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{"url":"https:\/\/ir.ymlib.yonsei.ac.kr\/handle\/22282913\/97093","text":"### BROWSE\n\n156 256\n\nCited 0 times in\n\n\ud55c\uad6d\uc778 \ub300\ub3d9\ub9e5 \uadfc\ubd80\uc758 \ud574\ubd80\ud559\uc801 \uad6c\uc870\n\nOther Titles\nAnatomical Structures of the Aortic Root in Koreans\nAuthors\n\uac15\ubbfc\uc6c5\u00a0\u00a0;\u00a0\u00a0\ub098\uba85\ud6c8\u00a0\u00a0;\u00a0\u00a0\uc720\uc7ac\ud5cc\u00a0\u00a0;\u00a0\u00a0\uc815\uc778\ud601\u00a0\u00a0;\u00a0\u00a0\uae40\uc218\uc77c\u00a0\u00a0;\u00a0\u00a0\uae40\uc2dc\uc6b1\u00a0\u00a0;\u00a0\u00a0\uc774\uc601\u00a0\u00a0;\u00a0\u00a0\uc784\uc2b9\ud3c9\u00a0\u00a0;\u00a0\u00a0\uc720\uc7ac\ud604\nCitation\nKorean Journal of Thoracic and Cardiovascular Surgery (\ub300\ud55c\ud749\ubd80\uc678\uacfc\ud559\ud68c\uc9c0), Vol.40(5) : 321-328, 2007\nJournal Title\nKorean Journal of Thoracic and Cardiovascular Surgery (\ub300\ud55c\ud749\ubd80\uc678\uacfc\ud559\ud68c\uc9c0)\nISSN\n2233-601X\nIssue Date\n2007\nAbstract\nBackground: It is very important to determine the surgical anatomy of the aortic root when performing spreading aortic root preserving heart surgery. This study focuses on the surgical aspect of the aortic root anatomy by performing dissection of Korean cadavers. Material and Method: The subjects were 62 cadavers. We measured the intercommissural distances, heights of the sinuses and the circumference of the sinotubular junction and the aortic annulus. Result: The mean age of death was 61.3 years. The intercommissural distance for the right coronary sinus was $0.73{\\pm}2.23mm$, that for the non coronary sinus was $19.34{\\pm}2.03mm$, and that for the left coronary sinus was $18.58{\\pm}2.15mm$. The height of sinus was $20.59{\\pm}2.48mm$ for the right coronary sinus, $18.61{\\pm}2.26mm$ for the non coronary sinus and $17.95{\\pm}19mm$ for the left coronary sinus. The circumference of the sinotubular junction was $70.73{\\pm}5.94mm$ and that of the aortic annulus was $77.94{\\pm}5.63mm$. There is no correlation between age and STJ, aortic annulus and the ratio of STJ of aortic annulus respectively (p=0.920, p=0.111, p=0.073). The tilting angle of the sinotubular junction and aortic annulus is from $2.03^{\\circ}$ to $7.77^{\\circ}$ $(mean=4.90^{\\circ})$. Conclusion: The intercommissural distance and the height of the sinus were largest in the right coronary sinus, and the position of the sinotubular junction to the aortic annulus is obliquely tilted levo-posteriorly.\nFiles in This Item:","date":"2020-02-29 00:21:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6588576436042786, \"perplexity\": 3327.4718757550204}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875148163.71\/warc\/CC-MAIN-20200228231614-20200229021614-00170.warc.gz\"}"}
| null | null |
"King of the World" is a song by the American rock band Weezer, released as a single on January 14, 2016 for the band's self-titled 2016 album, along with a music video.
Composition
Mark Beaumont of NME said that the song, as well as "LA Girlz", are "noble throwbacks to the quirky grunge of Weezer's debut".
Reception
Bill Bodkin at Pop-Break opined "Hearkening back to the sound of the Blue and Green albums, and even some of the Red album, "King of the World" kicks ass with classic Weezer crunch." Bodkin also praises Rivers Cuomo's songwriting for the song, stating that he writes with "hope and feel-good vibes". Chris DeVille at Stereogum described the song as "[feeling] like '90s Weezer with room to breathe."
Live performances
On March 31, 2016, "King of the World" was performed on The Tonight Show Starring Jimmy Fallon in celebration of the release of Weezer's 2016 album. Cuomo was dressed as Elvis Presley during the performance. The song was performed the following day on Good Morning America, where a photo was shot with the band and several Major League Baseball mascots.
Music video
A music video for "King of the World" was released on January 14, 2016, and was directed by Scantron Films. The video features a bearded man in a king's crown and cape, running along a boardwalk, causing havoc, eventually encountering a police officer, who attempts to arrest him.
Charts
Weekly charts
Year-end charts
References
2016 singles
Black-and-white music videos
Songs written by Rivers Cuomo
Weezer songs
|
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"redpajama_set_name": "RedPajamaWikipedia"
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Q: Minecraft NEI it is stuck in recipe mode I tried everything. How to get it to cheat mode?
I'm just playing FTB Infinity and my cheat mode For NEI dose not work. I could really use some help. If anyone would know how to fix it I need the answer. I have image of my config if that helps
A: World settings override global settings, go into the NEI options menu, click inventory, then toggle the global button on the top left hand corner to world. Then change it to cheat mode.
A: Launch FTB Infinity, go into single player/multi player then press E. There will be a button in the bottom left hand corner of the screen that says "Options", click that. Then click inventory, then click the recipe mode button until it says cheat mode.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 3,125
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\section{\large Introduction}
\label{intro}
The odd nature of time arises from a large number of reasons, first of all the discrepancy between the satisfactory description of dynamics in terms of time evolution, and the fundamental timelessness of general relativity and canonical quantum gravity \cite{DeWitt67,Elze04,Rovelli94II,Rovelli06,Isham06,Anderson07,Albrecht07}.
Several contradictions emerge when different theories are compared.
Classical and quantum mechanics
use time as an external parameter to generate a strongly continuous
unitary group of transformations $U(1)$ \cite{ReedSimon72}. Some
theories like the special and the general relativity, quantum field
theory, and canonical quantum gravity, use time as the negative metric
signature coordinate of a 4-dimensional differential manifold. It
is the coordinate associated to a smooth foliation of codimension
1 called topological time \cite{Itin04}, to which observables and
operators are associated according to locality. Time coordinate
is subject with space coordinate to Lorentz or
more general transformations \cite{Rovelli06}.
In metrology, clock time is a metric
time operatively defined at rest. In statistical physics,
time is associated to the concept of irreversibility and its flow
to the entropy of a closed system \cite{Lebowitz93,Lebowitz93II, Rovelli93}.
Such different views are mainly due to implicit assumptions of properties that the system under investigation is supposed to own, such as time reversal symmetry, covariance, locality, monotonical growth of entropy and others.
When considering the problem of the nature of time, there are implicitly two distinct problems to address. The first (i) it is about whether or not time must be included in the list of fundamental quantities of Nature, well defined at all the possible energy and length scales. The second is (ii) the description of the emergence of time metrology based on operatively defined clocks, and the explanation of time evolution experienced by an observer.
The present work strongly supports the view that time is not a fundamental quantity of Nature. I will show that Hamiltonian mechanics, which governs the dynamics of generalized coordinates or quantum fields, is rigorously well defined without the concept of time. As a consequence, it becomes apparently even more difficult to solve the problem (ii), basically because time disappears from the list of observable quantities, but there is a field of metrology entirely devoted to time and frequency measurements (capable to quantify the common experience of time flow).
A good theoretical model capable to be predictive and satisfactory without time answers only partly to the problem of the Nature of Time: it says what time is not \cite{Rovelli06}. In the present work the answer to (i) is addressed, with a particular attention in the definition of the time parameter in a Hamiltonian system in terms of other quantities. Next, also the second part (ii) of the problem is addressed, in order to account the experimentally measured and experienced clock time. The connection between the experimental clock time and the theoretical parameter time is the main goal of the present contribute. There is a recurrent misleading identification between clock time as measured by time and frequency references in a laboratory, and the parameter with respect to which entropy and disorder grow. The observation that thermodynamics leads towards equilibrium states has nothing to do with the concept that motion of its constituents can be described as a function of a clock time. The latter is operatively defined by the prescriptions for realizing the clock itself.
Therefore, clock time requires to be investigated by looking at the general properties of suitable subsystems acting as clocks according to their properties instead of thermodynamic properties of statistical states.
The fact that time is not a measurable quantity can be clarified as follows \cite{DeWitt67,Gerlach69,Banks85,Halliwell85,Brout89,Elze04}. One observes that a
clock measures with some uncertainty an hypothetical 'true' external
time $t$ as a classical quantity $T_{i}(t)$ where the index $i$ spans the clocks. The other observable
quantities $O_{j}(t)$ are detected as $O_{j}(T_{1})$, $O_{j}(T_{2})$,.... where $j$ spans the observables.
However, the clock used to label the dynamical quantities of the system is in turn object of a measurement which establishes its value, accuracy and stability, by means of another clock. Consequently, being the second clock subject to the same check by the first, a two-clock time measurement is required to determine the fractional
frequency stability from the Allan variance of both and a reference standard
is expressed by $T_{2}(T_{1})$ and $T_{1}(T_{2})$,\cite{Rovelli06,Levine99}
without any explicit use of $t$. In other words, a clock is not capable to measure such hypothetical external parameter time, but only self consistent quantities assisted by the recursive definition of $period$.
In the following we addresses the problem of explaining the macroscopic correspondence of clock time with parameter time of dynamics and we provide a universal definition of time for a Hamiltonian system in terms of generalized coordinates change in the phase space.
I will concentrate on a variational
approach which enables the introduction of time in a physical theory in two steps. The method provides a parameter, called parameter time, which does not correspond to a specific observable quantity. However it can be put in correspondence with measurable quantities via cyclic phenomena. This is achieved by dividing a system in opportune subsystems. The present approach partially recalls the distinction presented in Ref.\cite{Elze04} between parametric (proper) time and discrete physical time. Differently from there, here no compactified extra-dimensions are required to introduce a detector operator, neither a lapse function or other parameters to appear in the Lagrange function.
In Section II the definition of parameter time in the framework
of timeless Hamiltonian theories is presented. Time emerges as the
natural parameter after one imposes a variational principle on a timeless
action. The approach is applied in the subsection 2.1 to classical mechanics,
and extended to quantum field theory in the subsection 2.2. Section III is devoted to connect the parametric time with clock metric time measured by means of realistic devices.
In Section IV the results are compared with other approaches and discussed.
\section{\large Parameter Time in a Hamiltonian Timeless Scenario}
\label{sec:1}
The Maupertuis(-Euler-Lagrange-Jacobi) \cite{Arnold,Landau} action principle generates the dynamics without explicitly
using time (for a review on the Maupertuis principle, see
\cite{Gray99}) in the Hamilton theory. Furthermore, we consider only parameter independent Hamiltonians, consistently with the hypothesis of dealing with a closed system.
Such assumptions allow to express the variational principle, the Hamiltonian and the generalized coordinates in a timeless framework. The imposition of both the variational principle and the stationarity of the Hamiltonian individuate a special parametrization among all the possible parametrizations. Such parametrization is useful to describe dynamics. In the following the corresponding parameter is indicated by $\sigma$ and corresponds to the parameter $\tau$ of Ref. \cite{Arnold}, and to parameter time $t$ in ordinary Hamiltonian theory. The main difference from the latter is given by its derivation in a timeless framework. The capability of defining Hamiltonian mechanics without the concept of time will require consequently that some extra hypothesis are assumed in order to provide a definition of clock time. Its correspondence with the parameter $\sigma$ is defined and discussed in the next section.
\subsection{\large Parameter time in classical mechanics}
\label{sec:2}
To derive parameter time from the Maupertuis principle, we have to recast in the timeless framework the ordinary derivation of Hamilton equations of motion from a variational principle on asynchronous varied trajectories.
Contrarily from Lagrangian formalism, expressed by $n$ independent coordinates $q_{i}$ and their time derivative $\dot{q_{i}}$, the Hamiltonian approach allows to set first
order differential equations by virtue of the independence of the
momenta $p_{i}$ from the coordinates $q_{i}$. The number of independent
coordinates raises to $2n$.
The time independent Hamiltonian $H(\mathbf{p},\mathbf{q})$ is a function
of the generalized three dimensional coordinates $\mathbf{p}$ and
$\mathbf{q}$. The independence of $H$ from $t$ reduces the degrees of freedom to $2n-1$. It is necessary to assume that it exists a set of trajectories
in the coordinates space $\mu$ for which $H$ is constant.
A generic parametrization of the points
of the trajectories is set by a label $\lambda$ so that $q_{i}=q_{i}(\lambda)$
and $p_{i}=p_{i}(\lambda)$ where all such functions belong to $C^{2}$
on the interval $[\lambda_{A},\lambda_{B}]$$\in\mathbb{R}$. The
Hamiltonian $H(\mathbf{p},\mathbf{q})$ does not depend explicitly
on $\lambda$. In order to impose a variational principle
on the trajectory it is now considered a variation that is normally
used to impose asynchronous varied trajectories in canonical formalism
to derive Hamilton equation from the Maupertuis principle.
A new parametrization $\sigma$ of
the generalized coordinates and of $\lambda$ is
now defined, under the condition that $\frac{d\lambda}{d\sigma}\neq0$
on $[\sigma_{A},\sigma_{B}]$.
The parametrization $\lambda$ is generic and arbitrary under the given assumptions, necessary only as a starting point. On the contrary the parametrization $\sigma$ is strictly related to the stationarity of the Hamiltonian.
Such distinction is therefore fundamental in the present derivation and it represents a subtle principle and technical difference from the approach of Ref.\cite{Barbour94, Barbour94II}
The stationarity of the action is imposed:
\begin{equation}
A=\int p_{i}dq_{i}
\end{equation}
where the Einstein summation on the repeated indexes is adopted and $i=1,2,3$. The Maupertuis variational principle reads
\begin{equation}
\delta A=\delta\int p_{i}dq_{i}=0
\end{equation}
The imposition of the stationarity of the action is given by the variation
of the trajectories. Neglecting as usual second order perturbations and integrating by parts where necessary, one
has:
\begin{equation}
d\sigma=\left(\frac{\partial H}{\partial p_{i}}\right)^{-1}dq_{i}=-\left(\frac{\partial H}{\partial q_{i}}\right)^{-1}dp_{i}
\end{equation}
under the hypothesis that $\left(\frac{\partial H}{\partial p_{i}}\right) \ne 0$ and $\left(\frac{\partial H}{\partial q_{i}}\right) \ne 0$.
They differ from the Hamilton equations since $\sigma$ does not represent the macroscopic metric time. On the contrary, it only represents the natural parameterization of the system imposed by the energy conservation.
We now briefly discuss the results before extending to quantum field theory.
First, there is one natural parameterization ($\sigma$) which can be
defined by virtue of the properties of the Hamiltonian along the trajectory,
common among all the conjugated coordinate pairs $(p_{i},q^{i})$. Since
all the information about the trajectory and the Hamiltonian are assumed
to be known, the only free parameter of such equation is the label
$\sigma$ which can be identified with local parameter time of dynamics. Notice that
such parameter is not the observable quantity measured by clocks.
Second, the parameter $\sigma$ provides the measurement of the change of
the system along the trajectory. The ratio between the amount of change
of the conjugate variables $dq_{i}$ and $dp_{i}$ during motion is
weighted by the ratio between $\left(\frac{\partial H}{\partial p_{i}}\right)^{-1}$
and $-\left(\frac{\partial H}{\partial q^{i}}\right)^{-1}$. The quantity
$d\sigma$ measures the amount of change along the two generalized
coordinates when energy conservation holds.
\subsection{\large Parameter time emerging in Quantum Field Theory }
\label{sec:3}
The most convenient formalism to extend the action principle to general relativity and to quantum mechanics is the extended presymplectic approach \cite{Rovelli06}. There, dynamics is expressed on the unparameterized curve $\gamma$ in the relativistic configuration space $C=\mathbb{R}\times C_0$, where $C_0$ is the $m$-dimensional space of coordinates $q^i$, which extremizes the integral
\begin{equation}
A[\gamma]=\int_{\gamma}\theta
\end{equation}
where
\begin{equation}
\theta=p_i dq^i + p_t dt
\end{equation}
is the natural one-form defined on the cotangent space $T$*$C$ and the constraint
\begin{equation}
H(q^i,t,p_i,p_t)=0
\end{equation}
where $H$ is the relativistic Hamiltonian. In the extended presymplectic formalism, the variational principle reads:
\begin{equation}
\delta A[\gamma]=\delta \int_{\gamma}\theta=0
\end{equation}
Such principle allows a quantum extension, which goes beyond the scopes of the present section.
Both the lagrangian and the extended presymplectic formalism consider time as a part of the manifold where physics is defined. Time $t$ or $x_0$ assumes a role comparable to that of space, even when starting with an unparameterized curve as happens in presymplectic approach. Technically, since the action admits invariance under reparameterization of time (spacetime in relativistic domain), it does not represent a problem.
Here, in order to avoid the use of the concept of time, the configuration space is only $C_0$ instead of $C=\mathbb{R}\times C_0$ and the extended configuration space will only include fields and their conjugate momenta (generalized fields).
The quantum field theory is generally given in terms of (anti)commutation
relations. The physical content of a theory is well expressed under
the manifestly covariant Lagrangian formalism, but the physics can
be totally described in terms of S-matrix formalism, after one has
imported the physical content in the Hamiltonian approach. Here, in
order to ensure the continuity with the previous analysis, quantum
field theory is considered in Hamiltonian formalism. A Hamiltonian
operator $H=\int d^{3}x\mathcal{H}$ is given, where $\mathcal{H}$
is the Hamiltonian density. The Hamiltonian operator $H$ acts as
a constraint for quantum field dynamics. In quantum field theory,
such a constraint corresponds to being on the mass shell.
The action, in terms of a
quantum fields $\psi_{i}(x)$ and the conjugate coordinates $\pi_{i}(x)$,
can be re-expressed as:
\begin{equation}
A=\int d^{3}x\int d\psi_{i}\pi_{i}
\end{equation}
where the Einstein summation on the repeated indexes is adopted. The
roman index spans on the space dimensions 1, 2, and 3. To define time
as the natural parameterization of change in the generalized coordinate space $\mu_{Q}$, the points of the trajectories $f(q_{i},p_{i})=0$ are
replaced in QFT by space configurations of the generalized field $Q=\left(\psi_{i}(\mathbf{x}),\pi_{i}(\mathbf{x})\right)$
in $\mu_{Q}$ . In the classical case neighboring
position and momentum states are associated to the parameter $\sigma$,
while in QFT $\sigma$ labels the generalized field with support in $\mathcal{\mathbb{R}}^{3}$. Two arrays of fields variate the quantum fields and their conjugate fields respectively.
As in the previous case, the extremality of the action is obtained under the condition that:
\begin{equation}
d\sigma=\left(\frac{\delta\mathcal{H}}{\delta\pi_{i}}\right)_{\psi_{i}}^{-1}d\psi_{i}(\mathbf{x})=-\left(\frac{\delta\mathcal{H}}{\delta\psi_{i}}\right)_{\pi_{i}}^{-1}d\pi_{i}(\mathbf{x})
\end{equation}
Parameter time can be defined as the rate of change of the fields $\psi_{i}$
and their conjugate variables $\pi_{i}$ via the factor $\left(\frac{\delta\mathcal{H}}{\delta\pi_{i}}\right)_{\psi_{i}}^{-1}$and
$-\left(\frac{\delta\mathcal{H}}{\delta\psi_{i}}\right)_{\pi_{i}}^{-1}$
respectively. The parameter $\sigma$ belongs to $\mathbb{R}$ by
construction. The parameterization of the field distribution is locally
achieved by tagging neighboring configurations with the parameter
$\sigma$. Parameter time is therefore built in analogy with the
classical case even in the microscopic limit when small variations
of the fields are considered. Such a construction is compatible with
all the canonical Hamiltonian theories and naturally provides parameter
time in timeless models. An important remark is that parameter time is
by construction defined for those particles which are on the mass shell.
\section{\large Clock time}
\label{sec:4}
$\sigma$ has the property of providing a privileged parameterization suitable for describing dynamics, but it is not an observable quantity. In order to explain the macroscopic experience of time in complex systems, an observable quantity $T$ is built. $T$ realizes an experimentally measurable discrete approximation of $\sigma$.
Since (metric) time is operatively defined by clock standards based on the period of an oscillator, it is only defined in such systems complex enough to contain a subsystem acting as such a clock.
However, the concept of $period$ has to be relaxed to the concept of $cycle$ in the phase space $\mu$ or in $\mu_{Q}$.
Indeed, the definition of periodicity implicitly assumes that an external time is available in order to compare a period with the next one, which is meaningless in a timeless framework.
Defining the $clock$ $time$ $T$, measured for example by atomic clocks, corresponds to label simultaneous occurences in the phase space of two or more subsystems where one is identified as the clock. The clock corresponds to the cyclic subsystem, as defined below. The macroscopic time measured by a subsystem is a function of the subsystem itself. It is a matter of the experimentalist to choose suitable cyclic subsystems (macroscopic clocks) in order to provide a good approximation of the parameter time $\sigma$. The dynamics of the $i$-th observable $O_i$ will consequently be expressed by the simple law involving $\sigma$:
\begin{equation}
O_i (T) \cong O_i (\sigma)
\end{equation}
Let's consider a Hamiltonian system $S$ separable in two independent
subsystems $S_{1}$ and $S_{2}$, so that all the states are represented
by factorized (eigen)states of their respective Hamiltonian $\psi_{1}\otimes\psi_{2} \in H_{1}\otimes H_{2}$ where $H_1$ and $H_2$ are the Hilbert spaces of the subsystems 1 and 2 respectively.
From the previous analysis, the system $S$ owns a unique natural parameter
time $\sigma$ which is well defined also separately for the two subsystems by construction.
We now define the properties required by the system $S_1$ to act as a clock in $S$ in order to describe dynamics in $S_2$.
For a given $\bar{\sigma}$, a state $\psi \in H_{1}\otimes H_{2}$ consists of the tensor product of the state $\psi_{1}(\bar{\sigma}) \in H_1$ and the state $\psi_{2}(\bar{\sigma}) \in H_2$.
We say that $\bar{\psi_{1}}$ has multiplicity $\kappa_{AB}$ on the interval $(\sigma_A,\sigma_B)$ if there are $\kappa_{AB}$ values of $\tilde{\sigma}_i \in (\sigma_A,\sigma_B)$ such that $\psi_{1}(\tilde{\sigma}_i)=\bar{\psi_{1}}$ where $i \in (0,\kappa_{AB})$.
We say that the subsystem $S_1$ is cyclic in the phase space if
\begin{enumerate}
\item {its path in the phase space is closed, }
\item {its velocity $|dQ/d\sigma |\neq 0$ and it is smooth, }
\item {the multiplicity $\kappa_{AB}$ of a state vector in the System 1 monotonically grows with the interval $(\sigma_A,\sigma_B)$ and it tends to infinity when $\sigma_A \rightarrow - \infty \wedge \sigma_B\rightarrow + \infty$.}
\end{enumerate}
The second requirement grants that the realizations of two contiguous states occur along the $\sigma$
axis by respecting the order of the parameter $\sigma$. The third requirement that the clock never stops and its velocity in the phase space is enough to grant that the number of cycles is not finite.
Given the interval $(\sigma_A,\sigma_B)$, it is now defined the set $\Omega(\sigma_A,\sigma_B) \subset H_2$:
\begin{equation}
\Omega(\sigma_A,\sigma_B)=\left\{\psi_2 (\sigma) \in H_2 | \sigma \in \left(\sigma_A,\sigma_B\right) \right\}
\end{equation}
An arbitrary origin $\sigma_0$ is fixed for the parameter time. We associate to such origin the arbitrary initial states $\bar{\psi}_1 = \psi_1 (\sigma_0)$ and $\bar{\psi}_2 = \psi_2 (\sigma_0)$.
Macroscopic time duration $T^{(S_1)}$ of the interval $(\sigma_A,\sigma_B)$ measured by the cyclic subsystem $S_1$ is given by the number $k_{AB}$ of states $\psi_2(\sigma) \in \Omega$ so that $\psi_1(\sigma)=\bar{\psi}_1$. More explicitly, one has
\begin{equation}
T_{AB}^{(S_1)} \equiv k_{AB}
\end{equation}
A good clock has the property of being $stable$ (small standard deviation) and $accurate$ (high Q factor of the resonance associated to the clock) \cite{Levine99,Heavner05}. Since the accuracy refers to the arbitrary resonance frequency of the time standard (for example the Cesium resonance frequency), the present analysis considers only the requirement of stability.
Given a target standard deviation $\Sigma$ required in an experiment performed on the subsystem $S_2$ in the interval $(\sigma_A,\sigma_B)$, for an integration time $\tau$, the clock has to fulfill the following prescription:
\begin{equation}
\epsilon \equiv E^2\left[T_{i,i+1}^{(S_1)}\right]< \Sigma
\end{equation}
where $E^2$ is the standard deviation and
\begin{equation}
\sigma_{i+1}=\sigma_{i}+\tau
\end{equation}
where $i=0...N_{AB}$ with $N_{AB}=(\sigma_B-\sigma_A)/\tau$.
The definition of clock metric time loses of validity for time intervals $T^{(S_1)}$
comparable with the clock period, and for shorter time intervals.
Under such hyphotesis, dynamics of observables in the interval $(\sigma_A,\sigma_B)$ is approximated by the discrete valued equations:
\begin{equation}
x_{\rho} (T_i) \cong x_{\rho} (\sigma_i \pm \epsilon)=x_{\rho} (\sigma_i ) \pm O_{x\rho}[\sigma_i,\epsilon]
\end{equation}
\begin{equation}
p_{\rho} (T_i) \cong p_{\rho} (\sigma_i \pm \epsilon)=p_{\rho} (\sigma_i ) \pm O_{p\rho}[\sigma_i,\epsilon]
\end{equation}
where $\rho=1,2,3$, and $O_{x\rho}[\sigma,\epsilon]$ and $O_{p\rho}[\sigma,\epsilon]$ are higher order quantities in $\epsilon$. Such equations provide the bridge between parameter time of Hamiltonian timeless formalism, and the experimentally defined clock time experienced by observers.
\section{\large Discussion}
\label{sec:5}
Two consequences of the present interpretation of the clock role in the description of the evolution of another subsystem are briefly discussed.
The first consequence deals with the unavoidable semiclassicality of the measurement of a quantum system. Since clock time is by definition fundamentally discrete and it depends on the specific fabrication of the clock, a (macroscopic) measurement of time below one cycle (period) of the time standard is meaningless. At the present time the most advanced available clock technology is given by single ion atomic clocks based on $Al^{+}/Hg^{+}$ with a fractional uncertainty of about $1-2\times10^{17}$ \cite{Itano06}. Adopting such view it implies for example that Planck time scale is an extrapolation, an extention of the concept of clock time beyond its field of definition.
Following the terminology of Kofler and Brukner \cite{Kofler07}, macrorealism (property of a system of being in one or more macroscopically distinct states) and classical (or semiclassical) laws emerge out of quantum physics under the restriction of coarse-grained measurements. The description of time evolution of a system is necessarily semiclassic because the observer is tracking time with a macroscopic system whose fluctuations dominate on the short time scale. Indeed, $T$ is expected to fail as a good approximation of $\sigma$ in the fast decoherence process which occurs during a measurement.
The second point deals with the clock ambiguity problem, where clock is treated as a subsystem \cite{Albrecht07} like in the present approach. Though, an important distinction connected to the role of the parameter time is done. There, gauge invariance transforms one parameterization into another, so they are all equivalent. This implies that a complex system can be separated in many ways in a part which constitutes the clock, and the rest. Such property reveals the assumption that parameter time and clock time are considered to coincide. Such approach assumes consequently that parameter time is an observable quantity, contrarily to the argument presented in the Introduction on the non-observability of parameter time. Furthermore, gauge invariance, in the most general case, treats spacetime as a whole, while we maintained the two conceptually distinct in our study. Causality emerges only when the stationarity of energy is imposed, so time becomes part of 4-manifold spacetime \cite{Minguzzi06}.
\section{\large Conclusion }
\label{sec:6}
The problem of the Nature of Time consists of two parts: whether or not time is a fundamental quantity of Nature, and how clock time does emerge in the laboratory measurement in spite of a (timeless) theoretical and conceptual framework according to which parameter time is not observable. This work addresses both the two issues by providing an explicit Hamiltonian framework, entirely developed without the concept of time, and by defining cyclic subsystems capable to account the (discrete) definition of clock time used in time and frequency metrology.
By restricting
the attention to closed systems the Hamiltonian is time independent
and the action principle can be expressed in terms of only the conjugate
variables (Maupertuis action principle) without the concept
of time. The assumption of being on the mass shell, or equivalently that the stationarity of energy holds, along the trajectory in the phase space, provides a parameterization which gives the ratio of change
of conjugate variables
(generalized coordinates $q_{i}$ and $p_{i}$ or generalized quantum
fields $\psi_{i}$ and $\pi_{i}$). Since all the observables are
expressed in terms of such variables, $\sigma$ parameterizes the whole algebra of observables. In order to well approximate with an observable quantity the parameter time $\sigma$ for which the description of dynamics is simple, we introduce the clock time $T$.
Clock time (also called physical time) measured by macroscopic clocks is a coarse grained discrete
quantity which can be defined in a system $S$ complex enough to contain a subsystem $S_1$ cyclic in the phase space.
The cyclic subsystem acts as a clock reference used for the operative definition of time. Such metric clock time consists of a discrete approximation of the parameter $\sigma$. The stability is a function of the cyclic subsystem adopted to be the clock. The wanted stability is reached by adopting prescriptions in terms of standard deviation on a given integration time.
In particular, if one considers a system containing
a subsystem which corresponds to a cyclic phenomenon, the configurations
of the rest (the subsystem $S_2$) can be put in correspondence with complete contiguous cycles on the orbit
of the cyclic phenomenon in its phase space.
$T$ is therefore suitable to quantify the change
of the other observables defined in $S$, by virtue of its capability of discretize $\sigma$ at the desired precision, according to the characteristic of the experimental apparatus.
To conclude, the present work provides a unitary framework capable to account the timelessness of Nature at a fundamental level, and to explain how clock time can be defined in metrology and experiments, consistently with the dynamics of relations between variables and parameter time evolution itself.
\begin{acknowledgements}
I gratefully acknowledge E. Minguzzi, Y. Itin, P. Zenczykowski, and H. T. Elze for the helpful suggestions
to improve the first and the second version of the manuscript, and to J. Barbour for the useful criticisms.
\end{acknowledgements}
\bibliographystyle{spphys}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
Modern Tokamaks and future designs for power-plant scale Tokamaks,
have plasmas with a separatrix at the plasma's edge.
To determine whether these Tokamak plasmas are susceptible to an ideal
Magnetohydrodynamic Peeling mode (External kink) instability, a simple
model was generalised from a cylindrical to toroidal Tokamak geometry
in the first part\cite{part1} to this paper.
The conclusions from part (I) were:
(a.) Peeling-mode stability is determined by the value of $\Delta'$, a
poloidally averaged measure of the discontinuity in the radial
derivative of the perturbation to the magnetic field at the
plasma-vacuum interface (the separatrix), and (b.) that regardless of
the sign of $\delta W$ the growth rate could still be vanishingly
small.
To determine the consequences of part (I) for the stability of the
Peeling mode, this paper evaluates $\Delta'$.
It is essential that in this calculation any divergences due to the
X-point are incorporated and not accidentally removed or accentuated
by the discretisation of space that is usually required by numerical
modelling, and ideally that $\Delta'$ is calculated exactly.
That is the purpose of this paper.
This paper and part (I) are summarised in Ref. \cite{PRL}.
By using analytical methods to study Peeling mode stability in a
plasma equilibrium with a separatrix boundary (and an X-point), we
hope to avoid difficulties that are encountered with numerical
studies, and to gain a better understanding of the physical factors that
affect the plasma's stability.
The techniques developed here may also find further applications to
related problems using different models of the plasma.
The results from this and future work are intended to provide
understanding and tests, that will assist the development of codes for
stability calculations that incorporate more advanced models of
plasma physics and Tokamak geometry.
It is also hoped that the methods developed in this paper
will have applications outside of plasma physics.
{\bf Outline}: We review conformal transformations in Section
\ref{CT1}, and describe the Karman-Trefftz transformation\cite{Milne}
in Section \ref{KT1}.
The Karman-Trefftz transformation provides an example of a
transformation from a circular boundary to a separatrix boundary with
an X-point.
Section \ref{CP1} reviews how a complex potential may be defined and
used to calculate how the vacuum magnetic field will
transform under a mapping from a system with a circular boundary, to
one with a separatrix.
For a large aspect ratio system, the vacuum energy is calculated for
both a circular cross-section and a separatrix cross-section in
Section \ref{D}, obtaining the vacuum energy for a separatrix
cross-section in terms of a sum of Fourier co-efficients.
The Fourier co-efficients are determined by the plasma-vacuum boundary
conditions, and Section \ref{BCsec} discusses how these boundary
conditions are modified by a conformal transformation.
This is where we have departed from the conventional textbook
applications of conformal transformations, that require the boundary
conditions to be zero.
Instead the transformed boundary conditions presented in Section
\ref{BCsec} provide analytic expressions to determine the Fourier
co-efficients in terms of the straight field line angle, that is not
yet known.
The straight field line angle is calculated in terms of an equilibrium
vacuum field and the conformal transformation, in Section \ref{strt1}.
Section \ref{EQ1} calculates an equilibrium vacuum field for both the
circular boundary and the separatrix boundary systems, subsequently
allowing analytic expressions for the safety factor $q$ and the
straight field line angle to be obtained at the plasma-vacuum
boundary.
At this point all the analytic expressions needed to calculate the
vacuum energy have been obtained.
Section \ref{More} investigates how other quantities (in addition to
the field and the boundary conditions), change under a
conformal transformation.
These expressions are tested in Section \ref{recalc}, by
re-calculating the vacuum energy from a surface integral
representation given in the first part to this paper.
It is noted that this calculation infers the value for $\Delta'$
in terms of the vacuum energy $\delta W_V$.
Reassured with the results from Section \ref{recalc}, Section
\ref{secDelta'} calculates $\Delta'$ directly, and confirms the same
answer in terms of $\delta W_V$ found in the previous section.
At this stage we have analytic expressions for $\delta W_V$ and
$\Delta'$ in terms of a sum of Fourier coefficients, and analytic
expressions for the Fourier coefficients.
Section \ref{Sum1} calculates $\delta W_V$ by evaluating the sum of
Fourier coefficients in a number of ways, finding the same result
as for an equivalent calculation in a circular cross-section system,
and independent of the details of the separatrix geometry.
Section \ref{Conc1} provides a discussion that compares and extends
previous work, considers the mode structure, and summarises the paper.
\section{Conformal Transformations}\label{CT1}
A conformal transformation $w(z)$ (e.g. see Ref. \cite{Spiegel}),
is an analytic function between
complex planes $z \rightarrow w(z)$. It has the property that the
angle (and direction of the angle), between curves in the plane from
which they are mapped, is retained between the resulting curves in the
plane onto which it is mapped.
This angle-preserving property ensures that the unit normal to a line
in one plane will map to a vector normal to the mapped line, with a
consequence that a boundary condition of $0=\vec{n}_z.\vec{B}_z$
will be transformed to a boundary condition of $0= \vec{n}_w
. \vec{B}_w$ for the boundary in the $w(z)$ plane.
More generally, if the arc length around a boundary contour in the
$z$-plane is parameterised by $\alpha$, then it will be shown in
Section \ref{BCsec} that
$\left. \vec{n}_w . \nabla_w V(w) \right|_\alpha = \left|
\frac{dz}{dw} \right| \left. \vec{n}_z . \nabla_z
V(z)\right|_{\alpha}$.
An important property of conformal maps is that a
function that satisfies Laplace's equation,
will continue to do so after a conformal transformation. In
other words, if $\nabla^2 V(z)=0$, then provided $w(z)$ is a conformal
transformation, $\nabla^2_w V(z(w))=0$ also.
Riemann's mapping theorem indicates the existence of a conformal
transformation from a circle to any closed region. So provided a
suitable transformation may be found, and provided the boundary
conditions map in a simple enough way (as they often do), then it is
possible to
find solutions in complicated geometries by solving a
problem with a simple circular boundary.
\section{Karman-Trefftz transformation}\label{KT1}
A mapping that may be used to take us from a circle to a shaped cross
section with an X-point,
is the Karman-Trefftz transformation\cite{Milne}.
The Karman-Trefftz transformation is a generalisation of the Joukowski
transformation that is well known for its use in aerodynamics
calculations for the lift from an airplane wing. It maps from a domain
that surrounds a circle
containing the point $z=-l$ and whose edge passes through $z=l$, to
w(z), with
\begin{equation}\label{KTtransform}
\left( \frac{w+n l}{w-nl} \right) = \left( \frac{z+l}{z-l} \right)^n
\end{equation}
For simplicity we will restrict ourselves to domains of $z$ that are
symmetric about the real axis, so that $w(z)$ is also symmetric about
the real axis with a boundary that has
\begin{equation}
z=-a +(a+l)e^{i\alpha}
\end{equation}
with $a>0$, so that $\alpha$ will parameterise the boundary curve (in
both the $z$ and $w(z)$ planes).
Note that $\alpha$ is not the argument of either
$z$ or $w(z)$ (if we are centred on $z=a$ in the z-plane, $\alpha$ is
the poloidal angle).
If $n=3/2$, then the cusp-like point at $z=l$ becomes an X-point (with
a $\pi/2$ interior angle at the joining surfaces in the $w(z)$
plane)\cite{Milne}, and $n=2$ produces the Joukowski
transformation.
By making $l/a$ arbitrarily small we make the X-point region
arbitrarily localised, a situation similar to that described by
Webster\cite{Webster08}.
This may be seen by rearranging Eq. \ref{KTtransform} to give
\begin{equation}
w(z)=-nl \left( \frac{(z-l)^n + (z+l)^n}{(z-l)^n - (z+l)^n} \right)
\end{equation}
and writing it as an asymptotic expansion in $l/z$, with
\begin{equation}
w(z)=z + l \left( \frac{l}{z} \right)\frac{5}{12} \left\{ 1 +
\frac{7}{60} \left( \frac{l}{z} \right)^2 + \frac{13}{300} \left(
\frac{l}{z} \right)^4 + .. \right\}
\end{equation}
So provided $\left| l/z \right|$ is sufficiently small then $w(z)=z$.
In summary, the Karman-Trefftz transformation provides an explicit
representation of a transformation from a circular boundary (with
$z=-a+(a+l)e^{i\alpha}$), to a shaped boundary with an X-point.
\section{The complex potential}\label{CP1}
We will need to know how the vacuum magnetic field (the gradient of a
potential) is transformed as we move from a circular cross-section to
the X-point geometry.
This is most easily accomplished by
representing the magnetic field as a complex number whose real and
imaginary parts are interpreted as its $x$ and $y$ components, and by
defining a complex potential $\Omega$ in terms of the magnetic
potential $V$.
The complex representation for the magnetic field is given in terms of
the magnetic potential $V$, with
\begin{equation}
B_z = \frac{\partial V}{\partial x} + i \frac{\partial V}{\partial y}
\end{equation}
The complex potential $\Omega$ is defined in terms of $V$ and the
conjugate function of $V$, such that $\Omega$ is analytic and
satisfies the Cauchy-Riemann equations.
Specifically,
\begin{equation}
\Omega(z)=V(z)+i\psi(z)
\end{equation}
with $\psi$ the conjugate function of $V$. Then the Cauchy-Riemann
conditions are satisfied, with
\begin{equation}
\begin{array}{l}
\frac{\partial V}{\partial x} = \frac{\partial \psi}{\partial y}
\\
\frac{\partial V}{\partial y} = - \frac{\partial \psi}{\partial x}
\end{array}
\end{equation}
The Cauchy-Riemann conditions may be used to show that
\begin{equation}
B_z = \overline{ \frac{d\Omega}{dz} }
\end{equation}
and hence the field in the transformed system $w(z)$ may now easily be
found from
\begin{equation}\label{Bw}
B_w = \overline{ \frac{d\Omega}{dw} }
= \overline{ \frac{d\Omega}{dz} \frac{dz}{dw} }
= B_z \overline{ \frac{dz}{dw} }
\end{equation}
\section{The vacuum energy}\label{D}
Working in terms of the complex magnetic field and the
complex potential, we have
\begin{equation}\label{dWBw}
\delta W_V = \int \left| B_w \right|^2 dw_x dw_y
\end{equation}
where the integral extends from the boundary that is parameterised by
$\alpha$ at $w(z(\alpha))$, to infinity.
To evaluate the integral we use Eq. \ref{Bw} so that
$|B_w|^2 =
\left| { \frac{d\Omega}{dz} } \right|^2
\left| { \frac{dz}{dw} } \right|^2 $
and we change coordinates back to the circular cross-section
coordinates, with $dw_x dw_y = \frac{\partial (w_x,w_y)}{\partial
(x,y)}dxdy$ where $ \frac{\partial (w_x,w_y)}{\partial (x,y)} =
\left|\frac{dw}{dz}\right|^2 $ because $w(z)$ is an analytic
function\cite{Spiegel}.
So when we change into the $z$ coordinates (for the purpose of
evaluating the integral Eq. \ref{dWBw}), the factors of
$\left| \frac{dw}{dz} \right|^2$ and $\left| \frac{dz}{dw} \right|^2$
cancel to give
\begin{equation}
\delta W_V = \int \left| B_z \right|^2 dx dy
\end{equation}
In a similar way it may be shown that $\oint_{l_w} B_w . dl_w =
\oint_{l_z} B_z . dl_z$, reflecting the fact that the same total
current is contained within $l_z$ and $l_w$.
Hence the vacuum energy may be found in terms of the vacuum energy for
a solution with a circular boundary, which is much easier to
calculate. The actual values of $\delta W_V$ and $B_z$ are determined
by the plasma-vacuum boundary conditions, that will be modified by the
transformation. The mapping of the boundary condition and the
resulting boundary condition are obtained in Section \ref{BCsec}, but
for the present we will obtain the general solution in terms of the
Fourier coefficients that the boundary conditions will determine.
The vacuum field has $\nabla \wedge \vec{B}_V = \nabla . \vec{B}_V
=0$, so we may write $\vec{B}_V=\nabla V$ with $\nabla^2 V=0$.
We will start from a co-ordinate system with a circular cross section toroidal
geometry, then subsequently obtain a 2D problem by taking the large
aspect ratio limit. In this co-ordinate system
\begin{equation}
\nabla^2 V = \frac{1}{r} \frac{\partial}{\partial r} r\frac{\partial
V}{\partial r} + \frac{1}{r^2} \frac{\partial^2 V}{\partial
\alpha^2} + \frac{1}{R^2} \frac{\partial^2 V}{\partial \phi^2}
\end{equation}
with $r$, $\alpha$, $\phi$ the radial coordinate, poloidal and
toroidal angle respectively.
Writing
\begin{equation}\label{Vexp}
V = \sum_{p=-\infty}^{p=\infty} e^{ip\alpha - in\phi} V_p(r)
\end{equation}
and then projecting out the Fourier components, requires
\begin{equation}
0 = \frac{1}{r} \frac{\partial }{\partial r} r \frac{\partial
V_p}{\partial r} - p^2 V_p(r) - n^2 \left( \frac{r}{R} \right)^2
V_p(r)
\end{equation}
which for any given (finite) $n$ becomes 2-dimensional when $r/R\rightarrow 0$,
leaving
\begin{equation}
0 = r^2 \frac{\partial^2 V_p}{\partial r^2} + r \frac{\partial
V_p}{\partial r} - p^2 V_p
\end{equation}
that has solutions that tend to zero as $r\rightarrow \infty$, of $V_p
= a_p \left(\frac{r_a}{r} \right)^{|p|}$, where the $a_p$ will be
determined by the boundary conditions, and $r_a$ denotes the radial
position of the plasma-vacuum surface.
Note that it is only because the large aspect ratio
limit makes the problem 2-dimensional, that we are able to use
conformal transformations in the calculation.
The vacuum energy is
\begin{equation}\label{dWV1}
\delta W_V
=\frac {1}{2} \int \left| B_1^V \right|^2 \vec{dr}
\end{equation}
Into which we now substitute $V$, with
\begin{equation}\label{Vvac}
V = \sum_{p=-\infty}^{p=\infty} e^{ip\alpha - in\phi} a_p
\left(\frac{r_a}{r}\right)^{|p|}
\end{equation}
and integrate with respect to $\phi$ and $\alpha$, with
$r=r_a$ at the surface, to get
\begin{equation}\label{FdWV}
\delta W_V = 2\pi^2 R \sum_{p\neq0} |p| \left| a_p \right|^2
\end{equation}
Here and in the remainder of this article $R$ will be taken as a
typical measure of the major radius that is approximately constant and
independent of the poloidal angle.
$R$ is identical to the $R_0$ of the first part to this paper, but
because the rest of this paper considers a large aspect ratio limit,
we will simply write $R$ as opposed to $R_0$.
\section{Boundary conditions}\label{BCsec}
The boundary conditions in the $w$-plane determine $\vec{n}.\vec{B}$
in terms of the plasma perturbation.
We will write $\vec{n}.\vec{B}$ in the $w$-plane as $n_w.B_w$.
However, to obtain the co-efficients $a_p$ we need to know
$\vec{n}.\vec{B}$ in
the $z$-plane (that we write as $n_z.B_z$).
Therefore we need to know how $\vec{n}.\vec{B}$ is transformed as we
map between
the $z$-plane where the boundary is circular with $z(\alpha)=-a
+(a+l)e^{i\alpha}$,
and the $w$-plane whose boundary is shaped and contains an X-point.
This is calculated next.
Recall that the real and imaginary components are considered as
orthogonal vector components. Then the tangent vector $t_w(\alpha)$ of
the surface traced by $w(z(\alpha))$ is simply given by
\begin{equation}
t_w(\alpha) = \frac{
\frac{\partial w(z(\alpha )) }{\partial \alpha} }{
\left| \frac{\partial w}{\partial \alpha} \right|}
\end{equation}
However, using the fact that $w(z)$ is an analytic function, so that
$\frac{\partial w}{\partial \alpha} =\frac{dw}{dz} \frac{\partial
z}{\partial \alpha}$, then
\begin{equation}
t_w(\alpha) = \frac{\frac{\partial w(z(\alpha))}{\partial
\alpha}}{\left|\frac{\partial w}{\partial \alpha }\right|}
=
\frac{\frac{dw}{dz(\alpha)}
\frac{\partial z(\alpha)}{\partial \alpha}}
{\left|\frac{dw}{dz(\alpha)}\right|\left|
\frac{\partial z}{\partial \alpha} \right|}
= \frac{\frac{dw}{dz}}{\left| \frac{dw}{dz} \right|}t_z
\end{equation}
where the tangent $t_z$ of $z(\alpha)$ in the $z$-plane is again
simply given by $t_z(\alpha)=\frac{\partial z}{\partial \alpha}/\left|
\frac{\partial z}{\partial \alpha} \right|$.
To obtain the unit normals we rotate the tangent vector by $-\pi/2$,
by simply multiplying by $-i$. Hence
\begin{equation}
n_w=-it_w=\left(\frac{w'(z)}{|w'(z)|}\right)(-it_z)=\frac{w'(z)}{|w'(z)|}n_z
\end{equation}
We already know that $B_w = B_z \overline{\frac{dz}{dw}}$, so to obtain how
$n_w.B_w$ transforms we need to simplify
\begin{equation}
n_w.B_w = \left( \frac{w'(z)}{|w'(z)|} n_z \right) . \left( B_z
\overline{\frac{dz}{dw}} \right)
\end{equation}
where the dot product refers to the sum of the product of the real
parts, plus the product of the imaginary parts (examples may be found
in Appendix \ref{complexidentities}).
We will use $\overline{\frac{dz}{dw}}= \overline{1/\frac{dw}{dz}}
= \frac{dw}{dz}/\left| \frac{dw}{dz}\right|^2$, and write
$n_z=e^{i\theta_z}$, $B_z=r_Be^{i\theta_B}$, and
$\frac{dw}{dz}=r_{w'}e^{i\theta_{w'}}$, to give
\begin{equation}
\begin{array}{ll}
n_w.B_w
&= \frac{w'(z)}{|w'(z)|} n_z . B_z \frac{w'(z)}{|w'(z)|^2}
\\
&= \frac{r_B}{r_{w'}} \left[ e^{i\theta_{w'}} e^{i\theta_z}
. e^{i\theta_B}e^{i\theta_{w'}} \right]
\\
&= \frac{r_B}{r_{w'}} \left[ \cos(\theta_{w'} + \theta_z) + i \sin(
\theta_{w'} + \theta_z ) \right] .
\left[ \cos(\theta_{w'}+\theta_B) + i \sin(\theta_{w'} + \theta_B)
\right]
\\
&= \frac{r_B}{r_{w'}} \left[ \cos(\theta_{w'} + \theta_z)
\cos(\theta_{w'} + \theta_B)
\right. + \left.
\sin( \theta_{w'} + \theta_z ) \sin( \theta_{w'} + \theta_B ) \right]
\\
&= \frac{r_B}{r_{w'}} \cos( \theta_z - \theta_B )
\\
&= \frac{r_B}{r_{w'}} e^{i\theta_z} . e^{i\theta_B}
\\
&= \frac{1}{r_{w'}} n_z . B_z
\\
&= \frac{n_z.B_z}{\left| \frac{dw}{dz} \right|}
\end{array}
\end{equation}
This calculation is repeated by an alternative method in Section \ref{More}.
Knowing how $\vec{n}.\vec{B}$ transforms between $z$ and the $w(z)$
plane, we now return to the plasma-vacuum boundary conditions.
As shown in part (I), the plasma-vacuum boundary condition is
\begin{equation}\label{EPSBC}
\left. \nabla \psi . \vec{B} \right|_{edge} =
\left. \nabla \psi . \vec{B}^V \right|_{edge}
\end{equation}
where "edge", refers to the equilibrium position of the surface.
Because $\vec{B}_0 . \nabla \xi_{\psi}=\nabla \psi . \vec{B}_1$, we
therefore require that
\begin{equation}
\left. \vec{B}_0. \nabla \xi_{\psi} \right|_{edge} =
\left. \nabla \psi . \vec{B}_1^V \right|_{edge}
\end{equation}
For a single Fourier mode in straight field line co-ordinates
$\xi_{\psi}=\xi_m(\psi)e^{im\theta}$,
with $\theta=\frac{1}{q}\int^{\chi} \nu d\chi'$, $q=\frac{1}{q} \oint
\nu d\chi'$, $\nu =\frac{IJ_{\chi}}{R^2}$, $J_{\chi}$ the Jacobian of
the orthogonal $\chi$, $\psi$, $\phi$ co-ordinate system, with $\psi$
the poloidal flux, $\chi$ the poloidal angle, and $\phi$ the toroidal
angle, and $I(\psi)$ is the flux function for which $\vec{B}= I \nabla
\phi + \nabla \phi \wedge \nabla \psi$.
After taking derivatives $\vec{B} . \nabla \xi_{\psi}$ then gives
\begin{equation}
\nabla \psi . \vec{B}_1^V = \left( im - inq \right) \frac{I}{qR^2}
\xi_m e^{im\theta -in\phi}
\end{equation}
This may alternately be written as
\begin{equation}
n_w . B_w = im\Delta \frac{\xi_m}{RB_p} \frac{I}{qR^2} e^{im\theta -in\phi}
\end{equation}
where $\Delta=\frac{m-nq}{m}$.
Now we transform into the $z$ coordinates, transforming both
$n_w.B_w=n_z.B_z/|w'(z)|$ and $B_p=B_{pz}/|w'(z)|$, to get
\begin{equation}\label{nzBC}
n_z . B_z = im\Delta \frac{\xi_m}{R} \frac{|w'(z)|^2}{|B_{pz}|}
\frac{I}{qR^2} e^{im\theta -in\phi}
\end{equation}
Using Eq. \ref{Vexp} along with $V_k = a_k \left( \frac{r_a}{r}
\right)^{|k|}$, and that $n_z.B_z = \vec{e}_r . \nabla V$, then gives
\begin{equation}
\sum_{k=-\infty}^{\infty} a_k \frac{-|k|}{r_a} e^{ik\alpha} =
im\Delta \frac{\xi_m}{R} \frac{|w'(z)|^2}{|B_{pz}|} \frac{I}{qR^2}
e^{im\theta}
\end{equation}
From which the Fourier coefficients are easily obtained by multiplying
by $\frac{e^{-ip\alpha}}{2\pi}$ and integrating from $\alpha = -\pi$
to $\pi$, to give
\begin{equation}\label{ap1}
a_p = - \left(\frac{\Delta}{|p|} \right) \frac{\xi_m}{R}
\frac{1}{2\pi} \oint im \frac{|w'(z)|^2}{|B_{pz}|} \frac{Ir_a}{qR^2}
e^{im\theta -ip\alpha} d\alpha
\end{equation}
In the following section we will see that
\begin{equation}
\theta(\alpha)=\frac{Ir_a}{qR^2} \int^{\alpha}
\frac{|w'(z)|^2}{|B_{pz}|} d\alpha
\end{equation}
which will allow us to integrate by parts once to get
\begin{equation}\label{ap}
a_p = - \left(\frac{ip}{|p|} \right) \Delta \frac{\xi_m}{R}
\frac{1}{2\pi} \oint e^{im\theta -ip\alpha} d\alpha
\end{equation}
To evaluate the coefficients $a_p$, we will need an expression for
$\theta(\alpha)$, this is addressed in the following 2 Sections.
\section{The straight field-line angle}\label{strt1}
In the absence of equilibrium skin currents,
the plasma's equilibrium field $\vec{B}_0$ equals the vacuum's
equilibrium field $\vec{B}_0^V$ at the surface between the plasma and
the vacuum, with
$\left. \vec{B}_0 \right|_{edge}= \left. \vec{B}_0^V \right|_{edge}$.
Therefore provided that we know the equilibrium
vacuum field at the surface, then we also know the plasma's
field at the surface. Consequently, if we know the vacuum field at the
surface, then it is possible to calculate the straight field-line
variable at the surface. Firstly we note that
\begin{equation}\label{sfl1}
\theta = \frac{1}{q}\int^{\chi} \nu d\chi
=\frac{I}{qR^2} \int^{\chi} \frac{J_{\chi}B_p d\chi}{B_p}
=\frac{I}{qR^2} \int^{l} \frac{dl}{B_p}
\end{equation}
An element of arc length parallel to the tangent vector, $dl_w$ has
\begin{equation}
dl_w = \frac{\partial w}{\partial \alpha} d\alpha
= \frac{dw}{dz} \frac{\partial z}{\partial
\alpha} d\alpha
= \frac{dw}{dz} dl_z
\end{equation}
Hence an element of arc length $|dl_w|$ transforms such that
$|dl_w|=\left|\frac{dw}{dz}\right| |dl_z|= \left| \frac{dw}{dz}
\right| r_a d\alpha$, for a circular cross section of radius $r_a$ in
the $z$-plane.
Using this plus $|B_w|=|B_z|/|w'(z)|$, we may write Eq. \ref{sfl1} as
\begin{equation}\label{theta1}
\theta(\alpha) =
\frac{I}{qR^2} \int^{l_w} \frac{|dl_w|}{|B_{pw}|}
=\frac{Ir_a}{qR^2} \int^{\alpha} \left| \frac{dw}{dz}
\right|^2 \frac{d\alpha}{|B_{pz}|}
\end{equation}
Hence if we know the equilibrium field, then we can obtain an analytical
expression for the straight field-line coordinate as a function of
$\alpha$ in the $z$-plane.
\section{Equilibrium vacuum field}\label{EQ1}
The equilibrium vacuum field must (i) have a potential that satisfies
Laplace's equation in the vacuum region, (ii) have $\vec{n}
. \vec{B}_0=0$ at the plasma-vacuum boundary (including at the
strongly shaped X-point containing equilibrium), and (iii) have the
field $B_0=0$ at the X-point. The first part is most easily satisfied
- we can take a solution that satisfies Laplace's equation and
$n_z.B_z=0$ for a circular cross section, and after a conformal
transformation to a shaped cross-section we will still have
$n_w.B_w=0$ and a potential that satisfies Laplace's equation.
To obtain a field with $B_p=0$ at the X-point, we follow a procedure
that is equivalent to that when applying the Kutta condition to obtain
the flow around an airplane wing (using a conformal transformation).
Essentially, in the z-plane we combine a homogeneous
horizontal field and a circulating field, such that the field becomes
zero at a single point on the circular boundary. This is physically
equivalent to imposing an external horizontal field, and then driving
a current through the plasma. Mathematically it corresponds to taking
a complex potential in the $z$-plane of
\begin{equation}\label{omegaz}
\Omega = B_{p0}\left\{ i(z+a) - i \frac{(a+l)^2}{(z+a)} -2i(a+l) \ln
(z+a) \right\}
\end{equation}
with a boundary at
\begin{equation}\label{zsep}
z=-a +(a+l)e^{i\alpha}
\end{equation}
with $\alpha \in [0,2\pi]$, $B_{p0}$ a dimensional constant, and the
radius of the circular boundary $r_a = (a+l)$.
The sign of $B_{p0}$ determines the direction of the circulation of
$B_{pz}$, clockwise ($B_{p0}<0$) or anticlockwise ($B_{p0}>0$), and
the field $B_{pz}$ is obtained from
$B_{pz}=\overline{\frac{d\Omega}{dz}}$, with
\begin{equation}
B_{pz} = \overline{\frac{d\Omega}{dz}} = i\frac{(z-l)^2}{(z+a)^2} B_{p0}
\end{equation}
which at the separatrix given by Eq. \ref{zsep} gives $\left| B_{pz}
\right|= 2B_{p0} (1-\cos(\alpha) )$.
To consider an outermost flux surface that is just inside the
separatrix, we may instead consider
\begin{equation}
z = -a + (a+l-\epsilon)e^{i\alpha}
\end{equation}
with $\epsilon \ll l$. Then for $\epsilon \ll l \ll a$ we have
\begin{equation}\label{Bpzalpha}
\left| B_{pz} \right| \simeq 2 B_{p0} \left( 1 - \cos(\alpha) +
\frac{\epsilon^2}{2a^2} \right)
\end{equation}
so instead of $B_{pz}=0$ at the X-point (that is located at
$\epsilon=0$ and $\alpha=0$), we have $B_{pz}=\epsilon^2/a^2$.
Notice that we have retained the singular perturbation in $\epsilon$
(singular in that although formally $\epsilon^2/a^2 \ll \epsilon/a$,
it is the term in $\epsilon^2/a^2$ that qualitatively alters
$|B_{pz}|$ by preventing it from being zero), further details are
given in Appendix \ref{w'xpt}.
The field in the transformed space is given by $B_{pw}= B_{pz}
\overline{\frac{dz}{dw}}$, although we shall not need this here. Plots of
the equilibrium are given in Figure \ref{magneticfieldplots}.
\begin{figure}[tpb!]
\begin{center}
\resizebox{125mm}{!}
{\includegraphics[angle=0.]{Contourplots.eps}}
\caption{
The figure shows contour plots of the imaginary part of the complex
potential Eq. \ref{omegaz} for the magnetic field, with $a=l=1$.
Such plots give streamlines of the magnetic field\cite{Spiegel}.
The plot on the left is equivalent to a
combination of a vertical field and that produced by a current through
$z=i$, with a
field strength such that $B_z=0$ at the bottom of the plasma vacuum
surface.
The plot on the right is obtained from $\Omega(z)$ by a conformal
transformation, writing $\Omega(z(w))$ and calculating plots of
$\Omega(z(w))$ in the $w$-plane.
$z(w)$ is obtained from Eq. \ref{KTtransform} with $n=3/2$ and $l=1$,
that gives
$z(w) = \frac{(w+3/2)^{2/3} + (w-3/2)^{2/3}}{(w+3/2)^{2/3} - (w-3/2)^{2/3}}$.
For the plots the domain of $z$ and $w$ were rotated by substituting
with $iz$ and $iw$ respectively, so that the X-point is seen at the
bottom of the figure.
}\label{magneticfieldplots}
\end{center}
\end{figure}
As we approach the separatrix the behaviour of $\theta(\alpha)$ is
dominated by the
zeros in $w'(z)$ and $B_{pz}$ that occur near the X-point. Near the
X-point it may be shown (in appendix \ref{w'xpt}), that for the case
of $n=3/2$ with field lines crossing perpendicularly to each other,
that $|w'(z)|^2$ is given by
\begin{equation}\label{w'zsq}
\left| w'(z) \right|^2 \simeq \frac{1}{\sqrt{2}}\left(\frac{3}{2}
\right)^4 \frac{a}{l}
\sqrt{ 1 - \cos(\alpha) + \frac{\epsilon^2}{2 a^2} }
\end{equation}
We may use Eqs. \ref{Bpzalpha} and \ref{w'zsq} to calculate $q$, with
\begin{equation}
q=\frac{1}{2\pi}\oint \nu d\chi
=\frac{1}{2\pi} \frac{Ir_a}{R^2}
\oint
\frac{|w'(\alpha )|^2}{|B_{pz}|} d\alpha
= \frac{1}{2\pi} \frac{Ir_a}{R^2B_{p0}}
\frac{1}{\sqrt{c_a}} \oint
\frac{d\alpha}{ \sqrt{ 1-\cos{\alpha} + \epsilon^2/2a^2}}
\end{equation}
and $\sqrt{c_a}=2\sqrt{2}\left(\frac{2}{3}\right)^4 \left(\frac{l}{a}
\right)$.
Similarly for $\theta(\alpha)$ we have from Eqs. \ref{theta1},
\ref{Bpzalpha}, and \ref{w'zsq}, that
\begin{equation}
\theta(\alpha) = \frac{1}{q} \frac{Ir_a}{R^2B_{p0}}
\frac{1}{\sqrt{c_a}} \int^{\alpha}_{-\pi}
\frac{d\alpha}{ \sqrt{ 1-\cos{\alpha} + \epsilon^2/2a^2}}
\end{equation}
Because the integral is dominated by the divergence at the X-point
where $\alpha=0$, $q$ may be approximated by
\begin{equation}\label{thetaapprox}
q \simeq \frac{1}{2\pi} \frac{Ir_a}{R^2B_{p0}} \frac{1}{\sqrt{c_a}} \oint
\frac{d\alpha}{ \sqrt{ \frac{\alpha^2}{2} + \frac{\epsilon^2}{2a^2} }}
\simeq \frac{c_t\sqrt{2}}{2\pi} 2 \ln \left( \frac{2a\pi}{\epsilon} \right)
\end{equation}
with $c_t \equiv \frac{Ir_a}{R^2B_{p0}}\frac{1}{\sqrt{c_a}}$, and similarly
\begin{equation}
\theta(\alpha) \simeq \frac{c_t}{q} \int^{\alpha}_{-\pi}
\frac{d\alpha}{ \sqrt{ \frac{\alpha^2}{2} + \frac{\epsilon^2}{2a^2}
}}
=\frac{c_t\sqrt{2}}{q} \ln \left(
\frac{ \frac{a\alpha}{\epsilon} + \sqrt{\frac{a^2\alpha^2}{\epsilon^2}
+1} }{ -\frac{a\pi}{\epsilon} + \sqrt{ \frac{a^2\pi^2}{\epsilon^2}
+ 1} }
\right)
\end{equation}
Therefore as we approach the separatrix, with $\epsilon \rightarrow
0$, $q$ has a logarithmic divergence with $q \sim -\ln(\epsilon)$ that
is
typical for a Tokamak plasma near the separatrix. Hence the
qualitative features of $\theta(\alpha)$
could have reasonably been postulated without showing that they also
arise from a vacuum field whose potential satisfies Laplace's
equation, has $n_w.B_{pw}=0$ on the separatrix, and with
$B_{pz}\rightarrow 0$ at the X-point; but it is reassuring to know
that this is also the case.
It is interesting to note that in this model for the equilibrium
field, the angle at which the field lines meet at the X-point
determines how strongly the divergence is there.
For example, if instead of meeting at $\pi/2$ the lines make a cusp
(tending to parallel as they meet), then $q$ is finite
(A cusp is obtained by taking $n=2$ in the Karman-Trefftz
transformation, Eq. \ref{KTtransform}).
\section{More transformed quantities}\label{More}
Now we start to return to our problem of calculating $\Delta'$, by
firstly calculating $\delta W_V$ from
its surface integral representation that is given in part (I), with
\begin{equation}\label{EPSdWV}
\delta W_V = \pi \oint J_{\chi} d\chi \left( \frac{i}{n} \right)
\frac{\nabla \psi .\vec{B}_1^*}{B^2} \left[
R^2 B_p^2 \frac{i}{n} \frac{\partial}{\partial \psi} \left( \nabla
\psi .\vec{B}_1 \right)
\right]
\end{equation}
where the integral is over the plasma surface. This
requires us to know how $\vec{n}.\nabla$ transforms.
The calculation is done partly to reassure us that the transformed
quantities are correct, but also because it is a simple step to
subsequently obtain $\Delta'$.
First we calculate how $\nabla_w$ transforms. We have
\begin{equation}
\nabla_z f(z) = \left( \frac{\partial}{\partial x} + i
\frac{\partial}{\partial y} \right) f(z)
\end{equation}
and
\begin{equation}
\nabla_w f(z(w)) = \left( \frac{\partial}{\partial w_x} + i
\frac{\partial}{\partial w_y} \right) f(z(w))
\end{equation}
where $z=x+iy$ and $w=w_x + iw_y$.
Now we will use the chain rule to expand $\nabla_z f(z(w))$, noting
that because $w(z)$ is an analytic function it satisfies the
Cauchy-Riemann equations,
\begin{equation}
\begin{array}{c}
\frac{\partial w_x}{\partial x} = \frac{\partial w_y}{\partial y}
\\
\frac{\partial w_x}{\partial y} = - \frac{\partial w_y}{\partial x}
\end{array}
\end{equation}
and in addition, $\frac{\partial w}{\partial x}=\frac{dw}{dz}$.
This gives,
\begin{equation}
\begin{array}{ll}
\nabla_z f(z(w)) &=
\frac{\partial f}{\partial w_x} \frac{\partial w_x}{\partial x} +
\frac{\partial f}{\partial w_y} \frac{\partial w_y}{\partial x} + i
\frac{\partial f}{\partial w_x} \frac{\partial w_x}{\partial y} +i
\frac{\partial f}{\partial w_y} \frac{\partial w_y}{\partial y}
\\
&= \frac{\partial w_x}{\partial x} \left(
\frac{\partial f}{\partial w_x} + i \frac{\partial f}{\partial w_y}
\right)
+ \frac{\partial w_y}{\partial x} \left( -i \frac{\partial f}{\partial
w_x} + \frac{\partial f}{\partial w_y} \right)
\\
&= \frac{\partial w_x}{\partial x} \left(
\frac{\partial f}{\partial w_x} + i \frac{\partial f}{\partial w_y}
\right)
- i \frac{\partial w_y}{\partial x} \left( \frac{\partial f}{\partial
w_x} + i \frac{\partial f}{\partial w_y} \right)
\\
&= \left( \frac{\partial}{\partial x} \left(w_x - iw_y \right) \right)
\left( \frac{\partial f}{\partial w_x} + i \frac{\partial f}{\partial
w_y} \right)
\\
&= \overline{\frac{\partial w}{\partial x}} \nabla_w f
= \overline{\frac{dw}{dz}} \nabla_w f
\end{array}
\end{equation}
Hence we have $\nabla_w = \overline{\frac{dz}{dw}} \nabla_z$.
Now we consider the transformation of $\vec{n}.\nabla$.
In calculating how more complicated expressions transform, the author
has found the following identities useful, whose derivations are given
in Appendix \ref{complexidentities}.
\begin{equation}
a.bc=a\bar{b}.c=\bar{a}b.\bar{c}
\end{equation}
\begin{equation}\label{id2}
ab.cd = (a.c)(b.d) + (ia.c)(b.id)
\end{equation}
For example, to calculate $n_w . B_w$ we use Eq. \ref{id2} along with
the results of Section \ref{BCsec} to give
\begin{equation}
\begin{array}{ll}
\vec{n}.\vec{B} = n_w.B_w
&= \left( \frac{w'(z)}{|w'(z)|}n_z \right) . \left(
\overline{\frac{dz}{dw}}
B_z\right)
\\
&= \frac{1}{|w'(z)|} \left[
\left(n_z.B_z \right) \left(\frac{dw}{dz}.
\overline{\frac{dz}{dw}} \right)
+ \left(in_z
.B_z\right)\left(\frac{dw}{dz}.i
\overline{\frac{dz}{dw}}\right)
\right]
\\
&=\frac{n_z.B_z}{\left| w'(z) \right|}
\end{array}
\end{equation}
as before. In the last step we used,
\begin{equation}
\begin{array}{c}
\frac{dw}{dz}.\overline{\frac{dz}{dw}}=1
\\
\frac{dw}{dz}.i\overline{\frac{dz}{dw}}=0
\end{array}
\end{equation}
that may easily be confirmed by writing $\frac{dw}{dz}=\alpha+i\beta$,
so that $\overline{\frac{dz}{dw}}=\overline{1/(\alpha+i\beta)}$, and
multiplying out.
A similar calculation for $\vec{n}.\nabla$ gives
\begin{equation}
\begin{array}{ll}
\vec{n}.\nabla = n_w.\nabla_w
&= \frac{w'(z)}{|w'(z)|} n_z . \overline{\frac{dz}{dw}} \nabla_z
\\
&= \frac{1}{|w'(z)|} \left[
\left( \overline{\frac{dz}{dw}} . \frac{dw}{dz} \right) \left( n_z
. \nabla_z \right)
+ \left(i\overline{\frac{dz}{dw}} . \frac{dw}{dz} \right) \left( n_z
. i\nabla_z \right)
\right]
\\
&= \frac{n_z.\nabla_z}{|w'(z)|}
\end{array}
\end{equation}
\section{Recalculating $\delta W_V$}\label{recalc}
Firstly we re-express Eq. \ref{EPSdWV} using our transformed
quantities, then we show this gives us the same result, Eq. \ref{FdWV}
from Section \ref{D}, before showing how the calculation easily
generalises to give us $\Delta'$ in terms of $\delta W_V$.
Using
\begin{equation}
\left. \nabla \psi . \vec{B}_1^V \right|_{edge} = \left. \nabla \psi
. \vec{B}_1 \right|_{edge} = \left. \vec{B} . \nabla \xi_{\psi}
\right|_{edge}
\end{equation}
gives us
\begin{equation}
\left. \nabla \psi . \vec{B}_1^V \right|^*_{edge} = - \xi^*_m(\psi_a)
\left( \frac{I}{qR^2} \right) (im) \left( \frac{m-nq}{m} \right)
e^{-im\theta +in\phi}
\end{equation}
This is substituted into Eq. \ref{EPSdWV}, to give
\begin{equation}
\delta W_V = \pi \Delta \oint J_{\chi} d\chi \left( \frac{m}{nq}
\right) \left( \frac{I}{R^2B^2} \right)
\xi_m^* e^{-im\theta +in\phi} \left[ R^2 B_p^2 \frac{i}{n}
\frac{\partial}{\partial \psi} \nabla \psi . \vec{B}_1 \right]
\end{equation}
where $\Delta = \frac{m-nq}{m}$. Then using
$\frac{\partial}{\partial \psi} = \frac{\vec{n}.\nabla}{RB_p} $,
we get
\begin{equation}
\begin{array}{l}
\delta W_V = \pi \Delta \left( \frac{m}{nq} \right) \frac{\xi_m^*}{R}
\oint dl \left( \frac{I}{B^2} \right)
e^{-im\theta +in\phi} \frac{i}{n} \vec{n}.\nabla \left( RB_p \vec{n}
. \vec{B}_1 \right)
\end{array}
\end{equation}
where we used $dl=J_{\chi}B_p d\chi$.
Now we will transform this equation into coordinates in which the
plasma has a circular cross-section, using:
\begin{equation}\label{transforms}
\begin{array}{l}
dl_w = \left| w'(z) \right| dl_z = \left| w'(z) \right| r_a d\alpha
\\
\left| B_{pw} \right| = \frac{ \left| B_{pz} \right| }{ \left| w'(z) \right| }
\\
n_w . B_w = \frac{n_z . B_z}{\left| w'(z) \right|}
\\
n_w . \nabla_w = \frac{n_z . \nabla_z}{\left| w'(z) \right|}
\end{array}
\end{equation}
After using the chain rule to expand the term in $n_z.\nabla_z$, we
have
\begin{equation}\label{dWV4}
\begin{array}{l}
\delta W_V = \pi \Delta \left( \frac{m}{nq} \right) \xi_m^* \oint
d\alpha \frac{r_aI}{B^2} e^{-im\theta(\alpha) +in\phi}
\\
\frac{i}{n} \left[
\frac{\left| B_{pz} \right| }{\left| w'(z) \right|^2} n_z . \nabla_z
\left( n_z . B_z \right) +
n_z . B_z n_z . \nabla_z \left(
\frac{\left|B_{pz}\right|}{\left|w'(z)\right|^2} \right)
\right]
\end{array}
\end{equation}
The $n_z.\nabla_z$ operator acting on $n_z.B_z$ will produce a term of
order $n$ larger than $n_z.B_z$. Thus usually we would neglect the
second term. However here we need to be careful that there are no
geometrically driven divergences, this is done in Appendix \ref{app1},
where it is confirmed that the term is of order $\frac{1}{n}$ smaller
and may be neglected.
Hence if we retain only the leading order term in $n$, then
rearranging the expression slightly we have,
\begin{equation}
\delta W_V = \pi \Delta \left( \frac{m}{nq} \right)^2 \xi_m^* \oint
d\alpha e^{-im\theta(\alpha) +in\phi}
\frac{r_aI}{B^2}
\frac{iq}{m}
\frac{\left| B_{pz} \right| }{\left| w'(z) \right|^2} n_z . \nabla_z
\left( n_z . B_z \right)
\end{equation}
After comparison with Eq. \ref{theta1} for $\theta(\alpha)$, this may
be written as
\begin{equation}
\delta W_V = \pi \Delta \left( \frac{m}{nq} \right)^2 \xi_m^*
\left(\frac{r_a^2I^2}{R^2B^2} \right)
\oint d\alpha
\frac{-e^{-im\theta(\alpha) +in\phi}}{im\theta'(\alpha)}
n_z . \nabla_z \left( n_z . B_z \right)
\end{equation}
Using Eq. \ref{Vvac},
\begin{equation}
n_z. \nabla_z \left( n_z . B_z \right) = \left. \frac{\partial^2
V}{\partial r^2} \right|_{r=r_a}
= \sum_{p\neq 0} e^{ip\alpha -in\phi} a_p \frac{|p|(|p|+1)}{r_a^2}
\end{equation}
Giving
\begin{equation}
\delta W_V = \pi \Delta \left( \frac{m}{nq} \right)^2 \xi_m^*
\left(\frac{r_a^2I^2}{R^2B^2} \right)
\sum_{p\neq 0} a_p \frac{|p|(|p|+1)}{r_a^2}
\oint d\alpha \mbox{ }
\frac{-e^{-im\theta(\alpha)+ip\alpha} }{im\theta'(\alpha)}
\end{equation}
Next we observe that
\begin{equation}\label{keyeq}
\oint d\alpha \frac{1}{im\theta'(\alpha)} e^{ip\alpha -im\theta(\alpha)}
= \oint d\alpha \frac{1}{ip} e^{ip\alpha -im\theta(\alpha)}+ \mbox{O} \left(
\frac{\left(\frac{\epsilon}{a}\right) \ln \left( \frac{\epsilon}{a}
\right) }{n} \right)
\end{equation}
that we will justify below, and that appears to be the key result
linking the high $n$ and $q$ calculations at arbitrary cross-section,
to the circular cross section result.
Integrating by parts we get
\begin{equation}
\oint d\alpha \frac{e^{ip\alpha -im\theta(\alpha)}}{im\theta'(\alpha)}
= \oint d\alpha \frac{e^{ip\alpha -im\theta(\alpha)}}{ip} +
\frac{1}{ip} \oint d\alpha
\frac{\theta''(\alpha)}{im(\theta'(\alpha))^2} e^{ip\alpha -
im\theta(\alpha)}
\end{equation}
To estimate the second term we notice that
$|e^{ip\alpha-im\theta(\alpha)}| \leq 1$. Then taking $\theta(\alpha)$
as given by Eq. \ref{thetaapprox} then we find
\begin{equation}
\begin{array}{ll}
\left|
\oint \frac{\theta''(\alpha)}{im (\theta'(\alpha))^2} e^{ip\alpha
-im\theta(\alpha)}
\right|
& \lesssim
\left|
\frac{q}{mc_t} \oint
\frac{\frac{\alpha}{2}}{\sqrt{ \frac{\alpha^2}{2} + \frac{\epsilon^2}{2a^2
} }} d\alpha
\right|
\\
& =
\frac{q}{mc_t} \sqrt{2}
\left( \frac{\epsilon}{a} \right)
\left| \ln \left( \frac{2\pi a}{\epsilon} \right)
\right|
\\
& \sim
\frac{
\left( \frac{\epsilon}{a} \right)
\left| \ln \left( \frac{\epsilon}{a} \right) \right| }{n}
\end{array}
\end{equation}
where the integral is easily obtained by substituting
$\frac{a\alpha}{\epsilon} = \mbox{Sinh}(u)$.
Hence for $\epsilon /a \sim 1$ the term is of order $1/n$ and may be
neglected, and as $\epsilon /a \rightarrow 0$ the term also tends to
zero, and hence may be neglected.
Thus in the high-n limit we get
\begin{equation}\label{dW3}
\delta W_V = \pi \Delta \left( \frac{m}{nq} \right)^2 \xi_m^*
\left(\frac{r_a^2I^2}{R^2B^2} \right)
\sum_{p\neq 0} a_p \frac{i|p|(|p|+1)}{pr_a^2}
\oint d\alpha \mbox{ } e^{-im\theta(\alpha) +ip\alpha}
\end{equation}
Taking the complex conjugate of Eq. \ref{ap} and rearranging, we get
\begin{equation}
\oint d\alpha e^{-im\theta(\alpha)+ip\alpha} = i a_p^* \frac{|p|}{p}
\frac{R}{\xi_m^*} \frac{2\pi}{\Delta}
\end{equation}
which upon substitution into Eq. \ref{dW3}, gives
\begin{equation}\label{dW4}
\delta W_V = 2\pi^2 R \left( \frac{m}{nq} \right)^2
\left(\frac{I^2}{R^2B^2} \right)
\sum_{p\neq 0} \left( \left| p \right| + 1 \right) \left| a_p\right|^2
\end{equation}
For the high $m$,$n$ limit considered here, we expect the $|a_p|^2$
coefficients to be largest for $p\sim m\sim nq$, and hence in the high
$m$, $n$ limit we expect
\begin{equation}\label{dW5}
\begin{array}{l}
\delta W_V = 2\pi^2 R
\sum_{p\neq 0} \left| p \right| \left| a_p\right|^2
\end{array}
\end{equation}
where we have also taken $I^2/(R^2B^2) \simeq 1$.
Hence we have re-obtained Eq. \ref{FdWV} from the high-$n$ expression
given in part (I).
This gives us confidence in the reliability of the calculations.
In addition $[| n_z . \nabla_z (n_z.B_z)|]$ may be estimated by
approximating the plasma as a vacuum and solving Laplace's equation to
approximate and obtain the perturbed field both inside and outside the
plasma respectively, and correctly matching the fields at the
plasma-vacuum boundary.
Then we find that $[| n_z . \nabla_z
(n_z.B_z)|]=2n_z.\nabla_z(n_z.B_z)$. Hence the above calculation may
be used to infer that the term $-\pi |\xi_m|^2 \Delta^2 \Delta'$
appearing in Eq. 70 of the first part to this paper, is equal to
$2\delta W_V$, where
\begin{equation}
\Delta' \equiv \left[ \left| \frac{1}{2\pi}
\oint dl RB_p \frac{I^2}{R^2B^2} \frac{\frac{\partial}{\partial \psi}
\nabla \psi .\vec{B}_1 } {\nabla \psi . \vec{B}_1} \right| \right]
\end{equation}
Hence to evaluate $\Delta'$, we need solely evaluate $\delta W_V$.
\section{Directly calculating $\Delta'$}\label{secDelta'}
Here we show how to calculate $\Delta'$ directly, using the same
assumptions as in Section \ref{recalc}.
For simplicity in all that follows we will take $I^2/R^2 B^2\simeq 1$,
and firstly use Eq. \ref{EPSBC}, that
$\left. \nabla \psi . \vec{B}_1^V \right|_{edge}= \left. \nabla \psi
.\vec{B}_1 \right|_{edge}$ to write
\begin{equation}
\Delta' = \left[ \left| \frac{1}{2\pi}
\oint dl RB_p \frac{I^2}{R^2B^2} \frac{\frac{\partial}{\partial \psi}
\nabla \psi .\vec{B}_1 } {\nabla \psi . \vec{B}_1} \right| \right]
= \frac{1}{2\pi}
\oint dl RB_p \frac{ \left[ \left| \frac{\partial}{\partial \psi}
\nabla \psi .\vec{B}_1 \right| \right] }{\nabla \psi .\vec{B}_1}
\end{equation}
Next we use Eqs. \ref{transforms}, to obtain
\begin{equation}
\Delta' = \frac{1}{2\pi}
\oint r_a d\alpha \frac{1}{n_z . B_z} \left[ \left| n_z . \nabla
\left( n_z . B_z \right) \right| \right]
\end{equation}
Making the usual approximation that treats the perturbed field near
the plasma's edge as behaving the same as in a vacuum, and also using
Eq. \ref{Vvac}, gives
\begin{equation}
\left[ \left| n_z . \nabla_z \left( n_z . B_z \right) \right| \right]
= 2 \left. \frac{\partial^2 V}{\partial r^2} \right|_{r=r_a}
= 2 \sum_{p \neq 0} e^{ip\alpha -in\phi} \frac{a_p}{r_a^2} |p| \left(
|p| + 1 \right)
\end{equation}
We also have
\begin{equation}
\begin{array}{ll}
n_z . B_z = \left. \frac{\partial V}{\partial r} \right|_{r=r_a}
&= - \sum_{p\neq 0} e^{ip\alpha-in\phi} a_p \frac{|p|}{r_a}
\\
&= \Delta \frac{\xi_m}{R} \frac{e^{-in\phi}}{r_a} \sum_{p\neq 0}
e^{ip\alpha} \frac{1}{2\pi} \oint im\theta'(\beta)
e^{im\theta(\beta)-ip\beta} d\beta
\\
&= \Delta \frac{\xi_m}{R} \frac{e^{-in\phi}}{r_a}
\theta'(\alpha) e^{im\theta(\alpha)}
\end{array}
\end{equation}
where we used Eq. \ref{theta1} that implies $\theta'(\alpha) =
\frac{Ir_a}{qR^2} \frac{|w'|^2}{|B_{pz}|}$, Eq. \ref{ap1}, and that
$\theta'(\alpha) e^{im\theta(\alpha)} = \sum_{p\neq0}
e^{ip\alpha} \frac{1}{2\pi} \oint \theta'(\beta) e^{im\theta(\beta)
-ip\beta} d\beta$.
Note that the above equation can be obtained more directly from
Eq. \ref{nzBC}.
Using the above results, after some cancellations we obtain
\begin{equation}
\begin{array}{ll}
\Delta' &= \frac{2}{2\pi} \oint d\alpha \frac{R}{\xi_m \Delta}
\frac{e^{-im\theta(\alpha)}}{im\theta'(\alpha)} \sum_{p\neq 0}
e^{ip\alpha} a_p |p| \left( |p| + 1 \right)
\\
&= \frac{1}{\pi} \frac{R}{\xi_m \Delta} \sum_{p\neq 0} a_p |p|^2 \oint
d\alpha \frac{e^{-im\theta(\alpha)+ip\alpha} }{im\theta'(\alpha)}
+ \frac{1}{\pi} \frac{R}{\xi_m \Delta} \sum_{p\neq 0} a_p |p| \oint
d\alpha \frac{e^{-im\theta(\alpha)+ip\alpha} }{im\theta'(\alpha)}
\end{array}
\end{equation}
Then using the result Eq. \ref{keyeq} of Section \ref{recalc}, that
\begin{equation}
\begin{array}{ll}
\oint d\alpha \frac{e^{-im\theta(\alpha)}}{im\theta'(\alpha)} e^{ip\alpha}
&= \frac{1}{ip} \oint e^{-im\theta + ip\alpha} + \mbox{O} \left(
\frac{\left( \frac{\epsilon}{a} \right) \left| \ln \left(
\frac{\epsilon}{a} \right) \right| }{n}
\right)
\end{array}
\end{equation}
we obtain the first term as
\begin{equation}
\begin{array}{ll}
\frac{R}{\pi\xi_m \Delta} \sum_{p\neq 0} a_p |p|^2 \oint d\alpha
\frac{e^{-im\theta(\alpha)+ip\alpha} }{im\theta '(\alpha)}
&= \frac{2}{|\xi_m |^2}
\frac{R^2}{\Delta^2}
\sum_{p \neq 0} a_p |p|
\frac{(-ip)}{|p|} \Delta \frac{\xi_m^*}{R} \frac{1}{2\pi}
\oint e^{ip\alpha -im\theta}
\\
&= \frac{2 R^2}{|\xi_m|^2 \Delta^2} (-1) \sum_{p\neq 0} |p| |a_p|^2
\end{array}
\end{equation}
Using Eq. \ref{ap} for $a_p$, and the same approximations as above,
the second term gives
\begin{equation}
\begin{array}{ll}
\frac{R}{\pi \xi_m \Delta} \sum_{p\neq 0} |p| a_p \oint d\alpha
\frac{e^{-im\theta(\alpha) +ip\alpha}}{im\theta'(\alpha)}
&= \frac{R}{\pi \xi_m \Delta} \sum_{p\neq 0} |p| a_p \frac{1}{ip} \oint
e^{-im\theta +ip\alpha} + \mbox{O}\left(\frac{1}{nq} \right)
\\
&= -\frac{1}{\pi} \sum_{p\neq 0} \oint e^{im\theta(\beta)-ip\beta}
d\beta \frac{1}{2\pi} \oint e^{im\theta(\alpha)-ip\alpha} d\alpha
\\
&= -\frac{1}{\pi} \oint d\alpha e^{-im\theta(\alpha)} \sum_{p\neq 0}
e^{ip\alpha} \frac{1}{2\pi} \oint e^{im\theta(\beta)-ip\beta} d\beta
\\
&= -\frac{1}{\pi} \oint e^{-im\theta(\alpha)} e^{im\theta(\alpha)} d\alpha
= - 2
\end{array}
\end{equation}
Hence using all of the above, and Eq. \ref{FdWV} for $\delta W_V$,
\begin{equation}
\Delta' = -2 \left\{ \frac{R^2}{|\xi_m|^2 \Delta^2} \sum_{p\neq 0}
|p| |a_p|^2 + 1 \right\}
= -2 \left( \frac{\delta W_V}{2\pi^2\frac{|\xi_m|^2}{R} \Delta^2}
\right) \left\{ 1 + \mbox{O} \left( \frac{1}{\delta W_V} \right)
\right\}
\end{equation}
(Later we will find that $\sum_{p\neq0} |p||a_p|^2 = m \frac{|\xi_m|^2
\Delta^2}{R^2}$ and hence that $\Delta' \simeq - 2 m$.)
\section{Evaluating the sum}\label{Sum1}
We have found that the vacuum energy $\delta W_V$ and $\Delta'$ are
both determined from $\sum_{p=-\infty}^{\infty} |p||a_p|^2$,
that we may in principle evaluate using our analytical expression for
$a_p$.
We do that here.
Firstly note that if $|p|$ is replaced with $p$ in Eq. \ref{FdWV},
then we may easily resum the series, because
\begin{equation}\label{exactsum}
\begin{array}{ll}
\sum_{p=-\infty}^{\infty} p |a_p|^2
&= \Delta^2 \frac{|\xi_m|^2}{R^2} \sum_{p=-\infty}^{\infty}
p \frac{1}{2\pi} \oint d\alpha \mbox{ } e^{im\theta (\alpha)-ip\alpha}
\frac{1}{2\pi}
\oint d\beta \mbox{ } e^{-im\theta (\beta) +ip\beta}
\\
&= \frac{\Delta^2}{2\pi} \frac{|\xi_m|^2}{R^2}
\oint d\beta \mbox{ } e^{-im\theta (\beta)} \sum_{p=-\infty}^{\infty}
e^{ip\beta}
\frac{1}{2\pi} \oint d\alpha \mbox{ } p \mbox{ } e^{im\theta(\alpha)-ip\alpha}
\\
&= \frac{\Delta^2}{2\pi} \frac{|\xi_m|^2}{R^2}
\oint d\beta \mbox{ } e^{-im\theta (\beta)} \mbox{ } \sum_{p=-\infty}^{\infty}
e^{ip\beta}
\frac{1}{2\pi} \oint d\alpha \mbox{ } m\theta'(\alpha)
e^{im\theta(\alpha)-ip\alpha}
\\
&= \frac{\Delta^2}{2\pi} \frac{|\xi_m|^2}{R^2} m
\oint d\beta \mbox{ } e^{-im\theta (\beta)} \theta'(\beta) e^{im\theta(\beta)}
= \Delta^2 \frac{|\xi_m|^2}{R^2} m
\end{array}
\end{equation}
where in going from lines $3$ to $4$ we integrated by parts, and in
going from lines $4$ to $5$ we note that $\theta'(\beta)
e^{im\theta(\beta)}= \sum_{p\neq 0} e^{ip\beta}
\frac{1}{2\pi} \oint \theta'(\alpha)
e^{im\theta(\alpha)} e^{-ip\alpha} d\alpha$.
We might expect the values of the coefficients to be peaked for values
of $p\sim m \sim nq \gg 1$, and so it is likely that
$\sum_{p=-\infty}^{\infty} |p||a_p|^2 \simeq
\sum_{p=1}^{\infty}p|a_p|^2 \simeq \sum_{p=-\infty}^{\infty}p|a_p|^2$.
This has been confirmed by calculating the sums using a saddle point
approximation.
(The details of the calculation are too long to be included here)
It is instructive to recalculate the result using a simple
model for $\theta (\alpha)$, that encapsulates the fact that as we
approach the
separatrix (with $q\rightarrow \infty$ and the local field line pitch
$\nu$ becoming increasingly peaked near the X-point), the function
$\theta(\alpha)$ becomes increasingly similar to a step function.
In this simple model we take $q \sim \frac{1}{\delta}$, $\delta \ll
1$, and $\nu \sim q \sim \frac{1}{\delta}$ when $\alpha \in
(-\delta,\delta)$, this leads to a
very simple model for $\theta(\alpha)$, with
\begin{equation}
\theta(\alpha)=
\left\{
\begin{array}{ll}
0 & \alpha \in (-\pi, -\delta)
\\
\left( \frac{\alpha+\delta}{2\delta} \right) 2\pi & \alpha \in
(-\delta, \delta )
\\
2\pi &\alpha \in (\delta, \pi)
\end{array}
\right.
\end{equation}
So that
\begin{equation}
e^{im\theta(\alpha)}=
\left\{
\begin{array}{ll}
1 & \alpha \in (-\pi, -\delta)
\\
e^{im\left( \frac{\alpha+\delta}{2\delta} \right) 2\pi} & \alpha \in
(-\delta, \delta )
\\
1 &\alpha \in (\delta, \pi)
\end{array}
\right.
\end{equation}
Because $\theta(\alpha)$ is piecewise linear, it is easy to evaluate
$\oint d\alpha e^{im\theta(\alpha) - ip\alpha}$, that gives
\begin{equation}
a_p = -i \frac{p}{|p|} \Delta \frac{\xi_m}{R} m
\frac{\sin(p\delta)}{p(p\delta-m\pi)}
\end{equation}
Hence
\begin{equation}
\begin{array}{ll}
\delta W_V &= 2\pi^2 R \sum_{p\neq 0} |p| |a_p|^2
\\
&= 2\pi^2 R \frac{|\xi_m|^2}{R^2} \Delta^2 m^2 \sum_{p\neq 0}
\frac{\sin^2(p\delta)}{|p|(p\delta-m\pi)^2}
\\
& \simeq \frac{2\pi^2}{R} |\xi_m|^2 \Delta^2 m^2 \left(
\int_1^{\infty} \frac{\sin^2(p\delta)}{|p|(p\delta-m\pi)^2} dp +
\int_{1}^{\infty} \frac{\sin^2(p\delta)}{|p|(p\delta+m\pi)^2} dp
\right)
\\
& \rightarrow \frac{2\pi^2}{R} |\xi_m|^2 \Delta^2 m^2 \frac{1}{m}
\mbox{ for } m \gg 1
\end{array}
\end{equation}
the same as was obtained previously by the saddle point
approximation.
To obtain this result we used
\begin{equation}
\sum_{p=1}^{\infty} \frac{\sin^2(p\delta)}{|p|(p\delta-m\pi)^2} \simeq
\int_1^{\infty} \frac{\sin^2(p\delta)}{|p|(p\delta - m\pi)^2} dp =
\frac{1}{m} + \mbox{O} \left( \frac{\ln (\delta)}{m^2} \right) =
\frac{1}{m} + \mbox{O} \left( \frac{\delta \ln (\delta)}{m} \right)
\end{equation}
and
\begin{equation}
\sum_{p=-\infty}^{-1} \frac{\sin^2(p\delta)}{|p|(p\delta-m\pi)^2} \simeq
\int_1^{\infty} \frac{\sin^2(p\delta)}{|p|(p\delta + m\pi)^2} dp =
\mbox{O} \left( \frac{\ln (\delta)}{m^2} \right)=
\mbox{O} \left( \frac{\delta \ln (\delta)}{m} \right)
\end{equation}
where we also used $m\sim nq \sim \frac{n}{\delta}$.
Notice that the integrals do not diverge at $p\delta-m\pi=0$, because
$\sin(p\delta)=\sin(p\delta-m\pi)=0$ for $p\delta-m\pi=0$, this would
not be the case if $m$ were not an integer.
Hence not only do we find agreement with the calculation using the
saddle point approximation, but we again find that
\begin{equation}
\sum_{p=-\infty}^{\infty} |p| |a_p|^2 \rightarrow
\sum_{p=-\infty}^{\infty} p|a_p|^2
\mbox{ as } m\sim nq \rightarrow \infty
\end{equation}
Therefore both methods suggest that provided $m\gg 1$ and $n\gg1$, then
\begin{equation}
\sum_{p=-\infty}^{\infty} |p| |a_p|^2 \rightarrow
\sum_{p=-\infty}^{\infty} p|a_p|^2 = \Delta^2 \frac{|\xi_m|^2}{R^2} m
\end{equation}
In addition notice
that for $m\sim nq \gg 1$ the result of neither approximation methods
involve $\delta$ (or $\epsilon$), suggesting that the result may be
generic and independent of the detailed form of $\theta(\alpha)$.
Returning to the calculation of $\Delta'$, Sections \ref{recalc} and
\ref{secDelta'} showed that at leading order
\begin{equation}
\Delta' = -2 \left( \frac{\delta W_V}{2\pi^2\frac{|\xi
|^2}{R}\Delta^2} \right)
\end{equation}
and that using the above results gives
\begin{equation}
\Delta' = - 2 m
\end{equation}
In addition, the work described above suggests that the result is generic
for perturbations with $n\gg 1$, regardless of whether the plasma
cross-section is circular, or shaped with a separatrix boundary that
contains an X-point.
\section{Discussion}\label{Conc1}
\subsection{Scope \& Purpose of the Calculation}
In the first part to this paper we started from the simplest model
used to study Peeling modes, that considers Peeling modes in a
cylindrical plasma at marginal stability, then generalised it to a
toroidal plasma.
According to the model, the energy principle's $\delta W$
is determined by the value of $\Delta'$, that is a normalised measure
of the jump in the gradient of the normal component of the perturbed
magnetic field.
In this second part we have restricted ourselves to systems for which
the vacuum magnetic field may be treated as being approximately two
dimensional, as is the case for a sufficiently large aspect ratio
Tokamak.
This allows us to use a conformal transformation in our calculations,
and at high toroidal mode number we have obtained analytic expressions
for the vacuum energy and $\Delta'$, whenever the plasma is perturbed
by a radial displacement consisting of a Fourier mode in straight
field line co-ordinates.
These expressions remain valid for a plasma cross-section that
approximates a separatrix with an X-point, and appear to be generic,
independent of the exact form for $\theta(\alpha)$.
Because it is possible to do this analytically, there is the
possibility of making similar analytic progress with other linear
plasma instabilities whose plasma equilibria have a separatrix with
an X-point.
Such calculations can provide physical understanding and useful tests
during the development of codes to study the stability of more general
geometry Tokamak plasmas, either giving confidence in a code or
indicating its limitations.
\subsection{(In)Stability of the Ideal MHD Peeling Mode?}
According to the model developed in part (I) and our calculation here
of $\Delta' = -2m$, we can now examine the model's predictions.
According to the model developed in part (I), for the trial function
used by Laval et al\cite{Laval}, stability is determined by the sign
of
\begin{equation}
\delta W = -2\pi^2 \frac{|\xi_m|^2}{R} \Delta
\left[ \Delta \Delta' + \hat{J} \right]
\end{equation}
with
\begin{equation}
\Delta=\frac{m-nq}{nq}
\end{equation}
\begin{equation}
\hat{J}= \frac{1}{2\pi} \oint dl \frac{I}{RB_p}
\frac{\vec{J}.\vec{B}}{B^2}
\end{equation}
\begin{equation}
\hat{\Delta}'= \left[ \left|
\frac{1}{2\pi} \oint dl R B_p \frac{I^2}{R^2B^2}
\frac{\frac{\partial}{\partial \psi} \nabla \psi
.\vec{B}_1}{\nabla \psi .\vec{B}_1} \right| \right]
\end{equation}
and as mentioned in Section \ref{D}, $R=R_0$ is constant for the large
aspect ratio limit considered here.
$\delta W$ is minimised with respect to $\Delta$ (or equivalently, a
particular choice of toroidal mode number), finding
$\Delta=-\hat{J}/(2\Delta')$.
Then using our result of $\Delta' = -2m$ we get
\begin{equation}\label{minimiseddW}
\delta W = -\left(\frac{\pi}{2}\right)^2 \frac{|\xi|^2}{R}
\left( \frac{\hat{J}^2}{m} \right)
\end{equation}
When checking the dimensions of $\delta W$ it is essential to remember
that $\xi_m = (\nabla \psi .\vec{\xi})_m \sim rRB_p $, then because
$B^2 \sim p$ is an energy per unit volume, $\delta W \sim r^2 RB_p^2$
and hence has units of energy.
For the process of minimisation $n$ was treated as a continuous
variable (that for $m\sim nq \gg 1$ is a reasonable approximation).
Now we consider two cases in turn, firstly $\nabla \phi . \vec{J}=0$,
for which
\begin{equation}
\hat{J} = \frac{1}{2\pi} \oint dl \frac{I}{RB_p} \frac{-I'B_p^2}{B^2}
= \frac{1}{2\pi} \oint dl B_p \frac{-II'}{RB^2}
\sim 1
\end{equation}
So that although $\delta W <0$ for all $m$, because $m\sim nq
\rightarrow \infty$ then $\delta W \sim \frac{1}{m} \rightarrow 0$.
For the case of $\nabla \phi . \vec{J} \neq 0$ however,
\begin{equation}\label{Jhatphi}
\hat{J} = \frac{1}{2\pi} \oint dl \frac{I}{RB_p} \frac{\vec{J}.\vec{B}}{B^2}
= \frac{1}{2\pi} \oint \frac{dl}{B_p} \frac{I}{R^2} R
\frac{\vec{J}.\vec{B}}{B^2}
\simeq \left\langle R \frac{\vec{J}.\vec{B}}{B^2} \right\rangle
\frac{1}{2\pi} \oint \frac{dl}{B_p} \frac{I}{R^2}
= q \left\langle R \frac{\vec{J}.\vec{B}}{B^2} \right\rangle
\end{equation}
where $\left\langle \right\rangle$ denotes the poloidal average, and
$q= \frac{1}{2\pi} \oint \frac{I}{R^2} \frac{dl}{B_p}$ is the safety
factor\cite{Laval}.
The divergence in $\frac{\nu}{B_p^2}$ at the X-point will mean that
the poloidal location of the X-point will affect the value of
$\hat{J}$ (as a function of $q$), this is also the case for Mercier
stability and is discussed by Webster\cite{Webster08}.
Then using Eqs. \ref{minimiseddW}, \ref{Jhatphi}, and $m\sim nq$ we have
\begin{equation}
\begin{array}{ll}
\delta W &\sim - \left( \frac{\pi}{2} \right)^2 \frac{|\xi_m|^2}{R}
\frac{q}{n}
\langle R \frac{\vec{J}.\vec{B}}{B^2} \rangle^2 < 0
\end{array}
\end{equation}
Therefore if $\delta W<0$ is taken to indicate instability, the
result would indicate that the Peeling mode remains unstable near a
separatrix.
However as observed in the first part of this paper the growth rate
$\gamma$ has
$\gamma^2 = - \delta W /\int \vec{dr} \rho_0 |\xi |^2$, with $\int
\vec{dr} \rho_0 |\xi |^2$ diverging at a rate proportional to
$q'(\psi)$.
This gives $\ln(\gamma)= -\frac{1}{2} \ln (q'/q)$ for the limit of a
separatrix with $q$ and $q'$ tending to infinity, so that although
$\delta W$ is non-zero and negative, the mode will be marginally
stable.
This is similar to the calculation of the Mercier coefficient by
Webster\cite{Webster08}, where $D_M$ is found from the ratio of two
diverging quantities, with $D_M\sim q/q'\rightarrow 0$ as the
separatrix is approached.
Note that these results for $\Delta'$ and the growth rate appear to be
generic and independent of the detailed forms of $\theta(\alpha)$ or
the radial structure of the mode.
\subsection{Is the Mode Physically Acceptable?}
Because we require a poloidal mode number $m\sim nq$, then near the
separatrix where $q \rightarrow \infty$ we also require $m\rightarrow
\infty$.
This raises the question: Is the mode physically acceptable?
To answer this we reconsider the trial function, that has $\xi \sim
e^{im\theta(\alpha)}$, with
$\theta(\alpha)=\frac{1}{q}\int^{\alpha} \nu(\alpha) d\alpha$.
When $m\sim nq$ the trial function becomes
$\xi \sim e^{in\int^{\alpha} \nu d\alpha}$, from which we can see
that because $\nu$ is of order one and well behaved everywhere except
near the x-point (where it diverges to infinity), then so also is the
mode's structure.
As discussed in part (I),
the divergence in $m\sim nq \rightarrow \infty$ is
only manifested in close proximity to the x-point where the divergence
in $\nu$ causes the mode to oscillate increasingly
rapidly as the x-point is approached.
Elsewhere $\nu$ is typically of order $1$, and the mode structure is
like that for a finite
mode number, oscillating at a modest rate of order $n\nu \ll nq$, and
only weakly affected by the proximity of the flux surface to the
separatrix.
Hence the mode has a simple structure everywhere except for a region
close to the x-point where it oscillates so rapidly that MHD would no
longer be applicable.
The resulting mode structure is consistent with
observations of ELMs\cite{Kirk}, that show filamentary structures
that follow the magnetic field lines, and whose poloidal structure
near the X-point is difficult to determine.
\subsection{Previous Analytical Work}\label{analytical1}
As mentioned at the outset, Laval et al\cite{Laval}
considered a trial function consisting of a single Fourier mode in a
straight field line co-ordinate, that is resonant at a rational
surface in the vacuum just outside the plasma's surface.
For that trial function, they found that
for a positive non-zero current at the plasma edge, $\delta
W <0$, and suggested therefore that the Peeling mode would be
unstable for a non-zero positive current at the plasma's edge.
On the basis of the sign of $\delta W$ our study also finds this, but
our study also suggests that the growth rate will asymptote to
zero as the
outermost flux surface approximates a separatrix, so that the mode will
be marginally stable.
Lortz\cite{Lortz} also considered Peeling mode stability, in toroidal
plasmas with shaped cross-sections, using a systematic calculation
with a trial function whose resonant surface is inside the plasma.
An advantage of the calculation by Lortz\cite{Lortz}, is that the
radial structure of the mode is considered.
An unfortunate complication for this discussion is that in the
ordering scheme of Lortz\cite{Lortz}, the vacuum energy can be
neglected.
This was not the case for our calculation in Part (I)\cite{part1}, or
of Laval et al\cite{Laval}.
Nonetheless, we will consider the predictions of this calculation in
the limit where the outermost flux surface approximates a separatrix.
Connor et al\cite{Connor} review the calculation of Lortz\cite{Lortz},
and use it to consider trial functions with resonant surfaces both
inside and outside the plasma.
They find that stability of the Peeling mode
requires\cite{Connor}
\begin{equation}\label{HastieStab}
1-4D_M > \left( 2 \frac{S}{P} - 1 \right)^2
\end{equation}
where the Mercier co-efficient $D_M \equiv -Q/P$, and $P$, $Q$, $S$
are defined as,
\begin{equation}
\begin{array}{c}
P = 2\pi (q')^2 \left[
\oint \frac{J_{\chi}B^2}{R^2B_p^2} d\chi
\right]^{-1}
\\
Q = \frac{p'}{2\pi}
\oint \frac{\partial J_{\chi}}{\partial \psi} d\chi
- \frac{(p')^2}{2\pi}
\oint \frac{J_{\chi}}{B_p^2} d\chi
+ Ip' \oint \frac{J_{\chi}}{R^2B_p^2} d\chi
\left[
\oint \frac{J_{\chi}B^2}{R^2B_p^2} d\chi
\right]^{-1}
\times \left[
\frac{Ip'}{2\pi} \oint \frac{J_{\chi}B^2}{R^2B_p^2} - q'
\right]
\\
S=P + q' \oint \frac{j_{\parallel} B}{R^2B_p^2} J_{\chi} d\chi \left[
\oint \frac{J_{\chi} B^2}{R^2 B_p^2} d\chi
\right]^{-1}
\end{array}
\end{equation}
Substituting $D_M \equiv - Q/P$ into Eq. \ref{HastieStab}, gives the
stability requirement
\begin{equation}\label{SCon1}
\frac{Q}{P} - \left( \frac{S}{P} \right)^2 + \frac{S}{P} > 0
\end{equation}
Substituting for $P$, $Q$, and $S$, allows Eq. \ref{SCon1} to be
simplified to
\begin{equation}\label{SCon2}
p' \left[
\frac{\partial }{\partial \psi}
\frac{1}{2\pi} \oint J_{\chi} d\chi
- p' \frac{1}{2\pi}
\oint \frac{J_{\chi}}{B_p^2} d\chi \right]
+I' \left[
q' -
2p' \frac{1}{2\pi} \oint \frac{\nu}{B_p^2} d\chi
- I' \frac{1}{2\pi} \oint \frac{J_{\chi}B^2}{R^2B_p^2}
\right]>0
\end{equation}
This may be simplified further by noting that because
$\nu=IJ_{\chi}/R^2$, and in a large aspect ratio ordering where
$R$ is taken as approximately constant,
\begin{equation}
\frac{\partial}{\partial \psi} \frac{1}{2\pi} \oint J_{\chi} d\chi
= \frac{\partial}{\partial \psi} \frac{1}{2\pi} \oint \frac{\nu R^2}{I} d\chi
= \frac{R^2}{I} \frac{1}{2\pi} \oint \frac{\partial \nu}{\partial
\psi} d\chi
- R^2 \frac{I'}{I^2} \frac{1}{2\pi} \oint \nu d\chi
\end{equation}
and
\begin{equation}
q' = \frac{1}{2\pi} \oint \frac{\partial \nu}{\partial \psi} d\chi
\end{equation}
The Grad-Shafranov equation in $\psi$, $\chi$, $\phi$
co-ordinates, has
\begin{equation}\label{dnudpsi}
\frac{\partial \nu}{\partial \psi} = \frac{\nu}{B_p^2} \left\{
- \frac{\partial}{\partial \psi} \left( p+ B^2 \right) +
\frac{R^2B^2}{I} \frac{\partial}{\partial \psi} \left( \frac{I}{R^2}
\right)
\right\}
\end{equation}
Therefore if $R$ is taken as approximately constant (as would be the
case either
in a large aspect ratio limit or if we are sufficiently close to the
separatrix that the integral is dominated by the divergence at the
X-point and $R$ may be approximated by its value $R=R_X$ there), then
using Eq. \ref{dnudpsi}, Eq. \ref{SCon2} simplifies to a condition
for stability of
\begin{equation}
\begin{array}{ll}
0&< \frac{1}{2\pi} \oint \frac{\nu}{B_p^2} \frac{I}{R^2} d\chi \left\{
\left( -p' - \frac{II'}{R^2} \right) \frac{\partial}{\partial \psi}
\left( 2p + B^2 \right) \right\}
\\&=
\left( \nabla \phi . \vec{J} \right) I \frac{1}{2\pi}
\oint \frac{\nu}{R^2B_p^2}
\frac{\partial}{\partial \psi} \left( 2p + B^2 \right) d\chi
\end{array}
\end{equation}
This may be simplified further still, to
\begin{equation}
0< \frac{1}{2\pi}
\oint \frac{\nu}{B_p^2} \frac{I}{R^2}
\left( - \nabla \phi . \vec{J} \right)
\left[ 2 \left( \nabla \phi . \vec{J} \right)
- \frac{\partial B_p^2}{\partial \psi} \right]d\chi
\end{equation}
Because the integrals are dominated by the divergence of $\nu/B_p^2$
at the X-point and $\frac{\partial B_p^2}{\partial \psi} <0$
at the X-point, then based on the formulation of
Lortz\cite{Lortz,Connor}, then
provided $\nabla \phi . \vec{J}>0$ the negative
expression clearly indicates instability to the Peeling mode.
However, if we allow $\nabla \phi . \vec{J}$ to be negative, then the
formulation of Lortz\cite{Lortz,Connor} also suggests that
stability is possible provided
\begin{equation}
0 <
\frac{1}{2\pi} \oint \frac{\nu}{B_p^2} \left[ -2\left| \nabla \phi
. \vec{J} \right| - \frac{\partial B_p^2}{\partial \psi} \right]
d\chi
\end{equation}
Therefore in principle there is a range of negative current values at
the plasma edge for which the Peeling mode is stable.
The appendix calculates $\partial B_p^2/\partial \psi$ near the
X-point for a standard and a ``snowflake''\cite{Ryutov} divertor.
Interestingly, whereas a conventional X-point has a range of negative
current values for which the Peeling mode is stable, in the limit of
an exact ``snowflake'' X-point (with flux surfaces meeting at an angle
of $\pi/3$), the range of values of negative current for which the
Peeling mode is stable, tends to zero.
Whether this observation will have consequences for the plasma
behaviour in a ``snowflake'' X-point geometry remains to be seen, but
it is a qualitative difference between a conventional X-point and that
produced with a ``snowflake'' divertor.
As mentioned previously, the calculation
also considered the radial structure of the mode, with $\xi \sim
x^{\lambda_{\pm}}$, $x$ a radial co-ordinate, and
\begin{equation}
\lambda_{\pm} = - \frac{1}{2} \pm \sqrt{ \frac{1}{4} + \frac{Q}{P} }
\end{equation}
As we approach the separatrix, Webster\cite{Webster08} shows that
$D_M=-\frac{Q}{P} \rightarrow 0$, giving
\begin{equation}
\begin{array}{l}
\lambda_{+} \simeq \frac{Q}{P} \rightarrow 0
\\
\lambda_{-} \simeq -1
\end{array}
\end{equation}
and mode structures of $\xi_{-} \sim \frac{1}{x}$ and $\xi_{+} \sim x^{Q/P}$.
For the perturbations to satisfy the boundary condition of a mode
amplitude that tends to zero in the plasma, this requires us to use
$\xi_{-}$ for resonances outside the plasma (the ``external'' Peeling
mode), and $\xi_{+}$ for resonances inside the plasma (the
``internal'' Peeling mode).
It should be noted that a
potential problem with the analysis of Lortz et al\cite{Lortz} when
applied to Peeling modes, that the mode is taken to be sufficiently
localised that the equilibrium quantities (that include $q$ and $q'$),
are approximately constant.
This is almost certainly not the case near a separatrix.
\subsection{Summary}
We have started from a simple model for the Peeling mode, at
marginal stability in cylindrical geometry, and in Part (I) of
this paper generalised it to toroidal Tokamak geometry.
A conclusion of Part (I) is that Peeling mode
stability is determined by the value of $\Delta'$, a normalised
measure of the discontinuity in the gradient of the normal component
of the perturbed magnetic field at the plasma-vacuum boundary.
Therefore this paper evaluated $\Delta'$ in a large aspect
ratio Tokamak geometry with a separatrix and X-point, but in such a
way that the effect of the X-point is captured exactly, without
encountering the usual discretisation errors present in most numerical
methods.
This was possible by generalising the method of conformal
transformations beyond textbook presentations, that require a boundary
condition of either the function or its normal derivative to be zero.
Here we observe that even if the field's normal derivative is
non-zero at the boundary, it is still possible to use the conformal
transformation method.
In this case instead of obtaining an exact analytic solution (as would
be the case if its normal derivative were zero on the boundary), the
2-dimensional problem is reduced to a 1-dimensional problem that may
subsequently be solved exactly or approximated.
The approach avoids the errors that may arise due to the
discretisation of space near an X-point, that are necessarily
present in most numerical methods.
This paper also calculated analytical expressions
for physically realistic examples of the equilibrium vacuum magnetic
field, and the straight field line angle at the plasma-vacuum
boundary.
These and other results are likely to find opportunities for
application elsewhere.
It is found that a radial plasma perturbation
consisting of a single Fourier mode in straight field line
co-ordinates with a high toroidal mode number $n$, in a plasma
equilibrium with a separatrix and an x-point, will produce the same
change in the vacuum energy as the equivalent perturbation in a
cylindrical equilibrium, with $\delta W_V = 2\pi^2 \frac{|\xi_m|^2}{R}
\Delta^2 m$.
It also results in the same value for $\Delta'$, with $\Delta'=-2m$,
where $m$ is the poloidal mode number.
Despite our trial function requiring $m\sim nq \rightarrow \infty$, we
observe that the trial function has $\xi \sim e^{in \int^{\chi} \nu
d\chi}$ that is physically well behaved for all but a highly
localised region near the X-point where MHD will fail to apply.
Therefore we believe the trial function is physically acceptable even
for a separatrix boundary.
Previous work by Lortz\cite{Lortz} and Connor et al\cite{Connor} was
considered for an outer flux surface that tends to a
separatrix with an X-point.
Like Laval et al\cite{Laval} their work predicts the Peeling mode to be
unstable if there is a positive current at the plasma's edge, and it
also finds a well behaved radial structure for the mode.
Interestingly, for a conventional X-point there is predicted to be a
range of small but negative edge-current for which the Peeling mode is
stable, but in the limit of an exact snowflake divertor this range
shrinks to zero size - a qualitative difference between a
conventional and a snowflake divertor.
A limitation of the Lortz calculation is that it
approximates the equilibrium quantities as constant on the length
scale of the plasma instability, this is not necessarily the case for
$q$ or $q'$ near a separatrix, and therefore the results
when applied to a separatrix case should be
treated with caution.
Likewise, as noted in Part (I), there are potential limitations to the
high-$n$ ordering form of $\delta W$ used here, and this should be
investigated in future work.
Thus we have developed a simple model for the Peeling mode, and found
that despite $\delta W <0$, the growth rate $\gamma$ tends to
zero as the outermost flux surface tends to a separatrix with an
X-point.
As the outermost flux surface approaches a separatrix, the growth rate
falls with $\ln(\gamma/\gamma_A) = - \frac{1}{2} \ln (q'/q)$; this has
subsequently been confirmed with ELITE (S. Saarelma, private
communication), leading us to believe that the effect of a separatrix
on the high toroidal mode number ideal MHD model is now understood.
The ideal MHD prediction of marginal
stability at the separatrix means that other non-ideal terms
such as resistivity, non-linear terms, or terms neglected in the
high-$n$ analysis, will play a role in determining the
eventual stability.
In general it is hoped that
the methods and results contained in this paper will provide new
tools for studying plasmas in separatrix geometries, and have
potential applications in future studies of plasma stability and more
generally outside of plasma physics.
\begin{acknowledgments}
Thanks to Jack Connor for suggesting that a conformal
transformation might help, for suggesting the method of a saddle-point
approximation, and for many helpful discussions and suggestions.
Thanks to Jim Hastie and Chris Gimblett, for helpful discussions and
encouragement.
Thanks to A. Thyagaraja for pointing me to
Milne-Thompson's book (where I encountered the Karman-Trefftz
transformation).
Thanks to Samuli Saarelma for calculations with ELITE, and Tim Hender
for reading and commenting on this paper.
Thanks to D.D. Ryutov for questioning how stability might be different
with a snowflake divertor, and directing me to Ref. \cite{Ryutov}.
This work was jointly funded by the United Kingdom Engineering and
Physical Sciences Research Council, and by the European Community under
the contract of Association between EURATOM and UKAEA. The views and
opinions expressed herein do not necessarily reflect those of the
European Commission.
\end{acknowledgments}
\section{$w'(z)$ near the X-point}\label{w'xpt}
To obtain $w'(z)$ we differentiate both sides of Eq.
\ref{KTtransform} with
respect to $z$, and rearrange the resulting expression to get
\begin{equation}
w'(z) = \left( w(z) + nl \right)^2
\frac{ \left( z-l \right)^{n-1} }{ \left( z+ l \right)^{n+1} }
\end{equation}
and hence
\begin{equation}\label{w'a1}
\left| w'(z) \right|^2 = \left| w(z) +nl \right|^4
\frac{ \left| z- l \right|^{2(n-1)} }{\left| z+ l \right|^{2(n+1)}}
\end{equation}
We have deliberately obtained an implicit expression for $
\left|w'(z)\right|^2$, with $w'(z)$ given in terms of $w(z)$.
This is because one cannot simply expand $w'(z)$ in powers of
$\epsilon$, because the expansion will give the incorrect answer as
$\epsilon \rightarrow 0$ compared with the exact result for $\epsilon$
small but non-zero (i.e. $\epsilon \neq 0$ is a ``singular
perturbation'').
Instead by obtaining $|w'(z)|^2$ implicitly in the form given by
Eq. \ref{w'a1}, we need solely be careful with the term
$|z-l|^{2(n-1)}$, because $|w+nl|^4$ and $|z+l|^{2(n+1)}$ are well
behaved when expanded in $\epsilon$, as $\epsilon \rightarrow 0$.
Near the X-point,
\begin{equation}
\begin{array}{l}
\left| w(z) + nl \right|^4 = (2nl)^4 + \mbox{O} \left( \epsilon \right)
\\
\left| z + l \right|^{2(n+1)} = (2l)^{2(n+1)} + \mbox{O} \left(
\epsilon \right)
\end{array}
\end{equation}
We need to be more careful with $|z-l|^{2(n-1)}$, that with $z=-a +
(a+l-\epsilon)e^{i\alpha}$ gives
\begin{equation}
\begin{array}{ll}
\left| z - l \right|^2 &=
\left( (a+l-\epsilon)\cos(\alpha) -a + l \right)^2 + (a+l+\epsilon)^2
\sin^2(\alpha)
\\
&= 2(a+l)(a+l-\epsilon) \left[ 1 - \cos(\alpha) +
\frac{\epsilon^2}{2(a+l)(a+l-\epsilon)} \right]
\\
& \simeq 2a^2 \left[ 1 - \cos(\alpha) + \frac{\epsilon^2}{2a^2} \right]
\end{array}
\end{equation}
where we have retained the term that provides the singular
perturbation that prevents $(z-l)^2$ becoming zero for $\epsilon \neq
0$, but neglected all the lower order terms that modify the answer by
of order $\epsilon/a$ and $l/a$. In principle some plasma
cross-sections might require the retention of terms of order $l/a$,
but here we neglect them so as to keep algebraic details to a minimum.
Therefore at leading order we have
\begin{equation}
\left| w'(z) \right|^2
\simeq
\frac{(2nl)^4}{(2l)^{2(n+1)}}
(2a^2)^{(n-1)}
\left( 1 - \cos ( \alpha ) + \frac{\epsilon^2}{2a^2} \right)^{(n-1)}
\end{equation}
that for the case we are most interested in here with $n=3/2$
(corresponding to an X-point with a $\pi/2$ interior angle), we have
\begin{equation}
\left| w'(z) \right|^2
\simeq
\frac{(3l)^4}{(2l)^5}
\left( 2a^2 \right)^{1/2}
\left( 1-\cos (\alpha ) + \frac{\epsilon^2}{2a^2} \right)^{1/2}
= \frac{1}{\sqrt{2}}
\left( \frac{3}{2} \right)^4
\left( \frac{a}{l} \right)
\sqrt{ 1 -\cos (\alpha ) + \frac{\epsilon^2}{2a^2} }
\end{equation}
which is Eq. \ref{w'zsq}.
\section{Some identities involving complex numbers}\label{complexidentities}
In the following we write $a=a_x+ia_y$, $b=b_x+ib_y$, $c=c_x+ic_y$,
and $d=d_x+id_y$, and remind the reader that the dot product refers to
the sum of, the product of the real parts plus the product of the
imaginary parts. For example $a.b=a_xb_x+a_yb_y$. Multiplication is as
usual for complex numbers, for example $ab=a_xb_x-a_yb_y + i (a_xb_y +
a_y b_x)$. Then we find
\begin{equation}
\begin{array}{ll}
ab . cd &= \left[ (a_xb_x -a_yb_y)+i(a_xb_y+a_yb_x) \right] .
\left[ (c_xd_x - c_yd_y)+i(c_xd_y +c_yd_x) \right]
\\
&= (a_xb_x -a_yb_y)(c_xd_x - c_yd_y) +
(a_xb_y +a_yb_x)(c_xd_y + c_yd_x)
\\
&= a_x \left[ b_x(c_xd_x-c_yd_y) + b_y(c_xd_y+c_yd_x) \right] +
a_y \left[ b_x(c_xd_y+c_yd_x) -b_y(c_xd_x-c_yd_y) \right]
\\
&= a_x \left[ d_x(b_xc_x+b_yc_y) + d_y(b_yc_x-c_yb_x) \right] +
a_y \left[ d_y(b_xc_x+b_yc_y) + d_x(b_xc_y-b_yc_x) \right]
\\
&= (a_xd_x+a_yd_y)(b_xc_x+b_yc_y) +
(a_xd_y-a_yd_x)(b_yc_x-c_yb_x)
\\
&= (a.d)(b.c) + (ia.d)(ic.b)
\end{array}
\end{equation}
and similarly for $a.bc$,
\begin{equation}
\begin{array}{ll}
a.bc &=
a.\left( ( b_xc_x-b_yc_y ) + i (b_xc_y+b_yc_x) \right)
\\&=
a_xb_xc_x - a_xb_yc_y + a_yb_xc_y + a_yb_yc_x
\\&=
c_x (a_xb_x +a_yb_y) + c_y (a_yb_x - a_xb_y )
\\&=
(c_x + i c_y) . \left( (a_xb_x+a_yb_y) + i (a_yb_x-a_xb_y) \right)
\\&=
(c_x + i c_y) . \left( a_x (b_x-ib_y) + a_y (b_y+ib_x) \right)
\\&=
(c_x + i c_y) . \left( a_x (b_x-ib_y) - ia_y (-b_x+ib_y) \right)
\\&=
c . \left( (a_x+ia_y) (b_x-ib_y) \right)
\\&=
c . a\bar{b} = a\bar{b} . c
\end{array}
\end{equation}
\section{2nd term is order $\frac{1}{m}$ smaller than $\delta
W_V$}\label{app1}
Here it is shown that the second term in Eq. \ref{dWV4}, here written
as $\delta W_G$, is of order $1/m$ smaller than $\delta W_V$.
The term we are interested in is
\begin{equation}\label{dWG}
\delta W_G = \pi \Delta \left( \frac{m}{nq} \right) \xi_m^* \oint
d\alpha \frac{r_aI}{B^2} e^{-im\theta(\alpha) +in\phi}
\frac{i}{n} \left[
\left( n_z . B_z \right) n_z . \nabla_z \left(
\frac{\left|B_{pz}\right|}{\left|w'(z)\right|^2} \right)
\right]
\end{equation}
Using Eq. \ref{w'zsq} and Eq. \ref{Bpzalpha} we get
\begin{equation}\label{A4}
\begin{array}{ll}
n_z.\nabla_z \left( \frac{\left| B_{pz} \right| }{\left| w'(z)
\right|^2} \right)
&= -B_{p0} \frac{\partial}{\partial \epsilon} \left[ 2\sqrt{2} \left(
\frac{2}{3} \right)^4 \frac{l}{a} \sqrt{ 1 - \cos(\alpha) +
\epsilon^2/2a^2} \right]
\\
&= -B_{p0} \left( \frac{\epsilon}{a^2} \right)\sqrt{2} \left(
\frac{2}{3} \right)^4
\left( \frac{l}{a} \right)
\frac{1}{\sqrt{ 1 -\cos(\alpha) + \epsilon^2/2a^2}}
\\
&= - \left( \frac{\epsilon}{a^2} \right) c_a \frac{R^2B_{p0}^2}{Ir_a}
q\frac{\partial \theta}{\partial \alpha}
\end{array}
\end{equation}
where $c_a=\left[2\sqrt{2}\left(\frac{2}{3}\right)^4 \left(
\frac{l}{a} \right)\right]^2$ and is a constant.
Substituting the above Eq. \ref {A4} into Eq. \ref{dWG}, gives
\begin{equation}
\delta W_G = \pi \Delta \left( \frac{m}{nq} \right) \xi_m^*
\left( \frac{-\epsilon}{a^2} \right) c_a \frac{iq}{n}
\oint d\alpha
\frac{R^2B_{p0}^2}{B^2} e^{-im\theta(\alpha) +in\phi}
\left[
n_z . B_z \frac{\partial \theta}{\partial \alpha}
\right]
\end{equation}
Integrating by parts then gives
\begin{equation}\label{e5}
\delta W_G = \pi \Delta \left( \frac{m}{nq} \right) \xi_m^*
\left( \frac{-\epsilon}{a^2} \right) c_a \frac{q}{nm}
\oint d\alpha
\frac{R^2B_{p0}^2}{B^2} e^{-im\theta(\alpha) +in\phi}
\left[
\frac{\partial n_z . B_z }{\partial \alpha}
\right]
\end{equation}
Using $n_z.B_z=\frac{\partial V}{\partial r}$ and Eq. \ref{Vvac}
we get
\begin{equation}
\left. \frac{\partial}{\partial \alpha} \left( n_z . B_z \right)
\right|_{r_a} =
\sum_p \frac{-ip|p|a_p}{r_a} e^{ip\alpha -in\phi}
\end{equation}
Substituting this into \ref{e5} we get
\begin{equation}\label{e6}
\delta W_G = \pi \Delta \left( \frac{m}{nq} \right) \xi_m^*
\left( \frac{-\epsilon}{a^2} \right) c_a \frac{q}{nm} \frac{R^2B_{p0}^2}{B^2}
\sum_{p\neq0} \frac{-a_p|p|ip }{r_a}
\oint d\alpha \mbox{ }
e^{-im\theta(\alpha) +in\phi}
\end{equation}
Where the poloidal dependence in $\frac{R^2}{B^2}$ has been neglected
due to the large aspect ratio Tokamak ordering.
Using Eq. \ref{ap} we get
\begin{equation}
\oint e^{-im\theta +ip\alpha} d\alpha = -i a_p^* \frac{|p|}{p}
\frac{R}{\Delta \xi_m^*} 2\pi
\end{equation}
which may be inserted into \ref{e6}, and after some cancellations gives
\begin{equation}\label{e7}
\delta W_G = 2\pi^2R
\left( \frac{\epsilon}{a^2} \right) c_a \frac{R^2B_{p0}^2}{B^2} \frac{1}{n^2}
\frac{1}{r_a} \sum_{p\neq0} |p|^2 \left| a_p \right|^2
\end{equation}
The sum $\sum_{p\neq0} |p|^2 \left| a_p \right|^2$ may be evaluated
arbitrarily accurately, as is indicated next. We use the alternative
expression for $a_p$ given by Eq. \ref{ap1}, of
\begin{equation}
a_p = - \frac{\Delta}{|p|} \frac{\xi_m}{R} \frac{1}{2\pi} \oint im
\theta'(\alpha) e^{im\theta -ip\alpha} d\alpha
\end{equation}
which gives
\begin{equation}
\begin{array}{ll}
\sum_p |p|^2 \left|a_p \right|^2
& = \Delta^2 \frac{\left| \xi_m \right|^2}{R^2} \sum_{p\neq0}
\left( \frac{1}{2\pi} \oint d\beta \mbox{ } m \theta'(\beta)
e^{-im\theta(\beta)+ip\beta} \right)
\times
\\
& \left( \frac{1}{2\pi} \oint d\alpha \mbox{ } m \theta'(\alpha)
e^{im\theta(\alpha)-ip\alpha} \right)
\\
& = \Delta^2 \frac{\left| \xi_m \right|^2}{R^2}
\frac{m^2}{2\pi}
\oint d\beta \mbox{ } \theta'(\beta) e^{-im\theta(\beta)}
\sum_{p\neq 0} e^{+ip\beta}
\frac{1}{2\pi} \oint d\alpha \mbox{ } \theta'(\alpha)
e^{im\theta(\alpha)}e^{-ip\alpha}
\\
& = \Delta^2 \frac{\left| \xi_m \right|^2}{R^2}
\frac{m^2}{2\pi}
\oint \theta'(\beta) e^{-im\theta(\beta)} \theta'(\beta)
e^{im\theta(\beta)} d\beta
\\
& = \Delta^2 \frac{\left| \xi_m \right|^2}{R^2}
\frac{m^2}{2\pi}
\oint \left( \theta'(\beta) \right)^2 d\beta
\end{array}
\end{equation}
where in the penultimate line, we resum the Fourier series by noting
that $\theta'(\beta) e^{im\theta(\beta)}=
\sum_{p\neq 0} e^{ip\beta} \frac{1}{2\pi} \oint
e^{-ip\alpha} \theta'(\alpha) e^{im\theta(\alpha)}$.
We
approximate $\oint \left( \theta'(\beta) \right)^2 d\beta$, using the
analytic expression for $\theta(\beta)$ obtained from
Eqs. \ref{theta1}, \ref{Bpzalpha}, and \ref{w'zsq}, obtaining
\begin{equation}
\oint \left( \theta'(\beta) \right)^2 d\beta
= \frac{1}{q^2} \frac{1}{c_a} \frac{I^2r_a^2}{R^4B_{p0}^2} \oint
\frac{d\beta}{1-\cos(\beta) + \epsilon^2/2a^2}
\simeq \frac{1}{q^2} \frac{1}{c_a} \frac{I^2r_a^2}{R^4B_{p0}^2}
\frac{2\pi a}{\epsilon }
\end{equation}
Hence taking $r_a=a+l \simeq a$ we get
\begin{equation}
\begin{array}{ll}
\delta W_G &\simeq \left( 2\pi^2 R \frac{\epsilon}{a^2} c_a
\frac{R^2B_{p0}^2}{B^2}
\frac{1}{n^2} \right)
\left( \Delta^2
\frac{\left| \xi_m \right|^2}{R^2} \frac{m^2}{2\pi} \right)
\left(
\frac{I^2r_a^2}{R^4B_{p0}^2} \frac{1}{c_a} \frac{1}{q^2} \frac{2\pi
a}{\epsilon}
\right)
\\
&= 2\pi^2 \frac{\left| \xi_m \right|^2}{R} \Delta^2
= \left( \frac{1}{m} \right) \delta W_V
\end{array}
\end{equation}
where the last line uses the expression for $\delta W_V \sim \Delta^2
m$ obtained from the main text. Hence, we may neglect this term
compared with $\delta W_V$.
\section{Modifications to Peeling Mode Stability for a ``Snowflake''
divertor}
The integrals that determine the stability of Peeling modes, are
dominated by the divergence in $\nu/B_p^2$ that occurs near the
X-point in the separatrix.
This allows the integrals to be estimated by expanding the poloidal
flux functions in the vicinity of the X-point, as has been done in
Ref. \cite{Ryutov} for both a standard X-point and a ``snowflake''
divertor's X-point.
Near a standard X-point, Eq. 15 of Ref. \cite{Ryutov} gives
\begin{equation}\label{phiX}
\Phi = \left( \frac{\tilde{B}}{L} \right) \left( \frac{x^2-z^2}{2}
\right)
\end{equation}
with $\tilde{B}$ having the poloidal field's dimensions and $L$ a
length scale,
and the poloidal magnetic field given by\cite{Ryutov}
$B_x=-\frac{\partial \Phi}{\partial z}$ and $B_z=\frac{\partial
\Phi}{\partial x}$, leading to
\begin{equation}
B_p^2 = \frac{\tilde{B}^2}{L^2} \left( x^2 + z^2 \right)
\end{equation}
Note that near the X-point, $\nabla \psi \rightarrow R_X \nabla
\Phi$, with $R_X$ the major radius at the X-point.
Therefore near the X-point $\frac{\partial B_p^2}{\partial \psi} =
\frac{1}{R_X} \frac{\partial B_p^2}{\partial \Phi}$, and we
calculate $\partial B_p^2/\partial \psi$ from
\begin{equation}
\begin{array}{ll}
\frac{\partial B_p^2}{\partial \Phi} &=
\frac{\nabla \Phi . \nabla \left( B_p^2 \right)}{\left| \nabla \Phi
\right|^2}
\\
&= \left( \frac{\tilde{B}}{L} \right) 2 \frac{(x^2-z^2)}{(x^2+z^2)}
\end{array}
\end{equation}
and use Eq. \ref{phiX} to substitute for $x^2=2 \left( \frac{\Phi
L}{\tilde{B}} \right) + 2 z^2$, giving $\partial B_p^2/\partial
\Phi$ at constant $\Phi$ as
\begin{equation}
\frac{\partial B_p^2}{\partial \Phi} = 2 \left(
\frac{\tilde{B}}{L} \right)
\frac{\left( \Phi L/\tilde{B} \right) }{
\left( z^2 + \left( \Phi L/\tilde{B} \right) \right)}
\end{equation}
and similarly
$B_p^2 = \left( \tilde{B}^2/L^2 \right) 2
\left( z^2 + \Phi L/\tilde{B} \right)$.
The minimum value of $z$ is at $x=0$, giving
$z_{min}^2=-2\Phi L/\tilde{B}$, and
\begin{equation}
\left.
\frac{\partial B_p^2}{\partial \Phi} \right|_{z_{min}}
= -2 \left( \frac{\tilde{B}}{L} \right)
\end{equation}
For a ``snowflake'' divertor, Eq. (2) of Ref. \cite{Ryutov} gives
\begin{equation}
\Phi = \frac{AI}{c} \left( x^2 z - \frac{z^3}{3} \right)
\end{equation}
with I the plasma current, A has dimensions of the inverse cube of the
length scale over which the poloidal magnetic field varies near an
X-point, and c is the speed of light.
In a similar way to the calculation for the standard X-point, this
leads to
\begin{equation}
\frac{\partial B_p^2}{\partial \Phi}
= \left( \frac{AI}{ c} \right)
\frac{ 4z \left( \frac{c\Phi}{AI} \right)}{
\left( \frac{c\Phi}{AI} \right) + z^3}
\end{equation}
\begin{equation}
B_p^2 = \left( \frac{A^2I^2}{c^2} \right)
\left( \frac{4}{3} z^2 + \left( \frac{c\Phi}{AI} \right) \frac{1}{z}
\right)^2
\end{equation}
with $z_{min}^3 = - 3 \Phi c/AI $, giving
\begin{equation}
\left.
\frac{\partial B_p^2}{\partial \Phi}
\right|_{z_{min}}
=
\left( \frac{AI}{c} \right) \frac{4}{3}
\left( \frac{c\Phi}{AI} \right)^{1/3}
\end{equation}
that tends to zero as we approach a separatrix.
Therefore, whereas a conventional X-point leads to a Peeling mode
stability boundary for which the Peeling mode can be stable for a
range of small but non-zero negative current at the plasma's edge,
this window of stability tends to zero size for a snowflake
divertor.
Finally, for a ``snowflake plus'' divertor\cite{Ryutov}, a similar
calculation using the Eqs. of Ref. \cite{Ryutov} finds $\partial
B_p^2/\partial \Phi \sim \sqrt{\frac{I-I_{d0}}{I_{d0}}}$ with $I$ the
current in the divertor coils and $I_{d0}$ the divertor coil current
required for an exact snowflake divertor.
In that case the range of values of negative edge-current for which
Peeling modes are stable, tends to zero as $I \rightarrow I_{d0}$.
|
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Contracts and Leases
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John H. McKinley
John Richard Conger Jr.
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The Hazards of Garbage Truck Accidents
Thu 30th Jul, 2020 Personal Injury
Garbage trucks are an essential part of our communities. Unfortunately, this also means that garbage truck accidents can happen just about anywhere. Because of the size and weight of these vehicles, an accident with a garbage truck can result in catastrophic injuries or even death.
Victims of garbage truck accidents in and around Stockton, CA, and Central Valley, CA, are encouraged to contact a truck accident lawyer at McKinley, Conger, Jolley & Galarneau, LLP, for help-seeking compensation through an insurance claim or lawsuit.
The Dangers of Garbage Truck Accidents
Garbage trucks may not seem threatening, but their size and weight can make a collision very dangerous. When a small car–or worse, a pedestrian or bicyclist–gets in an accident with a garbage truck, it can lead to devastating injuries or fatalities.
Because of their size, garbage trucks are not always easily maneuverable and can take longer to stop when braking. Not only does the size and maneuverability of garbage trucks increase the dangers of accidents but also the way in which garbage trucks operate. When collecting trash, garbage trucks frequently pull over and reenter traffic, which can lead to major accidents.
For example, if a bicyclist speeds by while a garbage truck is pulled over to collect trash and the truck driver doesn't see the bicyclists when he pulls back onto the road, it can lead to a serious accident.
What Are Some Causes of Garbage Truck Accidents?
Garbage trucks are a part of daily life and can be seen operating throughout Stockton and nearby areas. Although most encounters with garbage trucks are uneventful, some people will be involved in an accident.
There are many possible causes of garbage truck accidents, but some of the most notable include:
Large blind spots or poor visibility
Improper maintenance of garbage trucks
Negligence in servicing garbage trucks
Driver failure to follow traffic laws
Types of Injuries in Garbage Truck Accidents
An accident with a garbage truck may result in minor to severe, life-threatening injuries. For some, these injuries can lead to permanent disability, severely impacting their quality of life and their ability to earn a living.
The extent of injuries largely depends on the details of the accident but may include:
Bone fractures
Spinal cord damage
Loss of limbs
Contact the Attorneys of McKinley, Conger, Jolley & Galarneau, LLP
Victims who have been injured in a garbage truck accident may be entitled to financial compensation for lost wages, hospital bills, and other damages related to their injuries. If you or a loved one have been injured in an accident and would like to learn more about your options for pursuing compensation, please call 209-477-8171 or schedule a consultation online with one of our truck accident lawyers today.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,651
|
\section{Introduction}
Social interaction patterns such as contacts and mixing patterns among individuals have
a direct impact on diverse phenomena studied in various research
areas. Clear-cut examples are the transmission of infectious diseases by
the respiratory or close-contact route, and collective
opinion formation. The availability of representative data on such
patterns has long been a concern since it used to be notoriously
difficult to collect it.
The available methods usually rely on surveys
and paper-diary methodologies~\cite{Mossong:2008} which are often
slow, inaccurate, and intrusive. Novel
technologies, however, afford new and promising means of collecting
this essential data.
Contact patterns data is indeed much needed. Recent studies of e-mail \cite{Ebel:2002} and
cellular phone call exchanges~\cite{Onnela:2007,Gonzales:2008},
collaboration networks~\cite{Newman:2001}, sexual contact
networks~\cite{Liljeros:2001}, and mobility by air
travel~\cite{Barrat:2004}, have revealed the presence of complex
properties and heterogeneities. In particular, the number of
interaction partners from one individual to the other is subject to
large fluctuations that have non-trivial
consequences on the dynamical processes taking place on these
networks~\cite{Pastor:2004,Boccaletti:2006,BBVbook}. A detailed
characterization of these structures is therefore of utmost importance
for the understanding of many phenomena, and crucially depends on
the availability of representative empirical data.
While important progress has been achieved in the last decade or so,
more is bound to come, for most of these recent characterizations of
complex networks focused on static configurations in which the temporal
dimension was not considered, mostly because of lack of data. Examples
of properties that arise in this temporal dimension are duration,
frequency, concurrency, and causality. For instance, if an individual A meets first
B then C, an information or a virus can spread from B to A and
then to C, but not from C to B. In the image of a static network in
contrast, the links allow propagation in both cases. The fact that
these static networks are in fact ``summaries'' of many different
interactions that do not occur simultaneously, might conceal important
insights. The few cases in which temporal aspects have been considered
in more detail, indeed revealed important
consequences~\cite{Eckmann:2004,Barabasi:2005,Holme:2005,Vazquez:2006,Iribarren,Kostakos,Vazquez:2007,Hui:2005,Scherrer:2008}.
Several more recent studies have demonstrated the potential of using
novel technologies such as Bluetooth and Wifi for collecting data on
both the structural and temporal aspects of social interaction
patterns~\cite{Hui:2005,Eagle:2006,Kostakos,Scherrer:2008}.
However, their spatial resolution in these is at best of the order of
$10$ meters, and the temporal resolution of the order of 2-5 minutes.
Moreover, these technologies detect local proximity between devices, which does
not imply a priori a social interaction between the individuals
carrying these devices.
Finally, these studies concern small groups and are not
easily reproducible.
In this paper, we present a novel experimental
framework based on active RFID devices that overcomes these
limitations. We discuss a recently performed pilot study, and a data
analysis that highlights the main advantages of this new data collection
technique. Non-technical accounts and supplementary material can be
found on the website of the SocioPatterns project
\cite{sociopatterns}.
\section{Methodology, experiments, data}
\subsection{Active RFID-based experimental framework}
The proposed experimental framework aims at measuring the contact
patterns of a group of
interacting individuals in a spatially bounded setting, such as a
set of offices or a conference. The participants are asked to carry
small RFID tags~\cite{finkenzeller2003rh}, henceforth called
\emph{beacons}. These beacons continuously broadcast small data
packets which are received by a number of stations and relayed
through a local network to a server. The stations are installed at
fixed locations in the environment. The beacons and stations we used
were created by and obtained from the OpenBeacon
project~\cite{openbeacon}.
RFID tags acting as beacons can be used to deploy indoors locative
systems~\cite{ni2004lil} that track the location of the tags.
Problems related to multiple path, phase fluctuations, etc. tend however to
limit the precision of the spatial localization of the tags.
Because of this, locative technologies are typically not viable, at low cost,
to infer face-to-face contact between individuals wearing RFID tags.
Moving from \emph{contact inference} to direct \emph{contact detection}
enabled us to bypass these limitations. To this end, we leveraged the
OpenBeacon active RFID platform~\cite{openbeacon} and operate the
RFID tags a bi-directional fashion. That is, tags no longer act
as simple beacons that passively emit
signals to be received and processed by a centralized post-processing
set-up. They rather exchange messages in a peer-to-peer fashion to
sense their neighborhood and assess directly contacts with nearby tags.
A high spatial resolution of less then $1-2$ meters is attained by
using very low radio power levels for the contact
sensing. Furthermore, assuming that the subjects wear the tags on
their chest, the body effectively acts as a shield for the sensing
signals. This way, contacts are detected only when participants
actually face one another. If a sensed contact persists for a few
seconds, then given the short range and the face-to-face requirement,
it is reasonable to assume that the experiment is able to detect an
ongoing social contact (as e.g. a conversation).
After the beacons detect a contact, they broadcast a report
message at a higher power level. These reports are received by the
stations and relayed to the monitoring infrastructure.
The reports are stored with a time stamp, the \textsc{id} of the relaying
station and the \textsc{id} of the tags which participate in the contact
event (up to $4$ simultaneous contacts are recorded, using the current hardware).
After a suitable tuning of the system parameters, we can
easily record individual contacts in a crowded room with just a small
number of receiving stations.
The raw data series is made with an
effective sampling frequency under one second. In many instances of
data processing, we applied a coarse graining filter using time windows of
$20$ seconds (see next section).
This value is chosen in order to minimize
statistical errors, and corresponds moreover to a typical timescale for
social interactions.
We finally note that messages between tags
and/or stations are encrypted and that the entire data management is
completely anonymous.
\subsection{Visualization}
The pilot studies we conducted so far were accompanied with publicly
displayed, dynamic visualizations of the contacts between individuals.
This is achieved by defining a contact network in which the
beacons/persons are nodes and the contacts are edges. Two different
types of visualizations can be displayed, providing
snapshots respectively of the \emph{instantaneous} state of the network, and
of the \emph{cumulative} state
since a given time (e.g. the start of the experiment, or the start of
the day in a multi-day experiment). The 'instantaneous' visualization
additionally displays marks for the stations, which are positioned in
a fixed layout. The location of the beacon marks in the visualization
is driven by a force-directed layout algorithm. Springs are associated
with both the explicitly shown contact edges and the edges between
beacons and stations, which are not shown. The rest length of these
springs is inversely proportional to the strength of the respective
contact or beacon-station proximity estimations.
The model is regularly updated based on the live data feed, and the
view is updated after each iteration of the algorithm, up to 25 times
per second. The result is a continuously morphing network
representation in which the marks of beacons that are in contact try
to occupy adjacent positions, and to move towards the
marks of the closest stations.
The other visual encodings are as follows: Edge thickness and
transparency encode contact strength; Beacon mark size encodes the
number of contacts reported by the beacon; Station mark size encodes
the number and proximity of the closest beacons. The main network
view is furthermore flanked by a side-bar with various data points and
charts, which are dynamically updated as well.
Figure~\ref{bt2_snapshot} shows a snapshot of the
visualization. Sample movies can be viewed on the
website of the Sociopatterns project~\cite{sociopatterns}.
These visualizations were primarily developed to visually
follow and inspect the ongoing experiment and as an aid in explaining
it, but we also introduced certain affordances. One such feature
consists of enabling the participants to tap their beacon by pressing
a button. The visualization immediately reacts by highlighting the
corresponding beacon mark and temporarily showing a small table with
some detailed contextual data in the side-bar. Other affordances that
were effectively exploited by the participants are the localization of
people and the identification of an observed but unknown contact
partner of a known person.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{bt2.png}
\caption{A snapshot of the visualization. The main view shows the
instantaneous state of the contact network at a given time during
lunch hour. The beacons are labeled with their \textsc{id}s, and can
also be labeled with available metadata (such as e.g. the actual names
of the persons). White edges represent contacts. The beacons are
positioned near the stations where their signals are
received.
The yellow circle behind beacon 4532
highlights a tap while some related data is shown in the
side-bar.
}
\label{bt2_snapshot}
\end{figure}
\subsection{A pilot study}
We have deployed our measuring infrastructure in a pilot experiment of
limited size. The experiment took place during the workshop ``Facing
the challenge of Infectious Diseases'' at the ISI Foundation on
October $13-17$, $2008$. Participants to the workshop were offered to
volunteer to participate to the experiment, and a large part agreed.
This allowed us to gather data in a very dynamical context with
periods of high social interaction (coffee and lunch breaks) and other
periods in which the participants sit together but (almost) do not
interact in a pairwise fashion. The experiment involved about $50$
attendees over four days. We placed reporting stations in the main
areas in which people were expected to be during the sessions and
breaks~--~namely the conference room, the bar (where coffee breaks
were taking place) and the cafeteria area (where lunch was served). We
also put a station in the lobby which is also suited for discussions
(see Figure \ref{fig:map}). Figure \ref{bt2_snapshot}
presents a snapshot of the visualization obtained during the lunch break
of the third day of the conference, showing a number of beacons in the
cafeteria area where people have lunch, while others are having coffee at the bar.
The new firmware proved to be as much reliable in a real-world
setting as it appeared to be in our preliminary experiments. The measuring
infrastructure received approximately $2 \cdot 10^6$ data packets
per day from the various beacons, among which $5 \cdot 10^5$ packets
reporting a contact. Around $150$ Mb of raw compressed data were
processed.
Some caveats have to be reported: because of
technical issues (some beacons had to be changed during the
experiment, some batteries failed and had to be replaced), some
beacons disappeared from the data for few hours. Moreover,
beacons were obviously tracked only when
within range of the stations. We will see in the next section
that, despite these issues corresponding to sampling problems,
the data analysis reveals interesting patterns and shows the large
potential of our experimental setup.
\begin{figure}[t]
\includegraphics[width=0.4\textwidth]{fig_beacon}
\hspace{1cm}
\includegraphics[width=0.4\textwidth]{fig_map_exp3_grayscale}
\caption{Left: photo of a beacon (Courtesy of M. Meriac \protect\cite{openbeacon}).
Right: map of the experiment premises. The circles denote the positions of the
reporting stations.}
\label{fig:map}
\end{figure}
\section{Results of the pilot study}
\subsection{Contacts characterization}
Let us first focus on the analysis of the contacts between individuals.
We define as a ``contact event'' between two beacons A and B the exchange of
at least one data packet between the two
beacons in a $20s$ time-window. We then define as the duration of
the contact A-B the time during which packets are exchanged between them at least every
$20s$. The contact is considered as broken whenever more
than $20s$ occur without a packet exchange.
The choice of a $20s$ window is based on the frequency with which
packets are sent by beacons, and
corresponds to a reasonable time-scale for
social interaction (e.g. encounter, brief conversation, etc.).
Given this definition, we can measure both the duration of each
contact and the intervals between two contacts. Figure
\ref{fig:durations}(left) shows the distribution of the contact
durations obtained using the whole dataset collected during the four
conference days. A very broad distribution is observed, close to a
power-law with exponent $\simeq -2$. Qualitatively, this behavior is
not unexpected: there are comparatively few long-lasting contacts and
a multitude of brief contacts. A similar result has been reported for
the duration of contacts between Bluetooth devices
\cite{Scherrer:2008}, with different exponents depending on the
experimental set-up. Our measurements, however, achieve higher spatial
and temporal accuracy than previous studies, and reliably select
face-to-face interactions at close range, allowing to detect social
interactions of a conversational type. These measurements clearly show that
no characteristic time of interaction can be determined but that these
interactions can occur on many different timescales.
We checked the robustness of the reported behavior along several
lines. First, we verified that the distribution is the same over
different periods of time: few hours, a whole day, or the whole
conference. We also checked that it is invariant across randomly
selected groups of individuals (see Figure \ref{fig:durations}(left)),
showing that each individual has a broad distribution of contact
durations. The obtained global heterogeneity therefore stems from an
heterogeneity of the contact patterns of each individual, and not from
an heterogeneity due to the difference of behavior among individuals.
The distribution of contact durations remains unchanged, even by
assuming a stricter definition of contact. The left panel of
Figure~\ref{fig:durations} shows the result obtained by defining
stronger contacts as the ones in which at least $5$ data packets are
exchanged in a $20s$ window (instead of $1$ data packet only for the
standard definition of a contact event).
\begin{figure}[t]
\includegraphics[width=\textwidth]{fig3}
\caption{Left: Distribution of contact durations obtained by
considering: all contact events; the stronger contacts only
(i.e. exchange of at least $5$ data packets between 2 beacons in a
$20s$ time window); the contacts of two individuals selected at
random. Right: Distribution of time intervals between contacts:
involving at least one common beacon; involving the same pair of
beacons; removing 20 randomly selected beacons; considering stronger
contacts only. }
\label{fig:durations}
\end{figure}
Let us now turn to the inter-contact time intervals, for which
previous studies \cite{Hui:2005,Scherrer:2008} have also reported
broad distributions. Time intervals between contacts can in fact be
defined in three different ways. One can measure the time between any
two reported contact events, regardless of the involved
beacons, thus yielding a characterization of
the global dynamic/social activity of the group under study. We
observe a broad distribution close to a power-law with exponent $-2.5$
(not shown, see the Sociopatterns project website~\cite{sociopatterns}).
Different measures, which are more important in relation
with spreading processes, focus on (i) the time intervals
between two contacts involving a given particular beacon, and (ii)
the time intervals between two contacts involving the same pair of
beacons. Figure \ref{fig:durations}(right) displays these
distributions, showing that also in this case a broad behavior
is obtained.
This behavior is robust with respect to possible (heavy) data loss, as
shown by the distribution obtained by removing the data coming from $20$
randomly selected beacons that represent more than $30\%$ of the whole
dataset. The stronger contact definition
also yield similar results.
The results presented here are not unexpected, since bursty behaviors
and broad distributions of events durations or inter-event intervals
have been reported in other studies on human behavior. It is
nonetheless striking to observe that our experimental setup based on a
new technology aimed at contact detection yields high quality data
within a relatively small experiment, in agreement with the expected
behavior. We foresee that larger experiments will allow to obtain
larger statistics and to investigate more in detail the social and
dynamical aspects of contact and mixing patterns, through a detailed
characterization of the links (intermittency, persistence, etc.) and
of the nodes (role inference, inclusion of background information,
etc.).
\subsection{Social networks}
\begin{figure}[t]
\includegraphics[width=\textwidth]{fig4v2}
\caption{
Number of beacons in the conference room as a function of
time, during the third day of the conference. Left: Average degree
$\langle k \rangle$ in the instantaneous contact network computed over
time windows of $20s$. Right: Number of pairs (2-cliques), triangles
(3-cliques), and $4$-cliques in the contact network. Note that we
consider the maximal cliques, i.e. that the three edges of a triangle
are not counted in the number of pairs. }
\label{fig:timeline}
\end{figure}
The data on social contacts can be used to build aggregated
networks of interactions between individuals on any timescale larger
than the time resolution. Individuals are the nodes of the network, and a link
of weight $w$ exists between two individuals if $w$ contact events have taken place between them in the
chosen time interval.
Let us first focus on ``instantaneous'' networks, constructed on short
timescales. Figure \ref{fig:timeline} shows the number of beacons in
the conference room as a function of time\footnote{Although a precise
tracking of the beacons' locations is a difficult task, it is easy to
know in which room each beacon is.},
during the third day of
the conference, which was divided into four sessions, separated by two
coffee breaks and a lunch break (indicated by the gray areas in the
Figure). The data, averaged over time windows of $20s$, clearly shows
the attendance of each session, in which most beacons are in the
conference room, whereas the breaks are identified by the small number
of beacons remaining in the conference room. The left panel also
displays the evolution of the average number of contacts per
individual during $20s$ periods. Strikingly, the number of contacts
per participant is low when the attendance in the conference room is
high, whereas a clear increase is observed during each break, clearly
signalling that most social interactions occur during the coffee and
lunch breaks, though some contacts may occur during the sessions when
people typically talk and discuss with their immediate neighbors.
This is further highlighted in the right panel of Figure
\ref{fig:timeline}, where we display, together with the attendance
curve in the conference room, the number of $2-$, $3-$ and $4-$
cliques in the contact networks aggregated over $20s$ time
windows. Note that we consider here maximal cliques, so that the edges
of a triangle are not counted as $2-$cliques, or that the $4$
triangles forming a $4-$clique are not counted in the number of
$3$-cliques. A fluctuating number of pairs is observed during the
session, corresponding most probably to participants turning towards
their neighbours, and peaks are observed at the beginning and end of
each session and in fact of each talk, when participants have indeed
more activity. $3-$ and $4-$ cliques are observed almost exclusively
during the breaks, as expected since many discussions take place in
small groups. It is worth to mention that the small number of
$3-$cliques observed during the sessions correspond to small groups of
participants remaining in the coffee break area for discussions even
after the beginning of the session.
The results illustrated in Figure~\ref{fig:timeline} are clearly
expected, since social interactions obviously take place during the
breaks. However, they point to the ability of our experimental setup
of resolving the mixing patterns by directly detecting the contact
events. A less elaborate setup, based on the inference of contact
events by spatial proximity, would show a large number of cliques (or
worse, a unique large clique) during the meeting session where
participants are physically close. In addition,
Figure~\ref{fig:timeline}(right) clearly shows how this technology is
able to detect interactions between $3$ or $4$ people, and not only
pairwise interactions.
\begin{figure}[t]
\includegraphics[width=\textwidth]{net_alldays_lin_norms_country_shapes_th500_100v4}
\caption{Social network of contacts between individuals (represented
by nodes), aggregated over the whole duration of the conference
(larger graph), and for each of the days (smaller panels). The size
of each node is proportional to its strength (given by the sum of
the weights of its links \protect\cite{Barrat:2004}), and the width
of each link is proportional to its weight. The color of each node
corresponds to the individual's country of affiliation, and the
shape to his/her academic position. For clarity, only links with
weight larger than 100 are reported (50 for the smaller panels). As
visible from the smaller panels, different interaction patterns are
obtained for different days. }
\label{fig:net}
\end{figure}
The data can also be used to construct aggregated networks on longer
timescales, for example for a single day or for the whole duration of
the experiment. The aggregated network becomes then denser as the
aggregation time increases, with an average degree ranging from a
value close to $20$ for the network aggregated over one day, to
approximately $40$ for the whole experiment duration, showing that
most participants have interacted with each other, which is in fact
one of the aims of a small-scale conference. The aggregated networks
are interesting in that they show broad distributions of the weights
(given by the number of packets exchanged between two beacons) which
are a proxy for the effective duration of a social
interaction. Without going into a detailed network analysis, we
provide in Figure~\ref{fig:net} a visualization of the networks of
social interactions obtained by aggregating the data for each day of
the conference (smaller panels) and for its entire duration (larger
graph), with the heterogeneity of links weights and nodes strengths
clearly visible.
\subsection{Contagion processes}
\begin{figure}[b!]
\includegraphics[width=\textwidth]{fig_spread_day5}
\caption{Left: Evolution of the number of 'Infected' individuals when a
single Infectious is introduced at the beginning of the day. For each
contact, the transmission probability is $0.01w$ for
each $20s$ time window, where $w$ is the number of packets exchanged
between the two beacons in contact during this time window. Right:
illustration of the contagion events in the population of beacons as a function
of time, for $20\%$ initially immune individuals. Black lines
indicate the infection from one beacon to another, as they occur in time.
}
\label{fig:spread}
\end{figure}
The dynamic network of contacts provides a realistic setting to
perform simulations of contagion processes in the population of
individuals, such as rumour or information spreading,
opinion formation, or epidemic processes. Particularly relevant is
the application to the spread of infectious diseases transmitted by
the respiratory or close-contact route (as for example influenza,
SARS, etc.). Models of epidemic spread on contact networks usually
rely on static configurations of networks where the aspects of
concurrency and causality are not taken into account. The data
collected with our experimental setup can be used for an emulation of
a contagion process among individuals where all topological and
temporal heterogeneities are considered.
Here we present a very simple example of a contagion process aimed at
showing the feasibility of such studies. We consider the basic
Susceptible-Infected (SI) model in which individuals are classified
in two mutually exclusive compartments, Susceptible (i.e. able to
contract a disease) and Infectious (i.e. infected and able to
transmit the infection)~\cite{Maybook}. The emulation is performed
on the contact data of the third conference day. At the beginning of
the day, a randomly selected individual is considered as infectious.
During each time window of $20s$, each contact between a susceptible
and an infectious individual can result in the contagion of the
susceptible that contracts the infection with probability
$0.01w$ (where $w$ is the number of packets exchanged between
the beacons of the individuals, i.e. a measure of the intensity and
duration of the social interaction). Some individuals are set as
immune since the start of the emulation, allowing for individuals who
are not susceptible to the disease and can never become infectious.
Figure~\ref{fig:spread} displays the number of infectious individuals
as a function of time for a single realization of the stochastic
model, and for different percentages of initially immune
individuals. An interesting pattern is observed, in agreement with
the previous analysis: most contagion events occur during the coffee
and lunch breaks, where social interactions are more likely to
occur. The right panel displays a schematic visualization of the
propagation dynamics, shown as a tree in which each newly
infected beacon is represented as a red disk at the time of its contamination, with lines going from
the infecting beacon to the infected one for each contagion event.
While this model is overly simplistic and does not aim to reproduce a
given realistic epidemic scenario, it offers the possibility of
studying simple contagion processes on a realistic dataset, and
provides a proof of concept showing how the data gathered through our
experimental set-up in proper settings (as e.g. larger social events)
can have a crucial value to understand and predict the impact of
infectious diseases.
\section{Conclusions and perspectives}
In this paper, we presented a novel experimental set-up which can
be used to gather information on social interactions of
individuals. The measures are based on active RFID devices, called
beacons, that individuals can wear
as badges. When two beacons are close enough
(typically one meter apart), they can exchange messages and relay them
to the measuring infrastructure. The very low power used for the
exchanged messages and the absorption of the used frequencies by
the human body ensure that contacts are detected only when
individuals face each other as in a real social contact. This allows us
to obtain data at very high spatial and temporal resolution, as
shown in a pilot experiment performed during a recent
conference. Here we presented some results of the corresponding
data analysis, showing the resolving power of experimental setup, able to
discriminate between social interaction and simple physical proximity.
We measured the distributions of the duration of social contacts
between individuals and of the intervals between contacts,
and found broad behaviors. Moreover, we showed how our experimental
setup can be used to construct social networks by aggregating the
contacts over the required timescale.
Our experimental set-up paves the way for a number of developments and
applications. Clearly, more experimental work is
needed
to obtain
larger statistics on contacts durations or frequencies, and to characterize
dynamically evolving social networks. The hardware and
software could also be upgraded to contain additional information
on the individuals and their interactions.
The presented set-up will also allow to study various dynamical
phenomena taking place on dynamically evolving contact networks,
as briefly illustrated above. Contagion processes,
such as rumour spreading, opinion formation, propagation of respiratory
or close-contact infections, take place on the dynamical network of
social contacts among individuals. Gathering data on social contacts
will allow a better modeling and understanding of the spread of viruses and information.
\vspace*{0.1cm}
\noindent
\small{V.C. is partially funded by the European Commission contract
n. ERC--2007--Stg204863 (EpiFor). A.V. is partially funded by the
NIH-NIDA-21DA024259-01 award.}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,671
|
G. P. PUTNAM'S SONS
_Publishers Since 1838_
Published by the Penguin Group
Penguin Group (USA) LLC
375 Hudson Street
New York, New York 10014
USA • Canada • UK • Ireland • Australia • New Zealand • India • South Africa • China
penguin.com
A Penguin Random House Company
Copyright © 2014 by Linda Tirado
Penguin supports copyright. Copyright fuels creativity, encourages diverse voices, promotes free speech, and creates a vibrant culture. Thank you for buying an authorized edition of this book and for complying with copyright laws by not reproducing, scanning, or distributing any part of it in any form without permission. You are supporting writers and allowing Penguin to continue to publish books for every reader.
ISBN 978-0-698-17528-0
Penguin is committed to publishing works of quality and integrity. In that spirit, we are proud to offer this book to our readers; however, the story, the experiences, and the words are the author's alone.
Version_1
For Tom, who can't say I didn't warn him
# Contents
Title Page
Copyright
Dedication
Foreword by Barbara Ehrenreich
Introduction
1 It Takes Money to Make Money
2 You Get What You Pay For
3 You Can't Pay a Doctor in Chickens Anymore
4 I'm Not Angry So Much as I'm Really Tired
5 I've Got Way Bigger Problems Than a Spinach Salad Can Solve
6 This Part Is About Sex
7 We Do Not Have Babies for Welfare Money
8 Poverty Is Fucking Expensive
9 Being Poor Isn't a Crime—It Just Feels Like It
10 An Open Letter to Rich People
Afterword
Acknowledgments
# Foreword
By Barbara Ehrenreich
I've been waiting for this book for a long time. Well, not this book, because I never imagined that the book I was waiting for would be so devastatingly smart and funny, so consistently entertaining and unflinchingly on target. In fact, I would like to have written it myself—if, that is, I had lived Linda Tirado's life and extracted all the hard lessons she has learned. I am the author of _Nickel and Dimed_ , which tells the story of my own brief attempt, as a semi-undercover journalist, to survive on low-wage retail and service jobs. Tirado is the real thing.
After my book came out in 2001, I spent over ten years on the road talking about it at union conferences, church gatherings, and mostly on college campuses. I did this partly for the money because I had lost my best-paying journalistic job in 1997, and then a few years later the media decided that writers no longer needed to be paid at all, as if writing involves no caloric expenditure whatsoever.
But I also did it because I was on a mission. People often asked how my work for _Nickel_ _and Dimed_ changed me, and I think they meant how did it make me, as a middle-class person, more aware of the poor. Well, I didn't need that much more awareness since I was born into the lower rung of the working class and managed to re-land in it by becoming a single mother and then marrying a warehouse worker when I was in my thirties. So my stint as a low-wage worker/journalist had only one major effect on me: It moved me from concern about the exploitation of low-wage workers—to something more like rage.
I had expected to experience material deprivation in my life at $7 an hour (the equivalent of about $9 today), and I certainly did. The fact that I had some built-in privileges like a working car (I got a Rent-A-Wreck in each of the cities where I worked so I wouldn't end up writing a book about waiting for buses) only made the deprivation part more shocking. Here I was—in good health, with no small children in my care—working full-time, sometimes more than one job at time, sometimes to the point where my legs felt like rubber, and I was living in a dump and dining at convenience stores or Wendy's.
What I had not expected was the daily humiliation, the insults and what seemed like mean-spirited tricks. To be poor is to be treated like a criminal, under constant suspicion of drug use and theft. It means having no privacy, since the boss has the legal right to search your belongings for stolen items. It involves being jerked around unaccountably, like the time Wal-Mart suddenly changed my schedule, obliterating the second job I had lined up. It means being ordered to "work through" injuries and illness, like the debilitating rash I once acquired from industrial-strength cleaning fluids.
And what was most amazing to me: Being a low-wage worker means being robbed by the very employer who is monitoring _you_ so insistently for theft. You can be forced to work overtime without pay or made to start working forty-five minutes before the time clock starts ticking. If you do the math, you may find that a few more hours have been shaved off your paycheck each week by the corporation's computers.
But when I made my way from campus to campus, telling my stories about work and urging students to take an interest in all the low-wage workers who were making their education possible every day—the food service workers, janitors, clerical workers, and adjunct faculty—I was invariably asked the question that boils down to: What's wrong with these people? Meaning the workers, not their bosses.
Typically, the questioner would be a frat boy who had taken Econ 101, a course which exists, as far as I can see, for the sole purpose of convincing young people that the existing class structure is just, fair, and unchangeable anyway. If there's nothing wrong with our economic arrangements, then the only remaining question is: Why do "these people" have children, lack savings, fail to go to college, eat junk food, smoke cigarettes, or whatever else is imagined to be holding them back?
So when I came across Linda Tirado's blog about six months ago, I felt an enormous wave of vindication. Even—or, perhaps, especially—her admission that she smokes cigarettes hit me like a gust of fresh air. She tells what it's like to be a low-wage worker for the long term, with an erratically employed husband and two small children to raise and support. She makes all the points I have been trying to make in my years of campaigning for higher wages and workers' rights: That poverty is not a "culture" or a character defect; it is a shortage of money. And that that shortage arises from grievously inadequate pay, aggravated by constant humiliation and stress, as well as outright predation by employers, credit companies, and even law enforcement agencies.
But let me get out of the way now. She can tell this so much better than I can.
# Introduction
In the fall of 2013, I was in my first semester of school in a decade. I had two jobs; my husband, Tom, was working full-time; and we were raising our two small girls. It was the first time in years that we felt like maybe things were looking like they'd be okay for a while.
After a particularly grueling shift at work, I was unwinding online when I saw a question from someone on a forum I frequented: _Why do poor people do things that seem so self-destructive?_ I thought I could at least explain what I'd seen and how I'd reacted to the pressures of being poor. I wrote my answer to the question, hit post, and didn't think more about it for at least a few days. This is what it said:
WHY I MAKE TERRIBLE DECISIONS, OR, POVERTY THOUGHTS
There's no way to structure this coherently. They are random observations that might help explain the mental processes. But often, I think that we look at the academic problems of poverty and have no idea of the why. We know the what and the how, and we can see systemic problems, but it's rare to have a poor person actually explain it on their own behalf. So this is me doing that, sort of.
Rest is a luxury for the rich. I get up at 6 a.m., go to school (I have a full course load, but I only have to go to two in-person classes), then work, then I get the kids, then I pick up my husband, then I have half an hour to change and go to Job 2. I get home from that at around 12:30 a.m., then I have the rest of my classes and work to tend to. I'm in bed by 3. This isn't every day, I have two days off a week from each of my obligations. I use that time to clean the house and soothe Mr. Martini and see the kids for longer than an hour and catch up on schoolwork. Those nights I'm in bed by midnight, but if I go to bed too early I won't be able to stay up the other nights because I'll fuck my pattern up, and I drive an hour home from Job 2 so I can't afford to be sleepy. I never get a day off from work unless I am fairly sick. It doesn't leave you much room to think about what you are doing, only to attend to the next thing and the next. Planning isn't in the mix.
When I was pregnant the first time, I was living in a weekly motel for some time. I had a minifridge with no freezer and a microwave. I was on WIC. I ate peanut butter from the jar and frozen burritos because they were 12/$2. Had I had a stove, I couldn't have made beef burritos that cheaply. And I needed the meat, I was pregnant. I might not have had any prenatal care, but I am intelligent enough to eat protein and iron whilst knocked up.
I know how to cook. I had to take Home Ec to graduate high school. Most people on my level didn't. Broccoli is intimidating. You have to have a working stove, and pots, and spices, and you'll have to do the dishes no matter how tired you are or they'll attract bugs. It is a huge new skill for a lot of people. That's not great, but it's true. And if you fuck it up, you could make your family sick. We have learned not to try too hard to be middle class. It never works out well and always makes you feel worse for having tried and failed yet again. Better not to try. It makes more sense to get food that you know will be palatable and cheap and that keeps well. Junk food is a pleasure that we are allowed to have; why would we give that up? We have very few of them.
The closest Planned Parenthood to me is three hours. That's a lot of money in gas. Lots of women can't afford that, and even if you live near one you probably don't want to be seen coming in and out in a lot of areas. We're aware that we are not "having kids," we're "breeding." We have kids for much the same reasons that I imagine rich people do. Urge to propagate and all. Nobody likes poor people procreating, but they judge abortion even harder.
Convenience food is just that. And we are not allowed many conveniences. Especially since the Patriot Act passed, it's hard to get a bank account. But without one, you spend a lot of time figuring out where to cash a check and get money orders to pay bills. Most motels now have a no-credit-card-no-room policy. I wandered around SF for five hours in the rain once with nearly a thousand dollars on me and could not rent a room even if I gave them a $500 cash deposit and surrendered my cell phone to the desk to hold as surety.
Nobody gives enough thought to depression. You have to understand that we know that we will never not feel tired. We will never feel hopeful. We will never get a vacation. Ever. We know that the very act of being poor guarantees that we will never not be poor. It doesn't give us much reason to improve ourselves. We don't apply for jobs because we know we can't afford to look nice enough to hold them. I would make a super legal secretary, but I've been turned down more than once because I "don't fit the image of the firm," which is a nice way of saying "gtfo, pov." I am good enough to cook the food, hidden away in the kitchen, but my boss won't make me a server because I don't "fit the corporate image." I am not beautiful. I have missing teeth and skin that looks like it will when you live on B12 and coffee and nicotine and no sleep. Beauty is a thing you get when you can afford it, and that's how you get the job that you need in order to be beautiful. There isn't much point trying.
Cooking attracts roaches. Nobody realizes that. I've spent a lot of hours impaling roach bodies and leaving them out on toothpick spikes to discourage others from entering. It doesn't work, but is amusing.
"Free" only exists for rich people. It's great that there's a bowl of condoms at my school, but most poor people will never set foot on a college campus. We don't belong there. There's a clinic? Great! There's still a copay. We're not going. Besides, all they'll tell you at the clinic is that you need to see a specialist, which, seriously? Might as well be located on Mars for how accessible it is. "Low-cost" and "sliding scale" sound like "money you have to spend" to me, and they can't actually help you anyway.
I smoke. It's expensive. It's also the best option. You see, I am always, always exhausted. It's a stimulant. When I am too tired to walk one more step, I can smoke and go for another hour. When I am enraged and beaten down and incapable of accomplishing one more thing, I can smoke and I feel a little better, just for a minute. It is the only relaxation I am allowed. It is not a good decision, but it is the only one that I have access to. It is the only thing I have found that keeps me from collapsing or exploding.
I make a lot of poor financial decisions. None of them matter, in the long term. I will never not be poor, so what does it matter if I don't pay a thing and a half this week instead of just one thing? It's not like the sacrifice will result in improved circumstances; the thing holding me back isn't that I blow five bucks at Wendy's. It's that now that I have proven that I am a Poor Person that is all that I am or ever will be. It is not worth it to me to live a bleak life devoid of small pleasures so that one day I can make a single large purchase. I will never have large pleasures to hold on to. There's a certain pull to live what bits of life you can while there's money in your pocket, because no matter how responsible you are you will be broke in three days anyway. When you never have enough money it ceases to have meaning. I imagine having a lot of it is the same thing.
Poverty is bleak and cuts off your long-term brain. It's why you see people with four different babydaddies instead of one. You grab a bit of connection wherever you can to survive. You have no idea how strong the pull to feel worthwhile is. It's more basic than food. You go to these people who make you feel lovely for an hour that one time, and that's all you get. You're probably not compatible with them for anything long term, but right this minute they can make you feel powerful and valuable. It does not matter what will happen in a month. Whatever happens in a month is probably going to be just about as indifferent as whatever happened today or last week. None of it matters. We don't plan long term because if we do we'll just get our hearts broken. It's best not to hope. You just take what you can get as you spot it.
I am not asking for sympathy. I am just trying to explain, on a human level, how it is that people make what look from the outside like awful decisions. This is what our lives are like, and here are our defense mechanisms, and here is why we think differently. It's certainly self-defeating, but it's safer. That's all. I hope it helps make sense of it.
While I was thinking that maybe a couple of people would read my essay, lightning struck. A lot of people started to share it. Someone suggested that I submit it for posting on the main page of the website we hung out on. That wasn't uncommon, so I did. The next thing I knew, the world had turned upside down. The Huffington Post ran my essay on its front page, _Forbes_ ran it, _The Nation_ ran it.
After the original piece went viral, I got a lot of emails from people who told me that they did not agree; they did not cope in the same ways. That's fair, and true. Keep it in mind. What was neither fair nor true was the criticism I received inferring that I was the wrong sort of poor. A lot of this criticism seemed to center on the fact that I was not born into poverty, as though that were the only way someone might find herself unable to make rent. And yet we have a term for it: downward mobility. We have homeless PhDs and more than one recently middle-class person on food stamps. Poverty is a reality to more people than we're willing to admit.
Overall, though, the response was overwhelmingly one of solidarity. I got thousands of emails from people saying that they understood exactly what I was trying to describe, that they felt the same way. They told me their stories—the things that bothered them and how they were dealing with life. It's not just me who feels this way, not by a long shot. Poor people talk about these things, but no one's listening to us. We don't usually get a chance to explain our own logic. The original piece that you just read, and this book, are simply that: explanations. I am doing what I can to walk you through what it is to be poor. To be sure, this is only one version. There are millions of us; our experiences and reactions to them are as varied as our personalities and backgrounds.
I haven't had it worse than anyone else, and actually, that's kind of the point. This is just what life is for roughly a third of the country. We all handle it in our own ways, but we all work in the same jobs, live in the same places, feel the same sense of never quite catching up. We're not any happier about the exploding welfare rolls than anyone else is, believe me. It's not like everyone grows up and dreams of working two essentially meaningless part-time jobs while collecting food stamps. It's just that there aren't many other options for a lot of people.
In fact, the Urban Institute found that half of Americans will experience poverty at some point before they're sixty-five. Most will come out of it after a relatively short time, 75 percent in four years. But that still leaves 25 percent who don't get out quickly, and the study also found that the longer you stay in poverty, the less likely it becomes that you will ever get out.
Most people who live near the bottom go through cycles of being in poverty and being just above it—sometimes they're just okay and sometimes they're underwater. It depends on the year, the job, how healthy you are. What I can say for sure is that downward mobility is like quicksand. Once it grabs you, it keeps constraining your options until it's got you completely.
I slid to the bottom through a mix of my own decisions and some seriously bad luck. I think that's true of most people. While it can seem like upward mobility is blocked by a lead ceiling, the layer between lower-middle class and poor is horrifyingly porous from above. A lot of us live in that spongy divide.
I got here in a pretty average way: I left home at sixteen for college, promptly behaved as well as you'd expect a teenager to, and was estranged from my family for over a decade. I quit college when it became clear that I was taking out loans to no good effect; I wasn't ready for it yet. I chased a career simply because it was the first opportunity available rather than because it was sensible.
And I also had medical bills. I had bouts of unemployment, I had a drunken driver total my car. I had everything I owned destroyed in a flood.
So it's not just one or the other: nature or nurture, poor or not poor. Poverty is a potential outcome for all of us.
This is a huge societal problem, and we're just starting to come to grips with all the ways that a technological revolution and globalization have vastly increased inequality. You cannot blame your average citizen for those things. Nor can you blame individual companies—it is how we, collectively, have decided to do things.
We got here partially because of bad policy decisions and partially because of factors nobody could have foreseen. Telling an individual company to do better is a lot like telling an individual poor person to save more—true and helpful, but not so easy in practice. Most companies, like most people, aren't the top 1 percent. They are following the market, not driving it. Besides which, any asshole with money can buy and run a company. They're not all smart enough to figure out long-term investments in human capital.
I am not, for all my frustration, opposed to capitalism. Most Americans, poor ones included, aren't. We like the idea that anyone can succeed. What I am opposed to is the sort of capitalism that sucks the life out of a whole bunch of the citizenry and then demands that they do better with whatever they have left. If we could just agree that poor people are doing the necessary grunt work and that there is dignity in that too, we'd be able to make it less onerous.
Put another way: I'm not saying that _someone_ doesn't have to scrub the toilets around here. All I'm saying is that maybe instead of being grossed out by the very idea of toilets, you could thank the people doing the cleaning, because if not for them, you'd have to do it your damn self.
In this book I have been careful to obscure identifying details about people. Most of the people I've worked for have long since turned over themselves and work elsewhere now. Just in case, however, I have changed places, personal details, and names as needed to protect people's privacy. Nobody, myself included, thought that I'd be writing a book.
A note about the definitions of certain words used in this book: These are my definitions, but I'll tell you what they are up front. _Poverty_ is when a quarter is a fucking miracle. _Poor_ is when a dollar is a miracle. _Broke_ is when five bucks is a miracle. _Working class_ is being broke, but doing so in a place that might not be run-down. _Middle class_ is being able to own some toys and to live in a nice place—and by "nice," I don't mean fancy; I mean that you can afford to buy your own furniture and not lease it and that while you still worry about bills, you aren't constantly worried about homelessness. And _rich_ is anything above that.
This book is not exhaustive, but it is a collection of some of the emotions and experiences I've had while trying to get back to the starting line. Some of these are illogical. Some are counterintuitive. Some are contradictory. That's because I am a human being, and we are all of those things.
There are also many things that I am not. Instead of attempting to point out how people who are different from me are in many ways far more disadvantaged than I have ever been in every instance I can think of (because that should be clear unless you have the peripheral vision of a racehorse), I will just say this: Here is how I have felt, _as me_ : as a relatively young person who is perceived as white, who is naturally sociable, who is intelligent and well-spoken, who was taught well and as a result loves learning things, who is able to lift objects up to fifty pounds repeatedly. And many times, with all of that going for me, I still saw no hope. I cannot begin to imagine how much harder it is for someone who faces more discrimination than I have or who grew up without these basic tools that I am lucky enough to have. Keep that in mind too.
I would lastly like this to be clear: I haven't spent a lot of time talking about the good things in my life—my loves and interests and friends. Those exist, because—again—I'm human. Those things are common to all humans, and for now you're interested in the things that are unique to the poor, and how we cope with them. I've focused on the things I've been most often criticized for in my life and explained the motivations as I see them. I'm here to tell you why _this_ person does what she does.
So take a tour with me through some of the aspects of life that poverty impacts and on which poor people are judged: our work ethics (or lack thereof), our sex lives (definitely way too much of that), our coping mechanisms (naughty poor people), our health practices (I know, you still can't believe that I smoke). And so on. Stick with me. It won't always be easy, but maybe you'll learn something about the lives of your fellow Americans in the process.
And truthfully? What I'm really hoping is that you'll learn something about yourself and that maybe you'll start thinking a bit differently.
So now, the book. Thank you for being open-minded. If you've made it this far (I planted some test profanity in here just to make sure we're on the same page), you might understand what I'm on about.
# 1
It Takes Money to Make Money
When I tried to come up with my worst, most exemplary terrible job to start off this chapter, I found myself a little bit stuck. Let's just say I have an embarrassment of riches to choose from. But here's one:
I was in my mid-twenties, married, childless. My husband and I lived in a small town in the mountains at the time. I was working as a bartender. If you have ever wondered where frat boys go to die, it's to grown-up fraternal organizations. They have their own members-only bars and pretty much feel like they can do whatever they want inside their members-only walls. And that included, at this place, violating the physical and mental boundaries of those of us serving them their drinks.
During NASCAR races, we would have dozens of people sitting around drinking Bud Light and arguing the relative merits of Junior. (For the uninitiated, that's Dale Earnhardt Jr. There is a raging debate about the relative merits of Junior versus his daddy, although both are beloved. If you are in the country and you mention Junior by his given name, you will have immediately outed yourself as city folk.) I had two bosses: One was nearing eighty and mostly just wanted to drink copious amounts of rotgut while pretending to manage the books. The other was in his prime—or at least he'd never lived a day in his life in which he didn't think he looked amazing. (Really, he was balding and portly and had a molester mustache. Let's call him M.M. for short.)
M.M. liked to remark on how young I was and then "accidentally" brush against the parts of me that didn't usually see daylight. He wasn't there every day, but when he was, I could look forward to being asked every twenty minutes or so whether I'd be willing to service him sexually. The fact that his wife was often within earshot mattered not at all, because _of course_ he was only joshing, proving how virile he still was. Except that the women who did sleep with him got the better shifts. Funny, right?
I wasn't desperate enough to do that for an extra $20 or $30 a day. (There is definitely an amount of money you could pay me to have sex with a skeevy old dude, but I'm fairly certain I've priced myself well out of M.M.'s financial reach.) Instead, I'd make my minimum wage and maybe another $10 or $20 in tips, leaving me with a grand total of enough income to qualify me for state aid.
So I picked up a second job waiting tables. The thing about working for tips is that you're supposed to always make at least the minimum wage. The federal minimum for waiting tables is $2.13 an hour (some states do have higher minimums for tipped employees, but only about half of them). If you don't make enough in tips to bump you up to the federal minimum wage of $7.25, the restaurant is supposed to kick in the difference. Corporate restaurants are too protective of their bottom lines to allow a single useless employee, so they typically send waitstaff home as soon as they can. The smaller mom-and-pop places, where there might be only a single waitperson on staff for hours at a time when it's slow, might have their employees do deep cleaning and other things when there are no customers. So there you are, working constantly but getting paid only $2.13 an hour. No matter how good a waitress you are, you probably won't make two or three hours' worth of minimum wage out of tips from your only table in hours. These aren't the kind of restaurants where generous patrons just take it into their heads to overtip. And if you remind your boss that he is supposed to top you off to $7.25, then you run the risk of finding yourself with reduced hours or fired altogether. So you pretty much keep your mouth shut about that.
My second job: I made $4 an hour or so at that one, because new people get the slow shifts. Hours would go by in which not a single customer walked in. Let me run you through the math on this one: On an average day, I'd work for six or seven hours. I might make $50. Some days were better, some worse. I couldn't take the busy dinner shifts at the restaurant even after I was trained, because I was at my first job until the middle of dinner rush and the restaurant needed its dinner staff in the door by midafternoon to prep. Meanwhile, I missed special events like picnics and the like at the bar that I might have made money on because I was at the restaurant.
And this is how it goes. Every time I've had more than one job, I've missed out on as much cash as I've made because of scheduling issues. Getting a second job wouldn't be worth it at all except for the fact that those special events and extra hours are never guaranteed in advance. If it's a week with no extra shifts, or with bad weather that keeps customers home, you're stuck. So you hedge that bet by finding another shitty job.
The most I've seen anyone manage at once was four jobs: bartending, dancing, waitressing, and teaching yoga. I've held down up to three: tending bar, waiting tables, and working as a voter registration canvasser. It nearly killed me, and I still didn't break twenty grand that year.
I think that most liberal Americans don't have too hard a time believing that it's difficult to make ends meet when you're making minimum wage. But I also think that people in both parties get hung up on the minimum wage as some kind of miracle line of demarcation—as if making more than the minimum puts you on easy street. Meanwhile, millions of people are making above minimum wage—so they don't get counted as making the minimum. And do you know what they're making? Instead of $7.25 an hour, they're getting $7.35 an hour. Maybe even $7.50! In many places in America—think fast-food restaurants, dollar stores, gas stations—most of the employees make under $8 or $9. And these employees are not all kids. So when you hear or participate in these discussions about minimum wage statistics, assume that the vast majority of service workers are making within a stone's throw of minimum wage. Our ladder's rungs are set close together, and there are so many of them that it takes us forever to climb it. My husband worked for the same restaurant for nearly two years before he broke $7.75. Was he making minimum? No. But the difference between minimum wage and $7.75 is just around $1,040 a year if you're working full-time, which is pretty rare.
So there you are, working all the time, bringing home so little, and very often getting behind. But your landlord doesn't care that you're working as hard as you can, that there aren't more hours for you to work. The only thing that matters to your landlord is whether or not you have the money for the rent. I've had a landlord tell me that I could be turning tricks if I really cared about paying my bills, that clearly the only reason I was broke is that I wasn't trying hard enough, that he had no patience for people who couldn't simply get along in life. He actually dispensed all of this as though it were helpful advice rather than a series of insults. And that was _after_ the begging, after I'd already debased myself, already explained that my hours got cut for the slow season and they hadn't warned me in time for me to find another job.
This is my bottom line point about work and poverty: It's far more demoralizing to work and be poor than to be unemployed and poor. I have never minded going without when I wasn't working. It sucks not to be able to find a job, but you expect to be tired and pissed off and to never be able to leave your house when you're flat broke. Working your balls off, begging for more hours, hustling every penny you can, and still not being able to cover your electric bill with any regularity is soul-killing.
The popular conception of minimum wage workers is that they're mostly teenagers working part-time. That would be because the Bureau of Labor Statistics, on its website, is pretty clear that about half of workers making the minimum or below are under the age of twenty-five. But that same BLS website will tell you that about half of workers making the minimum or below are _not_ under the age of twenty-five. That's 800,000 adults over the age of twenty-five working at minimum wage or below. Or, if you prefer, about 25,000 more people than live in all of San Francisco.
As I've pointed out already, a lot of adults are getting just pennies over the minimum wage—and I'd argue that your average adult does his job, however lowly, a damn sight better than most teenagers. And when you think about how insignificant a raise of even fifty cents above the minimum turns out to be, it's hard not to feel devalued—as if the sum of your accomplishments as an adult amounts to some nickels and dimes.
But let's put that frustration aside and talk about what it actually means to make minimum wage.
Working for minimum wage (or, as we've already established, close to it) means that making a long-term budget is an exercise in wishful thinking. You just have however much money you have until you run out, and you pay whatever bill is most overdue first. When I was working in Ohio at a fast-food joint, I'd generally get about twenty-five hours in a week. That was paid at $7.50, making my weekly check $187.50. My husband, working forty hours at the same place, brought home $300. We made about $25,000 or so between us, working every week of the year. That's a little over $9,000 above the poverty line for a family of two, or an extra $200 or so a week. We made ends meet, but barely. Not well enough to ever really feel comfortable or rest or take a day off without feeling guilty. And we were at the top of the bottom third of households that year, meaning that approximately one-third of the America population is living on the same sort of budget.
Or, for some, a much smaller one. The yearly income of a forty-hour-a-week minimum-wage worker is $15,080. So if you're paying half of that for housing, you're left with $7,540 to live on.
Yearly.
That's $628 per month, or $314 per paycheck, for everything else—food, clothes, car payments, gas. If you're lucky, you get all that money to live on. But who's lucky all of the time, or even most of the time? Maybe you get sick and lose your job. Even if you land a new job, that measly $314 is all you've got to last you until your paychecks at the new place start up. Or what if, God forbid, the car breaks down or you break a bone?
But all right, let's increase that salary. Let's be kind and bump it up well above the median fast-food worker's pay. If you're doing okay, making, say, $10 an hour, that's $20,800. That leaves you $10,400 to live on annually, $867 monthly, $433 per paycheck. Before taxes. (Which, by the way, we pay plenty of.) Not that $100 doesn't make a giant difference, but it's not like you're rolling around in money like Scrooge McDuck simply because you're earning better than the absolute least that can be legally paid.
Of course, those scenarios are if you are absolutely jacked, with half of your income going to rent. If we go with the old one-third recommendation, then your disposable income by paycheck rises a bit, to $418 for those making minimum and $577 if you're at double digits.
So, let's go with the more generous number. Say you make $10 an hour and you pay a third of that in rent. That's going to give you $1,066 a month to spend. You pay your utilities and for gas to get to work. Food and household stuff. Maybe you now have $500 left. And that's assuming, of course, that you have no medical bills or prescriptions or debts. And that's before taxes.
The truth is that what you've got left from all that work you've been doing is about $10 per day to spend on anything other than the barest necessities—and that's based on the premise that you live in a shitty apartment, eat cheaply, and work full-time with no missed days. Then, if you do all of those things and you are unburdened by debt and medical issues, you can do any number of things with your free time! You can rent a movie and buy microwave popcorn. You can drive to the nicer section of town and have fancy coffee. With $10 a day to spend at whim, the world is your oyster. Hell, you could even buy a can of oysters.
I'm hoping that I'm not being too subtle here, because this is what it comes down to: The math doesn't fucking work. You can't thrive on this sort of money. Period. You can survive. That's it.
—
There is something even worse than minimum wage. It's called temp work. I bet that the majority of Americans—unless they've experienced it for themselves—would be shocked to find out that companies regularly hire temps to work full-time hours, but because they hire these workers through temporary work agencies, they have to pay no benefits and offer no job security. To save a buck, companies will regularly hire such workers for years— _years_. __ And they do it because it's cheaper than hiring labor directly, and they are legally entitled to do so. The laws in this country are so weak that we're actually way behind South Korea (!) in temp worker protections.
So when financially comfortable people with health insurance and paid sick leave and all kinds of other benefits that pad their wallets and make their lives easier and healthier think that the poor are poor because somehow we lack the get-up-and-go to change our circumstances . . . well, I'm not sure my reaction is printable.
I regularly thank the gods that I don't have much experience working in the temp industry. I've got friends that do, though, and it's pretty awful. You get to work for a company full-time, as anything from a janitor to an attorney, but you don't get any benefits and they sure as hell aren't telling you to count on keeping this paycheck. They don't guarantee anything. You might have worked there for years, but as long as they keep hiring you through the agency, they can save on pesky things like raises and promotions. One plant I lived near used to hire a revolving number of temp workers whom they laid off after ninety days—the point at which a temp worker is supposed to get permanent job status. Then after three weeks of unemployment, the plant hired them again.
That factory isn't in town anymore. It had gotten a break from the local government, making its first years there tax-free. And wouldn't you know it, after the tax break expired, the company decided that the plant wasn't profitable enough and closed it. A temporary factory that hired temporary workers.
Who says capitalism isn't cruel?
# 2
You Get What You Pay For
As far as I'm concerned, I earn my wages with my scars. Anything above and beyond that is me doing my employers a favor. And I'm not inclined to do favors for people who treat me poorly. See, we work in insane conditions. Dangerous, even. Most kitchens in the middle of the summer are intolerable, with temperatures well into the triple digits. I've seen people sent to the hospital with heatstroke. A lot of us will run into the freezer for a few minutes until we cool down. I'm not a doctor and I can't say for sure, but I'm fairly certain that going from overheated to a minus-5 environment can't be healthy.
My arms and hands are covered in scars from the fryers. Oil at nearly 400 degrees doesn't tickle when it hits your skin, and you can't avoid the spatter entirely. I've burned my hands because the oven gloves had worn through and the owners were too cheap to spring for another pair. I've sliced my fingers open nearly to the bone when knives have slipped. I've dropped equipment on my feet because it was so busy I didn't have time to wash the grease off my hands. I've hurt myself in more ways than I can count because that was how I got my seven or eight bucks an hour.
Stuff like that is unavoidable; it's the nature of the work. We know and understand that when we take the jobs. Any dangerous job is like that; we're not stupid. The point is more that the risk is devalued—that our injuries, rather than being seen as a sign of our willingness to literally bleed for our employers, are seen as a liability.
The kitchen scars are more dramatic, but the emotional toll of retail is the worst. The conditions are patently impossible. I've been expected to spend three hours per shift stocking in the back, while also being told that the register was never to be left unattended. I've been told to always have coffee ready for customers and that it should always be fresh, and in the same breath been told that I was going through too much coffee. My section of the store is always supposed to be neat, but there's only one of me and over three hundred square feet to cover, and there are shoppers everywhere and not enough racks for all this shit to begin with.
My shoe size actually changed with the quality of the jobs I've had. The better ones let me sit down sometimes. At the not-so-nice ones, I've stood for eight to ten hours, and my feet have gotten so swollen that my shoes don't fit.
The mandatory cheerleading is why I never worked for Wal-Mart. Apparently this has changed now, but during employee meetings, they used to require their people to actually cheer. With pelvic thrusting. (Go watch the YouTube videos. It must be seen to be believed.) In those not long ago days, if you didn't wiggle your ass with sufficient vigor, you'd find yourself on the wrong side of management and then brought to the front to lead the cheer yourself. Sure, give me a W and an A and an L and a squiggly (or I guess now it's an asterisk since they rebranded), and I will happily shove them straight up your ass. Friends of mine will swear that they never got demerits until after they upset management by lacking enthusiasm. (To be fair to Wal*Mart, my friends weren't actually let go because they wouldn't wiggle enough. They can't prove causation. It's just that they didn't start getting demerits until they stopped wiggling.)
At work, I'm often told what words to say, and I will be written up if I deviate from the script or combine two steps to save time. In retail, we must acknowledge a customer who comes within a set radius of us with a certain tone and tenor in our voices. In telemarketing, our every word might be scripted. In fast food, we're typically given three greetings to choose from. At one large fast-food chain (let's call it LFC for short), the choices were these:
1) Welcome to LFC, how can I help you?
2) Welcome to LFC, would you like to try a delicious chicken meal today for only $4.99?
3) Welcome to LFC, what can we make fresh for you today?
The company even sent in undercover customers to make sure we stayed on script.
All of our actions are carefully dictated to us. I assume this is because employers think we have monkey brains and are incapable of making decisions. This means that they're paying me to pretend I'm not me and also that I care about you.
And as long as we're on the topic of insane things your bosses can do, you should be aware that you have no legal right to take breaks in America. Go ahead, Google it. Some states mandate breaks. Some farmwork has a federal break mandate. But overall, you've no right to demand a lunch break or a break at all. That's all at the discretion of your employer.
Some people have the luxury of asking themselves whether a job fulfills their career hopes and ambitions. I've got my own metric to gauge the fabulosity of a job: Does that job require me to keep my boss informed of the inner workings of my gastrointestinal system, or am I allowed to go to the bathroom at will? It's physically uncomfortable to hold it forever, and it sucks to stand by for the okay like a dog waiting for someone to open the door. But for me, the indignity of the whole thing is less about the potential bladder infections. It's more what the requirement for that kind of notification reveals about the tone of the place. In my experience, the jobs where the boss regulates your urinary tract also tend to demand a bunch of other degrading stuff.
—
We all know that a lot of folks think that poor people are lazy and incompetent. They think we get fired from jobs because we don't know how to behave, or we're always late, or we just don't care. But what rich people don't realize is how unbelievably easy it is to get fired. And a lot of times what gets you fired is that you're working more than one job.
Whenever you are working for the kind of place that has a corporate office, you're typically given the fewest possible hours—definitely less than full-time, because then they'd have to pay you benefits. (Full-time is often in the twenty-eight- to thirty-two-hours-a-week range, to boot.) But even though your employer might schedule you for twenty hours a week, you might wind up working ten, or thirty. It depends on how busy it is—when it's slow, they send you home, and when it's busy, they expect you to stay late. They also expect you to be able to come in to cover someone's shift if a co-worker gets sick at the last minute. Basically, they're expecting you to be available to work all the time. Scheduling is impossible.
At one chain, I was required to sign a contract stating that I was an at-will employee, that I would be part-time with no benefits, and that if I took another job without permission, I would be subject to termination because the company expected me to be able to come in whenever they found it necessary. And yes, this is legal in the United States of America.
It's unavoidable; even I have had to admit the impossibility of this system and let people go, one an employee that I actually liked very much. Competent, friendly, good sense of humor. But her other boss simply would not post the schedule far enough in advance for me to give the woman any hours. If the workweek started Monday, the schedule at her other job went up Sunday night. I tried to do my scheduling a week or more in advance, and when I called the other restaurant to discuss the issue, the manager told me that she didn't actually feel any need to change her routines and that it was my problem to deal with. I simply had to let the woman go, because her other boss wanted the availability.
How is that legal, you ask? Well, a huge number of jobs in this country—and a crazy high percentage of the jobs that poor people hold down—are considered at-will. Sometimes you'll sign a paper stating that you understand what that means, sometimes not. It depends on the sophistication and size of the business hiring you. What "at-will" means is that your boss can decide that your eyes are too brown one day and let you go on the spot. As long as they're not in violation of civil rights law, they don't have to give you a reason, and they can decide that anything is a fireable offense. I've been fired because my boss made a mistake on some paperwork. I've been fired because I had the flu. I've been fired because I wouldn't sleep with someone. I've been fired because I _did_ sleep with someone. I once saw a stripper fired because she couldn't afford breast implants and the club manager didn't find her natural breasts alluring enough to dance topless for drunken construction workers.
So let's break this down: You're poor, so you desperately need whatever crappy job you can find, and the nature of that crappy job is that you can be fired at any time. Meanwhile, your hours can be cut with no notice, and there's no obligation on the part of your employer to provide severance regardless of why, how, or when they let you go. And we wonder why the poor get poorer?
Of course not every firing is part of an intricate plot by the plutocrats. I've also been fired for calling off work too much ("calling off work," for those unfamiliar with the vernacular, just means that you call your boss to say you're not coming in). Usually I've called off because I was legitimately sick, because I rarely miss work more than I can help. But sometimes it was because my car wouldn't start or because I just couldn't face it. It doesn't matter what you say, and your boss doesn't care; the point is whether you do it too much, not whether your reasons are legit.
I admit it—I've been fired for doing some stupid shit. I've been fired for consistent tardiness because I simply didn't care, and more than once because I gave my boss the finger. And as a manager, I've fired people for being dumbasses—stuff like showing up to work too hung over to stand up straight. Once I had to fire a guy because he went and got knuckle tattoos. I've even fired someone for relentless creepiness. That was the one time I thanked God for at-will states. He wasn't a terrible worker, and there was nothing to point to, but he did brush his groin with his hand once too often while looking at the girls up front.
Idiot pranks are risky too. One kid I worked with got bored and built a castle out of cardboard boxes in the parking lot. They fired him a) because it made the company "look unprofessional," and b) for "time theft." I've seen someone get fired, no shit, because he didn't want to wear buttons proclaiming him proficient at cleaning and other menial tasks. I barely made it through the day without mentioning TPS reports. (If you don't know what those are, drop everything and go watch _Office Space_ right now.)
Mostly, I've fired people because they didn't care about the things that do matter to me. I've never cared any more for the owners of the companies I've worked for than they have for me, but I will kill myself for my co-workers. A lot of us do that. When we work through fevers and injuries and bone weariness, it's for the money but also because if we don't, we know that we'll be leaving our co-workers holding the bag. However bad the shift is, with a man down, it'll be that much worse on whoever's left. There's a siege mentality in the service industry in particular; you go through hell together. If you tap out and go home, you're leaving your co-workers to deal with more customers with even fewer hands. And that means that they're more likely to get fired themselves—because if customers start complaining about the service, the boss doesn't really care that you're covering for someone who's out sick. So you bet your sweet ass that if you work for me and I see you being dead weight, I'll get rid of you.
All of this is not to cast myself as some kind of paragon of work perfection. I'm a terrible corporate manager, every time I tried it. My employees loved me, but I made a lousy guardian of profit margins. My first loyalty is to my co-workers. Then the customers. And then, in a distant third, the company.
For example, when I found out that some of my employees had themselves a fantastic gig pulling the expired salad and bruised or unusable produce out of the Dumpster and taking it home, I started making sure that the food was disposed of _next to_ the trash rather than in it. This, you should know, was highly against the rules on everyone's part.
I figured if I got busted, I'd just say that I was trying to keep track of how much got thrown away to help me order properly the next time. I'm not sure what the company would have done if they'd found out; most companies simply don't want to know about stuff like that because even they don't want to be that harsh, but liability exists. No restaurant can knowingly allow anyone to eat expired food, even if it's obviously still sound. With that said, companies also discourage letting employees eat unservable food because they assume that a worker would have bought food instead of just going without, and heaven knows it's a sin to lose potential profits from workers! Only, most people don't buy their food half-off at their own stores; most people just drink more on hungry shifts when they can't eat. I always figured that my cooks would probably not be doing their best work if they were salivating every time some food finished cooking. And I just couldn't live with myself letting these guys look longingly at the burgers they were flipping as if they were Victorian street urchins lusting after a hot roll in a bakery window.
If one of my people was hungry, I gave them food. I'd send parents home with boxes of expired chicken nuggets for their kids. My bosses, of course, generally hated dealing with me. It's been a pattern. I don't really blame them—their jobs sucked as much as mine did and I was a huge pain in their asses. One of my favorite bosses once told me that he hated having to explain the _why_ of everything to me, but I considered it my job to be able to explain the why to the people who reported to me. If hours were getting cut or pay frozen, I damn well was going to give them a reason that made sense. If we were going to lay off a quarter of the staff, I'd better be able to explain it.
I know a lot of people think that I'm supposed to be a good little worker bee and do my part to help move the wheels of capitalism. I just don't see what's in it for me anymore beyond my little paycheck. Think about it this way: At my earning peak, I made approximately nineteen cents a minute before taxes.
So when I go out of my way to work hard, I'm not doing it for my bosses, I'm doing it for my co-workers. There's definitely a mutual covering of asses going on in the lower classes. (Hey, why should the upper classes do all the ass covering?) I've even tracked down babysitters for employees who'd lost their child care and couldn't afford to lose their shift as well. Instead of letting an employee call off work and winding up shorthanded to boot, I called around until I found a cashier who was more than happy to babysit for a few hours for some extra cash. I loaned the cook the money to pay the cashier, and everyone got something they needed. We do shit like that a lot.
We'd never survive otherwise.
—
Once I'm home from my shift, I try not to be short-tempered with my husband, whose fault my bad mood decidedly isn't. In turn, he tries not to be short-tempered with me. Working at a low-wage job means getting off work and having just enough mental energy to realize what you could be doing with your life . . . if only you could work up the will to physically move.
And honestly, I wouldn't even mind the degradations of my work life so much if the privileged and powerful were honest about it. If they just admitted that this is simply impossible. Instead, we're told to work harder and be grateful we have jobs, food, and a roof over our heads. And for fuck's sake, we are. But in exchange for all that work we're doing, and all our miserable work conditions, we're not allowed to demand anything in return. No sense of accomplishment, or respect from above, or job security. We are expected not to feel entitled to these things. Being poor while working hard is fucking crushing. It's living in a nightmare where the walls just never stop closing in on you.
I resent the fuck out of it every time my schedule's been cut and then I've been called in for tons of extra hours, as though my time weren't worth anything, just so that my boss can be sure not to pay me for a minute that I'm not absolutely necessary. I resent signing away my ability to get a second job and being told that I can't work more than twenty-eight hours a week either.
The result of all of this? I just give up _caring_ about work. I lose the energy, the bounce, the willingness. I'll perform as directed, but no more than that. I've rarely had a boss who gave me any indication that he valued me more highly than my uniform—we were that interchangeable—so I don't go out of my way for my bosses either. The problem I have isn't just being undervalued—it's that it feels as though people go out of their way to make sure you know how useless you are.
I'd been working for one company for over a year when I injured myself at work in November and had to go on leave for two months because I couldn't stand for long. So I wasn't invited to the company Christmas party. I went as a co-worker's date and watched as everyone got their Christmas bonuses. I didn't get one; I was technically not in the managerial position and thus didn't qualify. The fact that I'd worked the rest of the year didn't count.
What really got me, though, was when the owner of the company thanked the woman who was filling in for me for working so hard all year. He didn't recognize me at all.
With unwavering support like that, it's not really a mystery why I've rarely felt huge personal drive to make more money for the people signing my checks. I'm as loyal as they pay me to be, basically. Most of the people I know are the same way. It's only logical. See, if we perform really well, give it a full 120 percent, we might make shift manager. That's a whole extra $2 an hour. For that $2 or so, we get to be in the direct line of fire for the profit margins. We get to be held responsible for things outside our control.
And we get to be stuck.
If you're working at your typical service job, shift manager is about as high as you can get, because for every four or six shift managers, there's only one general manager position. But let's say that the company treats everyone so poorly that turnover is high. Then you might make assistant or even general manager, at which point you'll earn somewhere between $20,000 and $35,000 in exchange for physically punishing, emotionally draining eighty- or ninety-hour weeks. (Salaries in cities are generally higher, but both companies I worked for capped out in the mid-$30,000 range.) I'll put it this way: As general manager for a chain restaurant, I got eight days of maternity leave after I had my second daughter. Unpaid.
It's not like we don't wish for more, but really, what's the better option? School is an investment that doesn't make sense for people who aren't the academic sort. You have to pay cash money for it, you can't hold down as many hours at work, it's harder to find work because your schedule's inflexible, and dear God the cost of textbooks is enough to kill you. Hell, I _am_ the academic sort, and for many years school wasn't a good investment for me. It left me in debt with nothing to show for it.
Before I moved into the service economy, I tried to make a more fulfilling, less backbreaking living working in political organizing. To be clear: The jobs that I worked at in politics weren't exactly highly paid either. They were typically in the $8 to $10 hourly range. I laughed my ass off when people went digging through my financial history after my essay on poverty was published and found the Federal Election Commission filings of my political pay. (Pro tip for amateur PI sorts: Those numbers? That's how much I got for the _whole year_ , __ not per paycheck. Seriously, how much do you really think they pay someone to knock on doors or coordinate other people doing it?) The dark truth of many fulfilling, creative jobs and industries is that you are expected to accept very little pay at the start, just for the privilege of learning the ropes and working your way up. And that's fine if you've got Mom and Dad helping you. But if not, you tend not to go into those fields. Which means that the people who do go into those fields are often pretty privileged; not many Congressional staffers come out of the lower class.
And it's not just about how little you are paid in fields like politics. It's also the stuff you're expected to do in addition. For example, there are constant training sessions during the off-season. Most of them cost money and are held in Washington, DC. All of my friends who still work in politics went to them. I didn't. All of my friends who took short-term, low-pay jobs with people who could be mentors are still working in politics. I had to turn those jobs down—the ones I was offered, anyway. Often I didn't even bother to send in my résumé in the first place, because I knew I couldn't afford to work for so little. Mostly, I found myself perpetually stuck on the bottom rung, watching people I'd started out with vault above me because they weren't doing anything but this and they could afford to take the financial hits while they were paying their dues.
Here's another thing the poor can't afford: unpaid internships. I've had to turn down offers that might have improved my circumstances in the long run because I just couldn't afford to work for nothing. Again, the people who can afford unpaid internships are getting help from home—in my world, everyone else has to work for a living. And this means that we're being cut out of all that potential networking too. That's at least one reason why I've never had much of a professional network—I never had the chance to build one. Accepting an unpaid internship, or one of those internships that basically pays you lunch money, is for people who don't have to pay the rent.
Because I've always been in a take-what-you-can-get situation, I've wound up working the sorts of jobs that people consider beneath them. And yet people still wonder why we, working at the bottom, aren't putting our souls into our jobs. In turn, I wonder about people who think that those who are poor shouldn't demand reciprocity from their employers. We should devote ourselves to something that doesn't benefit us more than it absolutely has to? We're meant to care about their best interests, but they don't have to care about ours? If you're going to put as little as possible into my training and wages, if you're going to make sure that I can't get enough hours to survive in order to avoid giving me health care, and generally make sure that I'm as uncomfortable as possible at any given time just to make sure I know my place, then how can you expect me to care about your profit margin?
Remember, you get what you pay for.
# 3
You Can't Pay a Doctor in Chickens Anymore
Excruciating should be defined in the dictionary as an exposed nerve. Once I killed nearly a whole bottle of vodka in the space of a night, and I'm not a frequent drinker. I was at least six shots gone before the pain started to fade into blessed numbness.
It took me a few years of the long slide into poverty to cotton on to the unavailability of anything besides crisis medical care. I'd come from a home in which we went to the doctor when we needed to. Dad had benefits. It never occurred to me as a kid to question it. And it took me a while as an adult to understand that without benefits, which no longer come standard-issue with your average job like they used to, hospital administrators would rather you die on the street than sully their expensive sheets. (And the sheets are expensive. Like the Tylenol. Whole books have been written about that, and I can't do the subject justice in a few paragraphs, but don't think we don't know that they charge us triple for those lifesaving medications because we are not rich enough to have other rich people negotiate better prices for us.)
Being healthy and being poor are generally mutually exclusive conditions. We all have physical weaknesses, but a rich person gets these tended to before they get out of control. Poor people don't have that luxury. So it's pretty enraging to poor people when rich people, who get preventive care and can afford vitamins and gym memberships, look down on us as if we don't have a clue how to take care of our bodies. We know—we just can't afford it.
—
Dentistry is one of the things we are most lacking in. And it's one of the most glaring marks of poverty. I watch the tooth-bleaching ads and cringe, because I know exactly what I'm being pegged as. Incapable. Uneducated. Oblivious. What I should be pegged as: uninsured, and until recently, uninsurable.
I did get some dental surgery once. I had five teeth pulled and a partial denture built so that at least I would have front teeth. I think I was nearly twenty-six at the time. I made an appointment, which took all the force of will I had. I got in the chair at the office, and promptly listened to forty-five straight minutes of the most upsetting, judgmental lecture I'd ever received in my life. This woman, the dentist, decided that I must be on meth. (I'd like to make this point clear: I have never in my life done meth. Ever. Other drugs, sure, but not this one in particular. It seems to me that because I have failed so much, been weak so often, I am prouder of those things I have managed to avoid. It's doubly bad, then, to be accused of the things you _haven't_ done.)
Never mind that I had none of the other signs of being a meth addict; my skin, while not exactly in great shape, lacks the _huge fucking sores_ you get while on meth. My face, while much slimmer in recent years, isn't skeletal. I'm sometimes a bit energetic, but I'm never tweaked-out twitchy. In short, calling me a meth user because I have bad teeth is about as valid as calling me a genius because I'm a fast reader.
This dentist had come to her decision, though, no matter what I said. She made a point of telling me that they didn't make dentures as discolored as I'd need and that I'd have to get used to having everyone see how dark my teeth were in comparison with these shiny white front teeth I'd have on the right side. She told me all this, with her poky metal shit in my mouth, and I wondered whether she was intentionally hitting the sore spots. I'm sure she dispensed actual medical advice at some point, but I stopped listening. Instead, I wondered whether she'd bother to take out all the bone fragments that needed removing or whether she'd just let them heal over and cause me trouble. I wondered how many people came back for this kind of idiocy.
So I had my surgery, got a denture plate in place of my front teeth, and never went back. Call it weakness, call it cowardice, it'd be true. There is a shred of dignity that I will not let go of. I will not intentionally put myself in that situation again.
And that's why I don't like dentists. I have never in my life felt more attacked, more vulnerable, trashier than I did in that dentist's chair. At least when people on the Internet call you a meth user, you can console yourself with the fact that these people are idiots, as evidenced by the fact that they have nothing better to do than cast aspersions at strangers online. When a dentist does it, drill in hand, it's impossible not to worry that maybe that person is a serial killer, and fuck that. Not doing it again. Not even risking it. And it's not like there's a huge pool of dentists out there who will treat someone like me on a payment plan. I can't just shop around until I find one with a decent bedside manner.
My denture from that surgery broke about two years later. It just snapped while I was trying to eat a hamburger, separating the plate that fits on the roof of my mouth from the actual visible teeth part. I superglued it together for a while, until it wore down around the raw edges and wouldn't fit properly. Now I just use a lot of dental paste and try to never consume anything in front of another human.
So that was kind of awful. Worse, my teeth are actually one of the things I can honestly say aren't my fault. My destroyed teeth are the result of a car accident nearly a decade ago, in which the other driver was drunk and high and had been busted for those things so many times they'd revoked his license. There was no question of liability.
I was in the passenger seat because I hate driving in cities and always let others take that honor if possible, and my jaw hit the dash so hard I exploded the airbag. Over time it became clear that I had nearly exploded my jaw along with it.
I had car insurance, sure, but it only covered liability and uninsured drivers. (Thank God for _that_ extra five bucks in coverage a month!) I needed a car to get to work. So when the insurance company offered me a settlement check, I didn't think twice about signing the waiver (which, it turned out, meant that I had no right to future damages). I took it and bought another car. I didn't realize that check would be it—that there was no more money coming to take care of the damage the other driver had done to _me_. I thought they were just separate claims or something. I'd never filed a major insurance claim before; I had no idea what I was doing.
So that's how I found myself with a mouthful of fucked-up teeth and no resources to deal with them. Truthfully, even if I'd known what that waiver meant, I'm not sure that I'd have made a different decision. If it was a choice between my teeth and my car, I had to choose the car. I could survive with bad teeth, but I'd starve and lose my apartment without a car to get me to and from work. That said, I never would have imagined that dental care wouldn't be something I'd have access to for nearly a decade.
So I got the car (which turned out to be a lemon, _because of course it did_ ) __ and kept working, and over the years my teeth have continued to decay. I've brushed, flossed, rinsed religiously. And the cavities spread regardless. I bought a Waterpik. I bought made-for-TV mouth-cleaning tools.
Nothing helps.
My teeth are, since my story went viral, a thing I now talk about. But until the moment that I went full fuck-you gutterpunk and took them out for the whole Internet's viewing to underscore the effect of my dental problems, I hid them. I spent years learning to speak with my mouth closed, learning how to fake eat in public when I couldn't avoid it. I rarely told anyone when my mouth was hurting. It's not like I have an option now, but there's nothing that shames me more than acknowledging that I have failed at this too—this basic idea of keeping your own bones and enamel to yourself, of _having_ them at all. Nothing is worse than eating in public, because I mostly can't eat with my broken denture in. I usually eat alone, at night, tearing off bits of food and bolting them down without chewing whenever my stomach tells me that it can't wait any longer. There is no joy in food for me anymore; it is a necessary evil, something I consume to stay alive but lacking in anything like taste or texture. I don't eat much.
I've lost a lot of weight. People keep asking me how I've done it, and I always wonder what I should say. Mostly, I tell them that it's just losing baby fat now that I am out of my twenties. Sometimes I seriously consider telling them that they really ought to try a nice strong periodontal disease (it does _wonders_ for your thighs!).
I don't smile. Someone found a picture of me smiling from back in 2006, before my front teeth went and a wisdom tooth cracked off. It is one of the last times I smiled on camera, if not _the_ last. I don't allow people to take my picture anymore because nobody can ever just take a picture. Everyone wants you to grin like an insane person. They will cajole and wheedle and bring the whole group photo to a screeching halt until you finally, shamefully, admit that you can't, that you don't want a picture of you like this to exist. Or you have to be an ass, irrationally angry about a seemingly innocuous request. That'll get you out of it too. I actually don't mind being in pictures and I wish I had more to remember my friends and milestones with, but I've spent the better part of a decade telling everyone that I have a huge aversion, that it's best not to ask or expect, because I don't want to deal with the inevitable "Smile!"
Actually, never smiling has had an interesting impact on my life. I can't repress laughing with my friends, the people who are safe, who can see a broken mouth and not notice it. But among people who don't know me, about half of my jokes fall flat, because I am not doing the human thing and grinning my way through, making clear that my dry observation is meant to be amusing rather than cutting. So I learned to stop telling jokes, because while I have a lively sense of humor, I can't properly express it with my face.
It even messes with my relationships. My husband, for obvious reasons, would like to kiss me. I, for obvious reasons, feel like kissing is the anti-sex; once I have been reminded that I have teeth, I cease feeling anything like alluring.
My teeth have become one of my most hated obsessions. I'm constantly reminding myself to keep my fucking mouth shut (which has its side benefits in that never shutting up has been a problem for me in my life) and to make sure my denture is adjusted properly so I don't have weird sunken-mouth lips. I have two broken-to-the-point-of-missing teeth that are visible on the right top side, and I use cotton wadding to cover that as far as the basic "something vaguely whitish that has mass" concerns. I worry at my teeth with my tongue, testing which are still sound enough to masticate should I be caught in a rare public eating situation. I take prophylactic ibuprofen so the swelling doesn't get out of control. There's no good way to predict the swelling, and once it's started, the pain isn't quite the worst, but your productivity is pretty much gone for the day. As soon as the swelling sets in, there isn't much you can do besides hold ice to your cheek and pray.
When I was in acute pain, before I learned better, I used to go to urgent care or the ER. A lot of urgent cares won't dispense painkillers. My guess has always been that they assume you're an addict or a seller. In the ER, I think they figure that the wait and the bills are enough to deter most abusers, so they'll give you a day or maybe two of real no-shit medicine to get you through a few days' work. To get any sort of actual medicinal regimen, you have to have an actual doctor, a general practitioner. I don't have time to chase down a doctor's appointment when I'm in pain.
So why, I am asked, have I simply not gone to one of the free dental clinics? Well, because they aren't exactly flinging their doors open. I've researched some programs, looking for anyone who could help. Sometimes I am too rich, because I have a job at all. Sometimes I live in the wrong county, and the grant providing the funds is restricted to residents of the next county. A few times I've been unable to take off enough time from work to make it to where the clinic is, much less to do it for the multiple visits required to complete the job. Twice I've been told that they don't do critical cases, only basic cleanings and fillings, both of which are laughably inadequate at this point. So I have carried on, hoping to get dental insurance at some point. What I refused to confront or articulate for years was that it was likely I'd simply wind up being one of those gross people with no teeth. Probably by the age of thirty-five.
But rationality rarely enters into health care. Mostly, at least for me, medicine has been a patchwork of what's around when I really can't avoid seeking care for a second longer. And most of my interactions with the health care industry have pretty much made me want to avoid it all the more from then on. ER visits usually involve waiting for hours and then being handed a couple of ibuprofen for my trouble. And the whole time I'm waiting for those ibuprofen, I get to wonder what the bill's going to come out to and whether I should stay and wait longer or just go home and hope for the best.
—
Look, I'm not stupid. I can _be_ stupid, but I'm usually fairly savvy. I can read at a college level. I can do complex math problems given enough time and scratch paper. But I had trouble finding medical care.
Well, scratch that. I have had trouble finding _decent_ medical care. It's why I didn't have prenatal care for my oldest daughter. I found out I was pregnant in October, days before the last election I ever worked on. I had a suspicion I was pregnant—I mean, that's why I'd peed on seven pregnancy sticks, all of which had turned out positive. But I couldn't bring myself to believe the results, since I'd been told so many times that it was practically impossible for me to get pregnant. I made a command decision that all those store-bought tests had to have been defective. So I went to the local Planned Parenthood and requested a blood test. But after the nurse heard about all the tests I'd already taken she just laughed and went straight for an ultrasound. Sure enough, within three seconds she told me that I was already six weeks gone.
I didn't think about the pregnancy much to begin with; I had a job to finish, then we'd sort out what to do next. I knew that I'd be facing weeks of unemployment after Election Day, and I could sort out prenatal and baby products and such then. The pregnancy prompted our decision to send my husband to school; we'd been thinking about it since he came home from Iraq, and it seemed as good a time as any to have a guaranteed income. The GI Bill, along with paying for tuition, pays a living stipend. It would just cover all of our expenses if we were careful. I could stay home with the baby until I was ready to go back to work, and then we'd be in a decent position until he graduated. The stipend wasn't so much that we wouldn't qualify for Medicaid, so the birth itself would be covered.
It didn't exactly go according to plan. First, we qualified for Medicaid, and I started looking for an OB. There weren't a ton of doctors accepting new Medicaid patients. Planned Parenthood doesn't do prenatal care. I found my clinic through a flyer, advertising that it did in fact accept Medicaid and was enrolling NOW! In the waiting room for my first appointment, I realized that I was at a faith-based clinic. It was a church ministry.
Now, normally I'm cool with the Jesus folks doing the poor-people tending. It's sort of their mandate, and I honestly do not care about the religious beliefs of anyone willing to make sure my kid gestates properly. But there are the charities that happen to be church-run, and then there are the church charities. I was at one of the latter. That distinction is important: Some ministries are set up by churches to provide a service, and some seem to be set up to proselytize, tacking the service on as an afterthought.
When I showed up, I was ushered into an office, where I did the initial paperwork and learned about all the things the woman helping me praised Jesus for. Her pencil didn't break, praise Jesus. The weather was decent, praise Jesus. I honestly do not know what was in the paperwork she was walking me through; I was much too fascinated by this person who was nearly finished with the third page, praise Jesus.
After that, I was taken to an exam room, where I was greeted by a lovely young woman who took my blood pressure and asked me if I had a church home. She was followed by a nurse who told me that Jesus had a plan for this baby and congratulated me on making the decision to bear it. I asked about maybe getting another ultrasound—my weird hormones and the sudden ability to bear children had me freaking out that this kid wasn't viable, and I was terrified of coming to terms with having a child only to discover that it wouldn't make it. But I was told that they only did ultrasounds in the third trimester unless there was a problem.
And that was the end of my appointment. No reassurance, no actual medical advice, no real exam. Just some routine tests and the clear message that Jesus wanted me to have this baby. I, certain that Jesus also wanted me to have an ultrasound and pretty sure that I could manage a pregnancy just as well without that sort of help, never went back. There didn't seem to be much point in returning to a place that gave no better advice than to drink a lot of water and not get into a hot tub, which were both helpfully bullet-pointed on the packet of papers they sent home with me.
I did take a few stabs at finding a different clinic. The ones with open spots didn't take Medicaid, and the ones that accepted Medicaid were full. So instead, I read a lot of books, called all of my old friends who had kids, and compulsively Googled things to find out whether they were normal or whether I should present myself at the ER. Eventually, I did just that when my daughter finally decided to arrive.
Any hospital in a large city is used to random pregnant women showing up to give birth. I think, though, that most of them have a doctor. They wanted to know who mine was, and I told them that I was pretty sure whoever was on call that night would be my doctor.
I actually don't remember most of the process. I was in a room, then another room, and I was kind of too busy being in labor to really care what was happening. Tom took care of the paperwork; we gave them the Medicaid card and that was pretty much it. Then I had a baby. I think the process was probably streamlined given that there was going to be a baby soon whether or not the paperwork was done, and they much preferred that I give birth in the birthing room instead of in the waiting room, where it would be rather hard to clean up afterward.
We were visited by social workers a lot in the next days. I don't know what all I filled out; they showed up at random times. If I was awake, they had me fill out paperwork. If I was asleep, they woke me and had me fill out paperwork. I'd failed to plan ahead and bring pay stubs with me, which the lady was kind of miffed about, so I had to bring those in later.
—
I've been called crazy. It's not untrue. I suffer from a syndrome called We Don't Know What the Fuck Your Damage Is. That is to say, I've been diagnosed with so many things that it's impossible to tell what's likely the real problem. I mean, clearly I struggle. Things that are simple for most people don't come naturally to me. I have trouble bending my will to anyone, including myself. I'm reckless and impulsive. I'm irrational and prone to anger when I am in certain moods, and those moods occur more frequently than you'd expect. I go through depressive phases, lasting days or months, and I can destroy my entire life through sheer inattention in three weeks flat, because even if you can't muster the energy or the will to open your mail, they still want the money for the bills. At times I'm an insomniac, and at others I can't get out of bed. None of these things are typically so bad for me as to be unmanageable, but managing them sometimes doesn't leave much room for anything else.
It's not like mental health clinics are thick on the ground, like the people who need their services. Being poor in and of itself is an aggravating factor in a lot of mental illnesses; the stress is pretty brutal. If you're already kind of fragile, it can be really rough. I won't say that a good clinic for poor people doesn't exist somewhere, but I've never found a mental health professional who was willing or able to deal with only the parts I needed to fix, the insomnia and paralysis and depression (at least some of which is situational, and I'll discuss that more in Chapter 4).
When I have sought treatment for these things, the professionals seem to only want to talk about my anger. They talk about my fatalism, my caustic outlook. They see these things as problems to be fixed. Personally, I think that anger is the only rational response to my world sometimes, but when you're asking for services, you don't get to pick what they treat you for. Either you agree with them or you're labeled uncooperative and kicked out of the program.
The last time I found myself really struggling and went in, they told me that I would have to spend hours in treatment each week. And that was the only option. It was either that or no treatment at all. So I chose the latter. Now, to be fair, I showed up in a right state. I was having a bit of a meltdown because I was terrible at my job and putting in too many hours to be failing that hard, and my husband was having a rough patch, and the kids were sick, and I had just realized again that this was it, this was life, this was how it was going to be until I died. The best I could hope for was that not all of these things would happen at once too often. I can see them thinking I was seriously this critical all the time.
I went to the clinic hoping that I could develop a relationship with a therapist who would then be able to prescribe me the drugs that have made me competent and invulnerable, the ones that stave off emotional disaster so that I can simply get through the crunch. Even at my most breakdowny, I generally realize that I am reacting irrationally. What I need from the mental health system that I have never been able to get is just enough support to maintain.
What I need, what would probably actually improve my life outcome, is someone who I can call, can see frequently for short stretches when I've hit a rough patch, and can then not call when I'm okay. Someone who knows my history and won't question it when I call and ask, apropos of nothing, for something to help me sleep or avoid panic attacks. I need someone who's worked with me for long enough to understand that I don't really like medicine and that if I'm asking, it's serious. In short, I need the kind of mental health support that many people with quality insurance take for granted.
When I've had the guts to see a doctor about an ailment, I haven't had the access, and when I've had the access, I haven't had the guts. Until quite recently, I was scared to death that if a doctor ever did find something really wrong with me, I'd be _completely_ uninsurable, so I never went to the ER for anything that wasn't obvious and small, like a bad flu or potentially broken ankle.
Mostly, if I'm honest, I've been scared of the _look_. It's in doctor's offices and around social workers where I get the lectures, the judgments, the stares. People treat me like I'm a fucking idiot, as though I am incapable of noticing this rather large problem, rather than incapable of addressing it before it becomes such a large problem.
I came in for a fair bit of judgment over a cyst I developed. Doctors assumed I was just too ignorant to notice it, rather than the truth, which was that I lacked insurance and it wasn't life-threatening. I promise you, I was aware of the cyst I had for years. You can look up the gory details—it's called a pilonidal abscess. I think it's due to a tailbone injury I had as a teenager. A couple of times a year, my ass would swell and I'd smell like a rotting corpse for a few days in addition to the rather painful fact that I couldn't sit or stand in any position that didn't add to the pressure of the infection. It wasn't until just after my first daughter was born, when I had three months of Medicaid left, that I could have it excised. Prior to that, it landed me in the ER more than once, and every time, I'd be told patronizingly that I could simply have it taken care of, and probably should. Every time, I asked the doctor if they'd be willing to do the surgery at the rate I could afford; while I didn't have any takers, it did, at least, ensure that I didn't have to hear yet another explanation that surgery exists—as if it were something I'd never heard of simply because it was something I couldn't have.
Preventative medicine, man, it's a miracle. You can go to orthodontists and surgeons and eye doctors and rehab facilities after you throw out your back so that you don't wind up bedridden and debilitated. You get antibiotics and painkillers and blood pressure medicine.
Seriously, vision care alone is a miracle that only happens to the rich, never mind the rest. They don't get deep forehead wrinkles at thirty from spending their twenties squinting, they don't get headaches that cause them to take a large amount of ibuprofen every day—which, as all the bottles are pretty clear on, can't be good for you. They can see something in the first second it comes into their field of vision instead of five seconds later. The glasses and the decent food and the orthodontists—all of those things require money.
There is a price point for good health in America, and I have rarely been able to meet it. I choose not to pursue treatment if it will cost me more than it will gain me, and my cost-benefit is done in more than dollars. I have to think of whether I can afford any potential treatment emotionally, financially, and timewise. I have to sort out whether I can afford to change my life enough to make any treatment worth it—I've been told by more than one therapist that I'd be fine if I simply reduced the amount of stress in my life. It's true, albeit unhelpful. Doctors are fans of telling you to sleep and eat properly, as though that were a thing one can simply do.
Now, I'm not saying the system doesn't work at all. I've had lifesaving treatment, like when my throat swelled so much they had to put a tube in it to keep me breathing. I've got friends who can leave their houses only because they found a program to get them a wheelchair. Many people have needs that the system is built to meet, and it does that fairly efficiently to the extent that there's money.
The trouble is that we've left so many holes in the safety net Moby-Dick could swim through it. The system can't support everyone who needs the help, and it's led to a pastiche of half-finished treatments and conflicting diagnoses. We have the technology. Maybe we can start using it? There are a lot of us that would be awfully pleased to get some antibiotics.
# 4
I'm Not Angry So Much as I'm Really Tired
Almost nothing is more degrading than standing in a welfare line. The people who are looking at you know exactly how much money you make, because they know how poor you have to be to qualify. And the workers are either lovely or the worst human beings you'd ever care to meet. I had a caseworker who called just to check in because she knew I'd gotten a new job. And I had one who ignored me completely, just had me sit silently at her desk until she needed me to verify my information. Then she ignored me some more, and then she told me I could go. I left, with no idea what had just happened. I called the state to find out what changes she'd made to my file the next day rather than speak up during that incredibly effective stonewall.
I've felt the poorest with the people who were supposed to be helping me. I get that their jobs suck and they're overworked, but I go out of my way to not be another asshole customer. I have my paperwork and a list of questions ready to go. I have all my references, my pay stubs, medical bills, everything. Indexed. Sometimes I don't have a document, but then it's on my list of questions, to find out what I can use as a substitute. But often, none of that matters, because I am poor and asking for the benefits that I am qualified for and entitled to as a citizen, and in some people's eyes that makes me less than human.
Often enough, I _feel_ less than human—or less than the human that I know myself to be. For example, I love to read. I'm a naturally curious person, apt to ask uncomfortable questions without realizing it because I just want to know something. But I don't read when I'm working at minimum wage or near it. I'm too tired. I fall asleep because the effort of moving my eyes across the page and processing information is simply too much; my brain won't allow me to use what little energy I have left on frivolities like self-improvement. It just wants me to stare blankly at a wall or flickering screen until I pass out.
Understand that when I say I am tired and in the same breath bitch about a lack of hours at work, it's because I'm counting the totality of the shit that I have to deal with while being poor. It is super-inconvenient, all the time.
There's one episode of my life in particular that was just the worst. I was working two jobs, with no car. I lived two miles from one job and three miles from the other. It wasn't an inhuman amount of mileage; some people run that for fun. But then afterward, some people go home and relax.
So I'd get up in the morning, walk to work at about five a.m., and wait tables from six to about noon. I'd be home by about one, at which point I'd pass out unless I had errands to run. Then I'd get up at six, shower and fix my hair for the bar, walk three miles, tend bar until one or two in the morning, and either beg a ride from a co-worker or walk home. I'd get home at two or three, unwind, take a short nap, and start all over again.
Now, nobody can maintain that forever, and if I'd been lucky enough to get that many hours, I'd have been doing okay. The problem was that both these jobs were weekends or prime eating-out days only—three or sometimes four days each. So I'd spend Monday recuperating from the weekend, Tuesday trying to find better work (which also required more than a few miles of walking around dropping off applications), Wednesday taking care of the house, and Thursday taking a spare shift from one or the other job.
In other words, my commuting time was comparable to a typical suburbanite's: one, maybe two hours. Except mine was on foot, and it was to jobs at which I was on my feet all day. It's why I've never felt much need to exercise; I spend hours each day lifting heavy things and bending into impossible positions to get through stockrooms. I've stood and repeated so many times that I can assemble a cheeseburger in twenty seconds flat, assuming it's got multiple toppings. Less, if it's simple. You get plenty of miles in while running around a retail store or factory floor.
So I was either working or walking to or from work, about sixty hours every week. How did I spend my remaining time? Well, remember that I was walking. I lived in a fairly central location, but it was just about a mile or two from anything I needed, like a grocery store or a Laundromat. I did laundry twice over the weekend because I could make my clothes last two shifts but not three, so that was six hours. I went grocery shopping once a week, so that was four hours. I slept, so that was around fifty. I spent eight hours or so every week looking for work locally. I went to the unemployment office once a week to check the job boards, and that was five hours. I generally picked up a spare shift on Thursdays, so that was another six or so. I showered at least twice a day, what with all the walking, so that was about seven hours a week gone to washing or drying myself. And that leaves about three hours a day for everything else.
I was always and forever dreading the next time I'd have to get off the couch. I would finally sit down, and I would realize that if I had any hope of waking up at a reasonable hour tomorrow, I really did have to be in bed in three hours, and the dishes still needed to be done, and the toilet needed to be scrubbed, and I'd promised someone I'd make them dinner because I owed them and they got sick and called in the favor.
When some wealthier people sense an unwillingness in lower-paid workers to move faster than they absolutely have to, or to do much of anything with their free time, it's because we are marshaling our resources. We're not lazy, we're stockpiling leisure while we can. I can't tolerate more mental exercise after a full day of logistics and worry. Full capacity just isn't an option.
We start the day with a deficit. Most poor people don't wake up feeling refreshed and rested. When I wake up in the morning, I'm in pain. If it's ragweed or wood-burning season, I wake up with insane headaches. If I'm spared that, there's still my aching back, stiff from a night on a mattress that was worn out long ago. There's not a moment in my life that my mouth doesn't hurt; my tongue is raw from touching broken teeth and my jaw isn't any happier about them. (I fully realize that some of the trouble is that I don't know how bad I feel. There's no baseline, no normal "healthy" to compare an average day with.)
I'm not trying to say that only poor people feel pain. The point here is that life is a bit peachier if you have medicine or are under a doctor's supervision to treat these things. Allergies are less severe if you get allergy shots. My headaches are partially due to my jaw-teeth trouble. I realize the aging process would suck enough on its own—I'm generally less than pleased to have it helped along on a daily basis because I don't have enough money to seek proper medical attention. For fuck's sake, a decent _mattress_ can be considered a contributor to an optimal health outcome.
But poor people wake up knowing that today, no matter how physically shitty we may feel, we can't call in sick or slack off at our desk surfing the Internet. We have to go to our crappy jobs no matter what. We will feel guilty about the bills and the dishes and we will firmly put them out of our mind as we march out the door in our polyester uniform shirts. Or worse, we will have to find something to do with our endless unemployed hours.
Sometimes, that's all the day is, just another gray nothing. Other times, it's already a bad day and people just have to fucking push me. I've got a bit of a temper, and I have trouble holding my tongue when I'm pretty sure someone's being an asshole. My record from waking up to losing it is in the neighborhood of an hour. Mostly I make it through a whole day, but sometimes it's just not in the cards. The night before my record-setting morning, I'd made it home from work at ten p.m. and passed out by eleven. I'd been working extra and was short on sleep to begin with. My boss called at five a.m. wanting me to come in. I drank some coffee and dragged my sorry ass out the door, and when I showed up, he was mad that it had taken me half an hour to come in. He'd been under the impression that when I said, "I'll be there," I meant that I'd use my teleportation device instead of the beater car I had at the time. I blew it off, figuring that he was just in a bad mood. But he simply couldn't let it go—every time someone complained about this or that setup not being done properly, he said that if only I'd been there on time we'd have made it.
I lost it. Completely. This is the version of what I said that I can best remember through my blistering rage: "If you think I'm so goddamned terrible, why did you call me in? Did you not realize that I'd be on a fourteen-hour shift and that I was running on a few miserable fucking hours of sleep? WHAT IS WRONG WITH YOU, YOU INCOMPETENT FUCKING ASSHOLE?" And I said all this in my outdoor voice. In front of customers. I spent the afternoon looking for work, as I was newly unemployed.
Being poor is something like always being followed around by violins making "tense" movie music. You know that commercial where the band Survivor follows a guy around playing "Eye of the Tiger"? Yeah, it's like that, but the musicians are invisible and they're playing the shower scene from _Psycho_. Nobody likes being harried, but for a lot of us it starts upon waking and doesn't let up until we crash at night. Eventually, you just know that something bad is going to happen. That's not paranoia or pessimism; it's reality.
When my story went viral, I got a lot of blowback from people demanding to know how I dared to have children while I was living in a weekly motel. Well, I'll tell you: That's not how we started out the pregnancy. The VA didn't end up paying us the living stipend that we'd expected so we'd gotten a cheap apartment. That was fine, for the short term. Until one day, when I was heavily pregnant, a summer storm flooded our apartment and destroyed everything we owned.
The landlord hadn't paid for proper maintenance on the storm drains, and they backed up. We didn't have family in the area, so we went to stay at the motel while we sorted out the damage. We'd been in touch with maintenance, who'd assured us that they'd take care of the water.
What we hadn't realized was that the landlord's version of "taking care of it" was having the guys run a Shop-Vac for a while and then set up some box fans. This was to take care of a flood that was feet deep. The water soaked into the concrete walls so thoroughly that when we stopped in a few days later, you could see the mold growing to above your head.
We didn't have enough money to pay for both the motel and our rent. We called the landlord to get a new apartment, maybe one that wasn't toxic, and were told that the apartment was fine now that it was dry. We called the health department and the press, neither of whom cared much. The health department guy, in all fairness, happened to not be in charge of this particular issue and couldn't tell me who was. But he agreed that we definitely shouldn't live there, especially not with a baby.
The result? The landlord sued for eviction because we weren't paying the rent on our flooded apartment. Cue the movie violins. Something as simple as a summer storm can mean disaster. So I learned to simply expect that if things felt like they were going rather too well, something would come along to knock me back into reality.
Gruff attitudes are rife among people with low-wage jobs. And it's no wonder, really, considering the lives we lead. Yet many of our employers actually seem to think it's reasonable to require unfeigned good cheer in their employees, and this I don't get. It doesn't make sense to hire people at wages that guarantee they'll be desperate and then be disappointed when they're not always capable of pretending otherwise. Look, I don't like walking into a gas station or fast-food joint or box store and dealing with a bunch of sullen idiots either. But people don't seem to stop to wonder why we're uniformly so pissed off and unhelpful. I think you'll find that the happier employees are in general, the happier they are at work. It isn't rocket science. My guess is that, like me, a huge number of poor people are depressed. Anger is one of the few emotions that can penetrate depression. It's strong enough to punch through the haze, so a whole lot of people like me hold on to our anger. We cherish it. The alternative, at least for me, is a sort of dreary nothing. Anger and depression make for a cute couple, right?
—
Regardless of our mood, we're never fully checked into work because our brains are taken up with at least one and sometimes all of the following: 1) calculating how much we'll make if we stay an extra hour, 2) worrying we'll be sent home early because it's slow and theorizing how much we will therefore lose, 3) placing bets on whether we will be allowed to leave in time to make it to our other job or pick up our kids. Meanwhile, we spend massive amounts of energy holding down the urge to punch something after the last customer called us an idiot. People don't have any compunction about insulting service workers, but it's amazing how quickly they'll complain about your attitude if you're not sufficiently good-natured about it.
Our jobs are as much emotional labor as they are physical. What they are not, what they are never allowed to be, is mentally engaging. So we're trying to zombie out to survive. We're not allowed to deviate from policy even if the policy is kind of stupid and counterproductive. Nobody is interested in our thoughts, opinions, or the contributions we might be able to make—they want robots.
Our survival mechanisms are the things that annoy the customers most. Next time you see someone being "sullen" or "rude," try being nice to them. It's likely you'll be the first person to do so in hours. Alternatively, ask them an intelligent question. I used to come alive when someone legitimately wanted to know what I'd recommend. I knew everything about my products, having stared at all the boxes while I restocked them, but people rarely wanted me to tell them about anything more than the price.
What's guaranteed to be counterproductive for you is demanding better service with a superior attitude. We'll perform better service. But we'll be sure to hand you the shirt that we know is stained, or the meat that's within the technical limit of servable but will probably taste less than optimal. And we'll do it with a shit-eating grin on our face and well-wishes on our lips, just like you demand but refuse to pay a single extra penny for.
If you want us to be happy to serve you, make it worth our while and be pleasant. Next time you're in a low-wage place, try walking up to an employee and saying, "I'm sorry to disturb you, I know you have work, but could you tell me where this thing I need is?" I guarantee you, _that_ is how you get service from a demoralized staff. Respect their workload. There is no low-wage employer in the world that doesn't expect a ton of chores finished in a shift besides customer service. Don't just expect that millions of people are by nature pleased to grovel at the feet of your twenty dollars. Humans in general aren't built that way, and Americans in particular. We're supposed to have a stubborn streak of pride, remember?
—
In Cincinnati, I lived just under two miles from the closest grocery store that carried the sort of formula my daughter could tolerate. She was insanely colicky, so I used to spend my free time walking her around the city, letting the vibration of the stroller lull her into farting an incredible amount before she finally, blessedly, fell asleep. I went to the store most days, buying only what we absolutely needed, because I couldn't fit much more in the stroller. I still love to wander, because if nobody knows where I am, then nobody can ask me for anything or call me about an unpaid bill. And I get angry out of all proportion when someone disturbs my peace, because it is so rare that I actually feel light and free.
I don't get much of my own time, and I am vicious about protecting it. For the most part, I am paid to pretend that I am inhuman, paid to cater to both the reasonable and unreasonable demands of the general public. So when I'm off work, please feel free to go fuck yourself. The times that I am off work, awake, and not taking care of life's details are few and far between. It's the only time I have any autonomy. I do not choose to waste that precious time worrying about how you feel. Worrying about you is something they pay me for; I don't work for free. You don't get to demand this ten minutes from me too. This is mine, and my family's.
I actually don't mind, on feminist grounds, when men tell me to smile. I can see why women would, but I've worked in bars and I've worked in strip clubs and I've learned that you can commodify anything, including sex and pretend love and faked respect and false empathy. "Smile," coming from a man, is just the opening chatter to me at this point. It is a sign that this particular man has nothing original to say and is probably kind of a dick.
I do mind the smile-on-command directive on class grounds. Listen here, buster. It's not my fucking job to decorate your world, not unless you're willing to make it so. Sure, I'll smile. That'll be five bucks.
I feel bad about my reactions sometimes, because I can't always stop them even when they're directed at someone who's having the same sort of day as I am. I was once at a store and could not for the life of me find the fucking diapers. I wandered the length and breadth of the place—nothing. I was exhausted, completely finished. Some poor woman who worked there stepped into my field of vision. I meant to ask where the diapers were stocked like a normal human being. What came out instead was "Why did you people hide the fucking diapers?" I couldn't tell you how that made it from brain to mouth. It just happens sometimes. So when I am on the receiving end of customers' misery, I'm never sure whether to actually be mad at the customers. Maybe they tried to be polite and just didn't have the energy. Because when they were at work, someone else came in, and so on, and so on, and so on.
Maybe it's because, as I mentioned earlier, I spend a lot of the time depressed. Always have, always will. Give me medicine, I get less upset about being depressed, but the fact of it never leaves. Sometimes I am clinically, trouble-getting-out-of-bed depressed. Other times, I am just low-level, drag-myself-through-my-day depressed. Some people might call me pessimistic because I always expect disaster to occur. But looking at my life, I think that's bull. When I expect doom? That's what I call reality.
Mostly, I ignore the depression. I developed a caustic sense of humor. I discovered mosh pits to vent. I listen to seriously angry music. When that doesn't work, I soothe the emptiness with terrible food and old jazz. If that doesn't work and I can afford it, I go in and see someone about getting some medicine for a few weeks. That means making appointments any place I think I might be able to get in, assuming that I'll be turned down for service, and showing up to them all until I find someone who's willing to do me a solid and give me a week or two of anti-anxiety medicine. If I can't find anyone to do that, I just sort of check out for a while.
Those times, I can't get past the part of the day where you're supposed to put on pants. I'll stare at the pants. I will tell myself to put on the pants. I will get stern with myself about them. And then I'll lose a few hours to a discussion with myself about how much I actually really do deserve all the punishments I will heap upon me if I do not put on the pants. When I zone back in again, the sun will be down and it will blessedly be time for bed again.
Sometimes I can convince my boss that I have a terrible flu. Sometimes I just don't show up, and those times it's half and half whether I've got a job to go back to; it depends on how understaffed they are. Sometimes I haven't been employed in the first place.
Not all poor people are chemically depressed, but a lot of us are situationally depressed at any given time. And that's because our lives are depressing. I realize that might at first sound simplistic, but I don't think it's a lot more complicated than that.
When I think of myself and all the poor people I know, there is only one person who I would have called irrepressibly sunny. Her name was Melissa, and she seemed indefatigable. Nothing, and I mean not eviction, not being without electricity, not being called names—nothing brought this woman down. She once told me that even when she felt terrible, she liked being a bright spot. I'd known her for six months when her kid got in trouble and the school intimated that it was because she wasn't doing enough for him. And that's what finally broke her. She got into a terrible funk, withdrawn and silent unless you forced something out of her. She started noticing all the things that were wrong in her world, and that was the end. She was one of us.
That's the worst, watching someone lose hope. I'm not swelled with it personally, but I always like to see people who aren't only pretending to be in a good mood, people who are truly optimistic about life. Those people are contagious, even to a curmudgeon like me. It's heart-wrenching to watch that fade, like watching a star die or something. I can't think of anything poetic and tragic enough to describe it.
—
I recognize that the attitude that I fall into—hell, that I cultivate—as a ward against the instability of being poor isn't always helpful to me. But it's not as if I can just go in and out of it, like putting on or taking off my makeup. The attitude I carry as a poor person is my armor, and after so many years of fighting and clawing and protecting myself and my family from impending disaster, that armor has become a permanent part of me.
Take a walk through any impoverished neighborhood. You will hear the word "pussy" a lot. A lot. It's just how some people talk. "Suck my dick," a man will say jauntily to his friends. Or angrily to his friends. Or randomly to women passing on the street. "Fucking pussy" is a popular phrase too, as in "you're a" or "I need some." Street cant isn't something that poor Americans came up with magically a year after the Pilgrims got here. It's a product of environments in which everyone's always posturing just a bit, just in case. A lot of times it means absolutely nothing.
But there is always the potential that as you are walking down the street, some sort of altercation will erupt within feet of you. Maybe someone is angry with a cashier because their card was declined, and they start yelling about disrespect and ass-kicking and what they ought to do. Maybe a homeless person will loudly and suddenly commence complaining about whatever it is that is bothering them that day. Maybe a mercurial couple will have a disagreement in their own attention-seeking fashion.
I was sitting in a Denny's recently, drinking coffee and trying to finish writing a chapter of this book. The table next to me had a few kids, two men and a woman, all under twenty. And the table behind me had two people in it, one of whom took it into his head that he'd been insulted by Table 1 somehow. Next thing you know, everyone's out of their seats throwing insults back and forth, tossing gauntlet after gauntlet, trying to goad a fight. I wound up taking the aggressive dude outside to smoke while we waited for his friend to grab their food and leave. Someone else talked down the people who really had been confronted for zero reason.
That was a random Tuesday. I've been to the same Denny's more than once, and I expect to just drink my coffee. But you never know when you're going to be talking down an idiot. It doesn't happen all the time, and it's not like most trips to the store aren't rather boring and mundane. It's just that it _could_ happen at any time in the environments where everyone is always tense and worried and stressed. It _does_ happen with some frequency. And it's best to be prepared for the eventuality.
Being poor in the country requires a toughness. We have to be capable of changing our own damned tires and putting shims on a starter. We chop wood and catch or grow food. Country poor is not even going to the thrift store, because it's miles away. It's getting up and dealing with the animals and the crops (if you have them) before you go to work. It's expecting at any moment to break down at the side of the road because your truck is so old it doesn't have a computer in it anywhere. And there's no public transportation in the country. If you don't have a working car, you're hoofing it. Rain or snow.
So yeah, out of necessity poor people walk around being just a bit rough and tumble, a bit sharp-edged. We proudly declare that we are rednecks, we wear boots and have weapons with which to defend ourselves and we are doing well enough on our own, thank you. Or we scream that we are from streets somewhere, that we will take no shit, that our neighborhood doesn't have a place for weakness in it and it makes us hard like warriors.
It also makes me say "fuck." A lot. It's my vernacular as a matter of habit, and I developed it as a defense mechanism. Saying "fuck," especially as a woman, is the quickest and easiest way to assert that you aren't to be fucked with, or at least that you're pretending well enough. It's a tough word, a vulgar word, something you don't say comfortably if you're scared of public disapprobation or muggers.
That's the upside of it for me. The downside is that it doesn't go over well when it slips into situations where it's inappropriate and it might even come across as threatening. I know this affects how I'm treated when I engage with the upper classes, but it's a habit that's practically subconscious.
I walk with a tiny swagger. Many people who have lived in the not-so-nice parts of cities do this to varying degrees because it tells people from a distance that we know how to handle ourselves, and that we are streetwise enough to make a challenging target. It's also unconscious in me at this point. To middle- and upper-class people, it's one more thing that sets me apart, that sends the unintended signal that I don't belong in nicer company.
My tough demeanor was at first something I cultivated as a survival mechanism. But after a while it became more natural. It's a lot like hiding my teeth—I got so good at it that I didn't notice after a while. I stopped noticing anything, stopped registering things as inappropriate or odd. And I stopped noticing when I was being inappropriate or odd, to the extent that I ever knew.
Looking back, I can see where the crossing of the ways happened. I started to lose contact with the middle class at the same time that I became comfortable in the lower one. You can bridge both worlds, but only if you're consciously doing it and you're not too tired. Otherwise you revert. I'm perfectly capable of holding an intellectually stimulating discussion like a human being. But my friends will tell you that they can tell how tired I am by how frequently I replace polite words and phrases with profane or aggressive ones.
And I don't just have problems with playing the part of someone who gives a shit about the niceties; I have difficulty looking the part. That costs money.
I'm not going to claim that I had sterling self-esteem before I started seeing my economic status written all over my unmoisturized face. I was an awkward, overweight kid who liked books and chess. I was a nerd who missed the makeup and fashion years. But being poor sucked right out of me what little self-regard I might have had. Rich people complain when they have bad hair days or fat days. I have "fryer grease in my hair" days, and "not a single article of clothing makes me look like anything but shit" days. I'm not even going to bother explaining how bad teeth and bad skin might also get you pegged as less valuable, less worthy of respect. You're reading a book voluntarily, you're smart enough to figure it out. But those are only the _big_ visible markers. There are a whole lot of small ones. If the average rich person had to walk around for a day wearing a polyester work uniform, they'd need Xanax.
Poverty, or poor, or working class—whatever level of not enough you're at—you feel it in a million tiny ways. Sometimes it's the condescension, sometimes it's that you're itchy. I don't think people who have never been poor quite understand that.
I like to use jeans as an example. I just bought my first decent pair, the exorbitant $70 kind. It's like some kind of fucking miracle. I didn't know denim wasn't supposed to be uncomfortable. And I'd heard about jeans making your butt look amazing, but I'd never believed it. The kind you can buy at Wal-Mart come in two styles: mom jeans and low-cut skinny jeans meant for middle schoolers because no grown woman could get into them. Regardless of style, they are heavy, the fabric is the rugged we-mine-coal thickness, and once they stretch across your unfortunate lower abdomen, you're fucked. They'll hide the curves you like and prominently display the ones you'd rather nobody noticed.
Assuming, of course, that they fit at all. I have one favorite pair of jeans, which I've had for so long that they've gone soft from washing. I've worn them when I was a size 12 and when I was a size 16. If I wash them in really hot water and then throw them in a hot dryer, they'll shrink enough that I can belt them to stay up when I'm skinny. And if I wash them in cold and stretch the waistband while they're drying, they'll expand enough that I can zip them when I'm on the top side of my usual range of sizes. So yeah, when I put them on, I am wearing pants, but they're the kind that make you look weirder than you would just leaving the house without any pants on in the first place. At least if you're pantsless you're given the room to be crazy. Bad pants just mean bad taste for most people.
I'd never had occasion to walk into a makeup store until recently, when I was going on camera and desperately needed something for my face. I figured that if I went to a special makeup place, they could help me choose foundation and maybe lipstick, because I never know what color to get. The salesperson not only helped me with that, but she also hooked me up with free makeup application lessons, and gave me more free shit than I could have imagined. Samples of this, samples of that, here try this face cream. Just because I walked in with twenty bucks. It's insane, the perks you get at specialty stores.
I've heard my whole life that I should spend wisely, invest in my appearance, that it will make people take me more seriously. Buy a few key pieces, the style authorities say, which would be great if I could ever scrape together $300 of disposable income to spend on a suit. A $20 bottle of makeup, okay, I can do that every now and then. I've got $50 sometimes, but it's still not enough to buy a suit with. If I could put away $20 a week in a little piggy bank marked "nice suit for Linda," then I'd have enough to buy it about fifteen weeks from now. And who am I kidding? By the time I have $40 saved, I can think of ten other things that $40 could be spent on. Stuff like milk and toilet paper. How often am I really going to wear a suit, and how important might that suit be the one time I need it?
More than once I've shown up to a professional event wearing something entirely inappropriate. I've gone too casual to formal events, and I've gone the other way too. I'll show up to a casual event in heels. I don't have the time or resources for style handbooks and fashion magazines, and I don't get the social cues and niceties. And even if I did get them, I couldn't afford them.
Let me clarify: I'm not saying that all poor people don't know how to dress. There are certainly those among us who do better than others in this area (we have our share of aspiring fashion designers who watch _Project Runway_ and all those makeover shows, and you can learn to put on any kind of eye shadow in the world on YouTube). But even if I knew what to wear, I couldn't afford it. I once wore a suit two sizes too small because I'd gained some weight and didn't have anything else that fit. It didn't occur to me until hours into the thing that the only people speaking seriously with me were men looking for company that evening. What I actually had to say was never heard. Then there was the black-tie event to which I wore a light summer dress. Of course I knew it was the wrong season for it, but it was the nicest thing I had. I wasn't taken particularly seriously at that event either.
So, if first impressions are as important as everyone says they are, what do you think my chances are of getting a professional job if I'm competing against someone who dresses the part? I guarantee you that even if that other job candidate is a little less qualified than me, the boss is going to feel more comfortable hiring the person who she's not afraid will stick out like a (poor) sore thumb at the weekly meeting with the CEO.
I didn't really realize that I was fully lower class in both sensibilities and presentation until I found myself at what was the last of my professional social engagements. I was attempting to resurrect something like a career during the worst part of our stay in Ohio, when we weren't getting our GI Bill stipend, and I thought maybe I could scrape something up. I was invited out to dinner by a bunch of old political work colleagues, and I found myself with nothing to say. I had no insights on the new restaurants or movies or bars, nothing that you typically reach for to make conversation. Every single addition I could have made would have been inappropriate: I couldn't have talked about my neighbor getting in a fight with his truck while he was drunk because it wouldn't start and he thought punching it might help. (His roommate had disabled the thing. Friends don't let friends drink and drive, and smart friends let friends punch the truck instead of them.) I couldn't talk about which food banks were best for produce and which for diapers. I also didn't order any food or drinks, which was pointed out repeatedly by the waiter. ("Are you _sure_ you won't be ordering? Can I tempt you with this/that/the other?") I finally had to leave the table, track him down, explain that I couldn't afford anything on the menu, and ask could he _please_ stop making a huge deal out of it? And after that, I never called any of those colleagues again. Nor did they call me.
I understand why that happened. But what I don't understand is why people who walk into a fast-food restaurant often seem to think I should put on the same smile and elegant demeanor they could expect at Saks or the bank where they put their money. I think the sorts of people who honestly think that service workers should be more smiley and gracious just don't get it. They don't get it because they can take so much for granted in their own lives—things like respect, consideration, and basic fairness on the job. Benefits. Insurance. They're used to the luxury of choosing the most aesthetically pleasing item on the shelf, of caring what color their car is rather than simply whether it runs or not. They don't understand how depressing it is to be barely managing your life at any given moment of the day. So forgive me if I don't tell you to have a pleasant day with unfeigned enthusiasm when I hand you your fucking hamburger. You'll have to settle for the fake sort.
In my world, we don't have the time or the energy to bullshit about our feelings or worry about anyone else's. When I've found myself in professional situations, I'm driven nearly to distraction by how much fucking effort is wasted making sure we all feel nice and fuzzy and comfortable. I don't get that; it's not part of work to me. And it keeps me from getting ahead. If someone asks me my opinion on something, I simply give it. I don't bother spending five minutes talking about the weather and how lovely your shirt is first. I am thinking about the question I was asked. I figure nobody's getting paid to win the office nice competition. And it's amazing to me that some of the same people who can walk by a homeless person without even blinking are obsessed with what everyone thinks of them at work. Meanwhile I know that if I wasted half as much time in my service jobs talking about my feelings as I have in my professional life, I'd be out of work and lying right next to that homeless guy my white-collar friends just skirted past.
Maybe feelings are something that only professional people are allowed to have. My friends and I know that no one gives a shit about ours. We're constantly told to know our place and not make a fuss about the insane conditions we're expected to deal with, both at home and at work. And yeah, this discussion about attitude is coming back to the subject of work a lot, because guess what? It's what we spend a huge percentage of our lives on. And how we're treated there isn't something we can just shake off when we leave. It becomes a part of us, just like that armor we wear.
But still we're told to keep smiling, and to be grateful for the chance to barely survive while being blamed for not succeeding. Whether or not that's actually true isn't even relevant; that's what it feels like. Unwinnable. Sisyphean.
Responsible poverty is an endless cycle of no. No, you can't have that. You can't do that, can't afford that, can't eat that, can't choose that. This is off-limits, and that is not for you, and this over here is meant for different kinds of people. More than once I've spent money I couldn't really afford simply to state that I _could_ , __ if only to myself. Just to say it.
To be told that you deserve nothing more than that, are entitled to nothing more, is enraging. If poverty is supposed to be like prison, then why don't we kill two birds with one stone and put prisoners in all the low-wage jobs? All the private prisons would be wildly profitable, and the poor people would deserve their poverty because it would be their punishment.
Sure, we can beat the odds. Sometimes we can climb out of it. You're reading this book by a service worker, after all. But the irony of my success here is that I didn't get this chance because I worked my balls off for some asshole who thought me ungrateful for my sub-living wage. You're reading this book by me because lightning struck, because my story went viral. And by definition, that can't happen for everyone. You can hope for your one real shot, but you sure as hell don't plan for it. It hurts too much to plan and plan again and keep waiting for the magic day.
So that's been my American dream. And it's reality for millions of us, the people who are looking grumpy behind the counter. Our bodies hurt, our brains hurt, and our souls hurt. There's rarely anything to smile about.
# 5
I've Got Way Bigger Problems Than a Spinach Salad Can Solve
We all cope in our own special ways. I smoke. My friend drinks. In fact, I'm highly confident in betting that you and many of your friends cope by drinking as well. Come home from a long day at work, and what do you do? Pop open a beer? Or a bag of potato chips? Or maybe you take a Valium when you're feeling stressed out. Or get a massage. Or go to your gym and sit in the sauna room.
Why are other people's coping mechanisms better than poor people's? Because they're prettier. People with more money drink better wine out of nicer glasses. And maybe they get a prescription for benzos from their own personal on-call psychiatrist instead of buying a pack of cigarettes. They can buy whatever they like and it's okay, because retail therapy is a recognized course of treatment for the upper classes. Poor people don't have those luxuries. We smoke because it's a fast, quick hit of dopamine. We eat junk because it's cheap and it lights up the pleasure centers of our brain. And we do drugs because it's an effective way to feel good or escape something.
I get that poor people's coping mechanisms aren't cute. Really, I do. But what I don't get is why other people feel so free in judging us for them. As if our self-destructive behaviors therefore justify and explain our crappy lives.
Newsflash: It goes both ways. Sometimes the habits are a reaction to the situation.
And now I have to add one big caveat: Sometimes, sure, the stupid shit we do does explain our crappy lives. Are there meth addicts out there from nice middle-class homes who ended up homeless and far worse off than I've ever been? Absolutely. And if you want to believe that addiction is a person's fault and not a disease, then you can go right on ahead and judge that person for having brought about his own downfall.
But unless you're prepared to convince me that smoking and smoking alone keeps me poor, then please, spare me the lecture. I know it's bad for me. I'm addicted, not addled. There are reasons that I smoke, and they're reasonable ones. They keep me awake, they keep me going. Do they poison my lungs and increase my chances of getting cancer? Obviously. Does that stop me? No. Because the cost-benefit isn't a simple _I like it_ versus _I'll possibly live longer_. It's _I will be able to tolerate more_ versus _I will perpetually sort of want to punch something_. __
I once talked to a neighbor about the fact that people who lived on our block were statistically likely to die earlier than the people who lived five blocks over in the wealthy neighborhood. He told me that it was just life, it was the way it was. He'd stopped questioning it. So if you already figure you're going to die early, what's the motivation for giving up something that helps get you through the here and now?
Look, I'm not saying that getting in a cage match or smoking copiously and with glee is exactly good for my longevity. But I don't much see the point in worrying about the end of my life if stress will kill me first. If I don't vent, don't perform some kind of self-medicating, there won't be an old age anyway. I'll wind up dead or in jail or institutionalized when I finally lose it.
Let me be clear: I am not all poor people. Of course there are wholesome people in every class. There are poor people who would never dream of doing anything as déclassé as using drugs. This whole book could be called _You Can't Put an Entire Third of the Country into One Group of Behaviors_.
Often, those folks who are unlike me are religious. I tend to think of religion as the same sort of thing as smoking—a soothing ritual that brings someone a moment of peace. But if I don't want to be judged for my habits, I'm sure as hell not going to judge anyone else for theirs. That's why I always defend religious people against those assholes who act like they're too good for anything so magical as religion. We all think magically about something.
So, on the one hand, sure, poor people have been known to engage in some unhealthy behaviors. It's not as though we, the unwashed masses, are doing anything that _everyone_ doesn't do. It's not like drug and alcohol and cigarette sales just stop once a consumer hits $75,000 a year in income or something. It's a bit galling, actually, to be lectured about my self-destructive habits by someone who's fighting his own hangover. You're still getting drunk, friends, whether enjoying a bottle of Bordeaux or drinking a can of Mickey's. But it seems that the disapprobation of excessive drinking is meant mostly for those of us on the rotgut end of the scale.
I think the reason for this is that people are less moralistic about the vices themselves than they are about the cost of the vices. The logic is that if you've got excess money and throw it away on booze and cigarettes, then that's your business. But if you're poor, then that's a sin and a shame. Because if you're poor, rich people assume you're on welfare, or you're getting food stamps or some other social services. Once you take a penny from the government, a morality clause goes into effect, where you're never allowed to have anything that you might actually enjoy. It's the hair shirt of welfare.
I have trouble understanding why taking a few grand a year in food stamps is somehow magically different than taking trillions as a bailout. Food stamps cost $76.4 billion for 2013, compared with trillions, possibly hundreds of those, for the banks. And that's just _one_ instance of handouts for the upper parts of society; it's not like the feds handed cash to the banks and the rich are otherwise left to muddle on alone in the wilderness.
I do not see a difference, the way many people do, in the federal money. Whether you are getting your benefits in the form of SNAP cards or deductions, it's the same thing. There is this money that you otherwise would not have had, that the government gives you. Stimulus spending can happen in proactive or passive ways; whether it's a block grant or a tax break, it's still the government investing money in a thing because it wants to ease some burden for someone somewhere or to encourage or discourage certain behaviors. It wants people to not starve? Food stamps. It wants people to buy houses? Interest deductions.
The one difference? Rich people get way more from the government than poor people do—see above-referenced mortgage interest, capital gains, light inheritance taxes, retirement savings breaks—but the poor are the only ones getting shamed for it. You want to know how I could justify relaxing sometimes while I was on benefits? The same way you justify blowing a reckless amount of money on a really nice dinner while you take a business deduction because you talked about work for ten minutes.
People bitch about double taxation, where corporations are taxed for their profits and then they give money to their shareholders, who are also taxed. This is apparently hugely unfair, and the only reasonable solution is apparently to exempt people from having to pay taxes on their dividends. Because some kinds of income just don't count as income? Because someone, somewhere, already paid a tax on this particular individual dollar? By the same logic, I shouldn't be asked to pay payroll taxes because my bosses already paid taxes on it too.
Capital gain, by definition, is money you make for the simple fact of having money. That's it. No work, no nothing. Just have some money, wait for it to grow, and then you have more money. Which you clearly should not have to pay taxes on, because that would be unfair. Somehow.
This, of course, is nothing like unemployment, where an employer pays a tax for every employee, and then if I pull unemployment, I have to pay tax on that as well. But sure, keep thinking that we've got all the cushy non-taxation going on down here in the lower classes.
—
All humans chase good feelings. It's just that people with money chase them in ways specific to the upper classes, which makes it okay. You can't argue that a pair of expensive shoes or an expensive steak is actually something you need. It's just something that makes you feel good.
According to a study published in _Science_ magazine, which is a place I trust about science things, your brain actually has less capacity when you're poor. The theory is that so much of your brain is taken up with poverty-related concerns that there's simply less bandwidth available for other things, like life. It's not the only study like that.
At Princeton, they've found that the effect on the brains of poor people from the stress about money alone is equivalent to losing a bunch of IQ points. And they've also found that if you remove the stress, our brains snap back and perform at the same levels you'd expect to see in a wealthier test-subject pool. The same goes for the short-term memory impairment and trouble with complexities—skip a night of sleep and tell me how well you're performing the next day; you'd be functioning on about the same level we do every day. We're not dumb—we're conserving energy.
They're even starting to find similarities between people in poverty and soldiers with PTSD.
Poor people didn't need to wait for the science to know this, though. We feel it. We could have told you that being always tired and distracted wasn't great for higher cognitive activity. I stopped thinking in higher concepts, gradually. I feel stupid when I realize how long it's been since I thought about anything beyond what I had to get through to keep everything moving along: no philosophy, no music, no literature. We know we're not at capacity, and it rankles. So we fix it, best as we can. I know a few veterans, dealing with mild to moderate cases of PTSD, who have turned into potheads. It keeps them from getting too jumpy, keeps their memories from being too sharp. I hear that bankers like coke to stay focused. College kids take Ritalin to study.
I flirt with addiction, drinking too much coffee and smoking too much, but I've never let myself go there because I think it'd be too much of a relief and I'd never be able to come back voluntarily. And if I were dragged back, I'd face a lifetime of having to say no to one more thing that I knew would make me feel good. I doubt I'd do well with that. I'm not particularly strong that way.
Self-medication is a thing that exists. We fake rest and nutrition like we fake everything else to make it through the day. Mostly, we do it with chemical assistance. I smoke because it keeps me calm, because it keeps me awake, because it keeps me from feeling hungry, because it gives me five minutes to myself, because it just feels good and I like it.
Have you ever felt tempted to go to one of those places where you can pay to smash china? I never have, but then I never saw a reason to pay to smash things. I just did it. It feels good, really good, to break things when you're frustrated. It doesn't actually solve anything, but for a second you feel better. I like breaking glass. It's therapeutic. It was my favorite part of working as a picture framer; we had to smash the flawed glass into tiny bits for disposal. More than once, I popped in to help on my day off just to smash things. It's the same logic that explains mosh pits.
One day, when I have nothing but free time, I will start a mosh pit for old people. I quit jumping into them only when I started to realize that I'd become the creepy old person in the corner. For years, though, mosh pits were my anger therapy of choice.
Sex is also therapeutic when it's blissfully mindless. Orgasms for orgasms' sake. It makes your muscles relax, your headaches lessen. It makes the stress go away for however long it lasts. It's kind of amazing to have some outlet, somewhere, that you don't have to work for; that's the whole point of having a fuckbuddy. It's effort-free. As long as you're attracted enough that sex is a possibility and you feel safe, that's all that matters. Sex, done properly, makes you feel wonderfully accepted.
It's different from love. Maybe in the upper classes it's called a fling, but down here where I live it's a pressure release, and no love or imitation Hollywood romance or delusions of long-term commitment are required. It's not like I fuck everyone within arm's reach, but I don't expect to fall in love with everyone I've ever been infatuated with either. It's just nice to be in a pleasant spot for a while, that's all.
—
The coping that I and many of my friends do via medication isn't just about emotional relief. For me at least, it's just as much about physical pain management. I've stopped paying attention to how much ibuprofen I take in a day. More than I should, certainly. A reckless amount, even. I'm a pill popper, just not the narcotic sort. I start my day with ibuprofen and cold medicine, because I get sinus headaches from pretty much every part of nature and my jaw is always killing me. B12 for energy, vitamin C as a prophylactic measure. The ibuprofen starts to wear off in a couple hours, so I take some more. Repeat as necessary. Add in a pot of coffee and maybe a guilt-ridden switch to naproxen in the afternoon for pain management, plus whatever nicotine I get in there. And if I absolutely have to sleep well, I wind up taking something that says "p.m." on it, whatever that might be. If the pain is bad, as it often is for people with serious back injuries and dental problems like mine, alcohol or some kind of narcotics might be taken too. That, friends, is what pain management looks like outside the health care system.
Miraculously, I'm not dead yet, and as far as I know, my liver hasn't started to fail. My husband comes from healthy stock, the sort of people who maybe keep a bottle of aspirin around for emergencies. He was horrified at my intake, to the point that he once asked me to try not to take anything for a while to see if it would reset things for me. After a couple days I wound up in bed trying not to breathe too much because moving made the headache worse, and he's never mentioned it since.
I know that any actual cure of my chronic pain would have to at least partly involve lifestyle changes that simply haven't ever been logistically possible. Any kid who watches _Sesame Street_ can tell you that it's important to sleep well, drink lots of water, and eat a balanced diet. And I can guarantee you that I can drink lots of water. The other two are trickier, if not mostly impossible.
A balanced diet is one more detail to throw at me, and for years my diet consisted of whatever food at work had become expired for service most recently—sometimes beef, sometimes chicken. And when I got home, I ate dinner only when I was absolutely starving. I ate food that I was craving, because it made me feel better. Healthy food, sad to say, just doesn't work as well as a pan of brownies when it comes to soothing yourself.
I've got way bigger problems than a spinach salad can solve.
A human body doesn't care if acute stress is caused by almost getting your electricity shut off or by a looming deadline on a million-dollar contract. The reason that poor people wind up coping in ways that seem pointlessly self-destructive is that all the constructive stuff costs money. I can't afford to join a gym. I can't just pay a shrink to listen to me vent. I can't go shopping or find an acupuncturist or a good masseuse or whatever else it is that the people above me do to cope. I can't pay someone to make my back relax when I have strained it, and we don't get to take it easy when it happens if we want to keep our hours at work.
Our bodies are no longer our temples. We can't afford for them to be. I have agreed, more than once, to let people have parts of my body for money. I have observed, lying on a bed to sell my plasma for twenty bucks, that it's the modern-day opium den—people languid on medical tables instead of couches, staring at the closest TV or watching in fascination as their own blood is separated in the machine.
But I have only so many body parts I can spare. Only so much blood.
There are millions of us who have had enough of this. We have waited. We have been patient. We have coped. And we've survived, which we'll continue to do. Humans are amazingly resilient.
The question is, how can the rest of the country live knowing that so many of us have to live like this?
# 6
This Part Is About Sex
I'm writing a chapter about sex, so I'm trying to remember the names of everyone I've slept with. I don't think it's possible; sobriety hasn't always been involved. I never bought the idea that sex is actually immoral. God made me human, so I tend to think he doesn't expect me to act like an angel, if in fact angels don't mess around. And I really don't understand why rubbing genitals with someone is immoral. With all the evil in the world, we're really going to judge people who make each other feel good?
Being poor is isolating. You're constantly being rude to friends and family because you never have time to talk, never have time to hang out. Never have the money to do anything, not even to reciprocate a birthday present. You don't ever have anything new happening—no news to share unless you're getting married or having a baby. You lose the most interesting parts of yourself to the demands of survival. I got so boring when I was at my worst that even _I_ didn't want to hang out with myself. Why on earth would I invite anyone I liked to come over and stare at walls with me?
For me, sex has been a logical fix for that problem. It doesn't require conversation, no personality necessary. Just some skill and willingness and a partner with the same two things. It's catharsis without any baggage or investment. Sex is kind of magic that way; if you tell a woman she is beautiful, and you do it when you are as unguarded as you can possibly be, she will believe you, and it will stick with her. If you tell a man he is wanted, and you do it when you are making that very clear, he will remember your words longer than you do. You can fix people a little bit, plus there are orgasms and cuddling. I couldn't design better therapy.
Sex is fun. It's fun for rich people, it's fun for poor people. But there are two possible reasons for having sex that I think tend to be way more important to poor people than to rich people: 1) The chemical rush of sex is a great way to forget about your problems for a little while, and 2) sex is completely free.
Let's talk about the endorphin rush first. It's not just the thrill of an orgasm that I'm talking about. It's the physical comfort and feeling of a little pleasure in your body. Few things are more isolating than financial desperation. Sure I have my friends to talk to, but while we commiserate about the practical—the unpaid bills or the car troubles—we rarely talk about our feelings. We shy away from them. And when I come home from a long day at work, it's a guarantee that my husband has had just as sucky a day. If we want physical comfort and a loosening of the back muscles, it's only going to happen while we're having sex.
Given that the reason that I'm often in need of relaxation has to do with the lack of money, it's an added bonus that sex is also free. Entertainment costs. Movies, bowling, whatever you can think of that nice folks do on dates that don't involve sex—that's all a luxury. When you have nothing in your wallet and nothing else to do, sex is really good for killing time. I've spent more than one afternoon in bed because it was the only entertaining option I had. Given the choice between a) sex minus boredom, and b) celibacy plus boredom, I think we all know which one is preferable.
—
Wealthier people don't seem to understand it when some poor person pairs up with some other poor person who maybe isn't so perfect. Maybe doesn't have the greatest teeth, or the most steady employment, or the best attitude about the world. They seem to think that for every Julia Roberts, there's a Richard Gere just waiting to catapult her into respectability. It's only among the wealthy that most people could potentially model for clothing catalogs. Marry up as a life strategy—sure! In real life, Julia would have married a recently laid-off cab driver.
We choose from what's available, after all. It's not like laureates and models are thick on the ground, and Richard Gere isn't going to show up to whisk me out of the strip club anytime soon. So I wind up with people who are as flawed as I am; people who work where I do and shop where I do and socialize where I do. It doesn't lend itself to meeting a millionaire and running off to a happily-ever-after in the Hamptons, or even the suburbs.
That doesn't mean we're indiscriminate. We do not simply drop trou and rut like animals upon spotting another human that we might be able to fuck. We have sex for the same reasons rich people do—we are in love, we liked someone's smile, someone made us laugh. Sometimes they're cute and there's a spark.
Of course the kind of cliché downward spiral about poor women is that once things get really bad, they have nothing left to sell but their bodies. That's probably the worst thing most rich people can imagine a poor person having to sink to. Well, that and starving to death. But don't we all trade sex for something? Even rich people do that—just ask one of those women you see with a big fat diamond on her finger and a boring and unattractive husband to go with it.
Living rent-free is a pretty good incentive for adding a sexual element to an existing friendship. More than once, someone has offered me a place to live when I needed one, and then kind of let me know we'd be having sex. It wasn't a power imbalance; it was just an understanding that, value for value, this was the deal. If I didn't like it, I could leave and no harm done. I could probably still have crashed for a day or two, just not long-term. It's sex as currency. Cutting the bills by moving in with someone you've only just started dating is less sexual than it is practical. If you have found someone who you get along with, who you enjoy the company of, and it's likely to last at least a few months, it just makes sense to move in together. There is no shame in it, and nor should there be.
I've been in less comfortable sorts of sex-as-barter scenarios at work, but I've never had to accept them. I could always quit or get fired. I was young when the offers were made and didn't have kids to feed or extended family counting on me. I was lucky; it never worked on me because I had other options.
That said, the situation isn't always as gross as that. Sex, as a commodity, isn't traded so explicitly and openly as "here is cash, now please fuck me" in all cases. Sometimes, it's a quid pro quo. Sometimes it's even between friends. I don't see a problem with that; it's a human need, and filling it thus has economic value. Related: If you want to have some fun, ask a free-market religious conservative whether you should restrict prostitution, given that there's a clear market demand for it.
And look, you can't blame people for leading with their assets. My occasional forays into the sex industry have convinced me that breasts really are magic. I got bigger tips as a bartender in a strip club when I wore a corset. We all exploit our advantages. There's something about a corset that turns an otherwise reasonable bar patron into something resembling a monkey. A very well-tipping monkey, to be fair.
The act of putting on a corset is enough to negate any dental problems, weight issues, or personality flaws. Guys would just see that three inches where a very specific kind of fat folds together and boom—instant idiots.
The girls who actually took their bras off made the real money, relatively speaking—it was more than I made but still not enough. If there was cash within five feet of a topless woman, it was often hers for the asking. The only reason I never did it myself is that while my breasts are big, they're also kind of wonky. And also I can't dance. And I would not be able to keep my temper through what I saw those girls deal with. So I kept to my spot behind the bar. Guys actually thought I'd be impressed when they told me that they liked me best out of all the women at the club because real honest women wouldn't strip, that it was beneath them to like a stripper. Amazing. Some guys will moralize at you _while they're getting a lap dance_. These guys were conflicted about their own sexual moral systems and they blamed us for it, which led to insanely entertaining scenes of dancer rage. You'd see a girl storming out of the lap dance area and a guy leaving just as mad. And she'd tell you that he'd been rude, demanded some seriously inappropriate, um, dance moves, and then told her she was going to hell. It didn't happen often, but it was gold every single time.
—
It's ridiculous to suggest that poor people should behave more appropriately about sexual matters than anyone else does. I am fairly certain I could walk into any swanky bar and find well-off people who are hoping for a night's fling. I can say with near certainty that most high-end sex clubs cater to wealthy patrons.
I like to remind people that everyone's parents fucked. Sex isn't dirty, isn't abnormal, shouldn't be a source of shame. Sadly, we as a society are a bit more conflicted about it than that. And for some reason, we moralize more at the poor about sex than we do at the population in general.
Living in low-income neighborhoods, I've seen sexual health campaigns aimed at slut-shaming us into celibacy. They talk about things like self-esteem and value and all the usual abstinence arguments. They assume that our bodies are a gift that we should bestow selectively on others, rather than the one thing that can never be anything but our own. Even if we do share it, it is ours irrevocably.
These are the bodies that hold the brains we're supposed to shut off all day at work, the same bodies that aren't important enough to heal. These are the bodies that come with the genitalia that we should be so protective of? I really don't understand the logic.
You can't tell us that our brains and labor and emotions are worth next to nothing and then expect us to get all full of intrinsic worth when it comes to our genitals. Either we're cheap or we're not.
Make up your fucking mind.
# 7
We Do Not Have Babies for Welfare Money
I never expected to be a parent. I've got wonky hormones, and pregnancy was supposed to be a non-option for me; I was as surprised as anyone when I wound up getting pregnant. But once my husband and I had our oldest daughter, we decided we wanted a second child. Our kid needed a playmate, needed to learn to share, needed someone to join forces with against us. My husband's brother is only a few years younger than he is, and I'm forever hearing happy-childhood-with-Andy stories. As an only child, I have favorite childhood memories of times when I was utterly alone, sitting in a tree with a book. So Tom and I decided that for our child, we preferred the former.
So I never understand it when people want to know why poor people have kids. I don't think having kids is a money question—why does anyone have a second child, or a third? Because their family feels unfinished. We have two children now, and we're done. We feel done. But we didn't feel that way before our youngest came along. That's why we had a second child. Why do rich people have kids? Do they sit around looking at their bank statements and decide it's a good time to procreate? So yeah, poor people get to have kids too. Deal with it.
But what about all those unplanned pregnancies that you're tut-tutting over? Let's talk about those first, and the whole subject of birth control. Then we'll go on to discuss what we do with our babies once we have them.
—
I mentioned that I thought kids weren't exactly likely for me and my husband. A lot of people in my situation would have taken their chances and skipped birth control altogether. But I had a firm belief, instilled in me by my girl heroes from the 1990s, that I should simply be on birth control on principle. Just in case. It was a feminist act, somehow. And I fucking hated it.
The pills made my moods uncontrollable. My periods came nice and regularly, but they were suddenly insane-flood level instead of anything manageable. I'd switch brands or types of birth control, only to discover some fresh hell.
Did you know that if you forget one crucial pill, just one day, you can wind up pregnant anyway? As it turns out, the odds of medicine working are much lower if you don't actually take it. I'm a forgetful person, to put it very nicely. The Pill and I didn't get along well.
Since I was with one person who had been tested since he'd been with anyone, and as I was in the same situation, I just sort of stopped bothering at all. Call it magical thinking, or trusting vague assurances from doctors, but I really didn't think I'd wind up pregnant, because I have never had normal lady parts. I also think I'd have been more vigilant if we didn't have some notions about having kids in the future. I thought maybe we'd adopt some foster kids, actually. We both came from families in which you married and then had kids, and that was likely what we'd do. We were having a lot of fun as a couple and weren't in any hurry to get to the next step, but in skipping birth control, we weren't actually risking more than bringing on a planned future.
So that's how I ended up pregnant without meaning to. Did that happen because I'm poor? Maybe. If I'd had the luxury of having a regular gynecologist who made it her mission to find a reliable form of birth control for me that didn't mess me up mentally and physically, then I almost certainly wouldn't have gotten pregnant when I did. And that's the thing about gynecology in this country—we seem to care about women's bodies only once they are pregnant.
Just like every other sector of health care, access to family-planning services is heavily dependent on income. But a good portion of the unplanned pregnancies I've seen in my circles weren't the result of an active lack of concern for the outcome or even access to contraception. Rather, the condom broke, the pills didn't work, someone miscounted. And then there are people like me who just thought they'd never get pregnant. That all happens to plenty of rich people too. Likewise those cases where the heat of the moment really just sort of blew their brains away for a minute and no birth control got used at all—I don't think that's really something you can say belongs to any one group of people.
I've got zero trouble walking into a Planned Parenthood clinic. I cannot say the same for a lot of women I know. There's a stigma attached to walking into a place that a lot of demagogues associate with abortions. I actually enjoyed my brush with the protesters; they kept telling me, "You don't have to do this" and "You have options." Since I was arriving at the clinic for my first ultrasound to make sure that the baby was healthy, I had a ball pretending to be outraged that they obviously wanted me to abort my baby. You know how sometimes you leave and you figure they'll get it in about five minutes, and you have won utterly? I totally got to do that with the protesters.
I've actually led a bit of a charmed life when it comes to family planning, at least compared with most poor women. That's because I don't give a fuck whether people think I'm a bit of a whore and I've generally lived near enough to cities that clinics aren't too far away. I've usually had a car and enough cash to spare. But man, that does not mean it's easy. First, the fees are unpredictable. I've paid ten bucks for a month of pills, and I've paid fifty. It depends on the funding of the clinic you visit.
And that's just the pills; you have to take off work, have a car that'll make the trip, and pay for gas to get there. In rural areas, it might be a few hours to the closest clinic. Of course, most people have a doctor in their hometown. But they might not have a low-income clinic, and even if they do, it might not do birth control. Some women don't want to get birth control locally, because we've actually been pretty successful at slut-shaming Pill users, as though there's no use for them beyond their contraceptive value. Lots of women would be ashamed to be discovered as medicated harlots.
I know a woman who has been married for five years. She and her husband are in college, hoping to start a family—just maybe in a few years instead of now. She will not visit a Planned Parenthood in Utah or ask for the Pill from her gynecologist, because she is terrified that someone will find out. And that's in Utah, where the dominant religion is all for married people using birth control. I can't imagine what it's like someplace that whore pills and birth control are synonymous.
Look, it's not like condoms are superexpensive. But they're not free, and we don't run across those bowls that you see on college campuses. And God help you if you're poor and allergic to latex. Then you just don't get to have sex.
—
Okay, so for whatever reason—whether you wanted kids or your birth control didn't work—congratulations, you're pregnant! Now what?
Here's a big secret from a poor person: Having a baby is expensive only if you want it to be. Let's go back to the rich-people-looking-at-the-bank-statement thing: A lot of rich people look for a new house or a new apartment before they even get pregnant. Because, the thinking goes, they must have a nursery, or they must have a second or third or fourth bedroom. God forbid kids should share a room. But kids don't care. Kids know what they know. Babies are happy in a drawer in their parents' bedroom, and if kids are used to sleeping in the same room with their brother or sister, then they are happy with the company. Sure they'll fight over space at some point, but I don't care how big your house is—your kids will fight over space. So let's just dismiss the whole idea that kids require a big real estate investment.
The idea of privacy among nuclear family members is actually pretty new. Parents used to share a bed with their kids— _and still expand their families somehow_. If people could manage to perpetuate the species with their toddlers thrashing around in the same space, well, I'm not going to bitch about having to share space too much.
Now let's get down to the real basics of what kids need. Sure, you can buy disposable diapers or spend forty bucks a pop on bespoke organic cotton for Junior to poop on. Or you can tear up old T-shirts. Babies don't really even care whether their butts are covered; we do that to avoid the cleaning up that would be needed otherwise. It's quite satisfying to buy thrift store T-shirts with logos of things you hate for a quarter a pop and tear them up so your baby can do her worst.
What I think people are talking about when they say that kids are expensive is either stuff that's so unattainable that we'd never have kids at all if we waited for it to come along, or stuff that's entirely unnecessary. When I got pregnant, I started reading up on the latest in parenting, which I'd really not been paying attention to at all. And I was mostly pretty appalled. There are whole articles in women's magazines about how to politely turn down hand-me-downs, like this is a major problem for some people. The idea of turning down hand-me-downs is so crazy to me that I don't even know where to start. Your kid will be able to use this stuff for only a few months. And kids absolutely massacre clothes. My youngest can be sitting in the middle of the living room with nothing in her reach, and within five minutes her clothes can pick up a stain from something that I'm not even sure is in the same room with her. For what possible reason, short of family photos or weddings, would you pay retail for something you can get free or secondhand?
Until your kid is old enough to start begging for toys (and this is one of the reasons why I don't have cable: no toy commercials, no begging), the only truly essential expenses you have to incur are for food and medical care.
I've talked about what it's like to be impoverished and pregnant, but the bare fact remains that I wound up pregnant and _then_ hit full impoverishment. I had to figure out how I'd make it work. What I figured was this: Kids can eat pretty much everything adults can, and they don't eat nearly as much until they hit puberty. And thank God, WIC would cover formula until we were back on our feet.
I'm not being dismissive of hunger. I have known hungry children. The ones I've met have uniformly come from families that were overextended, that had cousins and close family friends crashing in their living rooms, or that had some medical emergency or long-term unemployment. There was always an actual external reason for their hunger. I've never met a parent that simply didn't bother to feed their kid. (I've no doubt they exist, but I've never met one—I think they're probably about as common as serial killers and receive as much publicity too.) The key is that it wasn't the having of the kids that was the backbreaking straw for these families.
Children, themselves, do not actually require much. Two families living on one poor person's income, though, or one family on no income, is impossible no matter how little they need. It's ridiculous to make the argument that people should be able to predict every possible downturn in their lives in advance. Poor people are not uniquely psychic. Just like rich people don't think, wow, maybe we shouldn't have kids because we might have an acrimonious custody battle someday, poor people don't decide not to have kids because they think, wow, maybe Aunt Jane will lose her job and have to come live with us with all her kids. And I'm using the extended-family example, but it could be any disaster—illness, whatever. The point is that people don't plan their lives around certain disaster. People who do are called paranoid.
And there are resources for families so their kids don't go hungry, ideally. On a daily level, there's WIC for formula, and once babies have outgrown that, they're old enough to eat whatever you're eating. I've heard critiques of that too—that you shouldn't have kids if you're not going to feed them a healthy diet (which apparently consists of organic kale and quinoa, because it's not like poor kids have never seen bananas or apples). What we eat is generally fine for our kids, at least according to the food safety people. And people do tend to buy healthier food when they have kids in the house, from what I've seen. I know we do. My kids eat a lot of fruit. A lot. My three-year-old is obsessed with all the different kinds of fruit in the world; we go to grocery stores and she picks it out and we call it a good choice. She eats chicken nuggets and fries, sure, but not constantly. Mostly, she eats a bit of my sandwich or her dad's noodles or whatever it is we're making.
I can promise you that I did not buy much fruit before the kids came along. I rarely bought anything perishable outside of the requisite coffee creamer and milk. So yes, believe it or not, poor people do sometimes make smart decisions just for their children's sake.
Yet hunger is still a real thing. I've been there. I didn't qualify for food stamps at one point because on paper we were said to be getting a living stipend from the VA. I know this sounds crazy, but hear me out and maybe (if you don't already) you will finally understand why being poor and qualifying for benefits is not the same as being poor and actually getting benefits. I've briefly mentioned before that we didn't get a stipend from the VA that we'd been promised. So here's what happened: Basically, we were awarded a certain amount of money through the GI Bill. Because of a paperwork error on the part of the university, the VA never actually mailed us the living stipend I've mentioned. And they knew this, and acknowledged this, as did the folks at SNAP to whom we applied for food stamps. However, simply because the government _said_ that we should be getting $1,200, we were disqualified from receiving food stamps. Despite the fact that everyone involved agreed that the money was theoretical and that we didn't actually have it. We eventually got it cleared up, but it was one more thing to deal with.
When I think about that, I hope that the people who want to make sure that they weren't feeding a single person who isn't abject are happy. I know I certainly felt better about the state of the country watching my husband being thanked for his service by the people telling him that they'd be rejecting him for food aid. So, dear voters and policymakers who are very, very afraid that a poor person might illicitly have a decent steak for their birthday: Thanks for the months of ramen.
I have the solution to hungry children in America. Nobody wants to do it, but here goes: Fucking feed people. Cancel the programs where we pay farmers not to farm, and grow food. Buy it from them and use it in schools. Create real jobs. Fund SNAP. Stop calling it welfare and start calling it something that describes what it is: a subsidy like any other so that the people actually moving this huge wheel of capitalism can live decent, maybe basic but still pleasant lives. Hunger: solved.
—
Now let's talk about health care, which obviously kids need. There's a thing called the Children's Health Insurance Program. Its income standards aren't as stringent as Medicaid's. Most kids who need it qualify. It's not the same as being able to take your kid to the doctor for every sniffle, but if they have a serious fall or a scary spike in temperature, you can call a nurse. We can also take them to the ER, which is where the pediatrician tells you to go if it's an acute situation anyway. In general, poor parents know what to do for minor maladies, and we know our kids will be taken care of if it's catastrophic and we're not working at a place that offers insurance.
But just because poor people can't afford to be hysterical about our kids' health doesn't mean we're blasé. I was sitting on a park bench one time when my daughter came screaming over to demand that I kiss her knee better. She'd scraped it somehow—it looked worse than it was.
So I kissed it and sent her back to play. Another mom, clearly in a different tax bracket, turned to me to ask whether I needed to borrow her antiseptic; I said we were fine. And then she told me how she'd really like to be able to be so nonchalant about her baby being hurt. I wasn't sure whether she was trying to be cutting or really meant it; either way, my brain started demanding I cross the vast gulf between "not making a big fuss over a skinned knee" and "nonchalant about my child being hurt." They're different things. Way different things. So, rich lady who thinks I'm nonchalant? Mind your own business. And maybe sometimes, when your little princess skins her knee, send her back out to play instead of acting as though she'd just lost a limb.
It's good for them, you know.
—
Once we move past all the daily subsistence-level stuff that we really need to worry about, the question seems to change from "Why do you have kids you can't afford to take care of?" to "Why don't you take care of your kids in the exact manner of which I approve?"
Take college, for example. Many rich people look at poor people and think it's disgusting that we can't afford to give our kids a good education. Or maybe they think we deserve it for being poor. Either way, I know they think that our kids' educations are suffering because of our class status. As if we haven't thought about this or don't have a plan. Well, newsflash, we do. My husband and I will do what our parents did. We'll make sure our kids are curious, well-read people. We'll make sure they get good grades. We'll make them study. And when it comes time, we'll see what sorts of scholarships and grants they qualify for and then probably take out loans for the rest. I guess I don't really see how my plan is that much different from a wealthier person's until you hit the "we'll just pay for it" level. Is it really the end of the world to go to a state school?
I think there's also a judgment leveled at poor parents that we give our kids a terrible quality of life, as if our children are deeply conscious of their poverty on a daily basis. And certainly, there are a lot of people who've been plunged into circumstances so bad that they can't keep their kids ignorant of it. But lots of people are just struggling to get by, and they're doing so without irreparably harming their children. Maybe things aren't picture-perfect all the time, but I don't see the value in that anyway. I promise that if, like I did, you paint rainbows on your kids' walls, it'll be a decade or so before they realize that it's crooked and definitely not a professional job. You might notice. I notice. My kids? They think it's a pretty fucking cool rainbow.
That's what I love about kids—everything is magic for them. If you tell them the world is awesome, and you make sure awesome things sometimes happen, they will totally go with you on that one. Awesome, to a kid, is a rainbow wall or a tickle fight or a hiding place in the closet. They don't realize that collecting every My Little Pony is awesome unless you tell them it is. Class, the relative having of things—that stuff doesn't come up until later. Until we put it there. Kids will not notice worn spots in their clothes until they are socialized to. They simply don't get social constructs, class included. We raise our children to believe whatever we decide they should. And like most poor people, I'll raise my kids to be resourceful and aware. At some point, I will discuss class with them, just like every parent will discuss the real world with their children.
The point is that my kids are loved and they know it. I've heard a lot of young women give that reason for having their babies, actually: love. I come from a culture where the girls marry young and the families are big. It's just how we roll in Utah. I've actually seen people have numbered jerseys made for their kids when they made a full team's worth. That is considered cute where I come from.
I've heard it said that poor people have kids because they want someone to love them unconditionally. But I think it's more nuanced than that. I think that the stereotypical teenage wannabe mom who gives that reason wants someone she can safely love, someone who is predictable and steady and will stick around no matter what. And yes, it's sad for a young girl to feel that way. That said, I can understand it. In Utah, where we often marry young and have babies young, young women might think, "I may as well get started being an adult and having a family." Is it the wisest course? No. But it's not crazy. It's not even unrealistic. It's not like these girls have brilliant futures in the Ivy League that they're passing up to have babies; those are typically reserved for the children of brilliant Ivy Leaguers. They are deciding to have their toddlers while they themselves are young and have the energy. And plenty of people, no matter where they are from, simply have love to give.
What really riles me is this idea that poor people are somehow inherently more selfish when we have children. There are plenty of rich people who have kids for exactly the same reasons I just described—because they want someone who will love them unconditionally, and with whom they can share that kind of all-encompassing love. But somehow, because they have money, rich people are entitled to feel that way without being derided. Let's not kid ourselves, though, that it's any less selfish or self-centered.
What, after all, is Baby Gap and its ilk appealing to, if not parents who enjoy dressing up their kids as little mini-mes? Trust me, no infant has a serious desire to wear a cable-knit V-neck sweater with a collared shirt underneath, no matter how adorable they look. Preschool prep classes? Not meant for the kid's self-esteem. So the problem of childbearing as an extension of your own personal brand sort of transcends social class.
—
Truth time: We do not breed for sweet, sweet government cheese.
I understand that some people will say, "But you just said that everything was cool because of welfare!" I can see how one might come to that objection if you're only working on what you have read in this book so far. But trusting that you will always at least be able to feed your family, even if it is at food banks and with SNAP, is a whole different ball game from actively deciding to have a child specifically for the money.
Okay, quick lesson time. Welfare isn't a thing. That is to say, welfare is a lot of things as opposed to one thing. And each of these things has different requirements. It's not hard to qualify for some things, relatively speaking. If you're starving, you can pretty much count on qualifying for SNAP or food bank services. Now, access to those things can be sketchy, but that's a different point. The point here is that food benefits can be spent only on food; the benefit card blocks anything that isn't approved. Cash benefits, the ATM-withdrawal kind of welfare—money that you can use on rent, gas, the water bill, clothing—are actually damn near impossible to qualify for. And to get them, you've got to jump through a lot of extra hoops. Cash benefits are the ones tied to work or looking for work or training for work or working for the state.
If you are desperate enough to be breeding for cash benefits, you are for all practical purposes having kids in order to be poor enough for the government to give you a full-time job. See, the reason everyone says that you get more money for having kids is that your benefits are determined by both your income and household size. So, to make it an income stream, you have to decrease your non-benefit income and/or increase your household size sufficiently. That, I think, is probably pretty rare. And if you think about it for a few seconds, I think you will see how ridiculous the whole idea of it is. It would be like breaking your leg so you can go to the hospital because they'll feed you while you're there.
I definitely have told that joke once or twice—that I was having kids for the sweet, sweet government cheese—but hello? I _meant_ it as a joke. Granted, I do think there are many stupid people out there. There are stupid rich people and there are stupid poor people. The stupid rich people think that welfare queens are breeding like rabbits. And sure, there are probably a few people out there who did not realize even after Kid One that kids are a giant pain in the ass. Maybe a few of those idiots thought they'd make an easy paycheck by having another kid. But I'd argue that there are a lot fewer of these poor idiots than those rich idiots think there are. And by a "lot fewer," I mean a statistically insignificant number of poor people are doing this. Can I prove this? No. But nor can I prove that people aren't breaking their legs just to get some lunch.
I'm not even certain how people think it's possible that someone would have kids for welfare benefits. Do these people not have kids of their own? Did they manage to sleep through the colic somehow, or did they simply block it out? I mean, if you're going to pay me in multiple tens of thousands of dollars a year to have a kid, okay, maybe it's worth thinking about. But a few thousand dollars extra, best-case scenario, and that's my entire income and I'll still be living this desperate life? Yeah, no, I'll pass on that deal.
To accept the idea that someone would have babies just for the money, you have to assume that they see their children as stock rather than as kids. The more temperate assumption that follows is that poor people neglect their kids. But what wealthier people view as neglect is pretty shallow stuff and to me is just a matter of taste. My kids have dirty faces sometimes. Unless we're going somewhere or expecting someone, my husband and I really don't bother making sure they're spotless. It's a losing battle with toddlers. Our kids are both fascinated with baths, so they're always pretty clean overall, but their faces and hands are a different story. Most people I know are the same way; we just don't have the time or energy to chase little kids down and demand that they keep their hands clean. They're little, they're _supposed_ to ignore the lawn in favor of the mud puddles.
That said, I'd be horrified if they left the house like that. I think most people are that way in secret, inclined to be a bit lax at home. It doesn't speak to your actual parenting skills, I think, although the state of your children's faces when nobody's watching is a clear indicator of whether or not you lean toward OCD.
I'm not saying that there are no standards, just that maybe some of them could use some loosening. And of course there's a line you're just not supposed to cross. There's being a bit lax, and then there's being legitimately a kind of awful parent. Don't worry, poor people disapprove of those people in Wal-Mart screaming empty threats at their kids at top volume too.
But I think it's a little misguided when wealthier people turn up their noses at the parenting style of poor people who don't necessarily treat their children like precious china that would break if looked at sideways. I'm not preparing our kids for a gentle world, full of interesting and stimulating experiences. I'm getting them ready to keep their damn mouths shut while some idiot tells them what to do. I'm preparing them to keep a sense of self when they can't define themselves by their work because the likeliest scenario is that (unlike doctors and lawyers and bankers) they will not want to. I'm getting them ready to scrap and hustle and pursue happiness despite the struggle.
I think a lot of what people see as bad parenting is simply that our kids have different expectations. It wouldn't make any sense to take wealthy kids and prepare their brains for drudge work. And it doesn't make much sense to take poor kids and prepare them to seek fulfillment from work. That's not how it goes for us. If they find it, that's fantastic. But odds are, they will work just as many zombie jobs as they will good ones.
I'll teach my kids to be curious, to learn stuff for themselves because learning is kind of awesome for its own sake, to find what interests them and get obsessive about it. But learning and thinking is only a hobby for the working class, and I think it's best they're prepared. You never know what their lives will be. The happiest people are the ones who can simply block out the worst of it.
—
When I gave birth to my oldest daughter, I was visited more than once in the hospital by authorities because I'd had no prenatal care. They asked invasive are-you-an-idiot questions. All the questions seemed designed to make me look like an unfit parent: One of them was "Do you have a job?" My answer was "Not at the moment." (True—I'd quit work just before giving birth and didn't intend to look for more work for another two weeks.) Another was "Do you have a permanent home?" My answer to that was also no. (This was during the time when our apartment was flooded and we were fighting with our landlord about our housing.) I was asked my education level (college dropout), and I was even asked if anyone in my home had a diagnosed mental disorder. Yep. After all this, I was pretty convinced they weren't going to let me take my daughter home. Luckily, I dodged that bullet. She was healthy enough, and it's not like I was the first uninsured woman to get pregnant in a century or anything.
A neighbor of mine was investigated because she was at work _too much_. And they asked her the same sorts of questions, meant to find fault: How many hours are you gone? Have you considered cutting back? She said that they spoke a lot about marriage during her sessions, like making her boyfriend a permanent fixture was some kind of panacea. (The dude was useless, and as far as I could tell, he was her one indulgence.) The whole time these authorities were shaming her for working so much, she was thinking about nannies. See, if she'd been wealthy enough to hire a nanny, it wouldn't matter how much she was gone. She was talking to the authorities only because she didn't get paid enough.
I knew a guy, single dad, who had two girls. One in particular decided to be a hell-raiser; she started fights at school, stole beer from the fridge and gave it to her friends (of course on school property), and generally made herself a giant pain in his ass. I don't think anyone really blamed the girl; being motherless at thirteen can't be easy, and her mom had died not too long previously. It was the sort of situation that, I imagine, wealthy kids get some extra consideration, maybe some therapy for. Instead, they sent the dad to jail for contributing to the delinquency of a minor. Needless to say, he lost his kids. He's been spending every penny he's got ever since, trying to regain custody.
Are these irresponsible parents who deserve to have their kids taken away—or to have even the threat of that held over their heads? No. They're just poor people who love their kids and are doing the best that they can for them with limited resources. So let's stop saying that poor people are irresponsible parents and start admitting that society doesn't seem to believe that if you are poor you are entitled to be a parent at all.
Given how easy it is to lose our kids, it's no wonder that many poor people avoid any brushes with authority. We've learned how truly defenseless we are, so we just stay away. And what's the biggest authority in most children's lives? School.
My kids are still little, but I am not looking forward to dealing with a school once they hit that age. I'm afraid my kid's going to repeat something she heard at home between me and her dad. For example, our endless _South Park_ references. What if someone hears her say something from the episode in which Cartman feeds a kid his own dead parents to make up for a pubic hair scam and assumes that we're teaching our kid about the joys of revenge via forced cannibalism? Is a woman from social services going to show up at my door and start asking questions about my salary and employment? Will it matter that she hasn't actually seen this happening, only heard us reference it in passing?
I've got it relatively easy here. I was well educated through much of my childhood. I don't have to feel awkward going to a parent-teacher meeting for my kid. I don't have to deal with a language barrier. I don't have to deal with getting the shaming that single parents so frequently come in for: Your child needs you home, you're not doing enough, you have to find more hours in the day or you're a bad parent.
When I was living in California, a Spanish-speaking neighbor asked me to read her a letter from her kids' school. The letter was full of impressive words. Words like "responsibility" and "consequences" and "requirements." She had been ducking the school for weeks because they'd required her son to participate in some fund-raising program and he owed the school money for not hitting his minimum sales. She didn't have it, so she stopped answering their calls. When she got this letter, which was a wordy "what we're up to" newsletter deal, she thought it was a collection notice. I tried to explain that he'd get to go to school regardless of a $20 debt, but I couldn't convince her. She simply didn't believe me. And the truth is, given how badly I've seen poor people treated by whatever system they're forced to deal with, I didn't really believe me either.
What it comes down to, then, is the idea that the _very same situations and behaviors_ are treated completely differently depending on how nice your stuff is. Kid gets into a fight at school? If he's black and poor, he's going to jail. If he's rich and white, he's going to military school. Was your daughter busted with drugs? If she's poor, she's getting charged. If she's rich, she'll go to a nice rehab facility for however long propriety demands. The only reason it looks like our kids misbehave more is that we can't afford to cover up for them when they do.
During World War II, we had government-sponsored day care facilities. It was generally acknowledged that single-parent households, which the families left behind by the soldiers were, needed extra support. Maybe, and this is just a thought, we could do that again. Child-care crisis solved. Plus, it's another jobs program.
I'm not saying that poor kids have the same opportunities as rich kids. They don't. And that's bullshit. But that is not the same thing as saying that the poor are not capable of being decent, loving parents of decent people.
Besides. If we don't keep having kids, who do you think is going to work in tomorrow's restaurants? _Your_ kids?
# 8
Poverty Is Fucking Expensive
I once lost a whole truck over a few hundred bucks. It had been towed, and when I called the company, they told me they'd need a few hundred dollars for the fee. I didn't have a few hundred dollars. So I told them when I got paid next and that I'd call back then.
It was a huge pain in the ass for those days. It was the rainy season, and I wound up walking to work, adding another six miles or so a day to my imaginary pedometer. It was my own fault that I'd been towed, really, and I spent more than a couple of hours hating myself. I finally made it to payday, and when I went to get the truck, they told me that I now owed over a thousand dollars, nearly triple my paycheck. They charged a few hundred dollars a day in storage fees. I explained that I didn't have that kind of money, couldn't even get it. They told me that I had some few months to get it together, including the storage fee for however long it took me to get it back, or that they'd simply sell it. They would, of course, give me any money above and beyond their fees if they recovered that much.
I was working two jobs at the time. Both were part-time. Neither paid a hundred bucks a day, much less two.
I wound up losing my jobs. So did my husband. We couldn't get from point A to point B quickly enough, and we showed up to work late, either soaked to the skin or sweating like pigs, one too many times. And with no work, we wound up losing our apartment.
It's amazing that the things which are absolute crises for me are simple annoyances for people with money. Anything can make you lose your apartment, because any unexpected problem that pops up, like they do, can set off that Rube Goldberg device.
One time I lost an apartment because my roommate got a horrible flu that we suspected was maybe something worse because it lingered forever—she missed work, and I couldn't cover her rent. Once it was because my car broke down and I missed work. Once it was because I got a week's unpaid leave when the company wanted to cut payroll for the rest of the month. Once my fridge broke and I couldn't get the landlord to fix it, so I just left. Same goes for the time that the gas bill wasn't paid in a utilities-included apartment for a week, resulting in frigid showers and no stove. That's why we move so much. Stuff like that happens.
Because our lives seem so unstable, poor people are often seen as being basically incompetent at managing their lives. That is, it's assumed that we're not unstable because we're poor, but rather that we're poor because we're unstable. So let's talk about just how fucking impossible it is to keep your life from spiraling out of control when you have no financial cushion whatsoever. And let's also talk about the ways in which money advice is geared only toward people who actually have money in the first place.
I once read a book for people in poverty, written by someone in the middle class, containing real-life tips for saving pennies and such. It's all fantastic advice: Buy in bulk, buy a lot when there's a sale, hand-wash everything you can, make sure you keep up on vehicle and indoor-filter maintenance.
Of course, very little of it was actually practicable. Bulk buying in general is cheaper, but you have to have a lot of money to spend on stuff you don't actually need yet. Hand-washing saves on the utilities, but nobody actually has time for that. If I could afford to replace stuff before it was worn out, vehicle maintenance wouldn't be much of an issue, but you really can't rinse the cheap filters again and again—quality costs money up front. In the long term, it makes way more sense to buy a good toaster. But if the good toaster is thirty bucks right now, and the crappiest toaster of them all is ten, it doesn't matter how many times I have to replace it. Ten bucks it is, because I don't have any extra tens.
It actually costs money to save money.
And it even costs more to get to your money if you're poor. One of the reasons that Wal-Mart is so popular among the serving class is that it costs three bucks to cash your paycheck—flat fee. And they let you keep all of it but for that fee. Banks, on the other hand, are a giant pain in the ass. I loathe them, actually. Not fire-of-a-thousand-suns level, but I don't enjoy being in them. They seem to me to exist only to take your money. I've heard that wealthy people don't have to pay fees for everything, but if you're poor and don't have so much money to put in the bank, then you fall below their minimum balances and even accessing your money can cost you money.
Banks are useless to me. If you run low on cash, they take some more money just to punish you for not having enough money, and then they charge you $25 because, now that they've taken your money, you actually have negative money. That's nearly 10 percent of your _next_ paycheck already. Besides which, banks are generally across town because they don't put banks in the places poor people live. They're always closed by the time you make it there from work, and the tellers always start being that kind of superior polite when they see your account balance.
I don't have a bank account for one reason: I am paranoid. I want my money, what I have of it, near me at all times. Otherwise, somebody might take it. I've had bank accounts just so that I could receive direct deposits from my employer. But since prepaid cards for payroll came out, I simply haven't needed a bank. They charge you $10 up front to set up the card and $5 a month in fees. The end.
I know that banks are where you go to get a loan, and that if you put your money in the bank and it stays there forever, you get good rates on things, but I don't get large loans and I don't have cash to just leave somewhere, so that doesn't really help me. And I can't get small loans there either.
That's why poor people pay insane interest rates. No matter what sort of credit rating you have, if your car's water pump goes out, you can't get a $300 loan from a bank. When something like that happens, some small emergency that I can't actually afford until the next paycheck, I've generally had three options: a payday loan, borrowing from friends, or doing without. My friends aren't always exactly flush themselves, leaving me with two choices: make it to work or not. When I've lived in the country or cities without good public transportation, making it to work has generally meant payday loans.
I'm kind of torn about payday lenders—the storefront small lenders that everyone's up in arms about. The way these places work is pretty simple: You give them some kind of collateral, like a postdated check, if you have a checking account, or a car title. In the nicer ones, you don't need collateral, but you have to give them more paperwork about your income and a whole list of people you know. They call every one of them, and if all the references check, you get the loan.
Then you are allowed to borrow somewhere between $100 and $1,000 usually, and you pay an extortionate APR. Like, hundreds and hundreds of percentage points. But because the loan is so short, it's a relatively small amount of money in practice. If you can just pay it back with your next check and make it, then you're fine.
The reason people are up in arms, though, is that typically that's not the case. Most people can't take that full hit the next pay period either, so they roll over the loan. Then they wind up getting stuck and basically paying rent for the use of the money until they can pay it off. Worse, if anything happens before they do, then they have to take out another loan to cover _that_ , __ and they can't do that at the same place. So then they owe two of these places money.
And payday lenders are brutal about getting it back. If any one of my employees was in default, we'd dread answering the phone. They'd call constantly. And they call for years. Meanwhile, clearly it's usury to charge 400 percent APR.
So I should be wholeheartedly against them, right? But the thing is, I'm not. Because they do serve a purpose that no one else does for poor people. I don't think in terms of annual APRs when I'm getting a payday loan. I think of it as a $15 poor tax. Every time we need to borrow $100 for a week, it costs $15. Are these places preying on the weak? Yep. Is it less moral than huge banks preying on the same demographic? Probably not, and those assholes have never bailed me out of a tight spot before. The payday places, evil empire though they are and all, actually do fill a niche where there's a real need. I've used them in the aforementioned water pump scenario and once when I got the flu and missed three days' work on a week I couldn't afford a short paycheck. Once it was because my husband's birthday was two days before payday and I'd put in extra shifts, so the expense was doable. I considered waiting until after, but his birthday was a day that we both had off, something nearly impossible to manage in an average week. We'd both requested it off months in advance, and I hadn't bothered to count ahead and remember which pay period to ask for extra hours. It was totally worth $15.
I figure that at some point it will occur to someone, somewhere, that the reason there are so many payday loan places is that there are so many people whose checks simply will not last a whole pay period unless everything goes perfectly, and that people who have things like perfect weeks aren't the sorts of people who've ever cashed a check at Wal-Mart at three a.m. because they ran out of the napkins they'd been using as toilet paper for two days. Those people will find a less shitty way of doing business; perhaps someone can start a nonprofit bank that charges minimum fees or something. For now, we have our fees to pay.
I put furniture rental in the payday loan column because rental places are in the business of letting poor people have nice things for more than retail. The rental is simple; it's just making twelve easy payments of $99.99 for something that might actually cost closer to $1,000 if you paid it all at once. You are renting to own, so there's no risk; you just pay them when the bill's due, and when you're done, you own some furniture. In the meantime, you have some furniture, which is handier than the saving-up thing because sometimes you actually need a bed. Plus furniture rental places are pretty decent about you missing a week or two if you're having a rough patch, provided you generally pay on time and it doesn't happen too often. They get more interest that way.
Our economy seems to be run on credit, and it really doesn't serve poor people well. I get running credit checks on employees that will be working with cash or jewels or incredibly expensive bits of duck or something. But you can find job listings informing you that you'll need a credit check to be a receptionist or lawn guy. I guess maybe you could theoretically bribe an indebted receptionist for company secrets, but what's a gardener going to do? Not mow the lawn? I don't understand what credit—which is purportedly to see whether you're financially stable—has to do with whether you can mow grass. And I _really_ don't get what being poor has to do with being a good driver, but I know that if you've got good credit, you get cheaper car insurance. This basically ensures that rich people pay less for car insurance than poor people do. Which I hope we all can agree is both ironic and tragic.
This is the part where people say, "But credit isn't _just_ an indicator of finance! It's an indicator of trustworthiness and character!"—which would be fine if so many people of perfectly wonderful character weren't poor. Some of us are excited to do our very best every day. Lots of people who are lacking in resources are, you know, average people. Normal, with typical characters.
The real reason poor people have bad credit is that life is more expensive than we can tolerate. Again, see medical bills: not fucking likely we're going to have the money for those anytime soon. The vast majority of the poor people I know have terrible credit, and this affects every aspect of our lives. Whether or not you're currently doing okay, if you've got a poor credit score, you're going to have trouble finding anyone to rent to you. So poor people tend to be scraping the bottom of the barrel when they're looking for a new place to live—they're basically moving into the places that no one else wants.
It just adds insult to injury that for people who don't have enough money to buy something, landlords require that you cough up the equivalent of three months' rent—first, last, and security deposit—right at the outset. And that's just to get the keys to the place. Then you have to pay deposits on all the utilities to get them turned on. You might see a new tenant with no electricity for a couple of weeks and no gas for another pay period after that. I've done it. I just stayed with friends for the first few weeks I had the apartment. Then I moved in when the power went on. It's a last resort, when the schedules don't match up and you need to move your stuff before you can cover the power bill.
All this really rocked me when I got out into the big wide world, actually; you've got to come up with $1,000 or more as a security deposit just to move into a different shitty apartment. And this is for a $400-a-month studio—in addition to the first and last months' rent. And good luck getting that security deposit back when you move out. The landlord will argue that you put the cracks in the wall and that's it. I've lived in buildings where residents would actually warn new renters about it, because no one could recall a single security deposit ever being refunded. Sure, if you have the benefit of parents who will co-sign the lease for you, then you can possibly avoid having to pay such a high deposit. But I didn't have that option. Many of us don't.
There are housing voucher programs, of course. There's a subsidized housing program called Section 8, which seems to be pretty much the only game in town no matter where I've lived, excepting some religious charities. Basically, the government gives you a rent voucher if you qualify, and you get a list of approved apartments to pick from. I've sometimes wished I lived in Section 8 units, because the management company has to make sure that the doors and windows and all the appliances work properly in those apartments. I lived in a mixed building once, half Section 8 and half self-pay. Those of us paying cash found that we got less maintenance done on our apartments because the government wasn't picking up the tab for part of the rent and therefore wasn't insisting on regular inspections. The people who have the feds making sure their apartments are at least basically maintained live in . . . well, the places aren't falling apart. It's actually one of the things the government inspects for—visible cracks in the walls or ceilings.
The waiting list is typically long for subsidized housing. Eight years in DC, three in Houston. I've never seen one under two years. And I've never found it worth getting on the list, because I am unlikely to live in the same county and have a two-year-long bad spell. If your income changes while you're on the list, you're supposed to call and tell them. Then you're off the list. Unless you know for certain that you will not be doing any better for at least a couple of years, it's not even worth filling out the paperwork.
We can do better than this. We choose not to.
—
It is impossible to be good with money when you don't have any. Full stop. People tell me to save, not to buy luxuries like basic entertainment or communication or expensive food like hamburgers or pretty much any seafood according to Fox News (Dear _The Daily Show_ : More of those segments, please), that those things are reserved for people better than me—read people with disposable income. And to the people who say that, I have only the wise words of Dick Cheney: Go fuck yourself.
If I'm saving my spare $5 a week, in the best-case scenario I will have saved $260 a year. For those of you who think in calendar quarters: $65 per quarter in savings. If you deny yourself even small luxuries, that's the fortune you'll amass. Of course you will never manage to actually save it; you'll get sick at least one day and miss work and dip into it for rent. Gas prices will spike and you'll need it to get to work. You'll get a tear in your work pants that you can't patch. Something, I guarantee you, will happen in three months.
When I have a few extra dollars to spend, I can't afford to think about next month—my present-day situation is generally too tight to allow me that luxury. I've got kids who are interested in their quality of life right now, not ten years from now. My whole family can be completely content for hundreds of hours for that money. Would some rich people think it was scandalous that a poor person would spend money on a game system? Probably, but that rich person can go to hell. Escape is the thing I value most, and it's a thing we'll sacrifice for.
When it comes to money, I think in value, not in sums. If I run a hundred dollars short, I can call in the loans and get my rent together, or just run up against the grace period for late payments. Or possibly I will be sort of fucked; it depends on whether or not I find a solution to the short-term problem. The only rational thing to do, really, is try to enjoy yourself as much as you can, if this is to be your life.
Here's the thing: We know the value of money. We work for ours. If we're at $10 an hour, we earn 83 cents, before taxes, every five minutes. We know exactly what a dollar's worth; it's counted in how many more times you have to duck and bend sideways out the drive-through window. Or how many floors you can vacuum, or how many boxes you can fill.
—
It's impossible to win, unless you are very lucky. For you to start to do better, something has to go right—and stay that way for long enough for you to get on your feet. I've done well in years that I had a job I didn't mind terribly and that paid me well enough to get into an apartment that met all the basic standards. I've done less well in years where I didn't have steady work. The trouble's been that my luck simply hasn't held out for long enough; it seems like just when I've caught up, something happens to set me back again. I've been fortunate enough that it's rarely compounded, and I've stayed at under sea level for short periods instead of long-term. But I've stared long-term in the face long enough to have accepted it as a real possibility. It's only an accident and a period of unemployment away.
It feels like I'm always climbing up the same hill, always trying to make it to neutral. And I don't have the stamina of Sisyphus to keep me going.
# 9
Being Poor Isn't a Crime—It Just Feels Like It
I think that I might be a felon. My crime? Moving from Ohio to Utah. We were getting food stamps in Ohio, and I called the state to let them know to shut down our account. We applied for food stamps when we got to Utah and were approved. But it turns out Ohio didn't actually shut our account. They kept giving us money instead. Utah told us that we needed to call Ohio back, which we did. We were assured that the mistake was fixed. We called back Utah, who told us that we were still Ohio recipients. The state of Utah called Ohio. Three times. After that, our caseworker told us she didn't have time to deal with those people.
The thing is, it's no wonder Utah couldn't get through to Ohio—there's one welfare office for all of Hamilton County, which includes Cincinnati. One. For Cincinnati. That would be like having one Starbucks for all of Manhattan, or one tiny dog store for all of Los Angeles.
Now, it's illegal to use welfare benefits if you are not a resident of the state issuing them. So we found ourselves with a food stamp card that had an ever-increasing balance that we couldn't use. Every time we called, Ohio's worker would confirm that yes, there was a file in which they could see that we'd been asking them to stop giving us money for months. They'd apologize. None of them could figure out what the problem was. Each of them assured us that they'd fixed the problem.
We used the Utah benefits for a few months, until we got on our feet. And then we got the bill. As it turns out, even though Utah was perfectly aware that we couldn't help Ohio's clerical errors, and that we'd spent dozens of hours trying to get them to fix it, the law still holds us responsible for the duplication of benefits and calls it fraud. We were responsible for paying back the state.
Of course I called the state when we got the letter. Our caseworker apologized the entire time she was telling us how completely fucked we were. We'd only just gotten on our feet and we now owed the state more money than we made in a month. Oh, and while we were dealing with that hit, we'd be unable to get any additional help, because now that we were just on our feet, we didn't qualify for anything.
So it's no wonder that I don't have a lot of respect for authority or authoritative institutions. I'm so used to seeing people being punished for things they haven't done wrong, I'm pretty much always half sure I'm in violation of a law. And I'm not even being particularly paranoid in saying that—in a country where loitering is considered a crime, cops can pretty much arrest you at will. Refusing to tell cops anything they want to know is also criminally punishable should you run into a cop who's willing to stretch the meaning of "obstruction" or "impeding." You don't have to be robbing a bank to be a criminal. You just have to be poor and down on your luck and fall asleep on a park bench. I was recently on a college campus and saw at least three kids passed out on benches or at tables. I was tempted to call campus security to report the scourge of people resting. It turns out that whether sleeping on a public bench is a crime or not depends entirely on whether you have enough money to look like you have a place to sleep.
Another funny thing: It's incredibly easy to pick up a misdemeanor while actively trying not to get a DUI. If you walk home from the bar because you're drunk, or if you stay home in the first place but drink in your front yard, you are publicly intoxicated. Never mind that your front yard is where the afternoon and evening shade are and that you are very clearly just hanging out with your friends who are all of legal age, or that you misjudged and are wasted but can get home safely enough if you simply put one foot in front of the other down the correct roads. The fact that you left your car at the bar knowing that you shouldn't drive and you don't have cab fare will not be a mitigating factor. Your one solid bit of judgment that evening will potentially be punished severely.
People seem to be increasingly afraid of the poor—building gated communities and taking separate entrances—but it's not like criminal behavior as we think of it has suddenly skyrocketed. We've just made more shit illegal. And once you have a criminal conviction, best of motherfucking luck getting a job if unemployment is above zero. I've seen people get criminal records for stuff that you really wouldn't expect. You know that level of criminality where you just sort of shake your head, like toilet-papering a house or jaywalking? It's still criminal. I worked with a woman whose son, maybe thirteen or so, was in juvenile detention for rapping loudly outside after curfew. Now, I'm not saying it isn't annoying to have some kid outside being rowdy at midnight. I'm just saying that it's a bit crazy to send the kid to jail for it.
We have decided to lock people up for social deviancy these days. We tell ourselves that we're not running debtors' prisons, that this isn't Dickensian England, because we rarely lock people up for the simple fact of not having money. Instead, we lock them up for not paying court fines, or because poor people should know better than to be poor publicly, and because the cost of doing routine business in this country is the same whether you're rich or poor. And for the poor, that cost is way too high.
For example, I'd say my car is registered about three-quarters of the time, because sometimes the $50 it costs to renew the registration is more than I have to spare. At times, that's been more than a day's wages for me. And yes, I can go to jail for driving without proper registration. But if I'm too broke to renew, then I better get my ass to work, so I have to drive . . . and I'm guessing that you see where I'm going with this. In short: I'm fucked. Insurance I'm better about, because my life was upended by an uninsured driver. But I've been without it too—insurance companies aren't like the power company. They don't negotiate dates and payment plans. If you can't make your premium, you'll simply be uninsured until you can.
When I'm driving uninsured—because I have to get to work or buy toilet paper, the only two reasons the car moves in that situation—I take back roads and shop on the edge of town to avoid density and thereby lessen my likelihood of being in a fender bender. That'll get you sent to jail too, even if you're willing and able to pay the damages out-of-pocket.
The degree to which an accident or a traffic ticket could destroy my financial security—what little of it I have ever had—has made me a super-defensive driver. I don't take chances. I drive at precisely two miles over the limit, which is generally the sweet spot of not getting pulled over for speeding. If I drive more slowly than the limit, I worry that I'll be pulled over for curiosity's sake.
My policy of avoiding law enforcement is magnified when I'm behind the wheel of the car. It's my mission to appear as average as possible. Never stand out and never get hassled—unless we're in Arizona. (My husband is half Puerto Rican. He's pretty tan. We avoid Arizona like the plague. We've no interest in being asked to present his papers.)
So I go out of my way to give a wide berth to police and authority figures in general. There's no sense tempting fate. I'm sure most of them are lovely people, but I have no reason to trust anyone who has any sort of power over me. You can never be sure what they'll judge you for, and judgment has a nasty habit of turning into investigation.
For a long time, I have believed that most people think that poor people are criminals. Sound paranoid? Hear me out: Assuming you work in an office or white-collar environment, does your boss search your bag for stolen Post-its on your way out the door at night? No, I didn't think so. But I've had to surrender my bag at the end of my shift so security could search it and make sure I didn't swipe a box of pens or something. They did it to everyone, even each other. What kind of message does that send me? That I'm trusted? Or respected? Yeah, probably not. Instead, it tells me that my bosses think that if I have to work this crap job, then I'm definitely a thief. Or that they think I am so underpaid that I might steal out of necessity.
From there, it doesn't feel like much of a leap to conclude that rich people have written off an entire swath of America as trashy, careless, immoral, and irresponsible. And sure, some of us are. But some rich people are too. And if you had your bag searched every night, I guarantee you that you'd be sorely tempted to steal a few Post-it notes, just out of spite, if only to prove that you were smarter than they were.
—
This is a generalization, and I am once again going to take an opportunity to say that this is me talking; other people will feel differently about this. But overall, I think that most poor people have too many disasters in their own immediate future to worry about to be concerned about whatever natural or political disasters might be occurring way outside their circle.
I have a hippie friend. She's been known to dig through my trash for cans and drive them across the country to her favorite recycling center. I think she's crazy. It's not that I don't care about global warming or the environment; it's that there's only so far out of my way I'm willing to go. I don't really have the time or energy to worry about macro concerns.
Overconsumption is a concern for people who've made it to regular consumption.
I know people who are poor and environmentalists. It's not that poverty is guaranteed to make you callous, but being poor means that you are inherently unwasteful. Poor people just can't afford to buy a ton of extraneous shit and then throw it away barely used. So I don't really see a need to make the environment My Issue. I tend to eat food that is rejected by other people. (There are places you can buy nearly expired food for cheap.) I don't buy many things new at the store, because I can't generally afford it. I shop at thrift stores, where I can buy an almost-working bread machine for $2 and fix a wire. I combine all my errands into a single trip as a matter of course, because running to a store is generally more than a half-hour commitment and I want to save on gas.
I do not care about the whales. I'm unfussed about owls. I could give you a lot of reasons why I don't consider myself an environmentalist, but it mostly comes down to this: my issue is people, in the micro. Once we've hit the part where my own species is mostly taken care of, I'll start to worry about African rhinos. Until then, I'll just keep restraining myself from punching people when they look me in the face and argue that an ecosystem somewhere is more important than homelessness. It's not unimportant, and I'm glad someone is keeping an eye on those things, but right now it is nineteen degrees outside and there are some human beings that I am more concerned about saving just at the moment.
Poor people are busy keeping a roof over their own heads so that they, too, don't join the unhoused ranks. And that's about all that many of us have got time to be concerned about. Environmental concerns, campaign financing, civic engagement writ large—these are luxury worries for people with time and influence.
Do I wish that poor people were a little more politically engaged? Sure. I think that would help, and it sure as hell couldn't hurt. But I also get why people aren't beating down the doors of the polling places. For one, we can't keep track of whether we're supposed to bring a DNA sample or a urine sample this time to prove our identity and residency. It keeps changing. For another, the hours and polling locations in poorer neighborhoods keep getting cut for some reason. It is definitely not at all a conscious effort to repress the poor (read likely Democrat) vote. At all. Ever. (Dear GOP: You guys might want to police your people. They keep openly saying that your goal is to repress the vote of the poor.)
Additionally, at this point elections are mostly held for the benefit of people who devotedly follow politics. Everyone else kind of figures it's a done deal. Most districts are gerrymandered to the point of safety one way or another. Voting doesn't really enter into it, because no matter who stays home or heads out, more people in X party will vote.
Look, I'm not saying these are good reasons for not voting, but they're reasons that I can wrap my brain around.
What's harder for some people to understand is why poor people so often vote against our own self-interest. Even I have a difficult time with that one. Steinbeck said that we'd never be a socialist country because there were no poor Americans, only temporarily embarrassed millionaires. A lot of people really do think that way. I was raised by one. My dad talks like he's part of the top 5 percent. (Spoiler: He isn't.)
I have a Republican friend and every time we get into politics and the economy, he tells me that I simply don't understand the American dream. He says it doesn't make sense to punish the people you're trying to join. He is fairly certain that in the next decade or two, he will be worried about capital gains. He works at Wal-Mart. He's nearing thirty. No degree, no real résumé, no particular ambition to do anything. Just a firm conviction that someday he'll have a fantastic high-powered career doing . . . something. He's not sure what, only that this is America and anyone can make it. While he's waiting, he'll be protecting his future interests at the ballot box.
But voting isn't always about money. My friend Rachel is a lovely woman. She's actually kind of a liberal on the money stuff but she's a strict Southern Baptist. She's also a firm GOP voter. She tells me that she's always been poor, she always will be poor, and it doesn't really matter to her whether or not the rich people get richer. At least, not in any way that's really going to affect her day-to-day. Systemically, sure, she'll give you that the economic policies advocated by her candidates are actually not great for her, but since she lives in a non-union area, it's much more important to her to have a candidate that's firm on the Second Amendment and abortion. Those things matter to her, in a real way, every day. She thinks about them, she knows people affected by them.
I tend to think that the economic policies aren't going to change much no matter how badly we want them to, but I'm sure that all my friends should be able to get married to whomever they wish, and I like the idea that I can get birth control without having to ask the blessing of the Republican leadership. Plus, I can't stomach supporting people who honestly think poor people are getting the long end of the stick. People that oblivious shouldn't be in charge of the free world, on principle.
So that's why I encourage everyone to vote for my guys. But I'm not about to judge a poor person who couldn't give a shit about any of it. That person hasn't been given a whole lot of proof that her vote will matter anyway; voting hasn't resulted in policy shifts toward a more equitable distribution of government services. Our schools are still worse, our roads less maintained, our police less friendly. And we simply don't give a fuck about quantitative easing or who might manage the prime index, because we do not have money and so those concerns are entirely irrelevant to us.
Poor people have gotten the message loud and clear: The powers that be are not concerned about us. Meanwhile, wealthier people get all exercised about a poor person dropping a cigarette butt on a city sidewalk, as if this is proof that poor people _just don't care_. __ Let's take that theory a step further. When powerful people stick a waste treatment plant in that same poor person's backyard, does that mean that rich people _just don't care_? I'm not even going to bother answering that one, because I think I already did.
Personally, I don't litter. It's not because I particularly feel any responsibility to the environment or anything. The reason I don't litter is that first, it's an insane ticket to pay if you don't have to, and second, it's one of the areas of my life where I get a bit fuck-you and refuse to live down to the expectations of rich people that I don't give a shit where I throw my trash. Besides, some poor asshole has to pick it up, and I try not to make peoples' jobs worse on principle.
It's always been interesting to me that we're expected to care about beautifying the roads or streets. I don't, really. Not until the places that I live get the same maintenance resources as the places where the mansions are.
If you wonder why I am angry sometimes, why I don't always feel a sense of human kinship with people wealthier than me, that's a pretty good example right there. They don't feel any toward me, and I'm under no obligation to be the bigger person. It seems like I'm expected to have the oblige, but I never get the noblesse. And yeah, no. I won't be doing that.
—
And now I'll finally say it: Some stereotypes exist for a reason. The bald front lawn and truck with no wheels, the pile of tires—these are all images that come to mind when you think of poor people. In fact, I am the proud owner of a tire pile, inherited from previous owners of my house. I can understand why you don't find that aesthetically pleasing. Hell, I don't find it aesthetically pleasing. But what I can't understand is why you'd judge the person who's too poor to pay the water bill to spray that dead lawn, or pay the mechanic's bill to fix that truck, or take the time off work to do something about those tires. (I saved up once and put in two rosebushes. They died because I was away at work too much to water them.)
Like most poor people, I have rented for most of my life, and some of my landlords have maintained my apartments so appallingly that I'm not exactly motivated to drop money I don't have on improving their property. When I finally did buy a house, I had enough to cover the mortgage but not to put money into something as frivolous as landscaping. My yard is pretty much dirt with some grass sprinkled here and there. I estimated the cost of putting grass in: It came to about a paycheck and a half, before we even considered the water bill.
While I'm on the topic, let me tell you about my house. You see, I have terrible, awful credit, mostly due to medical and student debt. There's no way in hell I'd find a mortgage. So when I was living in the trailer and got pregnant again, we needed space. I had my biological dad living with me, my husband, and one kid already, in a single-wide.
So we went looking for a place to rent, like you do. And what we found was nothing affordable. The only places we could have made rent on were either in student housing, which is not where you live if you're trying to get an infant to sleep, or were so beaten down that they would actually be unsafe, because you really shouldn't let babies play on surfaces with exposed nails.
So we asked my parents for help co-signing. What wound up happening is that they could refinance their own home for less than they could get a separate mortgage; they refinanced, paid cash from that for my house, and I pay _their_ mortgage because it's sort of _my_ mortgage. Understand that we are discussing a house that didn't even approach $100,000 here, so the monthly payments are reasonable. Better than any place we could find to rent. Those are the contortions that those of us who are lucky enough to have family help—something that I have only recently had the luxury of—have to go through in order to participate in the economy.
Now back to the subject of maintaining that house. Yard care, which I hear is a relaxing pastime for many, is just another chore that I don't have time in the day for. There's no point paying for grass seed if you don't have a decent lawnmower and you never have an afternoon off to mow it.
So, okay, the ugly-lawn stereotype, I own that one and I don't really care what people think of me on that score. But the stereotype about bugs attaching themselves to poor people because we're dirty? That one pisses me off. I would like to take this opportunity to correct a common misimpression: You do not have to be a sloppy housekeeper to get bugs. That is some classist bullshit, right there. I've lived in places with roaches; they were there before me, and I'll place a public bet that the exact same roaches are still living there years after the fact. I tried everything. We stopped eating at home for two weeks so that there wouldn't be a single scrap of food in the place—they stayed. We put down poison—they stayed. We tried to smash them all—they wore down our resistance through sheer numbers. It was like being part of a single scout unit and finding an entire army just beyond the ridge; you've got no chance.
Roaches are nearly impossible to kill without repeated professional extermination treatments, and those ain't free. They live in walls and under woodwork; if there is a single crack in your apartment they can come in at will. Seriously, call your local exterminator and ask him if it is possible to stop a roach infestation with half a can of Raid in an apartment with cracked walls and a leaky sink. Start a timer from the end of your question and see how long it takes for him to stop laughing.
Bedbugs and lice like rich people as much as they like poor people. But if you're a poor person with either of those things, you will be judged. The only difference between a poor person with lice and a rich person with lice is that a rich person pays someone else to pick the nits out of her kid's hair. And if you're a poor person unlucky enough to get bedbugs, holy hell does your life suck. There isn't an effective pesticide for bedbugs—well, okay, there are two, but they're so toxic you can't spray them in your living space and then keep living there. Bedbugs can live for months without any sort of sustenance, and they also can live in ductwork and other places that you can't see when you're deciding whether or not to move into a place. You can't stop them once someone's introduced them into a building without some serious and expensive effort, and you can pick those things up on the bus, or at a gas station, or in a rented car, or at the airport, or generally anywhere in public.
Flies are inevitable when there are holes in your screens during the summer and your AC sucks or is nonexistent and you have to keep the windows open. They're easier to control through simple cleaning and some vigilant swatting than cockroaches, but they're a normal annoyance and a simple fact of life. That said, it's considered trashy to have flypaper up. You can't even win when you're clearly deploying effective containment measures.
Rodents living in holes in the walls of poor people's houses is such a common thing that mice were the entire supporting cast of Disney's _Cinderella_. Similar mice have starred in more than one children's movie since then. If you live in an older building, you'll get mice somewhere in it. I guess the upside is that you can pretend you're Cinderella, but I wouldn't hold out hope for any glass slippers coming your way.
_Being poor_ : that's how you get ants. Having household pests isn't a result of a sloppy, irresponsible nature. It's a result of being broke. It's insulting and priggish to insist otherwise, especially if you're someone who actually pays someone to come to your home to clean for you.
Hey, I'm not blaming people for having those luxuries—I'd love to have them too. I've often thought that I need a wife. Or maybe a staff. I'm not really sure what would solve the problem, which is that there's always a time crunch. There just aren't enough minutes in the day for me to earn enough money and keep up on life's details and clean my house and maintain my yard and have a marriage and hang out with my kids. So my husband and I rank those things in order of importance by visibility: Are we the only people who see or have to live with this? Yes? Then who cares?
I really wish I were one of those naturally neat people. I'm not. I'm a natural slob. It takes some serious routine to get me to keep my house clean as a matter of course, but I'm normally too fucking tired when I get off work to clean, besides which I've been cleaning up after people all day. I'm rarely in the mood to carry on with that another couple hours when I only have eight hours off between shifts. My feet hurt, and my back is sore, and if I'd like both sleep and a shower, then wiping the grease off the oven isn't even on my list of priorities.
I always have way more stuff that I can neatly store. Anyone who has ever gone without can relate to this. Who knows when you might need something and can't afford to buy it? So I rarely throw anything away if I can store it and maybe use it in the future. Stained shirts might be useful rags for the one time in my life I get some furniture polish and motivation at the same time. My stash of ruined T-shirts made great diapers when my kids were babies. I've torn apart two broken coffeemakers to make one working one. You never throw anything away if one of the parts is working, because you might need that part eventually.
I tend to buy in bulk when I have the cash or if there's a really good sale. Right now there are probably ten bottles of laundry detergent in my closet, because I found it so cheap. I go to discounters and wait until the snacks actually expire, at which point they're ten cents or a quarter for a whole bag of chips. Granted, the only reason they sit around that long is that they're off-brand and actually kind of gross (I have seen chips that were supposed to taste like BBQ ranch and cheddar and sour cream all at once, which I think we can all agree is just the worst thing humans have invented), but you can give them to the kids and they'll never notice. Or you can have a couple beers and you won't really care either.
—
I guess some people would call all this kind of shameless. And that's what this whole discussion about civics, and citizenship, and personal responsibility comes down to: self-respect, or a perceived lack thereof. Most privileged people have enough compassion to feel badly for people who don't have money. But unfortunately, a not-insignificant percentage of advantaged people have a hard time understanding that shame is a luxury item, because there is a point at which things are so bad that you lose all sense of shame.
Shameless is admitting that you're poor and asking for money. It's being brazen. It's having sex in public because you've got nowhere else to go. It's openly selling drugs when that's what you do for a living. I'm not going to try to defend hard-core drug dealers. They're indefensible—unless they are on TV, in which case we are fascinated by them. But most "drug dealers," in fact, are people who essentially share weed with their friends at cost. They're not looking to morally flatten their neighborhoods; they just don't see anything wrong with people getting a little high instead of a little drunk. And pushing dime bags is enough to pay a bill or two, keep your phone or gas on, and keep your car moving.
That's desperation. And I'll tell you something else shamelessness can lead you to: selling your food stamps. Is that illegal? Yup. Is it understandable? Yup. If you are willing to live on nothing but ramen, you'll have at least $20 left over on your food stamp card. You can then, completely hypothetically and I have never done this, engage in a transaction with a neighbor. They get food, and in return you get $10 for your gas tank. Your neighbor will do you this favor so that you will take them in the car you now have gas for to cash their paycheck, which they need to do to replace the $10 they just gave you for gas anyway. That's what we mean by hustling; you have to figure out who's good for what at any given time so that you can find rides and babysitters and small loans. You also need everyone to know what you can be counted on for, because that is your bartering token.
Is that shameless? Maybe. Shameless is something that happens when you have been pushed beyond shame, when you have nothing left to lose. If you will shortly be homeless, what have you got to lose by begging in the street? Maybe you will avert the disaster. If not, you've simply gotten a head start on your new station in life.
"Trashy" is a word that has two meanings. It can mean classless, hitting _Maury_ levels of public airing of personal behavior. Or it can mean unkempt, which is largely a function of how much time and money you have to spend on maintaining your house and person.
Trashy, the insult, means that you embody the poor-white-person stereotypes. Trashy is what you call people who have brought their eighteen-month-old to the restaurant and are letting him gleefully tear paper napkins and tortillas apart and scatter the pieces on the floor around him like so much confetti. Trashy is talking loudly on your phone in the bathroom. Trashy is using your outside voice to have personal conversations in public areas that are decidedly inside.
My husband, who's from the West Virginia part of Ohio, says that in the sticks where he's from, you can always tell a trashy person because their chickens are out. If you build a chicken coop out of reclaimed fencing and duct tape, you're not necessarily trashy. But you'd better damn well keep your chickens in that coop and off the road.
Okay, so we've established this: Poverty isn't pretty. We can't afford to dress nicely. Our yards are a mess. We don't really care about your political pet projects. But do you know what we really do care about? Each other. And I'm going to make a big leap here that I am very comfortable with: Poor people are, as a rule, a bit more generous. We understand what it might be like to have to beg even if we have never done it ourselves. In fact, there's data to back me up. The latest research shows that people of low socioeconomic status are more likely to be altruistic than their higher-class counterparts. In 2011, the bottom 20 percent of earners gave a higher percentage of their wealth away than the top 20 percent.
I'll put it to you this way: If good citizenship consists of a well-ordered life, then we poor people make terrible citizens. But if it means being willing to help out your fellow human beings, I'd say we're right out in front waving a flag and waiting for everyone else to get on the bandwagon.
# 10
An Open Letter to Rich People
Dear Rich People,
I know that nobody understands you. I want to help. I have, for all my faults, always been rather compassionate to people who are in real pain.
I know that you understand what I mean when I say that sometimes I feel so unappreciated that I just can't be bothered to care. See? There. I feel your pain.
So to make it easier, I have some observations, some advice. Because if there's anything a poor person knows about, it's how to survive in this fucked-up world.
And seriously? You people are doing it wrong.
##
## 1. WORK
What is it with you people and your meetings? I've been allowed to sit in on a few of them recently. I don't know how you stand them. Suddenly, the insane rules you people make us live with seem inevitable. See, until I started sitting in on the meetings, I couldn't see a single reason for programs that had contradictory rules or relief programs that were practically inhumane in their lack of realism. Now I realize it's because every meeting results in nobody having a clue what they've actually done. They've been devoting only 10 percent of their brains to the meeting itself, the remainder being occupied with fantasies of mayhem and whatever song they last heard. Here are my observations from one such meeting:
* I'd have been fired from all my regular jobs if I made my bosses repeat themselves this much.
* WE HAVE BEEN OVER THIS SO MANY TIMES ALREADY!
* If time is money, how does this world function?
* Holy balls, the flattery.
* So many people not paying attention right now.
* Why are they reading the handout to us? I think everyone here can read.
* Is it possible that there is actually no point to this meeting?
I'm not kidding, rich people. You can email me for a copy of the notes.
I've just been sitting through meetings wondering when the work would get done. What I have discovered is this: In every one of them, someone opens by talking about how we want to be respectful of everyone's time and right to speak, along with a plea to keep comments short. Then everyone sort of tunes out while the agenda is being read. Some talking is done by whoever is running things, mostly follow-up from the last meeting.
Then the fun begins. Someone will rise, bring up a good point. Someone else will clarify. A third will ask a relevant question. And then—and here's the part that gets me every time—someone will ask a question that makes it perfectly clear that they weren't paying the slightest bit of attention during the last ten minutes or so. _And nobody calls them out for it_. As long as the question or observation is worded just a little differently, it counts as a new contribution. What the fuck, rich people? Time is money, unless that time is being spent repeating things that have been established already?
Worse are the endless reassurances. "I don't want you to think I'm opposed, because it's a fantastic idea you had to buy ten crocodiles and set them loose in a school as a publicity stunt, but I just don't think it'll work for us." Why on earth do you people not just tell each other when your ideas suck? Why the self-esteem dance? You guys, you're allowed to have bad ideas and irrelevant points. It does not make you a terrible human being. Maybe you should just accept that and then you don't have to cover any criticism, even the most gentle, in five minutes of apology. Maybe we could borrow some of your apology time for our workdays, and then both of our problems would be solved.
By the end of the meeting, which inevitably has run over by at least twenty minutes, nobody is entirely sure what's been accomplished, but everyone feels like their concerns were heard. I have come to the conclusion that business meetings are like group therapy for the wealthy. Everyone sits around looking at each other and waiting for it to be their turn to speak so that they can zone out for the remainder of the time they aren't allowed to leave the room.
The meetings are what made me realize that you guys slack off at work too. It's just that you don't call it slacking off (and that you all have office doors to close so no one can see you playing solitaire or shopping online).
So, rich people, now that we've established that your work ethic and approach to your job are not exactly unassailable, how about you get off your high horse about how we poor people do our jobs? Also:
* Please stop equating our jobs. I am not saying that you put in no effort, that you're not tired or overburdened or anything. I just think that we should delineate between the jobs where you can pee at will and the ones where you can't.
* For the love of God, please stop telling us that outrageous salaries are justified because some people are just worth that much. You guys can totally pretend that anyone can possibly justify earning thousands of dollars every minute. Just stop demanding that we pretend with you, that's all. You guys are supergood at excluding us from conversations. Maybe make that one of them. Just let me know when you start gossiping and I'll rejoin the conversation. I bet someone got laid.
* Maybe you could hire us? I hear rich people complaining about being overworked. I hear poor people complaining about being unemployed. I feel like there's a solution here. You know we work cheap, right? You could totally pay me $10 that one time to run your errands for you or write that standard report that's a pain because it's such rote work. _We are highly trained in rote work._
## 2. CIVICS
This is a big one for me. See, civics is the study of citizenry, its burdens and responsibilities and privileges. It's more than whether or not you, as a class, vote frequently. It's about whether or not you'd want to live in the nation you've created; if you were born tomorrow into the lower classes, would you be quite so sure that America is the land of opportunity? (See what I did there? That's _philosophy_. __ I am trying to speak in your language here, rich people. Because I care deeply about how your day is going today.)
Do I think rich people are highly hypocritical in this area. Um, yeah. Shall I delineate further?
* If you're the makers, _what do you make_? __ I make food and fill boxes and exchange goods for money. Please find a different word, rich people, besides _makers_. Maybe you could try "magicians," because you can create money where there was none before. And then please teach me how to do that too.
* I know it's a pipe dream, but maybe you guys can just admit that we all get shit (see entitlements, roads, tax credits, crop subsidies, fire departments) from the government and move on with your lives?
* It's relatively easy to keep a neighborhood looking nice if the local government actually maintains the roads and medians and signs. If they are too busy making sure that the already-nice sections of town stay that way, they do not have time to come improve the not-nice parts. This is why we laugh when you wonder why we live in run-down areas. It's because when public service cuts happen, they never happen in the bougie neighborhoods. You should know that, given that it's being reported in all your media outlets.
* Your dogs do not belong in restaurants even if they are supercute. I swear to God, the number of tiny dogs I've seen in inappropriate places is at least ten times higher than the number of times I've gotten laid in my life. And, newsflash: Only service animals are allowed in restaurants. That's actually a public health concern. I don't get why you're allowed to decide you're completely above the law simply because you found a purse to fit your dog into.
## 3. ATTITUDE
So, okay, sometimes I have a shitty attitude. I'll give you that. But at least I'm not often entitled. People in the upper classes are so used to having everything done for them that they get sort of irrational and start to feel like you're personally attacking them for not being honestly pleased to see them. It's a bit off-putting, to say the least, to have someone sweep in like that.
* If you think poor people are entitled, try denying a rich person with an attitude some service they think they've earned. It's like grief—there are phases. Anger and denial are first. Then comes "do you understand how fucked you are if I don't get the thing I want?" Followed by "I demand to see your manager" and "I've never been treated so poorly in my life." The final stage is bargaining, where they try to give you extra money because all of life is like valet service to them, and an extra five bucks can change the world.
* If that doesn't convince you, try wearing stained or unintentionally torn (professionally torn is fine and thus useless for these purposes) clothes and sitting on a stoop somewhere. Note how many rude comments or nasty glares you get from well-dressed people. Being rich is like being white, you guys. It's not that sometimes your life doesn't suck even if you're white. It's that you're not allowed to complain about the two times being white is unhandy, because all of your alternatives are much unhandier. Your other options are any race or ethnicity but white, all of whom face normal human shitty existence _and_ racism of the entrenched or overt variety. It's the same thing being rich. I'm not saying that sometimes you don't get the short end of the stick. All I'm saying is that you look ridiculous whining about how you just can't make ends meet on $200,000 because you have to spend so much money to survive. You come off as petulant and incapable of managing the slightest taste of reality when the raising of the capital gains tax back to what you paid under Clinton is cast as a brimstone-filled apocalypse. Sometimes you just have to bite your tongue and keep your mouths shut to avoid looking like assholes.
* Barack Obama caused a flap because he told rich people that they weren't the sole factors in their own success. You are not allowed to do that, because wealthy people are far too precious to face the idea that they didn't do it all themselves, or spring out of the womb, fully formed, as hotshot entrepreneurs or whatever they want us to see them as. I cannot fathom actually thinking that the entire world must collaborate to hide reality from me, and on top of that hubris, being upset when someone dares to speak a distasteful truth. You guys have got to get tougher than that.
## 4. HEALTH
I have no idea what a wealthy person's health care experience is typically like. I've never had that. But I do know that some of the things I see more comfortable folks doing look pretty stupid, and I tend to trust the people with the advanced degrees and years of experience when it comes to how things like cars or bodies work. At least I do if what they want me to do is reasonable and attainable. I only ignore the stuff that's out of reach. You guys, though—seriously, why even bother going to the doctor at all if you think you know everything?
* I am so sorry, rich people. It has to suck to have enough money to stay healthy, because then you don't have an excuse for aging. You have to _maintain_.
* On the other hand, some of the shit you people will pay for blows my mind. Like lotion with actual pearls ground up in it. Actual. Pearls. I stopped at a mall cart to ask about the stuff. It's obscenely expensive. I think that's because you're literally smearing semiprecious materials all over your face.
* You seriously need to control yourselves with the surgical anti-aging. You're starting to look . . . weird. At least we in the lower classes rarely have to live with botched plastic surgery. Very few poor women have someone over-collagenate their lips or paralyze their foreheads. Poverty has its privileges, and one of them is not having to worry about where the line between beauty standard and malpractice lawsuit is.
* We use home remedies because they are cheap, not because they are superior to all of Western medicine. If you can afford a real doctor and you prefer an herbalist, you have lost all sense of reason.
* You guys pay people actual cash money for the privilege of becoming physically exhausted. Has it occurred to you all that you could probably run, for free, on the streets—that you do not actually need to pay money to a gym for the privilege of running on a treadmill? I said that to a wealthy woman once, and she told me that she preferred to work out in air-conditioning. It is possible that I am fundamentally misunderstanding something here, but I thought that sweating was a _good_ thing when you're trying to lose weight?
* Concierge doctors. I am totally cool with people having on-call physicians. But I do think it makes you look like assholes to have your own special VIP offices. Doctors do that, you know; they have a regular office and waiting room for regular patients, and a swanky spa setup for the boutique patients. _It is the same doctor._ You are not getting the benefit of more expertise, he's just kissing your ass more in a slightly more refined setting. If you make the (valid) argument that you get more time, as well, I will just say this: Can you please hire special nurses to listen to your worries about this discolored spot you just discovered on your arm? There are already not enough doctors to go around. I promise you, a talented nurse is as good as a doctor in most cases.
## 5. COPING
I am certain that you have stress, rich people. Nobody's life is perfect. I am equally certain that your stress and my stress are only similar in that they are called the same thing. I take plenty of shit for my habits and vices; what I simply cannot stand by and allow to happen is for you to escape with no notice. I am sorry, guys, but I'm forcing you out of the human closet.
* You know who smokes? Rich people and poor people. You know what that means? _Rich people smoke too._ I'm not kidding, I've seen them at it. I even loaned my lighter to a couple of them, just so I could touch their hands and verify that they weren't holograms or something. With as much shit as I've taken in my life for having such a nasty, wasteful, stupid habit, I'd assumed that wealthy people would be much too good for something so déclassé. But nope, they're on the streets getting cancer with the rest of us. I think I'm done hearing about why poor people smoke. I don't know, why do rich people smoke? I'm willing to bet that our rationales are pretty fucking similar.
* You guys look pretty ridiculous talking about our drug and alcohol use while swanky rehab centers are doing a thriving business. It might behoove you to just admit that addiction is terrible and can hit anyone; otherwise we're probably going to have to start pointing out your raging prescription drug abuse problem. And you wouldn't want that; as it turns out, it's kind of embarrassing when people accuse you of copious drug use.
## 6. SEX
Tell me, how many of you were virgins when you got married? So, our sex lives are up for discussion how again? For all the concern about underprivileged people fucking with reckless abandon, you guys sure don't seem to hold yourselves to a higher standard.
* I know this argument has been made everywhere. But it's valuable. So here it is: You cannot cut access to birth control and then act surprised when people get pregnant. I am fairly certain that few wealthy people walk around with that infamous cheap aspirin between their knees. Poor people are allowed to fuck sometimes too! And we do! Because we're human! _Just like you!_
* You really need to start using condoms or something. Your STD rates are pretty much the same as ours. It's hard to listen to you guys on public health issues when you're getting the clap as often as we are.
* I know that we, the lower classes, tend to speak more frankly and openly than you guys do, as we lack a proper sense of rich-person propriety. So it is very possible that you do not know much about BDSM, and that would explain the success of the _Fifty Shades_ franchise. But I worry about you without any plainspoken poor people to tell you what's what, so please listen closely: _You need a safe word._ Do not, rich people, attempt bondage on your own. Please find a high-end sex club for your wanton romps.
## 7. PARENTING
I disapprove of about as many of the upper class's child-rearing habits as they do of mine. Rich and poor are different, you see, and as such, we value different things. I have trouble with the way you're raising your kids. They're not all special precious unicorns, destined to cure cancer. And if you tell them that they are, they feel entitled to act as though it were true.
You can stop this cycle, rich people. Just teach your kids that they're human like everyone else. Maybe a special snowflake, but one that will still get in trouble if they misbehave on the playground. I have faith in your ability to heal the next generation. I am counting on you, rich people. Don't let me down.
* One word: nannies. You cannot call anyone out on their parenting skills if they're doing as much of the parenting as you are—or more of it. It's great that you hired someone with advanced degrees and multiple languages to sing Junior to sleep—more power to you. But I don't see the difference between hiring a nanny or two so you can attend to the rest of life and dropping your kid with a sitter for the same reason. _It's the same thing._
* And the kids' accessories! I know I already talked a bit about this, but how much shit do you actually think an average toddler really needs? I have a weakness for bouncy balls and coloring books, and my kids get a ton of those. You know what they don't have? Anything that says Giorgio Armani on it. Because it's fucking silly to put designer anything on a kid.
* We feel bad for your kids, rich people. Your kids aren't allowed to be kids. Your kids have tutors by the time they're three and start taking standardized tests in preschool. Your kids have parents who seriously think it's a bad idea to just let them play with sticks and rocks, who think that's actually objectively bad parenting. Loosen up a bit. They'll survive it, and so will you.
* I promise you, you don't need a titanium stroller. You just don't. I thought I had the Range Rover of strollers when I got a normal-size one instead of the folding metal-pipe travel kind. But then I recently spent some time in upscale neighborhoods, and I realized that I had been wrong. I'd had the midsize stroller; the super-big ones come with not just a place for your kid but a place for your groceries and an attached activity center for Junior and wheels with extra shocks. I had the perverse impulse to ask a woman how much hers had run, and she told me. After that, I am assuming that this stroller also picks up the dry cleaning and will murmur sweet nothings into your ear on command. I've bought cars for less than half what an expensive stroller runs.
* Science disapproves of your antibacterial-spray fetish. Kids need to develop immunities, you see, which they do partially from coming into contact with germs. Not to mention, you're actually creating superbugs, bacteria that are resistant to our killing methods. I'm gonna be pissed if I get some superflu because you were afraid Johnny might catch a cold, that's all I'm saying.
* I am seriously disappointed in you for bringing back measles with the anti-vaccination kick. And whooping cough. Get on that, rich people. You need to self-police. Seriously, guys, I'm a mother. I understand wanting to protect your children. All I'm saying is that maybe, you could protect the kids from the mumps. Maybe we can start there.
## 8. PRACTICALITIES
I hope that at this point you are feeling like maybe you hadn't thought this whole stratification thing through all the way. You guys don't really ever talk to us and have no idea what our daily lives are like. But we watch and notice what you do when you are politely ignoring us. And I have some parting words of wisdom: When you think of your stacks of cash, remember that they are gifts, simple things put into your lives to make them easier. You get to have those things. Fucking enjoy them or pass them to the left, man.
* You guys completely take the little things for granted. If you are sleepy while you are driving, you just pull over and find a hotel. If your car breaks down, you call a shop. If you are sick, you go to a doctor. If you break a heel, you get a new pair of shoes. Appreciate that, assholes.
* Money doesn't buy happiness. It buys ease. You can make your life pleasant and enjoyable, get yourself a decent mattress and thus a decent night's sleep. Will it make you happy? Not a chance. But it doesn't hurt.
* If you guys are so good with money, then what do financial planners do? Put another way, maybe you're good with money because you're paying someone to sort out the details?
* Warranties are awesome. They only come on things you buy new. This is why all our shit is broken and yours isn't; you get a grace period after you buy something in which you can be pretty sure you won't have to buy it again, because if it breaks it's under warranty.
* As long as you keep holding me accountable for not making it when I was well under the national median income, I'll hear no whining about how difficult it is to find good help. (Pro tip on the help, rich people: Treat us fairly, pay us decently, and make it clear that you give half a fuck whether we live or die. We'll kill ourselves for you.)
And there you have it, rich people. I hope it helps.
#
Afterword
You've got a thousand more questions than you did when you started the book, don't you? When did we start reliving the Gilded Age? What do you mean they can fire you for no reason? Why bother trying at all if poor people are so fucked from the start?
Well, because we don't have an option. Millions of people every day aren't feeling particularly hopeful that today will be the day it all turns around—but we still look for a job that's marginally better than what we've got. Just in case. When all of your options are as bad as the next, you take your pick and, yes, you hope for the best. Sometimes those decisions turn out to be less than great. Occasionally that's on me. I'm only human, after all, and I make mistakes. But as often as not, the poor outcome was destined from the start. You can't choose between a terrible option and a dreadful option and come out of it whistling a happy tune. You can try to dismiss my depiction of poverty as being representative of just one person's experience, but I am not an aberration. Millions of people have had to shake their asses for Wal-Mart.
Hopefully that last paragraph answered some of your questions. I'm sorry that I don't know the answers to all of them. But I know exactly how you can find out: Ask someone.
There are poor and working-class people everywhere, guys. You can just have a conversation with one, like a real human being. Give it a try. You'll like it. We're entertaining. We have to be; we're stuck entertaining each other because cable is ridiculously expensive.
I don't claim to be an expert. I don't know what we do to solve the problems of stratification. What I do know is that we can and have to do better than this. We're so far behind the curve on these issues that we're having a public fight about whether or not the poor are too comfortable. (Hi, Paul Ryan!) It's not fucking pleasant to be poor. It's not a free ride, a gentle swing in the hammock. It's what's left when you've lost everything, when you're fighting to survive as opposed to fighting to get ahead.
If you feel that something must be done before the villagers find their pitchforks, here is what you can do: Stop being a dick to service workers whenever possible. Start filling out those stupid surveys when someone's done their job well, because they really do make us get a quota of them. Stop pretending you're doing us a favor or performing some high moral duty by refusing to tip. And start admitting that you need us as much as we need you.
And the next time you feel as though you're shouldering more than your fair share of society's burdens, ask yourself: How badly do I have to pee right now, and do I need permission?
# Acknowledgments
Mollie Glick, at Foundry, decided to be my agent and I wish her nothing but best-sellers in the future. I additionally hope that the next person she decides to make into an author has more idea what she is doing than I did. Amy Einhorn has a wicked sense of humor and is an amazing editor, and any praise you care to name should go in her direction. Thankfully, she put me in touch with Peternelle van Arsdale, who not only knows where to find good food but is adept at pulling half-formed thoughts from your brain and turning them into sense. Rodney Staton deserves thanks for patient questioning and teaching while I tried to get my brain in order.
I'd also like to thank:
Sara Benincasa, for keeping me posted and sending me into the best sales pitch in history; Alexis Welby, for being incredibly patient with me in general and also for an insane amount of stress tolerance; Kirsten Neuhaus, for coordinating details through time zones and making it work somehow; and all the people at Foundry who worked on my stuff that I don't even know about. Emily Brown and Katie Grinch, for taking my calls even when I had that tone and emailing me things endlessly when I lost the last thing in my inbox. And the people at Penguin: Ivan Held and Kate Stark, Andrea Ho and Lisa Amoroso, Linda Rosenberg, Meredith Dros, and Maureen Klier, as well as all the people I don't know to name, because all of you spent time making this thing come together. I won't pretend to have a clue how, but I really appreciate it. Finally, Liz Stein, who picked up the baton and ran with it like a pro.
Barbara Ehrenreich, who spoke for me without knowing it years ago, and whose encouragement came at just the right time.
John Oliver and Andy Zaltzman, for Hotties from History.
To everyone I have met along the way: You are all amazing in some way. I'm sorry for the times I have not been my best self, and grateful for the times you have been yours. Mostly, I am probably glad to have met and hung out with you. Four of you can seriously go fuck yourselves.
To my parents: Thank you for making me read. That getting-me-to-adulthood-alive thing was pretty hairy. I mean, looking back, _I'd_ have put a leash on me too. Sorry about the tattoos. I'm still not ruling out another one. And to my children: I damn well waited until _I_ was eighteen. You'll rule everything out until I'm not legally responsible for your stupidity. I love you, but sadly for you, I love you too much to let you be stuck at seventeen forever. That would be hell.
Nancy Stalnaker, Crystal Corrigan, and Jacob Leonard, for things they know about as well as general awesomeness: You're all ninjas. Ryan Clayton: The inscription was right. I can't say it better than that. Brianne Grebil: You renewed some much-needed faith in humanity. Thank you for random awesomeness.
Tom: I don't even think there are words. Thank you for giving me the time I needed to write in, keeping the kids from destroying my work, and insisting on silly cartoons when I needed them. You're the best, and the Independents will be on my playlist until I die.
Chritter, Slay Belle, and all the other mods in the places I was hanging out last fall: You're the best. Internet people in general: I have learned more about the world from interacting with you in the last few years than I had in my entire life. If ever I conduct myself correctly and with grace, it's because I'm thinking of the stuff you all had the patience to teach me. And if ever I land a hell of a one-liner, it's because I learned from the best.
Finally, to everyone who has read this and known exactly what I was talking about: You have earned more than you think you have. It is your right to demand it, and you do not need to ask for favors. I hope that you get a decent gig and get on top of things soon. You work for your paycheck, but you have earned dignity and respect. That is yours, and fuck anyone who tries to tell you otherwise.
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| 9,254
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\section{Introduction}
The diffusion--reaction system with two initially separated diffusing particles of spices $A$ and $B$ reacting according to the formula $m'A+n'B\rightarrow P({\rm inert})$ has been intensively studied during past years \cite{11,12,17,13,14,15,ckd,ara,18,19,cd,22,24}. As the diffusion-reaction equations describing the system are nonlinear, it is difficult to solve them and their general solutions remain unknown (except of very special cases). Thus, to simplify the calculations one usually uses various approximations, such as the quasistationary approximation \cite{11,12,17}, the scaling method \cite{11,13,14,15,ckd}, or the perturbation one \cite{18,19}. Using these methods, there were derived characteristic functions of the system which include: the time evolution of the reaction front $x_f(t)$, the width of the reaction zone $W_{\rm R}(t)$ or the width of the depletion zone $W_{\rm Dep}(t)$ \cite{11,12,13,14,15} which all appear to be the power functions of time $f(t)=K t^\gamma$. The results were confirmed by numerical calculations and simulations \cite{14,15,17}. However, as the methods of extracting the power functions are not based on analytical solutions of subdiffusion-reaction equations (not even on their approximately forms) the proportionality coefficient $K$ is unknown. The coefficient carries dynamical information about the system e.g. how the diffusion coefficient influences the process. As far as we know, there were only a few attempts to determine of $K$ by means of the quasistationary approximation \cite{12,17}.
The situation is more complicated in the case of the subfiffusion system since the equations describing the system contain a derivative of fractional order.
Subdiffusion occurs in systems where mobility of particles is
significantly hindered due to internal structure of the medium, as in porous media or gels \cite{mk,kdm}. The subdiffusion is characterized by a time dependence of the mean square displacement of transported particle $\left\langle \Delta x^2\right\rangle=2D_\alpha t^{\alpha}/\Gamma\left(1+\alpha\right)$, where $D_\alpha$ is the subdiffusion coefficient measured in the units $m^2/s^{\alpha}$ and $\alpha$ is the subdiffusion parameter which obeys $0<\alpha<1$. For $\alpha=1$ one deals with the normal diffusion.
Since no explicit solutions of the nonlinear (sub)diffusion-reaction equations are known, one commonly considers a simplified system, for example a one in which diffusion coefficients of both reactants are assumed to be equal to each other. There is assumed that the some characteristic functions are the same in the system with any simplified assumptions \cite{27a,10}.
The problem is to choose a method to study the subdiffusion-reaction equations. The scaling method dose not allow one to determine $K$ unless special conditions are taken into account. The perturbation method is of small efficiency because the first order correction is often insufficient, while the higher order corrections are hard to obtain even in the case of normal diffusion. So, there are a few problems solved using this method \cite{18,19}. An alternative method is the quasistationary one. In the case of normal diffusion-reaction system it is based on the assumption that process proceeds so slowly that changes of concentration of transported substance are small in some regions \cite{12,17}. Since the subdiffusion process is significantly slower than the normal diffusion one, we expect that the quasistationary approximation is also applicable to the subdiffusive case. Thus, we adopt the method in this study. The scaling method and the quasistationary approximation one are often treated as equivalent to each other. We note however that the equivalence holds only in the long time limit \cite{10}. At shorter times applicability of the quasistationary method does not imply applicability of scaling one and vice versa (this problem will be discussed in \cite{kosztlew}).
In this paper we find that the time evolution of the reaction front is given by the formula $x_{f}(t)=K t^{\alpha/2}$
for a system with arbitrary non--zero values of the subdiffusion coefficients. The coefficient $K$ fulfills the special equation derived in this paper. Our analytical results are confirmed by the numerical solutions of subdiffusion--reaction equations.
\section{\label{sys}The system}
A real system is usually three--dimensional, but we assume that it
is homogeneous in the plane perpendicular to the $x$ axis.
Therefore, we involve only one space variable $x$ into considerations. The subdiffusion--reaction equations are
\begin{equation}\label{eq2}
\frac{\partial }{\partial t}C_{i}(x,t)=D_{i}\frac{\partial^{1-\alpha}}{\partial t^{1-\alpha}}\frac{
\partial^{2}}{\partial x^{2}}C_{i}(x,t)-d_iR_{\alpha}(x,t),
\end{equation}
where $i=A,B$, $C_{i}$ denotes the concentration of the diffusing
particles of species $i$, $D_{i}$ -- the subdiffusion
coefficient, $d_A=m$, $d_B=n$, the parameters $m$ and $n$ occur in the reaction term (Eq.~(\ref{eq4}) below); the Riemann--Liouville fractional time derivative is defined for the
case of $0<\alpha<1$ as
\begin{displaymath}
\frac{d^\alpha f(t)}{dt^\alpha}=\frac{1}{\Gamma(1-\alpha)}\frac{d}{d t}\int_{0}^{t}d\tau\frac{f(\tau)}{(t-\tau)^\alpha}.
\end{displaymath}
Throughout this paper we assume that both of the reactants are mobile $D_{A},D_{B}>0$. Let us note that the choice of the reaction term is not obvious \cite{9,27a,10,sbsl,s,hw,10a}. The reaction term, which we involve into considerations and which was used to study the subdiffusion--reaction system in \cite{27a,10}, is
\begin{equation}\label{eq3}
R_{\alpha}(x,t)=\frac{\partial^{1-\alpha}}{\partial
t^{1-\alpha}}R(x,t) ,
\end{equation}
where the term $R(x,t)$ within the mean field approximation reads
\begin{equation}\label{eq4}
R(x,t)=kC_A^{m}(x,t)C_B^{n}(x,t),
\end{equation}
$k$ is the reaction rate and the parameters $m$ and $n$ are determined experimentally.
We assume that the particles of reactants $A$ and $B$ are initially separated from
each other. Thus, the initial conditions are
\begin{eqnarray}
C_A(x,0)&=&\left\{ \begin{array}{cc}
C_{0A}, & x<0 \\
0, & x>0
\end{array} \right.,\label{eq9a}\\
C_B(x,0)&=&\left\{ \begin{array}{cc}
0, & x<0 \\
C_{0B}, & x>0
\end{array} \right. .\label{eq9b}
\end{eqnarray}
It was observed \cite{11,12,17,13,14,15} that when
the process starts, there appear characteristic regions (see
Fig.~\ref{Fig.1}): the depletion zone ${\rm `Dep'}$, which is
defined as a region where the concentrations are significantly
smaller than the initial ones ($C_A\ll C_{0A}$ and $C_B\ll
C_{0B}$), the reaction region where the production of
particles $P$ is significant ($R(x,t)>0$), and the
diffusion region ${\rm `Dif'}$,
where the reaction term $R(x,t)$ is close
to zero and the particle transport appears to be almost subdiffusive
(i.e. without chemical reactions).
\begin{figure}[h!]
\centering
\includegraphics[scale=0.95]{Kosztolowicz_1.EPS}
\caption{\label{Fig.1}Schematic view of the system under
considerations; $x_{f}(t)$ is the reaction front, ${\rm `Dep'}$ and
${\rm `Dif'}$ denote the depletion zone and the
diffusion region, respectively.}
\end{figure}
For the normal diffusion the widths of the depletion zone $W_{{\rm
Dep}}$ and the reaction region $W_{{\rm R}}$ grow as the power
functions of time \cite{11,12,17,13,14,15,ckd}, $W_{{\rm Dep}}\sim t^{\theta}$, with $\theta=1/2$, and $W_{{\rm R}}\sim t^{\mu}$, where $\mu<\theta$. The value of parameter $\mu$ depends on the system under study.
For the system where the reactants $A$ and $B$ are mobile, there is
$\mu=1/6$ and for the system with a mobile reactant $A$ and a
static reactant $B$ we have $\mu=(m-1)/2(m+1)$, where $m$ is the parameter occurring in the
reaction term $R$ in Eq.~(\ref{eq4}) \cite{13} (see also
\cite{ckd,cd}). As was reported in \cite{ara,10}, $W_{\rm R}$ evolve in time according to the power functions also for the subdiffusive--reaction system with $\mu =\alpha/6$.
An important characteristics of the system under consideration is the time evolution of the reaction front $x_{f}(t)$. It is defined as a point where the reaction term
$R(x,t)$ reaches its maximum $R(x_{f}(t),t)=max$ or, as argued in \cite{14}, for $R\sim C_AC_B$ it is defined by the relation $C_A(x_f(t),t)=C_B(x_f(t),t)$ and in more general situation by $C_A(x_f(t),t)/m=C_B(x_f(t),t)/n$ \cite{ckd}. Unfortunately, these definitions are difficult to apply for the numerically obtained concentrations. In the following, we use the definition of the reaction front as
\begin{equation}\label{defxf}
x_f(t)=\frac{\int x R(x,t)dx}{\int R(x,t)dx} .
\end{equation}
Although the relations defining the reaction front are not always equivalent to each other, all of them provide to $x_f$ lying inside the reaction region, and in the long time limit the definitions lead to the power functions of time.
For the normal diffusion there is the dependence \cite{11,12,17,13,14,15}
\begin{equation}\label{xfg}
x_{f}(t)\sim t^{\gamma},
\end{equation}
with $\gamma=1/2$. It was shown in \cite{10} by
means of the scaling method that the relation (\ref{xfg}) with $\gamma=\alpha/2$ holds for
the subdiffusive system where the subdiffusion coefficients of
reactants are equal to each other.
\section{Quasistatic approximation}
The quasistatic approximation assumes that the concentration profile is
a slowly varying function of time in a given region. Thus,
the time derivation is small and consequently, the r.h.s. of (sub)diffusion
equation (\ref{eq2}) is also small in the region. It requires
\begin{equation}\label{r2}
\frac{\partial^{1-\alpha}}{\partial t^{1-\alpha}}\frac{\partial^2}{\partial x^2}C_{A,B}(x,t)\approx R_\alpha(x,t).
\end{equation}
Since the reaction term is relatively large in the reaction zone,
the quasistatic approximation holds in this zone under the condition
\begin{equation}\label{r3}
D\frac{\partial^{1-\alpha}}{\partial t^{1-\alpha}}
\frac{\partial^{2}}{\partial x^{2}}C_{A,B}(x,t)\gg\frac{\partial }{\partial t}C_{A,B}(x,t).
\end{equation}
We note that the condition (\ref{r3}) is fulfilled when the concentration profiles are given in the scaling form \cite{10}.
In the diffusive region where $R_\alpha(x,t)\approx 0$, the quasistatic approximation
is applicable when the concentration is a linear function of $x$, as the r.h.s. of
Eq. (\ref{eq2}) then vanishes. The regions outside the reaction zone, where the
concentration linearly varies with $x$, determine the borders of the quasistatic
region. The solution of subdiffusion equation without chemical reactions works here.
In the studies of the normal diffusion with reactions, one introduces the quasistatic approximation referring to the equilibration time $\tau_{\rm F}$ \cite{12,ckd,cd}. The parameter $\tau_{\rm F}$ is of order of the average time which is needed for the substance to spread over the interval of length $W_{\rm R}$ when the substance flows from outside of the interval.
For the normal diffusion-reaction system this parameter was estimated from the relation $\left\langle\Delta x^2\right\rangle\sim t$. Taking $\left\langle\Delta x^2\right\rangle\sim W^2_{\rm R}$ and $t\sim \tau_F$ one gets $\tau_{\rm F}\sim W_{\rm R}^2$. For the subdiffusive system the relation $\left\langle\Delta x^2\right\rangle\sim t^\alpha$ provides
\begin{equation}\label{tf}
\tau_{\rm F}\sim W_{\rm R}^{2/\alpha}.
\end{equation}
As for the normal diffusion case, let us assume that the relative change of the flux $J$ fulfills the relation $dJ/J=dt/\tau_{\rm J}$ which gives
\begin{equation}\label{tj}
(\tau_{\rm J})^{-1}\sim \frac{d({\rm log}J)}{dt}.
\end{equation}
The balance between the subdiffusion term and the reaction one is achieved when the equilibration time $\tau_{\rm F}$ of the reaction region is negligibly small comparing to the time ($\tau_{\rm J}$) of relative change of the flux in the long time limit. So, the quasistatic approximation is applicable when
\begin{equation}\label{r4}
\frac{\tau_{\rm F}}{\tau_{\rm J}}\rightarrow_{t\rightarrow\infty}0.
\end{equation}
The quasistatic region is usually defined as a region where at least one of the conditions (\ref{r2}), (\ref{r3}) or (\ref{r4}) is fulfilled. As far as we know, the equivalence of these definitions have not been proven yet. In our considerations we use the relation (\ref{r2}) as the definition of quasistatic approximation and we further show that the conditions (\ref{r4}) is fulfilled when Eq.~(\ref{r2}) is assumed.
\section{Time evolution of $W_{{\rm R}}$ and $W_{{\rm Dep}}$}\label{wrwd}
As in the normal diffusion reaction system, to find the widths of appropriate region we assume that the parameters $p=D_B/D_A$ and $q=C_{0B}/C_{0A}$ are the irrelevant parameters of the system. This means that the results obtained for $p=1$ and/or $q=1$ are qualitatively equivalent to the one with $p\neq 1$ and/or $q\neq 1$, except of very few obvious cases (for example, when $p=q=1$ the reaction front does not move). In this section we derive the time evolution of the widths of the reaction region $W_{\rm R}$ and the depletion zone $W_{{\rm Dep}}$ under condition $p=q=1$.
At first we argue that the assumption $\mu<\theta$ is correct not
only for the diffusive but for the subdiffusive systems as well.
In \cite{10} there was found
that $\theta = \alpha/2$ and $\mu = \alpha/6$ by means of the simplified scaling method. We confirm the above relation, using the method
already applied to the normal diffusion--reaction system
\cite{24}.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.95]{Kosztolowicz_2.EPS}
\caption{The symmetrical system with static reaction front
$x_{f}(t)=0$. The continuous lines denote the concentrations $C$ for
the system under considerations, the dashed one -- for the system
with the fully absorbing wall $C_{{\rm abs}}$ located at $x=0$.}\label{Fig.2}
\end{figure}
Let us assume that the initial concentrations and subdiffusion
coefficients of both reactants are equal to each other $C_{0A}=C_{0B}\equiv C_{0}$ and $D_{A}=D_{B}\equiv D$.
Due to the symmetry of the system, the reaction front will not
change its position $x_{f}(t)=0$. Proceeding as for the system with the normal diffusion
\cite{24}, we assume to further simplify of the calculations that
the concentrations of $A$ and $B$ particles can be given as
\begin{displaymath}
C_A(x,t)=C_{A \rm abs}(x,t)+\delta C_A(x,t),
\end{displaymath}
\begin{displaymath}
C_B(x,t)=C_{B \rm abs}(x,t)+\delta C_B(x,t),
\end{displaymath}
where $C_{A {\rm abs}}$ and $C_{B {\rm abs}}$ are the solutions of the pure subdiffusive equation in the system with the perfectly absorbing wall located at $x=0$ and
$\delta C_A$ and $\delta C_B$ are the corrections (see Fig.~\ref{Fig.2}). Symmetry of the
system ensures that $C_A(x,t)=C_B(-x,t)$, which provides $\delta
C_A(x,t)=\delta C_B(-x,t)$. For a perfectly absorbing wall
placed at $x=0$, the concentration profiles vanish at the wall
$C_{A \rm abs}(0,t)=C_{B \rm abs}(0,t)=0$. After calculations, we obtain
\begin{eqnarray}\label{eq27}
\lefteqn{C_{A \rm abs}(x,t)=}\nonumber\\
& & C_{0}\left[1 -\frac{2}{\alpha}H^{1 0}_{1 1}
\left(\left(\frac{-x}{\sqrt{Dt^{\alpha}}}\right)^{2/\alpha}
\left| \begin{array}{cc}
1 & 1 \\
0 & 2/\alpha
\end{array}\right. \right)
\right],
\end{eqnarray}
for $x<0$, and
\begin{eqnarray}\label{eq28}
\lefteqn{C_{B \rm abs}(x,t)=}\nonumber\\
&&C_{0}\left[1-\frac{2}{\alpha} H^{1 0}_{1 1}
\left(\left(\frac{x}{\sqrt{Dt^{\alpha}}}\right)^{2/\alpha}
\left| \begin{array}{cc}
1 & 1 \\
0 & 2/\alpha
\end{array}\right. \right)
\right],
\end{eqnarray}
for $x>0$, where $H$ denotes the Fox function, which can be
expressed as the series \cite{23}
\begin{equation}\label{eq29}
H^{1 0}_{1 1} \left(u \left|
\begin{array}{cc}
1 & 1 \\
p & q
\end{array}
\right. \right)= \frac{1}{q}u^{p/q}
\sum^\infty_{j=0}\frac{(-1)^{j}}{j!\Gamma(1-p/q-j/q)}u^{j/q}.
\end{equation}
Substituting $C(x,t)\equiv C_i(x,t)$, $\delta C(x,t)\equiv \delta C_i(x,t)$, $C_{\rm abs}(x,t)\equiv C_{i \rm abs}(x,t)$, where $i=A$ for $x<0$ and $i=B$ for $x>0$,
to the subdiffusion--reaction equation and taking into account that
$C_{\rm abs}$ fulfills the subdiffusion equation without
chemical reactions, we get
\begin{eqnarray*}
\frac{\partial }{\partial t}\delta
C(x,t)&=&\frac{\partial^{1-\alpha}}{\partial
t^{1-\alpha}}\left[D\frac{
\partial^{2}}{\partial x^{2}}\delta C(x,t)\right.\\
& & \left. -k\left[C_{\rm abs}(x,t) +\delta C(x,t) \right]\delta
C(x,t)\right].
\end{eqnarray*}
The limits of the reaction region occur for $x$ where
$\delta C$ is close to zero. In this region one can neglect the
term $(\delta C)^{2}$ in the above equation. Moreover, in the long
time limit we can approximate the Fox functions present in
Eqs.~(\ref{eq27}) and (\ref{eq28}) by the expression
$C_{\rm abs}(x,t)=a|x|/t^{\alpha/2}$, where
$a=C_{0}k/\Gamma(1-\alpha/2)\sqrt{D}$. So, we get
\begin{equation}\label{eq32}
\frac{\partial }{\partial t}\delta
C(x,t)=\frac{\partial^{1-\alpha}}{\partial
t^{1-\alpha}}\left[ D\frac{
\partial^{2}}{\partial x^{2}}\delta C(x,t)
-\frac{a|x|}{t^{\frac{\alpha}{2}}}\delta C(x,t)\right].
\end{equation}
As in the normal diffusion--reaction system \cite{24}, we assume that
\begin{equation}\label{eq32a}
\frac{\partial}{\partial t}\delta C(x,t)=0 .
\end{equation}
From Eqs.~(\ref{eq32}), (\ref{eq32a}) and the relation \cite{20}
\begin{equation}\label{pochtni}
\frac{d^{\beta}t^{\nu}}{dt^{\beta}}=
\frac{\Gamma(\nu+1)}{\Gamma(\nu+1-\beta)}t^{\nu-\beta},\qquad \nu>-1,
\end{equation}
we obtain
\begin{equation}\label{eq34}
D\frac{
\partial^{2}}{\partial x^{2}}\delta C(x,t)-\frac{a|x|}{t^{\frac{\alpha}{2}}}\delta
C(x,t)=\frac{A(x)}{t^{\alpha}},
\end{equation}
where $A(x)$ is the arbitrary function of $x$ only. In the long time limit the r.h.s. of Eq. (\ref{eq34}) can be neglected. Let us note that it is another justification to equal the r.h.s. of above equation to zero. To determine the function $A(x)$ we observe that $\delta C$ has a significant value in a finite region limited by the
depending on time points $-g(t)$ and $g(t)$, which lie inside of the
depletion zone (see Fig.~\ref{Fig.2}). Thus, we have
\begin{displaymath}\label{eq34a}
\delta C(-g(t),t)\approx\delta C(g(t),t)\approx 0
\end{displaymath}
and
\begin{displaymath}
\left. \frac{\partial^{2}\delta C(x,t)}{\partial
x^{2}}\right|_{x=-g(t)}\approx\left. \frac{\partial^{2}\delta
C(x,t)}{\partial x^{2}}\right|_{x=g(t)}\approx 0 .
\end{displaymath}
Since the left hand side of the Eq. (\ref{eq34}) is close to
zero for $|x|>g(t)$, the additional boundary conditions are
\begin{equation}\label{eq34c}
A(-g(t))=A(g(t))=0 .
\end{equation}
The above equations appear to be the boundary conditions for the
function $A$, which cannot depend on time (in contrary to the
boundary conditions (\ref{eq34c})). Thus, there is the only
solution $A(x)\equiv const\equiv 0$.
Solving the equation
(\ref{eq34}) with the right side equal to zero, we find
\begin{displaymath}
\delta C(x,t)=f(t){\rm Ai}\left(\lambda
\frac{x}{t^{\alpha/6}}\right) ,
\end{displaymath}
where ${\rm Ai}$ denotes the Airy function, which can be
approximated by the following expression for large~$u$
\begin{displaymath}
{\rm Ai}(u)\simeq
\frac{1}{2\sqrt{\pi}u^{1/4}}\exp\left[-\frac{2u^{3/2}}{3}\right].
\end{displaymath}
To obtain the function $f(t)$
we assume that it is the power function of time $f(t)\sim
t^{\lambda}$. Putting the function $f$ to Eq.~(\ref{eq32}) and using
(\ref{eq32a}), we obtain $\lambda=-\frac{\alpha}{3}$. Comparing
Eq.~(\ref{eq32}) with (\ref{eq2}) and (\ref{eq3}), we get
\begin{equation}\label{eq37}
R(x,t)=\frac{a|x|}{t^{\frac{\alpha}{2}}}\delta C(x,t) .
\end{equation}
Substituting Eq.~(\ref{eq37}) to Eq.~(\ref{eq32}) we obtain
\begin{equation}\label{eq38}
R(x,t)\sim
t^{-2\alpha/3}\left(\frac{x}{t^{\alpha/6}}\right)^{3/4}
\exp\left[-\frac{2}{3}\left(\frac{\lambda
x}{t^{\alpha/6}}\right)^{3/2}\right].
\end{equation}
As the width of the reaction region is defined by the relation \cite{ckd}
\begin{equation}\label{defWR}
W_{\rm R}^2(x,t)=\frac{\int (x-x_f(t))^2 R(x,t) dx}{\int R(x,t) dx} .
\end{equation}
it is easy to see that substituting (\ref{eq38}) to (\ref{defWR}) with $x_f\equiv0$, we obtain $W_{{\rm R}}\sim t^{\alpha/6}$.
Since the width of the depletion zone is defined by the conditions
$C_{i}\ll C_{0i}$, $i=A,B$, from Eqs. (\ref{eq27}) and
(\ref{eq28}) we get $W_{{\rm Dep}}\sim
t^{\alpha/2}$. Thus, the relation $\mu<\theta$ is fulfilled for
the system where the subdiffusion coefficients of the reactants
are equal to each other. We assume that this relation holds for the system
with any non--zero values of the subdiffusion coefficients.
\section{Concentration profile in ${\rm Dif}$ region}\label{wdif}
Since $W_{\rm R}\ll W_{{\rm Dep}}$,
the reaction region plays a role of a partially absorbing wall
with respect to the depletion zone. We find the concentration
profiles in the region outside the reaction one as a solution of
the subdiffusion equation in the system with partially absorbing
wall.
We find here the solutions of the subdiffusion equation without
chemical reactions (Eq. (\ref{eq2}) with $R_{\alpha}(x,t)\equiv0$) for the system with partially
absorbing wall. To calculate the concentration profiles, we use the
integral formula
\begin{equation}\label{eq41}
C(x,t)=\int G(x,t;x_{0})C(x_{0},0)dx_{0},
\end{equation}
where $G(x,t;x_{0})$ denotes the Green's function for the
subdiffusion equation. From a macroscopic point of
view, the Green's function is interpreted as a concentration
profile of the $N$ particles (divided by $N$) which are
instantaneously produced and start from the position $x_{0}$ at an
initial moment $t=0$. It is also interpreted as a probability
density of finding a particle in a point $x$ at time $t$ under
the condition that the particle is located in the position $x_{0}$
at the initial moment $t=0$.
There is a problem to set the boundary conditions at the partially
absorbing wall. To obtain the Green's function one can use the
method of images. The standard method of images has been applied for
the diffusive system with fully absorbing or fully reflecting
wall \cite{25}. Then, one replaces the wall by a fictitious
instantaneous point source of the particles (IPS) in such a manner
that the concentration profile generated by all IPS behaves as in
the system with the wall. In the system with the fully reflecting wall,
the flux vanishes at the wall. In this case the Green's function can
be obtained by replacing the wall by the auxiliary IPS of the same
strength in the position symmetric to the initial point $x_{0}$ with
respect to the wall
\begin{equation}\label{eq42a}
G(x,t;x_{0})=G_{0}(x,t;x_{0})+G_{0}(x,t;-x_{0}) ,
\end{equation}
where $G_{0}$ denotes the Green's function for homogeneous system. In the case of fully absorbing wall the concentration vanishes at the wall. The Green's function is
then a difference of IPS placed at $x_{0}$ and $-x_{0}$, which
gives
\begin{equation}\label{eq42}
G(x,t;x_{0})=G_{0}(x,t;x_{0})-G_{0}(x,t;-x_{0}).
\end{equation}
Sometimes the boundary conditions are not given explicitly by an
equation, but they are postulated in a heuristic form. In such a
case there is a possibility to use the generalized method of images
to find the Green's functions. Such a procedure was used to find the
Green's functions for the system with partially permeable wall
\cite{26} where the Green's function was obtained from
Eq.~(\ref{eq42a}) by reducing the IPS located at $-x_{0}$ by the
factor controlled by the permeability of the wall.
For the system with partially absorbing wall we start with a
physical condition, which can be stated as: \textit{if during a
given time interval $N$ particles reach the wall, the fraction
$\rho$ of them will be absorbed while $1-\rho$ will go through}.
The parameter $\rho$ is assumed to be a constant characterizing
the wall. Such a situation appears when the partially absorbing
wall is simulated by another IPS of the strength reduced by a
factor $\rho$. So, the Green's functions are as follows
\begin{equation}\label{eq43}
G_{A\,{\rm Dif}}(x,t;x_{0})=G_{0\,A}(x,t;x_{0})
-\rho_{A}G_{0\,A}(x,t;-x_{0}),
\end{equation}
and
\begin{equation}\label{eq44}
G_{B\,{\rm Dif}}(x,t;x_{0})=G_{0\,B}(x,t;x_{0})
-\rho_{B}G_{0\,B}(x,t;-x_{0}),
\end{equation}
where
\begin{equation}\label{eq45}
G_{0\,i}(x,t;x_{0})=\frac{1}{\alpha|x-x_{0}|} H^{1 0}_{1 1}
\left(\left(\frac{|x-x_{0}|}{\sqrt{D_{i}t^{\alpha}}}\right)
^{\frac{2}{\alpha}}\left|
\begin{array}{cc}
1 & 1 \\
1 & 2/\alpha
\end{array}\right. \right) ,
\end{equation}
for $i=A,B$. Using the integral formula (\ref{eq41}) and initial
conditions (\ref{eq9a}) and (\ref{eq9b}), we find (for details of the calculations
see the Appendix A)
\begin{eqnarray}\label{eq46}
\lefteqn{C_{A\,{\rm Dif}}(x,t)=C_{0A}-\frac{2}{\alpha}\eta_{A}}\nonumber \\
&&\times H^{1 0}_{1 1}
\left(\left(\frac{-x}{\sqrt{D_{A}t^{\alpha}}}\right)^{2/\alpha}
\left| \begin{array}{cc}
1 & 1 \\
0 & 2/\alpha
\end{array}\right. \right),
\end{eqnarray}
where
\begin{equation}\label{eq46a}
\eta_{A}=C_{A0}(1+\rho_{A})/2 ,
\end{equation}
and
\begin{eqnarray}\label{eq47}
\lefteqn{C_{B\,{\rm Dif}}(x,t)=C_{0B}-\frac{2}{\alpha}\eta_{B}}\nonumber \\
&&\times H^{1 0}_{1 1}
\left(\left(\frac{x}{\sqrt{D_{B}t^{\alpha}}}\right)^{2/\alpha}
\left| \begin{array}{cc}
1 & 1 \\
0 & 2/\alpha
\end{array}\right. \right),
\end{eqnarray}
where
\begin{equation}\label{eq47a}
\eta_{B}=C_{B0}(1+\rho_{B})/2 .
\end{equation}
Let us note that when $C_{0A}=C_{0B}=C_0$ and $D_A=D_B$, we obtain $\rho_A=1$ and $\rho_B=1$ from Eqs.~(\ref{eq27}) and~(\ref{eq28}). The subdiffusive fluxes are given by the formula
\begin{equation}\label{eq10}
J_{i}(x,t)=-D_{i}\frac{\partial^{1-\alpha}}{\partial
t^{1-\alpha}}\frac{\partial C_{i}(x,t)}{\partial x}.
\end{equation}
Using Eqs.~(\ref{eq46}) and (\ref{eq47}), we obtain
\begin{eqnarray}\label{eq48}
\lefteqn{J_{A\,{\rm Dif}}(x,t)=\frac{2}{\alpha}\sqrt{D_{A}}\eta_{A}\left(\frac{\sqrt{D_{A}}}{-x}\right)^{\frac{2}{\alpha}-1}}\nonumber \\
&&\times H^{1 0}_{1 1}
\left(\left(\frac{-x}{\sqrt{D_{A}t^{\alpha}}}\right)^{2/\alpha}
\left| \begin{array}{cc}
1 & 1 \\
-1+2/\alpha & 2/\alpha
\end{array}\right. \right),
\end{eqnarray}
\begin{eqnarray}\label{eq49}
\lefteqn{J_{B\,{\rm Dif}}(x,t)=-\frac{2}{\alpha}\sqrt{D_{B}}\eta_{B}\left(\frac{\sqrt{D_{B}}}{x}\right)^{\frac{2}{\alpha}-1}}\nonumber \\ &&\times H^{1 0}_{1 1}
\left(\left(\frac{x}{\sqrt{D_{B}t^{\alpha}}}\right)^{2/\alpha}
\left| \begin{array}{cc}
1 & 1 \\
-1+2/\alpha & 2/\alpha
\end{array}\right. \right).
\end{eqnarray}
In the following we use the shorter notation for the fluxes (\ref{eq48}) and (\ref{eq49})
\begin{eqnarray}
\label{linja} J_{A \rm Dif}&=&\frac{\sqrt{D_A}\eta_A}{t^{1-\alpha/2}}Q\left(\frac{-x}{\sqrt{D_A t^{\alpha}}}\right) ,\\
\label{linjb} J_{B \rm Dif}&=&-\frac{\sqrt{D_B}\eta_B}{t^{1-\alpha/2}}Q\left(\frac{x}{\sqrt{D_B t^{\alpha}}}\right) ,
\end{eqnarray}
where
\begin{equation}\label{defQ}
Q(z)=\frac{\alpha}{2}\sum^{\infty}_{k=0}\frac{1}{k!\Gamma(\alpha(1-k)/2)}(-z)^k.
\end{equation}
\section{Time evolution of the reaction front}\label{terf}
In this section we derive the time evolution of the reaction front within the quasistatonary approximation. The derivation is based on three assumptions, which are expected to hold in the long time limit.
\begin{enumerate}[(1)]
\item\label{zalozenie1} We assume that the characteristic functions evolve in time according to the formulas
\begin{equation}\label{zalWR}
W_R\sim t^{\alpha/6} ,
\end{equation}
\begin{equation}\label{zalDEP}
W_{{\rm Dep}}\sim t^{\alpha/2} ,
\end{equation}
\begin{equation}\label{zalxf}
x_f(t)\sim t^{\alpha/2} .
\end{equation}
The relations were derived in \cite{10} by means of the scaling method for the system where the subdiffusion coefficients of both reactants are equal to each other. The relation (\ref{zalWR}) was also found in \cite{ara} by means of the Monte Carlo simulations. Le us note that in Sec.~\ref{wrwd} we have shown that the relations (\ref{zalWR}) and (\ref{zalDEP}) are fulfilled for the system where $p=q=1$. The relation (\ref{zalxf}) will be also confirmed \textit{a posteriori} in this section.
\item\label{zalozenie2} The region, where the quasistatic approximation works, extends beyond the reaction zone. Therefore there is the region defined by the relation
\begin{equation}\label{defOVE}
W_R(t)\ll|x-x_{f}(t)|\ll W_{\rm Dep}(t) ,
\end{equation}
where the quasistatic approximation region overlaps with the diffusion one.
\item\label{zalozenie3} In the diffusion region the concentrations are given by Eqs. (\ref{eq46})--(\ref{eq47a}) with the parameters $\rho_A$ and $\rho_B$, which can be larger than unity.
\end{enumerate}
Starting with the above assumptions, we show at first the following
\begin{enumerate}[(a)]
\item\label{z1} The concentration profiles (\ref{eq46}) and (\ref{eq47}) extended to the reaction region vanish at the points which are identified with the point $x_z$ defined in Fig.~(\ref{Fig.1}) and by Eq.~(\ref{defxz}). In the long time limit the point $x_z$ is localized so close to $x_f$ that $x_z$ can be replaced by $x_f$.
\item\label{z2} The fluxes $J_A$ and $J_B$ flowing into the reaction region from the left and from the right side, respectively, are assumed to be balanced in such a way that $m$ particles $A$ and $n$ particles $B$ flow into the reaction region in the time unit.
\end{enumerate}
After showing that the conditions (\ref{z1}) and (\ref{z2}) hold, we use Eqs.~(\ref{eq46}) and (\ref{eq47}) to derive a relation describing the time evolution of the reaction front.
As mentioned earlier, we are guided by the procedure already used for the normal diffusion--reaction systems \cite{12}.
The quasistatic state can be defined by the following equations
\begin{displaymath}
\frac{\partial^{1-\alpha}}{\partial t^{1-\alpha}}\left[D_{A}\frac{\partial^{2}}{\partial x^{2}}C_{A}(x,t)-mR(x,t)\right]=0,
\end{displaymath}
and
\begin{displaymath}
\frac{\partial^{1-\alpha}}{\partial t^{1-\alpha}}\left[D_{B}\frac{\partial^{2}}{\partial x^{2}}C_{B}(x,t)-nR(x,t)\right]=0,
\end{displaymath}
which combined provide
\begin{displaymath}
\frac{\partial^{1-\alpha}}{\partial t^{1-\alpha}}\frac{
\partial^{2}}{\partial x^{2}}\Psi(x,t)=0 ,
\end{displaymath}
where
\begin{equation}\label{defPsi}
\Psi(x,t)\equiv\frac{1}{m}D_{A}C_{A}(x,t)-\frac{1}{n}D_{B}C_{B}(x,t) .
\end{equation}
Using the formula (\ref{pochtni}), we find
\begin{equation}\label{eq54}
\Psi(x,t)=E(x)t^{-\alpha}+F(t)x+G(t).
\end{equation}
Applying the operator $\frac{\partial^{1-\alpha}}{\partial t^{1-\alpha}}\frac{\partial}{\partial x}$ to Eq.~(\ref{eq54}), we obtain
\begin{equation}\label{eq55}
\frac{\partial^{1-\alpha}}{\partial t^{1-\alpha}}F(t)=\frac{1}{n}J_{B}(t)-\frac{1}{m}J_{A}(t).
\end{equation}
The function $\Psi$ changes its sign in the
reaction zone from positive where $C_{B}\simeq 0$ to negative
where $C_{A}\simeq 0$. Thus, there is the point $x_{z}(t)$ which lies
inside the reaction zone, where the function $\Psi$ is equal to
zero. Therefore,
\begin{equation}\label{defxz}
\Psi(x_{z}(t),t)=0.
\end{equation}
Since $x_{f}(t)$ also lies inside the reaction zone there is
\begin{equation}\label{gwiazdka}
|x_{z}(t)-x_{f}(t)|\leq\Omega t^{\alpha/6} ,
\end{equation}
where $\Omega$ is a positive constant. After simple calculations, we get
\begin{equation}\label{eq57}
\Psi(x,t)=\frac{E(x)-E(x_{z}(t))}{t^{\alpha}}+F(t)(x-x_{z}(t)) .
\end{equation}
Let us now consider the region where the region of diffusion approximation overlaps with the one of the quasistatic approximation for $x<x_f(t)$. The region occurs for such $x$ that the condition
\begin{equation}\label{defOve}
-W_{\rm Dep}(x,t)\ll x-x_f(t) \ll -W_{\rm R}(x,t) ,
\end{equation}
is fulfilled. Here $C_{A}\approx C_{A \rm Dif}$, $C_{B}\approx 0$, $J_{A}\approx J_{A \rm Dif}$, and $J_{B}\approx 0$. So, we get from Eq.~(\ref{defPsi})
\begin{equation}\label{PsiOve}
\Psi(x,t)=\frac{1}{m}D_AC_{A \rm Dif}(x,t) ,
\end{equation}
and from Eq.~(\ref{eq55})
\begin{equation}\label{fjad}
\frac{\partial^{1-\alpha}}{\partial t^{1-\alpha}}F(t)=-\frac{1}{m}J_{A \rm Dif}(t) .
\end{equation}
Let us note that $\Psi$ is given by the function of the variable $x/t^{\alpha/2}$ only (see Eq.~(\ref{eq46})). Therefore we deduce that
\begin{equation}\label{defE}
E(x)=ax^2 ,
\end{equation}
\begin{equation}\label{defF}
F(t)=\frac{b}{t^{\alpha/2}} ,
\end{equation}
and
\begin{equation}\label{defG}
G(t)=c ,
\end{equation}
where $a$, $b$, $c$ are constants.
We denote
\begin{equation}\label{defx}
x_f(t)-x=\epsilon(t).
\end{equation}
It is obvious that
\begin{displaymath}
\Omega_1 t^{\alpha/6}\ll \epsilon(t) \ll \Omega_2 t^{\alpha/2} ,
\end{displaymath}
where $\Omega_1$ and $\Omega_2$ are positive constants.
When $t\rightarrow\infty$ the inequality provides $t^{\alpha/6}/\epsilon(t)\rightarrow 0$ and
\begin{equation}\label{epsilon}
\epsilon(t)/t^{\alpha/2}\rightarrow 0 .
\end{equation}
Combining Eqs. (\ref{eq46}), (\ref{eq57}), (\ref{PsiOve}) and (\ref{defE})--(\ref{defx}) we obtain
\begin{eqnarray}\label{ala}
D_A\left[C_{0A}-\frac{2}{\alpha}\eta_A H^{1 0}_{1 1}
\left(\left(\frac{\epsilon(t)-x_f(t)}{\sqrt{D_{A}t^{\alpha}}}\right)^{2/\alpha}
\left| \begin{array}{cc}
1 & 1 \\
0 & 2/\alpha
\end{array}\right. \right)\right]&=&\nonumber \\
a\frac{(x_f(t)-\epsilon(t))^2-x_z^2(t)}{t^\alpha}
+b\frac{x_f(t)-\epsilon(t)-x_z(t)}{t^{\alpha/2}} .&&
\end{eqnarray}
Since in the long time limit $(x_f(t)-x_z(t)-\epsilon(t))/t^{\alpha/2}\rightarrow 0$ (see Eqs.~(\ref{gwiazdka}) and (\ref{epsilon})), from Eq.~(\ref{ala}) we get
\begin{equation}\label{albb}
C_{0A}-\frac{2}{\alpha}\eta_A H^{1 0}_{1 1}
\left(\left(\frac{-x_f(t)}{\sqrt{D_{A}t^{\alpha}}}\right)^{2/\alpha}
\left| \begin{array}{cc}
1 & 1 \\
0 & 2/\alpha
\end{array}\right. \right) = 0 .
\end{equation}
Similar considerations performed in the region
\begin{displaymath}
W_R(x,t)\ll x-x_f(t)\ll W_{\rm Dep}(x,t)
\end{displaymath}
provide
\begin{displaymath}
\Psi(x,t)=-\frac{1}{n}D_BC_{B \rm Dif}(x,t) ,
\end{displaymath}
and
\begin{equation}\label{aaaa}
\frac{\partial^{1-\alpha}}{\partial t^{1-\alpha}}F(t)=\frac{1}{n}J_{B \rm Dif}(t) ,
\end{equation}
which gives
\begin{equation}\label{abaa}
C_{0B}-\frac{2}{\alpha}\eta_B H^{1 0}_{1 1}
\left(\left(\frac{x_f(t)}{\sqrt{D_{B}t^{\alpha}}}\right)^{2/\alpha}
\left| \begin{array}{cc}
1 & 1 \\
0 & 2/\alpha
\end{array}\right. \right) = 0 .
\end{equation}
From Eqs.~(\ref{fjad}) and (\ref{aaaa}) we obtain
\begin{equation}\label{rowj}
\frac{1}{m}J_{A \rm Dif}=-\frac{1}{n}J_{B \rm Dif} ,
\end{equation}
and from Eqs.~(\ref{linja}), (\ref{linjb}) and (\ref{rowj}) we get
\begin{equation}\label{rsl}
\frac{1}{m}\sqrt{D_A}\eta_A Q\left(\frac{-x_f(t)}{\sqrt{D_A t^{\alpha}}}\right)=\frac{1}{n}\sqrt{D_B}\eta_B Q\left(\frac{x_f(t)}{\sqrt{D_B t^{\alpha}}}\right).
\end{equation}
Combining Eqs. (\ref{albb}), (\ref{abaa}), (\ref{rsl}) and using the identity \cite{27}
\begin{equation}\label{przekH}
H^{1 0}_{1 1}
\left(z^{2/\alpha}\left| \begin{array}{cc}
1 & 1 \\
0 & 2/\alpha
\end{array}\right. \right)=\frac{\alpha}{2}
H^{1 0}_{1 1}
\left(z\left| \begin{array}{cc}
1 & \alpha/2 \\
0 & 1
\end{array}\right. \right) ,
\end{equation}
we have
\begin{equation}\label{rowfi}
\Phi\left(\frac{-x_f(t)}{\sqrt{D_A t^{\alpha}}}\right)=\frac{n}{m}\frac{\sqrt{D_A}C_{0A}}{\sqrt{D_B}C_{0B}} \Phi\left(\frac{x_f(t)}{\sqrt{D_B t^{\alpha}}}\right) ,
\end{equation}
where $\Phi(z)\equiv
H^{1 0}_{1 1}
\left(z\left| \begin{array}{cc}
1 & \alpha/2 \\
0 & 1
\end{array}\right. \right)/{Q(z)}$.
It is clear that there is only one point $x_f$ which for given $t$ fulfills the definition of reaction front. The solution of the Eq. (\ref{rowfi}) is
\begin{equation}\label{xkt}
x_{f}(t)=Kt^{\alpha/2} ,
\end{equation}
where coefficient $K$ is the solution of the following equation
\begin{equation}\label{rowk}
\Phi\left(\frac{-K}{\sqrt{D_A}}\right)=\frac{n}{m}\frac{\sqrt{D_A}C_{0A}}{\sqrt{D_B}C_{0B}} \Phi\left(\frac{K}{\sqrt{D_B}}\right) .
\end{equation}
Thus, the time evolution of the reaction front is the power
function with the exponent depending on the subdiffusion parameter $\alpha$ only; the subdiffusion coefficients $D_A$ and $D_B$ controll the parameter $K$. Eqs. (\ref{xkt}) and (\ref{rowk}) are the main result of our paper.
The procedure developed in this paper is a extension of the one
already used for the normal diffusion case \cite{12}. Repeating our consideration for $\alpha=1$ we obtain the results identical with those from \cite{12}. Our formula (\ref{xkt}) with $K$ given by Eq.~(\ref{rowk})
is a generalization of Eq.~(21) in \cite{10}.
\section{Numerical solutions}
To verify correctness of our procedure, we compare the analytical functions which are derived in the previous sections with numerical solutions. We show that there exists the quasistatic approximation zone where, as required by Eqs.~(\ref{eq54}) and (\ref{defE})--(\ref{defG}), the function $\Psi$ is parabolic with respect to $x$. We also show that there exists the region of overlap of the diffusion zone and the quasistatic one; in this region $C_{A \rm Dif}$ or $C_{B \rm Dif}$ are the linear functions of $x$.
\subsection{Numerical procedure}\label{np}
As we show in the Appendix B, assuming that the functions $C_{A}$ and
$C_{B}$ and their second derivatives with respect to the space
variable are limited, Eq.~(\ref{eq2}) is equivalent to
\begin{equation}\label{eq14}
\frac{^{C}\partial^{\alpha}}{\partial
t^{\alpha}}C_i(x,t)=D_{i}\frac{
\partial^{2}}{\partial x^{2}}C_i(x,t)-d_i R(x,t),
\end{equation}
where $i=A,B$, $d_A=m$, $d_B=n$, with the Caputo fractional time derivative, which is defined
for $0<\alpha<1$ as \cite{20a}
\begin{displaymath}
\frac{^{C}d^{\alpha}f(t)}{d
t^{\alpha}} =\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}d\tau
\frac{f(\tau)}{d \tau}(t-\tau)^{-\alpha}.
\end{displaymath}
Throughout this paper we denote the Riemann-Liouville fractional derivative without any additional index as $d^{\alpha}f(t)/dt^{\alpha}$, others kinds of the fractional derivatives are labeled by indexes $C$ for the Caputo fractional derivative and $GL$ for the Gr\"{u}nwald-Letnikov one.
In the papers \cite{gmmp,yuste} there were presented the procedures of the numerical solving of the subdiffusion equation without chemical reaction, when one can use the equation with Riemann-Liouville as well as Caputo fractional time derivative. The situation is different in the case of the subdiffusion-reaction equations. In the numerical calculations the fractional derivative is replaced by series. In the case of Eqs.~(\ref{eq2}) and (\ref{eq3}) with Riemann-Liouville fractional derivative we have relatively complicated expression under the derivative, whereas in Eq.~(\ref{eq14}) the fractional derivative acts only on concentrations. It caused that the numerical procedure based on Eq.~(\ref{eq14}) is more convenient to use, at least in our opinion.
To numerically solve the normal diffusion equation one usually
substitutes the time derivative by the backward difference
$\frac{\partial f(t)}{\partial t}\simeq \frac{f(t)-f(t-\Delta
t)}{\Delta t}$. In the presented procedure we proceed in a similar
way. We use the Gr\"{u}nwald-Letnikov fractional
derivative which is defined as a limit of a fractional-order
backward difference \cite{20a}
\begin{equation}\label{gl}
\frac{^{GL}d^{\alpha}f(t)}{d
t^{\alpha}}=\lim_{\Delta t\rightarrow 0}(\Delta
t)^{-\alpha}\sum_{r=0}^{[\frac{t}{\Delta t}]}(-1)^{r} \left(
\begin{array}{c}
\alpha\\
r
\end{array} \right)f(t-r\Delta t),
\end{equation}
where $\alpha>0$, $[z]$ means the integer part of $z$ and
\begin{eqnarray*}
\left( \begin{array}{c}
\alpha\\
r
\end{array} \right)&=&\frac{\Gamma(\alpha+1)}{r!\Gamma(\alpha-r+1)}\nonumber\\
&=&\frac{\alpha(\alpha-1)(\alpha-2)\cdot\ldots\cdot
[\alpha-(r-1)]}{1\cdot 2\cdot 3\cdot\ldots\cdot r}.
\end{eqnarray*}
When the function $f(t)$ of positive argument has continuous
derivatives of the first order, the
Riemann-Liouville fractional derivative is equivalent to the
Gr\"{u}nwald-Letnikov one for any parameter $\alpha$ ($0<\alpha<1$) \cite{20a}. So, we have
\begin{equation}\label{glrl}
\frac{d^{\alpha}f(t)}{d
t^{\alpha}}=\frac{^{GL}d^{\alpha}f(t)}{d
t^{\alpha}}.
\end{equation}
The relation between Riemann-Liouville and Caputo derivatives is
more complicated and reads as
\begin{equation}\label{rlc}
\frac{d^{\alpha}f(t)}{\partial
t^{\alpha}}=\frac{^{C}d^{\alpha}f(t)}{d
t^{\alpha}}+\Phi_{1-\alpha}(t)f(0),
\end{equation}
where
\begin{equation}\label{phi}
\Phi_{q+1}(t)=\left\{
\begin{array}{cc}
\frac{t^{q}}{\Gamma(q+1)}&t>0\\
0&t\leq 0
\end{array}
\right. .
\end{equation}
From Eqs. (\ref{gl})-(\ref{phi}) we can express the Caputo
fractional derivative in terms of the fractional-order backward
difference
\begin{eqnarray}\label{glc}
\frac{^{C}d^{\alpha}f(t)}{d t^{\alpha}}&=&\lim_{\Delta
t\rightarrow 0}(\Delta t)^{-\alpha}\sum_{r=0}^{[\frac{t}{\Delta
t}]}(-1)^{r}\left(
\begin{array}{c}
\alpha\\
r
\end{array}
\right)f(t-r\Delta t)\nonumber\\&&-\frac{1}{t^{\alpha}\Gamma(1-\alpha)}f(0).
\end{eqnarray}
The standard way to approximate of the fractional derivative,
which is useful for numerical calculations, is to omit the limit
in Eq.~(\ref{glc}) and to change the infinite series to the finite one
\begin{eqnarray}\label{glca}
\frac{^{C}d^{\alpha}f(t)}{d t^{\alpha}}&\simeq&(\Delta
t)^{-\alpha}\sum_{r=0}^{L}(-1)^{r} \left(
\begin{array}{c}
\alpha\\
r
\end{array}
\right)f(t-r\Delta t)\nonumber \\ &&-\frac{1}{t^{\alpha}\Gamma(1-\alpha)}f(0),
\end{eqnarray}
where the memory length $L$ is a natural number of arbitrary chosen value less then (or equal) $[t/\Delta t]$.
Subdiffusion is a process with the memory. According to the short memory principle, the fractional derivative is approximated by the fractional derivative with moving lower limit $t-L$, where $L$ is the 'memory length' equals to a certain amount of time steps \cite{20a}. However, in the paper \cite{lewkoszt} there was shown that the numerical solutions of subdiffusion equation with the boundary conditions (\ref{eq9a}) and (\ref{eq9b}) are in agreement with the analytical one only when the memory length is closed to actual account of time steps contrary to `short memory principle'. So, in numerical calculations we take the memory length $L$ equals to the actual number of time steps $t_s$.
Substituting Eq.~(\ref{glca}) to Eq.~(\ref{eq14}), using the
following approximation of the second order derivative
\begin{displaymath}
\frac{d^{2}f(x)}{d x^{2}}\simeq \frac{f(x+\Delta
x)-2f(x)+f(x-\Delta x)}{(\Delta x)^{2}},
\end{displaymath}
we obtain
\begin{widetext}
\begin{eqnarray}\label{alg}
C_i(x,t)&=&-\sum_{r=1}^{L}(-1)^{r}\frac{\alpha(\alpha-1)(\alpha-2)
\cdot\ldots\cdot[\alpha-(r-1)]}{1\cdot 2\cdot 3\cdot\ldots\cdot r
}C_i(x,t-r\Delta t)+\frac{(\Delta t)^{\alpha}}{t^{\alpha}\Gamma(1-\alpha)}C_i(x,0)\nonumber \\
&& +D_{i}\frac{(\Delta
t)^{\alpha}}{(\Delta x)^{2}}[C_i(x+\Delta x,t-\Delta
t)-2C_i(x,t-\Delta t)+C_i(x-\Delta x,t-\Delta t)]\\
&&-d_i k(\Delta t)^{\alpha}C_A^m(x,t-\Delta t)C_B^n(x,t-\Delta t) \nonumber,
\end{eqnarray}
\end{widetext}
for $i=A,B$, $d_A=m$ and $d_B=n$.
\subsection{Numerical results}\label{nr}
Here we compare the analytical results with the numerical ones. In all figures there are presented functions calculated for the system where $\alpha=0.5$, $D_A=0.025$, $D_B=0.0125$, $C_{0A}=2$, $C_{0B}=1$, $k=1$, $m=n=1$. For numerical calculations we take $\Delta x=0.2$ and $\Delta t=0.05$ (all quantities are given in the arbitrary units). Additionally, in Figs.~(\ref{Fig.5}) and (\ref{Fig.6}) we plot the borders of the reaction zone $(x_f-W_R/2, x_f+W_R/2)$ calculated for the time $5000$. The position of the reaction front was calculated from the discrete version of Eq.~(\ref{defxf})
\begin{equation}\label{xf}
x_f(t)=\frac{\sum_i x_i R(x_i,t)}{\sum_i R(x_i,t)} ,
\end{equation}
and equals to $0.71$ for $t=5000$. The width of the reaction region calculated from discrete version of Eq.~(\ref{defWR})
\begin{displaymath}
W_{\rm R}^2(t)=\frac{\sum_i(x_i-x_f(t))^2 R(x_i,t)}{\sum_i R(x_i,t)} ,
\end{displaymath}
equals to $0.38$ for $t=5000$. Thus the reaction region occupies the interval $(0.52;0.90)$.
From Eq. (\ref{xf}) we find that
\begin{equation}\label{1}
x_f(t)=0.0838t^{0.251}
\end{equation}
This relation is very close to the relation (\ref{xkt}) with $K$ calculated from Eq.~(\ref{rowk}) which reads
\begin{equation}\label{2}
x_f(t)=0.0825t^{0.25}.
\end{equation}
In Figs.~\ref{Fig.3} and \ref{Fig.4} there are presented the concentration profiles $C_A$ and $C_B$ obtained numerically according to the formula~(\ref{alg}) and the functions given by Eqs.~(\ref{eq46}) and (\ref{eq47}) with $\rho_A=0.40$ and $\rho_B=3.64$, respectively. We observe a quite good agreement of the analytical and numerical functions in the diffusion region.
In Fig.~\ref{Fig.5} we present the function $\Psi(x,t)$ calculated numerically and its parabolic approximation $\Psi(x,t)=0.297(x/t^{\alpha/2})^2-0.168(x/t^{\alpha/2})+0.015$. We note that $\Psi$ is satisfactorily approximated by the parabolic function of $x$. The region where $\Psi$ is parabolic determines the quasistatic approximation region.
In Fig.~\ref{Fig.6} we present the numerical solutions of the subdiffusion--reaction equations and their linear approximations calculated from the formulas $C_A(x,t)\approx-0.816x+0.616$ and $C_B(x,t)\approx 0.620x-0.490$, respectively. As seen, the linear approximation of $C_A$ and $C_B$ is satisfactory outside the reaction region. This statement confirms correctness the quasistationary approximation in the region enclosing the reaction region.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.84]{Kosztolowicz_3.EPS}
\caption{\label{Fig.3}The symbols represent the numerical solutions of subdiffusion-reaction equation, the continuous lines are assigned to theoretical functions $C_{A {\rm Dif}}$ for the times given in the legend.}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.84]{Kosztolowicz_4.EPS}
\caption{\label{Fig.4}The symbols represent the numerical solutions of subdiffusion-reaction equation, the continuous lines are assigned to theoretical functions $C_{B {\rm Dif}}$ for the times given in the legend.}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.84]{Kosztolowicz_5.EPS}
\caption{\label{Fig.5}The function $\Psi$ (symbols) obtained numerically for the times given in the legend and their parabolic approximations inside the quasistatic approximation region (continuous line); the vertical lines represent the borders of the reaction zone calculated for $t=5000$.}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.82]{Kosztolowicz_6.EPS}
\caption{\label{Fig.6}The concentration profiles $C_A$ and $C_B$ obtained numerically (squares) calculated for time $5000$ and their linear approximations (dashed lines), the vertical lines represent the borders of the reaction zone.}
\end{figure}
We conclude this section by saying that our numerical results support the postulates of the quasistatic approximation.
\section{Final remarks}\label{fr}
Using the quasistationary approximation and utilizing the solution of the subdiffusion--reaction equations in the diffusive region, we show that the time evolution of the reaction front for the subdiffusion--reaction
system is a power function (\ref{xkt}) with the exponent $\alpha/2$ and the coefficient $K$ is controlled by the subdiffusion coefficients of the system.
The function $x_f\sim t^{\alpha/2}$ can be obtained by means of the scaling method. However, it is very hard within this method to find an explicit expression of the parameter $K$ for the case of $D_A\neq D_B$.
We note that in this paper we consider the process of subdiffusion controlls chemical reactions. It means that the reactions which proceed relatively fast when compared to the characteristic time of meeting of the particles of $A$ and $B$ \cite{11}. Under such assumption the quasistatic approximation works and the time evolution of the reaction front does not depend on the detailed form of the reaction term (expect of dependence of the parameters $m$ and $n$). This happens because the form of $R$ does not change the relation $W_R\sim t^{\alpha/6}$. Thus, the width of the reaction zone appears to be relatively small in comparison with the width of the quasistatic approximation region. The time evolution of $x_f$ is determined by the dynamics of transport of the particles to the reaction zone and it depends on the parameters $m$, $n$, $D_A$, $D_B$, $C_{0A}$ and $C_{0B}$ only. This statement is particularly important for the subdiffusion--reaction systems where the reaction term is not uniquely defined (as the fractional derivative can be involved into this term in a few ways \cite{27a,10,9,sbsl,s,hw,10a}).
As far as we know, the time evolution of the reaction front has not been measured experimentally in a subdiffusive system with two mobile reactants. For this reason we can compare the functions (\ref{eq46}) and (\ref{eq47}) with experimental data obtained for a subdiffusive system without chemical reactions. Our theoretical and the experimental functions presented in \cite{dsdoww} are qualitatively similar to each other if we take the units commonly used in real systems where $x$ is given in $10^{-2} m$, $t$ in $sec$, $D_A$ and $D_B$ are of the order $10^{-8} m^2/s^\alpha$. Since $K$ is controlled by the subdiffusion coefficients of reactants, the method presented in this paper can be used for extracting the subdiffusion parameter from experimental data. The numerical calculations show that if we take the subdiffusion coefficients of the order maintained above and we assume that $C_{0A}/C_{0B}$ is of the order of $1$, we obtain $K\sim 10^{-2}$ $m/s^{\alpha/2}$ from Eq.~(\ref{rowk}).
The quasistatic approximation in a normal diffusion system applies to a region where the equilibrium time $\tau_{\rm F}$ of the reaction region is negligibly small comparing to the characteristic time of change of the flux $\tau_{\rm J}$ in the long time limit \cite{12,ckd,cd}. Let us note that this fact is fulfilled in the subdiffusive--recation system.
Since $W_{\rm R}\sim t^{\alpha/6}$, we have $\tau_{\rm F}\sim t^{1/3}$ form (\ref{tf}). Taking the definition (\ref{tj}), which for the subdiffusion flux $J\sim 1/t^{1-\alpha/2}$ gives $\tau_{\rm J}\sim 1/t$ (see Eqs. (\ref{linja}) and (\ref{linjb})), we get $\tau_{\rm F}/\tau_{\rm J}\rightarrow_{t\rightarrow\infty}0$ for any value of the subdiffusive parameter $\alpha$. So, the assumptions adopted in our paper agree with the quasistationary condition (\ref{r4}).
The function $\Psi$ is approximated by parabolic function in the region where the quasistatic approximation region overlaps with the diffusion one. However, we expect that there are departures from this approximation in a region located within the reaction zone where the reaction term is significantly different from zero. It is because of the concentrations $C_A$ and $C_B$ have different scaling properties in that region. We expect that the width of that region is so narrow, as compared with the width the quasistatic approximation one, that the departure form parabolic approximation is hard to observe on the plots presented in our paper. We note that the possibility of occurring this departure does not influence our main results.
\begin{acknowledgments}
The authors wish to express their thanks
to Stanis{\l}aw Mr\'{o}wczy\'{n}ski for fruitful discussion and
critical comments on the manuscript. This paper was supported by
Polish Ministry of Education and Science under Grant No. 1 P03B 136
30.
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3
|
\section{Datasets and Experimental Setup}
This section describes the used data, the experimental setup, and the baselines.
\subsection{Datasets}
\label{Datasets}
\begin{table*}[ht!]
\centering
\begin{tabular}{lccccc}
& OlpBench & ReVerb45K & ReVerb20K & FB15K237 & WN18RR \\ \hline
\#entities & $2.4M$ &$27K$&$11.1K$&$14.5K$ & $41.1K$ \\
\#relations & $961K$ & $21.6K$ & $11.1K$ & $237$ & $11$\\
\#entity clusters & N/A & $18.6K$ & $10.8K$ & N/A & N/A\\ \hline
\#train triples & $30M$ & $36K$ & $15.5K$ & $272K$ & $86.8K$ \\
\#valid triples & $10K$ &$3.6K$&$1.6K$&$17.5K$ & $3K$ \\
\#test triples & $10K$ & $5.4K$ & $2.4K$&$20.5K$ & $3K$ \\ \hline
\end{tabular}
\caption{Statistics of datasets. Only ReVerb45K and ReVerb20K come with gold entity clusters. FB15K237 and WN18RR are canonicalized, and OlpBench is too large to allow for a manual annotation of gold clusters.}
\label{DataStatistics}
\end{table*}
We use the following five datasets to test our methods
(their statistics are given in Table~\ref{DataStatistics}):
\smallskip
\noindent \textbf{OlpBench}~\cite{olpbench} is a large-scale OKBC dataset collected from English Wikipedia that we use for pre-training.
The dataset comes with multiple training and validation sets, however, we only use the training set with \textsc{Thorough} leakage removal and \textsc{Valid-Linked} validation set, since their data have the highest quality.
\smallskip
\noindent\textbf{ReVerb45K}~\cite{cesi} and \textbf{Re\-Verb\-20K} (\citealp{care}, adapted from~\citealp{CanonicalizingOKB}), are small-scale OKBC datasets, obtained via the ReVerb OIE tool~\cite{reverb}.
They additionally come with gold clusters of entities, and information on which knowledge-base entities refer to the same real-world entity; this information can be used to improve the accuracy of the evaluation.
We find that $99.9\%$ of test samples in ReVerb20K and $99.3\%$ of test samples in ReVerb45K contain at least one entity or relation that did not appear in OlpBench, hence requiring out-of-distribution generalization.
Two entities or relations were considered different if their textual representations differed after the pre-processing (lowercase and removal of redundant whitespace).
If two inputs still differ after these steps, the potential canonicalization has to be performed by the model.
\smallskip
\noindent \textbf{FB15K237}~\cite{fb15k237} and \textbf{WN18\-RR}~\cite{conve} are the most commonly used datasets for knowledge base completion, collected from Freebase and WordNet knowledge bases, respectively.
Both datasets have undergone a data cleaning to remove test leakage through inverse relations, as reported by~\citet{conve}.
We highlight that WN18RR differs from other datasets in its content.
While other datasets describe real-world entities and were often collected from Wikipedia or similar sources, WN18RR consists of information on words and linguistic relations between them, creating a major domain shift between pre-training and fine-tuning.
\subsection{Experimental Setup}
In our experiments, we observe the performance of the randomly-initialized and pre-trained \textsc{Gru} and \textsc{NoEncoder} variant of each of the three KBC models.
Pre-training was done with \textsc{Gru} encoders on OlpBench.
To be able to ensure a completely fair comparison, we make several simplifications to the models, e.g., assuming that entities and relations in TuckER have the same dimension and that all dropout rates within TuckER and ConvE are equal.
While these simplifications can result in a performance drop, they allow us to run exactly the same grid search of hyperparameters for all models, excluding human factor or randomness from the search.
We describe the details of our experimental setup with all hyperparameter choices in Appendix~\ref{ExperimentalSetupAppendix}.
\subsection{Baselines}
\label{baselines}
We compare our work to a collection of baselines, re-implementing and re-training them where appropriate.
\paragraph*{Models for Knowledge Base Completion.}
We evaluate all three of our main KBC models, ConvE, TuckER, and $5^\star$E with and without encoders.
We include results of these models, obtained by related work and additionally compare to KBC models from related work, such as BoxE~\cite{boxe}, ComplEx~\cite{Complex}, TransH~\cite{transh}, and TransE~\cite{transe}.
We highlight that numbers from external work were usually obtained with more experiments and broader hyperparameter search compared to experiments from our work, where we tried to guarantee exactly the same environment for a large number of models.
\paragraph*{Models for Knowledge Base Canonicalization.}
\citet{care} use external tools for knowledge base canonicalization to improve the predictions of KBC models, testing multiple methods to incorporate such data into the model.
The tested methods include graph convolution neural networks~\cite{gcn}, graph attention neural networks~\cite{gat}, and newly introduced local averaging networks (LANs).
Since LANs consistently outperform the alternative approaches in all their experiments, we use them as the only baseline of this type.
\paragraph*{Transfer Learning from Larger Knowledge Bases.}
\citet{biggraph} release pre-trained embeddings for the entire WikiData knowledge graph, which we use to initialize the TuckER model.
We intialize the entity and relation embeddings of all entities that we can find on WikiData.
Since pre-trained WikiData embeddings are only available for dimension $d=200$, we compare them to the pre-trained and randomly initialized \textsc{NoEncoder\_TuckER} of dimension $d=200$ for a fair comparison.
We do not re-train WikiData with other dimensions due to the computational resources required.
We do not use this baseline on KBC datasets, since we could not avoid a potential training and test set cross-contamination.
WikiData was constructed from Freebase and is linked to WordNet, the knowledge bases used to construct the FB15k237 and WN18RR datasets.
\paragraph*{Transfer Learning from Language Models.}
Pre-trained language models can be used to answer KB queries~\cite{lama,yao2019kgbert}.
We compare our results to \textsc{KG-Bert} on the FB15K237 and WN18RR datasets~\citep{yao2019kgbert} and to \textsc{Okgit} on the ReVerb45K and ReVerb20K datasets~\citep{okgit}.
Using large transformer-based language models for KBC can be slow.
We estimate that a single round of evaluation of \textsc{KG-Bert} on the ReVerb45K test set takes over 14 days, not even accounting for training or validation.
\citet{yao2019kgbert} report in their code repository that the evaluation on FB15K237 takes over a month\footnote{\url{https://github.com/yao8839836/kg-bert/issues/8}}.
For comparison, the evaluation of any other model in this work takes up to a maximum of a couple of minutes.
Not only is it beyond our resources to perform equivalent experiments for \textsc{KG-Bert} as for other models, but we also consider this approach to be completely impractical for link prediction.
\citet{okgit} combine \textsc{Bert} with an approach by~\citet{care}, taking advantage of both knowledge base canonicalization tools and large pre-trained transformers at the same time.
Their approach is more computationally efficient, since only $\langle h,r\rangle$ are encoded with \textsc{Bert}, instead of entire triplets, requiring by a magnitude fewer passes through the transformer model.
This is the strongest published approach to OKBC.
\section{Introduction}
A knowledge base (KB) is a collection of facts, stored and presented in a structured way that allows a simple use of the collected knowledge for applications.
In this paper, a \emph{knowledge base} is a finite set of triples $\langle h,r,t\rangle$, where \emph{h} and \emph{t} are \emph{head} and \emph{tail} entities, while \emph{r} is a binary relation between them.
Manually constructing a knowledge base is tedious and requires a large amount of labor.
To speed up the process of construction, facts can be extracted from unstructured text automatically, using, e.g., open information extraction (OIE) tools, such as ReVerb~\cite{reverb} or more recent neural approaches~\cite{supervisedoie,oie_comparison}.
Alternatively, missing facts can be inferred from existing ones using \emph{knowledge base completion (KBC)} algorithms, such as ConvE~\cite{conve}, TuckER~\cite{tucker}, or 5$^\star$E~\cite{5stare}.
It is desirable to use both OIE and knowledge base completion approaches to automatically construct KBs.
However, automatic extractions from text yield uncanonicalized entities and relations.
An entity such as ``the United Kingdom'' may also appear as ``UK'', and a relation such as ``located at'' may also appear as ``can be found in''.
If we fail to connect these occurrences and treat them as distinct entities and relations, the performance of KBC algorithms drops significantly~\cite{care}.
If our target data are canonicalized, collecting additional uncanonicalized data from unstructured text is not guaranteed to improve the performance of said models.
An illustration of a knowledge base can be found on Figure~\ref{task_diagram}.
\begin{figure}
\centering
\begin{tikzpicture}[
xscale = 2.2,
yscale = 0.9,
Block/.style = {
fill = green!40,
text = black,
inner sep = 2mm,
rounded corners = 1mm
},
Vect/.style = {
fill = blue!40,
text = black,
inner sep = 2mm,
rounded corners = 1mm
},
Arrow/.style = {
->,
>=stealth
}
]
\node[Block, minimum width = 55mm, minimum height=9mm ] at (0.87,0) (inBlock){};
\node[Vect, minimum width = 12mm] at (0.5,0) (UK_full){the United Kingdom};
\node[Vect, minimum width = 12mm] at (1.8,0) (UK){UK};
\node[Vect, minimum width = 12mm] at (0,2) (Alan){Alan Turing};
\node[Vect, minimum width = 12mm] at (2,2) (Europe){Europe};
\draw[Arrow] (Alan) -- node [left,midway] {lived in} (UK_full);
\draw[Arrow] (UK) -- node [right,midway] {part of} (Europe);
\draw[Arrow,dotted] (Alan) -- node [above,midway] {lived in (?)} (Europe);
\end{tikzpicture}
\caption{An example of a small knowledge base with the fact whether Alan Turing lived in Europe missing. If the knowledge base is canonicalized, ``the United Kingdom'' and ``UK'' are known to be the same entity. If the knowledge base is uncanonicalized or open, this information may not be given.}
\label{task_diagram}
\end{figure}
Open knowledge base completion (OKBC) aims to mitigate this problem and to predict unseen facts even when real-world entities and relations may appear under several different names.
Existing work in the area overcomes this either by learning a mapping from entity and relation names to their embeddings~\cite{olpbench} or by using external tools for knowledge base canonicalization~\cite{cesi}, and using the obtained predictions to enhance the embeddings of the entities~\cite{care,okgit}.
In this work, we follow the first of these two approaches.
We pre-train RNN-based encoders that encode entities and relations from their textual representations to their embeddings jointly with a KBC model on a large OKBC benchmark.
We use this pre-trained KBC model and encoders to initialize the final model that is later fine-tuned on a smaller dataset.
More specifically, KBC parameters that are shared among all inputs are used as an initialization of the same parameters of the fine-tuned model.
When initializing the input-specific embeddings, we introduce and compare two approaches:
Either the pre-trained entity and relation encoders are also used and trained during the fine-tuning, or they are used in the beginning to compute the initial values of all entity and relation embeddings, and then dropped.
We evaluate our approach with three different KBC models and on five datasets, showing consistent improvements on most of them.
We show that pre-training turns out to be particularly helpful on small datasets with scarce data by achieving SOTA performance on the ReVerb20K and ReVerb45K OKBC datasets~\cite{care,cesi} and consistent results on the larger KBC datasets FB15K237 and WN18RR~\cite{fb15k237,conve}.
Our results imply that even larger improvements can be obtained by pre-training on a larger corpus.
The code used for the experiments is available at \url{https://github.com/vid-koci/KBCtransferlearning}.
We highlight that \citet{care} and \citet{olpbench} use the term ``open knowledge graph embeddings'' for OKBC.
We use the term ``open knowledge base completion'' to also include KBs with non-binary relations, which cannot be viewed as graphs, and to consider methods that may not produce embeddings.
Our main contributions are briefly as follows:
\begin{compactitem}
\item We introduce a novel approach for the transfer of knowledge between KBC models that works on both open and regular knowledge bases without the need for entity or relation matching.
\item We show that pre-training on a large OKBC corpus improves the performance of these models on both KBC and OKBC datasets.
\item We obtain improvements over state-of-the-art approaches on $3$ of $5$ observed datasets, with the difference being particularly significant on the smallest datasets (e.g., $0.058$ absolute increase of MRR and $65\%$ MR decrease on the ReVerb20K dataset).
\end{compactitem}
\section{Model for Transfer Learning}
In this section, we introduce the architecture of our model and how a pre-trained model is used to initialize the model for fine-tuning.
The model consists of two encoders, one for entities and one for relations, and a KBC model.
Given a triple $\langle h,r,t\rangle$, the entity encoder is used to map the head $h$ and the tail $t$ into their vector embeddings $\mathbf{v}_h$ and $\mathbf{v}_t$, while the relation encoder is used to map the relation $r$ into its vector embedding $\mathbf{v}_r$.
These are then used as the input to the KBC algorithm of choice to predict their score (correctness), using the loss function, as defined by the KBC model.
The two parts of the model are architecturally independent of each other and will be described in the following paragraphs.
An illustration of our approach is given in Figure~\ref{model_diagram}.
\subsection{Encoders}
\label{ModelEncodersSection}
We use two types of mappings from an entity to a low-dimensional vector space.
The first approach is to assign each entity and relation its own embedding, initialized randomly and trained jointly with the model.
This is the default approach used by most KBC models, however, to distinguish it from the RNN-based approach, we denote it \emph{NoEncoder}.
The second approach that we test is to use an RNN-based mapping from the textual representation of an entity or relation (name) to its embedding.
We use the GloVe word embeddings~\cite{glove} to map each word into a vector, and then use them as the input to the entity encoder, implemented as a GRU~\cite{gru}.
To separate it from the NoEncoder, we call this the GRU encoder throughout this paper.
\citet{olpbench} test alternative encoders as well, but find that RNN-based approaches (LSTMs, in their case) perform the most consistently across the experiments.
The size of the NoEncoder grows linearly with the number of entities, while the size of the GRU encoder grows linearly with the vocabulary.
For large knowledge bases, the latter can significantly decrease the memory usage, as the size of the vocabulary is often smaller than the number of entities and relations.
\paragraph*{Transfer between datasets.}
Here, pre-training is always done using GRU encoders, as the transfer from the NoEncoder to any other encoder requires entity matching, which we want to avoid.
When fine-tuning is done with a GRU encoder, its parameters are initialized from the pre-trained GRU parameters.
The same applies for the vocabulary, however, if the target vocabulary includes any unknown words, their word embeddings are initialized randomly.
For the initialization of the NoEncoder setup, the pre-trained GRU is used to generate initial values of all vector embeddings by encoding their textual representations.
Any unknown words are omitted, and entities with no known words are initialized randomly.
An equivalent process is used for relations.
During our preliminary experiments, we have also tried pre-training the encoders on the next-word-prediction task on English Wikipedia, however, that turned out to have a detrimental effect on the overall performance compared to randomly-initialized GRUs ($2-3\%$ MRR drop and slower convergence).
That line of experiments was not continued.
\subsection{Knowledge Base Completion Models}
We use three models for knowledge base completion, ConvE~\cite{conve}, Tuck\-ER~\cite{tucker}, and 5$^\star$E~\cite{5stare}, chosen for their strong performance on various KBC benchmarks.
In the following paragraphs, we briefly introduce the models.
We assume that vectors $\mathbf{v}_h$, $\mathbf{v}_r$, and $\mathbf{v}_t$, obtained with encoders of any type, are the input to these models.
TuckER assigns a score to each triple by multiplying the vectors with a core tensor $\mathcal{W} \in \mathbb{R}^{d_e\times d_e \times d_r}$, where $d_e$ is the dimension of entities, and $d_r$ is the dimension of relations.
Throughout our work, we make the simplifying assumption that $d_e=d_r$ to reduce the number of hyperparameters.
During transfer, $\mathcal{W}$ from the pre-trained model is used to initialize $\mathcal{W}$ in the fine-tuned model.
ConvE assigns a score to each triple by concatenating $\mathbf{v}_h$ and $\mathbf{v}_r$, reshaping them into a 2D matrix, and passing them through a convolutional neural network (CNN).
The output of this CNN is a $d_e$-dimensional vector, which is multiplied with $\mathbf{v}_t$ and summed with a tail-specific bias term $b_t$ to obtain the score of the triple.
During transfer, the parameters of the pre-trained CNN are used as the initialization of the CNN in the fine-tuned model.
Bias terms of the fine-tuned model are initialized at random, since they are entity-specific.
5$^\star$E models consider $\mathbf{v}_h$ and $\mathbf{v}_t$ to be complex projective lines and $\mathbf{v}_r$ a vector of $2\times 2$ complex matrices.
These correspond to a relation-specific M\"{o}bius transformation of projective lines.
We refer the reader to the work of~\citet{5stare} for the details.
Unlike in ConvE and TuckER, there are no shared parameters between different relations and entities.
Pre-training thus only serves as the initialization of the embeddings.
During the time of evaluation, the model is given a triple with a missing head or tail and is used to rank all the possible entities based on how likely they appear in place of the missing entity.
Following~\citet{conve}, we transform head-pre\-dic\-tion samples into tail-prediction samples by introducing reciprocal relations $r^{-1}$ for each relation and transforming $\langle ?,r,t\rangle$ into $\langle t,r^{-1},?\rangle$.
Following~\citet{care}, the name of the reciprocal relation is created by adding the prefix ``inverse of''.
During our preliminary experiments, we have also experimented with BoxE~\cite{boxe}, however, we have decided not to use it for further experiments, since it was much slower to train and evaluate, compared to other models.
A single round of training of BoxE with GRU encoders on OlpBench takes over $24$ days, which we could not afford.
\section{Related Work}
In recent years, there have been several approaches to improve the performance of KBC algorithms through data augmentation, commonly through various levels of connection with unstructured text.
\citet{NTN}, for example, use pre-trained word embeddings to initialize their entity embeddings.
\citet{dkrl} make use of an encoder that generates an embedding given a description of the entity, and they show that their approach generalizes even to previously unseen entities.
\citet{yao2019kgbert} make use of a large-scale pre-trained transformer model to classify whether a fact is true.
They rely on costly pre-training and do not generate embeddings for entities that could be used to incorporate background knowledge into natural language understanding systems~\cite{ernie}.
Previous attempts at open knowledge base completion are tied to existing work on the canonicalization of knowledge bases.
To canonicalize open knowledge bases, automatic canonicalization tools cluster entities using manually defined features~\cite{CanonicalizingOKB} or by finding additional information from external knowledge sources~\cite{cesi}.
\citet{care} use clusters obtained with these tools to augment entity embeddings for KBC.
We note that~\citet{care} use RNN-based encoders to encode relations, but not to encode entities.
\citet{olpbench}, on the other hand, introduce a model with RNN-based encoders for both entities and relations, similarly to our approach, however, they do not transfer beyond the introduced OlpBench dataset.
Finally, \citet{okgit} use both KB canonicalization tools and large-scale pre-trained model \textsc{Bert}, combining their predictions to make a more informed decision.
The development of methods that improve predictions of KBC algorithms through data augmentation or transfer is tied to the advances in OIE and KBC methods.
However, these are beyond the scope of this project; see the works by~\citet{oiesurvey} and~\citet{OldDog} for an overview.
\section{Experimental Results}
This section contains the outcome of pre-training and fine-tuning experiments.
In the second part of this section, we additionally investigate an option of zero-shot transfer.
For each model, we report its mean rank (MR), mean reciprocal rank (MRR), and Hits at 10 (H@10) metrics on the test set.
We selected $N=10$ for comparison, since it was the most consistently Hits@N metric reported in related work.
We report the Hits@N performance for other values of N, the validation set performance, the running time, and the best hyperparameters in Appendix~\ref{FullResultsAppendix}.
All evaluations are performed using the filtered setting, as suggested by~\citet{transe}.
When evaluating on uncanonicalized datasets with known gold clusters, ReVerb20K and ReVerb45K, the best-ranked tail from the correct cluster is considered to be the answer, following~\citet{care}.
The formal description of all these metrics can be found in Appendix~\ref{MetricsAppendix}.
\paragraph*{Pre-training results.}
\begin{table}
\centering
\begin{tabular}{l@{}ccc@{}}
& MR & MRR & H@10 \\ \hline
\textsc{Gru\_TuckER} & $\mathbf{57.2K}$ &$.053$&$.097$ \\
\textsc{Gru\_ConvE} & $\mathbf{57.2K}$ & $.045$ & $.086$\\
\textsc{Gru\_$5^\star$E} & 60.1K &$\mathbf{.055}$&$\mathbf{.101}$ \\ \hline
\cite{olpbench} & -- & $.039$ & $.070$\\ \hline
\end{tabular}
\caption{Comparison of pre-trained models on OlpBench with previously best result. The best value in each column is written in \textbf{bold}.}
\label{OlpBenchResults}
\end{table}
The performance of the pre-trained models on OlpBench is given in Table~\ref{OlpBenchResults}.
Our models obtain better scores than the previously best approach based on ComplEx~\cite{ComplexOrig}, however, we mainly attribute the improvement to the use of better KBC models.
\paragraph*{OKBC results.}
\begin{table*}
\centering
\begin{tabular}{@{}c@{}c@{\ }|ccc|ccc@{}}
& & \multicolumn{3}{c}{ReVerb20K} & \multicolumn{3}{c}{ReVerb45K} \\
Model & Pre-trained? & MR & MRR & Hits@10 & MR & MRR & Hits@10 \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_TuckER}} & no & $2611$ &$.196$&$.267$ & $5692$ & $.109$ & $.138$ \\
& yes & $\mathit{303}$ &$\mathit{.379}$&$\mathit{.540}$ & $\mathit{780}$ & $\mathit{.299}$ & $\mathit{.453}$ \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_ConvE}} & no & $1419$ &$.282$&$.380$ & $2690$ & $.232$ & $.333$ \\
& yes & $\mathit{227}$ &$\mathit{.400}$&$\mathit{.568}$ & $\mathit{666}$ & $\mathit{.345}$ & $\mathit{.500}$ \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_$5\star$E}} & no & $2301$ &$.228$&$.334$ & $3460$ & $.152$ & $.212$ \\
& yes & $\mathit{780}$ &$\mathit{.249}$&$\mathit{.363}$ & $\mathit{3279}$ & $\mathit{.189}$ & $\mathit{.261}$ \\ \hline
\multirow{2}{*}{\textsc{GRU\_TuckER}} & no & $581$ &$.364$&$.505$ & $1398$ & $.302$ & $.420$ \\
& yes & $\mathit{245}$ &$\mathit{.397}$&$\mathit{.558}$ & $\mathit{706}$ & $\mathit{.331}$ & $\mathit{.477}$ \\ \hline
\multirow{2}{*}{\textsc{GRU\_ConvE}} & no & $334$ &$.387$&$.540$ & $824$ & $.343$ & $.488$ \\
& yes & $\boldsymbol{\mathit{184}}$ &$\mathit{.409}$&$\mathit{.573}$ & $\mathit{600}$ & $\mathit{.357}$ & $\mathit{.509}$ \\ \hline
\multirow{2}{*}{\textsc{GRU\_$5\star$E}} & no & $395$ &$.390$&$.546$ & $836$ & $.357$ & $.508$ \\
& yes & $\mathit{202}$ &$\boldsymbol{\mathit{.417}}$&$\boldsymbol{\mathit{.586}}$ & $\boldsymbol{\mathit{596}}$ & $\boldsymbol{\mathit{.382}}$ & $\boldsymbol{\mathit{.537}}$ \\ \hline \hline
$\blacklozenge$ \textsc{Okgit(ConvE)} & yes$^\blacktriangle$ & $527$ & $.359$ & $.499$ & $773.9$ & $.332$ & $.464$ \\
$\dagger$ \textsc{CaRe(ConvE, LAN)} & no & $973$ & $.318$ & $.439$ & $1308$ & $.324$ & $.456$ \\
$\dagger$ \textsc{TransE} & no & $1426$ & $.126$ & $.299$ & $2956$ & $.193$ & $.361$ \\
$\dagger$ \textsc{TransH} & no & $1464$ & $.129$ & $.303$ & $2998$ & $.194$ & $.362$ \\ \hline \hline
\multirow{2}{*}{\textsc{NoEncoder\_TuckER}\textsubscript{$d=200$}} & no & $2855$ & $.184$ & $.248$ & $5681$ & $.109$ & $.138$ \\
& yes &$329$ & $.369$ & $.528$ & $805$ & $.275$ & $.426$ \\ \hline
\textsc{BigGraph\_TuckER}\textsubscript{$d=200$} & yes$^\blacktriangle$ & $1907$ & $.215$ & $.291$ & $2285$ & $.234$ & $.337$ \\ \hline
\end{tabular}
\caption{Comparison of different models with and without pre-training on the OKBC benchmarks ReVerb20K and ReVerb45K. The scores of each model are reported with and without pre-training, with the better of the two written in \textit{italics}. Separated from the rest with two lines, are previous best results on the datasets and TuckER results with $d=200$ for a fair BigGraph comparison. The best overall value in each column is written in \textbf{bold}.
Results denoted with $\dagger$ and $\blacklozenge$ were taken from~\cite{care} and~\cite{okgit}, respectively.\\
$^\blacktriangle$ Unlike all other models with a \textit{yes} entry, \textsc{BigGraph\_TuckER} and \textsc{Okgit} were not pre-trained on OlpBench, but on pre-trained WikiData embeddings or a masked language modelling objective, respectively.}
\label{ReVerbResults}
\end{table*}
Our results of the models on ReVerb20K and ReVerb45K are given in Table~\ref{ReVerbResults}.
All models strictly improve their performance when pre-trained on OlpBench.
This improvement is particularly noticeable for NoEncoder models, which tend to overfit and achieve poor results without pre-training.
However, when initialized with a pre-trained model, they are able to generalize much better.
$5^\star$E seems to be an exception to this, likely because there are no shared parameters between relations and entities, resulting in a weaker regularization.
GRU-based models do not seem to suffer as severely from overfitting, but their performance still visibly improves if pre-trained on OlpBench.
Finally, our best model outperforms the state-of-the-art approach by \citet{okgit} on ReVerb20K and ReVerb45K.
Even when compared to pre-trained \textsc{Gru\_ConvE}, which is based on the same KBC model, \textsc{Okgit(ConvE)} and \textsc{CaRe(ConvE,LAN)} lag behind.
This is particularly outstanding, because the \textsc{Bert} and \textsc{RoBERTa} language models, used by \textsc{Okgit(ConvE)}, received by several orders of magnitude more pre-training on unstructured text than our models, making the results more significant.
Similarly, the initialization of models with BigGraph seems to visibly help the performance, however, they are in turn outperformed by a \textsc{NoEncoder} model, initialized with pre-trained encoders instead.
This indicates that our suggested pre-training is much more efficient, despite the smaller computational cost.
\paragraph*{KBC results.}
\begin{table*}
\centering
\begin{tabular}{@{}c@{}c@{\ \ }|@{\ }c@{\ \ }c@{\ \ }c|@{\ }c@{\ \ }c@{\ \ }c@{}}
& & \multicolumn{3}{c}{FB15K237} & \multicolumn{3}{c}{WN18RR} \\
Model & pre-trained? & MR & MRR & Hits@10 & MR & MRR & Hits@10 \\ \hline
\multirow{2}{*}{\textsc{TuckER}} & no & $166$ &$.358$&$.545$ & $4097$ & $\mathit{.468}$ & $.528$ \\
& yes & $\mathit{151}$ &$\mathit{.363}$&$\mathit{.550}$ & $\mathit{3456}$ & $.467$ & $\mathit{.529}$ \\ \hline
\multirow{2}{*}{\textsc{ConvE}} & no & $212$ &$.320$&$.504$ & $6455$ & $.429$ & $.479$ \\
& yes & $\mathit{200}$ &$\mathit{.325}$&$\mathit{.510}$ & $\mathit{5792}$ & $\mathit{.435}$ & $\mathit{.486}$ \\ \hline
\multirow{2}{*}{\textsc{$5\star$E}} & no & $152$ &$.353$&$.539$ & $\mathit{2450}$ & $\mathit{.492}$ & $\mathit{.583}$ \\
& yes & $\boldsymbol{\mathit{143}}$ &$\mathit{.357}$&$\mathit{.544}$ & $2636$ & $\mathit{.492}$ & $.582$ \\ \hline \hline
\cite{conve} \textsc{ConvE} & no & $244$ & $.325$ & $.501$ & $4187$ & $.43$ & $.52$ \\
\cite{OldDog} \textsc{ConvE} & no & -- & $.339$ & $.536$ & -- & $.442$ & $.504$ \\
\cite{tucker} \textsc{TuckER} & no & -- & $.358$ & $.544$ & -- & $.470$ & $.526$ \\
\cite{5stare} \textsc{$5\star$E} & no & -- & $\boldsymbol{.37}$ & $\boldsymbol{.56}$ & -- & $\boldsymbol{.50}$ & $\boldsymbol{.59}$ \\ \hline
\cite{yao2019kgbert} \textsc{KG-Bert} & yes & $153$ & -- & $.420$ & $\boldsymbol{97}$ & -- & $.524$ \\
\cite{boxe} \textsc{BoxE} & no & $163$ & $.337$ & $.538$ & $3207$ & $.451$ & $.541$ \\
\cite{Complex} \textsc{ComplEx} & no & -- & $\boldsymbol{.37}$ & $\boldsymbol{.56}$ & -- & $.48$ & $.57$ \\
\cite{dura} \textsc{ComplEx-Dura} & no & -- & $\boldsymbol{.371}$ & $\boldsymbol{.560}$ & -- & $.491$ & $.571$ \\
\cite{dura} \textsc{Rescal-Dura} & no & -- & $.368$ & $.550$ & -- & $\boldsymbol{.498}$ & $.577$ \\ \hline
\end{tabular}
\caption{Comparison of different models with and without pre-training on the KBC benchmarks FB15K237 and WN18RR. The scores of each model are reported with and without pre-training, with the better of the two written in \textit{italics}. Separated from the rest with two lines, we first list previous scores obtained with the ConvE, $5^\star$E, and TuckER models, followed by other well-performing models in the literature. The best overall value in each column is written in \textbf{bold}.}
\label{fb15kResults}
\end{table*}
To evaluate the impact of pre-training on larger canonicalized knowledge bases, we compare the performance of models on FB\-15\-K237 and WN18RR.
For brevity, we treat the choice of an encoder as a hyperparameter and report the better of the two models in Table~\ref{fb15kResults}. Detailed results are given in Appendix~\ref{FullResultsAppendix}.
Pre-trained models outperform their randomly initialized counterparts as well, however, the differences are usually smaller.
We believe that there are several reasons that can explain the small difference, primarily the difference in the size.
Best models on FB15K237 and WN18RR only made between $3$ and $12$ times more steps during pre-training than during fine-tuning.
For comparison, this ratio was between $250$ to $1000$ for ReVerb20K.
The smaller improvements on FB15K237 and WN18RR can also be explained by the domain shift, as already described in Section~\ref{Datasets}.
Table~\ref{fb15kResults} additionally includes multiple recently published implementations of ConvE, $5^\star$E, and TuckER, as well as other strong models in KBC.
We note that the comparison with all these models should be taken with a grain of salt, as other reported models were often trained with a much larger hyperparameter space, as well as additional techniques for regularization (e.g., \textsc{Dura}~\citep{dura} or label smoothing~\citep{tucker}) and sampling (e.g., self-adversarial sampling~\citep{boxe}).
Due to the large number of observed models and baselines, we could not expand the hyperparameter search without compromising the fair evaluation of all compared models.
In Appendix~\ref{FullResultsAppendix}, we also report the best hyperparameters for each model.
We highlight that pre-trained models usually obtain their best result with a larger dimension compared to their randomly-initialized counterparts.
Pre-training thus serves as a type of regularization, allowing us to fine-tune larger models.
We believe that training on even larger pre-training datasets, we could obtain pre-trained models with more parameters and even stronger improvements across many KBC datasets.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{Figure_1.png}
\caption{Comparing convergence of the overall best randomly initialized (blue) and pre-trained (green) models on ReVerb45K and FB15K237.
Pre-trained models converge in fewer training steps despite a smaller learning rate.}
\label{figure_convergence}
\end{figure}
\begin{table*}[ht]
\centering
\begin{tabular}{cc|ccc|ccc}
& & \multicolumn{3}{c}{ReVerb20K} & \multicolumn{3}{c}{ReVerb45K} \\
Model & Pre-trained? & MR & MRR & Hits@10 & MR & MRR & Hits@10 \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_TuckER}} & no & $4862$ &$.001$&$.002$ & $8409$ & $.001$ & $.001$ \\
& yes & $\mathit{1486}$ &$\mathit{.012}$&$\mathit{.026}$ & $\mathit{1737}$ & $\mathit{.019}$ & $\mathit{.041}$ \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_ConvE}} & no & $4600$ &$.002$&$\mathit{.004}$ & $8464$ & $.001$ & $\mathit{.001}$ \\
& yes & $\mathit{1480}$ &$\mathit{.003}$&$.000$ & $\mathit{1962}$ & $\mathit{.003}$ & $\mathit{.001}$ \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_$5\star$E}} & no & $4795$ &$.001$&$.000$ & $8445$ & $.001$ & $.002$ \\
& yes & $\mathit{1422}$ &$\mathit{.005}$&$.000$ & $\mathit{1701}$ & $\mathit{.014}$ & $\mathit{.011}$ \\ \hline
\end{tabular}
\caption{Comparison of the zero-shot performance of different models on the OKBC benchmarks ReVerb20K and ReVerb45K. The scores of each model are reported with and without pre-training on OlpBench, with the better of the two written in \textit{italics}. Note that the model that was not pre-trained is equivalent to a random baseline.}
\label{0shotResults}
\end{table*}
Not only does pre-training allow us to train larger models, pre-trained models also require fewer training steps to obtain a similar performance, as seen in Figure~\ref{figure_convergence}.
The experiments in the figure were done with the best hyperparameter setup for both pre-trained and randomly initialized model.
Even though the pre-trained models were fine-tuned with a smaller learning rate, they converge in fewer steps, which can be attributed to the pre-training.
Finally, we highlight that language modelling as pre-training hardly justifies the enormous computational cost.
On the majority of metrics, \textsc{KG-Bert} performs worse than models with fewer parameters and less pre-training, with a notable exception of a remarkable MR on the WN18RR dataset.
\subsection{Zero-Shot Experiments}
To understand better what kind of knowledge is transferred between the datasets, we investigate the zero-shot performance of pre-trained models on ReVerb20K and ReVerb45K, where the impact of pre-training was the strongest.
If the improvement of pre-training was mainly due to the direct memorization of facts, the zero-shot performance should already be high without fine-tuning.
The results of the zero-shot evaluation are given in Table~\ref{0shotResults}.
We report the performance of all models with and without pre-training, the latter being equivalent to the random baseline.
Note that since no fine-tuning takes place, the choice of the encoder for the evaluation does not matter.
We thus chose to only evaluate one of them.
Observing the results, we see that pre-training the models results in a lower MR, but not necessarily a much higher MRR.
Even when the increase of MRR happens, the difference is much smaller than when comparing fine-tuned models in Table~\ref{ReVerbResults}.
This implies that the improvement induced by pre-training likely does not happen only due to the memorization of facts from OlpBench.
On the other hand, the MR of pre-trained models is comparable or even better than the MR of randomly initialized NoEncoder models, fine-tuned on the ReVerb datasets, reported in Table~\ref{ReVerbResults}.
Hence, pre-trained models carry a lot of ``approximate knowledge'', which is consistent with earlier remarks on pre-training serving as a type of regularization.
Knowing that the ReVerb20K and ReVerb45K test sets consist of facts that contain at least one previously unseen entity or relation, this can be seen as out-of-distribution generalization.
Comparing the zero-shot MRR results with the OlpBench results implies that while OKBC models are capable of out-of-domain generalization to unseen entities and relations, there is still space for improvement.
\section{Summary and Outlook}
In this work, we have introduced a novel approach to transfer learning between various knowledge base completion datasets.
The main strength of the introduced method is the ability to benefit from pre-training on uncanonicalized knowledge bases, constructed from facts, collected from unstructured text.
Scaling the introduced method up would let us train large-scale pre-trained models that have already shown to be incredibly successful in natural language processing.
We tested our method on $5$ different datasets, showing that pre-training improves the performance of models.
Pre-training turned out to be particularly beneficial on small-scale datasets, where we were able to obtain the most significant gains, e.g., $6\%$ absolute increase of MRR and $65\%$ decrease of MR over the previously best method on ReVerb20K, despite not relying on large pre-trained models like \textsc{Bert}.
There are several directions of future work, such as scaling of pre-training to larger models and datasets and investigating the impact of the encoder architecture.
\section*{Acknowledgments}
The authors would like to thank Ralph Abboud for his helpful comments on the paper manuscript.
This work was supported by the Alan Turing Institute under the EPSRC
grant EP/N510129/1, the AXA Research Fund, the ESRC
grant "Unlocking the Potential of AI for English Law", and
the EPSRC Studentship OUCS/EPSRC-NPIF/VK/1123106.
We also acknowledge the use of the EPSRC-funded Tier 2
facility JADE (EP/P020275/1) and GPU computing support
by Scan Computers International Ltd.
\section{Metrics}
\label{MetricsAppendix}
This section contains a formal description of the metrics that were used to score the models.
Let $\langle h,r,t \rangle$ be some test triple.
Given $\langle h,r,? \rangle$, the model is used to rank all entities in the knowledge base from the most suitable to the least suitable tail, so that the entity no.\ $1$ is the most likely tail according to the model.
All ranks are reported in the ``filtered setting'', which means that all other known correct answers, other than $t$, are removed from this list to reduce noise.
Let $r_t$ be the position or \textit{rank} of the correct tail in this list.
In ReVerb20K and ReVerb45K, there may be entities from the same cluster as $t$, equivalent to $t$.
In this case, $r_t$ is the lowest of their ranks.
The \textit{reciprocal rank} of an example is then defined as $\frac{1}{r_t}$.
The same process is repeated with the input $\langle ?,r,t\rangle$ and the head entity.
The \textit{mean rank (MR)} of the model is the average of all ranks across all test examples, both heads and tails.
The \textit{mean reciprocal rank (MRR)} is the average of all reciprocal ranks, both heads and tails.
The \textit{Hits@N} metric tells us how often the rank was smaller or equal to $N$.
The related literature most commonly uses $1$, $3$, and $10$ for values of~$N$, however, $5$, $30$, and $50$ are also occasionally reported.
\section{Detailed Experimental Setup}
\label{ExperimentalSetupAppendix}
Following~\citet{OldDog}, we use 1-N scoring for negative sampling and cross-entropy loss for all models.
The Adam optimizer~\cite{adam} was used to train the network.
We follow~\citet{conve} and~\citet{tucker} with the placement of batch norm and dropout in ConvE and TuckER, respectively, however, we simplify the setup by always setting all dropout rates to the same value to reduce the hyperparameter space.
To find the best hyperparameters, grid search is used for all experiments to exhaustively compare all options.
Despite numerous simplifications, we find that our re-implementations of the baselines (non-pre-trained NoEncoder models) perform comparable to the original reported values.
The experiments were performed on a DGX-1 cluster, using one Nvidia V100 GPU per experiment.
\paragraph*{Pre-training setup.}
Pre-training was only done with GRU encoders, as discussed in Section~\ref{ModelEncodersSection}.
Due to the large number of entities in the pre-training set, 1-N sampling is only performed with negative examples from the same batch, and batches of size $4096$ were used, following~\citet{olpbench}.
The learning rate was selected from \{$1\cdot 10^{-4}, 3\cdot 10^{-4}$\}, while the dropout rate was selected from \{$0.2, 0.3$\} for ConvE and \{$0.3, 0.4$\} for TuckER.
For 5$^\star$E, the dropout rate was not used, but N3 regularization was~\cite{Complex}, its weight selected from \{$0.1$,$0.03$\}.
For TuckER, models with embedding dimensions $100$, $200$, and $300$ were trained.
We saved the best model of each dimension for fine-tuning.
For ConvE, models with embedding dimensions $300$ and $500$ were trained, and the best model for each dimension was saved for fine-tuning.
Following~\citet{care}, we use a single 2d convolution layer with $32$ channels and $3\times 3$ kernel size.
When the dimension of entities and relations is $300$, they were reshaped into $15\times20$ inputs, while the $20\times25$ input shapes were used for the $500$-dimensional embeddings.
For 5$^\star$E, models with embedding dimensions $200$ and $500$ were trained, and the best model for each dimension was saved for fine-tuning.
Following~\citet{olpbench}, we trained each model for $100$ epochs.
Testing on the validation set is performed each $20$ epochs, and the model with the best overall mean reciprocal rank (MRR) is selected.
\paragraph*{Fine-tuning setup.}
Fine-tuning is performed in the same way as the pre-training, however, the models were trained for $500$ epochs, and a larger hyperparameter space was considered.
More specifically, the learning rate was selected from \{$3\cdot 10^{-5}, 1\cdot 10^{-4}, 3\cdot 10^{-4}$\}.
The dropout rate was selected from \{$0.2, 0.3$\} for ConvE and \{$0.3, 0.4$\} for TuckER.
The weight of N3 regularization for the 5$^\star$E models was selected from \{$0.3,0.1,0.03$\}.
The batch size was selected from \{$512, 1024, 2048, 4096$\}.
The same embedding dimensions as for pre-training were considered.
\section{Full Results}
\label{FullResultsAppendix}
This appendix contains detailed information on the best performance of all models. Table~\ref{DetailedFullResults} contains the detailed information on the performance of the best models both on validation and test sets of the datasets.
Table~\ref{HyerparameterTable} includes information on the best hyperparameter setups and approximate training times.
Note that the given times are approximate and are strongly affected by the selection of the hyperparameters as well as external factors.
\begin{table*}
\centering
\tiny
\begin{tabular}{@{}c@{}c@{\ }|@{\ \ }c@{\ \ \ \ }c@{\ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ }c@{\ \ }|@{\ }c@{\ \ \ \ }c@{\ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ \ }c@{\ \ \ \ }c@{}}
& & \multicolumn{8}{c}{OlpBench validation} & \multicolumn{8}{c}{OlpBench test} \\
Model & Pre-trained? & MR & MRR & H@1 & H@3 & H@5 & H@10 & H@30 & H@50 & MR & MRR & H@1 & H@3 & H@5 & H@10 & H@30 & H@50 \\ \hline
\textsc{GRU\_TuckER} & no & $\mathbf{55.6K}$ &$.058$&$.0.033$ & $.060$ & $.076$ & $.104$ & $.163$ & $.195$ & $\mathbf{57.2K}$ & $.053$ & $.029$ & $.054$ & $.070$ & $.097$ & $.155$ & $.189$ \\
\textsc{GRU\_ConvE} & no & $57.6K$ &$.047$&$.025$ & $.048$ & $.063$ & $.090$ & $.147$ & $.179$ & $\mathbf{57.2K}$ & $.045$ & $.022$ & $.047$ & $.060$ & $.086$ & $.140$ & $.173$ \\
\textsc{GRU\_$5\star$E} & no & $60.1K$ &$\mathbf{.060}$&$\mathbf{.034}$ & $\mathbf{.061}$ & $\mathbf{.078}$ & $\mathbf{.109}$ & $\mathbf{.171}$ & $\mathbf{.205}$ & $59.9K$ & $\mathbf{.055}$ & $\mathbf{.030}$ & $\mathbf{.056}$ & $\mathbf{.075}$ & $\mathbf{.101}$ & $\mathbf{.160}$ & $\mathbf{.194}$ \\ \hline \hline
\multicolumn{18}{c}{}\\
& & \multicolumn{8}{c}{ReVerb20K validation} & \multicolumn{8}{c}{ReVerb20K test} \\
Model & Pre-trained? & MR & MRR & H@1 & H@3 & H@5 & H@10 & H@30 & H@50 & MR & MRR & H@1 & H@3 & H@5 & H@10 & H@30 & H@50 \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_TuckER}} & no &$2532$ & $.196$ & $.160$ & $.207$ & $.230$ & $.265$ & $.318$ & $.352$ & $2611$ &$.196$&$.157$ & $.212$& $.236$& $.267$& $.332$& $.362$ \\
& yes & $354$ & $.367$& $.283$ & $.400$ & $.457$ & $.529$ & $.646$ & $.702$ & $303$ &$.367$ & $.295$ & $.412$ & $.466$ & $.540$ & $.659$ & $.714$ \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_ConvE}} & no & $1376$ &$.279$& $.233$ &$.301$ &$.329$ &$.364$& $.423$& $.458$ & $1419$ & $.282$ & $.227$&$.308$ &$.338$ &$.380$&$.442$&$.475$ \\
& yes & $292$ &$.392$& $.312$ & $.423$ & $.477$&$.546$&$.658$&$.706$ & $227$ & $.400$ & $.313$&$.440$& $.491$ & $.568$ &$.680$ &$.729$ \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_$5\star$E}} & no & $2213$ &$.230$& $.180$ &$.243$ &$.279$ &$.325$& $.421$& $.468$ & $2301$ & $.228$ & $.174$&$.243$ &$.278$ &$.334$&$.429$&$.473$ \\
& yes & $773$ &$.249$& $.194$ & $.263$ & $.299$&$.353$&$.466$&$.528$ & $780$ & $.249$ & $.191$&$.264$& $.302$ & $.363$ &$.471$ &$.532$ \\ \hline
\multirow{2}{*}{\textsc{GRU\_TuckER}} & no & $646$ &$.344$&$.277$& $.366$ &$.409$ &$.470$ & $.564$ &$.609$ & $598$ & $.357$ & $.283$ & $.391$ & $.434$ &$.493$ &$.594$ &$.639$ \\
& yes & $283$ &$.386$&$.305$ &$.416$ &$.465$ &$.543$ & $.660$ & $.707$ & $245$ & $.397$ & $.315$ & $.429$ &$.485$ & $.558$ & $.674$ & $.720$ \\ \hline
\multirow{2}{*}{\textsc{GRU\_ConvE}} & no & $333$ &$.377$&$.305$ &$.405$ &$.447$ &$.516$ &$.635$& $.683$ & $334$ & $.387$ & $.305$ & $.421$ & $.471$ & $.540$ & $.648$ & $.690$ \\
& yes & $\mathbf{229}$ & $.397$ & $.317$ & $.428$ &$.477$ & $.554$ & $.671$ &$.726$ & $\mathbf{184}$ & $.409$ & $.326$ & $.442$ & $.499$ & $.573$ &$.687$ & $.737$\\ \hline
\multirow{2}{*}{\textsc{GRU\_$5\star$E}} & no & $430$ &$.379$& $.302$ &$.411$ &$.454$ &$.523$& $.639$& $.685$ & $395$ & $.390$ & $.311$&$.422$ &$.475$ &$.546$&$.650$&$.697$ \\
& yes & $243$ &$\mathbf{.404}$& $\mathbf{.318}$ & $\mathbf{.440}$ & $\mathbf{.492}$&$\mathbf{.569}$&$\mathbf{.690}$&$\mathbf{.747}$ & $202$ & $\mathbf{.417}$ & $\mathbf{.330}$&$\mathbf{.455}$& $\mathbf{.512}$ & $\mathbf{.586}$ &$\mathbf{.701}$ &$\mathbf{.748}$ \\ \hline \hline
\multicolumn{18}{c}{}\\
& & \multicolumn{8}{c}{ReVerb45K validation} & \multicolumn{8}{c}{ReVerb45K test} \\
Model & Pre-trained? & MR & MRR & H@1 & H@3 & H@5 & H@10 & H@30 & H@50 & MR & MRR & H@1 & H@3 & H@5 & H@10 & H@30 & H@50 \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_TuckER}} & no &$5097$ & $.094$ & $.077$ & $.097$ & $.109$ & $.126$ & $.160$ & $.177$ & $5135$ &$.103$&$.088$ & $.106$ & $.116$ & $.131$ & $.164$ & $.182$ \\
& yes & $839$ & $.305$& $.228$ & $.335$ & $.386$ & $.453$ & $.559$ & $.610$ & $780$ &$.299$ & $.220$ & $.332$ & $.386$ & $.453$ & $.557$ & $.607$ \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_ConvE}} & no & $2741$ &$.235$& $.180$ & $.262$ & $.293$ & $.336$ & $.406$ & $.436$ & $2690$ & $.232$ & $.179$ & $.256$ & $.288$ & $.333$ & $.400$ & $.434$ \\
& yes & $632$ &$.353$& $.274$ & $.384$ & $.438$ & $.506$ & $.611$ & $.662$ & $666$ & $.345$ & $.267$ & $.378$ & $.432$ & $.500$ & $.601$ & $.644$ \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_$5\star$E}} & no & $3653$ &$.148$& $.116$ & $.151$ & $.174$ & $.210$ & $.289$ & $.324$ & $3460$ & $.152$ & $.123$ & $.152$ & $.174$ & $.212$ & $.289$ & $.327$ \\
& yes & $3233$ &$.187$& $.151$ & $.195$ & $.222$ & $.258$ & $.320$ & $.349$ & $3279$ & $.189$ & $.153$ & $.198$ & $.224$ & $.261$ & $.325$ & $.357$ \\ \hline
\multirow{2}{*}{\textsc{GRU\_TuckER}} & no & $1386$ &$.304$&$.242$ & $.330$ & $.369$ & $.423$ & $.504$ & $.545$ & $1398$ & $.302$ & $.242$ & $.323$ & $.365$ & $.420$ & $.506$ & $.546$ \\
& yes & $761$ &$.336$&$.263$ & $.368$ & $.413$ & $.473$ & $.570$ & $.613$ & $706$ & $.331$ & $.258$ & $.359$ & $.412$ & $.477$ & $.575$ & $.618$ \\ \hline
\multirow{2}{*}{\textsc{GRU\_ConvE}} & no & $776$ &$.351$&$.273$ & $.387$ & $.437$ & $.501$ & $.593$ & $.631$ & $824$ & $.343$ & $.268$ & $.374$ & $.424$ & $.488$ & $.584$ & $.626$ \\
& yes & $\mathbf{584}$ & $.364$ & $.285$ & $.400$ & $.452$ & $.518$ & $.620$ & $.665$ & $600$ & $.357$ & $.277$ & $.393$ & $.444$ & $.509$ & $.607$ & $.651$\\ \hline
\multirow{2}{*}{\textsc{GRU\_$5\star$E}} & no & $843$ &$.362$& $.287$ & $.394$ & $.446$ & $.511$ & $.607$ & $.649$ & $836$ & $.357$ & $.278$ & $.394$ & $.443$ & $.508$ & $.605$ & $.647$ \\
& yes & $603$ &$\mathbf{.385}$& $\mathbf{.305}$ & $\mathbf{.421}$ & $\mathbf{.474}$ & $\mathbf{.542}$ & $\mathbf{.642}$ & $\mathbf{.683}$ & $\mathbf{596}$ & $\mathbf{.382}$ & $\mathbf{.302}$ & $\mathbf{.416}$ & $\mathbf{.467}$ & $\mathbf{.537}$ & $\mathbf{.636}$ & $\mathbf{.676}$ \\ \hline \hline
\multicolumn{18}{c}{}\\
& & \multicolumn{8}{c}{FB15K237 validation} & \multicolumn{8}{c}{FB15K237 test} \\
Model & Pre-trained? & MR & MRR & H@1 & H@3 & H@5 & H@10 & H@30 & H@50 & MR & MRR & H@1 & H@3 & H@5 & H@10 & H@30 & H@50 \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_TuckER}} & no &$160$ & $.366$ & $.276$ & $.399$ & $.464$ & $.547$ & $.675$ & $.727$ & $166$ &$.358$&$.265$ & $.393$ & $.458$ & $.545$ & $.669$ & $.725$ \\
& yes & $142$ & $\mathbf{.369}$& $\mathbf{.277}$ & $\mathbf{.403}$ & $\mathbf{.468}$ & $\mathbf{.555}$ & $\mathbf{.680}$ & $\mathbf{.732}$ & $151$ &$\mathbf{.363}$ & $\mathbf{.269}$ & $\mathbf{.398}$ & $\mathbf{.464}$ & $\mathbf{.550}$ & $\mathbf{.678}$ & $\mathbf{.733}$ \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_ConvE}} & no & $200$ &$.326$& $.238$ & $.356$ & $.419$ & $.508$ & $.634$ & $.690$ & $212$ & $.320$ & $.230$ & $.351$ & $.417$ & $.504$ & $.634$ & $.690$ \\
& yes & $189$ &$.332$& $.242$ & $.363$ & $.428$ & $.513$ & $.647$ & $.703$ & $200$ & $.325$ & $.233$ & $.355$ & $.421$ & $.510$ & $.645$ & $.703$ \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_$5\star$E}} & no & $144$ &$.358$& $.267$ & $.393$ & $.456$ & $.542$ & $.670$ & $.723$ & $152$ & $.353$ & $.260$ & $.389$ & $.454$ & $.539$ & $.666$ & $.721$ \\
& yes & $\mathbf{137}$ &$.362$& $.270$ & $.398$ & $.462$ & $.548$ & $.672$ & $.727$ & $\mathbf{143}$ & $.357$ & $.264$ & $.393$ & $.459$ & $.544$ & $.670$ & $.727$ \\ \hline
\multirow{2}{*}{\textsc{GRU\_TuckER}} & no & $\mathbf{137}$ &$.355$&$.262$ & $.391$ & $.454$ & $.539$ & $.671$ & $.724$ & $144$ & $.350$ & $.256$ & $.386$ & $.451$ & $.538$ & $.666$ & $.723$ \\
& yes & $\mathbf{137}$ &$.357$&$.264$ & $.393$ & $.454$ & $.542$ & $.670$ & $.726$ & $\mathbf{143}$ & $.354$ & $.260$ & $.391$ & $.453$ & $.538$ & $.668$ & $.724$ \\ \hline
\multirow{2}{*}{\textsc{GRU\_ConvE}} & no & $161$ &$.340$&$.249$ & $.372$ & $.433$ & $.521$ & $.653$ & $.708$ & $166$ & $.334$ & $.242$ & $.368$ & $.431$ & $.519$ & $.650$ & $.705$ \\
& yes & $151$ & $.331$ & $.241$ & $.361$ & $.423$ & $.515$ & $.652$ & $.708$ & $157$ & $.327$ & $.237$ & $.359$ & $.422$ & $.513$ & $.646$ & $.704$\\ \hline
\multirow{2}{*}{\textsc{GRU\_$5\star$E}} & no & $145$ &$.351$& $.256$ & $.387$ & $.453$ & $.540$ & $.669$ & $.725$ & $150$ & $.345$ & $.249$ & $.380$ & $.449$ & $.536$ & $.667$ & $.725$ \\
& yes & $143$ &$.351$& $.257$ & $.386$ & $.451$ & $.537$ & $.667$ & $.722$ & $145$ & $.348$ & $.254$ & $.384$ & $.450$ & $.536$ & $.666$ & $.720$ \\ \hline \hline
\multicolumn{18}{c}{}\\
& & \multicolumn{8}{c}{WN18RR validation} & \multicolumn{8}{c}{WN18RR test} \\
Model & Pre-trained? & MR & MRR & H@1 & H@3 & H@5 & H@10 & H@30 & H@50 & MR & MRR & H@1 & H@3 & H@5 & H@10 & H@30 & H@50 \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_TuckER}} & no & $3701$ & $.469$ & $.437$ & $.483$ & $.499$ & $.524$ & $.571$ & $.595$ & $4097$ &$.468$&$.435$ & $.483$ & $.502$ & $.528$ & $.576$ & $.595$ \\
& yes & $3332$ & $.470$& $.439$ & $.479$ & $.500$ & $.529$ & $.583$ & $.604$ & $3456$ &$.467$ & $.434$ & $.480$ & $.500$ & $.529$ & $.583$ & $.604$ \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_ConvE}} & no & $6337$ &$.429$& $.403$ & $.437$ & $.455$ & $.479$ & $.512$ & $.530$ & $6455$ & $.429$ & $.404$ & $.440$ & $.455$ & $.479$ & $.512$ & $.530$ \\
& yes & $5678$ &$.433$& $.407$ & $.442$ & $.460$ & $.483$ & $.523$ & $.542$ & $5793$ & $.435$ & $.408$ & $.444$ & $.461$ & $.486$ & $.527$ & $.545$ \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_$5\star$E}} & no & $2527$ &$\mathbf{.492}$& $\mathbf{.450}$ & $\mathbf{.505}$ & $\mathbf{.531}$ & $.574$ & $.649$ & $\mathbf{.680}$ & $\mathbf{2450}$ & $\mathbf{.492}$ & $.448$ & $\mathbf{.507}$ & $\mathbf{.534}$ & $\mathbf{.583}$ & $\mathbf{.652}$ & $.680$ \\
& yes & $2576$ &$\mathbf{.492}$& $\mathbf{.450}$ & $.501$ & $.528$ & $\mathbf{.575}$ & $\mathbf{.651}$ & $\mathbf{.680}$ & $2636$ & $\mathbf{.492}$ & $\mathbf{.450}$ & $.504$ & $.533$ & $.582$ & $\mathbf{.652}$ & $\mathbf{.683}$ \\ \hline
\multirow{2}{*}{\textsc{GRU\_TuckER}} & no & $3201$ &$.456$&$.428$ & $.464$ & $.484$ & $.506$ & $.553$ & $.574$ & $3262$ & $.456$ & $.429$ & $.465$ & $.483$ & $.505$ & $.551$ & $.573$ \\
& yes & $2637$ &$.457$&$.422$ & $.469$ & $.490$ & $.523$ & $.579$ & $.609$ & $2790$ & $.455$ & $.418$ & $.469$ & $.494$ & $.524$ & $.577$ & $.600$ \\ \hline
\multirow{2}{*}{\textsc{GRU\_ConvE}} & no & $4376$ &$.431$&$.407$ & $.439$ & $.456$ & $.477$ & $.518$ & $.538$ & $4474$ & $.428$ & $.402$ & $.438$ & $.455$ & $.474$ & $.512$ & $.533$ \\
& yes & $5983$ & $.400$ & $.376$ & $.410$ & $.422$ & $.444$ & $.475$ & $.493$ & $6128$ & $.399$ & $.375$ & $.409$ & $.423$ & $.441$ & $.473$ & $.491$\\ \hline
\multirow{2}{*}{\textsc{GRU\_$5\star$E}} & no & $\mathbf{2444}$ &$.456$& $.418$ & $.463$ & $.492$ & $.533$ & $.598$ & $.629$ & $2545$ & $.452$ & $.413$ & $.462$ & $.489$ & $.527$ & $.595$ & $.629$ \\
& yes & $2971$ &$.426$& $.388$ & $.440$ & $.464$ & $.494$ & $.548$ & $.575$ & $3068$ & $.420$ & $.379$ & $.437$ & $.459$ & $.488$ & $.547$ & $.573$ \\ \hline \hline
\end{tabular}
\caption{Full results on both validation and test set of all datasets. In addition to the metrics reported in the paper, we also report H@N for $N\in\{1,3,5,10,30,50\}$, which appeared in related work. The best value in each column is written in \textbf{bold}.}
\label{DetailedFullResults}
\end{table*}
\begin{table*}
\centering
\scriptsize
\begin{tabular}{cc|cccccc}
& & \multicolumn{6}{c}{OlpBench} \\
Model & Pre-trained? & dimension & learning rate & batch & dropout & N3 weight & time \\ \hline
\textsc{GRU\_TuckER} & no & $300$ & $1\cdot 10^{-4}$ & $4096$ & $0.3$ & -- & $5$ days\\
\textsc{GRU\_ConvE} & no & $500$ & $1\cdot 10^{-4}$ & $4096$ & $0.2$ & -- & $5$ days\\
\textsc{GRU\_$5\star$E} & no & $500$ & $1\cdot 10^{-4}$ & $4096$ & -- & $0.03$ & $12$ days\\ \hline \hline
\multicolumn{8}{c}{}\\
& & \multicolumn{6}{c}{ReVerb20K} \\
Model & Pre-trained? & dimension & learning rate & batch & dropout & N3 weight & time \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_TuckER}} & no & $300$ & $3\cdot 10^{-5}$ & $512$ & $0.4$ & -- & $30$ min \\
& yes & $300$ & $3\cdot 10^{-4}$ & $512$ & $0.3$ & -- &$30$ min \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_ConvE}} & no & $300$ & $3\cdot 10^{-4}$ & $1024$ & $0.3$ & -- & $30$ min\\
& yes & $500$ & $1\cdot 10^{-4}$ & $512$ & $0.2$ & -- &$30$ min \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_$5\star$E}} & no & $200$ & $1\cdot 10^{-3}$ & $512$ & -- & $0.3$ & $30$ min\\
& yes & $500$ & $3\cdot 10^{-4}$ & $512$ & -- & $0.03$ &$30$ min \\ \hline
\multirow{2}{*}{\textsc{Gru\_TuckER}} & no & $300$ & $3\cdot 10^{-3}$ & $1024$ & $0.4$ & -- & $30$ min \\
& yes & $300$ & $3\cdot 10^{-5}$ & $2048$ & $0.4$ & -- & $30$ min\\ \hline
\multirow{2}{*}{\textsc{Gru\_ConvE}} & no & $300$ & $3\cdot 10^{-5}$ & $512$ & $0.3$ & -- & $30$ min\\
& yes & $500$ & $3\cdot 10^{-5}$ & $512$ & $0.3$ & -- & $30$ min\\ \hline
\multirow{2}{*}{\textsc{Gru\_$5\star$E}} & no & $200$ & $3\cdot 10^{-4}$ & $2048$ & -- & $0.1$ & $30$ min\\
& yes & $500$ & $1\cdot 10^{-4}$ & $1024$ & -- & $0.1$ & $30$ min\\ \hline \hline
\multicolumn{8}{c}{}\\
& & \multicolumn{6}{c}{ReVerb45K} \\
Model & Pre-trained? & dimension & learning rate & batch & dropout & N3 weight & time \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_TuckER}} & no & $100$ & $1\cdot 10^{-3}$ & $512$ & $0.4$ & -- & $2.5$h\\
& yes & $300$ & $3\cdot 10^{-4}$ & $4096$ & $0.3$ & -- & $2.5$h\\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_ConvE}} & no & $300$ & $3\cdot 10^{-4}$ & $512$ & $0.3$ & -- & $2$h\\
& yes & $500$ & $1\cdot 10^{-4}$ & $2048$ & $0.3$ & -- & $2.5$h\\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_$5\star$E}} & no & $200$ & $1\cdot 10^{-3}$ & $2048$ & -- & $0.3$ & $3$h\\
& yes & $200$ & $1\cdot 10^{-4}$ & $512$ & -- & $0.1$ & $3$h\\ \hline
\multirow{2}{*}{\textsc{Gru\_TuckER}} & no & $300$ & $1\cdot 10^{-3}$ & $512$ & $0.4$ & -- & $3$h \\
& yes & $300$ & $3\cdot 10^{-4}$ & $2048$ & $0.4$ & -- & $3$h\\ \hline
\multirow{2}{*}{\textsc{Gru\_ConvE}} & no & $300$ & $1\cdot 10^{-4}$ & $4096$ & $0.3$ & -- &$2$h \\
& yes & $500$ & $3\cdot 10^{-4}$ & $2048$ & $0.3$ & -- & $2.5$h\\ \hline
\multirow{2}{*}{\textsc{Gru\_$5\star$E}} & no & $500$ & $3\cdot 10^{-4}$ & $1024$ & -- & $0.03$ &$3$h \\
& yes & $500$ & $3\cdot 10^{-4}$ & $2048$ & -- & $0.1$ & $3$h\\ \hline \hline
\multicolumn{8}{c}{}\\
& & \multicolumn{6}{c}{FB15K237} \\
Model & Pre-trained? & dimension & learning rate & batch & dropout & N3 weight & time \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_TuckER}} & no & $200$ & $3\cdot 10^{-4}$ & $1024$ & $0.4$ & -- & $9.5$h\\
& yes & $300$ & $3\cdot 10^{-5}$ & $2048$ & $0.4$ & -- & $9.5$h\\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_ConvE}} & no & $300$ & $3\cdot 10^{-4}$ & $2048$ & $0.3$ & -- &$8.5$h \\
& yes & $500$ & $3\cdot 10^{-4}$ & $512$ & $0.3$ & -- & $9$h\\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_$5\star$E}} & no & $500$ & $1\cdot 10^{-4}$ & $512$ & -- & $0.3$ &$12$h \\
& yes & $500$ & $1\cdot 10^{-4}$ & $512$ & -- & $0.3$ & $12$h\\ \hline
\multirow{2}{*}{\textsc{Gru\_TuckER}} & no & $100$ & $3\cdot 10^{-4}$ & $512$ & $0.4$ & -- & $13$h\\
& yes & $200$ & $1\cdot 10^{-4}$ & $1024$ & $0.4$ & -- & $13$h\\ \hline
\multirow{2}{*}{\textsc{Gru\_ConvE}} & no & $300$ & $3\cdot 10^{-4}$ & $512$ & $0.3$ & -- & $11$h\\
& yes & $500$ & $3\cdot 10^{-4}$ & $512$ & $0.3$ & -- & $12$h\\ \hline
\multirow{2}{*}{\textsc{Gru\_$5\star$E}} & no & $500$ & $1\cdot 10^{-4}$ & $1024$ & -- & $0.1$ & $24$h\\
& yes & $500$ & $3\cdot 10^{-4}$ & $512$ & -- & $0.1$ & $24$h\\ \hline \hline
\multicolumn{8}{c}{}\\
& & \multicolumn{6}{c}{WN18RR} \\
Model & Pre-trained? & dimension & learning rate & batch & dropout & N3 weight & time \\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_TuckER}} & no & $100$ & $1\cdot 10^{-3}$ & $512$ & $0.3$ & -- & $7.5$h\\
& yes & $100$ & $1\cdot 10^{-3}$ & $512$ & $0.3$ & -- & $7.5$h\\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_ConvE}} & no & $300$ & $1\cdot 10^{-3}$ & $512$ & $0.3$ & -- &$8$h \\
& yes & $300$ & $1\cdot 10^{-3}$ & $512$ & $0.3$ & -- & $8$h\\ \hline
\multirow{2}{*}{\textsc{NoEncoder\_$5\star$E}} & no & $200$ & $1\cdot 10^{-3}$ & $512$ & -- & $0.3$ &$8$h \\
& yes & $500$ & $1\cdot 10^{-3}$ & $1024$ & -- & $0.3$ & $8$h\\ \hline
\multirow{2}{*}{\textsc{Gru\_TuckER}} & no & $100$ & $1\cdot 10^{-3}$ & $512$ & $0.3$ & -- & $6$h\\
& yes & $100$ & $1\cdot 10^{-3}$ & $1024$ & $0.4$ & -- & $6$h\\ \hline
\multirow{2}{*}{\textsc{Gru\_ConvE}} & no & $300$ & $1\cdot 10^{-3}$ & $1024$ & $0.3$ & -- & $8.5$h\\
& yes & $500$ & $1\cdot 10^{-3}$ & $2048$ & $0.3$ & -- & $8.5$h\\ \hline
\multirow{2}{*}{\textsc{Gru\_$5\star$E}} & no & $500$ & $3\cdot 10^{-4}$ & $512$ & -- & $0.1$ & $8.5$h\\
& yes & $500$ & $1\cdot 10^{-3}$ & $1024$ & -- & $0.3$ & $8.5$h\\ \hline \hline
\end{tabular}
\caption{Best hyperparameter setups and training time of best models for all datasets.}
\label{HyerparameterTable}
\end{table*}
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"redpajama_set_name": "RedPajamaArXiv"
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"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
One of the differences between field theories and mechanical systems
is that field theories have an infinite number of degrees of freedom,
which makes possible spontaneous symmetry breaking. More generally, we
know, by now, that also dynamical
symmetry breaking plays an important role in modern physics. In
condensed matter physics, for example, such a mechanism is used to describe
superconductivity, by the condensate of Cooper pairs, while
for particle physics it is the main source
of hadron masses and governs the low energy hadron dynamics.
Connected with dynamical symmetry breaking comes the problem
of restoration of broken symmetries for some sufficiently high temperature.
As it is well known, a phase transition occurs when there is a
singularity in the free energy or one of its derivatives.
The fact that almost all phenomena studied in theories near the
transition point exhibit scaling, that is, a power-law behavior
between two mesurable quantities, leads naturally to the classification
of different conformal field theories in universality classes,
which are itself determined by a set of numbers usually called
critical indices.
A large number of conformal field theories is known in
two dimensions and none is firmly established in four. The dimensionality
$(d=3)$ is between these extremes and contains two well established
families of conformal field theories. The first family contains the
usual Ising, XY and Heisenberg critical $\sigma$ models
and is characterized by a particular set of exponents
called scalar critical exponents.
The second family contains various critical four fermion models
\cite{Gat-Kovner-Rosenstein} and the set of exponents that
characterizes it is called chiral critical exponents.
The critical properties of the great majority of phase transitions
in three euclidean dimensions (magnetic systems, superconductors etc.)
are quite accurately described by the first family of conformal field
theories while the finite-temperature phase transitions in certain $(3+1)$
dimensional quantum field theories are argued to belong to these same
universality classes \cite{Svetitsky-Yaffe}.
When studying the finite-temperature chiral
restoration in QCD one is usually gided by the concepts of dimensional
reduction and universality. From the dimensional reduction point of view,
a hot field theory can be regarded as a static field theory at zero
temperature in a lower dimension \cite{Jourjine}.
For example, four dimensional QCD with $N_{f}$ light quarks
near the transition can be described by the three dimensional linear $\sigma$ model with the same global symmetry \cite{Wilczek}.
On the other hand, as the classification in universality classes is done
by the computation of critical indices and
those critical indices are infrared (IR) sensitive, we must understand
the role of each degree of freedom in this context. This can be easily
done with the use of effective field theory methods in which the
integration over different energy scales gives a theory for
the degrees of freedom which are the real responsible for infrared
divergencies \cite{Braaten}. Dimensional reduction is based in the
zero temperature Appelquist-Carazone decoupling theorem
\cite{Appelquist-Carazone}, where a low energy theory is constructed
by the integration over heavy fields in the functional integral.
According to standard dimensional reduction arguments,
the fermions themselves, even if they
are massless at zero temperature, do not influence the nature of phase
transition at finite-temperature. It is rather their bosonic composites,
the Goldstone bosons, which are of importance. This follows directly from
the universality of second order phase transitions \cite{Amit}, in
which the commonly held assumption
is that all the possible universality classes (or equivalently,
conformal field theories) are variations of the $\sigma$ model and one
need only match the correct symmetry-breaking pattern. As a consequence,
the chiral transition of four dimensional QCD, with $N_{f}=2$ flavors,
should lie in the same universality class as a three dimensional $O(4)$
magnet. Similarly, other models, e.g. four-fermion theories in $d$
dimensions such as the Gross-Neveu model (GN model) \cite{Gross-Neveu}
with discrete symmetries or the Nambu-Jona-Lasinio model (NJL model)
\cite{Nambu-Jona-Lasinio} with continous chiral symmetry,
are expected to be in the same universality class of a $(d-1)$ dimensional
Ising or Heisenberg magnet, respectively.
Recently, however, it was pointed out that there exist different $(d=3)$
conformal field theories with the same symmetry-breaking pattern
\cite{Gat-Kovner-Rosenstein}.
They are exactly the four-fermion interaction models of the
Nambu-Jona-Lasinio type.
Physically, this corresponds to the fact that on the chirally symmetric
side of the phase transition there are $N$ massless fermions whose effect
is felt even in the IR fixed point, just like the Goldstone bosons.
The presence of more than one universality class in $(d=3)$ makes the
procedure of dimensionally reducing a quantum field theory ambigous
and it is now uncertain to which
conformal field theory the $(d+1)$ finite-temperature quantum
field theory will reduce.
The argument if favor of the bosonic universality class goes as follows.
At finite-temperature, the $(d=4)$ fermion reduces to a collection of
$(d=3)$ massive fermions and there is no zero mode for which the
Matsubara frequency vanishes. Nevertheless,
even if a single massive field does not influence the phase transition, the cumulative effects of an infinite number of such fields may
have an appreciable impact. In order to see whether or not this happens,
all harmonics should be summed and their cumulative effects studied.
It is the purpose of this paper to discuss the assumptions underlying this
analysis and to determine to which universality class the simplest generalization of the Gross-Neveu model with continous chiral symmetry,
belongs. Besides being an interesting theoretical model, it is also
believed that, when properly extended to incorporate continous chiral
symmetry, four-fermion models are more realistic as effective theories
of QCD than the linear $\sigma$ model, especially at scales where
quark structure is important.
The paper is organized as follows. In section II, we present the model. In
section III we compute the critical exponents at zero temperature while, in
section IV, we obtain, after dimensional reduction, the new critical exponents
that determines the universality class of the reduced theory. In section V, we
analyse the vacuum structure at finite-temperature. Conclusions are given in
section VII. In the appendix, we compute the thermal renormalization group
functions that controlls thedependence of the thermal counterterms on the
temperature. In this paper we use $h\!\!\!\slash=c=1$.
\section{The chiral-continous GN model}
We are interested in studying the behavior of a multiplet of
N fermions coupled with a pair of composite self-interacting pseudo-scalar
and scalar fields, $\sigma(x)$ and $\pi(x)$ respectively,
in such a way that the euclidean functional action reads
\begin{equation}
S_{E}=\int d^{d}{\bf x}
\left\{
-\bar{\psi}
\left( \partial\!\!\!\slash+
g_{B}\left(\sigma+i\gamma_{S}\pi
\right)
\right)
\psi+
\frac{1}{2}(\partial_{\mu}\sigma)^{2}+
\frac{1}{2}(\partial_{\mu}\pi)^{2}-
\frac{1}{2}m_{B}^{2} \left( {\sigma}^{2}+{\pi}^{2} \right) +
\frac{\lambda_{B}}{4} \left( {\sigma}^{2}+{\pi}^{2} \right)^{2}
\right\}.
\label{initial-action}
\end{equation}
This model has continous chiral symmetry
\begin{equation}
\left\{ \begin{array}{ll}
\bar{\psi} \rightarrow \bar{\psi}e^{i\alpha \gamma_{s}} \\
\psi \rightarrow e^{i\alpha \gamma_{s}} \psi,
\end{array}
\right.,
\end{equation}
and $\sigma \rightarrow -\sigma$, $\pi \rightarrow \pi$ with $\alpha$
being a constant and we use for the $\gamma_{s}$ the following representation
$\gamma_{S}=
\left(
\begin{array}{ll}
1 & 0 \\
0 & -1
\end{array}
\right)
$.
In order to have an uniquely defined vacuum (see fig. 1), we must choose a
prefered direction in the space of components, say $\left< \sigma \right>=v$
and $\left< \pi \right> = 0$, and shift our fields as
\begin{equation}
\left\{ \begin{array}{ll}
\sigma(x) \rightarrow v + \sigma(x) \\
\pi(x) \rightarrow \pi(x)
\end{array}
\right.,
\end{equation}
where $v=\frac{\sqrt{m_{B}}}{\lambda_{B}}$. In this new picture the euclidean
action becomes
\begin{eqnarray}
S_{E} & = & \int d^{d}{\bf x}
\left\{
-\frac{1}{2}m_{B}^{2}v^{2}+
\frac{\lambda_{B}}{4}v^{4}-
\bar{\psi}
\left( \partial\!\!\!\slash+
g_{B}\left(v+\sigma+i\gamma_{S}\pi
\right)
\right)
\psi+
\frac{1}{2}(\partial_{\mu}\sigma)^{2}+
\frac{1}{2}(\partial_{\mu}\pi)^{2} \right. \\
& - &
\left.
\frac{1}{2}m_{\sigma}^{2}\sigma^{2}-
\frac{1}{2}m_{\pi}^{2}\pi^{2}-
{\lambda_{B}}v{\sigma^{3}}-
{\lambda_{B}}v{\sigma}{\pi^{2}}+
\frac{\lambda_{B}}{4} \left( {\sigma}^{2}+{\pi}^{2} \right)^{2}
\right\},
\label{shifted-action}
\end{eqnarray}
with $m_{\sigma}=m_{B}^{2}-3\lambda_{B}v^{2}$ and
$m_{\pi}=m_{B}^{2}-\lambda_{B}v^{2}$. If we use the definition of $v$ we can
see that $m_{\pi}=0$, as it should be since the $\pi$ field is a Goldstone
boson. Although new interaction terms have arised, it is easy to prove that
this renormalizable model needs only usual wave function, mass and coupling
constant renormalization to render the theory finite.
\begin{figure}[th]
\centerline{\epsfysize=3.5in\epsffile{pyu1.eps}}
\caption[region]
{\small\sf{Mexican hat shape for the zero temperature effective potential
in the tree level. The vacuum is infinitely degenerate and one
of the degrees of freedom will become a Goldstone boson after
the definition of an unique ground-state.}}
\end{figure}
The finite-temperature version of this model is obtained, in the Matsubara
formalism, by the compactification of the imaginary time dimension together
with the imposition of (anti)-periodic boundary conditions to
(fermionic)-bosonic fields \cite{Jourjine} as
\begin{equation}
\bar{\psi}({\bf x},\tau)=
\frac{1}{\sqrt{\beta}}
\sum_{n=-\infty}^{\infty}\bar{\psi}_{n}({\bf x})e^{i\omega_{n}\tau},
\end{equation}
with $\omega_{n}=\frac{2\pi}{\beta}(n+\frac{1}{2})$ and
\begin{equation}
\sigma({\bf x},\tau)=\frac{1}{\sqrt{\beta}}
\sum_{n=-\infty}^{\infty}\sigma_{n}({\bf x})e^{i\omega_{n}\tau},
\end{equation}
with $\omega_{n}=\frac{2\pi n}{\beta}$.
Since global aspects of the euclidean manifold do not affect local properties
of a quantum field theory we will, again, need only those previously
mentioned renormalization constants to render the theory finite.
\section{Critical exponents at zero temperature}
The chiral-continous Gross-Neveu model, in contrast to the discrete
Gross-Neveu model, is renormalizable
in four dimensions and it can be described, beyound the tree level,
by all the techniques developed for the $(\Phi^{2})^{2}$ theory:
renormalization group equations and large N expansion, for example.
To investigate the vacuum structure of the theory, we must consider the
leading order density $V_{eff}$ evaluated for constant fields
$\left< \sigma \right>=v$ and $\left< \pi \right>=0$
\cite{Rosenstein-report}.
This gives the gap equation (a cut-off $\Lambda$ is implied)
\begin{equation}
\frac{\partial V_{eff}}{\partial \sigma}=0 \Longrightarrow
-m_{B}^{2}+\lambda_{B}v^{2}=
2g_{B}^{2}\int^{\Lambda}\frac{d^{d}q}{(2\pi)^{d}}
\frac{1}{q^{2}+g_{B}^{2}v^{2}}.
\end{equation}
We define the critical coupling $g_{c}$ as
\begin{equation}
-m_{B}^{2}=2g_{c}^{2}\int^{\Lambda}\frac{d^{d}q}{(2\pi)^{d}}
\frac{1}{q^{2}},
\end{equation}
so that the gap equation takes the form
\begin{equation}
\frac{1}{2}
\left(
m_{B}^{2}t-\frac{\lambda_{B} v^{2}}{t+1}
\right)
=
\int^{\Lambda}\frac{d^{d}q}{(2\pi)^{d}}
\frac{g_{B}^{2}v^{2}}{q^{2}(q^{2}+g_{B}^{2}v^{2})},
\label{gap-equation}
\end{equation}
where $t=\left(\frac{g_{B}^{2}}{g_{c}^{2}}-1 \right)$ is the reduced coupling.
This form for the gap equation is particularly well suited for extracting
critical indices since the problem
reduces to counting the infrared divergences on the right hand side
\cite{Zinn-Justin}. As previously said, the
critical indices are the responsible for the power-law behavior
between two mesurable quantities as we approach the critical coupling.
In this sense
we will compute the critical exponents defined by
$\left. \left< \bar{\psi}\psi \right> \right|_{v \rightarrow 0}
\sim t^{\beta}$,
$\left. \left< \bar{\psi}\psi \right> \right|_{t \rightarrow 0}
\sim v^{1/\delta}$,
$\left. \left< \Sigma \right> \right|_{v \rightarrow 0}
\sim t^{\nu}$ etc,
and show that the introduction of a self-interaction
for the bosonic fields does not change the values of the exponents
obtained for the NJL model.
First, we can see that the critical indices for the NJL model are
recovered by simply putting $\lambda_{B}=0$ in the gap equation
(\ref{gap-equation}). Indeed, in this case the infrared behavior of
the integral with $v$ goes as
\begin{equation}
\int_{v}^{\Lambda}\frac{d^{d}q}{(2\pi)^{d}}
\frac{g_{B}^{2}v^{2}}{q^{2}(q^{2}+g_{B}^{2}v^{2})}
\sim
g_{B}^{2}v^{d-2}-g_{B}^{4}v^{d-4}+g_{B}^{6}v^{d-6}-...,
\label{integral-behavior}
\end{equation}
and the critical indices are simply $\beta=\nu=1/(d-2)$ and $\delta=d-1$,
since, to the order we are concerned, the self-energy and the
two point function coincide.
Now, let us show that even with $\lambda \neq 0$ we still have the same
critical indices. For $d<4$ this is rather obvious since the singularity
of the integral in (\ref{gap-equation}) allow us to neglect the $v^{2}$
term in the left hand side. However, if we are in four dimensions,
we can use the integral behavior (\ref{integral-behavior}) to obtain
\begin{equation}
m_{B}^{2}t-\frac{\lambda_{B} v^{2}}{t+1} \sim 2g_{B}^{2}v^{2},
\end{equation}
and compute the $\beta$ exponent from the behavior of
\begin{equation}
v^{2} \sim \frac{m_{B}^{2}t}{\frac{\lambda_{B}}{t+1}+2g_{B}^{2}}
\end{equation}
with $t$. This gives the mean field value $\beta=\frac{1}{2}$,
a characteristic of conformal field theories in dimensions greater
or equal to 4.
We conclude that the introduction of self-interaction for the composite
bosons, has not changed the structure of the IR singularities.
Also, we can see that, in $(d=4)$ all critical exponents are mean
field and there are no non-gaussian fixed points while, in three
dimensions the value $\beta=1$ tell us this is a chiral conformal
field theory of the Nambu-Jona-Lasinio type.
\section{Dimensional reduction and universality}
We will now consider the problem of computing critical indices when
a non zero temperature is introduced. We could do this in a similar way
to what we have done in the zero temperature case but we will, instead,
compute the fermionic determinant using a modified minimal subtraction
scheme at zero momentum and temperature $1/\beta$ because this is a more
suitable way of obtaining the functional form of the dimensionally
reduced theory.
As we have pointed out in the introduction, the reduced theory is an
effective theory for the zero modes of the original fields
\cite{Jourjine} (and see also \cite{Landsman}).
It can be explicitly obtained from the
initial partition funcion by integrating out the $\omega_{n} \neq 0$
Matsubara frequencies of the bosonic fields and, as we do not have
a zero mode in the case of fermions, the complete integration
over the $\psi$ and $\bar{\psi}$ fields. The resulting QFT will be
described by an euclidean partition function of the type
\begin{equation}
Z=\int {\cal D}{\sigma_{0}}{\cal D}{\pi_{0}} \,
exp{ \left\{
-\int d^{d-1}{\bf x}
\left(
{\cal L}_{eff}(\sigma_{0},\pi_{0})+\delta{\cal L}
\right)
\right\} },
\end{equation}
where $\delta{\cal L}$ includes all other local terms
not present in the original lagrangian density but which are
consistent with the symmetries and, in the lagrangian density
${\cal L}_{eff}(\sigma_{0},\pi_{0})$, the parameters will be, in
general, functions of the cut-off, the temperature and the bare
parameters.
Since we are interested in the computation of the finite-temperature
effective action at the leading order in the $1/N$ expansion it is
convenient to define
\begin{equation}
\left\{ \begin{array}{ll}
\tilde{g}=g_{B} \sqrt{N} \\
\tilde{m}^{2}=m_{B}^{2}N \\
\tilde{\lambda}=\lambda_{B} N.
\end{array}
\right.,
\end{equation}
and
\begin{equation}
\left\{ \begin{array}{ll}
\bar{m}_{1}=\tilde{g}_{B}\left( \sigma_{0}+i\pi_{0} \right) \\
\bar{m}_{2}=\tilde{g}_{B}\left( \sigma_{0}-i\pi_{0} \right),
\end{array}
\right.
\end{equation}
being the eigenvalues of
$\bar{M}\equiv g_{B}({\bf I}\sigma_{0}+i\gamma_{S}\pi_{0})$,
where ${\bf I}$ is the unit matrix.
At finite-temperature, the integration over the imaginary
time becomes a sum over Matsubara frequencies and if we remember that the
fields $\bar{\psi}$ and $\psi$ are anti-periodic in the time component, we obtain, after integrating out the fermionic degrees of freedom, the expression
for the fermionic determinant
\begin{equation}
trln{\left(\partial\!\!\!\slash+\bar{M}\right)}=
\frac{2}{\beta} \sum_{n=-\infty}^{\infty}
\int \frac{d^{d-1}{\bf k}}{(2\pi)^{d-1}}
\sum_{i=1}^{2}\ln{\left( {\bf k}^{2}+\omega_{n}^{2}+\bar{m}_{i}^{2}\right)},
\label{tr-ln-1}
\end{equation}
where, as usual, $\omega_{n}=\frac{2\pi}{\beta}(n+\frac{1}{2})$ for fermions,
or yet
\begin{equation}
trln{\left(\partial\!\!\!\slash+\bar{M}\right)}=
\frac{2}{\beta} \sum_{n=-\infty}^{\infty}
\int \frac{d^{d-1}{\bf k}}{(2\pi)^{d-1}}
\left\{
2\ln{\left( {\bf k}^{2}+\omega_{n}^{2} \right)}+
ln{\left( 1+\frac{\bar{m}_{1}^{2}}{{\bf k}^{2}+\omega_{n}^{2}}\right)}+
ln{\left( 1+\frac{\bar{m}_{2}^{2}}{{\bf k}^{2}+\omega_{n}^{2}}\right)}
\right\}.
\label{tr-ln-2}
\end{equation}
The first logarithim in the above expression is naturally absorved as an
overall normalization factor. The other two can be written, after
integration over ${\bf k}$ and summation over $n$, as a series
\begin{equation}
trln{\left(\partial\!\!\!\slash+\bar{M}\right)}=
2 \sum_{s=1}^{\infty} \frac{(-)^{s}{\tilde{g}}^{2s}}{s}
\left[ \left( \sigma_{0}+i\pi_{0} \right)^{2s}+
\left( \sigma_{0}-i\pi_{0} \right)^{2s}
\right]
\frac{(2\pi)^{d-1-2s}}{(4\pi)^{\frac{d-1}{2}}}
\frac{\Gamma\left( s-\frac{d-1}{2}\right)}{\Gamma(s)}
\frac{\zeta \left( 2s-(d-1),\frac{1}{2}\right)}{\beta^{d-2s}},
\label{tr-ln-3}
\end{equation}
where $\zeta$ is the analytic extension of the modified
Epstein-Hurwitz zeta function \cite{Elisalde-Romeo}.
In the limit of high temperature ($\beta \rightarrow 0$) the only
contributions to the effective action comming from expression
(\ref{tr-ln-3}) are those given by $s=1$ and $s=2$. Those contributions
are divergent so that thermal couterterms will be needed to make the
reduced theory finite in this limit \cite{Landsman}.
In this context, we join
the above divergences with those comming from the integration over
non-static bosonic modes \cite{Malbouisson-Silva-Neto-Svaiter} and
introduce the following counterterms (for $d=4-\varepsilon$)
\begin{equation}
\delta m^{2}=\frac{1}{24}\frac{1}{\beta^{2}}
\left( 4\tilde{\lambda}{\bf I}-\tilde{g}^{2}\gamma_{S} \right)+
\frac{\tilde{\lambda}\tilde{m}^{2}{\bf I}}{4\pi^{2}}
\left(
\frac{1}{\varepsilon}+1-\gamma-ln{\frac{2\beta\mu}{\sqrt{\pi}}}
\right),
\label{mass-counterterm}
\end{equation}
and
\begin{equation}
\delta \lambda=
\left[
\frac{10}{6}\frac{\lambda^{2}}{64\pi^{2}}{\bf I_{4}}+
\frac{{\tilde{g}^{4}}}{16\pi^{2}}
\left(
\begin{array}{ll}
1 & -3 \\
-3 & 1
\end{array}
\right)
\right]
\left(
-\frac{1}{\varepsilon}+\frac{\gamma}{2}+ln{\frac{2\beta\mu}{\sqrt{\pi}}}
\right),
\label{coupling-counterterm}
\end{equation}
where, as usual, $\gamma$ is the Euler number, $\mu$ is a
mass parameter introduced by dimensional regularization to give Green
functions their proper dimensions and we have defined, for simplicity
\begin{equation}
{\bf I_{4}}
=
\left(
\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}
\right).
\end{equation}
Since as $\beta \rightarrow 0$ we are in the disordered phase
($v=0$), we define
\begin{equation}
\phi=
\left(
\begin{array}{c}
\sigma_{0} \\
\pi_{0}
\end{array}
\right),
\end{equation}
and
\begin{equation}
\Phi=
\left(
\begin{array}{ll}
\sigma_{0} & 0 \\
0 & \pi_{0}
\end{array}
\right)
,
\end{equation}
so that the renormalized action for the $(d=4)$ becomes
\begin{equation}
S_{eff}({\sigma_{0}},{\pi_{0}})=\int d^{3}{\bf x}
\left\{
\frac{1}{2}\left( \partial_{i}\phi^{a} \right)^{2}+
\frac{1}{2}\phi^{a}m_{ab}^{2}(\beta)\phi^{b}+
\frac{1}{4}\phi^{a}
\Phi_{a\alpha}
\frac{\lambda_{\alpha \alpha^{'}}(\beta)}{\beta}
\Phi_{\alpha^{'}b}
\phi^{b}+
\frac{28}{9}
\frac{\tilde{\lambda}^{3}\zeta(3)}{2^{9}\pi^{4}}(\phi^{a}\phi^{a})^{3}
\right\},
\label{reduced-action}
\end{equation}
where, as usual,
\begin{equation}
\left\{
\begin{array}{ll}
m^{2}(\beta)=-\tilde{m}^{2}{\bf I}+\delta m^{2} \\
\lambda(\beta)=
\tilde{\lambda}{\bf I_{4}}+
\delta \lambda.
\end{array}
\right.
\label{renormalized-parameters}
\end{equation}
Now we are ready to start the computation of the critical indices for
the reduced theory. In this sense we will, again, set
$\left<\pi\right>=0$ and expand our field $\sigma$ around a constant
configuration $v$ to obtain the thermal effective potential
\begin{equation}
V_{eff}=
\frac{1}{2}m_{10}^{2}(\beta)v^{2}+
\frac{1}{4}\frac{\lambda_{10}(\beta)}{\beta}v^{4}+
\frac{28}{9}
\frac{\tilde{\lambda}^{3}(\beta)\zeta(3)}{2^{9}\pi^{4}}v^{6}.
\label{effective-potential}
\end{equation}
The first critical exponent one usualy computes is the $\beta$ exponent
which gives the power-law behavior of the order parameter with the
temperature. The critical temperature $\beta_{c}$ is
itself determined by the requirement of a vanishing thermal mass when
$\beta \rightarrow \beta_{c}$. In this sense the condition
$m_{10}(\beta_{c})=0$ gives
\begin{equation}
\beta_{c}^{2}=
\frac{1}{24}
\frac{\left( 4\tilde{\lambda}-{\tilde{g}}^{2}\right)}{\tilde{m}^{2}}.
\end{equation}
In order to have a critical temperature we will, for the moment, consider
the case $4\tilde{\lambda}>\tilde{g}^{2}$. Now, we can rewrite
$m_{10}^{2}(\beta)$ as
\begin{equation}
m_{10}^{2}(\beta)=
\frac{1}{24}\left( 4\tilde{\lambda}-\tilde{g}^{2} \right)
\left( \frac{1}{\beta^{2}}-\frac{1}{\beta_{c}^{2}} \right).
\end{equation}
As $\beta \rightarrow \beta_{c}$ we see that, by defining the reduced
temperature $\theta$ as
\begin{equation}
\beta=\frac{\beta_{c}}{1+\theta \beta_{c}},
\end{equation}
we obtain
\begin{equation}
m_{10}^{2}(\beta)=
\frac{1}{24}\left( 4\tilde{\lambda}-\tilde{g}^{2} \right)
\frac{2\theta}{\beta_{c}}+O(\theta^{2}) \sim a+b\theta,
\end{equation}
that is, $m_{10}^{2}(\beta)$ is linear in $\theta$, or equivalently,
we have for the first critical exponent the value $\beta=\frac{1}{2}$.
The second critical exponent, the $\delta$
critial exponent, is obtained from the inhomogenious gap equation,
when we consider an external source $J$ for the $\sigma$ field. In this case,
the gap equation becomes
\begin{equation}
m_{10}^{2}(\beta)v+\frac{\lambda_{10}(\beta)}{\beta}v^{3}+
\frac{28}{3}\frac{\tilde{\lambda}^{3}\zeta(3)}{2^{8}\pi^{4}}v^{5}=J,
\end{equation}
and, as we approach criticality $(m_{10}(\beta_{c}) \rightarrow 0)$,
the gap equation gives $\delta=3$, since $v^{3}$ dominates the $v^{5}$
term. Finally, as we are in the leading order in the $1/N$ expansion,
once again the two point function and the self-energy are the same.
This tells that the $\nu$ critical exponent is also equal to $\beta$
assuming the value $\frac{1}{2}$.
After all, to which universality class does the reduced theory belong?
As we have seen, all critical exponents are mean field
in the leading order of the $1/N$ expansion. Actually, one would get the
same critical exponents in this approximation in $d=3$
\cite{Rosenstein-Warr-Park}, although they would obviously not
satisfy hyperscaling relations in this dimension. Notice also that they
are almost as far from the scalar critical exponents $\beta \approx 0.3$
in three dimensions \cite{Amit}, as from the chiral critical exponents
computed in the previous section $\beta=1$ \cite{Gat-Vasilev}.
We would be tempted to conclude that dimensional reduction really spoiled
the critical behavior of thermodinamic quantities
\cite{Kocic-Kogut}, reforcing the thesis that this procedure is
ambigous. However, this sentence must be used with care.
The critical exponents we have just computed are related to an
effective field theory obtained from the integration over all fermions
and all nonstatic bosons. Since the critical indices come from the IR behavior
of mesurable quantities near the transition point and the integration we
have done gives IR finite results, it is reasonable to expect that the above
degrees of freedom will not improve mean field values for the exponents. In
order to recover non trivial values for the critical indices we must consider
the effects of the zero modes $\sigma_{0}$ and $\pi_{0}$ in the computation
of thermodynamic quantities, because they are the only responsible for
IR divergences in the finite-temperature version of this model.
Following the same steps of Kogut in \cite{Kocic-Kogut} we will consider
the critical behavior of the susceptibility $\chi$ for a $\sigma$ model
with action given by eq. (\ref{reduced-action}). Defining the critical
curvature $\mu^{2}_{c}$ as the point where the susceptibility diverges
\begin{equation}
\mu^{2}_{c}=\frac{\lambda_{10}(\beta)}{\beta}
\int^{\Lambda}\frac{d^{d-1}q}{(2\pi)^{d-1}}
\frac{1}{q^{2}},
\end{equation}
the expression for the inverse susceptibility can be recast into
\begin{equation}
\chi^{-1}
\left(
1+\frac{\lambda_{10}(\beta)}{\beta}\int^{\Lambda}
\frac{d^{d-1}q}{(2\pi)^{d-1}}
\frac{1}{q^{2}(q^{2}+\chi^{-1})}
\right)
=
\mu^{2}-\mu^{2}_{c}.
\label{susceptibility}
\end{equation}
Again, the extraction of the critical index $\gamma$ reduces to counting
the powers of the infrared singularities in the left hand side of the
previous equation. In three dimensions $(d-1=3)$ the second term in eq.
(\ref{susceptibility}) dominates the scaling region, giving the
zero temperature susceptibility exponent $\gamma=2/(d-3)$.
It is easy to obtain the other critical exponents
and they show the same type of behavior as $\gamma$. The reduced theory
has scalar critical indices and due to the dimensionality of the order
parameter we conclude that it belongs to the universality class
of the Heisenberg magnet, as reasonably expected.
\section{Vacuum structure at finite-temperature}
In the previous section, we defined the vacuum state in the $\sigma_{0}$
direction, breaking spontaneously a symmetry, and computed the critical
indices from the effective potential (\ref{effective-potential}), obtained
after dimensional reduction. Now comes the question. Is this really
a choice? As it will become clear soon, this is definitely not a choice.
If we still had the shape of a "mexican hat" for the effective potential
(see fig. 1), we would simply have to choose a prefered direction to be
the minimum configuration defining the vacuum of the theory. However,
differently from the zero temperature situation, this is no longer the case.
As we can see from expression (\ref{mass-counterterm}), the integration over
fermi fields give different thermal contributions for the masses of
$\sigma_{0}$ and $\pi_{0}$ and if we define the critical temperature as
the temperature in which we have a vanishing mass, we will conclude that,
since we have two different masses, we will have two critical temperatures
$\beta_{\sigma}^{-1}$ and $\beta_{\pi}^{-1}$ such that
\begin{figure}[hp]
\centerline{\epsfysize=4in\epsffile{pyu2.eps}}
\caption[region]
{\small\sf{Vacuum structure for $\beta>\beta_{\pi}$. We see that the ground state
is twice degenerate in the $\sigma_{0}$ direction while a false vacuum
(or metastable vacuum) can be identified in the $\pi_{0}$ direction.
After the identification of the true vacuum state we will no longer
have a Goldstone boson.}}
\end{figure}
\begin{equation}
\beta_{\sigma}^{2}=
\frac{1}{24}
\frac{\left( 4\tilde{\lambda}-{\tilde{g}}^{2}\right)}{\tilde{m}^{2}}
\end{equation}
and
\begin{equation}
\beta_{\pi}^{2}=
\frac{1}{24}
\frac{\left( 4\tilde{\lambda}+{\tilde{g}}^{2}\right)}{\tilde{m}^{2}}.
\end{equation}
Now, let us show that associated to each critical temperature
there is a second order phase transition and a symmetry restoring
phase. Indeed, if we go from zero temperature $(\beta = \infty)$ to a
temperature close to $\beta_{\pi}^{-1}$ the effective potential takes
the form of (fig. 2). There is a discrete degeneracy of the vacuum
showing that in the previous section we made a correct choice of the
ground-state.
If we continue increasing the temperature in such a way that
$\beta_{\sigma}<\beta<\beta_{\pi}$ we note (see fig. 3) that
the discrete symmetry in the $\pi_{0}$ direction is restored but
we continue with a broken phase in the $\sigma_{0}$ direction.
Complete restoration of symmetry (fig. 4) will only be achieved
when $\beta<\beta_{\sigma}$, and this is exactly the picture after
dimensional reduction since we have already taken the limit
$\beta \rightarrow 0$.
\begin{figure}[th]
\centerline{\epsfysize=4in\epsffile{pyu3.eps}}
\caption[region]
{\small\sf{Partial restoration of symmetry for $\beta_{\sigma}<\beta<\beta_{\pi}$.
The $\pi_{0}$ degree of freedom is now in the disordered phase while
we still have discrete breakdown of symmetry in the $\sigma_{0}$
direction.}}
\end{figure}
All the above discussion is valid only if $4\tilde{\lambda}>\tilde{g}^{2}$.
What happens when $4\tilde{\lambda}<\tilde{g}^{2}$? In this case
we will have only the $\beta_{\pi}$ critical temperature. This is to say,
the discrete symmetry on the $\pi_{0}$ direction will be restored while,
in the $\sigma_{0}$ direction, the minimum will become deeper and deeper
as we increase the temperature. We will not have a complete restoration of
symmetry and the effective potential will have the shape of (fig. 3).
\begin{figure}[th]
\centerline{\epsfysize=4in\epsffile{pyu4.eps}}
\caption[region]
{\small\sf{Complete disorder for $\beta<\beta_{\sigma}$. We see that, due to the
difference in the thermal masses of the static fields $\sigma_{0}$ and
$\pi_{0}$ this surface cannot be obtained by the revolution of a
generating curve.}}
\end{figure}
\section{Conclusions}
Dimensional reduction of the chiral-continous Gross-Neveu model was
performed giving an effective theory that belongs to the same universality
class of the Heisenberg magnet. In spite of the integration over fermions
and nonstatic bosons modes have given mean field results for the critical indices, we were able to explicitely determine the universality
class of the reduced theory by simply taking into account the effects of
the only relevant degrees of freedom in the computation of thermodynamic
quantities; the static Matsubara modes.
The presence of a coupling $i\gamma_{S}$ in the
original action is the responsible for the difference on the thermal
contributions for the masses of the pseudo-scalar and scalar fields
$\sigma_{0}$ and $\pi_{0}$. This difference has important implications.
First, we saw that if $4\tilde{\lambda}>\tilde{g}^{2}$
we have restoration of symmetry for some critical temperature, first
in the $\pi_{0}$ direction and after in the $\sigma_{0}$ direction,
while, if $4\tilde{\lambda}<\tilde{g}^{2}$ we have only partial restoration
of symmetry. The introduction of temperature in a theory with a continous
chiral symmetry has changed the role played by the $\pi_{0}$ field from the
zero temperature case, where it was a Goldstone boson, to the
finite-temperature case, where this is no longer true.
\section{Acknowledgment}
We are greatfull to C. de Callan, S. A. Dias and R. O. Ramos for
interesting discussions. This paper was suported by Conselho Nacional de
Desenvolvimento Cient\'{\i}fico e Tecnol\'ogico do Brasil (CNPq) and
Funda\c c\~ao Coordenadoria de Aperfei{\c c}oamento de Pessoal de
N\'{\i}vel Superior (CAPES).
\section*{Appendix A - Thermal RG flow}
We compute, for completeness, the thermal renormalization group
functions which controlls the dependence of the counterterms
on the temperature. As it is well known \cite{Landsman}, the transition
from a four-dimensional theory to an effective three-dimensional one is
renormalization point dependent. This is not surprising since the
Appelquist-Carazone decoupling theorem \cite{Appelquist-Carazone}
only holds if a particular class of renormalization prescriptions
is adopted, being optimal in BPHZ subtractions at zero momenta and
temperature $\beta$. Neverthless, we can write down the thermal
renormalization group equation which controls the dependence with
the temperature on the correlation functions and compute the
running quantities.
The independence of bare correlation functions on the choice of the
renormalization point, is expressed by the conditions
\begin{eqnarray}
\mu\frac{d}{d\mu}\Gamma^{(N)}=0, \\
T\frac{d}{dT}\Gamma^{(N)}=0,
\end{eqnarray}
and, for simplicity, we will work, in this section, with the temperature
$T$ instead of $\beta \equiv 1/T$.
Since we are only interested on the computation of thermal
renormalization group functions, we will consider solely
\begin{equation}
\left(
T_{0}\frac{\partial}{\partial T_{0}}+
\beta_{T}\frac{\partial}{\partial\lambda}+
\gamma_{T}m^{2}\frac{\partial}{\partial m^{2}}
\right)
\Gamma^{(N)}=0
\label{rg-equation}
\end{equation}
where
\begin{equation}
\beta_{T}=T_{0}\frac{\partial\lambda(T)}{\partial T_{0}},
\end{equation}
\begin{equation}
\gamma_{T}=m^{-2}T_{0}\frac{\partial m^{2}(T)}{\partial T_{0}},
\end{equation}
and
\begin{equation}
\Gamma^{(N)}=
\left.
\frac{\partial^{N}V_{\mbox{eff}}}{\partial\sigma_{0}^{N}}
\right|_{\sigma_{0}=0}.
\end{equation}
Using for the effective potential the expression (\ref{effective-potential})
we easily get
\begin{equation}
\beta_{T}=
\frac{1}{16\pi^{2}}
\left(
\frac{5}{12}\tilde{\lambda}^{2}+\tilde{g}^{4}
\right),
\label{beta-function}
\end{equation}
and
\begin{equation}
\gamma_{T}=
\frac{1}{12}
\left(
4\tilde{\lambda}-\tilde{g}^{2}
\right)
\frac{T^{2}}{m^{2}}+
\frac{\tilde{\lambda}}{4\pi^{2}}.
\label{gama-function}
\end{equation}
Equation (\ref{rg-equation}) can be solved by the method of characteristics.
One simply introduces a dilatation parameter $t$ and look for functions
$\lambda(t)$ (for fixed $g$) which satisfies
\begin{equation}
t\frac{d\lambda(t)}{dt}=\beta(\lambda(t)),
\end{equation}
with $\lambda(1)=\lambda$. If we now solve the above equation we get
\begin{equation}
\int_{\lambda}^{\lambda(t)}\frac{d\lambda^{'}}{\beta(\lambda^{'})}=ln{t}.
\label{running-coupling}
\end{equation}
To investigate the large $T$ limit we will have to study the behavior of the
effective coupling $\lambda(t)$ as $t$ goes to zero. Since $\beta(\lambda)$
is positive $\lambda(t)$ will decrease. Moreover, since the slope of
the $\beta$ function is also positive, the gaussian IR fixed point is
attractive.
We can solve a similar equation to $g$ and defining
$(\lambda^{*},g^{*})$ as the zero of the $\beta$ function
\begin{equation}
\beta(\lambda^{*},g^{*})=0,
\end{equation}
we see that, although the anomalous dimension for the $\sigma_{0}$ field
may change when we pass from the situation of total to the
situation of partial restoration of symmetry, the $\eta$
exponent defined as $\eta\equiv\eta(\lambda^{*},g^{*})$
remains unchanged $(\eta=0)$.
One may ask whether renormalization group improves the critical
indices after dimensional reduction or not. The answer is no and
this should not be surprising since the form of the ultraviolet and
thermal divergences that we had to renormalize are of the same type
as for a theory in four dimensions, where everything is mean field.
|
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| 3,964
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Covid-19 & Building Access Updates
Expand menu for About
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Amy Forsyth
Anna Chupa
Anthony Viscardi
Christine Ussler
Deirdre Murphy
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Jason Travers
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Lehigh University Art Galleries
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Linear Thinking
Jason Travers has been a faculty member in Art, Architecture and Design since 1999. Travers has instructed courses including Two-Dimensional Design, Drawing, Painting, Color Theory , Digital Foundations and Graphic Design. His work has been featured in numerous exhibitions both locally and nationally. Travers was selected as the Artist-in-Residence for Acadia National Park in Maine in summer 2014.
Current: The improvisational work I create strives to capture the transcendence of a fleeting moment. It is the mystery of the unknown that thrills me in this process. As an instructor, it is equally thrilling to see students share the same wonder of discovery.
As a young art student, I can recall my first visit to the first exhibition of work by the department faculty. I remember being confounded by the apparent disconnect that seemed to be prevalent in the work. These same professors who had profoundly influenced my development with heady intellectual dialogue and thoughtful critical analysis had by all appearances foregone all rational approaches in their own studio work. As an early proponent of craft and technical mastery as the pinnacle of fine art, I was perplexed by the sloppy abstraction and puzzling conceptual works that I encountered.
As my own work began to evolve during my graduate studies, the philosophical nature of the work was challenged by my critics. I had been trained to be a good student, employing the suggestions of my instructors without question and striving for a conclusive product that, in my mind, represented a successful piece of art. With my "academic" approach to art-making under fire, I was charged with the task of uncovering what was at the very heart of my desire to create. As my search gravitated toward a more visceral approach to process, I had to abandon the intellectual and logical or at least allow those sensibilities to function in a more subconscious way.
Essentially to progress artistically, we must forget what we already know and find fresh challenges in rediscovery. I think as a whole, the faculty in Art, Architecture and Design are unified in presenting the importance of process and creative discovery as the corner stone of our teaching philosophy, not just as a means to an end but as work of art in its own right. Although my foundation courses introduce a traditional approach to design theory, it is ultimately the first step in the creation of a unique and individual creative voice.
The "constructive imagination" that we strive to develop in our students provides a means of problem solving that is not just limited to the studio. It is a philosophy that encompasses the importance of all experiences, for better or worse, and the lessons and growth they provide. It is through this trial and error that we develop our unique vision and the ability to approach creative problem solving in our own distinctly personal way.
Students need to know that it is acceptable, and even imperative, to explore non-traditional ways to expand their creativity. It's this spirit of invention that energizes the classroom for both student and instructor alike. The excitement of a creative revelation in the classroom frequently results in a burst of inspiration in my own studio. The improvisational work I create strives to capture the transcendence of a fleeting moment. It is the mystery of the unknown that thrills me in this process. As an instructor, it is equally thrilling to see students share the same wonder of discovery.
The artist and teacher John Baldessari is quoted as saying that he prefers the term "younger artists" as opposed to students. By distinguishing our younger artists as peers, we welcome their insight and feed off of their passion. It is only our years of experience that separate us, as we share the same joys and frustrations in the creative process and continuously seek to find our own way by forging into the unknown.
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{"url":"https:\/\/blog.csdn.net\/Yasola\/article\/details\/78256950","text":"# HDU 5917 Instability\uff08Ramsey\u5b9a\u7406\uff09\n\nInstability\n\nTime Limit: 2000\/1000 MS (Java\/Others) Memory Limit: 65536\/65536 K (Java\/Others)\nTotal Submission(s): 592 Accepted Submission(s): 218\n\nProblem Description\nLong long ago, there was a prosperous kingdom which consisted of n cities and every two cites were connected by an undirected road.\n\nHowever, one day a big monster attacked the kingdom and some roads were destroyed. In order to evaluate the influence brought by the catastrophe, the king wanted to know the instability of his kingdom. Instability is defined as the number of the unstable subset of {1, 2,\u22ef,n}.\n\nA set S is unstable if and only if there exists a set A such that A\u2286S(|A|\u22653) and A is a clique or an independent set, namely that cites in A are pairwise connected directly or they are pairwise disconnected.\n\nArchaeologist has already restored themroads that were not destroyed by the monster. And they want you to figure out the instability.\n\nSince the answer may be tremendously huge, you are only required to write a program that prints the answer modulo 1000000007.\n\nInput\nThe first line contains only one integer T, which indicates the number of test cases.\n\nFor each test case, the first line contains two integers n (3\u2264n\u226450) and m (1\u2264m\u2264n(n\u22121)\/2), indicating the number of cities and the number of roads.\n\nThen the following are m lines, each of which contains two integers x and y, indicating there is a road between the city x and the city y.\n\nIt is guarenteed that there does not exist a road connecting the same city and there does not exist two same roads.\n\nOutput\nFor each test case, print a line \u201cCase #x: y\u201d, where x is the case number (starting from 1) and y is an integer indicating the instability modulo 1000000007.\n\nSample Input\n2\n4 3\n1 2\n2 3\n1 3\n3 0\n\nSample Output\nCase #1: 2\nCase #2: 1\n\nHint\n\n\u2022 In the first example, {1,2,3} and {1,2,3,4} , containing the subset {1,2,3} which is connected\ndirectly, are considered unstable.\n\u2022 In the second example, {1,2,3} is considered unstable because they are not pairwise connected\ndirectly.\n\nSource\n2016\u4e2d\u56fd\u5927\u5b66\u751f\u7a0b\u5e8f\u8bbe\u8ba1\u7ade\u8d5b\uff08\u957f\u6625\uff09-\u91cd\u73b0\u8d5b\n\nRecommend\nwange2014\n\n### \u9898\u76ee\u5927\u610f\n\n\u6709\u4e00\u4e2aV(3V50)$V(3 \\leqslant V \\leqslant 50)$\u4e2a\u70b9\u7684\u56fe\uff0c\u95ee\u6709\u591a\u5c11\u4e2a\u96c6\u5408\u81f3\u5c11\u5305\u542b\u4e00\u4e2a\u4e09\u5143\u73af\u6216\u4e09\u4e2a\u70b9\u4e4b\u95f4\u6ca1\u6709\u4efb\u4f55\u8fb9\u7684\u8865\u4e09\u5143\u73af\u3002\n\n### \u89e3\u9898\u601d\u8def\n\nRamsey\u5b9a\u7406\uff1a6 \u4e2a\u4eba\u4e2d\u81f3\u5c11\u5b58\u57283\u4eba\u76f8\u4e92\u8ba4\u8bc6\u6216\u8005\u76f8\u4e92\u4e0d\u8ba4\u8bc6\u3002\n\u6839\u636eRamsey\u5b9a\u7406\u53ef\u4ee5\u77e5\u9053\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u7b97\u51fan>5$n>5$\u7684\u96c6\u5408\u6570\u3002\u5bf9\u4e8e3V5$3 \\leqslant V \\leqslant 5$\u7684\u60c5\u51b5\uff0c\u7531\u4e8eV$V$\u53ea\u670950\uff0c\u6240\u4ee5\u53ef\u4ee5\u66b4\u529b\u6c42\u89e3\u3002\n\n### AC\u4ee3\u7801\n\n#include<iostream>\n#include<cstdio>\n#include<cstring>\n#include<string>\n#include<queue>\n#include<set>\n#include<vector>\n#include<map>\n#include<stack>\n#include<cmath>\n#include<algorithm>\nusing namespace std;\ntypedef long long LL;\n\nconst int MAXV=50+3;\nconst int MOD=1000000007;\nint V, E;\nbool maze[MAXV][MAXV];\n\nvoid init()\n{\nfor(int i=0;i<V;++i)\nfor(int j=0;j<V;++j)\nmaze[i][j]=false;\n}\n\ninline bool judge(int a, int b, int c)\/\/\u5224\u5b9a\u4e09\u4e2a\u70b9\u662f\u5426\u6784\u6210\u4e09\u5143\u73af\u6216\u76f8\u4e92\u72ec\u7acb\n{\nreturn maze[a][b] && maze[a][c] && maze[b][c] || !maze[a][b] && !maze[a][c] && !maze[b][c];\n}\n\ninline void mod_add(int &a, const int &b)\n{\na+=b;\nif(a>=MOD)\na-=MOD;\n}\n\nint main()\n{\nint T_T;\nscanf(\"%d\", &T_T);\nfor(int cas=1;cas<=T_T;++cas)\n{\nscanf(\"%d%d\", &V, &E);\ninit();\nfor(int i=0;i<E;++i)\n{\nint u, v;\nscanf(\"%d%d\", &u, &v);\n--u;\n--v;\nmaze[u][v]=maze[v][u]=true;\n}\nint ans=V>5?((1ll<<V)-1-V-V*(V-1)\/2-V*(V-1)*(V-2)\/6-V*(V-1)*(V-2)*(V-3)\/24-V*(V-1)*(V-2)*(V-3)*(V-4)\/120)%MOD:0;\/\/\u5982\u679cV>5\uff0c\u8ba1\u7b97\u51fan>5\u7684\u6240\u6709\u96c6\u5408\nfor(int i=0;i<V;++i)\nfor(int j=i+1;j<V;++j)\nfor(int k=j+1;k<V;++k)\n{\nif(judge(i, j, k))\n{\ncontinue;\n}\nfor(int x=k+1;x<V;++x)\n{\nif(judge(x, i, j) || judge(x, i, k) || judge(x, j, k))\n{\ncontinue;\n}\nfor(int y=x+1;y<V;++y)\nif(judge(y, i, j) || judge(y, i, k) || judge(y, i, x) || judge(y, j, k) || judge(y, j, x) || judge(y, k, x))\n}","date":"2019-04-24 22:08:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 4, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.21382936835289001, \"perplexity\": 4799.771370101093}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-18\/segments\/1555578663470.91\/warc\/CC-MAIN-20190424214335-20190425000335-00196.warc.gz\"}"}
| null | null |
State of The Arc
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2019 Summit of Hope
The Arc – Jefferson, Clear Creek & Gilpin Counties began in 1960 with a meeting of local parents of children with developmental disabilities. These parents wanted to create opportunities to raise their children at home instead of institutions, for their children to have access to education, and for their children to be valued in the community. From that initial meeting, the parents voted to form an affiliated chapter of the National Association for R* Children. We were granted 501(c)(3) status in 1961 and finalized our incorporation in 1963. Over the years we have gone through numerous name changes, especially as the stigma of the r-word increased. In 2011, our members voted to change our name to "The Arc – Jefferson, Clear Creek & Gilpin Counties" aligning with the naming standards of our national affiliate.
For nearly 50 years, the core values of our founding members have continued to inform all of our organization's activities. We have promoted reform in schools, the workplace, residential life, and in the broader community, all with the intent to promote independence and inclusion for people with developmental disabilities. The Arc – Jefferson, Clear Creek & Gilpin Counties promotes inclusion for people with developmental disabilities through advocacy, education, resources, community-building, policy change, and family support. We serve individuals of all ages who reside in Jefferson, Clear Creek and Gilpin Counties. We assist individuals and their caregivers with issues related to education, health care, residential options, employment, legal concerns, and systems navigation, as well as social, recreational, and religious inclusion.
*Editor's Note: Much has changed since The Arc in Jefferson County began its journey 50 years ago, including the language that we use to describe people with intellectual and developmental disabilities. Though we deeply believe in the use of People First language today, it was unheard of all those years ago. In the context of the articles below you will find that we have made some modifications to the language used. The words "retarded" and "retardation" are completely unacceptable today and have been replaced in the text by the use of "r*".
History Part 1: The Birth of a Movement
History Part 2: 1964-1979
History Part 4: The last 20+ Years
Thank You To Our 2019 Summit of Hope Sponsors
13949 W. Colfax Ave., Ste. 150
Denver West Building 1
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© The Arc - Jefferson, Clear Creek & Gilpin Counties 2020
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El cantón de Basilea-Campiña o Basilea-Campaña (en alemán: Basel-Landschaft; en francés: Bâle-Campagne; en italiano: Basilea Campagna; en romanche: Basilea-Champagna) es un cantón suizo cuya capital es Liestal.
Historia
El semicantón de Basilea-Campiña formaba, junto con el semicantón de Basilea-Ciudad, el cantón de Basilea. En la época romana el territorio de Basilea era un centro de actividades romano. En las cercanías se han encontrado restos de una población en la que vivieron unas 20.000 personas alrededor del año 200 d. C., llamada Augusta Raurica, que comprende entre otras cosas un anfiteatro y la reconstrucción de un pueblo romano. Actualmente el museo es visitado por cerca de 140.000 personas al año.
El semicantón de Basilea-Campiña es resultado de los territorios conquistados por la ciudad de Basilea, su antigua capital a finales del siglo XVI. Tras la llegada de Napoleón en 1798, la Campiña se alía con la ciudad. Desde el punto de vista económico, la Campiña dependía de la ciudad, quizás por el bajo nivel educativo en las áreas rurales. La ciudad de Basilea fue y sigue siendo el centro económico y cultural de los dos cantones.
En 1830, después de que llegase la calma a la región, las discusiones políticas comenzaron en el cantón. Algunos sectores de la población de la Campiña se sublevaron contra el poder central del cantón, alegando que el área rural había sido olvidada, por lo que el 26 de agosto de 1833 se consumó la separación en Basilea-Ciudad y Basilea-Campiña.
A principios de 1900 se empezó a gestar en los dos cantones un movimiento de reunificación cuando se industrializó la Campiña. Las dos mitades del cantón acordaron entonces la reunificación del cantón de Basilea, pero en 1969 la población de Basilea-Campiña votó en un referéndum a favor de mantener la independencia. Bajo un referéndum en 1994 el distrito de Laufen paso de pertenecer del Cantón de Berna al cantón de Basel - Landschaft.
Se cree que las diferencias económicas entre los dos cantones fueron la razón que hizo cambiar de opinión a la población.
Geografía
El semicantón de Basilea-Campiña se encuentra en el norte de Suiza. Limita con el semicantón de Basilea-Ciudad al norte, Soleura al sur, Jura al oeste y Argovia al este, con Alemania y Francia al norte. Las montañas del Jura lo atraviesan.
Economía
Los principales productos agrícolas del cantón son: fruta, productos caseros y cría de ganado bovino. Los sectores industriales más importantes son textil, metalúrgico y químico.
Desde el siglo XVII hasta finales del XX, la agricultura era de mucha importancia para el cantón; pero a partir de 1850, varias industrias, entre las cuales cabe destacar la química y la farmacéutica, se instalaron en Basilea, convirtiéndola en una de las zonas más ricas de Suiza.
Regiones
Las nueve comunas del distrito de Arlesheim pertenecían a la diócesis de Basilea. En 1792 las tropas francesas ocuparon el distrito y en 1793 estas tierras fueron anexadas a Francia. En 1815 tras el Congreso de Viena, el distrito se une a Basilea.
El distrito de Laufental tiene una historia parecida a la de Arlesheim. La diferencia es que en 1815, luego del congreso de Viena, el Laufental fue unido al cantón de Berna. En 1979, con la creación del Cantón del Jura, el distrito de Laufental se vuelve un exclave del cantón de Berna, por lo que en 1980 los habitantes del distrito deciden unirse al cantón de Basilea-Campiña, unión que entró en vigor el 1° de enero de 1994, tras un gran proceso administrativo.
Distrito de Arlesheim, con capital en Arlesheim
Distrito de Laufen, con capital en Laufen
Distrito de Liestal, con capital en Liestal
Distrito de Sissach, con capital en Sissach
Distrito de Waldemburgo, con capital en Waldenburg
Demografía
La población es de lengua alemana. La mayoría son protestantes.
Referencias
Enlaces externos
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\section{Introduction}
\label{Sec:Introduction}
The nuclear matter equation of state (EoS) was investigated recently with respect to various applications such as astrophysics of compact stars, the structure of nuclei, and heavy ion collisions (HIC).
The derivation of the EoS from a microscopic description has several difficulties:
One has to deal with a strongly interacting fermion system with arbitrary degeneracy.
Correlation effects, such as bound state formation and quantum condensates, have to be treated.
Effective nucleon-nucleon interactions are used, which are reconstructed from measured properties
and are possibly dependent on density and energy.
Different approximations are known which describe nuclear matter in limiting cases,
such as at low densities or at near-saturation density,
but a theory applicable in a wide region of temperature and density is needed.
Often, the processes under consideration are in non-equilibrium and are spatially inhomogeneous
so that the assumption of local thermodynamic equilibrium becomes questionable.
We investigate nuclear matter in thermodynamic equilibrium, confined in the volume $V$ at temperature $T$.
We are interested in the subsaturation region where the baryon density $n=n_n+n_p\leq n_{\rm sat}$
with the saturation density $n_{\rm sat} \approx 0.15$ fm$^{-3}$,
the temperature $T \stackrel{\scriptstyle <}{\phantom{}_{\sim}} 20$ MeV,
and the proton fraction $Y_p=n_p/n$ between 0 and 1.
As long as weak processes leading to $\beta$-equilibrium are suppressed, the number of neutrons and protons
$N_\tau = n_\tau V$ are independent variables, $\tau =n,p$.
This region of warm dense matter is of interest for nuclear structure calculations
and HIC
explored in laboratory experiments \cite{Natowitz}, and also for astrophysical applications.
For instance, core-collapse supernovae at the post-bounce stage are evolving within this region of the phase diagram \cite{Tobias},
and different processes such as neutrino emission and absorption, which strongly depend on the composition of warm dense matter,
influence the mechanism of supernova explosions.
Different approaches have been worked out to describe warm
nuclear matter.
At low densities, the formation of bound states is of relevance. The simple mass action law \cite{RMS,NSE}
gives already the nuclear statistical equilibrium (NSE). The exact behavior of the EoS
in the low-density limit is obtained
from the second virial coefficient \cite{BU,RMS,SRS,HS}. With increasing density, medium effects must be considered.
Other approaches have been worked out to treat dense nuclear systems.
For instance, the relativistic mean-field (RMF) approach \cite{Walecka} has been widely recognized to be powerful
in describing nuclear systems.
Properties near the saturation density are used to fix the parameters of the RMF approach,
see, e.g., Ref.~\cite{Typel1999}.
An alternative approach to describe the properties of nuclear matter near the saturation density
is based on the theory of normal Fermi liquids built up by Landau \cite{L56} which is designed for strongly degenerate systems.
The application of the Fermi-liquid approach to nuclear systems was developed by Migdal \cite{Mig}, see also \cite{Mig1}.
The approach describes low-lying excitations by several phenomenological Landau-Migdal parameters.
Pomeranchuk \cite{Pom58} has shown that Fermi liquids are stable only if some inequalities on the values of the Landau-Migdal parameters are fulfilled.
In a recent work \cite{KV16}, low-lying scalar excitation modes
in cold normal Fermi liquids have been investigated
for various values
of the scalar Landau-Migdal parameter $f_0$ in the particle-hole channel.
The stability of nuclear matter was then discussed.
After performing the bosonization of the local interaction, the possibility of Bose condensation of scalar quanta has been suggested that may result in the appearance of a novel metastable state in dilute nuclear matter.
The RMF approach as well as the Landau Fermi-liquid approach are based on a single-nucleon quasiparticle concept. In both cases it is not simple to introduce the formation of clusters, in particular of light elements with mass number $A \le 4$.
A generalization of the RMF approach has been proposed \cite{Typel,ferreira12,PCP15},
where light elements $^2$H ($d$), $^3$H ($t$), $^3$He ($h$), $^4$He ($\alpha$) are considered as new degrees of freedom in the effective Lagrangian.
The coupling constants of the light clusters to the meson fields are adapted from other theories,
in particular the quantum statistical (QS) approach.
Within the generalized RMF approach, the second virial coefficient is reproduced \cite{VT} fitting special terms in the EoS.
A similar problem arises also in the Landau Fermi-liquid approach. In Fig. 5 of Ref. \cite{KV16}, the energy as function of the baryon density is shown, taking into account the possibility of the Bose condensation of the scalar quanta.
Compared with the result of the original Fermi liquid theory, at low densities the energy might be shifted downwards due to Bose condensation. However, clustering is not included and the low-density virial expansion for finite temperature is not correctly reproduced. Until now, no systematic approach is known how to incorporate bound state formation in the Landau Fermi-liquid approach.
In contrast to these single-nucleon mean-field theories, a QS approach \cite{RMS} describes the formation of clusters
and their dissolution at increasing densities (Mott effect) in a systematic way.
Light elements are considered as quasiparticles in the corresponding few-nucleon spectral function.
Medium-dependent quasiparticle energies are given in Ref. \cite{R}.
The corresponding EoS \cite{Typel} interpolates between both limiting cases, the mean-field approach near the saturation density, and the low-density region where clusters are significant.
In the present work we will treat the problem how the Landau Fermi-liquid approach can be improved to include the formation of clusters. Although
the Fermi-liquid approach can be applied to non-equilibrium and inhomogeneous systems and to the systems with Cooper pairing, cf. \cite{Voskresensky:1993ud,Kolomeitsev:2010pm},
within this work we focus on equilibrium properties of the normal homogeneous nuclear matter.
We show in our dynamic structure factor approach that the low-density limit is correctly described. Formation of light clusters, in particular the second virial coefficient, are reproduced.
In detail, we\\
(i) investigate the polarization function (Fourier transform of the van Hove time-dependent pair correlation function) in relation to the EoS and establish the connection of the QS approach with the Fermi-liquid approach,\\
(ii) consider the dynamic structure factor as the central quantity which contains not only thermodynamic information, but also transport and kinetic properties,\\
(iii) introduce the formation of clusters and the inclusion of continuum correlations, \\
(iv) discuss the stability with respect to phase transitions.\\
Bosonization and Bose condensation together with metastability \cite{KV16} will be considered in a subsequent work.
We compare in this work two different approaches to the thermodynamical properties of nuclear matter:
Firstly, we consider the density equation of state based on the single-particle Green function,
and secondly the isothermal compressibility related to the dynamic structure factor, related to the density-density correlation function.
We investigate the applicability of the quasiparticle approximation (QPA) and the Fermi-liquid approach,
and compare results for the relativistic mean-field approximation with those of the Fermi-liquid approach.
Then, cluster formation is treated, and the Beth-Uhlenbeck formula is obtained in dilute warm nuclear matter
as well as the isothermal compressibility is considered.
Accurate results for the incompressibility are given valid in the low-density limit.
The paper is organized as follows.
In Sec. \ref{Isothermal} we elaborate the fluctuation-dissipation theorem.
Simplifying we consider cases of pure neutron matter and isospin-symmetric matter.
We relate the dynamic
and static structure factors, expressed in terms of
diagrams in the Matsubara or the Schwinger-Kadanoff-Baym-Keldysh techniques,
to the isothermal compressibility.
In Sec. \ref{Examples} and appendix \ref{sec:spectral} we consider special approximations, in particular the limiting cases of the perturbative loop diagram,
the one -- loop contribution with full and quasiparticle Green functions, the RMF approximation,
and the RPA resummation.
In Sec. \ref{sec:Fermi} we derive the static structure factor within the Fermi-liquid approach.
In Sec. \ref{RMF} we describe density-density correlations within the RMF approximation and recover the Landau-Migdal parameter in the scalar channel.
Generalizations to systems with arbitrary isospin composition are performed in Sec. \ref{arbitrary}.
Then in Sec. \ref{two} we study two-particle correlations.
We present the Beth-Uhlenbeck approach for the second virial coefficient and discuss the cluster decomposition of the polarization function.
In Sec. \ref{sec:NSE} we show how the nuclear statistical equilibrium model appears in the low-density limit.
We discuss the corrections at increasing density and give concluding remarks in Sec. \ref{Conclusions}.
In this work we use units $\hbar =c=k_{\rm B}=1$.
\section{Isothermal compressibility and the dynamic structure factor}
\label{Isothermal}
\subsection{The fluctuation-dissipation theorem}
\label{FDT}
Starting from a given nucleon-nucleon interaction potential, quantum statistics gives the possibility to derive thermodynamic potentials and related thermodynamic properties.
Using the technique of thermodynamic Green functions \cite{AGD},
different approaches are known to calculate an EoS.
For instance, one can evaluate expressions for the pressure $p(T,\mu_n,\mu_p)$,
or for the neutron and proton densities $n_\tau (T,\mu_n,\mu_p)$ for nuclear matter \cite{R}.
Here, $\mu_n, \mu_p$ denote the chemical potentials of neutrons and protons, respectively; $\tau =\{n,p\}$.
For instance, one can start from the expression for the
density of the fermion species (density EoS)
\begin{equation}
\label{neos}
n_\tau(T,\mu_n,\mu_p)= \frac{1}{V}\sum_{{\bf p},\sigma}\int_{-\infty}^\infty \frac{d \omega}{2 \pi}f(\omega-\mu_\tau)
A_\tau(\omega,{\bf p}; \mu_n,\mu_p)
\end{equation}
with the Fermi function
\begin{equation}
\label{Fermi}
f(z) =\frac{1}{e^{z/T}+1}
\end{equation}
and the single-particle spectral function $A_\tau(\omega,{\bf p}; \mu_n,\mu_p)$, see \cite{AGD}. In infinite matter we replace the sum over the single-particle states $p=\{ {\bf p},\sigma\}$ by $g_\tau V \int d^3p/(2 \pi)^3$, where $g_\tau$ is the spin degeneracy factor, $\bf p$ is the wave number vector.
Within the Matsubara Green function method, the spectral function is related to the self-energy, and systematic
approaches are available to calculate this EoS in an appropriate approximation.
This way, thermodynamic potentials have been derived for nuclear systems and further thermodynamic properties including composition, phase transitions, etc., have been considered.
Note that in the case of phase transition not only the baryon density $n$, but also the proton fraction $Y_p$ may be different in the different phases. The total baryon number is conserved, in our case $N=N_p+N_n$.
A related approach is based on the real time non-equilibrium Green function technique
\cite{KB,Keldysh}.
The Wigner transformed Green functions and corresponding self-energies
allow to describe slightly non-uniform systems involved in slow collective motions.
The technique is also convenient to describe systems in thermal equilibrium. Here, the Wick rotation is not required. All the non-equilibrium Green functions and self-energies are expressed via the retarded quantities.
General issues of the quantum kinetic approach were reviewed in \cite{Kolomeitsev:2013du}.
In Ref. \cite{Voskresensky:1987hm} the QPA in thermal equilibrium was applied within the non-equilibrium Green function technique. Within the QPA for the
nucleon Green functions the formalism has been applied to special problems in nuclear systems, cf. \cite{Voskresensky:2001fd}. The general formalism developed in \cite{Knoll:1995nz}
allows to account for the finite damping width of the
source particles due to their finite mean free path in matter.
To simplify the consideration we first consider the case of one baryon species that holds for the pure neutron matter (degeneracy factor $g=2$) and isospin-symmetric matter (degeneracy factor $g=4$ including isospin). Generalizations for systems with arbitrary isospin composition will be presented below.
In the present work,
the inclusion of few-nucleon correlations and the possible formation of bound states
is of interest. To evaluate thermodynamic properties of nuclear systems,
we investigate the isothermal compressibility $\kappa_{\rm iso}$,
\begin{equation}
\label{kappa}
\kappa_{\rm iso}(T,\mu)=-\frac{1}{V}\frac{\partial V}{\partial p}{\Big |}_T=
\frac{1}{n^2} \frac{\partial n}{\partial \mu}{\Big |}_T\,,
\end{equation}
where $\mu$ is the baryon chemical potential.
If this quantity is known, we can derive the EoS: we integrate
$n(T,\mu)$
at fixed
$T$ to obtain the pressure
\begin{equation}
\label{pressure}
p(T,\mu)=\int_{-\infty}^\mu n(T,\mu')d \mu'
\end{equation}
as thermodynamic potential.
The incompressibility is defined as
\begin{equation}
\label{incompr}
K(T, n)=\frac{1}{n\, \kappa_{\rm iso}}=n \frac{\partial \mu}{\partial n}\Big|_T.
\end{equation}
The isothermal compressibility is related to another fundamental quantity of the many-body
system, the dynamic structure factor.
\begin{equation}
S({\bf q},\omega)=\frac{1}{2 \pi V} \int_{-\infty}^\infty dt \langle \rho_{\bf{q}}^+(t) \rho_{\bf q}(0)\rangle e^{{\mathrm i} \omega t}\,.
\end{equation}
Here, the wave number dependent density fluctuation
\begin{equation}
\label{densityrhoq}
\rho_{\bf q}=\int d^3r\, e^{{\mathrm i} {\bf q} \cdot {\bf r}} \sum_\nu\psi_\nu^\dagger({\bf r}) \psi_\nu({\bf r})
\end{equation}
is the Fourier transform of the particle density. The intrinsic quantum numbers are denoted by
$\nu = \{\sigma, \tau\}$, and
$\psi_\nu^\dagger({\bf r}),\psi_\nu({\bf r})$ are the corresponding creation and annihilation operators.
The time dependence is according to the Heisenberg picture. The average $\langle \dots
\rangle={\rm Tr}\{ \rho_{\rm eq} \dots \}$ is performed with the equilibrium statistical operator, in our case the grand canonical statistical operator
\begin{equation}
\rho_{\rm eq}(T, \mu_\nu)
=\frac{e^{-(H-\sum \mu_\nu N_\nu)/T}}{{\rm Tr}\,e^{-(H-\sum \mu_\nu N_\nu)/T}},
\end{equation}
with the particle number operators $N_\nu=\int d^3r \,
\psi_\nu^\dagger({\bf r}) \psi_\nu({\bf r})$.
From the dynamic structure factor, the static structure factor,
\begin{equation}
S({\bf q})=\int \frac{d \omega}{2 \pi}\, S({\bf q},\omega),
\end{equation}
is derived,
describing equal-time fluctuations of the baryon density
$\hat n({\bf r})=\sum_\nu\psi_\nu^\dagger({\bf r}) \psi_\nu({\bf r})$.
The generalization to a multicomponent system where partial structure factors can be introduced, will be discussed below, see Eq. (\ref{partSF}).
In equilibrium systems where $n=\langle \hat n({\bf r})
\rangle$, the static structure factor can be rewritten in terms of the density derivative of the baryon chemical potential at fixed temperature \cite{LL1980}, Sect. XII,
\begin{eqnarray}\label{EqStstr0}
S({\bf q}\to 0)=(\langle \hat n^2({\bf r})
\rangle-n^2)/n=T\left\{\left(\frac{\partial \mu}{\partial n}\right){\Big |}_T\right\}^{-1}\,.
\end{eqnarray}
There are other quantities which are related to
$S({\bf q},\omega)$
so that the investigation of this quantity is of fundamental interest.
According to the fluctuation-dissipation theorem, the dynamic structure factor
is related to the response function $\chi({\bf q},\omega)$ defined by the density fluctuation induced by an external potential
$U({\bf q},\omega)$ as $\langle \rho_{\bf q}(\omega) \rangle=\chi({\bf q},\omega) U({\bf q},\omega)$.
The response function is related to the thermodynamic density-density Green function
\begin{equation}
L(1,2;1^+,2^+) = \frac{V}{{\mathrm i}^2}
\langle {\rm T}\{\psi^\dagger(1^+)\psi(1) \psi^\dagger(2^+) \psi(2) \}\rangle-
\frac{V}{{\mathrm i}^2} \langle \psi^+(1^+)\psi(1) \rangle \langle \psi^+(2^+) \psi(2) \rangle\,,
\end{equation}
with $1 = \{{\bf r}_1,\sigma_1,\tau_1\}= \{{\bf r}_1,\nu_1\}$. As well-known from the technique
of thermodynamic Green functions \cite{KB,AGD}
we introduce the Heisenberg-like dependence on the parameter $x$ according to
\begin{equation}
A(x)= e^{x (H-\sum \mu_\nu N_\nu)}Ae^{-x (H-\sum \mu_\nu N_\nu))};
\end{equation}
the ${\rm T}\{\dots \}$-product denotes the ordering of operators
with growing parameter values $x$
from right to left. After Fourier transformation with respect to spatial distances and the
parameter $x$,
\begin{equation}
\label{Lqo}
L({\bf q},{\mathrm i} z_\lambda)=\int d^3 r\int_0^{1/T} d x
e^{{\mathrm i} {\bf q} \cdot ({\bf r}_2-{\bf r}_1)}
e^{{\mathrm i} z_\lambda (x_2-x_1)} L(1,2;1^+,2^+)
\end{equation}
is a function defined at the bosonic Matsubara frequencies $z_\lambda=\pi \lambda T$,
where $\lambda = 0,\,\, \pm 2, \dots$ are the even numbers.
Analytical continuation from ${\mathrm i} z_\lambda$ into the whole complex $z$-plane
gives the corresponding damping width (spectral function of the van Hove-function $L$)
\begin{equation}
\label{Spectral}
\Gamma_{L} ({\bf q},\omega)={\mathrm i}[L({\bf q},\omega+{\mathrm i} 0)-L({\bf q},\omega-{\mathrm i} 0)]
=2 \,{\rm Im}\,L({\bf q},\omega-{\mathrm i} 0)\,.
\end{equation}
According to the fluctuation-dissipation theorem,
the response function $\chi({\bf q},\omega)$ which describes also transport and absorption
is related to density fluctuations according to
\begin{equation}
{\rm Im}\chi({\bf q},\omega)=-{\rm Im}L({\bf q},\omega-{\mathrm i} 0).
\end{equation}
As immediately seen from the spectral representation, the density-density correlation function is obtained from the width-function after multiplication with the Bose factor,
here
\begin{equation}\label{SL}
S({\bf q},\omega)=\frac{1}{\pi} \frac{1}{e^{ \omega/T}-1}{\rm Im}L({\bf q},\omega-{\mathrm i} 0).
\end{equation}
For the isothermal compressibility (\ref{kappa}) we find
\begin{eqnarray}\label{isot}
&&\kappa_{\rm iso}(T,\mu)
=\frac{1}{
n^2T} \lim_{{\bf q} \to 0} \int_{-\infty}^\infty \frac{d \omega}{\pi}
\frac{1}{e^{ \omega/T}-1}{\rm Im}L({\bf q},\omega-{\mathrm i} 0).
\end{eqnarray}
For $L({\bf q},{\mathrm i} z_\lambda)$ a systematic perturbation expansion with the Matsubara technique is available which can be represented by Feynman diagrams.
Below we show how bound state formation is treated this way.
\subsection{Real-time Green function technique}
\label{sec:nonequGF}
Another quantum statistical approach to calculate the isothermal compressibility (\ref{kappa}) starts from the real time non-equilibrium Green
function technique.
The four-dimensional nucleon current-current
auto-correlation function \cite{Voskresensky:1987hm,Knoll:1995nz} is given by the $-+$
component of their Wigner transformed self-energy
\begin{eqnarray} \label{Pi-+xq}\unitlength6mm
-{\mathrm i}\Pi^{\mu\nu}_{-+}(q;X) = \begin{picture}(4.3,.7) \put(0.2,0.2){\begin{picture}(5,1)\put(0,0){\oneloop}
\put(3.8,0.2){\makebox(0,0){$=$}} \end{picture} \int {\mathrm d}^4 \xi e^{{\mathrm i}
q\xi}\langle j^{\mu\dagger}(X-\xi/2) j^\nu(X+\xi/2)\rangle,
\end{eqnarray}
with $q^\alpha =(\omega, {\bf q})$, $X^\alpha =(x^\alpha +y^\alpha)/2$, $\alpha =0,1,2,3$. The dashed lines relate to a vector boson interacting with nucleons by a boson-two-fermion interactions, like in QED, while the
$-{\mathrm i}\Pi $-loop symbolically denotes the exact inclusion of all strong
interactions among the source particles. The bracket
$\langle\dots\rangle$ denotes a quantum ensemble average over the
source with quantum states and operators in the interaction picture.
The $\Pi_{-+}(q; X)$ and $ \Pi_{+-}(q; X)$ self energies have meaning of
the gain and loss terms in the Kadanoff-Baym quantum kinetic equation.
We use the convenient up and down $\{-,+\}$ contra-variant and co-variant notations of
\cite{Ivanov:1999tj}. They are introduced in Appendix \ref{A1} as well as relations between equilibrium two-point functions.
In the non-relativistic case considered here, the nucleon density-density correlator is determined as $-{\mathrm i}\Pi^{00}_{-+}$, calculated with $j_\mu =(\hat n, {\bf 0})$, $\hat n$ is the density of the fermion species under consideration, the bare vertices are taken as $V_0^{\pm} =(1, {\bf 0})$, $V_{0,\pm} =(\pm 1, {\bf 0})$. The Wigner
transform of $-{\mathrm i}\Pi^{00}_{-+}(x,y)$ has the meaning of the dynamic structure factor, \begin{eqnarray}
-{\mathrm i}\Pi^{00}_{-+}(q;X)=S(q;X)\,.
\end{eqnarray}
The static structure factor is then as follows
\begin{eqnarray} \label{stSpectr}
S({\bf q}\to 0; X)=(\langle \hat n^2 \rangle-\langle \hat n \rangle^2)/\langle \hat n \rangle=\int_{-\infty}^{\infty} \frac{{\mathrm d} q_0}{2\pi}[-{\mathrm i}\Pi^{00}_{-+}(q_0,{\bf q}\to 0; X)].
\end{eqnarray}
It has the meaning of the normalized variance of the fermion density in the medium.
In thermodynamic equilibrium, the dependence on $X$ disappears because of homogeneity
in space and time. Applying the first relation (\ref{eqrel}), cf. (\ref{EqStstr0}), we get the exact relation
\begin{eqnarray} \label{EqstSpectr}
S({\bf{q}}\to 0)=\lim_{{\bf{q}}\to 0}\int_{-\infty}^{\infty} \frac{{\mathrm d} q_0}{2\pi}\frac{\Gamma_{\rm B}(q_0,{\bf{q}})}{e^{q_0/T}-1}\,, \quad\Gamma_{\rm B}(q_0,{\bf{q}})=-2\mbox{Im}\Pi^{ R}(q_0,{\bf{q}})\,,
\end{eqnarray}
where $\Pi^R$ is the boson retarded self-energy, and $q_0=\omega$ as used above.
The boson width, $\Gamma_{\rm B}(q_0,{\bf{q}})$, in (\ref{EqstSpectr}) is non-zero only in the space-like region $q_0<|{\bf{q}}|$ and thus
\begin{eqnarray} \label{0EqstSpectr}
S({\bf{q}}\to 0)=T\lim_{{\bf{q}}\to 0}\int_{-\infty}^{\infty} \frac{{\mathrm d} q_0}{2\pi}\frac{\Gamma_{\rm B}(q_0,{\bf{q}})}{q_0}=T\mbox{Re}\Pi^R (0,{\bf{q}}\to 0) \,,
\end{eqnarray}
where we used the corresponding Kramers-Kronig relation.
Also, as follows from the definition of $G^{-+}$ fermion Green function, ${\mathrm i} G^{-+}=-\langle \psi_2^\dagger \psi_1\rangle$ and (\ref{neos}), (\ref{eqrel}) the fermion density is given by (see Eq. (\ref{neos}) and below)
\begin{eqnarray}\label{EQdistr}
n_\tau=g_\tau\int\frac{{\mathrm d}^4 p}{(2\pi)^4} \frac{A_\tau(p)}{e^{(p_0-\mu)/T}+1}\,,\quad A_\tau(p)=-2\mbox{Im}G^R_\tau(p)\,,
\end{eqnarray}
where $A_\tau(p)$ is the single-particle spectral function
which obeys the sum-rule
\begin{eqnarray}
\int_{-\infty}^{\infty} A_\tau( p) \frac{dp_0}{2\pi}=1\,,\quad \tau =n,p\,.
\end{eqnarray}
For pure neutron matter ($g=2$), the chemical potential $\mu=\mu_n$ is given by the neutron chemical potential. For isospin-symmetric matter, the total density is $n=n_n+n_p=2 n_n$ since the chemical potentials of neutrons and protons coincide, $\mu=\mu_n=\mu_p$, in total $g=4$. To get
coincidence with Eq. (\ref{neos}) we performed the variable shift $p_0\to p_0 -\mu$, so that the Green functions depend now on $p_0$ and occupations, on $p_0 -\mu$.
In the presence of the vector field $V=(V_0, {\bf V})$, like for the nucleon interacting with the $\omega$ meson, we should still perform the shift $\mu\to\mu-V_0$.
Concluding, first we used the Matsubara technique
and then we applied the non-equilibrium diagram technique. We see that the quantity $L({\bf{q}},\omega-{\mathrm i} 0)$ appeared in (\ref{Spectral}), (\ref{SL}) has the meaning of the advanced self-energy $\Pi^A$, $L({\bf q},\omega+i 0)=\Pi^R(q_0,{\bf{q}})$, $\Gamma_L =\Gamma_{\rm B}=-2\mbox{Im}\Pi^{R}(q_0,{\bf{q}})$
and
\begin{equation}
-{\mathrm i} \Pi^{00}_{-+}(\omega,{\bf q})
= \frac{\Gamma_L({\bf q},\omega)}{e^{\omega/T}-1} = \frac{2{\rm Im}\,L({\bf q},\omega-i 0)}{e^{ \omega/T}-1} \,.
\end{equation}
The real parts follow from the Kramers-Kronig relation.
In the long-wavelenth limit, we recover Eq. (\ref{isot}).
\section{Various approximations}
\label{Examples}
\subsection{Perturbation expansion: lowest order}
To become more familiar with the formalism we briefly discuss the trivial case of the
lowest order perturbation theory where all interactions are neglected,
and the ideal non-relativistic Fermi gas results.
In the simplest approximation (zeroth order of the interaction) we have the single-particle polarization loop result (see Fig. \ref{fig:1})
\begin{equation}\label{Lo}
L_0({\bf q},z)
g \int \frac{d^3 p}{(2\pi)^3}\frac{f(\epsilon^0_p)-f(\epsilon^0_{{\bf p}+{\bf q}})
}{z+\epsilon^0_{{ p}}-\epsilon^0_{{\bf p}+{\bf q}}}
\end{equation}
with $\epsilon^0_{{\bf p}}=\epsilon^0_p=E^0_p-\mu$, with $E^0_p=p^2/(2m)$ (the rest energy term is shifted to $\mu$).
For simplicity the index $\nu$ is dropped, the generalization from single-component to multi-component matter is trivial.
\begin{figure}[!th]
\includegraphics[width=6cm]{RMAfig1v2.pdf}
\caption{The one-loop perturbative contribution to the polarization function in Matsubara technique, $L_{0}({\bf q},{\mathrm i} z_\lambda)$ (zeroth order of interaction). }
\label{fig:1}
\end{figure}
Using (\ref{SL}), (\ref{Lo}) we get the result for the lowest order of perturbation theory
\begin{equation}
\label{S0q}
S_0({\bf q},\omega)= \frac{g}{e^{ \omega/T}-1}
\int \frac{d^3 p}{(2\pi)^3}\left[f(\epsilon^0_p)-f(\epsilon^0_{{\bf p}+{\bf q}})\right] \delta[\omega+\epsilon^0_{{ p}}-\epsilon^0_{{\bf p}+{\bf q}}]\,.
\end{equation}
With
\begin{equation}
\label{f1-f}
f(\epsilon^0_p)-f(\epsilon^0_{{\bf p}+{\bf q}})=
f(\epsilon^0_{{\bf p}+{\bf q}})\left[1-f(\epsilon^0_p)\right]
\left(e^{(\epsilon^0_{{\bf p}+{\bf q}}-\epsilon^0_{{p}})/T}-1 \right)
\end{equation}
we obtain for the resulting isothermal compressibility (\ref{isot}) after integration over $\omega$
\begin{eqnarray}
\label{eq:22}
&&
\kappa^{(0)}_{\rm iso}(T,\mu)
=\frac{g}{n^2T}
\int \frac{d^3 p}{(2\pi)^3}f(\epsilon^0_p) [1-f(\epsilon^0_p)].
\end{eqnarray}
The integral
is performed using integration by parts
\begin{eqnarray}
\int \frac{d^3 p}{(2\pi)^3}f(\epsilon^0_p) [1-f(\epsilon^0_p) ]=
-\frac{4 \pi}{(2 \pi)^3}\int_0^\infty p^2dp\frac{m T}{ p} \frac{df(\epsilon^0_p) }{dp}
=\frac{ m T}{2\pi^2 }\int_0^\infty dp\,f(\epsilon^0_p).
\end{eqnarray}
The same resulting expression
\begin{equation}
\kappa^{(0)}_{\rm iso}(T,\mu)=
g \frac{m}{2\pi^2 n^2}\int_0^\infty dp\,f(\epsilon^0_p)
\end{equation}
is obtained from the EoS of the ideal quantum gases. Neglecting all interactions we have
\begin{equation}
n^{(0)}(T, \mu)=\frac{1}{V} \sum_{p}\frac{1}{e^{ \epsilon^0_p/T}+1}=\frac{1}{V} \sum_{p} f(\epsilon^0_p)
=\frac{N}{V}\,,
\end{equation}
and $\partial n^{(0)}/\partial \mu= 1/(V T)\sum_p f(\epsilon^0_p) [1-f(\epsilon^0_p) ]$ in agreement with Eq. (\ref{eq:22}) with $\sum_p \to g V\int d^3p/(2 \pi)^3$. For symmetric matter, in addition to spin also isospin summation bas to be performed. The introduction of quasiparticles within the Matsubara Green function approach is discussed below in Sec. \ref{sec:mean}.
The lowest order perturbation theory where all interactions are neglected, i.e. the ideal Fermi gas, is often used as a system of reference.
To investigate the influence of the interaction on the incompressibility $K(T,n)$ (\ref{incompr}) in isospin-symmetric matter,
we introduce the excess quantity $\varphi_0(T,n)$ according to
\begin{equation}
\label{fi0}
K(T,n)=K^{(0)}(T,n)[1+\varphi_0(T,n)]
\end{equation}
where $K^{(0)}(T,n)=1/[n\kappa_{\rm iso}^{(0)}(T,n)]$ according to Eq. (\ref{eq:22}). Results for $\varphi_0(T,n)$ are presented in Sec. \ref{comp}.
\subsection{The single-particle mean-field approximation}
\label{sec:mean}
To give a simple example for the quasiparticle approach,
we discuss the lowest order with respect to the interaction where the mean-field (Hartree-Fock, HF) approximation is obtained. We start from the nonrelativistic Hamiltonian
\begin{equation}
H=\sum_1 \epsilon^0_1 \hat{a}^\dagger_1\hat{a}_1+H_{\rm int}, \qquad H_{\rm int}=\frac{1}{2} \sum_{12,1'2'} V(1,2;1',2') \hat{a}^\dagger_{1'} \hat{a}^\dagger_{2'}\hat{a}_2\hat{a}_1.
\end{equation}
Here $\hat{a}^\dagger_1, \hat{a}_1$ are creation and annihilation operators,
respectively, for the single-nucleon state $1=\{{\bf p}_1,\nu_1\}$ denoting wave number, spin, and isospin.
As above we consider firstly effective one-component systems (neutron matter, symmetric matter) where the summation over spin and isospin is replaced by the degeneracy factor $g =2$ or 4, respectively, so that only the momentum $\bf p$ remains to characterize the single-nucleon state. With the antisymmetrized interaction
we have for wave number ${\bf p}$ the quasiparticle energy in HF approximation ($1 \to {\bf p}, 2 \to {\bf k}$, spin and isospin are not given explicitly, but the Hartree term leads to the factor $g$)
\begin{eqnarray}
\label{EHF}
&&\epsilon^{\rm HF}_p = \epsilon^0_p+\int \frac{d^3k}{(2 \pi)^3} V({\bf p},{\bf k};{\bf p},{\bf k})_{\rm ex} f(\epsilon^{\rm HF}_k)
=\epsilon^0_p+\Delta_p^{\rm HF}, \\ && V({\bf p},{\bf k};{\bf p},{\bf k})_{\rm ex}=\sum_{\nu_k}^g \left[
V({\bf p},{\bf k};{\bf p},{\bf k})-V({\bf p},{\bf k};{\bf k},{\bf p})\delta_{\nu_k,\nu_p}\right]
\end{eqnarray}
As it is well-known from mean-field approximations, the self-energy $\Delta_p^{\rm HF}$ has no dependence on frequency $\omega$ so that
it is purely real. The spectral function follows as
$A^{\rm HF}(\omega , {\bf{p}})=2\pi \delta (\omega-\mu - \epsilon_{p}^0 -\Delta_p^{\rm HF})$.
The density EoS (\ref{neos}) relating $T, \mu$ to $n$,
reads in the HF QPA
\begin{equation}
\label{nHF}
n^{\rm HF}(T, \mu)=g \int \frac{d^3p}{(2 \pi)^3} \frac{1}{e^{\epsilon^{\rm HF}_p/T}+1}
=g \int \frac{d^3p}{(2 \pi)^3}f(\epsilon^{\rm HF}_p).
\end{equation}
We invert this relation so that the relation $\mu = \mu^{\rm HF}(T,n)$ is obtained.
>From this, the incompressibility (\ref{incompr})
\begin{equation}
\label{KTnB}
K(T, n)=n \frac{\partial \mu^{\rm HF}(T,n)}{\partial n}\Big|_T
\end{equation}
follows after performing $\partial /\partial n$ in Eq. (\ref{nHF}),
\begin{equation}
1=\frac{g}{2\pi^2T}\int_0^\infty p^2dp f(\epsilon^{\rm HF}_p)[1-f(\epsilon^{\rm HF}_p)] \left[1-\frac{\partial \Delta_p^{\rm HF}}{\partial \mu}\right] \frac{\partial \mu^{\rm HF}}{\partial n}\Big|_T.
\end{equation}
We obtain
\begin{equation}
\label{muHFn}
\frac{\partial \mu^{\rm HF}}{\partial n}\Big|_T=\left\{\frac{g}{2\pi^2 T}\int_0^\infty p^2dp
f(\epsilon^{\rm HF}_p)[1-f(\epsilon^{\rm HF}_p)]
\left[1- \int \frac{d^3k }{(2 \pi)^3T}V({\bf p},{\bf k};{\bf p},{\bf k})_{\rm ex}
f(\epsilon^{\rm HF}_k)[1-f(\epsilon^{\rm HF}_k)] \right]\right\}^{-1}.
\end{equation}
At low temperatures we replace ($k_{\rm F}=(6 \pi^2 n/g)^{1/3}$)
\begin{equation}
\label{Fermiint}
f(\epsilon^{\rm HF}_k)[1-f(\epsilon^{\rm HF}_k)]
=-\frac{\partial f(\epsilon^{\rm HF}_k)}{\partial k} \frac{m^* T}{k}
\approx \delta(k-k_{\rm F})\frac{m^* T}{ k},
\end{equation}
where $m^*=(d^2 \epsilon_p/dp^2)$ at $p=k_{\rm F}$ is the non-relativistic effective mass of the quasiparticle (the so called Landau effective mass).
We introduce the angular averaged interaction
\begin{equation}
\tilde V(k_F)=(1/2)\int_{-1}^1 dz
V(p_{\rm F}{\bf e}_p,k_{\rm F}{\bf e}_k;p_{\rm F}{\bf e}_p,k_{\rm F}{\bf e}_k)_{\rm ex}
\end{equation}
with the unit vector scalar product ${\bf e}_p \cdot {\bf e}_k =z$ (isotropic interaction).
In first order we find
\begin{equation}
\label{KHF}
K^{\rm HF}=n \frac{\partial \mu^{\rm HF}}{\partial n}=\frac{\hbar^2k_{\rm F}^2}{3m}
\left[1+\frac{m^*k_{\rm F} }{2 \pi^2
}\tilde V(k_{\rm F})\right].
\end{equation}
Now we show the alternative way to obtain the EoS in the HF QPA
starting from the dynamic structure factor.
The thermodynamic density-density Green function $L({\bf q},{\mathrm i} z_\lambda)$ (\ref{Lqo})
contains in addition to the zeroth order with respect to the interaction (\ref{Lo}) the following contributions
$L_1({\bf q}, {\mathrm i} z_\lambda)$ which are of first order with respect to the interaction:
\begin{figure}[!th]
\includegraphics[width=11cm]{L1newi.pdf}
\caption{The first-order perturbative contributions to the density auto-correlation function in Matsubara technique, $L_{1}({\bf q},{\mathrm i} z_\lambda)$ (first order of interaction).}
\label{fig:1a}
\end{figure}
The contributions (a) -- (d) of Fig. \ref{fig:1a} are contributions to the self-energy (SE)
in Hartree-Fock approximation. They are taken into account if we replace the free
single-nucleon propagator by the quasiparticle propagator with the spectral function
$A^{\rm HF}(\omega , {\bf{p}})$, see Eq. (\ref{EHF}). The evaluation is similar
to the evaluation of $L_0$, Eq. (\ref{Lo}),
\begin{equation}\label{L1SE}
L_1^{\rm SE}({\bf q},z)
g\int \frac{d^3 p}{(2\pi)^3}\frac{ f(\epsilon^{\rm HF}_p)-f(\epsilon^{\rm HF}_{{\bf p}+{\bf q}})
}{z+\epsilon^{\rm HF}_{p}-\epsilon^{\rm HF}_{{\bf p}+{\bf q}}}.
\end{equation}
The contribution (e) is a vertex correction,
and the exchange term (f) is a contribution to the screening equation.
The evaluation gives with quasiparticle propagators for the vertex (v) and screening (s) contribution
\begin{eqnarray}\label{L1v}
L_1^{\rm v,s}({\bf q},z)&=
g\int \frac{d^3 p}{(2\pi)^3}\int \frac{d^3 k}{(2\pi)^3}\frac{ f(\epsilon^{\rm HF}_p)-f(\epsilon^{\rm HF}_{{\bf p}+{\bf q}})}{z+\epsilon^{\rm HF}_{p}-\epsilon^{\rm HF}_{{\bf p}+{\bf q}}}
\frac{ f(\epsilon^{\rm HF}_k)-f(\epsilon^{\rm HF}_{{\bf k}+{\bf q}})}{z+\epsilon^{\rm HF}_{k}-\epsilon^{\rm HF}_{{\bf k}+{\bf q}}}
\nonumber \\ &&
\times V({\bf p}+{\bf q},{\bf k};{\bf p},{\bf k}+{\bf q})_{\rm ex}.
\end{eqnarray}
Performing the limit $z \to \omega+ {\mathrm i} 0$ we have
\begin{eqnarray}
S_1({\bf q},\omega)&=& \frac{1}{e^{\omega/T}-1}
g\int \frac{d^3 p}{(2\pi)^3}
\frac{\left( f(\epsilon^{\rm HF}_p)-f(\epsilon^{\rm HF}_{{\bf p}+{\bf q}})\right)
\left( f(\epsilon^{\rm HF}_k)-f(\epsilon^{\rm HF}_{{\bf k}+{\bf q}})\right)}{\epsilon^{\rm HF}_{p}-
\epsilon^{\rm HF}_{{\bf p}+{\bf q}}
+\epsilon^{\rm HF}_{k}-\epsilon^{\rm HF}_{{\bf k}+{\bf q}}}
\nonumber \\ &&
\times \left[\delta(\omega+
\epsilon^{\rm HF}_{p}-\epsilon^{\rm HF}_{{\bf p}+{\bf q}})+\delta(\omega+\epsilon^{\rm HF}_{k}-
\epsilon^{\rm HF}_{{\bf k}+{\bf q}}) \right] V({\bf p}+{\bf q},{\bf k};{\bf p},{\bf k}+{\bf q})_{\rm ex}.
\end{eqnarray}
After the limit ${\bf q} \to 0$ we arrive at the expression
\begin{eqnarray}
&&
\kappa^{(1)}_{\rm iso}(T,\mu)
=\frac{g}{n^2T}
\int \frac{d^3 p}{(2\pi)^3}f(\epsilon^{\rm HF}_p) [1-f(\epsilon^{\rm HF}_p)]\nonumber \\
&& \times \left[1-\frac{1}{T} \int \frac{d^3k }{(2 \pi)^3}V({\bf p},{\bf k};{\bf p},{\bf k})_{\rm ex}
f(\epsilon^{\rm HF}_k)[1-f(\epsilon^{\rm HF}_k)] \right]
\end{eqnarray}
in full agreement with Eq. (\ref{muHFn}) up to first order with respect to the interaction.
We conclude that we can reproduce the result (\ref{KTnB}) and the corresponding
relation (\ref{nHF}), starting from the density auto-correlation function or the dynamic
structure factor. As seen from the first-order calculation, the second way to
calculate the EoS using the dynamic structure factor is rather cumbersome compared with
the direct calculation of the density via the relation (\ref{neos}),
starting from the single-nucleon spectral function. However, the dynamic structure factor contains
much more
information not only about thermodynamic properties, but also on dynamic properties and
the response to external perturbations. Before solving the question how bound state formation can be implemented within the diagram expansion of $L({\bf q},\omega)$, see Sec. \ref{two}, we present the Fermi-liquid approach which is a very efficient approach to nuclear systems.
\section{Fermi-liquid approach}\label{sec:Fermi}
One possible way to evaluate the dynamic structure factor is the Fermi-liquid approach as worked out by
Landau and Migdal. It is based on the QPA using an effective nucleon-nucleon interaction.
This concept has been applied successfully to describe also the dynamic behavior of dense matter.
We outline some important results here. However, until now there exists no systematic approach to include
the formation of bound states. In the following Sec. \ref{two} we show how the dynamic structure has to be treated
to include the formation of clusters.
In the low-density limit, we can infer an interaction potential $V_{\tau\tau^{'}}(1,2;1',2')$
which reproduces the two-particle properties such as scattering phase shifts
and bound state formation.
Different potentials such as the local Yukawa potential
or the separable Yamaguchi potential
are known from the literature.
For dense systems they must be modified to obtain the known properties at saturation density $n_{\rm sat}$.
Empirical interactions such as the Skyrme interaction can be introduced,
another possibility are relativistic mean-field (RMF) expressions derived
from an effective Lagrangian containing interacting nucleon and meson fields.
Dense nuclear systems at low temperatures are degenerate. Because of the Pauli blocking,
interaction processes are possible only near the Fermi surface at $p_{\rm F}$, and the potential $V_{\tau\tau^{'}}(1,2;1',2')$ is of relevance only for such processes. Therefore only the direction $\bf n$ of the wave number,
${\bf p}\approx p_{\rm F} \cdot {\bf n}$, is changing owing to the interaction.
The Landau Fermi-liquid approach considers such processes and introduces
corresponding phenomenological Landau-Migdal interaction parameters.
In this sense the Fermi-liquid approach is a semi-empirical approach.
Being treated within the Fermi-liquid theory the 4-point (like-sign, if treated in terms of non-equilibrium Green function technique) particle-hole interaction $T_{\rm ph}$ is presented as the local interaction between particle-hole loops, cf. \cite{Voskresensky:1993ud},
\begin{eqnarray}
\label{vertices0}
\parbox{6cm}{\includegraphics[width=6cm]{Fig-18.pdf}}
\end{eqnarray}
or
\begin{eqnarray}
\hat T_{\rm ph}({\bf n}',{\bf n};q)=\hat \Gamma^\omega({\bf n}',{\bf n})+\langle \hat \Gamma^\omega({\bf n}',{\bf n}'' {\cal L}_{\rm ph}({\bf n}'';q)\hat T_{\rm ph}({\bf n}'',{\bf n};q)\rangle_{{\bf n}''}.\nonumber
\end{eqnarray}
The single-particle Green functions are taken in QPA,
the remaining background is encoded in the renormalized particle-hole interaction $\hat \Gamma^\omega({\bf n}',{\bf n})$.
The particle-hole propagator is given by the Lindhard function
\begin{equation}
{\cal L}_{\rm ph}({\bf n}'';q)=\int_{- \infty}^\infty \frac{d \epsilon}{2\pi i}\int_0^\infty \frac{dp \,\, p^2}{\pi^2} G(p_{\rm F +}) G(p_{\rm F -})\,,
\end{equation}
$p_{{\rm F}
\pm}=(\epsilon\pm\omega/2,p_{{\rm F}} \cdot{\bf{n}}\pm{\bf{k}}/2)$. The QPA and the improved quasiparticle approximation (IQPA) for the single-nucleon spectral function $A(p)$ are discussed in Appendix \ref{sec:quasi}.
The empty box, $\Gamma^{\omega}$, describes the $\delta$-functional interaction which can be expressed through the Landau-Migdal parameters as \cite{Mig,Mig1,Nozieres,GP-FL}
\begin{align}
\hat\Gamma^{\omega}({\bf{n}}\,',{\bf{n}}) =
\Gamma^{\omega}_0({\bf{n}}\,'{\bf{n}})\, \sigma'_0 \sigma_0 +
\Gamma^{\omega}_1({\bf{n}}\,'{\bf{n}})\,
(\vec{\sigma}' \cdot \vec{\sigma})\,.
\label{Gom-fullspin}
\end{align}
The matrices $\sigma_\mu$ with $\mu =0,\dots,3$ act on incoming
nucleons while the matrices $\sigma'_\mu$ act on outgoing nucleons;
$\sigma_0$ is the unity matrix and other Pauli matrices
$\sigma_{1,2,3}$ are normalized as ${\rm
Tr}\sigma_\mu\sigma_\nu=2\delta_{\mu\nu}$, and ${\bf{n}}\,',{\bf{n}}$ are unit vectors. We neglect here the spin-orbit
interaction, being suppressed for small transferred momenta
$|{\bf q}|\ll p_{\rm F}$ of our interest. The scalar and spin amplitudes in
Eq.~(\ref{Gom-fullspin}) can be expressed in terms of
dimensionless scalar and spin Landau-Migdal parameters
\begin{align}
{f}_{\tau',\tau}({\bf{n}}\,',{\bf{n}})
&={\cal N}\Gamma_{0,\tau',\tau}^\omega({\bf{n}}\,',{\bf{n}})\,, \nonumber\\
{g}_{\tau',\tau}({\bf{n}}\,',{\bf{n}}) &={\cal N}
\Gamma_{1,\tau',\tau}^\omega({\bf{n}}\,',{\bf{n}})\,.
\label{L-param}
\end{align}
The normalization factor
\begin{equation}
{\cal N}=g m_{\rm F}^*\,p_{\rm F}/(2\pi^2)
\end{equation}
contains the degeneracy factor $g =2$ for one type of non-relativistic fermions, like for the neutron matter, and $g =4$ for two types of fermions, like for the isospin-symmetric nuclear matter, $\tau',\tau$ relate to $n$ or $p$. In the case of a relativistic Fermi liquid one uses the normalization factor with $m^*_{\rm F}$ replaced by $E_{\rm F}=\sqrt{p_{\rm F}^2 +m^{*2}_{\rm F}}$. The baryon density, $n$, and the Fermi momentum, $p_{\rm F}$, are related as $n =g\,p_{\rm F}^3/(6\pi^2)$. Here we use normalization like in \cite{Nozieres,GP-FL}.
Note that another normalization on ${\cal N}(n_{\rm sat})$ instead of ${\cal N}(n)$ is used in \cite{Mig,Mig1}, where $n_{\rm sat}$ is the nuclear saturation density.
The generalization to a two-component system, e.g., to the nuclear
matter of arbitrary isotopic composition, is formally simple~\cite{Mig,Mig1}. Then, the amplitudes in ($nn$, $pp$, $np$ and $pn$) channels are in general different and equations for the partial amplitudes do not decouple. For the isospin-asymmetric case one also uses a symmetric normalization on $\sqrt{{\cal N}_{n} {\cal N}_{p}}$ with ${\cal N}_{\tau} =2 m_{\rm F}^*\,p_{\rm F,\tau}/\pi^2$, cf. \cite{Sawyer:1989nu}.
Bearing in mind the application to isospin-symmetric nuclear matter, we present
$\Gamma_0^\omega,\,\Gamma_1^\omega$ in Eqs.~(\ref{Gom-fullspin}) and (\ref{L-param}) as ~\cite{Mig,Mig1}
\begin{eqnarray}
\Gamma_0^\omega =({f}({\bf{n}}\,',{\bf{n}})+{f}'({\bf{n}}\,',{\bf{n}})\vec{\tau}' \cdot \vec{\tau})/{\cal N}\,,\quad
\Gamma_1^\omega =({g}({\bf{n}}\,',{\bf{n}})+{g}'({\bf{n}}\,',{\bf{n}})\vec{\tau}' \cdot \vec{\tau})/{\cal N}\,,
\end{eqnarray}
where $\vec{\tau}$ are the isospin Pauli matrices. In this parametrization, the quantities $f$ and $f'$ (and similarly $g$ and $g'$) are expressed through $f_{nn}$ and $f_{np}$ as $f=\frac{1}{2}(f_{nn}+f_{np})$ and $f'=\frac{1}{2}(f_{nn}-f_{np})$. For the neutron matter the parameters $f=f_{nn}$, $g=g_{nn}$ are the neutron-neutron Landau-Migdal scalar and spin parameters. For isospin-symmetric nuclear systems we have $f_{nn}=f_{pp}$ and $f_{np}=f_{pn}$ if Coulomb interaction is omitted.
The particle-hole interaction is adapted to known properties of nuclear matter so that it is parametrized in an empirical way. This can be done directly comparing with known properties near the saturation density $n_{\rm sat}$
of baryons.
The parameters $f({\bf{n}}\,',{\bf{n}})$ and $g({\bf{n}}\,',{\bf{n}})$ depend on the scattering angle and are expanded in Legendre polynomials,
\begin{eqnarray}
f_{\tau',\tau}({\bf{n}},{\bf{n}}')=\sum_l f_{l,\tau',\tau} P_l ({\bf{n}},{\bf{n}}')\,, \quad g_{\tau',\tau}({\bf{n}},{\bf{n}}')=\sum_l g_{l,\tau',\tau} P_l ({\bf{n}},{\bf{n}}')\,.
\end{eqnarray}
For the most important physical quantities only the $l=0,1$ harmonics contribute.
For instance, the incompressibility $K$ is related to the Landau-Migdal parameter $f_0$ as
\begin{equation}
K=n\frac{\partial \mu}{\partial n}{\Big |}_T =\frac{p_{\rm F}^2}{3E_{\rm F}} (1+f_0).
\end{equation}
The Landau effective mass of the nucleon quasiparticle can be expressed in terms of the Landau-Migdal parameter $f_1$, \begin{equation}
E_{\rm F}=\mu (1+f_1/3)\,.
\end{equation}
(Note that in the relativistic theory the particle energy at the Fermi surface plays the same role as the effective mass in the non-relativistic Fermi-liquid theory.)
The Landau-Migdal parameters can be extracted from the experimental data on atomic nuclei. Unfortunately, there are essential uncertainties in numerical values of some of
these parameters. These uncertainties are, mainly, due to attempts to get the best fit
to experimental data in each concrete case slightly modifying parametrization used
for the residual part of the $N-N$ interaction.
For example, basing on the analysis of Refs.~\cite{KS80}, with the normalization \cite{MSTV90} $C_0 =1/ N_0 =300$\,MeV$\cdot$fm$^{3}$ one gets ${f}_0 \simeq 0.25$, ${f}_0' \simeq 0.95$, ${g}_0 \simeq 0.5$, ${g}_0' \simeq 1.0$, see Table 3 in~\cite{Mig1}. The parameters ${g}_0 $, ${g}_0'$ are rather slightly density dependent whereas ${f}_0 $ and ${f}_0'$ depend on the density essentially. For ${f}_0 (n)$ Ref.~\cite{Mig} suggested to use a linear density dependence, then with above given parameters we have ${f}_0 (n)=-2.5+2.75\, n/n_{\rm sat}$. For strongly isospin-asymmetric matter, e.g. for neutron matter, there are no data from which the Landau-Migdal parameters can be extracted. In this case the parameters are calculated within a chosen model for the $N-N$ interaction, see the review \cite{Backman:1984sx}.
We are interested in the description of the scalar interaction channel. Then the empty box is
\begin{eqnarray}
\Gamma^\omega \equiv F (\theta)\,,\,\,\, \theta ={\bf{p}}\cdot {\bf{p}}{\,'}/ |{\bf{p}}||{\bf{p}}{\,'}|\,.
\end{eqnarray}
Simplifying our considerations we will retain only the zeroth harmonics $F_0$ and $f_0$.
For isospin-symmetric matter we take the combinations $F_0 =(F_{0,nn}+F_{0,np})/2$ and $f_0=(f_{0,nn}+f_{0,np})/2$ but for pure neutron matter $ F_{0,nn}$ and $f_{0,nn}$. Then (\ref{vertices0}) produces
\begin{eqnarray}
T^{\rm ph}_{- -}=F_0/[1+F_0 \Pi_{0,- -}^{N=1,00}]\,,
\end{eqnarray}
$\Pi_{0,- -}^{N=1,00}$ corresponds to the first one $- +$ loop term in $(G^{-+}_{\rm F}G^{+-}_{\rm F})$ loop expansion, cf. \cite{Knoll:1995nz} and Appendix \ref{sec:spectral}. In the standard Fermi-liquid approach for equilibrium systems, Eq. (\ref{vertices0}) is treated as equation for the retarded quantities \cite{Voskresensky:1993ud}. Resummation yields
$$T^{R}_{\rm ph} =F_0/[1+F_0 \Pi_{0}^{R}].$$
With the Fermi-liquid interaction follows
\begin{eqnarray}\label{Reself}
\mbox{Re}\Pi^R (0,{\bf{q}}\to 0)=\mbox{Re}\Pi^R_0 (0,{\bf{q}}\to 0)/[1+F_0\mbox{Re}\Pi^R_0 (0,{\bf{q}}\to 0)]\,.
\end{eqnarray}
Thus from (\ref{0EqstSpectr}) and (\ref{Reself}) we derive
\begin{eqnarray}\label{munF}
S({\bf{q}}\to 0)=
\frac{T}{\partial \mu/\partial n}=
\frac{T\,\mbox{Re}\Pi^R_0 (0,{\bf{q}}\to 0)}{[1+F_0\mbox{Re}\Pi^R_0 (0,{\bf{q}}\to 0)]}\,.
\end{eqnarray}
For $T\to 0$ one has $\mbox{Re}\Pi^R_0 (0,{\bf{q}}\to 0)={\cal N}$ and using (\ref{EqStstr0}) we derive ordinary relation for the Fermi liquid
\begin{equation}
\frac{\partial \mu}{\partial n}=\frac{1}{{\cal N}}+F_0 =\frac{1+f_0}{{\cal N}}.
\end{equation}
\section{Density-density correlations within relativistic mean-field approximation}\label{RMF}
\subsection{Self--consistent Hartree approximation}
To simplify our considerations we continue the study of pure neutron or isospin-symmetric matter.
The QPA and IQPA are discussed in the Appendix \ref{sec:spectral}.
Within the self-consistent Hartree approximation $\Sigma$ depends on $n$ and $\mu$ via the dependence of the Dirac effective fermion mass $m^* (n)$
and $f$ on $\mu -V_0$. Then Eq. (\ref{dmun}) being treated within the IQPA becomes
\begin{eqnarray}\label{dmunH}
\frac{\partial \mu}{\partial n}-\frac{\partial V_0}{\partial n}=\frac{T+ h_s^{\rm IQPA} (\partial m^* /\partial n)}{ h^{\rm IQPA}}\,,
\end{eqnarray}
\begin{eqnarray}
h^{\rm IQPA}=g\int\frac{{\mathrm d}^3 p}{(2\pi)^3}f(E_p -\mu +V_0)(1-f(E_p -\mu +V_0))\,,
\end{eqnarray}
\begin{eqnarray}\label{hsiqpa}
h^{\rm IQPA}_s=g\int\frac{{\mathrm d}^3 p}{(2\pi)^3} \frac{m^*}{E_p}f(E_p -\mu +V_0)(1-f(E_p -\mu +V_0)).
\end{eqnarray}
With $m^*$ not depending on $\mu$ and taking $V_0 =F_0 n$ we recover (\ref{munF}), now in the IQPA.
Reference \cite{Matsui} calculated the Landau-Migdal parameter $f_0$ in the non-linear Walecka model
with the RMF Lagrangian for the nucleons interacting with $\omega_0$ and $\sigma$ mean fields of $\omega$ and $\sigma$ mesons.
The generalization to include the $\rho_0^3$ meson field of the $\rho$ meson is straightforward. In this approach the nucleon distribution is given by
\begin{eqnarray}
f =\frac{1}{\mbox{exp}[(E_p -\mu+V_0)/T]+1}\,,\quad E_p=\sqrt{m^{\,*2}+{\bf{p}}^{\,2}}\,,\quad V_0 =\frac{g^2_\omega n}{m^2_\omega}\,,
\end{eqnarray}
$g_{\omega }$ is the $\omega N$ coupling, $m_\omega$ is the $\omega$ meson mass,
and the Dirac effective nucleon mass obeys the equation
\begin{eqnarray}\label{efmass}
m^* =m-\frac{g^2_\sigma}{m^2_\sigma}\int \frac{g{\mathrm d}^3 p}{(2\pi)^3}f \frac{m^*}{E_p}\,,
\end{eqnarray}
$g_{\sigma}$ is the $\sigma N$ coupling, $m_\sigma$ is the $\sigma$ meson mass.
Taking $\partial/\partial n$ in (\ref{efmass}) we find
\begin{eqnarray}
\frac{\partial m^*}{\partial n}=-\frac{g^2_\sigma}{m^2_\sigma}\left(\frac{\partial \mu}{\partial n}- \frac{\partial V_0}{\partial n}\right)\frac{m^* h_s}{T+\frac{g^2_\sigma}{m^2_\sigma} h_{s1}}\,,
\end{eqnarray}
the integral $ h_{s1}/T$ is reduced to
\begin{eqnarray}
\frac{ h_s}{T}=g \int_{m^*}^{\infty}\frac{ E_p {\mathrm d} E_p f}{2\pi^2 \sqrt{E_p^2 -m^{\,*2}}}\,,
\end{eqnarray}
\begin{eqnarray}
\frac{h_{s1}}{T}= g \int_{m^*}^{\infty}\frac{ (E_p^2 -2m^{\,*\,2}) {\mathrm d} E_p f}{2\pi^2 \sqrt{E_p^2 -m^{\,*2}}}\,,
\end{eqnarray}
and
\begin{eqnarray}
\frac{ h}{T} =g \int_{m^*}^{\infty}\frac{(2E_p^2 -m^{\,*\,2}) {\mathrm d} E_p f}{2\pi^2 \sqrt{E_p^2 -m^{\,*2}}}\,.
\end{eqnarray}
With these quantities at hand using (\ref{hsiqpa}) we find
\begin{eqnarray}\label{dmunHMatsui}
S({\bf{q}}\to 0)=\frac{T}{\partial \mu/\partial n}=\frac{h}{1+F_0^T h/T}\,,
\end{eqnarray}
where the quantity $F_0^T$ has the meaning of the zero harmonic of the scalar Landau-Migdal parameter for $T\neq 0$,
\begin{eqnarray}\label{MatsuiT}
F_0^T = \frac{g^2_\omega }{m^2_\omega}-\frac{g^2_\sigma }{m^2_\sigma}\frac{m^{\,*\,2} h_{s}^2}{h^2 [1+\frac{g^2_\sigma }{m^2_\sigma}(\ h_{s1}/T +m^{\,*\,2} h_s^2/hT)]}\,.
\end{eqnarray}
The latter result generalizes the result of Matsui \cite{Matsui} for $T\neq 0$.
For $T=0$
\begin{eqnarray}
& h_s/T =\frac{g p_{\rm F}}{2\pi^2}\,,\quad h/T= \frac{g E_{\rm F} p_{\rm F}}{2\pi^2}\,\quad E_{\rm F}=\sqrt{p_{\rm F}^2 +m^{*\,2}}\,,\\
& h_{s1}/T =-\frac{p_{\rm F}^3}{E_{\rm F}}+\frac{3}{2} E_{\rm F} p_{\rm F}-\frac{3}{2}m^{*\,2}\ln \frac{E_{\rm F}+p_{\rm F}}{m^*}\,,\nonumber
\end{eqnarray}
and the result (\ref{MatsuiT}) coincides with that of \cite{Matsui}. (Note that for $f_0$ and $F_0$ we use notations different from \cite{Matsui}.)
With $\partial \mu/\partial n$ at hand we may reconstruct the EoS, e.g. Eq.~(\ref{pressure}) for the pressure.
\subsection{Generalizations to isospin-asymmetric system}\label{arbitrary}
For a multi-component system, we have to generalize the expressions given in Sec. \ref{FDT} introducing partial structure factors for the constituents. We use the relation, cf. \cite{Sawyer:1989nu},
\begin{equation}
\label{partSF}
\langle\rho_\tau(0)\rho_{\tau'}(0)\rangle-\langle\rho_\tau (0)\rangle\langle\rho_{\tau'}(0)\rangle =T\frac{\partial \langle\rho_\tau (0)\rangle }{\partial \mu_{\tau'}}
=TV\frac{\partial n_\tau }{\partial \mu_{\tau'}}
\end{equation}
where $\rho_\tau (q \to 0)=N_\tau$ is the particle number
according to Eq. (\ref{densityrhoq}).
Introducing
\begin{eqnarray}
&S_{nn}({\bf q}\to 0)=\langle\rho^{\dagger}_n({\bf{q}})\rho_n({\bf{q}})\rangle_{{\bf{q}\to 0}}/n_n\,,\quad S_{np}({\bf q}\to 0)=\langle\rho^{\dagger}_n({\bf{q}})\rho_p({\bf{q}})\rangle_{{\bf{q}\to 0}}/\sqrt{n_n n_p}\,,\\
&S_{pp}({\bf q}\to 0)=\langle\rho^{\dagger}_p ({\bf{q}})\rho_p({\bf{q}})\rangle_{{\bf{q}\to 0}}/n_p\,,\quad S_{pn}({\bf q}\to 0)=\langle\rho^{\dagger}_p({\bf{q}})\rho_n({\bf{q}})\rangle_{{\bf{q}\to 0}}/\sqrt{n_p n_n}\,,
\end{eqnarray}
the static structure factor can be presented as \cite{Burrows:1998cg}
\begin{equation}
S({\bf q}\to 0)=S_{nn}({\bf q}\to 0)+S_{pp}({\bf q}\to 0)+2S_{np}({\bf q}\to 0)\,,
\end{equation}
provided that $S_{np}=S_{pn}$.
Thus
\begin{equation}\label{Stf}
S_{\tau,\tau'}({\bf q}\to 0)=T\left(\frac{\partial n_\tau}{\partial \mu_{\tau'}}\right){\Big |}_T
\end{equation}
and
\begin{equation}
\kappa_{\rm iso}(T,\mu)=
\frac{1}{n^2} \left[\left(\frac{\partial n_n}{\partial \mu_n}\right){\Big |}_T+ 2\left(\frac{\partial n_n}{\partial \mu_p}\right){\Big |}_T+\left(\frac{\partial n_p}{\partial \mu_p}\right){\Big |}_T\right]\,,
\end{equation}
provided $\left(\frac{\partial n_n}{\partial \mu_p}\right){\big |}_T=
\left(\frac{\partial n_p}{\partial \mu_n}\right){\big |}_T$.
Within the RMF approach one may find the values of the Landau-Migdal parameters in the scalar channel in a wide region of density, temperature, and asymmetry. For example
we can consider the RMF approach which claims to reproduce the known empirical data
and extrapolates for a wide range of temperatures and densities. Here
\begin{equation}
\label{RMF1}
n_\tau^{\rm RMF}(T,\mu_n,\mu_p)= \frac{1}{V} \sum_pf[E_\tau^{\rm RMF}({\bf p})-\mu_\tau]\,
\end{equation}
where the relativistic chemical potential $\mu_\tau$ includes the rest mass. The relativistic quasiparticle energies are given as
\begin{equation}
\label{RMF2}
E_\tau^{\rm RMF}({\bf p};T,n_n,n_p)=\sqrt{[m_\tau -S (T,n_n, n_p)]^2+{\bf p}^2}+V_\tau (T,n_n,n_p
\,.
\end{equation}
\subsection{Landau-Migdal parameters and DD2-RMF model}
Values for $S(T,n,Y_p)$ and $V_\tau(T,n,Y_p)$ according to the Typel DD2-RMF EoS \cite{Typel,Typel1999} are given in Ref. \cite{R}.
{}From $\delta E_\tau =\sum_{\tau'} F_{\tau \tau'} \delta n_{\tau'}$ we find
$F^{\tau \tau'}_0 =(\partial E_\tau/\partial n_{\tau'})_{p_\tau=p_{{\rm F},\tau}}$.
So we have
\begin{equation}
F_0^{nn}=-\frac{(m_n-S)(\partial S/\partial n_n)}{\sqrt{[m_n -S (T,n_n, n_p)]^2+ p_{{\rm F},n}^2}}+\frac{\partial V_n}{\partial n_n}\,,
\end{equation}
\begin{equation}
F_0^{pp}=-\frac{(m_p-S)(\partial S/\partial n_p)}{\sqrt{[m_p -S (T,n_n, n_p)]^2+ p_{{\rm F},p}^2}}+\frac{\partial V_p}{\partial n_p}\,,
\end{equation}
\begin{equation}
F_0^{np}=-\frac{(m_n-S)(\partial S/\partial n_p)}{\sqrt{[m_n -S (T,n_n, n_p)]^2+ p_{{\rm F},n}^2}}+\frac{\partial V_n}{\partial n_p}\,,
\end{equation}
\begin{equation}
F_0^{pn}=-\frac{(m_p-S)(\partial S/\partial n_n)}{\sqrt{[m_p -S (T,n_n, n_p)]^2+ p_{{\rm F},p}^2}}+\frac{\partial V_p}{\partial n_n}\,,
\end{equation}
\begin{figure}[!th]
\includegraphics[width=10cm]{MIFI2017a.pdf}
\caption{Landau parameter $f_0(n)$ at $T=0$ for neutron matter ($Y_p=0$) and symmetric matter ($Y_p=0.5$), derived from the DD2-RMF approximation, see Appendix \ref{DD2}.}
\label{fig:0}
\end{figure}
Fig. \ref{fig:0} shows the Landau-Migdal parameter $f_0(n,Y_p)$.
For symmetric matter, $f_0(n,0.5)$ a region of instability ($f_0(n)\le -1$) occurs
between density values $n= 0.000502$ and $ 0.09528$ fm$^{-3}$ (spinodal instability).
For neutron matter $f_0(n,0)$, no thermodynamic instability occurs. For details see Appendix \ref{DD2}.
A more general discussion how to relate RMF and Fermi-liquid theories is given in Ref. \cite{Matsui}.
\section{Two-particle correlations}\label{two}
\subsection{Cluster decomposition and inclusion of bound states}
We investigated the dynamic structure factor as a fundamental property
of the many-nucleon system which allows also to derive the thermodynamic properties,
in particular the EoS. We gave an extended discussion of the mean-field approximation
and showed the relation to the Landau-Migdal Fermi-liquid theory. These semiempirical
approaches, based on some parameter values to characterize the interaction,
are very efficient to describe nuclear systems, at least, near the saturation density and at low excitation energies.
At low temperatures the system is degenerate, and part of correlations are suppressed
because of Pauli blocking. The quasiparticle concept is appropriate to describe excitations
at these conditions.
A problem arises when we are going to low density matter, because the strong interaction
between the nucleons is no longer blocked out. Below the so-called Mott density,
bound states can be formed. Properties of nuclear systems are significantly influenced
by these few-body correlations. However, as a quantum effect, bound states are
not easily included in a mean-field approach which works with single-particle properties.
Similarly, within the Thomas-Fermi or the Fermi-liquid model the incorporation of
bound-state formation is difficult, see Ref. \cite{KV16}. In this work, we outline
the method how to include bound-state formation into the theory of the dynamic structure factor
for nuclear systems.
The QS approach to nuclear systems cannot describe bound state formation
in any finite order of perturbation theory but only after summation of infinite
orders of contributions. In the present section, we investigate the formation of clusters in warm nuclear matter in the
low-density limit, for $n/\Lambda^3 \ll1, \Lambda^2 =2\pi/mT$ in the ladder approximation where we first neglect in-medium effects on the single-particle Green functions.
In Fig. \ref{fig:G2} we present the two-particle Green
function such a ladder approximation. The QPA including medium effects will be discussed
in Sec. \ref{comp}.
\begin{figure}[!th]
\includegraphics[width=11cm]{G2a_v2.pdf}
\caption{Two-particle Green function in ladder approximation. }
\label{fig:G2}
\end{figure}
We have with $G^{(0)}_1(1,{\mathrm i} z_\nu)=1/({\mathrm i} z_\nu- \epsilon_1)$
\begin{equation}
\label{ladd}
G_2^{\rm ladd}(1,2;1',2',{\mathrm i} z_\mu)=\sum_\nu G^{(0)}_1(1,{\mathrm i} z_\nu) G^{(0)}_1(2,{\mathrm i} z_\mu-{\mathrm i} z_\nu)
\left[\delta_{1,1'}\delta_{2,2'}-\delta_{1,2'}\delta_{2,1'}+\sum_{34}V(1,2;3,4)
G_2^{\rm ladd}(3,4;1',2', {\mathrm i} z_\mu)\right]
\end{equation}
The ladder sum leads to a Schr\"odinger equation which describes
the quantum two-body problem. This so-called chemical picture
introduces bound states as new quasiparticles in a systematic way,
taking the corresponding sums of ladder diagrams into account.
In the low-density limit where medium effects on the single-particle Green functions are neglected,
we have in the representation with respect to the eigenstates of the two-nucleon system
$ \psi_{\nu,{\bf P}}({\bf p}_1,{\bf p}_2) $ with the energy
eigenvalues $ E^0_{\nu,{\bf P}} $ that can both be obtained as a solution of the free two-nucleon Schr{\"o}dinger equation
\begin{equation}
\left( \frac{p_1^2}{2m}+\frac{p_2^2}{2m}-E^0_{\nu,{\bf P}}\right) \psi_{\nu,{\bf P}}({\bf p}_1,{\bf p}_2)+\sum_{p_1',p_2'} V({\bf p}_1,{\bf p}_2;{\bf p}'_1,{\bf p}'_2) \psi_{\nu,{\bf P}}({\bf p}_1',{\bf p}_2')=0,
\end{equation}
the expression
\begin{equation}
\label{G2op}
G_2^{\rm ladd}(1,2;1',2', {\mathrm i} z_\mu)=\sum_{\nu.{\bf P}} \psi_{\nu,{\bf P}}({\bf p}_1,{\bf p}_2)\frac{1}{{\mathrm i} z_\mu -E^0_{\nu,{\bf P}}}
\psi^*_{\nu,{\bf P}}({\bf p}'_1,{\bf p}'_2),
\end{equation}
where ${\bf P}$ denotes the c.m. momentum and $\nu$ the intrinsic state of the two-body system.
The proof is easily given by insertion in Eq. (\ref{ladd}). Medium effects in the single-particle channel are included considering
the single-nucleon propagator $ G_1(1,{\mathrm i} z_\nu) $ in QPA and taking Pauli blocking
terms into account, see Refs. \cite{RMS,R,SRS} where an in-medium Schr{\"o}dinger equation is derived.
Our prescription to include cluster formation (bound states) into a systematic many-body approach
is to consider ladder propagators in addition to single-nucleon propagators (chemical picture).
Within a consistent approach, double counting of diagrams has to be avoided.
\subsection{The Beth-Uhlenbeck formula}
\label{sec:BU}
In this section we consider the low density limit of the nuclear matter EoS, assuming $n/\Lambda^3 \ll1, \Lambda^2 =2\pi/mT$ and using the ladder approximation where the Beth-Uhlenbeck formula is obtained for the second virial coefficient. To include cluster formation in the EoS, we consider firstly the density EoS (\ref{neos}) with the single-particle spectral function obtained from the cluster decomposition of the self-energy shown in Fig. \ref{fig:SE}, see Refs. \cite{RMS,SRS}. To calculate the nucleon self-energy $\Sigma(1,z)$ we introduce the few-particle $T_A$ matrices describing the
$A$ - nucleon cluster in the low-density limit.
\begin{figure}[!th]
\includegraphics[width=10cm]{diag4.pdf}
\caption{Cluster decomposition of the single-nucleon self-energy.}
\label{fig:SE}
\end{figure}
In particular we have
\begin{equation}
T_2^{\rm ladd}(1,2;1',2', {\mathrm i} z_\mu)=V(1,2;1'',2'') G_2^{\rm ladd}(1'',2'';1',2', {\mathrm i} z_\mu)[ G_{2}^{0}(1',2', {\mathrm i} z_\mu)]^{-1}\,
\end{equation}
with $G_{2}^{0}$ being the contribution to $G_2^{\rm ladd}$ in zeroth order of interaction. The $T_A$-matrix is the amputated part of the $G_A$ function, the propagator
of the $A$-nucleon cluster, in analogous manner to $G_2$.
The evaluation of the self-energy is straightforward, see \cite{RMS,SRS}.
As shown there, the spectral function is obtained from the imaginary part of the self-energy, and the density in the Beth-Uhlenbeck (BU) approach
follows according (\ref{neos}), (\ref{EQdistr}) as
\begin{equation}
\label{BU0}
n^{\rm BU}(T, \mu)=\frac{1}{V} \sum_{\bf p}f(\epsilon_p^0)+\frac{2}{V} \sum_{\nu,{\bf P}}
f_{\rm B}(\epsilon^0_{\nu,{\bf P}})-\frac{2}{V} \sum_{{\bf p}_1,{\bf p}_2}f_{\rm B}(\epsilon_{p_1}^0+\epsilon_{p_2}^0).
\end{equation}
Here, $f_{\rm B}(z)=[\exp(z/T)-1]^{-1}$ denotes the Bose distribution, with the free two-particle energies $\epsilon^0_{\nu,{\bf P}}=E^0_{\nu,{\bf P}}-\mu_1-\mu_2$. As already said, $\nu$ describes the intrinsic state of the two-body system, in particular bound and scattering states. The last term in (\ref{BU0}) describes the free particle contribution contained in the scattering states which must be subtracted.
Considering only the bound state part of the sum over the intrinsic quantum number $\nu$, see Eq. (\ref{G2op}),
the mass action law (nuclear statistical equilibrium, NSE) is reproduced for the two-nucleon bound-state formation.
In contrast to the NSE, the sum over the intrinsic quantum number $\nu$ in (\ref{BU0}) includes also the scattering states.
This is inevitable to obtain the correct second virial coefficient, which determines the second order of
density in the EoS $\mu(T,n)$. For the scattering states, we introduce, in addition to the intrinsic spin state $\alpha$,
the relative momentum ${\bf p}_{\rm rel}$
instead of the intrinsic quantum number $\nu$. Similarly, ${\bf p}_1,{\bf p}_2$ is
replaced by ${\bf P}, {\bf p}_{\rm rel}$ in the last contribution of Eq. (\ref{BU0}).
We transform to the energy $E_{\rm rel}=p_{\rm rel}^2/m$ and replace the summation over ${\bf p}_{\rm rel}$
by the integral over $E_{\rm rel}$ where the density of states $D(E_{\rm rel})$ is introduced.
For the free two-nucleon propagator,
the density of states is simply given by $D^0(E_{\rm rel})=Vm \,p_{\rm rel}/(4 \pi^2)$.
For the interacting two-particle system,
the density of states is related to the scattering phase shift $\delta_{\alpha, {\bf P}}(E_{\rm rel})$
in the channel $\alpha$ (describing the spin/isospin state), see \cite{LL1980}, Sect. VII, so that we have for the scattering part
\begin{equation}
D^{\rm scat}_{\alpha, {\bf P}}(E_{\rm rel})=g_\alpha \left(V \frac{m \,p_{\rm rel}}{4 \pi^2}+\frac{\partial}{\partial E_{\rm rel}} \delta_{\alpha, {\bf P}}(E_{\rm rel}) \right),
\end{equation}
where $g_\alpha$ denotes the degeneracy factor such as $g_\alpha=3$ for the spin-triplet channel where the deuteron is formed.
Altogether, from the QS approach we obtain the
Beth-Uhlenbeck formula \cite{SRS} (the first part of $D^{\rm scat}_{\alpha,{\bf P}}(E_{\rm rel})$ is compensated by the free nucleon contribution $D^0(E_{\rm rel}) $),
\begin{equation}
\label{BU}
n^{\rm BU}(T, \mu)=\frac{1}{V} \sum_{\bf p}f(\epsilon_p^0)+\frac{2}{V} \sum_{\alpha,{\bf P}}\int_{-\infty}^\infty \frac{dE_{\rm rel}}{\pi}
f_{\rm B}\left(E_{\rm rel}+\frac{P^2}{4m}-\mu_1-\mu_2 \right) D^{\rm BU}_{\alpha, {\bf P}}(E_{\rm rel}),
\end{equation}
where we implemented the contribution of the bound states (second term of the right-hand side of (\ref{BU0}) so that
\begin{equation}
\label{DBU}
D^{\rm BU}_{\alpha, {\bf P}}(E_{\rm rel})=g_\alpha \left(\sum_{\nu'} \pi \delta(E_{\rm rel}-E^0_{\alpha \nu', {\bf P}})
+\frac{\partial}{\partial E_{\rm rel}} \delta_{\alpha, {\bf P}}(E_{\rm rel}) \right)
\end{equation}
is the density of states which contains the bound state energy
$E^0_{\alpha \nu', {\bf P}}=E^0_{\alpha \nu'}+{\bf P}^2/4m$ and scattering phase shift $\delta_{\alpha, {\bf P}}(E)$ as function of the
energy $E$ of relative motion. For arbitrary mass numbers $A$, the intrinsic quantum number $\nu'$ denotes ground as well as possible excited bound sates in the spin/isospin channel $\alpha$.
Instead of a separate sum over bound states, cf. Eq. (\ref{BU0}),
we included the bound state contribution as $ \pi \delta(E-E^0_{\alpha \nu', {\bf P}})$ in the density of states,
contributing at negative values of the general variable $E_{\rm rel}$.
Note that for Breit-Wigner resonances, see \cite{LL3}, Chapter XVII,
\begin{eqnarray}
\mbox{tan} \delta_{\rm R} =-\frac{\Gamma_{\rm R}}{2M}\,,\quad {\rm{and}}\quad \frac{\partial \delta_{\rm R}}{\partial E} = \frac{A}{2} =\frac{\Gamma_{\rm R}/2}{M^2 +\Gamma_{\rm R}^2/4}\,,
\end{eqnarray}
where $M=E-E_{\rm R}\,,$ $E=E_{\rm R}-{\mathrm i} \Gamma_{\rm R}/2\,,$
cf. Appendix \ref{A1}, Eq. (\ref{eqrel1}). More general relations can be found in
\cite{Kolomeitsev:2013du}.
The Beth-Uhlenbeck formula is an exact expression for the second virial coefficient $b$ defined by
the virial expansion of the EoS \cite{BU,R,SRS} (valid for $n_\tau \Lambda^3 \ll 1$). At given $T$, the pressure is considered as function of the density, and the prefactors of a power expansion are denoted as virial coefficients $b$. This power expansion can also be done for the chemical potential, and the inversion gives the relations
\begin{eqnarray}\label{bnnp}
&&n_n(T,\mu_n,\mu_p)=\frac{2}{\Lambda^3} \left[b_n(T) e^{\mu_n/T}+2b_{nn}(T)e^{2 \mu_n/T}+2 b_{np}(T) e^{(\mu_n+\mu_p)/T}
+\dots \right],
\nonumber \\
&&n_p(T,\mu_n,\mu_p)=\frac{2}{\Lambda^3} \left[b_p(T) e^{\mu_p/T}+2b_{pp}(T)e^{2 \mu_p/T}+2 b_{pn}(T) e^{(\mu_n+\mu_p)/T}
+\dots \right],
\end{eqnarray}
where $ \Lambda^2=2 \pi /m T $. As well known, in the low-density limit all systems behave at finite $T$ like ideal classical gases so that $b_n(T)=b_p(T)=1$.
The second virial coefficient contains the effects of degeneration as well as interaction terms.
In particular, we have after performing integration by parts and using the Levinson theorem
\begin{equation}\label{bnn}
b_{nn}(T)=-\frac{1}{2^{5/2}}+ \frac{1}{2^{1/2} \pi T} \int_0^\infty dE e^{-E/(2T)} \delta_{nn}(E)
\end{equation}
and \begin{equation}\label{bnp}
b_{np}(T)=\frac{3}{2^{1/2}} [e^{E^0_d/T}-1]+ \frac{1}{2^{3/2} \pi T} \int_0^\infty dE e^{-E/(2T)} \delta_{np}(E)\,.
\end{equation}
The deuteron (binding energy $E^0_d=2.225$ MeV) arises in the isospin-singlet, spin-triplet channel so that the degeneracy factor is $g_d=3$. The scattering phase shifts $\delta_{nn}(E),\delta_{np}(E)$ are given by the contributions of the different channels, see also \cite{R,SRS,HS} for details. Assuming symmetric matter we have $b_{pp}=b_{nn},b_{pn}=b_{np} $ provided Coulomb effects are disregarded.
From (\ref{Stf}) and (\ref{bnnp}) we find relations between the partial static structure factors and the virial coefficients (\ref{bnn}), (\ref{bnp}):
\begin{eqnarray}
&&S_{nn}(T,\mu_n,\mu_p)=
\frac{2}{\Lambda^3}\left[b_n(T) e^{\mu_n/T}+4b_{nn}(T)e^{2 \mu_n/T}+2 b_{np}(T) e^{(\mu_n+\mu_p)/T}
+\dots \right]\,, \\
&&S_{np}(T,\mu_n,\mu_p)=\frac{4}{\Lambda^3}b_{pn}e^{(\mu_n+\mu_p)/T}+...\,,\nonumber\\
&&S_{pp}(T,\mu_n,\mu_p)=
\frac{2}{\Lambda^3}\left[b_{p}(T)e^{\mu_p/T}+4b_{pp}(T)e^{2\mu_p/T}+
2b_{pn}(T)e^{(\mu_n+\mu_p)/T}+\dots \right]\,.\nonumber
\end{eqnarray}
One can also express $S_{nn}$, $S_{np}$, $S_{pp}$ in terms of $n_n$ and $n_p$ variables. Simple explicit expressions are obtained for the second order with respect to density [the second virial coefficient of $S_{\tau, \tau'}(T,n_n,n_p)$].
Using the virial expansion of the density EoS (\ref{bnnp}) we can also perform the virial expansion of the incompressibility (\ref{incompr}) and, using Eq. (\ref{fi0}), for the excess quantity $\varphi_0(T,n)$. For symmetric matter ($Y_p=0.5)$ we obtain the virial expansion
\begin{equation}
\label{virphi}
\varphi_0(T,n)=-\frac{n \Lambda^3}{2}(b_{nn}+b_{np}-b_{nn}^{(0)}) + {\cal O}(n^2)= \varphi^{(1,{\rm sym})}_0(T)n+ {\cal O}(n^2),
\end{equation}
where $b_{nn}^{(0)}=-2^{-5/2}$ is the contribution of the ideal Fermi gas. Using the second virial coefficients of Ref. \cite{HS}, values for $\varphi^{(1)}_0(T)$ are given in Tab. \ref{tab:1}.
We add the corresponding results for neutron matter ($Y_p=0$)
\begin{equation}
\label{virphin}
\varphi^{(1,{\rm neut})}_0(T)=-\Lambda^3\left(b_{nn}+\frac{1}{2^{5/2}}\right)\,.
\end{equation}
\begin{table}[ht]
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$T$ & $\varphi^{(1,{\rm sym})}_0(T)$ [fm$^3$]& $\varphi^{(1,{\rm neut})}_0(T)$ [fm$^3$]\\
\hline
1& -41785.9 & -1954.2 \\
2& -4888.2 & -713.2 \\
3& -1815.3 & -390.6 \\
4& -966.2 & -254.3 \\
5& -606.7 & -182.3 \\
6& -421.5 & -138.7 \\
7& -311.8 & -110.1 \\
8& -241.3 & -90.26 \\
9& -193.6 & -75.80 \\
10& -159.3 & -64.85\\
\hline
\end{tabular}
\caption{\label{tab:1}%
Excess virial term $\varphi^{(1)}_0(T)$, Eq. (\ref{virphi}), of the incompressibility (\ref{fi0}) of nuclear matter
for different values of $T$. Results for symmetric matter $\varphi^{(1,{\rm sym})}_0(T)$ and neutron matter $\varphi^{(1,{\rm neut})}_0(T)$ using the virial coefficients of \cite{HS}.}
\end{table}
The second virial coefficient is a benchmark for the low-density behavior of $n(T, \mu)$.
The EoS (\ref{BU}) is derived within the Green function approach
if the self-energy is taken in the two-nucleon ladder (binary collision) approximation.
It can be extended to higher densities if the medium modifications are taken into account,
for instance in mean-field QPA \cite{SRS}.
In particular, disappearance of
bound states at increasing density because of Pauli blocking is described, see \cite{R}.
Within the real-time Green function technique, the corresponding improvements are described in \ref{sec:quasi}.
\subsection{Derivation of the Beth-Uhlenbeck formula from the dynamic structure factor}
After demonstrating firstly the inclusion of cluster formation via the normalization condition (\ref{neos}), we investigate a second approach to the EoS starting from the dynamic structure factor calculating the isothermal compressibility (\ref{isot}).
A cluster decomposition of the van Hove function $L$ (\ref{Spectral}) can be performed which gives the possibility to include the
contribution of clusters (nuclei). This can be done as shown in \cite{RD,RMA93}.
To reproduce the Beth-Uhlenbeck formula which accurately describes the contribution of two-nucleon correlations in the low-density limit,
we search for the diagrams in the evaluation of the density-density correlation function which are necessary to reproduce this result. Higher order terms of interaction in the perturbation expansion, see Fig.~\ref{fig:1a}, will not produce bound state contributions.
To account for the formation of bound states, we have to add diagrams
where the single nucleon propagators in the one-loop diagram, Fig.~\ref{fig:1}, are replaced
by two-nucleon propagators as described by ladder sums \cite{RD,RMA93}. This leads us to
the cluster decomposition of the density-density Green function given tentatively in Fig. \ref{fig:Lcl}, in
analogy to the cluster decomposition of the self-energy, Fig. \ref{fig:SE}.
\begin{figure}[!th]
\includegraphics[width=16cm]{Lcluster1.pdf}
\caption{Cluster decomposition of the polarization function. Cluster-particle propagation
is symbolized by the multiple particle propagators, see Eq. (\ref{G2op}),
and the vertex is introduced according to Eq. (\ref{Matrix}).}
\label{fig:Lcl}
\end{figure}
The loop diagram formed by free single-particle propagators (\ref{Lo}),
which describes the low-density limit, is completed by loop diagrams
consisting of the two-particle propagator (\ref{G2op}), the three-particle propagator, etc.,
which are given by the corresponding ladder Green functions $G_A$. We find
\begin{equation}
\label{Lclu}
L({\bf q},{\mathrm i} z_\lambda)=L_0({\bf q},{\mathrm i} z_\lambda)+L^{(0)}_2({\bf q},{\mathrm i} z_\lambda)+\dots
\end{equation}
The first-order contributions of the perturbation expansion $L_1({\bf q},z)$, see Fig. \ref{fig:1a},
are contained in $L^{(0)}_2({\bf q},z)$.
An accurate description of the cluster decomposition has to avoid unconnected diagrams and double counting.
A more detailed discussion is given below.
We consider here only the contribution of two-particle correlations $L^{(0)}_2({\bf q},{\mathrm i} z_\lambda)$.
The two-particle propagator in eigen-representation reads
\begin{equation}
\label{clusterprop}
\langle \nu,{\bf P}|G_2(z)|\nu',{\bf P}' \rangle = \frac{1}{z-E^0_{\nu,P}} \delta_{\nu\nu'} \delta_{{\bf P},{\bf P}'}.
\end{equation}
The two-particle vertex which is related to the density fluctuation describes, for instance,
the coupling to the interaction propagator. The coupling of the cluster to the interaction is given by the matrix element given in Fig. \ref{fig:3}. The crosses denote amputation, i.e. multiplication with $[G_2^{(0)}]^{-1}$.
\begin{figure}[!th]
\includegraphics[width=8cm]{RMAfig31.pdf}
\caption{Vertex matrix element for the two-particle contribution $L^{(0)}_2({\bf q},{\mathrm i} z_\lambda)$
to the polarization function.}
\label{fig:3}
\end{figure}
Using the two-nucleon eigen-states $|\nu,{\bf P}\rangle$ with intrinsic quantum number $\nu$ and c.m. momentum ${\bf P}$, we define this matrix element according to
\begin{equation}
\label{Matrix}
M_{\nu \nu'}({\bf q})=\langle \nu,{\bf P}|M({\bf q}, {\mathrm i} z_\lambda, {\mathrm i} z_\mu) |\nu',{\bf P}+{\bf q} \rangle = \sum_{{\bf p}_1,{\bf p}_2} \psi^*_{\nu,{\bf P}}(p_1,p_2)
[\psi_{\nu',{\bf P}+{\bf q}}({\bf p}_1+{\bf q},{\bf p}_2)+\psi_{\nu', {\bf P}+{\bf q}}({\bf p}_1,{\bf p}_2+{\bf q})].
\end{equation}
For the two-particle contribution to the van Hove function we obtain (see Fig. \ref{fig:2})
\begin{equation}
L^{(0)}_2({\bf q}, z)=\sum_{\nu\nu',{\bf P}}\frac{f_{\rm B}(\epsilon^0_{\nu,{\bf P}})-f_{\rm B}(\epsilon^0_{\nu', {\bf P}+{\bf q}})}{z+\epsilon^0_{\nu,{\bf P}}-\epsilon^0_{\nu', {\bf P}+{\bf q}}}|M_{\nu\nu'}({\bf q})|^2 -\sum_{{\bf p}_1,{\bf p}_2}
\frac{f_{\rm B}(\epsilon^0_{p_1}+\epsilon^0_{p_2})-f_{\rm B}(\epsilon^0_{{\bf p}_1+{\bf q}}+\epsilon^0_{p_2})}{z+\epsilon^0_{p_1}-\epsilon^0_{{\bf p}_1-{\bf q}}}\,.
\end{equation}
\begin{figure}[!th]
\includegraphics[width=8cm]{RMAfig2c.pdf}
\caption{Two-particle contribution $L^{(0)}_2({\bf q},{\mathrm i} z_\lambda)$ to the polarization function. The cluster propagator is given by Eq. (\ref{clusterprop}). The free two-particle contribution
has to be subtracted because it is a disconnected diagram.}
\label{fig:2}
\end{figure}
After inserting this contribution in the expression for the dynamic structure factor as discussed for the single nucleon contribution,
we transform $[f_{\rm B}(\epsilon^0_{\nu,P})-f_{\rm B}(\epsilon^0_{\nu',{\bf P}+{\bf q}})]f_{\rm B}(\epsilon^0_{\nu,P}-\epsilon^0_{\nu',{\bf P}+{\bf q}})=f_{\rm B}(\epsilon^0_{\nu,P}) [1+f_{\rm B}(\epsilon^0_{\nu',{\bf P}+{\bf q}})]$.
In the limit ${\bf q} \to 0$ the expression $f_{\rm B}(\epsilon^0_{\nu,P}) [1+f_{\rm B}(\epsilon^0_{\nu',P})]$ follows.
The matrix elements (\ref{Matrix}) are simplified in the limit ${\bf q} \to 0$. Using the completeness relation $ \sum_{{\bf p}_1,{\bf p}_2} |{\bf p}_1,{\bf p}_2 \rangle \langle {\bf p}_1,{\bf p}_2 |= {\rm I}$ the summation over ${\bf p}_1,{\bf p}_2$
gives
\begin{equation}
\lim_{{\bf q} \to 0}M_{\nu\nu'}({\bf q})=2 \langle \nu,{\bf P}|\nu',{\bf P}\rangle=2 \delta_{\nu,\nu'}.
\end{equation}
The two-nucleon contribution to the dynamic structure factor (\ref{SL}) and the isothermal compressibility (\ref{isot}) is calculated
using
\begin{equation}
\label{iso2}
\kappa^{(2)}_{\rm iso}(T,\mu)
=\frac{1}{n^2T}\left\{ \sum_{\nu,{\bf P}} f_{\rm B}(\epsilon^0_{\nu,P}) [1+f_{\rm B}(\epsilon^0_{\nu,P})] -\sum_{{\bf p}_1,{\bf p}_2} f_{\rm B}(\epsilon^0_{p_1}+\epsilon^0_{p_2}) [1+ f_{\rm B}(\epsilon^0_{p_1}+\epsilon^0_{p_2})]\right\}\,.
\end{equation}
Quite similar as in the derivation of the Beth-Uhlenbeck formula in Sec. \ref{sec:BU}, the second contribution
owing to the free nucleon states compensates the divergent part of the scattering states in the first contribution of Eq. (\ref{iso2}),
caused by the disconnected diagrams arising from the zeroth order of the ladder sum for the two-particle propagator.
Selecting only the bound state part of $\kappa^{(2)}_{\rm iso}(T,\mu)$, a mass action law is obtained as known from the NSE.
As demonstrated in Sec. \ref{sec:BU}, the summation over $\nu$ can be replaced by an integral over $E_{\rm rel}$ where the density of
states $D^{\rm BU}_{\alpha, {\bf P}}(E_{\rm rel})$ (\ref{DBU}) appears, and the summation over the (spin, isospin) channels $\alpha$.
The following expression for the isothermal compressibility is observed
\begin{eqnarray}
\label{kapBU}
&&\kappa^{(\rm BU)}_{\rm iso}(T,\mu_n,\mu_p)=\frac{1}{V n^2T}
\left\{ \sum_{\bf p} f(\epsilon_p^0) (1-f(\epsilon_p^0))
+ \sum_{\alpha, {\bf P}} \int_{-\infty}^\infty \frac{d E}{\pi} f_{\rm B}\left( \epsilon^0_{\alpha,P} \right) \left[1+f_{\rm B}\left(\epsilon^0_{\alpha,P} \right)\right]
D^{\rm BU}_{\alpha, {\bf P}}(E)\right\}\,.
\end{eqnarray}
It is easily shown that this expression is consistent with the Beth-Uhlenbeck formula (\ref{BU}) given above.
\subsection{The nuclear statistical equilibrium model}
\label{sec:NSE}
Starting from the dynamic structure factor and the isothermal compressibility, we presented the way
how to implement the formation of clusters. The cluster decomposition of the polarization function $L({\bf q}, z) \approx L_0({\bf q}, z)+L^{(0)}_2({\bf q}, z)+\dots+L^{(0)}_A({\bf q}, z)$
gives the possibility to include also larger clusters, i.e. in addition to the deuteron also the other light elements $^3$H, $^3$He, and $^4$He as well as larger clusters, characterized by the mass number $A$ and the intrinsic quantum state $\nu$
(including spin and isospin as well as excitation level).
Note that this cluster decomposition gives the correct virial limit if in addition to the bound states also scattering states are taken into account. For instance, $L^{(0)}_2({\bf q}, z)$
produces also the mean-field quasiparticle shifts shown in Fig. \ref{fig:1a}.
We have to consider the contribution $L^{(0)}_A({\bf q}, z)$ containing the $A$-particle propagator. This leads to the corresponding $A$-nucleon in-medium wave equation, see \cite{R}. We will consider here only the low-density limit where the solution of the
$A$-nucleon wave equation gives as bound states the nuclei with mass number $A$.
We also restrict us to only the bound state contribution neglecting contributions of the continuum.
The calculation of the compressibility is along to the former case, Eqs. (\ref{eq:22}), (\ref{iso2}) and gives
\begin{eqnarray}
\kappa^{(\rm NSE)}_{\rm iso}(T,\mu_n,\mu_p)&=&\frac{1}{V n^2T}
\left\{ \sum_{A=1,3,\dots}\sum_{\nu}^{\rm bound}g_{A,\nu} \sum_{\bf P} f(\epsilon^0_{A,\nu,{ P}}) [1- f(\epsilon^0_{A,\nu,{ P}})]
\right. \nonumber \\ && \left. + \sum_{A=2,4,\dots}\sum_{\nu}^{\rm bound} g_{A,\nu} \sum_{\bf P} f_{\rm B}(\epsilon^0_{A,\nu,{ P}}) [1+ f_{\rm B}(\epsilon^0_{A,\nu,{ P}})]\right\}\,.
\end{eqnarray}
with the Fermi function for odd numbers $A$ and the Bose function for even $A$.
$g_{A\nu}$ denotes the degeneracy of the cluster with mass number $A$ and the intrinsic quantum state $\nu$.
The binding energies $ E^0_{A,\nu,{ P}} $
of that clusters determine $\epsilon^0_{A,\nu,{ P}} =E^0_{A,\nu,{ P}}-Z \mu_p-(A-Z) \mu_n$, $Z$ denotes the proton number.
It is easily shown that this result corresponds to the standard result \cite{RMS}
\begin{eqnarray}
\label{kapNSE}
n^{(\rm NSE)}(T,\mu_n,\mu_p)&=&\frac{1}{V }
\sum_{A=1,3,\dots}\sum_{\nu}^{\rm bound}g_{A,\nu} \sum_{\bf P} f(\epsilon^0_{A,\nu,{ P}})
+\frac{1}{V }\sum_{A=2,4,\dots}\sum_{\nu}^{\rm bound} g_{A,\nu} \sum_{\bf P} f_{\rm B}(\epsilon^0_{A,\nu,{ P}}).
\end{eqnarray}
Thus we are convinced that clusters are correctly implemented in the alternative approach which is based on
the evaluation of the dynamic structure factor. Not the improvement of the single nucleon quasiparticle approach,
but the cluster expansion of the density-density Green function leads to the consistent treatment of bound nuclei.
The inclusion of scattering states becomes complex for $A >2$, see \cite{R}.
A cluster-virial expansion which treats the cluster-scattering states has been discussed in Ref. \cite{clustervirial}.
\section{Compressibility including cluster formation}
\label{comp}
Considering the isothermal compressibility as the key quantity of the present work,
we find the expression $\kappa^{(\rm BU)}_{\rm iso}(T,\mu_n,\mu_p)$ (\ref{kapBU})
in the low-density limit which tells us that the cluster contributions enter additively
the low-density limit, see also Eq. (\ref{kapNSE}).
A challenge is the extension of the cluster decomposition of the density-density Green function to higher densities where
self-energy and Pauli blocking must be included. We will not investigate this problem here but give only some brief comments.
For the single-nucleon contribution (first term in Eq. (\ref{kapBU})), the quasiparticle picture, Sec. \ref{sec:quasi}, can be introduced which allows to describe also
nuclear systems
near the saturation density.
This quasiparticle concept has been proven to be very efficient to describe matter at high densities.
It has been demonstrated in Sec. \ref{sec:mean} that the quasiparticle concept can also be introduced in the dynamic structure factor approach derived from the density-density Green function. However, in comparison with the
density EoS approach (\ref{neos}), more effort is needed within the density-density Green function approach, see Sec. \ref{sec:mean}, and in addition to the self-energy, also vertex terms have to be considered.
However, the latter approach opens the access to further quantities such as dynamic properties and transport processes. As pointed out in this work, similar concepts known from the density-EoS approach (\ref{neos})
can also applied to the cluster decomposition of $L$ (\ref{Lclu}) so that both approaches are equivalent.
Of interest is what we expect for the compressibility if cluster formation is taken into account.
We give some results valid for dilute warm nuclear matter according to Ref. \cite{R}.
The solution of the effective $A$-nucleon in medium wave equation leads to medium-dependent shifts of the bound state energies.
This can be interpreted as medium-dependent quasiparticle energies of the bound states.
Starting from the normalization condition which gives the density as function of $T, \mu_\tau$,
a generalized Beth-Uhlenbeck formula accounting for in-medium corrections was found \cite{SRS}
which includes quasiparticle-like bound states
as well as in-medium scattering states,
\begin{eqnarray}
\label{kapgBU}
&&\kappa^{(\rm BU)}_{\rm iso}(T,\mu_n,\mu_p)=\frac{1}{V n^2T}
\left\{ \sum_{\bf p} f(\epsilon_p) (1-f(\epsilon_p))
+ \sum_{\alpha, {\bf P}} \int_{-\infty}^\infty \frac{d E}{\pi} f_{\rm B}\left( \epsilon_{\alpha,P} \right) \left[1+f_{\rm B}\left(\epsilon_{\alpha,P} \right)\right]
D^{(\rm gBU)}_{\alpha, {\bf P}}(E)\right\}\,
\end{eqnarray}
with (after partial integration and using the Levinson theorem, see \cite{R})
\begin{equation}
\label{DgBU}
D^{\rm gBU}_{\alpha , {\bf P}}(E_{\rm rel})=g_\alpha \frac{1}{T}\left(\sum^{\rm bound}_\nu \pi [\delta(E_{\rm rel}-E_{\alpha \nu { P}})-\delta(E_{\rm rel})]
+[ \delta_{\alpha, { P}}(E_{\rm rel})-\frac{1}{2}\sin(2\delta_{\alpha ,{ P}}(E_{\rm rel}))] \right)\,.
\end{equation}
In contrast to the ordinary Beth-Uhlenbeck formula (\ref{kapBU}),
the free single particle energies $E^0_p$ are replaced by the quasiparticle energies $E_p(T,n,Y_p)$
which are depending on temperature and densities.
However, caution is needed to avoid double counting.
If low-order contributions of the ladder sum have been used already
to define the quasiparticles such as the Hartree-Fock shifts,
they have to be eliminated from the two-particle contributions.
This is the reason for the appearance of the sin-term at the end of Eq. (\ref{DgBU})
which is obtained from a consistent treatment of the optical theorem \cite{SRS}.
In addition, the generalized Beth-Uhlenbeck formula contains also medium-modified
binding energies $E_{\alpha \nu { P}}(T,n,Y_p)$ and scattering phase shifts $\delta_{\alpha ,{ P}}(E_{\rm rel};T,n,Y_p) $ depending on $T,n_\tau$ which are obtained from an in-medium Schr{\"o}dinger equation.
An important fact is the disappearance of bound states at increasing density because of Pauli blocking.
Using the density EoS of dilute warm nuclear matter obtained in \cite{R},
results for the excess contribution to the incompressibility $\varphi_0(T,n)$ (\ref{fi0}) are shown in Fig. \ref{fig:9} for $T=5$ MeV as function of the density. The region of spinodal instability is given
by the condition $\varphi_0(T,n) < -1$. Within the RMF approximation, a smooth behavior is obtained, the region of instability is shown. For zero temperature, in this RMF approximation the result for $f_0$ shown in Fig. \ref{fig:0}
is obtained. To investigate the influence of light cluster ($d, t, h, \alpha$) formation, the excess quantity
$\varphi_0(T,n)$ is also shown as obtained from the quantum statistical approach, extending the generalized
Beth-Uhlenbeck formula (\ref{kapgBU}) by including all cluster with $A\le 4$, see Ref. \cite{R}.
Strong deviations are found in the low-density region owing to the formation and dissolution of clusters. This is already seen from the virial expansion (\ref{virphi}) of the excess quantity $\varphi_0(T,n)$.
The evaluation of the term linear in the density for symmetric matter at $T=5$ MeV within the DD2-RMF approximation
(see Appendix \ref{DD2}) gives $\varphi_0^{\rm RMF}(T=5\, {\rm MeV},n)=-165.5\, n\, {\rm fm}^3+{\cal O}(n^2)$.
This value is only slightly different from the zero-temperature result $\varphi_0^{\rm RMF}(T=0,n)=-202.4\, n\, {\rm fm}^3+{\cal O}(n^2)$. An accurate result for symmetric matter at $T=5$ MeV,
\begin{equation}
\varphi_0(T=5\,{\rm MeV},n)=-606.73\, n\, {\rm fm}^3+{\cal O}(n^2)
\end{equation}
is obtained using the data for the second virial coefficients \cite{HS}, see also Tab. \ref{tab:1}. As shown in Fig. \ref{fig:9}, the RMF QPA fails to reproduce this benchmark.
In particular, two regions of spinodal instability are obtained, and in between a region of metastability occurs.
This result is also directly seen from the EoSs shown in \cite{R}. The influence of cluster formation is significant for densities below 0.05 fm$^{-3}$. The mass fraction of $\alpha$ particles is large near the baryon density 0.01 fm$^{-3}$, see \cite{R} for the composition at $T=5$ MeV. At higher densities the bound states disappear because of Pauli blocking, and the RMF QPA is applicable.
Coming back to the second approach to the thermodynamics of nuclear matter,
the investigation of the dynamic structure factor is expected to give results
for the compressibility identical with the density EoS approach (\ref{neos}) as shown in the low-density limit, but more effort is needed to treat the corrections at higher density. However, this alternative approach is able to give interesting properties with respect to the dynamic behavior of nuclear systems
not discussed in the present work.
\begin{figure}[!th]
\includegraphics[width=10cm]{MIFI2017phi1.pdf}
\caption{Excess contribution $\varphi_0(T, n)$ to the incompressibility, Eq. (\ref{fi0}), at $T=5$ MeV for symmetric matter ($Y_p=0.5$). The DD2-RMF approximation, see App. \ref{DD2}, is compared with the result accounting for light cluster formation \cite{R}.}
\label{fig:9}
\end{figure}
\section{Discussion and conclusions}
\label{Conclusions}
Our aim is to find the relation between the well-established Landau Fermi-liquid approach to nuclear systems and other
approaches such as the QS approach based on the normalization condition. In particular, we are interested in the problem how
the formation of light clusters such as $d,t,h,\alpha$ can be described. Previous work such as \cite{KV16} pointed out this problem.
There, it was not clear how the formation of bound states can be included in an approach where nucleons are described as quasiparticles.
To solve this problem we propose to consider a fundamental quantity, the density-density correlation function which is related to the
dynamic structure factor. The compressibility is included as limiting case and can be used to derive the nuclear matter EoS.
In addition, several other properties are related to the dynamic structure factor, equilibrium as well as non-equilibrium. Therefore, the density-density correlation function is an
important quantity by itself, and the systematic quantum-statistical treatment of cluster formation is an
actual problem. We found a solution of this problem performing a cluster decomposition of the density-density correlation function, i.e. using the concept of the chemical picture. Ladder sums describing few-body correlations are
considered as new element of a diagram representation of the thermodynamic Green functions. A cluster decomposition of
the relevant quantities, such as the self-energy or the polarization function, allows for the description of bound-state formation.
We show that this alternative approach leads to the same results known from the QS approach using the density EoS (\ref{neos}).
In particular, the exact results for the second virial coefficient, the Beth-Uhlenbeck formula, are obtained in both approaches.
Considering the isothermal compressibility as the key quantity of the present work,
we find the expression $\kappa^{(\rm BU)}_{\rm iso}(T,\mu_n,\mu_p)$ (\ref{kapBU})
in the low-density limit which tells us that the cluster contributions enter additively
the low-density limit, see also Eq. (\ref{kapNSE}). The excess contribution to the incompressibility $\varphi_0(T,n)$ (\ref{fi0}) is an
important quantity describing the interaction effects as well as the cluster formation in nuclear matter.
The relation to the Landau Fermi-liquid approach is found as follows: In the high-density (near the saturation density),
zero temperature case the nuclear system is degenerate, correlations are taken into account by the quasiparticle picture,
and bound state formation is suppressed because of Pauli blocking.
The dynamic structure factor and the isothermal compressibility are obtained from the QPA described in
Sec. \ref{sec:quasi}. Instead of the knowledge of the full interaction, only special Landau-Migdal parameters are needed.
We gave the parameter values comparing with well accepted parametrizations
of nuclear matter properties within the RMF, in particular DD2.
However, to include bound state formation what is essential in the low-density region,
we have to go beyond the Landau Fermi-liquid approach. Further diagrams for the dynamic
structure factor must be considered which appear in a cluster decomposition of the polarization function. They can be added to the
quasiparticle contribution to the isothermal compressibility. However, double counting has to be avoided.
Similar expressions may also be derived from the density-density Green function
approach. The medium-modified solutions of the two-nucleon problem have to be implemented in the cluster decomposition of the polarization function.
The relation of the single-quasiparticle contribution to the Fermi-liquid approach opens possibilities to treat non-equilibrium, inhomogeneous
processes in nuclear systems so that it is of relevance to have a consistent
description of equilibrium properties. However, this is sufficient only for parameter values
where cluster formation can be neglected. We have shown that additional contributions to the polarization function are obtained considering the two-particle clusters.
In particular, we have shown how the second virial coefficient appears.
It is rather difficult to include density effects for arbitrary cluster size $A$.
We expect similar results as the generalized Beth-Uhlenbeck formula.
It was not the aim of this work
to reproduce all sophisticated results obtained in the QS approach \cite{R} until now,
starting from the normalization condition (\ref{neos}).
We only demonstrated how to proceed with the density-density Green function approach to include
bound state formation, and standard results were reproduced like the second virial coefficient.
This is of interest because the polarization function is related to many physical properties,
including excitations and transport properties. In particular, the dynamic structure factor
is obtained, and it has been shown how the influence of clustering can be considered
using the cluster decomposition of the polarization function. \\
{\bf Acknowledgements.} D.N.V. and D.B. were supported by the
Russian Science Foundation, Grant No. 17-12-01427. D.N.V. was also supported by the Ministry of Education and Science of the Russian Federation within the state assignment, project No 3.6062.2017/BY.\\
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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MissionU founder Adam Braun will become WeGrow's COO, working alongside WeGrow CEO Rebekah Neumann.
"We say 'superpowers,' not 'power,' because we all have many," says Neumann, who sips a hot chocolate at WeWork headquarters while seated on a tangerine and pink sofa–a color scheme selected by her children.
Neumann, who is married to WeWork cofounder and CEO Adam Neumann, unveiled her vision for WeGrow last November. (At the time, she was operating a seven-student pilot.) Today, she is expanding that vision to include online learning, with the acquisition of higher education alternative MissionU. As part of the deal, MissionU founder Adam Braun, who previously founded nonprofit Pencils of Promise, will join WeWork as WeGrow's COO.
MissionU, which will keep staff on hand for the next few months as it winds down operations, raised $3 million in October 2016 and an additional $8.5 million in funding last September. WeWork, which declined to put a price on the deal, funded the acquisition with stock.
While WeGrow will be focused, for now, on children ages 3 through 10, MissionU's core offering and WeWork's other recent education investments hint at an expanding ambition.
MissionU, for example, targeted students ages 19-25, enrolling them in a one-year program designed as preparation for business intelligence jobs at hiring partners like Warby Parker. Instead of writing tuition checks, students committed to paying MissionU 15% of the pretax income for three years, assuming a salary of at least $50,000. MissionU built software that allowed for roughly 90% of the program to be delivered online.
WeWork made its first education acquisition, coding bootcamp Flatiron School, last October. "We are all students for life," Adam Neumann wrote in a blog post announcing the deal. Three months later, the company announced a joint partnership with 2U, which operates online graduate programs for universities including Berkeley and Yale. The deal grants 2U students access to WeWork common spaces, and provides WeWork members with $5 million in 2U scholarships.
The experiences of WeGrow's parents and mentors may offer the first clues as to how lifelong learning might infuse WeWork's global empire. For example, WeGrow plans to include a parent lounge in the WeGrow school space that architect Bjarke Ingels is designing. The lounge will host programming for parents, as well as offer them a place to work while their child is in class. Parents will also be encouraged to engage with some practices, like meditation, alongside their children.
Of the parents who have enrolled in WeGrow, one-third are WeWork employees, one-third are WeWork members, and one-third are from outside of the existing WeWork community.
|
{
"redpajama_set_name": "RedPajamaC4"
}
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\section{Introduction}\label{sec1}
The American Cancer Society estimates that there will be about 54,000 new cases and 11,230 deaths from cancer of the oral cavity and pharynx (throat) in the United States in 2022~\cite{siegel2022cancer}. The incidence rates increased by 0.8\% per year from 2009 to 2018 while the mortality rate increased by 0.4\% per year from 2010 to 2019~\cite{siegel2022cancer}.
Oropharyngeal cancer is one of the commonest types of head and neck cancer, and was traditionally treated with either open surgery or radiotherapy. TORS is a new minimally invasive technique that allows surgeons to remove oropharyngeal cancers with fewer side effects, to help preserve the function of patients after treatment~\cite{chow2020head}.
TORS, however, is challenging because surgeons need to work in a severely constrained environment, through the patient's mouth, superior to the retracted tongue, without haptic feedback, requiring a profound knowledge of the oropharynx and parapharynx anatomy~\cite{moore2012transoral}.
Image-guidance has the potential to improve TORS through anatomy and cancer visualization.
Current TORS image guidance approaches mostly use cross-sectional images to visualize preoperative computerized tomography (CT) or magnetic resonance imaging (MRI)~\cite{desai2008transoral,pratt2018transoral,chan2020augmented}.
CT is the most common intraoperative imaging modality. Comparisons between preoperative and intraoperative CT show that the former is insufficient to visualize the deformed anatomy, while the latter can improve fiducial localization accuracy and task efficiency~\cite{ma2017intraoperative,kahng2019improving}.
A tri-planar CT-based surgical navigation system is presented in~\cite{shi2022surgical}.
Deformable registration from preoperative images to cone-beam CT to augment anatomy structures and surgical planning data in stereoscopic video is presented in ~\cite{liu2015augmented}.
However, intra-operative CT does introduce radiation risk.
US can provide real-time intra-operative imaging for TORS without radiation.
In most previous research, the US transducer was placed in the patient's oral cavity.
Intra-oral US interferes with the TORS robot tools. These must either be removed~\cite{goepfert2015trans,clayburgh2016intraoperative}
or a miniaturized US transducer with limited scanning range must be used instead~\cite{liu2020abstract,green2020integrated,chang2021real}.
Shen~\cite{shen2019framework} has proposed an intraoral 3D US-guided AR system for transoral surgery, using a head-mounted display with AR displaying the tumor model segmented in the 3D US to guide the tumor resection in an {\em ex vivo} tongue. To the best of our knowledge, this is the most similar system to our study, but it does not solve the problem of removing robot tools from the patient's mouth to insert the US transducer. Therefore, to mitigate this problem, we explore a different US transducer placement whose feasibility for TORS is still unknown.
Transcervical US, which places the US transducer outside the patient's neck, can provide continuous US imaging during TORS.
Transcervical 3D US has been used in head and neck procedures such as neck fine-needle puncture~\cite{schipper2014ultrasound} and neck tumor dissection~\cite{snyder2014neck},
as well as in oral cancer diagnosis~\cite{klimek1998three,rebol2008volume,hong2015efficiency}, but its feasibility in TORS has not been studied.
While US is accessible and safe, US images are usually harder to interpret than CT or MRI.
Furthermore, US imaging is subject to a severe tradeoff between imaging resolution and depth; larger depths require lower frequencies for penetration and therefore will have lower image resolution.
US-preoperative image registration can combine the benefits of different image modalities to assist surgical planning. In particular, MRI provides high soft-tissue contrast and cancer imaging that can be exploited by MRI-US registration.
MRI-US registration and real-time US have been used in other surgeries such as prostatectomy~\cite{mathur2019feasibility,kalia2021preclinical}. We hypothesize that they can also be applied to TORS, but the feasibility of MRI-3D US registration in the oropharynx is not well studied. Chen {\em et al.}~\cite{chen2022Feasibility} conducted a feasibility study of MRI-3D US registration for TORS; their results show that the 3D images acquired by a low-frequency 3D transducer might not have sufficient resolution for image registration.
In this research, we propose a proof-of-concept AR system for TORS utilizing transcervical 3D US. To the best of our knowledge, our contributions include: (1) the first work evaluating the feasibility of MRI and freehand 3D US registration in the oropharynx, (2) the first work comparing the differences between two types of 3D US of the oropharyngeal region; (3) the first AR system utilizing both US and MRI in TORS.
\section{Materials and Methods}\label{sec:system}
Fig.~\ref{fig:surgical_worflow} shows the proposed US-guided AR workflow. In the planning stage, preoperative MRI and US are collected for surgical planning and feasibility evaluation of US-guided TORS. Just before surgery, the AR system is calibrated and a preoperative freehand US scanning and MRI-US registration are carried out for an initial AR display of the anatomy model. During surgery, the patient's tongue is retracted, and the surgeon can use the US transducer to scan the tumor side and visualize the US. A US-to-US intraoperative-to-preoperative registration generates the deformation field to update the anatomy model in AR. In this preliminary study, we evaluate the feasibility of the first and the second stage: MRI-US registration and system calibration.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{fig/workflow.png}
\caption{The proposed workflow of the AR-guided TORS.}
\label{fig:surgical_worflow}
\end{figure}
\subsection{System Hardware}\label{sec:system-hardware}
Fig.~\ref{fig:system} shows the components of our system. The da Vinci surgical system (Intuitive Surgical, Sunnyvale, CA) includes a patient side cart and a surgeon's console, which provides the surgeon with an endoscope view.
The US system is a BK3500 with a 14L3 linear 2D transducer (BK Medical, Burlington, MA).
A Polaris Spectra (Northern Digital, ON, Canada) tracks the US transducer.
To enable US probe maneuvering by the surgeon using the da Vinci's third manipulating arm, we designed a novel 3D-printed US transducer holder that allows the surgeon to grasp and move it with the da Vinci ProGrasp in the console, as in Fig.~\ref{fig:holder}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig/system.png}
\caption{The hardware systems and the transformations in the system calibration to transform from the US image frame to the camera image frame.}
\label{fig:system}
\end{figure}
\subsection{System Calibration} \label{sec:system-calibration}
Eq.~\ref{eq:transform} shows the transformations needed to project the US volume onto the camera image ($^iT_{US}$). The endoscopic camera (ECM) location $^{DV}T_{ECM}$ can be read from the da Vinci robot API and the marker on the probe can be tracked by the optical tracker $^{OT}T_{Probe}$. Therefore, the system calibration requires us to find the transformation from US image to the marker on the probe $^{Probe}T_{US}$, from optical tracker to robot $^{DV}T_{OT}$, and the camera projection matrix $M$.
\begin{equation}
^{i}T_{US} =M\ ^{ECM}T_{DV} \ ^{DV}T_{OT} \ ^{OT}T_{Probe}\ ^{Probe}T_{US} \label{eq:transform}
\end{equation}
\textbf{US to Optical Tracker Calibration.}
We used the PLUS toolkit~\cite{Lasso2014a} and SlicerIGT~\cite{ungi2016open} for US-tracker calibration. A 3D-printed marker is attached to the US transducer to localize it in the tracker coordinate frame $OT$.
PLUS enables collecting time-synchronized transducer locations and US images. We first run the temporal calibration to correct the time offset, then we collect a tracked US video sequence imaging the tip of a tracked stylus pointer. The transformation $^{Stylus}T_{tip}$ is known after a pivot calibration. We then use the fiducial registration in 3D Slicer to manually select the stylus tip location in the US images and to estimate a similarity transformation $^{Probe}T_{US}$
\[
\ ^{Probe}T_{US} = {\arg\min}_T \|^{OT}T_{Stylus}\ ^{Stylus}T_{tip} - \ ^{OT}T_{Probe}T\ ^{US}T_{tip}\|_2
\]
where $Probe$ is the coordinate frame of the marker attached to the US transducer.
\textbf{Robot to Optical Tracker Calibration.}
We attach a customized marker with a known marker coordinate origin to the robot tooltip and collect the corresponding points in the tracker and robot coordinate frames. We then estimate a rigid transformation between the corresponding points for $ ^{OT}T_{DV}$.
\[
^{OT}T_{DV} = {\arg\min}_T \|^{OT}T_{marker} - T\ ^{DV}T_{Tip}\|_2
\]
\textbf{Camera Projection Matrix Calibration.}
We use the camera projection calibration algorithm in~\cite{kalia2021preclinical} to estimate the camera projection matrix $M$, which includes the hand-eye calibration matrix (the transformation from ECM to the camera world coordinate system) and the camera intrinsic matrix. The algorithm automatically detects fourteen key points on the robot tool in the robot and image coordinates. By moving the robot tool we will get corresponding spatial 3D and 2D points to estimate $M$.
\begin{figure}
\centering
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=\textwidth]{fig/holder.jpg}
\caption{}\label{fig:holder}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{fig/calibrationv2.png}
\caption{}\label{fig:calibration}
\end{subfigure}
\caption{(a) Image showing the 3D-printed holder and how the robot grasps the US transducer. (b) Images of the system calibration. From left to right: (1) the tracked US transducer and stylus; (2) a US image of the stylus tip; (3) a robot tool holding our designed marker for robot-tracker calibration. }
\label{fig:us_calib}
\end{figure}
\subsection{MRI-3D US Registration}
For MRI-US registration we need to find a spatial transformation $^{MRI}T_{US}$. After the registration, we can map the MRI volume or the anatomy model generated from the MRI to the camera image by $^{i}T_{MRI} = \ ^{i}T_{US}\ ^{US}T_{MRI}$.
To simplify the transformation in the AR system, we assume $^{MRI}T_{US}$ is an affine transformation. We first manually rigidly pre-register the 3D US volume and axial T1 MRI based on the US transducer position and anatomy structures. Other MRI sequences can be used during this process. The Linear Correlation of Linear Combination (LC2)-based optimization~\cite{wein2013global} (ImFusion GmbH, Munich, Germany) is used to refine the affine transformation $^{US}T_{MRI}$. We evaluate the MRI-US registration for two different types of 3D US. We first used a 3D US transducer (xMATRIX X6-1, Philips Healthcare, Bothell, WA) to acquire 3D US of the oropharynx in five healthy volunteers and four patients with oropharyngeal cancer, referring the scanning protocol in Coquia {\em et al.}~\cite{coquia2015visualization}.
We then used the BK3500 US machine, 14L3 linear transducer and the optical tracker in Section \ref{sec:system-hardware} and PLUS to record the tracked US image sequences and reconstruct freehand 3D US for three healthy volunteers.
\section{Experiments and Results}\label{sec:experiments}
\subsection{System Evaluation}
To evaluate the accuracy of the robot-tracker calibration, we collected 100 pairs of points and used 50\% to estimate $^{DV}T_{OT}$ and 50\% for testing. The mean projection error is 1.73 mm in the training data and 1.84 mm in the test data.
We then measure the stylus-based calibration error by using 25 pairs of points to estimate $^{probe}T_{US}$ and the mean projection error is 0.59 mm. We used another 17 fiducial pairs for testing and the mean projection error is 0.78 mm.
We evaluated the system accuracy $^{i}T_{US}$ in a water-bath experiment, whose workflow is in Fig.~\ref{fig:grid_projection_exp}. We used a 3D-printed structure and nylon wires to build 25 grid points, with nominal grid size of 10~mm, within the accuracy of the 3D printer.
We used freehand 3D US to image the grids and thresholded by intensity to segment the wires. We then projected the wires from the US to the camera and labeled the grids. We put the structure in three different locations and collected 75 data points. The image size is $540\times960$, and the mean projection error is in Table~\ref{tab:grid_projection_mse}.
The final system demonstration is in Fig.~\ref{fig:AR_render}, projecting the anatomy models segmented from the MRI to the camera with $^iT_{MRI}$ on a human subject. For the mannequin, we used the US to scan the surface and we registered the surface with the skin surface in MRI.
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{fig/projectionv2.png}
\caption{The water-bath experiment. From left to right: (1) original camera image; (2) 3D freehand US volume and wire segmentation; (3) projected segmentation in the camera image; (4) the labeled wires
}
\label{fig:grid_projection_exp}
\end{figure}
\begin{table}[]
\centering
\caption{Mean projection error of the points projected from US to the camera.}
\label{tab:grid_projection_mse}
\begin{tabular}{|c|c|c|}
\hline
& Left Camera & Right Camera \\
\hline
Projection Error (pixel) & $27.85\pm10.52$ & $26.81\pm9.03$\\
\hline
\end{tabular}
\end{table}
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{fig/AR_render.jpg}
\includegraphics[width=0.4\textwidth]{fig/hv_AR_render.jpg}
\caption{Images of the AR guidance system viewed in the camera frame.}
\label{fig:AR_render}
\end{figure}
\subsection{Preliminary Evaluation of MRI-US Registration}
MRI was acquired using a 3T Philips Elition MRI (Philips Healthcare, Bothell, WA) for healthy volunteers, with axial resolution in T1 images of 0.35 to 0.45~mm in the transverse plane and 3.6 to 4.5~mm in the axial direction.
For participants with oropharyngeal cancer a Siemens Magnetom Espree 1.5T MRI (Siemens Healthcare, Erlangen, Germany) was used with a resolution of 0.375~mm in the transverse plane and 3.6~mm in the axial direction. A radiologist (EP) manually selected anatomical landmarks in US and MRI to evaluate the target registration error (TRE).
For the 3D transducer-collected US, the elevation angle is $45-90^\circ$ and the image depth is 7-9 cm. The frequency range is 1-6MHz. The spatial resolution is around $0.3\times0.2\times0.4 \text{mm}^3$. The TRE for MRI-3D probe-collected US is in Table~\ref{tab:TRE} and Fig.~\ref{fig:philips_registration} shows two registration examples. For freehand 3D US, the image depth is 6~cm at 9~MHz and the US volume spacing is 0.1~mm for three axes. We scanned the participant's neck from the common carotid to the submandibular gland, as in Fig.~\ref{fig:freehand_registration}.
We segmented the carotids that appeared in both MRI and US and used Slicer Vascular Modeling Toolkit \cite{hahn1909integration} to extract the centerlines. The TRE and the
average distance between the centerlines are in Table~\ref{tab:freehand_evaluation}: we first interpolate each vessel branch into 1000 points and then calculate the average distance between the corresponding points.
\begin{table}[]
\centering
\caption{The target registration error (TRE) (in mm) for patients or healthy volunteers (HV), and the average TRE for all the cases \cite{chen2022Feasibility}.}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Plane & \multicolumn{2}{c|}{Transverse} & \multicolumn{2}{c|}{Axial} & \multicolumn{2}{c|}{Total} \\
\hline
TRE (mm) & HVs & Patients & HVs & Patients & HVs & Patients \\
\hline
By Group & 7.29±6.50 & 6.45±3.47 & 3.04±4.33 & 5.94±6.23 & 8.26±7.41 & 9.63±5.91 \\
\hline
All & \multicolumn{2}{c|}{6.89±5.32} & \multicolumn{2}{c|}{4.39±5.49} & \multicolumn{2}{c|}{8.90±6.79} \\
\hline
\end{tabular}
\label{tab:TRE}
\end{table}
\begin{table}[]
\centering
\caption{The target registration error (TRE) and the average distance of the vessel centerlines for MRI-freehand 3D US registration.}\label{tab:freehand_evaluation}
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{} & \multicolumn{3}{c|}{TRE (mm)} & \multirow{2}{*}{Average Distance (mm)} \\
& Transverse & Axial & Total & \\
\hline
HV1 & 3.78±0.47 & 5.40±1.80 & 6.63±1.73 & 2.15 \\
\hline
HV2 & 3.18±0.84 & 0.89±1.56 & 3.61±1.01 & 2.13 \\
\hline
HV3 & 5.48±2.15 & 3.45±2.81 & 7.33±0.86 & 2.67 \\
\hline
Average & 4.34±1.88 & 2.88±2.79 & 5.85±2.05 & 2.32±0.25 \\
\hline
\end{tabular}
\end{table}
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{fig/hv2_p1_with_feature.png}
\caption{Examples of MRI-US registration using a 3D US probe \cite{chen2022Feasibility}.}
\label{fig:philips_registration}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.13\textwidth}
\includegraphics[width=\textwidth]{fig/scanning-path.png}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.375\textwidth}
\includegraphics[width=\textwidth]{fig/freehand-registration-us-plane.png}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\textwidth]{fig/freehand-registration-mri-plane.png}
\caption{}
\end{subfigure}
\caption{Examples of MRI-freehand 3D US registration. (a) the scanning trajectory (models from \cite{Lasso2014a}); (b) examples of 2D US, MRI-US overlay and re-sliced MRI; (c) MRI-US overlay of the reconstructed 3D US, MRI, re-sliced US, vessel segmentation and centerlines (yellow: MRI, red: US).}
\label{fig:freehand_registration}
\end{figure}
\section{Discussion}
We evaluate the accuracy of the system calibration in different steps. The US-tracker calibration and the robot-tracker calibration achieve a promising performance, and the water bath experiment shows that our system achieves comparative results with the previous system~\cite{kalia2021preclinical} with an additional optical tracker. Compared with the previous system, our system provides more degrees of freedom for the US transducer. This calibration technique can also be easily generalized to US guidance for other surgeries. The final system accuracy from the US to the camera is approximately 27 pixels, or 2.5\% of the image diagonal.
This error can be introduced by the calibration error and noise in the optical tracking, and the wire segmentation and point labeling error in the freehand US. Wire segmentation is difficult because the wires cause artifacts in the US images and the noise in tracking introduces jitter in the reconstructed US.
We evaluate the feasibility of MRI-US registration for two different 3D US. The 3D transducer with low frequency can image the deeper oropharyngeal region, making the tongue surface and larynx easily visible for pre-registration. However, the details of the oropharynx are not clear, making the refinement and evaluation difficult.
Freehand 3D US with higher frequency provides more details in the near-surface region and can scan a broader region. The common carotid artery, external carotid artery, and internal carotid artery are reliable landmarks for MRI-freehand 3D US registration. A future improvement is to detect these vessels automatically, with classical or learning-based techniques and use anatomy prior for pre-registration \cite{zeng2022learning}.
However, freehand 3D US has the disadvantage that the marker must be visible to the optical tracker during the scanning, making the process more complicated. The tracking reliability can be improved by better device space arrangement and careful marker design.
We find that the tissue deformation introduced by the forces applied on the participants during the US scanning increase the registration difficulty and error, and the visual offset in the AR display. Fig.~\ref{fig:AR_render} shows that the anatomy displayed on the mannequin is more realistic than on the human because the mannequin has a non-deformable surface that is similar to MRI. The participant's poses are different in MRI and US scanning, and their tissue is deformable, making the registration less accurate. In a real surgery scene, the deformation will be more severe because the tongue is retracted. However, for the surgeon to recall where the vessels are located presents an additional mental load because it is complicated and patient-dependent. Therefore, even with the visual offset, our AR display can be valuable to surgical decisions intraoperatively. This result also demonstrates the importance of real-time US to provide intraoperative information. A future improvement is to use deformable registration to improve registration accuracy, and also explore the feasibility of intraoperative US-preoperative US registration to correct the anatomy model.
\section{Conclusion}
We demonstrate the first proof-of-concept for a transcervical US-guided AR system for TORS. We conduct a feasibility study of transcervical 3D US-MRI registration for two different types of 3D US, showing that 3D US has the potential to visualize the anatomy intraoperatively. We propose a new AR system for TORS and evaluate the system accuracy. Our results show that 3D US transcervical imaging is a promising approach for image guidance in TORS.
\bmhead{Acknowledgments}
We would like to thank the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Charles Laszlo Chair in Biomedical Engineering held by Dr. Salcudean. We would like to thank David Black, Nicholas Rangga and Angela Li for their help in CAD design.
\section*{Declarations}
\begin{itemize}
\item Competing interests: The authors have no conflicts of interest.
\item Ethics approval: Institutional ethics approval was obtained for this study. All procedures in studies involving human participants were in accordance with the ethical standards of the institutional and national research committee.
\item Consent to participate: Informed consent was obtained from all individual participants in the study.
\item Data and code availability: Data and code are not publicly available.
\end{itemize}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,055
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Q: How much router packet loss is acceptable? I have had issues with my internet connection. My ISP has several times dropped my internet connection all together and it has been down for several days. When calling support they analyse and call city fiber owner and then they come back telling me it works fine. But I never get any explanation of the root cause. They often say it might be my bad and caused by household inner problems.
I am tired of it and decided to upgrade. I bought two "Netgear ProSafe Plus GS116Ev2" switches and two "ASUS ZenWiFi AX (XT8) AV6600". Most of the time it works pretty fine, but sometimes I drop signal during video calls or movie watching. Last week the connection went down completely and was gone for 5 days. Then suddenly it all works again. This is not acceptable as I now work from home full time. Support was of no help.
I have been an IT professional developer for 30 years but I am not very skilled in networks. I downloaded PingPlotter and started to monitor. I typically get good results from all hops in the trace route, EXCEPT for local router and ISP "central router"(?) that show very high packet loss of 60-90%.
Is that a problem? What should be my next step?
Typical plotter result:
A: General packet loss should be less than 1%.
However, what your ping plot shows is not general packet loss, it's just a couple routers that don't bother responding to pings very often. Note that only those two routers have high rates of ping losses. If they were generally dropping all types of traffic at that high of a packet loss rate, all the hops after them would show the same or higher packet loss, as true packet loss is cumulative across the path.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,402
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Craugastor inachus est une espèce d'amphibiens de la famille des Craugastoridae.
Répartition
Cette espèce est endémique du Guatemala. Elle se rencontre de 500 à dans les bassins du Motagua et du río Salamá.
Publication originale
Campbell & Savage, 2000 : Taxonomic reconsideration of Middle American frogs of the Eleutherodactylus rugulosus group (Anura: Leptodactylidae): a reconnaissance of subtle nuances among frogs. Herpetological Monographs, , .
Liens externes
Notes et références
Anoure (nom scientifique)
Craugastorinae
Faune endémique du Guatemala
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,595
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The 2nd October 2016 is designated by the United Nations as the International Day of Non-Violence, and is the same date as the birthday of Mahatma Gandhi, who was the leader of the Indian independence movement and pioneer of the philosophy and strategy of non-violence.
Every year on the International Day of Non-Violence, we re-commit ourselves to the cause of peace, as exemplified by the life of Mahatma Gandhi who was born on this day 147 years ago.
We know that a culture of non-violence begins with respect for others, but it does not end there. To nurture peace, we must respect nature. I am pleased this year's International Day of Non-Violence puts the focus on sustainability and the environment.
Today that commitment is reflected in a momentous way. India is depositing its instrument of ratification to the Paris Agreement on Climate Change. What better way to commemorate Mahatma Gandhi and his legacy for people and planet.
I warmly congratulate India for its climate leadership, and for building on the strong momentum we see from all corners of the globe for the agreement to enter into force as quickly as possible this year. India's ratification of the agreement moves the world an important step closer toward achieving that goal.
I urge all countries to complete their domestic processes for ratification and also strive in all activities to achieve progress through non-violence. This is essential to building a safer, healthier and more peaceful world.
To read more about the International Day of Non-Violence, and the principal of non-violence or non-violent resistance, head to www.un.org/en/events/nonviolenceday.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,668
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Shakib likely to miss NZ tour | The Asian Age Online, Bangladesh
January 27, 2021 January 27, 2021 omarmallick 0 Comments
Shakib Al Hasan went off the field injured while bowling his fifth over against West Indies during 3rd ODI in Chattogram on Monday. -Getty
Bangladesh's top all-rounder Shakib Al Hasan is a major doubt for Bangladesh's tour of New Zealand next month because of injury and personal reasons.Shakib, who made a return to the team following a ban, was superb in the recently concluded ODI series against West Indies. He made his presence felt with both bat and ball. However, a groin injury suffered on Monday, coupled with personal reasons could see him be absent.
He scored 113 runs and claimed six wickets to win the player of the series award. But Shakib limped off the field during Monday's 120-run win with a groin problem. He was unable to complete his bowling spell.Shakib said he would wait at least a day before making an assessment but admitted "it doesn't look very good at the moment".
In the third ODI, Shakib pulled up while bowling his fifth over, and left the field after receiving the physio's attention, leaving Soumya Sarkar to complete the over.Media reported that he might also skip the New Zealand series to be with his wife, who is expecting a child, in the United States.
Bangladesh had been counting on Shakib for the tour in February and March as they have never beaten the Kiwis in New Zealand.They will play three one-day internationals and three Twenty20 internationals during the series. Shakib said he would make an announcement after Bangladesh's two Tests against the West Indies. The first starts in Chittagong on February 3.
Bangladesh thrashed a second-string West Indies outfit 3-0 in the ODIs. Notably, Shakib registered bowling figures of 4/8, 2/30, and 0/12. With the bat, he got scores of 19, 43*, and 51. Shakib now has 266 career ODI scalps at 29.72. He has also scored 6,436 runs at 38.08.
In the third ODI, he registered his 48th career ODI half-century. On Monday, the 33-year-old became the only cricketer to register the double of 6,000 runs and 300 wickets in a single country. He has scored 6,045 international runs in Bangladesh, besides leading the wickets tally (across formats) 336.
← Charge-sheet rape-murder accused Dihan Feb 11 | The Asian Age Online, Bangladesh
Teletalk's network up-gradation for 4G, 5G progressing: PM tells JS | The Asian Age Online, Bangladesh →
'Shakib will shine from start'
November 23, 2020 November 23, 2020 omarmallick 0
Tigers get three-day break as Lankan tour gets postponed
September 27, 2020 September 27, 2020 omarmallick 0
Mahmudullah pleased with bowlers performance in first warm-up game | The Asian Age Online, Bangladesh
January 15, 2021 January 15, 2021 omarmallick 0
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,574
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{"url":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/cpaa.2017012","text":"# American Institute of Mathematical Sciences\n\n\u2022 Previous Article\nQuasineutral limit for the quantum Navier-Stokes-Poisson equations\n\u2022 CPAA\u00a0Home\n\u2022 This Issue\n\u2022 Next Article\nTrudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity\nJanuary\u00a0 2017,\u00a016(1):\u00a0253-272. doi:\u00a010.3934\/cpaa.2017012\n\n## Higher order asymptotic for Burgers equation and Adhesion model\n\n 1 Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Mangalore-575 025, India 2 School of Mathematical Sciences, National Institute of Science Education and Research, Bhimpur-Padanpur, Via-Jatni, Khurda-752050, Odisha, India 3 Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Mangalore-575 025, India\n\nManas R. Sahoo, E-mail address: manas@niser.ac.in\n\nReceived\u00a0 April 2016 Revised\u00a0 September 2016 Published\u00a0 November 2016\n\nThis paper is focused on the study of the large time asymptotic for solutions to the viscous Burgers equation and also to the adhesion model via heat equation. Using generalization of the truncated moment problem to a complex measure space, we construct asymptotic N-wave approximate solution to the heat equation subject to the initial data whose moments exist upto the order $2n+m$ and $i$-th order moment vanishes, for $i=0, 1, 2\\dots m-1$. We provide a different proof for a theorem given by Duoandikoetxea and Zuazua [3], which plays a crucial role in error estimations. In addition to this we describe a simple way to construct an initial data in Schwartz class whose $m$ moments are equal to the $m$ moments of given initial data.\n\nCitation: Engu Satynarayana, Manas R. Sahoo, Manasa M. Higher order asymptotic for Burgers equation and Adhesion model. Communications on Pure & Applied Analysis, 2017, 16 (1) : 253-272. doi: 10.3934\/cpaa.2017012\n##### References:\n\nshow all references\n\n##### References:\n [1] Thierry Cazenave, Fl\u00e1vio Dickstein, Fred B. Weissler. Universal solutions of the heat equation on $\\mathbb R^N$. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1105-1132. doi: 10.3934\/dcds.2003.9.1105 [2] Tai Nguyen Phuoc, Laurent V\u00e9ron. Initial trace of positive solutions of a class of degenerate heat equation with absorption. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2033-2063. doi: 10.3934\/dcds.2013.33.2033 [3] Abra\u00e3o D. C. Nascimento, Leandro C. R\u00eago, Raphaela L. B. A. Nascimento. Compound truncated Poisson normal distribution: Mathematical properties and Moment estimation. Inverse Problems & Imaging, 2019, 13 (4) : 787-803. doi: 10.3934\/ipi.2019036 [4] Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\\mathbb{R}_{+}^{n}$. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934\/dcdsb.2018319 [5] Rui Liu. Some new results on explicit traveling wave solutions of $K(m, n)$ equation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 633-646. doi: 10.3934\/dcdsb.2010.13.633 [6] C. Br\u00e4ndle, E. Chasseigne, Ra\u00fal Ferreira. Unbounded solutions of the nonlocal heat equation. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1663-1686. doi: 10.3934\/cpaa.2011.10.1663 [7] Perikles G. Papadopoulos, Nikolaos M. Stavrakakis. Global existence for a wave equation on $R^n$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 139-149. doi: 10.3934\/dcdss.2008.1.139 [8] Kazuhiro Ishige, Tatsuki Kawakami. Asymptotic behavior of solutions for some semilinear heat equations in $R^N$. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1351-1371. doi: 10.3934\/cpaa.2009.8.1351 [9] Arturo de Pablo, Guillermo Reyes, Ariel S\u00e1nchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 643-662. doi: 10.3934\/dcds.2013.33.643 [10] Xiangfeng Yang, Yaodong Ni. Extreme values problem of uncertain heat equation. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1995-2008. doi: 10.3934\/jimo.2018133 [11] Luz de Teresa, Enrique Zuazua. Identification of the class of initial data for the insensitizing control of the heat equation. Communications on Pure & Applied Analysis, 2009, 8 (1) : 457-471. doi: 10.3934\/cpaa.2009.8.457 [12] Caihong Chang, Qiangchang Ju, Zhengce Zhang. Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5991-6014. doi: 10.3934\/dcds.2020256 [13] Antonio Greco, Antonio Iannizzotto. Existence and convexity of solutions of the fractional heat equation. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2201-2226. doi: 10.3934\/cpaa.2017109 [14] Peter Pol\u00e1\u010dik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2020\u00a0 doi: 10.3934\/dcds.2020136 [15] Pavol Quittner. The decay of global solutions of a semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 307-318. doi: 10.3934\/dcds.2008.21.307 [16] David Henry, Octavian G. Mustafa. Existence of solutions for a class of edge wave equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1113-1119. doi: 10.3934\/dcdsb.2006.6.1113 [17] Dmitriy Chebanov. New class of exact solutions for the equations of motion of a chain of $n$ rigid bodies. Conference Publications, 2013, 2013 (special) : 105-113. doi: 10.3934\/proc.2013.2013.105 [18] Pengyan Ding, Zhijian Yang. Attractors of the strongly damped Kirchhoff wave equation on $\\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2019, 18 (2) : 825-843. doi: 10.3934\/cpaa.2019040 [19] Bei Gong, Zhen-Hu Ning, Fengyan Yang. Stabilization of the transmission wave\/plate equation with variable coefficients on ${\\mathbb{R}}^n$. Evolution Equations & Control Theory, 2020\u00a0 doi: 10.3934\/eect.2020068 [20] Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934\/dcdss.2017064\n\n2019\u00a0Impact Factor:\u00a01.105","date":"2020-10-30 04:47:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5160196423530579, \"perplexity\": 3211.874555409972}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-45\/segments\/1603107907213.64\/warc\/CC-MAIN-20201030033658-20201030063658-00091.warc.gz\"}"}
| null | null |
package net.sf.jabref.model.entry;
import org.junit.Test;
import static org.junit.Assert.assertEquals;
public class FieldNameTest {
@Test
public void testOrFieldsTwoTerms() {
assertEquals("aaa/bbb", FieldName.orFields("aaa", "bbb"));
}
@Test
public void testOrFieldsThreeTerms() {
assertEquals("aaa/bbb/ccc", FieldName.orFields("aaa", "bbb", "ccc"));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,516
|
{"url":"https:\/\/tex.stackexchange.com\/questions\/131346\/how-to-set-tikz-scatter-marker-weight","text":"# How to set Tikz scatter marker weight?\n\nI'm using tikz to produce scatter plots as shown in this question. I can define the marker size with \\tikzset{mark size=8} but now I'm looking for a way to make the line of the marker thicker in a similar way. Is this possible with \\tikzset?\n\nYou can set the size of the line width for the markers using \\tikzset{mark options={line width=3pt}}:\n\n\\documentclass{article}\n\n\\usepackage{pgfplots}\n\n\\begin{document}\n\\tikzset{mark options={mark size=5, line width=3pt}}\n\n\\begin{tikzpicture}\n\\begin{axis}","date":"2022-08-09 15:02:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9781150221824646, \"perplexity\": 1788.3378684400957}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882570977.50\/warc\/CC-MAIN-20220809124724-20220809154724-00276.warc.gz\"}"}
| null | null |
Q: What exactly happens when you browse a website in your browser? What happens behind the scenes when we type www.cnn.com in a browser and how does information gets displayed on the screen?
A technical explanation would be highly appreciated.
A: Missing so far from the other answers is what happens on the CNN side:
*
*A machine at CNN receives the message from your computer asking for the page.
*It redirects this request to one of the many computers CNN is using for its web site (the reason for this is that, this way, you can spread the work of putting together the response web pages over many computers)
*The CNN computer gets your request and it responds with a web page that's probably almost entirely precomputed, but maybe it'll change a few things before sending it to you (could be the ad at the top, maybe headline news). Sometimes computers assemble the page from lots of small components every time they get a request; not sure what CNN does
*the response makes its way through the network to your computer, which then displays it.
*the response didn't include the images: your computer then sends another request for the images, and pretty much the same scenario happens.
A: This obviously isn't a technical explanation, but it's a cute visual aid (from the excellent Vladstudio.com) that may be helpful to some:
A: How the web works: HTTP and CGI explained
Also a great explanation from CERN - alma mater of Web: How the web works
A: *
*Browser splits what you type (the URL) into a hostname and a path.
*Browser forms an HTTP request to ask for the data at the given hostname and path.
*Browser performs DNS lookup to resolve the hostname into an IP address.
*Browser forms a TCP/IP connection to the computer specified via the IP address. (This connection is actually formed out of many computers, each passing the data along to the next.)
*Browser sends the HTTP request down the connection to the given IP address.
*That computer receives the HTTP request from the TCP/IP connection and passes it to the web server program.
*Web server reads the hostname and path and finds or generates the data that you've asked for.
*Web server generates an HTTP response containing that data.
*Web server sends that HTTP response back down the TCP/IP connection to your machine.
*Browser receives the HTTP response and splits it into headers (describing the data) and the body (the data itself).
*Browser interprets the data to decide how to display it in the browser - typically this is HTML data that specifies types of information and their general form.
*Some of the data will be metadata that specifies further resources that need to be loaded, such as style sheets for detailed layout, or inline images, or Flash movies. This metadata is specified again as a URL, and this whole process repeats for each one until all are loaded.
A: Browser: "Ok, so, I have a user requesting this address: www.cnn.com. I figure since there are no slashes or anything, this is a direct request of a main page. There was also no protocol or port defined, so I'll assume it's HTTP and going to port 80... oh well, first things first. Hey DNS, pal, wake up! Where is this www.cnn.com hiding at?"
DNS: "Right... wait a sec, I'll ask the ISP servers. Ok, it looks like 157.166.226.25."
Browser: "Ok. Internet Protocol Suite, your turn! Call 157.166.226.25, please. Send them this HTTP header. It's asking for the basic structure and content of their main page so I know what else to fetch... oh well, not that you'd care about this I guess. "
TCP/IP: "What do you mean my turn? Like I wasn't just working my back off right there for the DNS? God, what does it take to get a bit of appreciation here..."
Browser: ...
TCP/IP: "Yeah, yeah... Connecting... I'll just ask the gateway to forward it. You know, it isn't all that easy, I'll have to divide your pretty request there into multiple parts so it reaches the end, and assemble any stuff they send back from all the thousands of packages I get... ah, right, you don't care. Figures."
Meanwhile, at the CNN headquarters, a message finally ends up at the door of the Web Server.
CNN Web Server: "Nzhôô! A customer! He wants news! The Front Page! How about it?"
CNN Server Side Script Engine: "Right, will do! Front page, right?"
CNN Database Server: "Yey! Work for me! What content do you need?"
CNN Server Side Script Engine: "... um, sorry DB, I have a copy of front page right here in my cache, no need to compile anything. But hey, take this user ID and store it, I'll send it to the customer too, so we know who we're talking to later on."
CNN Database Server: "Yey!"
Back at the user's computer...
TCP/IP: "Ooookay, here comes the reply. Oh boy, why do I have a feeling this'll be a big one..."
Browser: "Uh, wow... this has all sorts of javascript code... bunch of images, couple of forms... Right, this'll take a while to render. Better get to it. Hey, IP system, there's a bunch more stuff you'll need to get. Let's see I need a few stylesheets from i.cdn.turner.com - via HTTP and ask for the file /cnn/.element/css/2.0/common.css. And then get some of those scripts at i.cdn.turner.com too, I'm counting six so far..."
TCP/IP: "I get the picture. Just give me the server addresses and all that. And wrap that file stuff within the HTTP request, I don't want to deal with it."
DNS: "Checking the i.cdn.turner.com... hey, bit of trivia, it's actually called cdn.cnn.com.c.footprint.net. IP is 4.23.41.126"
Browser: "Sure, sure... wait a sec, this'll take a few nsec to process, I'm trying to understand all this script..."
TCP/IP: "Hey, here's the CSS you asked for. Oh, and... yeah, those additional scripts also just came back."
Browser: "Whew, there's more... some sort of video ad!"
TCP/IP: "Oh boy, what fun that sounds like..."
Browser: "There's all sorts of images too! And this CSS looks a bit nasty... right, so if that part goes there, and has this line at the top... how on earth would that fit anymore... no, I'll have to stretch this a bit to make it... Oh, but that other CSS file overrides that rule... Well, this one ain't going to be an easy piece to render, that's for sure!"
TCP/IP: "Ok, ok, stop distracting me for a sec, there's a lot to do here still."
Browser: "User, here's a small progress report for you. Sorry, this all might take a few secs, there's like 140 different elements to load, and going at 16 so far."
One or two seconds later...
TCP/IP: "Ok, that should be all. Hey, listen... sorry I snapped at you earlier, you managing there? This sure seems like quite the load for you too."
Browser: "Phew, yeah, it's all these websites nowdays, they sure don't make it easy for you. Well, I'll manage. It's what I'm here for."
TCP/IP: "I guess it's quite heavy for all of us these days... oh, stop gloating there DNS!"
Browser: "Hey user! The website's ready - go get your news!"
A: The first step is the DNS (Domain Name Server) lookup. It uses the DNS servers specified in your network settings (or given to you by DHCP) to lookup the top domain (cnn.com) and then ask that domain's nameserver for the IP address of the subdomain specified (www.cnn.com).
After it has the IP address, your browser begin communications with the web server. This is done using the specified protocol (which usually defaults to HTTP 1.1). A 'GET' request for '/' is made to the server, which responds with the HTML document contents and the appropriate headers (which tell the browser of the document's content-type, HTML, and other information). Then the browser parses the document and finds any URLs which it needs to embed in the page (like images or linked stylesheets) and does GET requests on each of those.
The browser also usually automatically makes a GET request for '/favicon.ico' (to display the little CNN icon next to the site title).
Your browser will also likely specify in its request headers that it wants the response content to be compressed, using the gzip algorithm. This makes the file download much smaller, if the server supports it. This is all transparent to you, even though it's like downloading a ZIP file and unzipping it.
When you reload the page, your browser checks if that page is already cached in your system, and if so, it does an HTTP request just for the header of the document, and checks its modified date. If this date is later than its cached copy, it requests the full document contents again and refreshes the page. Otherwise it just uses your local copy.
A: The info you just asked could fill a couple dozen books. But here is my attempt to explain it: Your browser tells your OS to find the IP address of cnn.com. Then your OS asks a DNS server for the IP address for cnn.com. The IP is sent to the broswer which contacts the IP address and requests the page. cnn.com then sends you and html page. The browsers parses the html and sends the information to the HTML renderer. The browser then tells the OS what to display on the screen.
A: Jeff Moser had an excellent technical analysis of an HTTPS request on his blog: The First Few Milliseconds of an HTTPS Connection.
A: There is a really cool video by the "Sendung mit der Maus" (a very popular German children's TV show that explains technology for children):
Die Sendung mit der Maus - Wie funktioniert das Internet (How the Internet works).
In German only, unfortunately, but funny even w/o the text. Men with funny helmets play the IP packets, and the data is written onto paper cards. Classic :-).
BTW, the explanations are actually fairly good.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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from __future__ import division, print_function, absolute_import
import glob
from os.path import join, dirname
def configuration(parent_package='', top_path=None):
from numpy.distutils.misc_util import Configuration, get_numpy_include_dirs
from numpy.distutils.misc_util import get_info as get_misc_info
from numpy.distutils.system_info import get_info as get_sys_info
from distutils.sysconfig import get_python_inc
config = Configuration('spatial', parent_package, top_path)
config.add_data_dir('tests')
# qhull
qhull_src = list(glob.glob(join(dirname(__file__), 'qhull',
'src', '*.c')))
inc_dirs = [get_python_inc()]
if inc_dirs[0] != get_python_inc(plat_specific=1):
inc_dirs.append(get_python_inc(plat_specific=1))
inc_dirs.append(get_numpy_include_dirs())
cfg = dict(get_sys_info('lapack_opt'))
cfg.setdefault('include_dirs', []).extend(inc_dirs)
def get_qhull_misc_config(ext, build_dir):
# Generate a header file containing defines
config_cmd = config.get_config_cmd()
defines = []
if config_cmd.check_func('open_memstream', decl=True, call=True):
defines.append(('HAVE_OPEN_MEMSTREAM', '1'))
target = join(dirname(__file__), 'qhull_misc_config.h')
with open(target, 'w') as f:
for name, value in defines:
f.write('#define {0} {1}\n'.format(name, value))
config.add_extension('qhull',
sources=['qhull.c'] + qhull_src + [get_qhull_misc_config],
**cfg)
# cKDTree
ckdtree_src = ['query.cxx',
'build.cxx',
'globals.cxx',
'cpp_exc.cxx',
'query_pairs.cxx',
'count_neighbors.cxx',
'query_ball_point.cxx',
'query_ball_tree.cxx',
'sparse_distances.cxx']
ckdtree_src = [join('ckdtree', 'src', x) for x in ckdtree_src]
ckdtree_headers = ['ckdtree_decl.h',
'cpp_exc.h',
'ckdtree_methods.h',
'cpp_utils.h',
'rectangle.h',
'distance.h',
'distance_box.h',
'ordered_pair.h']
ckdtree_headers = [join('ckdtree', 'src', x) for x in ckdtree_headers]
ckdtree_dep = ['ckdtree.cxx'] + ckdtree_headers + ckdtree_src
config.add_extension('ckdtree',
sources=['ckdtree.cxx'] + ckdtree_src,
depends=ckdtree_dep,
include_dirs=inc_dirs + [join('ckdtree', 'src')])
# _distance_wrap
config.add_extension('_distance_wrap',
sources=[join('src', 'distance_wrap.c')],
depends=[join('src', 'distance_impl.h')],
include_dirs=[get_numpy_include_dirs()],
extra_info=get_misc_info("npymath"))
config.add_extension('_voronoi',
sources=['_voronoi.c'])
return config
if __name__ == '__main__':
from numpy.distutils.core import setup
setup(maintainer="SciPy Developers",
author="Anne Archibald",
maintainer_email="scipy-dev@scipy.org",
description="Spatial algorithms and data structures",
url="https://www.scipy.org",
license="SciPy License (BSD Style)",
**configuration(top_path='').todict()
)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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Q: Do these java classes have correct structures to map from a JSON file? I'm parsing a JSON file into a java code from a particular URL. It looks easy. But, for some reasons the server respond with null for some key-values that I'll show below. I'm quite sure there something wrong over these structures (on the java sides).
Here is the json structure.
And here are the codes (the getters-setters have been provided I just don't show them to make it clear)
class JSON {
List<Result> result;
}
class Results{
String gender;
Name name;
Location location
}
class Name{
String title;
String first;
String last;
}
class Location {
String city;
String state;
String country;
String postcode;
}
I'm not sure where exactly I missed it. Are those classes have correct structures relative to the JSON structure?
A: It seems like you need to have results instead of result in JSON class
A: I saw your Java Class and JSON response in object Location there is not Inner Object street.
A: I see one problem,
The String postcode seems to be a number in the JSON above.
A: The issue is User defined class reference null
Try this one and create HAS-A reference to Object
Initialize the objects using constructor()
class JSON {
List<Result> result;
public JSON(){
result = new ArrayList()
}
}
class Results{
String gender;
Name name;
Location location
public Results(){
Name = new Name();
Location = new Location();
}
}
class Name{
String title;
String first;
String last;
}
class Location {
String city;
String state;
String country;
String postcode;
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,794
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Pulp are set to reissue their 1983 debut album 'It' along with their 1987 LP 'Freaks' and 1992's 'Separations'.
Pulp's frontman Jarvis Cocker recently blamed the tabloids for contributing to Amy Winehouse's death. He claimed the constant media attention the late singer received on a daily basis forced her to turn to drink and drugs and he went through a similar situation during the height of his fame during the Britpop era.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 4,326
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Have you seen a chance to be part of the action? Sign ups are available for most actions and events.
Want to be part of the team that makes church happen? Check out the different ways you can serve.
Difference-makers are not just those who put feet to the mission. Every goal is made possible by the generous hearts of givers.
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{
"redpajama_set_name": "RedPajamaC4"
}
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{"url":"https:\/\/www.lowermyhumidity.com\/8il5a\/paschen-series-colors-607993","text":"# paschen series colors\n\nAt least, that's how I like to think about it. To do this, you only need to calculate the shortest wavelength in the series. Observations of H\u03b1, iron, and oxygen lines in B, Be, and shell stars We carried out a spectroscopic survey of several B, Be, and shell starsin optical and near-infrared regions. Part of the Balmer series is in the visible spectrum, while the Lyman series is entirely in the UV, and the Paschen series and others are in the IR. Paschen series are the series of lines in the spectrum of the hydrogen atom which corresponds to transitions between the state with principal quantum number n = 3 and successive higher states. Crores) - Balmer .Balmer Lawrie \u2026 The Lyman series lies in the ultraviolet, whereas the Paschen, Brackett, and Pfund series lie in the infrared. Hydrogen Spectral Series: If so, to what color do they correspond? Calculate the wavelength of the lowest-energy line in the Lyman series to three significant figures. Electrons can only occupy specific energy levels in an atom. Pre lab Questions Let's examine the Paschen Series of transitions and practice calculating the photon wavelengths produced by these transitions: A. The Paschen series constitutes the transitions of electrons from to . 1) are called the Lyman series, but the energy released is so large that the spectral lines are in the ultraviolet region of the spectrum. Imgur. google_ad_client = \"ca-pub-0644478549845373\"; (a) Calculate the wavelengths of the first three lines in this series. Lyman, Balmer, and Paschen series. #n_i = 4\" \" -> \" \" n_f = 3# In this transition, the electron drops from the fourth energy level to the third energy level. The wavelengths of the Paschen series for hydrogen are given by {eq}1\/\\lambda = R_H (1\/3^2 - 1\/n^2) {\/eq}, n = 4, 5, 6, . \u25a1\u200b. The Paschen series constitutes the transitions of electrons from to . Prepared By: Sidra Javed 3. Correct answers: 2 question: The Paschen series is analogous to the Balmer series, but with m = 3. For this reason, we refer to n=1n=1n=1 as the ground state of the electron. During transition, an electron absorbs\/releases energy is in the form of light energy. Since the energy level of the electron of a hydrogen atom is quantized instead of continuous, the spectrum of the lights emitted by the electron via transition is also quantized. Paschen series : German - English translations and synonyms (BEOLINGUS Online dictionary, TU Chemnitz) En=\u22121312n2\u00a0kJ\/mol.E_n=-\\frac{1312}{n^2}\\text{ kJ\/mol}.En\u200b=\u2212n21312\u200b\u00a0kJ\/mol. For instance, we can fix the energy levels for various series. RE= -2.178 x 10-18J\u00a0 (it is negative because energy is being emitted), l = ( 6.626 x 10\u00a0- 34\u00a0J s) (3.0 x 108\u00a0m\/s)\/E, c= 3.0 x 108\u00a0m\/s ;l = wavelength (m) ;v= frequency (s-1). Correct answers: 2 question: The Paschen series is analogous to the Balmer series, but with m = 3. For atoms other than hydrogen, we simply multiply \u22121312n2\u00a0kJ\/mol-\\frac{1312}{n^2}\\text{ kJ\/mol}\u2212n21312\u200b\u00a0kJ\/mol or \u221213.6n2\u00a0eV-\\frac{13.6}{n^2}\\text{ eV}\u2212n213.6\u200b\u00a0eV by Zeff2,Z_{\\text{eff}}^2,Zeff2\u200b, where ZeffZ_{\\text{eff}}Zeff\u200b refers to the effective nuclear charge. These wavelengths are in the visible light spectrum (wavelengths 750nm- 450nm). Paschen series is displayed when electron transition takes place from higher energy states(n h =4,5,6,7,8,\u2026) to n l =3 energy state. (C) n=3\u2192n=2n=3\\rightarrow n=2n=3\u2192n=2 Johan Rydberg use Balmers work to derived an equation for all electron transitions in a hydrogen atom. For a single electron instead of per mole, the formula in eV (electron volts) is also widely used: Indeed, comparing the similarities of atoms was how the table was designed originally. The Lyman series lies in the ultraviolet, whereas the Paschen, Brackett, and Pfund series lie in the infrared. Forgot password? (A) n=2\u2192n=1n=2\\rightarrow n=1n=2\u2192n=1 The energy of the photon that is emitted is categorised into the Paschen, Balmer and Lyman series. . If an electron falls from any n\u22652n\\ge2n\u22652 to n=1,n=1,n=1, then the wavelength calculated using the Rydberg formula gives values ranging from 91 nm to 121 nm, which all fall under the domain of ultraviolet. The shortest wavelength of next series, i.e., Brackett series overlap with Paschen series. We call this the Balmer series. Using the Rydberg formula, we can compute the wavelength of the light the electron absorbs\/releases, which ranges from ultraviolet to infrared. Since the energy level of the electron of a hydrogen atom is quantized instead of continuous, the spectrum of the lights emitted by the electron via transition is also quantized. In which region of electromagnetic spectrum of lymen and balmer series of hydrogen spectrum falls ? E=h\u03bd=hc\u03bb,E=h\\nu=h\\frac{c}{\\lambda},E=h\u03bd=h\u03bbc\u200b, So when you look at the line spectrum of hydrogen, it's kind of like you're seeing energy levels. radiation. The H{\\alpha} emission strength of the stars in our sample show a steady decrease from late-B type to Ae stars, suggesting that the disc size may be dependent on the spectral type. Electron transition from n\u22654n\\ge4n\u22654 to n=3n=3n=3 gives infrared, and this is referred to as the Paschen series. Therefore spectral lines can be thought of the \"fingerprints\" of an element, and be used to identify an element. Note that nnn refers to the principal quantum number. 30 - A wavelength of 4.653 m is observed in a hydrogen... Ch. What are synonyms for Paschen? Figure $$\\PageIndex{4}$$: A schematic of the hydrogen spectrum shows several series named for those who contributed most to their determination. In this section we will discuss the energy level of the electron of a hydrogen atom, and how it changes as the electron undergoes transition. This chemistry video tutorial focuses on the bohr model of the hydrogen atom. Jahann Balmer in 1885 derived an equation to calculate the visible wavelengths that the hydrogen spectrum displayed. 30 - Do the Balmer and Lyman series overlap? Paschen series is displayed when electron transition takes place from higher energy states(n h =4,5,6,7,8,\u2026) to n l =3 energy state. Bohr named the orbits as K\u00a0(n=1),L\u00a0(n=2),M\u00a0(n=3),N\u00a0(n=4),O\u00a0(n=5),\u22ef\\text{K }(n=1), \\text{L }(n=2), \\text{M }(n=3), \\text{N }(n=4), \\text{O }(n=5), \\cdotsK\u00a0(n=1),L\u00a0(n=2),M\u00a0(n=3),N\u00a0(n=4),O\u00a0(n=5),\u22ef in order of increasing distance from the nucleus. Paschen and Balmer Lines in Active Galactic Lyman, Ba\/mer and Paschen series. Also, you can\u2019t see any lines beyond this; only a faint continuous spectrum.Furthermore, like the Balmer\u2019s formula, here are the formulae for the other series: Lyman Series. Using the properties of DeBroglie waves, we can calculate the wavelength and frequency of the following formula: So, this is called the Balmer series \u2026 Antonyms for Paschen. Example $$\\PageIndex{1}$$: The Lyman Series. If you assume the energy levels of an atom to be a staircase; if you roll a ball down the stairs the ball only has a few \"steps\" that it can stop on. For layman\u2019s series, n1 would be one because it requires only first shell to produce spectral lines. The line with the longest wavelength within a series corresponds to the electron transition with the lowest energy within that series. This is because the electrons on the orbit are \"captured\" by the nucleus via electrostatic forces, and impedes the freedom of the electron. The energy of the photon EEE absorbed\/released during the transition is equal to the energy change \u0394E\\Delta E\u0394E of the electron. See how the characteristic spectra of different elements are produced, and configure your own element's energy states to produce light of different colors. The significance of the numbers in the Rydberg equation. Pfund Series Citing this page: Generalic, Eni. These electrons are falling to the 2nd energy level from higher ones. The so-called Lyman series of lines in the emission spectrum of hydrogen corresponds to transitions from various excited states to the n = 1 orbit. 30 - (a) Which line in the Balmer series is the first... Ch. Synonyms for Paschen in Free Thesaurus. Jahann Balmer in 1885 derived an equation to calculate the visible wavelengths that the hydrogen spectrum displayed. Any given sample of hydrogen gas gas contains a large number of molecules. New user? Paschen Series. \u25a1E_{\\infty}-E_1=13.6\\text{ eV}.\\ _\\squareE\u221e\u200b\u2212E1\u200b=13.6\u00a0eV. At least that's how I like to think about it 'cause you're, it's the only real way you can see the difference of energy. This is the only series of lines in the electromagnetic spectrum that lies in the visible region. Observe that the energy level is always negative, and increases as n.n.n. You will have #1\/(lamda_1) = R * (1\/3^2 - 1\/4^2)# The second transition in the Paschen series corresponds to As this was discovered by a scientist named Theodore Lyman, this kind of electron transition is referred to as the Lyman series. Projected rotational velocities (vsini) have been measured for 216 B0-B9stars in the rich, dense h and \u03c7 Persei double cluster and comparedwith the distribution of rotational velocities for a sample of fieldstars having comparable ages (t~12-15 Myr) and masses (M~4-15Msolar). ... A color television tube also generates some x rays when its electron beam strikes the screen. Paschen series are the series of lines in the spectrum of the hydrogen atom which corresponds to transitions between the state with principal quantum number n = 3 and successive higher states. So, when you look at the line spectrum of hydrogen, it's kind of like you're seeing energy levels. I have one question in.. 1908 \u2013 Paschen found the IR lines with m = 3. Here is the equation: R= Rydberg Constant 1.0974x107\u00a0m-1;\u00a0\u00a0 \u03bb\u00a0is the wavelength;\u00a0 n\u00a0is equal to the energy level (initial and final), If we wanted to calculate energy we can adjust R by multipling by h (planks constant) and c (speed of light). En=\u221213.6n2\u00a0eV.E_n=-\\frac{13.6}{n^2}\\text{ eV}.En\u200b=\u2212n213.6\u200b\u00a0eV. Alright, so, energy is quantized. Hence in the figure above, the red line indicates the transition from n=3n=3n=3 to n=2,n=2,n=2, which is the transition with the lowest energy within the Balmer series. reactivity series \u2192 reaktivni niz. The transitions called the Paschen series and the Brackett series both result in spectral lines in the infrared region because the energies are too small. Ideally the photo would show three clean spectral lines - dark blue, cyan and red. The figure below shows the electron energy level diagram of a hydrogen atom. The wavelengths of the Paschen series for hydrogen are given by {eq}1\/\\lambda = R_H (1\/3^2 - 1\/n^2) {\/eq}, n = 4, 5, 6, . Bohr\u2019s model was a tremendous success in explaining the spectrum of the hydrogen atom. https:\/\/thefactfactor.com\/facts\/pure_science\/physics\/hydrogen-spectrum\/9122 When electrons change energy states, the amount of energy given off or absorbed is equal to a. hc b ... has to be transferred all at once and have enough energy, and only certain colors of light work. It is quite obvious that an electron at ground state must gain energy in order to become excited. Transitions, called the Paschen series and the Brackett series, lead to spectral lines in \u2026 . Title: Microsoft PowerPoint - 1M_06_HEmission Author: HP_Owner Created Date: 4\/14\/2008 7:20:14 AM Produce light by bombarding atoms with electrons. Using Balmer-Rydberg equation to solve for photon energy for n=3 to 2 transition. For instance, we can fix the energy levels for various series. Since a longer wavelength means smaller energy, the red line correspond to the transition which emits the lowest energy within the Balmer series, which is n=3\u2192n=2.n=3\\rightarrow n=2.n=3\u2192n=2. The electromagnetic force between the electron and the nuclear proton leads to a set of quantum states for the electron, each with its own energy. Obviously, a positive energy change means that the electron absorbs energy, while a negative energy change implies a release of energy from the electron. All the wavelength of Paschen series falls in the Infrared region of the electromagnetic spectrum. Title: Microsoft PowerPoint - 1M_06_HEmission Author: HP_Owner Created Date: 4\/14\/2008 7:20:14 AM B Star Rotational Velocities in h and \u03c7 Persei: A Probe of Initial Conditions during the Star Formation Epoch? Therefore our answer is (D). Ch. Brackett Series. 1 0. mandeep. Wavelength (nm) Relative Intensity: Transition: Color or region of EM spectrum: Lymann Series: 93.782 ... 6 -> 1 : UV: 94.976 ... 5 -> 1 : UV: 97.254 ... 4 -> 1 Calculate the wavelengths of the first three members in the Paschen series. Their formulas are similar to Balmer\u2019s except that the constant term is the reciprocal of the square of 1, 3, 4, or 5, instead of 2, and the running number n begins at \u2026 This is why you get lines and not a \"rainbow\" of colors when electrons fall. The transitions are named sequentially by Greek letter: n = 4 to n = 3 is called Paschen-alpha, 5 to 3 is Paschen-beta, 6 to 3 is Paschen-gamma, etc. The lines that appear at 410\u00a0nm, 434\u00a0nm, 486\u00a0nm, and 656\u00a0nm. These are wavelengths in the infrared (wavelengths 1mm-750nm). Likewise, an electron at a higher energy level releases energy as it falls down to a lower energy level. We call this the Balmer series. The energy change during the transition of an electron from n=n1n=n_1n=n1\u200b to n=n2n=n_2n=n2\u200b is Chemistry. The value, 109,677 cm -1 , is called the Rydberg constant for hydrogen. Recall that the energy level of the electron of an atom other than hydrogen was given by En=\u22121312n2\u22c5Zeff2\u00a0kJ\/mol.E_n=-\\frac{1312}{n^2}\\cdot Z_{\\text{eff}}^2\\text{ kJ\/mol}.En\u200b=\u2212n21312\u200b\u22c5Zeff2\u200b\u00a0kJ\/mol. The Bohr model was later replaced by quantum mechanics in which the electron occupies an atomic orbital rather than an orbit, but the allowed energy levels of the hydrogen atom remained the same as in the earlier theory. Each energy state, or orbit, is designated by an integer, n as shown in the figure. The Lyman series lies in the ultraviolet, whereas the Paschen, Brackett, and Pfund series lie in the infrared. So, when you look at the line spectrum of hydrogen, it's kind of like you're seeing energy levels. Lyman series, Balmer series, Paschen series. $\\begingroup$ You got pretty close to a decent (if crude) answer - but instead of focusing on the mass of the atom, look at where it is on the periodic table. So, this is called the Balmer series \u2026 1\u03bb=R(1n12\u22121n22)\u00a0m\u22121,\\frac{1}{\\lambda}=R\\left(\\frac{1}{n_1^2}-\\frac{1}{n_2^2}\\right)\\text{ m}^{-1},\u03bb1\u200b=R(n12\u200b1\u200b\u2212n22\u200b1\u200b)\u00a0m\u22121, The shortest wavelength of next series, i.e., Brackett series overlap with Paschen series. Log in here. Electron transition from n \u2265 4 n\\ge4 n \u2265 4 to n = 3 n=3 n = 3 gives infrared, and this is referred to as the Paschen series. (a) Calculate the wavelengths of the first three lines in this series. Because, it's the only real way you can see the difference of energy. Brackett Series. Combining this formula with the \u0394E\\Delta E\u0394E formula above gives the famous Rydberg formula: The Paschen series would be produced by jumps down to the 3-level, but the diagram is going to get very messy if I include those as well - not to mention all the other series with jumps down to the 4-level, the 5-level and so on. Which of the following electron transitions corresponds to the turquoise line (\u03bb\u2248485\u00a0nm)(\\lambda\\approx485\\text{ nm})(\u03bb\u2248485\u00a0nm) in the figure above? The Balmer series is basically the part of the hydrogen emission spectrum responsible for the excitation of an \u2026 This transition to the 2nd energy level is now referred to as the \"Balmer Series\" of electron transitions. Paschen Series (to n=3) n=4 to n=3: 1.06 x 10-19: 1.875 x 10-6: 1875: Infrared: n=5 to n=3: 1.55 x 10-19: 1.282 x 10-6: 1282: Infrared: Balmer Series (to n=2) n=3 to n=2: 3.03 x 10-19: 6.56 x 10-7: 656: visible: n=4 to n=2: 4.09 x 10-19: 4.86 x 10-7: 486: visible: n=5 to n=2: 4.58 x 10-19: 4.34 x 10-7: 434: visible: n=6 to n=2: 4.84 x 10-19: 4.11 x 10-7: 411: visible: Lyman Series ( to n=1) n=2 to n=1 30 - Show that the entire Paschen series is in the... Ch. Because, it's the only real way you can see the difference of energy. \u2192 Download high quality image. Passing it through a prism separates it. Projected rotational velocities (vsini) have been measured for 216 B0-B9stars in the rich, dense h and \u03c7 Persei double cluster and comparedwith the distribution of rotational velocities for a sample of fieldstars having comparable ages (t~12-15 Myr) and masses (M~4-15Msolar). In other words, the wavelength \u03bb\\lambda\u03bb can only take on specific values since n1n_1n1\u200b and n2n_2n2\u200b are integers. \u25a1_\\square\u25a1\u200b. When such a sample is heated to a high temperature or an electric discharge is passed, the [\u2026] n is the principa\/ quantum Turnover (in Rs. c. diffraction of light. If the electron is in any other shell, we say that the electron is in excited state. Since the energy level of the electron of a hydrogen atom is quantized instead of continuous, the spectrum of the lights emitted by the electron via transition is also quantized. \/\/-->, Energy, Wavelength and Electron Transitions. Calculate the longest and shortest wavelengths for the Paschen series and determine the photon energies corresponding to these wavelengths. E\u221e\u2212E1=13.6\u00a0eV. Hence, taking n f = 3,we get: \u1e7d= 1.5236 \u00d7 10 6 m \u20131. Their formulas are similar to Balmer\u2019s except that the constant term is the reciprocal of the square of 1, 3, 4, or 5, instead\u2026 \u2192 Download high quality image. where R=1.097\u00d7107\u00a0m\u22121R=1.097\\times10^7\\text{ m}^{-1}R=1.097\u00d7107\u00a0m\u22121 is the Rydberg constant. Spectrum White light is made up of all the colors of the visible spectrum. We call this the Balmer series. google_ad_height = 90; What part(s) of the electromagnetic spectrum are these in? Already have an account? 30 - (a) Which line in the Balmer series is the first... Ch. Previous Question Next Question. The energy level of the electron of a hydrogen atom is given by the following formula, where nnn denotes the principal quantum number: The Balmer series or Balmer lines in atomic physics, is the designation of one of a set of six different named series describing the spectral line emissions of the hydrogen atom.. As you I just discussed in the Spectral Lines page, electrons fall to lower energy levels and give off light in the form of a spectrum. (D) n=4\u2192n=2n=4\\rightarrow n=2n=4\u2192n=2, Observe that the red line has the longest wavelength within the Balmer series.","date":"2021-03-08 00:16:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7227016091346741, \"perplexity\": 983.7867158810969}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178381230.99\/warc\/CC-MAIN-20210307231028-20210308021028-00293.warc.gz\"}"}
| null | null |
I believe I have identified an issue with the current C7 kernel when running under 2012 R2 Hyper-V in a Hyper-V cluster. Every time the VM is live-migrated between nodes, the kernel will panic with a null pointer dereference every time - please see the log provided below.
Interestingly, this issue is resolved if the Microsoft-provided LIS4 release and its kmod-microsoft-hyper-v-4.0.11-20150728.x86_64.rpm package are installed, suggesting the issue resides in the C7-shipped hv_ modules.
I have a number of partial kdumps available and test machines available to assist with debugging this issue further.
live-migrate VM between cluster nodes. VM will panic each time.
3: console reports Buffer I/O error on dm0 and CPU#1 stuck for 22s. XFS dismounts FS owing to I/O error on /dev/sda. Console locks up. Manually reset.
3: XFS metadata I/O error xfs_trans_read_buf_map error 5 numblks 1, soft lockup - CPU3 stuck for 23s. Console locks up, manually reset.
4: XFS metadata I/O error xfs_trans_read_buf_map error 5 numblks 1, soft lockup - CPU0 stuck for 23s. Console locks up, manually reset.
3.10.0-229.7.2 was released under RHSA-2015-1137 (https://rhn.redhat.com/errata/RHSA-2015-1137.html). None of the listed security fixes appear to relate, and the CentOS changelog for same only lists "Debranding changes". I suspect this regression exists upstream, as I can also replicate it with the latest OpenVZ 7 kernel which is based from 3.10.0-229.7.2.
Thanks for your investigation. Please post these findings upstream at RHs bugzilla (which can be done without a service contract). Could you then also please cross-reference the bugzilla entries. Once it gets fixed in the upstream kernel, CentOS will inherit the fix.
Thanks. As kernel bugs are - as usual - marked private by RH, could you please report back on changes from time to time?
My bug (RHBZ1267591) has been closed as a duplicate of the pre-existing RHBZ1242390.
Can you access 1242390? If so, please still post some progress, if you find the time.
Sadly not, but I have had confirmation that 1242390 has also been resolved and rolled into 7.2. The dev I have been discussing this with has said he expects a 7.1 point release at some point, but there is no guarantee if or when this will happen.
7.2 is currently in beta phase (only for RH customers). So maybe this fix will not take to long to be released.
For anyone interested, I have just tested with kernel 3.10.0-229.20.1, and this problem remains unfixed upstream.
I will repeat testing and update this bug should any further pre-7.2 errata kernels be released.
I presume 3.10.0-229.20.1 is the last one before 7.2. Could you confirm that the upstream BZ dealing with this issue is 1242390 ?
Hyper-V servers running 2012-R2, with Failover Clustering.
I have had direct contact from TUV, and confirmation this is fixed in the release 7.2 kernel, but also in 3.10.0-229.24.1 for 7.1 which I can't find errata for but guess must be soon to ship.
The kernel for 7.2 (1511) is allready in the CR repo. You could give that a try. for 229.24.1 , that now sounds like a z-stream release, which CentOS does not reproduce.
How is your issue with the release of 7 (1511) and the new kernel?
The 7.2.1511 release of CentOS has resolved issue #9538 for me.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,055
|
package com.github.kompot.play2sec.authentication.providers.password
import play.api.mvc._
import play.api.data.Form
import UsernamePasswordAuthProvider._
import play.api.Logger
import play.api.libs.json._
import play.api.libs.functional.syntax._
import com.github.kompot.play2sec.authentication.{PlaySecPlugin, MailService}
import com.github.kompot.play2sec.{authentication => atn}
import com.github.kompot.play2sec.authentication.user.NameIdentity
import scala.concurrent.{ExecutionContext, Future}
import ExecutionContext.Implicits.global
import com.typesafe.plugin._
import com.github.kompot.play2sec.authentication.providers.AuthProvider
import scala.Some
import play.api.mvc.SimpleResult
abstract class UsernamePasswordAuthProvider[V, UL <: UsernamePasswordAuthUser,
US <: UsernamePasswordAuthUser, UR <: UsernamePasswordAuthUser,
L <: (String, String), S<: (String, String), R <: String]
(implicit app: play.api.Application) extends AuthProvider(app) {
protected val mailService: MailService
override protected def requiredSettings = List(
s"$CFG_MAIL.$CFG_MAIL_DELAY",
s"$CFG_MAIL.$CFG_MAIL_FROM.$CFG_MAIL_FROM_EMAIL"
)
override def onStart() {
super.onStart()
}
override val key = UsernamePasswordAuthProvider.PROVIDER_KEY
override def authenticate[A](request: Request[A], payload: Option[Case]) =
payload match {
case Some(SIGNUP) => processSignup(request)
case Some(LOGIN) => processLogin(request)
case Some(RECOVER_PASSWORD) => processRecover(request)
case Some(VERIFIED_EMAIL(email)) => processVerifiedEmail(request, email)
case _ =>
Future.successful(new LoginSignupResult(com.typesafe.plugin.use[PlaySecPlugin].login.url))
}
private def processVerifiedEmail[A](request: Request[A], email: String): Future[LoginSignupResult] = {
val authUser = buildLoginAuthUser(getVerifiedEmailTuple(email), request)
Future.successful(new LoginSignupResult(authUser))
}
private def processRecover[A](request: Request[A]): Future[LoginSignupResult] = {
val login: R = getRecover(request)
val authUser: UR = buildResetPasswordAuthUser(login, request)
sendResetPasswordEmail(request, authUser)
Future.successful(
new LoginSignupResult(use[PlaySecPlugin].passwordRecoveryRequestSuccessful(request))
)
}
private def processLogin[A](request: Request[A]): Future[LoginSignupResult] = {
import LoginResult._
val login: L = getLogin(request)
val authUser: UL = buildLoginAuthUser(login, request)
Logger.info("processLogin, authUser.expires = " + authUser.expires)
for {
r <- loginUser(authUser)
} yield {
Logger.info("Login result is " + r)
r match {
// The email of the user is not verified
// should not allow to log in
case USER_UNVERIFIED =>
new LoginSignupResult(use[PlaySecPlugin].userLoginUnverified(request))
// The user exists and the given password was correct
case USER_LOGGED_IN =>
new LoginSignupResult(authUser)
// don't expose this - it might harm users privacy if anyone
// knows they signed up for our service
// forward to login page
case WRONG_PASSWORD | NOT_FOUND =>
new LoginSignupResult(use[PlaySecPlugin].userNotFound(request))
// TODO replace with error fallback URL
case _ =>
new LoginSignupResult("/")
}
}
}
private def processSignup[A](request: Request[A]): Future[LoginSignupResult] = {
import SignupResult._
val signup: S = getSignup(request)
val authUser: US = buildSignupAuthUser(signup, request)
for {
r <- signupUser(authUser, request)
} yield {
Logger.info("Signup result is " + r)
r match {
case USER_EXISTS =>
new LoginSignupResult(use[PlaySecPlugin].userExists(request))
case USER_EXISTS_UNVERIFIED | USER_CREATED_UNVERIFIED =>
// User got created as unverified
// Send validation email
sendVerifyEmailMailing(request, authUser)
new LoginSignupResult(use[PlaySecPlugin].userSignupUnverified(request))
case USER_CREATED =>
new LoginSignupResult(authUser)
case _ =>
// TODO replace with some configurable URL
new LoginSignupResult("/")
}
}
}
override val isExternal = false
private def getSignup[A](request: Request[A]): S = getSignupForm.bindFromRequest()(request).get
private def getLogin[A](request: Request[A]): L = getLoginForm.bindFromRequest()(request).get
private def getRecover[A](request: Request[A]): R = getResetPasswordForm.bindFromRequest()(request).get
private def sendVerifyEmailMailing[A](request: Request[A], user: US) {
val subject = getVerifyEmailMailingSubject(user, request)
val record = generateSignupVerificationRecord(user)
val body = getVerifyEmailMailingBody(record, user, request)
mailService.sendMail(subject, Array(getEmailName(user)), body)
}
private def sendResetPasswordEmail[A](request: Request[A], user: UR) {
val subject = getResetPasswordEmailMailingSubject(user, request)
val record = generateResetPasswordVerificationRecord(user)
val body = getResetPasswordEmailMailingBody(record, user, request)
mailService.sendMail(subject, Array(getEmailName(user.email, "")), body)
}
def getEmailName(user: US): String = {
val name = user match {
case identity: NameIdentity => identity.name
case _ => ""
}
getEmailName(user.email, name)
}
def getEmailName(email: String, name: String): String = mailService.getEmailName(email, name)
protected def buildLoginAuthUser[A](login: L, request: Request[A]): UL
protected def buildSignupAuthUser[A](signup: S, request: Request[A]): US
protected def buildResetPasswordAuthUser[A](recover: R, request: Request[A]): UR
protected def getLoginForm: Form[L]
protected def getSignupForm: Form[S]
protected def getResetPasswordForm: Form[R]
protected def getVerifiedEmailTuple(email: String): L
protected def loginUser(authUser: UL): Future[LoginResult.Value]
protected def signupUser[A](user: US, request: Request[A]): Future[SignupResult.Value]
protected def generateSignupVerificationRecord(user: US): V
protected def generateResetPasswordVerificationRecord(user: UR): V
protected def getVerifyEmailMailingSubject[A](user: US, request: Request[A]): String
protected def getVerifyEmailMailingBody[A](verificationRecord: V, user: US, request: Request[A]): String
protected def getResetPasswordEmailMailingSubject[A](user: UR, request: Request[A]): String
protected def getResetPasswordEmailMailingBody[A](verificationRecord: V, user: UR, request: Request[A]): String
}
object UsernamePasswordAuthProvider {
val PROVIDER_KEY = "email"
val CFG_MAIL = "mail"
val CFG_MAIL_FROM_EMAIL = "email"
val CFG_MAIL_DELAY = "delay"
val CFG_MAIL_FROM = "from"
def handleLogin[A](request: Request[A]): Future[SimpleResult] =
atn.handleAuthentication(PROVIDER_KEY, request, Some(LOGIN))
def handleSignup[A](request: Request[A]): Future[SimpleResult] =
atn.handleAuthentication(PROVIDER_KEY, request, Some(SIGNUP))
def handleRecoverPassword[A](request: Request[A]): Future[SimpleResult] =
atn.handleAuthentication(PROVIDER_KEY, request, Some(RECOVER_PASSWORD))
def handleVerifiedEmailLogin[A](request: Request[A], email: String): Future[SimpleResult] =
atn.handleAuthentication(PROVIDER_KEY, request, Some(VERIFIED_EMAIL(email)))
}
sealed trait Case
case object SIGNUP extends Case
case object LOGIN extends Case
case object RECOVER_PASSWORD extends Case
case class VERIFIED_EMAIL(email: String) extends Case
object SignupResult extends Enumeration {
val USER_EXISTS = Value
val USER_CREATED_UNVERIFIED = Value
val USER_CREATED = Value
val USER_EXISTS_UNVERIFIED = Value
val UNKNOWN_ERROR = Value
}
object LoginResult extends Enumeration {
val USER_UNVERIFIED = Value
val USER_LOGGED_IN = Value
val NOT_FOUND = Value
val WRONG_PASSWORD = Value
val UNKNOWN_ERROR = Value
}
// TODO: Used?
trait UsernamePassword {
def getEmail: String
def getPassword: String
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,541
|
"""Usage: %prog [options] [<commitref>]*
If no <commitref>'s are supplied, it defaults to HEAD.
Calculates the generation number for one or more commits in a git repo.
Generation number of a commit C with parents P is defined as:
generation_number(C, []) = 0
generation_number(C, P) = max(map(generation_number, P)) + 1
This number can be used to order commits relative to each other, as long as for
any pair of the commits, one is an ancestor of the other.
Since calculating the generation number of a commit requires walking that
commit's entire history, this script caches all calculated data inside the git
repo that it operates on in the ref 'refs/number/commits'.
"""
from __future__ import print_function
from __future__ import division
import binascii
import collections
import logging
import optparse
import os
import struct
import sys
import tempfile
import git_common as git
import subprocess2
CHUNK_FMT = '!20sL'
CHUNK_SIZE = struct.calcsize(CHUNK_FMT)
DIRTY_TREES = collections.defaultdict(int)
REF = 'refs/number/commits'
AUTHOR_NAME = 'git-number'
AUTHOR_EMAIL = 'chrome-infrastructure-team@google.com'
# Number of bytes to use for the prefix on our internal number structure.
# 0 is slow to deserialize. 2 creates way too much bookeeping overhead (would
# need to reimplement cache data structures to be a bit more sophisticated than
# dicts. 1 seems to be just right.
PREFIX_LEN = 1
# Set this to 'threads' to gather coverage data while testing.
POOL_KIND = 'procs'
def pathlify(hash_prefix):
"""Converts a binary object hash prefix into a posix path, one folder per
byte.
>>> pathlify('\xDE\xAD')
'de/ad'
"""
if sys.version_info.major == 3:
return '/'.join('%02x' % b for b in hash_prefix)
else:
return '/'.join('%02x' % ord(b) for b in hash_prefix)
@git.memoize_one(threadsafe=False)
def get_number_tree(prefix_bytes):
"""Returns a dictionary of the git-number registry specified by
|prefix_bytes|.
This is in the form of {<full binary ref>: <gen num> ...}
>>> get_number_tree('\x83\xb4')
{'\x83\xb4\xe3\xe4W\xf9J*\x8f/c\x16\xecD\xd1\x04\x8b\xa9qz': 169, ...}
"""
ref = '%s:%s' % (REF, pathlify(prefix_bytes))
try:
raw = git.run('cat-file', 'blob', ref, autostrip=False, decode=False)
return dict(struct.unpack_from(CHUNK_FMT, raw, i * CHUNK_SIZE)
for i in range(len(raw) // CHUNK_SIZE))
except subprocess2.CalledProcessError:
return {}
@git.memoize_one(threadsafe=False)
def get_num(commit_hash):
"""Returns the generation number for a commit.
Returns None if the generation number for this commit hasn't been calculated
yet (see load_generation_numbers()).
"""
return get_number_tree(commit_hash[:PREFIX_LEN]).get(commit_hash)
def clear_caches(on_disk=False):
"""Clears in-process caches for e.g. unit testing."""
get_number_tree.clear()
get_num.clear()
if on_disk:
git.run('update-ref', '-d', REF)
def intern_number_tree(tree):
"""Transforms a number tree (in the form returned by |get_number_tree|) into
a git blob.
Returns the git blob id as hex-encoded string.
>>> d = {'\x83\xb4\xe3\xe4W\xf9J*\x8f/c\x16\xecD\xd1\x04\x8b\xa9qz': 169}
>>> intern_number_tree(d)
'c552317aa95ca8c3f6aae3357a4be299fbcb25ce'
"""
with tempfile.TemporaryFile() as f:
for k, v in sorted(tree.items()):
f.write(struct.pack(CHUNK_FMT, k, v))
f.seek(0)
return git.intern_f(f)
def leaf_map_fn(pre_tree):
"""Converts a prefix and number tree into a git index line."""
pre, tree = pre_tree
return '100644 blob %s\t%s\0' % (intern_number_tree(tree), pathlify(pre))
def finalize(targets):
"""Saves all cache data to the git repository.
After calculating the generation number for |targets|, call finalize() to
save all the work to the git repository.
This in particular saves the trees referred to by DIRTY_TREES.
"""
if not DIRTY_TREES:
return
msg = 'git-number Added %s numbers' % sum(DIRTY_TREES.values())
idx = os.path.join(git.run('rev-parse', '--git-dir'), 'number.idx')
env = os.environ.copy()
env['GIT_INDEX_FILE'] = idx
progress_message = 'Finalizing: (%%(count)d/%d)' % len(DIRTY_TREES)
with git.ProgressPrinter(progress_message) as inc:
git.run('read-tree', REF, env=env)
prefixes_trees = ((p, get_number_tree(p)) for p in sorted(DIRTY_TREES))
updater = subprocess2.Popen(['git', 'update-index', '-z', '--index-info'],
stdin=subprocess2.PIPE, env=env)
with git.ScopedPool(kind=POOL_KIND) as leaf_pool:
for item in leaf_pool.imap(leaf_map_fn, prefixes_trees):
updater.stdin.write(item.encode())
inc()
updater.stdin.close()
updater.wait()
assert updater.returncode == 0
tree_id = git.run('write-tree', env=env)
commit_cmd = [
# Git user.name and/or user.email may not be configured, so specifying
# them explicitly. They are not used, but requried by Git.
'-c', 'user.name=%s' % AUTHOR_NAME,
'-c', 'user.email=%s' % AUTHOR_EMAIL,
'commit-tree',
'-m', msg,
'-p'] + git.hash_multi(REF)
for t in targets:
commit_cmd.extend(['-p', binascii.hexlify(t).decode()])
commit_cmd.append(tree_id)
commit_hash = git.run(*commit_cmd)
git.run('update-ref', REF, commit_hash)
DIRTY_TREES.clear()
def preload_tree(prefix):
"""Returns the prefix and parsed tree object for the specified prefix."""
return prefix, get_number_tree(prefix)
def all_prefixes(depth=PREFIX_LEN):
if sys.version_info.major == 3:
prefixes = [bytes([i]) for i in range(255)]
else:
prefixes = [chr(i) for i in range(255)]
for x in prefixes:
# This isn't covered because PREFIX_LEN currently == 1
if depth > 1: # pragma: no cover
for r in all_prefixes(depth - 1):
yield x + r
else:
yield x
def load_generation_numbers(targets):
"""Populates the caches of get_num and get_number_tree so they contain
the results for |targets|.
Loads cached numbers from disk, and calculates missing numbers if one or
more of |targets| is newer than the cached calculations.
Args:
targets - An iterable of binary-encoded full git commit hashes.
"""
# In case they pass us a generator, listify targets.
targets = list(targets)
if all(get_num(t) is not None for t in targets):
return
if git.tree(REF) is None:
empty = git.mktree({})
commit_hash = git.run(
# Git user.name and/or user.email may not be configured, so specifying
# them explicitly. They are not used, but requried by Git.
'-c', 'user.name=%s' % AUTHOR_NAME,
'-c', 'user.email=%s' % AUTHOR_EMAIL,
'commit-tree',
'-m', 'Initial commit from git-number',
empty)
git.run('update-ref', REF, commit_hash)
with git.ScopedPool(kind=POOL_KIND) as pool:
preload_iter = pool.imap_unordered(preload_tree, all_prefixes())
rev_list = []
with git.ProgressPrinter('Loading commits: %(count)d') as inc:
# Curiously, buffering the list into memory seems to be the fastest
# approach in python (as opposed to iterating over the lines in the
# stdout as they're produced). GIL strikes again :/
cmd = [
'rev-list', '--topo-order', '--parents', '--reverse', '^' + REF,
] + [binascii.hexlify(target).decode() for target in targets]
for line in git.run(*cmd).splitlines():
tokens = [binascii.unhexlify(token) for token in line.split()]
rev_list.append((tokens[0], tokens[1:]))
inc()
get_number_tree.update(preload_iter)
with git.ProgressPrinter('Counting: %%(count)d/%d' % len(rev_list)) as inc:
for commit_hash, pars in rev_list:
num = max(map(get_num, pars)) + 1 if pars else 0
prefix = commit_hash[:PREFIX_LEN]
get_number_tree(prefix)[commit_hash] = num
DIRTY_TREES[prefix] += 1
get_num.set(commit_hash, num)
inc()
def main(): # pragma: no cover
parser = optparse.OptionParser(usage=sys.modules[__name__].__doc__)
parser.add_option('--no-cache', action='store_true',
help='Do not actually cache anything we calculate.')
parser.add_option('--reset', action='store_true',
help='Reset the generation number cache and quit.')
parser.add_option('-v', '--verbose', action='count', default=0,
help='Be verbose. Use more times for more verbosity.')
opts, args = parser.parse_args()
levels = [logging.ERROR, logging.INFO, logging.DEBUG]
logging.basicConfig(level=levels[min(opts.verbose, len(levels) - 1)])
# 'git number' should only be used on bots.
if os.getenv('CHROME_HEADLESS') != '1':
logging.error("'git-number' is an infrastructure tool that is only "
"intended to be used internally by bots. Developers should "
"use the 'Cr-Commit-Position' value in the commit's message.")
return 1
if opts.reset:
clear_caches(on_disk=True)
return
try:
targets = git.parse_commitrefs(*(args or ['HEAD']))
except git.BadCommitRefException as e:
parser.error(e)
load_generation_numbers(targets)
if not opts.no_cache:
finalize(targets)
print('\n'.join(map(str, map(get_num, targets))))
return 0
if __name__ == '__main__': # pragma: no cover
try:
sys.exit(main())
except KeyboardInterrupt:
sys.stderr.write('interrupted\n')
sys.exit(1)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,533
|
Gendelman is a surname. Notable people with the surname include:
Howard E. Gendelman (born 1954), American physician
Ofir Gendelman (born 1971), Israeli diplomat
Zvi Gendelman (born 1956), Israeli politician
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,204
|
{"url":"http:\/\/internetdo.com\/2022\/12\/solving-lesson-6-page-29-math-learning-topic-10-kite\/","text":"## Solving Lesson 6 Page 29 Math Learning Topic 10 \u2013 Kite>\n\nTopic\n\nProve $${n^n} > {(n + 1)^{n \u2013 1}}$$ for all $$n \\in \\mathbb{N}*,n \\ge 2.$$\n\nSolution method \u2013 See details\n\nInductive method: Prove that the statement is true with $$n \\ge p$$\n\nStep 1: Check the statement is true for $$n = p$$\n\nStep 2: Suppose the proposition is true for natural numbers $$n = k \\ge p$$ and prove the statement true for $$n = k + 1.$$ Conclusion.\n\nDetailed explanation\n\nStep 1: When $$n = 2$$ we have $${2^2} > {(2 + 1)^{2 \u2013 1}}$$ or $$4 > 3$$ obviously true\n\nSo the inequality holds for $$n = 2$$\n\nStep 2: With k being an arbitrary positive integer that the inequality is true, we must prove the inequality is true for k+1, that is:\n\n$${(k + 1)^{k + 1}} > {(k + 1 + 1)^{k + 1 \u2013 1}}$$ or $${(k + 1)^{k + 1} } > {(k + 2)^k}$$\n\nIndeed, by the assumption of induction we have:\n\n$${k^k} > {(k + 1)^{k \u2013 1}}$$\n\nI guess\n\n$${k^k}{(k + 1)^{k + 1}} > {(k + 1)^{k \u2013 1}}{(k + 1)^{k + 1}} = {( k + 1)^{k \u2013 1 + k + 1}} = {(k + 1)^{2k}}$$\n\nWhere $${(k + 1)^{2k}} = {\\left[ {{{(k + 1)}^2}} \\right]^k} = {\\left( {{k^2} + 2k + 1} \\right)^k} > {\\left( {{k^2} + 2k} \\right)^k}$$\n\n$$\\Rightarrow {k^k}{(k + 1)^{k + 1}} > {\\left( {{k^2} + 2k} \\right)^k} = {\\left[ {k.(k + 2)} \\right]^k} = {k^k}. {(k + 2)^k}$$\n\n$$\\Rightarrow {(k + 1)^{k + 1}} > {(k + 2)^k}$$\n\nSo the inequality holds for k+1. Thus, by the principle of mathematical induction, the inequality holds for all $$n \\in \\mathbb{N}*$$.","date":"2023-01-28 06:03:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8717767000198364, \"perplexity\": 292.94712866417285}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499524.28\/warc\/CC-MAIN-20230128054815-20230128084815-00705.warc.gz\"}"}
| null | null |
Free Radical
By Jeremy McCarter
Photo: Brigitte Lacombe
It began in the spring with the raccoons. In March, staffers of the Public Theater unlocked the control booth of the Delacorte Theater in Central Park, as they have every year for nearly half a century, and evicted the critters that had nested there for the winter. Fourteen hectic weeks of building and rehabilitating later, Macbeth is about to open. Summer for the New Yorkiest theater in New York has arrived.
There were only four or five raccoons this spring, which makes it a better year than most. And there are other reasons for fresh enthusiasm as well. After a year in which the Public's new artistic director, Oskar Eustis, inherited some shows from his predecessor, George C. Wolfe, this summer's lineup is Eustis's alone. Plus the institution's fiscal health is less perilous than it has been in a while. Five years ago, the Public faced one of the gravest crises in its crisis-riddled history. A pair of commercial wipeouts—On the Town and The Wild Party—plus the strain of 9/11 led to deficits, layoffs, pay cuts, and a reduced production slate. Fran Reiter began and ended a brief tenure as executive director that staffers regarded as disastrous; two board members resigned. Kenneth Lerer, the board chairman at the time, and his wife loaned the theater money to meet its payroll—twice. He says he feared for its survival.
Since arriving in 2002, executive director Mara Manus has led an effort to balance the books, allowing the organization to advance with confidence. But where should it go? With its reach, history, and unique mix of uptown glamour and downtown ideals, the Public has long occupied a pivotal place in New York culture, but New York is not what it was when Joseph Papp, the Public's founder, outdueled Robert Moses to bring Shakespeare to the park. At a complicated time, the place has a uniquely complicated leader. Eustis is bold and cautious, radical and judicious. As his first full summer season prepared to open, he offered a highly detailed look at his plans and how he intends to realize them.
When he enters a room, it's easy to see why so many people call Oskar Eustis, with affection, Clintonian. On a recent Sunday morning, he dropped by the last run-through of Macbeth before the show moved to the park. He placed a hand on this actor's shoulder, that one's elbow. For the show's star, Liev Schreiber, he had a big bear hug. Tall and bearded, Eustis looks a little like a bear himself.
Eustis likes to say he is the first person to run this theater who considers it his profession to run theaters, and a look at his previous gig offers some clues to how he does it. At Trinity Rep in Providence, he presided over a remarkable financial recovery, winning over civic leaders and major funders to eliminate a $3 million accumulated deficit. The seasons weren't especially cutting-edge—an annual production of A Christmas Carol was de rigueur—but some shows were modestly inventive. He produced My Fair Lady, but in a stripped-down, two-piano staging. He put on the obligatory Shakespeare histories, but as a massive multipart cycle, "The Henriad." Eustis's personal taste is more adventurous than these choices, but they suited the audience, which doubled while he was there.
It's telling that, when asked what made him the proudest, he names not an artistic triumph but an M.F.A. program that links Trinity Rep to Brown University. When pressed for ideas for the Public, he prominently mentions establishing closer ties with NYU and starting endowed chairs for playwrights in-house. These are not the priorities of a flashy impresario; they are long-term ideological, infrastructural commitments. Eustis considers himself, above all, an institution-builder. "That means translating dramaturgical skills to the institutional level, making somebody else's vision come to life," he says. "They feel like the same thing to me." He responds deeply to other people's work. During much of the Macbeth rehearsal, Eustis strokes his beard, occasionally scribbling notes on a legal pad. When Macbeth's henchmen kill Macduff's wife and young son, he audibly sniffles back tears.
The next morning, the stage crew at the Delacorte is watching heavy clouds roll over Central Park, again. Already this week a sudden downpour has washed a coat of paint off the half-built set, and with the actors soon to arrive, time is short. Seventy people are now working in shifts from 8 A.M. to 4 A.M. "We give everybody lots of Gatorade, sunscreen, and bug spray," notes Ruth Sternberg, the Public's director of production.
The torrents, the time pressure, the marauding fauna: It has always been this way for the Public, where ambitions constitutionally exceed means. The inexhaustible Papp believed that money followed art, and when he needed funds, he charmed rich patrons or badgered city fathers to keep the doors open. Then he hit a gusher: While other New York nonprofits were building their infrastructure slowly, the Public opened A Chorus Line, which poured some $40 million into the place after it transferred to Broadway in 1975. When the show closed in 1990, just months before Papp's death, his successors had to get by without the resource that had kept the Public going for a generation. The avant-garde director JoAnne Akalaitis lasted only a year and a half; though Wolfe banked a lot of money on transfers like Bring In 'Da Noise, Bring In 'Da Funk, the 2001 crisis wiped out all those gains. Manus arrived facing a half-million-dollar annual deficit.
Liev Schreiber as Macbeth, at a preview in Central Park last week. Photo: Michael Daniel/The Public Theater
Eustis isn't interested in continuing the boom-and-bust cycle, and anyway, he wouldn't be allowed to. When the board hired Manus, it gave the executive director authority equal to that of the artistic director, in part to impose fiscal discipline. A major shakeup followed her arrival—firings, resignations, a complete turnover in the development office—as did scattered complaints about her brusque style. But she's got the money flowing again. Some sponsors who had turned away have been wooed back, and individual giving has increased 80 percent. There are now 70 donor events a year, up from 30, with one almost every night in the park. She's also received promises from the city of $13 million toward a $16.5 million renovation of the building on Lafayette Street. New features include a lobby redesign, a patrons' lounge, a café, and replacements for the medieval bathrooms.
But with a business executive (who has scant theater background) newly ascendant, and an artistic executive who's really fond of infrastructure, what happens to the Public's freewheeling vibe, what Wolfe calls its "wonderful dance" of order and anarchy? Several artists who have worked there worry that he will opt for safe choices, importing a regional-theater mind-set to the Public; as one puts it, "Organizational concerns might start to trump artistic concerns." In other words, under Eustis, the competent manager and savvy fund-raiser, the place could become prosperous but dull—the antithesis of the Public's usual profile.
"I firmly believe there's a way to combine the raffishness with a stable infrastructure," says Eustis. In the Public's best days, that was its specialty; Papp drew crowds and kept the energy level high by putting on diversified work—a lot of it. To that end, Eustis wants to boost the production volume. While Macbeth readies for its opening, he has jammed Lafayette Street with David Hare's extended Iraq-war docudrama Stuff Happens; two new plays incubated under Wolfe, José Rivera's School of the Americas and Diana Son's Satellites; and a workshop of King Lear starring Kevin Kline. Eustis also volunteered to host the Hip-Hop Theater Festival, even though Sternberg warned that the staff was stretched too thin to manage it. When he overruled her, she proposed he make T-shirts for the crew that read I'M SORRY ABOUT JUNE, COMRADE over his signature. (Though he today calls himself a "radical Democrat," Eustis was a red-diaper baby, and he still calls lots of people "comrade": actors, staff, the receptionist …)
Volume isn't everything, of course. The work needs to be varied enough that the lobby crowd looks, as Wolfe always says, "like a subway station." So Eustis is moving in two directions at once. He wants first to reconnect with the history of what is now the longest-running major theater in town. Hare is already back, and John Guare will soon return for the first time since 1977, with a play slated to star Jeffrey Wright and Mos Def. He is even planning to have Akalaitis workshop Bacchae.
Assuming that the brand-name talent will deliver, he still faces a second, and greater, challenge: finding the new writers and actors who will define his Public. Eustis can make plenty of bets, since at the Public, only 35 percent of revenue is earned, the other 65 being contributed—the opposite of most places. Success depends on whether those bets turn out to be any good—if his taste suits the place. Last year, he made a smart call in producing Rinne Groff's The Ruby Sunrise, even if his own direction was so-so. Next season brings new plays by Julia Cho and actor–writer–slam poet Daniel Beaty, among others.
He's also bringing in his old collaborator Tony Kushner, with whom he's had a long and sometimes difficult relationship (see "Backstory"). In August, Eustis will bring Kushner's new translation of Brecht's Mother Courage to the park. It will be directed, in a small-world sort of way, by Wolfe.
If Stuff Happens is his ideal downtown show, Mother Courage may be the model for the park. It's inextricably rooted in the world today, which is key to his vision for all Public shows. It also brings a great actress, Meryl Streep, to a stage that Eustis wants to make an essential destination for great actors. (Brian Dennehy turns up next summer; he's "leaning towards" playing Falstaff.) And it marks Kushner's Delacorte debut—his official debut, anyway. Three years ago, he rewrote a speech—in Elizabethan verse—that Wolfe and director Mark Wing-Davey dropped into Henry V to ground the play more completely in our Iraq misadventure. Nobody spotted it.
The weather is so beautiful on the night of Macbeth's final dress rehearsal you almost forget the raccoons and bugs and storms. As 200 or so invited guests take their seats, Eustis has a big hug for Sam Mendes, a "recent friend." Joe Papp liked to make brave, go-for-broke gambles; on nights like this, you realize how resoundingly his biggest one paid off.
For all his modest stands and impersonal choices, Eustis has been nurturing a big move of his own, one that might out-Papp even Papp. He has spoken in the past of his belief in "radical accessibility." When I ask him to elaborate, he speaks deliberately. "There should be nobody economically excluded from seeing this work. I don't know the best way to do it, but we do have a very successful model in the park: We give them all away."
For decades, steep ticket prices have hampered every attempt to reform the New York theater: high cost creates risk, which limits audiences and excludes the young, which leads to conservative programming, which saps energy and diversity, which sticks you with the overpriced superannuated mess we're in today. If Eustis could somehow make every ticket free, that particular Gordian knot would be cut. Is that really what he intends?
He won't say those words, but he does add that people throughout the organization are "discussing how to execute" a plan for dramatically expanded access. Warren Spector, chairman of the board, agrees that ticket availability is a priority, as long as lowering the price doesn't seem to devalue the work. Manus agrees, but isn't sold on free tickets: Beyond the financial concern, she says, they would need a strategy to ensure that a broader audience was really being served.
If Eustis even comes close to this ideal, he wouldn't just be tending Joe Papp's legacy, he'd be markedly advancing it. In his pursuit of "democratization," the watchword for all his efforts, he'd prove that the Public still has an essential role to play in New York's aesthetic, political, and civic life—a radical one.
Two decades ago, on the very first night of his very first show, Tony Kushner heard a stranger join his characters in singing "The Internationale"—one who knew every word. "I thought some ancient lefty had wandered in," he recalls. It was Oskar Eustis. Eventually, Eustis would commission the play that became Angels in America, helping Kushner shape it and co-directing it in L.A. But just before the play's Broadway transfer, after one too many fights, the playwright replaced Eustis with George Wolfe. Now all three are working together in the park. "It's intense," says Eustis, "but not peculiar."
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,455
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Tiggarstavar (Plagiobothrys) är ett släkte av strävbladiga växter. Tiggarstavar ingår i familjen strävbladiga växter.
Dottertaxa till Tiggarstavar, i alfabetisk ordning
Plagiobothrys acanthocarpus
Plagiobothrys arizonicus
Plagiobothrys armeriifolius
Plagiobothrys austiniae
Plagiobothrys australasicus
Plagiobothrys bracteatus
Plagiobothrys calandrinioides
Plagiobothrys canescens
Plagiobothrys chorisianus
Plagiobothrys collinus
Plagiobothrys congestus
Plagiobothrys corymbosus
Plagiobothrys distantiflorus
Plagiobothrys elachanthus
Plagiobothrys figuratus
Plagiobothrys foliosus
Plagiobothrys fulvus
Plagiobothrys germainii
Plagiobothrys glaber
Plagiobothrys glomeratus
Plagiobothrys glyptocarpus
Plagiobothrys gracilis
Plagiobothrys greenei
Plagiobothrys hirtus
Plagiobothrys hispidus
Plagiobothrys humilis
Plagiobothrys humistratus
Plagiobothrys hystriculus
Plagiobothrys infectivus
Plagiobothrys jonesii
Plagiobothrys kingii
Plagiobothrys kunthii
Plagiobothrys lamprocarpus
Plagiobothrys leptocladus
Plagiobothrys linifolius
Plagiobothrys lithocaryus
Plagiobothrys macbridei
Plagiobothrys mexicanus
Plagiobothrys mollis
Plagiobothrys myosotoides
Plagiobothrys nothofulvus
Plagiobothrys oppositifolius
Plagiobothrys orientalis
Plagiobothrys orthostatus
Plagiobothrys parishii
Plagiobothrys pedicellaris
Plagiobothrys plebejus
Plagiobothrys plurisepalus
Plagiobothrys polycaulis
Plagiobothrys pratensis
Plagiobothrys pringlei
Plagiobothrys procumbens
Plagiobothrys pulchellus
Plagiobothrys pygmaeus
Plagiobothrys reticulatus
Plagiobothrys salsus
Plagiobothrys scouleri
Plagiobothrys scriptus
Plagiobothrys shastensis
Plagiobothrys stipitatus
Plagiobothrys strictus
Plagiobothrys tenellus
Plagiobothrys tener
Plagiobothrys tenuifolius
Plagiobothrys torreyi
Plagiobothrys trachycarpus
Plagiobothrys uliginosus
Plagiobothrys uncinatus
Plagiobothrys verrucosus
Bildgalleri
Källor
Externa länkar
Strävbladiga växter
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,045
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Adélard, mort en 903, était un membre de la dynastie des Popponides (l'ancienne maison de Babenberg) en Franconie, fils de († 886), margrave en Frise, et de Judith (Ingeltrude), possiblement une fille du margrave Évrard de Frioul et petite-fille de l'empereur Louis le Pieux.
Comte en Franconie, il meurt au cours de la querelle sanglante des Babenberg avec la dynastie des Conradiens. En 902, les deux parties se sont affrontées dans une bataille aux portes du cháteau de Bamberg où Adélard a été vaincu et arrêté. Selon les chroniques de Réginon de Prüm, son adversaire Gebhard veut se venger et le fit exécuter après la Diète de Forchheim l'année suivante.
Maison de Babenberg au haut Moyen Âge
Personnalité du haut Moyen Âge par nom
Popponides
|
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| 543
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One of the best pastimes people in a city like Mumbai will enjoy is trekking. Take a break from the bustle and dust of the crowded streets to walk in pristine purity of gushing streams and lonely peaks, and revel in the wonderful exercise of walking through the day.
Motor yacht Bella Christy presents a rare opportunity to buy the newest Pacific Mariner cruiser in turnkey condition. Launched in 2010 and immediately receiving acclaim for her luxurious superyacht features and excellent seaworthiness, the 85 foot yacht for sale Bella Christy has the amenities, luxuries, and technologies of a much larger yacht.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 921
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// Copyright (c) 2013 Jonathan Magnan (http://zzzportal.com) All rights reserved. Licensed under MIT License (MIT) License can be found here: https://zextensionmethods.codeplex.com/license
using System.Collections.Generic;
namespace Oragon.Architecture.Extensions
{
public static partial class OragonExtensions
{
#region Public Methods
/// <id>4C6147B5-7CCB-4E45-9611-11B86ADA8CB8</id>
/// <summary>
/// Concatenates all the elements of a IEnumerable, using the specified separator between each element.
/// </summary>
/// <typeparam name="T">Generic type parameter.</typeparam>
/// <param name="this">An IEnumerable that contains the elements to concatenate.</param>
/// <param name="separator">
/// The string to use as a separator. separator is included in the returned string only if value has more than one element.
/// </param>
/// <returns>
/// A string that consists of the elements in value delimited by the separator string. If value is an empty array, the method returns String.Empty.
/// </returns>
public static string Join<T>(this IEnumerable<T> @this, string separator)
{
return string.Join(separator, @this);
}
#endregion Public Methods
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,954
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\section{Conclusions}\label{sec:conclusions}
In this paper we have presented a finite length analysis of multi-slot type frameless ALOHA.
The analysis is exact, but its evaluation is computationally expensive and only feasible for moderate number of users.
The analysis has been extended to derive continuous approximations of the expected ripple size and its standard deviation.
The approximations have been proven to be tight and have also been compared with simulation results.
It has also been shown how these approximations can be used to accurately estimate the packet error rate in frameless ALOHA, making it possible to analyze the performance of frameless ALOHA for large contention periods.
Finally, it has been shown how the performance of frameless ALOHA can be substantially improved by introducing feedback and adapting the slot access probability dynamically using the approximate analysis.
\section{Approximation Using Differential Equations}\label{sec:diff_eqs}
The analysis presented in Section~\ref{sec:analysis} can provide the exact probability mass function of the number of unresolved users, however, its evaluation is feasible only for a moderate number of users and a small number of slot classes. In some situations, such as when running a computer search to optimize a parameter, it is preferable to have an approximate analysis which can be efficiently evaluated.
Having this in mind,
in this section we derive continuous approximations of the first and second moment of the ripple and clouds. These approximations are easy to evaluate and can be used to obtain an insight in the decoding process.
Furthermore, we show how these approximations can be used to estimate the packet error rate.
\input{./state_gen_func}
\subsection{First Moment}\label{sec:diff_eqs_first}
Denote by $C_h(u)$ and $R(u)$ the expected number of slots in the $h$-th cloud and the ripple, respectively. We have
\begin{equation}\label{eq:c_u_h_def}
C_h(u) := \sum_{ \mathcal{S}} \cc{h} \, \pcccru
\end{equation}
and
\begin{equation}\label{eq:r_u_def}
R(u) : = \sum_{ \mathcal{S} } (\r-1) \, \pcccru
\end{equation}
where we recall that $\mathcal{S}$ is the set of valid decoder states given in \eqref{eq:valid_states}.
It is possible to express $C_h(u)$ and $R(u)$ as the first-order derivatives of the state generating function \eqref{eq:theorem_new} in Theorem~\ref{theorem_state_gen_fun}, evaluated at $\mathbf{1}$, where $\mathbf{1}$ is the all one vector. In particular, we have
\begin{align}\label{eq:c_u_h_def_2}
C_h(u) &:=
\frac{\partial}{\partial\xc{h}} P_u (\xc{1},\xc{2},\hdots, \xc{k},\y)|_{\mathbf{1}}
\end{align}
and
\begin{align}\label{eq:r_u_def_2}
R(u) &: = \frac{\partial}{\partial \y} P_u (\xc{1},\xc{2},\hdots, \xc{k},\y)|_{\mathbf{1}}.
\end{align}
Let us first focus on $C_h(u)$. From \eqref{eq:c_u_h_def_2} we have
\begin{align}\label{eq:c_h_der}
C_h(u-1) &= (1-\puc{h}) C_h(u)
- (1-\puc{h}) \frac{\partial}{\partial \xc{h}} P_u \left( 1-\puc{1}, \hdots, 1-\puc{k}, \frac{1}{u} \right)
\end{align}
In \cite{Maatouk:phd} it was shown that, for $r\geq 6$, the drift term \eqref{eq:c_h_der} is $\mathcal{O}(1/n^2)$, leading to
\begin{align}\label{eq:c_h_der2}
C_h(u-1) &= (1-\puc{h}) C_h(u) + \mathcal{O}(1/n^2).
\end{align}
In a similar way, the expression for $R(u)$ can be obtained differentiating both sides of the recursion in \eqref{eq:theorem_new} with respect to $\y$ and evaluating the expression at $\mathbf{1}$,
\begin{align}\label{eq:r_der}
R(u-1) &= \left( 1 - \frac{1}{u} \right) R(u)+ \sum_{h=1}^{k} \puc{h} C_h(u) - P_u( \mathbf{1}) \\
&+ P_u(1-\puc{1},1-\puc{2},\hdots,1-\puc{k}, 1/u).
\end{align}
We again make use of a result in \cite{Maatouk:phd}, where it was shown that for $\r \geq 5$ the residual term
\[
- P_u( \mathbf{1}) + P_u(1-\puc{1},1-\puc{2},\hdots,1-\puc{k}, 1/u)
\]
can be approximated as $-1+ \mathcal{O}(1/n^2)$.
Recall that in frameless ALOHA, due to the contention mechanism, in a slot of type $h$, users are active with a probability $\paccessc{h}$. This induces a binomial slot degree distribution given by eq.~\eqref{eq:omega_h_j}.
If for a fixed contention period length, we look at the contention graph of frameless ALOHA from the perspective of slots, this is equivalent to saying that slots choose their neighbours uniformly at random and \emph{without replacement} (a user can not be active multiple times in a slot).
Following \cite{Karp2004}, \cite{Maatouk:2012} the frameless ALOHA decoder can be approximated by introducing the assumption that slots nodes choose their neighbors \emph{with replacement} in the bipartite graph representation of the contention process, i.e., the same user node can be chosen several times in the same slot. If a slot node has an odd number of edges connected to a user node, the user node will be active in that slot. If the number of edges is even $\{0,2,\hdots \}$, the user is not active in that slot. This approximation results in a slight decrease of the slot access probability $\paccessc{h}$. Nevertheless, intuitively the approximation becomes tighter as the number of users $n$ increases, since it becomes less and less likely that a slot chooses several times the same user.
Under the replacement assumption, the expression of $\puc{h}$ becomes:
\begin{equation}\label{eq:puc_x}
\puc{h} = \frac{1}{n} f_{h}\left(\frac{u}{n}\right) - \frac{1}{n^2} g_h \left( \frac{u}{n} \right)= \frac{1}{n} f_h \left(\frac{u}{n}\right) + \mathcal{O}(1/n^2)
\end{equation}
where
\[
f_h(x) = \frac{x \Omega_h''(1-x)}{1- x \, \Omega_h'(1-x) - \Omega_h(1-x)},
\]
\[
g_h(x) = \frac{f_h(x)}{x},
\]
and $\Omega_h(x)$ is the generator polynomial of the slot degree distribution of slots of type $h$,
\[
\Omega_h(x) = \sum_d \Omega_{h,d} \, x^d.
\]
From \eqref{eq:c_h_der2} and \eqref{eq:puc_x} we obtain the following difference equation for the $h$-th cloud
\begin{align}\label{eq:c_h_diff}
C_h(u) - C_h(u-1) = f_{h}\left(\frac{u}{n}\right) C_h(u) + \mathcal{O}(1/n^2)
\end{align}
Similarly, from \eqref{eq:r_der} and \eqref{eq:puc_x} we obtain
\begin{align}
R(u) - R(u-1) & = \frac{1}{u} R(u) - \sum_{h=1}^{k} f_{h}\left(\frac{u}{n}\right) C_h(u) +1
+ \mathcal{O}(1/n^2) \label{eq:r_diff}
\end{align}
Let us now define the expected normalized size of the clouds and ripple respectively as
\[
C_h(\xi):= C_h(u)/m
\] and
\[
R(\xi) := R(u)/ m.
\]
Making use of these definitions, and assuming $m = (1+\epsilon) n$, we can divide both sides of \eqref{eq:c_h_diff} by $m$ to obtain
\begin{align}\label{eq:c_h_diff_xi}
C_h(\xi) - C_h(\xi-1/n) = f_{h}(\xi) C_h(\xi) + \mathcal{O}(1/n^3)
\end{align}
and the same can be done for \eqref{eq:r_diff} leading to
\begin{align}\label{eq:r_diff_xi}
R(\xi) - R(\xi-1/n) &= \frac{1}{\xi n} R(\xi) - \sum_{h=1}^{k} f_{h}(\xi) C_h(\xi)
+ \frac{1}{n(1+\epsilon)} +\mathcal{O}(1/n^3).
\end{align}
As shown in \cite{Maatouk:phd}, it is possible to approximate $C_h(\xi)$ $R(\xi)$ respectively by $\hat C_h(\xi)$ $\hat R(\xi)$, which are the solutions to the following differential equations
\begin{align}\label{eq:c_h_diff_x}
\hat C_h'(\x) = f_{h}(\x) \hat C_h(\x)
\end{align}
and
\begin{align}\label{eq:r_diff_xx}
\hat R'(\x) = \frac{\hat R(\x)}{\x} - \sum_{h=1}^{k} f_{h}(\x) \hat C_h(\x) + \frac{1}{1+\epsilon}.
\end{align}
These approximations are tight during almost all the decoding process except for the last few users that are decoded. In particular, it was shown in \cite{Maatouk:phd} that as long as $u$ is a constant fraction of $n$, we have
\begin{equation}\label{eq:cloud_approx_error}
C_h(u) / m = \hat C_h (u/k) + \mathcal{O}(1/m)
\end{equation}
and
\begin{equation}\label{eq:ripple_approx_error}
R(u) / m = \hat R (u/k) + \mathcal{O}(1/m).
\end{equation}
Thus, the approximations become tighter for increasing $m$.
Finally, the expression for $\hat C_h(\x)$ and $\hat R(\x)$ are obtained by solving the differential equations \eqref{eq:c_h_diff_x} and \eqref{eq:r_diff_xx}, giving rise to the following solutions \cite{Maatouk:2012}
\[
\hat C_h(x) = \mathring{c}_{h} \big( 1- x \, \Omega_h'(1-x) - \Omega_h(1-x) \big)
\]
\[
\hat R(x) = x \left( \sum_{h=1}^{k} \mathring{c}_{h} \Omega_h'(1-x) + \frac{n}{m} \log(x) + \mathring{r} \right)
\]
where the values of the parameters $ \mathring{c}_{h} $ and $ \mathring{r} $ are determined by the initial conditions to the differential equations. In particular, for the clouds we have
\begin{align}
C_h(n) &= \sum_{\mathcal{S}} \cc{h} \, \pcccrn = \nslotc{h} \left( 1 - \Omega_{h,0} - \Omega_{h,1} \right) \left( 1 - \prod_{l=1}^{k} \big(1-\Omega_{l,1} \big)^{\nslotc{h}}\right).
\end{align}
Hence, by imposing $\hat C_h(x=1) = C_h(n) / m$, we obtain
\begin{equation}\label{eq:ch0}
\mathring{c}_{h} = \frac{\nslotc{h}}{m} \left( 1 - \prod_{l=1}^{k} \big(1-\Omega_{l,1} \big)^{\nslotc{l}}\right).
\end{equation}
For the ripple we have
\begin{align}
R(n) &= \sum_{\mathcal{S}} (\r-1) \, \pcccrn
= \sum_{h=1}^{k} \nslotc{h}\Omega_{h,1} - 1 + \prod_{h=1}^{k} \big( 1- \Omega_{h,1}\big)^{\nslotc{h}} .
\end{align}
Thus, imposing $\hat R(x=1) = R(n)/ m$ yields
\begin{align}\label{eq:r0}
\mathring{r} &= \sum_{h=1}^{k} \frac{\nslotc{h}}{m} \Omega_{h,1} \prod_{h=1}^{k} \big( 1-\Omega_{h,1} \big)^{\nslotc{h}}
- \frac{1}{m} \left( 1- \prod_{h=1}^{k} \big( 1-\Omega_{h,1} \big)^{\nslotc{h}}\right) .
\end{align}
\subsection{Second Moment}\label{sec:diff_eqs_second}
In the following, we shall focus on the variance of the ripple, $\sigma_R(u)$. By definition, we have
\[
\sigma_R(u) = \sum_{ \mathcal{S}} (r-1)^2 \pcccru - R(u)^2.
\]
Experimentally, we made the observation that the distribution of the ripple $\mathsf{R}_u$ is approximately binomial (see Fig.~\ref{fig:ripple_dist}). Under the assumption that $\mathsf{R}_u$ is binomially distributed with parameters $m$ and $\rho=R(u)/m$, a continuous approximation of the variance of the normalized ripple $\sigma_R^2(u)/m^2$ is given by
\[
\hat \sigma_R^2(x) = \frac{\rho (1-\rho)m}{m^2} = \frac{\rho (1-\rho)}{m} .
\]
\subsection{Numerical Results}\label{sec:diff_eqs_num}
Fig.~\ref{fig:ripple_cloud_mean_std} shows the expected normalized ripple and clouds sizes, as well as the normalized standard deviation of the ripple and their continuous approximation. The setting considered is $n= 400$ users, $k=2$ slot types, $\nslotc{1} = \nslotc{2} = 380$ slots, $\betac{1}=2.5$ and $\betac{2}=3$.
We can observe how the continuous approximation $\hat R(x)$ is very close to the actual expected normalized ripple size $R(u)/m$. There exists also a tight match between the normalized clouds $C_1(u)/m$ and $C_2(u)/m$ and their continuous approximations $\hat C_1(c)$ and $\hat C_2(x)$. We can also observe how the match between $\sigma_R(u)/m$ and $\hat \sigma_R(x)$ is tight.
Fig.~\ref{fig:approx_error_scaling} shows the absolute approximation error for $R$, $\sigma_R^2$, $C_1$ and $C_2$ as a function of $n$ for a setting with $k=2$ slot types, $\betac{1}=2.5$ and $\betac{2}=3$, ${\nslotc{1}/n = \nslotc{2}/n = 19 / n}$. In particular, every subfigure shows the error at 4 different decoding instants ($u={n/4, n/2, 3 n/4, n }$). All curves in the figure were obtained applying Theorem~\ref{theorem:class}. We can observe how in all cases the approximation error decreases as the number of users $n$ increases. The approximation error for the expected ripple and cloud sizes decreases approximately as $1/n$, which is in line with the error terms in \eqref{eq:cloud_approx_error} and \eqref{eq:ripple_approx_error}. It is also remarkable how the approximation error for the ripple variance $\sigma_R^2$ also decreases as the number of users $n$ increases, although this approximation was heuristically derived.
\begin{figure}[t]
\subfloat[Normalized expected cloud]{
\includegraphics[width=0.48\columnwidth]{figures/ripple_cloud_mean_std}
\label{fig_a}
}
\subfloat[Normalized expected ripple]{
\includegraphics[width=0.48\columnwidth]{figures/ripple_cloud_std}
\label{fig_b}
}
\caption{Normalized expected ripple and clouds, as well as the normalized standard deviation of the ripple and their continuous approximations for $n=400$ users, $k=2$ slot types, $\nslotc{1} = \nslotc{2} = 380$ slots, $\betac{1}=2.5$ and $\betac{2}=3$. }
\label{fig:ripple_cloud_mean_std}
\end{figure}
\begin{figure}[t]
\subfloat[$R$]{
\includegraphics[width=0.45\columnwidth]{figures/ripple_scaling}
\label{fig_R_scaling}
}
\subfloat[$\sigma_R^2$]{
\includegraphics[width=0.45\columnwidth]{figures/ripple_var_scaling}
\label{fig_R_var_scaling}
}
\subfloat[$C_1$]{
\includegraphics[width=0.45\columnwidth]{figures/cloud_1_scaling}
\label{fig_C_1_scaling}
}
\subfloat[$C_2$]{
\includegraphics[width=0.45\columnwidth]{figures/cloud_2_scaling}
\label{fig_C_2_scaling}
}
\caption{Absolute approximation error for $k=2$ slot types, $\betac{1}=2.5$ and $\betac{2}=3$, ${\nslotc{1}/n = \nslotc{2}/n = 19 / n}$
as a function of $n$. The different curves represent different decoding instants ($u={n/4, n/2, 3 n/4, n }$). Subfigures (a), (b), (c) and (d) represent the approximation error for $R$, $\sigma_R^2$, $C_1$ and $C_2$, respectively.}
\label{fig:approx_error_scaling}
\end{figure}
\subsection{Approximation to the PER}\label{sec:diff_eqs_per}
We shall now see how the continuous approximations derived in this section can be used in order to estimate the expected packet error rate, ${\mathsf{P}}$, i.e., the probability that a user is not resolved. In particular, we shall make use of the continuous approximations of the expected ripple $\hat R(x)$ and of its standard deviation $\hat \sigma_R(x)$.
Fig.~\ref{fig:ripple_dist} shows the distribution of the ripple for $n= 400$ users, $\nslotc{1} = \nslotc{2} = 380$, $\betac{1}=2.5$ and $\betac{2}=3$ at 4 different instants during the decoding process. In particular, the figure shows the probability mass function of the ripple cardinality when $u$ users are undecoded conditioned to the decoding process not failing for $u' >u$. That is,
\[
\frac{\Pr\{\Ripple_u=\r_u\}} { 1 - \sum_{u'=u+1}^{m} {{\mathsf{P}}}_{u'} } = \frac{\Pr\{\Ripple_u=\r_u\}} { \sum_{\r_u=0}^{m} \Pr\{\Ripple_u = \r_u \} }.
\]
Moreover, the figure also shows the probability density function of Gaussian random variables with mean $m \hat R(x)$ and standard deviation $\sqrt m * \hat \sigma_R(x)$. We make several observations in relation to this figure. The first is that the ripple cardinality has a bell shaped distribution across the whole decoding process, which resembles a Gaussian distribution. The second is that a Gaussian curve with mean $m \hat R(x)$ and standard deviation $\sqrt m * \hat \sigma_R(x)$ is reasonably close to the actual distribution of the ripple.
In particular, we can observe how the Gaussian approximation is tight at the beginning and the end of the decoding process, but it is not as tight in the middle of the decoding process, owing to the aforementioned overestimation of the standard deviation of the ripple size. We would also like to remark that, while random variable $\Ripple_u$ is discrete, we are approximating it by a continuous (Gaussian) random variable.
\begin{figure}[t]
\centering
\subfloat[u=n=400]{
\includegraphics[width=0.41\columnwidth]{figures/ripple_dist_1}
\label{fig_first_case}
}
\subfloat[u=300]{
\includegraphics[width=0.41\columnwidth]{figures/ripple_dist_2}
\label{fig_second_case}
}
\subfloat[u=200]{
\includegraphics[width=0.41\columnwidth]{figures/ripple_dist_3}
\label{fig_third_case}
}
\subfloat[u=100]{
\includegraphics[width=0.41\columnwidth]{figures/ripple_dist_4}
\label{fig_fourth_case}
}
\caption{Probability mass function of the ripple cardinality for $n= 400$ users, $k=2$ slot types, $\nslotc{1} = \nslotc{2} = 380$ slots, $\betac{1}=2.5$ and $\betac{2}=3$ at different decoding instants. The solid line shows a Gaussian distribution with mean $m \hat R(x)$ and standard deviation $\sqrt m * \hat \sigma_R(x))$.}
\label{fig:ripple_dist}
\end{figure}
Based on this observation, we can approximate the probability that exactly $u$ users remain unresolved after the end of the contention period, ${{\mathsf{P}}}_u$, as
\begin{equation}\label{eq:per_u_approx}
\hat {\mathsf{P}}_u = \mathtt{Q} \left( \frac{\hat R(u/m) } { \hat \sigma_R(u/m) } \right) \left( 1- \sum_{l=u+1}^{n} \hat {\mathsf{P}}_l \right)
\end{equation}
for $u< n$ and
\begin{equation}\label{eq:PMF_ru_norm}
\hat {\mathsf{P}}_n = \mathtt{Q} \left( \frac{\hat R(n/m)} { \hat \sigma_R(n/m) }\right)
\end{equation}
for $u = n$,
where $\mathtt{Q}$ is the tail distribution function of the standard normal distribution. The first term in \eqref{eq:per_u_approx} corresponds to the probability that a Gaussian distribution with mean $\hat R(u/m)$ and standard deviation $\hat \sigma_R(u/m)$ takes a negative value. Hence, this first term corresponds to the probability that our estimation of the ripple is negative. The second term in \eqref{eq:per_u_approx} implies we are conditioning to the decoding process in the previous decoding stages ($n$ down to $u + 1$) being successful. An alternative way of interpreting this is that, according to our definition of ${{\mathsf{P}}}_u$, we have
$\sum_{u=0}^{n} {{\mathsf{P}}}_u =1$
and by introducing the second term we are enforcing
$\sum_{u=0}^{n} \hat {{\mathsf{P}}}_u =1$.
Thus, we can approximate the expected packet error rate ${\mathsf{P}}$ as
\begin{equation}\label{eq:hatper}
\hat {\mathsf{P}} = \sum_{u=0}^{n} u /n \, \hat {\mathsf{P}}_u .
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=0.50\columnwidth]{figures/per_approx}
\caption{Packet error rate ${\mathsf{P}}$ and its approximation $\hat {\mathsf{P}}$ as a function of $m/n$ for $n= 400$ users, $k=2$ slot types, $\nslotc{1} = \nslotc{2} = m/2$, $\betac{1}=2.5$ and $\betac{2}=3$. }
\label{fig:per_approx}
\end{figure}
Fig.~\ref{fig:per_approx} show the packet error rate ${\mathsf{P}}$ as well as its approximation $\hat {\mathsf{P}}$ as a function of $m/n$, for $n= 400$ users, $k=2$ slot types, $\nslotc{1} = \nslotc{2} = m/2$, $\betac{1}=2.5$ and $\betac{2}=3$. The packet error rate ${\mathsf{P}}$ has been obtained using \eqref{eq:per} and applying Theorem~\ref{theorem:class}, which is exact, whereas the results for $\hat {\mathsf{P}}$ have been obtained using \eqref{eq:hatper}.
We can observe how the approximation of the packet error rate is tight. The fact that the estimation of the packet error rate is tight despite the fact that the standard deviation of the ripple is overestimated in the middle of the decoding process can be attributed to the fact that the dominating error event is that decoding stops at the end of the decoding process (when almost all users have been recovered). As we saw in Fig.~\ref{fig:ripple_dist}, the estimate of the standard deviation of the ripple in this regime (at the end of the decoding process) is tight.
\section{Dynamic Feedback}\label{sec:dynamic}
\begin{figure}[t]
\centering
\includegraphics[width=0.99\columnwidth]{figures/contention_adaptive}
\caption{An example of contention in frameless ALOHA with dynamic feedback with four users, where the contention subperiods have length $t=3$ slots. In this example the slot access probability for the first contention subperiod, $\paccessc{1}$, happens to be set too high. As a consequence, all the slots in the first contention subperiod present collisions. After the first $t=3$ slots the \ac{AP} notifies the users to use $\paccessc{2}$ in the next contention subperiod, which spans slots 4, 5 and 6. However, after the second contention subperiod still no users can be decoded. The \ac{AP} then notifies the users of the slot access probability to be used in the next contention subperiod, $\paccessc{3}$. Finally, in the third contention subperiod a singleton slot is received, which allows the \ac{AP} to decode all users leveraging on \ac{SIC}. The \ac{AP} notifies all users that a new contention period starts.}
\label{fig:frameless_adaptive}
\end{figure}
In this section show how the approximation of the analysis presented in this paper can be used together with feedback in order to improve the performance of frameless ALOHA.\footnote{The idea of using feedback in frameless ALOHA to adapt the slot access probability has also been explored in \cite{Ogata:2019}, where the slot access probability is adapted after every slot in order to keep the expected number of unresolved \emph{active} users per slot constant. However, the setting considered in \cite{Ogata:2019} is different to the one considered in this paper, since it assumes that collisions of size two can be detected and resolved by employing ZigZag decoding.}
We assume that a perfect and zero-delay feedback channel exists from the access point to the terminals. This channel is used to notify users of the value of the slot access probability which has to be used in upcoming slots.
We also assume that the contention period is divided into subperiods of duration $t$ slots, as depicted in Fig.~\ref{fig:frameless_adaptive}. All slots belonging to the same subperiod are of the same type, i.e., they are characterized by the same slot access probability.
At the end of the $k$-th subperiod (after $k \cdot t$ slots have elapsed), the \ac{AP} runs the decoding algorithm to decode as many users as possible (until the ripple is empty).
At this stage, denote by $u$ the number of users which are yet unresolved, and by $\S{u}$ the decoder state, with
\[
\S{u}:=( \cuc{1}, \cuc{2}, \cdots, \cuc{k}, \r_u=0 )
\]
where $\cuc{i}$ is the number of slots in the $i$-th cloud.
Next, based on the number of unresolved users $u$, the decoder state $\S{u}$, and using as latency target a contention period duration of $\nslot_T$ slots, the decoder computes the slot access probability to be used in the next subperiod, $\paccessc{k+1}$, and signals its to the users.
In particular, in order to determine the value of $\paccessc{k+1}$ to be used, the \ac{AP} assumes that the remaining slots up to $\nslot_T$ will be of class $k+1$. Thus, it sets
$\nslotc{k+1}= \nslot_T - k \, t$.
The \ac{AP} now needs to determine the initial state of the frameless ALOHA decoder, given $\nslotc{k+1}$, $u$ and $\S{u}$. The first step is determining the
slot degree distribution of the slots of class $k+1$, $\Omega_{k+1,j}'$, taking into account that $u$ out of $n$ users have already been decoded. Let us denote by $\mathsf{I}$ the random variable associated to the number of users which are active in a slot and by $\mathtt{i}$ its realization. Let us also denote by $\mathsf{J}$ the random variable associated to the number of \emph{unresolved} users which are active in a slot and by $\j$ its realization. We have
\begin{align} \label{eq:de_dist_virtual}
\Omega_{k+1,\j}' &= \Pr\{\mathsf{J}=\j\}= \sum_{\mathtt{i}=j}^{u} \Pr\{\mathsf{I}=\mathtt{i}\} \Pr\{\mathsf{J}=\j| \mathsf{I}=\mathtt{i}\} \\
&= \sum_{\mathtt{i}=\j}^{u} { n \choose \mathtt{i} } \left(\paccessc{k+1} \right)^\mathtt{i} \left( 1 - \paccessc{k+1} \right)^{n - \mathtt{i}}
{ \mathtt{i} \choose \j } \left(\frac{u}{n} \right)^\j \left( 1 - \frac{u}{n} \right)^{\mathtt{i}- \j}.
\end{align}
Finally, the initial state distribution of the frameless ALOHA decoder, $\S{u}'$, is given by
\begin{align
\Pr \{ \S{u}' = \cuc{1}, \cuc{2}, \cdots, \cuc{k+1}, \r_u \} &=
\frac{\nslotc{k+1} !}{\c_{k+1,u}! \, \r_{u}!\, (\nslotc{k+1}-\c_{k+1,n}-\r_{u})!} \\
& \mkern-20mu \times \left( 1-\Omega_{k+1,1}- \Omega_{k+1,0}\right)^{\c_{k+1,u}}\, {\Omega_{k+1,1}}^{\r_{u}} \, {\Omega_{k+1,0}}^{\nslotc{k+1} -\c_{k+1,u}-\r_{u}}
\end{align}
for $\Cuc{1}=\cuc{1}$, $\Cuc{2}=\cuc{2}$, \dots, $\Cuc{k}=\cuc{k}$, and for all non-negative $\c_{k+1, u},\r_{k+1, u}$ such that $\c_{k+1, u}+\r_{k+1, u} \leq \nslotc{k+1}$, with $\nslotc{k+1}= \nslot_T - k \, t$. Otherwise, we have
\[
\Pr \{ \S{u}' = \cuc{1}, \cuc{2}, \cdots, \cuc{k+1}, \r_u \} = 0 .
\]
Hence, we initialize the $\Cuc{1}$, $\Cuc{2}$ \dots, $\Cuc{k}$ according to the decoder state at the \ac{AP} (which is deterministically known), and $\Cuc{k+1}$ and $\Ripple_u$ are initialized according to a multinomial distribution, similarly as in \eqref{eq:init_cond_finite}.
Thus, $\paccessc{k+1}$ is obtained formally as
\[
\paccessc{k+1} = \underset{\paccessc{k+1}}{\arg \min} \, {\mathsf{P}} \left(n, \nslot_T, u, \Omega_{k+1,\j}',\S{u}'\right)
\]
where we have stressed that ${\mathsf{P}}$ is a function of $n$ , $\nslot_T$, $u$, $\Omega_{l,\j}'$ and $\S{u}'$. Note that $\Omega_{l,\j}'$ depends implicitly on $\paccessc{k+1}$ through \eqref{eq:de_dist_virtual}, and that ${\mathsf{P}}$ is obtained by using Theorem~\ref{theorem:class}.
Hence, it is in principle possible to run a computer search to find the value of $\paccessc{k+1}$ which minimizes the packet error rate.
However, running such a search is computationally complex because it would require evaluating Theorem~\ref{theorem:class} multiple times\footnote{If we let $m/n$ be a constant, the number of unresolved users $u$ be a constant fraction of $n$ and the number of slot classes $k$ be a constant, we have that the number of possible decoder states when $u$ users are unresolved is $\mathcal{O}(n^{k+1})$. Furthermore, from a given decoder state when $u$ users are unresolved, the number to states that the decoder can transit to is also $\mathcal{O}(n^{k+1})$. This is a clear indicator that the complexity of the analysis does not scale well with the number of users.}. Instead, we propose minimizing the approximation of the packet error rate $\hat {\mathsf{P}}$ derived in Section~\ref{sec:diff_eqs_per}. This approach is suboptimal given the fact that we are minimizing an approximation of ${\mathsf{P}}$, but it makes the search for $\paccessc{k+1}$ much faster. Thus, we obtain $\paccessc{k+1}$ as
\[
\paccessc{k+1} = \underset{\paccessc{k+1}}{\arg \min} \, \hat {\mathsf{P}} \left(n, \nslot_T, u, \Omega_{k+1,\j}',\S{u}'\right)
\]
where we remark again that that $\Omega_{l,\j}'$ depends implicitly on $\paccessc{k+1}$ through \eqref{eq:de_dist_virtual}.
Fig.~\ref{fig:per_dynamic} shows the packet error rate ${\mathsf{P}}$ as a function of the duration of the contention period in slots $m$, for a system with $n=50$ users. Five curves are shown. The first one shows the performance of a static setting with $\beta=2.94$, which is the value that minimizes ${\mathsf{P}}$ for $m=100$.
This first curve was obtained from \eqref{eq:per} after applying the finite length analysis in Theorem~\ref{theorem:class} for the particular case with only one slot type.
The figure also shows the performance of a dynamic frameless ALOHA scheme in which $\beta$ is changed dynamically every $t$ slots in order to minimize $\hat {\mathsf{P}}$ at $\nslot_T=100$. In particular, four different values of $t$ are considered, $5$, $10$, $20$ and $50$ slots.
The corresponding curves were obtained by means of simulations: for every value of $m$, the simulation run until $250$ unsuccessful contention periods had been collected (a contention period is successful only if all $n$ users are correctly decoded). Finally, the figure shows also the performance of an IRSA scheme with degree distribution $\Delta(x)=0.25x^2+0.6x^3+0.15x^8$, taken from \cite{L2011}.
We can observe how the introduction of dynamic feedback decreases considerably the packet error rate. Furthermore, as one would expect, the performance of the dynamic scheme improves as $t$ decreases, that is, as the number of feedback messages increases. The curve for $t=20$ is particularly interesting, since one can observe how ${\mathsf{P}}$ improves after every feedback and then it degrades as time elapses; the effect is clearly visible around $m=80$. In the simulated setting, it seems that sending feedback every $t=10$ slots may represent a good trade-off in terms of performance vs number of feedback messages sent.
Finally, if we compare the performance of IRSA to that of frameless ALOHA, we can observe that, for the latency target of $\nslot_T=100$, IRSA outperforms static frameless ALOHA, as well as dynamic frameless ALOHA with $t=50$, which only uses one feedback message.
However, if we allow for more feedback messages, dynamic frameless ALOHA outperforms IRSA. For example, dynamic frameless ALOHA with $t=10$ outperforms IRSA by more than two orders of magnitude.
The results in Fig.~\ref{fig:per_dynamic} indicate how the introduction of feedback can greatly improve the performance of frameless ALOHA. Additional improvements of the performance could be achieved by assuming a more capable receiver, such as one that can distinguish collisions of size 2 \cite{Ogata:2019} or a receiver which can decode collisions of size $\ell>1$ \cite{lazaro2016finite}.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\columnwidth]{figures/per_dynamic}
\caption{Packet error rate ${\mathsf{P}}$ as a function of $m$ for $n= 50$ users frameless ALOHA in a static and dynamic setting as well as for IRSA. Static frameless ALOHA uses $\beta=2.94$. The different curves for dynamic frameless ALOHA represent settings in which $\beta$ is changed dynamically every $5$, $10$, $20$ and $50$ slots such that the $\hat {\mathsf{P}}$ at the end of contention-period is minimized, and the target contention period duration is set to $\nslot_T=100$ slots. For IRSA the degree distribution used is $\Delta(x)=0.25x^2+0.6x^3+0.15x^8$, taken from \cite{L2011}.}
\label{fig:per_dynamic}
\end{figure}
\section{State Generating Func}
\input{./proof_finite_length_analysis}
\input{./state_gen_func_proof}
\section{Initial Conditions for Dynamic Feedback}
\input{./appendix_dynamic_feedback}
\end{appendices}
\section*{Acknowledgments}
The authors would like to thank Gianluigi Liva for pointing out the possibility of applying the analysis tools of LT codes to frameless ALOHA.
\input{finite_fless_A.bbl}
\end{document}
\section{Finite-Length Analysis}
For the sake of convenience and without loss of accuracy, we shall assume that the receiver works iteratively.
If the ripple is empty, the receiver simply stops. Otherwise, it carries out the following steps:
\begin{itemize}
\item Selects at random one of the slots in the ripple;
\item Resolves the user that was active in that slot (i.e., decodes its packet);
\item Cancels the interference contributed by the resolved user from all other slots in which its packet replicas were transmitted. This may cause some slots to leave the cloud and enter the ripple. Furthermore, some slots from the ripple may become degree zero and leave the ripple. These last slots correspond to the slots in the ripple in which the resolved user was active.
\end{itemize}
Thus, in each iteration, the reception algorithm either fails, or exactly one user gets resolved.
These assumptions are made to ease the analysis and have no impact on the performance.
\label{sec:analysis}
Following the approach in \cite{Karp2004,lazaro:inact2017,lazaro:SCC2017}, the iterative reception of frameless ALOHA with $k$ different slot types is represented as a finite state machine with state
\[
\S{u} :=( \Cuc{1}, \Cuc{2}, \cdots, \Cuc{k}, \Ripple_u )
\]
i.e., the state comprises the cardinalities from the first to $k$-th cloud and the ripple at the reception step in which $u$ users are unresolved.
Each iteration of the reception algorithm corresponds to a state transition.
The following proposition establishes a recursion that can be used to determine the state distribution.
\begin{theorem}\label{theorem:class}
Given that the decoder is in state ${\S{u}=(\cuc{1}, \cuc{2}, \cdots, \cuc{k}, \r_u )}$, where $u$ users are unresolved, and $\r_u>0$ (i.e., the ripple is not empty), the probability of the receiver transitioning to state ${\Pr \{\S{u-1}= \s{u-1}\}}$, where $u-1$ users are unresolved, is given by
\begin{align}
\Pr & \{ \S{u-1}= ( \s{u} + \pmb{\mathtt{w}} )| \S{u} = \s{u} \} =
\binom{\r_u-1}{\mathtt{a}_u-1} \left(\frac{1}{u}\right)^{\mathtt{a}_u-1} \times \\
& \left( 1- \frac{1}{u} \right)^{\r_u-\mathtt{a}_u} \mathlarger{\prod}_{h=1}^{h=k} { \binom{\cuc{h}}{\buc{h}} {\puc{h}}^{\buc{h}} (1-\puc{h})^{\cuc{h}-\buc{h}} }
\label{eq:prob_transition}
\end{align}
with
\[\s{u} =(\cuc{1}, \cuc{2}, \cdots, \cuc{k}, \r_u ) \]
\[{\pmb{\mathtt{w}} =(-\buc{1}, -\buc{2}, \cdots , -\buc{k}, \sum_{h=1}^{k} \buc{h} - \mathtt{a}_u ) }\]
and
\vspace{-2ex}
\begin{align}
\puc{h} = \frac{ \mathlarger {\sum}\limits_{d=k+1}^{n} \Omega_{h,d} \, \frac{d}{n} \binom{d-1}{k} \frac{\binom{u-1}{k}}{\binom{n-1}{k}} \frac{\binom{n-u}{d-k-1}}{\binom{n-k-1}{d-k-1}} }
{ 1 - \mathlarger{\sum}\limits_{h=0}^{k} \,\,
\mathlarger{\sum}\limits_{d=h}^{n} \Omega_{h,d} \, \frac{\binom{u}{h} \binom{n-u}{d-h}}{\binom{n}{d}} }
\label{eq:pu_theorem_class}
\end{align}
for $0 \leq \buc{h} \leq \cuc{h}$, $1 \leq \mathtt{a}_u \leq \r_u$.
\end{theorem}
\begin{IEEEproof}
See Appendix~\ref{app:proof_theorem_class}.
\end{IEEEproof}
Recall that out of the $m$ slots in the contention period, exactly $\nslotc{1}, \nslotc{2}, ... \nslotc{k}$ belong to slot type $1,2, ...,k$.
We focus on slots of type $h$, of which there are $\nslotc{h}$.
The initial state distribution corresponds to a multinomial with $\nslotc{h}$ experiments (slots) and three possible outcomes for each experiment: the slot being in the cloud, the ripple or having degree 0, with respective probabilities $(1-\Omega_{h,1} - \Omega_{h,0} )$, $\Omega_{h,1}$ and $\Omega_{h,0}$.
Denoting by $\mathtt{R}_{h, n}$ the random variable associated to the number of slots of type $h$ of reduced degree 1 when all $n$ users are still undecoded, we have
\begin{align}\label{eq:init_cond_finite}
\Pr\{(\mathtt{C}_{h, n} = \c_{h,n}, \mathtt{R}_{h,n} = \r_{h, n} \} &=
\frac{\nslotc{h}!}{\c_{h,n}! \, \r_{h,n}!\, (\nslotc{h}-\c_{h,n}-\r_{h,n})!} \\
&\times \left( 1-\Omega_{h,1}- \Omega_{h,0}\right)^{\c_{h,n}}\, {\Omega_{h,1}}^{\r_{n}} \, {\Omega_{h,0}}^{\nslotc{h}-\c_{h,n}-\r_{h,n}}
\end{align}
for all non-negative $\c_{h, n},\r_{h, n}$ such that $\c_{h, n}+\r_{h, n} \leq \nslotc{h}$.
If we observe that, when all $n$ users are still undecoded, the total number of degree one slots $\mathtt{R}_{n}$ is given by
\[
\mathtt{R}_{n} = \sum_{h=1}^{k} \mathtt{R}_{h, n}
\]
we can obtain from \eqref{eq:init_cond_finite} the initial state distribution of the receiver.
By applying recursively Theorem~\ref{theorem:class} and initializing as described the finite state machine, one obtains the state probabilities.
Let us denote by ${{\mathsf{P}}}_u$ the probability that exactly $u$ users remain unresolved after a contention period of $m$ slots. Obviously, the event that exactly $u$ users remain unresolved corresponds to the event that the user resolution ends at stage $u$.
The probability of this event is simply the probability that the ripple is empty when $u$ users are still unresolved. Formally we have
\begin{align}
\label{eq:peru}
{{\mathsf{P}}}_u &= \Pr\{ \Ripple_u =0\} = \sum_{\cuc{1}} \sum_{\cuc{2}} \hdots \sum_{\cuc{k}} \Pr\{\S{u} =(\cuc{1},\cuc{2},\cdots, \cuc{k}, 0) \}
\end{align}
where the summations is taken over all possible values of $\cuc{h}$, $h=1, ..., k$.
Thus, by applying Theorem~\ref{theorem:class} and then using \eqref{eq:peru}, one obtains the \ac{PMF} of the number of unresolved users, given number of users $n$ and the duration of the contention period\footnote{Note that ${{\mathsf{P}}}_u$ implicitly depends on the initial state distribution that is obtained through \eqref{eq:init_cond_finite}, while \eqref{eq:init_cond_finite} depends on the number of slots of a given type $m^{(h)}$, $h=1,2,...,k$, and thereby on the total number of slots $m$.} $m$.
The expected packet error rate ${\mathsf{P}}$ , i.e., the probability that a user is not resolved, can also be derived from Theorem~\ref{theorem:class}. In particular, we have
\begin{align}
{\mathsf{P}} &= \sum_{u=1}^{n} \frac{u}{n} {{\mathsf{P}}}_u
= \sum_{u=1}^{n} \sum_{\cuc{1}} \sum_{\cuc{2}} \hdots \sum_{\cuc{k}} \frac{u}{n}\Pr\{\S{u} =(\cuc{1},\cuc{2},\cdots, \cuc{\c}, 0) \} \label{eq:per}.
\end{align}
Hence, the expected throughput is simply the ratio of the expected number of resolved users over the number of slots in the contention period. i.e.,
\begin{align}
\mathsf{T} &= \frac{n ( 1-{\mathsf{P}} ) }{ m }.
\end{align}
As an example, in Fig.~\ref{fig:example_p_dist} we show the \ac{PMF} of the number of undecoded users $u$, i.e., ${\mathsf{P}}_u$ for $u=1,...,n$, when $n=50$ and $m = 60$, for frameless ALOHA with (i) a single slot type with mean initial slot degree $\beta=2.68$, and (ii) two slot types, with $\nslotc{1}=50$ slots of the first type, $\nslotc{2}=10$ slots of the second type, with mean initial degrees $\betac{1}=3$ and $\betac{2}=5$, respectively.\footnote{Recall that slot access probability of a type $h$ is $\paccessc{h} = \betac{h} / n$.}
The figure shows analytical results according to Theorem~\ref{theorem:class} and the outcome of Monte Carlo simulations.
It can be observed that the match is tight down to simulation error (100,000 contention periods were simulated).
In this particular example, we can observe how ${\mathsf{P}}_u$ has a bimodal distribution.
Thus, there are two points in the decoding process in which the ripple has a higher probability of becoming empty. For the example in Fig.~\ref{fig:example_p_dist}, the expected packet error rates for the contention with one and two slot classes correspond to $0.264$ and $0.555$, respectively.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\columnwidth]{figures/pmf_}
\caption{Examples of probability mass function of the number of undecoded users $u$ for $n=50$, $m = 60$.}
\label{fig:example_p_dist}
\end{figure}
\section{Introduction}\label{sec:Intro}
Random access protocols find applications in scenarios where there is a number of users sharing a common transmission medium and there exists uncertainty regarding which users are active.
They can be used both in the initial phase of grant-based access, where the active users contend with metadata in order to reserve the uplink resources for the subsequent data transmissions, or in grant-free access, where the active users contend directly with packets containing data.
The former approach forms the basis of mobile cellular access, e.g.,~\cite{TS36.321}.
However, the latter approach has been gaining research momentum recently, e.g.,~\cite{R1-1808304}, due its lower signaling overhead which makes it suitable for systems such as \ac{IoT}, where the amount of exchanged data is small but the number of contending users may be large~\cite{Laya2014}.
The first and still widely used random access protocols are ALOHA and slotted ALOHA \cite{R1975}.
Assuming a collision channel model, these two protocols offer low peak throughput ($1/2e$ and $1/e$, respectively) and high \ac{PER} even for low channel load.
However, it has been shown how the introduction of \ac{SIC} at the receiver can lead to higher performance by leveraging on coding-theoretic tools \cite{CGH2007,L2011}.
In practice, this implies storing and processing the receiver waveform and leads to higher receiver complexity.
The results presented in \cite{L2011} inspired a strand of works that applied various concepts from codes-on-graphs to design SIC-enabled slotted ALOHA schemes~\cite{PLC2011,LPLC2012,SPV2012,PSLP2014,JBVC2015,SGB2017}, which are usually referred to by using the umbrella term of \emph{coded slotted ALOHA}.
Frameless ALOHA~\cite{SPV2012,SP2013} is a version of SIC-enabled slotted ALOHA that exploits ideas originating from the rateless-coding framework \cite{luby02:LT}.
In its original version, frameless ALOHA is characterized by a contention period that consists of a number of slots that is not defined a priori, and by a slot access probability for the users to independently transmit their packets in a slot.
An asymptotic optimization of the slot access probability that maximizes the expected throughput was performed in \cite{SPV2012}, while a similar optimization in a cooperative, multi-base station scenario was recently considered in \cite{OIA2018}.
In \cite{Ogata:2019}, an evolution of frameless ALOHA was proposed which makes use of feedback, ZigZag decoding, and a dynamical variation of the slot access probability to improve performance.
A joint assessment of the optimal slot access probability and the contention termination criteria in non-asymptotic, i.e., finite-length scenarios were assessed by means of simulations in \cite{SP2013}.
The finite-length performance of frame-based coded slotted ALOHA schemes has been studied in \cite{ivanov:floor,SGB2017,graell:2018,fereydouniannon,MGS2018}. Here we remark that, due to the fact that in such schemes every user selects a random set of slots in a frame in which to transmit, interdependencies among slots are introduced; thus, the finite-length analytical model is in essence intractable, and one has to resort to approximations. The error floor of coded slotted ALOHA over the collision channel in the finite length regime was studied in \cite{ivanov:floor}. A frame asynchronous coded slotted ALOHA scheme was analyzed in \cite{SGB2017,graell:2018}. The scaling laws of coded slotted ALOHA were derived in \cite{fereydouniannon}. In \cite{MGS2018}, the performance of several advanced random access protocols was compared over the collision channel and over the additive white Gaussian noise channel.
The focus of this paper on finite-length analysis of frameless ALOHA and of its reliability-latency performance with and without feedback. In particular, the considered feedback takes the form of an update of the slot access probability of frameless ALOHA, which is the default degree of freedom that can be used to optimize performance.
This feedback induces multiple slot types in the contention, which substantially expands the design space, but also poses some challenges in terms of modeling and analytical performance optimization.
%
The analysis presented in the paper characterizes statistically the reliability of muli-slot type frameless ALOHA for a predefined latency target, which in our context translates to a fixed number of slots in the contention period. The motivation of this approach stems from the fact that mission-critical services typically feature a given latency budget, which is also reflected in the proverbial reliability requirement of $10^{-5}$ under the latency of 1~ms in the context of 5G ultra-reliable low-latency service category~~\cite{TR38.913}.\footnote{We also remark that extending indefinitely the contention period in frameless ALOHA will always be beneficial in terms of reliability; this is a general feature of the rateless-coding framework. However, a practical design target always involves limited latency, which in this case maps to limited number of
overhead slots.}
%
The feedback scheme proposed in this paper exploits the flexibility that frameless ALOHA offers with respect to the frame-based schemes. Specifically, the considered scheme assesses the reliability at predefined intermediate checkpoint within a contention period and uses feedback to drive the contention process towards maximizing the number of decoded users, and thereby the reliability, given the latency target.
%
In particular, the feedback sent by the access point to the users consists of an update on the slot access probability that should be subsequently used. This feedback induces a multi-slot type contention, since a new slot type is created whenever the slot access probability is changed.
The proposed scheme adapts to the reliability-latency performance observed at multiple points, which allows for a finer control of the contention process.
This paradigm change is a clear indicator of an enlarged design space compared to the frame-based designs that focus on just a single time instant, placed at the end of the latency budget.
Finally, we show that the proposed feedback scheme outperforms frame-based schemes.
This paper builds on preliminary results from \cite{lazaro:SCC2017} and \cite{stefanovic:globecom2017}. In \cite{lazaro:SCC2017} an exact finite-length analysis of frameless ALOHA was presented for the case in which all slots are statistically identical.
This analysis was then extended in \cite{stefanovic:globecom2017} to multiple slot types, sketching how the analysis could be changed to accommodate this extension without actually providing a full proof.
The contributions in this paper are the following.
We present in detail an exact, self-contained finite-length analysis of multi-slot type frameless ALOHA. Due to computational complexity, the analysis is only feasible for moderate number of users. This has been the motivation to extend the analysis towards deriving continuous approximations of the expected ripple size (the number of slots containing only one transmission) and its standard deviation.
Based on these approximations, we propose a method to estimate the \ac{PER} with very low complexity.
Finally, we exploit this estimation of the packet error rate to propose a feedback based scheme in which the slot access probability of frameless ALOHA is adapted dynamically.\footnote{Note that, although the paper ultimately proposes the approximate method in order to deal with the computational complexity, the nature of the approximation is substantially different from the approximations employed in \cite{ivanov:floor,SGB2017,graell:2018,fereydouniannon,MGS2018}. Specifically, in our case, the exact model and analysis of the problem are at hand, as shown in the paper, and one can, in principle, exactly assess the quality of the approximation. In case of the methods employed in \cite{ivanov:floor,SGB2017,graell:2018,fereydouniannon,MGS2018}, the analysis is from the very beginning approximate due to the interdependencies among slot selections by the users.}
We investigate the performance of the dynamic version of the scheme and show that it achieves a rather favorable performance, which progressively improves with the number of feedback opportunities at the expense of an increased computational burden.
The remainder of the paper is organized as follows.
Section~\ref{sec:II} provides a brief overview of frameless ALOHA and describes the system model.
Section~\ref{sec:analysis} presents the finite-length analysis, which can be used to obtain the exact probability mass function of the number of unresolved users for a given duration of the contention period.
In Section~\ref{sec:diff_eqs}, a low complexity approximate analysis of the frameless ALOHA decoder is presented, which is then used to estimate the packet error rate.
Section~\ref{sec:dynamic} shows how it is possible to largely improve the performance of frameless ALOHA by introducing feedback and applying the analysis derived in this paper.
Finally, Section~\ref{sec:conclusions} concludes the paper.
\section{Proof of Theorem~\ref{theorem:class}}
\label{app:proof_theorem_class}
\begin{figure}[t]
\centering
\includegraphics[width=0.5\columnwidth]{figures/ripple_cloud_fig
\caption{Evolution of the ripple and the $h$-th cloud through decoding and removal of replicas. Resolution of a single user, i.e., decoding a replica of its packet and removal of the other replicas, causes $\mathtt{A}_u$ slots to leave the ripple and $\Buc{h}$ slots to leave the $h$-th cloud and enter the ripple.}
\label{fig:proof}
\end{figure}
Let us start by remarking that, given the fact that every users decides whether to transmit or not in a slot independently from other users, all users are independent and statistically identical. Furthermore, all slots are mutually independent. Thus, if we look at the decoder when $u$ users are unresolved, the set of unresolved users is obtained by selecting $u$ users at random among the total of $n$ users.
In particular, the proof analyzes the variation of the clouds and ripple sizes in the transition from $u$ to $u-1$ unresolved users. Since we assume $\r_u>0$, in the transition from $u$ to $u-1$ unresolved users, exactly 1 user is resolved. All the edges coming out from the resolved user are erased from the decoding graph. As a consequence, some slots might leave the cloud and enter the ripple if their reduced degree becomes one, and other slots will leave the ripple if their reduced degree decreases from 1 to 0.
Let us first focus of the number of slots leaving $\cloudc{h}{u}$ and entering $\ripple{u-1}$ in the transition, denoted by $\buc{h}$ and with associated random variable given by $\Buc{h}$. Due to the nature of frameless ALOHA, in the decoding graph the neighbor users of a slot are selected uniformly at random and without replacement\footnote{In fact, it is users who choose their neighbor slots uniformly at random and without replacement.}. Thus, random variable $\Buc{h}$ is binomially distributed with parameters $\c_{h,u}$ and $\puc{h}$, being $\puc{h}$ the probability of a generic slot $\slot$ of type $h$ leaving $\cloudc{h}{u}$ to enter $\ripple{u-1}$,
\begin{equation}
\puc{h} := \Pr \{ \slot \in \ripple{u-1} | \slot \in \cloudc{h}{u} \}= \frac { \Pr \{ \slot \in \ripple{u-1}\, , \, \slot \in \cloudc{h}{u} \} } { \Pr \{ \slot \in \cloudc{h}{u} \}}.
\label{eq:pu_prob}
\end{equation}
We shall first focus on the numerator of \eqref{eq:pu_prob} and we shall condition it to the slot having degree $d$,
${\Pr \{ \slot \in \ripple{u-1}\, , \, \slot \in \cloudc{h}{u} | \deg(\slot)= d \}}$. This corresponds to the probability that one of the $d$ edges of slot $\slot$ is connected to the user being resolved at the transition, one edge is connected to one of the $u-1$ unresolved users after the transition and the remaining $d-2$ edges are connected to the $n-u$ resolved users before the transition. In other words, slot $\slot$ must have \emph{reduced} degree $2$ \emph{before} the transition and \emph{reduced} degree $1$ \emph{after} the transition.
It is easy to see how this probability, conditioned to $ \deg(\slot)= d$, corresponds to
\begin{align}\label{eq:z_and_l_d}
\Pr \{ \slot \in \ripple{u-1}\, , \, & \slot \in \cloudc{h}{u} | \deg(\slot)= d \} =
\frac{d}{n} (d-1)\frac{u-1}{n-1} \frac{\binom{n-u}{d-2}}{\binom{n-2}{d-2}}
\end{align}
for $d\geq 2$. In the complementary case, $d < 2$, it is obvious that the slot cannot enter the ripple. Thus, for $d<2$ we have
\[
\Pr \{ \slot \in \ripple{u-1}\, , \, \slot \in \cloudc{h}{u} | \deg(\slot)= d \} = 0.
\]
Let us now concentrate on the denominator of \eqref{eq:pu_prob}, that corresponds to the probability that a slot $\slot$ is in the $h$-th cloud when $u$ users are still unresolved. This is equivalent to the probability of slot $\slot$ not being in the ripple or having reduced degree zero (all edges connected to resolved users). Hence, we have
\begin{align}
\Pr \{ \slot \in \cloudc{h}{u}\}&= 1 - \mathlarger{\sum}_{d=1}^{n} \Omega_{h,d} u\frac{\binom{n-u}{d-1}}{\binom{n}{d}} - \mathlarger{\sum}_{d=0}^{n} \Omega_{h,d} \frac{\binom{n-u}{d}}{\binom{n}{d}}
\label{eq:z}
\end{align}
where the first summation on the right hand side corresponds to the probability of a slot being the ripple, and the second summation corresponds to the probability of a slot having reduced degree zero.
Inserting \eqref{eq:z_and_l_d} and \eqref{eq:z} in \eqref{eq:pu_prob}, the expression of $\puc{h}$ in \eqref{eq:pu_theorem_class} is obtained, and the variation in size of the clouds is determined, i.e., the \acp{PMF} of the random variables $\Buc{h}$ are obtained.
We focus next on the variation in size of the ripple in the transition from $u$ to $u-1$ unresolved users.
In this transition, some slots enter the ripple (in total $\sum_{h}\buc{h}$ slots), but there are also slots leaving the ripple.
Let us denote by $\mathtt{a}_u$ the number of slots leaving the ripple in the transition from $u$ to $u-1$ unresolved users, and let us refer to the associated random variable as $\mathtt{A}_u$. Assuming that the ripple is not empty\footnote{If the ripple is empty, $\r_u=0$, no slots can leave the ripple. Moreover, decoding stops, so there is no transition.}, the decoder will select uniformly at random one slot from the ripple, that we denote as $\slot$. The only neighbour of $\slot$, $c$ will get resolved and all slots in the ripple that are connected to $c$ will leave the ripple in the transition. Hence, slot $\slot$ will leave the ripple, and, additionally, the remaining $\r_u-1$ slots in the ripple will leave the ripple independently with probability $1/u$, which is the probability that they have $c$ as neighbour. Thus, the probability mass function of $\mathtt{A}_u$ is given by
\begin{align}
\Pr\{\mathtt{A}_u=\mathtt{a}_u & |\Ripple_u=\r_u\}
\binom{\r_u-1}{\mathtt{a}_u-1} \left(\frac{1}{u}\right)^{\mathtt{a}_u-1} \mkern-3mu \left( 1- \frac{1}{u} \right)^{\r_u-\mathtt{a}_u}.
\end{align}
Note that, by definition, we have that
${\r_{u-1}= \r_u -\mathtt{a}_u + \sum_{h=1}^{k} \buc{h}} $
and
${\c_{h,u-1} = \cuc{h} - \buc{h}}$.
Hence, the probability of transiting from state
$\s{u} =(\cuc{1}, \cuc{2}, \cdots, \cuc{k}, \r_u ) $
to state $\s{u} + \pmb{\mathtt{w}}$, with $\pmb{\mathtt{w}}$ given by
\[{\pmb{\mathtt{w}} =(-\buc{1}, -\buc{2}, \cdots , -\buc{k}, \sum_{h=1}^{k} \buc{h} - \mathtt{a}_u ) }\] corresponds to
\begin{align}\label{eq:joint}
\Pr \{\Buc{1}=\buc{1}, \Buc{2}=\buc{2}, \cdots, \Buc{k}=\buc{k}, \mathtt{A}_u = \mathtt{a}_u\}.
\end{align}
The proof is finalized by observing that the joint distribution in \eqref{eq:joint}, due to independence of $\Buc{h}$, $h=1,2,\cdots, k$, and $\mathtt{A}_u$, is obtained as the product of the \acp{PMF} of the individual random variables, which yields \eqref{eq:prob_transition}.
\vspace{-0.8cm}
\hspace{15cm}$\blacksquare$
\subsection{State generating Functions}\label{sec:state_gen_fun}
Our starting point to derive approximations of the distribution of the ripple and clouds is writing down the state generating function of the frameless ALOHA decoder, which is simply the probability generating function\footnote{The probability generating function is a representation of the probability mass function
using power series.} of the random variable associated to the state of the frameless ALOHA decoder.
Following the works in \cite{Karp2004,shokrollahi2009theoryraptor,Maatouk:2012}, which analyze the iterative LT decoding process, let us define the state generating function of the frameless ALOHA decoder as
\begin{align}
P_u (\xc{1},\xc{2},\dots, \xc{k},\y) & :=
\sum_{ \mathcal{S} } \pcccru \, {\xc{1}}^{\cc{1}} \, {\xc{2}}^{\cc{2}} \dots {\xc{k}}^{\cc{k}} \,\y^{r-1}.
\end{align}
where $\mathcal{S}$ represents the set of all valid decoder states,
\begin{align} \label{eq:valid_states}
\mathcal{S} = \Big\{ (\cc{1}, \cc{2}, \hdots, \cc{k}, \r) | \cc{1} \geq 0, \cc{2} \geq 0, \hdots \cc{k} \geq 0, \r \geq 0, \sum_{i=1}^{k} \cc{i} +\r \leq m \Big\}.
\end{align}
The following theorem establishes a recursion for the state generating function.
\begin{theorem}\label{theorem_state_gen_fun}
Consider a contention period with $m$ slots of $k$ different types and $n$ contending users, which in a slot of type $h$ are active with probability $p_{h}$. For $u=n, n-1,\hdots,1$, we have
\begin{align}\label{eq:theorem_new}
& P_u (\xc{1},\xc{2},\hdots, \xc{k},\y) = \\
&\frac{1}{\y} \Bigg[ P_u \Bigg( \xc{1}(1-\puc{1}) + \y \puc{1}, \xc{2}(1-\puc{2}) + \y \puc{2},
\hdots, \xc{k}(1-\puc{k}) + \y \puc{k}, \frac{1}{u} + \y \left( 1 -\frac{1}{u} \right)\Bigg) \\
& - P_u \left( \xc{1} (1-\puc{1}),\xc{2} (1-\puc{2}), \hdots, \xc{k} (1-\puc{k}), \frac{1}{u}\right)\Bigg]
\end{align}
where $\puc{h}$ is given by
\[
\puc{h} = \frac{ \mathlarger {\sum}\limits_{d=2}^{n-u+2} \Omega_{h,d} \, d (d-1)\frac{1}{n} \frac{u-1}{n-1} \frac{\binom{n-u}{d-2}}{\binom{n-2}{d-2}} }
{ 1 - \mathlarger{\sum}\limits_{d=1}^{\myop{n-u+1}} \Omega_{h,d} \,u \frac{\binom{n-u}{d-1}}{\binom{n}{d}}
- \mathlarger{\sum}\limits_{d=0}^{\myop{n-u}} \Omega_{h,d} \frac{\binom{n-u}{d}}{\binom{n}{d}} }
\]
and initial condition given by
\begin{equation}
\P_{n} (\xc{1},\xc{2},\hdots, \xc{k},\y) =
\mkern-2mu \frac{1}{y} \prod_{h=1}^{k} \big( (1-\Omega_{h,1}-\Omega_{h,0}) \x + \Omega_{h,1} \y + \Omega_{h,0} \big)^
\mkern-2mu
- \big( (1-\Omega_{h,1} - \Omega_{h,0}) \x + \Omega_{h,0} \big)^m . \label{eq:p_k}
\end{equation}
\end{theorem}
\begin{IEEEproof}
See Appendix~\ref{sec:theorem_state_proof}.
\end{IEEEproof}
Theorem~\ref{theorem_state_gen_fun} can be used to derive the probability of decoding failure in a similar way as done for the LT decoder in \cite{shokrollahi2009theoryraptor}. In particular, we have that the probability that the decoder fails when exactly $u$ users are undecoded is $1-P_u(\mathbf{1})$, where $\mathbf{1}$ is the all one vector. This corresponds to the probability that the ripple is empty. Hence, the probability that exactly $u$ users remain unresolved after a contention period of $m$ slots, ${{\mathsf{P}}}_u$, corresponds to:
\[
{{\mathsf{P}}}_u = 1-P_u(\mathbf{1}).
\]
\section{Background and System Model}
\label{sec:II}
\subsection{Background: Frameless ALOHA}
\label{sec:background}
\begin{figure}[t]
\centering
\includegraphics[width=0.85\columnwidth]{figures/contention}
\caption{An example of contention in Frameless ALOHA. All three users randomly and independently decide on a slot basis whether to transmit or not. Slot 1 and slot 2 are collision slots; the colliding transmissions can not be decoded and the \ac{AP} stores the slots (i.e., the signals observed in them) for later use. Slot 4 is a singleton slot and the \ac{AP} decodes a replica of the packet of user 2 from it. The \ac{AP} also learns that a replica of packet of user 2 occurred in slot 1, and removes (cancels) it from the stored signal. Slot 1 now becomes singleton and a replica of the packet of user 1 becomes decoded. In the same manner, the successive process of replica removal and decoding of a new packet replica occurs in slot 2. As all three users have become resolved, the \ac{AP} terminates the contention period after slot 4, and starts a new one.}
\label{fig:frameless_example}
\end{figure}
Frameless ALOHA \cite{SPV2012} can be regarded as a variant of slotted ALOHA with \ac{SIC} that is inspired by rateless codes \cite{luby02:LT}.
The time in frameless ALOHA is divided into equal-length slots and slots are organized into contention periods, whose length is a-priori not known.
In order to transmit a packet, users must first wait until a new contention period starts.
Next, in each slot of the contention period, every contending user transmits a replica of its packet with a predefined slot-access probability. This happens independently from the transmission in other slots and independently of the actions of any other contending user.
Furthermore, the assumption is made that each packet replica contains information about the slots in which the other replicas of the same packet are placed.
This could be accomplished by, e.g., enriching the packet header with the seed of a random number generator and the transmit decisions are determined by the output of this generator. Hence, when a packet replica is decoded, it provides information about the timing of all other replicas.
The \acf{AP} is required to store the waveform of the whole contention period and processes slots sequentially, leveraging \ac{SIC} to remove the interference caused by replicas of decoded packets. Since every packet contains a pointer to the location of all its replicas, when a packet is successfully decoded, the receiver can determine the location of all its replicas and remove them from the received waveform. This reduces the interference in those slots containing replicas of the decoded packets and it may enable the receiver to decode more packets within those slots.
This process is repeated until no more packets can be decoded.
This \ac{SIC} process is not exclusive to frameless ALOHA, but common to all SIC-enabled slotted ALOHA schemes. The peculiarity of frameless ALOHA is the fact that the contention period (also known as frame) duration need not be a priori defined. The \ac{AP} can trigger the \ac{SIC} process after the reception of every slot, and decide whether to end the contention period and start a new one, or alternatively to let the contention period continue.
This decision is made according to a predefined criterion, e.g., whether the target throughput has been reached and/or a predefined fraction of users have been resolved \cite{SP2013}.
The start or termination of a contention period can be signaled to users by means of a beacon signal transmitted by the access point \cite{SP2013}.
An example of contention period in frameless ALOHA is depicted in Fig.~\ref{fig:frameless_example}.
\subsection{System Model}
\label{sec:sysmodel}
We denote by $n$ the number of users contending for access to a single access point.
The duration of the contention period in slots is denoted by $m$; note that $m$ is not a-priori fixed.
Furthermore, we shall assume that $k$ different slot types exists.
In particular, we assume that, out of the total $m$ slots, exactly $\nslotc{1}, \nslotc{2}, ... \nslotc{k}$ are of type $1,2, ...,k$.
Slots of type $h$, are characterized by a slot access probability $\paccessc{h}$, given by $\paccessc{h} = \frac{\betac{h}}{n}$, which is equal for all users.
Hence, in a slot of type $h$, every contending user will be active (send a packet) with probability $\paccessc{h}$, independently from its transmission in others slots, and from the actions of the other users.
It is easy to verify that $\betac{h}$ is the mean number of users that transmitted in a slot of type $h$, and, thus, equal to the expected number of transmissions contained in the slot.
\begin{figure}[t]
\centering
\includegraphics[width=0.63\columnwidth]{figures/ripple_and_clouds_}
\caption{The clouds and the ripple.}
\label{fig:ripple_cloud}
\end{figure}
A collision channel model will be assumed. Hence, singleton slots, i.e., slots containing a single transmission, are decodable with probability $1$, and collision slots, i.e., slots containing two or more transmissions, are not decodable with probability $1$.
Perfect interference cancellation will be assumed, i.e., the removal of replicas from the slots leaves no residual transmission power.\footnote{This assumption is reasonable for practical interference cancellation methods and moderate to high signal-to-noise ratios \cite{L2011}.}
In order to model the successive interference cancellation process at the receiver, we introduce the following definitions:
\vspace{-4mm}
\begin{mydef}[Initial slot degree] The initial slot degree is the number of transmissions originally occurring in the slot.
\end{mydef}
\vspace{-6mm}
\begin{mydef}[Reduced slot degree] The reduced slot degree is the current number of unresolved transmissions in the slot, over the iterations of the reception algorithm.
\end{mydef}
\vspace{-6mm}
\begin{mydef}[Ripple] The ripple is the set of slots of reduced degree 1, and it is denoted by $\mathscr{R}$.
\end{mydef}
\vspace{-4mm}
\noindent The cardinality of the ripple, $|\mathscr{R}|$ is denoted by $\r$ and its associated random variable as $\mathsf{R}$.
\vspace{-4mm}
\begin{mydef}[$h$-th cloud] The $h$-th cloud, $\cloudsetc{h}$, is the set of slots of type $h$ with reduced degree $d > 1$.
\end{mydef}
\vspace{-4mm}
\noindent The cardinality of the $h$-th cloud, $| \cloudsetc{h}| $, is denoted by $\cc{h} $ and the corresponding random variable as $\Cloudc{h}$.
Upon reception, the reduced degree of a slot is equal to its initial degree, since all users that are active in the slot are unresolved. During the decoding process, the reduced degree of a slot is decreased by $1$ whenever one of the unresolved users which are active in the slot is decoded and its interference is cancelled.
Whenever the reduced degree of a slot in the $h$-th cloud becomes $1$, the slot leaves the $h$-th cloud and enters the ripple. Note, however, that when the reduced degree of a slot decreases from, say, $3$ to $2$, then the slot remains in the $h$-th cloud. Similarly, when the only unresolved user in a slot belonging to the ripple is resolved, the reduced degree of the slot becomes $0$ and the slot leaves the ripple. The slots of reduced degree $0$ are not of further use in the analysis, and are thus not considered explicitly. The transition of slots between the clouds and ripple is depicted in Fig.~\ref{fig:ripple_cloud}.
In order to keep track of the temporal variation of the ripple and cloud sizes as the \ac{SIC} process progresses, we introduce a subscript $u$, which corresponds to the number of unresolved users. Thus, initially we have $u=n$, and every time a user is resolved, $u$ is decreased by 1 until we have decoded all users, when $u=0$. Thus, the ripple when $u$ users are unresolved is denoted by $\ripple{u}$, its cardinality by $\r_{u}$ and the random variable associated to it by $\mathsf{R}_{u}$. Similarly, when $u$ users are unresolved, $\cloudc{h}{u}$, $\cuc{h}$ and $\Cloudc{h,u}$ denote the $h$-th cloud, its cardinality and the associated random variable, respectively.
Let us consider the example in Fig.~\ref{fig:frameless_example}, and assume that there are two slot types. Further, let us assume that slot 1 and 3 are of type 1, whereas slot 2 and 4 are of type 2. Initially slots 1 and 2 have degree $2$, hence they belong to the $1$-st cloud $\cloudsetc{1}$ and $2$-nd cloud $\cloudsetc{2}$, respectively. Similarly, slot 4 is initially in the ripple, $\mathscr{R}$, since it has a degree of one. In the first decoding step, the only active user in slot 4, user 2, gets resolved and the replica transmitted by user 2 in slot 1 is cancelled. As a consequence, slot 4 leaves the ripple, while slot 1 leaves the $1$-st cloud and enters the ripple. Hence, the $1$-st cloud becomes empty.
In the next step, user 1 is resolved, slot 1 leaves the ripple, and slot 2 leaves the $2$-nd cloud and enters the ripple. Thus, the $2$-nd cloud becomes empty.
Finally, user 3 is resolved, and slot 2 leaves the ripple. In this case, all 3 users could be resolved.
Finally, we denote the slot degree distribution of slot type $h$ by $\mathbf{\Omega_{h} }=\left\{ \Omega_{h,1}, \Omega_{h,2},...,\Omega_{h,n} \right\}$, $h=1,2,...,k$, where $\Omega_{h,j}$ is the probability that a slot of type $h$ has initial degree $j$.
It is straightforward to verify that $\Omega_{h,j}$, $j=1,2,...,n$, is given by
\begin{equation}\label{eq:omega_h_j}
\Omega_{h,j} = { n \choose j } \left(\paccessc{h} \right)^j \left( 1 - \paccessc{h} \right)^{n - j}
= { n \choose j }\left( \frac{\betac{h}}{n} \right ) ^j \left( 1 - \frac{\betac{h}}{n} \right)^{n - j}.
\end{equation}
Hence, the probability that a slot of type $h$ initially contains no transmission at all is $\Omega_{h,0}$, the probability that it initially belongs to the ripple is $\Omega_{h,1}$, and the probability that it initially belongs to the $h$-th cloud is $( 1 - \Omega_{h,0} - \Omega_{h,1} ) $.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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{"url":"https:\/\/physics.stackexchange.com\/questions\/2328\/does-it-contradict-special-relativity-that-an-electron-beam-in-a-television-pict","text":"# Does it contradict special relativity that an electron beam in a television picture tube can move across the screen faster than the speed of light?\n\nWhile looking at some exercises in my physics textbook, I came across the following problem which I thought was quite interesting:\n\nIt is possible for the electron beam in a television picture tube to move across the screen at a speed faster than the speed of light.\n\nWhy does this not contradict special relativity?\n\nI suspect that it's because the television is in air, and light in air travels slower than light in a vacuum. So I suppose they're saying the the electron could travel faster in air than the speed of light in air, like what causes Cherenkov radiation?\n\n\u2022 You could also just consider a person shining a laser pointer at a distant wall. As you spin around, the spot of the laser pointer moves on the wall with a speed dependent on the distance to the wall. In principle, the wall could be so far away that the spot moves faster than the speed of light. But the light is still moving at the speed of light (in air, or whatever). The spot is not really an object - unless you are the inmate trying to escape from the insane asylum on a beam of light! \u2013\u00a0Greg P Dec 28 '10 at 22:17\n\u2022 @Greg oh! move across the screen... so is it talking about the picture itself? I thought it was saying the beam from the electron gun was moving faster than light \u2013\u00a0wrongusername Dec 28 '10 at 22:21\n\u2022 Yes. It is something I remember from an intro relativity book. It means the actual spot (yes, the image) moving across the screen. Otherwise, I don't get the point of the question. The electrons themselves don't move faster than light. It is just an illusion of something moving faster than the speed of light. \u2013\u00a0Greg P Dec 28 '10 at 22:31\n\u2022 There were some other 'paradoxes' where objects seem to move at superluminal speeds. Particularly one from astrophysics which seemed interesting...perhaps someone can remember it for me. \u2013\u00a0Greg P Dec 28 '10 at 22:35\n\u2022 @GregP: en.wikipedia.org\/wiki\/Superluminal_motion has some descriptions of common examples. Although that might be a good question to ask on the site. \u2013\u00a0David Z Dec 28 '10 at 23:47\n\nThis is an example of what is sometimes called the \"Marquee Effect.\" Think of the light bulbs surrounding an old-fashioned movie theater marquee, where the light bulbs turn on in sequence to produce the illusion, from a distance, of a light source which is moving around the the marquee.\n\nThere is no limit on how short the time interval is between one light turning on and the next turning on, so the perceived light source position can move arbitrarily fast, but in fact nothing is actually moving at all.\n\nIn the case of the television screen, the phosphors on the screen can be lit in rapid sequence, but the electrons in the beam do not ever need to move at (or even near) the speed of light.\n\nMore generally, there are loads of examples of some imaginary or conceptual \"object\" moving faster than light, but in all these cases there is nothing actually moving at all. A classic example is the intersection point of two nearly parallel lines, which moves very rapidly as the angle between the lines changes. In this case it is obvious that the moving \"object\" isn't moving at all, but its still a good example of a case where you can discuss something moving faster than light without there being any violation of physical law.\n\nThis is a conflation of phase velocity, and group velocity. The beam can be seen to move from say left to right at higher than c, but no information or particles are traveling that fast. Information is being transmitted from the electron gun to the phosphor at well under the speed of light.\n\nIt has nothing to do with the media it is embedded in. The information is going from the electron gun to the screen, not from one location on the screen to another.\n\n\u2022 How come the beam is a wave? \u2013\u00a0wrongusername Dec 28 '10 at 22:12\n\u2022 @wrongusername: A beam of electrons behaves like a wave beam, the same is behaviour is verified even in much larger particles (such as small atoms). The reason lies in Quantum Mechanics, and I can't get into it here. \u2013\u00a0Malabarba Dec 28 '10 at 22:47\n\u2022 All of this is mostly irrelevant though, as the wave nature of the beam has nothing to do with your question. This could happen with virtually anything that moves. \u2013\u00a0Malabarba Dec 28 '10 at 22:48\n\u2022 Replace the electron beam with a marshmallow gun. People think of quantum theory and waves when they hear \"electron\" but not with \"marshmallows\". Of course, it may be hard to actually create a series of marshmallow collisions on a distant wall appearing to move faster than c, but heck this is only a thought experiment, so imagine if you have a powerful enough gun... \u2013\u00a0DarenW Mar 5 '11 at 21:17\n\nThere is no contradiction in \"processes\" made up of sequences of independent events such as these because there is no causal relationship between the individual events - in this case the flights of the individual electrons and their ultimate collision with the screen - that make up the process.\n\nIf the screen is far enough from the source and the scan so fast then the sequence of electron collisions moves across the screen at faster than $c$. In this case, the order of the collisions on the screen is reversed when observed from an inertial frame moving fast enough against the direction of motion of the scan. This is simply a manifestation of relativity of simultaneity. But this is not a problem, because each of the electron flights is causally independent, and therefore there is no possibility of a violation of causality wrought by the frame's motion.\n\nIt is, however, a problem if the sub-events making up the process are causally related. This would be so if the motion were that of lone electron, where its being ($B$) at each instant is causally related to its being at any former instant ($A$) - $A$ is then a cause of $B$. One can then find a relatively moving frame, for example, where each electron would fly back into the cathode, undergo inverse thermionic emission and become a conduction electron again.\n\nIndeed the above is the primary reason why we postulate that such faster than light motion is forbidden in relativity: it would violate causality and we make this postulate to uphold a causal description of the universe.\n\nHere's another example from Griffith's book \"Introduction to Electrodynamics\" which illustrates phenomena where what we see is not what we observe. The apparent speed can be much greater than the speed of light. This speed is just what we see, an illusion, and it's the result of our inability sometimes to see the actual direction of movement of an distant object w.r.t. us and the fact that the light needs some finite time to get to our eyes.\n\nProblem 12.6 Every 2 years, more or less, The New York Times publishes\n\nan article in which some astronomer claims to have found an object\n\ntraveling faster than the speed of light. Many of these reports\n\nresult from a failure to distinguish what is seen from what is\n\nobserved--that is, from a failure to account for light travel time.\n\nHere's an example: A star is traveling with speed $v$ at an angle $\\theta$ to\n\nthe line of sight (Fig. 12.6). What is its apparent speed across the\n\nsky'?\n\n(Suppose the light signal from $b$ reaches the earth at a time At after\n\nthe signal from a, and the star has meanwhile advanced a distance $\\Delta s$\n\nacross the celestial sphere; by \"apparent speed\" I mean $\\Delta s\/\\Delta t$.) What\n\nangle $\\theta$ gives the maximum apparent speed? Show that the apparent\n\nspeed can be much greater than $c$, even if $v$ itself is less than $c$.\n\nIt can be easily shown that the apparent speed in this example is:\n\n$u_{app}=\\frac{v\\sin\\theta}{1-\\frac{v}{c}\\cos\\theta}$\n\nTo find the angle $\\theta$ that gives the maximum apparent speed we just differentiate and solve, for $\\theta$, the equation:\n\n$\\frac{d u_{app}}{d\\theta}=0 \\Leftrightarrow \\theta_{max}=\\cos^{-1}(\\frac{v}{c})$\n\nAt this angle, $u_{app}=\\frac{v}{\\sqrt{1-v^2\/c^2}}=\\gamma v$\n\nThis result shows that when $v\\to c$, $u_{app}\\to \\infty$, even though $v<c$.\n\n\u2022 This is not quite the same issue as in the question, though. \u2013\u00a0David Z Jul 17 '11 at 23:37\n\u2022 I think this is exactly the same issue. The electrons gun changes its direction, lets say by $\\theta=\\phi$. If we suppose that the electrons are emitted once at $\\theta=0$ and once at $\\theta=\\phi$, we get an triangle. While the beam at $\\theta=0$ travels to the screen, the electron gun rotates by $\\phi$ and emits the second beam. So the time difference between the arrival of two beams can be very small. \u2013\u00a0AndyK Jul 17 '11 at 23:55\n\u2022 Yes, but in the case of the electron gun (and the light bulbs, and the laser pointer, etc.), the light is being emitted by two completely different objects. Nothing actually moves even close to the speed of light. In fact, nothing has to move at all, in the case of the light bulbs. But your example with the star involves light being emitted by the same object at two separate points. Without motion, there is no superluminal effect. That's why they're different phenomena. \u2013\u00a0David Z Jul 18 '11 at 0:05\n\nThe electron beam in a typical television moves with a speed no higher than $10^6 \\mathrm{m\/s}$, which is much less than the speed of light. I wonder what building-sized cathode ray tube television the textbook author has access to if he's made that claim.\n\nIn any case, even if the beam did move faster than the speed of light, that wouldn't cause any problem. Quoting part of the treatment of going faster than light in the Physics FAQ, section 3. Shadows and Light Spots.\n\nThink about how fast a shadow can move. If you project the shadow of your finger using a nearby lamp onto a distant wall and then wag your finger, the shadow will move much faster than your finger. If your finger moves parallel to the wall, the shadow's speed will be multiplied by a factor $D\/d$ where $d$ is the distance from the lamp to your finger, and $D$ is the distance from the lamp to the wall. The speed can even be much faster than this if the wall is at an angle to your finger's motion. If the wall is very far away, the movement of the shadow will be delayed because of the time it takes light to get there, but the shadow's speed is still increased by the same ratio. The speed of a shadow is therefore not restricted to be less than the speed of light.\n\nOthers things that can go FTL include the spot of a laser that has been aimed at the surface of the Moon. Given that the distance to the Moon is 385,000 km, try working out the speed of the spot if you wave the laser at a gentle speed. You might also like to think about a water wave arriving obliquely at a long straight beach. How fast can the point at which the wave is breaking travel along the beach?\n\nThese are all examples of things that can go faster than light, but which are not physical objects. It is not possible to send information faster than light on a shadow or light spot, so FTL communication is not possible in this way. This is not what we mean by faster than light travel, although it shows how difficult it is to define what we really do mean by faster than light travel.\n\nThe electron beam in a television is very similar. The electrons are emitted from one point in the cathode ray tube, and then the direction of the beam is modified by a varying electric field. The electron beam can move from one side of the television to the other in less than $10^{-5} s$, but the beam doesn't carry any information from a side of the screen to the other.","date":"2019-08-20 19:35:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5392948389053345, \"perplexity\": 273.0420877140295}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027315558.25\/warc\/CC-MAIN-20190820180442-20190820202442-00296.warc.gz\"}"}
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Q: Как удалить всю строку из входного файла использую данные лишь из одного столбца? Нужно удалить данные всей строки, используя данные столбца "idsot".
У меня есть код для парсинга файла, использовал DictReader.
import csv
file = open('sotrudnikii.csv', 'r', encoding='utf-8')
read = csv.DictReader(file, delimiter=';', quotechar='"')
manager = sorted(read, key=lambda x: int(x['idsot']), reverse=False)
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,113
|
Q: notepad++ removing first word in every line i am looking for a way to delete the first word in every line in notepad++.
The list looks like this:
jon52_rs - £5
caneflagi - £5
Tehhguam - £5
Clapu - £5-05
and i want this:
- £5
- £5
- £5
- £5.05
or ideally only the numbers (5,5,5,5.05). I tried this with one method, however it deleted all the comma numbers like 5.05.
Thanks
A: You can simply use..
Find: ^\S+[ ]
Replace:
A: In Notepad++
Find: ^\w+\s+(.*)
Replace: \1
^ beginning of the line
\w word
\s space
(.*) argument - following any number of characters
\1 Keep the first argument
A:
I am looking for a way to delete the first word in every line in notepad++.
It should work.
^(\w+)(.*)
demo
ideally only the numbers
try this one if you need only numbers:
^(.*)£(.*)
demo
In Notepad++
Find what : ^(\w+)(.*)
Replace with $2 or \2
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,310
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{"url":"https:\/\/zbmath.org\/?q=an%3A1281.62026","text":"## Estimation with quadratic loss.(English)Zbl\u00a01281.62026\n\nProc. 4th Berkeley Symp. Math. Stat. Probab. 1, 361-379 (1961).\nFollowing the introduction, in Section 2 of this paper is given a new proof by the authors of the result of [C. Stein, in: Proc. 3rd Berkeley Sympos. Math. Statist. Probability 1, 197\u2013206 (1956; Zbl\u00a00073.35602)] that the usual estimator of the mean of a multivariate normal distribution with the identity as covariance matrix is inadmissible when the loss is the sum of squares of the errors in the different coordinates if the dimension is at least three. An explicit formula is given for an estimator, still inadmissible, whose risk is never more than that of the usual estimator and considerably less near the origin. Other distributions and other loss functions are considered later in Section 2.\nIn Section 3 the general problem of admissibility of estimators for problems with quadratic loss is formulated and a sufficient condition for admissibility is given and its relation to the necessary and sufficient condition [C. Stein, Ann. Math. Stat. 26, 518\u2013522 (1955; Zbl\u00a00065.11703)] is briefly discussed.\nIn Section 4 theorems are given which show that under weak conditions Pitman\u2019s estimator for one or two location parameters is admissible when the loss is taken to be equal to the sum of squares of the errors. Conjectures are discussed for the more difficult problem where unknown location parameters are also present as nuisance parameters, and Blackwell\u2019s example is given.\nIn Section 5 a problem in multivariate analysis is given where the natural estimator is not even minimax although it has constant risk. These are related to the examples of one of the authors quoted by J. Kiefer [Ann. Math. Stat. 28, 573\u2013601 (1957; Zbl\u00a00080.13004)] and E. L. Lehmann [Testing statistical hypotheses. New York: John Wiley & Sons; London: Chapman & Hall (1959; Zbl\u00a00089.14102), pp. 231 and 338].\nIn Section 6 some unsolved problems are mentioned.\nThe results of Section 2 were obtained by the two authors working together. The remainder of the paper is the work of C. Stein.\nFor the entire collection see [Zbl\u00a00101.34803].\n\n### MSC:\n\n 62J07 Ridge regression; shrinkage estimators (Lasso) 62H12 Estimation in multivariate analysis 62C15 Admissibility in statistical decision theory","date":"2022-12-08 13:18:26","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8674184679985046, \"perplexity\": 493.5872725684672}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446711336.41\/warc\/CC-MAIN-20221208114402-20221208144402-00680.warc.gz\"}"}
| null | null |
__author__ = "Nils Tobias Schmidt"
__email__ = "schmidt89 at informatik.uni-marburg.de"
from threading import Event
import threading
class StopThread(threading.Thread):
''' Extends the `Thread` with an `Event` and the `terminate` method
like the `multiprocessing` api offers it.
Calling it will trigger the `Event`.
Just implement your cleanup code for this event.
'''
def __init__(self, *args, **kwargs):
super(StopThread, self).__init__(*args, **kwargs)
self.shall_terminate_event = Event()
def terminate(self):
''' Immitate the `processing` API and offer a way to do some clean up in the `Thread`. '''
self.shall_terminate_event.set()
def shall_terminate(self):
''' Can be queried to know if the `Thread` shall do some cleanup '''
return self.shall_terminate_event.is_set()
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{
"redpajama_set_name": "RedPajamaGithub"
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Neuroprosthesis Lab
Biophysical Models
Optogenetics has been a revolution in the way we communicate with the nervous system. It derives from the 2003 PNAS paper by Nagel and co-workers which described the genetic insertion of channelrhodopsin-2 into cells to make them light sensitive. Channelrhodopsin-2 is an optically active channel protein from an alga that allows for transmission of cations across cell membranes. As cations such as sodium are key to neuroelectronic function, inserting this optically active channel unto cell membranes render them light sensitive.
The ability to use gene therapy techniques to photosensitize specific cells is what makes this technique so powerful. can be used to target specific cells types. For example, specifically photosensitizing inhibitory excitory cells allows for much more powerful neural control. Furthermore, the technique has been demonstrated in animals from invertebrates to non-human primates, and is now undergoing clinical trials in humans.
In recent years there has been considerable focus to develop new variants of channelrhodopsin – different light sensitivity as well both inhibitory and excitory variants. However, one of the key challenges is that Channelrhodopsin-2 has a very high light requirement for activation – typically defined as 0.7mW/mm2 on the cell. This generates both technical and regulatory challenges.
What is the optimal stimulus method from an engineering perspective?
What is the light requirement at different Tissue depth?
Will intense light creation by implantables lead to unacceptable tissue heating?
Will the blue light cause photochemical damage to the tissue?
Figure 1: Results from biophysical modelling. (Left) The 4-state model of channelrhodopsin (Middle) Optical penetration of light into tissue (Right) Thermal modelling of LED probes
Optimal stimulus: In our papers by Nikolic et al. and Grossman et al., we combined a 4-state model of channelrhodopsin with a Hodgkin Huxley model of Channelrhodopsin-2. There are some detailed conclusions that can be drawn for any specific system. But a key conclusion is that as there is a low-efficiency light-adapted state for channelrhodopsin; short, high-intensity pulses prove most efficient for stimulus. Models available upon request.
Light transmission through tissue: Channelrhodopsin-2 encoded cells are considered to have a threshold of 0.7mW/mm2. This "threshold" is defined as what is required to achieve 50% stimulus in dissociated cells. So there is still some effect even below this level. In order to achieve this, light must traverse through tissue, which will tend to be backscattered. Furthermore, for non-collimated emitters such as LEDs, the emitted light will have an emission angle thus spreading light over a wide space. Na et all, therefore, developed a Monte Carlo model to explore the 3D decay profile. Models available upon request
Tissue heating effects: Light can be created externally and guided to the target tissue via optic fibres, or it can be generated locally. The latter allows for electronic multiplexing, but although best-in-class LEDs can be up to 80% efficient, micro LEDs used in optical probes have varied in wall plug efficiencies from 1%-30% efficiency. As the rest of the energy gets converted to heat, there is a possibility of local hot spots. Using a COMSOL finite element model of brain probes, we found that the thermal emission of the LEDs over a short duration should not exceed 5.2mW. Models available upon request
Photochemical degradation: In the 2009 study by Degenaar et al. and the 2018 study by Soltan et al., we explored the limits to blue light excitation of tissue from the regulatory perspective. Specific stimuli regimes need to be put in place to ensure that for 470nm light, the average irradiance in any 10,000 s period should not exceed 0.05mW/mm2. Calculations are available upon request.
Ahmed Soltan, John Martin Barrett, Pleun Maaskant, Niall Armstrong, Walid Al-Atabany, Lionel Chaudet, Mark Neil, Evelyne Sernagor and Patrick A head mounted device stimulator for optogenetic retinal prosthesis Journal of Neural Engineering, Volume 15, Number 6 Oct 2018
Na Dong, Ahmed Soltan, Rolando Berlinguer Palmini Nikhil Ponon, Anthony O'Neil, Andrew Trevelyan, Patrick Degenaar, Xiaohan Sun, "Opto-electro-thermal optimisation of optoelectronic probes for optogenetic neural stimulation" Submitted to Journal of Biophotonics – accepted and in revision with minor corrections to grammar DOI:10.1002/jbio.201700358
Grossman N, et al. Modeling Study of the Light Stimulation of a Neuron Cell With Channelrhodopsin-2 Mutants. IEEE T. on Biomedical Engineering 2011, 58(6), 1742-1751.
Nikolic K, Loizu J, Degenaar P, Toumazou C. A stochastic model of the single photon response in Drosophila photoreceptors. Integrative Biology 2010, 2, 354-370.
Nikolic K, et al. Photocycles of channelrhodopsin-2. Photochemistry and Photobiology 2009, 85(1), 400-411.
Degenaar P., Optobionic vision: a new genetically enhanced light on retinal prosthesis. Journal of Neural Engineering 2009, 6(3), 035007.
Nikolic K, et al. Noise reduction in analogue computation of Drosophila photoreceptors. Journal of Computational Electronics 2008, 7(3), 458-461.
Merz Court
Newcastle Univeristy
NE1 7RU, UK
Merz Reception
merz.reception@newcastle.ac.uk
Address: 3rd floor, Merz Court,
School of Engineering,
Newcastle University, NE1 7RU, UK
Copyright © by Neuroprosthesis lab | Newcastle University
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: meaning of index in texture atlas files I am following libgdx tutuorial on texture atlases. This is an excerpt from a texture atlas file. What is the meaning of index parameter and in what situations it is useful to the programmer? All texture region have it and it is the same namely -1, in all of them.
prehistoric.png
format: RGBA8888
filter: Nearest,Nearest
repeat: none
background
rotate: false
xy: 2, 2
size: 1280, 720
orig: 1280, 720
offset: 0, 0
index: -1
trex
rotate: false
xy: 1286, 479
size: 179, 243
orig: 179, 243
offset: 0, 0
index: -1
caveman
rotate: false
xy: 1286, 319
size: 83, 156
orig: 83, 156
offset: 0, 0
index: -1
A: From the Javadoc:
The number at the end of the original image file name, or -1 if none.
When sprites are packed, if the original file name ends with a number, it is stored >as the index and is not considered as part of the sprite's name.
I should also add, that this index is also used on findRegion(String name, int index) method which returns the first region found with the specified name and index.
A: It's typically used for animations. You can append frame numbers to the file names of each frame of animation before you pack them into the atlas, i.e. run0.png, run1.png, run2.png, etc. During texture packing, the number is removed from the sprite's name and used as its index. Then you can load the animation all at once:
animation = new Animation(0.1f, atlas.findRegions("run"));
The index is -1 when the original file name did not end in a number.
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Kommer ist der Familienname folgender Personen:
Björn Kommer (1912–2000), deutscher Denkmalpfleger
Björn R. Kommer (* 1942), deutscher Kunsthistoriker und Museumsdirektor
Detlev Kommer (1947–2005), deutscher Psychotherapeut und Gründungspräsident der Bundespsychotherapeutenkammer
Franz Kommer (* 1901), Bremer Bürgerschaftsabgeordneter (BDV)
Franziska Kommer (* 1999), deutsche Tennisspielerin
Gerd Kommer, deutscher Investmentbanker und Autor
Rudolf Kommer (1886–1943), rumänischer Journalist und Impresario
Siehe auch:
Commer (Begriffsklärung)
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Q: jquery function in cakephp2 I'm beginner on both jquery and cakephp. I have tried to run a jquery function into my page index and It doesn't work.
this function nest a ticker in my principal page index from another page called ticker.
cakephp path used are like that SessionModels/ticker.ctp and SessionModels/index.ctp.
when I run my page ticker.ctp alone, it runs without a problem displaying a ticker but when I have tried to nest it in a <div id="ticker"> </div> with my jquery function in index.ctp, it doesn't do anything .
this is my javascript/ jquery function :
<script type="text/javascript" > function ticker(){
$.ajax ({
type:'GET',
url:'/SessionModels/ticker/',
success:function(e){
$('#ticker').html(e);
},
complete : function(e){
setTimeout ("ticker()", 3000);
}
});
}
$(document).ready(function(){
ticker();
});</script>
I have added as I have seen also in cakephp tutorial this code in my controller
public $helpers = array('Js');
this is the content of my ticker.ctp
<marquee >
<?php foreach ($sessions as $session): ?>
<span style="color:#333333; font-size:12px; font-weight:bolder; font-family:Arial; vertical-align: text-top;"><?php echo $session['Dictionnary']['ticker']; ?>
</span>
<?php echo $session['SessionModel']['_open']; ?>
<?php echo $session['SessionModel']['_close']; ?>
<?php endforeach; ?>
</marquee>
and this is the content of my controller:
class SessionModelsController extends Controller {
public $helpers = array('Js');
public $components = array('RequestHandler');
public function index() {
$this->set('sessions', $this->SessionModel->find('all', array('contain' => 'Dictionnary.stock_name')));
}
public function ticker() {
$this->set('sessions', $this->SessionModel->find('all', array('contain' => 'Dictionnary.ticker')));
}
}
as I have said , it works perfectly without a problem but nesting the ticker.ctp in index.ctp is the problem.
thank you for the help
A: Http Status Code 304 'not modified' may indicate that your browser is using a cached version of the request instead of retrieving new data. This is probably caused by 'cache' headers being sent by your server.
jQuery.ajax() has a 'cache' option that prevents browser caching for the AJAX request and forces reloading the page by adding a random time stamp to each request.
Add the 'cache' option to your .ajax options and set it to 'false' to use it;
function ticker(){
$.ajax ({
type:'GET',
url:'/SessionModels/ticker/',
cache:false,
success:function(e){
$('#ticker').html(e);
},
complete : function(e){
setTimeout(ticker, 3000);
}
});
}
Read the documentation here jQuery.ajax()
To check if it the ticker() function is actually running, try temporarily changing it to this;
function ticker(){
alert('tick');
setTimeout(ticker, 3000);
}
I tested this in my situation and it works without problems, however, you should make sure that jQuery is loaded before this. Since you're using this code inside a View, it's possible you're loading jQuery afterwards (just before the closing </body> tag)
A better option is to use the buffer of the JsHelper and output the buffered script in you layout;
/**
* JsHelper will append this block to the $(document).ready()
*/
$script = "
function ticker() {
$.ajax ({
type:'GET',
url:'/timeline/qty_breakdown/',
cache:false,
success:function(e){
$('#ticker').html(e);
},
complete : function(e){
setTimeout (ticker, 3000);
}
});
}
ticker();
";
$this->Js->buffer($script);
Then, inside your layout (just before the closing </body> tag):
echo $this->Js->writeBuffer();
This will automatically wrap your script inside a $(document).ready(...)
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\section{Introduction}
The last few years have seen a renewal of interest \cite{kyotoconf}-
\cite{hss3}
in the low energy scalar
sector of QCD. Many physicists now believe in the
existence of the light, broad $I=J=0$ resonance, sigma in the
500-600 MeV region as well as a light broad $I,J=1/2,0$ resonance,
kappa in the 700-900 MeV region. Together with the well established
$f_0(980)$ and $a_0(980)$ scalar resonances, these comprise
a putative nonet of ``elementary particles".
Furthermore, this nonet seems likely to have a quark
structure like $qq{\bar q}{\bar q}$ rather than the conventional
$q{\bar q}$ \cite{Jaffe}. This of course raises the question
of where are the conventional $q{\bar q}$ p-wave scalars
expected in the quark model. Arguments have been given
\cite{mixing} that
the experimental data are better fit when the two scalar
nonets mix with each other and the resulting ``level
repulsion", pushes the conventional scalars to higher
masses than otherwise expected.
In order to further explore the feature of mixing between
$q{\bar q}$ type and $qq{\bar q}{\bar q}$ type states it seems
interesting to consider a linear SU(3)$\times$ SU(3)
sigma model
which contains also the pseudoscalar nonet partners of these two
scalar nonets. Parenthetically, we remark that while the non-
linear sigma model \cite{GL,CW} and its extension to the chiral
perturbation theory program \cite{CPT} are
often more efficient for systematic calculations,
linear sigma models have a very long history of furnishing
important insights into the nature of strong hadron dynamics.
The SU(2) linear sigma model was first given in ref. \cite{GL}.
It was used as a basis for understanding the current algebra
treatment of $\pi\pi$ scattering
near threshold in ref. \cite{w}. The SU(3) version was given
in the first of ref. \cite{l}. A detailed application to the low energy
pseudoscalar mass spectrum was given \cite{SU1} before QCD in which,
among other things, it
was shown how a U(1)$_A$ violating term natural in the
SU(3) model
could solve the $\eta^\prime$ problem. Such a term was later
discovered to arise from instanton effects \cite{tH1}. The
connection was pointed out in ref. \cite{MS} and emphasized by
't Hooft \cite{tH2}.
The model containing two different chiral nonets
to be discussed here was proposed in section V of
ref. \cite{BFMNS01} and an initial treatment, neglecting flavor
symmetry breaking, was given. A discussion, taking
the flavor symmetry breaking into account has very recently
been presented in ref. \cite{nr}. Actually, it turns out that
the model is very complicated since many different terms can be included
and various assumptions about the nature of the symmetry
breaking can be made. In this paper we will set up
the formalism for treating consequences of the model
which hold (at tree level) just due to the symmetry structure of
the model and will give a numerical treatment using what might be the
simplest choice of symmetry breaking terms.
Section II begins with a review of the flavor transformation
properties of the two chiral nonet fields, $M$ and $M'$ which are used
in the model. Each contains nine pseudoscalar
and nine scalar fields. Under chiral SU(3)$_L\times$
SU(3)$_R$ transformations
both fields transform in an identical manner. Thus a chiral Lagrangian
which respects only this symmetry cannot directly distinguish between
a ``two quark" (i.e. $q{\bar q}$) or a ``four quark" scalar, for
example. However, it is noted again that the U(1)$_A$
transformation
actually counts the number of quarks in these mesons and provides
a way to distinguish them. In order to make use of this, the Lagrangian
should of course be set up appropriately.
We implement this by requiring that the Lagrangian mock up the
anomalous U(1)$_A$ equation of the underlying QCD and
that the
analogs of the quark mass terms also mock up the
U(1)$_A$ transformation properties of the
quark mass terms in the underlying theory.
A reasonable
initial thought on which terms to include in the Lagrangian
is to restrict it to be renormalizable. It is noted, with details
in Appendix A, that the renormalizable $M-M'$ Lagrangian has
however
very many more terms than does the renormalizable single $M$
Lagrangian. An alternate way, which still satisfies generality,
is to consider any number of terms, renormalizable or not,
and just use the information which follows from the
symmetry behavior of the Lagrangian.
In order to exploit this symmetry information we derive,
in section III, vector type and axial vector type ``generating
equations" for the model. These can be differentiated with respect to
the fields to yield many tree level Ward identities which
are independent of the number of symmetric terms included in the
Lagrangian. In addition to the analog of ``two quark" condensates
which occur in the single $M$ model, the present model also brings
``four quark" condensates into the picture.
In section IV, we derive predictions
for the mass spectrum which follow from this symmetry approach.
The characteristic feature is mixing between``two quark" and
``four quark" mesons with the same quantum numbers. Assuming
isospin invariance, predictions are made for the $\pi-\pi'$
mixing sector, the $K-K'$ mixing sector, the strange
scalar $\kappa-\kappa'$ mixing sector and the sector
involving mixing of the four isocalar pseudoscalars ($\eta$
type particles). It is shown how to formulate the first three
of these mixing sectors in a parallel and economical way.
In section V, the mass spectrum relations are compared
with experiment. First the three $2\times 2$ mixing sectors
are treated. The inputs are taken to be the six masses of
the well known and not so well known particles, the pion and kaon
decay constants and a model parameter denoted $x_\pi$, which is
the squared mass of the unmixed (or ``bare") pion. These are enough to
determine all the relevant parameters of these three systems.
The pseudoscalar mixing is very sensitively dependent on
$x_\pi$; as it
increases from the experimental value, $m_\pi^2$ the
four quark components
of the pion and the kaon increase. On the other
hand, the
scalar $\kappa$ has a large four quark component. This feature
thus provides some support for a more exotic
structure of the low lying scalars.
Another interesting feature of the present model,
discussed in this section, is that it
permits one to estimate the strength of a
four quark vacuum condensate. Finally, section V contains
a brief summary, the connection with other results
on the same model and directions for future work.
\section{Symmetries and Lagrangian}
First, let us briefly review \cite{BFMNS01} the fields of the model
and their
transformation properties.
The schematic structure for the matrix $M(x)$ realizing a $q \bar q$ composite
in terms of quark fields $q_{aA}(x)$ can be written
\begin{equation}
M_a^b = {\left( q_{bA} \right)}^\dagger \gamma_4 \frac{1 + \gamma_5}{2} q_
{aA},
\label{M}
\end{equation}
where $a$ and $A$ are respectively flavor and color indices. Our convention
for matrix notation is $M_a^b \rightarrow M_{ab}$. Then $M$
transforms under chiral SU(3)$_L \times $ SU(3)$_R$ as
\begin{equation}
M \rightarrow U_L M U_R^\dagger,
\label{Mchiral}
\end{equation}
where $U_L$ and $U_R$ are unitary, unimodular matrices associated with the
transformations on the left handed
($q_L = \frac{1}{2}\left( 1 + \gamma_5 \right) q$) and right
handed ($q_R = \frac{1}{2}\left( 1 - \gamma_5 \right) q$) quark projections.
For the discrete transformations charge congugation $C$ and parity $P$
one verifies
\begin{equation}
C: \quad M \rightarrow M^T, \quad \quad P: \quad M({\bf x})
\rightarrow M^{\dagger}(-{\bf x}).
\label{MCP}
\end{equation}
The U(1)$_A$ transformation acts as $q_{aL}
\rightarrow e^{i\nu} q_{aL}$, $q_{aR} \rightarrow e^{-i\nu} q_{aR}$ and
results in:
\begin{equation}
M \rightarrow e^{2i\nu} M.
\label{MU1A}
\end{equation}
Next, consider the $qq{\bar q}{\bar q}$ type fields.
One interesting model \cite{Isgur}
postulates that the light scalars are ``molecules''
made out of two pseudoscalar mesons. The chiral realization of this
picture would result in the following schematic structure:
\begin{equation}
M_a^{(2)b} = \epsilon_{acd} \epsilon^{bef}
{\left( M^{\dagger} \right)}_e^c {\left( M^{\dagger} \right)}_f^d.
\label{M2}
\end{equation}
One can verify that $M^{(2)}$ transforms exactly in the same way
as $M$ under SU(3)$_L \times$ SU(3)$_R$, $C$ and $P$.
Under U(1)$_A$ it transforms as
\begin{equation}
M^{(2)} \rightarrow e^{-4i\nu} M^{(2)},
\end{equation}
which differs from Eq. (\ref{MU1A}).
Another interesting approach \cite{Jaffe} to explaining the light
scalar mesons was formulated by Jaffe in the framework of the MIT bag
model. It was observed that the spin-spin (hyperfine) piece of the
one gluon exchange interaction between quarks gives an exceptionally
strong binding to an s-wave $qq\bar q \bar q$ scalar state.
The scalar states of this
type may be formally written as bound states of a ``dual quark'' and
``dual antiquark''. There are two possibilities if the dual antiquark
is required to belong to a $\bar 3$ representation of flavor SU(3).
In the first case it belongs to a $\bar 3$ of color and is a spin singlet.
This has the schematic chiral realization,
\begin{eqnarray}
L^{gE} = \epsilon^{gab} \epsilon^{EAB}q_{aA}^T C^{-1} \frac{1 +
\gamma_5}{2} q_{bB}, \nonumber \\
R^{gE} = \epsilon^{gab} \epsilon^{EAB}q_{aA}^T C^{-1} \frac{1 - \gamma_5}{2}
q_{bB},
\end{eqnarray}
where $C$ is the charge conjugation matrix of the Dirac theory. A suitable
form for the $M$ matrix is:
\begin{equation}
M_g^{(3)f} = {\left( L^{gA}\right)}^\dagger R^{fA}.
\end{equation}
$M^{(3)}$ can be seen to transform in the same way as $M^{(2)}$ under
SU(3)$_L \times$ SU(3)$_R$, $C$, $P$ and U(1)$_A$. In
the second case
the dual antiquark belongs to a $6$ representation of color and has spin 1.
It has the corresponding schematic chiral realization:
\begin{eqnarray}
L_{\mu \nu,AB}^g = L_{\mu \nu,BA}^g = \epsilon^{gab} q^T_{aA} C^{-1}
\sigma_{\mu \nu} \frac{1 + \gamma_5}{2} q_{bB}, \nonumber \\
R_{\mu \nu,AB}^g = R_{\mu \nu,BA}^g = \epsilon^{gab} q^T_{aA} C^{-1}
\sigma_{\mu \nu} \frac{1 - \gamma_5}{2} q_{bB},
\end{eqnarray}
where $\sigma_{\mu \nu} = \frac{1}{2i} \left[ \gamma_\mu, \gamma_\nu \right]
$. This choice leads to an $M$ matrix
\begin{equation}
M_g^{(4) f} = {\left( L^{g}_{\mu \nu,AB}\right)}^\dagger R^{f}_{\mu \nu,AB},
\end{equation}
where the dagger operation includes a factor ${(-1)}^{\delta_{\mu 4} +
\delta_{\nu 4}}$. $M^{(4)}$ also transforms like $M^{(2)}$ and $M^{(3)}$
under all of SU(3)$_L \times$ SU(3)$_R$, $C$, $P$ and
U(1)$_A$. The
specific form favored by the MIT bag model calculation actually corresponds to
a particular linear combination of $M^{(3)}$ and $M^{(4)}$. Furthermore one
can verify that $M^{(2)}$ in Eq. (\ref{M2}) is related by a Fierz
transformation to a linear combination of $M^{(3)}$ and $M^{(4)}$.
Thus only two of $M^{(2)}$, $M^{(3)}$ and $M^{(4)}$ are linearly
independent. In any event, at the present effective Lagrangian level,
there are no quantum numbers to distinguish $M^{(2)}$, $M^{(3)}$,
and $M^{(4)}$ from each other so we may as well just denote an arbitrary
linear combination of them to be our $qq{\bar q}{\bar q}$ field,
$M^{\prime}$. Note that $M$ and $M^{\prime}$ are distinguished
from each other by their different U(1)$_A$
transformation properties.
These fields may be decomposed into hermitian scalar (S) and pseudoscalar ($\phi$)
nonets as,
\begin{eqnarray}
M &=& S +i\phi, \nonumber \\
M^\prime &=& S^\prime +i\phi^\prime.
\label{sandphi}
\end{eqnarray}
We will be interested in the situation where non-zero vacuum values
of the diagonal components of $S$ and $S'$ may exist. These will be
denoted by,
\begin{equation}
\left< S_a^b \right> = \alpha_a \delta_a^b,
\quad \quad \left< S_a^{\prime b} \right> =
\beta_a \delta_a^b.
\label{vevs}
\end{equation}
In the iso-spin invariant limit, $\alpha_1=\alpha_2$ and
$\beta_1=\beta_2$ while in the SU(3) invariant limit,
$\alpha_1=\alpha_2=\alpha_3$ and $\beta_1=\beta_2=\beta_3$.
The Lagrangian density which defines our model is
\begin{equation}
{\cal L} = - \frac{1}{2} {\rm Tr}
\left( \partial_\mu M \partial_\mu M^\dagger
\right) - \frac{1}{2} {\rm Tr}
\left( \partial_\mu M^\prime \partial_\mu M^{\prime \dagger} \right)
- V_0 \left( M, M^\prime \right) - V_{SB},
\label{mixingLsMLag}
\end{equation}
where $V_0(M,M^\prime) $ stands for a general function made
from SU(3)$_L \times$ SU(3)$_R$
(but not necessarily U(1)$_A$) invariants
formed out of
$M$ and $M^\prime$. Furthermore $V_{SB}$ is taken to be
a flavor symmetry breaking term which should mock up the quark
mass terms which perform this function in the fundamental
QCD Lagrangian. Other physical particles (including glueballs)
could be added for more realism, but Eq. (\ref{mixingLsMLag}) is already
quite complicated.
To get an initial indication of what is happening in this kind of
model the drastically simplified case
where the quark mass effective term, $V_{SB}$ is absent
and where $V_0$ is simply given by:
\begin{equation}
V_0 = -c_2 {\rm Tr} \left( M M^\dagger \right)
+ c_4 {\rm Tr} \left( M M^\dagger M M^\dagger \right)
+ d_2 {\rm Tr} \left( M^\prime M^{\prime \dagger} \right)
+ e {\rm Tr} \left( M M^{\prime \dagger} + M^\prime M^\dagger \right),
\label{mixingpot}
\end{equation}
was treated in sec.V of ref. \cite{BFMNS01}.
Here $c_2$, $c_4$ and $d_2$ are positive real constants. The $M$ matrix
field is chosen to have a wrong sign mass term so that there will be
spontaneous breakdown of chiral symmetry. A pseudoscalar octet is thus
massless. The mixing between the $M$ and $M^\prime$ is controlled by the
parameter $e$. The first feature found for this simplified model was that
the analog, $\langle{S'}^a_a\rangle$ of the $qq{\bar
q}{\bar q}$ condensate
in QCD acquired a small non-zero value due to the mixing between
$S$ and $S^\prime$. The main question is the level ordering. Since
the light pseudoscalars (e.g. $\pi^+ =\phi^2_1$) are naturally
identified, before mixing, with the $q{\bar q}$ field M, one
wonders whether the two quark rather than the four quark scalars
aren't the lightest ones. It was found however that it is
natural (but not unique) in the model to have the energy level pattern
in ascending order- pseudoscalar Nambu-Goldstone boson with
primarily $q{\bar q}$ structure, scalar with primarily $qq{\bar q}{\bar
q}$ structure, pseudoscalar with primarily $qq{\bar q}{\bar q}$ structure
and scalar with primarily $q{\bar q}$ structure. These refer
to degenerate octets which are each mixtures of $M$ and $M^\prime$
states. This seems to be similar to the expected experimental
pattern and gives us some motivation to proceed further.
The next question is what terms to include in the
Lagrangian Eq. (\ref{mixingLsMLag}). A natural first attempt
would be to consider a renormalizable model in which $V_0$ contains
all the SU(3)$\times$ SU(3) invariant terms up to four
powers of the
fields.
These are listed in Appendix A. It is seen that there are 21 terms of
this
type. This is a rather large number and while not impossible
to handle suggests trying another tack.
We will just allow $V_0$ to contain all possible terms which are
SU(3)$_L\times$ SU(3)$_R$ symmetric and use the
information provided by
this symmetry. This is more general and also allows for
non-renormalizable terms. The price to be paid is that we only get
information which follows just from the symmetry structure.
In an earlier treatment \cite{SU1} of the single chiral nonet case, it was
found
that the results obtained were essentially those
which could be obtained from
the ``current algebra" approach.
Furthermore, we will try to make use of the fact
that $M$ and $M^\prime$ have different U(1)$_A$
transformation properties.
We thus demand that the Lagrangian without $V_{SB}$ mock up
the anomalous U(1)$_A$ equation of QCD,
\begin{equation}
\delta {\cal L} =G,
\label{anomaly}
\end{equation}
where $\delta$ denotes the axial U(1) variation and
$G$ is proportional to the product of the QCD field strength
tensor and its dual. This can be achieved by making all
of the terms in $V_0$, except for a limited number,
U(1)$_A$
invariant. The special terms will be constructed to satisfy Eq.
(\ref{anomaly}). An example of a term which is not
U(1)$_A$
invariant is the mixing term used in
the simplified model above:
${\rm Tr}\, (M^{\prime}M^{\dagger})+ {\rm h.c.}$. However
a
mixing term of the type:
\begin{equation}
\epsilon_{abc}\epsilon^{def}M^a_dM^b_eM^{{\prime}c}_f +
{\rm h.c.}
\label{invtmixing}
\end{equation}
is U(1)$_A$ invariant and hence possibly the most
important one.
An SU(3)$_L\times$ SU(3)$_R$ invariant but not
U(1)$_A$
invariant term which mocks up Eq. (\ref{anomaly}) can be
seen \cite{U1A} to be
\begin{equation}
{\cal L}_{anom}= \frac{iG}{12} \,{\rm ln}\,(\frac{{\rm
det}\,
M}{{\rm det}\, M^{\dagger}}).
\label{anomterm}
\end{equation}
Here, $G$ is being formally considered as an effective pseudoscalar
glueball field in the effective Lagrangian. To get an $\eta'(958)$
mass term in the effective lagrangian framework one can \cite{U1A}
include a wrong sign mass term for G: $cG^2/2$ in the Lagrangian
which of course does not change the flavor symmetry structure.
Then integrating out $G$ yields the effective $\eta(960)$
mass term:
\begin{equation}
{\cal L}_\eta =-c_3[{\rm ln}\, (\frac{{\rm det}\,
M}{{\rm det}\,
M^{\dagger}})]^2,
\label{etaprmass}
\end{equation}
where $c_3=-1/(288c)$. The nature of this term becomes more
apparent when one goes to the non-linear realization
where $M\rightarrow \alpha_1 {\rm
exp}\, (i\phi/\alpha_1)$.
For the present paper we shall consider this to be the only
SU(3)$_L\times$ SU(3)$_R$ invariant but not U(1)$_A$
invariant term. However, it is not at all unique when
we consider a model with two chiral nonets. For example
one can also include something like the non- U(1)$_A$
invariant mixing term
${\rm Tr}\, (M^{\prime}M^{\dagger})+ {\rm h.c.}$ by
writing
a candidate Lagrangian piece:
\begin{equation}
\frac{iG}{12}[\gamma_1 {\rm ln}\, (\frac{{\rm det}\,
M}{{\rm det}
M^{\dagger}})
+\gamma_2
{\rm ln} \,
(\frac{{\rm Tr}\,
(MM'^{\dagger})}{{\rm Tr}\, (M'M^{\dagger})}],
\label{altanom}
\end{equation}
and proceeding as above. In order to properly mock up
the anomaly in this case it is necessary \cite{hsuss}
that the real numbers $\gamma_1$ and $\gamma_2$ satisfy
\begin{equation}
\gamma_1+\gamma_2=1.
\label{anomnorm}
\end{equation}
The generalization to more than two such terms is evident. It
may be noted that the $M-M'$ mixing term resulting from
Eq. (\ref{altanom}) mixes only the pseudoscalar fields
and not the scalar ones.
Finally, let us consider the flavor symmetry breaking terms.
To get more restrictions, we assume that such a term
should mock up both the SU(3)$_L\times$ SU(3)$_R$ and
U(1)$_A$
transformation properties of the quark mass terms in the fundamental
QCD Lagrangian. It is convenient to introduce a diagonal matrix,
\begin{equation}
A = {\rm diag} (A_1,A_2,A_3),
\label{defineA}
\end{equation}
which is proportional to the diagonal matrix
made from the three
light quark masses, $diag(m_u,m_d,m_s)$ (See \cite{MS} for further
details). Then, from Eq. (\ref{M}), we note an obvious
choice for a flavor symmetry breaking term,
\begin{equation}
V_{SB}=-{\rm Tr}\, [A(M+M^{\dagger})]=-2 {\rm Tr}\, (AS),
\label{vsb}
\end{equation}
which transforms like $(3,3^*)+(3^*,3)$ under
SU(3)$_L\times$ SU(3)$_R$.
Under the U(1)$_A$ transformation of Eq. (\ref{MU1A}), it
goes to
$-e^{2i\nu} {\rm Tr}\, (AM) + {\rm {\rm h.c.}}$. Note
that the
similar
simple possibility,
$-2 {\rm Tr} \, (A S')$
does not correctly mock up the U(1)$_A$ transformation
property of the QCD mass term.
However Eq. (\ref{vsb}) is not at all unique in correctly mocking
up the quark mass term. An interesting term which
does mock up the quark mass term also involves
mixing and has the form,
\begin{equation}
\epsilon_{abc}\epsilon^{def}A_d^aM_e^b {M'}_f^c +
{\rm {\rm h.c.}}
\label{sbmixer}
\end{equation}
This term mixes both scalars and pseudoscalars but with
opposite signs.
For what follows, it is convenient to record the behaviors
of the fields under infinitesimal transformations. Let us write the
infinitesimal vector (L+R) and axial vector (L-R) transformations
of $\phi$ and $S$ as,
\begin{eqnarray}
\delta_V \phi=[E_V,\phi], \quad \quad \delta_A \phi=-i[E_A,S]_+,
\nonumber \\
\delta_V S=[E_V,S], \quad \quad \delta_A S=i[E_A,\phi]_+.
\label{inftrans}
\end{eqnarray}
Here, unitarity demands that the infinitesimal matrices
obey,
\begin{equation}
E_V^{\dagger}=-E_V, \quad \quad E_A^{\dagger}=-E_A.
\label{unitarity}
\end{equation}
If we demand that the transformations be unimodular, so
that the U(1)$_A$ transformation is not included
(the U(1)$_V$ transformation is trivial for mesons), we
should
also impose ${\rm Tr}\, (E_A)=0$. However we will not do
this so the effects of
U(1)$_A$ will also be included. The transformation
properties
of the $qq{\bar q}{\bar q}$ type fields are:
\begin{eqnarray}
\delta_V\phi'=[E_V,\phi'], \quad \quad \delta_A \phi'=-i[E_A,S']_+
+2iS' {\rm Tr} \,(E_A),
\nonumber \\
\delta_V S'=[E_V,S'], \quad \quad \delta_A S'=i[E_A,\phi']_+
-2i\phi' {\rm Tr} (E_A).
\label{inftranspr}
\end{eqnarray}
The extra terms for the axial transformations reflect the different
U(1)$_A$ transformation properties of $M$ and $M'$.
\section{Generating equations}
We shall consider, in this paper, tree level predictions for the
Lagrangian of Eq.(\ref{mixingLsMLag}) in which the only
U(1)$_A$ violating term in $V_0$ is that of
Eq.(\ref{etaprmass}).
The only term in $V_{SB}$ will be taken to be the simplest one given
in Eq. (\ref{vsb}). In this minimal picture, there is no symmetry breaking
associated with the $qq{\bar q}{\bar q}$ fields in $M'$. The symmetry
breaking in the physical states (which contain two
quark as well as four quark components)
is due to the mixing terms which, as we have
already seen in Eq. (\ref{invtmixing}),
can be invariant under SU(3)$\times$ SU(3)$\times$
U(1)$_A$.
The method of treatment, as used earlier \cite{SU1} to discuss the model
containing only the field $M$, is based on two
generating equations which reflect the invariance of $V_0$ under
vector and axial vector transformations. Differentiating them once,
relates two point vertices (masses) with one point vertices.
Differentiating them twice relates three point vertices
(trilinear couplings) with masses and so on. These are essentially
tree level Ward identities.
Under the infinitesimal vector and axial vector transformations
we have,
\begin{eqnarray}
\delta_VV_0&=&{\{} {\rm Tr}\, (\frac{\partial
V_0}{\partial \phi}\delta_V \phi
+\frac{\partial V_0}{\partial S}\delta_VS)
+(\phi,S)\rightarrow(\phi',S'){\}} =0 ,
\nonumber \\
\delta_AV_0&=& {\{}{\rm Tr}\, (\frac{\partial
V_0}{\partial \phi}\delta_A \phi
+\frac{\partial V_0}{\partial S}\delta_AS)
+(\phi,S)\rightarrow(\phi',S'){\}} =- {\cal L}_\eta ,
\label{V0invariance}
\end{eqnarray}
wherein the non-zero value of the axial variation equation
reflects the presence in $V_0$ of the single $U(1)_A$ non-invariant
term of Eq. (\ref{etaprmass}). Using Eqs. (\ref{inftrans}) and (\ref{inftranspr})
as well as the arbitrariness of the variations $E_V$ and $E_A$ yields
the matrix
generating equations,
\begin{eqnarray}
&&{\{}[\phi,\frac{\partial V_0}{\partial \phi}]+
[S,\frac{\partial V_0}{\partial S}] + (\phi,S)\rightarrow(\phi',S'){\}}
=0,
\nonumber \\
&&{\{}[\phi,\frac{\partial V_0}{\partial S}]_+ -
[S,\frac{\partial V_0}{\partial \phi}]_+ +
(\phi,S)\rightarrow(\phi',S'){\}}=
1[2 {\rm Tr}\, (\phi'\frac{\partial V_0}{\partial S'}-
S'\frac{\partial V_0}{\partial \phi'}) -
8c_3 i\, {\rm ln}\, (\frac{{\rm det}\, M}{{\rm det}\,
M^{\dagger}})],
\label{geneqs}
\end{eqnarray}
where, in addition, the form of Eq. (\ref{etaprmass}) was used.
To get constraints on the particle masses we will differentiate
these equations once with respect to each of the four matrix fields:
$\phi,\phi',S,S'$ and evaluate the equations in the ground state.
Thus we also need the ``minimum" condition,
\begin{equation}
\langle\frac{\partial V_0}{\partial S}\rangle +
\langle\frac{\partial V_{SB}}{\partial S}\rangle=0,
\quad \quad \langle\frac{\partial V_0}{\partial
S'}\rangle + \langle\frac{\partial V_{SB}}{\partial
S'}\rangle=0. \label{mincond}
\end{equation}
Using our present choice of Eq. (\ref{vsb}) as the only flavor symmetry breaker
and Eq. (\ref{vevs}), this becomes
\begin{equation}
\langle\frac{\partial V_0}{\partial S_a^a}\rangle = 2A_a,
\quad \quad
\langle\frac{\partial V_0}{\partial {S'}_a^a}\rangle = 0.
\label{firstderiv}
\end{equation}
Now let us differentiate successively the vector generating equation
with respect
to $S_a^b$ and to ${S'}_a^b$. This gives with the help of
Eq.(\ref{firstderiv}),
the following two relations:
\begin{eqnarray}
(\alpha_a - \alpha_b)
\langle
{
{\partial^2 V_0} \over {\partial S_b^a \partial S_a^b}
}
\rangle
+ (\beta_a - \beta_b)
\langle
{
{\partial^2 V_0} \over {\partial {S'}_b^a \partial S_a^b}
}
\rangle
&=& 2(A_a-A_b),
\nonumber \\
(\alpha_a - \alpha_b)
\langle
{
{\partial^2 V_0} \over {\partial S_b^a \partial {S'}_a^b}
}
\rangle
+ (\beta_a - \beta_b)
\langle
{
{\partial^2 V_0} \over {\partial {S'}_b^a \partial{S'}_a^b}
}
\rangle
&=& 0 .
\label{scalarmasses}
\end{eqnarray}
The first of these equations relates the mass mixing transition
with the
unprimed scalar squared masses while the second of these relates the
mass mixing transition
with the primed scalar squared masses. It may be seen
that information is obtained only for particles with different
upper and lower SU(3) tensor indices. In the isospin invariant
limit (where $\alpha_1=\alpha_2$ etc.), information will be
obtained only for the kappa type particles (e.g. $\kappa^+=S_1^3$
when mixing is neglected). If isospin violation information is inserted,
information may be obtained also about the isovector scalars
like $a_0^+(980)$ (which is represented by $S_1^2$ when mixing is
neglected).
Next, let us differentiate successively the axial vector generating
equation with respect to $\phi$ and to $\phi'$. It is neater to write
the results first for the case when fields with different upper and lower
tensor indices are involved:
\begin{eqnarray}
(\alpha_a + \alpha_b)
\langle
{
{\partial^2 V_0} \over {\partial \phi_b^a \partial \phi_a^b}
}
\rangle
+ (\beta_a + \beta_b)
\langle
{
{\partial^2 V_0} \over {\partial {\phi'}_b^a \partial \phi_a^b}
}
\rangle
&=& 2(A_a+A_b),
\nonumber \\
(\alpha_a + \alpha_b)
\langle
{
{\partial^2 V_0} \over {\partial {\phi'}_b^a \partial {\phi}_a^b}
}
\rangle
+ (\beta_a + \beta_b)
\langle
{
{\partial^2 V_0} \over {\partial {\phi'}_b^a \partial {\phi'}_a^b}
}
\rangle
&=& 0
\label{offdiagpsmasses}
\end{eqnarray}
Next, let us write the corresponding equations
for the case when the upper and lower tensor indices on each field
are the same.
\begin{eqnarray}
\alpha_b
\langle
{
{\partial^2 V_0} \over {\partial \phi_a^a \partial \phi_b^b}
}
\rangle
+ \beta_b
\langle
{
{\partial^2 V_0} \over {\partial {\phi}_a^a \partial {\phi'}_b^b}
}
\rangle
&=& \sum_g \beta_g
\langle
{
{\partial^2 V_0} \over {\partial {\phi}_a^a \partial {\phi'}_g^g}
}
\rangle
-\frac{8c_3}{\alpha_a},
\nonumber \\
\alpha_b
\langle
{
{\partial^2 V_0} \over {\partial {\phi'}_a^a \partial \phi_b^b}
}
\rangle
+ \beta_b
\langle
{
{\partial^2 V_0} \over {\partial {\phi'}_a^a \partial {\phi'}_b^b}
}
\rangle
&=& \sum_g \beta_g
\langle
{
{\partial^2 V_0} \over {\partial {\phi'}_a^a \partial {\phi'}_g^g}
}
\rangle.
\label{pscalarmasses}
\end{eqnarray}
Note that the axial generating equation provides information on
the masses of all the pseudoscalars. Further differentiations
will relate a large number of trilinear and quadrilinear
coupling constants to the meson masses and to the quark mass
coefficients, $A_a$.
To fully characterize the system we will also require some
knowledge of the axial vector and vector currents \cite{SU1} obtained by
Noether's method:
\begin{eqnarray}
(J_\mu^{axial})_a^b &=&(\alpha_a+\alpha_b)\partial_\mu\phi_a^b +
(\beta_a+\beta_b)\partial_\mu{\phi'}_a^b+ \cdots,
\nonumber \\
(J_\mu^{vector})_a^b &=&i(\alpha_a-\alpha_b){\partial_\mu} S_a^b +
i(\beta_a-\beta_b)\partial_\mu {S'}_a^b+ \cdots,
\label{currents}
\end{eqnarray}
where the three dots stand for terms bilinear in the fields.
\section{Predictions for mass spectrum}
Here we consider the predictions for the mass
spectrum of the model with the Lagrangian given in
Eq. (\ref{mixingLsMLag}),
whose potential contains any SU(3)$_L\times$
SU(3)$_R\times$ U(1)$_A$
invariant terms whatsoever, amended with the
SU(3)$_L\times$ SU(3)$_R$
but not $U(1)_A$ invariant term of Eq. (\ref{etaprmass})
as well as the term, Eq.(\ref{vsb})
which transforms exactly like the QCD quark mass term. A
characteristic feature is mixing between fields with the same quantum
numbers. Specifically, there is information about mixing between
$\pi$ and $\pi'$, between $K$ and $K'$, between $\kappa$
and $\kappa'$ and among among the four $\eta$ type (isosinglet)
states.
We will take these up in turn. Note that we will be
working
in the isotopic spin invariant limit \cite{ispinviolation}.
\subsection{The $\pi-\pi'$ system}
For compactness let us denote,
\begin{eqnarray}
x_\pi &=& \frac{2A_1}{\alpha_1},
\nonumber \\
y_\pi &=&\langle \frac{\partial^2V}{\partial
{\phi'}_2^1\partial{\phi'}_1^2}
\rangle,
\nonumber \\
z_\pi&=& \frac{\beta_1}{\alpha_1}.
\label{xyzpi}
\end{eqnarray}
Here we have introduced the total potential $V=V_0+V_{SB}$.
However, since the second derivatives of $V_{SB}$ vanish
with our present choice of flavor symmetry breaker
we may just use $V_0$. Substituting $a=1, b=2$ into
both of Eqs. (\ref{offdiagpsmasses}) enables us to
write the (non-diagonal) matrix of squared $\pi$
and $\pi'$ masses as:
\begin{equation}
(M_\pi^2)=\left[ \begin{array}{c c}
x_\pi +z_\pi^2 y_\pi & -z_\pi y_\pi
\nonumber \\
-z_\pi y_\pi & y_\pi
\end{array} \right] .
\label{Mpi}
\end{equation}
It is clear that $z_\pi$ is a measure of the mixing
between $\pi$ and $\pi'$ since the matrix becomes diagonal
in the limit when $z_\pi$ is set to zero. So we see that
$x_\pi$ would be the squared pion mass in the single M
model and $y_\pi$ represents the squared mass of
the ``bare" $\pi'$. Denoting the eigenvalues of this
matrix by $m_\pi^2$ and $m_{\pi'}^2$, we read off the product
and sum rules:
\begin{eqnarray}
m_\pi^2m_{\pi'}^2=x_\pi y_\pi,
\nonumber \\
m_\pi^2 + m_{\pi'}^2= x_\pi + y_\pi (1+z_{\pi}^2).
\label{sumrules}
\end{eqnarray}
Assuming that the values of $m_\pi$ and $m_{\pi'}$ are known,
the first of these equations expresses $y_\pi$ in terms of
$x_\pi$. Then the second of these equations
also expresses $z_{\pi}^2$ in terms of $x_\pi$.
The value of $x_\pi$ is not known but its range
is restricted to be,
\begin{equation}
m_\pi^2 \leq x_\pi \leq m_{\pi'}^2.
\label{xrange}
\end{equation}
This range may be derived by expressing
$z_\pi^2$ in terms of $x_\pi$ as mentioned and
requiring $z_\pi^2\geq0$.
The transformation between the diagonal
fields (say $\pi^+$ and $\pi'^+$) and the
original pion fields is defined as:
\begin{equation}
\left[
\begin{array}{c} \pi^+ \\
\pi'^+
\end{array}
\right]
=
\left[
\begin{array}{c c}
{\rm cos}\, \theta_\pi & -{\rm sin} \,
\theta_\pi
\nonumber \\
{\rm sin} \, \theta_\pi & {\rm cos} \,
\theta_\pi
\end{array}
\right]
\left[
\begin{array}{c}
\phi_1^2 \\
{\phi'}_1^2
\end{array}
\right].
\label{mixingangle}
\end{equation}
The explicit diagonalization gives an expression for the
mixing angle $\theta_\pi$:
\begin{equation}
{\rm tan}\, (2\theta_\pi)=\frac{-2y_\pi
z_\pi}{y_\pi(1-z_\pi^2)-x_\pi},
\label{thetasubpi}
\end{equation}
which evidently is also known, up to a sign choice for
$z_\pi$, once $x_\pi$ is specified.
The mixing angle, $\theta_\pi$ can also be connected to the
experimentally known value of the pion decay constant (i.e.
the amplitude for the $\pi^+$ meson to decay to two leptons).
Substituting the expressions from Eq. (\ref{mixingangle}) for $\phi_1^2$
and ${\phi'}_1^2$ in terms of the physical fields $\pi^+$
and $\pi'^+$ into Eq. (\ref{currents}) yields,
\begin{eqnarray}
(J_\mu^{axial})_1^2 &=&F_\pi\partial_\mu \pi^+ + F_{\pi'}\partial_\mu
\pi'^+
+\cdots,
\nonumber \\
F_\pi &=&(\alpha_1+\alpha_2){\rm cos}\, \theta_\pi -
(\beta_1+\beta_2){\rm sin}\, \theta_\pi,
\nonumber \\
F_{\pi'} &=&(\alpha_1+\alpha_2){\rm sin}\, \theta_\pi +
(\beta_1+\beta_2){\rm cos}\, \theta_\pi.
\label{Fpis}
\end{eqnarray}
We can then obtain $\alpha_1$ (in the isospin invariant limit) as,
\begin{equation}
\alpha_1=\frac{F_\pi}{2({\rm cos}\, \theta_\pi-z_\pi
{\rm sin}\, \theta_\pi)}.
\label{alpha1fromF}
\end{equation}
We then successively obtain $A_1$ from the definition of $x_\pi$,
Eq. (\ref{xyzpi}) and $\beta_1$ from the definition of $z_\pi$,
Eq, (\ref{xyzpi}).
To sum up, specifying $x_\pi$ and the experimental quantities
$m_\pi, m_{\pi'}$ and $F_\pi$ determines all the other parameters of the
$\pi-\pi'$ system.
\subsection{The $K-K'$ system}
The treatment of this system is almost exactly analogous to that
of the $\pi-\pi'$ system above when one defines the
analogous variables,
\begin{eqnarray}
x_K &=& \frac{2(A_3+A_1)}{\alpha_3+\alpha_1},
\nonumber \\
y_K &=&\langle \frac{\partial^2V}{\partial{\phi'}_3^1\partial{\phi'}_1^3}
\rangle,
\nonumber \\
z_K&=& \frac{\beta_3+\beta_1}{\alpha_3+\alpha_1}.
\label{xyK}
\end{eqnarray}
Substituting $a=1, b=3$ into
both of Eqs. (\ref{offdiagpsmasses}) enables us to
write the (non-diagonal) matrix of squared $K$
and $K'$ masses as:
\begin{equation}
(M_K^2)=\left[ \begin{array}{c c}
x_K +z_{K}^2 y_K & -z_K y_K
\nonumber \\
-z_K y_K & y_K
\end{array} \right] .
\label{MK}
\end{equation}
This is observed to be identical to the expression for $(M_{\pi}^2)$
in Eq. (\ref{Mpi}) when one simply substitutes everywhere $K$ for $\pi$
and $K'$ for $\pi'$. Similarly, the four equations (\ref{sumrules}),
(\ref{xrange}), (\ref{mixingangle}) and (\ref{thetasubpi}) continue to
hold when one substitutes everywhere $K$ for $\pi$ and $K'$ for $\pi'$.
Similarly, the $K^+$ decay constant, $F_K$ is now defined from,
\begin{eqnarray}
(J_\mu^{axial})_1^3 &=&F_K\partial_\mu K^+ + F_{K'}\partial_\mu
K'^+
+\cdots,
\nonumber \\
F_K &=&(\alpha_1+\alpha_3){\rm cos}\, \theta_K -
(\beta_1+\beta_3){\rm sin}\, \theta_K,
\nonumber \\
F_{K'} &=&(\alpha_1+\alpha_3){\rm sin}\, \theta_K +
(\beta_1+\beta_3){\rm cos}\, \theta_K.
\label{FKs}.
\end{eqnarray}
We can then obtain $\alpha_3 +\alpha_1$ (in the isospin invariant limit)
as,
\begin{equation}
\alpha_3+\alpha_1=\frac{F_K}{{\rm cos}\, \theta_K-z_K
{\rm sin}\, \theta_K}.
\label{strangealph}
\end{equation}
We then successively obtain $A_3+A_1$ from the definition of $x_K$
and $\beta_3 + \beta_1$ from the definition of $z_K$.
To sum up, specifying $x_K$ and the experimental quantities
$m_K, m_{K'}$ and $F_K$ determines all the other parameters of the
$K-K'$ system.
\subsection{The $\kappa-\kappa'$ system}
Again, we can treat this system in an exactly analogous way to
the $\pi-\pi'$ and $K-K'$ cases if we define the analogous
quantities:
\begin{eqnarray}
x_\kappa &=& \frac{2(A_3-A_1)}{\alpha_3-\alpha_1},
\nonumber \\
y_\kappa &=&\langle \frac{\partial^2V}{\partial {S'}_3^1\partial {S'}_1^3}
\rangle,
\nonumber \\
z_\kappa &=& \frac{\beta_3-\beta_1}{\alpha_3-\alpha_1}.
\label{xyzkappa}
\end{eqnarray}
In this case, however, the vector generating equations in Eqs.
(\ref{scalarmasses}) with the choices $a=1$ and $b=3$ are used.
The transformation between the diagonal and original strange
scalar fields is given by,
\begin{equation}
\left[
\begin{array}{c} \kappa^+ \\
\kappa'^+
\end{array}
\right]
=
\left[
\begin{array}{c c}
{\rm cos} \, \theta_\kappa & -{\rm sin}\,
\theta_\kappa
\nonumber \\
{\rm sin}\, \theta_\kappa & {\rm cos}\,
\theta_\kappa
\end{array}
\right]
\left[
\begin{array}{c}
S_1^3 \\
{S'}_1^3
\end{array}
\right],
\label{kappamixingangle}
\end{equation}
where the mixing angle is determined by the diagonalization:
\begin{equation}
tan(2\theta_\kappa)=\frac{-2y_\kappa z_\kappa}
{y_\kappa(1-z_\kappa^2)-x_\kappa}.
\label{thetasubkappa}
\end{equation}
We may define $\kappa$ ``decay constants' as,
\begin{eqnarray}
F_\kappa &=&(\alpha_3-\alpha_1){\rm cos}\, \theta_\kappa
-
(\beta_3-\beta_1){\rm sin}\, \theta_\kappa,
\nonumber \\
F_{\kappa'} &=&(\alpha_3-\alpha_1) {\rm
sin}\, \theta_\kappa +
(\beta_3-\beta_1){\rm cos}\, \theta_\kappa,
\label{Fkappas}
\end{eqnarray}
although there is no direct experimental information
available about them.
Now let us consider the $\pi-\pi'$, $K-K'$ and
$\kappa-\kappa'$ systems together. Using the first two
we can get all of $A_1, A_3, \alpha_1,\alpha_3, \beta_1,
\beta_3$ from the experimental masses of $\pi,\pi',K,K'$,
the experimental decay constants $F_\pi,F_K$
and the assumed values of $x_\pi$ and $x_K$, as seen above.
This means that $x_\kappa$ and $z_\kappa$ may be read off
directly from Eqs. (\ref{xyzkappa}) while $y_\kappa$
can be found from the
product rule $m_\kappa^2m_{\kappa'}^2=x_\kappa y_\kappa$
if $m_\kappa$ and $m_{\kappa'}$ are furnished. Thus all
the parameters of the $\kappa-\kappa'$ system are known,
given the input masses and the values of $x_\pi$ and
$x_K$. However we have not yet made use of the sum rule
analogous to the second of Eqs. (\ref{sumrules}). This
provides another way to calculate $z_\kappa$ so we get the consistency
condition:
\begin{equation}
(\frac{\beta_3-\beta_1}{\alpha_3-\alpha_1})^2= \frac{x_\kappa(m_\kappa^2
+m_{\kappa'}^2-x_\kappa)}{m_\kappa^2m_{\kappa'}^2} -1
\label{consistentcond}
\end{equation}
Since the quantities in this equation depend on both $x_\pi$ and $x_K$,
the solution can determine the value of $x_K$ for each choice of $x_\pi$.
In other words, if $x_\pi$
is specified, the parameters of the $\pi-\pi'$, the $K-K'$ and
the $\kappa-\kappa'$ systems are all determined in the present model.
\subsection{The $\eta$ system}
This system is more complicated because, even in the
isotopic spin invariant limit, there are four different $I=0$
pseudoscalars which can mix with each other. These may be put together
as a column vector according to,
\begin{equation}
\Phi_0 =
\left[
\begin{array}{c}
\frac{\phi_1^1+\phi_2^2}{\sqrt{2}} \\
\phi^3_3\\
\frac{{\phi'}_1^1+{\phi'}_2^2}{\sqrt{2}} \\
{\phi'}^3_3
\end{array}
\right].
\label{etabasis}
\end{equation}
The part of the Lagrangian describing the masses of the $I=0$
pseudoscalars is then:
${\cal L} = -(1/2) \Phi_0^T (M^2_\eta) \Phi_0$, where $(M^2_\eta)$
is a symmetric $4\times4$ matrix. Relations among the matrix elements
follow by using both of Eqs. (\ref{pscalarmasses}). These connect
the transition masses both to the ``bare" unprimed particle masses
and to the ``bare" primed particle masses. The use of isospin
invariance relations like the ones given in Appendix B may also be useful.
Eventually, the matrix elements of $(M^2_\eta)$ depend on
four new quantities in addition to the ones appearing in the above
three subsystems. The resulting matrix elements are listed below:
\begin{eqnarray}
\left( M^2_\eta \right)_{11} &=&
{ {2 A_1} \over \alpha_1} - { {16 c_3}\over \alpha_1^2 }
- { {\beta_1^2 m_\pi^2 m_{\pi'}^2} \over {2 A_1
\alpha_1} }
+ 2 \left( {\beta_1 \over \alpha_1 } \right)^2
\langle { {\partial^2 { V}}
\over
{(\partial {\phi'}^1_1)^2}
}
\rangle
+ 4 \left( { {\beta_1\beta_3} \over {\alpha_1^2} }
\right)
\langle { {\partial^2 { V} }
\over
{\partial {\phi'}^1_1 \partial {\phi'}^3_3}
}
\rangle
\nonumber \\
&&+ 2 \left( {\beta_3\over \alpha_1} \right)^2
\langle { {\partial^2 { V} }
\over
{\partial {\phi'}^3_3 \partial
{\phi'}^3_3}
}
\rangle
\nonumber \\
\left( M^2_\eta \right)_{12} &=&
- { {8 \sqrt{2} c_3} \over {\alpha_1 \alpha_3} }
- { {\beta_1^2 m_\pi^2 m_{\pi'}^2} \over {\sqrt{2} A_1
\alpha_3} }
+ \left(
{ {2\sqrt{2} \beta_1^2} \over {\alpha_1 \alpha_3} }
\right)
\langle { {\partial^2 { V} }
\over
{(\partial {\phi'}^1_1)^2}
}
\rangle
+ \left(
{ {2\sqrt{2}\beta_1\beta_3}
\over
{\alpha_1\alpha_3} }
\right)
\langle { {\partial^2 { V} }
\over
{\partial {\phi'}^1_1 \partial {\phi'}^3_3}
}
\rangle
\nonumber \\
\left( M^2_\eta \right)_{13} &=&
- { {\beta_1 m_\pi^2 m_{\pi'}^2} \over {2 A_1 } }
+ 2 \left( {\beta_1 \over \alpha_1 } \right)
\langle { {\partial^2 { V} }
\over
{(\partial {\phi'}^1_1)^2}
}
\rangle
+ 2 \left( { {\beta_3} \over {\alpha_1} }
\right)
\langle { {\partial^2 { V} }
\over
{\partial {\phi'}^1_1 \partial {\phi'}^3_3}
}
\rangle
\nonumber \\
\left( M^2_\eta \right)_{14} &=&
\left( {\sqrt{2}\beta_1 \over \alpha_1 } \right)
\langle { {\partial^2 { V} }
\over
{\partial {\phi'}^1_1 \partial {\phi'}^3_3}
}
\rangle
+ \left( { {\sqrt{2}\beta_3} \over {\alpha_1} }
\right)
\langle { {\partial^2 { V} }
\over
{\partial {\phi'}^3_3 \partial
{\phi'}^3_3}
}
\rangle
\nonumber \\
\left( M^2_\eta \right)_{22} &=&
{ {2 A_3} \over \alpha_3} - { {8 c_3}\over \alpha_3^2 }
- { {\alpha_1\beta_1^2 m_\pi^2 m_{\pi'}^2} \over {A_1
\alpha_3^2} }
+ 4 \left( {\beta_1 \over \alpha_3 } \right)^2
\langle { {\partial^2 { V} }
\over
{(\partial {\phi'}^1_1)^2}
}
\rangle
\nonumber \\
\left( M^2_\eta \right)_{23} &=&
- { {\alpha_1\beta_1 m_\pi^2 m_{\pi'}^2} \over {\sqrt{2}
A_1 \alpha_3} }
+ \left(
{ {2\sqrt{2} \beta_1} \over {\alpha_3} }
\right)
\langle { {\partial^2 { V} }
\over
{(\partial {\phi'}^1_1)^2}
}
\rangle
\nonumber\\
\left( M^2_\eta \right)_{24} &=&
\left( { {2\beta_1} \over {\alpha_3} }
\right)
\langle { {\partial^2 { V} }
\over
{\partial {\phi'}^1_1 \partial {\phi'}^3_3}
}
\rangle
\nonumber \\
\left( M^2_\eta \right)_{33} &=&
- { {\alpha_1 m_\pi^2 m_{\pi'}^2} \over {2
A_1} }
+ 2
\langle { {\partial^2 { V} }
\over
{(\partial {\phi'}^1_1)^2}
}
\rangle
\nonumber \\
\left( M^2_\eta \right)_{34} &=&
\sqrt{2}
\langle { {\partial^2 { V} }
\over
{\partial {\phi'}^1_1 \partial {\phi'}^3_3}
}
\rangle
\nonumber \\
\left( M^2_\eta \right)_{44} &=&
\langle { {\partial^2 { V} }
\over
{\partial {\phi'}^3_3 \partial
{\phi'}^3_3}
}
\rangle
\label{big4by4}
\end{eqnarray}
The four new quantities are $c_3$, discussed earlier, and the
``bare" primed squared masses:
\begin{equation}
\langle { {\partial^2 { V} }
\over
{(\partial {\phi'}^1_1)^2}
}
\rangle, \hskip 1cm
\langle { {\partial^2 { V} }
\over
{\partial {\phi'}^1_1 \partial {\phi'}^3_3}
}
\rangle, \hskip 1cm
\langle { {\partial^2 { V} }
\over
{\partial {\phi'}^3_3 \partial
{\phi'}^3_3}
}
\rangle.
\nonumber \\
\end{equation}
These four quantities may be found by inputing the masses of four
isosinglet pseudoscalars. The net result is that all four systems
discussed will be completely described if all the experimental masses
and the decay constants, $F_\pi,F_K$
are specified together with an assumed value for $x_\pi$.
\section{Comparison with experiment and discussion}
In the preceding section we gave the tree level
formulas resulting
from the $M-M'$ model with any SU(3)$_L \times$ SU(3)$_R
\times$
U(1)$_A$ invariant terms together with
a single ``instanton" type term which mocks up the
U(1)$_A$
anomaly and the simplest structure which mocks up the
quark mass terms. Isotopic spin invariance was also assumed.
Information is provided for only the pseudoscalar nonets
and the strange scalar particles. Information about
the scalar isotriplets can be obtained by including isospin
violation effects while information about the scalar
isosinglets requires either assuming some specific form
for the invariant interaction terms or computing other
physical quantities. These will be discussed elsewhere.
Now we will input the experimental masses to try to
learn what the model has to say about the quark structure
of the various mesons being described. In particular we are
interested in the mixing angles like $\theta_\pi$,
governing admixtures of $q{\bar q}$ and $qq{\bar q}{\bar q}$
in the physical states and the four quark ``condensate"
strengths, $\beta_a$ which are associated with this mixing
in the present model.
The well known lowest pseudoscalar nonet masses and decay
constants will be taken, for definiteness (considering the ambiguity
as to which member of a non trivial isospin multiplet to choose),
to be:
\begin{eqnarray}
m_\pi = 0.137 \, {\rm GeV} &,& \, m_K = 0.496 \, {\rm
GeV}, \nonumber \\
m_\eta = 0.548 \, {\rm GeV}&,& \, m_{\eta^\prime} =
0.958 \, {\rm GeV},
\nonumber \\
F_\pi = 0.131\, {\rm GeV}&,& \ F_K = 0.160 \, {\rm
GeV} .
\label{inputs}
\end{eqnarray}
Next, let us consider what are the suitable experimental
inputs for the masses of the excited mesons, $\pi',K'.{\kappa}'$
and for the $\kappa$ meson itself. In the latest Review of particle
properties \cite{rpp} there are two dotted (i.e. considered
established) candidates for excited pions below 2 GeV: the $\pi(1300)$
and the $\pi(1800)$. These particles could have four quark
components and/or radially excited two quark components.
In fact, judging
from an investigation of excited baryons \cite{baryons}, it is likely
that both types are present. Clearly, however, for our present
investigation it seems reasonable to
assume that the four quark component is the dominant one
and to
choose the lower mass object
as the more suitable one. Similarly there are two undotted (non
established)
excited kaon candidates: the K(1460) and the K(1830). We will again
choose the lower value. As candidates for an excited strange scalar
there is a dotted $K_0^*(1430)$ and an undotted $K_0^*(1950)$
and we again choose the lower value. In the case of the low mass
strange scalar there is an undotted $K_0^*(800)$ candidate,
which we will interpret, with the help of \cite{BFSS1}, to be closer
to 900 MeV. We summarize these choices:
\begin{eqnarray}
m_{\pi'} = 1.30 \, {\rm GeV} &,& \, m_{K'} = \, 1.46
\, {\rm
GeV}, \nonumber \\
m_\kappa = 0.90 \, {\rm GeV}&,& \, m_{\kappa^\prime} =
1.42 \, {\rm GeV}.
\label{excitedinputs}
\end{eqnarray}
For the excited $\eta$ type pseudoscalar particles the Review
of particle properties lists, below 2 GeV, the possible masses
(all in GeV):
\begin{equation}
1.294,\hskip1cm 1.410, \hskip1cm 1.476, \hskip1cm 1.760.
\label{excitedetas}
\end{equation}
The first three of these are dotted but the fourth is undotted.
Here it seems more difficult to a priori choose which are
most relevant
so we shall study all possible pairings in a systematic way.
First let us discuss the $\pi-\pi'$, $K-K'$ and $\kappa-
\kappa'$ systems. After using the inputs of Eq. (\ref{inputs}),
all features of these systems in our model will,
as already discussed, be determined
by specifying $x_\pi$. Table \ref{systpara}
shows the predicted physical parameters for three values of
$x_\pi$. For orientation we note that in the chiral model with
a single field, M one has
\begin{eqnarray}
\alpha_1 \rightarrow F_{\pi}/2=0.0655 \, {\rm GeV}
&,& \alpha_3 \rightarrow
F_K-\alpha_1=0.0945 \, {\rm GeV},
\nonumber \\
A_1 \rightarrow \frac{\alpha_1}{2}m_\pi^2=6.15
\times 10^{-4} \,
{\rm GeV} ^3&,&
A_3 \rightarrow \frac{F_K}{2}m_K^2=0.01866 \, {\rm
GeV}^3,
\nonumber \\
\beta_1 \rightarrow 0 &,& \beta_2 \rightarrow 0.
\label{oneMlimit}
\end{eqnarray}
The single M model corresponds to the choice $x_\pi=m_\pi^2$.
Increasing $x_\pi$ has the effect of increasing the admixture of
the ``four quark" field component in the physical pion.
The ``quark mass ratio", $A_3/A_1$ = 30.3
in the single M model is not very different
from the value of 31.2 obtained using the values in the $x_\pi=
0.019 \, {\rm Gev}^2$ column. The ${\bar q}q$ meson
condensates
$\alpha_1$ and $\alpha_3$ are also very similar. Of course
the ``four quark" meson condensates $\beta_1$
amd $\beta_2$ are zero without $M'$. Despite the similarities,
the $6.4^o$ mixing angle already corresponds to about an 11
percent ``four quark" admixture in the physical pion
wave function.
Considering that the accuracy of current algebra predictions
for low energy pion physics is roughly ten percent, it
seems that this choice of $x_\pi$ is the most plausible one.
One sees from the second and third columns
that relatively small increases in $x_\pi$ lead to
large increases in four quark admixture for the pion and
the kaon. Interestingly, the behavior of the four quark
admixture in the strange scalar meson $\kappa$ is quite different.
When the pseudoscalars are closer to pure ``two quark" states
in the model the scalar has a large four quark admixture
($34.1^o$, with the choice
of $x_\pi$ in the first column). Thus the result is
consistent with having a fairly large four quark component
in the light scalars.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{c||c|c|c}
\hline \hline
& $x_\pi$=0.019\, (GeV$^2$)&
$x_\pi$=0.021 (GeV$^2$)&
$x_\pi$=0.022 (GeV$^2$)
\\
\hline
\hline
$\theta_\pi$ (deg.) & $ - 6.4$
& $- 19.1$
& $- 22.7$
\\
\hline
$\theta_k$ (deg.)&
$- 11.2$ &
$- 22.9$ &
$- 26.2$
\\
\hline
$\theta_\kappa$ (deg.) &
$34.1$ &
$28.1$ &
$26.5$
\\
\hline
$A_1 ({\rm GeV}^3)$ &
$6.19 \times 10^{-4}$ &
$6.51 \times 10^{-4}$ &
$6.66 \times 10^{-4}$
\\
\hline
$A_3 ({\rm GeV}^3)$ &
$1.94 \times 10^{-2}$ &
$2.07 \times 10^{-2}$ &
$2.12 \times 10^{-2}$
\\
\hline
$\alpha_1$ (GeV) &
$6.51 \times 10^{-2}$ &
$6.20 \times 10^{-2}$ &
$6.06 \times 10^{-2}$
\\
\hline
$\alpha_3$ (GeV)&
$9.24 \times 10^{-2}$ &
$8.83 \times 10^{-2}$ &
$8.69 \times 10^{-2}$
\\
\hline
$\beta_1$ (GeV) &
$7.18 \times 10^{-3}$ &
$2.12 \times 10^{-2}$ &
$2.50 \times 10^{-2}$
\\
\hline
$\beta_3$ (GeV)&
$2.03 \times 10^{-2}$ &
$3.38 \times 10^{-2}$ &
$3.74 \times 10^{-2}$
\\
\hline
\hline
\end{tabular}
\end{center}
\caption[]{$\theta_\pi, \theta_K$ and $\theta_\kappa$ are
respectively the ``four quark" admixtures in the $\pi, K$
and $\kappa$ states. $A_1, A_3$ represent the quark mass
parameters while $\alpha_1, \alpha_3$ and $\beta_1, \beta_3$
represent respectively the two and four quark condensate
strengths. These are plotted as functions of the assumed
``bare" pion squared mass, $x_\pi$.
}
\label{systpara}
\end{table}
The analogs of the two quark condensates $\alpha_1=\alpha_2$ and
$\alpha_3$ are approximately equal, in agreement with the
usual assumption that
the vacuum is approximately SU(3) symmetric. The analogs of the
four quark condensates in this model are roughly an order of magnitude
smaller than the similarly normalized two quark condensates. They are
furthermore seen to deviate appreciably from SU(3) symmetry.
It should be noted, as discussed in ref, \cite{BFSS2} for example,
that the tensor indices for the primed mesons really correspond
to ``dual quark" or diquark indices in accordance with,
\begin{equation}
Q_a \sim \epsilon_{abc}{\bar q}^b{\bar q}^c.
\label{dualquark}
\end{equation}
Thus in terms of the usual quarks,
\begin{equation}
\beta_1 \sim \langle{\bar d}d{\bar s}s\rangle,\hskip1cm
\beta_2 \sim \langle{\bar u}u{\bar s}s\rangle,
\hskip1cm \beta_1 \sim \langle{\bar d}d{\bar u}u\rangle.
\label{interpretbetas}
\end{equation}
Now consider the mixing of the four $\eta$ type
fields in the model. The basis is given in Eq.(\ref{etabasis})
while the elements of the $4\times 4$ mass squared matrix
are given in Eq.(\ref{big4by4}).
The orthogonal transformation matrix, $K$ which relates
the mass eigenstate fields, $\Phi$ to the original ones is defined by
\begin{equation}
\Phi_0=K \Phi.
\label{diagonalize}
\end{equation}
As discussed in the previous section, there are, after using
the symmetry information, four new unknown parameters characterizing
the $\eta$ system. Thus taking the four mass eigenvalues
from experiment could in principle determine, together with results
from the $\pi-\pi'$, $K=K'$ and $\kappa-\kappa'$ systems,
everything about the $\eta$ system for a given value of $x_\pi$.
However there is no guarantee that there will be an exact solution
for all choices of experimental parameters. This is the case,
in fact, so we will search numerically for a
choice of ``theoretical" masses which will best fit the
experimental inputs. The criterion for goodness of fit
will be taken to be the smallness of the quantity:
\begin{equation}
\chi \equiv \sum_{i} |m_i^{\rm exp.} - m_i^{\rm theo.}|\, / \,m_i^{\rm
exp.}.
\label{chi}
\end{equation}
As shown in Eq. (\ref{excitedetas}), there are three established
candidates and one
not yet established candidate below 2 GeV for the two excited
$\eta$ states. This yields six possible scenarios for choosing
them. The quantity $\chi$ for each choice is shown in
Table \ref{etascenarios} for three values of the parameter
$x_\pi$. It may be observed that the fits typically get worse
with increasing $x_\pi$, so it is reasonable to consider
the choice 0.019 GeV${}^2$ for this quantity as we did
previously. The smallest values of $\chi$ are found for
scenarios 5 and 6. However these both involve the $\eta(1760)$
state which is the one not yet established. The smallest
value of $\chi$ using only established states
is scenario 2. This case corresponds to an exact fit
with eta type masses in GeV (experimental values in
parentheses for comparison):
\begin{eqnarray}
&& 0.533 (0.548), \hskip2cm 0.963 (0.958),
\nonumber \\
&& 1.327 (1.294), \hskip2cm 1.716 (1.476).
\label{exactetas}
\end{eqnarray}
\begin{table}[htbp]
\begin{center}
\begin{tabular}{c||c|c|c}
\hline \hline
Scenario &
$x_\pi$=0.019 (GeV$^2$)&
$x_\pi$=0.021 (GeV$^2$)&
$x_\pi$=0.022 (GeV$^2$)
\\
\hline
\hline
1:${\{}\eta(1295), \eta(1405){\}}$ &
$6.23 \times 10^{-2}$ &
$3.99 \times 10^{-1}$ &
$5.08 \times 10^{-1}$
\\ \hline
2:${\{}\eta(1295), \eta(1475){\}}$ &
$2.85 \times 10^{-2}$ &
$3.39 \times 10^{-1}$ &
$4.44 \times 10^{-1}$
\\
\hline
3:${\{}\eta(1295), \eta(1760){\}}$ &
$2.35 \times 10^{-2}$ &
$1.37 \times 10^{-1}$ &
$2.28 \times 10^{-1}$
\\
\hline
4:${\{}\eta(1405), \eta(1475){\}}$ &
$8.28 \times 10^{-2}$ &
$3.63 \times 10^{-1}$ &
$4.49 \times 10^{-1}$
\\
\hline
5:${\{}\eta(1405), \eta(1760){\}}$ &
$1.50 \times 10^{-2}$ &
$1.62 \times 10^{-1}$ &
$2.38 \times 10^{-1}$
\\
\hline
6:${\{}\eta(1475), \eta(1760){\}}$ &
$2.84 \times 10^{-2}$ &
$1.78 \times 10^{-1}$ &
$2.68 \times 10^{-1}$
\\
\hline
\hline
\end{tabular}
\end{center}
\caption[]{A goodness of fit quantity,
$\chi \equiv \sum_{i} |m_i^{\rm exp.} - m_i^{\rm theo.}|\, / \,m_i^{\rm
exp.}$, where the $m_i$ are the four mass eigenvalues of the $\eta$ type
fields, is given for 6 possible scenarios and for three values of
$x_\pi$. Each scenario corresponds to a choice of $\eta$ type fields
including the $\eta(548)$ and the $\eta(958)$ as well as the two
listed in the left hand column.}
\label{etascenarios}
\end{table}
The detailed content of all the $\eta$ mass eigenstates
can be read off from the matrix $K^{-1}$. For scenario 2
we have,
\begin{equation}
K^{-1}=\left[ \begin{array}{c c c c}
-0.570 & 0.750 & -0.023 & 0.333
\nonumber \\
-0.329 & -0.573 & 0.142 & 0.737
\nonumber \\
0.704 & 0.267 & -0.309 & 0.581
\nonumber \\
0.267 & 0.192 & 0.940 & 0.088
\end{array} \right] .
\label{Kinverse}
\end{equation}
Thus, in the present model there is an 89 percent
probability ($(K^{-1})_{11}^2 +(K^{-1})_{12}^2$) that
the $\eta(548)$ is a quark-antiquark state and
an eleven percent probability that it is a four quark
state.
As expected, the $\eta(548)$ is most likely to be
in an ${\bar s}s$ state. In the case of the $\eta(958)$,
there is a 44 percent probability for it to be in a quark
antiquark state. There is a 54 percent probability for
it to be in the four quark state ${\phi'}_3^3$. This situation has
some plausibility since in terms of ordinary quarks, the
latter state has the content ${\bar u}u{\bar d}d$ and
it should be most energetically favorable to bind a four
quark state made without strange quarks.
The other scenarios which don't employ the
unconfirmed $\eta(1760)$ state (numbers
1 and 4) have contents very similar to the one in
Eq. (\ref{Kinverse}). On the other hand the three
scenarios employing the $\eta(1760)$ have a
rather different content, which seems unusual:
scenarios 3, 5 and 6 make the $\eta(958)$ almost
completely ${\phi'}^3_3$.
In scenario 2, which seems the most reasonable choice,
we notice that the $\eta(1295)$ has a 43 percent
probability
of being in a four quark state while the $\eta(1475)$ has
an 89 percent probability of being in a four quark state.
To sum up, the value $x_\pi$ =0.019 GeV$^2$ leads to
fairly small four quark content in the light pseudoscalars-
$\pi, K, \eta$ at the same time that the light scalar $\kappa$
has an appreciable four quark component. The ``excited"
$\eta$'s are predominantly four quark states. The $\eta(960)$
is mainly two quark in content but has a non trivial four quark
piece.
The results obtained here provide supporting evidence for the
feature, illustrated in the first treatment of this
model \cite{BFMNS01},
that the lightest scalars, unlike the lightest pseudoscalars,
have appreciable four quark components. That model neglected
quark masses and used the simplified choice of terms shown
in Eq. (\ref{mixingpot}).
The more recent treatment of ref. \cite{nr},
includes two additional invariant terms
beyond those in Eq. (\ref{mixingpot})
(although not all the renormalizable terms
shown in Appendix A) as well as four
types of quark mass splitting terms.
Our results for the present treatment, where quark
masses are included and which holds for any
possible SU(3)$_L \times$
SU(3)$_R \times$
U(1)$_A$ conserving terms, are also in
qualitative agreement for the $\pi$-type, K-type, $\eta$-type
and $\kappa$-type states with that treatment.
Roughly, this may be expected
since the present approach includes any choice of invariant terms.
However, we only used here the single quark mass splitting
term of Eq. (\ref{vsb}). Thus the results seem
qualitatively robust with respect to the treatment of the
mass splittings.
An interesting feature of our model is the presence of
``four quark" condensates as signaled by the non-zero
values of the $\beta_a$. To make a rough estimate of what this
corresponds to in quark language we proceed as follows.
In ref. \cite{MS} it was pointed out that the mass formulas
of the single M linear sigma model could be transformed to the
``current algebra" ones \cite{ca} by the replacements:
\begin{equation}
A_a=m_a \Lambda^2, \hskip1cm \alpha_a=-\frac{\langle{\bar
q}_a
q_a\rangle}
{2\Lambda^2},
\label{2vev}
\end{equation}
where the $m_a$ are the (``current" type) quark masses and
$\Lambda$ is the QCD scale factor. Taking $A_1$ = $6.19
\times
10^{-4}\, {\rm GeV}^2$ from the left column of Table
\ref{systpara} and $m_1
\approx$ 5 MeV we get $\Lambda \approx 0.35$ GeV (and
$\langle{\bar q}_a q_a\rangle
\approx -0.016 \, {\rm GeV}^3$). In the case of the four
quark condensate,
as one sees from the discussion in the Introduction, there are several
ways to couple the four quarks together to make scalars. We
are assuming that one such way has been selected. For that case,
it is reasonable to expect, on dimensional grounds, that
\begin{equation}
|\langle{\bar d} d {\bar s}s\rangle| \sim
\Lambda^5
\beta_1 \approx 4 \times 10^{-5}\,
{\rm GeV}^6.
\label{4vev}
\end{equation}
In comparing the scalar masses with experiment
there are expected to be, as discussed in the first four sections
of ref. \cite{BFMNS01}, non-negligible corrections
due to the use of unitary models for the pseudoscalar-
pseudoscalar scattering based on this
Lagrangian. We plan to report on this elsewhere. This should also enable
us to study the isosinglet scalar masses. For both isosinglet scalars
and pseudoscalars, the inclusion of possible glueball states is another
interesting topic we plan to pursue. The additional symmetry
breaking terms like those in Eqs. (\ref{altanom})
and (\ref{sbmixer}) seem also
to be worth investigating.
\section*{Acknowledgments}
\vskip -.5cm
We are happy to thank A. Abdel-Rehim, D. Black, M. Harada,
S. Moussa, S. Nasri and F. Sannino for many helpful
related discussions.
The work of A.H.F. has been supported by the 2004
Crouse Grant from the School of Arts and Sciences, SUNY
Institute of Technolgy.
The work of R.J. and J.S. is supported in part by the U. S. DOE under
Contract no. DE-FG-02-85ER 40231.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,618
|
using FluentValidation;
using SOURCE.BackOffice.Web.ViewModels.Pages.Document;
using System;
namespace SOURCE.BackOffice.Web.ViewModels.Validators.Document
{
public class PieceJointeViewModelValidator : AbstractValidator<PieceJointeViewModel>
{
public PieceJointeViewModelValidator()
{
RuleFor(x => x.Libelle)
.Must(LinkLibelleNotEmpty)
.WithMessage("UPLOAD_VALIDATION_LIBELLE_OBLIGATOIRE");
RuleFor(x => x.Url)
.Must(LinkUrlNotEmpty)
.WithMessage("UPLOAD_VALIDATION_URL_OBLIGATOIRE");
}
private bool LinkLibelleNotEmpty(PieceJointeViewModel instance, string Libelle)
{
return instance.IsFile || !string.IsNullOrWhiteSpace(Libelle);
}
private bool LinkUrlNotEmpty(PieceJointeViewModel instance, string Url)
{
return instance.IsFile || !string.IsNullOrWhiteSpace(Url);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 259
|
General Electric Theater est une série d'anthologie américaine présentée par Ronald Reagan et diffusée à la radio et à la télévision sur CBS entre le et le . La série était sponsorisée par le département des relations publiques de General Electric.
Radio
Après une audition le intitulé The Token avec Dana Andrews, l'émission de radio remplace durant l'été le The Bing Crosby Program et fait ses débuts sur CBS le , avec Ronald Colman dans Random Harvest. Avec des invités comme Cary Grant, Irene Dunne, Van Johnson, Jane Wyman, William Holden, Alan Young, Dorothy McGuire, John Hodiak, Ann Blyth, James Mason, Joan Fontaine et Judy Garland, la série a continué jusqu'au . Jaime del Valle produit et réalise l'émission, avec Ken Carpenter à la présentation et Wilbur Hatch pour la musique.
Télévision
La version télévisée du programme, produit par MCA-TV/Revue Studios est diffusée tous les dimanches soir du au . Chacun des 200 épisodes diffusés est une adaptation d'un roman, d'une nouvelle, d'une pièce, d'un film ou d'un magazine de fiction. Une exception est cependant faite en 1954 avec l'épisode Music for Christmas, qui présente le directeur de chorale et son groupe The Pennsylvanians qui jouent des musiques de Noël.
Le , Ronald Reagan fait ses débuts en tant que présentateur de l'émission. General Electric a ajouté un présentateur pour assurer une continuité dans le format de série d'anthologie, c'est-à-dire une série où seul le thème fait le lien entre des épisodes indépendants, sans personnages récurrents. La série est présente dans le classement des 30 meilleures audiences de 1953 à 1960 : pour la saison 1953-1954, pour la saison 1954-1955, pour la saison 1955-1956, pour la saison 1956-1957, pour la saison 1957-1958, pour la saison 1958-1959, pour la saison 1959-1960 et pour la saison 1960-1961.
Acteurs
Notes et références
Liens externes
Ronald Reagan
Série télévisée américaine des années 1960
Série télévisée américaine des années 1950
Série télévisée de CBS
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,779
|
use um::sqltypes::{
SQLCHAR, SQLHANDLE, SQLHDBC, SQLHSTMT, SQLHWND, SQLINTEGER, SQLRETURN, SQLSMALLINT, SQLULEN,
SQLUSMALLINT, SQLWCHAR
};
pub const SQL_WCHAR: SQLSMALLINT = -8;
pub const SQL_WVARCHAR: SQLSMALLINT = -9;
pub const SQL_WLONGVARCHAR: SQLSMALLINT = -10;
pub const SQL_C_WCHAR: SQLSMALLINT = SQL_WCHAR;
extern "system" {
pub fn SQLConnectW(
connectionHandle: SQLHDBC,
serverName: *const SQLWCHAR,
nameLength1: SQLSMALLINT,
userName: *const SQLWCHAR,
nameLength2: SQLSMALLINT,
authentication: *const SQLWCHAR,
nameLength3: SQLSMALLINT,
) -> SQLRETURN;
pub fn SQLDescribeColW(
statementHandle: SQLHSTMT,
columnNumber: SQLUSMALLINT,
columnName: *mut SQLWCHAR,
bufferLength: SQLSMALLINT,
nameLength: *mut SQLSMALLINT,
dataType: *mut SQLSMALLINT,
columnSize: *mut SQLULEN,
decimalDigits: *mut SQLSMALLINT,
nullable: *mut SQLSMALLINT,
) -> SQLRETURN;
pub fn SQLExecDirectW(
statementHandle: SQLHSTMT,
statementText: *const SQLWCHAR,
textLength: SQLINTEGER,
) -> SQLRETURN;
pub fn SQLGetDiagRecW(
handleType: SQLSMALLINT,
handle: SQLHANDLE,
recNumber: SQLSMALLINT,
sqlstate: *mut SQLWCHAR,
nativeError: *mut SQLINTEGER,
messageText: *mut SQLWCHAR,
bufferLength: SQLSMALLINT,
textLength: *mut SQLSMALLINT,
) -> SQLRETURN;
pub fn SQLDriverConnectW(
hdbc: SQLHDBC,
hwnd: SQLHWND,
szConnStrIn: *const SQLWCHAR,
cchConnStrIn: SQLSMALLINT,
szConnStrOut: *mut SQLWCHAR,
cchConnStrOutMax: SQLSMALLINT,
pcchConnStrOut: *mut SQLSMALLINT,
fDriverCompletion: SQLUSMALLINT,
) -> SQLRETURN;
pub fn SQLConnectA(
connectionHandle: SQLHDBC,
serverName: *const SQLCHAR,
nameLength1: SQLSMALLINT,
userName: *const SQLCHAR,
nameLength2: SQLSMALLINT,
authentication: *const SQLCHAR,
nameLength3: SQLSMALLINT,
) -> SQLRETURN;
pub fn SQLDescribeColA(
statementHandle: SQLHSTMT,
columnNumber: SQLUSMALLINT,
columnName: *mut SQLCHAR,
bufferLength: SQLSMALLINT,
nameLength: *mut SQLSMALLINT,
dataType: *mut SQLSMALLINT,
columnSize: *mut SQLULEN,
decimalDigits: *mut SQLSMALLINT,
nullable: *mut SQLSMALLINT,
) -> SQLRETURN;
pub fn SQLExecDirectA(
statementHandle: SQLHSTMT,
statementText: *const SQLCHAR,
textLength: SQLINTEGER,
) -> SQLRETURN;
pub fn SQLGetDiagRecA(
handleType: SQLSMALLINT,
handle: SQLHANDLE,
recNumber: SQLSMALLINT,
sqlstate: *mut SQLCHAR,
nativeError: *mut SQLINTEGER,
messageText: *mut SQLCHAR,
bufferLength: SQLSMALLINT,
textLength: *mut SQLSMALLINT,
) -> SQLRETURN;
pub fn SQLDriverConnectA(
hdbc: SQLHDBC,
hwnd: SQLHWND,
szConnStrIn: *const SQLCHAR,
cchConnStrIn: SQLSMALLINT,
szConnStrOut: *mut SQLCHAR,
cchConnStrOutMax: SQLSMALLINT,
pcchConnStrOut: *mut SQLSMALLINT,
fDriverCompletion: SQLUSMALLINT,
) -> SQLRETURN;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 33
|
Anxiety, anxious, nervous, loneliness, fear, panic, PTSD related to vata.
Sadness, sad, depression, couch-potato, home-body related to kapha.
see also majja, mano vaha srotas, brain, mind, sensation, meditation, Ayurvedic psychology, five skandhas, samjna.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,568
|
\section{Introduction}
Since their discovery, neutrinos have been an invaluable probe of physics `beyond the Standard Model' (BSM). Indeed the observation of flavour changes in neutrino oscillations, which require non-zero neutrino masses, is arguably one of the strongest pieces of evidence for new BSM physics. This observation behooves particle physicists to probe, to the greatest degree possible, all aspects of the neutrino sector of the SM.
As neutrino oscillations arise as a result of massive neutrino propagation from the point of production to the point of detection, one may ask if these massive fermions are propagating from one point to another as expected. In this work we aim to develop a coherent framework in which this question may be posed theoretically and answered experimentally.\footnote{Note that very recently Ref.~\cite{Gherghetta:2022ynp} appeared, with broadly similar motivations although very different specific considerations.}
In any free or interacting quantum field theory the propagator (two-point function) for a fermion may be captured by a K\"all\'en-Lehmann representation. Note that this is a non-perturbative representation, not reliant on any perturbative expansion, but only on the basic axioms of quantum mechanics. As a result, this representation may be included in any phenomenological description of neutrino oscillations, whether QM-like or QFT-like, and it will capture any QFT-compatible BSM modifications of neutrino propagation from production to detection, as illustrated schematically in Fig.~\ref{fig:KL}. To this end, in Section~\ref{sec:KLTheory} we generalise the usual neutrino oscillation framework to include the more general K\"all\'en-Lehmann form, propagating this general form all the way through to generalised formul{\ae} for oscillation probabilities of neutrino appearance and disappearance.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{Graphs/feyn.png}
\caption{A schematic Feynman diagram for a general neutrino propagation process. The double lines show the external states at the production and detection vertices. The elliptical blob symbolises the K\"all\'en-Lehmann propagator. In the case of canonical oscillations, this reduces to the usual Feynman propagator. }
\label{fig:KL}
\end{figure}
To illustrate the utility of this approach more concretely we provide two simple BSM scenarios in which the neutrino propagator becomes `broad'. The first is the previously studied `Pseudo-Dirac' neutrino model \cite{Wolfenstein:1981kw,Petcov:1982ya,Bilenky:1983wt,Kobayashi:2000md,deGouvea:2009fp,Anamiati:2017rxw,Anamiati:2019maf,Martinez-Soler:2021unz}, where neutrino masses are effectively Dirac and a small amount of lepton number violation leads to a small splitting of mass eigenstates. A second model generalises this setup further, through a fermionic clockwork-inspired deconstruction of fermions on a circle, to give a broad band of states in lieu of a single neutrino mass eigenstate. The mapping of both models into the K\"all\'en-Lehmann representation of the propagator is developed and their impact on observable neutrino oscillation data is investigated. Motivated by these models and the fact that an almost limitless zoology of models is in principle realisable, we then present a simple `top-hat' phenomenological ansatz which captures the dominant oscillation features of more complete microscopic models that lead to band-like neutrino spectral functions.
Armed with this formalism and utilitarian phenomenological ansatz, we first discuss how observations of neutrinos from distant sources can aide in testing these scenarios in Section~\ref{sec:Decoherence}. Then, in Section~\ref{sec:Experiments} we take the next step to see how well we currently understand BSM effects in neutrino propagation as it pertains to Earth-based experiments. In practical terms we do this by determining how well KamLAND, Daya Bay, T2K and NOvA measurements constrain non-SM contributions to neutrino propagation, finding that KamLAND has been particularly powerful. On the other hand, the upcoming JUNO experiment will provide an even more powerful and complementary probe, breaking flat directions that presently exist. Finally, Section~\ref{sec:Conclusions} offers some concluding remarks.
\section{Neutrino Propagation from K\"all\'en-Lehmann}\label{sec:KLTheory}
There is a broad literature concerning neutrino oscillation amplitudes, ranging from textbook quantum-mechanical derivations to involved quantum field theory treatments. There is little to be gained in repeating these analyses here, thus we focus on the core novel ingredient of this work, wherein the key addition to typical treatments, reviewed for example in \cite{Beuthe:2001rc,Kopp}, is to replace the free neutrino propagator and matrix elements found in QFT treatments
\begin{equation}
\sum_j U^*_{\alpha j} U_{\beta j} \frac{\slashed p+m_j}{p^2-m_j^2+i \epsilon} ~~,
\end{equation}
by the more general K\"all\'en-Lehmann propagator for spin-1/2 particles
\begin{equation}
\label{eq:KL}
G^{\text{KL}}_{\alpha \beta}(p^2) = i \int_0^{\infty} d\mu^2 \frac{\slashed p\tilde{\rho}_{\alpha \beta}(\mu^2)+\rho_{\alpha \beta}(\mu^2)}{p^2 - \mu^2 + i\epsilon} ~~,
\end{equation}
where $\alpha$ and $\beta$ denote the flavour eigenstate at the point of production and detection respectively.
This captures any new-physics effects on the propagation of neutrinos consistent with the axioms of QFT and, essentially, promotes the sum over mass eigenstates to a continuum integral. The $\rho$ functions here encode both the density of states, and the matrix elements describing the overlap between the mass and interaction eigenstates. One recovers the usual Feynman propagator if $\tilde{\rho}(\mu^2)$ and $\rho(\mu^2)/\mu$ are identical and comprise delta functions $\delta(\mu^2-m_j^2)$. More general scenarios such as neutrino mixing with hidden sector states, including for strongly-coupled hidden sectors, can be described by functions exhibiting a discrete or continuous density of states.
Since the propagator above and the spectral functions $\rho_{\alpha\beta}(\mu^2)$ and $\tilde{\rho}_{\alpha\beta}(\mu^2)$ are expressed in the interaction basis, whose eigenstates need not coincide with those of the vacuum Hamiltonion, the spectral functions need not be real nor obey the usual positivity requirements satisfied by K\"all\'en-Lehmann functions.
An exception is in the case $\alpha=\beta$, when the mapping between mass- and interaction-bases depends only on real, positive factors such as $|U_{\alpha j}|^2$. In this case, the spectral functions are required to be real, and both $\tilde{\rho}_{\alpha\alpha}(\mu^2)$ and $(\mu \tilde{\rho}_{\alpha\alpha}(\mu^2) - \rho_{\alpha\alpha}(\mu^2) )$ are positive-definite.
In all scenarios of interest, the neutrinos are ultra-relativistic. In this limit, chirality and helicity eigenstates coincide and the spin structure can be factored out of the two-point function. All the relevant oscillation physics is fully captured by the scalar amplitude which itself can be represented via the scalar K\"all\'en-Lehmann propagator. In practice this amounts to setting $\tilde{\rho}(\mu^2) = 0$ in Eq.~\ref{eq:KL}. Cataloguing the BSM possibilities for the neutrino sector thus fundamentally reduces to the study of a single scalar function, $\rho(\mu^2)$, which will be the object of interest in what follows.
Following the standard QFT calculations with this modification, the probability for neutrino flavour transitions for propagating neutrinos of momentum $\bold{p}$ is given by
\begin{equation}
\label{prob}
P_{\alpha\beta} = \left | \int_0^{\infty} d\mu^2 e^{-i \sqrt{\mu^2 + \bold{p}^2}L}\rho_{\alpha\beta}(\mu^2) \right | ^2 ~~.
\end{equation}
In this statement we have applied the approximation $L\approx T$, and are explicitly working in the limit in which the full QFT calculation reproduces that obtained in QM. This requires the additional assumption that the neutrino at the production and detection vertices can be approximated as a plane wave, or equivalently that the neutrino wavepackets maintain coherence over the entire distance of interest, $L$.
In order to expand $\sqrt{\mu^2 + \bold{p}^2}$, we must assume that all coherently-propagating neutrinos are relativistic. This requires that $\rho_{\alpha\beta}(\mu^2)$ does not have support for large $\mu^2$, which is expected for light, oscillating neutrinos with large (above ${\sim}$MeV) energies. We thus have that $\sqrt{\mu^2 + \bold{p}^2} \approx E + \mu^2/(2E)$, and can express the transition probability as
\begin{equation}
P_{\alpha\beta} = \left | \int_0^{\infty} d\mu^2 e^{-i \frac{\mu^2 L}{2E} }\rho_{\alpha\beta}(\mu^2) \right | ^2 ~~.
\end{equation}
This expression for the transition amplitude is simply the Fourier transform of the scalar spectral function.
\subsection{Two-Flavour Example}
To make concrete headway we now turn to two-flavour mixing as characterised by the unitary matrix
\begin{equation}
U = \begin{pmatrix}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{pmatrix} ~~,
\end{equation}
which rotates flavour states $(\nu_\alpha, \nu_\beta)$ into mass eigenstates $(\nu_1, \nu_2)$. For example, the $\nu_\alpha$ survival probability thus follows from the spectral density
\begin{equation}
\rho_{\alpha\alpha} = \delta(\mu^2 - m^2_1) \cos^2 \theta- \delta(\mu^2- m^2_2)\sin^2 \theta ~~,
\end{equation}
giving
\begin{equation}
P\left(\nu_\alpha \to \nu_\alpha\right) = 1 - \sin^2 2\theta \sin^2 \left( \frac{\Delta m_{12}^2 L}{4E} \right) ~~,
\end{equation}
where $\Delta m_{21}^2 \equiv m_2^2 - m_1^2$. Similarly, the $\nu_\alpha \rightarrow \nu_\beta$ ($\alpha\neq\beta$) transition probability follows from
\begin{equation}
\rho_{\alpha\beta} = \sin \theta \cos \theta \left(\delta(\mu^2 - m^2_1) - \delta(\mu^2- m^2_2) \right) ~~,
\end{equation}
giving
\begin{equation}
P\left(\nu_\alpha \to \nu_\beta\right) = \sin^2 2\theta \sin^2 \left( \frac{\Delta m_{12}^2 L}{4E} \right) ~~.
\end{equation}
We see that the predictions from this formalism map to textbook expressions in this standard case.
\subsection{Pseudo-Dirac Neutrinos}\label{subsec:PD}
A simple and well-known possibility that converts a single neutrino mass eigenstate into multiple states is that of pseudo-Dirac neutrinos \cite{Wolfenstein:1981kw,Petcov:1982ya,Bilenky:1983wt,Kobayashi:2000md,deGouvea:2009fp,Anamiati:2017rxw,Anamiati:2019maf,Martinez-Soler:2021unz}. Structurally, the model consists of Dirac neutrinos, preserving a lepton-number symmetry, supplemented by a small source of explicit lepton-number violation which can be naturally small. This symmetry breaking thus splits the Dirac neutrinos into two Majorana mass eigenstates. Explicitly, the usual Lagrangian is
\begin{equation}
\lambda \nu_R L H + h.c.,
\end{equation}
which (after electroweak symmetry breaking) generates a Dirac mass $m_D = \lambda v/\sqrt{2}$. The small lepton-number violating Majorana mass term for the right-handed neutrino is
\begin{equation}
\frac{1}{2} M \nu_R^2 +h.c.
\end{equation}
In the limit $m_D \gg M$, this has physical mass eigenvalues
\begin{equation}
M_{\pm}=m_D \pm \frac{M}{2} ~~,
\end{equation}
and the mixing angle between the EW gauge eigenstate and the mass eigenstates is
\begin{equation}
\tan (2 \theta) = \frac{2 m_D}{M} ~~.
\end{equation}
When $m_D \gg M$ the mixing angle is approximately maximal, $\theta \approx \pi/4$. This furnishes a basic example where the mass eigenstate that propagates from production to detection is, microscopically, multiple states of different mass.
This can be generalised to three flavours by the introduction of 3 right-handed neutrinos, $\nu_{Ri }$, for $i \in \{1,2,3\}$. For simplicity, we take the alignment limit where the $3 \times 3$ Dirac and Majorana mass matrices can be simultaneously diagonalised. We shall henceforth refer to the diagonal entries of these as $m_{D_i}$ and $M_{i}$ respectively. The $6 \times 6$ unitary matrix describing the rotation of the $\{\nu_{eL},\nu_{eR},\nu_{\mu L},\nu_{\mu R},\nu_{\tau L},\nu_{\tau R}\}$ flavour states into the mass eigenstates may be conveniently written as a product of $ 3 \times 3 $ and $2 \times 2$ matrices, defined as
\begin{equation}
O_{\alpha \beta a b} = U_{\alpha \beta} R_{a b} (\theta_i)~~,
\end{equation}
where $U_{\alpha \beta}$ is the usual leptonic mixing matrix and $R_{ab}$ is a $2 \times 2$ rotation matrix with a rotation angle $\tan 2\theta_i = 2 m_{D_i}/M_{i}$.
The spectral density for $\alpha$ to $\beta$ flavour transitions is
\begin{align}
\rho_{\alpha \beta} (\mu^2) = \sum_{i = 1}^{3} U^*_{\alpha i} U_{\beta i} \left[ \cos^2(\theta_i)\delta(\mu^2 - M_{i,-}^2) + \sin^2(\theta_i)\delta(\mu^2 - M_{i,+}^2) \right] ~~,
\end{align}
where $M_{i,+}, M_{i,-}$ refer to the masses of the physical Majorana mass eigenstates into which the $i^{\textnormal{th}}$ Dirac neutrino splits. Motivated by this, we now generalise further to a band of states.
\subsection{A Band of Neutrinos}\label{subsec:Band}
One way a band of neutrino states can be realised is through a `clockwork'-inspired~\cite{Choi:2015fiu,Kaplan:2015fuy,Giudice:2016yja} fermion ring. Consider $N$ identical copies of the Standard Model enjoying a large translation symmetry in theory space. These different sectors could have renormalisable (hence relevant at low energies) couplings to one another through one of three portals: Higgs, kinetic mixing and neutrino. We will focus on the latter and suppose the following ring of couplings for a single flavour of neutrinos
\begin{equation}
\mathcal{L}_\lambda = \frac{\lambda}{q} \left[ \sum_{j=1}^{j=N-1} L_j H_j (\psi_j-q \psi_{j+1}) + L_N H_N (\psi_N-q \psi_{1}) \right] ~~,
\end{equation}
where the $\psi$ are SM-neutral Weyl fermions, $\lambda$ a small Yukawa coupling, and we assume $q>1$. Upon electroweak symmetry breaking in all of the sectors the collective neutrino mass matrix, for one active flavour, becomes
\begin{equation}
M_\nu = \frac{m_D}{q}
\begin{pmatrix}
1 & -q & 0 & \cdots & & 0 \cr
0 & 1 & -q & \cdots & & 0 \cr
0 & 0 & 1 & \cdots & & 0 \cr
\vdots & \vdots & \vdots & \ddots & &\vdots \cr
0 & 0 & 0 &\cdots & 1 & -q \cr
-q & 0 & 0 &\cdots & 0 & 1
\end{pmatrix} \ ~~,
\label{psimass}
\end{equation}
where $m_D = \lambda v/\sqrt{2}$. The physical eigenvalues of this mass matrix are
\begin{equation}
m_j^2 = \frac{m_D^2}{q^2} \left(1+q^2 - 2 q \cos \left(\frac{2 \pi j}{N} \right) \right) ~~.
\end{equation}
We thus have a band of $N$ mass eigenstates centred at $\sim m_D$ with a breadth of $\sim m_D/q$. The overlap between these states and the flavour eigenstates is given by the elements of the rotation matrix
\begin{equation}
R_{jk} = \frac{\cos\left( \frac{2 \pi j k}{N} \right) + \sin\left( \frac{2 \pi j k}{N} \right)}{\sqrt{N }} ~~.
\end{equation}
Thus, for instance, the overlap between the $j^{\text{th}}$ mass eigenstate of a given generation and the interaction eigenstate in that sector of the ring is given by the elements $R_{1j}$.
This can be further generalised to include 3 active generations of neutrino in each copy of the SM, with (site-independent) couplings $\lambda_{\alpha}$ where $\alpha \in \{1,2,3\}$. We take $q$ to be the same for each generation. In our conventions, the spectral density describing transitions from the $\alpha$ to $\beta$ flavour eigenstates is then
\begin{equation}
\rho_{\alpha \beta} = \sum_{i=1}^{3} U^*_{\alpha i }U_{\beta i}\left(\sum^{N}_{j= 1} R_{1 j}^2\delta(\mu^2 - m_{ij}^2)\right) ~~,
\end{equation}
where the $m_{ij}$ refer to the masses of the $N$ physical mass eigenstates of the $i^{\textnormal{th}}$ generation. This model essentially replaces any would-be SM Dirac neutrino mass eigenstate, of mass $\sim m_D$, by a band of states spread about this mass scale.
Some comments are in order. This model serves only to illustrate that such a scenario is possible, but is not intended to advertise the model as being particularly strongly motivated in its own right. Furthermore, many details of the model are not necessary in order to realise the same qualitative scenario including, for instance, the translation symmetry that enforces equal $q$ at each site which could be softly broken.
Finally, a brief comment on cosmology. If each sector were truly identical then it would be necessary that only the SM sector is reheated at the end of inflation, otherwise the neutrinos and photons of the hidden sectors would presumably lead to inconsistencies with cosmological observations. It may also be the case that the reheating temperature is necessarily low to avoid over-populating the other neutrino states.
\subsection{Theory-Space Perspective}\label{subsec:TheorySpace}
To understand the phenomenology of these more exotic scenarios it is instructive to consider the process in terms of the `theory-space' sites, borrowing terminology from dimensional deconstruction \cite{Arkani-Hamed:2001kyx,Arkani-Hamed:2001nha}. Consider an $N$-site band model. Only one of the sites corresponds to an active, and potentially measurable neutrino. The remaining $N-1$ sites are sterile. When we talk of physically measurable neutrino oscillations, we refer to the probability of starting in the active flavour eigenstate of the ($\alpha$) generation and being measured in the active flavour eigenstate of the ($\beta$) generation at some later time, however here we will focus on a single-generation case.
Since the starting active flavour eigenstate is some superposition of the $N$ mass eigenstates through which the system evolves, there will be a generally non-zero probability of being in one of the other $N -1$ sites when a neutrino is detected. In general, the higher the value of $N$, the lower the detection probabilities become due to the greater number of available sites to which the system has evolved during propagation.
To illustrate this, we consider the specific case of $N=6$, corresponding to one active neutrino site and 5 sterile sites. In the main panel of Fig.~\ref{fig:ths} we consider initialising the system in the active flavour state, which we number as state 1, and plot the overlap of the time-dependent state with the $i^{\textnormal{th}}$ site eigenstate as a function of the measurement time. Due to the discrete rotational symmetry in theory space the curves for $ i = 5$ and $ i = 6$ overlap exactly with those of states $i = 3$ and $i=2$ respectively, so they are not visible on the plot. The inset images show the overlap with each site at a time $t_j$ and thus how the measurement probability is distributed between the $6$ sites. When viewed chronologically they illustrate the flow of probability in theory-space, which repeats periodically. As $N$ is increased, the time taken to return to the starting distribution in which the system is in the active state increases, as there are a greater number of sites for the probability to flow through first.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{Graphs/thspace.pdf}
\caption{The flow of probability around the ring of fermions for $N = 6$ sites and $q = 10^3$. The neutrino is initialised at $t=0$ in the active, $i = 1$ state and, as time evolves, may overlap with the other $N -1$ sites, before returning to a maximum of being detected in the $n =1$ state again after a time $T$. The probability that the system is in the $i^{\textnormal{th}}$ site on measurement after a time $t$ is shown in the main figure, in units of the cycle period $T$. The circular bar charts illustrate, to scale, the flow of probability around the sites, plotted in the same colours as in the main figure, at a number of instances. The inner and outer concentric dashed circles denote measurement probabilities of 0.5 and 1 respectively.}
\label{fig:ths}
\end{figure}
\subsection{A Phenomenological Ansatz}\label{subsec:Pheno}
The two scenarios considered above are just examples of the rich landscape of BSM possibilities for the neutrino sector. Whilst it is of course possible to construct the relevant spectral function for any given model, calculate the transition probabilities, and extract the bounds from experimental data, this process would need to be undertaken separately for each individual model under consideration. Given the number and range of theoretical possibilities, a comprehensive survey is not only cumbersome, but fundamentally not feasible. Experimental analyses are thus typically limited to a handful of the simplest models.
We find that over the energies probed by existing oscillation experiments, the $n$-flavour oscillation probability distributions arising from both the pseudo-Dirac and band models can be sufficiently mimicked by a spectral function consisting of $n$ top-hat functions. With respectively discrete and quasi-continuous spectral functions, the pseudo-Dirac and band cases span a broad landscape of plausible spectral functions and it is thus reasonable to expect the top-hat set-up to be capable of reproducing neutrino oscillation probability distributions for a broad range of microscopic scenarios. In this section, we will thus derive the general form of the transition probability generated by top-hat functions.
To this end, we begin by focusing on three-neutrino oscillations, replacing the delta functions (three at $m_i^2$) of the canonical spectral function with top-hat functions of (generally different) breadths $b_i$ centred at these values. Explicitly, we parametrise the spectral function as
\begin{equation}
\label{eq:hat}
\rho_{ee}(\mu^2) = \left\{ \begin{array}{ll}
\frac{1}{b_1}|U_{e1}|^2 = \frac{1}{b_1} \cos^2 \theta_{12}\cos^2 \theta_{13} & \qquad,\qquad m_1^2 - \frac{b_1}{2} \leq \mu^2 \leq m_1^2 + \frac{b_1}{2} \\
\frac{1}{b_2}|U_{e2}|^2 = \frac{1}{b_2}\cos^2\theta_{13}\sin^2\theta_{12} & \qquad,\qquad m_2^2 - \frac{b_2}{2} \leq \mu^2 \leq m_2^2 + \frac{b_2}{2} \\
\frac{1}{b_3} |U_{e3}|^2 = \frac{1}{b_3}\sin^2 \theta_{13} &\qquad,\qquad m_3^2 - \frac{b_3}{2} \leq \mu^2 \leq m_3^2 + \frac{b_3}{2} \\
0 &\qquad,\qquad \mathrm{otherwise}
\end{array} \right\} ~~.
\end{equation}
This is illustrated schematically in Fig. \ref{fig:sd}. We note that current measurements of the leptonic mixing matrix indicate $|U_{e3}|^2 \ll |U_{e 1}|^2, |U_{e 2}|^2$.\begin{figure}[t]
\begin{center}
\begin{tikzpicture}
\draw [<->] (1.6,-1) -- (0.8,-1);
\draw [<->] (4.4,-1) -- (3.6,-1);
\draw [<->] (2.8,-1) -- (2,-1);
\draw (0.8,-0.7) -- (0.8,-1.3);
\draw (1.6,-0.7) -- (1.6,-1.3);
\draw (2,-0.7) -- (2,-1.3);
\draw (2.8,-0.7) -- (2.8,-1.3);
\draw (3.6,-0.7) -- (3.6,-1.3);
\draw (4.4,-0.7) -- (4.4,-1.3);
\draw [fill=cyan] (0.8,0) rectangle (1.6,2.6);
\draw [fill=cyan] (2,0) rectangle (2.8,2.3);
\draw [fill=cyan] (3.6,0) rectangle (4.4,1);
\draw [<-] (0,3) node [left] {$\rho(\mu^2)$} -- (0,0);
\draw [<-] (5,0) node [below] {$\mu^2$} -- (0,0) node[left]{0};
\draw [very thick,black] (0.8,0) -- (0.8,2.6) -- (1.6,2.6)
-- (1.6,0) ;
\draw [ very thick,black] (2,0) -- (2,2.3) -- (2.8,2.3)
-- (2.8,0) ;
\draw [ very thick,black] (3.6,0) -- (3.6,1) -- (4.4,1)
-- (4.4,0) ;
\draw [<-,very thick] (1.2,3) -- (1.2,0);
\draw [<->,thick,red] (1.2,3.1) -- (2.4,3.1);
\draw [<->,thick, red] (2.4,3.1) -- (4,3.1);
\node[above,red] at (1.8,3.1) {$\Delta m_{2 1}^2$};
\node[above,red] at (3.2,3.1) {$\Delta m_{3 2}^2$};
\node[below] at (2.4,0) {$m_2^2$};
\node[below] at (1.2,0) {$m_1^2$};
\node[below] at (4.0,0) {$m_3^2$};
\draw [<-,very thick] (2.4,3) -- (2.4,0);
\draw [<-,very thick] (4,3) -- (4,0);
\node[below] at (1.2,-1) {$b_1$};
\node[below] at (2.4,-1) {$b_2$};
\node[below] at (4,-1) {$b_3$};
\end{tikzpicture}
\end{center}
\caption{Sketch of the form of the top-hat $\rho_{s}(\mu^2)$ for 3 flavour oscillations as defined in Eq. \ref{eq:hat}. The bold arrows represent the $\delta$-function eigenstates of the standard scenario. }
\label{fig:sd}
\end{figure}
Upon taking the Fourier transform, we arrive at the amplitude
\begin{equation}
iA_{ee} = \sum_{i=1}^{3} \textnormal{sinc}\left(\frac{Lb_i}{4E}\right)|U_{ei}|^2 e^{\frac{-iLm_i^2}{2E}} ~~,
\end{equation}
and probability
\begin{align}\label{eq:prob}
P\left(\nu_e \to \nu_e\right) = \left( \sum_i \mathrm{sinc}\left(\alpha_i\right) |U_{ei}|^2\right)^2 - 4\sum_{i<j} |U_{ei}|^2 |U_{ej}|^2 \mathrm{sinc}\left(\alpha_i\right) \mathrm{sinc}\left(\alpha_j\right)\sin^2\left(\Delta_{ji}\right) ~~,
\end{align}
where $\Delta_{ji} \equiv \Delta m_{ji}^2 L/(4E)$ and $\alpha_{i} \equiv b_i L/(4E)$. We note that since we are treating neutrino propagation in vacuum, the probability for antineutrino oscillations, $\overline\nu_e \to \overline\nu_e$, is identical.
For the case of equal breadths, $b_1 = b_2 = b_3 = b$, this simplifies to
\begin{eqnarray}
\label{eq:prop}
P\left(\nu_e \to \nu_e\right) & = & \textnormal{sinc}^2\left(\frac{Lb}{4E}\right)\bigg[1-\sin^2 2\theta_{12} \cos^4 \theta_{13} \sin^2 \Delta_{21} \nonumber \\
& & - \sin^2 2\theta_{13}\left(\cos^2\theta_{12}\sin^2\Delta_{31} + \sin^2\theta_{12}\sin^2\Delta_{32} \right) \bigg] ~~,
\end{eqnarray}
which we identify as the standard probability expression modulated by a factor of $\textnormal{sinc}^2\left(\frac{Lb}{4E}\right)$. We emphasise that these expressions only depend on the relative spacing of the states in $\mu^2$ and not on their absolute value.
\begin{figure}[h]
\centering
\subfloat[Comparison of 3-flavour oscillation probabilities generated by a density of states comprised of 3 top-hat functions with fractional breadths $\tilde{b}_i$ = 0.005, and a pseudo-Dirac density of states with masses $(M_1, M_2, M_3) = (1.23, 1.21, 1.39$) $\times 10^{-4} $eV, selected by performing a least squares fit to the top-hat probability distribution. The lower panel shows the fit residuals.]{\includegraphics[width=0.65\textwidth]{Graphs/pd05a.pdf} }
\qquad
\subfloat[A plot of the spectral function $\rho({\mu^2})$ for these two models with the same parameters as in (a). Also shown is the triple $\delta$-function density of states corresponding to the canonical scenario. The heights of the top-hat functions relative to each other are plotted to scale but the vertical extent of the (formally infinite) $\delta$-functions for the pseudo-Dirac and conventional models are intended for illustrative purposes only.]{\includegraphics[width=0.65\textwidth]{Graphs/pd05b.pdf} }
\caption{Evaluation of the top-hat phenomenological ansatz defined in Sec.~\ref{subsec:Pheno} to capture the oscillation behaviour of pseudo-Dirac models as detailed in Sec.~\ref{subsec:PD}.}
\label{fig:0.5pd}%
\end{figure}
\begin{figure}[h]
\centering
\subfloat[Comparison of 3-flavour oscillation probabilities generated by a density of states comprised of 3 top-hat functions with fractional breadths $\tilde{b}_i$ = 0.005, and a $N=10$ band density of states. The value of $q = 987.2$ was selected using a least squares fit to the top-hat probability distribution.]{\includegraphics[width=0.65\textwidth]{Graphs/band05a.pdf} }
\qquad
\subfloat[A plot of the spectral function $\rho({\mu^2})$ for these two models with the same parameters as in (a). Also shown is the triple $\delta$-function density of states corresponding to the canonical scenario. The heights of the top-hat functions relative to each other are plotted to scale but the vertical extent of the (formally infinite) $\delta$-functions for the band states and conventional model are intended for illustrative purposes only.]{\includegraphics[width=0.65\textwidth]{Graphs/band05b.pdf} }
\caption{Evaluation of the top-hat phenomenological ansatz defined in Sec.~\ref{subsec:Pheno} to capture the oscillation behaviour of band models as defined in Sec.~\ref{subsec:Band}.}
\label{fig:0.5band}%
\end{figure}
For the purpose of this exercise we assume a normal neutrino mass ordering and set $m_3$ to have an absolute value of 0.1 eV. We then fix $m_1$ and $m_2$ according to $m_i^2 = m_3^2 - \Delta m_{3i}^2$ for $i \in \{1,2\}$. We use the current best fit values of the $\Delta m_{ij}^2$ and the mixing angles as determined by existing experiments\footnote{Explicitly we take $\sin^2\theta_{12} = 0.3$, $\sin^2\theta_{13} = 2.2\times 10^{-2}$, $\Delta m_{21}^2 = 7.4\times 10^{-5}$ eV$^2$, and $\Delta m_{31}^2 = 2.5\times 10^{-3}$ eV$^2$\cite{Esteban:2020cvm}.}. We parametrise the breadths of the top-hat states according to a fractional value relative to the mass squared, $\tilde{b}_i \equiv b_i/m_i^2$. For reference, we highlight that at $\tilde{b}_1$ = 0.01, $b_1 \approx \Delta m_{12}^2$ and the spectral gap between the top-hats of state 1 and state 2 vanishes.
We now seek to demonstrate the capability of the top-hat to capture the phenomenology of the specific microscopic models introduced earlier. To approach this, we generate the (anti-)electron survival probability distribution as a function of baseline length over energy ($L/E$) for a given set of top-hats breadths $(b_1,b_2,b_3$), and perform a least squares fit of the pseudo-Dirac and band probability distributions over the range $L/E \in [5,30] $ km/MeV, treating the set ($M_1,M_2,M_3$) as free parameters in the former and fixing $N$ and fitting for $q$ in the latter. As an illustration, we consider a top-hat set up with fractional breadths $\tilde{b}_i$ = 0.005 which, for reference, corresponds to $b_1 \approx 0.5 \Delta m_{12}^2$. Fig.~\ref{fig:0.5pd} (a) compares the probability distribution for this set up with that generated by the best fit of the pseudo-Dirac model. The best fit parameters are $(M_1,M_2,M_3) = (1.23,1.21,1.39)\times 10^{-4}$ eV. The residuals for the fit are shown in the lower panel and show a disagreement of less than 0.1\% over the domain probed by JUNO. In Fig.~\ref{fig:0.5pd} (b) we plot the density of states for these models with the parameters required for matched probability distributions as detailed above. Also shown is the triple $\delta$-function spectrum of the canonical scenario. Whilst the heights of the top-hat states relative to each other are plotted to scale, the vertical extent of the (formally infinite) $\delta$-functions for both the pseudo-Dirac model and the standard case are for illustrative purposes only. We observe that probability matching occurs when the breadth of the top-hat function for a given state is roughly twice the pseudo-Dirac mass splitting of that generation.
In Fig.~\ref{fig:0.5band} (a) we show the agreement between the same top-hat density of states, with the best fit of the band model for $N = 10$, which is achieved for $q = q_* = 987.2$. Once again we see an excellent agreement of within 0.1\% over the JUNO energy range. As seen in the comparison of the spectral densities for the matched cases shown in Fig.~\ref{fig:0.5band} (b), the breadth of the top-hat of a given state should slightly exceed the breadth of the band. Note that whilst there are 10 mass eigenstates for each generation, degeneracies of the mass values of some states means that they do not appear as distinct lines on the plot.
Whilst we have merely shown matching to a single choice of top-hat breadths here, we have found that it is always possible to tune the model parameters to achieve an excellent agreement of the probability distributions for any set of top-hat breadths, and thus from the reverse perspective, to be able to find a choice of top-hat breadths which reproduce the probability distributions of these models for any given set of parameters.
\subsection{The Top-Hat Landscape}
Given their ability to capture the probability distributions of some specific theoretical models, it serves to explore the phenomenological space that can be spanned by top-hat spectral densities in this way. We will address this by exploring the effect on the transition probability of different top-hat breadths. We initially consider the case where only one of the three states has a finite breadth, and the remaining two are $\delta$-functions. In Fig.~\ref{fig:change1} we plot the probability distributions generated by setting $\tilde{b} = 0.03$ for the non-zero breadth state. This comprises of $\mathcal{O}( 1 )$ amplitude oscillations driven by $\Delta m_{12}^2$ on which a smaller amplitude, higher frequency, oscillation driven by $\Delta m_{13}^2$ is superimposed. We note that the impact of broadening the third top-hat in the spectral function is to damp the amplitude of the $\Delta m_{13}^2$ oscillations. As apparent from Eq.~\ref{eq:prop}, modifications to the large amplitude $\Delta m_{12}^2$ oscillations arise from the sinc terms in $b_1$ and $b_2$, and thus occur on broadening of the breadths of the first and second top-hat states.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{Graphs/diff-widths.pdf}
\caption{Plots of the anti-electron neutrino survival probability as a function of $L/E$ for the 3-flavour top-hat density of states set up as defined in Sec.~\ref{subsec:Pheno} for the case where only one of the three states has a finite breadth, of fractional value $\tilde{b} = 0.03$ and the remaining two are $\delta$-functions. Also shown for comparison is the probability distribution for the standard scenario in which the density of states comprises of 3 $\delta$-functions ($b_i = 0$). }
\label{fig:change1}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.75\textwidth]{Graphs/comparison-widths.pdf}
\caption{Plots of the anti-electron neutrino survival probability as a function of $L/E$ for the 3-flavour top-hat density of states as defined in Sec.~\ref{subsec:Pheno} with $\tilde{b}_{1}=\tilde{b}_{2}=\tilde{b}_{3}=\tilde{b}$, for various values of $\tilde{b}$. The case $\tilde{b}_i = 0$ corresponds to the standard scenario in which the density of states is comprised of 3 $\delta$-functions. }
\label{fig:comp}
\end{figure}
To gain a handle on the top-hat breadths required to produce measurable deviations from the standard case, we consider the impact on the probability distribution of setting $\tilde{b}_{1}=\tilde{b}_{2}=\tilde{b}_{3}=\tilde{b}$, such that the breadths of the three states relative to their central mass-squared are equal. We plot the transition probability for increasing values of $\tilde{b}$ in Fig.~\ref{fig:comp}. Very little deviation from the standard probability distribution occurs for $\tilde{b}$ $\lesssim$ 0.002, the point at which $b_1 \approx b_2$ becomes sizeable ($\sim$ 20\%) relative to $\Delta m_{12}^2$. By nature of their larger amplitude, the greatest overall modifications to the probability distribution will arise from modifications to the $\Delta m_{12}^2$ oscillations and thus the degree of deviation from the standard probability distribution is largely controlled by the comparative sizes of $b_1 \approx b_2 $ and $\Delta m_{12}^2$. Increasing $\tilde{b}_1$ (or $\tilde{b}_2$) both increases the frequency, and decreases the extent of the central peak, of (one of) the sinc functions modulating the $\Delta m_{21}^2$ oscillations. If $b_1 \ll \Delta m_{21}^2$, the entire energy range probed by JUNO lies approximately at the central peak of the modulating sinc function. In the opposing regime, the JUNO energy range falls in the tails of the sinc function and probability is driven towards zero. If $b_1$ and $\Delta m_{21}^2$ are of the same order, the probed energy range lies on the falling edge of the sinc function central peak and we see sizeable corrections to the probability, which increase with $L/E$. For reference, we note that for the set of masses used in this figure, $b_1 \approx \Delta m_{21}^2$ when $\tilde{b}_1 \approx 0.01$ .
\section{The Decoherence Limit}\label{sec:Decoherence}
When propagating over long distances the neutrino wave packets will ultimately decohere, leading to asymptotic neutrino detection probabilities. One may recall this from the standard two-neutrino treatment, wherein for $\Delta m_{12}^2 L\gg 4E$, there are many oscillations such that \begin{equation}
\sin^2 \left( \frac{\Delta m_{12}^2 L}{4E} \right) \to \frac{1}{2} ~~,
\end{equation}
and the survival probability asymptotes to
\begin{equation}
P\left(\nu_\alpha \to \nu_\alpha\right) = 1 - \frac{1}{2} \sin^2 2\theta ~~.
\end{equation}
Due to our simplifying ansatz of flavour alignment, these same limits factorise within the various scenarios considered above, when $L/E$ is much larger than any mass-squared-separation scale of interest for the model at hand.
In this limit, coherence is lost among the mass eigenstates and simplified expressions for the various models may be found.
For the pseudo-Dirac case, assuming all mass eigenstates having splittings such that coherence is lost amongst them, the probabilities become
\begin{eqnarray}
P_{\alpha\beta} & = & \sum_{i = 1}^{3} (U^*_{\alpha i} U_{\beta i})^2 \left[ \cos^4(\theta_i) + \sin^4(\theta_i) \right] \nonumber \\
& \to & \frac{1}{2} \sum_{i = 1}^{3} (U^*_{\alpha i} U_{\beta i})^2 ~~,
\end{eqnarray}
where in the final expression we have employed the limit $M\ll m_D$, thus $\theta_i \to \pi/4$. Importantly, note that this is a factor $1/2$ smaller than the corresponding standard three-neutrino result in the decoherence limit.
Now consider the band model, again with a band splitting for each mass eigenstate that is great enough for coherence to be lost amongst the band. In this case the probabilities become
\begin{eqnarray}
P_{\alpha\beta} & = & \sum_{i = 1}^{3} (U^*_{\alpha i} U_{\beta i})^2 \sum^{N}_{j= 1} R_{1 j}^4 \nonumber \\
& \to & \frac{3}{2 N} \sum_{i = 1}^{3} (U^*_{\alpha i} U_{\beta i})^2 ~~.
\end{eqnarray}
The emerging pattern is physically intuitive. Given a long-enough propagation distance the individual mass wave packets separate. In these flavour-aligned models, what would have been one mass eigenstate is replaced by $N$ individual separated states, of which an effective $N-1$ are sterile and undetectable. The overall numerical coefficient depends on the specific details of the model, however the suppression of the resulting signal universally scales inversely proportionally to the number of available states.
Since the top-hat model effectively corresponds to an infinite number of states, one would expect that in the decoherence limit the various detection probabilities would asymptotically vanish. In this case the electron neutrino survival probability asymptotes to
\begin{eqnarray}
\label{eq:prop}
P\left(\nu_e \to \nu_e\right) & \to & \sum_i \frac{1}{2} \left(\frac{4E}{Lb_i}\right)^2 |U_{ei}|^4 ~~,
\label{eq:thdec}
\end{eqnarray}
which indeed vanishes asymptotically as an inverse quadratic of the length scale. This will be important when we come to consider experimental constraints. Furthermore, the mixing angles are also important. For instance, one has
\begin{equation}
|U_{e1}|^4 \approx 0.5 ~~~~,~~~~ |U_{e2}|^4 \approx 0.09 ~~~~,~~~~ |U_{e3}|^4 \approx 5 \times 10^{-4} ~~~,
\end{equation}
hence, depending on the situation, it may be that constraints on the survival probability of the total electron neutrino flux will only likely significantly constrain broadening in the lightest neutrinos.
In the following subsections, we discuss how observations of neutrinos from very distant sources can, in principle, place strong constraints on broadened neutrinos through nonzero $b_i$. In contrast, Sections~\ref{sec:Experiments} and~\ref{sec:Constraints} will demonstrate how terrestrial, neutrino-oscillation focused measurements, are an interesting avenue to potentially discover a nonzero $b_i$.
\subsection{Supernova Constraints}
Let us first consider the longest baseline constraints arising from the detection of SN1987A neutrinos by Kamiokande-II \cite{Kamiokande-II:1987idp,Hirata:1988ad}, IMB \cite{Bionta:1987qt,IMB:1988suc}, and Baksan \cite{Alekseev:1988gp}. Constraints on the pseudo-Dirac case were considered in Ref.~\cite{Martinez-Soler:2021unz}, which informs our comments here. In Ref.~\cite{Martinez-Soler:2021unz} it was found that constraints were not strong in the decoherence limit since the flux reduction by a factor of 2 could be accommodated by a doubling of the supernova energy. Stronger limits were found for mass splittings in the $L/E$-range corresponding to SN1987A as this modifies the energy-dependence of the spectrum. However, in this work we are interested in large mass splittings $\Delta m^2 \gg 10^{-19} \text{ eV}^2$, such that the full decoherence limit is reached. We would thus expect that in the band model there would be similar flexibility, such that $N=3$ would be acceptable, and perhaps even larger. On the contrary, for the top-hat ansatz we would expect a significant suppression of the signal for a universal breadth of $b \gg 10^{-19} \text{ eV}^2$.
There is, however, the aforementioned caveat which concerns the universality of the breadth. We will illustrate this with the top-hat case, although similar aspects apply to the other two models. Due to the smallness of $|U_{e2}|^4$ and $|U_{e3}|^4$, from Eq.~\ref{eq:thdec} it appears there would be no strong constraint on $b_2$ and $b_3$, since a reduction of the flux due to decoherence in these modes would not be sufficient to generate tension with observations. On the contrary, we expect a limit in the ballpark of $b_1 \lesssim 10^{-19} \text{ eV}^2$ applies, otherwise the neutrino flux would be too greatly depleted. The analysis of Ref.~\cite{Martinez-Soler:2021unz} assumed universal splittings, however it would be interesting to repeat this analysis under non-universal assumptions, especially with application to the models considered here.
\subsection{Solar Constraints}
The case for solar neutrinos is somewhat more complex. While matter effects are important, it is still ultimately the element $|U_{ei}|^4$ which controls the magnitude of electron neutrino disappearance on broadening the $i^{\textnormal{th}}$ mass eigenstate. However, the overall fluxes are measured with greater precision and the physics of production understood with greater certainty. As a result, one again expects the most significant constraints on $b_1$, however constraints on $b_2$ may also be relevant. Estimating that strong constraints arise whenever $L b_i/E \ll 1$, then for typical solar neutrino energies and the Earth-Sun baseline one expects the limits to be in the region of $b_1 \lesssim 10^{-12} \text{ eV}^2$.
Solar constraints on the pseudo-Dirac case were studied in detail in Ref.~\cite{deGouvea:2009fp}, where indeed limits which would roughly correspond to $b_1 \lesssim 10^{-12} \text{ eV}^2$ were found, however in this case constraints in the region of $b_2 \lesssim 10^{-11} \text{ eV}^2$ also arise, due to the well-measured neutrino flux which gives sensitivity to effects at the $|U_{e2}|^4$-level. Which width is most strongly constrained is determined from an interplay between the precision of measurement and the magnitude broadening effect which, since it is controlled by the parameter combination $\sim b_j L/E$, is stronger at lower energies for a given fixed width and baseline.
\subsection{Atmospheric \& Astrophysical Constraints}
Due to the typical baseline and energies involved, one expects constraints from atmospheric neutrinos to be at the level of $b_i \lesssim 10^{-4} \text{ eV}^2$. However, as we will demonstrate in Section~\ref{sec:Experiments}, future long-baseline reactor antineutrino experiments will probe smaller breadths for all $b_i$, and thus we will not consider atmospheric constraints further here.
Finally, the observation of extragalactic neutrinos at neutrino observatories~\cite{IceCube:2013low,IceCube:2014stg} can provide an additional handle on neutrinos traveling great distances. Measuring the ratios of different flavours of the neutrinos upon arrival at Earth~\cite{IceCube:2015gsk,IceCube-Gen2:2020qha} can, in principle, help in constraining many of the models discussed here, however, such constraints would be subject to uncertainties on the overall neutrino flux, among others. Nevertheless, as precision improves~\cite{Song:2020nfh}, the BSM power of these measurements should be considered in more detail.
\subsection{Summary}
It is clear that astrophysical probes of neutrino oscillations allow for baselines that go deep into the decoherence regime. Indeed, due to the form of the PMNS matrix they lead to very strong constraints on the breadth of the lightest mass eigenstate, at the scale of $b_1 \lesssim 10^{-19} \text{ eV}^2$ and slightly weaker constraints at the level of $b_2 \lesssim 10^{-11} \text{ eV}^2$. However, they do not probe $b_3$ with the same power due to the smallness of $|U_{e3}|^2$. Moreover, these observations all probe the physics of broadened neutrinos in the decoherence (classical) regime as opposed to making measurements where the propagating neutrinos maintain coherence.
As a result, in Section~\ref{sec:Experiments}, we turn to terrestrial probes of neutrino breadths for all mass eigenstates, for their novelty in probing the quantum interference effects of broadened neutrinos, and also as a complementary probe to the methods discussed in this section, subject to a very different set of measurement techniques and assumptions.
\section{Terrestrial Experiments for Constraining Spectral Functions}
\label{sec:Experiments}
Having discussed the strengths and weaknesses of very long-baseline, astrophysical constraints on neutrino breadths in Section~\ref{sec:Decoherence}, we now shift our focus to terrestrial neutrino oscillation experiments. Section~\ref{sec:KLTheory} established our phenomenological description of the neutrino spectral functions; now we explore these experiments and their ability to test this scenario. In developing this phenomenological approach, we have focused on the case where neutrinos are propagating for long proper times (large $L/E_\nu$) in vacuum, where their interactions with any matter along the path of propagation can be neglected. To date, the experiments consistent with this assumption are those measuring oscillations of electron antineutrinos $\overline\nu_e$ produced in nuclear reactors. These oscillations, with $E_\nu \sim 1{-}10$ MeV, have been measured at a variety of baseline lengths, allowing for world-leading measurements of both mass-squared splittings $\Delta m_{21}^2$ and $\Delta m_{31}^2$ in the standard framework. A second class of experiments measure the oscillations of $\nu_\mu$ produced at ${\sim}$ GeV energies in proton-induced neutrino beams, travelling hundreds of kilometers. These $\nu_\mu$ disappearance experiments offer comparable sensitivity to $\Delta m_{31}^2$ and operate in a similar $L/E_\nu$ regime to the reactor antineutrino experiments.
\paragraph{Reactor Antineutrino $\overline\nu_e$ Experiments}
The most precise measurements of reactor antineutrinos are those from KamLAND~\cite{KamLAND:2008dgz,KamLAND:2010fvi,KamLAND:2013rgu}, with baselines of $L\approx 200$ km, and Daya Bay~\cite{DayaBay:2012fng,DayaBay:2013yxg,DayaBay:2015ayh,DayaBay:2016ggj,DayaBay:2018yms}, with baselines $L \approx 1$ km. These two correspond to measurements at $L/E \approx$ 40 km/MeV and 0.5 km/MeV, respectively. Due to the hierarchicy between $\Delta m_{21}^2 \approx 7.5 \times 10^{-5}$ eV$^2$ and $\Delta m_{31}^2 \approx 2.5 \times 10^{-3}$ eV$^2$, KamLAND is sensitive to oscillations driven by $\Delta m_{21}^2$ (where oscillations due to $\Delta m_{31}^2$ have averaged out over the energy uncertainty of its detector) and Daya Bay is sensitive to oscillations driven by $\Delta m_{31}^2$ (where the oscillations due to $\Delta m_{21}^2$ have yet to develop significantly at the Daya Bay $L/E$ ranges). Because of this, we expect that when studying the generalised spectral functions described in Section~\ref{subsec:Pheno}, KamLAND will be sensitive to nonzero $b_1$ and $b_2$ on the order of $\Delta m_{21}^2$ and that Daya Bay will be sensitive to $b_3$ on the order of $\Delta m_{31}^2$.
In simulating KamLAND, we develop our analysis to match the results of the most recent collaboration results in Ref.~\cite{KamLAND:2013rgu}. Our simulation of Daya Bay is modified from the analysis of Ref.~\cite{Arguelles:2022bvt} (see Ref.~\cite{Akhmedov:2022bjs} for further discussion), developed to match the official results from Ref.~\cite{DayaBay:2016ggj}.
We also consider the possibility of testing these phenomenological spectral functions in the future, namely by the JUNO~\cite{JUNO:2015zny,JUNO:2022mxj} experiment. JUNO is a medium-baseline reactor experiment that will measure antineutrino oscillations with $L \approx 50$ km, in the $L/E$ regime between that tested by Daya Bay and KamLAND. This will allow JUNO to simultaneously measure oscillations driven by the two mass-squared splittings in a precise way. Previous analyses, including Refs.~\cite{Abrahao:2015rba,Porto-Silva:2020gma,deGouvea:2020hfl,Huber:2021xpx,Basto-Gonzalez:2021aus,Marzec:2022mcz}, have demonstrated that JUNO is an impressive discovery ground for BSM physics -- here, we demonstrate that in our broad-neutrino framework, JUNO will be sensitive to all three spectral-function breadths $b_i$ simultaneously, and should exhibit impressive capability in searching for nonzero $b_i$'s. To simulate JUNO we use the same analysis described in Refs.~\cite{Ellis:2020ehi,Ellis:2020hus}, modified to accommodate our scenario. We refer the reader to Refs.~\cite{Ellis:2020ehi,Ellis:2020hus} and references therein for more detail.
The relevant $L/E$ ranges of these three reactor antineutrino experiments are displayed in Fig.~\ref{fig:PeeLE}.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{Graphs/Pee_Experiments.pdf}
\caption{Oscillation probability for reactor antineutrinos as a function of $L/E$ for the standard three-neutrino case (grey) and including nonzero spectral-function breadths as indicated in the legend (blue). We shade the regions of $L/E$ probed by existing/future experiments Daya Bay (red), JUNO (purple), and KamLAND (green). \label{fig:PeeLE}}
\end{figure}
Each experiment's $L/E$ range is shown as a shaded box, with Daya Bay, JUNO, and KamLAND in red, purple, and green, respectively. We also show the oscillation probability $P(\overline\nu_e \to \overline\nu_e)$ as a function of $L/E$ that is/can be measured by these three experiments. We display oscillation probabilities for two cases: grey for the standard three-neutrino scenario\footnote{Assuming the same values for the mass splittings and mixing angles as stated in Sec.~\ref{subsec:Pheno}.} and in blue where we additionally include nonzero spectral-function breadths, $b_1 = b_2 = 0.1\Delta m_{21}^2$ and $b_3 = 0.1 \Delta m_{31}^2$.
Here, the advantage of exploring these effects at large $L/E$ is clear, as in Figs.~\ref{fig:0.5pd}-\ref{fig:comp}. Because KamLAND operated at such large $L/E$, we would expect powerful sensitivity. However, not present in this figure (but present in our simulations) are the effects of finite energy resolution by the respective detectors. For instance, KamLAND has larger energy uncertainty than JUNO will and therefore is not as sensitive to fast, $\Delta m_{31}^2$-driven oscillations in its range of $L/E$ (and does not have sensitivity to $b_3$). Thus, despite its lower $L/E$, we expect JUNO to be the most powerful of these three.
\paragraph{Long-baseline $\nu_\mu$ Disappearance Experiments}
Throughout this work, we are interested in scenarios where neutrino propagation in vacuum is a suitable description. While modern-day (and future) long-baseline experiments measuring $P(\nu_\mu \to \nu_e)$ require the consideration of neutrino interactions with matter for an accurate calculation of oscillation probabilities, the disappearance probability $P(\nu_\mu \to \nu_\mu)$ and its CP-conjugate are insensitive to standard matter effects for the baselines/energies of interest.
For that reason, and for complete comparison against the tests from reactor antineutrino measurements, we include adapted simulations of the T2K~\cite{T2K:2021xwb} and NOvA~\cite{NOvA:2021nfi} experiments from Ref.~\cite{deGouvea:2022kma} to account for these effects in long-baseline $\nu_\mu \to \nu_\mu$ and $\overline\nu_\mu \to \overline\nu_\mu$ oscillations. The oscillation probabilities follow from Eq.~\eqref{eq:prob} with the substitution $|U_{ei}|^2 \to |U_{\mu i}|^2$. We also consider future long-baseline experiments DUNE~\cite{DUNE:2015lol,DUNE:2020ypp} and Hyper-Kamiokande~\cite{Hyper-KamiokandeProto-:2015xww,Hyper-Kamiokande:2018ofw}, which we will comment on in Section~\ref{sec:Constraints}.
In the context of Fig.~\ref{fig:PeeLE}, all of these experiments are situated at a similar $L/E$ to Daya Bay. Therefore, we expect sensitivity to $b_3$ but not competitive with what JUNO will have to offer in the coming decade, due to JUNO's larger $L/E$ and powerful energy resolution.
\section{Current \& Future Constraints on Spectral Functions}
\label{sec:Constraints}
\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{Graphs/CurrentConstraints_w1w2w3.pdf}
\caption{Current constraints from KamLAND (blue), Daya Bay (green), and long-baseline $\nu_\mu$ disappearance measurements from T2K and NOvA (orange) at $1$, $2$, and $3\sigma$ CL (dotted, dashed and solid respectively) on the reduced breadths $b_i$, relative to the overall neutrino-masses-squared when we assume $m_1 = 10^{-2}$ eV.\label{fig:CurrentWConstraints}}
\end{figure}
In this section, we provide the current constraints on the breadths $b_1$, $b_2$, and $b_3$. For simplicity, we will focus on the scenario in which the neutrino masses follow the normal ordering $m_3 > m_2 > m_1$. We will present results in terms of the dimensionless $\tilde{b}_i \equiv b_i/m_i^2$. We choose $m_1 = 10^{-2}$ eV as a benchmark for this presentation. Given the discussion in Section~\ref{sec:KLTheory}, we expect that the constraints on $b_i$ from these experiments would be largely unchanged if we considered the inverted mass ordering $m_2 > m_1 > m_3$, however the dimensionless $\tilde{b}_i$ would given the change in the overall $m_i^2$.
When analysing current data, to estimate the constraints on $\tilde{b}_i$, we fix the standard oscillation parameters to their best-fit values as stated in Sec.~\ref{subsec:Pheno}. For our analysis of T2K and NOvA's $\nu_\mu$ disappearance channels, we allow $\Delta m_{31}^2$ and $\sin^2\theta_{23}$ to vary independently.
We present current constraints on the three $\tilde{b}_i$ in Fig.~\ref{fig:CurrentWConstraints}. Notably, we find that current data from KamLAND, Daya Bay, T2K, and NOvA are consistent with $b_1 = b_2 = b_3 = 0$. Here, we compare $1\sigma$ (dotted), $2\sigma$ (dashed), and $3\sigma$ (dashed) constraints for the different experiments, KamLAND in blue, Daya Bay in green, and a combined analysis of T2K and NOvA in orange. As expected from our discussion in Section~\ref{sec:Experiments}, we find that KamLAND has strong sensitivity to $\tilde b_1$ and $\tilde b_2$, but no sensitivity to $\tilde{b}_3$. In contrast, Daya Bay and the long-baseline $\nu_\mu$ disappearance measurements are able to constrain $\tilde b_3$.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{Graphs/FutureConstraints_w1w2w3.pdf}
\caption{Expected future constraints by JUNO (purple) in the absence of a new-physics signal in the parameter space of $\tilde{b}_i$, compared against the current constraints from KamLAND (blue). Compared to Fig.~\ref{fig:CurrentWConstraints}, the data range here is so narrow that the constraints from Daya Bay, T2K, and NOvA do not appear.\label{fig:FutureWConstraints}}
\end{figure}
In contrast, future projections on $\tilde{b}_i$ from JUNO are displayed in Fig.~\ref{fig:FutureWConstraints}, compared against KamLAND's constraints. Note that we have changed the axes ranges here such that the Daya Bay, T2K, and NOvA constraints are no longer visible. Additionally, we have explored the capability of the future Hyper-Kamiokande and DUNE measurements of $\nu_\mu$ disappearance in this context; their sensitivities are both also outside the range shown in Fig.~\ref{fig:FutureWConstraints}.
\begin{table}[h]
\caption{Current and future (expected) terrestrial constraints on $\tilde b_i \equiv b_i/m_i^2$ and $b_i$ at $1\sigma$ confidence.\label{table:UpperLimits}}
\begin{center}
\begin{tabular}{c||c|c||c|c||c|c}
& $\tilde b_1$ & $b_1$ [eV$^2$] & $\tilde b_2$ & $b_2$ [eV$^2$] & $\tilde b_3$ & $b_3$ [eV$^2$] \\ \hline\hline
Current& $5.7 \times 10^{-2}$ & $5.7\times 10^{-6}$ & $1.6\times 10^{-1}$ & $2.8 \times 10^{-5}$ & $5.3\times 10^{-1}$ & $1.4\times 10^{-3}$ \\ \hline
Future& $5.1\times 10^{-2}$ & $5.1\times 10^{-6}$ & $4.0\times 10^{-2}$ & $7.0 \times 10^{-6}$ & $1.3\times 10^{-2}$ & $3.3\times 10^{-5}$ \\ \hline
\end{tabular}
\end{center}
\end{table}
Taking this set of constraints, we can derive $1\sigma$ upper limits on $\tilde b_i$ as well as $b_i$ given our benchmark $m_1 = 10^{-2}$ eV, which we present in Table~\ref{table:UpperLimits}. While the $\tilde{b}_i$ are useful for dimensionless comparisons, the future sensitivity on the absolute $b_i$ are notable in their own right, demonstrating sensitivity to meV-scale phenomena.
\section{Conclusions}\label{sec:Conclusions}
For decades the neutrino sector has provided us with a unique window through which to study the hidden world of matter. Indeed it remains a theoretically well-motivated location to hunt for new physics. With new neutrino oscillation observatories planned for the near future, it is timely to explore the diverse theoretical and experimental BSM landscape of neutrino oscillations.
In light of this, we have proposed a new general framework to organise future explorations of the neutrino sector, capturing new physics effects on neutrino propagation in a single spectral function. We demonstrated how this language both reproduces conventional neutrino flavour oscillation calculations, and is simultaneously capable of describing the phenomenology of more exotic theoretical models including those with discrete and continuous mass spectra. The relevant phenomenological features can in both cases be mimicked by a `toy' mass-spectrum comprising three top-hat functions with a model-specific choice of breadths. We emphasise that this top-hat set up should not be considered a concrete theoretical model, but moreover as a convenient phenomenological ansatz offering the possibility of model-independent analyses of experimental data. Instead of having to sequentially re-interpret searches for different theoretical models, one can equivalently constrain the `breadth' of neutrino spectral functions and thus probe the nature of neutrino propagation effects directly. In the instance of a positive hint for a non-zero neutrino breadth, whilst there may not be a unique invertible mapping from a given set of top-hat breadths to the true mass spectrum, a preference for top-hat functions of non zero breadth is likely to indicate the presence of extra states in the mass-spectrum, and thus the existence of additional sterile neutrinos.
After discussing how long-distance neutrino measurements can test this non-zero breadth in Section~\ref{sec:Decoherence}, we demonstrated the utility of this approach with terrestrial experiments in Sections~\ref{sec:Constraints}, where we explored the landscape of existing and future oscillation experiments and compared their capacity to probe new physics in neutrino propagation. We found that the long-baseline anti-electron neutrino oscillation experiment KamLAND constrains the breadths of the two lower mass states to remarkable sensitivity but provides no information on the breadth of the third state. Daya Bay and current long-baseline $\nu_\mu$ disappearance searches close this gap somewhat, but current constraints are comparatively weak. The near-future mid-baseline anti-electron neutrino experiment JUNO is projected to significantly improve upon the sensitivities of existing searches, most noticeably for that of the third, highest mass, state.
We highlight that whilst current data is consistent with the predictions of the conventional 3-neutrino model, the possibility of finite breadths are by no means excluded, particularly for the highest mass state, $m_3$. Finding new physics in the neutrino sector is possible, as testified by the rich and diverse range of BSM scenarios considered in literature. Armed with the tools to probe the neutrino mass spectrum in a general manner, this framework offers a useful and previously unexploited manner by which to harness the capabilities of future experiments to answer the question: How broad is a neutrino?
\acknowledgments
We are very grateful to Carlos A. Arg{\'u}elles, Toni Bert\'olez-Mart\'inez, and Jordi Salvado for providing Daya Bay simulation code from Ref.~\cite{Arguelles:2022bvt}, as well as Joachim Kopp for valuable conversations and comments on this draft. HB acknowledges partial support from the STFC Consolidated HEP grants ST/P000681/1 and ST/T000694/1 and thanks members of the Cambridge Pheno Working Group for helpful discussions.
\bibliographystyle{apsrev4-1_title}
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REGHA, Onajite
Mrs Onajite Regha is the Chief Executive Officer/Executive Secretary of the Electronics Payment Providers Association of Nigeria and the initiator of the association which she co-founded with top industry leaders. She has a degree in Mass Communication, a certificate...
ROBERTS, Olufunmilayo Ajike
Mrs Funmilayo Ajike Roberts was called to the Nigerian Bar as a Solicitor and Advocate of the Supreme Court of Nigeria in June 1982 after obtaining her Bachelor of Laws Degree in the Second Class Upper Division from the University of Ife, Ile-Ife, Osun State...
NWACHUKWU, Dr Margaret Ifeanyi
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OYEKANMI, (Mrs.) Ayodele Olufunmilola
LAWSON, (Mrs.) Omotunde Olayinka
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Cette page concerne les évènements survenus en 2021 en Finlande :
Évènement
Pandémie de Covid-19 en Finlande
: La région de Laponie enregistre la température la plus chaude du siècle.
Sport
Championnat de Finlande de hockey sur glace 2020-2021
Championnat de Finlande de hockey sur glace 2021-2022
Championnat de Finlande de football 2021
9- : Championnat du monde juniors de combiné nordique à Lahti.
16- : Championnat d'Europe de badminton par équipes mixtes à Vantaa.
5- : Arctic Race of Norway (cyclisme)
- : Organisation du championnat d'Europe masculin de volley-ball (avec la Pologne, la République Tchèque et l'Estonie).
Culture
Sortie de film
Compartiment n° 6
The Innocents
Création
à Tampere.
Décès
, musicien.
, musicien.
, animateur de radio et de télévision.
, musicien.
Sinikka Nopola, écrivaine et journaliste.
, actrice.
Notes et références
2021 en Finlande
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\section{Introduction}
\label{sec:intro}
The study of the boundary between quantum and classical mechanics raised as one of the most interesting
and challenging issues for the last thirty years~\cite{Zurek91,*HarochePT98,Schlosshauerbook,Schlosshauer05,*Schlosshauer2014}.
Motivated by the necessity of exploring problems like the measurement of quantum properties or the classical limit for a specific quantum model,
different ways to treat the so-called \textit{open quantum systems} has been proposed~\cite{Schlosshauerbook,Breuerbook}.
In particular, the density matrix formalism~\cite{Fano57} becomes an important theoretical frame to explore multipartite
systems with mixed quantum and classical features. From the wide set of problems linked with open systems,
the analysis of the dynamics of a quantum system in contact with \textit{reservoirs}, larger physical systems,
stands as a fundamental quest. Speaking specifically of the ``know-how", master equations become adequate to circumvent the task,
being deduced by tracing out the variables of the reservoirs~\cite{Schlosshauerbook,Breuerbook}. The integro-differential
equations obtained after the application of several approximations permits to summarize the effect of reservoirs via a
Lindblad operator~\cite{Gorini76,Lindblad76}. Extensions has been made in order to include memory effects,
known as non-markovian approaches~\cite{zhang2012general,BreuerRMP16,*Scorpo17,*Bernardes2017},
which has been observed in carefully prepared experimental setups~\cite{Li11,*liu2011experimental}.
On the other hand, since the seminal work of Jauho, Wingreen and Meir~\cite{Jauho94} about non-equilibrium quantum transport,
a wealth of theoretical and experimental works have investigated the transport phenomena under the action of time-vary fields~\cite{platero2004photon,*cota2005ac,souza2007spin,*souza2007transient,trocha2010beating,*perfetto2010correlation,Assuncao13,*odashima2017time}.
Recently, open quantum systems out-of-equilibrium have been theoretically investigated in quantum dots attached to leads in the presence of
photonic or phononic fields~\cite{liu2014photon,*kulkarni2014cavity,*hartle2015effect,*purkayastha2016out,*agarwalla2016tunable,*reichert2016dynamics,*mann2016dissipative}.
In this specific context, a method to deal with transport problems in semiconductor nanoestrutures has been developed by W.-M. Zhang
and co-workers~\cite{Yang17,*Xiong15,Zhang12prl,Jin10}, mixing the density operator formalism with nonequilibrium Green functions. From the point of view of the treatment of time-memory effects, this method is powerful because its direct application on the description of non-markovian setups. Still, the approach requires some familiarity with Keldysh non-equilibrium Green function technique.
Here, we present a formalism that offers an alternative path with immediate application on the study of dynamics of a general configuration of open quantum systems far from equilibrium. Using the Fermion-to-Qubit (FTQ) mapping, we provide a straightforward recipe to construct both, multi-partite Hamiltonian and Lindbladians, as tensor products of Pauli matrices. The FTQ mapping also sets automatically the complete computational basis to analyze the dynamics, written in terms of occupied and non-occupied states. The formalism presented here opens the possibility of future applications in the context of fermionic quantum computation~\cite{Bravyi02}.
The paper is organized as follows: in Sec.~\ref{sec:genform}, we set the foundations of our formalism, starting with the definition of a general form for fermionic operators. It is deduced an expression for a generic reservoir-system coupling, which is the key behind the construction of super-operators for open quantum dynamics. Section~\ref{sec:noninteract} presents the deduction of Lindbladian super-operators, for the case of non-interacting reservoirs considering a markovian condition. Section~\ref{sec:application} is devoted to the discussion of an application of our formalism on the context of transport phenomena. We focus on the behavior of electrons on charged quantum molecules, being probed by a nearby narrow conduction channel describing the action of a charge sensor. The behavior of populations, the entanglement dynamics, and the purity permits to conclude that the probe induces dephasing, a decoherence process which acts over the quantum dynamics of the coupled molecules. In Sec.~\ref{sec:summary} we summarize our results.
\section{Fermion-to-qubit mapping and the general Hamiltonian}
\label{sec:genform}
Consider the open multipartite system illustrated in Fig.~\ref{fig:system} with $N$ subsystems in space $\mathcal{S}$, in contact with $M$ reservoirs, each with $K_n$ inner states, defined in a space denoted as $\mathcal{R}$. We use $i$ as the index of the $i$-th subsystem in $\mathcal{S}$ so $i=1,2..,N$. In the reservoir space, we use two indexes: $n$, which labels the $n$-th reservoir, with $n=1,...,M$, and $k$, indicating the $k$-th state with $k=1,...,K_n$. The dimension of the whole, system and reservoirs, is given by $D=N+\sum_{n=1}^{M} K_n$.
The general Hamiltonian can be written as $H(t)=H_0+V(t)$ where
\begin{eqnarray}
\label{eq:Hamilterms}
H_0&=&H_\mathcal{S} \otimes I_\mathcal{R}^{\otimes \left(D-N\right)} + I_\mathcal{S}^{\otimes N} \otimes H_{\mathcal{R}},\nonumber\\
V(t)&=&\sum_{i=1}^{N}\sum_{n=1}^{M}\sum_{k=1}^{K_n}u_n(t)V_{i,(n,k)}d^{\dagger}_{(n,k)}d_i+\mathrm{h. c.}
\end{eqnarray}
where $H_{\mathcal{S}(\mathcal{R})}$ is the Hamiltonian of the multipartite system (reservoirs)
without coupling and the term $V(t)$ describes the coupling as a hopping process:
a particle is annihilated ($d_i$) at $\mathcal{S}$ at the same time that it is created ($d_{(n,k)}^\dagger$)
in $\mathcal{R}$ and vice versa. Inside the coupling term, the function $u_n(t)$ is a time-dependent parameter~\cite{Jauhobook}, and $V_{i,(n,k)}$ provides the coupling strength between system and reservoirs.
\begin{figure}[tb]
\centering\includegraphics[width=1\linewidth]{figure1}
\caption{Illustration of a generic setup composed of a multipartite system $\mathcal{S}$ interacting with $M$ reservoirs.}
\label{fig:system}
\end{figure}
Now we apply the FTQ mapping to this general system.
The operator $d_m$ is defined by using the Jordan-Wigner transformation~\cite{Schallerbook} as~\footnote{This tool has been applied in problems involving interacting quantum dynamics, one being the classic work of Haldane~\cite{Haldane80}
who show the equivalence between the $XXZ$ model for a Heisenberg-Ising chain and interacting spinless fermions.
Other example of the application of this transformation on the physics of strong correlated systems are the use
of a generalized of the transformation in the solution of some cases of lattice models~\cite{Batista01}.}
\begin{equation}
d_m=\sigma_z^{\otimes\left(m-1\right)}\otimes\sigma_{-}^{(m)}\otimes I^{\otimes\left(D-m\right)},
\label{eq:mapdm}
\end{equation}
where the index $m$ now runs over both, $\mathcal{S}$ and $\mathcal{R}$ indexes, $\sigma_z$ is a Pauli matrix, $I$ is the $2\times2$ identity matrix, and $O^{\otimes L}$ indicates a succession of $L$ tensorial products of operator $O$. The creation operator $d^{\dagger}_m$ is obtained by replacing $\sigma_-^{(m)}$ by $\sigma_+^{(m)}$ with $\sigma_{\pm}=(\sigma_x \pm i \sigma_y)/2$. It is straightforward to prove that $d_m$ and $d_m^\dagger$ follows the anticommutation relations $\left\{d_m,d^{\dagger}_l\right\}=\delta_{m,l}$, $\left\{d_m,d_l\right\}=\left\{d^{\dagger}_m,d^{\dagger}_l\right\}=0$.
The explicit form for $V(t)$ is now written as:
\begin{eqnarray}
\label{eq:V}
V(t)&=&\sum_{i,n,k} u_n (t)V_{i,(n,k)}\left(S_i \otimes \sigma_z^{\otimes K_1} \otimes \sigma_z^{\otimes K_2} \otimes \cdot \cdot \cdot \otimes \right. \\
&&\left. \sigma_z^{\otimes K_{n-1}} \otimes R_{n,k}^\dagger \otimes I^{\otimes K_{n+1}} \cdot \cdot \cdot \otimes I^{\otimes K_M}\right) +\mathrm{h. c.},\nonumber
\end{eqnarray}
where
\begin{eqnarray}
\label{eq:SRoperators}
S_i&=&I^{\otimes\left(i-1\right)}\otimes\sigma_-^{(i)}\otimes\sigma_z^{\otimes\left(N-i\right)},\\
R_{n,k}^{\dagger}&=&\sigma_z^{\otimes k-1} \otimes \sigma_+^{(k)} \otimes I^{\otimes K_n - k}\nonumber.
\end{eqnarray}
The $S_i$ ($R_{n,k}^{\dagger}$) operators run only over the subspace $\mathcal{S}$ ($n$-th subspace of $\mathcal{R}$), and they preserve the fermionic anticommutation relations.
In order to describe the quantum evolution of the entire $D$ dimensional system, we write the Von Neumann equation
in the interaction picture, $\dot{\rho}(t)= \mathcal{L}(t) \rho_0 + \int_{0}^t dt_1 \mathcal{L}(t) \mathcal{L}(t_1)\rho(t_1)$
where $\mathcal{L}(t)$ is the Liouvillian superoperator,
$\mathcal{L}(t)\rho(t_1)=-i [V_I(t),\rho(t_1)]$ ($\hbar=1$), and $V_I(t)$
is the coupling term defined as $V_I(t)=e^{iH_0t}V(t)e^{-iH_0t}$
with $e^{iH_0t}=e^{iH_\mathcal{S}t}\otimes e^{iH_\mathcal{R}t}$.
This transformation applies over the tensorial product inside
Eq.~(\ref{eq:V}) resulting on a tensorial product between system operators $S_i(t)=e^{iH_\mathcal{S}t}S_ie^{-iH_\mathcal{S}t}$
and similar terms for the reservoirs. Notice that, at the moment,
we are treating the full form of the system-reservoir interaction,
using a mathematical tool to distinguish the system from the reservoirs without a loss of generality.
\section{Non-interacting reservoirs and the Markov approximation}
\label{sec:noninteract}
Let us assume the Born approximation, $\rho(t)=\rho_\mathcal{S}(t) \otimes \rho_{\mathcal{R}}$,
where $\rho_\mathcal{S}(t)$ is the reduced density matrix
of multipartite system and $\rho_{\mathcal{R}}$ are the
density matrices for reservoirs, thus
$\rho_{\mathcal{R}}=\rho_1 \otimes \rho_2 \otimes \cdot \cdot \cdot \otimes \rho_M$.
We now consider the effect of non-interacting reservoirs, set in thermodynamical equilibrium, each described by
Hamiltonian $H_{\mathcal{R}}=\bigoplus_{n=1}^M H^{(n)}$ with $H^{(n)}=\bigoplus_{k=1}^{K_n}H^{(n,k)}=\bigoplus_{k=1}^{K_n} \varepsilon_n^{(k)} (\sigma_+ \sigma_-)^{(k)}$ and $\varepsilon_n^{(k)}$ being the energy of the $k$-th mode of reservoir $n$.
The density matrix for each reservoir is a mixed state described by
$\rho_n = \frac{1}{Z} \mathrm{Exp}[-\beta \bigoplus_{k=1}^{K_n} \epsilon_{n}^{(k)} (\sigma_+ \sigma_{-})^{(k)}]$,
where $\beta=1/(k_B T)$, $\epsilon_{n}^{(k)}$ is the free particle energy measured
from the chemical potential $\mu_n$, and $Z$ is the partition function.
After taking the partial trace over reservoirs degrees of freedom and
ignoring the null terms, we find
\begin{widetext}
\begin{eqnarray}
\label{eq:fulldotrho}
\dot{\rho}_\mathcal{S}(t)&=&-\frac{1}{2\pi} \int_0^t dt_1 \sum_{i,j,n,k} \large{\{}\Gamma_{i,j}^{n,k}(t,t_1)S_i(t)S_j^\dagger(t_1) \rho_\mathcal{S}(t_1)
\mathrm{Tr}_{\mathcal{R}}[ \rho_1 \otimes \rho_2 \otimes \cdot \cdot \cdot \otimes \rho_{n-1} \otimes
R_{n,k}^\dagger(t) R_{n,k}(t_1) \rho_n \otimes \cdot \cdot \cdot \otimes \rho_{M} ]\\
& +&\Gamma_{i,j}^{n,k}(t_1,t)S_i^\dagger(t) S_j(t_1) \rho_\mathcal{S}(t_1)
\mathrm{Tr}_{\mathcal{R}}[ \rho_1 \otimes \rho_2 \otimes \cdot \cdot \cdot \otimes \rho_{n-1} \otimes
R_{n,k}(t) R_{n,k}^\dagger(t_1) \rho_n \otimes \cdot \cdot \cdot \otimes \rho_{M}]\nonumber \\
&-&\Gamma_{i,j}^{n,k}(t_1,t) S_j(t_1) \rho_\mathcal{S}(t_1) S_i^\dagger(t)
\mathrm{Tr}_{\mathcal{R}}[\sigma_z^{\otimes K_1} \rho_1 \sigma_z^{\otimes K_1} \otimes\cdot \cdot \cdot \otimes \sigma_z^{\otimes K_{n-1}} \rho_{n-1} \sigma_z^{\otimes K_{n-1}}\otimes R_{n,k}^\dagger(t_1) \rho_n R_{n,k}(t) \otimes \rho_{n+1}\cdot \cdot \cdot \otimes \rho_{M}]\nonumber \\
&-&\Gamma_{i,j}^{n,k}(t,t_1)S_j^\dagger (t_1) \rho_\mathcal{S}(t_1) S_i(t)
\mathrm{Tr}_{\mathcal{R}}[ \sigma_z^{\otimes K_1} \rho_1 \sigma_z^{\otimes K_1} \otimes \cdot \cdot \cdot \otimes \sigma_z^{\otimes K_{n-1}} \rho_{n-1} \sigma_z^{\otimes K_{n-1}}\otimes R_{n,k}(t_1) \rho_n R_{n,k}^\dagger(t) \otimes \rho_{n+1} \cdot \cdot \cdot \otimes \rho_{M}] \nonumber\\
&+& h.c. \large{\}},\nonumber
\end{eqnarray}
\end{widetext}
where $R^{(\dagger)}_{n,k}(t)=e^{iH^{(n,k)}t}R^{(\dagger)}_{n,k}e^{-iH^{(n,k)}t}$ and $ \Gamma_{i,j}^{n,k}(t,t_1) = 2 \pi V_{i,(n,k)} V_{j,(n,k)}^\star u_n(t) u_n^\star(t_1)$. The equation for the reduced density matrix of the system is written as
\begin{eqnarray}
\label{eq:rhodotbftint}
\dot{{\rho}}_\mathcal{S}(t)&=&-\frac{1}{2\pi} \sum_{i,j}\sum_{n,k}\Big\{f_{n,k} \int_0^t dt_1\Gamma_{i,j}^{n,k}(t,t_1)e^{i \varepsilon_n^{(k)} (t-t_1)} \\
&&\times\left[{S}_i(t){S}_j^\dagger(t_1){\rho}_\mathcal{S}(t_1)-{S}_j^\dagger (t_1) {\rho}_\mathcal{S}(t_1) {S}_i(t)\right]\nonumber \\
&&+(1-f_{n,k})\int_0^t dt_1\Gamma_{i,j}^{n,k}(t_1,t) e^{-i \varepsilon_n^{(k)} (t-t_1)}\nonumber\\
&&\times \left[{S}_i^\dagger(t) {S}_j(t_1) {\rho}_\mathcal{S}(t_1)-{S}_j(t_1) {\rho}_\mathcal{S}(t_1) {S}_i^\dagger(t)\right]+ h.c. \Big\},\nonumber
\end{eqnarray}
after using the Baker-Hausdorff Lemma on the calculation of $\mathrm{Tr}_n \{ R_{n,k} (t_1) \rho_n R_{n,k}^\dagger (t) \} = f_{n,k} e^{i \varepsilon_n^{(k)} (t-t_1)}$ and $\mathrm{Tr}_n \{ R_{n,k}^\dagger (t_1) \rho_n R_{n,k} (t) \} = (1-f_{n,k})e^{-i\varepsilon_n^{(k)}(t-t_1)}$, where $f_{n,k}=1/[1+e^{\beta \epsilon_n^{(k)}}]$
is the Fermi distribution function for reservoir
$n$~\footnote{Additionally, $\mathrm{Tr}_n\{ \rho_n \}=\mathrm{Tr}_n\{ \sigma_z^{\otimes K_n} \rho_n \sigma_z^{\otimes K_n } \}=1$}.
At this point, we consider the wide-band limit~\cite{Jauhobook}.
We set $V_{i,(n,k)}=V_{i,n}$, meaning that all inner states on reservoir $n$ have the same coupling with the system.
The sum over $k$ turns into an integral, $\sum_k[...]\rightarrow \int[...] \mathfrak{D}_n(\varepsilon)d\varepsilon$,
where the density of states is assumed constant, $\mathfrak{D}_n(\varepsilon)=\mathfrak{D}_n$.
The bias voltage $eV$ is given by $\mu_n-\mu_m =eV$, where the source (drain) chemical potential
is $\mu_n$ ($\mu_m$). If $eV$ is high and considering low values of temperature,
we can assume that $f_{n(m),k}=f_{n(m)}=1(0)$ resulting in,
\begin{equation}
\label{rhoint2}
\dot{{\rho}}_{S}(t)=-\frac{1}{2} \sum_{i,j,n} \Gamma_{i,j}^{(n)}(t) \left[F_n(t,i,j)+G_n(t,i,j)\right],
\end{equation}
with
\begin{eqnarray}
\label{rhoint2p2}
F_n(t,i,j)&=& f_n\left[ {S}_i (t){S}_j^\dagger (t){\rho}_\mathcal{S} (t)-{S}_i^\dagger (t) {\rho}_\mathcal{S} (t) {S}_j (t)\right.\\
&&\left.+{\rho}_\mathcal{S} (t) {S}_j (t) {S}_i^\dagger (t) - {S}_j^\dagger (t) {\rho}_\mathcal{S} (t) {S}_i (t)\right]\nonumber \\
G_n(t,i,j)&=&\left(1-f_n\right)\left[{S}_i^\dagger (t) {S}_j (t) {\rho}_\mathcal{S} (t) - {S}_i (t) {\rho}_\mathcal{S} (t) {S}_j^\dagger (t)\right.\nonumber\\
&&\left. +{\rho}_\mathcal{S} (t) {S}_j^{\dagger} (t) {S}_i(t) - {S}_j(t) {\rho}_\mathcal{S} (t) {S}_i^{\dagger} (t)\right],\nonumber
\end{eqnarray}
and $\Gamma_{i,j}^{(n)}(t)=2\pi V_{i,n}V^{*}_{j,n}\mathfrak{D}_n\left|u_n(t)\right|^2$.
We apply the $\mathbf{vec}$ operation~\cite{Havel03} to both sides of Eq.~(\ref{rhoint2}),
obtaining $\frac{d}{dt} \mathbf{vec}[{\rho}_\mathcal{S}] (t) = - \frac{1}{2} {\mathbf{\Gamma}}(t) \mathbf{vec}[ {\rho}_\mathcal{S}] (t)$,
where ${\mathbf{\Gamma}}(t)$ is $N^2\times N^2$ supermatrix defined as
\begin{eqnarray}
\label{eq:supmatgamma}
{\mathbf{\Gamma}}(t) &=& \sum_{i,j,n} \Gamma^{(n)}_{i,j}(t) \left( f_n\left\{I_\mathcal{S} \otimes{S}_i (t) {S}_j^\dagger (t)- {S}^T_j (t) \otimes {S}_i^\dagger (t) \right.\right.\nonumber \\
&&\left.\left. +[{S}_j (t){S}_i^\dagger(t)]^T \otimes I_\mathcal{S} - {S}^T_i(t)\otimes{S}_j^\dagger(t)\right\}\right.\nonumber\\
&&\left.+(1-f_n)\left\{ I_\mathcal{S} \otimes {S}_i^\dagger (t){S}_j (t) - {S}_j^{\dagger T} (t)\otimes{S}_i (t)\right.\right.\nonumber \\
&&\left.\left. +[{S}_j^\dagger(t){S}_i (t)]^T \otimes I_\mathcal{S} - {S}_i^{\dagger T}(t)\otimes{S}_j (t) \right\}\right),
\end{eqnarray}
where the superscript $T$ means matrix transposition.
Writing the reduced density matrix in the Schr\"odinger picture,
$\mathbf{vec}[\rho^s_\mathcal{S}] (t) = [\mathrm{Exp}(i H_\mathcal{S}^T t) \otimes \mathrm{Exp}(-i H_\mathcal{S} t) ] \mathbf{vec}[ {\rho}_\mathcal{S}] (t)$,
and taking the time derivative we find the differential equation,
\begin{equation}\label{eq:drhodt}
\frac{d}{dt}\mathbf{vec}[\rho^s_\mathcal{S}] (t) = \mathcal{L} (t) \mathbf{vec}[\rho^s_\mathcal{S}] (t),
\end{equation}
where the superoperator $\mathcal{L}$ is given by
\begin{equation}
\label{eq:rhodotshro}
\mathcal{L} (t)= \mathcal{L}_0 - \frac{1}{2} \sum_{i,j,n}\Gamma^{(n)}_{i,j}(t)\left[f_n\mathcal{L}^{+}_{ij}+\left(1-f_n\right)\mathcal{L}^{-}_{ij}\right],
\end{equation}
with
\begin{eqnarray}
\label{eq:rhodotshrop2}
\mathcal{L}_0 &=& -i (I_\mathcal{S} \otimes H_\mathcal{S} - H_\mathcal{S}^T \otimes I_\mathcal{S})\nonumber\\
\mathcal{L}^{+}_{ij}&=&I_\mathcal{S} \otimes S_iS^{\dagger}_j+S_iS^{\dagger}_j\otimes I_\mathcal{S}-S^{\dagger}_i\otimes S^{\dagger}_j-S^{\dagger}_j\otimes S^{\dagger}_i\nonumber\\
\mathcal{L}^{-}_{ij}&=&I_\mathcal{S} \otimes S^{\dagger}_i S_j+S_i^{\dagger}S_j\otimes I_\mathcal{S}-S_i\otimes S_j-S_j\otimes S_i.\nonumber
\end{eqnarray}
Eq. (\ref{eq:drhodt}) has the formal solution
\begin{equation}\label{rhointegrated}
\mathbf{vec}[\rho^s_S] (t) = \mathcal{T} e^{\int_0^t d\tau \mathcal{L}(\tau)} \mathbf{vec}[\rho^s_S] (0),
\end{equation}
where $\mathcal{T}$ is the chronological time-ordering operator.
Writing the superoperators above in terms of Pauli matrices, we arrive to
\begin{widetext}
\begin{eqnarray}
\label{eq:Lijsigma}
\mathcal{L}^{\pm}_{i=j}&=&I^{\otimes (N+i-1)} \otimes (\sigma_{\mp} \sigma_{\pm})^{(i)} \otimes I^{\otimes (N-i)}+ I^{\otimes (i-1)} \otimes (\sigma_{\mp} \sigma_{\pm})^{(i)} \otimes I^{\otimes (2N-i)}\\
&&-2 I^{\otimes (i-1)} \otimes \sigma_{\pm}^{(i)} \otimes \sigma_z^{\otimes (N-i)}\otimes I^{\otimes (i-1)} \otimes \sigma_{\pm}^{(i)} \otimes \sigma_z^{\otimes (N-i)},\nonumber\\
\mathcal{L}^{\pm}_{i<j}&=&\pm I^{\otimes (N+i-1)} \otimes \sigma_{\mp}^{(i)}\otimes\sigma_z^{\otimes (j-i-1)}\otimes \sigma_{\pm}^{(j)} \otimes I^{\otimes (N-j)}
\pm I^{\otimes (i-1)} \otimes \sigma_{\mp}^{(i)}\otimes\sigma_z^{\otimes (j-i-1)}\otimes \sigma_{\pm}^{(j)} \otimes I^{\otimes (2N-j)}\nonumber\\
&-&I^{\otimes (i-1)} \otimes \sigma_{\pm}^{(i)} \otimes \sigma_z^{\otimes (N-i)} \otimes I^{\otimes (j-1)} \otimes \sigma_{\pm}^{(j)} \otimes \sigma_z^{\otimes (N-j)}
-I^{\otimes (j-1)} \otimes \sigma_{\pm}^{(j)} \otimes \sigma_z^{\otimes (N-j)} \otimes I^{\otimes (i-1)} \otimes \sigma_{\pm}^{(i)} \otimes \sigma_z^{\otimes (N-i)},\nonumber\\
\mathcal{L}^{\pm}_{i>j}&=&\pm I^{\otimes (N+j-1)} \otimes \sigma_{\pm}^{(j)}\otimes\sigma_z^{\otimes (i-j-1)}\otimes \sigma_{\mp}^{(i)} \otimes I^{\otimes (N-i)}\pm I^{\otimes (j-1)} \otimes \sigma_{\pm}^{(j)}\otimes\sigma_z^{\otimes (i-j-1)}\otimes \sigma_{\mp}^{(i)} \otimes I^{\otimes (2N-i)}\nonumber \\
&-&I^{\otimes (i-1)} \otimes \sigma_{\pm}^{(i)} \otimes \sigma_z^{\otimes (N-i)} \otimes I^{\otimes (j-1)} \otimes \sigma_{\pm}^{(j)} \otimes \sigma_z^{\otimes (N-j)}-I^{\otimes (j-1)} \otimes \sigma_{\pm}^{(j)} \otimes \sigma_z^{\otimes (N-j)} \otimes I^{\otimes (i-1)} \otimes \sigma_{\pm}^{(i)} \otimes \sigma_z^{\otimes (N-i)},\nonumber
\end{eqnarray}
\end{widetext}
Eqs.~(\ref{eq:Lijsigma}) provide the recipe to construct Lindbladian operators for fermionic systems,
based on fermion-to-qubit mapping.
In order to apply Eqs.~(\ref{eq:Lijsigma}), it is enough to specify $N$,
the total number of sites or levels being considered in the system.
Because the expressions are explicit tensorial products, they make numerical implementations
very straightforward.
\section{Quantum dynamics on coupled quantum molecules}
\label{sec:application}
In this section, we proceed to apply the formalism in the context of charged quantum dots~\cite{Kobayashi95,Tarucha96}. In this physical setup, metallic gates are used to confine electrons within a small region of AlGaAs-GaAs~\cite{Fujisawa98}. Manipulating the chemical potential of sources and drains, both described as electronic reservoirs, charges can be introduced inside the nanoestructure~\cite{Oosterkamp98}. Coherent tunneling between two adjacent quantum dots, which form an artificial molecule, permits the codification of information in a qubit, once it is possible to define a two-level system~\cite{Hayashi03}. From the point of view of quantum information processing, two-qubit operations are necessary for the implementation of a universal set of quantum gates. The Coulomb interaction between charges in two molecules~\cite{Shinkai09,Shinkai07} provides a rich dynamics, enough to implement quantum gates~\cite{Fujisawa11}, and the creation of maximally entangled states~\cite{Fanchini10,Oliveira15}.
We are interested on analyze the quantum dynamics of the coupled quantum molecules, under the effect of a small open quantum system. The last consists on an extra quantum dot with a source and a drain which can be used to charge or discharge the nanoestructure. The configuration of the complete physical system is shown in Fig.~\ref{fig:systemapp}. When a charge occupies the electronic level of this fifth dot, the electrostatic interaction between the charge and the electron in the molecule works as a capacitive probe. The tunneling between probe and molecules is forbidden, so there is no loss of electronic population of the molecule.
Using the formalism of Sec.~\ref{sec:genform}, it is straightforward to obtain the $16$D complete computational basis for the two quantum molecules system, denoted as $\mathcal{M}$. The basis is composed not only of the two-qubit subspace
$\mathcal{M}_{2\mathrm{QB}}=\left\{\ket{1010},\ket{0101},\ket{1001},\ket{0110}\right\}$, but also of others like
the one-particle subspace
$\mathcal{M}_1=\left\{\ket{1000},\ket{0100},\ket{0010},\ket{0001}\right\}$,
the two-particles per molecule subspace
$\mathcal{M}_2=\left\{\ket{1100},\ket{0011}\right\}$,
the three-particle subspace $\mathcal{M}_3=\left\{\ket{1110},\ket{1101},\ket{0111},\ket{1011}\right\}$,
the four-particle state $\ket{1111}$, and the vacuum $\ket{0000}$. The basis allows the description of closed and open quantum dynamics, as the effect of sources, which take an initial vacuum state to some occupied state (eventually the full-occupation state, $|1111\rangle$), or drains, that take any occupied state to the vacuum state, $|0000\rangle$.
For this specific application, we consider that a previous process of initialization prepared the two-qubit system in one of the four state of subspace $\mathcal{M}_{2\mathrm{QB}}$. The Hamiltonian for the two qubits on the coupled quantum molecules is written as
\begin{eqnarray}
H_\mathcal{M}&=&H_0+H_T+H_U\nonumber\\
&=&\bigoplus_{i=1}^4 \varepsilon_i^{(\mathcal{M})}P^{(i)}_++H_T+H_U
\end{eqnarray}
where
\begin{eqnarray}
\label{eq:hsystem}
H_T&=&-\Delta_{12}\left[\sigma^{(1)}_-\otimes\sigma^{(2)}_+\otimes I^{\otimes 2}+\sigma^{(1)}_+\otimes\sigma^{(2)}_-\otimes I^{\otimes 2}\right]\nonumber\\
&-&\Delta_{34}\left[I^{\otimes 2}\otimes\sigma^{(3)}_{-}\otimes\sigma^{(4)}_{+}+I^{\otimes 2}\otimes\sigma^{(3)}_{+}\otimes\sigma^{(4)}_{-}\right],\nonumber\\
H_U&=&U_a\left[P^{(1)}_{+}\otimes I \otimes P^{(3)}_{+} \otimes I + I \otimes P^{(2)}_{+} \otimes I \otimes P^{(4)}_{+}\right]\nonumber \\
&+&U_b\left[P^{(1)}_{+} \otimes I^{\otimes 2}\otimes P^{(4)}_{+}+I \otimes P^{(2)}_{+} \otimes P^{(3)}_{+} \otimes I\right], \nonumber
\end{eqnarray}
with $P_{+}=\sigma_{+}\sigma_{-}=\ket{+}\bra{+}$ and the index $1$ to $4$ describes a specific dot on quantum molecules. The term $H_0$ is the uncoupled term with the electronic energies $\varepsilon_i^{(\mathcal{M})}$, while $H_{T}$ and $H_{U}$ terms describe electronic tunneling inside each molecule and the Coulomb interactions between electrons, respectively.
\begin{figure}[tb]
\centering\includegraphics[width=0.7\linewidth]{figure2}
\caption{Scheme of the coupled quantum molecules with a probe: the first (second) molecule is composed by dots $1$ and $2$ ($3$ and $4$). Tunneling is allowed inside the molecules, described by parameter $\Delta_{ij}$. The electrostatic interactions between electrons inside the molecules, with parameter $U_{a(b)}$, provide the coupling between the molecules. The probe consists of a narrow channel provided by dot $5$, attached to a source (S) and a drain (D), coupled capacitive with $U_{\mathrm{p}}$ as illustrated.}
\label{fig:systemapp}
\end{figure}
When the probe is introduced, the full Hamiltonian reads as
\begin{equation}
\label{eq:hfull}
H_{\mathrm{full}}=H_\mathcal{S}\otimes I+I^{\otimes 4}\otimes \varepsilon_5P^{(5)}_+ +U_{\mathrm{p}}P^{(1)}_+\otimes I^{\otimes 3}\otimes P^{(5)}_+,
\end{equation}
where the last term describes the capacitive coupling between system and the extra dot with parameter $U_{\mathrm{p}}$.
To describe the action of both, source and drain of charge on the $5$-th dot, we use Eq.~(\ref{eq:Lijsigma}) to obtain the Lindblad term
\begin{eqnarray}
\mathcal{L}_{55}^{\pm} = && I^{\otimes 9} \otimes (\sigma_{\mp} \sigma_{\pm})^{(5)} + I^{\otimes 4} \otimes (\sigma_{\mp} \sigma_{\pm})^{(5)} \otimes I^{\otimes 5} \nonumber \\
&& -2 I^{\otimes 4} \otimes \sigma_{\pm}^{(5)} \otimes I^{\otimes 4} \otimes \sigma_{\pm}^{(5)}.
\label{eq:lindblad5dot}
\end{eqnarray}
Our goal is to check the quantum dynamics at the specific condition for generation of maximally entangled states~\cite{Oliveira15}. We start solving numerically Eq.~(\ref{eq:drhodt}), considering the terms of Eq.~(\ref{eq:lindblad5dot}). The solution, $\rho_{\mathcal{S}_5}(t)$, describes the dynamics of system and probe. Then, by tracing out the $5$-th dot, the behavior of the two-qubit system $\mathcal{S}$ (dots $1$ to $4$) is described by the reduced density matrix,
\begin{equation}
\rho_{\mathcal{M}}(t)=\mathrm{Tr}_{5} [\rho_{\mathcal{S}_5}(t)].
\end{equation}
The average occupation of $i$-th dot ($i=1...4$) is given by
\begin{equation}
\langle \hat{N}_i \rangle = \mathrm{Tr} \{ \hat{N}_i \rho_{\mathcal{M}}(t) \},
\end{equation}
where the operator $\hat{N}_i$ is defined as
\begin{equation}
\hat{N}_i=I^{\otimes i-1} \otimes P_+^{(i)} \otimes I^{\otimes 4-i}.
\end{equation}
To quantify the probabilities of occupation for each state on the two-qubit subspace $\mathcal{M}_{2\mathrm{QB}}$, we calculate the cross-population averages defined as,
\begin{eqnarray}
\label{eq:populations2qb}
P_{\ket{1001}}&=&\langle \hat{N}_1 (I^{\otimes 4}-\hat{N}_2) (I^{\otimes 4}-\hat{N}_3) \hat{N}_4 \rangle,\\
P_{\ket{0110}}&=&\langle (I^{\otimes 4}-\hat{N}_1) \hat{N}_2 \hat{N}_3 (I^{\otimes 4}-\hat{N}_4) \rangle,\nonumber\\
P_{\ket{1010}}&=&\langle \hat{N}_1 (I^{\otimes 4}-\hat{N}_2) \hat{N}_3 (I^{\otimes 4}-\hat{N}_4) \rangle\nonumber\\
P_{\ket{0101}}&=&\langle (I^{\otimes 4}-\hat{N}_1) \hat{N}_2 (I^{\otimes 4}-\hat{N}_3) \hat{N}_4 \rangle\nonumber.
\end{eqnarray}
As initial condition, we assume that the initialization process prepares the state $\ket{\Psi(0)}=|1001\rangle$ so $P_{|1001\rangle}(0)=1$ ($\langle \hat{N}_1 \rangle = \langle \hat{N}_4 \rangle = 1$). The effect of the probe over the evolution of cross-populations, Eqs.~(\ref{eq:populations2qb}), is shown in Fig.~\ref{fig:crosspop}, considering the physical parameters used to obtain maximally entangled states~\cite{Oliveira15}. For the sake of comparison, we include in the inset the evolution of the same quantities when the probe is turned off, considering a shorter time scale. The periodic coherent dynamics from Ref.~\cite{Oliveira15} is recovered from our approach, where a Bell state is dynamically generate at the times when $P_{|1001\rangle}=P_{|0110\rangle}=0.5$, while $P_{|1010\rangle}=P_{|0101\rangle}=0$.
When the probe is turned on, as a current passes through the narrow conduction channel in dot 5, the effect is to induce an attenuation of the coherent oscillations of populations $P_{|1001\rangle}$ and $P_{|0110\rangle}$, black and red lines on Fig.~\ref{fig:crosspop}(a) respectively. Additionally, we note the increase of populations $P_{|1010\rangle}$ and $P_{|0101\rangle}$, as can be seen from green and blue lines on Fig.~\ref{fig:crosspop}(b). Calculations of the value of population for states in subspaces of $\mathcal{M}$ different from $\mathcal{M}_{2\mathrm{QB}}$ shows that the charges remains confined at the quantum molecules, as expected. At long times, the population for each state on subspace $\mathcal{M}_{2\mathrm{QB}}$ tend to $0.25$ in the stationary regime.
\begin{figure}[tb]
\centering\includegraphics[width=1\linewidth]{figure3}
\caption{Time evolution of cross-populations, Eqs.~(\ref{eq:populations2qb}), for the states belonging to the subspace $\mathcal{M}_{2QB}$, considering probe coupling $U_{\mathrm{p}}=3J$ and $\Gamma_{55}^{(5)}=J$. (a) $P_{\ket{1001}}$ (black line) and $P_{\ket{0110}}$ (red line); (b) $P_{\ket{1010}}$ (green line) and $P_{\ket{0101}}$ (blue line). Inset: Time evolution of all cross-populations (same color choices) when the probe is turned off ($U_{\mathrm{p}}=0$). Physical parameters are those for creation of Bell states: $\Delta_{12}=\Delta_{34}=\Delta=\sqrt{3}/8 J$, $\varepsilon_{i}^{(\mathcal{M})}=0$, $U_a=(3/4)J$, $U_b=(1/4)J$. }
\label{fig:crosspop}
\end{figure}
To check the dynamics of populations inside each molecule, we calculate the occupations for the single $i$-th quantum dot, $\langle \hat{N}_i \rangle$.
The results for the first molecule (dots $1$ and $2$) are shown in Fig.~\ref{fig:moleculepop}(a), for the same initial condition and physical parameters of Fig.~\ref{fig:crosspop}. If probe is turned off, the occupations $\langle \hat{N}_1 \rangle$ and $\langle \hat{N}_2 \rangle$, shown with dashed lines, develop periodic population inversions, which is a signature of the coherent tunneling between the dots inside the molecule. The same behavior, not shown here, is obtained for the second molecule (dots $3$ and $4$). Once the probe is turned on, the coherent dynamics of $\langle \hat{N}_i \rangle$ is attenuated, as observed in Fig.~\ref{fig:crosspop} for cross-populations. It becomes clear that the second molecule is less affected by the probe, once the oscillations of $\langle N_3\rangle$ and $\langle N_4\rangle$ survive longer than those for the first molecule.
\begin{figure}[tb]
\centering\includegraphics[width=1\linewidth]{figure4}
\caption{Time evolution of the populations of the quantum dots in coupled quantum molecules, for the same physical parameters on Fig~\ref{fig:crosspop}: (a) $\langle \hat{N}_1 \rangle$ (black line) and $\langle \hat{N}_2 \rangle$ (red line), corresponding to the molecule coupled with the probe. We also plot the behavior of both, $\langle \hat{N}_1 \rangle$ (dashed black line) and $\langle \hat{N}_2 \rangle$ (dashed red line) when the probe is turned off ($U_{\mathrm{p}}=0$). (b) $\langle \hat{N}_3 \rangle$ (blue line) and $\langle \hat{N}_4 \rangle$ (green line), corresponding to the second molecule.}
\label{fig:moleculepop}
\end{figure}
The exact nature of the asymptotic state is the question that rises from the analysis presented above. The occupations of the dots are not able to distinguish between quantum superpositions and mixed states, so it is convenient to analyze the quantum dynamics using the tools of quantum information. Because the physical conditions used for our calculations are the same for the dynamical generation of Bell states, it is interesting to check the behavior of the degree of entanglement in the system. In order to fulfill this task, we use the concurrence as defined by Wooters~\cite{Wootters98}, which is a measurement of entanglement degree between two-qubits. Considering a generic density matrix in a two qubit space $\rho_{\mathrm{2QB}}$, an auxiliary Hermitian operator~\cite{Hill97} $R$ is defined as
\begin{equation}
R=\sqrt{\sqrt{\rho_{\mathrm{2QB}}}\;\widetilde{\rho}_{\mathrm{2QB}}\sqrt{\rho_{\mathrm{2QB}}}},
\end{equation}
where $\widetilde{\rho}_{\mathrm{2QB}}=(\sigma_y \otimes \sigma_y)\rho_{\mathrm{2QB}}^\star (\sigma_y \otimes \sigma_y)$, is the spin-flipped matrix with $\rho_{\mathrm{2QB}}^\star$ being the complex conjugate of $\rho_{\mathrm{2QB}}$. The concurrence is written as
\begin{equation}
C=\mathrm{max}(0,\lambda_1-\lambda_2-\lambda_3-\lambda_4),
\end{equation}
where $\lambda_i$ are the eigenvalues of the operator $R$ in decreasing order. For our application, we construct a $4\times 4$ density operator using only the terms on $\rho_{\mathcal{M}}$ related with states of subspace $\mathcal{M}_{\mathrm{2QB}}$. This can be done once the dynamics keeps the other state of the complete basis empty.
Because it is our interest to establish the purity of the coupled molecules, we calculate the linear entropy of the evolved density matrix $\rho_{\mathcal{M}}$, which is defined as
\begin{equation}
S(t)=1-\mathrm{Tr}\left[\rho_{\mathcal{M}}^{2}(t)\right].
\label{eq:linearentro}
\end{equation}
For bipartite quantum systems, the linear entropy works as an entanglement quantifier. Nevertheless, if the quantum system of interest is coupled with an open system, the linear entropy acts as a measurement of purity. If the state of the quantum system remains as a pure quantum state, the linear entropy value is $S=0$. If the quantum system goes to any kind of mixed state $S\neq 0$ with a maximum value $S_{\mathrm{max}}$ given by the expression
\begin{equation}
S_{\mathrm{max}}=1-\frac{1}{d},
\end{equation}
where $d$ is the dimension of the Hilbert space of the quantum system. This value is associated with the statistical mixture of all elements of the basis.
Both quantities are shown in Fig.~\ref{fig:concentro}. Dashed lines in Fig.~\ref{fig:concentro} illustrate the case when the probe is turned off ($U_{\mathrm{p}}=0$). The concurrence, Fig.~\ref{fig:concentro}(a), shows its periodic evolution from a separable ($C=0$) to an entangled Bell state ($C=1$), as discussed in Ref.~\cite{Oliveira15}. For all evolution, the linear entropy in Fig.~\ref{fig:concentro}(b) has a value $S=0$ (dashed line over the $Jt$ axis) which is consistent with the fact that the coupled molecules are a closed system.
This situation changes drastically when the probe is turned on. Red line on Fig.~\ref{fig:concentro} shows a case when the capacitive coupling is comparable with the electrostatic interaction between the electrons inside the molecules so $U_{\mathrm{p}}=J$. The probe changes the concurrence dynamics, Fig.~\ref{fig:concentro}(a), being the main features the lack of periodicity together with the decreasing of the entanglement degree. Around $Jt\sim 90$, occurs a sudden death of entanglement~\cite{Yu06a,Yu06b}, which were demonstrate experimentally in optical setups~\cite{Almeida07} through indirect measurements of concurrence. This phenomenon is an abrupt fall of entanglement to zero, which could be recovered (rebirth) after some time. The purity, red line in Fig.~\ref{fig:concentro}(b), increases smoothly with time, showing that the quantum state inside the molecules becomes a mixed state as times evolves.
As we increase $U_{\mathrm{p}}$, the behavior of concurrence basically shares the same characteristics discussed above with some slight differences. The first is the decrease of temporal scale for the first sudden death and rebirth in Fig.~\ref{fig:concentro}(a). The second is the definitive suppression of the entanglement degree, meaning the asymptotic state has $C=0$. Concerning the purity evolution, green and blue lines in Fig.~\ref{fig:concentro}(b), the time scale to attain the asymptotic mixed state decreases as the coupling parameter $U_{\mathrm{p}}$ increases, revealing the irreversible loss of quantum information on the coupled molecules due to the action of the probe. The results obtained for $U_{\mathrm{p}}=3J$, blue line, confirms the nature of the asymptotical behavior: the value of $S$ is $S_{\mathrm{max}}=1-1/4=0.75$ ($d=4$) which means the system goes to a statistical mixture of all states on $\mathcal{M}_{\mathrm{2QB}}$.
\begin{figure}[tb]
\centering\includegraphics[width=1\linewidth]{figure5}
\caption{Dynamics of (a) concurrence ($C$) and (b) linear entropy ($S$) for several values of coupling $U_{\mathrm{p}}$, for the same choice of physical parameters used on Fig.\ref{fig:crosspop}: $U_{\mathrm{p}}=0$ (black dashed lines), $U_{\mathrm{p}}=J$ (red lines), $U_{\mathrm{p}}=2J$ (green lines) and $U_{\mathrm{p}}=3J$ (blue lines).}
\label{fig:concentro}
\end{figure}
The results presented above characterize a process of decoherence, induced by the action of the probe on the dynamics of the charges inside the coupled quantum molecules. All these aspects together become signatures that the probe induces dephasing on the quantum system without the loss of particles. Additionally, the behavior of both, concurrence and linear entropy, shows that the initial pure system evolves to a statistical mixture when the system is probed.
\section{Summary}
\label{sec:summary}
In this work, we present a formalism for the treatment of the interaction between quantum systems in contact with reservoirs based on a fermion-to-qubit map. The formalism has a flexibility which permits the analysis of general configurations of multipartite systems coupled with multiple reservoirs. We focuss on the obtention of a master equation, where Linblandian operators keep the structure of fermion-to-qubit mapping. The success on the demonstration of such a form of master equation brings all the advantages of Jordan-Wigner transformation to problems of quantum information processing, as used for strong-correlated systems. Specifically, it is possible to treat problem where reservoirs can act as sources and drains of particles. In the particular case of non-interacting reservoirs prepared as thermal states, the method provides expressions for Lindblad super-operators that can be straightforwardly use on numerical implementations of non-equilibrium problems.
To illustrate, we apply our formalism to the problem of dynamics of two electrons inside quantum molecules in contact with a probe, the last being an open system. The probe is a narrow transmission channel, being an open quantum dot attached to source and drain leads. By using the general equation for Lindblad super-operators, we obtain a reduced density matrix for the coupled quantum molecules. Our calculations of populations, concurrence and linear entropy let us to conclude that the probe induces dephasing, which makes the system lose the ability to generate entangled Bell states as time evolves. It is worth to remark an interesting feature induced by the probe: the apparition of sudden deaths and rebirths of entanglement. This sudden death is usually explained as caused by the action of quantum noise over the composite entangled bipartite system. Additionally, the system evolves to an asymptotic state, being a statistical mixture of the four elements of the subspace $\mathcal{M}_{2\mathrm{QB}}=\left\{\ket{1010},\ket{0101},\ket{1001},\ket{0110}\right\}$, indicated by a linear entropy compatible to the number of states in a reduced Hilbert space.
\section{Acknowledgments}
This work was supported by CNPq (grant 307464/2015-6), FAPEMIG (grant APQ-01768-14) and the Brazilian National Institute of Science and Technology of Quantum Information (INCT-IQ).
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Mark Fish sets social media ablaze with "all lives matter" remarks
Soccer24 on 23 Jul, 2020
Bafana Bafana legend Mark Fish has sparked fury on microblogging site Twitter following a series of posts in which he repeatedly said "all lives matter."
Fish's remarks come at a time when the sporting world is trying to fight racial injustice through the #BlackLivesMatter campaign, an initiative which was fuelled by the killing of American citizen George Floyd by a police officer in Minneapolis, Minnesota on the 25th of May.
The former Orlando Pirates defender however has a radical view on the issue.
"Sorry to offend many people…I certainly am not a racist but I am a firm believer in all people matter so all lives matter…no matter our past…focus on today and the future….if you don't stand for something you will always fall for everything…have a fantastic evening !!," he noted in a tweet.
"If you stand for nothing you will fall for everything….All Lives matter," he wrote in another post.
Fish's posts have attracted massive backlash, with South African politician Julius Malema even telling the retired footballer that he is "stupid" and a "fool."
Real Madrid one hand on the LaLiga title
Plenty on the line as Barca, Atletico clash
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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{"url":"http:\/\/au.mathworks.com\/help\/fininst\/hullwhite1f-class.html?nocookie=true","text":"# HullWhite1F class\n\nCreate Hull-White one-factor model\n\n## Description\n\nThe Hull-White one-factor model is specified using the zero curve, alpha, and sigma parameters for the equation\n\n$dr=\\left[\\theta \\left(t\\right)-a\\left(t\\right)r\\right]dt+\\sigma \\left(t\\right)dW$\n\nwhere:\n\ndr is the change in the short-term interest rate over a small interval.\n\nr is the short-term interest rate.\n\n\u0398(t) is a function of time determining the average direction in which r moves, chosen such that movements in r are consistent with today's zero coupon yield curve.\n\n\u03b1 is the mean reversion rate.\n\ndt is a small change in time.\n\n\u03c3 is the annual standard deviation of the short rate.\n\nW is the Brownian motion.\n\n## Construction\n\n`OBJ = HullWhite1F(ZeroCurve,alpha,sigma)` constructs an object for a Hull-White one-factor model.\n\nFor example:\n\n```Settle = datenum('15-Dec-2007'); CurveTimes = [1:5 7 10 20]'; ZeroRates = [.01 .018 .024 .029 .033 .034 .035 .034]'; CurveDates = daysadd(Settle,360*CurveTimes,1); irdc = IRDataCurve('Zero',Settle,CurveDates,ZeroRates); alpha = .1; sigma = .01; HW1F = HullWhite1F(irdc,alpha,sigma);```\n\n## Properties\n\nThe following properties are from the `HullWhite1F` class.\n\n`ZeroCurve`\n\n`ZeroCurve` is specified using the output from `IRDataCurve` or `RateSpec`. This is the zero curve used to evolve the path of future interest rates.\n\nAttributes:\n\n `SetAccess` `public` `GetAccess` `public`\n\n`Alpha`\n\nMean reversion specified either as a scalar or function handle which takes time as an input and returns a scalar mean reversion value.\n\nAttributes:\n\n `SetAccess` `public` `GetAccess` `public`\n\n`Sigma`\n\nVolatility specified either as a scalar or function handle which takes time as an input and returns a scalar mean volatility.\n\nAttributes:\n\n `SetAccess` `public` `GetAccess` `public`\n\n## Methods\n\n simTermStructs Simulate term structures for Hull-White one-factor model\n\n## Definitions\n\n### Hull-White One-Factor Model\n\nThe Hull-White model is a single-factor, no-arbitrage yield curve model in which the short-term rate of interest is the random factor or state variable. No-arbitrage means that the model parameters are consistent with the bond prices implied in the zero coupon yield curve.\n\n## Copy Semantics\n\nValue. To learn how value classes affect copy operations, see Copying Objects in the MATLAB\u00ae documentation.\n\n## Examples\n\ncollapse all\n\n### Construct a Hull-White One-Factor Model\n\nConstruct a Hull-White one-factor model.\n\n```Settle = datenum('15-Dec-2007'); CurveTimes = [1:5 7 10 20]'; ZeroRates = [.01 .018 .024 .029 .033 .034 .035 .034]'; CurveDates = daysadd(Settle,360*CurveTimes,1); irdc = IRDataCurve('Zero',Settle,CurveDates,ZeroRates); alpha = .1; sigma = .01; HW1F = HullWhite1F(irdc,alpha,sigma) ```\n```HW1F = HullWhite1F with properties: ZeroCurve: [1x1 IRDataCurve] Alpha: @(t,V)inAlpha Sigma: @(t,V)inSigma ```\n\nUse the `simTermStructs` method with the `HullWhite1F` model to simulate term structures.\n\n```SimPaths = simTermStructs(HW1F, 10,'nTrials',100); ```\n\n## References\n\nHull, J. Options, Futures, and Other Derivatives, Prentice-Hall, 2011.","date":"2015-05-07 07:42:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.775938868522644, \"perplexity\": 4010.8773601163307}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-18\/segments\/1430460671221.78\/warc\/CC-MAIN-20150501061111-00037-ip-10-235-10-82.ec2.internal.warc.gz\"}"}
| null | null |
\section{Introduction}
Within the accepted cosmological paradigm -- $\Lambda$ cold dark matter ($\Lambda$CDM) --
approximately 85\% of the matter in the Universe is thought to be dark
\citep{komatsu11,planck1,planckcp}. As such, understanding the nature of this component
is of the upmost importance. While the precise properties of this exotic
matter are still unknown, various predictions about its behaviour and mass
distribution within galaxies have been made by both cosmological and particle
physics models.
While it has experienced great success on large scales, this cosmological
model has run into some difficulty adequately explaining a number of
observations made on smaller scales. In particular, a number of mismatches
between observation and theory with regard to the smallest, most dark matter
dominated galaxies we are able to observe - the dwarf spheroidals (dSphs) -
have been defined. The missing satellite problem has floated around for some
time now \citep{klypin99,moore99} and refers to the dearth of observed
luminous subhalos around the Milky Way (MW) and Andromeda (M31), compared to
the vast number of dark matter subhalos seen within dark matter only
simulations. The scope of this problem has somewhat lessened over the years,
as the community seems largely satisfied that this can be resolved with future
observations and a better understanding of the physics underlying galaxy
formation. Firstly, we do not expect all subhalos seen within the simulations
to have enough mass to accrete and retain the gas required to efficiently form
stars. As such, a lower mass of $V_{\rm max}\sim10-15{\rm\,km\,s^{-1}}$ is placed on
luminous galaxy formation \citep{penarrubia08b,koposov09}. Secondly, our
observations are currently incomplete (both areally and in terms of surface
brightness). By considering and correcting for the completeness of current
surveys of the halos of these galaxies
(\citealt{tollerud08,koposov09,walsh09}; Martin et al, in prep.) the number of
observed vs. predicted subhalos can be brought into much better agreement.
Another observed and as yet unresolved tension is the ongoing `cusp-core'
debate, which refers to the shape of the dark matter profile of galactic halos
as radius tends to zero. With central mass to light ratios of typically
$M/L>10{\rm\,M_\odot}/{\rm\,L_\odot}$ (e.g \citealt{mateo98,walker09b,tollerud12,collins13}),
one can treat the stars contained within dSphs as massless tracers of the dark
matter potential, and their small scales (half-light radii of
$r_{\rm{half}}\sim100-1000$s pc) allow us to probe their mass profiles in the
very centers of their halos. This allows us to test the predictions from
cosmological dark matter only simulations of halo mass profiles; namely that
these are steeply cusped (i.e., the density dramatically increases for
decreasing radius, \citealt{navarro97}). Increasingly, observations of dwarf
spheroidals (and other low surface brightness galaxies) show evidence for
constant density cores in the centers of galaxy halos
(e.g. \citealt{deblok02,deblok03,deblok05,walker11,amorisco12,jardel12}). Whether
this tension can be resolved by appealing to baryonic processes, such as
feedback from star formation or tidal stripping is something that is currently
being debated (e.g. \citealt{zolotov12,brooks12,garrison13}) and a theme we
shall return to later on.
Related to the cusp-core problem is the `too big to fail' (TBTF) problem,
which was originally identified by \citet{read06b} and has received much
attention recently from others (e.g. \citealt{kolchin11,kolchin12}). With the
limited kinematic data currently available for dSph galaxies, it is not
possible to accurately measure the slopes of their density profiles in many
cases, but from measurements of their central velocity dispersion, $\sigma_v$,
one can get a good grasp on the central masses, i.e. the mass within the
2-dimensional half-light radius, $r_{\rm half}$, of these systems
\citep{walker09b,wolf10}, and compare these with those of simulated
subhalos. Such an exercise was undertaken by \citet{kolchin11} using the
Aquarius set of simulations \citep{springel08}, and they found that each
MW-like Aquarius halo they studied had of order 10 subhalos with central
masses that were significantly higher than those of the MW dSphs. This means
either the most massive subhalos within MW systems do not necessarily form
stars, or that we are missing some crucial physics from these models (either
baryonic, or with respect to the properties of dark matter itself) that can
explain this discrepancy.
Each of these problems are currently being investigated by observers and
theorists alike, with proposed solutions ranging from fiddling with baryonic
physics (star formation, feedback, tidal forces etc.) to redefining the
cosmological paradigm (e.g. modified Newtonian dynamics, warm dark matter,
self interacting dark matter). From the observers point of view, one obvious
avenue has been to extend our sample of objects whose kinematics are well
measured. Obtaining the necessary kinematic information with which to study
the central masses of dSphs (i.e. radial velocities of individual stars within
these systems) is exceptionally challenging, meaning the majority of studies
thus far have focused on the 20 surrounding our own Galaxy. It has only been
within the last decade that telescopes capable of measuring kinematics of
extragalactic dSphs have become available. Now, thanks to several recent
papers
(e.g. \citealt{kalirai10,collins10,collins11b,collins13,tollerud12,tollerud13}),
we can add to this sample a further 25 dSphs from the Andromeda system, more
than doubling our sample size. Two of these works in particular, Tollerud et
al. (2012, henceforth T12) and Collins et al. (2013, henceforth C13) have
demonstrated that the majority of the M31 systems have very similar central
masses to their MW counterparts, which would imply that the self-same tensions
discussed above for the MW also apply to the M31 dSph system. In addition,
they highlighted a number of M31 dSphs whose masses appear lower than would be
expected when comparing with expectations based on Milky Way dSphs, casting
some doubt on the notion that all dSph galaxies are hosted within dark matter
halos whose central mass profiles are universal, i.e., behave in a
statistically similar way as a function of radius
\citep{mateo98,strigari08,walker09b,wolf10}.
In this work, we revisit the idea of universal mass profile of
\citet{walker09b} for the dSph population by including the M31 objects into
the analysis. In \S~\ref{sect:results} we show that a singular mass profile,
be it a \citet{navarro97} cusp (NFW) or a constant density core, provides a
poor fit to the Local Group dSphs, and instead we advocate a statistical range
of best fit mass profiles that track the scatter in mass for a given
half-light radius in this population. We then compare these findings with
numerical simulations, demonstrating that the mismatches discussed above do
not simply go away with a larger sample of systems. We identify a number of
unusual systems in M31 whose masses may pose a challenge to our understanding
of galaxy formation and evolution in \S~\ref{sect:outliers}. Finally, in
\S~\ref{sect:discussion}, we go on to discuss how the proposed solutions to
these problems stack up to the observations, before we conclude in
\S~\ref{sect:conclusions}.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=0.9\hsize]{fig1.pdf}
\caption{Half-light radius vs. velocity dispersion for MW (red triangles)
and M31 dSphs (blue circles). Overlaid are best fitting NFW and core
mass profiles to these data. Open symbols represent MW dSphs that are
too faint to be observed in M31, and hence are excluded from the
fits. $\sim50\%$ of all observations are inconsistent with these fits,
undermining the notion that all dSphs are embedded in halos that follow
a universal density profile.}
\label{fig:universal}
\end{center}
\end{figure}
\section{Data}
\label{sect:data}
As dSphs are largely dispersion supported systems with little or no evidence
of rotation, their velocity dispersions, in combination with their half-light
radii, can be utilized to estimate their central masses. For the Milky Way
population, we rely on the compilation of kinematic and structural properties
formed by \citet{walker09b}, although we exclude the tidally disrupting
Sagittarius galaxy (Sgr) from further analysis, as it is currently not in
equilibrium. The compilation from \citet{walker09b} were used within that
work to define the universal mass profile for dSph galaxies, which we shall
discuss further below. Since then, three Galactic dSphs have benefitted from
further study of their kinematics; Hercules \citep{aden09}, B\"ootes I
\citep{koposov11} and Segue 2 \citep{kirby13}. In all cases, the velocity
dispersions (and hence, calculated masses) have reduced.
For the Andromeda dSphs, we take our kinematics and structural properties from
the final table in C13, which is a compilation of the best velocity
dispersions from that work (those of Andromeda VI, XI, XVII, XIX, XX, XXII,
XXIII, XXV, XXVI, XXVIII and XXX [Casseopia II]), and from those presented by
T12 (Andromeda I, III, V, VII, IX, X, XIII, XIV, XV and XVIII). For And XVI
and XXI, we use newly derived values for the velocity dispersions of these
objects that have been made from much larger samples of member stars
($\sigma_v=5.6\pm1.0$ and $\sigma_v=5.4\pm0.9$ for XVI and XXI respectively,
Collins et al. in prep). The velocity dispersion for Andromeda (And) II is
taken from \citet{ho12}, and that of And XXIX is taken from
\citet{tollerud13}. The velocity dispersion for And XII is unresolved, so we
omit that from our study here. Owing to difficult observing conditions, the
velocity dispersion of And XXIV is not well constrained, so we omit this value
too. Finally, as And XXVII is likely a heavily disrupted system whose
kinematics and structural properties are not well constrained (C13, Martin et
al., in prep), we also remove this system from our analysis. This leaves us
with a sample of 25 M31 dSph galaxies for which velocity dispersions have been
reliably estimated. The structural properties are taken from
\citet{mcconnachie12} and Martin et al. (in prep., for the M31 dwarf galaxies that fall
in the PAndAS footprint), updated based on the
revised distances to the Andromeda dSphs presented in \citet{conn12b}.
\section{Results}
\label{sect:results}
\subsection{A universal mass profile?}
\label{sect:ump}
\begin{figure*}
\begin{center}
\includegraphics[angle=0,width=0.3\hsize]{fig2a.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig2b.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig2c.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig2d.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig2e.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig2f.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig2g.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig2h.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig2i.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig2j.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig2k.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig2l.pdf}
\caption{{ \bf Top row:} 2 dimensional likelihood contours for the three
free parameters ($V_{\rm max}$, $\sigma_{V_{\rm max}}$ and $R_S$) in
the NFW mass profile fits to the MW dSphs (red dashed contours), M31
dSphs (blue dot-dashed contours) and all Local Group dSphs (solid black
contours). The contours represent the 1 and $2\sigma$ (i.e. 68\% and
95\%) confidence intervals for these values. {\bf Second row:} The
resulting 1 dimensional marginalized likelihoods for $V_{\rm max}$,
$\sigma_{V_{\rm max}}$ and $R_S$ (from left to right) for the MW, M31
and full sample. Horizontal dashed lines represent the 1, 2 and 3$\sigma$
(i.e. 68\%,95\% and 99.7\%) confidence intervals, derived assuming a
Gaussian probability distribution. The best fit $\sigma_{V_{\rm max}}$
and $R_S$ agree quite well between the MW and M31 case, however the
values of $V_{\rm max}$ for the MW and M31 are marginally inconsistent,
at the level of $1\sigma$, with the M31 dSphs favoring fits with lower
central masses. {\bf Third and fourth rows:} As above six panels, but
for cored density profile fits. Again, the best fit values of $V_{\rm
max}$ for the MW and M31 are discrepant at the level of $1\sigma$.}
\label{fig:allML}
\end{center}
\end{figure*}
We focus our analysis here on the inclusion of the masses of the M31 dSphs
into the universal density profile of \citet{walker09b}. In their work, the
authors were spurred on by the earlier results of \citet{strigari08} that
showed that all of the MW dSphs for which kinematic data were available were
consistent with having the same mass contained within a radius of 300 pc
(roughly $1\times10^7{\rm\,M_\odot}$) despite spanning 6 decades in
luminosity. \citet{strigari08} used this result to argue that it was possible
that all dSphs inhabited a universal dark matter halo, where
the density as a function of radius was identical, irrespective of the number
of stars the halo hosted. However, \citet{wolf10} demonstrated
that extrapolations to both larger and smaller radii than the true half light
radius are extremely uncertain in cases where the velocity anisotropy is
unknown, and this is true for all the Local Group dSphs. For objects with
$r_{\rm half}<<300$~pc one has to extrapolate to radii inhabited by no
tracers, where tidal stripping may have removed the outer dark matter envelope
\citep{penarrubia08b}. That means that for some galaxies extrapolating out to
300~pc could over-estimate the enclosed mass by several orders of
magnitude. In the interest of trying to measure a more meaningful
mass for these objects to determine whether dSphs truly resided within a
universal halo, \citet{walker09b} measured the velocity
dispersion, and hence mass, within the half-light radius of the MW
dSphs. Then, by treating each velocity dispersion measurement from MW dSphs as
a measurement of the velocity dispersion at a given radius (the half-light
radius of the dSph in question) within a single dark matter halo, they could
map out the velocity dispersion profile for this singular halo. In particular,
they tested the cosmologically motivated Navarro Frenk White (NFW,
\citealt{navarro97}) density profile:
\begin{equation}
r_{\rm half}\sigma_v^2=\frac{2\eta
R_SV_{\mathrm{max}}^2}{5}\times\left[\frac{\mathrm{ln}(1+r_{\mathrm{half}}/R_S)-\frac{r_{\mathrm{half}}/R_S}{1+r_{\mathrm{half}}/R_S}}{\rm{ln}(1+\eta)-\frac{\eta}{1+\eta}}\right],
\label{eqn:nfw}
\end{equation}
\noindent where $V_{\rm max}$ is the maximum circular velocity of the halo,
$R_S$ is the scale radius of the halo and $\eta=2.16$. They also used a cored
density profile where:
\begin{equation}
\begin{aligned}
r_{\mathrm{half}}\sigma_v^2=\frac{2\eta R_SV_{\mathrm{max}}^2}{5(\mathrm{ln}[1+\eta])+\frac{2}{1+\eta}-\frac{1}{2(1+\eta)^2}-\frac{3}{2}}\times\\
\left[\mathrm{ln}(1+r_{\rm half}/R_S)+\frac{2}{1+r_{\rm half}/R_S}-\frac{1}{2(1+r_{\rm half}/R_S)^2}-\frac{3}{2}\right],
\end{aligned}
\label{eqn:core}
\end{equation}
\noindent with $\eta=4.42, \alpha=1$ and $\gamma=0$. The results of this study
showed that the MW dSphs were consistent with having formed with a universal
mass profile, although the authors noted that there was significant
scatter about this relation, a factor of 2 greater than expected from the
observational uncertainties alone. Later that same year, a revised study of the mass
of the Hercules dSph \citep{aden09}, which provided a better treatment of the
contaminating foreground population, determined a much lower value for the
velocity dispersion of this object ($3.72\pm0.91{\rm\,km\,s^{-1}}$ vs. $5.1\pm0.9{\rm\,km\,s^{-1}}$
from \citealt{simon07}). With their revised value, they showed that the mass
of Hercules was not consistent with the universal mass profile. As Hercules is
likely significantly affected by tides \citep{aden09,martin10}, this is
perhaps not unexpected. Similarly, an analysis of the B\"ootes I dSph by
\citet{koposov11}, who used multi-epoch observations taken with the VLT and
implemented an enhanced data reduction approach to measure extremely precise
radial velocities, measured a velocity dispersion of
$\sigma_v=4.6^{+0.8}_{-0.6}{\rm\,km\,s^{-1}}$, significantly lower than that of
$\sim6.5{\rm\,km\,s^{-1}}$ reported in previous studies. This also renders the B\"ootes I
dSph inconsistent with the universal mass profile.
In \citet{walker09b}, velocity dispersions for only 2 M31 dSphs (And II and
IX) were available. We therefore fit NFW and cored density profiles
(equations~\ref{eqn:nfw} and~\ref{eqn:core}) to the velocity dispersions of
the entire dSph population with $L>2\times10^4{\rm\,L_\odot}$ (ensuring we probe the
same luminosity regime in both the MW and M31), to see how well these
populations can be fit with a single density profile. Both profiles have two
free parameters of interest to fit, the circular velocity of the halo, $V_{\rm
max}$ and the scale radius $R_S$. To constrain these values, we use a
maximum likelihood fitting routine to determine the most probable values for
these parameters by maximising the likelihood function, $\mathcal{L}$, defined as:
\begin{equation}
\begin{aligned}
\mathcal{L}(\{r_{h,i},\sigma_{v,i},\delta_{\sigma_v,i}\}|V_{\rm max},R_S)=\prod_{i=0}^{N} \frac{1}{\sqrt{2\pi\delta_{\sigma_v,i}^2}}\\{\rm exp}\left[-\frac{(\sigma_{\mathrm{profile}}-\sigma_{v,i})^2}{2\delta_{\sigma_v,i}^2}\right]
\end{aligned}
\label{eqn:maxlike}
\end{equation}
\noindent where $\sigma_{\mathrm{profile}}$ is the velocity dispersion as
predicted by equations~\ref{eqn:nfw} and \ref{eqn:core} for a dSph with
half-light radius $r_{h,i}$; $\sigma_{v,i}$ is the measured velocity
dispersion of the $i^{{\rm th}}$ dSph and $\delta_{\sigma_v,i}$ is the
uncertainty on the measured dispersion. We include only the uncertainty in
$\sigma_v$ in our method, neglecting that of the half-light radius, as the
velocity dispersion parameter has a greater impact on the mass profile, as it
is proportional to the square of $\sigma_v$, depending only linearly on
$r_{\rm half}$..
We show the results of this fit in Fig.~\ref{fig:universal}. The red triangles
represent MW dSphs brighter than $L=2\times10^4{\rm\,L_\odot}$, while open triangles
represent those fainter than this cut. The blue circles are the M31 dSphs. The
magenta dot-dashed line shows our best fit NFW profile to the whole population
(with $V_{\rm max}=14.7\pm0.5{\rm\,km\,s^{-1}}$ and $R_S=876\pm284$~pc), whilst the cyan
dashed line is the best fit core profile (with $V_{\rm max}=14.0\pm0.4{\rm\,km\,s^{-1}}$
and $R_S=242\pm124$~pc) where the best fit $V_{\rm max}$ ($R_S$) parameter is
determined by marginalizing the 2D maximum likelihood contours over $R_S$
($V_{\rm max}$). In all cases, the quoted uncertainties are derived by
assuming that the likelihood functions have a Gaussian-like distribution,
allowing us to project the marginalized maximum likelihood contours to the
value at which $2\mathrm{ln}(\mathcal{L})$ has decreased by the square of the
confidence interval of interest, which in this case is the 1$\sigma$
(i.e. 68\%) confidence interval. Clearly, neither of these mass profiles is a
good fit for many of the Local Group dSphs. This is statistically demonstrated
by the reduced $\chi^2$ values for these fits ($\chi^2=4.1$ for both
profiles). Of the 39 objects, 24 are outliers at $>1\sigma$, with $\sim1/5$ of
the population being outliers at the $3\sigma$ level.
\subsection{Scatter about an average mass profile}
\label{sect:scatter}
From the above analysis, it is clear that the scatter about the best fitting
profiles is significant, and well beyond what we can hope to explain with
measurement uncertainties. But do we really expect that all low luminosity galaxies
should reside in dark matter halos with identical density profiles? The dark
matter subhalos produced in, for example, the Aquarius simulations
\citep{springel08} demonstrate a range of possible values of $V_{\rm max}$ and
$R_{s}$ for these objects. Further, work by e.g. \citet{zolotov12,brooks12}
and \citet{veraciro13} demonstrate that the infall time, host mass and
presence of baryons can all effect the dark matter structures of subhalos. As
such, scatter in density profiles is completely expected, and differences
between the satellite populations of the Milky Way and Andromeda might also be
seen that could tell us about the evolutionary histories of the two systems.
To investigate this, we introduce a mass-scatter term, $\sigma_{V_{\rm max}}$,
into our maximum likelihood fitting algorithm (equation~\ref{eqn:maxlike})
replacing $\delta_{\sigma_v,i}$ with $\delta_{\mathrm{tot},i}$ which is the combination of the
measured uncertainty in the velocity dispersion measurements and the mass-scatter
term, such that $\delta_{\mathrm{tot},i}=\sqrt{\delta_{\sigma_v,i}^2+\sigma_{V_{\rm max}}^2}$.
\begin{figure*}
\begin{center}
\includegraphics[angle=0,width=0.45\hsize]{fig3a.pdf}
\includegraphics[angle=0,width=0.45\hsize]{fig3b.pdf}
\includegraphics[angle=0,width=0.45\hsize]{fig3c.pdf}
\includegraphics[angle=0,width=0.45\hsize]{fig3d.pdf}
\caption{{\bf Top left:} $\sigma_v$ vs. $r_{\rm half}$ for all MW (red
triangles) and M31 (blue circles) dSph galaxies. Overlaid are the best
fit NFW profiles for the MW (red shaded region) and M31 (blue shaded
region). The dashed lines represent the average fit while the shaded
regions indicate the parameter space allowed by the introduction of the
scatter term, $\sigma_{V_{\rm max}}$, convolved with the uncertainties
in the fit parameters. At large $r_{\rm half}$ (higher mass) the
profiles of these populations begin to diverge, with the M31 fit turning
over at $r_{\rm half}\sim600{\rm\,pc}$ while the MW profile continues to
rise. {\bf Top right:} As top left, but with the best fit cored density
profiles overlaid. Again, the M31 profile is seen to turn over before
that of the MW profile. {\bf
Bottom left and right: } As top panels, but now the best fit profiles for NFW
(left) and core (right) are determined after excluding And XIX, XXI and
XXV. The removal of these objects results in best fit cored mass profile
parameters that agree extremely well for MW and M31 dSphs.}
\label{fig:fitsall}
\end{center}
\end{figure*}
If the mass profiles of the Andromeda and Milky Way dSphs are truly similar
within their inner regions, our algorithm should find best fit values of
$V_{\rm max}$, $R_S$ and $\sigma_{V_{\rm max}}$ that are broadly consistent
when fitting the two populations separately and as a whole. In
Figure~\ref{fig:allML} we show the likelihood contours for these
parameters. In the top three panels we overlay the $1\sigma$ and $2\sigma$
confidence interval contours (again, defined as the region of parameter space
where $2\mathrm{ln}(\mathcal{L})$ decreases by the square of the confidence interval
in question) for the NFW fits to the Milky Way (red dashed), Andromeda (blue
dot-dashed) and the full sample (black solid) for our 3 free parameters
(marginalized over the 3rd parameter not displayed in each 2D subplot), with
solid points representing the best fit values in each case. In the second row
of subplots, we show the one dimensional marginalised relative likelihood functions for
$V_{\rm max}$, $R_S$ and $\sigma_{V_{\rm max}}$ for each of these fits. The
lower 6 panels show the same, but for the cored fits. In the NFW case, while
the best fit values for $R_S$ in each case seem dramatically different for the
MW and M31 at first glance with $R_S=1034^{+1508}_{-524}{\rm\,pc}$\ and $R_S=322^{+247}_{-143}{\rm\,pc}$\ respectively, their
uncertainties are such that they agree within $1\sigma$. The amount of scatter
in mass at a given radius is also very similar, with $\sigma_{V_{\rm max}}=2.9^{+0.7}_{-0.5}\kms$\ and $\sigma_{V_{\rm max}}=3.9^{+0.7}_{-0.6}\kms$\
respectively for the MW and M31. The preferred values for $V_{\rm max}$
($V_{\rm max}=18.4^{+2.9}_{-3.1}\kms$\ and $V_{\rm max}=12.8^{+1.4}_{-1.2}\kms$\ respectively), however, are marginally less
consistent, with M31 preferring a lower value of $V_{\rm max}$ (and hence,
lower masses) than the MW system. For the core profiles, we get a similar
result, with the values for $R_S$ ($R_S=253^{+143}_{-99}{\rm\,pc}$\ and $R_S=142^{+72}_{-53}{\rm\,pc}$), and $\sigma_{V_{\rm
max}}$ ($\sigma_{V_{\rm max}}=2.9^{+0.8}_{-0.6}\kms$ and $\sigma_{V_{\rm max}}=3.8^{+0.7}_{-0.6}\kms$) for the MW and M31 agreeing within $1\sigma$,
and marginally inconsistent values of $V_{\rm max}$ ($V_{\rm max}=16.2^{+2.8}_{-2.1}\kms$\ and $V_{\rm max}=12.8^{+1.3}_{-1.1}\kms$).
In the top two panels of Fig.~\ref{fig:fitsall}, we overplot the best fit relations from this
analysis in the $r_{\rm half}-\sigma_v$ plane. In the left panel, we show the
best fit NFW profile for the MW (red line) and M31 (blue line) dSphs, with the
best fit core profiles in the right panel. The shaded regions represent the
scatter we derived convolved with the $1\sigma$ uncertainties for $V_{\rm max}$ and
$\sigma_{V_{\rm max}}$. Both the MW NFW and Core profiles provide an excellent
fit to all the dSphs barring the Hercules dSph. However, as it is likely to be
highly tidally disturbed \citep{aden09,martin10}, this is not too
surprising. For the M31 fits, we see 3 systems that are outliers at the
$\sim2-3\sigma$ level: And VI, VII and XXV. And XXV has previously been
identified as an unusually low mass system in C13, so this inconsistency is
perhaps not unexpected. However, And VI and VII are thought to represent
fairly typical satellites, with velocity dispersions similar to their MW
counterpart Fornax, which has a comparable half-light radius to these two
objects.
\begin{figure*}
\begin{center}
\includegraphics[angle=0,width=0.3\hsize]{fig4a.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig4b.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig4c.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig4d.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig4e.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig4f.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig4g.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig4h.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig4i.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig4j.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig4k.pdf}
\includegraphics[angle=0,width=0.3\hsize]{fig4l.pdf} \caption{As Fig.~\ref{fig:allML}, but with three significant low mass
outliers, And XIX, XXI and XXV, omitted from the fits. The removal of
these objects results in best fit NFW mass profile parameters that agree
extremely well for MW and M31 dSphs. }
\label{fig:outML}
\end{center}
\end{figure*}
The differences in the preferred $V_{\rm max}$ has a striking visual effect on
the resulting best fit profiles for the MW and M31. While in both the NFW and
Core case, the relations for MW and M31 populations track each other well at
the smaller radius (lower mass) end (albeit with greater scatter in M31), at
larger radii there appears to be a divergence between the two systems. Both
the NFW and core profiles for M31 begin to turnover at $\sim600$~pc, while the
MW profile continues to rise (turning over at $\sim1200$~pc in the cored
case). This turnover radius is interesting, as there are only 3 MW dSphs with
half-light radii $\gta600$~pc, one of which is the tidally disrupting
Sagittarius (Sgr) and thus is excluded from our fits, the other two being
Fornax and Sextans. In M31, there are 7 galaxies (And I, II, VII, XIX, XXI,
XXIII and XXV), 3 of which (And XIX, XXI and XXV) are curiously very low mass
for their size. In C13, they were measured to be $3\sigma$ outliers to
the best-fit mass profiles of \citet{walker09b} and, as can be seen in
Fig~\ref{fig:fitsall}, they are significant outliers to the
best fit MW relation ($2,2.5$ and $3\sigma$ respectively). In addition,
despite having very similar half-light radii, Sgr ($r_{\rm
half}=1550\pm50$~pc) and And XIX ($r_{\rm half}=2072^{+1092}_{-422}$~pc)
have very different velocity dispersions ($\sigma_v=11.7\pm0.7{\rm\,km\,s^{-1}}$ and
$\sigma_v=4.7^{+1.6}_{-1.4}{\rm\,km\,s^{-1}}$).
To see if it is these outliers driving the differences between the MW and M31
relations, we repeat the same fits performed above, but without all systems
that were found to lie at or greater than $3\sigma$ from the best fit profiles
of \citet{walker09b}. This included the three M31 outliers, and additionally
the MW dSphs, Hercules and CVn I, which are also outliers to the
\citet{walker09b} relation. In Fig.~\ref{fig:outML} we show the same contours
as in Fig.~\ref{fig:allML} for NFW and core profile fits, only this time with
these outliers omitted. The exclusion of Hercules and CVn I from the MW fits
has only a slight effect on these profiles, but removing the low mass M31
outliers is more substantial. The agreement between $R_s$, $V_{\rm max}$ and
$\sigma_{V_{\rm max}}$ is significantly better (as shown in
Table~\ref{tab:profiles}). In the lower two panels of Fig.~\ref{fig:fitsall},
we show these best fit profiles in the $r_{\rm half}-\sigma_v$ plane. Barring
the 5 excluded dSphs and the M33 satellite, And XXII \citep{chapman12}, all
the Local Group objects have velocity dispersions that are well described by
the M31 and MW relations. The best fit profiles to the whole Local Group
system are shown in Fig.~\ref{fig:fitLG}, and have preferred values for the
NFW (core) parameters of $R_S=664^{+412}_{-232}{\rm\,pc}$\ ($R_S=225^{+70}_{-55}{\rm\,pc}$), $V_{\rm max}=16.2^{+2.6}_{-1.7}\kms$\ ($V_{\rm max}=15.6^{+1.5}_{-1.3}\kms$) and $\sigma_{V_{\rm max}}=2.9^{+0.5}_{-0.4}\kms$\
($\sigma_{V_{\rm max}}=2.8^{+0.5}_{-0.4}\kms$). Therefore, whilst dSph galaxies do not live within dark matter halos
with identical density profiles, the vast majority do inhabit statistically
similar halos with a well defined mass range at any given radius.
In Fig.~\ref{fig:fitLG} we show the mass within the half-light radii of the
dSphs (calculated using the \citealt{walker09b} mass estimator, where $M_{\rm
half}=580r_{\rm half}\sigma_v^2$, tabulated in Table~\ref{tab:ml}) with the best fit NFW and Core relations
when excluding the outliers. At all radii, the total scatter is less than half
a magnitude in mass. For example, at $r_{\rm half}=10{\rm\,pc},100{\rm\,pc}$ and 1000 pc,
the average masses from the cored profile are $M_{\rm
half}=2.3\times10^4{\rm\,M_\odot},1.1\times10^6{\rm\,M_\odot}$ and $5.4\times10^7{\rm\,M_\odot}$, and
the scatter (i.e. half the distance outlined by the shaded band) is $M_{\rm
scatter}=1.2\times10^4{\rm\,M_\odot},0.6\times10^6{\rm\,M_\odot}$ and $2.5\times10^7{\rm\,M_\odot}$,
which is $\sim50\%$ of the average mass in each case. The numbers for the best
fit NFW profile are almost identical.
Our decision to exclude And XIX, XXI and XXV was based on their designation as
significant ($>3\sigma$) low mass outliers in C13, and to the derived MW
profile. There are several other potentially low mass systems that were
identified in C13 (namely And XXII) and T12 (And XIV, And XV and XXII
also), which we did not exclude, simply because their associated uncertainties
place them much closer to the regime of expected mass from the MW system. If
they too were shown to be truly low mass with subsequent observations, this
would imply that the M31 dwarf spheroidal systems do have greater scatter
towards lower masses in their mass profiles compared with the MW.
\begin{deluxetable*}{lccccccccc}
\tabletypesize{\footnotesize}
\tablecolumns{10}
\tablewidth{0pt}
\tablecaption{Best fit parameters from mass profile fits to MW and M31 dSph
data using NFW and cored profiles.}
\tablehead{
\colhead{Model} & & \colhead{Full} & & & \colhead{M31} & & &
\colhead{MW} & \\ & \colhead{$V_{\rm max}$}
& \colhead{$R_S$ } & \colhead{$\sigma_{V_{\rm max}} $} & \colhead{$V_{\rm max}$}
& \colhead{$R_S$ } & \colhead{$\sigma_{V_{\rm max}} $} & \colhead{$V_{\rm max}$}
& \colhead{$R_S$ } & \colhead{$\sigma_{V_{\rm max}}$}\\ &
\colhead{(${\rm\,km\,s^{-1}}$)} & \colhead{(pc)} & \colhead{(${\rm\,km\,s^{-1}}$)} & \colhead{(${\rm\,km\,s^{-1}}$)} & \colhead{(pc)} & \colhead{(${\rm\,km\,s^{-1}}$)} &\colhead{(${\rm\,km\,s^{-1}}$)} & \colhead{(pc)} & \colhead{(${\rm\,km\,s^{-1}}$)} }
\startdata
NFW & $13.6^{+1.3}_{-1.0}$ & $408^{+221}_{-143}$ & $3.6\pm0.5$ & $12.8^{+1.4}_{-1.2}$ & $322^{+247}_{-143}$ & $3.9^{+0.7}_{-0.6}$ & $18.4^{+2.9}_{-3.1}$ & $1034^{+1508}_{-524}$ & $2.9^{+0.7}_{-0.5}$\\
NFW (minus outliers) &$16.2^{+2.6}_{-1.7}$ & $664^{+412}_{-232}$ & $2.9^{+0.5}_{-0.4}$ & $16.7^{+3.5}_{-2.4}$ & $790^{+828}_{-349}$ & $3.2^{+0.7}_{-0.6}$ & $18.7^{+4.9}_{-4.1}$ & $708^{+1816}_{-391}$ & $2.4^{+0.7}_{-0.5}$\\
Core &$13.5^{+1.1}_{-0.9}$ & $165^{+58}_{-47}$ & $3.5^{+0.5}_{-0.4}$ & $12.8^{+1.3}_{-1.1}$ & $142^{+72}_{-53}$ & $3.8^{+0.7}_{-0.6}$ & $16.2^{+2.8}_{-2.1}$ & $253^{+143}_{-99}$ & $2.9^{+0.8}_{-0.6}$\\
Core (minus outliers) &$15.6^{+1.5}_{-1.3}$ & $225^{+70}_{-55}$ & $2.8^{+0.5}_{-0.4}$ & $15.7^{+2.1}_{-1.7}$ & $257^{+108}_{-88}$ & $3.2^{+0.7}_{-0.6}$ & $15.9^{+3.1}_{-2.1}$ & $208^{+119}_{-82}$ & $2.5^{+0.8}_{-0.6}$\
\enddata
\label{tab:profiles}
\end{deluxetable*}
\subsection{Comparing the observational scatter to simulations}
\label{sect:obssims}
Briefly, we compare the best fit values of $V_{\rm max}$ and the scatter in
this term with recent cosmological and semi-analytical models to deduce
whether the values we statistically obtain for the Local Group dSphs compare
favorably with our theoretical understanding of galaxy formation and
evolution.
If we naively compare to dark-matter only simulations, such as the subhalos in
the Aquarius simulations \citep{springel08} of 6 MW-mass dark matter halos, we
find that our measured values of $V_{\rm max}$ are lower than would be
expected. The same discrepancy was pointed out by \citet{kolchin12}. While the
MW dSphs have $12\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} V_{\rm max}\lta25{\rm\,km\,s^{-1}}$, they found at least 10 subhalos
in each Aquarius host with $V_{\rm max}>25{\rm\,km\,s^{-1}}$. This discrepancy is referred
to as the `Too Big To Fail' problem, and would seem to persist when
including M31 dSphs.
If we instead compare with models where baryons are taken into account, do we
still see such inconsistencies? In \citet{rashkov12}, dark matter subhalos
from the high resolution Via Lactea II simulations are populated with baryons
at the time of infall into their host halo by dynamically tagging dark matter
particles as stars. These systems are then traced until $z=0$, where their
final properties are compared to observations. These simulations are able to
reproduce many observed properties of MW dSphs (such as velocity dispersions,
sizes, metalicities, number count etc.), and that the present day values of
$V_{\rm max}$ for the 10 most luminous subhalos are more compatible with
observations, having $10<V_{\rm max}<40{\rm\,km\,s^{-1}}$ ($\sim50\%$ of which are less
than $20{\rm\,km\,s^{-1}}$). Our average $V_{\rm max}$ plus scatter term gives a
statistical range for the $V_{\rm max}$ of the Local Group dSphs of
$\sim12-22{\rm\,km\,s^{-1}}$. So while the bulk of their sample is consistent, there remain
too many high mass dSph satellites to be fully consistent. We can also
compare our calculated values of $R_{S}$ with those of the \citet{rashkov12}
simulations. The range of $R_{\rm max}$ (which is the radius at which the
circular velocity of the halo is at a maximum, i.e., equal to $V_{\rm max}$)
for their 10 most luminous subhalos ranges from $\sim790-5400$ pc (assuming
an NFW profile). According to \citet{penarrubia08a}, $R_{\rm max}\sim2R_S$,
so this corresponds to scale radii for the \citet{rashkov12} halos of
$395{\rm\,pc}\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} R_s\lta2700$~pc which is consistent with the scale radius of
$R_S=664^{+412}_{-232}{\rm\,pc}$\ that we find for our combined NFW profile (with outliers excluded),
suggesting that these subhalos are similarly dense to the Local Group
dSphs.
\citet{bovill11a} model satellites within the Local Volume from reionization
until today, tracing the merger histories and tidal interactions of these
objects as they merge to form more massive galaxies. As with the
\citet{rashkov12} study, they are able to reproduce many of the observed
properties of MW and M31 dSphs. For satellites with similar luminosities to
those we fit in this work (i.e., $L\gta10^4{\rm\,L_\odot}$) they measure $10\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} V_{\rm
max}\lta30{\rm\,km\,s^{-1}}$ which is, again, largely consistent with the range of
$V_{\rm max}$ we find. In this instance, the \citet{bovill11a} model produces
more bright, massive satellites than we see in the Local Group. They discuss
this in \citet{bovill11b} as the ``missing bright satellite''
problem. However, for the systems with comparable luminosities, there is
significant overlap in their masses.
From these comparisons, we are content that the best fit profiles to the MW,
M31 and total Local Group dSph we have derived are not hugely at odds with
predictions from simulations. Some tension remains at the higher end of the
subhalo mass range, as the simulations we compare with identify at least a few
subhalos with greater $V_{\rm max}$ than are compatible with observations.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=0.9\hsize]{fig5a.pdf}
\includegraphics[angle=0,width=0.9\hsize]{fig5b.pdf} \caption{ The best
fit NFW (magenta) and core (cyan) relations to the whole Local Group
population as seen in the $r_{\rm half}-\sigma_v$ (top) and $r_{\rm
half}-M_{\rm half}$ (bottom) planes. We see that the velocity
dispersions and masses for all the dSphs, barring the excluded
outliers and 3 further objects (discussed further in the text) agree
with the fits to the whole Local Group population within their
uncertainties. }
\label{fig:fitLG}
\end{center}
\end{figure}
\section{The mass-to-light ratios and circular velocities of Local
Group dSphs}
\label{sect:outliers}
\begin{deluxetable}{lccc}
\tabletypesize{\footnotesize}
\tablecolumns{4}
\tablewidth{0pt}
\tablecaption{The masses, mass-to-light ratios, and $V_{\rm max}$ values for
Local Group dSphs as derived in this work}
\tablehead{
\colhead{Name} & \colhead{$M_{\rm half}$} & \colhead{$[M/L]_{\rm half}$}
&$V_{c,1/2}$ \\ & \colhead{($\times10^7{\rm\,M_\odot}$)}& \colhead{$({\rm\,M_\odot}/{\rm\,L_\odot}$)} & \colhead{$({\rm\,km\,s^{-1}})$} \\ }
\startdata
AndI& 5.05$\pm1.35$ & 22.4$\pm8.5$ & 16.1$\pm4.4$ \\
AndII& 3.31$\pm0.7$ & 27.6$\pm10.8$ & 12.3$\pm2.6$ \\
AndIII& 1.95$\pm0.46$ & 39.0$\pm12.9$ & 14.7$\pm3.7$ \\
AndV& 2.30$\pm0.40$ & 78.0$\pm19.5$ & 16.6$\pm3.3$ \\
AndVI& 4.67$\pm0.9$ & 27.5$^{+7.61}_{-6.85}$ & 19.5$^{+4.2}_{-3.9}$ \\
AndVII& 7.61$\pm0.9$ & 9.0$\pm1.6$ & 20.5$\pm2.7$ \\
AndIX& 2.25$\pm0.7$ & 302$\pm132$ & 17.2$\pm6.0$ \\
AndX& 0.46$\pm0.2$ & 61.2$^{+52}_{-49}$ & 10.1$^{+5.1}_{-4.3}$ \\
AndXI& 0.41$^{+0.3}_{-0.2}$ & 165.5$^{+ 196}_{- 142}$ & 12.0$^{+11.0}_{-8.1}$ \\
AndXIII& 0.26$^{+0.22}_{-0.16}$ & 126.6$^{+ 153}_{- 108}$ & 9.2$^{+10.1}_{-6.4}$ \\
AndXIV& 0.42$\pm0.2$ & 41.6$\pm28.2$& 8.4$\pm5.2$ \\
AndXV& 0.22$\pm0.11$ & 9.0$^{+7.1}_{-7.0}$ & 6.3$^{+3.4}_{-3.3}$ \\
AndXVI& 0.24$^{+0.08}_{-0.06}$ & 11.6$^{+3.9}_{-2.9}$ & 8.8$^{+3.2}_{-2.7}$ \\
AndXVII& 0.13$^{+0.33}_{-0.19}$ & 12.82$^{+ 44.79}_{- 26.38}$ & 4.5$^{+11.2}_{-4.5}$ \\
AndXVIII& 1.5$\pm0.5$ & 44.8$\pm27.1$ & 15.3$\pm6.1$ \\
AndXIX& 2.7$^{+1.9}_{-1.2}$ & 118.0$^{+ 124.2}_{- 85.2}$ & 7.4$^{+6.6}_{-3.8}$ \\
AndXX& 0.30$^{+0.28}_{-0.17}$ & 213.0$^{+ 282.2}_{- 171.0}$ & 11.2$^{+12.0}_{-7.0}$ \\
AndXXI& 1.2$^{+0.5}_{-0.4}$ & 29.8$^{+18.7}_{- 16.5}$ & 7.1$^{+3.1}_{-2.7}$ \\
AndXXII& 0.10$^{+0.11}_{-0.08}$ & 69.7$^{+ 102.2}_{- 82.4}$ & 4.4$^{+4.7}_{-3.9}$ \\
AndXXIII& 3.4$^{+0.8}_{-0.7}$ & 68.4$^{+ 46.4}_{- 46.1}$ & 11.2$^{+2.8}_{-2.6}$ \\
AndXXV& 0.33$^{+0.19}_{-0.18}$ & 10.2$^{+ 10.0}_{-9.5}$ & 4.7$^{+2.9}_{-2.6}$ \\
AndXXVI& 0.83$^{+0.72}_{-0.46}$ & 279.8$^{+ 383}_{- 277.6}$ & 13.7$^{+ 15.7}_{-9.6}$ \\
AndXXVIII& 0.53$^{+0.36}_{-0.27}$ & 50.5$^{+ 51.0}_{- 39.1}$ & 10.4$^{+7.7}_{-5.8}$ \\
AndXXIX& 0.68$^{+0.23}_{-0.22}$ & 67.8$^{+ 38.6}_{- 37.5}$ & 9.0$^{+3.4}_{-3.2}$ \\
AndXXX& 2.1$^{+2.0}_{-1.2}$ & 300.0$^{+ 433.1}_{- 302.3}$ & 18.6$^{+17.7}_{-11.4}$ \\
Scl& 1.3$\pm0.3$ & 18.2$\pm9.8$ & 14.5$\pm3.9$ \\
LeoT& 0.58$\pm0.22$ & 196.9$\pm120.0$ & 11.8$\pm5.1$ \\
UMaI& 2.6$^{+1.2}_{-1.1}$ &3731$^{+ 2577}_{- 2524}$ & 18.8$^{+8.9}_{-8.5}$ \\
LeoIV& 0.13$\pm0.10$ & 299.1$\pm54.2$ & 5.2$\pm4.0$ \\
Com& 0.09$\pm0.03$ & 510.8$pm309.0$ & 7.3$\pm2.2$ \\
CVnII& 0.09$\pm0.03$ & 229.9$^{+ 158}_{- 152}$ & 7.3$^{+3.0}_{-2.6}$ \\
LeoI& 1.2$\pm0.3$ & 7.1$\pm3.3$ & 14.5$\pm3.5$ \\
LeoII& 0.38$\pm0.07$ & 12.9$\pm5.2$ & 10.4$\pm2.2$ \\
Car& 0.6$\pm0.2$ & 50.7$\pm28.9$ & 10.4$\pm3.0$ \\
UMi& 1.5$\pm0.3$ & 146.6$\pm76.5$ & 15.0$\pm2.9$ \\
Dra& 0.94$\pm0.18$ & 69.7$\pm22.0$ & 14.4$\pm3.0$ \\
For& 5.3$\pm0.6$ & 7.6$\pm2.5$ & 18.5$\pm2.4$ \\
Sex& 2.5$\pm0.71$ & 120.4$\pm74.4$ & 12.5$\pm4.2$ \\
Boo& 0.30$^{+0.08}_{-0.06}$ & 198.0$^{+ 83.4}_{- 69.1}$ & 7.3$^{+2.0}_{-1.6}$ \\
CVnI& 1.9$\pm0.2$ & 164.3$\pm31.2$ & 12.0$\pm1.4$ \\
Herc& 0.26$^{+0.11}_{-0.10}$ & 145.6$^{+ 95.8}_{- 89.7}$ & 5.8$^{+2.8}_{-2.4}$ \\
LeoV& 0.04$\pm0.05$ & 197.5$\pm339.1$ & 3.8$\pm4.4$ \\
Wil1& 0.03$\pm0.02$ & 536.2$\pm613.8$ & 6.8$\pm4.6$ \\
UmaII& 0.26$\pm0.10$ &1319.1$\pm961$ & 9.0$\pm3.9$ \\
Seg& 0.02$\pm0.01$ &1374.7$^{+ 1453.47}_{- 1234.16}$ & 5.8$^{+3.9}_{-2.8}$ \\
Seg2& 0.02$\pm0.02$ & 536.4$\pm588.7$ & 4.1$^{+0.9}_{-4.1}$ \\
\enddata
\label{tab:ml}
\end{deluxetable}
In Fig.~\ref{fig:ML}, we plot the masses contained within the half-light radii
($M_{\rm half}$) of all the Local Group dSphs as a function of their
luminosity within the half-light radius ($L_{\rm half}$), where the points are
colour coded as in previous figures. The values themselves can be found in
Table~\ref{tab:ml}. Additionally we overplot lines of constant mass-to-light ratio (with
$[M/L]_{\rm half}=1,10,100$ and $1000$). It can be seen that the majority of
these objects (including two of our outliers, And XIX and XXI, labeled in
plot) have $[M/L]_{\rm half}\gta10$, indicating that their dynamical masses
are much higher than can feasibly be explained by the mass of their baryons
alone (although see recent work in predicting the velocity dispersions and
mass-to-light ratios of M31 dSphs using MOND, without dark matter by
\citealt{mcgaugh13}). This is typically ascribed to the presence of dark
matter halos in these objects, whose mass dominates that of their baryons. The
green shaded region in this plot represents the parameter space in this
framework typically inhabited by globular cluster systems of the MW
\citep{rejkuba07}, whose masses can be explained by their stellar content
alone, without invoking dark matter.
Interestingly, we see a few objects on this plot whose mass-to-light ratios
are consistent (within $1\sigma$ uncertainties) with those of the Galactic
globular clusters, suggesting that they possess little or no dark matter. In a
couple of cases, the very large uncertainties on current measurements mean
that this overlap is not significant, and will likely disappear with future
observations. But there are two objects, And XV and XXV, that are particular
noteworthy. The masses of And XV and XXV are derived from sample sizes of
$\sim30$ stars. The potential implication of this is that these galaxies
contain very little dark matter, which would be quite unexpected for objects
of their sizes and luminosities.
In Fig.~\ref{fig:vmax} we plot the circular velocities measured within
the half-light radius (a good proxy for the central mass of these objects)
of the Local Group dSphs as a function of their half-light radii. We derive
$V_{c, 1/2}$ from the measured masses within the half-light radius using the
relationship between circular velocity and mass:
$V_{c,1/2}=\sqrt{\frac{GM_{\rm half}}{r_{\rm half}}}$. These values are
tabulated in Table~\ref{tab:ml}. The shaded regions overplotted represent
the circular velocity profiles of Aquarius subhalos (taken from
\citealt{kolchin12}) and are labeled with their maximum circular velocity in
each case. Subhalos with maximum circular velocities below $10{\rm\,km\,s^{-1}}$
(i.e. below the cyan profile in Fig.~\ref{fig:vmax}) are proposed to be too
low mass to form luminous galaxies, as their star formation is highly
suppressed due to inefficient gas cooling, causing them to remain
essentially dark ($V_{\rm max, limit}\sim10{\rm\,km\,s^{-1}}$, \citealt{koposov09}). The
red and blue curves in the figure are representative of the $\sim10$
most-massive subhalos seen in DM only simulations where we would naively
expect the most luminous dwarf galaxies in the Local Group to reside. When
including the M31 dSphs, we see that there are now a number of systems that
may be consistent with living in such massive halos. However, many of these
systems are the less luminous objects ($-6>M_V>-8$), where our measurement
uncertainties are large. For the brighter M31 dSphs whose velocity
dispersions (and hence, masses) are well resolved, there are only 2 objects
(And VI and And VII) that may inhabit halos with maximum circular velocities
of $24{\rm\,km\,s^{-1}}$ or greater. As such, the TBTF problem would seem to be present
in Andromeda as well as the Milky Way.
The previously discussed outliers from C13 and T12 (Hercules, And XIV, XV, XVI, XIX, XXI
and XXV) are all labeled in Fig.~\ref{fig:vmax}, as is the MW dSph,
Bo\"otes I (Boo I). These objects again stand out as
they fall tentatively shy of the pre-reionization star formation
threshold. If their halos have always been so low mass, they should never have
been able to form stars. Given the large uncertainties, all but 4 of these
outliers are (just) consistent with this lower limit,and thus, not of great
concern. But And XXII and XXV in M31, and Herc and Boo I in the MW, all fall
below this threshold, even when taking their uncertainties into account. This
implies that, in order for us to observe these systems now, their masses must
have been higher in the past, and have been reduced by some physical process
during their evolution. We discuss this further in \S~\ref{sect:discussion}.
Combined, these low mass-to-light ratios and lower than predicted masses,
highlight the ongoing tensions between observations and theory. It is clear
that, if the predictions from the $\Lambda$CDM paradigm are to be reconciled with our
observations, we must investigate avenues that can lower the masses of dark
matter subhalos over the course of their cosmic evolution. In the next
section, we discuss numerous possibilities for this that have been put forth
recently, and comment on their ability to reproduce our findings within the
Local Group.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=0.9\hsize]{fig6.pdf}
\caption{Mass within the half-light radius ($M_{\rm half}$) as a function
of luminosity within the half-light radius ($L_{\rm half}$) for Local
Group dSphs. Points are colour coded as in previous figures. The dashed
lines represent mass-to-light ratios of $[M/L]_{\rm half}=1,10,100$ and
$1000{\rm\,M_\odot}/{\rm\,L_\odot}$. The green shaded region indicates the parameter space
typically inhabited by simple stellar systems (i.e., those without dark
matter). It is interesting to note that there are a number of M31
objects, namely And XV and XXV, that are consistent with this
regime. }
\label{fig:ML}
\end{center}
\end{figure}
\begin{figure*}
\begin{center}
\includegraphics[angle=0,width=0.9\hsize]{fig7.pdf}
\caption{ Circular velocities within the half-light radius($V_{c,1/2}$) for
Local Group dSphs, as derived from their velocity dispersions. The
shaded lines are circular velocity profiles of subhalos from the
Aquarius simulations (taken from \citealt{kolchin12}), and are labeled
with their maximum circular velocities. Subhalos with maximum circular
velocities below $10{\rm\,km\,s^{-1}}$ (i.e. profiles below the cyan shaded curve)
are not thought to be massive enough to efficiently cool their hydrogen
and form stars. A number of dSphs (particularly Herc and And XXV) have
circular velocities that see them preferentially residing in halos that fall
below this low mass cut-off for star formation.}
\label{fig:vmax}
\end{center}
\end{figure*}
\section{Explaining the masses of the Local Group dSphs - observations
vs. theory}
\label{sect:discussion}
From the analysis in \S~\ref{sect:results} and \S~\ref{sect:outliers} we still
see some discrepancies between the masses of subhalos in simulations, and the
masses we infer from observations of the subhalos within the Local Group. At
face value, it seems that we expect to observe luminous satellites around MW
mass halos with higher central masses than we do. One explanation for this
missing massive satellite problem could simply be that at these low halo
masses ($M_{\rm halo}<10^{10}{\rm\,M_\odot}$) star formation becomes increasingly
stochastic, so that the luminosity of a subhalo does not necessarily correlate
with the mass of the subhalo \citep{kuhlen12,kuhlen13}. Other solutions appeal
to physical processes affecting the evolution of dwarf galaxies, and can be
broadly assigned to three categories: The effect of tidal interactions with
the host galaxy, the effect of stellar feedback on the mass profiles of
galaxies and the true mass of the host system. Finally, some have also
appealed to the modification of the current cosmological paradigm,
$\Lambda$CDM, either via the properties of the dark matter itself (e.g. warm
dark matter, \citealt{anderhalden13} or self interacting dark matter,
\citealt{rocha13,zavala13}), or via the modification of Newtonian gravity
(i.e. MOND, \citealt{mcgaugh13}) to remove the need for dark matter
altogether. In the subsequent sections, we discuss the physical processes that
might be responsible for lowering the central masses of the whole Local Group
population, and why this effect might be more pronounced in some objects
(i.e. Boo I, Herc, And XIX, XXI and And XXV) than others.
\subsection{Tides}
The notion that the lower central masses observed in the outlying dSphs is the
result of some physical process that has moved them away from a more typical
mass is appealing, and the possibility has been briefly discussed in other
works, particularly \citet{collins11b} and C13. There, the authors point out
that dSphs that are more extended at a given luminosity, such as the extreme
outliers, And XIX, XXI and XXV, also tend to have lower central masses. One of
the MW low mass outliers, the MW Herc object, is also already thought to be a
tidally disrupting system, transitioning into a stellar stream
\citep{martin10}.
Outside of the Local Group, a number of tidally disrupting dwarf galaxy
systems have recently been observed whose half-light radii are also much more
extended than would be expected from the $r_{\rm half}-L$ relation
\citep{brasseur11b}, resulting in very low surface brightnesses. These include
NCG 4449B ($M_V=-13.4$ and $r_{\rm half}=2.7$~kpc,
\citealt{rich12,delgado12}), and the Hydra dwarf galaxy, HCC-087 ($M_V=-11.6$
and $r_{\rm half}=3.1$~kpc, \citealt{koch12}). Their closest analog within the
Local Group is And XIX ($M_V=-9.3$, $r_{\rm half}=2.1^{+1.0}_{-0.4}{\rm\,kpc}$,
Martin et al. in prep.) making it an outlier to the \citet{brasseur11b} relation,
as well as falling below the mass expectation for a galaxy of its radial
scale. Taking these examples into account, perhaps
these properties (low surface brightness and/or low mass) are indicators of a
system undergoing significant tidal interaction with its host.
Mass loss of subhalos from interactions with their hosts has also been
studied in numerical models, both in a dark matter only context (e.g.,
\citealt{tormen98,klypin99,ghigna00,hayashi03,zentner03,kravtsov04,kazantzidis04a})
and with the inclusion of baryons (e.g.,
\citealt{penarrubia10,donghia10,zolotov12,brooks12}). In all cases, as these
systems orbit their host, their dark matter is stripped, lowering their
densities and masses at all radii. After the dark matter is removed, the
stars reach a dynamic equilibrium with their lower density potential,
causing a drop in the central mass. In simulations where baryonic physics
are included, the mass losses from subhalos as a result of tidal
interactions with a host are more pronounced than for the dark matter
only case. Further, the size of the mass reduction increases with earlier infall
times and more radial orbits. In \citet{zolotov12}, they demonstrated that a
subhalo accreted at $z>6$~Gyr in an SPH simulation would experience a
greater reduction in its mass than is seen with a dark matter only set
up. Similarly, the mass of subhalos on radial orbits in the SPH simulation
also experience a more significant drop in mass than their dark matter only
counterparts. In all cases, the presence of a massive baryonic disk in the
host galaxy (such as those hosted by the Galaxy and M31) reduces the masses
of the satellite population at a much greater rate than in the dark matter
only case.
One could therefore argue that the outliers seen in this study, such as
Hercules, And XIX, XXI and XXV, may have fallen in to their host galaxies
earlier, and onto more radial orbits where they interact more significantly
with their host, leading to a more pronounced mass loss. It is difficult to
properly model the orbital properties of these objects, but recent work
by \citet{watkins13} modelled the orbital properties of M31 dSphs by combining
the timing argument with phase-space distribution functions. This work found
no evidence to suggest that the M31 outliers are on very radial orbits, nor do
they seem to have experienced particularly close passages with M31 itself,
perhaps ruling out this option.
A prime example of a tidally disrupting dSph within the MW is the Sgr
dSph. This object is currently undergoing violent tidal disruption, yet it has
a velocity dispersion that is entirely consistent with the best fit NFW and
cored mass profiles to both the MW alone and to the full Local Group, perhaps
arguing against the mechanism we have outlined above. However, Sgr is
currently near the pericenter of its orbit, only $\sim20$ kpc from the
Galactic center \citep{law10}. The outliers we refer to are located further
out ($D_{host}>70{\rm\,kpc}$ for all outliers, \citealt{martin10,koposov11,conn13}),
and so we do not expect them to be currently experiencing significant tidal
distortions, rather that their past interactions with their host have removed
more mass from their centers than their more `typical' counterparts.
In summary, numerical models have demonstrated that tidal mechanisms are able
to lower the masses of dSphs, and could explain the lower than expected masses
of the Local Group outliers, Herc, And XIV, XV, XVI, XIX, XXI and XXV if they
have experienced more significant past interactions with their host.
\subsection{Feedback from star formation and supernova}
For many years, kinematic studies of low surface brightness galaxies have
shown that the mass profiles of these objects are less centrally dense than
expected. They are more compatible with flatter, cored halo functions, rather
than the cuspier NFW profiles seen in simulations
(e.g. \citealt{flores94,deblok02,deblok03,deblok05}). Many have argued that
this is a result of bursty, energetic star formation and supernova
(SN) within these galaxies. These processes drive mass out from the center of the halo, flattening the high
density cusp into a lower density core, leading to a lower central mass than
predicted by pure dark matter simulations
(e.g. \citealt{navarro96b,dekel03,read05,mashchenko06,pontzen12,governato12,maccio12}). Could
the lower than expected central masses of the Local Group dSphs also be caused
by feedback?
\citet{zolotov12} and \citet{brooks12} compared a dark matter only simulation
with a smooth particle hydrodynamic (SPH) simulation of a MW type galaxy in a
cosmological context to see whether the inclusion of baryons and feedback in
the latter can produce satellite galaxies with lower central masses and
densities. For galaxies with a stellar mass $M_*>10^7{\rm\,M_\odot}$ ($M_V\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$}-12$) at
the time of infall, feedback can reduce the central mass of dSph
galaxies. Below this mass, the galaxies have an insufficient total mass to
retain enough gas beyond reionization to continue with the significant, bursty
star formation processes required to remove mass from their centers. The Local
Group outliers discussed above have $M_V\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$}-10$, so unless they have been
significantly tidally stripped by their hosts after falling in, i.e.,
experienced {\it total} (dark matter plus baryonic) mass losses of greater
than $\sim90\%$ \citep{penarrubia08b}, feedback cannot explain their current
masses. This is supported by the findings of \citet{garrison13}, where they
model the dynamical effect of SN feedback on the mass distribution of dark
matter halos. To match the current observed central masses in MW dSph
galaxies, one would need to deposit 100\% of the energy resulting from $40,000$
SN directly to the dark matter halos, which is greater than the expected total
number of SN to have ever occurred in the majority of these systems. The work
of \citet{penarrubia12} also support the findings of these works, namely that
fainter dSphs should not be able to significantly lower their central masses
via feedback, and if they were able, they would serve to exacerbate the
missing satellite problem.
In \citet{dicintio13}, they also study the effect of feedback in subhalos on
their mass profiles, using a suite of galaxies from the MaGICC project
\citep{stinson10,stinson13}. Their findings show that the mass of stars formed
per halo mass is the most important factor for the shaping of the central mass
profiles of galaxies. Objects with stellar-to-halo mass ratios of $M_*/M_{\rm
halo}\lta0.01$ are not able to alter their dark matter distributions, but as
this fraction increases, so does the ability to flatten their central mass
profiles. They find that this process is maximally efficient in galaxies with
$M_*\approx10^{8.5}{\rm\,M_\odot}$, and below this, the central dark matter slope
increases once more. As such, their findings are similar to those of
\citet{zolotov12}. This seeming consensus on the amount of baryons
required to efficiently reshape the dark matter mass profile of a dwarf
galaxies via feedback means that, in principle, based on their current luminosities only 4
of the dSphs discussed in this work (And II, And VII, Fornax,
and Leo I, \citealt{mcconnachie12}) have enough explosive energy at
their disposal to reduce their central densities with feedback alone. For the
remaining MW and M31 objects, another mechanism, such as tides, would need
to be invoked to explain their low masses.
\subsection{Host mass}
From a plethora of works
(e.g. \citealt{moore99,ghigna00,kravtsov04b,zentner05,vandenbosch05,zheng05,giocolo08,springel08}),
we know that the number of subhalos within a host halo, scales with the mass
of the host halo itself. Therefore, when comparing the mass of halos we
observe within the MW or Andromeda with those found in simulations, it is
important that we select a simulated galaxy of the same mass. Unfortunately,
in the case of both M31 and the MW, the total masses of these systems are
actually quite uncertain, ranging from $\sim0.7-2.7\times10^{12}{\rm\,M_\odot}$ for the
Milky Way (e.g., \citealt{wilkinson99,xue08,li08,watkins10,piffl13}) and
$\sim0.8-2.2\times10^{12}{\rm\,M_\odot}$ (e.g.,
\citealt{evans00a,evans00b,li08,guo10,watkins10}), making this difficult. From
the point of view of abundance matching, a galaxy as luminous as the MW should
be hosted by a halo with an even higher mass than these estimates
($\sim3\times10^{12}{\rm\,M_\odot}$, \citealt{behroozi13}), which could imply that our
Galaxy is a significant outlier when compared with the bulk of the galaxies
within the Universe.
\citet{veraciro13} investigated the effect of varying the mass of the host
halo on both the number and dynamics of subhalos using the Aquarius
simulations \citep{springel08}. They found that they were able to match both
these quantities when using a simulated halo whose mass was consistent with
the lower bound of observational constraints for the MW,
$8\times10^{11}{\rm\,M_\odot}$. This immediately eliminates the TBTF problem, as the
most massive simulated subhalos have $V_{\rm
max}\lta25{\rm\,km\,s^{-1}}$. \citet{dicintio12} also find that they can match the number
and dynamics of MW satellites using halos from the CLUES simulations with
masses of $5-7\times10^{11}{\rm\,M_\odot}$ \citep{gottlober10,dicintio12}. Thus,
if the virial mass of the MW is at the lower end of current observational
estimates, the masses we measure for its subhalos would be much more inline
with predictions from numerical simulations. However, it is worth noting
that such a low mass for the MW would lessen the probability of hosting
massive satellites like the LMC and SMC, and may just replace the missing
massive satellite problem with a found massive satellite problem
(e.g. \citealt{kolchin11b,busha11}). It is also in conflict with a recent
estimate of the mass of the MW from \citet{kolchin13} who use the Aquarius
simulations to demonstrate that the 3D space motion of the Leo I dSph puts a
lower limit of $\sim1\times10^{12}{\rm\,M_\odot}$ on the mass of the MW at a
confidence level of 95\%. \citet{veraciro13} also find that the Andromeda
satellites can be best matched if the host mass is $1.77\times10^{12}{\rm\,M_\odot}$,
implying that the mass of M31 is roughly twice that of the MW. This value is
compatible with the best estimate for the Andromeda mass when using the full
dwarf galaxy satellite population as a tracer (L. Watkins, private
communication). If Andromeda is more massive than the MW, it could explain
the fact that M31 has more (in number terms) massive non-dSph satellites
(NGC147, NGC 185, NGC 205, M32 and M33) than the MW (LMC and SMC). It may
also explain the low mass outliers, particularly the statistically
significant And XIX, XXI and XXV. In particular, the very low value of
$V_{c,1/2}$ we derive for And XXV strongly indicates that it is currently
residing in a dark matter subhalo with a maximum circular velocity below the
mass limit expected for luminous galaxy formation, suggesting it has
experienced some physical process that has lowered its mass significantly
over the course of its evolution. A more massive host would imply that the
dSphs (which are more susceptible to tidal disruption than the more massive
satellites) within this system have experienced greater tidal forces over
the course of their evolution, which could lower their masses below the SF
threshold of $10-15{\rm\,km\,s^{-1}}$ \citep{penarrubia08b,koposov09}.
Precisely pinning down the correct viral masses of the MW and M31 is clearly
an important step towards better understanding the masses of the dwarf
galaxies we observe within the Local Group in a cosmological context. Without
precise mass estimates, it is difficult for us to quantify discrepancies with
theoretical expectations, like those of the TBTF problem, and might help us to
explain the differences between the masses of dSphs we see around M31 and the MW.
\section{Conclusions}
\label{sect:conclusions}
The relatively high dark-to-stellar mass ratios of dSph galaxies single them
out as excellent probes of the behaviour of dark matter on the smallest of
scales. Comparisons of the masses of the MW dSphs with expectations from
cosmological simulations have revealed several discrepancies, most notably the
issue of cuspy vs. cored central densities and a dearth of luminous high mass
subhalos around the MW compared with dark matter only simulations (the `too
big to fail' problem). In this paper we have expanded these analyses by
including the dSph satellites of M31 in the comparisons.
We revisit the notion that all dSph are embedded in dark matter halos that
follow a universal density profile in their centers \citep{walker09b} by
fitting NFW and cored mass profiles to the full sample of MW and M31 dSphs in
$r_{\rm half}-\sigma_v$ space. We find that no singular profile provides a
good fit to the data, but that their masses are instead described by a range
of halo profiles with a well defined scatter as a function of half light
radius. We find that when comparing fits for solely MW dSphs to solely M31
dSphs, the latter prefer significantly lower masses for a given size than the
former. We demonstrate that this offset is driven by 3 low mass outliers in
M31, whose half-light radius place them in a region of parameter space with
very few MW dSphs for comparison (And XIX, XXI, and XXV). Once these outliers
are removed, we find that the two populations agree exceptionally well,
following mass profiles with similar average values of $V_{\rm max}$ and
$R_S$, and a scatter in mass that equates to $\sim50\%$ of the average mass at
any specific radius.
We also derive the $V_{c, 1/2}$ values for each dSph directly from their
velocity dispersions and find them to be in good agreement with our fitted
ranges, with the exception of the 3 excluded outliers. Further inspection of
these values, plus the mass-to-light ratios of the population reveal a number
of interestingly low mass systems. The most significant of these are And XV,
XIX, XXI and XXV from M31 and Herc and Boo I from the MW. In particular, the
central mass of And XXV is so low that if it had always been this way, it
would never have formed stars. And yet, now; it is clearly luminous. By
comparing the properties of these objects with those of observed tidally
disrupting dwarf galaxies, we postulate that tides are a candidate mechanism
for lowering the masses of these objects, especially when combined with
stellar feedback at early epochs that can reduce the central masses of these
galaxies before they fall into their host.
When comparing the computed values of $V_{c,1/2}$ from our observed sample
with the circular velocity profiles of subhalos within simulations, we still
see an offset between the most massive simulated subhalos, and the most
massive dSphs in both the MW and M31, described as the TBTF problem. We argue
that, as this problem was defined via comparisons with a dark matter only
simulation of a MW type halo that neglects the effects on the mass profiles of
dSphs from baryonic processes (such as feedback and tides), and may not be
directly comparable to the MW and M31 (given the uncertainties on
observational measurements of their masses), it is difficult to quantify how
serious or significant this problem truly is. By running simulations with
baryonic processes included, and precisely determining the masses of the MW
and M31, we will better be able to assess whether there is truly a missing
massive satellite problem. As such, the masses of Local Group dSphs should be
thought of as a constraint for more complex simulations that include a wide
range of physical processes that are not currently well understood.
\section*{Acknowledgments}
We would like to thank the anonymous referee for their helpful comments and
suggestions, as well as a thorough discussion surrounding the `Too Big To
Fail' problem.
The research leading to these results has received funding from the European
Research Council under the European Union's Seventh Framework Programme (FP 7)
ERC Grant Agreement n. [321035].
R.I. gratefully
acknowledges support from the Agence Nationale de la Recherche though
the grant POMMME (ANR 09-BLAN-0228).
G.F.L. and N.B. gratefully acknowledge financial support for his
ARC Future Fellowship (FT100100268) and through the
award of an ARC Discovery Project (DP110100678).
A.K. thanks the Deutsche Forschungsgemeinschaft for funding from Emmy-Noether
grant Ko 4161/1.
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Q: two MPPT tracker on the same solar panel string? I would like to know if possible to use two distinct MPPT on the same solar panel string.
A: I'm going to guess yes, depending on which trackers you use, but this is probably not a recommended mode of operation. From the point of view of the solar panels, you can think of an MPPT as a variable resistor which sets its load to whatever gives the highest amount of power. In theory, having two of them is just like having two variable resistors in parallel. If everything works properly, each one will get half the power (and the resistances will be double).
However, the problem comes when you take the feedback into account. MPPT's use a variety of different algorithms to get the right "resistance" value. There could be modes where the MPPT's are "fighting" each other, resulting in improper power point tracking or even undefined operation. Unless you can find a MPPT that's designed to work with other MPPTs, I would probably avoid it. Why do you need this anyway?
also see https://meta.stackexchange.com/questions/66377/what-is-the-xy-problem
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NRL's two-conference proposal is heating up. Gorden Tallis thinks it's unfair
Gorden Tallis is not a fan of the proposed two-conference system in a revamped NRL season because he believes it will create an unfair playing field.
Matt Encarnacion
AAPApril 7, 20209:38am
The Broncos could lose their four points under a revamped competition structure.Source:News Corp Australia
Tune into our new show Fox League Live on Channel 502 every day at 5pm and on Saturday at 3pm.
Splitting the NRL competition into two conferences would be unfair for some teams, according to former Kangaroos star Gorden Tallis.
A radical proposal to isolate two 'bubbles' of teams, in NSW and Queensland, for a 14-week regular season is believed to be gathering steam at league central.
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One conference would include eight Sydney-based teams, while the remaining eight sides would be quarantined in the sunshine state.
There are suggestions points earned from the two rounds of games completed before the coronavirus-enforced shutdown could be scrapped.
St George Illawarra, based one hour south of Sydney in Wollongong, could be asked to cross the border along with Newcastle Knights.
Each team would face each other twice before a four-week finals series, taking the entire season to 18 weeks, not including State of Origin.
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The idea is understood to be one of a handful of scenarios the innovation committee will consider when it meets on Thursday.
Only two of the six teams who won their opening two games of the season come out of Sydney: Parramatta and Penrith.
The other four are Newcastle, Brisbane, Canberra and Melbourne, all four of whom would be allocated to the non-Sydney conference.
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Tallis, who played 16 Tests for Australia, is not convinced by the two- conference plan.
"I don't believe in two conferences. I really don't," he said on Fox League Live.
"If you're a Queenslander, and you're going to go to Sydney and play in front of an empty stadium, then you've got to go.
"Everybody's got to play each other once, but there can't be two conferences.
Gorden Tallis is concerned a conference system would favour the Sydney teams.Source:News Corp Australia
"I don't think it's fair on some teams that would be in the conferences, because one conference will be weaker than the other. There's no doubt about that."
Tallis also dismissed the notion of resuming the season with State of Origin, saying that after a long lay-off it would dilute the standard of the rivalry.
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"What makes Origin so special is the standard that it's played at, what the players put their body through," he said.
"And it's the best product that we can get on our television set. So you don't want to dilute that by not having everybody fit, fighting and raring to go."
Originally published asNRL's two-conference proposal is heating up. Tallis thinks it's unfair
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The Ugly Truth About Biofuels
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Brian Westenhaus
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New Pressure Cooking Technique Turns Algae to Crude Oil in a Minute
By Brian Westenhaus - Nov 01, 2012, 7:07 PM CDT
University of Michigan engineers lead by Phil Savage, an Arthur F. Thurnau professor and a professor of chemical engineering can "pressure-cook" algae for as little as a minute and transform an unprecedented 65% into bio crude oil.
Savage said, "We're trying to mimic the process in nature that forms crude oil with marine organisms." It's quite a reversal from the natural method thought to take millions of years to make crude oil.
The Michigan team's findings will be presented today, Nov. 1 at the 2012 American Institute of Chemical Engineers Annual Meeting in Pittsburgh.
Savage's organism of choice is the ocean-going green marine micro-alga of the genus Nannochloropsis.
The simplified explanation of the most successful process so far to make their one-minute biocrude is Savage and Julia Faeth, a doctoral student in Savage's lab, fill a steel pipe connector with 1.5 milliliters of wet algae, cap it and plunge it into 1,100-degree Fahrenheit sand. The small volume ensures that the algae is heated through, but with only a minute to warm up, the algae's temperature should have just grazed the 550-degree mark before the team pulls the reactor back out.
Previously, Savage and his team heated the algae for times ranging from 10 to 90 minutes. They saw their best results, with about half of the algae converted to biocrude, after treating it for 10 to 40 minutes at 570 degrees.
Why are the one-minute results so much better? Savage and Faeth won't be sure until they have done more experiments, but they have some ideas.
Savage said, "My guess is that the reactions that produce biocrude are actually must faster than previously thought." Faeth suggests that the fast heating might boost the biocrude by keeping unwanted reactions at bay.
Faeth explains, "For example, the biocrude might decompose into substances that dissolve in water, and the fast heating rates might discourage that reaction."
Related Article: Putin Plays Down Russia's Deadly Dependence on Oil & Gas Revenues
Another point the team makes is that shorter reaction times mean that the reactors don't have to be as large.
"By reducing the reactor volume, the cost of building a biocrude production plant also decreases," Faeth said, though both she and Savage cautioned that they couldn't say for sure whether the new method is faster and cheaper until the process is further developed.
This news is a major breakthrough because current commercial makers of algae-based fuel first dry the algae and then extract the natural oil. But at over $20 per gallon, this fuel production process is a long way from the gas pump.
Savage points out, "Companies know that that approach is not economical, so they are looking at approaches for using wet algae, as are we."
At the very crux of the breakthrough is a major advantage. The wet method doesn't just extract the existing fat from the algae – it also breaks down proteins and carbohydrates. The Michigan minute method did this so successfully that the oil contained about 90 percent of the energy in the original algae. Breakthrough, indeed.
Savage remarks with the obvious, "That result is near the upper bound of what is possible."
It's not a done scientific solution – or not quite yet, anyway.
Related Article: Biofuels Company Teams Up with Enzyme Producer to Try and Kickstart Industry
Before biocrude can be fed into the existing refinery system as petroleum, it needs pre-refining to get rid of the extra oxygen and nitrogen atoms that abound in living things and follow along in the reaction forming the bio oil.
The Savage lab is already developing better methods for this segment of biofuel production, breaking the record with a biocrude that was 97 percent carbon and hydrogen earlier this year. We'll have to wait as a paper on this work is currently under review.
The research, "The Effects of Heating Rate and Reaction Time on Hydrothermal Liquefaction of Microalgae," was funded by the Emerging Frontiers in Research and Innovation program of the National Science Foundation. It looks like money well spent.
The university is already at work pursuing patent protection for the intellectual property, and is seeking commercialization partners to help bring the technology to market. Call quickly.
It's a bit humbling to consider the effort, innovation, intuition and creativity launched to corner a competitive algae fuel product then to have a breakthrough appear that is so simple of a process. The research is very likely to be replicated and see some commercial headway quite soon.
Should the full process show a competitive to crude oil produced price the issue will move to optimal algae mass per area. How far algae and other organisms can get to in making fuels per area is anybody's guess now. What the potential organism list could include is another fascinating question.
Congratulations Professor Savage! Let's hope the process can go to scale very economically. We're wondering if you're working to process algae to natural gas as well?
By. Brian Westenhaus
Source: Breakthrough! Algae to Oil In a Minute
Biofuels Company Teams Up with Enzyme Producer to Try and Kickstart Industry
Algal Biofuel is Currently Unsustainable, but Technology can Change That
Durwood M. Dugger on November 02 2012 said:
The energy required to heat the algal mass, is far greater than the energy stored in the algal mass. This has been the second most significant hurdle in algal biofuels inability to demonstrate economic feasibility - lack of or negative ROE. The first major hurdle is algae biofuels lack of sustainability (http://www.reuters.com/article/2012/10/24/us-usa-biofuels-algae-idUSBRE89N1Q820121024).
While the heat process may be academically interesting, in the real world it means that you need a free high volume heat source above 500 degrees and a free bio-available nutrient (phosphorus) source for sustainability, and then you might make a killing with algae biofuels. Neither circumstances seem very probable and totally improbable at the scales required to be a significant part of offsetting our energy deficit.
The heat process seems to be accomplishing two tasks, it ruptures the algae cells releasing the lipids and flares off water vapor increasing the lipid to water ratio of the yield. Since the article doesn't give much information about the final chemical form of the lipids, we don't know how much, if any of the additionally required lipid stabilization might have occurred.
What is clear is the high level of heat energy in-put required to make a lower energy yield product. So bottom line - the process doesn't do anything to offset the other big energy/costs sinks in algae production - growout, harvesting, and de-watering. The heat process's primary contribution (at a very high energy cost) is cell rupturing and final stage drying. The product would still have to be further stabilized to have any value as fuel.
If you had the two critical commodities to make the heat process work - free heat and nutrients, they would be still be more economically attractive applied to more direct higher value processes and products such as direct electrical generation (from the heat) and food (using the nutrients). If anything the high energy input here makes algae biofuel production even less sustainable and less economically feasible.
Eddie Holman on November 02 2012 said:
Mr Dugger presents a valid ROE balance analysis, yet his oversight of possible creative solutions to the high energy input equation leads me to inquire: In the Wright Brothers experiment, was the pre-flight energy imbalance negative? Isn't that the case even today? Work the equation the other way my friend, and
MrColdWaterOfRealityMan on November 03 2012 said:
Nothing changes the fact that algae is simply an inefficient solar collector. Adding heat to cook it may convert it, but can not possibly improve the net energy picture, which, when all the calculations are done, is likely to prove low, or negative.
Ed on November 12 2012 said:
In my opinion, the heat for this process is not going to be a problem at all. Set up a solar concentrating collector in Arizona and you've got yourself huge amounts of heat for free. We get above 110 degree ambient temperature here all summer without any solar concentration. 550 degrees should be what, 5x suns of solar concentration? I bet pumping wet algae through a pipe running through the focal point of a 5x parabolic trough solar concentrator would be easier than trying to plunge their capped steel pipe into a pile of hot sand.
RH on November 22 2012 said:
Why not run the piping through molten salt heated by CSP, the then you have an industrial plant producing electricity and bio fuel at the same time!!!!
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,371
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Przyłęk (do 1954 gmina Grabów nad Wisłą) – gmina wiejska w województwie mazowieckim, w powiecie zwoleńskim, z siedzibą w Przyłęku. W latach 1975–1998 gmina położona była w województwie radomskim.
Według danych z 30 czerwca 2004 gminę zamieszkiwało 6545 osób. Natomiast według danych z 31 grudnia 2019 roku gminę zamieszkiwały 6154 osoby.
Struktura powierzchni
Według danych z roku 2002 gmina Przyłęk ma obszar 130,89 km², w tym:
użytki rolne: 79%
użytki leśne: 11%
Gmina stanowi 22,91% powierzchni powiatu.
Demografia
Dane z 30 czerwca 2004:
Piramida wieku mieszkańców gminy Przyłęk w 2014 roku.
Sołectwa
Andrzejów, Babin, Baryczka, Grabów nad Wisłą, Helenów, Ignaców, Kulczyn, Krzywda, Lipiny, Lucimia, Łagów, Łaguszów, Ławeczko Nowe, Ławeczko Stare, Mierziączka, Mszadla Dolna, Mszadla Nowa, Mszadla Stara, Okrężnica, Pająków, Przyłęk, Rudki, Stefanów, Szlachecki Las, Wysocin, Wólka Łagowska, Wólka Zamojska, Załazy, Zamość Nowy, Zamość Stary.
Miejscowości bez statusu sołectwa: Borowiec, Kulczyn (osada) i Ruda.
Sąsiednie gminy
Chotcza, Janowiec, Policzna, Puławy, Wilków, Zwoleń
Przypisy
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,758
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<header> fajl log kdc </header> Il-fajl log kdc. <hr>
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{
"redpajama_set_name": "RedPajamaGithub"
}
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using System;
using System.Collections.Generic;
using BenchmarkDotNet.Attributes;
using BenchmarkDotNet.Jobs;
namespace EnumsNET.PerfTestConsole
{
[SimpleJob(RuntimeMoniker.Net48), SimpleJob(RuntimeMoniker.NetCoreApp50)]
public class DictionaryBenchmarks
{
private readonly Dictionary<Type, string> _typeDictionary;
private readonly DictionarySlim1<Type, string, TypeEqualityComparer> _typeDictionarySlim1;
private readonly DictionarySlim2<Type, string> _typeDictionarySlim2;
private readonly DictionarySlim3<Type, string> _typeDictionarySlim3;
private readonly Dictionary<int, string> _intDictionary;
private readonly DictionarySlim1<int, string, DefaultEqualityComparer<int>> _intDictionarySlim1;
private readonly DictionarySlim2<int, string> _intDictionarySlim2;
private readonly DictionarySlim3<int, string> _intDictionarySlim3;
public DictionaryBenchmarks()
{
_typeDictionary = new Dictionary<Type, string>(11)
{
{ typeof(AttributeTargets), nameof(AttributeTargets) },
{ typeof(Base64FormattingOptions), nameof(Base64FormattingOptions) },
{ typeof(ConsoleColor), nameof(ConsoleColor) },
{ typeof(ConsoleKey), nameof(ConsoleKey) },
{ typeof(ConsoleModifiers), nameof(ConsoleModifiers) },
{ typeof(ConsoleSpecialKey), nameof(ConsoleSpecialKey) },
{ typeof(DateTimeKind), nameof(DateTimeKind) },
{ typeof(DayOfWeek), nameof(DayOfWeek) },
{ typeof(EnvironmentVariableTarget), nameof(EnvironmentVariableTarget) },
{ typeof(GCCollectionMode), nameof(GCCollectionMode) },
{ typeof(GCNotificationStatus), nameof(GCNotificationStatus) }
};
_typeDictionarySlim1 = new DictionarySlim1<Type, string, TypeEqualityComparer>(_typeDictionary, _typeDictionary.Count);
_typeDictionarySlim2 = new DictionarySlim2<Type, string>(_typeDictionary, _typeDictionary.Count);
_typeDictionarySlim3 = new DictionarySlim3<Type, string>(_typeDictionary, _typeDictionary.Count);
_intDictionary = new Dictionary<int, string>(11);
for (var i = 0; i < 11; ++i)
{
_intDictionary.Add(i, i.ToString());
}
_intDictionarySlim1 = new DictionarySlim1<int, string, DefaultEqualityComparer<int>>(_intDictionary, _intDictionary.Count);
_intDictionarySlim2 = new DictionarySlim2<int, string>(_intDictionary, _intDictionary.Count);
_intDictionarySlim3 = new DictionarySlim3<int, string>(_intDictionary, _intDictionary.Count);
}
[Benchmark(Baseline = true)]
public bool Type_Dictionary() => _typeDictionary.TryGetValue(typeof(DayOfWeek), out _);
[Benchmark]
public bool Type_DictionarySlim1() => _typeDictionarySlim1.TryGetValue(typeof(DayOfWeek), out _);
[Benchmark]
public bool Type_DictionarySlim2() => _typeDictionarySlim2.TryGetValue(typeof(DayOfWeek), out _);
[Benchmark]
public bool Type_DictionarySlim3() => _typeDictionarySlim3.TryGetValue(typeof(DayOfWeek), out _);
[Benchmark]
public bool Int_Dictionary() => _intDictionary.TryGetValue(7, out _);
[Benchmark]
public bool Int_DictionarySlim1() => _intDictionarySlim1.TryGetValue(7, out _);
[Benchmark]
public bool Int_DictionarySlim2() => _intDictionarySlim2.TryGetValue(7, out _);
[Benchmark]
public bool Int_DictionarySlim3() => _intDictionarySlim3.TryGetValue(7, out _);
internal sealed class DictionarySlim1<TKey, TValue, TKeyComparer>
where TKey : notnull
where TKeyComparer : struct, IEqualityComparer<TKey>
{
public readonly struct Entry
{
public readonly int Next;
public readonly TKey Key;
public readonly TValue Value;
public Entry(int next, TKey key, TValue value)
{
Next = next;
Key = key;
Value = value;
}
}
private readonly int[] _buckets;
internal readonly Entry[] _entries;
public int Count { get; }
public DictionarySlim1(IEnumerable<KeyValuePair<TKey, TValue>> dictionary, int count)
{
Count = count;
var size = HashHelpers.PowerOf2(count);
var buckets = new int[size];
var entries = new Entry[size];
var i = 0;
if (dictionary != null)
{
foreach (var pair in dictionary)
{
var value = pair.Key;
ref int bucket = ref buckets[value.GetHashCode() & (size - 1)];
entries[i] = new Entry(bucket - 1, value, pair.Value);
bucket = i + 1;
++i;
}
}
_buckets = buckets;
_entries = entries;
}
internal bool TryGetValue(TKey key, out TValue value)
{
var entries = _entries;
for (var i = _buckets[key.GetHashCode() & (_buckets.Length - 1)] - 1; i >= 0; i = entries[i].Next)
{
if (default(TKeyComparer).Equals(key, entries[i].Key))
{
value = entries[i].Value;
return true;
}
}
value = default!;
return false;
}
}
internal sealed class DictionarySlim2<TKey, TValue>
where TKey : notnull
{
public readonly struct Entry
{
public readonly int Next;
public readonly TKey Key;
public readonly TValue Value;
public Entry(int next, TKey key, TValue value)
{
Next = next;
Key = key;
Value = value;
}
}
private readonly int[] _buckets;
internal readonly Entry[] _entries;
public int Count { get; }
public DictionarySlim2(IEnumerable<KeyValuePair<TKey, TValue>> dictionary, int count)
{
Count = count;
var size = HashHelpers.PowerOf2(count);
var buckets = new int[size];
var entries = new Entry[size];
var i = 0;
if (dictionary != null)
{
foreach (var pair in dictionary)
{
var value = pair.Key;
ref int bucket = ref buckets[value.GetHashCode() & (size - 1)];
entries[i] = new Entry(bucket - 1, value, pair.Value);
bucket = i + 1;
++i;
}
}
_buckets = buckets;
_entries = entries;
}
internal bool TryGetValue(TKey key, out TValue value)
{
var entries = _entries;
for (var i = _buckets[key.GetHashCode() & (_buckets.Length - 1)] - 1; i >= 0; i = entries[i].Next)
{
if (EqualityComparer<TKey>.Default.Equals(key, entries[i].Key))
{
value = entries[i].Value;
return true;
}
}
value = default!;
return false;
}
}
internal sealed class DictionarySlim3<TKey, TValue>
where TKey : notnull
{
public readonly struct Entry
{
public readonly int Next;
public readonly int HashCode;
public readonly TKey Key;
public readonly TValue Value;
public Entry(int next, int hashCode, TKey key, TValue value)
{
Next = next;
HashCode = hashCode;
Key = key;
Value = value;
}
}
private readonly int[] _buckets;
internal readonly Entry[] _entries;
public int Count { get; }
public DictionarySlim3(IEnumerable<KeyValuePair<TKey, TValue>> dictionary, int count)
{
Count = count;
var size = HashHelpers.PowerOf2(count);
var buckets = new int[size];
var entries = new Entry[size];
var i = 0;
if (dictionary != null)
{
foreach (var pair in dictionary)
{
var value = pair.Key;
var hashCode = value.GetHashCode();
ref int bucket = ref buckets[hashCode & (size - 1)];
entries[i] = new Entry(bucket - 1, hashCode, value, pair.Value);
bucket = i + 1;
++i;
}
}
_buckets = buckets;
_entries = entries;
}
internal bool TryGetValue(TKey key, out TValue value)
{
var entries = _entries;
var hashCode = key.GetHashCode();
for (var i = _buckets[hashCode & (_buckets.Length - 1)] - 1; i >= 0; i = entries[i].Next)
{
if (hashCode == entries[i].HashCode && EqualityComparer<TKey>.Default.Equals(key, entries[i].Key))
{
value = entries[i].Value;
return true;
}
}
value = default!;
return false;
}
}
internal struct DefaultEqualityComparer<T> : IEqualityComparer<T> where T : struct, IEquatable<T>
{
public bool Equals(T x, T y) => x.Equals(y);
public int GetHashCode(T obj) => obj.GetHashCode();
}
internal struct TypeEqualityComparer : IEqualityComparer<Type>
{
public bool Equals(Type x, Type y) => x.Equals(y);
public int GetHashCode(Type obj) => obj.GetHashCode();
}
internal static partial class HashHelpers
{
public static int PowerOf2(int v)
{
if ((v & (v - 1)) == 0 && v >= 1)
{
return v;
}
var i = 4;
while (i < v)
{
i <<= 1;
}
return i;
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,044
|
Q: ASP.NET - hiding BoundField input I have a simple connection to a database achieved through a DetailsView control in ASP.NET. One of the fields is a password so I'd like its content to be hidden while typing. My first thought was TextBox's TextMode="Password" but that failed miserably since BoundField class doesn't implement it. I'm guessing it must be done programmatically through an event-handler but I simply don't know how.
Here's the markup I've written so far, if needed:
<body>
<form id="form1" runat="server">
<div class="register" runat="server">
<asp:DetailsView ID="InsertData" runat="server" HorizontalAlign="Center"
Height="100px" Width="170px"
AutoGenerateRows="False" DataSourceID="usersConnectionString"
DefaultMode="Insert" OnItemCommand="Button_click" OnItemInserted="Insert_click">
<Fields>
<asp:BoundField DataField="userName" HeaderText="Name"
SortExpression="userName" />
<asp:BoundField DataField="userPassword" HeaderText="Password"
SortExpression="userPassword" />
<asp:BoundField DataField="userEmail" HeaderText="Email"
SortExpression="userEmail" />
<asp:CommandField ShowInsertButton="True" />
</Fields>
</asp:DetailsView>
<asp:SqlDataSource ID="usersConnectionString" runat="server"
ConnectionString="<%$ ConnectionStrings:usersConnectionString %>"
InsertCommand="INSERT INTO user(userName, userPassword, userEmail) VALUES (@userName, @userPassword, @userEmail)">
<InsertParameters>
<asp:Parameter Name="userName" Type="String" />
<asp:Parameter Name="userPassword" Type="String" />
<asp:Parameter Name="userEmail" Type="String" />
</InsertParameters>
</asp:SqlDataSource>
<asp:Label ID="Message_label" ForeColor="red" Visible="false" runat="server" Text="[messageLabel]"></asp:Label>
</div>
</form>
</body>
A: The best thing to do in this is to make an ItemTemplate and within that, have a textbox control and set its input mode/type to be Password.
The ItemTemplate allows you to add normal/custom user controls to your liking outside of the standard BoundField columns
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,245
|
Джованни да Милано (ит. Giovanni da Milano; род. ок. 1320-25 гг – ум. после 1369г) – живописец итальянского проторенессанса.
Биография
О жизни и творчестве Джованни сохранилось крайне мало документальной информации. Прозвище «Джованни из Милана» свидетельствует о ломбардском происхождении живописца, но в ломбардских архивных документах его имя не обнаружено. Источником информации о нём являются исключительно флорентийские архивы.
Несмотря на то, что свои произведения он подписывал «Джованни да Милано», в официальных документах сообщается, что происходил он из Каверсаччо, небольшого городка, который находится в провинции Комо. Дату рождения художника относят приблизительно к 1320-25 годам, основываясь на том, что в списке флорентийской Гильдии врачей и аптекарей (Арте деи Медичи э дельи Специали) его имя впервые появляется в 1346 году. По всей вероятности, навыки художественного ремесла Джованни приобрёл в Ломбардии до того, как приехал во Флоренцию, сформировавшись в кругу художников, перенимавших передовые художественные новации после того, как Милан посетил выдающийся мастер Джотто ди Бондоне.
После того, как 17 октября 1346 года «Йоханнес де Медиолано» был зарегистрирован во Флоренции среди прочих приезжих художников, он какое-то время работал там, однако уже в 1348 году покинул город, вероятно, вследствие разразившейся эпидемии чумы, выкосившей в середине XIV века примерно треть населения Европы. Исследователи предполагают, что именно в период этого отъезда он создал фреску «Мадонна с младенцем и святыми» в люнете Оратория Мадонна делле Грацие в Мендризио (Швейцария) – первое известное произведение художника.
В начале 1350-х годов Джованни возвращается в Тоскану. В 1355-60 годах он расписал полиптих для Госпиталя в Прато (Спедале делла Мизерикордия), в котором продемонстрировал свои способности в изображении человеческой жестикуляции, элементов природы и разных мелких деталей одежды.
В 1362 году его имя вновь появляется в списке флорентийской Гильдии врачей и аптекарей. В 1363 году в местной налоговой ведомости (Portata dell'Estimo) он числится как житель флорентийского округа Сан Пьер Маджоре. Вероятно, художник решил надолго остаться во Флоренции, поэтому в 1366 году он для себя и своих детей приобретает флорентийское гражданство. К этому периоду относится ещё один сохранившийся документ от 1365 года; в нём руководство Орсанмикеле соглашается продлить художнику время для завершения росписи капеллы Ринуччини в церкви Санта Кроче – единственный фресковый цикл, правда, незавершённый, сохранившийся из всего монументального творчества мастера. В это же время им была создана картина «Пьета» (ок. 1365г., ныне в Галерее Академии, Флоренция). Вазари сообщает, что во время проживания во Флоренции Джованни да Милано сотрудничал с Таддео Гадди, однако его версия не имеет документального подтверждения. Этот период считается расцветом творчества Джованни.
Последнее упоминание имени художника относится к 1369 году, когда Джованни да Милано оказался в списке среди лучших современных ему мастеров, отправившихся расписывать залы Ватиканского дворца в Риме – Джоттино, Джованни и Аньоло Гадди и Бартоломео Булгарини (росписи не сохранились). Предположительно фрески были выполнены в связи с тем, что папа Урбан V решил перенести папский престол из Авиньона в Рим, поэтому папский дворец надлежало привести в соответствующее столь высокому статусу состояние. Учёные полагают, что Джованни да Милано был руководителем приглашённых художников, поскольку он был самым старшим, самым опытным и самым влиятельным мастером в этой группе.
Имя художника "Джованни из Комо" ("de Comes"), упоминается в документе от 1375 года, сообщающем о ремонтных работах в Монтекассино, но имеет ли оно отношение к Джованни да Милано - неизвестно. После этого имя художника было забыто на несколько веков и возвращено из небытия только благодаря исследованиям в XIX веке.
Творчество
Историки искусства отмечают, что, соединив ломбардскую художественную традицию с достижениями эпохи Джотто, Джованни да Милано способствовал выработке нового художественного языка, ставшего неотъемлемой частью готической живописи периода её расцвета. Ломбардская культура в целом, и живопись в частности, стали особенно интенсивно развиваться после 1277 года, когда к власти в Ломбардии пришёл клан Висконти. Появились фресковые росписи и иллюстрированные миниатюрами книги, в которых местные художники вырабатывали свой оригинальный язык, имевший местные отличительные черты. После того, как Джотто побывал в Падуе, создав фрески в знаменитой капелле Скровеньи, местные художники стали перенимать его способы передачи объёма и пространства. Один из его последователей – Джусто де Менабуои, работал в Милане и Падуе, создав значительные фресковые циклы. Джованни да Милано впитал все эти художественные наработки и развил их далее в русле нарождающейся позднеготической идиомы. С помощью особого мазка кисти он придавал фигурам объём, но в то же время удлинял их, и обряжал в элегантные, шитые золотом наряды. Он стал ключевой фигурой в художественной жизни второй половины XIV века, каким-то образом предвосхитив позднеготическую культуру, и став центром чрезвычайно интересного художественного процесса, в котором соединились ломбардская и тосканская традиции.
Произведения
Ранние работы
К ранним работам художника исследователи относят те произведения, в которых видят сильное ломбардское влияние и первые самостоятельные шаги. Кроме фрески «Мадонна с младенцем и святыми» в люнете Оратория Мадонна делле Грацие в Мендризио (Швейцария), выполненной в 1348-50гг, это в основном небольшого формата произведения, атрибуция которых носит не бесспорный характер:
1. «Христос с Богородицей и святые» (Лондон, Национальная галерея). Работу датируют по-разному, от 1345-50 до 1360-64 годов.
2. Несколько изображений «Распятия Христа» (Оксфорд, Художественная галерея церкви Христа; Амстердам, Гос. музей; рисунок «Распятия» из Гос. музеев Берлина, и «Распятие» из Частного собрания) – все они датируются 1350-ми – 60ми годами.
3. «Архангел Михаил» (1355-60гг; Частное собрание)
4. «Мадонна с младенцем, святыми и евангельскими сюжетами» (ок. 1355г., Рим, Галерея Корсини), икона, размером 55х87 см, на которой художник изобразил, кроме Мадонны, «Благовещение», «Рождество», «Распятие», «Оплакивание Христа» и также 6 святых: Св. Николая, Стефана (или Лаврентия), Якова, Евстафия, Екатерину Александрийскую и Маргариту.
Полиптих из Прато
Благодаря сохранившейся надписи известно, что заказчиком полиптиха был «Брат Франческо», которого ныне уверенно ассоциируют с Франческо де'Тьери, священником, работавшим администратором Спедале делла Мизерикордия в Прато с 1332 по 1372 год. В надписи сохранилось и имя автора «Johannes de Mediolano». Принадлежность алтарной картины Госпиталю Мизерикордия подтверждается присутствием на ней св. Варнавы, которому был посвящён госпитальный храм.
Полиптих состоит из пятнадцати деревянных панелей: «Мадонна с младенцем на троне» - в центре, по сторонам от неё святые Екатерина Александрийская, Бернар Клервосский, Варфоломей и Варнава. В среднем ряду: «Мученичество св. Екатерины», «Явление Богородицы св. Бернару», «Благовещение», «Мученичество св. Варфоломея», «Мученичество св. Варнавы». На картинах пределлы: «Рождество», «Поклонение волхвов», «Введение во храм» и «Моление о чаше», «Поцелуй Иуды», «Путь на Голгофу».
Согласно документам, ранее, приблизительно с 1367 года, полиптих стоял в алтаре мужского лазарета госпиталя – Pellegrino nuovo, но впоследствии был удалён оттуда, вероятно в ходе ремонтных работ 1692-1702 годов. До середины XIX в., когда он был перенесён в помещение, которым пользовался comissario (администратор) госпиталей Прато, научным кругам о нём не было ничего известно; тогда же пропали боковые пилястры полиптиха.
Недавно найденные документы проливают свет на некоторые обстоятельства создания произведения. В 1355 году Франческо деТьери получил разрешение епископа Пистойи использовать завещанное наследство братьев Никколо и Эджидио ди Сальвато Мати для завершения оформления госпитальной капеллы. Во время эпидемии чумы 1348 года эти братья занялись строительством капеллы, посвященной Св. Бернарду и Богородице. Живопись полиптиха была связана со всеми этими событиями, и поэтому же св. Бернард изображён на нём непосредственно справа от Богородицы.
Иконографические особенности произведения говорят в пользу того, что оно было создано ок. 1355 года или чуть позже. На нем изображён несколько архаичный трон Богородицы и видно сильное влияние ломбардской культуры, выраженное в живом рассказе сюжетов и внимании к изображению элементов природы.
Полиптих из Пизы
О создании этого произведения не сохранилось никаких документов, поэтому его атрибуция Джованни да Милано основана на стилистическом анализе и косвенных данных. От полиптиха до нас дошли всего пять фрагментов: центральная панель «Мадонна с младенцем на троне» (ныне находится в Скандиччи (Флоренция) в храме Сан Бартоломео ин Туто; боковые панели: «Св. Антоний –аббат» (Уильямстаун, Колледж искусства) и «Св. Франциск» (Лувр, Париж); а также два навершия с изображением Благовещения (Ангел Благовеста и Благовестуемая Мария – оба в Музее Сан Маттео, Пиза).
Полиптих был написан для ордена Умилиатов. Связи Джованни да Милано с этим орденом подтверждаются тем точно установленным фактом, что он написал для Умилиатов большой полиптих, основная часть которого хранится ныне во Флоренции (Полиптих Оньиссанти). Этот орден имел множество отделений в Ломбардии, откуда происходил художник. Все эти соображения привели исследователей к мысли, что полиптих мог быть исполнен для пизанской церкви Сан Торпе – местного храма ордена Умилиатов. С другой стороны, присутствие св. Франциска может свидетельствовать в пользу того, что полиптих мог быть исполнен для какого-то францисканского храма Пизы.
Перемещение полиптиха из алтаря и его расформирование могло произойти в XVIII или XIX веках, когда монашеские ордена подвергались обструкции. Его датировка тоже остается предметом дискуссии. Предполагают, что он мог быть создан около 1360 года. Между Пизой и Флоренцией в 1353-65 годах шла война, и заказ ломбардскому художнику расписать полиптих мог быть дружеским жестом, отражающим связи между домом Висконти и Пизой, возникшие в середине XIV века.
У исследователей не вызывает сомнений рука Джованни да Милано в создании всех сохранившихся частей полиптиха, за исключением фигуры Ангела Благовеста, которую, возможно, написал какой-то местный пизанский художник. Отмечается также, что в Пизе в тот период работало много сиенских художников, и сиенский готический стиль отражён в произведениях художника, исполненных в этом городе.
Полиптих Оньиссанти
Это крупный полиптих, расписанный для флорентийской церкви Оньиссанти (ц. Всех Святых), которая принадлежала ордену Умилиатов. Историк искусства Джорджо Вазари приписал его кисти Джованни да Милано, основываясь на подписи, которая когда-то была на раме полиптиха, но рама не сохранилась. Полиптих сначала украшал главный алтарь храма, затем был перемещён в боковую капеллу в том же храме, а затем, приблизительно в конце XVII – нач. XVIII в. размонтирован. В итоге от него сохранились пять из шести боковых панелей и пять малых картин нижнего яруса (все в Галерее Уффици, Флоренция), обрезанная центральная панель со сценой «Коронования Марии» (Национальный Музей Изящных искусств, Буэнос Айрес), а также один из семи пинаклей с изображением «Троицы и святых Павла и Иоанна Богослова» (Частное собрание). На боковых панелях художник изобразил Св. Екатерину Александрийскую и Св. Лючию, Св. Стефана и Св. Лаврентия, Иоанна Крестителя и Св. Луку, Св. Петра и Св. Бенедикта, Св. Якова-старшего и Св. Григория. На пяти малых картинах нижнего яруса изображены хоры: хор девственниц, хор мучеников, хор апостолов, хор патриархов и хор пророков.
Учёные до сих пор спорят о времени его создания; наибольшую поддержку имеет точка зрения, что полиптих был расписан в 1360-х годах – в период расцвета творчества художника. Любопытной особенностью полиптиха являются изображения картин сотворения мира в верхней части панелей среднего яруса. Реставрационные работы, проведенные в 2006-08 годах, открыли зрителям превосходные яркие краски, которыми оперировал мастер. Структура и конструкция полиптиха очень близки полиптиху из Прато, который художник расписал в 1355 году.
Неизвестный полиптих («Полиптих Милан-Турин-Лондон-Париж»)
Полиптих был реконструирован из разных приписанных кисти Джованни да Милано отдельных частей, однако, не полностью, т.к. в предложенной реконструкции не хватает правой части и одной картины пределлы. Его название "Милан-Турин-Лондон-Париж" связано с тем, что происхождение этого полиптиха неизвестно, а его части находятся в разных собраниях живописи Милана, Турина, Лондона и Парижа. О его происхождении учёные высказали несколько гипотез. По одной версии он мог принадлежать Камальдульскому монастырю Санта Мария дельи Анджели во Флоренции, откуда в Лондонскую Национальную галерею попали три навершия полиптиха с изображением Богоматери, Апокалиптического Христа и Иоанна Крестителя. По другой версии, он происходит из ц. Санта Кроче во Флоренции, поскольку Джорджо Вазари упоминает находившийся там полиптих кисти Джованни да Милано, располагавшийся в алтаре св. Герардо из Вилламаньи. Третья версия гласит, что полиптих мог быть создан для флорентийской церкви Сан Сальваторе делль'Арчивесковадо (Архиепископская церковь Св. Сальваторе), поскольку оттуда происходит его центральная панель, изображающая Христа-судью. Складное епископское кресло, на котором восседает Христос, по сути является символом епископства, в связи с чем исследователи полагают, что полиптих мог быть заказан именно для этой церкви.
Кроме перечисленных частей полиптиху предположительно принадлежат две картины пределлы из частного собрания: «Неверие Фомы» и «Воскресение». Вероятно, изначально это был триптих, однако боковая панель с изображениями святых из Галереи Сабауда, Турин, делает возможным допущение, что это был пентаптих. Хронологически его располагают в период расцвета творчества мастера – в 1360-е годы.
Капелла Ринуччини
Капелла, расположенная во флорентийской церкви Санта Кроче, первоначально принадлежала богатому семейству Гвидалотти, а не Ринуччини. В 1351 году Лапо ди Лизио Гвидалотти завещал 700 флоринов на строительство и роспись капеллы. Распорядителем этих больших средств стало Общество Святой Девы Марии дель Орто, санкционировавшее начало работ в 1365 году.
Джорджо Вазари ошибочно считал автором росписи капеллы Таддео Гадди. Авторство Джованни да Милано в 1864 году установил известный историк искусства Кавальказелле. В дальнейшем его мнение было подтверждено обнаруженным документом от 26 мая 1365 года, сообщающим, что художнику Джованни да Милано, с которым заключил контракт Фра Лодовико ди Джованни, хранитель библиотеки Санта Кроче, выделяется дополнительное время для завершения работ над фресками. Несколько дней спустя были перечислены 50 флоринов частичной предоплаты за предстоящие работы. Исследователи предполагают, что в ходе росписей возник какой-то конфликт, и автор прервал работу над фресками до их завершения. Фрески были завершены в 1370-71 годах, когда попечительство над капеллой перешло к семейству Ринуччини. В итоге все картины нижнего яруса на обеих стенах и фигуры нижних святых, расположенные над входом, были написаны неизвестным художником, которого называли «Мастер Ринуччини» до тех пор, пока Лючано Беллози в 1973 году не определил, что это был Маттео ди Пачино.
Капелла Ринуччини посвящена Рождеству Марии и св. Марии Магдалине. Программа росписей представляет собой эпизоды из их жития. На северной стене изображены: «Изгнание Иоакима из храма», «Сон Иоакима», «Встреча Иоакима и Анны у Золотых ворот», «Рождество Марии», «Введение Марии во храм», «Обручение Марии». На южной стене: «Пир в доме Симона Фарисея», «Христос у Марии и Марфы», «Воскрешение Лазаря», «Noli me tangere», «Чудесное нахождение ребёнка», «Воскрешение жены правителя по молитвам Марии Магдалины». Свод капеллы украшен изображениями четырёх пророков и медальоном с благословляющим Христом, который написан на дереве и прикреплён к потолку. Над входом располагаются полуфигуры двенадцати апостолов со скрижалями, на которых изображён символ веры. На пилястрах слева фигуры св. Франциска и св. Антония, справа – св. Людовик Тулузский и св. Андрей из Ананьи.
Исследователи отмечают, что тщательно продуманное изображение храмовой архитектуры (особенно на фреске «Изгнание Иоакима из храма») свидетельствует о влиянии живописи Таддео Гадди. Библейским историям, изложенным во фресках, присущ мягкий, спокойный тон повествования. Особое внимание художник уделяет жанровым сценкам и разным бытовым деталям («Рождество Марии», «Пир в доме Симона Фарисея»). Отмечается также, что в лучших изображениях женщин («Рождество Марии» и «Изгнание Иоакима из храма»), художник сумел передать как их юную красоту, так и элегантность одежд (декольтированные платья с открытыми плечами). Вообще, экспертами особо отмечается эта особенность творчества Джованни да Милано – изображение модных одежд, в которые он обряжал святых дев не только на фресках, но и в полиптихах. Сцену «Воскрешение Лазаря» Джованни да Милано решил в ключе, существенно отличающемся от традиционной версии. Некоторые исследователи полагают, что столь необычное решение могло послужить причиной серьёзных разногласий с заказчиком, вынудивших художника прервать работу над фресками.
Источники
При написании статьи были использованы следующие книги:
Джорджо Вазари. Жизнеописания наиболее знаменитых живописцев, ваятелей и зодчих. Москва. 1996. т. I. Стр. 450-451, 457
Прокопп М., Итальянская живопись XIV века. Будапешт, 1988 стр. 42-43
Энциклопедический словарь живописи (под ред. М. Лаклотта). М. 1997, стр. 283-284
Пешке И. Монументальная живопись эпохи Джотто в Италии 1280-1400. М. 2003, стр. 350-361
Giovanni da Milano. Giunti, Firenze, 2008.
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M. Boskovits, Govanni da Milano., Firenze 1966;
M. Gregori, Govanni da Milano alla cappella Rinuccini, Milano 1966;
M. Boskovits, Notes sur Govanni da Milano, in Revue de l'art, XI (1971), pp. 55–58;
M. Gregori, Govanni da Milano, storia di un polittico, in Paragone, XXIII (1972), 265, pp. 3–35;
L. Bellosi, Buffalmacco e il Trionfo della Morte, Torino 1974, passim;
R. Fremantle, Florentine Gothic painters. From Giotto to Masaccio, London 1975, pp. 181–191;
Govanni da Milano a cura di L. Cavadini, Valmorea 1980;
M. Gregori, Alcune osservazioni non marginali su G. da M. e sulla situazione della pittura del Trecento in Lombardia, ibid., pp. 7–18;
E. Skaug, The Rinuccini Tondo. An 18th century copy or a 14th century original?, in Atti del Convegno sul restauro delle opere d'arte, Firenze 1976, I, Firenze 1981, pp. 333–339;
R.C. Proto Pisani, Un inedito di Govanni da Milano la tavola di S. Bartolo in Tuto, in Bollettino d'arte, s. 6, LXVIII (1983), 19, pp. 49–58;
E. Skaug, Punch-marks. What are they worth? Problems of Tuscan workshop interrelationships in the mid-fourteenth century: the Ovile Master and G. da M., in La pittura nel XIV e XV secolo. *Il contributo dell'analisi tecnica alla storia dell'arte. Atti del XXIV Congresso internazionale di storia dell'arte… Bologna 1979, Bologna 1983, pp. 253–282;
C. Volpe, Il lungo percorso del "dipingere dolcissimo e tanto unito", in Storia dell'arte italiana, V, Torino 1983, pp. 289, 298-301;
G. Ragionieri, Pittura del Trecento a Firenze, in La pittura in Italia. Il Duecento e il Trecento, I, Milano 1986, pp. 301–303;
M. Boskovits, Pittura e miniatura a Milano: Duecento e primo Trecento, in Il millennio ambrosiano, a cura di C. Bertelli, III, Milano 1989, pp. 26–29;
P.P. Donati, Un inedito affresco di G. da M., in Prospettiva, 1988-89, nn. 53-56, pp. 173–176;
L. Bellosi, Govanni da Milano inDiz. della pittura e dei pittori, II, Torino 1990, pp. 607 s.;
D. Pescarmona, Como e Canton Ticino, in La pittura in Lombardia. Il Trecento, Milano 1992, pp. 108–114;
C. Quattrini, Govanni da Milano, ibid., pp. 416 s.;
C. Travi, Il Trecento, in La pittura a Como e nel Canton Ticino dal Mille al Settecento, a cura di M. Gregori, Cinisello Balsamo 1994, pp. 10–18, 254-367;
M. Gregori, Govanni da Milano in Enc. dell'arte medievale, VI, Roma 1995, pp. 728–738;
D. Gordon, Govanni da Milano in The Dictionary of art, XII, New York 1996, pp. 702 s.;
Pittura a Milano dall'Alto Medioevo al tardo gotico, a cura di M. Gregori, Milano 1997, ad indicem.
Marco Ciatti e Maria Pia Mannini (a cura di),Giovanni da Milano: il polittico di Prato, Arte & restauri, 2, Prato 2001.
Daniela Parenti e Angelo Tartuferi (a cura di), Giovanni da Milano. Capolavori del gotico fra Lombardia e Toscana, catalogo della mostra Firenze 2008, Giunti, Firenze 2008.
Laura Facchin, Opere di artisti svizzeri alla Galleria Sabauda. Una prima indagine, in Giorgio Mollisi (a cura di), Svizzeri a Torino nella storia, nell'arte, nella cultura, nell'economia dal Cinquecento ad oggi, «Arte&Storia», anno 11, numero 52, ottobre 2011, Edizioni Ticino Management, Lugano 2011
Художники по алфавиту
Художники Италии
Персоналии по алфавиту
Художники Италии XIV века
Художники готики
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,758
|
Q: Sync pyserial with arduino serial output Is there a simple method to sync the data stream of pyserial serial connection with arduino? The simple piece of code I run gives weird values from time to time which I can only assume is due to a bit shift or garbage data.
For example, some runs i get a data array with the correct output:
[4.697265625, 4.7021484375, 4.6923828125, 4.6826171875, 4.677734375, 4.66796875, 4.658203125, ....]
Then some runs I would get data which looks like this:
[1.6408595718088704e-28, 2.524370304146793e-28, 1.640838386579482e-28, 2.524370304146793e-28, 1.640838386579482e-28 ... ]
I assume the garbage data is due to out of sync bytes being sent (ie the python program is receiving the first byte of the next data packet and bytes 2,3,4 of the first packet).
Has anyone encountered this before and have a solution?
Here is the code I use to send the value across the serial port:
Arduino
void loop(){
double val = 5.0*(analogRead(1))/1024.0;
//Serial.println(val);
sendToPc(&val);
}
void sendToPc(double* data){
byte* byteData = (byte*)(data);
Serial.write(byteData, 4);
}
The the simple python code to access it.
Python code
self.handle.reset_input_buffer()
self.handle.reset_output_buffer()
while(True):
self.handle.readinto(self.rawData)
rawData is a byte array of size 4 bytes.
Thanks in advance.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 92
|
{"url":"https:\/\/includestdio.com\/7223.html","text":"# keyword argument \u2013 Proper way to use **kwargs in Python\n\n## The Question :\n\n481 people think this question is useful\n\nWhat is the proper way to use **kwargs in Python when it comes to default values?\n\nkwargs returns a dictionary, but what is the best way to set default values, or is there one? Should I just access it as a dictionary? Use get function?\n\nclass ExampleClass:\ndef __init__(self, **kwargs):\nself.val = kwargs['val']\nself.val2 = kwargs.get('val2')\n\n\n\nA simple question, but one that I can\u2019t find good resources on. People do it different ways in code that I\u2019ve seen and it\u2019s hard to know what to use.\n\n498 people think this answer is useful\n\nYou can pass a default value to get() for keys that are not in the dictionary:\n\nself.val2 = kwargs.get('val2',\"default value\")\n\n\n\nHowever, if you plan on using a particular argument with a particular default value, why not use named arguments in the first place?\n\ndef __init__(self, val2=\"default value\", **kwargs):\n\n\n\n277 people think this answer is useful\n\nWhile most answers are saying that, e.g.,\n\ndef f(**kwargs):\nfoo = kwargs.pop('foo')\nbar = kwargs.pop('bar')\n...etc...\n\n\n\nis \u201cthe same as\u201d\n\ndef f(foo=None, bar=None, **kwargs):\n...etc...\n\n\n\nthis is not true. In the latter case, f can be called as f(23, 42), while the former case accepts named arguments only \u2014 no positional calls. Often you want to allow the caller maximum flexibility and therefore the second form, as most answers assert, is preferable: but that is not always the case. When you accept many optional parameters of which typically only a few are passed, it may be an excellent idea (avoiding accidents and unreadable code at your call sites!) to force the use of named arguments \u2014 threading.Thread is an example. The first form is how you implement that in Python 2.\n\nThe idiom is so important that in Python 3 it now has special supporting syntax: every argument after a single * in the def signature is keyword-only, that is, cannot be passed as a positional argument, but only as a named one. So in Python 3 you could code the above as:\n\ndef f(*, foo=None, bar=None, **kwargs):\n...etc...\n\n\n\nIndeed, in Python 3 you can even have keyword-only arguments that aren\u2019t optional (ones without a default value).\n\nHowever, Python 2 still has long years of productive life ahead, so it\u2019s better to not forget the techniques and idioms that let you implement in Python 2 important design ideas that are directly supported in the language in Python 3!\n\n86 people think this answer is useful\n\nI suggest something like this\n\ndef testFunc( **kwargs ):\noptions = {\n'option1' : 'default_value1',\n'option2' : 'default_value2',\n'option3' : 'default_value3', }\n\noptions.update(kwargs)\nprint options\n\ntestFunc( option1='new_value1', option3='new_value3' )\n# {'option2': 'default_value2', 'option3': 'new_value3', 'option1': 'new_value1'}\n\ntestFunc( option2='new_value2' )\n# {'option1': 'default_value1', 'option3': 'default_value3', 'option2': 'new_value2'}\n\n\n\nAnd then use the values any way you want\n\ndictionaryA.update(dictionaryB) adds the contents of dictionaryB to dictionaryA overwriting any duplicate keys.\n\n56 people think this answer is useful\n\nYou\u2019d do\n\nself.attribute = kwargs.pop('name', default_value)\n\n\n\nor\n\nself.attribute = kwargs.get('name', default_value)\n\n\n\nIf you use pop, then you can check if there are any spurious values sent, and take the appropriate action (if any).\n\n43 people think this answer is useful\n\nUsing **kwargs and default values is easy. Sometimes, however, you shouldn\u2019t be using **kwargs in the first place.\n\nIn this case, we\u2019re not really making best use of **kwargs.\n\nclass ExampleClass( object ):\ndef __init__(self, **kwargs):\nself.val = kwargs.get('val',\"default1\")\nself.val2 = kwargs.get('val2',\"default2\")\n\n\n\nThe above is a \u201cwhy bother?\u201d declaration. It is the same as\n\nclass ExampleClass( object ):\ndef __init__(self, val=\"default1\", val2=\"default2\"):\nself.val = val\nself.val2 = val2\n\n\n\nWhen you\u2019re using **kwargs, you mean that a keyword is not just optional, but conditional. There are more complex rules than simple default values.\n\nWhen you\u2019re using **kwargs, you usually mean something more like the following, where simple defaults don\u2019t apply.\n\nclass ExampleClass( object ):\ndef __init__(self, **kwargs):\nself.val = \"default1\"\nself.val2 = \"default2\"\nif \"val\" in kwargs:\nself.val = kwargs[\"val\"]\nself.val2 = 2*self.val\nelif \"val2\" in kwargs:\nself.val2 = kwargs[\"val2\"]\nself.val = self.val2 \/ 2\nelse:\nraise TypeError( \"must provide val= or val2= parameter values\" )\n\n\n\n31 people think this answer is useful\n\nSince **kwargs is used when the number of arguments is unknown, why not doing this?\n\nclass Exampleclass(object):\ndef __init__(self, **kwargs):\nfor k in kwargs.keys():\nif k in [acceptable_keys_list]:\nself.__setattr__(k, kwargs[k])\n\n\n\n14 people think this answer is useful\n\nHere\u2019s another approach:\n\ndef my_func(arg1, arg2, arg3):\n... so something ...\n\nkwargs = {'arg1': 'Value One', 'arg2': 'Value Two', 'arg3': 'Value Three'}\n# Now you can call the function with kwargs like this:\n\nmy_func(**kwargs)\n\n\n\n12 people think this answer is useful\n\nYou could do something like this\n\nclass ExampleClass:\ndef __init__(self, **kwargs):\narguments = {'val':1, 'val2':2}\narguments.update(kwargs)\nself.val = arguments['val']\nself.val2 = arguments['val2']\n\n\n\n12 people think this answer is useful\n\nFollowing up on @srhegde suggestion of using setattr:\n\nclass ExampleClass(object):\n__acceptable_keys_list = ['foo', 'bar']\n\ndef __init__(self, **kwargs):\n[self.__setattr__(key, kwargs.get(key)) for key in self.__acceptable_keys_list]\n\n\n\nThis variant is useful when the class is expected to have all of the items in our acceptable list.\n\n12 people think this answer is useful\n\nI think the proper way to use **kwargs in Python when it comes to default values is to use the dictionary method setdefault, as given below:\n\nclass ExampleClass:\ndef __init__(self, **kwargs):\nkwargs.setdefault('val', value1)\nkwargs.setdefault('val2', value2)\n\n\n\nIn this way, if a user passes \u2018val\u2019 or \u2018val2\u2019 in the keyword args, they will be used; otherwise, the default values that have been set will be used.\n\n5 people think this answer is useful\n\nIf you want to combine this with *args you have to keep *args and **kwargs at the end of the definition.\n\nSo:\n\ndef method(foo, bar=None, *args, **kwargs):\ndo_something_with(foo, bar)\nsome_other_function(*args, **kwargs)\n\n\n\n1 people think this answer is useful\n\n@AbhinavGupta and @Steef suggested using update(), which I found very helpful for processing large argument lists:\n\nargs.update(kwargs)\n\n\n\nWhat if we want to check that the user hasn\u2019t passed any spurious\/unsupported arguments? @VinaySajip pointed out that pop() can be used to iteratively process the list of arguments. Then, any leftover arguments are spurious. Nice.\n\nHere\u2019s another possible way to do this, which keeps the simple syntax of using update():\n\n# kwargs = dictionary of user-supplied arguments\n# args = dictionary containing default arguments\n\n# Check that user hasn't given spurious arguments\nunknown_args = user_args.keys() - default_args.keys()\nif unknown_args:\nraise TypeError('Unknown arguments: {}'.format(unknown_args))\n\n# Update args to contain user-supplied arguments\nargs.update(kwargs)\n\n\n\nunknown_args is a set containing the names of arguments that don\u2019t occur in the defaults.\n\n0 people think this answer is useful\n\nAnother simple solution for processing unknown or multiple arguments can be:\n\nclass ExampleClass(object):\n\ndef __init__(self, x, y, **kwargs):\nself.x = x\nself.y = y\nself.attributes = kwargs\n\ndef SomeFunction(self):\nif 'something' in self.attributes:\ndosomething()\n\n\n\n0 people think this answer is useful\n\n**kwargs gives the freedom to add any number of keyword arguments. One may have a list of keys for which he can set default values. But setting default values for an indefinite number of keys seems unnecessary. Finally, it may be important to have the keys as instance attributes. So, I would do this as follows:\n\nclass Person(object):\nlisted_keys = ['name', 'age']\n\ndef __init__(self, **kwargs):\n_dict = {}\n# Set default values for listed keys\nfor item in self.listed_keys:\n_dict[item] = 'default'\n# Update the dictionary with all kwargs\n_dict.update(kwargs)\n\n# Have the keys of kwargs as instance attributes\nself.__dict__.update(_dict)","date":"2021-01-23 13:53:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.24854107201099396, \"perplexity\": 5894.285479454966}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703538082.57\/warc\/CC-MAIN-20210123125715-20210123155715-00722.warc.gz\"}"}
| null | null |
layout: video-embed
title: "Simplicity Matters by Rich Hickey"
date: 2014-09-03
categories: design functional rich-hickey
video-url : https://www.youtube.com/embed/rI8tNMsozo0
---
Great talk on how simplicity should be a design goal rather an afterthought.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,480
|
Playmobil 5103 prehistoric bear with cavemen figures 5103 made by Geobra, Germany 2010-2011. Includes two figures approx 2,95 inch 7,5 cm tall, bear, cave and accessories. This item comes mint in sealed box, never played with.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,075
|
Multi-purpose centre including markets, gym and sporting activities. Offers a wide range of activities for the community including many junior sport programs for children aged 3-12 years.
Hall for hire (grey building under the bridge).
1 main room upstairs and 2 meeting rooms downstairs.
Currently utilised by 11 clubs.
Capacity: Non denominational chapel - up to 100 people. Suitable for weddings and meetings.
Part of Cobbin Farm Homestead.
The Geelong Conference Centre provides a premium, tranquil and distraction-free environment.
This facility is managed by a community group.
The Lara RSL building was built in 1865 and is one of the oldest buildings still in use in Lara. It has been the headquarters of the Lara RSL since 1950 and is open for visitors on request.
The timeless Australian story of wool combined with exciting contemporary exhibitions, presented in an historic bluestone woolstore just one block from Geelong's vibrant new waterfront.
Vibrant arts centre situated on the Bellarine Peninsula in Drysdale.
Available for hire for community and professional events.
A medium-sized hall with kitchen is available for community groups, meetings and other functions.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,447
|
Between 1992 and 1996, 587 wild red-legged partridges (Alectoris rufa) from 16 Spanish provinces were examined to study the variations of helminth communities in this game species across a broad geographical area. The survey revealed 13 species of helminth parasites. Dicrocoelium sp., Rhabdometra nigropunctata, and Cheilospirura gruweli were the most common species, whereas Raillietina bolivari, Choanotaenia infundibulum, Tetrameres sp., and Capillaria anatis were the most rare. Subulura suctoria, Heterakis gallinarum, Heterakis tenuicaudata, Capillaria contorta, Trichostrongylus tenuis, and Raillietina tetragona occurred with intermediate frequencies. The abundance of C. gruweli, S. suctoria, H. tenuicaudata, T. tenuis, and R. tetragona was inversely correlated to latitude and directly correlated to yearly mean temperature, whereas the abundance of Dicrocoelium sp. was directly correlated to latitude and inversely correlated to yearly mean temperature. The abundance of R. tetragona was inversely correlated to latitude and yearly mean humidity. The number of helminths per partridge and the number of helminth species per partridge were lower in young birds than in adults. Partridge body condition was inversely correlated to abundance of C. contorta. Richer infracommunities were linked to richer component communities. At the infracommunity level, total number of helminths per partridge and number of helminth species per partridge were inversely correlated to latitude and directly correlated to yearly mean temperature. At the component community level, both species richness and diversity (Simpson's index) were inversely correlated to latitude and directly correlated to mean temperature. Across the broad geographical range of the study area, the helminth parasite communities of red-legged partridges had marked geographical variation in their structure. Our results suggest that this variation is determined by the distribution of both intermediate and definitive hosts. We discuss the implications of this variation for the hypothesis that supplementary releases of captive-bred partridges for sport hunting can affect the helminth fauna of wild red-legged partridges.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,145
|
package org.jboss.narayana.compensations.internal;
import org.jboss.narayana.compensations.api.CompensationHandler;
import org.jboss.narayana.compensations.api.ConfirmationHandler;
import org.jboss.narayana.compensations.api.TransactionLoggedHandler;
/**
* @author paul.robinson@redhat.com 19/04/2014
*/
public interface BAController {
void beginBusinessActivity() throws Exception;
void closeBusinessActivity() throws Exception;
void cancelBusinessActivity() throws Exception;
void completeBusinessActivity(boolean isException) throws Exception;
boolean isBARunning();
Object suspend() throws Exception;
void resume(Object context) throws Exception;
Object getCurrentTransaction() throws Exception;
ParticipantManager enlist(Class<? extends CompensationHandler> compensationHandlerClass,
Class<? extends ConfirmationHandler> confirmationHandlerClass,
Class<? extends TransactionLoggedHandler> transactionLoggedHandlerClass) throws Exception;
ParticipantManager enlist(CompensationHandler compensationHandler, ConfirmationHandler confirmationHandler,
TransactionLoggedHandler transactionLoggedHandler) throws Exception;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,793
|
Q: How to pass an array of checked/unchecked checkbox values to PHP email generator? I have added a checkbox to a form that the user can dynamically add rows to. You can see the form here.
I use an array to pass the values for each row to a PHP email generator, and all works fine for other inputs, but I can't get the checkbox to work. The checkbox input currently looks like this:
<input type="checkbox" name="mailing[]" value="Yes">
Then in the PHP I have this:
$mailing = trim(stripslashes($_POST['mailing'][$i]));
But it is not working as expected, i.e. I am only seeing 'Yes' for the first checkbox checked, and nothing for subsequent checkboxes that are checked.
One further issue is that I would like the value 'No' to be generated for unchecked checkboxes.
Could someone help with this?
Thanks,
Nick
Form:
<form method="post" action="bookingenginetest.php">
<p>
<input type="checkbox" name="mailing[]" value="Yes">
<label>Full Name:</label> <input type="text" name="name[]">
<label>Email:</label> <input type="text" name="email[]">
<label>Telephone:</label> <input type="text" name="telephone[]">
<span class="remove">Remove</span>
</p>
<p>
<span class="add">Add person</span><br /><br /><input type="submit" name="submit" id="submit" value="Submit" class="submit-button" />
</p>
</form>
Cloning script:
$(document).ready(function() {
$(".add").click(function() {
var x = $("form > p:first-child").clone(true).insertBefore("form > p:last-child");
x.find('input').each(function() { this.value = ''; });
return false;
});
$(".remove").click(function() {
$(this).parent().remove();
});
});
A: $mailing = array();
foreach($_POST as $v){
$mailing[] = trim(stripslashes($v));
}
To handle unchecked boxes it would be better to set each checkbox with a unique value:
<input type="checkbox" name="mailing[1]" value="Yes">
<input type="checkbox" name="mailing[2]" value="Yes">
or
<input type="checkbox" name="mailing[a]" value="Yes">
<input type="checkbox" name="mailing[b]" value="Yes">
Then have a list of the checkboxes:
$boxes = array(1,2,3);
$mailing = array();
$p = array_key_exists('mailing',$_POST) ? $_POST['mailing'] : array();
foreach($boxes as $v){
if(array_key_exists($v,$p)){
$mailing[$v] = trim(stripslashes($p[$v]));
}else{
$mailing[$v] = 'No';
}
}
print_r($mailing);
You could also use this with a number of checkboxes instead:
$boxes = 3;
$mailing = array();
$p = array_key_exists('mailing',$_POST) ? $_POST['mailing'] : array();
for($v = 0; $v < $boxes; $v++){
if(array_key_exists($v,$p)){
$mailing[$v] = trim(stripslashes($p[$v]));
}else{
$mailing[$v] = 'No';
}
}
print_r($mailing);
A: Here's my solution:
With an array of checkboxes in the html like so...
<input type="hidden" name="something[]" value="off" />
<input type="checkbox" name="something[]" />
<input type="hidden" name="something[]" value="off" />
<input type="checkbox" name="something[]" />
<input type="hidden" name="something[]" value="off" />
<input type="checkbox" name="something[]" />
I then fix the posted array with this function...
$_POST[ 'something' ] = $this->fixArrayOfCheckboxes( $_POST[ 'something' ] );
function fixArrayOfCheckboxes( $checks ) {
$newChecks = array();
for( $i = 0; $i < count( $checks ); $i++ ) {
if( $checks[ $i ] == 'off' && $checks[ $i + 1 ] == 'on' ) {
$newChecks[] = 'on';
$i++;
}
else {
$newChecks[] = 'off';
}
}
return $newChecks;
}
This will give me an array with values of either 'on' or 'off' for each (and every) checkbox.
Note that the hidden input MUST be BEFORE the checkbox input in order for the function to work right.
A: Change the value for each checkbox to something unique:
<input type="checkbox" name="mailing[]" value="Yes-1">
<input type="checkbox" name="mailing[]" value="Yes-2">
etc. In order to do this from your jQuery code, add another line that assigns the value to the new checkbox:
x.find('input:checkbox').each(function() { this.value='Yes-'+n; });
You'll have to define n on the initial page load. Assuming you start with only one "person", just add right above your $(".add").click handler:
var n=1;
And then:
*
*in your $(".add").click handler, increment the value of n
*in your $(".remove").click handler, decrement the value of n
A: To get checked and unchecked both values in POST place a hidden field with exactly the same name but with opposite value as compared to the original chkbox value such that in this case the value will be '0'
<input type="hidden" name="chk_name[]" value="0" />
<input type="checkbox" name="chk_name[]" value="1"/>
<input type="hidden" name="chk_name[]" value="0" />
<input type="checkbox" name="chk_name[]" value="1"/>
<?php
function getAllChkboxValues($chk_name) {
$found = array(); //create a new array
foreach($chk_name as $key => $val) {
//echo "KEY::".$key."VALue::".$val."<br>";
if($val == '1') { //replace '1' with the value you want to search
$found[] = $key;
}
}
foreach($found as $kev_f => $val_f) {
unset($chk_name[$val_f-1]); //unset the index of un-necessary values in array
}
final_arr = array(); //create the final array
return $final_arr = array_values($chk_name); //sort the resulting array again
}
$chkox_arr = getAllChkboxValues($_POST['chk_name']); //Chkbox Values
echo"<pre>";
print_r($chkox_arr);
?>
A: Here's my solution:
<span>
<input class="chkBox" onchange="if($(this).is(':checked')){$(this).parent().find('.hidVal').prop('disabled',true);}else{$(this).parent().find('.hidVal').prop('disabled', false);}" type="checkbox" checked name="session[]" value="checked_value_here" />
<input type="hidden" class="hidVal" name="session[]" value="un_checked_value_here" />
</span>
<span>
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"redpajama_set_name": "RedPajamaStackExchange"
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Your suitors won't be able to catch up when you wear this Sonic The Hedgehog Tails Adult Womens Costume. Form fitting dress is made of traditional character colors, spikes, and tail and comes with gloves, ears, and shoe covers revamping the look of this traditional Sega character.
This costume is fantastic! It fits well, I probably could have gotten a small so it would be tighter, but the medium is comfortable and easy to run around in. Mind you, I'm 5' 2" and 130 pounds with my measurements at 38-30-34.
Guest Services Response:Thank you for reviewing your purchase. We're so happy to see that you enjoyed your purchase so much! Stop by our webpage, click the live chat option atop our page and connect with a representative. They can help you figure out which videogame character you should emulate next.
Costume came with missing accessories and a stain on the sleeve. A bit disconcerning and frustrating to not get everything that you paid for.
Guest Services Response:We're terribly sorry to hear that your costume arrived in that condition! Thank you for reviewing your costume with us. We understand our costumers have come to expect quality items from our stores. Please feel free to contact us via the live chat option atop our webpage so we can better assist you with your dilemma.
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{
"redpajama_set_name": "RedPajamaC4"
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Home » Serve » Recognition » Dartmouth Alumni Award
Heiyab F. Tessema '04 Th'05 Th'06
Recipient of the 2017-18 Dartmouth Young Alumni Distinguished Service Award
The first member of your family born in the United States, your childhood included time in Illinois and Maryland before your family landed in New York when you were 12. Inspired to apply to Dartmouth by a sign on your college counselor's door, you were accepted and invited to campus for Dimensions. That weekend visit was the beginning of several fantastic friendships that sustained you through college and beyond.
Once matriculated, you quickly immersed yourself in your studies. Your rigorous academic schedule focused on engineering studies modified with economics. You were grateful for the camaraderie of the Integrated Math and Physical Sciences group and the additional support you received through the Dartmouth Society of Black Engineers. You spent a memorable term as an exchange student at Morehouse College in Atlanta, Georgia. Ultimately, you earned your bachelor of engineering and masters in engineering management.
Outside the classroom, you prioritized community service. As a member of Alpha Phi Alpha, the historically black fraternity, you took on a variety of leadership positions. Your role as vice president of the Afro-American Society was significant to you, and you were pleased to be part of this long-established group, through which you met many members of BADA (the Black Alumni at Dartmouth Association). A particular highlight was your involvement with the Dartmouth Alliance for Children of Color, which provided mentorship for children from the Upper Valley. You vividly remember walking by candlelight across the winter landscape of the campus each year, holding vigil in remembrance of Martin Luther King, Jr. A member of the senior honors society Casque and Gauntlet, you co-founded the InterCommunity Council, which brought together student leaders from across campus. You also enjoyed hosting a radio show and playing intramural sports.
Upon graduation, you joined Savin Engineers, P.C. as a professional environmental engineer. Looking for work with a quicker pace, you transitioned into software technology as lead business architect for Agora Group Inc. after a few years. Now an engagement manager and product lead for Oliver Wyman, you assist clients in working through complex business and technical issues.
Despite your highly demanding work schedule, you prioritize volunteering for Dartmouth. You've served as an alumni interviewer, as the regional coordinator and treasurer for BADA, as an alumni councilor, and as a current member of the Executive Committee of the Association of Alumni. In your work with BADA, you've particularly enjoyed bringing students and alumni together, and you fondly remember your work planning BADA's 40th Reunion in 2012, which delighted you with its tremendous turnout and passionate alumni engagement. During your term on the Alumni Council, you relished meeting alumni of all backgrounds and hearing new ideas, and found interaction with students through the Student Affairs Committee particularly meaningful. In a similar vein, you also helped establish an alumni association for your fraternity, Alpha Phi Alpha, to continue building powerful bonds between students and alumni.
The mentoring process is crucial to you. Cognizant of the opportunity gap for women and minorities in technology, you've focused on issues such as public health by serving as COO of the Ethiopian Global Initiative; helped create access, awareness, and opportunities for top black and Latinx engineering talent with Code2040; and provided tutoring and programs for the rising leaders programs. Often on campus, you consider yourself an unofficial mentor to STEM students.
Your wonderful family visited campus often during your undergraduate days, and they appreciate this next phase of dedicated and extensive volunteerism for Dartmouth. Hobbs, for your devotion to your community and meaningful contributions to Dartmouth students and alumni alike, we are grateful and pleased to present you with the Dartmouth Young Alumni Distinguished Service Award.
Dartmouth Alumni Award
Class Awards
Club and Group Awards
Wearers of the Green
History: Alumni Award
History: Young Alumni Award
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Come visit and take a look. The farmers market is located in Collingswood, New Jersey at Between Collins & Irvin Ave. Along High Speed Line. Call to learn more about its assortment of fruits, local specialties, vegetables, crafts and organic food. Hours are May 2-November 21 Saturday, 8:00 a.m.- 12:00 noon. Takes SFMNP. Click the Edit link if you visit this market to tell us more about what is offered.
If you've shopped at Collingswood Farmers Market, tell us what you think of the market.
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1 Which are the Best Shoe Shine Kit on the Internet?
2.2 Why should people buy it?
3.2 Why should people buy it?
4.2 Why should people buy it?
5 4. FootFitter Classic Shoe Brush Set, 9 Piece Polishing Kit for Men!
5.2 Why should people buy it?
Which are the Best Shoe Shine Kit on the Internet?
Seven shoe shine brush kit is very portable. It one of the best shoe shine kits in the market because of the elegant travel case and convenient tools.
· It features the best accessories with horsehair bristle brush.
· There are advanced cleaning tools with black and brown polishes.
There are large, medium and small sized bristles so you can apply the shoe polishes without any problem. The box also comes with compact tools. You can find two bristles, two cleaning cloths, two polishes and more in a compact leather case.
· There are two sponge daubers with two compact brushes.
· The compact case is very travelling friendly so that you can carry anywhere.
· The brushes may shed if you are not careful.
PU leather case will give you a sharp look without any compromise on accessories. There is a PU leather case to carry all the 11 tools simultaneously.
· Sleek and elegant shoe shining kit with PU leather case.
· It also includes every piece of gears for leather shoes.
There are eleven kits in a single box that is why it is one of the best shoe shine kits. You can find a shoe brush, one metal shoe horn, one shine cloth, two shoes clear and more in the PU leather case.
· The shoe polishes are of high quality, so they will not attract dust.
· It also comes with lifetime warranty.
· The front latch is cheap because it will come out.
There nine pieces of wooden tools for shoe shining. There is a wooden box with two sets of wooden brushes. You shall love the KIWI paste tins.
The brushes are very useful because of the horsehair bristles. One should buy this because they are the best shoe shine kits on the market.
· There are two soft micro-fiber clothes with wooden handle.
· It is very easy to clean and apply polish because of the wooden handle.
· The polish smells like cheap hair wax.
4. FootFitter Classic Shoe Brush Set, 9 Piece Polishing Kit for Men!
The FootFitter is one of the best shoe shine kits on the market. It offers professional cleaning with four-way suede and nubuck cleaner. The triangle shaped tool offers deep cleaning from three different sides.
The horsehair bristles offer efficient cleaning. Apart from that, the triangle shaped brush offers deep cleaning in corners.
There is nine shoe shining kits for men. There are two different brushes in the box. You can apply different colored polish without any problem.
· The bristle brushes are well made, and they will last longer than a normal brush.
· There will be no shedding of brushes.
· The clasp of the box can break if you are not careful.
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"redpajama_set_name": "RedPajamaC4"
}
| 5,170
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Yaddo est une colonie d'artistes située à Saratoga Springs dans l'État de New York aux États-Unis. Son but est de développer la créativité des artistes en mettant à leur disposition un environnement favorable au travail. Elle offre des résidences aux artistes de tous horizons : chorégraphie, peinture, cinéma, littérature, musique, théâtre, photographie, gravure, sculpture, et art vidéo.
Histoire
La propriété est achetée en 1881 par l'homme d'affaires et sa femme, l'écrivain . La première bâtisse est détruite lors d'un incendie en 1893, faisant place à la maison actuelle.
Yaddo est un mot qui aurait été inventé par Christina, l'un des enfants des Trask, qui cherchait à prononcer Shadow (ombre).
Colonie d'artistes
En 1900, sans héritier après la mort prématurée de leurs quatre enfants, Spencer Trask décide de reconvertir la maison familiale en résidence d'artistes, en hommage à son épouse. Il meurt dans un accident de train le Jour de l'an 1909. Katrina Spencer se remarie avec le banquier et philanthrope , et tous deux décident d'investir dans le projet de Spencer Corporation of Yaddo. Les premiers artistes s'y installent à partir de 1926. Le succès de Yaddo convaincra par la suite Katrina Trask de faire don d'une de leurs propriétés comme maison de repos pour les ouvrières textiles de la ville de Troy (New York), la .
Période du maccarthysme
En 1947, au cours de la deuxième peur rouge du maccarthysme, la paix et le calme que cultive Yaddo sont mis à mal avec l'accusation d'espionnage qui frappe l'écrivain et journaliste Agnes Smedley, qui y réside depuis 1943. En 1949 éclate l'affaire Robert Lowell, lorsque ce poète américain tente d'évincer Elizabeth Ames, Directrice de Yaddo depuis 1926, au prétexte qu'elle est soupçonnée par le FBI de complicité d'espionnage avec Agnes Smedley pour le compte de l'Union soviétique. Mais celle-ci est finalement disculpée et restera directrice de Yaddo jusqu'à sa mort en 1977.
Reconnaissance et période actuelle
Plus de ont séjourné à Yaddo, parmi lesquels Hannah Arendt, Milton Clark Avery, James Baldwin, Leonard Bernstein, Truman Capote, Carson McCullers, John Cheever, Aaron Copland, Kenneth Fearing, Jonathan Franzen, Daniel Fuchs, Philip Guston, Patricia Highsmith, Langston Hughes, Ted Hughes, Alfred Kazin, Jacob Lawrence, Robert Lowell, Sylvia Plath, Henry Roth, Philip Roth, Katherine Anne Porter, Mario Puzo, Clyfford Still, Virgil Thomson, Colm Tóibín, Flannery O'Connor, Anne Truitt ou David Foster Wallace.
L'écrivain américain John Cheever déclarait :
Le sculpteur Daniel Chester French a réalisé une statue commémorative de Spencer Trask, The Spirit of Life, que l'on peut voir dans le parc de Saratoga.
Les artistes qui ont séjourné à Yaddo ont remporté 61 Prix Pulitzer, 56 National Book Awards, 22 National Book Critics Award, un Prix Nobel de littérature, et de nombreux autres prix. Plusieurs livres d'auteurs de Yaddo ont été adaptés au cinéma.
En 2013, Susan Unterberg et A. M. Homes, d'anciennes résidentes, sont nommées co-présidentes.
Après plus de 100 ans d'histoire, Yaddo accepte désormais de recevoir des contributions sous forme de dons ou de souscriptions pour des projets spécifiques, dans le souci de préserver ce lieu d'inspiration de la communauté artistique. Lors de la célébration de son centième anniversaire, Yaddo a reçu d'importantes contributions de la Spencer Trask & Company et de son dirigeant actuel .
La romancière Patricia Highsmith a fait don de sa propriété de 3 millions de dollars à la communauté.
Visites
On peut se promener dans le parc entre les bosquets de pins, les pelouses, les lacs artificiels et leurs canards, la roseraie. Seul lieu que les touristes sont autorisés à visiter, le parc est conçu à l'image des jardins italiens que les Trask avaient pu visiter en Europe. Il est orné de nombreuses sculptures et statues, d'une pergola, ou encore d'un cadran solaire qui porte l'inscription
Yaddo fait partie du Registre national des monuments historiques américain.
Notes et références
Voir aussi
Article connexe
MacDowell Colony
Liens externes
An Artists' Retreat Reaches Out, nytimes.com.
Art dans l'État de New York
Création artistique
Institution artistique
National Historic Landmark dans l'État de New York
National Historic Landmark en 2013
Registre national des lieux historiques en 2013
Comté de Saratoga
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\section{Introduction}
\label{sec:intro}
The first detections of coalescing binary black hole (BBH)
systems~\cite{Abbott:2016blz,Abbott:2016nmj} inaugurated the field of
gravitational wave (GW) astronomy.
Beside BBHs, binary neutron stars (BNS) are one of the expected
sources for future GW detections~\cite{Aasi:2013wya,Abbott:2016ymx}.
In contrast to BBH mergers, it is expected that BNS mergers
produce electromagnetic (EM) signals,
as kilonovae (also called macronovae),
radio flares or short gamma-ray bursts (SGRBs).
While SGRBs are powered by collimated highly relativistic
outflows, e.g.,~\cite{Paczynski:1986px,Eichler:1989ve,Soderberg:2006bn},
kilonovae are transient emissions in the optical or
near-infrared band, e.g.,~\cite{Tanvir:2013pia,Yang:2015pha,Jin:2016pnm},
produced by the radioactive decay of r-process nuclei
in the neutron-rich material ejected during the merger.
Additionally, mildly and sub- relativistic outflows can generate
synchrotron radiation (radio flares) even years after the merger
of the two neutron stars, see e.g.,~\cite{Nakar:2011cw}.
One possibility to study BNS mergers are
numerical relativity (NR) simulations. Those simulations
allow to describe the system even beyond the merger of the two stars solving
Einsteins field equations. Over the last years more microphysical descriptions
have been included, e.g., realistic equation of states (EOSs),
neutrino transport, magnetic fields.
It also has become a common approach to extract information from
NR simulations about the unbound material ejected from the system
and use these information to estimate possible EM counterparts.
However, the computation of ejecta and lightcurves is still challenging.
While current state-of-the art numerical simulations cover
the last $10-20$ orbits before and up to $\sim 50$ms after the merger,
it is computationally too expensive to study the dynamical ejected material
longer than a fraction of a second.
But, it is possible to use relativistic simulations
as initial conditions and either assume free expansion of the
ejecta material, e.g.,~\cite{Goriely:2011vg},
evolution on a fixed spacetime background, e.g.,~\cite{Rosswog:2013kqa,Grossman:2013lqa},
or use radiative transfer Monte-Carlo simulations, e.g.,~\cite{Tanaka:2013ana,Hotokezaka:2013kza}.
Our work is complementary to most previous studies, we will use a
large set of numerical relativity data obtained from
different groups to derive phenomenological fits
relating the binary parameters to the ejecta properties.
Knowing the basic properties of the
ejecta allows to give estimates
on the expected kilonovae and radio flares.
In general, the time between a GW detection and the observation
of the corresponding kilonovae (about a few days) is not long enough
to perform full NR simulations which have typical run times of weeks to months.
Therefore, NR simulations can only be used for comparison once GW and EM observations
finish. The advantage of the phenomenological model proposed in this article
is that even before the EM follow up observations start first estimates of the kilonovae
properties can be given.
Furthermore, after the kilonovae has been detected, the model can be used
to reduce the part of the BNS parameter space which
has to be covered by full NR simulations.
\section{Employed Dataset}
\label{sec:data}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\textwidth]{EOS_overview.pdf}
\caption{Mass vs.~radius relations (left) and mass vs.~compactness relations (right)
for all EOSs used in this work.
Tabulated EOSs are marked with dashed lines, piecewise polytropes with solid lines.
The markers refer to configurations employed in this work.}
\label{fig:EOS}
\end{center}
\end{figure}
Over the last years numerical relativity (NR) has made a tremendous progress and a large number of
groups have studied the merger process of BNSs, see e.g.,~\cite{Faber:2012rw,Baiotti:2016qnr}
and references therein.
Despite the computation of the emitted GW signal, the investigation of
ejected material and EM counterparts went into the focus of research.
Combining published work from different groups enables us to obtain
an NR catalog to derive fitting formulas
for important ejecta quantities. In this article we use results
from~\cite{Hotokezaka:2012ze,Bauswein:2013yna,Dietrich:2015iva,Lehner:2016lxy,Sekiguchi:2016bjd,Dietrich:2016hky},
where the mass, kinetic energy, and velocity of the ejecta are reported.
The data set combines results based on grid structured
codes~\cite{Hotokezaka:2012ze,Dietrich:2015iva,Lehner:2016lxy,Sekiguchi:2016bjd,Dietrich:2016hky}
with results employing a SPH code~\cite{Bauswein:2013yna}
under conformal flatness approximation
and it includes simplifies EOSs,
tabulated EOS as well as simulations with and without neutrino treatment.
In total $172$ simulations have been considered.
Although simulation techniques are continuously improved and
higher accuracy is achieved,
the characterization of ejecta is still challenging and results have to be assigned with
large uncertainties.
Considering the accuracy of the NR data points,
quantities as the mass and kinetic energy have uncertainties which range between $\sim 10\%$
up to even $\sim 100\%$, see e.g.,~appendix~A of~\cite{Hotokezaka:2012ze}
and table~III of~\cite{Dietrich:2016hky},
where multiple resolutions have been employed.
In general one finds that the fractional
uncertainty is larger for lower massive ejecta.
In addition to the uncertainty of the results employing the same numerical code also
differences between different implementations/codes exist.
For some cases those discrepancies are quite large (up to a factor of $\sim 5$ in extreme cases) and
they also depend on the implementation of thermal effects and if
neutrino cooling or transport is included in the simulations.
Those differences can produce systematic uncertainties. We try to minimize selection
effects by including a large number of simulations produced by
a variety of numerical codes.
In the future crosschecks among different codes employing the same physical systems will be needed
for a better estimate of systematic errors.
In our work, we restrict our analysis to dynamical ejecta. Ejecta produced
after BH formation are not included, but will contribute to the
total amount of ejecta and to the corresponding EM signals,
see e.g.,~\cite{Metzger:2016pju}.
Thus, our results can be seen as lower bounds for the luminosity
of EM observables.
Furthermore, while some of our data points were computed by NR
simulations including neutrinos and tabulated EOSs,
the effect of magnetic fields is not studied,
although magnetic fields will influence the binary
dynamics shortly around and after merger
and lead to mass ejection by
magnetic winds.
The complete dataset is reported in table~\ref{tab:overview}, where a simulation number
is assigned to every data point (first column).
In total we consider 23 different EOSs (shown in figure~\ref{fig:EOS}).
Most EOSs are represented by a piecewise polytrope fitted to a zero-temperature EOS (straight lines),
see e.g.,~\cite{Read:2009yp}. An additional thermal contribution
to the pressure according to $p_{\rm th} = \rho \epsilon (\Gamma_{\rm th} - 1)$
is added for the evolution, where $\rho$ is the rest-mass density and $\epsilon$
the internal energy. The parameter
$\Gamma_{\rm th}$ is also reported in table~\ref{tab:overview}.
Some simulations use full tabulated EOSs (dashed lines),
which we denote as full in table~\ref{tab:overview}.
Simulations with tabulated EOSs and neutrino treatment are denoted with fullN.
In addition to the parameters describing the binary, we report the mass of the
ejected material $M_{\rm ej}$, the kinetic energy $T_{\rm ej}$, the average velocity
inside the orbital plane $v_\rho$, the average velocity perpendicular to the
orbital plane $v_z$, and the total velocity $v_{\rm ej}$.
\begin{small}
\renewcommand{\arraystretch}{0.9}
\begin{longtable}{ll|cccc|ccccc}
\caption{ \label{tab:overview} NR data used in this work. Columns refer to:
The data ID, cf.~e.g.,~figure~\ref{fig:Mej},
mass of the first star $M_1$,
mass of the second star $M_2$,
$\Gamma_{\rm th}$ modeling thermal effects for piecewise polytropic EOS,
ejecta mass $M_{\rm ej}$,
kinetic energy of the ejecta $T_{\rm ej}$,
average velocity inside the orbital plane $v_{\rho}$,
average velocity perpendicular to the orbital plane $v_z$,
total average ejecta velocity $v_{\rm ej}$.
In cases where $v_\rho$ and $v_z$ are given, we
estimate the total ejecta velocity as
$v_{\rm ej} = \sqrt{v_\rho^2+v_z^2}$.
Note that in~\cite{Sekiguchi:2016bjd}
the ejecta velocity was estimated based on
$T_{\rm ej} = M_{\rm ej} v^2_{\rm ej}/2$,
consequently we use this relation to compute the kinetic energy
not stated in~\cite{Sekiguchi:2016bjd}.}\\
$\#$ & Ref & EOS & $M_1$ & $M_2$ & $\Gamma_{\rm th}$ & $M_{\rm ej}$ & $T_{\rm ej}$ & $v_{\rho}$ & $v_{z}$ & $v_{\rm ej}$\\
& & &$[M_\odot]$&$[M_\odot$]& &$[10^{-3}M_\odot]$ & $[10^{50}$erg]& $[c]$ & $[c]$ & $[c]$ \\
\hline
1 & ALF2 & \cite{Dietrich:2016hky} & 1 & 1.75 & 1.75 & 36 & 12.69 & 0.18 & 0.03 & 0.18 \\
2 & ALF2 & \cite{Dietrich:2016hky} & 1.167 & 1.75 & 1.75 & 25 & 10.73 & 0.19 & 0.06 & 0.2 \\
3 & ALF2 & \cite{Dietrich:2016hky} & 1.1 & 1.65 & 1.75 & 24 & 7.5 & 0.17 & 0.07 & 0.18 \\
4 & ALF2 & \cite{Dietrich:2016hky} & 1 & 1.5 & 1.75 & 21 & 4.8 & 0.15 & 0.07 & 0.17 \\
5 & ALF2 & \cite{Dietrich:2016hky} & 1.222 & 1.527 & 1.75 & 7.5 & 3.93 & 0.17 & 0.12 & 0.21 \\
6 & ALF2 & \cite{Hotokezaka:2012ze} & 1.2 & 1.5 & 1.8 & 5.5 & 3 & 0.21 & 0.1 & 0.23 \\
7 & ALF2 & \cite{Hotokezaka:2012ze} & 1.25 & 1.45 & 1.8 & 3 & 1.5 & 0.2 & 0.1 & 0.22 \\
8 & ALF2 & \cite{Hotokezaka:2012ze} & 1.3 & 1.4 & 1.8 & 1.5 & 0.8 & 0.16 & 0.11 & 0.19 \\
9 & ALF2 & \cite{Hotokezaka:2012ze} & 1.4 & 1.4 & 1.8 & 2.5 & 1.5 & 0.21 & 0.13 & 0.25 \\
10 & ALF2 & \cite{Dietrich:2016hky} & 1.375 & 1.375 & 1.75 & 3.4 & 1.36 & 0.17 & 0.1 & 0.2 \\
11 & ALF2 & \cite{Hotokezaka:2012ze} & 1.35 & 1.35 & 1.8 & 2.5 & 1.5 & 0.22 & 0.12 & 0.25 \\
12 & ALF2 & \cite{Hotokezaka:2012ze} & 1.3 & 1.3 & 1.8 & 2 & 1 & 0.19 & 0.1 & 0.21 \\
13 & APR4 & \cite{Hotokezaka:2012ze} & 1.2 & 1.5 & 2 & 7.5 & 5.5 & 0.24 & 0.12 & 0.27 \\
14 & APR4 & \cite{Hotokezaka:2012ze} & 1.2 & 1.5 & 1.8 & 8 & 5.5 & 0.23 & 0.11 & 0.25 \\
15 & APR4 & \cite{Hotokezaka:2012ze} & 1.2 & 1.5 & 1.6 & 9 & 5 & 0.2 & 0.1 & 0.22 \\
16 & APR4 & \cite{Hotokezaka:2012ze} & 1.3 & 1.6 & 1.8 & 2 & 1.5 & 0.24 & 0.08 & 0.25 \\
17 & APR4 & \cite{Hotokezaka:2012ze} & 1.2 & 1.4 & 1.8 & 3 & 2 & 0.21 & 0.12 & 0.24 \\
18 & APR4 & \cite{Hotokezaka:2012ze} & 1.25 & 1.45 & 1.8 & 7 & 4.5 & 0.22 & 0.11 & 0.25 \\
19 & APR4 & \cite{Hotokezaka:2012ze} & 1.3 & 1.5 & 1.8 & 12 & 8.5 & 0.23 & 0.12 & 0.26 \\
20 & APR4 & \cite{Hotokezaka:2012ze} & 1.3 & 1.4 & 1.8 & 8 & 5 & 0.19 & 0.12 & 0.22 \\
21 & APR4 & \cite{Hotokezaka:2012ze} & 1.25 & 1.35 & 1.8 & 5 & 3 & 0.18 & 0.1 & 0.21 \\
22 & APR4 & \cite{Hotokezaka:2012ze} & 1.4 & 1.5 & 1.8 & 0.6 & 0.9 & 0.35 & 0.12 & 0.37 \\
23 & APR4 & \cite{Hotokezaka:2012ze} & 1.45 & 1.45 & 1.8 & 0.1 & 0.1 & 0.29 & 0.13 & 0.32 \\
24 & APR4 & \cite{Hotokezaka:2012ze} & 1.4 & 1.4 & 1.8 & 14 & 10 & 0.22 & 0.15 & 0.27 \\
25 & APR4 & \cite{Hotokezaka:2012ze} & 1.35 & 1.35 & 2 & 5 & 3 & 0.19 & 0.13 & 0.23 \\
26 & APR4 & \cite{Hotokezaka:2012ze} & 1.35 & 1.35 & 1.8 & 7 & 4 & 0.19 & 0.12 & 0.22 \\
27 & APR4 & \cite{Hotokezaka:2012ze} & 1.35 & 1.35 & 1.6 & 11 & 6 & 0.19 & 0.13 & 0.23 \\
28 & APR4 & \cite{Hotokezaka:2012ze} & 1.3 & 1.3 & 1.8 & 2 & 1 & 0.19 & 0.1 & 0.21 \\
29 & H4 & \cite{Dietrich:2016hky} & 1 & 1.75 & 1.75 & 40 & 12.51 & 0.17 & 0.02 & 0.17 \\
30 & H4 & \cite{Dietrich:2016hky} & 1.167 & 1.75 & 1.75 & 14 & 4.65 & 0.18 & 0.05 & 0.19 \\
31 & H4 & \cite{Dietrich:2016hky} & 1.1 & 1.65 & 1.75 & 17 & 4.83 & 0.17 & 0.04 & 0.17 \\
32 & H4 & \cite{Dietrich:2016hky} & 1 & 1.5 & 1.75 & 27 & 8.04 & 0.17 & 0.03 & 0.17 \\
33 & H4 & \cite{Dietrich:2016hky} & 1.222 & 1.527 & 1.75 & 6.6 & 3.04 & 0.18 & 0.11 & 0.21 \\
34 & H4 & \cite{Hotokezaka:2012ze} & 1.2 & 1.5 & 2 & 4 & 2 & 0.21 & 0.09 & 0.23 \\
35 & H4 & \cite{Hotokezaka:2012ze} & 1.2 & 1.5 & 1.8 & 3.5 & 2 & 0.21 & 0.09 & 0.23 \\
36 & H4 & \cite{Hotokezaka:2012ze} & 1.2 & 1.5 & 1.6 & 4.5 & 2 & 0.19 & 0.1 & 0.21 \\
37 & H4 & \cite{Hotokezaka:2012ze} & 1.2 & 1.4 & 1.8 & 2.5 & 1 & 0.19 & 0.1 & 0.21 \\
38 & H4 & \cite{Hotokezaka:2012ze} & 1.25 & 1.45 & 1.8 & 2 & 1.5 & 0.19 & 0.1 & 0.21 \\
39 & H4 & \cite{Hotokezaka:2012ze} & 1.3 & 1.5 & 1.8 & 3 & 2 & 0.19 & 0.1 & 0.21 \\
40 & H4 & \cite{Hotokezaka:2012ze} & 1.3 & 1.4 & 1.8 & 0.7 & 0.4 & 0.18 & 0.1 & 0.21 \\
41 & H4 & \cite{Hotokezaka:2012ze} & 1.25 & 1.35 & 1.8 & 0.6 & 0.3 & 0.18 & 0.1 & 0.21 \\
42 & H4 & \cite{Hotokezaka:2012ze} & 1.4 & 1.4 & 1.8 & 0.3 & 0.2 & 0.17 & 0.13 & 0.21 \\
43 & H4 & \cite{Dietrich:2016hky} & 1.375 & 1.375 & 1.75 & 3.4 & 1.59 & 0.19 & 0.1 & 0.21 \\
44 & H4 & \cite{Hotokezaka:2012ze} & 1.35 & 1.35 & 2 & 0.4 & 0.2 & 0.2 & 0.1 & 0.22 \\
45 & H4 & \cite{Hotokezaka:2012ze} & 1.35 & 1.35 & 1.8 & 0.5 & 0.2 & 0.19 & 0.11 & 0.22 \\
46 & H4 & \cite{Hotokezaka:2012ze} & 1.35 & 1.35 & 1.6 & 0.7 & 0.4 & 0.21 & 0.11 & 0.24 \\
47 & H4 & \cite{Hotokezaka:2012ze} & 1.3 & 1.3 & 1.8 & 0.3 & 0.1 & 0.16 & 0.1 & 0.19 \\
48 & MS1 & \cite{Hotokezaka:2012ze} & 1.2 & 1.5 & 1.8 & 3.5 & 1.5 & 0.19 & 0.1 & 0.21 \\
49 & MS1 & \cite{Hotokezaka:2012ze} & 1.25 & 1.45 & 1.8 & 1.5 & 0.8 & 0.19 & 0.11 & 0.22 \\
50 & MS1 & \cite{Hotokezaka:2012ze} & 1.3 & 1.4 & 1.8 & 0.6 & 0.2 & 0.17 & 0.09 & 0.19 \\
51 & MS1 & \cite{Hotokezaka:2012ze} & 1.4 & 1.4 & 1.8 & 0.6 & 0.2 & 0.13 & 0.09 & 0.16 \\
52 & MS1 & \cite{Hotokezaka:2012ze} & 1.35 & 1.35 & 1.8 & 1.5 & 0.6 & 0.14 & 0.08 & 0.16 \\
53 & MS1 & \cite{Hotokezaka:2012ze} & 1.3 & 1.3 & 1.8 & 1.5 & 0.5 & 0.15 & 0.08 & 0.17 \\
54 & MS1b & \cite{Dietrich:2016hky} & 0.944 & 1.944 & 1.75 & 65 & 21.45 & 0.18 & 0.02 & 0.18 \\
55 & MS1b & \cite{Dietrich:2016hky} & 1 & 1.75 & 1.75 & 49 & 15.19 & 0.17 & 0.03 & 0.17 \\
56 & MS1b & \cite{Dietrich:2016hky} & 1.167 & 1.75 & 1.75 & 24 & 7.69 & 0.18 & 0.05 & 0.19 \\
57 & MS1b & \cite{Dietrich:2016hky} & 1.1 & 1.65 & 1.75 & 26 & 7.33 & 0.17 & 0.04 & 0.17 \\
58 & MS1b & \cite{Dietrich:2016hky} & 1 & 1.5 & 1.75 & 32 & 7.87 & 0.16 & 0.03 & 0.16 \\
59 & MS1b & \cite{Dietrich:2016hky} & 1.222 & 1.527 & 1.75 & 4.8 & 1.64 & 0.15 & 0.11 & 0.19 \\
60 & MS1b & \cite{Dietrich:2016hky} & 1.375 & 1.375 & 1.75 & 2.3 & 0.39 & 0.13 & 0.06 & 0.14 \\
61 & SLy & \cite{Dietrich:2016hky} & 1 & 1.75 & 1.75 & 24 & 8.94 & 0.19 & 0.03 & 0.19 \\
62 & SLy & \cite{Dietrich:2016hky} & 1.167 & 1.75 & 1.75 & 6.5 & 5.54 & 0.25 & 0.11 & 0.27 \\
63 & SLy & \cite{Dietrich:2016hky} & 1.1 & 1.65 & 1.75 & 16 & 7.69 & 0.19 & 0.11 & 0.22 \\
64 & SLy & \cite{Dietrich:2016hky} & 1 & 1.5 & 1.75 & 18 & 9.12 & 0.19 & 0.12 & 0.22 \\
65 & SLy & \cite{Dietrich:2016hky} & 1.222 & 1.527 & 1.75 & 18 & 8.4 & 0.16 & 0.11 & 0.19 \\
66 & SLy & \cite{Dietrich:2016hky} & 1.375 & 1.375 & 1.75 & 16 & 4.83 & 0.17 & 0.1 & 0.2 \\
67 & ALF2 & \cite{Dietrich:2015iva} & 1.25 & 1.45 & 1.75 & 3.9 & 0.8 & - & - & 0.15 \\
68 & ALF2 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 3.8 & 3.36 & - & - & 0.28 \\
69 & ALF2 & \cite{Dietrich:2015iva} & 1.35 & 1.35 & 1.75 & 3.5 & 0.7 & - & - & 0.15 \\
70 & ALF2 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 4.49 & 3.8 & - & - & 0.27\\
71 & ALF4 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 5.7 & 6.07 & - & - & 0.3\\
72 & ALF4 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 7.4 & 7.65 & - & - & 0.29\\
73 & APR & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 5.96 & 6.37 & - & - & 0.31\\
74 & APR & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 7.38 & 7.9 & - & - & 0.3\\
75 & APR3 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 4.65 & 4.69 & - & - & 0.3\\
76 & APR3 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 6.15 & 5.5 & - & - & 0.27\\
77 & DD2 & \cite{Bauswein:2013yna} & 1.2 & 1.8 & full & 17.08 & 6.72 & - & - & 0.17\\
78 & DD2 & \cite{Bauswein:2013yna} & 1.35 & 2 & full & 6.41 & 9.64 & - & - & 0.31\\
79 & DD2 & \cite{Bauswein:2013yna} & 1.35 & 1.8 & full & 14.85 & 9.48 & - & - & 0.21\\
80 & DD2 & \cite{Bauswein:2013yna} & 1.2 & 1.6 & full & 10.9 & 6.39 & - & - & 0.2\\
81 & DD2 & \cite{Lehner:2016lxy} & 1.18 & 1.54 & fullN & 1.3 & 0.76 & - & - & 0.3\\
82 & DD2 & \cite{Bauswein:2013yna} & 1.2 & 1.5 & full & 8.79 & 4.97 & - & - & 0.2\\
83 & DD2 & \cite{Bauswein:2013yna} & 1.5 & 1.8 & full & 18.84 & 15.52 & - & - & 0.25\\
84 & DD2 & \cite{Lehner:2016lxy} & 1.25 & 1.47 & fullN & 0.42 & 0.29 & - & - & 0.3\\
85 & DD2 & \cite{Sekiguchi:2016bjd} & 1.25 & 1.45 & fullN & 5 & 1.61 & - & - & 0.19\\
86 & DD2 & \cite{Bauswein:2013yna} & 1.2 & 1.35 & full & 3.17 & 2.06 & - & - & 0.2\\
87 & DD2 & \cite{Bauswein:2013yna} & 1.35 & 1.5 & full & 3.57 & 3.13 & - & - & 0.25\\
88 & DD2 & \cite{Sekiguchi:2016bjd} & 1.3 & 1.4 & fullN & 3 & 0.87 & - & - & 0.18\\
89 & DD2 & \cite{Bauswein:2013yna} & 2 & 2 & full & 0.25 & 0.25 & - & - & 0.25\\
90 & DD2 & \cite{Bauswein:2013yna} & 1.8 & 1.8 & full & 1.37 & 1.63 & - & - & 0.26\\
91 & DD2 & \cite{Bauswein:2013yna} & 1.6 & 1.6 & full & 7.8 & 7.4 & - & - & 0.27\\
92 & DD2 & \cite{Bauswein:2013yna} & 1.5 & 1.5 & full & 5.38 & 4.66 & - & - & 0.26\\
93 & DD2 & \cite{Lehner:2016lxy} & 1.36 & 1.36 & fullN & 0.43 & 0.31 & - & - & 0.3\\
94 & DD2 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 2.57 & 3.31 & - & - & 0.34\\
95 & DD2 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.8 & 2.26 & 2.61 & - & - & 0.32\\
96 & DD2 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 2.72 & 2.9 & - & - & 0.3\\
97 & DD2 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & full & 3.07 & 2.18 & - & - & 0.22\\
98 & DD2 & \cite{Sekiguchi:2016bjd} & 1.35 & 1.35 & fullN & 2 & 0.46 & - & - & 0.16\\
99 & DD2 & \cite{Bauswein:2013yna} & 1.2 & 1.2 & full & 3.09 & 1.37 & - & - & 0.17\\
100 & ENG & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 5.29 & 5.01 & - & - & 0.29\\
101 & ENG & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 6.32 & 5.3 & - & - & 0.26\\
102 & Glenh3 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 1.08 & 0.62 & - & - & 0.23\\
103 & Glenh3 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 1.69 & 0.9 & - & - & 0.22\\
104 & GS2 & \cite{Bauswein:2013yna} & 1.2 & 1.5 & full & 10.69 & 6.14 & - & - & 0.18\\
105 & GS2 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & full & 2.74 & 2.16 & - & - & 0.19\\
106 & H3 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 1.43 & 1.15 & - & - & 0.27\\
107 & H4 & \cite{Dietrich:2015iva} & 1.25 & 1.45 & 1.75 & 6 & 2.8 & - & - & 0.23\\
108 & H4 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 1.28 & 1.09 & - & - & 0.27\\
109 & H4 & \cite{Dietrich:2015iva} & 1.35 & 1.35 & 1.75 & 0.6 & 0.5 & - & - & 0.3\\
110 & H4 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 1.93 & 1.64 & - & - & 0.27\\
111 & MPA1 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 3.64 & 3.6 & - & - & 0.3\\
112 & MPA1 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 4.48 & 4.35 & - & - & 0.29\\
113 & MS1 & \cite{Dietrich:2015iva} & 1.25 & 1.45 & 1.75 & 5.8 & 1.2 & - & - & 0.15\\
114 & MS1 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 1.17 & 0.98 & - & - & 0.27\\
115 & MS1 & \cite{Dietrich:2015iva} & 1.35 & 1.35 & 1.75 & 0.7 & 0.2 & - & - & 0.18\\
116 & MS1 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 2.38 & 1.19 & - & - & 0.21\\
117 & MS1b & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 1.67 & 1.26 & - & - & 0.25\\
118 & MS1b & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 3.64 & 1.85 & - & - & 0.21\\
119 & MS2 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 0.81 & 0.65 & - & - & 0.26\\
120 & NL3 & \cite{Bauswein:2013yna} & 1.2 & 1.8 & full & 15.68 & 5.75 & - & - & 0.15\\
121 & NL3 & \cite{Bauswein:2013yna} & 1.35 & 2 & full & 12.85 & 7.62 & - & - & 0.2\\
122 & NL3 & \cite{Bauswein:2013yna} & 1.35 & 1.8 & full & 18.81 & 11.31 & - & - & 0.21\\
123 & NL3 & \cite{Bauswein:2013yna} & 1.2 & 1.6 & full & 9.96 & 5.57 & - & - & 0.19\\
124 & NL3 & \cite{Bauswein:2013yna} & 1.2 & 1.5 & full & 7.95 & 4.5 & - & - & 0.19\\
125 & NL3 & \cite{Bauswein:2013yna} & 1.5 & 1.8 & full & 8.1 & 4.94 & - & - & 0.21\\
126 & NL3 & \cite{Lehner:2016lxy} & 1.25 & 1.47 & fullN & 2.3 & 1.22 & - & - & 0.25\\
127 & NL3 & \cite{Bauswein:2013yna} & 1.35 & 1.5 & full & 2.72 & 2.25 & - & - & 0.24\\
128 & NL3 & \cite{Bauswein:2013yna} & 1.2 & 1.35 & full & 4.25 & 2.74 & - & - & 0.21\\
129 & NL3 & \cite{Bauswein:2013yna} & 2 & 2 & full & 1.91 & 2.18 & - & - & 0.29\\
130 & NL3 & \cite{Bauswein:2013yna} & 1.8 & 1.8 & full & 9.08 & 7.25 & - & - & 0.24\\
131 & NL3 & \cite{Bauswein:2013yna} & 1.6 & 1.6 & full & 3.74 & 2.59 & - & - & 0.22\\
132 & NL3 & \cite{Bauswein:2013yna} & 1.5 & 1.5 & full & 1.7 & 1.04 & - & - & 0.2\\
133 & NL3 & \cite{Lehner:2016lxy} & 1.36 & 1.36 & fullN & 0.015 & 0.01 & - & - & 0.45\\
134 & NL3 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 1.57 & 2.03 & - & - & 0.34\\
135 & NL3 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.8 & 1.6 & 2.99 & - & - & 0.32\\
136 & NL3 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 1.86 & 1.98 & - & - & 0.3\\
137 & NL3 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & full & 2.09 & 0.98 & - & - & 0.18\\
138 & NL3 & \cite{Bauswein:2013yna} & 1.2 & 1.2 & full & 2.15 & 0.91 & - & - & 0.17\\
139 & SFHo & \cite{Bauswein:2013yna} & 1.2 & 1.8 & full & 5.78 & 10.08 & - & - & 0.34\\
140 & SFHo & \cite{Bauswein:2013yna} & 1.35 & 1.8 & full & 11.76 & 16.22 & - & - & 0.31\\
141 & SFHo & \cite{Bauswein:2013yna} & 1.2 & 1.6 & full & 16.91 & 11.1 & - & - & 0.21\\
142 & SFHo & \cite{Bauswein:2013yna} & 1.2 & 1.5 & full & 13.39 & 8.94 & - & - & 0.22\\
143 & SFHo & \cite{Bauswein:2013yna} & 1.5 & 1.8 & full & 6.34 & 14.4 & - & - & 0.42\\
144 & SFHo & \cite{Lehner:2016lxy} & 1.25 & 1.47 & fullN & 2.2 & 1.8 & - & - & 0.25\\
145 & SFHo & \cite{Sekiguchi:2016bjd} & 1.25 & 1.45 & fullN & 11 & 5.66 & - & - & 0.24\\
146 & SFHo & \cite{Bauswein:2013yna} & 1.2 & 1.35 & full & 5.44 & 3.86 & - & - & 0.22\\
147 & SFHo & \cite{Bauswein:2013yna} & 1.35 & 1.5 & full & 18.73 & 13.34 & - & - & 0.23\\
148 & SFHo & \cite{Sekiguchi:2016bjd} & 1.3 & 1.4 & fullN & 6 & 2.15 & - & - & 0.2\\
149 & SFHo & \cite{Sekiguchi:2016bjd} & 1.33 & 1.37 & fullN & 9 & 3.55 & - & - & 0.21\\
150 & SFHo & \cite{Bauswein:2013yna} & 1.8 & 1.8 & full & 0.17 & 0.24 & - & - & 0.29\\
151 & SFHo & \cite{Bauswein:2013yna} & 1.6 & 1.6 & full & 1.13 & 1 & - & - & 0.21\\
152 & SFHo & \cite{Bauswein:2013yna} & 1.5 & 1.5 & full & 4.1 & 4.13 & - & - & 0.27\\
153 & SFHo & \cite{Lehner:2016lxy} & 1.36 & 1.36 & fullN & 3.4 & 1.8 & - & - & 0.25\\
154 & SFHo & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 2.96 & 3.37 & - & - & 0.32\\
155 & SFHo & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.8 & 3.26 & 4.18 & - & - & 0.34\\
156 & SFHo & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 3.82 & 4.14 & - & - & 0.3\\
157 & SFHo & \cite{Bauswein:2013yna} & 1.35 & 1.35 & full & 4.83 & 3.61 & - & - & 0.23\\
158 & SFHo & \cite{Sekiguchi:2016bjd} & 1.35 & 1.35 & fullN & 11 & 4.76 & - & - & 0.22\\
159 & SFHo & \cite{Bauswein:2013yna} & 1.2 & 1.2 & full & 1.88 & 1.26 & - & - & 0.21\\
160 & SFHx & \cite{Bauswein:2013yna} & 1.2 & 1.5 & full & 14.67 & 7.91 & - & - & 0.19\\
161 & SFHx & \cite{Bauswein:2013yna} & 1.35 & 1.35 & full & 6.16 & 4.36 & - & - & 0.22\\
162 & SLy & \cite{Dietrich:2015iva} & 1.25 & 1.45 & 1.75 & 6.5 & 5.1 & - & - & 0.3\\
163 & SLy & \cite{Dietrich:2015iva} & 1.35 & 1.35 & 1.75 & 12.2 & 7.1 & - & - & 0.26\\
164 & SLy4 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 3.99 & 3.75 & - & - & 0.29\\
165 & SLy4 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 6.4 & 5.53 & - & - & 0.27\\
166 & TM1 & \cite{Bauswein:2013yna} & 1.2 & 1.5 & full & 8.66 & 3.94 & - & - & 0.17\\
167 & TM1 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 2 & 1.37 & 2.02 & - & - & 0.36\\
168 & TM1 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.8 & 1.33 & 1.77 & - & - & 0.34\\
169 & TM1 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & 1.5 & 1.53 & 1.86 & - & - & 0.32\\
170 & TM1 & \cite{Bauswein:2013yna} & 1.35 & 1.35 & full & 1.67 & 0.74 & - & - & 0.16\\
171 & TMA & \cite{Bauswein:2013yna} & 1.2 & 1.5 & full & 10.21 & 6.4 & - & - & 0.2\\
172 & TMA & \cite{Bauswein:2013yna} & 1.35 & 1.35 & full & 2.05 & 1.19 & - & - & 0.18\\
\end{longtable}
\end{small}
\section{Ejecta properties}
\label{sec:fits}
\subsection{Ejecta mass}
Considering EM signals from BNS mergers, one of the most important quantities
influencing the luminosity of kilonovae and radio flares is the mass of
the material ejected from the system.
The authors in \cite{Foucart:2012nc,Kawaguchi:2016ana} proposed
fitting formulas for the disk and ejecta mass for BHNS systems.
To our knowledge no fit for the mass of the ejected material for BNS mergers exists to date.
Our fitting formula
\begin{equation}
\frac{M_{\rm ej}^{\rm fit}}{10^{-3}M_\odot} = \left[ a \left(\frac{M_2}{M_1}\right)^{1/3} \left(\frac{1-2 C_1}{C_1}\right)+
b \left(\frac{M_2}{M_1}\right)^{n} + c \left(1 - \frac{M_1}{M^*_{1}}\right) \right] M_1^* + (1\leftrightarrow 2) + d. \label{eq:Mej_fit}
\end{equation}
is an extension of the work done for BHNS systems to a system consisting of two neutron stars.
We denote the mass in isolation of the i-th star as $M_i$,
the baryonic mass as $M_i^*$, and the compactness as $C_i$.
Let us emphasize that although it has been shown that for BNS mergers
a significant part of the ejecta is produced by shocks,
e.g.,~\cite{Hotokezaka:2012ze}, \eqref{eq:Mej_fit} gives a robust estimate
for the ejecta for almost all considered configurations.
For our data we obtain the following fitting parameters:
\begin{equation}
a = -1.35695 , \quad b = 6.11252 ,\quad c = -49.43355 ,\quad d = 16.1144,\quad n = -2.5484. \label{eq:par:Mej}
\end{equation}
The left panels of figure~\ref{fig:Mej} show our results for the ejecta mass.
In the upper panel we present $M_{\rm ej}$
for the numerical simulation (blue circles) and
for our fitting formula $M_{\rm ej}^{\rm fit}$ (red crosses).
Both quantities are plotted as a function of the simulation-ID
introduced in table~\ref{tab:overview}.
The bottom panel shows the absolute residual $\Delta M_{\rm ej} =
M_{\rm ej}^{\rm fit}-M_{\rm ej}$. We include as shaded regions the
1$\sigma$ ($\Delta M_{\rm ej}^{1\sigma} = 4.4 \times 10^{-3}M_\odot$)
and $2\sigma$ confidence intervals.
Our model function has an average residual of
$\Delta \bar{M}_{\rm ej} = 2.9\times10^{-3}M_\odot$,
which corresponds to a fractional error of $\sim 72\%$.
Overall, because of the difficulties computing the
ejecta properties, see section~\ref{sec:data},
$\Delta \bar{M}_{\rm ej}$ is of the same order as
the numerical uncertainty of the NR data points and therefore can
be considered as a possible estimate.
Additionally, we present the results obtained from the fit
in Fig.~\ref{fig:fitMej}, where the absolute and relative difference between the NR data
and the fit are shown as a function of the mass ratio and the compactnesses of the stars.
Obviously for equal mass setups the relative difference is larger because of the
smaller ejecta mass. Those setups also have the highest NR uncertainty.
Considering the influence of the compactnesses, we find that for
larger compactness of the lighter star
the absolute error increases.
Let us also mention the possibility of obtaining fits for the ejecta mass
(and other quantities) which are independent of the compactness of
the stars and solely depend on the mass and tidal deformability,
i.e.~on quantities directly accessible by a GW observation without assuming an EOS.
One possibility might be the usage of quasi-universal compactness-Love
relations as mentioned in~\cite{Yagi:2016bkt} to substitute the
compactness in \eqref{eq:Mej_fit}, also the baryonic mass
could be represented by the gravitational mass with introducing
deviations to the NR only slightly larger than those of the current
fits~\footnote{We thank Nathan~K.~Johnson-McDaniel for pointing this out.}.
We are not following this approach here, since it did not allowed a better
representation of the NR data and we tend to stay closer to the work previously
presented for BHNSs systems.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\textwidth]{Fits.pdf}
\caption{From left to right: ejecta mass $M_{\rm ej}$, kinetic energy of the ejecta $T_{\rm ej}$, and
velocity of the ejecta $v_{\rm ej}$.
The top panels show the NR data and the results obtained by our phenomenological fits.
The bottom panels show the absolute difference between the fit and the NR data, as shaded regions
we also include the 1-$\sigma$ and 2-$\sigma$ confidence interval.}
\label{fig:Mej}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\textwidth]{Fit_Mej.pdf}
\caption{Difference between the ejecta mass of the NR simulation and the proposed fit.
Top panels show the absolute difference
$\Delta M_{\rm ej} = M_{\rm ej}^{\rm NR} - M_{\rm ej}^{\rm fit}$
between the fit and the NR data and
bottom panels the relative difference
$2 \Delta M_{\rm ej}/(M_{\rm ej}^{\rm NR} + M_{\rm ej}^{\rm fit}$ . }
\label{fig:fitMej}
\end{center}
\end{figure}
\subsection{Kinetic energy}
To estimate the kinetic energy of the ejecta we use a similar approach as for the
unbound mass, i.e.,
\begin{equation}
\frac{T_{\rm ej}^{\rm fit} } { 10^{50} {\rm erg}} = \left[ a \left(\frac{M_2}{M_1}\right)^{1/3} \left(\frac{1-2 C_1}{C_1}\right)+
b \left(\frac{M_2}{M_1}\right)^{n} + c \left(1 - \frac{M_1}{M^*_{1}}\right)
\right] M_1^* + (1\leftrightarrow 2) + d . \label{eq:Tej_fit}
\end{equation}
The fitting parameters for the kinetic energy are:
\begin{equation}
a = -1.94315, \quad b = 14.9847,\quad c = -82.0025, \quad d = 4.75062, \quad n = -0.87914. \label{eq:par:Tej}
\end{equation}
The average residual between our fit and the pure NR data is
$\Delta \bar{T}_{\rm ej} = 1.74\times10^{50}$erg,
which corresponds to a difference of 79\%.
Thus, the kinetic energy is slightly worse
represented by our fit than the ejecta mass.
The middle panels of figure~\ref{fig:Mej} represent our
results for the kinetic energy, where again the
1$\sigma$ and $2\sigma$ intervals are included
($\Delta T_{\rm ej}^{1\sigma} = 2.4 \times 10^{50}$erg ).
\subsection{Ejecta velocities}
For the velocity we simplify our fitting function and restrict our analysis to the first
66 data points in table~\ref{tab:overview}. For these data points the velocities inside the orbital plane
and perpendicular to it are given. For BHNSs it is known that
the velocity depends linearly on the mass ratio of the system, see~\cite{Kawaguchi:2016ana}.
It was shown in~\cite{Dietrich:2016hky} that the same functional dependence holds for BNSs with
high mass ratio or systems employing a stiff EOS. However, shock produced ejecta have a
higher velocity component orthogonal to the orbital plane
and should be included for a reliable estimate.
Thus, we introduce an EOS dependent fitting function
by including a first order polynomial depending on the compactness
$(1+ c \ C_{1,2})$, which leads to
\begin{align}
v_\rho & = \left[ a \left(\frac{M_1}{M_2}\right) \left(1 + c\ C_1\right) \right] + (1\leftrightarrow 2) + b . \label{eq:vrho_fit}
\end{align}
The parameters are:
\begin{equation}
a = -0.219479, \quad b = 0.444836, \quad c = -2.67385. \label{eq:par:vrho}
\end{equation}
Employing these parameters the NR data are represented with an average
error of $\Delta \bar{v}_\rho = 0.020$, which corresponds to a
percentile difference of $13\%$.
The same expression is used for the velocity orthogonal to the orbital plane:
\begin{align}
v_z & = \left[ a \left(\frac{M_1}{M_2}\right) \left(1 + c\ C_1\right) \right] + (1\leftrightarrow 2) + b . \label{eq:vz_fit}
\end{align}
As discussed, e.g.,~\cite{Hotokezaka:2012ze},
torque produced ejecta have much smaller velocities perpendicular
to the orbital plane than inside the orbital plane.
Thus, mostly shock driven ejecta cause large velocities orthogonal to the orbital plane.
The parameters we obtain for $v_z$ are:
\begin{equation}
a = -0.315585, \quad b = 0.63808, \quad c = -1.00757\label{eq:par:vz}
\end{equation}
with average residuals of $\Delta v_z = 0.013$ and a fractional difference of $33\%$.
The fractional difference is larger than for $v_\rho$ since the
absolute value of the velocities is smaller.
From $v_{\rho}$ and $v_{z}$ we estimate the total ejecta velocity as
\begin{equation}
v_{\rm ej} = \sqrt{v_\rho^2+v_z^2}. \label{eq:vej}
\end{equation}
To check our description of $v_{\rm ej}$ we compare all data points
(including the remaining 105 data points for which only
the total ejecta velocity $v_{\rm ej}$ is known) to our fits.
In total we obtain average residuals of
$\Delta \bar{v}_{\rm ej} = 0.036$
and an average percentile uncertainty of 15\%.
Figure~\ref{fig:Mej} (right panels) shows the ejecta velocities.
We find that the residuals are smaller for the 66 data points which we used to obtain
the fits of $v_\rho,v_z$ than for the remaining 105 data points.
Overall one sees that the phenomenological fit slightly underestimates the velocity.
\subsection{Other quantities}
\subsubsection{Geometry:}
\label{sec:geometry}
The geometry of the ejecta can be extracted from NR simulations
by considering 3D volume data of the density,
but those data are not accessible for most of the
configurations presented in table~\ref{tab:overview}.
Thus, we want to present in the following a model for homogeneously
distributed material inside an annular sector moving with the velocity $v_{\rm ej}$.
Inside the $\rho-z$-plane the ejecta is distributed in a circular sector
with a polar opening angle $2 \theta_{\rm ej}$.
The ejected material has an azimuthal opening angle of $\phi_{\rm ej}$.
Under the assumption that the ejecta consists of
particles moving radially outward with velocity $v_{\rm ej}$, we obtain by averaging over
all particles the following equations for $v_\rho$ and $v_z$:
\begin{equation}
v_\rho \approx v_{\rm ej} \frac{\sin{(\theta_{\rm ej})}}{\theta_{\rm ej}} , \quad v_z \approx v_{\rm ej} \frac{1-\cos{(\theta_{\rm ej})}}{\theta_{\rm ej}}. \label{eq:theta1}
\end{equation}
For a non-zero, but small $\theta_{\rm ej}$ one gets
\begin{equation}
\frac {\theta_{\rm ej}^3}{24} + \frac{\theta_{\rm ej}}{2} - \frac{v_z}{v_\rho} \approx 0,
\end{equation}
which can be solved for $\theta_{\rm ej}$:
\begin{equation}
\theta_{\rm ej} \approx \frac{ - 2^{4/3} v_\rho^2 + 2^{2/3}
(v_\rho^2 ( 3 v_z + \sqrt{ 9 v_z^2 + 4 v_\rho^2}))^{2/3} }
{(v_\rho^5(3 v_z + \sqrt{9 v_z^2 + 4v_\rho^2}))^{1/3}}. \label{eq:theta_fit}
\end{equation}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.95\textwidth]{geometry_final+.png}
\caption{2D density plots with rest mass $\rho$ shown from blue to
red with increasing density and
the unbound material $\rho_u$ shown brown to green with increasing density.
Geometric units are employed.
We use the velocity as extracted from the numerical simulation and show
$\theta_{\rm ej}$ and $\phi_{\rm ej}$ as approximated from \eqref{eq:theta_fit} and
\eqref{eq:phi_fit}.
Left: Simulations \#66
(SLy,1.375$M_\odot$,1.375$M_\odot$)
Right: Simulation \#55 (MS1b,1.000$M_\odot$,1.750$M_\odot$). }
\label{fig:geometry}
\end{center}
\end{figure}
In contrast to the opening angle $\theta_{\rm ej}$, it is more difficult
from our current results to estimate the azimuthal
angle $\phi_{\rm ej}$. In~\cite{Kawaguchi:2016ana} was assumed that
BHNS setups have an azimuthal angle of $\phi_{\rm ej} \approx \pi$.
This is in agreement with high mass ratio BNS mergers employing stiff EOSs~\cite{Dietrich:2016hky},
i.e.~for setups where torque is the dominant ejection mechanism.
Contrary if shock ejecta are present, e.g.~for softer EOSs,
the azimuthal angle even increases up to $2 \pi$,
i.e.~there exists a correlation between $\theta_{\rm ej}$ and $\phi_{\rm ej}$.
Assuming that the opening angles vary between
$\theta_{\rm ej} \in [\pi/8,3 \pi/8]$ and
$\phi_{\rm ej} \in [\pi,2\pi]$, and that $\theta_{\rm ej}$ and $\phi_{\rm ej}$ are linearly correlated, we obtain
\begin{equation}
\phi_{\rm ej} = 4 \theta_{\rm ej} + \frac{\pi}{2}. \label{eq:phi_fit}
\end{equation}
To test our approximations, we present snapshots of the density profile
in the $x$-$y$ and $x$-$z$ plane for the simulations \#55 and \#66 in figure~\ref{fig:geometry}.
We show the rest-mass density $\rho$ (color bar ranging from blue to red)
and the unbound rest mass density $\rho_u$ (color bar ranging from brown to green).
The two cases present two rather extreme setups, namely a stiff EOS
with a large mass ratio and a soft EOS for an equal mass system.
In figure~\ref{fig:geometry} we also include the approximations for $\theta_{\rm ej}$
and $\phi_{\rm ej}$ obtained from \eqref{eq:theta_fit} and \eqref{eq:phi_fit}.
The examples show that the geometry of the higher density ejecta regions can be
described reasonably well with our model.
\subsubsection{Composition:}
\label{sec:composition}
Caused by different ejecta mechanisms the composition and electron fraction of the ejecta
varies depending on the EOS, mass ratio, and total mass.
As pointed out in the literature, unbound material ejected due to torque in the tidal tail of the NSs has
a low electron fraction, see e.g.,~\cite{Rosswog:2015nja}.
Contrary ejecta produced via shock heating have overall
a broader range in electron fraction, e.g.,~\cite{Sekiguchi:2016bjd}.
Table~\ref{tab:Ye} shows the fraction of data from table~\ref{tab:overview} for which we also know the
average electron fraction.
Note that the electron fraction of the ejected material varies significantly
among different implementations for the neutrino transport,
e.g.,~\cite{Radice:2016dwd,Palenzuela:2015dqa,Lehner:2016lxy}
find overall smaller electron fractions of the unbound material
than reported in~\cite{Sekiguchi:2016bjd}.
Consequently the presented results have to be taken with care and
the following should be regarded as a qualitative
discussion.
\begin{table}[t]
\caption{ \label{tab:Ye} Columns refer to:
The data ID as in table~\ref{tab:overview},
the mass of the first star $M_1$,
the mass of the second star $M_2$,
the ejecta mass $M_{\rm ej}$,
the kinetic energy of ejecta $T_{\rm ej}$,
the ejecta velocity $v_{\rm ej}$, and
the electron fraction $Y_e$.
All setups have been simulated in~\cite{Sekiguchi:2016bjd}.}
\begin{center}
\begin{small}
\begin{tabular}{l|ccc|cccc}
$\#$ & EOS & $M_1$ & $M_2$ & $M_{\rm ej}$ & $T_{\rm ej}$ & $v_{\rm ej}$ & $Y_e$\\
& &$[M_\odot]$&$[M_\odot$]& $[10^{-3}M_\odot]$ & $[10^{50}$erg]& $[c]$ & \\
\hline
85 & DD2 & 1.25 & 1.45 & 5 & 1.61 & 0.19 & 0.2 \\
88 & DD2 & 1.3 & 1.4 & 3 & 0.87 & 0.18 & 0.26 \\
98 & DD2 & 1.35 & 1.35 & 2 & 0.46 & 0.16 & 0.3 \\
145 & SFHo & 1.25 & 1.45 & 11 & 5.66 & 0.24 & 0.18 \\
148 & SFHo & 1.3 & 1.4 & 6 & 2.15 & 0.2 & 0.27 \\
149 & SFHo & 1.33 & 1.37 & 9 & 3.55 & 0.21 & 0.3 \\
158 & SFHo & 1.35 & 1.35 & 11 & 4.76 & 0.22 & 0.31 \\
\end{tabular}
\end{small}
\end{center}
\end{table}
Figure~\ref{fig:Ye} summarized the important results from table~\ref{tab:Ye}.
As shown in figure~\ref{fig:EOS} the DD2 EOS is softer than SFHo.
Considering the left panel of figure~\ref{fig:Ye} we observe that for both EOSs
an increasing mass ratio leads to a smaller electron fraction.
This is expected since more ejecta are produced due to torque independent of the EOS.
The right panel shows the dependence between the ejecta mass and the electron fraction.
For all setups more massive ejecta are produced for the softer EOS, e.g., for $q=1$ more
than five times more mass is ejected for the SFHo EOS.
For this mass ratio the dominant ejection mechanism for SFHo
is shock heating, which seems to be suppressed for increasing
mass ratios. Thus, the ejecta mass and the electron fraction decreases
for increasing $q$ (see also the explanation in~\cite{Sekiguchi:2016bjd}).
Interestingly is that while for DD2 $Y_e(M_{\rm ej})$ is monotonic, this is not true for
SFHo, where beyond a mass ratio of $q\approx1.1$ the ejecta mass is growing again.
We propose that for $q> 1.1$ also SFHo setups become dominated by torque produced ejecta
and shocks are suppressed.
Finalizing our consideration of the composition,
we want to present a fit for the electron fraction as a function of the
mass ratio for a total mass of $M=2.7M_\odot$ for the data of~\cite{Sekiguchi:2016bjd}:
\begin{equation}
Y_{e} = 0.306 - 0.318 (q-1) - 2.568 (q-1)^2 \label{eq:fit:Ye}.
\end{equation}
The fit is shown as a black dashed line in figure~\ref{fig:Ye} (left panel).
To generalize~\eqref{eq:fit:Ye} to different
total masses and higher mass ratios more simulations
including realistic microphysical treatments are required.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\textwidth]{Ye.pdf}
\caption{Left panel: Electron fraction $Y_e$ as a function of the mass ratio $q$.
Right panel: Electron fraction $Y_e$ as a function of the ejecta mass $M_{\rm ej}$.
We present data for two different EOSs: SFHo (blue dashed dotted line ) and
the stiffer DD2 (red solid line).
In the left panel we also include as a black dashed line the fit of \eqref{eq:fit:Ye}.}
\label{fig:Ye}
\end{center}
\end{figure}
\subsubsection{Spin effects:}
\label{sec:spin}
Let us also briefly comment on the effect of the star's intrinsic rotation on the
ejecta quantities. We summarize in tab.~\ref{tab:Spin} the spinning configurations
of~\cite{Dietrich:2016lyp}.
Figure~\ref{fig:Spin} visualizes these data and shows the influence of the mass ratio
and of the spin of the secondary (less massive star)
on the ejecta mass.
The figure shows two distinct effects
(i) for an increasing mass ratio more material becomes unbound (as already discussed above),
(ii) if the spin of the secondary star is aligned to the orbital angular momentum
(positive) then the ejecta mass increases even further.
As pointed out in~\cite{Dietrich:2016lyp}
spin aligned to the orbital angular momentum enhances the ejection,
while contrary antialigned spin leads to lower massive ejecta.
This can be understood by considering the fluid velocity inside the tidal tail,
which at lowest order can be approximated as the sum of the orbital fluid velocity
and the fluid velocity connected to the intrinsic rotation of the star.
In cases where the individual star also has spin parallel to the orbital
angular momentum the fluid velocity inside the tail is higher and consequently material
gets unbound and leaves the system.
This effect becomes most prominent for systems for which material ejection is caused
by torque, e.g.~by unequal mass systems.
Because in unequal mass systems the mass ejection happens mostly from
the tidal tail of the lower massive star,
the determining quantity is the spin of the secondary star $\chi_2$
as shown in figure~\ref{fig:Spin}.
\begin{table}[t]
\caption{ \label{tab:Spin} Overview about the spinning simulations taken from~\cite{Dietrich:2016lyp}.
The columns refer to: EOS, individual masses $M_{1,2}$, dimensionless spins of the stars $\chi_{1,2}$,
the ejecta mass $M_{\rm ej}$, kinetic energy of the ejecta $T_{\rm ej}$,
velocity inside the orbital plane $v_\rho$ and perpendicular to it $v_z$.}
\begin{small}
\begin{center}
\begin{tabular}{ccccc|cccc}
EOS & $M_1$ & $\chi_1$ & $M_2$ & $\chi_2$ & $M_{\rm ej}$ & $T_{\rm ej}$ & $v_{\rho}$ & $v_{z}$ \\
& $[M_\odot]$ & & $[M_\odot$]& & $[10^{-3}M_\odot]$ & $[10^{50}$erg]& $[c]$ & $[c]$ \\
\hline
ALF2 & 1.375 & 0.102 & 1.375 & -0.102 & 4.1 & 0.55 & 0.12 & 0.07 \\
ALF2 & 1.375 & 0.102 & 1.375 & 0.000 & 2.0 & 0.36 & 0.13 & 0.05 \\
ALF2 & 1.375 & 0.102 & 1.375 & 0.102 & 1.6 & 0.32 & 0.16 & 0.05 \\
ALF2 & 1.528 & 0.104 & 1.223 & -0.102 & 4.5 & 1.7 & 0.15 & 0.11 \\
ALF2 & 1.528 & 0.104 & 1.222 & 0.000 & 5.5 & 2.1 & 0.16 & 0.13 \\
ALF2 & 1.528 & 0.104 & 1.223 & 0.102 & 6.7 & 2. & 0.16 & 0.08 \\
ALF2 & 1.651 & 0.107 & 1.100 & -0.101 & 11 & 3.6 & 0.18 & 0.05 \\
ALF2 & 1.651 & 0.107 & 1.100 & 0.000 & 14 & 4.1 & 0.18 & 0.04 \\
ALF2 & 1.651 & 0.107 & 1.100 & 0.101 & 24 & 7.5 & 0.18 & 0.04 \\
H4 & 1.375 & 0.100 & 1.375 & -0.100 & 1.5 & 0.62 & 0.16 & 0.10 \\
H4 & 1.375 & 0.100 & 1.375 & 0.000 & 0.7 & 0.23 & 0.17 & 0.10 \\
H4 & 1.375 & 0.100 & 1.375 & 0.100 & 2.0 & 0.78 & 0.15 & 0.07 \\
H4 & 1.528 & 0.100 & 1.223 & -0.100 & 4.1 & 1.7 & 0.17 & 0.09 \\
H4 & 1.528 & 0.100 & 1.222 & 0.000 & 6.4 & 3.2 & 0.18 & 0.08 \\
H4 & 1.528 & 0.100 & 1.223 & 0.100 & 7.8 & 3.0 & 0.18 & 0.11 \\
H4 & 1.651 & 0.101 & 1.100 & -0.099 & 9.5 & 2.4 & 0.17 & 0.03 \\
H4 & 1.651 & 0.101 & 1.100 & 0.000 & 19 & 5.5 & 0.17 & 0.03 \\
H4 & 1.651 & 0.101 & 1.100 & 0.099 & 27 & 7.5 & 0.17 & 0.02 \\
\end{tabular}
\end{center}
\end{small}
\end{table}
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\textwidth]{spin_2.pdf}
\caption{Ejecta mass for the spinning configurations of table~\ref{tab:Spin}
as a function of the mass ratio $q$ and the spin of
the secondary star $\chi_2$ for the ALF2 EOS (left)
and the H4 EOS (right).}
\label{fig:Spin}
\end{center}
\end{figure}
\section{Kilonovae}
\label{sec:kilonovae}
It is expected that the ejected material is heated up
because of the radioactive decay of r-process elements
and consequently triggers EM emission called kilo- or macronovae, see among
others~\cite{Li:1998bw,Metzger:2010sy,Roberts:2011xz,Goriely:2011vg,Tanvir:2013pia,Korobkin:2012uy,Grossman:2013lqa,Tanaka:2013ana,Tanaka:2016sbx,Rosswog:2016dhy}
and for overview articles~\cite{Fernandez:2015use,Metzger:2016pju}.
Up to date there are three possible kilonovae candidates
for which a connection to a GRB has been made:
GRB 050709~\cite{Jin:2016pnm}, GRB 060614~\cite{Yang:2015pha},
GRB 130603B~\cite{Tanvir:2013pia}.
The most likely origin of these kilonovae candidates
are compact binary mergers.
\subsection{Peak quantities}
\begin{figure}[t]
\includegraphics[width=\textwidth]{macro.png}
\caption{Kilonovae properties: upper panel shows the time when the
peak luminosity is reached; middle panels show the corresponding luminosity,
and the bottom panel the corresponding temperature.
We present results for four different EOSs, from left to right:
APR4, MPA1, MS1b, NL3, i.e., the compactness is from left to right decreasing,
see figure~\ref{fig:EOS}. The quantities are given in terms of the
individual masses of the stars $M_1,M_2$.}
\label{fig:macro}
\end{figure}
Based on the work of~\cite{Grossman:2013lqa} we will present some important kilonovae properties.
The time $t_{\rm peak}$ at which the peak in the near-infrared occurs,
the bolometric luminosity at this time $L_{\rm peak}$, and the corresponding temperature
$T_{\rm peak}$ are given as:
\numparts \begin{align}
t_{\rm peak} & = 4.9 \ {\rm days} \times \left( \frac{M_{ej}}{10^{-2} M_\odot} \right)^{\frac{1}{2}}
\left( \frac{\kappa}{10 {\rm cm^2 g^{-1}} } \right)^{\frac{1}{2}}
\left( \frac{v_{\rm ej}}{0.1} \right)^{-\frac{1}{2}} , \label{eq:tpeak} \\
L_{\rm peak}& = 2.5 \cdot 10^{40} {\rm erg \, s^{-1}} \times \left( \frac{M_{ej}}{10^{-2} M_\odot} \right)^{1-\frac{\alpha}{2}}
\left( \frac{\kappa}{10 {\rm cm^2 g^{-1}} } \right)^{-\frac{\alpha}{2}}
\left( \frac{v_{\rm ej}}{0.1} \right)^{\frac{\alpha}{2}} , \label{eq:Lpeak} \\
T_{\rm peak} & = 2200 {\rm K} \times \left( \frac{M_{ej}}{10^{-2} M_\odot} \right)^{-\frac{\alpha}{8}}
\left( \frac{\kappa}{10 {\rm cm^2 g^{-1}} } \right)^{-\frac{\alpha+2}{8}}
\left( \frac{v_{\rm ej}}{0.1} \right)^{\frac{\alpha-2}{8}} . \label{eq:Kpeak}
\end{align} \endnumparts
In \cite{Grossman:2013lqa} the authors assume that the energy release due to
the radioactive decay is proportional to $\sim t^{-\alpha}$ with
$\alpha=1.3$. We set the average opacity to $\kappa =
10~{\rm cm^2 g^{-1}}$~\footnote{Notice that
as shown in e.g.,~\cite{Tanaka:2013ana,Kasen:2013xka}
the typical opacity for a kilonovae is significantly higher than for
typical supernovae explosions, which is caused by the
presence of lanthanides.
The exact value of the opacity depends on the composition of the
material, which is not included in our models.}.
In figure~\ref{fig:macro} we present $t_{\rm peak},L_{\rm peak},T_{\rm peak}$
for four different EOSs as a function of the individual masses $M_1,M_2$.
We find for all setups that an increasing mass-ratio increases $t_{\rm peak}$, $L_{\rm peak}$ and
decreases $T_{\rm peak}$.
Furthermore an increasing total mass leads to a decreasing
$t_{\rm peak}$.
Considering the influence of the EOS, softer EOSs lead to more luminous kilonovae
in particular for equal mass merger. This can be explained by smaller ejecta mass
caused by the absence of shock driven ejecta for stiff EOSs. For systems close to
equal mass the temperature of the kilonovae is higher.
Interesting is also that for equal mass systems the luminosity and the temperature have
saddle points, see middle and lower panels.
This means that keeping the mass ratio fixed
a local extrema exist for which the luminosity becomes maximal
and that also a local extrema exists for
which the temperature becomes minimal. Both points do not have to coincide.
It would be interesting to test with further NR simulations whether such a saddle
point exists or is just an artifact of the employed fit.
\subsection{Time evolution}
\subsubsection{Luminosity:}
To determine the luminosity of the kilonovae, we follow the discussion
of~\cite{Kawaguchi:2016ana}, which we briefly summarize below.
As described in section~\ref{sec:geometry} the ejecta is modeled as a
partial sphere in the latitudinal and longitudinal direction.
We further assume that the material is homogeneously distributed inside the ejecta and that
photons purely escape from the latitudinal edge. This agrees with the assumptions
made in~\cite{Kawaguchi:2016ana} and also gives valid results for BNS mergers
as shown below.
Considering that the optical depth increases with decreasing density,
the whole region becomes visible after
\begin{equation}
t_c = \sqrt{\frac{\theta_{\rm ej} \kappa M_{\rm ej}}{2 \phi_{\rm ej} (v_{\rm max}-v_{\rm min})}},
\end{equation}
with $v_{\rm max}$, $v_{\rm min}$ being the maximum
and the minimum speed of the ejecta.
The mass of the photon escaping region is then given by
$M_{\rm obs} = M_{\rm ej} (t/t_c)$ for times $t<t_c$.
In \cite{Korobkin:2012uy,Wanajo:2014wha}
was shown that the specific heating for energy release caused by
radioactive decay can be approximated by
$
\dot{\epsilon} \approx \dot{\epsilon}_0 \left(\frac{t}{\rm 1 day}\right)^{-\alpha}.
$
This allows to write the bolometric luminosity as
\begin{equation}
L(t) = (1 + \theta_{\rm ej}) \epsilon_{\rm th} \dot{\epsilon}_0 M_{\rm ej}
\begin{cases}
\frac{t}{t_c} \left(\frac{t}{1 \ {\rm day}}\right)^{-\alpha}, \quad t \leq t_c \\
\left(\frac{t}{1\ \rm day}\right)^{-\alpha}, \quad t > t_c \\
\end{cases}, \label{eq:Lbol}
\end{equation}
where we will use $\dot{\epsilon}_0=1.58 \times 10^{10} {\rm erg \ g^{-1}\ s^{-1}}$ and $\alpha=1.3$
for our considerations~\footnote{Note that as discussed in~\cite{Kawaguchi:2016ana} \eqref{eq:Lbol}
also used the assumption of a small opening angle $\theta_{\rm ej}$ which is valid for BHNSs
but might be violated for BNS systems. However, figure~\ref{fig:Lbol}
reveals that reasonable results are also obtained for BNS systems with larger opening angles,
see e.g., SLy (1.35,1.35).}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\textwidth]{Lbol.pdf}
\caption{Comparison of the bolometric luminosity given by \eqref{eq:Lbol} (dashed lines)
and a radiative transfer simulation (solid lines). The results of the radiative transfer simulation
was presented in~\cite{Tanaka:2013ana,Tanaka:2013ixa} and is public available at~\cite{Tanaka_web}.
The legend characterizes the EOS and the individual masses of the NSs are given in solar masses.}
\label{fig:Lbol}
\end{center}
\end{figure}
In figure~\ref{fig:Lbol} a comparison between \eqref{eq:Lbol}
and the radiative transfer simulations of~\cite{Tanaka:2013ana,Tanaka_web}
is presented. One sees remarkable agreement between the simple
model function and the radiative transfer simulations.
As input variables for \eqref{eq:Lbol},
we have used the stated ejecta masses from~\cite{Tanaka_web}.
This is necessary since $L_{\rm bol}$ depends strongly on $M_{\rm ej}$
such that a difference in $M_{\rm ej}$ produces a large difference in
$L_{\rm bol}$ and a comparison would not test the assumptions made for \eqref{eq:Lbol},
but how \eqref{eq:Mej_fit}
approximates this particular setup.
Furthermore, $v_{\rm min}$ is set to 0.02,
$v_{\rm max} = 2 v_{\rm ej} - v_{\rm min}$,
and $\theta_{\rm ej}$ and $\phi_{\rm ej}$ are
chosen according to \eqref{eq:theta_fit} and \eqref{eq:phi_fit}.
Figure~\ref{fig:Lbol} proves that \eqref{eq:Lbol}, which was
originally proposed for BHNS setups in~\cite{Kawaguchi:2016ana}
also allows to describe BNS mergers and the time evolution of the kilonovae.
\subsubsection{Lightcurves:}
From the given luminosity the bolometric magnitude can be computed according to:
\begin{equation}
M_{\rm bol} \approx 4.74 -2.5 \log_{10} \left( \frac{L_{\rm bol}}{L_\odot}\right), \label{eq:Mbol}
\end{equation}
with $L_\odot$ denoting the bolometric luminosity of the sun.
To compute the magnitude in each wavelength,
we have to know the spectra of the kilonovae.
One possible approach to compute the spectra is by considering
the effective temperature of the photosphere
\begin{equation}
T \approx \left( \frac{L(t)}{\sigma S(t)}\right)^{1/4},
\end{equation}
with $S(t)$ being the surface of the latitudal edge, and to assume
that the spectrum of a kilonovae can be approximated by
a pseudo black body spectrum, e.g.,~\cite{Kasen:2013xka}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\textwidth]{Mx.pdf}
\caption{Bolometric corrections for the ugriz-bands (left) and
KHJ-bands (right) as a function of the rescaled time
$t'=t [{\rm days}](0.01M_\odot/M_{\rm})^{1/3.2}$.
We use public available results of~\cite{Tanaka_web}
and show them as dashed and dot-dashed lines.
The average of the available data for each individual band is shown
as a black solid line and a fit of the average is visible as a
red solid line. The parameters for the fit are given in
\eqref{eq:BC_z}-\eqref{eq:BC_J}.}
\label{fig:BC}
\end{center}
\end{figure}
Another approach enabling us to compute the spectrum
are bolometric corrections (BC) as discussed in~\cite{Kawaguchi:2016ana}.
The final magnitude in each band (denoted by the subscript $X$) is then given by
\begin{equation}
M_X(t) = M_{\rm bol}(L(t)) - BC_X(t).
\end{equation}
To compute the bolometric corrections we use the
public available light curves of~\cite{Tanaka_web}.
It was shown in~\cite{Kawaguchi:2016ana}
that the time evolution of the BCs for BHNSs agrees once
the elapsed time is rescaled by
$t'= t \cdot (10^{-2} M_\odot/M_{\rm ej})^{1/3.2}$.
Figure~\ref{fig:BC} shows that the same rescaling
can be used for BNS data.
We present for five different setups~\cite{Tanaka_web} the BCs for the ugriz-band
in the left and for the KHJ-band in the right panel.
The difference among the different setups of the BC is about 1 magnitude.
To obtain the final BC, we average the results of all five configurations
(black solid line) and fit
the average with a polynomial (red solid lines)
\begin{equation}
BC_X = a_0 + a_1 t' + a_2 t'^2 + a_3 t'^3 + a_4 t'^4 .
\end{equation}
The final parameters for the polynomials fits are
\numparts \begin{eqnarray}
BC_z: & (1.072,0.3646,-0.1032,0.00368,0.0000126) & t' \in [2,15] \label{eq:BC_z}\\
BC_i: & (0.6441,0.0796,-0.122,0.00793,-0.000122) & t' \in [2,15] \\
BC_r: & (-2.308,1.445,-0.5740,0.0531,-0.00152) & t' \in [2,15] \\
BC_g: & (-6.195,4.054,-1.754,0.2246,-0.009813) & t' \in [2,8.5] \\
BC_u: & (40.01,-56.79, 25.73,-5.207,0.3813) & t' \in [2,5] \\
BC_K: & (-7.876,3.245, -0.3946,0.0216,-0.000443) & t' \in [2,15] \\
BC_H: & (-2.763,1.502,-0.2133,0.0128,-0.000288) & t' \in [2,15] \\
BC_J: & (-1.038,1.348,-0.2364,0.0137,-0.000261) & t' \in [2,15] \label{eq:BC_J}
\end{eqnarray} \endnumparts
As an example we compare the lightcurves obtained from the discussed model and
computed with the radiative MC code of~\cite{Tanaka:2013ana,Tanaka_web}
for two systems: one equal mass system employing a soft EOS (SLy $(1.35M_\odot,1.35M_\odot$) )
and one unequal masses case with a stiffer EOS (H4 $(1.20M_\odot,1.50M_\odot)$ ).
As for figure~\ref{fig:Lbol} we use here the ejecta mass stated in~\cite{Tanaka_web} to
compute the bolometric luminosities.
Figure~\ref{fig:lightcurve} shows that after applying the BCs,
the MC results and those obtained by the simple model agree well.
Additionally, we also include lightcurves computed with the public available code
of~\cite{Kawaguchi_web} (thin dot dashed lines), which was developed for BHNS mergers
and which shows a larger disagreement to the MC results.
The difference between the MC simulation and the model presented here is smaller
because of the particular choice of the BCs.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\textwidth]{M_bands.pdf}
\caption{Absolute Magnitudes in the ugridz-bands (left panels) and JHK-bands (right panels)
for the equal mass SLy (1.35,1.35) and the unequal mass H4 (1.20,1.50) setups.
The solid lines represent the data reported in~\cite{Tanaka:2013ana,Tanaka_web}.
The dashed lines represent data obtained from \eqref{eq:Lbol} including
the computed BC corrections.
We also include as a thin dashed dotted line results obtained with the
public available code of~\cite{Kawaguchi_web}.}
\label{fig:lightcurve}
\end{center}
\end{figure}
\section{Radio flares}
\label{sec:radio}
\begin{figure}[t]
\includegraphics[width=\textwidth]{radio.png}
\caption{Radio flares properties: upper panel shows the time once the
peak in the radio band is observable after the merger of the two neutron stars;
lower panel shows the radio fluency at this time.
We present results for four different EOSs, from left to right:
APR4, MPA1, MS1b, NL3, i.e., the compactness is from left to right decreasing,
see figure~\ref{fig:EOS}. The quantities are given in terms of the
individual masses of the stars $M_1,M_2$.}
\label{fig:radio}
\end{figure}
In addition to kilonovae, it is possible that
sub-relativistic outflows produce radio flares
with peak times of a few month up to years
after the merger of the compact binary.
In order to estimate the radio emission,
we use the model of~\cite{Nakar:2011cw}.
The strongest signal is expected at a time
\begin{align}
t_{\rm peak}^{\rm rad} & = 1392 \ {\rm days} \times \left( \frac{T_{\rm ej} }{10^{49}{\rm erg}} \right)^{\frac{1}{3}}
\left( \frac{n_0}{\rm cm^{-3}}\right)^{-\frac{1}{3}}
\left( \frac{v_{\rm ej}}{0.1} \right)^{-\frac{5}{3}}
\end{align}
after the merger of the system.
The radio fluence at this time is
\begin{align}
{F^\nu}^{\rm rad}_{\rm peak} & = 0.3 \ {\rm mJy} \times
\left( \frac{T_{\rm ej} }{10^{49}{\rm erg}} \right)
\left( \frac{n_0}{\rm cm^{-3}}\right)^{\frac{p+1}{4}}
\left( \frac{\epsilon_B}{\rm 0.1}\right)^{\frac{p+1}{4}} \nonumber \\
& \times \left( \frac{\epsilon_e}{\rm 0.1}\right)^{p-1}
\left( \frac{v_{\rm ej}}{1} \right)^{\frac{5p-7}{2}}
\left( \frac{D}{10^{27} {\rm cm}} \right)^{-2} \left( \frac{\nu_{\rm obs}}{1.4 {\rm GHz}} \right)^{-\frac{p-1}{2}}
\end{align}
for an observation frequency $\nu_{\rm obs}$ higher than the
self-absorption and synchrotron peak frequency at a distance $D$.
The parameters $\epsilon_B$ and $\epsilon_e$, both set to $0.1$,
determine how efficient the energy of the blast wave is transfered to the magnetic field and to
electrons.
$n_0$ denotes the surrounding particle density
and is set to
$0.1 {\rm cm^{-3}}$~\footnote{Notice that the overall uncertainty
on the density of the surrounding material is rather large.
To constrain the EOSs or extract the binary parameters
from radio observations strict bounds on $n_0$ will be needed.}.
Additionally we assume $p=2.3$ and $\nu_{\rm obs}=1.4 {\rm GHz}$,
as done in~\cite{Nakar:2011cw}.
In figure~\ref{fig:radio} we present for four different EOSs
the expected peak time $t_{\rm peak}$ (upper panel) and radio fluence
${F^\nu}^{\rm rad}_{\rm peak}$ (lower panel).
We find that for an increasing total mass the peak time
$t_{\rm peak}^{\rm rad}$ decreases while the
peak fluency ${F^\nu}^{\rm rad}_{\rm peak}$ increases.
For larger mass ratios the peak fluency is largest.
Considering different EOSs we find significant differences.
In general the observable peak time in the radio band,
i.e.~$t_{\rm peak}^{\rm rad}$, happens later for
softer EOSs, for those setups also the peak fluency is higher.
\section{Conclusion}
\label{sec:conclusion}
\subsection{Summary}
\label{sec:summary}
In this work we have derived fitting functions for the main
ejecta properties from binary neutron star mergers,
namely the mass, kinetic energy, and velocity of the unbound material.
Our work is (as a first step) restricted to dynamical
ejecta for which a large number of numerical simulation data are available.
In total we use a sample of 172 numerical simulations
of binary neutron star mergers to derive our fits.
The high number of data points
allows to cover a large region of the possible
binary neutron star parameter
space including 23 different EOSs, total masses between $2.4 M_\odot$ and $4 M_\odot$,
and mass ratios between $q=1.0$ and $q\approx2.1$.
The residual errors of the fitting functions are of the order
of the uncertainty of the numerical relativity results.
Additionally, we presented estimates for the geometry of the ejected material
and compared those with numerical relativity simulations.
We found that the high density region of the ejected material can be
approximated by a three dimensional annular sector, i.e.~a crescent-like structure.
Using the results of~\cite{Sekiguchi:2016bjd} we also discussed the influence
of the EOS and mass ratio on the electron fraction inside the ejected material,
where in general softer and higher mass ratio configurations
are characterized by lower electron fractions.
Following~\cite{Dietrich:2016lyp} we presented
how the intrinsic rotation on the individual neutron
stars affects the ejecta mass, where we found in particular that for high mass ratios
the aligned spin of the lower star increases the amount of the ejected material.
Based on estimated ejecta properties we studied possible electromagnetic observables
for binary neutron star mergers. In particular, we have focused on the possibility
of the formation of kilonovae and radio flares.
Considering kilonovae, analytical models have been employed to determine the
time when the kilonovae is brightest as well as the corresponding luminosity and
temperature. While these estimates just represent the properties of the EM counterpart
at a fixed time, we also used the model proposed in~\cite{Kawaguchi:2016ana}
to derive the time evolution of the luminosity and light curve.
We checked the model against radiative transfer simulations of~\cite{Tanaka_web}
and found good agreement.
Finally, we estimated the peak time and peak fluency of the radio flares produced after
the binary neutron star merger.
Those flares will be observable month up to years after the merger.
\subsection{Consequences for future observations}
\label{sec:consequences}
The first two GW detections GW150914 and GW151226 have proven
that pipelines for EM follow studies are in place and work reliably.
Detailed informations can be found in~\cite{Abbott:2016gcq} and references therein.
However, in case of an upcoming GW detection of a BNS system an estimate
about corresponding kilonovae and radio flares may support follow up studies.
Once a GW is detected the first parameter estimates for the binary properties are produced
within the first minutes after the detection.
This time is small enough to allow observations in the
visible, near-infrared, and radio band.
On a practical term it is important to point out that the time between
the GW detection and the kilonovae observation is too short to perform full NR simulations,
which typically have run times of the order of weeks to months.
Thus, once the first knowledge about the properties of the binary is available
phenomenological formulas, as presented here, are needed to obtain estimates for possible EM counterparts.
After the kilonovae observation NR simulations with microphysical descriptions as
neutrinos transport, tabulated EOS, and magnetic fields can be performed
to obtain more reliable results. At this stage,
our estimates help to reduce the region in the parameter
space which have to be covered by NR simulations.
Notice that the situation is different for radio flares, which are detectable
on the order of years after the merger. Full-NR simulations
for a variety of parameters can be performed between the detection of the GWs and
the observation of the radio signal. \\
Overall, our work represents a first step towards a systematic combination between binary parameters
accessible from gravitational wave observations and electromagnetic counterparts for a large
range of the binary neutron star parameter space. In the future even more setups have to be included
testing extreme corners of the parameter space. Furthermore,
a detailed microphysical description in numerical simulations will help
to account for other effects as e.g.,~magnetic fields and the ejecta
produced by the disk wind after the formation of the merger remnant.
\ack
We thank Sebastiano Bernuzzi, Brett Deaton, Francois Foucart, Kyohei Kawaguchi,
Nathan~K.~Johnson-McDaniel, David Radice, Masaomi Tanaka
for comments and fruitful discussions.
It is a pleasure to also thank Matthias Hempel who kindly gave us the
EOS tables for cold neutron stars in beta-equilibrium.
We are grateful to Masaomi Tanaka for making his
Monte Carlo simulation data public available and to
Kyohei Kawaguchi for making his code to compute lightcurves for
BNS systems available.
Parts of the presented results relied on simulations
performed on SuperMUC at the LRZ (Munich) under
the project number pr48pu, Jureca (J\"ulich)
under the project number HPO21, Stampede
(Texas, XSEDE allocation - TG-PHY140019).
\section*{References}
\bibliographystyle{iopart-num}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,100
|
{"url":"https:\/\/www.homebuiltairplanes.com\/forums\/threads\/what-should-i-build.19624\/page-2","text":"# What Should I Build?\n\n### Help Support HomeBuiltAirplanes.com:\n\n#### cluttonfred\n\n##### Well-Known Member\nHBA Supporter\nI am going to go out on a limb and suggest a plane that meets your criteria including mild aerobatic capability except only the wings are all aluminum, the fuselage and tail are fabric-covered 4130 tubing: John Monnett's Sonerai II in the low-wing, stretch model with your choice of conventional or tricycle gear. It's still a great design in terms of bang for the buck, still supported by Steve Bennet at Great Plains and you could probably even contact Monnett himself if you had a question. And you could meet your performance goals with an economical 2180cc VW engine from Great Plains or AeroVee.\n\nS P E C I F I C A T I O N S\n\nSPAN 18' 8\"\nLENGTH 20' 4\"\nHEIGHT 5' 5\"\nENGINE 2180CC ONLY\nFUEL CAPACITY - STD. 10 GALLONS\nFUEL CAPACITY - OPT. 6 GALLONS\nEMPTY WEIGHT (No Starter) 540 LBS.\nGROSS WEIGHT 1150 LBS.\nWING AREA 84 SQ. FT.\nSEATS 2 TANDEM\n\nP E R F O R M A N C E\n\nDESIGN LIMIT AT FULL GROSS +\/- 4 G's\nAEROBATIC LIMIT SOLO W\/755 LBS. +\/- 6 G's\nTAKE OFF DISTANCE 900 FT.\nSTALL SPEED 50 MPH\nLANDING SPEED 60 MPH\nMANEUVERING SPEED 115 MPH\nCRUISING SPEED AT 75% 140 MPH\nVNE 200 MPH\nRANGE W\/45 MINUTE RESERVE 245 SM\nRATE OF CLIMB AT GROSS 700 FPM\n\n#### Direct C51\n\n##### Well-Known Member\nDo you think the Sonex is fast enough for your liking? They seem slow to me.\n\nThere are some pretty draggy designs which are faster, just by using a larger hp engine (without getting into engine choice). If you are going with a conversion, can you not bump up the power slightly without adding too much extra cost?\n\nSimilarly, composite materials, even fibreglass, are much more streamlined and faster as a result. Do you just love working with metal, or are you just a little cost-constrained?\nThe Sonex with 120HP cruises about 170KTAS at 75%, 150KTAS if you run 60% or so. That is the fastest all aluminum 2 seater on 120HP that I know of. Getting more power out of a conversion is not easy, cheap, or good for the engine. Aluminum and rivets fit my skills a lot better than composites. I also do not have a climate controlled workspace.\n\nI have somewhat similar requirments as you and am leaning towards an rv-7. The sonex seemed appealing at first but it's just too small. I figure with a basic panel and interior, used engine, and fixed pitch prop the cost difference between the two isn't huge. My fear with the sonex was growing out of it, possibly before even finishing it. The rv offers a lot more capability and room to upgrade as needs might change or the bank account starts getting too full.\n\n#### Georden\n\n##### Well-Known Member\nNot sure the prices on the Subaru or rotary options, I'd imagine all in with psru they aren't substantially cheaper than the lycoming, but overhaul would be much less.\n\nIt it takes a certain amount of power to carry a specific load with a given performance. The sonex achieves near rv\nperformance by sacrificing load carrying ability. I'm not trying to argue for or against the rv for your needs, just saying in my case the sonex was really tempting, but just couldn't really do what i wanted to do. I'd rather spend 60k on a basic rv that can do what I want to do than 45k on a sonex that can't. I'd really prefer to have a wittman tailwind, but didn't suggest it as it's not rated for aerobatics, and in my case I think I'd be a lot more likely to finish the build starting with a kit rather than a pile of raw materials and I don't feel comfortable buying someone half finished project which the only way to get a tailwind \"kit\"\n\n#### Battson\n\n##### Well-Known Member\nThe Sonex with 120HP cruises about 170KTAS at 75%, 150KTAS if you run 60% or so. That is the fastest all aluminum 2 seater on 120HP that I know of. Getting more power out of a conversion is not easy, cheap, or good for the engine. Aluminum and rivets fit my skills a lot better than composites. I also do not have a climate controlled workspace.\n170KTAS? Hahahaha... right, must be an Oops. The day one of those overtakes me (more than double the hp) I will eat my propeller covers. I have flown beside one, they aren't \"super fast\". I would say \"nippy\" and fun.\n\nBut it's not an RV6A, which I have also flown.\nThat was properly fast, with only 160hp, so same ballpark. We were doing 160-170KIAS at 500ft.\nPlus they can carry something. The Sonex is really quite small, you couldn't take much with you. Definitely an airplane you could out-grown in a months not years, if you fly 200hrs a year.\n\nI am not saying don't consider Sonex - hell, I would love one, they are cool. Just remember - you get what you pay for in aviation. Speed costs, payload costs, performance costs. There are no short-cuts or wonders of design, without associated cost - and the cost is always directly proportional to the performance.\n\nI thought I would fly 50hrs a year, it will probably be closer to 300 in reality. I would hate to have invested emotionally and physically in a plane which I grew out of too soon.\n\nYou can fly light in a slightly bigger aircraft.\nYou cannot fly heavy in a smaller aircraft.\n\nLast edited:\n\n#### cluttonfred\n\n##### Well-Known Member\nHBA Supporter\nThe William Wynne Corvair conversions in the 100-120 hp range are well-proven and solid. I wonder if the best way forward given your goals would be to find a part-built RV kit with good workmanship and power it with a Corvair?\n\n#### Battson\n\n##### Well-Known Member\nIf you're pioneering a conversion, I bet I never see it fly. This is based on decades of promises and fails. ( it's also a cheater bet. Like always betting a random horse will lose )\nI have seen this too.\nOne fellow I met, trialled one particular 8 cylinder auto engine with his own conversion, then swapped for another 8 cylinder engine after the first one failed, I think overheated?? I can't recall. Last I heard, his engine cuts out about 6 seconds after take-off due to some problem with the computer, then clicks back in a few seconds later.\nHe is a braver man than I.\nHe has also dead-stick landed more engine failures than anyone else I have met...\n\nThe worlds needs inventors.\n\n#### Direct C51\n\n##### Well-Known Member\nBattson - What do you mean by outgrow? I've been flying recreationally for 10 years and professionally for 8. I think I know what my needs are. I'm not sure how a few months of flying a Sonex is all of a sudden going to change that. This is really the silliest suggestion I've heard.\n\nGeordan - $45k for a Corvair Sonex is really high, and$60k for an RV-7 is somewhat low. Sure you can skew the numbers and make it sound better. A Sonex is more like $35k. I think putting a Corvair in to an RV goes against my entire mission. I am not looking for the roomiest airplane that flies on 120HP with the best payload. My mission is simply the fastest 2 seater on 120HP. I don't think an RV is going to perform well, or be very fast on 120HP. This whole exercise is basically asking which 2 seat aluminum airplane has the least drag, which means fastest speed, which really means most efficient. Of course weight comes in to play, more weight means bigger wings, more parasite drag, more induced drag... Last edited: #### cluttonfred ##### Well-Known Member HBA Supporter My mission is simply the fastest 2 seater on 120HP. Personally, I am not a RV fan, too expensive for me and they don't really tickle my imagination. But given your mission parameters including desired materials, engine and budget, I still think a second-hand RV kit is the way to go, though I'd probably revise my recommendation to an RV-4. There are other options out there that would also work if you are willing to scratch-build -- T18, Midget Mustang, etc. #### Direct C51 ##### Well-Known Member How does an RV-4 meet my mission? Seriously? Read what you quoted of me. An RV-4 is designed for 150HP - 180HP. It says right on their website. I just said my mission is the fastest 2 seater on 120HP and you think an RV-4 is faster than a Sonex on 120HP? #### rv6ejguy ##### Well-Known Member A Sonex would seem to be the way to go if you fit and have enough baggage space. BTW there are at about 30 times more Subarus flying than Corvairs- very successful if done right. How fast can a 120hp Sonex be? 201 KTAS with a turbocharged 912... My friend flew a Q2 with DD turbo Subaru for over 800 hours, 190 KTAS at 12,500 feet on 5.5 GPH. #### Direct C51 ##### Well-Known Member I find it ironic that the RV owner is one of the only ones not trying to sell me on an RV. I really like RVs and had intended to build one until reality hit and I realized that I just cannot resonably afford to build or fly one. The Subaru is a good auto conversion, but a little more complicated than I would like. Having a PSRU and cooling system adds too much complexity to a simple airplane like the Sonex in my opinion. The Corair is pretty similar to an air cooled flat airplane motor. That picture you posted is awesome, but I think had you shown the right side, the VSI might be a bit telling. Not to mention it is 33 MPH over design Vne. Impressive nevertheless. #### rv6ejguy ##### Well-Known Member I find it ironic that the RV owner is one of the only ones not trying to sell me on an RV. I really like RVs and had intended to build one until reality hit and I realized that I just cannot resonably afford to build or fly one. The Subaru is a good auto conversion, but a little more complicated than I would like. Having a PSRU and cooling system adds too much complexity to a simple airplane like the Sonex in my opinion. The Corair is pretty similar to an air cooled flat airplane motor. That picture you posted is awesome, but I think had you shown the right side, the VSI might be a bit telling. Not to mention it is 33 MPH over design Vne. Impressive nevertheless. I've looked at a Sonex myself and I read your post about costs. The aircraft is level in that screenshot: From the owner: 147KTS - 3000ft with only 30\". Normally I cruise at 32-35\" 5000rpm @ 8000-10.000ft with gives me 170KTS TAS I did warn him about the Vne...! The same guy had a turbo VW in the Sonex before, cruise was over 150 knots at altitude. He has also done EFI and EFI turbo Jabiru 2200s in Sonex airframes. Turbos are magic if you want speed up high. #### Victor Bravo ##### Well-Known Member HBA Supporter IMHO you have two valid choices, the Tailwind and the Vari-EZ. Both represent compromises in order to get most of your desired results. The professional pilot (airline) in the hangar next to me just bought a used Wittman Tailwind, which was very well built and properly set up. He just did several two-way runs with a good GPS to figure out how fast he is going and not going, because like all experienced pilots he does not trust the pitot-static system. On a relatively stock 125 HP engine, he is just under 200 MPH at full throttle. At cruise power, he is at about 180. A very clean Tailwind, but no wheel pants yet. When he puts the wheel pants on, these speeds will likely go up by 4-7 mph. An O-200 powered Tailwind would still be a 170 MPH airplane if you keep it light and pay attention to aerodynamic cleanup. There was a recent article within the last year in Sport Aviation about the Tailwind. For 50 years, it has been the hands-down winner by far in the \"best bang for the buck\" category. I realize that this airplane represents a significant change (construction materials) from your original airplane choice parameters. However, I have to agree with several others here that in order to meet your most important requirements, you may have to allow your thinking to be flexible in one or two areas. Scratchbuilding the Tailwind will require that you learn two different building skills (welding and wood). You will need to find a hangar to keep it in. It is not a hard acro airplane. In exchange for these three concessions you will get the following benefits: - The speed you want, from the engine size you want. - You will be at, or under, your target budget of$30K, a very rare feat in homebuilt aviation!\n- Two seats, and a surprising amount of room compared to most other small homebuilts.\n- A solid network of builders and mentors. Not as many as RV of course, but plenty.\n- A little more pride and unique-ness when you roll up to the flight line at an airshow.\n\nYou can also save significant time and money by getting a half-finished Tailwind project of course.\n\nA PROPERLY built and clean Vari-EZ will cruise at 180-190 MPH on 100 HP. With a 120HP Corvair (if it balances on the airplane) it should do another 10-12. You can scratchbuild a Vari-EZ, or buy a half-built one, for under $30K if you don't get carried away with avionics. Once again, you are going to compromise on the material, but you will achieve several benefits: - If built per plans, your loops and rolls and half-Cubans are no problem. - You will achieve very VERY high levels of cruise fuel efficiency and XC speed. - Your non-pilot passengers will feel like you gave them a ride in an F-104. - There is a group of educated owners, builders, and experts to mentor you. Both of these airplanes will be very very hard to match or beat with sheet aluminum and realistically stay within your budget\/engine size. #### autoreply ##### Well-Known Member Have a good look at my first suggestion in post 15. Doesn't have some of the caveats of the Sonex, but seems in the same price range. #### flyvulcan ##### Well-Known Member Log Member Perhaps the old Lancair 235 might work for you, if you can squeeze into it. It was designed around the O235 which is around the hp that your Corvair is. I don't know whether the Corvair would fit but you could look into it. The Lancair 235 fits the bill with \"higher\" speeds, probably more so than the Sonex and possibly on a par with the RVs. I've seen different speeds quoted for the Lancair 235 so speaking with owners who won't exaggerate would be required. There is a decent looking kit advertised for sale in Florida for$11900. You can see the ad at 1994 Lancair 235 Kit The Aircraft Pages.\n\nMaybe this might appeal to you.\n\nGood luck with your search and build.","date":"2021-05-16 11:51:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2534785866737366, \"perplexity\": 3879.8874035523418}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243991269.57\/warc\/CC-MAIN-20210516105746-20210516135746-00184.warc.gz\"}"}
| null | null |
Carroll County Crime
Union Bridge man charged with assaulting two people, resisting arrest
By Mary Grace Keller
Carroll County Times |
James J. Jones (Carroll County Sheriff's Office)
A Union Bridge man faces assault charges after he allegedly attacked two people and tangled with a police officer who tried to arrest him.
James J. Jones, 39, was charged with resisting arrest and three counts of second-degree assault, online court records show. He is being held without bond after a Monday bond review.
According to charging documents, Carroll County Sheriff's Office responded to a residence in Union Bridge at about 8:42 p.m. Jan. 3. A woman there told police that Jones grabbed her neck, put her in a chokehold, pushed her to the floor, climbed on top of her and squeezed her neck with his hands. Another person entered the bedroom and tried to intervene by hitting Jones with wrapping paper, according to charging documents.
Jones then grabbed the person by the back of the neck, shoved him into a bathroom and held him against the bathroom sink, police allege. The woman came into the bathroom, Jones let go of the male, then punched the woman's mouth multiple times, grabbed her by the back of the neck, and shoved her face into the sink, charging documents read. Police saw blood in the bathroom and cuts and dried blood on the woman's face, according to charging documents.
Jones removed the handcuffs placed on him by the deputy, who then wrapped their arms around Jones to contain him, police allege. Jones pulled away and both he and the deputy fell over the railing on the front porch and landed in the front yard, according to charging documents. Jones tried to pull away but was eventually arrested, charging documents state.
Once Jones was inside the police vehicle he kicked the door when the deputy tried to shut it, causing the door to hit the deputy's knee, according to charging documents. Jones was taken to central booking.
"He did not assault [the woman]," Jones' attorney Stephen Bourexis said in an interview Friday.
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Daily arrest report for those arrested in Carroll County Jan. 15, 2020
Jones has a court date set for March 4.
Most Read • Carroll County Crime
Top Maryland news
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,441
|
@interface FLTabBarView ()
@property (nonatomic, strong) FLTabBarMessageItemView *messageView;
@property (nonatomic, strong) FLTabBarContactsItemView *contactsView;
@property (nonatomic, strong) FLTabBarDynamicItemView *dynamicView;
@end
@implementation FLTabBarView
- (instancetype)initWithFrame:(CGRect)frame {
if (self = [super initWithFrame:frame]) {
[self setupItem];
}
return self;
}
- (void)setupItem {
self.messageView = [[FLTabBarMessageItemView alloc] initWithOrientation:FLSelected title:@"消息" type:FLTabBarMessage];
self.messageView.tag = 0;
[self addSubview:self.messageView];
[self.messageView mas_makeConstraints:^(MASConstraintMaker *make) {
make.left.bottom.top.equalTo(self);
// make.width.greaterThanOrEqualTo(0);
}];
UITapGestureRecognizer *tapA = [[UITapGestureRecognizer alloc] initWithTarget:self action:@selector(didTapView:)];
[self.messageView addGestureRecognizer:tapA];
self.contactsView = [[FLTabBarContactsItemView alloc] initWithOrientation:FLLeft title:@"联系人" type:FLTabBarContacts];
self.contactsView.tag = 1;
[self addSubview:self.contactsView];
[self.contactsView mas_makeConstraints:^(MASConstraintMaker *make) {
make.left.equalTo(self.messageView.mas_right);
make.top.bottom.equalTo(self);
make.width.equalTo(self.messageView);
}];
UITapGestureRecognizer *tapB = [[UITapGestureRecognizer alloc] initWithTarget:self action:@selector(didTapView:)];
[self.contactsView addGestureRecognizer:tapB];
self.dynamicView = [[FLTabBarDynamicItemView alloc] initWithOrientation:FLLeft title:@"我的" type:FLTabBarDynamic];
self.dynamicView.tag = 2;
[self addSubview:self.dynamicView];
[self.dynamicView mas_makeConstraints:^(MASConstraintMaker *make) {
make.left.equalTo(self.contactsView.mas_right);
make.top.bottom.right.equalTo(self);
make.width.equalTo(self.contactsView);
}];
UITapGestureRecognizer *tapC = [[UITapGestureRecognizer alloc] initWithTarget:self action:@selector(didTapView:)];
[self.dynamicView addGestureRecognizer:tapC];
}
- (void)didTapView:(UITapGestureRecognizer *)tap {
UIView *view = tap.view;
FLTabBarItemView *tapView;
if ([view isKindOfClass:[FLTabBarItemView class]]) {
tapView = (FLTabBarItemView *)view;
}
else {
return;
}
if (tapView.orientation == FLSelected) {
return;
}
if (_delegate && [_delegate respondsToSelector:@selector(fl_tabBarView:shoulSelectItemAtIndex:)]) {
BOOL should = [_delegate fl_tabBarView:self shoulSelectItemAtIndex:tapView.tag];
if (!should) {
return;
}
}
if ([tapView isEqual:self.messageView]) {
self.contactsView.orientation = FLLeft;
self.dynamicView.orientation = FLLeft;
}
else if ([tapView isEqual:self.contactsView]) {
self.messageView.orientation = FLRight;
self.dynamicView.orientation = FLLeft;
}
else if ([tapView isEqual:self.dynamicView]) {
self.messageView.orientation = FLRight;
self.contactsView.orientation = FLRight;
}
tapView.orientation = FLSelected;
if (_delegate && [_delegate respondsToSelector:@selector(fl_tabBarView:didSelectItemAtIndex:)]) {
[_delegate fl_tabBarView:self didSelectItemAtIndex:tapView.tag];
}
}
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,265
|
I was woken up yesterday by the pitter patter of little feet and an excited little voice, "Hey guys, my bunny is awake. It's time to get up! It's Christmas!" It's hard not to get caught up in his excitement. Tired, I stumbled downstairs to get some breakfast ready for the kid and feed the dog.
We had planned on having our own family Christmas on Christmas Eve morning, before the festivities with my parents in the afternoon. Just us. Our little team. As we sat down to open our gifts for each other, I opened the blinds and as if it was ordered up for me, it was snowing the most perfect, light amazing snowfall. With every present my son opened, they were all (even his new bed sheets) welcomed with, "This is awesome!" and "WOW!" After all the presents were opened, my husband and I exchanged a hug, kiss and Merry Christmas and as I turned, Jack ran up to hug me, jumping up into a full arms and legs bear hug. He kissed me on the cheek, looked me in the eyes and said, "Merry Christmas, Mommy."
When you are infertile, it is moments like those that you can only dream about. Moments like those that make every ounce of struggle and fight to bring them into this world worth everything.
Overall, I have had a very enlightening holiday season. I discovered things about myself that have really helped to bring everything full circle. I didn't give myself enough credit for my weight loss. I took a lot of things for granted. I was so inside myself that I didn't realize how many people I was pushing away- including my husband. This clarity? It has utterly changed my life. I am so overflowing with love and happiness that I wish I could bottle it up and gift it to others.
My Christmas was about as close to perfect as it could get. A kid whose child-like wonder is contagious. Seeing him perform in his first Christmas program a week ago was a moment I do not ever want to forget. A moment I waited and wanted ever since I knew I wanted to be a mom.
Singing with my best friend, regardless of my lingering pneumonia and tempo mis-cue on this song, still so much fun. So grateful to her for giving me the opportunity to do one of the things I love the most.
If there is one thing that has become the most clear to me over the last month, it is that being happy, positive and enjoying life is much more fun. Life is entirely way too short to spend time being unhappy or trying to solve things that are beyond our control.
I am beyond grateful for everything that I have. The life that I live. The incredible people I am surrounded by. The special people who inspire me every single day. They are always in my thoughts and hold the most important pieces of my heart.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 311
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,425
|
\section{Introduction}
The purpose of this paper is to investigate the asymptotic behavior of the difference
\begin{eqnarray}\label{Delta_rho}
\Delta \rho ^{X,Y}_\alpha := \rho _\alpha (X + Y) - \rho _\alpha (X)
\end{eqnarray}
as $\alpha \rightarrow 1$, where $X$ and $Y$ are fat-tailed random variables (loss variables) and
$(\rho _\alpha )_{0 < \alpha < 1}$ is a family of risk measures.
The case where $\rho _\alpha $ is an $\alpha$-percentile value-at-risk (VaR),
has been treated in \cite {Kato_IJTAF}, where it was shown that
the asymptotic behavior of $\Delta \mathrm {VaR}^{X,Y}_\alpha $ drastically changes
according to the relative magnitudes of the thicknesses of the tails of $X$ and $Y$
(the definition of the VaR is given in (\ref {def_VaR}) in the next section).
In this paper, we study a progressive case in which
$\rho _\alpha $ is given as a parameterized spectral risk measure,
and we obtain similar results as in \cite {Kato_IJTAF}.
In particular, we find that if $X$ and $Y$ are independent and if the tail of $X$ is sufficiently fatter than that of $Y$, then
$\Delta \rho ^{X,Y}_\alpha $ converges to the expected value $\mathop {\rm E} [Y]$ as $\alpha \rightarrow 1$
whenever $(\rho _\alpha )_{0 < \alpha < 1}$ are spectral risk measures converging to
a risk measure of the worst case scenario. That is, whenever
\begin{eqnarray}\label{conv_worst_case}
\rho _\alpha (Z) \ \mathop {\longrightarrow }_{\alpha \rightarrow 1} \ \mathop {\rm ess\;sup}_{\omega }Z(\omega )
\end{eqnarray}
for each loss random variable $Z$ in some sense.
Our result does not require any specific form for $\rho _\alpha $,
implying that this property is robust.
Furthermore, assuming some technical conditions for the probability density functions of $X$ and $Y$,
we study the asymptotic behavior of the Euler contribution, defined as
\begin{eqnarray}\label{EC}
\rho ^\mathrm {Euler}_\alpha (Y | X+Y) = \frac{\partial }{\partial h}\rho _\alpha (X + hY)\Big |_{h = 1}
\end{eqnarray}
(see Remark 17.1 in \cite {Tasche2008}),
and show that $\Delta \rho ^{X, Y}_\alpha $ is asymptotically equivalent to $\delta \rho ^\mathrm {Euler}_\alpha (Y | X+Y)$
as $\alpha \rightarrow 1$.
Here, $\delta \in (0, 1]$ is a constant
determined according to the relative magnitudes of the thicknesses of the tails of $X$ and $Y$.
We now briefly review the financial background for this study.
In quantitative financial risk management,
it is important to capture tail loss events by using adequate risk measures.
One of the most standard risk measures is the VaR.
The Basel Accords, which provide a set of recommendations for regulations in the banking industry,
essentially recommend using VaR as a measure of risk capital for banks.
VaRs are indeed simple, useful, and
their values are easy to interpret.
For instance, a yearly $99.9\%$ VaR calculated as $x_0$
means that the probability of a risk event
with a realized loss larger than $x_0$ is $0.1\%$.
In other words, an amount $x_0$ of risk capital is sufficient
to prevent a default with 99.9\% probability.
The meaning of the amount $x_0$ is therefore easy to understand.
However, VaRs are often criticized for their lack of subadditivity
(see, for instance, \cite {Acerbi, Acerbi-Tasche1, Embrechts}).
VaRs do not reflect the risk diversification effect.
The expected shortfall (ES) has been proposed
as an alternative risk measure that is coherent (in particular, subadditive) and tractable, with the risk amount at least that of the corresponding VaR.
Note that there are various versions of ES, such as
the conditional value-at-risk (CVaR),
the average value-at-risk (AVaR),
the tail conditional expectation (TCE),
and the worst conditional expectation (WCE).
These are all equivalent under some natural assumptions
(see \cite {Acerbi-Nordio-Sirtori, Acerbi-Tasche1, Acerbi-Tasche2, Artzner-Delbaen-Eber-Heath}).
It should be noted that the Basel Accords have also considered recently the adoption of ESs as a minimal capital requirement, in order to better capture market tail risks (see for instance \cite {BCBS2012, BCBS2016}).
A spectral risk measure (SRM) has been proposed as a generalization of ESs,
in \cite {Acerbi}.
SRMs are characterized by a weight function $\phi$ that represents
the significance of each confidence level for the risk manager.
SRMs are equivalent to comonotonic law-invariant coherent risk measures
(see Remark \ref {rem_SRM} in the next section).
VaRs and ESs as risk measures depend on a confidence level parameter $\alpha \in (0, 1)$.
We let $\mathrm {VaR}_\alpha $ (resp., $\mathrm {ES}_\alpha $) denote the VaR (resp., ES) with confidence level $\alpha $.
When $\alpha $ is close to $1$, the values of $\mathrm {VaR}_\alpha $ and $\mathrm {ES}_\alpha $ are
increasing without bound as in (\ref {conv_worst_case}).
The parameter $\alpha $ corresponds to the risk aversion level of the risk manager.
Higher values of $\alpha $ indicate that
the risk manager is more risk-averse and evaluates the tail risk as more severe.
In this paper, we consider a family $(\rho _\alpha )_{0 < \alpha < 1}$ of
SRMs parameterized by the confidence level $\alpha $.
we make a mathematical assumption that intuitively implies situation (\ref {conv_worst_case}) and investigate the asymptotic behaviors of (\ref {Delta_rho}) and (\ref {EC})
as $\alpha \rightarrow 1$,
when the tail probability function of $X$ (resp., $Y$) is regularly varying with index $-\beta $ (resp., $-\gamma $).
Our main theorem asserts that the asymptotic behaviors of (\ref {Delta_rho}) and (\ref {EC})
strongly depend on
the relative magnitudes of $\beta $ and $\gamma$.
Note that our results include
the case $\rho _\alpha = \mathrm {ES}_\alpha $,
the inclusion of which was discussed as a future task in \cite {Kato_IJTAF}.
The rest of this paper is organized as follows.
In Section \ref {sec_pre}, we prepare the basic settings and introduce the definitions for SRMs based on confidence level.
In Section \ref {sec_main}, we give our main results.
We numerically verify our results in Section \ref {sec_numerical}.
Finally, Section \ref {sec_conclusion} summarizes our studies.
Throughout the main part of this paper, we assume that $X$ and $Y$ are independent. The more general case where $X$ and $Y$ are not independent is studied in Appendix \ref {sec_add}.
All proofs are given in Appendix \ref {sec_proofs}.
\section{Preliminaries}\label{sec_pre}
Let $(\Omega , \mathcal {F}, P)$ be a standard probability space and let
$\mathscr {L}_+$ denote a set of non-negative random variables defined on $(\Omega , \mathcal {F}, P)$.
For each $Z\in \mathscr {L}_+$, we denote by $F_Z$ the distribution function of $Z$
and by $\bar{F}_Z$ its tail probability function; that is,
$F_Z(z) = P(Z\leq z)$ and $\bar{F}_Z(z) = P(Z > z)$.
Moreover, for each $\alpha \in (0, 1)$, we define
\begin{eqnarray}\label{def_VaR}
\mathrm {VaR}_\alpha (Z) = \inf \{ z\in \mathbb {R}\ ; \ P(Z \leq z) \geq \alpha \} .
\end{eqnarray}
Note that $\mathrm {VaR}_\alpha (Z)$ is exactly the left-continuous version
of the generalized inverse function of $F_Z$.
We now introduce the definition of SRMs.
\begin{defn} \
\begin{itemize}
\item [{\rm (i)}]A Borel measurable function $\phi: [0, 1) \longrightarrow [0, \infty )$ is called an admissible spectrum
if $\phi $ is right-continuous, non-decreasing, and satisfies
\begin{eqnarray}\label{cond_normalization}
\int ^1_0\phi (\alpha )d\alpha = 1.
\end{eqnarray}
\item [{\rm (ii)}] A risk measure $\rho: \mathscr {L}_+ \longrightarrow [0, \infty )$ is called an SRM
if there is an admissible spectrum $\phi $ such that $\rho = M_\phi $, where
\begin{eqnarray*}
M_\phi (Z) = \int ^1_0\mathrm {VaR}_\alpha (Z)\phi (\alpha )d\alpha , \ \ Z\in \mathscr {L}_+.
\end{eqnarray*}
\end{itemize}
\end{defn}
\begin{remark}\label{rem_SRM}
SRMs are law-invariant, comonotonic, and coherent risk measures.
However, as shown in \cite{Foellmer-Schied, Kusuoka, Jouini-Schachermayer-Touzi}, if $(\Omega , \mathcal {F}, P)$ is atomless,
then
for any law-invariant comonotonic convex risk measure $\rho $, there is a probability measure $\mu $ on $[0, 1]$ such that
\begin{eqnarray}\label{Kusuoka-rep}
\rho (Z) = \int ^1_0\mathrm {ES}_\alpha (Z)\mu (d\alpha ),
\end{eqnarray}
for each $Z\in L^\infty (\Omega , \mathcal {F}, P)$. This is due to the generalized Kusuoka representation theorem (Theorem 4.93 in \cite {Foellmer-Schied}),
where $\mathrm {ES}_\alpha (Z)$ is the $\alpha $-percentile expected shortfall of $Z$:
\begin{eqnarray}\label{def_ES}
\mathrm {ES}_\alpha (Z) = \frac{1}{1 - \alpha }\int ^1_\alpha \mathrm {VaR}_u(Z)du.
\end{eqnarray}
Moreover, such a $\rho $ is always coherent and satisfies the Fatou property \cite {Jouini-Schachermayer-Touzi}.
Furthermore, representation (\ref {Kusuoka-rep}) can also be rewritten as $\rho (Z) = M_{\phi _\mu }(Z)$, where
\begin{eqnarray*}
\phi _\mu (\alpha ) = \int ^1_0\frac{1}{1 - u}1_{[0, \alpha ]}(u)\mu (du).
\end{eqnarray*}
Here, it is easy to see that $\phi _\mu $ is non-negative, non-decreasing, right-continunous, and satisfies
\begin{eqnarray*}
\int ^1_0\phi _\mu (\alpha )d\alpha = \int ^1_0\frac{1}{1-u}\int ^1_01_{\{ 0 \leq u \leq \alpha \}}d\alpha \mu (du) = 1,
\end{eqnarray*}
meaning that $\phi _\mu $ is an admissible spectrum (see \cite {Shapiro}).
Therefore, any law-invariant comonotonic convex (or coherent) risk measure is completely
characterized as an SRM.
Arguments similar to those above, replacing $L^\infty (\Omega , \mathcal {F}, P)$ with
$L^p(\Omega , \mathcal {F}, P)$, where $1\leq p < \infty $,
can be found in \cite {Pflug-Roemisch, Shapiro}.
\end{remark}
Next, we introduce a family $(\rho _\alpha )_{0 < \alpha < 1}$ of SRMs parameterized by the confidence level $\alpha $.
\begin{defn} Let $(\phi _\alpha )_{0 < \alpha < 1}$ be a familly of admissible spectra and let $\rho _\alpha = M_{\phi _\alpha }$.
Then $(\rho _\alpha )_{0 < \alpha < 1}$ is called a set of confidence-level-based spectral risk measures (CLBSRMs) if
\begin{eqnarray}\label{def_CLBSRM}
\Phi _\alpha \ \longrightarrow ^\mathrm {w} \ \delta _1, \ \ \alpha \rightarrow 1,
\end{eqnarray}
where $\Phi _\alpha $ is a probability measure on $[0, 1]$ defined by $\Phi _\alpha (du) = \phi _\alpha (u)du$ and
$\delta _1$ is the Dirac measure with unit mass at $1$.
\end{defn}
Condition (\ref {def_CLBSRM}) formally implies (\ref {conv_worst_case}).
Indeed, if $Z\in \mathscr {L}_+$ is a bounded random variable with a distribution function that is continuous and strictly increasing on $[0, z^*]$,
where $z^* = \mathop {\rm esssup}_{\omega }Z(\omega )$,
then the function $u\mapsto \mathrm {VaR}_u(Z)$ is bounded and continuous, so that (\ref {def_CLBSRM}) gives
\begin{eqnarray*}
\rho _\alpha (Z) = \int ^1_0\mathrm {VaR}_u(Z)\Phi _\alpha (du) \longrightarrow z^*, \ \ \alpha \rightarrow 1,
\end{eqnarray*}
where we recognize $\mathrm {VaR}_1(Z) = F^{-1}_Z(1) = z^*$.
Moreover, we see that
\begin{lemma}\label{lem_pointwise}
Relation $(\ref {def_CLBSRM})$ is equivalent to
\begin{eqnarray}\label{conv_pointwise}
\phi _\alpha (u) \longrightarrow 0, \ \ \alpha \rightarrow 1 \ \mbox { for each } \ u\in [0, 1).
\end{eqnarray}
\end{lemma}
We now give some examples of CLBSRMs. \\
\noindent
{\it Example 1.} Expected Shortfalls
$(\mathrm {ES}_\alpha )_{0 < \alpha < 1}$ defined by (\ref {def_ES}) is a typical example of a CLBSRM.
The corresponding admissible spectra are given as
\begin{eqnarray*}
\phi ^{\mathrm {ES}}_\alpha (u) = \frac{1}{1-\alpha }1_{[\alpha , 1)}(u).
\end{eqnarray*}
It is easy to see that (\ref {def_CLBSRM}) does hold.
Indeed, for any bounded continuous function $f$ defined on $[0, 1]$, we see that
\begin{eqnarray*}
\frac{1}{1 - \alpha }\int ^1_\alpha f(u)du =
\int ^1_0f(u + \alpha (1 - u))du \longrightarrow f(1), \ \ \alpha \rightarrow 1
\end{eqnarray*}
due to the bounded convergence theorem.
Equivalently, we can also check that $(\mathrm {ES}_\alpha )_\alpha $ satisfies (\ref {conv_pointwise}).
$\mathrm {ES}_\alpha $ is characterized as the smallest law-invariant coherent risk measures that are greater than or equal to $\mathrm {VaR}_\alpha $ \cite {Kusuoka}.
Note that if the distribution function of the target random variable $Z$ is continuous, then
$\mathrm {ES}_\alpha (Z)$ coincides with $\mathrm {CVaR}_\alpha (Z)$, where
\begin{eqnarray*}
\mathrm {CVaR}_\alpha (Z) = \mathop {\rm E} [Z\ | \ Z\geq \mathrm {VaR}_\alpha (Z)]
\end{eqnarray*}
(see \cite {Acerbi-Tasche2} for details). \\
\noindent
{\it Example 2.} Exponential/Power SRMs
An admissible spectrum $\phi $ corresponding to an SRM $M_\phi $ represents the
preferences of a risk manager for each quantile of the loss distribution.
Therefore, the form taken by $\phi $ corresponds to the manager's risk aversion,
which is also described in terms of utility functions in classical decision theory.
Recently, the relation between expected utility functions and SRMs has been studied,
though it has not been entirely resolved.
Here we introduce some examples of SRMs based on specific utility functions.
The exponential utility function is a typical example of tractable utility functions
\begin{eqnarray*}
U_\gamma (p) = -\frac{e^{-\gamma p}}{\gamma },
\end{eqnarray*}
where $p$ denotes the
profit-and-loss ($p > 0$ indicating profit)
and $\gamma $ characterizes the degree of risk preference.
We focus on the case $0 < \gamma < \infty $ so that $U_\gamma $ describes a risk-averse utility function.
We transform the parameter $\gamma $ into the confidence level $\alpha \in (0, 1)$ using
$\alpha = (2/\pi )\tan ^{-1}\gamma $.
Note that the original parameter $\gamma $ can be recovered using the inverse
$\gamma = t_\alpha := \tan (\pi \alpha /2)$.
The exponential utility of the loss $l$ with confidence level $\alpha $ is then given as
$U_{t_\alpha }(-l) = -e^{lt_\alpha }/t_\alpha $.
Cotter and Dowd \cite {Cotter-Dowd} have proposed
an SRM $\rho ^{\mathrm {EXP}}_\alpha = M_{\phi ^{\mathrm {EXP}}_\alpha }$ based on the exponential utility by constructing
an admissible spectrum
$\phi ^{\mathrm {EXP}}_\alpha (u) = -\lambda U_{t_\alpha }(-u)$
for some $\lambda > 0$, so that
$\phi ^{\mathrm {EXP}}_\alpha (u)$ satisfies (\ref {cond_normalization}).
Then, $\lambda $ must be set as
$t_\alpha ^2/(e^{t_\alpha } - 1)$,
giving
\begin{eqnarray*}
\phi ^{\mathrm {EXP}}_\alpha (u) =
\frac{t_\alpha e^{-t_\alpha (1-u)}}{1-e^{-t_\alpha }}.
\end{eqnarray*}
Note that the theoretical validity of the above method is still unclear.
Other methods to adequately construct SRMs from exponential utility functions
have been discussed in \cite{Brandtner-Kuersten, Sriboonchitta-Nguyen-Kreinovich, Waechter-Mazzoni},
but no definite answer has been reached.
In particular, it is pointed out in \cite {Brandtner-Kuersten} that there exists
no general consistency between expected utility theory and SRM-decision making.
In any case, we can easily verify that $(\phi ^\mathrm {EXP}_\alpha )_\alpha $ as defined above satisfies
(\ref {def_CLBSRM})--(\ref {conv_pointwise}), which implies that
$(\rho ^\mathrm {EXP}_\alpha )_\alpha $ is actually a CLBSRM.
Similarly to the above, an SRM $\rho ^\mathrm {POW}_\alpha = M_{\phi ^\mathrm {POW}_\alpha }$ based on the power utility function has been studied in \cite {Dowd-Cotter-Sorwar}.
After changing the risk aversion parameter to the confidence level $\alpha \in (0, 1)$ as above,
$\phi ^\mathrm {POW}_\alpha $ is given as
\begin{eqnarray*}
\phi ^\mathrm {POW}_\alpha (u) = \frac{u^{\alpha /(1-\alpha )}}{1-\alpha }.
\end{eqnarray*}
We can also verify that $(\rho ^\mathrm {POW}_\alpha )_{0 < \alpha < 1}$ is a CLBSRM. \vspace{3mm}
We now introduce some notations and definitions used in asymptotic analysis and extreme value theory.
Let $f$ and $g$ be positive functions defined on $[x_0, x_1)$, where $x_0\in [0, \infty )$ and $x_1 \in (x_0, \infty ]$.
We say that $f$ and $g$ are asymptotically equivalent (denoted as $f\sim g$) as $x\rightarrow x_1$
if
$\lim _{x\rightarrow x_1}f(x)/g(x) = 1$.
When $x_1 = \infty $,
we say that $f$ is regularly varying with index $k\in \mathbb {R}$ if
it holds that $\lim _{x\rightarrow \infty }f(tx)/f(x) = t^k$ for each $t > 0$.
Moreover, we say that $f$ is ultimately decreasing if
$f$ is non-increasing on $[x_2, \infty )$ for some $x_2 > 0$.
For more details, we refer the reader to \cite {Bingham-Goldie-Teugels, Embrechts-Klueppelberg-Mikosch}.
\section{Main results}\label{sec_main}
Our main purpose is
to investigate the property of (\ref {Delta_rho}) for a CLBSRM $(\rho _\alpha )_{0 < \alpha < 1}$ and
random variables $X, Y\in \mathscr {L}_+$ whose
distributions are fat-tailed.
To consider this case,
we assume that $\bar{F}_X$ and $\bar{F}_Y$ are regularly varying functions with indices $-\beta $ and $-\gamma $, respectively.
That is, $\bar{F}_X(x), \bar{F}_Y(x) > 0$ for each $x\geq 0$ and
\begin{eqnarray}\label{regular_variation}
\lim _{x\rightarrow \infty }\frac{\bar{F}_X(tx)}{\bar{F}_X(x)} = t^{-\beta }, \ \
\lim _{x\rightarrow \infty }\frac{\bar{F}_Y(tx)}{\bar{F}_Y(x)} = t^{-\gamma }, \ \ t > 0
\end{eqnarray}
for some $\beta , \gamma > 0$.
In \cite {Kato_IJTAF},
we study the asymptotic property of (\ref {Delta_rho}) as $\alpha \rightarrow 1$ when
$\rho _\alpha = \mathrm {VaR}_\alpha $.
The results display
the following five patterns:
(i) $\beta + 1 < \gamma $, (ii) $\beta < \gamma \leq \beta + 1$,
(iii) $\beta = \gamma $,
(iv) $\gamma < \beta \leq \gamma + 1$, and
(v) $\gamma + 1 < \beta $.
In cases (iv) and (v), we consider the difference $\Delta \mathrm {VaR}^{Y, X}_\alpha $ instead of
$\Delta \mathrm {VaR}^{X, Y}_\alpha $, and the results are restated consequences of cases (i) and (ii).
Hence, we assume here that $\beta \leq \gamma $ and focus on cases (i)--(iii) only.
We further assume that $\beta > 1$.
This assumption guarantees the integrability of $X$ and $Y$
(see, for instance, Proposition A3.8 in \cite {Embrechts-Klueppelberg-Mikosch}).
Let $(\rho _\alpha )_{0 < \alpha < 1}$ be a CLBSRM with a family of admissible spectra $(\phi _\alpha )_{0 < \alpha < 1}$.
Here we assume that
\begin{eqnarray}\label{ass_bdd_phi}
\phi _{\alpha }(1-) = \lim _{u\rightarrow 1}\phi _\alpha (u) < \infty
\end{eqnarray}
for each $\alpha \in (0, 1)$.
Then, Lemma A.23 in \cite {Foellmer-Schied} implies that
\begin{eqnarray*}
\rho _\alpha (X + Y) \leq \phi _\alpha (1-)\int ^1_0\mathrm {VaR}_u(X+Y)du =
\phi _{\alpha }(1-)(\mathop {\rm E} [X] + \mathop {\rm E} [Y]) < \infty
\end{eqnarray*}
for each $0 < \alpha < 1$.
This immediately implies that $\rho _\alpha (X), \rho _\alpha (Y) < \infty $.
Furthermore, by (17.9b) and Proposition 17.2 in \cite {Tasche2008}, we see that
\begin{eqnarray}\label{ineq_general}
\Delta \rho ^{X, Y}_\alpha \leq \rho ^\mathrm {Euler}_\alpha (Y | X+Y) \leq \rho _\alpha (Y),
\end{eqnarray}
where $\rho ^\mathrm {Euler}(Y | X+Y)$ is given by (\ref {EC}) if
$\rho _\alpha (X + hY)$ is continuously differentiable in $h$.
Note that inequality (\ref {ineq_general}) holds for each $0 < \alpha < 1$ whenever $\rho _\alpha $ is coherent.
Our main purpose in this section is to investigate in detail the asymptotic behavior of $\Delta \rho ^{X, Y}_\alpha $, as well as $\rho ^\mathrm {Euler}_\alpha (Y | X + Y)$ if it is defined, as $\alpha \rightarrow 1$.
To clearly state our main results, we establish the following conditions,
which are assumed to hold in Section 4 of \cite {Kato_IJTAF}.
\begin{itemize}
\item[ \mbox{[C1]}] \ $X$ and $Y$ are independent.
\item[ \mbox{[C2]}] \ There is some $x_0 \geq 0$ such that
$F_X$ has a positive, non-increasing density function $f_X$ on $[x_0, \infty )$; that is,
$F_X(x) = F_X(x_0) + \int ^x_{x_0}f_X(y)dy, \ \ x \geq x_0$.
\item[ \mbox{[C3]}] \ The function $x^{\gamma - \beta }\bar{F}_Y(x)/\bar{F}_X(x)$ converges to
some real number $k$ as $x\rightarrow \infty $.
\end{itemize}
Let us adopt the notation
\begin{eqnarray}\label{def_M_bar}
\bar{M}(\alpha ) =
\left\{
\begin{array}{ll}
\mathop {\rm E} [Y] & \mbox { if } \beta + 1 < \gamma , \\
\frac{k}{\beta }\int ^1_0\mathrm {VaR}_u(X)^{\beta + 1 - \gamma }\phi _\alpha (u)du& \mbox { if } \beta < \gamma \leq \beta + 1, \\
\{ (1 + k)^{1/\beta } - 1 \}\rho _\alpha (X)& \mbox { if } \beta = \gamma
\end{array}
\right.
\end{eqnarray}
for $0 < \alpha < 1$.
Note that $\bar{M}(\alpha )$ is finite for each fixed $\alpha \in (0, 1)$
(see Corollary \ref {cor_finiteness_M} in Appendix \ref {sec_proofs}).
Our main results are the two following theorems.
\begin{theorem}\label{th_main} \ Assuming {\rm [C1]--[C3]},
$\Delta \rho ^{X, Y}_\alpha \sim \bar{M}(\alpha )$ as $\alpha \rightarrow 1$.
\end{theorem}
Formally, assertions (i)--(iii) of Theorem 4.1 in \cite {Kato_IJTAF}
are the same as the assumptions of Theorem \ref {th_main},
by setting $\Phi _\alpha = \delta _{\alpha }$.
That is, we have $\Delta \mathrm {VaR}^{X, Y}_\alpha \sim \bar{f}(\alpha )$ as $\alpha \rightarrow 1$,
where
\begin{eqnarray}\label{def_f_bar}
\bar{f}(\alpha ) =
\left\{
\begin{array}{ll}
\mathop {\rm E} [Y] & \mbox { if } \beta + 1 < \gamma , \\
\frac{k}{\beta }\mathrm {VaR}_\alpha (X)^{\beta + 1 - \gamma }& \mbox { if } \beta < \gamma \leq \beta + 1, \\
\{ (1 + k)^{1/\beta } - 1 \}\mathrm {VaR}_\alpha (X)& \mbox { if } \beta = \gamma .
\end{array}
\right.
\end{eqnarray}
Theorem \ref {th_main} justifies the following relation:
\begin{eqnarray*}
\Delta \rho ^{X, Y}_\alpha = \int ^1_0\Delta \mathrm {VaR}^{X, Y}_u\phi _\alpha (u)du
\sim
\int ^1_0\bar{f}(u)\phi _\alpha (u)du = \bar{M}(\alpha ), \ \ \alpha \rightarrow 1.
\end{eqnarray*}
Note that condition [C3] is not required for Theorem \ref {th_main} when $\beta + 1 < \gamma $.
Moreover, when $\beta + 1 < \gamma $,
Theorem \ref {th_main} implies that
$\Delta \rho ^{X, Y}_\alpha $ converges to $\mathop {\rm E} [Y]$ as $\alpha \rightarrow 1$.
The limit $\mathop {\rm E} [Y]$ does not depend on the forms of $(\phi _\alpha )_\alpha $,
so this result is robust.
The second main result is as follows.
\begin{theorem}\label{th_equivalence_Euler_EL}
Assume {\rm {[C1]}} and {\rm {[C3]}}. Moreover,
assume that
\begin{itemize}
\item [{\rm [C4]}]$X$ and $Y$ have positive, continuous, and ultimately decreasing density functions $f_X$ and $f_Y$, respectively, on $[0, \infty )$.
\end{itemize}
Under these assumptions, $\rho ^\mathrm {Euler}_\alpha (Y | X + Y) \sim \bar{M}(\alpha )/\delta $ as $\alpha \rightarrow 1$, where
$\delta $ is a positive constant given by
\begin{eqnarray}\label{def_delta}
\delta =
\left\{
\begin{array}{cl}
1 & \mbox { \rm if } \beta + 1 < \gamma , \\
k / (\mathop {\rm E} [Y]\beta + k\gamma )& \mbox { \rm if } \beta + 1 = \gamma , \\
1 / \gamma & \mbox { \rm if } \beta < \gamma < \beta + 1, \\
\{ 1 + k - (1+k)^{1-1/\beta }\} / k& \mbox { \rm if } \beta = \gamma .
\end{array}
\right.
\end{eqnarray}
\end{theorem}
Theorems \ref {th_main} and \ref {th_equivalence_Euler_EL} together imply that
if $X$ and $Y$ are independent, and if
$F_X$ and $F_Y$ have adequate density functions,
then
\begin{eqnarray}\label{asy_behavior_marginal_Euler}
\Delta \rho ^{X, Y}_\alpha \sim \delta \rho ^\mathrm {Euler}_\alpha (Y | X + Y), \ \ \alpha \rightarrow 1.
\end{eqnarray}
Note that $\delta $ is always smaller than or equal to $1$,
so that (\ref {asy_behavior_marginal_Euler}) is consistent with inequality (\ref {ineq_general}).
In particular, if $\beta + 1 < \gamma $, then the asymptotic equivalence between
the marginal risk contribution $\Delta \rho ^{X, Y}_\alpha $
and
the Euler contribution $\rho ^\mathrm {Euler}_\alpha (Y | X + Y)$ is justified (see (17.10) in \cite {Tasche2008}).
Note that $\Delta \rho _\alpha ^{X, Y}$ is always larger than or equal to $\mathop {\rm E} [Y]$
so long as the random vector $(X, Y)$ satisfies a suitable
technical condition, such as Assumption (S) in \cite {Tasche2000}.
(Here, we modify some conditions of the original version of Assumption (S)
to facilitate focusing on non-negative random variables.)
Indeed, because $\rho _\alpha $ is a convex risk measure,
the function $r(h):= \rho _\alpha (X + hY)$ is convex.
Thus, we get
\begin{eqnarray}\label{lower_bound}
\Delta \rho _\alpha ^{X, Y} = r(1) - r(0) \geq r'(0) = \mathop {\rm E} [Y],
\end{eqnarray}
where the last equality in the above relation is obtained from (see (5.12) in \cite {Tasche2000})
\begin{eqnarray}\label{derivative_VaR}
\frac{\partial }{\partial h}\mathrm {VaR}_\alpha (X+hY) = \mathop {\rm E} [Y | X+hY = \mathrm {VaR}_\alpha (X+hY)]
\end{eqnarray}
and
\begin{eqnarray*}
\frac{\partial }{\partial h}\Big |_{h=0}\rho _\alpha (X + hY) =
\int ^1_0\frac{\partial }{\partial h}\Big |_{h=0}\mathrm {VaR}_u(X + hY)\phi _\alpha (u)du = \mathop {\rm E} [Y]
\end{eqnarray*}
due to the dominated convergence theorem.
Therefore,
if $\beta + 1 < \gamma $, then
\begin{eqnarray*}
\mathop {\rm E} [Y] \leq \Delta \rho ^{X, Y}_\alpha \sim \rho ^\mathrm {Euler}_\alpha ( Y | X + Y) \longrightarrow \mathop {\rm E} [Y], \ \ \alpha \rightarrow 1.
\end{eqnarray*}
In Section \ref {sec_numerical},
we numerically verify the above relation.
Note that we can also verify a version of Assumption (S) under [C4].
\begin{remark}\label{rem_main} \
\begin{itemize}
\item [(i)] If $F_X$ is continuous, then
$F_X(X)$ has a uniform distribution on $(0, 1)$
(see, for instance, Lemma A.21 in \cite {Foellmer-Schied}).
Therefore, $\bar{M}(\alpha )$ with $\beta < \gamma \leq \beta + 1$ is rewritten as
\begin{eqnarray*}
\bar{M}(\alpha ) = \frac{k}{\beta }\mathop {\rm E} \hspace{0mm}^{Q^X_\alpha }[X^{\beta + 1 - \gamma }],
\end{eqnarray*}
where $\mathop {\rm E} \hspace{0mm}^{Q^X_\alpha }$ denotes the expectation operator
with respect to the probability measure $Q^X_\alpha $ defined as
\begin{eqnarray}\label{def_Q}
\frac{dQ^X_\alpha }{dP} = \phi _\alpha (F_X(X)).
\end{eqnarray}
Note that we have
$\rho _\alpha (X) = \mathop {\rm E} \hspace{0mm}^{Q^X_\alpha }[X]$,
and so $Q^X_\alpha $ represents the risk scenario that attains the maximum in the following
robust representation of $\rho _\alpha (X)$:
\begin{eqnarray*}
\rho _\alpha (X) = \max _{Q\in \mathscr {Q}}\mathop {\rm E} \hspace{0mm}^Q[X],
\end{eqnarray*}
where $\mathscr {Q}$ is a set of probability measures on $(\Omega , \mathcal {F})$.
Also note that if $\rho _\alpha = \mathrm {ES}_\alpha $, then $Q^X_\alpha $ is given by
\begin{eqnarray*}
\frac{dQ^X_\alpha }{dP} = \frac{1}{1 - \alpha }1_{\{ X \geq \mathrm {VaR}_\alpha (X) \}},
\end{eqnarray*}
and therefore
\begin{eqnarray*}
\mathop {\rm E} \hspace{0mm}^{Q^X_\alpha }[X^{\beta + 1 - \gamma }] =
\mathop {\rm E} [X^{\beta + 1 - \gamma } | X\geq \mathrm {VaR}_\alpha (X)].
\end{eqnarray*}
Until the end of Remark \ref {rem_main},
we assume that $F_X$ and $F_Y$ are continuous.
\item [(ii)]In fact, we can relax the independence condition [C1] so that
$X$ may weakly depend on $Y$ within the negligible joint tail condition
(see Remark A.1 in \cite {Kato_IJTAF}).
In this case, under some additional assumptions such as [A5] and [A6] in \cite {Kato_IJTAF},
we can make the same assertion as in Theorem \ref {th_main},
where the value $\mathop {\rm E} [Y]$ in the definition (\ref {def_M_bar}) of $\bar{M}(\alpha )$ is replaced
by $\mathop {\rm E} \hspace{0mm}^{Q^X_\alpha }[Y]$.
In particular, if $\beta + 1 < \gamma $,
then
\begin{eqnarray}\label{case_weak_dependent}
\Delta \rho ^{X, Y}_\alpha \sim
\mathop {\rm E} \hspace{0mm}^{Q^X_\alpha }[Y], \ \ \alpha \rightarrow 1.
\end{eqnarray}
Indeed, our proof in Appendix \ref {sec_proofs} also works
by applying Theorem A.1 in \cite {Kato_IJTAF} instead of Theorem 4.1.
Note that we need some additional condition to have that
\begin{eqnarray}\label{cond_liminf}
\liminf _{\alpha \rightarrow \infty }\mathop {\rm E} \hspace{0mm}^{Q^X_\alpha }[Y] > 0
\end{eqnarray}
(see Proposition \ref {prop_lower_bdd} in Appendix \ref {sec_proofs}).
\item [(iii)] As mentioned in Appendix A.1 of \cite {Kato_IJTAF},
we can get another version of Theorem A.1 by switching the roles of $X+Y$ and $X$ and
by imposing modified (though somewhat artificial) mathematical conditions such as [A5'] and [A6'] in \cite {Kato_IJTAF}.
In particular, if $\beta + 1 < \gamma $, we see that
\begin{eqnarray}\label {asy_eq_VaR}
\Delta \mathrm {VaR}_\alpha ^{X, Y} \sim
\mathop {\rm E} [Y | X + Y = \mathrm {VaR}_\alpha (X + Y)], \ \ \alpha \rightarrow 1
\end{eqnarray}
and then (by the same proof as Theorem \ref {th_main} with (\ref {asy_eq_VaR}))
\begin{eqnarray}\label{asy_eq_contribution}
\Delta \rho _\alpha ^{X, Y} \sim
\mathop {\rm E} \hspace{0mm}^{Q^{X+Y}_\alpha }[Y] = \rho ^\mathrm {Euler}_\alpha (Y | X + Y), \ \ \alpha \rightarrow 1
\end{eqnarray}
under some assumptions.
Here, $Q^{X+Y}_\alpha $ is a probability measure defined by $(\ref {def_Q})$
with replacing $X$ by $X+Y$.
If $X$ and $Y$ are independent (with natural assumptions on the density functions), then
(\ref {asy_behavior_marginal_Euler}) implies that (\ref {asy_eq_contribution}) is also true.
Here, note that the last equality of (\ref {asy_eq_contribution}) is obtained by
(\ref {EC}), (\ref {derivative_VaR}), and the dominated convergence theorem.
Indeed, we have
\begin{eqnarray}\label{Euler_rho}
\rho ^\mathrm {Euler}_\alpha (Y | X + Y) =
\int ^1_0\mathop {\rm E} [Y | X + Y = \mathrm {VaR}_u(X + Y)]\phi _\alpha (u)du =
\mathop {\rm E} \hspace{0mm}^{Q^{X+Y}_\alpha }[Y]
\end{eqnarray}
because $F_{X+Y}(X+Y)$ is uniformly distributed on $(0, 1)$.
In Appendix \ref {sec_add},
we will show that
under some technical conditions that are more natural than
both [A5]--[A6] and [A5']--[A6'] in \cite {Kato_IJTAF},
relations (\ref {case_weak_dependent}) and (\ref {asy_eq_contribution}) simultaneously hold
in the case $\beta + 1 < \gamma $, even if $X$ and $Y$ are dependent.
Note that if $\rho _\alpha = \mathrm {ES}_\alpha $, then
\begin{eqnarray*}
\mathop {\rm E} \hspace{0mm}^{Q^{X+Y}_\alpha }[Y] = \mathrm {ES}^\mathrm {Euler}_\alpha (Y | X + Y) =
\mathop {\rm E} [Y | X + Y \geq \mathrm {VaR}_\alpha (X + Y)],
\end{eqnarray*}
which is known as the component CVaR (also known as the CVaR contribution) and widely used, particularly
in the practice of credit portfolio risk management
(see for instance \cite {Andersson-et-al, Kalkbrener-Kennedy-Popp, Puzanova-Duellmann}).
\end{itemize}
\end{remark}
\section{Numerical analysis}\label{sec_numerical}
In this section, we numerically investigate the behavior of $\Delta \rho _\alpha ^{X, Y}$.
Throughout this section, we assume that the distributions of $X$ and $Y$ are
given as $\mathrm {GPD}(\xi _X, \sigma _X)$ and $\mathrm {GPD}(\xi _Y, \sigma _Y)$, respectively,
with $\xi _X, \xi _Y \in (0, 1)$ and $\sigma _X, \sigma _Y > 0$, where
$\mathrm {GPD}(\xi , \sigma )$ denotes the generalized Pareto distribution
whose distribution function is given by
$1 - ( 1 + \xi x/\sigma ) ^{-1/\xi }$, $x\geq 0$.
Then, $\bar{F}_X$ and $\bar{F}_Y$ satisfy (\ref {regular_variation}) with $\beta = 1/\xi _X$ and $\gamma = 1/\xi _Y$.
Note that condition [C3] is satisfied with
\begin{eqnarray*}
k = \left( \frac{\sigma _Y}{\xi _Y}\right) ^{1/\xi _Y}\left( \frac{\sigma _X}{\xi _X}\right) ^{-1/\xi _X}
\end{eqnarray*}
(see (5.2) in \cite {Kato_IJTAF}).
Also note that $\mathrm {VaR}_\alpha (X)$ and $\mathrm {VaR}_\alpha (Y)$ are analytically solved as
\begin{eqnarray*}
\mathrm {VaR}_\alpha (X) =
\frac{\sigma _X}{\xi _X}\left\{ (1-\alpha )^{-\xi _X} - 1 \right\} , \ \
\mathrm {VaR}_\alpha (Y) =
\frac{\sigma _Y}{\xi _Y}\left\{ (1-\alpha )^{-\xi _Y} - 1 \right\} .
\end{eqnarray*}
We numerically compute
$\Delta \mathrm {VaR}^{X, Y}_\alpha , \Delta \mathrm {ES}^{X, Y}_\alpha ,
\Delta \rho ^{\mathrm {EXP}, X, Y}_\alpha , \Delta \rho ^{\mathrm {POW}, X, Y}_\alpha $,
and $\mathrm {ES}^\mathrm {Euler}_\alpha $, where we let
$\mathrm {ES}^\mathrm {Euler}_\alpha = \mathrm {ES}^\mathrm {Euler}_\alpha (Y | X + Y)$ for brevity.
In all calculations, we fix $\sigma _X = 100$ and $\sigma _Y = 80$.
For $\xi _X$ and $\xi _Y$, we examine several patterns to study each of the following three cases:
(i) $\beta + 1 < \gamma $, (ii) $\beta < \gamma \leq \beta + 1$, and (iii) $\beta = \gamma $. \\
\noindent
{\it Case }(i) \ $\beta + 1 < \gamma $
We set $\xi _X = 0.5$ and $\xi _Y = 0.1$.
Hence, $\beta = 2$ and $\gamma = 10$, so that $\beta + 1 < \gamma $ holds.
Figure \ref {fig_1_1} shows the graphs of
$\Delta \mathrm {ES}^{X, Y}_\alpha ,
\Delta \rho ^{\mathrm {EXP}, X, Y}_\alpha , \Delta \rho ^{\mathrm {POW}, X, Y}_\alpha $,
and $\mathrm {ES}^\mathrm {Euler}_\alpha $.
These values are always larger than $\mathop {\rm E} [Y]$ whenever $\alpha \in (0, 1)$,
and they converge to $\mathop {\rm E} [Y]$ for both $\alpha \rightarrow 0$ and $\alpha \rightarrow 1$.
Indeed,
\begin{eqnarray}\label{limit_alpha_zero}
\lim _{\alpha \rightarrow 0}\Delta \rho ^{X, Y}_\alpha = \mathop {\rm E} [X + Y] - \mathop {\rm E} [X] = \mathop {\rm E} [Y]
\end{eqnarray}
holds because
$\phi ^{\mathrm {ES}}_\alpha (u), \phi ^{\mathrm {EXP}}_\alpha (u), \phi ^{\mathrm {POW}}_\alpha (u)
\longrightarrow 1$, $\alpha \rightarrow 0$ for each $u\in [0, 1)$.
The limit as $\alpha \rightarrow 1$ is a consequence of Theorem \ref {th_main}.
Moreover, the forms of these graphs are unimodal.
That is, the function
$\alpha \rightarrow \Delta \rho ^{X, Y}_\alpha $
incereases on $(0, \alpha _0)$
and decreases on $(\alpha _0, 1)$ for some $\alpha _0\in (0, 1)$.
Intuitively, the values of $\Delta \rho ^{X, Y}_\alpha $ seem to become large
as $\alpha $ increases because
a larger $\alpha $ implies a greater risk sensitivity.
However, our result implies that the impact of adding loss variable $Y$
into the prior risk profile $X$ is maximized at some $\alpha _0 < 1$.
Figure \ref {fig_1_2} shows the relation between
$\Delta \mathrm {ES}^{X, Y}_\alpha $ and $\Delta \mathrm {VaR}^{X, Y}_\alpha $.
We see that $\Delta \mathrm {ES}^{X, Y}_\alpha $ takes a maximum at $\alpha = \alpha _0$,
where $\alpha _0$ is a solution to
\begin{eqnarray}\label{alpha_VaR_ES}
\Delta \mathrm {VaR}^{X, Y}_{\alpha _0} = \Delta \mathrm {ES}^{X, Y}_{\alpha _0}.
\end{eqnarray}
Indeed, we have the following result.
\begin{proposition}\label{prop_max_ES}
If there is a unique solution $\alpha _0\in (0, 1)$ to $(\ref {alpha_VaR_ES})$,
then $\max _{0 < \alpha < 1}\Delta \mathrm {ES}^{X, Y}_\alpha = \Delta \mathrm {ES}^{X, Y}_{\alpha _0}$.
\end{proposition}
Note that unlike the case of SRMs,
$\Delta \mathrm {VaR}^{X, Y}_\alpha $ takes a value smaller than $\mathop {\rm E} [Y]$ if $\alpha $ is small.
This is because VaR is not a convex risk measure,
so the relation (\ref {lower_bound}) is not guaranteed for $\rho _\alpha = \mathrm {VaR}_\alpha $.
In particular, we observe that
\begin{eqnarray}\label{limit_VaR_zero}
\lim _{\alpha \rightarrow 0}\Delta \mathrm {VaR}^{X, Y}_\alpha =
\mathop {\rm essinf}(X+Y) - \mathop {\rm essinf}X = 0.
\end{eqnarray}
\noindent
{\it Case }(ii) \ $\beta < \gamma \leq \beta + 1$
Figure \ref {fig_2_1} shows the approximation errors, defined as
\begin{eqnarray}\label{def_error}
\mathrm {Error}_\alpha = \frac{\bar{M}(\alpha )}{\Delta \rho ^{X, Y}_\alpha } - 1
\end{eqnarray}
with $\xi _X = 2/3$ ($\beta = 1.5$) and $\xi _Y = 0.5$ ($\gamma = 2$).
We see that $\mathrm {Error}_\alpha $ is close to $0$ as $\alpha \rightarrow 1$
for each case of $\rho _\alpha = \mathrm {ES}_\alpha , \rho ^{\mathrm {EXP}}_\alpha , \rho ^{\mathrm {POW}}_\alpha $.
Moreover, we numerically verify the assertion of Theorem \ref {th_equivalence_Euler_EL}
for $\rho _\alpha = \mathrm {ES}_\alpha $ in Figure \ref {fig_2_1_1}.
We observe that $\bar{M}_\alpha / \mathrm {ES}^\mathrm {Euler}_\alpha $ converges to
$\delta = 1/\gamma = \xi _Y = 0.5$ as $\alpha \rightarrow 1$.
By contrast, the convergence speed of $\mathrm {Error}_\alpha $ as $\alpha \rightarrow 1$
decreases if the tails of $X$ and $Y$ are less fat-tailed.
Figure \ref {fig_2_2} shows $\mathrm {Error}_\alpha $ with $\xi _X = 2/7$ ($\beta = 3.5$) and $\xi _Y = 0.25$ ($\gamma = 4$).
We find that $\mathrm {Error}_\alpha $ decreases as $\alpha $ tends to $1$,
but the gap between $\mathrm {Error}_\alpha $ and $0$ is still large, even in the case $\alpha = 0.999$.
\ \\
\noindent
{\it Case} (iii) \ $\beta = \gamma $
Finally, we look at the case $\xi _X = \xi _Y = 0.7$.
The results are summarized in Figures \ref {fig_3_1} and \ref {fig_3_1_1}.
We see that $\mathrm {Error}_\alpha $ approaches $0$ as $\alpha \rightarrow 1$
for each case of $\rho _\alpha = \mathrm {ES}_\alpha , \rho ^{\mathrm {EXP}}_\alpha , \rho ^{\mathrm {POW}}_\alpha $.
We also confirm that $\bar{M}(\alpha )/ \mathrm {ES}^\mathrm {Euler}_\alpha $ converges to
$\delta = \{1 + k - (1+k)^{1-1/\beta }\} / k \approx 0.870$ as $\alpha \rightarrow 1$.
Similarly to Case (ii),
the convergence speed of $\mathrm {Error}_\alpha $ decreases as the tails of $X$ and $Y$
become thinner.
Figure \ref {fig_3_2} shows the graph of $\mathrm {Error}_\alpha $ with $\xi _X = \xi _Y = 0.3$.
The approximation error tends to zero as $\alpha \rightarrow 1$,
but remains smaller than $-20\% $ even when $\alpha = 0.999$.
\section{Conluding remarks}\label{sec_conclusion}
In this paper, we have studied the asymptotic behavior of the difference
between $\rho _\alpha (X + Y)$ and $\rho _\alpha (X)$ as $\alpha \rightarrow 1$
when $\rho _\alpha $ is a parameterized SRM satisfying (\ref {conv_worst_case}).
We have shown that $\Delta \rho _\alpha ^{X, Y}$ is asymptotically equivalent to $\bar{M}(\alpha )$ given by (\ref {def_M_bar}),
whose form changes according to
the relative magnitudes of the thicknesses of the tails of $X$ and $Y$.
In particular, for $\beta + 1 < \gamma $,
we found the convergence $\lim _{\alpha \rightarrow 1}\Delta \rho _\alpha ^{X, Y} = \mathop {\rm E} [Y]$
for general CLBSRMs $(\rho _\alpha )_\alpha $.
Moreover, we also found that
$\Delta \rho _\alpha ^{X, Y}\sim \delta \rho ^{\mathrm {Euler}}_\alpha (Y | X+Y)$ as $\alpha \rightarrow 1$
for a constant $\delta \in (0, 1]$ given by (\ref {def_delta}).
This clarifies the asymptotic relation between the marginal risk contribution and the Euler contribution.
Our numerical results in the case $\beta + 1 < \gamma $ showed that
$\Delta \rho _\alpha ^{X, Y}$ is not increasing but is unimodal with respect to $\alpha $, which implies that the impact of $Y$ in the portfolio $X+Y$ does not always increase with $\alpha $.
Interestingly, this phenomenon is inconsistent with intuition.
Our results essentially depend on the assumption that $X$ and $Y$ are independent.
However,
the dependence structure of the loss variables $X$ and $Y$ plays
an essential role in financial risk management.
The case of dependent $X$ and $Y$ for $\rho _\alpha = \mathrm {VaR}_\alpha $ has already been studied in Section A.1 of \cite {Kato_IJTAF}. As mentioned in Remark \ref {rem_main}, we have now generalized this result to the case of CLBSRMs.
However, we require the somewhat strong assumption that $X$ and $Y$ are not strongly dependent on each other.
With the additional analysis in Appendix \ref {sec_add} below,
we will see that our main results still hold
for a general dependence structure if $\beta + 1 < \gamma $,
but that they are easily violated if $\beta \leq \gamma \leq \beta + 1$.
In future work, we will continue to study the asymptotic behavior of
$\Delta \rho _\alpha ^{X, Y}$ as $\alpha \rightarrow 1$,
without the independence condition.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,891
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{"url":"https:\/\/maleadt.github.io\/LLVM.jl\/stable\/lib\/interop\/","text":"# Julia\/LLVM interop\n\nThis section lists functionality for connecting Julia with LLVM.jl, e.g. emitting code for the Julia JIT or creating types that are compatible with Julia's global state.\n\n## Base functionality\n\nLLVM.Interop.JuliaContextFunction\nJuliaContext()\nJuliaContext(f::Function)\n\nReturns the (session-bound) LLVM context used by the Julia compiler. This only works on Julia 1.5 or below; starting with Julia 1.6 there is no global context. On those versions, you need to use the do-block version of this function to create a temporary context, and pipe it through instead of assuming and accessing a single global contex. The do-block version also works on 1.5 and below, where it just returns the global context.\n\nsource\nBase.convertMethod\nconvert(LLVMType, typ::Type, ctx::Context; allow_boxed=true)\n\nConvert a Julia type typ to its LLVM representation. Fails if the type would be boxed.\n\nsource\nLLVM.Interop.create_functionFunction\ncreate_function(rettyp::LLVMType, argtyp::Vector{LLVMType}, [name::String])\n\nCreate an LLVM function, given its return type rettyp and a vector of argument types argtyp. The function is marked for inlining, to be embedded in the caller's body. Returns both the newly created function, and its type.\n\nsource\nLLVM.Interop.call_functionFunction\ncall_function(f::LLVM.Function, rettyp::Type, argtyp::Type, args::Expr)\n\nGenerate a call to an LLVM function f, given its return type rettyp and a tuple-type for the arguments. The arguments should be passed as an expression yielding a tuple of the argument values (eg. :((1,2))), which will be splatted into the call to the function.\n\nsource\n\n## Inline assembly\n\nLLVM.Interop.@asmcallMacro\n@asmcall asm::String [constraints::String] [side_effects::Bool=false]\nrettyp=Nothing argtyp=Tuple{} args...\n\nCall some inline assembly asm, optionally constrained by constraints and denoting other side effects in side_effects, specifying the return type in rettyp and types of arguments as a tuple-type in argtyp.\n\nsource","date":"2021-05-12 20:29:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.18697728216648102, \"perplexity\": 6383.68060123879}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243989705.28\/warc\/CC-MAIN-20210512193253-20210512223253-00450.warc.gz\"}"}
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Business of Esports TV: VR Surgery
(Livestream 140)
In this segment, we discuss Osso VR spending $27M to develop a surgery game.
The Business Of Esports brings you news, debates, and all the information you need to know about the gaming sector, the world's fastest-growing market. With Paul "The Profit" Dawalibi leading the charge, and a variety of special guests, BoE TV is the only place to find insider information on the esports industry!
Check out the full livestream here:
Paul Dawalibi 12:15
Let's talk about gaming, intersecting with other things. I have a couple of stories where gaming is intersecting with other industries. And I think they're interesting stories. And, and the first one here is this from TechCrunch. And the headline is Osso VR. Osso, I think it's pronounced Osso VR raises 27 million to turn surgery into a video game, basically, and I sort of, I don't love the wave. It's already the the article starts kind of down on VR. The the author Lucas says virtual reality did not turn into the ultimate office replacement telepresence machine during the pandemic. And it wasn't for lack of trying. But some startups focused on employee training and VR have found added validation, blah, blah, blah. So they're talking about Osso VR, VR startup focused on medical training. And the idea here is that you will be able to train surgeons without having you know, to use corpses and other things, you know, other more other real life alternatives in terms of surgical training, and that they will be able to do it all in VR. This was a big round 27 million Series B, led by GSR Ventures Signal Fire, additional participation from Signal Fire Kaiser Permanente Ventures, Anorak Ventures, among others. So, pretty interesting. They've got some corporate VCs in there, like Kaiser Permanente. But we're seeing gaming, and in this case, VR intersecting with health care in a very real way. anyone think this is a big trend? anyone think this will you know, who disagrees with even the beginning of this article, which is, you know, VR has sort of failed to solve the telepresence office telepresence problem.
William Collis 14:04
I don't know if I disagree with that. I think if like, I think, like, if you think of a moment for VR to have a watershed, that's not game driven. It was the COVID era, right? Like you had real need and you for people to have alternate solutions to have more emerge, like to have more communication and better richer communication from at home. Right. And you saw runs on people were going out and buying hardware to enable this there were like popular camera models for you know, like online learning were sold out. microphones were sold out, right, like, VR did not sell out, right? There wasn't a run on the index, so people could use it for you know, so I, I agree that I think VR had a moment where it was called to see if it could be, you know, something valuable for at least if it was at some time now to be an immersive office software. And it didn't answer right, I think. But that does not mean at all that VR doesn't have Tons of awesome applicable use cases, training being one of them. I think it's a good example. And I think this is a case where look like, it's difficult to train doctors, right. And doctors make a lot of money. So anything that simplifies or reduces the resource costs to the training, there's probably a high willingness to pay there on the other side. I think it makes sense. You know, like, I think this is sort of the thing where like, there's a lot of trainings that get put into VR that I wonder about, but this one seems a bit more intuitive, particularly because people told me anecdotally, but like surgeons, like amateur, like, first year, med students come in now way better at, you know, basic surgery or basic surgical skills than they used to, because of games, right. Like, I think it's well known in the medical industry that people have better hand eye because a lot of surgeons are coming in having game today. So I don't know, it's interesting, but it seems like there's a real synergy there.
Jimmy Baratta 15:52
sorry. Go ahead. Jimmy.
I just wanted to add to that, and kind of just add two anecdotes that I had in my life, which I think are parallel as well, as, you know, when when YouTube was really on the up and up, I had a friend of mine that was in medical school, and he was working with some investors and creating a platform where they would film surgeries and put them on a shareable site like that. So to wilms point, you know, these med students that are coming in being a little bit more advanced, or you know, having that know how, additionally, and this is an in medical, but a friend of mine owns about 40, fully immersive VR rigs, you know, like those kind of like space pods that you hop into 20 of them in LA 20 of them in London, he has a deal with one of the history museums in Great Britain. And you can do one of those King Tut walkthroughs. And, you know, in Egypt, and I just, you know, I love seeing stuff like that, because it's this application of essentially, what was gaming tech, but more so in history or in science. And so when I read this, I kind of disagreed with that first sentence. I mean, well, it's hard, you know, because I know all of those VR rigs couldn't be used during COVID because of health and safety issues. And because they were on alone here or there. And I think that that was more shot at the individual Oculus type units, right. But I all I wanted to say with my anecdotes was I've seen really awesome integrations or overlaps or what have you, in science in history in other areas, utilizing tech that essentially was created for gamers. And I think this is awesome. I can't wait to see where it goes in five years.
There's no question that the tech VR tech is good enough to to drive some of these applications like training surgeons, right. My question is, are there psychological barriers that remain? In other words,
Jimmy Mondal 17:35
Would you want to be operated on like, let's say you had a serious surgery? Right? And and somehow you have you know, that before you go in for surgery, the doctor who's operating on you, he or she trained entirely in VR, right?
There's no
I can see it now. Like, Oh, don't worry, like Dr. John is a pro he has over 2000 hours in Surgeon Simulator. Like he's good.
Suregon Simulator 3000
Yeah, like, oh, like that's like a pilot flying you who's only ever flown in Microsoft, you know, like,
my view is that eventually that psychological barrier probably will disappear like 50 years from now I guarantee people won't care that their surgeon only operated in VR before on on them because it it mentally I think will have been accepted. The question at do you guys feel there's a big barrier today the Jimmy M maybe
I think there's a barrier for the doctors themselves, right? Because I feel like there's just a difference in what you're operating on. You know, like the the tactile nature, I don't want to go into descriptives. But there's a lot so I think Yeah, like probably 90 10% would be great. Like, that'd be a great distribution. And I as a patient would feel comfortable with that when I was like, okay, he spent 2000 hours in surgeon simple 3000. And then the next, you know, 1000 hours doing real people like fine or whatever,
but like to the flip side to like, there is a first surgery, right? Like, it's not like every doctor goes into the market having done 10th out like there has to be a first one right? I guess if it's going to be my doctor's first surgery. I prefer that they've done 1000 hours in Surgeon Simulator, then like they're just starting Good luck.
At least with VR, at least at the VR training. You could have a very detailed readout about your surgeon right like he's time played 635 hours and scores. Like Like,
what clan is he in?
What do you do if you got your
weight? He didn't get the like so him backup accomplishments. What he's missing that, Dr. Lindsay? You're gonna say something?
Lindsay Poss 19:53
Oh, yeah, I was just going to say that to William's point. I don't think anyone's just goes in cold on their first surgery, I'm pretty sure there's assistance for quite a bit from an experienced surgeon. So just so you know, William, if you ever have to get surgery, it's not gonna be like their first rodeo all by themselves.
But my point is my point, though, is I'd rather have them had the VR training in addition to everything else, right, rather. So it doesn't have to be a substitution, it can just be another tool in a toolkit.
Yeah, and I completely agree with and I think, even though it's fun to make fun of the analytics they could receive, we all know that any feedback a computer gives is going to be vastly more accurate than a human feedback staring at it. So I mean, to be quite honest, I'd rather have robots doing my surgeries because the margin of error for robots there is quite slim. But if the best I can get as a robot training human how to do my surgery, then I suppose I'd take it.
It's true. Owen says I think VR has a lot more growth to be done. I think some say it has failed because expectations were too high at launch, from marketing as well as consumer.
Who here thinks expectations as part of the problem with VR?
Yeah, Gameboy VR was such a letdown. I've never
No, is it that like we were promised flying cars, and this is what we got? Like, is it that kind of effect? Does anyone think expectations are actually part of the problem?
maybe to some extent,
in the gaming world, I definitely have noticed that. I don't know how that translates here to science. But I think that's a great observation.
I definitely
I think that you first?
Well, I definitely I think VR was marketed as like, when the first generation stuff was coming out is like, this is the watershed moment, right? Like it's ready, it's going to be immersive. And in some sense, I think maybe the displays were very good, even from the beginning. But I think VR taught us there was a bigger problem to be solved in just the displays. And I think the industry made such a big Wow, it is true when you put up your VR set it's impressive. You definitely are like, oh, wow, this is different. Like, you're ready. But the problem is there's other gaps that hold it back. And I think the marketing was disingenuous about, hey, the displays are great, but we didn't figure out how you should move.
Yeah, we didn't tell you that you needed a nuclear generator in your backyard to be able to run this thing like 10 frames a second. Sorry, good luck.
Robert says I've created 3D stereoscopic visualization or suites. And VR is important for training the surgeons on neurosurgery for example. Wow, interesting. My surgeon would have to be ranked diamond at a minimum says Wade.
But you could get a discount at gold though.
Those guys would just tell other guys we just held back by the other surgeons in their training guys.
I promise man, it wasn't me. It was my team. It was my team
Robert says AR headsets have overlays for surgery such as carpal tunnel to advise through the operation that
I love that
See, that's cool. Right? That is really cool. Robert also says developed in Unity, which is where the game architecture systems have such involved industrial and medical applications. No, like, no question. The medical applications I think are fascinating.
0 comments on "Business of Esports TV: VR Surgery"
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,574
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La retama amarilla (Retama sphaerocarpa L.) es un arbusto perteneciente a la familia de las fabáceas. Es originaria del Norte de África y de la península ibérica. Muy frecuente en casi toda la Península, excepto en los Pirineos, Cornisa Cantábrica, Galicia y buena parte de Portugal.
Descripción
Es un arbusto que puede alcanzar 3 m de altura; generalmente desprovisto de hojas, grisáceo y muy ramificado. Posee (o no) las hojas alternas, linear lanceoladas, tempranamente caedizas. Las flores son papilionáceas, muy pequeñas de 5-8 mm de longitud, agrupadas en racimos. Cáliz de 2 a 3,5 mm, bilabiado; el labio superior profundamente bífido, y el inferior dividido en 3 dientecitos agudos. Legumbre más o menos ovoidea, con el mucrón muy poco marcado, de color pajizo. Es fácil de distinguir por sus hojas lineares y, sobre todo, por la cantidad de flores amarillas que produce. Su producción es tan grande, que puede dar la impresión de que es una planta que solo tiene tronco y un montón de pétalos.
Hábitat
Crece en matorrales seriales producidos por la degradación de los encinares y en pinares.
Distribución
Nativa del noroeste de África y de la península ibérica. Es una especie común en casi toda la Península, a excepción del norte y buena parte de Portugal. Es xerófila, tolerante a los fríos invernales y a los calores estivales; puede vegetar tanto en suelos calizos como en silíceos desde 0 a 400 msnm. Puede formar matorrales muy extensos, especialmente por Aragón, La Mancha, sur de Extremadura y Andalucía, donde pasta generalmente el ganado ovino, por lo general en encinares degradados.
Principios activos
En la corteza y las ramas hallamos un alcaloide llamado retamina y otro denominado d-esparteína, estructuralmente afín. Su uso interno es desaconsejeble por su toxicidad.
Uso tradicional
Los peregrinos del camino de Santiago recogían esta planta para tratar afecciones del aparato respiratorio en fase aguda y en las fiebres eruptivas. En Galicia tradicionalmente se ha empleado para hacer escobas destinadas a barrer. En el pasado fue utilizada para calentar los hornos de las tahonas.
Taxonomía
Retama sphaerocarpa fue descrita por (L.) Boiss. y publicado en Voyage botanique dans le midi de l'Espagne 2: 144. 1840.
Etimología
Retama: nombre genérico que deriva de Retáma, -ae f. – del árabe andalusí ratama (ár. culto ratam); castellano: retama f. = nombre de no pocas genísteas, como las retamas propiamente dichas –Retama monosperma (L.) Boiss., R. sphaerocarpa (L.) Boiss. y R. raetam (Forssk.) Webb– y especies varias de los géneros Cytisus L., Genista L. y Spartium L.
sphaerocarpa: epítetocompuesto por las palabras griegas σφαιρα —sphera— y καρποϛ —karpos— que significa "con frutos esféricos".
Citología
Números cromosomáticos de Retama sphaerocarpa (Fam. Leguminosae) y táxones infraespecificos: n=24; 2n=48
Sinonimia
Lygos sphaerocarpa (L.) Heywood in Feddes Repert. 79: 53 (1968)
Boelia sphaerocarpa (L.) Webb, Otia Hispan. ed. 2 21, tab. 15 (1853)
Genista sphaerocarpos (L.) Lam., Encycl. 2: 616 (1788)
Spartium sphaerocarpum L., Mant. Pl. 571 (1771)
Retama atlantica Pomel in Bull. Soc. Sci. Phys. Algérie 11: 172 (1874)
Retama sphaerocarpa var. mesogaea (Webb) Willk. in Willk. & Lange, Prodr. Fl. Hispan. 3: 419 (1877)
Boelia sphaerocarpa var. mesogaea Webb, Otia Hispan. ed. 2 21 (1853)
Nombres comunes
Castellano: chinastra, chinestra, escoba alta, escoba florida, escobeta, giniestra, jinestrera, jinestrón, lluvia de oro, retama, retama amarilla, xesta, xestas, retama blanca, retama borde, retama común, retama de bolas, retama de monte, retamón, xinestra.
Referencias
Voyage botanique dans le midi de l'Espagne 2:144. 1840
USDA, ARS, National Genetic Resources Program. Germplasm Resources Information Network - (GRIN) [Data from 07-Oct-06].
Bibliografía
Greuter, W. et al. (Eds.) (1989) Med-Checklist Vol. 4 (published)
Vicioso, C. (1953) Genisteas Espaniolas 2. Min. de Agric. Madrid No. 72
Polunin, O. & Smythles, B.E. (1973) Fls. of South-West Europe. Oxford Univ. Press
Heywood, V.H. & Ball, P.W. (1968) Leguminosae. In: Flora Europaea Vol. 2. ed. Tutin, T.G. et al.
Burkart, A. (1952) Acme Agency, Buenos Aires 569 pp Las Leguminosas Argentinas
Quezel, P. & Santa, S. (1962) Nouvelle flore de l'Algerie et des regions desertiq. merid. 1
Negre, R. (1961) Petite flore des regions arides du Maroc occidentale. 1. CNRS.
Maire, R. (Quezel, P., Ed.) (1987) Flore de l'Afrique du Nord, Vol. 16. Dicots. Leguminosae, part.
Dominguez, E. (1987) Pap. in Flora Vasc. de Andalucía Occ. Ketres, Barcelona
Ozenda, P. (1977) Flore du Sahara, Ed. 2. Edns. CNRS, Paris.
Polunin, O. (1976) Trees and Bushes of Europe. Oxford Univ. Press
Enlaces externos
sphaerocarpa
Plantas medicinales
Flora de la cuenca mediterránea
Endemismos iberomagrebíes
Plantas descritas en 1840
Plantas descritas por Linnaeus
Plantas descritas por Boissier
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,259
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Q: datatable: automatical refresh after a specified time Could you recommend me some easy way to automatically refresh of dataTable each e.g. 10 minutes?
<rich:dataTable id="table" var="item" value="#{bean.model}">
<rich:columnGroup>
<rich:column><h:outputText value="#{item.id}"/></rich:column>
<rich:column><h:outputText value="#{item.name}"/></rich:column>
...
</rich:columnGroup>
<f:facet name="footer">
<h:commandButton id="load" action="#{bean.loadData}"/>
</f:facet>
</rich:dataTable>
I've tried a few ways but it always had a catch, so I'm looking for some common method..
A: Try Richfaces a4j:poll (link)
<h:form>
<a4j:poll id="poll" interval="600000" reRender="table"/>
</h:form>
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,224
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Toronto, ON (April 17, 2019) – Atlific Hotels is delighted to announce that Lake Louise Inn has received the prestigious Employer of Choice designation for its outstanding achievements in human resources practices in 2018. The accolade was awarded to the hotel last night by The Alberta Hotel and Lodging Association (AHLA) at its 99th Annual Convention and Trade Show.
The Employer of Choice designation is bestowed upon properties that go above and beyond standard human resources practices to create safe, fair and positive workplace environments for their staff.
A distinguished force in the Canadian hotel management industry, Atlific Hotels has over 60 years of experience managing highly acclaimed properties across the country. With over 55 dynamic hotels in their roster, Atlific strives to train and inspire its team members to provide both an ethical and exceptional guest service experience.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Anyone asking you to work on a business plan while you're still searching for a virtuous business model isn't an investor you want. Who you raise money from is a key question - not every dollar is the same.
Good money comes when you can reverse the balance of power. Raising funds is like a mix between playing poker and buying a house. You need to go all in with your own cards. But you can't keep a poker face in front of the landlord, because it could be a long-term relationship, and that should be based on honesty.
Oussama Ammar is the co-founder of The Family. At The Family, we helped our portfolio startups to raise $500M+. Time to show you the key lessons we've taken away from it all.
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{
"redpajama_set_name": "RedPajamaC4"
}
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\section{Introduction and main results}
In recent years, dynamical systems with different
non-uniform hyperbolicity conditions
have been studied. Speaking about one-dimensional (real or complex) dynamics,
such restrictions are often put on
the derivatives at critical values of the map (see last Section).
Simplest and most studied are unicritical polynomial maps $f(z)=z^d+c$
and exponential maps $E(z)=a\exp(z)$. In the first two results, we
prove that for each such polynomial or exponential
map without sinks, but otherwise arbitrary,
there is always
a certain expansion along the critical orbit.
\begin{theorem}\label{uni}
Let $f(z)=z^d+c$, where $d\ge 2$ and $c\in \C$. Assume that $c$ does not belong
to the basin of an attracting cycle.
Then
$$\chi_-(f, c)=\liminf_{n\to\infty} \frac{1}{n}\log |Df^n(c)|\ge 0.$$
\end{theorem}
Theorem~\ref{uni} has been known before for real $c$
(more generally, for S-unimodal maps of an interval)~\cite{nowsan}.
\begin{theorem}\label{theo:exp}
Let $E(z)=a\exp(z)$, where $a\in \C
\setminus \{0\}$. Assume that $0$ does not belong to the basin of an attracting
cycle. Then
$$\chi_-(E, 0)=\liminf_{n\to\infty} \frac{1}{n}\log |DE^n(0)|\ge 0.$$
\end{theorem}
These two theorems are special cases of the following theorem.
Let $\sU=\sU_{V,V'}$ be the set of all holomorphic maps $f:V\to V'$ between open sets $V\subset V'\subset \C$,
for which there exists a unique point $c=c(f)\in V'$
and a positive number $\rho=\rho(f)$ with the following properties:
\begin{enumerate}
\item[(U1)] $f: V\setminus f^{-1}(c)\to V'\setminus \{c\}$ is an unbranched covering map;
\item[(U2)] for each $n=0,1,\ldots$, $f^n(c)$ is well defined and $B(f^n(c),\rho(f))\subset V'$.
\end{enumerate}
\begin{theorem}\label{thm:main}
For any $f\in \sU_{V,V'}$, if $c(f)$ does not belong to the basin of an attracting cycle, then $\chi_{-}(c(f))\ge 0$.
\end{theorem}
See next Section for the proof and the last Section for some applications.
On the other hand, we have
\begin{theorem}\label{thm:typ}
Let $f(z)=z^d+c$, where $d\ge 2$ and $c\in \C$. Assume that the Julia set
$J(f)$ of $f$ has a positive area. Then for almost every $z\in J(f)$, there exists
$$\chi(f, z)=\lim_{n\to\infty} \frac{1}{n}\log |Df^n(z)|=0.$$
\end{theorem}
Quadratic polynomials with a positive area Julia set do exist~\cite{BC}.
For the proof of Theorem~\ref{thm:typ}, see Section~\ref{typ}. To show that $\chi_{-}(f, z)\ge 0$
for a typical point we introduce the notion of a {\em slowly recurrent point} $z$ for a map $f\in \sU_{V,V'}$
and prove that $\chi_{-}(f, z)\ge 0$ for such $z$, see next Section.
In Section~\ref{typ} we show that for $f(z)=z^d+c$ a.e. point of $J(f)$ is slowly recurrent.
In the opposite direction,
that any $z\in J(f)$ with a positive upper Lyapunov exponent $\chi_{+}(f, z)$ is not a density point of $J(f)$
is an immediately consequence of the following general fact:
\begin{theorem}\label{thm:up}
Let $g$ be a polynomial of degree at least $2$.
For every $\lambda>1$ there exist $\rho>0$, a positive integer $N$ and
$\alpha>0$ as follows. Suppose
$$\chi_+(g, z)=\limsup_{n\to\infty} \frac{1}{n}\log |Dg^n(z)|>2\log \lambda$$
for some $z\in J(g)$.
Then there exists a
subset $H$ of the positive
integers such that the upper density of $H$ is at least $\alpha$,
and for every $n\in H$,
if $V_n$ denotes a connected component of $g^{-n}(B(g^{n}(z),\rho))$
which contains the point $z$ then $V_n\subset B(z, \lambda^{-n}\rho)$ and
the map $g^{n}: V_n\to B(g^{n}(z),\rho)$ is at most $N$-critical.
\end{theorem}
For the proof of a yet more general version, see Section~\ref{up}.
See also Corollary~\ref{renorm}.
Finally, for the unicritical polynomials we have the following
\begin{theorem}\label{thm:backward}
Suppose $f(z)=z^d+c$ has no an attracting cycle in $\C$.
Let $\bar x=\{x_{-n}\}_{n=0}^\infty$, $x_0=0$, $f(x_{-n})=x_{-(n-1)}$, $n>0$,
be a backward orbit of $0$. Then
$$\chi_-^{back}(f, \bar x):=\liminf_{n\to\infty} \frac{1}{n}\log |Df^n(x_{-n})|\ge 0.$$
\end{theorem}
This theorem can be deduced from Theorem~\ref{uni} by modifying the proof of Propsition 1 in \cite{GS}.
(We leave the details to an interested reader.) In Section~\ref{up},
we shall provide a proof based on a modification of our argument in Section~\ref{1}.
\section{Proof of Theorem~\ref{thm:main}}\label{1}
Let $f: V\to V'$ be a map in $\sU$ and let $c=c(f), \rho=\rho(f)$.
Furthermore,
let $AB(f)$ denote the union of the basin of attracting cycles of $f$.
So when $f$ has no attracting cycle, $AB(f)=\emptyset$.
We need two lemmas for the proof of Theorem~\ref{thm:main}.
In the first Lemma
a general construction is introduced which is used also
later on. Throughout the proofs, the Koebe principle applies.
\begin{lemma} \label{lem:return}
Assume that $c$ is not a periodic point. Given $\lambda>1$ there exists $\delta_0$, such
that for each $\delta\in (0, \delta_0)$,
if $n\ge 1$ is the first entry time of $z\notin
AB(f)$ into $\overline{B(c;\delta)}$, then
\begin{itemize}
\item if either $|z-c|\le \delta$ or there is no
neighborhood of $z$ such that $f^n$ maps it conformally onto
$U_n=B(f^n(z), |f^n(z)-c|)$, then
\begin{equation}\label{lambdadelta}
|Df^{n}(z)|\ge \lambda^{-n}\frac{|f^n(z)-c|}{\max\{\delta, |z-c|\}};
\end{equation}
\item otherwise, i.e., if $|z-c|>\delta$ and $f^{n}$ maps a neighborhood of $z$ conformally onto
$U_n$, then
\begin{equation}\label{nolambdadelta}
|Df^{n}(z)|\ge \frac{|f^n(z)-c|}{12|z-c|}.
\end{equation}
\end{itemize}
\end{lemma}
\begin{proof}
Let $\delta\in (0,\rho/2]$. Let $n\ge 1$ and $z\in \overline{B(c,\delta)}\setminus AB(f)$ be as in the Lemma, and write $z_i=f^i(z)$.
Let $\{\tau_i\}_{i=0}^n$ be a sequence of positive numbers with the following properties:
\begin{enumerate}
\item $\tau_n=|z_n-c|$ and $U_n=B(z_n, \tau_n)$;
\item for each $0\le i< n$, $\tau_{i}$ be the maximal number such that
\begin{itemize}
\item $0< \tau_{i}\le \tau_{i+1}$ and
\item $f^{n-i}$ maps a neighborhood $U_i$ of $z_i$ conformally onto $B(z_n, \tau_{i})$.
\end{itemize}
\end{enumerate}
Let $$\mathcal{I}=\{0\le i<n: \tau_{i}<\tau_{i+1} \}$$ and let $$N=\# \mathcal{I}.$$
Note that for each $i\in \mathcal{I}$, $c\in \partial f(U_i)$.
Since $f^{n}$ maps $U_0$ conformally onto $B(z_n, \tau_0)$, by the Koebe $\frac{1}{4}$ Theorem, we have
\begin{equation}\label{eqn:dertau0}
|Df^{n}(z)|\ge \frac{\tau_0}{ 4\eps_0}, \text{ where } \eps_0=\dist(z_0, \partial U_0).
\end{equation}
{\bf Claim 1. } There exists a universal constant $K>1$ such that for each $i\in\mathcal{I}$, we have
$\tau_{i+1}\le K \tau_i$. Moreover,
\begin{equation}\label{eqn:diamU1}
\eps_0\le 2 \delta+|z-c|\le 3 \max\{\delta, |z-c|\}.
\end{equation}
\begin{proof}[Proof of Claim 1]
We first note that for each $0\le i<n$, $U_{i}\not\supset \overline{B(c, 2\delta)}$
for otherwise,
$$f^{n-i}(U_{i})= B(z_n, \tau_{i})\subset \overline{B(c, 2\delta)}\subset U_{i},$$
which implies by the Schwarz lemma that $z_{i}$, hence $z_0$, is contained in the basin of an attracting cycle of $f$, a contradiction!
The inequality (\ref{eqn:diamU1}) follows.
Now let $i\in \mathcal{I}$. Then $i<n-1$ and $U_{i+1}\supset \partial f(U_i)\ni c$, so
\begin{equation}\label{diam}
\diam (f(U_i))\ge |c-z_{i+1}|\ge \delta.
\end{equation}
Since $U_{i+1}\not\supset \overline{B(c,2\delta)}$,
it follows that $\textrm{mod}(U_{i+1}\setminus f(U_i))$ is bounded from above by a universal constant.
Since $f^{n-i-1}: U_{i+1}\to B(z_n, \tau_{i+1})$ is a conformal map, we have
$$\textrm{mod}(U_{i+1}\setminus f(U_i))=\log \frac{\tau_{i+1}}{\tau_i}.$$
Thus $\tau_{i+1}/\tau_i$ is bounded from above by a universal constant.
\end{proof}
By (\ref{eqn:dertau0}), it follows that
\begin{equation}\label{genin}
|Df^n(z_0)|\ge \frac{|\tau_n|}{\max\{\delta, |z-c|\}} (12 K^N)^{-1}.
\end{equation}
Since $c$ is not a periodic point,
$$C(\delta)=\inf\{m\ge 1: \exists z\in B(c,2\delta) \text{ such that } f^m(z)\in B(c,2\delta)\}\to \infty$$
as $\delta\to 0$. Thus given $\lambda>1$, there is $\delta_0>0$ such that when $\delta\in (0,\delta_0]$ we have
$$12K\le \lambda^{C(\delta)}.$$
For $i<i'$ in $\mathcal{I}\cup \{n-1\}$, we have $w:=f^{n-i'-1}(c)\in B(c;2\delta)$ and $f^{i'-i}(w)=f^{n-i-1}(c)\in B(c,2\delta)$,
$i'-i\ge C(\delta)$. Thus $n\ge C(\delta) N.$
Consider several cases. If $N\ge 1$, then
$$\lambda^n\ge \lambda^{C(\delta)N}\ge 12 K^N.$$
By~(\ref{genin}), the inequality~(\ref{lambdadelta}) holds in this case.
If $|z-c|\le \delta$,
since $z\in B(c;2\delta)$ and $f^n(z)\in B(c,2\delta)$ we have $n\ge C(\delta)$.
Hence, if $|z-c|\le \delta$ and $N=0$,
$$12\le \lambda^{C(\delta)}\le \lambda^n.$$
By~(\ref{genin}), then~(\ref{lambdadelta}) holds again.
Finally, if $N=0$, by~(\ref{genin}), the inequality~(\ref{nolambdadelta}) holds.
\end{proof}
\begin{lemma}\label{lem:awayfrom0}
There exists $M=M(f)>1$ and given $\lambda>1$ and $\delta\in (0, 1)$ there exists
$\kappa=\kappa(\delta, \lambda)$ such that whenever $z\notin AB(f)$,
$|f^j(z)-c|\ge \delta$ holds for $0< j\le n$
and $B(f^n(z),\delta)\subset V'$, we have
$$|Df^n(z)|\ge \frac{\kappa \lambda^{-n}}{\max\{M, |z-c|\}}.$$
\end{lemma}
\begin{proof} Fix $\lambda$ and $\delta$.
We define the numbers $\tau_i$, domains $U_i$, $0\le i\le n$ and the number $\eps_0$
as in the proof of Lemma~\ref{lem:return}, with the only difference that
we start with $\tau_n=\delta$.
Let us show that there exists $M=M(f)$ such that
\begin{equation}\label{eqn:Ui2}
U_i\not\supset \overline{B(c, M+\delta)}\text{ for each }0\le i<n.
\end{equation}
If $V\not=\C$, we define
$M=\dist(c, \partial V)+1$. Then~(\ref{eqn:Ui2}) is obvious since $U_i$ must be in $V$.
If $V=\C$, then also $V'=\C$. In this case,
$f: \C\to \C$ is either a polynomial or a transcendental entire function.
It always has a (finite) periodic orbit $P$ (this fact is trivial for polynomials, and was proved by Fatou for
entire functions). Define $M=\max\{|w-c|: w\in P\}+1$.
Then~(\ref{eqn:Ui2}) holds,
for otherwise, we would have that $B(z_n, \delta)=f^{n-i}(U_i)\supset f^{n-i}(P)=P$, hence,
$B(z_n, \delta)\subset B(c, M+\delta)$.Then
$\overline{f^{n-i}(U_i)}=\overline{B(z_n, \delta)}\subset \overline{B(c, M+\delta)}\subset U_i$,
which would then imply $z_i\in AB(f)$ and hence $z\in AB(f)$, a contradiction.
It follows that
\begin{equation}\label{eqn:eps0second}
\eps_0\le M+|z-c|+\delta\le 3\max\{M, |z-c|\}.
\end{equation}
To complete the proof, we need to consider the following set of indexes
$$\mathcal{I}_\lambda=\{i\in
\mathcal{I}: \lambda \tau_i\le \tau_{i+1}\}.$$
For each $i\in\mathcal{I}_\lambda$, $\diam (f(U_i))\ge |z_{i+1}-c|\ge \delta$.
Since $f^{n-i-1}$ maps $f(U_i)$ onto $B(z_n, \tau_i)$ with a distortion which depends merely on
$\lambda$, there exists $\alpha=\alpha(\lambda)>0$ such that
\begin{equation}\label{eqn:alpha}
B(z_{i+1}, \alpha \delta) \subset f(U_i).
\end{equation}
Moreover, by (\ref{eqn:Ui2}), there exists $K=K(\delta, \lambda)>0$ such that
\begin{equation}\label{eqn:K}
\frac{\tau_{i+1}}{\tau_i}=e^{\mod(U_{i+1}\setminus f(U_i))}\le K.
\end{equation}
Furthermore, by (\ref{eqn:Ui2}) and $\tau_{i+1}\ge \lambda \tau_i$,
there exists a constant $D=D(\delta,\lambda)>0$ such that
\begin{equation}\label{eqn:D}
|z_{i+1}-c|\le D.
\end{equation}
Let us prove that there exists $m_0=m_0(\lambda,\delta)$ such that $\#\mathcal{I}_\lambda\le m_0.$
To this end, let $i(0)<i(1)<\cdots<i(m-1)$ be all the elements of $\mathcal{I}_\lambda$. For each $0\le j\le j'<m$, we have
$$\mod(U_{i(j')+1}\setminus f^{i(j')-i(j)+1}(U_{i(j)}))=
\log \frac{\tau_{i(j')+1}}{\tau_{i(j)}}\ge \lambda\cdot (j'-j+1).$$
By (\ref{eqn:Ui2}),
it follows that there exists $m_1=m_1(\delta,\lambda)$ such that for $0\le j<j'<m$ with $j'-j\ge m_1$,
$\diam (f^{i(j')-i(j)+1}(U_{i(j)}))\le \alpha\delta/2$. For such $j, j'$,
since $z_{i(j)+1}\not\in AB(f)$, $f^{i(j')-i(j)+1}(U_{i(j)})$ is not properly contained in $f(U_{i(j)})$,
and thus by (\ref{eqn:alpha}), we have $|z_{i(j)+1}-z_{i(j')+1}|\ge \alpha\delta/2$.
In particular, the distance between
any two distinct points in the set
$\{z_{i(km_1)}: 0\le k< m/m_1\}$
is at least $\alpha\delta/2$. By (\ref{eqn:D}),
the last set is contained in a bounded set $\overline{B(c,D)}$,
thus its cardinality is bounded from above by a constant. Thus $m=\#\mathcal{I}_\lambda$ is bounded from above.
It follows that
$$\tau_0\ge \tau_n K^{-m_0}\lambda^{-(n-m_0)}\ge \delta K^{-m_0}\lambda^{-n}. $$
So
$$|Df^{n}(z)|\ge \frac{\tau_0}{4 \epsilon_0}\ge\frac{\kappa \lambda^{-n}}{\max\{M, |z-c|\}},$$
where $\kappa=\delta K^{-m_0} 4^{-1}$.
\end{proof}
As a corollary of Lemmas~\ref{lem:return}-\ref{lem:awayfrom0}, we have
\begin{lemma}\label{lem:pass}
Assume that $c$ is not a periodic point.
Given $\lambda>1$ and $\sigma\in (0,1)$ there exists $C=C(\lambda, \sigma)>0$
such for every $z_0\notin AB(f)$ and $s\ge 1$ whenever
$B(f^s(z_0),\sigma)\subset V'$, we have:
if $z_0=c$, then
$$|Df^s(c)|\ge C \lambda^{-s},$$
and if $z_0\not=c$, then
$$|Df^s(z_0)|\ge C \frac{\min\{1, \inf_{i=0}^s |f^i(z_0)-c|\}}{|z_0-c|} \lambda^{-s}.$$
\end{lemma}
\begin{proof}
Fix $\lambda>1$ and $\sigma>0$, let
$\delta_0=\delta_0(\lambda)\in (0,\sigma]$ be given by Lemma~\ref{lem:return} and let
$\tilde\kappa=\kappa(\delta_0(\lambda), \lambda)$ be given by
Lemma~\ref{lem:awayfrom0}.
Define $\epsilon_0=\inf_{i=0}^s |f^i(z_0)-c|$.
If $\epsilon_0> \delta_0$, we are done by
Lemma~\ref{lem:awayfrom0} with $C=\tilde\kappa$.
Otherwise let $s_0\in \{0,...,s\}$ be minimal such that $|f^{s_0}(z_0)-c|=\eps_0$
and $s_{max}\in \{s_0,...,s\}$ be maximal such that $|f^{s_{max}}(z_0)-c|\le \delta_0$.
We show that
\begin{equation}\label{s0s}
Df^{s-s_0}(f^{s_0}(z_0))|\ge \frac{\tilde\kappa}{M} \lambda^{-(s-s_0)}.
\end{equation}
Indeed, if $s_{\max}=s_0$, we are done by Lemma~\ref{lem:awayfrom0}, where we put $z=f^{s_0}(z_0)$ and $\delta=\delta_0$.
Let $s_0<s_{max}$.
Define $\eps_1=\inf_{i=s_0+1}^{s_{max}} |f^i(z_0)-c|$ and let $s_1\in \{s_0+1,...,s_{max}\}$ be minimal
such that $|f^{s_1}(z_0)-c|=\eps_1$. If $s_1=s_{max}$ then we stop. Otherwise, define
$\eps_2=\inf_{i=s_1+1}^{s_{max}} |f^i(z_0)-c|$ and let $s_2\in \{s_1+1, \ldots, s_{max}\}$
be minimal such that $|f^{s_2}(z_0)-c|=\eps_2$. Repeating that argument,
we obtain a sequence of positive numbers $0<\eps_1\le \eps_2\le \cdots\eps_k\le \delta_0$
and a sequence of integers $0\le s_0<s_1<s_2<\cdots<s_k=s_{max}$.
Applying Lemma~\ref{lem:return} to $z=f^{s_0}(z_0)$, $\delta=\eps_1$ and $n=s_1-s_0$, we obtain
$$|Df^{s_1-s_0}(f^{s_0}(z_0))|\ge \lambda^{-(s_1-s_0)}.$$
For each $i=2,3,\ldots, k$, applying Lemma~\ref{lem:return} to $z=f^{s_{i-1}}(z_0)$, $\delta=\eps_{i}$ and $n=s_i-s_{i-1}$, we obtain
$$|Df^{s_i-s_{i-1}}(f^{s_{i-1}}(z_0))|\ge \lambda^{-(s_i-s_{i-1})}.$$
Therefore
$$|Df^{s_{max}-s_0}(f^{s_0}(z_0))|=\prod_{i=1}^k |Df^{s_i-s_{i-1}}(f^{s_{i-1}}(f^{s_0}(z_0)))|
\ge \lambda^{-(s_{max}-s_0)}.$$
If $s_{max}<s$, we can further apply Lemma~\ref{lem:awayfrom0}
with $z=f^{s_{max}}(z_0)$, $\delta=\delta_0$ and $n=s-s_{max}$:
$$|Df^{s-s_{max}}(f^{s_{max}}(z_0))|\ge \frac{\tilde\kappa}{M} \lambda^{-(s-s_{max})}.$$
Thus~(\ref{s0s}) follows:
$$|Df^{s-s_0}(f^{s_0}(z_0))|\ge \lambda^{-(s_{max}-s_0)}\frac{\tilde\kappa}{M} \lambda^{-(s-s_{max})}=
\frac{\tilde\kappa}{M}\lambda^{-(s-s_0)}.$$
This proves the Lemma if $s_0=0$ (including the case $z_0=c$).
If $s_0>0$, we apply Lemma~\ref{lem:return} for $z=z_0$, $\delta=\epsilon_0$
and $n=s_0$:
$$|Df^{s_0}(z_0)|\ge \lambda^{-s_0} \frac{\epsilon_0}{12 \max\{\epsilon_0, |z_0-c|\}}=\lambda^{-s_0}
\frac{\epsilon_0}{12 |z_0-c|}.$$
Combining this with~(\ref{s0s}) we obtain finally:
$$|Df^s(z_0)|\ge C \frac{\inf_{i=0}^s |f^i(z_0)-c|}{|z_0-c|} \lambda^{-s},$$
where $C=\tilde\kappa/(12M)$.
\end{proof}
Theorem~\ref{thm:main} follows at once from Lemma~\ref{lem:pass}.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
We may certainly assume that $c$ is not periodic. Fix $\lambda>1$.
Applying Lemma~\ref{lem:pass} with $z_0=c$, we find $C>0$ such that for each $s\ge 1$,
$$|Df^s(c)|\ge C \lambda^{-s}.$$
Hence, $\chi_{-}(f, c)\ge -\log\lambda$ for every $\lambda>1$
\end{proof}
Let us formulate another consequence of Lemma~\ref{lem:pass}.
\begin{definition} Let $z\notin AB(f)$ such that the forward orbit of $z$ is well-defined.
We call $z$
{\em (exponentially) slowly recurrent} if
for any $\alpha>0$, $|f^n(z)-c|\ge e^{-\alpha n}$ holds for every large $n$.
\end{definition}
As in the proof of Theorem~\ref{thm:main}, Lemma~\ref{lem:pass} implies:
\begin{lemma}\label{lem:slexp}
If $z$ is a slowly recurrent point and there exists $\delta>0$
such that $B(f^n(z), \delta)\subset V'$ for every $n$,
then $\chi_{-}(f,z)\ge 0$.
\end{lemma}
\section{Expansion along the orbit}\label{up}
Let us fix a polynomial $g$ of degree at least $2$.
Theorem~\ref{thm:up} of the Introducton is an immediate corollary
of the following.
\begin{theorem}\label{thm:positiv}
For every $\lambda>1$ and $\epsilon_0>0$ there exist $\rho>0$,
$N, \tilde n\in \Z^+$ and
$\alpha>0$ (which depend on $g$ and $\lambda$, $\epsilon_0$ only) as follows.
Let $z\in J(g)$ and $m\in \Z^+$. Assume
\begin{equation}\label{chi+}
\frac{1}{m}\log |Dg^m(z)|>\epsilon_0+\log \lambda.
\end{equation}
Then there exists a
subset $H_m$ of the set $\{1,2,...,m\}$ such that the following hold:
\begin{enumerate}
\item[(a)] $\#\ H_m\ge \alpha m$,
\item[(b)] for every $n\in H_m$,
if $V_n$ is a connected component of $g^{-n}(B(g^{n}(z),\rho))$
which contains the point $z$ then
the map $g^{n}: V_n\to B(g^{n}(z),\rho)$ is at most $N$-critical,
\item[(c)] $V_n\subset B(z, \lambda^{-n}\rho)$ whenever $n\in H_m$ and $n\ge \tilde n$.
\end{enumerate}
\end{theorem}
This result has already been referred to in~\cite{GPR}.
\begin{remark}\label{ratexp}
Theorem~\ref{thm:positiv} as well as its proof (see below) hold for rational functions as well
(derivatives are then
taken in the spherical distance).
\end{remark}
We need some preparations for the proof.
\begin{definition} We say that $m$ is a {\em $\lambda$-hyperbolic time} for $z$
if $$|Dg^{m-i}(g^i(z))|\ge \lambda^{m-i}$$ holds for each $i=0,1,\ldots, m-1$.
\end{definition}
Let $W_\delta^k(z)$ , $k>0$, denote the connected component of $g^{-k}(B(f^k(z),\delta))$ which contains $z$.
Next Lemma is known as the ``telescope lemma'' of~\cite{P0} (see also~\cite{PRS}, Lemma 2.3
for a simplified method and~\cite{GPR} for $C^1$ multimodal interval maps):
\begin{lemma}\label{lem:hyp}
Let $\lambda>1$ and $\epsilon>0$.
There exist $C=C(\lambda,\epsilon)$ and
$\delta_0=\delta_0(\lambda,\epsilon)>0$ as follows.
Assume that $s$ is a
$\lambda e^{\epsilon}$-hyperbolic time for $z\in J(g)$. Then, for
every $\delta\in (0, \delta_0]$ and every $i=0,...,s-1$,
\begin{equation}\label{backall}
W_\delta^{s-i}(g^i(z))\subset B(g^i(z), C \delta \lambda^{-(s-i)}).
\end{equation}
\end{lemma}
\
Denote by $Crit'$ the set of critical points of $g$ which lie in the Julia set $J(g)$.
Recall the definition of ``shadow''~\cite{PR}.
Fix $z\in J(g)$. For $n\in {\bf N}$, set $\varphi(n)=-\log d(g^n(z), Crit')$.
By~\cite{DPU}, there exists $C_g>0$, which depends only on $g$
such that
\begin{equation}\label{psibound}
\sum_{k=0}^{n-1}{\bf '} \ \varphi(k)\le C_g n, \ \ \ n=1,2,...
\end{equation}
where $\sum {\bf '}$ denotes the summation over all but at most $M=\#\ Crit'$ indexes.
Given $K>0$, define a ``shadow'' $S(j, K)$ of
$j\in {\bf N}$ to be the following interval of the real line:
$$S(j, K)=(j, j+K \varphi(j)].$$
For any $N\in \Z^+$, let $A(N, K)$ be a set of all
$n\in \Z^+$ such that there are at most $N$ integers $j$ so that
$n\in S(j, K)$.
Let us denote by $l(S)$ the lenght of an interval $S\subset {\bf R}$.
{\bf Claim 1.}
For every $m\in \Z^+$,
$$\frac{\#\ \{A(N, K)\cap \{1,..., m\}\}}{m}\ge 1-\frac{C_g K}{N}.$$
\begin{proof}[Proof of Claim 1]
Let $1\le n_1<...<n_r\le m$ be all $n$ with the property that
$n$ lies in at least $N+1$ ``shadows''. Then obviously
$\sum_{j=0}^{m-1}{\bf '} \ l(S(j, K))\ge N r$.
On the other hand, by~(\ref{psibound}),
$${\sum_{j=0}^{m-1}}{\bf '} \ l(S(j, K))=\sum_{j=0}^{m-1}{\bf '} \
\varphi(j)\le C_g m.$$
Therefore, $r\le \frac{C_g K}{N} m$.
\end{proof}
Given $r>0$, we denote by $G(N, r)$ the set of all
$n\in {\bf N}$ such that the map
$g^n: W_r^n(z)\to B(g^n(z), r)$ is at most $N$-critical.
Then we have
{\bf Claim 2.}
Suppose that $n$ is an $\lambda e^\epsilon$-hyperbolic time for $z$, for some $\epsilon>0$.
Set $K_0=1/\log\lambda$. There exists $\tilde\delta=\tilde\delta (\lambda, \epsilon)$
such that for every $\delta\in (0, \tilde\delta]$ and $N\in \Z^+$,
if $n\in A(N, K_0)$, then $n\in G(N, \delta)$.
\begin{proof}[Proof of Claim 2.]
Let $C=C(\lambda, \epsilon)$ and $\delta_0=\delta_0(\lambda, \epsilon)$
be taken from Lemma~\ref{lem:hyp}. Define $\tilde\delta=\min\{\delta_0, 1/C\}$.
Let $\delta\le \tilde\delta$.
If $1\le j<n$ is such that $W_\delta^j(g^{n-j}(z))\cap Crit'\not=\emptyset$, then
$d(g^{n-j}(z), Crit')\le \diam(W_\delta^j(g^{n-j}(z)))\le C \delta \lambda^{-j}$.
Hence,
$$\varphi(n-j)\ge j\log\lambda + \log\frac{1}{C \delta}\ge
\frac{j}{K_0}.$$
It means that $n\in S(n-j, K_0)$.
Now, assume that $g^n: W_\delta^n(z)\to B(g^n(z), \delta)$ is {\em at least}
$N+1$-critical. By the preceding consideration,
$n$ belongs to at least $N+1$ different
``shadows'' $S(n-j, K_0)$. Therefore, $n\notin A(N, K_0)$.
Thus $n\in A(N, K_0)$ implies $n\in G(N, \delta)$.
\end{proof}
The last ingredient of the proof is the Pliss Lemma~\cite{Pliss}:
\begin{lemma}\label{pliss}
Let $0<b_1<b_2\le B$ and $\theta=(b_2-b_1)/(B-b_1)$. Given real numbers
$a_1,...,a_r$ satisfying $\sum_{j=1}^r a_j\ge b_2 r$ and $a_j\le B$ for all
$1\le j\le r$, there are $l>\theta r$ and $1<n_1<...<n_l\le r$ such that
$\sum_{i=n+1}^{n_j} a_i\ge b_1(n_j-n)$ for each $0\le n<n_j$, $j=1,...,l$.
\end{lemma}
\begin{proof}[Proof of Theorem~\ref{thm:positiv}]
Let $z\in J(g)$, $m\in \Z^+$ and
$$\frac{1}{m}\log|Dg^{m}(z)|>\epsilon_0+ \log\lambda.$$
Let $\epsilon=\epsilon_0/8$. Consider the set
$$\tilde H_m=\{n\in \{1,...,m\}: n \mbox{ is a }
\lambda e^{4\epsilon} -\mbox{hyperbolic time for }
z\}.$$
By the Pliss Lemma~\ref{pliss}, there exists $\theta=\theta(\lambda, \epsilon)$
such that
\begin{equation}\label{hmk}
\frac{\#\ \tilde H_m}{m}>\theta, \ k\ge 1.
\end{equation}
Let $K_0=1/\log(\lambda e^{2\epsilon})$ and let
$\tilde\delta=\tilde\delta(\lambda, 2\epsilon)$ be the constant from Claim 2.
We define
$$\rho=\tilde\lambda, \ N=[2 C_f K_0/\theta]+1 \mbox{ and }
\alpha=\theta/2.$$
Consider the corresponding sets $A(N, K_0)$ and $G(N, \rho)$.
By Claim 1,
\begin{equation}\label{cl1}
\frac{\#\ \{A(N, K_0)\cap \{1,..., m\}\}}{m}\ge 1-\frac{C_g K_0}{N}>1-\frac{\theta}{2}
\end{equation}
while by Claim 2,
\begin{equation}\label{cl2}
A(N, K_0)\cap \tilde H_m\subset G(N, \rho).
\end{equation}
Then~(\ref{hmk}) and~(\ref{cl1}) give us
\begin{equation}\label{cl3}
\#\ \{A(N, K_0)\cap \tilde H_m\}>\frac{\theta}{2} m.
\end{equation}
Hence, by~(\ref{cl2}) we must have:
\begin{equation}\label{g}
\frac{\#\ \{G(N, \rho)\cap \tilde H_m\}}{m}>\alpha.
\end{equation}
Denote $H_m=G(N, \rho)\cap \tilde H_m$.
Then the properies (a) and (b)
hold for $n\in H_m$. Furthermore, by Lemma~\ref{lem:hyp},
for every $n\in H$, $\diam V_n\le 2 C (\lambda e^{2\epsilon})^{-n}$, where $C=C(\lambda, 2\epsilon)>0$.
Hence, $\diam V_n\le \lambda^{-n}$, for every $n\ge \tilde n$,
where $\tilde n=\tilde n(\lambda, \epsilon)$. Thus (c)
holds too.
\end{proof}
\begin{remark}\label{notdens}
Theorem~\ref{thm:positiv}
implies the following fact which is announced in the Abstract: every point $z$ of a polynomial Julia set $J(g)$ satisfying
$\chi_{+}(g, z)>0$ is not a Lebesgue density point of $J(g)$.
(By Remark~\ref{ratexp}, this holds as well
for rational functions with nowhere dense Julia sets.)
In fact $J(g)$ is even {\it upper mean porous} at such $z$
which means:
there are $r>0$ and a
subset of $\Z^+$ of a positive upper density such that
for every $j$ in this subset $B(z,2^{-j})$
contains a ball disjoint from $J(g)$ of radius $r2^{-j}$.
The proof is the same as in~\cite{PR} in the case of $g$ satisfying
Collet-Eckmann condition.
\end{remark}
\section{Unicritical polynomials}\label{typ}
In this Section, $f(z)=z^d+c$, where $d\ge 2$ and $c\in\C$.
Recall that a point $z\in J(f)$ is {\em slowly recurrent}
if for any $\alpha>0$, $|f^n(z)|\ge e^{-\alpha n}$ holds for every large $n$.
Next fact is crucial.
\begin{lemma}\label{lem:slarea}
Almost every point $z\in J(f)$ is slowly recurrent.
\end{lemma}
\begin{proof}
If the critical point $0$ is not in the Julia set, then $0$ is attracted by either an attracting cycle or a parabolic cycle,
and it is well known that the Julia set has measure zero. In the following, we assume that $0\in J(f)$.
We need the following fact which is a particular case of~\cite{P}, Lemma 1.
{\bf Claim 1.} There is a constant $K=K(d)>0$ such that for any $\eps>0$, any $w\in J(f)$
and any integer $s\ge 1$, if $|w|<\eps$ and $|f^s(w)|<\eps$,
then $$s\ge C(\eps):= K\log(\eps^{-1}).$$
Now we fix $\alpha>0$
and consider $E_n=\{z\in J(f): |f^n(z)|<e^{-2\alpha n}\}$. Let $V$ be a component
of $f^{-n}(B(0, e^{-\alpha n}))$.
If $0\le j_1<j_2\le n$ are such that
$f^{j_1}(V)$ and $f^{j_2}(V)$ both contain $0$, then $j_2-j_1$ is a return time of
$f^{j_1}(0)\in B(0, e^{-\alpha n})$ into the ball. By Claim 1, $j_2-j_1\ge C(e^{-\alpha n})=\tilde C\alpha n$
for some constant $\tilde C=\tilde C(d)>0$. Thus
$\#\ \{0\le j\le n: f^j(V)\ni 0\}$ is bounded by
$1+1/(\alpha C)$. It means that the map
$f^n: V\to B(0, e^{-\alpha n})$ is a (branched) cover with a degree which is
uniformly bounded by some $D=D(d, \alpha)$.
Since $E_n\cap V\subset f^{-n}(B(0, e^{-2\alpha n}))$, it follows
from a version of the Koebe distortion theorem for multivalent maps, see e.g.~\cite{PR}, Lemma 2.1,
that there exists $\alpha'>0$, which depends only on $\alpha$ and $D$
such that $\area(V\cap E_n)/\area(V)\le e^{-\alpha' n}$.
As $V\subset f^{-n}(B(0, e^{-\alpha n}))\subset B(0, 3)$,
$\area(E_n)$ is exponentially small with $n$ and thus $\sum_{n=1}^\infty \area(E_n)<\infty$.
By the Borel-Cantelli lemma, a.e. $z$ is
contained in only finitely many $E_n$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:typ}]
By Lemma~\ref{lem:slexp}, $\chi_{-}(z)\ge 0$ for every slowly recurrent $z\in J(f)$ and using
Lemma~\ref{lem:slarea}, $\chi_{-}(z)\ge 0$ for almost every $z\in J(f)$.
And $\chi_{+}(z)\le 0$ for a.e. $z$ because each $z$ with
$\chi_{+}(z)> 0$ is not a Lebesgue density point by Theorem~\ref{thm:up}.
\end{proof}
We shall now consider backward orbits of the critical point and prove Theorem~\ref{thm:backward}
of the Introduction. We first prove the following variation of Lemma~\ref{return} for polynomial maps
which gives a better estiamte.
\begin{lemma} \label{lem:returnpoly}
Let $f(z)=z^d+c$ where $d\ge 2$ and $c\in\C$. Assume that $f$ has no an attracting cycle.
Then for each $\lambda>1$ there is $\delta_0>0$
such that for each $\delta\in (0,\delta_0)$, if $z\not\in B(0,\delta)$ and $n\ge 1$
is the minimal positive integer such that $|f^n(z)|\le \delta$, then
$$|Df^n(z)|\ge \frac{\delta}{12|z|}\lambda^{-n}.$$
Moreover, if $|z|=\delta$, then $$|Df^n(z)|\ge \lambda^{-n}.$$
\end{lemma}
\begin{proof} Fix $\lambda$ and $\delta$. Define the numbers $\tau_i$, domains $U_i$, $0\le i\le n$,
the index set $\mathcal{I}$ and the number $\eps_0$ as in the proof of Lemma~\ref{lem:return},
with the only difference that we start with $U_n=B(z_n,\delta)$, so $\tau_n=\delta$.
As $f$ has no attracting cycle in $\C$, $U_i\not\supset \overline{B(0,2\delta)}$ for each $0\le i\le n$.
Thus the conclusion of Claim 1 holds with $c$ replaced by $0$ in the current setting.
So we obtain similar to (\ref{genin}) the following estimate:
$|Df^n(z)|\ge \delta/(12|z| K^{N}).$
Arguing as in the last part of the proof of Lemma~\ref{lem:return}, we show that $K^{N}\le \lambda^{n}$,
provided that $\delta$ is small enough.
In the case $|z|=\delta$, we have $12 K^{N}\le \lambda^{n}.$ Thus the lemma holds.
\end{proof}
As an immediate corollary, we have
\begin{lemma}\label{lem:closereturn}
Let $f(z)=z^d+c$ where $d\ge 2$ and $c\in\C$. Assume that $f$ has no an attracting cycle.
Then for each $\lambda>1$ there is $\delta_0>0$ such that for each $z\in\C$ and $n\ge 1$
with $|f^n(z)|\le \delta_0$ and with $|f^n(z)|\le |f^j(z)|$ for each $0\le j<n$, we have
$$|Df^n(z)|\ge \min\left(\frac{\delta_0}{12|z|},1\right) \lambda^{-n}.$$
\end{lemma}
\begin{proof}
Let $\delta_0$ be given by Lemma~\ref{lem:returnpoly} and let $\delta_0'=\min(|z|,\delta_0)$.
By the assumption, there is a sequence of integers $1\le n_1<n_2<\cdots<n_k=n$ such that
\begin{itemize}
\item $n_1$ is the minimal positive integer such that $|f^{n_1}(z)|\le \delta_0'$;
\item $n_{j+1}$ is the minimal integer with $n_{j+1}>n_j$
and $|f^{n_{j+1}}(z)|\le |f^{n_j}(z)|$ for each $j=1,\ldots, k-1$.
\end{itemize}
If $|z|\ge \delta_0$, then $n_1$ is the minimal positive integer such that $|f^{n_1}(z)|\le \delta_0$, so
by Lemma~\ref{lem:returnpoly}, we have
$$|Df^{n_1}(z)|\ge \frac{\delta_0}{12|z|}\lambda^{-n}.$$
If $|z|<\delta_0$, then $|z|=\delta_0'\in (0,\delta_0]$ and $n_1$
is the minimal integer such that $|f^{n_j}(z)|\le \delta_0'$,
so by the latter inequality of Lemma~\ref{lem:returnpoly}, we have
$$|Df^{n_1}(z)|\ge \lambda^{-n_1}.$$
For each $1\le j<n$, putting $\delta_j:=|f^{n_j}(z)|\in (0,\delta_0]$, $n_{j+1}-n_j$
is the minimal positive integer such that |$f^{n_{j+1}-n_j}(f^{n_j}(z))|\le \delta_j$,
so by the latter inequality in Lemma~\ref{lem:returnpoly} again, we have
$$|Df^{n_{j+1}-n_j}(f^{n_j}(z))|\ge \lambda^{-(n_{j+1}-n_j)}.$$
Therefore, if $|z|\ge \delta_0$, then
$$|Df^n(z)|=|Df^{n_1}(z)|\prod_{j=1}^{k-1} |Df^{n_{j+1}-n_j}(f^{n_j}(z))|\ge \frac{\delta_0}{12|z|}\lambda^{-n},$$
and if $|z|<\delta_0$, then
$$|Df^n(z)|\ge \lambda^{-n}.$$
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:backward}]
Fix $\lambda>1$ and let $\delta_0$ be given by Lemma~\ref{lem:closereturn}.
Let $\bar{x}=\{x_{-n}\}_{n=0}^\infty$ be a backward orbit of $0$.
Then for each $n$, applying Lemma~\ref{lem:closereturn} to $z=x_{-n}$, we obtain
$$|Df^n(x_{-n})|\ge K_n \lambda^{-n},$$
where $K_n=\min \left(\delta_0/(12|x_{-n}|), 1\right).$ Clearly $K_{n}$ is bounded from below by a positive constant depending only on $f$. Thus
$$\chi_-^{back}(f, \bar x) \ge -\log \lambda.$$
Since this holds for all $\lambda>1$, $\chi_-^{back}(f,\bar x)\ge 0$.
\end{proof}
\section{Some applications and remarks}
Let us note the following special case of Theorem~\ref{thm:main}:
\begin{theorem}\label{rat}
Let $g$ be a rational function on the Riemann sphere of degree at least two.
Given a critical value $c$ of $g$,
define its postcritical
set $P(c)=\overline{\cup_{n\ge 0}g^n(c)}$.
Assume that $c_0$ is a critical value of $g$ not in the basin of
an attracting cycle, such that $P(c_0)$ is disjoint
from the union $X$ of the postcritical sets of
all other critical values of $g$.
Then
$$\chi_-(g, c_0)=\liminf_{n\to\infty} \frac{1}{n}\log \|Dg^n(c_0)\|\ge 0,$$
where $\|\cdot\|$ denote the norm in the spherical metric.
\end{theorem}
\begin{proof}
By means of a M\"obius conjugacy, we may assume that $\infty\in X$, so that the orbit of $c_0$
lies in a compact subset of $\C$ and
$\chi_-(g,c_0)$ can be calculated using the Euclidean metric instead of the spherical metric.
Then define $V'=\overline{\C}\setminus X$ and $V=g^{-1}(V')$, and apply Theorem~\ref{thm:main} to $g\in \sU_{V, V'}$.
\end{proof}
An immediate corollary of Theorem~\ref{uni} along with
Remark 13 of~\cite{lanal} is as follows:
\begin{coro}\label{ruelle}
Assume that the map $f(z)=z^d+c$ has no attracting cycles.
Then the power series
$$F(t)=1+\sum_{n=1}^\infty\frac{t^n}{Df^n(c)}$$
has the radius of convergence at least $1$, and
\begin{equation}\label{fr}
F(t)\not=0 \text{ for every } |t|<1.
\end{equation}
\end{coro}
\begin{remark}The function $F(t)$
should be interpreted as ``Fredholm determinant''
of the operator $T: \phi\mapsto \sum_{w: f(w)=z}\frac{\phi(w)}{Df(w)^2}$
acting in a space of functions $\phi$, which are analytic outside of
$J(f)$ and locally integrable on the plane.
Then~(\ref{fr}) reflects the fact that $T$ is
a contraction operator in this space.
Note that this operator plays, in particular, an important
role (after Thurston)
in the problem of stability in holomorphic dynamics.
\end{remark}
Another consequence of Theorem~\ref{thm:main} is that:
\begin{coro}\label{cor:pollike}
Let $g_i: V_i\to V_i'$, $i=0,1$, be two mappings
in the class $\sU$ which are quasi-conformally conjugated
(i.e., there exists a q-c map $h: \C\to \C$ such that $h(V_0)=V_1$, $h(V_0')=V_1'$, and $h\circ g_0=g_1\circ h$ on $V_0$).
Assume that $\omega_{g_0}(c(g_0))$ (the $\omega$-limit set of the point $c(g_0)$ by the map
$g_0$) is compactly contained in $V_0$. If, for a subsequence $n_k\to \infty$,
$\lim_{k\to\infty} \frac{1}{n_k}\log |Dg_0^{n_k}(c(g_0))|=0$,
then also $\lim_{k\to\infty} \frac{1}{n_k}\log |Dg_1^{n_k}(c(g_1))|=0$.
\end{coro}
\begin{proof}
Normalize the maps in such a way that $c(g_0)=c(g_1)=0$ and $h(1)=1$.
As in~\cite{mss}, one can include $g_0$ and $g_1$ in a family $g_\nu$ of quasi-conformally conjugated maps of the class
$\sU$, with $c(g_\nu)=0$,
which depends holomorphically on
$\nu\in D_r=\{|\nu|<r\}$, for some $r>1$. Namely, if $\mu=\frac{\partial h}{\partial \bar z}/\frac{\partial h}{\partial z}$ is complex dilatation
of $h$, then, for every $\nu\in D_r$, where $r=||\mu||_{\infty}^{-1}$, let $h_\nu$ be the unique q-c homeomorphism of $\C$ with complex dilatation
$\nu \mu$, which leaves the points $0,1$ fixed (in particular, $h_0=id$ and $h_1=h$). Then we can define domains $V_\nu=h_\nu(V_0)$,
$V_\nu'=h_\nu(V_0')$, and the map
$g_\nu=h_\nu\circ g_0\circ h_\nu^{-1}: V_\nu\to V_\nu'\in \sU$ with $c(g_\nu)=0$.
As $\omega_{g_\nu}(0)=h_\nu(\omega_{g_0}(c(g_0))$ is compactly contained in $V_\nu=h_\nu(V_0)$,
by the Schwarz lemma and a compactness argument,
given a compact subset $K$ of the disk $D_r$, there exists $C$, such that
$|Dg_\nu(g_\nu^i(0))|\le C$ for every $\nu\in K$ and every $i\ge0$.
Then $u_k(\nu)=n_k^{-1}\log |Dg_\nu^{n_k}(0)|$ is a sequence
of harmonic functions in $D_r$, which is bounded on compacts. On the other hand, by Theorem~\ref{thm:main},
every limit value of the sequence $\{u_k\}$ is non-negative, and, by the assumption,
$u_k(0)\to 0$.
According to the Minimum Principle, $u_k(\nu)\to 0$ for any $\nu$.
\end{proof}
\begin{remark}\label{qcinv}
In particular, the Collet-Eckmann condition
$\chi_-(g, c(g))>0$ is a quasi-conformal invariant.
In fact, it is even a topological invariant~\cite{PR}.
\end{remark}
The following is a consequence of Theorem~\ref{thm:up}:
\begin{coro}\label{renorm}
Let $g$ be a polynomial which is infinitely-renormalizable around a critical
value $c$.
If $\chi_{+}(g, c)>0$, then
$J(g)$ is not locally-connected.
\end{coro}
Indeed,
let $J_n$, $n\ge 1$, be a sequence of ``small'' Julia sets such that $c\in J_n$ and let $p_n\to \infty$
be their corresponding periods. Theorem~\ref{thm:up} implies that there is a sequence of integers
$m_n\to \infty$ such that $\inf_{n} \diam g^{m_n}(J_n)>0$. Hence, a Hausdorff limit point of the sequence
of compacts $g^{m_n}(J_n)$, $n\ge 1$, must be a non-trivial wandering subcontinuum of $J(g)$.
By Theorem 3.2 of~\cite{bl} (see also~\cite{lback}),
$J(g)$ cannot be locally-connected.
\
{\bf Some motivations and historical remarks.}
In 1-dimensional dynamics often the asymptotic behaviour of derivatives
along typical trajectories is reflected in the asymptotic behaviour of
derivatives along critical trajectories. E.g. hyperbolicity is equivalent to
the attraction of all critical trajectories to attracting periodic orbits.
This implies
$\chi_-(c)< 0$ for all critical values $c$. This paper provides converse
theorems.
Another theory is that the `strong non-uniform hyperbolicity condition'
saying that there is
$\chi>0$ such that for all probability invariant measures $\mu$ on Julia set
$J$ for a rational map $g$\; $\chi_\mu(g):=\int \log |g'|\,d\mu\ge \chi$, is
equivalent to so called Topological Collet-Eckmann conditionsee, see e.g.~\cite{PRS}.
The latter in
presence of only one critical point in $J$ (and no parabolic orbits) is equivalent to Collet-Eckmann
condition, see Remark~\ref{qcinv}.
A motivation to Theorem~\ref{uni} has been the theorem saying that for all $\mu$ as
above $\chi_\mu(g)\ge 0$, see~\cite{P}.
In particular for $\mu$-almost every $x\in J$, for any $\mu$,\;
$\chi(g,x)\ge 0$.
This suggested the question whether critical values also have this property
(under appropriate assumptions).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,161
|
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