to zero, $$\lim_{\text{Im}(\varepsilon)\rightarrow 0_{-}} W_{L}(\varepsilon) =G_L^{A}(\varepsilon)V_L.$$ The selection of $W_{R}$ is similar. Intuitively, the subspace $W$ obtained this way represents the solution of the corresponding TDSE with initial conditions supported in the extended device region $\widetilde{C}$. Combining the subspaces $W_L$ and $W_R$, we have $$\label{eq: W'}
W = \left[
\begin{array}{cll}
W_L &
\begin{matrix}
0 \\
0\\
\end{matrix} &
\begin{matrix}
0 \\
0\\
\end{matrix}
\\
0& I_{n_{C}} &0 \\
\begin{matrix}
0 \\
0\\
\end{matrix} &
\begin{matrix}
0 \\
0\\
\end{matrix} &
W_R \\
\end{array} \right].$$ We notice in passing that unlike the basis $V_L$ amd $V_R$, the basis $W_L$ and $W_R$ do not have compact support.
We now examine the specific form of the reduced model . With the specific choices of the subspaces (Eqs and ), one can simplify the matrix $M$ in Eq as follows, $$\label{eq: mass}
M_{LL} = \left(V^{*}_{L}W_{L}\right)^{-1} = \varepsilon I - H_{\Gamma_{\!L},\Gamma_{\!L}}(t) - \Sigma_{L}(t,\varepsilon)
=: \varepsilon I - H_{\text{eff},L}(t,\varepsilon),$$ and similarly, $$M_{RR} = \varepsilon I - H_{\text{eff},R}(t,\varepsilon).$$ Here $\Sigma_{\alpha}$ is the self energy [@brandbyge2002density; @popov2004tunnel; @danielewicz1984quantum; @xue2002first] contributed by the left ($\alpha=L$) or right ($\alpha=R$) lead, $$\label{eq: selfenergyL}
\Sigma_{\alpha}(t,\varepsilon)=H_{\Gamma_{\!\alpha},\alpha} (\varepsilon I- H_{\alpha,\alpha}(t))^{-1} H_{\alpha,\Gamma_{\!\alpha}},$$ and $H_{\text{eff},\alpha}$ is the effective Hamiltonian associated with $\Omega_{\Gamma_{\alpha}}$ [@meier1999non], $$H_{\text{eff},\alpha}(t,\varepsilon)=H_{\Gamma_{\!\alpha},\Gamma_{\!\alpha}}(t)+\Sigma_{\alpha}(t,\varepsilon).$$
Overall, the effective Hamiltonian $H_\text{eff}$ in is simplified to, $$H_\text{eff}(t):=H_c(t)+\Sigma(t,\varepsilon),$$ where ${H}_{c}$ is the Hamiltonian restricted in the extended center region $\Omega_{\widetilde{C}}$, $$\begin{aligned}
H_c(t) := H(t) \large|_{\Omega_{\widetilde{C}}\times\Omega_{\widetilde{C}}},
\end{aligned}$$ and $\Sigma$ is a block-wise diagonal matrix that incorporates the self-energies of two leads, $$\Sigma(t,\varepsilon) = \left[
\begin{array}{ccc}
\Sigma_{L}(t,\varepsilon) & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & \Sigma_{R}(t,\varepsilon)\\
\end{array}
\right].$$
The self-energy involves the inverse of a large-dimensional (or infinite-dimensional) matrix. Similar to the inversion in , it can be efficiently computed using a recursive algorithm, which has been well documented[@williams1982green; @sancho1984quick; @sancho1985highly]. The self-energy only needs to be computed once for constant external potential and for periodic external potentials, it can be pre-computed for one period.
Let [$\rho_{c}$]{} be the density-matrix restricted in the extended center region $\Omega_{\widetilde{C}}$, [[*i.e.*]{}]{}, $$\begin{aligned}
\rho_c(t) := \rho(t)|_{\Omega_{\widetilde{C}}\times\Omega_{\widetilde{C}}} = D(t) + \rho_{c}(0).
\end{aligned}$$ The reduced model for this part of the density-matrix can now be written as, $$\label{eq: reducedLvN}
i \frac{d}{dt} \rho_c(t) = [H_\text{eff}(t), \rho_c(t)] + {\Theta_c}(t),$$
With our choice of the subspaces, the reduced dynamics is driven by the effective Hamiltonian $H_\text{eff}$. The non-homogeneous term ${\Theta_c}$ embodies the effect of the potential, $$\label{eq: theta'}
\begin{aligned}
\Theta_{c}(t) = M^{*}\widetilde{V}^{*}\left(\varepsilon^{*}I-H\right)^{-1}\Theta(t) \left(\varepsilon I-H\right)^{-1}\widetilde{V}M,
\end{aligned}$$ where $M$ is computed from Eq and $\widetilde{V}$ is in the form of $$\widetilde{V} =
\left(
\begin{array}{ccc}
V_{L} & -H_{LC} & 0 \\
-H_{C\Gamma_{\!L}} V_{L}^{*}(\varepsilon-H_{LL})V_{L} & \varepsilon - H_{CC} & -H_{C\Gamma_{\!R}} V_{R}^{*}(\varepsilon-H_{RR})V_{R} \\
0 & -H_{RC} & V_{R}
\end{array}
\right).$$
The practical implementation of the reduced model hinges on the availability of efficient algorithms to compute (i) the self-energy ; (ii) the initial density-matrix in the center and contact region; and (iii) the non-homogeneous term . The computation of the self-energy and the initial density-matrix, as previously discussed, can be computed using the selective inversion techniques, which is applicable for problems that can be cast into the form of where the Green’s function is accompanied by sparse vectors. As for the non-homogenous term, we find that $V_{\alpha}$ and $H_{\alpha,C}, \alpha=L,R$ have non-zeros elements only associated with those degrees of freedom in the domain $\widetilde{\Omega}$, which implies the sparsity of $\widetilde{V}$. Upon closer inspection, we find that the product of inverse matrices in $\Theta(t)$, [[*i.e.*]{}]{}, $\left(\varepsilon^{*}I-H\right)^{-1}\Theta(t) \left(\varepsilon I-H\right)^{-1}$, can be written as a sum of single matrix inverses (partial fractions), provided that $\varepsilon$ is in the resolvent of $H$ and $\text{Im}(\varepsilon)\ne 0$. For example, we have, $$\left(zI - H\right)^{-1} \left(\varepsilon I-H\right)^{-1} =\frac1{{\varepsilon}-z} \Big( \left(zI - H\right)^{-1} - \left(\varepsilon I-H\right)^{-1}\Big).$$ Consequently, all those blocks can be written in the general form , and one compute $\Theta_{c}$ efficiently by using the selective inversion techniques [@lin2011selinv].
Properties of the reduced models
--------------------------------
### The Hermitian property of $\rho_{c}(t)$
The projection method produces an approximation of the density-matrix in the extended center region, leading to an open quantum-mechanical model that can be subsequently used to predict the current. The influence from the infinite leads, through the self-energy, has been implicitly incorporated into the effective Hamiltonian. By taking the Hermitian of the reduced model , and noticing the anti-Hermitian property of the term $\widetilde\Theta$, we find that $\rho_{c}^{*}$ also satisfies with initial condition $\rho_{c}^{*}(0)$. As $\rho_{c}(0)$ is Hermitian, and in light of the uniqueness of the solution, we obtain the Hermitian property for $\rho_{c}(t)$.
### The stability of the reduced models
Next, let us turn to the analysis of stability. Since the stability of linear non-homogeneous system is implied by the stability of homogeneous system, we focus on the homogeneous case in Eq to study its stability. The problem can be addressed as the stability of a finite system $X(t)$, $$\label{eq: stability}
i\frac{d}{dt} X(t) = A(t)X(t) - X(t)A^{*}(t), \quad X(0) = \rho_{c}(0),$$ where $A=H_{c}+\Sigma^{*}$. Since $\rho_{c}(0)$ has an eigen-decomposition $\rho_{c}(0) = \sum_{\ell} n_{\ell} \psi^{0}_{\ell}\psi^{0*}_{\ell}$, it is not difficult to verify that $X(t) = \sum_{\ell} n_{\ell}\psi_{\ell}(t)\psi_{\ell}^{*}(t)$ is the solution of Eq if $\psi_{\ell}(t)$ satisfies $$\label{eq: waveequation}
i\frac{d}{dt} \psi_{\ell}(t) = A(t)\psi_{\ell}(t), \quad \psi_{\ell}(0) = \psi_{\ell}^{0}.$$ It suffices to analyze the stability of Eq .
There exists a decomposition $A(t)=A_{1}(t)+ i A_{2}(t)$, where $A_{1},A_{2}$ are real-valued symmetric matrices and $A_{2}$ is determined from $\Sigma$ due to the Hermitian property of $H_{c}$. Further computation yields, $$A_{2}(t) =
\left(
\begin{array}{ccc}
\widetilde{\Phi}_{L}\Lambda_{L}(t)\widetilde{\Phi}_{L}^{*} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & \widetilde{\Phi}_{R}\Lambda_{R}(t)\widetilde{\Phi}_{R}^{*} \\
\end{array}
\right),$$ where $\widetilde{\Phi}_{\alpha} = H_{\Gamma_{\!\alpha},\alpha}\Phi_{\alpha}$ and $\Phi_{\alpha}$ is the eigenvectors of $H_{\alpha,\alpha}$. Thanks to the special form of $\Sigma$, one can compute that $\Lambda_{\alpha}$ is a real diagonal matrix, in the form, $\Lambda_{\alpha}=\text{diag}(\lambda^{\alpha}_{1},\lambda^{\alpha}_{2},\cdots,\lambda^{\alpha}_{n})$, with $$\lambda^{\alpha}_{\ell} = \text{Im}\left(\frac{1}{\varepsilon^{*}-\mu^{\alpha}_{\ell}}\right),$$ where $\mu^{\alpha}_{\ell}$ is the eigenvalue of $H_{{\alpha},{\alpha}}$.
To ensure the stability, it is enough to require that $A_{2}$ has only non-positive eigenvalues [@brauer1966perturbations], [[*i.e.*]{}]{}, $$\lambda^{\alpha}_{\ell} = \text{Im}\left(\frac{1}{\varepsilon^{*}-\mu^{\alpha}_{\ell}}\right) = \frac{\text{Im}(\varepsilon)}{|\varepsilon^{*}-\mu^{\alpha}_{\ell}|^{2}} \le 0.$$ This confirms that when $\varepsilon$ has negative imaginary part, the stability of is guaranteed.
Higher order subspace projections {#sec: highorder}
---------------------------------
The Galerkin-Petrov projection method can be extended to higher order, by expanding the subspaces $V_{L/R}$ and $W_{L/R}$ to higher dimensions. Here we provide two options to extend the current subspaces.
[**Expanding the contact region.** ]{} One straightforward approach is to keep the choices of $V$ and $W$ according to and , but increase the size of the region $\Omega_\Gamma$ to increase the subspace. Through numerical tests, we observe that this is a rather simple alternative, and it captures steady state current with subspaces of relatively small dimensions $n_\Gamma$.
[**Block Krylov subspaces.**]{} Another approach, as motivated by the block Krylov techniques [@ma2019coarse] for large-dimensional dynamical systems, is to expand the subspace $V_{L}$ to the block Krylov subspace, $$\label{eq: Krylov-V}
V_{L,m}=\left[ V_L \;\; H_{LL}V_L \; \cdots\; H_{LL}^{m-1}V_L \right] =: \mathcal{K}_{m}\left(H_{LL};V_{L} \right).$$ The corresponding $W_{L,m}$ has a similar structure, $$\label{eq: Krylov-W}
W_{L,m} = \left[ W_L \;\; V_L \; \cdots\; H_{LL}^{m-2}V_L \right] =: \mathcal{K}_{m}\left(H_{LL};W_{L} \right).$$
The Krylov subspaces are composed of a generating matrix and a starting block. In order to keep the additional blocks full rank, we pick $V_L$ based on the interaction range in $H_{LL}$. For example, if $H_{LL}$ is based on a one-dimensional nearest-neighbor Hamiltonian, then we pick $n_\Gamma=1$ to define $V_L$, which would be a one-dimensional vector; We pick $n_\Gamma =2$ for a next nearest neighbor Hamiltonian, etc.
Numerical Experiments and Discussions {#sec: results}
=====================================
To test the reduction method, we consider a one-dimensional two-lead molecular junction model within a TB setting. We follow the setup in Zelovich et al. [@zelovich2014state]. More specifically, in the computation, the leads are represented by two finite atomic chains with increasing lengths ( $n_{L}$ and $n_{R}$ respectively) to mimic an infinite dimensional system and eliminate the finite size effect. The extended molecule with length $n_{C}$ is represented by a finite atomic chain coupled with both leads. Here, the atomic unit is used throughout the paper if not stated otherwise.
Initially, the system is configured in thermodynamic equilibrium, with all single-particle levels occupied up to the Fermi energy $\varepsilon_{F}=0.3$. The on-site energy is taken as $\alpha=2$, and the hopping integral between nearest neighbors is $\beta=-1$. At time $t = 0_+$, a bias potential is switched on in the electrodes. With the computed density-matrix, we study the bond current through the molecular junction to monitor the dynamics, using the formula [@zelovich2014state] $$I(t) = 2\beta \text{Im}[\rho_{j,j+1}(t)].$$ For the time propagation of the density-matrix, we use the fourth-order Runge-Kutta scheme to solve the full model , as well as . We fix the size of the center region $n_{C}=20$ and simulate the system under two different types of external potentials: (1) constant biased potential: $U_{L/R}= \mp \frac{\delta U}2$ to mimic direct current (DC) circuit; (2) time-dependent potential: A sinusoidal signal in the left lead, $U_L=\sin \omega t,$ to mimic an alternating current (AC).
In principle, the bath size needs to be infinite to model the two semi-infinite leads; but in computations, one can only treat a system of finite-size and expect the system to reach a steady state in the limit as the bath size goes to infinity. First, we examine such size effect by varying $n_L$/$n_R$ and observing the current in the center region. More specifically, we run direct simulations using $n_{L}=n_{R}=200,500,1000,2000$. Our results (Figure \[fig: currentfull\]) suggest that, for the constant potential case, the electric current gradually develops into a steady state until the propagating electronic waves reach the ends of the leads and get reflected toward the bridge. As we extend the leads size to $n_{L}=n_{R}=1000$, the backscattering effect occurs much later and is no longer observed within the time window of our simulation. For the dynamic potential case, we observe periodic changes of the electric current. Size effects become insignificant when the size is increased to $n_{L}=n_{R}=500$ over the duration of the simulation. We point out that this effort of using sufficiently large bath size is only to generate a faithful result from the full model , to examine the accuracy of the reduced model .
![(Color online) The finite size effect on the electric current. The figures show the time evolution of the currents through a junction coupled with leads of different lengths. Top: constant bias potential $U_{L} = -U_{R}=0.1$. Bottom: dynamic potential $U_{L}=0.2\sin(0.05t), U_{R}=0$.[]{data-label="fig: currentfull"}](current_full.eps)
Next we compute the transient current of the DC circuit (case 1) from the effective reduced models and compare it with the current from the full model to evaluate the accuracy of the reduction method. We also examine the different choices of increasing the subspaces (as discussed in section \[sec: highorder\]). In particular, in Figure \[fig: DC\] we show the numerical results from using the subspaces and , and we choose the dimension $n_\Gamma$ from 1 to 10. First we notice that no recurrent phenomenon is observed, which can be attributed to the non-homogeneous term $\Theta(t)$ as well as the self-energy in Eq , since they take into account the influence from the bath. The results improve as we expand the subspace, $V_{\alpha}$ and $W_{\alpha},\alpha=L/R$ in Eq . The steady state current has already been well captured by the reduced model with dimensions $n_\Gamma=2$, while the transient results improve as we expand $n_{\Gamma}$, and we arrive at a very satisfactory result when $n_{\Gamma}=4$.
![(Color online) The simulation of the DC circuit (constant bias $U_{L}=-U_{R}=0.1$). The figure shows the time history of the current from the reduced model with different subspace dimensions, compared to the result from the full model . The subspaces are chosen from and by extending the contact region $\Omega_{\Gamma}$ with parameter $\varepsilon=0.3-0.1i$. The inset shows the transient stage of the current. []{data-label="fig: DC"}](current_reduced_const.eps){width="48.00000%"}
We also tested the Krylov subspaces according to and . The subspaces can be expanded by increasing $m$. The steady state is well captured when $m=3,$ the transient requires higher order approximations. Our observation is that in order to achieve the same accuracy, we need larger subspaces than the previous approach. On the other hand, the Krylov subspace approach is more robust in the regime where $\text{Im}(\varepsilon)$ is close to zero.
![(Color online) The results from the simulation of the DC circuit (constant bias $U_{L}=-U_{R}=0.1$). The figure shows the time evolution of the current from the reduced model , generated by the block Krylov subspaces and for various choices of dimensions (Nsub=$m$), with parameter $\varepsilon=0.3-0.01i$. The results are compared to the result from the full model . The inset shows the transient stage of the current.[]{data-label="fig: DC-krylov"}](current_ext.eps){width="48.00000%"}
Another important factor that plays a role in the reduced model is the selection of the parameter $\varepsilon$, which can be viewed as an interpolation point for the self-energy. Therefore, we study the dependence of $\varepsilon$ in the reduced models, by observing the electric current at steady state for various different choices of ${\varepsilon}$. For the imaginary part, we require $\text{Im}({\varepsilon})$ to be strictly less than zero to ensure that the self-energy is well defined and has the stability assurance. We start with $\text{Im}({\varepsilon})=0.1$. When $|\text{Im}(\varepsilon)|$ is further decreased ($<0.01$), the electric current exhibits oscillations around the true value of the steady state. For the real part of ${\varepsilon}$, the optimal value appears around the Fermi energy. See Figure \[fig: DC’\]. This suggests that ${\varepsilon}$ should be around the Fermi level with small imaginary part, although when the imaginary part is too small, the numerical robustness might be affected.
![(Color online) The example of DC circuit with constant bias ($U_{L}=-U_{R}=0.1$). The Figure shows the steady-state current predicted by the reduced model using various choices of the parameter $\varepsilon$ with $n_{\Gamma}=1$.[]{data-label="fig: DC’"}](epsilon.eps){height="33.00000%"}
Finally, we turn to the example of the AC circuit. Since a time-dependent external potential is imposed, $H_{c}$ and $\Theta_{c}$ in Eq are time-dependent as well. They need to be evaluated at each time step. Due to the periodic property, it suffices to pre-compute $H_{c}(t)$ and $\Theta_{c}(t)$ within one time period. As shown in Figure \[fig: AC\], a periodic electric current has been reproduced by the reduced model , and the accuracy also improves as we expand the subspace size $n_{\Gamma}$. The electric current is already well captured when $n_{\Gamma}=4$.
![(Color online) The example of an AC circuit with time-dependent potential $U_{L}(t)=0.2\sin(0.05t), U_{R}=0$. This figure displays the time evolution of the currents from the reduced models with different subspace dimensions $n_{\Gamma}$, compared to that from the full model. The parameter $\varepsilon = 0.3-0.1i$ is used.[]{data-label="fig: AC"}](current_reduced_sine.eps){width="48.00000%"}
Summary {#sec: summary}
=======
We have proposed to formulate the quantum transport problem in a molecular junction coupled with infinite baths as a reduced-order modeling problem. The goal is to derive a finite quantum system with open boundary conditions. Motivated by the works [@sanchez2006molecular; @zelovich2014state; @zelovich2015molecule], we work with the density-matrix, and obtain reduced Liouville-von Neumann equations for the center and contact regions. The reduced equations are derived using a systematic projection formalism, together with appropriate choices of the subspaces. Numerical experiments have shown that the reduced model is very effective in capturing the steady-state electric current as well as the transient process of the electric current. The accuracy increases as we expand the contact regions in the reduced model.
In order to demonstrate the reduction procedure, we have considered a one-dimensional junction system. But the validity of the projection approach is not restricted to the one-dimensional system. It can be applied to general coupled system-bath dynamics that require model reduction due to the computational complexity. The extension to systems that are of direct practical interest is underway. Another possible extension is the data-driven implementation of reduced-order modeling. In this case, rather than computing the matrices in the reduced models from the underlying quantum mechanical models, they are inferred from observations [@benner2015survey; @ma2019coarse].
Self-consistency has not been included in the Liouville-von Neumann equation, especially the Coulomb potential, which in the linear response regime, leads to a dense matrix [@yabana2006real] from the Hartree term. This creates considerable difficulty for the reduce-order modeling since the partition is no longer reasonable. However, the Coulomb and exchange correlation are known to be important for the Coulomb blockade phenomena [@kurth2010dynamical]. This difficulty in the modeling of quantum transport has also been pointed out in [@kurth2005time; @ullrich2011time]. In practice, this is often dealt with by solving Poisson’s equation in a relatively larger domain with Dirichlet boundary conditions [@taylor2001ab]. We will address this issue under the framework of reduced-order modeling in separate works.
This research was supported by NSF under grant DMS-1619661 and DMS-1819011.
|
---
abstract: 'We discuss the construction of low-energy tight-binding Hamiltonians for condensed matter systems with a strong coupling to the quantum electromagnetic field. Such Hamiltonians can be obtained by projecting the continuum theory on a given set of Wannier orbitals. However, different representations of the continuum theory lead to different low-energy formulations, because different representations may entangle light and matter, transforming orbitals into light-matter hybrid states before the projection. In particular, a multi-center Power-Zienau-Woolley transformation yields a dipolar Hamiltonian which incorporates the light-matter coupling via both Peierls phases and a polarization density. We compare this dipolar gauge Hamiltonian and the straightforward Coulomb gauge Hamiltonian for a one-dimensional solid, to describe sub-cycle light-driven electronic motion in the semiclassical limit, and a coupling of the solid to a quantized cavity mode which renormalizes the band-structure into electron-polariton bands. Both descriptions yield the same result when many bands are taken into account, but the dipolar Hamiltonian is more accurate when the model is restricted to few electronic bands, while the Coulomb Hamiltonian requires fewer electromagnetic modes.'
author:
- Jiajun Li
- Denis Golez
- Giacomo Mazza
- 'Andrew J. Millis'
- Antoine Georges
- Martin Eckstein
bibliography:
- 'apssamp01.bib'
title: 'Electromagnetic coupling in tight-binding models for strongly correlated light and matter'
---
Introduction
============
In recent years, many condensed matter experiments have explored the intriguing phenomena which arise when matter is driven by electromagnetic fields far beyond the linear response regime [@Basov2017]. On the one hand, this includes highly nonlinear electron dynamics induced by [*classical*]{} electromagnetic fields, such as light-driven Bloch oscillations [@Schubert2014] and high-harmonic generation, or the engineering of entirely new states, so-called Floquet phases, under strong time-periodic driving [@Oka2009; @Wang2013; @McIver2018]. Even more intriguing proposals for hybrid light-matter states have been made for the case when the [*quantum*]{} nature of the electromagnetic field becomes relevant [@Schachenmayer2015; @Kiffner2019; @Schlawin2019; @Sentef2018b; @Mazza2019; @Andolina2019; @Wang2019; @Curtis2019; @Kiffner2019b; @Orgiu2015], through structuring the photon modes using a cavity. Cavity quantum electrodynamics (QED) has reached the regime of ultra-strong coupling between few emitters and photons [@Frisk-Kockum2019] and demonstrated the possibility to control chemical reactions [@Thomas2019]. First experiments with condensed matter systems in cavities have led to tantalizing observations, including an enhancement of the superconducting transition temperature of a material through coupling to vacuum fluctuations [@Anoop2019].
The rich physics of complex condensed matter systems is largely understood in terms of minimal tight-binding models, which describe interacting electron systems on a lattice with only a few valence orbitals per site. The recent developments in cavity QED, therefore, call for tight-binding models which incorporate the electromagnetic field, and can thus provide a low-energy description for the ultra-strong light-matter coupling in solids, complementary to first-principle approaches [@Ruggenthaler2014; @Flick2018; @Schaefer2018; @Nielsen2018]. However, while the continuum formulation of quantum electrodynamics is textbook knowledge [@LoudonBook; @TannoudjiBook], electromagnetic coupling when the electronic hamiltonian is projected to a restricted low-energy model raises subtle issues. The most straightforward approach to derive few level models in atomic physics or few band models in condensed matter is to project the continuum theory on a subset of orbitals. However, while the exact theory is invariant under canonical transformations, the accuracy of a projection to a subset of orbitals in general depends on the representation resulting from the choice of canonical variables. For example, in a representation in which the canonical field variable is the macroscopic displacement field rather than the microscopic electric field, the matter orbitals are hybrid light-matter objects, the matter-field coupling appears differently, and the accuracy of a truncation to a small number of orbitals will change. The differences in the projected light-matter Hamiltonians have been recently discussed very actively in relation to the derivation of few level models for individual atoms in cavity QED [@Bernardis2018; @Bosman2017; @Gely2017; @Vukics2014; @Bernardis2018b; @Di-Stefano2019], motivated in part by long-standing debates on fundamental questions regarding the interpretation of the superradiant phase transition [@Dicke1954; @Rzaewski1975; @Keeling2007]. In the solid, one should expect a similar dependence of the reduced low-energy Hamiltonian on the starting point of the projection. The Wannier orbital onto which the projection is performed has a different meaning in different representations , and one may consider which choice best represents the physics within a minimal set of bands.
Tight-binding models with a coupling to the electromagnetic field have a long history in semiclassical description, where the tight-binding Hamiltonians $H[\bm A, \phi]$ is written in terms of the scalar and vector potentials $\bm A(\bm r,t)$ and $\phi(\bm r,t)$. The most widely used minimal semiclassical Hamiltonian for electrons with charge $q$ in one band is obtained by the Peierls substitution [@Peierls1933], $$\begin{aligned}
\label{jhedvqx}
H=\sum_{\alpha,\alpha'} t_{\alpha,\alpha'} e^{iq\chi_{\alpha,\alpha'}}\,c_{\alpha}^\dagger c_{\alpha'} + \sum_{\alpha}q\phi_\alpha c_{\alpha}^\dagger c_{\alpha}.\end{aligned}$$ Here $c_{\alpha}^\dagger$ ($c_{\alpha}$) are creation (annihilation) operators for an electron in a Wannier orbital localized at site $\bm R_\alpha$ of a given lattice, $t_{\alpha,\alpha'}$ denotes the tunnelling matrix elements in the absence of electromagnetic fields, and the Peierls phase factors are given in terms of the vector potential by $$\begin{aligned}
\chi_{\alpha,\alpha'} = \int_{\bm R_{\alpha'}}^{\bm R_{\alpha}} d\bm r \cdot \bm A(\bm r),\end{aligned}$$ where the integral is taken along a straight line. It must be emphasized that the gauge in Eq. is not fixed, but the Hamiltonian defines a gauge theory in which the physics remains invariant under the transformation $$\begin{aligned}
\label{fghjkl01}
\bm A
\to\bm A+\bm \nabla \Lambda,
\,\,
\phi
\to\phi-\partial_t \Lambda,
\,\,\,
c_{\alpha}
\to c_\alpha e^{iq\Lambda(\bm R_\alpha)}, \end{aligned}$$ with an arbitrary function $\Lambda(\bm r,t)$. In this aspect, the Peierls Hamiltonian fundamentally differs from any naively projected continuum Hamiltonian, such as a projection of the Coulomb gauge Hamiltonian onto a subset of bands. An elegant way to fix the light-matter coupling matrix elements in more general (multi-band) semiclassical Hamiltonians is in fact to request the existence of such a gauge structure [@Boykin2001]. Gauge-invariant semiclassical Hamiltonians like Eq. turn out to be not only conceptually elegant but also powerful in practice, as strong-field phenomena in solids, such as optically driven Bloch oscillations, can be captured even within a single-band approximation.
In contrast to the semiclassical description, recent studies regarding the quantum light-matter coupling often rely on a linearized light-matter coupling or on a projection of the Coulomb gauge Hamiltonian on the valence bands. This is certainly valid when the coupling is not too strong, but must be carefully reconsidered in the ultra-strong coupling regime. One cannot easily quantize a semiclassical description by replacing the classical fields $\bm A$ by quantum fields, because the semiclassical approximation misses field-induced interactions and Lamb-shifts in the solid, but one may impose that the semiclassical approximation to the [*projected*]{} quantum theory should lead to the known semiclassical tight-binding descriptions. Among the class of quantum light-matter Hamiltonians which are derived in this paper from the continuum theory, one particular representation, often referred to in the quantum optics literature as “the dipolar gauge", results in a light-matter coupling via Peierls phases and inter-band dipolar matrix elements, which has the known semiclassical limit and seems to provide an accurate few-band representation of the physics.
The article is organized as follows. In Sect. \[sec:222\], we introduce the quantum light-matter Hamiltonian in Coulomb gauge and discuss the general formalism of unitary transformations of the light-matter coupled theory. The multi-center Power-Zienau-Woolley (PZW) transformation is introduced to obtain the general quantum Hamiltonian with Peierls phases and inter-band dipolar matrix elements, to which we refer, following the convention, as the dipolar gauge Hamiltonian . The semiclassical limit of the dipolar gauge Hamiltonian then does indeed both have the gauge invariance given by Eq. , and it leads to a faster convergence to the full description as the number of electron bands taken into account is increased. In Sec. \[seconeband\], we exemplarily consider in detail the example of a one-dimensional solid, both to analyze the strongly driven semiclassical dynamics, and to evaluate the light-dressed electron polariton band-structure in a cavity. Sect. \[sec:555\] provides a conclusion and outlook.
Tight-binding models in Coulomb and dipolar gauge {#sec:222}
=================================================
Continuum light-matter Hamiltonian in Coulomb gauge {#seckqhjxnla}
---------------------------------------------------
In Coulomb gauge, the electromagnetic field is expressed in terms of the transverse vector potential $\bm A(\bm r)$ ($\bm \nabla\cdot\bm A=0$), and its canonical conjugate variable $\bm \Pi(\bm r)$ is related to the electric field (see below) [@LoudonBook; @TannoudjiBook]. The Hamiltonian is split in a light and matter contribution as $$\begin{aligned}
\label{oossoo}
H^C&=H_{el}^C+H_{em}.\end{aligned}$$ Here $H^C$ denotes the minimal coupling Hamiltonian for electrons with charge $q$ in the continuum, $$\begin{aligned}
H_{el}^C =
\int d^3\bm r\,
\psi_{\bm r}^\dagger
\frac{(-i\bm \nabla -q \bm A(\bm r))^2}{2m}
\psi_{\bm r}
+H_{latt}
+
H_{int}^C,
\label{jjjjxsjsjsd}\end{aligned}$$ where $\psi_{\bm r}^\dagger$ and $\psi_{\bm r}$ are creation and annihilation operators for electrons at point $\bm r$ (the spin index is suppressed throughout the paper for simplicity); $H_{int}^C$ is the instantaneous Coulomb interaction, and $H_{latt}$ is the lattice potential, which is taken to be a given external potential if the nuclei approximately remain at fixed positions. The second part $H_{em}$ of the Hamiltonian describes the energy of the transverse electromagnetic fields in the empty cavity, $$\begin{aligned}
H_{em}
&=
\frac{1}{2}
\int d^3\bm r
\Big[
\frac{1}{\epsilon_0\epsilon(\bm r)}\bm \Pi^2+
\frac{1}{\mu_0}(\bm \nabla \times \bm A)^2
\Big].
\label{lwcbas}\end{aligned}$$ Here the space-dependent dielectric function $\epsilon(\bm r)$ accounts for lossless media, such as partially transparent mirrors, which define an arbitrary cavity environment. It does not yet include the part of the matter which is treated explicitly within a microscopic description below. The longitudinal fields are not independent dynamical degrees of freedom, but fixed by the charge distribution, and their energy is included in the long-range Coulomb interaction $H^C_{int}$[@LoudonBook; @TannoudjiBook].
The resulting Heisenberg equations of motion for $\boldsymbol{A}$ and $\boldsymbol{\Pi}$ can be derived using the canonical commutation relation of $\bm A$ and $\bm \Pi$ (Appendix \[AppendixA\]) and are the transverse components of the Maxwell equations with the current operator $$\begin{aligned}
\label{jhqvsv}
\bm j_C(\bm r)
&=
-\frac{\delta H^{C}_{el}}{\delta \bm A(\bm r)},\end{aligned}$$ if the canonical variables are related to the electric and magnetic field as $\bm B (\bm r)= \bm \nabla\times \bm A (\bm r)$ and $$\begin{aligned}
\label{kcwbslxNAZ}
\bm \Pi(\bm r) =
- \epsilon_0\epsilon(\bm r) \bm E^T(\bm r).\end{aligned}$$ (Here and in the following, superscripts $T$ and $L$ refer to transverse and longitudinal components of a vector field, respectively.) The current operator satisfies the continuity equation $\bm \nabla \cdot \bm j + \partial_t \rho=0$ with the microscopic charge density $$\begin{aligned}
\label{sdfghzfghj}
\rho(\bm r)=\psi_{\bm r}^\dagger \psi_{\bm r}.\end{aligned}$$ Note that with Eq. , $H_{em}$ takes the standard form $\frac12\int d^3 \bm r [\epsilon \epsilon_0(\bm E^T)^2+\bm B^2/\mu_0] $ for the energy stored in the transverse modes. The longitudinal components of the electric field is constrained by the charge distribution, $\bm\nabla\cdot \bm E^L =\rho-\rho_{\rm background}$.
In the derivation of low-energy Hamiltonians, one must restrict the Hilbert space to certain energy bands in the solid and certain modes of the electromagnetic field. The selection of modes is an essential part in defining the low energy Hamiltonian, and we adopt the following general notation:
For the matter, one can assume the existence of an electronic single-particle basis of localized orbitals $w_{\alpha}(\bm r)$, each centered around a position $\bm R_\alpha$. Typically, these will be Wannier orbitals $w_{\bm R,n}(\bm r)\equiv w_{n}(\bm r-\bm R)$, where $\alpha=(\bm R,n)$ labels the orbital $n$ and the lattice site, but in practice we only assume that they are sufficiently localized on the atomic scale and mutually orthogonal $$\begin{aligned}
&\int d^3\bm r\,w_{\alpha}(\bm r)^* w_{\alpha'}(\bm r) = \delta_{\alpha,\alpha'}.\end{aligned}$$ Because of the orthogonality, the corresponding creation and annihilation operators $c_{\alpha}^\dagger$ and $c_{\alpha}$ for electrons in the field-independent Wannier orbitals satisfy canonical anticommutation relations, and field operators can then be expanded like $$\begin{aligned}
\label{xqQSKWKQK}
\psi_{\bm r}
= \sum_{\alpha} w_{\alpha}(\bm r) c_{\alpha},
\,\,\,\,\,
c_{\alpha}
=
\int d^3\bm r \,w_{\alpha}(\bm r)^* \psi_{\bm r}.\end{aligned}$$ The expansion of the electromagnetic field into a set of transverse modes is written as $$\begin{aligned}
\label{abkxlnm;z}
&\bm A(\bm r)
=
\sum_{\nu}
\bm {\phi}_\nu(\bm r) Q_{\nu},
\\
\label{hhhshwh}
&\bm \Pi(\bm r)
=
\sum_{\nu}
\bm\phi_{\nu}(\bm r)^*
\eta_\nu \epsilon(\bm r)
\,\Pi_\nu,\end{aligned}$$ where the operators $Q_{\nu}$ and $\Pi_\nu$ denote the canonical variables, $[Q_\nu,\Pi_{\nu'}]=i\delta_{\nu,\nu'}$, and the factor $\epsilon(\bm r)$ and an arbitrary additional rescaling factor $\eta_\nu $ have been introduced for later convenience. The mode functions are supposed to be orthogonal (with respect to $\eta_\nu \epsilon(\bm r)$) $$\begin{aligned}
&\int d^3\bm r\,
\eta_\nu \epsilon(\bm r)
\, \bm{\phi}_{\nu}(\bm r)^* \cdot \bm {\phi}_{\nu'}(\bm r)
=
\delta_{\nu,\nu'},\end{aligned}$$ and provide a complete set of transverse functions, with the inverse transformation $$\begin{aligned}
Q_\nu&=\int d^3\bm r \,\eta_\nu \epsilon(\bm r) \bm A(\bm r)\cdot\bm \phi^*_\nu(\bm r),
\\
\label{hhhshwhinv}
\Pi_\nu&=\int d^3\bm r \,\bm \Pi(\bm r) \cdot\bm\phi_\nu(\bm r).\end{aligned}$$ For example, the modes can be taken as normal modes of the resonator Hamiltonian $H_{em}$ \[Eq. \] so that $H_{em} = \sum_\nu \frac{\omega_\nu}{2} (Q_\nu^2 + \Pi_\nu^2)$, or from a suitable multi-mode approximation [@Lentrodt2018] of an open cavity. Using Maxwell equations for the free cavity and Eq. , the diagonalization is achieved by solving the generalized eigenvalue problem $$\begin{aligned}
\label{nm,mnm,m}
\omega_\nu^2\mu_0 \epsilon_0\epsilon(\bm r) \bm b_\nu(\bm r) =- \nabla^2 \bm b_\nu(\bm r)\end{aligned}$$ for transverse mode functions $\bm b_\nu(\bm r)$ that are orthogonal with respect to $\epsilon(\bm r)$, $\int d^3 \bm r \,\epsilon(\bm r)\bm b_\nu(\bm r) \cdot \bm b_{\nu'}(\bm r) =\delta_{\nu,\nu'}$. The fields can thus be expanded as and with hermitian $Q_\nu$, $\Pi_\nu$ and real functions $\bm \phi_\nu(\bm r) \equiv \bm b_\nu(\bm r)/\sqrt{\epsilon_0\omega_\nu}$, $\eta_\nu\equiv \epsilon_0\omega_\nu$.
General gauge transformations {#sec:hauge}
-----------------------------
Before the projection to a low-energy manifold, one can choose the explicit form of light-matter coupling by performing a canonical transformation. The resulting representation implies a certain definition of the polarization density, and it specifies the definitions of physical fields, such as the macroscopic electric field, in terms of canonical field variables. The transformation therefore modifies the form of light-matter coupling [@TannoudjiBook; @LoudonBook] and can significantly affect the quality of few-bands approximations. In the literature these transforma- tions are often referred to as gauge transformations and we will use this terminology in what follows. In this section, we recapitulate the general formulation of such a gauge transformation in terms of a unitary transformation which mixes light and matter.
The representation of the Coulomb gauge Hamiltonian in the Wannier basis can be denoted by the formal expression $$\begin{aligned}
H^{C}[c,c^\dagger,\bm\Pi,\bm A]
\equiv
H^{C}_{el}[c,c^\dagger,\bm\Pi,\bm A]
+
H_{em}[\bm\Pi,\bm A],\nonumber\end{aligned}$$ where the square brackets indicate the dependence on the canonical variables. As discussed in the last section, in the Coulomb gauge, the canonical field $\boldsymbol{\Pi}$ is identified with the transverse electric field, while the free charge corresponds to the total particle number operator $\rho(\boldsymbol{r})=\psi^\dag_{\boldsymbol{r}}\psi_{\boldsymbol{r}}$, and no bound charge is present. This can be changed by a general unitary transformation $\mathcal{W}$, where the transformed Hamiltonian reads $$\begin{aligned}
\mathcal{W}\, H^{C}
\mathcal{W}^\dagger
&=
H^{C}[\bar c,\bar c^\dagger,\bar {\bm \Pi},\bar {\bm A} ]
\equiv
H^{\mathcal {W}}[c,c^\dagger, {\bm \Pi}, {\bm A} ].
\label{jhvaxkLZ}\end{aligned}$$ Here the expression of the transformed operators $\bar O = \mathcal{W} O \mathcal{W}^\dagger \equiv \bar O [c,c^\dagger,\bm\Pi,\bm A]$, for $O=c,c^\dagger,\bm \Pi,\bm A$ depends on both matter and fields in general, and the last step simply re-expresses the Hamiltonian in terms of $c,c^\dagger, {\bm \Pi}, {\bm A}$, so that $H^{C}$ and $H^{\mathcal {W}}$ have a different functional dependence on the canonical variables in general.
We emphasize that both in Coulomb gauge and in the $\mathcal{W}$ gauge, the symbols $c,c^\dagger, {\bm \Pi}, {\bm A}$ will be used to denote a set of operators which satisfy the canonical (anti)-commutation relations, and thus serve to construct the Hilbert space. In contrast, gauge-invariant physical observables such as the microscopic electromagnetic fields $\bm E$ and $\bm B$ itself do not depend on the gauge, but their representation in term of the canonical variables does, which will be denoted by a subscript $\mathcal{W}$ or $C$. (We will also introduce a set of operators which are defined differently for each gauge, such as the polarization.) If $X$ is a gauge-invariant observable, its representation in $\mathcal{W}$ gauge is obtained by $\mathcal{W}
X_C\mathcal{W}^\dagger
=
X_C[\bar c,\bar c^\dagger, \bar {\bm \Pi},\bar {\bm A}]
\equiv
X_\mathcal{W}[c,c^\dagger, {\bm \Pi}, {\bm A}]$ (just as the transformation of $H$). Written for the microscopic fields, the above discussion implies that the canonical variables $c,c^\dagger, {\bm \Pi}, {\bm A}$ correspond to different physical quantities in different gauges.
For the discussion of the solid, it is useful to consider a rather general class of gauge transformation which mix light and matter in the form of a linear mapping of the matter operators, $$\begin{aligned}
\label{lcnaxs}
\bar c_{\alpha}[c,\bm A]
\equiv
\mathcal{W}
c_{\alpha}
\mathcal{W}^\dagger
=
\sum_{\alpha'}
W[\bm A]_{\alpha,\alpha'} c_{\alpha'}.\end{aligned}$$ Here the matrix $W[\bm A]$ depends on the field operator $\bm A$ only (not on $\bm \Pi$), and it is unitary in terms of the matter indices. One can show that this relation for the electron operators already fixes the transformation of the electromagnetic fields: Because the transformation depends only on $\bm A$, one has $ \mathcal{W} {\bm A} \mathcal{W}^\dagger = {\bm A}$. Furthermore, the transformation of $\bm \Pi$ can always be represented as a shift by the transverse component of a field $\bm P_ \mathcal{W} (\bm r)$ $$\begin{aligned}
\label{ergbnm01}
\mathcal{W} {\bm \Pi} \mathcal{W}^\dagger &= {\bm \Pi} + \bm P^T_{\mathcal{W}}(\bm r),\end{aligned}$$ which will be identified as polarization density. In Appendix \[sechdjd\] we show that the polarization density that corresponds to the general transformation is a simple quadratic form in the matter operators, $$\begin{aligned}
\label{jgkxalsz;}
\bm P_{\mathcal{W}}(\bm r)
&=
\sum_{\alpha,\alpha'}
\bm M(\bm r)_{\alpha,\alpha'}
\,c_{\alpha}^\dagger
c_{\alpha'},
\\
\label{jgkxalsz;01}
\bm M(\bm r)
&=
-iW[\bm A]^\dagger \frac{\delta}{\delta \bm A(\bm r)}W[\bm A].\end{aligned}$$ By transforming $\bm \Pi$ in the free field Hamitonian $H_{em}$ to $\bm \Pi+\bm P_{\mathcal{W}}^T$, one then arrives at a transformed Hamiltonian which contains both a new light-matter interaction $\sim \bm P\cdot\bm \Pi$, and an induced interaction $\sim \bm P\cdot \bm P$. In summary, the Hamiltonian can be written as $$\begin{aligned}
H_{\mathcal{W}}=H_{el,\mathcal{W}} + H_{em} + H_{PP} + H_{EP},\end{aligned}$$ where $H_{el,\mathcal{W}}$ is obtained by applying the transformation and to $H_{C,el}$, and $$\begin{aligned}
H_{EP}
&=
\frac{1}{2}\int d^3\bm r
\frac{1}{\epsilon_0\epsilon(\bm r)}
\big[\bm \Pi(\bm r)\cdot \bm P^T_{\mathcal{W}}(\bm r)+h.c.\big],
\label{jlxlxwxww01}
\\
\label{jlxlxwxww}
H_{PP}
&=
\frac{1}{2}\int d^3\bm r
\frac{1}{\epsilon_0\epsilon(\bm r)}\bm P^T_{\mathcal{W}}(\bm r)^2.\end{aligned}$$ (In the second line, $\bm \Pi$ and $\bm P^T_{\mathcal{W}}$ do not commute because $\bm P^T_{\mathcal{W}}$ can depend on the vector potential.)
With the Hamiltonian $H_{\mathcal{W}}$, Heisenberg equations can be identified with the [*macroscopic*]{} Maxwell equation for the transverse fields with the current density $$\begin{aligned}
\label{jhqvsv01}
\bm J_{\mathcal{W}}
=
-\frac{\delta (H_{el,\mathcal{W}}+H_{EP}+H_{PP})}{\delta \bm A(\bm r)},\end{aligned}$$ if $\bm \Pi$ is identified with a displacement field $\bm D^T_\mathcal{W}$, $$\begin{aligned}
\label{fffafaffa}
&\bm \Pi(\bm r) =
- \epsilon_0\epsilon(\bm r) \bm E_\mathcal{W}^T(\bm r)
- \bm P^T_{\mathcal{W}}(\bm r) \equiv -\bm D^T_{\mathcal{W}}(\bm r) .\end{aligned}$$ The current $\bm J_\mathcal{W}$ satisfies the continuity equation with a charge density $$\begin{aligned}
\label{hhhahha}
\rho_{macr,\mathcal{W}}
=
\rho_\mathcal{W}
+
\bm \nabla \cdot \bm P_\mathcal{W},\end{aligned}$$ which we term “macroscopic charge density”, because the related current is the source term in the macroscopic Maxwell equations. Comparison with the microscopic continuity equation shows that $\bm J_{\mathcal{W}}$ is related to the microscopic current by $$\begin{aligned}
\label{cghghvgggs}
\bm j_\mathcal{W} = \bm J_{\mathcal{W}} + \partial_t \bm P_\mathcal{W},\end{aligned}$$ showing again the gauge-dependent separation of the charge into a macroscopic charge density and a polarization charge $-\bm \nabla \cdot \bm P_\mathcal{W}$.
### Semiclassical approximation {#semiclassical-approximation .unnumbered}
The semiclassical approximation corresponds to replacing the electromagnetic field by its expectation value, while leaving the quantum description of the matter. It can be obtained by decoupling the square of the operator $\bm \Pi + \bm P^T_{\mathcal{W}}(\bm r)$ in the light-matter Hamiltonian, analogous to a mean-field decoupling $AB\to A\langle B\rangle+\langle A\rangle B - \langle A\rangle\langle B\rangle$ of the products $\bm \Pi\cdot \bm P_\mathcal{W}$ and $\bm P_\mathcal{W}^2$ in Eqs. and . The resulting equations, written in terms of the microscopic fields $\bm B(\bm r,t)$ and $\bm E(\bm r,t)$, are the classical Maxwell equations for the transverse fields with the microscopic current $\langle \bm j\rangle$ and charge as a source term. Matter is described by the semiclassical Hamiltonian, $$\begin{aligned}
H_{sc,\mathcal{W}}=H_{el,\mathcal{W}}[c^\dagger,c,\bm A(\bm r,t)] -\int d^3\bm r
\bm E^T(\bm r,t)
\cdot \bm P^T_{\mathcal{W}}(\bm r),
\label{jlxlxwxww03}\end{aligned}$$ up to the constant term $\propto \int \langle \bm P^T_{\mathcal{W}}(\bm r)\rangle^2$. In these equations, the vector potential is still transverse, $\bm E^T(\bm r,t)=-\partial_t \bm A(\bm r,t)$. The current is given by Eq. , with the macroscopic charge contribution Eq. , $\bm J_{sc,\mathcal{W}}= -\frac{\delta H_{sc,\mathcal{W}}}{\delta \bm A(\bm r)}$, and the polarization charge contribution $\partial_t \langle\bm P_{\mathcal{W}}(\bm r)\rangle$.
One possible requirement for the construction of tight-binding light-matter Hamiltonians is to find a gauge transformation such that the semi-classical Hamiltonian after projection to subset of bands has an explicit gauge structure as defined by Eq. .
### Restriction of the cavity modes {#restriction-of-the-cavity-modes .unnumbered}
Often it is useful to express the fields using a general mode expansion and . With the replacement [^1] $$\begin{aligned}
\label{fghjfgfhhgh}
\frac{\delta }{\delta \bm A(\bm r)}
=
\sum_{\nu}
\eta_\nu \epsilon(\bm r)
\bm\phi_{\nu}^*(\bm r)
\frac{\delta }{\delta Q_\nu}\end{aligned}$$ in Eq. , we obtain the expansions of the polarization density $\bm P^T_\mathcal{W}(\bm r)$ in terms of the mode functions $\bm \phi$, $$\begin{aligned}
\label{kvxlnzA;}
\bm P^T_{\mathcal{W}}(\bm r)
&=
\sum_{\nu}
\eta_\nu \epsilon(\bm r)
\bm\phi_{\nu}^*(\bm r)
P^T_{\mathcal{W},\nu},
\\
\label{kvxlnzA03}
P^T_{\mathcal{W},\nu}
&=
\sum_{\alpha,\alpha'}
c_{\alpha}^\dagger
(M_\nu)_{\alpha,\alpha'}
c_{\alpha'},
\\
\label{fghjkljh}
M_\nu
&=
-i
W[\bm A]^\dagger
\frac{\delta }{\delta Q_\nu}
W[\bm A].\end{aligned}$$ The current operator is expanded in an analogous manner. Within this expansion, the light-matter Hamiltonian becomes $$\begin{aligned}
\label{jbvbjbx01}
H_{EP}
&=\frac{1}{2\epsilon_0}
\sum_{\nu}\eta_\nu
\big(\Pi_\nu^\dagger P_{\mathcal{W},\nu}^{T} + h.c.\big),
\\
\label{jbvbjbx02}
H_{PP}
&=\frac{1}{2\epsilon_0}
\sum_{\nu}\eta_\nu
(P_{\mathcal{W},\nu}^{T})^\dagger P_{\mathcal{W},\nu}^{T}.\end{aligned}$$
This equation is particularly useful when the number of modes is restricted. For example, a coarse graining of the fields can formally be achieved by truncating the relevant modes $\nu$ to a low energy subspace, i.e., introducing a momentum cutoff, or one can restrict the modes to few normal modes of the cavity resonator or a suitable multi-mode approximation [@Lentrodt2018]. It is important to note that such a truncation changes the dipolar interaction $H_{PP}$ and the light-matter coupling $H_{EP}$ in a consistent manner. For example, a restriction to a single mode which is homogeneous over the solid is consistent with an all to all interaction $\propto \bar P^2$, where $\bar P$ is a volume averaged polarization (see examples in Sec. \[seconeband\]).
PZW transformation and dipolar gauge
------------------------------------
For the description of a single atom in a cavity at strong coupling, one often uses the PZW transformation to change the light-matter coupling from the form $\boldsymbol{A}\cdot\boldsymbol{p}$ to $\boldsymbol{E}\cdot\boldsymbol{r}$, which is suitable when electrons are localized close to an atomic center ($\bm r=\bm 0$). In second quantization, the unitary transformation reads $$\begin{aligned}
\label{,acbskcslc}
\mathcal{W}_\text{PZW}
&=
\exp\Big(-iq\int d^3
\bm r
\, \chi(\bm r,\bm 0)\,
\psi_{\bm r}^\dagger\psi_{\bm r}
\Big),\end{aligned}$$ where $\chi(\bm r,\bm r')$ is the line integral over the vector potential along a straight path, $$\begin{aligned}
\label{jjwjwkqblq}
\chi(\bm r,\bm r')
&=
\int_{\bm r'}^{\bm r} d\bm s \cdot \bm A(\bm s).\end{aligned}$$ The PZW transformation is particularly useful as a starting point for the multipolar expansion of the atom-field interaction, where an electron remains localized close to a given atomic center. In the solid, the choice of a fixed origin is however not very convenient, as it explicitly breaks the spatial translational invariance. For the derivation of the semiclassical Peierls Hamiltonian , Luttinger introduced a similar [*multi-center PZW transformation*]{} [@Luttinger1951]. The analog for the quantum case is the definition of field-dependent hybrid light-matter orbitals $$\begin{aligned}
\label{sbxalgg}
\tilde w_{\alpha}(\bm r)
=
e^{-iq\chi(\bm r,\bm R_\alpha)} w_{\alpha}(\bm r),\end{aligned}$$ where the phase due to the vector potential for each orbital is defined relative to the center of the Wannier orbital. However, these orbitals are not orthogonal. It is easy to see that the overlap matrix instead reads $$\begin{gathered}
\label{ghgsgw}
\int d^3\bm r\,\tilde w_{\alpha}(\bm r)^\dagger \tilde w_{\alpha'}(\bm r)
\\=
e^{-iq\chi_{\alpha,\alpha'}}
\int d^3\bm r\,w_{\alpha}(\bm r)^*
e^{iq\Phi(\bm R_{\alpha'},\bm r,\bm R_\alpha)}
w_{\alpha'}(\bm r),\end{gathered}$$ where $\Phi(\bm R_{\alpha'},\bm r,\bm R_\alpha)$ is the magnetic flux through the oriented triangle $\bm R_{\alpha'}\to\bm r\to\bm R_{\alpha}\to\bm R_{\alpha'}$, and we have used the shorthand notation for the Peierls phase $\chi_{\alpha,\alpha'}=\chi(\bm R_\alpha,\bm R_{\alpha'})$. Because the Wannier orbitals are exponentially localized, $\Phi $ is of the order of the magnetic flux per lattice plaquette, and the corresponding deviation of the overlap matrix of order $q\Phi/\hbar$ is typically much smaller than one. For example, for classical electromagnetic waves in vacuum, even an almost atomically strong electric field amplitude $1~$MV/cm implies only a magnetic field of order $0.3$ Tesla, and that the flux $\Phi_0$ through a plaquette of size $10^{-19}m^2$ gives $e\Phi_0/\hbar \sim10^{-4}$. One could re-orthogonalize the field-dependent orbitals order by order in the magnetic flux, which would finally lead to a light-matter Hamiltonian including an explicit interaction with the magnetic field (magnetic dipolar interactions). While this is possible, in the present manuscript we neglect all magnetic field-dependent matrix elements of that order (magnetic flux per plaquette), thereby obtaining a Hamiltonian containing only the dominant electric dipolar terms (“electric dipole approximation”).
From now on, we therefore assume that the overlap is given by the identity $\delta_{\alpha,\alpha'}$. The orthogonality implies that one can construct annihilation (creation) operators $\bar c_{\alpha}=\int d^3\bm r \,\tilde w_{\alpha}(\bm r)^* \psi_{\bm r}$ and ($\bar c_{\alpha}^\dagger$) for electrons in the hybrid orbitals which satisfy canonical anti-commutation relations. The transformation from $c$ to $\bar c$ is therefore unitary and of the type , and all properties derived in the previous section apply. In particular, we can directly determine the corresponding Hamiltonian $H_{\rm Dip}$, and the polarization operator $\bm P_{\rm Dip}$. (The subscript refers to “dipolar” or PZW gauge.) For better readability, we have shifted a rather straightforward derivation of the polarization operator and the Hamiltonian in dipolar gauge to the Appendix \[pppqqqppp01\] and \[pppqqqppp02\], and summarize the results here.
All derivations use the approximation that the vector potential varies weakly on the atomic scale, which is consistent with the electric approximation above and corresponds to neglecting magnetic dipolar and electric quadrupolar matrix elements. We do not make, however, the approximation that the fields vary little over the full crystal (i.e., treating the crystal as a big molecule), as this would neglect the momentum-dependence of the field modes inside the solid from the outset. The slow variation of the fields is usually justified by an energy separation. In the mode expansion of the electromagnetic fields, modes which vary on some short scale $\lambda_c$ can be disregarded because the corresponding energy is large compared to the relevant electronic transitions in the solid (coarse graining). In the solid, $\lambda_c$ could be a UV wavelength such that $\hbar\omega_c=2\pi\hbar c/\lambda_c$ is the order of several $eV$, but still $\lambda_c$ spans many lattice spacings.
For later use, it is convenient to represent both the Hamiltonian and the polarization using the mode expansion and rather than the continuum, as in Eqs. and . For the expansion coefficients we get $$\begin{aligned}
\label{gegege}
P_\mathcal{{\rm Dip},\nu}
&=
q
\sum_{\alpha,\alpha'}
c_{\alpha}^\dagger c_{\alpha'}
\tilde D^\nu_{\alpha,\alpha'},\end{aligned}$$ where we introduced the dipolar matrix elements $$\begin{aligned}
\label{cghjhbjknjkjn}
\bm D_{\alpha,\alpha'}
&=
\int d^3{\bm r}\,
w_{\alpha}(\bm r)^*
\,\bm r\,
w_{\alpha'}(\bm r),\end{aligned}$$ which are then projected on the mode functions and dressed by a Peierls phase $\chi_{\alpha,\alpha'}$, $\tilde D^{\nu}_{\alpha,\alpha'}=\big(
\bm \phi_\nu(\bm R_{\alpha,\alpha'})
\cdot
\bm D_{\alpha,\alpha'}\big)e^{iq\chi_{\alpha,\alpha'}}$; $\bm R_{\alpha,\alpha'}=(\bm R_{\alpha,}+\bm R_{\alpha'})/2$ is the position of the bond $(\alpha,\alpha')$. The Hamiltonian is given by (see App. \[pppqqqppp02\]) $$\begin{aligned}
\label{fghjfghjkfghj01}
H_{\rm Dip} = H_{em} + H_{{\rm Dip},el} + H_{EP}+H_{PP}.\end{aligned}$$ Here $H_{{\rm Dip},el}$ is obtained by dressing all matrix elements in the field free Hamiltonian $H_{el,C}[\bm A=0]$ with Peierls factors, i.e., an operator $O_C=c_{\alpha}^\dagger O_{\alpha,\alpha'} c_\alpha$ becomes $O_{\rm Dip}=c_{\alpha}^\dagger e^{iq\chi_{\alpha,\alpha'}}O_{\alpha,\alpha'} c_\alpha$. This replacement holds both for single-particle terms and for two-particle terms (interactions) which can be written as products of such operators. The dipolar light-matter interaction $H_{EP}$ is obtained by inserting the polarization operator into the general expression , $$\begin{aligned}
\label{fghjfghjkfghj02}
H_{EP}
&=
\sum_{\nu}
\frac{q\eta_\nu}{2\epsilon_0}
\sum_{\alpha,\alpha'}
\Big(
\Pi_\nu^\dagger
c_{\alpha}^\dagger c_{\alpha'} \tilde D^{\nu}_{\alpha,\alpha'}+h.c.\big).\end{aligned}$$ Analogously, we obtain the $\bm P^2$ term , $$\begin{gathered}
\label{fghjfghjkfghj03}
H_{PP}
=
\sum_{\nu}
\frac{q^2\eta_\nu}{2\epsilon_0}
\sum_{\alpha,\alpha'}
c_{\alpha}^\dagger c_{\alpha'}
\big(\tilde D^{\nu}(\tilde D^{\nu})^\dagger\big)_{\alpha,\alpha'}
\\
+
\sum_{\nu}
\frac{q^2\eta_\nu}{2\epsilon_0}
\sum_{\alpha,\alpha',\beta,\beta'}
c_{\alpha}^\dagger
c_{\beta'}^\dagger
c_{\beta}
c_{\alpha'}
\tilde D^{\nu}_{\alpha,\alpha'}
(\tilde D^{\nu})^\dagger_{\beta',\beta}.\end{gathered}$$ We have written $H_{PP}$ in normal-ordered form, in order to indicate that the first term is a renormalization of a band structure which is relevant even for the case of a single electron.
We remark that the dipolar Hamiltonian has a gauge structure which makes it invariant under a shift $\bm A\to\bm A+\bm \nabla \Lambda$ and a simultaneous transformation $c_\alpha\to c_\alpha e^{iq\Lambda(\bm R_\alpha)}$. This implies that the current satisfies the continuity equation with the charge density $$\begin{aligned}
\rho_{macr,{\rm Dip}}(\bm r)
=
q\sum_{\alpha}
\delta(\bm r-\bm R_\alpha)
c_{\alpha}^\dagger
c_{\alpha}.\end{aligned}$$ This clarifies the separation of the macroscopic and polarization charges according to Eq. in the dipolar gauge.
Finally, the semiclassical approximation in the dipolar gauge becomes $$\begin{gathered}
H_{sc,{\rm Dip}}=H_{el,{\rm Dip}}[c^\dagger,c,\bm A(\bm r,t)]
\\
-
q
\sum_{\alpha,\alpha'}
\bm E(\bm R_{\alpha},t)\cdot \bm D_{\alpha,\alpha'}
e^{iq\chi_{\alpha,\alpha'}(t)}
\,
c_{\alpha}^\dagger c_{\alpha'}
\Big],
\label{jlxlxwxww03ggg}\end{gathered}$$ where the Peierls factors $\chi_{\alpha,\alpha'}(t)$ are calculated using the field $\bm A(\bm r,t)$. We note that, when a scalar potential term $H_\phi=q\int d^3\bm r \phi(\bm r) \rho_{macr,{\rm Dip}}(\bm r)=q\sum_{\alpha} c_{\alpha}^\dagger c_\alpha \phi(\bm R_\alpha)$ is added, the semiclassical Hamiltonian is invariant under the gauge transformation Eq. , because the vector potential enters only via the Peierls phases of the hopping and inter-band dipolar matrices. Since the gauge shift in Eq. does not mix operators from different Wannier orbitals, the gauge structure is preserved even after projection onto a subset of orbitals. This shows that the semiclassical limit of the dipolar Hamiltonian, even after truncation, falls into the class of models which have a simple gauge structure [@Boykin2001], as defined by Eq. .
One-dimensional solid {#seconeband}
=====================
In this section, we will systematically compare the convergence of few-band approximations in the Coulomb and dipolar gauge by numerically solving a specific model. The example we choose is a one-dimensional solid, driven by quantum and classical fields which are polarized along the direction of the solid (Fig. \[figcavity\]). Driving the electrons in the system with strong [*classical*]{} fields leads to phenomena such as nonlinear Bloch oscillations and dynamical localization (band narrowing), which are in part captured already in a suitable single-band model and the Peierls substitution. These effects should have an analog when the solid is strongly coupled to a quantized cavity mode with polarization along the material, which hybridizes with the electronic bands to form electron-polariton bands.
![ The setup studied in Sec. \[seconeband\]. An electron is subject to a periodic potential $V(x)$ along the $x$ direction and confined to $z=y=0$. In the quantum case, we take into account only one cavity mode with constant amplitude and polarization along the chain. In the classical case, the electron is driven by a time-dependent field $\bm E(t)$, again polarized along the chain.[]{data-label="figcavity"}](figure01.pdf){width="22.00000%"}
For simplicity, the discussion is restricted to a single electron, deferring the more involved case of induced interactions to subsequent work. The potential is taken to be sinusoidal, with periodicity $a$ along the direction ($x$) of the solid. The field-free electronic continuum Hamiltonian then reads (in first quantization) $$\begin{aligned}
\label{modelAAA}
H_{el}=\frac{p_x^2}{2}+2V_0\cos\left(Gx\right),\end{aligned}$$ where $G=2\pi$, and $V_0$ sets the periodic potential along the solid. Here we have set $\hbar=1$, $a=1$, and $m=1$ (the electron mass), so that length is measured in units of $a$ and energy in units of $\frac{\hbar^2}{ma^2}$. Eigenstates of the un-driven system are Bloch bands $\langle x|k,m\rangle=\phi_{k,m}(x)$ with quasi-momentum $k\in[-\pi,\pi)$ and band energy $\epsilon_{k,m}$, and $m=0,1,2,...$. The band structure and dipolar matrix elements are obtained determined using a plane wave representation of the Bloch states (for details, see Appendix \[app:fghjkl\]).
Semiclassical case: Nonlinear Bloch oscillations
------------------------------------------------
### Formulation
{width="99.00000%"}
We first discuss the description of light-driven dynamics in classical laser fields. The vector potential $A(t)$ is taken to be homogeneous over the solid (i.e., the laser spot extends over many lattice spacings), corresponding to an electric field $E(t)=-\partial_t A(t)$. Only the polarization $A\equiv A_x$ along the solid is considered. One thus must solve for the electron dynamics using the time-dependent semiclassical Hamiltonian $$\begin{aligned}
H(t)=\frac{\left(p_x-qA(t)\right)^2}{2}+2V_0\cos\left(Gx\right).\end{aligned}$$ The exact time-dependent calculation can be carried out straightforwardly in the plane-wave basis. To implement the few-band cutoff in the Coulomb gauge, we expand the Hamiltonian in the Bloch basis, $\langle k,m | H(t) | k',m'\rangle=\delta_{kk'}H^{\rm C}_{k;m,m'}(t)$, and neglect all bands with $m>m_{\rm max}$. The matrix elements are given by $$\begin{aligned}
H^{\rm C}_{k;m,m'}&=
\Big[\epsilon_{k;m}+\frac{q^2A(t)^2}{2}\Big]\delta_{m,m'}-qA(t)p_{k;m,m'}, \end{aligned}$$ with the matrix element $p_{k;m,m'}=-i\langle \phi_{k,m}|\partial_x|\phi_{k,m'}\rangle$ of the bare momentum operator. The dipolar gauge is directly obtained by projecting the semiclassical Hamiltonian to a Bloch-basis. When going from the Wannier representation of Eq. to the Bloch representation, the single-particle Hamiltonian and the dipolar matrix $D_{\alpha,\alpha'}\to \delta_{k,k'} D_{k;m,m'}$ become diagonal in momentum (see Appendix for the evaluation of $D$), and the multiplication with the phase factor $\chi_{\alpha,\alpha'}$ corresponds to the Peierls substitution $k\to k-qA(t)$. One finally arrives at $\langle k,m | H^{\rm Dip}| k',m'\rangle=\delta_{kk'}H^{\rm Dip}_{k;m,m'}$, with $$\begin{aligned}
\label{ghjkl}
H^{\rm Dip}_{k;m,m'}=
\epsilon_{k-qA(t);m}\delta_{m,m'}-qE(t)D_{k-qA(t);m,m'}.\end{aligned}$$ The exact current operator is $$\begin{aligned}
\label{cvkbjkjb}
j_x=-\frac{\delta H}{\delta A} = q(p_x -qA(t)).\end{aligned}$$ In Coulomb gauge, its matrix elements read $\langle k,m|j_x|k',m'\rangle = \delta_{kk'} q (p_{k;m,m'}-qA(t)\delta_{m,m'})$. In dipolar gauge, the current has two components, see Eq. . One is the macroscopic current, $\langle k,m | J^{\rm Dip} |k',m'\rangle =-\delta_{k,k'} \,\delta H^{\rm Dip}_{k;m,m'} /\delta A = q\,\delta_{k,k'}\,\partial_k H^{\rm Dip}_{k;m,m'}$, which would enter as a source term in the macroscopic Maxwell equations, and the other one is the current of the bound charges, given by the time-derivative of the local polarization $dP^{\rm Dip}/dt$. The polarization operator $P$ has matrix elements $\langle k,m|P^{\rm Dip}|k',m'\rangle = \delta_{kk'}qD_{k-qA(t),m,m'}$. The current and polarization density are understood to be coarse grained over the solid.
### Results
To compare the exact results with the few-band approximations in different gauges, we prepare an initial state with $k=0,m=0$ in all three situations and apply an oscillating vector potential $A(t)=A_0\sin(\omega t)$. The resulting currents are compared in Fig. \[fig\_curr\]a-c for different band numbers $m_{\rm max}$. We choose $qA_0=5.0$ and $\omega=1.0$. The potential is $V_0=10$, which corresponds to well-separated bands, see Fig. \[fig\_curr\]g.
In the exact solution, the current exhibits multiple oscillation periods: On the timescale $2\pi$, which corresponds to the period of $A(t)$, the current has a non-sinusoidal time-dependence (non-linear Bloch oscillations), in addition to fast oscillations which have a frequency of the order of the band splitting. In dipolar gauge, it is the free charge current $J^{\rm Dip}$ (yellow solid lines in Fig. \[fig\_curr\]d-f) which gives rise to these nonlinear Bloch oscillations: Due to the large amplitude $A_0$, the mechanical momentum $k-qA(t)$ in dipolar gauge passes the boundary of the first Brillouin zone $[-\pi,\pi)$ during the time-evolution, leading to a reversal of the current. The fast oscillation in the current is instead attributed to the bound-charge current $dP^{\rm Dip}/dt$ (see Fig. \[fig\_curr\]d-f). The free charge current is converged even in the one-band approximation, while the description of the inter-band polarization current $dP^{\rm Dip}/dt$ is reasonably approximated with a minimal two-band cutoff. In Coulomb gauge, the fast (inter-band) oscillations appear as well when more than one band is taken into account, but the nonlinear Bloch oscillations are less well represented. In particular the one-band cutoff in Coulomb gauge results only in a sinusoidal time-dependence in the current. As the number of bands is increased, both the dipolar and the Coulomb gauge description converge, but the results show that the dipolar gauge is advantageous over the Coulomb gauge for a few-band approximation in a gapped lattice system.
The agreement of two-band dipolar approximation with the exact result becomes slightly worse at later times (Fig. \[fig\_curr\]b). This is expected, as the electron is successively excited to higher bands in a process closely related to Landau Zener tunnelling. In order to investigate this excitation process in more detail, we consider the initial state $k=0,m=0$ subject to a constant electric field, $A(t)=-Et$, for a case with a smaller gap ($V_0=1$). Figure \[fig\_curr\]h) shows the time-evolution of the probability for the electron to stay in the first band, obtained within the two-band approximation. In the dipolar gauge, states with different band index $m$ are mixed via the dipolar matrix element. Around time $t=\pi/E$, one has $k-qA(t)=-\pi$ (for $k=0$), so that the system passes through an anti-crossing where the energy difference $|\epsilon_{k-A(t),1}-\epsilon_{k-A(t),2}|$ is minimal. The reduction $\Delta P$ of the occupation probability in the first band during the traverse of the anticrossing can be related to Landau-Zener tunnelling [@niu1998], with $\Delta P=\exp[-\frac{\pi}{2}E_g^2/(2\pi E/a)]$ (black arrows in Fig. \[fig\_curr\]h). We note that due to the gauge-dependence of momentum this probability does not correspond to the projection of the exact wave function to the equilibrium basis (eigenstates of Bloch momentum $k$), but fits well the exact result projected onto a basis which is defined by the eigenstates of gauge-invariant Bloch momentum $k_m=k-A(t)$. The two-band calculation in Coulomb gauge gives a completely different result for either of the two basis sets. In fact, in the Coulomb gauge the physical energy bands, defined with gauge-invariant Bloch momentum $k_m$, are time-dependent superpositions of several states in the bare momentum basis. During time-evolution, many orbitals are therefore needed to recover the gauge-invariant current and electron occupation in the lowest physical band. This clearly shows the advantage of preserving the gauge structure under the few-band approximation, which is satisfied by the dipolar Hamiltonian.
Quantum case: Electron-polariton bands
--------------------------------------
### Formulation
As a second problem, we consider the one-dimensional chain coupled to the quantized modes of a perfectly isolated resonator. We assume that the resonator Hamiltonian $H_{em}$, Eq. has been diagonalized in the form $H_{em} = \sum_\nu \frac{\omega_\nu}{2} (Q_\nu^2 + \Pi_\nu^2)$, with canonical variables $Q_\nu$ and $\Pi_\nu$, and use the mode expansion described around Eq. . In the following we take into account only one mode ($\nu\equiv0$). The mode function $\bm b_0(\bm r)$ is assumed to be constant throughout the solid, and $b_0$ denotes the corresponding component of the mode function along the chain direction. Here the one mode approximation is intended to facilitate an exact comparison of the dipolar and Coulomb gauge. A quantitative solution of a given cavity setup may require more than one cavity mode, depending on the cavity geometry.
With one mode and the expansion , the Peierls factors $q\chi_{\alpha,\alpha'}$ in the dipolar Hamiltonian for sites at distance $|\bm R_\alpha - \bm R_\alpha'|=na$ become $ngQ_0$, with the dimensionless coupling constant $$\begin{aligned}
\label{vbnm00000}
g= \frac{q b_0 a}{\hbar\sqrt{\omega_0 \epsilon_0}},\end{aligned}$$ where $\hbar$ is restored for concreteness. With some manipulation, the dipolar light-matter Hamiltonian is then obtained from Eqs. - , $$\begin{aligned}
\label{fhgjklkjhbvhbkl02}
H^{\rm Dip} &= \frac{\omega_0}{2}(Q_0^2+\Pi_0^2)+H_0 + H_{EP} +H_{PP},\end{aligned}$$ with $$\begin{aligned}
H_0&=\sum_{k,m}
c_{k,m}^\dagger c_{k,m}
\epsilon_{k-gQ_0;m},
\\
H_{EP}
&=
\frac{g \omega_0 }{2}
\sum_{k,m,m'}
\Big[
c_{k,m}^\dagger c_{k,m'}
\Pi_0
D_{k-gQ_0;m,m'}
+
h.c.
\Big]
\\
H_{PP}
&=
\frac{g^2\omega_0 }{2}
\sum_{k,m,m'}
c_{k,m}^\dagger c_{k,m'}
\big(D_{k-gQ_0}^2\big)_{m,m'}.\end{aligned}$$ ($D_{k-gQ_0}^2$ implies a matrix multiplication in band indices.) Here only the first term of $H_{PP}$ \[Eq. \] has been included, because the second contribution is an electron-electron interaction which vanishes for the case of one electron.
Similar to the semiclassical case, the projected Coulomb Hamiltonian is obtained as $$\begin{aligned}
\label{fhgjklkjhbvhbkl02}
&H^{\rm C} = \frac{\omega_0}{2}(Q_0^2+\Pi_0^2) +
\sum_{k,m,m'}
c_{k,m}^\dagger c_{k,m'}
h^{\rm C}_{k;m,m'},
\\
&h^{\rm C}_{k;m,m'}=
\delta_{m,m'}\big[\epsilon_{k;m}
+\frac{g^2}{2} Q_0^2\big]
-
gQ_0
p_{k;m,m'}.\end{aligned}$$ Below we compute the spectrum of the two Hamiltonians as a function of the band cutoff $m_{\rm max}$ and the coupling constant $g$. The wave function at momentum $k$ is expanded as $$\begin{aligned}
|\chi_k\rangle
=
\sum_{m=0}^{m_{\rm max}}
\sum_{n=0}^{n_{\rm max}}
\chi_{k;m,n}
|k,m;n\rangle,\end{aligned}$$ where $|k,m;n\rangle=\frac{1}{\sqrt{n!}}\big(a_0^\dagger\big)^n c_{k,m}^\dagger |0\rangle$ is the basis state with one electron in band $m$ and $n$ photons; $a_0^\dagger=\frac{1}{\sqrt{2}}(Q_0-i\Pi_0)$ is the photon creation operator. The numerical cutoff $n_{\rm max}$ in the number of photons is taken large enough so that the spectrum is converged (up to $n_{\rm max}=100$, depending on the parameters). The dipolar Hamiltonian can become highly nonlinear in $Q_0$, and thus couple many photon states. Instead of a Taylor expansion in $Q_0$, we therefore calculate matrix-elements directly in the $Q_0$ representation; for $X_k\equiv \epsilon_{k}, D_k$, $$\begin{aligned}
\label{ggegegeg}
\langle n|X_{k-gQ_0}|n'\rangle_{\rm ph}
=
\int dQ \,\psi^{\rm ph}_n(Q)\psi^{\rm ph}_{n'}(Q)X_{k-gQ},\end{aligned}$$ where the eigenfunctions $\psi^{\rm ph}_n(Q) =e^{-Q^2/2}H_n(Q)/\sqrt{\pi}$ are given in terms of Hermite polynomials $H_n(Q)$.
![ Electron-polariton band structure in different regimes. a) Photon energy resonant between bands, $V_0=5$, $\omega_0=20$. Dashed lines correspond to $g=0$, colored lines to $g=1$. The color of the lines indicates the photon number expectation value $n_{\rm photon}=\langle \chi_{k;m;n}| a_0^\dagger a_0 |
\chi_{k;m;n} \rangle$ in the states. b) Well-separated bands, photon energy off-resonant between bands, $V_0=10$, $\omega_0=1$. The lowest two electron-polariton bands are shown for different values of the coupling $g$. []{data-label="fig2"}](figure03.pdf){width="0.9\columnwidth"}
### Results
We first analyze the exact band structure of the light-matter coupled system (i.e., the band and photon number cutoff is taken large enough such that the results are converged and identical in both gauges). Figure \[fig2\]a illustrates the case of well isolated bands and a photon energy which is resonant to the transition between the first and second band ($V_0=5$ and $\omega_0=20$). For $g=0$, the bands are given by the bare bands with $n=0,1,2,...$ photons, $|k,m;n\rangle$, with energy $E_{k,m;n}^{(0)}=\epsilon_{k,0}+(n+\tfrac12)\omega_0$ (black dashed lines). For $g>0$, the wave function $|k,m\;n\rangle$ adiabatically evolves into a hybridized electron-polariton band $|\chi_{k,m;n}\rangle$ with energy $E_{k,m;n}$. To illustrate the hybridization, the bands in Fig. \[fig2\]a are colored according to the photon number expectation value $\langle \chi_{k;m;n}| a_0^\dagger a_0 | \chi_{k;m;n} \rangle$. The hybridization opens a gap at the level crossings of the bands $E_{k,1;0}$ and $E_{k,0;1}$.
![ Convergence of the spectrum with the band cutoff $m_{\rm max}$ for the off-resonant case, for $V_0=10$, $\omega_0=1$. a) Position of the lowest band. b) Width of the lowest (zero-photon) band $E_{k,0;0}$. c) Width of the one-photon band $E_{k,0;1}$. In all cases, the dipolar gauge is converged for $m_{\rm max}\ge 2$, so that symbols for $m_{\rm max}=2,4,6$ are indistinguishable. []{data-label="fig3"}](figure04.pdf){width="0.9\columnwidth"}
Figure \[fig2\]b shows a different parameter regime, with well isolated bands and off-resonant photon energy (same parameters $V_0=10$, $\omega_0=1$ as in Fig. \[fig\_curr\]). Here we focus only on the dependence of the lowest two bands $E_{k,0;0}$ and $E_{k,0;1}$ on the coupling $g$. The first electron band $E_{k,1;0}$ is out of scale, c.f., Fig. \[fig\_curr\]d). With increasing $g$, one observes both a shift of the bands and a renormalization of the dispersion in $k$. The shift of the bands is, to a large extent, given by the $g^2$ terms in the Hamiltonians, which derives from the $\bm A^2$ term in Coulomb gauge, and from the $H_{PP}$ term in dipolar gauge. This shows that it is crucial to consistently take into account the dipolar interactions $H_{PP}$ when switching between different gauges [@Keeling2007]. The band narrowing can be interpreted as the quantum analog of dynamical localization [@Dunlap1986]. This is most easily understood in dipolar gauge, where the narrowing is already accurately described in the one-band approximation (see below): In the classical case, when a system is driven with a time-periodic high-frequency field corresponding to $Q \to Q(t) = Q_{\rm max}\cos(\omega t)$, the narrowing arises from a time-average of the band structure over one period, $$\begin{aligned}
\label{flippp}
\bar \epsilon_{k,0} = \frac{1}{T}\int_0^T dt\, \epsilon_{k-gQ(t),0}. \end{aligned}$$ In the limit of well-separated bands, where the band has the form $ \epsilon_{k,0}\approx -2J\cos(k)$, this leads to a renormalization $\bar \epsilon_{k,0} = \epsilon_{k,0} \mathcal{J}_0(gQ_{\rm max})$ of the band given by the zeroth order Bessel function. Equation can be transformed to an average of $\epsilon_{k-gQ,0}$ over the classical probability of finding the oscillator at $Q$, $$\begin{aligned}
\label{flipppp}
\bar \epsilon_{k,0} = \frac{1}{\pi}\int dQ \frac{\theta(|Q_{\rm max}|-|Q|)}{\sqrt{Q_{\rm max}^2-Q^2}} \epsilon_{k-gQ,0}.\end{aligned}$$ In the quantum case, the corresponding probability distribution is determined by the photon wave function. If additional inter-band dipolar terms are neglected, the lowest electron-polariton band becomes \[c.f. Eq. \] $$\begin{aligned}
E_{k,0;0} \approx \int \frac{dQ}{\pi} e^{-Q^2} \epsilon_{k-gQ,0} + \frac{\omega_0}{2}.\end{aligned}$$ (The dipolar term leads to an admixture of higher photon number states to the electron-polariton.)
![ Electron-polariton wavefunction. The color shows the overlap $|\langle k,n;m|\chi\rangle|^2$ with the bare state $|k;m;n\rangle$ of band index $m$, photon number $n$. a) and b) Off resonant case $V_0=10$, $\omega_0=1$. c) and d) Resonant case $V_0=5$, $\omega_0=20$. []{data-label="fig4"}](figure05.pdf){width="1\columnwidth"}
In order to systematically analyze the convergence with the band cutoff $m_{\rm max}$, we focus on the renormalization of the lowest electron-polariton band in the large gap case ($V_0=10$, $\omega_0=1$). In Fig. \[fig3\]b and c we plot the splitting $E_{k=0,0;n}-E_{k=\pi,0;n}$ for the lowest two bands $n=0,1$ as a function of the coupling $g$, obtained in dipolar and Coulomb gauge with different band cutoff $m_{\rm max}$. The splitting can be taken as a measure of the band narrowing. For $n=1$, one even observes a sign change, analog to the band flipping for large arguments of the Bessel function in the classical case . For both $n=1$ and $n=0$, the band renormalization is essentially correct within the one-band picture in the dipolar gauge, while it is entirely missed in Coulomb gauge, where for $m_{\rm max}=1$ the bands are only shifted by the $g^2$ term. In Coulomb gauge, the band cutoff must be increased for larger coupling, consistent with the behavior in the semiclassical case, Fig. \[fig\_curr\].
In Fig. \[fig3\]a we plot the position of the lowest band $E_{k=0,0;0}$. In the one-band approximation the shift is overestimated in Coulomb gauge and underestimated in dipolar gauge. This can be explained already to second order in $g$: In Coulomb gauge, the band shift arrises from the $g^2 Q_0^2$ contribution ($\bm A^2$ term), which is already present in the one-band approximation, and is modified due to higher order contributions from the inter-band matrix elements $p_{k;m,m'}$ when more bands are taken into account. In the dipolar gauge, dipolar matrix elements vanish between states of the same band because of inversion symmetry, $D_{k;m,m}=0$. The band shifts arising due to $H_{PP}$ therefore become relevant only when more than one band is taken into account.
The above result clearly show that the dipolar gauge is advantageous at least in the case of isolated bands. However, it should be noted that this statement includes only the convergence with the number of bands, not the number of photons. It is interesting to analyze the structure of the exact electron-polariton wave function in Coulomb gauge and dipolar gauge. Although the energy and the expectation value of gauge invariant observables is the same in both cases, the wave functions differ, as the two gauges correspond to a different light-matter basis. In Fig. \[fig4\] we plot the overlap $|\langle k,m;n|\chi_{k,0;0}\rangle|^2$ of the lowest electron-polariton state at $k=0$ and $g=2$, both in Coulomb gauge and in dipolar gauge. It is evident that the dipolar gauge is more localized in the band index, while the Coulomb gauge is more localized in the photon number. This can be expected because for large $g$ the dipolar gauge Hamiltonian becomes highly nonlinear in $Q_0$ and thus directly couples states with very different photon numbers. This observation is even more prominent at a level anti-crossing, see Fig. \[fig4\]c and d.
conclusion {#sec:555}
==========
We have investigated few-band approximations to the light-matter Hamiltonian within different gauges, extending previous studies on the comparison between different gauges from atomic systems [@Bernardis2018; @Di-Stefano2019] to the solid. The general idea is to obtain tight-binding models from a projection of the continuum theory. When this projection is performed in different gauges, it effectively amounts to a projection to different light-matter hybrid states, because gauge transformations mix light and matter and therefore change the physical meaning of the bare electronic orbitals $c_\alpha$. This shows that the gauge choice can be crucial in order to derive the most accurate few band approximation.
In particular, we have used a multi-center Power-Zienau-Woolley transformation to derive the dipolar gauge Hamiltonian, which features coupling to the vector potential via the Peierls phase factors and direct inter-band coupling through dipolar matrix elements coupling electronic states to the electric field. The dipolar Hamiltonian consistently exhibits advantages over the Coulomb-gauge Hamiltonian for the description of both the semi-classical dynamics and of quantum effects such as the band renormalization due to electron-polariton formation. Beyond that quantitative advantage, the semiclassical limit of the dipolar Hamiltonian has a particularly simple gauge structure [@Boykin2001] as defined by Eq. . The formalism also provides an alternative derivation applicable to the (already nontrivial) light-matter coupling in the semiclassical limit when more than one band, strong correlations, and strong laser fields beyond linear response have to be taken into account [@Golez2019].
In dipolar gauge, the relevant dipolar matrix elements may be determined ab-initio, chosen ad hoc to build some minimal model, or fitted to describe linear optical properties; in either case, the derivation of the dipolar-gauge Hamiltonian implies that the same dipolar matrix elements determine both the linear $\bm E\cdot \bm P$ light-matter interaction and the $\bm P^2$ interaction term when the modes of the cavity or the band is truncated. The importance of consistently keeping both terms has recently been highlighted in the context of atomic physics [@Schaefer2019]. This consistency can also allow to perform further gauge transformations within the truncated model, changing, e.g., back to a Hamiltonian which contains only the vector potential but is nevertheless equivalent to the dipolar gauge. This procedure can be considered as implementing a non-linear truncation within the Coulomb gauge [@Di-Stefano2019].
The particular model which is used in the paper to compare dipolar and Coulomb gauge is, of course, rather simplistic. In particular, unless a certain resonance condition is at play, one would expect that all cavity modes, and not just one, contribute to the vacuum renormalization of the electron bands. On a fundamental level, it will be interesting to systematically take into account infinitely many cavity modes and thus perform the continuous limit from free space to the closed cavity. In the present manuscript, the restriction to one cavity mode rather serves to facilitate a comparison with the exact solution, while keeping physics qualitatively correct: The most dominant effect is a vacuum-induced band narrowing, which provides the quantum analog of dynamical localization in the classical case [@Dunlap1986]. That this effect is opposite to the cavity activated transport [@Schachenmayer2015] in a band which is non-dispersive at zero light-matter coupling.
At strong coupling, it is challenging to deal with the interacting problem of many electrons and photons, in any gauge. One can integrate out the photon field to obtain a description of the solid with induced retarded interactions, which can be dealt with (non-equilibrium) Green’s function techniques. For this task, a gauge is favorable in which a linear coupling dominates. In Coulomb gauge, the coupling $\bm A \cdot \bm p$ is linear only in the weak coupling limit when the diamagnetic term is neglected. In the dipolar gauge, in contrast, the coupling is linear ($\bm E\cdot\bm P$) when the intra-band Peierls phase can be neglected. This may be true, e.g., when the bands are are weakly dispersive in the direction of the field polarization.
At strong coupling, even the basis set to which the projection is performed may itself be optimized. Together with the induced retarded interactions, this is a [*downfolding* ]{} problem which is somewhat similar in nature to deriving few band models with a screened retarded interaction in correlated electron systems [@Biermann2014]. It will be interesting to see whether techniques developed in this context, can be transferred to strongly coupled light-matter systems.
We acknowledge discussions with D. Jaksch, C. Schäfer, and Ph. Werner. M.E. and J. Li were supported by the ERC starting grant No. 716648. GM and AG acknowledge the support of the European Research Council (ERC-319286-‘QMAC’). GM acknowledges support from the FNS/SNF Ambizione Grant PZ00P2-186146. AJM was supported by the Energy Frontier Research Center on Programmable Quantum Materials funded by the U.S. Department of Energy (DOE), Oce of Science, Basic Energy Sciences (BES), under award \# DE-SC0019443. The Flatiron Institute is a division of the Simons Foundation.
Maxwell equations {#AppendixA}
=================
In this section we derive the Heisenberg equations of motion for the Hamiltonian . The canonical variables satisfy canonical communication relations ($\hbar=1$) $$\begin{aligned}
\label{,xsan;Z}
[A_j(\bm r'),\Pi_l(\bm r)]
=
i\delta^T_{jl}(\bm r-\bm r'),\end{aligned}$$ with the transverse $\delta$-function $\delta^T_{jl}(\bm r-\bm r')$ which defines the projection on the transverse component of a field $\bm X=(X_1,X_2,X_3)$, $$\begin{aligned}
X^T_l(\bm r)
&= \sum_{j}\int d^3\bm r'\,\delta^T_{lj}(\bm r-\bm r') X_j(\bm r'),
\\
\delta^T_{lj}(\bm r)
&=
\frac{1}{V}\sum_{\bm q} e^{i\bm q\bm r}
\Big(
1-\frac{q_lq_j}{\bm q^2}
\Big).\end{aligned}$$ To derive Heisenberg equations $\dot O = i [H^{\rm C},O]$, it will be useful to note that the general commutator of an observable $O[\bm A]$ which depends only on $\bm A$ with the field $\bm \Pi$ is equivalent to the functional derivative $$\begin{aligned}
\label{cwjlXA}
[\mathcal{O}[\bm A],\bm \Pi(\bm r)]
=
i\Big(\frac{\delta \mathcal{O}[\bm A]}{\delta \bm A(\bm r)}\Big)^T.\end{aligned}$$ This is seen as follows. A generic operator $\mathcal{O}[\bm A]$ which is a functional of $\bm A$ is a sum of terms $$\begin{gathered}
\mathcal{O}[\bm A]=
\int d^3\bm r_1 \cdots
d^3\bm r_n
\sum_{l_1,...,l_n}
K_{l_1,...,l_n}(\bm r_1,...,\bm r_n)\,\,\,\times
\\
\times \,\,\,
A_{l_1}(\bm r_1)\cdots A_{l_n}(\bm r_n),\end{gathered}$$ so that $$\begin{gathered}
\frac{\delta \mathcal{O}[\bm A]}{\delta A_l(\bm r)}=
\sum_{j=1}^n
\int d^3\bm r_1 \cdots
d^3\bm r_n
\sum_{l_1,...,l_n}
\delta(\bm r-\bm r_j)
\delta_{l_j,l}
\,\,\,\,\times\\
\times\,\,\,\,
K_{l_1,...,l_n}(\bm r_1,...,\bm r_n)
\prod_{a\neq j}
A_{l_a}(\bm r_a).\end{gathered}$$ With the commutator $$\begin{aligned}
[A_{l_1}(\bm r_1)
&
\cdots A_{l_n}(\bm r_n), \Pi_m(\bm r)]
\\
&=
\sum_{j=1}^n
[A_{l_j}(\bm r_j), \Pi_m(\bm r)]
\prod_{a\neq j}
A_{l_a}(\bm r_a)
\\
&=
i
\sum_{j=1}^n
\delta^T_{m,l_j}(\bm r-\bm r_j)
\prod_{a\neq j}
A_{l_a}(\bm r_a)\end{aligned}$$ we get $$\begin{aligned}
[\mathcal{O}[\bm A],\Pi_m(\bm r)]
=
i\int d^3\bm r'
\sum_{l}
\delta^T_{m,l}(\bm r-\bm r')
\frac{\delta \mathcal{O}[\bm A]}{\delta A_l(\bm r')},
\nonumber\end{aligned}$$ from which Eq. follows.
With Eq. , one gets the resulting Heisenberg equations of motion for $\bm \Pi$ and $\bm \nabla\times\bm A$, $$\begin{aligned}
\partial_t \bm \Pi(\bm r,t) &= i[H^{\rm C},\bm \Pi(\bm r,t)]
\nonumber
\\
&=
\label{ghjknbnm}
-\frac{1}{\mu_0}\bm\nabla\times\bm\nabla\times \bm A
-\Big(\frac{\delta H^{\rm C}_{el}[\bm A]}{\delta \bm A(\bm r)}\Big)^T,
\\
\partial_t \bm B(\bm r,t)
&=
\partial_t \bm \nabla\times A(\bm r,t)
=
\nabla\times
i[H^{\rm C},\bm A(\bm r,t)]
\nonumber\\
&=
\bm \nabla\times \frac{1}{\epsilon_0\epsilon(\bm r)} \bm \Pi,\end{aligned}$$ which correspond to the transverse Maxwell components of Maxwell equations with the identification and .
Polarization operator Eq. for a general gauge {#sechdjd}
==============================================
To derive the polarization operator Eq. , we note that the unitary matrix can be written in the form $$\begin{aligned}
\label{lqxn;maz}
W[\bm A]=e^{-iS[\bm A]},\end{aligned}$$ where $ S$ is hermitian. The $S$-matrix need not be derived in closed form, but instead one can always use a coupling constant integral to derive a representation in terms of an ordered exponential $$\begin{aligned}
\label{gsgggeg}
W[\bm A]= T_\lambda e^{-i \int_0^1 d\lambda K[\bm A,\lambda]},\end{aligned}$$ with a hermitian matrix $K[\bm A,\lambda]$. Here $T_{\lambda}$ orders operators with larger $\lambda$ to the left. Although the final equation does not depend on the parametrization and can thus be derived using either of the two representations or , we chose the representation in the following. We first note that the transformation Eq. with is generated by a $\mathcal{W}$ of the form $$\begin{aligned}
\mathcal{W}
&=
\bar T_{\lambda}
e^{i\int_0^1 d\lambda \mathcal{K}(\lambda)},\,\,\,\,\,
\mathcal{K}(\lambda)
=
\sum_{\alpha,\alpha'}
c_{\alpha}^\dagger
K(\lambda)_{\alpha,\alpha'}
c_{\alpha'},
\label{axn;MZ}\end{aligned}$$ where $\bar T_{\lambda}$ orders operators with larger $\lambda$ to the right. This is again shown by the coupling constant integral. With $\mathcal{W}(\lambda):= \bar T_{\lambda} e^{i\int_0^\lambda d\lambda' \mathcal{K}(\lambda')}$ and $\bar c_\alpha(\lambda) :=\mathcal{W}(\lambda) c_\alpha \mathcal{W}(\lambda)^\dagger$, $$\begin{aligned}
\nonumber
\frac{d}{d\lambda}
\bar c_\alpha(\lambda)
&=
i\mathcal{W}(\lambda)
[\mathcal{K}(\lambda),c_\alpha]
\mathcal{W}(\lambda)^\dagger
\\
&=
i\sum_{\beta,\beta'}
K(\lambda)_{\beta,\beta'}
\mathcal{W}(\lambda)
[
c_{\beta}^\dagger
c_{\beta'},c_\alpha]
\mathcal{W}(\lambda)^\dagger
\nonumber
\\
&=
-i\sum_{\beta,\beta'}
K(\lambda)_{\beta,\beta'}
\mathcal{W}(\lambda)
c_{\beta'}
\delta_{\beta,\alpha}
\mathcal{W}(\lambda)^\dagger
\nonumber
\\
&=
-i\sum_{\beta'}
K(\lambda)_{\alpha,\beta'}
\bar c_{\beta'}(\lambda).\end{aligned}$$ With $\bar c_\alpha(0)=c_\alpha$, this equation is integrated to $\bar c_\alpha(1)=\sum_{\beta} \big[T_\lambda e^{-i\int_{0}^1 d\lambda} K(\lambda)\big]_{\alpha,\beta} c_{\beta} $, and thus gives Eq. with Eq. . We now use $\mathcal{W}$ to transform the fields. Because $\mathcal{W}$ does not depend on $\bm \Pi$, the transformation $ \mathcal{W} {\bm A} \mathcal{W}^\dagger = {\bm A}$ follows immediately. To derive Eq. , we again use a coupling constant integral, with $\bar {\bm \Pi}(\bm r;\lambda) := \mathcal{W}(\lambda){\bm \Pi}(\bm r)\mathcal{W}(\lambda)^\dagger$, $$\begin{aligned}
\frac{d}{d\lambda}
&\bar {\bm \Pi}(\bm r;\lambda)
=
\mathcal{W}(\lambda)
\big[
i\mathcal{K}(\lambda),{\bm \Pi}(\bm r)
\big]
\mathcal{W}(\lambda)^\dagger
\nonumber\\
&=
i
\sum_{\alpha,\alpha'}
\mathcal{W}(\lambda)
c_{\alpha}^\dagger
\big[
K[\bm A;\lambda]_{\alpha,\alpha'},{\bm \Pi}(\bm r)
\big]
c_{\alpha'}
\mathcal{W}(\lambda)^\dagger
\nonumber\\
&
\stackrel{\eqref{cwjlXA}}{=}
-\sum_{\alpha,\alpha'}
\mathcal{W}(\lambda)
c_{\alpha}^\dagger
\Big[
\frac{\delta K[\bm A;\lambda]_{\alpha,\alpha'}}{\delta \bm A(\bm r)}
\Big]^T
c_{\alpha'}
\mathcal{W}(\lambda)^\dagger
\nonumber\\
&=
-\sum_{\alpha,\alpha'}
\bar c_{\alpha}^\dagger
\Big[
\frac{\delta K[\bm A;\lambda]_{\alpha,\alpha'}}{\delta \bm A(\bm r)}
\Big]^T
\bar c_{\alpha'}
\nonumber\\
&=
-\sum_{\alpha,\alpha'}
c_{\alpha}^\dagger
\Big[
W(\lambda)^\dagger
\frac{\delta K[\bm A;\lambda]}{\delta \bm A(\bm r)}
W(\lambda)
\Big]^T_{\alpha,\alpha'}
c_{\alpha'}.\end{aligned}$$ This equation can be directly integrated to Eq. with $$\begin{aligned}
\label{hqxbjlnkza}
\bm P^T_{\mathcal{W}}(\bm r)
&=
\sum_{\alpha,\alpha'}
c_{\alpha}^\dagger
\bm M(\bm r)^T_{\alpha,\alpha'}
c_{\alpha'}
\\
\label{hqxbjlnkza01}
\bm M(\bm r)
&=
-
\int_0^1d\lambda\,
W(\lambda)^\dagger
\frac{\delta K[\bm A;\lambda]}{\delta \bm A(\bm r)}
W(\lambda).\end{aligned}$$ The matrix $\bm M$ can be further simplified by noting that $$\begin{aligned}
&\frac{d}{d\lambda}
\Big[W(\lambda)^\dagger
\frac{\delta }{\delta \bm A(\bm r)}
W(\lambda)
\Big]
\nonumber\\&=
\frac{dW(\lambda)^\dagger}{d\lambda}
\frac{\delta }{\delta \bm A(\bm r)}
W(\lambda)
+
W(\lambda)^\dagger
\frac{\delta }{\delta \bm A(\bm r)}
\frac{dW(\lambda)}{d\lambda}
\nonumber\\
&\stackrel{\eqref{gsgggeg}}{=}
i
W(\lambda)^\dagger
K^\dagger
\frac{\delta }{\delta \bm A(\bm r)}
W(\lambda)
-i
W(\lambda)^\dagger
\frac{\delta }{\delta \bm A(\bm r)}
K(\lambda)W(\lambda)
\nonumber\\
&=
i W(\lambda)^\dagger
K^\dagger
\frac{\delta }{\delta \bm A(\bm r)}
W(\lambda)
-i
W(\lambda)^\dagger
K(\lambda)
\frac{\delta }{\delta \bm A(\bm r)}
W(\lambda)
\nonumber\\
&\qquad-i
W(\lambda)^\dagger
\frac{\delta K(\lambda)}{\delta \bm A(\bm r)}
W(\lambda).\end{aligned}$$ Because $K$ is hermitian the first terms cancel, and $$\begin{aligned}
&
W(\lambda)^\dagger
\frac{\delta K(\lambda)}{\delta \bm A(\bm r)}
W(\lambda)
=
i\frac{d}{d\lambda}
\Big[
W(\lambda)^\dagger
\frac{\delta }{\delta \bm A(\bm r)}
W(\lambda)
\Big].\end{aligned}$$ This is inserted into Eq. and integrated to $$\begin{aligned}
\bm M[\bm A]
&=
-i
W(1)^\dagger
\frac{\delta }{\delta \bm A(\bm r)}
W(1)
+i
W(0)^\dagger
\frac{\delta }{\delta \bm A(\bm r)}
W(0).
\nonumber\end{aligned}$$ The second term in the sum vanishes because $W(0)_{\alpha,\alpha'}=\delta_{\alpha,\alpha'}$ is independent of $\bm A$, and thus Eq. follows.
Polarization operator in the dipolar gauge {#pppqqqppp01}
==========================================
### The matrix W, Eq. , for the dipolar gauge {#the-matrix-w-eq.-for-the-dipolar-gauge .unnumbered}
In order to derive the Polarization $\bm P$ in the dipolar approximation, we first explicitly write down the matrix $W[\bm A]$ \[Eq. \] corresponding to the transformation from Wannier orbitals $c_\alpha$ to the hybrid field matter orbitals $\bar c_\alpha$, and the use the general expression for the polarization.
The orbitals $\bar w_{\alpha}(\bm r)\approx \tilde w_{\alpha}(\bm r)$ \[Eq. \] can be used to define hybrid light-matter field operators, $$\begin{aligned}
\label{weicqbx;'}
\bar c_\alpha
=
\int d^3\bm r\,
\bar w_{\alpha}(\bm r)^\dagger\,
\psi(\bm r),\end{aligned}$$ which satisfy canonical anti-commutation relations $$\begin{aligned}
[\bar c_{\alpha}^\dagger,\bar c_{\alpha'}]_+=\delta_{\alpha,\alpha'}\end{aligned}$$ because of the orthogonality. This implicitly defines a unitary transformation $\mathcal{W}$ such that $$\begin{aligned}
\label{kaskb}
\bar c_\alpha
=
\mathcal{W}
c_\alpha
\mathcal{W}^\dagger.\end{aligned}$$ With Eq. , the field operator transforms as $$\begin{aligned}
\bar \psi(\bm r)
&\equiv
\mathcal{W}
\psi(\bm r)
\mathcal{W}^\dagger
\stackrel{\eqref{xqQSKWKQK}}{=}
\sum_{\alpha} w_{\alpha}(\bm r) \bar c_{\alpha}
\nonumber\\
&
\stackrel{\eqref{weicqbx;'}}{=}
\sum_{\alpha}
\int d^3\bm r'\,
w_{\alpha}(\bm r)
\bar w_{\alpha}(\bm r')^\dagger\,
\psi(\bm r')
\nonumber\\
&
\stackrel{\eqref{xqQSKWKQK}}{=}
\sum_{\alpha,\alpha'}
\int d^3\bm r'\,
w_{\alpha}(\bm r)
\bar w_{\alpha}(\bm r')^\dagger\,
w_{\alpha'}(\bm r') c_{\alpha'}
\nonumber\\
&\approx
\sum_{\alpha,\alpha'}
\int d^3\bm r'\,
w_{\alpha}(\bm r)
e^{iq\chi(\bm r,\bm R_\alpha)}
w_{\alpha}(\bm r')^*\,
w_{\alpha'}(\bm r') c_{\alpha'}
\nonumber\\
&=
\sum_{\alpha}
e^{iq\chi(\bm r,\bm R_\alpha)}
w_{\alpha}(\bm r)
c_{\alpha}.
\label{jaxblnJSBDLna}\end{aligned}$$ A projection of the first and last line on $w_\alpha^*$ gives $$\begin{aligned}
\label{gggsgsgsgs}
&\bar c_{\alpha}
=
\sum_{\alpha'}
W[\bm A]_{\alpha,\alpha'}c_{\alpha'}
\\
\label{scn;axz'mlxa}
&W[\bm A]_{\alpha,\alpha'}
=
\int d^3{\bm r}\,
w_{\alpha}(\bm r)^*
e^{iq\chi(\bm r,\bm R_{\alpha'})}
w_{\alpha'}(\bm r).\end{aligned}$$ As $W$ is unitary in the matter indices (up to magnetic corrections), this is a transformation precisely of the form , so that all properties from the previous section App. \[sechdjd\] can be reused to determine the corresponding Hamiltonian $H_\mathcal{W}$, and the polarization operator $\bm P_\mathcal{W}$.
### The polarization density Eq. {#the-polarization-density-eq. .unnumbered}
Using Eqs. and we have $$\begin{aligned}
&\bm M(\bm r)_{\alpha,\alpha'}
=
-i
\sum_{\alpha''}
(W[\bm A]^\dagger)_{\alpha,\alpha''}
\frac{\delta}{\delta \bm A(\bm r)}
W[\bm A]_{\alpha'',\alpha'}
\nonumber\\
&=
-i
\sum_{\alpha''}
W[\bm A]_{\alpha'',\alpha}^\dagger
\frac{\delta}{\delta \bm A(\bm r)}
W[\bm A]_{\alpha'',\alpha'}
\nonumber\\
&\stackrel{\eqref{scn;axz'mlxa}}{=}
-i
\sum_{\alpha''}
\int d^3{\bm r'}\,
w_{\alpha''}(\bm r')
e^{-iq\chi(\bm r',\bm R_{\alpha})}
w_{\alpha}(\bm r')^*
\,\,\,\times\nonumber\\&\,\,\,\,\times\,\,\,\,
\frac{\delta}{\delta \bm A(\bm r)}
\int d^3{\bm r''}\,
w_{\alpha''}(\bm r'')^*
e^{iq\chi(\bm r'',\bm R_{\alpha'})}
w_{\alpha'}(\bm r'').\end{aligned}$$ As the derivative acts only on $\chi(\bm r'',\bm R_{\alpha'})$, we can contract the $\sum_{\alpha''} w_{\alpha''}(\bm r')w_{\alpha''}(\bm r'')^*=\delta(\bm r''-\bm r')$, leading to $$\begin{gathered}
\bm M(\bm r)_{\alpha,\alpha'}
=
q\int d^3{\bm r'}\,
w_{\alpha}(\bm r')^*
e^{-iq\chi(\bm r',\bm R_{\alpha})}
\,\,\,\,\times\\\times\,\,\,\,
\frac{\delta \chi(\bm r',\bm R_{\alpha'})}{\delta \bm A(\bm r)}
e^{iq\chi(\bm r',\bm R_{\alpha'})}
w_{\alpha'}(\bm r').\end{gathered}$$ With the electric approximation $e^{-iq\chi(\bm r',\bm R_{\alpha})} e^{iq\chi(\bm r',\bm R_{\alpha'})} \approx e^{iq\chi_{\alpha,\alpha'}}$, this gives $$\begin{aligned}
\label{ggsgggsg}
\bm M(\bm r)_{\alpha,\alpha'}
&=
q
e^{iq\chi_{\alpha,\alpha'}}
\int d^3{\bm r'}\,
w_{\alpha}(\bm r')^*
\frac{\delta \chi(\bm r',\bm R_{\alpha'})}{\delta \bm A(\bm r)}
w_{\alpha'}(\bm r').\end{aligned}$$ To further simplify this expression, we use the expansion of $\bm A$ within the integral for $\chi$, and perform the derivative . This leads to the expansion , where the expansion coefficients are given by $$\begin{aligned}
\label{maintext:ggsgggsg}
(\bm M_{\nu})_{\alpha,\alpha'}
&=
qe^{iq\chi_{\alpha,\alpha'}}
\int d^3{\bm r'}\,
w_{\alpha}(\bm r')^*
\chi^{\nu}_{\alpha'}(\bm r')
w_{\alpha'}(\bm r'),\end{aligned}$$ with the straight path integral over the mode function, $$\begin{aligned}
\chi^{\nu}_{\alpha'}(\bm r)
=
\int_{\bm R_{\alpha'}}^{\bm r}
d\bm r' \cdot \bm \phi_\nu(\bm r').\end{aligned}$$ Further we assume that the relevant mode functions vary little over one lattice spacing (this corresponds to the electric dipolar approximation), and thus replace $\bm \phi_\nu(\bm r')\approx \bm \phi_\nu(\bm R_{\alpha,\alpha'})$, with the position $\bm R_{\alpha,\alpha'}\equiv \frac{\bm R_{\alpha}+\bm R_{\alpha'}}{2}$ of the bond $(\alpha,\alpha')$. This gives $$\begin{aligned}
\label{App-gegege}
\bm P_\mathcal{W,\nu}
&=
q
\sum_{\alpha,\alpha'}
c_{\alpha}^\dagger c_{\alpha'}
e^{iq\chi_{\alpha,\alpha'}}
\big(
\bm \phi_\nu(\bm R_{\alpha,\alpha'})
\cdot
\bm D_{\alpha,\alpha'}\big),\end{aligned}$$ within the expansion , with the dipolar Matrix elements $$\begin{aligned}
\label{hbwhkqjs}
\bm D_{\alpha,\alpha'}
&=
\int d^3{\bm r}\,
w_{\alpha}(\bm r)^*
(\bm r-\bm R_{\alpha'})
w_{\alpha'}(\bm r).
\\
\label{cghjhbjknjkjn}
&=
\int d^3{\bm r}\,
w_{\alpha}(\bm r)^*
\,\bm r\,
w_{\alpha'}(\bm r).\end{aligned}$$ (The second line follows from the orthogonality of the Wannier orbitals.) This is precisely Eq. . The expression is analogous to a coarse graining of the polarization density $$\begin{gathered}
\label{gegege01}
\bm P_\mathcal{W}(\bm r)
=
q
\sum_{\alpha,\alpha'}
c_{\alpha}^\dagger c_{\alpha'}
e^{iq\chi_{\alpha,\alpha'}}
\bm D_{\alpha,\alpha'}
\delta(\bm r- \bm R_{\alpha,\alpha'}).\end{gathered}$$
Electronic Hamiltonian in dipolar gauge {#pppqqqppp02}
=======================================
Using the expansion of the field operators $\bar \psi(\bm r)$ in $\mathcal{W}$ gauge, the kinetic energy of the continuum Hamiltonian is written as $$\begin{aligned}
\mathcal{W}
H_{el,C}^{(0)}
\mathcal{W}^\dagger
&=
\int d^3\bm r
\,\,
\bar \psi (\bm r)^\dagger
\frac{(-i\bm \nabla -q \bm A(\bm r))^2}{2m}
\bar \psi(\bm r)
\nonumber\\
&=
\sum_{\alpha,\alpha'}
c_{\alpha}^\dagger c_{\alpha'}
\bar h^{kin}_{\alpha,\alpha'},\end{aligned}$$ with $$\begin{gathered}
\bar h^{kin}_{\alpha,\alpha'}
=
\sum_{\alpha,\alpha'}
c_{\alpha}^\dagger c_{\alpha'}
\int d^3\bm r
\,\,
w_{\alpha}(\bm r)^*
e^{-iq\chi(\bm r,\bm R_\alpha)}
e^{iq\chi(\bm r,\bm R_{\alpha'})}
\,\,\,\times\\\times\,\,\,\,
\frac{(-i\bm \nabla + q[\bm \nabla \chi(\bm r,\bm R_{\alpha'})]-q \bm A(\bm r))^2}{2m}
w_{\alpha'}(\bm r).\end{gathered}$$ From Eq. it follows that [@LoudonBook] $$\begin{aligned}
\label{lxqnla}
\bm \nabla
\chi(\bm r,\bm 0)=
\bm A(\bm r)
-
\int d^3\bm r'\,\bm \theta(\bm r,\bm r')\times\bm B(\bm r'),\end{aligned}$$ with $$\begin{aligned}
\bm \theta(\bm r,\bm r')
=
-\int_0^1 ds \, s\, \bm r' \delta(\bm r - s\bm r').\end{aligned}$$ Hence the PZW transformation removes the vector potential from the kinetic energy, up to small magnetic terms of the order of the flux per lattice plaquette. Neglecting again these magnetic contributions, and consistently using the electric approximation $e^{-iq\chi(\bm r,\bm R_\alpha)}e^{iq\chi(\bm r,\bm R_{\alpha'})}\approx e^{iq\chi_{\alpha\alpha'}}$ for the phase factors, we have $$\begin{aligned}
\bar h^{kin}_{\alpha,\alpha'}
=
e^{iq\chi_{\alpha,\alpha'}}
\int d^3\bm r
\,
w_{\alpha}(\bm r)^*
\frac{(-i\bm \nabla )^2}{2m}
w_{\alpha'}(\bm r),\end{aligned}$$ i.e., the matrix element is given by the field-free matrix element dressed by a Peierls phase.
To show that the analogous Peierls substitution can be made for all other matrix elements, it suffices to consider the transformation of an operator which is a function of position, $O=\int d^3\bm r\,\,\psi (\bm r)^\dagger O(\bm r)\psi(\bm r)$. This covers both matrix elements of the lattice potential and of the Coulomb interaction, the latter involving a pair of such operators. Using again the expansion , the transformed operator is written as $\mathcal{W} O \mathcal{W}^\dagger = \sum_{\alpha,\alpha'}
c_{\alpha}^\dagger c_{\alpha'} \bar o_{\alpha,\alpha'}$, with $$\begin{aligned}
\bar o_{\alpha,\alpha'}
&=
\int d^3\bm r
\,\,
w_{\alpha}(\bm r)^*
e^{-iq\chi(\bm r,\bm R_\alpha)}
O(\bm r)
e^{iq\chi(\bm r,\bm R_{\alpha'})}
w_{\alpha'}(\bm r).\end{aligned}$$ With the electric approximation, $e^{-iq\chi(\bm r,\bm R_\alpha)}e^{iq\chi(\bm r,\bm R_{\alpha'})}\approx e^{iq\chi_{\alpha\alpha'}}$, this expression becomes the Peierls substitution $\bar o_{\alpha,\alpha'}=e^{iq\chi_{\alpha,\alpha'}}o_{\alpha,\alpha'}$ of the field-free matrix elements $$\begin{aligned}
o_{\alpha,\alpha'}
&=
\int d^3\bm r
\,\,
w_{\alpha}(\bm r)^*
O(\bm r)
w_{\alpha'}(\bm r).\end{aligned}$$
Details of the one-dimensional solid {#app:fghjkl}
====================================
In this section we provide some details for the solution of the model system defined by Eq. . Eigenstates of the un-driven Hamiltonian $$\begin{aligned}
H=\frac{p_x^2}{2}+2V_0\cos\left(Gx\right).\end{aligned}$$ are Bloch bands $\langle x|k,m\rangle=\phi_{k,m}(x)$ with quasi-momentum $k\in[-\pi,\pi)$ and band energy $\epsilon_{k,m}$, and band index $m=0,1,2,3,...$ To determine these functions, we use a plane wave representation of the Bloch states, $\phi_{k,m}(x)=\frac{1}{\sqrt{L_x}}\sum_{n}u_{k,m}(n)e^{i(k+nG)x}$, leading to the eigenvalue problem $$\begin{gathered}
\epsilon_{k,m} u_{k,m}(n)=\frac{(k+nG)^2}{2}u_{k,m}(n)
\\+V_0[u_{k,m}(n+1)+u_{k,m}(n-1)].\end{gathered}$$ In the numerical calculation the summation in the plane-wave expansion restricted to $n\in \left[-\frac{n_{\rm max}}{2},\frac{n_{\rm max}}{2} \right]$ with a cutoff $n_{\rm max}=50$. Similarly, the exact time-dependent calculation can be carried out straightforwardly on the plane-wave basis, with the ansatz $\psi_{k}(x,t)=\frac{1}{\sqrt{L_x}}\sum_{n}u_{k}(n,t)e^{i(k+nG)x}$, with $$\begin{gathered}
i\dot u_{k}(n)=\frac{(k+nG-qA(t))^2}{2}u_{k}(n)
\\
+V_0[u_{k}(n+1)+u_{k}(n-1)].\end{gathered}$$
Dipolar matrix elements in the Bloch basis become diagonal in momentum $k$ and can be determined as $$\begin{aligned}
D_{k;m,m'}=i\sum_{n}u^*_{k,m}(n)\partial_k u_{k,m'}(n),\end{aligned}$$ which is is well-defined provided the phases of complex $u_{k,m}$’s are fixed. The bare momentum operator is given by $$\begin{aligned}
p_{k;m,m'}=k\delta_{mm'}+\sum_{n} u^*_{k,m}(n) nG u_{k,m'}(n).\end{aligned}$$
[^1]: Note that a dependence on $\bm A$ implies a dependence on all $Q_\nu$, and the functional derivative Eq. becomes $[\mathcal{O}[\bm A],\Pi_\nu]=i\Big(\frac{\delta \mathcal{O}[\bm A]}{\delta Q_\nu}\Big)^T$.
|
---
abstract: 'We have observed an $N$-body/Smoothed Particle Hydrodynamics simulation of a Milky Way like barred spiral galaxy. We present a simple method that samples $N$-body model particles into mock $Gaia$ stellar observations and takes into account stellar populations, dust extinction and $Gaia''s$ science performance estimates. We examine the kinematics around a nearby spiral arm at a similar position to the Perseus arm at three lines of sight in the disc plane; $(l,b)=(90,0), (120,0)$ and $(150,0)$ degrees. We find that the structure of the peculiar kinematics around the co-rotating spiral arm, which is found in [@KHGPC14], is still visible in the observational data expected to be produced by $Gaia$ despite the dust extinction and expected observational errors of $Gaia$. These observable kinematic signatures will enable testing whether the Perseus arm of the Milky Way is similar to the co-rotating spiral arms commonly seen in $N$-body simulations.'
bibliography:
- 'ref2.bib'
date: 'Submitted to MNRAS: $22^{nd}$ December 2014.'
title: 'The stellar kinematics of co-rotating spiral arms in Gaia mock observations'
---
\[firstpage\]
methods: $N$-body simulations — methods: numerical — galaxies: structure — galaxies: kinematics and dynamics — The Galaxy: structure
Introduction {#intro-sec}
============
The spiral features visible in many galaxies have long been the subject of debate. Although it has been almost a century since the resolution of the “great debate” of [@SC21], when it was argued over whether these beautiful spiral structures were nebulae within our galaxy or galaxies in their own right, the mechanisms which generate them are still uncertain. One of the problems with developing a comprehensive theory of spiral arms is the so called “winding dilemma". It is known from observations of disc galaxies that the stars in the inner region have a higher angular velocity than those in the outer region. Therefore the spiral structure should “wind up" relatively quickly if the spiral arms rotate at the mean rotation velocity of the stars [e.g. @W96], contrary to observations of many “grand design” spiral galaxies. A proposed solution to the winding dilemma is given by spiral density wave theory [@LS64] which treats the spiral structure as a density wave which can rotate rigidly as a feature with a constant pattern speed and thus be long lived.
However, no $N$-body simulations have yet been able to reproduce these long lived stable spiral arms, despite the increase in computational power and resolution which has occurred in recent years [e.g. @S11; @DB14]. Recent work has shown spiral modes and waves which survive over multiple rotations [@QDBMC11; @RDL13; @SC14] while the spiral arm features in the stellar mass are short-lived but recurrent [e.g. @SC84; @CF85; @B03; @FBSMKW11; @GKC12; @GKC12-2; @GKC13; @BSW13; @RFetal13; @DVH13] including in galaxies with a central bar [e.g. @GKC12-2], implying that the large spiral arms visible in external galaxies may only $appear$ to be rigid structures extending over the disc, while in fact being made of transient reforming features.
The interpretation of the transient and recurrent spiral arm features observed in $N$-body simulations is still in debate. For example, [@MFQDCVEB12] showed for the first time (by studying the time evolution of the disc power spectrum) that spiral wave modes in $N$-body simulations can last for as long as 1 Gyr, which can justify treating the wave modes as quasi-stationary structure, and the transient and recurrent spiral arm features can be explained by the superposition of different modes with different pattern speeds [see also @RDQW12; @SC14]. On the other hand, [@GKC12; @DVH13; @BSW13] demonstrated non-linear growth of the spiral arm features due to similar but different (in terms of evolution) mechanisms from swing-amplification [@T81], which could be difficult to explain with the linear superposition of the wave modes.
Our position within the Milky Way gives us a unique view of these spiral structures seen in external galaxies, but it comes with its own set of problems which we must overcome when studying them. The location and kinematics of the gaseous component of the arms may be determined from HI and CO observations [e.g. @DHT01; @NS03; @KK09]. However to observe the kinematics of the stellar component in and around the spiral arms we must look through the disc plane, which carries the heaviest levels of dust and gas, and thus high levels of extinction.
Dust extinction has long been a problem for Milky Way model construction. Although there are reasonably reliable extinction maps for extra galactic sources whose extinction by the interstellar medium of the Milky Way can be corrected as a function $A_\lambda(l,b)$ [e.g. @SFD98], three dimensional extinction mapping for sources within the Milky Way i.e. $A_\lambda(l,b,d)$ is more challenging. There are three dimensional extinction maps for individual sections of the sky [e.g. @DS01; @MRRSP06; @HBJ14; @SM14] and two dimensional maps have been extended to three dimensions [e.g. @DCL03]. However a truly Galactic 3D extinction map does not yet exist [@RB13]. The European Space Agency (ESA)’s $Gaia$ mission will help us map the stellar structure and kinematics of the Milky Way, and help constrain extinction at the same time [@BJetal13].
$Gaia$, which was launched on the 19th December 2013 will provide detailed astrometric [e.g. @LLHOBH12], spectroscopic [e.g. @Ketal11] and photometric [e.g. @Jetal10] information for around one billion stars in the Milky Way. Detailed information on $Gaia$ scientific accuracies is available in, for example, [@dB12]. Synthetic $Gaia$ mock data have already been used to demonstrate different applications of the real $Gaia$ data set. For example, [@AMAFR14] use three tracer populations (OB, A and Red Clump stars) with the $Gaia$ selection function, errors and dust extinction, and demonstrated that the $Gaia$ mock data can recover the parameters of the Galactic warp. [@RGFAAA14] examine the Galactic bar in the $Gaia$ observable space using Red Clump tracers with the $Gaia$ selection function, errors and dust extinction combined with selected Red Clump stars from the Apache Point Observatory Galactic Evolution Experiment [APOGEE DR10, e.g. @Aetal14] showing the value of combining data from complimentary surveys. In [@HK13] we show that we can recover the large scale structure of the Galactic disc with our Made-to-Measure Galaxy modelling code, [@HKM13; @HK12; @HK13], and make a good estimation of the patten speed of the bar, using tracer populations of M0III and Red Clump stars with the $Gaia$ selection function, errors and dust extinction.
There exist full mock catalogues of $Gaia$ stars, e.g. the $Gaia$ Universe Model Snapshot () which provides a view of the Besançon Galaxy model as seen from $Gaia$ [@Rea12], taking into account dust extinction while assuming there are no observational errors. This detailed prediction of $Gaia$ observations gives an excellent indication of the volume and quality of data which will become available from $Gaia$, predicting 1.1 billion observable stars, almost 10,000 times more than from its predecessor $Hipparcos$. can be extended through the $Gaia$ Object Generator () [@Xetal14] to simulate intermediate and final catalogue data including the introduction of realistic astrometric, photometric and spectroscopic observational errors to the catalogue based upon $Gaia$ science performance estimates. While these mock data provide an excellent example of the capabilities of $Gaia$, the Besançon galaxy model is an axisymmetric model and a kinematic model not a dynamical model. Although $Gaia$ will not provide accelerations, the kinematics it will provide are from a dynamical system, the Milky Way. Thus it is important for our purpose to generate catalogues from fully dynamical models with non-axisymmetric structures, such as spiral arms and a bar, which for example $N$-body disc galaxy models can provide.
Therefore we propose here to create mock $Gaia$ observations from an $N$-body model using a population synthesis code such as [@SBJB11], or the methodology presented in [@PCK12] or [@LWCKHFC14]. is a flexible population synthesis code for generating a synthetic stellar catalogue from an $N$-body or an analytical galaxy model over wide sections of the sky, with a sampling scheme which generates a smoothly distributed sample of stars. Synthetic catalogues generated from dynamical Galaxy models are essential for preparing to exploit the real $Gaia$ catalogue and can be used to determine whether certain features within the Milky Way will be visible to $Gaia$.
In our previous work [@KHGPC14] we examined the kinematics of both the stellar and gas components around a transient, co-rotating spiral arm in a simulated barred spiral galaxy similar in size to the Milky Way. Although this arm is transient, similar arms recur during the evolution of the galaxy. We made predictions of observable kinematic signatures that may be visible in the Milky Way’s Perseus arm if it is also a transient, recurrent and co-rotating spiral arm. We then compared our simulation with data from APOGEE and the maser sources from [@Retal14] measured by the Bar and Spiral Structure Legacy (BeSSeL) survey and the Japanese VLBI Exploration of Radio Astronomy (VERA), finding tentative agreement between our simulation and the observations. Owing to the low number of maser sources and the lack of distance information for the APOGEE stars no firm conclusions could be drawn; however it is encouraging to see similar features in both, including the possible signatures of a co-rotating spiral arm.
In this paper we build upon the previous work by generating a stellar sample with different populations from the simulation data in @KHGPC14 and making mock observations of these stars taking into account the expected $Gaia$ science performance estimates. The aim is not to make further predictions about the kinematics of transient, recurrent and co-rotating spiral arms but rather to examine whether these signatures, remain visible in the $Gaia$ data if they exist in the Milky Way.
Simulation {#sim}
==========
We use the simulated galaxy which is presented in @KHGPC14 and [@GKC14b]. The details of the numerical simulation code, and the galaxy model are described in [@KHGPC14]. We briefly describe the galaxy model in this section. The galaxy is set up in isolated conditions, and consists of a gas and stellar disc but no bulge component. The discs are embedded in a static dark matter halo potential [@RK12; @KHGPC14]. The dark matter halo mass is $M_{\rm dm}=2.5 \times 10^{12}$ $\rm M_{\odot}$, and the dark matter density follows the density profile from [@NFW97], with a concentration parameter of $c=10$. The stellar disc is assumed to follow an exponential surface density profile with the initial mass of $M_{\rm d,*} = 4.0 \times 10^{10}$ $\rm M_{\odot}$, a radial scale length of $R_{\rm d,*} = 2.5$ kpc and a scale height of $z_{\rm d,*} = 350$ pc. The gas disc is set up following the method of @SDMH05, and has an exponential surface density profile with the scale length of $R_{d,g} = 8.0$ kpc. The total mass of the gas is $10^{10}$ $\rm M_{\odot}$. The simulation comprises $10^6$ gas particles and $4 \times 10^6$ star particles; therefore each particle has a mass of $10^4$ $\rm M_{\odot}$. The resolution is sufficient to minimise numerical heating from Poisson noise [@FBSMKW11; @S13]. We employ a minimum softening length of $158$ pc (equivalent to a Plummer softening length of $53$ pc) with the spline softening and variable softening length for gas particles as suggested by @PM07.
The radial profile of the mean metallicity of stars and gas is initially set by $\mathrm{[Fe/H]} (R) = 0.2 - 0.05(R/1 \text{ kpc})$, and the metallicity distribution function at each radius is centred on the mean metallicity value with the dispersion set to a Gaussian distribution of $0.05$ dex for the gas and $0.2$ dex for the stars. The stellar ages are set randomly between 0 and 10 Gyr for stars present at the beginning of the simulation.
The simulation was run for 1 Gyr from the initial conditions with the $N$-body smoothed particle hydrodynamics code, [e.g. @KG03; @RK12; @BKW12; @KOGBC13; @KGBGR14] without the inclusion of any continuous external inflow of gas for simplicity. In this paper we use the same snapshot of the galaxy as used in [@KHGPC14] which is taken at $t=0.925$ Gyr, as this snapshot shows a spiral arm at a similar location to the Perseus arm from the Milky Way (see Fig. \[galaxy\]).
![Snapshot of the simulated galaxy from @KHGPC14 which is also used in this paper. The left (right) panel shows the face-on view of the star (gas) particle distribution. The solid line indicates the position of the spiral arm identified. The observer is assumed to be located at $(x,y)=(-8, 0)$ kpc. Three line-of-sight directions ($l_{\rm LOS}=90, 120$ and 150 deg) are highlighted with the dotted lines. The galaxy is rotating clockwise.[]{data-label="galaxy"}](snap_c.ps){width="\hsize"}
$Gaia$ mock catalogue {#Model}
=====================
\[Kawata14\] In [@KHGPC14] the kinematics of the spiral arm shown in Fig. \[galaxy\] are examined at three lines of sight $l_{\rm LOS}=90, 120$ and 150 deg, with $b_{\text{los}}=0$ because of the lower extinction relative to other lines of sight in the plane. Predictions are made of the observational signatures of co-rotating spiral arms notably the difference in kinematic structure between the trailing near side and leading far side of the spiral arm. In general, in [@KHGPC14] (as also shown in [@GKC14]) the stars in the trailing near side rotate slower because they tend to be at the apo-centre and migrate outward, and the stars in the leading far side rotate faster as they tend to be at the peri-centre and migrate inward. There are however some stars which follow the opposite trend, leading to multiple populations seen in the rotational velocity in the leading far side; one faster, and one slower than the single population in the trailing near side. These features which will be discussed later may be caused by the co-rotation resonance of the spiral arm, and are visible at different galactic longitudes because the arm in the simulation co-rotates at all the examined radial range. However, in [@KHGPC14], the spiral arm kinematics are examined using the full, error and extinction free $N$-body data and thus such trends, when present, are easy to identify.
In this Section we describe how we generate a sample of stars from the $N$-body model of [@KHGPC14] to produce a mock $Gaia$ catalogue. It is worth noting that the population synthesis code, [@SBJB11] provides a tool to generate stellar populations from $N$-body simulation data. However, because we plan to combine such a tool with our Made-to-Measure Galaxy modelling code, , we have developed our own simplified version of , a population synthesis code called , (Stellar Numbers And Parameters Determined Routinely And Generated Observing $N$-body Systems). uses the same isochrones and extinction map as , but uses a different and more simplistic process to generate the stellar catalogue which is described in Section \[Synth\]. allows us to add the expected Gaia errors more easily, and enables us to track the link between sampled stars and their parent $N$-body particle for our future studies, e.g. modelling of the Galactic disc by fitting tracers from multiple stellar populations, and identifying radially migrating stars and non-migrating stars trapped by the spiral arm [@GKC14].
Extinction {#Ex}
----------
We use the extinction map of the Milky Way taken from [@SBJB11], which is a 3D polar logarithmic grid of the dust extinction constructed using the method presented in @BKF10 and the dust maps from @SFD98. The same extinction is applied in [@HK13] and more detail is given there. In an update from @HK13 we follow the correction to the Schlegel $E_{B-V}$ presented in @Setal14 such that $$E_{B-V}=E_{B-V}\biggl(0.6+0.2\biggl(1-\text{tanh}\biggl(\frac{E_{B-V}-0.15}{0.1}\biggr)\biggr)\biggr).$$ This correction is made as it has been suggested [e.g. @AG99; @YFS07] that the reddening is overestimated by the maps from [@SFD98] by $\sim$1.3-1.5 in regions with high extinction with $A_V>0.5$ ($E_{B-V}>0.15$). This correction reduces extinction by $\sim40\%$ for low latitude high extinction regions but has minimal effect on high latitude low extinction regions.
Population Synthesis: {#Synth}
----------------------
The goal of this population synthesis code is to split each $N$-body particle from the galaxy simulation into an appropriate number of stellar particles creating a mock catalogue of observable stars from our $N$-body model. We must choose an IMF and a set of isochrones with which to work. We choose a Salpeter IMF [@S55] where the IMF, $\Phi(m)$, is defined in each mass interval d$m$ as $$\Phi(m)\text{d}m=Am^{-(x+1)}\text{d}m,$$ where $x=1.35$ is the Salpeter index, and $A$ is a constant for normalisation in the desired mass range. We set this constant as $$A_i=m_i\biggl(\int_{m_{\star,\text{min}}}^{m_{\star,i,\text{max}}}m^{-x}\text{d}m\biggr)^{-1},$$ where $m_i$ is the $N$-body particle mass, $m_{\star,i,\text{max}}$ is the maximum initial mass of any surviving star and $m_{\star,\text{min}}$ is the minimum stellar mass to be considered. We make use of the Padova isochrones [e.g. @BBCFN94; @MGBGSG08], although the choice of isochrones (and IMF) may be substituted with others with no change to the methodology.
It is worth noting that the Padova isochrones are available only for stellar masses above 0.15 $M_{\odot}$. for example uses the isochrones from @CBAH00 to extend the mass limit down to 0.07 $M_{\odot}$, which is the hydrogen mass burning limit. We set our lower limit on stellar mass as $m_{\star,\text{min}}=0.1$ $M_{\odot}$ to correspond with the simulation from [@KHGPC14] and extrapolate from the Padova isochrones for $0.1\leq M_{\odot}\leq0.15$. It is relatively safe to do this because all such stars lie on the main sequence. Additionally these exceedingly faint stars will not be visible at the distance of the spiral arms which are the focus of this work.
As discussed in Section \[sim\] each $N$-body star particle in the simulated galaxy has been assigned an age and metallicity within the chemodynamical code , then it is made to evolve. When we examine the snapshot, each particle is matched to its nearest isochrone in both metallicity and age from the grid of isochrones which are extracted from . Once an isochrone is selected, we identify $m_{\star,i,\text{max}}$ from the isochrone. We then determine how many stars to sample from the $N$-body particle by integrating the IMF over the desired mass range; $$N_s=A\int_{m_{\star,i,<V_{\text{lim}}}}^{m_{\star,i,\text{max}}}m^{-(x+1)}\text{d}m,$$ where $m_{\star,i,<V_{\text{lim}}}$ is minimum mass required for the star particle to be brighter than our apparent magnitude selection limit, $V_{\text{lim}}$, taking into account the extinction value at the position of the parent particle. Stars smaller than $m_{\star,i,<V_{\text{lim}}}$ are not used in the subsequent analysis, to save on computational time.
We then randomly sample stellar masses from the section of the isochrone $N_s$ times. We have weighted the random selection by the IMF using the equation $$m_{\star}=(R m_{\star,i,\text{max}}^{-x}+(1-R)m_{\star,i,<V_{\text{lim}}}^{-x})^{\frac{1}{-x}},$$ where $R$ is a random number between 0 and 1.
The isochrones are comprised of discrete stellar data, and therefore we then interpolate within the nearest isochrone values of $M_V$ and $V-I_c$ to determine $M_{V_{\star}}$ and $V-I_{c\star}$ for the generated $m_{\star}$. At this stage we assume the generated stars have the same position and velocity as their parent particles.
Observational Errors {#Error}
--------------------
Having generated the visible stellar catalogue we then add observational errors based upon the $Gaia$ Science Performance estimates[^1]. We use the post launch error estimates approximated from the estimates in pre-launch performance by Mercè Romero-Gómez [e.g. @RGFAAA14], provided through the Gaia Challenge collaboration[^2]. We assume the position and velocity of the Sun is known. We locate the observer at ($-8$,0,0) kpc as shown in Fig. \[galaxy\], and the motion of the Sun is assumed to be 228 km s$^{-1}$. For this work, while generating the stellar catalogue we produced stars only brighter than $V_{\text{lim}}\leq16$ mag, which is well within $Gaia's$ $m_G\leq20$ mag magnitude limit for the astrometry. However, because we are interested in the Galactic radial and rotation velocity for the stars, which requires the full 6D phase space information, we chose the lower magnitude limit where $Gaia$ RVS can produce the reasonably accurate line-of-sight velocity. Note that the errors are added to the parallax, proper motion and line-of-sight velocities.
A full description of the method to add the pre-launch $Gaia$ error is available in [@HK13]. However the $Gaia$ science performance estimates have been revised after launch, and as such a correction must be made. The error in parallax has increased, and although it has little effect for stars with $m_V\leq16$ mag which we work with in this paper, the coefficients within the equation to describe the pre-launch parallax performance (provided by Kazi, Antoja & DeBruijne (Oct. 2014) by fitting to the new estimations on the $Gaia$ science performance web page) are revised to $$\begin{aligned}
\sigma_{\pi}&=&(-11.5+706.1z+32.6z^2)^{1/2} \nonumber \\ & &\times(0.986+(1-0.986)(V-I_c)),
\label{sigpi}\end{aligned}$$ where $$z=\text{max}(10^{0.4(12-15)},10^{0.4(G-15)}),
\label{zmax}$$ correcting also the typo for equations (\[sigpi\]) and (\[zmax\]) in [@HK13].
Additionally, because of the loss of spectroscopic accuracy by $\sim1.5$ mag in the RVS post launch performance we also apply a correction to the error function for the end of mission radial velocity. We change the table[^3] of values $a$ and $b$, again determined by fitting the revised performance estimates on the $Gaia$ science performance web page, for the equation $$\sigma_{v_r} = 1 + b\text{e}^{a(V-14)},$$ where $a$ and $b$ are constants dependant on the spectral type of the star. The new table along with the code to add the $Gaia$ error is available online[^4].
Results {#R}
=======
As discussed in Section \[Kawata14\], it was shown in [@KHGPC14] that in general the stars in the trailing near side of the spiral arm rotate slower than average because they tend to be at the apo-centre, and the stars in the leading far side of the spiral arm rotate faster than average as they tend to be at the peri-centre. However, there are groups of stars which follow different trends leading to multiple populations which will be discussed later. It is important to determine whether such features will still be visible in the $Gaia$ catalogue, not just the error and extinction-free $N$-body model. In this Section we show the result of sampling these $N$-body data into stellar data, first looking at the properties of the resulting mock stellar catalogue, and then examining the spiral arm kinematics with the stellar data taking into account dust extinction and $Gaia$ science performance estimates.
Population synthesis {#population-synthesis}
--------------------
In this section we describe the stellar catalogue produced by , and show the resulting colour magnitude diagram (CMD) varying the area of the sky coverage. Fig. \[CMD\] shows the CMD for stars generated by from particles within a square region of $\pm2$ deg (upper) and $\pm5$ deg (lower) around $(l,b) = (90,0)$ deg. The upper panel of Fig. \[CMD\] shows clearly the individual stellar isochrones because there are only a small number of $N$-body particles in the selected region, and each particle has only one age and metallicity. These problems are resolved when smoothing is applied in the phase space distribution and age-metallicity distribution [e.g. @SBJB11]. However, as discussed in Section \[Synth\] we deliberately avoid this smoothing to maintain the clear particle-star relation. The lower panel of Fig. \[CMD\] shows no such discrete structure, as there are sufficiently many particles to cover a broad range of stellar ages and metallicities in the CMD. Therefore, care is required with the resolution of the $N$-body simulation and the selection function if we discuss in detail the stellar population distribution in the CMD. However, this is unlikely to affect the study in this paper.
![Colour magnitude diagram for stars generated by from particles within a square region of $\pm2$ deg (upper) and $\pm5$ deg (lower) around $(l,b) = (90,0)$ deg. Stars with apparent magnitude of $m_V\leq16$ only are included.[]{data-label="CMD"}](HR2_c.ps){width="\hsize"}
Observable Spiral Arm Kinematics
--------------------------------
In this section we examine if the possible kinematic signatures of co-rotating transient and recurrent spiral arms identified in [@KHGPC14] will be visible in the $Gaia$ data even given the dust extinction in the disc and $Gaia's$ science performance accuracy. A detailed analysis of the kinematics themselves is the focus of [@KHGPC14], while this work is concerned with the visibility of this kinematic structure in the $Gaia$ data. We examine the rotational velocities of the stars in the catalogue for different distances because in [@KHGPC14] we found the rotation velocity is most affected by the transient co-rotating spiral arm. Then we calculated the Probability Density Function (PDF) of the rotation velocity of stars behind and in front of the spiral arm using Kernel Density Estimation (KDE) which we are using as a desirable alternative to histograms [e.g. @W06].
Fig. \[Crot\] shows a smoothed contour plot of the galactocentric rotational velocity against distance for particles and stars within a square region of $\pm5$ degrees around $(l,b)=(90,0)$ (left), $(l,b)=(120,0)$ (middle) and $(l,b)=(150,0)$ (right). This compares the kinematics of the underlying $N$-body model (upper) with the stellar catalogue generated with , before (middle) and after (lower) the addition of the errors from the $Gaia$ science performance estimates. Owing to the high percentage of low mass and luminosity stellar types which would dominate the selected region and saturate the plot at small distances, we have made cuts to our sample to visualise the underlying kinematic structure from the stellar catalogue. We have first cut the sample of stars in all three lines of sight with absolute magnitude, $M_V\leq-1$, calculated from the apparent magnitude $m_V$ and observed distance $d_{\text{obs}}$, assuming the dust extinction is known. We then cut with $\sigma_{v_{\text{los}}}/(v_{\text{los}}\times d_{\text{obs}})\leq0.015$ kpc$^{-1}$ to select the stars with lower error in the line of sight velocities at a smaller distance to generate similar quantities of data at different distance scales. This is purely for illustration purposes and we are not suggesting that this is the best possible selection function. The upper panels of Fig. \[Crot\] show the different kinematic structure in the $N$-body model at the different lines of sight. These are the same data as those shown in the top panels of Fig. 4 from [@KHGPC14]. Note that the density colour scale for the $N$-body data is different from the stellar data in the middle and lower panels.
The middle row of panels of Fig. \[Crot\] show the velocities of the selected stars, which appear slightly different from those of the $N$-body data owing to the selection function. While the general shape of the distribution has been recovered, at $(l,b)=(90,0)$ deg (middle left) the fast rotating stars within the arm dominate the density scale and wash out the rest of the plot slightly. At $(l,b)=(120,0)$ deg (middle), although there is some saturation around $220$ km s$^{-1}$ the kinematic structure is clearly visible and is a good match to the particle data. Similarly at $(l,b)=(150,0)$ deg (middle right), despite the lower number of counts, the kinematic structure is clearly shown.
The lower panels of Fig. \[Crot\] show the error affected rotation velocity and distance for the selected stars taking $Gaia$ science performance estimates into account. The rotation velocity is calculated from the observed parallax, proper motion and line of sight velocities. At $(l,b)=(90,0)$ (lower left) the shape of the distribution remains relatively unchanged, with the main loss in accuracy occurring around $d_{\text{obs}}\approx7-10$ kpc. The recovery of the kinematic structure around the spiral arm around $d_{\text{obs}}\approx4$ kpc remains almost identical to the case without observational errors. At $(l,b)=(120,0)$ (lower middle) the visible loss of accuracy is again in the outer region of $d_{\text{obs}}\approx7-10$ kpc, with the region containing the spiral arm remaining very similar to that of the error free case. At $(l,b)=(150,0)$ (lower right), the entire distribution remains very similar to the middle right panel, the case without $Gaia$ like observational errors.
{width="\hsize"}
{width="\hsize"}
Fig. \[Hrot\] shows the PDF’s, with a KDE bandwidth of 4, for the rotational velocity of the stars in the catalogue within a square region of $\pm5$ degrees around $(l,b)=(90,0)$ (left), $(l,b)=(120,0)$ (middle) and $(l,b)=(150,0)$ (right) in the trailing near side, between 1 and 2 kpc closer than the centre of the arm (upper) and leading far side between 1 and 2 kpc further than the centre of the arm (lower). Note that these distance bins were chosen as they show the discussed structure most clearly; the same features are present closer to the arm but are less clear. The centre of the arm was determined to be at $d=4.0$ kpc at $(l,b)=(90,0)$, $d=3.4$ kpc at $(l,b)=(120,0)$ and $d=3.3$ kpc at $(l,b)=(150,0)$. Note that Fig. \[Hrot\] uses all the stars with $m_V\leq16$, not applying the selection function used for illustration purposes in Fig. \[Crot\]. At all three lines of sight Fig. \[Hrot\] shows a clear difference in the distribution of velocities for the ‘true’ data (black solid) when comparing the different observed distances, as shown in [@KHGPC14]. This is a positive outcome considering the loss of data from the dust extinction. When comparing the ‘true’ (black solid) stellar catalogue data with the stellar data taking into account dust extinction and $Gaia$’s expected errors (red dashed) a general smoothing out of the structure is evident in the ‘observed’ data. The upper panels of Fig. \[Hrot\] showing the trailing near side of the arm show very similar PDF’s when comparing the true and observed stellar data, whereas the lower panels showing the leading far side show an information loss, especially at $(l,b)=(90,0)$, where the three peaks are no longer resolved. This is to be expected because of the higher distances and therefore additional extinction; however at $(l,b)=(120,0)$ and $(150,0)$ even on the far side of the spiral arm the structure within the distribution is still clearly visible.
When comparing the ‘observed’ data in Fig. \[Hrot\] in front and behind the spiral arms, we see a clear difference in the PDF at all three lines of sight. In each case, the PDF in the trailing near side of the spiral arm forms a single central peak similar to the mean rotation velocity, with a small tail towards faster rotation velocities whereas the leading far side of the spiral arm shows a broader distribution of velocities with a peak velocity faster than the peak for the trailing near side. The difference is particularly apparent at $(l,b)=(120,0)$ deg where the leading far side shows two clear peaks, one faster and one slower than the single peak in the trailing near side. This bimodal distribution can also be seen in the lower middle panel of Fig. \[Crot\] between 4.39 and 5.39 kpc (although note that Fig. \[Crot\] uses a different selection function). Also at $(l,b)=(150,0)$ deg the single broad peak in the trailing near side is easily distinguishable from the leading far side which shows three peaks. These three peaks are also partially visible in the lower right panel of Fig. \[Crot\] between 4.29 and 5.29 kpc. These features all match those observed in [@KHGPC14] despite the addition of dust extinction and observational errors to the data.
In general, as shown in [@GKC14], the stars in the leading side rotate faster as they tend to be at peri-centre phase and migrating inward, and stars in the trailing side rotate slower as they tend to be at apo-centre phase and migrating outward. This explains the single large peak in the trailing side, and the largest peak on the leading side which has a higher rotational velocity than the single peak on the trailing side as shown in Fig. \[Hrot\]. However, when the transient spiral arm starts forming, stars which are close to the arm on the trailing side and are close to the peri-centre phase, are accelerated towards the arm, passing through and then slowing down as they reach the apo-centre on the leading side as discussed in [@KHGPC14]. These stars correspond to the ‘slower’ peaks visible in the lower panels of Fig. \[Hrot\]. Similarly, the stars which are close to the arm and close to the apo-centre phase on the leading side are decelerated by the arm, and are overtaken by the arm. Then they are accelerated again by the arm once they are on the trailing side at peri-centre phase, which corresponds to the small tail present at high velocities in the upper panels of Fig. \[Hrot\]. The difference in the rotation velocity distribution between the leading and trailing side of the spiral arm seen in Figs. \[Crot\] and \[Hrot\] is that the latter population is smaller than the former. It appears that it is easier for stars to escape from the arm on the leading side than the trailing side. From our analysis of $N$-body simulations this appears to be a common feature of transient and co-rotating spiral arms.
[@CQ12] propose that the radial overlap of multiple longer-lived patterns moving at different pattern speeds can reproduce the transient spiral features, which when strong enough can lead to radial migration away from the co-rotation radius associated with co-rotating spiral arms as seen, for example in [@GKC12; @GKC12-2]. In such a scenario, the spiral arm features are co-rotating, which may give rise to the co-existence of many inner and outer Lindblad resonances in a range of radii and lead to the features visible in Figs. \[Crot\] and \[Hrot\]. However, further analysis of the spiral arms in $N$-body simulations is required before drawing firm conclusions on the mechanism that generates such kinematic signatures, which we will tackle in future studies.
From Figs. \[Crot\] and \[Hrot\] we find that $Gaia's$ scientific accuracy ought to be sufficient to examine the kinematic structure of the nearby spiral arms in the Milky Way, even on the far side of the arm. Fig. \[Hrot\] shows clear differences in the kinematics in the leading and trailing sides of the spiral arm, notably the difference in the number and locations of the peaks, and the small high velocity tail present in the trailing near side. The comparison between the middle and lower panels of Fig. \[Crot\] shows little difference, implying that the observational error from $Gaia$ will have limited effect on our ability to study the kinematics of the spiral arms. Further examination of galaxy models constructed using the different theories of spiral arm formation will be essential to determine the distinct kinematic signatures of each theory.
Summary {#SF}
=======
We observed our $N$-body/SPH simulation of a Milky Way like barred spiral galaxy to create a mock $Gaia$ stellar catalogue, with particular interest in the stellar kinematics in and around the spiral arms. We focused on the same three lines of sight in the disc plane as [@KHGPC14], $(l,b)=(90,0), (120,0)$ and $(150,0)$ deg and analysed the galactocentric rotational and line of sight velocities of the selected stars as a function of the distance from the observer. In agreement with existing literature on $N$-body spiral galaxy simulations the spiral arm features seen in the stellar mass in our model are transient, recurrent and co-rotating, i.e. the spiral arm is rotating at the circular velocity of the stars at the selected lines of sight.
We show that the structure in the kinematics identified in [@KHGPC14] remains visible after the inclusion of dust extinction and observational errors based upon $Gaia$ science performance estimates. Although the inclusion of these observational effects makes the trends less clear, they are still observable in the mock $Gaia$ data in front of, inside and behind the spiral arm. The structure on the trailing near side is relatively unchanged, whereas the structure on the leading far side is, unsurprisingly, more affected, although the bi-modal (or more) and broader distribution of the rotation velocities is still clearly visible. Because we believe that these kinematic signatures are indications of transient and co-rotating spiral arms owing to the co-rotation resonance at all radii, we predict they should be visible in the $Gaia$ data at different longitudes if the Milky Way’s Perseus arm is also a transient and co-rotating spiral arm.
Encouraged by the success of this study, we intend to repeat the analysis with simulated galaxies which use different theories of spiral structure formation, for example test particle simulations [e.g. @MQ08; @MBSB10; @MF10; @FSF14; @Aetal14-2] and $N$-body simulations with a fixed spiral arm potential [e.g. @WBS11]. From these analyses we expect to make predictions of the kinematic signatures of different spiral arm theories, which can be tested by the $Gaia$ stellar catalogue.
Acknowledgements {#acknowledgements .unnumbered}
================
We gratefully acknowledge the support of the UK’s Science & Technology Facilities Council (STFC Grant ST/H00260X/1 and ST/J500914/1). The calculations for this paper were performed on Cray XT4 at Center for Computational Astrophysics, CfCA, of the National Astronomical Observatory of Japan and the DiRAC facilities (through the COSMOS consortium) including the COSMOS Shared Memory system at DAMTP, University of Cambridge operated on behalf of the STFC DiRAC HPC Facility. This equipment is funded by BIS National E-infrastructure capital grant ST/J005673/1 and STFC grants ST/H008586/1, ST/K00333X/1 & ST/J001341/1. The authors acknowledge the use of the IRIDIS High Performance Computing Facility, and associated support services at the University of Southampton. We would also like to thank PRACE for the use of the Cartesius facility. This work was carried out, in part, through the $Gaia$ Research for European Astronomy Training (GREAT-ITN) network. The research leading to these results has received funding from the European Union Seventh Framework Programme (\[FP7/2007-2013\] under grant agreement number 264895). We would also like to thank Mercè Romero-Gómez and Francesca Figueras for providing the subroutine to calculate the $Gaia$ performance errors, including the update to post launch estimates, Sanjib Sharma for providing the extinction maps and isochrones and Sami Niemi for the suggestion of using KDE’s to visualise the velocity distributions.
\[lastpage\]
[^1]: http://www.cosmos.esa.int/web/Gaia/science-performance
[^2]: http://astrowiki.ph.surrey.ac.uk/dokuwiki/doku.php
[^3]: http://www.cosmos.esa.int/web/Gaia/table-5
[^4]: https://github.com/mromerog/Gaia-errors
|
---
author:
- Liran Rotem
bibliography:
- 'C:/Users/Liran/Dropbox/citations/library.bib'
title: 'A letter: The log-Brunn-Minkowski inequality for complex bodies'
---
We will use the following terminology: A real body $K\subseteq\RR^{n}$ is the unit ball of a norm $\left\Vert \cdot\right\Vert $ on $\RR^{n}$, i.e. a convex, origin symmetric, compact set with non-empty interior. Similarly, a complex body $K\subseteq\CC^{n}$ is the unit ball of a norm $\left\Vert \cdot\right\Vert $ on $\CC^{n}$. By identifying $\CC^{n}\simeq\RR^{2n}$ we see that every complex body is also a real body, but not vice versa. In fact, a complex body $K\subseteq\CC^{n}$ is a real body which is also symmetric with respect to complex rotations, i.e. if $z\in K$ implies that $e^{i\theta}z\in K$ for all $\theta\in\RR$.
For a real body $K$, the support function of $K$ is defined as $h_{K}(\theta)=\left\Vert \theta\right\Vert _{K}^{\ast}=\sup_{x\in K}\left\langle x,\theta\right\rangle $. Given two such bodies $K$ and $T$ and a number $0\le\lambda\le1$, we define the logarithmic mean of $K$ and $T$ by $$L_{\lambda}(K,T)=\left\{ x\in\RR^{n}:\ \left\langle x,\theta\right\rangle \le h_{K}(\theta)^{1-\lambda}h_{T}(\theta)^{\lambda}\text{ for all }\theta\in\RR^{n}\right\} ,$$ where $\left\langle \cdot,\cdot\right\rangle $ denotes the standard Euclidean inner product.
The log-Brunn-Minkowski inequality states that $\left|L_{\lambda}(K,T)\right|\ge\left|K\right|^{1-\lambda}\left|T\right|^{\lambda}$, where $\left|\cdot\right|$ denotes the (Lebesgue) volume. It was conjectured by Böröczky, Lutwak, Yang and Zhang ([@Boroczky2012]), who proved it for $K,T\subseteq\RR^{2}$. Saroglou proved ([@Saroglou2014]) that the inequality holds when $K$ and $T$ are $n$-dimensional real bodies which are unconditional with respect to the same basis.
The goal of this note is to explain why the log-Brunn-Minkowski inequality holds for complex bodies:
\[thm:main-thm\]For complex bodies $K,T\subseteq\CC^{n}$ and $0\le\lambda\le1$ we have $\left|L_{\lambda}(K,T)\right|\ge\left|K\right|^{1-\lambda}\left|T\right|^{\lambda}$.
Theorem \[thm:main-thm\] will follow from a result of Cordero-Erausquin ([@Cordero-Erausquin2002]). In his work, Cordero-Erausquin proved a generalization of the Blaschke-Santaló inequality in the complex case. Specifically, he proved that for complex bodies $K,T\subseteq\CC^{n}$ we have $$\left|K\cap T\right|\left|K^{\circ}\cap T\right|\le\left|B_{2}^{2n}\cap T\right|,\tag{\ensuremath{\ast}}\label{eq:gen-santalo}$$ where $K^{\circ}$ is the polar body to $K$ and $B_{2}^{2n}\subseteq\CC^{n}$ is the unit Euclidean ball. As a side note we remark that proving the same inequality for general real bodies is an open problem – see [@Klartag2007c] for a partial result and a short discussion.
Cordero-Erausquin’s proved the inequality (\[eq:gen-santalo\]) as a corollary of a general theorem about complex interpolation - see Theorem \[thm:complex-lc\] below. The main point of this letter is the observation that the same general theorem also implies Theorem \[thm:main-thm\]. This was apparently known to Cordero-Erausquin himself, but not to other researchers in the community who haven’t studied the complex case carefully. Theorem \[thm:main-thm\] may be a strong indication that the log-Brunn-Minkowski conjecture is true in general. Alternatively, it may indicate the existence of a rich theory of geometric inequalities in the complex case.
Let us briefly recall the definition of complex interpolation. We will give the construction for the finite-dimensional case, following the presentation of [@Cordero-Erausquin2002], and refer the reader to [@Bergh1976] for a more detailed account. Set $S=\left\{ z\in\CC:\ 0<\re z<1\right\} $, and define $$\FF=\left\{ f:\overline{S}\to\CC^{n}:\ \begin{array}{l}
f\text{ is bounded and continuous on \ensuremath{\bar{S}} and analytic on }S\\
\text{such that }{\displaystyle \lim_{t\to\pm\infty}f(it)=\lim_{t\to\pm\infty}f(1+it)=0}
\end{array}\right\} .$$ Given two norms $\left\Vert \cdot\right\Vert _{0}$ and $\left\Vert \cdot\right\Vert _{1}$ on $\CC^{n}$, we define a norm on $\FF$ by $$\left\Vert f\right\Vert _{\FF}=\max\left\{ \sup_{t\in\RR}\left\Vert f(it)\right\Vert _{0},\sup_{t\in\RR}\left\Vert f(1+it)\right\Vert _{1}\right\} .$$ Finally, for $\lambda\in[0,1]$, we define the interpolated norm $\left\Vert \cdot\right\Vert _{\lambda}$ by $$\left\Vert x\right\Vert _{\lambda}=\inf\left\{ \left\Vert f\right\Vert _{\mathcal{F}}:\ f\in\FF,\ f(\lambda)=x\right\} .$$
It is not hard to see that for $\lambda=0,1$ we recover the original norms $\left\Vert \cdot\right\Vert _{0},\left\Vert \cdot\right\Vert _{1}$. The only other result we will need from the standard theory of complex interpolation is the following:
\[prop:interp-bound\]Let $\left\Vert \cdot\right\Vert _{0}$,
This inequality, with its simple proof, may be found for example in [@Pisier1989] as equation $\left(7.26\right)^{\ast}$.
If $K$ is the unit ball of $\left\Vert \cdot\right\Vert _{0}$ and $T$ is the unit ball of $\left\Vert \cdot\right\Vert _{1}$ we will write $C_{\lambda}(K,T)$ for the unit ball of $\left\Vert \cdot\right\Vert _{\lambda}$. Proposition \[prop:interp-bound\] implies that $h_{C_{\lambda}(K,T)}(z)\le h_{K}(z)^{1-\lambda}h_{T}(z)^{\lambda}$ for all $z\in\CC^{n}$, and hence $C_{\lambda}(K,T)\subseteq L_{\lambda}(K,T)$.
The main theorem of [@Cordero-Erausquin2002] is the following:
\[thm:complex-lc\]The function $\lambda\longmapsto\left|C_{\lambda}(K,T)\right|$ is log-concave on $[0,1]$.
It is now easy to deduce Theorem \[thm:main-thm\], as $$\left|L_{\lambda}(K,T)\right|\ge\left|C_{\lambda}(K,T)\right|\ge\left|C_{0}(K,T)\right|^{1-\lambda}\cdot\left|C_{1}(K,T)\right|^{\lambda}=\left|K\right|^{1-\lambda}\left|T\right|^{\lambda}.$$
|
---
abstract: |
In this work we suggest a new model for generating random satisfiable $k$-CNF formulas. To generate such formulas – randomly permute all $2^k\binom{n}{k}$ possible clauses over the variables $x_1,\ldots,x_n$, and starting from the empty formula, go over the clauses one by one, including each new clause as you go along if after its addition the formula remains satisfiable. We study the evolution of this process, namely the distribution over formulas obtained after scanning through the first $m$ clauses (in the random permutation’s order).
Random processes with conditioning on a certain property being respected are widely studied in the context of graph properties. This study was pioneered by Ruciński and Wormald in 1992 for graphs with a fixed degree sequence, and also by Erdős, Suen, and Winkler in 1995 for triangle-free and bipartite graphs. Since then many other graph properties were studied such as planarity and $H$-freeness. Thus our model is a natural extension of this approach to the satisfiability setting.
Our main contribution is as follows. For $m \geq cn$, $c=c(k)$ a sufficiently large constant, we are able to characterize the structure of the solution space of a typical formula in this distribution. Specifically, we show that typically all satisfying assignments are essentially clustered in one cluster, and all but $e^{-\Omega(m/n)} n$ of the variables take the same value in all satisfying assignments. We also describe a polynomial time algorithm that finds ${\emph{whp}}$ a satisfying assignment for such formulas.
author:
- 'Michael Krivelevich [^1], Benny Sudakov [^2] and Dan Vilenchik'
title: On the random satisfiable process
---
Introduction
============
Constraint satisfaction problems play an important role in many areas of computer science, e.g. computational complexity theory [@Cook71], coding theory [@Gallager], and artificial intelligence [@Pearl], to mention just a few. The main challenge is to devise efficient algorithms for finding satisfying assignments (when such exist), or conversely to provide a certificate of unsatisfiability. One of the best known examples of a constraint satisfaction problem is $k$-SAT, which is the first to be proven as NP-complete. Although satisfactory approximation algorithms are known for several NP-hard problems, the problem of finding a satisfying assignment (if such exists) is not amongst them. In fact, H[å]{}stad [@Hastad01] proved that it is NP-hard to approximate MAX-3SAT (the problem of finding an assignment that satisfies as many clauses as possible) within a ratio better than 7/8.
In trying to understand the inherent hardness of the problem, many researchers analyzed structural properties of formulas drawn from different distributions. One such distribution is the **uniform** distribution where instances are generated by picking $m$ clauses uniformly at random out of all $2^k\binom{n}{k}$ possible clauses. Although many problems still remain unsolved, in general this distribution seems to be quite well understood (at least for some values of $m$ and $k$). This is also true for the **planted $k$-SAT** model, where one first fixes some assignment $\psi$ to the variables and then picks $m$ clauses uniformly at random out of all $(2^k-1)\binom{n}{k}$ clauses satisfied by $\psi$. Comparatively, much less is known for variants of these distributions where extra conditions are imposed. These conditions distort the randomness in such a way that the “standard" methods and tools employed to analyze the original distributions are a-priori of little use in the new setting. Our work concerns the latter.
Our Contribution {#sec:ourCont}
----------------
In this work we suggest a new model for generating random satisfiable $k$-CNF formulas. To generate such formulas – randomly permute all $2^k\binom{n}{k}$ possible clauses over the variables $x_1,\ldots,x_n$, and starting from the empty formula, go over the clauses one by one, including each new clause as you go along if after its addition to the formula, the formula remains satisfiable. We study the evolution of this process, namely the distribution over formulas obtained after scanning through the first $m$ clauses (in the random permutation’s order); we use ${{\cal{P}}^{{\rm sat}}_{n,m}}$ to denote this distribution. Clearly, for every $m$, all formulas in ${{\cal{P}}^{{\rm sat}}_{n,m}}$ are satisfiable (as every clause is included only if the so-far obtained formula remains satisfiable).
Random processes with conditioning on a certain property being respected are widely studied in the context of graph properties. This study was pioneered by Ruciński and Wormald in 1992 [@RucinskiW92] for graphs with a fixed degree sequence, and also by Erdős, Suen, and Winkler in 1995 for triangle-free and bipartite graphs [@ErdosMaxGraph]. Since then many other graph properties were studied such as planarity [@PlanarGraphProcess], $H$-freeness [@HFreeGraphs] and also the property of being intersecting in the context of hypergraphs [@IntersectingHyper]. Thus our model is a natural extension of this approach to the satisfiability setting. The main difficulty when dealing with these restricted processes is that the edges of the random graph (and the clauses of the random $k$-CNF formula) are no longer independent due to conditioning. Thus the rich methods that have been developed to understand the “classical" random graph models, $G_{n,p}$ for example, do not carry over, at least not immediately, to the restricted setting.
Quite frequently in restricted random processes, the typical size of a final graph or formula (after all edges/clauses have been scanned) is a fascinating subject of study. This is however [*not*]{} the case here, as it is quite easy to see that deterministically the final random formula will have $(2^k-1){n\choose k}$ clauses and a unique satisfying assignment. Therefore, in the setting under consideration here the process itself (i.e. a typical development of a restricted random formula and of its set of satisfying assignments as the number of scanned clauses $m$ grows) is much more interesting than the final result, and indeed in this paper we will study the development of a random satisfiable formula.
As it turns out, if $m$ is chosen so that almost all $k$-CNF formulas with $m$ clauses over $n$ variables are satisfiable, then ${{\cal{P}}^{{\rm sat}}_{n,m}}$ is statistically close to the uniform distribution over such formulas since ${\emph{whp}}$ none of the $m$ clauses will be rejected (writing ${\emph{whp}}$ we mean with probability tending to $1$ as $n$ goes to infinity). Therefore if this is the case, then the clauses are practically independent of each other, and the “usual" techniques apply. Remarkable phenomena occurring in the uniform distribution are [**phase transitions**]{}. With respect to the property of being satisfiable, such a phase transition takes place too. More precisely, there exists a threshold $d=d(n,k)$ such that almost all $k$-CNF formulas over $n$ variables with $m$ clauses such that $m/n>d$ are not satisfiable, and almost all formulas with $m/n<d$ are [@Friedgut]. Thus, while ${{\cal{P}}^{{\rm sat}}_{n,m}}$ is statistically close to the uniform distribution for $m/n$ below the threshold, it is not clear how does a typical ${{\cal{P}}^{{\rm sat}}_{n,m}}$ instance look like when crossing this threshold (which is conjectured to be roughly 4.26 for 3SAT), and whether there exists a polynomial time algorithm for finding a satisfying assignment for such instances. In this work we analyze ${{\cal{P}}^{{\rm sat}}_{n,m}}$ when $m/n$ is some sufficiently large constant *above* the satisfiability threshold. The first part of our result is characterizing the structure of the solution space of a typical formula in ${{\cal{P}}^{{\rm sat}}_{n,m}}$. By the “solution space" of a formula we mean the set of all satisfying assignments (which is a subset of all $2^n$ possible assignments). Formally,
\[thm:StructOfSolutionSpace\] Let ${F}$ be random $k$-CNF from ${{\cal{P}}^{{\rm sat}}_{n,m}}$, $m/n \geq c$, $c=c(k)$ a sufficiently large constant. Then ${\emph{whp}}$ ${F}$ enjoys the following properties:
1. All but ${e^{-\Omega(m/n)}}n$ variables are frozen.
2. The formula induced by the non-frozen variables decomposes into connected components of at most logarithmic size.
3. Letting $\beta({F})$ be the number of satisfying assignments of ${F}$, we have $\frac{1}{n}\log\beta(F)={e^{-\Omega(m/n)}}$.
By a [*frozen variable*]{} we mean a variable that takes the same value in all satisfying assignments. Notice that the third item in Theorem \[thm:StructOfSolutionSpace\] follows directly from the first. One immediate corollary of this theorem is:
Let ${F}$ be random $k$-CNF from ${{\cal{P}}^{{\rm sat}}_{n,m}}$, $m/n \geq c\log n$, $c=c(k)$ a sufficiently large constant. Then ${\emph{whp}}$ ${F}$ has only one satisfying assignment.
The corollary follows from the third item in Theorem \[thm:StructOfSolutionSpace\] since ${e^{-\Omega(m/n)}}=o(n^{-1})$ for $m/n\geq c\log n$, and therefore $\log \beta(F)=o(1)$, or in turn, $\beta(F)=1+o(1)$.
The characterization given by Theorem \[thm:StructOfSolutionSpace\] is in sharp contrast with the structure of the solution space of ${{\cal{P}}^{{\rm sat}}_{n,m}}$ formulas with $m/n$ just below the threshold. Specifically, the *conjectured* picture, some supporting evidence of which was proved rigorously for $k\geq 8$ [@AchiRicciTers06; @ClusteringPhysicists; @AminAchi], is that typically random $k$-CNF formulas in the near-threshold regime have an exponential number of [**clusters**]{} of satisfying assignments. While any two assignments in distinct clusters disagree on at least ${\varepsilon}n$ variables, any two assignments within one cluster coincide on $(1-{\varepsilon})n$ variables. Furthermore, each cluster has a linear number of frozen variables (frozen w.r.t. all satisfying assignments within that cluster). This structure seems to make life hard for most known SAT heuristics. One explantation seems to be that the algorithms do not “steer” into one cluster but rather try to find a “compromise” between the satisfying assignments in distinct clusters, which actually is impossible.
Complementing this picture *rigorously*, we show that a typical formula in ${{\cal{P}}^{{\rm sat}}_{n,m}}$ (in the above-threshold regime) can be solved efficiently. Formally,
\[thm:PlytimeAlg\] There exists a deterministic polynomial time algorithm that ${\emph{whp}}$ finds a satisfying assignment for $k$-CNF formulas from ${{\cal{P}}^{{\rm sat}}_{n,m}}$, $m/n\geq c$, $c=c(k)$ a sufficiently large constant.
Our proof of Theorem \[thm:PlytimeAlg\] is constructive in the sense that we explicitly describe the algorithm.
\[rem:depOnk\] Observe that in both theorems we have $m/n \geq c(k)$, $c$ some function of $k$. We assume that $k$ is fixed, and therefore $c(k)$ is some constant. The true dependency is given by $c(k)=c_02^k$ where $c_0$ is some moderate universal constant, say 100. The exponential dependency on $k$ is somewhat inevitable as the satisfiability threshold itself scales exponentially with $k$ (asymptotically $2^k\ln 2$). In this work we do not go to such fine details as determining the constant $c_0$, though this task is perhaps manageable.
\[rem:k-col\] Another natural problem to study is $k$-colorability. Similar to random $k$-CNF formulas, the random graph $G_{n,p}$ also goes through a phase transition w.r.t. the property of being $k$-colorable, as $np$ grows. Analogously to the random $k$-CNF process that we defined, one can consider a restricted random graph process. Specifically, randomly order all $\binom{n}{2}$ edges of the graph, go over them in that order and include each new edge as long as the resulting graph remains $k$-colorable. Some of the results that we have for $k$-SAT extend to the $k$-colorability process. A more thorough discussion is given in Section \[sec:k-col\].
Related Work and Techniques
---------------------------
Almost all polynomial-time heuristics suggested so far for random instances (either SAT or graph optimization problems) were analyzed when the input is sampled according to a planted-solution distribution, or various semi-random variants thereof. Alon and Kahale [@AlonKahale97] suggest a polynomial time algorithm based on spectral techniques that ${\emph{whp}}$ properly $k$-colors a random graph from the planted $k$-coloring distribution (the distribution of graphs generated by partitioning the $n$ vertices into $k$ equally-sized color classes, and including every edge connecting two different color classes with probability $p=p(n)$), for graphs with average degree greater than some constant. In the SAT context, Flaxman’s algorithm, drawing on ideas from [@AlonKahale97], solves ${\emph{whp}}$ planted 3SAT instances where the clause-variable ratio is greater than some constant. Also [@TechReport; @WP; @ExpectedPoly3SAT] address the planted 3SAT distribution.
On the other hand, very little work was done on non-planted distributions, such as ${{\cal{P}}^{{\rm sat}}_{n,m}}$. In this context one can mention a work of Chen [@Chen03] who provides an *exponential* time algorithm for the uniform distribution over satisfiable $k$-CNF formulas with exactly $m$ clauses where $m/n$ is greater than some constant. Ben-Sasson et al. [@EBSPlanted] also study this distribution but with $m/n=\Omega(\log n)$, a regime where the uniform distribution and the planted distribution essentially coincide (since typically there is only one satisfying assignment), and leave as an open question whether one can characterize the regime $m/n=o(\log n)$. This question was resolved in [@UniformSAT] (and in [@UniformCol] for the uniform distribution over $k$-colorable graphs).
While some of the ideas suggested in these works have proven to be instrumental for our setting, most of their analytical methods break when considering ${{\cal{P}}^{{\rm sat}}_{n,m}}$. In ${{\cal{P}}^{{\rm sat}}_{n,m}}$ not only do clauses depend on each other (unlike the planted distribution where clauses are chosen independently), but the order in which they are introduced also plays a role (which is not the case in the uniform distribution studied in [@UniformSAT], although the clauses are not chosen independently). Therefore we had to come up with new analytical tools that might be of interest in other settings as well.
Paper’s Structure
-----------------
The rest of the paper is structured as follows. In Section \[sec:PropOfRandomInst\] we discuss relevant structural properties that a typical formula in ${{\cal{P}}^{{\rm sat}}_{n,m}}$ possesses, the proofs of some properties are postponed to Sections \[sec:ExpndrProof\] and \[sec:ProofOfPropSizeOfConnectedComp\]. One consequence of this discussion will be a proof of Theorem \[thm:StructOfSolutionSpace\]. We then prove Theorem \[thm:PlytimeAlg\] in Section \[sec:ProofOfThm\] by presenting an algorithm and showing that it meets the requirements of Theorem \[thm:PlytimeAlg\]. In Section \[sec:k-col\] we discuss the $k$-colorability setting (mentioned in Remark \[rem:k-col\]) more elaborately, and concluding remarks are given in Section \[sec:Discussion\].
To simplify the presentation we shall address, in what follows, only the case $k=3$. The case of general $k$ easily follows from the same arguments (taking $m/n \geq c(k)$, $c(k)$ as mentioned in Remark \[rem:depOnk\]).
Properties of a Random ${{\cal{P}}^{{\rm sat}}_{n,m}}$ Instance {#sec:PropOfRandomInst}
===============================================================
This section contains the technical part of the paper. In it we analyze the structure of a typical formula in ${{\cal{P}}^{{\rm sat}}_{n,m}}$. Here and throughout we think of $m$ as $cn$, $c$ at least some sufficiently large constant.
Preliminaries and Techniques
----------------------------
When analyzing some structural properties of a random instance in ${{\cal{P}}^{{\rm sat}}_{n,m}}$ it will be more convenient to analyze the same property under a somewhat different distribution, and then to go back to ${{\cal{P}}^{{\rm sat}}_{n,m}}$ (maybe pay some factor in the estimate).
The variation we consider is ${{\cal{P}}^{{\rm sat}}_{n,p}}$ and is defined as follows: permute at random all possible $M=8\binom{n}{3}$ clauses, go over the clauses in the permutation’s order and include each clause with probability $p=m/M$ if also its addition leaves the instance satisfiable. Let ${{\cal{P}}_{n,p}}$ be defined similarly, just without the conditioning (i.e., all clauses chosen at random are included in the formula, thus making it not necessarily satisfiable).
${{\cal{P}}^{{\rm sat}}_{n,m}}={{\cal{P}}^{{\rm sat}}_{n,p}}|\{\text{ exactly $m$ clauses
were chosen}\}$.
To generate ${F}$ in ${{\cal{P}}^{{\rm sat}}_{n,m}}$ one first picks a random permutation of the clauses and then scans one by one the first $m$ clauses, skipping clauses whose addition will make the instance unsatisfiable. The key point is to notice that any ordered $m$-tuple of clauses is equally likely to be chosen as the first $m$ clauses. This is exactly the case in ${{\cal{P}}^{{\rm sat}}_{n,p}}$ when conditioning on the fact that exactly $m$ clauses were chosen – any set of $m$ clauses is equally likely, and also any permutation of them.
\[lem:Main2RandSATnp\] Set $M=8\binom{n}{3}$. For any property $A$, if $p=m/M$ then $Pr^{{{\cal{P}}^{{\rm sat}}_{n,m}}}[A] \leq
O(\sqrt{m})\cdot Pr^{{{\cal{P}}^{{\rm sat}}_{n,p}}}[A]$.
Let $X$ be a random variable counting the number of clauses whose coin toss was successful. $X$ is distributed $Binom(8 \binom{n}{3},p)$, and therefore $E[X]=m$. Standard calculations show that $Pr[X = m] = \Omega(m^{-0.5})$.
$$Pr^{{{\cal{P}}^{{\rm sat}}_{n,m}}}[A]=Pr^{{{\cal{J}}^{{\rm sat}}_{n,p}}}[A|X=m]=\frac{Pr^{{{\cal{J}}^{{\rm sat}}_{n,p}}}[A \wedge X=m]}{Pr^{{{\cal{J}}^{{\rm sat}}_{n,p}}}[X=m]} \leq O(\sqrt{m})\cdot Pr^{{{\cal{P}}^{{\rm sat}}_{n,p}}}[A].$$
\[rem::RedfinigWhp\] In the remainder of the section we analyze ${{\cal{P}}^{{\rm sat}}_{n,p}}$ instead of ${{\cal{P}}^{{\rm sat}}_{n,m}}$. When we use the expression “with high probability" (abbreviated ${\emph{whp}}$) we will always mean with probability $1-o(m^{-1/2})$. Lemma \[lem:Main2RandSATnp\] will then imply that we can switch back to ${{\cal{P}}^{{\rm sat}}_{n,m}}$ and still the property holds with probability $1-o(1)$. We will actually prove that all the properties hold with probability $1-o(n^{-3})$ which is always at least $1-o(m^{-1/2})$ since $m=O(n^3)$.
The Discrepancy Property {#sec:discrepacny}
------------------------
A well known result in the theory of random graphs is that a random graph ${\emph{whp}}$ will not contain a small yet unexpectedly dense subgraph. This is also the case for ${{\cal{P}}_{n,p}}$ (when considering the graph induced by the formula). In general, discrepancy properties play a fundamental role in the proof of many important structural properties such as expansion, the spectra of the adjacency matrix, etc., and indeed in our case the discrepancy property plays a major role both in the algorithmic perspective and in the analysis of the clustering phenomenon. The following discussion rigorously establishes the above stated fact.
\[def:proportional\] We say that a 3CNF formula ${F}$ on $n$ variables is **$\rho$-proportional** if there exists *no* set $U$ of variables such that:
- $|U|\leq n/10^6$,
- There are at least $\rho\cdot |U|$ clauses in ${F}$ each containing at least two variables from $U$.
(We say that a clause $C$ contains a variable $x$ if $x$ appears in $C$ either as $x$ or as $\bar{x}$, in this context we do not differentiate between the two cases).
\[prop:NoDenseSubgraphs\] Let ${F}$ be distributed according to ${{\cal{P}}_{n,p}}$ with $n^2p\geq d$, $d$ a sufficiently large constant, and set $\rho=n^2p/5500$. Then ${\emph{whp}}$ ${F}$ is $\rho$-proportional.
To see how Proposition \[prop:NoDenseSubgraphs\] corresponds to the random graph context, consider the graph induced by the formula ${F}$ (the vertices are the variables, and two variables are connected by an edge if there exists some clause containing them both) and observe that every clause that contains at least two variables from $U$ contributes an edge to the subgraph induced by $U$. Thus if we have many such clauses, this subgraph will be prohibitively dense. Since ${F}$ is random so is its induced graph, and therefore the latter will typically not occur.
The probability that a random formula ${F}$ in ${{\cal{P}}_{n,p}}$ contains a set $U$ of variables of size $u$ that violates proportionality is at most (using the union bound): $$\sum_{u=1}^{n/10^6}
\binom{n}{u}\cdot\binom{8n\binom{u}{2}}{un^2p/5500}\cdot
p^{un^2p/5500}=o(n^{-3}).$$ The first term accounts for the possible ways of choosing the variables of $U$, the second is to choose the $un^2p/5500$ clauses that contain at least two variables from $U$ (out of at most $8n\binom{u}{2}$ possible ones), and the last term is just the probability of the chosen clauses to actually appear in $F$. To bound this sum we use the fact that $u \leq n/10^6$, the fact that $n^2p$ can be arbitrarily large (constant), and the following standard estimate for the binomial coefficient: $$\binom{n}{x}\leq \left(\frac{en}{x}\right)^x.$$
\[cor:NoDensSub\] Let ${F}^*$ be distributed according to ${{\cal{P}}^{{\rm sat}}_{n,p}}$ with $n^2p\geq d$, $d$ a sufficiently large constant. Then ${\emph{whp}}$ ${F}^*$ is $n^2p/5500$-proportional.
The corollary follows easily by observing that the proportionality property is monotonically decreasing.
Crude Characterization of the Solution Space’s Structure
--------------------------------------------------------
In this section we make the first step towards proving Theorem \[thm:StructOfSolutionSpace\] (clustering). We give a rather crude characterization of the structure of the solution space of a typical instance in ${{\cal{P}}^{{\rm sat}}_{n,p}}$. This characterization will be refined in the sequel.
\[def:concentrated\] A 3CNF ${F}$ is called $r$-**concentrated** if every two satisfying assignments $\psi_1,\psi_2$ of ${F}$ are at Hamming distance at most $r$ from each other.
\[prop:Concentration\] Let ${F}^*$ be distributed according to ${{\cal{P}}^{{\rm sat}}_{n,p}}$ with $n^2p \geq d$, $d$ a sufficiently large constant, let $\rho=30/(n^2p)$ then ${\emph{whp}}$ ${F}^*$ is $\rho n$-concentrated.
An immediate corollary of this proposition is that typically all satisfying assignments of a ${{\cal{P}}^{{\rm sat}}_{n,p}}$ instance can be enclosed in a ball of radius $30/(np)$ in $\{0,1\}^n$. This gives a “first-order" characterization of the structure of the solution space.
Fix two assignments $\varphi$ and $\psi$ at distance $\alpha n$, and let us bound $Pr[\varphi \text{ and } \psi \text{
satisfy } F^*]$. Assume w.l.o.g. that, say, $\varphi$ is the all-TRUE assignment. We shall now upper bound the probability of a set of clauses in ${{\cal{P}}_{n,p}}$ that may result in an instance $F^*$ that is satisfied by both assignments. In particular a clause of the form $C_1=(x \vee \bar{y} \vee \bar{z})$, where $x$ is a variable on which $\varphi$ and $\psi$ disagree, and $y,z$ are variables on which both agree, cannot be chosen to ${{\cal{P}}_{n,p}}$. Let us call such a clause a type 1 clause. If a type 1 clause appears is included, then either it is included in $F^*$, and then $\psi$ cannot be a satisfying assignment, or it is rejected and then $\varphi$ is already at this point not a satisfying assignment. The same applies for clauses of the form $C_2=(s \vee w \vee t)$, where on all three variables, $s,w,t$, both assignments disagree – call them type 2. It remains to upper bound the probability of a ${{\cal{P}}_{n,p}}$ instance that does not contain type 1 and type 2 clauses. There are $\alpha n\binom{(1-\alpha)n}{2}$ type 1 clauses and $\binom{\alpha n}{3}$ type 2 clauses. The probability of none being chosen is $$\label{eq:ConcentrationEq}
(1-p)^{\alpha n\binom{(1-\alpha)n}{2}+ \binom{\alpha n}{3}} \leq \exp\{-p \cdot \left(\alpha n\binom{(1-\alpha)n}{2}+ \binom{\alpha n}{3}\right)\}.$$ If $30/n^2p \leq \alpha \leq 1/2$ then $$p \cdot \alpha n\binom{(1-\alpha)n}{2} \geq p \alpha n \cdot
n^2/8 \geq 3n.$$ If $\alpha \geq 1/2$ then
$$p \cdot \binom{\alpha n}{3} \geq n \cdot n^2p/48 \geq 3n.$$ In the last inequality we use the fact that $n^2p$ can be arbitrarily large (specifically, greater than 144). In any case, the expression in (\[eq:ConcentrationEq\]) is at most $5^{-n}$. Since we have no more than $4^n$ ways of choosing the pair $\varphi,\psi$, we deduce using the union bound that ${\emph{whp}}$ no such “bad" pair exists.
The Core Variables {#sec:CoreVertices}
------------------
We describe a subset of the variables, referred to as the *core variables*, which plays a crucial role in the understanding of ${{\cal{P}}^{{\rm sat}}_{n,p}}$. A variable is said to be frozen in ${F}$ if in every satisfying assignment it takes the same value. The notion of a core captures this phenomenon. In addition, a core typically contains all but a small (though constant) fraction of the variables. This implies that a large fraction of the variables is frozen, a fact which must leave imprints on various structural properties of the formula. These imprints allow efficient heuristics to recover a satisfying assignment of the core. A second implication of this is an upper bound on the number of possible satisfying assignments, and on the distance between every such two. Thus the notion of a core plays a key role in obtaining a characterization of the cluster structure of the solution space.
Let us now proceed with a rigorous definition of a core. Before doing so, we take a long detour on expanding sets.
(support) Given a 3CNF formula ${F}$ and some assignment $\psi$ to the variables, we say that a variable $x$ **supports** a clause $C$ (in which it appears) w.r.t. $\psi$ if $x$ is the only variable whose literal evaluates to true in $C$ under $\psi$.
\[def:expanding\](expanding set) Given a 3CNF formula ${F}$ and an assignment $\psi$ to the variables (not necessarily satisfying), a set of variables $Z$ is called **$t$-expanding** in ${F}$ w.r.t. $\psi$ if every variable $x\in Z$ supports at least $t$ clauses in ${F}[Z]$ w.r.t. $\psi$.
${F}[Z]$ stands for the subformula of ${F}$ containing the clauses where all three variables belong to $Z$. The following proposition illustrates the usefulness of Definition \[def:expanding\].
\[prop:UniqueOfSat\] Let ${F}$ be a 3CNF formula on $n$ variables and let $Z$ be a $t$-expanding set w.r.t. some assignment $\psi$. If in addition:
- $\psi$ satisfies ${F}$,
- ${F}$ is $n/10^6$-concentrated (Definition \[def:concentrated\]),
- ${F}$ is $t$-proportional (Definition \[def:proportional\]),
then the variables in $Z$ are frozen in ${F}$.
By contradiction, let $\psi$ be the satisfying assignment w.r.t. which $Z$ is defined and let $\psi'$ be some satisfying assignment of ${F}$ such that there exists a non-empty set $U \subseteq Z$ of variables for which $\forall x\in U,\psi(x)\neq\psi'(x)$ (if for every $\psi'$ it holds that $U=\emptyset$ then we are done). Take $x\in U$ and consider all the clauses that $x$ supports w.r.t. $\psi$ in ${F}[Z]$. It must be that every such clause contains at least another variable $y$ on which $\psi$ and $\psi'$ disagree (since every such clause is satisfied by $\psi'$ but the literal corresponding to $x$ is false under $\psi'$). Therefore $y$ belongs to $U$ by definition. We conclude that there exists a set $U$ of variables and $t\cdot |U|$ clauses each containing at least two variables from $U$ (no clause was counted twice since the supporter of a clause is unique by definition). Further, we assumed that ${F}$ is $n/10^6$-concentrated and therefore $|U|\leq n/10^6$. Combining the latter two facts we derive a contradiction to the $t$-proportionality of ${F}$.
\[prop:ExpanderSize\] Let ${F}$ be distributed according to ${{\cal{P}}_{n,p}}$ with $n^2p\geq d$, $d$ a sufficiently large constant. Then ${\emph{whp}}$ there exists an integer $t=t(n,p)>0$, a set $Z$ of variables, and an assignment $\psi$ such that:
- $Z$ is $t$-expanding w.r.t. $\psi$,
- $|Z|=(1-{e^{-\Omega(n^2p)}})n$.
- ${F}$ is $t/10$-proportional,
- $\psi$ satisfies ${F}^*$,
- ${F}^*$ is $n/10^6$-concentrated,
The complete proof of this proposition is deferred to Section \[sec:ExpndrProof\].
\[cor:FrozenInSatFormula\] The set $Z$ promised in Proposition \[prop:ExpanderSize\] is frozen in ${F}^*$, and furthermore $Z$ is $t$-expanding w.r.t. every satisfying assignment of ${F}^*$.
To see why $Z$ is frozen, let $S$ be the set of clauses in ${F}$ that are supported w.r.t. $\psi$. First observe that ${F}^*$ is $t$-proportional as it is a subformula of ${F}$ (and ${F}$ is $t/10$-proportional and therefore also $t$-proportional). Furthermore $S$ is contained in ${F}^*$. This is because $\psi$ is a satisfying assignment of ${F}^*$ throughout the entire generating process, thus every clause in $S$ that arrives is not rejected. Therefore $Z$ is also $t$-expanding in $F^*$ w.r.t. $\psi$. Finally apply Proposition \[prop:UniqueOfSat\] to ${F}^*$. The second part of the corollary is immediate from the fact that $Z$ is frozen.
\[def:SelfContained\](self-contained sets) Given a 3CNF formula ${F}$ we say that a set of variables $Z$ is **$r$-self-contained** in ${F}$ if every variable $x\in Z$ appears in at most $r$ clauses in ${F}\setminus{F}[Z]$.
Finally, we are ready to define a core.
\[def:core\](core) A set of variables ${{\cal{H}}}$ is called a $t$-**core** of ${F}$ w.r.t. an assignment $\psi$ if ${{\cal{H}}}$ is $t$-expanding in $F$ w.r.t $\psi$ and also $(t/3)$-self-contained in ${F}$.
The property of being self-contained is necessary for the algorithmic part (the proof of Theorem \[thm:PlytimeAlg\], at least as our analysis proceeds).
\[prop:CoreSize\] Let ${F}^*$ be distributed according to ${{\cal{P}}^{{\rm sat}}_{n,p}}$ with $n^2p\geq d$, $d$ a sufficiently large constant. Then ${\emph{whp}}$ there exists an integer $t=t(n,p)>0$, a satisfying assignment $\psi$ of $F^*$, and a $t$-core ${{\cal{H}}}$ w.r.t. $\psi$ such that:
- $|{{\cal{H}}}|=(1-{e^{-\Omega(n^2p)}})n$,
- ${{\cal{H}}}$ is frozen in ${F}^*$,
- ${F}^*$ is $t/10$-proportional.
The proof of this proposition is best understood in the context of the proof of Proposition \[prop:ExpanderSize\]. Therefore the proof appears in Section \[sec:CoreSizeProof\].
\[rem:MaximalityOfCore\] Observe that if there exist two $t$-cores ${{\cal{H}}}_1$ and ${{\cal{H}}}_2$ that satisfy the conditions of Proposition \[prop:CoreSize\], then also their union ${{\cal{H}}}_1 \cup {{\cal{H}}}_2$ is a $t$-core (since the core variables are frozen). Therefore we may speak of a unique maximal $t$-core. From now on, when we refer to a $t$-core, we mean the maximal one. Note that this maximal core is also frozen by Proposition \[prop:UniqueOfSat\]. Therefore it can serve as a $t$-core for [*any*]{} satisfying assignment of $F$ and thus is effectively uniquely defined by the formula.
Satellite Variables {#sec:Sattelite}
-------------------
In this section we isolate another set of variables which we call satellite variables. As it turns out, to prove Theorems \[thm:StructOfSolutionSpace\] and \[thm:PlytimeAlg\], it is enough to distinguish between core and satellite variables and all other variables in $V$. Let us start with a formal definition of a satellite variable.
\[def:Satellite\] Given a formula $F$ with a core set ${{\cal{H}}}$ w.r.t. to an assignment $\varphi$, a variable $x$ is called a *$0$-satellite* with respect to ${{\cal{H}}}$ if $x \in {{\cal{H}}}$. A variable $x$ is called an *$i$-satellite* if $F$ contains a clause of the form $(x \vee \ell_{z_1} \vee \ell_{z_2})$ or $(\bar{x} \vee \ell_{z_3} \vee \ell_{z_4})$ where for every $j=1,2,3,4$, $z_j$ is a $b$-satellite for $b<i$, and $\varphi(\ell_{z_j})=FALSE$, moreover at least one of $z_j$ is an $(i-1)$-satellite. We say that $x$ is a satellite variable if it is $b$-satellite for some number $b \geq 1$.
In this definition, $\ell_z$ stands for a literal corresponding to a variable $z$ (i.e. $\ell=z$ or $\ell=\bar{z}$). Observe that if ${{\cal{H}}}$ is frozen in $F$ then ${{\cal{H}}}\cup {{\cal{S}}}$ is frozen as well (this follows from a simple inductive argument).
Before we formally state the property involving the satellite variables we introduce some additional notation. The connected components of a formula ${F}$ are the sub-formulas ${F}[C_1],\ldots,{F}[C_k]$, where $C_1,C_2,\ldots,C_k$ are the connected components in the graph $G_{F}$ induced by ${F}$ (the vertices of $G_{F}$ are the variables, and two variables are connected by an edge if there exists some clause containing them both). Given a set of variables $A$ and an assignment $\varphi$ we denote by ${F}_{out}(A,\varphi)$ the subformula of ${F}$ which is the outcome of the following procedure: set the variables in $A$ according to $\varphi$ and simplify ${F}$ (by simplify we mean remove every clause that contains a TRUE literal, and remove FALSE literals from the other clauses).
\[prop:SizeOfConnectedComp\] Let ${F}^*$ be distributed according to ${{\cal{P}}^{{\rm sat}}_{n,p}}$ with $n^2p\geq d$, $d$ a sufficiently large constant. There ${\emph{whp}}$ exists an integer $t=t(n,p)>0$, a satisfying assignment $\psi$ of $F^*$, and a $t$-core ${{\cal{H}}}$ w.r.t. $\psi$ such that:
- $|{{\cal{H}}}| \geq (1-{e^{-\Omega(n^2p)}})n$.
- ${F}^*$ is $t/10$-proportional.
- Let ${{\cal{S}}}$ be its satellite variables, ${{\cal{H}}}\cup {{\cal{S}}}$ are frozen in ${F}^*$,
- The largest connected component in ${F}^*_{out}({{\cal{H}}}\cup {{\cal{S}}},\psi)$ is of size at most $\log n$.
The new addition compared with Proposition \[prop:CoreSize\] is the fact that we characterize the structure of the formula induced by the variables not in ${{\cal{H}}}\cup {{\cal{S}}}$.
Our proof strategy is the following. Expose the first part of the random formula $F$ and consider a $t$-core ${{\cal{H}}}$ promised ${\emph{whp}}$ by Proposition \[prop:CoreSize\]. We look at a “large" connected component outside the core (if none exists then we are done) and consider the following “shattering" procedure. Expose the second part of the random formula, and suppose for the time being that the core does not change (even if new clauses are included in $F^*$). Let $x$ be a non-core variable after the first part, which lies in a spanning tree of a large connected component. The key observation is that when resuming the random clause process, $x$ becomes a satellite variable with high (constant) probability, in which case the spanning tree splits into parts. Since the tree is large, it contains many variables $x$, and therefore with very high probability at least one of them will become a satellite variable and shatter the tree. Finally, it remains to upper bound the number of possible large trees vs. the probability that such a tree does not survive. The complete proof is given is Section \[sec:ProofOfPropSizeOfConnectedComp\].
One problem with the approach we just described is that we assumed that the core ${{\cal{H}}}$ established after the first round does not change when resuming the random clause process. This is not necessarily the case as for example some core variables may violate the self-containment property and be removed, and this may cause a chain reaction of other variables leaving the core (maybe their support is too small, or they violate the self-containment requirement). However, ${\emph{whp}}$ all the variables the are removed from the core when resuming the random clause process remain satellite variables, and furthermore there are very few such variables.
In several papers which studied planted-solution distributions, for example [@AlonKahale97; @flaxman], a similar notion of a core appears (without the notion of satellite variables), and an analysis of the structure of the instance ($k$-colorable graph or $k$-CNF formula) induced on the non-core variables is also given. The main difference from our setting is the fact that the planted distribution is a product space, and therefore it was possible to prove that the core variables are distributed similarly to a uniformly random set of variables. In our case establishing such a property is a more challenging task. As it turns out, the approach that we take – defining the satellite variables – simplifies considerably the proof of this property.
The Majority Vote {#sec:Majority}
-----------------
Given a 3CNF formula ${F}$ and a variable $x$ we let $N^+(x)$ be the set of clauses in ${F}$ in which $x$ appears positively (namely, as the literal $x$), and $N^{-}(x)$ be the set of clauses in which $x$ appears negatively (that is, as $\bar{x}$). The Majority Vote assignment over ${F}$, which we denote by MAJ, assigns every $x$ according to the sign of $|N^+(x)|-|N^-(x)|$ (TRUE if the difference is positive and FALSE otherwise).
\[prop:MajVoteSuccRate\] Let ${F}^*$ be distributed according to ${{\cal{P}}^{{\rm sat}}_{n,p}}$ with $n^2p\geq d$, $d$ a sufficiently large constant. Then ${\emph{whp}}$ every satisfying assignments of ${F}^*$ differs from MAJ on at most ${e^{-\Omega(n^2p)}}n$ variables.
Consider the following two-step procedure to generate $F$: in the first step go over the $M=8\binom{n}{3}$ clauses and toss a coin with success probability $p_1$. We take the clauses that were chosen and put them first, ordered at random. Call $F_1$ this first part (and respectively define $F_1^*$ in our standard way, i.e., by scanning sequentially the clauses of $F_1$ and including those whose addition leaves the formula satisfiable.). Observe that $F_1$ is distributed according to ${\cal{P}}_{n,p_1}$. Then in the second round, every clause that was not chosen in the first round is included with probability $p_2$, and the chosen clauses are ordered at random and then concatenated after $F_1$. Call $F_2$ this last part. At the end of this subsection we prove that $F=F_1 \cup F_2$ is distributed according to ${{\cal{P}}_{n,p}}$ when $p=p_1 +(1-p_1)p_2$. Therefore we may think of $F$ as generated in two steps (with the suitable choice of $p_1,p_2$). We will use this technique to prove several other properties as well.
Let $d_0$ be the constant promised in Proposition \[prop:CoreSize\], and choose $d \geq 200d_0$. Set $p_1 = p/200$. By the choice of $d_0$ and Proposition \[prop:CoreSize\] ${\emph{whp}}$ all but ${e^{-\Omega(n^2p)}}n$ variables are frozen in $F_1^*$, and w.l.o.g assume that they all take the value TRUE. Further observe that ${\emph{whp}}$ at this point all but ${e^{-\Omega(n^2p)}}n$ variables appear in no more than say $n^2p/30$ clauses (in $F_1$ distributed according to ${{\cal{P}}_{n,p_1}}$, and therefore also in $F_1^*$). This is because every variable $x$ is expected to appear in $F_1$ in $p_1\cdot 8\binom{n}{2}\leq 4n^2p_1 = n^2p/50$ clauses. These appearances are independent (binomially distributed), therefore one can apply the Chernoff bound for example to bound the probability that $x$ appears in more than $n^2p/30$ clauses, which will be ${e^{-\Omega(n^2p)}}$. This in turn gives that the expected number of such variables is ${e^{-\Omega(n^2p)}}n$. To obtain concentration around this value, consider an ordering on the $M$ clauses and let $X_i$ be an indicator random variable which is 1 iff clause $i$ appeared in the first round. Let $f(X_1,X_2,\ldots,X_M)$ be a function which counts the number of variables that appear in more than $n^2p/30$ clauses in $F$. As claimed, $E[f]={e^{-\Omega(n^2p)}}n$, and $f$ satisfies the Lipschitz condition with difference 3: for every $i$ and every two assignments $a=(a_1,\ldots,a_{i-1},a_i,a_{i+1},\ldots,a_M)$ and $a'=(a_1,\ldots,a_{i-1},a'_i,a_{i+1},\ldots,a_M)$ of values to $X_1,\ldots,X_M$ (that possibly differ on the $i^{th}$ coordinate), it holds that $|f(a)-f(a')| \leq 3$ (every clause contains three variables). Using the method of bounded differences (e.g., Theorem 7.4.3 of [@TheProbMethod]) it follows that $f$ is concentrated around its expected value.
Let $Z$ be then the set of frozen variables that appear in at most $n^2p/30$ clauses of ${F}_1$. Recall that we have assumed w.l.o.g. that they all froze to TRUE. By the above discussion together with Proposition \[prop:CoreSize\] ${\emph{whp}}$ $$|Z|\geq (1-{e^{-\Omega(n^2p)}})n-{e^{-\Omega(n^2p)}}n
\geq 0.999n.$$ Now let us consider the second iteration of coin flips. Fix $x \in Z$, observe that every clause containing $x$ positively, if chosen in the second round will be included in $F^*$. There are at least $4\binom{|Z|-1}{2}- n^2p/30$ such clauses with the other two variables from $Z$ – call them “good" clauses. As for clauses where $x$ appears negatively, and the other two variables are in $Z$, there are only at most $3\binom{|Z|-1}{2}$ clauses such that if chosen will be included (since one way of negating the variables in $Z$ results in a FALSE clause on frozen variables) – call them “bad" clauses. In addition there are at most $8(n-|Z|)n$ clauses, containing $x$ and at least one variable outside $Z$, that we don’t say anything about, but let us adversarially assume that $x$ appears in all of them negatively, and if chosen are included in $F^*$ (they are also part of the bad clauses).
In expectation, $p_2\cdot \left(4\binom{|Z|-1}{2}-n^2p/30\right)
\geq 1.8n^2p$ good clauses containing $x$ will be chosen in the second round, and $p_2\cdot \left(3\binom{|Z|-1}{2}+8(n-|Z|)n\right) \leq 1.6n^2p$ bad clauses. (Recall that $199p/200 \le p_2\le p$.)
Suppose that in the $n^2p/30$ clauses from the first round also $x$ appears negatively. To conclude, for the majority vote of $x$ to be wrong it must have been the case that the number of good clauses containing $x$ or the number of bad clauses containing $x$ deviates by at least $(1.8-1.6-1/30)n^2p/2$ from its expectation. But since both are binomially distributed with expectation $\Theta(n^2p)$, this happens with probability ${e^{-\Omega(n^2p)}}$. Using the linearity of expectation all but ${e^{-\Omega(n^2p)}}n$ of the variables in $Z$ are expected to have a “proper" gap. To obtain concentration around this value we use again the method of bounded differences, similarly to hat has been used earlier in the proof. Finally observe that $|Z|\geq (1-{e^{-\Omega(n^2p)}})n$, and therefore $|Z|-{e^{-\Omega(n^2p)}}n = (1-{e^{-\Omega(n^2p)}})n$ as required.
### Justifying the two-step distribution.
Let ${{\cal{P}}_{n,p_1,p_2}}$ be the distribution of the two-step process. For brevity, set ${\cal P}_1={{\cal{P}}_{n,p}}$, ${\cal P}_2={{\cal{P}}_{n,p_1,p_2}}$. Let us now prove that ${\cal P}_1$ and ${\cal P}_2$ are identical for $p=p_1+(1-p_1)p_2$. Let $\sigma$ be an ordered list of $|\sigma|=r$ clauses. Then $$Pr_{{\cal P}_1} [\mbox{ get list $\sigma$}]=
\frac{p^{r}(1-p)^{M-r}}
{r!}.$$ On the other hand, $$\begin{aligned}
Pr_{{\cal P}_2} [\mbox{ get list $\sigma$}] &=&
\sum_{i=0}^r \frac{p_1^i(1-p_1)^{M-i}}{i!}\,\cdot\,
\frac{p_2^{r-i}(1-p_2)^{M-r}}{(r-i)!}\\
&=& (1-p_1)^M p_2^r (1-p_2)^{M-r}\sum_{i=0}^r
\frac{\left(\frac{p_1}{1-p_1}\right)^i}{i!}\,\cdot\,
\frac{\left(\frac{1}{p_2}\right)^i}{(r-i)!}\\
&=& (1-p_1)^M p_2^r (1-p_2)^{M-r} \frac{1}{r!}\sum_{i=0}^r
{r\choose i} \left(\frac{p_1}{(1-p_1)p_2}\right)^i\\
&=& (1-p_1)^M p_2^r (1-p_2)^{M-r} \frac{1}{r!}
\left(1+\frac{p_1}{(1-p_1)p_2}\right)^r\\
&=& \frac{((1-p_1)(1-p_2))^{M-r}(p_1+p_2-p_1p_2)^r}{r!}\,.\end{aligned}$$
Choosing $p_1,p_2$ to satisfy $p_1+p_2-p_1p_2=p$, we conclude that the distributions ${\cal P}_1$ and ${\cal P}_2$ are indeed identical.
Proof of Theorem \[thm:StructOfSolutionSpace\]
----------------------------------------------
Theorem \[thm:StructOfSolutionSpace\] follows from Proposition \[prop:SizeOfConnectedComp\] which implies that all but ${e^{-\Omega(n^2p)}}n$ of the variables are frozen. Therefore, there are at most $\exp\{{{e^{-\Omega(n^2p)}}n}\}$ possible ways to set the assignment of the remaining variables. Furthermore, Proposition \[prop:SizeOfConnectedComp\] describes the formula induced by the non-frozen variables.
Proof of Theorem \[thm:PlytimeAlg\] {#sec:ProofOfThm}
===================================
In this section we prove that the algorithm , which is described in Figure \[fig:SATAlg\], meets the requirements of Theorem \[thm:PlytimeAlg\]. The main principles underlying were designed with the planted distribution in mind (see [@flaxman] for example). An additional ingredient that we add is a unit-clause-propagation step. Given a 1-2-3-CNF formula (namely a formula which contains clauses of size 1,2 and 3), the unit-clause-propagation is the following simple heuristic:
> *while there exists a clause of size 1, set the variable appearing in this clause in a satisfying manner, remove this clause and all other clauses satisfied by this assignment, and remove the FALSE literals of the variable from other clauses.*
We say that ${F}^*$ is *typical* in ${{\cal{P}}^{{\rm sat}}_{n,p}}$ if Propositions \[prop:SizeOfConnectedComp\] and \[prop:MajVoteSuccRate\] hold. The discussion in Section \[sec:PropOfRandomInst\] guarantees that indeed ${\emph{whp}}$ ${F}^*$ is typical. Therefore, to prove Theorem \[thm:PlytimeAlg\] it suffices to consider a typical ${F}^*$ and prove that (always) finds a satisfying assignment for ${F}^*$. As the parameter $t$ for we use the $t$ promised in Proposition \[prop:SizeOfConnectedComp\].
We let ${{\cal{H}}}$ be the $t$-core promised in Proposition \[prop:SizeOfConnectedComp\], ${{\cal{S}}}$ its satellite variables, and $\varphi$ be the satisfying assignment w.r.t. which ${{\cal{H}}}$ is defined. In all the following propositions we assume ${F}^*$ is typical (we don’t explicitly state it every time for the sake of brevity).
\[prop:ReassigmentCorrect\] Let $\psi_1$ be the assignment defined in line 7 of . Then $\psi_1$ agrees with $\varphi$ on the assignment of all variables in ${{\cal{H}}}$.
Let $B_i$ be the set of core variables whose assignment in $\pi_i$ disagrees with $\varphi$ at the beginning of the $i^{th}$ iteration of the main for-loop – line 2 in . It suffices to prove that $|B_{i+1}|\leq |B_i|/2$ (if this is true, then after $\log n$ iterations $B_{\log n}=\emptyset$). Observe that by Proposition \[prop:MajVoteSuccRate\], $|B_0|\leq n/10^7$ (as the Majority Vote error-rate ${e^{-\Omega(n^2p)}}$ can be made arbitrarily small). By contradiction, assume that not in every iteration $|B_{i+1}|\leq |B_i|/2$, and let $j$ be the first iteration violating this inequality. Consider a variable $x\in B_{j+1}$. If also $x \in B_j$, this means that $x$’s assignment was not flipped in the $j^{th}$ iteration, and therefore, $x$ supports at least $2t/3$ clauses w.r.t. $\pi_j$. Since ${{\cal{H}}}$ is $t/3$-self-contained, at least $2t/3-t/3 = t/3$ of these clauses contain only core variables. Since the literal of $x$ is true in all these clauses, but in fact should be false under $\varphi$, each such clause must contain another variable on which $\varphi$ and $\pi_j$ disagree, that is another variable from $B_j$. If $x\notin B_j$, this means that $x$’s assignment was flipped in the $j^{th}$ iteration. This is because $x$ supports less than $2t/3$ clauses w.r.t. $\pi_j$. Since $x$ supports at least $t$ clauses w.r.t. $\varphi$ ($t$-expanding property of the core), it must be that in at least $t-2t/3=t/3$ of them, the literal of some other core variable evaluates to TRUE (not FALSE as it should be in $\varphi$). Letting $U=B_j \cup B_{j+1}$, there are at least $t/3\cdot|B_{j+1}|$ clauses containing at least two variables from $U$ (every clause is counted exactly once as the supporter of a clause is unique). Using our assumption, $|B_{j+1}|\geq |B_j|/2$, we obtain $|U|=|B_j \cup B_{j+1}|\leq |B_j|+|B_{j+1}|\leq
3|B_{j+1}|$, therefore $t/3\cdot|B_{j+1}| \geq (|U|/3)\cdot t/3 = (t/9)|U|$. Finally,
- $|B_j|\leq n/10^7$ (because $B_0$ is already small enough, and by our assumption the sets $B_1,B_2,\ldots B_j$ only decrease in size),
- $|B_{j+1}|$ may exceed $n/10^7$, in which case we consider w.l.o.g. only the first $n/10^7$ variables (this is in line with our assumption $|B_{j+1}|\geq |B_j|/2$),
- $|U|\leq 3|B_{j+1}|\leq 3n/10^7\leq n/10^6$,
- there are $t|U|/9$ clauses containing two variables from $U$.
The last two items contradict the $t/10$-proportionality of ${F}^*$.
\[prop:UnassigmentCorrect\] Let $\xi$ be the partial assignment defined in line 12 of . Then all assigned variables in $\xi$ are assigned according to $\varphi$, and all the variables in ${{\cal{H}}}$ are assigned.
By Proposition \[prop:ReassigmentCorrect\], $\psi_1$ coincides with $\varphi$ (the satisfying assignment w.r.t. which ${{\cal{H}}}$ is defined) on ${{\cal{H}}}$. Furthermore, by the definition of $t$-core, every core variable supports at least $t$ clauses w.r.t. $\varphi$, and also w.r.t. $\psi_1$ (the assignment at hand before the unassignment step begins). Hence all core variables survive the first round of unassignment. By induction it follows that the core variables survive all rounds. Now suppose by contradiction that not all assigned variables are assigned according to $\varphi$ when the unassignment step ends. Let $U$ be the set of variables that remain assigned when the unassignment step ends, and whose assignment disagrees with $\varphi$. Every $x\in U$ supports at least $t$ clauses w.r.t. to $\xi$ (the partial assignment defined in line 12 of ), but each such clause must contain another variable on which $\xi$ and $\varphi$ disagree (since $\varphi$ satisfies this clause). Thus, we have $t\cdot |U|$ clauses each containing at least two variables from $U$ (again no clause is counted twice as the support of a clause is unique). Since $U\cap {{\cal{H}}}=\emptyset$ (by the first part of this argument) and $|{{\cal{H}}}|\geq
(1-{e^{-\Omega(n^2p)}})n$ it follows that $|U|\leq {e^{-\Omega(n^2p)}}n <
n/10^6$, contradicting the $t/10$-proportionality of ${F}^*$.
\[prop:UnitClauseCorrect\] By the end of the unit-clause propagation step all the variables which get assigned are assigned according to $\varphi$, furthermore the set of satellite variables ${{\cal{S}}}$ is assigned.
The proof is by induction on the iterations of the unit clause propagation. The base case are clauses of the form $(x \vee \ell_z \vee \ell_y)$ where $\ell_z,\ell_y$ are FALSE literals under $\xi$ and $x$ is unassigned. By the previous proposition, $\xi$ can be extended to a satisfying assignment of $F$, but every such extension must set $x=TRUE$. This is exactly what the unit clause propagation does. The step of the induction is proven similarly to the base case.
Now to the satellite variables. The previous proposition gives that ${{\cal{H}}}$ remains assigned according to $\varphi$. By the definition of satellite variables, ${{\cal{S}}}$ will be set in the unit clause propagation (the $i$-satellite variables will be set in iteration $i$ of the unit-clause propagation).
\[prop:ExhastiveSearchCorrect\] The exhaustive search, Step 5 of , completes in polynomial time with a satisfying assignment of ${F}^*$.
By Proposition \[prop:UnitClauseCorrect\], the partial assignment at the beginning of the exhaustive search step is partial to the satisfying assignment $\varphi$ of the entire formula. Therefore the exhaustive search will succeed. Further observe that the unassigned variables are outside of ${{\cal{H}}}\cup {{\cal{S}}}$. Proposition \[prop:SizeOfConnectedComp\] then guarantees that the running time of the exhaustive search will be at most polynomial.
\
Theorem \[thm:PlytimeAlg\] follows.
Proof of Proposition \[prop:ExpanderSize\] {#sec:ExpndrProof}
==========================================
Let $F$ be the random ${{\cal{P}}_{n,p}}$ instance, $F^*$ be its satisfiable part. We divide the process of generating $F$ into two steps like in the proof of Proposition \[prop:MajVoteSuccRate\]: in the first round go over the $M=8\binom{n}{3}$ clauses and toss a coin with success probability $p_1=p/2$. Take the clauses that were chosen and put them first ordered at random. In the second round, every clause that was not chosen, is included with probability $p_2$, $p_2$ satisfies $p_1+(1-p_1)p_2=p$; then the included clauses are ordered at random and concatenated after the first part. Observe that this distribution is identical to ${{\cal{P}}^{{\rm sat}}_{n,p}}$ as explained before.
Let $t$ be such that $F$ (and hence also $F_1$) is ${\emph{whp}}$ $t$-proportional (we can choose $t=n^2p/5500$ as asserted in Proposition \[prop:NoDenseSubgraphs\]). Also take $n^2p$ sufficiently large so that ${F}^*$ is ${\emph{whp}}$ $n/10^6$-concentrated (as required by Proposition \[prop:ExpanderSize\], and as promised to be the case ${\emph{whp}}$ by Proposition \[prop:Concentration\]).
Fix $\psi$ to be some assignment (not necessarily a satisfying assignment of $F^*$), and let $B_\psi$ be a random variable counting the number of variables whose support in $F_1$ w.r.t. $\psi$ is smaller than $502t$. A bound of the sort $Pr[B_\psi >
n/10^7]=o(2^{-n})$ would be very useful as we can then take the union bound over all possible assignments $\psi$. Fix some variable $x$, and w.l.o.g. assume $x$ is TRUE in $\psi$. There are $\binom{n-1}{2}$ clauses that $x$ supports w.r.t. $\psi$, each included w.p. $p_1$. Therefore in expectation $x$ supports at least $n^2p_1/3= n^2p/6$ clauses. Since the support of $x$ is distributed binomially, the probability that $x$ supports less than $t$ clauses in $F_1$ w.r.t. $\psi$ is at most $e^{-n^2p/50}$ (say, use the Chernoff bound). Finally observe that the set of clauses that $x$ supports is disjoint from the set of clauses that $y \neq x$ supports. Therefore, the probability that there are at least $n/10^7$ such variables is at most ${n\choose {n/10^7}}e^{-(n^2p/50) \cdot (n/10^7)} < 3^{-n}$ for sufficiently large $n^2p$.
In particular, ${\emph{whp}}$ every $\psi$ that satisfies ${F}_1^*$ has the desired property. Let now $\psi$ be a satisfying assignment of ${F}_1^*$ such that $B_\psi\leq n/10^7$, and consider the following procedure which, as we shall prove, produces a large $500t$-expanding set $Z$ in ${F}_1^*$ (and therefore also in ${F}^*$ which contains ${F}_1^*$). When using the notation ${F}[A]$ for a formula ${F}$ and a set of variables $A$ we mean all clauses in ${F}$ in which all three variables belong to $A$.
Clearly, $Z$ is $500t$-expanding in ${F}_1$ (by the construction). It remains to prove that $Z$ is large. By our assumption on $B_\psi$ step 1 removes at most $n/10^7$ variables, let $A$ be those variables. It remains to prove that in the iterative step not too many variables were removed. Suppose by contradiction that in the iterative step more than $n/10^7$ variables were removed, and consider iteration $j=n/10^7$ and the set $W=\{a_1,\ldots,a_{j}\}$ ($a_i \in W$ is defined in line 2 of Figure \[fig:ExpandingSet\]). Every $a_i \in W$ appears in more than $502t-500t=2t$ clauses in which at least another variable belongs to $U=W\cup A$ (by the choice of $Z_0$ and the condition in line 2 that caused $a_i$ to be removed). Therefore, by iteration $n/10^7$, the set $U$ contains at most $n/10^7+n/10^7
\leq n/10^6$ variables, and there are more than $2t \cdot |W| \geq 2t \cdot |U|/2= t|U|$ clauses containing at least two variables from $U$ (no clause is counted twice as the support of a clause is unique). This contradicts the $t$-proportionality of $F_1$. To conclude, $|Z|\geq
\left(1-10^{-6}\right)n\geq 0.99n$ as required. Observe that $|W| \geq |U|/2$ by our assumption on the size of $A$ and by the choice of $j=n/10^7$.
It follows that ${\emph{whp}}$ for every satisfying assignment $\psi$ of $F_1^*$ there exists a $500t$-expanding set $Z$ of variables of cardinality $|Z|\ge 0.99n$. W.l.o.g. we can take $Z$ to be maximal such set.
Observe that $Z$ and $F_1^*$ satisfy the conditions of Proposition \[prop:UniqueOfSat\] (that is, $\psi$ is a satisfying assignment, $F_1^*$ is $t$-proportional and $n/10^6$-concentrated) and therefore $Z$ is frozen in $F_1^*$; w.l.o.g. assume that all variables in $Z$ froze to TRUE. Since all variables of $Z$ are frozen in $F_1^*$, we can take [*the same*]{} $Z$ for every satisfying assignment $\psi$ of $F_1^*$.
So let $Z$ be as above, $|Z|\ge 0.99n$. Now we consider the second round of coin tosses, call the chosen clauses $F_2$. We prove that after adding them, with probability $1-o(2^{-n})$ $Z$ extends to a $t$-expanding set $Z'$, $Z\subseteq Z'$, of the required size ($|Z'|\geq (1-{e^{-\Omega(n^2p)}})n$). Fix some variable $x \notin Z$ and observe that $x$ supports $\binom{|Z|}{2}$ clauses, where $x$ appears without negation and the other two variables are in $Z$ and appear as negated. Since $x \notin Z$, we know that in the first iteration at most $500t$ such clauses were included. In expectation, $F_2$ contains at least $p_2\left(
\binom{|Z|}{2}-500t\right) \geq n^2p/5 \geq 1000t$ such clauses (this is due to $p_2\ge p/2$). If indeed at least $500t$ clauses are included then $Z \cup \{x\}$ is a $500t$-expanding set. The probability that less than $500t$ of them were included is ${e^{-\Omega(n^2p)}}$ (again, Chernoff bound). We can argue similarly about the number of clauses in $F_2$, containing $x$ and two variables from $Z$, where all three variables appear as negated.
Call a variable $x$ [*good*]{} if it participates it at least $500t$ clauses in $F_2$ where the other two variables are from $Z$ and are negated and $x$ is not negated, and also in at least $500t$ clauses in $F_2$ where the other two variables from $Z$ and all three variables are negated; otherwise $x$ is called [*bad*]{}. Observe that for every good $x$, for every satisfying assignment $\psi$ of $F_1^*\cup F_2^*$, we can add $x$ to $Z$, regardless of whether $\psi$ sets $x$ to TRUE or FALSE. The above argument shows that the expected number of bad variables is ${e^{-\Omega(n^2p)}}n$. Applying standard concentration techniques, we can derive that ${\emph{whp}}$ the number of bad variables is ${\emph{whp}}$ ${e^{-\Omega(n^2p)}}n$ as well.
And so we have proven that ${\emph{whp}}$ there exists a $500t$-extending set $Z'$ (which contains $Z$) and $|Z'|=|Z|+(1-{e^{-\Omega(n^2p)}})|V\setminus Z| \geq (1-{e^{-\Omega(n^2p)}})|V|$.
For conclusion, we have shown that there exists a $500t$-expanding set $Z'$ in ${F}^*$ of cardinality $|Z'|=(1-{e^{-\Omega(n^2p)}})n$ w.r.t. $\psi$, where $\psi$ is some satisfying assignment of ${F}^*$ (in fact this is true w.r.t. all satisfying assignments of $F^*$ by the frozenness property). Scaling everything down (setting $t'=500t$), $Z'$ is $t'$-expanding and (at least) $t'/500$-proportional. This completes the proof of the proposition.
\[rem:t/500-prop\] Note that here we proved $t'/500$-proportionality, which is stronger than what we are required to prove ($t'/10$-proportionality). In general, we prefer clear and shorter presentation over optimizing the constants in the proofs. Later we will use this slackness in other proofs that rely on this one.
Proof of Proposition \[prop:CoreSize\] {#sec:CoreSizeProof}
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Let $Z$ be the $t$-expanding set promised by Proposition \[prop:ExpanderSize\]. Consider the procedure in Figure \[fig:Core\], which shall produce a $t'$-core (for $t'=10t/11$). Recall that using the notation ${F}[A]$ for formula ${F}$ and set of variables $A$ we mean all clauses in ${F}$ in which all three variables belong to $A$.
First let us explain why indeed ${{\cal{H}}}$ is a $t'$-core. By its construction ${{\cal{H}}}$ is $10t/11$-expanding (or $t'$-expanding). Further, ${{\cal{H}}}$ is $t/11$-self-contained, or $t'/10$-self-contained (which also implies $t'/3$-self-contained as $1/3>1/10$).
\[rem:t/6-self-contained\] By the definition of a core we are required to prove only $t'/3$-self-contained, but we shall need this slackness in the proof of Proposition \[prop:SizeOfConnectedComp\].
It remains prove that $|{{\cal{H}}}|\geq(1-{e^{-\Omega(n^2p)}})n$. By Proposition \[prop:ExpanderSize\], $|H_0| \geq (1-e^{-n^2p/c_1})n$ for some constant $c_1>0$ independent of $n,p$. Let $A=V\setminus H_0$, and note that $|A| \leq e^{-n^2p/c_1}n$. Suppose that the iterative procedure (line 2) removed more than $e^{-n^2p/c_1}n$ variables. Consider iteration $j=e^{-n^2p/c_1}n$ and the set $W=\{a_1,\ldots,a_j\}$ ($a_i \in W$ is defined in line 2 of Figure \[fig:Core\]). Define $U = W \cup A$. One possibility for the removal of $a_i$ is that it appears in at least $t/11$ clauses in which at least another variable belongs to $U$. Another is that $a_i$ supports less than $10t/11$ clauses w.r.t. ${F}[H_{i}]$. In the latter case $a_i$ must support at least $t-10t/11=t/11$ clauses in ${F}\setminus {F}[H_{i}]$ (by the choice of $a_i\in Z$). In any case $a_i$ appears in at least $t/11$ clauses with at least another variable from $U$. Therefore, by iteration $j$, there exists a set $U$ containing at most $2e^{-n^2p/c_1}n<< n/10^6$ variables, and there are at least $(t/33) \cdot |W| \geq (t/33) \cdot (|U|/2) = t|U|/66$ clauses containing at least two variables from it (we divide $t/11$ by 3 as a clause could have been counted three times). This however contradicts the $t/500$-proportionality of $F$ (recall that in Proposition \[prop:ExpanderSize\], when proving the existence of a $t$-expanding set $Z$, we actually proved that ${F}$ is $t/500$-proportional – Remark \[rem:t/500-prop\]).
Finally, the core variables are frozen as they are a subset of $Z$, and $Z$ – the $t$-expanding set – is frozen.
Proof of Proposition \[prop:SizeOfConnectedComp\] {#sec:ProofOfPropSizeOfConnectedComp}
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Let $F$ be the random ${{\cal{P}}_{n,p}}$ instance, $F^*$ its satisfiable part. We divide the process of generating $F$ into two steps like in the proof of Proposition \[prop:MajVoteSuccRate\]: in the first round go over the $M=8\binom{n}{3}$ clauses and toss a coin with success probability $p_1=p/2$. Take the clauses that were chosen and put them first. In the second round, every clause that was not chosen is included with probability $p_2$, where $p_2$ satisfies $p_1+(1-p_1)p_2=p$. Let $F_1^*$ be the part of $F^*$ that corresponds to the first iteration ($F_1^*$ is distributed according to ${{\cal{P}}^{{\rm sat}}_{n,p_1}}$). Let $F_2$ be the clauses that were chosen in the second round.
By Proposition \[prop:CoreSize\], if we take $n^2p$ to be sufficiently large, then $F_1^*$ has ${\emph{whp}}$ a $t$-core ${{\cal{H}}}$ w.r.t. to a satisfying assignment $\psi$ with the following properties (the last property did not appear in Proposition \[prop:CoreSize\], we define and justify it immediately after):
- ${F}_1$ is $t/500$-proportional (Remark \[rem:t/500-prop\]).
- ${{\cal{H}}}$ is $t/10$-self-contained and not only $t/3$-self-contained (Remark \[rem:t/6-self-contained\]).
- $|{{\cal{H}}}| \geq (1-{e^{-\Omega(n^2p)}})n$.
- ${{\cal{H}}}$ is frozen.
- $F_1^*$ is bounded.
We say that a formula $F$ is *bounded* if no variable appears in more than $n$ clauses. In $F_1$ every variable is expected to appear in $O(n^2p)$ clauses, and we may assume that $n^2p = O(n^{1/2})$ (if not, then in particular ${\emph{whp}}$ ${{\cal{H}}}=V$ and the entire discussion in this section is unnecessary). Standard calculations then show that ${\emph{whp}}$ no variable appears in more than $n$ clauses of $F_1$.
We now discuss what happens to ${{\cal{H}}}$ in the second round, that is when adding $F_2$. We will be interested in large connected components of $F_1$ whose vertices are not in ${{\cal{H}}}$ (Proposition \[prop:SatteliteShatter\]), and also in vertices that may leave ${{\cal{H}}}$ due to $F_2$ (Propositions \[prop:Core’\] and \[prop:IfNotCoreThenSat\]). The key to understanding the transformation that ${{\cal{H}}}$ and the connected components undergo lies in the notion of satellite variables.
First observe that $F_1$ is ${\emph{whp}}$ $t/500$-proportional, and therefore also is $F_2$ (as they are almost identically distributed, and there is enough slackness in the choice of constants to accommodate this difference). Hence ${\emph{whp}}$ $F=F_1 \cup F_2$ is $t/250$-proportional (and so is $F^*$). Assume that this is the case.
\[prop:Core’\] If after the second round $F^*$ remains $t/250$-proportional, then there exists a satisfying assignment $\psi$ of $F^*$ and a set ${{\cal{H}}}'\subseteq {{\cal{H}}}$ of variables which is a $t/2$-core of $F^*$ w.r.t. $\psi$. Furthermore, $|{{\cal{H}}}'| \geq (1-{e^{-\Omega(n^2p)}})n$.
We call a variable $x \in {{\cal{H}}}$ *dirty* if in $F_2$ there exists a clause $C$ containing $x$ and some variable not in ${{\cal{H}}}$. Let $D$ be the set of dirty variables. For a specific $x$, there are ${e^{-\Omega(n^2p)}}n^2$ clauses such that if chosen to $F_2$ will make $x$ dirty. The probability that any of them appears is at most $p_2 \cdot {e^{-\Omega(n^2p)}}n^2 = {e^{-\Omega(n^2p)}}$ (since ${e^{-\Omega(n^2p)}}$ is much smaller than $n^2p_2$ for sufficiently large $p$). Linearity of expectation gives $E[|D|]=e^{-\Omega(n^2p)}n$. Also observe that $D$ satisfies the Lipschitz condition with difference 3 (as every new clause can effect 3 new variables). Therefore also concentration is obtained. Let us assume from now on that indeed $|D|={e^{-\Omega(n^2p)}}n$.
Consider ${{\cal{H}}}$ after scanning $F_2$ (to complete $F^*$) and set ${{\cal{H}}}_0 = {{\cal{H}}}\setminus D, i=0$. Very similarly to the procedure in Figure \[fig:Core\], consider the following iterative procedure:
> *while there exists $x \in {{\cal{H}}}_i$ s.t. $x$ supports less than $t/2$ clauses in $F[{{\cal{H}}}_i]$ w.r.t. $\psi$, or appears in more than $t/6$ clauses where some variable belongs to $V \setminus {{\cal{H}}}_i$, define ${{\cal{H}}}_{i+1}={{\cal{H}}}_i \setminus \{x\}, i=i+1$.*
Set $\ell=e^{-cn^2p}n$, where $c$ is some constant satisfying $|D| \leq e^{-cn^2p}n$. Suppose that the iterative process reached iteration $\ell$, and let $W_\ell$ be the set of variables that were removed in iterations $1 \ldots \ell$, let $U=W_\ell \cup D$, and observe that $|W_\ell| \geq |U|/2$ by our choice of $c$. Take $x\in W_\ell$, if $x$ was removed in iteration $i$ because it appeared in more than $t/6$ clauses where some variable belongs to $V \setminus {{\cal{H}}}_i$, then since $x$ was part of ${{\cal{H}}}$ to begin with, and ${{\cal{H}}}$ was $t/10$-self-contained, then $x$ must appear in at least $t/6-t/10=t/15$ clauses in which at least another variable belongs to $U$. If $x$ was removed because it supports less than $t/2$ clauses in $F[{{\cal{H}}}_i]$, then again, $x$ was part of ${{\cal{H}}}$, and therefore it supports at least $t$ clauses in $F[{{\cal{H}}}]$, and hence it must support (and, in particular, appear in) at least $t-t/2=t/2$ clauses in which some variable belongs to $U$. At any rate, every $x \in W_\ell$ appears in at least $t/15$ clauses in which at least another variable belongs to $U$. Finally,
- there are $t|W_\ell|/15\cdot 1/3 \geq t|U|/90$ clauses containing at least two variables from $U$ (we divide by 3 as every clause might have been over–counted up to 3 times, and we use the fact that $|W_\ell| \geq |U|/2$),
- $|U| = |D|+|W_\ell| = {e^{-\Omega(n^2p)}}n < n/10^6$ (we used our estimate on $|D|$, and the fact that we look at the iterative process until iteration $\ell$, therefore $|W| \leq \ell = {e^{-\Omega(n^2p)}}n$).
Combining these two facts contradicts the $t/250$-proportionality of $F^*$. Therefore if we let $W$ denote the set of variables that were removed in the iterative step, in all iterations, then ${\emph{whp}}$ $|W| \le\ell$. Now set $t'=t/2$, and let ${{\cal{H}}}'={{\cal{H}}}\setminus \{D \cup W \}$. We have shown that the set ${{\cal{H}}}'$ is a $t'$-core of the required size. Further, $F^*$ is (at least) $t'/10$-proportional as required by Proposition \[prop:SizeOfConnectedComp\].
Finally observe that ${{\cal{H}}}$ is frozen and hence ${{\cal{H}}}' \subseteq {{\cal{H}}}$ is frozen too. Therefore although ${{\cal{H}}}'$ is defined w.r.t. $\psi$, it will be a core of $F^*$ regardless of which satisfying assignments survive at the end (as it will be a core w.r.t. all $F^*$’s satisfying assignments, and at least one is guaranteed to survive).
\[prop:IfNotCoreThenSat\] Let ${{\cal{S}}}'$ be the satellite variables of ${{\cal{H}}}'$. If $F^*$ is $t/10$-proportional then ${{\cal{H}}}\setminus {{\cal{H}}}' \subseteq {{\cal{S}}}'$.
Let $A={{\cal{H}}}\setminus {{\cal{H}}}'$. Let ${{\cal{S}}}'$ be all the satellite variables of ${{\cal{H}}}'$, and by contradiction assume that the set $B= A \setminus {{\cal{S}}}'$ is non-empty. Every $x$ in $B$ belongs to ${{\cal{H}}}$ and therefore supports at least $t$ clauses where the other two variables appear in ${{\cal{H}}}$. Observe that in none of these $t$ clauses the other two variables are in ${{\cal{H}}}'\cup {{\cal{S}}}'$ (as otherwise $x$ is in ${{\cal{S}}}'$). Therefore we have found a set $B$, $|B| = {e^{-\Omega(n^2p)}}n \leq n/10^6$, for which there are at least $t|B|$ clauses containing two variables from $B$. This contradicts the $t/10$-proportionality of $F^*$.
In the proof of Proposition \[prop:SizeOfConnectedComp\] we consider two “types" of satellite variables. The first type, which we just met, are the variables in ${{\cal{H}}}\setminus {{\cal{H}}}'$. The second type, which we will make use of in the proof of Proposition \[prop:SatteliteShatter\] ahead, are satellite variables of ${{\cal{H}}}$ whose “job" is to shatter the large connected components in the formula induced by variables not in ${{\cal{H}}}$ (when exposing the second part of $F$). In some sense these two types represent competing processes. The one is variables leaving ${{\cal{H}}}$, but still remaining satellite variables, the other is new variables attaching to ${{\cal{H}}}$ as satellite variables.
Recall our notation ${F}_{out}(A,\varphi)$ ($A$ a set of variables, $\varphi$ an assignment) which stands for the subformula of ${F}$ which is the outcome of the following procedure: set the variables in $A$ according to $\varphi$ and simplify ${F}$ (by simplify we mean remove every clause that contains a TRUE literal, and remove FALSE literals from the other clauses). The connected components of a formula ${F}$ are the sub-formulas ${F}[C_1],\ldots,{F}[C_k]$, where $C_1,C_2,\ldots,C_k$ are the connected components in the graph $G_{F}$ induced by ${F}$ (the vertices of $G_{F}$ are the variables, and two variables are connected by an edge if there exists some clause containing them both).
\[prop:SatteliteShatter\] Let ${{\cal{H}}}$ be a $t$-core of $F_1^*$, let ${{\cal{S}}}$ be the set of all satellite variables of ${{\cal{H}}}$ in $F^*$, and let $\psi$ be a satisfying assignment of $F^*$. Then the largest connected component in $F^*_{out}[{{\cal{H}}}\cup {{\cal{S}}},\psi]$ is ${\emph{whp}}$ of size at most $\log n$.
First let us show why Proportion \[prop:SatteliteShatter\] completes the proof of Proposition \[prop:SizeOfConnectedComp\]. Since the proposition is true for $F$ it is true, by monotonicity, for $F^*$. We take ${{\cal{H}}}'$ for the core to be given by Proposition \[prop:SizeOfConnectedComp\], and denote by ${{\cal{S}}}'$ its satellite variables. Observe that (under the assumption of proportionality) ${{\cal{H}}}\subseteq {{\cal{H}}}' \cup {{\cal{S}}}'$, and hence by the definition of satellite variables ${{\cal{S}}}\subseteq {{\cal{S}}}'$. In particular ${{\cal{H}}}\cup {{\cal{S}}}\subseteq {{\cal{H}}}' \cup {{\cal{S}}}'$.
Proof of Proposition \[prop:SatteliteShatter\]
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Let us refine the process of generating $F$: first we generate $F_1$ (and $F_1^*$), and fix ${{\cal{H}}}$ according to $F_1^*$. Then in the second round ($F_2$) first toss the coins of clauses $C$ s.t. at most one literal in $C$ belongs to ${{\cal{H}}}$, call $J \subseteq F_2$ the set of clauses that were chosen. Finally toss the coins of the other clauses (the ones that were not picked in the first step and contain at least two variables from ${{\cal{H}}}$), call $K \subseteq
F_2$ the set of clauses that were chosen. In this new terminology $F=F_1 \cup J \cup K$, and set $F'=F_1 \cup J$. To prove Proposition \[prop:SatteliteShatter\] it suffices to consider only trees of size $\log n$ in $F'$. This is because $(a)$ every connected component of size at least $\log n$ contains a tree of size $\log
n$, and $(b)$ only the clauses of $F'$ may contribute edges to the connected components of $F^*_{out}[{{\cal{H}}}\cup {{\cal{S}}},\psi]$.
We will prove Proposition \[prop:SatteliteShatter\] as follows: fix an arbitrary tree $T$ on $r$ vertices, and let $V(T)$ denote its set of vertices. The following two conditions are necessary for $T$ to belong to $F^*_{out}[{{\cal{H}}}\cup {{\cal{S}}},\psi]$:
- $A=\{$there exists a subformula of $F'$ that induces $T\}$,
- $B=\{$the clauses in $K$ do not prevent the following from holding: $V(T)\cap {{\cal{S}}}=\emptyset \}$.
The probability that $F^*_{out}[{{\cal{H}}}\cup {{\cal{S}}},\psi]$ contains a tree of size at least $r$ is at most $$\sum_{T:|V(T)|=r}Pr[A \wedge B]=\sum_{T:|V(T)|=r}Pr[A]\cdot Pr[B|A] \leq \left(\max_{T:|V(T)|=r} Pr[B|A]\right)
\cdot \left(\sum_{T:|V(T)|=r}Pr[A]\right) \equiv q \cdot h.$$ Our next goal is to bound $q$ and $h$, and then to show that $q \cdot
h=o(n^{-3})$ for $r = \log n$. In fact we shall prove that $q \cdot h =o(n^{-\Omega(n^2p)})$ for $r=\log n$. The next two lemmas establish the desired bounds (we use $d=n^2p$).
\[lem:SatelliteLem1\] $h=\sum_{T:|V(T)|=r}Pr[A]\leq n(100d)^r$.
\[lem:SatelliteLem2\] $q=\max_{T:|V(T)|=r} Pr[B|A] \leq e^{-dr/8}$.
To conclude, for $r=\log n$, $$q \cdot h \leq n (100d)^{\log n} \cdot n^{-d/8} \leq n^{1+\log(100d)-d/8}=o(n^{-\Omega(d)}).$$ The last equality is true since $d/8 >> 1+\log(100d)$ for sufficiently large $d=n^2p$. We shall now prove the two lemmas.
[**Proof of Lemma \[lem:SatelliteLem1\].**]{} The quantity $h$ to be estimated is obviously the expected number of trees of size $r=\log n$ induced by a formula $F'\subseteq F$ and is therefore at most the expected number of such trees induced by $F$ itself. We thus estimate from above the latter quantity.
Let $T$ be a fixed tree on $r$ variables (a tree in the regular graph sense), and let $F_T$ be a fixed collection of clauses such that each edge of $T$ is induced by some clause of $F_T$ – we call such $F_T$ an *inducing* set of clauses. We say that a clause set $F_T$ is *minimal* w.r.t. $T$ if by deleting a clause from $F_T$, $T$ is not induced by the new formula anymore. By the definition of minimality, $|F_T| \leq |E(T)| = |V(T)|-1$ (as $T$ is a tree). In our argument we shall be interested only in $(T,F_T)$ s.t. $F_T$ is a minimal set of clauses that induces $T$.
Given a tree $T$ of size $r$, we estimate the number of ways to extend $T$ to a minimal inducing set $F_T$. Every clause in $F_T$ can cover either one or two edges of $T$ (it cannot cover three edges or we have a cycle in $T$). Following the argument in [@flaxman], let $N_{T,s}$ be the number of ways to pair $2s$ edges of $T$ to form $s$ clauses in $F_T$ that cover two edges. There are 8 ways to set the polarity of variables in every clause of $F_T$ (and there are $r-1-s$ such clauses), and at most $n^{r-1-2s}$ ways to choose the third variable in the $r-1-2s$ clauses that cover exactly one edge. Using this terminology, the expected number of $r$-trees induced by a random formula $F$, generated according to ${{\cal{P}}_{n,p}}$ with $n^2p=d$, is at most: $$\begin{aligned}
\label{eq:UnionBound}
\sum_{r-trees}\sum_{s=0}^{r/2}N_{T,s}8^{r-1-s}n^{r-1-2s}\left(\frac{d}{n^2}\right)^{r-1-s}
\leq \sum_{r-trees}\left(\sum_{s=0}^{r/2}N_{T,s}\right)(8d)^{r}n^{1-r}\,.\end{aligned}$$ Our next task is to obtain useful upper bounds on the sum $\sum_{s=0}^{r/2}N_{T,s}$. To this end let us fix a degree sequence $(d_1, . . . , d_r)$ for $T$, and consider the following procedure for *properly* pairing edges. By proper we mean that every pair of edges can be covered by a 3CNF clause; for example, we cannot pair the edges $(x_1,x_2)$ and $(x_3,x_4)$ as they result in a 4CNF clause. For each vertex, we specify a permutation of the edges incident to that vertex. Then we iterate through the vertices, and for each vertex, we iterate through the edges and pair up each unpaired edge with the edge given by the permutation associated with the current vertex (and leave the edge unpaired if the permutation sends the edge to itself). Any pairing of edges which can be covered by clauses can be generated this way by choosing the permutations to transpose each pair of edges to be covered by a single clause and to leave fixed all the other edges. Since there are $d_i!$ different permutations for vertex $i$, we have $$\sum_{s=0}^{r/2}N_{T,s} \leq \prod_{i=1}^r d_i!.$$ A classical result by Prüfer is that the number of $r$-trees with degree sequence $(d_1, . . . , d_r)$ equals $\binom{r-2}{d_1-1,\ldots,d_r-1}$ (see, for example, [@LovaszBook], Section 4.1, p. 33). There are $\binom{n}{r}$ ways to choose the $r$ vertices of the tree. So (\[eq:UnionBound\]) is at most $$\sum_{d_1+\ldots_+d_r=2(r-1)}\binom{n}{r}\binom{r-2}{d_1-1,\ldots,d_r-1}\left(\prod_{i=1}^r d_i!\right)(8d)^{r}n^{1-r}
\leq \sum_{d_1+\ldots_+d_r=2(r-1)}\left(\prod_{i=1}^r d_i\right)
(8d)^rn.$$ By convexity, for $(d_1, . . . , d_r)$ with $d_1 + . . . + d_r = 2(r - 1)$, the product $\prod_{i=1}^r d_i$ is maximized when $d_1 =\ldots = d_r$, and so $\prod_{i=1}^r d_i \leq 2^r$. The number of ways to choose positive integers $(d_1, . . . , d_r)$ so that $d_1 + . . . + d_r = 2(r - 1)$ is $\binom{2r-3}{r-1}$ which is less than $2^{2r}$. Hence, the expected number of $r$-trees induced by a random formula $F$ is at most $n\cdot 2^{2r}\cdot 2^r\cdot (8d)^r \le n(100d)^r$. $\blacksquare$
[**Proof of Lemma \[lem:SatelliteLem2\].**]{} Fix a tree $T$ in $F'=F_1 \cup J$ on $r$ vertices (recall that $J$ is the set of clauses that contain at most one variable from ${{\cal{H}}}$), and consider the set of clauses that have at least two variables in ${{\cal{H}}}$, which we now toss their coins (we use $K$ to denote the set of clauses that were chosen among the latter).
Assume w.l.o.g. that the assignment $\psi$, w.r.t. which ${{\cal{H}}}$ is defined, is the all-TRUE assignment. Look at a variable $x \in V(T)$. We call a clause $(x \vee \bar{z}_1
\vee \bar{z}_2)$, $(\bar{x} \vee \bar{z}_3 \vee \bar{z}_4)$, where the $z_i$’s are some variables in ${{\cal{H}}}$, a type 1, respectively type 2, clause. If clauses of both types appear in $K$ then $x$ surely belongs to ${{\cal{S}}}$ (and therefore $V(T) \cap {{\cal{S}}}\neq \emptyset$). We call $x \notin {{\cal{H}}}$ *elusive* if at least one of the two types of clauses didn’t appear in $K$.
Set $\rho=1-{e^{-\Omega(n^2p)}}$, since $|{{\cal{H}}}| \geq \rho n$ there are at least $\binom{\rho n}{2} \geq
(\rho n)^2/3$ clauses of type 1. We assume that $F_1$ is bounded and hence every variable appears in at most $n$ clauses, therefore at most $n$ clauses of type 1 have been included in $F_1$. An identical argument applies to clauses of type 2. Note also that the clauses of $J$ cannot belong to any of the types. Therefore the probability that no clause of type 1 belongs to $K$ is at most $(1-p_2)^{(\rho n)^2/3-n} \leq e^{-d/7}$ (here we use: $d=n^2p$ is large, $p_2=(p-p_1)/(1-p_1)$, $p_1=p/2$). The same is true by symmetry for clauses of type 2. Let $E_x$ be the event that $x$ is elusive, and let $P_i$ be the event that no clause of type $i$ for $x$ appeared, $i=1,2$ (namely, $E_x =
P_1 \vee P_2$). $$Pr[E_x] = Pr[P_1 \vee P_2] \leq Pr[P_1]+Pr[P_2]\leq 2e^{-d/7}
\leq e^{-d/8}.$$
Further observe that for $x \neq y$ the events $E_x$ and $E_y$ are independent as they involve disjoint sets of clauses (each variable supports its own set of clauses). Recall the events $A,B$ which were defined above. In this terminology we just upper bounded the probability of $B$ given $A$, and therefore the following is true: $$Pr[B|A] \leq \left(e^{-d/8}\right)^r = e^{-dr/8}.$$ Since our upper bound on $Pr[B|A]$ only depends on the fact that $|V(T)|=r$, then also $$q=\max_{T:|V(T)|=r} Pr[B|A]\leq e^{-dr/8}.$$ $\blacksquare$
$k$-Colorability {#sec:k-col}
================
In this section we will discuss, in a high level fashion, how one can obtain similar results to the ones we have for $k$-SAT for the random graph process (of $k$-colorability). Before we start our discussion let us recall the algorithm due to Alon and Kahale for coloring $k$-colorable graphs [@AlonKahale97]. The first step of the algorithm is a spectral step; specifically a $k$-coloring of the graph (not necessarily proper) is obtained by looking at some eigenvectors of the graph (that hopefully reflect in some sense a proper $k$-coloring). Then, this initial $k$-coloring is refined using a series of combinatorial steps (very similar to our Steps 2–4 in Algorithm ), until possibly a proper $k$-coloring is reached (or the algorithm fails). The algorithm was analyzed on graphs drawn from the planted distribution first defined at [@Kuc77] (the distribution is defined by the following procedure: partition the vertex set into $k$ color classes of size $n/k$ each: $V_1,V_2,\ldots,V_k$; next, include every $V_i-V_j$ edge with probability $p$). The algorithm was shown to find ${\emph{whp}}$ a proper $k$-coloring of the graph when $np\geq ck^2$, $c$ some sufficiently large constant.
It is possible to prove that the algorithm works also for graphs drawn from our distribution for the same edge density (maybe the constant $c$ is different). The main challenge is to reprove the spectral properties of the graph. The basic idea is to notice that ${\emph{whp}}$ every $k$-coloring has all of its color classes of linear size, and also to prove discrepancy properties (similar, yet more elaborate, to Proposition \[prop:NoDenseSubgraphs\]). Another crucial ingredient in the proof is establishing a similar notion of a core (Definition \[def:core\]).
Unfortunately, at this point we are still unable to answer a seemingly much simpler question: how many edges will such a graph typically contain by the end of the process? We expect the answer to be about $\binom{k}{2}\left(\frac{n}{k}\right)^2$ – which would corresponds to the case where a unique final $k$-coloring is nearly balanced.
Discussion {#sec:Discussion}
==========
As we already mentioned, only a vanishing proportion of $k$-CNFs with $m$ clauses over $n$ variables are satisfiable when $m/n$ is above the threshold. In recent years, several papers studied different distributions over satisfiable 3CNF formulas in the above threshold regime, more precisely some sufficiently large constant factor above the threshold. In particular, [@flaxman] considered the planted 3SAT distribution, and [@UniformSAT] addressed the planted and uniform distributions, both papers developing new analytical and algorithmic techniques. Our work joins this line of research by studying a new distribution over satisfiable 3CNF formulas, and once again introducing new analytical ideas to face the intricacies of ${{\cal{P}}^{{\rm sat}}_{n,m}}$. Furthermore, one interesting conclusion emerges from combining [@flaxman],[@UniformSAT] and our result. In all three distributions the instances show basically the same uni-cluster structure of the solution space, and the same algorithm solves them all. This gives rise to the following question: does forcing (in some “natural" way) the unlikely event of being satisfiable in the above threshold regime generally result in the structure suggested by Theorem \[thm:StructOfSolutionSpace\] (for clause-variable ratio greater than some sufficiently large constant)? This question has been answered positively for the planted and uniform distributions, and in this paper for the random satisfiable 3CNF process.
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[^1]: Research supported in part by a USA-Israel BSF Grant, and by a grant from the Israel Science Foundation, and by Pazy Memorial Award.
[^2]: Research supported in part by NSF CAREER award DMS-0546523 and a USA-Israeli BSF grant.
|
---
author:
- |
R. Amorim[^1] and J. Barcelos-Neto[^2]\
Instituto de Física\
Universidade Federal do Rio de Janeiro\
RJ 21945-970 - Caixa Postal 68528 - Brasil
title: '**BV quantization of a vector-tensor gauge theory with topological coupling**'
---
We use the BV quantization method for a theory with coupled tensor and vector gauge fields through a topological term. We consider in details the reducibility of the tensorial sector as well as the appearance of a mass term in the effective vectorial theory .
PACS: 03.70.+k, 11.10.Ef, 11.15.-q
Introduction
============
The quantization method due to Batalin and Vilkoviski (BV) [@BV1; @BV2] has been shown to be a powerful functional procedure to deal with a wide variety of gauge theories. In the BV scheme, covariance, reducibility, openness of gauge algebras and the presence of gauge anomalies are taken into account in a very natural way.
In this work we consider the BV method to quantize the coupled vector-tensor gauge theory present by Allen and collaborators [@Allen] and recently treated by Lahiri [@Lahiri] from the canonical point of view. It might be opportune to mention that tensor gauge theories have attracted much attention, formerly connected with the theory of strings, but also with cosmic strings, vortices and black holes [@tensor]. Another use of tensorial gauge theories can be seen in connection with the appearance of a mass term for vector gauge fields, in the context of an effective theory [@Lahiri]. As tensor gauge theories are reducible, the BV scheme is a natural quantization procedure for them.
In the work of ref. [@Lahiri], it was not directly taken into account the question of reducibility of constraints related to the tensor gauge fields. This is an important point because if one use that all the constraints are independent, the theory has zero tensor degrees of freedom, what is not actually true [@Kaul]. This point is naturally considered here and the integration over the tensorial sector can be done without infinities, leading to an effective massive vectorial theory. We mention that in references [@Lahiri], the presence of this mass term was considered at classical level. Here, we investigate how it comes quantically as a pole in the propagator of the gauge fixed effective vector theory.
Our work is organized as follows. In Sec. 2 we discuss the BV quantization of a theory with topological coupling of vector and tensor gauge fields. The obtainment of the effective mass for the vector gauge field is achieved at Sec. 3. We leave Sec. 4 for some concluding remarks.
Vector coupled to tensor gauge fields
=====================================
The theory we are going to deal is described by the following Lagrangian density [@Allen; @Lahiri]
$${\cal L}=-\,\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}
-\,\frac{1}{12}\,H_{\mu\nu\rho}\,H^{\mu\nu\rho}\,
+{m\over4}\,\epsilon^{\mu\nu\rho\lambda}\,
B_{\mu\nu}F_{\rho\lambda}\,,
\label{2.1}$$
where $F_{\mu\nu}$ and $H_{\mu\nu\rho}$ are totally antisymmetric tensors written in terms of the potentials $A_{\mu}$ and $B_{\mu\nu}$ (also antisymmetric) through the curvature tensors
$$\begin{aligned}
F_{\mu\nu}&=&\partial_\mu A_{\nu}-\partial_\nu A_{\mu}\,,
\label{2.2}\\
H_{\mu\nu\rho}&=&\partial_\mu B_{\nu\rho}
+\partial_\rho B_{\mu\nu}+\partial_\nu B_{\rho\mu}\,.
\label{2.3}\end{aligned}$$
In (\[2.1\]), $\epsilon^{\mu\nu\rho\lambda}$ is the totally antisymmetric symbol and m is a mass parameter. It is easy to see, by using the (coupled) Euler-Lagrange equations for $A^\mu$ and $B^{\mu\nu}$, as well as the Jacobi identity, that $F_{\mu\nu}$ satisfy a massive Klein-Gordon equation, with mass parameter given by the factor $m$ appearing in (\[2.1\]).
We observe that both $F_{\mu\nu}$ and $H_{\mu\nu\rho}$ are invariant under the gauge transformations
$$\begin{aligned}
\delta A^\mu&=&\partial^\mu\Lambda\,,
\label{2.4}\\
\delta B^{\mu\nu}&=&\partial^\mu\Lambda^\nu
-\partial^\nu\Lambda^\mu\,,
\label{2.5}\end{aligned}$$
where $\Lambda$ and $\Lambda^\mu$ are (before fixing the gauge) generic functions of spacetime. The reducible character of this theory is seem from the fact that if we choose the gauge parameter $\Lambda^\mu$ as the gradient of some scalar $\Omega$ we have that $B^{\mu\nu}$ does not change under the gauge transformation (\[2.5\]). Actually, in this situation
$$\begin{aligned}
\delta B^{\mu\nu}&=&(\partial^\mu\partial^\nu
-\partial^\nu\partial^\mu)\,\Omega\,,
\nonumber\\
&=&0\,.
\label{2.6}\end{aligned}$$
Expressions (\[2.4\]) and (\[2.5\]) can be rewritten as
$$\begin{aligned}
\delta A^\mu(x)&=&\int d^4y\,R^\mu(x,y)\,\Lambda(y)\,,
\label{2.7}\\
\delta B^{\mu\nu}(x)&=&\int d^4y\,
R^{\mu\nu}_\rho(x,y)\,
\Lambda^\rho(y)\,,
\label{2.8}\end{aligned}$$
where
$$\begin{aligned}
R^\mu(x,y)&=&\partial^\mu\,\delta(x-y)\,,
\label{2.9}\\
R^{\mu\nu}_\rho(x,y)&=&\partial\,^{[\mu}\delta(x-y)\,
\delta^{\nu]}_\rho
\label{2.10}\end{aligned}$$
are the generators of the gauge symmetries, and
$$Z^\mu(x,y)=\partial^\mu\,\delta(x-y)
\label{2.11}$$
is the reducibility operator [@BV1; @BV2] corresponding to (\[2.6\]) since
$$\int d^4y\,Z^\rho(x,y)\,R^{\mu\nu}_\rho(y,z)=0\,.
\label{2.11a}$$
In expression (\[2.10\]), as well as in some forthcoming ones, the antisymmetrization is performed in a normalized way, i.e. $K^{[\mu\nu]}=(K^{\mu\nu}-K^{\nu\mu})/2$ for some $K^{\mu\nu}$ tensor.
We are now ready to write down the solution of the classical master equation. From expressions (\[2.9\] - \[2.11\]) we notice that the algebra of the gauge generators is closed and Abelian. To fix the notation, it is convenient to distinguish between both sectors (vectorial and tensorial) of the theory. Let us so generically denote by $c$ and $d$ the ghosts associated respectively to the vectorial and to the tensorial sectors. In this way, the solution we are looking for reads [@BV1; @BV2]
$$\begin{aligned}
S&=&S_0+\int d^4xd^4y\,\Bigl[i\,A^*_\mu(x)\,
\partial^\mu_x\,\delta(x-y)\,c(y)
\nonumber\\
&&\phantom{S_0}+2\,i\,B^*_{\mu\nu}\,\partial_x^{[\mu}\,
\delta(x-y)\,\delta^{\nu]}_\rho\,d^\rho(y)
+d^*_\rho(x)\,\partial^\rho_x\,\delta(x-y)\,d(y)\Bigr]\,,
\nonumber\\
&=&S_0+\int d^4x\,\Bigl[i\,A^*_\mu\,\partial^\mu c
+i\,B^*_{\mu\nu}\,\partial^{\mu}d^{\nu}
+d^*_\mu\,\partial^\mu d\Bigr]\,,
\label{2.12}\end{aligned}$$
where $S_0$ is the action corresponding to the Lagrangian (\[2.1\]) and $A^\ast_\mu$, $B^\ast_{\mu\nu}$, $c^\ast_\mu$, $d^\ast_\mu$ are antifields of the BV formalism.
To fix the gauge freedom of the vectorial sector, we introduce the gauge-fixing fermionic functional
$$\psi=-\int d^4x\,\,\bar c\,\Bigl(\partial^\mu A_\mu
-\frac{\alpha}{2}\,b\Bigr)+\cdots\,.
\label{2.13}$$
and extend the classical action to
$$\bar S=S+\int d^4x\,\bar c^*\,b+\cdots
\label{2.14}$$
Here, dots are representing the corresponding quantities for the tensorial sector which we are going to discuss soon. After substituting the antifields $A_\mu^\ast$ and $\bar c^\ast$ by the derivatives of $\psi$ with respect to $A_\mu$ and $\bar c$ we arrive, for the vectorial sector, at the usual Faddeev-Popov expression for the vacuum functional with covariant Gaussian gauge-fixing.
The fixation of the tensorial counterpart is not so simple. First of all we need to have a covariant gauge fixing with the same reducibility character of the gauge freedom of $H_{\mu\nu\rho}$. It may be chosen as
$$\Xi^\nu=\partial_\mu\,B^{\mu\nu}\,.
\label{2.15}$$
We notice that $\Xi^\mu$ is divergenceless. To implement (\[2.15\]) we first observe that in (\[2.13\]) and (\[2.14\]) we have used the ghosts $c$, $c^*$, $\bar c$ and $\bar c^*$ to fix the vectorial gauge invariance of (\[2.1\]). As we have already used the unbarred quantities $d^\mu$, $d_\mu^*$, $d$ and $d^*$ in the tensorial sector of (\[2.12\]), we expect to use the additional pairs $\bar d_\mu$, $\bar d^{*\mu}$, $\bar d$ and $\bar d^*$ in a similar fashion. This gives the tensorial sector of the BV action and the corresponding fixing fermion functional
$$\begin{aligned}
\bar S&=&\cdots\,+\int d^4x\,\bar d^*_\mu\,e^\mu\,,
\label{2.16}\\
\psi&=&\cdots\,-\int d^4x\,\Bigl[\bar d_\mu\,
\Bigl(\partial_\nu B^{\nu\mu}
-{\beta\over2}\,e^\mu\Bigr)
+\bar d\,\partial_\mu\,d^\mu\Bigr]\,,
\label{2.17}\end{aligned}$$
where $e^\mu$ is an auxiliary field that plays a similar role as $b$. As we can observe, the reducibility of the original theory has not been completely fixed yet. Even though the ghost gauge invariance
$$d^\mu\longrightarrow d^\mu +\partial^\mu\,\zeta
\label{2.18}$$
has been fixed in (\[2.17\]), there remains to consider the further invariance
$$\bar d^\mu\longrightarrow\bar d^\mu +\partial^\mu\,\bar\zeta
\label{2.19}$$
of the complete action. This has already been discussed by Henneaux and Teitelboim in ref.[@BV2]. The solution comes by introducing further pairs ($\eta$, $\eta^*$) and ($f$, $f^*$) in the theory. This is achieved by adding the term $-\int d^4x\,\eta\,\partial^\mu\,\bar
d_\mu$ to the fixing fermion $\psi$ and $i\int d^4x\,\eta^*f$ to the action. In these expressions, $f$ can be considered as the ghost corresponding to symmetry (\[2.19\]) and $\eta$ can be seen as an auxiliary field, on the same footing as $b$ and $b^\mu$.
Putting the considerations above all together, we can write the complete BV action as
$$\begin{aligned}
\bar S=S_0&\!\!+\!\!&\int d^4x\,
\Bigl(i\,A^*_\mu\,\partial^\mu c+\bar c^*\,b
+i\,B^*_{\mu\nu}\,\partial^\mu d^\nu
\nonumber\\
&\!\!+\!\!&d^*_\mu\,\partial^\mu d+\bar d^*_\mu\,e^\mu
+i\,\bar d^*\,\bar f+i\,\eta^*f\Bigr)
\label{2.20}\end{aligned}$$
and the complete gauge fixing fermion functional as
$$\begin{aligned}
\psi=-\int d^4x\,\Bigl[\bar c\,
\Bigl(\partial_\mu\,A^\mu -{\alpha\over 2}\,b\Bigl)
&+&\bar d_\mu\,\Bigl(\partial_\nu\,B^{\nu\mu}
-{\beta\over2}\,e^\mu\Bigr)
\nonumber\\
&+&\bar d\,\partial_\mu\,d^\mu
+\eta\,\partial^\mu\,\bar d_\mu\Bigr]\,.
\label{2.21}\end{aligned}$$
At this stage, it might be elucidative to present a table of parity and ghost numbers for fields and antifields of the theory.
$$\begin{aligned}
\epsilon\,[A^\mu,B^{\mu\nu},b,d,\bar d,f^*,\bar f^*,e^\mu,c^*,\bar c^*,d^*_\mu,
\bar d^{*\mu},\eta]&=&0\,,
\nonumber\\
\epsilon\,[A^*_\mu,B^*_{\mu\nu},b^*,d^*,\bar d^*,f,\bar f,e^*_\mu,
c,\bar c,d^\mu,\bar d_{\mu},\eta^*]&=&1\,,
\label{2.22}\\
\nonumber\\
gh(d^*)&=&-3\,,
\nonumber\\
gh(c^*,d^*_\mu,\bar d,f^*)&=&-2\,,
\nonumber\\
gh(A^*_\mu,B^*_{\mu\nu},\bar c,b^*,\bar d_\mu,e^*_\mu,\eta^*,\bar f)&=&-1\,,
\nonumber\\
gh(A^\mu,B^{\mu\nu},\bar c^*,b,\bar d^{*\mu},e^\mu,\eta,\bar f^*)&=&0\,,
\nonumber\\
gh(c,d^\mu,\bar d^*,f)&=&1\,,
\nonumber\\
gh(d)&=&2\,.
\label{2.23}\end{aligned}$$
Making use of the tables above, we observe that action (\[2.20\]) actually has even Grassmamnian parity and ghost number zero, as expected.
With the gauge-fixing functions, antifields are obtained as usual
$$\begin{aligned}
A_\mu^*&=&{\delta \psi\over{\delta A^\mu }}
\,\,=\,\,\partial_\mu\bar c\,,
\nonumber\\
B_{\mu\nu}^*&=&{\delta\psi\over{\delta B^{\mu\nu}}}
\,\,=\,\,\partial\,_{[\mu}\,\bar d_{\nu]}\,,
\nonumber\\
\bar c^*&=&{\delta\psi\over{\delta\bar c}}
\,\,=\,\,-\,\partial_\mu A^\mu
+{\alpha\over2}\,b\,,
\nonumber\\
d_\mu^*&=&{\delta\psi\over{\delta d^\mu}}
\,\,=\,\,\partial_\mu\,\bar d\,,
\nonumber\\
\bar d^{*\mu}&=&{\delta\psi\over{\delta\bar d_\mu}}
\,\,=\,\,-\,\partial_\nu B^{\nu\mu}+{\beta\over2}\,e^\mu+\partial_\mu\eta\,,
\nonumber\\
\bar d^*&=&{\delta\psi\over{\delta\bar d}}
\,\,=\,\,-\,\partial_\mu\,d^\mu\,,
\nonumber\\
\eta^*&=&{\delta \psi\over{\delta\eta}}
\,\,=\,\,-\,\partial^\mu\,\bar d_\mu \,.
\nonumber\\
b^*&=&{\delta \psi\over{\delta b}}
\,\,=\,\,{\alpha\over2}\,\bar c \,.
\nonumber\\
e_\mu^*&=&{\delta \psi\over{\delta e^\mu}}
\,\,=\,\,{\beta\over2}\,\bar d_\mu \,.
\label{2.24}\end{aligned}$$
Now, the combination of (\[2.23\]) and (\[2.24\]) leads to
$$\begin{aligned}
\bar S\,\,=\,\,S_0&\!+\!&\int d^4x\,
\Bigl[i\,\partial_\mu\bar c\,\partial^\mu c
+\Bigl(\partial_\mu A^\mu-{\alpha\over2}\,b\Bigr)\,b
+i\,\partial\,_{[\mu}\bar d_{\nu]}\,
\partial^{\mu} d^{\nu}
\nonumber\\
&\!+\!&\partial_\mu\bar d\partial^\mu d
+\Bigl(\partial_\nu B^{\nu\mu}-{\beta\over2}\,e^\mu
-\partial^\mu\eta\Bigr)\,e_\mu
-i\,\partial_\mu d^\mu\bar f + i\,f\partial^\mu\bar d_\mu\Bigr]\,.
\label{2.25}\end{aligned}$$
Once the process of construction of the gauge fixed BV action is done and as we know that the theory is free of anomalies [@BV2], the next step for the BV quantization is to write the vacuum functional (or its external current dependent generalizations)
$$Z=\int\,[d\mu]\,\exp\{i\bar S\}\,,
\label{2.26}$$
where $\bar S$ is given by (\[2.25\]) and $[d\mu]$ represents the Liouville measure for all the fields appearing there. If we introduce external currents, we are able to generate all the Green’s functions of the theory in a perturbative scheme, by the usual derivative expansion in the external sources. An interesting point we are going to discuss in the next section is how the functional integrations over the antisymmetric tensor field can be performed in a closed way, leading to an effective massive theory for the vectorial field.
Massive vectorial effective theory
==================================
As we have mentioned in the last section, the classical equations of motion for the fields $A^\mu$ and $B^{\mu\nu}$ can be manipulated in order to show that $F^{\mu\nu}$ satisfies a massive Klein-Gordon equation. The same kind of feature could be obtained for $H^{\mu\nu\rho}$, after eliminating $F^{\mu\nu}$. We can make two comments: the first one is that these results are obviously classical. The second one is that are not the fields themselves that satisfy massive Klein-Gordon equations, but their curvature tensors. To see how corresponding features appear at quantum level, we are going to obtain an effective theory, first by functionally integrating over the auxiliary fields $b$, $e^\mu$ and $\eta$ to introduce Gaussian fixing terms in the action . After that, we get for the $B^{\mu\nu}$ depending part of the gauge-fixed effective action
$$\begin{aligned}
\bar S_B&=&\int d^4x\,\Bigl[-\,\frac{1}{12}\,
H_{\mu\nu\rho}\,H^{\mu\nu\rho}\,
-{1\over{2\beta}}\,\bigl(\partial_\nu B^{\nu\mu}\bigr)^2
+{m\over4}\,\epsilon^{\mu\nu\rho\lambda}\,
B_{\mu\nu}F_{\rho\lambda}\Bigr]\,,
\nonumber\\
&=&\int d^4x\,\Bigl[{1\over4}\,B_{\nu\lambda}\,
O^{\nu\lambda}_{\alpha\beta}\,B^{\alpha\beta}
+{1\over2}\,D^{\nu\lambda}\,B_{\nu\lambda}\Bigr]\,.
\label{3.1}\end{aligned}$$
In order to simplify the notation we have defined the operator
$$O^{\mu\nu}_{\alpha\beta}={1\over2}\,
\delta^{\mu\nu\lambda}_{\alpha\beta\gamma}\,
\partial_\lambda\,\partial^\gamma
+{2\over\beta}\,\delta\,^{[\mu}_{[\alpha}
\partial^{\nu]}\partial_{\beta]}
\label{3.2}$$
and the dual tensor
$$D^{\mu\nu}={m\over2}\,\epsilon^{\mu\nu\rho\lambda}\,
F_{\rho\lambda}\,.
\label{3.3}$$
In expression (\[3.2\]), $\delta^{\mu\nu\lambda}_{\alpha\beta\gamma}=6\,
\delta\,^\mu_{[\alpha}\delta^\nu_\beta\delta^\lambda_{\gamma]}$ values 1 if $\mu,\nu,\lambda$ are an even permutation of $\alpha,\beta,\gamma$.
To perform the integral in the $B$’s we need the inverse of $O^{\mu\nu}_{\alpha\beta}$ given by (\[3.2\]), in such a way that
$$O^{\mu\nu}_{\alpha\beta}\,(O^{-1})_{\mu\nu}^{\rho\sigma}
=\delta_{[\alpha}^\rho\,\delta_{\beta]}^\sigma\,.
\label{3.4}$$
It is just a matter of algebraic calculation to show that
$$(O^{-1})_{\mu\nu}^{\rho\sigma}={1\over2\Box}\,
\Bigl[\delta_{\mu\nu}^{\rho\sigma}
-{4\over\Box}\,\Bigl(1-{\beta\over2}\Bigr)\,
\delta_{[\mu}^{[\rho}\partial_{\nu]}\partial^{\sigma]}\Bigr]\,.
\label{3.5}$$
With the aid of (\[3.5\]), we can see that after functionally integrating over $B^{\mu\nu}$, the part of the effective action which had depended of the tensorial field acquires the form
$$\begin{aligned}
\bar S_B&=&-\,{1\over4}\int d^4x\,D^{\mu\nu}\,
(O^{-1})_{\mu\nu}^{\rho\sigma}\,D_{\rho\sigma}\,,
\nonumber\\
&=&-\,{1\over4}\int d^4x\,D_{\mu\nu}\,
\Bigl[{1\over\Box}\,\delta^\nu_\sigma
-{2\over{\Box^2}}\,\Bigl((1-{\beta\over2}\Bigr)\,
\partial_\sigma\partial^\nu\Bigr]\,D^{\mu\sigma}\,.
\label{3.6}\end{aligned}$$
By using the explicit form of $D_{\mu\nu}$ given by (\[3.3\]) and the Jacobi identity satisfied by $F_{\mu\nu}$, we observe that the term proportional to $(1-{\beta\over2})$ in (\[3.6\]) vanishes identically. The final result reads
$$\begin{aligned}
\bar S_B&=&-\,{m^2\over4}\int d^4x\,F_{\mu\nu}\,
{1\over\Box}\,F^{\mu\nu}\,,
\nonumber\\
&=&-\,{m^2\over2}\int d^4x\,\Bigl(A_\mu\,A^\mu
+{\partial_\mu A^\mu\over\Box}\Bigr)\,
\partial_\nu A^\nu\,.
\label{3.7}\end{aligned}$$
At last we have that the complete effective action can be written as
$$\bar S=\int d^4x\,\Bigl[{1\over2}\,A_\mu\,
\bigl(\Box - m^2\bigr)\,A^\mu
-\,{1\over2}\,\partial_\mu\,A^\mu\,
\Bigl((1-{1\over\alpha} -{m^2\over\Box}\Bigr)\,
\partial_\nu\,A^\nu\Bigr] + S_{ghost}\,,
\label{3.8}$$
where $S_{ghost}$ represents the ghost dependent part of the effective action which appears in the vacuum functional (\[2.26\]).
The operator which acts on (the quadratical part of) $A^\mu$ in (\[3.7\]), written in momentum space, reads
$$G_{\mu\nu}=-\,\Bigl[\bigl(k^2+m^2)\,\eta_{\mu\nu}
-\,\Bigl(1-{1\over\alpha}+{m^2\over k^2}\Bigr)\,
k_\mu k_\nu\Bigr]\,.
\label{3.9}$$
Its inverse gives the propagator for the vectorial field in momentum space. It is just given by
$$K_{\mu\nu}=-\frac{1}{k^2+m^2}\,
\Bigl[\eta_{\mu\nu}+\Bigl(\frac{\alpha-1}{k^2}+
\frac{m^2}{k^4}\Bigr)\,k_\mu k_\nu\Bigr]\,,
\label{3.10}$$
which presents a pole in $k_0=\vec k^2+m^2$, showing that at quantum level the elimination of the tensorial fields is also traduced by the introduction of a mass term for the vectorial field. Observe however that here it becomes clear that is the vectorial field itself that effectively acquires a mass.
Conclusion
==========
With the aid of the Batalin and Vilkoviski procedure, we have constructed the vacuum functional for a field theoretical model consisting of vector and tensor gauge fields interacting through a topological term. The reducibility of the tensorial sector was taken properly into account. After a covariant gauge-fixing was implemented, we were able to integrate over the tensorial fields, obtaining a massive vectorial effective theory as output. The propagator for the vectorial bosons have been calculating, showing the expected pole at $\vec p\,^2+m^2$.
As a final comment, we observe that the introduction of non-Abelian gauge symmetries is possible for a corresponding extended model [@Bar]. Its BV formulation, as well as some phenomenological related consequences for this non-Abelian version of the present model are under study and will be presented elsewhere [@BR].
[**Acknowledgment:**]{} This work is supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq (Brazilian Research Agency).
[30]{} I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B102 (1981) 27; Phys. Rev. D28 (1983) 2567. General and nice reviews on the BV method can be found in M. Henneaux and C. Teitelboim, [*Quantization of gauge systems*]{} (Princeton University Press, Princeton, 1992); F. De Jonghe, Ph.D. Thesis, Leuven (1993); J. Gomis, J. París and S. Samuel, [*Antibracket, antifields and gauge-theory quantization*]{}, Preprint CCNY-HEP-94/03. T. J. Allen, M. J. Bowick and A. Lahiri, Mod. Phys. Lett. A6 (1991) 559 A. Lahiri, Mod. Phys. Lett. A8 (1993) 2403 See, for example, M.B. Green, J.H. Schwarz and E. Witten, [*Superstring Theory*]{} (Cambridge University Press, Cambridge, 1987) as well as references appearing in [@Lahiri]. R.K. Kaul, Phys. Rev. D18 (1978) 1127. J. Barcelos-Neto and M.B.D. Silva, [*Geometric formulation for antisymmetric tensor gauge field theories*]{}, Preprint UFRJ-IF-FPC-002/95. R. Amorim and J. Barcelos-Neto, [*Massive vector gauge theories*]{}, Preprint UFRJ-IF-FPC-006/95. R. Amorim and J. Barcelos-Neto, work in progress.
[^1]: Electronic mails: ift01001 @ ufrj and amorim @ ifsu1.ufrj.br
[^2]: Electronic mails: ift03001 @ ufrj and barcelos @ vms1.nce.ufrj.br
|
---
abstract: 'We present an [*ab initio*]{} approach for evaluating a free energy profile along a reaction coordinate by combining logarithmic mean force dynamics (LogMFD) and first-principles molecular dynamics. The mean force, which is the derivative of the free energy with respect to the reaction coordinate, is estimated using density functional theory (DFT) in the present approach, which is expected to provide an accurate free energy profile along the reaction coordinate. We apply this new method, first-principles LogMFD (FP-LogMFD), to a glycine dipeptide molecule and reconstruct one- and two-dimensional free energy profiles in the framework of DFT. The resultant free energy profile is compared with that obtained by the thermodynamic integration method and by the previous LogMFD calculation using an empirical force-field, showing that FP-LogMFD is a promising method to calculate free energy without empirical force-fields.'
author:
- Makoto Nakamura
- Masao Obata
- Tetsuya Morishita
- Tatsuki Oda
title: 'An ab initio approach to free-energy reconstruction using logarithmic mean force dynamics'
---
\[sec:introduction\][INTRODUCTION]{}
====================================
Free energy is a significant physical property for estimating thermodynamic stability. It is desirable to estimate free energy as accurately as possible. Such free energy estimation is becoming important in a variety of research fields; in particular, biological molecules including proteins or interfaces of nano-scale materials have been raised as a target for free energy calculations. It is thus desirable to develop methods that improve the accuracy and efficiency in free energy calculations using molecular simulation. The free energy in molecular systems has often been evaluated for a given constraint (reaction pass).[@Ciccotti2004] Such a constraint is usually specified by using a set of reaction coordinates,[@Sprik1998] for example, distances between molecules, bond angles, and dihedral angles, etc.
In order to get free energy landscapes, various techniques \[thermodynamic integration (TI)[@Kirkwood1935], free energy perturbation,[@Zwanzig1954] umbrella sampling,[@Torrie1977] and so on\] have been developed so far. Although TI is derived from the statistical mechanics faithfully, some difficulties have been pointed out; poor sampling which could come from a breakdown of the ergodicity and numerical integration as postprocessing.
To overcome these difficulties, free energy calculation methods based on mean force dynamics (MFD) have been proposed.[@Rosso2002; @Laio2002] In MFD, a set of $L$ reaction coordinates (collective variables), ${\bf X} \equiv \{X_{1}, X_{2}, \cdots, X_{L} \}$, is regarded as a set of fictitious dynamical variables, and their trajectories are designed to be generated by hypothetical dynamical equations. Morishita [*et al.*]{}[@Morishita2012; @Morishita2013] have recently introduced a logarithmic form of the free energy along ${\bf X}$ \[$F({\bf X})$\] to enable us to easily sample rare events in MFD calculations. This method is called logarithmic mean force dynamics (LogMFD), in which the free energy can be estimated on-the-fly.
The evaluation of mean force (MF), i.e., slope of $F({\bf X})$ with respect to ${\bf X}$, in such a method based on MFD can be improved by incorporating first-principles (FP) molecular dynamics (MD),[@Car1985] replacing the classical MD using empirical force fields. FPMD allows us to include effects of the electronic state explicitly; for example, bond-formation or bond-breaking, which may considerably influence the free energy profiles in molecular systems.
In this paper, we have developed first-principles MFD in the framework of LogMFD, namely, first-principles LogMFD (FP-LogMFD). We reconstruct the free energy landscape for a molecular system of glycine dipeptide using FP-LogMFD. This demonstration indicates that FPMD can be incorporated into LogMFD of multi-dimensional ${\bf X}$-systems and that the scheme developed here is found to be promising for the free energy reconstruction using [*ab initio*]{} techniques. The successful combination of LogMFD and FPMD is indebted to the efficiency for sampling rare events in LogMFD. The logarithmic form introduced in LogMFD suppresses the effective energy barriers for the dynamical variables ${\bf X}$. This makes it possible to sample configurations with higher energy, as frequently as those with much lower energy. This feature also makes it possible to improve the accuracy of the MF by increasing the number of statistical samples (FPMD steps).
In the next section, we review the LogMFD method briefly and demonstrate how to incorporate FPMD into LogMFD. In Sec. III, we present the free energy profile with respect to the dihedral angles in glycine dipeptide molecule. In Sec. IV, we will discuss the entropic contribution and numerical accuracy in the present results by comparing it with the result previously obtained using an empirical force field. Finally, Sec. V summarizes this paper.
\[sec:theory\]Methods
=====================
\[subsec:theory\]Equations of mean force dynamics
-------------------------------------------------
We present a brief review for LogMFD. This review would be a good introduction to our new scheme which employs a non-empirical approach. We consider a system of $N$ atoms with a given temperature $T_{\rm ext}$, and aim to reconstruct the free energy profile $F({\bf X})$ with respect to ${\bf X}$. Each reaction coordinate $X_{p}(\{ {\bf R}_{I} \})$ is generally a function of the atomic coordinates $\{ {\bf R}_{I}\}$, where $p$ and $I$ specify the $p$’th reaction coordinate and the $I$’th atom, respectively. In MFD, however, ${\bf X}$ are regarded as dynamical variables, being independent of $\{ {\bf R}_{I}\}$. We now consider the following postulated Hamiltonian for ${\bf X}$;
$$\begin{aligned}
H_{\rm MFD} & = &
\sum_{p}^{L}\frac{1}{2}M_{p} \dot{X}_{p}^{2} + F({\bf X}),
\label{HMFD}\end{aligned}$$
where the first and second terms on the right-hand-side are the kinetic and potential energies, respectively, for $X_{p}$ ($\dot{X}_{p}$ means the velocity $d X_{p}/dt$) and $M_{p}$ is the fictitious mass for $X_{p}$. The equation of motion for ${X_{p}}$ is thus obtained as,
$$\begin{aligned}
M_{p} \ddot{X}_{p} &=& -
\frac{\partial F({\bf X})}{\partial X_{p}},
\label{eqmf}\end{aligned}$$
where $-{\partial F({\bf X})/\partial X_{p}}$ is the MF. The solution for this equation of motion fulfills the conservation law, i.e., $H_{\rm MFD}$ can be seen as a constant of motion, as long as the MF is accurately evaluated.
Several methods based on MFD have been proposed thus far, which provide us free energy profiles with respect to reaction coordinates and allow us to discuss many kinds of physics involving the reaction coordinates. Metadynamics[@Laio2002] has been introduced utilizing the concept of MFD, and has been applied to a variety of systems including biosystems to sample rare events and to reconstruct free energy profiles. Morishita [*et al.*]{}[@Morishita2012; @Morishita2013] have proposed LogMFD in which $F({\bf X})$ in Eq. (\[HMFD\]) is replaced with a logarithmic form of $F({\bf X})$, and have demonstrated several improvements in the free energy calculation.
In LogMFD, the following Hamiltonian is introduced instead of Eq. (\[HMFD\]); $$\begin{aligned}
H_{\rm LogMFD} & = & \sum_{p}^{L} \frac{1}{2} M_{p}\dot{X}_{p}^{2} +
\gamma{\mathrm {log}} \{ \alpha F({\bf X})+ 1 \},
\label{HMFD1}\end{aligned}$$ where $\gamma$ and $\alpha$ are positive constant parameters, which are chosen to effectively reduce the energy barriers experienced by ${\bf X}$. The resultant equation of motion for $X_{p}$ is given as, $$\begin{aligned}
M_{p} \ddot{X}_{p} &=& - \left(\frac{\alpha \gamma}{\alpha F({\bf X})+1}\right)
\frac{\partial F({\bf X})}{\partial X_{p}}.
\label{eqmfdeta0}\end{aligned}$$ In practice, ${\bf X}$ can be thermostatted in LogMFD calculations, and the equation of motion is slightly modified as follows; $$\begin{aligned}
M_{p} \ddot{X}_{p} &=& - \left(\frac{\alpha \gamma}{\alpha F({\bf X})+1}\right)
\frac{\partial F({\bf X})}{\partial X_{p}}
- M_{p} \dot{X}_{p} \dot{\eta},
\label{eqmfd} \\
Q_{\eta} \ddot{\eta} &=& \sum_{p} M_{p} \dot{X}_{p}^{2} -
L k_{\rm B} T_{X},
\label{eqmfdeta}\end{aligned}$$ where $\eta$ is the thermostat variable which controls the temperature of ${\bf X}\ $, $Q_{\eta}$ is the mass for $\eta$, and $k_{\rm B}$ is Boltzmann’s constant. With a single Nosé-Hoover thermostat [@Nose1984; @Hoover1985] as in Eqs. (\[eqmfd\]) and (\[eqmfdeta\]), the following pseudo Hamiltonian is a constant of motion instead of $H_{\rm LogMFD}$;[@Morishita2012; @Morishita2013] $$\begin{aligned}
\hat{H}_{\rm LogMFD} & = & \sum_{p}^{L} \frac{1}{2} M_{p}\dot{X}_{p}^{2} +
\gamma{\mathrm {log}} \{ \alpha F({\bf X})+ 1 \}
\nonumber \\
& & +\frac{1}{2}Q_{\eta} \dot{\eta}^{2}+ L k_{\rm B}T_{X} \eta.
\label{HMFD2}\end{aligned}$$ Note that $T_{X}$ is not necessarily the same as the temperature for atoms, $T_{\rm ext}$. The heights of the energy barriers on $\gamma \log \{ \alpha F({\bf X})+1 \}$ are much lower than those on $F({\bf X})$. This reduction of the barrier height enables the coordinate $X_{p}$ to easily cross the barriers at a moderate temperature of $T_{X}$, allowing us to evaluate the free energy associated with rare events.
$\partial F({\bf X})/\partial X_p$ is obtained as an ensemble average and, practically, can be estimated as a time-averaged quantity from a thermostatted MD or Monte Carlo (MC) simulation at a given temperature $T_{\rm ext}$ with a given potential $\Phi$ for the $N$-atom system and a set of fixed reaction coordinates ${\bf X}$; $$\begin{aligned}
\frac{\partial F({\bf X})}{\partial X_{p}}
& = &
\frac{1}{Z}\int d{\bf R}
\left[
\frac{\partial \Phi({\bf R}) }{\partial X_{p}}
\right]_{\bf X}
e^{-\Phi({\bf R})/k_{\rm B}T_{\rm ext}}
\nonumber \\
& \simeq &
\frac{1}{\tau} \int^{\tau}_{0} dt
\left[
\frac{\partial \Phi({\bf R}(t)) }{\partial X_{p}}
\right]_{\bf X},
\label{canonical}\end{aligned}$$ where $$\begin{aligned}
Z & = & \int d{\bf R} e^{-\Phi({\bf R})/k_{\rm B} T_{\rm ext}}.
\label{pf}\end{aligned}$$ Here, $\tau$ is the simulation time period, and the $[\ \ \ ]_{\bf X}$ represents the ensemble average under the set of constraints. In the MF estimation, it is expected that the canonical MD or MC simulation provides the canonical distribution under the constraint on ${\bf X}$. The MF is, in our approach, evaluated using thermostatted FPMD. The potential energy $\Phi({\bf R})$ and the details of the MF evaluation will be discussed later on.
We need to know $F({\bf X})$ to calculate the force on $X_{p}$ in Eq. (\[eqmfd\]), however, $F({\bf X})$ itself is the quantity we want to obtain. This problem can be solved using the conserved quantity, $H_{\rm LogMFD}$ or $\hat{H}_{\rm LogMFD}$ \[Eq. (\[HMFD1\]) or (\[HMFD2\])\]. Using this conservation law, $F({\bf X})$ can be directly evaluated with $\hat{H}_{\rm LogMFD}$ (when we employ a single Nosé-Hoover thermostat) whose value needs to be set at the beginning of the LogMFD run;[@Morishita2012; @Morishita2013] $$\begin{aligned}
F({\bf X}) & = & \frac{1}{\alpha}
\left[ {\rm exp}
\left\{ \frac{1}{\gamma}
\left(\hat{H}_{\rm LogMFD} - \sum_{p}^{L}
\frac{1}{2}M_{p} \dot{X}_{p}^{2}
\right.
\right.
\right.
\nonumber \\
& & \left.
\left.
\left.
- \frac{1}{2}
\textcolor{black}{
Q_{\eta}
}
\dot {\eta}^{2} - L k_{\rm B}T_{X} \eta
\right)
\right\}
- 1
\right] .
\label{freeenergy}\end{aligned}$$ It is required that $F({\bf X}) > 0$ at any ${\bf X}$ to enhance the sampling in the ${\bf X}$ subspace. This requirement can be actually fulfilled by using appropriate values for $\hat{H}_{\rm LogMFD}$; $\hat{H}_{\rm LogMFD}$ should be larger than the sum of the initial kinetic energy for ${\bf X}$ and the initial terms for $\eta$. See Ref. for details. Equation (\[freeenergy\]) indicates that the $F({\bf X})$ is successively obtained along the dynamics of $\{X_{p}\}$, namely, “on-the-fly". which overcomes some drawbacks of the TI method. decompose the ${\bf X}$ subspace into many bins with a finite width, implying a possible missing of remarkable characters in the free energy due to a discretized mesh. In contrast, LogMFD provides $F({\bf X})$ with much higher resolution than TI, since LogMFD generates almost continuous [**X**]{}-trajectories and the $F({\bf X})$ trajectories (this will be illustrated in Fig. \[every\]).
The flow chart of the LogMFD method is displayed in Fig. \[chart\]. To update $\{ X_{p} \}$, the most important quantity is the MF, which is evaluated using thermostatted FPMD in our approach. Details of our FPMD approach are presented in the next subsection; Car-Parrinello molecular dynamics (CP-FPMD)[@Car1985] with double Nosé-Hoover thermostats.[@Blochl1992; @Morishita1999]
![\[chart\] Flow chart of first-principles LogMFD.](fig1.eps){width="5cm"}
\[subsec:meanforce\]First-principles mean force
-----------------------------------------------
In MFD methods, the MF needs to be estimated as accurately as possible at the temperature $T_{\rm ext}$. The MF from first-principles could improve the accuracy of the free energy profiles. We now address two technical issues associated with the evaluation of the MF in our approach. Firstly, the constraint on ${\bf X}$ during the FPMD run is discussed. In order to impose a constraint on atomic coordinates, one may employ the SHAKE method in which a holonomic constraint is realized,[@SHAKE] although complex equations should be solved for the Lagrange multiplier. Alternatively, the harmonic potential method can also be utilized, allowing us to use Eq. (\[canonical\]) without any correction terms. In this work, we chose the latter method. to keep a given temperature for the system and to generate the canonical distribution for the atomic trajectories, we employ thermostats.
Newly developed thermostats,[@Morishita2010] as well as the original thermostat,[@Nose1984; @Hoover1985] can also be used in conjunction with FPMD in which the Born-Oppenheimer(BO) surface is strictly searched in the time evolution[@Payne1992] or CP-FPMD.[@Car1985]
In our FPMD approach, the following energy is considered: $$\begin{aligned}
E_{\rm tot} & = & E_{\rm BP} + E_{\rm hp},
\label{etot} \\
E_{\rm hp} & = & \sum_{p} \frac{1}{2} k_{p} (\tilde{X}_{p}( \{ {\bf R}_{I} \} ) - X_{p})^{2},
\label{hp} \\
E_{\rm BP} & = & \sum_{i} m_{\varphi}
\langle
\dot{\varphi}_{i} | \dot{\varphi}_{i}
\rangle
+ \frac{1}{2} \sum_{I} M_{I} \dot{{\bf R}}_{I}^{2}
\nonumber \\
& & + \frac{1}{2} Q_{\rm e}\dot{x}_{\rm e}^{2} + 2E_{\rm kin}^{0} x_{\rm e}
\nonumber \\
& & + \frac{1}{2} Q_{R}\dot{x}_{R}^{2} + g k_{\rm B}T x_{R}
\nonumber \\
& & + E_{\rm fp}[\{ \varphi_{i} \},\{ {\bf R}_{I} \}],
\label{bp}\end{aligned}$$ where $E_{\rm BP}$ and $E_{\rm hp}$ are the energy in the Blöchl-Parrinello(BP) method[@Blochl1992] and the harmonic potentials for the constraint, respectively, and $E_{\rm fp}$ represents the potential energy in the system of electrons and ions (see Eq. (5) in Ref. ). $x_{R}$ ($x_{\rm e}$) is the dynamical variable for the thermostat and $Q_{R}$ ($Q_{\rm e}$) is the corresponding mass for $x_{R}$ ($x_{\rm e}$). $g$ is the number of ionic degrees of freedom. The quantity $\tilde{X}_{p}( \{ {\bf R}_{I} \} )$ in Eq. (\[hp\]) is constructed from the current atomic coordinates, which is tightly constrained to $X_{p}$ according to $E_{\rm hp}$ \[Eq. (12)\]. To this end, the constant $k_{p}$ is chosen to be a large value. The atomic forces come from the contributions of $E_{\rm fp}$, the thermostat, and the constraint $E_{\rm hp}$. These contributions result in the following equation of motion for ${\bf R}_{I}$; $$\begin{aligned}
M_{I}\ddot{\bf R}_{I} &=& {\bf F}_{I}^{\rm fp} - M_{I}\dot{\bf R}_{I}\dot{x}_{R}
\nonumber \\ & & - \sum_{p} k_{p} (\tilde{X}_{p}( \{ {\bf R}_{I} \} )-X_{p})
\frac{\partial \tilde{X}_{p}( \{ {\bf R}_{I} \} )}
{\partial {\bf R}_{I}}.
\label{cp4}\end{aligned}$$ The equations of motion for the wavefunction ($\varphi_{i}$) and the heat baths ($x_{\rm e}$ and $x_{R}$) are not changed from the original BP method by introducing the constraint, implying a less effort for converting a conventional computational code to the present one.
According to Eq. (\[canonical\]), the first-principles mean force is obtained as a time average of $-\partial E_{\rm tot}/\partial X_{p}$: $$\begin{aligned}
-\frac{\partial F({\bf X})}{\partial X_{p}} & \sim & k_{p}
\langle
\tilde{X}_{p}( \{ {\bf R}_{I} \} )-X_{p}
\rangle ,
\label{mf}\end{aligned}$$ where $\langle \ \ \ \rangle$ represents a canonical ensemble or a time average. This formula is general as far as the atomic configuration samples the canonical distribution under the required constraint. This implies that the relation of Eq. (\[mf\]) is also useful in the TI method, as explained in Appendix A. In order to show an achievement of the constraint and the temperature control, we present typical time evolution of the reaction coordinate $X_{p}$ in the next section.
In CP-FPMD, the fictitious kinetic energy of the wave function, namely, the first term in Eq. (\[bp\]), should follow the dynamics of $\{ {\bf R}_{I} \}$ as quickly as possible.[@Blochl1992] To this end, it is important to find an appropriate $E_{\rm kin}^{0}$. For a given atomic configuration, (i) we start a CP-FPMD run with the system exactly on the BO surface. (We converge the electronic state to the BO surface beforehand.) Then (ii) we perform the CP-FPMD run for a few tens of MD steps without the heat baths (may be better, without the constraint). During this period, the system slightly leaves the exact BO surface. (iii) If the temperature of the system reaches $T_{\rm st}$ within the given period, then we set the value of the kinetic energy of the wave functions at the moment as $E_{\rm kin}^{0}$. Due to a practical reason for stabilizing the simulation, $T_{\rm st}$ is taken as $\sim 0.85 T_{\rm ext}$. (iv) When the system does not reach an appropriate temperature, we introduce a new atomic configuration by distorting the previous atomic configuration. We restart the process from (i). After setting $E_{\rm kin}^{0}$, we switch on the thermostats in the BP method accompanied with the constraint of Eq. (\[hp\]). The rest of the FPMD steps are used for the MF evaluation of Eq. (\[mf\]). In the computation mentioned above, the initial atomic configurations for each of the series of the CP-FPMD runs were taken from the atomic configuration in the CP-FPMD runs previously done. We will detail the procedure to perform FP-LogMFD calculations, including parameter settings, in the next section.
![\[glycine\] Atomic structure of the glycine dipeptide molecule with atomic specification. Two dihedral angles, $\phi$ and $\psi$, are formed by the atomic series C(2)-N(1)-C(3)-C(4) and N(1)-C(3)-C(4)-N(2), respectively. The figure displays the atomic configuration with .](fig2.eps){width="6cm"}
\[sec:demonstration\]NUMERICAL DEMONSTRATION
============================================
\[subsec:molecule\]Molecular configuration
------------------------------------------
To illustrate our [*ab initio*]{} approach to free energy reconstruction, we consider the free energy profile of glycine dipeptide molecule \[2-(Acetylamino)-N-methylacetamide\] in vacuum, as shown in Fig. \[glycine\]. The atoms are specified by the symbol with numbering of C(1), C(2), $\cdots$ , O(1), O(2), .. etc.. from the left-hand-side of the figure. The two dihedral angles are labeled as $\phi$ and $\psi$, which are formed by the atomic series C(2)-N(1)-C(3)-C(4) and N(1)-C(3)-C(4)-N(2), respectively. In other words, these angles are formed by the plane of N(1)-C(3)-C(4) and the plane associated with the peptide bond (-OCNH-). In nature, the latter plane in proteins has usually observed as the [*trans*]{}-form rather than as the [*cis*]{}-form.[@text-book] Actually, in our calculation, the [*cis*]{}-form of the right-hand-side of the peptide bonds in Fig. \[glycine\] is higher in energy by 2.1 kcal/mol than the form presented in Fig. \[glycine\].
In this section, we demonstrate the application of FP-LogMFD to the glycine dipeptide molecule. We have obtained the free energy landscape $F(\phi, \psi)$ with respect to the dihedral angles $\phi$ and $\psi$ at room temperature 300 K ($=T_{\rm ext}$). First, FP-LogMFD with the fixed dihedral angle $\phi$ was performed to set the parameters required and to obtain the one-dimensional free energy profile along $\psi$. Then, the FP-LogMFD runs in the $\phi$-$\psi$ space were performed, revealing the details of the two-dimensional free energy landscape. We have also performed TI calculation to reconstruct the one-dimensional free energy profile, which is compared to the FP-LogMFD result for benchmarking.
\[subsec:parameter\]Parameter setting
-------------------------------------
For the CP-FPMD runs, we have used the plane wave basis set and density functional theory with the generalized gradient approximation(GGA).[@KS; @Perdew92] The energy cutoffs of 25 and 250 Ry are taken for electronic wave function and charge density, respectively.[@Pasquarello1992] The ultrasoft pseudopotentials are used.[@Vanderbilt1990] The $\Gamma$-point sampling is adopted for the molecular system placed in a cubic box with the dimension of 20 a.u.(10.58 Å). For the canonical FPMD simulation in the framework of the BP method, the time step is set to 10 a.u. ($\sim$ 0.24 fs). This is a typical value for the CP method.[@Car1985] The parameters for $Q_{R}$, $Q_{\rm e}$, and $m_{\varphi}$ are set to $5 \times 10^{5}$ a.u., $5 \times 10^{3}$ a.u., and 200 a.u., respectively. $E_{\rm kin}^{0}$ in Eq. (\[bp\]) is automatically determined by the anzatz described before (see Sec. \[subsec:meanforce\]). Figure \[kineticenergy\] presents the time evolution of the kinetic energy of the wave functions and the instantaneous temperature of the molecular system (proportional to the kinetic energy of atoms). In the present calculations, $E_{\rm kin}^{0}$ was set to be 0.0049 a.u.
![\[kineticenergy\] Time evolution of the instantaneous temperature (green full curve, left scale) and the kinetic energy of the wave functions (red dashed curve, right scale) in the FPMD run at $T_{\rm ext}=300$ K with the constraints The horizontal line indicates the values of $T_{\rm ext}$ and $E_{\rm kin}^{0}$. The arrow indicates the time step at which the constraint on the dihedral angles and the temperature control is turned on.](fig3.eps){width="7.5cm"}
In this demonstration, the following harmonic potentials are employed to constraint $\tilde{\phi}$ and $\tilde{\psi}$; $$\begin{aligned}
E_{\rm hp} & = & \frac{1}{2} k_{\phi}
(\tilde{\phi}( \{ {\bf R}_{I} \} ) - \phi)^{2}
+\frac{1}{2} k_{\psi}
(\tilde{\psi}( \{ {\bf R}_{I} \} ) - \psi)^{2},
\label{harmonicpotential}\end{aligned}$$ where $\phi$ and $\psi$ are the target dihedral angles \[$X_{p}$ in Eq. (\[hp\])\] and $\tilde{\phi}$ and $\tilde{\psi}$ are the temporal ones determined from the instantaneous molecular configuration \[$\tilde{X}_{p}$ in Eq. (\[hp\])\]. Both of $k_{\phi}$ and $k_{\psi}$ are taken to be 2.4 a.u./${\rm rad}^{2}$ (). Figure \[dihed\] shows typical time evolution of $\tilde{\phi}$ and $\tilde{\psi}$ with . This figure indicates that the temporal $\tilde{\phi}$ and $\tilde{\psi}$ fluctuate around the respective given value, implying the constraint to be imposed correctly.
![\[dihed\] Time evolution of the instantaneous dihedral angles, (a) $\tilde{\phi}$ and (b) $\tilde{\psi}$, in the FPMD run with the constraints, (horizontal lines).](fig4-1.eps "fig:"){width="7.5cm"} ![\[dihed\] Time evolution of the instantaneous dihedral angles, (a) $\tilde{\phi}$ and (b) $\tilde{\psi}$, in the FPMD run with the constraints, (horizontal lines).](fig4-2.eps "fig:"){width="7.5cm"}
![\[meanforce\] Time evolution of the instantaneous force, (a) $k_{\phi}(\tilde{\phi}-\phi)$ and (b) $k_{\psi}(\tilde{\psi}-\psi)$, accompanied with the horizontal lines which indicates the mean forces, $-\partial F/\partial \phi$ and $-\partial F/\partial \psi$.](fig5-1.eps "fig:"){width="7.5cm"} ![\[meanforce\] Time evolution of the instantaneous force, (a) $k_{\phi}(\tilde{\phi}-\phi)$ and (b) $k_{\psi}(\tilde{\psi}-\psi)$, accompanied with the horizontal lines which indicates the mean forces, $-\partial F/\partial \phi$ and $-\partial F/\partial \psi$.](fig5-2.eps "fig:"){width="7.5cm"}
The time evolution of the temporal force is presented in Fig. \[meanforce\]. Averaging over 500 steps (from the 21th step to 520th step), the mean forces acting on $\phi$ and $\psi$ were estimated to be , respectively (the fluctuations are limited to ). The accuracy of the MF strongly depends on the number of MD steps, defined as $N_{\rm BP}$. In fact, we found that the decrease of $N_{\rm BP}$ (from 500 steps to 300 steps) deteriorated the MF, and the resultant free energy profile became much worse, compared to those by $N_{\rm BP}=500$. $N_{\rm BP}$ was thus set to 500 steps in the present study.
Evaluation of the MF is also needed in the TI method. The MF at each of the grid points in the TI calculation was obtained by averaging the instantaneous forces from a set of many FPMD runs with Eqs. (\[etot\]) and (\[hp\]). The successive simulation started with random atomic distortions from the previous atomic configuration. In the present study, we have performed 120 FPMD runs, each consisting of 600 FPMD steps, i.e., 72,000 FPMD steps in total for each grid point of the reaction coordinate. 60,000 FPMD steps out of the 72,000 steps were devoted to estimation of the MF at a single grid point.
For a set of given coordinates $(\phi, \psi)$, as shown in Fig. \[chart\], $-\partial F/\partial \phi$ and $-\partial F/\partial \psi$ were estimated for the hypothetical dynamics given by Eqs. (\[eqmfd\]) and (\[eqmfdeta\]) with $T_{X}=300$ K in FP-LogMFD. A single Nosé-Hoover thermostat[@Nose1984; @Morishita2010; @Hoover1985] was used. In the FP-LogMFD runs, the variables of $\phi$, $\psi$, and $\eta$ were updated using a time step of 1 $\tau$, with the masses of and $Q_{\eta}=L k_{\rm B}T_{\rm X} \tau_{\eta}^{2}$ with $\tau_{\eta}=50$ $\tau$, where $\tau$ represents the time unit. \[the time unit can, in fact, be arbitrarily chosen, e.g., $\tau$=1 fs, since the dynamics of $\phi$ has nothing to do with the resultant $F(\phi)$.\] The parameters of $\alpha$ and $\gamma$, which determines the degree of the effective reduction of the free energy barriers, were taken as $\alpha=3$ (kcal/mol)$^{-1}$ and $\gamma=1/\alpha$, with this value of $\gamma$ corresponding to 170 K. After solving Eqs. (\[eqmfd\]) and (\[eqmfdeta\]), the conversion to $F(\phi, \psi)$ was performed using Eq. (\[freeenergy\]) with $\hat{H}_{\rm LogMFD}=1$ kcal/mol. $\hat{H}_{\rm LogMFD}$ should be set to ensure $\alpha F_{\rm min}(\phi, \psi) + 1 > 0$, where $F_{\rm min}$ is the minimum of the free energy. Note however that there is, in principle, no upper limit for the value of $\hat{H}_{\rm LogMFD}$.[@Morishita2013]
The validity of the LogMFD results mainly depends on the accuracy of the MF, which influences the conservation of $\hat{H}_{\rm LogMFD}$ \[Eq. (\[HMFD2\])\]. As was already mentioned, the quality of the MF can be controlled by $N_{\rm BP}$ and the mass parameter $M_{\phi(\psi)}$.[@Morishita2013] The increase of $M_{\phi(\psi)}$, which reduces (suppresses) the velocity of the dynamical variables, results in a more accurate profile for the MF, and thus, the free energy profile. We found, by decreasing the $M_{\phi(\psi)}$ by the factor ten, that the difference between the LogMFD and TI results becomes 0.22 kcal/mol from 0.18 kcal/mol on average. In the present system, the periodicity with respect to $\phi$ and $\psi$ can be available for checking the accuracy of simulations.
\[subsec:1d\]One dimensional profile
------------------------------------
![\[every\] (a) MF with respect to the dihedral angle $\psi$ in the glycine dipeptide molecule, constraining the other dihedral angle $\phi$ to . The blue curve and red symbols indicate the MF calculated from the LogMFD and TI calculations, respectively. (b) The magnified profile of the MF (vibrational curve) around , with a smooth curve showing the profile obtained by averaging over ten MFD time steps. The arrow indicates the width of the mesh used in the TI calculation, showing that the MF around this $\psi$ range is approximated by only a single grid-point result. ](fig6-1.eps "fig:"){width="7.5cm"} ![\[every\] (a) MF with respect to the dihedral angle $\psi$ in the glycine dipeptide molecule, constraining the other dihedral angle $\phi$ to . The blue curve and red symbols indicate the MF calculated from the LogMFD and TI calculations, respectively. (b) The magnified profile of the MF (vibrational curve) around , with a smooth curve showing the profile obtained by averaging over ten MFD time steps. The arrow indicates the width of the mesh used in the TI calculation, showing that the MF around this $\psi$ range is approximated by only a single grid-point result. ](fig6-2.eps "fig:"){width="7.5cm"}
![\[logmfd-ti\] Free energy profiles with respect to the dihedral angle $\psi$ in the glycine dipeptide molecule, constraining the other dihedral angle $\phi$ to , obtained from the LogMFD (blue curve) and TI (red dots) calculations. The logarithmic energy ($\gamma \log (\alpha F(\psi)+1$) ) (magenta curve) is also presented for comparison, indicating a substantial reduction of the free energy barrier.](fig7.eps){width="7.5cm"}
For demonstrating the free energy evaluation using FP-LogMFD, we performed FP-LogMFD simulations for the dynamical variable $\psi$ while keeping $\phi$ to be . In Fig. \[every\](a), the MF profiles from the LogMFD and TI calculations are presented, showing the LogMFD result is in good agreement with the TI result. Figure \[every\](a) also shows that there are regions where the MF drastically varies in a narrow range, e.g., . In Fig. \[every\](b), the magnified profile in the range of indicates that, although the data by LogMFD shows a vibrational behavior, the MF averaged over 10 MFD steps varies smoothly. This behavior of the MF in LogMFD is remarkable when the profile exhibits a rapid variation. As shown in Fig. \[every\](b), a set of uniformly sparse grid points is only used in the TI method due to a limited computational resources. LogMFD thus can provide missing data in between each of the grid points in the TI calculations without much additional computational cost.
Figure \[logmfd-ti\] shows the free energy profiles obtained by the LogMFD and TI methods. Each of the free energy profiles is shifted to have the same value (5 kcal/mol) at for comparison in Fig. \[logmfd-ti\]. LogMFD runs were initiated at (around the minimum) to either direction (with increasing or decreasing $\psi$) with $T_{\rm X}=300$ K and were ended at after passing through the periodic boundary at . It should be remarked that the value of estimated when was sampled for the first time is almost the same as the estimated when was sampled the second time, indicating the energy dissipation, which degrades the accuracy of $F(\psi)$, is negligible.
There is the maximum at , and the minimum at in the profile, as shown in Fig. \[logmfd-ti\]. We stress here that the dynamics for the reaction coordinate $\psi$ was very smooth, even the large energy barrier exists. The difference between the minimum and maximum free energy approximately amounts to 8 kcal/mol, corresponding to about 4024 K. This energy difference was entirely suppressed by the logarithmic form. Figure \[logmfd-ti\] also shows the effective potential curve of $\gamma \log ( \alpha F(\psi) + 1 )$, indicating that the actual energy barrier for $\psi$ became $\sim$ 0.8 kcal/mol, comparable to 402 K. Such a substantial reduction of the energy barrier can be controlled by the parameters ($\alpha$ and $\gamma$).
![\[distance\] Atomic distances for O(1)-H(7), O(1)-O(2), and O(1)-N(2) as a function of $\psi$.](fig8.eps){width="7.5cm"}
Before proceeding to the two dimensional landscape, we discuss the one-dimensional free energy profile in more detail. As pointed out, there are the minimum and maximum in the profile. The former and latter are related to a hydrogen bond and a rendezvous of a pair of the oxygen atoms in the peptide bonds, respectively. Other characteristic properties are found around , where the free energy shows . We consider that this is due to breaking of the hydrogen bond which is formed around . This consideration is supported by the fact that the MF in the corresponding part almost zero (see Fig. \[every\]). Atomic distances as a function of $\psi$ are shown in Fig. \[distance\]. From this figure, the free energy minimum in Fig. \[logmfd-ti\] is found to appear around the minimum distance of O(1)-H(7) and O(1)-N(2), while the energy maximum appears around the minimum distance of O(1)-O(2). The latter case may correspond to a large electric dipole state for the molecule. The distance of 2.5 $\sim$ 3 Å for O(1)-H(7) at is out of the range of the hydrogen bonding, where the MF is $\sim$ 0. From this, we consider that an energy of about 3 kcal/mol is gained by the hydrogen bond (see Fig. \[logmfd-ti\]). This energy is comparable to a typical bonding energy of the hydrogen bond (3 $\sim$ 10 kcal/mol) reported in a literature.[@text-book]
\[subsec:2d\]Two dimensional profile
------------------------------------
![\[freemap2\] (a) Free energy contour map $F(\phi,\psi)$ and (b)–(e) typical atomic configurations at three stable states (C5 and C7 atomic configurations) and an unstable state. The white bullets indicate the positions for the stable states. A pass way that approximately connects these stable states with a straight line is displayed in a white dashed line (see also Fig. \[feC5C7\]). ](fig9.eps){width="8.0cm"}
![\[feC5C7\] The one dimensional free energy profile along the line that approximately connects the C5 and C7 states. The full and dotted curves represents the bare and symmetrized plot of $F(\phi,\psi)$.](fig10.eps){width="7.5cm"}
For constructing the two dimensional free energy profile, the dynamical equations for both of $\phi$ and $\psi$ were used. Supposing that the free energy minimum in the $\phi$-$\psi$ space may be lower than that in the one-dimensional $\psi$ space at , $\hat{H}_{\rm LogMFD}$ was increased to 1.2 kcal/mol to shift the baseline of the free energy landscape. The temperature $T_{X}$ and $\alpha$ were chosen to be ; $T_{X}=200$ K and $\alpha$ =2 (kcal/mol)$^{-1}$ , while the other parameters took the same values as used in the one dimensional FP-LogMFD calculations. The two-dimensional FP-LogMFD runs were started from the minimum of the one dimensional profile of $F(\psi)$ and were extended to four directions. The branch off can be performed from one simulation to others. These simulations can be performed independently, implying that parallel treatment is highly effective in LogMFD.[@Morishita2013]
Figure \[freemap2\] shows the free energy contour map in the $\phi$-$\psi$ plane with typical molecular configurations. The glycine dipeptide molecule has an intrinsic mirror symmetry in its atomic geometry. Atomic structures which are related to each other by the mirror operation with respect to the N(1)-C(3)-C(4) plane has the same energy in gas phase. This feature should also be seen in the free energy landscape $F(\phi, \psi)$. Therefore, the statistical errors can be reduced by symmetrizing the two-dimensional free energy with respect to the point of (there is the inversion symmetry in the map). A non-symmetrized free energy profile along the white dashed line in Fig. \[freemap2\] is presented in the last paragraph in this subsection.
The free energy landscape (Fig. \[freemap2\]) shows that there are three stable states (three energy valleys) and a series of unstable states (energy mountains). The most stable state appears around , whose atomic configuration is presented in Fig. \[glycine\] (or Fig. \[freemap2\](d)). This is assigned to the C5 configuration[@Cheam1989] and is stabilized by the hydrogen bond, the five-membered ring, and the configuration with separated oxygen atoms (almost zero electric dipole). The other two stable states, found around , are assigned to the C7 configuration and are also stabilized with the hydrogen bond, the seven-membered ring, and the configuration with moderately separated oxygen atoms (small electric dipole). The free energy for the C7 configuration is higher by 0.58 kcal/mol than that for the C5 configuration. This energy difference is quite small and comparable to 290 K. The total (internal) energy computation also indicates that the C5 configuration is either lower in energy than the C7 by 0.38 kcal/mol, while the work by the quantum chemistry calculation reports that the C5 is The atomic configuration of the most unstable state is presented in Fig. \[freemap2\](b). This instability comes from an assemble of oxygen atoms in the molecule (implying a large electric dipole). The energy barrier measured from the bottom of the free energy landscape (highest energy mountain) amounts to 26 kcal/mol, corresponding to 13000 K and to 730 K with $ \gamma {\rm log} (\alpha F+1)$. Again, LogMFD enables to sample such higher energy configuration in the same footing used around the ground state.
It is interesting to see the transition from the most stable state to another stable state. The one dimensional free energy profile roughly linking the C7 and C5 configurations is presented in Fig. \[feC5C7\], as a typical energy profile. For simplicity, the pass way of reaction coordinate was assumed to be along the straight line which connects the two states near the C5 and C7 states in the $\phi$-$\psi$ plane, as specified in Fig. \[freemap2\]. From Fig. \[feC5C7\], the energy barrier between C5 and C7 configurations is estimated to be about 3 kcal/mol when measured from the C5 configuration. The energy differences between the C5 and C7 states shown in Fig. \[feC5C7\] are about 1 kcal/mol. These values are slightly larger than the value reported above (0.58 kcal/mol) because of the approximate pass way (this approximation causes the uncertainty of about 0.4 kcal/mol).
\[subsec:efficiency\]Computational efficiency
---------------------------------------------
In constructing the free energy profile (Fig. \[logmfd-ti\]), 4 $\times$ 10$^{6}$ FPMD steps were devoted in the FP-LogMFD calculation, while 7.2 $\times$ 10$^{6}$ FPMD steps were needed in the TI calculation. About 45 % of the computational cost was saved. This demonstrates a good efficiency of LogMFD in the computational cost. In addition, in the course of the construction of the two dimensional profile (Fig. \[freemap2\]), we carried out a set of LogMFD runs which, in total, sampled 1.2 $\times$ 10$^{8}$ FPMD steps (configurations). Even though the accuracy in the two dimensional profile may be slightly reduced, the computational cost is only 30 times larger than that in the one dimensional calculation.
\[sec:results\]DISCUSSIONS
==========================
![\[dft\] Total (internal) energy for the glycine dipeptide molecule as a function of $\psi$ keeping , which was obtained from the density functional theory (DFT) calculation with (purple triangle symbols) or without (green circle symbols) the van der Waals(vdW) correction. The blue asterisks denote the difference between the free energy obtained by LogMFD and that by TI, while the red squares denote the difference between the free energy by LogMFD and the internal energy without the vdW correction. ](fig11.eps){width="7.5cm"}
As mentioned in Sec. \[subsec:molecule\], the peptide bond takes the [*trans*]{}- or [*cis*]{}-form. In our simulations, the [*trans*]{}-form has been entirely observed and the statistical sampling of the [*cis*]{}-form has been missed. This is because the barrier between the [*trans*]{}- and [*cis*]{}-forms may be extremely high, and also because the reaction coordinates chosen in the present LogMFD calculations may not be suitable for sampling the [*cis*]{}-form. If one needs to sample the [*cis*]{}-form, incorporation of additional reaction coordinates is of use, which is easily realized in LogMFD.
It is interesting to see the contribution of the entropy in the free energy. We have calculated the total energy (the internal energy) as a function of $\psi$ keeping , as shown in Fig. \[dft\]. The grid points used in the calculation of the internal energy are the same as used in the TI calculation. The internal energy profile is very similar to the free energy profile (Fig. \[logmfd-ti\]). The difference between the free energy and the internal energy is found to be within 0.5 kcal/mol (if the energy scale is adjusted to give zero entropy at ), implying a small contribution from the entropy. We roughly estimated the uncertainty of the free energy as $\sim$ 0.4 kcal/mol, which is comparable to the variation of the entropy with $\psi$. It is thus considered that the entropic contribution is hardly changed with $\psi$. This is not surprising because the number of possible conformations in the present system is relatively small, which does not significantly depend on the dihedral angles. Also, the glycine dipeptide molecule is in vacuum, not in a solvent. We however stress that LogMFD is able to unveil the variation of the entropy, if any, which is, for example, seen in our preliminary calculations for a model system of protein-G consisting of 56 amino acids.[@Isobe2001]
The free energy profile for the glycine dipeptide molecule was previously obtained using an empirical force field.[@Morishita2012; @Morishita2013] The profile is similar to that obtained using FP-LogMFD in this work, indicating the validity of the empirical force-field to some extent. There are, however, some differences in the profile. As pointed out in Sec. \[subsec:1d\], we observe the around . This behavior is also seen in the internal energy profiles (Fig. \[dft\]) in the FP-LogMFD approach. In fact, the explicit inclusion of the van der Waals interaction[@Dion2004; @Cooper2010; @Obata2013] into calculations does not change the overall profile of the internal energy . It is thus considered that the is not attributed to an inappropriate DFT description, while the linear behavior observed around in the previous results may come from insufficient transferability of the empirical force field.
\[sec:summary\]SUMMARY
======================
We have demonstrated that the [*ab initio*]{} based MF can be incorporated into the LogMFD method, which improves the reliability and accuracy in the free energy calculation. FP-LogMFD has been applied to reconstruction of the free energy landscapes of the glycine dipeptide molecule, and the C5 and C7 conformations have been identified as the ground and metastable conformations, respectively. It has been confirmed that the substantial reduction of the free energy barriers, thanks to the logarithmic form, enables us to efficiently reconstruct the free energy profile, which was found to agree well with that obtained by the TI method. The free energy profile from the first-principles approach indicates that the empirical force field for the glycine dipeptide molecule is sufficient to obtain the overall profile of the free energy landscape.
The LogMFD method allows us not only to easily sample rare events, but also to reconstruct the free energy profile “[*on-the-fly*]{}" without suffering from the problems such as how to arrange the grid points or how to perform the numerical integration (as postprocessing) in TI. It has been demonstrated in the present study that free energy profiles using [*ab initio*]{} force field can be reconstructed with less computational cost than is needed in the TI method. The FP-LogMFD method developed here is thus a promising tool for reconstructing free energy profiles, especially those in which accurate descriptions for interatomic interactions are required.
The computation in this work was done using the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo and the facilities of the Research Center for Computational Science, National Institutes of Natural Sciences, Okazaki, Japan. This work was partly supported by Grant-in-Aid for Scientific Research from JSPS/MEXT (Grant Nos. 22104012, 22340106, 23510120 and 24740297) and the Computational Materials Science Initiative (CMSI), Japan.
\[sec:appendix\]Thermodynamic integration
=========================================
In order to check the result of the LogMFD calculations, thermodynamic integration (TI) was also performed for comparison using the same computational conditions for the first-principles MD calculations (see Sec. \[subsec:parameter\]). When one carries out a long-time CP-FPMD simulation, the energy tends to flow to the electronic degrees of freedom from the ionic degrees of freedom. Consequently, the lift from the BO surface of the electronic wave functions becomes obvious, and finally, the simulation may break down.[@Pastore1991] However, one can, instead, perform multiple short FPMD runs and the mean force profile in the TI calculation can be constructed by averaging over the configurations from all of these short runs at a given set of $\phi$ and $\psi$. Figure \[aveti\] represents the convergence behavior of the MF with several fixed $\psi$, as a function of the number of statistical samplings. we decided to use 60,000 FPMD steps in total to estimate the MF at each grid point of $\psi$ in our TI calculation.
![\[aveti\] Cumulative averages of the mean force in the TI calculation for several dihedral angles ( ) under the condition .](fig12.eps){width="7.5cm"}
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---
abstract: 'We compute the $h_1$-localized cohomology of the motivic Steenrod algebra over ${\ensuremath{\mathbb{C}}}$. This serves as the input to an Adams spectral sequence that computes the motivic stable homotopy groups of the $\eta$-local motivic sphere. We compute some of the Adams differentials, and we state a conjecture about the remaining differentials.'
address:
- |
Department of Mathematics\
University of Kentucky\
Lexington, KY 40506, USA
- |
Department of Mathematics\
Wayne State University\
Detroit, MI 48202, USA
author:
- 'Bertrand J. Guillou'
- 'Daniel C. Isaksen'
title: 'The $\eta$-local motivic sphere'
---
[^1]
\[section\] \[thm\][Proposition]{} \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Conjecture]{} \[thm\][Remark]{} \[thm\][Example]{} \[thm\][Definition]{} \[thm\][Notation]{}
Introduction
============
Consider the Hopf map $\eta:{\ensuremath{\mathbb{A}}}^2\setminus\{0\} {\longrightarrow}{\ensuremath{\mathbb{P}}}^1$ that takes $(x,y)$ to $[x:y]$. In motivic homotopy theory, ${\ensuremath{\mathbb{A}}}^2\setminus\{0\}$ and ${\ensuremath{\mathbb{P}}}^1$ are models for the motivic spheres $S^{3,2}$ and $S^{2,1}$ respectively. Therefore, $\eta$ represents an element of the stable motivic homotopy group $\pi_{1,1}$.
Computations of motivic stable homotopy groups share many similarities to the classical computations, but the motivic computations also exhibit “exotic" non-classical phenomena. One of the first examples is that $\eta$ is not nilpotent, i.e., $\eta^k$ is non-zero for all $k$ [@Morel].
Working over ${\ensuremath{\mathbb{C}}}$ (or any algebraically closed field of characteristic zero), we have an Adams spectral sequence for computing motivic stable $2$-complete homotopy groups with good convergence properties [@Morel2] [@DI] [@HKO]. The $E_2$-page of this spectral sequence is the cohomology $\operatorname{Ext}_A$ of the motivic Steenrod algebra $A$. In the Adams spectral sequence, $\eta$ is detected by the element $h_1$ of $\operatorname{Ext}_A$. The failure of $\eta$ to be nilpotent is detected by the fact that $h_1^k$ is a non-zero permanent cycle for all $k$.
A further investigation of the motivic Adams $E_2$-page reveals a number of other classes that survive $h_1$-localization, i.e., classes $x$ such that $h_1^k x$ is non-zero for all $k$. The first few are $c_0$ in the $8$-stem, $P h_1$ in the $9$-stem, $d_0$ in the $14$-stem, and $e_0$ in the $17$-stem. A fairly predictable pattern emerges, involving classes that all map non-trivially to the cohomology of motivic $A(2)$ [@IMotA(2)].
However, in the 46-stem, a surprise occurs. The class $B_1$ is $h_1$-local. This element is not detected by the cohomology of motivic $A(2)$. At this point, it has become clear that an algebraic computation of the $h_1$-localized cohomology $\operatorname{Ext}_A[h_1^{-1}]$ is an interesting and non-trivial problem.
The first goal of this article is calculate $\operatorname{Ext}_A[h_1^{-1}]$. We will show that it is a polynomial algebra over ${\ensuremath{\mathbb{F}}}_2 [h_1^{\pm 1}]$ on infinitely many generators.
\[thmA\] The $h_1$-localized algebra $\operatorname{Ext}_A[h_1^{-1}]$ is a polynomial algebra over ${\ensuremath{\mathbb{F}}}_2[h_1^{\pm 1}]$ on generators $v_1^4$ and $v_n$ for $n\geq 2$, where:
1. $v_1^4$ is in the 8-stem and has Adams filtration 4.
2. $v_n$ is in the $(2^{n+1} - 2)$-stem and has Adams filtration 1.
Although it is simple to state, this is actually a surprising answer. The elements $d_0$, $e_0$, and $e_0 g$ are indecomposable elements of $\operatorname{Ext}_A$ that are all $h_1$-local. From consideration of the May spectral sequence, or from the cohomology of $A(2)$, one might expect to have the relation $e_0^3 + d_0 \cdot e_0 g = 0$.
In the terms of Theorem \[thmA\], $e_0^3 + d_0 \cdot e_0 g$ corresponds to $h_1^9 v_3^3 + h_1^9 v_2^2 v_4$. Theorem \[thmA\] says that this expression is non-zero after $h_1$-localization, so it is non-zero before localization as well. The only possibility is that $e_0^3 + d_0 \cdot e_0 g$ equals $h_1^5 B_1$.
The relation $e_0^3 + d_0 \cdot e_0 g = h_1^5 B_1$ in $\operatorname{Ext}_A$ is hidden in the motivic May spectral sequence. As described in [@Istems], it is tightly connected to the classical Adams differential $d_3(h_1 h_5 e_0) = h_1^2 B_1$ [@BJM Corollary 3.6].
Our $h_1$-local calculation surely implies other similarly exotic relations in higher stems. We do not yet possess a sufficiently detailed understanding of $\operatorname{Ext}_A$ in that range, so we cannot identify any explicit examples with certainty. However, we expect to see a hidden relation $e_0 (e_0 g)^2 + d_0 \cdot e_0 g^3 = h_1^5 \cdot g^2 B_1$ in $\operatorname{Ext}_A$ in the 91-stem. Here, we anticipate that $e_0 g^3$ and $g^2 B_1$ are indecomposable elements.
So far, we have only discussed the entirely algebraic question of computing $\operatorname{Ext}_A[h_1^{-1}]$, which informs us about the structure of $\operatorname{Ext}_A$. But $\operatorname{Ext}_A[h_1^{-1}]$ is also the $E_2$-page of an Adams spectral sequence that converges to the $2$-complete motivic stable homotopy groups of the $\eta$-local motivic sphere $S^{0,0}[\eta^{-1}]$, which is the homotopy colimit of the sequence $$\xymatrix@1{
S^{0,0} \ar[r]^\eta &
S^{-1,-1} \ar[r]^\eta &
S^{-2,-2} \ar[r]^\eta &
\cdots.
}$$ In order to compute $\pi_{*,*} S^{0,0}[\eta^{-1}]$, we thus only need to compute Adams differentials on $\operatorname{Ext}_A[h_1^{-1}]$ . There are indeed non-trivial differentials. In the non-local case, we know that $d_2(e_0) = h_1^2 d_0$ and $d_2(e_0 g) = h_1^2 e_0^2$ [@Istems]. This implies the analogous $h_1$-local differentials $d_2(v_3) = h_1 v_2^2$ and $d_2(v_4) = h_1 v_3^2$.
Unfortunately, we have not been able to identify all of the Adams differentials. We expect the answer to turn out as stated in Conjecture \[mainConj\].
\[mainConj\] For all $k \geq 3$, there is an Adams differential $d_2 (v_k ) = h_1 v_{k-1}^2$.
Conjecture \[mainConj\] has the following immediate consequences.
\[mainConj2\]
1. The $E_\infty$-page of the $h_1$-local Adams spectral sequence is $${\ensuremath{\mathbb{F}}}_2[h_1^{\pm 1}][v_1^4,v_2]/v_2^2.$$
2. The motivic stable homotopy groups of the $\eta$-local motivic sphere are $$\pi_{*,*}(S^{0,0}[\eta^{-1}]) {\cong}{\ensuremath{\mathbb{F}}}_2[\eta^{\pm 1},\mu,\epsilon]/\epsilon^2,$$ where $\eta$ has degree $(1,1)$; $\mu$ has degree $(9,5)$; and $\epsilon$ has degree $(8,5)$.
Given the Adams differentials proposed in Conjecture \[mainConj\], we can compute that the $h_1$-local $E_3$-page is equal to ${\ensuremath{\mathbb{F}}}_2[h_1^{\pm 1}][v_1^4,v_2]/v_2^2$. For degree reasons, there are no possible higher differentials, so this expression is also equal to the $h_1$-local $E_\infty$-page.
There are no possible hidden extensions in $E_\infty$, so we obtain $\pi_{*,*}(S^{0,0}[\eta^{-1}])$ immediately.
The $h_1$-localization of the element $Ph_1$ of $\operatorname{Ext}_A$ is $h_1 v_1^4$, and $\mu$ is the standard notation for the element of $\pi_{9,5}$ detected by $P h_1$. The $h_1$-localization of the element $c_0$ of $\operatorname{Ext}_A$ is $h_1^2 v_2$, and $\epsilon$ is the standard notation for the element of $\pi_{8,5}$ detected by $c_0$.
We present one more consequence of Conjecture \[mainConj\].
\[thmANSS\] The following are equivalent:
1. Conjecture \[mainConj\] holds.
2. At $p=2$, the $\alpha_1$-localization of the classical Adams-Novikov spectral sequence $E_2$-page is a free ${\ensuremath{\mathbb{F}}}_2 [\alpha_1^{\pm 1}]$-module with basis consisting of elements of the form $\alpha_{k/b}$ for $k = 1$ and $k \geq 3$.
Here $b$ is an integer that depends on $k$. The point is that $\alpha_{k/b}$ is the generator of the classical Adams-Novikov $E_2$-page in degree $(2k-1,1)$.
Recall that $\tau$ is an element of $\pi_{0,-1}$. Because $\tau \eta^4$ is zero, we know that the $\eta$-localization $C\tau[\eta^{-1]}$ of the cofiber $C\tau$ of $\tau$ splits as $S^{0,0}[\eta^{-1}] \vee S^{1,-1}[\eta^{-1}]$. Therefore, $\pi_{*,*}(C\tau[\eta^{-1}])$ is the same as two copies of $\pi_{*,*}(S^{0,0}[\eta^{-1}])$.
We explain in [@Istems] that $\pi_{*,*} (C\tau)$ is equal to the classical Adams-Novikov $E_2$-page. Therefore, $\pi_{*,*} (C\tau[\eta^{-1}])$ is the same as the $\alpha_1$-localization of the Adams-Novikov $E_2$-page.
The origin of this work lies in the first author’s attempt to analyze the Adams spectral sequence beyond the 45-stem. The $h_1$-local calculations discussed in this article are a helpful tool in the analysis of Adams differentials. We expect that the $h_1$-local calculations will continue to be a useful tool in the further analysis of Adams differentials. For example, our work leads us to anticipate an Adams differential $d_2(e_0g^3) = h_1^2(e_0g)^2$ in the 77-stem.
In this article, we are working exclusively in motivic homotopy theory over ${\ensuremath{\mathbb{C}}}$. A natural extension is to consider $h_1$-local and $\eta$-local calculations over other fields. Preliminary calculations over ${\ensuremath{\mathbb{R}}}$ show that the picture is more complicated. We plan to explore this in more detail in future work.
The questions studied in this article become trivial in the classical situation, where $h_1^4$ is zero in the cohomology of the classical Steenrod algebra, and $\eta^4$ is zero in the classical 4-stem. This is consistent with the principle that $\tau$-localization corresponds to passage from the motivic to the classical situations [@Istems], and that $\tau h_1^4$ and $\tau \eta^4$ are both zero motivically.
However, $\eta$ is not nilpotent in ${\ensuremath{\mathbb{Z}}}/2$-equivariant stable homotopy groups. The equivariant analogues of our calculations are interesting open questions.
Our work raises the question of why Nishida’s nilpotence theorem [@Nishida] fails in motivic homotopy theory. One might wonder whether $\eta$ in $\pi_{1,1}$ can be non-nilpotent because it has “simplicial dimension" $0$. However, this cannot be the full explanation since the element $\mu$ of $\pi_{9,5}$ detected by $P h_1$ is also not nilpotent, and $\mu$ has simplicial dimension $4$. In fact, the elements $\mu_{8k+1}$ of $\pi_{8k+1,4k}$ detected by $P^k h_1$ are all not nilpotent. We expect that these are the only elements of $\pi_{*,*}$ that fail to be nilpotent.
Organization
------------
Section \[sec:background\] contains a review of the motivic Steenrod algebra and sets our notation. Section \[sec:CohomA\] computes $\operatorname{Ext}_A[h_1^{-1}]$, as stated in Theorem \[thmA\]. For completeness, we also discuss $\operatorname{Ext}_{A(2)}[h_1^{-1}]$.
Section \[sec:MaySS\] discusses the same computation from the point of view of the motivic May spectral sequence. The point of this section is that it allows us to analyze in Section \[sec:LocMap\] the localization map $\operatorname{Ext}_A{\longrightarrow}\operatorname{Ext}_A[h_1^{-1}]$ in detail through a range. This leads to some hidden relations in $\operatorname{Ext}_A$ that are needed in [@Istems]. The localization map is essential for deducing information about Adams differentials in $\operatorname{Ext}_A$ from Adams differentials in $\operatorname{Ext}_A[h_1^{-1}]$. We also consider the May spectral sequence and the localization map for $A(2)$ in Sections \[sec:MaySS\] and \[sec:LocMap\]. These sections are intended to be read in conjunction with the charts in [@GI].
Section \[sec:ASS\] gives some computations of Adams differentials in support of Conjecture \[mainConj\]. We also discuss the role of a speculative “motivic modular forms" spectrum.
Much of the data for our computations, especially regarding the May spectral sequence, is given in tables to be found in Section \[sec:Tables\].
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors thank Haynes Miller for a conversation that led to the proof of Theorem \[thmA\].
Background {#sec:background}
==========
We continue with notation from [@Istems] as follows:
1. ${\ensuremath{\mathbb{M}}}_2$ is the motivic cohomology of ${\ensuremath{\mathbb{C}}}$ with ${\ensuremath{\mathbb{F}}}_2$ coefficients.
2. $A$ is the mod 2 motivic Steenrod algebra over ${\ensuremath{\mathbb{C}}}$, and $A_{*,*}$ is its dual.
3. $A(n)$ is the ${\ensuremath{\mathbb{M}}}_2$-subalgebra of $A$ generated by ${\ensuremath{\mathrm{Sq}}}^1$, ${\ensuremath{\mathrm{Sq}}}^2$, ${\ensuremath{\mathrm{Sq}}}^4$, …, ${\ensuremath{\mathrm{Sq}}}^{2^n}$, and $A(n)_*$ is its dual.
4. $\operatorname{Ext}_A$ is the trigraded ring $\operatorname{Ext}_A({\ensuremath{\mathbb{M}}}_2,{\ensuremath{\mathbb{M}}}_2)$.
5. More generally, $\operatorname{Ext}_B$ is the trigraded ring $\operatorname{Ext}_B({\ensuremath{\mathbb{M}}}_2, {\ensuremath{\mathbb{M}}}_2)$ for any Hopf algebra $B$ over ${\ensuremath{\mathbb{M}}}_2$.
6. $A_{{\mathrm{cl}}}$ is the mod 2 classical Steenrod algebra, and $A^{{\mathrm{cl}}}_*$ is its dual.
7. $A(n)_{{\mathrm{cl}}}$ is the ${\ensuremath{\mathbb{M}}}_2$-subalgebra of $A_{{\mathrm{cl}}}$ generated by ${\ensuremath{\mathrm{Sq}}}^1$, ${\ensuremath{\mathrm{Sq}}}^2$, ${\ensuremath{\mathrm{Sq}}}^4$, …, ${\ensuremath{\mathrm{Sq}}}^{2^n}$, and $A(n)^{{\mathrm{cl}}}_*$ is its dual.
8. $\operatorname{Ext}_{A_{\mathrm{cl}}}$ is the bigraded ring $\operatorname{Ext}_{A_{{\mathrm{cl}}}}({\ensuremath{\mathbb{F}}}_2,{\ensuremath{\mathbb{F}}}_2)$.
9. More generally, $\operatorname{Ext}_B$ is the trigraded ring $\operatorname{Ext}_B({\ensuremath{\mathbb{F}}}_2, {\ensuremath{\mathbb{F}}}_2)$ for any Hopf algebra $B$ over ${\ensuremath{\mathbb{F}}}_2$.
The following two theorems of Voevodsky are the starting points of our calculations.
${\ensuremath{\mathbb{M}}}_2$ is the bigraded ring ${\ensuremath{\mathbb{F}}}_2[\tau]$, where $\tau$ has bidegree $(0,1)$.
Our main object of study will be a localization of $\operatorname{Ext}_A$. It will be more convenient for us to work with the dual $A_{*,*} = \operatorname{Hom}_{{\ensuremath{\mathbb{M}}}_2}(A,{\ensuremath{\mathbb{M}}}_2)$.
[@V2] [@V3 Theorem 12.6] The dual motivic Steenrod algebra $A_{*,*}$ is generated as an ${\ensuremath{\mathbb{M}}}_2$-algebra by $\xi_i$ and $\tau_i$, of degrees $(2(2^i-1),2^i-1)$ and $(2^{i+1}-1,2^i-1)$ respectively, subject to the relations $$\tau_i^2 = \tau \xi_{i+1}.$$ The coproduct is given on the generators by the following formulae, in which $\xi_0 = 1$: $$\Delta(\tau_k) = \tau_k\otimes 1 + \sum_i \xi_{k-i}^{2^i}\otimes \tau_i$$ $$\Delta(\xi_k) = \sum_i \xi_{k-i}^{2^i} \otimes \xi_i.$$
The quotient $A_{*,*}/\tau=A_{*,*}\otimes_{{\ensuremath{\mathbb{M}}}_2}{\ensuremath{\mathbb{F}}}_2$ is analogous to the odd-primary classical dual Steenrod algebra, in the sense that there is an infinite family of exterior generators $\tau_i$ and an infinite family of polynomial generators $\xi_i$. On the other hand, the localization $A_{*,*}[\tau^{-1}]$ is analogous to the mod 2 classical dual Steenrod algebra, which has only polynomial generators $\tau_i$.
$\operatorname{Ext}$ groups
---------------------------
We are interested in computing a localization of $\operatorname{Ext}_A$. Before localization, this is a trigraded object. In [@Istems], classes in $\operatorname{Ext}_A$ are described in degrees of the form $(s,f,w)$, where:
1. $f$ is the Adams filtration, i.e., the homological degree.
2. $s+f$ is the internal degree, corresponding to the first coordinate in the bidegrees of $A$.
3. $s$ is the stem, i.e., the internal degree minus the Adams filtration.
4. $w$ is the motivic weight.
In the cobar complex, $\xi_1$ represents an element $h_1$ of $\operatorname{Ext}_A$ in degree $(1,1,1)$. Because we will invert $h_1$, it is convenient to choose a new grading that is more $h_1$-invariant. Except where otherwise noted, we will use the grading $(t, f, c)$, where:
1. $t = s - w$ is the Milnor-Witt stem.
2. $f$ is the Adams filtration.
3. $c = s +f - 2w$ is the Chow degree [@Istems], which turns out to be a convenient grading for calculational purposes.
The terminology “Milnor-Witt stem" arises from the work of Morel [@Morel], which describes the motivic stable homotopy groups $\pi_{s,w}$ with $s-w=0$ in terms of Milnor-Witt $K$-theory.
The terminology “Chow degree" arises from the fact that the grading $s+f-2w$ is a natural index from the higher Chow group perspective on motivic cohomology [@Bloch].
The $h_1$-local cohomology of $A$ {#sec:CohomA}
=================================
The goal of this section is to compute $\operatorname{Ext}_A[h_1^{-1}]$ explicitly. We will accomplish this by expressing $A_{*,*}$ as a series of extensions of smaller Hopf algebras.
\[defn:Bi\] For each $i\geq -1$, let $B_i$ be the subalgebra of $A_{*,*}$ generated by the elements $\xi_k$ and also by $\tau_0,\dots,\tau_i$.
In particular, $B_{-1}$ is a polynomial ${\ensuremath{\mathbb{M}}}_2$-algebra on the elements $\xi_k$.
\[lem:B-1\] $\operatorname{Ext}_{B_{-1}}[h_1^{-1}]$ is isomorphic to ${\ensuremath{\mathbb{M}}}_2[h_1^{\pm 1}]$.
We have an isomorphism $B_{-1} {\longrightarrow}A_*^{\mathrm{cl}}\otimes_{{\ensuremath{\mathbb{F}}}_2}{\ensuremath{\mathbb{M}}}_2$ of Hopf algebras. Under this mapping, the element $h_1$ in $\operatorname{Ext}_{B_{-1}}$ corresponds to $h_0$ in $\operatorname{Ext}_{A_{{\mathrm{cl}}}}$. Adams’s vanishing line of slope 1 [@A] implies that $\operatorname{Ext}_{A_{{\mathrm{cl}}}}[h_0^{-1}]$ is isomorphic to ${\ensuremath{\mathbb{F}}}_2[h_0^{\pm 1}]$.
We proceed to compute $\operatorname{Ext}_{B_n}[h_1^{-1}]$ inductively via a Cartan-Eilenberg spectral sequence [@CE §XVI.6], [@R A1.3.14] for the extension of Hopf algebras $$B_{n-1}{\longrightarrow}B_n {\longrightarrow}E(\tau_n),$$ where $E(\tau_n)$ is an exterior algebra on the generator $\tau_n$. The extension is cocentral, so the spectral sequence takes the form $$E_2 {\cong}\operatorname{Ext}_{B_{n-1}}\otimes \operatorname{Ext}_{E(\tau_n)} {\cong}\operatorname{Ext}_{B_{n-1}}[v_n] \Rightarrow \operatorname{Ext}_{B_n}.$$ The $E_2$-page of this spectral sequence has four gradings: three from $\operatorname{Ext}_{B_n}$ and one additional Cartan-Eilenberg grading associated with the filtration involved in construction of the spectral sequence. However, we will suppress the Cartan-Eilenberg grading because we won’t need it for bookkeeping purposes.
The class $v_n$ has degree $(2^n-1,1,1)$ in the $E_2$-term. The differentials take the form $$d_r:E_r^{(t,f,c)} {\longrightarrow}E_r^{(t-1,f+1,c)}.$$
\[MainExtProp\]
1. $\operatorname{Ext}_{B_0}[h_1^{-1}] {\cong}{\ensuremath{\mathbb{M}}}_2[h_1^\pm,v_0]$.
2. $\operatorname{Ext}_{B_1}[h_1^{-1}] {\cong}{\ensuremath{\mathbb{F}}}_2[h_1^\pm,v_1^4]$.
3. $\operatorname{Ext}_{B_n}[h_1^{-1}] {\cong}{\ensuremath{\mathbb{F}}}_2[h_1^\pm,v_1^4,v_2,\dots,v_n]$ for all $n \geq 2$.
In each case, $v_i$ has degree $(2^i-1,1,1)$.
When $n=0$, Lemma \[lem:B-1\] says that the Cartan-Eilenberg spectral sequence takes the form $$E_2 {\cong}{\ensuremath{\mathbb{M}}}_2[h_1^{\pm 1}][v_0] \Rightarrow \operatorname{Ext}_{B_0}[h_1^{-1}].$$ Since $\tau_0$ is primitive, the spectral sequence collapses at $E_2$. We conclude that $\operatorname{Ext}_{B_0}[h_1^{-1}]{\cong}{\ensuremath{\mathbb{M}}}_2[h_1^{\pm 1},v_0]$ with $v_0$ in degree $(0,1,1)$.
Taking now $n=1$, the computation of $\operatorname{Ext}_{B_0}[h_1^{-1}]$ in the previous paragraph tells us that the spectral sequence takes the form $$E_2 {\cong}{\ensuremath{\mathbb{M}}}_2[h_1^{\pm 1},v_0,v_1] \Rightarrow \operatorname{Ext}_{B_1}[h_1^{-1}],$$ with $v_0$ in degree $(0,1,1)$ and $v_1$ in degree $(1,1,1)$. The coproduct formula $$\Delta(\tau_1) = \tau_1\otimes 1 + \xi_1\otimes \tau_0 + 1\otimes \tau_1$$ gives rise to the differential $d_2(v_1) = h_1v_0$. It follows that $E_3 {\cong}{\ensuremath{\mathbb{M}}}_2[h_1^\pm, v_1^2]$. There is next a differential $d_3(v_1^2) = \tau h_1^3$, which can be verified by the cobar complex calculation $$d( \tau_1|\tau_1 | \xi_1 + \xi_1 | \tau_0\tau_1 | \xi_1 + \tau_1\xi_1 | \tau_0 | \xi_1 + \xi_1^2 | \tau_0 | (\tau_1+\tau_0\xi_1)) = \tau \xi_1 | \xi_1 | \xi_1 | \xi_1.$$ The class $v_1^4$ in degree $(4,4,4)$ cannot support any higher differentials for degree reasons, and we have $$\operatorname{Ext}_{B_1}[h_1^{-1}] {\cong}E_\infty=E_4 {\cong}{\ensuremath{\mathbb{F}}}_2[h_1^\pm,v_1^4].$$
For $n\geq 2$, the argument is by induction, using a Cartan-Eilenberg spectral sequence at every turn. Each of these spectral sequences collapses at $E_2$ since there are no possible values for differentials on $v_n$.
\[ExtAThm\] $$\operatorname{Ext}_A[h_1^{-1}] {\cong}{\ensuremath{\mathbb{F}}}_2[h_1^\pm,v_1^4,v_2,v_3,\dots],$$ where $h_1$ has degree $(0,1,0)$; $v_1^4$ has degree $(4,4,4)$; and $v_n$ has degree $(2^n-1,1,1)$ for $n \geq 2$.
Since $A$ is $\operatorname*{colim}B_n$, $\operatorname{Ext}_A$ equals $\operatorname*{colim}\operatorname{Ext}_{B_n}$. The calculation follows from part (3) of Proposition \[MainExtProp\].
Theorem \[ExtAThm\] implies that the part of $\operatorname{Ext}_A [h_1^{-1}]$ in Chow degree zero is equal to ${\ensuremath{\mathbb{F}}}_2[h_1^{\pm 1}]$. Another more direct argument for this observation uses the isomorphism between $\operatorname{Ext}_{A_{\mathrm{cl}}}$ and the Chow degree zero part of $\operatorname{Ext}_A$ [@Istems]. This isomorphism takes classical elements of degree $(s,f)$ to motivic elements of degree $(s,f,0)$. The classical calculation $\operatorname{Ext}_{A_{\mathrm{cl}}} [h_0^{-1} ] ={\ensuremath{\mathbb{F}}}_2 [h_0^{\pm 1}]$ corresponds to the Chow degree zero part of $\operatorname{Ext}_A [h_1^{-1}]$.
The $h_1$-local cohomology of $A(2)$ {#sec:CohomA(2)}
------------------------------------
For completeness, we will also explicitly calculate $\operatorname{Ext}_{A(2)}[h_1^{-1}]$, where $A(2)$ is the ${\ensuremath{\mathbb{M}}}_2$-subalgebra of $A$ generated by ${\ensuremath{\mathrm{Sq}}}^1$, ${\ensuremath{\mathrm{Sq}}}^2$, and ${\ensuremath{\mathrm{Sq}}}^4$.
Dual to the inclusion $A(2){\hookrightarrow}A$ is a quotient map $$A_{*,*} {\longrightarrow}A(2)_{*,*} {\cong}{\ensuremath{\mathbb{M}}}_2[\xi_1,\xi_2,\tau_0,\tau_1,\tau_2]/(\xi_1^4,\xi_2^2,\tau_0^2=\tau\xi_1,\tau_1^2=\tau\xi_2,\tau_2^2)$$ We filter $A(2)_{*,*}$ by sub-Hopf algebras $$B_{-1}(2)\subseteq B_0(2) \subseteq B_1(2)\subseteq A(2)_{*,*}$$ where $B_{-1}(2)$ is the subalgebra generated by $\xi_1$ and $\xi_2$; $B_0(2)$ is generated by $\xi_1$, $\xi_2$, and $\tau_0$; and $B_1(2)$ is generated by $\xi_1$, $\xi_2$, $\tau_0$, and $\tau_1$. The notation is analogous to the notation in Definition \[defn:Bi\].
\[lem:B-1(2)\] $\operatorname{Ext}_{B_{-1}(2)}[h_1^{-1}]$ is isomorphic to ${\ensuremath{\mathbb{M}}}_2[h_1^{\pm 1},a_1]$, where $a_1$ has degree $(4,3,0)$.
We have an isomorphism $B_{-1}(2) {\longrightarrow}A(1)_*^{\mathrm{cl}}\otimes_{{\ensuremath{\mathbb{F}}}_2} {\ensuremath{\mathbb{M}}}_2$. Under this mapping, the element $h_1$ in $\operatorname{Ext}_{B_{-1}(2)}$ corresponds to $h_0$ in $\operatorname{Ext}_{A(1)_{{\mathrm{cl}}}}$. It is well-known that $\operatorname{Ext}_{A(1)_{{\mathrm{cl}}}}[h_0^{-1}]$ is isomorphic to ${\ensuremath{\mathbb{F}}}_2 [ h_0^{\pm 1}, a^{{\mathrm{cl}}} ]$, where $a^{{\mathrm{cl}}}$ has degree $(4,3)$ (for example, see [@R Theorem 3.1.25]).
\[CohomBi(2)\]
1. $\operatorname{Ext}_{B_0(2)}[h_1^{-1}]{\cong}{\ensuremath{\mathbb{M}}}_2[h_1^{\pm},a_1,v_0]$.
2. $\operatorname{Ext}_{B_1(2)}[h_1^{-1}] {\cong}{\ensuremath{\mathbb{F}}}_2[h_1^\pm,a_1,v_1^4]$.
3. $\operatorname{Ext}_{A(2)}[h_1^{-1}] {\cong}{\ensuremath{\mathbb{F}}}_2[h_1^\pm,a_1,v_1^4,v_2]$.
In each case, $a_1$ has degree $(4,3,0)$; $v_0$ has degree $(0,1,1)$; $v_1^4$ has degree $(4,4,4)$; and $v_2$ has degree $(3,1,1)$.
The proof uses a series of Cartan-Eilenberg spectral sequences as in the proof of Proposition \[MainExtProp\], given Lemma \[lem:B-1(2)\] as the starting point.
The classes $a_1$, $v_1^4$, and $h_1^2 v_2$ correspond respectively to the classes $u$, $P$, and $c$ in [@IMotA(2) Theorem 4.13].
Using the structure of $\operatorname{Ext}_{A(2)_{{\mathrm{cl}}}}[h_0^{-1}]$, similar arguments show that $$\operatorname{Ext}_{A(3)}[h_1^{-1}] {\cong}{\ensuremath{\mathbb{F}}}_2[h_1^{\pm 1},g,b_{31},v_1^4,v_2,v_3],$$ where $g$ has degree $(8,4,0)$ and $b_{31}$ has degree $(12,2,0)$. Using the structure of $\operatorname{Ext}_{A(3)_{{\mathrm{cl}}}}[h_0^{-1}]$, $$\operatorname{Ext}_{A(4)}[h_1^{-1}] {\cong}{\ensuremath{\mathbb{F}}}_2[h_1^{\pm 1},g^2,\Delta_1,b_{41},v_1^4,v_2,v_3,v_4],$$ where $g^2$ has degree $(16,8,0)$, $\Delta_1$ has degree $(24,4,0)$, and $b_{41}$ has degree $(28,2,0)$.
The $h_1$-local motivic May spectral sequence {#sec:MaySS}
=============================================
Although Theorem \[ExtAThm\] gives a complete description of $\operatorname{Ext}_A[h_1^{-1}]$, it unfortunately tells us very little about the localization map $\operatorname{Ext}_A \rightarrow \operatorname{Ext}_A[h_1{^{-1}}]$. The problem is that the proof of Theorem \[ExtAThm\] is incompatible with the motivic May spectral sequence approach to $\operatorname{Ext}_A$, as carried out in [@Istems].
A detailed understanding of the localization map allows for the transfer of information from the well-understood $\operatorname{Ext}_A[h_1^{-1}]$ to the much more complicated $\operatorname{Ext}_A$. In this section we carry out the computation of the $h_1$-localized motivic May spectral sequence. This will allow us to obtain information about the localization map in Section \[sec:LocMap\].
We recall the details of the motivic May spectral sequence from [@DI]. This spectral sequence has four gradings: three from $\operatorname{Ext}_A$ and one additional May grading associated with the filtration involved in construction of the spectral sequence. We will grade this spectral sequence in the form $(m-f, t, f, c)$, where $m$ is the May grading, $f$ is the Adams filtration, $t = s-w$ is the Milnor-Witt stem, and $c = s+f-2w$ is the Chow degree.
The $E_1$-page is a polynomial algebra over ${\ensuremath{\mathbb{M}}}_2$ on generators $h_{ij}$ for $i\geq 1$, $j\geq 0$, where
1. $h_{i0}$ has degree $(i-1, 2^{i-1}-1, 1, 1)$.
2. $h_{ij}$ has degree $(i-1, 2^{j-1}(2^i-1) - 1, 1, 0)$ for $j>0$.
Note in particular that $h_{i0}$ has Chow degree $1$, while $h_{ij}$ has Chow degree $0$ for $j > 0$. In a sense, this wrinkle in the gradings is the primary source of “exotic" motivic phenomena that do not appear in the classical situation.
The $d_1$-differential is given by the classical formula $$d_1(h_{ij}) = \sum_{0<k<i} h_{kj}h_{i-k,j+k}.$$
The $h_1$-local $E_1$-term {#sec:AMayE1}
--------------------------
Consider the $h_1$-localization $E_1[h_1^{-1}]$ of the May $E_1$-page. In order to simplify the calculation, we introduce the following notation.
In $E_1[h_1^{-1}]$, define
1. $h_{n0}'$ to be $h_{n0} + h_1^{-1} h_{20}h_{n-1,1}$.
2. $h_{n-1,2}'$ to be $h_{n-1,2}+h_1^{-1}\sum_{k=2}^{n-1} h_{k,1}h_{n-k,k}$.
We may replace the algebra generators $h_{n0}$ and $h_{n-1,2}$ by $h_{n0}'$ and $h_{n-1,2}'$ to obtain another set of algebra generators for $E_1[h_1^{-1}]$ that will turn out to be calculationally convenient.
\[defn:F1-G1\]
1. Let $F_1$ be the ${\ensuremath{\mathbb{M}}}_2[h_1^{\pm 1}]$-subalgebra of $E_1[h_1^{-1}]$ on polynomial generators $h_0$, $h_{20}$, $h_{n1}$ for all $n \geq 2$, and $h'_{n2}$ for all $n \geq 1$.
2. Let $G_1$ be the ${\ensuremath{\mathbb{F}}}_2$-subalgebra of $E_1[h_1^{-1}]$ on polynomial generators $h_{n0}'$ for $n \geq 3$ and $h_{ij}$ for $i \geq 1$ and $j \geq 3$.
Note that $F_1$ is a differential graded subalgebra of $E_1[h_1^{-1}]$ since $d_1(h_{20})=h_1h_0$ and $d_1(h_{n1})= h_1h_{n-1,2}'$. The generators of $F_1$ are indicated in the figure below as the elements that are outside of the shaded region. Note also that $G_1$ is a differential graded subalgebra of $E_1[h_1^{-1}]$ because $d_1(h_{n,0}') = \sum_{j=3}^{n-1} h_{n-j,j}h_{j,0}'$ if $n\geq 3$. The generators of $G_1$ are indicated in the figure below as the elements in the shaded regions.
\[prop:E1-split\] $E_1[h_1{^{-1}}]$ splits as a tensor product $F_1 \otimes_{{\ensuremath{\mathbb{F}}}_2} G_1$.
This follows immediately from the definitions, using that $h_{n0}'$ and $h_{n2}'$ can be used as algebra generators in place of $h_{n0}$ and $h_{n2}$.
(0,0)(10,7) (1.8,4)(4.6,0.2)(5.9,0.2)(2.7,4.4) (5.5,6.5)(9.2,6.5)(9.1,0.6)(5.02,5.63) (1,6)[$h_0$]{} (2.5,6)[$h_1$]{} (4,6)[$h_2'=h_2$]{} (5.5,6)[$h_3$]{} (7,6)[$h_4$]{} (8.5,6)[$h_5$]{} (1.7,5)[$h_{20}$]{} (1.7,5.1)(1,5.8) (3.2,5)[$h_{21}$]{} (3.2,5.1)(4,5.8) (4.7,5)[$h_{22}'$]{} (6.2,5)[$h_{23}$]{} (7.7,5)[$h_{24}$]{} (2.5,4)[$h_{30}'$]{} (4,4)[$h_{31}$]{} (4,4.1)(4.7,4.8) (5.5,4)[$h_{32}'$]{} (7,4)[$h_{33}$]{} (8.5,4)[$h_{34}$]{} (3.2,3)[$h_{40}'$]{} (4.7,3)[$h_{41}$]{} (4.7,3.1)(5.5,3.8) (6.2,3)[$h_{42}'$]{} (7.7,3)[$h_{43}$]{} (4,2)[$h_{50}'$]{} (5.5,2)[$h_{51}$]{} (5.5,2.1)(6.2,2.8) (7,2)[$h_{52}'$]{} (8.5,2)[$h_{53}$]{} (4.7,1)[$h_{60}'$]{} (6.2,1)[$h_{61}$]{} (6.2,1.1)(7,1.8) (7.7,1)[$h_{62}'$]{} (1.8,4)(4.6,0.2) (2.7,4.4)(5.9,0.2) (2.25,4.2)[0.48]{}[24]{}[205]{} (5.5,6.5)(9.2,6.5) (5.02,5.63)(9.1,0.6) (5.5,5.93)[0.57]{}[90]{}[215]{}
We will now show that the perhaps obscurely defined subalgebra $G_1$ is isomorphic to the familiar classical May $E_1$-page.
\[prop:cl-G1\] Let $E_1^{{\mathrm{cl}}}$ be the $E_1$-page of the classical May spectral sequence. Consider the algebra map $S:E_1^{{\mathrm{cl}}}{\longrightarrow}G_1$ determined by
1. $S(h_{n0}) = h_{n+2,0}'$ for all $n \geq 1$.
2. $S(h_{nk}) = h_{n,k+2}$ for all $n \geq 1$ and $k \geq 1$.
The map $S$ is an isomorphism of differential graded algebras.
We need to check that $S$ preserves the May $d_1$ differential. This is a straightforward computation.
The $h_1$-local $E_2$-term
--------------------------
We now have a good understanding of $E_1[h_1^{-1}]$ from Propositions \[prop:E1-split\] and \[prop:cl-G1\]. Next we analyze the $h_1$-localization $E_2[h_1^{-1}]$ of the motivic May $E_2$-page. Since localization is exact, $E_2[h_1^{-1}]$ is isomorphic to the cohomology of the differential graded algebra $E_1[h_1^{-1}]$.
$E_2[h_1^{-1}]$ is isomorphic to $${\ensuremath{\mathbb{M}}}_2[h_1^{\pm}][b_{20},b_{21},b_{31},b_{41},\dots] \otimes_{{\ensuremath{\mathbb{F}}}_2} G_2,$$ where $G_2$ is the cohomology of the differential graded algebra $G_1$ from Definition \[defn:F1-G1\].
This follows from the splitting given in Proposition \[prop:E1-split\]. The calculation of the cohomology of the subalgebra $F_1$ from Definition \[defn:F1-G1\] is straightforward.
Because of Proposition \[prop:cl-G1\], we know that $G_2$ is isomorphic to the classical May $E_2$-page. May’s original calculation [@May] of the classical $E_2$-term in stems below $156$ gives us complete understanding of $G_2$ in a much larger range because the map $S$ from Proposition \[prop:cl-G1\] approximately quadruples degrees.
Generators and relations for $E_2[h_1^{-1}]$ up to the Milnor-Witt 66-stem can be found in Tables \[E2GenTable\] and \[E2RelTable\], where we use the following notation.
For an element $x$ in the classical May spectral sequence, let ${\mathbf{x}}$ be the element $S(x)$ of the $h_1$-localized motivic May spectral sequence from Proposition \[prop:cl-G1\].
According to this notation, the classical element $c_0 = h_1h_0(1)$ may be written as $c_0 = h_1^2{\mathbf{h_0}}$, so that the elements $c_0$ and ${\mathbf{h_0}}$ are practically interchangeable. As the primary goal of our computation is to relate the answer to $\operatorname{Ext}_A$, we will most often choose to work with $c_0$. However, we will opt instead to use ${\mathbf{h_0}}$ when it illuminates the structure of the $h_1$-localized motivic May spectral sequence, especially in Section \[subsctn:May-diff\].
See also Table \[E4NotationTable\] for additional notation used on later pages. The names of many classes in the May spectral sequence have been chosen to agree with the notation of [@Istems], and we denote the remaining new classes by $y_n$.
The $h_1$-local May differentials {#subsctn:May-diff}
---------------------------------
We now understand $E_2[h_1^{-1}]$ in a very large range of dimensions. The next step is to compute the higher differentials to obtain $E_\infty[h_1^{-1}]$.
\[prop:May-d2\] Table \[E2GenTable\] gives the values of the May $d_2$ differential on the multiplicative generators of $E_2[h_1^{-1}]$ through the Milnor-Witt 66-stem.
As discussed in [@DI §5], the $d_2$ differential of the motivic May spectral is easy to determine from the classical $d_2$ differential; the formulas are the same, except that powers of $\tau$ must sometimes be inserted to balance the weights. This, combined with the fact that $h_1$-localization kills the classes $h_0$ and $h_2$, leads to the values in Table \[E2GenTable\].
The values of the May $d_2$ differential given in Table \[E2GenTable\] allow us to compute $E_4[h_1^{-1}]$ directly. A chart of $E_4[h_1^{-1}]$ through the Milnor-Witt 66-stem is given in [@GI]. Now we proceed to the higher May differentials and the higher $h_1$-localized pages of the motivic May spectral sequence.
For $r \geq 4$, some values for the May $d_r$ differentials are given in Tables \[d4MayTable\]–\[higherMayTable\]. The May $d_r$ differentials are zero on all other multiplicative generators of $E_r[h_1^{-1}]$ through the Milnor-Witt 66-stem.
Most of the differentials are forced by the known structure of $\operatorname{Ext}_A[h_1^{-1}]$ given in Theorem \[ExtAThm\]. For example, Theorem \[ExtAThm\] implies that $h_1^k v_3 v_4$ are the only non-zero elements in the Milnor-Witt 22-stem with Chow degree 2. We know that $h_1^k e_0 \cdot e_0 g$ survive the May spectral sequence to detect these elements. However, the element $\phi$ in the May $E_4$-page is also in the Milnor-Witt 22-stem with Chow degree 2. Therefore, it cannot survive the May spectral sequence. The only possibility is that $d_6(\phi)$ equals $h_1 c_0^2 h_5$.
There are a handful of more difficult cases, which are handled individually in the following lemmas.
\[lem:d4\]
1. $d_4({\mathbf{P}}) = {\mathbf{h_0^4h_3}}$.
2. $d_4({\mathbf{\Delta}}) = {\mathbf{h_4P}}$.
For the first formula, we use Nakamura’s squaring operations [@Nak] in the May spectral sequence to compute that $$\begin{split}
d_4({\mathbf{P}}) &= d_4 ({\ensuremath{\mathrm{Sq}}}_0({\mathbf{b_{20}}})) = {\ensuremath{\mathrm{Sq}}}_1 d_2({\mathbf{b_{20}}}) = {\ensuremath{\mathrm{Sq}}}_1({\mathbf{h_0^2h_2}}) \\
&= {\ensuremath{\mathrm{Sq}}}_1({\mathbf{h_0^2}}){\ensuremath{\mathrm{Sq}}}_0({\mathbf{h_2}}) + {\ensuremath{\mathrm{Sq}}}_0({\mathbf{h_0^2}}){\ensuremath{\mathrm{Sq}}}_1({\mathbf{h_2}}) = 0 + {\mathbf{h_0^4}}{\mathbf{h_3}}.
\end{split}$$ The proof for the second formula is similar: $$\begin{split}
d_4({\mathbf{\Delta}}) &= d_4 ({\ensuremath{\mathrm{Sq}}}_0({\mathbf{b_{30}}})) = {\ensuremath{\mathrm{Sq}}}_1 d_2({\mathbf{b_{30}}}) = {\ensuremath{\mathrm{Sq}}}_1({\mathbf{h_3b_{20}}}) \\
&= {\ensuremath{\mathrm{Sq}}}_1({\mathbf{h_3}}){\ensuremath{\mathrm{Sq}}}_0({\mathbf{b_{20}}}) + {\ensuremath{\mathrm{Sq}}}_0({\mathbf{h_3}}){\ensuremath{\mathrm{Sq}}}_1({\mathbf{b_{20}}}) = {\mathbf{h_4}}{\mathbf{P}}+ 0.
\end{split}$$
\[lem:d6\] $d_6(c_0g\phi\Delta_1) = h_1^7 D_3' \Delta_1$.
The class $D_3'\Delta_1$ cannot survive by Theorem \[ExtAThm\]. There are no classes for it to hit, and the only other differential that could possibly hit $h_1^7 D_3'\Delta_1$ is $d_{10}(h_1^{-8} c_0g^4\phi)$. But $d_{10}(g^4\phi) = h_1^{13}gy_{45}$, so $c_0g^4\phi$ is a $d_{10}$-cycle.
\[lem:d8\]
1. $d_8(\Delta_1{\mathbf{P}}) = h_1^{-6} c_0^2 e_0^2h_6$.
2. $d_8({\mathbf{P^2}})= {\mathbf{h_0^8h_4}}$.
Start with the relation $c_0^2 \cdot\Delta_1 \phi = h_1^{2} e_0 \cdot \Delta_1 B_1$. We will show in Lemma \[lem:d10\] that $\Delta_1 B_1$ is a permanent cycle. Therefore, $c_0^2 \Delta_1 \phi$ is a permanent cycle. However, $\Delta_1 \phi$ cannot survive by Theorem \[ExtAThm\], so we must have $d_{10}(\Delta_1\phi) = h_1^3 e_0^2 h_6$. Since $d_{10}(c_0^2 \Delta_1 \phi)$ must be zero, it follows that $c_0^2 e_0^2 h_6 $ must be zero in $E_{10}[h_1^{-1}]$. The only possibility is that $d_8 (\Delta_1 {\mathbf{P}} ) = h_1^{-6} c_0^2 e_0^2 h_6$. This establishes the first formula.
For the second formula, we use Nakamura’s squaring operations [@Nak] as in the proof of Lemma \[lem:d4\] to compute $$\begin{split}
d_8({\mathbf{P^2}}) &=d_8({\ensuremath{\mathrm{Sq}}}_0({\mathbf{P}})) = {\ensuremath{\mathrm{Sq}}}_1 d_4({\mathbf{P}}) =
{\ensuremath{\mathrm{Sq}}}_1( {\mathbf{h_0^4h_3}}) \\
&= {\mathbf{h_0^8h_4}} + {\ensuremath{\mathrm{Sq}}}_1({\mathbf{h_0^4}}){\mathbf{h_3^2}} =
{\mathbf{h_0^8h_4}}.
\end{split}$$
\[lem:d10\] $d_{10}(\Delta_1 B_1) = 0$.
We know that $d_{16}(e_0g^4) = h_1^{16} e_0h_6$, yet $c_0^2\cdot e_0g^4 = h_1^{2} e_0^2 \cdot e_0g^3$ is a permanent cycle. So $c_0^2e_0 h_6$ must be zero in $E_{16}[h_1^{-1}]$. The only possibilities are that $d_{10}(\Delta_1B_1) = h_1 c_0^2e_0h_6$ or $d_{14}(g^3B_1) = h_1^{9} c_0^2e_0 h_6$.
In the notation of Theorem \[ExtAThm\], $c_0$, $e_0$, $e_0 g$ and $e_0 g^3$ correspond to $h_1^2v_2$, $h_1^3 v_3$, $h_1^7 v_4$, and $h_1^{15} v_5$ respectively. Since these elements are algebraically independent, we conclude that the relation $ h_1^2e_0 (e_0 g)^2 + c_0^2 \cdot e_0 g^3 = 0$ in $E_\infty[h_1^{-1}]$ must be resolved by $$h_1^2 e_0 (e_0 g)^2 + c_0^2 \cdot e_0 g^3 = h_1^7 g^2 B_1$$ in $\operatorname{Ext}_A[h_1^{-1}]$.
Similarly, the relation $e_0^2 \cdot e_0g^3 + (e_0g)^3 = 0$ in $E_\infty[h_1^{-1}]$ must be resolved in $\operatorname{Ext}_A[h_1^{-1}]$ by $$e_0^2 \cdot e_0g^3 + (e_0g)^3 = x,$$ where $x$ is either $h_1^{13} \Delta_1 B_1$ or $h_1^5 g^3 B_1$. Multiply the first hidden relation by $e_0^2$, multiply the second hidden relation by $c_0^2$, and add to obtain $$h_1^2 e_0^3 (e_0 g)^2 + c_0^2 (e_0 g)^3 = h_1^7 e_0^2 \cdot g^2 B_1 + c_0^2 x.$$
Again using Theorem \[ExtAThm\], the left side of this last relation is not zero, so the right side is also not zero. This implies that $x$ cannot be $h_1^5 g^3 B_1$, since $h_1^7 e_0^2 \cdot g^2 B_1$ equals $h_1^5 c_0^2\cdot g^3 B_1$ in the May spectral sequence with no possible hidden extension.
Therefore, $x$ must be $h_1^{13} \Delta_1 B_1$, and $\Delta_1 B_1$ must survive the May spectral sequence.
$E_\infty[h_1^{-1}]$ and $\operatorname{Ext}_A[h_1^{-1}]$
---------------------------------------------------------
The May differentials given in Section \[subsctn:May-diff\] allow us to compute $E_\infty[h_1^{-1}]$ explicitly through the Milnor-Witt 66-stem. See [@GI] for a chart of this calculation.
The final step is to pass from $E_\infty[h_1^{-1}]$ to $\operatorname{Ext}_A[h_1^{-1}]$ by resolving hidden extensions.
\[prop:HiddenRels\] Table \[ExtRelTable\] lists some relations in $\operatorname{Ext}_A$ that are hidden in $E_\infty[h_1^{-1}]$. Through the Milnor-Witt 66-stem, all other hidden relations are multiplicative consequences of these relations.
Arguments for the relations involving $g^2B_1$ and $\Delta_1 B_1$ were given already in the proof of Lemma \[lem:d10\]. The other relations in Table \[ExtRelTable\] are established similarly.
Finally, we have calculated $\operatorname{Ext}_A[h_1^{-1}]$ through the Milnor-Witt 66-stem with the May spectral sequence and obtained the same answer as in Theorem \[ExtAThm\]. Multiplicative generators for $\operatorname{Ext}_A[h_1^{-1}]$ through the Milnor-Witt 66-stem are given in Table \[ExtGenTable\].
The $h_1$-local May spectral sequence for $A(2)$ {#sec:A2MayE1}
------------------------------------------------
We sketch here the calculation of the $h_1$-localized May spectral sequence over $A(2)$. The $E_1$-term is a polynomial algebra on the generators $h_0$, $h_1$, $h_2$, $h_{20}$, $h_{21}$, and $h_{30}$.
Then $d_1(h_{20}) = h_0 h_1$ and $d_1(h_{21}) = h_1 h_2$. As in Section \[sec:AMayE1\], we replace $h_{30}$ by $h_{30}' = h_1^{-1} h_0(1)
= h_{30} + h_1^{-1} h_{20} h_{21}$, so that $d_1 (h_{30}') = 0$.
The $E_2$-page is then the polynomial algebra ${\ensuremath{\mathbb{M}}}_2 [ h_1^{\pm 1}, b_{20}, b_{21}, h_{30}' ]$, and the only differential is $d_2 ( b_{20} ) = \tau h_1^3$.
It follows that $E_3$ is given by ${\ensuremath{\mathbb{F}}}_2 [ h_1^{\pm 1}, b_{20}^2, b_{21}, h_{30}' ]$. No more differentials are possible, and $E_3=E_\infty$.
Note that $b_{20}^2$, $b_{21}$, and $h_{30}'$ correspond respectively to $v_1^4$, $h_1^{-1} a_1$, and $v_2$ in the notation of Proposition \[CohomBi(2)\].
The localization map {#sec:LocMap}
====================
The calculation of $\operatorname{Ext}_A$ is given in [@Istems] up to the $70$-stem. In this section, we will use the May spectral sequence analysis of $\operatorname{Ext}_A[h_1^{-1}]$ from Section \[sec:MaySS\] to determine the localization map $$L:\operatorname{Ext}_A {\longrightarrow}\operatorname{Ext}_A [h_1^{-1}]$$ in the same range. A detailed understanding of the localization map is essential for transfer of information between the localized and non-localized situations.
Table \[LocztnTable\] lists some values of the localization map $L: \operatorname{Ext}_A \rightarrow \operatorname{Ext}_A[h_1^{-1}]$ on multiplicative generators of $\operatorname{Ext}_A$. Through the 70-stem, the localization map is zero on all generators of $\operatorname{Ext}_A$ not listed in Table \[LocztnTable\].
Note that $\operatorname{Ext}_A[h_1{^{-1}}]$ is concentrated in degrees $(t,f,c)$ such that $t-c$ is even. Many of the generators of $\operatorname{Ext}_A$ are in degrees $(t,f,c)$ such that $t-c$ is odd. Therefore, all of these generators must map to $0$ in the localization.
The values of $L$ on $u$, $v$, $u'$, $v'$, and $U$ follow from applying the May $E_4$ relation $h_1^4\Delta = d_0^2 + Pg$ to the May descriptions of these classes.
The remaining values are again determined by their May descriptions, together with the value of $L(B_1)$, which follows from the relation $h_1^7B_1 = h_1^2e_0^3 + c_0^2 \cdot e_0g$ established in Proposition \[prop:HiddenRels\].
Table \[LocztnTable\] gives values for the localization map in two forms. First, it uses the notation from Theorem \[ExtAThm\] involving the elements $v_n$. Second, it uses a different notation for the generators of $\operatorname{Ext}_A[h_1^{-1}]$ given in Table \[ExtGenTable\] that is more compatible with the standard notation for $\operatorname{Ext}_A$.
With a detailed understanding of the localization map in hand, we can establish some hidden relations in $\operatorname{Ext}_A$ that are needed in [@Istems].
The following hidden extensions hold in $\operatorname{Ext}_A$:
1. $e_0^3 + d_0 \cdot e_0g = h_1^5B_1 $.
2. $d_0 v + e_0 u = h_1^3 x'$.
3. $e_0 u' + d_0 v' = h_1^2 c_0 x'$.
Table \[LocztnTable\] says that $L( e_0^3 + d_0 \cdot e_0g)$ equals $h_1^9 v_3^3 + h_1^9 v_2^2 v_4$, which is non-zero. It follows that $e_0^3+d_0 \cdot e_0g$ is non-zero in $\operatorname{Ext}_A$. From the calculation in [@Istems], the only possibility is that it equals $h_1^5 B_1$. This establishes the first formula.
The argument for the second formula is similar. Table \[LocztnTable\] says that $L(d_0 v + e_0 u )$ equals $h_1^6 v_1^4 v_2^2 v_4 + h_1^6 v_1^4 v_3^3$, which is non-zero. It follows that $d_0 v + e_0 u$ is non-zero in $\operatorname{Ext}_A$, and the only possibility is that it equals $h_1^3 x'$.
For the third formula, Table \[LocztnTable\] says that $L(e_0 u' + d_0 v')$ equals $h_1^7 v_1^4 v_2 v_3^3 + h_1^7 v_1^4 v_2^3 v_4$, which is non-zero. It follows that $e_0 u' + d_0 v'$ is non-zero in $\operatorname{Ext}_A$. There are several possible non-zero values for $e_0 u' + d_0 v'$. However, $e_0 u' + d_0 v'$ must be annihilated by $\tau$ because both $u'$ and $v'$ are. Then $h_1^2 c_0 x'$ is the only possible value.
The localization map for $A(2)$
-------------------------------
For completeness, we also describe the localization map $$\operatorname{Ext}_{A(2)}{\longrightarrow}\operatorname{Ext}_{A(2)}[h_1^{-1}].$$ The calculation of $\operatorname{Ext}_{A(2)}$ is given in [@IMotA(2)].
Table \[LocztnTable2-A(2)\] lists some values of the localization map $L: \operatorname{Ext}_{A(2)} {\longrightarrow}\operatorname{Ext}_{A(2)}[h_1^{-1}]$ on multiplicative generators of $\operatorname{Ext}_{A(2)}$. The localization map is zero on all generators of $\operatorname{Ext}_{A(2)}$ not listed in Table \[LocztnTable2-A(2)\].
The generators for $\operatorname{Ext}_{A(2)}$ are given in [@IMotA(2) Table 7]. The values of $L$ follow by comparison of the localized and unlocalized May spectral sequences for $A(2)$.
Now consider the diagram $$\xymatrix{
\operatorname{Ext}_{A} \ar[r] \ar[d] & \operatorname{Ext}_{A}[h_1^{-1}] \ar[d] \\
\operatorname{Ext}_{A(2)} \ar[r] & \operatorname{Ext}_{A(2)}[h_1^{-1}]
}$$ in which the horizontal maps are localizations and the vertical maps are induced by the inclusion $A(2) {\longrightarrow}A$. Given that $\operatorname{Ext}_A[h_1^{-1}]$ and $\operatorname{Ext}_{A(2)}[h_1^{-1}]$ are computed explicitly in Theorem \[ExtAThm\] and Proposition \[CohomBi(2)\], one might expect that the map $\operatorname{Ext}_A [h_1^{-1}] {\longrightarrow}\operatorname{Ext}_{A(2)} [h_1^{-1}]$ would be easy to determine. The obvious guess is that this map takes $v_1^4$ to $v_1^4$, takes $v_2$ to $v_2$, and takes $v_n$ to $0$ for $n \geq 3$. However, the Cartan-Eilenberg spectral sequences of Section \[sec:CohomA\] hide some of the values of this map.
\[lem:restriction\] The map $\operatorname{Ext}_A[h_1^{-1}] {\longrightarrow}\operatorname{Ext}_{A(2)} [h_1^{-1}]$ takes $v_1^4$, $v_2$, $v_3$, $v_4$, $v_5$, and $v_6$ to $v_1^4$, $v_2$, $h_1^{-3} a_1 v_2$, $h_1^{-9} a_1^3 v_2$, $h_1^{-21} a_1^7 v_2$, and $h_1^{-45} a_1^{15} v_2$.
This follows from the May spectral sequence calculations of Section \[sec:MaySS\]. The given values for $\operatorname{Ext}_A[h_1^{-1}] {\longrightarrow}\operatorname{Ext}_{A(2)} [h_1^{-1}]$ are apparent on the May $E_\infty$-pages. We are using that $a_1$ is represented by $a_1=h_1b_{21}$.
Lemma \[lem:restriction\] suggests an obvious conjecture on the complete description of the map $\operatorname{Ext}_A [h_1^{-1}] {\longrightarrow}\operatorname{Ext}_{A(2)} [h_1^{-1}]$.
\[conj:restriction\] The map $\operatorname{Ext}_A [h_1^{-1}] {\longrightarrow}\operatorname{Ext}_{A(2)} [h_1^{-1}]$ takes $v_1^4$ to $v_1^4$ and takes $v_n$ to $h_1^{-3(2^{n-2}-1)} a_1^{2^{n-2} - 1} v_2$ for $n \geq 2$.
The Adams spectral sequence for $S[\eta^{-1}]$ {#sec:ASS}
==============================================
Recall that $\operatorname{Ext}_A$ is the $E_2$-page for the motivic Adams spectral sequence that converges to the $2$-complete motivic stable homotopy groups $\pi_{*,*}$ of the motivic sphere $S^{0,0}$. The element $h_1$ in $\operatorname{Ext}_A$ detects the motivic Hopf map $\eta$ in $\pi_{1,1}$.
Let $S^{0,0}[\eta^{-1}]$ to be the homotopy colimit of the sequence $$S^{0,0} \xrightarrow{\eta} S^{-1,-1} \xrightarrow{\eta}S^{-2,-2}\xrightarrow{\eta} \dots.$$
The homotopy groups $\pi_{*,*}(S^{0,0}[\eta^{-1}])$ are then the target of an $h_1$-localized Adams spectral sequence whose $E_2$-page is $\operatorname{Ext}_A[h_1^{-1}]$. However, we must consider convergence. A priori, there could be an infinite family of homotopy classes linked together by infinitely many hidden $\eta$-multiplications. These classes would not be detected in $\operatorname{Ext}_A[h_1^{-1}]$. But this cannot occur, as the argument of [@A] carries over readily to the motivic setting to establish a vanishing line of slope $1$ in $\operatorname{Ext}_A$.
In the $(s,f,w)$-grading, the Adams differentials behave according to $$d_r:E_r^{s,f,w} {\longrightarrow}E_r^{s-1,f+r,w}.$$ In the $h_1$-invariant grading, this becomes $$d_r:E_r^{t,f,c} {\longrightarrow}E_r^{t-1,f+r,c+r-1}.$$
\[prop:Adams-d2\] The Adams $d_2$ differential for $S^{0,0}[\eta^{-1}]$ takes the following values.
1. $d_2(v_1^4) = 0$.
2. $d_2(v_2) = 0$.
3. $d_2(v_3) = h_1v_2^2$.
4. $d_2(v_4) = h_1v_3^2$.
The first two formulas follow immediately because there are no possible non-zero values. The third formula follows from the Adams differential $d_2(e_0) = h_1^2 d_0$ in the unlocalized case [@Istems], together with the fact that the localization map takes $c_0$ and $e_0$ to $h_1^2 v_2$ and $h_1^3 v_3$. The fourth formula follows from the Adams differential $d_2(e_0 g) = h_1^2 e_0^2$ in the unlocalized case [@Istems], together with the fact that the localization map takes $e_0$ and $e_0 g$ to $h_1^3 v_3$ and $h_1^7 v_4$.
Proposition \[prop:Adams-d2\] suggests an obvious conjecture for the values of the Adams $d_2$ differential on the rest of the generators of $\operatorname{Ext}_A[h_1^{-1}]$. See Conjecture \[mainConj\] for an explicit statement.
Motivic modular forms and Adams differentials
---------------------------------------------
In the classical case, the topological modular forms spectrum ${\mathit{tmf}}$ is a spectrum whose ${\ensuremath{\mathbb{F}}}_2$-cohomology is equal to the quotient $A_{{\mathrm{cl}}}//A(2)_{{\mathrm{cl}}}$. This implies that $\operatorname{Ext}_{A(2)_{{\mathrm{cl}}}}$ is the $E_2$-page of the Adams spectral sequence converging to the $2$-complete homotopy groups of ${\mathit{tmf}}$.
One might speculate that there is a motivic spectrum ${\mathit{mmf}}$ (called “motivic modular forms") whose motivic ${\ensuremath{\mathbb{F}}}_2$-cohomology is isomorphic to $A//A(2)$. Then $\operatorname{Ext}_{A(2)}$ would be the $E_2$-page of the motivic Adams spectral sequence converging to the $2$-complete motivic homotopy groups of ${\mathit{mmf}}$. However, no such motivic spectrum is known to exist. See [@NSO] for one piece of the program for constructing ${\mathit{mmf}}$.
In any case, we assume for the rest of this section that ${\mathit{mmf}}$ does exist, and we explore some of the computational consequences.
\[lem:d2-a1\] Suppose that ${\mathit{mmf}}$ exists. Then, in the Adams spectral sequence $$\operatorname{Ext}_{A(2)}\Rightarrow \pi_{*,*}({\mathit{mmf}}),$$ there is an Adams differential $d_2(a_1)=h_1^2c_0$.
Since $d_2(e_0) = h_1^2 d_0$ in the Adams spectral sequence for $S^{0,0}$, it follows that $d_2(e_0) = h_1^2 d_0$ in the Adams spectral sequence for ${\mathit{mmf}}$ as well.
We have the relation $c_0 a_1 = h_1^2 e_0$ in $\operatorname{Ext}_{A(2)}$ [@IMotA(2)]. Therefore, $a_1$ must support a differential, and $h_1^2 c_0$ is the only possible value.
Note that the element $a_1$ was called $u$ in [@IMotA(2)].
Suppose that ${\mathit{mmf}}$ exists. Then Conjecture \[mainConj\] is equivalent to Conjecture \[conj:restriction\].
The existence of ${\mathit{mmf}}$ ensures that the Adams $d_2$ differential is compatible with the map $r: \operatorname{Ext}_A {\longrightarrow}\operatorname{Ext}_{A(2)}$, so that $d_2(r(v_n)) = r(d_2(v_n)$. For degree reasons, $r(v_n)$ is either equal to $h_1^{-3(2^{n-2}-1)} a_1^{2^{n-2}-1} v_2$, or it is zero. Also for degree reasons, $d_2(v_n)$ is either equal to $h_1 v_{n-1}^2$, or it is zero.
Suppose that Conjecture \[mainConj\] holds. Then $r(d_2(v_n))$ equals $h_1 r(v_{n-1})^2$. We may assume by induction that $r(v_{n-1}^2)$ equals $h_1^{-3(2^{n-2}-2)} a_1^{2^{n-2} -2} v_2^2$. In particular, this shows that $r(d_2(v_n))$ is non-zero. But $r(d_2(v_n))$ equals $d_2 (r(v_n))$, so $r(v_n)$ must also be non-zero. This establishes Conjecture \[conj:restriction\].
Now suppose that Conjecture \[conj:restriction\] holds. Then $d_2(r(v_n))$ is equal to $h_1^{-3(2^{n-2}-2)+1} a_1^{2^{n-2}-2} v_2^2$ because of the differential $d_2(a_1) = h_1^2 c_0$ from Lemma \[lem:d2-a1\]. But $d_2(r(v_n))$ equals $r(d_2(v_n))$, so $d_2(v_n)$ must also be non-zero. This establishes Conjecture \[mainConj\].
Tables {#sec:Tables}
======
[|C|C|C|C|]{} & & (m-f,t,f,c) & d\_2\
& h\_1\^[-2]{}c\_0 = h\_[30]{} + h\_1\^[-1]{} h\_[20]{}h\_[21]{} & (2,3,1,1) &\
& h\_3 & (0,3,1,0) &\
& h\_4 & (0,7,1,0) &\
& h\_[40]{}\^2+ h\_1\^[-2]{} h\_[20]{}\^2h\_[31]{}\^2 & (6,14,2,2) &\
& h\_5 & (0,15,1,0) &\
& h\_1\^[-1]{} h\_0(1,3) = h\_[50]{} h\_3 + h\_1\^[-1]{} h\_[20]{}h\_[31]{}h\_[23]{} & (4,18,2,1) &\
& + h\_[40]{} h\_[23]{} +h\_1\^[-1]{} h\_[20]{}h\_[41]{}h\_3 & &\
& b\_[23]{} = h\_[23]{}\^2 & (2,22,2,0) &\
& h\_[50]{}\^2 + h\_1\^[-2]{} h\_[20]{}\^2h\_[41]{}\^2 & (8,30,2,2) &\
& h\_6 & (0,31,1,0) &\
& h\_3(1) = h\_4h\_[33]{}+h\_[23]{}h\_[24]{} & (2,34,2,0) &\
& b\_[24]{} = h\_[24]{}\^2 & (2,46,2,0) &\
& b\_[33]{} = h\_[33]{}\^2 & (4,54,2,0) &\
& h\_[60]{}\^2 + h\_1\^[-2]{} h\_[20]{}\^2h\_[51]{}\^2 & (10,62,2,2) &\
& h\_7 & (0,63,1,0) &\
b\_[20]{} & b\_[20]{} = h\_[20]{}\^2 & (2,2,2,2) & h\_1\^3\
b\_[21]{} & b\_[21]{} = h\_[21]{}\^2 & (2,4,2,0) & h\_1\^2\
b\_[31]{} & b\_[31]{} = h\_[31]{}\^2 & (4,12,2,0) & b\_[21]{}\
b\_[41]{} & b\_[41]{} = h\_[41]{}\^2 & (6,28,2,0) & b\_[31]{}\
b\_[51]{} & b\_[51]{} = h\_[51]{}\^2 & (8,60,2,0) & b\_[41]{}\
[|C|C|C|]{} & (m-f,t,f,c)\
& (2,6,2,1)\
& (0,10,2,0)\
+ & (6,21,3,2)\
& (0,22,2,0)\
+ & (4,25,3,1)\
& (4,33,3,1)\
+ & (8,36,4,2)\
& (4,37,3,1)\
+ & (2,37,3,0)\
& (0,46,2,0)\
& (8,48,4,2)\
+ & (2,49,3,0)\
& (6,52,4,1)\
+ & (8,60,4,2)\
+ & (6,64,4,1)\
& (2,65,3,0)\
[|C|C|C|]{} & & (m-f,t,f,c)\
P & b\_[20]{}\^2 & (4,4,4,4)\
e\_0 & b\_[21]{}h\_0(1) & (4,7,4,1)\
g & b\_[21]{}\^2 & (4,8,4,0)\
B & b\_[21]{} + b\_[31]{} & (8, 18, 4, 2)\
B\_1 & c\_0 B & (10, 21, 7, 3)\
& h\_1 b\_[21]{}B & (10,22,7,2)\
\_1 & b\_[31]{}\^2 & (8,24,4,0)\
D\_4 & g+ b\_[21]{}b\_[31]{} & (8,26,6,1)\
& & (12,28,4,4)\
s\_1 & h\_1 [h\_4\^2]{}b\_[21]{}b\_[31]{}+ h\_1 g[b\_[23]{}]{} + h\_1\^[-1]{} [h\_3h\_5]{}b\_[21]{}\^3 & (6,30,7,0)\
D\_3’ & h\_1\^4 & (10,32,8,3)\
y\_[34]{} & [h\_5]{}\^2b\_[21]{} + h\_1\^2 [h\_3(1)]{} & (2,34,4,0)\
y\_[35]{} & [h\_4]{}b\_[41]{} & (6,35,3,0)\
& & (8,36,4,2)\
& & (8,37,3,2)\
& & (6,40,4,1)\
& & (4,44,4,0)\
y\_[45]{} & g + \_1 & (12,45,7,2)\
y\_[60]{} & h\_1\^8 + b\_[21]{} b\_[31]{} g & (14,60,12,2)\
& & (16,60,4,4)\
y\_[61]{} & b\_[31]{} + b\_[31]{} & (8,61,5,1)\
& + b\_[41]{} + b\_[21]{} &\
y\_[64]{} & & (10,64,4,2)\
[|C|C|C|]{} & (m-f,t,f,c) & d\_4\
g & (4,8,4,0) & h\_1\^4 [h\_4]{}\
\_1 & (8,24,4,0) & g[h\_5]{}\
& (12,28,4,4) &\
y\_[35]{} & (6,35,3,0) & y\_[34]{}\
& (8,37,3,2) &\
& (16,60,4,4) &\
b\_[41]{} & (14,65,5,2) & h\_1\^2 y\_[64]{}\
[|C|C|C|]{} & (m-f,t,f,c) & d\_6\
& (10,22,7,2) & h\_1 c\_0\^2h\_5\
[c\_0]{}g\^3 & (14,27,15,1) & h\_1\^[10]{} D\_4\
[h\_4]{}g\^3 & (12,31,13,0) & h\_1\^7 s\_1\
[c\_0]{}g& (16,33,14,3) & h\_1\^7 D\_3’\
[h\_4]{}g& (14,37,12,2) & h\_1\^9\
[h\_4]{}gD\_4 & (12,41,11,1) & h\_1\^8\
[h\_4]{}gs\_1 & (10,45,12,0) & h\_1\^9\
[h\_4]{}g\^3\_1 & (20,55,17,0) & h\_1\^7 s\_1\_1\
[c\_0]{}g\_1 & (24,57,18,3) & h\_1\^7 D\_3’\_1\
g\^2y\_[45]{} & (20,61,15,2) & h\_1\^4y\_[60]{}\
[h\_4]{}g\_1 & (22,61,16,2) & h\_1\^9 \_1\
[c\_0]{}g & (28,61,18,7) & h\_1\^7 D\_3’\
[h\_4]{}gb\_[41]{} & (20,65,14,2) & h\_1\^9 b\_[41]{}\
[h\_4]{}gD\_4\_1 & (20,65,15,1) & h\_1\^8 \_1\
[|C|C|C|]{} & (m-f,t,f,c) & d\_8\
g\^2 & (8,16,8,0) & h\_1\^8 [h\_5]{}\
\_1\^2 & (16,48,8,0) & g\^2[h\_6]{}\
\_1 & (20,52,8,4) & h\_1\^[-6]{} [c\_0]{}\^2e\_0\^2[h\_6]{}\
\^2 & (24,56,8,8) &\
[c\_0]{}\_1y\_[35]{} & (16,62,10,1) & h\_1\^6y\_[61]{}\
[|C|C|C|]{} & (m-f,t,f,c) & d\_[10]{}\
\_1& (18,46,11,2) & h\_1\^3e\_0\^2[h\_6]{}\
& (22,50,11,6) & h\_1\^[-7]{}[c\_0\^6h\_6]{}\
[c\_0]{}g\^6 & (26,51,27,1) & h\_1\^[18]{}\_1D\_4\
g\^4 & (26,54,23,2) & h\_1\^[13]{}gy\_[45]{}\
g\^2\^2 & (28,60,22,4) & h\_1\^[12]{}c\_0D\_4\
g\^2\_1& (26,62,19,2) & h\_1\^[13]{}\_1\
[|C|C|C|]{} & (m-f,t,f,c) & d\_[12]{}\
g\^6 & (24,63,25,0) & h\_1\^[12]{}[h\_4h\_6]{}g\^3\
\^2 & (20,44,14,4) & h\_1\^[2]{}[c\_0\^4h\_6]{}\
[c\_0]{}g\^4& (28,57,26,3) & h\_1\^[19]{}D\_4\
e\_0g\_1& (26,61,19,3) & h\_1\^8[h\_6]{}e\_0\
[c\_0]{}g\^2\^2 & (30,63,25,5) & h\_1\^[16]{}D\_3’\
g\^2\_1 D\_4 & (24,66,18,1) & h\_1\^8[h\_6]{}gD\_4\
[|C|C|C|]{} & (m-f,t,f,c) & d\_[14]{}\
g\^2& (18,38,15,2) & h\_1\^[9]{}[c\_0\^2h\_6]{}\
e\_0g\^6 & (28,55,28,1) & h\_1\^[21]{}b\_[41]{}D\_4\
g\^4e\_0& (30,61,27,3) & h\_1\^[18]{}b\_[41]{}D\_3’\
[|C|C|C|C|]{} & (m-f,t,f,c) & d\_[r]{} &\
g\^4 & (16,32,16,0) & d\_[16]{} & h\_1\^[16]{}[h\_6]{}\
e\_0\^2g\^5 & (28,54,28,2) & d\_[18]{} & h\_1\^[21]{}[h\_6]{}\
g\^8 & (32,64,32,0) & d\_[32]{} & h\_1\^[32]{}[h\_7]{}\
[|C|C|]{} & (t,f,c)\
h\_1\^2 e\_0\^3 + c\_0\^2 e\_0g = h\_1\^7 B\_1 & (21,14,3)\
h\_1\^2 e\_0(e\_0g)\^2 + c\_0\^2e\_0g\^3 = h\_1\^7 g\^2B\_1 & (37,22,3)\
h\_1\^4 e\_0\^4 e\_0g + c\_0\^4e\_0g\^3 = h\_1\^[14]{} B\_1& (43,28,5)\
e\_0\^2 e\_0g\^3 + (e\_0g)\^3 = h\_1\^[13]{} \_1 B\_1 & (45,24,3)\
h\_1\^6 e\_0\^7 + h\_1\^4 c\_0\^2e\_0\^4 e\_0g + h\_1\^2 c\_0\^4e\_0(e\_0g)\^2 + c\_0\^6 e\_0g\^3 = h\_1\^[23]{} B\_1 & (49,34,7)\
[|C|C|C|C|]{} & & (t,f,c) & d\_2\
c\_0 & h\_1\^2 v\_2 & (3,3,1) &\
P & v\_1\^4 & (4,4,4) &\
e\_0 & h\_1\^3 v\_3 & (7,4,1) & c\_0\^2\
e\_0 g & h\_1\^7 v\_4 & (15,8,1) & h\_1\^2 e\_0\^2\
e\_0g\^3 & h\_1\^[15]{} v\_5 & (31,16,1) & h\_1\^2 (e\_0g)\^2 ?\
e\_0g\^7 & h\_1\^[31]{} v\_6 & (63,32,1) & h\_1\^2(e\_0g\^3)\^2 ?\
[CCCHCC]{} & & (s,f,w) & (s-w,) & &\
P\^kh\_1 & & (1,1,1)+k(8,4,4) & (4k,4k) & h\_1v\_1\^[4k]{} & h\_1P\^k\
P\^kc\_0 & & (8,3,5)+k(8,4,4) & (3,1) +k(4,4) & h\_1\^2v\_1\^[4k]{}v\_2 & P\^kc\_0\
P\^kd\_0 & & (14, 4, 8)+k(8,4,4) & (6,2)+k(4,4) & h\_1\^2v\_1\^[4k]{}v\_2\^2 & h\_1\^[-2]{}P\^kc\_0\^2\
P\^ke\_0 & & (17, 4, 10)+k(8,4,4) & (7,1)+k(4,4) & h\_1\^3v\_1\^[4k]{}v\_3 & P\^ke\_0\
e\_0 g & & (37,8,22) & (15,1) & h\_1\^7v\_4 & e\_0g\
u & h\_1 d\_0& (39,9,21) & (18,6) & h\_1\^[3]{}(v\_1\^4v\_3\^2 + v\_2\^6) & h\_1\^[-3]{}(Pe\_0\^2+c\_0\^6)\
P\^k u & & (39,9,21)+k(8,4,4) & (18,6)+k(4,4) & h\_1\^[3]{}v\_1\^[4k]{}(v\_1\^4v\_3\^2 + v\_2\^6) & h\_1\^[-3]{}P\^k(Pe\_0\^2+c\_0\^6)\
v & h\_1 e\_0& (42,9,23) & (19,5) & h\_1\^[4]{}(v\_1\^4v\_4 + v\_2\^4v\_3) & h\_1\^[-3]{}(Pe\_0g+h\_1\^[-4]{}c\_0\^4e\_0)\
P\^k v & & (42,9,23)+k(8,4,4) & (19,5)+k(4,4) & h\_1\^[4]{}v\_1\^4(v\_1\^4v\_4 + v\_2\^4v\_3) & h\_1\^[-3]{}P\^k(Pe\_0g+h\_1\^[-4]{}c\_0\^4e\_0)\
B\_1 & c\_0B & (46,7,25) & (21,3) & h\_1\^[4]{}(v\_2\^2v\_4+v\_3\^3) & h\_1\^[-7]{}c\_0\^2e\_0g + h\_1\^[-5]{}e\_0\^3\
u’ & c\_0 d\_0 & (46,11,25) & (21,7) & h\_1\^[4]{}(v\_1\^4v\_2v\_3\^2 + v\_2\^7) & h\_1\^[-4]{}(Pc\_0e\_0\^2 + h\_1\^[-6]{}c\_0\^7)\
P\^k u’ & & (46,11,25)+k(8,4,4) & (21,7)+k(4,4) & h\_1\^[4]{}v\_1\^[4k]{}(v\_1\^4v\_2v\_3\^2 + v\_2\^7) & h\_1\^[-4]{}P\^k(Pc\_0e\_0\^2 + h\_1\^[-6]{}c\_0\^7)\
v’ & c\_0 e\_0& (49,11,27) & (22,6) & h\_1\^[5]{}(v\_1\^4v\_2v\_4 + v\_2\^5v\_3) & h\_1\^[-4]{}(Pc\_0e\_0g + h\_1\^[-4]{}c\_0\^5e\_0)\
P\^k v’ & & (49,11,27)+k(8,4,4) & (22,6)+k(4,4) & h\_1\^[5]{}v\_1\^[4k]{}(v\_1\^4v\_2v\_4 + v\_2\^5v\_3) & h\_1\^[-4]{}P\^k(Pc\_0e\_0g + h\_1\^[-4]{}c\_0\^5e\_0)\
B\_8 & h\_0(1)B\_1 & (53,9,29) & (24,4) & h\_1\^[5]{}(v\_2\^3v\_4+v\_2v\_3\^3) & h\_1\^[-6]{}c\_0(h\_1\^[-2]{}c\_0\^2e\_0g + e\_0\^3)\
x’ & h\_0(1) B P & (53,10,28) & (25,7) & h\_1\^[3]{}v\_1\^4(v\_2\^2v\_4+v\_3\^3) & h\_1\^[-6]{}P(h\_1\^[-2]{}c\_0\^2e\_0g+e\_0\^3)\
B\_[21]{} & h\_0(1)\^3 B & (59,10,32) & (27,5) & h\_1\^[5]{}(v\_2\^4v\_4+v\_2\^2v\_3\^3) & h\_1\^[-8]{}c\_0\^2(h\_1\^[-2]{}c\_0\^2e\_0g + e\_0\^3)\
B\_[22]{} & b\_[21]{}d\_0 B & (62,10,34) & (28,4) & h\_1\^[6]{}(v\_2\^2v\_3v\_4+v\_3\^4) & h\_1\^[-6]{}e\_0(h\_1\^[-2]{}c\_0\^2e\_0g + e\_0\^3)\
U & \^2 h\_1\^2 d\_0 & (64,14,34) & (30,10) & h\_1\^[4]{}(v\_1\^8v\_3v\_4+v\_2\^[10]{}) & h\_1\^[-6]{}(P\^2e\_0e\_0g + h\_1\^[-10]{}c\_0\^[10]{})\
P\^2 x’ & & (69,18,36) & (33,15) & h\_1\^[3]{}v\_1\^[12]{}(v\_2\^2v\_4+v\_3\^3) & h\_1\^[-6]{}P\^3(h\_1\^[-2]{}c\_0\^2e\_0g+e\_0\^3)\
[CCCHC]{} & & (s,f,w) & (s-w,) &\
P & & (8,4,4) & & v\_1\^4\
c & & (8,3,5) & & h\_1 v\_2\
u &h\_1b\_[21]{} & (11,3,7) & & a\_1\
d & & (14,4,8) & & h\_1\^2 v\_2\^2\
e & & (17,4,10) & & a\_1 v\_2\
g & & (20,4,12) & & h\_1\^[-2]{} a\_1\^2\
h\_1 & & (25,5,13) & & h\_1\^[-5]{} v\_1\^4 a\_1\^2 + h\_1 v\_2\^4\
c & & (32,7,17) & & h\_1\^[-5]{} v\_1\^4 v\_2 a\_1\^2 + h\_1 v\_2\^5\
u & & (35,7,19) & & h\_1\^[-6]{} v\_1\^4 a\_1\^3 + v\_2\^4 a\_1\
\^2 & & (48,8,24) & & h\_1\^[-12]{} v\_1\^8 a\_1\^4 + v\_2\^8\
[NSO]{}
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[^1]: The first author was supported by Simons Collaboration Grant 282316. The second author was supported by NSF grant DMS-1202213.
|
---
abstract: |
We review a long-standing difficulty in some semiclassical models of vacuum and vacuum decay. Surprisingly enough these models, careless of their transparent formulation, are affected by, both, technical and conceptual issues. After proving some general results that are relevant for, both, the Euclidean and Lorentzian sectors of their dynamics, we briefly highlight their importance in connection with the issues discussed before, arguing that their solution might be interesting in our quest for quantum gravity.\
------------------------------------------------------------------------
\
\
------------------------------------------------------------------------
author:
- |
Stefano Ansoldi[^1]\
[*Department of Physics, University of Kyoto, Kyoto, JAPAN and*]{}\
[*International Center for Relativistic Astrophysics (I.C.R.A), Pescara, ITALY and*]{}\
[*Istituto Nazionale di Fisica Nucleare (I.N.F.N.), Sezione di Trieste, ITALY*]{}
title: |
Vacuum and semiclassical gravity:\
a difficulty and its bewildering significance[^2]
---
Introduction
============
The study of vacuum properties and decay is a fascinating subject, and it becomes even more interesting when its interplay with gravitation is considered. A natural context to study properties of vacuum is early universe cosmology, where vacuum energy density plays a dominant role. For this reason, early after the first studies of vacuum and vacuum decay [@bib:VacuumDecay], gravity entered the scene [@bib:PhReD1980..21..3305L; @bib:Bubbles01; @bib:PhReD1987..35..1747G; @bib:Bubbles02]: in most models thin relativistic shells are used to describe the dynamics of vacuum bubbles. Classical models already reveal interesting properties, but they tend to be affected by issues, in connection with stability and the presence of singularities. To avoid these problems, quantum models (mostly semiclassical) have been developed [@bib:NuPhy1990B339...417G; @bib:QuantumModels]: vacuum decay is then described in the WKB approximation as spacetime tunnelling driven by the thin shell. Despite the earliest of these works date back to more than 30 years ago, issues left open in the original formulation [@bib:NuPhy1990B339...417G] have not yet been solved [@bib:NuPhy1990B339...417G; @bib:QuantumTroubles]. Here we review some of these issues from a general perspective. We also point out that, although the existing literature mostly focuses on applications relevant for early universe cosmology, the open problems are of a more general nature and can be recognized as a general difficulty in the semiclassical quantization of the (general relativistic) shell system. After a preliminary review of some background material in Sec.$\:$\[sec:predef\], we prove in Sec.$\:$\[sec:genres\] a collection of general properties of shell dynamics in spherically symmetric, but otherwise arbitrary, configurations. Then, in Sec.$\:$\[sec:tunpro\], we present in the perspective of these results some issues related with the WKB description of the tunnelling process; a concluding discussion follows in Sec.$\:$\[sec:dissec\]. Apart from the included bibliography, additional references can be also found in [@bib:ClQuG2002..19..6321A; @bib:ExtraBiblio].
\[sec:predef\]Background, conventions and notations
===================================================
Throughout the paper curvature conventions follow [@bib:Freem1970...1..1279W], with Greek indices taking the values $0,1,2,3$, lowercase Latin indices taking the values $1, 2, 3$, and uppercase latin indices the values $0, 1$. We consider two parts ${\mathcal{M}} _{{{\scriptscriptstyle{}(-)}}}$ and ${\mathcal{M}} _{{{\scriptscriptstyle{}(+)}}}$ of two spacetimes and assume that they have a common timelike part $\Sigma$ in their boundaries. Various quantities related to the submanifolds ${\mathcal{M}} _{{{\scriptscriptstyle{}(-)}}}$, ${\mathcal{M}} _{{{\scriptscriptstyle{}(+)}}}$ and $\Sigma$ are defined in the table below and in Fig.$\:$\[fig:geoquadef\], panel \[a\].
--------------------------------------------------------- ---------- ------------------------------------------ -------------------------------------------------------------- ------------------------------------------------------------------------------------------- --------------- ------------------------------------------------------------------------------------------------------ --------------------------------------------------
[Submanifold]{} [Dim.]{} [Holonomic]{} [Metric]{} [Signature]{} [Covariant]{} [Matter]{}
\[-2mm\] [system]{} [basis]{} [components]{} [Derivative]{} [Content]{}
\[-5mm\] ${\mathcal{M}} _{{{\scriptscriptstyle{}(-)}}}$ $4$ $x ^{{{\scriptscriptstyle{}(-)}}\alpha}$ $\partial ^{{{\scriptscriptstyle{}(-)}}} _{\mu}$ ${}^{{{\scriptscriptstyle{}(-)}}}\!g _{\mu \nu} (x ^{{{\scriptscriptstyle{}(-)}}\alpha})$ $2$ ${}^{{{\scriptscriptstyle{}(-)}}}\!\nabla _{\mu}$, $(\dots) ^{{{\scriptscriptstyle{}(-)}}} _{; \mu}$ ${}^{{{\scriptscriptstyle{}(-)}}}\!T _{\mu \nu}$
${\mathcal{M}} _{{{\scriptscriptstyle{}(+)}}}$ $4$ $x ^{{{\scriptscriptstyle{}(+)}}\alpha}$ $\partial ^{{{\scriptscriptstyle{}(+)}}} _{\mu}$ ${}^{{{\scriptscriptstyle{}(+)}}}\!g _{\mu \nu} (x ^{{{\scriptscriptstyle{}(+)}}\alpha})$ $2$ ${}^{{{\scriptscriptstyle{}(+)}}}\!\nabla _{\mu}$, $(\dots) ^{{{\scriptscriptstyle{}(+)}}} _{; \mu}$ ${}^{{{\scriptscriptstyle{}(+)}}}\!T _{\mu \nu}$
$\Sigma$ $3$ $\xi ^{a}$ ${\mbox{\textbf{\textit{e}}}} _{(a)} = \partial _{\xi ^{a}}$ $g _{m n} (\xi ^{a})$ $1$ ${}^{{{\scriptscriptstyle{}(3)}}}\!\nabla _{a}$, $(\dots) _{| a}$ $S _{m n}$
--------------------------------------------------------- ---------- ------------------------------------------ -------------------------------------------------------------- ------------------------------------------------------------------------------------------- --------------- ------------------------------------------------------------------------------------------------------ --------------------------------------------------
In the above setup, we *locally* define the embedding of $\Sigma$ in ${\mathcal{M}} _{{{\scriptscriptstyle{}(\pm)}}}$ as $x ^{{{\scriptscriptstyle{}(\pm)}}\mu} = F ^{\mu} _{{{\scriptscriptstyle{}(\pm)}}} (\xi ^{a})$. Moreover we assume $
{}^{{{\scriptscriptstyle{}(-)}}}\!g _{\mu \nu} (x ^{{{\scriptscriptstyle{}(-)}}\rho})
e ^{\mu} _{(m)}
e ^{\nu} _{(n)} | _{x ^{\rho} = F _{{{\scriptscriptstyle{}(-)}}} ^{\rho} (\xi ^{a})} =
{}^{{{\scriptscriptstyle{}(+)}}}\!g _{\mu \nu} (x ^{{{\scriptscriptstyle{}(+)}}\sigma}) e ^{\mu} _{(m)} e ^{\nu} _{(n)} | _{x ^{\sigma} = F _{{{\scriptscriptstyle{}(+)}}} ^{\sigma} (\xi ^{a})} =
g _{m n} (\xi ^{a}),
$ so that the metric on $\Sigma$ is well defined (by assumption, it is also non degenerate). Various geometric quantities, as for instance the normal[^3] to $\Sigma$, ${\mbox{\textbf{\textit{n}}}}$, or the extrinsic curvature of $\Sigma$, $K _{ij} \stackrel{\mathrm{def.}}{=} - n _{\alpha} e _{(j)} ^{\beta} \nabla _{\beta} e _{(i)} ^{\alpha}$, are calculated with respect to the embedding of $\Sigma$ in ${\mathcal{M}} _{{{\scriptscriptstyle{}(-)}}}$ or ${\mathcal{M}} _{{{\scriptscriptstyle{}(+)}}}$. We specify this using “${\scriptstyle(\pm)}$” as a sub/superscript or using “$| _{{{\scriptscriptstyle{}(\pm)}}}$”. The *jump* of quantities across $\Sigma$, is shortly denoted as “$[(\dots{})]$”; consistently, square brackets *are not* used with any other meaning.
![\[fig:geoquadef\][Definition of the geometrical quantities (panel \[a\]) which are important to characterize a *junction* according to the notations and conventions defined in Sec.$\:$[\[sec:predef\]]{} on page . In the spherically symmetric case the dynamics of the system can be reduced to a one-dimensional effective classical problem [@bib:PhReD1987..35..1747G; @bib:PhReD1989..40..2511S; @bib:NuPhy1990B339...417G; @bib:ClQuG1997..14..2727S]. All the properties of the solutions can be qualitatively obtained from diagrams like the one in panel \[b\]: turning points are $R _{1}$ and $R _{2}$ (see also the zoomed in area), zeroes of $f _{{{\scriptscriptstyle{}(\pm)}}}$ are $\hat{R} ^{{{\scriptscriptstyle{}(\pm)}}}$, points at which the signs $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$ vanish are $\bar{R} ^{{{\scriptscriptstyle{}(\pm)}}}$ (where $f _{{{\scriptscriptstyle{}(\pm)}}}$ are tangent to $V (R)$). Three trajectories are possible, two classical ones (the *bounded* $[0 , \hat{R} _{1}]$ and the *unbounded* or *bounce* $[ \hat{R} _{2} , + \infty)$) and a tunnelling one, $[ \hat{R} _{1} , \hat{R} _{2} ]$. The two grayed areas highlight a $T ^{{{\scriptscriptstyle{}(-)}}}$ region (on the left) and a $T ^{{{\scriptscriptstyle{}(+)}}}$ region (on the right), respectively. As discussed in the text, the vanishing of $\epsilon _{{{\scriptscriptstyle{}(+)}}}$ at $\bar{R} ^{{{\scriptscriptstyle{}(+)}}}$ indicates a difficulty in the description of the tunnelling process.]{}](fig000a "fig:"){width="7cm"} ![\[fig:geoquadef\][Definition of the geometrical quantities (panel \[a\]) which are important to characterize a *junction* according to the notations and conventions defined in Sec.$\:$[\[sec:predef\]]{} on page . In the spherically symmetric case the dynamics of the system can be reduced to a one-dimensional effective classical problem [@bib:PhReD1987..35..1747G; @bib:PhReD1989..40..2511S; @bib:NuPhy1990B339...417G; @bib:ClQuG1997..14..2727S]. All the properties of the solutions can be qualitatively obtained from diagrams like the one in panel \[b\]: turning points are $R _{1}$ and $R _{2}$ (see also the zoomed in area), zeroes of $f _{{{\scriptscriptstyle{}(\pm)}}}$ are $\hat{R} ^{{{\scriptscriptstyle{}(\pm)}}}$, points at which the signs $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$ vanish are $\bar{R} ^{{{\scriptscriptstyle{}(\pm)}}}$ (where $f _{{{\scriptscriptstyle{}(\pm)}}}$ are tangent to $V (R)$). Three trajectories are possible, two classical ones (the *bounded* $[0 , \hat{R} _{1}]$ and the *unbounded* or *bounce* $[ \hat{R} _{2} , + \infty)$) and a tunnelling one, $[ \hat{R} _{1} , \hat{R} _{2} ]$. The two grayed areas highlight a $T ^{{{\scriptscriptstyle{}(-)}}}$ region (on the left) and a $T ^{{{\scriptscriptstyle{}(+)}}}$ region (on the right), respectively. As discussed in the text, the vanishing of $\epsilon _{{{\scriptscriptstyle{}(+)}}}$ at $\bar{R} ^{{{\scriptscriptstyle{}(+)}}}$ indicates a difficulty in the description of the tunnelling process.]{}](fig000b "fig:"){width="7cm"}
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The dynamics of $\Sigma$ and of its energy matter content in the spacetime manifold $\overline{{\mathcal{M}} _{{{\scriptscriptstyle{}(-)}}} \cup {\mathcal{M}} _{{{\scriptscriptstyle{}(+)}}}}$ (the overbar denotes the closure) is determined by Israel junction conditions [@bib:IsraelJunctions], which relate the jump of the extrinsic curvature of $\Sigma$ to the surface stress-energy tensor $S _{mn}$ and its trace $S$: $$[K _{ij}] \equiv K _{ij} ^{-} - K _{ij} ^{+}
=
8 \pi \left( S _{ij} - \frac{g _{ij}}{2} S \right)
.
\label{eq:isrjuncon}$$ In particular, we are interested in the spherically symmetric reduction of the above equations, i.e. we consider the special case in which $\Sigma$ describes the spherically symmetric evolution of a sphere. A general spherically symmetric four dimensional spacetime can always be considered as a product of a two sphere ${\mathbb{S}} ^{2}$ and of another, again two dimensional, Lorentzian manifold ${\mathbb{M}} _{2}$. Therefore, the metric ${}^{(4)}\!{\mbox{\textbf{\textit{g}}}}$ on the four dimensional manifold can always be decomposed in the form $${}^{(4)}\!{\mbox{\textbf{\textit{g}}}} = \left( \matrix{{\mbox{\textbf{\textit{g}}}} _{{\mathbb{M}} _{2}} & {\mathbb{O}} \cr {\mathbb{O}} & {\mbox{\textbf{\textit{g}}}} _{{\mathbb{S}} ^{2}}} \right) , \quad
{\mbox{\textbf{\textit{g}}}} _{{\mathbb{M}} _{2}} = \left( \matrix{ \gamma _{11} (x _{1} , x _{2}) & \gamma _{12} (x _{1} , x _{2}) \cr \gamma _{12} (x _{1} , x _{2}) & \gamma _{22} (x _{1} , x _{2}) } \right) , \quad
{\mbox{\textbf{\textit{g}}}} _{{\mathbb{S}} ^{2}} = \rho ^ {2} ( x _{1} , x _{2} ) \left( \matrix{ 1 & 0 \cr 0 & \sin ^{2} \Theta} \right) ,$$ where we denote with $(x _{1} , x _{2})$ coordinates in ${\mathbb{M}} _{2}$ and with $(\Theta , \Phi)$ the usual coordinates in ${\mathbb{S}} ^{2}$. Thanks to general covariance, we can subject ${\mbox{\textbf{\textit{g}}}} _{{\mathbb{M}} _{2}}$ to two conditions so that the four dimensional metric can be locally described by only two functions: a natural choice is $\rho (x _{1} , x _{2})$ plus another function coming from ${\mathbb{M}} _{2}$, which can be wisely chosen to be the invariant $\Delta = \gamma ^{AB} \rho _{,A} \rho _{,B}$ (the modulus square of the vector normal to the $\rho (x _{1}, x _{2}) = \mathrm{const.}$ surfaces). If $\Delta > 0$ we are in what is called an $R$-region, whereas if $\Delta < 0$ we are in a $T$-region [@bib:TRRegions]. When we apply the above decomposition in ${\mathcal{M}} _{{{\scriptscriptstyle{}(\pm)}}}$ we can conveniently use $\Delta _{{{\scriptscriptstyle{}(\pm)}}}$ to describe various properties of the dynamics of the shell in a coordinate invariant way [@bib:ClQuG2001..18..2195O].
In the following it is also useful to further specialize ${}^{{{\scriptscriptstyle{}(\pm)}}} g _{\mu \nu}$ and $g _{m n}$, defined before, as follows.
[Manifold]{} [Metric components]{} [Matter content]{}
--------------------------------------------------------- ----------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------
\[-5mm\] ${\mathcal{M}} _{{{\scriptscriptstyle{}(-)}}}$ $(t _{{{\scriptscriptstyle{}(-)}}} , \theta , \phi , r _{{{\scriptscriptstyle{}(-)}}})$ $\mathrm{diag}( - f _{{{\scriptscriptstyle{}(-)}}} (r _{{{\scriptscriptstyle{}(-)}}}) , r _{{{\scriptscriptstyle{}(-)}}} ^{2} , r _{{{\scriptscriptstyle{}(-)}}} ^{2} \sin ^{2} \theta , 1 / f _{{{\scriptscriptstyle{}(-)}}} (r _{{{\scriptscriptstyle{}(-)}}}) )$ spherically symm.
${\mathcal{M}} _{{{\scriptscriptstyle{}(+)}}}$ $(t _{{{\scriptscriptstyle{}(+)}}} , \theta , \phi , r _{{{\scriptscriptstyle{}(+)}}})$ $\mathrm{diag}( - f _{{{\scriptscriptstyle{}(+)}}} (r _{{{\scriptscriptstyle{}(+)}}}) , r _{{{\scriptscriptstyle{}(+)}}} ^{2} , r _{{{\scriptscriptstyle{}(+)}}} ^{2} \sin ^{2} \theta , 1 / f _{{{\scriptscriptstyle{}(+)}}} (r _{{{\scriptscriptstyle{}(+)}}}) )$ spherically symm.
$\Sigma$ $(\tau , \theta , \phi)$ $\mathrm{diag}(-1 , R (\tau) ^{2} , R (\tau) ^{2} \sin ^{2} \theta$ ) $M (R)$
We now see that $\mathrm{sign} ( \Delta _{{{\scriptscriptstyle{}(\pm)}}}) = \mathrm{sign} ( f _{{{\scriptscriptstyle{}(\pm)}}})$, so that if $f _{{{\scriptscriptstyle{}(\pm)}}} < 0$ we are in a $T ^{{{\scriptscriptstyle{}(\pm)}}}$-region of ${\mathcal{M}} _{{{\scriptscriptstyle{}(\pm)}}}$, whereas if $f _{{{\scriptscriptstyle{}(\pm)}}} > 0$ we are in an $R ^{{{\scriptscriptstyle{}(\pm)}}}$-region of ${\mathcal{M}} _{{{\scriptscriptstyle{}(\pm)}}}$. Moreover, after using all the freedom to fix the coordinate systems, $R (\tau)$ is the only degree of freedom ($\tau$ is the proper time of an observer comoving with $\Sigma$) describing the dynamics of the shell and only one junction condition is non-trivial: $$R
\left[ \epsilon \sqrt{\dot{R} ^{2} + f (R)} \right]
=
M (R)
\quad \mathrm{where} \quad
\left . n ^{\alpha} \partial _{\alpha} r _{{{\scriptscriptstyle{}(\pm)}}} \right | _{\Sigma}
=
\epsilon _{{{\scriptscriptstyle{}(\pm)}}} \sqrt{ \dot{R} ^{2} + f _{{{\scriptscriptstyle{}(\pm)}}} (R)}
;
\label{eq:juncon}$$ an overdot denotes a derivative with respect to $\tau$, $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$ are signs (we will come back later to their meaning) and $M (R)$ describes the matter energy content of the shell, after its equation of state has been specified. The only nontrivial junction condition (\[eq:juncon\]) can then be rewritten in the form $$\dot{R} ^{2} + V (R) = 0
\quad \mathrm{where} \quad
V (R) = - \frac{(R ^{2} f _{{{\scriptscriptstyle{}(-)}}} (R) + R ^{2} f _{{{\scriptscriptstyle{}(+)}}} (R) - M ^{2} (R)) ^{2} - 4 R ^{4} f _{{{\scriptscriptstyle{}(-)}}} (R) f _{{{\scriptscriptstyle{}(+)}}} (R)}{4 M ^{2} (R) R ^{2}}
.
\label{eq:claeffequ}$$ A closed form expression for the $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$ signs not involving $\dot{R}$ can also be obtained starting from (\[eq:juncon\]): $$\epsilon _{{{\scriptscriptstyle{}(\pm)}}}
=
\mathrm{sign} \{ M (R) ( R ^{2} f _{{{\scriptscriptstyle{}(-)}}} (R) - R ^{2} f _{{{\scriptscriptstyle{}(+)}}} (R) \mp M ^{2} (R) ) \}
.
\label{eq:epssig}$$ Eqs.$\:$(\[eq:claeffequ\]) and (\[eq:epssig\]) form a set of equations which is completely equivalent to (\[eq:juncon\]) (see also Prop.$\:$\[prop:radno\_res\] below). *Classical solutions* of the junction condition can exist only in the region $V (R) \leq 0$ and their turning points are solutions of the equation $V (R) = 0$. We call *tunnelling trajectories* the solutions in the inverted potential $- V (R)$. Also notice, that the junction condition (\[eq:isrjuncon\]) is a first order equation, (it contains $\dot{R}$ but not $\ddot{R}$) and it is not the *equation of motion* of the system but a first integral of it [@bib:ClQuG1997..14..2727S; @bib:LagFormShells]. The second order equation of motion can be obtained from an effective Lagrangian, $L _{\mathrm{EFF}}$, that describes the dynamics of the *only* remaining degree of freedom $R (\tau)$. If $$H _{\mathrm{EFF}} (R , \dot{R})
=
R
\left[
\epsilon \sqrt{\dot{R} ^{2} + f}
\right]
-
M (R)
\: \: \: \: \: \: \mathrm{and} \: \: \: \: \: \:
P _{\mathrm{EFF}} (R , \dot{R})
=
R
\left[
\tanh ^{-1}
\left(
\frac{\dot{R}}{\epsilon \sqrt{\dot{R} ^{2} + f}}
\right) ^{\mathrm{sgn} (f)}
\right]
\label{eq:effmom}$$ are the effective (super)hamiltonian and effective momentum, respectively, we have that the $L _{\mathrm{EFF}} = P _{\mathrm{EFF}} \dot{R} - H _{\mathrm{EFF}}$ holds and the second order equation of motion is given by $$\frac{d}{d \tau}
\left(
\frac{\partial L _{\mathrm{E FF}}}{\partial \dot{R}}
\right)
-
\frac{\partial L _{\mathrm{EFF}}}{\partial R}
=
0
.$$ Moreover, $H _{\mathrm{EFF}} \equiv 0$ is a constraint on the system. We also anticipate that in this setup the expression for the *Euclidean momentum*, i.e. the momentum along a tunnelling trajectory, is $$P ^{\mathrm{(e)}} _{\mathrm{EFF}} (R , R')
=
R
\left[
\arctan
\left(
\frac{R'}{\epsilon \sqrt{f (R) - (R') ^{2}}}
\right)
\right]
=
- \imath P _{\mathrm{EFF}} (R , \dot{R})
.
\label{eq:euceffmom}$$ The last equality in the equation above suggests that the Euclidean system can be obtained by Wick rotating the classical one[^4]. This can be proved deriving the Euclidean junction condition [@bib:NuPhy1990B339...417G].
\[sec:genres\]Some general results
==================================
We now prove some general results to provide toeholds for the discussion of the problems that will emerge in the Euclidean sector. Some of them appear already in the literature, but not in a systematic exposition. These results also provide simple, but useful consistency checks.
The junction condition and the effective potential are invariant under the relabelling “$+ \leftrightarrow -$”, whereas the signs change as $\epsilon _{{{\scriptscriptstyle{}(\pm)}}} \rightarrow - \epsilon _{{{\scriptscriptstyle{}(\mp)}}}$.
*Proof:* the validity of the above result for $V (R)$ and $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$ is manifest from (\[eq:claeffequ\]) and (\[eq:epssig\]). Then the invariance of the junction condition (\[eq:juncon\]) immediately follows. $\square$
Another property directly related to the algebraic structure of (\[eq:juncon\]), is the following.
\[prop:f\_RbigV\_R\] The relations $V(R) \leq f _{{{\scriptscriptstyle{}(\pm)}}} (R)$ always hold. Moreover, if $V(\bar{R}) = f _{{{\scriptscriptstyle{}(\pm)}}} (\bar{R})$, then $V (R)$ and $f _{{{\scriptscriptstyle{}(\pm)}}} (R)$ are tangent at $R = \bar{R}$.
*Proof:* the first result follows immediately after some algebra, since $$f _{{{\scriptscriptstyle{}(\pm)}}} (R) - V (R)
=
\frac{\left( R ^{2} f _{{{\scriptscriptstyle{}(-)}}} - R ^{2} f _{{{\scriptscriptstyle{}(+)}}} \mp M ^{2} (R) \right) ^{2}}{4 M ^{2} (R) R ^{2}}
=
a _{{{\scriptscriptstyle{}(\pm)}}} ^{2} (R)
\geq 0
.
\label{eq:radargsemposdef}$$ This, together with the first derivative of the equation above, $f ' _{{{\scriptscriptstyle{}(\pm)}}} (R) - V ' (R) = 2 a (R) a ' (R)$ completes the proof of the second result, because $f _{{{\scriptscriptstyle{}(\pm)}}} (\bar{R}) = V (\bar{R})$, i.e. $a _{{{\scriptscriptstyle{}(\pm)}}} (\bar{R}) = 0$, implies $f ' _{{{\scriptscriptstyle{}(\pm)}}} (\bar{R}) = V ' (\bar{R})$. $\square$
Two immediate consequences then follow.
\[prop:sigandtan\] Along a classical trajectory the signs $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$ vanish at the points at which the potential $V (R)$ is tangent to the metric function $f _{{{\scriptscriptstyle{}(\pm)}}} (R)$ and only at these points.
*Proof:* the result follows from (\[eq:epssig\]) and (\[eq:radargsemposdef\]), which show that $\epsilon _{{{\scriptscriptstyle{}(\pm)}}} = \mathrm{sign} (a _{{{\scriptscriptstyle{}(\pm)}}} (R))$. $\square$
\[prop:radandsol\] The conditions about the well-definiteness of the two radicals in the junction condition (\[eq:juncon\]) do not impose any restriction on the classical/tunnelling solutions. \[prop:radno\_res\]
*Proof:* from what we have seen above, if the effective (classical looking) equation (\[eq:claeffequ\]) is satisfied, the arguments of the radicals, $\dot{R} ^{2} + f _{{{\scriptscriptstyle{}(\pm)}}} (R) = f _{{{\scriptscriptstyle{}(\pm)}}} (R) - V (R)$, are always nonnegative by Prop.$\:$\[prop:f\_RbigV\_R\]. $\square$
Up to this point we have systematized some results concerning the mutual relationships of the metric functions $f _{{{\scriptscriptstyle{}(\pm)}}}$, the potential $V$ and the signs $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$. Some additional results are now obtained about the relative positions of the zeroes of $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$, $V$ and the positions of the zeroes of $f _{{{\scriptscriptstyle{}(\pm)}}}$.
\[prop:sigchanegf\_R\] The signs $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$ can change either i) along a classical solution of (\[eq:juncon\]) in a region in which $f _{{{\scriptscriptstyle{}(\pm)}}} \leq 0$, or ii) along a tunnelling trajectory.
*Proof:* let us assume that at the point $R = \bar{R}$ we have $\epsilon _{{{\scriptscriptstyle{}(\pm)}}} (\bar{R}) = 0$; this implies $a _{{{\scriptscriptstyle{}(\pm)}}} (\bar{R}) = 0$ by Prop.$\:$\[prop:sigandtan\]. If we are along a classical solution of (\[eq:juncon\]) because of (\[eq:claeffequ\]) we must have $V (\bar{R}) \leq 0$, i.e. $f _{{{\scriptscriptstyle{}(\pm)}}} (\bar{R}) \leq 0$. If instead we have $f _{{{\scriptscriptstyle{}(\pm)}}} (\bar{R}) > 0$, then we also have $V (\bar{R}) >0$, i.e. we are along a tunnelling trajectory. $\square$
An analogous result can then be proved for the turning points of the classical solutions.
\[prop:turpoiposf\_R\] A turning point $R _{0}$ of (\[eq:juncon\]) always satisfies $f _{{{\scriptscriptstyle{}(\pm)}}} (R _{0}) \geq 0$.
*Proof:* we know that in general $f _{{{\scriptscriptstyle{}(\pm)}}} (R) - V (R) \geq 0$; thus at $R _{{\scriptscriptstyle{0}}}$, where $V(R _{{\scriptscriptstyle{0}}}) = 0$, we have $f _{{{\scriptscriptstyle{}(\pm)}}} (R _{{\scriptscriptstyle{0}}}) \geq 0$. $\square$
Related to this result is the following one, which details the regularity properties of $P _{\mathrm{EFF}}$.
\[prop:effmomdef\] The effective momentum $P _{\mathrm{EFF}}$ is always well defined on a classically allowed trajectory, with the exception of the points at which $f _{{{\scriptscriptstyle{}(\pm)}}}$ vanish.
*Proof:* by Prop.$\:$\[prop:radno\_res\] the radicals that appear in the expression of $P _{\mathrm{EFF}}$ are always well defined along a classical solution of the junction condition. Critical points of $P _{\mathrm{EFF}}$ can thus only appear either if they vanish or because of the presence of the inverse hyperbolic tangent, whose argument must be in the interval $(-1,+1)$. Two complementary subcases can be singled out.\
$f _{{{\scriptscriptstyle{}(\pm)}}} \neq 0)$ Under this condition we can rule out the first possibility, since along a classically allowed trajectory $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$ can vanish only if $f _{{{\scriptscriptstyle{}(\pm)}}} < 0$, when the radical is in the numerator. Troubles with the inverse hyperbolic tangent are quickly excluded as well. If $f _{{{\scriptscriptstyle{}(\pm)}}} > 0$, then the exponent of its argument is $+1$, and we can forget about it. At the same time the absolute value of the numerator is lower than the one of the denominator, so the momentum is well defined in this case. If $f _{{{\scriptscriptstyle{}(\pm)}}} < 0$, then the absolute value of the numerator *is* bigger than the one of the denominator, but the ratio is raised to the power $-1$ and still there is no problem.\
$f _{{{\scriptscriptstyle{}(\pm)}}} = 0)$ This case is more subtle; if the corresponding $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$ sign is non vanishing, then the absolute value of the argument of the inverse hyperbolic tangent is equal to $1$ and the momentum has a logarithmic, i.e. integrable, divergence. If, instead, also $\epsilon _{{{\scriptscriptstyle{}(\pm)}}} = 0$, a *case by case* analysis is required.\
We can, thus, conclude that the momentum can be *not well defined* only at the points where $f _{{{\scriptscriptstyle{}(\pm)}}} = 0$; if $\epsilon _{{{\scriptscriptstyle{}(\pm)}}} \neq 0$ it has a logarithmic divergence; a *case by case* analysis has instead to be done if $\epsilon _{{{\scriptscriptstyle{}(\pm)}}} = 0$. $\square$
The regularity of the Euclidean effective momentum can be analyzed in general as well.
\[prop:euceffmomdef\] The Euclidean effective momentum $P ^{(\mathrm{e})} _{\mathrm{EFF}}$ is always well defined along a tunnelling trajectory in the sense that, at most, it can have discontinuities.
*Proof:* from a quick inspection of (\[eq:euceffmom\]) and from the result in Prop.$\:$\[prop:radno\_res\] we see that the only trouble to the momentum can come from vanishing radicals in the denominator, i.e. from vanishing $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$. This can indeed happen along a tunnelling trajectory (Prop.$\:$\[prop:sigchanegf\_R\]) and if this happens *inside* it, e.g. at $\bar{R}$, the argument of the corresponding inverse tangent function tends to $\pm \infty$ there. If $\lim _{R \to \bar{R} ^{+}} \neq \lim _{R \to \bar{R} ^{-}}$ in the standard branch of the $\arctan$ function a discontinuity appears. This discontinuity can be eliminated by choosing another appropriate branch, although this will, in general, affect the value of $P ^{(\mathrm{e})} _{\mathrm{EFF}}$ at, at least, one turning point. If $\bar{R}$ coincides with a turning point, i.e. it occurs not *inside* but *at the boundary* of a tunnelling trajectory, a *case by case* analysis is again required. $\square$
The physical content of these results can be expressed in general terms using the concepts of $R ^{{{\scriptscriptstyle{}(\pm)}}}$ and $T ^{{{\scriptscriptstyle{}(\pm)}}}$ introduced before. We have, in fact, proved that:
1. the $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$ signs can only vanish either i) in the closure of a $T ^{{{\scriptscriptstyle{}(\pm)}}}$ region along a classical solution of the junction condition (\[eq:juncon\]) or ii) along a tunnelling trajectory;
2. turning points can only be present in the closure of an $R ^{{{\scriptscriptstyle{}(\pm)}}}$ region; this second fact is coherent with the fact that in the Euclidean description of the manifolds ${\mathcal{M}} _{{{\scriptscriptstyle{}(\pm)}}}$, there is no region corresponding to the $T ^{{{\scriptscriptstyle{}(\pm)}}}$ ones; loosely speaking, the shell has *nowhere* to tunnel from one of these regions!
These results are valid for arbitrary spherically symmetric junctions, independently from the matter content of spacetime and/or of the shell. It is also noteworthy to stress the following particular case, which happens when a turning point is exactly on the boundary of an $R$-region of one of the two spacetimes (i.e. at a point where at least one of $f _{{{\scriptscriptstyle{}(\pm)}}}$ vanishes). In this case one of the $f _{{{\scriptscriptstyle{}(\pm)}}}$ is zero, but $V$ is also zero, so the corresponding $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$ sign is zero too. We call this the *exceptional case*, but we will not consider it further here.
\[sec:tunpro\]Tunnelling problems
=================================
The above results can be used to obtain general insight non only about the classical dynamics of the system, but, especially, about the semiclassical one. Here we will not discuss the interesting possibilities of using WKB methods to determine the semiclassical stationary states [@bib:SemicState; @bib:ClQuG2002..19..6321A], but we will, instead, concentrate on the tunnelling process. In particular, it is possible to use the Euclidean momentum (\[eq:euceffmom\]) to calculate the value of the tunnelling action (i.e of the probability) $$S _{\mathrm{TUN}} = \int _{R _{1}} ^{R _{2}} P _{\mathrm{EFF}} ^{\mathrm{(e)}} (R) d R
\quad \mathrm{where} \quad
P _{\mathrm{EFF}} ^{\mathrm{(e)}} (R) = \left . P _{\mathrm{EFF}} ^{\mathrm{(e)}} (R , R') \right | _{R ' = \sqrt{V (R)}}
;$$ $P _{\mathrm{EFF}} ^{\mathrm{(e)}} (R)$ is the Euclidean Momentum *evaluated along a tunnelling trajectory*. This has been done for various configurations and results in agreement with other independent calculations have been obtained (for instance the results by Coleman and de Luccia [@bib:PhReD1980..21..3305L] and by Parke [@bib:PhLeB1983.121...313P], can be reproduced). Unfortunately things do not always work out so smoothly: problems arise when at least one of the $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$ signs vanishes along the tunnelling trajectory. These problems relate i) to the ones connected with the difficulty to build the Euclidean manifold interpolating between the classical spacetime configuration described by the pre- and post-tunnelling solutions of the junction condition [@bib:NuPhy1990B339...417G] and ii) to the ones connected with the difference in the tunnelling description given by canonical and path-integral methods [@bib:NuPhy1990B339...417G]. It is convenient to summarize these issues using as a definite model, the case in which we have a de Sitter/Schwarzschild junction ( $f _{{{\scriptscriptstyle{}(+)}}} (r _{{{\scriptscriptstyle{}(+)}}}) = 1 - \chi ^{2} r _{{{\scriptscriptstyle{}(+)}}} ^{2}$, $f _{{{\scriptscriptstyle{}(-)}}} (r _{{{\scriptscriptstyle{}(-)}}}) = 1 - 2 m / r _{{{\scriptscriptstyle{}(-)}}}$) by a matter shell with equation of state $p = - \sigma$, $\sigma$ being the tension of the shell (i.e. $M (R) = 4 \pi \sigma R ^{2}$).
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\
![\[fig:firtuncas\][A case of tunnelling where no problems appear: in \[a\] and \[b\] the Euclidean parts of spacetime participating in the junction are the grayed areas. Initial and final slices are labelled (i) and (f) respectively. The Euclidean trajectory of the shell is the solid curve. Initial and final Euclidean times and the normal are also shown.]{}](fig001a "fig:"){width="4.5cm"} ![\[fig:firtuncas\][A case of tunnelling where no problems appear: in \[a\] and \[b\] the Euclidean parts of spacetime participating in the junction are the grayed areas. Initial and final slices are labelled (i) and (f) respectively. The Euclidean trajectory of the shell is the solid curve. Initial and final Euclidean times and the normal are also shown.]{}](fig002a "fig:"){width="4.5cm"} ![\[fig:firtuncas\][A case of tunnelling where no problems appear: in \[a\] and \[b\] the Euclidean parts of spacetime participating in the junction are the grayed areas. Initial and final slices are labelled (i) and (f) respectively. The Euclidean trajectory of the shell is the solid curve. Initial and final Euclidean times and the normal are also shown.]{}](fig003a "fig:"){width="4.5cm"}
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To this end, in Fig.$\:$\[fig:firtuncas\] we first analyze a situation free from troubles. In panels \[a\] and \[b\] we see that the normal is always transverse to the constant $r _{{{\scriptscriptstyle{}(\pm)}}}$ surfaces. According to the definitions of $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$, they then do not vanish, so that *via* Prop.$\:$[\[prop:euceffmomdef\]]{} we know that the momentum does not have any troubles along the tunnelling trajectory, as shown in panel \[c\].
A different situation is, instead, shown in Fig.$\:$\[fig:sectuncas\].
![\[fig:sectuncas\][A case of tunnelling when one part of the spacetime participating in the junction is problematic. If we proceed as in the case of Fig. [\[fig:firtuncas\]]{} we encounter soon a point $P$ after which, the determination of the part of spacetime participating in the junction becomes unclear (see the text for details).]{}](fig001b "fig:"){width="4.5cm"} ![\[fig:sectuncas\][A case of tunnelling when one part of the spacetime participating in the junction is problematic. If we proceed as in the case of Fig. [\[fig:firtuncas\]]{} we encounter soon a point $P$ after which, the determination of the part of spacetime participating in the junction becomes unclear (see the text for details).]{}](fig002b "fig:"){width="4.5cm"} ![\[fig:sectuncas\][A case of tunnelling when one part of the spacetime participating in the junction is problematic. If we proceed as in the case of Fig. [\[fig:firtuncas\]]{} we encounter soon a point $P$ after which, the determination of the part of spacetime participating in the junction becomes unclear (see the text for details).]{}](fig003b "fig:"){width="4.5cm"}
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The de Sitter part of the junction in panel \[a\] is fine as before, but the Schwarzschild one in panel \[b\] has a peculiar feature: there is a point $P$ along the trajectory at which the normal is not transverse to the constant $r _{{{\scriptscriptstyle{}(-)}}}$ surface, so that $\epsilon _{{{\scriptscriptstyle{}(-)}}} (P) = 0$. Thus, we have some difficulty in identifying the part of the Schwarzschild spacetime participating in the junction, since the small area which is both *dark-grayed* and *crossed-hatched* is covered twice by the evolution of the Euclidean spacetime slice. This difficulty in identifying the instanton is reflect by the momentum plot in panel \[c\]: a discontinuity appears, as expected from Prop.$\:$[\[prop:euceffmomdef\]]{}. Notice that if we naively take the union of the grayed regions as the part of spacetime participating in the junction, a strange boundary (the $AB$ line) appears[^5].
The problem shown above is not typical of the Schwarzschild patch and can appear also in the de Sitter one, as shown in Fig.$\:$\[fig:thituncas\].
------------------------------------------------------------------------
\
![\[fig:thituncas\][The problem which appeared in Fig.$\:$[\[fig:sectuncas\]]{} is generic and can affect both spacetimes, separately, or at once, as in this case. Now, in panel \[a\] some problems appear in connection with the existence of point $Q$, whereas in panel \[b\] the situation maintains the same difficulty encountered in Fig.$\:$[\[fig:sectuncas\]]{}. Two discontinuities of $P _{\mathrm{EFF}} ^{(\mathrm{e})} (R)$ are present (panel \[c\]).]{}](fig001c "fig:"){width="4.5cm"} ![\[fig:thituncas\][The problem which appeared in Fig.$\:$[\[fig:sectuncas\]]{} is generic and can affect both spacetimes, separately, or at once, as in this case. Now, in panel \[a\] some problems appear in connection with the existence of point $Q$, whereas in panel \[b\] the situation maintains the same difficulty encountered in Fig.$\:$[\[fig:sectuncas\]]{}. Two discontinuities of $P _{\mathrm{EFF}} ^{(\mathrm{e})} (R)$ are present (panel \[c\]).]{}](fig002c "fig:"){width="4.5cm"} ![\[fig:thituncas\][The problem which appeared in Fig.$\:$[\[fig:sectuncas\]]{} is generic and can affect both spacetimes, separately, or at once, as in this case. Now, in panel \[a\] some problems appear in connection with the existence of point $Q$, whereas in panel \[b\] the situation maintains the same difficulty encountered in Fig.$\:$[\[fig:sectuncas\]]{}. Two discontinuities of $P _{\mathrm{EFF}} ^{(\mathrm{e})} (R)$ are present (panel \[c\]).]{}](fig003c "fig:"){width="4.5cm"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In both diagrams, at $Q$ in panel \[a\] and at $P$ in panel \[b\], the normal to the shell become non-transverse to the constant $r _{{{\scriptscriptstyle{}(\pm)}}}$ surfaces. At these points the corresponding signs $\epsilon _{{{\scriptscriptstyle{}(\pm)}}}$ vanish and the momentum develops a discontinuity, as shown in panel \[c\]. It is non trivial to build the junction and we face again a double covering with inverted normal direction in the Schwarzschild patch during the evolution of the initial slice (i) into the final one (f).
Concerning the last two cases, we notice, as we did at the end of the proof of Prop.$\:$\[prop:euceffmomdef\], that, in view of the form of the effective momentum $P _{\mathrm{EFF}} ^{\mathrm{(e)}}$, it is certainly possible to choose appropriate branches of the inverse tangent functions to *cure* the discontinuity: nevertheless, this spoils the vanishing of the momentum at one of the turning points. It would, moreover, be interesting, to understand these possibilities in connection with the structure of the Euclidean spacetime diagram.
\[sec:dissec\]Discussion and Conclusion
=======================================
Remembering the results presented in Sec.$\:$\[sec:genres\], we would like to point out that the problems discussed in the previous section under very specific settings, are in fact general ones. In our opinion this fact has not yet been properly recognized. We would also like to make clear that, although the problems with the effective momentum *could* be solved by arguing that it is the effective theory that has intrinsic limits, *we think that this partial solution would be rather unsatisfactory*. The main point we are trying to make here is that the problems of the effective formulation are *closely tied* to the geometric properties of the Euclidean solution to the junction condition and to the *Euclidean structure of the spacetime patches* that participate in the junction. Moreover, the difficulties described above *can* be absent, and *when they are absent*, perfectly consistent results are obtained. In this respect, it becomes even more suggestive to draw the following picture. The idea of bubbles/shells tunnelling has been originally developed to overcome some weak points in the description provided by purely classical models of vacuum bubbles using general relativistic shells: in particular, the problem of initial singularity, i.e. the fact that exponentially expanding solutions giving rise to a baby universe, classically, have a singularity in their past. Tunnelling (i.e. the use of quantum effects) solves this issue: in fact, we can start with a solution regular in the past (the bounded solution, which classically would never grow enough) and have it tunnel into an infinitely expanding one (the unbounded/bounce solution) which has the late time behavior we are interested in, but cannot exist at early times. We are, thus *using quantum effects to circumvent the consequences of classical singularity theorems*. The same idea has been employed also in a logically opposite direction using different spacetimes to build up the junction: then the reversed tunnelling process can describe a collapsing shell of matter [@bib:ShellCollapse], which classically doomed to crash in a future singularity, is, instead, saved by quantum effects, again avoiding the fate prescribed by classical singularity theorems. Another fact which has not yet been appreciated is that, in view of the simple but general results of Sec.$\:$\[sec:genres\], this kind of processes is affected by the same difficulties, again related to a hard to interpret behavior in the Euclidean sector. In this sense the more general formulation of the problems shown in Figs.$\:$\[fig:firtuncas\] and \[fig:sectuncas\], that can be obtained in terms of the results of Sec.$\:$\[sec:genres\], shows that they are *very general issues which appear when we try to use semiclassical quantum effects to circumvent the consequences of classical singularity theorems*. As we said, this issues are not restricted to applications to the cosmological (baby-universes) scenario, as it is often believed, but represent instead another manifestation of the intrinsic difficulty in the interplay between the properties of general relativity and those of quantum theory. It seems to us a great opportunity, given to us by the intuitive and beautiful geometric character of Israel junction conditions, that these issues can appear at a technically rather simple level, offering us the possibility to concentrate our attention on their physical/geometric significance. This study is still work in progress, which at present is focused on finding general geometric criteria i) to identify the appearance of the above issues in the junction tunnelling process and ii) to characterize the difficulties in the Euclidean sector in terms of intuitive properties of the effective theory. Additional results will be reported (hopefully soon) elsewhere.
#### Acknowledgements. {#acknowledgements. .unnumbered}
[The author would like to thank A. Guth for some stimulating discussion on the subject of this work, which is partly supported by an Invitation Fellowship of the *Japan Society for the Promotion of Science* and has been partly supported by a grant of the *Fulbright Commission*.]{}
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[^1]: Email: `ansoldi@trieste.infn.it`; Webpage: `http://www-dft.ts.infn.it/\simansoldi`; Mailing address: Dipartimento di Matematica e Informatica, Università degli Studi di Udine, via delle Scienze 206, I-33100 Udine (UD), ITALY.
[^2]: Submitted to the Proceedings of Science (`http://pos.sissa.it`.)
[^3]: We assume that it points from ${\mathcal{M}} _{{{\scriptscriptstyle{}(-)}}}$ to ${\mathcal{M}} _{{{\scriptscriptstyle{}(+)}}}$; moreover, since $\Sigma$ is timelike, *the normal is also transverse*.
[^4]: $R '$ denotes the derivative of $R$ with respect to $\tau ^{(\mathrm{e})} = - \imath \tau$, so that $\dot{R} = \imath R '$. As far as the spacetime structure is concerned, we also have to Wick rotate the time coordinates in ${\mathcal{M}} _{{{\scriptscriptstyle{}(\pm)}}}$, $t _{{{\scriptscriptstyle{}(\pm)}}} ^{(\mathrm{e})} = - \imath t _{{{\scriptscriptstyle{}(\pm)}}}$, to obtain the corresponding Euclidean manifolds.
[^5]: There are reasonable proposals [[@bib:NuPhy1990B339...417G]]{} to deal with this problem, but we will not discuss them here. The point we would like to stress, focusing on the problem rather than the possible solutions, is that particular care must be used when dealing with the Euclidean junction.
|
---
author:
- |
L. Flodén, P. Jonasson, M. Olsson Lindberg, T. Lobkova\
(Department of Quality Technology and Management,Mechanical\
Engineering and Mathematics,Mid Sweden University)
title: '**Homogenization of monotone parabolic problems with an arbitrary number of spatial and temporal scales** '
---
**1. Introduction**
The mathematical theory of nonlinear partial differential equations plays an important role in e.g. applied mathematics and physics. In this paper we present a homogenization result for the general monotone parabolic problem with multiple spatial and temporal scales$$\begin{aligned}
\partial _{t}u^{\varepsilon }(x,t)-\nabla \cdot a\left( \frac{x}{\hat{\varepsilon}_{1}},\ldots ,\frac{x}{\hat{\varepsilon}_{n}},\frac{t}{\check{\varepsilon}_{1}},\ldots ,\frac{t}{\check{\varepsilon}_{m}},\nabla
u^{\varepsilon }(x,t)\right) \!\!\! &=&\!\!\!f(x,t)\,\text{\thinspace in\ }\Omega _{T}, \notag \\
u^{\varepsilon }(x,t)\!\!\! &=&\!\!\!0\,\text{\thinspace \thinspace on
\thinspace }\partial \Omega \!\!\times \!\!(0,T),\,\,\,\,\,{}~~~~
\label{first one} \\
u^{\varepsilon }(x,0)\!\!\! &=&\!\!\!u^{0}(x)\,\text{\thinspace in\ }\Omega ,
\notag\end{aligned}$$where $f\in L^{2}(\Omega _{T})$ and $u^{0}\in L^{2}(\Omega )$. Here $\Omega $ is an open bounded set in $\mathbb{R}
^{N}$ with smooth boundary and $\Omega _{T}=\Omega \times (0,T)$. We let $Y=(0,1)^{N}$ and $S=(0,1)$ and we assume that $a$ is $Y$-periodic in the $n$ first variables and $S$-periodic in the following $m$ variables. Finally we let $\hat{\varepsilon}_{k}$ for $k=1,\ldots ,n$ and $\check{\varepsilon}_{j}$ for $j=1,\ldots ,m$ be scale functions depending on $\varepsilon $ that tend to zero as $\varepsilon $ does, where the scales are assumed to fulfil certain conditions of separatedness.
The homogenization of (\[first one\]) means studying the asymptotic behavior of the corresponding sequence of solutions $u^{\varepsilon }$ as $\varepsilon $ tends to zero and finding the limit problem $$\begin{aligned}
\partial _{t}u(x,t)-\nabla \cdot b(x,t,\nabla u) &=&f(x,t)\text{ in }\Omega
_{T}, \\
u(x,t) &=&0\text{ on }\partial \Omega \times (0,T), \\
u(x,0) &=&u^{0}(x)\text{ in }\Omega ,\end{aligned}$$which admits the function $u$, the limit of $\left\{ u^{\varepsilon
}\right\} $, as its unique solution. Here $b$ is characterized by local problems, one for each microscopic spatial scale. For more informative texts on homogenization theory we suggest e.g. [@Al1], [@CoDo] and [LNW]{}.
The main tools to carry out the homogenization process for (\[first one\]) are multiscale convergence and very weak multiscale convergence in the evolution setting. Here very weak multiscale convergence, see e.g. [FHOP]{} and [@FHOLP], is the key to handling the difficulties that appear when rapid time oscillations are present. The nonlinearity of the problem is treated by applying the perturbed test functions method.
Homogenization results for linear parabolic equations with oscillations in one spatial scale and one temporal scale were studied by using asymptotic expansions in [@BLP]. In [@Ho1] parabolic problems containing fast oscillations in space as well as in time were treated for the first time applying two-scale convergence methods. Parabolic homogenization problems have also been investigated in e.g. [@FlOl1] and [@FHOLP1] for different choices of fixed scales. Linear parabolic problems with an arbitrary number of scales in both space and time were homogenized in [FHOLP]{}. Homogenization results for monotone, not necessarily linear, problems have been presented in e.g. [@FlOl], [@NgWo], [@FHOS], [@Wo1] and [@Wo2]. The case with one spatial microscale and an arbitrary number of temporal scales was treated by Persson in [@Per1].
The paper is organized in the following way. In Section 2 we give some preparatory theory concerning multiscale and very weak multiscale convergence. In Section 3 we present the homogenization result for ([first one]{}) and in the last section we look at a special case of ([first one]{}) to illustrate the use of the presented result.
We let $F_{\sharp }(Y)$ be the space of all functions in $F_{loc}(\mathbb{R}
^{N})$ which are the periodic repetition of some function in $F(Y)$. We also let $Y_{k}=Y$ for $k=1,\ldots ,n,$ $Y^{n}=Y_{1}\times \cdots \times Y_{n}$ (the nN-dimensional open unit cell), $y^{n}=y_{1},\ldots ,y_{n}$ (corresponding spatial multivariable), $dy^{n}=dy_{1}\cdots dy_{n},$ $S_{j}=S $ for $j=1,\ldots ,m,$ $S^{m}=S_{1}\times \cdots \times S_{m}$ (the m-dimensional open unit cell), $s^{m}=s_{1},\ldots ,s_{m}$ (corresponding local temporal multivariable), $ds^{m}=ds_{1}\cdots ds_{m}$ and $\mathcal{Y}_{n,m}=Y^{n}\times S^{m}$, where we interpret $\mathcal{Y}_{0,m}$ as $S^{m}$. We let $\hat{\varepsilon}_{k}(\varepsilon )$, for $k=1,\ldots ,n,$ and $\check{\varepsilon}_{j}(\varepsilon )$, $j=1,\ldots ,m,$ be strictly positive functions such that $\hat{\varepsilon}_{k}(\varepsilon )$ and $\check{\varepsilon}_{j}(\varepsilon )$ go to zero when $\varepsilon $ does. We also use the notations $\hat{\varepsilon}^{n}=\hat{\varepsilon}_{1},\ldots ,\hat{\varepsilon}_{n}$ and $\check{\varepsilon}^{m}=\check{\varepsilon}_{1},\ldots ,\check{\varepsilon}_{m}$ and furthermore $\frac{x}{\hat{\varepsilon}^{n}}$ denotes $\frac{x}{\hat{\varepsilon}_{1}},\ldots ,\frac{x}{\hat{\varepsilon}_{n}}$ and, similarly, by $\frac{t}{\check{\varepsilon}^{m}}$ we mean $\frac{t}{\check{\varepsilon}_{1}},\ldots ,\frac{t}{\check{\varepsilon}_{m}}$.
**2. Multiscale and very weak multiscale convergence**
In [@Ng1] Nguetseng presented a new homogenization technique based on a certain type of convergence which has become known as two-scale convergence. This was extended in [@AlBr] to so-called multiscale convergence, which allows use of multiple scales and makes it possible to capture numerous types of spatial microscopic oscillations. Below we define evolution multiscale convergence i.e., the concept has been further developed to include temporal oscillations, see [@FHOLP].
\[Defintion multiscale\]A sequence $\left\{ u^{\varepsilon }\right\} $ in $L^{2}\left( \Omega _{T}\right) $ is said to $(n+1,m+1)$-scale converge to $u_{0}\in L^{2}\left( \Omega _{T}\times \mathcal{Y}_{n,m}\right) $ if $$\begin{gathered}
\int_{\Omega _{T}}u^{\varepsilon }(x,t)v\left( x,t,\frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}}\right) dxdt \\
\rightarrow \int_{\Omega _{T}}\int_{\mathcal{Y}_{n,m}}u_{0}(x,t,y^{n},s^{m})v(x,t,y^{n},s^{m})~dy^{n}ds^{m}dxdt\end{gathered}$$for any $v\in L^{2}\left( \Omega _{T};C_{\sharp }\left( \mathcal{Y}_{n,m}\right) \right) $. We write $$u^{\varepsilon }\left( x,t\right) \overset{n+1,m+1}{\rightharpoonup }u_{0}(x,t,y^{n},s^{m}).$$
Next we define some concepts regarding relations between scale functions.
We say that the scales in a list $\left\{ \varepsilon _{1},\ldots
,\varepsilon _{n}\right\} $ are separated if$$\lim_{\varepsilon \rightarrow 0}\frac{\varepsilon _{k+1}}{\varepsilon _{k}}=0$$for $k=1,\ldots ,n-1$ and that the scales are well-separated if there exists a positive integer $l$ such that$$\lim_{\varepsilon \rightarrow 0}\frac{1}{\varepsilon _{k}}\left( \frac{\varepsilon _{k+1}}{\varepsilon _{k}}\right) ^{l}=0$$for $k=1,\ldots ,n-1$.
\[Lists of scales\]Let $\left\{ \hat{\varepsilon}_{1},\ldots ,\hat{\varepsilon}_{n}\right\} $ and $\left\{ \check{\varepsilon}_{1},\ldots ,\check{\varepsilon}_{m}\right\} $ be lists of (well-)separated scales. Collect all elements from both lists in one common list. If from possible duplicates, where by duplicates we mean scales which tend to zero equally fast, one member of each pair is removed and the list in order of magnitude of all the remaining elements is (well-)separated, the lists $\left\{ \hat{\varepsilon}_{1},\ldots ,\hat{\varepsilon}_{n}\right\} $ and $\left\{ \check{\varepsilon}_{1},\ldots ,\check{\varepsilon}_{m}\right\} $ are said to be jointly (well-)separated.
We give the two following theorems, which state a compactness result for $\left( n+1,m+1\right) $-scale convergence and a characterization of multiscale limits for gradients, respectively.
\[thmultiskale\]Let $\left\{ u^{\varepsilon }\right\} $ be a bounded sequence in $L^{2}\left( \Omega _{T}\right) $ and suppose that the lists $\left\{ \hat{\varepsilon}_{1},\ldots ,\hat{\varepsilon}_{n}\right\} $ and $\left\{ \check{\varepsilon}_{1},\ldots ,\check{\varepsilon}_{m}\right\} $ are jointly separated. Then there exists a $u_{0}$ in $L^{2}\left( \Omega
_{T}\times \mathcal{Y}_{n,m}\right) $ such that, up to a subsequence, $$u^{\varepsilon }\left( x,t\right) \overset{n+1,m+1}{\rightharpoonup }u_{0}\left( x,t,y^{n},s^{m}\right) .$$
See Theorem 2.66 in [@Per2] or Theorem A.1. in [@FHOLP].
The space $W_{2}^{1}(0,T;H_{0}^{1}(\Omega ),L^{2}(\Omega ))$ that appears in the theorem below is the space of all functions in $L^{2}(0,T;H_{0}^{1}(\Omega ))$ such that the time derivative belongs to $L^{2}(0,T;H^{-1}(\Omega ))$.
\[gradkarmulti\]Let $\left\{ u^{\varepsilon }\right\} $ be a bounded sequence in $W_{2}^{1}(0,T;H_{0}^{1}\left( \Omega \right) ,L^{2}\left(
\Omega \right) )$ and suppose that the lists $\left\{ \hat{\varepsilon}_{1},\ldots ,\hat{\varepsilon}_{n}\right\} $ and $\left\{ \check{\varepsilon}_{1},\ldots ,\check{\varepsilon}_{m}\right\} $ are jointly well-separated. Then, up to a subsequence,$$\begin{aligned}
u^{\varepsilon }\left( x,t\right) &\rightarrow &u\left( x,t\right) \text{ in
}L^{2}(\Omega _{T}), \\
u^{\varepsilon }\left( x,t\right) &\rightharpoonup &u\left( x,t\right) \text{
in }L^{2}(0,T;H_{0}^{1}\left( \Omega \right) )\end{aligned}$$and$$\nabla u^{\varepsilon }\left( x,t\right) \overset{n+1,m+1}{\rightharpoonup }\nabla u\left( x,t\right) +\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\left(
x,t,y^{j},s^{m}\right)$$where $u\in W_{2}^{1}(0,T;H_{0}^{1}\left( \Omega \right) ,L^{2}\left( \Omega
\right) )$ and $u_{j}\in L^{2}(\Omega _{T}\times \mathcal{Y}_{j-1,m};H_{\sharp }^{1}(Y_{j})/\mathbb{R}
)$ for $j=1,\ldots ,n$.
See Theorem 2.74 in [@Per2] or Theorem 4 in [@FHOLP].
Multiscale convergence is very useful for homogenization of problems involving rapid oscillations on several micro levels. Unfortunately, we can only use this for sequences that are bounded in the $L^{2}$-norm but when rapid time oscillations are present we encounter sequences that do not possess this boundedness. Multiscale convergence has a large class of test functions and the limit captures both the global trend and the microscopic oscillations. If we downsize this class so that it only captures the microscopic fluctuations then it becomes possible to apply it to certain sequences that do not have to be bounded in any Lebesgue space. This is the idea behind so-called very weak multiscale convergence. A first compactness result of very weak multiscale convergence type was given in [@Ho1], see also [@NgWo], [@FHOP] and [@FHOLP2].
A sequence $\left\{ w^{\varepsilon }\right\} $ in $L^{1}(\Omega _{T})$ is said to $(n+1,m+1)$-scale converge very weakly to $w_{0}\in L^{1}(\Omega
_{T}\times \mathcal{Y}_{n,m})$ if$$\begin{gathered}
\int_{\Omega _{T}}w^{\varepsilon }(x,t)v_{1}\left( x,\frac{x}{\hat{\varepsilon}_{1}},\ldots ,\frac{x}{\hat{\varepsilon}_{n-1}}\right)
v_{2}\left( t,\frac{t}{\check{\varepsilon}_{1}},\ldots ,\frac{t}{\check{\varepsilon}_{m}}\right) c\left( \frac{x}{\hat{\varepsilon}_{n}}\right) dxdt
\\
\rightarrow \int_{\Omega _{T}}\int_{\mathcal{Y}_{n,m}}w_{0}(x,t,y^{n},s^{m})v_{1}(x,y^{n-1})v_{2}(t,s^{m})c(y_{n})dy^{n}ds^{m}dxdt\end{gathered}$$for any $v_{1}\in D(\Omega ;C_{\sharp }^{\infty }(Y^{n-1})),$ $v_{2}\in
C_{\sharp }^{\infty }\left( Y_{n}\right) /\mathbb{R}
$ and $c\in D(0,T;C_{\sharp }^{\infty }\left( S^{m}\right) )$ where $$\int_{Y_{n}}w_{0}(x,t,y^{n},s^{m})~dy_{n}=0.$$We write$$w^{\varepsilon }\left( x,t\right) \underset{vw}{\overset{n+1,m+1}{\rightharpoonup }}w_{0}(x,t,y^{n},s^{m}).$$
The following theorem is essential for the homogenization of (\[first one\]).
\[T vw\]Let $\left\{ u^{\varepsilon }\right\} $ be a bounded sequence in $W_{2}^{1}(0,T;H_{0}^{1}\left( \Omega \right) ,L^{2}\left( \Omega \right) )$ and assume that the lists $\left\{ \hat{\varepsilon}_{1},\ldots ,\hat{\varepsilon}_{n}\right\} $ and $\left\{ \check{\varepsilon}_{1},\ldots ,\check{\varepsilon}_{m}\right\} $ are jointly well-separated. Then there exists a subsequence such that$$\frac{u^{\varepsilon }(x,t)}{\varepsilon _{n}}\overset{n+1,m+1}{\underset{vw}{\rightharpoonup }}u_{n}(x,t,y^{n},s^{m}),$$where, for $n=1,2,\ldots ,$ $u_{n}\in L^{2}(\Omega _{T}\times \mathcal{Y}_{n-1,m};H_{\sharp }^{1}(Y_{n})/\mathbb{R}
)$.
See Theorem 2.78 in [@Per2] or Theorem 7 in [@FHOLP].
**3. The homogenization result**
We study the homogenization of the problem$$\begin{aligned}
\partial _{t}u^{\varepsilon }(x,t)-\nabla \cdot a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon
}(x,t)\right) &=&f(x,t)\text{ in }\Omega _{T}, \notag \\
u^{\varepsilon }(x,t)\!\! &=&\!\!0\text{ on }\partial \Omega \!\!\times
\!\!(0,T), \label{equation1} \\
u^{\varepsilon }(x,0)\!\! &=&u^{0}(x)\text{ in }\Omega , \notag\end{aligned}$$where $f\in L^{2}(\Omega _{T})$ and $u^{0}\in L^{2}(\Omega )$. Here we assume that$$a:\mathbb{R}
^{nN}\times
\mathbb{R}
^{m}\times
\mathbb{R}
^{N}\rightarrow
\mathbb{R}
^{N}$$satisfies the following structure conditions, where $C_{0}$ and $C_{1}$ are positive constants and $0<\alpha \leq 1$:
1. $a(y^{n},s^{m},0)=0$ for all $(y^{n},s^{m})\in
\mathbb{R}
^{nN}\times
\mathbb{R}
^{m}$.
2. $a(\cdot ,\cdot ,\xi )$ is $\mathcal{Y}_{n,m}$-periodic in $(y^{n},s^{m})$ and continuous for all $\xi \in
\mathbb{R}
^{N}$.
3. $a(y^{n},s^{m},\cdot )$ is continuous for all $(y^{n},s^{m})\in
\mathbb{R}
^{nN}\times
\mathbb{R}
^{m}$.
4. $(a(y^{n},s^{m},\xi )-a(y^{n},s^{m},\xi ^{\prime }))\cdot (\xi
-\xi ^{\prime })\geq C_{0}\left\vert \xi -\xi ^{\prime }\right\vert ^{2}$ for all
$\ \ \ (y^{n},s^{m})\in
\mathbb{R}
^{nN}\times
\mathbb{R}
^{m}$ and all $\xi ,\xi ^{\prime }\in
\mathbb{R}
^{N}$.
1. $\left\vert a(y^{n},s^{m},\xi )-a(y^{n},s^{m},\xi ^{\prime
})\right\vert \leq C_{1}(1+\left\vert \xi \right\vert +\left\vert \xi
^{\prime }\right\vert )^{1-\alpha }\left\vert \xi -\xi ^{\prime }\right\vert
^{\alpha }$ for all
$\ \ \ (y^{n},s^{m})\in
\mathbb{R}
^{nN}\times
\mathbb{R}
^{m}$ and all $\xi ,\xi ^{\prime }\in
\mathbb{R}
^{N}$.
Finally we assume that the lists $\left\{ \hat{\varepsilon}^{n}\right\} $ and $\left\{ \check{\varepsilon}^{m}\right\} $ in ([equation1]{}) are jointly well-separated.
In order to formulate the theorem below in a neat way we define some numbers determined by how the scales functions present are related to each other. We define $d_{i}$ and $\rho _{i}$, $i=1,\ldots ,n$, as follows:
1. If $$\underset{\varepsilon \rightarrow 0}{\lim }\frac{\check{\varepsilon}_{1}}{(\hat{\varepsilon}_{i})^{2}}=0,$$then $d_{i}=m$. If$$\underset{\varepsilon \rightarrow 0}{\lim }\frac{\check{\varepsilon}_{j}}{(\hat{\varepsilon}_{i})^{2}}>0\text{ and }\underset{\varepsilon \rightarrow 0}{\lim }\frac{\check{\varepsilon}_{j+1}}{(\hat{\varepsilon}_{i})^{2}}=0$$for some $j=1,\ldots ,m-1$, then $d_{i}=m-j$. If$$\underset{\varepsilon \rightarrow 0}{\lim }\frac{\check{\varepsilon}_{m}}{(\hat{\varepsilon}_{i})^{2}}>0,$$then $d_{i}=0$.
2. If$$\underset{\varepsilon \rightarrow 0}{\lim }\frac{(\hat{\varepsilon}_{i})^{2}}{\check{\varepsilon}_{j}}=C,$$$0<C<\infty ,$ for some $j=1,\ldots ,m$ we say that we have resonance and we let $\rho _{i}=C$, otherwise $\rho _{i}=0$.
This means that $d_{i}$ is the number of temporal scales faster than the square of the spatial scale in question and $\rho
_{i} $ indicates whether there is resonance or not.
We are now prepared to give and prove the main theorem of the paper. Here $W_{2_{{\large \sharp }}}^{1}(S;H_{\sharp }^{1}(Y)/\mathbb{R}
,L_{\sharp }^{2}(Y)/\mathbb{R}
)$ denotes the space of all functions $u$ such that $u\in L_{\sharp
}^{2}(S;H_{\sharp }^{1}(Y)/\mathbb{R}
)$ and $\partial _{s}u\in L_{\sharp }^{2}(S;(H_{\sharp }^{1}(Y)/\mathbb{R}
)^{\prime })$.
\[first inline\]Let $\left\{ u^{\varepsilon }\right\} $ be a sequence of solutions in $W_{2}^{1}(0,T;H_{0}^{1}(\Omega ),L^{2}(\Omega ))$ to ([equation1]{}). Then it holds that$$u^{\varepsilon }(x,t)\rightarrow u(x,t)\text{ in }L^{2}(\Omega _{T}),
\label{equation 2}$$$$u^{\varepsilon }(x,t)\rightharpoonup u(x,t)\text{ in }L^{2}(0,T;H_{0}^{1}(\Omega )) \label{equation3}$$and$$\nabla u^{\varepsilon }(x,t)\overset{n+1,m+1}{\rightharpoonup }\nabla u(x,t)+\overset{n}{\underset{j=1}{\sum }}\nabla _{y_{j}}u_{j}\left(
x,t,y^{j},s^{m-d_{j}}\right) ,$$where $u\in W_{2}^{1}(0,T;H_{0}^{1}(\Omega ),L^{2}(\Omega ))$ is the unique solution to$$\begin{aligned}
\partial _{t}u(x,t)-\nabla \cdot b(x,t,\nabla u) &=&f(x,t)\text{ in }\Omega
_{T}, \\
u(x,t) &=&0\text{ on }\partial \Omega \times (0,T), \\
u(x,0) &=&u^{0}(x)\text{ in }\Omega \end{aligned}$$with$$b(x,t,\nabla u)=\int_{\mathcal{Y}_{n,m}}a\left( y^{n},s^{m},\nabla
u(x,t)+\!\dsum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\!\left(
x,t,y^{j},s^{m-d_{j}}\right) \right) dy^{n}ds^{m},$$where $u_{i}\in L^{2}(\Omega _{T}\times \mathcal{Y}_{i-1,m-d_{i}};H_{\sharp
}^{1}(Y_{i})/\mathbb{R}
)$ for $i=1,\ldots ,n$. Here $u_{i}$, for$i=1,\ldots ,n$, are the unique solutions to the system of local problems$$\begin{gathered}
\rho _{i}\partial _{s_{m-d_{i}}}u_{i}(x,t,y^{i},s^{m-d_{i}}) \notag \\
-\nabla _{y_{i}}\!\cdot \!\!\int_{S_{m-d_{i}+1}}\!\!\cdots
\!\int_{S_{m}}\!\int_{Y_{i+1}}\!\!\cdots \!\int_{Y_{n}}\!a\!\left(
y^{n},s^{m},\nabla u(x,t)+\!\dsum\limits_{j=1}^{n}\nabla
_{y_{j}}u_{j}\!\left( x,t,y^{j},s^{m-d_{j}}\right) \!\!\right)
\label{equation5} \\
\times dy_{n}\cdots dy_{i+1}ds_{m}\cdots ds_{m-d_{i}+1}=0 \notag\end{gathered}$$if we assume that $u_{i}\in L^{2}(\Omega _{T}\times \mathcal{Y}_{i-1,m-d_{i}-1};W_{2_{{\large \sharp }}}^{1}(S_{m-d_{i}};H_{\sharp
}^{1}(Y_{i})/\mathbb{R}
,L_{\sharp }^{2}(Y_{i})/\mathbb{R}
))$ when $\rho _{i}\neq 0$.
The lists $\left\{ \hat{\varepsilon}^{n}\right\} $ and $\left\{ \check{\varepsilon}^{m}\right\} $ of scales are jointly well-separated and $\left\{
u^{\varepsilon }\right\} $ is bounded in $W_{2}^{1}(0,T;H_{0}^{1}\left(
\Omega \right) ,L^{2}\left( \Omega \right) )$, see Proposition 3.16 in [Per2]{}, which means that Theorem \[gradkarmulti\] is applicable and hence, up to a subsequence, $$\begin{aligned}
u^{\varepsilon }(x,t) &\rightarrow &u(x,t)\text{ in }L^{2}(\Omega _{T}), \\
u^{\varepsilon }(x,t) &\rightharpoonup &u(x,t)\text{ in }L^{2}(0,T;H_{0}^{1}(\Omega ))\end{aligned}$$and $$\nabla u^{\varepsilon }(x,t)\overset{n+1,m+1}{\rightharpoonup }\nabla u(x,t)+\overset{n}{\underset{j=1}{\sum }}\nabla _{y_{j}}u_{j}(x,t,y^{j},s^{m}),$$where $u\in W_{2}^{1}(0,T;H_{0}^{1}\left( \Omega \right) ,L^{2}\left( \Omega
\right) )$ and $u_{j}\in L^{2}(\Omega _{T}\times \mathcal{Y}_{j-1,m};H_{\sharp }^{1}(Y_{j})/\mathbb{R}
)$ for $j=1,\ldots ,n$.
The weak form of (\[equation1\]) reads: find $u^{\varepsilon }\in
W_{2}^{1}(0,T;H_{0}^{1}\left( \Omega \right) ,L^{2}\left( \Omega \right) )$ such that$$\begin{gathered}
\int_{\Omega _{T}}-u^{\varepsilon }(x,t)v(x)\partial _{t}c(t)+a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla
u^{\varepsilon }(x,t)\right) \cdot \nabla v(x)c\left( t\right) dxdt \notag
\\
=\int_{\Omega _{T}}f(x,t)v(x)c(t)dxdt \label{svag form}\end{gathered}$$for all $v\in H_{0}^{1}\left( \Omega \right) $ and $c\in D\left( 0,T\right) $. By choosing $\xi ^{\prime }=0$ in $(v)$ we have $$\left\vert a(y^{n},s^{m},\xi )\right\vert \leqslant C_{1}(1+\left\vert \xi
\right\vert )^{1-\alpha }\left\vert \xi \right\vert ^{\alpha }$$and since$$C_{1}(1+\left\vert \xi \right\vert )^{1-\alpha }\left\vert \xi \right\vert
^{\alpha }<C_{1}(1+\left\vert \xi \right\vert )^{1-\alpha }(1+\left\vert \xi
\right\vert )^{\alpha }$$we obtain$$\left\vert a(y^{n},s^{m},\xi )\right\vert \leqslant C_{1}(1+\left\vert \xi
\right\vert ). \label{flyttning}$$The boundedness of $\left\{ u^{\varepsilon }\right\} $ in $L^{2}(0,T;H_{0}^{1}(\Omega ))$ together with (\[flyttning\]) gives, up to a subsequence, that $$a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon }(x,t)\right) \overset{m,n}{\rightharpoonup }a_{0}(x,t,y^{n},s^{m})$$for some $a_{0}\in L^{2}(\Omega _{T}\times \mathcal{Y}_{n,m})$ due to Theorem \[thmultiskale\]. We let $\varepsilon $ tend to zero in (\[svag form\]) and obtain$$\begin{gathered}
\int_{\Omega _{T}}-u(x,t)v(x)\partial _{t}c(t)+\left( \int_{\mathcal{Y}_{n,m}}a_{0}(x,t,y^{n},s^{m})dy^{n}ds^{m}\right) \cdot \nabla v(x)c(t)~dxdt
\notag \\
=\int_{\Omega _{T}}f(x,t)v(x)c\left( t\right) ~dxdt, \label{homog_probl}\end{gathered}$$which is the homogenized problem if we can prove that $$a_{0}\left( x,t,y^{n},s^{m}\right) =a\left( y^{n},s^{m},\nabla
u+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\right)$$with $u$ and $u_{j}$ as given in the theorem. To characterize $a_{0}$ we will use the system of local problems (\[equation5\]), and deriving this will be our next aim.
In (\[svag form\]) we will use test functions defined according to the following. Let $r_{\varepsilon }=r(\varepsilon )$ be a sequence of positive numbers tending to zero as $\varepsilon $ does. Fix $i=1,\ldots ,n$ and choose$$v(x)=r_{\varepsilon }v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right)$$and$$c(t)=c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right)
,~\lambda =1,\ldots ,m$$with $v_{1}\in D\left( \Omega \right) ,\,v_{j}\in C_{\sharp }^{\infty
}\left( Y_{j-1}\right) $ for $j=2,\ldots ,i,$ $v_{i+1}\in C_{\sharp
}^{\infty }\left( Y_{i}\right) /\mathbb{R}
,$$c_{1}\in D\left( 0,T\right) $ and$\ c_{l}\in C_{\sharp }^{\infty }\left(
S_{l-1}\right) $ for $l=2,\ldots ,\lambda +1$. We get$$\begin{gathered}
\int_{\Omega _{T}}-u^{\varepsilon }(x,t)v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \\
\times \left( r_{\varepsilon }\partial _{t}c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right) \right. \\
+\left. \sum_{l=2}^{\lambda +1}\frac{r_{\varepsilon }}{\check{\varepsilon}_{l-1}}c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
\partial _{s_{l-1}}c_{l}\left( \frac{t}{\check{\varepsilon}_{l-1}}\right)
\cdots c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right)
\right) \\
+a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon }\left( x,t\right) \right) \cdot \left(
r_{\varepsilon }\nabla v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \right.
\\
+\left. \sum_{j=2}^{i+1}\frac{r_{\varepsilon }}{\hat{\varepsilon}_{j-1}}v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots \nabla
_{y_{j-1}}v_{j}\left( \frac{x}{\hat{\varepsilon}_{j-1}}\right) \cdots
v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \right) \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right) dxdt \\
=\int_{\Omega _{T}}f(x,t)r_{\varepsilon }v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right) dxdt.\end{gathered}$$Applying Theorem \[gradkarmulti\] and the definition of $r_{\varepsilon }$, we may let $\varepsilon \rightarrow 0$ and get $$\begin{gathered}
\lim_{\varepsilon \rightarrow 0}\int_{\Omega _{T}}-u^{\varepsilon
}(x,t)v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots
v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \\
\times \left( \sum_{l=2}^{\lambda +1}\frac{r_{\varepsilon }}{\check{\varepsilon}_{l-1}}c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots \partial _{s_{l-1}}c_{l}\left( \frac{t}{\check{\varepsilon}_{l-1}}\right) \cdots c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right) \right) \\
+a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon }\left( x,t\right) \right) \\
\cdot \sum_{j=2}^{i+1}\frac{r_{\varepsilon }}{\hat{\varepsilon}_{j-1}}v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots \nabla
_{y_{j-1}}v_{j}\left( \frac{x}{\hat{\varepsilon}_{j-1}}\right) \cdots
v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right) dxdt=0\end{gathered}$$if we omit the terms passing to zero. Rewriting we obtain$$\begin{gathered}
\lim_{\varepsilon \rightarrow 0}\int_{\Omega _{T}}-\frac{1}{\hat{\varepsilon}_{i}}u^{\varepsilon }(x,t)\sum\limits_{l=2}^{\lambda +1}\frac{r_{\varepsilon
}\hat{\varepsilon}_{i}}{\check{\varepsilon}_{l-1}}v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \notag \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
\partial _{s_{l-1}}c_{l}\left( \frac{t}{\check{\varepsilon}_{l-1}}\right)
\cdots c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right)
\notag \\
+a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon }\left( x,t\right) \right) \label{3equat} \\
\cdot \sum\limits_{j=2}^{i+1}\frac{r_{\varepsilon }}{\hat{\varepsilon}_{j-1}}v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots \nabla
_{y_{j-1}}v_{j}\left( \frac{x}{\hat{\varepsilon}_{j-1}}\right) \cdots
v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \notag \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right) ~dxdt=0,
\notag\end{gathered}$$where we have factored out $\frac{1}{\hat{\varepsilon}_{i}}$ from the first sum to make it obvious that it is possible to pass to the limit by means of very weak $(i+1,\lambda +1)$-scale convergence. Suppose that $\{\frac{r_{\varepsilon }\hat{\varepsilon}_{i}}{\check{\varepsilon}_{\lambda }}\}$ and $\{\frac{r_{\varepsilon }}{\hat{\varepsilon}_{i}}\}$ are bounded. This implies that$$\frac{r_{\varepsilon }\hat{\varepsilon}_{i}}{\check{\varepsilon}_{\lambda -j}}\rightarrow 0,\text{ }j=1,\ldots ,\lambda -1$$and$$\frac{r_{\varepsilon }}{\hat{\varepsilon}_{i-j}}\rightarrow 0,~j=1,\ldots
,i-1$$as $\varepsilon \rightarrow 0$ due to the fact that the scales are separated. Hence, under these assumptions (\[3equat\]) turns into$$\begin{gathered}
\lim_{\varepsilon \rightarrow 0}\int_{\Omega _{T}}-\frac{1}{\hat{\varepsilon}_{i}}u^{\varepsilon }(x,t)\frac{r_{\varepsilon }\hat{\varepsilon}_{i}}{\check{\varepsilon}_{\lambda }}v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right)
\notag \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
\partial _{s_{\lambda }}c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right) +a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon }\left( x,t\right) \right)
\label{springboard} \\
\cdot \frac{r_{\varepsilon }}{\hat{\varepsilon}_{i}}v_{1}(x)v_{2}\left(
\frac{x}{\hat{\varepsilon}_{1}}\right) \cdots v_{i}\left( \frac{x}{\hat{\varepsilon}_{i-1}}\right) \nabla _{y_{i}}v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \notag \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right) dxdt=0,
\notag\end{gathered}$$which will be our springboard when deriving both the independencies of the local time variables in the corrector functions and the local problems. This will be done for the two different cases nonresonance and resonance.
**Case 1:** Nonresonance $\left( \rho _{i}=0\right) $. First we derive the independencies for $d_{i}>0$. Let $\lambda $ successively be $m,\ldots
,m-d_{i}+1$. If $r_{\varepsilon }=\frac{\check{\varepsilon}_{\lambda }}{\hat{\varepsilon}_{i}}$ we have from the chosen values of $\lambda $ and the meaning of $d_{i}$ that$$\frac{r_{\varepsilon }\hat{\varepsilon}_{i}}{\check{\varepsilon}_{\lambda }}=1$$and$$\frac{r_{\varepsilon }}{\hat{\varepsilon}_{i}}=\frac{\check{\varepsilon}_{\lambda }}{\left( \hat{\varepsilon}_{i}\right) ^{2}}\rightarrow 0
\label{two differ cas2}$$as $\varepsilon \rightarrow 0$. Hence, we may use (\[springboard\]) for this choice of $r_{\varepsilon }$ and we have $$\begin{gathered}
\lim_{\varepsilon \rightarrow 0}\int_{\Omega _{T}}-\frac{1}{\hat{\varepsilon}_{i}}u^{\varepsilon }(x,t)v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
\partial _{s_{\lambda }}c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right) +a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon }\left( x,t\right) \right) \\
\cdot \frac{\check{\varepsilon}_{\lambda }}{\left( \hat{\varepsilon}_{i}\right) ^{2}}v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right)
\cdots v_{i}\left( \frac{x}{\hat{\varepsilon}_{i-1}}\right) \nabla
_{y_{i}}v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right) dxdt=0.\end{gathered}$$We let $\varepsilon $ tend to zero and obtain, due to Theorem \[T vw\] and (\[two differ cas2\]), that$$\begin{gathered}
\int_{\Omega _{T}}\int_{\mathcal{Y}_{i,\lambda }}-u_{i}(x,t,y^{i},s^{\lambda
})v_{1}(x)v_{2}(y_{1})\cdots v_{i+1}(y_{i}) \\
\times c_{1}(t)c_{2}(s_{1})\cdots \partial _{s_{\lambda }}c_{\lambda
+1}(s_{\lambda })dy^{i}ds^{\lambda }dxdt=0\end{gathered}$$and by the variational lemma we have$$\int_{S_{\lambda }}-u_{i}(x,t,y^{i},s^{\lambda })\partial _{s_{\lambda
}}c_{\lambda +1}(s_{\lambda })ds_{\lambda }=0$$almost everywhere for all $c_{\lambda +1}\in C_{\sharp }^{\infty
}(s_{\lambda })$. This means that $u_{i}$ is independent of $s_{m-d_{i}+1},\ldots ,s_{m}$.
We proceed by deriving the local problems and for this purpose we choose $r_{\varepsilon }=\hat{\varepsilon}_{i}$ and $\lambda =m-d_{i},$ where $d_{i}\geq 0$. Since $d_{i}\geq 0$ and $\rho _{i}=0$ we conclude that $$\frac{r_{\varepsilon }\hat{\varepsilon}_{i}}{\check{\varepsilon}_{\lambda }}=\frac{(\hat{\varepsilon}_{i})^{2}}{\check{\varepsilon}_{m-d_{i}}}\rightarrow
0$$as $\varepsilon \rightarrow 0$ and $$\frac{r_{\varepsilon }}{\hat{\varepsilon}_{i}}=1,$$which means that (\[springboard\]) is valid and we get $$\begin{gathered}
\lim_{\varepsilon \rightarrow 0}\int_{\Omega _{T}}-\frac{1}{\hat{\varepsilon}_{i}}u^{\varepsilon }(x,t)\frac{\left( \hat{\varepsilon}_{i}\right) ^{2}}{\check{\varepsilon}_{m-d_{i}}}v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
\partial _{s_{m-d_{i}}}c_{m-d_{i}+1}\left( \frac{t}{\check{\varepsilon}_{m-d_{i}}}\right) +a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon }\left( x,t\right) \right) \\
\cdot v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots
v_{i}\left( \frac{x}{\hat{\varepsilon}_{i-1}}\right) \nabla
_{y_{i}}v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
c_{m-d_{i}+1}\left( \frac{t}{\check{\varepsilon}_{m-d_{i}}}\right) dxdt=0.\end{gathered}$$As $\varepsilon \rightarrow 0$ we obtain$$\begin{gathered}
\int_{\Omega _{T}}\int_{\mathcal{Y}_{n,m}}a_{0}(x,t,y^{n},s^{m}) \\
\cdot v_{1}(x)v_{2}(y_{1})\cdots v_{i}(y_{i-1})\nabla _{y_{i}}v_{i+1}(y_{i})
\\
\times c_{1}(t)c_{2}(s_{1})\cdots
c_{m-d_{i}+1}(s_{m-d_{i}})dy^{n}ds^{m}dxdt=0\end{gathered}$$and, finally,$$\begin{gathered}
\int_{S_{m-d_{i}+1}}\cdots \int_{S_{m}}\int_{Y_{i}}\cdots
\int_{Y_{n}}a_{0}(x,t,y^{n},s^{m}) \label{case 1} \\
\cdot \nabla _{y_{i}}v_{i+1}(y_{i})dy_{n}\cdots dy_{i}ds_{m}\cdots
ds_{m-d_{i}+1}=0 \notag\end{gathered}$$almost everywhere for all $v_{i+1}\in H_{\sharp }^{1}\left( Y_{i}\right) /\mathbb{R}
$, which is the weak form of the local problem in this nonresonance case.
**Case 2:** Resonance $(\rho _{i}=C)$. As in the first case we begin with the independencies for $d_{i}>0$. Again, let $\lambda $ successively be $m,\ldots ,m-d_{i}+1$. Now choose $r_{\varepsilon }=\frac{\check{\varepsilon}_{\lambda }}{\hat{\varepsilon}_{i}}$ directly implying that$$\frac{r_{\varepsilon }\hat{\varepsilon}_{i}}{\check{\varepsilon}_{\lambda }}=1$$and $$\frac{r_{\varepsilon }}{\hat{\varepsilon}_{i}}=\frac{\check{\varepsilon}_{\lambda }}{\left( \hat{\varepsilon}_{i}\right) ^{2}}\rightarrow 0$$when $\varepsilon \rightarrow 0$, by the restriction of $\lambda $ and the definition of $d_{i}$ and $\rho _{i}$. Thus, (\[springboard\]) turns into$$\begin{gathered}
\lim_{\varepsilon \rightarrow 0}\int_{\Omega _{T}}-\frac{1}{\hat{\varepsilon}_{i}}u^{\varepsilon }(x,t)v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
\partial _{s_{\lambda }}c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right) +a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon }\left( x,t\right) \right) \\
\cdot \frac{\check{\varepsilon}_{\lambda }}{\left( \hat{\varepsilon}_{i}\right) ^{2}}v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right)
\cdots v_{i}\left( \frac{x}{\hat{\varepsilon}_{i-1}}\right) \nabla
_{y_{i}}v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
c_{\lambda +1}\left( \frac{t}{\check{\varepsilon}_{\lambda }}\right) dxdt=0\end{gathered}$$and a passage to the limit gives$$\begin{gathered}
\int_{\Omega _{T}}\int_{\mathcal{Y}_{i,\lambda }}-u_{i}(x,t,y^{i},s^{\lambda
})v_{1}(x)v_{2}(y_{1})\cdots v_{i+1}(y_{i}) \\
\times c_{1}(t)c_{2}(s_{1})\cdots \partial _{s_{\lambda }}c_{\lambda
+1}(s_{\lambda })dy^{i}ds^{\lambda }dxdt=0.\end{gathered}$$Hence,$$\int_{S_{\lambda }}-u_{i}(x,t,y^{i},s^{\lambda })\partial _{s_{\lambda
}}c_{\lambda +1}(s_{\lambda })ds_{\lambda }=0$$almost everywhere for all $c_{\lambda +1}\in C_{\sharp }^{\infty }\left(
S_{\lambda }\right) $, and thus $u_{i}$ is independent of $s_{\lambda }$.
To extract the local problem we choose $r_{\varepsilon }=\hat{\varepsilon}_{i}$ and $\lambda =m-d_{i}$, where $d_{i}\geq 0$, which gives $$\frac{r_{\varepsilon }\hat{\varepsilon}_{i}}{\check{\varepsilon}_{\lambda }}=\frac{\left( \hat{\varepsilon}_{i}\right) ^{2}}{\check{\varepsilon}_{m-d_{i}}}\rightarrow \rho _{i}$$as $\varepsilon \rightarrow 0$ and $$\frac{r_{\varepsilon }}{\hat{\varepsilon}_{i}}=1$$and from (\[springboard\]) we then have$$\begin{gathered}
\lim_{\varepsilon \rightarrow 0}\int_{\Omega _{T}}-\frac{1}{\hat{\varepsilon}_{i}}u^{\varepsilon }(x,t)\frac{\left( \hat{\varepsilon}_{i}\right) ^{2}}{\check{\varepsilon}_{m-d_{i}}}v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
\partial _{s_{m-d_{i}}}c_{m-d_{i}+1}\left( \frac{t}{\check{\varepsilon}_{m-d_{i}}}\right) +a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon }\left( x,t\right) \right) \\
\cdot v_{1}(x)v_{2}\left( \frac{x}{\hat{\varepsilon}_{1}}\right) \cdots
v_{i}\left( \frac{x}{\hat{\varepsilon}_{i-1}}\right) \nabla
_{y_{i}}v_{i+1}\left( \frac{x}{\hat{\varepsilon}_{i}}\right) \\
\times c_{1}(t)c_{2}\left( \frac{t}{\check{\varepsilon}_{1}}\right) \cdots
c_{m-d_{i}+1}\left( \frac{t}{\check{\varepsilon}_{m-d_{i}}}\right) dxdt=0.\end{gathered}$$Letting $\varepsilon $ tend to zero and applying Theorem \[T vw\] we obtain $$\begin{gathered}
\int_{\Omega _{T}}\int_{\mathcal{Y}_{n,m}}-\rho
_{i}u_{i}(x,t,y^{i},s^{m-d_{i}})v_{1}(x)v_{2}(y_{1})\cdots v_{i+1}(y_{i}) \\
\times c_{1}(t)c_{2}(s_{1})\cdots \partial
_{s_{m-d_{i}}}c_{m-d_{i}+1}(s_{m-d_{i}})+a_{0}(x,t,y^{n},s^{m}) \\
\cdot v_{1}(x)v_{2}(y_{1})\cdots v_{i}(y_{i-1})\nabla _{y_{i}}v_{i+1}(y_{i})
\\
\times c_{1}(t)c_{2}(s_{1})\cdots
c_{m-d_{i}+1}(s_{m-d_{i}})dy^{n}ds^{m}dxdt=0\end{gathered}$$and hence, we end up with$$\begin{gathered}
\int_{S_{m-d_{i}}}\cdots \int_{S_{m}}\int_{Y_{i}}\cdots \int_{Y_{n}}-\rho
_{i}u_{i}(x,t,y^{i},s^{m-d_{i}})v_{i+1}(y_{i}) \notag \\
\times \partial
_{s_{m-d_{i}}}c_{m-d_{i}+1}(s_{m-d_{i}})+a_{0}(x,t,y^{n},s^{m})
\label{locCas2} \\
\cdot \nabla _{y_{i}}v_{i+1}(y_{i})c_{m-d_{i}+1}(s_{m-d_{i}})dy_{n}\cdots
dy_{i}ds_{m}\cdots ds_{m-d_{i}}=0 \notag\end{gathered}$$almost everywhere for all $v_{i+1}\in H_{\sharp }^{1}\left( Y_{i}\right) /\mathbb{R}
$ and $c_{m-d_{i}+1}\in C_{\sharp }^{\infty }\left( S_{m-d_{i}}\right) $, the weak form of the local problem in this second case.
What remains is to characterize $a_{0}$ and to this end we use perturbed test functions, see [@Ev1] and [@Ev2], according to$$\begin{gathered}
p^{k}\left( x,t,y^{j},s^{m}\right) \\
=p^{k,0}\left( x,t\right) +\sum_{j=1}^{n}p^{k,j}\left(
x,t,y^{j},s^{m-d_{j}}\right) +\delta c(x,t,y^{n},s^{m}),\end{gathered}$$where $p^{k,0}\in D\left( \Omega _{T}\right) ^{N}$, $p^{k,j}\in D(\Omega
_{T};C_{\sharp }^{\infty }(\mathcal{Y}_{j,m-d_{j}}^{N}))$ for $j=1,\ldots ,n$, $c\in D(\Omega _{T};C_{\sharp }^{\infty }\left( \mathcal{Y}_{n,m}\right)
)^{N}$ and $\delta >0$. We choose these sequences such that$$p^{k,0}\left( x,t\right) \rightarrow \nabla u(x,t)\text{ in }L^{2}(\Omega
_{T})^{N},$$$$p^{k,j}\left( x,t,y^{j},s^{m-d_{j}}\right) \rightarrow \nabla
_{y_{j}}u_{j}\left( x,t,y^{j},s^{m-d_{j}}\right) \text{ in }L^{2}(\Omega
_{T}\times \mathcal{Y}_{j,m-d_{j}})^{N}$$and such that they converge almost everywhere to the same limits as $k\rightarrow \infty $, see p. 388 in [@Kuf]. We introduce the notation $$p_{\varepsilon }^{k}\left( x,t\right) =p^{k}\left( x,t,\frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}}\right) .$$Using property $\left( iv\right) $ we get $$\left( a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon }\right) -a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},p_{\varepsilon }^{k}\right) \right) \cdot
(\nabla u^{\varepsilon }(x,t)-p_{\varepsilon }^{k}(x,t))\geq 0$$and integration and expansion leads to$$\begin{gathered}
\int_{\Omega _{T}}a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon }\right) \cdot \nabla u^{\varepsilon
}(x,t)-a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon }\right) \cdot p_{\varepsilon }^{k}(x,t) \notag
\\
-a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},p_{\varepsilon }^{k}\right) \cdot \nabla u^{\varepsilon }(x,t)
\label{star2} \\
+a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},p_{\varepsilon }^{k}\right) \cdot p_{\varepsilon }^{k}(x,t)dxdt\geq 0.
\notag\end{gathered}$$Due to Theorem 30.A (c) in [@ZeiIIB] we may replace $vc$ with $u^{\varepsilon }$ in (\[svag form\]) and get another way of expressing the first term in (\[star2\])* *and hence it can be written as$$\begin{gathered}
\int_{\Omega _{T}}f\left( x,t\right) u^{\varepsilon }\left( x,t\right)
-a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},\nabla u^{\varepsilon }\right) \cdot p_{\varepsilon }^{k}\left( x,t\right)
\notag \\
-a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},p_{\varepsilon }^{k}\right) \cdot \nabla u^{\varepsilon }\left( x,t\right)
+a\left( \frac{x}{\hat{\varepsilon}^{n}},\frac{t}{\check{\varepsilon}^{m}},p_{\varepsilon }^{k}\right) \cdot p_{\varepsilon }^{k}\left( x,t\right)
\text{ }dxdt \label{bluttan} \\
-\int_{0}^{T}\left\langle \partial _{t}u^{\varepsilon }\left( t\right)
,u^{\varepsilon }\left( t\right) \right\rangle _{H^{-1}\left( \Omega \right)
,H_{0}^{1}\left( \Omega \right) }\text{ }dt\geq 0. \notag\end{gathered}$$We note that $p^{k}$, $a\left( y^{n},s^{m},p^{k}\right) $ and their product are admissible test functions and since $$-\lim_{\varepsilon \rightarrow 0}\inf \!\int_{0}^{T}\!\!\left\langle
\partial _{t}u^{\varepsilon }\left( t\right) ,u^{\varepsilon }\left(
t\right) \right\rangle _{H^{-1}\left( \Omega \right) ,H_{0}^{1}\left( \Omega
\right) }dt\leq -\!\int_{0}^{T}\!\!\left\langle \partial _{t}u\left(
t\right) ,u\left( t\right) \right\rangle _{H^{-1}\left( \Omega \right)
,H_{0}^{1}\left( \Omega \right) }\!dt$$(see p. 12–13 in [@NgWo1]) we get, up to a subsequence, that$$\begin{gathered}
\int_{\Omega _{T}}\int_{\mathcal{Y}_{n,m}}f\left( x,t\right) u\left(
x,t\right) -a_{0}\left( x,t,y^{n},s^{m}\right) \cdot p^{k}\left(
x,t,y^{n},s^{m}\right) \notag \\
-a\left( y^{n},s^{m},p^{k}\right) \cdot \left( \nabla u\left( x,t\right)
+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\left(
x,t,y^{j},s^{m-d_{j}}\right) \right) \notag \\
+a\left( y^{n},s^{m},p^{k}\right) \cdot p^{k}\left( x,t,y^{n},s^{m}\right)
\text{ }dy^{n}ds^{m}dxdt \label{epsigrans} \\
-\int_{0}^{T}\left\langle \partial _{t}u\left( t\right) ,u\left( t\right)
\right\rangle _{H^{-1}\left( \Omega \right) ,H_{0}^{1}\left( \Omega \right) }\text{ }dt\geq 0 \notag\end{gathered}$$when $\varepsilon $ tends to zero. We proceed by letting* *$k$ tend to infinity. From the choice of $p^{k}$ we have that$$p^{k}\left( x,t,y^{n},s^{m}\right) \rightarrow \nabla
u(x,t)+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\left(
x,t,y^{j},s^{m-d_{j}}\right) +\delta c(x,t,y^{n},s^{m})$$in $L^{2}\left( \Omega _{T}\times \mathcal{Y}_{n,m}\right) ^{N}$ and almost everywhere in $\Omega _{T}\times \mathcal{Y}_{n,m}$. Furthermore$$a\left( y^{n},s^{m},p^{k}\right) \rightarrow a\left( y^{n},s^{m},\nabla
u+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}+\delta c\right)$$almost everywhere in $\Omega _{T}\times \mathcal{Y}_{n,m}$ and hence$$\begin{gathered}
a\left( y^{n},s^{m},p^{k}\right) \cdot p^{k}\left( x,t,y^{n},s^{m}\right)
\rightarrow a\left( y^{n},s^{m},\nabla u+\sum\limits_{j=1}^{n}\nabla
_{y_{j}}u_{j}+\delta c\right) \\
\cdot \left( \nabla u(x,t)+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\left(
x,t,y^{j},s^{m-d_{j}}\right) \right) +\delta c(x,t,y^{n},s^{m})\end{gathered}$$almost everywhere in $\Omega _{T}\times \mathcal{Y}_{n,m}$. When we pass to the limit in (\[epsigrans\]) we will use Lebesgue’s generalized majorized convergence theorem for the third and fourth term where we go through the details for the fourth term. Choosing $\xi =p^{k}$ in (\[flyttning\]) we have that$$\left\vert a\left( y^{n},s^{m},p^{k}\right) \right\vert \leq
C_{1}(1+\left\vert p^{k}(x,t,y^{n},s^{m})\right\vert ). \label{begr}$$Successively applying Cauchy-Schwarz inequality and (\[begr\]) we get$$\begin{aligned}
\left\vert a\left( y^{n},s^{m},p^{k}\right) \cdot
p^{k}(x,t,y^{n},s^{m})\right\vert &\leq &\left\vert a\left(
y^{n},s^{m},p^{k}\right) \right\vert \left\vert
p^{k}(x,t,y^{n},s^{m})\right\vert \\
&\leq &C_{1}(1+\left\vert p^{k}(x,t,y^{n},s^{m})\right\vert )\left\vert
p^{k}(x,t,y^{n},s^{m})\right\vert \\
&=&C_{1}\left( \left\vert p^{k}(x,t,y^{n},s^{m})\right\vert +\left\vert
p^{k}(x,t,y^{n},s^{m})\right\vert ^{2}\right) .\end{aligned}$$Letting $k\rightarrow \infty $, we have$$\begin{gathered}
\int_{\Omega _{T}}\int_{\mathcal{Y}_{n,m}}\left\vert
p^{k}(x,t,y^{n},s^{m})\right\vert +\left\vert
p^{k}(x,t,y^{n},s^{m})\right\vert ^{2}dy^{n}ds^{m}dxdt \\
\rightarrow \int_{\Omega _{T}}\int_{\mathcal{Y}_{n,m}}\left\vert \nabla
u(x,t)+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\left(
x,t,y^{j},s^{m-d_{j}}\right) +\delta c(x,t,y^{n},s^{m})\right\vert \\
+\left\vert \nabla u(x,t)+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\left(
x,t,y^{j},s^{m-d_{j}}\right) +\delta c(x,t,y^{n},s^{m})\right\vert
^{2}dy^{n}ds^{m}dxdt\end{gathered}$$and hence, by Lebesgue’s generalized majorized convergence theorem we conclude that$$\begin{gathered}
\int_{\Omega _{T}}\int_{\mathcal{Y}_{n,m}}a\left( y^{n},s^{m},p^{k}\right)
\cdot p^{k}\left( x,t,y^{n},s^{m}\right) dy^{n}ds^{m}dxdt \\
\rightarrow \int_{\Omega _{T}}\int_{\mathcal{Y}_{n,m}}a\left(
y^{n},s^{m},\nabla u+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}+\delta
c\right) \\
\cdot \left( \nabla u(x,t)+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\left(
x,t,y^{j},s^{m-d_{j}}\right) +\delta c(x,t,y^{n},s^{m})\right)
dy^{n}ds^{m}dxdt.\end{gathered}$$Thus, as $k$ tends to infinity in (\[epsigrans\]) we find that$$\begin{gathered}
\int_{\Omega _{T}}\int_{\mathcal{Y}_{n,m}}f(x,t)u(x,t)-a_{0}(x,t,y^{n},s^{m})
\\
\cdot \left( \nabla u(x,t)+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\left(
x,t,y^{j},s^{m-d_{j}}\right) +\delta c(x,t,y^{n},s^{m})\right) \\
-a\left( y^{n},s^{m},\nabla u+\sum\limits_{j=1}^{n}\nabla
_{y_{j}}u_{j}+\delta c\right) \\
\cdot \left( \nabla u(x,t)+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\left(
x,t,y^{j},s^{m-d_{j}}\right) \right) +a\left( y^{n},s^{m},\nabla
u+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}+\delta c\right) \\
\cdot \left( \nabla u(x,t)+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\left(
x,t,y^{j},s^{m-d_{j}}\right) +\delta c(x,t,y^{n},s^{m})\right)
dy^{n}ds^{m}dxdt \\
-\int_{0}^{T}\left\langle \partial _{t}u(t),u(t)\right\rangle
_{H^{-1}(\Omega ),H_{0}^{1}(\Omega )}dt\geq 0,\end{gathered}$$where some terms vanishes directly and we have$$\begin{gathered}
\int_{\Omega _{T}}\int_{\mathcal{Y}_{n,m}}f(x,t)u(x,t)-a_{0}(x,t,y^{n},s^{m})
\notag \\
\cdot \left( \nabla u(x,t)+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\left(
x,t,y^{j},s^{m-d_{j}}\right) +\delta c(x,t,y^{n},s^{m})\right)
\label{Assttar2} \\
+a\left( y^{n},s^{m},\nabla u+\sum\limits_{j=1}^{n}\nabla
_{y_{j}}u_{j}+\delta c\right) \cdot \delta c(x,t,y^{n},s^{m})dy^{n}ds^{m}dxdt
\notag \\
-\int_{0}^{T}\left\langle \partial _{t}u(t),u(t)\right\rangle
_{H^{-1}(\Omega ),H_{0}^{1}(\Omega )}dt\geq 0. \notag\end{gathered}$$If we replace $vc$ by $u$ in (\[homog\_probl\]) we get $$\begin{gathered}
\int_{0}^{T}\left\langle \partial _{t}u,u\right\rangle _{H^{-1}(\Omega
),H_{0}^{1}(\Omega )}dt+\int_{\Omega _{T}}\left( \int_{\mathcal{Y}_{n,m}}a_{0}(x,t,y^{n},s^{m})dy^{n}ds^{m}\right) \cdot \nabla u(x,t)dxdt
\notag \\
=\int_{\Omega _{T}}f(x,t)u(x,t)dxdt \label{Astar3}\end{gathered}$$and with (\[Astar3\]) in (\[Assttar2\]) we obtain$$\begin{gathered}
\int_{\Omega _{T}}\int_{\mathcal{Y}_{n,m}}\sum\limits_{j=1}^{n}-a_{0}(x,t,y^{n},s^{m})\cdot \nabla _{y_{j}}u_{j}\left(
x,t,y^{j},s^{m-d_{j}}\right) \notag \\
-a_{0}(x,t,y^{n},s^{m})\cdot \delta c(x,t,y^{n},s^{m}) \label{twostars} \\
+a\left( y^{n},s^{m},\nabla u+\sum\limits_{j=1}^{n}\nabla
_{y_{j}}u_{j}\right) \cdot \delta c(x,t,y^{n},s^{m})dy^{n}ds^{m}dxdt\geq 0.
\notag\end{gathered}$$Using the local problems we will eliminate the first $n$ terms in ([twostars]{}). We study them one at the time by letting $j$ successively be equal to $1,\ldots ,n$. If $\rho _{j}=0$ we use the local problem (\[case 1\]) from case 1 with $i=j$ and the corresponding term vanishes directly. If $\rho _{j}\neq 0$ then, by assumption, $u_{j}\in L^{2}(\Omega _{T}\times
\mathcal{Y}_{j-1,m-d_{j}-1},W_{2_{{\large \sharp }}}^{1}(S_{m-d_{j}};H_{\sharp }^{1}(Y_{j})/\mathbb{R}
,L_{\sharp }^{2}(Y_{j})/\mathbb{R}
))$, which implies that $u_{j}(x,t,y^{j-1})\in W_{2_{{\large \sharp }}}^{1}(S_{m-d_{j}};H_{\sharp }^{1}(Y_{j})/\mathbb{R}
,L_{\sharp }^{2}(Y_{j})/\mathbb{R}
)$. Then, from (\[locCas2\]) with $i=j$, we obtain that $$\begin{gathered}
\rho _{j}\partial _{s_{m-d_{j}}}u_{j}\left( x,t,y^{j},s^{m-d_{j}}\right) \\
=\nabla _{y_{j}}\!\cdot \!\left( \!\int_{S_{m-d_{j}+1}}\!\!\ldots
\!\int_{S_{m}}\!\int_{Y_{j+1}}\!\!\ldots
\!\int_{Y_{n}}\!\!\!a_{0}(x,t,y^{n},s^{m})dy_{n}\cdots dy_{j+1}ds_{m}\cdots
ds_{m-d_{j}+1}\!\right)\end{gathered}$$i.e. $\!\int_{S_{m-d_{j}+1}}\!\ldots \!\int_{S_{m}}\!\int_{Y_{j+1}}\!\ldots
\!\int_{Y_{n}}\!{\small -}a_{0}(x,t,y^{n},s^{m})dy_{n}\cdots
dy_{j+1}ds_{m}\cdots ds_{m-d_{j}+1}\nabla _{y_{j}}$ in (\[twostars\]) can be replaced with the derivative $\rho _{j}\partial _{s_{m-d_{j}}}u_{j}$. Thus, Corollary 4.1 in [@NgWo] yields that the term in question vanishes. What remains of (\[twostars\]) is$$\begin{gathered}
\int_{\Omega _{T}}\int_{\mathcal{Y}_{n,m}}\left(
-a_{0}(x,t,y^{n},s^{m})+a\left( y^{n},s^{m},\nabla
u+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}+\delta c\right) \right) \\
\cdot \delta c(x,t,y^{n},s^{m})dy^{n}ds^{m}dxdt\geq 0.\end{gathered}$$Dividing by $\delta $ and passing to the limit in the sense of letting $\delta $ tend to zero, we deduce that$$a_{0}\left( x,t,y^{n},s^{m}\right) =a\left( y^{n},s^{m},\nabla
u+\sum\limits_{j=1}^{n}\nabla _{y_{j}}u_{j}\right) .$$Finally, by the uniqueness of $u$, the whole sequence converges and the proof is complete.
**4. An illustrative example**
In this section we investigate a specific nonlinear parabolic problem with a number of rapid spatial and temporal scales, some of which are not powers of $\varepsilon $. More precisely we consider the (3,4)-scaled problem
$$\begin{array}[t]{rcl}
\!\!\partial _{t}u^{\varepsilon }(x,t)\!-\!\nabla \!\cdot a\!\left( \frac{x}{2\sqrt{\varepsilon }},\!\frac{x}{\varepsilon ^{2}},\!\frac{t}{e^{\varepsilon
}-1},\!\frac{t}{\ln (1+\varepsilon ^{2})},\!\frac{t}{\varepsilon ^{3}\ln
\left( 1+\frac{1}{\varepsilon }\right) },\!\nabla u^{\varepsilon
}(x,t)\right) \!\!\!\! & \!=\!\!\! & \!\!\!f(x,t)\text{ in\ }\Omega _{T}, \\
u^{\varepsilon }(x,t)\!\!\!\! & \,\!{}\!=\! & \!\!\!0\text{ on\ }\partial
\Omega \!\!\times \!\!(0,T),\! \\
u^{\varepsilon }(x,0)\!\!\!\! & \,\!=\! & \!\!\!u^{0}(x)\text{ in }\Omega .\end{array}$$
To apply Theorem \[first inline\] we must be reassured that the two lists $\left\{ 2\sqrt{\varepsilon },\varepsilon ^{2}\right\} $ and $\left\{
e^{\varepsilon }-1,\ln (1+\varepsilon ^{2}),\varepsilon ^{3}\ln \left( 1+\frac{1}{\varepsilon }\right) \right\} $ are jointly well-separated. It holds that $$\lim_{\varepsilon \rightarrow 0}\frac{1}{2\sqrt{\varepsilon }}\left( \frac{\varepsilon ^{2}}{2\sqrt{\varepsilon }}\right) ^{1}=0,$$$$\lim_{\varepsilon \rightarrow 0}\frac{1}{e^{\varepsilon }-1}\left( \frac{\ln
(1+\varepsilon ^{2})}{e^{\varepsilon }-1}\right) ^{3}=0$$and$$\lim_{\varepsilon \rightarrow 0}\frac{1}{\ln (1+\varepsilon ^{2})}\left(
\frac{\varepsilon ^{3}\ln \left( 1+\frac{1}{\varepsilon }\right) }{\ln
(1+\varepsilon ^{2})}\right) ^{3}=0,$$which implies that both the spatial and temporal scales are well-separated. Moreover,$$\lim_{\varepsilon \rightarrow 0}\frac{\ln (1+\varepsilon ^{2})}{\varepsilon
^{2}}=1,$$so we can remove duplicates and make the joint list $\!\left\{ 2\sqrt{\varepsilon },e^{\varepsilon }\!-\!1,\varepsilon ^{2},\varepsilon ^{3}\ln
\left( 1+\frac{1}{\varepsilon }\right) \right\} \!$, which is well-separated. According to Definition \[Lists of scales\] this shows that our lists of scales are jointly well-separated. For the rest we assume that our problem fulfils the assumptions of Theorem \[first inline\].
To begin with, from Theorem \[first inline\] we know that the convergence results (\[equation 2\]) and (\[equation3\]) hold, i.e. that$$u^{\varepsilon }(x,t)\rightarrow u(x,t)\text{ in }L^{2}(\Omega _{T})$$and$$u^{\varepsilon }(x,t)\rightharpoonup u(x,t)\text{ in }L^{2}(0,T;H_{0}^{1}(\Omega )).$$
To determine the independencies and make the local problems more precise, we need to identify which values of $d_{i}$ and $\rho _{i}$ to use. We recall that $d_{i}$ is the number of temporal scales faster than the square of the spatial scale in question and $\rho _{i}$ indicates whether there is resonance or not. Let us start with the slowest spatial scale, i.e. $i=1$. To find $d_{1}$ we investigate on the basis of (I) how the first spatial scale is related to the temporal scales present in the problem. We have $$\lim_{\varepsilon \rightarrow 0}\frac{e^{\varepsilon }-1}{\left( 2\sqrt{\varepsilon }\right) ^{2}}=\frac{1}{4}>0$$and $$\lim_{\varepsilon \rightarrow 0}\frac{\ln (1+\varepsilon ^{2})}{(2\sqrt{\varepsilon })^{2}}=0,$$which means that $d_{1}=2$. For the scale in question we have resonance since $$\lim_{\varepsilon \rightarrow 0}\frac{\left( 2\sqrt{\varepsilon }\right) ^{2}}{e^{\varepsilon }-1}=4,$$i.e., $\rho _{1}=4$ according to (II). For $i=2$ we obtain$$\lim_{\varepsilon \rightarrow 0}\frac{\varepsilon ^{3}\ln \left( 1+\frac{1}{\varepsilon }\right) }{\left( \varepsilon ^{2}\right) ^{2}}=\infty$$and hence $d_{2}=0$. We also observe that $$\lim_{\varepsilon \rightarrow 0}\frac{\left( \varepsilon ^{2}\right) ^{2}}{\varepsilon ^{3}\ln \left( 1+\frac{1}{\varepsilon }\right) }=0,$$which means that $\rho _{2}=0$.
Now from Theorem \[first inline\] we have$$\nabla u^{\varepsilon }(x,t)\overset{3,4}{\rightharpoonup }\nabla
u(x,t)+\nabla _{y_{1}}u_{1}(x,t,y_{1},s_{1})+\nabla
_{y_{2}}u_{2}(x,t,y^{2},s^{3}),$$where $u\in W_{2}^{1}(0,T;H_{0}^{1}(\Omega ),L^{2}(\Omega ))$, $u_{1}\in
L^{2}(\Omega _{T};W_{2_{{\large \sharp }}}^{1}(S_{1};H_{\sharp }^{1}(Y_{1})/\mathbb{R}
,L_{\sharp }^{2}(Y_{1})/\mathbb{R}
))$ and $u_{2}\in L^{2}(\Omega _{T}\times \mathcal{Y}_{1,3};H_{\sharp
}^{1}(Y_{2})/\mathbb{R}
)$. Here $u$ is the unique solution to$$\begin{aligned}
\partial _{t}u(x,t)-\nabla \cdot b(x,t,\nabla u) &=&f(x,t)\text{ in }\Omega
_{T}, \\
u(x,t) &=&0\text{ on }\partial \Omega \times (0,T), \\
u(x,0) &=&u^{0}(x)\text{ in }\Omega\end{aligned}$$with $$\begin{gathered}
b(x,t,\nabla u) \\
=\int_{Y_{2,3}}a(y^{2},s^{3},\nabla u\left( x,t\right) +\nabla
_{y_{1}}u_{1}(x,t,y_{1},s_{1})+\nabla
_{y_{2}}u_{2}(x,t,y^{2},s^{3}))dy^{2}ds^{3}\end{gathered}$$and we have the two local problems$$\begin{gathered}
4\partial _{s_{1}}u_{1}(x,t,y_{1},s_{1})-\nabla _{y_{1}}\cdot
\int_{S_{2}}\int_{S_{3}}\int_{Y_{2}}a(y^{2},s^{3},\nabla u\left( x,t\right)
\\
+\nabla _{y_{1}}u_{1}(x,t,y_{1},s_{1})+\nabla
_{y_{2}}u_{2}(x,t,y^{2},s^{3}))dy_{2}ds_{3}ds_{2}=0\end{gathered}$$and$$-\nabla _{y_{2}}\cdot a(y^{2},s^{3},\nabla u\left( x,t\right) +\nabla
_{y_{1}}u_{1}(x,t,y_{1},s_{1})+\nabla _{y_{2}}u_{2}(x,t,y^{2},s^{3}))=0.$$
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---
abstract: |
The first goal of this study is to quantify the magnitude and spatial variability of air quality changes in the U.S. during the COVID-19 pandemic. We focus on two pollutants that are federally regulated, nitrogen dioxide () and fine particulate matter (). is emitted during fuel combustion by all motor vehicles and airplanes. is emitted by airplanes and, among motor vehicles, mostly by diesel vehicles, such as commercial heavy-duty diesel trucks. Both and are also emitted by conventional power plants, although almost exclusively by coal power plants. Observed concentrations at all available ground monitoring sites (240 and 480 for and , respectively) were compared between April 2020, the month during which the majority of U.S. states had introduced some measure of social distancing (e.g., business and school closures, shelter-in-place, quarantine), and April of the prior five years, 2015–2019, as the baseline. Large, statistically-significant decreases in concentrations were found at more than 65% of the monitoring sites, with an average drop of 2 parts per billion (ppb) when compared to the mean of the previous five years. The same patterns are confirmed by satellite-derived column totals from NASA OMI, which showed an average drop in 2020 by 13% over the entire country when compared to the mean of the previous five years. concentrations from the ground monitoring sites, however, were not significantly lower in 2020 than in the past five years and were more likely to be higher than lower in April 2020 when compared to the previous five years.
The second goal of this study is to explain the different responses of these two pollutants, i.e., was significantly reduced but was nearly unaffected, during the COVID-19 pandemic. The hypothesis put forward is that the shelter-in-place measures affected people’s driving patterns most dramatically, thus passenger vehicle emissions were reduced. Commercial vehicles (generally diesel) and electricity demand for all purposes remained relatively unchanged, thus concentrations did not drop significantly. To establish a correlation between the observed changes and the extent to which people were actually sheltering in place, thus driving less, we use a mobility index, which was produced and made public by Descartes Labs. This mobility index aggregates cell phone usage at the county level to capture changes in human movement over time. We found a strong correlation between the observed decreases in concentrations and decreases in human mobility, with over 4 ppb decreases in the monthly average where mobility was reduced to near zero and around 1 ppb decrease where mobility was reduced to 20% of normal or less. By contrast, no discernible pattern was detected between mobility and concentrations changes, suggesting that decreases in personal-vehicle traffic alone may not be effective at reducing pollution.
author:
- 'Cristina L. Archer$^1$'
- Guido Cervone
- Maryam Golbazi
- Nicolas Al Fahel
- Carolynne Hultquist
bibliography:
- 'covid19.bib'
date: 'Received: date / Accepted: date'
title: 'Changes in air quality and human mobility in the U.S. during the COVID-19 pandemic'
---
Introduction {#intro}
============
Worldwide, about 91% of the population is exposed to poor air quality. The World Health Organization (WHO) estimates that on average, 4.2 million people die each year from causes directly attributed to air pollution [@whodeath]. Nitrogen dioxide () is one of a group of highly reactive gases known as nitrogen oxides (). can irritate the human respiratory system and is also harmful to ecosystems by the formation of nitric acid and acid rain [@epano2; @lin2011]. is another harmful air pollutant that consists of microscopic particles with a diameter smaller than 2.5 $\mu$m. These particles can pose a great risk to human health because they can penetrate into human lungs and even the bloodstream; is also often associated with poor visibility [@epapm]. and are both primary (i.e., they can be directly emitted into the atmosphere) and secondary (i.e., they can also form after chemical reactions in the atmosphere) pollutants. High concentrations of both are not necessarily found where their emissions are highest, due to processes such as chemical reactions, transport, or diffusion. and are the main focus of this paper because they are among the seven “criteria" pollutants that are regulated at the federal level by the U.S. Environmental Protection Agency (EPA) via the National Ambient Air Quality Standards (NAAQS). The novel coronavirus disease (SARS-CoV-2/COVID-19, COVID-19 hereafter for brevity) was first identified in Wuhan, China, on December 30, 2019 [@who; @chan2020]. Cases started to spread initially in China but quickly expanded to other countries across the world. COVID-19 was declared a global pandemic in March 2020 [@whoEurope]. At the time of this study, over 9 million people have been affected by the virus, with over 470,000 deaths in 213 countries and territories [@worldometer; @JHU]. COVID-19 first reached the U.S. in February 2020 and since then it has caused over 120,000 deaths in the span of a few months [@cdc; @JHU]. The death rate of COVID-19 is significantly higher among people with cardiovascular and respiratory illnesses [@acc], which are also strongly connected with air pollution [@isaifan2020]. Furthermore, new studies suggest that higher concentrations of air pollutants result in a higher risk of COVID-19 infection [@yongjian2020] and mortality [@wu2020].
In the U.S., social distancing measures were implemented state by state with the goal of limiting the spread of the pandemic. In general, closure or non-physical interaction options (e.g., delivery only) were implemented for schools, restaurants, and public places of gathering. Businesses, workers, and types of activities that were deemed essential during the pandemic either continued operating under strict protection measures (e.g., personal protective equipment (PPE), masks) or switched to online work. Non-essential businesses requiring physical presence and interaction closed completely (e.g., hair salons, bars, gyms). The extent of social distancing measures, seriousness of the implementation, and the degree of compliance varied throughout the U.S. Most states announced some level of social distancing orders starting in mid-March, 2020 [@bbc], often including a mandatory quarantine for people diagnosed with or showing symptoms of the coronavirus. By the beginning of April, almost all states had a mandatory shelter-in-place or lockdown order [@nystats]. Hereafter, lockdown and shelter-in-place will be used interchangeably. The social distancing measures have led to drastic changes in mobility and energy use and therefore changes in emissions of pollutants.
Globally, the COVID-19 outbreak is forcing large changes in economic activities [@NCAR]. In China, following the strict social distancing measures, transportation decreased noticeably and, as a result, China experienced a drastic decrease in atmospheric pollution, specifically [@NCAR], , and [@manuel2020; @NCAR; @nasa] concentrations in major urban areas. However, emissions from residential heating and industry remained steady or slightly declined [@chen2020]. Using satellite data, Zhang et al. [@zhang2020nox] and the National Center for Atmospheric Research [@NCAR] reported a 70% and 50% decrease in concentrations in Eastern China, respectively. Bao and Zhang [@bao2020] showed an average of 7.8% decrease in the Air Quality Index over 44 cities in northern China. Bawens et al. [@Bauwensetal2020] and Shi and Brasseur [@ShiBra20] reported an increase in concentrations in the same region. Chen et al. [@chen2020], reported that and concentrations were decreased by 12.9 and 18.9 $\mu$g/m$^3$, respectively. They estimated that this improvement in the air quality of China avoided over 10,000 - and -related deaths during the pandemic, which could potentially outnumber the confirmed deaths related to COVID-19 in China [@chen2020]. Other researchers also have proposed that the improvements of air quality during the pandemic might have saved more lives than the coronavirus has taken [@dutheil2020; @gfeed]. Likewise, Isaifan [@isaifan2020] argues that the shutting down of industrial and anthropogenic activities caused by COVID-19 in China may have saved more lives by preventing air pollution than by preventing infection.
European countries, such as France and Italy, experienced a sharp reduction in their air pollution amid COVID-19 [@esa]. In Brazil, a significant decrease in concentrations and, to a lower extent, in levels was observed, while ozone levels were higher due to a decrease in concentrations in -limited locations [@dantas2020; @nakada2020]. The same findings were observed in Kazakhstan and Spain, respectively [@kerimray2020; @tobias2020changes]. Likewise, Iran [@nemati2020] and India [@mahato2020; @sharma2020] reported noticeable improvements in air pollution during the pandemic. Le et al. [@le2020] looked into the impacts of the forced confinement on emissions and concluded that global emissions decreased by 17% by early April compared to the average level in 2019. They believe that the yearly-mean emissions would decrease by 7% if restrictions remain by the end of 2020.
In the United States, as a result of social distancing, states started to experience a dramatic decrease in personal transportation and mobility in general [@gao2020]. Personal vehicle transportation decreased by approximately 46% on average nationwide, while freight movement only decreased by approximately 13% [@inrixtraffic]. Air traffic decreased significantly as well [@businessinsider]. On-road vehicle transportation is a main source of emissions [@nei]. Airports too are usually hot spots for pollution [@nasa].
The Houston Advanced Research Center (HARC) [@harc] analyzed the daily averages of hourly aggregated concentrations of benzene, toluene, ethylbenzene, and xylenes (BTEX) across six stations in Houston, USA. They reported a decrease in BTEX levels in the atmosphere while an intensified formation on in the region. Similarly, the New York Times reported huge declines in pollution over major metropolitan areas, including Los Angeles, Seattle, New York, Chicago, and Atlanta using satellite data [@NYtimes].
While a noticeable number of studies have looked into the correlation between lockdown measures amid COVID-19 and air quality in different countries, none has evaluated air quality for the entire United States. The goals of this study are to investigate the magnitude and spatial variability of air quality ( and ) changes in the U.S. during the COVID-19 pandemic and to understand the relationships between mobility and changes. An innovative aspect of this study is that we use an extensive database of ground monitoring stations for and (Section \[sec:AQdata\]) and a third-party high-resolution mobility dataset derived from cellular device movement (Section \[sec:mob\]). In addition, we included satellite-retrieved information to increase the spatial data coverage (Section \[sec:sat\]). Whereas most studies rely only on a comparison to 2019, we consider five prior years (2015–2019) to provide a more robust measure.
Data {#sec:data}
====
Air quality data {#sec:AQdata}
----------------
Criteria pollutant concentration data, originally measured and quality-checked by the various state agencies, are centrally collected and made available to the public by the EPA through their online Air Quality System (AQS or AirData) platform [@AQS]. For , the reported concentrations are one-hour averages, thus 24 records are reported daily (if no records are missing). For , the reported concentrations are 24-hour averages, thus one value is reported per day. The AQS pre-generated files are updated twice per year: once in June, to capture the complete data for the prior year, and once in December, to capture the data for the summer. The daily files, containing daily-average and daily-maximum of one-hour concentrations and 24-hour-average of concentrations, were downloaded for the years 2015–2019. At the time of this study (May 2020), however, the pre-generated files for April 2020 were not yet available.
For the year 2020 only, the data source was the U.S. EPA AirNow program [@Airnow], which collects real-time observations of criteria pollutants from over 2,000 monitoring sites operated by more than 120 local, state, tribal, provincial, and federal agencies in the U.S., Canada, and Mexico. As stated on AirNow website, “these data are not fully verified or validated and should be considered preliminary and subject to change." Of the two types of files available from AirNow, namely AQObsHourly and Hourly, AQObsHourly files were downloaded for March and April 2020 because of their smaller file size (they are updated once per hour, as opposed to multiple times). Texas and New York do not feed measurements to Airnow, thus their 2020 data were downloaded directly from their state websites [@TXNO2; @NYNO2].
The NAAQS for and are based on the comparison of a “design value", which is a specific statistic of measured concentrations over a specific time interval, against a threshold value as follows:
- : annual mean of 1-hour concentrations may not exceed 53 parts per billion (ppb);
- : 98th percentile of 1-hour daily maximum concentrations, averaged over 3 years, may not exceed 100 ppb;
- : annual mean of 24-hour concentrations, averaged over 3 years, may not exceed 12 $\mu$g/m$^3$;
- : 98th percentile of 24-hour concentrations, averaged over 3 years, may not exceed 35 $\mu$g/m$^3$.
Clearly it is not possible to calculate the design values as early as April because neither the annual average nor the 98th percentile can be calculated after only four months. As such, in this study we will use a simple monthly average as the representative metric to compare the concentrations in April 2020 to those in the previous five Aprils.
An air quality station, whether measuring or , was used in this study only if it reported both in 2020 (through AirNow) and in the five years prior (through AQS). In addition, only air quality stations that were reporting at least 75% of the time were retained. Note that not all -measuring sites also measure , and vice versa. Of the 426 and 882 sites that measured and , respectively, in April 2020, only 271 and 819 reported at least 75% of the time, and ultimately only 201 and 480 reported and also in April 2015–2019 for at least 75% of the time. These are the sites that we will focus on in this study and that are shown in Figures \[fig:NO2map\] and \[fig:PM25map\].
Satellite data {#sec:sat}
--------------
Satellite observations for were acquired using the OMI instrument flying onboard the NASA AURA satellite, and were downloaded using the NASA GIOVANNI portal [@giovanni]. Specifically, the Nitrogen Dioxide Product (OMNO2d) was used, which is a Level-3 global gridded product at a 0.25x0.25 spatial resolution provided for all pixels where cloud fraction is less than 30%. The product comes in two variants, the first measuring the concentration in the total column and the second the concentration only in the troposhere. For this work, the latter measurements were used.
The satellite-derived column totals at the pixels of the ground monitoring sites are well correlated with the concentrations recorded at the ground monitoring sites in all years, with R-square values varying between 0.76 and 0.80. As an example, we show the correlation between the two in 2016 and 2020 in Figure \[fig:sat-stations\]. As such, we can use satellite-derived column totals to: 1) confirm the results obtained from the ground monitoring sites and 2) analyze pixels where no ground monitoring sites are available.
Mobility data {#sec:mob}
-------------
Mobility measures aim to capture general patterns of observed movement and most available data products today utilize mobile device activity as a proxy. While policy makers set social distancing guidance, there are various policies enacted and various degrees to which policies are followed throughout the country. We seek to observe actual patterns of movement by using a dataset developed by Descartes Labs [@descartes] that provides an aggregated mobility measure based on anonymized and/or de-identified mobile device locations. Mobility is essentially a statistical representation of the distance a typical member of a given population moves in a day. Descartes Labs calculated the farthest distance apart recorded by smartphone devices utilizing select apps (with location reporting enabled) for at least 10 uses a day, spread out over at least 8 hours in a day, with a day defined as 00:00 to 23:59 local time [@Warren2020]. The maximum distance for each qualifying device is tied to the origin county in which the anonymized user is first active each day. Aggregated results at the county level are produced as a statistical measure of general travel behavior.
Mobility data are ultimately provided as percent of normal, i.e., the ratio of aggregated mobility during each day of the COVID-19 pandemic over that of the baseline (17 February–7 March 2020). Note that the baseline period is in late winter 2020, whereas the period of focus in this paper is April 2020, in spring. As such, a fraction of the differences in mobility may be due to differences in weather and/or climate rather than to COVID-19 restrictions. We did not attempt to correct for this type of bias. Another caveat, noted by the the producers of the data [@Warren2020], is that the raw data used to calculate mobility are available for only a small fraction of the total number of devices (a few percent at most), thus the resulting mobility may or may not truly represent the average behaviour in each county. Nonetheless, the effects of these sampling errors are expected to be small. The mobility data are made freely available by Descartes Labs at the U.S. county level [@gitdata].
Results {#sec:results}
=======
Observed air quality changes {#sec:conc}
----------------------------
In the rest of this paper, we will compare the monthly-average of the pollutant of interest – or – during the month of April 2020 to the average of the five monthly-averages during April of the years 2015 through 2019. There are two reasons for this choice. First, using five years to establish a reference is more meaningful than, say, using just the year 2019, because year-to-year variability can occur regardless of the pandemic. In fact we found that, in general, the year 2019 was relatively clean when compared to the previous five, thus a comparison between April 2020 and April 2019 may under-estimate the true impact of COVID-19. Second, although the monthly-average is not the design value for either or , it is a value that is representative of the overall air quality during the entire month of April. Alternative metrics, such as the monthly maximum, are more representative of extreme circumstances, like wildfires, that are not necessarily associated to COVID-19.
Starting with , the April 2020 averages were generally below the April 2015–2019 average at the ground monitoring sites, as most sites lay below the 1:1 line in Figure \[fig:scatter\]b. In addition, 65% of the sites were characterized by concentrations in 2020 that were lower than those in all of the previous five years (for the month of April). Only a few sites (5 in total, $<2\%$) experienced concentrations in 2020 that were higher than those in all of the previous five years (for the month of April). The average drop in concentrations was -2.02 ppb (Tables \[tab:NO2stats\] and \[tab:stats\]).
The same pattern is confirmed in the satellite-derived data. Out of the 227 pixels with ground monitoring sites, a total of 127 (56%) exhibited lower in 2020 than in the previous five years and only 5% higher (Figure \[fig:scatter\]b). Once all 14,706 pixels with valid satellite retrievals all over the country are considered, a similar pattern of lower column totals in 2020 than in the five previous years emerges from these data too (Figure \[fig:scatter\]c), but with 28% of the pixels lower in 2020 than in the previous five years and 5% higher (for the month of April, Table \[tab:stats\]).
In terms of spatial variability, Figure \[fig:NO2map\] shows that, although reductions were recorded all over the country, the highest decreases were observed in California and the Northeast, where the shelter-in-place measures started earlier (March 11 for California, the earliest in the country, and March 22 for New York, third earliest [@nystats]) and lasted longer (both states still have major restrictions in place as of June 10, 2020 [@WaPost]). Noticeable exceptions were North Dakota and Wyoming, where either no significant decreases or actual small increases in concentrations were observed. North Dakota enforced no shelter-in-place measures and in Wyoming only the city of Jackson implemented a stay-at-home order as of April 20, 2020 [@nystats]. However, as discussed in Section \[sec:mob\], actual people’s mobility, as opposed to state ordinances, is a better metric to understand the real effect of COVID-19 on air quality because not everybody in all counties followed the state- or county-restrictions all the time.
Figure \[fig:NO2map\] was useful because it included actual concentrations measured near the ground. However, the spatial coverage was sparse and urban areas were over-sampled compared to rural areas. This weakness is addressed via the NASA OMI satellite data, which are shown in Figure \[fig:NO2sat\] as the difference between the monthly average of column total in 2020 and that in 2015–2019 for the month of April. The regions with low coverage of ground concentration of and mobility in the Midwest are generally characterized by near-normal column totals. The Northeast hotspot of low mobility is also a hotspot of low , consistent with [@Bauwensetal2020], although it is surrounded by patches of above-normal values that were not detectable from the ground monitoring stations. The Los Angeles area is another hotspot of decreases, as for low mobility. For , the ground monitoring stations depict a completely different response to COVID-19. Whereas most sites were laying below the 1:1 line (Figure \[fig:scatter\]a), the majority of sites laid above it (Figure \[fig:scatter\]b), indicating an overall increase in monthly-average in the country in April 20202 with respect to the previous five years. Only 18% of the sites reported concentrations of that were lower in 2020 than in the previous five years (in the month of April), while 24% of the stations reported the highest levels in 2020 compared to the previous five years (for the month of April). The average increase in concentrations was small, +0.05 $\mu$g/m$^3$ (Tables \[tab:PM25stats\] and \[tab:stats\]).
In summary, we report a large decrease (-2.02 ppb, or 27%) in monthly-average concentrations across the U.S. ground monitoring stations, confirmed by the satellite-derived column total decrease of 7.1 $\times 10^{14}$ molecules/cm$^2$ (or 24%) at the pixels of the ground monitoring stations during April of 2020 when compared to April of the previous five years. When all the pixels with valid data were included, a drop of 2.4$\times 10^{14}$ molecules/cm$^2$ (or 13%) during April of 2020 was observed when compared to April of the previous five years (Table \[tab:stats\]). The monthly-average of , however, increased slightly on average (+0.05 $\mu$g/m$^3$ when compared to the previous five-year average) during the same period (Table \[tab:stats\]). In the next Section \[sec:stat\], we try to explain the reasons for these differences.
Observed mobility changes {#sec:mobchanges}
-------------------------
Time series of mobility data at the counties with ground monitoring sites are shown in Figure \[fig:spaghetti\]a and at the counties with ground monitoring sites in Figure \[fig:spaghetti\]b. Only a few counties had both types of monitoring sites, thus the counties included in the two figures are generally different. Yet, the patterns are very similar. First of all, mobility on average dramatically dropped starting in the second half of March, reaching values around 20% by April, and then started to recover in May, as some states reopened for business or relaxed the shelter-in-place measures [@WaPost]. Second, a distinct minimum in mobility during the month of April is clearly visible, which confirms that this month was the most relevant for air quality impacts from COVID-19. There is some variability around this general behaviour, but nonetheless only a few counties barely reached normal mobility in April. Lastly, the typical traffic reduction during the weekends is confirmed in the mobility data, regardless of the pandemic. This adds confidence to the use of mobility data as proxies for people’s actual behaviours.
In terms of spatial variability, changes in mobility during COVID-19 in the U.S. were not uniform, although in general mobility was reduced in most states (Figure \[fig:mobility\_avg\]a). Note the high count of non-valid data in many counties in the Midwest (Figure \[fig:mobility\_avg\]b), possibly due to low population density. However, the ground monitoring stations of both and are generally located in counties with high data availability. In general, the strongest decreases in mobility are found around large urban areas throughout the country, e.g., the Northeast corridor from Washington D.C. to Boston; the San Francisco and Los Angeles areas in California; Seattle in the Northwest; and Chicago. A few isolated counties experienced increases in mobility (in red in Figure \[fig:mobility\_avg\]a). Wyoming stands out as one of the few states with no significant decreases in mobility, consistent with the lack of shelter-in-place measures [@nystats].
Relationships between air quality and mobility changes {#sec:stat}
------------------------------------------------------
To better interpret the relationship between mobility and the air pollutant of interest, either or , the mobility data were divided into bins, based on the monthly-average (in April 2020) of the mobility in the county where each ground monitoring site was located. For most cases, there was only one ground monitoring site per county. But in some cases, such as Los Angeles county in California for or Maricopa county in Arizona for , multiple monitoring sites were located in the same county and therefore they were all paired to the same mobility value. The change in monthly-average concentration of the pollutant between April 2020 and the five previous Aprils was then calculated for each mobility bin.
Starting with , there is a clear relationship with mobility (Figure \[fig:bins\]a). Large and negative changes in concentrations, of the order of -4 ppb, were found at locations where mobility was basically halted, i.e., where it was less than 1% of normal in April 2020, as in full lock down. As mobility increased, the benefits decreased, although not linearly. For example, decreases by 2–3 ppb in concentrations occurred where mobility was restricted but not to a full lock down (i.e., between 1% and 20% of normal). Past 20%, the changes in concentrations were still negative and significant, but not large, less than 1 ppb on average. This suggests that responds modestly to changes in mobility that are not large, but then, if mobility is reduced dramatically (i.e., by at least 80%, thus it is down to 20% of normal), large decreases in can occur.
With respect to , there is no obvious relationship between the reductions in mobility and the observed concentrations (Figure \[fig:bins\]b). Only for the most extreme mobility reductions, i.e., the bin with $<$1% mobility, which indicates that the entire population was sheltering at home for the entire month of April, concentrations decreased by about 1 $\mu$g/m$^3$. After the first bin, as mobility increased, both increasing and decreasing concentrations of were found, with large standard deviations and no discernible pattern. We conclude that the changes in were not directly caused by changes in people’s mobility.
How can we reconcile the clear relationship of with mobility with the lack thereof for ? The hypothesis we put forward is that the shelter-in-place measures affected mostly people’s driving patterns, thus passenger vehicle – mostly fueled by gasoline – emissions were reduced and so were the resulting concentrations of . Commercial vehicles (generally diesel) and electricity demand for all purposes (often provided by coal-burning power plants), however, remained relatively unchanged, thus concentrations did not drop significantly and did not correlate with mobility. To test this hypothesis, in a subsequent study we will use a photochemical model, coupled with a numerical weather prediction model, which we will run with and without emissions from diesel vehicles, while keeping everything else the same. The difference between the concentrations of the pollutants in the two cases will be attributable to diesel traffic alone. Similarly, we will be able to reduce emissions from other sectors, to reflect the effect of COVID-19 on other aspects of life, such as air traffic or business closures.
Conclusions and future work
===========================
This study analyzed the effects of COVID-19 on air quality, more specifically and fine particulate concentrations, in the U.S. Although different states introduced different levels of shelter-in-place and social distancing measures at different times, by the beginning of April 2020 all states but a few had adopted some restrictions. As such, the analysis focused on the month of April 2020, which was compared to April of the previous five years, 2015–2019.
Two types of measurements were used, and concentrations from the ground monitoring stations – maintained by the states – and satellite-derived column totals in the troposphere. Although the two measurements are not identical, they are strongly correlated with one another because the near-ground concentrations of are the dominant contributors to the tropospheric column total.
To quantify social distancing, we used the mobility index calculated and distributed by Descartes Labs. Their algorithms account for people’s maximum distance travelled in a day by tracing the user’s location multiple times a day while using selected apps. Mobility is represented as a percent value, such that 100% means normal conditions, i.e., those during the period of 17 February – 7 March 2020.
We found that levels decreased significantly in April 2020 when compared to April of the five previous years, by up to 8 ppb in the monthly average at some locations. On average over all U.S. monitoring sites, the decrease in levels was between 24% (from satellite) and 27% (from ground stations). The decreases in were largest where mobility was reduced the most, with a direct, although not linear, relationship between the two. In terms of spatial variability, hotspots of reduced concentrations in the Northeast and California coincided perfectly with hotspots of reduced mobility. Vice versa, states where social distancing measures were minimal experienced the smallest reduction in , e.g., Wyoming and North Dakota.
By contrast, the concentrations of did not decrease significantly during the same period and even reached unprecedented high values at about a fifth of the sites. In addition, changes in concentrations were not correlated with changes in people’s mobility, neither spatially nor as aggregated statistics.
We propose that the different response to reduced people’s mobility between and could be explained by the fact that commercial vehicles (including delivery trucks, buses, trains), generally diesel fueled, remained more or less in circulation, while passenger vehicles, gasoline fueled, dropped dramatically due to COVID-19. emissions are much larger from diesel than from gasoline vehicles. In addition, other sources of emissions, like power plants, did not decrease. We plan to verify this hypothesis in a subsequent study using a photochemical model coupled with a numerical weather prediction model, as described in Section \[sec:stat\].
As far as we know, this is the first study to use ground monitoring stations to assess the effects of COVID-19 on air quality in the U.S. Satellite-derived column totals have been used in a few previous studies, but none looked at the correlation between the two types of measurements. Another innovation of this study is the use of mobility data, which are an excellent proxy for actual people’s behaviour, as opposed to the state or county regulations, which may or may not be fully followed by people.
This analysis has also numerous limitations. First of all, we paired mobility data and pollutant concentrations at the county level, thus we implicitly assumed that the measured concentration and county-average mobility were representative of the entire county. For large counties, especially those with also low population density, this assumption may not hold. The second implicit assumption of our pairing is that local mobility affects local pollution only and, vice versa, that local pollution is affected by local mobility only. In other words, we are neglecting the effects of transport and chemical reactions, which could cause either an increase or a decrease of pollution regardless of the local mobility change in the county of interest. For example, consider the case that the prevailing wind is such that a county is located downwind of an airport. If the airport was shutdown during the pandemic, that county would see a reduction of and concentrations even if no social distancing measures were in place. Another limitation is that we looked at people’s mobility as the only factor explaining concentration changes, whereas emissions changed also in response to business and school closures, air traffic reductions, among many others sources. Lastly, we focused on two pollutants only, and , because of time constraints; future work will include other regulated compounds, such as ozone and carbon monoxide.
The authors would like to thank Descartes Labs for providing the mobility data free of charge. The air quality and satellite-derived data were provided by numerous federal and state agencies, listed in the references.
Conflict of interest {#conflict-of-interest .unnumbered}
====================
On behalf of all authors, the corresponding author states that there is no conflict of interest.
State N. Sites 2015 2016 2017 2018 2019 2020
---------------------- ---------- ------- ------- ------- ------- ------- -------
Arizona 4 12.49 11.68 13.58 12.82 10.33 9.06
California 56 9.73 9.23 8.79 8.85 8.13 5.92
Colorado 7 11.37 9.87 9.39 9.94 9.5 8.25
Connecticut 3 11.71 11.13 11.72 10.34 8.43 6.54
District Of Columbia 1 9.19 8.74 7.43 8.25 8.46 7.26
Florida 7 5.65 4.23 4.73 5.55 5.34 4.77
Georgia 2 11.39 12.15 10.83 11.62 10.8 10.83
Hawaii 1 3.04 2.99 4.59 3.21 3.72 2.85
Indiana 4 11.49 9.43 8.32 9.89 8.2 7.2
Iowa 1 1.42 2.33 1.66 2.08 1.9 1.69
Kansas 4 6.10 4.86 4.4 5.98 5.16 4.47
Kentucky 1 13.94 14.95 13.08 13.18 17.43 11.25
Maine 2 4.30 3.67 3.99 4.03 3.39 2.83
Maryland 5 10.43 10 8.57 8.84 8.11 6.74
Massachusetts 8 9.47 9.08 7.15 8.45 6.47 5.15
Michigan 2 9.37 8.28 8.37 8.59 7.39 6.79
Minnesota 2 7.51 5.95 7.25 9.88 6.53 5.08
Mississippi 1 3.91 4.25 4.28 3.85 3.79 2.68
Missouri 6 10.81 9.17 8.82 8.62 8.64 6.92
Montana 1 0.54 0.61 0.9 0.68 0.46 0.36
Nevada 2 8.48 7.93 9.77 10.09 8.53 6.76
New Jersey 8 13.80 13.39 12.33 12.48 12.38 8.03
New Mexico 8 4.68 4.56 4.48 4.74 5.22 3.58
New York 5 13.64 12.56 12.05 12.42 11.3 7.55
North Carolina 3 6.04 6.29 6.05 6.11 6.58 4.4
North Dakota 6 2.91 1.98 2.44 2.59 2.16 1.95
Ohio 5 14.60 12.13 10.84 11.17 11.28 8.25
Oklahoma 3 8.89 8.43 7.02 7.38 7.38 6.02
Oregon 2 12.39 11.59 10.69 9.42 9.98 7.77
Pennsylvania 1 12.87 12.89 11.74 10.81 8.7 10.39
Rhode Island 2 14.03 13.5 10.23 13.37 10.15 8.47
South Carolina 2 5.59 6.03 6.73 6.62 5.84 4.92
Texas 40 5.80 6.23 5.23 6.41 5.95 5.26
Utah 7 3.42 2.95 4.5 4.15 2.92 2.32
Vermont 2 7.32 6.12 6.23 6.16 5.74 4
Virginia 9 5.77 5.05 4.92 5.25 5.42 4.04
Washington 2 16.80 18.17 14.25 13.32 11.36 9.52
Wisconsin 2 12.50 11.11 9.61 11.36 9.75 8.63
Wyoming 13 1.69 1.62 1.61 1.55 1.31 1.33
: Monthly-average of concentrations (ppb) by state in the month of April of the years 2015–2020.[]{data-label="tab:NO2stats"}
State N. Sites 2015 2016 2017 2018 2019 2020
---------------------- ---------- ------- ------ ------ ------ ------ -------
Alabama 4 7.94 7.56 8.82 7.26 7.64 8.59
Alaska 4 3.45 3.57 4.36 3.43 4.45 3.83
Arizona 13 5.67 5.82 6.9 8.13 5.03 4.75
Arkansas 4 7.01 7.12 8.12 6.98 7.51 7.54
California 61 7.06 6.65 6.35 7.93 6.31 5.26
Colorado 8 5.31 3.65 4.63 5.87 5.26 5.21
Connecticut 8 4.8 5.54 3.81 5.61 5.32 5.75
Delaware 3 5.73 5.65 6.21 6.48 5.91 7.09
District Of Columbia 1 6.21 6.68 7.11 8 6.33 4.67
Florida 16 7.01 7 7.41 7.68 6.39 9.52
Georgia 10 7.87 8.31 8.52 8.25 8.83 8.44
Hawaii 7 5.96 4.72 6.68 4.26 3.2 3.32
Idaho 5 4.47 5.51 4.21 4.43 3.55 5.72
Illinois 14 8.23 7.67 7.06 8.03 7.43 8.11
Indiana 15 7.5 8.21 6.17 6.64 6.21 8.47
Iowa 9 7.78 6.97 6.18 7.09 5.65 8.71
Kansas 3 6.34 6.21 6.85 8.31 8.77 11.47
Kentucky 12 6.97 7.03 6.27 6.48 6.77 8.04
Louisiana 4 7.81 7.06 8.43 7.52 6.84 7.74
Maine 6 5.16 5.64 4.93 4.46 3.63 4.05
Maryland 10 6.93 6.75 5.74 6.46 4.33 4.81
Massachusetts 9 4.67 5.08 3.02 5.6 4.68 5.71
Michigan 11 6.7 6.7 5.52 6.48 6.33 7.1
Minnesota 18 5.09 5.66 5.06 6.22 4.88 5.73
Mississippi 7 7.98 7.49 8.32 8.27 6.97 10.15
Missouri 13 7.65 6.3 6.46 7.33 7.33 6.6
Montana 11 5.02 4.57 4.07 4.81 3.7 4.21
Nebraska 2 8.02 6.65 9.82 9.77 6.1 8.17
Nevada 6 6.16 4.81 4.57 5.25 3.23 4.11
New Hampshire 5 4.3 4.2 3.18 4.11 3.34 4.01
New Jersey 3 5.69 7.27 7.32 7.17 6.09 5.84
New Mexico 5 7.34 5.05 7.02 7.93 5.58 4.87
New York 7 5.32 4.99 4.38 4.93 4.92 4.5
North Carolina 13 6.78 7.25 7.16 6.53 6.4 5.41
North Dakota 6 4.65 3.24 4.33 5.26 3.58 4.2
Ohio 12 7.49 7.86 5.55 6.82 6.68 6.99
Oklahoma 9 7.32 7.31 7.52 8.78 7.71 8.19
Oregon 12 5.3 4.94 4.28 5.14 4.24 5.17
Pennsylvania 24 7.41 7.01 7.64 6.99 6.52 6.97
Rhode Island 5 4.84 5.2 4.56 6.01 3.3 4.08
South Carolina 6 7.33 6.95 7.95 6.64 6.6 6.62
South Dakota 8 6.65 4.93 5.35 5.54 3.68 5.01
Tennessee 11 6.41 6.68 6.8 6.45 6.5 6.23
Texas 11 10.04 8.84 9.13 8.79 8.76 10.27
Utah 7 5.6 3.3 3.47 4.59 3.03 3.79
Vermont 4 4.32 4.24 3.27 4.69 4.48 4.73
Virginia 6 5.74 5.76 6.33 5.31 5.75 5.66
Washington 11 5.57 6.5 3.53 4.01 4.17 5.11
Wisconsin 18 5.45 7.47 4.58 5.46 6.68 7.49
Wyoming 3 4.04 2.75 2.73 3.3 2.57 1.73
: Monthly-average of concentrations ($\mu$g/m$^3$) by state in the month of April of the years 2015–2020.[]{data-label="tab:PM25stats"}
2015 2016 2017 2018 2019 2020
--------------------------------------------- ------ ------ ------ ------ ------ ------
**NASA OMI**
at all pixels (10$^{16}$molecules/cm$^2$) 0.16 0.15 0.14 0.13 0.14 0.12
at ground sites (10$^{16}$molecules/cm$^2$) 0.32 0.31 0.28 0.29 0.29 0.23
**Ground monitoring stations**
(ppb) 8.16 7.69 7.22 7.62 7.0 5.52
($\mu$g/m$^3$) 6.52 6.36 6.02 6.60 5.82 6.31
: Average air quality measurements in April of the years 2015–2020 from NASA OMI and ground monitoring stations.[]{data-label="tab:stats"}
a)![Scatter plots of monthly-average concentrations (ppb) from the ground monitoring sites versus monthly-average column totals ($10^{16}$ molecules/cm$^2$) retrieved from the NASA OMI satellite at the pixels of the ground monitoring sites during (a) April 2016 and (b) April 2020. []{data-label="fig:sat-stations"}](NO2_Sat_AQS_2016-min.png "fig:"){width="8cm"} b)![Scatter plots of monthly-average concentrations (ppb) from the ground monitoring sites versus monthly-average column totals ($10^{16}$ molecules/cm$^2$) retrieved from the NASA OMI satellite at the pixels of the ground monitoring sites during (a) April 2016 and (b) April 2020. []{data-label="fig:sat-stations"}](NO2_Sat_AQS_2020-min.png "fig:"){width="8cm"}
![Scatter plots of: (a) and (b) observed concentrations at ground monitoring sites, and column totals from NASA OMI satellite at (c) pixels of the ground monitoring sites and (d) all pixels during April of 2020 (y-axis) versus April of the previous five years 2015–2019 (x-axis). Blue-filled markers represent sites for which the values in April 2020 were lower than in any April of 2015–2019; red-filled markers represent sites for which the values in April 2020 were higher than in any April of 2015–2019. []{data-label="fig:scatter"}](ScatterPlots-min.png){width="\textwidth"}
![Difference in monthly-average concentrations (ppb) between April 2020 and the five previous Aprils (2015–2019). Negative values (blue) indicate a decrease in concentrations in April 2020, vice versa positive values (red) indicate an increase.[]{data-label="fig:NO2map"}](map_NO2.png){width="\textwidth"}
![Difference in monthly-average concentrations ($\mu$g/m$^3$) between April 2020 and the five previous Aprils (2015–2019). Negative values (blue) indicate a decrease in concentrations in April 2020, vice versa positive values (red) indicate an increase.[]{data-label="fig:PM25map"}](map_PM25.png){width="\textwidth"}
![Difference between the NASA OMI monthly-average column totals ($10^{16}$ molecules/cm$^2$) in 2020 and in 2015–2019 for the month of April. The “hotspot" of reduced in the Northeast is apparent. []{data-label="fig:NO2sat"}](map_OMI_Difference-min.pdf){width="\textwidth"}
a)![Mobility, expressed as percent of normal, at the locations of the ground stations monitoring (a) and (b) during March–May 2020. The average of all the stations is shown in red.[]{data-label="fig:spaghetti"}](lines_Mobility_NO2-min "fig:"){width="8cm"} b)![Mobility, expressed as percent of normal, at the locations of the ground stations monitoring (a) and (b) during March–May 2020. The average of all the stations is shown in red.[]{data-label="fig:spaghetti"}](lines_Mobility_PM25-min.pdf "fig:"){width="8cm"}
a)![Spatial distribution of (a) mobility, expressed as percent of normal, in April 2020 and (b) mobility data availability, expressed as number of missing days, during March–May 2020.[]{data-label="fig:mobility_avg"}](map_mobility_average.png "fig:"){width="\textwidth"} b)![Spatial distribution of (a) mobility, expressed as percent of normal, in April 2020 and (b) mobility data availability, expressed as number of missing days, during March–May 2020.[]{data-label="fig:mobility_avg"}](map_mobility_NADays.png "fig:"){width="\textwidth"}
a)![Changes in monthly-average concentrations of (a) and (b) near ground monitoring sites during April of 2020 versus April of the previous five years as a function of mobility index bins in April 2020. []{data-label="fig:bins"}](NO2_Mobility_Bins-min.png "fig:"){width="12cm"} b)![Changes in monthly-average concentrations of (a) and (b) near ground monitoring sites during April of 2020 versus April of the previous five years as a function of mobility index bins in April 2020. []{data-label="fig:bins"}](PM25_Mobility_Bins-min.png "fig:"){width="12cm"}
|
[Deriving Gauge Symmetry and Spontaneous Lorentz Violation]{}
[**J.L. Chkareuli**]{}$^{1}$, [**C.D. Froggatt**]{}$^{2}$[** and H.B. Nielsen**]{}$^{3}$
$^{1}$[*E. Andronikashvili*]{} [*Institute of Physics and* ]{}
[*I. Chavchavadze Tbilisi State University, 0177 Tbilisi, Georgia\
*]{}
$^{2}$[*Department of Physics and Astronomy, Glasgow University,*]{}\
[*Glasgow G12 8QQ, Scotland*]{}\
$^{3}$[*Niels Bohr Institute, Blegdamsvej 17-21, DK 2100 Copenhagen,* ]{}
[*Denmark*]{}
[**Abstract**]{}
We consider a class of field theories with a four-vector field $A_{\mu }(x)$ in addition to other fields supplied with a global charge symmetry - theories which have partial gauge symmetry in the sense of only imposing it on those terms in the Lagrangian density which have derivatives as factors in them. We suppose that spontaneous Lorentz invariance breaking occurs in such a theory due to the four-vector field taking a non-zero vacuum expectation value. Under some very mild assumptions, we show that this Lorentz violation is not observable and the whole theory is practically gauge invariant. A very important presupposition for this theorem is that an initial condition is imposed on the no-derivative expressions corresponding to the early Universe being essentially in a vacuum state. This condition then remains true forever and can be interpreted as a gauge constraint. We formulate the conditions under which the spontaneous Lorentz violation becomes observable. Spontaneously broken Lorentz invariance could be seen by some primordially existing or created fossil charges with the property of moving through the Universe with a fixed velocity.
Introduction
============
It is by now a rather old idea to seek to obtain the photon as the Nambu-Goldstone boson for spontaneous Lorentz invariance violation (SLIV) [@bjorken]. If, for instance, a four-vector field $A_{\mu }$ takes on in vacuum a non-zero temporally and spatially constant value we have an example of the SLIV [@alan; @jackiw; @cfn]. A priori one would then expect in theories with such spontaneous breakdown to see in practice violation of physical Lorentz invariance. It is the purpose of the present article to point out that under some rather mild assumptions, however, one [*one does not find this breaking of Lorentz invariance to be recognizable experimentally*]{}[^1].
A very simple way to achieve the SLIV is to introduce into a usual QED Lagrangian density a potential term $P(A^{2})$ for the four-vector field $%
A_{\mu }(x)$ (so that $A^{2}(x)=A^{\mu }(x)A_{\mu }(x)$) which then after having been transferred to the Hamiltonian density in the usual way could have its minimum for a non-zero value of $A_{\mu }$[@alan; @cfn]. In this way a vacuum apparently without Lorentz symmetry can rather easily come about $$A_{\mu }(x)=a_{\mu }(x)+n_{\mu }V$$where $n_{\mu }$ is an properly oriented unit Lorentz vector with $%
n^{2}=n_{\mu }n^{\mu }=\pm 1$ for time-like and space-like Lorentz violation, respectively, while $V$ is a proposed SLIV scale.
At first one could easily come to the belief that in such models with SLIV one will find scatterings that indeed reflect the special direction in which the symmetry is broken and thus that one obtains a theory that for all practical purposes breaks Lorentz invariance. However, we deliver a theorem telling that really in a very large class of cases such models [*do not show Lorentz symmetry breaking*]{} but rather behave as an ordinary Lorentz and gauge invariant theory manifesting the SLIV only in some noncovariant gauge choice.
In section 2 we shall put up the type of model which we consider: a rather general type of field theory with a four-vector field $A_{\mu }(x)$ which although it comes to play the role of the four vector potential in electrodynamics is a priori [*not* ]{}assumed to obey full gauge invariance - only terms in the Lagrangian with derivatives are assumed to be gauge invariant. And we put forward and prove rather closely related versions of our theorem.
In sections 3 and 4 we argue how it is possible at all to avoid our theorem first by means of initial conditions with what is essentially immovable charges, and then by allowing extra terms in the Lagrangian which give no way to this theorem to work.
In section 5 we make a discussion of that the presuppositions of our theorem - especially concerning the need for gauge symmetry of the derivative containing terms - are quite reasonable to require in order to avoid bottomlessness of the Hamiltonian and/or to avoid domain walls in cosmology.
Finally, in section 6 we present a resume and conclude.
The theorem
===========
Now we should have in mind that it is a priori the philosophy of the present article to work with theories that are [*not*]{} a priori gauge invariant. This means that we do not impose gauge invariance on the whole Lagrange density ${\cal L}(x)$. Instead, we shall impose gauge symmetry only on the vector field kinetic term or, in general, on the part of our Lagrangian density having derivatives (see for more disussion section 5). Most importantly this means that the kinetic term for the four-vector field $%
A_{\mu }(x)$ to be the usual one, $-\frac{1}{4}F_{\mu \nu }^{2}$, so as to see that, in spite of that one would have expected to find effects of Lorentz breaking (stemming, say, from some vector field constraint $A^{2}$ $=
$ $V^{2}$ where $V$ is the SLIV scale[@nambu; @ac]), it turns out to deliver only simple free Maxwell equations. We shall generalize our type of models to include matter fields such as Dirac fields or Weyl fields and a charged scalar that could potentially be used as a Higgs field $\phi $ and also we include into consideration as minimal QED couplings so the non-minimal ones with dimensionful coupling constants. With such matter fields our assumption of the terms with derivatives to be gauge invariant means that most of the terms in the matter field Lagrangian density should actually have the usual gauge invariant form as it takes place for the conventional minimal electrodynamics[^2]. However, for the general vector field potential $P(A^{2})$ and the non-minimal couplings in the Lagrangian many new gauge breaking terms, including the terms with derivatives, could in principle appear in it through the radiative corrections. We propose that all these terms are very small being substanially suppressed by the high SLIV scale $V$ which is usually thought to be close to the Planck mass $M_{P}$. The simplest choice for possible “large” terms could be then that they were only allowed to depend on the non-derivative Lorentz invariant field combinations from each type of field involved $A^{2}$, $|\phi |^{2}$ and $\overline{\psi }\psi $ ($%
\overline{\psi }\gamma _{5}\psi $) separately but not on their mutual contractions, like as the contraction of the field $A_{\mu }$ with the fermion current $\overline{\psi }\gamma ^{\mu }\psi $ apart from its usual occurrence in the minimal QED[^3]. The complicated high-dimension operators such the four-fermion current-current term etc. are also ignored. This simplest choice for gauge breaking terms is then happened to be a base for the following theorem to work.
: Consider a Lorentz invariant Abelian field theory with a priori a [*global* ]{}charge conservation symmetry only, while gauge symmetry is not imposed in as far we allow for terms containing the squared four vector field $A^{2}$ in combination with globally charge symmetric but derivative free combinations of the matter fields. This means that the theory has in addition to only fully gauge invariant terms - as usual electrodynamics - a (seemingly) gauge breaking term
$${\cal L}_{br}(A^{2},|\phi |^{2},\overline{\psi }\psi ,\overline{\psi }\gamma
_{5}\psi ) \label{gf}$$
depending only on the globally phase transformation invariant combinations without any derivatives and only on $A_{\mu }$ via $A^{2}$. Provided now that the Universe should have started with initial conditions so as to ensure in early times the vanishing of the partial derivative of the gauge non-invariant part ${\cal L}_{br}$
$$\frac{\partial {\cal L}_{br}}{\partial A^{2}}=0\quad \label{gc}$$
and that $A_{\mu }(x)$ equals a non-zero constant in these asymptotic early times (i.e. a Lorentz symmetry breaking vacuum), then the theory is indeed interpretable as a Lorentz and gauge invariant theory with a non-covariant gauge choice properly fixed in the theory.
The basic idea in the proof of this theorem is that once the Universe gets started in a state in which the gauge condition (\[gc\]) is satisfied, this condition will go on being fulfilled. In fact, we can easily see that by varying $A_{\mu }$ in the whole Lagrangian density, which now is of the form
$${\cal L}={\cal L}_{inv}+{\cal L}_{br}$$
where the terms ${\cal L}_{inv}$ are gauge invariant terms the equation of motion obtained becomes
$$\partial _{\mu }F^{\mu \nu }=2A^{\nu }\frac{\partial {\cal L}_{br}}{\partial
A^{2}}+j_{matter}^{\nu }. \label{gme}$$
Here $j_{matter}^{\nu }$ is the current coming from the matter fields $\psi $ and $\phi $ other than the four-vector one $A_{\mu }$. With the assumption of requiring global charge conservation - or phase transformation symmetry - for the matter fields we get that the current $j_{matter}^{\nu }$ becomes conserved $\partial _{\nu }j_{matter}^{\nu }=0$, and since the lefthand side of the equation (\[gme\]) is divergence free due to the antisymmetry of $%
F_{\mu \nu }$, we thus arrive at the conclusion that the vector field current
$$j_{A}^{\nu }=A^{\nu }\frac{\partial {\cal L}_{br}}{\partial A^{2}}
\label{ja}$$
from the non-gauge invariant term becomes itself separately conserved.
Let us first consider the case of the vacuum background $A^{\nu }$ in the early times were time-like. Then if we start from the mentioned ”gauge condition” (\[gc\]) in early times the conservation of the vector field current $j_{A}^{\nu }$ (\[ja\]) comes to say that the partial derivative of the gauge non-invariant part of the Lagrangian $\frac{\partial {\cal L}%
_{br}}{\partial A^{2}}$ does not vary in the direction of the four vector field $A_{\nu }$. In fact, the requirement of the separate conservation of $%
j_{A}^{\mu }$ implies that
$$(\partial _{\mu }A^{\mu })\frac{\partial {\cal L}_{br}}{\partial A^{2}}%
+A^{\mu }\partial _{\mu }\frac{\partial {\cal L}_{br}}{\partial A^{2}}=0$$
and thus if $\frac{\partial {\cal L}_{br}}{\partial A^{2}}=0$ at the start, the derivative of $\frac{\partial {\cal L}_{br}}{\partial A^{2}}$ in the direction of the the field $A^{\mu }$, namely the second term in this equation, stays equal to zero. So, this partial derivative $\frac{\partial
{\cal L}_{br}}{\partial A^{2}}$ must take the same value all along a curve laid out to follow the $A^{\nu }$ field direction by having these fields as tangents. This means that once it is zero at the beginning it must remain zero along these curves. However, if we start in a vacuum state having Lorentz invariance spontaneously broken through the vector field $A^{\nu }$ vacuum expectation value ($<A^{\nu }>$ $=n^{\nu }V$ with $n^{2}=1$ for the time-like SLIV), such a beginning with the gauge condition (\[gc\]) fulfilled is basically enforced. One namely then has a non-zero but constant $A^{\nu }$ leading to the $F_{\mu
\nu }=0$. Since the matter current $j_{matter}^{\nu }$ is always zero in vacuum, it follows that also the current $\ \ j_{A}^{\nu }=0.$Then, with with $A^{\nu }\neq 0$ we conclude that the factor $\frac{\partial {\cal L}%
_{br}}{\partial A^{2}}$ in the current $j_{A}^{\nu }$ (\[ja\]) should also be zero. But once we have $\frac{\partial {\cal L}_{br}}{\partial A^{2}}$ zero at the start, it follows from the equations of motion that this gauge condition is satisfied forever.
So we have seen that a very mild initial condition can enforce the special gauge condition (\[gc\]) and the vanishing of the current $j_{A}^{\nu }$ at all times. Thus the potential possibility for seeing e.g. Coulomb fields in the $F_{\mu \nu }$ around the $j_{A}^{\nu }$ charges, which could reveal the non-gauge invariant properties of the $j_{A}^{\nu }$ current, is in fact prevented. That is to say that an observer, who only has access directly to the usual $F_{\mu \nu }$ fields but not to the $A^{\nu }$ field, could not hope to gain access to the gauge dependent features of $A^{\nu }$ indirectly via the $j_{A}^{\nu }$ anymore, because this current would remain zero.
It should be remarked that for the SLIV to be non-observable we had to assume the special gauge condition (\[gc\]) as an initial condition, due to the Universe being at first in a vacuum state. Otherwise, by allowing initial deviations from the vacuum state, we would get some true observable breaking of Lorentz invariance. However, such alternative initial conditions means that there are a kind of fossilsof places where the gauge condition (\[gc\]) is not fulfilled. Actually such deviations from the vacuum inspired gauge would mean that the current $j_{A}^{\nu }$ were not zero but flowed in the direction of the $A^{\nu }$ field. In the coordinate system in which the vacuum $A^{\nu }$ had only the time component $A^{0}$ this would mean fossil charges at rest forever. Having such charges at one’s disposal the physical Lorentz violation would actually be accessible. In fact the very finding of some resting charge of this type, which is only able to follow the direction of the essentially vacuum four vector field $A^{\nu }$, would by itself constitute a strong breaking of Lorentz invariance: these charges do not move at all in the special frame. They could be observed by means of the Coulomb fields, say, by which they would be surrounded. But it should be stressed that we proved above that these fossil resting charges cannot be produced, if the condition (\[gc\]) is fulfilled all over the Universe and there are no charges of this type to begin with. They can only come from the start, but cannot be produced.
For simplicity we treated above only the case that the vacuum expectation value of the $A^{\mu }$ field was pointed out in a timelike direction. Actually, we mainly used that case to ensure that the curves having the $%
A^{\mu }$ as tangents in space-time would come from the past and run into future, so that an assumption about the partial derivative $\frac{\partial
{\cal L}_{br}}{\partial A^{2}}$ being zero in the early Universe were sufficient to ensure it once we had proved that it did not vary along these curves. Note however that the frames in which the assumed initial conditions $\frac{\partial {\cal L}_{br}}{\partial A^{2}}=0$ are satisfied actually form a special non-Lorentz invariant set for the case of a space-like SLIV ($%
<A^{\mu }>$ $=n^{\mu }V$ with $n^{2}=-1$). In such a frame, a curve having a corresponding space-like vacuum expectation value $<A^{\mu }>$ as a tangent will generically have a component along the time direction associated with a non-zero $<A^{0}>$. Thus the proof of our theorem follows in the same way for the space-like as in the time-like case.
Counter examples by initial conditions
======================================
The obvious question to ask is whether we can make the SLIV become observable by not having the initial conditions with $\frac{\partial {\cal L}%
_{br}}{\partial A^{2}}=0$ which was needed in the theorem. So let us think about rather small deviations from this condition having a perturbatively small $\frac{\partial {\cal L}_{br}}{\partial A^{2}}$. Even in this case the conservation of the current $j_{A}^{\mu }$ following from the basic equation (\[gme\]) would enforce the partial derivative $\frac{\partial {\cal L}%
_{br}}{\partial A^{2}}$ to remain approximatively constant along the curves tangential to the $A_{\mu }$ field. So we would indeed find that such perturbatively small deviations from zero would remain constant along these curves. In the approximate vacuum situation with a background vacuum with $%
A_{\mu }\approx V_{\mu }$ being constant in space-time ($V_{\mu }=n_{\mu }V$ where $V$ is a proposed SLIV scale) we could, for instance, in the $V_{\mu }$ timelike case talk about the frame in which the spatial components of $%
V_{\mu }$ are zero as the frame specified by $V_{\mu }$ and we would simply have the curves tangential to $A^{\mu }$ or approximately to $V^{\mu }$ are the timelike curves meaning the time tracks of resting particles. In this situation the partial derivative $\frac{\partial {\cal L}_{br}}{\partial
A^{2}}$ will stay approximately constant as function of time, but could (while we still consider it small) vary as a function of space. In the first approximation we see from the expression (\[ja\]) that $j_{A}^{\mu }$ now represents a charge distribution $V^{0}\frac{\partial {\cal L}_{br}}{%
\partial A^{2}}$ which does not change with time. This charge distribution is via the Maxwell equations - our equations of motion derived by the variation of $A_{\mu }$ - observable, as are usual charges, e.g. via the Coulomb field they cause. But now since $A_{\mu }$ or $V_{\mu }$ is neither gauge nor Lorentz invariant in the SLIV case such observation of these fields via the Maxwell equations means that these symmetries are effectively broken and, therefore, SLIV becomes physically observable. Thus our theorem does not work if the initial condition that the partial derivative $\frac{%
\partial {\cal L}_{br}}{\partial A^{2}}$ be zero is not fulfilled. So, this condition mentioned in the theorem is quite needed.
We should really think of the deviations from zero of this $\frac{\partial
{\cal L}_{br}}{\partial A^{2}}$ as representing charge density that must be fossil in the sense that we could not obtain it unless it were there to begin with. One would therefore be reasonable to call the charge density $%
\rho _{fossil}=V^{0}\frac{\partial {\cal L}_{br}}{\partial A^{2}}$ a fossil charge.We also remark that it does not change as time goes on and thus just rests where it is to begin with. So we could call it [*the fossil resting charge density*]{}. It is just via this resting fossil charge density that the Lorentz non-invariance can come into the theories which in other respects obey the conditions of our theorem but only lacks to obey the initial condition requirements.
One should of course seek experimentally to look for such resting fossil charges. If one had single quanta of them, one might observe them as particles moving with a remarkably fixed velocity. If this velocity were larger than the velocity of an atomic electron, such a fossil charge would produce ionization tracks over long distances in an emulsion without stopping. This would be a characteristic signal and its non-observation puts a severe upper bound on the density of resting fossil charges with such a velocity. However one might guess that their velocity would follow the cosmic microwave background. In which case the velocity observed in practice would be that of the earth relative to this microwave background, which is of the order of 400 km/s [@pdg]. Unfortunately this velocity is too small to produce ionization and the resting fossil charges would not be so easily observed.
In the case of a space-like $<A^{\mu }>$, the fossil charges would look like tachyons passing by. They would then give rise to Cerenkov radiation and would also make tracks in emulsions. The non-observation of such effects again places a severe upper bound on any fossil charge density.
Counter example by allowing extra terms
=======================================
The assumptions made in our theorem above might seem a bit arbitrary and not so simple, and indeed are not exceedingly beautiful. Apart from the necessary requirement to have the gauge symmetric $F_{\mu \nu }^{2}$ form for the vector field kinetic terms that we discuss in the next section, we can wonder about why we could not let in the Lagrangian ${\cal L}_{br}$ the gauge violating high-dimension operator terms which, for instance, could depend also on a extra construction $A_{\mu }\overline{\psi }\gamma ^{\mu }\psi$ etc. in addition to its dependence on $A^{2}$, $\overline{\psi }\psi$ and $|\phi |^{2}$. The point, however, is that with such a type of term, say $|\phi |^{2}A_{\mu }%
\overline{\psi }\gamma ^{\mu }\psi$, would indeed lead to a model in which the SLIV were observable - of course only through the non-renormalizable terms that now had to be used. Nevertheless, it would in this case be a genuine Lorentz non-invariant theory to live in, even if one starts with the above SLIV vacuum. We can indeed see that if we allow the ${\cal L}_{br}$ to depend also on $A^{\mu }\overline{\psi }\gamma _{\mu }\psi $ so as to become a function of the type ${\cal L}_{br}(A^{2},A^{\mu }\overline{\psi }\gamma
_{\mu }\psi ,\overline{\psi }\psi ,|\phi |^{2})$ we get the new current $%
j_{A}^{\mu }$ coming from the variation of $A^{\mu }$ in the gauge symmetry breaking part ${\cal L}_{br}$
$$j_{A}^{\mu }=A^{\mu }\frac{\partial {\cal L}_{br}}{\partial A^{2}}+\overline{%
\psi }\gamma ^{\mu }\psi \frac{\partial {\cal L}_{br}}{\partial (A^{\nu }%
\overline{\psi }\gamma _{\nu }\psi )} \label{cur}$$
and thus the requirement of it being divergenceless $\partial _{\mu
}j_{A}^{\mu }=0$ would now no longer immediately lead to that the partial derivative $\frac{\partial {\cal L}_{br}}{\partial A^{2}}$ would stay zero along curves tangentially following the $A^{\mu }$ field even at first being zero at such a curve in the early times. Rather we now have an extra term$$\overline{\psi }\gamma ^{\mu }\psi \frac{\partial {\cal L}_{br}}{\partial
(A^{\nu }\overline{\psi }\gamma _{\nu }\psi )}$$ in the current $j_{A}^{\mu }$ (\[cur\]) and when we require its divergenceless $\partial _{\mu }j_{A}^{\mu }=0$ we get the divergence of this extra term into the equation then obtained $$(\partial _{\mu }A^{\mu })\frac{\partial {\cal L}_{br}}{\partial A^{2}}%
+A^{\mu }\partial _{\mu }\frac{\partial {\cal L}_{br}}{\partial A^{2}}+%
\overline{\psi }\gamma ^{\mu }\psi \partial _{\mu }\frac{\partial {\cal L}%
_{br}}{\partial (\overline{\psi }\gamma _{\nu }\psi A^{\nu })}+(\partial
_{\mu }\overline{\psi }\gamma ^{\mu }\psi )\frac{\partial {\cal L}_{br}}{%
\partial (\overline{\psi }\gamma ^{\nu }\psi A_{\nu })}=0 \label{div1}$$This means that even if we started with the $\frac{\partial {\cal L}_{br}}{%
\partial A^{2}}$ equal to zero it would not stay zero because of the third term in the equation (\[div1\]) $$\overline{\psi }\gamma ^{\mu }\psi \partial _{\mu }\frac{\partial {\cal L}%
_{br}}{\partial (\overline{\psi }\gamma _{\nu }\psi A^{\nu })}\text{ \ }
\label{div2}$$while the last term in it vanishes because the fermion current $\overline{%
\psi }\gamma ^{\mu }\psi $ is conserved. However, provided we arrange for a fermion current flow in the direction of a non-zero gradient multiplier being in the term (\[div2\]), then the $\frac{\partial {\cal L}_{br}}{%
\partial A^{2}}$ will vary even if it were initially zero. In fact, since the term $(\partial _{\mu }A^{\mu })\frac{\partial {\cal L}_{br}}{\partial
A^{2}}$ in the equation (\[div1\]) is initially zero, there is no way to avoid the variation of the partial derivative $\frac{\partial {\cal L}_{br}}{%
\partial A^{2}}$ along the curves having the $A^{\mu }$ fields as tangents to be non-zero (see in the above).
Remember that we think of deviations of the $\frac{\partial {\cal L}_{br}}{%
\partial A^{2}}$ from zero as representing fossil charge. Thus, in the above case, fossil resting charge can indeed be produced by a suitable field configuration even if there were no fossil charges to begin with. So, with the extra dependence of the ${\cal L}_{br}$ on the $\overline{\psi }\gamma ^{\mu }\psi A_{\mu }$, the SLIV should in principle be observable even in a world that had started out without any so-called fossil charges.
Why partial gauge symmetry?
===========================
A crucial assumption in our theorem, that was designated as the partial gauge invariance, is that the kinetic term for the vector field $A_{\mu }$ should take the standard gauge symmetric $F_{\mu \nu }^{2}$ form being also (at least approximately) stable against radiative corrections stemming from the gauge breaking terms ${\cal L}_{br}$ (\[gf\]). Although we are not able to single out this $F_{\mu \nu }^{2}$ form on any symmetry ground there are two serious arguments in favor of such a choice.
The first argument is related to the physical requirement that the kinetic part of the Hamiltonian should be bounded from below. A priori, one should consider, instead of the $F_{\mu \nu }^{2}$, kinetic terms of the general form, $a(\partial _{\mu }A_{\nu })^{2}+b(\partial _{\mu }A^{\mu })^{2}$, where $a$ and $b$ are arbitrary constants. Note then that the first term in this form looks like as the kinetic terms of four scalar fields with an exception that the $A_{0}$ component has the wrong metric, if we choose $a$ negative so that the spatial components get the right sign for getting positive energy density. So, the $A_{0}$ containing term always gives a negative contribution to the Hamiltonian unless there is a way to get rid of the $\partial _{0}A_{0}$ term from the Lagrangian and the spatial derivative on $A_{0}$ terms $%
(\partial _{i}A_{0})^{2}$, which also comes with the wrong sign in our term with coefficient $a$. For the momentum $\Pi ^{\mu }(x)$ conjugate to $A_{\mu }(x)$, $$\Pi ^{\mu }=2(a\partial ^{0}A^{\mu }+bg^{0\mu }\partial _{\nu }A^{\nu })$$the Hamiltonian density becomes in a usual way just
$${\cal H}(x)=a[(\partial _{0}A_{\mu })^{2}+(\nabla A_{\mu })^{2}]+b[(\partial
_{0}A_{0})^{2}-(\nabla \cdot \vec{A})^{2}] \nonumber$$
which, for $b\neq -a$, can be written in the form $${\cal H}(x)=\frac{(\Pi _{0}+2b\nabla \cdot \vec{A})^{2}}{a+b}-a\vec{\Pi}%
^{2}+a(\nabla \cdot A_{0})^{2}-a(\nabla \vec{A})^{2}-b(\nabla \cdot \vec{A}%
)^{2}$$ So, the only Lorentz covariant way to get rid of the $(\partial
_{0}A_{0})^{2}$ is to have the equality $b=-a$ . Alternatively we can get this kinetic $A_{0}$ term with the right sign by choosing $b$ bigger than the positive $-a$ in the above form. But the gradient energy term from $A_{0}
$, i.e. $(\nabla A_{0})^{2}$ still would come with the negative coefficient $%
a$ and thus formally there would be no bottom to the Hamiltonian anyway. Now the equation of motion for $A_{0}$ may be written $$(a+b)\partial _{0}\partial ^{0}A_{0}-a\bigtriangleup A_{0}-b\partial
_{0}\nabla \cdot \vec{A}=0$$In the limit $a=-b$ this equation reduces to a constraint
$$-a\bigtriangleup A_{0}-b\partial _{0}\nabla \cdot \vec{A}=0$$
which is in fact the usual Gauss constraint $\nabla \cdot \overrightarrow{E}%
=0$. After partial integration in the action and throwing away boundary terms, this case $a=-b$ leads to the standard kinetic term. Now it is well-known from conventional QED that, if this Gauss constraint is inserted into the Hamiltonian, positivity of the kinetic energy can be guaranteed. However, there is no such constraint available for the general form of the kinetic term. Thus boundedness from below of the Hamiltonian requires a kinetic term of the standard gauge invariant $F_{\mu \nu }^{2}$ form.
There is another, more phenomenological argument for the absence of a gauge non-invariant addition to the vector field kinetic term. This is that the presence of the gauge non-invariant addition in the vector field kinetic term
$${\cal L}_{kin}=-\frac{1}{4}F_{\mu \nu }^{2}-\frac{\beta }{2}(\partial _{\mu
}A^{\mu })^{2} \label{kin}$$
(where $\beta $ is some positive constant) immediately leads to the domain wall solution for the SLIV, caused by a standard quartic polynomial in $%
A^{2} $ contained in ${\cal L}_{br}$ (\[gf\]) $$P(A^{2})=n^{2}\frac{M^{2}}{2}A_{\mu }^{2}-\frac{\lambda }{4}(A_{\mu
}^{2})^{2},\text{ }M^{2}>0,\text{ \ }n^{2}=n_{\mu }n^{\mu }=\pm 1\text{ \ }$$where the vector field mass squared $n^{2}M^{2}$ may be positive or negative for time-like or space-like Lorentz violation, respectively. Actually, one can find that the vector field $A_{\mu }$ develops a domain wall solutions $%
V(n\cdot x)$
$$A_{\mu }=a_{\mu }+n_{\mu }V(n\cdot x),\text{ \ }n\cdot x=n_{\mu }x^{\mu }
\label{vev}$$
$$V(n\cdot x)=\sqrt{\frac{M^{2}}{\lambda }}\tanh \left[ \frac{(n\cdot x)M}{%
\sqrt{2\beta }}\right]$$with walls being oriented orthogonal to the SLIV direction $n_{\mu }$. Indeed, it is clear that the standard kinetic $F_{\mu \nu }^{2}$ term in the ${\cal L}_{kin}$ (\[kin\]) is invariant under substitution (\[vev\]), while the second term in it leads to the differential equation for $V(n\cdot
x)$ $$\beta (\partial \cdot n)^{2}V+M^{2}V-\lambda V^{3}=0$$whose solution is just given by Eq.(\[vev\]). Note that these domain walls are topologically stable since for both time-like SLIV (kink in a time) and space-like SLIV (kink in a preferred space direction) cases the system possesses the disconnected vacua with expectation values $\pm \sqrt{\frac{%
M^{2}}{\lambda }}$, respectively. Actually, any rotation from one vacuum to another would necessarilly intersect the wall.
These walls could unavoidably lead to a wall-dominated Universe in the early times and, therefore, to its immediate collapse. Thus we have to exclude the presence of the corresponding gauge non-invariant addition to the $F_{\mu
\nu }^{2}$ kinetic term (taking constant $\beta =0$ in the ${\cal L}_{kin}$ (\[kin\])) in this case on phenomenological grounds.
Conclusion
==========
We have discusssed an influence of the spontaneous Lorentz violation on the origin of the gauge internal symmetry in the general Abelian vector field theory. We formulated the theorem which seems to govern this mechanism and found the conditions under which the SLIV becomes physically observable in terms of some primordially existing or created fossil charges with the property moving through the Universe with a fixed velocity. The partial gauge symmetry, namely gauge symmetry of the derivative containing terms in the Lagrangian was found to be crucial for our consideration, first of all the gauge symmetry for vector field kinetic term to avoid the botomlessness of the Hamiltonian and exclude the domain wall solution in the SLIV theory that would otherwise lead to the wall-dominated Universe in the early times and, therefore, to its immediate collapse. As a result with gauge invariant kinetic term, one comes in the minimal theory to the exactly QED case with the non-covariant gauge choice as the only SLIV effect.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Rabi Mohapatra for useful discussions. One of us (C.D.F.) wants to thank the Niels Bohr Fund for support to visit Niels Bohr Institute.
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V.A. Kostelecky and S. Samuel, Phys. Rev. D [**39** ]{}(1989) 683; V.A. Kosteletsky and R. Potting, Nucl. Phys. B[** 359** ]{}(1991) 545 (1991); D. Colladay and V. A. Kostelecky, Phys. Rev. D[**58**]{} (1998) 116002; R. Bluhm and V.A. Kostelecky, Phys. Rev. D [**71** ]{}(2005) 065008.
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J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen, Phys. Rev. Lett. [**87** ]{}(2001)[** **]{}091601; J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen, Nucl. Phys. B [**609** ]{}(2001) 46; Per Kraus and E.T. Tomboulis, Phys. Rev. D [**66** ]{}(2002) 045015; A. Jenkins, Phys. Rev. D [**69** ]{}(2004) 105007.
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[^1]: Note that previous direct calculations of Lorentz violating processes in the tree [@nambu] and one-loop approximation[@ac] for QED with the non-linear vector field constraint $A^{2}$ $=$ $V^{2}$ (where $V$ is the SLIV scale, see below), which appear to be completely cancelled, provide explicit examples of this theorem. On the other hand, many counter examples with physical Lorentz violation also exist when underlying assumptions for this theorem are not satisfied[@alan; @jackiw].
[^2]: Note that minimal (dimensionless coupling) vector-fermion field Lagrangian satisfying the additional global symmetry requirements including charge conjugation, parity reflection and fermion number conservation automtically appears as the conventional QED provided the partial gauge symmetry, namely gauge symmetry of the vector field kinetic term in the Lagrangian is presupposed.
[^3]: Possible operators of dimension 4 or less include a potential term $P(A^{2})$ of fourth order in $A_{\mu }$ and a seagull term $|\phi |^{2}A^{2}$ with an arbitrary coefficient.
|
---
abstract: 'Nonlinear regression models addressing both efficacy and toxicity outcomes are increasingly used in dose-finding trials, such as in pharmaceutical drug development. However, research on related experimental design problems for corresponding active controlled trials is still scarce. In this paper we derive optimal designs to estimate efficacy and toxicity in an active controlled clinical dose finding trial when the bivariate continuous outcomes are modeled either by polynomials up to degree 2, the Michaelis-Menten model, the Emax model, or a combination thereof. We determine upper bounds on the number of different doses levels required for the optimal design and provide conditions under which the boundary points of the design space are included in the optimal design. We also provide an analytical description of the minimally supported D-optimal designs and show that they do not depend on the correlation between the bivariate outcomes. We illustrate the proposed methods with numerical examples and demonstrate the advantages of the $D$-optimal design for a trial, which has recently been considered in the literature.'
author:
- |
[Holger Dette, Katrin Kettelhake, Kirsten Schorning ]{}\
[Ruhr-Universität Bochum ]{}\
[Fakultät für Mathematik ]{}\
[44780 Bochum, Germany ]{}\
[e-mail: holger.dette@ruhr-uni-bochum.de ]{}\
- |
[Weng Kee Wong ]{}\
[Department of Biostatistics ]{}\
[UCLA School of Public Health ]{}\
[Los Angeles, CA 90095-1772 ]{}\
[e-mail: wkwong@ucla.edu ]{}\
- |
[Frank Bretz ]{}\
[Statistical Methodology ]{}\
[Novartis Pharma AG ]{}\
[4002 Basel, Switzerland ]{}\
[e-mail: frank.bretz@novartis.com]{}
bibliography:
- 'drug.bib'
title: 'Optimal designs for active controlled dose finding trials with efficacy-toxicity outcomes '
---
Keywords and Phrases: Active controlled trials, dose finding, optimal design, admissible design, Emax model, Equivalence theorem, Particle swarm optimization, Tchebycheff system.
Introduction
============
There is a vast literature on optimal design of experiments, with applications ranging across many disciplines. [@atkinson1996] showed the usefulness of optimal designs for real applications using examples from agriculture, animal breeding studies, accelerated life-testing experiments and computer experiments. [@berger2005] gave examples of the varied disciplines that increasingly use optimal design ideas for scientific investigations. Most of the literature focuses on optimal designs for models with a univariate outcome. In practice, however, drug trials are often conducted to measure multiple outcomes that are likely to be correlated. For instance, pharmaceutical dose-finding trials invariably have bivariate outcomes involving efficacy and toxicity. As a motivating example consider a randomized controlled clinical trials for hypertensive patients treated with an angiotensin-converting-enzyme (ACE) inhibitor. In such trials change from baseline in the sitting blood pressure is a frequent efficacy outcome. However, patients starting on an ACE inhibitor usually have a modest reduction in glomerular filtration rate (GFR) that stabilizes after several days. Because this decrease may be significant in conditions of decreased renal perfusion, the renal function should be closely monitored over the first few days in those patients \[see [@sidnav2014] and [@talipi2015]\]. Thus, the amount of decrease in GFR from baseline is a common outcome to assess unwanted side effects.
Several papers have addressed design problems that incorporate both efficacy and toxicity of a drug. [@fancha2004] proposed using a continuous ratio model for a trinomial outcome, where the outcome of a patient may be classified as “no reaction” when neither toxicity nor efficacy occurs, “efficacy” for efficacy without toxicity, and “adverse reaction” for toxicity. [@heisemyers1996] used the Gumbel bivariate binary quantal response model to study efficacy and toxicity. In their example, patients were continuously monitored whether they experienced a toxicity event or a treatment benefit. The authors investigated locally D- and Q-optimal designs, where D-optimal designs were constructed for estimating all model parameters and Q-optimal designs were obtained by maximizing the probability of efficacy without toxicity at a selected dose level. More recently, [@ireland2013] applied c-optimal designs to the bivariate Emax model for continuous efficacy and toxicity outcomes. The author determined the dose level providing the best possible combination of efficacy and toxicity, based on a pre-specified clinical utility index \[see [@carrothers2011]\].
Adaptive dose finding trials incorporating both efficacy and safety have been investigated as well. For example, [@thall2004] and [@drafed2006] found adaptive designs for dose-finding based on efficacy-toxicity outcome using a Gumbel bivariate logistic regression or a Cox bivariate model. [@drafewu2008] proposed new designs for selecting drug combinations for a bivariate Probit correlated model based on an efficacy-toxicity outcome profile of a drug using Bayesian, minimax and adaptive methods. More recently, [@talili2013] considered a joint model with mixed correlated toxicity and efficacy outcomes, with one discrete and the other continuous. Using outcomes constructed with Archimedean Copula, they extended the continual reassessment method to find the optimal dose for Phase III trials based on both efficacy and toxicity considerations. Some advantages and disadvantages of adaptive designs have been discussed in [@detborbre2013].
Recently, the use of active controls instead of placebo in clinical trials has received considerable attention in the literature \[see for example [@pct], [@ethics], [@helbenfri2014; @helbenzinknefri2015] among many others\], and design issues for univariate outcomes have been discussed in [@detkisbenbre2014; @detketbre2015]. To our best knowledge, the problem of determining optimal designs for active controlled trials with bivariate mean outcomes has not been considered in the literature so far. In the present paper we derive optimal designs when the bivariate outcomes are efficacy and toxicity measures and are modeled using combinations of the linear, quadratic, Michaelis-Menten and Emax model as possible mean functions. We note that the Emax model is especially flexible in its shape and commonly applied in dose-finding trials. In particular, the Emax model can be justified through the relationship of drug-receptor interactions and therefore deduced from the chemical equilibrium equation \[see e.g. [@borou2002]\].
In Section \[sec2\] we introduce the model and provide some technical background. In Section \[sec3\] we consider various models for efficacy and toxicity outcomes without an active control and provide upper bounds on the required number of doses for each combination of the possible mean functions in the bivariate model. We also state sufficient conditions under which the boundary points of the design space are included as support points of the optimal design and determine the minimally supported optimal designs analytically. In Section \[sec4\] we apply these results to design active-controlled dose finding trials. In Section \[sec5\] we report the optimal designs for a real trial with bivariate outcome using particle swarm optimization. We conclude with a discussion in Section \[sec6\] and provide the technical proofs for our main results in Section \[sec7\].
Optimal designs for bivariate outcome {#sec2}
=====================================
We consider a dose-finding trial investigating both the efficacy and toxicity of a new drug under investigation. Our goal is to find an optimal design for collecting data for the two outcomes at different dose levels. Given a statistical model defined on a given dose interval of interest, say $\mathcal{D}= [L,R] \subset {\mathbb{R}}_0^+$, and a given design criterion, the design problem is to determine the optimal number of doses, $k$, the dose levels $d_1, \ldots, d_k$ from the dose interval ${\mathcal{D}}$, and the number of patients assigned to each dose. In practice, the total sample size, say $n_1$, is determined either by standard power considerations or by requirements on the precision of estimating the dose-response curves. That is, for a given value of $n_1$, the optimal design needs to specify the number of patients $n_{1i}$ at each dose level $d_i$ subject to $\sum_{i=1}^k n_{1i} = n_1$. Note that we use the index “$1$” here in the notation (i.e. $n_1,n_{1i},\ldots) $ since in later sections we consider active controlled clinical trials with a second sample denoted by the index “$2$”.
Let $Y_{ij} $ be the two-dimensional outcome variable at dose level $d_i$ from subject $j$ and assume that $$\label{eq1}
Y_{ij}=(Y^e_{ij},Y^t_{ij})^T \sim \mathcal{N}_2(\eta_1(d_i,\theta_1),\Sigma_1), \qquad j=1,\dots,n_{1i}, i=1, \ldots, k.$$ The regression function $$\eta_1(d,\theta_1)=(\eta_1^e(d,\theta_1^e),\eta_1^t(d,\theta_1^t))^T \in {\mathbb{R}}^2$$ describes the expected efficacy ($\eta_1^e$) and toxicity ($\eta_1^t$) at dose level $d \in {\mathcal{D}}$, where the $(s_1^e+1)$- and $(s_1^t+1)$-dimensional vectors $\theta_1^e$ and $\theta_1^t$ define the parameters of the model $\eta_1^e$ and $\eta_1^t$, respectively. The parameter $\theta_1=((\theta_1^e)^T,(\theta_1^t)^T)^T$ varies in a compact parameter space, say $\Theta_1 \subset {\mathbb{R}}^{s_1}$, where $s_1=s_1^e+s_1^t+2$. The unknown covariance matrix is $$\Sigma_1 = {\mathrm{Cov}}(Y) = \begin{pmatrix} \sigma_e^2 & \rho \sigma_e \sigma_t \\ \rho \sigma_e \sigma_t & \sigma_t^2 \end{pmatrix},$$ where $-1<\rho<1$ denotes the correlation between the two outcome variables and the variances of the random variables $Y^e_{ij}$ and $Y^t_{ij}$ are given by $\sigma_e^2$ and $\sigma_t^2$, respectively. We assume that $\eta_1$ is continuously differentiable with respect to the parameter $\theta_1$ and denote by $$\begin{aligned}
f_{e}(d) &=& \frac{\partial}{\partial \theta_1^e} \eta_1^e(d,\theta_1)=(f_0^e(d),\ldots,f_{s_1^e}^e(d))^T ~,~
f_{t}(d) = \frac{\partial}{\partial \theta_1^t} \eta_1^t(d,\theta_1)=(f_0^t(d),\ldots,f_{s_1^t}^t(d))^T\end{aligned}$$ the gradients of the two mean responses with respect to $\theta_1^e$ and $ \theta_1^t$, respectively. Following [@fatu2001], the Fisher information matrix is given by the $s_1 \times s_1$-matrix $$\begin{aligned}
{\mathcal{I}}_1(d,\theta_1) &=& (\tfrac{\partial}{\partial \theta_1} \eta_1(d,\theta_1) )^T \Sigma_1^{-1} ( \tfrac{\partial}{\partial \theta_1} \eta_1(d,\theta_1))
=\begin{pmatrix} f_{e}(d) & \mathbf{0}_{s_1^e+1 } \\ \mathbf{0}_{s_1^t+1 } & f_{t}(d) \end{pmatrix} \Sigma_1^{-1} \begin{pmatrix} f_{e}^T(d) & \mathbf{0}_{s_1^t+1}^T \\ \mathbf{0}_{ s_1^e+1}^T & f_{t}^T(d) \end{pmatrix} \\
&=& \frac{1}{\sigma_e^2\sigma_t^2(1-\rho^2)} F(d).
\end{aligned}$$ Here, $\mathbf{0}_d$ is the $d$-dimensional vector with all entries equal to $0$ and $$\begin{aligned}
\label{fmat}
F(d)&=& \frac{1}{\sigma_e^2\sigma_t^2(1-\rho^2)} \begin{pmatrix} \sigma_t^2 \mathcal{F}_1 & -\rho\sigma_e \sigma_t \mathcal{F}_2 \\
-\rho\sigma_e \sigma_t \mathcal{F}_2^T & \sigma_e^2 \mathcal{F}_3 \end{pmatrix}\end{aligned}$$ is defined through $$\begin{aligned}
\mathcal{F}_1 &=& f_e(d) f_e^T(d) \in {\mathbb{R}}^{s_1^e+1 \times s_1^e+1} , \quad \mathcal{F}_3 = f_t(d) f_t^T(d) \in {\mathbb{R}}^{s_1^t+1 \times s_1^t+1} , \\
\mathcal{F}_2 &=& f_e(d) f_t^T(d) \in {\mathbb{R}}^{s_1^e+1 \times s_1^t+1} .\end{aligned}$$ Note that we have suppressed the dependency of the matrices $F$, $\mathcal{F}_1$, $\mathcal{F}_2$ and $\mathcal{F}_3$ on the parameter $\theta_1$ in our notation.
Throughout this paper we consider approximate designs in the sense of [@kiefer1974], which are defined as probability measures with finite support on the design space ${\mathcal{D}}$. If an approximate design $\xi$ has $k$ support points, say $d_1,\dots,d_k$, with corresponding positive weights $\omega_1, \ldots, \omega_k$, such that $\sum_{i=1}^k \omega_i = 1$, and $n_1$ observations can be taken, a rounding procedure is applied to obtain integers $n_{1i}$, $i=1,\ldots,k$, from the possibly rational numbers $\omega_i n_1$ \[see [@pukrie1992]\]. The information matrix $M_1(\xi,\theta_1)$ of a design $\xi$ is defined by the $s_1 \times s_1$ matrix $$\label{Mone}
M_1(\xi,\theta_1) = \int_{\mathcal{D}}{\mathcal{I}}_1(d,\theta_1) d\xi(d) = \sum_{i=1}^k \frac{\omega_i}{\sigma_e^2\sigma_t^2(1-\rho^2)} F(d_i),$$ where the matrix $F(d)$ is defined in .
If observations are taken according to an approximate design $\xi$ it can be shown that, under standard regularity conditions, the maximum likelihood estimator $\hat \theta_1$ is asymptotically normally distributed, that is $$\sqrt{n_1}(\hat \theta_1 - \theta_1) \stackrel{ \mathcal{D}}{\longrightarrow} \mathcal{N}_{s_1} (\mathbf{0}, M_1^{-1}(\xi,\theta_1)),$$ as $n_1 \to \infty$, where the symbol $\stackrel{ \mathcal{D}}{\longrightarrow}$ means convergence in distribution. Consequently designs that make the information matrix $M_1(\xi,\theta_1)$ large in some sense are appropriate. There are several design criteria used in practice. An important example is Kiefer’s $\phi_p$-criterion \[see [@kiefer1974]\]. To be precise, let $ p \in [-\infty,1)$ and let $ K \in \mathbb{R}^{s_1 \times m}$ be a given matrix of full column rank. A design $\xi^*$ is called locally $\phi_p$-optimal for estimating the linear combination $K^T \theta_1$ if it maximizes the concave functional $$\phi_p(\xi) = \Bigl(\frac {1}{m} {\mathrm{tr}}(K^T M_1^- (\xi,\theta_1)K)^{-p} \Bigr)^{\frac {1}{p}}$$ among all designs $\xi$ satisfying Range$(K) \subset$ Range$(M_1(\xi,\theta_1))$, i.e. $K^T \theta_1$ is estimable by the design $\xi$. Here, ${\mathrm{tr}}(A)$ and $A^-$ denote the trace and a generalized inverse of the matrix $A$, respectively.
One key advantage of working with approximate designs is that convex optimization theory can be applied if the design criterion is a concave functional. As a consequence, a general equivalence theorem is available to verify whether a design is optimal among all designs. Any concave functional has its own equivalence theorem but collectively they all have a similar form. For each member of Kiefer’s $\phi_p$-criterion, a direct application of Theorem 7.14 in [@pukelsheim2006] yields the following result.
\[equithm\] Let K be a $s_1\times m$ matrix of full column rank. If $p \in (-\infty , 1)$, a design $\xi^*$ with ${\rm} Range(K) \subset {\rm}Range (M_1(\xi^*,\theta_1))$ is locally $\phi_p$-optimal for estimating the linear combination $K^T \theta_1$ if and only if there exists a generalized inverse $G$ of the information matrix $M_1(\xi^*,\theta_1)$, such that $$\label{aequ}
{\mathrm{tr}}\bigl( {\mathcal{I}}_1(d,\theta_1)GK( C_K(\xi^*))^{p+1} K^TG^T\bigr) ~-~ {\mathrm{tr}}( C_K(\xi^*))^{p}\leq 0$$ holds for all $d \in {\mathcal{D}}$, where $C_K(\xi^*) =(K^TM^-_1(\xi^*,\theta_1)K)^{-1}$. If $p = -\infty $, a design $\xi^*$ with Range$(K) \subset$ Range$(M_1(\xi^*,\theta_1))$ is locally $\phi_{-\infty}$-optimal for estimating the linear combination $K^T \theta_1$ if and only if there exists a generalized inverse $G$ of the information matrix $M_1(\xi^*,\theta_1)$ and a non-negative definite matrix $E \in \mathbb{R}^{m \times m}$ with ${\rm tr} (E) =1$, such that $$\label{aequ1}
{\mathrm{tr}}\big( {\mathcal{I}}_1 (d,\theta_1) GK C_K(\xi^*) E C_K(\xi^*) K^TG^T\big) - \lambda_{\min}(C_K(\xi^*)) \leq 0.$$ holds for all $d \in \mathcal{D}$. Moreover, if the design $\xi^*$ is $\phi_p$-optimal, there is equality in the above inequalities.
The function on the left hand side of (\[aequ\]) or (\[aequ1\]) is a function of the dose $d$ and is sometimes called the sensitivity function of the design $\xi^*$. In practice, one plots the sensitivity function over the entire dose range and checks whether it is bounded above by zero. If it does, the design $\xi^*$ is optimal; otherwise it is not. In addition, the sensitivity function, along with the equivalence theorem, can be used to provide a lower bound on the efficiency of any design. For example, if $p > - \infty$, one can show that the $\phi_p$-efficiency of a design $\xi$ can be bounded from below, that is $$\label{effbound}
\mbox{eff}_p(\xi) = \frac {\phi_p(\xi)}{\sup_\eta \phi_p(\eta)} \geq \frac { {\mathrm{tr}}( C_K(\xi))^{p}} {\max_{d \in \mathcal{D}} {\mathrm{tr}}( {\mathcal{I}}_1(d,\theta_1)GK( C_K(\xi))^{p+1} K^TG^T) }$$ \[see [@dett:1996]\]. Moreover, characterizations of the type or are also useful to find optimal designs analytically if the model is not too complicated. However, regression models with multivariate outcome are complex and in practice optimal designs have to be found numerically \[see [@chang1997], [@atasei2007] or [@sagnol2011] among others\].\
For such calculations, sharp bounds on the number of support points of the optimal designs are useful, because they can substantially reduce the complexity of the optimization problem.
In order to derive such upper bounds we follow [@karstud1966] and call a design $\xi_1$ admissible if there does not exist a design $\xi_2$, such that $M_1(\xi_1,\theta_1) \neq M_1(\xi_2,\theta_1)$ and $$ M_1(\xi_1,\theta_1) \leq M_1(\xi_2,\theta_1)$$ with respect to the Loewner ordering. In other words, the information matrix of an admissible design cannot be improved and numerical optimization can be restricted to the class of admissible designs. The characterization of the number of support points of admissible designs has found considerable attention in the recent literature \[see [@yangstuf2009; @yangstuf2012b], [@yang2010], [@detmel2011] or [@dettscho2013]\]. These authors obtained substantially smaller bounds on the number of support points of optimal designs than provided by the classical approach using Caratheodory’s theorem \[see [@pukelsheim2006] for example\].
In the following we demonstrate that the results in the above references can in fact be proved under weaker assumptions than usually made in the literature. For this purpose we will make use of the theory of Tchebycheff systems \[see [@karstud1966]\]. A set of $k+1$ continuous functions $u_0,\ldots,u_k \colon [L,R] \to {\mathbb{R}}$ is called a Tchebycheff system on the interval $[L,R]$ if the inequality $\det (u_i(d_j))^k_{i,j=0}>0$ holds for all $L \leq d_0 < d_1 < \ldots < d_k \leq R$. We define the index $I(\xi)$ of a design $\xi$ on the interval $[L,R]$ as the number of support points, where interior support points are counted by one and the support points at the boundary of the interval $[L,R]$ are counted by one half.
Note that we can rewrite the information matrix $M_1(\xi,\theta_1)$ in the form $$\label{Monepsi}
M_1(\xi,\theta_1) = \begin{pmatrix} \int_L^R \psi_{11}(d)d\xi(d) & \ldots & \int_L^R \psi_{1s_1}(d)d\xi(d) \\ \vdots & & \vdots \\ \int_L^R \psi_{s_11}(d)d\xi(d) & \ldots & \int_L^R \psi_{s_1s_1}(d)d\xi(d) \end{pmatrix},$$ where we ignore the dependence of the functions $\psi_{ij}$ on the parameter $\theta_1$. We now define $\psi_0(d) \equiv 1$ and choose a basis, say $\{ \psi_0,\ldots,\psi_k \}$, for the space ${\rm span} (\{\psi_{ij} | 1 \leq i,j \leq s_1 \} \cup \{ 1\} )$. We further assume that $\psi_k$ is one of the diagonal elements of the matrix $M_1(\xi,\theta_1)$, does not coincide with any of the other elements $\psi_{ij}$ and that $\{ \psi_0,\ldots,\psi_{k-1} \}$ is a basis of the space $$\mbox{span}\big( \{ \psi_{ij} \mid i,j \in \{1,\ldots,s_1\}; \ \ \psi_{ij} \neq \psi_k \} \cup \{1\} \big) .$$ Our next result, Theorem \[detmelmod\], is a more general version of Theorem $3.1$ in [@detmel2011] that is specific to our problem here. The proof is quite similar to the one given in this reference and is omitted for the sake of brevity. Theorem \[detmelmod\] yields better bounds on the number of support points of optimal designs obtained from the current literature; an example is given at the end of Section \[sec71\].
\[detmelmod\]
- If $\{\psi_0,\psi_1,\ldots,\psi_{k-1}\} $ [and]{} $ \{\psi_0, \psi_1,\ldots,\psi_{k}\}$ are Tchebycheff systems on the interval ${\mathcal{D}}$, then for any design $\xi$ there exists a design $\xi^+$ with at most $\tfrac{k+2}{2}$ support points, such that $M_1(\xi^+,\theta_1) \geq M_1(\xi,\theta_1)$. If the index of the design $\xi$ satisfies $I(\xi) < \frac{k}{2}$, then the design $\xi^+$ is uniquely determined in the class of all designs $\eta$ satisfying $$\label{bed}
\int_L^R \psi_i(d) d\eta(d) = \int_L^R \psi_i(d)d\xi(d), \quad i=0,\ldots,k-1$$ and coincides with the design $\xi$. Otherwise, in the case $I(\xi)\geq\tfrac{k}{2}$, the following two assertions are valid.
1. If $k$ is odd, then $\xi^+$ has at most $\tfrac{k+1}{2}$ support points and $\xi^+$ can be chosen such that its support contains the point $R$.
2. If $k$ is even, then $\xi^+$ has at most $\tfrac{k}{2}+1$ support points and $\xi^+$ can be chosen such that its support contains the points $L$ and $R$.
- If $\{\psi_0,\psi_1,\ldots,\psi_{k-1}\} $ [and]{} $ \{\psi_0, \psi_1,\ldots,-\psi_{k}\} $ are Tchebycheff systems, then for any design $\xi$ there exists a design $\xi^-$ with at most $\tfrac{k+2}{2}$ support points, such that $M_1(\xi^-,\theta_1) \geq M_1(\xi,\theta_1)$. If the index of the design $\xi$ satisfies $I(\xi) < \frac{k}{2}$ then the design $\xi^-$ is uniquely determined in the class of all designs $\eta$ satisfying and coincides with the design $\xi$. Otherwise, in the case $I(\xi)\geq\tfrac{k}{2}$, the following two assertions are valid.
1. If $k$ is odd, then $\xi^-$ has at most $\tfrac{k+1}{2}$ support points and $\xi^-$ can be chosen such that its support contains the point $L$.
2. If $k$ is even, then $\xi^-$ has at most $\tfrac{k}{2}+1$ support points.
We note that Theorem \[detmelmod\] provides information about the admissible designs. For example, consider the case $(A2)$ with $k=2m$ for some $m \in \mathbb{N}$. Any design $\xi$ with index $I(\xi) \geq m$ can be improved with respect to the Loewner ordering by a design with at most $m+1$ support points that includes the boundary points $L$ and $R$. It follows that admissible designs are designs with index $<m$ and designs with $m+1$ support points that include the boundary points $L$ and $R$ of the design space.
Optimal designs for placebo-controlled dose finding trials {#sec3}
==========================================================
In this section we study optimal designs for several nonlinear regression models which are commonly used in placebo-controlled dose-finding trials with joint efficacy-toxicity outcomes. In particular we use Theorem \[detmelmod\] to derive bounds on the number of support points of optimal designs and explicit expressions for minimally supported designs. The proofs of the results presented here can be found in the Appendix.
Bounds on the number of support points
--------------------------------------
In order to determine bounds for the number of support points of optimal designs we note that the mapping $M \to (K^TM^-K)^{-1}$ is increasing with respect to the Loewner ordering on the set of all $s_1 \times s_1$-matrices satisfying Range$(K) \subset$ Range$(M)$ \[see [@pukelsheim2006]\]. That is, if $$M_1 \geq M_2 \quad \Rightarrow \quad (K^TM_1^-K)^{-1} \geq (K^TM_2^-K)^{-1},$$ for all matrices $M_1, M_2$ satisfying the range inclusion. It therefore follows that the information matrix $(K^TM^-(\xi,\theta_1)K)^{-1}$ of a non-admissible design can be improved with respect to the Loewner ordering. Because the $\phi_p$-criteria are monotone, we have $\phi_p(\xi) \leq \phi_p(\xi^*)$ for any design $\xi$, where $\xi^*$ is either $\xi^+$ or $\xi^-$ as given in Theorem \[detmelmod\]. This conclusion is true, whenever the assumptions of Theorem \[detmelmod\] are satisfied. The following results show that this is in fact the case for many of the commonly used dose response models with a bivariate outcome and give upper bounds on the number of support points of such designs.
\[thmefftoxLIN\] Assume that the model for efficacy is given by $\eta_1^e(d,\theta_1)=\vartheta_0^e + \vartheta_1^e d$ and that $\xi$ is an arbitrary design on the dose range ${\cal D} = [L,R]$.
- If $\eta_1^t(d,\theta_1)=\vartheta_0^t + \vartheta_1^t d$, there exists a design $\xi^*$ with at most two support points, such that $M_1(\xi^*,\theta_1)\geq M_1(\xi,\theta_1)$. If the index of the design $\xi$ satisfies $I(\xi)\geq 1$, $\xi^*$ can be chosen such that the support of $\xi^*$ contains the points $L$ and $R$.
- If $\eta_1^t(d,\theta_1)=\vartheta_0^t + \vartheta_1^t d + \vartheta_2^t d^2$, there exists a design $\xi^*$ with at most three support points, such that $M_1(\xi^*,\theta_1)\geq M_1(\xi,\theta_1)$. If the index of the design $\xi$ satisfies $I(\xi)\geq 2$, $\xi^*$ can be chosen such that the support of $\xi^*$ contains the points $L$ and $R$.
- If $\eta_1^t(d,\theta_1)$ is given by a Michaelis-Menten model, that is $\eta_1^t(d,\theta_1)=\tfrac{\vartheta_1^t d}{\vartheta_2^t + d}$, there exists a design $\xi^*$ with at most four support points, such that $M_1(\xi^*,\theta_1)\geq M_1(\xi,\theta_1)$. If the index of the design $\xi$ satisfies $I(\xi)\geq 3$, $\xi^*$ can be chosen such that the support of $\xi^*$ contains the points $L$ and $R$.
- If $\eta_1^t(d,\theta_1)$ is given by an Emax-model, that is $\eta_1^t(d,\theta_1)=\vartheta_0^t +\tfrac{\vartheta_1^t d}{\vartheta_2^t + d}$, there exists a design $\xi^*$ with at most four support points, such that $M_1(\xi^*,\theta_1)\geq M_1(\xi,\theta_1)$. If the index of the design $\xi$ satisfies $I(\xi)\geq 3$, $\xi^*$ can be chosen such that the support of $\xi^*$ contains the points $L$ and $R$.
\[rem0\] [Note that the bounds provided by Theorem \[thmefftoxLIN\] are not necessarily sharp. For example, if $\eta^t_1$ is the Emax and $\eta^e_1$ is the linear model, then by the first part of Theorem \[thmefftoxLIN\](d) one does not decrease the information (with respect to the L[oe]{}wner ordering) by considering only designs with at most four support points. Any design with four support points or three support points in the interior of the dose range has index $\geq 3$ and can therefore be further improved by a design with at most four support points including the boundary points $L$ and $R$. As one requires at least three different dose levels to estimate the parameters in the Emax model, it follows that one can restrict the search of optimal designs to three point designs with at least one boundary point as support point (as the index should be smaller than or equal to $5/2$) or to four point designs containing both boundary points in its support.]{}
\[thmefftoxQUAD\] Assume that the model for efficacy is given by $\eta_1^e(d,\theta_1)=\vartheta_0^e + \vartheta_1^e d + \vartheta_2^e d^2$ and let $\xi$ denote an arbitrary design on the dose range ${\cal D} = [L,R]$.
- If $\eta_1^t(d,\theta_1)=\vartheta_0^t + \vartheta_1^t d+\vartheta_2^t d^2$, there exists a design $\xi^*$ with at most three support points, such that $M_1(\xi^*,\theta_1)\geq M_1(\xi,\theta_1)$. If the index of the design $\xi$ satisfies $I(\xi)\geq 2$, $\xi^*$ can be chosen such that the support of $\xi^*$ contains the points $L$ and $R$.
- If $\eta_1^t(d,\theta_1)=\tfrac{\vartheta_1^t d}{\vartheta_2^t + d}$, there exists a design $\xi^*$ with at most five support points, such that $M_1(\xi^*,\theta_1)\geq M_1(\xi,\theta_1)$. If the index of the design $\xi$ satisfies $I(\xi)\geq 4$, $\xi^*$ can be chosen such that the support of $\xi^*$ contains the points $L$ and $R$.
- If $\eta_1^t(d,\theta_1)=\vartheta_0^t + \tfrac{\vartheta_1^t d}{\vartheta_2^t + d}$, there exists a design $\xi^*$ with at most five support points, such that $M_1(\xi^*,\theta_1)\geq M_1(\xi,\theta_1)$. If the index of the design $\xi$ satisfies $I(\xi)\geq 4$, $\xi^*$ can be chosen such that the support of $\xi^*$ contains the points $L$ and $R$.
\[thmefftoxMM\] Assume that the model for efficacy is given by $\eta_1^e(d,\theta_1)=\tfrac{\vartheta_1^e d}{\vartheta_2^e + d}$ and let $\xi$ denote an arbitrary design on the dose range ${\cal D} = [L,R]$.
- If $\eta_1^t(d,\theta_1)=\tfrac{\vartheta_1^t d}{\vartheta_2^t + d}$ with $\vartheta_2^e \neq \vartheta_2^t$, there exists a design $\xi^*$ with at most five support points, such that $M_1(\xi^*,\theta_1)\geq M_1(\xi,\theta_1)$. If the index of the design $\xi$ satisfies $I(\xi)\geq 4$, $\xi^*$ can be chosen such that the support of $\xi^*$ contains the point $R$.
- If $\eta_1^t(d,\theta_1)=\vartheta_0^t+\tfrac{\vartheta_1^t d}{\vartheta_2^t + d}$ with $\vartheta_2^e \neq \vartheta_2^t$, there exists a design $\xi^*$ with at most five support points, such that $M_1(\xi^*,\theta_1)\geq M_1(\xi,\theta_1)$. If the index of the design $\xi$ satisfies $I(\xi)\geq 4$, $\xi^*$ can be chosen such that the support of $\xi^*$ contains the points $L$ and $R$.
\[thmefftoxEMAX\] Assume that the model for efficacy is given by $\eta_1^e(d,\theta_1)=\vartheta_0^e + \tfrac{\vartheta_1^e d}{\vartheta_2^e + d}$ and let $\xi$ denote an arbitrary design on the dose range ${\cal D} = [L,R]$. If $\eta_1^t(d,\theta_1)=\vartheta_0^t + \tfrac{\vartheta_1^e d}{\vartheta_2^e + d}$ with $\vartheta_2^e \neq \vartheta_2^t$, there exists a design $\xi^*$ with at most five support points, such that $M_1(\xi^*,\theta_1)\geq M_1(\xi,\theta_1)$. If the index of the design $\xi$ satisfies $I(\xi)\geq 4$, $\xi^*$ can be chosen such that the support of $\xi^*$ contains the points $L$ and $R$.
\[rem1\] [The remaining cases can be obtained by interchanging the roles of $\eta^e$ and $ \eta^t$ in Theorem \[thmefftoxLIN\] - \[thmefftoxEMAX\]. For example, consider the case, where $\eta_1^e(d,\theta_1)$ is the Emax and $\eta_1^t(d,\theta_1)$ the Michaelis-Menten model with $\vartheta_2^e \neq \vartheta_2^t$. Then it follows from Theorem \[thmefftoxMM\](b) that for any design $\xi$ there exists a design $\xi^*$ with at most five support points, such that $M_1(\xi^*,\theta_1)\geq M_1(\xi,\theta_1)$. Moreover, if the index of the design $\xi$ satisfies $I(\xi)\geq 4$, then $\xi^*$ can be chosen such that the support of $\xi^*$ contains $L$ and $R$. The other cases are obtained in the same way. ]{}
Minimally supported $D$-optimal designs {#sec32}
---------------------------------------
For a design $\xi$ let $\# \, \rm{supp}(\xi)$ be the number of its support points and let $$m^* = \min \{ \# \,{ \rm supp} (\eta) \mid \det (M_1(\eta,\theta_1)) > 0, \ \eta \ \mbox{design on} \ \mathcal{D}\}$$ be the minimal number of support points required for a design with a non-singular information matrix in model . A design $\xi$ is called minimally supported if $\det (M_1(\xi,\theta_1))>0$ and the number of support points is given by $m^*$. Minimally supported designs are useful if, for example, a drug under investigation may be only available at few dose levels.\
In general, the optimal designs have to be found numerically for complex models and even then many of the current algorithms may not work well. However, if one restricts the search to minimally supported designs, the optimization problem can be greatly simplified which may then allow us to determine locally $D$-optimal designs. In some cases these minimally supported optimal designs may not be optimal among all designs \[see Section \[sec5\] below for some examples\] so that an equivalence theorem must be used to confirm its optimality among all designs or its efficiency should be evaluated using the estimate . Before we present analytically derived minimally supported designs for model for different efficacy-toxicity regression models, we give a result about the general structure of these designs.
\[equallyweighted\] If the number of parameters in the mean function of the efficacy model is the same as the number of parameters in the mean function of the toxicity model, i.e. $s^e_1 = s^t_1$, the minimally supported locally D-optimal design for model is a uniform design. Moreover, its support points do not depend on the entries in the covariance matrix $\Sigma_1$.
The following result provides minimally supported $D$-optimal designs for several commonly used dose-response models. Its proof makes use of Theorem \[equallyweighted\], which reduces the optimization problem to the determination of the support points.
\[thmefftoxminiLIN\]
- Assume that the model for efficacy is given by $\eta_1^e(d,\theta_1)=\vartheta_0^e + \vartheta_1^e d$.
- If $\eta_1^t(d,\theta_1)=\vartheta_0^t + \vartheta_1^t d$, the minimally supported D-optimal design is a two-point design with equal masses at the points $L$ and $R$.
- If $\eta_1^t(d,\theta_1)=\tfrac{\vartheta_1^t d}{\vartheta_2^t + d}$, the minimally supported D-optimal design is a two-point design with equal masses at the points $L \lor \frac{1}{2}(\sqrt{R^2+10 R \vartheta_2^t+9 (\vartheta_2^t)^2}-R-3 \vartheta_2^t)$ and $R$.
- Assume that the model for efficacy is given by $\eta_1^e(d,\theta_1)=\vartheta_0^t + \vartheta_1^t d+\vartheta_2^t d^2$.
- ****If $\eta_1^t(d,\theta_1)=\vartheta_0^t + \vartheta_1^t d+\vartheta_2^t d^2$, the minimally supported D-optimal design is a three-point design with equal masses at the points $L$, $\tfrac{L+R}{2}$ and $R$.
- If $\eta_1^t(d,\theta_1)=\vartheta_0^t + \tfrac{\vartheta_1^t d}{\vartheta_2^t + d}$, the minimally supported D-optimal design is a three-point design with equal masses at the points $L$, $\sqrt{(L+\vartheta_2^t) (R+\vartheta_2^t)}-\vartheta_2^t$ and $R$.
- Assume that the model for efficacy is given by $\eta_1^e(d,\theta_1)=\tfrac{\vartheta_1^e d}{\vartheta_2^e + d}$.
- If $\eta_1^t(d,\theta_1)=\vartheta_0^t + \vartheta_1^t d$, the minimally supported D-optimal design is a two-point design with equal masses at the points $L \lor \frac{1}{2}(\sqrt{R^2+10 R \vartheta_2^e+9 (\vartheta_2^e)^2}-R-3 \vartheta_2^e)$ and $R$.
- If $\eta_1^t(d,\theta_1)=\tfrac{\vartheta_1^t d}{\vartheta_2^t + d}$, the minimally supported D-optimal design is a two-point design with equal masses at the optimal points $L \lor \frac{\sqrt{ R \vartheta_2^e \vartheta_2^t (R+\vartheta_2^e+\vartheta_2^t)+ (\vartheta_2^e\vartheta_2^t)^2}- \vartheta_2^e \vartheta_2^t}{(R+\vartheta_2^e+\vartheta_2^t)}$ and $R$.
- Assume that the model for efficacy is given by $\eta_1^e(d,\theta_1)=\vartheta_0^e + \tfrac{\vartheta_1^e d}{\vartheta_2^e + d}$.
- If $\eta_1^t(d,\theta_1)=\vartheta_0^t + \vartheta_1^t d+\vartheta_2^t d^2$, the minimally supported D-optimal design is a three-point design with equal masses at the points $L$, $\sqrt{(L+\vartheta_2^e) (R+\vartheta_2^e)}-\vartheta_2^e$ and $R$.
- If $\eta_1^t(d,\theta_1)=\vartheta_0^t + \tfrac{\vartheta_1^t d}{\vartheta_2^t + d}$, the minimally supported D-optimal design is a three-point design with equal masses at the points $L$, $\frac{\sqrt{(L+\vartheta_2^e) (L+\vartheta_2^t) (R+\vartheta_2^e) (R+\vartheta_2^t)}+L R-\vartheta_2^e \vartheta_2^t}{L+R+\vartheta_2^e+\vartheta_2^t}$ and $R$.
Active-controlled dose-finding trials {#sec4}
=====================================
The use of active controls instead of placebo in clinical trials has received considerable attention in the literature \[see [@pct] and [@ethics] among many others\]. In active controlled dose-finding trials patients are randomized to receive either one of several doses of the new drug or an active control (a marketed drug administered at a specific dose level). Inference issues for active-controlled dose-finding trials were investigated only more recently \[see, for example, [@helbenfri2014; @helbenzinknefri2015]\]. [@detkisbenbre2014; @detketbre2015] investigated optimal design problems for such trials by determining the optimal number of different dose levels, the individual dose levels within the dose range under investigation and the allocation ratios of patients at each dose level and the active control. Despite the increasing importance of such trials \[see [@compare]\], there is virtually no work on developing optimal designs for active-controlled dose-finding trials with efficacy-toxicity outcomes, especially given the fact that designs for placebo controlled trials do not extend directly to active-controlled trials \[see [@detkisbenbre2014]\]. Our goal in this section is to design an active-controlled dose-finding trial with a pre-determined total number of patients $N$ by determining the optimal number $k$ of different dose levels for the new drug, their individual dose levels $d_1,\ldots, d_k$, and the optimal number $n_1$ of patients to be assigned to the new drug, along with the allocation scheme across the recommended doses. The remaining number $n_2 = N-n_1$ of patients are assigned to the the active control, which is assumed to be available at a fixed dose level $C$. In terms of approximate designs, we have designs of the form $$\label{xitilde}
\tilde \xi = \begin{pmatrix} (d_1,0) & \ldots & (d_k,0) & (C,1) \\ \tilde\omega_1 & \ldots & \tilde\omega_k & \tilde\omega_{k+1} \end{pmatrix}~,$$ where $\tilde\omega_i$ denotes the proportion of patients assigned treated at the $i^{th}$ dose level of the new drug, $i=1,\ldots,k$ and $ \tilde\omega_{k+1}$ the proportion of patients treated with the active control, that is $n_2 \approx \tilde \omega_{k+1}N$. Here the second component of a design points in specifies if patients receive the new drug (“$0$”) or the active control (“$1$”). Note that the approximate design $\tilde \xi $ induces an approximate design of the form $$\label{xi}
\xi = \begin{pmatrix} d_1 & \ldots & d_k \\ \omega_1 & \ldots & \omega_k \end{pmatrix},$$ for the new drug defining $\omega_i = \tilde \omega_i / (1-\tilde\omega_{k+1})$. Extending the statistical model from [@detkisbenbre2014] to the efficacy-toxicity outcomes considered here, we have $$\begin{aligned}
\label{eq2}
Y_{ij} =(Y_{ij} ^e,Y_{ij} ^t)^T \sim \mathcal{N}_2(\eta_1(d_i,\theta_1),\Sigma_1) ~; j=1,\ldots , n_{1i} ,\\
Z_j =(Z_j^e,Z_j^t)^T \sim \mathcal{N}_2(\eta_2(\theta_2),\Sigma_2)~;j=1,\ldots ,n_2,
\label{eq2a}\end{aligned}$$ where $Y_{ij}$ denotes the outcome from the $j$th patient treated with the new drug at dose level $d_i$, and $Z_j$ the outcome from the $j$th patient treated with the active control. The two-dimensional vector $\eta_2 (\theta_2) $ is the expected outcome, and $\theta_2 $ a parameter which varies in a compact parameter space, say $\Theta_2$, and $\Sigma_2$ is a $2 \times 2$ covariance matrix. The function $\eta_2: \Theta_2 \to {\mathbb{R}}^2$ is assumed to be continuously differentiable. Assuming that all observations are independent, it can be shown that the information matrix of a design $\tilde \xi$ defined in has a block-structure of the form $$\label{M}
M(\tilde\xi, \theta) = \begin{pmatrix} (1-\tilde\omega_{k+1})M_1(\xi,\theta_1) & \bf{0} \\ \bf{0} & \tilde\omega_{k+1} {\mathcal{I}}_2(\theta_2)\end{pmatrix},$$ where $\theta=(\theta_1^T,\theta_2^T)^T$ and $${\mathcal{I}}_2(\theta_2) = (\tfrac{\partial}{\partial \theta_2} \eta_2(\theta_2) )^T \Sigma_2^{-1} ( \tfrac{\partial}{\partial \theta_2} \eta_2(\theta_2))$$ is the Fisher information matrix corresponding to the active control. Following [@detketbre2015] locally optimal designs for active-controlled dose-finding trials can be obtained from locally optimal designs for ordinary dose-finding trials. We extend this result to the class of admissible designs in Theorem \[thethmadm\], whose proof can be found in the Appendix.
\[thethmadm\] If $\xi$ is an admissible design of the form in model and $\tilde\omega_{k+1} \in (0,1)$, the design $\tilde \xi$ defined in is an admissible design for the model with an active control .
We now characterize admissible designs for various regression functions in the model with an active control. For this purpose we apply Theorem \[thethmadm\] to the results from Section \[sec3\]. We illustrate the methodology in an example with the Michaelis-Menten and Emax model. The other models discussed in Section \[sec3\] can be considered in a similar way.
\[exefftoxAC\] [Suppose that the mean outcome for toxicity is given by an Emax model. We consider two situations, where the efficacy outcome is first modeled by an Emax model and in the second case, is modeled by the Michaelis-Menten model. For the first case, it follows from Theorem \[thmefftoxEMAX\] (d) that admissible designs in trials without an active control have at most five support points. By Theorem \[thethmadm\], we conclude that admissible designs in active-controlled trials are of the form with at most six support points and a positive weight $\tilde \omega_6 \in (0,1)$ for the active control. Similarly, for the second case, it follows from Theorem \[thmefftoxMM\] (b) that there exists an admissible design for the corresponding active-controlled trial with at most six support points with a positive weight $\tilde \omega_6 \in (0,1)$ for the active control. Moreover, the dose levels for the new drug include the boundary points $L$ and $R$ of the dose range.]{}
In a similar way, $\phi_p$-optimal designs for active-controlled trials with efficacy-toxicity outcomes can be obtained. For this purpose we state the following result which can be proved in a similar way as Theorem $1$ in [@detketbre2015] using the block-structure of the matrix $M(\tilde \xi,\theta)$ in .
\[probD\] Let $\xi^*$ denote the locally $\phi_p$-optimal design of the form in the dose-response model with masses $ w^*_1 , \ldots , w_k^*,$ at the points $d^*_1, \ldots , d^*_k$, respectively. The design $\tilde \xi^* $ with masses $ \tilde w^*_1= \rho_p({1+\rho_p} )^{-1} w^*_1 , \ldots , \tilde w^*_k = \rho_p({1+\rho_p} )^{-1} w_k^*,$ and $ \tilde w^*_{k+1} = ({1+\rho_p} )^{-1}$ at the points $(d^*_1,0), \dots , (d^*_k,0)$ and $ (C,1)$, respectively, is locally $\phi_p$-optimal in the dose-response model with an active control , where $$\label{rho}
\rho_p =
\begin{cases}
\frac {({{\mathrm{tr}}}[\{ {\mathcal{I}}^{-1}_2(\theta_2) \}^{-p}])^{1/(p-1)}}
{({{\mathrm{tr}}}[\{M^{-1}_1(\tilde \xi^*, \theta_1)\}^{-p}])^{1/(p-1)}} & \mbox{if } p \in (-\infty ,1) \setminus \{0\} \\
\frac{s_1}{2} & \mbox{if } p =0 \\
\frac{\lambda_{{\min}} ({\mathcal{I}}_2(\theta_2) ) }{\lambda_{{\min}} (M_1(\tilde \xi^*, \theta_1)) }
& \mbox{if } p =- \infty \\
\end{cases} ~.$$
We note that Proposition \[probD\] can be extended to construct minimally supported designs. In particular, any minimally supported $\phi_p$-optimal design of the form for the dose response model yields a minimally supported $\phi_p$-optimal design for the dose response model with an active control by the transformation described in Proposition \[probD\]. We conclude this section by constructing minimally supported $D$-optimal designs for some of the models considered in Section \[sec32\].
\[exefftoxACD\]
Assume that the effect of the drug on efficacy and toxicity are both studied using Emax models. The minimally supported $D$-optimal design for model with an active control can be obtained from Theorem \[thmefftoxminiLIN\] part (4b) and Proposition \[probD\]. We set $s_1=6$ and Theorem \[thmefftoxminiLIN\] provides the support points of the minimally supported $D$-optimal design for the dose-response model . Proposition \[probD\] yields $\tilde \omega^*_4 =1/4$ for the proportion of patients treated with the active control. Additionally, the minimally supported $D$-optimal design for model with an active control allocates the rest of the patients equally to the new drug at $3$ dose levels given by
$$L, ~\frac{\sqrt{(L+\vartheta_2^e) (L+\vartheta_2^t) (R+\vartheta_2^e) (R+\vartheta_2^t)}+L R-\vartheta_2^e \vartheta_2^t}{L+R+\vartheta_2^e+\vartheta_2^t}
\mbox{ and } ~R.$$ In a similar manner explicit results for the other models considered in Section \[sec32\] can be obtained (and are omitted for space considerations).
Examples {#sec5}
========
We now apply our results from previous sections and construct optimal designs for active controlled trials for three examples. In the first one we determine the locally $D$-optimal design for a particular scenario of the motivating example in the introduction. The second example compares the $D$-optimal design with the $E$-optimal design, which is another type of optimal design sometimes used for making inference on the model parameters. The third example contrasts $D$-optimal designs with minimally supported $D$-optimal designs with recommendations on their use in practice from a statistical viewpoint.
If the optimal designs are not minimally supported they usually have to be determined numerically and several algorithms have been proposed in the literature for this purpose. The optimal designs presented in this section are found using particle swarm optimization (PSO), which is a prominent member of the class of nature-inspired metaheuristic algorithms. PSO has been widely used to solve hard and large dimensional optimization problems in engineering and computer science, and it has only been used recently to find optimal designs \[see [@kimli11], [@chen2014] or [@phoa]\]. For space consideration, we omit details on PSO and refer the interested reader to [@qiu] and [@wongmix] for details and illustrations.
\[ex51\]
[@talipi2015] used an Emax-model with parameters $\theta_1^e=(2.5,14.5,0.2)^T$ for the mean efficacy outcome and an exponential model $\eta_1^t(d,\theta_1^t)=0.163 + 0.037 e^{(3.3\log(6)d)}$ to model the toxicity effects \[see Table 1 in this reference\]. As described in Section 3.3.1 of [@talipi2015], they used a uniform design to allocate patients to the dose levels $0$, $0.05$, $0.2$, $0.4$, $0.6$, $0.8$, and $1$, respectively. For the error distribution in model they assumed a two dimensional centered normal distribution with parameters $\rho=0.4$, $\sigma_e=7$ and $\sigma_t=8$. We simulated data according to model with sample sizes $n_1=350$ for the new drug and fitted an Emax and the quadratic model for efficacy and toxicity, respectively. The quadratic model was used, because it yields a similar shape as the exponential model and minimally supported designs are explicitly available for the combinations of the Emax and a quadratic model. The fits of both regression models to the simulated data are shown in Figure \[figneu\]. The estimates for the parameters are given by $\hat \theta_1^e=(2.588,15.64,0.26)$ and $\hat \theta_1^t=(0.24,-11.632,25.11)$ for the Emax and quadratic model, while the estimates for the covariance are obtained as $\hat \rho=0.387$, $\hat \sigma_e=7.272$ and $\hat \sigma_t=8.311$. We used this information to determine a locally $D$-optimal design for the active controlled trial. Note that we do not require information from the model for the active control for this purpose as we are calculating $D$-optimal designs \[see Proposition \[probD\]\].
![*Fit of an Emax (efficacy) and a quadratic model to the data generated by a model discussed in [@talipi2015].*[]{data-label="figneu"}](Eff_Emaxfit.pdf "fig:"){width="6.5cm"} ![*Fit of an Emax (efficacy) and a quadratic model to the data generated by a model discussed in [@talipi2015].*[]{data-label="figneu"}](Tox_Quadfit.pdf "fig:"){width="6.5cm"}
By Theorem \[thmefftoxQUAD\](c) and Theorem \[thethmadm\], we only need to consider designs with at most six support points. We first used the PSO algorithm to generate the locally $D$-optimal design for model and in the second step, applied Proposition \[probD\] to determine the locally optimal design for the model with an active control. The results are shown in Table \[optdesignsmotivatingexample\]. The locally D-optimal design has five support points and is therefore not minimally supported. The minimally supported D-optimal design can be obtained from Theorem \[thmefftoxminiLIN\] (4a) and is shown in the right part of Table \[optdesignsmotivatingexample\]. The optimality of the design for the new drug was checked by Theorem \[equithm\]. Figure \[equivData\] displays the sensitivity function of the locally D-optimal and the minimally supported D-optimal design. The results confirm its optimality and its non-optimality, respectively. The $D$-efficiency of the minimally supported designs is given by $0.9886$. We note that the lower bound for the $D$-efficiency of the minimally supported optimal design does not need the knowledge of the locally $D$-optimal design and is given by $0.9532$.
The good performance of the minimally supported design is also confirmed by calculating the $D$-efficiency of the uniform design used in [@talipi2015] relative to our locally $D$-optimal design and the minimally supported $ D$-optimal design. These relative efficiencies are $0.575$ and $0.581$, respectively, showing that the performance of the design implemented by [@talipi2015] could be substantially improved by using locally $D$-optimal designs.
[|c|c|]{} $D$-optimal design & minimally supported D-optimal design\
$(0,0)$ $(0.18,0)$ $(0.49,0)$ $(1,0)$ $(C,1)$
--------- ------------ ------------ --------- ---------
$0.09$ $0.16$ $0.16$ $0.09$ $0.5$
: *Locally $D$-optimal design and minimally supported D-optimal design for a situation discussed in [@talipi2015]. The efficacy is modeled by an Emax model and the toxicity by a quadratic model.*[]{data-label="optdesignsmotivatingexample"}
&
$(0,0)$ $(0.31,0)$ $(1,0)$ $(C,1)$
--------- ------------ --------- ---------
$0.25$ $0.25$ $0.25$ $0.25$
: *Locally $D$-optimal design and minimally supported D-optimal design for a situation discussed in [@talipi2015]. The efficacy is modeled by an Emax model and the toxicity by a quadratic model.*[]{data-label="optdesignsmotivatingexample"}
\
[|c|c|c|]{} $\rho$ & $D$-optimal design & $E$-optimal design\
$0.1$ &
$(0,0)$ $(23.84,0)$ $(150,0)$ $(C,1)$
--------- ------------- ----------- ---------
$0.16$ $0.31$ $0.31$ $0.22$
: *Locally $D$- and $E$-optimal designs for an active-controlled trial, where the efficacy is modeled by an Emax model and the toxicity by a Michaelis-Menten model. The parameters in the two models are $\theta_1^e = (0, 0.466, 25)^T, \theta_1^t = (300,50)^T, \sigma_e=0.2, \sigma_t=20 ,\sigma_e^{AC}=0.2, \sigma_t^{AC}=29.8$ and $\rho \in \{0.1, 0.5, 0.8\}$.*[]{data-label="optdesignsexample"}
&
$(0,0)$ $(19.08,0)$ $(150,0)$ $(C,1)$
--------- ------------- ----------- ---------
$0.22$ $0.47$ $0.25$ $0.06$
: *Locally $D$- and $E$-optimal designs for an active-controlled trial, where the efficacy is modeled by an Emax model and the toxicity by a Michaelis-Menten model. The parameters in the two models are $\theta_1^e = (0, 0.466, 25)^T, \theta_1^t = (300,50)^T, \sigma_e=0.2, \sigma_t=20 ,\sigma_e^{AC}=0.2, \sigma_t^{AC}=29.8$ and $\rho \in \{0.1, 0.5, 0.8\}$.*[]{data-label="optdesignsexample"}
\
$0.5$ &
$(0,0)$ $(23.84,0)$ $(150,0)$ $(C,1)$
--------- ------------- ----------- ---------
$0.16$ $0.31$ $0.31$ $0.22$
: *Locally $D$- and $E$-optimal designs for an active-controlled trial, where the efficacy is modeled by an Emax model and the toxicity by a Michaelis-Menten model. The parameters in the two models are $\theta_1^e = (0, 0.466, 25)^T, \theta_1^t = (300,50)^T, \sigma_e=0.2, \sigma_t=20 ,\sigma_e^{AC}=0.2, \sigma_t^{AC}=29.8$ and $\rho \in \{0.1, 0.5, 0.8\}$.*[]{data-label="optdesignsexample"}
&
$(0,0)$ $(19.37,0)$ $(150,0)$ $(C,1)$
--------- ------------- ----------- ---------
$0.15$ $0.49$ $0.31$ $0.05$
: *Locally $D$- and $E$-optimal designs for an active-controlled trial, where the efficacy is modeled by an Emax model and the toxicity by a Michaelis-Menten model. The parameters in the two models are $\theta_1^e = (0, 0.466, 25)^T, \theta_1^t = (300,50)^T, \sigma_e=0.2, \sigma_t=20 ,\sigma_e^{AC}=0.2, \sigma_t^{AC}=29.8$ and $\rho \in \{0.1, 0.5, 0.8\}$.*[]{data-label="optdesignsexample"}
\
$0.8$ &
$(0,0)$ $(23.84,0)$ $(150,0)$ $(C,1)$
--------- ------------- ----------- ---------
$0.16$ $0.31$ $0.31$ $0.22$
: *Locally $D$- and $E$-optimal designs for an active-controlled trial, where the efficacy is modeled by an Emax model and the toxicity by a Michaelis-Menten model. The parameters in the two models are $\theta_1^e = (0, 0.466, 25)^T, \theta_1^t = (300,50)^T, \sigma_e=0.2, \sigma_t=20 ,\sigma_e^{AC}=0.2, \sigma_t^{AC}=29.8$ and $\rho \in \{0.1, 0.5, 0.8\}$.*[]{data-label="optdesignsexample"}
&
$(0,0)$ $(18.65,0)$ $(150,0)$ $(C,1)$
--------- ------------- ----------- ---------
$0.11$ $0.51$ $0.33$ $0.05$
: *Locally $D$- and $E$-optimal designs for an active-controlled trial, where the efficacy is modeled by an Emax model and the toxicity by a Michaelis-Menten model. The parameters in the two models are $\theta_1^e = (0, 0.466, 25)^T, \theta_1^t = (300,50)^T, \sigma_e=0.2, \sigma_t=20 ,\sigma_e^{AC}=0.2, \sigma_t^{AC}=29.8$ and $\rho \in \{0.1, 0.5, 0.8\}$.*[]{data-label="optdesignsexample"}
\
\[ex52\] [Consider a situation where the efficacy outcome is described by an Emax model and a Michaelis-Menten model is used for the toxicity outcome. The nominal parameter values are $\theta_1=(0, 0.466, 25, 300, 50)^T$, and the dose interval is $\mathcal{D}=[0,150]$. We chose $\sigma_e=0.2$, $\sigma_t=20$ and various values for the correlation in the covariance matrix are considered. By Theorem \[thmefftoxMM\](b) and Theorem \[thethmadm\], we only need to consider designs with at most six support points. We applied the PSO algorithm to generate the locally $D$- and $E$-optimal designs for model and Proposition \[probD\] to determine the locally optimal designs for the dose finding trial with an active control. The results are shown in Table \[optdesignsexample\] for various values of the correlation $\rho$. By definition, an $E$-optimal design minimizes the maximum eigenvalue of the inverse of the information matrix, whereas a $D$-optimal design minimizes the volume of the confidence ellipsoid for the parameter. The locally $D$- and $E$-optimal designs for the active controlled trial have four support points and are therefore minimally supported. Consequently, the support points of the $D$-optimal designs do not depend on the elements of the covariance matrix $\Sigma_1$, as predicted by Theorem \[equallyweighted\]. On the other hand, the interior support points of the $E$-optimal design are slightly changing with the correlation $\rho$. The optimality of both designs was checked by Theorem \[equithm\] and Figure \[equivD\] displays the sensitivity functions of the designs that confirm their optimality for $\rho=0.1$. ]{}
\[ex53\] [Assume that the efficacy outcome is described by a quadratic model and the toxicity outcome by an Emax-model, where the nominal values of the model parameters are given by $\theta_1=(0.5, 0.01, 0.1,0.1, 2.4, 1.2)^T$. The dose interval is $\mathcal{D}=[0,7]$ and we chose $\sigma_e=0.1$ and $\sigma_t=0.4$. It follows from Theorem \[thmefftoxQUAD\](d) and Theorem \[exefftoxAC\] that only designs with at most six support points have to be considered. The locally $D$-optimal designs are determined in the same way as described in Example \[ex51\] and \[ex52\] and the results are listed in the left part of Table \[optdesignsexamplezwei\] for different values of the correlation. Note that the $D$-optimal designs are not minimally supported and the support points and weights depend on the correlation. The minimally supported $D$-optimal designs can be found by an application of Theorem \[thmefftoxminiLIN\] and do not depend on $\rho$ \[see the right part of Table \[optdesignsexamplezwei\]\]. The optimality of the numerically calculated $D$-optimal designs was checked by Theorem \[equithm\] and the corresponding sensitivity functions are displayed in Figure \[equiv\] for different values of the correlation, that is $\rho=0.1, 0.5$ and $0.9$. We observe that all designs calculated by the metaheuristic PSO-algorithm are in fact $D$-optimal. Moreover, the efficiencies of the minimally supported designs are given by $0.96, 0.81$ and $0.34$ for the case $\rho = 0.1, 0.5,$ and $0.9$, respectively. From the efficiencies we see that the minimally supported designs are only efficient if the efficacy and toxicity outcomes are nearly uncorrelated. For a strong correlation between efficacy and toxicity minimally supported designs cannot be recommended. Finally, we note that the values of the lower bounds in for these $3$ minimally supported optimal designs are $0.87, 0.67$ and $0.18$.]{}
[|c|c|c|]{} $\rho$ & optimal & minimally supported $D$-optimal\
$0.1$ &
$(0,0)$ $(0.86,0)$ $(3.58,0)$ $(7,0)$ $(C,1)$
--------- ------------ ------------ --------- ---------
$0.225$ $0.15$ $0.15$ $0.225$ $0.25$
: *Locally $D$-optimal design (left) and the minimally supported $D$-optimal designs (right). The efficacy and toxicity are modeled by a quadratic model with parameter $\theta_1^e=(0.5, 0.01, 0.1)^T$ and Emax-model with parameter $\theta_1^t = (0.1, 2.4, 1.2)^T$, respectively. The elements in the covariance matrix are $\sigma_e=0.1, \sigma_t= 0.4$ and various correlation values.*[]{data-label="optdesignsexamplezwei"}
&
$(0,0)$ $(1.94,0)$ $(7,0)$ $(C,1)$
--------- ------------ --------- ---------
$0.25$ $0.25$ $0.25$ $0.25$
: *Locally $D$-optimal design (left) and the minimally supported $D$-optimal designs (right). The efficacy and toxicity are modeled by a quadratic model with parameter $\theta_1^e=(0.5, 0.01, 0.1)^T$ and Emax-model with parameter $\theta_1^t = (0.1, 2.4, 1.2)^T$, respectively. The elements in the covariance matrix are $\sigma_e=0.1, \sigma_t= 0.4$ and various correlation values.*[]{data-label="optdesignsexamplezwei"}
\
$0.5$ &
$(0,0)$ $(0.8,0)$ $(3.73,0)$ $(7,0)$ $(C,1)$
---------- ----------- ------------ ---------- ---------
$0.2175$ $0.1575$ $0.1575$ $0.2175$ $0.25$
: *Locally $D$-optimal design (left) and the minimally supported $D$-optimal designs (right). The efficacy and toxicity are modeled by a quadratic model with parameter $\theta_1^e=(0.5, 0.01, 0.1)^T$ and Emax-model with parameter $\theta_1^t = (0.1, 2.4, 1.2)^T$, respectively. The elements in the covariance matrix are $\sigma_e=0.1, \sigma_t= 0.4$ and various correlation values.*[]{data-label="optdesignsexamplezwei"}
&
$(0,0)$ $(1.94,0)$ $(7,0)$ $(C,1)$
--------- ------------ --------- ---------
$0.25$ $0.25$ $0.25$ $0.25$
: *Locally $D$-optimal design (left) and the minimally supported $D$-optimal designs (right). The efficacy and toxicity are modeled by a quadratic model with parameter $\theta_1^e=(0.5, 0.01, 0.1)^T$ and Emax-model with parameter $\theta_1^t = (0.1, 2.4, 1.2)^T$, respectively. The elements in the covariance matrix are $\sigma_e=0.1, \sigma_t= 0.4$ and various correlation values.*[]{data-label="optdesignsexamplezwei"}
\
$0.9$ &
$(0,0)$ $(0.7,0)$ $(3.99,0)$ $(7,0)$ $(C,1)$
--------- ----------- ------------ --------- ---------
$0.21$ $0.165$ $0.165$ $0.21$ $0.25$
: *Locally $D$-optimal design (left) and the minimally supported $D$-optimal designs (right). The efficacy and toxicity are modeled by a quadratic model with parameter $\theta_1^e=(0.5, 0.01, 0.1)^T$ and Emax-model with parameter $\theta_1^t = (0.1, 2.4, 1.2)^T$, respectively. The elements in the covariance matrix are $\sigma_e=0.1, \sigma_t= 0.4$ and various correlation values.*[]{data-label="optdesignsexamplezwei"}
&
$(0,0)$ $(1.94,0)$ $(7,0)$ $(C,1)$
--------- ------------ --------- ---------
$0.25$ $0.25$ $0.25$ $0.25$
: *Locally $D$-optimal design (left) and the minimally supported $D$-optimal designs (right). The efficacy and toxicity are modeled by a quadratic model with parameter $\theta_1^e=(0.5, 0.01, 0.1)^T$ and Emax-model with parameter $\theta_1^t = (0.1, 2.4, 1.2)^T$, respectively. The elements in the covariance matrix are $\sigma_e=0.1, \sigma_t= 0.4$ and various correlation values.*[]{data-label="optdesignsexamplezwei"}
\
Conclusions and further research {#sec6}
================================
In this paper we investigated the optimal design problem for active controlled trials with bivariate outcomes. Upper bounds on the number of support points of locally optimal have been derived, which are used to reduce the dimensionality of the corresponding optimization problems. We also determined minimally supported $D$-optimal designs explicitly for specific combinations of models for the efficacy and toxicity and note that in general the optimal designs for active controlled clinical trials with bivariate outcomes are not minimally supported. Nevertheless, it is demonstrated that for the models under consideration the minimally supported $D$-optimal designs are rather efficient, provided that the correlation between efficacy and toxicity is weak. Our results demonstrate that statistical inference in clinical trials with bivariate outcomes can be improved substantially by the appropriate use of efficient designs.
This paper discusses locally optimal designs, which require a-priori information about the unknown model parameters if they appear in the model in a nonlinear way \[see [@chernoff1953]\]. When preliminary knowledge regarding the unknown parameters of a nonlinear model is available, and the application of locally optimal designs is well justified \[see for example [@debrpepi2008]\]. Locally optimal designs are typically used as benchmarks for commonly used designs \[see the discussion in Example \[ex51\]\]. Additionally, locally optimal designs serve as basis for constructing optimal designs with respect to more sophisticated optimality criteria, which are robust against a misspecification of the unknown parameters; see [@pronwalt1985] or [@chaver1995], [@dette1997] among others. An interesting direction for future research is to further develop the methodology introduced in the present paper to address uncertainty in the preliminary information for the unknown parameters.
[**Acknowledgements**]{} The authors would like to thank Martina Stein, who typed parts of this manuscript with considerable technical expertise. This work has been supported in part by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823) of the German Research Foundation (DFG). Kettelhake, Schorning and Wong were partially supported by a grant from the National Institute Of General Medical Sciences of the National Institutes of Health under Award Number R01GM107639. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
Appendix {#sec7}
========
Proof of Theorems \[thmefftoxLIN\], \[thmefftoxQUAD\], \[thmefftoxMM\] and \[thmefftoxEMAX\] {#sec71}
--------------------------------------------------------------------------------------------
We present the proof of Theorem \[thmefftoxEMAX\] only for the case, where the effect of the drug on efficacy and toxicity is modeled by an Emax model. In this case the gradient of the outcome with respect to the parameter is given by $$\begin{aligned}
\frac{\partial}{\partial \theta_1}\eta_1(d,\theta_1) &=& \begin{pmatrix} 1
& \frac{d}{\vartheta_2^e+d} & -\frac{\vartheta_1^e d}{(\vartheta_2^e+d)^2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{d}{\vartheta_2^t+d} & -\frac{\vartheta_1^t d}{(\vartheta_2^t+d)^2}
\end{pmatrix}.
\end{aligned}$$ It is easy to see that there exists a full column rank matrix $L \in {\mathbb{R}}^{6 \times 10} $ which does not depend on the variable $d$ such that $$\frac{\partial}{\partial \theta_1}\eta_1(d,\theta_1) = \begin{pmatrix}
\nu^T(d) &0 \\ 0 & \nu^T(d) \end{pmatrix} L^T~,$$ where the vector $\nu (d)$ is defined by the linearly independent functions in the gradient, i.e. $$\nu(d)=(1,\tfrac{1}{\vartheta_2^e+d},\tfrac{1}{(\vartheta_2^e+d)^2},\tfrac{1}{\vartheta_2^t+d},\tfrac{1}{(\vartheta_2^t+d)^2})^T \in {\mathbb{R}}^5.$$ Consequently, we obtain for the information matrix in the representation $$\label{MLSigma}
M_1(\xi, \theta_1) = L \int_{{\mathcal{D}}} \begin{pmatrix} \nu(d)\nu^T(d) & \mathbf{0} \\ \mathbf{0} & \nu(d)\nu^T(d) \end{pmatrix} d \xi(d) L^T,$$ where the matrix $\mathbf{0} $ denotes a $ 5 \times 5$ square matrix with all entries $0$ and $$\label{matmat}
\nu(d) \nu^T(d) = \begin{pmatrix}
1 & \tfrac{1}{\vartheta_2^e+d} & \tfrac{1}{(\vartheta_2^e+d)^2} & \tfrac{1}{\vartheta_2^t+d} & \tfrac{1}{(\vartheta_2^t+d)^2} \\
\tfrac{1}{\vartheta_2^e+d} & \tfrac{1}{(\vartheta_2^e+d)^2} & \tfrac{1}{(\vartheta_2^e+d)^3} & \tfrac{1}{(\vartheta_2^e+d)(\vartheta_2^t+d)} & \tfrac{1}{(\vartheta_2^e+d)(\vartheta_2^t+d)^2} \\
\tfrac{1}{(\vartheta_2^e+d)^2} & \tfrac{1}{(\vartheta_2^e+d)^3} & \tfrac{1}{(\vartheta_2^e+d)^4} & \tfrac{1}{(\vartheta_2^e+d)^2(\vartheta_2^t+d)} & \tfrac{1}{(\vartheta_2^e+d)^2(\vartheta_2^t+d)^2} \\
\tfrac{1}{\vartheta_2^t+d} & \tfrac{1}{(\vartheta_2^e+d)(\vartheta_2^t+d)} &\tfrac{1}{(\vartheta_2^e+d)^2(\vartheta_2^t+d)} & \tfrac{1}{(\vartheta_2^t+d)^2} & \tfrac{1}{(\vartheta_2^t+d)^3} \\
\tfrac{1}{(\vartheta_2^t+d)^2} & \tfrac{1}{(\vartheta_2^e+d)(\vartheta_2^t+d)^2} & \tfrac{1}{(\vartheta_2^e+d)^2(\vartheta_2^t+d)^2} & \tfrac{1}{(\vartheta_2^t+d)^3} & \tfrac{1}{(\vartheta_2^t+d)^4} \end{pmatrix}.$$ Now Theorem 14.2.9 in [@harv1997] shows that an improvement with respect to the Loewner ordering can be obtained by improving the common block $$\int_{{\mathcal{D}}} \nu(d)\nu^T(d)d \xi(d)$$ in the matrix . For this purpose we now use Theorem \[detmelmod\]. The functions $\psi_0(d)=1$ and $$\begin{aligned}
\psi_1(d) &=&\tfrac{1}{\vartheta_2^e+d}, ~\psi_2(d) = \tfrac{1}{(\vartheta_2^e+d)^2},~ \psi_3(d) = \tfrac{1}{(\vartheta_2^e+d)^3},~ \psi_4(d) = \tfrac{1}{(\vartheta_2^e+d)^4} , \\
\psi_5(d) &=& \tfrac{1}{\vartheta_2^t+d}, ~\psi_6(d) = \tfrac{1}{(\vartheta_2^t+d)^2}, ~\psi_7(d) = \tfrac{1}{(\vartheta_2^t+d)^3}, \psi_8(d) = \tfrac{1}{(\vartheta_2^t+d)^4}.\end{aligned}$$ fulfill the conditions specified in the paragraph before Theorem \[detmelmod\]. It follows by an application of Theorem 1.1 in Chapter IX of [@karstud1966] that the sets $\{\psi_0 , \ldots , \psi_7\}$ and $\{\psi_0 , \ldots , \psi_8\}$ are Tchebycheff systems and Theorem \[detmelmod\] is applicable with $k=8$. Part (A2) of this result yields that there exists a design $\xi^*$ with at most five support points including $L$ and $R$ such that $$\int_{{\mathcal{D}}} \nu(d)\nu^T(d)d \xi(d) \leq \int_{{\mathcal{D}}} \nu(d)\nu^T(d)d \xi^*(d),$$ and the assertion follows. We note that an application of Theorem $3.1$ in [@detmel2011] is not possible because the different functions from matrix do not form a Tchebycheff system.
$\Box$
Proof of Theorem \[equallyweighted\]
------------------------------------
Let $\xi$ be a minimally supported design of the form . As $s^e_1 = s^t_1$ we have $k=s^t_1+1$. Considering the Cholesky decomposition $\Sigma^{-1}_1 = \tilde \Sigma \tilde \Sigma^T$ of the inverse of the covariance matrix $\Sigma_1$ we obtain for the information matrix $M_1(\xi,\theta_1)$ the representation $$\begin{aligned}
M_1(\xi,\theta_1) \nonumber
&=& \sum_{i=1}^k \omega_i (\tfrac{\partial}{\partial \theta_1} \eta_1(d_i,\theta_1) )^T \tilde\Sigma \tilde \Sigma^T ( \tfrac{\partial}{\partial \theta_1} \eta_1(d_i,\theta_1)) \\
&=& G^T \mbox{Diag} (\omega_1, \omega_1,\ldots \omega_k, \omega_k) G, \label{quadmat}\end{aligned}$$ where the matrix $G$ is defined by $$\begin{aligned}
\label{quadmat1}
G= \begin{pmatrix} \tilde\Sigma^T (\tfrac{\partial}{\partial \theta_1} \eta_1(d_1,\theta_1) ) \\ \vdots \\ \tilde\Sigma^T (\tfrac{\partial}{\partial \theta_1} \eta_1(d_k,\theta_1) ) \end{pmatrix} ~=~ (I_k \otimes \tilde\Sigma^T ) \begin{pmatrix} (\tfrac{\partial}{\partial \theta_1} \eta_1(d_1,\theta_1) ) \\ \vdots \\
(\tfrac{\partial}{\partial \theta_1} \eta_1(d_k,\theta_1) ) \end{pmatrix}~\in ~{\mathbb{R}}^{2k\times 2k} .\end{aligned}$$ and $A \otimes B$ denotes the Kroecker product of the matrices $A$ and $ B$. Now $$\det (M_1(\xi,\theta_1) ) = (\det G)^2 \prod^k_{i=1} w^2_i$$ and consequently, the minimally supported $D$-optimal design must have equal weights. Moreover, the representation $$\det (G) = \big( \det ( \tilde\Sigma ) \big) ^k \det \Big( \big (\tfrac{\partial}{\partial \theta_1} \eta_1(d_j,\theta_1) \big)_{j=1,\ldots ,k}
\big)$$ shows that the support points of the minimally supported $D$-optimal design do not depend on the elements of the matrix $\Sigma_1$. This completes the proof of Theorem \[equallyweighted\]. $\Box$
Proof of Theorem \[thmefftoxminiLIN\]
-------------------------------------
We show only the proof of part 1(b) as the proofs for other cases are similar. If a linear and a Michaelis-Menten model are used to describe the effect of the drug on efficacy and toxicity, at least two support points, say $d_1, d_2$, are necessary to guarantee invertibility of the information matrix. From Theorem \[equallyweighted\] it follows $\omega_1^*=\omega_2^*=\tfrac{1}{2}$. Consider now the determinant of the information matrix of a design $\xi$ with equal weights at the points $d_1$ and $d_2$, then it follows by a straightforward calculation that $$\label{det}
\det(M_1 (\xi,\theta_1)) = \frac{{\vartheta_1^t}^2 d_1^2 d_2^2 (d_1-d_2)^4}{16 \left(\rho^2-1\right)^2 \sigma_e^4 \sigma_t^4 (\vartheta_2^t+d_1)^4 (\vartheta_2^t+d_2)^4}.$$ If we assume w.l.o.g. that $d_1 < d_2$, then the right hand side of is a monotone function of $d_2$. Consequently the right boundary point $R$ is one of the optimal support points, that is $d_2=R$. Maximizing the remaining expression with respect to the point $d_1$ in the interval $[L,R] $ gives $$d_1=
L \lor \frac{1}{2}(\sqrt{R^2+10 R \vartheta_2^t+9 (\vartheta_2^t)^2}-R-3 \vartheta_2^t),$$ which proves the result. $\Box$
Proof of Theorem \[thethmadm\]
------------------------------
Assume that $\tilde \xi$ is not admissible, that is there exists a design $$\tilde \eta = \begin{pmatrix} (\overline{d}_1,0) & \ldots & (\overline{d}_l,0) & (C,1) \\ \overline{\omega}_1 & \ldots & \overline{\omega}_l & \overline{\omega}_{l+1} \end{pmatrix}$$ such that $M(\tilde \eta,\theta_1) \neq M(\tilde\xi,\theta_1)$ and $M(\tilde \eta,\theta_1) \geq M(\tilde\xi,\theta_1)$. This yields immediately $\overline{\omega}_{l+1} \geq \tilde\omega_{k+1}$ and $$(1-\overline{\omega}_{l+1}) M_1(\eta,\theta_1) \geq (1-\tilde{\omega}_{k+1}) M_1(\xi,\theta_1),$$ where $\eta$ denotes the design with masses $\tfrac{\overline{\omega}_1}{1-\overline{\omega}_{l+1}}, \ldots,\tfrac{\overline{\omega}_{l}}{1-\overline{\omega}_{l+1}}$ at the points $\overline{d}_1,\ldots,\overline{d}_l$, respectively. Therefore we obtain $$(1-\overline{\omega}_{l+1}) M_1( \eta,\theta_1) \geq (1-\tilde \omega_{k+1}) M_1(\tilde \xi,\theta_1) \geq (1-\overline{\omega}_{l+1}) M_1( \xi,\theta_1).$$ Because the design $ \xi$ is admissible we have $M_1( \eta,\theta_1) = M_1( \xi,\theta_1)$. Using the block structure of the information matrix and the assumption that the design $\tilde \xi$ is not admissible it follows that $$(\overline{\omega}_{l+1}-\tilde \omega_{k+1})M_1( \xi,\theta_1) \leq 0 \ \ \mbox{and} \ \ (\tilde \omega_{k+1}-\overline{\omega}_{l+1}){\mathcal{I}}(\theta_2) \leq 0.$$ This yields $\overline{\omega}_{l+1}=\tilde \omega_{k+1}$ and $M(\tilde \eta, \theta_1)=M(\tilde \xi, \theta_1)$, which is a contradiction to the assumption that the design $\tilde \xi$ is not admissible. The desired result follows. $\Box$
|
---
abstract: 'We report on wide-field optically detected magnetic resonance imaging of nitrogen-vacancy centers (NVs) in type IIa polycrystalline diamond. These studies reveal a heterogeneous crystalline environment that produces a varied density of NV centers, including preferential orientation within some individual crystal grains, but preserves long spin coherence times. Using the native NVs as nanoscale sensors, we introduce a 3-dimensional strain imaging technique with high sensitivity ( $< 10^{-5}$ Hz$^{-1/2}$) and diffraction-limited resolution across a wide field of view.'
address: '$^1$Dept. of Electrical Engineering and Computer Science, MIT, Cambridge, MA 02139'
author:
- 'Matthew E. Trusheim$^1$ and Dirk Englund$^1$'
bibliography:
- 'library.bib'
title:
- 'Wide-Field Strain Imaging with Preferentially-Aligned Nitrogen-Vacancy Centers in Polycrystalline Diamond '
- 'Supplemental Material for Wide-Field Strain Imaging with Preferentially-Aligned Nitrogen-Vacancy Centers in Polycrystalline Diamond'
---
Introduction
============
Recent years have seen rapid advances in the development of quantum memories and sensors based on solid-state spin systems. At the forefront of these spin systems is the nitrogen vacancy center in diamond (NV), which has an electron spin triplet ground state with exceptionally long coherence time even at room temperature [@Bar-Gill2013]. Measuring these spins by optically detected magnetic resonance (ODMR) has enabled high performance sensing of electromagnetic fields [@maze_nanoscale_2008; @Dolde2011], temperature [@Kucsko2013a] and pressure[@Doherty2014]. The highest-sensitivity experiments have used single-crystal diamond grown through chemical vapor deposition. However, such samples remain expensive and small (on the scale of millimeters), which limits the adoption and scaling of NV sensing techniques. Polycrystalline diamond (PCD) presents an attractive alternative as it can be grown on the wafer scale and at lower cost, with the potential for long spin coherence times[@Balmer2009; @Jahnke2012]. However, a major concern with PCD has been that the diverse crystal structure, including grain boundaries and varied growth regimes, could lead to inconsistent and poor NV properties. In this work, we employ wide-field ODMR spectroscopy [@LeSage2013; @Steinert2013; @DeVience2015a; @Chen2013] to characterize the properties of hundreds of individually-resolved NVs across a field of view of $> 300$ $\upmu$m$^2$ and to use those centers as high-resolution nanoscale sensors of local crystal strain. These studies reveal that NVs in PCD are naturally preferentially aligned in some crystal grains, have spatially varying concentration, can be strongly strained near grain boundaries, and exhibit consistently long spin coherence times. Using these NVs, we introduce a method for wide-field strain imaging. We produce detailed, 3-dimensional strain maps of PCD structure with diffraction-limited resolution and sensitivity below $10^{-5}$ Hz$^{-1/2}$, outperforming traditional optical strain measurement techniques such as Raman imaging[@Kato2012109]. These studies show the application of NVs for high-resolution strain imaging, and demonstrate the viability and potential advantages of polycrystalline diamond for quantum sensing and information processing.
The NV is an electronic spin-1 system consisting of a substitutional nitrogen atom adjacent to a vacancy in the diamond lattice [@Doherty2012]. The ground-state spin can be coherently manipulated by microwave fields, as well as initialized and detected through optical illumination because the m$_s = \pm1$ sublevels are dark states that emit reduced fluorescence. The ground state Hamiltonian describing the system is :
$$H = \frac{1}{\hbar^2}[(D_{gs}+\mathcal{E} _z)S_z^2-\mathcal{E} _x(S_x^2-S_y^2)\\+ \mathcal{E} _y(S_xS_y+S_yS_x)] +\frac{g\mu_b}{\hbar}\mathbf{S}\cdot \mathbf{B}$$
Where $D_{gs}$ = 2.87 GHz is the spin-spin interaction energy, $\mathbf{\mathcal{E} } = \mathbf{d_{gs}}\cdot(\mathbf{E}+\mathbf{\sigma})$ is the energy of interaction with external electric ($\mathbf{E})$ and strain ($\mathbf{\sigma}$) fields, $\mathbf{B}$ is the magnetic field at the location of the NV, $d_{gs}$ is the NV electric dipole moment, $g$ the electron g-factor, and $\mu_b$ the Bohr magneton. Strain or electric fields along the $\vec{z}$ direction defined by the NV axis (Figure 1a) correspond to shifts in lattice spacing that preserve the NV’s trigonal symmetry and affect both $m_s=\pm1$ spin levels equally, while the levels are split in the presence of non-axial strains ($\mathcal{E}_x, \mathcal{E}_y$) that break this symmetry. For low transverse magnetic fields $B_{\perp} = \sqrt{B_x^2+B_y^2} << D_{gs}$, the ground state transition frequencies are
$$\hbar\omega_{\pm} = D_{gs}+\mathcal{E} _z \pm \sqrt{\mathcal{E} _{\perp}^2+(g\mu_bB_z)^2}$$
Where $\mathcal{E} _{\perp}=\sqrt{\mathcal{E} _x^2+\mathcal{E} _y^2} $ is the perpendicular effective electric field. This expression indicates two limiting regimes corresponding to the dominance of the $B_z$ or $\mathcal{E} _{\perp}$ terms. In the high-field regime, $B_z \gg \mathcal{E} _{\perp}$, the NV resonance frequency is not sensitive to changes in $\mathcal{E} _{\perp}$ in first order. A large magnetic bias field therefore allows probing of NV orientation based on magnetic field projection along the NV axis. In the low-field regime, $ B_z \ll\mathcal{E} _{\perp}$, the NV is sensitive to changes in $\mathcal{E} _{\perp}$ but not $\mathbf{B}$. Operating in this regime by eliminating the bias magnetic field enables probing of local strain through measurement of both $\mathcal{E}_z$ and $\mathcal{E}_\perp$. The change in energy per unit strain $ \mathbf{\mathcal{E}} (\mathbf{\sigma})$ is anisotropic and has been measured in separate cantilever-based experiments [@Teissier2014; @Ovartchaiyapong2014] as well as under isotropic pressure[@Doherty2014]. For this work, we assume a shift of $\mathcal{E}_\perp = 20.6$ GHz and $\mathcal{E}_z= 9.38$ GHz derived from the mean of the reported values.
{width="17cm"}
Wide-Field ODMR
===============
To probe the properties of NVs in PCD, we performed wide-field ODMR measurements using a custom-built fluorescence microscope (Figure 1a). Optical illumination is provided by a 532 nm green laser modulated by a double-pass acousto-optic modulator, and resulting NV fluorescence is spectrally filtered (650 nm long-pass) and collected on an electron-multiplying CCD camera. Spin manipulation is accomplished by applying microwaves through a 15$\mu$m copper wire placed nearby to the area of interest, while the external magnetic field is controlled by three orthogonal current-controlled electromagnets in addition to a permanent rare-earth magnet. The measurements were performed at room temperature without external stabilization or control.
We investigated a type-IIa polycrystalline diamond provided by Element6, grown by chemical vapor deposition with a nitrogen concentration $<$ 50 ppb. We first imaged the sample in fluorescence. Individual crystal grains vary in size from $~10-1000$ $\upmu$m$^2$ in area, and are easily identified in fluorescence imaging by their boundaries (Figure 1c,d). These grain boundaries fluoresce brightly across the visible band, likely due to a high density of optically-active lattice traps and amorphous carbon[@Jahnke2012], and indicate the transition between two growth regimes. Individual NV centers within individual PCD grains are visible under 100 x magnification (Fig 1b,c). The measurements show an NV density that varies significantly within and between grains, from roughly $\sim 0.1$ NV/$\upmu$m$^2$ to densities approaching $ 1$ NV/$\upmu$m$^2$. This heterogeneity contrasts with the homogenous NV distribution in single-crystal CVD diamond.
![a) Wide-field fluorescence image of NVs in polycrystalline diamond. Each fluorescent spot corresponds to a single NV, and the background corresponds to many out-of-focus NVs. Bright grain boundaries are visible on the edges of the image. b) ODMR spectrum averaged over the field of view. Dimming corresponds to driving between the m$_s$ = 0 and m$_s$ = $-1$ spin sublevels of different geometric classes of NVs. c,d) ODMR contrast images corresponding to the resonance marked with the green (red) arrow in b. The geometric classes of NVs is spatially distinct.[]{data-label="fig:spectrum"}](fig2_resub.pdf){width="12cm"}
We then investigated NV properties in the high magnetic field regime. An external magnetic field of $\sim$ 100 G was applied and the ODMR spectrum of NVs within the field of view obtained under continuous-wave microwave and optical excitation (Figure 2a,b). The observed resonance frequencies correspond to specific geometric classes of NVs $\{i\}$, each with a defined angle to the external magnetic field resulting in a different resonance frequency $\hbar\omega_i = D_{gs}+g\mu_bB_{z(i)}$. Interestingly, only three distinct NV orientations appear in this region, rather than the four possible classes corresponding the four crystallographic (111) directions. In Figure 2c,d, we map the location of NV geometric classes $\{i=1,2\}$ by showing ODMR at frequencies $\omega_i$. This shows spatial separation of NV classes, likely by growth region as no intersecting grain boundary is observed, with the lower region only containing NVs at a single frequency. This effect persists independent of the direction of the applied permanent magnetic field (Supplemental Material), and across grains in the PCD. Preferential orientation of NVs through engineered growth has been theoretically predicted[@Miyazaki2014a; @Karin2014a] and observed previously in single crystal diamond with controlled growth along {110}[@Pham2012a; @Edmonds2012], {111}[@fukui2014; @Tahara2015] and {113}[@lesik2015preferential]. These results indicate that preferential alignment occurs naturally in PCD, possibly due to predominant {110} and {111} grain textures.
{width="17cm"}
Strain Imaging
==============
We next turn to strain mapping of the PCD by wide-field continuous-wave ODMR spectroscopy. These measurements must be performed in the low magnetic field regime where $B_z \ll \mathcal{E}_\perp$; to achieve this, we canceled external magnetic fields using three-axis electromagnets. Figure 3b shows a representative ODMR spectrum. From double-Lorentzian fits to these low-field spectra, taken in parallel across the field of view with a total measurement time of 150 seconds, we extracted $(D_{gs}+\mathcal{E} _z)$ and $\mathcal{E}_\perp$, following Equation 2, and converted them to strain. The resulting strain maps are shown in Figures 3c and 3d, respectively. Axial strain values relative to a $D_{gs}$ parameter of 2.87 GHz are shown. For non-axial strain, the minimum detectable resonance splitting is set by the magnetic field induced from the NV-site $^{14}$N nuclear spin[@Dolde2011], allowing for absolute calibration of a zero value. The possible presence of multiple NV orientations could lead to a worst-case underestimation of strain through both the axial and non-axial parameters by a factor of two. Figure 3c indicates a maximal axial strain of $6\cdot10^{-4}$ at the grain boundary which relaxes towards the center of the grain, while Figure 3d indicates a maximum non-axial strain of $1.8\cdot10^{-4}$. These strains relax to their minimum value over a distance of 24 $\upmu$m from the grain boundary, setting a relevant length scale of the use of PCD in device design.
The sensitivity of this technique can be characterized by the 68% confidence interval on the fitted resonance frequencies, which corresponds to the measurement standard deviation and depends primarily on the detected photon rate. Due to the non-uniform illumination across the field of view as well as background fluorescence near grain boundaries, the sensitivity varies spatially. The mean 68% confidence interval on the two resonance frequencies for each 320 x 320 nm$^2$ pixel in the field of view is displayed in Figure 4. We achieve a median per-pixel ODMR spectral resolution of 245 kHz, resulting in an axial strain measurement precision of $2.7\cdot10^{-5}$ (nonaxial $1.2\cdot10^{-5}$) and a corresponding sensitivity of $3.21\cdot10^{-4}$ Hz$^{-1/2}$ (non-axial $1.5\cdot10^{-4}$ Hz$^{-1/2}$) for a 150 second measurement. To characterize a best-case sensitivity without the sample-specific background near the grain boundaries, we focus on the high-SNR regions corresponding to in-focus individual NVs at the center of the field of view. In these regions, we achieve 68% confidence intervals of 79 kHz, corresponding to a measurement precision of $8.2\cdot10^{-6}$ (non-axial $3.8\cdot10^{-6}$) and sensitivity of $1.02\cdot10^{-4}$ Hz$^{-1/2}$ (non-axial $4.7 \cdot10^{-5}$ Hz$^{-1/2}$).
{width="17cm"}
Using this low magnetic field ODMR technique, we performed wide-field strain imaging in three dimensions across several crystal grains. These studies again reveal a strong strain gradient near grain boundaries. In the grain shown in Figure 3a-c, the D$_{gs}$ parameter corresponding to axial strain is lower near the boundary than in the center of the grain. This corresponds to tensile rather than compressive strain, while the reverse is true for the grain shown in Figure 3c,d. The non-axial strain at the center of the field of view is reduced with increased depth into the diamond, from a maximal value $ > 1\cdot10^{-4}$ near the surface to $ < 5\cdot10^{-5}$ at a depth of 3 $\upmu$m. As visible in the fluorescence profile (Figure 5a), the location and angle of the grain boundary varies with depth into the diamond, which is in turn reflected by rotation in the axial strain profile (Figure 5b,c). Here the strain is observed to relax within 10 $\upmu$m from the grain boundary. Further strain images, including extreme strain gradients and comparisons to reference single-crystal diamonds, are included in the Supplemental Material. These maps demonstrate the power of this technique for imaging crystal strain with high precision in three dimensions.
{width="17cm"}
Discussion
==========
Our studies reveal advantages and disadvantages of PCD for NV-based applications. The observation of preferential NV alignment within PCD grains, in combination with consistently high NV spin coherence times similar to single-crystal samples, can greatly improve the signal-to-noise of NV sensing applications (i.e a fourfold improvement in ODMR contrast)[@Pham2012a]; the large areas available for PCD diamond add to the usefulness in wide-field sensing applications, for example in biology[@Barry2000]. Strong strain gradients could also allow for sub-diffraction resolution of individual NVs using ODMR[@Chen2013]. In photonic devices, well-defined angular alignment of NVs can improve NV-dipole coupling to optical modes. PCD enables the production of wafer-scale photonic circuits in diamond, but roughness and scattering from grain boundaries have limited device performance compared to single-crystal diamond[@Rath2015a]. Large grains with low intra-grain roughness, such as those present in the sample characterized in this work, may reduce loss while still providing large-area substrates. Wide-field pre-characterization of the PCD could also allow for identification of and design compensation for grain boundaries.
Polycrystalline diamond could also serve as a natural environment for studying correlations between crystal strain and the spin and optical properties of embedded emitters. For example, high strain could break the orbital degeneracy of the SiV [@Muller2014a], changing spin and orbital coherence properties, or strain gradients could be employed to tune relative NV sensitivity to electric and magnetic fields[@Jamonneau2015]. PCD provides a substrate that is ultra-pure and enables long spin coherence times, in contrast to other highly strained host materials such as nanodiamond. The strain in PCD can be very large; the observed NV ground state spin strain shifts of over 8 MHz in the axial direction are higher than those reported in cantilever-based experiments[@Teissier2014; @Ovartchaiyapong2014] and would require pressures in the tens of GPa to produce externally.
Conclusion and Outlook
======================
The NV-based strain imaging technique introduced in this work reaches the optical diffraction limit and a high sensitivity of $10^{-5}$ Hz$^{-1/2}$ ($<$ 10 MPa), which outperforms more traditional strain imaging techniques such as Raman imaging[@Kato2012109] that are limited to absolute sensitivities $>$ 10 MPa in addition to being orders of magnitude slower [@Mermoux2004]. Through sectional imaging, this 3D imaging technique also offers an advantage over birefringence[@Pinto2009; @Hoa2014] or x-ray topography[@Umezawa2011523] strain imaging methods, which image whole-sample and near-surface strains, respectively. Although limited to diamond and other materials with optically accessible, strain-sensitive spin defects (e.g. silicon carbide[@Falk2014]), this technique has potential for precisely characterizing internal strains, such as those induced by geological formation[@Vlasov2013] or in strain-engineered devices, as well as for mapping externally applied strains. More advanced dynamical-decoupling sequences can increase axial strain sensitivity[@Brunner2013; @Toyli2013], which in turn could enable sub-diffraction strain imaging by resolving NVs in the spectral domain. While this work images individual NVs, higher-NV-density samples would increase sensitivity[@2008.NPhys.Taylor], potentially enabling the imaging of few-site dislocations in single-crystal diamond or increasing the field of view. By imaging NVs in different independent orientations, this technique additionally could provide full vector reconstruction of the local strain. Since the NV is a truly nanoscale sensor, operable over a wide range of temperatures[@Toyli2012] and pressures[@Doherty2014], this method can be used to perform ultra-high resolution strain mappings dynamically in previously unexplored regimes.\
Supplemental Material
=====================
See the supplemental material for a description extended discussions of preferential alignment and spin coherence times as well as additional strain imaging datasets.
Acknowledgements
================
The authors would like to thank Daniel Twitchen and Matthew Markham for helpful discussions, as well as C. Foy and H. Clevenson for their perspective on the manuscript. This work was supported in part by the Air Force Office of Scientific Research (AFOSR) MURI (FA9550-14-1-0052), the AFOSR Presidential Early Career Award (supervised by Gernot Pomrenke), the Army Research office MURI biological transduction program, and the Office of Naval Research (N00014-13-1-0316).
References
==========
Preferential Alignment under Varying Magnetic Fields
====================================================
One possible explanation for the appearance of less than four resonances in the NV ODMR spectrum under an external magnetic field is that fortuitous alignment of the field has resulted identical Zeeman shifts for different geometric classes of NV centers. To demonstrate that this is not the case, we vary the external field orientation relative to the sample and observe only a single resonance line under all magnetic field conditions, varying in frequency due to different magnetic field projection along the NV axis.
![NV ODMR under different angular alignments of the external magnetic field (left to right). Top Row: Fluorescence. Middle Row: ODMR spectra averaged across the entire field of view. Bottom Row: ODMR contrast maps at the NV resonance frequency for each magnetic field orientation. While the area imaged is constant both in PL and contrast, only one resonance line is seen for all magnetic field angles.[]{data-label="fig:spectrum"}](s0.png){width="19cm"}
NV Spin Coherence
=================
With the NV density observed to vary significantly across the PCD sample, a natural question is whether NV spin properties likewise change due to the apparent difference in paramagnetic defect density. To address this, we performed Hahn Echo measurements in regions with differing NV densities. Figure S2 shows the PL and spin data for each region represented in main text Figure 2f. The NV density was determined by counting the number of in-focus NV centers and dividing by the field of view, while the $T_2$ was computed through single-exponential fitting of revival decay envelope. We found that NV T$_2$ time is not strongly dependent on NV density, indicating that spin-bath correlation time does not significantly differ. In turn, this implies NV formation probability is dependent on crystal region.
{width="17cm"}
{width="17cm"}
Reference Measurements on Single-Crystal Diamond
================================================
We use a single-crystal type IIa diamond with $\sim1$ ppm N (element6) as a reference sample. At zero magnetic field, we observe overlapping $^{14}$N hyperfine transitions from all four NV geometric classes. The inner resonances $m_i = 0$ are split by non-axial strain, while the outer resonances $m_i = \pm1$ are degenerate for $\vec{B} = 0$. The linewidth of the outer resonances, $\sim300$ kHz, therefore sets the minimum detectable magnetic field (or alternately, the maximum residual field), while the separation of the inner resonances corresponds to the average strain across the field of view of $1.9(5)\cdot10^{-5}$.
{width="17cm"}
High Strain Gradients
=====================
We observe extremely high strain gradients of over one part in a thousand strain across 10 microns. This corresponds to pressures $ >$ 1 GPa. Remarkably, NV centers remained stable and displayed ODMR despite this wide change. Polycrystalline diamond could offer a testbed for the functionality of quantum devices in these extreme operational regimes.
![Highly strained PCD. a) NV PL. b) Axial and c) non-axial strain[]{data-label="fig:spectrum"}](s2.pdf){width="17cm"}
Strain Measurement in the Presence of Multiple NV Orientations
==============================================================
The description given in the main text is accurate in the single-NV case and generalizes directly to ensembles of NVs with identical orientation. In the case of multiple NV orientations, the resonance frequencies $\omega_{\pm}$ are different for each orientation as the projection of magnetic field and strain along the NV axis differs for each orientation. If the individual resonance lines for each orientation are resolvable the local field along and perpendicular to each NV direction can be reconstructed, in principle allowing for vector imaging. In the case that the resonance lines are not individually resolvable (i.e. the resonance linewidth is greater than the difference in the resonance frequencies between orientations), the observed resonance frequencies are the weighted average of the constituent resonance frequencies, where the weighting is given by the relative frequency of each orientation within the ensemble: $$\hbar\omega_{\pm,obs} = \frac{1}{N}\Sigma_{i=1}^N F_i\omega_{\pm,i} =\frac{1}{N}\Sigma_{i=1}^N F_i (D_{gs}+\mathcal{E} _{z,i} \pm \sqrt{\mathcal{E} _{\perp,i}^2+(g\mu_bB_{z,i})^2})$$
We determine the axial strain by taking the mean of the two resonance frequencies, and the non-axial strain by the difference, in the limit of no magnetic field. Because there is a linear relation between resonance frequency and strain in this limit, the observed strain is the weighted average of the true axial and non-axial strains for each orientation.
$$\mathcal{E} _{z,obs} = \frac{\hbar}{2}(\omega_{+,obs}+\omega_{-,obs}) = \frac{1}{N}\Sigma_{i=1}^N F_i (D_{gs}+\mathcal{E} _{z,i})$$
$$\mathcal{E} _{\perp,obs} = \frac{\hbar}{2}(\omega_{+,obs}-\omega_{-,obs}) = \frac{1}{N}\Sigma_{i=1}^N F_i \mathcal{E} _{\perp,i}$$
In this limit, therefore, we measure the strain components relative to the mean NV axial and non-axial moments.
|
---
abstract: 'In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on Banach space with locally monotone operators, which is a generalization of the classical result by J.L. Lions for monotone operators. In particular, we show that local monotonicity implies the pseudo-monotonicity. The main result is applied to various types of PDE such as reaction-diffusion equations, generalized Burgers equation, Navier-Stokes equation, 3D Leray-$\alpha$ model and $p$-Laplace equation with non-monotone perturbations.'
author:
- |
[**Wei Liu [^1]** ]{}\
[Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany]{}\
title: '[**Existence and Uniqueness of Solutions to Nonlinear Evolution Equations with Locally Monotone Operators**]{} [^2] '
---
AMS Subject Classification: 35K55, 34G20, 35Q30\
Keywords: nonlinear evolution equation; locally monotone; pseudo-monotone; Navier-Stokes equation; Leray-$\alpha$ model; Burgers equation; porous medium equation; $p$-Laplace equation; reaction-diffusion equation.
Main results
============
Consider the following Gelfand triple $$V \subseteq H\equiv H^*\subseteq V^*,$$ $i.e.$ $(H, \<\cdot,\cdot\>_H)$ is a real separable Hilbert space and identified with its dual space $H^*$ by the Riesz isomorphism, $V$ is a real reflexive Banach space such that it is continuously and densely embedded into $H$. If $\<\cdot,\cdot\>_V$ denotes the dualization between $V$ and its dual space $V^*$, then it follows that $$\<u, v\>_V=\<u, v\>_H, \ \ u\in H ,v\in V.$$ The main aim of this paper is to establish the existence and uniqueness of solutions for general nonlinear evolution equations $$\label{1.1}
u'(t)=A(t,u(t))+b(t), \ 0<t<T, \ u(0)=u_0\in H,$$ where $T>0$, $u'$ is the generalized derivative of $u$ on $(0,T)$ and $A:[0,T]\times V\rightarrow V^*, b:[0,T]\rightarrow V^*$ is measurable, $i.e.$ for each $u\in L^1([0,T]; V)$, $A(t,u(t))$ is $V^*$-measurable on $[0,T]$.
It’s well known that (\[1.1\]) has a unique solution if $A$ satisfies the monotone and coercivity conditions (cf. [@Li69; @KR79; @Z90]). The proof is mainly based on the Galerkin approximation and the monotonicity tricks. The theory of monotone operators started from the substantial work of Minty [@Mi62; @Mi63], then it was studied systematically by Browder [@Bro63; @Bro64] in order to obtain existence theorems for quasi-linear elliptic and parabolic partial differential equations. The existence results of Browder were generalized to more general classes of quasi-linear elliptic differential equations by Leray and Lions [@LL65], and Hartman and Stampacchia [@HS66]. We refer to [@Br73; @Li69; @Sh97; @Z90] for more detailed exposition and references.
One of most important extensions of monotone operator is the pseudo-monotone operator, which was first introduced by Brézis in [@Br68]. The prototype of a pseudo-monotone operator is the sum of a monotone operator and a strongly continuous operator ($i.e.$ a operator maps a weakly convergent sequence into a strongly convergent sequence). Hence the theory of pseudo-monotone operator unifies both the monotonicity arguments and the compactness arguments. For example, it can be applied to show the existence of solutions for general quasi-linear elliptic equations with lower order terms which satisfy no monotonicity condition (cf. [@Bro72; @Sh97; @Z90]).
This variational approach has also been adapted for analyzing stochastic partial differential equations (SPDE). The existence and uniqueness of solutions to SPDE was first developed by Pardoux [@Par75], Krylov and Rozovskii [@KR79], we refer to [@G; @RRW] for further generalizations. Within this framework many different types of properties have been established recently, $e.g.$ see [@L08b; @RZ] for the small noise large deviation principle, [@GM09] for discretization approximation schemes to the solution of SPDE, [@L08; @LW08; @W07] for the Harnack inequality and consequent ergodicity, compactness and contractivity for the associated transition semigroups, and [@L10; @BGLR; @GLR] for the invariance of subspaces and existence of random attractors for corresponding random dynamical systems.
In this work we establish the existence, uniqueness and continuous dependence on initial conditions of solutions to (\[1.1\]) by using the local monotonicity condition instead of the classical monotonicity condition. The analogous result for stochastic PDE has been established in [@LR10]. The standard growth condition on $A$ (cf. [@Li69; @KR79; @Z90]) is also replaced by a much weaker condition such that the main result can be applied to larger class of examples. One of the key observations is that we show the local monotonicity implies the pseudo-monotonicity (see Lemma \[L2.1\]), which may have some independent interests. The main result is applied to establish the existence and uniqueness of solutions for a large class of nonlinear evolution equations such as reaction diffusion equations, generalized Burgers type equations, generalized $p$-Laplace equations, 2-D Navier-Stokes equation and 3D Leray-$\alpha$ model of turbulence.
Suppose for fixed $\alpha>1, \beta\ge 0$ there exist constants $\delta>0$, $C$ and a positive function $f\in L^1([0,T]; \mathbb{R})$ such that the following conditions hold for all $t\in[0,T]$ and $v,v_1,v_2\in V$.
1. (Hemicontinuity) The map $ s\mapsto \<A(t,v_1+s v_2),v\>_V$ is continuous on $\mathbb{R}$.
2. (Local monotonicity) $$\<A(t,v_1)-A(t, v_2), v_1-v_2\>_V
\le \left(C+\rho(v_1)+\eta(v_2) \right) \|v_1-v_2\|_H^2,$$ where $\rho,\eta: V\rightarrow [0,+\infty)$ are measurable functions and locally bounded in $V$.
3. (Coercivity) $$2 \<A(t,v), v\>_V \le -\delta
\|v\|_V^{\alpha} +C\|v\|_H^2+ f(t).$$
4. (Growth) $$\|A(t,v)\|_{V^*} \le \bigg( f(t)^{\frac{\alpha-1}{\alpha}} +
C\|v\|_V^{\alpha-1} \bigg) \bigg( 1+ \|v\|_H^{\beta} \bigg).$$
\(1) If $\beta=0$ and $\rho=\eta\equiv 0$, then $(H1)-(H4)$ are the classical monotone and coercive conditions in [@Z90 Theorem 30.A] (see also [@Li69; @KR79; @PR07]). It can be applied to many quasi-linear PDE such as porous medium equation and $p$-Laplace equation (cf. [@Z90; @PR07]).
\(2) One typical form of $(H2)$ in applications is $$\rho(v)=\eta(v)=C\|v\|^\gamma,$$ where $\|\cdot\|$ is some norm on $V$ and $C,\gamma$ are some constants. The typical examples are 2-D Navier-Stokes equation on bounded or unbounded domain and Burgers equation, which satisfy $(H2)$ but do not satisfy the classical monotonicity condition ($i.e.$ $\rho=\eta\equiv0$). We refer to section 3 for more examples.
\(3) If $\rho\equiv 0$ in $(H2)$, then the existence and uniqueness of solutions to (\[1.1\]) with general random noise has been established in [@LR10] by using some different techniques.
\(4) $(H4)$ is also weaker than the following standard growth condition assumed in the literature (cf. [@KR79; @Z90; @PR07]: $$\label{growth}
\|A(t,v)\|_{V^*} \le f(t)^{\frac{\alpha-1}{\alpha}}+
C\|v\|_V^{\alpha-1} .$$ The advantage of $(H4)$ is, $e.g.$, to include many semilinear type equations with nonlinear perturbation terms. For example, if we consider the reaction-diffusion type equation, $i.e.$ $A(u)=\Delta
u+F(u)$, then for verifying the coercivity $(H3)$ we have $\alpha=2$. Hence (\[growth\]) implies that $F$ has at most linear growth. However, we can allow $F$ to have some polynomial growth by using $(H4)$ here. We refer to section 3 for more details.
Now we can state the main result, which gives a unified framework to analyze various classes of nonlinear evolution equations.
\[T1\] Suppose that $V \subseteq H$ is compact and $(H1)$-$(H4)$ hold, then for any $u_0\in H,\ b\in L^{\frac{\alpha}{\alpha-1}}([0,T];V^*)$ $(\ref{1.1})$ has a solution $$u\in L^\alpha([0,T];V)\cap C([0,T];H), \ u'\in L^{\frac{\alpha}{\alpha-1}}([0,T];V^*)$$ such that $$\<u(t), v\>_H=\<u_0,v\>_H + \int_0^t \<A(s,u(s))+b(s), v\>_V d s , \ t\in[0,T], v\in V.$$ Moreover, if there exist constants $C$ and $\gamma$ such that $$\label{c3}
\rho(v)+\eta(v) \le C(1+\|v\|_V^\alpha) (1+\|v\|_H^\gamma), \
v\in V,$$ then the solution of $(\ref{1.1})$ is unique.
\[R1\] (1) The proof is based on Galerkin approximation. Moreover, by the Lions-Aubin theorem (cf. [@Sh97 Chapter III, Proposition 1.3]), the compact embedding of $V\subseteq H$ implies the following embedding $$W^1_\alpha(0,T;V,H):=\{ u\in L^\alpha([0,T];V): u'\in L^{\frac{\alpha}{\alpha-1}}([0,T];V^*) \} \subseteq L^\alpha(0,T;H)$$ is also compact. Hence there exists a subsequence of the solutions of the Galerkin approximated equations (see (\[2.1\]) in Section 2) strongly converges to the solution of (\[1.1\]) in $L^\alpha(0,T;H)$.
\(2) One can easily see from the proof that the solution of $(\ref{1.1})$ is unique if all solutions of $(\ref{1.1})$ satisfy $$\int_0^T \left( \rho(u(s)) +\eta(u(s)) \right) d s<\infty.$$
\(3) The compact embedding $V \subseteq H$ is required in the main result. For (global) monotonicity one can easily drop this assumption. In fact, the classical monotonicity tricks only works in general for the operator satisfies $(H2)$ with $C=\rho=\eta=0$. For $C>0$ (but $\rho=\eta=0$) one can apply a standard exponential transformation to (\[1.1\]) to reduce the case $C>0$ to the case $C=0$. However, this kind of techniques does not work for the locally monotone case. In order to verify the pseudo-monotonicity of $A(t,\cdot)$, we have to split it into the sum of $A(t,\cdot)-cI$ and $cI$. And $I$ is pseudo-monotone if and only if the embedding $V \subset H$ is compact.
\(4) We can also establish a similar result for stochastic evolution equations in Hilbert space with additive noise: $$\label{SDE}
d X(t)=A(t, X(t))dt + B d N(t), \ t\ge 0, \ X(0)=x.$$ Here $A:[0,T]\times V\rightarrow V^*$ and $B\in L(U; H)$, where $U$ is another Hilbert space and $N(t)$ is a $U$-valued adapted stochastic process definded on a filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ (cf. [@GLR; @PR07]). By a standard transformation (substitution), (\[SDE\]) can be reduced to deterministic evolution equations with a random parameter which Thoerem \[T1\] can be applied to. This result and some further applications will be investigated in a separated paper.
Next result is the continuous dependence of solution of $(\ref{1.1})$ on $u_0$ and $b$.
\[T2\] Suppose that $V \subseteq H$ is compact and $(H1)$-$(H4)$ hold, $u_i$ are the solution of $(\ref{1.1})$ with $u_{i,0}\in H$ and $b_i\in L^{\frac{\alpha}{\alpha-1}}([0,T];V^*)\cap L^{2}([0,T]; H)$, $i=1,2$ respectively and satisfy $$\int_0^T\left( \rho(u_1(s))+\eta(u_2(s)) \right) d s<\infty.$$ Then there exists a constant $C$ such that $$\begin{split}
\|u_1(t)-u_2(t)\|_H^2
\le & \exp\left[\int_0^t \left(C+\rho(u_1(s))+\eta(u_2(s)) \right) d s \right]\\
& \ \ \cdot \left( \|u_{1,0}-u_{2,0}\|_H^2 +\int_0^t \|b_1(s)-b_2(s)\|_H^2 ds \right), \ t\in[0, T].
\end{split}$$
The paper is organized as follows. The proofs of the main results are given in the next section. In Section 3 we apply the main results to several concrete semilinear and quasi-linear evolution equations on Banach space.
Proofs of Main Theorems
=======================
Proof of Theorem \[T1\]
-----------------------
In order to make the proof easier to follow, we first give the outline of the proof for the reader’s convenience.
Step 1: Galerkin approximation; local monotonicity and coercivity implies the existence (and uniqueness) of solutions to the approximated equations;
Step 2: A priori estimates was obtained from coercivity;
Step 3: Verify the weak limits by using modified monotonicity tricks;
Step 4: Uniqueness follows from local monotonicity.
The main difficulty is in the third step. The classical monotonicity tricks does not work for locally monotone operators. The crucial part for overcoming this difficulty is the following result: every locally monotone operator is pseudo-monotone. Then by using some techniques related with pseudo-monotonicity one can establish the existence of solutions.
We first recall the definition of pseudo-monotone operator introduced first by Brézis in [@Br68]. We use the standard notation “$\rightharpoonup$” for weak convergence in Banach space.
The operator $A: V\rightarrow V^*$ is called pseudo-monotone if $v_n\rightharpoonup v$ in $V$ and $$\liminf_{n\rightarrow\infty} \<A(v_n), v_n-v\>_V\ge 0$$ implies for all $u\in V$ $$\<A(v), v-u\>_V \ge \limsup_{n\rightarrow\infty} \<A(v_n), v_n-u\>_V.$$
\(1) We remark that the definition of pseudo-monotone operator here coincides with the definition in [@Z90] (one should replace $A$ here by $-A$ in [@Z90] due to different form of the formulation for evolution equations).
\(2) The class of pseudo-monotone operators is stable under summation ($i.e.$ the sum of two pseudo-monotone operators is still pseudo-monotone) and strictly smaller than the class of operator of type (M) (cf. [@Z90; @Sh97]). And note that the class of operator of type (M) is not stable under summation. A counterexample can be found in [@Sh97].
\[condition P\] If $A$ is pseudo-monotone, then $v_n\rightharpoonup v$ in $V$ implies that $$\liminf_{n\rightarrow\infty} \<A(v_n), v_n-v\>_V\le 0.$$
If the conclusion is not true, then there exists $v_n\rightharpoonup
v$ in $V$ such that $$\liminf_{n\rightarrow\infty} \<A(v_n), v_n-v\>_V> 0.$$ Then we can extract a subsequence such that $v_{n_k}\rightharpoonup
v$ and $$\label{contradiction}
\lim_{k\rightarrow\infty} \<A(v_{n_k}), v_{n_k}-v\>_V> 0.$$ Then by the pseudo-monotonicity of $A$ we have for all $u\in V$ $$\<A(v), v-u\>_V \ge \limsup_{k\rightarrow\infty} \<A(v_{n_k}), v_{n_k}-u\>_V.$$ By taking $u=v$ we obtain $$\limsup_{k\rightarrow\infty} \<A(v_{n_k}), v_{n_k}-v\>_V\le 0,$$ which is a contradiction to (\[contradiction\]).
Hence the proof is completed.
We also recall a slightly modified definition of pseudo-monotone operator by Browder (cf. [@Bro77]): The operator $A: V\rightarrow V^*$ is called pseudo-monotone if $v_n\rightharpoonup v$ in $V$ and $$\liminf_{n\rightarrow\infty} \<A(v_n), v_n-v\>_V\ge 0$$ implies $$A(v_n)\rightharpoonup A(v)\ \ \text{and} \ \
\lim_{n\rightarrow\infty} \<A(v_n), v_n\>_V=\<A(v), v\>_V.$$ From this definition one clearly see the role of pseudo-monotone operator for verifying the limit of weakly convergent sequence under nonlinear operator.
If $A$ is bounded ($i.e.$ $A$ maps bounded set into bounded set), then it’s easy to show that these two definitions are equivalent by Proposition \[condition P\]. In particular, under the assumption of $(H4)$, these two definitions are equivalent.
\[L2.1\] If the embedding $V\subseteq H$ is compact, then $(H1)$ and $(H2)$ implies that $A(t,\cdot)$ is pseudo-monotone for any $t\in[0,T]$.
For simplicity, we denote $A(t,\cdot)$ by $A(\cdot)$ for any fixed $t\in[0,T]$.
Suppose $v_n\rightharpoonup v$ in $V$ and $$\label{2.0}
\liminf_{n\rightarrow\infty} \<A(v_n), v_n-v\>_V\ge 0,$$ then for any $u\in V$ we need to show $$\label{e1}
\<A(v), v-u\>_V \ge \limsup_{n\rightarrow\infty} \<A(v_n), v_n-u\>_V.$$
Given $u$ and the constant $C$ in $(H2)$, we take $$K=\|u\|_V+ \|v\|_V+ \sup_n \|v_n\|_V; \ \ C_1=\sup_{v:\|v\|_V\le 2K} \left(C+\rho(v)+\eta(v) \right)< \infty.$$ Since the embedding $V\subseteq H$ is compact, we have $v_n\rightarrow v$ in $V^*$ and $$\<C_{1}v, v-u\>_V = \lim_{n\rightarrow\infty} \<C_{1}v_n, v_n-u\>_V.$$ Hence for proving (\[e1\]) it’s sufficient to show that $$\<A_0(v), v-u\>_V \ge \limsup_{n\rightarrow\infty} \<A_0(v_n), v_n-u\>_V,$$ where $A_0=A-C_{1}I$ ($I$ is the identity operator).
Then $(H2)$ implies that $$\limsup_{n\rightarrow\infty} \<A_0(v_n), v_n-v\>_V\le \limsup_{n\rightarrow\infty}
\<A_0(v), v_n-v\>_V =0.$$ By (\[2.0\]) we obtain $$\label{2.2}
\lim_{n\rightarrow\infty} \<A_0(v_n), v_n-v\>_V= 0.$$ Let $z=v+t(u-v)$ with $t\in(0,\frac{1}{2})$, then the local monotonicity $(H2)$ implies that $$\<A_0(v_n)-A_0(z), v_n-z\>_V\le 0,$$ $i.e.$ $$t \<A_0(z), v-u\>_V-(1-t)\<A_0(v_n), v_n-v\>_V \ge t\<A_0(v_n), v_n-u\>_V-\<A_0(z), v_n-v\>_V.$$ By taking $\limsup$ on both sides and using (\[2.2\]) we have $$\<A_0(z), v-u\>_V\ge \limsup_{n\rightarrow\infty} \<A_0(v_n), v_n-u\>_V.$$ Then letting $t\rightarrow 0$, by the hemicontinuity $(H1)$ we obtain $$\<A_0(v), v-u\>_V \ge \limsup_{n\rightarrow\infty} \<A_0(v_n), v_n-u\>_V.$$ Therefore, $A$ is pseudo-monotone.
For some concrete operators, the local monotonicity $(H2)$ might be easier to check by explicit calculations than the definition of pseudo-monotonicity. Hence the above result can be also seen as a computable sufficient condition for the pseudo-monotonicity in applications.
The proof of Theorem \[T1\] is split into a few lemmas. Let $X:=L^\alpha([0,T];V)$, then $X^*=L^{\frac{\alpha}{\alpha-1}}([0,T];V^*)$. We denote by $W^1_\alpha(0,T;V, H)$ the Banach space $$W^1_\alpha(0,T;V, H)=\{ u\in X: u'\in X^* \},$$ where the norm is defined by $$\|u\|_W:=\|u\|_X+\|u'\|_{X^*}= \left(\int_0^T\|u(t)\|_V^\alpha dt\right)^{\frac{1}{\alpha}}+
\left(\int_0^T\|u'(t)\|_{V^*}^{\frac{\alpha}{\alpha-1}} dt\right)^{\frac{\alpha-1}{\alpha}}.$$ It’s well known that $W^1_\alpha(0,T;V, H)$ is a reflexive Banach space and it is continuously imbedded into $C([0,T];H)$ (cf. [@Z90]). Moreover, we also have the following integration by parts formula $$\begin{split}
\<u(t), v(t)\>_H -\<u(0), v(0)\>_H=\int_0^t& \<u'(s), v(s)\>_V ds
+ \int_0^t \<v'(s), u(s)\>_V ds,\\
& \ t\in[0,T], \ u,v\in W^1_\alpha(0,T; V,H).
\end{split}$$
We first consider the Galerkin approximation to (\[1.1\]).
Let $\{e_1,e_2,\cdots \}\subset V$ be an orthonormal basis in $H$ and let $H_n:=span\{e_1,\cdots,e_n\}$ such that $span\{e_1,e_2,\cdots\}$ is dense in $V$. Let $P_n:V^*\rightarrow H_n$ be defined by $$P_ny:=\sum_{i=1}^n \<y,e_i\>_V e_i, \ y\in V^*.$$ Obviously, $P_n|_H$ is just the orthogonal projection onto $H_n$ in H and we have $$\<P_nA(t,u), v\>_V=\<P_nA(t,u),v\>_H=\<A(t,u),v\>_V, \ u\in V, v\in H_n.$$
For each finite $n\in \mathbb{N}$ we consider the following evolution equation on $H_n$: $$\label{2.1}
u_n'(t)=P_nA(t,u_n(t))+P_n b(t), \ 0<t<T, \ u_n(0)=P_nu_0\in H_n.$$ It is easy to show that $P_nA$ is locally monotone and coercive on $H_n$ (finite dimensional space). According to the classical result of Krylov (cf. [@K99] or [@PR07 Theorem 3.1.1]), there exists a unique solution $u_n$ to (\[2.1\]) such that $$u_n\in L^\alpha([0,T];H_n)\cap C([0,T];H_n), \ u_n'\in L^{\frac{\alpha}{\alpha-1}}([0,T];H_n) .$$
\[l2.3\]
Under the assumptions of Theorem \[T1\], there exists a constant $K>0$ such that $$\|u_n\|_X+\sup_{t\in[0,T]}\|u_n\|_H+\|A(\cdot,u_n)\|_{X^*}\le K, \ n\ge 1.$$
By the integration by parts formula and $(H3)$ we have $$\begin{split}
& ~~~~\|u_n(t)\|_H^2-\|u_n(0)\|_H^2\\
&=2\int_0^t\<u_n'(s),u_n(s)\>_V d s\\
&=2\int_0^t\<P_nA(s,u_n(s))+P_nb(s),u_n(s)\>_V d s\\
&=2\int_0^t\<A(s,u_n(s))+b(s),u_n(s)\>_V d s\\
&\le\int_0^t\left(-\delta\|u_n(s)\|_V^\alpha+C\|u_n(s)\|_H^2+f(s)+\|b(s)\|_{V^*}\|u_n(s)\|_V \right) d s \\
&\le \int_0^t\left(-\frac{\delta}{2}\|u_n(s)\|_V^\alpha+C\|u_n(s)\|_H^2+f(s)+C_1\|b(s)\|_{V^*}^{\frac{\alpha}{\alpha-1}} \right) d s,
\end{split}$$ where $C_1$ is a constant induced from Young’s inequality.
Hence we have for $t\in [0,T]$, $$\|u_n(t)\|_H^2+\frac{\delta}{2}\int_0^t\|u_n(s)\|_V^\alpha d s\le \|u(0)\|_H^2 +
C\int_0^t\|u_n(s)\|_H^2 ds +\int_0^t \left(f(s)+C_1\|b(s)\|_{V^*}^{\frac{\alpha}{\alpha-1}} \right) d s.$$ Then by Gronwall’s lemma we have $$\|u_n(t)\|_H^2 \le e^{Ct}\left( \|u(0)\|_H^2
+\int_0^t e^{-Cs} \left(f(s)+C_1\|b(s)\|_{V^*}^{\frac{\alpha}{\alpha-1}} \right) d s \right), \ t\in[0, T].$$ $$\frac{\delta}{2}\int_0^t\|u_n(s)\|_V^\alpha d s\le e^{Ct}\left( \|u(0)\|_H^2 +
\int_0^t e^{-Cs}\left(f(s)+C_1\|b(s)\|_{V^*}^{\frac{\alpha}{\alpha-1}} \right) d s \right), \ t\in[0, T].$$ Therefore, there exists a constant $C_2$ such that $$\|u_n\|_{X}+\sup_{t\in[0,T]}\|u_n(t)\|_H \le C_2, \ n\ge 1.$$ Then by $(H4)$ there exists a constant $C_3$ such that $$\|A(\cdot, u_n)\|_{X^*} \le C_3, \ n\ge 1.$$ Hence the proof is complete.
Note that $X,X^*$ and $H$ are reflexive spaces, by the estimates in Lemma \[l2.3\], there exists a subsequence, again denote by $u_n$, such that as $n\rightarrow\infty$ $$\begin{split}
u_n &\rightharpoonup u\ \ \text{in}\ X \ (\text{also in} \ \ W^1_\alpha(0,T;V,H)); \\
A(\cdot,u_n)&\rightharpoonup w \ \ \text{in} \ X^*; \\
u_n(T) &\rightharpoonup z \ \ \text{in} \ H.
\end{split}$$
Recall that $u_n(0)=P_nu_0\rightarrow u_0$ in $H$ as $n\rightarrow\infty$.
Under the assumptions of Theorem \[T1\], the limit elements $u,w$ and $z$ satisfy $u\in W^1_\alpha(0,T;V,H)$ and $$u'(t)=w(t)+b(t), \ 0<t<T, \ u(0)=u_0, \ u(T)=z.$$
The proof is standard (cf. [@Z90 Lemma 30.5]), we include it here for the completeness. Recall the following integration by parts formula $$\<u(T), v(T)\>_H -\<u(0), v(0)\>_H=\int_0^T \<u'(t), v(t)\>_V dt
+ \int_0^T \<v'(t), u(t)\>_V dt, \ u,v\in W^1_\alpha(0,T; V,H).$$ Then, for $\psi\in C^\infty([0,T])$ and $v\in H_n$, by (\[2.1\]) we have $$\begin{split}
& \<u_n(T), \psi(T)v\>_H -\<u_n(0), \psi(0)v\>_H \\
=&\int_0^T \<u_n'(t), \psi(t)v\>_V dt
+ \int_0^T \<\psi'(t)v, u_n(t)\>_V dt\\
=&\int_0^T \<A(t,u_n(t))+b(t), \psi(t)v\>_V dt
+ \int_0^T \<\psi'(t)v, u_n(t)\>_V dt.
\end{split}$$ Letting $n\rightarrow\infty$ we obtain for all $v\in \bigcup_{n} H_n$, $$\label{part}
\<z, \psi(T)v\>_H -\<u_0, \psi(0)v\>_H
=\int_0^T \<w(t)+b(t), \psi(t)v\>_V dt
+ \int_0^T \<\psi'(t)v, u(t)\>_V dt.$$ Since $\bigcup_{n} H_n$ is dense in $V$, it’s easy to show that (\[part\]) hold for all $v\in V, \psi\in C^\infty([0,T])$.
If $\psi(T)=\psi(0)=0$, then we have $$\int_0^T \<w(t)+b(t), v\>_V \psi(t) dt
=- \int_0^T \<u(t), v\>_V\psi'(t) dt.$$ This implies that $u'=w+b, t\in(0,T)$. In particular, we have $u\in W^1_\alpha(0,T;V,H)$.
Then by the integration by parts formula we have $$\begin{split}
& \<u(T), \psi(T)v\>_H -\<u(0), \psi(0)v\>_H \\
=&\int_0^T \<u'(t), \psi(t)v\>_V dt
+ \int_0^T \<\psi'(t)v, u(t)\>_V dt \\
=&\int_0^T \<w(t)+b(t), \psi(t)v\>_V dt
+ \int_0^T \<\psi'(t)v, u(t)\>_V dt.
\end{split}$$ Hence by (\[part\]) we obtain $$\<u(T), \psi(T)v\>_H -\<u(0), \psi(0)v\>_H = \<z, \psi(T)v\>_H -\<u_0, \psi(0)v\>_H.$$ Then by choosing $\psi(T)=1, \psi(0)=0$ and $\psi(T)=0, \psi(0)=1$ respectively we obtain that $$u(T)=z, \ u(0)=u_0.$$ Hence the proof is complete.
Next lemma is very crucial for the proof of Theorem \[T1\]. The result basically says that $A$ is also a pseudo-monotone operator from $X$ to $X^*$, hence one can still use a modified monotonicity tricks to verify the limit of the Galerkin approximation as a solution to (\[1.1\]). The techniques used in the proof is inspired by the works of Hirano and Shioji [@H89; @S97].
\[L2.5\] Under the assumptions of Theorem \[T1\], suppose that $$\label{2.3}
\liminf_{n\rightarrow \infty} \int_0^T\<A(t,u_n(t)), u_n(t)\>_V d t \ge
\int_0^T \<w(t), u(t)\>_V d t,$$ then for any $v\in X$ we have $$\int_0^T\<A(t,u(t)), u(t)-v(t)\>_V d t \ge
\limsup_{n\rightarrow \infty}\int_0^T \<A(t,u_n(t)), u_n(t)-v(t)\>_V d t.$$
Since $W^1_\alpha(0,T;V,H)\subset C([0,T];H)$ is a continuous embedding, we have that $u_n(t)$ converges to $u(t)$ weakly in $H$ for all $t\in[0,T]$. Hence $u_n(t)$ also converges to $u(t)$ weakly in $V$ for all $t\in[0,T]$.
**Claim 1:** For all $t\in[0,T]$ we have $$\label{claim 1}
\limsup_{n\rightarrow\infty}\<A(t,u_n(t)),u_n(t)-u(t)\>_V \le 0 .$$
Suppose there exists a $t_0$ such that $$\limsup_{n\rightarrow\infty}\<A(t_0,u_n(t_0)),u_n(t_0)-u(t_0)\>_V > 0.$$ Then we can take a subsequence such that $$\lim_{i\rightarrow\infty}\<A(t_0,u_{n_i}(t_0)),u_{n_i}(t_0)-u(t_0)\>_V > 0.$$ Note that $u_{n_i}(t_0)$ converges to $u(t_0)$ weakly in $V$ and $A(t_0,\cdot)$ is pseudo-monotone, we have $$\<A(t_0,u(t_0)), u(t_0)-v\>_V \ge \limsup_{i\rightarrow\infty}\<A(t_0,u_{n_i}(t_0)),u_{n_i}(t_0)-v\>_V,
\ v\in V.$$ In particular, we have $$\limsup_{i\rightarrow\infty}\<A(t_0,u_{n_i}(t_0)),u_{n_i}(t_0)-u(t_0)\>_V\le 0,$$ which is a contradiction with the definition of this subsequence.
Hence (\[claim 1\]) holds.
By $(H3)$ and $(H4)$ there exists a constant $K$ such that $$\begin{split}
\<A(t,u_n(t)),u_n(t)-v(t)\>_V\le & -\frac{\delta}{2}\|u_n(t)\|_V^\alpha +K\left( f(t)
+\|u_n(t)\|_H^2 \right)\\
& +K\left(1+\|u_n(t)\|_H^{\alpha \beta} \right)\|v(t)\|_V^\alpha,\
v\in X.
\end{split}$$
Then by Lemma \[l2.3\], Fatou’s lemma, (\[2.3\]) and (\[claim 1\]) we have $$\begin{split}
0&\le \liminf_{n\rightarrow \infty}\int_0^T \<A(t,u_n(t)), u_n(t)-u(t)\>_V d t \\
&\le \limsup_{n\rightarrow \infty}\int_0^T \<A(t,u_n(t)), u_n(t)-u(t)\>_V d t \\
&\le \int_0^T \limsup_{n\rightarrow \infty} \<A(t,u_n(t)), u_n(t)-u(t)\>_V d t \le 0.
\end{split}$$
Hence $$\lim_{n\rightarrow \infty}\int_0^T \<A(t,u_n(t)), u_n(t)-u(t)\>_V d t = 0.$$
**Claim 2:** There exists a subsequence $\{u_{n_i}\}$ such that $$\label{claim 2}
\lim_{i\rightarrow \infty} \<A(t,u_{n_i}(t)), u_{n_i}(t)-u(t)\>_V = 0 \ \ \text{for}\ a.e. \ t\in[0,T].$$ Define $ g_n(t)= \<A(t,u_{n}(t)), u_{n}(t)-u(t)\>_V, \ t\in[0,T] $, then $$\lim_{n\rightarrow \infty}\int_0^T g_n(t) d t = 0, \ \ \limsup_{n\rightarrow\infty} g_n(t)\le 0,\ \ t\in[0,T].$$ Then by Lebesgue’s dominated convergence theorem we have $$\lim_{n\rightarrow \infty}\int_0^T g_n^+(t) d t = 0,$$ where $g_n^+(t):=max\{g_n(t), 0 \}$.
Note that $|g_n(t)|=2g_n^+(t)-g_n(t)$, hence we have $$\lim_{n\rightarrow \infty}\int_0^T |g_n(t)| d t = 0.$$ Therefore, we can take a subsequence $\{g_{n_i}(t)\}$ such that $$\lim_{i\rightarrow \infty} g_{n_i}(t) = 0 \ \ \text{for}\ a.e. \ t\in[0,T],$$ $i.e.$ (\[claim 2\]) holds.
Therefore, for any $v\in X$, we can choose a subsequence $\{u_{n_i}\}$ such that $$\lim_{i\rightarrow \infty}\int_0^T \<A(t,u_{n_i}(t)), u_{n_i}(t)-v(t)\>_V d t =
\limsup_{n\rightarrow \infty}\int_0^T \<A(t,u_n(t)), u_n(t)-v(t)\>_V d t;$$ $$\lim_{i\rightarrow \infty} \<A(t,u_{n_i}(t)), u_{n_i}(t)-u(t)\>_V = 0 \ \ \text{for}\ a.e. \ t\in[0,T].$$ Since $A$ is pseudo-monotone, we have $$\<A(t,u(t)), u(t)-v(t)\>_V \ge \limsup_{i\rightarrow \infty} \<A(t,u_{n_i}(t)), u_{n_i}(t)-v(t)\>_V,\ \ t\in[0,T].$$ By Fatou’s lemma we obtain $$\begin{split}
\int_0^T\<A(t,u(t)), u(t)-v(t)\>_V d t &\ge
\int_0^T\limsup_{i\rightarrow \infty} \<A(t,u_{n_i}(t)), u_{n_i}(t)-v(t)\>_V d t\\
&\ge \limsup_{i\rightarrow \infty}\int_0^T \<A(t,u_{n_i}(t)), u_{n_i}(t)-v(t)\>_V d t\\
&= \limsup_{n\rightarrow \infty}\int_0^T \<A(t,u_n(t)), u_n(t)-v(t)\>_V d t.
\end{split}$$ Hence the proof is complete.
**Proof of Theorem \[T1\]** (i) Existence: The integration by parts formula implies that $$\|u_n(T)\|_H^2-\|u_n(0)\|_H^2=2 \int_0^T \<A(t,u_n(t))+b(t), u_n(t)\>_V d t;$$ $$\|u(T)\|_H^2-\|u(0)\|_H^2=2 \int_0^T \<w(t)+b(t), u(t)\>_V d t.$$ Since $u_n(T)\rightharpoonup z$ in $H$, by the lower semicontinuity of $\|\cdot\|_H$ we have $$\liminf_{n\rightarrow \infty} \|u_n(T)\|_H^2\ge \|z\|_H^2=\|u(T)\|_H^2.$$ Hence we have $$\begin{split}
\liminf_{n\rightarrow \infty} \int_0^T\<A(t,u_n(t)), u_n(t)\>_V d t &\ge \frac{1}{2}
\left( \|u(T)\|_H^2 -\|u(0)\|_H^2 \right)-\int_0^T\<b(t),u(t)\>_V dt\\
&=\int_0^T \<w(t), u(t)\>_V d t.
\end{split}$$
By Lemma \[L2.5\] we have for any $v\in X$, $$\begin{split}
\int_0^T\<A(t,u(t)), u(t)-v(t)\>_V d t
&\ge\limsup_{n\rightarrow \infty}\int_0^T \<A(t,u_n(t)), u_n(t)-v(t)\>_V d t\\
&\ge\liminf_{n\rightarrow \infty}\int_0^T \<A(t,u_n(t)), u_n(t)-v(t)\>_V d t\\
&\ge \int_0^T \<w(t), u(t)\>_V d t- \int_0^T \<w(t), v(t)\>_V d t\\
&=\int_0^T \<w(t), u(t)-v(t)\>_V d t.
\end{split}$$ Since $v\in X$ is arbitrary, we have $A(\cdot,u)=w$ as the element in $X^*$.
Hence $u$ is a solution to (\[1.1\]).
\(ii) Uniqueness: Suppose $u(\cdot,u_0),v(\cdot,v_0)$ are the solutions to (\[1.1\]) with starting points $u_0,v_0$ respectively, then by the integration by parts formula we have for $t\in[0, T]$, $$\begin{split}
\|u(t)-v(t)\|_H^2&=\|u_0-v_0\|_H^2+ 2\int_0^t\< A(s,u(s))-A(s,v(s)),u(s)-v(s)\>_V ds\\
&\le \|u_0-v_0\|_H^2+ 2\int_0^t \left(C+\rho(u(s))+ \eta(v(s)) \right) \|u(s)-v(s)\|_H^2 ds.
\end{split}$$ By (\[c3\]) we know that $$\int_0^T \left(C+\rho(u(s))+ \eta(v(s)) \right) d s <\infty.$$ Then by Gronwall’s lemma we obtain $$\label{estimate of difference}
\|u(t)-v(t)\|_H^2\le\|u_0-v_0\|_H^2 \exp\left[2\int_0^t \left(C+\rho(u(s))+ \eta(v(s)) \right) d s \right], \ t\in[0, T].$$ In particular, if $u_0=v_0$, this implies the uniqueness of the solution of $(\ref{1.1})$.
Proof of Theorem \[T2\]
-----------------------
By $(H2)$ we have $$\begin{split}
\|u_1(t)-u_2(t)\|_H^2&=\|u_{1,0}-u_{2,0}\|_H^2+ 2\int_0^t\< A(s,u_1(s))-A(s,u_2(s)),u_1(s)-u_2(s)\>_V ds\\
& \ \ + 2\int_0^t\< b_1(s)-b_2(s),u_1(s)-u_2(s)\>_V ds\\
&\le \|u_{1,0}-u_{2,0}\|_H^2+ \int_0^t \| b_1(s)-b_2(s)\|_H^2 d s\\
&\ \ +\int_0^t \left(C+\rho(u_1(s))+ \eta(u_2(s)) \right) \|u_1(s)-u_2(s)\|_H^2 ds, \ t\in[0, T],
\end{split}$$ where $C$ is a constant.
Then by Gronwall’s lemma we have $$\begin{split}
\|u_1(t)-u_2(t)\|_H^2
\le & \exp\left[\int_0^t \left(C+\rho(u_1(s))+ \eta(u_2(s))\right) d s \right]\\
& \cdot \left( \|u_{1,0}-u_{2,0}\|_H^2 +\int_0^t
\|b_1(s)-b_2(s)\|_H^2 ds \right), \ t\in[0, T].
\end{split}$$
Application to examples
=======================
It’s obvious that the main results can be applied to nonlinear evolution equations with monotone operators ($e.g.$ porous medium equation, $p$-Laplace equation) perturbated by some non-monotone terms ($e.g.$ some locally Lipschitz perturbation). Moreover, we also formulate some examples where the coefficients are only locally monotone. For simplicity here we only formulate the examples where the coefficients are time independent, but one can easily adapt all these examples to the time dependent case.
Here we use the notation $D_i$ to denote the spatial derivative $\frac{\partial}{\partial x_i}$, $\Lambda \subseteq \mathbb{R}^d$ is an open bounded domain with smooth boundary. For standard Sobolev space $W_0^{1,p}(\Lambda)$ $(p\ge 2)$ we always use the following (equivalent) Sobolev norm: $$\|u\|_{1,p}:=\left(\int_\Lambda |\nabla u(x)|^p d x\right)^{\frac{1}{p}}.$$
As preparation we first give a lemma for verifying $(H2)$.
\[L3.1\] Consider the Gelfand triple $$V:=W_0^{1,2}(\Lambda)\subseteq H:=L^2(\Lambda) \subseteq W^{-1,2}(\Lambda)$$ and the operator $$A(u)=\Delta u+ \sum_{i=1}^d f_i(u)D_i u,$$ where $f_i$ ($i=1,\cdots,d$) are Lipschitz functions on $\mathbb{R}$.
\(1) If $d<3$, then there exists a constant $K$ such that $$2 \<A(u)-A(v), u-v\>_V
\le - \|u-v\|_V^2+ \left(K +K\|u\|_{L^4}^4 + K\|v\|_V^2 \right)\|u-v\|_H^2,\ u,v\in V.$$ In particular, if $f_i$ are bounded functions for $i=1,\cdots,d$, then we have $$2 \<A(u)-A(v), u-v\>_V
\le - \|u-v\|_V^2+ \left(K + K\|v\|_V^2 \right)\|u-v\|_H^2,\ u,v\in V.$$
\(2) For $d=3$ we have $$2 \<A(u)-A(v), u-v\>_V
\le - \|u-v\|_V^2+ \left(K +K\|u\|_{L^4}^8+ K\|v\|_V^4 \right)\|u-v\|_H^2,\ u,v\in V.$$ In particular, if $f_i$ are bounded functions for $i=1,\cdots,d$, we have $$2 \<A(u)-A(v), u-v\>_V
\le - \|u-v\|_V^2+ \left(K + K\|v\|_V^4 \right)\|u-v\|_H^2,\ u,v\in V.$$
\(3) If $f_i$ are bounded measurable functions on $\Lambda$ and independent of $u$ for $i=1,\cdots,d$, i.e. $$A(u)=\Delta u+ \sum_{i=1}^d f_i\cdot D_i u,$$ then for any $d\ge 1$ we have $$2 \<A(u)-A(v), u-v\>_V
\le - \|u-v\|_V^2+ K\|u-v\|_H^2,\ u,v\in V.$$
$(1)$ Since all $f_i$ are Lipschitz and linear growth, we have $$\begin{aligned}
& ~~~~ \<A(u)-A(v), u-v\>_V \\
&= - \|u-v\|_V^2+ \sum_{i=1}^d \int_\Lambda \left(f_i(u)D_i u-f_i(v)D_i v\right)\left(u-v\right) d
x\\
&= - \|u-v\|_V^2+ \sum_{i=1}^d \int_\Lambda \left(f_i(u)(D_i u-D_i
v) +D_iv( f_i(u) -f_i(v))\right)\left(u-v\right) dx\\
&\le - \|u-v\|_V^2+ \sum_{i=1}^d \bigg[ \left( \int_\Lambda (D_i
u-D_i v)^2 d x\right)^{1/2} \left( \int_\Lambda f_i^2(u)(u-v)^2 dx
\right)^{1/2}\\
&~~ +\left( \int_\Lambda (D_i v)^2 d x\right)^{1/2} \left(
\int_\Lambda \left(f_i(u)-f_i(v)\right)^2
(u-v)^2 dx \right)^{1/2} \bigg] \\
&\le - \|u-v\|_V^2+ K \|u-v\|_V \left(\int_\Lambda
\left(1+ u^4\right) dx \right)^{1/4} \left(\int_\Lambda
(u-v)^4 dx \right)^{1/4} + K \| v\|_V \left(\int_\Lambda
(u-v)^4 dx \right)^{1/2} \\
&\le - \|u-v\|_V^2+ K \|u-v\|_V^{3/2} \|u-v\|_H^{1/2}\left(1+\|u\|_{L^4} \right) + 2K \| v\|_V \|u-v\|_V
\|u-v\|_H \\
&\le - \frac{1}{2}\|u-v\|_V^2+ \left(K +K\|v\|_V^2+K\|u\|_{L^4}^4
\right)\|u-v\|_H^2, \ u,v\in V,\end{aligned}$$ where $K$ is a constant that may change from line to line, and we also used the following well known estimate on $\mathbb{R}^2$ (see [@MS02 Lemma 2.1]) $$\label{e3}
\|u\|_{L^4}^4 \le 2 \|u\|_{L^2}^2 \|\nabla u\|_{L^2}^2, \ u\in W_0^{1,2}(\Lambda) .$$
$(2)$ For $d=3$ we use the following estimate (cf. [@MS02]) $$\label{e4}
\|u\|_{L^4}^4 \le 4 \|u\|_{L^2} \|\nabla u\|_{L^2}^3, \ u\in W_0^{1,2}(\Lambda),$$ then the second assertion can be derived similarly by using Young’s inequality.
$(3)$ This assertion obviously follows from the estimates in $(i)$.
\[R2\] (1) If all $f_i$ are bounded, then the local monotonicity $(H2)$ also implies the coercivity $(H3)$.
\(2) If we write the operator in the following vector form $$A(u)=\Delta u+ \nabla \cdot \vec{F}(u),$$ where $\vec{F}(x)=(F_1(x),\cdots,F_d(x)): \mathbb{R}\rightarrow \mathbb{R}^d$ satisfies $$|\vec{F}(x)-\vec{F}(y)|\le C(1+|x|+|y|)|x-y|, \ x,y\in \mathbb{R}.$$ Then by using the divergence theorem (or Stokes’ theorem) one can show that for $d<3$, $$2 \<A(u)-A(v), u-v\>_V
\le - \|u-v\|_V^2+ \left(K +K\|u\|_{L^4}^4 + K\|v\|_{L^4}^4 \right)\|u-v\|_H^2,\ u,v\in V,$$ for $d=3$ we have $$2 \<A(u)-A(v), u-v\>_V
\le - \|u-v\|_V^2+ \left(K +K\|u\|_{L^4}^8 + K\|v\|_{L^4}^8 \right)\|u-v\|_H^2,\ u,v\in V.$$ And it’s also easy to show the coercivity $(H3)$ holds since we have $$\<\nabla \cdot \vec{F}(u), u\>_V=-\int_\Lambda \vec{F}(u)\cdot \nabla u dx=0,
\ u\in W_0^{1,2}(\Lambda).$$
The first example is a general semilinear equation on $\Lambda \subseteq \mathbb{R}$, which unifies the classical reaction-diffusion equation and Burgers equation.
\[E0\] Consider the following equation $$\label{Burges1}
u'= \frac{\partial^2 u}{\partial x^2} + \frac{\partial F}{\partial x}(u)+g(u)+h, \ u(0)=u_0\in L^2(\Lambda).$$ Suppose the following conditions hold for some constant $C>0$:
\(i) $F$ is a function on $\mathbb{R}$ satisfies $$\ |F(x)-F(y)| \le C(1+|x|+|y|)|x-y|, \ x,y\in \mathbb{R}.$$
\(ii) $g$ is a continuous function on $\mathbb{R}$ such that $$\label{c0}
\begin{split}
g(x)x &\le C(x^2+1), \ x\in \mathbb{R};\\
|g(x)| & \le C(|x|^3+1), \ x\in \mathbb{R};\\
(g(x)-g(y))(x-y)&\le C(1+|x|^t+|y|^t)(x-y)^2, \ x,y\in \mathbb{R},
\end{split}$$ where $t\ge 1$ is a constant.
\(iii) $h\in W^{-1,2}(\Lambda)$.
Then $(\ref{Burges1})$ has a solution $u\in W^1_2(0,T; W_0^{1,2}(\Lambda), L^2(\Lambda))$. Moreover, if $t\le 2$, then the solution of $(\ref{Burges1})$ is also unique.
We define the Gelfand triple $$V:=W_0^{1,2}(\Lambda)\subseteq H:=L^2(\Lambda) \subseteq W^{-1,2}(\Lambda)$$ and the operator $$A(u)=\frac{\partial^2 u}{\partial x^2} + \frac{\partial F}{\partial x}(u)+g(u), \ u\in V.$$ It is easy to show that $(H1)$ holds by the continuity of $F$ and $g$.
Similar to Lemma \[L3.1\], one can easily show that $$\begin{split}
& \<\frac{\partial F}{\partial x}(u)- \frac{\partial F}{\partial x}(v), u-v\>_V \\
=& -\int_\Lambda \( F(u)-F(v) \)
\left(\frac{\partial u}{\partial x}-\frac{\partial v}{\partial x} \right) dx \\
\le & \frac{1}{4}\|u-v\|_V^2+C\left( 1+\|u\|_{L^4}^4+\|v\|_{L^4}^4
\right
) \|u-v\|_H^2, \ u,v\in V.
\end{split}$$ By integration by parts formula we have $$\<\frac{\partial F}{\partial x}(u), u\>_V=0, \ u\in V.$$
(\[c0\]) and (\[e3\]) implies that $$\begin{split}
& \<g(u)-g(v),u-v\>_V \\
\le & C\left(1+\|u\|_{L^{2t}}^t+ \|v\|_{L^{2t}}^t \right)\|u-v\|_{L^4}^2\\
\le & \frac{1}{4}\|u-v\|_V^2+C\left(
1+\|u\|_{L^{2t}}^{2t}+\|v\|_{L^{2t}}^{2t} \right) \|u-v\|_H^2, \
u,v\in V.
\end{split}$$ Therefore, we have $$2 \<A(u)-A(v), u-v\>_V
\le - \|u-v\|_V^2+ C \left(1 + \|u\|_{L^4}^4 +\|u\|_{L^{2t}}^{2t} +
\|v\|_{L^4}^4 +\|v\|_{L^{2t}}^{2t} \right)\|u-v\|_H^2,\ u,v\in V,$$ $i.e.$ $(H2)$ holds.
Note that by (\[c0\]) we have $$\<g(u),u\>_V \le C\left(1+\|u\|_{H}^2 \right), \ u\in V,$$ hence $(H3)$ holds with $\alpha=2$.
By the Sobolev embedding theorem we have $$\|g(u)\|_{V^*}\le C\left(1+\|u\|_{L^3}^3\right)\le C\left(1+\|u\|_V\|u\|_{H}^{2}\right), \ u\in V,$$ $$\|\frac{\partial F}{\partial x}(u)\|_{V^*}\le \|F(u)\|_{H} \le
C\left(1+\|u\|_{L^4}^2\right)\le C\left(1+\|u\|_V\|u\|_{H}\right), \
u\in V.$$ Hence $(H4)$ also holds (with $\beta=2$).
Therefore, the assertions follow from Theorem \[T1\].
If we take $F(x)=x^2$ and $g=h=0$, then (\[Burges1\]) is the classical Burgers equation in fluid mechanics. If we take $g(x)=x-x^3$ and $F=h=0$, then (\[Burges1\]) is the well known reaction-diffusion equation.
\[E1\] Consider the following equation $$\label{Burges}
u'=\Delta u+ \sum_{i=1}^d f_i(u)D_i u+g(u)+h, \ u(0)=u_0\in L^2(\Lambda).$$ Suppose the following conditions hold for some constant $C>0$:
\(i) $f_i$ are bounded Lipschitz functions on $\mathbb{R}$ for $i=1,\cdots, d$;
\(ii) $g$ is a continuous function on $\mathbb{R}$ such that $$\label{c1}
\begin{split}
g(x)x &\le C(x^2+1), \ x\in \mathbb{R};\\
|g(x)| & \le C(|x|^r+1), \ x\in \mathbb{R};\\
(g(x)-g(y))(x-y)&\le C(1+|x|^t+|y|^t)(x-y)^2, \ x,y\in \mathbb{R},
\end{split}$$ where $r,t\ge 1$ are some constants.
\(iii) $h\in W^{-1,2}(\Lambda)$.
Then we have
\(1) if $d=2$, $r=\frac{7}{3}$ and $t=2$, $(\ref{Burges})$ has a unique solution $u\in W^1_2(0,T; W_0^{1,2}(\Lambda), L^2(\Lambda))$.
\(2) if $d=3$, $r=\frac{7}{3}$ and $t\le 3$, $(\ref{Burges})$ has a solution $u\in W^1_2(0,T; W_0^{1,2}(\Lambda), L^2(\Lambda))$. Moreover, if $t=\frac{4}{3}$, $f_i, i=1,2,3$ are bounded measurable functions on $\Lambda$ and independent of $u$, then the solution of $(\ref{Burges})$ is also unique.
We define the Gelfand triple $$V:=W_0^{1,2}(\Lambda)\subseteq H:=L^2(\Lambda) \subseteq W^{-1,2}(\Lambda)$$ and the operator $$A(u)=\Delta u+ \sum_{i=1}^d f_i(u)D_i u+g(u), \ u\in V.$$ By assumption (\[c1\]) we have $$\<g(u)-g(v),u-v\>_V
\le C\left(1+\|u\|_{L^{2t}}^t+ \|v\|_{L^{2t}}^t \right)\|u-v\|_{L^4}^2\\.$$ Then from (\[e3\]) or (\[e4\]) and Lemma \[L3.1\] we have for $d=2$ $$2 \<A(u)-A(v), u-v\>_V
\le - \|u-v\|_V^2+ K\left(1 +\|v\|_V^2 + \|u\|_{L^{2t}}^{2t}+ \|v\|_{L^{2t}}^{2t} \right)\|u-v\|_H^2,\ u,v\in V,$$ and for $d=3$, $$2 \<A(u)-A(v), u-v\>_V
\le - \|u-v\|_V^2+
K\left(1 +\|v\|_V^4 + \|u\|_{L^{2t}}^{4t}+ \|v\|_{L^{2t}}^{4t} \right)\|u-v\|_H^2,\ u,v\in V,$$ i.e. $(H2)$ holds.
Note that $$\<g(u),u\>_V \le C\left(1+\|u\|_{H}^2 \right), \ u\in V.$$ Then by Lemma \[L3.1\] and Remark \[R2\] we know that $(H3)$ holds with $\alpha=2$.
For $d=2,3$ we have $$\|g(u)\|_{V^*}\le C\left(1+\|u\|_{L^{6r/5}}^{r}\right), \ u\in V.$$ For $r= \frac{7}{3}$, by the interpolation theorem we have $$\|u\|_{L^{6r/5}}\le \|u\|_{L^2}^{4/7} \|u\|_{L^6}^{3/7}, \ u\in W_0^{1,2}(\Lambda) \subseteq L^6(\Lambda).$$ Then $$\|g(u)\|_{V^*}\le C\left(1+\|u\|_{L^{6r/5}}^{r}\right)\le C\left(1+
\|u\|_H^{4/3} \|u\|_{V} \right), \ u\in V.$$ Hence $(H4)$ holds.
The hemicontinuity $(H1)$ follows easily from the continuity of $f$ and $g$.
Therefore, all assertions follow from Theorem \[T1\].
In particular, if $d=3$ and $f_i, i=1,2,3$ are bounded measurable functions on $\Lambda$ and independent of $u$, then we have $$2 \<A(u)-A(v), u-v\>_V
\le - \|u-v\|_V^2+
K\left(1 + \|u\|_{L^{2t}}^{4t}+ \|v\|_{L^{2t}}^{4t}
\right)\|u-v\|_H^2,\ u,v\in V.$$ Since $t=\frac{4}{3}$, by the interpolation inequality we have $$\|u\|_{L^{2t}} \le \|u\|_{L^2}^{5/8} \|u\|_{L^{6}}^{3/8}, \ u\in V.$$ Therefore $$\|u\|_{L^{2t}}^{4t} \le C \|u\|_{H}^{10/3} \|u\|_{V}^{2}, \ u\in V.$$ Hence the solution of of $(\ref{Burges})$ is unique.
\(1) As we mentioned in Remark 1.1, the classical result for monotone operators can not be applied to the above example. The typical example of monotone perturbation is to assume all $f_i$ are independent of unknown solution $u$, $g$ is monotone ($e.g.$ Lipschitz) and has linear growth ($r=1$). However, here we allow $g$ is locally monotone ($e.g.$ locally Lipschitz) and has certain polynomial growth ($r>1$).
\(2) The boundedness of $f_i$ is only assumed in order to verify the coercivity $(H3)$. This assumption can be removed if we formulate (\[Burges\]) in vector form as explained in Remark \[R2\].
We may also consider the following quasi-linear evolution equations on $\mathbb{R}^d\ (d\ge 3)$.
Consider the Gelfand triple $$V:=W_0^{1,p}(\Lambda)\subseteq H:=L^2(\Lambda) \subseteq W^{-1,q}(\Lambda)$$ and the following equation on $\mathbb{R}^d$ for $p> 2$ $$\label{p-Laplace}
u'=\sum_{i=1}^d D_i\left(|D_iu|^{p-2} D_i u \right) +g(u)+h, \ u(0)=u_0\in L^2(\Lambda).$$ Suppose the following conditions hold:
\(i) $g$ is a continuous function on $\mathbb{R}$ such that $$\begin{split}\label{c2}
g(x)x &\le C(|x|^{\frac{p}{2}+1}+1), \ x\in \mathbb{R};\\
|g(x)| & \le C(|x|^{r}+1), \ x\in \mathbb{R};\\
(g(x)-g(y))(x-y)&\le C(1+|x|^t+|y|^t)|x-y|^{s}, \ x,y\in \mathbb{R},
\end{split}$$ where $C>0$ and $r,s,t\ge 1$ are some constants.
\(ii) $h\in W^{-1,q}(\Lambda)$, $p^{-1}+q^{-1}=1$.
Then we have
\(1) if $d<p$, $s=2$ and $r= p+1$, $(\ref{p-Laplace})$ has a solution. Moreover, if $t\le p$ also holds, then the solution is unique.
\(2) if $d>p$, $2<s<p$, $r=\frac{2p}{d}+p-1$ and $t\le \frac{p^2(s-2)}{(d-p)(p-2)}$, $(\ref{p-Laplace})$ has a solution. The solution is unique if $t\le \frac{p(p-s)}{p-2}$ also holds.
\(1) It’s well known that $\sum_{i=1}^d D_i\left(|D_iu|^{p-2} D_i u\right) $ satisfy $(H1)$-$(H4)$ (cf. [@L08; @L08b]). In particular, there exists a constant $\delta>0$ such that $$\label{e7}
\sum_{i=1}^d\< D_i\left(|D_iu|^{p-2} D_i u\right)- D_i\left(|D_iv|^{p-2} D_i v\right) , u-v \>_V
\le - \delta \|u-v\|_V^p, \ u,v\in W_0^{1,p}(\Lambda).$$
Recall that for $d<p$ we have the following Sobolev embedding $$W_0^{1,p}(\Lambda) \subseteq L^{\infty}(\Lambda).$$ Hence we have $$\begin{split}
\<g(u)-g(v),u-v\>_V & \le C \int_\Lambda \left(1+|u|^t+ |v|^t \right) |u-v|^2 d x\\
&\le C \left( 1+ \|u\|_{L^{\infty}}^t +\|v\|_{L^{\infty}}^t \right) \|u-v\|^2_{L^{2}}\\
&\le C \left( 1+ \|u\|_{V}^t +\|v\|_{V}^t \right) \|u-v\|_{H}^{2}, \ u,v\in V,
\end{split}$$ where $C$ is a constant may change from line to line.
Hence $(H2)$ holds.
Note that from (\[c2\]) we have $$\begin{split}
\<g(u), u\>_V & \le C \int_\Lambda (1+ |u|^{\frac{p}{2}+1})dx\\
& \le C\left(1+\|u\|_{L^\infty}^{p/2}\|u\|_H\right)\\
& \le \frac{\delta}{2} \|u\|_V^p + C\left(1+\|u\|_H^2\right), \ u\in V.
\end{split}$$ Therefore, $(H3)$ holds with $\alpha=p$ by (\[e7\]).
$(H4)$ follows from the following estimate: $$\|g(u)\|_{V^*} \le C\left(1+\|u\|_{L^{p+1}}^{p+1}\right)\le C\left(1+\|u\|_{L^{\infty}}^{p-1}\|u\|_{H}^2\right)
, \ u\in V.$$
Hence the assertions follow from Theorem \[T1\].
\(2) Note that for $d>p$ we have the following Sobolev embedding $$W_0^{1,p}(\Lambda) \subseteq L^{p_0}(\Lambda), \ p_0=\frac{dp}{d-p}.$$ Let $t_0=\frac{p(s-2)}{s(p-2)}\in(0,1)$ and $p_1\in(2,p_0)$ such that $$\frac{1}{p_1}=\frac{1-t_0}{2}+ \frac{t_0}{p_0}.$$ Then we have the following interpolation inequality $$\|u\|_{L^{p_1}}\le \|u\|_{L^{2}}^{1-t_0} \|u\|_{L^{p_0}}^{t_0}, \ u\in W_0^{1,p}(\Lambda).$$ Since $2<s<p$, it is easy to show that $s<p_1$.
Let $p_2=\frac{p_1}{p_1-s}$, then by assumption (\[c2\]) we have $$\label{e8}
\begin{split}
\<g(u)-g(v),u-v\>_V & \le C \int_\Lambda \left(1+|u|^t+ |v|^t \right) |u-v|^s d x\\
&\le C \left( 1+ \|u\|_{L^{tp_2}}^t +\|v\|_{L^{tp_2}}^t \right) \|u-v\|^s_{L^{p_1}}\\
&\le C \left( 1+ \|u\|_{L^{tp_2}}^t +\|v\|_{L^{tp_2}}^t \right) \|u-v\|_{L^{2}}^{s(1-t_0)} \|u-v\|_{L^{p_0}}^{st_0}\\
&\le \varepsilon \|u-v\|_{L^{p_0}}^{p} +
C_\varepsilon \left( 1+ \|u\|_{L^{tp_2}}^{tb} +\|v\|_{L^{tp_2}}^{tb} \right) \|u-v\|_{L^{2}}^{2}
,
\end{split}$$ where $\varepsilon, C_\varepsilon$ are some constants and the last step follows from the following Young inequality $$xy\le \varepsilon x^a +C_\varepsilon y^b, \ x,y\in\mathbb{R},\ a=\frac{p-2}{s-2},\ b=\frac{p-2}{p-s}.$$ By calculation we have $$\frac{s}{p_1}=\frac{p-s}{p-2}+\frac{p(s-2)}{p_0(p-2)},\ p_2=\frac{p_0(p-2)}{(p_0-p)(s-2)}.$$ Hence if $t\le \frac{(p_0-p)(s-2)}{p-2}$, then $$\|u\|_{L^{tp_2}}\le C \|u\|_{L^{p_0}} \le C \|u\|_{V}, \ v\in V.$$ Therefore, $(H2)$ follows from (\[e7\]) and (\[e8\]).
$(H3)$ can be verified for $\alpha=p$ in a similar way.
For $r=\frac{2p}{d}+p-1$, by the interpolation inequality we have $$\|g(u)\|_{V^*}\le C\left(1+ \|u\|_{L^{rp_0^\prime}}^r \right)
\le C\left( 1+\|u\|_{p_0}^{p-1}\|u\|_H^{\beta} \right), \ u\in V,$$ where $$\frac{1}{p_0}+\frac{1}{p_0^\prime},\ \ \beta=\frac{2p}{d}.$$ Therefore, $(H4)$ also holds.
Then all assertions follow from Theorem \[T1\].
One further generalization is to replace $\sum_{i=1}^d
D_i\left(|D_iu|^{p-2} D_i u\right)$ by more general quasi-linear differential operator $$\sum_{|\alpha|\le m} (-1)^{|\alpha|}D_\alpha A_\alpha(x,Du(x,t);t),$$ where $Du=(D_\beta u)_{|\beta|\le m}$. Under certain assumptions (cf. [@Z90 Proposition 30.10]) this operator satisfies the monotonicity and coercivity condition.
According to Theorem \[T1\], we can obtain the existence and uniqueness of solutions to this type of quasi-linear PDE with some non-monotone perturbations ($e.g.$ some locally Lipschitz lower order terms).
Now we apply Theorem \[T1\] to the Navier-Stokes equation.
Let $\Lambda$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary. It’s well known that by means of divergence free Hilbert spaces $V,H$ and the Helmhotz-Leray orthogonal projection $P_H$, the classical form of the Navier-Stokes equation can be formulated in the following form: $$\label{NSE}
u'=Au+B(u)+f,\ u(0)=u_0\in H,$$ where $$V=\left\{ v\in W_0^{1,2}(\Lambda,\mathbb{R}^2): \nabla \cdot v=0 \ a.e.\ \text{in} \ \Lambda \right\}, \
\|v\|_V:=\left(\int_\Lambda |\nabla v|^2 dx \right)^{1/2},$$ and $H$ is the closure of $V$ in the following norm $$\|v\|_H:=\left(\int_\Lambda | v|^2 dx \right)^{1/2}.$$ The linear operator $P_H$ (Helmhotz-Leray projection) and $A$ (Stokes operator with viscosity constant $\nu$) are defined by $$P_H: L^2(\Lambda, \mathbb{R}^2)\rightarrow H,\ \text{ orthogonal projection};$$ $$A:=W^{2,2}(\Lambda, \mathbb{R}^2)\cap V\rightarrow H, \ Au=\nu P_H \Delta u,$$ and the nonlinear operator $$B: \mathcal{D}_B\subset H\times V\rightarrow H, \ B(u,v)=- P_H\left[(u \cdot \nabla) v\right], B(u)=B(u,u).$$
It’s well known that by using the Gelfand triple $$V\subseteq H\equiv H^*\subseteq V^*$$ the following mappings $$A: V\rightarrow V^*, \ B: V\times V\rightarrow V^*$$ are well defined. In particular, we have $$\<B(u,v),w\>_V=-\<B(u,w),v\>_V, \ \<B(u,v),v\>_V=0,\ u,v,w\in V.$$
(2D Navier-Stokes equation) For $f\in L^2(0,T;V^*)$ and $u_0\in H$, $(\ref{NSE})$ has a unique solution.
The hemicontinuity $(H1)$ is easy to show since $B$ is a bilinear map.
Note that $ \<B(v),v\>_V=0$, it’s also easy to get the coercivity $(H3)$ $$\<Av+B(v)+f,v\>_V\le -\nu\|v\|_V^2+\|f\|_{V^*}\|v\|_V \le
-\frac{\nu}{2}\|v\|_V^2+C\|f\|_{V^*}^2, \ v\in V.$$ Recall the following estimate (cf. [@MS02 Lemma 2.1, 2.2]) $$\label{e2}
\begin{split}
|\<B(w),v\>_V| &\le 2 \|w\|_{L^4(\Lambda;\mathbb{R}^2)}\|v\|_V; \\
|\<B(w),v\>_V| &\le 2 \|w\|_V^{3/2} \|w\|_H^{1/2} \|v\|_{L^4(\Lambda;\mathbb{R}^2)}, v,w\in V. \\
\end{split}$$ Then we have $$\begin{split}
\<B(u)-B(v),u-v\>_V &=- \<B(u,u-v),v\>_V+ \<B(v,u-v),v\>_V \\
&= - \<B(u-v),v\>_V \\
&\le 2 \|u-v\|_V^{3/2} \|u-v\|_H^{1/2} \|v\|_{L^4(\Lambda;\mathbb{R}^2)} \\
& \le \frac{\nu}{2} \|u-v\|_V^{2} + \frac{32}{\nu^3} \|v\|_{L^4(\Lambda;\mathbb{R}^2)}^4 \|u-v\|_H^{2},
\ u,v\in V.
\end{split}$$ Hence we have the local monotonicity $(H2)$ $$\<Au+B(u)-Av-B(v),u-v\>_V
\le -\frac{\nu}{2} \|u-v\|_V^{2} + \frac{32}{\nu^3} \|v\|_{L^4(\Lambda;\mathbb{R}^2)}^4 \|u-v\|_H^{2}.$$
The growth $(H4)$ follows from (\[e2\]) and (\[e3\]).
Hence the existence of solution to (\[NSE\]) follows from Theorem \[T1\].
By definition any solution $u$ of (\[NSE\]) is a element in $L^2([0,T];V)$ and $C([0,T];H)$, then (\[e3\]) implies that $$\int_0^T \|u(t)\|_{L^4}^4 d t \le 2\sup_{t\in[0,T]}\|u(t)\|_H^2 \int_0^T \|u(t)\|_{V}^2 d t < \infty.$$ Hence the solution of (\[NSE\]) is also unique.
\(1) The main result can be also applied to some other classes of two dimensional hydrodynamical models such as magneto-hydrodynamic equations, the Boussinesq model for the Bénard convection and 2D magnetic Bénard problem. We refer to [@CM10] (and the references therein) for the details of these models. Note that the assumption $(C1)$ in [@CM10] implies a special type of local monotonicity (e.g. see (2.8) in [@CM10]).
\(2) For the 3D Navier-Stokes equation, we recall the following estimate (cf. [@MS02 (2.5)]) $$\|\psi\|_{L^4}^4\le 4 \|\psi\|_{L^2} \|\nabla \psi\|_{L^2}^3, \ \psi\in W_0^{1,2}(\Lambda; \mathbb{R}^3).$$ Then one can show that $$\begin{split}
\<B(u)-B(v),u-v\>_V &= - \<B(u-v),v\>_V \\
&\le 2 \|u-v\|_V^{7/4} \|u-v\|_H^{1/4} \|v\|_{L^4(\Lambda;\mathbb{R}^3)} \\
& \le \frac{\nu}{2} \|u-v\|_V^{2} + \frac{2^{12}}{\nu^7} \|v\|_{L^4(\Lambda;\mathbb{R}^3)}^8 \|u-v\|_H^{2},
\ u,v\in V.
\end{split}$$ Hence we have the following local monotonicity $(H2)$ $$\<Au+B(u)-Av-B(v),u-v\>_V
\le -\frac{\nu}{2} \|u-v\|_V^{2} + \frac{2^{12}}{\nu^7} \|v\|_{L^4(\Lambda;\mathbb{R}^3)}^8 \|u-v\|_H^{2}.$$ However, we only have the following growth condition in the 3D case: $$\|B(u)\|_{V^*}\le 2 \|u\|_{L^4(\Lambda; \mathbb{R}^3)}^2\le 4\|u\|_H^{1/2}\|u\|_V^{3/2}, \ u\in V.$$ Unfortunately, this is not enough to verify $(H4)$ in Theorem \[T1\].
Now we apply the main result to the 3D Leray-$\alpha$ model of turbulence, which is a regularization of the 3D Navier-Stokes equation. It was first considered by Leray [@L34] in order to prove the existence of a solution to the Navier-Stokes equation in $\mathbb{R}^3$. Here we use a special smoothing kernel in the 3D Leray-$\alpha$ model, which was first considered in [@CH05] (cf. [@CTV07] for more references). It has been shown in [@CH05] that the 3D Leray-$\alpha$ model compares successfully with empirical data from turbulent channel and pipe flows for a wide range of Reynolds numbers. This model has a great potential to become a good sub-grid-scale large-eddy simulation model of turbulence. The Leray-$\alpha$ model can be formulated as follows: $$\begin{split}\label{3D}
& u^\prime=\nu \Delta u-(v\cdot \nabla)u-\nabla p+f, \\
& \nabla\cdot u=0,\ u=v-\alpha^2 \Delta v
\end{split}$$ where $\nu>0$ is the viscosity, $u$ is the velocity, $p$ is the pressure and $f$ is a given body-forcing term.
By using the same divergence free Hilbert spaces $V, H$ (but in 3D) one can rewrite the Leray-$\alpha$ model into the following abstract form: $$\label{Leray}
u^\prime=Au+B(u,u)+f, \ u(0)=u_0\in H,$$ where $$Au=\nu P_H\Delta u, B(u,v)=-P_H\left[\left( \left(I-\alpha^2\Delta\right)^{-1}u \cdot \nabla \right)v \right].$$
(3D Leray-$\alpha$ model) For $f\in L^2(0,T;V^*)$ and $u_0\in H$, $(\ref{Leray})$ has a unique solution.
$(H1)$ holds obviously since $B$ is a bilinear map.
Note that $ \<B(u,v),v\>_V=0$, it’s also easy to get the coercivity $(H3)$: $$\<Av+B(v,v)+f,v\>_V\le -\nu\|v\|_V^2+\|f\|_{V^*}\|v\|_V \le
-\frac{\nu}{2}\|v\|_V^2+C\|f\|_{V^*}^2, \ v\in V.$$ Recall the following well-known estimate (cf. [@MS02 Lemma 2.1, 2.2]) $$\label{e5}
\begin{split}
& |\<B(u,v),w\>_V|\\
\le& c \|(I-\alpha^2\Delta)^{-1}u\|_{H}^{1/4} \|(I-\alpha^2\Delta)^{-1}u\|_{V}^{3/4}
\|v\|_{H}^{1/4} \|v \|_{V}^{3/4} \|w\|_V \\
\le & C \|u\|_H \|v\|_{H}^{1/4} \|v \|_{V}^{3/4} \|w\|_V, \ u, v,w\in
V,
\end{split}$$ where $c, C$ are some constants.
Then we have $$\begin{split}
& \<B(u,u)-B(v,v),u-v\>_V \\
=&- \<B(u,u-v),v\>_V+ \<B(v,u-v),v\>_V \\
=& - \<B(u-v, u-v),v\>_V \\
\le& C \|u-v\|_H^{5/4} \|u-v\|_V^{3/4} \|v\|_{V} \\
\le& \frac{\nu}{2} \|u-v\|_V^{2} + C_\nu \|v\|_{V}^{8/5}
\|u-v\|_H^{2}, \ u,v\in V.
\end{split}$$ Hence we have the local monotonicity $(H2)$: $$\<Au+B(u,u)-Av-B(v,v),u-v\>_V
\le -\frac{\nu}{2} \|u-v\|_V^{2} + C_\nu \|v\|_{V}^{8/5}
\|u-v\|_H^{2}.$$
Note that (\[e5\]) also implies that $(H4)$ holds.
Hence the existence and uniqueness of solutions to (\[Leray\]) follows from Theorem \[T1\].
The main result can also be applied to some other equations such as 3D Tamed Navier-Stokes equation, which is also a modified version of 3D Navier-Stokes equation. One may refer to [@LR10; @RZ09] for more details.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank Professors Phillipe Clément, Michael Röckner and Feng-yu Wang for their valuable discussions.
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[^1]: Corresponding author: wei.liu@uni-bielefeld.de
[^2]: Supported in part by DFG–Internationales Graduiertenkolleg “Stochastics and Real World Models”, the SFB-701 and the BiBoS-Research Center. The support of Issac Newton Institute for Mathematical Sciences in Cambridge is also gratefully acknowledged where part of this work was done during the special semester on “Stochastic Partial Differential Equations”.
|
---
abstract: 'We introduce a nonsymmetric real matrix which contains all the information that the usual Hermitian density matrix does, and which has exactly the same tensor product structure. The properties of this matrix are analyzed in detail in the case of multi-qubit (e.g. spin $= 1/2$) systems, where the transformation between the real and Hermitian density matrices is given explicitly as an operator sum, and used to convert the essential equations of the density matrix formalism into the real domain.'
author:
- 'Timothy F. Havel'
bibliography:
- 'phys.bib'
- 'math.bib'
- 'csci.bib'
- 'self.bib'
- 'nmr.bib'
title: The Real Density Matrix
---
[^1]
Prologue
========
The density matrix plays a central role in the modern theory of quantum mechanics, and an equally important role in its applications to optics, spectroscopy, and condensed matter physics. Viewed abstractly, it is a self-adjoint operator $\rho$ on system’s Hilbert space, the expectation values $0 \le \BRA{\psi}\,\rho\,\KET{\psi} \le 1$ of which give the probability of observing the system in the state $\KET{\psi}$. As a matrix, however, it is generally represented versus the operator (or “Liouville”) basis $\KET{i}\BRA{j}$ induced by a choice of a complete orthonormal basis $\{\KET{i}\mid i=0,1,\ldots\}$ in the underlying Hilbert space. These complex-valued matrices $\RHO \equiv [\BRA{i}\rho\KET{j}]_{i,j}$ are necessarily Hermitian and positive semi-definite. Their diagonal entries are the probabilities of these mutually exclusive basis states, whereas their off-diagonal entries prescribe the amounts by which the probabilities of their coherent superpositions deviate from the corresponding classical mixtures due to interference.
Another option is to use a operator basis the elements of which have rank exceeding one, so that it is not induced by any Hilbert space basis. The most common example here is the representation of operators on a two-dimensional Hilbert space by real linear combinations of Pauli matrices $\{ \SIG[0] (\equiv \MAT I_{2\LAB D}), \SIG[1], \SIG[2], \SIG[3] \}$. In this case the basis elements themselves are self-adjoint and so can be given a physical interpretation, e.g. as the components of the Bloch or Stokes vector [@Bloch:46; @FeyVerHel:57]. Although arbitrary bases of self-adjoint operators could be used, for multi-particle systems it is desirable that the overall basis be induced by identical bases on each particle’s Hilbert subspace. With the Pauli matrices this leads to the so-called *product operator* representation [@ErnBodWok:87], in which density operators are represented by linear combinations of all possible tensor (*ergo* Kronecker) products of the Pauli matrices, e.g. $\SIG[k]^1\SIG[\ell]^2 \equiv (\SIG[k] \otimes \SIG[0]) (\SIG[0] \otimes \SIG[\ell])$. In contrast to the Hermitian case, it has not been widely recognized that this tensor product structure is reflected by the real-valued coefficients in the expansion of any density operator in terms of product operators (also known as the *coherence vector* [@MahleWeber:98]). Thus if one properly arranges these coefficients in a matrix one obtains a *real* but *nonsymmetric* analog of the Hermitian density matrix with the *same* tensor product structure. This fact holds for any number of Hilbert spaces $\mathfrak H_k$ ($k = 1,2,\ldots$) of arbitrary (even infinite) dimension $L > 0$ and self-adjoint bases $\{ B^k_\ell \}_{\ell=1}^L$ for the space of bounded linear operators on each.
The purpose of this paper is show how, in the case of multi-qubit (*ergo* two-state quantum) systems, one can perform all the usual Hermitian density matrix calculations entirely with the real density matrix. While the formulae are more complicated in most cases than they are with the Hermitian density matrix, we argue that they are in many respects closer to the underlying physics than the Hermitian formulae, simply because the entries of the real density matrix correspond to (expectation values of) observables. Indeed, it is well-known that the single-qubit Pauli algebra is nothing but a complex matrix representation of the geometric (or Clifford) algebra of a three-dimensional Euclidean vector space [@HavelDoran:02]. This *real* algebra in turn has been demonstrated to be a concise but versatile formalism within which to analyze and teach much of modern physics [@Hestenes:03; @DoranLasen:03]. It makes a certain amount of sense to use a representation wherein the “reverse-even” entities in the algebra, i.e. scalars and vectors, are real while the less-familiar “reverse-odd” entities, i.e. “pseudo-scalars” and “pseudo-vectors”, are purely imaginary (cf. @Baylis:99). The drawback of our representation is that the matrices no longer form a representation of the underlying geometric algebra, i.e. the geometric product no longer corresponds to matrix multiplication. We will leave it to the community to decide if or when the advantages outweigh the disadvantages, and offer our results simply as the outcome of an intellectual exercise.
Metamorphosis {#sec:meta}
=============
We begin by introducing a bit of notation which will considerably simplify the remainder of our presentation. First, instead of the above bra-ket notation, let us write the $2\times2$ elementary matrices as $$\MAT E_{00} \leftrightarrow \KET0\BRA0 \,,\M \MAT E_{10} \leftrightarrow \KET1\BRA0 \,,\M \MAT E_{01} \leftrightarrow \KET0\BRA1 \,,\M \MAT E_{11} \leftrightarrow \KET1\BRA1 \,,$$ where $\KET0 \leftrightarrow \MAT e_0$, $\KET1 \leftrightarrow \MAT e_1$ denote an orthonormal basis for a two-dimensional Hilbert space. Then it is easily seen that, for any nonnegative integers $i, j \le M \equiv 2^N - 1$, the $(M+1)\times(M+1)$ elementary matrix $\MAT E_{ij}$ is the Kronecker product of $2\times2$ elementary matrices the indices of which are the bits $i_n, j_n \in \{ 0,\,1 \}$ in the binary expansions of $i, j$, respectively, i.e. $$\MAT E_{ij} ~\equiv~ \big[ \delta_{ik} \delta_{j\ell} \big]_{k,\ell=0}^{M,M} ~=~ \MAT E_{i_1j_1} \otimes\cdots\otimes \MAT E_{i_Nj_N} ~,$$ where the $\delta$’s are Kronecker deltas. In an analogous fashion, we will denote the usual $2\times2$ Pauli matrices by $$\MAT P_{00} \M[0.5]\equiv\M[0.5] \SIG[\,0] \,,\M \MAT P_{10} \M[0.5]\equiv\M[0.5] \SIG[\,1] \,,\M \MAT P_{01} \M[0.5]\equiv\M[0.5] \SIG[\,2] \,,\M \MAT P_{11} \M[0.5]\equiv\M[0.5] \SIG[\,3] \,.$$ In this notation it may readily be verified that the multiplication table among the Pauli matrices may be expressed succinctly as $$\MAT P_{ij\,} \MAT P_{k\ell} ~=~ \imath^{\,(i\ell-jk)(1-2ij)(1-2k\ell)}\, \MAT P_{(i+k-2ik),\M[0.05](j+\ell-2j\ell)} \quad\big( i,j,k,\ell \in \{0,\,1\} \big) ~,$$ where $\imath^2 = -1$. This indexing scheme may be extended to all Kronecker products of these matrices in the same way as for the $2\times2$ elementary matrices, i.e. $$\MAT P_{ij} ~=~ \MAT P_{i_1j_1} \otimes\cdots\otimes \MAT P_{i_Nj_N} \quad(0 \le i,j \le M) ~.$$ For example, if $M = 3$ we have $\MAT P_{01} = \SIG[0]\otimes\SIG[2]$, $\MAT P_{02} = \SIG[2]\otimes\SIG[0]$, and $\MAT P_{03} = \SIG[2]\otimes\SIG[2]\,$.
The Hermitian density matrix, of course, can be expanded relative to either the elementary matrix basis or the Pauli matrix basis, e.g. $$\RHO ~=~ {\sum}_{i,j=0}^{\,1,\,1}\; \rho_{ij}\, \MAT E_{ij} ~=~ \text{\large$\tfrac12$} ~ {\sum}_{i,j=0}^{1,\,1}\; \sigma_{ij}\, \MAT P_{ij} ~, \label{eq:expansion}$$ for a single qubit with $\rho_{ij} \in \FLD C$ but $\sigma_{ij} \in \FLD R$. Both of these bases are orthogonal relative to the Hilbert-Schmidt inner product, which is given by $\HIP{\MAT X}{\MAT Y} \equiv \TR(\MAT{XY}^\dag) \equiv 2^N\, \langle\, \MAT{XY}^{\dag \,} \rangle$ for any number of qubits $N > 0$. Thus there is a unique unitary superoperator $2^{{\ensuremath\M[0.075]}-1/2}\, \ALG U$ which carries the Pauli to the elementary matrix basis, where the factor of $\sqrt2$ comes from $\| \MAT P_{ij} \|^2 \equiv \HIP{\MAT P_{ij}}{\MAT P_{ij}} = 2\, \| \MAT E_{ij} \|^2$. This superoperator, moreover, is nearly self-adjoint since $\HIP{\MAT E_{ij}}{\MAT P_{k\ell}} = \HIP{\MAT E_{k\ell}}{\MAT P_{ij}}$ for all $0\le i,j,k,\ell \le 1$ with the sole exception of $\HIP{\MAT E_{10}}{\MAT P_{01}} = \imath\, \HIP{\MAT E_{01}}{\MAT P_{10}}$ and its complex conjugate. On applying $\ALG U$ to both sides of Eq. (\[eq:expansion\]), therefore, we obtain $$\ALG U(\RHO) ~=~ \text{\large$\tfrac12$} ~ {\sum}_{i,j=0}^{\,1,\,1}\; \rho_{ij}{\ensuremath\M[0.075]}\sqrt{{\imath}^{\,j{\ensuremath\M[0.075]}-{\ensuremath\M[0.075]}i}/2^{{\ensuremath\M[0.075]}|j-i{\ensuremath\M[0.075]}|}}\, \big({\ensuremath\M[0.075]}\MAT P_{ji} \:+\: {(-1)}^{(1-{\ensuremath\M[0.075]}i{\ensuremath\M[0.075]}){\ensuremath\M[0.075]}j}\, \MAT P_{ij{\ensuremath\M[0.075]}} \big) ~=~ {\sum}_{i,j=0}^{1,\,1}\; \sigma_{ij}\, \MAT E_{ij} ~.$$ We will take this as our definition of the *real density matrix* for a single qubit. Henceforth, we shall denote this by $$\begin{bmatrix} ~\sigma_{00}~&~\sigma_{01}~ \\ ~\sigma_{10}~&~\sigma_{11}~ \end{bmatrix} \M\equiv\M \SIG \M\equiv\M \ALG U(\RHO) \M=\M 2 \begin{bmatrix} \AVG{\RHO\, \SIG[0]} & \AVG{\RHO\, \SIG[2]} \\ \AVG{\RHO\, \SIG[1]} & \AVG{\RHO\, \SIG[3]} \end{bmatrix} ~.$$ Note that our choice of normalization gives $\sigma_{00} = 2{\ensuremath\M[0.075]}\langle\, \RHO\, \rangle = 1$, so that although $\ALG U$ is otherwise unitary the Hilbert-Schmidt norm is scaled by a factor of $\sqrt2$; specifically $2\, \|\RHO\M[0.1]\|^2 =\|\SIG\M[0.15]\|^2 \equiv 2{\ensuremath\M[0.075]}\langle \SIG^{\!\top\!} \SIG\M[0.1] \rangle = 1 + \sigma_{10}^2 + \sigma_{01}^2 + \sigma_{11}^2$.
Let us now look at some explicit representations of the superoperator $\ALG U$. To begin with, the mapping from the Pauli to the basis elementary is clearly $$\begin{aligned} \MAT E_{00} ~=\M[0.75] & \HALF\big( \MAT P_{00} + \MAT P_{11} \big) \\
\MAT E_{10} ~=\M[0.75] & \HALF\big( \MAT P_{10} - \imath\, \MAT P_{01} \big) \end{aligned} \qquad
\begin{aligned} \MAT E_{01} ~=\M[0.75] & \HALF\big( \MAT P_{10} + \imath\, \MAT P_{01} \big) \\
\MAT E_{11} ~=\M[0.75] & \HALF\big( \MAT P_{00} - \MAT P_{11} \big) ~, \end{aligned}$$ and hence (since coordinates are contravariant) $$\label{eq:usvd}
\KET{\SIG} ~\equiv~ \begin{bmatrix} 1\\ \sigma_{10}\\ \sigma_{01}\\ \sigma_{11} \end{bmatrix} \M[0.4]=\M[0.4] \begin{bmatrix} ~1~&0&0&0~\\ ~0~&1&0&0~\\ ~0~&0&-\imath&0~\\ ~0~&0&0&1~ \end{bmatrix} \M[-0.5] \begin{bmatrix} ~1&~0&0&1\\ ~0&~1&1&0\\ ~0&~1&-1&0\\ ~1&~0&0&\!-1 \end{bmatrix} \M[-0.5] \begin{bmatrix} \rho_{00}\\ \rho_{10}\\ \rho_{01}\\ \rho_{11} \end{bmatrix} \M[0.5]\equiv\M[0.6] \EMB{\ALG{VW}}\, \KET{\RHO} \M[0.5]\equiv\M[0.6] \EMB{\ALG U}\, \KET{\RHO} ~,$$ where we have factored the overall superoperator’s matrix $\EMB{\ALG U}$ into the product of a diagonal matrix $\EMB{\ALG V}$ and a purely real one $\EMB{\ALG W}$. An operator sum representation for the superoperator $\ALG W$ may be derived from the singular value decomposition of its *Choi matrix*, i.e. $$\begin{bmatrix} ~1&~0&~0&\!-1\\ ~0&~1&~1&0\\ ~0&~1&~1&0\\ ~1&~0&~0&\!-1 \end{bmatrix} ~=~ \begin{bmatrix} ~1&~0&\!-1&0\\ ~0&~1&0&1\\ ~0&~1&0&\!-1\\ ~1&~0&1&0 \end{bmatrix} \begin{bmatrix} ~1\:&\:0\:&\:0\:&\:0~\\ ~0\:&\:1\:&\:0\:&\:0~\\ ~0\:&\:0\:&\:0\:&\:0~\\ ~0\:&\:0\:&\:0\:&\:0~ \end{bmatrix} \begin{bmatrix} 1&0&~1&0\\ 0&1&~0&1\\ 0&1&~0&\!-1\\ \!-1&0&~1&0 \end{bmatrix}^{\displaystyle\top} ,$$ where the Choi matrix (left) is obtained simply by swapping certain pairs of the entries in $\EMB{\ALG W} \equiv \big[ w_{ij} \big]_{i,j=0}^{\,3,\,3}$ [@Havel!QPT:03]; specifically $w_{20} \leftrightarrow w_{01}$, $w_{30} \leftrightarrow w_{11}$, $w_{22} \leftrightarrow w_{03}$ and $w_{32} \leftrightarrow w_{13}$. Observe that the left-singular vectors associated with the nonzero singular values of $\EMB{\ALG W}$ can be written as “columnized” Pauli matrices, specifically $\KET{\MAT P_{00}}$ and $\KET{\MAT P_{10}}$ in the notation of Eq. (\[eq:usvd\]), while the corresponding right-singular vectors are $\KET{\MAT P_{11}}$ and $\KET{\MAT P_{10}\,}$. It follows that the operator sum form of $\ALG W$ is [@Havel!QPT:03 Proposition 3] $$\ALG W(\RHO) ~=~ \MAT P_{00}\, \RHO\, \MAT P_{11} \,+\, \MAT P_{10}\, \RHO\, \MAT P_{10} ~=~ \begin{bmatrix} \rho_{00} + \rho_{11} & \rho_{10} - \rho_{01} \\ \rho_{10} + \rho_{01} & \rho_{00} - \rho_{11} \end{bmatrix} ~.$$
Although an operator sum form for $\ALG V$ could be obtained by this same approach, since it is diagonal a more compact representation of its action may be obtained by packing its nonzero entries into a single $2\times2$ matrix which acts via the “entrywise” or *Hadamard product* “$\odot$” [@HaShViCo:01], as follows: $$\begin{split}
\SIG ~=~ \MAT Q \odot \big( \RHO\, \MAT P_{11} \,+\, \MAT P_{10}\, \RHO\, \MAT P_{10} \big) ~\equiv\M & \begin{bmatrix} ~1&-\imath~\\ ~1&~1~ \end{bmatrix} \odot \begin{bmatrix} \rho_{00} + \rho_{11} & \rho_{10} - \rho_{01} \\ \rho_{10} + \rho_{01} & \rho_{00} - \rho_{11} \end{bmatrix} \\
\equiv\M& \begin{bmatrix} ~\rho_{00} + \rho_{11} ~&~ \imath\, (\rho_{01} - \rho_{10})~ \\ ~\rho_{01} + \rho_{10} ~&~ \rho_{00} - \rho_{11}~ \end{bmatrix} ~.
\end{split}$$ Since $\ALG W$ is self-adjoint and the overall superoperator $\ALG U$ is unitary (up to a factor of $\sqrt2$), it is easily seen that the inverse $\ALG U^{-1}$ can be written as $$\RHO ~=~ \HALF \left( \big(\M[0.05] \OL{\MAT Q} \odot \SIG \big)\, \MAT P_{11} ~+~ \MAT P_{10}\, \big(\M[0.05] \OL{\MAT Q} \odot \SIG \big)\, \MAT P_{10}\, \right) ~,$$ where the overbar indicates the complex conjugate of all the matrix entries.
The beauty of this operator sum form for the superoperator $\ALG U$ is that the Hadamard product obeys the mixed product formula with the Kronecker product, $$\label{eq:mixed}
(\MAT A \otimes \MAT B) \odot (\MAT C \otimes \MAT D) ~=~ (\MAT A \odot \MAT C) \otimes (\MAT B \odot \MAT D) ~,$$ just like the usual matrix product does. Thus if we extend $\ALG U$ to factorable multi-qubit density matrices in the obvious way, $$\ALG U(\RHO^1 \otimes\cdots\otimes \RHO^N) ~\equiv~ \ALG U(\RHO^1) \otimes\cdots\otimes \ALG U(\RHO^N) ~,$$ and thence to arbitrary multi-qubit density matrices by linearity, we immediately obtain a general expression. Explicitly, in the case of two qubits, we get $$\begin{split}
\SIG \M\equiv\M & \SIG^1 \otimes \SIG^2 \M\equiv\M \ALG U(\RHO^1) \otimes \ALG U(\RHO^2) \\
=\M & \Big( \MAT Q \odot \big( \RHO^1\, \MAT P_{11} + \MAT P_{10}\, \RHO^1\, \MAT P_{10} \big)\! \Big) \otimes \Big( \MAT Q \odot \big( \RHO^2\, \MAT P_{11} + \MAT P_{10}\, \RHO^2\, \MAT P_{10} \big)\! \Big) \\
=\M & \big( \MAT Q \otimes \MAT Q \big) \odot \Big(\! \big( \RHO^1\, \MAT P_{11} + \MAT P_{10}\, \RHO^1\, \MAT P_{10} \big) \otimes \big( \RHO^2\, \MAT P_{11} + \MAT P_{10}\, \RHO^2\, \MAT P_{10} \big)\! \Big) \\
=\M & \big( \MAT Q \otimes \MAT Q \big) \odot \big( \begin{aligned}[t] & \!
(\RHO^1\, \MAT P_{11}) \otimes (\RHO^2\, \MAT P_{11}) + (\RHO^1\, \MAT P_{11}) \otimes (\MAT P_{10}\, \RHO^2\, \MAT P_{10}) ~\cdots \\
& +\, (\MAT P_{10}\, \RHO^1\, \MAT P_{10}) \otimes (\RHO^2\, \MAT P_{11}) + (\MAT P_{10}\, \RHO^1\, \MAT P_{10}) \otimes (\MAT P_{10}\, \RHO^2\, \MAT P_{10}) \big) \end{aligned} \\
=\M & \big( \begin{aligned}[t] & \!
\MAT Q \otimes \MAT Q \big) \odot \big( (\RHO^1 \otimes \RHO^2) (\MAT P_{11} \otimes \MAT P_{11}) + (\MAT P_{00} \otimes \MAT P_{10}) (\RHO^1 \otimes \RHO^2) (\MAT P_{11} \otimes \MAT P_{10}) ~\cdots \\
& \,+ (\MAT P_{10} \otimes \MAT P_{00}) (\RHO^1 \otimes \RHO^2) (\MAT P_{10} \otimes \MAT P_{11}) + (\MAT P_{10} \otimes \MAT P_{10}) (\RHO^1 \otimes \RHO^2) (\MAT P_{10} \otimes \MAT P_{10}) \big) \end{aligned} \\
\equiv\M & \MAT Q^{\otimes2} \odot \big( \RHO\, \MAT P_{33} + \MAT P_{10}\, \RHO\, \MAT P_{32} + \MAT P_{20}\, \RHO\, \MAT P_{31} + \MAT P_{30}\, \RHO\, \MAT P_{30} \big) ~, \label{eq:metamorph}
\end{split}$$ where $\RHO \equiv \RHO^1 \otimes \RHO^2$ and $\MAT Q^{\otimes N}$ ($N > 0$) denotes the $N$-fold Kronecker power of $\MAT Q$. It is readily verified that the general formula for $N$ qubits is $$\label{eq:you}
\ALG U(\RHO) ~=~ \MAT Q^{\otimes N} \odot {\sum}_{m\,=\,0}^{\,M}\, \MAT P_{m,\M[0.05]0}\, \RHO\, \MAT P_{M,\M[0.05](M-m)} ~,$$ where $\M[-0.05]\RHO\M[-0.05]$ isany (notnecessarilyfactorable)densitymatrix.Similarly,theinverseisgivenby $$\label{eq:uoy}
\ALG U^{-1}(\SIG) ~=~ 2^{-N}\, {\sum}_{m\,=\,0}^{\,M}\, \MAT P_{m,\M[0.05]0}\, \big(\M[0.05] \OL{\MAT Q}^{\M[0.1]\otimes N\M[-0.1]} \odot \SIG \big)\, \MAT P_{M,\M[0.05](M-m)} ~.$$ Evidently $\MAT Q^{\,\otimes N\!} \odot{\ensuremath\M[0.075]}\ALG U^{-1}(\SIG) \,=\, 2^{-N}{\ensuremath\M[0.075]}\ALG U({\ensuremath\M[0.075]}\OL{\MAT Q}^{\,\otimes N\!} \odot{\ensuremath\M[0.075]}\SIG {\ensuremath\M[0.075]})$.
Reality Check
=============
The following properties of the real density matrix $\SIG$ are worth noting explicitly:
- In addition to being real, it is nonsymmetric with one fixed element $\sigma_{00} = 1$;
- It contains all the same information that the Hermitian density matrix does (since they are related by the bijection $\ALG U$).
- It is diagonal if and only if the Hermitian density matrix is diagonal (which is why we defined $\sigma_{11}$ to be the coefficient of the diagonal Pauli matrix $\SIG[3]$).
- It has the same tensor product structure as the Hermitian density matrix, since (as shown by Eq. (\[eq:metamorph\])) $\ALG U$ maps Kronecker products to Kronecker products.
In addition to these nice analytic features, the real density matrix can also be quite useful for displaying the results of *quantum state tomography*: the determination of density matrices from experimental data. In most cases to date, the real or imaginary parts of the Hermitian density matrix have been displayed using two-dimensional bar graphs (see e.g [@NielsChuan:00 §7.7.4]). Although useful, such a plot must both omit information and exhibit redundant information. Real density matrices are definitely superior in this respect, and sometimes may also exhibit the underlying symmetry of a state more clearly. Bar graphs of the real density matrix are shown below for both the diagonal Hilbert space basis as well as the Bell basis, illustrating how easily these states may be distinguished. Further examples with experimental NMR data may be found in @HCLBFPTWBH:02.
(500,260) ( 0, 25)[{width="100pt" height="100pt"}]{} (125, 25)[{width="100pt" height="100pt"}]{} (250, 25)[{width="100pt" height="100pt"}]{} (375, 25)[{width="100pt" height="100pt"}]{} ( 30, 10)[$\KET{00}\BRA{00}$]{} (155, 10)[$\KET{01}\BRA{01}$]{} (280, 10)[$\KET{10}\BRA{10}$]{} (405, 10)[$\KET{11}\BRA{11}$]{} ( 0,155)[{width="100pt" height="100pt"}]{} (125,155)[{width="100pt" height="100pt"}]{} (250,155)[{width="100pt" height="100pt"}]{} (375,155)[{width="100pt" height="100pt"}]{} ( 5,140)[$\KET{00}\BRA{00} + \KET{11}\BRA{11}$]{} (130,140)[$\KET{00}\BRA{00} - \KET{11}\BRA{11}$]{} (255,140)[$\KET{01}\BRA{01} + \KET{10}\BRA{10}$]{} (380,140)[$\KET{01}\BRA{01} - \KET{10}\BRA{10}$]{}
The fact that the mapping between Pauli matrix coefficients and the entries of the Hermitian density operator preserves the tensor product structure has been noted earlier by @PitteRubin:00a (and without doubt by many other researchers as well). In our present notation, their observation was based upon the following simple relation: $$\begin{bmatrix} \sigma_{00} & \sigma_{10} \\ \sigma_{11} & \imath\sigma_{01} \end{bmatrix} ~=~ \begin{bmatrix} ~1~ & ~1~ \\ ~1~ & -1~ \end{bmatrix} \begin{bmatrix} \rho_{00} & \rho_{10} \\ \rho_{11} & \rho_{01} \end{bmatrix} ~.$$ While this is certainly a simpler relation than our operator sum, the “density matrix” on the right-hand side is not the usual Hermitian one, and the mapping between the two can be written explicitly only by using operator sums, supermatrices, or the like. Our goal here is to translate the usual operations and relations on Hermitian density matrices into the real domain, and the reordering of the entries of the real density matrix as above offers no advantage for this purpose.
As our first example of such a translation, let us show how the usual criterion for the purity of the Hermitian density matrix can be carried over to the real domain: $$\begin{aligned}
\M[-3] 1 \M=\M 2^N\, \langle\, \RHO^2\, \rangle \M=\M &
\left\langle\, \RHO \,{\sum}_{m\,=\,0}^{m=M}\, \MAT P_{m,0} \big( \OL{\MAT Q}^{\,\otimes N} \odot \SIG \big) \MAT P_{M,M-m}\right\rangle \notag \\ =\M &
\left\langle\! \big( \OL{\MAT Q}^{\,\otimes N} \odot \SIG \big) {\sum}_{m=0}^M\, \MAT P_{M,M-m}\, \RHO\, \MAT P_{m,0} \right\rangle \notag \\ =\M &
\left\langle\, {\sum}_{m\,=\,0}^{m=M}\, \MAT P_{m,0}\, \RHO\, \MAT P_{M,M-m}\, \big( \OL{\MAT Q}^{\,\otimes N} \odot \SIG \big)^{\!\dag} \right\rangle \\ \notag =\M &
\left\langle\! \big( \OL{\MAT Q}^{\,\otimes N} \odot \SIG \big) \big( \OL{\MAT Q}^{\,\otimes N} \odot \SIG \big)^{\!\dag} \right\rangle \\ \notag =\M &
\left\langle \big( \MAT Q^{\otimes N} \odot \OL{\MAT Q}^{\,\otimes N} \odot \SIG \big)^{\dag}\, \SIG\, \right\rangle ~=~ \big\langle\, \SIG^\top \SIG\, \big\rangle ~.\end{aligned}$$ In going to the last line, we have used the general relation $\langle (\MAT A \odot \MAT B)\, \MAT C^\dag \rangle = \langle (\MAT A \odot \MAT C)^\dag\, \MAT B{\ensuremath\M[0.075]}\rangle$ for arbitrary conformant matrices $\MAT A, \MAT B, \MAT C$ [@Lutkepohl:96]. This derivation easily generalizes to a formula for the ensemble-average expectation values of any observable with Hermitian matrix $\EMB\mu$ and corresponding real matrix $\EMB\nu = \ALG U(\EMB\mu)$, showing that $$\big\langle\, \EMB\mu \,\big|\, \RHO\, \big\rangle ~\equiv~ 2^{{\ensuremath\M[0.075]}N}\, \big\langle\, \EMB\mu\, \RHO\, \big\rangle ~=~\big\langle\, \EMB\nu^\top \SIG\, \big\rangle ~=~ 2^{{\ensuremath\M[0.075]}-N{\ensuremath\M[0.075]}} \big\langle\, \EMB\nu \,\big|\, \SIG \,\big\rangle ~.$$
We close this section by noting that the *partial trace* operation corresponds simply to extracting a principal submatrix of the real density matrix [@SomCorHav:98].
Life in the Real World
======================
While the expectation values of observables carry over to the real domain without significant complication, things become distinctly more challenging when it comes to integrating the equations of motion. In the case of a single qubit, it is readily verified that the commutator with an arbitrary Hamiltonian $\EMB\mu = \ALG U^{-1}(\EMB\nu)$ becomes $$\label{eq:1pc} \begin{split}
\big[\M[-0.25]\big[\, \SIG,\, \EMB\nu\, \big]\M[-0.25]\big] ~\equiv~ \ALG U\big( \big[\, \RHO,\, \EMB\mu\, \big] \big)
& ~=\M[.5] \imath\,\MAT P_{01} \big( \EMB\nu\, \MAT E_{11}\M[.1] \SIG \,-\,\SIG\, \MAT E_{11}\M[.1] \EMB\nu \big) \MAT P_{01} \\
& ~=\M[.5] \imath \begin{bmatrix} 0 & \sigma_{11\,} \nu_{10} -\sigma_{10\,} \nu_{11} \\ \sigma_{01\,} \nu_{11} - \sigma_{11\,} \nu_{01} & \sigma_{10\,} \nu_{01} - \sigma_{01\,} \nu_{10} \end{bmatrix} ~,
\end{split}$$ wherein the matrix entries are the components of the usual vector cross product.
This equation of motion is most simply integrated by considering the matrix representation of the commutation superoperator defined by $\EMB\nu$, which we henceforth assume without loss of generality has $\nu_{00} = 0$. Letting $\KET{\MAT X}$ denote the column vector of height $(M+1)^2$ obtained by stacking the columns of the $(M+1)\times(M+1)$ matrix $\MAT X$ on top of one another in left-to-right order, and applying the well-known identity $$\label{eq:oppr2prop}
\KET{ \MAT{AXB} } ~=~ \big( \MAT B^\top \otimes \MAT A \big)\, \KET{\MAT X}$$ (see e.g. [@Lutkepohl:96]), we find that[^2] $$\label{eq:form0} \begin{split}
\M[-1] \dot{\SIG} ~=~ \Big|\, \tfrac{\displaystyle\imath}2\, \big[\M[-0.25]\big[\, \SIG,\, \EMB\nu\, \big]\M[-0.25]\big] \Big\rangle ~=\M[0.5]
& \text{\large$\HALF$}\, \Big( \MAT P_{01} \otimes \MAT P_{01} \Big) \Big( \MAT P_{00} \otimes \EMB\nu\, \MAT E_{11} \,-\,\EMB\nu^\top \MAT E_{11} \otimes \MAT P_{00} \Big)\, \big|\, \SIG\, \big\rangle \\ =\M[0.5]
& \frac12 \begin{bmatrix} ~0&0&0&0\\ ~0&0&-\nu_{11}&\nu_{01}\\ ~0&\nu_{11}&0&-\nu_{10}\\ ~0&-\nu_{01}&\nu_{10}&0 \end{bmatrix}\! \begin{bmatrix} \sigma_{00}\\ \sigma_{10}\\ \sigma_{01}\\ \sigma_{11} \end{bmatrix} ~\equiv~ \HALF\, \EMB{\ALG R}_{\EMB\nu}\, \KET{\SIG}\, .
\end{split}$$ Since $\EMB{\ALG R}_{\EMB\nu}^{\,3} = -{\|\EMB\nu\|}^{2\,} \EMB{\ALG R}_{\EMB\nu\,}$ ($\|\EMB\nu\|^2 \equiv 2\M[0.05] \langle \EMB\nu^\top \EMB\nu \rangle \equiv \HIP{\EMB\nu}{\EMB\nu}$), this one-sided matrix differential equation is easily integrated by using the Cayley-Hamilton theorem to exponentiate the (lower-right $3\times3$ block of the) coefficient matrix [@NajfeHavel:95a], obtaining $$\begin{split}
& \big|\M[0.1]\SIG(t)\M[0.1]\big\rangle \M[.5]=\M[.5] \bigg( \MAT P_{00} \otimes \MAT P_{00} \,+\, \frac{\sin(\|\EMB\nu\|\M[.1]t/2\M[.1])}{\|\EMB\nu\|}\, \EMB{\ALG R}_{\EMB\nu} \,+\, \frac{1-\cos(\|\EMB\nu\|\M[.1]t/2\M[.1])} {\|\EMB\nu\|{\X[1.4]}^2}\, \EMB{\ALG R}_{\EMB\nu}^{\,2} \bigg)\, \big|\M[0.1]\SIG(0)\M[0.1]\big\rangle \\ =\M[.5]
& \Big|\, \SIG(0) \,+\, \imath\, \sin(\|\EMB\nu\|\M[.1]t/2\M[.1])\, \big[\M[-0.25]\big[\, \SIG(0),\, \hat{\EMB\nu}\, \big]\M[-0.25]\big] \,-\, \big( 1-\cos(\|\EMB\nu\|\M[.1]t/2\M[.1]) \big)\, \big[\M[-0.25]\big[ \big[\M[-0.25]\big[\, \SIG(0),\, \hat{\EMB\nu}\, \big]\M[-0.25]\big], \hat{\EMB\nu}\, \big]\M[-0.25]\big] \Big\rangle ,
\end{split}$$ wherein $\hat{\EMB\nu} \equiv \EMB\nu/\|\EMB\nu\|$ and it is readily shown that $$\label{eq:r2}
\EMB{\ALG R}_{\EMB\nu}^{\,2\,} ~=~ \KET{\EMB\nu} \BRA{\EMB\nu} ~-~ \|\EMB\nu\|^2\M[0.25] \big( \MAT P_{00} \otimes \MAT P_{00} \:-\: \MAT E_{00} \otimes \MAT E_{00} \big)$$ is $-\|\EMB\nu\|^2$ times the projection onto the plane orthogonal to the unit vector $\KET{\hat{\EMB\nu}} = \BRA{\hat{\EMB\nu}}^\top$. The whole formula can thus be expressed more geometrically as $$\KET{{\smash{\check\SIG\M[0.1]}}(t)\!} ~=~ \HIP{\hat{\EMB\nu}}{{\smash{\check\SIG\M[0.1]}}}\; \KET{\hat{\EMB\nu}} \,+\, \sin(\|\EMB\nu\|\M[.1]t/2\M[.1])\, \KET{\hat{\EMB\nu}} \times \KET{{\smash{\check\SIG\M[0.1]}}} \,+\, \cos(\|\EMB\nu\|\M[.1]t/2\M[.1])\, \KET{\hat{\EMB\nu}} \times \big({\ensuremath\M[0.075]}\KET{\hat{\EMB\nu}} \times \KET{{\smash{\check\SIG\M[0.1]}}} \big) ,~$$ where ${\smash{\check\SIG\M[0.1]}} \equiv \SIG(0) - \MAT E_{00}$ and “$\times$” is the cross product of the (last three components of the) vectors it connects. This is a standard expression for rotation of the three-dimensional vector $\KET{{\smash{\check\SIG\M[0.1]}}}$ about the axis $\KET{\hat{\EMB\nu}}$ by an angle $\|\EMB\nu\|\M[.1]t/2\M[0.05]$.
The extension of these formulae to general multi-particle commutators is not straightforward, since the tensor product of commutation superoperators is not simply related to the tensor products of their underlying operators. Nevertheless, we can give a reasonably simple formula for the two-particle commutator with a factorable Hamiltonian, which is the most important case in practice. This is based upon a geometric algebra expression for the commutator of a tensor product of two three-dimensional vectors (or equivalently in the present context, traceless $2\times2$ Hermitian matrices), which is derived in : $$\label{eq:doran}
2\, \big[ \MAT A \otimes \MAT C,\, \MAT B \otimes \MAT D \big] ~=~ \HIP{\MAT A}{\MAT B} \big( \MAT P_{00} \otimes [ \MAT C, \MAT D ] \big) ~+~ \big( [ \MAT A, \MAT B ] \otimes \MAT P_{00} \big) \HIP{\MAT C}{\MAT D} ~.$$ Letting $\MAT a \equiv \ALG U(\MAT A)$, etc. be the corresponding real matrices, this translates to: $$\label{eq:form1a}
2\, \big[\M[-0.25]\big[ \MAT a \otimes \MAT c ,\, \MAT b \otimes \MAT d \big]\M[-0.25]\big] ~=~ \HIP{\MAT a}{\MAT b} \big( \MAT E_{00} \otimes \M[.1][\M[-.15][{\ensuremath\M[0.075]}\MAT c, \MAT d \M[.1]]\M[-.15]] \big) ~+~ \big( \M[.1][\M[-.15][{\ensuremath\M[0.075]}\MAT a, \MAT b \M[.1]]\M[-.15]] \otimes \MAT E_{00} \big)\HIP{\MAT c}{\MAT d}$$ This formula is easily extended to the case in which $a_{00},\ldots,d_{00} \ne 0$ by multilinearity; in the following, however, we will need only the case in which $a_{00} = c_{00} = 1$, which introduces two additional terms: $$\label{eq:form1b} \begin{split}
\M[-.5] \big[\M[-0.25]\big[{\ensuremath\M[0.075]}\MAT E_{00} \otimes \MAT c ,\, \MAT b \otimes \MAT d {\ensuremath\M[0.075]}\big]\M[-0.25]\big] \M[.5]\leftrightarrow\M[.5] \HALF\, \big[{\ensuremath\M[0.075]}\MAT P_{00} \otimes \MAT C \,,\: \MAT B \otimes \MAT D \big] ~=~ \HALF\, \MAT B \otimes [\M[.1] \MAT C , \MAT D \M[.1]] \M[.5]\leftrightarrow\M[.5]
& \HALF\, \MAT b \otimes [\M[-.15][\M[.1] \MAT c, \MAT d \M[.1]]\M[-.15]] ~;
\\
\M[-.5] \big[\M[-0.25]\big[{\ensuremath\M[0.075]}\MAT a \otimes \MAT E_{00} ,\, \MAT b \otimes \MAT d {\ensuremath\M[0.075]}\big]\M[-0.25]\big] \M[.5]\leftrightarrow\M[.5]
\HALF\, \big[ \MAT A \otimes \MAT P_{00} \,,\: \MAT B \otimes \MAT D \big] ~=~ \HALF\, [\M[.1] \MAT A ,\, \MAT B \M[.1]] \otimes \MAT D \M[.25]\leftrightarrow\M[.5]
& \HALF\, [\M[-.15][\M[.1] \MAT a, \MAT b \M[.1]]\M[-.15]] \otimes \MAT d ~.
\end{split}$$ Finally, we shall need the general result [@Lutkepohl:96; @Havel!QPT:03]: $$\label{eq:form2}
\KET{ \MAT X \otimes \MAT Y } ~=~ \big( \MAT P_{00} \otimes \EMB{\ALG K}_{22} \otimes \MAT P_{00} \big)\, \big(\, \KET{ \MAT X } \otimes \KET{ \MAT Y } \big) ,$$ for $2\times2$ matrices $\MAT X$ & $\MAT Y$, where the two-particle *commutation matrix* $\EMB{\ALG K}_{22}$ is given by $$\EMB{\ALG K}_{22} ~=~ {\sum}_{i,\,j=0}^{\,3,\,3}\, \MAT E_{ij} \otimes \MAT E_{ji} ~=~ \begin{bmatrix} ~1~&~0~&~0~&~0~\\[-1ex] ~0~&~0~&~1~&~0~\\[-1ex] ~0~&~1~&~0~&~0~\\[-1ex] ~0~&~0~&~0~&~1~ \end{bmatrix} ~.$$
We are interested in the case that $\MAT a = \SIG^1$, $\MAT b = \EMB\nu^1$, $\MAT c = \SIG^2$ and $\MAT d = \EMB\nu^2$, i.e. we have a factorizable two-particle state $\SIG^1 \otimes \SIG^2$ evolving under a bi-axial interaction $\EMB\nu^1 \otimes \EMB\nu^2$. To express this more compactly, we define the matrix $\EMB{\ALG S}_{\EMB\nu^1}$ via $$\big\langle\, \EMB\nu^1\, \big|\, \SIG^1\, \big\rangle\, \big|\, \MAT E_{00}\, \big\rangle ~+~ \big|\, \EMB\nu^1\, \big\rangle ~=~ \begin{bmatrix} ~0~&\nu_{10}^1&\nu_{01}^1&\nu_{11}^1\\[-1ex] \nu_{10}^1&0&0&0\\[-1ex] \nu_{01}^1&0&0&0\\[-1ex] \nu_{11}^1&0&0&0 \end{bmatrix}\! \begin{bmatrix} \text{\small$1$}\\[-1ex] \sigma_{10}^1\\[-1ex] \sigma_{01}^1\\[-1ex] \sigma_{11}^1 \end{bmatrix} ~\equiv\M \EMB{\ALG S}_{\!\EMB\nu^1}\, \big|\, \SIG^1\, \big\rangle$$ with an analogous definition in $\HIP{\EMB\nu^2}{\SIG^2}\, \KET{\MAT E_{00}} + \KET{\EMB\nu^2} \equiv\, \EMB{\ALG S}_{\!\EMB\nu^2}\, \KET{\SIG^2}$. Then Eqs. (\[eq:form0\]), (\[eq:form1a\]), (\[eq:form1b\]) & (\[eq:form2\]) give us $$\begin{aligned}
\label{eq:ugly}
\M[-1]\partial_t\, \big|\, \SIG^1\otimes\SIG^2\, \big\rangle \M[.25]=\M[.5]
&\notag \Big|\M[0.2] \tfrac{\displaystyle\imath}4\M[0.1] \big[\M[-.25]\big[ \SIG^1 \otimes \SIG^2 ,\: \EMB\nu^1 \otimes \EMB\nu^2\, \big]\M[-.25]\big] \Big\rangle \\ =\M[.5]
& \big( \MAT P_{00} \otimes \EMB{\ALG K}_{22} \otimes \MAT P_{00} \big)\M[0.1] \tfrac{\displaystyle\imath}4
\begin{aligned}[t] \Big(
& \HIP{\SIG^1}{\EMB\nu^1}\, \big|\, \MAT E_{00}\, \big\rangle \otimes \big|\M[0.1] \big[\M[-.25]\big[ \SIG^2,\: \EMB\nu^2\, \big]\M[-.25] \big] \big\rangle ~+ \\
\M[-3] \cdots~ & \big|\M[0.1] \big[\M[-.25]\big[ \SIG^1,\: \EMB\nu^1\, \big]\M[-.25]\big] \big\rangle \otimes \big|\, \MAT E_{00}\, \big\rangle\, \HIP{\SIG^2}{\EMB\nu^2} ~+ \\
\M[-3] \cdots~ & \big|\, \EMB\nu^1\, \big\rangle \otimes \big|\M[0.1] \big[\M[-.25]\big[ \SIG^2 ,\: \EMB\nu^2 \big]\M[-.25]\big] \big\rangle ~+ \\
\M[-3] \cdots~ & \big|\M[0.1] \big[\M[-.25]\big[ \SIG^1 ,\: \EMB\nu^1 \big]\M[-.25]\big]\, \big\rangle \otimes \big|\, \EMB\nu^2\, \big\rangle \Big)
\end{aligned}
\\ =\M[.5]
&\notag \big( \MAT P_{00} \otimes \EMB{\ALG K}_{22} \otimes \MAT P_{00} \big) \,\tfrac14\, \big( \EMB{\ALG S}_{\EMB\nu^1} \otimes \EMB{\ALG R}_{\EMB\nu^2}
\begin{aligned}[t]
\,+\, \EMB{\ALG R}_{\EMB\nu^1} \otimes \EMB{\ALG S}_{\EMB\nu^2} \big)
\big(\M[0.1] \big|\M[0.1] \SIG^1\M[0.1] \big\rangle \otimes \big|\M[0.1] \SIG^2\M[0.1] \big\rangle \big) \end{aligned}
\\ =\M[.5]
&\notag \big( \MAT P_{00} \otimes \EMB{\ALG K}_{22} \otimes \MAT P_{00} \big) \,\tfrac14\,
\begin{aligned}[t]
\big( \EMB{\ALG S}_{\EMB\nu^1} \otimes \EMB{\ALG R}_{\EMB\nu^2} \,+\, \EMB{\ALG R}_{\EMB\nu^1} \otimes \EMB{\ALG S}_{\EMB\nu^2} \big) ~\cdots \M[4] & \\
\cdots~ \big( \MAT P_{00} \otimes \EMB{\ALG K}_{22} \otimes \MAT P_{00} \big)\, \big|\, \SIG^1 \otimes \SIG^2\, \big\rangle & ~.
\end{aligned}\end{aligned}$$
Since left or right multiplication of a $4$D column or row vector by $\EMB{\ALG R}_{\MAT x}$ gives the cross product of the last three components of that vector with $[ x_{10}, x_{01}, x_{11} ]$, it may be seen that $\EMB{\ALG S}_{\EMB\nu^1}$, $\EMB{\ALG R}_{\EMB\nu^1}$ are mutually annihilating (i.e. $\EMB{\ALG S}_{\EMB\nu^{1\,}} \EMB{\ALG R}_{\EMB\nu^1} = \EMB{\ALG R}_{\EMB\nu^1\,} \EMB{\ALG S}_{\EMB\nu^1} = \MAT 0$), and similarly for $\EMB{\ALG S}_{\EMB\nu^2}$, $\EMB{\ALG R}_{\EMB\nu^2}$. As a result, the two terms on the last line of Eq. (\[eq:ugly\]) commute and their exponential factorizes. It is moreover easily shown that $\EMB{\ALG S}_{\MAT x}^{\M[.1]3} = \|\MAT x\|^2 \EMB{\ALG S}_{\X[1.5]\MAT x}$ and $\EMB{\ALG R}_{\MAT x}^3 = -\|\MAT x\|^{2\,} \EMB{\ALG R}_{\X[1.5]\MAT x\,}$, so the overall integral is $$\label{eq:overall} \begin{split}
\big|\M[0.1] \SIG(t)\M[0.1] \big\rangle ~=~ \big( \MAT P_{00} \otimes \EMB{\ALG K}_{22} \otimes \MAT P_{00} \big)\, \MAT{Exp}\big( \EMB{\ALG S}_{\EMB\nu^1} \otimes \EMB{\ALG R}_{\EMB\nu^2} \M[0.25]t/4 \big)\, \MAT{Exp}\big( \EMB{\ALG R}_{\EMB\nu^1} \otimes \EMB{\ALG S}_{\EMB\nu^2} \M[0.25]t/4 \big) ~\cdots \M[2] & \\
\cdots~ \big( \MAT P_{00} \otimes \EMB{\ALG K}_{22} \otimes \MAT P_{00} \big)\, \big|\M[0.1] \SIG(0)\M[0.1] \big\rangle , &
\end{split}$$ where $\SIG(0) = \SIG^1 \otimes \SIG^2$ and (letting $\MAT P_{00}^{\otimes4}$ be the $2\times2$ identity tensored with itself $4$ times) $$\begin{split}
\M[-1] \MAT{Exp}\big(\,\EMB{\ALG S}_{\EMB\nu^1} \otimes \EMB{\ALG R}_{\EMB\nu^2} \M[.25]t/4 \big) \M[.5]=\M[.5]
\MAT P_{00}^{\otimes4} \,+\, \frac{\sin\!\big( \|\EMB\nu^1\| \|\EMB\nu^2\| \M[0.1]t/4 \big)}{\|\EMB\nu^1\| \|\EMB\nu^2\|}\, \big( \EMB{\ALG S}_{\EMB\nu^1} \otimes \EMB{\ALG R}_{\EMB\nu^2} \big) ~+ &
\\ \cdots~ \frac{1 - \cos\!\big(\|\EMB\nu^1\| \|\EMB\nu^2\| \M[0.1]t/4 \big)}{{\|\EMB\nu^1\|}^2 {\|\EMB\nu^2\|}^2}\, \big( \EMB{\ALG S}_{\EMB\nu^1} \otimes \EMB{\ALG R}_{\EMB\nu^2} \big)^2 &
\end{split}$$ with an almost identical expression for $\MAT{Exp}\big( \EMB{\ALG R}_{\EMB\nu^1} \otimes \EMB{\ALG S}_{\EMB\nu^2} \M[.25]t/4 \big)$. The square in the last term may be evaluated by combining Eq. (\[eq:r2\]) with $$\label{eq:s2}
\EMB{\ALG S}_{\MAT x}^{\M[.1]2} ~=~ \KET{\MAT x} \BRA{\MAT x} ~+~ \|\MAT x\|^2\M[0.25] \MAT E_{00} \otimes \MAT E_{00} ~.$$
Because $\EMB{\ALG S}_{\EMB\nu^1} \otimes \EMB{\ALG R}_{\EMB\nu^2}$ and $\EMB{\ALG R}_{\EMB\nu^1} \otimes \EMB{\ALG S}_{\EMB\nu^2}$ are mutually annihilating, the product of their exponentials expands to only five terms, two pairs of which have identical trigonometric coefficients. On pulling the right-hand column operator “$\KET{}$” back to the left in Eq. (\[eq:overall\]), essentially reversing what we did to derive the differential version in Eq. (\[eq:ugly\]), we obtain the integrated equation of motion we have been seeking: $$\label{eq:final2p} \begin{split}
\SIG(t) \M[.5]=\M[.5] \SIG^1 \otimes \SIG^2 \,+\, \sin\!\big( \|\EMB\nu^1\| \|\EMB\nu^2\| \,t/4 \big)\, \big[\M[-0.25]\big[ \SIG^1 \otimes \SIG^2 ,\, \hat{\EMB\nu}^1 \otimes \hat{\EMB\nu}^2 \M[.1]\big]\M[-0.25]\big] \:- \M[3.2] &
\\[0.5ex] \cdots~ \big(1 - \cos\!\big( \|\EMB\nu^1\| \|\EMB\nu^2\| \,t/4 \big) \big)\, \big[\M[-0.25]\big[ \big[\M[-0.25]\big[ \SIG^1 \otimes \SIG^2 ,\, \hat{\EMB\nu}^1 \otimes \hat{\EMB\nu}^2 \M[.1]\big]\M[-0.25]\big] ,\, \hat{\EMB\nu}^1 \otimes \hat{\EMB\nu}^2 \M[.1]\big]\M[-0.25]\big] \,.
\end{split}$$ Equations (\[eq:form1a\]–\[eq:form1b\]) tell us (more or less) what the geometric interpretation of the two particle commutator is, so we turn our attention to the double commutator. Geometric algebra shows that the double commutator of tensor products of the corresponding traceless $2\times2$ Hermitian matrices reduces to $$\begin{gathered}
2\, \big[ \big[ \MAT A \otimes \MAT C,\, \MAT B \otimes \MAT D \big],\, \MAT B \otimes \MAT D \big] \\
\begin{aligned}[t] =\M[.5]
& \big[ \HIP{\MAT A}{\MAT B} \big( \MAT P_{00} \otimes [ \MAT C, \MAT D ] \big) \,+\, \big( [ \MAT A, \MAT B ] \otimes \MAT P_{00} \big) \HIP{\MAT C}{\MAT D} \,,\: \MAT B \otimes \MAT D \big] \M[2] \\ =\M[.5]
& \HIP{\MAT A}{\MAT B}\, \MAT B \otimes [{\ensuremath\M[0.075]}[\MAT C,{\ensuremath\M[0.075]}\MAT D],{\ensuremath\M[0.075]}\MAT D {\ensuremath\M[0.075]}] \,+\, [{\ensuremath\M[0.075]}[\MAT A,{\ensuremath\M[0.075]}\MAT B],{\ensuremath\M[0.075]}\MAT B {\ensuremath\M[0.075]}] \otimes \MAT D\, \HIP{\MAT C}{\MAT D} ~,
\end{aligned} \end{gathered}$$ so we will be done once we figure out what the double commutator of $2\times2$ matrices is. On expanding the commutator and using the fact that for such matrices the anticommutator satisfies $\MAT{AB} + \MAT{BA} = \HIP{\MAT A}{\MAT B}\, \MAT P_{00\,}$, we get (including the real analogs): $$\label{eq:1pdc} \begin{split}
[\M[-0.12][{\ensuremath\M[0.075]}[\M[-0.12][\, \MAT a ,{\ensuremath\M[0.075]}\MAT b \,]\M[-0.12]] ,{\ensuremath\M[0.075]}\MAT b \,]\M[-0.12]] \M[.5]\leftrightarrow\M[.5] &
[{\ensuremath\M[0.075]}[{\ensuremath\M[0.075]}\MAT A,{\ensuremath\M[0.075]}\MAT B {\ensuremath\M[0.075]}],{\ensuremath\M[0.075]}\MAT B {\ensuremath\M[0.075]}] \M[.5]=\M[.5] \MAT{AB}^2 \,-\, 2\, \MAT{BAB} \,+\, \MAT B^2 \MAT A \\ =\M[.5] &
\MAT A{\ensuremath\M[0.075]}\| \MAT B \|^2 \,-\, 2\, (\MAT{BA} + \MAT{AB} - \MAT{AB}){\ensuremath\M[0.075]}\MAT B \\ =\M[.5] &
2\, \big( \MAT A{\ensuremath\M[0.075]}\| \MAT B \|^2 \,-\, \HIP{\MAT A}{\MAT B}{\ensuremath\M[0.075]}\MAT B \big) \M[.5]\leftrightarrow\M[.5] \MAT a\, \|{\ensuremath\M[0.075]}\MAT b {\ensuremath\M[0.075]}\|^2 \,-\, \HIP{\MAT a}{\MAT b}\, \MAT b ~.
\end{split}$$ Back-substitution of this and the corresponding expression for $[[\MAT C, \MAT D], \MAT D]$ into the preceding equation now yields: $$\begin{gathered}
\M[-1] \big[ \big[ \MAT A \otimes \MAT C,\, \MAT B \otimes \MAT D \big],\, \MAT B \otimes \MAT D \big] \\\begin{aligned}[t] &
\M[-.5]=\M[.5] \HIP{\MAT A}{\MAT B}\, \MAT B \otimes \big( \MAT C\, \|\MAT D\|^2 \,-\, \HIP{\MAT C}{\MAT D}\, \MAT D \big) \:+\: \big( \MAT A\, \|\MAT B\|^2 \,-\, \HIP{\MAT A}{\MAT B}\, \MAT B\big) \otimes \MAT D\, \HIP{\MAT C}{\MAT D} \\ &
\M[-.5]=\M[.5] \HIP{\MAT A}{\MAT B}\, \|\MAT D\|^2\, (\MAT B \otimes \MAT C) ~+~ \HIP{\MAT C}{\MAT D}\, \|\MAT B\|^2\, (\MAT A \otimes \MAT D) ~-~ 2\, \HIP{\MAT A}{\MAT B}\, \HIP{\MAT C}{\MAT D}\, (\MAT B \otimes \MAT D) .\M[-1]
\end{aligned} \end{gathered}$$ Since no matrix products occur in this expression, we may transliterate directly to the corresponding real expression including the additional terms arising from setting $a_{00} = c_{00} = 1$ (cf. Eq. (\[eq:form1b\])): $$\begin{gathered}
4\, \big[\M[-0.25]\big[ \big[\M[-0.25]\big[ (\MAT a + \MAT E_{00}) \otimes (\MAT c + \MAT E_{00}) ,\, \MAT b \otimes \MAT d \M[.1]\big]\M[-0.25]\big] ,\, \MAT b \otimes \MAT d \M[.1]\big]\M[-0.25]\big] \M[.5]= \\
\begin{aligned}[t]
\HIP{\MAT a}{\MAT b}\, \|\MAT{d}\|^2\, (\MAT b \otimes \MAT c) \:+\: \HIP{\MAT c}{\MAT d}\, \|\MAT b\|^2\, (\MAT a \otimes \MAT d)
~\cdots \M &
\\ -~ 2\, \HIP{\MAT a}{\MAT b}\, \HIP{\MAT c}{\MAT d}\, (\MAT b \otimes \MAT d) ~\cdots \M &
\\ +~ 2\, \big[\M[-0.25]\big[\, \MAT b \otimes [\M[-0.12][{\ensuremath\M[0.075]}\MAT c ,{\ensuremath\M[0.075]}\MAT d {\ensuremath\M[0.075]}{]\M[-0.13]]} \,+\, [\M[-0.12][{\ensuremath\M[0.075]}\MAT a ,{\ensuremath\M[0.075]}\MAT b {\ensuremath\M[0.075]}]\M[-0.12]] \otimes \MAT d ,\, \MAT b \otimes \MAT d \,\big]\M[-0.25]\big] \M[-1.5] & \M[1.5].
\end{aligned}\end{gathered}$$ The first of these additional terms (given on the last line of the above equation) may be evaluated via Eqs. (\[eq:form1a\]), (\[eq:form1b\]) & (\[eq:1pdc\]) as indicated below, $$\begin{split}
\M[-1] 2\, \big[\M[-0.25]\big[\, \MAT b \otimes [\M[-0.12][{\ensuremath\M[0.075]}\MAT c ,{\ensuremath\M[0.075]}\MAT d {\ensuremath\M[0.075]}{]\M[-0.15]]} ,\, \MAT b \otimes \MAT d \,\big]\M[-0.25]\big] \M[.5]=\M[.75] &
{\|{\ensuremath\M[0.075]}\MAT b {\ensuremath\M[0.075]}\|}^2 \big( \MAT E_{00} \otimes [\M[-0.12][{\ensuremath\M[0.075]}[\M[-0.12][{\ensuremath\M[0.075]}\MAT c ,{\ensuremath\M[0.075]}\MAT d {\ensuremath\M[0.075]}]\M[-0.13]] ,{\ensuremath\M[0.075]}\MAT d {\ensuremath\M[0.075]}]\M[-0.13]] \big) \:+\: \big({\ensuremath\M[0.075]}[\M[-0.12][{\ensuremath\M[0.075]}\MAT b ,{\ensuremath\M[0.075]}\MAT b {\ensuremath\M[0.075]}]\M[-0.13]] \otimes \MAT E_{00} \big)\, \big\langle{\ensuremath\M[0.075]}[\M[-0.12][{\ensuremath\M[0.075]}\MAT c ,{\ensuremath\M[0.075]}\MAT d {\ensuremath\M[0.075]}]\M[-0.13]] {\ensuremath\M[0.075]}\big|{\ensuremath\M[0.075]}\MAT d {\ensuremath\M[0.075]}\big\rangle \\ =\M[.75] &
{\|{\ensuremath\M[0.075]}\MAT b {\ensuremath\M[0.075]}\|}^2{\ensuremath\M[0.075]}\big(\, \MAT E_{00} \otimes ({\ensuremath\M[0.075]}{\|{\ensuremath\M[0.075]}\MAT d{\ensuremath\M[0.075]}\|}^2 {\ensuremath\M[0.075]}\MAT c \,-\, \HIP{\MAT c}{\MAT d}\, \MAT d {\ensuremath\M[0.075]}) \big) ~,
\end{split}$$ with an analogous expression for the remaining term. We now put in our previous values $\MAT a = {\smash{\check\SIG\M[0.1]}}^1 \equiv \SIG^1 - \MAT E_{00}$, $\MAT b = \EMB\nu^1$, $\MAT c = {\smash{\check\SIG\M[0.1]}}^2 \equiv \SIG^2 - \MAT E_{00}$ and $\MAT d = \EMB\nu^2$ and expand the result fully to get: $$\begin{gathered}
4\, \big[\M[-0.25]\big[ \big[\M[-0.25]\big[ \SIG^1 \otimes \SIG^2 ,\, \EMB\nu^1 \otimes \EMB\nu^2 \M[.1]\big]\M[-0.25]\big] ,\,\EMB\nu^1 \otimes \EMB\nu^2 \M[.1]\big]\M[-0.25]\big] \M[.5]= \\
\begin{aligned}[t]
\big\langle\, \SIG^1 \,\big|\, \EMB\nu^1 \big\rangle\, {\big\|\EMB\nu^2\big\|}^2 \, \big( \EMB\nu^1 \otimes {\smash{\check\SIG\M[0.1]}}^2{\ensuremath\M[0.075]}\big) \:+\: {\big\|{\ensuremath\M[0.075]}\EMB\nu^1\big\|}^2\, \big\langle\, \SIG^2 \,\big|\, \EMB\nu^2\, \big\rangle\, \big( {\smash{\check\SIG\M[0.1]}}^1 \otimes \EMB\nu^2{\ensuremath\M[0.075]}\big) ~\cdots \M &
\\ -~ 2\, \big\langle\, \SIG^1 \,\big|\, \EMB\nu^1\, \big\rangle\, \big\langle\, \SIG^2 \,\big|\, \EMB\nu^2\, \big\rangle\, \big( \EMB\nu^1 \otimes \EMB\nu^2{\ensuremath\M[0.075]}\big) ~\cdots \M &
\end{aligned} \\ \begin{aligned}[t]
+~{\big\|{\ensuremath\M[0.075]}\EMB\nu^1{\ensuremath\M[0.075]}\big\|}^2\, {\big\|{\ensuremath\M[0.075]}\EMB\nu^2{\ensuremath\M[0.075]}\big\|}^2{\ensuremath\M[0.075]}\big( \MAT E_{00} \otimes {\smash{\check\SIG\M[0.1]}}^2 \,+\, {\smash{\check\SIG\M[0.1]}}^1 \otimes \MAT E_{00} \big) ~\cdots \M[1.25] &
\\ -~ \Big( {\big\|{\ensuremath\M[0.075]}\EMB\nu^1{\ensuremath\M[0.075]}\big\|}^2\, \big\langle\, \SIG^2 \,\big|\, \EMB\nu^2\, \big\rangle\, \MAT E_{00} \otimes \EMB\nu^2 \:+\: \big\langle\, \SIG^1 \,\big|\, \EMB\nu^1\, \big\rangle\, {\big\|{\ensuremath\M[0.075]}\EMB\nu^2{\ensuremath\M[0.075]}\big\|}^2\, \EMB\nu^1 \otimes \MAT E_{00} \Big) . &
\end{aligned}\end{gathered}$$ Finally, we divide through by $\|\EMB\nu^1\|^2\, \|\EMB\nu^2\|^2$ to get the normalized “vectors” $\smash{\hat{\EMB\nu}}^1$ and $\smash{\hat{\EMB\nu}}^2$, replace the $\SIG$’s by ${\smash{\check\SIG\M[0.1]}}$’s inside the traces (which doesn’t change their values) and recombine terms to obtain: $$\begin{gathered}
4\, \big[\M[-0.25]\big[ \big[\M[-0.25]\big[ \SIG^1 \otimes \SIG^2 ,\, \hat{\EMB\nu}^1 \otimes \hat{\EMB\nu}^2 \M[.1]\big]\M[-0.25]\big] ,\, \hat{\EMB\nu}^1 \otimes \hat{\EMB\nu}^2 \M[.1]\big]\M[-0.25]\big] \M[.5]= \\
\begin{aligned}[t]
& \big( {\smash{\check\SIG\M[0.1]}}^1 \:-\: \big\langle\, {\smash{\check\SIG\M[0.1]}}^1 \,\big|\, \hat{\EMB\nu}^1\, \big\rangle\, \hat{\EMB\nu}^1 \big) \otimes \big( \big\langle\, {\smash{\check\SIG\M[0.1]}}^2 \,\big|\, \hat{\EMB\nu}^2\, \big\rangle\, \hat{\EMB\nu}^2 \:+\: \MAT E_{00} \big) ~\cdots \M[2] \\ +~
& \big( \big\langle\, {\smash{\check\SIG\M[0.1]}}^1 \,\big|\, \hat{\EMB\nu}^1\, \big\rangle\, \hat{\EMB\nu}^1 \:+\: \MAT E_{00} \big) \otimes \big( {\smash{\check\SIG\M[0.1]}}^2 \:-\: \big\langle\, {\smash{\check\SIG\M[0.1]}}^2 \,\big|\, \hat{\EMB\nu^2}\, \big\rangle\, \hat{\EMB\nu}^2 \big) ~.
\end{aligned}\end{gathered}$$ This has a fairly simple interpretation: The rejection of ${\smash{\check\SIG\M[0.1]}}^1$ from $\hat{\EMB\nu}^1$ is tensored with the projection of ${\smash{\check\SIG\M[0.1]}}^2$ onto $\hat{\EMB\nu}^2$ (plus the usual scalar part of $1$) and added to the projection of ${\smash{\check\SIG\M[0.1]}}^1$ onto $\hat{\EMB\nu}^1$ (plus the scalar part) tensored with the rejection of ${\smash{\check\SIG^2\M[0.1]}}$ from $\hat{\EMB\nu}^2$. This should be contrasted with the single commutator (Eq. (\[eq:form1a\])), wherein the inner and outer products of each pair are tensored together both ways and added.
Meta-Metamorphosis
==================
In the previous section we have shown how the Hamiltonians usually assumed for quantum computing with qubits can be integrated entirely within the real domain. For a single qubit, the results could be interpreted as a simple Bloch vector rotation. With a bi-axial interaction between two qubits, we also found that that the integrated expression had a reasonably nice geometric interpretation. Algebraically, however, it is usually easier to integrate in the Hermitian domain, simply because Hermitian matrices are easier to diagonalize. In this section, therefore, we shall derive formulae by which matrix representations of general superoperators can be translated from the Hermitian into the real domain, along with some specific examples of their utility.
Using the identity given in Eq. (\[eq:oppr2prop\]) together with our operator sum expressions for $\ALG U$ and its inverse (Eqs. (\[eq:you\]) & (\[eq:uoy\])), it is straightforward to show that an arbitrary superoperator $\ALG S$ with matrix representation $\EMB{\ALG S}$ acting on $\RHO$ via $\RHO \mapsto \EMB{\ALG S}\, \KET{\RHO}$ transforms into the real domain according to $$\label{eq:supopstfm}
\EMB{\ALG S} \M\stackrel{\ALG U}{\longleftrightarrow}\M 2^{-N\,} \EMB{\ALG Q} \bigg( \sum_{m'=0}^M\, \MAT P_{M,(M-m')} \otimes \MAT P_{m',0} \bigg) \EMB{\ALG S} \bigg( \sum_{m=0}^M\, \MAT P_{M,(M-m)} \otimes \MAT P_{m,0} \bigg) \OL{\EMB{\ALG Q}} ~,$$ where $\EMB{\ALG Q} \equiv \DMAT({\ensuremath\M[0.075]}\KET{\MAT Q^{\otimes N}})$. The superoperator $\ALG S$ may be written in operator sum form versus the basis of elementary matrices [@Havel!QPT:03] as $$\EMB{\ALG S}\, \KET{\RHO} ~=~ \bigg| \sum_{i,j=0}^{M,M}\, \sum_{k,\ell=0}^{M,M}\, s_{k\ell}^{ij}\, \MAT E_{ki\,} \RHO\M[0.05] \MAT E_{j\ell} \bigg\rangle ~=~ \sum_{i,j=0}^{M,M}\, \sum_{k,\ell=0}^{M,M}\, s_{k\ell}^{ij}\, \big( \MAT E_{\ell j} \otimes \MAT E_{ki} \big)\, \KET{\RHO} \,.$$ On substituting the second of these equations (sans $\KET\RHO$) into the first and rearranging things a bit, the transformed superoperator becomes $$2^{-N\,} \EMB{\ALG Q} \bigg( \sum_{i,j=0}^{M,M}\, \sum_{k,\ell=0}^{M,M}\, s_{k\ell}^{ij}\, \sum_{m,m'=0}^{M,M}\, \big( \MAT P_{M,(M-m')\,} \MAT E_{\X[1.3]\ell j\,} \MAT P_{M,(M-m)} \big) \otimes \big( \MAT P_{m',0\,} \MAT E_{ki\,} \MAT P_{m,0} \big)\! \bigg) \OL{\EMB{\ALG Q}} ~.$$ Unfortunately, because each factor in the above Kronecker product depends on both indices $m$ and $m'$, this cannot be regarded as a transformation of the elementary matrix basis into the real domain.
Better insight can be obtained by looking at how the Choi matrix $\EMB{Choi}(\EMB{\ALG S})$ of the superoperator transforms. This may be obtained from the propagating matrix $\EMB{\ALG S}$ simply by replacing Kronecker products of the elementary matrices by dyadic products of the corresponding columnized basis, but in the opposite order [@Havel!QPT:03]. This task is facilitated by expressing the left- and right-multiplication by diagonal matrices as a Hadamard product, using the well-known formula [@HaShViCo:01] $$\label{eq:wellknown}
\DMAT(\MAT a)\, \MAT X\, \DMAT^\dag(\MAT b) ~=~ (\MAT{ab}^\dag) \odot \MAT X ~,$$ which is essentially a special case of Eq. (\[eq:form2\]). This allows the transformed superoperator to be rewritten as $$\begin{gathered}
2^{-N\,} \Big( \big|\, \MAT Q^{\otimes N} \big\rangle\, \big\langle{\ensuremath\M[0.075]}\MAT Q^{\otimes N} \big| \Big) \odot \sum_{i,j=0}^{M,M}\, \sum_{k,\ell=0}^{M,M}\, s_{k\ell}^{ij} ~\cdots \\
\cdots \sum_{m,m'=0}^{M,M}\, \Big(\! \big( \MAT P_{M,(M-m')\,} \MAT E_{\X[1.3]\ell j\,} \MAT P_{M,(M-m)} \big) \otimes \big( \MAT P_{m',0\,} \MAT E_{ki\,} \MAT P_{m,0} \big)\! \Big) ~.\end{gathered}$$ The advantage of this form is that the Hadamard product commutes with the $\EMB{Choi}$ operator (since it rearranges the entries of the product’s operands identically), giving us $$\begin{aligned}
\M[-0.25] \EMB{Choi}(\EMB{\ALG S}) \M[0.25]\stackrel{\ALG U}{\longleftrightarrow}\M[0.5] &
2^{-N\,} \EMB{Choi}\Big( \big|\, \MAT Q^{\otimes N} \big\rangle\, \big\langle{\ensuremath\M[0.075]}\MAT Q^{\otimes N} \big| \Big) \odot \sum_{i,j=0}^{M,M}\, \sum_{k,\ell=0}^{M,M}\, s_{k\ell}^{ij} ~\cdots \notag\\ &
\cdots\, \sum_{m,m'=0}^{M,M}\, \EMB{Choi}\Big(\! \big( \MAT P_{M,(M-m')\,} \MAT E_{\X[1.3]\ell j\,} \MAT P_{M,(M-m)} \big) \otimes \big( \MAT P_{m',0\,} \MAT E_{ki\,} \MAT P_{m,0} \big)\! \Big) \notag\\ =\M[0.5] &
2^{-N\,} \big( \OL{\MAT Q}^{{\ensuremath\M[0.075]}\otimes N\!} \otimes \MAT Q^{\otimes N} \big) \odot \sum_{i,j=0}^{M,M}\, \sum_{k,\ell=0}^{M,M}\, s_{k\ell}^{ij} ~\cdots \notag\\ &
\cdots\, \sum_{m,m'=0}^{M,M}\, \Big( \big|\, \MAT P_{m',0\,} \MAT E_{ki\,} \MAT P_{m,0} \big\rangle \big\langle \MAT P_{M,(M-m')\,} \MAT E_{\X[1.3]\ell j\,} \MAT P_{M,(M-m)} \big| \Big) \notag\\ =\M[0.5] &
2^{-N\,} \big( \OL{\MAT Q}^{{\ensuremath\M[0.075]}\otimes N\!} \otimes \MAT Q^{\otimes N} \big) \odot \sum_{i,j=0}^{M,M}\, \sum_{k,\ell=0}^{M,M}\, s_{k\ell}^{ij}\sum_{m,m'=0}^{M,M}\, \big( \MAT P_{m,0} \otimes \MAT P_{m',0} \big)\, \KET{\MAT E_{ki\,}} ~\cdots\notag\\ &
\cdots~ \big\langle\, \MAT E_{\X[1.3]\ell j\,} \big| \big( \MAT P_{M,(M-m)} \otimes \MAT P_{M,(M-m')} \big) \\\notag =\M[0.5] &
2^{-N\,} \big( \OL{\MAT Q}^{{\ensuremath\M[0.075]}\otimes N\!} \otimes \MAT Q^{\otimes N} \big) \odot \sum_{m,m'=0}^{M,M}\, \big( \MAT P_{m,0} \otimes \MAT P_{m',0} \big)\, \sum_{i,j=0}^{M,M}\, \sum_{k,\ell=0}^{M,M}\, s_{k\ell}^{ij} ~\cdots \\\notag &
\cdots~ \big( \MAT E_{ij} \otimes \MAT E_{k\ell} \big)\big( \MAT P_{M,(M-m)} \otimes \MAT P_{M,(M-m')} \big) \\\notag \equiv\M[0.5] &
2^{-N\,} \big( \OL{\MAT Q}^{{\ensuremath\M[0.075]}\otimes N\!} \otimes \MAT Q^{\otimes N} \big) \odot \sum_{m=0}^{M'}\, \MAT P_{m,0}\; \EMB{Choi}(\EMB{\ALG S})\; \MAT P_{M',(M'-m)} ~,\end{aligned}$$ where $M' \equiv (M+1)^2 - 1 = 2^{2N} - 1$ and we have used the relation $\KET{\MAT E_{ki}}\, \BRA{\MAT E_{\ell j}} = (\MAT e_i \otimes \MAT e_k)(\MAT e_j \otimes \MAT e_\ell)^\top = \MAT E_{ij} \otimes \MAT E_{k\ell}$.
In other words, the Choi matrix of a superoperator maps into the real domain much like a density matrix on twice as many qubits. In fact we can write the *real* transformation matrix $\EMB{\ALG T}$, which acts on the real density matrix as $\EMB{\ALG T}\, \KET{\SIG}$, in the following compact form: $$\EMB{\ALG T} \M[0.7]=\M[0.7] 2^{-N}\, \EMB{Choi}\Big(\, \ALG U\big(\, \EMB{Choi}( \EMB{\ALG S} )\, \big) \Big)\, \OL{\EMB{\ALG Q}}^{\,2} ~.$$ Turning this around, we also find that we can express the Choi matrix of $\ALG S$ in the Pauli basis as $$\begin{split}
& 2^{-N}\; \EMB{Choi}(\EMB{\ALG S}) \M[0.7]=\M[0.7] \ALG U^{-1}\big( \EMB{Choi}( \EMB{\ALG T}{\ensuremath\M[0.075]}\EMB{\ALG Q}^2 {\ensuremath\M[0.075]}) \big) \\ \equiv\M &
\sum_{i,j=0}^{M,M} \sum_{k,\ell=0}^{M,M}\, \ALG U^{-1}\big({\ensuremath\M[0.075]}t_{k\ell}^{ij}\; \EMB{Choi}\big( (\MAT E_{\ell j} \otimes \MAT E_{ki}){\ensuremath\M[0.075]}\EMB{\ALG Q}^2 {\ensuremath\M[0.075]}\big) \big) \\ =\M &
\sum_{i,j=0}^{M,M} \sum_{k,\ell=0}^{M,M}\, t_{k\ell}^{ij}~ \ALG U^{-1}\Big( \EMB{Choi}(\MAT E_{\ell j} \otimes \MAT E_{ki}) \odot \EMB{Choi}\big(\, \KET{\MAT 1\, \MAT 1^\top} \BRA{\MAT Q^{\otimes N\!} \odot \MAT Q^{\otimes N}}\, \big)\! \Big) \\ =\M &
\sum_{i,j=0}^{M,M} \sum_{k,\ell=0}^{M,M}\, t_{k\ell}^{ij}\; \ALG U^{-1}\big( \MAT E_{ij} \otimes \MAT E_{k\ell} \big) \odot \big( \big( \MAT Q^{\otimes N\!} \odot \MAT Q^{\otimes N} \big) \otimes \big( \MAT 1{\ensuremath\M[0.075]}\MAT 1^\top \big) \big) \big) \\ =\M &
2^{-2N}\, \big( \big( \MAT Q^{\otimes N\!} \odot \MAT Q^{\otimes N} \big) \otimes \big( \MAT 1{\ensuremath\M[0.075]}\MAT 1^\top \big) \big) \odot \sum_{i,j=0}^{M,M} \sum_{k,\ell=0}^{M,M}\, t_{k\ell}^{ij}\; \big( \MAT P_{ij} \otimes \MAT P_{k\ell} \big) ~,
\end{split}$$ where $\MAT 1$ denotes a column vector of ones of the appropriate size.
It is time for our examples! We shall begin with operator sums for single qubit rotations, and go on to show how rotations about the $\SIG[3]$ axis as well as the $\SIG[3]^1\SIG[3]^2$ interaction between two qubits can be compactly described in the real domain using Hadamard products. We close by showing that this description also extends quite nicely to $\SIG[3]$ dephasing as well as nonunital relaxation back towards a nonrandom equilibrium state (i.e. $T_2$ and $T_1$ relaxation in NMR parlance).
Consider the Choi matrix of the propagator which rotates a single qubit (Eqs. (\[eq:form0\]–\[eq:r2\]) above): $$\begin{aligned}
& \EMB{Choi}\big( \MAT P_{00} \otimes \MAT P_{00} \,+\, \sin(\|\EMB\nu\|\M[.1]t/2\M[.1])\, \EMB{\ALG R}_{\widehat{\EMB\nu}} \,+\, (1-\cos(\|\EMB\nu\|\M[.1]t/2\M[.1]))\, \EMB{\ALG R}_{\widehat{\EMB\nu}}^{\,2} \big) \notag\\ =\M[0.5] &
\begin{bmatrix} ~1~&~0~&~0~&~1~\\ ~0~&~0~&~0~&~0~\\ ~0~&~0~&~0~&~0~\\ ~1~&~0~&~0~&~1~ \end{bmatrix} ~+~ \sin(\|\EMB\nu\|\M[.1]t/2\M[.1]) \begin{bmatrix} ~0&0&0&0\\ ~0&0&-\hat\nu_{11}&\hat\nu_{10}\\ ~0&\hat\nu_{11}&0&-\hat\nu_{10}\\ ~0&-\hat\nu_{01}&\hat\nu_{01}&0 \end{bmatrix} ~+~ \cdots \\\notag &
\cdots~ (1-\cos(\|\EMB\nu\|\M[.1]t/2\M[.1])) \begin{bmatrix} ~0&0&0& -\hat\nu_{10}^2-\hat\nu_{11}^2 \\ ~0&0& \hat\nu_{10}\hat\nu_{01} & \hat\nu_{01}\hat\nu_{11} \\ ~0& \hat\nu_{10}\hat\nu_{01} &0& \hat\nu_{01}\hat\nu_{11} \\ -\hat\nu_{01}^2-\hat\nu_{11}^2 & \hat\nu_{10}\hat\nu_{11} & \hat\nu_{10}\hat\nu_{11} & -\hat\nu_{01}^2-\hat\nu_{10}^2 \end{bmatrix} ~,\end{aligned}$$ where the “hat” on the $\nu$’s indicates normalization by $\|\EMB\nu\|$. A general operator sum can be derived from this matrix (as well as by expanding the implied commutators via Eq. (\[eq:1pc\])), but since this is a bit involved we shall restrict ourselves to rotations by an angle $\vartheta =\|\EMB\nu\|\M[0.1]t/2$ about the $\LAB x$, $\LAB y$ or $\LAB z$ coordinate axes. On substituting $\hat\nu_{10} = 1$ and $\hat\nu_{01} = \hat\nu_{11} = 0$ into the above Choi matrix we obtain the following singular value decomposition: $$\begin{split}
\begin{bmatrix} ~1~&~0~&~0~& \cos(\vartheta)\\ ~0~&~0~&~0~& \sin(\vartheta)\\ ~0~&~0~&~0~& -\sin(\vartheta)\\ ~1~&~0~&~0~& \cos(\vartheta) \end{bmatrix} \M[0.5]=\M[0.5] &
\begin{bmatrix} \cos(\vartheta/2)&\sin(\vartheta/2)\\ \sin(\vartheta/2)~&-\cos(\vartheta/2)\\ -\sin(\vartheta/2)~&~\cos(\vartheta/2)\\ \cos(\vartheta/2)&\sin(\vartheta/2) \end{bmatrix} \cdots \\ &
\cdots \begin{bmatrix} \cos(\vartheta/2)&0\\ 0&\sin(\vartheta/2) \end{bmatrix}\M[-0.25] \begin{bmatrix} ~1~&~0~&~0~&~1~\\ ~1~&~0~&~0~&-1~ \end{bmatrix} ~,
\end{split}$$ as may be readily verified using the usual half-angle formulae. The corresponding operator sum for rotation by $\vartheta$ about the $\LAB x$-axis is simply: $$\begin{split}
\ALG U\Big( \EMB e^{-\imath(\vartheta/2)\MAT P_{10}} \RHO\, \EMB e^{~\imath(\vartheta/2)\MAT P_{10}} \Big) \M[0.5]=\M[0.5] &
\cos(\vartheta/2) \begin{bmatrix} \cos(\vartheta/2)&-\sin(\vartheta/2)\\ \sin(\vartheta/2)&\cos(\vartheta/2) \end{bmatrix}\M[-0.5] \begin{bmatrix} \text{\small$1$}&\sigma_{01}\\ \sigma_{10}&\sigma_{11} \end{bmatrix} ~+~ \cdots \\ &
\sin(\vartheta/2) \begin{bmatrix} \sin(\vartheta/2)&\cos(\vartheta/2)\\ -\cos(\vartheta/2)&\sin(\vartheta/2) \end{bmatrix}\M[-0.5] \begin{bmatrix} \text{\small$1$}&\sigma_{01}\\ \sigma_{10}&\sigma_{11} \end{bmatrix}\M[-0.5] \begin{bmatrix} ~1~&~0~\\ ~0~&-1~ \end{bmatrix} .
\end{split}$$ In a similar fashion, it can be shown that the operator sum for a $\LAB y$-rotation is: $$\begin{split}
\ALG U\Big( \EMB e^{-\imath(\vartheta/2)\MAT P_{01}} \RHO\, \EMB e^{~\imath(\vartheta/2)\MAT P_{01}} \Big) \M[0.5]=\M[0.5] &
\cos(\vartheta/2) \begin{bmatrix} \text{\small$1$}&\sigma_{01}\\ \sigma_{10}&\sigma_{11} \end{bmatrix}\M[-0.5] \begin{bmatrix} \cos(\vartheta/2)&-\sin(\vartheta/2)\\ \sin(\vartheta/2)&\cos(\vartheta/2) \end{bmatrix} ~+~ \cdots \\ &
\sin(\vartheta/2) \begin{bmatrix} ~1~&~0~\\ ~0~&-1~ \end{bmatrix}\M[-0.5] \begin{bmatrix} \text{\small$1$}&\sigma_{01}\\ \sigma_{10}&\sigma_{11} \end{bmatrix}\M[-0.5] \begin{bmatrix} \sin(\vartheta/2)&\cos(\vartheta/2)\\ -\cos(\vartheta/2)&\sin(\vartheta/2) \end{bmatrix} .
\end{split}$$
For a $\LAB z$-rotation, on the other hand, the Choi matrix turns out to be rank $4$ with singular value decomposition: $$\begin{split} &
\begin{bmatrix} 1&0&0&\cos(\vartheta)\\ 0&0&-\sin(\vartheta)&0\\ 0&\sin(\vartheta)&0&0\\ \cos(\vartheta)&0&0&1 \end{bmatrix} \M[0.5]=\M[0.5]
\begin{bmatrix} ~1~&~0~&~0~&~1~\\ ~0~&~1~&-1~&~0~\\ ~0~&~1~&~1~&~0~\\ ~1~&~0~&~0~&-1~ \end{bmatrix} \cdots \\ &
\cdots \begin{bmatrix} \cos^2(\vartheta/2)&0&0&0\\ 0&\cos(\vartheta/2)\sin(\vartheta/2)&0&0\\ 0&0&\cos(\vartheta/2)\sin(\vartheta/2)&0\\ 0&0&0&\sin^2(\vartheta/2) \end{bmatrix}\M[-0.5] \begin{bmatrix} ~1~&~0~&~0~&~1~\\ ~0~&1~&-1~&~0~\\ ~0~&~1~&~1~&~0~\\ ~1~&~0~&~0~&-1~ \end{bmatrix} .
\end{split}$$ This corresponds to the operator sum $$\begin{split}
\ALG U\Big( \EMB e^{-\imath(\vartheta/2)\MAT P_{11}} \RHO\, \EMB e^{~\imath(\vartheta/2)\MAT P_{11}} \Big) \M[0.5]=\M[0.5] &
\cos^2(\vartheta/2)\, \SIG ~+~ \sin^2(\vartheta/2)\, \MAT P_{11\,} \SIG\, \MAT P_{11} ~+~ \cdots \\ &
\cdots~ \imath \cos(\vartheta/2)\sin(\vartheta/2) \big( \MAT P_{10\,} \SIG\, \MAT P_{01} ~+~ \MAT P_{01\,} \SIG\, \MAT P_{10} \big) ~,
\end{split}$$ which has the pleasant feature that the trigonometric functions occur as scalar factors in each term and not embedded in the operators. This enables us to use Eq. (\[eq:wellknown\]) to rewrite it in terms of the Hadamard product as follows: $$\begin{split}
\ALG U\Big( \EMB e^{-\imath(\vartheta/2)\MAT P_{11}} \RHO\, \EMB e^{~\imath(\vartheta/2)\MAT P_{11}} \Big) \M[0.5] &
=\M[0.5]\begin{bmatrix} 1&\cos(\vartheta)\\ \cos(\vartheta)&1 \end{bmatrix} \odot \begin{bmatrix} \text{\small$1$}&\sigma_{01}\\ \sigma_{10}&\sigma_{11} \end{bmatrix} ~+~ \cdots \\ \cdots~ \begin{bmatrix} ~0~&~1~\\ ~1~&~0~ \end{bmatrix}\M[-0.25] &
\left( \begin{bmatrix} 0&-\sin(\vartheta)\\ \sin(\vartheta)&0 \end{bmatrix} \odot \begin{bmatrix} \text{\small$1$}&\sigma_{01}\\ \sigma_{10}&\sigma_{11} \end{bmatrix} \right)\M[-0.25] \begin{bmatrix} ~0~&~1~\\ ~1~&~0~ \end{bmatrix} ~.
\end{split}$$
As our next example, wherein the Hadamard product enables even greater simplifications, consider an “Ising-type” interaction between two qubits of the form $\SIG[3] \otimes \SIG[3] = \MAT P_{33}$, which is also known as “weak scalar coupling” in NMR [@ErnBodWok:87]. An operator sum expression for this could be obtained by expanding the general formula given in Eq. (\[eq:final2p\]), but because this Hamiltonian is again diagonal in the $\SIG[3]$ eigenbasis a simpler expression can be obtained directly starting from the diagonal matrix of the corresponding propagator, i.e. $$\begin{split}
\big|{\ensuremath\M[0.075]}\EMB e^{-\imath \MAT P_{33} \pi J t / 2} \RHO\M[0.1] \EMB e^{\,\imath \MAT P_{33} \pi J t / 2}\M[0.1] \big\rangle \M[0.5]=\M[0.5] &
\big( \EMB e^{\,\imath \MAT P_{33} \pi J t / 2} \otimes \EMB e^{-\imath \MAT P_{33} \pi J t / 2} \big)\M[0.1] \KET{\RHO} \\ =\M[0.5] &
\DMAT\big(\M[0.1] \KET{\MAT J(t)} \big) \KET{\RHO} \M[0.5]\equiv\M[0.5] \EMB{\ALG J}(t)\, \KET{\RHO} ~,
\end{split}$$ where $$\MAT J(t) ~\equiv~ \begin{bmatrix} 1 & e^{\,\imath \pi J t} & e^{\,\imath \pi J t} & 1 \\ e^{-\imath \pi J t} & 1 & 1 & e^{-\imath \pi J t} \\ e^{-\imath \pi J t} & 1 & 1 & e^{-\imath \pi J t} \\ 1 & e^{\,\imath \pi J t} & e^{\,\imath \pi J t} & 1 \end{bmatrix} .$$ On transforming this into the real domain via Eq. (\[eq:supopstfm\]), we find that $\EMB{\ALG J}$ has been converted into the sum of the diagonal matrix $$\EMB{\ALG C}(t) ~\equiv~ \DMAT\M[-0.25]\left(\M[0.25] \begin{picture}(1,50)(0,50) \thicklines \put(0,10){\line(0,1){84}} \end{picture} \begin{bmatrix} 1 & \cos( \pi J t) & \cos( \pi J t) & 1 \\ \cos( \pi J t) & 1 & 1 & \cos( \pi J t) \\ \cos( \pi J t) & 1 & 1 & \cos( \pi J t) \\ 1 & \cos( \pi J t) & \cos( \pi J t) & 1 \end{bmatrix} \begin{picture}(10,50)(0,39) \thicklines \put(0,0){\line(1,6){7}} \put(0,84){\line(1,-6){7}} \end{picture} \M[-0.25]\right) ~,$$ and the anti-diagonal matrix $$\MAT P_{15,0}\M[0.25] \EMB{\ALG S}(t)\M[0.1] ~\equiv~ \DMAT\M[-0.25]\left(\M[0.25] \begin{picture}(1,50)(0,50) \thicklines \put(0,10){\line(0,1){84}} \end{picture} \begin{bmatrix} 0 & \sin( \pi J t) & \sin( \pi J t) & 0 \\ \sin( \pi J t) & 0 & 0 & -\sin( \pi J t) \\ \sin( \pi J t) & 0 & 0 & -\sin( \pi J t) \\ 0 & -\sin( \pi J t) & -\sin( \pi J t) & 0 \end{bmatrix} \begin{picture}(10,50)(0,39) \thicklines \put(0,0){\line(1,6){7}} \put(0,84){\line(1,-6){7}} \end{picture} \M[-0.25]\right) ~,$$ where the (self-inverse) left factor of $\MAT P_{15,0} = \SIG[1]^{\otimes4}$ simply reverses the order of the rows. The nonzero entries of the Choi matrix of $\EMB{\ALG C}(t) + \EMB{\ALG S}(t)$ turn out to comprise two $4\times4$ blocks along the diagonal, which are exactly the two matrices above, i.e. $$\begin{split}
\EMB{\ALG C}(t) ~=~ & \DMAT\big( \big| \EMB{\ALG E}_{\EMB{\ALG C}\,} \EMB{Choi}( \EMB{\ALG C}(t) ) \EMB{\ALG E}_{\EMB{\ALG C}\,}^\top \big\rangle \big) \\ \text{and}\quad
\MAT P_{15,0}\M[0.25] \EMB{\ALG S}(t) ~=~ & \DMAT\big( \big| \EMB{\ALG E}_{\EMB{\ALG S}\,} \EMB{Choi}( \EMB{\ALG S}(t) ) \EMB{\ALG E}_{\EMB{\ALG S}\,}^\top \big\rangle \big)\M[0.1] ~,
\end{split}$$ wherein $$\EMB{\ALG E}_{\EMB{\ALG C}} ~\equiv~ \sum_{i=0}^3 \MAT e_i (\MAT e_i\M[0.1] \otimes \MAT e_i)^\top \qquad\text{and}\qquad \EMB{\ALG E}_{\EMB{\ALG S}} ~\equiv~ \sum_{i=0}^3 \MAT e_i\M[0.1] (\MAT e_i \otimes \MAT e_{3-i})^\top$$ project out the rows / columns of their respective blocks.
Thus we can obtain the desired operator sum representation by computing the eigenvalues and eigenvectors of the $4\times4$ symmetric matrices $$\MAT C(t) ~\equiv~ \EMB{\ALG E}_{\EMB{\ALG C}\,} \EMB{Choi}( \EMB{\ALG C}(t) ) \EMB{\ALG E}_{\EMB{\ALG C}\,}^\top \quad\text{and}\quad \MAT S(t) ~\equiv~ \EMB{\ALG E}_{\EMB{\ALG S}\,} \EMB{Choi}( \EMB{\ALG S}(t) ) \EMB{\ALG E}_{\EMB{\ALG S}\,}^\top ~,$$ letting the operators’ matrices be the diagonal / anti-diagonal matrices formed from the entries of these eigenvectors, and multiplying each term in the sum by the corresponding eigenvalue. The results are $$\EMB{\ALG C}(t)\, \KET{\SIG} \M[0.5]=\M[0.5] \Big|\, \HALF\, \big( 1 + \cos(\pi J t) \big)\, \SIG ~+~ \HALF\, \big( 1 - \cos(\pi J t) \big)\, \MAT P_{33}\M[0.2] \SIG\M[0.3] \MAT P_{33} \,\Big\rangle$$ and $$\begin{split}
\! \EMB{\ALG S}(t)\, \KET{\SIG} \M[0.5]=\M[0.5] \HALF\, \sin(\pi J t)\, \Big|\, \MAT P_{30}\M[0.25] \DMAT\big( [\M[0.3]1,\,1,\,1,-1\M[0.1]] \big)\M[0.2] \SIG\M[0.3] \DMAT\big( [\M[0.3]1,\,1,\,1,-1\M[0.1]] \big)\M[0.1] \MAT P_{30} &
~\cdots \\ \cdots~ -~ \MAT P_{30}\M[0.25] \DMAT\big( [\M[0.1]-1,\,1,\,1,\,1\M[0.1]] \big)\M[0.2] \SIG\M[0.3] \DMAT\big( [\M[0.1]-1,\,1,\,1,\,1\M[0.1]] \big)\M[0.1] \MAT P_{30}\Big\rangle & ~.
\end{split}$$ By using Eq. (\[eq:wellknown\]) to replace these operator sums by Hadamard products and taking advantage of the symmetry of $\MAT S(t)$, however, we can obtain an even simpler expression, namely $$\label{eq:simpler}
\big( \EMB{\ALG C}(t) + \EMB{\ALG S}(t) \big)\M[0.1] \KET{\SIG} ~=~ \big| \MAT C(t) \odot \SIG \,-\, \MAT S(t) \odot (\MAT P_{30}\M[0.15] \SIG\M[0.15] \MAT P_{30}) \big\rangle ~.$$
Finally, we show how one can also use Hadamard products with the real density matrix to describe simple relaxation processes, in a manner similar to that described in @HaShViCo:01 for the usual Hermitian density matrix. For a single qubit undergoing $T_1$ (dissipation) and $T_2$ (decoherence) relaxation, the time derivative is given by: $$\partial_{t\,} \SIG(t) ~=~ -\MAT R \odot \SIG(t) ~\equiv~ -\begin{bmatrix} 0&1/T_2\\ 1/T_2&1/T_1 \end{bmatrix} \odot \begin{bmatrix} \text{\small$1$}&\sigma_{01}(t)\\ \sigma_{10}(t)&\sigma_{11}(t) \end{bmatrix} ~.$$ Assuming that these relaxation processes are uncorrelated, this can immediately be extended to any number of qubits using the fact that the Hadamard product satisfies the mixed product formula with the Kronecker product (Eq. (\[eq:mixed\])). In the case of two qubits relaxing with Hadamard relaxation matrices $\MAT R^1$, $\MAT R^2$, for example, we obtain $$\begin{split}
\partial_{t\,} \SIG(t) \M[0.5]=\M[0.5] & -\! \big( \MAT R^1 \otimes (\MAT{11}^\top) + (\MAT{11}^\top) \otimes \MAT R^2 \big) \odot \SIG(t) \\ =\M[0.5] &
- \begin{bmatrix} 0&1/T_2^2& 1/T_2^1&1/T_2^1+1/T_2^2\\ 1/T_2^2&1/T_1^2& 1/T_2^1+1/T_2^2&1/T_2^1+1/T_1^2\\ 1/T_2^1&1/T_2^1+1/T_2^2&1/T_1^1&1/T_1^1+1/T_2^2\\ 1/T_2^1+1/T_2^2&1/T_2^1+1/T_1^2&1/T_1^1+1/T_2^2&1/T_1^1+1/T_1^2 \end{bmatrix} \odot\, \SIG(t) ~.
\end{split}$$ where $\MAT 1$ is a $4\times1$ vector of $1$’s. The fact that uncorrelated $T_1$ as well as $T_2$ relaxation can be extended so easily to multiple spins in this way is actually a significant advantage of the real density matrix over the Hermitian, since in the latter case the diagonal terms are mixtures of terms decaying at differing rates, substantially complicating their treatment via Hadamard products [@HaShViCo:01].
When correlations are present, however, these advantages are largely lost, since then the off-diagonal entries of the real density matrix consist of mixtures of terms with differing decay rates (fortunately, $T_1$ relaxation is usually largely uncorrelated [@ErnBodWok:87]). Let us work through the case of two qubits in detail, assuming for simplicity that the $T_2$ relaxation processes at the two qubits are totally correlated and have the same rate $1/T_2$, as for example in an NMR gradient-diffusion experiment [@HaShViCo:01]. In this case the Hadamard relaxation matrix for the Hermitian density matrix has the form $$\MAT R ~=~ \frac1{T_2} \begin{bmatrix} ~0~&~1~&~1~&~4~\\ ~1~&~0~&~0~&~1~\\ ~1~&~0~&~0~&~1~\\ ~4~&~1~&~1~&~0~ \end{bmatrix} ~,$$ and the corresponding $16\times16$ diagonal relaxation superoperator $$\EMB{\ALG R} ~=~ \DMAT\big({\ensuremath\M[0.075]}\KET{\MAT R} \big)$$ is easily exponentiated into a diagonal matrix of survival probabilities for the entries of the (traceless part of the) Hermitian density matrix. In this case, however, it turns out to be almost as easy, but more revealing, to convert $\EMB{\ALG R}$ into the real domain and perform the integration there. The result of the first step is $$\EMB{\ALG R} ~\stackrel{\ALG U}{\longleftrightarrow}~ \frac1{T_2}
\left[ \begin{smallmatrix}
0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\\[0.25ex]
0\:&\:1\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\\[0.25ex]
0\:&\:0\:&\:1\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\\[0.25ex]
0\:&\:0\:&\:0\:&\:2\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\!-2\:&\:0\:&\:0\:&\:0\\[0.25ex]
0\:&\:0\:&\:0\:&\:0\:&\:1\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\\[0.25ex]
0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\\[0.25ex]
0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:2\:&\:0\:&\:0\:&\:2\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\\[0.25ex]
0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:1\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\\[0.25ex]
0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:1\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\\[0.25ex]
0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:2\:&\:0\:&\:0\:&\:2\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\\[0.25ex]
0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\\[0.25ex]
0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:1\:&\:0\:&\:0\:&\:0\:&\:0\\[0.25ex]
0\:&\:0\:&\:0\:&\!-2\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:2\:&\:0\:&\:0\:&\:0\\[0.25ex]
0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:1\:&\:0\:&\:0\\[0.25ex]
0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:1\:&\:0\\[0.25ex]
0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0\:&\:0
\end{smallmatrix} \right] ~,$$ which again has nonzero entries only on its diagonal and its anti-diagonal. Representing the action of this matrix on $\KET{\SIG}$ as a sum of a Hadamard product and a Hadamard product coupled with row/column inversion as in Eq. (\[eq:simpler\]), we obtain the real equation of motion: $$-T_2\, \partial_t\, {\SIG}(t) \M[.5]=\M[.3] \begin{bmatrix} ~0~&~1~&~1~&~2~\\ ~1~&~0~&~2~&~1~\\ ~1~&~2~&~0~&~1~\\ ~2~&~1~&~1~&~0~ \end{bmatrix} \!\odot{\ensuremath\M[0.075]}\SIG(t) ~+~ \begin{bmatrix} ~0~&~0~&~0~&-2~\\ ~0~&~0~&~2~&~0~\\ ~0~&~2~&~0~&~0~\\ -2~&~0~&~0~&~0~ \end{bmatrix} \odot\, \big( \MAT P_{30}\M[0.1] \SIG(t) \M[0.1]\MAT P_{30} \big)$$ It may readily be verified that the operations on $\SIG$ which occur in the two terms of this expression commute, and hence this equation can be integrated by exponentiating them separately. With the first term this leads to a simple Hadamard (entrywise) exponential [@HaShViCo:01], namely $$\MAT D(t) ~\equiv~ \MAT{Exp}_{{\ensuremath\M[0.075]}\odot\!} \left( -\frac{t}{T_2} \begin{bmatrix} ~0~&~1~&~1~&~2~\\ ~1~&~0~&~2~&~1~\\ ~1~&~2~&~0~&~1~\\ ~2~&~1~&~1~&~0~ \end{bmatrix} \right) ~=~ \begin{bmatrix} ~1~&e^{-t/T_2}&e^{-t/T_2}&e^{-2t/T_2} \\ e^{-t/T_2}&~1~&e^{-2t/T_2}&e^{-t/T_2}\\ e^{-t/T_2}&e^{-2t/T_2}&~1~&e^{-t/T_2}\\ e^{-2t/T_2}&e^{-t/T_2}&e^{-t/T_2}&~1~ \end{bmatrix} .$$ To exponentiate the second term, we note that the Hadamard product is with $2\, \MAT P_{03} = 2\, \SIG[2]\otimes\SIG[2]$ and resort briefly to superoperators in order to simplify the exponential as follows: $$\begin{split} &
\MAT{Exp}\big( -\!(2t/T_2)\, \DMAT({\ensuremath\M[0.075]}\KET{\MAT P_{03}} )\, (\MAT P_{30} \otimes \MAT P_{30})\, \big) \\[0.5ex] =\M &
{\sum}_{k=0}^\infty\, \big( -\!(2t/T_2)\, \DMAT({\ensuremath\M[0.075]}\KET{\MAT P_{03}} )\, (\MAT P_{30} \otimes \MAT P_{30}) \,\big)^k \big/\, k! \\[0.5ex] =\M &
\begin{aligned}[t] \MAT P_{00} \otimes \MAT P_{00} ~+~ \DMAT({\ensuremath\M[0.075]}\KET{\MAT P_{30}} ) \sum_{k=1}^\infty\, \frac{{(2t/T_2)}^{2k}}{2k!} ~\cdots & \\
-~ \DMAT( \KET{\MAT P_{03}} )\, & (\MAT P_{30} \otimes \MAT P_{30}) \sum_{k=0}^\infty\, \frac{{(2t/T_2)}^{2k+1}}{(2k+1)!}
\end{aligned} \\ =\M &
\begin{aligned}[t] \MAT P_{00} \otimes \MAT P_{00} ~+~ \DMAT({\ensuremath\M[0.075]}\KET{\MAT P_{30}} )\, (\cosh( 2t / T_2 ) \,-\, 1) ~\cdots & \\
-~ \DMAT({\ensuremath\M[0.075]}\KET{\MAT P_{03}} )\, & (\MAT P_{30} \otimes \MAT P_{30})\,\sinh( 2t / T_2 ) ~.
\end{aligned}
\end{split}$$ This derivation relies on the facts that $\MAT P_{30} \otimes \MAT P_{30}$ squares to the identity, $\DMAT(\KET{\MAT P_{03}})$ squares to $\DMAT(\KET{\MAT P_{30}})$, and each commutes with the other.
Going back to operator sum notation and abbreviating $\SIG \equiv \SIG(0)$, we thus obtain in all: $$\begin{aligned}
&
\SIG(t) \M[.5]=\M[.5] \begin{aligned}[t] \MAT D(t) \odot \big( \SIG \:+\: (\cosh(2t/T_2) - 1)\, \MAT P_{30} \odot \SIG ~\cdots &
\\ -~ \sinh & (2t/T_2)\,\MAT P_{03} \odot ({\ensuremath\M[0.075]}\MAT P_{30}\, \SIG\, \MAT P_{30} {\ensuremath\M[0.075]}){\ensuremath\M[0.075]}\big)
\end{aligned} \notag \\[1ex] =\M[.5] &
\begin{bmatrix}
\sigma_{00} & \sigma_{01}\, e^{-t/T_2} & \sigma_{02}\, e^{-t/T_2} & \Delta(t;\, \sigma_{03},\, \sigma_{30}) \\[.5ex]
\sigma_{10}\, e^{-t/T_2} & \sigma_{11} & \Delta(t;\, \sigma_{12}, -\sigma_{21}) & \sigma_{13}\, e^{-t/T_2} \\[.5ex]
\sigma_{01}\, e^{-t/T_2} & \Delta(t;\, \sigma_{21},\, -\sigma_{12}) & \sigma_{22} & \sigma_{23}\, e^{-t/T_2} \\[.5ex]
\Delta(t;\, \sigma_{30},\, \sigma_{03}) & \sigma_{31}\, e^{-t/T_2} & \sigma_{32}\, e^{-t/T_2} & \sigma_{33}
\end{bmatrix} ,\end{aligned}$$ where $\Delta(t;\, x,\, y) \equiv (\cosh(2t/T_2){\ensuremath\M[0.075]}x + \sinh(2t/T_2)\, y)){\ensuremath\M[0.075]}\exp(-2t/T_2)$. From this we see that the anti-diagonal entries decoher into mixtures with their symmetrically placed opposites in the real density matrix. These mixtures correspond to the real and imaginary parts of the $\rho_{12} = \bar\rho_{21}$ entries in the Hermitian density matrix, otherwise known as *zero-quantum coherences*, which are immune to correlated noise [@ErnBodWok:87].
Epilogue
========
We have seen that one can, with some effort, do pretty much everything with the real density matrix that one could with the usual Hermitian one. This may be useful as a didactic device, or in calculations with experimental (e.g. NMR) data where it is desirable to keep the experimentally measured values of the observables in sight at all times. This work is also a good demonstration of the power of Choi matrix decompositions as a means of finding operator sum representations of linear superoperators [@Havel!QPT:03].
Although the Hermitian density matrix is expected to be better suited, by and large, for the purposes of numerical calculations, it is worth emphasizing that for theoretical and/or expository purposes the compact but lucid notation of geometric algebra offers significant advantages over any matrix formalism. In this regard, we point out that @HavDorFur:03 have recently introduced a *parity-even* (rather than reverse-even, aka Hermitian) multi-qubit density operator via geometric algebra, which generalizes the *multi-particle space-time algebra* introduced for isolated systems to open multi-qubit systems. It is our hope that in due course such a geometric formulation may provide new insights into some of the conceptual problems that underlie quantum physics.
The existence of the real density matrix is further of some theoretical interest, since it provides a coordinate ring within which one can study the issues of entanglement and decoherence via invariant theoretic methods [@GraRotBet:98; @Makhlin:02]. There are intimate connections between invariant theory and geometric algebra, and it is often easier to automate symbolic computations in an invariant ring than it is at the more abstract level of geometric algebra [@Sturmfels:93; @Havel:97; @Havel:01].
The author thanks Nicolas Boulant, David Cory and Chris Doran for useful discussions. This work was supported by ARO grants DAAD19-01-1-0519, DAAD19-01-1-0678, by DARPA grant MDA972-01-1-0003, and by a grant from the Cambridge-MIT Institute, Ltd.
[^1]: corresponding author.
[^2]: The factor of $1/2$ here does not mean that the rotation is spinorial, but rather that the rate of rotation is $1/\sqrt2$ times the Hilbert-Schmidt norm of the Pauli matrix that generates it, while we pick up another factor of $1/\sqrt2$ on transforming to the real domain.
|
---
author:
- 'Shin-Yi Lin, Nagayoshi Ohashi, Jeremy Lim, Paul T. P. Ho, Misato Fukagawa, and Motohide Tamura'
bibliography:
- 'mnemonic.bib'
- 'abaur.bib'
title: Possible Molecular Spiral Arms in the Protoplanetary Disk of AB Aur
---
Introduction
============
Circumstellar disks of low and intermediate mass pre–main-sequence stars, namely T Tauri stars and Herbig Ae/Be stars, respectively, are generally considered to be the birth places of planetary systems [cf. @Zuckerman:2001]. Knowledge of the disk structure and kinematics is vital for understanding the physical environment of planet formation. Herbig Ae stars, in particular, are considered to be the progenitors of prototypical Vega-like stars, which are surrounded by debris-disks. These disks are believed to be produced by colliding planetesimals as well as perhaps planets. How does a circumstellar disk containing gas and dust evolve into a disk containing primarily dust? By observing the evolving circumstellar disks of Herbig Ae stars and looking for evidence for actual formation of (proto)planets, we hope to provide instructive clues to the process of planet formation. This work may provide a snapshot at or close to the starting point.
AB Aur [V = 7.06 $\pm$ 0.06, spectral type = A0 Ve + sh, d=144pc, @vandenAncker:1997] is one of the nearest and best studied Herbig Ae stars. With an age of 2 - 5 Myr [@DeWarf:2003], it not only has a bright dust disk, but is also rich in gas. It is thus an ideal place to study the initial condition of planet formation. The moderate inclination angle also makes it a desirable source for imaging. Previous millimeter wavelength observations with an angular resolution of 5$''$ made by [@Mannings:1997] with the OVRO interferometer in CO (1-0) revealed a velocity gradient along the major axis attributed to a disk in Keplerian motion. Optical imaging with the HST [@Grady:1999] showed a north-south asymmetry in the dust disk. Near-IR imaging with the Coronographic Imager with Adaptive Optics on the Subaru Telescope shows an even more complicated structure [@Fukagawa:2004 hereafter F04, see Figure \[Fig 1\]]. F04 identified at least two apparent spiral-like structures in the dust disk: a prominent inner arm with a radius of 230 AU from the east to northeast, and an outer arm with a radius of 330 AU from the south to northeast. We will use “the inner arm” and “the outer arm” to refer to these two arms respectively hereafter. AB Aur is one of the few stars around which prominent spiral structures have been discovered in the circumstellar dust [c.f. @Clampin:2003]. However, the near-IR image traces scattered light from the dust disk, and provides no information beyond the surface of the optically thick dust disk. The apparent spiral structures may just be surface features, and the existence of density spiral arms need to be verified. It would also be intriguing to measure the kinematics of the arms and to look for streaming motion, if any, along or across the spirals to examine the possibility of dynamical perturbations from underlying planetary bodies.
Imaging at millimeter or sub-millimeter wavelengths with higher angular resolution is able to trace the dust and molecular gas emission directly from the disk, allowing us to study both its structure and kinematics in greater detail. Since the best millimeter images available at the time we started our project was the 5$''$ OVRO observation [@Mannings:1997], we surmised that the complex disk morphology expected from the Subaru image were smoothed by the large synthesized beam. Here, we present submillimeter imagings of the dust and gas disk of AB Aur at a wavelength of 850 $\micron$ with the Submillimeter Array [SMA[^1], @Ho:2004], with an angular resolution of 1$''$ to unveil the hidden information.
Millimeter observations conducted at the same time as this work, by [@Corder:2005] and Piétu et al. (2005, hereafter P05) with comparable angular resolutions, revealed interesting results. [@Corder:2005] obtained images of the $^{13}$CO(1-0) line and 2.7 millimeter continuum emission at an angular resolution of $\sim$2$''$, and images of $^{12}$CO(1-0) and C$^{18}$O(1-0) line emission at lower angular resolutions with the OVRO. After subtracting a best-fit Gaussian source from their 2.7 mm dust continuum map, Corder et al. see weak residuals that appear to roughly coincide with the spiral arms seen in the near-IR. With 2$''$ angular resolution, the $^{13}$CO(1-0) emission is satisfactorily fit by a disk with radius of $\sim615$ AU in Keplerian rotation, as the early OVRO observation had claimed [@Mannings:1997]. Their $^{12}$CO(2-1) map observed at $\sim2''$, however, can not be fit by a pure-disk model. Corder et al. attributed the discrepancy to the contamination of a large scale envelope. [@Pietu:2005], observed the circumstellar disk of AB Aur with the PdBI in $^{13}$CO(2-1) and in continuum at 1.4 mm with the highest angular resolution currently available ($\sim0''\!\!.59$), as well as other CO isotopes at lower resolution. They discovered asymmetric distribution of dust continuum emission, which follows inwards along the spiral-like features. However, their results of the $\chi^2$-fittings of the $^{13}$CO gas kinematics showed a velocity profile shallower than Keplerian. On the other hand, the $^{12}$CO velocity law is much steeper than Keplerian. Piétu et al. also interpreted the non-Keplerian kinematics as the results of contamination from the extended envelope but not from the disk itself.
In contrast with the contemporary OVRO and PdBI observations, our work presents the highest resolution images of a higher transitional line ($J=3-2$) of $^{12}$CO. Because AB Aur is still embedded in a relatively extended envelope [e.g. @Grady:1999; @Semenov:2005], observations of molecular gas at its higher transitions (which, for the same molecule, traces denser and/or warmer gas), should suffer less contamination from the surrounding envelope and therefore better trace the disk. Moreover, flux density measurement at high frequency extends the calculation of the spectral index to the sub-mm regime. With the capability of the SMA to observe at higher frequency and high angular resolution, our goal is to unveil the hidden structure and kinematics of the possible spiral arms.
Observations
============
We observed AB Aur over three different tracks in the dust continuum at 345 GHz and the $^{12}$CO (3-2) rotational transition simultaneously with the SMA. The observations were made over the period from Dec 2003 to Sep 2004, with one compact and two extended array configurations. The projected shortest and longest baselines are $\sim$14 m and $\sim$220 m, respectively. Data obtained from seven of the eight antennae were usable. The resultant size of the synthesized beam was $1''\!\!.0\times~0''\!\!.72$ (144AU $\times$ 115AU) with natural weighting (P.A.$\sim$ -58$^{\circ}$). The correlater was configured to provide a narrow band of 512 channels over 104 MHz and hence a velocity resolution of 0.17 km s$^{-1}$ for the CO(3-2) line, and an overall bandwidth of 2 GHz in each of the lower and upper sidebands for measuring the dust continuum from the line-free channels.
Bandpass calibration that calibrates the instrumental response of different velocity channels was done with Jupiter and quasar 2232+117 at different observing epochs. Flux calibration was made with Uranus and the moon of Jupiter, Callisto. Amplitude and phase calibration was done with either the strong quasar 0359+509 or 3C84 on different observation epochs, which have an angular separation of 21.7$^{\circ}$ and 26.4$^{\circ}$ from AB Aur, respectively. We used either the weaker quasar 3C 111 or 3C 120, with angular separation of 10.7$^{\circ}$ and 25.6$^{\circ}$, respectively, from AB Aur, to verify the quality of the phase referencing from both 0359+509 and 3C 84. The maps of both 3C111 and 3C120 look like point sources at phase center, as would be expected for proper phase referencing, verifying that our positions are accurate to 0$''\!\!.1$. The same calibration scheme was applied to AB Aur. We used the software MIR, adapted for the SMA from an IDL-based package originally developed for the OVRO array [@Scoville:1993], for calibrating the data. We then used the NRAO AIPS package to produce images of both the continuum and molecular line.
Results
=======
Dust Continuum Emission
-----------------------
We measured a total flux density of 235 $\pm$ 42mJy for the dust continuum emission within a radius of $\sim$350AU from the central star. This is about 65$\%$ of the flux density observed by the JCMT [@Mannings:1994]. It is not surprising since the JCMT is sensitive to the extended envelope as well as the circumstellar disk while the extended envelope was resolved out by the SMA. Note that the JCMT primary beam is even smaller than the SMA primary beam, which implies that the missing flux may be a lower limit.
The dust continuum map made with natural weighting is superposed on the Subaru near-IR image in the left panel of Figure 1. The continuum emission has an overall size of 450AU $\times$ 270AU (P.A. 66$^{\circ}$), derived by a Gaussian fit to the data of the most compact array configuration, which only marginally resolved the disk. The emission does not peak at the stellar optical position, nor does it exhibit an intensity distribution decreasing monotonically in the radial direction, as is generally found for circumstellar disks associated with pre–main-sequence stars. Instead, the disk shows two distinct peaks - one the northeast direction (NE) and the other southwest (SW) - and an extension towards the northeast (labeled ET in Figure 1). The two peaks form a ring-like structure with a radius of 150 AU (measured at the maximum of the two peaks) centered at the stellar position. They also coincide with the inner region of the prominent inner arm inferred by F04 from the Subaru image, as suggested by P05. Compared with the 1.4 mm continuum image of the dust disk made by P05, we find that the SW peak is in agreement with the local column density enhancement $\sim$1$''$ west from the center in their 1.4 mm map, while there is no apparent peak (no local maxima) in that map at the NE position.
The northeast extension ET, which does not have a counterpart in the PdBI map, appears to be closely aligned with the outer region of the same spiral arm traced by the NE and SE peaks closer to the star. Because of missing short spacings with the SMA, large scale structures are not well sampled, which produces large-scale corrugated features at low levels in our map. We suppressed this effect by removing data points with uv distances smaller than 30k$\lambda$. The right panel of Figure \[Fig 1\] presents the resultant map superposed on the Subaru image. The northeast extension, which was a 5$\sigma$ detection, disappears in this image, suggesting that the ET feature is possibly part of a large scale disk or envelope. On the other hand, this extension is close to our CO spiral arms (see Section 3.2), suggesting that it may be an enhancement in a larger-scale disk, which is difficult to image against possible enhancements in a large-scale envelope.
Molecular Gas
-------------
$^{12}$CO (3-2) traces warmer and denser regions of the molecular gas disk as compared to the lower CO transitions. The JCMT observations of $^{12}$CO(3-2)/$^{13}$CO(3-2) line ratio by [@Thi:2001] show that the optical depth of $^{13}$CO is 0.21. Assuming a $^{12}$C/$^{13}$C isotopic ratio of 60, we find that the $^{12}$CO emission is optically thick, and will likely trace the gas temperature distribution close to the disk surface (c.f. P05). In contrast, the dust emission at 850 $\micron$ is likely to be optically thin and hence a good tracer of column density. The measurements of dust and CO emission are complementary, and together they provide a more complete picture of the properties of the disk. Note that the optical depth computed here with the observation of Thi et al. is an average of the disk and any envelope component, since the observation was done by single dish telescope with a resolution of $\sim$ 14”. The computed high optical depth does not imply that the disk is optically thick everywhere. At some locations, either or both the column density and excitation temperature may not be high enough to make $^{12}$CO (3-2) optically thick.
We detected significant emission ($>$3$\sigma$) in velocity channels between 4.1 km s$^{-1}$ to 8.0 km s$^{-1}$ with a peak brightness temperature of 34.3 $\pm$ 4.0K. We only detect 20% of the CO(3-2) flux measured by the JCMT [@Thi:2001], suggesting that the single-dish observation contains considerable emission contributed from the large scale envelope. The CO line emission did not suffer from the large-scale ripple effect as the dust emission, perhaps because in a given velocity channel the CO emission is not as extended as the dust. Therefore the inner 30k$\lambda$ data were retained for maximum sensitivity. The left panel of Figure \[Fig 3\] shows the integrated intensity map of CO over the velocity channels from 4.1 km s$^{-1}$ to 8.0 km s$^{-1}$, superposed on the mean velocity map in false color. These maps were made using “MOMNT” in AIPS to bring out non-random signals and suppress random noise fluctuations. The gas disk with a FWHM size of 530AU $\times$ 330AU (P.A. $\sim$ 66.7$^{\circ}$), derived from Gaussian fitting of the data of the lowest resolution data set, appears to be larger than the dust disk. By contrast with the dust disk, the gas disk exhibits a peak at the stellar position (right panel of Figure \[Fig 3\]). Since $^{12}$CO(3-2) is optically thick and the dust continuum at 850 $\micron$ is optically thin, this suggests that the central peak in the gas disk is a temperature effect. Indeed, the gas disk also shows a central depression in optically thinner lines, such as $^{13}$CO and C$^{18}$O (P05), consistent with this interpretation.
There is a secondary peak in the $^{12}$CO(3-2) intensity map coincident with the NE peak in the dust disk, which possibly traces the inner spiral arm (see the right panel of Figure \[Fig 3\]). The $^{12}$CO(2-1) emission observed by P05 does not peak at this position. This is most probably due to insufficient angular resolution of their $^{12}$CO(2-1) map. When we convolve our $^{12}$CO(3-2) map with the same beam size of the $^{12}$CO(2-1) map ($2''\!\!.0\times~1''\!\!.6$), the secondary peak disappears and the resultant map looks similar to the PdBI map.
As shown in the left panel of Figure \[Fig 3\], the largest velocity gradient is along the major axis, consistent with the pattern of a rotating disk. Therefore the molecular gas emission is interpreted as a large rotating disk as previous millimeter observations have shown [@Mannings:1997; @Corder:2005; @Pietu:2005]. Although the bulk motion of the gas disk looks agreeable with rotation, there are also velocity gradients along the minor axis. In particular, at the south end of the minor axis, there is an abrupt change in velocity, suggestive of non-circular motion.
Figure \[Fig 2\] shows channel maps of the $^{12}$CO(3-2) emission superposed on the Subaru image. We found a general velocity gradient along the major axis as seen in the MOM 1 map. However, the maps do not show a simple “butterfly” diagram as seen in the $^{13}$CO maps of OVRO [@Corder:2005] and PdBI (P05) at comparable angular resolution, or the $^{12}$CO maps at lower transitions at lower angular resolution. We found that in several velocity channels - mainly from 4.8 km s$^{-1}$ to 6.2 km s$^{-1}$ - the CO emission traces the inner spiral arm inferred by F04 from scattered light. The CO emission in the 8.0 km s$^{-1}$ channel coincides with the outer arm, but it is only of 4 sigma significance and is the only channel tracing the outer arm. There are also many channels, most notably at 5.0 km s$^{-1}$, 6.7 km s$^{-1}$ to 7.3 km s$^{-1}$, and 7.8 km s$^{-1}$, where the emission extends well outside of the near-IR images. These features suggest that the CO emission traces extended structure, presumably the envelope or the outer region of a much larger CO disk, as well as a more compact disk. A simple model to better delineate which parts of the emission in the channel maps show non-circular or non-Keplerian rotation will be discussed in the next section.
Discussion
==========
Disk Mass and SED
-----------------
Based on single dish observations, [@Acke:2004] deduced the power-law index $n$ in $\lambda F_{\lambda}
\propto \lambda ^{-n}$ for the spectral energy distribution (SED) of AB Aur to be as high as $4.32$, suggesting a particle size distribution dominated by relatively small grains and therefore little grain growth. However, the flux of dust continuum emission we observed is only 65% of the single dish measurement. This implies that the single dish data are also sensitive to the large scale envelope, where there may be indeed relatively little if any grain growth.
We derived a spectral index $n = 3.70 \pm 0.21$ by fitting the total flux density from interferometers with a typical angular resolution of 1$''$ at 850 $\micron$ (SMA), 2 mm (NMA; Ohashi et al. in preparation), 1.4 mm and 2.8 mm (PdBI, P05). The obtained value of $n$ is only 3 $\sigma$ different from the single dish result. Therefore there may be more grain growth than [@Acke:2004] thought, even though this number still suggests a relatively primitive disk [e.g. see @Pietu:2005]. Note that the power-law index is averaged over the entire disk, and given its observed structure at millimeter and submillimeter wavelengths is weighted toward the outer region. There may still be significant grain growth further in. We infer a disk mass of $0.0075 \pm 0.0030
{\rm{M}_\odot}$, assuming the emissivity $K(\nu) = 0.1(\nu / 10^{12} Hz)^{\beta}$ cm$^{2}$ g$^{-1}$ [@Beckwith:1990], a dust temperature of 26K derived from the average intensity of the CO (3-2) emission, and a gas-to-dust mass ratio of 100. Assuming the dust continuum emission to be completely optically thin, i.e. $ n = \beta + 3$ [@Beckwith:1990], the dust emissivity index $\beta$ was estimated to be 0.70 from the spectral index *n*. The disk mass is consistent with the OVRO observations [0.009${\rm{M}_\odot}$, @Corder:2005], but is smaller than the mass P05 obtained by a factor of 2. Note that the $\beta$ Piétu et al. derived ($\beta \sim 1.4$) from their 1.4mm and 2.8mm dust maps is without the optically thin assumption and hence is larger. If we use the same $\beta$ as Piétu et al. used, the derived disk mass is consistent (0.016${\rm{M}_\odot}$ $\pm$ 0.006${\rm{M}_\odot}$).
Kinematics of the Spiral Arms
-----------------------------
P05 has made a $\chi^2$ model fitting for the disk from their $^{12}$CO/$^{13}$CO observations at an angular resolution of around 1$''$. The deduced velocity laws all deviate from Keplerian motions. The radial velocity profile is shallower than Keplerian from $^{13}$CO, but steeper from $^{12}$CO. Their $^{13}$CO(2-1) results are probably the best data on which to base such a model, due to good signal-to-noise ratio, and the highest angular resolution currently available. In addition, being optically thin allows this line to sample the entire depth of the disk. Given the limitation of our data quality, our goal here is not to derive a dynamical model of the disk. Instead we intend to illustrate that the motion of the disk - in particular the part that traces the main spiral arm - clearly deviates from Keplerian. By comparing our data with a simple disk model with a power-law distributed intensity and Keplerian rotation, we attempt to identify which parts of the emission in the channel maps exhibit non-Keplerian motion. We use the best fit parameters derived from the $^{13}$CO(2-1) observations done by P05 to define our model: a central mass of 2.2 ${\rm{M}_\odot}$ and an inclination angle (*i*) of 42$^{\circ}$. The outer radius for the disk is assumed to be $530$ AU, as inferred by our $^{12}$CO data. We also fit a power-law distribution of $^{12}$CO(3-2) brightness temperature of $T = T_{100} (r/100AU)^{-0.47\pm
0.02}$, $T_{100} \sim 23.4 K$. Figure \[Fig 4\] shows the observed and the modeled channel maps. The observational data, plotting from 2$\sigma$ ($\sim$0.5Jy), are shown in color. The white contours indicate the 10%, 30%, and 70% intensity levels of emission of the model. Since the dynamical range of every velocity channel is never over 10, emission that falls within the outermost contours is consistent with the interpretation that the gas motion follows Keplerian rotation law. On the other hand, emission that falls outside this boundary must deviate significantly from the gas motion defined by our model disk.
As shown in Figure \[Fig 4\], most of the observed emission falls inside the boundary. This suggests that the bulk motion of the disk is consistent with Keplerian, and thus is consistent with being gravitationally bound by the central star. However, there are excess emission components which can not be explained by the Keplerian kinematics, especially in the velocity channels from 5.5 km s$^{-1}$ to 6.0 km s$^{-1}$. Moreover, the excess emission components are spatially consistent with the inner spiral arm. This is where the emission coincides with the inner spiral arm. The dynamical ranges of these four channels are no more than 3, suggesting that the deviation from Keplerian is probably even more substantial than this simple model comparison suggests. If we use a power law radial dependence for the velocity with an index of $-0.42$ which P05 derived from the $^{13}$CO(2-1) observation, the inner arm still falls outside the model boundary. The inner spiral arm seen in the scattered light image of F04 is therefore a physical arm which has different kinematics as compared with the bulk of the disk. There is also deviation from Keplerian motion in the 8.0 km s$^{-1}$ channel which might trace the outer arm. Note that even though the emission falls inside the outermost contours, the emission peaks often do not coincide with the predicted positions of emission maxima (Figure \[Fig 4\]). We made a similar comparison with the $^{13}$CO data of P05. The bulk motion of the emission falls inside the contours, and the emission maxima are consistent with the positions predicted by the model. This implies that what we are really seeing in $^{12}$CO is largely a temperature effect. The observed differences strongly suggest that our $^{12}$CO(3-2) map preferentially traces elevated gas temperatures at or close to the surface of the disk.
We show a position-velocity (P-V) diagram in Figure \[Fig 19\] to illustrate the position of the excess emission more clearly. Note that the cut along the major axis (P.A.$\sim67^{\circ}$) does not pass through the bulk of the emission tracing the main arm, and the non-Keplerian features are not prominent along this particular cut. Therefore, we have selected another cut with P.A. $\sim50^{\circ}$, which passes through the bulk of the emission in the arm, and compared the P-V diagram of the Keplerian model along the same cut. The P-V diagram clearly exhibits anomalous emission which can not be explained by the Keplerian motion at the position of the inner arm. This is consistent with what we found in the channel maps. Most importantly, the emission of the deviation is red-shifted with respect to the expected Keplerian motion. This red-shifted motion can be explained by possible streaming motion along or across the spiral arms. Spiral arms are the product of nonaxisymmetric gravitational potential (either due to the presence of a planet or due to self-excited gravitational instability), and hence would be expected to show radial motion in addition to the circular Keplerian motion. In the case of AB Aur, based on the Subaru image, the bright southeast side is the near side. Together with the information from our MOM1 map, the spiral arms are trailing, and the motion across the arms is radially outward. At P.A$\sim50^{\circ}$, as shown in Figure \[Fig 18\], the outward streaming motion on the spiral arm contributes an additional red-shifted component in the line-of-sight velocity, which accounts for the anomalous emission. However, on the premise that our model is correct, if the excess emission is indeed the result of outward streaming motion, the cause of the large magnitude outward motion ($\sim$ 1 km s$^{-1}$, from comparing the peak position of a Keplerian disk) may be enigmatic. We should be careful that the above statement provides only a qualitative argument about how the materials around the spiral arms deviate from Keplerian motion, and needs to be verified by a more detailed model.
We also take notice of other possibilities to interpret the non-Keplerian kinematics. For example, in the case of HL Tau, various motions of the circumstellar gas that may happen around an embedded young system, such as outflows [@Cabrit:1996] and infalls [@Hayashi:1993] in addition to the large-scale Keplerian rotation [@Sargent:1987], have been proposed to explain the complex kinematics. In AB Aur case, there is no evidence for any ongoing or recent outflows, or any ionized jet seen at centimeter wavelengths, or any evidence for molecular outflows. Although AB Aur is still embedded in a large scale envelope, we expect that the interferometer will filter out the extended structure larger than 13” (radius $>$900 AU). In addition, the fact that the CO (3-2) line is a better tracer of the warmer and denser region of the molecular gas, suggests that the non-Keplerian motions may be mainly from the disk itself. The anomalous emission coincides so well with the inner spiral arm, that the emission could come from the spiral arm.
Physical Nature of the Spiral Arms
----------------------------------
The disk of AB Aur exhibits great complexity in multi-wavelength studies by various authors. The SMA observation show clearly asymmetric structures and disturbed kinematics, which may interpret as a spiral arm in the high resolution $^{12}$CO(3-2) maps. In $^{13}$CO observations with better resolution and higher dynamical range done by PdBI (P05), the channel maps resemble a butterfly diagram, and the signature of a spiral arm is not as clearly seen as in $^{12}$CO(3-2). Although in some velocity channels there are extended structures which may be contributed by the inner spiral arm. On the other hand, in the $^{12}$CO(3-2) channel maps, the emission patterns are nothing like the standard butterfly diagram. Note that $^{12}$CO(3-2) is the higher transition of the main CO isotope, and it traces the emission closer to the warmer disk surface heated by the central star, while the optically thin $^{13}$CO(2-1) and $^{13}$CO(1-0) lines trace emission all the way to the cold midplane of the disk. Hence, the overall discrepancy of the channel map patterns, as well as the spiral arms in $^{12}$CO(3-2) may be due to temperature enhancement rather than column density effect. Moreover, this could be a “surface” effect. The $^{12}$CO emission is more optically thick and weighted towards the surface layers, whereas the $^{13}$CO emission is more weighted towards deeper cooler layers. The lack of dust coinciding with the spiral arm also supports this argument (right panel in Figure \[Fig 3\]).
From the optically thin dust continuum observation at 850 $\micron$, we also found a central depression and asymmetric local enhancement in density that forms a ring-like structure which may be the inward extension of the inner spiral, as P05 also suggest in their 1.4 mm map. This ring-like structure, however, should be due to the column density effect. The NE peak in the 850 $\micron$ dust continuum map is absent in the 1.4 mm dust map. This suggests that the SED at this position may be quite different from other parts of the disk. Grain growth or complexity in chemical composition resulting in different dust emissivity may be the cause.
In summary, the observational results may be interpreted as a slightly higher density distribution at the spiral position, where the density contrast against the rest of the disk is not high enough to be picked out in optically thin lines. In addition, the temperature at the position of the spiral arm may be higher than the rest of the disk. Gas in the arms may be compressed and heated. There may also be a larger scale height at the spiral position so that material therein near the disk surface will receive more direct radiation from the central star, and become warmer. Hence, optically thick lines such as $^{12}$CO(3-2), can be more sensitive to the spiral structure.
Excitation Mechanisms
---------------------
Both F04 and P05 discussed the possibility of exciting the spiral-like structure by the self-gravity of the disk. With the surface density derived from our dust continuum observation, we can estimate the Toomre’s Q parameter: $$Q \sim \frac{c_s \kappa}{\pi G \Sigma}~,$$ where c$_s$ is the sound speed, $\kappa$ is the effective angular velocity which is $\sim$ $\Omega$ (Keplerian angular veolcity), and $\Sigma$ is the surface density. We can thus evaluate the disk stability at least in the inner 200AU region of the dust disk where the emission extends out to the inner spiral arm and further. We fit a power-law distribution of CO (3-2) brightness temperature of T = T$_{100}$ (r/100AU)$^{-0.47\pm0.02}$. As a zeroth order estimation, we assume that the dust temperature is coupled to that of the gas. At the radius of 150AU, with a surface density of 2.74 $\pm$ 1.04 g cm$^{-2}$ (K $\sim$ 0.05 cm$^2$ s$^{-1}$; gas-to-dust ratio of 100) and $T_{gas}\sim T_{dust}\sim 20.3\pm6.6 K$, we derived a Q of 9.5 $\pm$ 4.0. The Q value we derive is about 2$\sigma$ larger than unity, the value at or below which the disk is gravitationally unstable. Note that P05 also found a Q value $\sim$ 11 at 100AU with a much smaller uncertainty, certainly higher than unity. Therefore the gravitational instability mechanism is not favored by current observations, even though we also note that the disk mass estimation we performed still has large ambiguity because of the missing flux and uncertainty of the dust emissivity.
[@Pietu:2005] discussed the possibility of excitation by a coeval stellar companion. A number of optical/infrared studies place stringent mass limits on any such stellar companion (see P05). Instead, the spiral arms may be excited by a giant planet or planets. A giant planet inside a circumstellar disk can excite density waves outward, and hence produce a prominent one-arm spiral from its position. Consequently, if there is a (forming) giant planet at about several tens AU to a hundred AU away from AB Aur, the inner arm can be formed and maintained by the density waves it excites. A giant planet with an orbit on a different plane from the circumstellar disk may also excite bending waves that puff up the spiral so that at the spiral position the tilted surface can receive more stellar radiation and become warmer, and hence can be detected by the higher transitions of CO. The relatively large inner (column density) depression in the dust and gas disk provides additional evidence for a giant planet. For example, the depression of emission in the central dust disk resembles the central cavity of Vega-type stars: the inner radii of the ring-like structure around AB Aur and the asymmetric structure around Vega-type stars are of the same order ($\sim$70 - 80AU). This phenomenon can also be associated with the existence of a giant planet at about several tens of AU to $\sim$ 100 AU to clear out a cavity. Note that the central depression of dust continuum does not have to imply a cavity (hole). For example, a similar structure can be created with the presence of a spiral arm or a ring which enhances local column density further out in the disk.
Summary
=======
Our SMA observations of the Herbig Ae star AB Aur, on the dust continuum emission at 850$\micron$ and the $^{12}$CO(3-2) line emission, resolve its circumstellar disk at an angular resolution of $1''\!\!.0 \times 0''\!\!.7$. The disk is much more complicated than ordinary Keplerian disks observed around other Herbig Ae stars. The observations reveal that:
1. The dust emission exhibits an asymmetric local density enhancement that forms a ring-like structure with a central depression. The derived disk mass is 0.0075 - 0.016 ${\rm{M}_\odot}$, for a dust emissivity index of 0.7 - 1.4. This is consistent with the results of [@Corder:2005] and [@Pietu:2005].
2. In addition to the gas motion delineated by the best fit disk model of P05, we also found gas components associated with the spiral arms identified in the near-IR image. Their motion deviates from the Keplerian kinematics expected from the stellar mass of AB Aur. Streaming motion along and across the spiral arm may account for the deviation.
3. The radial outward motions of the spiral arms as well as a central cavity within the disk, suggest the possible existence of a giant planet forming in the disk. This is consistent with a relatively small mass for the disk, which is probably not large enough to excite spiral arms due to gravitational instability.
Acknowledgement: We thank M. Momose, F. Shu, and C. Yuan for fruitful discussion. N.O. was supported in part by NSC grant NSC93-2112-M-001-042.
![Left: CO moment maps. The contours represent the CO integrated intensity (zeroth moment), with 0.2Jy Beam$^{-1}$ $\times$ km s$^{-1}$ (1$\sigma$) spacing. Here the “noise” of the final integrated map was estimated from integrating the r.m.s. noise of 6 channels with a bandwidth of $\sim$1km s$^{-1}$, because at any position the line width is smaller than 1km s$^{-1}$. The color scale map is the mean velocity (first moment) map from 4.2km s$^{-1}$ to 8.0km s$^{-1}$. Right: CO MOM0 map superposed on dust continuum map. The off-center peak of the CO is very close to the NE peak of the dust continuum emission. Crosses in both panels represent the stellar position. \[Fig 3\] ](f3.eps){width="\textwidth"}
![CO channel maps (in contours) with natural weighting. Contour spacing is 2$\sigma$ (or 0.5Jy Beam$^{-1}$). The background in every velocity channel is the near-IR image from Subaru (F04) Note that there is significant CO emission beyond the near-IR image. Note also that some velocity channels trace the spiral-like structures. \[Fig 2\]](f4.eps){width="\textwidth"}
![Observed and modeled CO(3-2) channel maps. The contours are the 10%, 30%, and 70% intensity levels of the modeled CO emission from a power-law distributed ($exponent =
-0.47$) intensity Keplerian disk. The model contours appear to be not exactly symmetric because it has been convolved with the same beam as the observation, with a PA of $\sim$ -58$^{\circ}$. Color maps are the observed CO emission, starting from 2$\sigma$ ($\sim$0.5Jy Beam$^{-1}$).\[Fig 4\]](f5.eps){width="\textwidth"}
![Position-velocity diagram along P.A.$\sim$50$^{\circ}$. The emission (in contours with -2$\sigma$, 2$\sigma$, 4$\sigma$, 6$\sigma$, and 8$\sigma$ intensity) lying outside the gray-shaded area (10% peak intensity) cannot be explained by the Keplerian motion. The hatched area indicates the position of the inner spiral arm. The systemic velocity is $\sim5.84~km~s^{-1}$. The position-velocity diagram has not been smoothed across the cut.\[Fig 19\]](f6.eps){width="80.00000%"}
![Diagram illustrating the motion on the spirals. N denotes the north. The yellow curve indicates the inner spiral arm. The deep blue arrow represents the direction of the Keplerian motion of the material going through the spiral. The red arrow indicates the outward radial motion. The light blue and pink arrows are the components along the line-of-sight of the Keplerian motion and the radial outward motion, respectively. \[Fig 18\]](f7.eps){width="50.00000%"}
[^1]: The Submillimeter Array (SMA) is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica.
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abstract: 'We report a refinement of Robertson-Schrödinger uncertainty relation via Wigner-Yanase skew information. Besides the well known quantum uncertainty arising from the noncommutativity of observables, there is classical uncertainty arising from the mixedness of the states that is quantified by the difference between the variance and the skew information. Our refined uncertainty relation for canonical observables is saturated by all the Gaussian states, pure or mixed, and thus provides an alternative measure for the non-Gaussianity of quantum states. Generalizations to the case of metric adjusted skew information are presented, unifying and refining most of previous results.'
author:
- Sixia Yu
- 'C.H. Oh'
title: 'Robertson-Schrödinger Uncertainty Relation Refined by Skew Information '
---
One of fundamental feature of quantum theory is Heisenberg’s uncertainty principle, which states that [*canonically conjugated observables can only be simultaneously determined with a characteristic uncertainty*]{} [@hur]. The mere existence of incompatible observables already leads to the quantum contextuality [@KS], another profound nonclassical feature of quantum theory. The tradeoffs between the accuracies of [*measuring*]{} or [*preparing*]{} different observables are expressed via various kinds of uncertainty relations, providing also a quantification of Bohr’s complementarity principle. Practically, uncertainty relations find numerous applications ranging from setting the fundamental limit of the accuracy of estimating some unknown parameters, as in the quantum metrology [@qmetro], to the detection of quantum entanglement [@ed; @edyu].
Soon after Heisenberg’s original qualitative derivation of uncertainty relation Kennard and Weyl [@kw] proposed the exact mathematical formulation of uncertainty relation for [*preparation*]{}. Interestingly an exact formulation of uncertainty relations for [*measurement*]{} was obtained only recently by Werner [@werner] via a joint-measurement approach and by Ozawa [@ozawa] via a measurement-disturbance approach. Schrödinger [@sur] refined the Heisenberg uncertainty relation by the correlations of two observables and Robertson [@rur] further generalized to the case of more than two observables. For $n$ observables $\{X_k\}_{k=1}^n$ Robertson-Schrödinger (RS) uncertainty relation reads$$\label{rsur}
\left|\sigma_X\right|\ge\left|i\delta_X\right|,$$ where $\sigma_X$ is the covariance matrix and $\delta_X$ is the matrix formed by commutators with matrix elements
$$\begin{aligned}
[[\sigma_X]]_{kj}&=&\frac12\langle X_k X_j+X_jX_k\rangle_\varrho-\langle X_k\rangle_\varrho\langle X_j\rangle_\varrho,\\{}
[[\delta_X]]_{kj}&=&\frac i2\langle[X_k,X_j]\rangle_\varrho,\end{aligned}$$
for $j,k=1,2,\ldots,n$ with $\langle O\rangle_\varrho={{\rm Tr}}\varrho O$ being the expectation value of an observable $O$ in the state $\varrho$. Here we have denoted by $|A|$ the determinant of a square matrix $A$. Both RS and Heisenberg uncertainty relations can be saturated. In the case of canonical observables, e.g., positions and momenta, Heisenberg’s uncertainty relation is saturated by the coherent states and a restrict family of squeezed states while the RS uncertainty relation Eq.(\[rsur\]) is saturated by all pure Gaussian states. The stronger the uncertainty relation, the larger is the family of the minimal uncertainty states.
Many efforts have been made to further refine RS uncertainty relation, e.g., by using skew information. Based on several nice properties such as convexity and additivity, Wigner and Yanase (WY) [@wy63] introduced their skew information $$I_\varrho(X^\dagger,X)=-\frac12{{\rm Tr}}[\sqrt\varrho,X^\dagger][\sqrt\varrho,X],$$ to quantify the information content of a quantum mechanical state $\varrho$ with respect to observables not commuting with (i.e., skew to) the conserved quantity $X$. Being a measure for the noncommutativity between a state $\varrho$ and an observable $X$, the skew information provides a measure of quantum uncertainty of $X$ in the state $\varrho$ and was used by Luo to derive a refinement of Heisenberg’s uncertainty relation for mixed state [@luo03; @luo05]. Furuichi [@F08] presented a refinement, taking into account the correlations, that is independent of RS uncertainty relation. Park [@park05] derived a refinement of Schrödinger’s uncertainty for two observables, which can be saturated by a mixed state. However Park’s approach is somewhat complicated and cannot be easily generalized to more than three observables.
In this Letter we report a genuine refinement of RS uncertainty relation for $n$ observables by the skew information. In terms of the skew information matrix $I_X$ with matrix elements $
[ [I_X]]_{kj}=I_\varrho(X_k,X_j)$ and $c_X=\sigma_X-I_X$, our refined RS uncertainty relation reads $$\begin{aligned}
\label{orur}
\left|\sigma_X+c_X|\cdot|\sigma_X-c_X\right|\ge|\delta_X|^2.\end{aligned}$$ The nontrivial refinement over the RS uncertainty relation Eq.(\[rsur\]) of the above uncertainty relation is shown explicitly by its two weaker versions as below $$\label{orur2}
\left|\sigma_{X}\right|^{\frac2n}-\left|\delta_{X}\right|^{\frac2n}\ge\left((\left| \sigma_{X}\right|^{\frac 1n}-\left| I_X\right|^{\frac1n}\right)^{2}\ge\left|c_X\right|^{\frac 2n}.$$ The additional uncertainty $|c_X|^{1/n}$ can be regarded as classical because it vanishes for pure states and is a concave function of the state. Then we shall show that our refined RS uncertainty relation Eq.(\[orur\]) for canonical observables is saturated by all the Gaussian states, pure or mixed, providing an alternative measure for the non-Gaussianity of a quantum state. Finally our refined uncertainty relation is generalized to the case of metric adjusted skew information, unifying and refining most of previous results.
Our main result is based on a simple observation that leads to RS uncertainty relation [@rur] as well as a simple derivation [@ghp09] of the dynamic uncertainty relation [@dyur], with a special case being $\sigma_X\ge I_X$ [@luo2]. From a set of $n$ observables $\{X_k\}$, by denoting $X_k^\prime=X_k-\langle X_k\rangle_\varrho$, we introduce a set of $2n$ operators $$Y_{k\pm}=\frac {\sqrt\varrho X_k^\prime\pm X_k^\prime\sqrt\varrho}{\sqrt2}:=\frac1{\sqrt2}[\sqrt\varrho,X_k^\prime]_\pm$$ with $\varrho$ being a given state. This set of operators has been used by Park to derive a refined RS uncertainty relation for two observables [@park05]. Let $L_X$ denote the $2n\times 2n$ matrix whose matrix elements are $[[L_X]]_{k\mu,j\nu}={{\rm Tr}}Y_{k\mu}^\dagger Y_{j\nu}$ for $j,k=1,2,\ldots,n$ and $\mu,\nu=\pm$. Explicitly we have
$$\begin{aligned}
{{\rm Tr}}(Y_{k\pm}^\dagger Y_{j\pm})&=&\pm\frac12{{\rm Tr}}[\sqrt\varrho,X_k^\prime]_\pm[\sqrt\varrho,X_j^\prime]_\pm,\\{}
{{\rm Tr}}(Y_{k+}^\dagger Y_{j-})&=&-\frac12\langle[X_k,X_j]\rangle_\varrho=i[[\delta_X]]_{kj}.\end{aligned}$$
The simple observation reads $L_X\ge0$ since $L_X$ can be regarded as the Gram matrix of $2n$ operators $Y_{k\mu}$ with respect to the inner product ${{\rm Tr}}X^\dagger Y$. When arranged in a block form, with each block matrix of size $n\times n$, the condition $L_X\ge0$ becomes $$\label{Lx}
L_X=\left(\begin{array}{cc} \sigma_X+c_X & i\delta_X\\
-i\delta_X&\sigma_X-c_X\end{array}\right)\ge0$$ which is the matrix form of our refined RS uncertainty relation. Using Schur complement condition for positive semidefinite, we obtain $$\label{lx2}
\sigma_X+c_X\ge\delta_X\frac1{\sigma_X-c_X}\delta_X.$$ If $\sigma_X-c_X$ has some zero eigenvalues we have only to understand its inverse appearing in Eq.(\[lx2\]) as being defined in its range, which contains the range of $\delta_X$ as $L_X\ge0$.
Starting from the matrix form one can obtain various scalar uncertainty relations expressed via various characteristics of the positive semidefinite matrix $L_X$, as proposed by Trifonov and Donev [@tr]. For example all the principal minors of $L_X$ must be nonnegative. As a special case, by taking the determinants of both sides of Eq.(\[lx2\]) we obtain immediately our refined RS uncertainty relation Eq.(\[orur\]). In the case of two observables stronger scaler uncertainty relations are possible. In fact the matrix form Eq.(\[lx2\]) can be equivalently characterized by the following set of scalar uncertainty relations (see Appendix 1)
$$\begin{aligned}
\label{21}
|\sigma_X|-|c_X|-\sqrt{(|\sigma_X|-|c_X|)^2-|L_X^+L_X^-|}\ge\delta^2,\\
\frac{L_a^+}{L_a^-}|L_X^-|\ge\delta^2 \quad (a=1,2),\label{22}\end{aligned}$$
where we have denoted $\delta=\langle[X_1,X_2]\rangle_\varrho/2$ and $L_X^{\pm}=\sigma_X\pm c_X$ are $2\times 2$ matrix with matrix elements denoted by $L_a^\pm=[[L_X^\pm]]_{aa}$ with $a=1,2$ and $L_{12}^\pm=[[L_X^\pm]]_{12}$. We note that uncertainty relation Eq.(\[21\]) is stronger than the uncertainty relation Eq.(\[orur\]) for two observables.
To derive the weaker versions Eq.(\[orur2\]) of our refined RS uncertainty relation we need to employ the Minkowski’s inequality for the determinants of positive semidefinite matrices: $ |A|^{\frac 1n}-|B|^{\frac 1n}\ge |A-B|^{\frac 1n}$ for two $n\times n$ Hermitian matrices $A\ge B\ge0$. The first inequality in Eq.(\[orur2\]) is obtained by applying Minkowski’s inequality for $A=2\sigma_X$ and $B=I_X$ together with our refined RS uncertainty relation Eq.(\[orur\]). The second inequality in Eq.(\[orur2\]) is obtained by applying Minkowski’s inequality one more time for $A=\sigma_X$ and $B=I_X$.
From the weaker but suggestive versions Eq.(\[orur2\]), especially the second one, it is tempting to introduce a two-dimensional uncertainty vector $(|\delta_X|^{1/n},|c_X|^{1/n})$ whose length provides the lower bound of the variance. The first component $|\delta_X|^{1/n}$ is quantum uncertainty since it arises from the non commutativity among observables. The second component $|c_X|^{1/n}$ can be regarded as a kind of classical uncertainty for two reasons. First, it comes from the mixing of the quantum states and vanishes for pure state. Second, it is a concave function of the state $\varrho$. This is because the WY skew information $I_\varrho(X^\dagger,X)$ is a convex function of the state $\varrho$ so that the skew information matrix $I_X$ is a convex matrix function of $\varrho$. As a result the classical uncertainty matrix $c_X$ is a concave matrix function of $\varrho$ and, due to Minkowski’s inequality, the classical uncertainty $|c_X|^{1/n}$ is a concave function of $\varrho$. This means that the more mixing of the state the larger is the classical uncertainty.
One of the main reasons why the quantity $|c_X|^{1/n}$ can be regarded as classical uncertainty is that even if those $n$ observables are commuting there is still a nontrivial lower bound for the variance that is arising from the mixedness of the quantum states, i.e., the uncertainty of which pure states. Also in the case of an odd number of observables the quantum uncertainty, as given by the determinants of the commutator matrix, also vanishes and the classical uncertainty provides a nontrivial bound for mixed states, similar to the case of dynamical uncertainty relation as noticed in [@ghp09].
Luo [@luo05; @luo3] also advocated a separation of the classical and quantum uncertainties and obtained a refinement of Heisenberg’s uncertainty relation for two observables [@luo05]. Taking $U_{X_a}^2=\sigma_{X_a}^2-c_{X_a}^2$ as a measure of the quantum uncertainty for each observable $X_{a}$ with $a=1,2$, Luo managed to prove that $U_{X_1}U_{X_2}\ge \delta^2$, which improves Heisenberg uncertainty relation $\sigma_{X_1}\sigma_{X_2}\ge\delta^2$ since $\sigma_X\ge U_X$. Our refined RS uncertainty relation Eq.(\[orur\]) improves that of Luo considering Cauchy’s inequality $$U_{X_1}U_{X_2}\ge\sqrt{\left|L_X^+L_X^-\right|}+|L_{12}^+L_{12}^-|,$$ and $|L_X^\pm|=L_{1}^\pm L_{2}^\pm-(L_{12}^\pm)^2$. Furuichi improved Luo’s result by showing $U_{X_1}U_{X_2}\ge \delta^2+(L_{12}^-)^2$ [@F08], which can be further refined by our uncertainty relations Eq.(\[22\]) (see Appendix 1). We believe that (without a proof) Park’s refinment [@park05] can also be derived from uncertainty relations Eq.(\[21\]) and Eq.(\[22\]).
Our refined RS uncertainty relation Eq.(\[orur\]) can also be saturated. Denoting $\Delta_G:=|L_X^+L_X^-|-|\delta_X|^2$ with $L_X^\pm=\sigma_X\pm c_X$ and from the inequality $\Delta_G\ge |L_X|$ it is clear that a necessary condition for our refined RS uncertainty relation Eq.(\[orur\]) to be attained is that $L_X$ has some zero eigenvalues. That is to say $2n$ operators $Y_{k\mu}$ are linearly dependent, i.e., there exist $2n+1$ complex numbers $a_k,b_k,c$ such that $\sum_k (a_k \sqrt\varrho X_k+ b_k X_k\sqrt\varrho) = c\sqrt\varrho,$ which amounts to requiring that the state $\varrho$ generates a linear transformation among observables $X_k$, e.g., $\varrho X_i\varrho^{-1}$ is a linear combination of $X_i$.
Consider an $n$-mode bosonic system or $n$ interacting quantum harmonic oscillators with their annihilation and creation operators, denoted collectively by $$\Lambda=(a^\dagger,a)=(a^\dagger_1,a^\dagger_2,\ldots,a^\dagger_n,a_1,a_2,\ldots,a_n),$$ satisfying $[a_j,a_k^\dagger]=\delta_{jk}$. Let $\varrho\propto e^{-\beta H}$ be the thermal state of a most general quadratic Hamiltonian $$\label{H}
H=\frac 12\Lambda NJ \Lambda^T,\quad J=\left(\begin{array}{cc}0&I_n\\-I_n&0\end{array}\right)$$ in which the transposition acts only on $2n\times 2n$ matrix without affecting the bosonic operators and $NJ$ is a $2n\times 2n$ symmetric matrix such that $H$ is Hermitian.
The correlation matrix $C_\Lambda$ of $2n$ operators $\Lambda$ in the thermal state $\varrho$, whose matrix elements are given by $[[C_\Lambda]]_{kj}={{\rm Tr}}\varrho\Lambda_k\Lambda_j$, can be readily calculated with the help of linear quantum transformation theory [@lqt]. From the commutators $[\Lambda^T,\Lambda]=J^T$ it follows immediately $[H,\Lambda]=\Lambda N$ and the identity $e^{-\beta H}\Lambda e^{\beta H}=\Lambda M$ with $M=e^{-\beta N}$, which is obtained by Heisenberg’s equation of motion. Suppose that $M-I_{2n}$ is invertible and it follows ${{\rm Tr}}\varrho\Lambda=0$. From identities
$$\begin{aligned}
\label{ct}
{{\rm Tr}}\varrho \Lambda_j\Lambda_k={{\rm Tr}}[[\Lambda M]]_j \varrho \Lambda_k=[[C_\Lambda M]]_{kj},\\
\label{cc}
{{\rm Tr}}\sqrt\varrho\Lambda_j\sqrt\varrho\Lambda_k={{\rm Tr}}[[\Lambda \sqrt M]]_j \varrho \Lambda_k=[[C_\Lambda \sqrt M]]_{kj}.\end{aligned}$$
it follows $C^T_\Lambda=C_\Lambda M$ and $c_{\Lambda}=C_\Lambda \sqrt M$. Considering $J=C^T_\Lambda-C_\Lambda=2\delta_\Lambda$ and $\sigma_\Lambda=(C^T_\Lambda+C_\Lambda)/2$ we obtain $$C_\Lambda=J\frac{I_{2n}}{M-I_{2n}},\quad\sigma_\Lambda\pm c_\Lambda=\frac12J\frac {\sqrt M\pm I_{2n}}{\sqrt M\mp I_{2n}},$$ from which it follows $|(\sigma_\Lambda+ c_\Lambda)(\sigma_\Lambda- c_\Lambda)|=|\delta_\Lambda|^2$, which also holds in the case of singular $M-I_{2n}$ since we can always perturb slightly the parameters in $N$ to ensure that the new $M-I_{2n}$ is invertible. Now we consider $2n$ canonical observables $X=(x,p)=\Lambda u$, e.g., positions and momenta in the case of harmonic oscillators, where $$u=\frac1{\sqrt2}\left(\begin{array}{cc}I_n&iI_n\\I_n&-i I_n\end{array}\right).$$ It turns out that $\sigma_X=u^T\sigma_\Lambda u$, $c_X=u^Tc_\Lambda u$, and $\delta_X=u^T\delta_\Lambda u$ so that our refined uncertainty relation Eq.(\[orur\]) for $2n$ canonical observables $X=\Lambda u$ is saturated by the thermal states of all the quadratic Hamiltonians.
On the other hand a Gaussian state is determined completely by its correlation matrix $C_\Lambda$, supposing ${{\rm Tr}}\varrho\Lambda=0$. Provided that $C_\Lambda$ is invertible, the matrix $M^\prime=C^{-1}C^T$ belongs to symplectic group, i.e., satisfies $M^{\prime T}J M^\prime=J$. Thus there exists an element $N^\prime$ in the symplectic algebra such that $M^\prime=e^{-N^\prime}$. The thermal state of the corresponding quadratic Hamiltonian Eq.(\[H\]) for $N^\prime$ at temperature $\beta=1$ has exactly the same correlation matrix $C_\Lambda$. By an argument of continuity, any Gaussian state can be approximated by the thermal state of a quadratic Hamiltonian so that all the Gaussian states, pure or mixed, saturate our refined RS uncertainty relation Eq.(\[orur\]). Thus the nonzero difference $\Delta_G$ signals the non-Gaussianity of a quantum state.
Let us now explore some generalizations of our refined RS uncertainty relation. For any bivariable function $g(x,y)$ and a given state $\varrho$ with eigensystem $\{\lambda_k, |\psi_k\rangle\}$ we introduce a generalized covariance matrix $\sigma_X(g)$, called $g$-covariance for short, with matrix elements
$$\begin{aligned}
[[\sigma_X({g})]]_{kj}={{\rm Tr}}X_k {{\mathcal J}}_\varrho^{g}(X_j),\\ {{\mathcal J}}_\varrho^g(Z)=\sum_{j,k} g(\lambda_j,\lambda_k)P_jZ P_k,\quad P_k=|\psi_k\rangle\langle\psi_k|.\end{aligned}$$
for a set of $n$ observables $\{X_k\}_{k=1}^n$. For examples the covariance matrix $\sigma_X$ corresponds to $g$-covariance $\sigma_X(g)$ with $g(x,y)=(x+y)/2$ while the commutator matrix $\delta_X$ corresponds to $g$-covariance $\sigma_X(\epsilon)$ with $\epsilon(x,y)=i(y-x)/2$ since ${{\rm Tr}}X_k {{\mathcal J}}_\varrho^{\epsilon}(X_j)=[[\delta_X]]_{kj}$. It is clear that $\sigma_X(cg)=c\sigma_X(g)$ for any complex number $c$ and $\sigma_X(g_1+g_2)=\sigma_X(g_1)+\sigma(g_2)$.
[*Observation 1*]{} If $g(x,y)\ge 0$ for $x,y\ge0$, since ${{\rm Tr}}Z^\dagger{{\mathcal J}}_\varrho^g(Z)\ge 0$ for an arbitrary operator $Z$, then we have $\sigma_X({g})\ge0$. As an immediate consequence we have $\sigma_X({g_1})-\sigma_X({g_2})=\sigma_X({g_1-g_2})\ge 0$ if two bivariable functions satisfying $g_1(x,y)\ge g_2(x,y)$ for $x,y\ge0$.
[*Observation 2*]{} For two arbitrary bivariable functions $g_a(x,y)$ with $a=1,2$ and two operators $Y,Z$ we have ${{\rm Tr}}{{\mathcal J}}^{g_1}_\varrho(Y)^\dagger{{\mathcal J}}^{g_2}_\varrho(Z)={{\rm Tr}}Y^\dagger{{\mathcal J}}^{g_1^*g_2}_\varrho(Z)$. As the Gram matrix of $2n$ observables $Y_{ka}={{\mathcal J}}_\varrho^{g_a}(X_k)$ with $k=1,2,\ldots, n$ and $a=1,2$ with respect to inner product ${{\rm Tr}}Y^\dagger Z$, the $2n\times 2n$ matrix $L_X^g$ matrix defined by $[[L_X^g]]_{ka,jb}={{\rm Tr}}Y_{ka}^\dagger Y_{jb}$ should be nonnegative, i.e., $$\label{grur}
L_X^g=\left(\begin{array}{cc} \sigma_X(|g_1|^2) & \sigma_X({g_1^*g_2})\\
\sigma_X({g_2^*g_1})&\sigma_X(|g_2|^2)\end{array}\right)\ge0.$$ This is a generalization of our refined uncertainty relation in matrix form Eq.(\[Lx\]). As an immediate application, for any three functions $g_\pm(x,y)\ge0,g_0(x,y)$ satisfying $g_+g_-\ge |g_0|^2$ it holds $$\label{urz}
|\sigma_X({g_+})|\cdot|\sigma_X({g_-})|\ge|\sigma_X({g_0})|^2$$ since $\sigma_X({g_+})\ge\sigma_X(|g_0|^2/g_-)$. Considering two functions $a_x,b_x$ such that $g_\pm(x,y)=(a_x\pm a_y)(b_x\pm b_y)\ge0$ and $g_+g_-\ge g_0^2$ where $g_0=\mu(a_xb_y-a_yb_x)$ for some constant $\mu$, e.g., as given in [@ko11], we have a modified commutator matrix $[[\sigma_X({g_0})]]_{kj}=\mu{{\rm Tr}}a_\varrho b_\varrho[X_k,X_j]$. The uncertainty relation Eq.(\[urz\]) improves those Heisenberg type of uncertainty relations, e.g., as in [@ko11], with a modified commutator matrix.
Immediate after its discovery, WY skew information was generalized by Dyson to a one-parametered family, called Wigner-Yanase-Dyson (WYD) skew information [@wy63]. Recently the skew information is further generalized by Hansen [@H06] to a most general family of skew information, called metric adjusted skew information or $f$-skew information, $$I_\varrho^f(X^\dagger,X)=\frac{f(0)}2\sum_{j,k}\frac{(\lambda_k-\lambda_j)^2}{\lambda_jf(\lambda_k/\lambda_j)}{{\rm Tr}}P_kX^\dagger P_j X$$ that is parametrized by the whole set ${{\mathcal F}}$ of regular symmetric operator monotone functions $f(x)$. A nonnegative function $f(x)$ for $x\ge0$ is operator monotone if $f(A)\le f(B)$ for any two Hermitian matrices satisfying $0\le A\le B$, symmetric if $xf(1/x)=f(x)$, normalized if $f(1)=1$, and regular if $f(0)>0$ [@petz96]. WYD skew information is the $f$-skew information corresponding to $$f_{\alpha}(x)=\frac{\alpha(1-\alpha)(1-x)^2}{(1-x^\alpha)(1-x^{1-\alpha})},\quad 0<\alpha\le \frac 12.$$ with $\alpha=1/2$ being the WY skew information. The $f$-skew information associated with $f_M(x)=(1+x)/2$ becomes the quantum Fisher information [@qfi], up to some constant factor, in certain cases. The $f$-skew information matrix $I_X^f$, whose matrix elements are given by $[[I_X^f]]_{kj}=I_\varrho^f(X_k,X_j)$, can be regarded as a $g$-covariance. In fact if we denote $m_f(x,y)=yf(x/y)$ for an arbitrary $f(x)$ then we have $I_X^f=\sigma_X({m_{f_*}})$ with $f_*(x)=f(0){(1-x)^2}/[2f(x)].$
As another application, for a regular $f\in {{\mathcal F}}$ we take $g_1=\sqrt{m_f}$ and $g_2=\epsilon/g_1$ in our observation 2. In this case we have $\sigma_X(g_1^2)=\sigma_X({m_f})$, $\sigma_X({|g_2|^2})=I_X^f/[2f(0)]$, and $ \sigma_X(\epsilon)=\delta_X$ and from Eq.(\[grur\]) it follows $$\label{g1}
|\sigma_X({m_f})|\cdot |I_X^f|\ge[2f(0)]^n|\delta_X|^2.$$ By denoting $\lambda_f=\min_{x\ge0}\left(1+x-f_*(x)\right)/[2f(x)]$, we have $2f_M(x)-f_*(x)\ge2 \lambda_f f(x)$ so that $2m_{f_M}-m_{f_*}\ge \lambda_fm_f$ and thus $2\sigma_X-I_X^f\ge2\lambda_f\sigma_X({m_f})$ due to observation 1. Thus we obtain $$\label{fur}
|\sigma_X-c_X^f|\cdot|\sigma_X+c_X^f|\ge[4\lambda_ff(0)]^n\left|\delta_X\right|^{2}$$ with $c_X^f=\sigma_X-I_X^f$ being the metric adjusted classical uncertainty matrix. For WYD skew information we have $\lambda_{f_\alpha}=1$ and in general $1-f(0)\le\lambda_f\le \min\{1,1/[4f(0)]\}$ (see Appendix 2). Generalized uncertainty relation Eq.(\[fur\]) refines those results in [@Yanagi11; @fy12]. By taking $f(x)=f_{\alpha=\frac12}(x)$ the metric adjusted skew information $I_X^f$ becomes the WY skew information and both two generalizations above reduce to our refined RS uncertainty relation Eq.(\[orur\]).
In summary we have derived such a strong refinement of RS uncertainty relation Eq.(\[orur\]) that all the Gaussian states, pure or mixed, become minimal uncertainty states. The nonzero difference $\Delta_G$ between two sides of our refined RS uncertainty relation Eq.(\[orur\]) provides therefore a natural measure for non-Gaussianity of quantum states. A classical uncertainty that arises from the mixing of pure states is identified and quantified by the difference between the variance and WY skew information. Generalizations Eq.(\[g1\]) and Eq.(\[fur\]) to the metric adjusted skew informations are also presented and corresponding minimal uncertainty states may have potential applications in quantum optics and quantum computational tasks, like Gaussian states. Also the applications in entanglement detection as well as quantum metrology may be expected. At last our refined uncertainty relation should be helpful to sharpen, e.g., Ozawa’s uncertainty relation for measurement and disturbances [@ozawa], which has been tested experimentally [@exps].
This work is supported by National Research Foundation and Ministry of Education, Singapore (Grant No. WBS: R-710-000-008-271) and NSF of China (Grant No. 11075227).
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[*Appendix 1 Proof of Eq.(\[21\]) and Eq.(\[22\]). —* ]{} In the case of two observables $\delta_X$ and $L_X^\pm=\sigma_X\pm c_X$ are two by two matrices. Since $\delta_X=\delta\sigma_y$ with $\delta=\langle [X_1,X_2]\rangle_\varrho/2$ and $\sigma_y$ being the second Pauli matrix and for an arbitrary two by two matrix $U$ it holds $\sigma_y U^T\sigma_y U=|U|$, the matrix form of uncertainty relation Eq.(\[lx2\]) becomes $|L_X^-|L_X^+- \delta^2 L_X^-\ge0$, whose diagonal elements are exactly Eq.(\[22\]) and whose determinant leads to inequality $$\delta^4-2A\delta^2+B\ge0$$ with $A=|\sigma_X|-|c_X|$ and $B=|L_X^+L_X^-|$. As a quadratic function of $\delta^2$, the above inequality leads to either Eq.(\[21\]) or $A+\sqrt{A^2-B}\le \delta^2$. However the second alternative is impossible because the refined RS uncertainty relation Eq.(\[orur\]) for $n=2$ leads to $A\ge\delta^2$. Note that Eq.(\[22\]) can be rewritten as $$\begin{aligned}
L_1^+L_2^-&\ge& \delta^2+\frac{L_1^+}{L_1^-}(L_{12}^-)^2\\
L_2^+L_1^-&\ge& \delta^2+\frac{L_2^+}{L_2^-}(L_{12}^-)^2\end{aligned}$$ from which it follows $$U_{X_1}U_{X_2}\ge\delta^2+\sqrt{\frac{L_1^+L_2^+}{L_1^-L_2^-}}(L_{12}^-)^2$$ which refines Furuichi’s result $U_{X_1}U_{X_2}\ge\delta^2+(L_{12}^-)^2$ [@F08] because $L_a^+\ge L_a^-$ due to the fact that $c_X\ge 0$ and $L_a^\pm=[[L_X^\pm]]_{aa}=\sigma_{X_a}\pm c_{X_a}$.
[*Appendix 2 Range of $\lambda_f$. —* ]{} Here we shall determine the range of $\lambda_f=\min_{x\ge 0}F(x)$ with $$F(x)=\frac1{2f(x)}\left(1+x-\frac{f(0)(1-x)^2}{2f(x)}\right)$$ for an arbitrary normalized symmetric operator monotone function $f(x)$. We have the upper bound $\lambda_f\le \min\{1,1/[4f(0)]\}$ because $F(1)=1$ and $F(0)=1/[4f(0)]$. To get the lower bound, we let $x_0\ge0$ achieve the minimum of $F(x)$, i.e., $\lambda_f=F(x_0)$. If we denote $z=(1+x_0)f(0)/[2f(x_0)]$, since $(1-x_0)^2\le (1+x_0)^2$, then we have $\lambda_f\ge z(1-z)/f(0)$. On the one hand, since $f\in {{\mathcal F}}$, we have $(1+x_0)/2\ge f(x_0)$, i.e, we have $ 1/2\ge z$, so that the quadratic function $z(1-z)$ is increasing. On the other hand since $f(x)$ is concave we have $f^\prime(x)\ge f^\prime(\infty)=\lim_{x\to \infty}f(x)/x=\lim_{x\to \infty}f(1/x)=f(0)$. As a consequence $f(x)-f(1)\ge f(0)(x-1)$ for $x\ge 1$. For $x\le 1$ we have $1/x\ge 1$ and $f(1/x)\ge 1+f(0)(1-x)/x$ which, together with $f(x)=xf(1/x)$, leads to $f(x)\ge x+f(0)(1-x)$. Thus we obtain $$f(x)\ge \frac{1+x}2-\frac{1-2f(0)}2|1-x|\ge f(0)(1+x).$$ As a result we obtain $z\ge f(0)$ so that $\lambda_f\ge 1-f(0)=\lambda_f^*$. We note that $\lambda_f^*\ge f(0)$ as $f(0)\le 1/2$. In general we have $F^\prime(1)=0$, i.e., $F(1)$ is always a local extremal point. Numerical evidences show that $\lambda_f=1$ whenever $4f(0)\le 1$ and we conjecture that $\lambda_f=\min\{1,1/[4f(0)]\}$ always holds. For the monotone operator function $f_{\alpha}(x)$ corresponding to Wigner-Yanase-Dyson skew information it holds $\lambda_{f_\alpha}=1$ because $\lambda_{f_\alpha}\ge 1$ follows from the inequality $$|x^{\alpha}-x^{1-\alpha}|\le(1- 2\alpha)|1-x|,\quad 0<\alpha\le \frac12.$$
For those $f$-skew informations corresponding to monotone functions $f(x)$ with $\lambda_f=1/[4f(0)]$, or equivalently $f(x)\le f(0)(1+\sqrt x)^2$, e.g., $f_M(x)=(1+x)/2$ and $f_{1/2}(x)=(1+\sqrt x)^2/4$, the uncertainty relation Eq.(\[fur\]) are refinements of RS uncertainty relation. However the one given in Eq.(\[orur\]) corresponding to $f_{\alpha=1/2}(x)$ is the strongest. This is because $I_X\le I_X^{f}$ and $c_X\ge c_X^{f}$ which means that $r_k\ge r_k^f$ with $r_k^{(f)}$ being the eigenvalues of $\frac1{\sqrt{\sigma_X}}c^{(f)}\frac1{\sqrt{\sigma_X}}$ arranged in decreasing order and thus $$\begin{aligned}
&&|\sigma_X-c_X^f|\cdot|\sigma_X-c_X^f|
=|\sigma_X|^2\prod_{k}(1-(r_k^f)^2)\cr
&\ge&|\sigma_X|^2\prod_{k}(1-r_k^2)
=|\sigma_X-c_X|\cdot|\sigma_X-c_X|.\end{aligned}$$ For those $f$-skew informations corresponding to monotone functions $f(x)$ with $\lambda_f\not =1/[4f(0)]$ the uncertainty relation Eq.(\[fur\]) may be independent of RS uncertainty relation.
|
---
abstract: 'Graphs triangulating the $2$-sphere are generically rigid in $3$-space, due to Gluck-Dehn-Alexandrov-Cauchy. We show there is a *finite* subset $A$ in $3$-space so that the vertices of each graph $G$ as above can be mapped into $A$ to make the resulted embedding of $G$ infinitesimally rigid. This assertion extends to the triangulations of any fixed compact connected surface, where the upper bound obtained on the size of $A$ increases with the genus. The assertion fails, namely no such finite $A$ exists, for the larger family of all graphs that are generically rigid in $3$-space and even in the plane.'
address: 'Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel'
author:
- Karim Adiprasito and Eran Nevo
title: Rigidity with few locations
---
Introduction
============
A theorem of Dehn asserts that the $1$-skeleton of every simplicial convex $3$-polytope is infinitesimally rigid [@Dehn]. Combined with Steinitz theorem, this gives Gluck’s result that the $1$-skeleton of any simplicial $2$-sphere is *generically* rigid in $\R^3$ [@Gluck], i.e., the locus of realizations that are not infinitesimally rigid is of codimension one in the configuration space of all possible locations. See [@Connelly:RigiditySurvey] and [@Pak-book] for further references and discussion.\
We ask the following question: How generic does the embedding of a generically rigid graph need to be to guarantee that it is infinitesimally rigid?\
We give a natural precise meaning to this meta question, and partially answer it for various families of graphs, including the one mentioned above.
Let us first recall some notions pertaining to infinitesimal rigidity: An embedding of a graph $G=(V,E)$ into $\R^d$ is any map $f: V\rightarrow \R^d$ such that $f(V)$ affinely spans $\R^d$; it defines a realization of the edges in $E$ by segments via linear extension, this realization is called the *framework* $f(G)$. A *motion* of $f(G)$ is any assignment of velocity vectors $a:V\rightarrow \R^d$ that satisfies $$\label {eq:infi-rigidity}
\langle a(v)-a(u),f(v)-f(u)\rangle=0$$ for every edge $uv \in E$. A motion $a$ is *trivial* if the relation (\[eq:infi-rigidity\]) is satisfied for [*every pair*]{} of vertices; otherwise $a$ is *nontrivial*. The framework $f(G)$ is *infinitesimally rigid* if all its motions are trivial. Equivalently, (\[eq:infi-rigidity\]) says that the velocities preserve infinitesimally the distance along an embedded edge, and if (\[eq:infi-rigidity\]) applies to all pairs of vertices then the velocities necessarily correspond to a rigid motion of the entire space.
We now arrive at the central definition of this note, quantifying the genericity of the embedding needed for an infinitesimally rigid embedding.
\[def:c\] Let $F$ be a family of graphs. We say $F$ is *$d$-rigid with $c$-locations* if there exists a set $A\subseteq \mathbb{R}^d$ of cardinality $c$ such that for any graph $G=(V(G),E(G))\in F$ there exists a map $f:V(G)\rightarrow A$ such that the framework $f(G)$ is infinitesimally rigid. Denote by $c_d(F)$ the minimal such $c$.
We are interested in the question whether a given infinite family of finite graphs, which is known to be generically infinitesimally $d$-rigid (namely, $d$-rigid with $\aleph_0$-locations), is also $d$-rigid with $c$-locations, for some *finite* $c$.
Clearly, the answer is yes iff *any* *generic* set $A$ will do in Definition \[def:c\], i.e. any $A$ where the $c\times d$ entries of its vectors are algebraically independent over the rational numbers. Perhaps surprisingly, we show:
\[thm:2-spheres\] Let $F(S^2)$ be the family of $1$-skeleta of all triangulations of the $2$-sphere. Then $F(S^2)$ is $3$-rigid with $76$-locations. Namely, $c_3(F(S^2))\leq 76$.
The phenomenon in Theorem \[thm:2-spheres\] generalizes to any surface, orientable or not: let $F(S)$ be the family of $1$-skeleta of all triangulations of compact connected surfaces, and let $F(g)$ be the subfamily when fixing the surface of genus $g$ (orientable or non-orientable genus). Fogelsanger proved that any graph in $F(S)$ is generically $3$-rigid [@Fogelsanger].
\[thm:S\_g\] For any $g$, $c_3(F(g))$ is finite.
In fact, Fogelsanger showed more generally that the $1$-skeleton of any *minimal cycle complex* (defined in the Preliminaries) of dimension $d\ge 2$ is generically $(d+1)$-rigid. Our results on surfaces extend to this context as well: let $FOG(b)$ be the family of $2$-dimensional minimal cycle complexes with *socle* of dimension at most $b$ in the Artinian reduction of their Stanley-Reisner ring over the reals. This provides a natural generalization of genus in terms of intersection homology, and we obtain:
\[thm:FOG\] For any $b>0$, $c_3(FOG(b))$ is finite.
We now consider the larger family of all generically $3$-rigid graphs. Some well-known open problems are to characterize this family by combinatorial means, and, concretely, whether there exists a deterministic polytime algorithm to decide if a given graph is generically $3$-rigid; see e.g. the survey [@Jackson-Jordan06:3Drigidity].
Let $F_d$ be the family of all generically d-rigid finite graphs. Note that for $d=1$, $F_1$ is the family of connected graphs, so considering a spanning tree for $G\in F_1$ shows $c_1(F_1)=2$. Perhaps not surprisingly, $F_d$ is quantitatively more complicated for any $d\ge 2$, as shown also by Fekete and Jordan [@Fekete-Jordan Sec.4] for $d=2$:
\[thm:no\_c(F\_d)\] For any $d\ge 2$ and any finite $c$, $F_d$ is *not* $d$-rigid with $c$-locations.
Let us remark that infinitesimal rigidity for (slightly) non-generic embeddings has been considered in the literature, for subfamilies of triangulated spheres and manifolds – in the centrally symmetric case and in the balanced case, see e.g. Stanley [@Stanley-CSpolytopes; @Stanley-balancedCM] (phrased in the language of face rings), for convex position, see e.g. Izmestiev and Schlenker [@Izmestiev-Schlenker], and for embeddings on small grids [@Fekete-Jordan] – however, the problem of embedding with a constant number of locations, seems to be new.
**Outline**: Preliminaries are given in Section \[sec:prelim\]. We prove Theorem \[thm:no\_c(F\_d)\] in Section \[sec:all\], Theorem \[thm:2-spheres\] in Section \[sec:2-spheres\], Theorem \[thm:S\_g\] in Section \[sec:surfaces\], and conclude with open questions in Section \[sec:conclude\].
Preliminaries {#sec:prelim}
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Complexes and polytopes.
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We set some notation: the *link* of a face $\sigma$ in a simplicial complex $X$ is the subcomplex $\operatorname{lk}_{\sigma}(X)=\{\tau\in X:\ \sigma\cap\tau=\emptyset,\ \sigma\cup\tau\in X \}$ and its (open) star is the filter $\operatorname{st}_{\sigma}(X)=\{\tau\in X: \sigma \subseteq \tau\}$; the set of all $l$-dimensional faces ($l$-faces for short) of $X$ is $X_l$ and its *$l$-skeleton* is the subcomplex $X_{\le l}=\cup_{i\le l}X_i$. The *geometric realization* of $X$ is denoted by $|X|$; we say $X$ is a *simplicial $d$-sphere / surface* if $|X|$ is a topological $d$-sphere / surface.
A $d$-dimensional polytope is *simplicial* if any face in its boundary is a simplex; it is *$k$-neighborly* if any subset of its vertices of size $k$ is the vertexset of some face of it; it is *stacked* if it is either a $d$-simplex, or can be obtained from a stacked $d$-polytope by gluing on one of its $(d-1)$-faces a $d$-simplex.
Rigidity
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The rigidity matrix of the framework $f(G)$, of a graph $G=(V,E)$ and an embedding $f:V\rightarrow \R^d$, is the $d|V|\times|E|$ real matrix where in the column of edge $vu\in E$ the entries in the $v$-rows are $f(v)-f(u)$, in the $u$-rows are $f(u)-f(v)$, and the other entries in this column are zero; denote this matrix by $R(f(G))$. When $|V|=n>d$, the rank of $R(f(G))$ is always $\le dn-\binom{d+1}{2}$, and equality holds iff $f(G)$ is infinitesimally rigid.
The space of *affine $2$-stresses*, or simply *stresses*, of a framework $f(G)$ of a graph $G=(V,E)$ is the real vector space $$\{(w_{\sigma})\in \mathbb{R}^{E}: \forall \tau\in V,\ \ \sum_{\tau \in \sigma}w_{\sigma}(f(\sigma\setminus\tau)-f(\tau))=\overrightarrow{0} \}.$$ The space of affine $2$-stresses of $f(G)$ is exactly the kernel of the rigidity matrix $R(f(G))$.
For $G$ the $1$-skeleton of a simplicial $2$-sphere with $n$ vertices, it has $3n-6$ edges, and thus for $f:V\rightarrow \mathbb{R}^3$, $f(G)$ is infinitesimally rigid iff its only affine $2$-stress is the trivial (all zero) stress.
All $d$-rigid graphs {#sec:all}
====================
Clearly, for $F'_d \subseteq F_d$ the subfamily of *minimally* generically infinitesimally $d$-rigid graphs, $c_d(F_d)=c_d(F'_d)$. For $d=2$, $F'_2$ is the family of the well studied Laman graphs.
Theorem \[thm:no\_c(F\_d)\] for $d=2$ was proved in [@Fekete-Jordan Sec.4], we give their short argument for completeness: Suppose by contradiction that $c=c_2(F'_2)$ is finite, and let $G$ be a Laman graph that is $2$-rigid with $c$-locations but not with $(c-1)$-locations. We may assume $|V(G)|=n>c$ (as any Laman graph is a strict subgraph of another Laman graph), so for any map $f:V(G)\rightarrow A$, $|A|=c$, $A \subset \mathbb{R}^2$, there exist two vertices $w,u\in V(G)$ with $f(w)=f(u)$. Let $G'$ be obtained from $G$ by adding for each pair of vertices $x,y \in V(G)$ a new vertex $v=v(x,y)$ and two new edges $vx$ and $vy$. Then $G'$ is Laman because $G'$ is obtained from $G$ by Henneberg moves, and these moves preserve the property of being Laman. Assume by contradiction that there exists a map $f_{G'}:V(G')\rightarrow A$ with $|A|=c$ and $f_{G'}(G')$ infinitesimally rigid. Consider $x,y \in V(G)$ with $f_{G'}(x)=f_{G'}(y)$ and the vertex $v=v(x,y)$. Let $a(v)$ be any non-zero vector perpendicular to $f_{G'}(x)-f_{G'}(v)=f_{G'}(y)-f_{G'}(v)$, and let $a(w)=0$ for every vertex $w\neq v$. Then $a$ is a non-trivial infinitesimal motion of $G'$, which is a contradiction.
**Case $d>2$**: First proof: Argue similarly to the $d=2$ case, when adding a new vertex $v(B)$ for any $d$-subset $B\subseteq V(G)$, and connecting $v(B)$ to all vertices in $B$, to obtain $G'$ from $G\in F'_d$, where $G$ requires $c=c_d(F'_d)$ locations.
Second proof: Restrict to repeated cones over Laman graphs, forming a subfamily $F''_d$ of $F'_d$, so $c_d(F''_d)\le c_d(F'_d)\le c_d(F_d)$. By the Cone Lemma (see e.g. [@TayWhiteWhiteley-Skel2Cone]), any $G\in F''_d$ is minimally generically infinitesimally $d$-rigid, and projection gives that $c_2(F'_2)\le c_d(F''_d)$, so by the $d=2$ case, $c_d(F_d)$ is infinite. $\square$
Graphs of simplicial $2$-spheres {#sec:2-spheres}
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The proof of Theorem \[thm:2-spheres\] requires the following couple of simple facts: let $G$ be a maximal planar graph on at least $5$ vertices, equivalently $G$ is the $1$-skeleton of a triangulation of the $2$-sphere different from the boundary of a tetrahedron.
\[lem:deg3,4,5\] $G$ has a vertex of degree $\in\{3,4,5\}$.
By Euler’s formula, the average degree is $<6$, and by maximality of $G$ it is $\ge 3$.
\[lem:starhole\] For any vertex $v\in G=(V,E)$, there is an edge $uv\in G$ such that the contraction of $v$ to $u$ yields a graph $G'=(V-\{v\},E')$ which is again the $1$-skeleton of a triangulation of the $2$-sphere.
The contraction as above is defined by the edgeset $E':=(E\setminus \{e\in E:\ v\in e\})\cup\{uw:\ vw\in E, w\neq u \}$. This lemma too should be known; as we failed to find a reference we provide a proof.
Let the vertices in $\operatorname{lk}_v(G)$ be $u_1,u_2,\ldots,u_t$ in the cyclic order. We need to show that for some $i$, $u_iu_j\notin G$ for any $j\neq i+1,i-1 \mod{t}$, namely that $vu_i$ is not part of a missing triangle of the sphere triangulation (a.k.a. separating triangle). Then $u=u_i$ is good.
This follows from planarity, and is even simpler to argue when $\deg(v)\in\{3,4,5\}$, which suffices for our purposes: if $\deg(v)=3$ then any $u_i$ is good. If $\deg(v)=4,5$, if $u_i$ is not good then w.l.o.g. by relabeling $(\mod t)$ $u_iu_{i+2}\in G$, so planarity shows that $u=u_{i+1}$ is good (and there exists another $u_j$ which is good as well).
The proof of Theorem \[thm:2-spheres\] follows by showing:
\[thm:24\] Let $A$ be a generic subset of $\R^3$, $|A|=76$. Then for any graph $G\in F(S^2)$ there exists a function $f_G:V(G)\rightarrow A$ such that
\(R) the framework $f_G(G)$ is infinitesimally $3$-rigid, and
\(C) for any subgraph $H\subseteq G$ of a subcomplex that triangulates a $4-$ or $5-$gon, the restriction of $f_G$ to $V(H)$ is injective. Equivalently, for any subcomplex which is a disc consisting of up to $3$ triangles, $f_G$ is injective on its vertices.
For Theorem \[thm:2-spheres\] we only need (R), however, for our inductive proof to work we require (C) as well. The theorem clearly holds if $|V(G)|\le 76$. Assume $|V(G)|>76$, let $v\in G$ be a vertex of degree $3,4$ or $5$ (it exists by Lemma \[lem:deg3,4,5\]), and let $uv\in G$ such that the contraction of $v$ to $u$ gives a smaller graph $G'\in F(S^2)$ ($u$ exists by Lemma \[lem:starhole\]). By the induction hypothesis, there exists a function $f_{G'}$ satisfying (R) and (C) for $G'$; we will show that $f_{G'}$ can be extended to a function $f_G$ as required. We just need to show that $A$ has enough room so that $f_G(v)$ can be defined so that (R) and (C) hold for $G$. We loose nothing by assuming that $f_{G'}$ has an image of smallest possible size.
To achieve (C), $f_G(v)$ just needs to avoid the values of $f_G$ on all the vertices (i) in the link $\operatorname{lk}_G(v)$, (ii) in triangles without $v$ that share an edge with triangles with $v$, and (iii) in triangles without $v$ that share an edge with triangles that share an edge with triangles with $v$. Thus, when $\deg(v)=3$ (resp. $4$; resp. $5$), $f_G(v)$ needs to avoid a set $N$ of at most $12$ (resp. $16$; resp. $20$) values in $A$, so if $|A|>20$, we can define $f_G(v)$ so that (C) is satisfied.
We now turn to the rigidity requirement (R). Let $c(G)$ (resp. $C(G)$) be the minimum size $c$ such that $G$ is $3$-rigid with $c$-locations (resp. and with $f_G:V(G)\rightarrow A$, $|A|=c$, satisfying property (C) as well). Thus $c(G)\leq C(G)$, hence it is enough to show that for $G,G'\in F(S^2)$ as above, $C(G)\le \max(C(G'),76)$.
Let $f_{G'}:V(G')\rightarrow A$ with $|A|=C(G')$ satisfy (R) and (C) of Theorem \[thm:24\].
**First**, we notice that $f_{G'}$ has an extension ${f}_G: V(G)\rightarrow A\cup\{a\}$, with ${f}_G(v)=a$ not necessarily in $A$, such that ${f}_G(G)$ is rigid. This follows by combining (C) with a closer look at Whiteley’s proof of the Contraction Lemma [@Whiteley-split]; rephrased here suitably.
\[prop:WhiteleylikeContraction\] Let $vu$ be an edge in $G=(V,E)$ and assume that $u$ and $v$ have two common neighbors $a$ and $b$. Contract $v$ to $u$ to obtain graph $G'=(V-v,E')$. Let $f:V-v \rightarrow \mathbb{R}^3$ such that
\(i) $f(G')$ is infinitesimally rigid, and
\(ii) $f(a),f(b),f(u)$ are all different.\
Then there exists an extension $f(v)$ of $f$ to $V$, such that the framework $f(G)$ is infinitesimally rigid.
To guarantee (ii) in our case, we need that $f_{G'}$ is injective on the restriction to the subgraph $H=\operatorname{lk}_v(G)$ of $G'$, so (C) for $f_{G'}$ implies (ii).
**Second**, we show that in fact $f_G(v)$ can be chosen in $A$ so that resulted $f_G$ is as required in the theorem. We need the following crucial fact:
\[prop:key\] Let $v\in G$ be a vertex of degree $\le b$, $A\subseteq \mathbb{R}^3$ a generic subset of size $\ge \binom{b+3}{3}$, $f: V(G)\rightarrow \mathbb{R}^3$ such that $f(V(G)\setminus\{v\})=A$ and $f(G)$ is infinitesimally rigid.
Then there exists another map $f': V(G)\rightarrow A$ that agrees with $f$ on $V(G)\setminus\{v\}$, and such that $f'(G)$ is infinitesimally rigid.
Consider the rigidity matrix $R(f_G(G);x,y,z)$ of the framework $f_G(G)$, where $f_G(v)$ is a vector with *variable* entries $(x,y,z)$. As we assumed that for some assignment $f_G(G)$ is infinitesimally rigid then, for $|V|\ge 4$, at least one of the determinants of $(3|V|-6)\times(3|V|-6)$-minors in $R(f_G(G);x,y,z)$ is nonzero; denote by $P$ such determinant. Then $P$ is a nonzero polynomial in the variables $x,y,z$ of degree at most $b$ (as $v$ is incident to at most $b$ edges). The dimension of the space of all such polynomials is $\binom{b+3}{3}$. Then, as $A$ is generic, no choice of different $\binom{b+3}{3}$ points in $A$ make the determinant $P$ vanish on each of the points. To see this, put the coefficients of the polynomial $P$ into a vector $U$. Consider the $\binom{b+3}{3}$ by $\binom{b+3}{3}$ matrix $M$, where the entries of the $j$-th row are the values of our $\binom{b+3}{3}$ monomials computed using the coordinates of the $j$-th point, $(x_j,y_j,z_j)$. The determinant of $M$ is then a non-zero polynomial in $x_1,\ldots, x_{\binom{b+3}{3}}, y_1,\ldots,y_{\binom{b+3}{3}}, z_1,\ldots,z_{\binom{b+3}{3}}$ with rational coefficients. Since our points are generic, it follows that $\det(M)\neq 0$. On the other hand, if the value of $P$ at each of our $\binom{b+3}{3}$ points is $0$, then $MU=0$. Hence $U=0$, a contradiction, as $P$ is a nonzero polynomial.
As $|A|\ge \binom{b+3}{3}$ there is a choice $f_G(v)\in A$ as desired.
Summarizing, to guarantee both (C) and (R) for $f_G(G)$, $f_G(v)$ needs to avoid at most $20+(\binom{8}{5}-1)=75$ values in $A$, which is possible for $|A|=76$.
This completes the proof that $c_3(F(S^2))\le 76$. $\ \ \square$\
Variations on the above proof give stronger upper bounds for subfamilies of $F(S^2)$, which we consider next. In the rest of this section we keep the notation used in Theorem \[thm:24\] and its proof.
\[prop:stacked\] The family of $1$-skeleta of *stacked* $d$-polytopes is $d$-rigid with $(d+1)$-locations, but not with $d$-locations.
In particular, for the subfamily $T\subseteq F(S^2)$ of graphs of stacked $3$-polytopes, $c_3(T)=4$.
For $v$ a vertex of degree $d$, its contraction gives the graph $G'$ of a stacked polytope. Letting $f_G(v)$ be different from the $f_G$-values of the neighbors of $v$ makes $f_G(G)$ rigid, by say the Gluing Lemma [@Asimow-Roth2; @Whiteley96-SomeMatroids], or simply by inspecting the rigidity matrix of $f_G$.
Note that the assertion of Proposition \[prop:stacked\] extends to any $d$-polytope whose $1$-skeleton contains the $1$-skeleton of a stacked $d$-polytope as a spanning subgraph. There are numerous such examples, e.g. [@Shemer; @Padrol], and in particular:
\[cor:2-nei\] The family of $1$-skeleta of *2-neighborly* $d$-polytopes is $d$-rigid with $(d+1)$-locations, but not with $d$-locations.
Next we consider the subfamily $Q\subseteq F(S^2)$ of graphs that can be reduced to the complete graph $K_4$ by always contracting vertices of degree $\le 4$, obtaining a smaller triangulation of the $2$-sphere at each step.
$c_3(Q)\le 10$.
In fact, any generic subset $A$ of $\R^3$ of size $10$ satisfies that for any graph $G\in Q$ there exists a function $f_G:\ V(G)\rightarrow A$ such that
(R’) the framework $f_G(G)$ is infinitesimally $3$-rigid, and
(C’) for any subgraph $H\subseteq G$ of a subcomplex that triangulates a $4$-gon, the restriction of $f_G$ to $V(H)$ is injective. Equivalently, for any subcomplex which is a disc consisting of $2$ triangles, $f_G$ is injective on its vertices.
Contract a vertex $v\in G$ of degree $\le 4$ to vertex $u$ so that $G'$ is in $Q$. Again, we extend $f_{G'}$ to $f_G$. For $(C')$ to hold for $f_G$, $f_G(v)$ needs to avoid a subset $N$ of at most $8$ points in $A$.
To achieve also $(R')$, we show that for any two points $x,y\in A\setminus N$, either $f_G(v)=x$ or $f_G(v)=y$ provides the desired extension.
Note that removing an edge from $G'$ creates a nontrivial infinitesimal motion, unique up to scaling, say $M_{G'}:V(G')\rightarrow\mathbb{R}^3$, so for the right choice of an edge $e$ this is the nontrivial motion for the induced framework of the graph $G-v$. Then $M_{G'}$ does not preserve the distance between the vertices of $e$ up to first order. Note that the vertices of $e$ belong to $\operatorname{lk}_v(G)$.
Tentatively define $f_G(v)=x$; it may allow an extension of the motion $M_{G'}$ on $G'$ to $G$ for suitable $M_G(v)=v_x$. Similarly when tentatively defining $f_G(v)=y$ and $M_G(v)=v_y$. Assume by contradiction that both options extend the motion $M_{G'}$. As $M_{G'}$ is unique (up to nonzero scalar multiplication), we will get a nontrivial infinitesimal motion $M$ on the octahedron $O$ with antipodal vertices $x,y$ and equator $4$-cycle $f_G(\operatorname{lk}_v(G))$, by setting $M(x)=v_x, M(y)=v_y$ and $M(f_G(u))=M_{G'}(u)$ for any $u\in \operatorname{lk}_v(G)$. As all $6$ vertices of $O$ are in different points of $\mathbb{R}^3$, and as $A$ is generic, by Gluck’s theorem $O$ is infinitesimally rigid; a contradiction.
Graphs of surfaces {#sec:surfaces}
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Barnette and Edelson [@BE88; @BE89] have shown that for any given compact surface $M$, the number of its *irreducible* triangulations, namely those where no edge can be contracted to give a (smaller) triangulation of $M$, is finite. More strongly, and useful for our purposes, Schipper [@Schipper91] has shown the following, using the Barnette-Edelson results.
([@Schipper91 Lem.9])\[lem:Schipper\] For any compact $2$-manifold $M$ there exists a constant $n_0(M)\in \mathbb{N}$ such that any triangulation $\Delta$ of $M$, with $n>n_0(M)$ vertices, contains a vertex $v$ of degree at most $6$ such that $v$ has a neighbor $u$ such that the contraction of $v$ to $u$ results in a (smaller) triangulation $\Delta'$ of $M$; equivalently, $vu$ is in no missing triangle of $\Delta$.
\[rem:n\_0(g)\] In fact, the argument in Schipper’s proof shows more: for some $0<\epsilon <1$, independent of $M$, at least $\epsilon n$ such vertices $v$ exist. However, $\{n_0(M)\}_M$ is of course unbounded, as the minimal number of vertices needed to triangulate a surface grows with the genus.
With Lemma \[lem:Schipper\] at hand, we can prove Theorem \[thm:S\_g\] in the same spirit of the proof of Theorem \[thm:24\].
Let $\Delta_{\leq 1}$ denote the $1$-skeleton of a triangulation $\Delta$. Let $n>n_0(M)$ and contract $v$ to $u$ as in Lemma \[lem:Schipper\]. We need to verify that infinitesimal rigidity of the framework $f_{\Delta'_{\le 1}}(\Delta'_{\le 1})$ implies the existence of an extension $f_{\Delta_{\le 1}}(v)\in A$ that makes $f_{\Delta_{\le 1}}(\Delta_{\le 1})$ infinitesimally rigid. For this we first use Proposition \[prop:WhiteleylikeContraction\]; in order to apply it we need the following condition to hold, which guarantees that condition (ii) in the proposition holds:
(C") any $2$-ball $B$ in $\Delta'$ made of at most $4$ triangles with a common vertex has $f_{\Delta'_{\le 1}}$ injective on the vertices of $B$.
(Indeed, the contraction of $v$ as in Lemma \[lem:Schipper\], replaces its star by such ball $B$.) For (C") to hold for $f_{\Delta_{\le 1}}(\Delta_{\le 1})$, $f_{\Delta_{\le 1}}(v)$ needs to avoid at most $48=6+6+12+24$ values in $A$.
Now, by Propositions \[prop:WhiteleylikeContraction\] and \[prop:key\] for $b=6$ we finish the proof as before: among any $\binom{9}{3}=84$ points in $A$, there will be a point $a$ such that $f_{\Delta_{\le 1}}(v)=a$ makes the rigidity matrix of $\Delta_{\le 1}$ full rank. Thus, by avoiding at most $48+83=131$ values in $A$, both (R) and (C") are guaranteed, given $n>n_0(M)$. Thus, $c_3(F(g))\le \max(n_0(M),132)$. $\square$
Note that when the (orientable or non-orientable) genus of $M$ tends to infinity, so does $n_0(M)$ (see Remark \[rem:n\_0(g)\]). We do not know if $\{c_3(F(g))\}_g$ is bounded.
Graphs of minimal cycles {#sec:FOG}
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Consider now a (strongly) minimal homology $2$-cycle, i.e. a simplicial complex of dimension $2$ that has but a single homology cycle in dimension $2$, and is minimal with that property.
A weakly minimal cycle that is not strongly minimal is a union of two smaller weakly minimal cycles that share a triangle.
A weakly minimal cycle is a weakly minimal cycle in characteristic zero.
The *degeneracy* of $\mu$ is the difference of the dimension of space of linear $2$-stresses and the dimension of the space of linear $1$-stresses. Algebraically, this quantity measures the distance of the underlying intersection ring from a Poincaré duality algebra. We denote it by $\delta(\mu)$.
The *singular set* of a minimal cycle $\mu$ is the subcomplex of faces at which the underlying complex of their star is *not* a manifold; as such, it is a subgraph of $\mu$, denote it by $S(\mu)$. For a vertex $v$ in particular, let us denote by $m_v$ the number of connected components in the link of $v$.
For a strongly minimal cycle $\mu$, we have $$\delta(\mu)\ =\ 3(\beta_1(|\mu|)-2\sum_{v\ \text{vertex in}\ \mu} (m_v-1)\ \ge\ \beta_1(|\mu|).$$
If $v$ is a vertex with $m_v\ge 2$, then we can disconnect $\mu$ at this vertex by introducting $m_v$ vertices $(v_1,\dots,v_{m_v})$ whose links are the connected components of the link of $v$ in $\mu$. This increases the first Betti number by $(m_v-1)$.
In this process, we have decreased $\delta(\mu)$ by $\sum_{v\ \text{vertex in}\ \mu} (m_v-1)$. The remaining cycle $\mu'$ is Buchsbaum, so in terms of its Betti number, [@Novik] gives $$\delta(\mu')\ = \ 3(\beta_1(|\mu'|).$$
[<span style="font-variant:small-caps;"></span> ]{}
For every $\delta_0 \ge 0$, there exists an $N=N(\delta_0)$ such that every minimal $2$-cycle with $\delta(\mu)\le \delta_0$ and number of vertices greater than $N$ has a vertex of degree $\le 6$
If all vertices of $\mu$ are of degree $7$ or higher, then $\delta(\mu)$ tends to infinity as the number of vertices grows.
[<span style="font-variant:small-caps;"></span> ]{} There are now two situations: Once we have identified a vertex $v$ of degree $6$ or less is contained in a contractible edge (i.e. an edge that is contained in no empty triangle.) In that case, we conclude We can contract such an edge, ultimately ending up with a number of locations depending only on $\delta_0$; so again finite locations suffice.
If an edge is not contractible, i.e., it is contained in an empty triangle $\Delta$, then following Fogelsanger, $\mu$ decomposes into two smaller minimal cycles, $\mu_0$ and $\mu_1$. Now, we note that $\mu_0$ and $\mu_1$ can only intersect in a graph. Moreover, no cycle in that graph, apart from $\Delta$ can be contractible in both $\mu_0$ and $\mu_1$ since $\mu$ is minimal. Hence, the dimension of linear degree $2$ stresses in their intersection is at most $2\beta_1(\mu)$. Hence, the number of degree $1$ stresses in the intersection of $\mu_0$ and $\mu_1$ is at most $3\delta(\mu)$.
Hence, we can decompose $\mu$ into strictly smaller cycles of smaller deficiency, and we can induct in this way.
Let us close the section by remarking that the dependence on degeneracy in Theorem \[thm:FOG\] is necessary.
There is no finite number of locations in $\R^3$ such that every minimal $2$-cycle can be realized with vertices in these locations.
The proof follows the idea in the first proof of Theorem \[thm:no\_c(F\_d)\].
Indeed, assume that $c$ locations are enough to guarantee that every minimal $2$-cycle can be realized in an infinitesimally rigid way. Consider a minimal $2$-cycle $\mu$ that requires $c$ locations.
Consider secondly the boundary of a tetrahedron $\Delta$, and mark the four vertices by $0,1,2$ and $3$. We subdivide the triangle $\{123\}$ with a triangle $\Gamma$ that bisects its edges, as well as the triangles incident to $0$. Denote the resulting subdivision of $\Delta$ by $\Delta'$. (Thus, $\Delta'$ has 7 vertices and 10 triangles.)
Totally order the triples of vertices in $\mu$. According to this order, for every triple of vertices of $\mu$, attach a new copy of $\Delta'$ along the vertices $1$, $2$ and $3$ to the currently constructed complex $\mu'$ (starting with the original $\mu$), then remove $\Gamma$ and some triangle of $\mu'$, and connect both along a simplicial tube.
The resulted complex $\mu''$ is also a minimal $2$-cycle, and contains the $1$-skeleton of $\mu$, thus also requires $c$ location; however it has more vertices. Repeating this process, can can assume $\mu$ has more than $c$ vertices, so it has a triple of vertices $T$ occupying at most two locations, and with the copy $w$ of $0\in \Delta$ corresponding to $T$, all $4$ vertices $T\cup\{w\}$ are contained in an affine plane $P$. Now, in $\mu''$, assign $w$ a nonzero velocity perpendicular to $P$, and zero velocity to all other vertices; this is a nontrivial motion on the $1$-skeleton of $\mu''$, a contradiction.
Planar Laman graphs {#sec:PlanarLaman}
===================
Consider the subclass of Laman graphs which are also planar, a.k.a. pointed pseudo-triangulations by the main result of [@planarMinimallyRigid05]. Unlike the class of all Laman graphs (see Theorem \[thm:no\_c(F\_d)\]), this subclass admits rigidity with few locations in the plane:
\[thm:PlanarLaman\] Let $A\subseteq \R^2$ be a generic subset of size $|A|=44$. Then for any planar Laman graph $G=(V,E)$ there exists a map $f:V(G)\rightarrow A$ for which the framework $f(G)$ is infinitesimally rigid.
[<span style="font-variant:small-caps;"></span> ]{}
The proof is by induction on the sequence of Henneberg moves used to construct $G$ from the single-edge graph $K_2=G_0$, denoted $G_0,G_1,\ldots,g_t=G$. Identify $A\cong A'\times [4]$ for some generic subset $A'\subseteq \R^2$ of size $|A'|=11$. Suppose, by induction, that the map $f_{i-1}=(\pi,c):V(G_{i-1})\rightarrow A'\times [4]$ both
\(R) makes $\pi(G_{i-1})$ infinitesimally rigid (hence the same holds for $f_{i-1}(G_{i-1})$), and
\(C) $c$ is injective (hence so is $f_{i-1}$) on the restriction to the the 2 or 3 vertices of $V(G_{i-1})$ involved in the next Henneberg move, i.e. to $N(v)$ the set of neighbors of the unique vertex $v$ in $V(G_i)\setminus V(G_{i-1})$.
We now extend $f_{i-1}$ to $f_i=(\pi',c'):V(G_i)\rightarrow A'\times [4]$ satisfying (R) and (C).
The Henneberg move of type (I) connects $v$ to any subset $N(v)$ of size two; then by avoiding any of the two locations $f_{i-1}(N(v))$
Concluding remarks {#sec:conclude}
==================
Regarding simplicial spheres, does a higher dimensional analog of Theorem \[thm:2-spheres\] hold? Namely,
Let $F(S^{d-1})$ be the family of $1$-skeleta of triangulations of the $(d-1)$-sphere. For $d>3$, is $F(S^{d-1})$ $d$-rigid with $c$-locations for some finite $c$? Namely, is $c_d(F(S^{d-1}))$ finite?
Regarding surfaces, does a uniform bound in Theorem \[thm:S\_g\] hold? Namely, for $F(S)$, the family of all graphs of compact connected surfaces, we ask:
Is $c_3(F(S))$ finite?
We remark that for the larger family of Fogelsanger’s minimal cycle complexes [@Fogelsanger], and even for the intermediate family which still contains $F(S)$, of complexes minimal with respect to containment among those supported by homology $2$-cycles, rigidity with few locations fails:
There is no finite set of locations in $\R^3$ such that every minimal $2$-cycle can be realized with vertices in these locations in an infinitesimally rigid way.
The proof follows the idea in the first proof of Theorem \[thm:no\_c(F\_d)\].
Indeed, assume that $c$ locations are enough to guarantee that every minimal $2$-cycle can be realized in an infinitesimally rigid way. Consider a minimal $2$-cycle $\mu$ that requires $c$ locations.
Consider secondly the boundary of a tetrahedron $\Delta$, and mark the four vertices by $0,1,2$ and $3$. We subdivide the triangle $\{123\}$ in some way that introduces exactly 3 new vertices, all in its interior, which form a triangle $\Gamma$ (in the interior of the triangle $\{123\}$). Denote the resulting subdivision of $\Delta$ by $\Delta'$. Totally order the triples of vertices in $\mu$. According to this order, for every triple of vertices of $\mu$, attach a new copy of $\Delta'$ along the vertices $1$, $2$ and $3$ to the currently constructed complex $\mu'$ (starting with the original $\mu$), then remove $\Gamma$ and some triangle of $\mu'$, and connect both along a simplicial tube.
The resulted complex $\mu''$ is also a minimal $2$-cycle, and contains the $1$-skeleton of $\mu$, thus also requires $c$ locations; however it has more vertices. Repeating this process, we can assume $\mu$ has more than $2c$ vertices, so it has a triple of vertices $T$ occupying at most one location, and with the copy $w$ of $0\in \Delta$ corresponding to $T$, all $4$ vertices $T\cup\{w\}$ are contained in an affine plane $P$. Now, in $\mu''$, assign $w$ a nonzero velocity perpendicular to $P$, and zero velocity to all other vertices; this is a nontrivial infinitesimal motion on the $1$-skeleton of $\mu''$, a contradiction.
Regarding Theorem \[thm:no\_c(F\_d)\], while the phenomenon of rigidity with few locations does not hold for Laman graphs in the plane, Walter Whiteley asked us whether it does hold for the subfamily of planar Laman graphs, see [@PlanarLaman2005 Thm.1] for an equivalent characterization via pointed pseudo-triangulations. We leave this question open.
Let $F$ be the family of planar Laman graphs. Is $c_2(F)$ finite?
Given that rigidity with few locations holds, it is interesting to find optimal bounds. For $F(S^2)$, note that the vertices of the octahedron must occupy 6 different locations in $\R^3$ in any infinitesimally rigid embedding, thus $c_3(F(S^2))\ge 6$.
\[prob:PlanarLaman\] For every surface $M$, what is the value $c_3(F(M))$? At least, find improved bounds.
In this note we considered infinitesimal rigidity with $c$-locations, for $c$ a constant. More generally, for a family $F$ of generically $d$-rigid graphs, let $F(n)$ be the subfamily of graphs in $F$ with at most $n$ vertices, and let $c_{d,F}(n)$ be the minimum $c$ such that $F(n)$ is $d$-rigid with $c$-locations. One can study the growth of the function $c_{d,F}(n)$.
For $F_3$, the family of all generically $3$-rigid graphs, what is the asymptotic growth of $c_{3,F_3}(n)$? Is it sublinear in $n$?
Let us remark that for $G$ a minimally $d$-rigid graph, the chromatic number $\chi(G)$ of $G$ is a lower bound for $c_{d,F_d}(n)$, as infinitesimal $d$-rigidity forces the vertices of any edge to occupy two different locations. However, as any induced subgraph $G'=(V',E')$ of $G$ supports no nontrivial stress, $G'$ must satisfy $|E'|\leq d|V'|$, hence $G$ is $(2d-1)$-degenerate, so $\chi(G)\leq 2d$, and we get no growth with $n$ in the lower bound on $c_{d,F_d}(n)$ by using the chromatic number.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Isabella Novik and Orit Raz for helpful discussions, and Bob Connelly, Louis Theran and Walter Whiteley for helpful comments and references. We thank the referee for valuable advice, resulted in greatly improved presentation. K.A. was supported by ERC StG 716424 - CASe and ISF Grant 1050/16. E.N. was partially supported by ISF grant 1695/15, by ISF-NRF joint grant 2528/16 and by ISF-BSF joint grant 2016288.
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|
---
abstract: 'A world sheet in anti-de Sitter space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds. We consider the family of lightlike hypersurfaces along spacelike submanifolds in the world sheet. The locus of the singularities of lightlike hypersurfaces along spacelike submanifolds forms the caustic of the world sheet. This notion is originally introduced by Bousso and Randall in theoretical physics. In this paper we give a mathematical framework for the caustics of world sheets as an application of the theory of graph-like Legendrian unfoldings.'
author:
- Shyuichi IZUMIYA
title: |
Caustics and Maxwell sets of world sheets\
in anti-de Sitter space
---
Introduction
============
In this paper we consider geometrical properties of caustics and Maxwell sets of world sheets in anti-de Sitter space as an application of the theory of Legendrian unfoldings [@Izumiya93; @Izumiya-Takahashi; @Izumiya-Takahashi2; @Izumiya-Takahashi3; @Graph-like; @GeomLag14] which is a special but an important case of the theory of wave front propagations [@Zak1]. Anti-de Sitter space is one of the Lorentz space forms with rich geometric properties. It is defined as a pseudo-sphere with a negative curvature in semi-Euclidean space with index 2 which admits the biggest symmetry in Riemannian or Lorentz space forms. Anti-de Sitter space plays important roles in theoretical physics such as the theory of general relativity, the string theory and the brane world scenario etc. It is one of the typical model of bulk spaces of the brane world scenario or the string theory (cf. [@Bousso; @Bousso-Randall; @KR01; @M98; @RS99; @W98]). On the other hand, one of the important objects in the theoretical physics is the notion of lightlike hypersurfaces (light-sheets in physics) because they provide good models for different types of horizons [@Ch; @MTW]. In [@IzuLight] we considered lightlike hypersurfaces along spacelike submanifolds with general codimension in anti-de Sitter space. lightlike hypersurfaces usually have singularities. We showed that lightlike hypersurfaces are wave fronts and applied the theory of Legendrian singularities [@Arnold1; @Zak] to obtaining geometric properties of the singularities of lightlike hypersrufaces.
A world sheet (or a brane) in anti-de Sitter space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds. Each spacelike submanifold is called a momentary space. Since a momentary space is a spacelike submanifold, we have a lightlike hypersurface along each momentary space as a consequence of [@IzuLight]. The set of singular values of a lightlike hypersurface is called the focal set along the momentary space. Since the world sheet is a one-parameter family of momentary spaces, we naturally consider the family of lightlike hypersurfaces along momentary spaces in the world sheet. The locus of the singularities (the focal sets) of lightlike hypersurfaces along momentary spaces is the caustic of the world sheet which was introduced by Bousso and Randall [@Bousso; @Bousso-Randall] in order to define the notion of holographic domains. In this paper we construct a mathematical framework for the caustic of a world sheet and investigate the geometric properties of the singularities of the caustics of world sheets. For the purpose, we apply the theory of graph-like Legendrian unfoldings [@Graph-like; @GeomLag14]. We also consider the notion of Maxwell sets (crease sets) of world sheets which play an important role in the cosmology [@Penrose; @Siino]. In their paper [@Bousso; @Bousso-Randall] the authors draw pictures on the simplest case (cf. [@Bousso-Randall Figures 2 and 3]). However, this case the caustic coincides with the Maxwell set (i.e. a line). In general, these sets are different, so that we consider both of them in this paper and emphasize that the Maxwell set of a world sheet is also an important subject.
On the other hand, caustics appear in several area in physics (i.e. geometrical optics [@Nye], the theory of underwater acoustics [@Brekhov] and the theory of gravitational lensings [@Peters] , and so on) and mathematics (i.e. classical differential geometry [@Bru-Gib; @Izu07; @Porteous] and theory of differential equations [@Hormander; @IzuHJ93], and so on [@Arnold-pink]). The notion of caustics originally belongs to geometrical optics. We can observe the caustic formed by the rays reflected at a mirror. One of the examples of caustics in the classical differential geometry is the evolute of a curve in the Euclidean plane which is given by the envelope of normal lines emanated from the curve. The ray in the Euclidean plane is considered to be a line, so that the evolute is the caustic in the sense of geometrical optics. Moreover, the singular points of the evolute correspond to the vertices of the original curve. The vertex is the point at where the curve has higher order contact with the osculating circle (i.e. the point where the curvature has an extremum). Therefore, the evolute provides important geometrical information of the curve. We have the notion of evolutes for general hypersurfaces in the Euclidean space similar to the plane curve case. In particular, there are detailed investigations on evolutes for surfaces in the Euclidean $3$-space [@Izu07; @Porteous]. Analogous to the Euclidean case, we can define the evolute of a hypersurface in Lorentz-Minkowski space [@Saloom; @Farid]. Since a world sheet is a timelike submanifold, we may consider the evolute of a timelike hypersurface in Lorentz-Minkowski space. However, the normal line is directed by a spacelike vector, so that the speed of the line exceeds the speed of the ray. Although the evolute of a timelike hypersurface is a caustic in the theory of Lagrangian singularities, it is not a caustic in the sense of physics. The situation in anti-de Sitter space is similar to that of Lorentz-Minkowski space. In a Lorentz manifold, the ray is directed by a lightlike vector, so that rays emanated from a spacelike submanifold forms a lightlike hypersurface. Moreover, we have no notions of the time constant in the relativity theory. Hence everything that is moving depends on the time. Therefore, we have to consider one parameter families of spacelike submanifolds (i.e. world sheets) in a Lorentz manifold, so that the notion of caustics by Bousso and Randall [@Bousso; @Bousso-Randall] is essential. For further theoretical investigation, we construct a mathematical (geometric) framework for the caustics and the Maxwell sets of world sheets in this paper.
We remark that the similar construction can be obtained for other Lorentz space forms (i.e. Lonrentz-Minkowski space and de Sitter space). For a general Lorentz manifold, the situation is different from the case of Lorentz space forms. In this case, we cannot construct explicit generating families for corresponding graph-like Legendrian unfoldings (cf. §6). However, we can apply the theory of graph-like Legendrian unfoldings by using the classical method of characteristics for the (singular) eikonal equation corresponding to the Lorentz metric. The detailed results will be appeared in elsewhere.
Semi-Euclidean space with index 2 {#sec:1}
=================================
In this section we prepare the basic notions on the semi-Euclidean (n+2)-space with index 2. For detailed properties of the semi-Euclidean space, see [@Oneil]. For any vectors ${\mbox{\boldmath $x$}}=(x_{-1}, x_0, x_1,\cdots,x_{n}), {\mbox{\boldmath $y$}}=(y_{-1}, y_0, y_1,\cdots,y_{n})\in \Bbb R^{n+2},$ the [*pseudo scalar product* ]{} of ${\mbox{\boldmath $x$}}$ and ${\mbox{\boldmath $y$}}$ is defined to be $\langle{\mbox{\boldmath $x$}},{\mbox{\boldmath $y$}}\rangle =-x_{-1} y_{-1}-x_0 y_0 +\sum_{i=1}^{n}x_i
y_i$. We call $(\Bbb R^{n+2}, \langle ,\rangle )$ a [*semi-Euclidean*]{} (n+2)-[*space with index 2*]{} and write $\Bbb
R^{n+2}_2$ instead of $(\Bbb R^{n+2},\langle ,\rangle )$. We say that a non-zero vector ${\mbox{\boldmath $x$}}$ in $\Bbb R^{n+2}_2$ is [*spacelike*]{}, [*null*]{} or [*timelike*]{} if $\langle{\mbox{\boldmath $x$}},{\mbox{\boldmath $x$}}\rangle>0,\langle{\mbox{\boldmath $x$}},{\mbox{\boldmath $x$}}\rangle =0$ or $\langle{\mbox{\boldmath $x$}},{\mbox{\boldmath $x$}}\rangle <0$ respectively. The norm of the vector ${\mbox{\boldmath $x$}}\in \Bbb R^{n+2}_2$ is defined to be $\|{\mbox{\boldmath $x$}}\|=\sqrt{|\langle{\mbox{\boldmath $x$}},
{\mbox{\boldmath $x$}}\rangle|}$. We define the [*signature*]{} of ${\mbox{\boldmath $x$}}$ by $${\rm sign} ({\mbox{\boldmath $x$}})=\left\{
\begin{array}{ccc}
1\qquad\quad ${\mbox{\boldmath $x$}}$\ \mbox{is\ spacelike}\\
\mbox{}0\qquad\quad {\mbox{\boldmath $x$}}\ \mbox{is\ null} \hspace*{\fill}\\
-1\qquad {\mbox{\boldmath $x$}}\ \mbox{is\ timelike}
\end{array}\right.$$ For a non-zero vector ${\mbox{\boldmath $n$}}\in \Bbb R^{n+2}_2$ and a real number $c$, we define a [*hyperplane with pseudo-normal* ]{}${\mbox{\boldmath $n$}}$ by $$HP({\mbox{\boldmath $n$}},c)=\{{\mbox{\boldmath $x$}}\in \Bbb R^{n+2}_2 |\langle{\mbox{\boldmath $x$}},{\mbox{\boldmath $n$}}\rangle=c\}.$$ We call $HP({\mbox{\boldmath $n$}},c)$ a [*Lorentz hyperplane*]{}, a [*semi-Euclidean hyperplane with index 2*]{} or a [*null hyperplane*]{} if ${\mbox{\boldmath $n$}}$ is [*timelike, spacelike or null* ]{}respectively.
We now define the [*Anti de Sitter $n+1$-space* ]{} (briefly, the [*AdS $n+1$-space*]{}) by $$AdS^{n+1}=\{{\mbox{\boldmath $x$}}\in \Bbb R_2^{n+2}\ |\ \langle{\mbox{\boldmath $x$}},{\mbox{\boldmath $x$}}\rangle =-1\}=H^{n+1}_1,$$ the [*unit pseudo $n+1$-sphere with index 2*]{} by $$S^{n+1}_2=\{{\mbox{\boldmath $x$}}\in {{\mathbb R}}_2^{n+2}\ |\ \langle{\mbox{\boldmath $x$}},{\mbox{\boldmath $x$}}\rangle =1\},$$ and the [*[(]{}closed[)]{} nullcone*]{} with vertex $\bm{\lambda}\in {{\mathbb R}}^{n+2}_2$ by\
$$\Lambda_{\bm{\lambda}}^{n+1}=\{{\mbox{\boldmath $x$}}\in {{\mathbb R}}_{2}^{n+2}|\langle{\mbox{\boldmath $x$}}-\bm{\lambda}, {\mbox{\boldmath $x$}}-\bm{\lambda}\rangle=0\}.$$ In particular we write $\Lambda ^*=\Lambda ^{n+1}_0\setminus \{{\mbox{\boldmath $0$}}\}$ and also call it the [*[(]{}open[)]{} nullcone*]{}. Our main subject in this paper is $AdS^{n+1}$. Since the causality of $AdS^{n+1}$ is violated, it is usually considered the universal covering space $\widetilde{AdS}^{n+1}$ of $AdS^{n+1}$ in physics which is called the [*universal Anti de Sitter space*]{}. We remark that the local structure of these spaces are the same. Since $AdS^{n+1}$ is a Lorentz space form, there exists a lightcone on each tangent space. Such a lightcone is explicitly expressed as follows: For any $\bm{\lambda} \in AdS^{n+1},$ we have a hyperplane $HP(\bm{\lambda},-1).$ This hyperplane is the tangent hyperplane of $AdS^{n+1}$ at $\bm{\lambda}.$ We can show that $$HP(\bm{\lambda},-1)\cap AdS^{n+1}=\Lambda ^{n+1}_{\bm{\lambda}}\cap AdS^{n+1}.$$ Therefore, $HP(\bm{\lambda},-1)\cap AdS^{n+1}$ is the lightcone in the tangent hyperplane $HP(\bm{\lambda},-1)$ of $AdS^{n+1}$ at $\bm{\lambda}.$ We write it by $LC^{AdS}(\bm{\lambda})$ and call an [*anti-de Sitter lightcone*]{} (briefly, an [*AdS-lightcone*]{}) at $\bm{\lambda}\in AdS^{n+1}$.
For any ${\mbox{\boldmath $x$}}_1,\cdots, {\mbox{\boldmath $x$}}_{n+1} \in {{\mathbb R}}^{n+2}_2$, we define a vector ${\mbox{\boldmath $x$}}_1\wedge \cdots\wedge {\mbox{\boldmath $x$}}_n$ by $${\mbox{\boldmath $x$}}_1\wedge\cdots\wedge {\mbox{\boldmath $x$}}_{n+1}= \vmatrix
-{\mbox{\boldmath $e$}}_{-1}&-{\mbox{\boldmath $e$}}_0&{\mbox{\boldmath $e$}}_1&\cdots&{\mbox{\boldmath $e$}}_{n}\vspace{2mm}\\
x^1_{-1}&x^1_0&x^1_1&\cdots&x^1_{n}\\
\vdots&\vdots&\vdots&\vdots&\vdots\\
x^{n+1}_{-1}&x^{n+1}_{0}&x^{n+1}_1&\cdots&x^{n+1}_{n}
\endvmatrix,$$ where $\{{\mbox{\boldmath $e$}}_{-1}, {\mbox{\boldmath $e$}}_0, {\mbox{\boldmath $e$}}_1,\cdots,{\mbox{\boldmath $e$}}_{n}\}$ is the canonical basis of $\Bbb R^{n+2}_2$ and ${\mbox{\boldmath $x$}}_i=(x^i_{-1}, x^i_0,
x^i_1,\cdots,x^i_{n})$. We can easily check that $$\langle{\mbox{\boldmath $x$}},\ {\mbox{\boldmath $x$}}_1\wedge\cdots\wedge
{\mbox{\boldmath $x$}}_{n+1}\rangle=\textrm{det}({\mbox{\boldmath $x$}}, {\mbox{\boldmath $x$}}_1,\cdots,{\mbox{\boldmath $x$}}_{n+1}),$$ so that ${\mbox{\boldmath $x$}}_1\wedge\cdots\wedge {\mbox{\boldmath $x$}}_{n}$ is pseudo-orthogonal to any ${\mbox{\boldmath $x$}}_i$ (for $i=1,\cdots,n$).
World sheets in in anti-de Sitter space
=======================================
In this section we introduce the basic geometrical framework for the study of world sheets in anti-de Sitter $n+1$-space. Consider the orientation of ${{\mathbb R}}^{n+2}_2$ provided by the condition that $\textrm{det}({\mbox{\boldmath $e$}}_{-1}, {\mbox{\boldmath $e$}}_0,{\mbox{\boldmath $e$}}_1,\cdots,{\mbox{\boldmath $e$}}_{n})>0.$ This orientation induces the orientation of $x_{-1}x_0$-plane, so that it gives a time orientation on $AdS^{n+1}$. If we consider the universal Anti de Sitter space $\widetilde{AdS}^{n+1},$ we can determine the future direction. The world sheet is defined to be a timelike submanifold foliated by a codimension one spacelike submanifolds. Here, we only consider the local situation, so that we considered a one-parameter family of spacelike submanifolds. Let $AdS^{n+1}$ be the oriented and time-oriented anti-de Sitter space. Let ${\mbox{\boldmath $X$}}:U\times I{\longrightarrow}AdS^{n+1}$ be a timelike embedding of codimension $k-1,$ where $U\subset {{\mathbb R}}^s$ ($s+k=n+2$) is an open subset and $I$ an open interval. We write $W={\mbox{\boldmath $X$}}(U\times I)
$ and identify $W$ and $U\times I$ through the embedding ${\mbox{\boldmath $X$}}.$ Here, the embedding ${\mbox{\boldmath $X$}}$ is said to be [*timelike*]{} if the tangent space $T_p W$ of $W$ at $p={\mbox{\boldmath $X$}}(u,t)$ is a timelike subspace (i.e., Lorentz subspace of $T_pAdS^{n+1}$) for any point $p\in W$. We write $\mathcal{S}_t={\mbox{\boldmath $X$}}(U\times\{t\})$ for each $t\in I.$ We call $\mathcal{S}=\{\mathcal{S}_t\ |t\in I\}$ a [*spacelike foliation*]{} on $W$ if $\mathcal{S}_t$ is a spacelike submanifold for any $t\in I.$ Here, we say that $\mathcal{S}_t$ is [*spacelike*]{} if the tangent space $T_p\mathcal{S}_t$ consists only spacelike vectors (i.e., spacelike subspace) for any point $p\in \mathcal{S}_t.$ We call $\mathcal{S}_t$ a [*momentary space*]{} of $\mathcal{S}=\{\mathcal{S}_t\ |t\in I\}$. For any $p={\mbox{\boldmath $X$}}(u,t)\in W\subset AdS^{n+1},$ we have $$T_pW=\langle {\mbox{\boldmath $X$}}_t(u,t),{\mbox{\boldmath $X$}}_{u_1}(u,t),\dots ,{\mbox{\boldmath $X$}}_{u_s}(u,t)\rangle _{{\mathbb R}},$$ where ${\mbox{\boldmath $X$}}_t=\partial {\mbox{\boldmath $X$}}/\partial t,{\mbox{\boldmath $X$}}_{u_j}=\partial {\mbox{\boldmath $X$}}/\partial u_j.$ We say that $(W,\mathcal{S})$ (or, ${\mbox{\boldmath $X$}}$ itself) is a [*world sheet*]{} if $W$ is time-orientable. Since $W$ is time-orientable, there exists a timelike vector field ${\mbox{\boldmath $v$}}(u,t)$ on $W$ [@Oneil Lemma 32]. Moreover, we can choose that ${\mbox{\boldmath $v$}}$ is adapted with respected to the time-orientation of $AdS^{n+1}.$ Here, we say that a timelike vector field ${\mbox{\boldmath $v$}}(u,t)$ on $W$ is [*adapted*]{} if $
\textrm{det}({\mbox{\boldmath $X$}}(u,t),{\mbox{\boldmath $v$}}(u,t),{\mbox{\boldmath $e$}}_1,\dots ,{\mbox{\boldmath $e$}}_{n})>0.
$ Let $N_p(W)$ be the pseudo-normal space of $W$ at $p={\mbox{\boldmath $X$}}(u,t)$ in ${{\mathbb R}}^{n+2}_2.$ Since $T_pW$ is a timelike subspace of $T_p{{\mathbb R}}^{n+2}_2,$ $N_p(W)$ is a $k$-dimensional Lorentz subspace of $T_p{{\mathbb R}}^{n+2}_2$. (cf.,[@Oneil]). On the pseudo-normal space $N_p(W),$ we have a $(k-1)$-dimensional spacelike subspace: $$N^{AdS}_p(W)= \{{\mbox{\boldmath $\xi$}}\in N_p(W)\ |\ \langle {\mbox{\boldmath $\xi$}},{\mbox{\boldmath $X$}}(u,t)\rangle =0\ \},$$ so that we have a $(k-2)$-unit sphere $$N^{AdS}_1(W)_p=\{{\mbox{\boldmath $\xi$}}\in N^{AdS}_p(W)\ |\ \langle {\mbox{\boldmath $\xi$}},{\mbox{\boldmath $\xi$}}\rangle =1\ \}.$$ Therefore, we have a unit spherical normal bundle over $W$: $$N^{AdS}_1(W)=\bigcup _{p\in W} N^{AdS}_1(W)_p.$$
On the other hand, we write $N_p(\mathcal{S}_t)$ as the pseudo-normal space of $\mathcal{S}_t$ at $p={\mbox{\boldmath $X$}}(u,t)$ in ${{\mathbb R}}^{n+2}_2.$ Then $N_p(\mathcal{S}_t)$ is a $k+1$-dimensional semi-Euclidean subspace with index $2$ of $T_p{{\mathbb R}}^{n+2}_2$ [@Oneil]. On the pseudo-normal space $N_p(\mathcal{S}_t),$ we have two kinds of pseudo spheres: $$\begin{aligned}
N_p(\mathcal{S}_t;-1)& = & \{{\mbox{\boldmath $v$}}\in N_p(\mathcal{S}_t)\ |\ \langle {\mbox{\boldmath $v$}},{\mbox{\boldmath $v$}}\rangle =-1\ \} \\
N_p(\mathcal{S}_t;1)&= & \{{\mbox{\boldmath $v$}}\in N_p(\mathcal{S}_t)\ |\ \langle {\mbox{\boldmath $v$}},{\mbox{\boldmath $v$}}\rangle =1\ \}.\end{aligned}$$ We remark that $N_p(\mathcal{S}_t;-1)$ is the $k$-dimensional anti-de Sitter space and $N_p(\mathcal{S}_t;1)$ is the $k$-dimensional pseudo-sphere with index $2.$ Therefore, we have two unit spherical normal bundles $N(\mathcal{S}_t;-1)$ and $N(\mathcal{S}_t;1)$ over $\mathcal{S}_t$. By definition, ${\mbox{\boldmath $X$}}(u,t)$ is one of the timelike unit normal vectors of $\mathcal{S}_t$ at $p={\mbox{\boldmath $X$}}(u,t),$ so that ${\mbox{\boldmath $X$}}(u,t) \in N_p(\mathcal{S}_t).$ Since $\mathcal{S}_t={\mbox{\boldmath $X$}}(U\times \{t\})$ is a codimension one spacelike submanifold in $W,$ there exists a unique timelike adopted unit normal vector field ${\mbox{\boldmath $n$}}^T(u,t)$ of $\mathcal{S}_t$ such that ${\mbox{\boldmath $n$}}^T(u,t)$ is tangent to $W$ at any point $p={\mbox{\boldmath $X$}}(u,t).$ It means that ${\mbox{\boldmath $n$}}^T(u,t)\in N_p(\mathcal{S}_t)\cap T_pW$ with $\langle {\mbox{\boldmath $n$}}^T(u,t),{\mbox{\boldmath $n$}}^T(u,t)\rangle =-1$ and $\det ({\mbox{\boldmath $X$}}(u,t),{\mbox{\boldmath $n$}}^T(u,t),{\mbox{\boldmath $e$}}_1,\dots ,{\mbox{\boldmath $e$}}_n ) >0.$ We define a $(k-2)$-dimensional spacelike unit sphere in $N_p(\mathcal{S}_t)$ by $$N^{AdS}_1(\mathcal{S}_t)_p[{\mbox{\boldmath $n$}}^T]=\{{\mbox{\boldmath $\xi$}}\in N_p(\mathcal{S}_t;1)\ |\ \langle {\mbox{\boldmath $\xi$}}, {\mbox{\boldmath $n$}}^T(u,t)\rangle =\langle {\mbox{\boldmath $\xi$}},{\mbox{\boldmath $X$}}(u,t)\rangle=0,p={\mbox{\boldmath $X$}}(u,t)\ \}.$$ Then we have a [*spacelike unit $(k-2)$-spherical bundle $N_1(\mathcal{S}_t)[{\mbox{\boldmath $n$}}^T]$ over $\mathcal{S}_t$ with respect to ${\mbox{\boldmath $n$}}^T$*]{}. Since we have $T_{(p,\xi)}N^{AdS}_1(\mathcal{S}_t)[{\mbox{\boldmath $n$}}^T]=T_p\mathcal{S}_t\times T_\xi N^{AdS}_1(\mathcal{S}_t)_p[{\mbox{\boldmath $n$}}^T],$ we have the canonical Riemannian metric on $N^{AdS}_1(\mathcal{S}_t)[{\mbox{\boldmath $n$}}^T]$ which we write $(G_{ij}((u,t),{\mbox{\boldmath $\xi$}}))_{1\leqslant i,j\leqslant n-1}.$ Since ${\mbox{\boldmath $n$}}^T$ is uniquely determined, we can write $N_1^{AdS}[\mathcal{S}_t]=N_1^{AdS}(\mathcal{S}_t)[{\mbox{\boldmath $n$}}^T].$ Moreover, we remark that $N_1^{AdS}(W)|\mathcal{S}_t=N_1^{AdS}[\mathcal{S}_t]$ for any $t\in I.$
We now define a map $\mathbb{NG}:N^{AdS}_1(W){\longrightarrow}\Lambda ^*$ by $\mathbb{NG}({\mbox{\boldmath $X$}}(u,t),{\mbox{\boldmath $\xi$}})={\mbox{\boldmath $n$}}^T(u,t)+{\mbox{\boldmath $\xi$}}$. We call $\mathbb{NG}$ an [*$AdS$-world nullcone Gauss image*]{} of $W={\mbox{\boldmath $X$}}(U\times I)$. A [*momentary nullcone Gauss image*]{} of $N_1^{AdS}[\mathcal{S}_t]$ is defined to be the restriction of the $AdS$-world nullcone Gauss image $$\mathbb{NG}(\mathcal{S}_t)=\mathbb{NG}|N_1^{AdS}[\mathcal{S}_t]:N_1^{AdS}[\mathcal{S}_t]{\longrightarrow}\Lambda ^*.$$ This map leads us to the notions of curvatures. Let $T_{(p,\xi)}N_1[\mathcal{S}_t]$ be the tangent space of $N_1[\mathcal{S}_t]$ at $(p,{\mbox{\boldmath $\xi$}}).$ Under the canonical identification $(\mathbb{NG}(\mathcal{S}_t)^*T{{\mathbb R}}^{n+2}_2)_{(p,{\mbox{\scriptsize \boldmath$\xi$}})}
=T_{({\mbox{\scriptsize \boldmath$n$}}^T(p)+{\mbox{\scriptsize \boldmath$\xi$}})}{{\mathbb R}}^{n+1}_1\equiv T_p{{\mathbb R}}^{n+2}_2,$ we have $$T_{(p,{\mbox{\scriptsize \boldmath$\xi$}})}N_1[\mathcal{S}_t]=T_p\mathcal{S}_t\oplus T_\xi S^{k-2}\subset T_pM\oplus N_p(\mathcal{S}_t)=T_p{{\mathbb R}}^{n+2}_2,$$ where $T_\xi S^{k-2}\subset T_\xi N_p(\mathcal{S}_t)\equiv N_p(\mathcal{S}_t)$ and $p={\mbox{\boldmath $X$}}(u,t).$ Let $$\Pi ^t :\mathbb{NG}(\mathcal{S}_t)^*T{{\mathbb R}}^{n+2}_2=TN_1[\mathcal{S}_t]\oplus {{\mathbb R}}^{k+1}
{\longrightarrow}TN_1[\mathcal{S}_t]$$ be the canonical projection. Then we have a linear transformation $$S_N (\mathcal{S}_t)_{(p,{\mbox{\scriptsize \boldmath$\xi$}})}=-\Pi^t_{\mathbb{NG}(\mathcal{S}_t)(p,\xi)}\circ d_{(p,\xi)}\mathbb{NG}(\mathcal{S}_t)
: T_{(p,\xi)}N^{AdS}_1[\mathcal{S}_t]{\longrightarrow}T_{(p,\xi)}N^{AdS}_1[\mathcal{S}_t],$$ which is called a [*momentary nullcone shape operator*]{} of $N^{AdS}_1[\mathcal{S}_t]$ at $(p,{\mbox{\boldmath $\xi$}}).$
On the other hand, we choose a pseudo-normal section ${\mbox{\boldmath $n$}}^S(u,t)\in N^{AdS}_1(W)$ at least locally. Then we have $\langle {\mbox{\boldmath $n$}}^S,{\mbox{\boldmath $n$}}^S\rangle =1$ and $\langle {\mbox{\boldmath $X$}}_t,{\mbox{\boldmath $n$}}^S\rangle = \langle {\mbox{\boldmath $X$}}_{u_i},{\mbox{\boldmath $n$}}^S\rangle =\langle {\mbox{\boldmath $n$}}^T,{\mbox{\boldmath $n$}}^S\rangle =0,$ so that the vector ${\mbox{\boldmath $n$}}^T (u,t)+ {\mbox{\boldmath $n$}}^S(u,t)$ is lightlike. We define a mapping $$\mathbb{NG}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S):U{\longrightarrow}\Lambda ^*$$ by $\mathbb{NG}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)(u)={\mbox{\boldmath $n$}}^T(u,t_0)+{\mbox{\boldmath $n$}}^S(u,t_0),$ which is called a [*momentary nullcone Gauss images of $\mathcal{S}_{t_0}={\mbox{\boldmath $X$}}(U\times \{t_0\})$ with respect to ${\mbox{\boldmath $n$}}^S.$*]{} Under the identification of $\mathcal{S}_{t_0}$ and $U\times\{t_0\}$ through ${\mbox{\boldmath $X$}},$ we have the linear mapping provided by the derivative of the momentary nullcone Gauss image $\mathbb{NG}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)$ at each point $p={\mbox{\boldmath $X$}}(u,t_0)$, $$d_p\mathbb{NG}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S):T_p\mathcal{S}_{t_0}{\longrightarrow}T_p{{\mathbb R}}^{n+1}_1= T_p\mathcal{S}_{t_0}\oplus N_p(\mathcal{S}_{t_0}).$$ Consider the orthogonal projection $\pi ^t:T_p\mathcal{S}_{t_0}\oplus
N_p(\mathcal{S}_{t_0})\rightarrow T_p\mathcal{S}_{t_0}.$ We define $$S_p(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)=-\pi^t\circ d_p\mathbb{NG}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S):T_p\mathcal{S}_{t_0}{\longrightarrow}T_p\mathcal{S}_{t_0}.$$ We call the linear transformation $S_{p}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)$ a [*momentary ${\mbox{\boldmath $n$}}^S$-shape operator*]{} of $\mathcal{S}_{t_0}={\mbox{\boldmath $X$}}(U\times \{t_0\})$ at $p={\mbox{\boldmath $X$}}(u,t_0).$ Let $\{\kappa
_{i}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)(p)\}_ {i=1}^s$ be the eigenvalues of $S_{p}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)$, which are called [ *momentary nullcone principal curvatures of $\mathcal{S}_{t_0}$ with respect to ${\mbox{\boldmath $n$}}^S$*]{} at $p={\mbox{\boldmath $X$}}(u,t_0)$. Then a [*momentary nullcone Gauss-Kronecker curvature of $\mathcal{S}_{t_0}$ with respect to ${\mbox{\boldmath $n$}}^S$*]{} at $p={\mbox{\boldmath $X$}}(u,t_0)$ is defined to be $$K_N(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)(p)={\rm det} S_{p}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S).$$ We say that a point $p={\mbox{\boldmath $X$}}(u,t_0)$ is a [*momentary ${\mbox{\boldmath $n$}}^S$-nullcone umbilical point*]{} of $\mathcal{S}_{t_0}$ if $$S_{p}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)=\kappa (\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)(p) 1_{T_{p}\mathcal{S}_{t_0}}.$$ We say that $W={\mbox{\boldmath $X$}}(U\times I)$ is [*totally ${\mbox{\boldmath $n$}}^S$-nullcone umbilical*]{} if any point $p={\mbox{\boldmath $X$}}(u,t)\in W$ is momentary ${\mbox{\boldmath $n$}}^S$-nullcone umbilical. Moreover, $W={\mbox{\boldmath $X$}}(U\times I)$ is said to be [*totally nullcone umbilical*]{} if it is totally ${\mbox{\boldmath $n$}}^S$-nullcone umbilical for any ${\mbox{\boldmath $n$}}^S.$ We deduce now the nullcone Weingarten formula. Since ${\mbox{\boldmath $X$}}_{u_i}$ $(i=1,\dots s)$ are spacelike vectors, we have a Riemannian metric (the [*first fundamental form* ]{}) on $\mathcal{S}_{t_0}={\mbox{\boldmath $X$}}(U\times\{t_0\})$ defined by $ds^2 =\sum _{i=1}^{s} g_{ij}du_idu_j$, where $g_{ij}(u,t_0) =\langle {\mbox{\boldmath $X$}}_{u_i}(u,t_0 ),{\mbox{\boldmath $X$}}_{u_j}(u,t_0)\rangle$ for any $u\in U.$ We also have a [*nullcone second fundamental invariant of $\mathcal{S}_{t_0}$ with respect to the normal vector field ${\mbox{\boldmath $n$}}^S $*]{} defined by $h _{ij}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S )(u,t_0)=\langle -({\mbox{\boldmath $n$}}^T +{\mbox{\boldmath $n$}}^S)
_{u_i}(u,t_0),{\mbox{\boldmath $X$}}_{u_j}(u,t_0)\rangle$ for any $u\in U.$ By the similar arguments to those in the proof of [@IzuSM Proposition 3.2], we have the following proposition.
Let $\{{\mbox{\boldmath $X$}}, {\mbox{\boldmath $n$}}^T,{\mbox{\boldmath $n$}}^S_1,\dots ,{\mbox{\boldmath $n$}}^S_{k-1}\}$ be a a pseudo-orthonormal frame of $N(\mathcal{S}_{t_0})$ with ${\mbox{\boldmath $n$}}^S_{k-1}={\mbox{\boldmath $n$}}^S.$ Then we have the following momentary nullcone Weingarten formulae [:]{} 1.5pt
[(a)]{} $\mathbb{NG}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)_{u_i}=\langle {\mbox{\boldmath $n$}}^T_{u_i},{\mbox{\boldmath $n$}}^S\rangle({\mbox{\boldmath $n$}}^T+{\mbox{\boldmath $n$}}^S)+\sum _{\ell =1}^{k-2}\langle ({\mbox{\boldmath $n$}}^T+{\mbox{\boldmath $n$}}^S)_{u_i},{\mbox{\boldmath $n$}}^S_\ell \rangle{\mbox{\boldmath $n$}}^S_\ell -\sum_{j=1}^{s}
h_i^j(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S ){\mbox{\boldmath $X$}}_{u_j}$
[(b)]{} $
\pi ^t\circ \mathbb{NG}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)_{u_i}=-\sum_{j=1}^{s}
h_i^j(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S ){\mbox{\boldmath $X$}}_{u_j}.
$
Here $\displaystyle{\left(h_i^j(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S )\right)=\left(h_{ik}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)\right)\left(g^{kj}\right)}$ and $\displaystyle{\left( g^{kj}\right)=\left(g_{kj}\right)^{-1}}.$
Since $\mathbb{NG}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)_{u_i}=d\mathbb{NG}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)({\mbox{\boldmath $X$}}_{u_i})$, we have $$S_p(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S )({\mbox{\boldmath $X$}}_{u_i}(u,t_0))=-\pi^t\circ \mathbb{NG}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)_{u_i}(u,t_0),$$ so that the representation matrix of $S_p(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S )$ with respect to the basis $$\{{\mbox{\boldmath $X$}}_{u_1}(u,t_0),{\mbox{\boldmath $X$}}_{u_2}(u,t_0),\dots ,{\mbox{\boldmath $X$}}_{u_s}(u,t_0)\}$$ of $T_p\mathcal{S}_{t_0}$ is $(h^i_j(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)(u,t_0)).$ Therefore, we have an explicit expression of the momentary nullcone Gauss-Kronecker curvature of $\mathcal{S}_{t_0}$ with respect to ${\mbox{\boldmath $n$}}^S$ by $$K_N (\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S )(u,t_0)=\frac{\displaystyle{{\rm det}\left(h_{ij}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S )(u,t_0)\right)}}
{\displaystyle{{\rm det}\left(g_{\alpha \beta}(u,t_0)\right)}}.$$ Since $\langle -({\mbox{\boldmath $n$}}^T +{\mbox{\boldmath $n$}}^S )(u,t),{\mbox{\boldmath $X$}}_{u_j}(u,t)\rangle =0$ we have $$h_{ij}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)(u,t)=\langle {\mbox{\boldmath $n$}}^T (u,t)+{\mbox{\boldmath $n$}}^S (u,t),{\mbox{\boldmath $X$}}_{u_iu_j}(u,t)\rangle.$$ Therefore the momentary nullcone second fundamental invariant of $\mathcal{S}_{t_0}$ at a point $p_0={\mbox{\boldmath $X$}}(u_0,t_0)$ depends only on the values ${\mbox{\boldmath $n$}}^T (u_0)+{\mbox{\boldmath $n$}}^S (u_0)$ and ${\mbox{\boldmath $X$}}_{u_iu_j}(u_0)$, respectively. Therefore, we write $$h_{ij}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)(u_0,t_0)=h_{ij}(\mathcal{S}_{t_0})(p_0,{\mbox{\boldmath $\xi$}}_0),$$ where $p_0={\mbox{\boldmath $X$}}(u_0,t_0)$ and ${\mbox{\boldmath $\xi$}}_0={\mbox{\boldmath $n$}}^S(u_0,t_0)\in N^{AdS}_1(W)_{p_0}.$ Thus, the momentary ${\mbox{\boldmath $n$}}^S$-shape operator and the momentary nullcone curvatures also depend only on ${\mbox{\boldmath $n$}}^T (u_0,t_0)+{\mbox{\boldmath $n$}}^S (u_0,t_0)$, ${\mbox{\boldmath $X$}}_{u_i}(u_0,t_0)$ and ${\mbox{\boldmath $X$}}_{u_iu_j}(u_0,t_0)$, independent of the derivation of the vector fields ${\mbox{\boldmath $n$}}^T$ and ${\mbox{\boldmath $n$}}^S .$ We may write $S_{p_0}(\mathcal{S}_{t_0};{\mbox{\boldmath $\xi$}}_0)=S_{p_0}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S),$ $\kappa _i(\mathcal{S}_{t_0},{\mbox{\boldmath $\xi$}}_0)(p_0)= \kappa _i(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)(p_0)$ $(i=1,\dots ,s)$ and $K_N(\mathcal{S}_{t_0},{\mbox{\boldmath $\xi$}}_0)(p_0)=K_N (\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)(p_0)$ at $p_0={\mbox{\boldmath $X$}}(u_0,t_0)$ with respect to ${\mbox{\boldmath $\xi$}}_0={\mbox{\boldmath $n$}}^S(u_0,t_0).$ We also say that a point $p_0={\mbox{\boldmath $X$}}(u_0,t_0)$ is [*momentary ${\mbox{\boldmath $\xi$}}_0$-nullcone umbilical*]{} if $S_{p_0}(\mathcal{S}_{t_0};{\mbox{\boldmath $\xi$}}_0)=\kappa _i(\mathcal{S}_{t_0})(p_0,{\mbox{\boldmath $\xi$}}_0)1_{T_{p_0}\mathcal{S}_{t_0}}$. The momentary space $\mathcal{S}_{t_0}$ is said to be [*totally momentary nullcone umbilical*]{} if any point $p={\mbox{\boldmath $X$}}(u,t_0)$ is momentary ${\mbox{\boldmath $\xi$}}$-nullcone umbilical for any ${\mbox{\boldmath $\xi$}}\in N^{AdS}_1(\mathcal{S}_{t_0})_p[{\mbox{\boldmath $n$}}^T]$. Moreover, we say that a point $p_0={\mbox{\boldmath $X$}}(u_0,t_0)$ is a [*momentary ${\mbox{\boldmath $\xi$}}_0$-nullcone parabolic point* ]{} of $W$ if $K_N (\mathcal{S}_{t_0};{\mbox{\boldmath $\xi$}}_0)(p_0)=0.$ Let $\kappa _N(\mathcal{S}_{t})_i(p,{\mbox{\boldmath $\xi$}})$ be the eigenvalues of the momentary nullcone shape operator $S_N(\mathcal{S}_{t}) _{(p,{\mbox{\scriptsize \boldmath$\xi$}})}$, $(i=1,\dots ,n-1)$. We write $\kappa _N(\mathcal{S}_t)_i(p,{\mbox{\boldmath $\xi$}})$, $(i=1,\dots ,s)$ as the eigenvalues belonging to the eigenvectors on $T_p\mathcal{S}_t$ and $\kappa _N(\mathcal{S}_t)_i(p,{\mbox{\boldmath $\xi$}})$, $(i=s+1,\dots n)$ as the eigenvalues belonging to the eigenvectors on the tangent space of the fiber of $N_1[\mathcal{S}_t].$
For $p_0={\mbox{\boldmath $X$}}(u_0,t_0)$ and ${\mbox{\boldmath $\xi$}}_0\in N^{AdS}_1[\mathcal{S}_{t_0}]_{p_0},$ we have $$\kappa _N(\mathcal{S}_{t_0})_i(p_0,{\mbox{\boldmath $\xi$}}_0)=\kappa _i(\mathcal{S}_{t_0},{\mbox{\boldmath $\xi$}}_0)(p_0),\ (i=1,\dots s),\ \kappa _N(\mathcal{S}_{t_0})_i(p_0,{\mbox{\boldmath $\xi$}}_0)=-1,\ (i=s+1,\dots n).$$
We call $\kappa _N(\mathcal{S}_{t})_i(p,{\mbox{\boldmath $\xi$}})=\kappa _i(\mathcal{S}_{t},{\mbox{\boldmath $\xi$}})(p)$, $(i=1,\dots ,s)$ the [*nullcone principal curvatures*]{} of $\mathcal{S}_{t}$ with respect to ${\mbox{\boldmath $\xi$}}$ at $p={\mbox{\boldmath $X$}}(u,t)\in W.$
[*Proof.* ]{}
Since $\{{\mbox{\boldmath $X$}}, {\mbox{\boldmath $n$}}^T,{\mbox{\boldmath $n$}}^S_1,\dots ,{\mbox{\boldmath $n$}}^S_{k-1}\}$ is a pseudo-orthonormal frame of $N(\mathcal{S}_t)$ and $${\mbox{\boldmath $\xi$}}_0={\mbox{\boldmath $n$}}^S_{k-1}({{\overline{u}}}_0,t_0)\in S^{k-2}=N_1[\mathcal{S}_{t_0}]_p,$$ we have $
\langle {\mbox{\boldmath $n$}}^T({{\overline{u}}}_0,t_0),{\mbox{\boldmath $\xi$}}_0\rangle =\langle {\mbox{\boldmath $n$}}^S_i({{\overline{u}}}_0,t_0),{\mbox{\boldmath $\xi$}}_0\rangle =0$ for $i=1,\dots ,k-2.$ Therefore, we have $$T_{\bm{\xi}_0}S^{k-2}=\langle {\mbox{\boldmath $n$}}^S_1({{\overline{u}}}_0,t_0),\dots ,{\mbox{\boldmath $n$}}^S_{k-2}({{\overline{u}}}_0,t_0)\rangle .$$ By this orthonormal basis of $T_{{\mbox{\scriptsize \boldmath$\xi$}}_0}S^{k-2},$ the canonical Riemannian metric $G_{ij}(p_0,{\mbox{\boldmath $\xi$}}_0)$ is represented by $$(G_{ij}(p_0,{\mbox{\boldmath $\xi$}}_0))=\left(
\begin{array}{cc}
g_{ij}(p_0) & 0 \\
0 & I_{k-2}
\end{array}
\right) ,$$ where $g_{ij}(p_0)=\langle {\mbox{\boldmath $X$}}_{u_i}({{\overline{u}}}_0,t_0), {\mbox{\boldmath $X$}}_{u_j}({{\overline{u}}}_0,t_0)\rangle $.
On the other hand, by Proposition 3.1, we have $$-\sum_{j=1}^s h^j_i(\mathcal{S}_{t_0},{\mbox{\boldmath $n$}}^S){\mbox{\boldmath $X$}}_{u_j}=\mathbb{NG}(\mathcal{S}_{t_0},{\mbox{\boldmath $n$}}^S)_{u_i}=
d_{p_0}\mathbb{NG}(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S)\left(\frac{\partial}{\partial u_i}\right),$$ so that we have $$S_{p_0}(\mathcal{S}_{t_0};{\mbox{\boldmath $\xi$}}_0)\left(\frac{\partial}{\partial u_i}\right)=\sum_{j=1}^s h^j_i(\mathcal{S}_{t_0};{\mbox{\boldmath $n$}}^S){\mbox{\boldmath $X$}}_{u_j}.$$ Therefore, the representation matrix of $S_{p_0}(\mathcal{S}_{t_0};{\mbox{\boldmath $\xi$}}_0)$ with respect to the basis $$\{{\mbox{\boldmath $X$}}_{u_1}({{\overline{u}}}_0,t_0),\dots ,{\mbox{\boldmath $X$}}_{u_s}({{\overline{u}}}_0,t_0),{\mbox{\boldmath $n$}}^S_1({{\overline{u}}}_0,t_0),\dots ,{\mbox{\boldmath $n$}}^S_{k-2}({{\overline{u}}}_0,t_0)\}$$ of $T_{(p_0,\bm{\xi}_0)}N_1[\mathcal{S}_{t_0}]$ is of the form $$\left(
\begin{array}{cc}
h^j_i(\mathcal{S}_{t_0},{\mbox{\boldmath $n$}}^S)(u_0,t_0) & * \\
0 & -I_{k-2}
\end{array}
\right).$$ Thus, the eigenvalues of this matrix are $\lambda _i=\kappa _i(\mathcal{S}_{t_0},{\mbox{\boldmath $\xi$}}_0)(p_0)$, $(i=1,\dots ,s)$ and $\lambda _i=-1,$ $(i=s+1,\dots ,n-1)$. This completes the proof.
$\Box$
Lightlike hypersurfaces along momentary spaces
==============================================
We define a hypersurface $
\mathbb{LH}_{\mathcal{S}_t}:N^{AdS}_1[\mathcal{S}_t]\times {{\mathbb R}}{\longrightarrow}AdS^{n+1}
$ by $$\mathbb{LH}_{\mathcal{S}_t}(((u,t),{\mbox{\boldmath $\xi$}}),\mu)={\mbox{\boldmath $X$}}(u,t)+\mu ({\mbox{\boldmath $n$}}^T(u,t)+{\mbox{\boldmath $\xi$}})={\mbox{\boldmath $X$}}(u,t)+\mu\mathbb{NG}(\mathcal{S}_t)((u,t),{\mbox{\boldmath $\xi$}}),$$ where $p={\mbox{\boldmath $X$}}(u,t),$ which is called a [*momentary lightlike hypersruface*]{} in anti-de Sitter space along $\mathcal{S}_t$. We remark that $\mathbb{LH}_{\mathcal{S}_t}(N^{AdS}_1[\mathcal{S}_t]\times{{\mathbb R}})$ is a lightlike hypersurface. Here a hypersurface is [*lightlike*]{} if the tangent space of the hypersurface at any regular point is a lightlike hyperplane.
We define a family of functions $H: U\times I\times AdS^{n+1}{\longrightarrow}{{\mathbb R}}$ on a world sheet $W={\mbox{\boldmath $X$}}(U\times I)$ by $
H((u,t),{\mbox{\boldmath $\lambda$}})=\langle {\mbox{\boldmath $X$}}(u,t) ,{\mbox{\boldmath $\lambda$}}\rangle +1.
$ We call $H$ the [*anti-de Sitter height function*]{} (briefly, AdS-height function) on the world sheet $W={\mbox{\boldmath $X$}}(U\times I).$ For any fixed $(t_0,{\mbox{\boldmath $\lambda$}}_0)\in I\times{{\mathbb R}}_2^{n+2},$ we write $h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}(u)=H((u,t_0),{\mbox{\boldmath $\lambda$}}_0).$
Let $W$ be a world sheet and $H: U\times I\times(AdS^{n+1}\setminus W)\to{{\mathbb R}}$ the AdS-height function on $W.$ Suppose that $p_0={\mbox{\boldmath $X$}}(u_0,t_0)\not={\mbox{\boldmath $\lambda$}}_0.$ Then we have the following$:$
[(1)]{} $h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}(u_0)=\partial h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}/\partial u_i(u_0)=0$, $(i=1,\dots ,s)$ if and only if there exist ${\mbox{\boldmath $\xi$}}_0 \in N^{AdS}_1[\mathcal{S}_{t_0}]_{p_0}$ and $\mu_0\in
{{\mathbb R}}\setminus \{0\}$ such that $
{\mbox{\boldmath $\lambda$}}_0 =\mathbb{LH}_{\mathcal{S}_{t_0}}(((u_0,t_0),{\mbox{\boldmath $\xi$}}_0),\mu_0).
$
[(2)]{} $h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}(u_0)=\partial h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}/\partial u_i(u_0)=
{\rm det}{\mathcal H}(h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)})(u_0)=0$ $(i=1,\dots ,s)$ if and only if there exist ${\mbox{\boldmath $\xi$}}_0 \in N_1[\mathcal{S}_{t_0}]_{p_0}$ such that $
{\mbox{\boldmath $\lambda$}}_0=\mathbb{LH}_{\mathcal{S}_{t_0}}(((u_0,t_0),{\mbox{\boldmath $\xi$}}_0),\mu_0)
$ and $1/{\mu_0}$ is one of the non-zero momentary nullcone principal curvatures $\kappa_N(\mathcal{S}_{t_0})_i((u_0,t_0),{\mbox{\boldmath $\xi$}}_0), (i=1,\dots ,s).$
[(3)]{} Under the condition [(2)]{}, ${\rm rank}\, {\mathcal H}(h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)})(u_0)=0$ if and only if $p_0={\mbox{\boldmath $X$}}(u_0,t_0)$ is a non-parabolic momentary ${\mbox{\boldmath $\xi$}}_0$-nullcone umbilical point.
[*Proof.* ]{}
\(1) We denote that $p_0={\mbox{\boldmath $X$}}(u_0,t_0).$ The condition $h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0})(u_0)=\langle {\mbox{\boldmath $X$}}(u_0,t_0),{{\mbox{\boldmath $\lambda$}}_0}\rangle+1 =0$ means that $$\begin{aligned}
\langle {\mbox{\boldmath $X$}}(u_0,t_0)-\lambda _0,{\mbox{\boldmath $X$}}(u_0.t_0)-{\mbox{\boldmath $\lambda$}}_0\rangle &=&\langle{\mbox{\boldmath $X$}}(u_0,t_0),{\mbox{\boldmath $X$}}(u_0,t_0)\rangle-2\langle{\mbox{\boldmath $X$}}(u_0,t_0),{\mbox{\boldmath $\lambda$}}_0\rangle+\langle{\mbox{\boldmath $\lambda$}}_0,{\mbox{\boldmath $\lambda$}}_0\rangle \\
&=&-2(1+\langle{\mbox{\boldmath $X$}}(u_0,t_0),{\mbox{\boldmath $\lambda$}}_0\rangle)=0,\end{aligned}$$ so that ${\mbox{\boldmath $X$}}(u_0,t_0)-{{\mbox{\boldmath $\lambda$}}_0}\in \Lambda^*.$ Since $\partial h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}/\partial u_i(u)=\langle {\mbox{\boldmath $X$}}_{u_i}(u,t_0), {{\mbox{\boldmath $\lambda$}}_0}\rangle $ and $\langle {\mbox{\boldmath $X$}}_{u_i},{\mbox{\boldmath $X$}}\rangle=0,$ we have $\langle {\mbox{\boldmath $X$}}_{u_i}(u,t_0),{\mbox{\boldmath $\lambda$}}_0\rangle=-\langle {\mbox{\boldmath $X$}}_{u_i}(u,t_0)-{\mbox{\boldmath $\lambda$}}_0\rangle$. Therefore, $\partial h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}/\partial u_i(u_0)=0$ if and only if ${\mbox{\boldmath $X$}}(u_0,t_0)-{{\mbox{\boldmath $\lambda$}}_0}\in N_{p_0}M.$ On the other hand, the condition $h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)} (u_0)=\langle {\mbox{\boldmath $X$}}(u_0,t_0),{\mbox{\boldmath $\lambda$}}_0\rangle+1=0$ implies that $\langle{\mbox{\boldmath $X$}}(u_0,t_0),{\mbox{\boldmath $X$}}(u_0,t_0)-{\mbox{\boldmath $\lambda$}}_0\rangle =0$. This means that ${\mbox{\boldmath $X$}}(u_0,t_0)-{\mbox{\boldmath $\lambda$}}_0\in T_{p_0}AdS^{n+1}.$ Hence $h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}(u_0)=\partial h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}/\partial u_i(u_0)=0$ $(i=1,\dots, s)$ if and only if ${\mbox{\boldmath $X$}}(u_0,t_0)-{{\mbox{\boldmath $\lambda$}}_0}\in N_{p_0}(\mathcal{S}_{t_0})\cap \Lambda^*\cap T_{p_0}AdS^{n+1}.$ Then we denote that ${\mbox{\boldmath $v$}}={\mbox{\boldmath $X$}}(u_0,t_0)-{{\mbox{\boldmath $\lambda$}}_0}\in N_{p_0}(\mathcal{S}_{t_0})\cap \Lambda^*\cap T_{p_0}AdS^{n+1}.$ If $\langle {\mbox{\boldmath $n$}}^T(u_0,t_0),{\mbox{\boldmath $v$}}\rangle =0,$ then ${\mbox{\boldmath $n$}}^T(u_0,t_0)$ belongs to a lightlike hyperplane in the Lorentz space $T_{p_0}AdS^{n+1},$ so that ${\mbox{\boldmath $n$}}^T(u_0,t_0)$ is lightlike or spacelike. This contradiction to the fact that ${\mbox{\boldmath $n$}}^T(u_0,t_0)$ is a timelike unit vector. Thus, $\langle {\mbox{\boldmath $n$}}^T(u_0,t_0),{\mbox{\boldmath $v$}}\rangle \not=0.$ We set $${\mbox{\boldmath $\xi$}}_0=\frac{-1}{\langle {\mbox{\boldmath $n$}}^T(u_0,t_0),{\mbox{\boldmath $v$}}\rangle}{\mbox{\boldmath $v$}}-{\mbox{\boldmath $n$}}^T(u_0,t_0).$$ Then we have $$\begin{aligned}
\langle {\mbox{\boldmath $\xi$}}_0,{\mbox{\boldmath $\xi$}}_0\rangle &=& -2\frac{-1}{\langle {\mbox{\boldmath $n$}}^T(u_0,t_0),{\mbox{\boldmath $v$}}\rangle} \langle {\mbox{\boldmath $n$}}^T(u_0,t_0),{\mbox{\boldmath $v$}}\rangle-1=1 \\
\langle {\mbox{\boldmath $\xi$}}_0,{\mbox{\boldmath $n$}}^T(u_0,t_0)\rangle &=& \frac{-1}{\langle {\mbox{\boldmath $n$}}^T(u_0,t_0),{\mbox{\boldmath $v$}}\rangle} \langle {\mbox{\boldmath $n$}}^T(u_0,t_0),{\mbox{\boldmath $v$}}\rangle+1=0.\end{aligned}$$ This means that ${\mbox{\boldmath $\xi$}}_0\in N_1[\mathcal{S}_{t_0}]_{p_0}.$ Since $-{\mbox{\boldmath $v$}}=\langle {\mbox{\boldmath $n$}}^T(u_0,t_0),{\mbox{\boldmath $v$}}\rangle({\mbox{\boldmath $n$}}^T(u_0,t_0)+{\mbox{\boldmath $\xi$}}_0),$ we have ${{\mbox{\boldmath $\lambda$}}_0}={\mbox{\boldmath $X$}}(u_0,t_0)+\mu_0\mathbb{NG}(\mathcal{S}_{t_0})((u_0,t_0){\mbox{\boldmath $\xi$}}_0)$, where $p_0={\mbox{\boldmath $X$}}(u_0,t_0)$ and $\mu_0=\langle {\mbox{\boldmath $n$}}^T(u_0,t_0),{\mbox{\boldmath $v$}}\rangle.$ For the converse assertion, suppose that ${\mbox{\boldmath $\lambda$}}_0={\mbox{\boldmath $X$}}(u_0,t_0)+\mu_0\mathbb{NG}(\mathcal{S}_{t_0})((u_0,t_0),{\mbox{\boldmath $\xi$}}_0).$ Then ${\mbox{\boldmath $\lambda$}}_0-{\mbox{\boldmath $X$}}(u_0,t_0)\in N_{p_0}(\mathcal{S}_{t_0}))\cap \Lambda ^*$ and $\langle{\mbox{\boldmath $\lambda$}}_0-{\mbox{\boldmath $X$}}(u_0,t_0),{\mbox{\boldmath $X$}}(u_0,t_0)\rangle=\langle \mu_0\mathbb{NG}(\mathcal{S}_{t_0})(p_0,{\mbox{\boldmath $\xi$}}_0),{\mbox{\boldmath $X$}}(u_0)\rangle=0.$ Thus we have ${\mbox{\boldmath $\lambda$}}_0-{\mbox{\boldmath $X$}}(u_0)\in N_{p_0}(\mathcal{S}_{t_0})\cap \Lambda ^*\cap T_{p_0}AdS^{n+1}.$ By the previous arguments, these conditions are equivalent to the condition that $h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}(u_0)=\partial h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}/\partial u_i(u_0)=0$ $(i=1,\dots, s)$.
\(2) By a straightforward calculation, we have $$\frac{\partial ^2 h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}}{\partial u_i\partial u_j}(u)
=\langle{\mbox{\boldmath $X$}}_{u_iu_j}(u,t_0),{\mbox{\boldmath $\lambda$}}_0\rangle.$$ Under the conditions ${{\mbox{\boldmath $\lambda$}}_0}={\mbox{\boldmath $X$}}(u_0)+\mu_0({\mbox{\boldmath $n$}}^T(u_0)+{\mbox{\boldmath $\xi$}}_0)$, we have $$\frac{\partial ^2 h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}}{\partial u_i\partial u_j}(u_0)
=\langle {\mbox{\boldmath $X$}}_{u_iu_j}(u_0,t_0),{\mbox{\boldmath $X$}}(u_0,t_0)\rangle +\mu_0\langle{\mbox{\boldmath $X$}}_{u_iu_j}(u_0,t_0), ({\mbox{\boldmath $n$}}^T(u_0,t_0)+{\mbox{\boldmath $\xi$}}_0)\rangle .$$ Since $\langle {\mbox{\boldmath $X$}}_{u_i},{\mbox{\boldmath $X$}}\rangle =0,$ we have $\langle{\mbox{\boldmath $X$}}_{u_iu_j},{\mbox{\boldmath $X$}}\rangle=-\langle {\mbox{\boldmath $X$}}_{u_i},{\mbox{\boldmath $X$}}_{u_j}\rangle.$ Therefore, we have $$\left(\frac{\partial ^2 h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)}}{\partial u_i\partial u_\ell}(u_0)\right)\left(g^{j\ell}(u_0,t_0)\right)
=\left(\mu_0 h^j_i(\mathcal{S}_{t_0})((u_0,t_0),{\mbox{\boldmath $\xi$}}_0)-\delta ^j_i\right).$$ Thus, ${\rm det}{\mathcal H}(h_{(t_o,{\mbox{\scriptsize \boldmath$\xi$}}_0)})(u_0)=0$ if and only if $1/\mu_0$ is an eigenvalue of $(h^i_j(\mathcal{S}_{t_0})((u_0,t_0),{\mbox{\boldmath $\xi$}}_0)),$ which is equal to one of the momentary nullcone principal curvatures $\kappa _N(\mathcal{S}_{t_0})_i((u_0,t_0),{\mbox{\boldmath $\xi$}}_0),$ $(i=1,\dots ,s)$.
\(3) By the above calculation, ${\rm rank}\, {\mathcal H}(h_{(t_0,{\mbox{\scriptsize \boldmath$\lambda$}}_0)})(u_0)=0$ if and only if $$(h^i_j(\mathcal{S}_{t_0})((u_0,t_0),{\mbox{\boldmath $\xi$}}_0))=\frac{1}{\mu _0}(\delta ^j_i),$$ where $1/\mu_0=\kappa _N(\mathcal{S}_{t_0})_i((u_0,t_0),{\mbox{\boldmath $\xi$}}_0),$ $(i=1,\dots ,s)$. This means that $p_0={\mbox{\boldmath $X$}}(u_0,t_0)$ is a non-parabolic momentary ${\mbox{\boldmath $\xi$}}_0$-nullcone umbilical point.
$\Box$
Graph-like big fronts
=====================
In this section we briefly review the theory of graph-like Legendrian unfoldings. Graph-like Legendrian unfoldings belong to a special class of big Legendrian submanifolds (for detail, see [@Izumiya93; @Izumiya-Takahashi; @Izumiya-Takahashi2; @Izumiya-Takahashi3; @Zakalyukin95]). Recently there appeared a survey article [@Graph-like] on the theory of graph-like Legendrian unfoldings. Let ${\mathcal F} :({{\mathbb R}}^k\times ({{\mathbb R}}^m\times{{\mathbb R}}),0)\to ({{\mathbb R}},0)$ be a function germ. We say that ${\mathcal F}$ is a [*graph-like Morse family of hypersurfaces*]{} if $
(\mathcal{F}, d_q\mathcal{F}):({{\mathbb R}}^k\times({{\mathbb R}}^m\times {{\mathbb R}}),0)\to ({{\mathbb R}}\times {{\mathbb R}}^k,0)$ is a non-singular and $(\partial {\mathcal F}/\partial t)(0)\not= 0,$ where $$d_q\mathcal{F}(q,x,t)=\left(\frac{\partial \mathcal{F}}{\partial q_1}(q,x,t), \dots ,
\frac{\partial \mathcal{F}}{\partial q_k}(q,x,t)\right).$$ Moreover, we say that ${\mathcal F}$ is [*non-degenerate*]{} if $(\mathcal{F}, d_q\mathcal{F})|_{{{\mathbb R}}^k\times({{\mathbb R}}^m\times \{0\})}$ is non-singular. For a graph-like Morse family of hypersurfaces $\mathcal{F},$ $\Sigma _*(\mathcal{F})=(\mathcal{F}, d_q\mathcal{F})^{-1}(0)$ is a smooth $m$-dimensional submanifold germ of $({{\mathbb R}}^k\times({{\mathbb R}}^m\times {{\mathbb R}}),0).$ We now consider the space of $1$-jets $J^1({{\mathbb R}}^m,{{\mathbb R}})$ with the canonical coordinates $(x_1,\dots ,x_m,t,p_1,\dots ,p_m)$ such that the canonical contact form is $\theta =dt-\sum_{i=1}^m p_idx_i.$ We define a mapping $\Pi:J^1({{\mathbb R}}^m,{{\mathbb R}}){\longrightarrow}T^*{{\mathbb R}}^m$ by $\Pi(x,t,p)=(x,p),$ where $(x,t,p)=(x_1,\dots, x_m,t,p_1,\dots ,p_m).$ Here, $T^*{{\mathbb R}}^m$ is a symplectic manifold with the canonical symplectic structure $\omega=\sum_{i=1}^m dp_i\wedge dx_i$ (cf. [@Arnold1]). We define a mapping $\mathscr{L}_{\mathcal{F}}:(\Sigma_{*}(\mathcal{F}),0) \to J^1({{\mathbb R}}^m,{{\mathbb R}})$ by $$\mathscr{L}_{\mathcal F}(q,x,t)=\left(x,t,-\frac{\displaystyle \frac{\partial \mathcal{F}}{\displaystyle\partial x_1}(q,x,t)}{\frac{\displaystyle \partial \mathcal{F}}{\displaystyle\partial t}(q,x,t)},\dots , -\frac{\displaystyle\frac{\partial \mathcal{F}}{\partial x_m}(q,x,t)}{\frac{\displaystyle\partial \mathcal{F}}{\displaystyle\partial t}(q,x,t)},\right).$$ It is easy to show that $\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))$ is a Legendrian submanifold germ (cf., [@Arnold1]), which is called a [*graph-like Legendrian unfolding germ.*]{} We call $\overline{\pi}|_{\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))}:\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})){\longrightarrow}{{\mathbb R}}^m\times {{\mathbb R}}$ a [*graph-like Legendrian map germ*]{}, where $\overline{\pi}:J^1({{\mathbb R}}^m,{{\mathbb R}}){\longrightarrow}{{\mathbb R}}^m\times{{\mathbb R}}$ is the canonical projection. We also call $W(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))=\overline{\pi}(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))$ a [*graph-like big front*]{} of $\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})).$ We say that ${\mathcal F}$ is a [*graph-like generating family*]{} of $\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})).$ Moreover, we call $W_t(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))=\pi _1(\pi _2^{-1}(t)\cap W(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))$ a [*momentary front*]{} for each $t\in ({{\mathbb R}},0),$ where $\pi _1:{{\mathbb R}}^m\times {{\mathbb R}}{\longrightarrow}{{\mathbb R}}^m$ and $\pi _2:{{\mathbb R}}^m\times {{\mathbb R}}{\longrightarrow}{{\mathbb R}}$ are the canonical projections. The [*discriminant set of the family*]{} $\{W_t(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))\}_{t\in ({{\mathbb R}},0)}$ is defined by the union of the [*caustic*]{} $$C_{\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))}=\pi _1( \Sigma (W (\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))))$$ and the [*Maxwell stratified set*]{} $$M_{\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))}=\pi _1(SI_{W(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))}),$$ where $\Sigma (W (\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))$ is the critical value set of $\overline{\pi}|_{\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))}$ and $SI_{W(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))}$ is the closure of the self intersection set of $W(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))).$
We now define equivalence relations among graph-like Legendrian unfoldings. Let ${\mathcal F} :({{\mathbb R}}^k\times ({{\mathbb R}}^m\times{{\mathbb R}}),0)\to ({{\mathbb R}},0)$ and ${\mathcal G} :({{\mathbb R}}^{k}\times ({{\mathbb R}}^m\times{{\mathbb R}}),0)\to ({{\mathbb R}},0)$ be graph-like Morse families of hypersurfaces. We say that $\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))$ and $\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G}))$ are [*Legendrian equivalent*]{} if there exist a diffeomorphism germ $\Phi:({{\mathbb R}}^m\times {{\mathbb R}},\overline{\pi}(p )){\longrightarrow}({{\mathbb R}}^m\times {{\mathbb R}},\overline{\pi}(p' ))$ and a contact diffeomorphism germ $\widehat{\Phi}:(J^1({{\mathbb R}}^m,{{\mathbb R}}),p){\longrightarrow}(J^1({{\mathbb R}}^m,{{\mathbb R}}),p')$ such that $\overline{\pi}\circ\widehat{\Phi}=\Phi\circ \overline{\pi}$ and $\widehat{\Phi}(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))=(\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G}))),$ where $p=\mathscr{L}_{\mathcal F}(0)$ and $p'=\mathscr{L}_{\mathcal G}(0).$ We also say that $\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))$ and $\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G}))$ are [*$S.P^+$-Legendrian equivalent*]{} if these are Legendrian equivalent by a diffeomorphism germ $\Phi:({{\mathbb R}}^m\times {{\mathbb R}},\overline{\pi}(p )){\longrightarrow}({{\mathbb R}}^m\times {{\mathbb R}},\overline{\pi}(p' ))$ of the form $\Phi (x,t)=(\phi _1(x),t+\alpha (x))$ and a contact diffeomorphism germ $\widehat{\Phi}:(J^1({{\mathbb R}}^m,{{\mathbb R}}),p){\longrightarrow}(J^1({{\mathbb R}}^m,{{\mathbb R}}),p')$ with $\overline{\pi}\circ\widehat{\Phi}=\Phi\circ \overline{\pi}.$ Moreover, graph-like big fronts $W(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))$ and $W(\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G})))$ are [*$S.P^+$-diffeomorphic*]{} if there exists a diffeomorphism germ $\Phi:({{\mathbb R}}^m\times {{\mathbb R}},\overline{\pi}(p )){\longrightarrow}({{\mathbb R}}^m\times {{\mathbb R}},\overline{\pi}(p' ))$ of the form $\Phi (x,t)=(\phi _1(x),t+\alpha (x))$ such that $\Phi(W(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))))=W(\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G})))$ as set germs. By definition, if $\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))$ and $\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G}))$ are $S.P^+$-Legendrian equivalent, then $W(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))$ and $W(\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G})))$ are $S.P^+$-diffeomorphic. The converse assertion holds generically [@Graph-like; @GeomLag14].
Suppose that the sets of critical points of $\overline{\pi}|_{\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))}, \overline{\pi}|_{\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G}))}$ are nowhere dense respectively. Then $\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))$ and $\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G}))$are $S.P^+$-Legendrian equivalent if and only if $
W(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))$ and $W(\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G})))
$ are $S.P^+$-diffeomorphic.
We remark that if $W(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))$ and $W(\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G})))$ are $S.P^+$-diffeomorphic by a diffeomorphism germ $\Phi:({{\mathbb R}}^m\times {{\mathbb R}},\overline{\pi}(p )){\longrightarrow}({{\mathbb R}}^m\times {{\mathbb R}},\overline{\pi}(p' ))$, then $$\Phi ( C_{\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))}\cup M_{\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))})=
C_{\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G}))}\cup M_{\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G}))}.$$
For a graph-like Morse family of hypersurfaces $\mathcal{F}:({{\mathbb R}}^k\times ({{\mathbb R}}^m\times{{\mathbb R}}),0)\to ({{\mathbb R}},0),$ by the implicit function theorem, there exist function germs $F:({{\mathbb R}}^k\times {{\mathbb R}}^m,0)\to ({{\mathbb R}},0)$ and $\lambda :({{\mathbb R}}^k\times ({{\mathbb R}}^m\times {{\mathbb R}}),0){\longrightarrow}{{\mathbb R}}$ with $\lambda (0)\not= 0$ such that ${\mathcal F}(q,x,t)
=\lambda(q,x,t)( F(q,x)-t).$ We have shown in [@Graph-like] that $\mathcal{F}$ is a graph-like Morse family of hypersurfaces if and only if $F$ is a Morse family of functions. Here we say that $F:({{\mathbb R}}^k\times{{\mathbb R}}^m,0){\longrightarrow}({{\mathbb R}},0)$ is a [*Morse family of functions*]{} if $$dF_q=\left(\frac{\partial F}{\partial q_1},\dots, \frac{\partial F}{\partial q_k}\right):({{\mathbb R}}^k\times{{\mathbb R}}^m,0){\longrightarrow}{{\mathbb R}}^k$$ is non-singular. We consider a graph-like Morse family of hypersurfaces $$\mathcal{F}(q,x,t)=\lambda (q,x,t)(F(q,x)-t).$$ In this case $
\Sigma _*(\mathcal{F})=\{(q,x,F(q,x))\in ({{\mathbb R}}^k\times ({{\mathbb R}}^m\times{{\mathbb R}}),0)\ |\ (q,x)\in C(F)\},
$ where $$C(F)=\left\{ (q,x)\in ({{\mathbb R}}^k\times{{\mathbb R}}^m,0)\ \Bigm|\ \frac{\partial F}{\partial q_1}(q,x)=\cdots =\frac{\partial F}{\partial q_k}(q,x)=0\ \right\}.$$ Moreover, we define a map germ $
L(F):(C(F),0){\longrightarrow}T^*{{\mathbb R}}^m
$ by $$L(F)(q,x)=\left(x,\frac{\partial F}{\partial x_1}(q,x),\dots ,\frac{\partial F}{\partial x_m}(q,x)\right)$$ It is known that $L(F)(C(F))$ is a Lagrangian submanifold germ (cf., [@Arnold1]) for the canonical symplectic structure. In this case $F$ is said to be a [*generating family*]{} of the Lagrangian submanifold germ $L(F)(C(F)).$ We remark that $\Pi (\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))=L(F)(C(F))$ and the graph-like big front $W(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))$ is the graph of $F|C(F).$ Here we call $\pi|_{L(F)(C(F))}:L(F)(C(F)){\longrightarrow}{{\mathbb R}}^m$ a [*Lagrangian map germ*]{}, where $\pi :T^*{{\mathbb R}}^m{\longrightarrow}{{\mathbb R}}^m$ is the canonical projection. Then the set of critical values of $\pi|_{L(F)(C(F))}$ is called a [*caustic*]{} of $L(F)(C(F))=\Pi(\mathscr{L}_{\mathcal{F}}(\Sigma _*(\mathcal{F})))$ in the theory of Lagrangian singularities, which is denoted by $C_{L(F)(C(F))}.$ By definition, we have $C_{L(F)(C(F))}=C_{\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))}.$
Let ${\mathcal F}, \mathcal{G} :({{\mathbb R}}^k\times ({{\mathbb R}}^m\times{{\mathbb R}}),0)\to ({{\mathbb R}},0)$ be graph-like Morse families of hypersurfaces. We say that $\Pi(\mathscr{L}_{\mathcal{F}}(\Sigma _*(\mathcal{F})))$ and $\Pi(\mathscr{L}_{\mathcal{G}}(\Sigma _*(\mathcal{G})))$ are [*Lagrangian equivalent*]{} if there exist a diffeomorphism germ $\Psi:({{\mathbb R}}^m, \pi\circ\Pi(p )){\longrightarrow}({{\mathbb R}}^m,\pi\circ\Pi(p' ))$ and a symplectic diffeomorphism germ $\widehat{\Psi}:(T^*{{\mathbb R}}^m,\Pi( p)){\longrightarrow}(T^*{{\mathbb R}}^m,\Pi( p'))$ such that $\pi\circ\widehat{\Psi}=\Psi\circ \pi$ and $\widehat{\Psi}(\Pi(\mathscr{L}_{\mathcal{F}}(\Sigma _*(\mathcal{F}))))=\Pi(\mathscr{L}_{\mathcal{G}}(\Sigma _*(\mathcal{G}))),$ where $p=\mathscr{L}_{\mathcal F}(0)$ and $p'=\mathscr{L}_{\mathcal G}(0).$ By definition, if $\Pi(\mathscr{L}_{\mathcal{F}}(\Sigma _*(\mathcal{F})))$ and $\Pi(\mathscr{L}_{\mathcal{G}}(\Sigma _*(\mathcal{G})))$ are Lagrangian equivalent, then the caustics $C_{\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))}$ and $C_{\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G}))}$ are diffeomorphic as set germs. The converse assertion, however, does not hold (cf. [@GeomLag14]). Recently, we have shown the following theorem (cf. [@Izumiya-Takahashi2; @Graph-like; @GeomLag14])
With the same notations as the above, $\Pi(\mathscr{L}_{\mathcal{F}}(\Sigma _*(\mathcal{F})))$ and $\Pi(\mathscr{L}_{\mathcal{G}}(\Sigma _*(\mathcal{G})))$ are Lagrangian equivalent if and only if $\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))$ and $\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G}))$ are $S.P^+$-Legendrian equivalent.
We have the following corollary of Proposition 5.1 and Theorem 5.2.
Suppose that the sets of critical points of $\overline{\pi}|_{\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))}, \overline{\pi}|_{\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G}))}$ are nowhere dense, respectively. Then $\Pi(\mathscr{L}_{\mathcal{F}}(\Sigma _*(\mathcal{F})))$ and $\Pi(\mathscr{L}_{\mathcal{G}}(\Sigma _*(\mathcal{G})))$ are Lagrangian equivalent if and only if $W(\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F})))$ and $W(\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G})))$ are $S.P^+$-diffeomorphic.
There are the notions of Lagrangian stability of Lagrangian submanifold germs and $S.P^+$-Legendrian stability of graph-like Legendrian unfolding germs, respectively. Here we do not use the exact definitions of those notions of stability, so that we omit to give the definitions. For detailed properties of such stabilities, see [@Arnold1; @Graph-like]. We have the following corollary of Theorem 5.2.
The graph-like Legendrian unfolding $\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))$ is $S.P^+$-Legendrian stable if and only if the corresponding Lagrangian submanifold $\Pi(\mathscr{L}_{\mathcal{F}}(\Sigma _*(\mathcal{F})))$ is Lagrangian stable.
Let ${\mathcal F}:({{\mathbb R}}^k\times ({{\mathbb R}}^m\times{{\mathbb R}}),0)\to ({{\mathbb R}},0)$ be a graph-like Morse family of hypersurfaces. We define $\overline{f}:({{\mathbb R}}^k\times {{\mathbb R}},0){\longrightarrow}({{\mathbb R}},0)$ by $\overline{f}(q,t)={\mathcal F}(q,0,t).$ For graph-like Morse families of hypersurfaces ${\mathcal F} :({{\mathbb R}}^k\times ({{\mathbb R}}^m\times{{\mathbb R}}),0)\to ({{\mathbb R}},0)$ and ${\mathcal G} :({{\mathbb R}}^{k}\times ({{\mathbb R}}^m\times{{\mathbb R}}),0)\to ({{\mathbb R}},0)$, we say that $\overline{f}$ and $\overline{g}$ are [*$S.P$-$\mathcal{K}$-equivalent*]{} if there exist a function germ $\nu :({{\mathbb R}}^k\times{{\mathbb R}},0){\longrightarrow}{{\mathbb R}}$ with $\nu(0)\not= 0$ and a diffeomorphism germ $\phi :({{\mathbb R}}^k\times {{\mathbb R}},0){\longrightarrow}({{\mathbb R}}^k\times {{\mathbb R}},0)$ of the form $\phi (q,t)=(\phi_1(q,t),t)$ such that $\overline{f}(q,t)=\nu (q,t)\overline{g}(\phi (q,t)).$ Although we do not give the definition of $S.P^+$-Legendrian stability, we give a corresponding notion for graph-like Morse family of hypersurfaces. We say that ${\mathcal F}$ is an [*infinitesimally $S.P^+$-$\mathcal{K}$-versal unfolding*]{} of $\overline{f}$ if $${\cal E}_{k+1}=\left\langle \frac{\partial \overline{f}}{\partial q_1},\dots, \frac{\partial \overline{f}}{\partial q_k},\overline{f} \right\rangle _{{\cal E}_{k+1}}+
\left\langle \frac{\partial \overline{f}}{\partial t} \right\rangle _{{{\mathbb R}}}+
\left\langle \frac{\partial \mathcal{F}}{\partial x_1}|_{{{\mathbb R}}^k\times\{0\}\times {{\mathbb R}}} ,\dots
,\frac{\partial \mathcal{F}}{\partial x_m}|_{{{\mathbb R}}^k\times\{0\}\times{{\mathbb R}}} \right\rangle _{{{\mathbb R}}},$$ where ${\cal E}_{k+1}$ is the local ${{\mathbb R}}$-algebra of $C^\infty$-function germs $({{\mathbb R}}^k\times{{\mathbb R}},0){\longrightarrow}{{\mathbb R}}.$ It is known the following theorem in [@Izu95; @Zakalyukin95].
The graph-like Legendrian unfolding $\mathscr{L} _\mathcal{F}(\Sigma _*(\mathcal{F}))$ is $S.P^+$-Legendre stable if and only if $\mathcal{F}$ is an infinitesimally $S.P^+$-${\cal K}$-versal unfolding of $\overline{f}.$
In [@Graph-like] we have shown the following theorem.
Let ${\mathcal F},\mathcal{G} :({{\mathbb R}}^k\times ({{\mathbb R}}^m\times{{\mathbb R}}),0)\to ({{\mathbb R}},0)$ be graph-like Morse families of hypersurfaces such that ${\mathscr{L}_{\mathcal F}(\Sigma _*({\mathcal F}))}, {\mathscr{L}_{\mathcal G}(\Sigma _*({\mathcal G}))}$ are $S.P^+$-Legendrian stable. Then the following conditions are equivalent[:]{}
[(1)]{} $\mathscr{L}_{\mathcal{F}}(\Sigma _*(\mathcal{F}))$ and $\mathscr{L}_{\mathcal{G}}(\Sigma _*(\mathcal{G}))$ are $S.P^+$-Legendrian equivalent,
[(2)]{} $\overline{f}$ and $\overline{g}$ are $S.P$-$\mathcal{K}$-equivalent,
[(3)]{} $\Pi(\mathscr{L}_{\mathcal{F}}(\Sigma _*(\mathcal{F})))$ and $\Pi(\mathscr{L}_{\mathcal{G}}(\Sigma _*(\mathcal{G})))$ are Lagrangian equivalent,
[(4)]{} $W(\mathscr{L}_{\mathcal{F}}(\Sigma _*(\mathcal{F})))$ and $W(\mathscr{L}_{\mathcal{G}}(\Sigma _*(\mathcal{G})))$ are $S.P^+$-diffeomorphic.
Unfolded lightlike hypersrufaces
================================
Returning to our situation, we have the following proposition.
Let $H$ be the AdS-height function on $W.$ For any $((u,t),{\mbox{\boldmath $\lambda$}})\in \Delta ^*H^{-1}(0),$ the germ of $H$ at $(u,{\mbox{\boldmath $\lambda$}})$ is a non-degenerate graph-like Morse family of hypersurfaces.
[*Proof.* ]{}
We denote that $${\mbox{\boldmath $X$}}(u,t)=(X_{-1}(u,t),X_0(u,t),X_1(u,t),\dots ,X_n(u,t))\ {\rm and}\
{\mbox{\boldmath $\lambda$}}=(\lambda _{-1},\lambda _0,\lambda _1,\dots ,\lambda _n).$$ We define an open subset $U_{-1}^+=\{{\mbox{\boldmath $\lambda$}}\in AdS^{n+1}\ |\ \lambda _{-1}>0\ \}$. For any ${\mbox{\boldmath $\lambda$}}\in U_{-1}^+,$ we have $$\lambda _{-1}=\sqrt{1-\lambda _0^2+\lambda _1^2+\cdots \lambda _n^2}.$$ Thus, we have a local coordinate of $AdS^{n+1}$ given by $(\lambda _0,\lambda _1,\dots ,\lambda _n)$ on $U_{-1}^+.$ By definition, we have $$H(u,t,{\mbox{\boldmath $\lambda$}})=-X_{-1}(u,t)\sqrt{1-\lambda _0^2+\sum_{i=1}^n \lambda _i^2}-X_0(u,t)\lambda_0+
X_1(u,t)\lambda_1 +\cdots
+X_n(u,t)\lambda _n.$$ We now prove that the mapping $$\Delta^*H|(U\times \{t\}\times U_{-1}^+)=\left(H, \frac{\partial H}{\partial u_1},\dots ,\frac{\partial H}{\partial
u_s}\right):U\times\{t\}\times U_{-1}^+{\longrightarrow}{{\mathbb R}}\times {{\mathbb R}}^s$$ is non-singular at $(u,t,{\mbox{\boldmath $\lambda$}})\in \Delta ^*H^{-1}(0)\cap (U\times\{t\}\times U_{-1}^+).$ Indeed, the Jacobian matrix of $\Delta ^*H|(U\times \{t\}\times U_{-1}^+)$ is given by
$$\left(
\begin{array}{ccccc}
&
X_{-1}\displaystyle{\frac{\lambda _0}{\lambda _{-1}}}-X_0 & -X_{-1}\displaystyle{\frac{\lambda _1}{\lambda _{-1}}}+X_1 & \cdots &
-X_{-1}\displaystyle{\frac{\lambda _n}{\lambda _{-1}}}-X_n \\
{\smash{\lower1.0ex\hbox{\bg A}}}& X_{-1u_1}\displaystyle{\frac{\lambda _0}{\lambda _{-1}}}-X_{0u_1} &
-X_{-1u_1}\displaystyle{\frac{\lambda _1}{\lambda _{-1}}}+X_{1u_1}& \cdots &-X_{-1u_1}\displaystyle{\frac{\lambda _n}{\lambda _{-1}}}-X_{nu_1}\\
&\vdots & \vdots & \ddots & \vdots \\
& X_{-1u_s}\displaystyle{\frac{\lambda _0}{\lambda _{-1}}}-X_{0u_s} &
-X_{-1u_s}\displaystyle{\frac{\lambda _1}{\lambda _{-1}}}+X_{1u_s}& \cdots &-X_{-1u_s}\displaystyle{\frac{\lambda _n}{\lambda _{-1}}}-X_{nu_s}
\end{array}
\right) ,$$ where $$\begin{aligned}
{\smash{\lower1.0ex\hbox{\bg A}}}=
\left(\!\!
\begin{array}{ccc}
\langle {\mbox{\boldmath $X$}}_{u_1} ,{\mbox{\boldmath $\lambda$}}\rangle & \!\! \cdots\!\! & \langle {\mbox{\boldmath $X$}}_{u_s},{\mbox{\boldmath $\lambda$}}\rangle \\
\langle {\mbox{\boldmath $X$}}_{u_1u_1},{\mbox{\boldmath $\lambda$}}\rangle & \!\! \cdots\!\! &
\langle {\mbox{\boldmath $X$}}_{u_1u_s},{\mbox{\boldmath $\lambda$}}\rangle \\
\vdots & \!\!\ddots\!\! & \vdots \\
\langle {\mbox{\boldmath $X$}}_{u_su_1},{\mbox{\boldmath $\lambda$}}\rangle &\!\! \cdots\!\! &
\langle {\mbox{\boldmath $X$}}_{u_su_s},{\mbox{\boldmath $\lambda$}}\rangle
\end{array}
\!\!
\right) .\end{aligned}$$
We now show that the rank of $${\smash{\lower1.0ex\hbox{\bg B}}}=
\left(
\begin{array}{cccc}
X_{-1}\displaystyle{\frac{\lambda _0}{\lambda _{-1}}}-X_0 & -X_{-1}\displaystyle{\frac{\lambda _1}{\lambda _{-1}}}+X_1 & \cdots &
-X_{-1}\displaystyle{\frac{\lambda _n}{\lambda _{-1}}}-X_n \\
X_{-1u_1}\displaystyle{\frac{\lambda _0}{\lambda _{-1}}}-X_{0u_1} &
-X_{-1u_1}\displaystyle{\frac{\lambda _1}{\lambda _{-1}}}+X_{1u_1}& \cdots &-X_{-1u_1}\displaystyle{\frac{\lambda _n}{\lambda _{-1}}}-X_{nu_1}\\
\vdots & \vdots & \ddots & \vdots \\
X_{-1u_s}\displaystyle{\frac{\lambda _0}{\lambda _{-1}}}-X_{0u_s} &
-X_{-1u_s}\displaystyle{\frac{\lambda _1}{\lambda _{-1}}}+X_{1u_s}& \cdots &-X_{-1u_s}\displaystyle{\frac{\lambda _n}{\lambda _{-1}}}-X_{nu_s}
\end{array}
\right)$$ is $s+1$ at $(u,t,{\mbox{\boldmath $\lambda$}})\in \Sigma _*(H).$ Since $(u,t,{\mbox{\boldmath $\lambda$}})\in \Sigma _*(H),$ we have $${\mbox{\boldmath $\lambda$}}={\mbox{\boldmath $X$}}(u,t)+\mu\left({\mbox{\boldmath $n$}}^T(u,t)+\sum _{i=1}^{k-1}\xi _i{\mbox{\boldmath $n$}}_i(u,t)\right)$$ with $\sum_{i=1}^{k-1}\xi ^2_i=1,$ where $\{{\mbox{\boldmath $X$}},{\mbox{\boldmath $n$}}^T,{\mbox{\boldmath $n$}}^S_1,\dots ,{\mbox{\boldmath $n$}}^S_{k-1}\}$ is a pseudo-orthonormal (local) frame of $N(M).$ Without the loss of generality, we assume that $\mu\not= 0$ and $\xi_{k-1}\not= 0.$ We denote that $${\mbox{\boldmath $n$}}^T(u,t)=^t\!\!(n^T_{-1}(u,t),n^T_0(u,t),\dots n^T_n(u,t)),\
{\mbox{\boldmath $n$}}_i(u,t)=^t\!\!(n^i_{-1}(u,t),n_0^i(u,t),\dots n^i_n(u,t)).$$
It is enough to show that the rank of the matrix $${\smash{\lower1.0ex\hbox{\bg C}}}=
\left(
\begin{array}{cccc}
X_{-1}\displaystyle{\frac{\lambda _0}{\lambda _{-1}}}-X_0 & -X_{-1}\displaystyle{\frac{\lambda _1}{\lambda _{-1}}}+X_1 & \cdots &
-X_{-1}\displaystyle{\frac{\lambda _n}{\lambda _{-1}}}-X_n \\
X_{-1u_1}\displaystyle{\frac{\lambda _0}{\lambda _{-1}}}-X_{0u_1} &
-X_{-1u_1}\displaystyle{\frac{\lambda _1}{\lambda _{-1}}}+X_{1u_1}& \cdots &-X_{-1u_1}\displaystyle{\frac{\lambda _n}{\lambda _{-1}}}-X_{nu_1}\\
\vdots & \vdots & \ddots & \vdots \\
X_{-1u_s}\displaystyle{\frac{\lambda _0}{\lambda _{-1}}}-X_{0u_s} &
-X_{-1u_s}\displaystyle{\frac{\lambda _1}{\lambda _{-1}}}+X_{1u_s}& \cdots &-X_{-1u_s}\displaystyle{\frac{\lambda _n}{\lambda _{-1}}}-X_{nu_s} \\
n^T_{-1}\displaystyle{\frac{\lambda _0}{\lambda _{-1}}}-n^T_0 & -n^T_{-1}\displaystyle{\frac{\lambda _1}{\lambda _{-1}}}+n^T_1 & \cdots &
-n^T_{-1}\displaystyle{\frac{\lambda _n}{\lambda _{-1}}}-n^T_n \\
n^1_{-1}\displaystyle{\frac{\lambda _0}{\lambda _{-1}}}-n^1_0 & -n^1_{-1}\displaystyle{\frac{\lambda _1}{\lambda _{-1}}}+n^1_1 & \cdots &
-n^1_{-1}\displaystyle{\frac{\lambda _n}{\lambda _{-1}}}-n^1_n \\
\vdots & \vdots & \ddots & \vdots \\
n^{k-2}_{-1}\displaystyle{\frac{\lambda _0}{\lambda _{-1}}}-n^{k-2}_0 & -n^{k-2}_{-1}\displaystyle{\frac{\lambda _1}{\lambda _{-1}}}+n^{k-2}_1 & \cdots &
-n^{k-2}_{-1}\displaystyle{\frac{\lambda _n}{\lambda _{-1}}}-n^{k-2}_n
\end{array}
\right)$$ is $n+1$ at $(u,t,{\mbox{\boldmath $\lambda$}})\in \Sigma _*(H).$ We denote that $${\mbox{\boldmath $a$}}_i=^t\!\!(x_i(u,t),x_{iu_1}(u,t),\dots x_{iu_s}(u,t),n^T_i(u,t),n^1_i(u,t),\dots ,n^{k-2}_i(u,t)).$$ Then we have $${\smash{\lower1.0ex\hbox{\bg C}}}=\left({\mbox{\boldmath $a$}}_{-1}\frac{\lambda_0}{\lambda _{-1}}-{\mbox{\boldmath $a$}}_0,-{\mbox{\boldmath $a$}}_{-1}\frac{\lambda _1}{\lambda _{-1}}+{\mbox{\boldmath $a$}}_1,\dots ,
-{\mbox{\boldmath $a$}}_{-1}\frac{\lambda _n}{\lambda _{-1}}+{\mbox{\boldmath $a$}}_n\right).$$ It follows that $$\begin{aligned}
\det {\smash{\lower1.0ex\hbox{\bg C}}}\!\!\!\!\!\!\!
&{}&=\frac{\lambda _{-1}}{\lambda _{-1}}\det ({\mbox{\boldmath $a$}}_0,{\mbox{\boldmath $a$}}_1,\dots,{\mbox{\boldmath $a$}}_n)+\frac{\lambda_0}{\lambda_{-1}}\det ({\mbox{\boldmath $a$}}_{-1}{\mbox{\boldmath $a$}}_{1},\dots ,{\mbox{\boldmath $a$}}_n)
\\
&{}&-\frac{\lambda _1}{\lambda _{-1}}(-1)\det({\mbox{\boldmath $a$}}_{-1},{\mbox{\boldmath $a$}}_0,{\mbox{\boldmath $a$}}_2,\dots ,{\mbox{\boldmath $a$}}_n)-\cdots
-\frac{\lambda_n}{\lambda _{-1}}(-1)^{n-1}\det({\mbox{\boldmath $a$}}_{-1}{\mbox{\boldmath $a$}}_0,{\mbox{\boldmath $a$}}_1,\dots ,{\mbox{\boldmath $a$}}_{n-1}).\end{aligned}$$ Moreover, we define $\delta _i=\det ({\mbox{\boldmath $a$}}_{-1},{\mbox{\boldmath $a$}}_0,{\mbox{\boldmath $a$}}_1,\dots,{\mbox{\boldmath $a$}}_{i-1},{\mbox{\boldmath $a$}}_{i+1},\dots ,{\mbox{\boldmath $a$}}_n)$ for $i=-1,0,1,\dots ,n$ and ${\mbox{\boldmath $a$}}=(-\delta _{-1},-\delta _0,-\delta _1,(-1)^2\delta _2,\dots ,(-1)^{n-1}\delta _n).$ Then we have $${\mbox{\boldmath $a$}}={\mbox{\boldmath $X$}}\wedge{\mbox{\boldmath $X$}}_{u_1}\wedge\cdots \wedge {\mbox{\boldmath $X$}}_{u_s}\wedge{\mbox{\boldmath $n$}}^T\wedge{\mbox{\boldmath $n$}}_1\wedge\cdots\wedge{\mbox{\boldmath $n$}}_{k-2}.$$ We remark that ${\mbox{\boldmath $a$}}\not=0$ and ${\mbox{\boldmath $a$}}=\pm\|{\mbox{\boldmath $a$}}\|{\mbox{\boldmath $n$}}_{k-1}.$ By the above calculation, we have $$\begin{aligned}
\det{\smash{\lower1.0ex\hbox{\bg C}}}\!\!\!\!\!\!\!&{}&=\left\langle\left(\frac{\lambda{-1}}{\lambda_{-1}},\frac{\lambda_0}{\lambda _{-1}},\dots,\frac{\lambda_n}{\lambda _{-1}}\right), {\mbox{\boldmath $a$}}\right\rangle =\frac{1}{\lambda_{-1}}\left\langle {\mbox{\boldmath $X$}}(u)+\mu\left({\mbox{\boldmath $n$}}^T(u)+\sum_{i=1}^{k-1}\xi_i{\mbox{\boldmath $n$}}_i(u)\right),{\mbox{\boldmath $a$}}\right\rangle \\
&{}&=\frac{1}{\lambda _{-1}}\times\pm\mu\xi_{k-1}\|{\mbox{\boldmath $a$}}\|=\pm \frac{\mu\xi_{k-1}\|{\mbox{\boldmath $a$}}\|}{\lambda _{-1}}\not= 0.\end{aligned}$$ Therefore the Jacobi matrix of $\Delta^*H$ is non-singular at $(u,t,{\mbox{\boldmath $\lambda$}})\in \Delta ^*H^{-1}(0).$
For other local coordinates of $AdS^{n+1}$, we can apply the same method for the proof as the above case. Therefore, the AdS-height function $H$ is a non-degenerate big Morse family of hypersurfaces.
On the other hand, we have $$\frac{\partial H}{\partial t}(u,t,{\mbox{\boldmath $\lambda$}})=\langle {\mbox{\boldmath $X$}}_t(u,t),{\mbox{\boldmath $\lambda$}}\rangle.$$ Since ${\mbox{\boldmath $\xi$}}\in N^{AdS}_1[\mathcal{S}_t]_p=N_1^{AdS}(W)_p$ and ${\mbox{\boldmath $X$}}_t(u,t)\in T_pW,$ we have $\langle {\mbox{\boldmath $X$}}_t(u,t) , {\mbox{\boldmath $\xi$}}\rangle =0.$ Moreover, we have $\langle {\mbox{\boldmath $X$}},{\mbox{\boldmath $X$}}\rangle =-1,$ so that $\langle{\mbox{\boldmath $X$}}_t(u,t),{\mbox{\boldmath $X$}}(u,t)\rangle =0.$ Therefore, for ${\mbox{\boldmath $\lambda$}}={\mbox{\boldmath $X$}}(u,t)+\mu({\mbox{\boldmath $n$}}^T(u,t)+{\mbox{\boldmath $\xi$}}),$ we have $$\frac{\partial H}{\partial t}(u,t,{\mbox{\boldmath $\lambda$}})=\langle {\mbox{\boldmath $X$}}_t(u,t),{\mbox{\boldmath $\lambda$}}\rangle
=\mu \langle {\mbox{\boldmath $X$}}_t(u,t),{\mbox{\boldmath $n$}}^T(u,t)\rangle.$$ We remark that ${\mbox{\boldmath $n$}}^T(u,t)$ is a timelike vector such that $\langle {\mbox{\boldmath $n$}}^T(u,t),{\mbox{\boldmath $X$}}_{u_i}(u,t)\rangle =0$, $(i=1,\dots s)$. Since $\{{\mbox{\boldmath $X$}}_t(u,t),{\mbox{\boldmath $X$}}_{u_1}(u,t),\dots {\mbox{\boldmath $X$}}_{u_s}(u,t)\}$ is a basis of the Lorentz space $T_pW$ and ${\mbox{\boldmath $n$}}^T(u,t)\in T_pW,$ we have $\langle {\mbox{\boldmath $X$}}_t(u,t),{\mbox{\boldmath $n$}}^T(u,t)\rangle \not= 0.$ Moreover, ${\mbox{\boldmath $\lambda$}}\notin W$ implies $\mu\not= 0.$ Thus we have $\partial H/\partial t(u,t)\not= 0$ for ${\mbox{\boldmath $\lambda$}}={\mbox{\boldmath $X$}}(u,t)+\mu({\mbox{\boldmath $n$}}^T(u,t)+{\mbox{\boldmath $\xi$}}).$ This completes the proof.
$\Box$
We also consider the local coordinate $U^+_{-1}$. Since $H$ is a non-degenerate graph-like Morse family of hypersurfaces, we have a non-degenerate graph-like Legendrian unfolding $$\mathscr{L} _H:\Sigma _*(H){\longrightarrow}J^1(U^+_{-1},I).$$ By definition, we have $$\frac{\partial H}{\partial \lambda _0}((u,t),{\mbox{\boldmath $\lambda$}}) \\
=X_{-1}(u)\displaystyle{\frac{\lambda _0}{\lambda _{-1}}}-X_0(u),\ \frac{\partial H}{\partial \lambda _i}((u,t),{\mbox{\boldmath $\lambda$}})=-X_{-1}(u)\displaystyle{\frac{\lambda _i}{\lambda _{-1}}}+X_i(u),$$ $(i=1,\dots , n)$ and $\partial H/\partial t((u,t),{\mbox{\boldmath $\lambda$}})=\langle {\mbox{\boldmath $X$}}_t(u,t),{\mbox{\boldmath $\lambda$}}\rangle.$ It follows that $$\begin{aligned}
&{}&\left[\frac{\partial H}{\partial t}((u,t),{\mbox{\boldmath $\lambda$}}):\frac{\partial H}{\partial \lambda _0}((u,t),{\mbox{\boldmath $\lambda$}}):\frac{\partial H}{\partial \lambda _1}((u,t),{\mbox{\boldmath $\lambda$}}):\cdots :\frac{\partial H}{\partial \lambda _n}((u,t),{\mbox{\boldmath $\lambda$}})\right]=
\\
&{}&[\langle {\mbox{\boldmath $X$}}_t,{\mbox{\boldmath $\lambda$}}\rangle:X_{-1}(u)\lambda _0-X_0(u)\lambda _{-1}:X_1(u)\lambda _{-1}-X_{-1}(u)\lambda _1:\cdots :X_n(u)\lambda _{-1}-X_{-1}(u)\lambda _n].\end{aligned}$$ We denote that $$D_i({\mbox{\boldmath $X$}},{\mbox{\boldmath $\lambda$}})=\det\begin{pmatrix}
X_{-1} & X_i \\
\lambda _{-1} & \lambda _i \\
\end{pmatrix},\
(i=0,1,\dots ,n).$$ Then we have $$\mathscr{L}_H((u,t),{\mbox{\boldmath $\lambda$}})=\left({\mbox{\boldmath $\lambda$}}, t,-\frac{D_0(({\mbox{\boldmath $X$}},{\mbox{\boldmath $\lambda$}})}{\langle {\mbox{\boldmath $X$}}_t,{\mbox{\boldmath $\lambda$}}\rangle},\frac{D_1(({\mbox{\boldmath $X$}},{\mbox{\boldmath $\lambda$}})}{\langle {\mbox{\boldmath $X$}}_t,{\mbox{\boldmath $\lambda$}}\rangle},\dots ,\frac{D_n(({\mbox{\boldmath $X$}},{\mbox{\boldmath $\lambda$}})}{\langle {\mbox{\boldmath $X$}}_t,{\mbox{\boldmath $\lambda$}}\rangle}\right),$$ where $$\Sigma _*(H)=\Bigl\{((u,t),{\mbox{\boldmath $\lambda$}})\ \Bigm|\ {\mbox{\boldmath $\lambda$}}=\mathbb{LH}_{\mathcal{S}_t}(((u,t),{\mbox{\boldmath $\xi$}}),\mu)\ ((p,{\mbox{\boldmath $\xi$}}),\mu)\in
N_1^{AdS}[\mathcal{S}_t]\times{{\mathbb R}}, p={\mbox{\boldmath $X$}}(u,t)\ \Bigr\}.$$ We observe that $H$ is a graph-like generating family of the non-degenerate graph-like Legendrian unfolding $\mathscr{L}_H(\Sigma _*(H))$. Proposition 4.1 asserts that the graph-like big front $W(\mathscr{L}_H(\Sigma _*(H))$ of the non-degenerate graph-like Legendrian unfolding $\mathscr{L}_H(\Sigma _*(H))$ is given by $$\Bigl\{({\mbox{\boldmath $\lambda$}},t)\in AdS^{n+1}\times I \Bigm| {\mbox{\boldmath $\lambda$}}=\mathbb{LH}_{\mathcal{S}_t}(((u,t),{\mbox{\boldmath $\xi$}}),\mu),{\mbox{\boldmath $\xi$}}\in N^{AdS}_1[\mathcal{S}_t]_p, p={\mbox{\boldmath $X$}}(u,t),\mu \in {{\mathbb R}}\ \Bigr\}.$$ We define a mapping $\mathbb{LH}:N_1^{AdS}(W)\times {{\mathbb R}}{\longrightarrow}AdS^{n+1}\times I$ by $$\mathbb{LH}({\mbox{\boldmath $X$}}(u,t),{\mbox{\boldmath $\xi$}},\mu)=(\mathbb{LH}_{\mathcal{S}_t}({\mbox{\boldmath $X$}}(u,t),{\mbox{\boldmath $\xi$}},\mu),t),$$ which is called an [*unfolded lightlike hypersruface*]{} of $W.$ We write $\mathbb{LH}_{(W,\mathcal{S})}=\mathbb{LH}(N_1^{AdS}(W)\times {{\mathbb R}}).$ Then we have $\mathbb{LH}_{(W,\mathcal{S})}=W(\mathscr{L}_H(\Sigma _*(H)),$ so that the image of the unfolded lightlike hypersruface of $W$ is the graph-like big front set of $\mathscr{L}_H(\Sigma _*(H)).$ Each momentary front is the lightlike hypersurface $\mathbb{LH}_{\mathcal{S}_t}(N_1^{AdS}[\mathcal{S}_t]\times{{\mathbb R}})$, which is called a [*momentary lightlike hypersruface*]{} along the momentary space $\mathcal{S}_t.$ By assertion (2) of Proposition 4.1, a singular point of the momentary lightlike hypersruface $\mathbb{LH}_{\mathcal{S}_t}(N_1^{AdS}[\mathcal{S}_t]\times{{\mathbb R}})$ is a point ${\mbox{\boldmath $\lambda$}}_0=\mathbb{LH}_{\mathcal{S}_{t_0}}(((u_0,t_0),{\mbox{\boldmath $\xi$}}_0,\mu_0)$ for $1/\mu _0 =\kappa _N(\mathcal{S}_{t_0})_i((u_0,t_0),{\mbox{\boldmath $\xi$}}_0),$ $i=1,\dots ,s.$ Then we have the following corollary of Proposition 4.1.
A singular point of $\mathbb{LH}_{(W,\mathcal{S})}$ is the point $({\mbox{\boldmath $\lambda$}}, t)\in AdS^{n+1}\times I$ such that $
{\mbox{\boldmath $\lambda$}}=\mathbb{LH}_{\mathcal{S}_{t}}(((u,t),{\mbox{\boldmath $\xi$}},\mu),$ where $1/\mu =\kappa _N(\mathcal{S}_{t})_i((u,t),{\mbox{\boldmath $\xi$}}),$ $i=1,\dots ,s.$
For a non-zero nullcone principal curvature $\kappa _N(\mathcal{S}_{t_0})_i((u_0,t_0),{\mbox{\boldmath $\xi$}}_0)\not= 0,$ we have an open subset $O_i\subset N_1^{AdS}(W)$ such that $\kappa _N(\mathcal{S}_t)_i({\mbox{\boldmath $X$}}(u,t),{\mbox{\boldmath $\xi$}})\not= 0$ for $({\mbox{\boldmath $X$}}(u,t),{\mbox{\boldmath $\xi$}})\in O_i.$ Therefore, we have a non-zero nullcone principal curvature function $\kappa _N(\mathcal{S})_i:O_i{\longrightarrow}{{\mathbb R}}$. We define a mapping $
\mathbb{LF}_{\kappa_N(\mathcal{S}_t)_i} :O_i\cap N_1^{AdS}[\mathcal{S}_t]{\longrightarrow}AdS^{n+1}
$ by $$\mathbb{LF}_{\kappa_N(\mathcal{S}_t)_i}({\mbox{\boldmath $X$}}(u,t),{\mbox{\boldmath $\xi$}})={\mbox{\boldmath $X$}}(u,t)+\frac{1}{\kappa_N(\mathcal{S}_t)_i({\mbox{\boldmath $X$}}(u,t),{\mbox{\boldmath $\xi$}})}\mathbb{NG}((u,t),{\mbox{\boldmath $\xi$}}).$$ We also define $$\mathbb{LF}_{\mathcal{S}_t}=\bigcup_{i=1}^s \left\{\mathbb{LF}_{\kappa_N(\mathcal{S}_t)_i}({\mbox{\boldmath $X$}}(u,t),{\mbox{\boldmath $\xi$}})\ |\ ({\mbox{\boldmath $X$}}(u,t),{\mbox{\boldmath $\xi$}})\in N_1^{AdS}[\mathcal{S}_t]\ \mbox{s.t.}\ \kappa _N(\mathcal{S}_t)_i({\mbox{\boldmath $X$}}(u,t),{\mbox{\boldmath $\xi$}})\not= 0\right\} .$$ We call $\mathbb{LF}_{\mathcal{S}_t}$ the [*momentary lightlike focal set*]{} along $\mathcal{S}_t={\mbox{\boldmath $X$}}(U\times\{t\})$ in $AdS^{n+1}.$ By definition, the momentary lightlike focal set along $\mathcal{S}_t={\mbox{\boldmath $X$}}(U\times\{t\})$ is the critical values set of the momentary lightlike hypersurface $\mathbb{LH}_{\mathcal{S}_t}(N_1^{AdS}[\mathcal{S}_t]\times{{\mathbb R}})$ along $\mathcal{S}_t$. Moreover, an [*unfolded lightcone focal set*]{} of $(W,\mathcal{S})$ is defined to be $$\mathbb{LF}_{(W,\mathcal{S})}=\bigcup _{t\in I} \mathbb{LF}_{\mathcal{S}_{t}}\times \{t\} \subset AdS^{n+1}\times I.$$ Then $\mathbb{LF}_{(W,\mathcal{S})}$ is the critical value set of $\mathbb{LH}$.
Contact with lightcones
=======================
In this section we consider the geometric meanings of the singularities of momentary lightlike hypersrufaces in Anti-de Sitter space from the view point of the theory of contact of submanifolds with model hypersurfaces in [@mont1]. We begin with the following basic observations.
Let ${\mbox{\boldmath $\lambda$}}_0\in AdS^{n+1}$ and $\mathcal{S}_{t_0}={\mbox{\boldmath $X$}}(U\times\{t_0\})$ a monetary space of $W={\mbox{\boldmath $X$}}(U\times I)$ without points satisfying $K_N (\mathcal{S}_{t_0})(p,{\mbox{\boldmath $\xi$}})= 0.$ Then $\mathcal{S}_{t_0}\subset
\Lambda ^{n+1}_{\lambda _0}\cap AdS^{n+1}$ if and only if ${\mbox{\boldmath $\lambda$}}_0=\mathbb{LF}_{\mathcal{S}_{t_0}}$ is the momentary lightcone focal set. In this case we have $\mathbb{LH}_{\mathcal{S}_{t_0}}(N^{AdS}_1[\mathcal{S}_{t_0}]\times {{\mathbb R}})\subset \Lambda ^{n+1}_{\lambda _0}\cap AdS^{n+1}$ and $\mathcal{S}_{t_0}={\mbox{\boldmath $X$}}(U\times \{t_0\})$ is totally momentary nullcone umbilical.
[*Proof.* ]{}
By Proposition 3.1, $K_N(\mathcal{S}_{t_0})(p_0,{\mbox{\boldmath $\xi$}}_0)\not= 0$ if and only if $$\{({\mbox{\boldmath $n$}}^T+{\mbox{\boldmath $n$}}^S), ({\mbox{\boldmath $n$}}^T+{\mbox{\boldmath $n$}}^S)_{u_1},
\dots , ({\mbox{\boldmath $n$}}^T+{\mbox{\boldmath $n$}}^S)_{u_{s}}\}$$ is linearly independent for $p_0={\mbox{\boldmath $X$}}(u_0,t_0)\in \mathcal{S}_{t_0}$ and ${\mbox{\boldmath $\xi$}}_0={\mbox{\boldmath $n$}}^S(u_0,t_0),$ where ${\mbox{\boldmath $n$}}^S:\times I{\longrightarrow}N_1^{AdS}[\mathcal{S}_{t_0}]$ is a local section. By the proof of the assertion (1) of Proposition 4.1, $\mathcal{S}_{t_0}\subset \Lambda ^{n+1}_{\lambda _0}\cap AdS^{n+1}$ if and only if $h_{{\mbox{\scriptsize \boldmath$\lambda$}}_0,t_0}(u)= 0$ for any $u\in U,$ where $h_{{\mbox{\scriptsize \boldmath$\lambda$}}_0,t_0}(u)=H(u,t_0,{\mbox{\boldmath $\lambda$}}_0)$ is the AdS-height function on $\mathcal{S}_{t_0}.$ It also follows from Proposition 4.1 that there exists a smooth function $\eta
:U\times N_1^{AdS}[\mathcal{S}_{t_0}]{\longrightarrow}{{\mathbb R}}$ and section ${\mbox{\boldmath $n$}}^S:U\times I{\longrightarrow}N_1^{AdS}[\mathcal{S}_{t_0}]$ such that $${\mbox{\boldmath $X$}}(u,t_0)={\mbox{\boldmath $\lambda$}}_0+\eta (u,{\mbox{\boldmath $n$}}^S(u,t_0))({\mbox{\boldmath $n$}}^T(u,t_0)\pm{\mbox{\boldmath $n$}}^S(u,t_0)).$$ In fact, we have $\eta (u,{\mbox{\boldmath $n$}}^S(u,t_0))=-1/\kappa _N(\mathcal{S}_{t_0})_i(p,{\mbox{\boldmath $\xi$}})$ $i=1,\dots, s$, where $p={\mbox{\boldmath $X$}}(u,t_0)$ and ${\mbox{\boldmath $\xi$}}={\mbox{\boldmath $n$}}^S(u,t_0).$ It follows that $\kappa _N(\mathcal{S}_{t_0})_i(p,{\mbox{\boldmath $\xi$}})=\kappa _N(\mathcal{S}_{t_0})_j(p,{\mbox{\boldmath $\xi$}}),$ so that $\mathcal{S}_{t_0}={\mbox{\boldmath $X$}}(U\times \{t_0\})$ is totally nullcone umbilical. Therefore we have $$\mathbb{LH}_{\mathcal{S}_{t_0}}(u,{\mbox{\boldmath $n$}}^S(u,t_0),\mu)={\mbox{\boldmath $\lambda$}}_0+(\mu +\eta (u,{\mbox{\boldmath $n$}}^S(u,t_0))({\mbox{\boldmath $n$}}^T(u,t_0)\pm{\mbox{\boldmath $n$}}^S(u,t_0)).$$ Hence we have $\mathbb{LH}_{\mathcal{S}_{t_0}}(N_1^{AdS}[\mathcal{S}_{t_0}]\times {{\mathbb R}})\subset \Lambda ^{n+1}_{\lambda _0}\cap AdS^{n+1}.$ By definition, the critical value set of $\mathbb{LH}_{\mathcal{S}_{t_0}}(N_1^{AdS}[\mathcal{S}_{t_0}]\times {{\mathbb R}})$ is the lightlike focal set $\mathbb{LF}_{\mathcal{S}_{t_0}},$ which is equal to ${\mbox{\boldmath $\lambda$}}_0$ by the previous arguments.
For the converse assertion, suppose that ${\mbox{\boldmath $\lambda$}}_0=\mathbb{LF}_{\mathcal{S}_{t_0}}.$ Then we have $${\mbox{\boldmath $\lambda$}}_0={\mbox{\boldmath $X$}}(u,t_0)+\frac{1}{\kappa _N(\mathcal{S}_{t_0})_i({\mbox{\boldmath $X$}}(u,t_0),{\mbox{\boldmath $\xi$}})}\mathbb{NG}(\mathcal{S}_{t_0})(u,t_0,{\mbox{\boldmath $\xi$}}),$$ for any $i=1,\dots ,s$ and $(p,{\mbox{\boldmath $\xi$}})\in N_1^{AdS}[\mathcal{S}_{t_0}],$ where $p={\mbox{\boldmath $X$}}(u,t_0).$ Thus, we have $$\kappa _N(\mathcal{S}_{t_0})_i({\mbox{\boldmath $X$}}(u,t_0),{\mbox{\boldmath $\xi$}})=\kappa _N(\mathcal{S}_{t_0})_j({\mbox{\boldmath $X$}}(u,t_0),{\mbox{\boldmath $\xi$}})$$ for any $i,j=1,\dots ,s.$ This means that $\mathcal{S}_{t_0}$ is totally momentary nullcone umbilical. Since $\mathbb{NG}(\mathcal{S}_{t_0})(u,t_0,{\mbox{\boldmath $\xi$}})$ is null for any $(u,{\mbox{\boldmath $\xi$}})$, we have ${\mbox{\boldmath $X$}}(U\times\{t_0\})\subset \Lambda ^{n+1}_{\lambda_0}\cap AdS^{n+1}.$ This completes the proof.
$\Box$
We now consider the relationship between the contact of a one parameter family of submanifolds with a submanifold and the $S.P$-${\mathcal K}$-classification of functions. Let $ U_i\subset {{\mathbb R}}^r$, ($i=1,2$) be open sets and $g_i:(U_i\times I, ({{\overline{u}}}_i,t_i)){\longrightarrow}({{\mathbb R}}^n,\bm{y}_i)$ immersion germs. We define $\overline{g}_i:(U_i\times I, ({{\overline{u}}}_i,t_i)){\longrightarrow}({{\mathbb R}}^n\times I,(\bm{y}_i,t_i))$ by $\overline{g}_i({{\overline{u}}},t)=(g_i({{\overline{u}}}),t).$ We denote that $(\overline{Y}_i,(\bm{y}_i,t_i))=\overline{g}_i(U_i\times I),(\bm{y}_i,t_i)).$ Let $f_i:({{\mathbb R}}^n,\bm{y}_i) {\longrightarrow}({{\mathbb R}},0)$ be submersion germs and denote that $(V(f_i),\bm{y}_i)=(f_i^{-1}(0),\bm{y}_i).$ We say that [*the contact of $\overline{Y}_1$ with the trivial family of $V(f_1)$ at $(\bm{y}_1,t_1)$*]{} is of the [*same type in the strict sense*]{} as [*the contact of $\overline{Y}_2$ with the trivial family of $V(f_2)$ at $(\bm{y}_2,t_2)$*]{} if there is a diffeomorphism germ $\Phi:({{\mathbb R}}^n\times I,(\bm{y}_1,t_1)) {\longrightarrow}({{\mathbb R}}^n\times I,(\bm{y}_2,t_2))$ of the form $\Phi (\bm{y},t)=(\phi_1(\bm{y},t),t+(t_2-t_1))$ such that $\Phi(\overline{Y}_1)=\overline{Y}_2$ and $\Phi(V(f_1)\times I) = V(f_2)\times I$. In this case we write $SK(\overline{Y}_1,V(f_1)\times I;(\bm{y}_1,t_1)) = SK(\overline{Y}_2,V(f_2)\times I;(\bm{y}_2,t_2))$. We can show one of the parametric versions of Montaldi’s theorem of contact between submanifolds as follows:
We use the same notations as in the above paragraph. Then the following conditions are equivalent:
[(1)]{} $
SK(\overline{Y}_1,V(f_1)\times I;(\bm{y}_1,t_1)) = SK(\overline{Y}_2,V(f_2)\times I;(\bm{y}_2,t_2))
$
[(2)]{} $f_1 \circ g_1$ and $f_2 \circ g_2$ are $S.P$-${\mathcal K}$-equivalent [(]{}i.e., there exists a diffeomorphism germ $\Psi :(U_1\times I,({{\overline{u}}}_1,t_1)){\longrightarrow}(U_2\times I,({{\overline{u}}}_2,t_2))$ of the form $\Psi ({{\overline{u}}},t)=(\psi_1 ({{\overline{u}}},t),t+(t_2-t_1))$ and a function germ $\lambda :(U_1\times I,({{\overline{u}}}_1,t_1)){\longrightarrow}{{\mathbb R}}$ with $\lambda ({{\overline{u}}}_1,t_1)\not= 0$ such that $(f_2\circ g_2)\circ \Phi ({{\overline{u}}},t) =\lambda ({{\overline{u}}},t)f_1\circ g_1({{\overline{u}}},t)$[).]{}
Since the proof of Proposition 7.2 is given by the arguments just along the line of the proof of the original theorem in [@mont1], we omit the proof here.
We now consider a function $
{\mathfrak h}_{\bm{\lambda}}:AdS^{n+1}{\longrightarrow}{\mathbb R}
$ defined by ${\mathfrak h}_{\bm{\lambda}}({\mbox{\boldmath $x$}})=\langle {\mbox{\boldmath $x$}},\bm{\lambda}\rangle +1,$ where $\bm{\lambda}\in AdS^{n+1}.$ For any $\bm{\lambda}_0\in AdS^{n+1}$, we have the Lorentzian tangent hyperplane $HP(\bm{\lambda}_0,-1)$ of de Sitter space $AdS^{n+1}$ at $\bm{\lambda}_0$, so that we have an AdS-lightcone $$\mathfrak{h}_{\bm{\lambda}_0}^{-1}(0)=AdS^{n+1}\cap HP(\bm{\lambda}_0,-1)=LC^{AdS}(\bm{\lambda}_0 ).$$ Moreover, we consider a point $\bm{\lambda}_0=\mathbb{LH}_{\mathcal{S}_{t_0}} ({\mbox{\boldmath $X$}}({{\overline{u}}}_0,t_0),{\mbox{\boldmath $\xi$}}_0,\mu _0).$ Then we have $$\mathfrak{h}_{\bm{\lambda}_0}\circ{\mbox{\boldmath $X$}}({{\overline{u}}}_0,t_0)=H((u_0,t_0),\mathbb{LH}_{\mathcal{S}_{t_0}} ({\mbox{\boldmath $X$}}({{\overline{u}}}_0,t_0),{\mbox{\boldmath $\xi$}}_0,\mu _0))=0.$$ By Proposition 4.1, we also have relations that $$\frac{\partial \mathfrak{h}_{\bm{\lambda}_0}\circ{\mbox{\boldmath $X$}}}{\partial u_i}({{\overline{u}}}_0,t_0)=\frac{\partial H}{\partial u_i}(({{\overline{u}}}_0,t_0),\mathbb{LH}_{\mathcal{S}_{t_0}} ({\mbox{\boldmath $X$}}({{\overline{u}}}_0,t_0),{\mbox{\boldmath $\xi$}}_0,\mu _0))=0.$$ for $i=1,\dots ,s.$ This means that the AdS-lightcone $\mathfrak{h}_{\bm{\lambda}_0}^{-1}(0)=LC^{AdS}(\bm{\lambda}_0)$ is tangent to $\mathcal{S}_{t_0}={\mbox{\boldmath $X$}}(U\times \{t_0\})$ at $p_0={\mbox{\boldmath $X$}}({{\overline{u}}}_0,t_0).$ The AdS-lightcone $LC^{AdS}(\bm{\lambda}_0)$ is said to be a [*tangent anti-de Sitter lightcone*]{} (briefly, a [*tangent AdS-lightcone*]{}) of $\mathcal{S}_{t_0}={\mbox{\boldmath $X$}}(U\times \{t_0\})$ at $p_0={\mbox{\boldmath $X$}}({{\overline{u}}}_0,t_0)$. We write that $LC^{AdS}(\mathcal{S}_{t_0};p_0,{\mbox{\boldmath $\xi$}}_0,\mu _0)=LC^{AdS}(\bm{\lambda}_0),$ where $\bm{\lambda}_0=\mathbb{LH}_{\mathcal{S}_{t_0}} ({\mbox{\boldmath $X$}}({{\overline{u}}}_0,t_0),{\mbox{\boldmath $\xi$}}_0,\mu _0).$ Then we have the following simple lemma.
Let ${\mbox{\boldmath $X$}}:U\times I{\longrightarrow}AdS^{n+1}$ be a world sheet in anti-de Sitter space. We consider two points $(p_1,{\mbox{\boldmath $\xi$}}_1,\mu_1),(p_2,{\mbox{\boldmath $\xi$}}_2,\mu_2)\in
N_1(\mathcal{S}_{t_0})\times {{\mathbb R}},$ where $p_i={\mbox{\boldmath $X$}}({{\overline{u}}}_i,t_0)$, $(i=1,2).$ Then $\mathbb{LH}_{\mathcal{S}_{t_0}}({\mbox{\boldmath $X$}}({{\overline{u}}}_1,t_0),{\mbox{\boldmath $\xi$}}_1,\mu_1))=\mathbb{LH}_{\mathcal{S}_{t_0}} ({\mbox{\boldmath $X$}}({{\overline{u}}}_2,t_0),{\mbox{\boldmath $\xi$}}_2,\mu_2))$ if and only if $$LC^{AdS}(\mathcal{S}_{t_0},p_1,{\mbox{\boldmath $\xi$}}_1,\mu_1)=LC^{AdS}(\mathcal{S}_{t_0},p_2,{\mbox{\boldmath $\xi$}}_2,\mu_2).$$
By the definition of unfolded lightlike hypersruface, $$\mathbb{LH}({\mbox{\boldmath $X$}}(\overline{u}_1,t_1),{\mbox{\boldmath $\xi$}}_1,\mu_1)=\mathbb{LH}({\mbox{\boldmath $X$}}(\overline{u}_2,t_2),{\mbox{\boldmath $\xi$}}_2,\mu_2)$$ if and only if $t_1=t_2$ and $\mathbb{LH}_{\mathcal{S}_{t_1}} ({\mbox{\boldmath $X$}}({{\overline{u}}}_1,t_1),{\mbox{\boldmath $\xi$}}_1,\mu_1)=\mathbb{LH}_{\mathcal{S}_{t_1}} ({\mbox{\boldmath $X$}}({{\overline{u}}}_2,t_1),{\mbox{\boldmath $\xi$}}_2,\mu_2)$. Eventually, we have tools for the study of the contact between world sheets and anti-de Sitter lightcones. Since we have $h_{\bm{\lambda}}({{\overline{u}}},t)=\mathfrak{h}_{\bm{\lambda}}\circ{\mbox{\boldmath $X$}}({{\overline{u}}},t),$ we have the following proposition as a corollary of Proposition 7.2.
Let ${\mbox{\boldmath $X$}}_i : (U\times I,({{\overline{u}}}_i,t_i)) {\longrightarrow}(AdS^{n+1},p_i)$ $(i=1,2)$ be world sheet germs with $W_i={\mbox{\boldmath $X$}}_i(U\times I)$ and $\bm{\lambda}_i=\mathbb{LH}_{\mathcal{S}_{t_i}} ({\mbox{\boldmath $X$}}({{\overline{u}}}_i,t_i),{\mbox{\boldmath $\xi$}}_i,\mu _i).$ Then the following conditions are equivalent:
[(1)]{} $SK(\overline{W}_1, LC^{AdS}(\mathcal{S}_{t_1},p_1,{\mbox{\boldmath $\xi$}}_1,\mu_1)\times I;(p_1,t_1))=
SK(\overline{W}_2,LC^{AdS}(\mathcal{S}_{t_2},p_2,{\mbox{\boldmath $\xi$}}_2,\mu_2)\times I;(p_2,t_2)),$
[(2)]{} $h_{1,\bm{\lambda}_1}$ and $h_{2,\bm{\lambda}_2}$ are $S.P$-$\mathcal{K}$-equivalent.\
Caustics and Maxwell sets of world sheets
=========================================
In this section we apply the theory of graph-like Legendrian unfoldings to investigate the singularities of the caustics and the Maxwell sets of world sheets. In [@Bousso; @Bousso-Randall] Bousso and Randall gave an idea of caustics of world sheets in order to define the notion of holographic domains. The family of lightlike hypersrufaces $\{\mathbb{LH}_{\mathcal{S}_{t}}(N^{AdS}_1[\mathcal{S}_t]\times{{\mathbb R}})\}_{t\in J}$ sweeps out a region in $AdS^{n+1}.$ A [*caustic*]{} of a world sheet is the union of the sets of critical values of lightlike hypersrufaces along momentary spaces $\{\mathcal{S}_t\}_{t\in I}.$ A [*holographic domain*]{} of the world sheet is the region where the light-sheets sweep out until [*caustics*]{}. So this means that the boundary of the holographic domain consists the caustic of the world sheet. The set of critical values of the lightlike hypersruface of a momentary space is the lightlike focal set of the momentary space. Therefore the notion of caustics in the sense of Bousso-Randall is formulated as follows: A [*caustic of a world sheet*]{} $(W,\mathcal{S})$ is defined to be $$\displaystyle{C(W,\mathcal{S})=\bigcup _{t\in I} \mathbb{LF}_{\mathcal{S}_{t}}}=\pi_1(\mathbb{LF}_{(W,\mathcal{S})}),$$ where $\pi_1:AdS^{n+1}\times I{\longrightarrow}AdS^{n+1}$ is the canonical projection. We call $C(W,\mathcal{S})$ a [*BR-caustic*]{} of $(W,\mathcal{S}).$ By definition, we have $\Sigma (W(\mathscr{L}_{H}(\Sigma _*(H)))=\mathbb{LF} _{(W,\mathcal{S})},$ so that we have the following proposition.
Let $(W,\mathcal{S})$ be a world sheet in $AdS^{n+1}$ and $H:U\times I\times (AdS^{n+1}\setminus W){\longrightarrow}{{\mathbb R}}$ the $AdS$-height function on $W.$ Then we have $C(W,\mathcal{S})=C_{\mathscr{L}_{H}(\Sigma _*(H))}.$
In [@Bousso; @Bousso-Randall] the authors did not consider the Maxwell set of a world sheet. However, the notion of Maxwell sets plays an important role in the cosmology which has been called a [*crease set*]{} by Penrose (cf. [@Penrose; @Siino]). Actually, the topological shape of the event horizon is determined by the crease set of lightlike hypersrufaces. Here, we write $M(W,S)=M_{\mathscr{L}_H(\Sigma _*(H))}$ and call it a [*BR-Maxwell set*]{} of the world sheet $(W,\mathcal{S}).$
Let ${\mbox{\boldmath $X$}}_i:(U\times I, ({{\overline{u}}}_i,t_i)){\longrightarrow}(AdS^{n+1},p_i)$, $(i=1,2)$ be germs of timelike embeddings such that $(W_i,\mathcal{S}_{i})$ are world sheet germs, where $W_i={\mbox{\boldmath $X$}}_i(U\times I).$ For $\bm{\lambda}_i=\mathbb{LH}_{\mathcal{S}_{t_i}} ({\mbox{\boldmath $X$}}({{\overline{u}}}_i,t_i),{\mbox{\boldmath $\xi$}}_i,\mu _i),$ let $H_i :(U\times I\times (AdS^{n+1}\setminus W_i) ,({{\overline{u}}}_i,t_i,\bm{\lambda}_i)){\longrightarrow}{{\mathbb R}}$ be $AdS$-height function germs. We also write $h_{i,\bm{\lambda}_i}({{\overline{u}}},t)=H_i({{\overline{u}}},t,\bm{\lambda}_i).$ Since $$W(\mathscr{L}_{H_i}(\Sigma _*(H_i)))=\mathbb{LH}_{(W_i,\mathcal{S}_i)},$$ we can apply Theorem 5.2 and Corollary 5.3 to our case. Then we have the following theorem.
Suppose that the set of critical points of $\overline{\pi}|_{\mathscr{L}_{H_i}(\Sigma _*(H_i))}$ are nowhere dense for $i=1,2$, respectively. Then the following conditions are equivalent[:]{}
[(1)]{} $(\mathbb{LH}_{(W_1,\mathcal{S}_1)}, \bm{\lambda}_1)$ and $(\mathbb{LH}_{(W_2,\mathcal{S}_2)}, \bm{\lambda}_2)$ are $S.P^+$-diffeomorphic,
[(2)]{} $\mathscr{L}_{H_1}(\Sigma _*(H_1))$ and $\mathscr{L}_{H_2}(\Sigma _*(H_2))$ are $S.P^+$-Legendrian equivalent,
[(3)]{} $\Pi(\mathscr{L}_{H_1}(\Sigma _*(H_1)))$ and $\Pi(\mathscr{L}_{H_2}(\Sigma _*(H_2))$ are Lagrangian equivalent.
We remark that conditions (2) and (3) are equivalent without any assumptions (cf. Theorem 5.2). Moreover, if we assume that $\mathscr{L}_{H_i}(\Sigma _*(H_i))$ are $S.P^+$-Legendrian stable, then we can apply Proposition 7.4 and Theorem 5.6 to show the following theorem.
Suppose that $\mathscr{L}_{H_i}(\Sigma _*(H_i))$ are $S.P^+$-Legendrian stable for $i=1,2,$ respectively. Then the following conditions are equivalent[:]{}
[(1)]{} $(\mathbb{LH}_{(W_1,\mathcal{S}_1)}, \bm{\lambda}_1)$ and $(\mathbb{LH}_{(W_2,\mathcal{S}_2)}, \bm{\lambda}_2)$ are $S.P^+$-diffeomorphic,
[(2)]{} $\mathscr{L}_{H_1}(\Sigma _*(H_1))$ and $\mathscr{L}_{H_2}(\Sigma _*(H_2))$ are $S.P^+$-Legendrian equivalent,
[(3)]{} $\Pi(\mathscr{L}_{H_1}(\Sigma _*(H_1)))$ and $\Pi(\mathscr{L}_{H_2}(\Sigma _*(H_2))$ are Lagrangian equivalent,
[(4)]{} $h_{1,\bm{\lambda}_1}$ and $h_{2,\bm{\lambda}_2}$ are $S.P$-$\mathcal{K}$-equivalent,
[(5)]{} $SK(\overline{W}_1, LC^{AdS}(\mathcal{S}_{t_1},p_1,\bm{\xi}_1,\mu_1)\times I;(p_1,t_1))=
SK(\overline{W}_2, LC^{AdS}(\mathcal{S}_{t_2},p_2,\bm{\xi}_2,\mu_2)\times I;(p_2,t_2)).$
By definition and Proposition 8.1, we have the following proposition.
If $\Pi(\mathscr{L}_{H_1}(\Sigma _*(H_1)))$ and $\Pi(\mathscr{L}_{H_2}(\Sigma _*(H_2))$ are Lagrangian equivalent, then BR-caustics $C(W_1,\mathcal{S}_1)$, $C(W_2,\mathcal{S}_2)$ and BR-Maxwell sets $M(W_1,\mathcal{S}_1)$, $M(W_2,\mathcal{S}_2)$ are diffeomorphic as set germs, respectively.
World hyper-sheets in $AdS^{n+1}$
=================================
In this section we consider the case when $k=2.$ For an open subset $U\subset {{\mathbb R}}^n,$ let ${\mbox{\boldmath $X$}}:U\times I{\longrightarrow}AdS^{n+1}$ be a timelike embedding such that $(W, \mathcal{S})$ is a world sheet. In this case $(W,\mathcal{S})$ is said to be a [*world hyper-sheet*]{} in $AdS^{n+1}.$ Since the pseudo normal space $N_p(W)$ is a Lorentz plane, $N_p^{AdS}(W)$ is a spacelike line, so that $N_1^{AdS}(W)_p$ comprises two points. For any $\bm{\xi}\in N_1^{AdS}(W)_p,$ we have $-\bm{\xi}\in N_1^{AdS}(W)_p.$ We define a pseudo normal section $\bm{n}^S({{\overline{u}}},t)\in N_1^{AdS}(W)_p$ for $p=\bm{X}({{\overline{u}}},t)$ by $$\bm{n}^S({{\overline{u}}},t)=\frac{{\mbox{\boldmath $X$}}({{\overline{u}}},t)\wedge {\mbox{\boldmath $X$}}_{u_1}({{\overline{u}}},t)\wedge \dots \wedge {\mbox{\boldmath $X$}}_{u_{n-1}}({{\overline{u}}},t)\wedge {\mbox{\boldmath $X$}}_t(u,t)}{\|{\mbox{\boldmath $X$}}({{\overline{u}}},t)\wedge {\mbox{\boldmath $X$}}_{u_1}({{\overline{u}}},t)\wedge \dots \wedge {\mbox{\boldmath $X$}}_{u_{n-1}}({{\overline{u}}},t)\wedge {\mbox{\boldmath $X$}}_t({{\overline{u}}},t)\|}.$$ Therefore the momentary nullcone Gauss images $$\mathbb{NG}(\mathcal{S}_{t_0},\pm\bm{n}^S):U{\longrightarrow}\Lambda ^*$$ are given by $\mathbb{NG}(\mathcal{S}_{t_0},\pm\bm{n}^S)({{\overline{u}}})=\bm{n}^T({{\overline{u}}},t_0)\pm \bm{n}^S({{\overline{u}}},t_0).$ Therefore we have the momentary nullcone shape operators $$S_N^\pm (\mathcal{S}_{t_0})_p=S_p(\mathcal{S}_{t_0};\pm\bm{n}^S)=-\pi ^t\circ d_p\mathbb{NG}(\mathcal{S}_{t_0},\pm\bm{n}^S):T_p\mathcal{S}_{t_0}{\longrightarrow}T_p\mathcal{S}_{t_0}.$$ It follows that we have momentary nullcone principal curvatures $$\kappa _N^\pm(\mathcal{S}_{t_0})_i( p)=\kappa _N(\mathcal{S}_{t_0})(p,\pm \bm{n}^S({{\overline{u}}},t_0)),\ ( i=1,\dots ,n-1).$$ Then the momentary lightlike hypersrufaces $\mathbb{LH}^\pm_{S_t}:U\times {{\mathbb R}}{\longrightarrow}AdS^{n+1}$ are given by $$\mathbb{LH}^\pm_{S_t}({{\overline{u}}},\mu)={\mbox{\boldmath $X$}}({{\overline{u}}},t)+\mu(\bm{n}^T({{\overline{u}}},t)\pm\bm{n}^S({{\overline{u}}},t))={\mbox{\boldmath $X$}}({{\overline{u}}},t)+\mu\mathbb{NG}(\mathcal{S}_{t},\pm\bm{n}^S)({{\overline{u}}}).$$ Moreover, the unfolded lightlike hypersrufaces $\mathbb{LH}^\pm:U\times {{\mathbb R}}{\longrightarrow}AdS^{n+1}\times I$ are given by $$\mathbb{LH}^\pm({{\overline{u}}},\mu)=(\mathbb{LH}^\pm_{S_t}({{\overline{u}}},\mu),t)=({\mbox{\boldmath $X$}}({{\overline{u}}},t)+\mu\mathbb{NG}(\mathcal{S}_{t},\pm\bm{n}^S)({{\overline{u}}}),t).$$ For the $AdS$-height function $H:U\times I\times AdS^{n+1}{\longrightarrow}{{\mathbb R}}$ on $(W,\mathcal{S}),$ $
\Sigma _*(H)=\Sigma ^+_*(H)\cup \Sigma ^-_*(H),
$ where $$\Sigma ^\pm_*(H)=\{(({{\overline{u}}},t),\bm{\lambda})\ |\ \bm{\lambda}=\mathbb{LH}^\pm_{\mathcal{S}_t}({{\overline{u}}},t,\mu), \mu\in {{\mathbb R}}\}.$$ Then the image of unfolded lightlike hypersrufaces is $$\mathbb{LH}_W=\mathbb{LH}^+(U\times {{\mathbb R}})\cup\mathbb{LH}^-(U\times {{\mathbb R}})=W(\mathscr{L}_H(\Sigma_*(H))),$$ which is the graph-like big front set of $\mathscr{L}_H(\Sigma_*(H)).$ The momentary lightlike focal sets along $\mathcal{S}_t$ are $$\mathbb{LF}^\pm_{\mathcal{S}_t}=\bigcup_{i=1}^{n-1} \left\{\mathbb{LF}^\pm_{\kappa^\pm_N(\mathcal{S}_t)_i}({{\overline{u}}},t)\bigm |
({{\overline{u}}},t)\in U\times I\ s.t.\ \kappa^\pm_N(\mathcal{S}_t)_i({\mbox{\boldmath $X$}}({{\overline{u}}},t))\not= 0\right\},$$ where $$\mathbb{LF}^\pm_{\kappa^\pm_N(\mathcal{S}_t)_i}({{\overline{u}}},t)={\mbox{\boldmath $X$}}({{\overline{u}}},t)+\frac{1}{\kappa^\pm_N(\mathcal{S}_t)_i({\mbox{\boldmath $X$}}({{\overline{u}}},t))}\mathbb{NG}(\mathcal{S}_{t_0},\pm\bm{n}^S)({{\overline{u}}}).$$ The unfolded lightcone focal set is $$\mathbb{LF}_{(W,\mathcal{S})}=\bigcup_{t\in I} \mathbb{LF}^+_{\mathcal{S}_t}\times \{t\}\cup \bigcup_{t\in I} \mathbb{LF}^-_{\mathcal{S}_t}\times \{t\}
\subset AdS^{n+1}\times I.$$ In this case the BR-caustic is $$C(W,\mathcal{S})=\pi_1(\mathbb{LF}_{(W,\mathcal{S})})=\bigcup_{t\in I}\mathbb{LF}^+_{\mathcal{S}_t}\cup \bigcup_{t\in I}\mathbb{LF}^-_{\mathcal{S}_t}.$$ Moreover, the BR-Maxwell set is $$M(W,\mathcal{S})=M_{\mathscr{L}_H(\Sigma_*(H))}=M_{\mathscr{L}_H(\Sigma^+_*(H))}\cup M_{\mathscr{L}_H(\Sigma^-_*(H))}.$$
World sheets in $AdS^3$
=======================
In this section we consider world sheets in the $3$-dimensional anti de Sitter space as an example. Let $(W, \mathcal{S})$ be a world sheet in $AdS^3$, which is parameterized by a timelike embedding $\bm{\Gamma} :J\times I{\longrightarrow}AdS^{3}$ such that $\mathcal{S}_t=\bm{\Gamma} (J\times \{t\})$ for $t\in I.$ In this case we call $\mathcal{S}_t$ a [*momentary curve*]{}. We assume that $s\in J$ is the arc-length parameter. Then $\bm{t}(s,t)=\bm{\gamma} _t'(s)$ is the unit spacelike tangent vector of $\mathcal{S} _t$, where $\bm{\gamma}_t(s)=\bm{\Gamma}(s,t).$ We have the unit pseudo-normal vector field $\bm{n}(s,t)$ of $W$ in $AdS^3$ defined by $$\bm{n}(s,t)=\frac{\bm{\Gamma}(s,t)\wedge \bm{t}(s,t)\wedge \bm{\Gamma}_t(s,t)}{\|\bm{\Gamma}(s,t)\wedge \bm{t}(s,t)\wedge \bm{\Gamma}_t(s,t)\|}.$$ The unit timelike normal vector of $\mathcal{S} _t$ in $TW$ is defined to be $\bm{b}(s,t)=\bm{\Gamma}(s,t)\wedge \bm{n} (s,t)\wedge \bm{t}(s,t).$ We choose the orientation of $\mathcal{S} _t$ such that $\bm{b}(s,t)$ is adopted (i.e. ${\rm det}\, (\bm{\Gamma}(s,t), \bm{b}(s,t), \bm{e}_1,\bm{e}_2)> 0$). Therefore, $\{\bm{\Gamma}(s,t),\bm{b}(s,t),\bm{n}(s,t),\bm{t}(s,t)\}$ is a [pseudo-orthonormal frame]{} along $W. $ On this moving frame, we can show the following [*Frenet-Serret type formulae*]{} for $S_t$: $$\setlength\arraycolsep{2pt}
\left\{
\begin{array}{ccl}
\displaystyle{\frac{\partial \bm{\Gamma}}{\partial s}(s,t)} &=& \bm{t}(s,t), \\
\displaystyle{\frac{\partial \bm{b}}{\partial s}(s,t)} &=& \tau_g(s,t) \bm{n}(s,t)-\kappa_g(s,t) \bm{t}(s,t), \\
\displaystyle{\frac{\partial \bm{n}}{\partial s}(s,t)} &=& \tau_g(s,t) \bm{b}(s,t)-\kappa_n(s,t) \bm{t}(s,t) , \\
\displaystyle{\frac{\partial \bm{t}}{\partial s}(s,t)} &=& \bm{\Gamma}(s,t)-\kappa_g(s,t) \bm{b}(s,t) + \kappa_n(s,t) \bm{n}(s,t),
\end{array}
\right.$$ where $\kappa_g(s,t) = \langle \frac{\partial \bm{t}}{\partial s}(s,t), \bm{b}(s,t) \rangle,$ $\kappa_n(s,t) = \langle \frac{\partial \bm{t}}{\partial s}(s,t), \bm{n}(s,t) \rangle,$ $\tau _g(s,t)= \langle \frac{\partial \bm{b}}{\partial s}(s,t),\bm{n}(s,t)\rangle.$ We call $\kappa _g(s,t)$ a [*geodesic curvature*]{}, $\kappa _n(s,t)$ a [*normal curvature*]{} and $\tau _g(s,t)$ a [*geodesic torsion*]{} of $\mathcal{S}_t$ respectively. Then $\bm{b}(s,t_0)\pm \bm{n}(s,t_0)$ are lightlike. We have the momentary lightlike hypersrufaces $\mathbb{LS}^\pm_{\mathcal{S}_{t_0}}:J\times\{t_0\}\times {{\mathbb R}}{\longrightarrow}AdS^3$ along $\mathcal{S}_{t_0}$ defined by $
\mathbb{LS}^\pm_{\mathcal{S}_{t_0}}((s,t_0),u)=\bm{\Gamma} (s,t_0)+u(\bm{b}(s,t_0)\pm \bm{n}(s,t_0)).
$ Here, we use the notation $\mathbb{LS}^\pm_{\mathcal{S}_{t_0}}$ instead of $\mathbb{LH}^\pm_{\mathcal{S}_{t_0}}$ because the images of these mappings are lightlike surfaces. We adopt $\bm{n}^T=\bm{b}$ and $\bm{n}^S=\bm{n}.$ By the Frenet-Serret type formulae, we have $$\frac{\partial (\bm{n}^T\pm \bm{n}^S)}{\partial s}(s,t)=\frac{\partial (\bm{b}\pm \bm{n})}{\partial s}(s,t)=\tau_g(s,t)(\bm{n}\pm\bm{b})(s,t)-(\kappa _g(s,t)\pm \kappa _n(s,t))\bm{t}(s,t).$$ Therefore, we have $\kappa ^\pm (\mathcal{S}_t)(s,t)=\kappa _g(s,t)\pm \kappa _n(s,t).$ It follows that $$\mathbb{LF}^\pm _{\mathcal{S}_{t_0}}=\left\{\bm{\Gamma}(s,t_0)+\frac{1}{\kappa _g(s,t_0)\pm \kappa _n(s,t_0)}(\bm{b}\pm \bm{t})(s,t_0)\bigm | s\in J, \kappa _g(s,t_0)\pm \kappa _n(s,t_0)\not= 0\right\}.$$ We consider the $AdS$-height function $H:J\times I\times AdS^3{\longrightarrow}{{\mathbb R}}$. Then we have $$\begin{aligned}
&{}& \frac{\partial H}{\partial s}(s,t,\bm{\lambda})=\langle \bm{t}(s,t),\bm{\lambda}\rangle, \\
&{}& \frac{\partial^2 H}{\partial s^2}(s,t,\bm{\lambda})=\langle (\bm{\Gamma}-\kappa _g\bm{b}+\kappa _n\bm{n})(s,t),\bm{\lambda}\rangle, \\
&{}& \frac{\partial^3 H}{\partial s^3}(s,t,\bm{\lambda})=\langle ((1+\kappa _g^2+\kappa _n^2)\bm{t}+(\kappa _n\tau _g-\kappa _g')\bm{b}+(\kappa _n'-\kappa _g\tau_g)\bm{n})(s,t),\bm{\lambda}\rangle.\end{aligned}$$ It follows that the following proposition holds. We write $H_{t_0}(s,\bm{\lambda})=H(s,t_0,\bm{\lambda}).$
[(1)]{} $H_{t_0}(s,\bm{\lambda})=\partial H_{t_0}/\partial s (s,\bm{\lambda})=0$ if and only if there exists $ u\in {{\mathbb R}}$ such that $\bm{\lambda}= \bm{\Gamma} (s,t_0)+u(\bm{b}(s,t_0)\pm \bm{n}(s,t_0))$
[(2)]{} $H_{t_0}(s,\bm{\lambda})=\partial H_{t_0}/\partial s (s,\bm{\lambda})=\partial^2 H_{t_0}/\partial s^2 (s,\bm{\lambda})=0$ if and only if $\kappa _g(s,t_0)\pm \kappa _n(s,t_0)\not= 0$ and $$\bm{\lambda}=\bm{\Gamma} (s,t_0)+\displaystyle{\frac{1}{\kappa _g(s,t_0)\pm \kappa _n(s,t_0)}}(\bm{b}(s,t_0)\pm \bm{n}(s,t_0)).$$
[(3)]{} $H_{t_0}(s,\bm{\lambda})=\partial H_{t_0}/\partial s (s,\bm{\lambda})=\partial^2 H_{t_0}/\partial s^2 (s,\bm{\lambda})=\partial^3 H_{t_0}/\partial s^3 (s,\bm{\lambda})=0$ if and only if $\kappa _g(s,t_0)\pm \kappa _n(s,t_0)\not= 0$, $((\kappa _n\pm\kappa_g)\tau _g\mp (\kappa _n'\pm\kappa _g'))(s_0,t_0)=0$ and $$\bm{\lambda}=\bm{\Gamma} (s,t_0)+\displaystyle{\frac{1}{\kappa _g(s,t_0)\pm \kappa _n(s,t_0)}}(\bm{b}(s,t_0)\pm \bm{n}(s,t_0)).$$
[(4)]{} $H_{t_0}(s,\bm{\lambda})=\partial H_{t_0}/\partial s (s,\bm{\lambda})=\partial^2 H_{t_0}/\partial s^2 (s,\bm{\lambda})=\partial^3 H_{t_0}/\partial s^3 (s,\bm{\lambda})=\partial^4 H_{t_0}/\partial s^4 (s,\bm{\lambda})=0$ if and only if $\kappa _g(s,t_0)\pm \kappa _n(s,t_0)\not= 0$, $((\kappa _n\pm\kappa_g)\tau _g\mp (\kappa _n'\pm\kappa _g'))(s_0,t_0)=((\kappa _n\pm\kappa_g)\tau _g\mp (\kappa _n'\pm\kappa _g'))'(s,t_0)=0$ and $$\bm{\lambda}=\bm{\Gamma} (s,t_0)+\displaystyle{\frac{1}{\kappa _g(s,t_0)\pm \kappa _n(s,t_0)}}(\bm{b}(s,t_0)\pm \bm{n}(s,t_0)).$$
[*Proof.* ]{}
Since we have the pseudo-orthonormal frame $\{\bm{\Gamma}(s,t),{\mbox{\boldmath $b$}}(s,t), {\mbox{\boldmath $n$}}(s,t), {\mbox{\boldmath $t$}}(s,t)\},$ there exist real numbers $\lambda, \mu, \nu \in {{\mathbb R}}$ such that $\bm{\lambda}=\xi\bm{\Gamma}(s,t)+
\lambda {\mbox{\boldmath $b$}}(s,t_0)+\mu {\mbox{\boldmath $n$}}(s,t_0) +\nu {\mbox{\boldmath $t$}}(s,t_0).$
\(1) The condition $\partial H_{t_0}/\partial s (s,\bm{\lambda})=0$ means that $\nu =0$. Moreover, the condition $H_{t_0}(s,\bm{x})=0$ means that $\xi=1$. Since $\langle \bm{\lambda},\bm{\lambda}\rangle =-1,$ we have $\lambda ^2-\mu ^2=0.$ It follows that $$\bm{\lambda}=\bm{\Gamma}(s,t_0)+
\mu ({\mbox{\boldmath $b$}}(s,t_0)\pm {\mbox{\boldmath $n$}}(s,t_0)).$$ We put $u=\mu $. This completes the proof of (1).
\(2) With the assumption that (1) holds, the condition $\partial^2 H_{t_0}/\partial s^2 (s,\bm{\lambda})=0$ means that $$0=\langle \bm{\Gamma}-\kappa _g\bm{b}+\kappa _n\bm{n},\bm{\lambda}\rangle=(\kappa _g\pm\kappa _n)u-1.$$ Therefore, we have $\kappa _g(s,t_0)\pm \kappa _n(s,t_0)\not= 0$ and $$\bm{\lambda}=\bm{\Gamma} (s,t_0)+\displaystyle{\frac{1}{\kappa _g(s,t_0)\pm \kappa _n(s,t_0)}}(\bm{b}(s,t_0)\pm \bm{n}(s,t_0)).$$ This completes the proof of (2).
\(3) By the similar arguments to the above cases, we have the assertion (3).
Moreover, if we calculate the $4$th derivative $\displaystyle{\frac{\partial^4 H_{t_0}}{\partial s^4}}$, then we have the assertion (4). Since those arguments are tedious, we omit the detail here.
$\Box$
According to the above proposition, we introduce an invariant defined by $$\sigma^\pm (s,t)=((\kappa _n\pm\kappa _g)\tau _g\mp(\kappa_n'\pm\kappa_g'))(s,t).$$
Suppose that $\kappa _g(s,t_0)\pm \kappa _n(s,t_0)\not= 0$ and we denote $\tau =+\ \mbox{or}\ -.$ Then the following conditions are equivalent[:]{}
[(1)]{} $\sigma ^\tau (s,t_0)\equiv 0$,
[(2)]{} $\{\bm{\lambda}^\tau_0\}=\mathbb{LF}^\tau_{\mathcal{S}_{t_0}}$,
[(3)]{} There exists $\bm{\lambda}_0\in AdS^3$ such that $\mathcal{S}_{t_0}\subset LC^{AdS}(\bm{\lambda}_0).$
[*Proof.* ]{}
We define $\bm{\ell }_\pm:I{\longrightarrow}AdS^3$ by $$\bm{\ell}_\pm (s)=\bm{\Gamma} (s,t_0)+\displaystyle{\frac{1}{\kappa _g(s,t_0)\pm \kappa _n(s,t_0)}}(\bm{b}(s,t_0)\pm \bm{n}(s,t_0)).$$ Then $\bm{\ell}_\pm (I)=\mathbb{LF}^\pm_{\mathcal{S}_{t_0}}.$ By a straightforward calculation, we have $$\bm{\ell}_\pm '(s)=-\frac{\sigma ^\pm (s,t_0)}{(\kappa _g(s,t_0)\pm \kappa _n(s,t_0))^2}({\mbox{\boldmath $n$}}(s,t_0)\pm {\mbox{\boldmath $b$}}(s,t_0)).$$ Therefore conditions (1) and (2) are equivalent. Suppose that (2) holds. Then we have $\bm{\lambda}_0^\tau=\bm{\ell}_\tau (s)$ for any $s\in I.$ Thus, we have $\bm{\Gamma} (s,t_0)\in \Lambda _{\bm{\lambda}^\tau_0}\cap AdS^3=LC^{AdS}({\bm{\lambda}^\tau_0})$ for any $s\in I$, so that (3) holds. Suppose that (3) holds. Then there exists a point $\bm{\lambda}_0\in AdS^3$ such that $\mathcal{S}_{t_0}\subset LC^{AdS}({\bm{\lambda}_0})=HP(\bm{\lambda}_0,-1)\cap AdS^3.$ This condition is equivalent to the condition that $\langle \bm{\Gamma}(s,t_0),\bm{\lambda}_0\rangle =-1$ at any $s\in I.$ Then $H_{t_0}(s,\bm{\lambda}_0)$ is constantly equal to zero. By the previous calculations, this is equivalent to the condition that $\{\bm{\lambda} _0 \}=\bm{\ell}_\tau (I)$ and (1) holds. This completes the proof.
$\Box$
We also have a classification of singularities of momentary lightlike hypersrufaces.
[(1)]{} The lightlike hypersruface $\mathbb{LS}^\pm_{\mathcal{S}_{t_0}}(I\times\{t_0\}\times {{\mathbb R}})$ at $\bm{\lambda}_0=\bm{\ell}_\pm (s_0)\in \mathbb{LF}^\pm_{\mathcal{S}_{t_0}}$ is local diffeomorphic to the cuspidaledge $\bm{CE}$ if $\sigma ^\pm (s_0,t_0)\not= 0,$
[(1)]{} The lightlike hypersruface $\mathbb{LS}^\pm_{\mathcal{S}_{t_0}}(I\times\{t_0\}\times {{\mathbb R}})$ at $\bm{\lambda}_0=\bm{\ell}_\pm (s_0)\in \mathbb{LF}^\pm_{\mathcal{S}_{t_0}}$ is local diffeomorphic to the swallowtail $\bm{SW}$ if $\sigma ^\pm (s_0,t_0)= 0$ and $\partial \sigma ^\pm/\partial s (s_0,t_0)\not= 0.$
Here, $\bm{CE}=\{(u,v^2,v^3)\in ({{\mathbb R}}^3,0)\ |\ (u,v)\in ({{\mathbb R}}^2,0)\ \}$ and $\bm{SW}=\{(3u^4+vu^2,4u^2+2uv,v)\in ({{\mathbb R}}^3,0)\ |\ (u,v)\in ({{\mathbb R}}^2,0)\ \}.$
In order to prove Theorem 10.3, we use some general results on the singularity theory for unfoldings of function germs. Detailed descriptions are found in the book [@Bru-Gib]. Let $F:({{{\mathbb R}}}\times{{{\mathbb R}}}^r,(s_0,x_0))\rightarrow {{{\mathbb R}}}$ be a function germ. We call $F$ an [*$r$-parameter unfolding*]{} of $f$, where $f(s)=F_{x_0}(s,x_0).$ We say that $f$ has an [*$A_k$-singularity*]{} at $s_0$ if $f^{(p)}(s_0)=0$ for all $1\leq p\leq k$, and $f^{(k+1)}(s_0)\ne 0.$ Let $F$ be an unfolding of $f$ and $f(s)$ has an $A_k$-singularity $(k\geq 1)$ at $s_0.$ We denote the $(k-1)$-jet of the partial derivative $\frac{\partial F}{\partial x_i}$ at $s_0$ by $j^{(k-1)}(\frac{\partial F}{\partial x_i}(s,x_0))(s_0)=\sum_{j=0}^{k-1} \alpha_{ji}(s-s_0)^j$ for $i=1,\dots ,r$. Then $F$ is called an [*$\mathcal{R}$-versal unfolding* ]{} if the $k\times{r}$ matrix of coefficients $(\alpha _{ji})_{j=0,\dots ,k-1;i=1,\dots ,r}$ has rank $k$ $(k\leq {r}).$ We introduce an important set concerning the unfoldings relative to the above notions. A [*$\ell$th-discriminant set*]{} of $F$ is $${\mathcal D}^\ell _F=\left\{x\in {{{\mathbb R}}}^r\Bigm|\exists s \ {\rm with }\ F=
\frac{\partial F}{\partial s}=\cdots =\frac{\partial^\ell F}{\partial s^\ell}= 0 \ {\rm at }\ (s,x)\right\}.$$ For $\ell =1,$ it is simply denoted by $\mathcal{D}_F,$ which is called a [*discriminant set*]{} of $F.$ Then we have the following classification (cf., [@Bru-Gib]).
Let $F:({{{\mathbb R}}}\times{{{\mathbb R}}}^r,(s_0,x_0))\rightarrow {{{\mathbb R}}}$ be an $r$-parameter unfolding of $f(s)$ which has an $A_k$ singularity at $s_0$. Suppose that $F$ is an $\mathcal{R}$-versal unfolding.
[(1)]{} If $k=2$, then ${\mathcal D}_F$ is locally diffeomorphic to $\bm{CE}\times {{{\mathbb R}}}^{r-2}$.
[(2)]{} If $k=3$, then ${\mathcal D}_F$ is locally diffeomorphic to $\bm{SW}\times {{{\mathbb R}}}^{r-2}$.
For the proof of Proposition 10.3, we have the following propositions. Let $\bm{\Gamma} :I\times J{\longrightarrow}W\subset {{\mathbb R}}^3_1$ be a world sheet with $\kappa _n(s,t)\pm \kappa _g(s,t)\not= 0$ and $H:I\times J\times {{\mathbb R}}^3{\longrightarrow}{{\mathbb R}}$ the $AdS$-height function on $\bm{\Gamma}.$ We define $h_{t_0,\bm{\lambda}_0}(s)=H_{t_0}(s,\bm{\lambda}_0)=H(s,t_0,\bm{\lambda}_0)$ and consider that $H_{t_0}$ is a $3$-parameter unfolding of $h_{t_0,\bm{\lambda}_0}.$
If $h_{t_0,\bm{\lambda}_0}$ has an $A_k$-singularity $(k=2,3)$ at $s_0,$ then $H_{t_0}$ is an $\mathcal{R}$-versal unfolding of $h_{t_0,\bm{\lambda}_0}.$
[*Proof.* ]{}
We write that $\bm{\Gamma}(s,t)=(X_0(s,t),X_1(s,t),X_2(s,t))$ and $\bm{\lambda}=(\lambda_{-1},\lambda _0,\lambda _1,\lambda _2).$ Then we have $$H_{t_0}(s,\bm{\lambda}_0)=-X_{-1}(s,t_0)\lambda_{-1} -X_0(s,t_0)\lambda _0+X_1(s,t_0)\lambda _1+X_2(s,t_0)\lambda _2+1.$$ Since $\bm{\lambda}\in AdS^3,$ we have $
-\lambda ^2_{-1}-\lambda ^2_0+\lambda ^2_1+\lambda ^2_2=-1.
$ Then we consider the local coordinates $(\lambda _0,\lambda _1,\lambda_2)$ of $AdS^3$ given by $\lambda _{-1}=\sqrt{1-\lambda^2_0+\lambda ^2_1+\lambda ^2_2}>0.$ Therefore, we have $$\frac{\partial H_{t_0}}{\partial \lambda _{0}}(s,\bm{\lambda}_0)=-X_0(s,t_0)+X_{-1}(s,t_0)\frac{\lambda_0}{\lambda _{-1}},\ \frac{\partial H_{t_0}}{\partial \lambda_i}(s,\bm{\lambda}_0)=X_i(s,t_0)-X_{-1}(s,t_0)\frac{\lambda_i}{\lambda _{-1}},\ i=1,2.$$ Thus we obtain $$\begin{aligned}
j^2\left(\frac{\partial H_{t_0}}{\partial \lambda_0}(s_0,\bm{\lambda}_0)\right)&{}&\!\!\!\!\!\!=-X_0(s_0,t_0)+X_{-1}(s_0,t_0)\frac{\lambda_0}{\lambda _{-1}}
\\
&{}&+\left(-\frac{\partial X_0}{\partial s}(s_0,t_0)+\frac{\partial X_{-1}}{\partial s}(s_0,t_0)\frac{\lambda_0}{\lambda _{-1}}\right)(s-s_0)\\
&{}&+\frac{1}{2}\left(-\frac{\partial^2 X_0}{\partial s^2}(s_0,t_0)+\frac{\partial^2 X_{-1}}{\partial s^2}(s_0,t_0)\frac{\lambda_0}{\lambda _{-1}}\right)(s-s_0)^2,\end{aligned}$$ $$\begin{aligned}
j^2\left(\frac{\partial H_{t_0}}{\partial \lambda_i}(s_0,\bm{\lambda}_0)\right)&{}&\!\!\!\!\!\!=X_i(s_0,t_0)-X_{-1}(s_0,t_0)\frac{\lambda_i}{\lambda _{-1}}\\
&{}&+\left(\frac{\partial X_i}{\partial s}(s_0,t_0)-\frac{\partial X_{-1}}{\partial s}(s_0,t_0)\frac{\lambda_i}{\lambda _{-1}}\right)(s-s_0)\\
&{}&+\frac{1}{2}\left(\frac{\partial^2 X_i}{\partial s^2}-\frac{\partial^2 X_{-1}}{\partial s^2}(s_0,t_0)\frac{\lambda_i}{\lambda _{-1}}\right)(s-s_0)^2,\end{aligned}$$ $i=1,2.$ We consider a matrix $$\mbox{\Large $A$}=\begin{pmatrix} -X_0+X_{-1}\frac{\lambda_0}{\lambda _{-1}} & X_1-X_{-1}\frac{\lambda_0}{\lambda _{-1}} & X_2-X_{-1}\frac{\lambda_0}{\lambda _{-1}}\\
-\frac{\partial X_0}{\partial s}+\frac{\partial X_{-1}}{\partial s}\frac{\lambda_0}{\lambda _{-1}} & \frac{\partial X_1}{\partial s}-\frac{\partial X_{-1}}{\partial s}\frac{\lambda_0}{\lambda _{-1}} &
\frac{\partial X_2}{\partial s}-\frac{\partial X_{-1}}{\partial s}\frac{\lambda_0}{\lambda _{-1}} \\
-\frac{\partial^2 X_0}{\partial s^2}+\frac{\partial^2 X_{-1}}{\partial s^2}\frac{\lambda_0}{\lambda _{-1}} & \frac{\partial^2 X_1}{\partial s^2}-\frac{\partial^2 X_{-1}}{\partial s^2}\frac{\lambda_1}{\lambda _{-1}} &
\frac{\partial^2 X_2}{\partial s^2}-\frac{\partial^2 X_{-1}}{\partial s^2}\frac{\lambda_2}{\lambda _{-1}}
\end{pmatrix}$$ at $(s_0,t_0).$ Then we have $$\det \mbox{\Large $A$}=\frac{1}{\lambda_{-1}}\left\langle \bm{\lambda}_0,\bm{\Gamma}(s_0,t_0)\wedge \frac{\partial \bm{\Gamma}}{\partial s}(s_0,t_0)\wedge
\frac{\partial^2 \bm{\Gamma}}{\partial s^2}(s_0,t_0)\right\rangle$$ We also have $$\frac{\partial \bm{\Gamma}}{\partial s}(s_0,t_0)=\bm{t}(s_0,t_0),\
\frac{\partial^2 \bm{\Gamma}}{\partial s^2}(s_0,t_0)=-\kappa _g(s_0,t_0)\bm{b}(s_0,t_0)+\kappa_n(s_0,t_0)\bm{n}(s_0,t_0).$$ By Proposition 10.1, we have $\bm{\lambda}_0=(\bm{\Gamma}+(\bm{b}\pm\bm{n})/(\kappa_g\pm\kappa_n))(s_0,t_0)$, so that $$\det \mbox{\Large $A$}=\frac{1}{\lambda_{-1}}\langle \bm{\lambda}_0,\kappa _g(s_0,t_0)\bm{n}(s_0,t_0)-\kappa _n\bm{b}(s_0,t_0)\rangle
=\pm\frac{1}{\lambda_{-1}}\not=0.$$ This means that $H_{t_0}$ is an $\mathcal{R}$-versal unfolding of $h_{t_0,\bm{\lambda}_0}.$
For other local coordinates of $AdS^3,$ we have the similar calculations to the above case.
$\Box$
[*Proof of Theorem 10.3.*]{} By (1) of Proposition 10.1, the discriminant set $D_{H_{t_0}}$ of the $AdS$-height function on $\mathcal{S}_{t_0}$ is the lightlike hypersruface along $\mathcal{S}_{t_0}.$ It also follows (3) and (4) of Proposition 10.1 that $h_{t_0,\bm{\lambda}_0}$ has an $A_2$-singularity (respectively, $A_3$-singularity) at $s_0$ if $\sigma ^\pm (s_0,t_0)\not= 0$ (respectively, $\sigma ^\pm (s_0,t_0)= 0$ and $(\sigma ^\pm)' (s_0,t_0)\not= 0$). By Proposition 10.5, $H_{t_0}$ is an $\mathcal{R}$-versal unfolding of $h_{t_0,\bm{\lambda}_0}$ for each case. Then we can apply the classification theorem (Theorem 10.4) to our situation. This completes the proof.
$\Box$
We remark that $D^2_{H_{t_0}}$ is the lightlike focal curve $\mathbb{LF}^\pm_{\mathcal{S}_{t_0}}$. Since the critical value set of the swallow tail is locally diffeomorphic to a [*$(2,3,4)$-cusp*]{} which is defined by $C=\{(t^2,t^3,t^4)\ |\ t\in {{\mathbb R}}\},$ we have the following corollary.
The lightlike focal curve $\mathbb{LF}^\pm_{\mathcal{S}_{t_0}}$ is locally diffeomorphic to a line if $\sigma ^\pm (s_0,t_0)\not= 0$. It is locally diffeomorphic to the $(2,3,4)$-cusp if $\sigma ^\pm (s_0,t_0)= 0$ and $(\sigma ^\pm)' (s_0,t_0)\not= 0$.
On the other hand, we now classify $S.P^+$-Legendrian stable graph-like Legendrian unfoldings $\mathscr{L}_H(\Sigma _*(H))$ by $S.P^+$-Legendrian equivalence. By Theorems 5.5 and 5.6, it is enough to classify $\overline{f}$ by $S.P$-$\mathcal{K}$-equivalence under the condition that $$\dim _{{{\mathbb R}}} \frac{{\cal E}_{1+1}}{\left\langle \frac{\partial \overline{f}}{\partial q},\overline{f} \right\rangle _{{\cal E}_{1+1}}+
\left\langle \frac{\partial \overline{f}}{\partial t} \right\rangle _{{{\mathbb R}}}} \leq 3.$$ In [@Izudoc; @Izu95] we have the following proposition.
With the above condition, $\overline{f}:({{\mathbb R}}\times{{\mathbb R}},0){\longrightarrow}({{\mathbb R}},0)$ with $\partial\overline{f}/\partial t(0)\not= 0$ is $S.P$-$\mathcal{K}$-equivalent to one of the following germs[:]{}
[(1)]{} $q,$
[(2)]{} $\pm t\pm q^2,$
[(3)]{} $\pm t + q^3,$
[(4)]{} $\pm t \pm q^4,$
[(5)]{} $\pm t + q^5.$
The infinitesimally $S.P^+$-$\mathcal{K}$-versal unfolding $\mathcal{F}:({{\mathbb R}}\times ({{\mathbb R}}^3\times {{\mathbb R}}),0){\longrightarrow}({{\mathbb R}},0)$ of each germ in the above list is given as follows (cf. [@Izu95 Theorem 4.2]):
\(1) $q$
\(2) $\pm t\pm q^2,$
\(3) $\pm t + q^3+x_0q,$
\(4) $\pm t\pm q^4+x_0q+x_1q^2,$
\(5) $\pm t+q^5+x_0q+x_1q^2+x_2q^3.$
By Theorem 5.6, we have the following classification.
Let $(W,\mathcal{S})$ be a world sheet in $AdS^3$ parametrized by a timelike embedding $\bm{\Gamma} :J\times I{\longrightarrow}AdS^3$ and $H:J\times I\times AdS^3{\longrightarrow}{{\mathbb R}}$ be the $AdS$-height squared function of $(W,\mathcal{S}).$ Suppose that the corresponding graph-like Legendrian unfolding $\mathscr{L}_H(\Sigma _*(H))\subset J^1(AdS^3,I)$ is $S.P^+$-Legendrian stable. Then the germ of the image of the unfolded lightlike hypersrufaces $\mathbb{LH}_W$ at any point is $S.P^+$-diffeomorphic to one of the following set germs in $({{\mathbb R}}^3\times {{\mathbb R}},0)$[:]{}
[(1)]{} $\{(u,v,w),0)\ |\ (u,v,w)\in ({{\mathbb R}}^3,0)\ \},$
[(2)]{} $\{(-u^2,v,w),\pm 2u^3)\ |\ (u,v,w)\in ({{\mathbb R}}^3,0)\ \},$
[(3)]{} $\{(\mp 4u^3-2vu,v,w),3u^3\pm vu^2)\ |\ (u,v,w)\in ({{\mathbb R}}^3,0)\ \},$
[(4)]{} $\{((5u^4+2vu+3wu^2,v,w),\pm(4u^4+vu^2+2wu^3))\ |\ (u,v,w)\in ({{\mathbb R}}^3,0)\ \}.$
[*Proof.* ]{}
For any $(s_0,t_0,\bm{\lambda}_0)\in J\times I\times AdS^3,$ the germ of $\mathscr{L}_H(\Sigma _*(GH)\subset J^1(AdS^3,I)$ at $\bm{z}_0=\mathscr{L}_H(s_0,t_0,\bm{\lambda}_0)$ is $S.P^+$-Legendrian stable. It follows that the germ of $h_{\bm{\lambda}_0}$ at $(s_0,t_0)$ is $S.P$-$\mathcal{K}$-equivalent to one of the germs in the list of Proposition 10.7. By Theorem 5.6, the graph-like Legendrian unfolding $\mathscr{L}_H(\Sigma _*(H))$ is $S.P^+$-Legendrian equivalent to the graph-like Legendrian unfolding $\mathscr{L}_{\mathcal{F}}(\Sigma_*(\mathcal{F}))$ where $\mathcal{F}$ is the infinitesimally $S.P$-$\mathcal{K}$-versal unfolding of one of the germs in the list of Proposition 10.7. It is also equivalent to the condition that the germ of the graph-like big front $W(\mathscr{L}_{\mathcal{F}}(\Sigma_*(\mathcal{F})))$ is $S.P^+$-diffeomorphic to the corresponding graph-like big front of one of the normal forms. For each normal form, we can obtain the graph-like big front. We only show that (5) in Proposition 10.7. In this case we consider $\mathcal{F}(q,x_0,x_1,x_2,t)=\pm t+q^5+x_0q+x_1q^2+x_2q^3.$ Then we have $$\frac{\partial \mathcal{F}}{\partial q}=5q^4+x_0+2x_1q+3x_1q^2,$$ so that the condition $\mathcal{F}=\partial \mathcal{F}/\partial q=0$ is equivalent to the condition that $$x_0=-(5q^4+x_0+2x_1q+3x_1q^2),\ t_0=\pm(4q^5+x_1q^2+2x_2q^3).$$ If we put $u=q, v=x_0,w=x_1,$ then we have $$W(\mathscr{L}_{\mathcal{F}}(\Sigma_*(\mathcal{F})))=\{((-(5u^4+2vu+3wu^2),v,w),\pm(4u^4+vu^2+2wu^3))| (u,v,w)\in ({{\mathbb R}}^3,0)\}.$$ It is $S.P^+$-diffeomorphic to the set germ of (4). We have similar calculations for other cases. We only remark here that we obtain the germ of (1) for both the germs of (1) and (2) in Proposition 10.7. Since $W(\mathscr{L}_{\mathcal{H}}(\Sigma_*(\mathcal{H})))=\mathbb{LH}_W,$ this completes the proof.
$\Box$
As a corollary, we have a local classification of BR-caustics in this case.
With the same assumption for the world sheet $(W,\mathcal{S})$ as Theorem 10.8, the BR-caustic $C(W,\mathcal{S})$ of $(W,\mathcal{S})$ at a singular point is locally diffeomorphic to the cuspidaledge $\bm{CE}$ or the swallowtail $\bm{SW}$.
[*Proof.* ]{}
The BR-caustic $C(W,\mathcal{S})$ of $(W,\mathcal{S})$ is the set of the critical values of $\pi_1\circ \overline{\pi}|_{\mathscr{L}_H(\Sigma _*(H))}.$ Therefore, it is enough to calculate the set of critical values of $\pi_1\circ \overline{\pi}|_{\mathscr{L}_{\mathcal{F}}(\Sigma_*(\mathcal{F}))}$ for each normal form $\mathcal{F}$ in Proposition 10.7. For (5) in Proposition 10.7, by the proof of Theorem 10.8 we have $$\Sigma_*(\mathcal{F})=\{(u,5u^4+2vu+3wu^2,v,w)\in ({{\mathbb R}}\times ({{\mathbb R}}^3\times {{\mathbb R}}),0)|(u,v,w)\in ({{\mathbb R}}^3,0)\}.$$ It follows that $$\pi_1\circ \overline{\pi}\circ \mathscr{L}_{\mathcal{F}}(u,5u^4+2vu+3wu^2,v,w)=(5u^4+2vu+3wu^2,v,w).$$ Then the Jacobi matrix of $f(u,v,w)=(5u^4+2vu+3wu^2,v,w)$ is $$J_f=\begin{pmatrix}
20u^3+2v+6wu & 0 & 0 \cr
2u & 1 & 0 \cr
3u^2 & 0 & 1 \cr
\end{pmatrix},$$ so that the set of critical values of $f$ is given by $$\{(-(15u^4+3wu^2),-10u^3-3wu,w)\in ({{\mathbb R}}^3,0)\ |\ (u,w)\in ({{\mathbb R}}^2,0)\}.$$ For a linear isomorphism $\psi:({{\mathbb R}}^3,0){\longrightarrow}{{\mathbb R}}^3,0)$ defined by $
\psi (x_0,x_1,x_2)=(-\frac{1}{5}x_0,-\frac{2}{5}x_1,\frac{3}{5}x_2),
$ we have $\psi(-(15u^4+3wu^2),-10u^3-3uw,w)=(3u^4+\frac{3}{5}wu^2,4u^3+\frac{6}{5}wu,\frac{3}{5}w).$ If we put $U=u, V=\frac{3}{5}w,$ then we have $(3U^4+VU^2,4U^3+2VU,V),$ which is the parametrization of $\bm{SW}.$ By the arguments similar to the above, we can show that the set of critical values of $\pi_1\circ \overline{\pi}|_{\mathscr{L}_{\mathcal{F}}(\Sigma_*(\mathcal{F}))}$ is a regular surface for (3) and is diffeomorphic to $\bm{CE}$ for (4) in Proposition 10.7, respectively. This completes the proof.
$\Box$
[Since a world sheet $(W,\mathcal{S})$ is a timelike surface in $AdS^3,$ we can define the [*$AdS$-evolute*]{} of $(W,\mathcal{S})$ by $$Ev^{AdS}_{(W,\mathcal{S})}=\displaystyle{\bigcup _{i=1}^2\left\{\frac{\pm 1}{\sqrt{\kappa ^2_i(u,t)-1}}(\kappa _i(u,t)\bm{X}(u,t)+\bm{n}^S(u,t))\ |\ (u,t)\in U\times I,\kappa ^2_i(u,t)>1\ \right\}},$$ where $\kappa _i(s,t)$ $(i=1,2)$ are the principal curvatures of $W$ at $p=\bm{X}(u,t)$ with respect to $\bm{n}^S$ (cf. [@timeads]). The $AdS$-evolute of a timelike surface has singularities in general. Actually, it is a caustic in the the theory of Lagrangian singularities. Similar to the notion of evolutes of surfaces in Euclidean space ${{\mathbb R}}^3$ (cf. [@Porteous]), the corank two singularities of the $AdS$-evolute appear at the umbilical points (i.e. $\kappa _1(u,t)=\kappa _2(u,t)$). The singularities of the $AdS$-evolute of a generic surface in $AdS^3$ are classified into $\bm{CE}$, $\bm{SW},$ $\bm{PY}$ or $\bm{PU}$, where $\bm{PY}=\{(u^2-v^2+2uv,-2uv+2uw,w)|w^2=u^2+v^2\}$ is the [*pyramid*]{} and $\bm{PU}=\{(3u^2+wv,3v^2+wu,w)|w^2=36uv\}$ is the [*purse*]{}. The pyramid and the purse of the $AdS$-evolute correspond to the umbilical points of the timelike surface in $AdS^3.$ So the singularities of BR-caustics of world sheets are different from those of the $AdS$-evolutes of surfaces. Since the singularities of BR-caustics are only corank one singularities, the pyramid and the purse never appeared in general. Moreover, the normal geodesic of a timelike surface is a spacelike curve, so that it is not a ray in the sense of the relativity theory. Therefore, the $AdS$-evolute of a timelike surface in anti-de Sitter space-time is not a caustic in the sense of physics. ]{}
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Shyuichi Izumiya, Department of Mathematics, Hokkaido University, Sapporo 060-0810,Japan
e-mail:[izumiya@math.sci.hokudai.ac.jp]{}
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abstract: 'A rare coincidence of scales in standard particle physics is needed to explain why $\Lambda$ or the negative pressure of cosmological dark energy (DE) coincides with the positive pressure $P_0$ of random motion of dark matter (DM) in bright galaxies. Recently Zlosnik et al. (2007) propose to modify the Einsteinian curvature by adding a non-linear pressure from a medium flowing with a four-velocity vector field $U^\mu$. We propose to check whether a smooth extension of GR with a simple kinetic Lagrangian of $U^\mu$ can be constructed, and whether the pressure can bend space-time sufficiently to replace the roles of DE, Cold DM and heavy neutrinos in explaining anomalous accelerations at all scales. As a specific proof of concept we find a Vector-for-$\Lambda$ model (${\mathbf V\!\Lambda}$-model) and its variants. With essentially [*no free parameters*]{}, these appear broadly consistent with the solar system, gravitational potentials in dwarf spiral galaxies and the bullet cluster of galaxies, early universe with inflation, structure formation and BBN, and late acceleration with a 1:3 ratio of DM:DE.'
author:
- |
HongSheng Zhao\
Scottish University Physics Alliance, University of St Andrews, KY16 9SS, UK
title: 'Coincidences of Dark Energy with Dark Matter – Clues for a Simple Alternative?'
---
The incompleteness of standard physics and Einstein’s General Relativity (GR) is evident from the smallness of the cosmological constant $\Lambda$ or the vacuum energy density ${\Lambda c^2 \over 8 \pi G} \sim (0.001{\rm eV})^4$, compared to the expected quantum pressure ${c^5 m_P^4 \over \hbar^3} \sim (10^{28}{\rm eV})^4$ at scales of the Planck mass $m_P=\sqrt{\hbar c \over G}$. Current speculations of the new physics of $\Lambda$ are as free as analogous speculations of the Pioneer Anomaly (Turyshev, Nieto, Anderson 2006), both represent acceleration discrepancies of order $\sim 7a_0$, driven by unidentified (likely unrelated) pressures $\sim 72P_0$, where $a_0 \equiv 1.2\AA/\sec^{2}$, and $P_0 \equiv {a_0^2 \over 8 \pi G}$ are scales of acceleration and pressure. On intermediate scales, galaxy clusters and spiral galaxies often reveal a discrepant acceleration of order $(0.1-2)a_0$. GR, if sourced primarily by baryons and photons with negligible mass density of neutrinos and other particles in the Standard Model or variations, appears an adequate and beautiful theory in the inner solar system, but appears increasingly inadequate in accounting for astronomical observations as we move up in scales from 100AU to 1 kpc to 1Gpc. The universe made of known material of positive pressure should show a de-accelerating expansion as an open universe, but instead it is turning into accelerating now, evidenced by much dimmer supervonae detected at redshift unity. A standard remedy to restore harmony with GR and fit successfully large scale observations (Spergel et al. 2006 and references therein) is to introduce a “dark sector”, in which two exotic components dominate the matter-energy budget of the Universe at the redshift $z$ with a split of $\Omega_{DE}:\Omega_{DM}=3:(1+z)^3$ approximately: a Dark Energy (DE) as a negative pressure and nearly homogeneous field described by unknown physics, and a Cold Dark Matter (DM) as a colissionless and pressureless fluid motivated by perhaps the MSSM physics. However, anticipating several new particles from the LHC, the success of this Concordance Model still gives little clue to the physics of governing the present $1:3$ ratio of its constituents. This ratio is widely considered improbable, because standard particle physics expects a ratio $1:10^{120}$. Here we speculate whether the $3:(1+z)^3$ ratio could come from a coincidence of scales of $a_0 \equiv 1.2\AA/\sec^{2}$ with the cosmological baryon energy density $\rho_b c^2 \sim 3.5 \times {(1+z)^3P_0}$.
[*A deeper link of DM and DE*]{}: It is curious that the distribution of DM in dwarf galaxies is extremely ordered, something that the cuspy $\Lambda$CDM halos are still struggling to explain even with the maximum baryonic feedback (Gnedin & Zhao 2002). For example, on galaxy scales the Newtonian gravity of DM $g_{DM}={V_c^2 \over R}-g_B$ and Newtonian gravity of baryons $g_B={GM_B \over R^2}$ have a tight correlation: \[ggb\] (g\^2/g\_B)\^[n]{} -g\^n a\_0\^n, g g\_[DM]{}+g\_B, where $n \ge 1$ (Zhao & Famaey 2006). This rule holds approximately at [*all radii*]{} R of all spiral galaxies of baryonic mass $M_B(R)$ and circular velocity $V_c(R)$ within the uncertainty of the stellar mass-to-light and object distance. For low surface brightness galaxies or at the very outer edge of bright spirals, the gravity $g$ is weaker than $a_0$, our empirical formula predicts $g^2/g_B = (V_c^2/R)^2/(GM_B/R^2) = V_c^4/(GM_B) \sim a_0$, which is essentially the normalisation of (baryonic) Tully-Fisher relation (McGaugh 2005). Bulges and central part of elliptical galaxies are dominated by baryons inside a transition radius where the baryon and DM contribute about equally to the rotation curve, Eq.(\[ggb\]) predicts $g_{DM}=g_B=a_0/2$; we can define a DM pressure $P_0 \equiv a_0 \times {a_0 \over 8\pi G}$ at transition by multiplying the local gravity $(g_{DM}+g_B)=a_0$ with the DM column density above this radius ${g_{DM} \over 4\pi G}=a_0/(8 \pi G)$. This scale $P_0$ appears on larger scales too. All X-ray clusters have gas pressure and the DM random energy density comparable to $P_0$. The amplitude of the scale $a_0$ appears in the $r^{-1}$ cusp of CDM halos too (Xu, Wu, Zhao 2007, Kaplinhat & Turner 2001). These can be understood since the last scattering shell at $z=1000$ has a thickness $2 L \sim 10$Mpc and contains typical potential wells of depth $c^2/N \sim (1000\kms)^2$ due to inflation, where $N \equiv 10^5$, hence the typical internal acceleration is $c^2/N/L \sim 0.2 a_0$. Also a DM sphere of radius 5Mpc turning non-linear now would fall in with an acceleration $\sim 200 \times H_0^2 \times 5Mpc\sim a_0$. While correlations of baryon and DM can generally be understood in a galaxy formation theory where DM and baryons interact, the unlimited freedoms of dark particles means a good spread of its concentration, hence the correlation would have substantial history-dependent variance from galaxies to galaxies and radii to radii. For example, DM is unexpected in Tidal Dwarf Galaxies, is observed for its $a_0$ acceleration (Gentile et al. 2007). The tightness of such hidden regulations on DM at all radii for all galaxies is anomalous, at least challenging in the standard framework.
It is even more curious that DM in various systems and DE are tuned to [*a common scale $P_0$, hence requiring a coincidence in two dark sectors. These empirical facts are unlikely random coincidences*]{} of the fundamental parameters of the dark sectors. Since all these anomalies are based on the gravitational acceleration of ordinary matter in GR, one wonders if the dark sectors are not just a sign of an overlooked possible field in the gravitational sector.
Continue along Zhao (2006), here we propose to investigate whether the roles of both DM and DE could be replaced by a vector field in a modified metric theory. This follows from two long lines of investigations pursued by Kostelecky, Jacobson, Lim and others on consequences of symmetry-breaking in string theory, and by Milgrom, Bekenstein, Sanders, Skordis and others driven by astronomical needs. These two independent lines were first merged by the pioneering work of Zlosnik et al.(2007). [*Existence of an explicit Lagrangian*]{} satisfying main constraints for the solar system, galaxy rotation curves and cosmological concordance ratio remains to be demonstrated.
[*Warming up to vector field*]{}: In Einstein’s theory of gravity, the slightly bent metrics for a galaxy in an uniform expanding background set by the flat FRW cosmology is given by $$\label{metric}
g_{\mu\nu} dx^{\mu}dx^{\nu}
=-(1+{2\Phi \over c^2}) d(ct)^2 + (1-{2\Psi \over c^2}) a(t)^2 dl^2$$ where $dl^2=\left(dx^2 +dy^2 + dz^2 \right)$ is the Euclidian distance in cartesian coordinates. In the collapsed region of galaxies, the metric is quasi-static with the potential $\Phi(t,x,y,z)=\Psi(t,x,y,z)$ due to DM plus baryon, which all follow the geodesics of $g_{\mu\nu}$.
Modified gravity theories are often inspired to preserve the Weak Equivalence Principle, i.e., particles or small objects still go on geodesics of above physical metric independent of their chemical composition. Unlike in Einstein’s theory, the Strong Equivalence Principle and CPT can be violated by, e.g., creating a preferred frame using a vector field. The Einstein-Aether theory of Jacobson & Mattingly (2001) is such a simple construction, where a unit vector field $U^{\mu}$ is designed to couple only to the metric but not matter directly. It has a kinetic Lagrangian with linear superposition of quadratic co-variant derivatives $\nabla (c^2U) \nabla (c^2U)$, where $c^2U^{\mu}$ is constrained to be a time-like four-momentum vector per unit mass by $-g_{\mu\nu}U^{\mu} U^\nu= 1.$ The norm condition means the vector field introduces up to 3 new degrees of freedom; e.g., a perturbation in the FRW metric (Eq.\[metric\]) has $c^2U_{\mu} \equiv g_{\mu\nu}c^2 U^{\nu} \approx (c^2+\Phi,{A_x \over c},{A_y \over c},{A_z \over c})$, containing a four-vector made of an electric-like potential $\Phi$ and three new magnetic-like potentials. But for spin-0 mode perturbations with a wavenumber vector ${\mathbf k}$, we can approximate $U_{\mu} - (1,{\mathbf 0}) \approx ({\Phi \over c^2},{{\mathbf k} V \over c})$, which contains just one degree of freedom, i.e., the flow potential $V(t,x,y,z)$. We expect an initial fluctuation of $c|{\mathbf k}|V \sim |\Phi| \sim c^2 N^{-1} \equiv 10^{-5}c^2$ can be sourced by a standard inflaton; the vector field tracks the spectrum of metric perturbation (Lim 2004).
Most recently Zlosnik et al. (2007) suggested to replace the linear $\lambda \nabla U \nabla U$ with a non-linear kinetic Lagrangian $ F(\lambda \nabla U\nabla U)$ to extend Jacobson’s framework. They showed this class of non-linear models is promising to produce the DE effect in cosmology and the DM-like effect in the weak field limit. Here we continue along the lines of the pioneering authors, but aim for a single Lagrangian with parameters in good match with basic observations of a range of scales.
[*A Simple Lagrangian for $\Lambda$*]{}: The difficulty of writing down a specific Lagrangian is that there are infinite ways to form pressure-like terms quadratic to co-variant derivatives of the vector field. Simplicity is the guide when choosing gravity since GR plus $\Lambda$CDM largely works. Let’s start with forming two pressure terms for any four-momentum-like field $A^{\mu}$ with a positive norm $m c^2 \equiv \sqrt{-g_{\alpha\beta}A^{\alpha}A^{\beta}}$ by $$\label{Z}
8\pi G {\cal J}(A) \equiv
{1 \over 3} \left({\nabla_\alpha A^\alpha \over m}\right)^2,~
8\pi G {\cal K}(A) \equiv {\nabla_\parallel A^\alpha \over m} {\nabla_\parallel A_\alpha \over m}$$ where the RHSs are co-variant with dimension of acceleration squared, and $\nabla_\parallel=A^\alpha \nabla_\alpha$ or $\nabla_\alpha$ stands for the co-variant derivative with space-time coordinates along the direction of the vector $A$ or the dummy index $\alpha$ respectively. From these we can generate two simpler pressure terms $K$ and $J$ of the unit vector field $U^\alpha$ by $$\label{KJgalaxy}
\begin{array}{cllcll}
J \equiv {\cal J}(U) &\sim&0, & K \equiv {\cal K}(U) &\sim&{|\nabla \Phi|^2 \over 8\pi G} ~\mbox{\rm in galaxies}\\
&\sim&{3 c^2H^2 \over 8\pi G}, & &\sim&0 ~\mbox{\rm in flat universe}
\end{array}\label{KJcosmo}$$ where the approximations hold for $U^\alpha$ with negligible spatial components and nearly flat metric (Eq.\[metric\]). Note the $J$ and $K$ are constructed so that we can control time-like Hubble expansion and space-like galaxy dynamics [*separately*]{}.[^1] The $K$-term, with a characteristic pressure scale ${a_0^2 \over 8\pi G} =P_0$ in galaxies, is the key for our model. The $J$-term, meaning critical density, has a characteristic scale $N^2P_0 \sim 10^{10} P_0$: at the epoch of recombination $z=1000$ when baryons, neutrinos, and photons contribute $\sim (8,3,5) \times 10^9 P_0$ respectively to the term $J={3c^2 H^2 \over 8 \pi G}$; so the epochs of equality and recombination nearly coincide.
Now we are ready to construct our total action $S=\int d^4x |-g|^{1\over 2} {\cal L}$ in physical coordinates, where the Langrangian density $$\label{action}
{\cal L} = {R \over 16 \pi G} + L_m + L_J + L_K + (U_\nu U^\nu \!\! +1) L^m, ~$$ where $R$ is the Ricci scalar, $L_m$ is the ordinary matter Lagrangian. For the vector field part, $L^m$ is the Lagrangian multiplier for the unit norm and we propose the new Lagrangian $$\label{LJ}\label{LK}
L_J = \int_{0}^{J} \!\! \lambda_N\left(\sqrt{|J| \over P_0}\right) d J, ~
L_K = \int_{\infty}^{K} \!\! \lambda_n\left(\sqrt{|K| \over P_0}\right) d K,$$ where the non-negative continuous functions $\lambda_i(x) = \left[0, \lambda(x)-\lambda(N)\right]_{\rm max},
~\lambda(x)=\left(1+{x \over i} \right)^{-n} $, where the subscript $i=$ either $n$ or $N$. Incidentally, $n=0$ gives GR. The cutoffs (e.g., with $n=\pm 1$) guarantee a bounded Hamiltonian with kinetic terms $L_K$ and $L_J$ always bounded between $\pm N^2P_0$ (e.g., in a lab near Earth $K \sim (10^{13}-10^{22}) P_0 > N^2P_0$, so $L_K=0$). The condition at the tidal boundary $K=J=0$ is well-behaved too (cf. eq.44-48 of Famaey et al. 2007 on Cauchy problem). Note $1-{dL_K \over dK}>\mu_{min} \equiv (1+N/n)^{-n} \sim 10^{-15}$ and $1 - {dL_J \over dJ} > \mu_B \equiv (1+N/N)^{-n} \sim 2^{-3}$.
Taking variations of the action with respect to the metric and the vector field, we can derive the modified Einstein’s equation (EE) and the dynamical equation for the vector field. The expressions are generally tedious (Halle 2007), but the results simplifies in the perturbation and matter-dominated regime that interest us. As anticipated in Lim (2004) the $ij$-cross-term of EE yields $\Psi-\Phi=0$ for all our models, which means incidentally twice as much deflection for light rays as in Newtonian. As anticipated in Dodelson & Liguri (2007), the $ti$-term EE can be casted into that of an unstable harmonic oscillator equation with a negative string constant $\dot{\dot{V}} + b_1 H \dot{V} - (1-\mu_B) b_2 H^2 V = S(\Phi, \dot{\Psi})$ if $(1-\mu_B)>0$, so we expect that $H V$ tracks $\Phi$. The tt-term of the EE takes the form $$\label{tt}
8 \pi G \rho = 3\mu_B H^2 + 2 \grad \cdot [(1-\lambda_n) \grad \Phi] - \Lambda_0 - Q(\dot{\Phi},\dot{V},V)$$ where we approximated $1-\lambda_N(x) \sim 2^{-n}=\mu_B$ as a constant in matter dominated regime where $J < N^2 P_0$ and the $Q$-term is zero for static galaxies and uniform FRW flat cosmology. So the tt-equation of Einstein reduces to the simple form $$\begin{aligned}
\label{poisson}
4 \pi G \rho &=& \grad^2\Phi - \grad \cdot \left[\lambda_n\left({|\grad \Phi|\over a_0}\right)\grad \Phi\right],~\mbox{\rm in galaxies} \\
{8 \pi G \bar{\rho} \over 3\mu_B} &=& H^2 - {\Lambda_0 \over 3\mu_B}, ~\mbox{\rm in matter-dominated FRW}
\label{hubble}\end{aligned}$$ Here the pressure from the vector field creates new sources for the curvature. The term ${\grad(\lambda_n(x)\grad \Phi) \over 4\pi G}$ in the Poisson equation acts as if adding DM for quasi-static galaxies. A cosmological constant in the Hubble equation is created by $$\label{cosmocst}
{\Lambda_0 c^2 \over 8\pi G} = -\!\!\int_{\infty}^{0} \!\!\! \lambda_n(x) d(P_0x^2) \approx {2(nP_0)^2 \over (n-1)(n-2)}$$ [*For binary stars and the solar system*]{}, $4 \pi G \rho - \grad^2\Phi \approx 0$ is true because the gravity at distances 0.3AU to 30AU from a Sun-like star is much greater than the maximum vector field gradient strength $Na_0$, so ${dL_K \over dK}=0$; in fact, $|\grad \Phi| \approx {G M_\odot \over r^2} \sim (10^9-10^5) a_0$, and the typical anomalous acceleration is $N a_0 \mu_{min} \sim 10^{-10}a_0$, well-below the current detection limit of $10^{-4}a_0$ (Soreno & Jezter 2006). This might explain why most tests of non-GR effects around binary pulsars, black holes and in the solar system yield negative results; Pluto at 40 AU and the Pioneer satellites at 100 AU might show interesting effects. Extrapolating the analysis of Foster & Jacobson (2006), we expect GR-like PPN parameters and gravitational wave speeds in the inner solar system.
[*Near the edges of galaxies*]{}, we recover the non-relativistic theory of Bekenstein & Milgrom (1984) with a function \[mu\] (x) 1-\_n(x) \~\_[min]{} + x, . Note that $\mu(x) \rightarrow x$ hence rotation curves are asymptotically flat except for a negligible correction $\mu_{min} \sim 10^{-15}$. In the intermediate regime $x=1$ our function with $1-\lambda_n(x) \sim (0.55-0.6)$ for $n=2-5$ respectively. Eq.(\[ggb\]) argues that galaxy rotation curves prefer a relatively sharper transition than $\mu(x)=x/(1+x)=0.5$ at $x=1$ (Famaey, Gentile, Bruneton, Zhao 2007) where we can identify $g_B/(g_{DM}+g_B) = \mu(x)$. So our model should fit observed rotation curves.
[*For the Hubble expansion:*]{} the vector field creates cosmological constant-like term ${\Lambda_0 c^2\over 8\pi G} \approx 9 P_0 $ below the zero-point of the energy density in the solar system because the zero point of our Lagrangian (Eq.\[LK\]) is chosen at $N^2 P_0 \le K <+\infty$. During matter domination, the contribution of matter $8 \pi G \rho$ and $\Lambda_0$ to the Hubble expansion $H^2$ (Eq.\[hubble\]) is further scaled-up because the effective Gravitational Constant $G_{eff}=G/\mu_B = 2^n G \ge G$ with GR being the $n=0$ special case. [^2] Coming back to the original issue of the $3:1$ ratio of matter density to our cosmological constant, Eq.(\[hubble\]) predicts that ${\Lambda_0 c^2 \over 8\pi G \mu_B}: {\bar{\rho}_b c^2 \over \mu_B}
\sim {9 P_0 \over \mu_B}: {4 (1+z)^3 P_0 \over \mu_B}$, which is close to the desired $3:(1+z)^3$ ratio. Adding neutrinos makes the explanation slightly poorer. So the DE scale is traced back to a separate coincidence of scale, i.e., the present baryon energy density $\bar{\rho}_b c^2 \sim 4 P_0$, where $P_0$ contains a scale $a_0$ for the anomalous accelerations on galactic scale. Our model predicts that [*DE is due to a constant of vacuum*]{}, preset by the modification parameter $n$ of the gravity; $n=0$ gives GR.
In our model, the effective DM (the dog) follows the baryons (the tail) throughout the universal $(1+z)^3$ expansion with a ratio set by $n$. To fit the $\Lambda$CDM-like expansion exactly, we note the Hubble equation for a flat FRW cosmology with vector field and standard mix of baryons, neutrinos and photons ${\Omega_b h^2 \over 0.02} \approx {\Omega_{\nu}h^2 \over 0.002}{0.07{\rm eV} \over m_\nu} \approx {\Omega_{ph} h^2 \over 0.000025} \sim 1$ yields at the present epoch \[Omega\] [\_b + \_+ \_[ph]{} \_B]{} = 1-[\_0 3\_B H\_0\^2]{} = \_[m]{}\^[CDM]{} The 2nd equality fixes $\mu_B^{-1}=2^{n}=(8-8.4)$ if we adopt $a_0/c \approx H_0/6 \approx 12$km/s/Mpc and $\Omega_{m}^{\Lambda CDM}=(0.25-0.3)$. The 1st equality would predict an uncertain but very small neutrino mass $m_{\nu} \sim \pm 0.3$eV.
The BBN also anchors any modification to GR. In the radiation-dominated era $|J| ={3c^2H^2 \over 8 \pi G} \gg N^2 P_0$, the dynamics is driven by \[raddom\] [8 G ]{} 3H\^2 - \_0 - \_[N]{}, , where ${\Lambda_{N}c^2 \over 8\pi G} = - \int_{0}^{\infty} \!\!\! \lambda_N(x) d(P_0 N^2 x^2)
= -N^2P_0/8$ for $n=3$ is a finite negative number, much smaller than the radiation pressure $\sim (z/1000)^3N^2P_0$. So the early universe is GR-like, especially the Hubble parameter at the BBN, insensitive to the precise value of $N^2 P_0$.
Note a more general version of our ${\mathbf V}$ector-for-$\Lambda$ model has a Lagrangian \[LKLJ\] L\_K+L\_J = \_K [K]{}(U\_K\^[1 ]{}) +\_J[J]{}(U\_J\^[1 ]{}) -P\_0 [V]{}(\_K,\_J), with four vector degrees in $U\lambda_J^{1 \over {\cal N}}$ and the scalar $\lambda_K/\lambda_J$; it is optional to replace $U\lambda_K^{1 \over {\cal N}}$ with $U\lambda_J^{1 \over {\cal N}}$ to reduce the total freedom to 4 as in Bekenstein’s TeVeS. Our simple model is equivalent to the special case of two non-dynamical scalar fields $\lambda_K$ and $\lambda_J$ with ${1 \over {\cal N}} \sim {1 \over N} \rightarrow 0$, hence ${\cal K}={\cal K}(U)=K$ and ${\cal J}={\cal J}(U)=J$ (Eq.\[Z\]). The potential is smooth with $P_0 {\cal V}(\lambda_K,\lambda_J)= \int_{\mu_{\rm min}}^1
\left[{\cal H}(\mu_{\rm min} + \lambda_K -\lambda) - {N^2 \over n^2} {\cal H}(\lambda-\lambda_J-2^{-n})\right] P_n d\lambda $, where $P_n \equiv (\lambda^{-{1 \over n}}-1)^2 n^2 P_0$ and ${\cal H}(y)$ is the Heaviside function of $y$. A vector field $A_{\mu} \approx (mc^2 + m\Phi, mc{\mathbf A})$ with a mass scale $m$ has a quantum degeneracy pressure limit $\sim {c^5 \over \hbar^3}m^4$. It is intriguing that our model suggests the existence of a zero point vacuum energy ${\Lambda_0 c^2 \over 8 \pi G} \sim P_0 {\cal V}(1,1) \sim 9P_0 \sim (0.001{\rm eV})^4$. And the (positive) radiation pressure at the epoch of baryon-radiation equality coincides with the cutoff energy density $P_0 {\cal V}(0,0) \sim -N^2 P_0 \sim -(0.3{\rm eV})^4$, and the vacuum-to-cutoff energy density ratio $\sim 9/N^2 \sim 10^{-9}$ coincides with the cosmic baryon-to-photon or baryon-to-neutrino number ratio $\eta \sim 3 \times 10^{-10}$ due to a tiny asymmetry with antibaryons. Can theories like quantum gravity and inflation explain [*these coincidences*]{}? Understanding these might give clues to how the four-vector potential of photons decouples from the baryon current vector, and decouples from our E&M-like vector field $A^\mu$ in spontaneous symmetry breaking in the string theory (Kostelecky & Samuel 1989, Carroll & Shu 2006, Ferreira et al. 2007).
[*Massive neutrinos are optional*]{} for our model because the $L_J$ term creates a massive neutrino-like effect in cosmology without affecting galaxy rotation curves. There are a few ways to create the impression of a fluid of 2eV neutrino in clusters of galaxies as well (Angus et al. 2007, Sanders 2005, Zlosnik et al. 2007). E.g., a general Lagrangian with ${\cal N} \sim n$ would have new dynamical freedoms $\mu \equiv 1-\lambda_K$ and $1-\lambda_J$, which satisfy second order differential equations in time in galaxies, reminiscent of fluid equations for DM. Then the Bekenstein-Milgrom $\mu$-function would acquire a history-dependent non-local relativistic correction of order ${c \over {\cal N} a_0 \tau} \sim 1$ if the temporal variation (relaxation) time scale $\tau$ of the scalar field $\lambda_K$ is comparable to the Hubble time. This dynamical correction is hard to simulate, but is most important at the tidal boundary of (merging) systems where a condensate of the dynamical freedoms $\lambda_K$ and $\lambda_J$ oscillate rapidly, could in principle act as an extra DM source to explain some outliers to the Bekenstein-Milgrom theory, e.g., the merging bullet cluster with its efficient lensing and high speed (Angus & McGaugh 2007). A dynamical field $\lambda_J$ is desirable as an inflaton to seed perturbations (Kanno & Soda 2004).
In summary, we demonstrate as a proof of concept that [*at least one alternative*]{} Lagrangian for the gravity (Eq.\[action\],\[LKLJ\]) can be sketched down to resemble the GR plus $\Lambda$CDM but with somewhat [*less-fining*]{} in terms of fitting several types of observations [*from dwarf spiral galaxies to the cosmic acceleration*]{}. The keys are a zero-point pressure scale $P_0$ at the edge of galaxies, and a universal convergence source term ${1-\mu_B \over 8\pi G} (c^2\nabla_\alpha U^\alpha)^2$ below the cutoff pressure $N^2P_0$, which is near the epoch of equality and the last scattering. However, the CMB should be sensitive to the $\mu_B \equiv 2^{-n}$ modification parameter. [^3] It should be feasible to falsify the present model and variations by simultaneous fits to the supernovae distances and the CMB.
HSZ acknowledges helpful comments from Anaelle Halle, Benoit Famaey, Tom Zlosnik, Pedro Ferreira, Constantinous Skordis, David Mota, Eugene Lim, Meng Su and the anonymous referee, and the support from KITP during the gravitational lensing program 2006.
[0]{} Angus G.W., McGaugh S. 2007, MNRAS, (arXiv0704.0381) Angus G.W., Shan H, Zhao H., Famaey B., 2007, ApJ, 654, L13 Bekenstein J., 2004, Phys. Rev. D., 70, 3509 Bekenstein J., & Milgrom M. (1984), ApJ, 286, 7 (BM84) Carroll S., Lim E. 2004, Phys. Rev. D., 70, 13525 Carroll S., Shu J. 2006, Phys. Rev. D., 73, 103515 Dodelson S. & Liguri M., Phy.Rev. Lett., 97, 1301 Famaey B., Gentile G, Bruneton J.P., Zhao H., 2007, Phys.Rev.D., 75, 3002 Ferreira P., Gripaios B.M., Saffari R., Zlosnik T.G., 2007, Phys. Rev. D., 75 d4014 Foster B.J. and Jacobson T., Phys. Rev. D 73, 064015 (2006) Gentile G., Famaey B., Tiret O., Combes F., Kroupa P., Zhao H., 2007, A&A Lett., 472, L25 Gnedin O. & Zhao H., 2002, MNRAS, 333, 299 Halle A., arXiv0710.3898 (and Zhao & Halle in preparation) Jacobson T. & Mattingly D., 2001, PRD, 64, 024028 Kanno S. & Soda J., 2006, PRD, in press (hep-th/0604192) Kostelecky V.A. and Samuel S., 1989, Phys. Rev. D 39, 683. Li B, Mota D.F., Burrow J., arXiv0709.4581 Lim E.A., Phys. Rev. D 71, 063504 (2005) McGaugh S. 2005, Phys. Rev. Letters, 95, 1302 Sanders R.H. 2005, MNRAS, 363, 459 Soreno M. & Jetzer, Ph, 2006, MNRAS, 371, 626 Spergel D.N. et al. 2007, ApJS, 170, 377 Xu B.X, Wu X.B., Zhao H., 2007, ApJ, 664, 198 Turyshev G.S., Nieto M.N., Anderson J.D., 2006, arxiv:gr-qc/0510081 Zlosnik T., Ferreira P., Starkman G. 2007, PRD, 75, d4017 Zhao H.S., Famaey B., 2006, ApJ, 638, L9 Zhao H.S., 2006, astro-ph/0610056, “Quantum to Cosmology: Fundamental Physics in Space”, proceedings editted by Ulf Isaelsson, Slava Turyshev
[^1]: A full study should include space-like terms $8\pi G K_{12} \equiv 2g^{\alpha\beta}
(c^2\nabla_\alpha U^\gamma) (c^2\nabla_\beta U_\gamma) - {2 \over 3}(c^2\nabla_\alpha U^\alpha)^2$ and $8\pi G K_{13} \equiv 2g^{\alpha\beta}
(c^2\nabla_\alpha U^\gamma) (c^2\nabla_\beta U_\gamma) - 2 (c^2\nabla_\alpha U^\beta)(c^2 \nabla_\beta U^\alpha)$ which change details of structure formation, PPN, and gravitational waves, which are beyond our goal here.
[^2]: While Carroll & Lim (2004) found a scale-down of G because they were interested in stable spin-0 modes with $(1-\mu_B)<0$ for a restricted class ($c_4=0 \ne c_1$) of Jacobson’s models, Dodelson & Liguri (2006) argue that an unstable growth of the vector field is helpful to structure growth in many gravity theories.
[^3]: In radiation-dominated era, the perturbed Poisson equation for radiation is approximately $ 16\pi G \delta \rho \approx 2|{\mathbf k}|^2 \Phi$, at very short-wavelength (hence large $x \sim |{\mathbf k}|\Phi/a_0 \gg 1$) , where $\Phi=\Psi$. But after recombination, $Q \sim 2 q |{\mathbf k}|^2\Phi \propto 2(1-\mu_B)|{\mathbf k}|^2\Phi$, resembling a dissipationless DM term $4 \pi G\delta\rho_{DM}$ to make the matter perturbation grow as $4 \pi G \delta{\rho} \approx \left(1-q \right) |{\mathbf k}|^2\Phi$.
|
---
abstract: 'Recent experimental studies have shown that confinement can profoundly affect self-organization in semi-dilute active suspensions, leading to striking features such as the formation of steady and spontaneous vortices in circular domains and the emergence of unidirectional pumping motions in periodic racetrack geometries. Motivated by these findings, we analyze the two-dimensional dynamics in confined suspensions of active self-propelled swimmers using a mean-field kinetic theory where conservation equations for the particle configurations are coupled to the forced Navier-Stokes equations for the self-generated fluid flow. In circular domains, a systematic exploration of the parameter space casts light on three distinct states: equilibrium with no flow, stable vortex, and chaotic motion, and the transitions between these are explained and predicted quantitatively using a linearized theory. In periodic racetracks, similar transitions from equilibrium to net pumping to traveling waves to chaos are observed in agreement with experimental observations and are also explained theoretically. Our results underscore the subtle effects of geometry on the morphology and dynamics of emerging patterns in active suspensions and pave the way for the control of active collective motion in microfluidic devices.'
author:
- Maxime Theillard
- 'Roberto Alonso-Matilla'
- David Saintillan
bibliography:
- 'rsc.bib'
title: Geometric control of active collective motion
---
Introduction
============
A common feature of many active matter systems is their ability to spontaneously self-organize into complex dynamic mesoscale structures above a certain density [@Marchetti13; @Saintillan13; @Saintillan15]. Such is the case of suspensions of motile bacteria [@Cisneros11; @Wensink12; @Dunkel13; @Gachelin14], cellular extracts [@Surrey01; @Schaller10; @Sanchez12], collections of colloidal rollers [@Bricard13; @Bricard15], shaken grains [@Kudrolli08; @Deseigne10], among many others. Particle-particle interactions, whether long-ranged such as hydrodynamic or electrostatic interactions, or short-ranged such as direct contact forces, are the drivers of self-organization [@Marchetti13]. The symmetries of these interactions along with their coupling with system geometry dictates the structure and morphology of the emerging patterns, which include: steady vortices [@Bricard15; @Tsang15], asters [@Surrey01], traveling bands [@Schaller10; @Bricard13], density shocks [@Lefauve14; @Tsang16], as well as more complex spatiotemporal chaotic patterns composed of unsteady jets and vortices [@Cisneros11].
Of interest to us in this work is the case of suspensions of hydrodynamically interacting slender self-propelled particles such as swimming bacteria [@Lauga09]. In these suspensions, particles exert dipolar stresses on the surrounding medium and also align in shear due to their elongated shape. The interplay between these two effects has been known to lead to hydrodynamic instabilities in the case of extensile particles or so-called pushers, which are thought to be responsible for the emergence of collective motion above a critical density [@Saintillan08; @Saintillan08b; @Subra09; @Baskaran09]. In large unconfined systems, the collective dynamics in the nonlinear regime takes the form of unsteady chaotic motions reminiscent of high-Reynolds-number turbulence, characterized by strong jets and vortices, enhanced swimming speeds and diffusivities, and efficient fluid mixing [@Saintillan12].
Only recently have interactions with boundaries and dynamics in confined geometries gained attention in experiments. In dilute systems, it is well known that self-propelled particles accumulate at boundaries [@Berke08; @Li09; @Vladescu14; @Figueroa15] as a result of both kinematic [@Li09; @Elgeti13; @Elgeti15; @Ezhilan15; @Ezhilan15b] and hydrodynamic mechanisms [@Berke08; @Spagnolie12; @Schaar15]. In complex geometries, transport of the particles along curved boundaries has also been exploited to design ratchets for concentrating microswimmers or directing their motion [@Galajda07; @Hulme08; @Lambert10; @Altshuler13; @Yariv14]. The case of semi-dilute to concentrated suspensions in confinement, however, has largely been unexplored but in a few studies. Wioland *et al.* [@Wioland13] first analyzed the flow inside small droplets of a dense bacterial suspension squeezed between two flat plates. Rather than observing chaotic motion as in bulk systems, they reported the emergence of a steady vortex; detailed observation of the bacterial velocity field in fact revealed a more complex structure with a counter-rotating boundary layer surrounding the vortex core. This vortex was subsequently captured by Lushi *et al.* [@Lushi14] in discrete particle simulations using a basic model accounting for dipolar hydrodynamic interactions as well as steric forces, where it was found that including hydrodynamic interactions is critical in order to correctly capture the counter-rotating boundary layer. Interactions between such vortices were also considered recently using a microfluidic lattice of circular chambers each containing one vortex and connected by junctions [@Wioland15]: in this case, hydrodynamic coupling was shown to produce synchronization on the scale of the lattice, with adjacent vortices rotating either in the same or opposite direction depending on the geometry of the junctions between chambers.
The case of periodic geometries such as circular channels and racetracks has also been studied, where spontaneous flows have been reported in both bacterial [@Wioland16] and sperm [@Creppy15] suspensions above a critical density. In the case of bacteria, Wioland *et al.* [@Wioland16] systematically studied the effect of geometry by varying the channel width. In very narrow channels, unidirectional flow takes place with a nearly parabolic velocity profile. Upon increasing channel width, flow patterns start exhibiting longitudinal oscillations leading to sinusoidal trajectories and eventually take the form of arrays of counter-rotating vortices. Longitudinal density waves were also reported in the case of dense semen [@Creppy15]. The observed transition to directed motion has been predicted in a number of models for active nematics [@Voituriez05; @Ravnik13], where extensile stresses were found to be the destabilizing factor leading to spontaneous flows. These models, however, neglected polarization, which plays an important role in setting the structure of the suspension in confined systems of self-propelled particles [@Ezhilan15]; they also assumed anchoring boundary conditions for the nematic order parameter field at the channel boundaries, whereas the distribution of particle orientations near the walls appears to be dependent on flow conditions in experiments [@Wioland13; @Wioland16]. A qualitative explanation for the transition can also be gleaned from recent studies on the effective rheology of active suspensions [@Gachelin13; @Lopez15; @Saintillan10; @Alonso16], where a decrease of the effective viscosity due to activity can lead at sufficiently high densities to a superfluid-like behavior in weak flows; this connection will be made clearer below.
In this paper, we use numerical simulations based on a continuum kinetic model together with linear stability analyses to predict and characterize transitions to spontaneous flows and collective motion in various two-dimensional microfluidic geometries, with the aim of explaining the experimental observations discussed above. The governing equations are presented in section 2 and consist of evolution equations for the concentration, polarization and nematic order parameter, which are coupled to the Navier-Stokes equations for the mean-field flow induced by the swimmers. Results from simulations and theory are then discussed in section 3, where both circular domains and periodic racetracks are considered. We conclude in section 4.
Model and simulation method
===========================
Continuum model
---------------
We consider a collection of active Brownian particles suspended in a Newtonian fluid of density $\rho$ and shear viscosity $\mu$. The particles swim with velocity $V_{s}$ and have constant translational and rotational diffusivities $d_{t}$ and $d_{r}$, respectively. As a result of their self-propulsion, they also exert a net force dipole on the suspending fluid with stresslet strength $\sigma_{0}$, which we assume to be negative as is the case for extensile swimmers such as bacteria and sperm [@Hatwalne04; @Drescher11]. The suspension, with mean number density $n$, is confined in a finite domain with characteristic dimension $H$, which will be defined more precisely later. Dimensional analysis of the governing equations identifies four relevant dimensionless groups: $$Re=\frac{\rho H^{2} d_{r}}{\mu}, \quad Pe_s=\frac{V_{s}}{2d_{r}H},\quad \alpha = \frac{\sigma_{0} n}{\mu d_{r}}, \quad \Lambda = \frac{d_{r}d_{t}}{V_{s}^{2}}.$$ The Reynolds number $Re$ is typically very small for active suspensions; it will be set to $10^{-6}$ in all the simulations and to zero in the stability results shown below. The swimming Péclet number $Pe_s$ denotes the ratio of the persistence length of swimmer trajectories to the size of the domain and is a measure of confinement. The activity parameter $\alpha$ compares the destabilizing effects of active stresses and of concentration to dissipative processes, namely viscosity and orientation decorrelation by rotational diffusion. Finally, $\Lambda$ is a swimmer-specific parameter comparing diffusive processes to the strength of self-propulsion: the limit of $\Lambda\rightarrow 0$ describes athermal swimmers, whereas $\Lambda\rightarrow \infty$ corresponds to Brownian particles that do not swim.
We adopt a two-dimensional continuum mean-field description of the active suspension based on the probability density function $\Psi(\mathbf{x},\mathbf{p},t)$ of finding a particle at position $\mathbf{x}$ with orientation $\mathbf{p}$ at time $t$, where $\mathbf{p}$ is a unit vector defining the swimming direction and orientation of the bacteria [@Saintillan08]. Following prior studies [@Saintillan13], we approximate $\Psi$ in terms of its first three orientational moments: $$\Psi(\mathbf{x},\mathbf{p},t)\approx \frac{1}{2\pi}\left[c(\mathbf{x},t)+2 \mathbf{p}\cdot\mathbf{m}(\mathbf{x},t)+4\mathbf{pp}:\mathbf{D}(\mathbf{x},t)\right], \label{eq:psiexpansion}$$ where $c$, $\mathbf{m}$, and $\mathbf{D}$ are defined as integrals over the unit circle $C$ of orientations: $$\begin{aligned}
c(\mathbf{x},t)&=\int_{{C}}\Psi(\mathbf{x},\mathbf{p},t)\,d\mathbf{p}, \\
\mathbf{m}(\mathbf{x},t)&=\int_{{C}}\mathbf{p}\,\Psi(\mathbf{x},\mathbf{p},t)\,d\mathbf{p}, \\
\mathbf{D}(\mathbf{x},t)&=\int_{{C}}\left(\mathbf{pp}-\frac{\mathbf{I}}{2}\right)\Psi(\mathbf{x},\mathbf{p},t)\,d\mathbf{p}. \end{aligned}$$ The zeroth moment $c$ is the local concentration, whereas the first and second moments $\mathbf{m}$ and $\mathbf{D}$ describe the local polarization and nematic alignment in the suspension, respectively. Starting from a Smoluchowski equation for $\Psi(\mathbf{x},\mathbf{p},t)$ [@Saintillan08], hierarchical evolution equations for the moments can be obtained, which are written as: $$\begin{aligned}
\partial_{t} c&=-\nabla\cdot\mathbf{F}_{c}, \label{eq:M12d}\\
\partial_{t} \mathbf{m}&=-\nabla\cdot\mathbf{F}_{m}+\tfrac{1}{2}\zeta \mathbf{E}\cdot\mathbf{m}-\mathbf{W}\cdot\mathbf{m}-\mathbf{m}, \label{eq:M22d}\\
\begin{split}
\partial_{t}\mathbf{D}&=-\nabla\cdot\mathbf{F}_{D}+\tfrac{1}{2}\zeta c\mathbf{E}+\tfrac{2}{3}\zeta\mathbf{E}\cdot\mathbf{D}-\tfrac{1}{3}\zeta \left(\mathbf{D}:\mathbf{E}\right)\mathbf{I} \\
&\quad+\mathbf{D}\cdot\mathbf{W}-\mathbf{W}\cdot\mathbf{D}-4\mathbf{D},
\end{split} \label{eq:M32d} \end{aligned}$$ where the dimensionless shape parameter $\zeta$ denotes Bretherton’s constant [@Bretherton62]; we set $\zeta=1$ in the present study as is adequate for slender swimmers. Variables have been made dimensionless using length scale $H$ and time scale $d_r^{-1}$. The source terms on the right-hand side of Eqs. (\[eq:M22d\])–(\[eq:M32d\]) arise from alignment and rotation by the rate-of-strain and vorticity tensors $\mathbf{E}$ and $\mathbf{W}$ of the disturbance velocity field $\mathbf{u}$, and from rotational diffusion which promotes relaxation towards isotropy with $\mathbf{m}=\mathbf{0}$ and $\mathbf{D}=\mathbf{0}$. The fluxes in the equations for $c$, $\mathbf{m}$, and $\mathbf{D}$ include contributions from advection by the flow, self-propulsion and translational diffusion, and are given by $$\begin{aligned}
\mathbf{F}_{c}&=\mathbf{u} \,c +2Pe_s \,\mathbf{m}-4\Lambda Pe_s^{2}\nabla c, \label{eq:Fc} \\
\mathbf{F}_{m}&=\mathbf{u}\,\mathbf{m}+2Pe_s\left(\mathbf{D}+c\tfrac{\mathbf{I}}{2}\right)-4\Lambda Pe_s^{2}\nabla\mathbf{m}, \label{eq:Fm}\\
\mathbf{F}_{D}&=\mathbf{u}\,\mathbf{D}+2Pe_s\left(\mathbf{T}-\mathbf{m}\tfrac{\mathbf{I}}{2}\right)-4\Lambda Pe_s^{2}\nabla \mathbf{D}, \end{aligned}$$ where the third-order tensor $\mathbf{T}$ is the third orientational moment and is related to the polarization according to the closure approximation implied by Eq. (\[eq:psiexpansion\]) as $$T_{ijk} =\frac{1}{4}\left(m_{i}\delta_{jk} +m_{j}\delta_{ik}+m_{k}\delta_{ij}\right).$$ Direct steric interactions between swimmers are neglected within this model. Their leading effect is expected to be an enhancement of local nematic alignment due to the elongated shape of the particles; this effect could easily be incorporated in our model using a nematic alignment potential as previously done by Ezhilan *et al.* [@Ezhilan13], though hydrodynamic interactions alone are sufficient to capture all the phenomenology observed in experiments.
Finally, the disturbance fluid velocity field $\mathbf{u}$ satisfies the incompressible Navier-Stokes equations forced by the divergence of the active stress tensor $\alpha\mathbf{D}$: $$\begin{aligned}
\nabla\cdot\mathbf{u}=0, \quad
Re \left(\partial_{t}\mathbf{u}+\mathbf{u}\cdot\nabla\mathbf{u}\right)=-\nabla p + \nabla^{2}\mathbf{u}+\alpha \nabla\cdot\mathbf{D}. \end{aligned}$$ Note that additional passive stresses also arise due to the inextensibility of the particles in the flow field they generate: we neglect those here as it can be shown that they only act to increase the Newtonian viscosity in the limit of weak flows relevant to the spontaneous flow transitions investigated here, and therefore only renormalize the value of $\alpha$ at the instability threshold. Including these stresses would be straightforward and the reader is referred to our previous work [@Saintillan10; @Alonso16] on the rheology of active suspensions for more details.
In all of our simulations and analysis, we enforce a no-slip boundary condition on the velocity $\mathbf{u}$ on the domain boundary $S$. The natural boundary condition for the particle distributions is a no-translational-flux condition on the probability density function $\Psi$, which translates into no-flux conditions on the orientational moments upon closure of the equations [@Ezhilan15]: $\mathbf{n}\cdot\mathbf{F}_{c}=\mathbf{n}\cdot\mathbf{F}_{m}=\mathbf{n}\cdot\mathbf{F}_{D}=0,$ where $\mathbf{n}$ is the local unit normal. These conditions express the balance between self-propulsion and translational diffusion in the wall-normal direction, and were shown to correctly capture particle distributions near boundaries in confined systems [@Ezhilan15].
Numerical approach
------------------
We solve the governing equations numerically using a hybrid finite-difference finite-volume framework [@Min06; @Mirzadeh2011; @Theillard13; @Guittet15]. The method is implemented on adaptive quadtree grids and the domain boundaries are represented using the level-set method. At each time step of the algorithm, the moment equations (\[eq:M12d\])–(\[eq:M32d\]) are solved semi-implicitly: the diffusive terms are treated implicitly and the advective terms are computed using a semi-Lagrangian approach for improved stability, whereas the remaining coupling terms are treated explicitly. Values of the concentration field are stored at the cell centers to improve its total conservation, while the polarization and nematic order parameter fields are stored at the mesh nodes for better accuracy. Knowledge of the second moment $\mathbf{D}$ allows one to calculate the divergence of the active stress $\alpha \nabla \cdot \mathbf{D}$, which is an input to the Navier-Stokes solver [@Guittet15] used to update the fluid velocity field. A detailed description of the algorithm will be presented elsewhere.
Results and discussion
======================
Circular disks
--------------
Motivated by the experiments of Wioland *et al.* [@Wioland13] in quasi-two-dimensional droplets, we first investigate the dynamics in circular domains, where we take the confinement length scale $H$ to be the radius of the disk. Our simulations in this case show that the collective self-organization depends critically on the level of activity (parameter $\alpha$) and degree of confinement (swimming Péclet number $Pe_s$). Specifically, three distinct phases illustrated in Fig. \[fig:stream\_plots\] are observed depending on the values of $\alpha$ and $Pe_s$: an axisymmetric equilibrium state with no fluid flow (phase I), an axisymmetric and steady double vortex (phase II), and a turbulent-like unsteady chaotic state (phase III). In some cases, more complex axisymmetric flow patterns can also be observed before the transition to chaos, including triple vortices as illustrated in Fig. \[fig:triplevortex\]; such flow patterns are only very rarely observed and we do not discuss them further. Transitions between the three regimes, which are characterized in more detail below, can be captured in a phase diagram in the $(\alpha,Pe_s)$-plane as shown in Fig. \[fig:phasediagram\](a), where we find that either increasing activity or decreasing confinement successively destabilizes phase I into phase II followed by phase III. While distinguishing between these states is straightforward by simple observation of the dynamics, we also introduce an order parameter as a quantitative measure: $$\Phi=\Big\langle \frac{2}{\pi}\int_{\Omega}\frac{|V_{\theta}|}{|\mathbf{V}|}d\mathbf{x}-1\Big\rangle_{t}, \label{eq:Phi}$$
![Axisymmetric triple vortex in which the net particle velocity ${V_\theta}$ changes sign twice across the disk radius; this state is only rarely observed and therefore not included in the phase diagram of Fig. \[fig:phasediagram\]. (a) Concentration profile and streamlines of the net particle velocity. (b) Radial profile of the azimuthal particle velocity averaged over the azimuthal direction. Parameter values for this simulation are $Pe_s=0.2$, $\alpha=-40$, and $\Lambda=0.2$. []{data-label="fig:triplevortex"}](Figure2.pdf){width="45.00000%"}
![Flow transitions in circular disks. (a) Phase diagram in the $(\alpha,Pe_s)$-plane for $\Lambda=0.1$ showing the transitions between phases I, II and III. The black curve shows the marginal stability for the equilibrium state of phase I as predicted by a linear stability analysis. (b) Order parameter $\Phi$ defined in Eq. (\[eq:Phi\]) as a function of activity $|\alpha|$ for three different values of $Pe_s$. $\Phi=-1$ corresponds to purely radial motion, $\Phi=+1$ to purely azimuthal motion, and $\Phi=\sqrt{2}-1$ to equal amounts of radial and azimuthal motion. []{data-label="fig:phasediagram"}](Figure3.pdf)
where $\mathbf{V}=2Pe_s\, \mathbf{m}(\mathbf{x})/c(\mathbf{x})+\mathbf{u}(\mathbf{x})$ is the net velocity of the active particles due to both swimming and advection by the flow, $V_\theta$ is its azimuthal component, $\Omega$ is the circular domain, and $\langle\cdot\rangle_t$ denotes a time average. Note that the velocity $\mathbf{V}$ is the same as that measured in experiments, which typically perform particle-image velocimetry based on swimmer displacements [@Wioland13; @Wioland16]. The order parameter in Eq. (\[eq:Phi\]) is defined such that $\Phi=-1$ for purely radial motion (phase I), $\Phi=+1$ for purely azimuthal motion (phase II), and $\Phi=\sqrt{2}-1$ for a system in which motion occurs equally along the radial and azimuthal directions (phase III). A plot of $\Phi$ vs $|\alpha|$ for various values of $Pe_s$ is shown in Fig. \[fig:phasediagram\](b), where it indeed jumps from $-1$ to $\approx +1$ as the transition from phase I to phase II occurs, before eventually decreasing to a value close to $\sqrt{2}-1$ as the chaotic state of phase III emerges. We now proceed to characterize the three phases in more detail.
*Phase I: Equilibrium state with no flow. —* This regime, illustrated in Fig. \[fig:stream\_plots\](a), occurs for low levels of activity (small $|\alpha|$) and strong levels of confinement (large $Pe_s$), and is characterized by the absence of hydrodynamic flow. As is known to be the case in dilute confined active suspensions [@Ezhilan15], particles tend to accumulate near the system boundaries and on average point towards the boundary, which leads to a net radial polarization $m_{r}(r)>0$ that reaches its maximum at the walls. The azimuthal polarization $m_{\theta}$ is zero in this case, as are off-diagonal components $D_{r\theta}$ and $D_{\theta r}$ of the nematic order tensor. An analytical solution for the first two moments can in fact be derived in this case by neglecting nematic alignment ($\mathbf{D}=\mathbf{0}$), which is a good approximation as shown by full numerical simulations. The solution, given in Appendix A, predicts a dimensionless characteristic thickness for the wall accumulation layer given by $$\Omega^{-1}=2\Lambda Pe_s \sqrt{2/(1+2\Lambda)},$$ which can also be interpreted as the length scale over which the effects of the boundary are screened by diffusive processes [@Yan15]. As first discussed by Ezhilan & Saintillan [@Ezhilan15], two interesting limiting cases are found. When $\Lambda\rightarrow 0$ (strong-propulsion limit), the thickness of the layer is set by the balance of swimming and translational diffusion and is given by $\Omega^{-1}H\approx \sqrt{2}\, d_t/V_s$; on the other hand, the weak-propulsion limit of $\Lambda\rightarrow \infty$ yields a thickness of $\Omega^{-1}H \approx \sqrt{d_t/d_r}$, which is a purely diffusive length scale.
![Double vortex flow in a circular disk (phase II): radial profiles of (a) the azimuthal fluid velocity $u_\theta$, (b) the azimuthal net particle velocity $V_\theta$, (c) the azimuthal polarization $m_\theta$, and (d) the active flow forcing $f_{\theta}=\alpha r^{-2} \partial_{r}(r^2 D_{r\theta})$ for different levels of activity, for $Pe_s=1$ and $\Lambda=0.1$. []{data-label="fig:profiles"}](Figure4.pdf)
![Linear stability of the equilibrium base state (phase I) in a circular disk. (a) Marginal stability curves in the $(\alpha,Pe_s)$-plane for the first three unstable modes at $\Lambda=0.25$. (b) Unstable eigenmodes for the net particle velocity $V_\theta$ for $Pe_s=0.5$, and $\Lambda=0.25$ at the onset of instability. (c) Marginal stability curves for the first unstable mode for different values of $\Lambda$ obtained by linear stability analysis (full lines); symbols show the marginal curve for the transition from phase I to phase II in numerical simulations. []{data-label="fig:stabilitydisk"}](Figure5.pdf)
*Phase II: Steady axisymmetric vortex. —* Upon decreasing confinement or increasing the level of activity, a first transition is observed from the equilibrium state to a steady axisymmetric double vortex shown in Fig. \[fig:stream\_plots\](b), as a consequence of the coupled effect of nematic alignment induced by hydrodynamic interactions and base-state heterogeneities in the concentration and radial polarization profiles. The direction of rotation in this case is arbitrary with equal probabilities for clockwise and counter-clockwise motions, though it is found to remain the same for the duration of the simulation. More details on this regime, which is identical to that reported in the experiments of Wioland *et al.* [@Wioland13], are shown in Fig. \[fig:profiles\], where profiles of the azimuthal components of the fluid velocity $u_\theta$, net particle velocity $V_\theta$, polarization $m_\theta$, and active flow forcing $f_{\theta}=\alpha r^{-2} \partial_{r}(r^2 D_{r\theta})$ are plotted for different parameter values. While the azimuthal fluid velocity $u_{\theta}$ always points in the same direction, the net particle velocity $V_\theta$ follows the fluid flow near the center of the domain but changes sign at a distance away from the boundary, indicating that particles near the circular wall move against the local flow in agreement with experiments and previous models [@Wioland13; @Lushi14]. Fig. \[fig:profiles\](c) in fact shows that the azimuthal polarization is negative, i.e. particles swim against the flow everywhere but only overcome it near the wall. This observation is consistent with the well-known phenomenon of upstream swimming, by which near-wall bacteria tend to swim against the fluid in pressure-driven channel flows of active suspensions [@Hill07; @Kaya12; @Kantsler14; @Ezhilan15; @Mathijssen16]. As the transition from phase I to phase II occurs, we also find that particle accumulation at the boundary is weakened, which is a consequence of the alignment of the particles with the hydrodynamic flow, which reduces the wall-normal polarization and therefore hinders the ability of the particles to swim towards the wall; a similar effect is again also known to occur in pressure-driven channel flows [@Figueroa15; @Ezhilan15].
Mechanistic insight into the formation of the double vortex can be gained by performing a linear stability analysis, in which we theoretically analyze the growth of axisymmetric perturbations to the equilibrium base state with no flow (phase I). Results from this analysis are summarized in Fig. \[fig:stabilitydisk\]. The analysis indeed reveals a linear instability of the equilibrium base state, with a hierarchy of unstable modes for which we plot the marginal stability curves in the $(\alpha,Pe_s)$-plane in Fig. \[fig:stabilitydisk\](a) and the net azimuthal velocity $V_\theta$ in Fig. \[fig:stabilitydisk\](b). The first unstable mode has a structure that is very similar to the nonlinear flow field of Fig. \[fig:profiles\](b), with a vortex core and a counter-rotating boundary layer near the domain wall. Subsequent modes, which become unstable at higher values of $|\alpha|$ as shown in Fig. \[fig:stabilitydisk\](a), exhibit more and more complex structures with alternating layers rotating in opposite directions. These modes are only very rarely observed in simulations, and an example of a triple vortex similar for the second unstable mode is shown in Fig. \[fig:triplevortex\]; in most cases, however, we find that the double vortex instead destabilizes directly into the chaotic state.
The linearized theory also sheds light on the mechanism for the transition, which can be summarized as follows: (i) At equilibrium (phase I), particles near the boundary have a net polarization towards the wall [@Ezhilan15]; (ii) A weak azimuthal flow perturbation causes these particles to rotate due to shear and align at an angle with respect to the radial direction, leading to a net shear nematic alignment $D_{r\theta}$ which is strongest near the wall, as well as an azimuthal polarization in the direction opposite to the fluid flow; (iii) Hydrodynamic disturbances induced by the swimming activity in the nematically-aligned region tend to reinforce the flow perturbation in the case of pushers via the active forcing term $f_{\theta}=\alpha r^{-2} \partial_{r}(r^2 D_{r\theta})$ in the $\theta$-momentum equation. In particular, it can be checked both in theory and simulations that suspensions of pullers for which $\alpha>0$ are always stable and only exhibit phase I. A more quantitative comparison between theory and simulations is shown in Fig. \[fig:stabilitydisk\](c), where the marginal stability curves for the first unstable mode for different values of $\Lambda$ are found to match the numerical transition from phase I to phase II. Fig. \[fig:stabilitydisk\](c) also shows the influence of the parameter $\Lambda$: increasing its value has a stabilizing effect due to diffusion, which smoothes out the wall accumulation layer responsible for driving the flow.
![Analytical prediction of the marginal stability curve. (a) Low-$Pe_s$ asymptote $\alpha^0_c$ for the critical value of the activity parameter leading to instability of the equilibrium state: the plot compares the theoretical prediction of Eq. (\[eq:lowPe\]) to the full numerical solution of the linear stability problem (LSA). (b) High-$Pe_s$ asymptote: the plot shows $(\alpha-\alpha^0_c)/\Lambda$ as a function of $Pe_s$ and confirms the scaling prediction of Eq. (\[eq:highPe\]) with fitting parameter $\beta\approx 150$. []{data-label="fig:stabilitydisk2"}](Figure6.pdf)
The low- and high-Péclet limits for the marginal stability of the equilibrium state can also be characterized more precisely as illustrated in Fig. \[fig:stabilitydisk2\]. In weakly confined systems ($Pe_s\rightarrow 0$), the marginal value of $\alpha$ for instability becomes independent of system size as shown by the constant asymptote in Fig. \[fig:stabilitydisk\](c): in this case, the instability is primarily driven by processes inside the accumulation layer and an approximate expression for the critical value of $\alpha$ is derived in Appendix B as $$\alpha^0_c \approx -\frac{32\Lambda}{1+2\Lambda} \qquad \mbox{as}\quad Pe_s\rightarrow 0, \label{eq:lowPe}$$ which matches the numerical solution of the linear stability problem excellently, especially for small values of $\Lambda$, as shown in Fig. \[fig:stabilitydisk2\](a). In the limit of strong confinement ($Pe_s\rightarrow \infty$), we expect translational diffusion to be the main stabilizing factor, which suggests an asymptote of the form $$\alpha^\infty_c\approx -\beta \Lambda Pe_s^2 \qquad \mbox{as}\quad Pe_s\rightarrow \infty, \label{eq:highPe}$$ where $\beta$ is an unknown constant. Plotting $(\alpha-\alpha^0_c)/\Lambda$ as a function of $Pe_s$ in Fig. \[fig:stabilitydisk2\](b) indeed collapses all the marginal stability curves onto a single power-law consistent with Eq. (\[eq:highPe\]), where the fitting parameter $\beta$ is found to depend very weakly on $\Lambda$ and to asymptote to $\beta\approx 150$ at high values of $\Lambda$. A composite approximation for the marginal stability curve can therefore be written as $$\alpha_c \approx -\frac{32\Lambda}{1+2\Lambda}-150 \Lambda Pe_s^2,$$ which provides an excellent fit to our numerical data over a wide range of parameter values. The high-$Pe_s$ asymptote of Eq. (\[eq:highPe\]) can also be used to define a critical domain diameter for the emergence a double vortex in strong confinement: $$D_c\approx 5\sqrt{\frac{6\mu d_t}{n\sigma_0}}.$$ Interestingly, this critical domain size only depends upon $d_t$ and does not involve $d_r$. The scaling with number density $D_c\sim n^{-1/2}$ also differs from the scaling of $n^{-1}$ for the critical system size for the onset of collective motion in bulk systems [@Hohenegger10].
*Phase III: Turbulent swirling state. —* As the level of activity keeps increasing and confinement is decreased, phase II becomes unstable itself leading to phase III, which is an unsteady chaotic state analogous to that observed in unbounded systems [@Saintillan08b; @Ezhilan13]. There is no clear structure to the flow in this case, which is instead characterized by local jets and vortices driven by active stresses. Note that the transition to phase III is not predicted by our linear theory, which can only account for axisymmetric disturbances. Based on previous analyses of unbounded active turbulence [@Saintillan08b; @Subra09; @Baskaran09], we hypothesize that the transition nonetheless results from a linear instability of the double vortex of phase II to two-dimensional disturbances, though a more detailed theoretical analysis remains to be done in this case.
Periodic channels: circular annuli
----------------------------------
![Flow transitions in circular annuli. (a) Phase diagram in the $(\alpha,Pe_s)$ plane for $\Lambda=0.5$ and $r_\mathrm{min}=1$, showing the transitions between phases I, II, III, and IV. The black curve shows the marginal stability for the equilibrium state of phase I as predicted by a linear stability analysis. (b) Mean azimuthal particle velocity $|\overline{V}_\theta|$ as a function of activity $|\alpha|$ for three different values of $Pe_s$. []{data-label="fig:phasediagannulus"}](Figure8.pdf)
We now turn our attention to the case of periodic channels and first focus on annulus geometries in which the two boundaries are concentric circles. We use the channel halfwidth as the characteristic length scale $H$, and introduce an additional parameter as the dimensionless inner radius $r_{\mathrm{min}}$. The phenomenology in this case is illustrated in Fig. \[fig:annulus\], where four distinct regimes are observed: an axisymmetric equilibrium state with no fluid flow (phase I), an axisymmetric spontaneously flowing state with net fluid pumping (phase II), a spontaneously flowing state with net fluid pumping and traveling density waves (phase III), and a turbulent-like chaotic state (phase IV). With the exception of phase III (traveling waves), these regimes are qualitatively similar to those found in circular disks. Transitions between the different states also show similar trends in the $(\alpha,Pe_s)$-plane, as depicted in the phase diagram of Fig. \[fig:phasediagannulus\](a) where phase III occupies a thin region between phases II and IV. Transitions are also characterized in Fig. \[fig:phasediagannulus\](b), where the absolute value of the mean azimuthal particle velocity $|\overline{V}_\theta|$ is plotted as a function of $|\alpha|$ for different Péclet numbers. As expected, the transition from equilibrium to net pumping is accompanied by a bifurcation from zero to a finite flow rate; the flow rate initially increases with activity up to the point when traveling waves appear, after which it start decreasing to reach zero in the chaotic state. We now discuss the various regimes in more detail.
![Equilibrium distribution (phase I) inside an annulus. (a) Concentration and (b) polarization profiles across the gap as functions of $Pe_s$, for $\Lambda=0.5$ and $r_\mathrm{min}=Pe_s$, which is equivalent to varying gap width in dimensional terms. (c)-(d) Transition from accumulation to depletion at $r=r_\mathrm{min}$ in the $(Pe_s,r_\mathrm{min})$-plane for $\Lambda=0.5$ and $1$, respectively. []{data-label="fig:basestateannulus"}](Figure9.pdf)
*Phase I: Equilibrium state with no flow. —* The equilibrium state shown in Fig. \[fig:annulus\](a) is very similar to that found in circular domains, and is characterized by particle accumulation and wall normal polarization at the outer boundary. The particle distribution at the inner boundary, however, shows a subtle dependence on parameter values and can either show accumulation (with $m_r<0$) or depletion (with $m_r>0$). This is illustrated in Fig. \[fig:basestateannulus\](a)-(b), where we plot radial concentration and polarization profiles across the gap for different channel widths (i.e. different values of $Pe_s$ and $r_\mathrm{min}$ at a fixed ratio of $Pe_s/r_\mathrm{min}$): in wide channels (small $Pe_s$) accumulation occurs at both boundaries, but a depletion is observed in narrow channels (large $Pe_s$) in which case attraction by the outer boundary dominates due to curvature effects and propagates across the entire gap due to diffusion in spite of the presence of the inner wall. Depletion at the inner boundary occurs in strongly diffusive systems or under strong confinement, where both the concentration and polarization profiles become linear across the gap. The transition between the two types of distributions is captured in Fig. \[fig:basestateannulus\](c) in terms of $Pe_s$ and $r_\mathrm{min}$ for $\Lambda=0.5$: as $r_\mathrm{min}$ decreases for a fixed gap width, the curvature of the outer boundary becomes more positive while that of the inner boundary becomes more negative, leading to a higher symmetry breaking between walls thus fostering depletion at the inner wall. The same transition is captured in Fig. \[fig:basestateannulus\](d) for $\Lambda=1$, where particle attraction by the outer wall is yet stronger as translational diffusion is enhanced.
![Linear stability of the equilibrium base state (phase I) in a circular annulus. (a) Marginal stability curves in the $(\alpha,Pe)$-plane for the first unstable modes at $\Lambda=0.5$ and $r_{\mathrm{min}}=1$ (b) Unstable eigenmodes for the net particle velocity $V_\theta$ for $Pe=0.5$, $\Lambda=0.5$, and $r_\mathrm{min}=1$ at the onset of instability. (c) Marginal stability curves for the first unstable mode in the $(\alpha,r_\mathrm{min})$ plane for different values of $Pe_s$ and for $\Lambda=0.5$. []{data-label="fig:annulusstability"}](Figure10.pdf)
*Phase II: Spontaneous flow with net pumping. —* As activity is increased or confinement is decreased, a first transition to an axisymmetric flowing state with net fluid pumping occurs (phase II). As previously seen in Fig. \[fig:phasediagannulus\](b), the mean azimuthal velocity $|\overline{V}_\theta|$ is non-zero in this regime and increases monotonically with the level of activity up to the point where traveling waves appear (phase III below). The transition to unidirectional flow is similar to that reported in both bacterial [@Wioland16] and sperm [@Creppy15] suspensions. It was also predicted in a few previous theoretical and numerical models [@Voituriez05; @Ravnik13], though these typically imposed anchoring boundary conditions on the nematic order parameter, which are not appropriate to describe suspensions of swimmers such as bacteria.
As in the case of the disk, the transition to spontaneous flow can be understood as a linear instability of the equilibrium base state (phase I) as analyzed more precisely in Fig. \[fig:annulusstability\]. Here again, an infinite series of unstable modes exists, which involve increasingly complex azimuthal flow fields with alternating layers rotating in opposite directions, for which we show the marginal stability curves and profiles of the net particle velocity in Fig. \[fig:annulusstability\](a)-(b). The first unstable mode, which causes the strongest pumping, is typically observed in simulations, though higher modes are also seen on rare occasions. As shown in Fig. \[fig:annulusstability\](b), upstream swimming generally occurs near the annulus boundaries; this is always true of the outer boundary, though it is in some cases suppressed near the inner boundary when accumulation does not occur there as explained in Fig. \[fig:basestateannulus\]. The dependence of the transition to net pumping on the inner radius $r_\mathrm{min}$ is illustrated in Fig. \[fig:annulusstability\](c): the marginal stability curve varies only weakly with $r_\mathrm{min}$ and has a non-monotonic dependence, which again reflects the change in the nature of the equilibrium base state found in Fig. \[fig:basestateannulus\]. In very large annuli ($r_{\mathrm{min}}\rightarrow \infty$), the effect of boundary curvature becomes negligible locally and the marginal stability curve asymptotes to that for a straight channel.
![Racetrack geometry: each boundary is composed of two straight sections (length $L$) and two half-circles (radii $r_\mathrm{min}$ and $r_\mathrm{min}+2H$). The distance between the two walls is $2H$, where the half-width $H$ is chosen for non-dimensionalization. []{data-label="fig:racegeom"}](Figure11.pdf)
*Phase III: Spontaneous flow with traveling waves. —* At yet higher levels of activity, net pumping persists but the azimuthal symmetry of the flow is lost and periodic traveling density waves appear as shown in Fig. \[fig:annulus\](c) as well as Fig. \[fig:racetrack\] below. Such waves were also observed in experiments on both bacterial [@Wioland16] and sperm suspensions [@Creppy15]. As illustrated in Fig. \[fig:phasediagannulus\](b), the average azimuthal velocity $|\overline{V}_\theta|$ in this regime systematically decreases with activity level, as more and more of the motion takes place in the radial direction. As for the other steady states discussed previously, this one was found to be stable over long time intervals. A full characterization of these waves is beyond the scope of the present work. As either $|\alpha|$ is increased or $Pe_s$ decreased, the waves become more intense up to the point when the chaotic state of phase IV emerges; this state is similar to phase III observed above in circular domains.
Periodic channels: racetracks
-----------------------------
For direct comparison with the experiments of Wioland *et al.* [@Wioland16], we consider as a final example racetrack geometries composed of straight sections of length $L$ connected by two half-annuli as shown in Fig. \[fig:racegeom\]. The characteristic length scale for non-dimensionalization is still taken to be the channel half-width $H$. We focus here on the dynamics in the straight sections; the various flow regimes in this case echo those found in circular annuli and are illustrated in Fig. \[fig:racetrack\](a)-(b), showing instantaneous flow patterns and corresponding mean velocity profiles. A transition to net pumping is first observed upon increasing channel width at a fixed value of $\alpha$, followed by the appearance of traveling waves. As the waves become stronger the flow takes the form of alternating counter-rotating vortices and eventually destabilizes into chaos. This phenomenology is identical to that observed in the experiments [@Wioland16] (see Fig. 4 of that reference). The mean velocity profiles are also consistent with the theoretically predicted unstable linear eigenmodes shown in Fig. \[fig:racetrack\](c) for the same parameter values. The last row in Fig. \[fig:racetrack\] demonstrates the possibility of a more complex flow regime that is only rarely observed: here the flow has a complex unsteady structure, but the mean velocity profile highlights two counter-flowing streams near the top and bottom walls and resembles the second unstable linear mode.
We finish by describing the relationship between the onset of spontaneous pumping in confinement and the effective rheology of the system. In recent work, Alonso-Matilla *et al.* [@Alonso16] calculated the effective relative viscosity $\eta_r$ in a dilute active suspension confined in a planar channel and subject to an applied pressure-driven flow, where $\eta_r$ is defined as the ratio of the flow rate in pure fluid by that in the presence of swimmers at a given pressure gradient. In agreement with previous experiments in the same geometry [@Gachelin13], suspensions of pushers were found to enhance the flow, i.e. decrease the effective viscosity of the suspension as a result of activity. This effect was found to be the strongest in the limit of vanishing flow strength, and the zero-flow-rate viscosity $\eta_{r}^0$ was further found to decrease with increasing $|\alpha|$, until it eventually reaches zero suggesting an apparent transition to superfluidity. The exact dependence of $\eta_r^0$ with activity is plotted in Fig. \[fig:rheology\](a), where it is found that the value of $|\alpha|$ for which $\eta_r^0$ reaches zero coincides precisely with the prediction of the linear theory for the marginal stability of the equilibrium state and onset of spontaneous flow. This provides an additional interpretation for the transition to pumping as a consequence of apparent superfluidity: as the resistance of the system to flow is effectively zero, a small perturbation in the fluid velocity can amplify at no cost leading to unidirectional flow. In the flowing state, the input of mechanical energy by the swimmers exactly balances viscous dissipation in the solvent. This interpretation is further borne out by Fig. \[fig:rheology\](b), which plots the mean flow rate in the spontaneous flow regime as a function of the length $L$ of the straight section of the racetrack. Remarkably, we observe that the flow rate in the pumping regime is completely independent of channel length, another signature of an effectively frictionless flow.
![(a) Zero-flow-rate relative velocity $\eta_r$ as a function of activity $|\alpha|$ in a suspension of pushers confined between two flat plates, obtained using the model of Alonso-Matilla *et al.* [@Alonso16] As activity increases, the relative viscosity decreases and reaches superfluidity at the critical value of $|\alpha|$ for the onset of spontaneous flows as predicted by our linear stability analysis (LSA). This calculation was performed in three dimensions for ease of comparison with the rheological model. (b) Average longitudinal velocity $|V_{\parallel}|$ as a function of channel length $L$ in periodic racetracks in the spontaneous flow regimes (phases II and III).[]{data-label="fig:rheology"}](Figure13.pdf)
Concluding remarks
==================
We have used a combination of numerical simulations and theory to explore the effect of confinement and geometry on the onset and structure of spontaneous flows in semi-dilute suspensions of active swimming microorganisms. A mean-field theory based on the coupled Smoluchowski and Navier-Stokes equations was used to describe the dynamical evolution of swimmer configurations. We solved these governing equations numerically in two dimensions and compared our numerical results with predictions from a linear stability analysis. Our results agree well with prior experimental studies and were able to capture and explain the spontaneous directed motions arising in pusher suspensions.
We first analyzed the dynamics of swimmers in circular disks, where three distinct flow regimes were found depending on the level of activity and degree of confinement: equilibrium with no flow, a steady double vortex, and a swirling chaotic state. The equilibrium state manifests at low levels of activity or strong confinement, where the spatial and orientational distribution of particles is axisymmetric and the net disturbance flow generated by the swimmers vanishes. Particles accumulate at the boundaries and on average are radially polarized towards the wall, reaching their maximum polarization at the boundaries. Increasing the level of activity or decreasing confinement destabilizes the system into the double vortex state, and a mechanism based on the shear alignment of the swimmers in the disturbance flow they generate inside the accumulation layer was uncovered. By the same shear alignment mechanism, swimmers were in fact shown to orient against the flow throughout the domain allowing them to swim upstream near the boundary, thus leading to the double vortex structure reported in experiments where bacterial velocities were measured. A further increase in the level of activity or a decrease in confinement originates a second transition to a turbulent-like chaotic state analogous to that observed in bulk systems.
We then turned our focus to swimmer dynamics in periodic geometries. Our simulations in circular annuli captured four flow regimes quite similar to those found in circular disks: an equilibrium state with no flow, an axisymmetric state with net fluid pumping, the emergence of traveling density waves, followed by a chaotic state. The transitions between regimes were again found to be governed by the level of activity and degree of confinement, with only a weak dependence on the inner radius dimension. Similar transitions were also observed in periodic racetracks, where we were able to connect the onset of spontaneous pumping with the effective rheology of the suspension. Specifically, the transition to net pumping was shown to occur at the same level of activity at which the zero-shear-rate viscosity becomes zero in a pressure-driven planar channel flow, suggesting that spontaneous flows in confinement are in fact a consequence of the apparent superfluidity of the system. This conclusion was further supported by observing that the net flow rate is independent of channel length in periodic racetracks.
Our numerical and theoretical results both underscore the subtle interplay between confinement, geometry, and activity in semi-dilute active suspensions. While this study focused on fairly simple geometries previously considered in experiments, we anticipate that a wealth of more complex flow regimes might arise in other types of geometries. Continuum modeling as performed in this work proves to be a valuable tool for the understanding and prediction of these flows and could also play a useful role in the design of microfluidic devices for bioengineering applications involving bacteria.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The authors thank Barath Ezhilan, Jérémie Palacci, Hugo Wioland, Zvonimir Dogic, and Michael Shelley for valuable discussions. Funding from NSF Grants CBET-1532652 and DMS-1463965 is gratefully acknowledged.
Appendix A: Analytical solutions for equilibrium states {#appendix-a-analytical-solutions-for-equilibrium-states .unnumbered}
=======================================================
If the nematic order tensor is neglected $(\mathbf{D}=\mathbf{0})$, which is a good approximation at equilibrium as shown by the full numerical solution, simple closed-form analytical solutions can be derived for the radial concentration and polarization profiles which we provide here. In axisymmetric geometries, the steady governing equations for $c(r)$ and $m_r(r)$ obtained by setting the $\mathbf{F}_{c}$ and $\mathbf{F}_{m}$ in Eqs. (\[eq:Fc\])–(\[eq:Fm\]) in the absence of flow simplify to: $$\begin{aligned}
-\frac{d}{dr}(rm_r)+2\Lambda Pe_s \frac{d}{dr}\left(r\frac{dc}{dr}\right)=0, \label{eq:ctheory} \\
-\frac{dc}{dr}+4\Lambda Pe_s \frac{d}{dr}\left[\frac{1}{r}\frac{d}{dr}(rm_r)\right]-m_r=0. \label{eq:mtheory} \end{aligned}$$ In a circular disk, the boundary conditions at $r=1$ are: $$\begin{aligned}
-m_r+2\Lambda Pe_s \frac{dc}{dr}=0, \label{eq:BCcr} \\
-\frac{c}{2}+2\Lambda Pe_s \frac{dm_r}{dr}=0, \label{eq:BCmr} \end{aligned}$$ and we also require that the solution remain bounded at $r=0$ and satisfy the normalization $$\int_{0}^{1}c(r) r dr = \frac{1}{2}.$$ Integrating Eq. (\[eq:ctheory\]) once easily shows that the boundary condition of Eq. (\[eq:BCcr\]) in fact applies everywhere across the gap and expresses the local balance between swimming and diffusive fluxes. Inserting Eq. (\[eq:BCcr\]) into Eq. (\[eq:mtheory\]) and manipulating then provides a modified Bessel equation for $m_r(r)$: $$r^2\frac{d^2 m_r}{dr^2}+r\frac{dm_r}{dr}-(1+\Omega^2 r^2)m_r=0,$$ where $$\Omega^2=\frac{1}{4Pe_s^2 \Lambda}\left(1+\frac{1}{2\Lambda}\right).$$ After applying boundary conditions, the concentration and polarization profiles are obtained as $$\begin{aligned}
c(r) = A_1 + A_2 I_0(\Omega r), \quad
m_r(r) = A_2 I_1(\Omega r), \label{eq:disksol} \end{aligned}$$ where the constants $A_1$ and $A_2$ are expressed in terms of incomplete Bessel functions as $$\begin{aligned}
A_1&=1-\frac{2}{\Omega}I_1(\Omega)A_2, \\
A_2&=\left[(4\Lambda^2 Pe_s^2-1)I_0(\Omega)+\frac{2}{\Omega}I_1(\Omega)+4\Lambda^2 Pe_s^2\Omega^2 I_2(\Omega)\right]^{-1}.\end{aligned}$$ The expression for $c(r)$ in Eq. (\[eq:disksol\]) is identical to that previously obtained by Yan & Brady [@Yan15]. The solution inside an annulus is also easily obtained by applying boundary conditions of Eqs. (\[eq:BCcr\])–(\[eq:BCmr\]) at both walls $r=r_\textrm{min}$ and $r_\mathrm{min}+2$ but is omitted here for brevity. The solution in a straight channel was also previously calculated by Ezhilan & Saintillan [@Ezhilan15].
Appendix B: Low- and high-$Pe_s$ marginal stability limits in a circular disk {#appendix-b-low--and-high-pe_s-marginal-stability-limits-in-a-circular-disk .unnumbered}
=============================================================================
The stability analysis is performed by perturbing the governing equations about the equlibrium state, which we denote by $(c^0,\mathbf{m}^0,\mathbf{D}^0)$. We focus here on the marginal stability, for which the growth rate is set to zero. In large domains, the effect of boundary curvature on the structure of the accumulation layer is negligible, which prompts us to use Cartesian coordinates. Upon linearization of the equations, we arrive at a coupled system for the variables $m'_x$ and $D'_{xy}$ (where $'$ refers to perturbations): $$\begin{aligned}
&0=-2 Pe_s \frac{dD'_{xy}}{dy}+4\Lambda Pe_s^2 \frac{d^2 m'_x}{dy^2}-\frac{3}{4}\alpha_c D'_{xy}m^0_y-m'_x, \\
&0=-\frac{1}{2}Pe_s\frac{dm'_x}{dy}+4\Lambda Pe^2_s \frac{d^2 D'_{xy}}{dy^2}-\alpha_c D'_{xy}\left(\frac{c^0}{4}+D^0_{yy}\right)-4D'_{xy}. \label{eq:evpb} v\end{aligned}$$ These constitute an eigenvalue problem for $(m'_x, D'_{xy})$ with corresponding eigenvalue $\alpha_c$.
In the low-$Pe_s$ limit, the dominant balance in Eq. (\[eq:evpb\]) is between the last two terms, which capture flow alignment and rotational diffusion. An estimate for $\alpha_c$ can then be obtained by balancing these two terms and by using the maximum values of $c^0$ and $D^0_{yy}$, which are attained at the walls: $$\alpha_{c}^0\approx \frac{-4}{\frac{c^0_{wall}}{4}+D^0_{yy,wall}}=-\frac{32\Lambda}{1+2\Lambda}\qquad \mbox{as}\quad Pe_s\rightarrow 0, \vspace{-0.15cm}$$ which indeed agrees with the full numerical solution of the eigenvalue problem as shown in Fig. \[fig:stabilitydisk2\](a).
In the high-$Pe_s$ limit (strong confinement), the dominant balance now takes place between the second and third terms in Eq. (\[eq:evpb\]), which describe translational diffusion and shear alignment. While the use of Cartesian coordinates is no longer justified in this case, the form of the equations still suggests a scaling of the type $$\alpha^\infty_c\approx -\beta \Lambda Pe_s^2 \qquad \mbox{as}\quad Pe_s\rightarrow \infty, \vspace{-0.15cm}$$ which is again supported by numerical data with $\beta\approx 150$ as shown in Fig. \[fig:stabilitydisk2\](b).
|
---
abstract: 'This paper addresses the general problem of domain adaptation which arises in a variety of applications where the distribution of the labeled sample available somewhat differs from that of the test data. Building on previous work by , we introduce a novel distance between distributions, *discrepancy distance*, that is tailored to adaptation problems with arbitrary loss functions. We give Rademacher complexity bounds for estimating the discrepancy distance from finite samples for different loss functions. Using this distance, we derive novel generalization bounds for domain adaptation for a wide family of loss functions. We also present a series of novel adaptation bounds for large classes of regularization-based algorithms, including support vector machines and kernel ridge regression based on the empirical discrepancy. This motivates our analysis of the problem of minimizing the empirical discrepancy for various loss functions for which we also give novel algorithms. We report the results of preliminary experiments that demonstrate the benefits of our discrepancy minimization algorithms for domain adaptation.'
author:
- |
Yishay Mansour\
Google Research and\
Tel Aviv Univ.\
`mansour@tau.ac.il` Mehryar Mohri\
Courant Institute and\
Google Research\
`mohri@cims.nyu.edu` Afshin Rostamizadeh\
Courant Institute\
New York University\
`rostami@cs.nyu.edu`
title: 'Domain Adaptation: Learning Bounds and Algorithms'
---
Introduction
============
In the standard PAC model [@valiant] and other theoretical models of learning, training and test instances are assumed to be drawn from the same distribution. This is a natural assumption since, when the training and test distributions substantially differ, there can be no hope for generalization. However, in practice, there are several crucial scenarios where the two distributions are more similar and learning can be more effective. One such scenario is that of *domain adaptation*, the main topic of our analysis.
The problem of domain adaptation arises in a variety of applications in natural language processing [@Dredze07Frustratingly; @Blitzer07Biographies; @jiang-zhai07; @chelba; @daume06], speech processing , computer vision [@martinez], and many other areas. Quite often, little or no labeled data is available from the *target domain*, but labeled data from a *source domain* somewhat similar to the target as well as large amounts of unlabeled data from the target domain are at one’s disposal. The domain adaptation problem then consists of leveraging the source labeled and target unlabeled data to derive a hypothesis performing well on the target domain.
A number of different adaptation techniques have been introduced in the past by the publications just mentioned and other similar work in the context of specific applications. For example, a standard technique used in statistical language modeling and other generative models for part-of-speech tagging or parsing is based on the maximum a posteriori adaptation which uses the source data as prior knowledge to estimate the model parameters [@roark03supervised]. Similar techniques and other more refined ones have been used for training maximum entropy models for language modeling or conditional models [@DellaPietra; @jelinek; @chelba; @daume06].
The first theoretical analysis of the domain adaptation problem was presented by , who gave VC-dimension-based generalization bounds for adaptation in classification tasks. Perhaps, the most significant contribution of this work was the definition and application of a distance between distributions, the $d_A$ distance, that is particularly relevant to the problem of domain adaptation and that can be estimated from finite samples for a finite VC dimension, as previously shown by . This work was later extended by who also gave a bound on the error rate of a hypothesis derived from a weighted combination of the source data sets for the specific case of empirical risk minimization. A theoretical study of domain adaptation was presented by , where the analysis deals with the related but distinct case of adaptation with multiple sources, and where the target is a mixture of the source distributions.
This paper presents a novel theoretical and algorithmic analysis of the problem of domain adaptation. It builds on the work of and extends it in several ways. We introduce a novel distance, the *discrepancy distance*, that is tailored to comparing distributions in adaptation. This distance coincides with the $d_A$ distance for 0-1 classification, but it can be used to compare distributions for more general tasks, including regression, and with other loss functions. As already pointed out, a crucial advantage of the $d_A$ distance is that it can be estimated from finite samples when the set of regions used has finite VC-dimension. We prove that the same holds for the discrepancy distance and in fact give data-dependent versions of that statement with sharper bounds based on the Rademacher complexity.
We give new generalization bounds for domain adaptation and point out some of their benefits by comparing them with previous bounds. We further combine these with the properties of the discrepancy distance to derive data-dependent Rademacher complexity learning bounds. We also present a series of novel results for large classes of regularization-based algorithms, including support vector machines (SVMs) [@ccvv] and kernel ridge regression (KRR) [@krr]. We compare the pointwise loss of the hypothesis returned by these algorithms when trained on a sample drawn from the target domain distribution, versus that of a hypothesis selected by these algorithms when training on a sample drawn from the source distribution. We show that the difference of these pointwise losses can be bounded by a term that depends directly on the empirical discrepancy distance of the source and target distributions.
These learning bounds motivate the idea of replacing the empirical source distribution with another distribution with the same support but with the smallest discrepancy with respect to the target empirical distribution, which can be viewed as reweighting the loss on each labeled point. We analyze the problem of determining the distribution minimizing the discrepancy in both 0-1 classification and square loss regression. We show how the problem can be cast as a linear program (LP) for the 0-1 loss and derive a specific efficient combinatorial algorithm to solve it in dimension one. We also give a polynomial-time algorithm for solving this problem in the case of the square loss by proving that it can be cast as a semi-definite program (SDP). Finally, we report the results of preliminary experiments showing the benefits of our analysis and discrepancy minimization algorithms.
In section \[sec:prelim\], we describe the learning set-up for domain adaptation and introduce the notation and Rademacher complexity concepts needed for the presentation of our results. Section \[sec:dist\] introduces the discrepancy distance and analyzes its properties. Section \[sec:bounds\] presents our generalization bounds and our theoretical guarantees for regularization-based algorithms. Section \[sec:alg\] describes and analyzes our discrepancy minimization algorithms. Section \[sec:exp\] reports the results of our preliminary experiments.
Preliminaries {#sec:prelim}
=============
Learning Set-Up
---------------
We consider the familiar supervised learning setting where the learning algorithm receives a sample of $m$ labeled points ${{\mathcal S}}= (z_1,
\ldots, z_m) = ((x_1, y_1), \ldots, (x_m, y_m)) \in (X \times Y)^m$, where $X$ is the input space and $Y$ the label set, which is ${\{0,
1\}}$ in classification and some measurable subset of $\Rset$ in regression.
In the *domain adaptation problem*, the training sample ${{\mathcal S}}$ is drawn according to a *source distribution* $Q$, while test points are drawn according to a *target distribution* $P$ that may somewhat differ from $Q$. We denote by $f\colon X \to Y$ the target labeling function. We shall also discuss cases where the source labeling function $f_Q$ differs from the target domain labeling function $f_P$. Clearly, this dissimilarity will need to be small for adaptation to be possible.
We will assume that the learner is provided with an unlabeled sample ${{\mathcal T}}$ drawn i.i.d. according to the target distribution $P$. We denote by $L\colon Y \times Y \to \Rset$ a loss function defined over pairs of labels and by ${{\cal L}}_Q(f, g)$ the expected loss for any two functions $f, g\colon X \to Y$ and any distribution $Q$ over $X$: ${{\cal L}}_Q(f, g) =
\operatorname*{\rm E}_{x \sim Q} [L(f(x), g(x))]$.
The domain adaptation problem consists of selecting a hypothesis $h$ out of a hypothesis set $H$ with a small expected loss according to the target distribution $P$, ${{\cal L}}_P(h, f)$.
Rademacher Complexity
---------------------
Our generalization bounds will be based on the following data-dependent measure of the complexity of a class of functions.
Let $H$ be a set of real-valued functions defined over a set $X$. Given a sample $S \!\in\! X^m$, the empirical Rademacher complexity of $H$ is defined as follows: $${\widehat}{\mathfrak{R}}_S(H) = \frac{2}{m} \operatorname*{\rm E}_\sigma \Big[\sup_{h \in H} \big|
\sum_{i=1}^m \sigma_i h(x_i) \big| \, \Big| S = (x_1, \ldots, x_m) \Big].$$ The expectation is taken over $\sigma = (\sigma_1, \ldots, \sigma_n)$ where $\sigma_i$s are independent uniform random variables taking values in ${\{-1, +1\}}$. The Rademacher complexity of a hypothesis set $H$ is defined as the expectation of ${\widehat}{\mathfrak{R}}_S(H)$ over all samples of size $m$: $${\mathfrak{R}}_m(H) = \operatorname*{\rm E}_S \big[ {\widehat}{\mathfrak{R}}_S(H) \big| |S| = m \big].$$
The Rademacher complexity measures the ability of a class of functions to fit noise. The empirical Rademacher complexity has the added advantage that it is data-dependent and can be measured from finite samples. It can lead to tighter bounds than those based on other measures of complexity such as the VC-dimension [@koltchinskii_and_panchenko].
We will denote by ${\widehat}R_S(h)$ the empirical average of a hypothesis $h
\colon X \to \Rset$ and by $R(h)$ its expectation over a sample $S$ drawn according to the distribution considered. The following is a version of the Rademacher complexity bounds by and . For completeness, the full proof is given in the Appendix.
\[th:rademacher\] Let $H$ be a class of functions mapping $Z = X \times Y$ to $[0, 1]$ and ${{\mathcal S}}= (z_1, \ldots, z_m)$ a finite sample drawn i.i.d. according to a distribution $Q$. Then, for any $\delta > 0$, with probability at least $1 - \delta$ over samples ${{\mathcal S}}$ of size $m$, the following inequality holds for all $h \in H$: $$R(h) \leq {\widehat}R(h) + {\widehat}{\mathfrak{R}}_{{\mathcal S}}(H) + 3 \sqrt{\frac{\log \frac{2}{\delta}}{2m}}.$$
Distances between Distributions {#sec:dist}
===============================
Clearly, for generalization to be possible, the distribution $Q$ and $P$ must not be too dissimilar, thus some measure of the similarity of these distributions will be critical in the derivation of our generalization bounds or the design of our algorithms. This section discusses this question and introduces a *discrepancy* distance relevant to the context of adaptation.
The $l_1$ distance yields a straightforward bound on the difference of the error of a hypothesis $h$ with respect to $Q$ versus its error with respect to $P$.
\[prop:l\_1\_bound\] Assume that the loss $L$ is bounded, $L \leq M$ for some $M > 0$. Then, for any hypothesis $h \in H$, $${\lvert{{\cal L}}_Q(h, f) - {{\cal L}}_P(h, f)\rvert} \leq M \, l_1(Q, P).$$
This provides us with a first adaptation bound suggesting that for small values of the $l_1$ distance between the source and target distributions, the average loss of hypothesis $h$ tested on the target domain is close to its average loss on the source domain. However, in general, this bound is not informative since the $l_1$ distance can be large even in favorable adaptation situations. Instead, one can use a distance between distributions better suited to the learning task.
Consider for example the case of classification with the 0-1 loss. Fix $h \in H$, and let $a$ denote the support of ${\lverth - f\rvert}$. Observe that ${\lvert{{\cal L}}_Q(h, f) - {{\cal L}}_P(h, f)\rvert} = {\lvertQ(a) - P(a)\rvert}$. A natural distance between distributions in this context is thus one based on the supremum of the right-hand side over all regions $a$. Since the target hypothesis $f$ is not known, the region $a$ should be taken as the support of $|h - h'|$ for any two $h, h' \in H$.
This leads us to the following definition of a distance originally introduced by \[pp. 271-272\] under the name of *generalized Kolmogorov-Smirnov distance*, later by as *the $d_A$ distance*, and introduced and applied to the analysis of adaptation in classification by and .
Let $A \subseteq 2^{{\lvertX\rvert}}$ be a set of subsets of $X$. Then, the *$d_A$-distance* between two distributions $Q_1$ and $Q_2$ over $X$, is defined as $$d_A(Q_1,Q_2) = \sup_{a \in A} {\lvertQ_1(a) - Q_2(a)\rvert}.$$
As just discussed, in 0-1 classification, a natural choice for $A$ is $A = H \Delta H = {\{{\lverth' - h\rvert}\colon h, h' \in H\}}$. We introduce a distance between distributions, *discrepancy distance*, that can be used to compare distributions for more general tasks, e.g., regression. Our choice of the terminology is partly motivated by the relationship of this notion with the discrepancy problems arising in combinatorial contexts [@chazelle].
Let $H$ be a set of functions mapping $X$ to $Y$ and let $L\colon Y \times Y \to \Rset_+$ define a loss function over $Y$. The discrepancy distance ${\mathrm{disc}}_L$ between two distributions $Q_1$ and $Q_2$ over $X$ is defined by $$\begin{gathered}
{\mathrm{disc}}_L(Q_1, Q_2) = \max_{h, h' \in H} \Big| {{\cal L}}_{Q_1}(h', h)
- {{\cal L}}_{Q_2}(h', h) \Big|.\end{gathered}$$
The discrepancy distance is clearly symmetric and it is not hard to verify that it verifies the triangle inequality, regardless of the loss function used. In general, however, it does not define a *distance*: we may have ${\mathrm{disc}}_L(Q_1, Q_2) = 0$ for $Q_1 \neq
Q_2$, even for non-trivial hypothesis sets such as that of bounded linear functions and standard continuous loss functions.
Note that for the 0-1 classification loss, the discrepancy distance coincides with the $d_A$ distance with $A = H \Delta H$. But the discrepancy distance helps us compare distributions for other losses such as $L_q(y, y' ) = |y - y' |^q$ for some $q$ and is more general.
As shown by , an important advantage of the $d_A$ distance is that it can be estimated from finite samples when $A$ has finite VC-dimension. We prove that the same holds for the ${\mathrm{disc}}_L$ distance and in fact give data-dependent versions of that statement with sharper bounds based on the Rademacher complexity.
The following theorem shows that for a bounded loss function $L$, the discrepancy distance ${\mathrm{disc}}_L$ between a distribution and its empirical distribution can be bounded in terms of the empirical Rademacher complexity of the class of functions $L_H = {\{x \mapsto L(h'(x),
h(x))\colon h, h' \in H\}}$. In particular, when $L_H$ has finite pseudo-dimension, this implies that the discrepancy distance converges to zero as $O(\sqrt{\log m/m})$.
Assume that the loss function $L$ is bounded by $M > 0$. Let $Q$ be a distribution over $X$ and let ${\widehat}Q$ denote the corresponding empirical distribution for a sample ${{\mathcal S}}= (x_1, \ldots ,
x_m)$. Then, for any $\delta > 0$, with probability at least $1 -
\delta$ over samples ${{\mathcal S}}$ of size $m$ drawn according to $Q$: $${\mathrm{disc}}_L(Q, {\widehat}Q) \leq {\widehat}{\mathfrak{R}}_{{\mathcal S}}(L_H) + 3 M \sqrt{\frac{\log \frac{2}{\delta}}{2m}}.$$
We scale the loss $L$ to $[0, 1]$ by dividing by $M$, and denote the new class by $L_H /M$. By Theorem \[th:rademacher\] applied to $L_H /M$, for any $\delta > 0$, with probability at least $1 -
\delta$, the following inequality holds for all $h, h' \in H$: $$\frac{{{\cal L}}_Q(h', h)}{M} \leq \frac{{{\cal L}}_{{\widehat}Q}(h', h)}{M} +
{\widehat}{\mathfrak{R}}_{{\mathcal S}}(L_H/M) + 3 \sqrt{\frac{\log \frac{2}{\delta}}{2m}}.$$ The empirical Rademacher complexity has the property that ${\widehat}{\mathfrak{R}}(\alpha
H) = \alpha {\widehat}{\mathfrak{R}}(H)$ for any hypothesis class $H$ and positive real number $\alpha$ [@bartlett]. Thus, ${\mathfrak{R}}_{{\mathcal S}}(L_H/M)
= \frac{1}{M}{\mathfrak{R}}_{{\mathcal S}}(L_H)$, which proves the proposition.
For the specific case of $L_q$ regression losses, the bound can be made more explicit.
\[cor:dis\_bound\] Let $H$ be a hypothesis set bounded by some $M \!>\! 0$ for the loss function $L_q$: $L_q(h, h') \leq M$, for all $h, h' \in H$. Let $Q$ be a distribution over $X$ and let ${\widehat}Q$ denote the corresponding empirical distribution for a sample ${{\mathcal S}}= (x_1, \ldots , x_m)$. Then, for any $\delta > 0$, with probability at least $1 - \delta$ over samples ${{\mathcal S}}$ of size $m$ drawn according to $Q$: $${\mathrm{disc}}_{L_q}(Q, {\widehat}Q) \leq 4q {\widehat}{\mathfrak{R}}_{{\mathcal S}}(H) + 3 M \sqrt{\frac{\log \frac{2}{\delta}}{2m}}.$$
The function $f\colon x \mapsto x^q$ is $q$-Lipschitz for $x \in [0, 1]$: $$|f(x') - f(x)| \leq q |x' - x|,$$ and $f(0) = 0$. For $L = L_q$, $L_H = {\{x \mapsto {\lverth'(x) -
h(x)\rvert}^q \colon h, h' \in H\}}$. Thus, by Talagrand’s contraction lemma [@talagrand], ${\widehat}{\mathfrak{R}}(L_H)$ is bounded by $2q {\widehat}{\mathfrak{R}}(H')$ with $H' \!=\! {\{x \mapsto (h'(x) - h(x))\colon h, h' \in H\}}$. Then, ${\widehat}{\mathfrak{R}}_{{\mathcal S}}(H')
$ can be written and bounded as follows $$\begin{gathered}
{\widehat}{\mathfrak{R}}_{{\mathcal S}}(H') = \operatorname*{\rm E}_\sigma \bigl[
\sup_{h, h'} \frac{1}{m} |\sum_{i = 1}^m \sigma_i (h(x_i) - h'(x_i))|\bigr] \\
{\begin{aligned}
& \leq \operatorname*{\rm E}_\sigma [ \sup_{h} \frac{1}{m} |\sum_{i = 1}^m \sigma_i
h(x_i)| ] + \operatorname*{\rm E}_\sigma [ \sup_{h'} \frac{1}{m} |\sum_{i = 1}^m \sigma_i
h'(x_i)| ] \\
& = 2 {\widehat}{\mathfrak{R}}_{{\mathcal S}}(H),
\end{aligned}}\end{gathered}$$ using the definition of the Rademacher variables and the sub-additivity of the supremum function. This proves the inequality ${\widehat}{\mathfrak{R}}(L_H) \leq 4q {\widehat}{\mathfrak{R}}(h)$ and the corollary.
A very similar proof gives the following result for classification.
\[cor:dis\_class\] Let $H$ be a set of classifiers mapping $X$ to ${\{0, 1\}}$ and let $L_{01}$ denote the 0-1 loss. Then, with the notation of Corollary \[cor:dis\_bound\], for any $\delta > 0$, with probability at least $1 - \delta$ over samples ${{\mathcal S}}$ of size $m$ drawn according to $Q$: $${\mathrm{disc}}_{L_{01}}(Q, {\widehat}Q) \leq 4 {\widehat}{\mathfrak{R}}_{{\mathcal S}}(H) + 3 \sqrt{\frac{\log \frac{2}{\delta}}{2m}}.$$
The factor of $4$ can in fact be reduced to $2$ in these corollaries when using a more favorable constant in the contraction lemma. The following corollary shows that the discrepancy distance can be estimated from finite samples.
\[cor:dis\_bound\_emp\] Let $H$ be a hypothesis set bounded by some $M \!>\! 0$ for the loss function $L_q$: $L_q(h, h') \!\leq\! M$, for all $h, h' \!\in\! H$. Let $Q$ be a distribution over $X$ and ${\widehat}Q$ the corresponding empirical distribution for a sample ${{\mathcal S}}$, and let $P$ be a distribution over $X$ and ${\widehat}P$ the corresponding empirical distribution for a sample ${{\mathcal T}}$. Then, for any $\delta > 0$, with probability at least $1 - \delta$ over samples ${{\mathcal S}}$ of size $m$ drawn according to $Q$ and samples ${{\mathcal T}}$ of size $n$ drawn according to $P$: $$\begin{gathered}
{\mathrm{disc}}_{L_q}(P, Q) \leq {\mathrm{disc}}_{L_q}({\widehat}P, {\widehat}Q) +\\
\qquad 4q \Big({\widehat}{\mathfrak{R}}_{{\mathcal S}}(H) + {\widehat}{\mathfrak{R}}_{{\mathcal T}}(H)\Big) +
3 M \Bigg(\sqrt{\frac{\log \frac{4}{\delta}}{2m}} + \sqrt{\frac{\log
\frac{4}{\delta}}{2n}}\Bigg).\end{gathered}$$
By the triangle inequality, we can write $$\begin{gathered}
{\mathrm{disc}}_{L_q}(P, Q) \leq {\mathrm{disc}}_{L_q}(P, {\widehat}P) + {\mathrm{disc}}_{L_q}({\widehat}P, {\widehat}Q) +\\
{\mathrm{disc}}_{L_q}(Q, {\widehat}Q).\end{gathered}$$ The result then follows by the application of Corollary \[cor:dis\_bound\] to ${\mathrm{disc}}_{L_q}(P, {\widehat}P)$ and ${\mathrm{disc}}_{L_q}(Q, {\widehat}Q)$.
As with Corollary \[cor:dis\_class\], a similar result holds for the 0-1 loss in classification.
Domain Adaptation: Generalization Bounds {#sec:bounds}
========================================
This section presents generalization bounds for domain adaptation given in terms of the discrepancy distance just defined. In the context of adaptation, two types of questions arise:
1. we may ask, as for standard generalization, how the average loss of a hypothesis on the target distribution, ${{\cal L}}_P(h, f)$, differs from ${{\cal L}}_{{\widehat}Q}(h, f)$, its empirical error based on the empirical distribution ${\widehat}Q$;
2. another natural question is, given a specific learning algorithm, by how much does ${{\cal L}}_P(h_Q, f)$ deviate from ${{\cal L}}_P(h_P,
f)$ where $h_Q$ is the hypothesis returned by the algorithm when trained on a sample drawn from $Q$ and $h_P$ the one it would have returned by training on a sample drawn from the true target distribution $P$.
We will present theoretical guarantees addressing both questions.
Generalization bounds
---------------------
Let $h_Q^* \in \operatorname*{\rm argmin}_{h \in H} {{\cal L}}_Q(h, f_Q)$ and similarly let $h_P^*$ be a minimizer of ${{\cal L}}_P(h, f_P)$. Note that these minimizers may not be unique. For adaptation to succeed, it is natural to assume that the average loss ${{\cal L}}_Q(h_Q^*, h_P^*)$ between the best-in-class hypotheses is small. Under that assumption and for a small discrepancy distance, the following theorem provides a useful bound on the error of a hypothesis with respect to the target domain.
\[th:gen\_bound\] Assume that the loss function $L$ is symmetric and obeys the triangle inequality. Then, for any hypothesis $h \in H$, the following holds $$\begin{gathered}
{{\cal L}}_P(h, f_P) \leq {{\cal L}}_P(h_P^*, f_P) + {{\cal L}}_Q(h, h_Q^*) + {\mathrm{disc}}(P, Q) \\
+ {{\cal L}}_Q(h_Q^*, h_P^*).
\label{eq:gen_bound}\end{gathered}$$
Fix $h \in H$. By the triangle inequality property of $L$ and the definition of the discrepancy ${\mathrm{disc}}_L(P, Q)$, the following holds $$\begin{aligned}
{{\cal L}}_P(h, f_P)
& \leq {{\cal L}}_P(h, h_Q^*) + {{\cal L}}_P(h_Q^*, h_P^*) + {{\cal L}}_P(h_P^*, f_P)\\
& \leq {{\cal L}}_Q(h, h_Q^*) + {\mathrm{disc}}_L(P, Q) + {{\cal L}}_P(h_Q^*, h_P^*) \\
& \quad + {{\cal L}}_P(h_P^*, f_P).\dqed\end{aligned}$$
We compare (\[eq:gen\_bound\]) with the main adaptation bound given by and : $$\begin{gathered}
{{\cal L}}_P(h, f_P) \leq {{\cal L}}_Q(h, f_Q) + {\mathrm{disc}}_L(P, Q) + \\
\min_{h \in H} \big({{\cal L}}_Q(h, f_Q) + {{\cal L}}_P(h, f_P) \big).
\label{bound-old-adap}\end{gathered}$$ It is very instructive to compare the two bounds. Intuitively, the bound of Theorem \[th:gen\_bound\] has only one error term that involves the target function, while the bound of (\[bound-old-adap\]) has three terms involving the target function. One extreme case is when there is a single hypothesis $h$ in $H$ and a single target function $f$. In this case, Theorem \[th:gen\_bound\] gives a bound of ${{\cal L}}_P(h, f) + {\mathrm{disc}}(P, Q) $, while the bound supplied by (\[bound-old-adap\]) is $2{{\cal L}}_Q(h, f) + {{\cal L}}_P(h, f) + {\mathrm{disc}}(P, Q)$, which is larger than $3 {{\cal L}}_P(h, f) + {\mathrm{disc}}(P, Q)$ when ${{\cal L}}_Q(h, f) \leq
{{\cal L}}_P(h, f)$. One can even see that the bound of (\[bound-old-adap\]) might become vacuous for moderate values of ${{\cal L}}_Q(h, f)$ and ${{\cal L}}_P(h, f)$. While this is clearly an extreme case, an error with a factor of 3 can arise in more realistic situations, especially when the distance between the target function and the hypothesis class is significant.
While in general the two bounds are incomparable, it is worthwhile to compare them using some relatively plausible assumptions. Assume that the discrepancy distance between $P$ and $Q$ is small and so is the average loss between $h_Q^*$ and $h_P^*$. These are natural assumptions for adaptation to be possible. Then, Theorem \[th:gen\_bound\] indicates that the regret ${{\cal L}}_P(h, f_P) -
{{\cal L}}_P(h_P^*, f_P)$ is essentially bounded by ${{\cal L}}_Q(h, h_Q^*)$, the average loss with respect to $h_Q^*$ on $Q$. We now consider several special cases of interest.
1. When $h_Q^* = h_P^*$ then $h^* = h_Q^* = h_P^*$ and the bound of Theorem \[th:gen\_bound\] becomes $$\label{eq:bound1}
{{\cal L}}_P(h, f_P) \leq {{\cal L}}_P(h^*, f_P) + {{\cal L}}_Q(h, h^*) + {\mathrm{disc}}(P, Q).$$ The bound of (\[bound-old-adap\]) becomes $$\begin{gathered}
{{\cal L}}_P(h, f_P) \leq {{\cal L}}_P(h^*, f_P) + {{\cal L}}_Q(h, f_Q) + \\{{\cal L}}_Q(h^*, f_Q) + {\mathrm{disc}}(P, Q),\end{gathered}$$ where the right-hand side essentially includes the sum of $3$ errors and is always larger than the right-hand side of (\[eq:bound1\]) since by the triangle inequality ${{\cal L}}_Q(h, h^*) \leq {{\cal L}}_Q(h, f_Q)$ $ +
{{\cal L}}_Q(h^*, f_Q)$.
2. When $h_Q^* = h_P^*=h^* \wedge {\mathrm{disc}}(P, Q) = 0$, the bound of Theorem \[th:gen\_bound\] becomes $${{\cal L}}_P(h, f_P) \leq {{\cal L}}_P(h^*, f_P) + {{\cal L}}_Q(h, h^*),$$ which coincides with the standard generalization bound. The bound of (\[bound-old-adap\]) does not coincide with the standard bound and leads to: $${{\cal L}}_P(h, f_P) \leq {{\cal L}}_P(h^*, f_P) + {{\cal L}}_Q(h, f_Q) + {{\cal L}}_Q(h^*, f_Q).$$
3. When $f_P \!\in\! H$ (consistent case), the bound of (\[bound-old-adap\]) simplifies to, $${\lvert{{\cal L}}_P(h, f_P) - {{\cal L}}_Q(h, f_P)\rvert} \leq {\mathrm{disc}}_L(Q, P),$$ and it can also be derived using the proof of Theorem \[th:gen\_bound\].
Finally, clearly Theorem \[th:gen\_bound\] leads to bounds based on the empirical error of $h$ on a sample drawn according to $Q$. We give the bound related to the 0-1 loss, others can be derived in a similar way from Corollaries \[cor:dis\_bound\]-\[cor:dis\_bound\_emp\] and other similar corollaries. The result follows Theorem \[th:gen\_bound\] combined with Corollary \[cor:dis\_bound\_emp\], and a standard Rademacher classification bound (Theorem \[th:rademacher\_classification\]) [@bartlett].
\[th:gen\_bound\_emp\] Let $H$ be a family of functions mapping $X$ to ${\{0, 1\}}$ and let the rest of the assumptions be as in Corollary \[cor:dis\_bound\_emp\]. Then, for any hypothesis $h \in H$, with probability at least $1 - \delta$, the following adaptation generalization bound holds for the 0-1 loss: $$\begin{gathered}
{{\cal L}}_P(h, f_P) - {{\cal L}}_P(h_P^*, f_P) \leq \\
{{\cal L}}_{{\widehat}Q}(h, h_Q^*) +
{\mathrm{disc}}_{L_{01}}({\widehat}P, {\widehat}Q) +
(4q + \frac{1}{2}) {\widehat}{\mathfrak{R}}_{{\mathcal S}}(H) + 4 q {\widehat}{\mathfrak{R}}_{{\mathcal T}}(H) + \\
4 \sqrt{\frac{\log \frac{8}{\delta}}{2m}} + 3 \sqrt{\frac{\log
\frac{8}{\delta}}{2n}} + {{\cal L}}_Q(h_Q^*, h_P^*).\end{gathered}$$
Guarantees for regularization-based algorithms
----------------------------------------------
In this section, we first assume that the hypothesis set $H$ includes the target function $f_P$. Note that this does not imply that $f_Q$ is in $H$. Even when $f_P$ and $f_Q$ are restrictions to $\operatorname{supp}(P)$ and $\operatorname{supp}(Q)$ of the same labeling function $f$, we may have $f_P \in H$ and $f_Q \not \in H$ and the source problem could be non-realizable. Figure \[fig:consistent\] illustrates this situation.
For a fixed loss function $L$, we denote by $R_{{\widehat}Q}(h)$ the empirical error of a hypothesis $h$ with respect to an empirical distribution ${\widehat}Q$: $R_{{\widehat}Q}(h) = {{\cal L}}_{{\widehat}Q}(h, f)$. Let $N\colon H \to \Rset_+$ be a function defined over the hypothesis set $H$. We will assume that $H$ is a convex subset of a vector space and that the loss function $L$ is convex with respect to each of its arguments. Regularization-based algorithms minimize an objective of the form $$F_{{\widehat}Q}(h) = {\widehat}R_{{\widehat}Q}(h) + \lambda N(h),$$ where $\lambda \geq 0$ is a trade-off parameter. This family of algorithms includes support vector machines (SVM) [@ccvv], support vector regression (SVR) [@vapnik98], kernel ridge regression [@krr], and other algorithms such as those based on the relative entropy regularization [@bousquet-jmlr].
We denote by $B_F$ the Bregman divergence associated to a convex function $F$, $$B_F(f \Arrowvert g) = F(f) - F(g) - {\left\langle f - g , \nabla
F(g) \right\rangle}$$ and define $\Delta h$ as $\Delta h = h' - h$.
\[lemma:stability\] Let the hypothesis set $H$ be a vector space. Assume that $N$ is a proper closed convex function and that $N$ and $L$ are differentiable. Assume that $F_{{\widehat}Q}$ admits a minimizer $h \in H$ and $F_{{\widehat}P}$ a minimizer $h' \in H$ and that $f_P$ and $f_Q$ coincide on the support of ${\widehat}Q$. Then, the following bound holds, $$B_N(h' \Arrowvert h) + B_N(h \Arrowvert h') \leq \frac{2{\mathrm{disc}}_L({\widehat}P, {\widehat}Q)}{\lambda}.$$
Since $B_{F_{{\widehat}Q}} = B_{{\widehat}R_{{\widehat}Q}} + \lambda B_{N}$ and $B_{F_{{{\widehat}P}}} =
B_{{\widehat}R_{{{\widehat}P}}} + \lambda B_{N}$, and a Bregman divergence is non-negative, the following inequality holds: $$\lambda \bigl(B_N(h' \Arrowvert h) + B_N(h \Arrowvert h')\bigr) \leq B_{F_{{\widehat}Q}}(h' \Arrowvert h) + B_{F_{{{\widehat}P}}}(h \Arrowvert h').$$ By the definition of $h$ and $h'$ as the minimizers of $F_{{\widehat}Q}$ and $F_{{{\widehat}P}}$, $\nabla_{{\widehat}Q} F (h) = \nabla_{{\widehat}P} F (h') = 0$ and $$\begin{gathered}
\lambda \big(B_{F_{{\widehat}Q}}(h' \Arrowvert h) + B_{F_{{{\widehat}P}}}(h \Arrowvert h')\big) \\
{\begin{aligned}
& = {\widehat}R_{{\widehat}Q}(h') - {\widehat}R_{{\widehat}Q}(h) + {\widehat}R_{{\widehat}P}(h) - {\widehat}R_{{\widehat}P}(h')\\
& = \big({{\cal L}}_{{\widehat}P}(h, f_P) - {{\cal L}}_{{\widehat}Q}(h, f_P)\big)\\
& \quad - \big({{\cal L}}_{{\widehat}P}(h', f_P) - {{\cal L}}_{{\widehat}Q}(h', f_P)\big)
\leq 2 {\mathrm{disc}}_L({\widehat}P, {\widehat}Q).
\end{aligned}}\end{gathered}$$ This last inequality holds since by assumption $f_P$ is in $H$.
We will say that a loss function $L$ is *$\sigma$-admissible* when there exists $\sigma \in \Rset_+$ such that for any two hypotheses $h, h' \in H$ and for all $x \in X$, and $y \in Y$, $$\big|L(h(x), y) - L(h'(x), y)\big| \leq \sigma \big|h(x) - h'(x)\big|.$$ This assumption holds for the hinge loss with $\sigma = 1$ and for the $L_q$ loss with $\sigma = q (2M)^{q - 1}$ when the hypothesis set and the set of output labels are bounded by some $M \in \Rset_+$: $\forall h
\in H, \forall x \in X, |h(x)| \leq M$ and $\forall y \in Y, |y| \leq
M$.
\[th:stability\] Let $K\colon X \times X \to \Rset$ be a positive-definite symmetric kernel such that $K(x, x) \leq \kappa^2 < \infty$ for all $x \in X$, and let $H$ be the reproducing kernel Hilbert space associated to $K$. Assume that the loss function $L$ is $\sigma$-admissible. Let $h'$ be the hypothesis returned by the regularization algorithm based on $N(\cdot) = {\lVert\cdot\rVert}_K^2$ for the empirical distribution ${\widehat}P$, and $h$ the one returned for the empirical distribution ${\widehat}Q$, and that and that $f_P$ and $f_Q$ coincide on $\operatorname{supp}({\widehat}Q)$. Then, for all $x \in X$, $y \in Y$, $$\big| L(h'(x), y) - L(h(x), y) \big| \leq \kappa \sigma \sqrt{\frac{{\mathrm{disc}}_L({\widehat}P, {\widehat}Q)}{\lambda}}.$$
For $N(\cdot) = {\lVert\cdot\rVert}_K^2$, $N$ is a proper closed convex function and is differentiable. We have $B_N(h' \Arrowvert h) =
{\lVerth' - h\rVert}_K^2$, thus $B_N(h' \Arrowvert h) + B_N(h \Arrowvert
h') = 2 {\lVert\Delta h\rVert}_K^2$. When $L$ is differentiable, by Lemma \[lemma:stability\], $$\label{eq:Delta_h}
2 {\lVert\Delta h\rVert}_K^2 \leq \frac{2{\mathrm{disc}}_L({\widehat}P, {\widehat}Q)}{\lambda}.$$ This result can also be shown directly without assuming that $L$ is differentiable by using the convexity of $N$ and the minimizing properties of $h$ and $h'$ with a proof that is longer than that of Lemma \[lemma:stability\].
Now, by the reproducing property of $H$, for all $x \in H$, $\Delta
h(x) = {\left\langle \Delta h , K(x, \cdot) \right\rangle}$ and by the Cauchy-Schwarz inequality, ${\lvert\Delta h(x)\rvert} \leq {\lVert\Delta h\rVert}_K (K(x, x))^{1/2}
\leq \kappa {\lVert\Delta h\rVert}_K$. By the $\sigma$-admissibility of $L$, for all $x \in X$, $y \in Y$, $${\lvertL(h'(x), y) - L(h(x), y)\rvert} \leq \sigma {\lvert\Delta h (x)\rvert} \leq \kappa \sigma {\lVert\Delta h\rVert}_K,$$ which, combined with (\[eq:Delta\_h\]), proves the statement of the theorem.
Theorem \[th:stability\] provides a guarantee on the pointwise difference of the loss for $h'$ and $h$ with probability one, which of course is stronger than a bound on the difference between expected losses or a probabilistic statement. The result, as well as the proof, also suggests that the discrepancy distance is the “right” measure of difference of distributions for this context. The theorem applies to a variety of algorithms, in particular SVMs combined with arbitrary PDS kernels and kernel ridge regression.
In general, the functions $f_P$ and $f_Q$ may not coincide on $\operatorname{supp}({\widehat}Q)$. For adaptation to be possible, it is reasonable to assume however that $$L_{{\widehat}Q}(f_Q(x), f_P(x)) \ll 1 \quad \text{and}
\quad L_{{\widehat}P}(f_Q(x), f_P(x)) \ll 1.$$ This can be viewed as a condition on the proximity of the labeling functions (the $Y$s), while the discrepancy distance relates to the distributions on the input space (the $X$s). The following result generalizes Theorem \[th:stability\] to this setting in the case of the square loss.
\[th:stability2\] Under the assumptions of Theorem \[th:stability\], but with $f_Q$ and $f_P$ potentially different on $\operatorname{supp}({\widehat}Q)$, when $L$ is the square loss $L_2$ and $\delta^2 = L_{{\widehat}Q}(f_Q(x), f_P(x)) \ll 1$, then, for all $x \in X$, $y \in Y$, $$\begin{gathered}
\big| L(h'(x), y) - L(h(x), y) \big| \leq \\ \frac{2 \kappa M}{\lambda}\Big(\kappa \delta +
\sqrt{\kappa^2 \delta^2 + 4 \lambda {\mathrm{disc}}_L({\widehat}P, {\widehat}Q)}\Big).\end{gathered}$$
Proceeding as in the proof of Lemma \[lemma:stability\] and using the definition of the square loss and the Cauchy-Schwarz inequality give $$\begin{gathered}
\lambda \big(B_{F_{{\widehat}Q}}(h' \Arrowvert h) + B_{F_{{{\widehat}P}}}(h \Arrowvert h')\big) \\
{\begin{aligned}
& = {\widehat}R_{{\widehat}Q}(h') - {\widehat}R_{{\widehat}Q}(h) + {\widehat}R_{{\widehat}P}(h) - {\widehat}R_{{\widehat}P}(h')\\
& = \big({{\cal L}}_{{\widehat}P}(h, f_P) - {{\cal L}}_{{\widehat}Q}(h, f_P)\big)\\
& \quad - \big({{\cal L}}_{{\widehat}P}(h', f_P) - {{\cal L}}_{{\widehat}Q}(h', f_P)\big)\\
& \qquad + 2 \operatorname*{\rm E}_{{\widehat}Q} [(h'(x) - h(x)) (f_P(x) - f_Q(x)]\\
& \leq 2 {\mathrm{disc}}_L({\widehat}P, {\widehat}Q) + 2 \sqrt{\operatorname*{\rm E}_{{\widehat}Q} [\Delta h(x)^2] \operatorname*{\rm E}_{{\widehat}Q}[L(f_P(x), f_Q(x))]}\\
& \leq 2 {\mathrm{disc}}_L({\widehat}P, {\widehat}Q) + 2 \kappa {\lVert\Delta h\rVert}_K \delta.
\end{aligned}}\end{gathered}$$ Since $N(\cdot) = {\lVert\cdot\rVert}_K^2$, the inequality can be rewritten as $$\lambda {\lVert\Delta h\rVert}_K^2 \leq {\mathrm{disc}}_L({\widehat}P, {\widehat}Q) + \kappa \delta {\lVert\Delta h\rVert}_K .$$ Solving the second-degree polynomial in ${\lVert\Delta h\rVert}_K$ leads to the equivalent constraint $${\lVert\Delta h\rVert}_K \leq \frac{1}{2 \lambda}\Big(\kappa \delta + \\
\sqrt{\kappa^2 \delta^2 + 4 \lambda {\mathrm{disc}}_L({\widehat}P, {\widehat}Q)}\Big).$$ The result then follows by the $\sigma$-admissibility of $L$ as in the proof of Theorem \[th:stability\], with $\sigma = 4M$.
Using the same proof schema, similar bounds can be derived for other loss functions.
When the assumption $f_P \in H$ is relaxed, the following theorem holds.
\[th:stability3\] Under the assumptions of Theorem \[th:stability\], but with $f_P$ not necessarily in $H$ and $f_Q$ and $f_P$ potentially different on $\operatorname{supp}({\widehat}Q)$, when $L$ is the square loss $L_2$ and $\delta' = L_{{\widehat}Q}(h_P^*(x), f_Q(x))^{1/2} + L_{{\widehat}P}(h_P^*(x), f_P(x))^{1/2} \ll 1$, then, for all $x \in X$, $y \in Y$, $$\begin{gathered}
\big| L(h'(x), y) - L(h(x), y) \big| \leq \\ \frac{2 \kappa M}{\lambda}\Big(\kappa \delta' +
\sqrt{\kappa^2 \delta'^2 + 4 \lambda {\mathrm{disc}}_L({\widehat}P, {\widehat}Q)}\Big).\end{gathered}$$
Proceeding as in the proof of Theorem \[th:stability2\] and using the definition of the square loss and the Cauchy-Schwarz inequality give $$\begin{gathered}
\lambda \big(B_{F_{{\widehat}Q}}(h' \Arrowvert h) + B_{F_{{{\widehat}P}}}(h \Arrowvert h')\big) \\
{\begin{aligned}
& = \big({{\cal L}}_{{\widehat}P}(h, h_P^*) - {{\cal L}}_{{\widehat}Q}(h, h_P^*)\big)\\
& \quad - \big({{\cal L}}_{{\widehat}P}(h', h_P^*) - {{\cal L}}_{{\widehat}Q}(h', h_P^*)\big)\\
& \qquad - 2 \operatorname*{\rm E}_{{\widehat}P} [(h'(x) - h(x)) (h_P^*(x) - f_P(x)]\\
& \qquad + 2 \operatorname*{\rm E}_{{\widehat}Q} [(h'(x) - h(x)) (h_P^*(x) - f_Q(x)]\\
& \leq 2 {\mathrm{disc}}_L({\widehat}P, {\widehat}Q) + 2 \sqrt{\operatorname*{\rm E}_{{\widehat}P} [\Delta h(x)^2] \operatorname*{\rm E}_{{\widehat}P}[L(h_P^*(x), f_P(x))]} \\
& + 2 \sqrt{\operatorname*{\rm E}_{{\widehat}Q} [\Delta h(x)^2] \operatorname*{\rm E}_{{\widehat}Q}[L(h_P^*(x), f_Q(x))]}\\
& \leq 2 {\mathrm{disc}}_L({\widehat}P, {\widehat}Q) + 2 \kappa {\lVert\Delta h\rVert}_K \delta'.
\end{aligned}}\end{gathered}$$ The rest of the proof is identical to that of Theorem \[th:stability2\].
Discrepancy Minimization Algorithms {#sec:alg}
===================================
The discrepancy distance ${\mathrm{disc}}_L({\widehat}P, {\widehat}Q)$ appeared as a critical term in several of the bounds in the last section. In particular, Theorems \[th:stability\] and \[th:stability2\] suggest that if we could select, instead of ${\widehat}Q$, some other empirical distribution ${\widehat}Q'$ with a smaller empirical discrepancy ${\mathrm{disc}}_L({\widehat}P, {\widehat}Q')$ and use that for training a regularization-based algorithm, a better guarantee would be obtained on the difference of pointwise loss between $h'$ and $h$. Since $h'$ is fixed, a sufficiently smaller discrepancy would actually lead to a hypothesis $h$ with pointwise loss closer to that of $h'$.
The training sample is given and we do not have any control over the support of ${\widehat}Q$. But, we can search for the distribution ${\widehat}Q'$ with the minimal empirical discrepancy distance: $$\label{eq:dis_min}
{\widehat}Q' = \operatorname*{\rm argmin}_{{\widehat}Q' \in {{\cal Q}}} {\mathrm{disc}}_L({\widehat}P, {\widehat}Q'),$$ where ${{\cal Q}}$ denotes the set of distributions with support $\operatorname{supp}({\widehat}Q)$. This leads to an optimization problem that we shall study in detail in the case of several loss functions.
Note that using ${\widehat}Q'$ instead of ${\widehat}Q$ for training can be viewed as *reweighting* the cost of an error on each training point. The distribution ${\widehat}Q'$ can be used to emphasize some points or de-emphasize others to reduce the empirical discrepancy distance. This bears some similarity with the reweighting or *importance weighting* ideas used in statistics and machine learning for sample bias correction techniques [@elkan; @bias] and other purposes. Of course, the objective optimized here based on the discrepancy distance is distinct from that of previous reweighting techniques.
We will denote by $S_Q$ the support of ${\widehat}Q$, by $S_P$ the support of ${\widehat}P$, and by $S$ their union $\operatorname{supp}({\widehat}Q) \cup \operatorname{supp}({\widehat}P)$, with $|S_Q| = m_0 \leq m$ and $|S_P| = n_0 \leq n$.
In view of the definition of the discrepancy distance, problem (\[eq:dis\_min\]) can be written as a min-max problem: $${\widehat}Q' = \operatorname*{\rm argmin}_{{\widehat}Q' \in {{\cal Q}}} \max_{h, h' \in H}
{\lvert{{\cal L}}_{{\widehat}P}(h', h) - {{\cal L}}_{{\widehat}Q'}(h', h)\rvert}.$$ As with all min-max problems, the problem has a natural game theoretical interpretation. However, here, in general, we cannot permute the $\min$ and $\max$ operators since the convexity-type assumptions of the minimax theorems do not hold. Nevertheless, since the max-min value is always a lower bound for the min-max, it provides us with a lower bound on the value of the game, that is the minimal discrepancy: $$\begin{gathered}
\label{eq:max-min}
\max_{h, h' \in H} \min_{{\widehat}Q' \in {{\cal Q}}} {\lvert{{\cal L}}_{{\widehat}P}(h', h) -
{{\cal L}}_{{\widehat}Q'}(h', h)\rvert}
\leq\\
\min_{{\widehat}Q' \in {{\cal Q}}} \max_{h, h' \in H} {\lvert{{\cal L}}_{{\widehat}P}(h', h) -
{{\cal L}}_{{\widehat}Q'}(h', h)\rvert}.\end{gathered}$$ We will later make use of this inequality. Let us now examine the minimization problem (\[eq:dis\_min\]) and its algorithmic solutions in the case of classification with the 0-1 loss and regression with the $L_2$ loss.
Classification, 0-1 Loss
------------------------
For the 0-1 loss, the problem of finding the best distribution ${\widehat}Q'$ can be reformulated as the following min-max program: $$\begin{aligned}
& \min_{\ Q'} \max_{a \in H \Delta H} \big| {\widehat}Q'(a) -
{\widehat}P(a) \big| \\
& \text{subject to} \quad \forall x \in S_Q, {\widehat}Q'(x) \geq 0 \wedge \sum_{x \in S_Q} {\widehat}Q'(x) = 1,\end{aligned}$$ where we have identified $H \Delta H = {\{{\lverth' - h\rvert}\colon h, h'
\in H\}}$ with the set of regions $a \subseteq X$ that are the support of an element of $H \Delta H$. This problem is similar to the min-max resource allocation problem that arises in task optimization [@kouvelis]. It can be rewritten as the following linear program (LP): $$\begin{aligned}
\min_{\ Q'} & \quad \delta \\
\text{subject to} & \quad \forall a \in H \Delta H, {\widehat}Q'(a) -
{\widehat}P(a) \leq \delta\\
& \quad \forall a \in H \Delta H, {\widehat}P(a) - {\widehat}Q'(a)
\leq \delta\\
& \quad \forall x \in S_Q, {\widehat}Q'(x) \geq 0 \wedge \sum_{x \in S_Q}
{\widehat}Q'(x) = 1.\end{aligned}$$ The number of constraints is proportional to $|H \Delta H|$ but it can be reduced to a finite number by observing that two subsets $a, a'
\!\in\! H \Delta H$ containing the same elements of $S$ lead to redundant constraints, since $$\big| {\widehat}Q'(a) - {\widehat}P(a) \big| = \big| {\widehat}Q'(a') - {\widehat}P(a') \big|.$$ Thus, it suffices to keep one canonical member $a$ for each such equivalence class. The necessary number of constraints to be considered is proportional to $\Pi_{H \Delta H}(m_0 + n_0)$, the shattering coefficient of order $(m_0 + n_0)$ of the hypothesis class $H \Delta H$. By the Sauer’s lemma, this is bounded in terms of the VC-dimension of the class $H \Delta H$, $\Pi_{H \Delta H}(m_0 + n_0)
\leq O((m_0 + n_0)^{VC(H \Delta H)})$, which can be bounded by $O((m_0
+ n_0)^{2 VC(H)})$ since it is not hard to see that $VC(H \Delta H)
\leq 2 VC(H)$.
In cases where we can test efficiently whether there exists a consistent hypothesis in $H$, e.g., for half-spaces in $\mathbb{R}^d$, we can generate in time $O((m_0+n_0)^{2d})$ all consistent labeling of the sample points by $H$. (We remark that computing the discrepancy with the 0-1 loss is closely related to agnostic learning. The implications of this fact will be described in a longer version of this paper.)
\
(a)\
\
(b)
Computing the Discrepancy in 1D
-------------------------------
We consider the case where $X = [0, 1]$ and derive a simple algorithm for minimizing the discrepancy for 0-1 loss. Let $H$ be the class of all prefixes (i.e., $[0,z]$) and suffixes (i.e., $[z,1]$). Our class of $H\Delta H$ includes all the intervals (i.e., $(z_1,z_2]$) and their complements (i.e., $[0,z_1]\cup (z_2,1]$). We start with a general lower bound on the discrepancy.
Let $U$ denote the set of *unlabeled regions*, that is the set of regions $a$ such that $a \cap S_Q = \emptyset$ and $a \cap S_P \neq
\emptyset$.
If $a$ is an unlabeled region, then ${\lvert{\widehat}Q'(a) - {\widehat}P(a)\rvert} =
{\widehat}P(a)$ for any ${\widehat}Q'$. Thus, by the max-min inequality (\[eq:max-min\]), the following lower bound holds for the minimum discrepancy: $$\label{eq:lower_bound}
\max_{a \in U} {\widehat}P(a) \leq
\min_{{\widehat}Q' \in {{\cal Q}}} \max_{h, h' \in H} {\lvert{{\cal L}}_{{\widehat}P}(h', h) -
{{\cal L}}_{{\widehat}Q'}(h', h)\rvert}.$$ In particular, if there is a large unlabeled region $a$, we cannot hope to achieve a small empirical discrepancy. In the one-dimensional case, we give a simple linear-time algorithm that does not require an LP and show that the lower bound (\[eq:lower\_bound\]) is reached. Thus, in that case, the $\min$ and $\max$ operators commute and the minimal discrepancy distance is precisely $\min_{a \in U} {\widehat}P(a)$.
Given our definition of $H$, the unlabeled regions are open intervals, or complements of these sets, containing only points from $S_P$ with endpoints defined by elements of $S_Q$.
Let us denote by $s_1, \ldots, s_{m_0}$ the elements of $S_Q$, by $n_i$, $i \in [1, m_0]$, the number of consecutive unlabeled points to the right of $s_i$ and $n=\sum n_i$. We will make an additional technical assumption that there are no unlabeled points to the left of $s_1$. Our algorithm consists of defining the weight ${\widehat}Q'(s_i)$ as follows: $${\widehat}Q'(s_i) = n_i/n.$$ This requires first sorting $S_Q\cup S_P$ and then computing $n_i$ for each $s_i$. Figure \[fig:1d\_example\] illustrates the algorithm.
\[prop:1d\_alg\] Assume that $X$ consists of the set of points on the real line and $H$ the set of half-spaces on $X$. Then, for any ${\widehat}Q$ and ${\widehat}P$, ${\widehat}Q'(s_i) = n_i/n$ minimizes the empirical discrepancy and can be computed in time $O((m + n) \log (m + n))$.
Consider an interval $[z_1, z_2]$ that maximizes the discrepancy of ${\widehat}Q'$. The case of a complement of an interval is the same, since the discrepancy of a hypothesis and its negation are identical. Let $s_i, \ldots, s_j \in [z_1, z_2]$ be the subset of ${\widehat}Q$ in that interval, and $p_{i'}, \ldots, p_{j'}\in [z_1, z_2]$ be the subset of ${\widehat}P$ in that interval. The discrepancy is $d=|\sum_{k=i}^j {\widehat}Q'(s_k) - \frac{j'-i'}{n}|$. By our definition of ${\widehat}Q'$, we have that $\sum_{k=i}^j {\widehat}Q'(s_k)= \frac{1}{n}\sum_{k=i}^j n_k$. Let $p_{i''}$ be the maximal point in ${\widehat}P$ which is less than $s_i$ and $j''$ the minimal point in ${\widehat}P$ larger than $s_j$. We have that $j'-i' = (i''-i') + \sum_{k=i}^{j-1} n_k + (j''-j'))$. Therefore $d=| (i''-i') +(j''-j')-n_j|= | (i''-i') - (n_j
-(j''-j'))|$. Since $d$ is maximal and both terms are non-negative, one of them is zero. Since $j'-j'' \leq n_j$ and $i''-i'\leq n_i$, the discrepancy of ${\widehat}Q'$ meets the lower bound of (\[eq:lower\_bound\]) and is thus optimal.
Regression, $L_2$ loss {#sec:l2}
----------------------
For the square loss, the problem of finding the best distribution can be written as $$\begin{aligned}
& \min_{{\widehat}Q' \in {{\cal Q}}} \max_{h, h' \in H} \Big| \operatorname*{\rm E}_{{\widehat}P}[(h'(x) - h(x))^2] - \operatorname*{\rm E}_{{\widehat}Q'}[(h'(x) - h(x))^2] \Big|.\end{aligned}$$ If $X$ is a subset of $\Rset^N$, $N \!>\! 1$, and the hypothesis set $H$ is a set of bounded linear functions $H = {\{{{{\mathbf x}}}\mapsto {{{\mathbf w}}}^\top
{{{\mathbf x}}}\colon {\lVert{{{\mathbf w}}}\rVert} \!\leq\! 1\}}$, then, the problem can be rewritten as $$\begin{aligned}
& \min_{{\widehat}Q' \in {{\cal Q}}} \max_{\substack{{\lVert{{{\mathbf w}}}\rVert} \leq 1\\{\lVert{{{\mathbf w}}}'\rVert} \leq 1}} \Big| \operatorname*{\rm E}_{{\widehat}P}[(({{{\mathbf w}}}' - {{{\mathbf w}}})^\top {{{\mathbf x}}})^2] - \operatorname*{\rm E}_{{\widehat}Q'}[(({{{\mathbf w}}}' - {{{\mathbf w}}})^\top {{{\mathbf x}}})^2]\Big| \nonumber\\
& = \min_{{\widehat}Q' \in {{\cal Q}}} \max_{\substack{{\lVert{{{\mathbf w}}}\rVert} \leq 1\\{\lVert{{{\mathbf w}}}'\rVert} \leq 1}} \Big| \sum_{{{{\mathbf x}}}\in S} ({\widehat}P({{{\mathbf x}}}) - {\widehat}Q'({{{\mathbf x}}}))[({{{\mathbf w}}}' - {{{\mathbf w}}})^\top {{{\mathbf x}}}]^2 \Big| \nonumber\\
& = \min_{{\widehat}Q' \in {{\cal Q}}} \max_{{\lVert{{{\mathbf u}}}\rVert} \leq 2} \Big| \sum_{{{{\mathbf x}}}\in S} ({\widehat}P({{{\mathbf x}}}) - {\widehat}Q'({{{\mathbf x}}}))[{{{\mathbf u}}}^\top {{{\mathbf x}}}]^2 \Big| \nonumber\\
\label{eq:34}
& = \min_{{\widehat}Q' \in {{\cal Q}}} \max_{{\lVert{{{\mathbf u}}}\rVert} \leq 2} \Big| {{{\mathbf u}}}^\top \big(\sum_{{{{\mathbf x}}}\in S} ({\widehat}P({{{\mathbf x}}}) - {\widehat}Q'({{{\mathbf x}}})) {{{\mathbf x}}}{{{\mathbf x}}}^\top\big) {{{\mathbf u}}}\Big|.\end{aligned}$$ We now simplify the notation and denote by ${{{\mathbf s}}}_1, \ldots, {{{\mathbf s}}}_{m_0}$ the elements of $S_Q$, by $z_i$ the distribution weight at point ${{{\mathbf s}}}_i$: $z_i = {\widehat}Q'({{{\mathbf s}}}_i)$, and by ${{{\mathbf M}}}({{{\mathbf z}}}) \in \Sset^N$ a symmetric matrix that is an affine function of ${{{\mathbf z}}}$: $${{{\mathbf M}}}({{{\mathbf z}}}) = {{{\mathbf M}}}_0 - \sum_{i = 1}^{m_0} z_i {{{\mathbf M}}}_i,$$ where ${{{\mathbf M}}}_0 = \sum_{{{{\mathbf x}}}\in S} P({{{\mathbf x}}}) {{{\mathbf x}}}{{{\mathbf x}}}^\top$ and ${{{\mathbf M}}}_i =
{{{\mathbf s}}}_i {{{\mathbf s}}}_i^\top$. Since problem (\[eq:34\]) is invariant to the non-zero bound on ${\lVert{{{\mathbf u}}}\rVert}$, we can equivalently write it with a bound of one and in view of the notation just introduced give its equivalent form $$\label{eq:36}
\min_{\substack{\| {{{\mathbf z}}}\|_1 = 1\\ {{{\mathbf z}}}\geq 0}} \max_{{\lVert{{{\mathbf u}}}\rVert} = 1} {\lvert{{{\mathbf u}}}^\top {{{\mathbf M}}}({{{\mathbf z}}}) {{{\mathbf u}}}\rvert}.$$ Since ${{{\mathbf M}}}({{{\mathbf z}}})$ is symmetric, $\max_{{\lVert{{{\mathbf u}}}\rVert} = 1} {{{\mathbf u}}}^\top {{{\mathbf M}}}({{{\mathbf z}}}) {{{\mathbf u}}}$ is the maximum eigenvalue $\lambda_{\max}$ of ${{{\mathbf M}}}({{{\mathbf z}}})$ and the problem is equivalent to the following maximum eigenvalue minimization for a symmetric matrix: $$\min_{\substack{\| {{{\mathbf z}}}\|_1 = 1\\ {{{\mathbf z}}}\geq 0}} \max{\{\lambda_{\max}({{{\mathbf M}}}({{{\mathbf z}}})), \lambda_{\max}(-{{{\mathbf M}}}({{{\mathbf z}}}))\}},$$ This is a convex optimization problem since the maximum eigenvalue of a matrix is a convex function of that matrix and ${{{\mathbf M}}}$ is an affine function of ${{{\mathbf z}}}$, and since ${{{\mathbf z}}}$ belongs to a simplex. The problem is equivalent to the following semi-definite programming (SDP) problem: $$\begin{aligned}
\label{eq:sdp}
\min_{{{{\mathbf z}}}, \lambda} & \quad \lambda\\
\text{subject to} & \quad \lambda {{{\mathbf I}}}- {{{\mathbf M}}}({{{\mathbf z}}}) \succeq 0\\
& \quad \lambda {{{\mathbf I}}}+ {{{\mathbf M}}}({{{\mathbf z}}}) \succeq 0\\
& \quad {{{\mathbf 1}}}^\top {{{\mathbf z}}}= 1 \wedge {{{\mathbf z}}}\geq 0.\end{aligned}$$ SDP problems can be solved in polynomial time using general interior point methods [@nesterov]. Thus, using the general expression of the complexity of interior point methods for SDPs, the following result holds.
\[prop:sdp\] Assume that $X$ is a subset of $\Rset^N$ and that the hypothesis set $H$ is a set of bounded linear functions $H = {\{{{{\mathbf x}}}\mapsto
{{{\mathbf w}}}^\top {{{\mathbf x}}}\colon {\lVert{{{\mathbf w}}}\rVert} \!\leq\! 1\}}$. Then, for any ${\widehat}Q$ and ${\widehat}P$, the discrepancy minimizing distribution ${\widehat}Q'$ for the square loss can be found in time $O(m_0^2 N^{2.5} + n_0N^2)$.
It is worth noting that the unconstrained version of this problem (no constraint on ${{{\mathbf z}}}$) and other close problems seem to have been studied by a number of optimization publications [@fletcher; @overton; @jarre; @helmberg; @alizadeh]. This suggests possibly more efficient specific algorithms than general interior point methods for solving this problem in the constrained case as well. Observe also that the matrices ${{{\mathbf M}}}_i$ have a specific structure in our case, they are rank-one matrices and in many applications quite sparse, which could be further exploited to improve efficiency.
Experiments {#sec:exp}
===========
This section reports the results of preliminary experiments showing the benefits of our discrepancy minimization algorithms. Our results confirm that our algorithm is effective in practice and produces a distribution that reduces the empirical discrepancy distance, which allows us to train on a sample closer to the target distribution with respect to this metric. They also demonstrate the accuracy benefits of this algorithm with respect to the target domain.
----- -----
(a) (b)
----- -----
Figures \[fig:ex\_illustration\](a)-(b) show the empirical advantages of using the distribution ${\widehat}Q'$ returned by the discrepancy minimizing algorithm described in Proposition \[prop:1d\_alg\] in a case where source and target distributions are shifted Gaussians: the source distribution is a Gaussian centered at $-1$ and the target distribution a Gaussian centered at $+1$, both with standard deviation 2. The hypothesis set used was the set of half-spaces and the target function selected to be the interval $[-1,1]$. Thus, training on a sample drawn form $Q$ generates a separator at $-1$ and errs on about half of the test points produced by $P$. In contrast, training with the distribution ${\widehat}Q'$ minimizing the empirical discrepancy yields a hypothesis separating the points at $+1$, thereby dramatically reducing the error rate.
----- -----
(a) (b)
----- -----
Figures \[fig:ex\_sdp\](a)-(b) show the application of the SDP derived in (\[eq:sdp\]) to determining the distribution minimizing the empirical discrepancy for ridge regression. In Figure \[fig:ex\_sdp\](a), the distributions $Q$ and $P$ are Gaussians centered at $(\sqrt{2},\sqrt{2})$ and $(-\sqrt{2},
-\sqrt{2})$, both with covariance matrix $2 {{{\mathbf I}}}$. The target function is $f(x_1, x_2) = (1 - |x_1|) + (1 - |x_2|)$, thus the optimal linear prediction derived from $Q$ has a negative slope, while the optimal prediction with respect to the target distribution $P$ in fact has a positive slope. Figure \[fig:ex\_sdp\](b) shows the performance of ridge regression when the example is extended to 16-dimensions, before and after minimizing the discrepancy. In this higher-dimension setting and even with several thousand points, using ([ http://sedumi.ie.lehigh.edu/]{}), our SDP problem could be solved in about 15s using a single 3GHz processor with 2GB RAM. The SDP algorithm yields distribution weights that decrease the discrepancy and assist ridge regression in selecting a more appropriate hypothesis for the target distribution.
Conclusion {#sec:conc}
==========
We presented an extensive theoretical and an algorithmic analysis of domain adaptation. Our analysis and algorithms are widely applicable and can benefit a variety of adaptation tasks. More efficient versions of these algorithms, in some instances efficient approximations, should further extend the applicability of our techniques to large-scale adaptation problems.
Proof of Theorem \[th:rademacher\]
==================================
Let $\Phi({{\mathcal S}})$ be defined by $\Phi({{\mathcal S}}) = \sup_{h \in H} R(h) - {\widehat}R(h)$. Changing a point of ${{\mathcal S}}$ affects $\Phi({{\mathcal S}})$ by at most $1/m$. Thus, by McDiarmid’s inequality applied to $\Phi({{\mathcal S}})$, for any $\delta > 0$, with probability at least $1 - \frac{\delta}{2}$, the following holds for all $h \in H$: $$\label{eq:mcd1}
\Phi({{\mathcal S}}) \leq \operatorname*{\rm E}_{{{\mathcal S}}\sim D}[\Phi({{\mathcal S}})] + \sqrt{\frac{\log \frac{2}{\delta}}{2m}}.$$ $\operatorname*{\rm E}_{{{\mathcal S}}\sim D}[\Phi({{\mathcal S}})]$ can be bounded in terms of the empirical Rade-macher complexity as follows: $$\begin{gathered}
\operatorname*{\rm E}_{{{\mathcal S}}}[\Phi({{\mathcal S}})] = \operatorname*{\rm E}_{{\mathcal S}}\big[ \sup_{h \in H} \operatorname*{\rm E}_{{{\mathcal S}}'}[R_{{{\mathcal S}}'}(h)] - R_{{\mathcal S}}(h) \big]\\
\begin{aligned}
& = \operatorname*{\rm E}_{{\mathcal S}}\big[ \sup_{h \in H} \operatorname*{\rm E}_{{{\mathcal S}}'}[R_{{{\mathcal S}}'}(h) - R_{{\mathcal S}}(h)] \big]\\
& \leq \operatorname*{\rm E}_{{{\mathcal S}}, {{\mathcal S}}'} \big[ \sup_{h \in H} R_{{{\mathcal S}}'}(h) - R_{{\mathcal S}}(h) \big]\\
& = \operatorname*{\rm E}_{{{\mathcal S}}, {{\mathcal S}}'} \big[ \sup_{h \in H} \frac{1}{m}
\sum_{i = 1}^m (h(x'_i) - h(x_i)) \big]\\
& = \operatorname*{\rm E}_{\sigma, {{\mathcal S}}, {{\mathcal S}}'} \big[ \sup_{h \in H} \frac{1}{m}
\sum_{i = 1}^m \sigma_i (h(x'_i) - h(x_i)) \big]\\
& \leq \operatorname*{\rm E}_{\sigma, {{\mathcal S}}'} \big[ \sup_{h \in H} \frac{1}{m}
\sum_{i = 1}^m \sigma_i h(x'_i) \big]
+ \operatorname*{\rm E}_{\sigma, {{\mathcal S}}} \big[ \sup_{h \in H} \frac{1}{m}
\sum_{i = 1}^m -\sigma_i h(x_i) \big]\\
& = 2 \operatorname*{\rm E}_{\sigma, {{\mathcal S}}} \big[ \sup_{h \in H} \frac{1}{m}
\sum_{i = 1}^m \sigma_i h(x_i) \big]
\leq 2 \operatorname*{\rm E}_{\sigma, {{\mathcal S}}} \big[ \sup_{h \in H} \big| \frac{1}{m}
\sum_{i = 1}^m \sigma_i h(x_i) \big| \big] \\
\label{eq:27}
& = {\mathfrak{R}}_m(H).
\end{aligned}\end{gathered}$$ Changing a point of ${{\mathcal S}}$ affects ${\mathfrak{R}}_m(H)$ by at most $2/m$. Thus, by McDiarmid’s inequality applied to ${\mathfrak{R}}_m(H)$, with probability at least $1 - \delta/2$, the following holds: $${\mathfrak{R}}_m(H) \leq {\widehat}{\mathfrak{R}}_{{\mathcal S}}(H) + \sqrt{\frac{2 \log \frac{2}{\delta}}{m}}.$$ Combining this inequality with Inequality (\[eq:mcd1\]) and the bound on $\operatorname*{\rm E}_{{{\mathcal S}}}[\Phi({{\mathcal S}})]$ above yields directly the statement of the theorem.
Rademacher Classification Bound
===============================
\[th:rademacher\_classification\] Let $H$ be a family of functions mapping $X$ to ${\{0, 1\}}$ and let $L_{01}$ denote the 0-1 loss. Let $Q$ be a distribution over $X$. Then, for any $\delta > 0$, with probability at least $1 -
\delta$, the following inequality holds for all samples ${{\mathcal S}}$ of size $m$ drawn according to $Q$: $${{{\cal L}}_{01}}_Q(h, h_Q^*) \leq {{{\cal L}}_{01}}_{{\widehat}Q}(h, h_Q^*) + {\widehat}{\mathfrak{R}}_{{\mathcal S}}(H)/2 + \sqrt{\frac{\log \frac{1}{\delta}}{2m}}.$$
Discrepancy Minimization with Kernels and $L_2$ loss
====================================================
Here, we show how to generalize the results of Section \[sec:l2\] to the high-dimensional case where $H$ is the reproducing kernel Hilbert space associated to a positive definite symmetric (PDS) kernel $K$.
\[prop:sdp\_kernel\] Let $K$ be a PDS kernel and let $H$ denote the reproducing kernel Hilbert space associated to $K$. Then, for any ${\widehat}Q$ and ${\widehat}P$, the problem of determining the discrepancy minimizing distribution ${\widehat}Q'$ for the square loss can be cast an SDP depending only on the Gram matrix of the kernel function $K$ and solved in time $O(m_0^2 (m_0 +
n_0)^{2.5} + n_0(m_0 + n_0)^2)$.
Let $\Phi\colon X \to H$ be a feature mapping associated with $K$. Let $p_0 = m_0 + n_0$. Here, we denote by $s_1, \ldots, s_{m_0}$ the elements of $S_Q$ and by $s_{m_0 + 1}, \ldots, s_{p_0}$ the element of $S_P$. We also define $z_i = {\widehat}Q'(s_i)$ for $i \in [1,
m_0]$, and for convenience $z_i = 0$ for $i \in [m_0 + 1, m_0 +
n_0]$. Then, by Proposition \[prop:sdp\], the problem of finding the optimal distribution ${\widehat}Q'$ is equivalent to $$\label{eq:48}
\min_{\substack{\| {{{\mathbf z}}}\|_1 = 1\\ {{{\mathbf z}}}\geq 0}} {\{\lambda_{\max}({{{\mathbf M}}}({{{\mathbf z}}})), \lambda_{\max}(-{{{\mathbf M}}}({{{\mathbf z}}}))\}},$$ where ${{{\mathbf M}}}({{{\mathbf z}}}) = \sum_{i = 1}^{p_0} ({\widehat}P(s_i) - z_i) \Phi(s_i)
\Phi(s_i)^\top$. Let ${{{\mathbf \Phi}}}$ denote the matrix in $\Rset^{N \times p_0}$ whose columns are the vectors $\Phi(s_1), \ldots, \Phi(s_{m_0 +
n_0})$. Then, observe that ${{{\mathbf M}}}({{{\mathbf z}}})$ can be rewritten as $${{{\mathbf M}}}({{{\mathbf z}}}) = {{{\mathbf \Phi}}}{{{\mathbf A}}}{{{\mathbf \Phi}}}^\top,$$ where ${{{\mathbf A}}}$ is the diagonal matrix $${{{\mathbf A}}}= \operatorname{diag}({\widehat}P(s_1) - z_1, \ldots, {\widehat}P(s_{p_0}) - z_{p_0}).$$ Fix ${{{\mathbf z}}}$. There exists $t_0 \in \Rset$ such that, for all $t \geq t_0$, ${{{\mathbf B}}}= {{{\mathbf A}}}+ t {{{\mathbf I}}}$ is a positive definite symmetric matrix. For any such $t$, let ${{{\mathbf N}}}'({{{\mathbf z}}})$ denote $${{{\mathbf N}}}'({{{\mathbf z}}}) = {{{\mathbf \Phi}}}{{{\mathbf B}}}{{{\mathbf \Phi}}}^\top.$$ Since ${{{\mathbf B}}}$ is positive definite, there exists a diagonal matrix ${{{\mathbf B}}}^{1/2} \!\in\! \Rset^{p_0 \times p_0}$ such that ${{{\mathbf B}}}= {{{\mathbf B}}}^{1/2}
{{{\mathbf B}}}^{1/2}$. Thus, we can write ${{{\mathbf N}}}'({{{\mathbf z}}})$ as ${{{\mathbf N}}}'({{{\mathbf z}}}) = {{{\mathbf Y}}}{{{\mathbf Y}}}^\top$ with ${{{\mathbf Y}}}= {{{\mathbf \Phi}}}{{{\mathbf B}}}^{1/2}$. ${{{\mathbf Y}}}{{{\mathbf Y}}}^\top$ and ${{{\mathbf Y}}}^\top {{{\mathbf Y}}}$ have the same characteristic polynomial modulo multiplication by $X^{N - p_0}$. Thus, since ${{{\mathbf \Phi}}}^\top {{{\mathbf \Phi}}}= {{{\mathbf K}}}$, the Gram matrix of kernel $K$ for the sample $S$, ${{{\mathbf N}}}'({{{\mathbf z}}})$ has the same same characteristic polynomial modulo multiplication by $X^{N - p_0}$ as $${{{\mathbf N}}}''({{{\mathbf z}}}) = {{{\mathbf Y}}}{{{\mathbf Y}}}^\top = {{{\mathbf B}}}^{1/2} {{{\mathbf K}}}{{{\mathbf B}}}^{1/2}.$$ Now, ${{{\mathbf N}}}''({{{\mathbf z}}})$ can be rewritten as ${{{\mathbf N}}}''({{{\mathbf z}}}) = {{{\mathbf Z}}}{{{\mathbf Z}}}^\top$ with ${{{\mathbf Z}}}=
{{{\mathbf B}}}^{1/2} {{{\mathbf K}}}^{1/2}$. Using the fact that ${{{\mathbf Z}}}{{{\mathbf Z}}}^\top$ and ${{{\mathbf Z}}}^\top {{{\mathbf Z}}}$ have the same characteristic polynomial, this shows that ${{{\mathbf N}}}'({{{\mathbf z}}})$ has the same characteristic polynomial modulo multiplication by $X^{N -
p_0}$ as $${{{\mathbf N}}}'''({{{\mathbf z}}}) = {{{\mathbf K}}}^{1/2} {{{\mathbf B}}}{{{\mathbf K}}}^{1/2}.$$ Thus, assuming without loss of generality that $N > p_0$, the following equality between polynomials in $X$ holds for all $t
\geq t_0$: $$\det(X{{{\mathbf I}}}- {{{\mathbf \Phi}}}{{{\mathbf B}}}{{{\mathbf \Phi}}}^\top) = X^{N - p_0}\det(X{{{\mathbf I}}}- {{{\mathbf K}}}^{1/2} {{{\mathbf B}}}{{{\mathbf K}}}^{1/2}).$$ Both determinants are also polynomials in $t$. Thus, for every fixed value of $X$, this is an equality between two polynomials in $t$ for all $t \geq t_0$. Thus, the equality holds for all $t$, in particular for $t = 0$, which implies that ${{{\mathbf M}}}({{{\mathbf z}}}) = {{{\mathbf \Phi}}}{{{\mathbf A}}}{{{\mathbf \Phi}}}^\top$ has the same non-zero eigenvalues as ${{{\mathbf M}}}'({{{\mathbf z}}}) = {{{\mathbf K}}}^{1/2} {{{\mathbf A}}}{{{\mathbf K}}}^{1/2}$. Thus, problem (\[eq:48\]) is equivalent to $$\label{eq:54}
\min_{\substack{\| {{{\mathbf z}}}\|_1 = 1\\ {{{\mathbf z}}}\geq 0}} {\{\lambda_{\max}({{{\mathbf M}}}'({{{\mathbf z}}})), \lambda_{\max}(-{{{\mathbf M}}}'({{{\mathbf z}}}))\}}.$$ Let ${{{\mathbf A}}}_0$ denote the diagonal matrix $${{{\mathbf A}}}_0 = \operatorname{diag}(P(s_1), \ldots, P(s_{p_0})),$$ and for $i \in [1, m_0]$, let ${{{\mathbf I}}}_i \in \Rset^{p_0 \times p_0}$ denote the diagonal matrix whose diagonal entries are all zero except from the $i$th one which equals one. Then, $${{{\mathbf M}}}'({{{\mathbf z}}}) = {{{\mathbf M}}}'_0 - \sum_{i = 1}^{m_0} z_i {{{\mathbf M}}}'_i$$ with ${{{\mathbf M}}}'_0 = {{{\mathbf K}}}^{1/2} {{{\mathbf A}}}_0 {{{\mathbf K}}}^{1/2}$ and ${{{\mathbf M}}}'_i = {{{\mathbf K}}}^{1/2} {{{\mathbf I}}}_i
{{{\mathbf K}}}^{1/2}$ for $i \in [1, m_0]$. Thus, ${{{\mathbf M}}}'({{{\mathbf z}}})$ is an affine function of ${{{\mathbf z}}}$ and problem (\[eq:54\]) is a convex optimization problem that can be cast as an SDP, as described in Section \[sec:l2\], in terms of the Gram matrix ${{{\mathbf K}}}$ of the kernel function $K$.
|
---
abstract: 'The low-rank approximation is a complexity reduction technique to approximate a tensor or a matrix with a reduced rank, which has been applied to the simulation of high dimensional problems to reduce the memory required and computational cost. In this work, a dynamical low-rank approximation method is developed for the time-dependent radiation transport equation in 1-D and 2-D Cartesian geometries. Using a finite volume discretization in space and a spherical harmonics basis in angle, we construct a system that evolves on a low-rank manifold via an operator splitting approach. Numerical results on four test problems demonstrate that the low-rank solution requires less memory than solving the full rank equations with the same accuracy. It was furthermore shown that the low-rank algorithm can obtain much better results at a moderate extra cost by refining the discretization while keeping the rank fixed.'
address:
- 'Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, 46545, USA'
- 'Karlsruhe Institute of Technology, Steinbuch Centre for Computing, Karlsruhe, Germany'
author:
- Zhuogang Peng
- Ryan McClarren
- Martin Frank
bibliography:
- 'mybibfile.bib'
title: 'A low-rank method for two-dimensional time-dependent radiation transport calculations'
---
Low-rank approximation, Radiation transport
Introduction
============
The numerical simulation of the radiation transport process is fundamental in a wide range of applications from supernovas to medical imaging, where accurate numerical solutions of a linear Boltzmann equation, also known as the radiative transfer equation (RTE), are required. In this equation, we are solving for the specific intensity, which is a seven-dimensional function that describes the movement of the flow of particles in terms of time, position, direction, and energy. Despite the rapid development of high-performance computing, the numerical solution of the RTE remains challenging due to the high computational costs and the large memory requirements caused by such a high-dimensional phase space. In this paper, we focus on reducing the memory required to solve the RTE to enable better performance on exascale-class computers [@Shalf2011].
Due to the rich phase space, computational methods for radiation transport require discretizations in energy and direction[^1], in addition to spatial and temporal discretizations. For the energy variable, the multigroup method is commonly used [@Pomraning]. The direction, or angular variables, can be treated by solving the equations along particular directions via the discrete ordinates or S$_N$ method [@LewisMiller]. An alternative technique employs a basis expansion using the natural basis for the sphere: spherical harmonics. The spherical harmonics or P$_N$ method uses a truncated expansion to approximate the angular dependence [@Mcclarren2008; @Laboure:2016hm; @Dargaville:2019fa]. Low-order approximations are also used such as flux-limited diffusion [@morel2000; @Olson:2000vq], simplified P$_N$ [@McClarren:2011ga; @Frank:2007ig; @Modest:2012df] along with hybrid methods such as quasi-diffusion [@anistratov1986solution]. Finally, stochastic methods based on the implicit Monte Carlo method [@Fleck:1971vj; @McClarren:2009bs; @Wollaber:2016hv; @SmedleyStevenson:2015gt] are an alternative approach, though often requiring significant computational cost.
Although the aforementioned methods require many degrees of freedom to describe the phase-space dependence of the solution, it is known that many transport problems require only a subspace of the full phase space (called a manifold in mathematical parlance) to describe the transport of particles. An example of this phenomenon is problems in the diffusion limit: these problems require only a linear dependence on angular variables. One can also formulate problems where this manifold over which the solution depends evolves over time: a beam entering a scattering medium would be described by a delta-function in space and angle at time zero, but eventually, relax to much smoother distribution that we could characterize using a simple basis expansion.
We desire to generalize this idea, and possibly automatically discover the manifold that describes the system evolution. We accomplish this task by expressing the solution to a transport problem as a basis expansion in space and angle and using techniques to determine what subspace of those bases are needed to describe the solution and how that subspace evolves. We use the dynamical low-rank approximation (DLRA) of Koch and Lubich to evolve time-dependent matrices by tangent-space projection [@Koch2007]. DLRA has been extended to tensors [@Koch2010], and further results can be found in [@Nonnenmacher2008]. DLRA has been used to reduce the computational complexity of quantum propagation [@Kloss2017] by restricting the evolution to lower-rank amongst other work [@Markovsky2008; @Jahnke2008; @Einkemmer2018]. The asymptotic analysis of the DLRA in a one-dimensional radiative transport equation with the backward Euler and Crank–Nicolson scheme was made in [@ding2019error]. In this work, we apply DLRA to neutral particle transport.
Here we give a brief mathematical introduction to the robust and accurate projector-splitting method developed by Lubich [@Lubich2014] to perform the DLRA for matrix differential equations of the form $$\frac{\partial}{\partial t} A(t) \equiv \dot{A}(t)= F(A(t)),$$ for $A(t) \in \mathbb{R}^{m \times n}$. DLRA seeks to find an approximating matrix $Y(t)=U(t)S(t)V^T(t)$, where $U(t)\in\mathbb{R}^{m\times r}$ , $V(t)\in\mathbb{R}^{n\times r}$ are orthonormal matrices and $S(t)\in\mathbb{R}^{r\times r}$ so that $Y(t)$ has rank $r$. Note that rank $r$ matrices are a manifold, $\mathcal{M}_r$, in the space $\mathbb{R}^{m \times n}$. We evolve the solution on $\mathcal{M}_r$ directly by inserting the ansatz for $Y(t)$ into the equation and projecting the right-hand side onto the tangent space of $\mathcal{M}_r$.
In this work, we develop a reduced system of the radiative transfer equation with this low-rank method, where the intensity only involves in its low-rank orthogonal bases rather the full phase space. We design a numerical scheme for the 2-D RTE with a finite volume discretization in space and a spherical harmonic (P$_N$) expansion in angle. We further demonstrate the memory saving features by comparing the low-rank results with full-rank solutions and benchmark results.
Derivation of Method
====================
We consider a energy-independent radiation transport equation $$\label{RadiativeTransfer}
\frac{1}{c} \frac{\partial \psi ({\bm{r}}, \hat{\Omega}, t)}{\partial t} + \hat{\Omega} {\bm{\cdot}}{{\bm{\nabla}}}\psi ({\bm{r}}, \hat{\Omega}, t) + {\sigma_\mathrm{t}}({\bm{r}}) \psi ({\bm{r}}, \hat{\Omega}, t)
= \frac{1}{4\pi} {\sigma_\mathrm{s}}({\bm{r}}) \phi({\bm{r}}, t) + Q({\bm{r}},t).$$ Here the angular flux $\psi({\bm{r}}, \hat{\Omega}, t)$ with units of particles per area per steradian per time is a function of position ${\bm{r}} \in D$ where $D$ is the computational domain, direction $\hat{\Omega}(\mu, \varphi)$ where $\mu$ is the cosine of the polar angle and $\varphi$ is the azimuthal angle, and time $t$. Note that $\hat{\Omega}$ is a unit vector. The total and isotropic scattering macroscopic cross-sections with units of inverse length are denoted as ${\sigma_\mathrm{t}}({\bm{r}})$ and ${\sigma_\mathrm{s}}({\bm{r}})$, respectively, $c$ is the particle speed and $S({\bm{r}},t)$ is a prescribed source with units of particles per volume per time. We also write the scalar flux, $\phi({\bm{r}},t)$, as the integral of the angular flux $$\phi({\bm{r}},t) = \int_{4 \pi}\psi({\bm{r}}, \hat{\Omega}, t) \, d \hat{\Omega}.$$
In this study we approximate the solution to Eq. using the form $$\label{LRA1}
\psi({\bm{r}},\hat{\Omega},t) \approx \sum_{i,j = 1}^r X_i({\bm{r}},t)S_{ij}(t)W_j(\hat{\Omega},t);$$ that is, we seek the best approximation with rank $r$ to the solution of Eq. (\[RadiativeTransfer\]), where we have written $X_i$ as an orthonormal basis for ${\bm{r}}$ and $W_j$ as an orthonormal basis for $\hat{\Omega}$ using the inner products $$\langle f,g\rangle _{{\bm{r}}} = \int_{D} f g\, d{\bm{r}}, \quad \langle f,g\rangle _{\hat{\Omega}} = \int_{4 \pi} f g \, d\hat{\Omega} .$$ Due to orthonormality we also have $\langle X_i, {X}_j \rangle_{{\bm{r}}} = \langle W_i, {W}_j \rangle_{\hat{\Omega}} = \delta_{ij}.$ We also use $
\bar{X} = \{X_1, X_2, ..., X_r\}
$ and $
\bar{W} = \{W_1, W_2, ..., W_r\}
$ as ansatz spaces. The expansion in Eq. is not unique and we add orthogonal constraints $\langle X_i, \dot{X}_j \rangle_{{\bm{r}}} = 0$ and $\langle W_i, \dot{W}_j \rangle_{\hat{\Omega}} = 0$ as gauge conditions, which identify $X$ and $W$ in Grassmann manifolds [@Edelman1998]. We now define orthogonal projectors using the bases: $$\label{Projection1}
P_{\bar{X}} g = \sum_{i=1}^{r}X_i \langle X_i g \rangle_{{\bm{r}}},$$ $$\label{Projection2}
P_{\bar{W}} g = \sum_{j=1}^{r}W_j \langle W_j g \rangle_{\hat{\Omega}}.$$
We apply the projectors to define a split of the original equations into three steps and each of these is solved for a time step: $$\begin{gathered}
\label{eq: MainEq1}
\partial_t \psi_1({\bm{r}}, \hat{\Omega},t) = P_{\bar{W}} \bigg( - \hat{\Omega} \cdot {{\bm{\nabla}}}\psi({\bm{r}}, \hat{\Omega}, t)
- \sigma_t({\bm{r}}, t) \psi ({\bm{r}}, \hat{\Omega}, t) \\
+ \frac{1}{4\pi} {\sigma_\mathrm{s}}({\bm{r}}, t) \phi({\bm{r}}, t)
+ Q({\bm{r}},t) \bigg),\end{gathered}$$
$$\begin{gathered}
\label{eq: MainEq2}
\partial_t \psi_2({\bm{r}}, \hat{\Omega},t) = - P_{\bar{X}} P_{\bar{W}} \bigg( - \hat{\Omega} \cdot {{\bm{\nabla}}}\psi({\bm{r}}, \hat{\Omega}, t)
- \sigma_t({\bm{r}}, t) \psi ({\bm{r}}, \hat{\Omega}, t) \\
+ \frac{1}{4\pi} {\sigma_\mathrm{s}}({\bm{r}}, t) \phi({\bm{r}}, t)
+ Q({\bm{r}},t) \bigg),\end{gathered}$$
$$\begin{gathered}
\label{eq: MainEq3}
\partial_t \psi_3({\bm{r}}, \hat{\Omega},t) = P_{\bar{X}} \bigg( - \hat{\Omega}\cdot {{\bm{\nabla}}}\psi({\bm{r}}, \hat{\Omega}, t)
- \sigma_t({\bm{r}}, t) \psi ({\bm{r}}, \hat{\Omega}, t) \\
+ \frac{1}{4\pi} {\sigma_\mathrm{s}}({\bm{r}}, t) \phi({\bm{r}}, t)
+ Q({\bm{r}},t) \bigg).\end{gathered}$$
The $\psi_2$ step uses $\psi_1$ as an initial condition, and the $\psi_3$ step uses $\psi_2$ as an initial condition. It can be shown that the above evolution is contained in the low-rank manifold $\mathcal{M}_r$ if the initial value is in $\mathcal{M}_r$ [@Lubich2014] because the right-hand side of each step remains in the tangent space $\mathcal{T} \mathcal{M}_r$.
The main advantage of this scheme comes from the fact that the only the low-rank components $X({\bm{r}}, t)$, $S(t)$, and $W(\hat{\Omega}, t)$ need to be stored during the time evolution rather than the full size solution $\psi({\bm{r}}, \hat{\Omega}, t)$. To demonstrate this we first formulate the projections (\[eq: MainEq1\] - \[eq: MainEq3\]) explicitly from time $t_0$ to $t_0+h$ where $h$ is the step size. The low-rank representation of $\psi({\bm{r}},\hat{\Omega},t_0)$ is given by the initial condition $$\label{eq: ini_cond}
\psi^{(0)}({\bm{r}},\hat{\Omega},t_0) = \sum_{i,j = 1}^r X^{(0)}_i({\bm{r}},t_0)S^{(0)}_{ij}(t_0)W^{(0)}_j(\hat{\Omega},t_0)$$ In the first projection the basis $W_j$ does not change with time and is evaluated at the initial value $W^{(0)}_j(\hat{\Omega})$. We simplify the notation by writing $K_j({\bm{r}},t) = \sum^{r}_{i} X_{i}({\bm{r}},t)S_{ij}(t)$, then becomes $$\label{eq: ini_cond2}
\psi^{(0)}({\bm{r}},\hat{\Omega},t_0) = \sum_{j=1}^r K^{(0)}_j({\bm{r}},t_0) W^{(0)}_j(\hat{\Omega}),$$ We plug this solution into Eq. and multiply by $W^{(0)}_\ell(\hat{\Omega})$ and integrate over $\mu$ and $\varphi$ to get $$\begin{gathered}
\label{eq:psi1}
\partial_t K_j = -\sum_{l=1}^{r} {{\bm{\nabla}}}K_l \, \langle \hat{\Omega} W_l W_j\rangle_{\hat{\Omega}} - {\sigma_\mathrm{t}}K_j
+ \frac{1}{4\pi} {\sigma_\mathrm{s}}\sum_{l=1}^{r} K_l \langle W_l\rangle_{\hat{\Omega}} \langle W_j\rangle_{\hat{\Omega}} \\+ Q \langle W_j\rangle_{\hat{\Omega}}.\end{gathered}$$ Equation resembles the standard P$_N$ equations, a point we will return to later. It is a system of advection problems coupled through the streaming term. Notice that $\psi$ has not been formulated in this equation.
We can then factor $K_j^{(1)}$, which is the solution of Eq. into $X_{i}^{(1)}$ and $S_{ij}^{(1)}$ using a QR decomposition. In the second step both the $X_i$ and $W_j$ are preserved. The initial condition to solve Eq. are $X_{i}^{(2)}({\bm{r}}) = X_{i}^{(1)}({\bm{r}}, t_0+h)$, $S_{i}^{(2)}(t_0) = S_{i}^{(1)}(t_0+h)$, and $W_{i}^{(2)}(\hat{\Omega}) = W_{i}^{(0)}(\hat{\Omega})$. Then, we can perform similar calculations on Eq. to get $$\begin{gathered}
\label{eq:s_eq}
\frac{d}{dt} S_{ij} = \sum_{kl}^{r} \langle {{\bm{\nabla}}}X_k \, X_l \rangle_{{\bm{r}}} S_{kl} \langle \hat{\Omega} W_l W_j \rangle_{\hat{\Omega}}
+\sum_{k}^{r} \langle \sigma_t X_k X_i \rangle_{{\bm{r}}} S_{kj} \\
- \frac{1}{4\pi} \sum_{kl}^{r} \langle \sigma_s X_k X_i \rangle_{{\bm{r}}}S_{kl}\langle W_l\rangle_{\hat{\Omega}} \langle W_j\rangle_{\hat{\Omega}} - \langle X_i Q \rangle_{{\bm{r}}} \langle W_j\rangle_{\hat{\Omega}}.\end{gathered}$$ We call the solution $S_{ij}^{(2)}$. Equation defines a set of $r^2$ ordinary differential equations. The solution is used to create an initial condition for Eq. , where $X_{i}^{(3)}({\bm{r}}) = X_{i}^{(1)}({\bm{r}}, t_0+h)$, $S_{i}^{(3)}(t_0) = S_{i}^{(2)}(t_0+h)$, and $W_{i}^{(3)}(\hat{\Omega}, t_0) = W_{i}^{(0)}(\hat{\Omega})$. Notice that $X_i$ does not change with time in this step.
Writing $L_i = \sum_j^r S_{ij}(t)W_{j}(\hat{\Omega}, t)$ we can multiply Eq. by a spatial basis function and integrate over space to get $$\begin{gathered}
\label{eq:L_eq}
\partial_t L_{i} = -\hat{\Omega} \sum_{k}^{r} \langle {{\bm{\nabla}}}X_k \, X_i \rangle_{{\bm{r}}} L_k - \sum_{k}^{r} \langle \sigma_t X_k X_i \rangle_{{\bm{r}}} L_{k}
+ \frac{1}{4\pi} \sum_{k}^{r} \langle \sigma_s X_k X_i \rangle_{{\bm{r}}} \langle L_k \rangle_{\hat{\Omega}} \\
+ \langle Q X_i \rangle_{{\bm{r}}},\end{gathered}$$ which evolves the solution in $\mu$ and $\varphi$ spaces. Upon factoring $L_i = S^{(3)}_{ij}(t)W_j^{(3)}(\hat{\Omega},t)$ using a QR decomposition we can write the low-rank solution as $\psi({\bm{r}}, \hat{\Omega}, t_0+h) = \sum_{i,j,=1}^r X_i^{(1)}({\bm{r}},t_0+h) S^{(3)}_{ij}(t_0+h)W_j^{(3)}(\hat{\Omega},t_0+h).$
Numerical Scheme
================
In this section the procedure outlined above of solving Eqs. , , and is implemented in one and two spatial dimensions with a first-order explicit time integrator [@Lubich2014] and a finite volume discretization in space. For the angular basis, we use a spherical harmonics expansion. In this section, we describe the numerical method to solve the 2D problem in detail.
Discretization details
----------------------
In the two dimensional system, we write the spatial variables as $x$ and $z$. The transport equation in this reduced geometry is $$\begin{gathered}
\label{eq: 2D_RadiativeTransfer}
\frac{1}{c} \frac{\partial \psi (x,z,\mu,\varphi, t)}{\partial t} + \mu \partial_z \psi(x, z, \mu, \varphi, t) + \sqrt{1 - \mu^2} \cos \varphi \, \partial_x \psi(x, z, \mu, \varphi, t) \\
+ {\sigma_\mathrm{t}}(x, z) \psi (x,z,\mu,\varphi, t)
= \frac{1}{4\pi} {\sigma_\mathrm{s}}(x, z) \phi(x, z, t) + Q(x,z,t).\end{gathered}$$ The projection system simplifies to $$\begin{gathered}
\label{eq: 2D_k_eq}
\partial_t K_j = -\sum_{l=1}^{r} \partial_z K_{l} \, \langle \mu W_j W_{l}\rangle_{\hat{\Omega}} -\sum_{l=1}^{r} \partial_x K_{l} \, \left\langle \sqrt{1-\cos \varphi ^2} \, W_j W_{l}\right\rangle_{\hat{\Omega}} - {\sigma_\mathrm{t}}K_j \\
+ \frac{{\sigma_\mathrm{s}}}{2} \sum_{l=1}^{r} K_{l} \langle W_{l} \rangle_{\mu} \langle W_j\rangle_{\mu} + \frac{\langle W_j\rangle_{\mu}}{2} Q,\end{gathered}$$
$$\begin{gathered}
\label{eq: 2D_s_eq}
\partial_t S_{ij} = \sum_{kl}^{r} \langle \partial_z X_k \, X_l \rangle_{{\bm{r}}} S_{kl} \left\langle \mu W_l W_j \right\rangle_{\hat{\Omega}} + \sum_{kl}^{r} \langle \partial_x X_k \, X_l \rangle_{{\bm{r}}} S_{kl} \langle \sqrt{1 - \cos \varphi^2} \,W_l W_j \rangle_{\hat{\Omega}}\\
+\sum_{k}^{r} \langle \sigma_t X_k X_i \rangle_{{\bm{r}}} S_{kj}
- \frac{1}{2} \sum_{kl}^{r} \langle \sigma_s X_k X_i \rangle_{{\bm{r}}} S_{kl} \langle W_l\rangle_{\hat{\Omega}} \langle W_j\rangle_{\hat{\Omega}} - \frac{1}{2}\langle X_i Q \rangle_{{\bm{r}}} \langle W_j\rangle_{\hat{\Omega}},\end{gathered}$$
$$\begin{gathered}
\label{eq: 2D_L_eq}
\partial_t L_{i} = -\mu \sum_{k}^{r} \langle \partial_z X_k \, X_i \rangle_{{\bm{r}}} L_k -\sqrt{1 - \cos \varphi^2} \sum_{k}^{r} \langle \partial_x X_k \, X_i \rangle_{{\bm{r}}} L_k - \sum_{k}^{r} \langle \sigma_t X_k X_i \rangle_{{\bm{r}}} L_{k} \\
+ \frac{1}{2} \sum_{k}^{r} \langle \sigma_s X_k X_i \rangle_{{\bm{r}}} \langle L_k \rangle_{\hat{\Omega}}
+ \frac{1}{2}\langle Q X_i \rangle_{{\bm{r}}}.\end{gathered}$$
The bases we use are arise from a finite volume discretization in space with a constant mesh area $\Delta x \Delta z$ and $N_x \times N_z$ zones, and the truncated at $N_l$ spherical harmonics in angle. To make orthonormal bases we define $$\label{eq: 2D_Discretization1}
X_i(t,x,z) = \sum_{p=1}^{N_x}\sum_{q=1}^{N_z}Z_{p q}(x,z)u_{p q i}(t)$$
$$\label{eq: 2D_Discretization2}
W_j(t, \mu, \varphi) = \sum_{l=0}^{N_l} \, \sum_{k=0}^{N_l}Y_{l}^{k}(\mu, \varphi) v_{l k j}(t)$$
Note that $Z_{pq}(x,z) = \frac{1}{\sqrt{\Delta x \, \Delta z}}$ with $x \in [x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}]$ and $z \in [z_{i-\frac{1}{2}},z_{i+\frac{1}{2}}]$, where $p$ and $q$ are the cell index. The spherical harmonics are defined as $$Y_l^k(\mu, \varphi) = \sqrt{\frac{2l+1}{4 \pi} \, \frac{(l-k)!}{(l+k)!}} \, P_l^k(\mu) \, e^{i \, k\varphi},$$ where $P_{l} ^k (\mu)$ is the associate Legendre polynomial. The negative $k$ are not necessary here because of the recursion properties of the spherical harmonics [@Brown2005]. Here $u_{pqi}$ and $v_{lkj}$ are components of the time dependent tensor $U(t) \in \mathbb{R}^{Nx \times Nz \times r}$ and $V(t) \in \mathbb{R}^{N_l \times (N_l+1) \times r}$.
We also notice that the computations in Eqs. , can be performed by the matrix multiplication instead of tensor operation. This is achieved by rearranging the matrices $Z \in \mathbb{R}^{N_x \times N_z}$ and $Y \in \mathbb{R}^{N_l \times N_l+1}$ into vectors and tensors $U$ and $V$ to matrices.
To make the notation consistent we denote the number of degrees of freedom in space as $m = N_x N_z$ and the angular degrees of freedom as $n = \frac{1}{2}(N_l+1)(N_l+2)$. In addition, the index of $U$ and $V$ conform to Eqs. and .
Consequently the new ansatz spaces $X$ and $W$ have the form $$X = UZ = \begin{bmatrix}
u_{1\,1\,1}&u_{1\,1\,2}&...&u_{1\,1\,r}\\...&...&...&...\\u_{N_x\,1\,1}&u_{N_x\,1\,2}&...&u_{N_x\,1\,r}\\u_{1\,2\,1}&u_{1\,2\,2}&...&u_{1\,2\,r}\\...&...&...&...\\u_{N_x \,N_z\,1}&u_{N_x \,N_z\,2}&...&u_{N_x \,N_z\,r}
\end{bmatrix} ^T
\begin{bmatrix}
Z_{1\,1}\\...\\Z_{N_x \, 1}\\Z_{1\,2}\\...\\Z_{N_x \, N_z}
\end{bmatrix}$$
$$W = V Y =
\begin{bmatrix}
v_{0\,0\,1}&v_{0\,0\,2}&...&v_{0\,0\,r}\\v_{0\,1\,1}&v_{0\,1\,2}&...&v_{0\,1\,r}\\v_{1\,1\,1}&v_{1\,1\,2}&...&v_{1\,1\,r}\\v_{0 \,2\,1}&v_{0\,2\,2}&...&v_{0 \,2\,r}\\...&...&...&...\\v_{N_l \,N_l\,1}&v_{N_l \,N_l\,2}&...&v_{N_l \,N_l\,r}
\end{bmatrix} ^T
\begin{bmatrix}
Y_0^0\\Y_1^{0}\\Y_1^{1}\\Y_2^{0}\\...\\Y_{N_l}^{N_l}
\end{bmatrix}$$
To solve Eqs. and we need to calculate spatial integration terms like $\langle \partial_z X_k \, X_l \rangle_{{\bm{r}}}$. Due to our use of finite volume method, the basis $X_i$ is discontinuous and piecewise constant in space. We apply integration by parts and obtain $$\label{eq: 2D_Term1}
\begin{aligned}
\langle \partial_z X_k \, X_l \rangle_{{\bm{r}}} = & \left \langle \partial_z \Big(\sum_p^{N_x} \sum_q^{N_z} Z_{pq} u_{pqk} \Big) \sum_{p'}^{N_x} \sum_{q'}^{N_z} Z_{p'q'} u_{p'q'k} \right \rangle_{{\bm{r}}} \\
=& \sum_{p'}^{N_x} \sum_{q'}^{N_z} u_{p'q'l} \left ( \left \langle \partial_z (Z_{p'q'} \sum_p^{N_x} \sum_q^{N_z} Z_{pq} u_{pqk}) \right \rangle _{{\bm{r}}} - \left \langle \cancelto{0}{\partial_z Z_{p'q'}} \quad \sum_{p'}^{N_x} \sum_{q'}^{N_z} Z_{p'q'} u_{pqk} \right \rangle _{{\bm{r}}} \right )\\
=& \sum_{p}^{N_x} \sum_{q}^{N_z} u_{pql} \left ( \frac{\{u\}_{p,q+\frac12,k} - \{u\}_{p,q-\frac12,k}}{2 \Delta z} \right)
\end{aligned}$$ Here, $\{u\}_{p,q+\frac12,k}$ denotes the value at the cell boundary between the spatial cells $p$ and $p+1$. We set the value at the cell boundary by the average of its left and right values, such as $\{u\}_{p,q+\frac12,k} = \frac12(u_{p,q+1,k}+u_{p,q,k})$.
For the angular terms, e.g., $ \langle \mu W_j W_{j'} \rangle_{\hat{\Omega}}$, we can use write the integral as $$\label{eq: 2D_Term2}
\begin{aligned}
\langle \mu W_j W_{j'} \rangle_{\hat{\Omega}}= & \left \langle \sum_{l=1}^{N_l} \, \sum_{k=-N_l}^{N_l}Y_{l}^{k}(\mu, \varphi) v_{l k j}(t) \sum_{l'=1}^{N_l} \, \sum_{k'=-N_l}^{N_l}Y_{l'}^{k'}(\mu, \varphi) v_{l' k' j'}(t) \right\rangle_{\hat{\Omega}} \\
= & \left\langle \sum_{l=1}^{N_l} \, \sum_{k=-N_l}^{N_l} \, \sum_{l'=1}^{N_l} \, \sum_{k'=-N_l}^{N_l}
v_{l k j}(t) \, \mu Y_l^k(\mu, \varphi) Y_{l'}^{k'}(\mu, \varphi) \, v_{l' k' j'}(t) \right\rangle_{\hat{\Omega}} \\
= & \sum_{l=1}^{N_l} \, \sum_{k=-N_l}^{N_l} \, \sum_{l'=1}^{N_l} \, \sum_{k'=-N_l}^{N_l}
v_{l k j}(t) \left \langle \mu Y_l^k(\mu, \varphi) Y_{l'}^{k'}(\mu, \varphi) \right \rangle_{\hat{\Omega}} v_{l' k' j'}(t)
\end{aligned}$$
There are two ways to calculate this term. One is to precompute the time-independent part $\langle \mu Y_l^k(\mu, \varphi) Y_{l'}^{k'}(\mu, \varphi) \rangle_{\hat{\Omega}}$ which forms a $n \times n$ matrix $C$. Thus Eqs. requires $\mathcal{O}(n^2 r)$ operations, which is affordable because usually $n$ is not large and $C$ is sparse. Alternatively, we could calculate $ \langle \mu W_j W_{j'} \rangle_\mu$ on-the-fly by choosing $\mathcal{O}(n)$ quadrature points in angle, and it requires $\mathcal{O}(n r^2)$ operations for all the $r^2$ entries. In this work we chose the first method which is more straightforward to implement.
Upwind scheme with slope reconstruction
---------------------------------------
We apply the standard upwinding technique to solve the advection equation Eqs. . For the space derivative term we use upwinding to write $$\label{eq: 2D_dK}
\begin{aligned}
\partial_z K_{j} \, \langle \mu W_j W_{l}\rangle_{\hat{\Omega}} =&\frac{1}{\Delta z}(K_{p, q+\frac{1}{2},j} - K_{p,q-\frac{1}{2},j}) \langle \mu W_j W_{l}\rangle_{\hat{\Omega}}\\
=&\frac{K_{p,q+1,j} - K_{p,q-1,j}}{2\Delta z} \, (V^T C V)_{jl}- \frac{K_{p,q+1,j}- K_{p,q,j} + K_{p,q-1,j}}{2\Delta z} \, (V^T \Sigma V)_{jl} \\
=& \frac{1}{\sqrt{\Delta x \Delta z}} \sum_{i=1}^r \sum_{p=1}^{N_x} \sum_{q=1}^{N_z} \Bigg ( \left ( \frac{u_{p,q+1,i} - u_{p,q-1,i}}{2 \Delta z} \right ) S_{ij} (V^T C V)_{jl} \\ &- \left ( \frac{u_{p,q+1,i} + u_{p,q-1,i} - 2u_{p,q,i}}{2 \Delta z} \right ) S_{ij} (V^T \Sigma V)_{jl} \Bigg )
\end{aligned}$$ where $\Sigma$ is a stabilization matrix that we take to be a diagonal matrix with the singular values of $C$. Other stabilization terms could be used, including Lax-Friedrichs where $V^T \Sigma V$ is replaced by a constant times an identity matrix. This method allows the spatial variables to be differentiated separately, so the same treatment can be applied to $\partial_x K_{l} \, \langle \sqrt{1-\cos \varphi ^2} \, W_j W_{l}\rangle_{\hat{\Omega}}$.
Additionally, the harmonic mean limiter is adopted to reconstruct this slope term $K_j$ for better accuracy. We define the slope of $z$ direction in the edge of the cell $(p, q)$ as $$m_{p,q}^+ = \frac{K_{p,q,j}-K_{p,q-1,j}}{\Delta z}, \quad \quad m_{p,q}^- = \frac{K_{p,q+1,j}-K_{p,q,j}}{\Delta z}.$$ Then the slope of this cell is $$m_{p,q} =
\begin{cases}
\frac{2 m_{p,q}^+ m_{p,q}^-}{m_{p,q}^+ + m_{p,q}^-} & \text{if $m_{p,q}^+ m_{p,q}^- > 0$}\\
0 & \text{otherwise}
\end{cases}$$ With the reconstructed slope $$K_{p,q,j}^+ = K_{p,q,j} + \frac{\Delta z}{2} m_{p,q}, \quad \quad K_{p,q,j}^- = K_{p,q,j} - \frac{\Delta z}{2} m_{p,q},$$ Eqs. can be modified $$\label{Term2_2nd}
\begin{aligned}
\partial_z K_{j} \, \langle \mu W_j W_{l}\rangle_{\hat{\Omega}} = & \frac{1}{\Delta z}(K_{p, q+\frac{1}{2},j} - K_{p,q-\frac{1}{2},j}) \langle \mu W_j W_{l}\rangle_{\hat{\Omega}} \\
&=\frac{1}{2\Delta z}\Big ((K_{p,q+1,j}^-+K_{p,q,j}^+ - K_{p,q-1,j}^+ - K_{p,q,j}^-)(V^T C V)_{jl}\\
&- (K_{p,q+1,j}^- - K_{p,q,j}^+ - K_{p,q,j}^- + K_{p,q-1,j}^+)(V^T \Sigma V)_{jl} \Big ),
\end{aligned}$$
Spherical harmonics expansions can yield oscillatory or negative solutions [@Mcclarren2008]. To address this issue we implemented angular filtering [@McClarren2010; @Radice2013; @Laboure:2016hm] which can significantly increase the performance of $P_n$ method in solving radiative transfer equation by removing the oscillations. We implemented the Lanczos filter into our explicit solver by using an equivalent equation approach [@Radice2013] and combined it with the low-rank approximation algorithm. The filtered equation has the form of $$\label{Filtered_RadiativeTransfer}
\begin{aligned}
\frac{1}{c} \frac{\partial \psi ({\bm{r}}, \hat{\Omega}, t)}{\partial t} + \hat{\Omega} {\bm{\cdot}}{{\bm{\nabla}}}\psi ({\bm{r}}, \hat{\Omega}, t) + {\sigma_\mathrm{t}}({\bm{r}}) \psi ({\bm{r}}, \hat{\Omega}, t) + \beta {\sigma_\mathrm{f}}({\bm{r}}, \hat{\Omega}) \psi({\bm{r}}, \hat{\Omega}, t)
\\
= \frac{1}{4\pi} {\sigma_\mathrm{s}}({\bm{r}}) \phi({\bm{r}}, t) + Q({\bm{r}},t).
\end{aligned}$$ where the free parameter $\beta$ is the filter strength, ${\sigma_\mathrm{f}}= \log \ \frac{\sin \eta}{ \eta} \frac{l}{l+1}$ and $\eta$ is the order of the [P$_N$]{} expansion and $l$ is the index of [P$_N$]{} moments.
Memory Reduction
----------------
The memory footprint required to compute the solution is based on storing the matrices $U$, $V$, and $S$. Therefore, the memory required is $$\label{2D_Memory}
\mathrm{memory} = 2(mr+r^2+nr),$$ where factor 2 assumes that we need to store the previous step solution as well as the new step. The full solution to this problem without splitting would require a memory footprint of $2mn$. Therefore, for $r \ll m,n$, there will be large memory savings.
Conservation
------------
The low-rank algorithm we have described does not conserve the number of particles. This loss of conservation is a result of information lost in the algorithm when restricting the solution to low-rank descriptions. We have addressed this by globally scaling the solution after each time step to correct for any particles lost. We can do this because we know the number of particles that are absorbed, travel across the boundary, and born from sources. This correction will not preserve higher moments of the solution, and it only preserves the zeroth-order moment in space and angle. This issue has also been addressed in [@Einkemmer2019], where a correction term calculated by the imposed conservation law is added to each splitting step.
Numerical Results
=================
We run the transport simulations with four test problems, including the plane source problem and Reed’s problem in 1D slab geometry in addition to the line source problem and lattice problem in 2D planar geometry. It should be noted that the full rank of the solution matrix only depends on the columns $n$ since the number of mesh zones $m$ is always larger than $n$, which is the number of angular basis functions. To find out the relations behind different low-rank cases, we fix the spatial resolution and vary the $n$ and the rank $r$ in the following simulations. The CFL condition with the formula $CFL = \frac{\Delta t}{\Delta z}$ in 1D and $CFL = \min(\frac{\Delta t}{\Delta x},\frac{\Delta t}{\Delta z})$ in 2D is set to $0.2$. The conservation fix is applied to the plane and line source problems unless specified otherwise. Additionally, we used one simulation to demonstrate that the low-rank algorithm is compatible with the filter, other than that all the results are unfiltered. Our implementation is written in Matlab, and our particle speeds are set to $c=1$.
[1]{}
![Solutions to the plane source problem using the low rank method compared to the analytic solution. Numerical smearing errors are observed in both edges due to numerical dissipation [@Raithby1999]. More accurate results can be obtained with the fixed rank but higher [P$_N$]{}order, which will be shown shortly.[]{data-label="PP_T1"}](./Figures/PP_T1_noscale.eps)
[1]{}
![Solutions to the plane source problem using the low rank method compared to the analytic solution. Numerical smearing errors are observed in both edges due to numerical dissipation [@Raithby1999]. More accurate results can be obtained with the fixed rank but higher [P$_N$]{}order, which will be shown shortly.[]{data-label="PP_T1"}](./Figures/PP_T1_scale.eps)
![Comparison of P$_7$ solutions of rank $4$ with and without a filter to the analytic solution at t=1.[]{data-label="PP_filter_T1"}](./Figures/PP_T1_filter.eps)
![Solutions to the plane source problem using the low rank method compared to the analytic solution at t=5.[]{data-label="PP_T5"}](./Figures/PP_T5.eps)
[1]{}
![The comparison of errors on the plane source problem with different memory usage are shown. Each dotted line represents the error with a fixed rank that varies the number of angular basis functions n. The bold dot denotes the full rank solution.[]{data-label="PP_results"}](./Figures/PP_T1_error.eps)
[1]{}
![The comparison of errors on the plane source problem with different memory usage are shown. Each dotted line represents the error with a fixed rank that varies the number of angular basis functions n. The bold dot denotes the full rank solution.[]{data-label="PP_results"}](./Figures/PP_T5_error.eps)
Plane source problem
--------------------
The plane source problem describes a plane of isotropically moving particles emitted at $t=0$ in a purely scattering medium with no source, i.e., ${\sigma_\mathrm{t}}= {\sigma_\mathrm{s}}= 1$ and $Q = 0$. The only spatial variable in this slab geometry is $z$; the initial condition is given by a delta function as $\psi(z,\mu,t) = \delta(z)/2$. For the simulation parameters, we set the spatial resolution to be $\Delta z = 0.01$, which corresponds to the number of mesh zones $m = 300$ for the $t=1$ solution and $m = 1200$ for $t=5$. $P_{23}$ solutions with different rank are compared. When used, the filter strength $\beta$ is set to $50$. The analytical benchmark solution that we compare to was given by Ganapol [@Ganapol2008].
Figure \[PP\_T1\] shows the solutions of varying rank and Legendre polynomial orders with and without the conservation fix where the zeroth moment is scaled. As can be seen in either case, the solution with rank $12$, which is half of the full rank, matches the analytic solution to the scale of the graph in the middle part of the problem. It also agrees well with the full rank solution. The rank $8$ solutions still capture the analytical solution well. However, the loss of conservation can be observed when the conservation fix is not applied. Rank $4$ is not sufficient for accurate results and suffers more for the conservation lost. As shown in Figure \[PP\_filter\_T1\], the low-rank solution can be improved by the filter: P$_7$ solutions of reduced rank improve when a filter is used. Figure \[PP\_T5\] presents the solution at $t = 5$, at which even the rank $8$ appears to be sufficient. At this later time, there are few remaining uncollided particles from the initial condition. Therefore, fewer angular degrees of freedom is needed.
For a more quantitative comparison, the root mean square (RMS) error of the numerical results with different $n$ and $r$ is shown in Figure \[PP\_results\]. In this figure, the colors for the dotted lines correspond to the rank used in a calculation, and different values of the $n$, the number of angular basis functions, are corresponding dots. For each color the value of $n$ ranges from $r$ to $100$. The large points are the value of the error using the standard full rank method with $r=n$. We can observe that the low-rank solution is more accurate than the full rank with the same memory usage. For example, the error of full rank solution $n=12$ at $t = 1$ using with a memory footprint of $7200$ is about $0.026$. With the same memory, the error can be reduced to $0.013$. We can also use $70\%$ of the memory to achieve the same accuracy. Increasing the resolution and rank will contribute to the accuracy of solutions. Given the way we performed this study with a fixed spatial mesh and time step and the conservation fix we used, we can see some error stagnation in the low-rank solution at $t=5$. Other numerical experiments indicate that increasing the number of spatial zones can further decrease the error.
![The material layout in Reed’s problem where the blank zone means vacuum[]{data-label="Reeds_layout"}](./Figures/Reeds.eps)
[1]{}
![Solutions to the Reed’s problem at t=5 using the low rank method compared to the high-degree full rank solution.[]{data-label="Reeds_results"}](./Figures/Reed_T5.eps)
[1]{} ![The comparison of errors for Reed’s problem with different memory usage are shown. Each dotted line represents the error with a fixed rank that varies the number of angular basis functions n. The bold dot denotes the full rank solution.[]{data-label="Reeds_error"}](./Figures/Reeds_T5_error.eps "fig:")
Reed’s problem
--------------
The second test problem is Reed’s problem, which is a multi-material problem, and its set-up is detailed in Figure \[Reeds\_layout\]. Because Reed’s problem does not have an analytical solution, a numerical result with a high degree of angular basis and full rank, where $\Delta z = 0.01$ and $P_{99}$ (corresponding to $m=1600$ and $n=100$) is set as a benchmark for memory analysis. Figure \[Reeds\_results\] shows that the rank $4$ solutions differ in the vacuum regions ($z \in (3,5)$ and $z \in (11,13)$) and the scattering region with source ($z \in (2,3)$ and $z \in (13,14)$), but the rank $8$ solution matches the P$_{99}$ reference solution well.
It can be observed in Figure \[Reeds\_error\] that the low-rank solutions (solid lines with small dots) can give solutions with comparable errors to the full rank solutions (large dots) with much larger memory. For example, the rank $10$ solutions obtain a solution error better than the full rank P$_{19}$ solution with less memory.
[.5]{}
![Solutions to the line source problem at $t = 1$ using rank $210$ compared to the analytic solution. []{data-label="LS_results1"}](./Figures/LS_T1_ana.eps){width="\columnwidth"}
[.5]{}
![Solutions to the line source problem at $t = 1$ using rank $210$ compared to the analytic solution. []{data-label="LS_results1"}](./Figures/LS_T1_r210P19.eps){width="\columnwidth"}
[.5]{}
![Solutions to the line source problem at $t = 1$ using rank $210$ compared to the analytic solution. []{data-label="LS_results1"}](./Figures/LS_T1_r210P29.eps){width="\columnwidth"}
[.5]{}
![Solutions to the line source problem at $t = 1$ using rank $210$ compared to the analytic solution. []{data-label="LS_results1"}](./Figures/LS_T1_r210P39.eps){width="\columnwidth"}
[1]{}
![Solutions to the line source problem at $t = 1$ using rank $210$ compared to the analytic solution. []{data-label="LS_results1"}](./Figures/LS_T1_r210.eps)
[1]{} ![Solutions to the line source problem at $t = 1$ using rank $136$ compared to the analytic solution at the cut $z=0$.[]{data-label="LS_results2"}](./Figures/LS_T1_r136.eps "fig:")
![The comparison of errors for the line source problem with different memory usage are shown. Each dotted line represents the error with a fixed rank that varies the number of angular basis functions $n$. The bold dot denotes the full rank solution.[]{data-label="LS_error"}](./Figures/LS_T1_error.eps)
Line source problem
-------------------
The line source problem is a natural extension of the plane source problem to 2-D, where the plane source becomes a line. In this problem we have ${\sigma_\mathrm{t}}= {\sigma_\mathrm{s}}= 1$, $Q = 0$ and the initial condition $\psi(x, z, \mu, \varphi, t) = \delta(x)\delta(z)$. We use a computational domain of $[-1.5,1.5] \times [-1.5,1.5]$ for the simulation time $t=1$, while the spatial grid is set to be $150 \times 150$.
The analytical solution from Ganapol [@Ganapol2008] and the rank 210 solutions with P$_{19}$, P$_{29}$ and P$_{39}$ are shown in Figure \[LS\_results1\], from which we can see that the P$_{15}$ solution which is full rank, has large oscillations. The results obtained with the same rank but more angular basis functions are much better, as we can see in Figure \[LS\_results1\]e where fixing the rank and increasing the expansion order improves the solution dramatically. Figure \[LS\_results2\] demonstrates that increasing the angular resolution while fixing the rank improves the solution considerably.
-------- --------- --------- --------- ---------
Memory 1625184 1681632 2487100 2571250
Error 0.2541 0.0829 0.1972 0.0629
-------- --------- --------- --------- ---------
In this figure, we note that the full rank P$_{15}$ solution has negative values, which are due to the oscillatory property of [P$_N$]{} discretizations [@Mcclarren2008]. This nonphysical phenomenon can be alleviated by increasing the order of [P$_N$]{} with low-rank approximation with only a modest increase in memory.
Figure \[LS\_error\] shows that memory saving is more significant in the 2D problem, where the number of spatial degrees of freedom is large. As we can see, the memory footprint is mostly dominated by rank. It indicates that the accuracy can be improved as we increase the P$_N$ order with few extra memory costs. For example, the error of the solution at rank 78 with the low-rank method is three times smaller than the full rank solution. It is also shown in Table \[comparsion\] that only $3.4\%$ extra memory is required for this accuracy increase. Alternatively, we can save $65\%$ of the memory by adopting a low-rank solution at rank 78 rather than a full rank solution at rank 210.
![The material layout in the Lattice problem. The blue zones are purely scattering region, the black are absorbing region and the yellow is also the scattering region with an isotropic source which is turned on at T$ = 0s$. The checkerboard is surrounded by vacuum.[]{data-label="Lattice_layout"}](./Figures/Lattice.eps)
[.5]{} ![Solutions to Lattice problem with P$_{15}$ and different rank.[]{data-label="Lattice_results"}](./Figures/LatticeP15r136.eps "fig:"){width="\columnwidth"}
[.5]{} ![Solutions to Lattice problem with P$_{15}$ and different rank.[]{data-label="Lattice_results"}](./Figures/LatticeP15r55.eps "fig:"){width="\columnwidth"}
[.5]{} ![Solutions to Lattice problem with P$_{15}$ and different rank.[]{data-label="Lattice_results"}](./Figures/LatticeP15r36.eps "fig:"){width="\columnwidth"}
[.5]{} ![Solutions to Lattice problem with P$_{15}$ and different rank.[]{data-label="Lattice_results"}](./Figures/LatticeP15r21.eps "fig:"){width="\columnwidth"}
[.5]{} ![Solutions to Lattice problem with P$_{15}$ and different rank.[]{data-label="Lattice_results"}](./Figures/LatticeP15r10.eps "fig:"){width="\columnwidth"}
[.5]{} ![Solutions to Lattice problem with P$_{15}$ and different rank.[]{data-label="Lattice_results"}](./Figures/LatticeP15convergence.eps "fig:"){width="\columnwidth"}
Lattice problem
---------------
As described in Figure \[Lattice\_layout\] , the lattice problem is a $7 \times 7$ checkerboard with purely scattering zones ${\sigma_\mathrm{t}}= {\sigma_\mathrm{s}}= 1$, purely absorbing zones ${\sigma_\mathrm{t}}= 10, {\sigma_\mathrm{s}}= 0$, and an isotropic source at the center where $Q = 1$ [@Brunner2005]. In this problem, we use a spatial grid of size $210 \times 210$ for the computational domain $[0,7] \times [0,7]$.
Figure \[Lattice\_results\] shows that the solution with different rank agrees well to each other except the absorbing zones, where high-rank solutions such as rank 55 and 36 contain more details than the rank 21 and 10 solutions. Additionally, low-rank solutions converge to the full rank as we increase the rank, which leads to more memory usage. As we can see that even with $25\%$ of the full rank memory the difference is still small.
Conclusion and Future Work
==========================
We have developed a practical algorithm to find the low-rank solution of the slab and planar geometry transport equation using explicit time integration. The method is based on projecting the equation to low-rank manifolds and numerically integrating with three steps. The numerical simulations show that on several test problems, the memory savings of the low-rank method can be on the order of a factor of 2-3. This study establishes a projection-based framework for the direct model decomposition of the radiation transport problems.
In future work will attempt to solve the conservation issues present in the low-rank method using a high-order low-order approach (such as quasi-diffusion), where the low-rank method is used only to create a closure. In this approach, the low-rank method is estimating a ratio of moments that should be insensitive to conservation issues. Furthermore, several research problems, including asymptotic preservation, the extension to energy-dependent transport equation should be explored, and other angular treatments, such as discrete ordinates.
References {#references .unnumbered}
==========
[^1]: The terminology direction and angle are typically used interchangeably to describe the angular component of the phase space.
|
---
abstract: 'Lithium chloride LiCl is widely used as a prototype system to study the strongly dissociated 1-1 electrolyte solution. Here, we combined experimental measurements and classical molecular dynamics simulations to study the ion conduction in this system. Ionic conductivities were reported at both 20$^\circ$C and 50$^\circ$C from experiments and compared to results from molecular dynamics simulations. The main finding of this work is that transference numbers of Li$^+$ and Cl$^-$ become comparable at high concentration. This phenomenon is independent of the force fields employed in the simulation and may be resulted from the ion-specific concentration dependence of mobility.'
address: 'Department of Chemistry-Ångström Laboratory, Uppsala, University, Lägerhyddsvägen 1, 75121 Uppsala, Sweden'
author:
- Are Yllö
- Chao Zhang
title: Experimental and molecular dynamics study of the ionic conductivity in aqueous LiCl electrolytes
---
Electrolyte solution ,Ionic conductivity ,Molecular dynamics ,Force fields ,Debye-Onsager theory
Introduction
============
Aqueous electrolytes play important roles in many areas of science and engineering, such as electrophysiology, electrochemistry and colloid science. Simple 1-1 electrolyte which is completely dissociated in dilute solution is often used as a prototype system to develop analytical theories such the well-known Debye-Hückel theory [@Fawcett:2004ww]. This tradition dates back to the beginning of Physical Chemistry and coins the early physical chemists as “Ionists” [@Servos:1996tl].
Lithium chloride (LiCl) as an example of these simple 1-1 electrolytes is of particular interest due to its very high solubility ($\sim$ 45 wt% at room temperature). The structure of LiCl solution has been extensively investigated by X-ray diffraction and neutron scattering experiments [@Tromp:1992hr; @Winkel:2011kn; @Ansell:2006dv] in together with reverse Monte Carlo and molecular dynamics simulations [@Ibuki:2009ca; @Harsanyi:2005ju; @Petit:2008tw; @Ibuki:2009ca; @Harsanyi:2011ea; @Pluharova:2013jo; @Aragones:2014ki; @Pethes:2017fr]. The synergy between experiments and simulations has been proven to be useful to gain a deeper understanding of solvation structures of Li$^+$ and Cl$^-$.
In the molecular dynamics simulation community, another interest of modeling LiCl solution was on developing various kinds of force-fields where cations and anions are commonly described by Lennard-Jones (LJ) potential and point charge [@Joung2008; @Li:2015gp]. Despite of its simplicity, this approach has been shown be capable to capture both single ion properties (such as the hydration free energy) to ion-ion interactions as reflected in radial distribution functions and the solubility [@Aragones:2014ki]. We refer interested readers to a recent work on this topic for a comprehensive overview and benchmarks [@Pethes:2017fr].
On the other hand, the dynamical and transport properties of these models were often overlooked. In particular, the ionic conductivity of LiCl calculated from molecular dynamics simulations has not been compared to experimental measurements at both room temperature and elevated temperature. This fact is somehow surprising, because the basic function of any electrolyte is to serve as an ionic conductor.
In this work, we carried out both experimental measurements and molecular dynamics simulations of the ionic conductivity in LiCl solutions. Ionic conductivities were reported at both 20$^\circ$C and 50$^\circ$C from experiments and compared to those calculated from molecular dynamics simulations using three different force-field models [@Pluharova:2013jo; @Joung2008; @Li:2015gp] (See Section 2 for details) and SPC/E water [@Berendsen:1987uu]. In addition to provide reference data for future force-field developing works, the main finding of our study is that transference numbers (i.e. the fractional contribution to the ionic conductivity) of Li$^+$ and Cl$^-$ become comparable at high concentration. This phenomenon is independent of the force fields employed in the simulation and can be explained by taking into account the ion-paring and ion-specific effects. The later imposes a challenge to the Debye-Onsager theory of the ionic conductivity.
Experimental and computational methods
======================================
Ionic conductivity measurements
-------------------------------
The conductivity measurement of LiCl at 2, 5, 10, 15, 20, 25, 30, 35 and 40 wt % were performed with an “InLab” conductivity meter (Mettler Toledo). The conductivity meter probe used is a 4 pole InLab 738-ISM by (Mettler Toledo) which has a sensitivity range from 0.01–1000 mS/cm and gives accurate measurements up to 100$^\circ$C. Before measuring, the probe was calibrated with a standardized 12.88 mS/cm potassium chloride (KCl) solution (Mettler Toledo). After the successful calibration of the instrument, the probe was lowered into respective solution. The measurement ran until both the conductivity and the temperature of the solution had equilibrated at a stable value. The mean of the five independent measurements were then noted as the final conductivity of that solution at 20$^\circ$C.
Similar measurements were then done at an elevated temperature of approximately 50$^\circ$C. The solutions were heated to 50$^\circ$C by placing them in a heated water bath with an external thermometer attached to a reference plastic container with deionized water. When the solution had reached the sought-after temperature, the measurements were carried out in the same way as before.
Molecular dynamics simulations
------------------------------
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Model $\sigma_{\textrm{Li,Li}} $ (nm) $\epsilon_{\textrm{Li,Li}} $\sigma_{\textrm{Cl,Cl}} $\epsilon_{\textrm{Cl,Cl}} $ (kJ/mol) $q_{\textrm{Li}}~/
$ (kJ/mol) $ (nm) q_{\textrm{Cl}}$ (e)
------------------------ --------------------------------- --------------------------------------------------------------------- ----------------------------------------------------------------------------- --------------------------------------- --------------------------------------------------------------------------
JC-S [@Joung2008] 0.1409 1.4089 0.4830 0.0535 $+1~/-1$
LI-IOD-S [@Li:2015gp] 0.2343 0.0249 0.3852 2.2240 $+1~/-1$
PL [@Pluharova:2013jo] 0.1800 0.0765 0.4100 0.4928 $+0.75~/-0.75$
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The initial cubic box containing simple point charge/extended (SPC/E) water molecules [@Berendsen:1987uu] and random distributed Li$^+$/Cl$^-$ ions was 2.963 nm for each side. Water molecules were kept rigid using the SETTLE algorithm [@Miyamoto1992]. The Ewald summation was implemented using the Particle Mesh Ewald (PME) [@Ewald] scheme and short-range cutoffs for the van der Waals and Coulomb interaction in the direct space are 1 nm.
Three force fields (ion models) for LiCl were chosen in this study which are Joung-Cheatham III (JC-S) [@Joung2008], Li-Song-Merz (LI-IOD-S) [@Li:2015gp] and Pluhařov' a -Mason-Jungwirth (PL) [@Pluharova:2013jo]. JC-S was parameterized against thermodynamic data such hydration free energy and lattice energy of salt crystal and has been validated for higher salt concentration [@Zhang:2010zh; @Zhang:2012fo] at room temperature. LI-IOD-S focus on the structural aspect and was fitted to the ion oxygen distance in the first solvation shell. PL-S was tuned by scaling down the point charge of each ion by the refractive index of liquid water in order to make up the missing electronic polarization. The corresponding LJ parameters and point charges of these three models are summarized in Table \[table:potential\]. In all cases, the Lorentz-Berthelot combination rule was used between two dissimilar non-bonded atoms.
Regarding the technical setting in simulations, the steepest descent algorithm was used for the energy minimization before the equilibration. The NVT (constant number of particles, constant volume and constant temperature) equilibration ran for 1 ns with the timestep of 2 fs. The temperature was then held in place using the Bussi-Donadio-Parrinello thermostat which preserves both thermodynamic and dynamic properties [@Bussi:2008wu]. The follow-up NPT (constant number of particles, constant pressure and constant temperature) simulations ran for 10 ns each and trajectories were collected every 0.5ps for conductivity calculation and structural analysis. During the NPT simulations, Parrinello-Rahman barostat [@M.1980] was employed with a reference pressure of 1.0 bar. This simulation protocol was used for LiCl solution at 2, 5, 10, 15, 20, 25, 30, 35 and 40 wt % and both 20$^\circ$C and 50$^\circ$C and all simulations were performed using GROMACS 4 package [@Hess2008]. This corresponds to simulations with following compositions in terms of number ratio $N_\text{salt}$/$N_\text{water}$: 6/694, 15/676, 30/646, 46/614, 61/584, 77/552, 94/518, 110/486, 127/452. Statistical errors were estimated using the standard deviation of observables from 5 equispaced segments in the full trajectory.
Results and discussion
======================
Ionic conductivity and transference number
------------------------------------------
The simplest way to calculate the ionic conductivity in molecular dynamics simulations is to use the Nernst-Einstein equation [@Nitzan:2013tk]:
$$\begin{aligned}
\sigma &=& \sigma_+ + \sigma_- \\
& = &\frac{q_+^{2}\rho D_+}{kT} +\frac{q_-^{2}\rho D_-}{kT}
\label{eq:nersteinstein}\end{aligned}$$
where $\sigma$ is the ionic conductivity of the solution, $\sigma_+$ and $\sigma_-$ are ionic conductivities for cation and anion respectively. $q_+$ and $q_-$ are point charges of ions in the model. $\rho$ is the number density of the salt, $k$ is Boltzmann constant and $T$ is the temperature.
$D_+$ and $D_-$ are self-diffusion coefficients of cation and anion respectively and computed from the corresponding mean squared displacement using the Einstein relation as follows:
$$\label{msd}
D_{+/-} = \lim_{t\rightarrow \infty}\frac{1}{6t}\frac{1}{N_\text{salt}}\sum_i^{N_\text{salt}}\langle[\mathbf{r}_{i,+/-}(t)-\mathbf{r}_{i,+/-}(0)]^2 \rangle$$
where $t$ is the time, $N_\text{salt}$ is the number of LiCl salt, $\mathbf{r}_{i, +/-}$ is the position of $i$th cation or anion, $\langle \cdots
\rangle $ indicates the ensemble average.
One should note that the the Nernst-Einstein equation holds only for non-interacting charged particles in a homogeneous and isotropic solvent. Thus, the ion-ion correlation is not taken into account in the formula. In other words, the ionic conductivity calculated using the Nernst-Einstein equation gives an upper bound of the actual value.
On the other hand, since $\sigma$ is a sum of individual contributions of cations and anions by construction, the transference number $t_{+/-}$ can be readily extracted as:
$$t_{+/-} = \frac{\sigma_{+/-}}{\sigma}$$
![Ionic conductivities vs. wt% of LiCl from MD simulations and experimental measurements at 20$^\circ$C a) and 50$^\circ$C b). Literature value at 20$^\circ$C is from Ref. [@CRC:99].[]{data-label="fig:mdcond"}](Fig1){width="\linewidth"}
![Molar conductivities vs. wt % of LiCl from MD simulations and experimental measurements at 20$^\circ$C a) and 50$^\circ$C b).[]{data-label="fig:mdmolcond"}](Fig2){width="\linewidth"}
From Fig. \[fig:mdcond\]a, we see that the results from JC-S is the one that comes closest to the measured and the literature values in the whole concentration range at 20$^\circ$C, although three ion models seem be equally well at lower concentrations. Our measured molar conductivity is also in accord with the result in a recent report [@Yim:2018gq]. At 50$^\circ$C (See Fig. \[fig:mdcond\]b), LI-IOD-S gives results which agrees best with measured values. JC-S overestimates the conductivity for lower to mid-range concentrations and underestimates it for higher concentrations. At both temperatures, PL significantly overestimates the conductivity from mid to high concentrations. Similar behavior of PL has been reported for the diffusion coefficient of Li$^+$ and Cl$^-$ recently [@Pethes:2017fr]. This is likely due to the fact that point charge of ions are scaled down in this model which leads to a much weaker ion-solvent interaction.
Both JC-S and LI-IOD-S manage to describe the parabola behavior of the ionic conductivity as a function of the concentration and to provide accurate estimates of the corresponding concentration at the conductivity maximum. The reason for the conductivity maximum comes from a tradeoff between the increase of number of charge transportors and the decrease of their mobility as the concentration goes up. When molar conductivities are plot instead (Fig. \[fig:mdmolcond\]), one can see clearly that the mobility of ions reduces as a function of the concentration. Results of JC-S and LI-IOD-S have better agreements with experiments while PL shows a much higher deviation in the mid-to-high concentration range.
![Transference numbers vs. wt % of LiCl at 20$^\circ$C a) and 50$^\circ$C b).[]{data-label="fig:trans"}](Fig3){width="\linewidth"}
Fig. \[fig:trans\] shows the transference numbers of Li$^+$ and Cl$^-$ of three models at both 20$^\circ$C and 50$^\circ$C. The chloride ions contributes a larger fraction of the electrical current (0.55 to 0.65) while lithium ions stand for a smaller fraction (0.45 to 0.35). However, this gap diminishes as the concentration increase and eventually the transference numbers become similar nearly the solubility limit.
Radial distribution function and ion-pairing
--------------------------------------------
The configurational distribution function $P(\mathbf{r}^N)$ can be reduced to its two-particle version as [@Chandler:1987tp]:
$$\rho(\mathbf{r}_1,\mathbf{r}_2)=N(N-1)\int d\mathbf{r}_3\int d\mathbf{r}_4\cdots\int
d\mathbf{r}_N P(\mathbf{r}^N)$$
which gives the joint probability distribution to find one particle at position $\mathbf{r}_1$ and any other particle at $\mathbf{r}_2$. Note that the factor $N(N-1)$ accounts for all possible pairs.
In an ideal gas, particles are uncorrelated. As a result, the $\rho(\mathbf{r}_1,\mathbf{r}_2)$ simply equals to $N(N-1)/V^2\approx
\rho^2$ where $\rho$ is the number density. This leads to the definition of the quantity $g(\mathbf{r}_1,\mathbf{r}_2)$ called the pair distribution function:
$$g(\mathbf{r}_1,\mathbf{r}_2)=\rho(\mathbf{r}_1,\mathbf{r}_2)/\rho^2$$
This quantity reflects the density deviation from the (uncorrelated) ideal gas.
For isotropic fluid, this function depends upon $|\mathbf{r}_1-\mathbf{r}_2|=r$, this makes $g(r)$ called a radial distribution function.
The coordination number, i.e. the number of neighbouring atoms within first minimum of the $g(r)$ from a central atom, is define as:
$$\label{eq:coordnum}
n =4\pi\rho\int_0^{r_{min}} x^2g(x)dx$$
The first peak of $g_{\text{Li}^+-\text{O}}$ steadily decreases for both JC-S and LI-IOD-S ion models at 20$^\circ$C with increasing LiCl concentration (Fig. \[fig:rdf-li-o\]). Similar trend was seen for $g_{\text{Cl}^--\text{H}}$ with LI-IOD-S (Fig. \[fig:rdf-cl-h\]). This is expected, since the coordinating hydrogen/oxygen atoms of water molecules are gradually replaced by the counter-ions, see Table \[table:cumnumrdf\]. The anomaly is that $g_{\text{Cl}^--\text{H}}$ at 20wt% has the highest first peak with JC-S. In the case of PL, the peak heights of $g_{\text{Li}^+-\text{O}}$ and $g_{\text{Cl}^--\text{H}}$ are not much modulated by the concentration.
Regarding the radial distribution function of Li$^+$-Cl$^-$, it goes up with increasing concentration (Fig. \[fig:rdf-li-cl\]) for JC-S and PL. An opposite trend was found in the case of LI-IOD-S. Despite that, coordination numbers between Li$^+$ and Cl$^-$ become larger with the concentration for all three ion models, which is a sign of ion-pairing.
![Radial distribution functions (RDFs) of Li$^+$-O RDF at 5, 10, 20, 30, 40 wt% of LiCl and the temperature of 20$^\circ$C.[]{data-label="fig:rdf-li-o"}](Fig4){width="\linewidth"}
![Radial distribution functions (RDFs) of Cl$^-$-H at 5, 10, 20, 30, 40 wt% of LiCl and the temperature of 20$^\circ$C.[]{data-label="fig:rdf-cl-h"}](Fig5){width="\linewidth"}
![Radial distribution functions (RDFs) of Li$^+$-Cl$^-$ at 5, 10, 20, 30, 40 wt% of LiCl and the temperature of 20$^\circ$C.[]{data-label="fig:rdf-li-cl"}](Fig6){width="\linewidth"}
We notice that the coordination number of Li$^+$-O and Cl$^-$-H in LI-IOD-S is more sensitive to the concentration, in contrast to other two ion models (Table \[table:cumnumrdf\]). This is in accord with Fig. \[fig:rdf-li-o\] and Fig. \[fig:rdf-cl-h\].
Although radial distribution functions at 50$^\circ$C have a similar concentration dependence (data not shown), coordination numbers of ion-water become smaller in most cases as shown in Table \[table:cumnumrdf\] which were expected, because hydration shells become less structured at elevated temperature. In contrast, all three ion models show that the cation-anion coordination number goes up with the temperature. This may be due to the fact that the dielectric constant of liquid water decreases with the temperature and the solvent screening is weaker accordingly.
wt% LiCl 5 10 20 30 40
------------------------- ------ ------ ------ ------ -----
JC-S: Li$^+$-O 4.19 4.16 4.10 3.87 3.2
\* 4.20 4.17 4.08 3.83 3.2
LI-IOD-S: Li$^+$-O 3.73 3.30 2.79 2.34 2.0
\* 3.63 3.2 2.67 2.27 2.0
PL: Li$^+$-O 3.95 3.88 3.65 3.28 2.8
\* 3.92 3.82 3.58 3.19 2.7
JC-S: Cl$^-$-H 6.81 6.84 6.85 6.57 5.4
\* 6.65 6.67 6.64 6.37 5.3
LI-IOD-S: Cl$^-$-H 5.88 5.29 4.50 3.79 3.3
\* 5.65 5.06 4.18 3.63 3.1
PL: Cl$^-$-H 5.56 5.50 5.11 4.65 3.9
\* 5.29 5.27 4.90 4.32 3.6
JC-S: Li$^+$-Cl$^-$ 0.00 0.01 0.03 0.18 0.8
\* 0.01 0.02 0.07 0.24 0.8
LI-IOD-S: Li$^+$-Cl$^-$ 0.62 0.96 1.35 1.75 2.0
\* 0.67 1.01 1.46 1.81 2.1
PL: Li$^+$-Cl$^-$ 0.05 0.12 0.35 0.71 1.2
\* 0.07 0.16 0.4 0.78 1.3
: Coordination numbers as defined in Eq. \[eq:coordnum\] at different concentrations of LiCl. The row starting with the model name shows the data at 20$^\circ$C and the row starting with $*$ shows the corresponding data at 50$^\circ$C. Data at 40wt% were rounded off to the first decimal to indicate a lower accuracy.[]{data-label="table:cumnumrdf"}
Ion-paring contribution to the ionic conductivity
-------------------------------------------------
We mentioned at the beginning of Section 3.1 that the ionic conductivity calculated from the Nernst-Einstein equation provides an upper bound and the actual conductivity is always smaller because of the ion-paring (ion-ion correlation). Near the solubility limit, the ionic conductivity may be reduced by 30% when ion-ion correlations are taken into considered in the calculation [@Chowdhuri:2001bl]. This is similar to our estimation based on the mean square charge displacement [@DeLeeuw:1981uc], which gives a value of 40% for LiCl. Therefore, the overshooting of PL in the ionic conductivity at high concentration as shown in Fig. 1 is not because of the missing of ion-pairing contribution in Eq. \[eq:nersteinstein\] but likely due to the down-scaling of the charge in the model (See Table \[table:potential\]).
One of the main observations in this study is that the transference number of the chloride ion becomes similar to that of the lithium ion. This is in accord to the tracer diffusion measurement reported in the literature [@Tanaka:1987kg]. The standard explanation for this phenomenon is that lithium and chloride ions pair up at high concentration and move together in a concerted manner.
![The normalized molar conductivities of Li$^+$ and Cl$^-$ vs. wt% of LiCl at 20$^\circ$C a) and 50$^\circ$C b).[]{data-label="fig:relative-cond"}](Fig7){width="\linewidth"}
Instead, we notice that the relative reduction of the Cl$^-$ conductivity with the increase of the concentration can be notably larger than that of the Li$^+$ conductivity in the case of PL (Fig. \[fig:relative-cond\]). . This observation is interesting because the PL model was adjusted to take into account the missing electronic polarization. In addition, we noticed that the difference in the concentration dependence between Li$^+$ and Cl$^-$ becomes more apparent at 50$^o$C because the dielectric constant goes down. Thus, it would be of interest to investigate the effect of polarization on the ion-specific concentration dependence of mobility.
The implication of this observation is twofold. Firstly, since the molar conductivity of the chloride ion at infinite dilution is larger than that of the lithium ion, therefore the ion-specific concentration dependence could lead to a crossover between cation transference numbers and anion transference number even without considering the ion-pairing. Secondly, the standard Debye-Onsager theory may need be expanded in order to consider ion-specific concentration dependence for mobility of Li$^+$ and Cl$^-$.
Conclusion
==========
In this work, we carried out both experimental measurements and molecular dynamics simulations of the ionic conductivity in LiCl solutions. Ionic conductivities were reported at both 20$^\circ$C and 50$^\circ$C from experiments and compared to those calculated from molecular dynamics simulations using three different ion models. In addition to provide reference data for future force-field developments, the main finding of our study is that transference numbers of Li$^+$ and Cl$^-$ become similar at high concentration. This phenomenon is independent of the force fields employed in the simulation and may be due to the ion-specific concentration dependence of mobility.
Acknowledgments
===============
C.Z. thanks Uppsala University for a start-up grant. Funding from the Swedish National Strategic e-Science program eSSENCE is also gratefully acknowledged.
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|
---
address: |
$^1$Physik-Department der Technischen Universit[ä]{}t M[ü]{}nchen,\
D-85747 Garching, Germany\
$^2$Laboratory of Radiation Physics, Institute of Solid State Physics,\
University of Latvia, LV 2169 Salaspils, Miera str. 31, Latvia\
E-mail: Anatoli.Afanasjev@Physik.TU-Muenchen.DE
author:
- 'A. V. Afanasjev$^{1,2}$ and P. Ring$^1$'
title: |
Cranked relativistic Hartree-Bogoliubov theory:\
Superdeformation in the $A\sim 190$ mass region
---
=cmr8
1.5pt
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
epsf
Despite the fact that superdeformation at high spin has been studied experimentally and theoretically for one and half decade a number of theoretical questions such as, for example, the underlying mechanism of identical bands and the role of pairing correlations in the regime of weak pairing in rotating nuclei, remains still not fully resolved and further development of theoretical tools is definitely required. During the last decade the relativistic mean field (RMF) theory [@R.96] became a standard microscopic tool of nuclear structure studies. Systematic investigation of SD rotating nuclei in the regime of weak pairing correlations in the $A\sim 150$ [@KR.93; @AKR.96; @Hung; @ALR.98] and $A\sim 60$ (see Ref. [@A60] and references therein) mass regions revealed that cranked relativistic mean field (CRMF) theory [@KR; @AKR.96], in which pairing correlations are neglected, provides an astonishingly accurate description of the properties of SD bands, such as moments of inertia, transition quadrupole moments $Q_t$, effective alignments $i_{eff}$, single-particle properties at superdeformation etc. In particular we have to keep in mind that this theory has only seven free parameters fitted to the properties of few spherical nuclei [@NL1].
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One should clearly recognize that the neglect of pairing correlations used in CRMF theory is an approximation because pairing correlations being weak are still present even at the highest rotational frequencies. Moreover, the rotational properties of nuclei at low and medium spin are strongly affected by pairing correlations. In order to describe such properties within the relativistic framework, cranked relativistic Hartree-Bogoliubov theory has been developed [@A190; @CRHB; @J1Rare]. This theory is an extension of CRMF theory to the description of pairing correlations in the rotating frame. In this theory the particle-hole channel is treated fully relativistically on the Hartree level, while the particle-particle channel is approximated by the best currently available non-relativistic interaction: the pairing part of the Gogny force. The use of this force has a clear advantage since it provides both an automatic cutoff of high-momentum components and, as follows from non-relativistic studies, an excellent description of pairing properties in finite nuclei. An additional feature of CRHB theory is that approximate particle number projection is performed by means of the Lipkin-Nogami method (further APNP(LN)) [@L.60; @N.64; @PNL.73]. The comparative study of pairing and rotational properties in the rare earth region performed within the frameworks of CRHB theory and non-relativistic cranked Hartree-Fock-Bogoliubov theory based on the finite range force of the Gogny type [@J1Rare] indicates that APNP(LN) plays a more important role in the relativistic calculations most likely reflecting the lower effective mass.
In Refs. [@A190; @CRHB] CRHB theory has been applied for a systematic investigation of the properties of SD bands in even-even nuclei of the $A\sim 190$ mass region. The calculations have been performed using the well established parameter sets NL1 [@NL1] for the RMF Lagrangian and D1S [@D1S] for the Gogny force. Fig. \[sysj2j1\] compares the experimental dynamic and kinematic moments of inertia with the ones obtained in the CRHB calculations. Since the SD bands in $^{190,192}$Hg, $^{196,198}$Pb and $^{198}$Po are not linked to the low spin level scheme, their ’experimental’ spin values and thus kinematic moments of inertia have been established based on the comparison with calculated kinematic moments of inertia. Note that the analysis of experimental and calculated effective alignments $i_{eff}$ between the bands in different nuclei confirms the present assignment of the spin values for unlinked bands. One can see that very good agreement exists in all the cases with an exception of the yrast SD band in $^{198}$Pb. The investigation of the structure of the SD bands in neighboring odd nuclei is needed for a better understanding of the problems seen in this nucleus.
Proton and neutron scalar density distributions for the yrast SD band in $^{192}$Hg calculated at rotational frequency $\Omega_x=0.1$ MeV are shown in Fig. \[dens\]. Both distributions show considerable variations as a function of the coordinate. These variations are caused by shell effects. It is interesting to mention that the maximal density is reached along the symmetry axis at the distance of $6-8$ fm from the center of nucleus. These density distributions correspond to a transition quadrupole moment $Q_t=19.6$ $e$b. A detailed comparison presented in Ref. [@A190] shows that the results of the CRHB calculations are within the error bars of available experimental data on transition quadrupole moments $Q_t$ for yrast SD bands of even-even nuclei in the $A\sim 190$ mass region.
In conclusion, cranked relativistic Hartree-Bogoliubov theory has been developed and applied for a systematic investigation of SD bands of even-even nuclei in the $A\sim 190$ mass region. Using well established parameter sets for the RMF Lagrangian and Gogny force the available experimental data is described very well without any new adjustable parameters. Further investigations of odd and odd-odd nuclei are needed for a deeper understanding of the properties of SD bands in CRHB theory. Such an investigation is in progress.
Acknowledgments {#acknowledgments .unnumbered}
===============
A.V.A. acknowledges support from the Alexander von Humboldt Foundation. This work is also supported in part by the Bundesministerium f[ü]{}r Bildung und Forschung under the project 06 TM 979.
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References {#references .unnumbered}
==========
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|
---
abstract: 'In this paper we consider the following question: Is it possible to construct all real root representations of a given quiver $Q$ by using universal extension functors, starting with a real Schur representation? We give a concrete example answering this question negatively.'
address: 'Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, U.K.'
author:
- Marcel Wiedemann
title: A remark on the constructibility of real root representations of quivers using universal extension functors
---
Introduction
============
Let $k$ be a field and let $Q$ be a (finite) quiver. We fix a representation $S$ with ${{\operatorname{\mathsf{End}}}}_{kQ} S=k$ and ${{\operatorname{\mathsf{Ext}}}}^1_{kQ} (S,S)=0$. In analogy to [@ringel Section 1] we consider the following subcategories of ${{\operatorname{\mathsf{rep}}}}_k Q$. Let ${{\mathfrak{M}}}^S$ be the full subcategory of all modules $X$ with ${{\operatorname{\mathsf{Ext}}}}^1_{kQ}(S,X)=0$ such that, in addition, $X$ has no direct summand which can be embedded into some direct sum of copies of $S$. Similarly, let ${{\mathfrak{M}}}_S$ be the full subcategory of all modules $X$ with ${{\operatorname{\mathsf{Ext}}}}^1_{kQ}(X,S)=0$ such that, in addition, no direct summand of $X$ is a quotient of a direct sum of copies of $S$. Finally, let ${{\mathfrak{M}}}^{-S}$ be the full subcategory of all modules $X$ with ${{\operatorname{\mathsf{Hom}}}}_{kQ} (X,S)=0$, and let ${{\mathfrak{M}}}_{-S}$ be the full subcategory of all modules $X$ with ${{\operatorname{\mathsf{Hom}}}}_{kQ} (S,X)=0$. Moreover, we consider $$\begin{aligned}
{{\mathfrak{M}}}_S^S = {{\mathfrak{M}}}^S \cap {{\mathfrak{M}}}_S,\quad {{\mathfrak{M}}}_{-S}^{-S}= {{\mathfrak{M}}}^{-S} \cap {{\mathfrak{M}}}_{-S}.\end{aligned}$$ According to [@ringel Proposition 1 & $1^*$ and Proposition 2], we have the following equivalences of categories $$\begin{aligned}
\overline{\sigma}_S&:& {{\mathfrak{M}}}^{-S} \to {{\mathfrak{M}}}^S/S, \\
\underline{\sigma}_S&:& {{\mathfrak{M}}}_{-S} \to {{\mathfrak{M}}}_S/S, \\
\sigma_S&:& {{\mathfrak{M}}}_{-S}^{-S} \to {{\mathfrak{M}}}^S_S/S,\end{aligned}$$ where ${{\mathfrak{M}}}^{S}/S$ denotes the quotient category of ${{\mathfrak{M}}}^{S}$ modulo the maps which factor through direct sums of copies of $S$, similarly for ${{\mathfrak{M}}}_S/S$ and ${{\mathfrak{M}}}^{S}_S/S$. We call the functor $\sigma_S$ [*universal extension functor*]{}. A brief description of these functors is given in Section \[notation\]. This paper is dedicated to the following question.
Let ${\alpha}$ be a positive non-Schur real root for $Q$ and let $X_{\alpha}$ be the unique indecomposable representation of dimension vector ${\alpha}$.
Does there exist a sequence of real Schur roots ${\beta}_1,\ldots,{\beta}_n\; (n\ge 2)$ such that $$\begin{aligned}
X_{\alpha}= \sigma_{X_{{\beta}_n}}\cdot \ldots \cdot \sigma_{X_{{\beta}_2}} (X_{{\beta}_1})\quad ?\end{aligned}$$ Here, $X_{{\beta}_i}$ denotes the unique indecomposable representation of dimension vector ${\beta}_i$.
One might reformulate the above question as follows. Is it possible to construct all real root representations of $Q$ using universal extension functors, starting with a real Schur representation?
One of the nice facts about the universal extension functor $\sigma_S$ is that it allows one to keep track of certain properties of representations. For instance, the functor $\sigma_S$ preserves indecomposable tree representations [@wiedemann Lemma 3.16] (for a definition of “tree representation” and background results we refer the reader to [@ringel_3 Introduction]) and, moreover, if we apply the functor $\sigma_S$ to a representation of known endomorphism ring dimension, we can easily compute the dimension of the endomorphism ring of the resulting representation [@ringel Proposition 3 & $3^*$]. Hence, if $X_{\alpha}=\sigma_{X_{{\beta}_n}}\cdot \ldots \cdot \sigma_{X_{{\beta}_2}} (X_{{\beta}_1})$ with ${\beta}_i\; (i=1,\ldots,n)$ real Schur roots, then $X_{\alpha}$ is a tree representation and one can easily compute $\dim {{\operatorname{\mathsf{End}}}}_{kQ} X_{\alpha}$.
Question $(\star )$ was first answered affirmatively by Ringel [@ringel Section 2] for the quiver
$Q(g,h): \xymatrix{ 1 \ar@/^1.8pc/[r]^{\mu_1}
\ar@/^0.9pc/[r]^*-<0.4pc>{\vdots}_{\mu_g} & 2 \ar@/^1.8pc/[l]^{\nu_h} \ar@/^0.9pc/[l]^*-<1.3pc>{\vdots}_{\nu_1}}$,
with $g,h\ge 1$. In [@wiedemann Theorem B] Question $(\star)$ was answered affirmatively for the quiver
$Q(f,g,h)$: $\xymatrix{ 1 \ar@<1.5ex>[r]^{{\lambda}_1} \ar@<-1ex>[r]^*-<0.4pc>{\vdots}_{{\lambda}_f} & 2 \ar@/^1.8pc/[r]^{\mu_1}
\ar@/^0.9pc/[r]^*-<0.4pc>{\vdots}_{\mu_g} & 3 \ar@/^1.8pc/[l]^{\nu_h} \ar@/^0.9pc/[l]^*-<1.3pc>{\vdots}_{\nu_1}}$,
with $f,g,h\ge 1$. More examples of real root representations which can be constructed using universal extension functors can be found in [@wiedemann-thesis Appendix].
Hence, there are quivers for which Question $(\star)$ can be answered affirmatively. The question is, can it be answered affirmatively in general? Unfortunately the answer is negative in general.
In Section \[counterexample\] we give a concrete example answering Question $(\star )$ negatively.
This paper is organized as follows. In Section \[notation\] we discuss further notation and background results and in Section \[counterexample\] we describe an example answering Question $(\star )$ negatively.
[**[Acknowledgements.]{}**]{} The author would like to thank his supervisor, Prof. W. Crawley-Boevey, for his continuing support and guidance. The author also wishes to thank Prof. C. Ringel for his interest in this work and for stimulating discussions.
Further Notation and Background Results {#notation}
=======================================
Let $k$ be a field. Let $Q$ be a finite quiver, i.e. an oriented graph with finite vertex set $Q_0$ and finite arrow set $Q_1$ together with two functions $h,t:Q_1 \to Q_0$ assigning head and tail to each arrow $a\in Q_1$. A representation $X$ of $Q$ is given by a vector space $X_i$ (over $k$) for each vertex $i\in Q_0$ together with a linear map $X_a: X_{t(a)} \to X_{h(a)}$ for each arrow $a\in Q_1$. Let $X$ and $Y$ be two representations of $Q$. A homomorphism $\phi: X \to Y$ is given by linear maps $\phi_i: X_i\to Y_i$ such that for each arrow $a\in Q_1$, $a:i\to j$ say, the square
$\xymatrix{ X_i \ar@<0.0ex>[r]^-{X_a} \ar@<0.0ex>[d]_{\phi_i} & X_j \ar@<0.0ex>[d]^-{\phi_j} \\
Y_i \ar@<0.0ex>[r]^-{Y_a} & Y_j }$
commutes.
A dimension vector for $Q$ is given by an element of ${\mathds{N}}^{Q_0}$. We will write $e_i$ for the coordinate vector at vertex $i$ and by ${\alpha}[i],\, i\in Q_0,$ we denote the $i$-th coordinate of ${\alpha}\in {\mathds{N}}^{Q_0}$. We can partially order ${\mathds{N}}^{Q_0}$ via ${\alpha}\ge {\beta}$ if ${\alpha}[i]\ge {\beta}[i]$ for all $i\in Q_0$. We define ${\alpha}>{\beta}$ to mean ${\alpha}\ge {\beta}$ and ${\alpha}\ne {\beta}$. If $X$ is a finite dimensional representation, meaning that all vector spaces $X_i\; (i\in Q_0)$ are finite dimensional, then ${\underline{\dim}\,}X= (\dim X_i)_{i\in Q_0}$ is the dimension vector of $X$. Throughout this paper we only consider finite dimensional representations. We denote by ${{\operatorname{\mathsf{rep}}}}_k Q$ the full subcategory with objects the finite dimensional representations of $Q$. The Ringel form on ${\mathds{Z}}^{Q_0}$ is defined by $$\begin{aligned}
\langle{\alpha},{\beta}\rangle = \sum_{i\in Q_0} {\alpha}[i]{\beta}[i] - \sum_{a\in Q_1}{\alpha}[t(a)]{\beta}[h(a)]\end{aligned}$$ Moreover, let $({\alpha},{\beta})=\langle{\alpha},{\beta}\rangle+\langle{\beta},{\alpha}\rangle$ be its symmetrization.
We say that a vertex $i\in Q_0$ is loop-free if there are no arrows $a:i\to i$. By a quiver without loops we mean a quiver with only loop-free vertices. For a loop-free vertex $i\in Q_0$ the simple reflection $s_i:{\mathds{Z}}^{Q_0}\to {\mathds{Z}}^{Q_0}$ is defined by $$\begin{aligned}
s_i({\alpha}):={\alpha}-({\alpha},e_i)e_i.\end{aligned}$$
A simple root is a vector $e_i$ for $i\in Q_0$. The set of simple roots is denoted by $\Pi$. The Weyl group, denoted by $W$, is the subgroup of $\textrm{GL}({\mathds{Z}}^n)$, where $n=|Q_0|$, generated by the $s_i$. By $\Delta^+_{\textrm{re}}(Q) := \{{\alpha}\in W(\Pi) : {\alpha}> 0\} $ we denote the set of (positive) real roots for $Q$.
We have the following remarkable theorem.
\[kac-theorem\] Let $k$ be a field, $Q$ be a quiver and let ${\alpha}\in \Delta^+_{\textrm{re}}(Q)$. There exists a unique indecomposable representation (up to isomorphism) of dimension vector ${\alpha}$.
For finite fields and algebraically closed fields the theorem is due to Kac [@kac Theorem 1 and 2]. As pointed out in the introduction of [@schofield], Kac’s method of proof showed that the above theorem holds for fields of characteristic $p$. The proof for fields of characteristic zero is due to Schofield [@schofield Theorem 9].
For a given positve real root ${\alpha}$ for $Q$ the unique indecomposable representation (up to isomorphism) of dimension vector ${\alpha}$ is denoted by $X_{\alpha}$. By a real root representation we mean an $X_{\alpha}$ for ${\alpha}$ a positive real root. A Schur representation is a representation with ${{\operatorname{\mathsf{End}}}}_{kQ} (X)=k$. By a real Schur representation we mean a real representation which is also a Schur representation. A positive real root is called a real Schur root if $X_{\alpha}$ is a real Schur representation.
We have the following useful formula: if $X,Y$ are representations of $Q$ then we have $$\begin{aligned}
\dim {{\operatorname{\mathsf{Hom}}}}_{kQ}(X,Y) - \dim {{\operatorname{\mathsf{Ext}}}}^1_{kQ} (X,Y) = \langle {\underline{\dim}\,}X, {\underline{\dim}\,}Y \rangle.\end{aligned}$$ It follows that ${{\operatorname{\mathsf{Ext}}}}^1_{kQ}(X_{\alpha},X_{\alpha})=0$ for ${\alpha}$ a real Schur root.
Universal Extension Functors {#universal}
----------------------------
We use this section to describe briefly how the functors $$\begin{aligned}
\overline{\sigma}_S&:& {{\mathfrak{M}}}^{-S} \to {{\mathfrak{M}}}^S/S, \\
\underline{\sigma}_S&:& {{\mathfrak{M}}}_{-S} \to {{\mathfrak{M}}}_S/S, \\
\sigma_S&:& {{\mathfrak{M}}}_{-S}^{-S} \to {{\mathfrak{M}}}^S_S/S,\end{aligned}$$ operate on objects.
The functor $\overline{\sigma}_S$ is given by the following construction: Let $X\in{{\mathfrak{M}}}^{-S}$ and let $E_1,\ldots,E_r$ be a basis of the $k$-vector space ${{\operatorname{\mathsf{Ext}}}}^1_{kQ} (S,X)$. Consider the exact sequence $E$ given by the elements $E_1,\ldots,E_r$ $$\begin{aligned}
E: 0\to X\to Z\to \bigoplus_r S \to 0.\end{aligned}$$ According to [@ringel Lemma 3] we have $Z\in {{\mathfrak{M}}}^S$ and we define $\overline{\sigma}_S(X):=Z$. Now, let $Y\in{{\mathfrak{M}}}_{-S}$ and let $E'_1,\ldots,E'_s$ be a basis of the $k$-vector space ${{\operatorname{\mathsf{Ext}}}}^1_{kQ}(Y,S)$. Consider the exact sequence $E'$ given by $E'_1,\ldots,E'_s$ $$\begin{aligned}
E': 0\to \bigoplus_s S\to U\to Y\to 0.\end{aligned}$$ Then we have $U\in {{\mathfrak{M}}}_S$ and we set $\underline{\sigma}_S(Y):=U$. The functor $\sigma_S$ is given by applying both constructions successively.
The inverse $\overline{\sigma}_S^{-1}$ is constructed as follows: Let $X\in {{\mathfrak{M}}}^{S}$ and let $\phi_1,\ldots,\phi_r$ be a basis of the $k$-vector space ${{\operatorname{\mathsf{Hom}}}}_{kQ}(X,S)$. Then by [@ringel Lemma 2] the sequence $$\begin{aligned}
0\to X^{-S}\to X \stackrel{(\phi_i)_i}{\longrightarrow} \bigoplus_r S\to 0\end{aligned}$$ is exact, where $X^{-S}$ denotes the intersection of the kernels of all maps $X\to S$. We set $\overline{\sigma}^{-1}_S (X):=X^{-S}$. Now, let $Y\in{{\mathfrak{M}}}_{S}$. The inverse $\underline{\sigma}_S^{-1}$ is given by $\underline{\sigma}_S^{-1}(Y):=Y/Y'$, where $Y'$ is the sum of the images of all maps $S\to Y$. The inverse $\sigma^{-1}_S$ is given by applying both constructions successively.
Both constructions show that $${\underline{\dim}\,}\sigma^{\pm 1}_S(X) = {\underline{\dim}\,}X - ({\underline{\dim}\,}X, {\underline{\dim}\,}S)\cdot{\underline{\dim}\,}S. \tag{$\dagger$}\label{refl-dim}$$
Moreover, we have the following proposition.
Let $X\in {{\mathfrak{M}}}^{-S}_{-S}$. Then $$\begin{aligned}
\dim {{\operatorname{\mathsf{End}}}}_{kQ} \sigma_S(X) = \dim {{\operatorname{\mathsf{End}}}}_{kQ}(X) + \langle {\underline{\dim}\,}X, {\underline{\dim}\,}S \rangle \cdot \langle {\underline{\dim}\,}S,{\underline{\dim}\,}X \rangle.\end{aligned}$$ Let $Y\in {{\mathfrak{M}}}^{S}_{S}$. Then $$\begin{aligned}
\dim {{\operatorname{\mathsf{End}}}}_{kQ} \sigma^{-1}_S(Y) = \dim {{\operatorname{\mathsf{End}}}}_{kQ}(Y) - \langle {\underline{\dim}\,}Y, {\underline{\dim}\,}S \rangle \cdot \langle {\underline{\dim}\,}S,{\underline{\dim}\,}Y \rangle.\end{aligned}$$
A negative and unpleasant example {#counterexample}
=================================
Let $k$ be a field and let $Q$ be a quiver. We recall Question $(\star )$ stated in the introduction.
Let ${\alpha}$ be a positive non-Schur real root for $Q$ and let $X_{\alpha}$ be the unique indecomposable representation of dimension vector ${\alpha}$.
Does there exist a sequence of real Schur roots ${\beta}_1,\ldots,{\beta}_n\; (n\ge 2)$ such that $$\begin{aligned}
X_{\alpha}= \sigma_{X_{{\beta}_n}}\cdot \ldots \cdot\sigma_{X_{{\beta}_2}} (X_{{\beta}_1})\quad ?\end{aligned}$$
We remark that in the case that $X_{\alpha}$ can be constructed in the above way we have ${\beta}_i<{\alpha}$ for $i=1,\ldots,n$.
In the following we give an explicit example of a non-Schur real root representations which cannot be constructed using universal extension functors.
We consider the quiver $Q$
(2,4.5)(-4.5,-4) (-1.5,-2)[$Q:$]{} $ \xymatrix{ 1 \ar@<.0ex>[dr]_a &2 \ar@<.0ex>[d]^b &3 \ar@<.0ex>[dl]^c \\
&4 \ar@<.0ex>[d]^d & \\
&5 \ar@<.0ex>[dl]_e \ar@<.0ex>[d]^f \ar@<.0ex>[dr]^g & \\
6 &7 &8 }$
and the real root ${\alpha}=(1,1,1,8,12,2,7,7)=s_8s_7s_5s_4s_8s_7s_5s_8s_7s_5s_6s_4s_5s_4s_1s_2s_3(e_4)$.
For the convenience of the reader we give an explicit description of the representation $X_{\alpha}$.
We start by considering the representation $X_{\alpha}$ over the field $k={\mathds{Q}}$. In this case, one can use the result [@crawley Proposition A.4] to construct the representation $X_{\alpha}$; we get
(2,5)(-2.2,-4) (0.5,-2)[$X_{\alpha}:$]{} $ \xymatrix{ &&k \ar@<.0ex>[dr]_{X_a} &k \ar@<.0ex>[d]^{X_b} &k \ar@<.0ex>[dl]^{X_c} \\
&& &k^8 \ar@<.0ex>[d]^{X_d} & \\
&& &k^{12} \ar@<.0ex>[dl]_{X_e} \ar@<.0ex>[d]^{X_f} \ar@<.0ex>[dr]^{X_g} & \\
&& k^2 &k^7 &k^7 }$
with
$$\begin{aligned}
X_a &=& \left[ \begin{array}{cccccccc}
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0
\end{array}
\right]^t, \\
& & \\
X_b &=& \left[ \begin{array}{cccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0
\end{array}
\right]^t, \\
& & \\
X_c &=& \left[ \begin{array}{cccccccc}
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1
\end{array}
\right]^t,\\
& & \\
X_d &=& \left[ \begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{array}
\right], \end{aligned}$$
$$\begin{aligned}
X_e &=& \left[ \begin{array}{cccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
\end{array}
\right], \\
& & \\
X_f &=& \left[ \begin{array}{cccccccccccc}
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{array}
\right], \\
& & \\
X_g &=& \left[ \begin{array}{cccccccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1
\end{array}
\right].\end{aligned}$$
In particular, we see that $X_{\alpha}$ is a tree representation.
The representation $X_{\alpha}$, as given above, is defined over every field $k$. Moreover, it is not difficult to see that ${{\operatorname{\mathsf{End}}}}_{k Q} (X_{\alpha})$ is local. Hence, the representation $X_{\alpha}$ is the unique indecomposable representation of dimension vector ${\alpha}$ over every field $k$.
Moreover, $\dim {{\operatorname{\mathsf{End}}}}_{k Q} (X_{\alpha}) =9$ so that $X_{\alpha}$ is not a real Schur representation.
\[main-theorem-2\] There exists no real Schur root ${\beta}$ with the following properties:
(i) $X_{\alpha}\in {{\mathfrak{M}}}^{X_{\beta}}_{X_{\beta}}$,
(ii) ${{\operatorname{\mathsf{Hom}}}}_{kQ}(X_{\alpha},X_{\beta})\ne 0$${{\operatorname{\mathsf{Hom}}}}_{kQ}(X_{\beta},X_{\alpha})\ne 0$.
If we had a sequence of real Schur roots ${\beta}_1,\ldots,{\beta}_n\; (n\ge 2)$ such that $ X_{\alpha}= \sigma_{X_{{\beta}_n}}\cdot \ldots\cdot \sigma_{X_{{\beta}_2}} (X_{{\beta}_1})$ then ${\beta}_n$ would have to satisfy conditions (i) and (ii). Note that condition (ii) merely states that $\sigma^{-1}_{X_{{\beta}_n}}(X_{\alpha})\ne X_{\alpha}$. Thus, once we have established the claim it is clear that $X_{\alpha}$ provides an example which answers Question $(\star )$ negatively.
We use the rest of this section to prove the above theorem. We show that there are no real Schur roots satisfying (i).
Condition (i) requires ${\beta}< {\alpha}$ by [@ringel Lemma 2] and $${{\operatorname{\mathsf{Ext}}}}^1_{kQ}(X_{\alpha},X_{\beta}) = 0 = {{\operatorname{\mathsf{Ext}}}}^1_{kQ}(X_{\beta},X_{\alpha}),$$ which implies that $\langle {\alpha},{\beta}\rangle \ge 0$ and $\langle {\beta},{\alpha}\rangle \ge 0$. Hence, we start by determining the set of real roots ${\beta}$ with the following properties:
1. ${\beta}< {\alpha}$,
2. $\langle {\alpha},{\beta}\rangle \ge 0$ and $\langle {\beta},{\alpha}\rangle \ge 0$.
These roots are potential candidates for a reflection. Using the arguments given in [@schofield_2 Section 6], it is easy to determine the real roots ${\beta}$ which satisfy (i’) and (ii’): both conditions imply that $s_{\alpha}({\beta})<0$ and, hence, if $s_{\alpha}=s_{i_1}\ldots s_{i_n}$ we get $s_{\alpha}({\beta}) = s_{i_1}\ldots s_{i_n}({\beta})<0$ if and only if ${\beta}=s_{i_n}\ldots s_{i_{m+1}}(e_{i_m})$ for some $m$. Thus, once we have written $s_{\alpha}$ as a product of the generators $s_i$ it is straightforward to find the real roots ${\beta}$ satisfying (i’) and (ii’). A decomposition of $s_{\alpha}$ into a product of the generators $s_i$ can be achieved as follows: if $s_i({\alpha})={\alpha}'<{\alpha}$ then $s_{\alpha}=s_is_{{\alpha}'}s_i$; this gives an algorithm to find a shortest expression of $s_{\alpha}$ in terms of the $s_i$.
Applying the above algorithm to the real root ${\alpha}$, we get the following potential candidates for a reflection $$\begin{aligned}
{\beta}_1&=&(0,0,0,1,2,0,1,1), \\
{\beta}_2&=&(0,1,1,4,7,1,4,4), \\
{\beta}_3&=&(1,0,1,4,7,1,4,4), \quad \textrm{and} \\
{\beta}_4&=&(1,1,0,4,7,1,4,4).\end{aligned}$$
We see that $\langle {\beta}_i,{\alpha}\rangle = 0 = \langle {\alpha},{\beta}_i \rangle$ for $i=2,3,4$, and hence the only reflection candidate is ${\beta}_1$. Note that ${\beta}_1$ is a real Schur root, and hence indeed a candidate for a reflection. However, ${\beta}_1$ does not satisfy condition (i), that is $X_{\alpha}\notin {{\mathfrak{M}}}^{X_{{\beta}_1}}_{X_{{\beta}_1}}$. Assume to the contrary that $X_{\alpha}\in {{\mathfrak{M}}}^{X_{{\beta}_1}}_{X_{{\beta}_1}}$. Then $\sigma^{-1}_{X_{{\beta}_1}} (X_{\alpha}) \in {{\mathfrak{M}}}^{-X_{{\beta}_1}}_{-X_{{\beta}_1}}$, that is $$\begin{aligned}
{{\operatorname{\mathsf{Hom}}}}_{kQ} (\sigma^{-1}_{X_{{\beta}_1}} (X_{\alpha}),X_{{\beta}_1}) = 0 = {{\operatorname{\mathsf{Hom}}}}_{kQ} (X_{{\beta}_1},\sigma^{-1}_{X_{{\beta}_1}} (X_{\alpha})).\end{aligned}$$ Using formula (\[refl-dim\]) from Section \[universal\], we get ${\gamma}_1:={\underline{\dim}\,}\sigma^{-1}_{X_{{\beta}_1}} (X_{\alpha}) = (1,1,1,3,2,2,2,2)$. The following diagram, however, shows that ${{\operatorname{\mathsf{Hom}}}}_{kQ}(X_{{\beta}_1},X_{{\gamma}_1})\ne 0$. The representation $X_{{\gamma}_1}$ can be constructed using the result [@crawley Proposition A.4] together with the same reasoning as for $X_{\alpha}$ to pass to any field $k$.
$ \xymatrix{ & X_{{\beta}_1}&&&&& X_{{\gamma}_1} & \\
0 \ar@<.0ex>[ddr] &0 \ar@<.0ex>[dd] &0 \ar@<.0ex>[ddl] &&&
k \ar@<.0ex>[ddr]_{\textrm{\tiny{$\begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}$}}} & k
\ar@<.0ex>[dd]^{\textrm{\tiny{$\begin{bmatrix} 0\\ 1 \\0 \end{bmatrix}$}}} &k \ar@<.0ex>[ddl]^{\textrm{\tiny{$\begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}$}}} \\
&&&&&&& \\
&k \ar@<.0ex>[dd]^{\textrm{\tiny{$\begin{bmatrix} 1\\ 1 \end{bmatrix}$}}} \ar@<.0ex>[rrrrr]^{\textrm{\tiny{$\begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}$}}} & &&&
&k^3 \ar@<.0ex>[dd]^{\textrm{\tiny{$\begin{bmatrix} 0&1&0 \\ 0&0&1 \end{bmatrix}$}}} & \\
&&&&&&& \\
&k^2 \ar@<.0ex>[ddl] \ar@<.0ex>[dd]^{\textrm{\tiny{$[1\, 0]$}}} \ar@<.0ex>[ddr]^{\textrm{\tiny{$[0\, 1]$}}} \ar@<.0ex>[rrrrr]^{\textrm{\tiny{$\begin{bmatrix} 0&0 \\0& 0 \end{bmatrix}$}}}& &&&
&k^2 \ar@<.0ex>[ddl]^{\textrm{id}} \ar@<.0ex>[dd]^{\textrm{id}} \ar@<.0ex>[ddr]^{\textrm{id}} \\
&&&&&&& \\
0 &k \ar@/^-2.5pc/[rrrrr]_{\textrm{\tiny{$\begin{bmatrix} 0&0 \end{bmatrix}$}}} &k \ar@/^-1.8pc/[rrrrr]^{\textrm{\tiny{$\begin{bmatrix} 0&0 \end{bmatrix}$}}} &&&
k^2 &k^2 &k^2}$
This is a contradiction, and hence $X_{\alpha}\notin {{\mathfrak{M}}}^{X_{{\beta}_1}}_{X_{{\beta}_1}}$ which completes the proof of the theorem and we see that, indeed, the representation $X_{\alpha}$ answers Question $(\star)$ negatively.
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A. Schofield, ‘General representations of quivers’, [*Pro. London Math. Soc.*]{} (3) 65 (1992) 46-64.
A. Schofield, ‘The field of definition of a real representation of $Q$’, [*Proc. American Math. Soc.*]{} 116 (1992) 293-295.
M. Wiedemann, ‘Quiver representations of maximal rank type and an application to representations of a quiver with three vertices’, [*Bull. London Math. Soc.*]{} 40 (2008) 479-492
M. Wiedemann, ‘On real root representations of quivers’, [*PhD thesis, in preparation*]{}
|
---
author:
- |
\
Department of Physics & Astronomy, University of the Western Cape, Cape Town, South Africa\
INAF - Istituto di Radioastronomia, via Gobetti 101, 40129 Bologna, Italy\
E-mail:
title: 'HELP-ing Extragalactic Surveys : The Herschel Extragalactic Legacy Project and The Coming of Age of Multi-Wavelength Astrophysics'
---
The HELP project & its science objectives {#intro.sec}
=========================================
How did galaxies form and evolve? This is one of the most challenging questions in astronomy today. Although astronomers now have a good understanding of the background cosmology and of the formation of the large-scale structure of the dark matter [@Springel2005; @Vogelsberger2014], the complex astrophysics that leads to the variety and numbers of galaxies observed within dark matter halos is still very poorly understood. Anecdotal clues to these questions can be found through focussed studies of individual galaxies. However, the fundamental requirement for rigorous testing of any theories of galaxy formation and evolution is a complete statistical audit or census of the stellar content and star formation rates of galaxies in the Universe at different times and as a function of the mass of the dark matter halos that host them. This audit requires many elements. We need unbiased maps of large volumes of the Universe made with telescopes that probe the different wavelengths at which the different physical processes of interest manifest themselves. We need catalogues of the galaxies contained within these maps with photometry estimated uniformly from field-to-field, from telescope-to-telescope and from wavelength-to-wavelength. We need to understand the probability of a galaxy of given properties appearing in our data sets. We need the machinery to bring together these various data sets and calculate the value-added physical data of primary interest, e.g. the distances (or redshifts), stellar masses, star-formation rates and the actual number densities of the different galaxy populations.
Since the advent of large-format detector arrays at optical and infrared wavelengths, many astronomical facilities have been undertaking ambitious programmes to observe large areas of the sky and study galaxy evolution in the distant Universe. ESA’s Herschel mission had a unique place in this context, probing the obscured star-formation history of the Universe, i.e. roughly 50% of all star formation activity, through observations at far-infrared and sub-millimeter wavelengths. Herschel extragalactic surveys were a major goal of Herschel and accounted for around 10% of the observing time available during Herschel’s 3.5 year science mission. However, the full exploitation of this dataset is complicated by Herschel’s large instrumental beam size (6-37" at 70-500 $\mu$m).
The Herschel Extragalactic Legacy Project (HELP, PI : Seb Oliver, University of Sussex, <http://herschel.sussex.ac.uk>) brings together several of the teams that have been undertaking ambitious coordinated multi-wavelength digital sky surveys to study large volumes of the distant Universe over the past decade. These observational projects are mature enough that we are now able to undertake the necessary homogenization and thus provide the first representative and comprehensive census of the galaxy populations in the distant Universe. HELP was therefore funded by the European Commission FP7-SPACE-2013-1 Scheme (Grant Agreement 607254) to produce over the 2014-2017 4-year funding period a database for the distant Universe as an accessible value-added resource for the astronomical community. HELP was conceived to assemble the ancillary data and develop the tools necessary to fully capitalize on Herschel’s legacy and to enable astronomers not directly involved with the mission to fully exploit the Herschel dataset and its multi-wavelength counterparts. We intend to provide a vast resource for studying the distant Universe, similar to the SDSS for the nearby Universe, as a lasting legacy of several major ground-based and space-based surveys and a solid foundation for future space missions and ground-based observatory projects.
The Herschel satellite & its mission {#herschel.sec}
====================================
Galaxies emit electromagnetic radiation over a very wide wavelength range, but most of it is absorbed by the Earth’s atmosphere and thus cannot be studied from the ground. With the development of satellite missions, however, astronomers have gradually been able to explore galaxy emission over the full electromagnetic spectrum, from $\gamma$-rays to radio waves, in an uninterrupted manner. Comparing the properties of galaxies observed at different wavelengths have thus enabled substantial progress in the understanding of the physical processes driving their formation and evolution.
The Herschel satellite [@Pilbratt2010] was developed by ESA and carried out its 3.5 year science mission between 2009 and 2013. Herschel was the first 4-m class space telescope and the first far-infrared and sub-millimeter telescope sensitive enough to detect galaxies in the distant Universe, vastly improving the state of observations in this poorly explored band. Herschel thus signaled the completion of an ambitious program jointly carried out by ESA and NASA to map galaxy evolution across most of the life of the Universe at all wavelengths. In particular, the Herschel imaging instruments SPIRE [@Griffin2010] and PACS [@Poglitsch2010] fully constrain the peak of the far-infrared and sub-millimeter background with six photometric channels covering the 70-500 $\mu$m wavelength range. Herschel thus allows us to thoroughly investigate the sources of the infrared background radiation and characterize their total obscured star formation as a function of cosmic time through systematic observing programs such as HerMES [@Oliver2012] and H-ATLAS [@Eales2010a]. However, the large size of the Herschel beam (6-37" across the 70-500 $\mu$m wavelength range) means that the objects that can be clearly seen as individual sources only make up a small fraction of the cosmic infrared background. In order to unlock the full information available in Herschel maps one must therefore develop new methods to estimate the short-wavelength sources most likely to contribute to emission in Herschel bands in a statistically rigorous manner. HELP therefore brings together leading scientists working on Herschel extragalactic surveys with experts in multi-wavelength data reduction, analysis and homogenization to make sure that we can effectively build upon existing best practices to deliver the highest-quality multi-wavelength datasets and selection functions.
HELP data products & Herschel’s scientific legacy {#df.sec}
=================================================
Many of the outstanding problems in galaxy evolution require a multi-wavelength approach to properly account for different physical processes over the full life-cycle of galaxies and black holes. Modeling tools to account for the Spectral Energy Distribution of galaxies have become sufficiently advanced that mis-matched photometry (i.e. photometry at different wavelengths which samples different areas of a galaxy) may be the dominant source of error. Thus in all wavebands it is now common to produce “aperture-matched” catalog. In this approach the source detection is carried out on either a single image or, for the greatest sensitivity, on a combined image, but then the photometry is carried out on all images with a common aperture or a common model for the galaxy light profile. However, statistical studies of galaxy populations also require a detailed understanding and modeling of the selection processes in the basic data products and the derived properties. These “selection functions” are seldom available for individual datasets and rarely, if ever, published for derived physical properties, but are crucial to estimate source number densities in a reliable manner [@Vaccari2010; @Eales2010b; @Burgarella2013; @Marchetti2015].
The starting point for our project will be the multi-wavelength images and catalogs that have been produced and publicly released by our team as well as by others based on more than a decade of concerted observational projects [@Lonsdale2003; @Mauduit2012; @Jarvis2013]. Our first task is to homogenize these data, so that photometry and calibration is consistent from catalog to catalog, field to field, and from wavelength to wavelength, building upon the framework developed as part of the Spitzer Data Fusion ([@Vaccari2010; @Marchetti2015], <http://www.mattiavaccari.net/df/>). The benchmark for this will be that we can determine accurate photometric redshifts, or distances, adopt consistent galaxy spectral templates and characterize the physical properties of the galaxies [@RowanRobinson2008; @RowanRobinson2013]. For this purpose we will also assemble available optical spectroscopy from, e.g. SDSS, 2dFGRS, GAMA, PRIMUS, VVDS, zCOSMOS and provide homogeneous reliability flags for both spectroscopic and photometric redshifts. We will thus produce a key input catalogs for future spectroscopic surveys with, e.g. WEAVE and WAVES.
HELP will thus produce and publicly release multi-wavelength datasets (catalogs and images) and selection functions as well as physical parameters for individual galaxies over the 1,000 deg$^2$ covered by the Herschel extragalactic surveys (See Figure \[sky-help.fig\]). The data products as well as the techniques and tools that we will produce will enable astronomers to fully capitalize on datasets provided by Herschel as well as other surveys (See Table \[surveys.tab\]. This will extend the kinds of scientific investigations made possible a decade ago in the nearby Universe by the SDSS into the early Universe and provide a lasting legacy for surveys and facilities in the future. In so doing, HELP will provide a complementary view to the AstroDeep project focusing on five small sky areas totaling 2 deg$^2$ and thus sampling fainter galaxies but not the full range of environments, rare objects and short-lived phases in galaxy evolution (see Figure \[footprint.fig\]).
Conclusion {#conclusion.sec}
==========
HELP (Herschel Extragalactic Legacy Project, <http://herschel.sussex.ac.uk>) will produce an unrivaled public database of multi-wavelength sky images, catalogs of individually detected sources and their physical properties as well as related selection functions over the 1,000 deg$^2$ of the extragalactic sky covered by the Herschel satellite. It will bring together in a concerted manner space-based and ground-based wide-area galaxy surveys for the first time, enabling highly-accurate statistical studies of galaxy properties and their evolution with cosmic time and providing a lasting legacy for the astronomical community to mine for decades to come and a template for future space science data curation projects in the petascale and exascale era.
[llll]{} Wavelength & Telescope / Instrument & Observing Band & Survey Project\
Ultraviolet & GALEX & FUV & NUV & DIS, MIS, AIS\
Visible & PS1, SDSS, DECAM, VST, CFHT, INTWFC & $ugrizy$ & PS1, SDSS, DES, ATLAS, KIDS, VOICE, INTWFS\
Near-Infrared & 2MASS, UKIRT, VISTA & $ZYJHK$ & UKIDSS, VIDEO, VIKING, VHS\
Mid-Infrared & IRAC, WISE & 3.6/4.5/5.8/8.0/12.0 $\mu$m & WISE, SWIRE, SERVS, S-COSMOS, SpUDS, SPLASH\
Far-Infrared & WISE, MIPS, PACS & 22/24/70/100/160 $\mu$m & WISE, SWIRE, PEP, HerMES\
(Sub-)Millimeter & SPIRE, SCUBA, SCUBA2, LABOCA, AzTEC, ACT, SPT & 250/350/500/850 $\mu$m & HerMES, SHADES, S2LS, ACT, SPT\
Radio & ATCA, GMRT, LOFAR, MeerKAT, JVLA & 0.6, 1.4, 3, 5 GHz & ATLAS, LOFAR, WODAN, MIGHTEE, VLASS\
{width="10.5cm"}
{width="10.5cm"}
Acknowledgements {#acknowledgements .unnumbered}
================
HELP is funded by the EC REA (FP7-SPACE-2013-1 GA 607254 - Herschel Extragalactic Legacy Project). Mattia Vaccari is also supported by the Square Kilometre Array South Africa Project, the South African NRF and DST (DST/CON 0134/2014) and Italy’s MAECI (PGR GA ZA14GR02 - Mapping the Universe on the Pathway to SKA).
[99]{} Burgarella D., Buat, V., Gruppioni, C., et al., 2013, A&A, 554, 70 Griffin M. J., Abergel, A., Abreu, A., et al., 2010, A&A, 518, L3 Eales S. A., Dunne, L., Clements, D., et al., 2010, PASP, 122, 499 Eales S. A., Raymond, G., Roseboom, I. G., et al., 2010, A&A, 518, L23 Jarvis M. J., Bonfield, D. G., Bruce, V. A., et al., 2013, MNRAS, 428, 1281 Lonsdale C. J., Smith, H. E., Rowan-Robinson, M., et al., 2003, PASP, 115, 897 Marchetti, L., Vaccari M., Franceschini, A., et al., 2015, MNRAS, in press, ArXiv `1511.06167` Mauduit J.-C., Lacy, M., Farrah, D., et al., 2012, PASP, 124, 714 Oliver S. J., Bock, J., Altieri, B., et al., 2012, MNRAS, 424, 1614 Pilbratt, G., Riedinger, J. R., Passvogel, T., et al. 2010, A&A, 518, L1 Poglitsch, A., Waelkens, C., Geis, N., et al. 2010, A&A, 518, L2 Rowan-Robinson M., Babbedge, T., Oliver, S., et al., 2008, MNRAS, 386, 697 Rowan-Robinson M., Gonzalez-Solares, E., Vaccari, M., Marchetti, L., 2013, MNRAS, 428, 1958 Schlegel, D. J., Finkbeiner, D. P., Davis, M. 1998, ApJ, 500, 525 Springel, V., White, S. D. M., Jenkins, A., et al., 2005, Nature., 435, 629 Vaccari M., Marchetti, L., Franceschini, A., et al., 2010, A&A, 518, L20 Vogelsberger, M., Genel, S., Springel, V., et al., 2014, Nature, 509, 177
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---
abstract: 'The anti-de Sitter space/conformal field theory correspondence (AdS/CFT) can potentially provide a complete formulation of string theory on a landscape of stable and metastable vacua that naturally give rise to eternal inflation. As a model for this process, we consider bubble solutions with de Sitter interiors, obtained by patching together dS and Schwarzschild-AdS solutions along a bubble wall. For an interesting subclass of these solutions the bubble wall reaches spacelike infinity in the black hole interior. Including the effects of perturbations leads to a null singularity emanating from this point. Such solutions are interpreted as states in a single CFT, and are shown to be compatible with holographic entropy bounds. The construction suggests de Sitter entropy be interpreted as the total number of degrees of freedom in effective field theory, with a novel adaptive stepsize cutoff.'
author:
- 'David A. Lowe'
bibliography:
- 'bubble.bib'
title: 'Some comments on embedding inflation in the AdS/CFT correspondence'
---
introduction
============
In recent years convincing evidence has accumulated that string theory has a vast landscape of consistent vacuum states, along with numerous long-lived metastable states that may well be relevant for realistic cosmology[@Kachru:2003aw]. It is important to develop tools in string theory that are capable of describing transitions between such states. Currently the AdS/CFT formulation of string theory [@Maldacena:1997re] is the most promising nonperturbative formulation of string theory. It is the purpose of the present paper to study how this landscape might be embedded in this framework, building on the earlier work of [@Alberghi:1999kd; @Freivogel:2005qh].
We begin by reviewing classical bubble solutions in asymptotically AdS space, following the original work of [@Blau:1986cw]. For many of the most interesting solutions, the bubble wall appears to reach AdS infinity in finite time. We show this situation is unstable to perturbations, and that instead a null singularity emanates from this point. We discuss how this is to be interpreted in terms of the CFT, and propose a new interpretation of de Sitter entropy compatible with this picture.
Review of classical bubble solutions
====================================
In order to obtain solutions describing bubbles with inflating interiors inside asymptotically anti-de Sitter space, we follow the procedure of Blau, Guendelman and Guth [@Blau:1986cw]. The basic idea is to consider a spherically symmetric bubble in the thin-wall limit. The interior of the bubble is modeled by a piece of pure de Sitter space and the exterior by Schwarzschild-anti de Sitter space. This reduces the problem to specifying the radial position of the bubble as a function of time which reduces to the one-particle potential scattering problem described in [@Alberghi:1999kd]. In [@Alberghi:1999kd] the case of general dimension and charge was considered. Here we will specialize to the case of vanishing charge and asymptotically $AdS_{4}$ spacetime. These solutions are similar to those considered in [@Blau:1986cw] and were studied in extensively in [@Freivogel:2005qh]. The main features of all the solutions can be found in the following two examples in figure \[fig:time symmetric\] and \[fig:time asymmetric\], that we will discuss in detail.
![Penrose diagrams showing a time symmetric bubble trajectory. The left diagram refers to de Sitter space, the right diagram refers to Schwarzschild-anti de Sitter space. The bubble trajectory is shown in blue. The right segment of Schwarzschild-anti de Sitter corresponds to the exterior of the bubble. The left segment of de Sitter corresponds to the interior.\[fig:time symmetric\]](tsym)
![Penrose diagrams showing a time asymmetric bubble trajectory.\[fig:time asymmetric\]](tasym)
It is important to understand how these solutions change when subject to small perturbations. One issue is whether the spherical symmetric approximation is valid. This was considered some time ago in the context of the original work by Blau et al. [@Garriga:1991ts; @Garriga:1991tb]. While perturbations do indeed grow in the de Sitter interior, these do not qualitatively change the behavior of the solutions. Another issue peculiar to the situation with negative cosmological constant is what happens to the asymptotically anti-de Sitter boundaries that appear on the left. This is investigated in appendix \[sec:Appendix:-Generic-perturbations\]. For the time asymmetric case, figure \[fig:time asymmetric\], the conclusion is that a null singularity emerges from the point at which the bubble wall hits spacelike infinity. When one goes beyond leading order in perturbation theory, we believe this singularity should become spacelike, becoming null as it approaches the bubble wall. This is shown in figure \[fig:Time-asymmetric-perturbed\].
Since these solutions are ultimately to be embedded in string theory, the de Sitter regions will be metastable. We will assume the anti-de Sitter vacuum under consideration is completely stable. There will therefore be tunneling between the two solutions. In particular, any timelike trajectory in the de Sitter interior will tunnel back to the vicinity of the anti de Sitter vacuum state, with a timescale exponentially smaller than the Poincare recurrence time of the de Sitter space. As shown in [@Coleman:1980aw; @Abbott:1985kr] these regions undergo a crunch. We have schematically shown this in figure \[fig:Time-asymmetric-perturbed\] as a future singularity in the de Sitter Penrose diagram.
![Time asymmetric perturbed bubble solution.\[fig:Time-asymmetric-perturbed\]](tasymp)
In fact, the situation is more complicated than this, because while a given timelike curve will eventually enter a crunching region, the volume of a given set of compact spacelike slices will increase sufficiently fast that there is always remnant de Sitter space left at arbitrarily late time (see for example [@Linde:1991sk]). However for the purpose of predicting experiments measured by a local observer, a locally defined measure, implicit in figure \[fig:Time-asymmetric-perturbed\], is most useful, rather than a volume weighted measure.
The same considerations apply to the time symmetric situation of figure \[fig:time symmetric\]. Once again, perturbations will generate a singularity emerging from the endpoint of the bubble wall, extending into the exterior Schwarzschild-anti de Sitter region. Of course, if one demands time symmetry, this will also extend outward from the starting point of the bubble wall. Likewise, on the interior, one must also take account of the fact that any timelike path will eventually tunnel to a crunch, so the asymptotic de Sitter future (and past) is effectively removed. This is shown in figure \[fig:Time-symmetric-perturbed\]. Solutions of the type shown in figure \[fig:time symmetric\] can only be obtained by a high degree of fine tuning. It seems unlikely such solutions would emerge from the underlying conformal field theory description for finite values of Planck’s constant.
![Time symmetric perturbed bubble solution.\[fig:Time-symmetric-perturbed\]](tsymp)
The version of the AdS/CFT correspondence advocated in [@Witten:1998qj] maps geometries with a fixed boundary conditions at infinity (modulo conformal transformations) to conformal field theory on the boundary. We have seen that when perturbations are taken into account the bubble solutions have only a single asymptotically AdS region, therefore we expect a description in terms of a single conformal field theory. The simplest interpretation is in terms of pure states in the CFT [^1]. That said, it is a subtle question how one constructs approximate local quantities behind the horizon for geometries such as figure \[fig:Time-asymmetric-perturbed\]. In the work of [@Hamilton:2006fh], where static black holes were consider, this was accomplished using analyticity, so that in general CFT correlators needed to be continued to complex values of space and time. For eternal Schwarzschild-AdS black holes, this can be reformulated in terms of the thermofield double formalism, and hence mixed states as in [@Freivogel:2005qh]. However unlike in [@Freivogel:2005qh], we no longer have another asymptotically AdS exterior region behind the horizon, so it is no longer possible to construct a precisely defined set of observables associated with this region. So there is no obvious inconsistency with a pure state description. We consider the implications of this picture and the quantum consistency of these solutions, once we have tackled the issue of interpreting de Sitter entropy in this framework.
de Sitter Entropy
=================
Each of the bubble solutions considered above share the property that the expanding de Sitter region is behind a black hole event horizon from the viewpoint of the AdS boundary. These solutions are to be mapped into states in the boundary CFT, so we should associate them with particular black hole microstates. Moreover, as emphasized in [@Freivogel:2005qh], these states will yield distinctive correlators which lead to the hope that there might be a practical way to select such states from a thermal ensemble. For large black holes in AdS (and many other examples in string theory) the Bekenstein-Hawking entropy $S_{BH}$ is identified with the logarithm of the number of microstates. This leads to the conclusion that the number of de Sitter bubble states should be bounded by $e^{S_{BH}}$.
However it is not entirely clear how to interpret the Gibbons-Hawking entropy of de Sitter space [@Gibbons:1977mu]. Some earlier discussions of this in the context of AdS/CFT can be found in [@Lowe:2004zs]. Another interesting attempt at interpreting de Sitter entropy in terms of effective field theory can be found in [@Cohen:1998zx]. Suppose we imagine effective field theory in a de Sitter background with an ultraviolet energy cutoff $\Lambda$ and an infrared length cutoff $L$. If we identify $L$ with the horizon size of our present universe and the number of possible states in the effective field theory with the Gibbons-Hawking entropy, we obtain $\Lambda=100$ MeV. A more stringent bound is obtained if we restrict attention to those states with small gravitational back-reaction, demanding that the total energy not be inside it’s own Schwarzschild radius. Again taking $L$ to be the horizon size today, we obtain $\Lambda=10^{-2.5}$ eV. Clearly there does not seem to be a conventional effective field theory description that extends from horizon size scales down to scales probed in accelerator experiments.
In this work we advocate a variant on this idea that receives support from some numerical coincidences. It was assumed above that the ultraviolet cutoff is a simple uniform cutoff everywhere in spacetime. However we know, for example from [@Hamilton:2005ju; @Hamilton:2006az; @Hamilton:2006fh], that there is a complex relation between a UV cutoff in the bulk and physics in the CFT on the boundary, and that this is in general state dependent. Since we don’t have a straightforward way to identify the de Sitter bubble states in the CFT, we will make the optimistic assumption that the bulk state is encoded using perhaps the most efficient coding one could imagine, an adaptive stepsize with many points appearing near aggregations of matter/energy, and few points appearing away from such regions. The current entropy of the universe is dominated by cosmic microwave background photons with $S_{CMB}\sim10^{85}$ [^2] and with Compton wavelengths of around $\lambda=1$mm. Current experiments have tested effective field theory out to about $\Lambda=1$ TeV, so an estimate of an upper bound on the log of number of possible states in effective field theory would be $$S_{tot,CMB}=(\lambda\Lambda)^{3}S_{CMB}=10^{130}.$$ Of course it might be something of an overestimate to require so many states associated with a single CMB photon. Taking into account correlations between different states would certainly lower the total number, but it is not clear how to estimate the magnitude of this effect. Suppose instead we estimate a lower bound by considering states associated with individual hydrogen atoms. There are approximately $10^{80}$ atoms in our horizon volume, with a size of $\lambda=5\times10^{-11}$m . Estimating the associated log of the number of states gives $$S_{tot,atoms}=(\lambda\Lambda)^{3}10^{80}=10^{103}.$$ So we see these crude estimates on the number of possible states in an adaptive stepsize effective field theory brackets the Gibbons-Hawking entropy$$S_{tot,atoms}<S_{GH}=10^{122}<S_{tot,CMB}$$ Henceforth we will adopt the view that the Gibbons-Hawking entropy counts the number of possible states in effective field theory, albeit one with a rather clever UV cutoff.
According to this picture, if we want to have a useful effective field theory description around a semiclassical de Sitter background, the total number of available states should saturate the Gibbons-Hawking entropy. This will be our criterion for deciding when a local observer would regard herself as living in a semiclassical de Sitter region. Most states in conformal field theory correspond to backgrounds without a geometric interpretation. Likewise one can expect many states where some observables correspond to a particular geometric background, but many other observables do not, causing an effective field theory description to break down. So this criterion leads to the conclusion that if the CFT states are to be identified with black hole microstates, one must respect the bound $$S_{BH}>S_{GH}\label{eq:enbound}$$ in order that the interior of the bubble have an interpretation as a semiclassical spacetime.
Another logical possibility is that the bubble solutions require extra degrees of freedom, beyond those present in a single CFT, as advocated in [@Freivogel:2005qh]. In that picture, the CFT on the left boundary of Schwarzschild-AdS is regarded as being deformed. The correlations between those degrees of freedom and those in the CFT on the right induce a mixed state reduced density matrix, when the left boundary CFT is traced over. These ideas were most precisely stated for solutions such as figure \[fig:time symmetric\], where the asymptotic AdS region is replaced by a segment of de Sitter space for a finite period of time. It was argued in [@Freivogel:2005qh] that this corresponds to a non-local perturbation of the left CFT. If one goes beyond the thin-wall limit, we expect similar physics to emerge from a suitable local irrelevant perturbation of the left CFT. In fact one could dispense entirely with the right boundary in this picture, and study the CFT deformation on the left corresponding to the bubble. This is closely related to the proposal of [@Gubser:1998iu; @Gubser:1998kv; @Intriligator:1999ai; @Danielsson:2000ze] where an irrelevant perturbation of $\mathcal{N}=4$ $SU(N)$ Yang-Mills was conjectured to be dual to gravity is asymptotically flat space. Related ideas where the asymptotically AdS structure is changed via CFT perturbations also appears in [@Hertog:2004rz; @Hertog:2005hu][^3]. If this picture was consistent, it would offer a very different interpretation of de Sitter entropy, since there are an unlimited number of UV degrees of freedom in the left CFT that might be associated with bubble geometries, and hence no apparent bound by the black hole entropy. However there is no strong evidence that effective field theories deformed by these irrelevant perturbations really have continuum limits, suggesting the UV completions are ill-defined. This is good news for AdS/CFT as a nonperturbative description of string theory, since in order for it to be complete and self-consistent, one should not need to add additional degrees of freedom every time a quantum fluctuation creates a region of spacetime causally disconnected from the boundary.
### Implications for de Sitter bubbles \[sub:Implications-for-de\] {#implications-for-de-sitter-bubbles-subimplications-for-de .unnumbered}
Time symmetric bubbles (and solutions related by diffeomorphism) always respect the bound $S_{GH}>S_{BH}$, [@Freivogel:2005qh]. So we conclude that when the microscopic description of the solution is taken into account via the CFT, there are not enough available states to represent a full semiclassical spacetime inside the bubble. These solutions are therefore spurious once quantum effects are taken into account.
All solutions where the bubble wall stays in a single region of the Schwarzschild-anti de Sitter Penrose diagram fall into this class. Quantum tunneling solutions involving smooth initial data, as in [@Farhi:1989yr], involve transitions between classical solutions of this type. We see therefore that the rate of tunneling up the potential vanishes in the AdS/CFT framework for quantum gravity. Another argument for the vanishing of this tunneling rate is given in appendix \[sec:Detailed-balance\].
On the other hand, time asymmetric solutions of the type shown in figure \[fig:Time-asymmetric-perturbed\] exist where the bound (\[eq:enbound\]) is satisfied. These solutions pass the quantum consistency condition described above. In classical relativity it is not clear whether a state emerging from singular initial data is physically meaningful. In a complete formulation of quantum gravity such as AdS/CFT this objection must be revisited. In the case at hand we have a set of solutions that involve an initial singularity, however all the indications are that these correspond to well-defined states in the CFT. We regard these black hole microstates as the most promising way to obtain a description of inflation embedded in the AdS/CFT correspondence.
I thank G.L. Alberghi and R. McNees for helpful discussions. This research is supported in part by DOE grant DE-FG02-91ER40688-Task A.
Generic \[sec:Appendix:-Generic-perturbations\]perturbations of bubble solutions
================================================================================
Various solutions in this paper, have the property that an asymptotically AdS region meets another asymptotic region at some instance in time. Here we will show that such behavior is highly non-generic, and that once perturbations are taken into account, the would-be Cauchy horizon that emanates from this intersection point instead becomes a null singularity, once leading order effects are considered. It is expected this null singularity will become spacelike, once higher order effects are incorporated.
To see this, we consider the following analog problem that is believed to capture the essential physics. We take neutral matter in the form of a scalar field of mass $m_{s}$ and set up the equations of motion in a spherically symmetric ansatz. The solutions should be qualitatively the same as matter with nontrivial angular momentum. It is helpful to set up the Einstein equations with the variables similar to those used in [@Poisson:1990eh]
$$ds^{2}=g_{ab}dx^{a}dx^{b}+r(x^{a})^{2}d\Omega^{2}$$ with the radius and time directions denoted by $x^{a}$, $a=1,2$. It is helpful to define $$\begin{aligned}
f & \equiv & g^{ab}r_{,a}r_{,b}\equiv1-2m(x^{a})/r\\
\kappa & \equiv & -m(x^{a})/r^{2}\\
T & = & T_{\, a}^{a}\,,\,\, P=T_{\,\theta}^{\theta}=T_{\,\phi}^{\phi}\end{aligned}$$ so that the Einstein equations become
$$\begin{aligned}
r_{;ab}+\kappa g_{ab} & = & -4\pi r(T_{ab}-g_{ab}T)\\
m_{,a} & = & 4\pi r^{2}(T_{a}^{\, b}-\delta_{a}^{\, b}T)r_{,b}\,.\end{aligned}$$
Here the stress energy tensor is defined to include matter contributions as well as the contribution from the cosmological constant. Conservation of the stress energy tensor yields the equation$$(r^{2}T^{ab})_{;b}=(r^{2})^{,a}P\,.$$ At this point we specialize to compute the behavior of the metric near the boundary of AdS, near point P where the bubble wall reaches the cylinder at spacelike infinity as shown in figure \[fig:time symmetric\]. Outside the future cone of P and exterior to the bubble, the mass function takes the form$$m(r)=m_{0}+\Lambda r^{3}/6$$ with the cosmological constant related to the AdS radius of curvature $R$ by $\Lambda=-3/R^{2}$. The main point is that if the geometry is no longer asymptotically AdS to the past of some point, generic matter perturbations will induce both the normalizable and non-normalizable modes with respect to asymptotically AdS geometry. Both of these perturbations become normalizable in the geometry to the past of point P. The stress energy of the “non-normalizable” modes diverges as $r\to\infty$ for $\Delta>3$$$\begin{aligned}
T_{tt} & \sim & r^{2(\Delta-3)}\\
T_{rr} & \sim & r^{2(\Delta-4)}\\
T_{\theta\theta} & \sim & T_{\phi\phi}\sim r^{2(\Delta-3)}\,.\end{aligned}$$ Here the conformal dimension $\Delta$ is related to the mass of the scalar field $m_{s}$ by $\Delta=3/2+\sqrt{9/4+m_{s}^{2}R^{2}}$. For sufficiently large $\Delta$ this will dominate over the cosmological constant contribution as $r\to\infty$. In this limit, one obtains the equation$$\Box m=-16\pi^{2}r^{3}T_{ab}T^{ab}\,.\label{eq:mlap}$$ Our strategy will be similar to that of [@Poisson:1990eh; @Bonanno:1994ma], namely we will treat the back-reaction of the stress energy on the geometry at leading order in perturbation theory around an asymptotically AdS geometry. This will lead to a curvature singularity as one approaches the future light-cone of the point P.
To proceed, we solve (\[eq:mlap\]) using Kruskal coordinates. Outside the future light-cone of point P, we can use the pure AdS metric near infinity$$ds^{2}=(1-\Lambda r^{2}/3)dudv+r^{2}d\Omega^{2}$$ where the new coordinates are related to the old coordinates by$$\begin{aligned}
u=-t+r^{*} & , & v=t+r^{*}\\
r^{*} & = & \sqrt{\frac{3}{\left|\Lambda\right|}}\arctan\left(\sqrt{\frac{\left|\Lambda\right|}{3}}r\right)\,.\end{aligned}$$ In these coordinates (\[eq:mlap\]) takes a simple form $$\frac{\partial^{2}m}{\partial u\partial v}\sim-cr^{4\Delta-9}$$ where $c$ is a constant dependent on the amplitude of the matter perturbation. The solution for $m$ is divergent as one approaches the future light-cone of the point P ($v=\frac{\pi}{2}\sqrt{\frac{\left|\Lambda\right|}{3}}$)$$m\sim\frac{1}{(v-\frac{\pi}{2}\sqrt{\frac{\left|\Lambda\right|}{3}})^{4\Delta-7}}\,.$$
This provides strong evidence for at least a null singularity extending out on the future light-cone of the point P. Eventually higher order terms will come to dominate and the perturbative analysis breaks down, as is expected in [@Poisson:1990eh; @Bonanno:1994ma]. It is expected that the null singularity becomes a true spacelike singularity in the full non-linear analysis.
\[sec:Detailed-balance\]Detailed balance
========================================
The implications of detailed balance for quantum tunneling from flat space to de Sitter space has been previously studied in [@guth]. Let us reconsider that argument in the present context. Detailed balance implies the transition probabilities between any two states $i,j$ are related by$$\Gamma_{ij}p_{i}=\Gamma_{ji}p_{j}$$ where $\Gamma_{ij}$ is the transition rate from state $i$ to state $j$ and $p_{i}$ is the equilibrium probability of state $i$. Consider $AdS_{4}$ in thermal equilibrium, with large radius of curvature $R$ and temperature $T$, in the regime where Schwarzschild-anti-de Sitter space dominates the canonical ensemble. Let us suppose there exist some states in this ensemble corresponding to a small region of the Schwarzschild-AdS spacetime tunneling off into a de Sitter bubble. We assume the initial size of the bubble is the smallest length scale in the problem. Applying detailed balance to a transition between two such states, we obtain
$$\Gamma_{up}=\Gamma_{CDL}e^{S_{GH}-S_{BH}}\thicksim e^{-aR^{2}T^{2}}\label{eq:rans}$$
where $\Gamma_{CDL}$ is the Coleman-De Luccia tunneling rate, $\Gamma_{up}$ is the rate of tunneling up the potential, $S_{BH}$ is the entropy of the Schwarzschild-AdS spacetime and $S_{GH}$ is the Gibbons-Hawking entropy of a single de Sitter bubble, and $a$ is a coefficient independent of $T$ [@Klebanov:1996un]. In this limit $S_{BH}\sim aR^{2}T^{2}$ which we take to be much larger than $S_{GH}$. We expect $\Gamma_{CDL}$ to be independent of $R$ and $T$ in this limit, since it should be dominated by local physics.
The temperature dependence of the upward tunneling rate can be estimated as follows. Suppose an excited state with energy $M$ is assembled, capable of tunneling to the de Sitter bubble. The probability of such a state will be $e^{-M/T-S_{BH}}$. The rate of upward transitions will then be$$\Gamma_{up}=\Gamma_{tunnel}e^{-M/T}\label{eq:wans}$$ where $\Gamma_{tunnel}$ and $M$ should be independent of $T$ and $R$ by locality. However the two rates (\[eq:rans\]) and (\[eq:wans\]) have different temperature dependencies, so cannot match. This implies the rate of quantum tunneling $\Gamma_{tunnel}$ must vanish. This provides an argument, independent to that given in section \[sub:Implications-for-de\], that quantum tunneling up the potential, as envisioned in [@Farhi:1989yr], does not happen. Nevertheless we do expect transitions between states of the type shown in figure \[fig:Time-asymmetric-perturbed\] and Schwarzschild-AdS geometries, and these should be governed by the rate (\[eq:rans\]).
[^1]: Arguments for a mixed state interpretation are presented in [@Freivogel:2005qh].
[^2]: The horizon entropy of black holes formed after the big bang can easily dominate over the entropy of CMB photons. We will choose to count entropy by not averaging over the possible internal states of these objects, so their contribution will remain subdominant.
[^3]: There the perturbation was by a wrong sign marginal operator. However this still needed a regulator to make the deformed CFT well-defined, which can be re-expressed as adding additional irrelevant operators.
|
---
abstract: 'A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, $\Delta_{{\mbox{\upshape {\tiny lex}}}}$ — an operation that transforms a monomial ideal of $S={{\bf k}}[x_i: i\in\N]$ that is finitely generated in each degree into a squarefree strongly stable ideal — is defined and studied. It is proved that (in contrast to the reverse lexicographic case) a squarefree strongly stable ideal $I\subset S$ is fixed by lexicographic shifting if and only if $I$ is a universal squarefree lexsegment ideal (abbreviated USLI) of $S$. Moreover, in the case when $I$ is finitely generated and is not a USLI, it is verified that all the ideals in the sequence $\{\Delta_{{\mbox{\upshape {\tiny lex}}}}^i(I)\}_{i=0}^{\infty}$ are distinct. The limit ideal $\overline{\Delta}(I)=\lim_{i\rightarrow\infty}\Delta_{{\mbox{\upshape {\tiny lex}}}}^i(I)$ is well defined and is a USLI that depends only on a certain analog of the Hilbert function of $I$.'
address: 'Department of Mathematics, University of Washington, Seattle, WA 98195-4350, email: \[babson,novik,thomas\]@math.washington.edu'
author:
- Eric Babson
- Isabella Novik
- Rekha Thomas
title: Reverse lexicographic and lexicographic shifting
---
Introduction
============
This paper deals with two problems related to [*algebraic shifting*]{} that were raised by Gil Kalai in [@K00].
Algebraic shifting is an algebraic operation introduced by Kalai [@BK], [@K91] that transforms a simplicial complex $\Gamma$ into a simpler ([*shifted*]{}) complex $\Delta(\Gamma)$, while preserving important combinatorial, topological and algebraic invariants such as face numbers, reduced Betti numbers and extremal algebraic Betti numbers. There are two versions of algebraic shifting — exterior and symmetric: the first one amounts to computing the (degree) reverse lexicographic generic initial ideal (${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}}$) of the Stanley-Reisner ideal of $\Gamma$ in the exterior algebra, while the second one amounts to computing ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}}$ in the symmetric algebra and then applying a certain “squarefree” operation $\Phi$. In this paper we consider only the symmetric version of algebraic shifting. We refer to this operation as [*revlex shifting*]{} and denote it by $\Delta_{{\mbox{\upshape {\tiny rl}}}}$. Clearly $\Delta_{{\mbox{\upshape {\tiny rl}}}}(\Gamma)\neq \Gamma$ if $\Gamma$ is not shifted. Among the many beautiful properties of revlex shifting is the fact that the converse statement holds as well, namely that $$\label{P5}
\Delta_{{\mbox{\upshape {\tiny rl}}}}(\Gamma)=\Gamma \quad \mbox{if } \Gamma \mbox{ is shifted,}$$ and hence that $\Delta_{{\mbox{\upshape {\tiny rl}}}}(\Delta_{{\mbox{\upshape {\tiny rl}}}}(\Gamma))=
\Delta_{{\mbox{\upshape {\tiny rl}}}}(\Gamma)$ for an arbitrary complex $\Gamma$. This result was stated in [@K91] and a somewhat hard proof was given in [@AHH]. Eq. (\[P5\]) along with the two problems on algebraic shifting posed by Gil Kalai [@K00 Problems 16 & 5] is the starting point of our paper.
In [@K00 Problem 16] Kalai asks if algebraic shifting can be axiomatized. In that direction we prove the following result. (We denote by $[n]$ the set $\{1, 2, \ldots, n\}$, and by $f(\Gamma)$ and $\beta_i(\Gamma)$, $i\geq 0$, the $f$-vector and the reduced simplicial Betti numbers of $\Gamma$ computed with coefficients in a field [[**k**]{}]{}, respectively.)
\[stable\] Let $\Delta$ be an operation that associates with every $n\geq 0$ and every simplicial complex $\Gamma$ on the vertex set $V=[n]$ a shifted simplicial complex $\Delta(\Gamma)$ on the same vertex set. Assume further that $\Delta$ satisfies the following properties:
1. $f(\Delta(\Gamma))=f(\Gamma)$;
2. $\Delta(\Gamma\ast\{n+1\})=\Delta(\Gamma)\ast\{n+1\}$;
3. if $\Gamma'\subseteq\Gamma$, then $\Delta(\Gamma')\subseteq \Delta(\Gamma)$;
4. $\sum_{i=0}^{\dim\Gamma}\beta_i(\Gamma)\leq
\sum_{i=0}^{\dim\Delta(\Gamma)}\beta_{i}(\Delta(\Gamma))$.
Then for every shifted complex $\Gamma$, $\Delta(\Gamma)=\Gamma$.
As a corollary we obtain a new and much simpler proof of Eq. (\[P5\]). (Here $\Gamma\ast\{n+1\}$ is [*the cone*]{} over $\Gamma$, that is, a simplicial complex on the vertex set $[n+1]$ whose set of faces consists of faces of $\Gamma$ together with $\{F\cup\{n+1\} : F\in \Gamma\}$.)
Problem 5 in [@K00] asks whether the property given by Eq. (\[P5\]) holds if one considers symmetric shiftings with respect to arbitrary term orders. Since in the case of exterior shiftings the answer is positive (as was shown by Kalai [@Ka90 Prop. 4.2]), one may expect to have the same result in the symmetric case as well. Here we consider (degree) lexicographic order, and denote the corresponding shifting operation by $\Delta_{{\mbox{\upshape {\tiny lex}}}}$. To our surprise we discover that only very few shifted complexes are fixed by lex shifting. Our results are summarized in Theorem \[Thm2\] below.
Denote by $\N$ the set of all positive integers. We say that an ideal $I\subset S={{\bf k}}[x_i: i\in\N]$ is a [*universal squarefree lexsegment ideal*]{} (abbreviated USLI) if it is finitely generated in each degree and is a squarefree lexsegment ideal of $S$. (Equivalently, an ideal $I$ of $S$ that is finitely generated in each degree is a USLI if $I\cap S_{[n]}$ is a squarefree lexsegment ideal of $S_{[n]}:={{\bf k}}[x_1, \ldots, x_n]$ for every $n$.) Thus, for example, the ideal $\langle x_1x_2, x_1x_3, x_1x_4x_5x_6x_7\rangle$ is a USLI, while the ideal $\langle x_1x_2, x_1x_3, x_2x_3\rangle$ is a squarefree lexsegment of $S_{[3]}$ but is not a squarefree lexsegment of $S$, and hence is not a USLI. A simplicial complex $\Gamma$ is a [*USLI complex*]{} if its Stanley-Reisner ideal, $I_\Gamma$, is a USLI. Recall that for a monomial ideal $J\subset S_{[n]}$ the [*(bi-graded) Betti numbers*]{} of $J$ are the invariants $\beta_{i,j}(J)$ that appear in the minimal free resolution of $J$ as an $S_{[n]}$-module. $$\label{alg_betti}
\ldots
\bigoplus_j S_{[n]}(-j)^{\beta_{i,j}(J)} \rightarrow \ldots
\rightarrow \bigoplus_j S_{[n]}(-j)^{\beta_{1,j}(J)} \rightarrow
\bigoplus_j S_{[n]}(-j)^{\beta_{0,j}(J)} \rightarrow J \rightarrow 0$$ Here $S_{[n]}(-j)$ denotes $S_{[n]}$ with grading shifted by $j$. Following [@Eis2], we define the [*$B$-sequence*]{} of $J$, $B(J):=\{B_j(J) : j\geq 1\}$, where $B_j(J):=\sum_{i=0}^j (-1)^i
\beta_{i,j}(J)$. (The $B$-sequence of an ideal contains the same information as its Hilbert series — see Section \[infinite\_section\] for more details as well as for the definition of the $B$-sequence for a monomial ideal of $S$ that is finitely generated in each degree.)
\[Thm2\]
1. A (finite) shifted simplicial complex $\Gamma$ satisfies $\Delta_{{\mbox{\upshape {\tiny lex}}}}(\Gamma)=\Gamma$ if and only if $\Gamma$ is a USLI complex. Moreover, if $\Gamma$ is not a USLI complex, then all the complexes in the sequence $\{\Delta_{{\mbox{\upshape {\tiny lex}}}}^i(\Gamma)\}_{i=0}^{\infty}$ are distinct. (Here $\Delta_{{\mbox{\upshape {\tiny lex}}}}^i(\Gamma)$ denotes the complex obtained from $\Gamma$ by $i$ consecutive applications of $\Delta_{{\mbox{\upshape {\tiny lex}}}}$.)
2. The [*limit*]{} complex $\overline{\Delta}_{{\mbox{\upshape {\tiny lex}}}}(\Gamma):=
\lim_{i\rightarrow\infty}\Delta_{{\mbox{\upshape {\tiny lex}}}}^i(\Gamma)$ is well defined and is a (usually infinite) USLI complex. Moreover, $\overline{\Delta}_{{\mbox{\upshape {\tiny lex}}}}(\Gamma)$ is the unique USLI complex whose Stanley-Reisner ideal has the same $B$-sequence as $I_\Gamma$.
The last part of the theorem implies that if two simplicial complexes $\Gamma_1$ and $\Gamma_2$ that have the same $h$-vector (up to possibly several zeros appended at the end), then $\overline{\Delta}_{{\mbox{\upshape {\tiny lex}}}}(\Gamma_1)=\overline{\Delta}_{{\mbox{\upshape {\tiny lex}}}}(\Gamma_2)$. Thus, in contrast to revlex shifting, the operation $\overline{\Delta}_{{\mbox{\upshape {\tiny lex}}}}$ forgets all the information that $\Gamma$ carries (including the dimension of $\Gamma$) except its $h$-numbers.
Our theorems establish for simplicial complexes, results similar in spirit to those in commutative algebra due to Bigatti-Conca-Robbiano [@BCR] and Pardue [@Pardue]. Theorem 4.3 in [@BCR] asserts that if $I$ is a strongly stable ideal in $S_{[n]}$ and $\mathcal{L}$ is a distraction matrix, then ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}}(D_{\mathcal L}(I))=I$, while Proposition 30 in [@Pardue] asserts that sufficiently (but finitely) many applications of the operation ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}} \circ D_{\mathcal L}$ to a monomial ideal $I\subset S_{[n]}$ results in the unique lexsegment ideal of $S_{[n]}$ having the same Hilbert function as $I$.
The structure of the paper is as follows. Section \[axiom\_section\] is devoted to the proof of Theorem \[stable\]. In Section \[revlex\_section\] after recalling basic facts and definitions related to generic initial ideals and revlex shifting we provide a short new proof of Eq. (\[P5\]). In Section \[USLI\_section\] we introduce and study the class of universal squarefree lexsegment ideals (USLIs) and the class of [*almost USLIs*]{} — the notions that play a crucial role in the proof of Theorem \[Thm2\]. Finally in Section \[infinite\_section\] we prove Theorem \[Thm2\]. We close with a brief discussion of arbitrary term orders.
Axiomatizing Algebraic Shifting {#axiom_section}
===============================
This section is devoted to the proof of Theorem \[stable\]. We start by reviewing several notions pertaining to simplicial complexes.
Denote the collection of all subsets of $[n]:=\{1, 2, \ldots, n\}$ by $2^{[n]}$. Recall that a simplicial complex $\Gamma$ on the vertex set $[n]$ is a collection $\Gamma\subseteq 2^{[n]}$ that is closed under inclusion. (We do not require that every singleton $\{i\}\subseteq[n]$ is an element of $\Gamma$.) The elements of $\Gamma$ are called faces and the maximal faces (under inclusion) are called [*facets*]{}. $F\in\Gamma$ is an [*$i$-dimensional face*]{} (or an $i$-face) if $|F|=i+1$. The [*dimension*]{} of $\Gamma$, $\dim \Gamma$, is the maximal dimension of its faces. The number of $i$-dimensional faces of $\Gamma$ is denoted by $f_i(\Gamma)$, and the sequence $f(\Gamma):=(f_{-1}(\Gamma), f_0(\Gamma), f_1(\Gamma),
\ldots f_{\dim\Gamma}(\Gamma))$ is called the $f$-vector of $\Gamma$. Another set of invariants associated with $\Gamma$ is the set of its reduced Betti numbers $\beta_i(\Gamma):=\dim_{{\bf k}}\widetilde{H}_i(\Gamma; {{\bf k}})$, where $\widetilde{H}_i(\Gamma; {{\bf k}})$ is the $i$-th reduced simplicial homology of $\Gamma$ with coefficients in a field ${{\bf k}}$.
A simplicial complex $\Gamma$ on the vertex set $[n]$ is called [*shifted*]{} if for every $F\in\Gamma$, $i\in F$, and $i<j\leq n$, the set $(F\setminus\{i\})\cup\{j\}$ is a face of $\Gamma$ as well. The Betti numbers of a shifted complex $\Gamma$ are combinatorial invariants and can be computed via the following well-known formula [@BK Thm. 4.3]:
\[betti\] If $\Gamma$ is a shifted complex on the vertex set $[n]$, then $$\beta_i(\Gamma)=|\{F\in\max(\Gamma) \,:\, |F|=i+1,\, n\notin \Gamma\}|,$$ where $\max(\Gamma)$ denotes the set of facets of $\Gamma$.
For a simplicial complex $\Gamma$ and a vertex $v$ of $\Gamma$ define the [*antistar of $v$ in $\Gamma$*]{} as ${\mbox{\upshape ast}\,}_{\Gamma} (v) = \{F\in\Gamma: v \notin F\}$. Define also the [*link of $v$ in $\Gamma$*]{} by ${\mbox{\upshape lk}\,}_\Gamma (v):=
\{F\in{\mbox{\upshape ast}\,}_{\Gamma}(v) : F\cup\{v\}\in\Gamma\}$. Note that if $\Gamma$ is a shifted complex on the vertex set $[n]$, then ${\mbox{\upshape lk}\,}_\Gamma (n)$ and ${\mbox{\upshape ast}\,}_\Gamma (n)$ are shifted complexes on $[n-1]$.
If $\Gamma$ is a simplicial complex on $V$ and $u\not\in V$, then the [*cone*]{} over $\Gamma$ with apex $u$ is a simplicial complex, denoted $\Gamma\ast \{u\}$, on the vertex set $V\cup\{u\}$ whose faces are all sets of the form $F\cup A$, where $F\in\Gamma$ and $A\subseteq \{u\}$. Thus for any vertex $v$ of $\Gamma$, $\Gamma={\mbox{\upshape lk}\,}_\Gamma(v)\ast\{v\} \cup {\mbox{\upshape ast}\,}_\Gamma(v)$ and ${\mbox{\upshape lk}\,}_\Gamma(v)\ast\{v\} \subseteq \Gamma \subseteq
{\mbox{\upshape ast}\,}_\Gamma(v)\ast\{v\}$.
Now we are ready to verify Theorem \[stable\] asserting that if $\Delta$ is an operation that associates with every $n\geq 0$ and every simplicial complex $\Gamma$ on the vertex set $V=[n]$ a shifted simplicial complex $\Delta(\Gamma)$ on the same vertex set, and if $\Delta$ satisfies the following properties:
1. $f(\Delta(\Gamma))=f(\Gamma)$;
2. $\Delta(\Gamma\ast\{n+1\})=\Delta(\Gamma)\ast\{n+1\}$;
3. if $\Gamma'\subseteq\Gamma$, then $\Delta(\Gamma')\subseteq \Delta(\Gamma)$;
4. $\sum_{i=0}^{\dim\Gamma}\beta_i(\Gamma)\leq
\sum_{i=0}^{\dim\Delta(\Gamma)}\beta_{i}(\Delta(\Gamma))$,
then for every shifted complex $\Gamma$, $\Delta(\Gamma)=\Gamma$.
[*Proof of Theorem \[stable\]: *]{} Fix a shifted complex $\Gamma$ on $n$ vertices. If $n=0$ or $n=1$ then $\Delta(\Gamma)=\Gamma$ by property (1). We proceed by induction on $n$. Since the link and the antistar of the vertex $n$ in $\Gamma$, $\Gamma'={\mbox{\upshape lk}\,}_\Gamma (n)$ and $\Gamma''={\mbox{\upshape ast}\,}_\Gamma (n)$, respectively, are shifted complexes on the vertex set $[n-1]$ and since $\Gamma'\ast \{n\}\subseteq \Gamma \subseteq \Gamma''\ast\{n\}$, the induction hypothesis together with properties (2) and (3) yield $$\Gamma'\ast \{n\}\subseteq \Delta(\Gamma)
\subseteq \Gamma''\ast\{n\}.$$ Therefore, $$\begin{aligned}
A &:=& \{F\in\max(\Delta(\Gamma)) \,:\, n\notin F\}
=\{F\in\max(\Delta(\Gamma)) \,:\, F\in \Gamma''\} \\
&\subseteq& \Gamma''\setminus\Gamma' \stackrel{(\star)}{=}
\{F\in\max(\Gamma) \,:\, n\notin F\}=:B,\end{aligned}$$ where $(\star)$ follows from the shiftedness of $\Gamma$: $$F\in\max(\Gamma)\cap 2^{[n-1]} \Longleftrightarrow
F\in\Gamma \mbox{ but } F\cup\{n\} \notin \Gamma \Longleftrightarrow
F\in {\mbox{\upshape ast}\,}_\Gamma(n)\setminus {\mbox{\upshape lk}\,}_\Gamma(n)=\Gamma''\setminus \Gamma'.$$ On the other hand, Lemma \[betti\] and property (4) imply that $$|A|=\sum_{i=0}^{\dim\Delta(\Gamma)}\beta_i(\Delta(\Gamma))\geq
\sum_{i=0}^{\dim\Gamma} \beta_i(\Gamma)=|B|,$$ and thus that $A=B$. Hence $\Delta(\Gamma)\supseteq A=\Gamma''\setminus \Gamma'$, and we infer that $$\Delta(\Gamma)\supseteq (\Gamma'\ast\{n\})\cup( \Gamma''\setminus \Gamma')=
\Gamma.$$ Since $f(\Gamma)=f(\Delta(\Gamma))$ by property (1), it follows that $\Delta(\Gamma)=\Gamma$.
Generic Initial Ideals and revlex shifting {#revlex_section}
==========================================
In this section we review basic facts and definitions related to generic initial ideals and revlex shifting. We also provide a new short proof of Eq. (\[P5\]) asserting that $\Delta_{{\mbox{\upshape {\tiny rl}}}}(\Gamma)=\Gamma$ for a shifted $\Gamma$. Let $S_{[n]}={{\bf k}}[x_1, \ldots, x_n]$ be the ring of polynomials in $n$ variables over a field ${{\bf k}}$ of characteristic zero, and let $\Gamma$ be a simplicial complex on the vertex set $[n]$. We recall that the [*Stanley-Reisner ideal*]{} of $\Gamma$ [@St] is the squarefree monomial ideal $I_\Gamma \subset S_{[n]}$ whose generators correspond to nonfaces of $\Gamma$: $$I_{\Gamma} := \langle \prod_{j=1}^k x_{i_j} \in S_{[n]}\,:\,
\{i_1<i_2<\ldots<i_k\}\notin\Gamma\rangle.$$ The Stanley-Reisner ideal of a shifted complex is called a [*squarefree strongly stable ideal*]{}. (Equivalently, a squarefree monomial ideal $I$ is squarefree strongly stable, if for every minimal generator $m$ of $I$ and for every $1\leq i<j$ such that $x_j|m$ but $x_i\not|m$, the monomial $mx_i/x_j$ lies in $I$.)
Let $\succ$ be a term order on $S_{[n]}$ that refines the partial order by degree where lower degree monomials are more expensive than higher degree monomials, and satisfies $x_1\succ\ldots\succ x_n$. Let $I\subset S_{[n]}$ be a homogeneous ideal such as the Stanley-Reisner ideal of $\Gamma$. Consider a generic $n\times n$ matrix $g$. Then $g$ acts on the set of linear forms of $S_{[n]}$ by $gx_j=\sum_{i=1}^{n}g_{ij}x_i$ and this action can be extended uniquely to a ring automorphism on $S_{[n]}$ that we also denote by $g$. Following [@Eis Thm. 15.18] define the [*generic initial ideal of*]{} $I$ with respect to $\succ$ as $${\mbox{\upshape Gin}}_\succ(I):={\mbox{\upshape in}}_\succ(gI),$$ where ${\mbox{\upshape in}}_\succ(gI)$ is the initial ideal of $gI$ with respect to $\succ$ in the sense of Gröbner basis theory. The same theorem in [@Eis] asserts that we can choose $g$ to be upper triangular and hence we assume from now on that $gx_j=\sum_{i=1}^jg_{ij}x_i$.
We briefly outline how to compute ${\mbox{\upshape Gin}}_\succ(I)$ (for a detailed description the reader is referred to [@Eis Thm. 15.18]).
An exterior monomial in $\bigwedge^l(S_{[n]})_d$ is an element of the form $m_1\wedge\ldots\wedge m_l$ where each $m_i$ is a monomial of $S_{[n]}$ of degree $d$ and $m_1\succ \ldots \succ m_l$. The extension of $\succ$ to an order on monomials of $\bigwedge^l(S_{[n]})_d$ is the order in which $m_1\wedge\ldots\wedge m_l \succ n_1\wedge\ldots\wedge n_l$ if for some $s$ we have that $m_s\succ n_s$ and $m_i=n_i$ for $i<s$. For a non-zero element $f$ of $\bigwedge^l(S_{[n]})_d$, define the [*initial term*]{} of $f$, ${\mbox{\upshape in}}_\succ(f)$, to be the $\succ$-largest monomial appearing in $f$ with nonzero coefficient when $f$ is written as a linear combination of (distinct) monomials.
Consider a generic $n\times n$ upper-triangular matrix $g$ and its action on $S_{[n]}$. Let $I_d$ be the $d$-th homogeneous component of a homogeneous ideal $I$, and let $f_1, \ldots, f_t$ be a basis of $I_d$. Then $g(f_1)\wedge \ldots \wedge g(f_t)\in\wedge^t(S_{[n]})_d$. Denote by $M_d=m_1\wedge \ldots \wedge m_t$ the monomial ${\mbox{\upshape in}}_\succ (g(f_1)\wedge \ldots \wedge g(f_t))$ and by $V_d$ the subspace of $(S_{[n]})_d$ spanned by $m_1, \ldots, m_t$.
\[gin\_constr\] ${\mbox{\upshape Gin}}_\succ(I)=\bigoplus V_d$.
Several basic properties of Gins are summarized in the following lemma.
\[cone\] Let $I\subset S_{[n]}$ be a homogeneous ideal. Then
1. ${\mbox{\upshape Gin}}_\succ(I)$ is a strongly stable monomial ideal (that is, if $m\in {\mbox{\upshape Gin}}_\succ(I)$, $x_j|m$ and $1\leq i <j$, then $x_im/x_j\in {\mbox{\upshape Gin}}_\succ(I)$ as well).
2. ${\mbox{\upshape Gin}}_\succ(I)$ and $I$ have the same Hilbert function (that is, $\dim_{{\bf k}}({\mbox{\upshape Gin}}_\succ(I)_d)=\dim_{{\bf k}}(I_d)$ for all $d$).
3. If $J\subseteq I$ is a homogeneous ideal of $S_{[n]}$, then ${\mbox{\upshape Gin}}_\succ(J)\subseteq {\mbox{\upshape Gin}}_\succ(I)$.
4. Let $\succ'$ be an extension of $\succ$ to a term order on $S_{[n+1]}$ satisfying $x_n\succ' x_{n+1}$. Then ${\mbox{\upshape Gin}}_{\succ'}(IS_{[n+1]})=({\mbox{\upshape Gin}}_\succ I) S_{[n+1]}.$ In particular, for a simplicial complex $\Gamma$ on $[n]$, ${\mbox{\upshape Gin}}_{\succ'}(I_{\Gamma\ast\{n+1\}})=({\mbox{\upshape Gin}}_\succ I_\Gamma) S_{[n+1]}.$
Part (1) is [@Eis Thm. 15.18 and Thm. 15.23]. Part (2) follows from [@Eis Thm 15.3]. Part (3) is obvious from the definitions. To prove part (4), consider a generic upper-triangular $(n+1)\times(n+1)$ matrix $\widetilde{g}$ and its left-upper $n\times n$ submatrix $g$. Then $g$ acts on $S_{[n]}$, $\widetilde{g}$ acts on $S_{[n+1]}$, and $\widetilde{g}x_i=gx_i$ for all $1\leq i\leq n$. Therefore for every (homogeneous) element $h$ of $I\subset S_{[n]}\subset S_{[n+1]}$, $\widetilde{g}h=gh$. Thus for $h\in I$, ${\mbox{\upshape in}}_{\succ'}(\widetilde{g}h)={\mbox{\upshape in}}_{\succ}(gh)$, implying that ${\mbox{\upshape Gin}}_\succ I \subseteq {\mbox{\upshape Gin}}_{\succ'}(IS_{[n+1]})$, and hence that $({\mbox{\upshape Gin}}_\succ I)S_{[n+1]}\subseteq {\mbox{\upshape Gin}}_{\succ'}(IS_{[n+1]})$. The lemma follows, since both the ideals $({\mbox{\upshape Gin}}_\succ I)S_{[n+1]}$ and ${\mbox{\upshape Gin}}_{\succ'}(IS_{[n+1]})$ have the same Hilbert function.
In the later sections we compare Gins of the same ideal $I$ computed with respect to different term orders. For that we need the following definition:
Let $I_1\neq I_2$ be two monomial ideals of $S_{[n]}$ and let $\succ$ be a term order. We say that $I_1\succ I_2$ if the largest monomial in the symmetric difference of $I_1$ and $I_2$ is in $I_1$. Equivalently, $I_1\succ I_2$ if the largest monomial in the symmetric difference of $G(I_1)$ and $G(I_2)$ is in $G(I_1)$, where $G(I_1)$ and $G(I_2)$ are the sets of minimal generators of $I_1$ and $I_2$ respectively.
One immediate observation is
\[gin>gin\] Let $\sigma$ and $\tau$ be two term orders on $S_{[n]}$. Then ${\mbox{\upshape Gin}}_{\sigma}(I)\geq_\sigma {\mbox{\upshape Gin}}_{\tau}(I)$ for any homogeneous ideal $I\subset S_{[n]}$.
Let $f_1, \ldots, f_t$ be a basis of $I_d$, and let $g$ be a generic $n \times n$ upper-triangular matrix. Since $M'_d:={\mbox{\upshape in}}_{>_{\tau}}(g(f_1)\wedge\ldots\wedge g(f_t))$ appears in $g(f_1)\wedge\ldots\wedge g(f_t)$ with a non-zero coefficient, it follows that $M_d:={\mbox{\upshape in}}_{\sigma}(g(f_1)\wedge\ldots\wedge g(f_t)) \geq_{\sigma} M'_d$ (for every $d\geq 0$). Proposition \[gin\_constr\] implies the lemma.
We remark that a stronger version of Lemma \[gin>gin\] was proved in [@Conca Cor. 1.6].
Another ingredient needed for defining revlex shifting is the notion of the squarefree operation. This is a bijection $\Phi$ between the set of all monomials in $\{x_i : i\in \N\}$ and the set of all squarefree monomials in $\{x_i : i\in \N\}$, defined by $$\Phi(\prod_{j=1}^k x_{i_j})=\prod_{j=1}^k x_{i_j+j-1}, \mbox{ where }
i_1\leq i_2\leq \ldots\leq i_k.$$ Note that for a monomial $m\in S_{[n]}$, $\Phi(m)$ may not belong to $S_{[n]}$. However the graded reverse lexicographic order has the following remarkable property [@K91 Lemma 6.3(ii)], [@AHH Lemma 1.1]: if $m$ is a minimal generator of ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}} I_\Gamma$ (where $\Gamma$ is a simplicial complex on $[n]$), then $\Phi(m)$ is an element of $S_{[n]}$. This leads to the following definition (due to Kalai):
\[equiv\] Let $\Gamma$ be a simplicial complex on the vertex set $[n]$. The reverse lexicographic shifting of $\Gamma$, $\Delta_{{\mbox{\upshape {\tiny rl}}}}(\Gamma)$, is a simplicial complex on $[n]$ whose Stanley-Reisner ideal is given by $$I_{\Delta_{{\mbox{\upshape {\tiny rl}}}}(\Gamma)}=\langle \Phi(m) \; : \; m\in
G({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}} I_\Gamma) \rangle,$$ where $G(I)$ denotes the set of the minimal generators of a monomial ideal $I$.
We now provide a new and simple proof of Eq. (\[P5\]) (due originally to Aramova, Herzog, and Hibi [@AHH]).
\[AHH\] The revlex shifting $\Delta_{{\mbox{\upshape {\tiny rl}}}}$ satisfies all the conditions of Theorem \[stable\]. Thus $\Delta_{{\mbox{\upshape {\tiny rl}}}}(\Gamma)=\Gamma$ for every shifted complex $\Gamma$.
It is well-known that (symmetric) revlex shifting satisfies all the conditions of Theorem \[stable\], except possibly for property (2) whose proof appears to be missing in the literature (for the exterior version of algebraic shifting it was recently verified by Nevo [@Nevo]): the fact that $\Delta(\Gamma)$ is a shifted simplicial complex follows from Lemma \[cone\](1); property (1) is [@K91 Lemma 6.3(i)]; property (3) is a consequence of Lemma \[cone\](3); property (4) follows from [@H Cor. 8.25] asserting that $\beta_i(\Gamma)=\beta_i(\Delta(\Gamma))$ for all $i$. To prove property (2) it suffices to check that $\Delta(\Gamma)$ and $\Delta(\Gamma\star \{n+1\})$ have the same set of minimal nonfaces (equivalently, $I_{\Delta(\Gamma)}\subset S_{[n]}$ and $I_{\Delta(\Gamma\star \{n+1\})}\subset S_{[n+1]}$ have the same set of minimal generators). This follows from Definition \[equiv\] and Lemma \[cone\](4).
#### **Remarks**
\(1) We note that to verify the inequality $\sum \beta_i(\Gamma)\leq \sum
\beta_i(\Delta_{{\mbox{\upshape {\tiny rl}}}}(\Gamma))$ one does not need to use the fact that $\beta_i(\Gamma)=\beta_i(\Delta_{{\mbox{\upshape {\tiny rl}}}}(\Gamma))$ for all $i$, which is a consequence of the deep result due to Bayer–Charalambous–Popescu [@BCP] and Aramova–Herzog [@AH] that revlex shifting preserves extremal (algebraic) Betti numbers. Instead one can use the standard flatness argument (see [@H Thm. 3.1]) to show that $\beta_{i,j}(I_\Gamma) \leq
\beta_{i,j}({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}}(I_\Gamma)) = \beta_{i,j}(I_{\Delta(\Gamma)})$ for all $i$, $j$, where the equality comes from the fact that $\Phi$ applied to (minimal generators of) a strongly stable ideal ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}}(I_\Gamma)$ preserves algebraic Betti numbers (see [@AHH Lemma 2.2]). The Hochster formula [@Hoc] then asserts that the reduced Betti numbers of a simplicial complex are equal to certain algebraic graded Betti numbers of its Stanley-Reisner ideal.
\(2) In algebraic terms, the statement of Theorem \[AHH\] translates to the fact that if $I\subset S_{[n]}$ is a squarefree strongly stable ideal, then $\Phi({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}}(I))=I$, where $\Phi({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}}(I)):=\langle \Phi(m): m \in G({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}}(I)) \rangle
$. Hence ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}}(I)=\langle\Phi^{-1}(\mu) : \mu\in G(I)\rangle$, that is, computing the revlex Gin of a squarefree strongly stable ideal $I$ simply amounts to applying $\Phi^{-1}$ to the minimal generators of $I$.
\(3) Our proof (as well as the original proof in [@AHH]) of the equation $\Phi({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}}(I))=I$ for a squarefree strongly stable ideal $I$ works only over a field ${{\bf k}}$ of characteristic zero. We however do not know of any counterexamples in the case of a field of positive characteristic.
Combinatorics of USLIs, almost USLIs, and lex Gins {#USLI_section}
==================================================
In this section we introduce and study the class of universal squarefree lexsegment ideals (USLIs) and the class of almost USLIs. These notions turn out to be crucial in the proof of Theorem \[Thm2\]. To allow for infinitely generated ideals (as we need in the following section) we consider the system of rings $S_{[n]}$, $n\in\N$, endowed with natural embeddings $S_{[n]}\subseteq S_{[m]}$ for $m\geq n$, and provide definitions suitable for the direct limit ring $S=\lim_{n\rightarrow\infty}S_{[n]}={{\bf k}}[x_i : i\in\N]$.
Recall that a squarefree monomial ideal $I\subset S$ ($I\subset S_{[n]}$, respectively) is a [*squarefree lexsegment ideal*]{} of $S$ ($S_{[n]}$, respectively) if for every monomial $m\in I$ and every squarefree monomial $m'\in S$ ($m'\in S_{[n]}$, respectively) such that $\deg(m')=\deg(m)$ and $m'>_{{\mbox{\upshape {\tiny lex}}}} m$, $m'$ is an element of $I$ as well.
\[USLI\] An ideal $L$ of $S$ (or of $S_{[n]}$) is a [*universal squarefree lexsegment ideal*]{} (abbreviated USLI) if it is finitely generated in each degree and $LS$ is a squarefree lexsegment ideal of $S$. Equivalently, an ideal $L=L(k_\bullet)$ (here $k_\bullet=\{k_i\}_{i\in\N}$ is a sequence of nonnegative integers) is a USLI with $k_i$ minimal generators of degree $i$ (for $i\in\N$) if and only if the set of minimal generators of $L$, $G(L)$, is given by $$G(L)=
\bigcup_{r=1}^{\infty}\left\{
(\prod_{j=1}^{r-1} x_{R_j})\cdot x_l \;:\; R_{r-1}+1
\leq l \leq R_r-1\right\}, \mbox{ where } R_j=j+\sum_{i=1}^j k_i.
$$
The easiest way to verify the description of the set $G(L)=\{m_1 >_{{\mbox{\upshape {\tiny lex}}}} m_2 >_{{\mbox{\upshape {\tiny lex}}}} \cdots \ >_{{\mbox{\upshape {\tiny lex}}}} m_s >_{{\mbox{\upshape {\tiny lex}}}} \cdots \}$ of a USLI $L$ is by induction on $s$. Indeed, if $m_1, \cdots, m_s$ satisfy the above description and $m_s=(\prod_{j=1}^{r-1} x_{R_j})\cdot x_l$, then there are two possibilities for $m_{s+1}$: either $\deg(m_{s+1})=\deg(m_s)=r$ (equivalently, $l<R_r-1$) or $\deg(m_{s+1})=r'>r$ (equivalently, $l=R_r-1$ and $k_i=0$ for all $r<i<r'$). In the former case, since $m_{s}>_{{\mbox{\upshape {\tiny lex}}}} m_{s+1}$ and since $m_s$ is the immediate lex-predecessor of $m':=(\prod_{j=1}^{r-1} x_{R_j})\cdot x_{l+1}$, it follows that $m'\geq_{{\mbox{\upshape {\tiny lex}}}} m_{s+1}\in L$ which together with $L$ being a USLI implies that $m'\in L$. Since $m'$ is not divisible by any of $m_1, \cdots, m_s$, this yields $m_{s+1}=m'$. The treatment of the latter case is similar: just observe that every squarefree monomial of degree $r'$ that is lex-smaller than $m':=(\prod_{j=1}^{r-1} x_{R_j})\cdot (\prod_{j=1}^{r'-r+1} x_{l+j})=
(\prod_{j=1}^{r'-1} x_{R_j})\cdot x_{R_{r'-1}+1}$ is divisible by at least one of $m_1, \ldots, m_s$ and hence is in $L-G(L)$, while $m'$ is not divisible by any of $m_1, \cdots, m_s$.
1. The ideal $\langle x_1x_2, x_1x_3,
x_2x_3\rangle$ (the Stanley-Reisner ideal of three isolated points) is a lexsegment in $S_{[3]}$, but is not a lexsegment in $S$, and hence is not a USLI.
2. The ideal $I = \langle x_1x_2, x_1x_3,
x_1x_4x_5x_6x_7\rangle$ is the USLI with $k_1 = 0, k_2 = 2, k_3 =
k_4 = 0, k_5 = 1$ and $k_i = 0$ for all $i > 5$. In this example, check that $R_1 = 1, R_2 = 4, R_3 = 5, R_4 = 6$ and $R_5 = 8$.
Note that every USLI is a squarefree strongly stable ideal, and hence is the Stanley-Reisner ideal of a shifted (possibly infinite) simplicial complex (we refer to such complex as a [*USLI complex*]{}). All complexes considered in this section are assumed to be finite.
The following lemma describes certain combinatorial properties of USLI complexes. This lemma together with Lemmas \[main\_lemma\] and \[Pardue\_lemma\] below provides a key step in the proof of Theorem \[Thm2\].
\[comb\_USLI\] Let $\Gamma$ be a USLI complex on the vertex set $[n]$ with $I_\Gamma=L(k_\bullet)$.
1. If $I_\Gamma\neq 0$ and $k_d$ is the last nonzero entry in the sequence $k_\bullet$, then $\Gamma$ has exactly $d$ facets. They are given by $$F_i=\left\{\begin{array}{ll}
\{R_j : 1\leq j \leq i-1\}\cup [R_i+1, n] & \mbox{ if $1\leq
i\leq d-1$,}\\
\{R_1, \ldots, R_{d-1}\}\cup [R_d, n] & \mbox{ if $i=d$.}
\end{array}\right.$$
2. If $\Gamma'$ is a shifted complex on $[n]$ such that $f(\Gamma)=f(\Gamma')$, then $\Gamma=\Gamma'$. (In other words every USLI complex is the only shifted complex in its $f$-class).
We verify part (1) by induction on $n+d+\sum k_i$. The assertion clearly holds if $d=1$ or if $\sum k_i=1$. For instance, if $d=1$ and $k_1=n$ (equivalently, $R_1=n+1$), then $F_1=[n+1, n]=\emptyset$ is the only facet of $\Gamma$.
Note that $R_d$ is the index of the first variable that does not divide any of the minimal generators of $I_\Gamma$. Thus if $R_d\leq n$, then $\Gamma={\mbox{\upshape lk}\,}_\Gamma(n)\star\{n\}$, and we are done by applying induction hypothesis to the USLI complex ${\mbox{\upshape lk}\,}_\Gamma(n)$. So assume that $R_d=n+1$. Then ${\mbox{\upshape lk}\,}_\Gamma(n)$ and ${\mbox{\upshape ast}\,}_\Gamma(n)$ are easily seen to be the USLI complexes on the vertex set $[n-1]$ whose Stanley-Reisner ideals are given by $L_1=L(k_1, \ldots, k_{d-2}, k_{d-1}+1)$ and $L_2=L(k_1, \ldots, k_{d-1}, k_d-1)$, respectively. Hence by induction hypothesis the complex ${\mbox{\upshape lk}\,}_\Gamma(n)\star\{n\}$ has exactly $d-1$ facets, namely the sets $F_1, \ldots, F_{d-1}$ from the list above. Now if $k_d>1$, then by induction hypothesis the facets of ${\mbox{\upshape ast}\,}_\Gamma(n)$ are the sets $F_1-\{n\}, \ldots,
F_{d-1}-\{n\}, F_d$. Since $\Gamma= ({\mbox{\upshape lk}\,}_\Gamma(n)\star\{n\}) \cup {\mbox{\upshape ast}\,}_\Gamma(n)$, it follows that $\max(\Gamma)=\{F_1, \ldots, F_d\}$. Similarly, if $k_d=1$ and $k_j$ is the last nonzero entry in the sequence $(k_1, \ldots, k_{d-1})$, then the facets of ${\mbox{\upshape ast}\,}_\Gamma(n)$ are the sets $F_1-\{n\}, \ldots,
F_{j-1}-\{n\}, F_d$, and the result follows in this case as well.
To prove part (2) we induct on $n$. The assertion is obvious for $n=1$. For $n>1$ we consider two cases.
[**Case 1:**]{} $R_d\leq n$. In this case $\Gamma={\mbox{\upshape lk}\,}_\Gamma(n)\star\{n\}$, so $\beta_i(\Gamma)=0$ for all $i$. Since among all squarefree strongly stable ideals with the same Hilbert function the squarefree lexsegment ideal has the largest algebraic Betti numbers [@AHHlex Thm. 4.4], and since by Hochster’s formula [@Hoc], $\beta_{n-i-1}(\Lambda)=\beta_{i-1,n}(I_\Lambda)$ for any simplicial complex $\Lambda$ on the vertex set $[n]$, it follows that $\beta_i(\Gamma')\leq\beta_i(\Gamma)=0$, and so $\beta_i(\Gamma')=0$ for all $i$. Since $\Gamma'$ is shifted, Lemma \[betti\] implies that all facets of $\Gamma'$ contain $n$. Thus $\Gamma'={\mbox{\upshape lk}\,}_{\Gamma'}(n)\star\{n\}$, and the assertion follows from induction hypothesis applied to ${\mbox{\upshape lk}\,}_\Gamma(n)$ and ${\mbox{\upshape lk}\,}_{\Gamma'}(n)$.
[**Case 2:**]{} $R_d=n+1$. In this case all facets of $\Gamma$ but $F_d$ contain vertex $n$ (this follows from part (1) of the Lemma), and we infer from Lemma \[betti\] that $$\beta_i(\Gamma)=\left\{\begin{array}{ll} 0, \mbox{ if $i\neq d-2$} \\
1, \mbox{ if $i= d-2$.} \\
\end{array}
\right.$$ Recall the Euler-Poincaré formula asserting that for any simplicial complex $\Lambda$, $$\sum_{j\geq -1}(-1)^j f_j(\Lambda)
= \sum_{j\geq -1}(-1)^j \beta_j(\Lambda)
=:\widetilde{\chi}(\Lambda).$$ Therefore, $\widetilde{\chi}(\Gamma')=\sum_{j\geq -1}(-1)^j f_j(\Gamma')=
\sum_{j\geq -1}(-1)^j f_j(\Gamma)=\widetilde{\chi}(\Gamma)=(-1)^{d-2}$, and hence not all Betti numbers of $\Gamma'$ vanish. The same reasoning as in Case 1 then shows that $\beta_i(\Gamma')=\beta_i(\Gamma)$ for all $i$. Applying Lemma \[betti\] once again, we obtain that $\Gamma'=({\mbox{\upshape lk}\,}_{\Gamma'}(n)\star\{n\})\cup\{ F'\}$, where $|F'|=d-1$ and $F'$ is the only facet of $\Gamma'$ that does not contain $n$. Thus $f({\mbox{\upshape lk}\,}_{\Gamma}(n))= f({\mbox{\upshape lk}\,}_{\Gamma'}(n))$ and $f({\mbox{\upshape ast}\,}_{\Gamma}(n))= f({\mbox{\upshape ast}\,}_{\Gamma'}(n))$, and so ${\mbox{\upshape lk}\,}_{\Gamma}(n)={\mbox{\upshape lk}\,}_{\Gamma'}(n)$ and ${\mbox{\upshape ast}\,}_{\Gamma}(n)={\mbox{\upshape ast}\,}_{\Gamma'}(n)$ (by induction hypothesis), yielding that $\Gamma=\Gamma'$.
We now turn to the class of [*almost USLIs*]{}. (Recall our convention that lower degree monomials are lex-larger than higher degree monomials.)
Let $I\subset S$ (or $I\subset S_{[n]}$) be a squarefree strongly stable monomial ideal with $G(I)=\{m_1>_{{\mbox{\upshape {\tiny lex}}}} \ldots >_{{\mbox{\upshape {\tiny lex}}}} m_l>_{{\mbox{\upshape {\tiny lex}}}} m_{l+1} \}$. We say that $I$ is [*an almost USLI*]{} if $I$ is not a USLI, but $L=\langle m_1, \ldots, m_l\rangle$ is a USLI. We say that a simplicial complex $\Gamma$ is [*an almost USLI complex*]{} if $I_\Gamma$ is an almost USLI.
As we will see in the next section (see also Lemma \[Pardue\_lemma\] below), what makes almost USLI complexes noninvariant under lex shifting is the following combinatorial property. (We recall that the [*regularity*]{} of a finitely generated stable monomial ideal $I$, ${\mbox{\upshape reg}}(I)$, is the maximal degree of its minimal generators.)
\[main\_lemma\] Let $\Gamma$ be an almost USLI complex. Then $|\max(\Gamma)|>{\mbox{\upshape reg}}(I_\Gamma)$.
Assume $\Gamma$ is a simplicial complex on $[n]$ with $G(I_\Gamma)=\{m_1>_{{\mbox{\upshape {\tiny lex}}}}\ldots>_{{\mbox{\upshape {\tiny lex}}}}m_l>_{{\mbox{\upshape {\tiny lex}}}}m_{l+1}\}$. We have to show that $|\max(\Gamma)|>\deg(m_{l+1})=:d$. We verify this by induction on $d$. To simplify the notation assume without loss of generality that every singleton $\{i\}\subset[n]$ is a vertex of $\Gamma$ (equivalently, $I_\Gamma$ has no generators of degree 1). If there are generators of degree 1 then the proof given below can be modified by letting the index $R_1$ play the role of the index $1$. As $I_\Gamma$ is an almost USLI, and so $\langle m_1, \ldots, m_l\rangle$ is a USLI, this leaves two possible cases:
[**Case 1:**]{} [*$m_1, \ldots, m_l$ are divisible by $x_1$, but $m_{l+1}$ is not divisible by $x_1$.*]{} Since $I_\Gamma$ is squarefree strongly stable, it follows that $m_{l+1}=\prod_{j=2}^{d+1}x_j$. In this case each set $F_i=[n]-\{1, i\}$, $i=2, \ldots, d+1$, is a facet of $\Gamma$. (Indeed the product $\prod\{x_j : j\in F_i\}$ is not divisible by $m_{l+1}$, and it is also not divisible by $x_1$, and hence by $m_1, \ldots, m_{l}$, implying that $F_i$ is a face. To show that $F_i$ is a maximal face observe that $F_i\cup \{i\}$ contains the support of $m_{l+1}$, and hence is not a face, but then shiftedness of $\Gamma$ implies that neither is $F_i\cup\{1\}$.) Since there also should be a facet containing $1$, we conclude that $\max(\Gamma)\geq d+1>\deg(m_{l+1})$, completing the proof of this case.
[**Case 2:**]{} [*All minimal generators of $I$ are divisible by $x_1$.*]{} In this case consider an almost USLI $I_\Gamma':=\langle x_1, m_1/x_1, \ldots, m_{l+1}/x_1 \rangle$. By induction hypothesis $\Gamma'$ has $s>\deg(m_{l+1})-1$ facets which we denote by $F_1, \ldots, F_s$. One easily verifies that $\max(\Gamma)=\left\{\{1\}\cup F_1, \ldots, \{1\}\cup F_s, [2,n]\right\},
$ and so $|\max(\Gamma)|=s+1>\deg(m_{l+1})$.
We close this section with an algebraic lemma that relates regularity of ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I_\Gamma)$ to the number of facets of $\Gamma$ (for an arbitrary complex $\Gamma$).
\[Pardue\_lemma\] For a (finite) simplicial complex $\Gamma$, ${\mbox{\upshape reg}}({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I_{\Gamma}))\geq |\max(\Gamma)|$.
This fact is a corollary of [@Pardue Lemma 23] applied to squarefree (and hence radical) ideal $I_\Gamma\in S_{[n]}$. For $\sigma\subseteq[n]$, we denote by $P_\sigma$ the (prime) ideal in $S_{[n]}$ generated by $\{x_j : j\notin\sigma\}$. It is well known that $I_\Gamma$ has the following prime decomposition: $
I_\Gamma=\cap_{\sigma\in\max(\Gamma)} P_\sigma.
$ Thus the variety of $I_\Gamma$, $\mathcal{V}(I_\Gamma)$, is the union (over $\sigma\in\max(\Gamma)$) of the irreducible subvarieties $\mathcal{V}(P_\sigma)$. Each such subvariety is a linear subspace of ${{\bf k}}^n$ of codimension $n-|\sigma|$. [@Pardue Lemma 23] then implies that the monomial $m:=\prod x_i^{r_i}$, where $r_i=|\{\sigma\in\max(\Gamma): |\sigma|=n-i\}|$, is a minimal generator of ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I_\Gamma)$. Hence ${\mbox{\upshape reg}}({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I_\Gamma))\geq \deg(m)=|\max(\Gamma)|$.
Lex shifting, $B$-numbers and the limit complex {#infinite_section}
===============================================
In this section after defining the notion of lexicographic shifting and the notion of $B$-numbers (a certain analog of the Hilbert function) we prove Theorem \[Thm2\]. We remark that extending the notion of algebraic shifting to an arbitrary term order $\succ$ is not entirely automatic since the $\Phi$-image of the set of minimal generators of ${\mbox{\upshape Gin}}_{\succ}(I_\Gamma)\subset S_{[n]}$, $G({\mbox{\upshape Gin}}_{\succ}(I_\Gamma))$, may not be a subset of $S_{[n]}$. This however can be easily corrected if one considers the system of rings $S_{[n]}$, $n\in\N$, endowed with natural embeddings $S_{[n]}\subseteq S_{[m]}$ for $m\geq n$, and makes all the computations in the direct limit ring $S=\lim_{n\rightarrow\infty}S_{[n]}={{\bf k}}[x_i : i\in\N]$. This is the approach we adopt here. We work with the class of monomial ideals $I\subset S$ finitely generated in each degree. Throughout this section we use the graded lexicographic term order on $S$.
\[gin\_def\] Let $I$ be a monomial ideal of $S$ that is finitely generated in each degree. Define $${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I):=\lim_{n\rightarrow\infty}\,
\left({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I\cap S_{[n]})\right)S,$$ where we consider $I\cap S_{[n]}$ as an ideal of $S_{[n]}$.
Since the $d$-th component of ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I\cap S_{[n]})$ depends only on the $d$-th component of $I\cap S_{[n]}$, or equivalently on the minimal generators of $I\cap S_{[n]}$ of degree $\leq d$, Lemma \[cone\](4) implies that ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I)$ is well-defined and that for every $d$ there is $n(d)$ such that $({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}I)_d=(({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I\cap S_{[n]}))S)_d$ for all $n\geq n(d)$. Thus ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I)$ is a monomial ideal finitely generated in each degree. (It is finitely generated if $I$ is.) Moreover, it follows from Lemma \[cone\](1) that ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I)$ is a strongly stable ideal.
Recall that the squarefree operation $\Phi$ takes monomials of $S$ to squarefree monomials of $S$. If $I\subset S$ is a monomial ideal finitely generated in each degree, we define $\Phi(I):=\langle \Phi(m) : m\in G(I)\rangle$, where $G(I)$ is the set of minimal generators of $I$.
Let $I$ be a homogeneous ideal of $S$ that is finitely generated in each degree. The [*lexicographic shifting*]{} of $I$ is the squarefree strongly stable ideal $\Delta_{{\mbox{\upshape {\tiny lex}}}}(I)=\Phi({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I))$. The [*$i$-th lexicographic shifting*]{} of $I$ is the ideal $\Delta_{{\mbox{\upshape {\tiny lex}}}}^i(I)$, where $\Delta_{{\mbox{\upshape {\tiny lex}}}}^i$ stands for $i$ successive applications of $\Delta_{{\mbox{\upshape {\tiny lex}}}}$. We also define the [*limit ideal*]{} $\overline{\Delta}(I):=\lim_{k\rightarrow\infty}\Delta_{{\mbox{\upshape {\tiny lex}}}}^k(I)$.
The rest of the section is devoted to the proof of Theorem \[Thm2\]. First however we digress and review several facts on algebraic Betti numbers (defined by Eq. (\[alg\_betti\])).
\[betti-prop\] Let $I$ and $J$ be monomial ideals of $S_{[n]}$.
1. If $I_j=J_j$ for all $0\leq j\leq j_0$, then $\beta_{i,j}(I)=\beta_{i,j}(J)$ for all $i$ and all $j\leq j_0$.
2. The Betti numbers of $I\subset S_{[n]}$ coincide with those of $IS_{[n+1]}\subset S_{[n+1]}$, that is, $\beta_{i,j}(I)=\beta_{i,j}(IS_{[n+1]})$ for all $i, j$.
Part (1) follows from the standard facts that $$\beta_{i,j}(I)=
\dim_{{\bf k}}{\mbox{\upshape Tor}}_i^{S_{[n]}}({{\bf k}}, I)_{j}=
\dim_{{\bf k}}{\mbox{\upshape Tor}}_i^{S_{[n]}}(I, {{\bf k}})_{j},$$ where we identify ${{\bf k}}$ with the $S_{[n]}$-module $S_{[n]}/\langle x_1, \ldots, x_n\rangle$. For part (2) note that if $\F$ is the free minimal resolution of $I$ over $S_{[n]}$, then $\F\otimes_{S_{[n]}} S_{[n+1]}$ is the free minimal resolution of $IS_{[n+1]}$ over $S_{[n+1]}$, yielding the lemma.
The above properties allow to extend the definition of the Betti numbers to the class of monomial ideals of $S$ that are finitely generated in each degree.
\[betti\_def\] Let $I\subset S$ be a monomial ideal finitely generated in each degree. Define $$\beta_{i,j}(I):=
\lim_{n\rightarrow\infty}\beta_{i,j}(I\cap S_{[n]}) \quad \mbox{for all }
i, j\geq0,$$ where we consider $I\cap S_{[n]}$ as an ideal of $S_{[n]}$.
We remark that since $I$ is finitely generated in each degree, for a fixed $j_0$ there exists $n_0$ such that $(I\cap S_{[n+1]})_j=((I\cap S_{[n]})S_{[n+1]})_j$ for all $0\leq j \leq j_0$ and $n\geq n_0$. Hence it follows from Lemma \[betti-prop\] that (for a fixed $i$) the sequence $\{\beta_{i,j_0}(I\cap S_{[n]})\}_{n\in\N}$ is a constant for indices starting with $n_0$, and thus $\beta_{i,j_0}(I)$ is well-defined.
The Betti numbers of strongly stable ideals (of $S_{[n]}$) were computed by Eliahou and Kervaire [@ElKer], and the analog of this formula for squarefree strongly stable ideals (of $S_{[n]}$) was established by Aramova, Herzog, and Hibi [@AHHlex]. Definition \[betti\_def\] allows to state these results as follows. (For a monomial $u$ define $m(u):=\max\{i : x_i|u\}$.)
\[EK\] Let $I\subset S$ be a monomial ideal finitely generated in each degree, let $G(I)$ denote its set of minimal generators, and let $G(I)_j=\{u\in G(I): \deg u=j\}$.
1. If $I$ is strongly stable, then $\beta_{i, i+j}(I)=\sum_{u\in G(I)_j} {m(u)-1 \choose i}$;
2. If $I$ is squarefree strongly stable, then $\beta_{i, i+j}(I)=\sum_{u\in G(I)_j} {m(u)-j \choose i}$. In particular, if $I=L(k_\bullet)$ is a USLI, then $\beta_{i, i+j}(I)=\sum_{l=1}^{k_j}{k_1+\ldots+k_{j-1}+l-1 \choose i}$.
Using the notion of the Betti numbers, one can define a certain analog of the Hilbert function — the $B$-numbers — of a monomial ideal $I$ of $S$ that is finitely generated in each degree.
\[B-definition\] Let $I\subset S$ (or $I\subset S_{[n]}$) be a monomial ideal finitely generated in each degree, and let $\beta_{i,j}(I)$ be its graded Betti numbers. Define $$B_j(I):=\sum_{i=0}^j (-1)^i\beta_{i,j}(I) \quad \mbox{ for all }
j\geq 0 \quad (\mbox{e.g., $B_0=0$ and $B_1(I)=|G(I)_1|$}).$$ The sequence $B(I):=\{B_j(I): j\geq 1\}$ is called the [**$B$-sequence**]{} of $I$.
It is well known and is easy to prove (see [@Eis2 Section 1B.3]) that for every $n\in\N$ the polynomial $\sum_j B_j(I\cap S_{[n]})x^j$ equals $(1-x)^n \text{Hilb}(I\cap S_n, x)$, where $\text{Hilb}(I\cap S_n, x)$ is the Hilbert series of $I\cap S_{[n]}$. In particular, if $\Gamma$ is a $(d-1)$-dimensional simplicial complex on $[n]$ and $I_{\Gamma}\subset S_{[n]}$ is its Stanley-Reisner ideal then $$\frac{1-\sum_j B_j(I_\Gamma)x^j}{(1-x)^n} = \text{Hilb}(S_{[n]}/I_\Gamma, x)
=\sum_{i=0}^{d} \frac{f_{i-1}(\Gamma)x^i}{(1-x)^i}=
\frac{\sum_{i=0}^d h_i(\Gamma)x^i}{(1-x)^{d}},$$ where $\{h_i(\Gamma)\}_{i=0}^d$ is the $h$-vector of $\Gamma$ [@St]. (Recall that $h_j=\sum_{i=0}^j (-1)^{j-i}{d-i \choose j-i}f_{i-1}$ for $0\leq j \leq d$. In particular, $h_1=f_0-d$.) Thus $\sum_j B_j(I_\Gamma)x^j=1-(1-x)^{h_1}\sum_i h_ix^i$ (if one assumes that $\{i\}\in\Gamma$ for every $i\in[n]$), and so the $h$-vector of $\Gamma$ defines the $B$-sequence of $I_\Gamma$.
The following lemma provides the analog of the “$f(\Gamma)=f(\Delta_{{\mbox{\upshape {\tiny rl}}}}(\Gamma))$-property".
\[cone2\] If $I\subset S$ is a monomial ideal that is finitely generated in each degree, then the ideals $I$ and $\Delta_{{\mbox{\upshape {\tiny lex}}}}(I)$ have the same $B$-sequence. In particular, if $I$ is finitely generated, then for a sufficiently large $n$, the ideals $I\cap S_{[n]}$ and $\Delta_{{\mbox{\upshape {\tiny lex}}}}(I)\cap S_{[n]}$ have the same Hilbert function (in $S_{[n]}$).
Since for every $n\in\N$ the ideals $I\cap S_{[n]}$ and ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I\cap S_{[n]})$ have the same Hilbert function (in $S_{[n]}$) (see Lemma \[cone\]), and since $B_i(I)=\lim_{n\rightarrow\infty} B_i(I\cap S_{[n]})$, the above remark implies that $B(I)=B({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I))$. Finally, since ${\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I)$ is a strongly stable ideal (Lemma \[cone\]), we infer (by comparing the two formulas of Lemma \[EK\]) that $\beta_{i,j}({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I))=\beta_{i,j}(\Phi{\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I))=\beta_{i,j}
(\Delta_{{\mbox{\upshape {\tiny lex}}}}(I))$ for all $i, j$, and so $B({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I))=B(\Delta_{{\mbox{\upshape {\tiny lex}}}}(I))$. The result follows.
Now we are ready to verify the first part of Theorem \[Thm2\]. In fact we prove the following slightly more general result.
\[main\] Let $I$ be a squarefree strongly stable ideal of $S$ finitely generated in each degree. Then $\Delta_{{\mbox{\upshape {\tiny lex}}}}(I)>_{{\mbox{\upshape {\tiny lex}}}} I$ unless $I$ is a USLI in which case $\Delta_{{\mbox{\upshape {\tiny lex}}}}(I)= I$. Moreover if $I$ is finitely generated and is not a USLI, then all ideals in the sequence $\{\Delta^i_{{\mbox{\upshape {\tiny lex}}}}(I)\}_{i\geq 0}$ are distinct.
There are several possible cases.
[**Case 1:**]{} $I=L(k_\bullet)$ is a USLI. To prove that $\Delta_{{\mbox{\upshape {\tiny lex}}}}(I)=I$, it suffices to show that for every $d\geq 1$, $\Delta_{{\mbox{\upshape {\tiny lex}}}}(L(k^{(d)})= L(k^{(d)})$, where $k^{(d)}:=\{k_1, \ldots, k_d, 0,0,\ldots\}$ is the sequence $k_\bullet$ truncated at $k_d$. But this is immediate from Lemmas \[comb\_USLI\](2) and \[cone2\]. Indeed, for $n=n(d)$ sufficiently large the simplicial complexes on the vertex set $[n]$ whose Stanley-Reisner ideals are given by $\Delta_{{\mbox{\upshape {\tiny lex}}}}(L(k^{(d)})\cap S_{[n]}$ and $L(k^{(d)})\cap S_{[n]}$, respectively, are shifted and have the same $f$-numbers. Since the second complex is a USLI complex, it follows that those complexes, and hence their ideals, coincide.
[**Case 2:**]{} $I=\langle m_1, \ldots, m_l, m_{l+1} \rangle$ is an almost USLI. Let $n$ be the largest index of a variable appearing in $\prod_{i=1}^{l+1}m_i$, and let $\Gamma$ be a simplicial complex on $[n]$ with $I_\Gamma = I\cap S_{[n]}$. Then $${\mbox{\upshape reg}}(\Delta_{{\mbox{\upshape {\tiny lex}}}}(I))={\mbox{\upshape reg}}({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I_\Gamma))
\stackrel{\mbox{\tiny {Lemma \ref{Pardue_lemma}}}}{\geq} |\max(\Gamma)|
\stackrel{\mbox{\tiny {Lemma \ref{main_lemma}}}}{>}{\mbox{\upshape reg}}(I_\Gamma)={\mbox{\upshape reg}}(I),$$ yielding that $\Delta_{{\mbox{\upshape {\tiny lex}}}}(I)\neq I$ in this case. Moreover, since by Eq. (\[P5\]), $\Phi({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}}(I_\Gamma))=I_\Gamma$ and since $\Phi$ is a lex-order preserving map, we infer from Lemma \[gin>gin\] that $\Phi({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny lex}}}}(I_\Gamma))\geq_{{\mbox{\upshape {\tiny lex}}}} \Phi({\mbox{\upshape Gin}}_{{\mbox{\upshape {\tiny rl}}}}(I_\Gamma))
=I_\Gamma$, and hence that $\Delta_{{\mbox{\upshape {\tiny lex}}}}(I)>_{{\mbox{\upshape {\tiny lex}}}} I$.
[**Case 3:**]{} I is squarefree strongly stable, but is not a USLI. In this case we sort $G(I)=\{m_1, \ldots, m_l, m_{l+1}, \ldots\}$ by graded lex-order and assume that $m_{l+1}$ is the first non-USLI generator of $I$. Let $I_1=\langle m_1, \ldots, m_l \rangle$ and let $I_2=\langle m_1, \ldots, m_{l+1} \rangle$. Then $I_1$ is a USLI, $I_2$ is an almost USLI, and $I_1\subset I_2\subseteq I$. Hence by the previous two cases $I_1=\Delta_{{\mbox{\upshape {\tiny lex}}}}(I_1)\subset\Delta_{{\mbox{\upshape {\tiny lex}}}}(I_2)$ and $\Delta_{{\mbox{\upshape {\tiny lex}}}}(I_2)>_{{\mbox{\upshape {\tiny lex}}}} I_2$, and so there exists a monomial $m$, $m_l>_{{\mbox{\upshape {\tiny lex}}}} m>_{{\mbox{\upshape {\tiny lex}}}} m_{l+1}$, such that $m \in G(\Delta_{{\mbox{\upshape {\tiny lex}}}}(I_2)) \subseteq G(\Delta_{{\mbox{\upshape {\tiny lex}}}}(I))$. Thus $\Delta_{{\mbox{\upshape {\tiny lex}}}}(I)>_{{\mbox{\upshape {\tiny lex}}}} I$.
Finally to show that for a finitely generated ideal $I$, all ideals in the sequence $\{\Delta^i_{{\mbox{\upshape {\tiny lex}}}}(I)\}_{i\geq 0}$ are distinct, it suffices to check that none of those ideals is a USLI. This is an immediate corollary of Lemmas \[comb\_USLI\](2) and \[cone2\].
Our next goal is to prove the second part of Theorem \[Thm2\]. To do that we fix a sequence of integers $B=\{B_j : j\geq 1\}$ and study the class $\M(B)$ of all monomial ideals $I\subset S$ that are finitely generated in each degree and satisfy $B(I)=B$.
There is at most one USLI in the class $\M(B)$.
Recall that a USLI $L=L(k_\bullet)$ is uniquely defined by its $k$-sequence $k_\bullet=\{k_i : i\geq 1\}$, where $k_i=\beta_{0,i}(L)=|G(L)_i|$. Recall also that $B(L)$ is a function of $k_\bullet$ (see Lemma \[EK\](2)), and so to complete the proof it suffices to show that this function is one-to-one, or more precisely that $k_j$ is determined by $k_1, \ldots, k_{j-1}, B_j$ (for every $j\geq 1$). And indeed, $$\begin{aligned}
k_j&=&\beta_{0,j}(L)=B_j-\sum_{i=1}^j (-1)^i\beta_{i,j}(L)
\quad (\mbox{by definition of } B_j)\\
&=& B_j-\sum_{i=1}^j(-1)^i \sum_{l=1}^{k_{j-i}}{k_1+\ldots+k_{j-i-1}+l-1 \choose i}
\quad (\mbox{by Lemma } \ref{EK}(2)).
$$
Now we are ready to prove (the slightly more general version of) the second part of Theorem \[Thm2\].
For every ideal $I\in\M(B)$, the limit ideal $\overline{\Delta}_{{\mbox{\upshape {\tiny lex}}}}(I)$ is well defined and is the unique USLI of $\M(B)$.
Fix $I\in \M(B)$. To show that $\overline{\Delta}_{{\mbox{\upshape {\tiny lex}}}}(I)$ is well defined, it suffices to check that for every $d\geq 0$, there exists $s=s(d)$ such that $$\label{stab}
G(\Delta^{s}_{{\mbox{\upshape {\tiny lex}}}}(I))_{\leq d}=G(\Delta^{s+1}_{{\mbox{\upshape {\tiny lex}}}}(I))_{\leq d}$$ (where $G(J)_{\leq d}:=\cup_{j\leq d} G(J)_j$), and hence that all ideals $\Delta^{i}_{{\mbox{\upshape {\tiny lex}}}}(I)$, $i\geq s$, have the same $d$-th homogeneous component. We verify this fact by showing that the collection of all possible sets of minimal generators $$\label{finite}
\mathcal{G}_{\leq d}:=\{ G(J)_{\leq d} :
J\in\M(B), J \mbox{ is squarefree strongly stable}\}
\quad \mbox{is finite}.$$ (This yields (\[stab\]), since all ideals $\Delta^{i}_{{\mbox{\upshape {\tiny lex}}}}(I)$, $i\geq 1$, are squarefree strongly stable, and since $\Delta^{i}_{{\mbox{\upshape {\tiny lex}}}}(I)\leq_{{\mbox{\upshape {\tiny lex}}}} \Delta^{i+1}_{{\mbox{\upshape {\tiny lex}}}}(I)$ by Theorem \[main\].) Eq. (\[finite\]) can be easily proved by induction. It clearly holds for $d=0$. Now if $J\in\M(B)$ is squarefree strongly stable, then by Lemma \[EK\](2) and Definition \[B-definition\], $$|G(J)_d|=\beta_{0,d}(J)=
B_d-\sum_{i=1}^{d}(-1)^i\sum_{u\in G(J)_{d-i}}{m(u)-(d-i) \choose i},$$ so assuming that the collection $\mathcal{G}_{\leq d-1}$ is finite, or equivalently that the set of integers $\{m(u): u\in G(J)_{\leq d-1}\in\mathcal{G}_{\leq d-1}\}$ is bounded (say by $n(d)$), we obtain that there exists a constant $g(d)$ such that $|G(J)_d|\leq g(d)$ for all squarefree strongly stable ideals $J\in\M(B)$. But then the squarefree strongly stable property implies that $m(u)< n(d)+g(d)+d$ for every $u\in G(J)_{\leq d}\in \mathcal{G}_{\leq d}$, and (\[finite\]) follows.
The second part of the statement is now immediate: indeed if $G(\Delta^s(I))_{\leq d} = G(\Delta^{s+1}(I))_{\leq d}$, then by Theorem \[main\], $G(\Delta^s(I))_{\leq d}= G(\overline{\Delta}(I))_{\leq d}$ is the set of minimal generators of a USLI.
Remarks on other term orders
============================
We close the paper by discussing several results and conjectures related to algebraic shifting with respect to arbitrary term orders. To this end, we say that an order $\succ$ on monomials of $S$ is a [*term order*]{} if $x_i\succ x_{i+1}$ for $i\geq 1$, $m\succ m'$ as long as $\deg(m)<\deg(m')$, and the restriction of $\succ$ to $S_{[n]}$ is a term order on $S_{[n]}$ for all $n\geq 1$. In addition, we restrict our discussion only to those term orders on $S$ that are compatible with the squarefree operation $\Phi$, that is, $\Phi(m)\succ\Phi(m')$ if $m\succ m'$.
Similarly to Definition \[gin\_def\], for a term order $\succ$ on $S$ and a homogeneous ideal $I\subset S$ that is finitely generated in each degree, we define $\Delta_\succ(I):=\Phi({\mbox{\upshape Gin}}_\succ(I))$. Thus $\Delta_\succ(I)$ is a squarefree strongly stable ideal that has the same $B$-sequence as $I$. (Indeed, the proof of Lemma \[cone2\] carries over to this more general case.)
We say that a squarefree monomial ideal $I\subset S$ is a [*US$\succ$I*]{} if for every monomial $m\in I$ and every squarefree monomial $m'$ such that $\deg(m)=\deg(m')$ and $m'\succ m$, $m'$ is an element of $I$ as well. Being US$\succ$I implies being squarefree strongly stable.
In view of Theorems \[Thm2\] and \[AHH\] it is natural to ask the following:
1. Does $\Delta_\succ(I)=I$ hold for every US$\succ$I I?
2. Is there a term order $\succ$ other than the lexicographic order for which the equality $\Delta_\succ(I)=I$ implies that $I$ is a US$\succ$I?
3. Is there a term order $\succ$ other than the reverse lexicographic order such that the equation $\Delta_\succ(I)=I$ holds for all squarefree strongly stable ideals $I$?
The next proposition answers the first question in the affirmative.
If $I$ is a US$\succ$I, then $\Delta_\succ(I)=I$ for every term order on $S$ that is compatible with $\Phi$.
Exactly as in the proof of Theorem \[main\] (see the last three lines of Case 2), one can show that $\Delta_\succ(I)\succeq I$. Hence either $\Delta_\succ(I)= I$, in which case we are done, or the $\succ$-largest monomial, $m$, in the symmetric difference of $G(\Delta_\succ(I))$ and $G(I)$ is an element of $G(\Delta_\succ(I))$. Since $I$ is a US$\succ$I, we obtain in the latter case that $G(\Delta_\succ(I))_i=G(I)_i$ for all $i<\deg(m)$ and $$G(I)_{i_0}=\{m'\in G(\Delta_\succ(I))_{i_0} : m'\succ m\}
\quad \mbox{ for } i_0=\deg(m),$$ that is, $G(I)_{i_0}$ is a strict subset of $ G(\Delta_\succ(I))_{i_0}$. This is however impossible, since it contradicts the fact that the ideals $I$ and $\Delta_\succ(I)$ have the same $B$-sequence.
The answer to the second question is negative as follows from the following result.
If $I$ is a USLI, then $\Delta_\succ(I)=I$ for all term orders $\succ$.
We omit the proof as it is completely analogous to that of Theorem \[main\], Case 1.
While we do not know the answer to the third question, we believe that it is negative. In fact it is tempting to conjecture that the following holds. Let $\succ$ be a term order on $S$ other than the (graded) reverse lexicographic order, and let $k\geq 2$ be the smallest degree on which $\succ$ and revlex disagree. Write $m_i$ to denote the $i$th squarefree monomial of $S$ of degree $k$ with respect to the revlex order. (It is a fundamental property of the revlex order that every squarefree monomial of $S$ of degree $k$ is of the form $m_i$ for some finite $i$.)
Let $i_0\geq 1$ be the smallest index for which $I_{i_0}:=\langle m_1, \cdots, m_{i_0}\rangle$ is not a US$\succ$I. Then $\Delta_{\succ}(I_{i_0})\neq I_{i_0}$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Aldo Conca for helpful discussions and to the anonymous referees for insightful comments.
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