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---
author:
- 'Quentin De Mourgues\'
title: |
\
\
of the KZB Classification Theorem
---
Rauzy-type dynamics are group actions on a collection of combinatorial objects. The first and best known example (the Rauzy dynamics) concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincaré map on compact orientable translation surfaces. The equivalence classes on the objects induced by the group action have been classified by Kontsevich and Zorich, and by Boissy through methods involving both combinatorics algebraic geometry, topology and dynamical systems. Our precedent paper [@DS17] as well as the one of Fickenscher [@Fic16] proposed an ad hoc combinatorial proof of this classification.
However, unlike those two previous combinatorial proofs, we develop in this paper a general method, called the labelling method, which allows one to classify Rauzy-type dynamics in a much more systematic way. We apply the method to the Rauzy dynamics and obtain a third combinatorial proof of the classification. The method is versatile and will be used to classify three other Rauzy-type dynamics in follow-up articles.
Another feature of this paper is to introduce an algorithmic method to work with the sign invariant of the Rauzy dynamics. With this method we can prove most of the identities appearing in the literature so far ([@KZ03],[@Del13] [@Boi13] [@DS17]...) in an automatic way.
The sign invariant {#sec.signinv}
==================
Arf functions for permutations {#ssec.arf_inv}
------------------------------
For $\s$ a permutation in $\kS_n$, let () = \# { 1i<j n | (i)<(j) } i.e. $\chi(\s)$ is the number of pairs of non-crossing edges in the diagram representation of $\s$.
Let $E=E(\s)$ be the subset of $n$ edges in $\cK_{n,n}$ described by $\s$. For any $I \subseteq E$ of cardinality $k$, the permutation $\s|_{I} \in \kS_k$ is defined in the obvious way, as the one associated to the subgraph of $\cK_{n,n}$ with edge-set $I$, with singletons dropped out, and the inherited total ordering of the two vertex-sets.
Define the two functions $$\begin{aligned}
A(\s)
&:=
\sum_{I \subseteq E(\s)} (-1)^{\chi(\s|_I)}
\ef;
&
\Abar(\s)
&:=
\sum_{I \subseteq E(\s)} (-1)^{|I|+\chi(\s|_I)}
\ef.\end{aligned}$$ When $\s$ is understood, we will just write $\chi_I$ for $\chi(\s|_I)$. The quantity $A$ is accessory in the forthcoming analysis, while the crucial fact for our purpose is that the quantity $\Abar$ is invariant in the $\perms_n$ dynamics.
In the following section, we define a technique to demonstrate identities of the arf invariant involving differents configurations.
Automatic proofs of Arf identitites {#ssec.arfcalcseasy}
-----------------------------------
We will *not* try to evaluate Arf functions of large configurations starting from scratch. We will rather compare the Arf functions of two (or more) configurations, which differ by a finite number of edges, and establish linear relations among their Arf functions. The method we develop here, gives an algorithm to find and check Arf identities.
In order to have the appropriate terminology for expressing this strategy, let us define the following: Given a permutation $\s$ define $\s_{k,\ell}$ to be a permutation with $k$ marks on its bottom line and $\ell$ marks on its top line. The marks are all at distinct positions and do not touch the corners of the permutation. These marks break the bottom (respectively top) line into $k+1$ open interval $P_{-,1},\ldots,P_{-,k+1}$ (respectively $\ell+1$ open interval $P_{+,1},\ldots,P_{+,\ell+1}$).
For example if $k=1,\ell=3$: $$\put(50,-30){$P_{-,1}$}\put(100,-30){$P_{-,2}$}\put(26,25){$P_{+,1}$}\put(60,25){$P_{+,2}$}\put(90,25){$P_{+,3}$}\put(125,25){$P_{+,4}$}
\s_{k,\ell}=\raisebox{-20pt}{\includegraphics[scale=2.5]{P05_Arf/figure/fig_arf_ex1.pdf}}$$
Let $\s_{k,\ell,E'}$ be the permutation obtained by adding a set of edges $E'$ on the marks of permutation $\s_{k,\ell}$ with the following convention: an edge $e\in E'$ is a pair $(i.x,j.y)$. The edge connects the $i$th bottom mark and the $j$th top mark, and it is ordered as the $x$th edge within the bottom mark and the $y$th edge within the top mark. Note that if $i=0$ of $i=k+1$ (likewise of $j$) this implies that the edge is connected to a corner of the permutation.
For example if $k=1,\ell=3$ and $E'=\{(0.1,2.2),(1.1,3.1),(1.2,1.1),(1.3,2.1),(2,1.2)\}$: $$\put(60,-30){$P_{-,1}$}\put(110,-30){$P_{-,2}$}\put(36,25){$P_{+,1}$}\put(70,25){$P_{+,2}$}\put(105,25){$P_{+,3}$}\put(135,25){$P_{+,4}$}
\s_{k,\ell,E'}=\raisebox{-20pt}{\includegraphics[scale=2.5]{P05_Arf/figure/fig_arf_ex2.pdf}}$$ We will define an algorithm that allows one to check if, for all $\s$, we have $\sum^n_{i=1} K_i\Abar(\s_{k,\ell,E^i})=0$ or $\sum^n_{i=1} K_iA(\s_{k,\ell,E^i})=0$ for some $k,\ell,(E^i)_i,(K_i)_i,n.$
\[def.637647\] Let $\s_{k,\ell,E'}$, $P_{-,1},\ldots,P_{-,k+1}$ and $P_{+,1},\ldots,P_{+,\ell+1}$ be as defined above. Then define the $m \times (k+1)(\ell+1)$ matrix valued in $\gf_2$ Q\_[e,ij]{} := {
[ll]{} 1 &\
0 &
. For $v \in \gf_2^{(k+1)(\ell+1)}$, let $|v|$ be the number of entries equal to $1$. Similarly, identify $v$ with the corresponding subset of $[(k+1)(\ell+1)]$. Given such a construction, introduce the following functions on $(\gf_2)^{(k+1)(\ell+1)}$ $$\begin{aligned}
A_{k,\ell,E'}(v)
&:=
\sum_{u \in (\gf_2)^{E'}}
(-1)^{\chi_u + (u,Qv)}
\ef;
&
\Abar_{k,\ell,E'}(v)
&:=
\sum_{u \in (\gf_2)^{E'}}
(-1)^{|u|+\chi_u + (u,Qv)}
\ef.\end{aligned}$$
The construction is illustrated in Figure \[fig.arf\_ex\_def\].
![\[fig.arf\_ex\_def\]The permutation $\s_{1,4,\{(0.1,0.1),(0.2,3.1),(0.3,1.1),(1.1,4.1),(1.2,2.1)\}}$. We cannot show the full matrix $Q$ for such a big example, but we can give one row, for the edge which has the label $e$ in the drawing. The row $Q_e$ reads $(Q_e)_{11, 12, \ldots, 15, 21, \ldots, 25}=(1,1,0,0,0,\,0,0,1,1,1)$.](FigFol/Figure4_fig_arf_ex_def.pdf)
Let us comment on the reasons for introducing such a definition. The quantities $A_{k,\ell,E'}(v)$ (respectively $\Abar_{k,\ell,E'}(v)$) do not depend on $\s$ and allows to sum together many contributions to the function $A(\s_{k,\ell,E'})$. Our goal is to have $E'$ of fixed size, while $E$ (the edge set of $\s
|
{
"pile_set_name": "ArXiv"
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---
abstract: 'We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the ‘Lie’ operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.'
address: |
Department of Mathematics, Room 2-089\
Massachusetts Institute of Technology\
Cambridge, MA 02139, USA
author:
- Michael Ching
title: |
Bar constructions for topological operads and\
the Goodwillie derivatives of the identity
---
Introduction {#introduction .unnumbered}
============
The motivation for this paper was an effort to construct an operad structure on the derivatives (in the sense of Tom Goodwillie’s homotopy calculus [@goodwillie:1990; @goodwillie:1991; @goodwillie:2003]) of the identity functor $I$ on the category of based spaces. Such an operad structure has been ‘known’ intuitively by experts for some time but, as far as the author knows, no explicit construction has previously been given. One piece of evidence for such a structure is the calculation, due to various people, of the homology of these derivatives. This homology is the suspension of the standard Lie operad and so is itself an operad. It is reasonable to ask, therefore, if there is an operad structure on the derivatives themselves[^1] that induces this structure on the homology.
Our construction is based on the partition poset model for the derivatives $\partial_*I$ described by Arone and Mahowald in [@arone/mahowald:1999]. They show that the derivatives are the dual spectra associated to certain finite complexes known as the partition poset complexes. In the present work we notice that these complexes are precisely the simplicial bar construction[^2] on the operad $P$ in based spaces with $P(n) = S^0$ for all $n$. Most of the paper is concerned with showing that such a bar construction has a natural cooperad structure.[^3] We do this by reinterpreting the bar construction in terms of spaces of trees. The cooperad structure then comes from a natural way to break trees apart. Taking duals, we get the required operad structure on the derivatives of the identity. In fact, we can view the derivatives of the identity as a cobar construction on the cooperad $Q$ in spectra with $Q(n) = S$, the sphere spectrum, for all $n$.
In the final part of the paper (Section \[sec:alg\]) we show that by taking homology we do indeed recover the ‘Lie’ operad structure on $H_*(\partial_*I)$. We do this by introducing spectral sequences for calculating the homology of the topological bar and cobar constructions. The $E^1$ terms of these spectral sequences can be identified with algebraic versions of the bar and cobar constructions, which in turn are related to the theory of Koszul duality for operads introduced by Ginzburg and Kapranov in [@ginzburg/kapranov:1994]. Our main result on this connection is that if the homology of a topological operad $P$ is Koszul, then the homology of the bar construction $B(P)$ is its Koszul dual cooperad. In our case of interest, we deduce that the induced operad structure on the homology of the derivatives of the identity is that of the Koszul dual of the cocommutative cooperad. This is precisely the ‘Lie’ operad structure referred to above.
Outline of the paper {#outline-of-the-paper .unnumbered}
--------------------
We now give a more detailed description of the paper. The first two sections are concerned with preliminaries. In Section \[sec:monoidal\] we recall the notions of symmetric monoidal and enriched categories and specify the categories we will be working with in this paper. These are symmetric monoidal categories that are enriched, tensored and cotensored over the category ${\mathcal{T}_{}}$ of based compactly-generated spaces (where ${\mathcal{T}_{}}$ is a symmetric monoidal category with respect to the smash product). It is to operads in these categories that we refer in the title when we say ‘topological operads’. We also require an extra condition that relates the symmetric monoidal structure to the tensoring over ${\mathcal{T}_{}}$. This condition (see Definition \[def:axiom\]) is crucial to our later constructions. The two main examples of categories satisfying our requirements are: based spaces themselves, and a suitable symmetric monoidal category of spectra, such as that of EKMM [@elmendorf/kriz/mandell/may:1997].
In Section \[sec:operads\] we recall the definitions of operads and cooperads. We should stress that the constructions of this paper apply only to what we call *reduced* operads and cooperads. These are $P$ with $P(0) = {\ast}$ and $P(1) = S$ the unit of the symmetric monoidal structure. The bar construction can still be defined for more general operads, but the cooperad structure described here does not seem to extend to such cases. In this section we also define modules and comodules over operads and cooperads respectively.
The real substance of the paper starts in Section \[sec:trees\]. Here we define the trees that will form the combinatorial heart of our description of the bar and cobar constructions. It is not a coincidence that these trees are the same species used by, for example, Getzler and Jones in their work [@getzler/jones:1994] on the bar constructions for algebraic operads and Koszul duality. We also describe what we call a *weighting* on a tree (Definition \[def:weighting\]), that is, a suitable assignment of lengths to the edges of the tree. The spaces $w(T)$ of weightings are at the heart of everything we do in this paper.
In Section \[sec:bardef\] we give our description of the bar construction on an operad in terms of such trees. If $P$ is an operad of based spaces, we can think of a point in the bar construction $B(P)$ as a weighted tree (that is, a tree with lengths assigned to the edges) with vertices labelled by points coming from the spaces $P(n)$. See Definition \[def:bar(operad)\] for a precise statement and Definition \[def:formal\_bar\] for a more formal approach. In Section \[sec:simpbar\] we show that what we have defined is isomorphic to the standard simplicial bar construction on an operad.
In Section \[sec:cooperad\] we concern ourselves with the cooperad structure on $B(P)$. This is given by the process of ‘ungrafting’ trees (see Definition \[def:grafting\] and beyond). This involves taking a weighted, labelled tree and breaking it up into smaller trees. Finding the right way to weight and label these smaller trees gives us the required cooperad structure maps.
One of the advantages of the way we have set up the theory is that the cobar construction on a cooperad is strictly dual to the bar construction on an operad. In Section \[sec:cobar\] we go through the definitions and results dual to those of Section \[sec:bar\].
The short section Section \[sec:dual\] is devoted to a simple but key result (Proposition \[prop:duality\]) that relates the bar and cobar constructions via a duality functor that reduces to Spanier–Whitehead duality in the case of spectra. This result says that, under the right circumstances, the dual of the bar construction on an operad $P$ is isomorphic to the cobar construction on the dual of $P$. This allows us, later on, to identify the derivatives of the identity as the cobar construction on a cooperad of spectra.
Before turning to our main example and application, we deal in Section \[sec:bar(modules)\] with the two-sided bar and cobar constructions. These include the bar construction for a module over an operad and, dually, the cobar construction for a comodule over a cooperad. To describe these requires a fairly simple generalization of much of the work we did in Sections \[sec:trees\]–\[sec:bar\], in particular, a more general notion of tree (see Definition \[def:gen\_trees\]).
Finally, in Section \[sec:application\] we are able to complete the main aim of this paper. We identify the partition poset complexes with a bar construction and deduce the existence of an operad structure on the derivatives of the identity functor (Corollary \[cor:operad\]). We also give examples of modules over the resulting operad, including, in particular, a module $M_X$ naturally associated to a based space $X$.
The last section of the paper Section \[sec:alg\] is concerned with the relationship of our work to the
|
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"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'In this article, we study the axialvector-diquark-axialvector-antidiquark type scalar, axialvector, tensor and vector $ss\bar{s}\bar{s}$ tetraquark states with the QCD sum rules. The predicted mass $m_{X}=2.08\pm0.12\,\rm{GeV}$ for the axialvector tetraquark state is in excellent agreement with the experimental value $(2062.8 \pm 13.1 \pm 4.2) \,\rm{MeV}$ from the BESIII collaboration and supports assigning the new $X$ state to be a $ss\bar{s}\bar{s}$ tetraquark state with $J^{PC}=1^{+-}$. The predicted mass $m_{X}=3.08\pm0.11\,\rm{GeV}$ disfavors assigning the $\phi(2170)$ or $Y(2175)$ to be the vector partner of the new $X$ state. As a byproduct, we obtain the masses of the corresponding $qq\bar{q}\bar{q}$ tetraquark states. The light tetraquark states lie in the region about $2\,\rm{GeV}$ rather than $1\,\rm{GeV}$.'
---
\
Zhi-Gang Wang [^1]\
Department of Physics, North China Electric Power University, Baoding 071003, P. R. China
PACS number: 12.39.Mk, 12.38.Lg
Key words: Tetraquark state, QCD sum rules
Introduction
============
Recently, the BESIII collaboration studied the process $J/\psi \to \phi \eta \eta^\prime$ and observed a structure $X$ in the $\phi\eta^\prime$ mass spectrum [@BES-2000]. The fitted mass and width are $m_X=(2002.1\pm 27.5 \pm 15.0)\,\rm{MeV}$ and $\Gamma_X=(129 \pm 17 \pm 7)\,\rm{MeV}$ respectively with assumption of the spin-parity $J^P=1^-$, the corresponding significance is $5.3\sigma$; while the fitted mass and width are $m_X=((2062.8 \pm 13.1 \pm 4.2)
\,\rm{MeV}$ and $\Gamma_X=(177 \pm 36 \pm 20)\,\rm{MeV}$ respectively with assumption of the spin-parity $J^P=1^+$, the corresponding significance is $4.9\sigma$. The $X$ state was observed in the $\phi\eta^\prime$ decay model rather than in the $\phi\eta$ decay model, they maybe contain a large $ss\bar{s}\bar{s}$ component, in other words, it maybe have a large tetraquark component. In Ref.[@Wang-Luo-Liu], Wang, Luo and Liu assign the $X$ state to be the second radial excitation of the $h_1(1380)$. In Ref.[@Cui-etal], Cui et al assign the $X$ to be the partner of the tetraquark state $Y(2175)$ with the $J^{PC}=1^{+-}$.
We usually assign the lowest scalar nonet mesons $\{f_0(500),a_0(980),\kappa_0(800),f_0(980) \}$ to be tetraquark states, and assign the higher scalar nonet mesons $\{f_0(1370),a_0(1450),K^*_0(1430),f_0(1500) \}$ to be the conventional ${}^3P_0$ quark-antiquark states [@Close2002; @ReviewAmsler2; @Maiani-Scalar]. In Ref.[@WangScalarNonet], we take the nonet scalar mesons below $1\,\rm{ GeV}$ as the two-quark-tetraquark mixed states and study their masses and pole residues with the QCD sum rules in details, and observe that the dominant Fock components of the nonet scalar mesons below $1\,\rm{ GeV}$ are conventional two-quark states. The light tetraquark states maybe lie in the region about $2\,\rm{GeV}$ rather than lie in the region about $1\,\rm{GeV}$.
In this article, we take the axialvector diquark operators as the basic constituents to construct the tetraquark current operators to study the scalar ($S$), axialvector ($A$), tensor ($T$) and vector ($V$) tetraquark states with the QCD sum rules, explore the possible assignments of the new $X$ state. We take the axialvector diquark operators as the basic constituents because the favored configurations from the QCD sum rules are the scalar and axialvector diquark states [@WangLDiquark; @Dosch-Diquark-1989], the current operators or quark structures chosen in the present work differ from that in Ref.[@Cui-etal] completely.
The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the $ss\bar{s}\bar{s}$ tetraquark states in section 2; in section 3, we present the numerical results and discussions; section 4 is reserved for our conclusion.
QCD sum rules for the $ss\bar{s}\bar{s}$ tetraquark states
==========================================================
We write down the two-point correlation functions $\Pi_{\mu\nu\alpha\beta}(p)$ and $\Pi(p)$ firstly, $$\begin{aligned}
\Pi_{\mu\nu\alpha\beta}(p)&=&i\int d^4x e^{ip \cdot x} \langle0|T\left\{J_{\mu\nu}(x)J_{\alpha\beta}^{\dagger}(0)\right\}|0\rangle \, , \\
\Pi(p)&=&i\int d^4x e^{ip \cdot x} \langle0|T\left\{J_0(x)J_0^{\dagger}(0)\right\}|0\rangle \, ,\end{aligned}$$ where $J_{\mu\nu}(x)=J_{2,\mu\nu}(x)$, $J_{1,\mu\nu}(x)$, $$\begin{aligned}
J_{2,\mu\nu}(x)&=&\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big\{s^{Tj}(x)C\gamma_\mu s^k(x) \bar{s}^{m}(x)\gamma_\nu C \bar{s}^{Tn}(x)+s^{Tj}(x)C\gamma_\nu s^k(x)\bar{s}^m(x)\gamma_\mu C \bar{s}^{Tn}(x) \Big\} \, , \nonumber\\
J_{1,\mu\nu}(x)&=&\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big\{s^{Tj}(x)C\gamma_\mu s^k(x) \bar{s}^{m}(x)\gamma_\nu C \bar{s}^{Tn}(x)-s^{Tj}(x)C\gamma_\nu s^k(x)\bar{s}^m(x)\gamma_\mu C \bar{s}^{Tn}(x) \Big\} \, , \nonumber\\
J_0(x)&=&\varepsilon^{ijk}\varepsilon^{imn}s^{Tj}(x)C\gamma_\mu s^k(x) \bar{s}^m(x)\gamma^\mu C \bar{s}^{Tn}(x) \, ,\end{aligned}$$ where the $i$, $j$, $k$, $m$, $n$ are color indexes, the $C$ is the charge conjugation matrix. Under charge conjugation transform $\widehat{C}$, the currents $J_{\mu\nu}(x)$ and $J_0(x)$ have the properties, $$\begin{aligned}
\widehat{C}\,J_{2,\mu\nu}(x)\,\widehat{C}^{-1}&=&+ \,J_{2,\mu\nu}(x)\, , \nonumber \\
\widehat{C}\,J_{1,\mu\nu}(x)\,\widehat{C}^{-1}&=&- \,J_{1,\mu\nu}(x)\, , \nonumber \\
\widehat{C}\,J_0(x)\,\widehat{C}^{-1}&=& +J_0(x) \, .\end{aligned}$$
The doubly-strange diquark operators $$\begin{aligned}
s^{Tj} C\Gamma s^k&=&\frac{1}{2}\Big(s^{Tj} C\Gamma s^k-s^{Tk} C\Gamma s^j \Big)=\frac{1}{2}\varepsilon^{ijk}s^{Tj} C\Gamma s^k\end{aligned}$$ with $\Gamma=\gamma_\mu$, $\sigma_{\mu\nu}$ in color antitriplet $\bar{3}_c$ and $$\begin{aligned}
s^{Tj} C\Gamma s^k&=&\frac{1}{2}\Big(s^{Tj} C\Gamma s^k+s^{Tk} C\Gamma s^j \Big)\end{aligned}$$ with $\Gamma=1$, $\gamma_{5}$, $\gamma_{\mu}\gamma_5$ in color sextet $6_c$ satisfy Fermi-Dirac statistics. On the other hand, the scattering amplitude for one-gluon exchange is proportional to $$\begin{aligned}
\left(\frac{\lambda^a}{2}\
|
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"pile_set_name": "ArXiv"
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---
abstract: |
Let $w$ be a word in the free group of rank $n \in \mathbb{N}$ and let $\mathcal{V}(w)$ be the variety of groups defined by the law $w=1$. Define $\mathcal{V}(w^*)$ to be the class of all groups $G$ in which for any infinite subsets $X_1, \dots, X_n$ there exist $x_i \in X_i$, $1\leq i\leq n$, such that $w(x_1, \dots,
x_n)=1$. Clearly, $\mathcal{V}(w) \cup \mathcal{F} \subseteq
\mathcal{V}(w^*)$; $\mathcal{F}$ being the class of finite groups. In this paper, we investigate some words $w$ and some certain classes $\mathcal{P}$ of groups for which the equality $\left(\mathcal{V}(w) \cup \mathcal{F}\right)\cap \mathcal{P}=
\mathcal{P} \cap \mathcal{V}(w^*)$ holds.
address: 'Department of Mathematics,University of Isfahan,Isfahan 81746-73441, Iran.'
author:
- Alireza Abdollahi
title: A combinatorial problem in infinite groups
---
[**Introduction and results**]{}
Let $w$ be a word in the free group of rank $n \in \mathbb{N}$ and let $\mathcal{V}(w)$ be the variety of groups defined by the law $w=w(x_1,\dots,x_n)=1$. P. Longobardi, M. Maj and A. Rhemtulla in [@LMR] defined $\mathcal{V}(w^*)$ to be the class of all groups $G$ in which for any infinite subsets $X_1, \dots, X_n$ there exist $x_i \in X_i$, $1\leq i\leq n$, such that $w(x_1,
\dots, x_n)=1$ and raised the question of whether $\mathcal{V}(w)
\cup \mathcal{F} =\mathcal{V}(w^*)$ is true; $\mathcal{F}$ being the class of finite groups. There is no example, so far, of an infinite group in $\mathcal{V}(w^*)\backslash\mathcal{V}(w)$. In fact the origin of this problem is the following observation:\
Let $G$ be an infinite group such that in every two infinite subsets of $G$ there exist two commuting elements, then $G$ is abelian. This is an immediate consequence of the answer of B. H. Neumman to a question of P. Erdös; B. H. Neumman proved that an infinite group $G$ is centre-by-finite if and only if every infinite subset of $G$ contains two distinct commuting elements [@N]. Since this first paper, problems of a similar nature have been the object of several articles (for example [@A2], [@A3], [@AT1], [@D1], [@D3], [@DRS], [@G], [@LW], [@LM2], [@LMMR], [@RH]).\
As far as we know, the equality $\mathcal{V}(w)\cup \mathcal{F}=\mathcal{V}(w^*)$ is known for the following words: $w=x^m$, $w=[x_1, \dots, x_n]$ [@LMR], $w=[x,y]^2$ [@LM], $w=[x,y,y]$ [@S1], $w=[x,y,y,y]$ [@S2], $w=(xy)^{-3}x^3y^3$ [@A1], $w=x_1^{\alpha_1}\cdots x_m^{\alpha_m}$ where $\alpha_1, \dots, \alpha_m$ are non-zero integers [@AT2], $w=(xy)^2(yx)^{-2}$ or $w=[x^m,y]$ where $m\in\{3,6\}\cup \{2^k
\;|\; k\in\mathbb{N}\}$ [@AT3], $w=[x^n,y][x,y^n]^{-1}$ where $n\in\{\pm 2,3\}$ [@Taeri] and $w=[x^m,y^m]$ or $w=(x_1^mx_2^m\cdots x_n^m)^2$ where $m\in\{2^k \;|\; k\in \mathbb{N}\}$ [@Bouk].\
In [@PS], P. Puglisi and L. S. Spiezia proved that every infinite locally finite group (or locally soluble group) in $\mathcal{V}([x,_ky]^*)$ is a $k$-Engel group; (recall that $[x,_ky]$ is defined inductively by $[x,
_0y]=x$ and $[x,_ky]=[[x,_{k-1}y],y]$ for $k\in\mathbb{N}$). In [@D2], C. Delizia proved the equality $\mathcal{V}(w)\cup
\mathcal{F}=\mathcal{V}(w^*)$ on the classes of hyperabelian, locally soluble and locally finite groups where $w=[x_1, \dots,
x_k, x_1]$ and $k$ is an integer greater than 2. Later G. Endimioni generalized these results by proving that every infinite locally finite or locally soluble group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$, where $w$ is a word in a free group such that finitely generated soluble groups in $\mathcal{V}(w)$ are nilpotent (see Theorem 3 of [@E]) (recall that the variety $\mathcal{V}([x_1, \dots,
x_k, x_1])$ ($k>2$) is exactly the variety of nilpotent groups of nilpotency class at most $k$ [@Mac] and every finitely generated soluble Engel group is nilpotent [@Gru].)\
We say that a group $G$ is locally graded if and only if every finitely generated non-trivial subgroup of $G$ has a non-trivial finite quotient. We proved in Theorem 4 of [@A3] that an infinite locally graded group in $\mathcal{V}([x_1,_kx_2]^*)$ is a $k$-Engel group. We generalize this result as Theorem A, below. In order to state our first result we need the following definition. Following [@KR] we say that a group $G$ is restrained if and only if $\left<x\right>^{\left<y\right>}=\left<x^{y^i}
\;|\; i\in \mathbb{Z}\right>$ is finitely generated for all $x,y\in G$. We show by Proposition 1 below, why the following theorem improves the above mentioned results.\
[**Theorem A.**]{} [*Let $w$ be a word in a free group such that every finitely generated residually finite group in $\mathcal{V}(w)$ is polycyclic-by-finite. Then every infinite finitely generated locally graded restrained group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$.*]{}\
G. Endimioni proved that every infinite locally nilpotent group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$, where $w$ is a word in a free group (see Theorem 1 of [@E]). The following theorem generalizes Theorem 1 of [@E].\
[**Theorem B.**]{} [*Let $w$ be a word in a free group and let $\mathcal{P}$ be a class of groups which satisfies the following conditions:\
(1) the class $\mathcal{P}$ is closed under taking subgroups.\
(2) every $\mathcal{P}$-group is soluble.\
(3) every infinite finitely generated ($\mathcal{P}$-by-finite)-group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$.\
Then every infinite residually \[(locally $\mathcal{P}$)-by-finite\] group in $\mathcal{V}(w^*)$ belongs to $\mathcal{V}(w).$\
*]{}
For example, the classes of nilpotent groups, polycyclic groups, abelian-by-nilpotent groups and soluble residually finite groups satisfy the assumptions of Theorem B.\
Here we also obtain some reductions in investigation of the equality $\mathcal{V}(w) \cup \mathcal{F}=\mathcal{V}(w^*)$ on certain classes of groups and certain words $w$. For example\
[**Theorem C.**]{} [*Let $w$ be a non-trivial word in a free group. Then every non-linear simple locally finite group does not belong to the class $\mathcal{V}(w^*)$.*]{}\
In [@E], G. Endimioni proved that if $w$ be a word in a free group such that finitely generated soluble groups in $\mathcal{V}(w)$ are polycyclic, then every finitely generated soluble group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$. Before stating our next result, we need a notation (see [@G1]). Let $\alpha$ be a non-zero element of some field of characteristic $p$. Denote the group generated by the
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'In recent years, a number of experimental studies have been conducted to investigate the mechanical behavior and damage mechanisms of articular cartilage under impact loading. Some experimentally observed results have been explained using a non-linear viscoelastic impact model. At the same time, there is the need of simple mathematical models, which allow comparing experimental results obtained in drop impact testing with impact loads of different weights and incident velocities. The objective of this study was to investigate theoretically whether the main features of articular impact could be qualitatively predicted using a linear viscoelastic theory or the linear biphasic theory. In the present paper, exact analytical solutions are obtained for the main parameters of the Kelvin–Voigt and Maxwell impact models. Perturbation analysis of the impact process according to the standard viscoelastic solid model is performed. Asymptotic solutions are obtained for the drop weight impact test. The dependence of the coefficient of restitution on the impactor parameters has been studied in detail.'
address: 'Institute of Mathematics and Physics, Aberystwyth University, Ceredigion SY23 3BZ, Wales, UK'
author:
- 'I.I. Argatov'
title: ' Mathematical modeling of linear viscoelastic impact: Application to drop impact testing of articular cartilage'
---
Impact contact problem ,blunt indenter ,asymptotic model ,coefficient of restitution
Nomenclature {#nomenclature .unnumbered}
============
--------------- ------------------------------------------------
$b$ damper coefficient
$D$ discriminant of the characteristic equation
$E_{\rm dyn}$ incremental dynamic modulus
$E_{\rm max}$ maximum incremental dynamic modulus
$E_{10}$ modulus at stresses of 10 MPa
$e_*$ coefficient of restitution
$F$ contact force
$F_M$ maximum contact force
$g$ gravitational acceleration
$h$ cartilage layer thickness
$h_0$ drop height of the impactor
$H_A$ aggregate modulus
$k$ stiffness coefficient
$k_1$, $k_2$ spring stiffnesses in the standard solid model
$k_0$ instantaneous stiffness
$k_\infty$ long-term stiffness
$m$ impactor mass
$t$ time variable
$t_c$ impact duration
$t_m$ time to maximum displacement
$t_M$ time to maximum contact force
$v_0$ initial impact velocity
$x$ displacement
$\dot{x}$ velocity
$\ddot{x}$ acceleration
$x_m$ maximum displacement
--------------- ------------------------------------------------
$${}$$
------------------------------ -------------------------------------------------------------------
$\beta$ damping coefficient in the Kelvin–Voigt model
$\beta_1$ real part of complex roots of the characteristic equation
$\Delta m$ percentage increase in mass of cartilage sample
$\epsilon$ strain
$\varepsilon_0$ non-dimensional parameter accounting for the gravitational effect
$\zeta$ loss factor in the Maxwell model
$\zeta_1$ imaginary part of complex roots of the characteristic equation
$\eta$ loss factor in the Kelvin–Voigt model
$\kappa$ cartilage permeability
$\varkappa_1$, $\varkappa_2$ spring stiffnesses in the standard solid model
$\lambda$ Lamé coefficient
$\lambda_1$ root of the characteristic equation
$\Lambda$ non-dimensional parameter in the standard solid model
$\mu$ Lamé coefficient
$\xi$ non-dimensional displacement
$\rho$ ratio of the long-term and instantaneous stiffnesses
$\sigma$ stress
$\tau$ non-dimensional time
$\tau_D$ typical diffusion time
$\tau_R$ relaxation time
$\Psi(\tau)$ dimensionless relaxation function
$\omega$ angular frequency of damped oscillations
$\omega_0$ angular frequency of undamped oscillations
------------------------------ -------------------------------------------------------------------
Introduction {#1dsSectionI}
============
Articular cartilage is a soft hydrated tissue covering the end of each bone at the joints. Cartilage has no known function other than maintaining mechanical competence of joints, allowing bones to move against one another without friction. But there is no need to underline its significance to health of a human body, since almost all the load transmitted by a human joint goes through the articular cartilage, and it prevents biomechanical damage caused by severe loading including impact loading. It is believed that severe articular impact can initiate post-traumatic arthritis [@JeffreyGregoryAspden1995; @QuinnAllen2001]. An impact loading of the joint constitutes the action of extremely high non-physiological loads applied very rapidly (for instance, due to a car accident, sports injury, or a fall from a height).
In recent years, a number of experimental studies have been conducted to investigate the mechanical behavior and damage mechanisms of articular cartilage under impact loading [@AtkinsonHautAltiero1998; @VerteramoSeedhom2007; @BurginAspden2008]. In particular, the experimental data on relative dissipation of the impact energy $\Delta E/E_0$ versus overall impactor energy $E_0$ obtained in [@Varga2007] were fitted with quadratic curves. Here, $E_0=mv_0^2/2$, $\Delta E=m(v_1^2-v_0^2)/2$, $v_0$ and $v_1$ are the initial impact and rebound velocities, respectively, $m$ is the impactor mass. Since, $v_1=-e_* v_0$, where $e_*$ is the coefficient of restitution, we easily get $\Delta E/E_0=1-e_*^2$. Thus, the experimental data and fitting curves for dissipation of the impact energy [@Varga2007] can be recalculated in terms of the coefficient of restitution as presented in Fig. \[Varga2007.pdf\], which shows a non-monotonic dependence of $e_*$ on $v_0$. Some experimentally observed results have been explained using a non-linear viscoelastic impact model [@Edelsten2010]. At the same time, there is the need of a simple mathematical model, which allows comparing experimental results obtained in drop impact testing with impact loads of different weights and incident velocities.
![Coefficient of restitution $e_*$ versus the impact velocity $v_0$ for articular cartilage samples of different thicknesses. Based on the experimental data and fitting curves obtained in [@Varga2007]. []{data-label="Varga2007.pdf"}](Varga2007.pdf)
A variety of mathematical models were suggested to describe the stress-strain response of articular cartilage that represents a multiphasic, structurally complex material possessing viscoelastic properties. It is long known that articular cartilage possesses viscoelastic properties [@HayesMockros1971; @Lau_et_al_2008], though there is no direct correspondence between viscoelastic parameters and parameters of the biphasic/poroelastic models of cartilage. The biphasic theory [@MowKueiLaiArmstrong1980], which models the tissue as a mixture of a solid phase and a fluid phase, has demonstrated very good agreement with experimental results in the creep and stress relaxation tests [@SoltzAteshian2000]. The objective of this study was to investigate theoretically whether the main features of articular impact observed in [@Varga2007; @Edelsten2010] could be qualitatively predicted using a linear viscoelastic theory or the linear biphasic theory.
The rest of the paper is organized as follows. In Sections \[1dsSection1\] and \[1dsSection2\], we consider in detail the viscoelastic Kelvin–Voigt and Maxwell impact models, respectively. Since some elements of the presented solutions are known in the literature, we pay a particular attention to the evaluation of the contact force, $F(t)$, and impactor displacement, $x(t)$, at the time moments $t_M$ and $t_m$, when the force and displacement reach their maxima, $F_M$ and $x_m$, respectively. In Section \[1dsSection3\], we outline a closed form solution of the impact equation in the case of standard solid model. In order to get analytical approximations, we consider the standard solid model as a perturbation of the Kelvin–Voigt (Section \[1dsSection4\]) or the Maxwell model (Section \[1dsSection5\]). In particular, simple analytical approximations are derived for the impact duration, $t_c$, and for the coefficient of restitution, $e_*$. In Sections \[1dsSection05\] and \[1dsSection06\], we consider the influence of the gravity effect on these parameters in the framework of the Kelvin–Voigt and Maxwell models for drop weight impact. In Section \[1dsSection07\], we develop an asymptotic model for the force-displacement relationship in the indentation problem for a thin biphasic layer corresponding to the conditions of the so-called blunt impact, when the specimen thickness is much smaller than the radius of a flat-ended cylindrical impactor. An example of application of the developed linear theory of viscoelastic impact for analyzing experimental data is given in Section \[1dsSection10\]. Finally, in Sections \[1dsSectionD\] and \[1dsSectionC\], we outline a discussion of the results obtained and formulate our conclusions.
Viscoelastic Kelvin–Voigt impact model {#1dsSection1}
======================================
In this section, the
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We first show that the pions produced at high $p_T$ in heavy-ion collisions over a wide range of high energies exhibit a scaling behavior when the distributions are plotted in terms of a scaling variable. We then use the recombination model to calculate the scaling quark distribution just before hadronization. From the quark distribution it is then possible to calculate the proton distribution at high $p_T$, also in the framework of the recombination model. The resultant $p/\pi$ ratio exceeds one in the intermediate $p_T$ region where data exist, but the scaling result for the proton distribution is not reliable unless $p_T$ is high enough to be insensitive to the scale-breaking mass effects.'
author:
- 'Rudolph C. Hwa$^1$ and C. B. Yang$^{1,2}$'
date: January 2003
title: 'Scaling Behavior at High $p_T$ and the $p/\pi$ Ratio'
---
Introduction
============
There are three separate and independent aspects about the hadrons produced at large transverse momentum $(p_T)$ in heavy-ion collisions at high energies that collectively contribute to a coherent picture to be addressed in this paper. One is the existence of a scaling behavior at large $p_T$ that we have found by presenting the data in terms of a new variable. Another is the issue about the surprisingly large proton-to-pion ratio at moderate $p_T$ ($\sim$ 2 - 3 GeV/c) discovered by PHENIX [@ts] in central $AuAu$ reactions at $\sqrt{s} =$ 130 and 200 GeV. The third issue concerns the hadronization process relevant for the formation of hadrons at large $p_T$ and the applicability of the recombination model [@dh]. It is our goal to show that, in light of the scaling behavior of the $\pi^0$ produced, the recombination mechanism naturally gives rise to a $p/\pi$ ratio that exceeds 1 in the $2 < p_T < 3$ GeV/c range.
Particle production in heavy-ion collisions at very high energies is usually described in terms of hydrodynamical flow [@hyd], jet production at high $p_T$ [@jet], thermal statistical model [@the], or a combination of various hadronization mechanisms [@all]. In none of the conventional approaches does one expect protons to be produced at nearly the same rate as the pions. If all hadrons with $p_T > 2$ GeV/c are regarded as products of jet fragmentation, then the known fragmentation functions of quark or gluon jets would suppress proton relative to pion by the sheer weight of the proton mass. Such a discrepancy from the observed data led some to regard the situation as an anomaly and proposed the gluonic baryon junction as a mechanism to enhance the proton production rate [@vg]. Their predictions remain to be checked by experiments.
The parton fragmentation functions have been used even at low $p_T$ in string models where the production of particles in hadronic collisions is treated as the fragmentation of diquarks, as done in the dual parton model [@dpm]. There has been a long-standing dichotomy on whether particle production in the fragmentation region can better be described by fragmentation [@dpm; @lpy] or recombination [@dh; @hy]. It is possible that the two pictures might be unified in a more comprehensive treatment of hadronization in the future. Here we extend the recombination model to the central region at large $p_T$. It should be recognized that an essential part of the recombination model is the determination of the distribution functions of the quarks and antiquarks that are to recombine. In the case of large-$p_T$ hadrons the underlying physics is undoubtedly hard collisions of partons and the associated radiation of gluons. If the parton distributions can be calculated just before hadronization, then the final step of recombination can readily be completed. If those distributions cannot be determined in pQCD, then the step between the initiating large-$p_T$ parton and the resultant hadrons may efficiently be described by a fragmentation function, determined phenomenologically from experiments. Thus in that sense the two approaches, recombination and fragmentation, are not contradictory, but complementary.
We state from the outset that no attempt will be made here to perform a first-principle calculation of the parton distributions at large $p_T$ before recombination. However, from the observed data on pion production in central $AuAu$ collisions at the Relativistic Heavy-Ion Collider (RHIC), it is possible to work backwards in the recombination model to determine the quark (and antiquark) distribution at large $p_T$. On the basis of the quark distributions inferred, it is then possible to calculate the proton distribution in the recombination model. The basic idea is that if there is a dense system of quarks and antiquarks produced in a heavy-ion collision whatever the dynamics responsible for them may have been (gluons having been converted to $q\bar{q}$ pairs before hadronization), then the formation of pions and protons (and whatever else) is prescribed by the recombination model without any arbitrariness in normalization and momentum dependence.
One limitation of the recombination model as it stands at present is that it is formulated in a frame-independent way in terms of momentum fractions and is therefore inapplicable to a system where the particle momenta are low and the mass effects are large. The physics of recombination is still valid at low momentum, but the details of the wave functions of the constituent quarks become important; they have not been built into the recombination function that takes the simplest form in the infinite momentum frame. Thus our calculation of particles produced at midrapidity is not reliable when $p_T$ is of the order of the masses of the hadrons under consideration. For protons we can trust the results only for $p_T > 3$ GeV/c. For pions the lower limit of validity can be pushed much lower.
Since our approach makes crucial use of the experimental data on the pion spectrum as the input, it is essential to relate the spectra determined at different energies to an invariant distribution so that the scale-invariant recombination model can be applied. To discover the existence of an invariant distribution with no theoretical prejudices is a problem worthy in its own right. Fortunately, that turns out to be possible. The analysis for that part of the study will be presented below first to emphasize its independence from the theoretical modeling of hadronization. It should be mentioned that the scaling of transverse mass spectra has been investigated recently [@sb]. The emphasis there has been on the dependences on the particle species and centrality for $m_T<3.8$ GeV, while our focus is on the dependence on energy ($17<\sqrt s<200
$ GeV) for $p_T<8$ GeV/c. Thus the two studies are complementary to each other.
A Universal Scaling Distribution
================================
The preliminary data of the $p_T$ distributions of $\pi^0$ produced at RHIC at $\sqrt{s} = 130$ and 200 GeV were shown by the PHENIX Collaboration at Quark Matter 2002 [@ddl] for central $AuAu$ collisions together with the WA98 data for $PbPb$ collisions at $\sqrt{s} = 17$ GeV [@rey]. They show that the level of the tail at large $p_T$ rises , as $\sqrt{s}$ is increased. We want to consider the possibility that the three sets of data points can be combined to form a universal curve.
The $\pi^0$ inclusive distributions at midrapidity are integrated over $\eta$ for a range of $\Delta \eta = 1$ so that the data points are given for the following quantity [@ddl]: $$\begin{aligned}
f(p_T, s) = {1 \over 2 \pi p_T}{dN \over dp_T}=
\int_{\Delta\eta} d\eta \left(2 \pi p_TN_{evt} \right)^{-1}
{d^2N_{\pi^0}
\over d p_T d\eta} .
\label{1}\end{aligned}$$ In comparing the PHENIX data with those of WA98 one should recognize that in addition to the difference in the colliding nuclei there is a slight mismatch in centrality (top 10% for PHENIX and top 12.7% for WA98) [@we].
To unify the three data sets it is natural to first consider a momentum fraction variable similar to $x_F$ in longitudinal momentum. However, so much momenta are taken by the other particles outside the $\Delta \eta = 1$ range, it is unwise to also use $\sqrt{s}/2$ as the scale to calculate the transverse momentum fraction. We assume that for every $\sqrt{s}$ there is a relevant scale $K$ to describe the $p_T$ behavior relative to that scale. Let us define $$\begin{aligned}
z = p_T/K,
\label{2}\end{aligned}$$ and transform $f(p_T, s)$ to a new function $\Phi (z,K)$, where $$\begin{aligned}
\Phi (z,K) = K^2 f(p_T, s) = {1 \over 2 \pi z} {dN \over dz} .
\label{3}\end{aligned}$$ We adjust $K$ for each $s$ and check whether all three data sets coalesce into one universal dependence on $z$, which we would simply label as $\Phi(z)$, if it is possible.
In Fig.
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We study the ground-state spin correlations in the gapless incommensurate regime of a $S=1/2$ $XXZ$ chain and a two-leg antiferromagnetic ladder under a magnetic field, in which the gapless excitations form a Tomonaga-Luttinger (TL) liquid. We calculate numerically the two-spin correlation functions and the local magnetization in the two models using the density-matrix renormalization-group method. By fitting the numerical results for an open $XXZ$ chain of 100 spins to correlation functions of a Gaussian model, we determine the TL-liquid parameter $K$ and the amplitudes of the correlation functions. The value of $K$ estimated from the fits is in excellent agreement with the exact value obtained from the Bethe ansatz. We apply the same method to the open ladder consisting of 200 spins and determine the dependence of $K$ on the magnetization $M$. The $K$-$M$ relation changes drastically depending on the ratio of the coupling constants in the leg and rung directions. We also discuss implications of these results to experiments on the nuclear spin relaxation rate $1/T_1$ and dynamical spin structure factors.'
address:
- |
Department of Earth and Space Science, Graduate School of Science, Osaka University,\
Toyonaka, Osaka 560-0043, Japan
- 'Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan'
author:
- 'T. Hikihara'
- 'A. Furusaki'
date: 'November 20, 2000'
title: 'Spin correlations in the two-leg antiferromagnetic ladder in a magnetic field'
---
[2]{}
INTRODUCTION
============
Spin ladder systems have been studied extensively over the past decade.[@review] There are reasons why the ladders have attracted so much attention. Firstly, they naturally interpolate one- and two-dimensional systems and may provide some hints to better understand the high-temperature superconductivity which occurs in square lattice ${\rm CuO_2}$ planes. Secondly, spin ladders themselves have interesting physics and deserve through investigation in their own right. One of their most surprising properties is that low-energy physics of spin ladders depends drastically on the number of legs. Spin-$1/2$ antiferromagnetic (AF) ladders, for example, have a finite gap in the spin excitation spectrum in the even-leg case, whereas they have no gap in the odd-leg case. The ground state of an even-leg ladder is a spin singlet and its properties can be understood from the short-range resonating-valence-bond picture.[@RVB] This spin-gap behavior has been observed experimentally on $S=1/2$ two-leg ladder compounds,[@exp1; @exp2] such as ${\rm SrCu_2O_3}$ and ${\rm Cu_2(C_5H_{12}N_2)_2Cl_4}$.
A gapless phase can appear in even-leg ladders when an external field $h$ is applied. If the field $h$ is larger than a critical field $h_{c1}$, which is equal to the spin gap, and if it is smaller than the saturation field $h_{c2}$, then the ground state has a nonzero magnetization $M$ and the energy gap between the ground state and the first-excited states vanishes. The gapless mode has been shown to be described as a Tomonaga-Luttinger (TL) liquid both in the strong- and weak-coupling limit,[@Chi-Gia; @Gia-Tsv; @Furu-Zhn] where the coupling in the rung-direction $J_\perp$ is much larger or much smaller than the one in the leg-direction $J_\parallel$, respectively. However, the $M$ dependence of the TL-liquid parameter $K$, which governs the spin correlations in long wave length, has been obtained analytically only in the strong-coupling limit and it remains as a nontrivial problem to determine $K$ for general $J_\perp/J_\parallel$. In the gapless phase the system shows incommensurate spin correlations since the Fermi wavenumber of Jordan-Wigner fermions is shifted from $\pi/2$ in the presence of the magnetic field which acts as a chemical potential for the fermions. The wavenumber $Q$ characterizing the incommensurability of the gapless mode varies continuously as $h$ increases. This gapless incommensurate (IC) phase is in fact in the same universality class as the one-dimensional $S = 1/2$ $XXZ$ model in a magnetic field, as we will see.
In this paper, we study low-energy properties of the $S = 1/2$ two-leg AF ladder in the gapless IC regime for broad range of $J_\perp / J_\parallel$. We show that the system is a TL liquid for arbitrary $J_\perp / J_\parallel$ and determine the $M$ dependence of $K$ numerically. To this end, we compute numerically the ground-state spin-correlation functions and the local magnetization in the open ladders using the density-matrix renormalization-group (DMRG) method [@White1; @White2] and extract the TL-liquid parameter by fitting the data to correlation functions obtained from the Abelian bosonization. This method was applied in our previous work [@CorAm] to the $S=1/2$ $XXZ$ chain at $h = 0$ and proved to be effective in determining both the TL-liquid parameter and amplitudes of correlation functions. In order to demonstrate the validity of the analysis in the gapless IC phase, we first apply it to the $S = 1/2$ $XXZ$ chain for $h > 0$. The model is exactly solvable by the Bethe ansatz and the TL-liquid parameter $K$ can be calculated for arbitrary value of $M$. It thus provides a good test ground to check accuracy of our method. We find that $K$ estimated from the DMRG data is in excellent agreement with the exact calculation. We then apply the same method to the two-leg ladders in a magnetic field. Our numerical data of correlation functions are fitted well for broad range of $J_\perp/J_\parallel$ to the formulas based on the bosonization approach, confirming that the gapless modes are in fact in the universality class of a TL liquid for arbitrary $J_\perp/J_\parallel$. The $M$ dependence of $K$ obtained in the large $J_\perp/J_\parallel$ limit agrees with the analytic result obtained through mapping to the $XXZ$ chain. In this limit $K$ is less than 1 for $0 < M < 1$. As $J_\perp/J_\parallel$ decreases, $K$ increases and become larger than 1 for intermediate values of $M$. Our numerical result indicates that $K$ takes a universal value 1 for any $J_\perp/J_\parallel$ in the limits $M\to0$ and $M\to1$.
The plan of the paper is as follows. We first review the Abelian bosonization approach to the $S=1/2$ $XXZ$ chain under a magnetic field in Sec. II A. The formulas of the spin correlations and the local magnetization in finite open chains are presented. In Sec. II B, we show numerical data for the $XXZ$ chain of $L=100$ sites obtained from the DMRG calculation and fit the data to the functions given in Sec. II A. In Sec. III A, we briefly review some relevant results of the previous analytic studies on the two-leg ladders in the strong- and weak-coupling limits. The DMRG data and the results of fitting on the open ladders with $L = 200$ sites are shown in Sec. III B. The $M$ dependence of $K$ for various values of $J_\perp/J_\parallel$ is obtained. Its implications to NMR and neutron scattering experiments are briefly mentioned. Finally, our results are summarized in Sec. IV.
$XXZ$ CHAIN
===========
Bosonization approach
---------------------
In this section, we consider spin-1/2 $XXZ$ chains with open ends in a magnetic field $h$. The Hamiltonian is $${\cal H}_{\rm ch} = {\cal H}_0 + {\cal H}_h \label{eq:Hchn}$$ with $$\begin{aligned}
{\cal H}_0 &=&
J \sum_{l=1}^{L-1} (\bbox{S}_l, \bbox{S}_{l+1} )_\Delta,
\nonumber \\
{\cal H}_h &=& - h \sum_{l=1}^L S^z_l, \nonumber\end{aligned}$$ where $\bbox{S}_l$ are $S = 1/2$ spin operators and $(\bbox{S}_l, \bbox{S}_{l'} )_\Delta =
S^x_l S^x_{l'} + S^y_l S^y_{l'} + \Delta S^z_l S^z_{l'}$. We assume the system size $L$ to be even throughout this paper and treat only the case where $J > 0$ and $0 \le \Delta \le 1$. We note that the Hamiltonian (\[eq:Hchn\]) for $-1 < \Delta\le 1$ can be solved exactly by Bethe ansatz for arbitrary values of $h$.
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We provide a quantitative version of the isoperimetric inequality for the fundamental tone of a biharmonic Neumann problem. Such an inequality has been recently established by Chasman adapting Weinberger’s argument for the corresponding second order problem. Following a scheme introduced by Brasco and Pratelli for the second order case, we prove that a similar quantitative inequality holds also for the biharmonic operator. We also prove the sharpness of both such an inequality and the corresponding one for the biharmonic Steklov problem.'
author:
- 'D. Buoso'
- 'L.M. Chasman'
- 'L. Provenzano'
bibliography:
- 'bibliography.bib'
title: On the stability of some isoperimetric inequalities for the fundamental tones of free plates
---
Introduction
============
The stability of isoperimetric inequalities is an important question that has gained significant interest in recent decades. For example, the celebrated Faber-Krahn inequality for the smallest eigenvalue of the Dirichlet Laplacian, $$\lambda_1(\Omega)\ge\lambda_1(\Omega^*),$$ can be improved in the following quantitative form: $$\lambda_1(\Omega)\ge\lambda_1(\Omega^*)(1+C\mathcal{A}(\Omega)^2),
\label{quantfk}$$ for some constant $C>0$. Here $\Omega\subset\mathbb{R}^N$ is a bounded open set, $N\geq2$, $\Omega^*$ is a ball such that $|\Omega|=|\Omega^*|$, and $\mathcal A(\Omega)$ is the so-called Fraenkel asymmetry of the domain $\Omega$ (see for the definition). Quantitative versions of the type have also been established for other isoperimetric inequalities involving eigenvalues of the Laplace operator, see, e.g., [@brascosteklov; @brasco2015; @brascopratelli].
Fewer isoperimetric inequalities have been established for eigenvalues of the biharmonic operator, namely for the first nontrivial eigenvalue of the Dirichlet (“clamped plate”) problem [@ashbaugh; @nadirashvili], of the Neumann (“free plate”) problem [@chasmanpreprint; @chasman], and of the Steklov problem introduced in [@buosoprovenzano] (see also [@buosoprovenzano0]). An isoperimetric inequality is still missing for another Steklov problem introduced in [@kuttler68], the conjectured optimizer being the regular pentagon (see, e.g., [@antunesgazzola; @bucurgazzola11] and the references therein).
Among these inequalities, the first one that has been given in quantitative form is the inequality for Steklov problem in [@buosoprovenzano], namely $$\lambda_2(\Omega)\le\lambda_2(\Omega^*)(1-C\mathcal{A}(\Omega)^2),
\label{quantitative_bp}$$ where $\lambda_2(\Omega)$ is the first nontrivial eigenvalue of the biharmonic Steklov problem $$\label{SteklovPDE}
\begin{cases}\Delta^2u-\tau\Delta u=0 &\text{in $\Omega$,}\\
\frac{\partial^2 u}{\partial \nu^2}= 0 &\text{on $\partial\Omega$,}\\
\tau\frac{\partial u}{\partial \nu}
-{\rm div}_{\partial\Omega}\Big(D^2u\cdot \nu\Big)-\frac{\partial\Delta u}{\partial \nu} = \lambda u &\text{on $\partial\Omega$,}
\end{cases}$$ where $\tau$ is a strictly positive constant.
In this paper we provide a quantitative form for the isoperimetric inequality for the first non-trivial eigenvalue of the following biharmonic Neumann problem: $$\label{NeumannPDE}
\begin{cases}\Delta^2u-\tau\Delta u=\lambda u &\text{in $\Omega$,}\\
\frac{\partial^2 u}{\partial \nu^2}= 0 &\text{on $\partial\Omega$,}\\
\tau\frac{\partial u}{\partial \nu}
-{\rm div}_{\partial\Omega}\Big(D^2u\cdot \nu\Big)-\frac{\partial\Delta u}{\partial \nu} = 0 &\text{on $\partial\Omega$.}
\end{cases}$$ We recall that for $N=2$, problem describes the transverse vibrations of an unconstrained thin elastic plate with shape $\Omega\subset \mathbb{R}^2$ when at rest. The constant $\tau$ represents the ratio of lateral tension to lateral rigidity and is taken to be non-negative.
When $\tau>0$ and $\Omega\subset\mathbb{R}^N$ is a smooth connected bounded open set, it is known that the spectrum of the Neumann biharmonic operator $\Delta^2-\tau\Delta$ consists entirely of non-negative eigenvalues of finite multiplicity, repeated according to their multiplicity: $$0=\lambda_1(\Omega)<\lambda_2(\Omega)\leq\cdots\leq\lambda_j(\Omega)\leq\cdots.$$ Note that since constant functions satisfy problem with eigenvalue $\lambda=0$, the first positive eigenvalue is $\lambda_2$, which is usually called the “fundamental tone” of the plate. In [@chasman], the author proved that $$\label{iso_neumann}
\lambda_2(\Omega)\leq \lambda_2(\Omega^*)$$ with equality if and only if $\Omega=\Omega^*$. The proof of inequality is based on Weinberger’s argument for the Neumann Laplacian, taking suitable extensions of the eigenfunctions of the ball as trial functions (see [@weinberger]). In [@brascopratelli], the authors carry out a more careful analysis of such an argument, improving Weinberger’s inequality to a quantitative form. In a similar way, we start from the proof of and improve the result to the quantitative inequality by means of this finer analysis.
The question of sharpness is another important issue that has to be addressed when dealing with quantitative isoperimetric inequalities. More precisely, given an inequality of the form $$%\label{quantitative_general}
\lambda_2(\Omega)\leq\lambda_2(\Omega^*)\left(1-\Phi({\rm dist}(\Omega,\mathcal B))\right),$$ where $\Phi$ is some modulus of continuity, ${\rm dist}(\cdot,\cdot)$ is a suitable distance between open sets and $\mathcal B$ is the family of all balls in $\mathbb R^N$, we say that it is sharp if there exists a family $\lbrace\Omega_{\varepsilon}\rbrace_{\varepsilon\in(0,\varepsilon_0)}$ such that ${\rm dist}(\Omega_{\varepsilon},\mathcal B)\rightarrow 0$, $\lambda_2(\Omega_{\varepsilon})\rightarrow\lambda_2(\Omega^*)$ as $\varepsilon\rightarrow 0$, and there exists contants $c_1,c_2>0$ which do not depend on $\varepsilon>0$ and $\Omega^*$ such that [$$c_1\Phi({\rm dist}(\Omega_{\varepsilon},\mathcal B))\leq 1-\frac{\lambda_2(\Omega_{\varepsilon})}{\lambda_2(\Omega^*)}\leq c_2\Phi({\rm dist}(\Omega_{\varepsilon},\mathcal B)),$$]{} as $\varepsilon\rightarrow 0$. Note that, in our case, the distance function is given by the Fraenkel asymmetry ${\rm dist}(\Omega,\mathcal B)=\mathcal A(\Omega)$ while the modulus of continuity is $\Phi(t)=Kt^2$, for some $K>0$. By means of the construction introduced in [@brascosteklov; @brascopratelli], we prove in Section \[sharpness\_neumann\] that the quantitative Neumann inequality is sharp.
It is worth noting that in the Neumann Laplacian case in [@brascopratelli], the authors try, as a first guess, to consider ellipsoids as the family $\lbrace\Omega_{\varepsilon}\rbrace_{\varepsilon\in(0,\varepsilon_0)}$, with the ball $\Omega_0$ being the maximizer. Unfortunately, this is not a good family to prove sharpness; this can be explained observing that different directions of perturbation behave in a different way with respect to the fundamental tone. In particular, some directions are not “good enough” to see the sharpness (cf. [@brascopratelli Remark 5.2]). This phenomenon can be observed in our case as well: therefore we need to restrict our analysis by excluding some directions. See (\[perturbation\]) and Remark \[directions\].
The Steklov problem is of particular interest despite its recent introduction, since in [@buosoprovenzano] the authors show that it has a very strict relationship with the Neumann problem . Using a mass perturbation argument, they prove that the Steklov problem can in fact be viewed as a limiting Neumann problem where the mass is distributed only on the boundary. Note that this construction was already performed in [@lambertiprozisaac] for the Laplace operator, obtaining similar results (see also [@dallarivaproz; @lambertiproz] for the computation of the topological derivative). Moreover, this justifies the fact of thinking of Steklov problems in terms of vibrating objects (plates or membranes) where the mass lies only on the boundary (see [@steklov]). The authors also prove the quantitative inequality by adapting an argument due to Brock (see [@brock]) for the Steklov Laplacian to the biharmonic case in the refined version of [@br
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'Observations of nearby galaxies have firmly established, over a broad range of galactic environments and metallicities, that star formation occurs exclusively in the molecular phase of the interstellar medium (ISM). Theoretical models show that this association results from the correlation between chemical phase, shielding, and temperature. Interstellar gas converts from atomic to molecular only in regions that are well shielded from interstellar ultraviolet (UV) photons, and since UV photons are also the dominant source of interstellar heating, only in these shielded regions does the gas become cold enough to be subject to Jeans instability. However, while the equilibrium temperature and chemical state of interstellar gas are well-correlated, the time scale required to reach chemical equilibrium is much longer than that required to reach thermal equilibrium, and both timescales are metallicity-dependent. Here I show that the difference in time scales implies that, at metallicities below a few percent of the Solar value, well-shielded gas will reach low temperatures and proceed to star formation before the bulk of it is able to convert from atomic to molecular. As a result, at extremely low metallicities, star formation will occur in a cold atomic phase of the ISM rather than a molecular phase. I calculate the observable consequences of this result for star formation in low metallicity galaxies, and I discuss how some current numerical models for H$_2$-regulated star-formation may need to be modified.'
author:
- 'Mark R. Krumholz'
title: Star Formation in Atomic Gas
---
Introduction
============
In present day galaxies, star formation is very well-correlated with the molecular phase of the interstellar medium (ISM) [@wong02a; @kennicutt07a; @leroy08a; @bigiel08a]. In contrast, in the inner parts of disks where there are significant molecular fractions, star formation correlates very poorly or not at all with the atomic ISM. At large galctocentric radii where the ISM becomes atomic-dominated star formation does begin to correlate with H <span style="font-variant:small-caps;">i</span>, but this appears to be only because H$_2$ itself becomes correlated with H <span style="font-variant:small-caps;">i</span>, and the H$_2$ forms stars in the same way regardless of where it is found within a galaxy [@bigiel10a; @schruba11a]. Strong association between star formation and H$_2$ and a lack of association with H <span style="font-variant:small-caps;">i</span> is also found down the lowest metallicity systems that have been measured, at roughly 20% of Solar [@bolatto11a]. In summary, all available observational data indicates that star formation occurs only where the hydrogen in the ISM has converted to H$_2$.
Theoretical models have explained these observations as resulting from a correlation between chemistry and temperature [@schaye04a; @krumholz11b; @glover12a]. Molecular hydrogen is not an important coolant in modern-day galaxies, and while carbon monoxide (which forms only when it is catalyzed by H$_2$ – @van-dishoeck86a) is, the C <span style="font-variant:small-caps;">ii</span> found in H <span style="font-variant:small-caps;">i</span> regions is almost as effective. However, H$_2$ is an excellent proxy for the presence of cold gas because both are sensitive to destruction by UV photons, which photodissociate H$_2$ and increase the temperature through the grain photoelectric effect. As a result, both H$_2$ and low temperature gas are found only in regions of high extinction where the UV photon density is far below its mean value in the ISM, and, conversely, any region that where the photodissociation rate is high enough to convert the bulk of the ISM to H <span style="font-variant:small-caps;">i</span> is also likely to be warm. Since low temperatures that remove thermal pressure support are a prerequisite for collapse into stars, this correlation between temperature and chemical state in turn induces a correlation between star formation and chemical state.
However, the correlation between H$_2$ and star formation must break down at sufficiently low metallicities. Before the first stars formed in the universe, and for a short time thereafter, there were no or very few heavy elements. As a result, forming H$_2$ was extremely difficult due to a lack of dust grain surfaces to catalyze the ${\rm H~\textsc{i}} \rightarrow {\rm H}_2$ reaction. Theoretical models of star formation in such environments indicate that H$_2$ fractions remain extremely small until the density rises so high (${\protect\raisebox{-0.5ex}{$\:\stackrel{\textstyle >}
{\sim}\:$}}10^9$ cm$^{-3}$) that H$_2$ can form via three-body reactions [@palla83a; @lepp84a; @ahn07a; @omukai10a]. The underlying physical basis for this result is a disconnect of timescales: the equilibrium chemical state the gas would reach after a very long time would be H$_2$-dominated, but the cooling and star formation times are short enough that the gas does not reach equilibrium before collapsing into a star.
While this result has been known for zero and extremely low metallicity systems for some time, the relationship between chemical state and star formation in intermediate metallicity regime, for which observations are at least in principle possible in the local universe, has received fairly little attention. @omukai10a consider the chemical evolution of collapsing gas cores with metallicities from 0 to Solar, and investigate under what circumstances they can form H$_2$. However, because their calculation starts with unstable, collapsing cores, it does not address the question of in what phase of the ISM one expects to find such collapsing regions in the first place, which is the central problem for understanding the observed galactic-scale correlation between ISM chemical state and star formation. @glover12b simulate the non-equilibrium chemical and thermal behavior of clouds with metallicities from 1% of Solar to Solar. They find that the bulk of the cloud material converts to H$_2$ before star formation in the high metallicity clouds but not in the lowest metallicity ones, indicating that the star formation - H$_2$ correlation should begin to break down at metallicities observable in nearby galaxies. However, given the computational cost of their simulations, they are able to explore a very limited number of cases, and it is unclear how general their results might be.
The goal of this paper is to go beyond the studies of @omukai10a and @glover12b by deriving general results about the correlation between chemical state and star formation over a wide range of environments and metallicities. I do not perform detailed simulations, such as those of @omukai10a and @glover12b, for every case. Instead, I rely on fairly simple models that can be integrated semi-analytically. The benefit of this approach is that it is the only way to survey a large parameter space, and thereby to answer the central questions with which I am concerned: under what conditions do we expect the correlation between star formation and H$_2$ to break down? When such a breakdown occurs, what is the governing physical mechanism that causes it? What are the resulting observational signatures? What are the implications of this breakdown for the models of star formation commonly adopted in studies of galaxy formation? In the remainder of this paper, I seek to answer these questions.
Model
=====
Consider spatially uniform gas characterized by a mean number density of H nuclei ${\overline{n}_{\rm H}}$, column density of hydrogen nuclei ${N_{\rm H}}$, metallicity $Z'$ relative to Solar, and temperature $T$. A fraction $f_{\rm H_2}$ of the H nuclei are locked in H$_2$ molecules. It is generally more convenient to characterize models by values of the visual extinction $A_V$ instead of ${N_{\rm H}}$. These two are related by $A_V/{N_{\rm H}}\approx 4.0 \times 10^{-22} Z'$ mag cm$^2$, with the normalization chosen as a compromise between the values for Milky Way extinction and the extinction curves of the Large and Small Magellanic Clouds adjusted to Milky Way metallicity.[^1]
Timescale Estimates
-------------------
We are interested in following the behavior of initially warm, atomic gas, and considering whether it will be able to cool to temperatures low enough to allow star formation, and how its chemical state will evolve as it does so. Before computing detailed evolutionary histories, it is helpful first to make a rough estimate of the timescales involved. In interstellar gas that is dense enough to be a candidate for star formation, but that is not yet molecular or forming stars, the dominant cooling process is emission in the \[C <span style="font-variant:small-caps;">ii</span>\] 158 $\mu$m line, which removes energy at a rate $$\Lambda_{\rm CII} \approx k_{\rm CII-H} {\delta_{\rm C}}k_B {T_{\rm CII}}{\mathcal{C}}{\overline{n}_{\rm H}}$$ per H atom, where $k_{\rm CII-H}\approx 8\times 10^{-10} e^{-{T_{\rm CII}}/T}$ cm$^3$ s$^{-1}$ is the rate coefficient for collisional excitation of C <span style="font-variant:small-caps;">ii</span> by H atoms, ${\delta_{\rm C}}\approx 1.1\times 10^{-4} Z'$ is the gas
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the Ollivier Ricci curvature sense. Our main result is a classification of all self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs, even-dimensional demi-cubes, Johnson graphs $J(2n,n)$, the Gosset graph and suitable Cartesian products). We also present a purely combinatorial reformulation of this result. We show that Bonnet-Myers sharpness implies Lichnerowicz sharpness. We also relate Bonnet-Myers sharpness to an upper bound of Bakry-Émery $\infty$-curvature, which motivates a general conjecture about Bakry-Émery $\infty$-curvature.'
author:
- 'D. Cushing'
- 'S. Kamtue'
- 'J. Koolen'
- 'S. Liu'
- 'F. Münch'
- 'N. Peyerimhoff'
title: 'Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature'
---
Introduction and statement of results
=====================================
A fundamental question in geometry is in which way local properties determine the global structure of a space. A famous result of this kind is the Bonnet-Myers Theorem [@My41] for complete $n$-dimensional Riemannian manifolds $M$ with $K = \inf {\rm Ric}_M(v) > 0$ (a condition on the local invariant ${\rm Ric}_M ={\rm Tr}(R_M)$), where the infimum is taken over all unit tangent vectors $v$ of $M$. Under this condition, $M$ is compact and its diameter satisfies $$\label{eq:BM_RG_ineq}
{\rm diam}(M) \le \pi \sqrt{\frac{n-1}{K}}.$$ Moreover, Cheng’s Rigidity Theorem [@Cheng75] states that this diameter estimate is sharp if and only if $M$ is the $n$-dimensional round sphere. Note that inequality can be reformulated as an upper bound on the infimum of the Ricci curvature in terms of the diameter, and this reformulation is the viewpoint we will assume in this paper.
In the discrete setting of graphs there are several analogs of Ricci curvature notions providing Bonnet-Myers type theorems (see, e.g., [@FS; @HLLY; @LMP; @LLY11; @Ol09], ...). In view of Cheng’s rigidity result, it is natural to ask for which graphs the Bonnet-Myers estimates is sharp. We call such graphs *Bonnet-Myers sharp* graphs. For example, in the case of Bakry-Émery $\infty$-curvature, Bonnet-Myers sharp graphs have been fully characterised and are only the hypercubes (see [@LMP2]).
The motivation of this paper is to study Bonnet-Myers sharpness with respect to another curvature notion, namely, *Ollivier Ricci curvature*. (In fact, we will consider a modification of Ollivier’s definition introduced in [@LLY11].) Henceforth, all graphs $G = (V,E)$ with vertex set $V$ and edge set $E$ will be simple (loopless without multiple edges) and edges can be identified with $2$-element subsets of $V$. In this paper, we will only formulate and derive our results for regular graphs, that is, all vertices have the same valency, even though similar questions can be posed for non-regular graphs.
Ollivier Ricci curvature $\kappa(x,y)$ is a notion based on optimal transport and is defined on pairs of different vertices $x,y \in
V$. The precise definition requires a longer introduction and is given in Subsection \[sec:OllivKant\]. Generally, $\kappa(x,y)$ is positive if the average distance between corresponding neighbours of $x$ and $y$ is smaller than $d(x,y)$. For now, we confine ourselves to provide a useful connection of this curvature with a particular combinatorial property, to provide the readers with some understanding of this notion. Note that this Proposition follows directly from Proposition \[prop:curvcalc0\] by choosing $m = \frac{2D}{L} - 2$.
\[prop:curvcalc\] Let $G=(V,E)$ be a $D$-regular graph of diameter $L$ and $e=\{x,y\} \in E$. Assume that $e$ is contained in precisely $\frac{2D}{L}-2$ triangles and there is a perfect matching between the neighbours of $x$ and the neighbours of $y$ which are not involved in these triangles. Then we have $$\kappa(x,y) = \frac{2}{L}.$$
Let us now state the discrete Bonnet-Myers Theorem for Ollivier Ricci curvature and introduce the associated notion of Bonnet-Myers sharpness for this curvature notion:
\[thm:DBM\] Let $G= (V,E)$ be a connected $D$-regular graph and $\inf_{x \sim y}
\kappa(x,y) > 0$. Then $G$ has finite diameter $L = {\rm diam}(G) <
\infty$ and $$\label{eq:BM_est}
\inf_{x \sim y} \kappa(x,y) \le \frac{2}{L}.$$ We say that such a graph $G$ is [**[*$\boldsymbol{(D,L)}$-Bonnet-Myers sharp*]{}**]{} (with respect to Ollivier Ricci curvature) if holds with equality.
Many of our results require the additional condition of self-centeredness. Note that a graph $G=(V,E)$ is called self-centered if, for every vertex $x \in V$, there exists a vertex $\overline{x} \in V$ such that $d(x,\overline{x}) = {\rm diam}(G)$ (see Subsection \[sec:graph\_notation\] for its definition). Let us now state the main results of this paper.
- *Cartesian products:* $G_1 \times G_2 \times \cdots \times G_k$ is Bonnet-Myers sharp if and only if all factors $G_i$ are Bonnet-Myers sharp and satisfy $$\label{eq:cartprod_cond0}
\frac{D_1}{L_1} = \frac{D_2}{L_2} =\cdots = \frac{D_k}{L_k},$$ where $D_i$ and $L_i$ are the vertex degrees and the diameters of the graphs $G_i$, respectively (see Theorem \[thm:cartprod\]).
- Every Bonnet-Myers sharp graph is Lichnerowicz sharp (see Theorem \[thm:lichn\]).
- *Classification of self-centered Bonnet-Myers sharp graphs:* Self-centered Bonnet-Myers sharp graphs are precisely the following ones: Hypercubes, cocktail party graphs, the Johnson graphs $J(2n,n)$, even-dimensional demi-cubes, the Gosset graph and Cartesian products of them satisfying (see Theorem \[thm:main\]).
- Self-centered $(D,L)$-Bonnet-Myers sharp graphs are Bakry-[É]{}mery $\infty$-curvature sharp in all vertices with normalized $\infty$-curvature value $\frac{1}{D} + \frac{1}{L}$ (see Theorem \[thm:aBM-BEcurv\]).
We provide more detailed information about these results in the next subsection. In particular, result (c) above is based on another result which can be reformulated in purely combinatorial terms. This combinatorial reformulation is derived in Subsection \[subsec:comb\_res\].
Our results on Bonnet-Myers sharp graphs
----------------------------------------
It is useful to know that the vertex degree $D$ and the diameter $L$ of a Bonnet-Myers sharp graph cannot be arbitrary:
\[thm:DL\_rel\] Any $(D,L)$-Bonnet-Myers sharp graph satisfies $L \le D$. Moreover $L$ must divide $2D$.
This theorem is proved in Section \[sec:self-centBMsh\].
Another fundamental result between local and global properties of a closed $n$-dimensional Riemannian manifold $M$ is Lichnerowicz’ Theorem [@Li58 p. 135] which states that, under the condition $K = \inf {\rm Ric}_M(v) > 0$, the smallest positive Laplace-Beltrami eigenvalue $\lambda_1(M)$ satisfies $$\frac{n}{n-1}K \le \lambda_1.$$ The associated rigidity result is Obata’s Theorem [@Ob62], which states that this eigenvalue estimate is sharp if and only if $M$ is the $n$-dimensional round sphere. There is a discrete analogue of Lichnerowicz’ Theorem for Ollivier Ricci Curvature and the normalized Laplacian $\Delta_G = D^{-1} A_G - {\rm Id}$ of an arbitrary graph $G=(V,E)$, where $A_G$ denotes the adjacency matrix of $G$ and $D$ is here a diagonal matrix whose entries are the valencies $d_x$ of the vertices $x \in V$. In the case
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'Cerium-doped manganite thin films were grown epitaxially by pulsed laser deposition at $720\,^\circ$C and oxygen pressure $p_{O_2}=1-25\,$Pa and were subjected to different annealing steps. According to x-ray diffraction (XRD) data, the formation of CeO$_2$ as a secondary phase could be avoided for $p_{O_2}\ge 8\,$Pa. However, transmission electron microscopy shows the presence of CeO$_2$ nanoclusters, even in those films which appear to be single phase in XRD. With O$_2$ annealing, the metal-to-insulator transition temperature increases, while the saturation magnetization decreases and stays well below the theoretical value for electron-doped La$_{0.7}$Ce$_{0.3}$MnO$_3$ with mixed Mn$^{3+}$/Mn$^{2+}$ valences. The same trend is observed with decreasing film thickness from 100 to 20nm, indicating a higher oxygen content for thinner films. Hall measurements on a film which shows a metal-to-insulator transition clearly reveal holes as dominating charge carriers. Combining data from x-ray photoemission spectroscopy, for determination of the oxygen content, and x-ray absorption spectroscopy (XAS), for determination of the hole concentration and cation valences, we find that with increasing oxygen content the hole concentration increases and Mn valences are shifted from 2+ to 4+. The dominating Mn valences in the films are Mn$^{3+}$ and Mn$^{4+}$, and only a small amount of Mn$^{2+}$ ions can be observed by XAS. Mn$^{2+}$ and Ce$^{4+}$ XAS signals obtained in surface-sensitive total electron yield mode are strongly reduced in the bulk-sensitive fluorescence mode, which indicates hole-doping in the bulk for those films which do show a metal-to-insulator transition.'
author:
- 'R. Werner'
- 'C. Raisch'
- 'V. Leca'
- 'V. Ion'
- 'S. Bals'
- 'G. Van Tendeloo'
- 'T. Chassé'
- 'R. Kleiner'
- 'D. Koelle'
bibliography:
- 'References.bib'
title: 'Transport, magnetic, and structural properties of La$_{0.7}$Ce$_{0.3}$MnO$_3$ thin films. Evidence for hole-doping'
---
Before sending further, pass this check list: \[ \] find and resolve all “??” and “” \[ \] Spell-check \[ \] order of using/defining abbreviations \[ \] citation order (use RefTest) \[ \] figure order & reference order \[ \] Check LaTeX output files (\*.log) for warnings \[ \] Check BibTeX output (screen) for warnings \[ \] update PACS \[ \] decide on color figures \[ \] Re-read paper in the morning
After completing this list, if you made at least one correction, re-do this check-list from the beginning, until no corrections will be done.
–> 61.05.cj X-ray absorption spectroscopy: EXAFS, NEXAFS, XANES, etc.
61.05.cp X-ray diffraction
–> 68.37.Lp Transmission electron microscopy (TEM)
68.55.J- Morphology of films
–> 71.30.+h Metal–insulator transitions and other electronic transitions
–> 72.60.+g Mixed conductivity and conductivity transitions
72.80.Ga Transition-metal compounds
–> 75.47.Lx Manganites
75.60.Ej Magnetization curves, hysteresis, Barkhausen and related effects
75.70.Ak Magnetic properties of monolayers and thin films
–> 81.15.Fg Laser deposition
Introduction {#Sec:Introduction}
============
Hole-doped manganese perovskite oxides La$_{1-x}A_x$MnO$_3$, where $A$ is a divalent alkaline earth metal, have been intensively studied over the last years due to the interesting interplay between charge, spin, orbital and structural degrees of freedom.[@Imada98; @Coey99; @Salamon01] Without doping, LaMnO$_3$ is an antiferromagnetic insulator due to the super-exchange between the Mn$^{3+}$ ions.[@Millis98] In the hole-doped manganites, the divalent ion introduces holes by changing some Mn valences from Mn$^{3+}$ to Mn$^{4+}$. The properties of the hole-doped manganites are determined by the interplay of Hund´s rule coupling and the Jahn-Teller distortion of the Mn$^{3+}$ ions.[@Millis95] Their behavior can be qualitatively described by the double-exchange model,[@Zener51; @Anderson55] describing the interaction between manganese ions with mixed valences (Mn$^{3+}$ and Mn$^{4+}$). The strong spin-charge coupling via the double-exchange interaction explains the correlation between the metal-to-insulator (MI) and ferromagnet-to-paramagnet (FP) transition. Close to the MI transition temperature $T_{MI}$ an external magnetic field can reduce the spin disorder and therefore enhance the electron hopping between the manganese ions with mixed valences. This results in a large resistivity drop, called colossal magnetoresistance.[@Jonker50]
By substitution of La with a tetravalent ion, like Ce,[@Mandal97; @Gebhardt99; @Ganguly00] Sn,[@Li99a] or Te,[@Tan03a] instead of a divalent one, some of the Mn$^{3+}$ ions become Mn$^{2+}$ with electronic structure t$^3_{2g}$e$^2_g$ (compared to the t$^3_{2g}$e$^1_g$ electronic structure for Mn$^{3+}$). Hence, an extra electron may be induced in the e$_g$-band. Since Mn$^{2+}$ is a non-Jahn-Teller ion, like Mn$^{4+}$, one might expect a similar magnetic interaction between the Mn$^{3+}$ and Mn$^{2+}$ ions as for the well known hole-doped case.[@Mitra03a]
The first attempts to achieve electron-doping by substituting La with Ce were reported by Mandal and Das.[@Mandal97] However, they found hole-doping in their bulk samples. Later on, it was revealed that the bulk samples are a multiphase mixture which leads to the hole-doped behavior.[@Ganguly00; @Philip99] Single phase [La$_{0.7}$Ce$_{0.3}$MnO$_3$]{} (LCeMO) thin films have been prepared without any CeO$_2$ impurities [@Mitra01a; @Raychaudhuri99] regarding x-ray diffraction (XRD) data. The films showed FP and MI transitions similar to the hole-doped manganites. Surface-sensitive X-ray photoemission spectroscopy revealed the existence of Mn$^{2+}$ and Mn$^{3+}$ valences,[@Mitra03a; @Han04] which was interpreted as evidence of electron-doping. However, Hall measurements and thermopower measurements on comparable samples showed a hole-type character. [@Wang06; @Zhao00; @Yanagida04; @Yanagida05] By Ganguly [*et al.*]{} [@Ganguly00] it was further questioned whether LaMnO$_3$ accepts Ce-doping at all. Those authors questioned the reports on single phase LCeMO-films and claimed the presence of multi-phase mixtures, consisting of hole doped La-deficient phases with cerium oxide inclusions. Certainly, the existence of electron-doped manganites could enable new types of spintronic devices, such as $p-n$ junctions based on doped manganites.[@Mitra01] This motivates further research in order to improve understanding of the basic properties of those materials.
In this paper we present the results of studies on transport, magnetic and structural properties of LCeMO thin films grown by pulsed laser deposition (PLD) and their dependence on deposition parameters, annealing procedures and film thickness. We combine a variety of different characterization techniques in order to clarify the nature of the FP and MI transition in our LCeMO thin films.
Experimental Details {#Sec:Experiment}
====================
A commercially available stoichiometric polycrystalline La$_{0.7}$Ce$_{0.3}$MnO$_3$ target was used for thin film growth by PLD on (001) SrTiO$_3$ (STO) substrates (unless stated otherwise). The target was ablated by using a KrF ($\lambda$ = 248 nm) excimer laser at a repetition rate of $2-5\,$Hz. The energy density on the target was $E_d=2\,{\rm J/cm}^2$, while the substrate temperature during deposition was kept at $T_s=720\,^{\circ}$C for all films for which data are presented below, except for sample K with slightly lower $T_s$ and $E_d$ (cf. Tab. \[tab:overview\]). The oxygen pressure $p_{O_2}$ during film growth was varied in the 1–25Pa range with the aim of yielding
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'Using methods from the theory of total positivity, we provide a full classification of attainable term structure shapes in the two-factor Vasicek model. In particular, we show that the shapes normal, inverse, humped, dipped and hump-dip are always attainable. In certain parameter regimes up to four additional shapes can be produced. Our results show that the correlation and the difference in mean-reversion speeds of the two factor processes play a key role in determining the scope of attainable shapes. The mathematical tools from total positivity can likely be applied to higher-dimensional generalizations of the Vasicek model and to other interest rate models as well.'
address: 'Institute for Mathematical Stochastics, TU Dresden'
author:
- 'Martin Keller-Ressel'
bibliography:
- 'references.bib'
title: 'Total positivity and the classification of term structure shapes in the two-factor Vasicek model'
---
Introduction
============
The term structure of interest rates – summarized in the form of the yield or forward curve – is one of the most fundamental economic indicators. Its shape encodes important information on the preferences for short- vs. long-term investments, the desire for liquidity and on expectations of central bank decisions and the general economic outlook. It is therefore a natural question – to be asked of any mathematical model of the term structure – which shapes of yield and forward curves the model is able to (re-)produce. Already in [@vasicek1977equilibrium] a paragraph is dedicated to this question, with Vasicek concluding that [`normal`]{} (increasing), [`inverse`]{} (decreasing) and [`humped`]{} (endowed with a single maximum) shapes can be attained in his single-factor model. The same classification of shapes has been shown to hold in the Cox-Ingersoll-Ross model and furthermore in all one-dimensional affine term structure models (including short-rate models with jumps), see [@cox1985theory Eq. (26)f], [@keller-ressel2008yield; @keller-ressel2018correction].\
It is also well-known, that in the Hull-White extended Vasicek model [@hull1990pricing] *any* initial term structure can be perfectly fitted and therefore that any shape of the term structure can be reproduced at the time of calibration. However, as time progresses, this initial shape will disappear and – due to ergodicity effects – the model will behave closer and closer to a Vasicek model with time-homogeneous coefficients. Therefore, even in view of Hull-White-extended models, the classification of attainable term structure shapes in time-homogeneous short-rate models is a reasonable and important question.\
Here, we provide for the first time a systematic classification of term structure shapes beyond the one-dimensional case. In our main result, Theorem \[thm:main\], we classify all attainable shapes for both the yield and forward curve in the two-dimensional Vasicek model. As expected, several additional shapes, such as a [`dipped`]{} curve, which are not attainable in the one-dimensional case become attainable in the two-factor model. We also give some stronger attainment results, showing for instance that also the locations of humps and dips can typically be chosen without restrictions.\
Our main mathematical tool is the theory of total positivity (see e.g. [@karlin1968total]), a theory linked to the variation-diminishing properties of certain matrices, function systems and integral kernels. Total positivity has broad applications in numerical interpolation, differential equations and stochastic processes. Within mathematical finance, it has been applied to study monotonicity and convexity of options prices [@kijima2002monotonicity] and to the principal-component-analysis of the term structure of interest rates [@salinelli2006correlation; @lord2007level]. Our application to the shape analysis of the term structure is new and fundamentally different from the results in [@salinelli2006correlation; @lord2007level]. While the results in this paper are limited to the two-dimensional Vasicek model, we are confident that the underlying theory can be applied to other multi-factor interest rate models as well.
Preliminaries
=============
Shapes of the term structure
----------------------------
In our terminology *term structure* refers to either the yield curve or the forward curve. The *shape* $\mathsf{S}$ of the term structure is defined by the number and sequence of local maxima or minima of the term structure curve. In common financial market terminology a local maximum is called a ‘hump’ and a local minimum a ‘dip’. As the term structure curves produced by the Vasicek model (or most other models) are smooth, it is clear that the shape of the term structure curve can be conveniently analyzed by considering its derivative: Any sign change of the derivative (from strictly positive to strictly negative or vice versa) corresponds to a local extremum of the term structure; the type of sign change (${\textup{\texttt{+}}}$ to ${\textup{\texttt{-}}}$ or ${\textup{\texttt{-}}}$ to ${\textup{\texttt{+}}}$) determines the type of the extremum (hump or dip). The basic shapes and their conventional names are listed in Table \[tab:shape\]. For ‘higher order’ shapes we use the letters `H` for a hump and `D` for a dip, e.g., the shape ${\texttt{HDH}}{}$ corresponds to a term structure with two local maxima, interlaced by a single local minimum.
\[tab:shape\]
Shape $\mathsf{S}$ of the term structure Description Sign sequence of derivative
------------------------------------------ -------------------------------------------------------- ---------------------------------------------------------------------------------
[`normal`]{} strictly increasing ${\bm{[}{\textup{\texttt{+}}}\bm{]}}$
[`inverse`]{} strictly descreasing ${\bm{[}{\textup{\texttt{-}}}\bm{]}}$
[`humped`]{} single local maximum ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$
[`dipped`]{} single local minimum ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$
[`HD`]{} hump-dip, i.e. local maximum followed by local minimum ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$
[`DH`]{}, [`HDH`]{}, etc. further sequences of multiple ‘dips’ and ‘humps’ ${\bm{[}\dotsc\bm{]}}$
: Shapes of the term structure
Sign sequences
--------------
We introduce several notions associated to a *sign sequence*. The primary purpose of a sign sequence will be to keep track of the number and the directions of sign changes of a numeric sequence or of a continuous function. The notion of a sign sequence appears implicitly in many of the results related to total positivity, however, the terminology introduced here is new.
(i) A **sign sequence** is a non-empty sequence of the symbols ${\textup{\texttt{+}}}$ and ${\textup{\texttt{-}}}$. Only finite sign sequences will be considered here. Also zeroes can be allowed; we comment on this later. We include sign sequences in square brackets and write e.g. $${\bm{[}{\textup{\texttt{+}}}\bm{]}}, \qquad {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}, \qquad {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$$ for some valid sign sequences.
(ii) A sign sequence is called **pure** if the signs ${\textup{\texttt{+}}}$ and ${\textup{\texttt{-}}}$ alternate, e.g., the sequences $${\bm{[}{\textup{\texttt{+}}}\bm{]}}, \qquad {\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$$ are pure. Any sign sequence can be reduced to a **pure sign sequence** by replacing blocks of ${\textup{\texttt{+}}}$’s by a single ${\textup{\texttt{+}}}$ and blocks of ${\textup{\texttt{-}}}$’s by a single ${\textup{\texttt{-}}}$, e.g. $${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}} \quad \text{reduces to} \quad {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}.$$ Note that this reduction preserves the number and direction of sign changes, which is our primary object of interest.
(iii) Two sign sequences are called **equivalent**, if they reduce to the same pure sequence. This defines an equivalence relation $\simeq$, e.g., $${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}} \simeq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}\bm{]}}.$$
(iv) In a similar way we can define a **subsequence** relation $\subseteq$ where we treat blocks of signs as if they were single signs. E.g. we have $${\bm{[}{\textup{\texttt
|
{
"pile_set_name": "ArXiv"
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abstract: |
Let $k$ be an algebraic closure of finite fields with odd characteristic $p$ and a smooth projective scheme ${\bf X}/W(k)$. Let ${\bf X}^0$ be its generic fiber and $X$ the closed fiber. For ${\bf X}^0$ a curve Faltings conjectured that semistable Higgs bundles of slope zero over ${\bf X}^0_{{{\mathbb C}}_p}$ correspond to genuine representations of the algebraic fundamental group of ${\bf
X}^0_{{{\mathbb C}}_p}$ in his $p$-adic Simpson correspondence [@Fa3]. This paper intends to study the conjecture in the characteristic $p$ setting. Among other results, we show that isomorphism classes of rank two semistable Higgs bundles with trivial chern classes over $X$ are associated to isomorphism classes of two dimensional genuine representations of $\pi_1({\bf X}^0)$ and the image of the association contains all irreducible crystalline representations. We introduce intermediate notions *strongly semistable Higgs bundles* and *quasi-periodic Higgs bundles* between semistable Higgs bundles and representations of algebraic fundamental groups. We show that quasi-periodic Higgs bundles give rise to genuine representations and strongly Higgs semistable are equivalent to quasi-periodic. We conjecture that a Higgs semistable bundle is indeed strongly Higgs semistable.
address:
- 'Institut für Mathematik, Universität Mainz, Mainz, 55099, Germany'
- 'School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China'
- 'Institut für Mathematik, Universität Mainz, Mainz, 55099, Germany'
author:
- Guitang Lan
- Mao Sheng
- Kang Zuo
title: 'Semistable Higgs bundles and representations of algebraic fundamental groups: Positive characteristic case'
---
\[section\] \[thm\][Theorem]{} \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Proposition]{} \[thm\][Addendum]{} \[thm\][Variant]{} \[thm\][Lemma and Definition]{} \[thm\][Construction]{} \[thm\][Notations]{} \[thm\][Question]{} \[thm\][Problem]{} \[thm\][Remark]{} \[thm\][Remarks]{} \[thm\][Definition]{} \[thm\][Claim]{} \[thm\][Assumption]{} \[thm\][Assumptions]{} \[thm\][Properties]{} \[thm\][Example]{} \[thm\][Conjecture]{} \[thm\][Proposition and Definition]{}
[^1]
Introduction
============
N. Hitchin [@Hitchin] introduced rank two stable Higgs bundles over a compact Riemann surface $X$ and showed that they correspond naturally to irreducible representations of the fundamental group $\pi_1(X)$ by solving a Yang-Mills equation, which generalizes the earlier works by Donaldson, Uhlenbeck-Yau for polystable vector bundles. Later C. Simpson obtained the full correspondence for any polystable Higgs bundles over arbitrary dimensional complex projective manifolds. In [@Fa3] G. Faltings established the correspondence between Higgs bundles and generalized representations of $\pi_1(X)$ over $p$-adic fields. He conjectured that semistable Higgs bundles under his functor shall correspond to usual $p$-adic representations of $\pi_1(X)$. In this paper we intend to study Faltings’s conjecture in the characteristic $p$ setting.\
Let $k$ be the algebraic closure of finite fields of odd characteristic $p$. Let ${\bf X}/W(k)$ be a smooth projective $W:=W(k)$-scheme and $X/k$ its closed fiber. In this paper, if not specified, a Higgs bundle over $X$ means a system of Hodge bundles $$(E=\oplus_{i+j=n}E^{i,j},\theta=\oplus_{i+j=n}\theta^{i,j}),$$ where $E$ is a vector bundle over $X$, $\theta$ is a morphism of ${{\mathcal O}}_X$-modules satisfying $$\theta^{i,j}: E^{i,j}\to
E^{i-1,j+1}\otimes \Omega_{X}, \quad \quad \theta\wedge \theta=0.$$ For simplicity, we assume throughout that $n\leq p-2$. Fix an ample divisor ${\bf H}\subset {\bf X}$ over $W$. The Higgs semistability of $(E,\theta)$ is referred to the $\mu$-semistability with respect to $H\subset X$, the reduction of ${\bf H}$.
There is a functor from the category of quasi-periodic Higgs-de Rham sequences of type $(e,f)$ to the category of crystalline representations of $\pi_1({\bf X'}^0)$ into ${\mathrm{GL}}({{\mathbb F}}_{p^f})$, where ${\bf X'}^0$ is the generic fiber of ${\bf X'}:={\bf
X}\times_{W}{{\mathcal O}}_K$ for a totally ramified extension $\mathrm{Frac}(W)\subset K$ with ramification index $e$. There is also a functor in the opposite direction. These two functors are equivalence of categories in the case $e=0$ and quasi-inverse to each other.
Consequently, we obtain the following
Under the above functors, there is one to one correspondence between the isomorphism classes of irreducible crystalline ${{\mathbb F}}_{p^f}$-representations of $\pi_1({\bf X}^0)$ and the isomorphism classes of periodic Higgs stable bundles of period $f$.
The leading term of a quasi-periodic Higgs-de Rham sequence is a quasi-periodic Higgs bundle. We show that
A quasi-periodic Higgs bundle is strongly Higgs semistable with trivial chern classes. Conversely, A strongly Higgs semistable bundle with trivial chern classes is quasi-periodic.
Strongly semistable vector bundles are strongly semistable Higgs bundles with trivial Higgs fields. As a semistable bundle need not be strongly semistable, the notion of strongly semistability should be replaced by the strongly Higgs semistability. The next result supports our viewpoint.
A rank two semistable Higgs bundle is strongly Higgs semistable.
We would like to make the following
A semistable Higgs bundle is strongly Higgs semistable.
As an application of the above results, we obtain the following
Any isomorphism class of rank two semistable Higgs bundles with trivial chern classes over $X$ is associated to an isomorphism class of crystalline representations of $\pi_1({\bf X}^0)$ into ${\mathrm{GL}}_2(k)$. The image of the association contains all irreducible crystalline representations of $\pi_1({\bf X}^0)$ into ${\mathrm{GL}}_2(k)$.
The plan of our paper is arranged as follows: in Section 2 we introduce the notions *strongly Higgs semistable bundles* which generalizes the notion of strongly semistable vector bundles in the paper [@LS] of Lange-Stuhler and *quasi-periodic Higgs bundles* which generalizes the notion of periodic Higgs subbundles introduced in [@SZ]. We show that a strongly Higgs semistable with trivial chern classes is equivalent to a quasi-periodic Higgs bundle, and a rank two semistable Higgs bundle is strongly Higgs semistable. We conjecture that semistable Higgs bundles of arbitrary rank are strongly Higgs semistable. In Section 3 we show in Theorem \[correspondence in the type (0,f) case\] that there is a one to one correspondence between the strict $p$-torsion category $\mathcal{MF}^{\nabla}_{[0,n],f}({\bf X}/W)$ of Faltings with endomorphism ${{\mathbb F}}_{p^f}$ and the category of periodic Higgs-de Rham sequences of type $(0,f)$. In Section 4, we extend the construction for periodic Higgs bundles to quasi-periodic Higgs bundles. In Section 5, we give some complements and applications of the above theory.\
[**Acknowledgements:**]{} Arthur Ogus has recently pointed to us that the inverse Cartier transform in the paper [@OV] for the nilpotent Higgs bundles coincides with the construction in [@LSZ]. Christopher Deninger has drawn our attention to the work [@Langer], and Adrian Langer has helped us understanding [@Langer]. We thank them heartily.
Strongly semistable Higgs bundles
=================================
In this paper, a vector bundle over $X$ means a torsion free coherent sheaf of ${{\mathcal O}}_X$-module. A Higgs-de Rham sequence over $X$ is a sequence of form $$\xymatrix{
& (H_0,\nabla_0)\ar[dr]^{Gr_{Fil_0}} && (H_1,\nabla_1)\ar[dr]^{Gr_{Fil_1}} \\
(E_0,\theta_0) \ar[ur]^{C_0^{-1}} & & (E_1,\theta_1) \ar[ur]^{C_0^{-1}}&&\ldots }$$ In the sequence, $C_0^{-1}$ is the inverse Cartier transform constructed in [@OV] (see also [@LSZ]). A. Ogus remarked that the exponential twisting of [@LSZ] is equivalent to the more general construction in [@OV] and the equivalence is implicitly implied by Remark 2.10 loc. cit.. $Fil_i$ is a decreasing filtration on $H_i$ with the property $Fil_i^0=H_i$ and $Fil_i^{n+1}=0$ and such that $\nabla_i$ obeys the Griffiths transversality with respect to
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---
abstract: 'The relationship between antiferromagnetic spin fluctuations and superconductivity has become a central topic of research in studies of superconductivity in the iron pnictides. We present unambiguous evidence of the absence of magnetic fluctuations in the non-superconducting collapsed tetragonal phase of CaFe$_2$As$_2$ via inelastic neutron scattering time-of-flight data, which is consistent with the view that spin fluctuations are a necessary ingredient for unconventional superconductivity in the iron pnictides. We demonstrate that the collapsed tetragonal phase of CaFe$_2$As$_2$ is non-magnetic, and discuss this result in light of recent reports of high-temperature superconductivity in the collapsed tetragonal phase of closely related compounds.'
author:
- 'J. H. Soh,$^{1}$ G. S. Tucker,$^{1}$ D. K. Pratt,$^{2}$ D. L. Abernathy,$^{3}$ M. B. Stone,$^{3}$ S. Ran,$^{1}$ S. L. Bud$^{\prime}$ko,$^{1}$ P. C. Canfield,$^{1}$ A. Kreyssig,$^{1}$ R. J. McQueeney$^{1}$ and A. I. Goldman$^{1}$'
title: 'The non-magnetic collapsed tetragonal phase of CaFe$_2$As$_2$ and superconductivity in the iron pnictides'
---
The $A$Fe$_2$As$_2$ ($A$ = Ba, Sr, Ca), or “122”, family of compounds has been one of the most widely studied classes of iron pnictide superconductors [@Johnston_2010; @PandG_2010; @CandB_2010; @Stewart_2011] in recent years, and a great deal of attention has been focused on CaFe$_2$As$_2$ [@Ni_2008; @PC_2009] in particular. At ambient pressure, the substitution of Co or Rh for Fe [@Kumar_2009; @Matusiak_2010; @Harnagea_2011; @Ran_2012; @Danura_2011] results in the suppression of antiferromagnetic (AFM) order and, over some range in substitution, superconductivity (SC) emerges with transition temperatures ($T_{\rm{c}}$) of up to $\approx$ 20 K. Under modest applied pressure Ca(122) manifests fascinating new behavior including a transition to an isostructural volume collapsed tetragonal (cT) phase that is generally believed to be non-magnetic and non-superconducting. The cT phase in Ca(122) is distinguished by a striking 9.5% reduction in the tetragonal **c** lattice parameter, with respect to the high-temperature ambient-pressure tetragonal (T) phase, along with the absence of the stripe-like magnetic order found for the low-temperature ambient-pressure orthorhombic phase [@Goldman_2008].
The first liquid media clamp-cell pressure measurements of Sn-flux solution-grown Ca(122) found traces of SC for applied pressures between roughly 0.25 and 0.9 GPa [@Milton_2008; @Park_2008]. These studies were rapidly followed by transport measurements and neutron diffraction experiments under hydrostatic pressure conditions using He gas pressure cells which showed: (i) no evidence of SC for P $< 0.6$ GPa [@Yu_2009] and; (ii) the existence of the cT structure for $P >$ 0.35 GPa at low temperatures [@Kreyssig_2008; @Goldman_2009]. That work demonstrated that the traces of SC originally found in the frozen liquid clamp-cell measurements probably resulted from significant non-hydrostatic pressure components generated during the transition to the cT phase, although the origin of the SC phase was not identified in these studies. Later experiments, utilizing uniaxial pressure, concluded that the T phase could be stabilized to low temperatures by the presence of non-hydrostatic pressure components and was likely the source of superconductivity in the original liquid clamp-cell measurements [@Prokes_2010].
Recently, superconductivity with $T_{\rm{c}}$ in excess of 45 K has been reported for the substitution of Sr [@Jeffries_2012] or selected rare earths ($R$) [@Lv_2011; @Saha_2012; @Ma_2013] for Ca, or co-doping by La and P [@Kudo_2013], and it has been proposed that these high $T_{\rm{c}}$ values are realized in the cT phase as well [@Saha_2012; @Jeffries_2012]. Since it is generally accepted that there is a close connection between SC in the iron pnictides and the presence of correlated AFM fluctuations in these compounds [@Johnston_2010; @PandG_2010; @CandB_2010; @Stewart_2011; @LandC_2010; @Dai_2012], the possibility of high values of $T_{\rm{c}}$ in the cT phase raises important questions regarding the nature of the cT phase, and the relationship between magnetic fluctuations and unconventional superconductivity in the iron pnictides. It is, therefore, important to clearly establish whether the cT phase of Ca(122) is, in fact, non-magnetic.
There is already evidence that the cT phase of Ca(122) is non-magnetic, consistent with the absence of unconventional superconductivity. First, as noted above, the low-temperature stripe-like AFM order is absent in the cT phase. However, alternative magnetic ground states for the cT phase have been proposed [@Yildirim_2009], and the origin of the suppression of magnetic order, whether it arises from a reduction in the iron moment, changes in the magnetic exchange, or a more subtle change in electronic structure has come under renewed scrutiny [@Jeffries_2012]. Furthermore, the absence of AFM order does not directly speak to the presence or absence of magnetic *fluctuations* in the cT phase. It is well known that strong AFM fluctuations remain after long-range magnetic order is lost in the iron pnictides at optimal doping [@Johnston_2010; @PandG_2010; @CandB_2010; @Stewart_2011; @LandC_2010; @Dai_2012].
Total energy calculations described in Reference predict that the cT phase is non-magnetic and this has been supported by other theoretical studies [@Colonna_2011; @Tomic_2012; @Widom_2013]. Our previous inelastic neutron scattering studies of the T [@Diallo_2010] and cT phases [@Pratt_2009] showed that, at least over a narrow range in momentum transfer (**Q**) close to the AFM wavevector, $\textbf{Q}_{\rm{stripe}}$, and energy transfers ($E$) less than 7 meV, the AFM fluctuations are suppressed, or absent, in the cT phase. Again, this result finds support in other experimental measurements [@Danura_2011; @Ma_2013]. But the narrow scope of the neutron measurements could not exclude the presence of correlated magnetic fluctuations at other positions in reciprocal space [@Yildirim_2009], or simply a change in the energy scale of the fluctuations as has been found, for example, in the well known volume collapse of Ce [@Loong_1987], or very recently in nonsuperconducting Ba(Fe$_{0.85}$Ni$_{0.15}$)$_2$As$_2$ [@Wang_2013]. A much wider view in both **Q** and $E$ must be obtained to clearly establish the presence or absence of magnetic fluctuations in the cT phase of Ca(122).
Here we present unambiguous evidence that the magnetic fluctuations in the non-superconducting cT phase of Ca(122) are absent via inelastic neutron scattering measurements using the ARCS time-of-flight (TOF) instrument [@Abernathy_2012] at the Spallation Neutron Source at Oak Ridge National Laboratory. This result provides clear evidence that the cT phase of Ca(122) is a non-magnetic metal, with no static or dynamic magnetic moment, and supports the view that spin fluctuations are a necessary ingredient for unconventional SC in the iron pnictides. The complete suppression of magnetism in the cT phase also provides a non-magnetic analog for a detailed study of the AFM fluctuation spectrum of the paramagnetic T phase out to energy transfers above 100 meV, and we use this to demonstrate that the dynamical susceptibility, $\chi^{\prime\prime}(\textbf{\rm{Q}},\omega)$, is well described by the model for short-range, over-damped anisotropic spin-correlations introduced in Reference .
The sample used in this study was a co-aligned set of 12 single crystals produced by solution growth using an FeAs flux [@Ran_2011]. The co-alignment provided a total sample mass of $\sim$1.5 grams and a sample mosaic of 1.5$^{\circ}$ full-width-at-half-maximum. As described in Reference , FeAs-flux samples quenched from the melt at 960$^{\circ}$C, or annealed at temperatures above 700$^{\circ}$ C, transform directly from the T phase into the cT structure at low temperature at ambient pressure; the strain field associated with a uniform distribution of fine-sized FeAs precipitates appears to play a key role in the ambient pressure transformation and can be used to systematically tune the behavior of the Ca(122) samples [@Gati_2012]. For the present measurements, the samples were as-grown, quenched from the melt at 960$^{\circ}$ C. Other than
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{
"pile_set_name": "ArXiv"
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abstract: 'Statistical techniques are used in all branches of science to determine the feasibility of quantitative hypotheses. One of the most basic applications of statistical techniques in comparative analysis is the test of equality of two population means, generally performed under the assumption of normality. In medical studies, for example, we often need to compare the effects of two different drugs, treatments or preconditions on the resulting outcome. The most commonly used test in this connection is the two sample $t$-test for the equality of means, performed under the assumption of equality of variances. It is a very useful tool, which is widely used by practitioners of all disciplines and has many optimality properties under the model. However, the test has one major drawback; it is highly sensitive to deviations from the ideal conditions, and may perform miserably under model misspecification and the presence of outliers. In this paper we present a robust test for the two sample hypothesis based on the density power divergence measure [@MR1665873], and show that it can be a great alternative to the ordinary two sample $t$-test. The asymptotic properties of the proposed tests are rigorously established in the paper, and their performances are explored through simulations and real data analysis.'
author:
- 'A. Basu'
- 'A. Mandal'
- 'N. Martin'
- 'L. Pardo'
bibliography:
- 'reference.bib'
date: 'September 28, 2014'
title: '**Robust Tests for the Equality of Two Normal Means based on the Density Power Divergence** '
---
**:** 62F35, 62F03.
: Robustness, Density Power Divergence, Hypothesis Testing.
Introduction: Motivation and Background
=======================================
In many scientific studies, often the main problem of interest is to compare different population groups. In medical studies, for example, the primary research problem could be to test for the difference between the location parameters of two different populations receiving two different drugs, treatments or therapy, or having two different preconditions. The normal distribution often provides the basic setup for statistical analyses in medical studies (as well as in other disciplines). Inference procedures based on the sample mean, the standard deviation and the one and two-sample $t$-tests are often the default techniques for the scenarios where they are applicable. In particular, the two sample $t$-test is the most popular technique in testing for the equality of two means, performed under the assumption of equality of variances. Its applicability in real life situations is, however, tempered by the known lack of robustness of this test against model perturbations. Even a small deviation from the ideal conditions can make the test completely meaningless and lead to nonsensical results. This problem is caused by the fact that the $t$-test is based on the classical estimates of the location and scale parameters (the sample mean and the sample standard deviation). Large outliers tend to distort the mean and inflate the standard deviation. This may lead to false results of both types, i.e. detecting a difference when there isn’t one, and failing to detect a true significance.
In this paper we are going to develop a class of robust tests for the two sample problem which evolves from an appropriate minimum distance technique in a natural way. This class of tests is indexed by two real parameters $%
\beta $ and $\gamma $, and we will constrain each of these parameters to lie within the $[0,1]$ interval. Our general minimum distance approach will allow us to study the likelihood ratio test in an asymptotic sense, as the likelihood ratio test is asymptotically equivalent to the test generated by the parameters $%
\beta =\gamma =0$. Normally we will work with the one parameter family of test statistics corresponding to $%
\beta =\gamma $; the outlier stability of the proposed tests increase with the tuning parameter $\gamma $.
Let $X$ and $Y$ be independent random variables whose distributions are modeled as normals having unknown means $\mu_1$ and $\mu_2$, respectively, with an unknown but common variance $\sigma^2$. We are interested in testing the null hypothesis $$H_{0}:\mu_1=\mu_2\text{ against }H_{1}:\mu_1\neq \mu_2,
\label{EQ:0}$$ under the above set up. It is well known that the exact two sample $t$-test (which is equivalent to the likelihood ratio test) rejects the null hypothesis in (\[EQ:0\]) if and only if $$t=\frac{\left\vert \bar{X}-\bar{Y}\right\vert }{S_{p}\sqrt{\frac{1%
}{n_1}+\frac{1}{n_2}}}>t_{\frac{\alpha }{2}}(n_1+n_2-2),$$ where $\bar{X}$ and $\bar{Y}$ are the sample means corresponding to the random samples $X_{1},X_{2},\ldots ,X_{n_1}$ and $%
Y_{1},Y_{2},\ldots ,Y_{n_2}$ obtained from the two distributions, $$S_{p}^{2}=\frac{(n_1-1)S_{1}^{2}+(n_2-1)S_{2}^{2}}{n_1+n_2-2},$$ $$S_{1}^{2}=\frac{1}{n_1-1}\sum_{i=1}^{n_1}\left( X_i-\bar{X}%
\right) ^{2},\quad S_{2}^{2}=\frac{1}{n_2-1}\sum_{i=1}^{n_2}\left( Y_{i}-%
\bar{Y}\right) ^{2},$$and $t_{\frac{\alpha }{2}}(n_1+n_2-2)$ is the $100(1-\frac{\alpha }{2})$-th quantile of the $t$-distribution with $n_1+n_2-2$ degrees of freedom. The $t$-test is the uniformly most powerful unbiased and invariant test for this hypothesis. Testing the equality of means of independent normal populations with unknown variances which are not necessarily equal, is referred to as the Behrens-Fisher problem.
In this paper we will use the density power divergence (DPD) measure [@MR1665873], which provides a natural robustness option for many standard inference problems. The density power divergence and its variants have been successfully used by many authors in a variety of inference problems; see, eg. [@MR1859416], [@MR2299175; @MR2466551], [@MR3011625; @basu2013], [@MR3117102]. However, the two sample problem requires a non-trivial extension of the currently existing techniques. Our purpose in this paper is to derive the asymptotic properties of the class of two sample tests based on the density power divergence and demonstrate their robust behavior in practical situations.
**Example 1 (Cloth Manufacturing data)**: In order to emphasize the need for applications early, we now present a motivational example. This example illustrates the use of quality control methods practiced in a clothing manufacturing plant. Levi-Strauss manufactures clothing from cloth supplied by several mills. The data used in this example (see Table [TAB:Staudte\_Sheather]{}) are for two of these mills and were obtained from the quality control department of the Levi plant in Albuquerque, New Mexico ([@lambert1987introduction], p. 86). In order to maintain the anonymity of these two mills we have coded them $A$ and $B$. A measure of wastage due to defects in cloth and so on is called *run-up*. It is quoted as percentage of wastage per week and is measured relative to computerized layouts of patterns on the cloth. Since the people working in the plant can often beat the computer in reducing wastage by laying out the patterns by hand, it is possible for run-up to be negative. From the viewpoint of quality control, it is desirable not only that the run-up be small but that the quality from week to week be fairly consistent. There are 22 measurements on run-up for each of the two mills and they are presented in Table \[TAB:Staudte\_Sheather\]. The $t$-test for the equality of the two means against the two-sided alternative has a $p$-value of 0.3428 and fails to reject the null hypothesis; however, when the presumed outliers (presented in bold fonts in Table \[TAB:Staudte\_Sheather\]) are removed from the dataset, the same two-sample $t$-test produces a $p$-value of 0.0308, leading to clear rejection. Choosing $\beta = \gamma$ to be the only parameter, the $p$-values of the DPD tests (to be developed in the next section) for testing the same hypotheses are presented in Figure [fig:Staudte\_Sheather\_book\_p\_val]{} as a function of $\gamma$. It is observed that the $p$-values of the tests with the full data and those with the outlier deleted data are practically identical for $\gamma = 0.2$ or larger, and lead to solid rejection. Thus, while the outliers mask the significance in case of the two sample $t$-test, the more robust DPD tests are able to capture the same.
-------- -------- --------- --------- ------------- --------- --------- ------------- ------------- --------- --------- --------
Mill A $0.12$ $1.01$ $-0.20$ $0.15$ $-0.30$ $-0.07$ $0.32$ $% $-0.32$ $-0.17$ $0.24$
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{
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| null | null | null |
---
abstract: |
We analytically calculate the dominant two-loop electroweak correction, of $\mathcal{O}(G_F^2m_t^4)$, to the partial width of the decay of a Higgs boson, with mass $M_H\ll m_t$, into a bottom-quark pair, and describe the most important conceptual and technical details of our calculation. As a by-product of our analysis, we also recover the $\mathcal{O}(\alpha_sG_Fm_t^{2})$ correction. Relative to the Born result, the $\mathcal{O}(G_F^2m_t^4)$ correction turns out to be approximately $+0.047\%$ and, thus, more than compensates the $\mathcal{O}(\alpha_sG_Fm_t^2)$ one, which amounts to approximately $-0.022\%$.
PACS numbers: 11.10.Gh, 12.15.Ji, 12.15. Lk, 14.80.Bn
author:
- |
Mathias Butenschön, Frank Fugel, Bernd A. Kniehl\
[II. Institut für Theoretische Physik, Universität Hamburg,]{}\
[Luruper Chaussee 149, 22761 Hamburg, Germany]{}
title: |
-3cm
DESY 07-003ISSN 0418-9833
hep-ph/0702215
January 2007
1.5cm $\mathcal{O}(G_F^2m_t^4)$ two-loop electroweak correction to Higgs-boson decay to bottom quarks
---
Introduction
============
The standard model (SM) of elementary particle physics predicts the existence of a last undiscovered particle, the Higgs boson, whose mass $M_H$ is a free parameter of the theory. The direct search for the Higgs boson at the CERN Large Electron-Positron Collider LEP 2 only led to a lower bound of $M_H>114$ GeV at 95% confidence level [@Barate:2003sz]. On the other hand, high-precision measurements, especially at LEP and the SLAC Linear Collider SLC, were sensitive to the Higgs-boson mass via electroweak radiative corrections. These indirect measurements yielded the value $M_H=\left(85^{+39}_{-28}\right)$ GeV and an upper limit of $M_H<166$ GeV at 95% confidence level [@LEPEWWG]. The vacuum-stability and triviality bounds suggest that $130{\rlap{\lower 3.5 pt \hbox{$\mathchar \sim$}} \raise 1pt
\hbox {$<$}}M_H{\rlap{\lower 3.5 pt \hbox{$\mathchar \sim$}} \raise 1pt
\hbox {$<$}}180$ GeV if the SM is valid up to the grand-unification scale (for a review, see Ref. [@Kniehl:2001jy]). For these reasons, one hopes to discover the Higgs boson at the CERN Large Hadron Collider (LHC), which will be capable of producing particles with masses up to 1 TeV. The first question after discovering a new scalar particle will be if it actually is the Higgs boson of the SM, or possibly some particle of an extended Higgs sector. Therefore, it is necessary to know the SM predictions for the production and decay rates of the SM Higgs boson with high precision. Its decay into a bottom-quark pair is of special interest, as it is by far the dominant decay channel for $M_H{\rlap{\lower 3.5 pt \hbox{$\mathchar \sim$}} \raise 1pt
\hbox {$<$}}140$ GeV (see, for instance, Ref. [@Kniehl:1993ay]).
At this point, we wish to summarise the current status of the calculations of radiative corrections to the $H\to b\overline{b}$ decay width in the so-called intermediate mass range, defined by $M_W\le M_H\le 2M_W$. The correction of order ${\cal O}(\alpha_s)$ was first calculated in Ref. [@Braaten:1980yq]. The complete one-loop electroweak correction was found in Ref. [@Kniehl:1991]. As for the ${\cal O}(\alpha_s^2)$ correction, the leading [@Gorishnii:1991zr] and next-to-leading [@Surguladze:1994gc] terms of the expansion in $m_b^2/M_H^2$ of the diagrams without top quarks are known. The diagrams containing a top quark can be divided into two classes. The diagrams containing gluon self-energy insertions were calculated exactly [@Kniehl:1994vq], while for the double-triangle contributions the four leading terms of the expansion in $M_H^2/m_t^2$ are known [@Chetyrkin:1995pd]. In Ref. [@Chetyrkin:1996sr], the ${\cal O}(\alpha_s^3)$ correction without top-quark contributions was calculated in the massless limit. The correction induced by th top quark was subsequently found in Ref. [@Chetyrkin:1997vj] using an appropriate effective field theory. As for the correction of order ${\cal O}(\alpha_s G_F m_t^2)$, the universal part, which appears for any Higgs-boson decay to a fermion pair, was calculated in Ref. [@Kniehl:1994ph] and the non-universal one, using a low-energy theorem, in Ref. [@KniehlSpira]. The latter result was independently found in Ref. [@Kwiatkowski:1994cu]. Apart from the Higgs-boson decay into a $t\overline{t}$ pair, only the one into a $b\overline{b}$ pair has such non-universal top-quark-induced contributions, as bottom is the weak-isospin partner of top. The universal and non-universal corrections of order ${\cal O}(\alpha_s^2 G_F m_t^2)$ were calculated in Refs. [@delu] and [@Chetyrkin:1996ke], respectively. Finally, also a result for the universal correction of order ${\cal O}(G_F^2 m_t^4)$ was published [@Djouadi].
In this paper, we calculate the complete correction of order ${\cal O}(G_F^2 m_t^4)$, including both the universal and non-universal contributions. To this end, we formally assume that $M_H\ll m_t$. This includes the intermediate mass range of the Higgs boson. Our result for the universal contribution in the on-mass-shell scheme agrees with the one found in Ref. [@Djouadi], after correcting an obvious mistake in the latter paper. The key results of our calculation were already presented in a brief communication [@prl]. Here, the full details are exhibited.
Our calculations are performed in ’t Hooft-Feynman gauge. We adopt the on-mass-shell scheme and regularise the ultraviolet divergences by means of dimensional regularisation, with $D=4-2\epsilon$ space-time dimensions and ’t Hooft mass scale $\mu$. We use the anti-commuting definition of $\gamma_5$. As a simplification, we take the Cabibbo-Kobayashi-Maskawa quark mixing matrix to be unity. The Feynman diagrams are generated and drawn using the program `FeynArts` [@Hahn:2000kx] and evaluated using the program `MATAD` [@MATAD], which is written in the programming language `FORM` [@FORM].
In order to check our calculations, we also rederive the correction of order ${\cal O}(\alpha_s G_F m_t^2)$. Our result agrees with Refs. [@Kniehl:1994ph; @KniehlSpira; @Kwiatkowski:1994cu]. Since this calculation follows the lines of the one leading to the ${\cal O}(G_F^2 m_t^4)$ correction, being actually simpler, we refrain from going into details with it.
This paper is organised as follows. In Section \[CapRenSchema\], we describe in detail the renormalisation procedure underlying our analysis. In Section \[CapOurCalc\], we present the details of our diagrammatic calculations. In Section \[CapNieder\], we explain how a part of our calculations can be checked through the application of a low-energy theorem. In Section \[Numerics\], we evaluate the ${\cal O}(G_F^2 m_t^4)$ corrections numerically and compare them with the ${\cal O}(\alpha_s G_F m_t^2)$ ones. We conclude with a summary in Section \[CapZusammenfassung\].
Renormalisation procedure {#CapRenSchema}
=========================
For the reader’s convenience, we present in this section the details of the renormalisation procedure which has to be carried out. We derive general expressions for the mass counterterms and wave-function renormalisation constants in the on-shell scheme, valid for any number of loops. Furthermore, we derive the tadpole renormalisation counterterms and describe the treatment of the corrections due to external legs. In our calculations, we do not need to consider electric-charge renormalisation constants, because, to the orders we consider here, there are no such contributions.
Before going into details
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We present DeepNav, a Convolutional Neural Network (CNN) based algorithm for navigating large cities using locally visible street-view images. The DeepNav agent learns to reach its destination quickly by making the correct navigation decisions at intersections. We collect a large-scale dataset of street-view images organized in a graph where nodes are connected by roads. This dataset contains 10 city graphs and more than 1 million street-view images. We propose 3 supervised learning approaches for the navigation task and show how A\* search in the city graph can be used to generate supervision for the learning. Our annotation process is fully automated using publicly available mapping services and requires no human input. We evaluate the proposed DeepNav models on 4 held-out cities for navigating to 5 different types of destinations. Our algorithms outperform previous work that uses hand-crafted features and Support Vector Regression (SVR) [@mcdonalds].'
author:
- |
Samarth Brahmbhatt\
Georgia Institute of Technology\
Atlanta USA\
[samarth.robo@gatech.edu]{}
- |
James Hays\
Georgia Institute of Technology\
Atlanta USA\
[hays@gatech.edu]{}
title: 'DeepNav: Learning to Navigate Large Cities'
---
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'For a finite cyclic $p$-group $G$ and a discrete valuation domain $R$ of characteristic $0$ with maximal ideal $pR$ the ${R[G]}$-permutation modules are characterized in terms of the vanishing of first degree cohomology on all subgroups (cf. Thm. A). As a consequence any ${R[G]}$-lattice can be presented by ${R[G]}$-permutation modules (cf. Thm. C). The proof of these results is based on a detailed analysis of the category of cohomological $G$-Mackey functors with values in the category of $R$-modules. It is shown that this category has global dimension $3$ (cf. Thm. E). A crucial step in the proof of Theorem E is the fact that a gentle $R$-order category (with parameter $p$) has global dimension less or equal to $2$ (cf. Thm. D).'
address:
- |
B. Torrecillas\
Departamento de Algebra y Análisis Matemático\
Universidad de Almería\
04071 Almería, Spain
- |
Th. Weigel\
Università di Milano-Bicocca\
U5-3067, Via R.Cozzi, 53\
20125 Milano, Italy
author:
- 'B. Torrecillas and Th. Weigel'
bibliography:
- 'gorenstein.bib'
title: 'Lattices and cohomological Mackey functors for finite cyclic p-groups'
---
[^1]
Introduction {#s:intro}
============
For a Dedekind domain $R$ and a finite group $G$ one calls a finitely generated left $R[G]$-module $M$ an [*$R[G]$-lattice*]{}, if $M$ - considered as an $R$-module - is projective. In this paper we focus on the study of $R[G]$-lattice, where $R$ is a discrete valuation domain of characteristic $0$ with maximal ideal $pR$ for some prime number $p$, and $G$ is a finite cyclic $p$-group. The study of such lattices has a long history and was motivated by a promissing result of F.-E. Diederichsen (cf. [@CR:met1 Thm. 34:31], [@died:ham]) who showed that for the finite cyclic group of order $p$ there are precisely three directly indecomposable such lattices up to isomorphism: the trivial $R[G]$-lattice $R$, the free $R[G]$-lattice $R[G]$, and the augmentation ideal $\omega_{{R[G]}}=\operatorname{ker}(R[G]\to R)$. A similar finiteness result holds for cyclic groups of order $p^2$ (cf. [@hr:rep1]). However, for cyclic $p$-groups of order larger than $p^2$ there will be infinitely many isomorphism types of such lattices; even worse, in general this classification problem is “wild” (cf. [@diet:rep1], [@diet:rep2], [@gud:wild]). If the $R[G]$-lattice $M$ is isomorphic to $R[\Omega]$ for some finite left $G$-set $\Omega$, $M$ will be called an [*$R[G]$-permutation lattice*]{}. The main purpose of this paper is to establish the following characterization of $R[G]$-permutation lattices for finite cyclic $p$-groups (cf. Cor. \[cor:hil90lat1\], Prop. \[prop:elequi\]).
Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$ for some prime number $p$, let $G$ be a finite cyclic $p$-group, and let $M$ be an $R[G]$-lattice. Then the following are equivalent.
- $M$ is an $R[G]$-permutation lattice,
- $H^1(U,\operatorname{res}^G_U(M))=0$ for all subgroups $U$ of $G$,
- $M_U$ is $R$-torsion free for all subgroups $U$ of $G$,
where $M_U=M/\omega_{R[U]}M$ denotes the $U$-coinvariants of $M$.
By a result of I. Reiner (cf. [@CR:met1 Thm. 34.31], [@rei:intcyc]), one knows that there are ${\mathbb{Z}}[C_p]$-lattices satisfying (ii), where $C_p$ is the cyclic group of order $p$, which are not ${\mathbb{Z}}[C_p]$-permutation lattices. Hence the conclusion of Theorem A does not hold for the ring $R={\mathbb{Z}}$.
Theorem A has a number of interesting consequences which we would like to explain in more detail. For a finite $p$-group $G$ it is in general quite difficult to decide whether a given ${R[G]}$-lattice $M$ is indeed an ${R[G]}$-permutation lattice. A sufficient criterion to the just mentioned problem was given by A. Weiss in [@weiss:rig] for an arbitrary finite $p$-group $G$ and the ring of $p$-adic integers $R={\mathbb{Z}}_p$. He showed that if for a normal subgroup $N$ of $G$ the ${\mathbb{Z}}_p[G/N]$-module $M^N$ of $N$-invariants is a ${\mathbb{Z}}_p[G/N]$-permutation module, and $\operatorname{res}^G_N(M)$ is a free ${\mathbb{Z}}_p[N]$-module, then $M$ is a ${\mathbb{Z}}_p[G]$-permutation module (cf. [@karp:ind Chap. 8, Thm. 2.6]). Theorem A extends A. Weiss’ result for cyclic $p$-groups in the following way (cf. Prop. \[prop:exweiss\]).
Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$ for some prime number $p$, let $G$ be a finite cyclic $p$-group, and let $N$ be a normal subgroup of $G$. Suppose that the $R[G]$-lattice $M$ is satisfying the following two hypothesis.
- $\operatorname{res}^G_N(M)$ is an $R[N]$-permutation module, and
- $M^N$ is an $R[G/N]$-permutation module.
Then $M$ is an ${R[G]}$-permutation module.
Although it seems impossible to describe all isomorphism types of directly indecomposable $R[G]$-lattices, where $R$ is a discrete valuation domain of characteristic $0$ with maximal ideal $pR$ and $G$ is a finite cyclic $p$-group, one can (re)present such lattices in a very natural way (cf. Thm. \[thm:preslat\]).
Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$ for some prime number $p$, let $G$ be a finite cyclic $p$-group, and let $M$ be an $R[G]$-lattice. Then there exist finite $G$-sets $\Omega_0$ and $\Omega_1$, and a short exact sequence $$\label{eq:preslat}
\xymatrix{
0\ar[r]& R[\Omega_1]\ar[r]&R[\Omega_0]\ar[r]&M\ar[r]&0
}$$ of $R[G]$-lattices.
The proof of Theorem A and Theorem C is based on the theory of [*cohomological Mackey functors*]{} for a finite group $G$. Mackey functors were first introduced by A.W.M. Dress in [@dress:mac]. Cohomological Mackey functors satisfy an additional identity (cf. [@pw:user]). The category of cohomological $G$-Mackey functors ${\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}})$ with values in the category of $R$-modules coincides with the category of contravariant functors of an $R^\circledast$-order category ${{\mathcal{M}}}_R(G)$ (cf. §\[ss:maccat\]). In case that $G$ is a cyclic $p$-group or order $p^n$, one has a [*unitary projection functor*]{} (cf. §\[ss:funRcat\]) $$\label{eq:unipro}
\pi\colon{\mathfrak{cMF}}_R(G)\longrightarrow{{\mathcal{G}}}_R(n,p)$$ which can be used to analyze the category ${\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}})$. Here ${{\mathcal{G}}}_R(n,p)$ denotes the [*gentle $R$-order category*]{} supported on $n+1$ vertices and parameter $p$ (cf. §\[
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
author:
- 'C. E. Ekuma'
- 'M. Jarrell'
- 'J. Moreno'
- 'D. Bagayoko'
title: |
\[Supplementary Information\]\
Re-examining the electronic structure of germanium: A first-principle study
---
DFT and Progress in band gap in materials
=========================================
Despite the great progress made possible by density functional theory (DFT), from 1964 to present, problems associated with obtaining theoretically the measured energy or band gaps, for finite and crystalline semiconductors, respectively, have persisted. Specifically, most DFT calculations, with emphasis on those utilizing local density approximation (LDA) and semi-local potentials, have led to semiconductor band gaps that are 30 – 50% smaller than their corresponding, measured values. Much effort has been deployed to find explanations of and remedies to this recalcitrant band gap problem. Perdew and Zunger [@PhysRevB.23.5048] introduced the self interaction correction (SIC) to local spin density (LSD) approximation calculations. While the exact functional for the ground state is self interaction free, these authors discussed corrections that appear to be needed for the description of finite systems, beginning with atoms, and of localized states in solids. This self interaction is argued to contribute to the underestimation of the band gaps of insulators by DFT calculations [@PhysRevB.23.5048]. Consequently, self interaction corrections (SIC) are expected to improve the agreement between calculated band gaps and measured ones, in addition to improving binding energies and bringing orbital energies closer to removal energies [@PhysRevB.23.5048; @PhysRevB.26.5445]. While self interaction corrections have led to some improvements in band gap calculations, they have not totally resolved the problem. Applications of SIC have mostly overestimated the band gap of semiconductors [@PhysRevB.52.R14316; @PhysRevB.54.5495]. According to Cohen [@Cohen2008], self interaction is well-defined only for one-electron systems.
According to the literature, a major source of the theoretical underestimation of band gaps consists of the derivative discontinuity of the exchange correlation energy, Exc [@PhysRevLett.51.1888; @PhysRevLett.49.1691; @PhysRevLett.51.1884; @PhysRevB.32.3883]. Perdew [@PhysRevLett.49.1691], following a thought experiment on a diatomic molecule, established the existence of a derivative discontinuity of the exchange correlation energy, i.e., a discontinuity in the exchange correlation potential, V$_{xc}$. Perdew and Levy [@PhysRevLett.51.1884] generalized this discontinuity to the case of semiconductors. They showed that the exchange correlation potential may jump by the discontinuity, $\Delta_{xc}$, when the number of electrons in the system under study increases by one. Band gaps calculated with a local density approximation (LDA) potentials, according to their findings, are to be augmented by this discontinuity in order to reproduce the corresponding, measured values. The authors suggested, without claiming to have a proof of it, that this discontinuity is a non zero (and positive) in real semiconductors and insulators. Sham and Schlüter [@PhysRevLett.51.1888] also found a derivative discontinuity of E$_{xc}$ in insulators. These authors, however, asserted that their work does not show whether or not this discontinuity is non zero in real insulators. Subsequent work by Sham and Schlüter [@PhysRevB.32.3883] derived the discontinuity of the functional derivative of E$_{xc}$ in insulators by considering an increase of the number of electrons by one. Cautiously, these authors concluded that the discrepancy between calculated and measured band gap is a measure of the discontinuity $\Delta_{xc}$ - given the results from several calculations – if the employed LDA potentials are assumed to be good approximations. The description of our method below indicates the strong possibility that some current LDA and GGA potentials may be very good approximations.
Despite its popular use to explain the disagreement between calculated and measured band gaps, the above discontinuity has not yet been established to be non zero in real semiconductors or insulators. Further, Sham and Schlüter [@PhysRevB.32.3883] underscored the fact that, in principle, DFT and Kohn Sham LDA hold only if the number of particle is kept constant. The question could arise whether or not the discontinuity, derived by considering a change of the number of particle, is strictly applicable to DFT or LDA calculations. From the preceding, it has not yet been established that DFT or LDA calculations cannot obtain the correct band gaps, despite the fact that presently known LDA potentials do not have a discontinuity and that most of the numerous, previous ab-initio DFT and LDA calculations did not.
Another presumed contributor to the band gap underestimation by theory stems from the use of local (LDA) and semi-local (GGA) potentials. The question naturally arises as to what extent the local and semi-local potentials fail to capture key feature of the exact one. We are aware of no definitive answer, given that the exact one is not known. We would have had to delve into this matter further if we were dealing with molecules or their dissociation. The solid state systems of interest to us, to judge by previous results obtained with our method [@Bagayoko2004; @Bagayoko2008], possible errors due to the use of local and semi-local potentials appear to be very small.
There exist several approaches that have been introduced to address the band gap problem. Review articles and books are the best sources for discussing these approaches and for examples of the many DFT calculations that led to band gaps much smaller than their corresponding, experimental counterparts. In contrast, a summary of results from BZW LDA calculations for over 10 materials show agreement between theory and experiment. Illustrative examples of discrepancies between theory and experiment follow. The case of Ge is summarized in this article. Some previous LDA, GGA, and GW calculations did not yield the measured band gap, from first principle. A table provided by Ekuma and Bagayoko [@Ekumab2011] shows a multitude of DFT calculations with vastly different band gaps for titanium dioxide. With the computational method described here, Ekuma and Bagayoko obtained the measured, direct gap and predicted an indirect one. For elemental silicon, Grüning [@Myrta2006] reported an LDA band gap of 0.7 eV, much smaller than the 1.25 eV they reported as the measured value. These authors also performed calculations with the exact exchange (EXX), EXX plus LDA, EXX plus the random phase approximation (RPA). The last approach or scheme yielded 0.6 eV, a gap smaller than the above LDA gap, while the first two led to 1.5 and 1.6 eV respectively, values much larger than the experimental one. With the original version of our method, Zhao et al. utilized an LDA potential to obtain a gap of 1.02 for Si, much closer to the experimental one. Generalized gradient approximation (GGA) calculations have led to improvements of calculated properties of materials, including lattice parameters. Specifically, Hao [@PhysRevB.85.014111] reported revised Tao-Perdew-Staroverov-Scuseria (revTPSS) meta-GGA calculated lattice parameters that are in agreement with experiment following a zero-point phonon correction, for over 50 materials. Despite this very significant success, most GGA and meta-GGA calculations, including the previous ones discussed here for Ge, have not produced band gaps in agreement with experiment.
From the above summary, historical overview of the band gap problem, it appears that the scientific community believes that the derivative discontinuity of the exchange correlation energy is the main source of the disagreement between DFT calculated energy and band gaps and their corresponding, measured ones. This belief led to the development of several schemes aimed at resolving the band gap problem. Except for the few, most of these schemes are ad hoc as they include adjustable parameters that vary with the material under study. The continuing growth in the number of these schemes seems be a problem in itself, the ad hoc nature of most of them does not lend itself to predictive capabilities from first principle, the aim of theory to inform and to guide experiment. The only exception to the above trend consists of the work of our group. This work has not yet gotten the attention of the community at large, presumably due to the strength of the above belief, on the one hand, and the preponderance of results that are explained with the discontinuity, on the other hand. As we previously noted [@Bagayoko2005], the situation resembles that of the Ptolemaic model of the solar system where epicycles were continually introduced to explain its disagreement with observations. The quintessential point in support of the our method, described below, is the following: For all DFT calculations of energy bands, the *minima of the occupied energies, which add up to yield the ground state energy of the electron system, are obtained from the theory if the “correct” ground state charge density is utilized, subject to the constraint that the number of particle is kept constant* [@PhysRev.136.B864; @PhysRev.140.A1133].
Most of the previous DFT calculations, including those with GGA and LDA potentials, have consisted of judicious selecting large basis set and of performing iterations to obtain self consistent eigenvalues of the Kohn-Sham type equation. It is assumed that the single basis set in question leads to the correct representation of the electronic cloud in the system under study, a system that can be drastically different from an atomic or ionic one. *In particular, as we recently pointed
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{
"pile_set_name": "ArXiv"
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| null | null | null |
---
abstract: 'High-frequency (up to $\omega = 6 \,10^4 \un{rad/s}$) rheological measurements combined with light scattering investigations show that an isotropic and multiconnected phase of surfactant micelles exhibits a terminal relaxation time of a few $\un{\mu s}$, much smaller than in solutions of entangled wormlike micelles. This result is explained in terms of the local hexagonal order of the microscopic structure and we discuss its relevance for the understanding of dynamic behaviour in related systems, such as wormlike micelles and sponge phases.'
author:
- 'D. Constantin[^1]'
- 'J.-F. Palierne'
- 'É. Freyssingeas'
- 'P. Oswald'
title: 'High-frequency rheological behaviour of a multiconnected lyotropic phase'
---
In recent years, experimental evidence was presented as to the existence of isotropic phases consisting of connected surfactant micelles [@danino1; @kato1; @kato2]. It has been proposed that they provide an intermediate structure between entangled wormlike micelles and sponge phases [@porte1; @drye1]. Indeed, experimental results [@porte1; @appell1; @khatory1] show that, in some ionic wormlike micellar systems, a dramatic decrease in both viscosity and relaxation time is induced by increasing the counterion concentration, feature that could be explained by the appearance of connections in the micellar network. On the theoretical side, models for the flow behaviour of these connected phases have been developed [@drye1; @lequeux1], and rheology data has been interpreted according to these models in order to characterize the appearance of connections, qualitatively [@narayanan1; @hassan1; @aitali1] or quantitatively [@in1]. Throughout this body of work, however, only the relaxation modes specific to polymer systems have been considered. This approach is certainly valid in dilute phases with not too many connections, but it must fail when the density of connections becomes important and in concentrated systems, where the micelles begin to interact (sterically or otherwise). How does the system behave then and which are the relevant concepts ?
In this Letter, we try to answer these questions by investigating a concentrated and highly connected isotropic phase of a nonionic surfactant/water mixture. We argue that, in the absence of reptation (suppressed by the connections), it can be short-range order (for a concentrated system) that dominates the rheological behaviour.
We employ high-frequency rheology and dynamical light scattering (DLS) to study the isotropic phase in the [ ]{}lyotropic mixture, where [ ]{}is the non-ionic surfactant hexa-ethylene glycol mono-n-dodecyl-ether, or (for the phase diagram see [@mitchell]). Its dynamic behaviour has already been investigated by measuring the shear viscosity [@strey1; @darrigo1], sound velocity and ultrasonic absorption [@darrigo1] as well as NMR relaxation rates [@burnell1], all pointing to the presence of wormlike micelles (at least above 10 % surfactant concentration by weight [@darrigo1]). In previous experiments [@sallen1; @constantin] we have shown that, for 50 % wt surfactant concentration, above the hexagonal mesophase, the isotropic phase has a structure consisting of surfactant cylinders that locally preserve the hexagonal order over a distance $d$ that varies from about $40 \, \un{nm}$ at $40 { {\,}^{\circ} \mbox{C}}$ to $25 \, \un{nm}$ at $60 { {\,}^{\circ} \mbox{C}}$. Between the cylinders there is a large number of thermally activated connections (with an estimated density $n \sim 10^{6} \, \un{\mu m^{-3}}$) [@constantin].
We prepared the [ ]{}mixture with 50.0 % [ ]{}weight concentration. The surfactant was purchased from Nikko Chemicals Ltd. and used without further purification. We used ultrapure water from Fluka Chemie AG. The mixture was carefully homogenized by repeatedly heating, stirring and centrifuging and then allowed to equilibrate at room temperature over a few days.
Rheology measurements were performed in a piezorheometer, the principle of which has been described in reference [@cagnon] : the liquid sample of thickness $60 \un{\mu m}$ is contained between two glass plates mounted on piezoelectric ceramics. One of the plates is made to oscillate vertically with an amplitude of about 1 nm by applying a sine wave to the ceramic. This movement induces a squeezing flow in the sample and the stress transmitted to the second plate is measured by the other piezoelectric element. The shear is extremely small : $\gamma \leq 10^{-4}$, so the sample structure is not altered by the flow. The setup allows us to measure the storage ($G'$) and loss ($G''$) shear moduli for frequencies ranging from $1$ to $6 \, 10^{4} \un{rad/s}$ with five points per frequency decade. The entire setup is temperature regulated within $0.05 { {\,}^{\circ} \mbox{C}}$ and hermetically sealed to avoid evaporation.
Ten temperature points in the isotropic phase have been investigated, from $38.85 { {\,}^{\circ} \mbox{C}}$ (transition temperature from the hexagonal phase) up to $48 { {\,}^{\circ} \mbox{C}}$. The results are displayed in figure \[fig1\]. For clarity, only curves corresponding to 40, 42, 44, 46, and $48 { {\,}^{\circ} \mbox{C}}$ are plotted. Values below $1 \un{Pa}$ (solid horizontal line) are not reliable, as the signal/noise ratio becomes poor. At low frequencies, the response is purely viscous; it is only above $\omega = 10^{3} \un{rad/s}$ that there is a noticeable increase in the value of the storage modulus $G'$.
On general grounds, the low-frequency behaviour of the storage and loss moduli in a fluid is [@ferry] : $G' \propto \omega ^2$ and $G'' \propto \omega$. The slope of $G'$ vs. $\omega$ yields the “zero-shear viscosity” $\eta _0$ and the two curves cross at a frequency $\omega=1/\tau$, where $\tau$ is the terminal relaxation time. The ratio $\eta _0 / \tau$ defines a shear modulus. If $\tau$ is the only relevant time scale in the system, the complex modulus $G^*(\omega) = G' + i G''$ has a simple analytical expression, known as the Maxwell model [@ferry] : $$\label{maxwell}
G^*(\omega) = \frac{i \omega \eta _0}{1+i \omega \tau} \, .$$ The relaxation time $\tau$ separates two regimes : for $\omega \tau \ll 1$, the system can be considered as a viscous fluid with viscosity $\eta _0$, while for $\omega \tau \gg 1$ it exhibits elasticity, with a shear modulus $G_{\infty} = \eta _0 / \tau$.
As shown in \[fig2\], we obtain robust results for the static viscosity $\eta _0$ and for the relaxation time $\tau$ (plotted vs. temperature in figure \[fig3\]).
The temperature variation of the parameters $\eta _0$ and $\tau$ can be described by Arrhenius laws; for the viscosity : $$\label{arrhenius}
\eta _0 (T) = \eta _0 (T^*) \exp \left [ \frac{E_{\eta}}{k_B} \left (\frac{1}{T}-\frac{1}{T^*}\right ) \right ] \, ,$$ yielding an activation energy $E_{\eta} = 35 \pm 1 \, k_B T$ (solid curve in figure \[fig3\]). For comparison, continuous shear measurements in a Couette rheometer (Haake, model RS100), give an activation energy $E_{\eta} = 31 \, k_B T$ [@sallen3]. The relaxation time has an activation energy $E_{\tau} = 38 \pm 6 \, k_B T$ (solid curve in figure \[fig3\]). Within experimental precision, $E_{\eta} = E_{\tau}$. The high-frequency elastic modulus is therefore constant in temperature : $$\label{eq:ginf}
G_{\infty} = \eta _0 / \tau = 44 \pm 6 \, 10^3 \, \un{Pa} \, .$$
The DLS setup uses an Ar laser ($\lambda = 514 \, \un{nm}$), delivering up to $1.5 \un{W}$, a thermostated bath of an index matching liquid (decahydronaphthalene, $n = 1.48$), a photomultiplier and a PC-controlled 256 channel Malvern correlator with sample times as fast as $0.1 \un{\mu s}$. The scattering vector $q$ varies in the range $4 \, 10^{6}$ – $3 \, 10^{7} \un{m^{-1}}$. The signal is monoexponential over the whole range. In figure \[fig5\] we show the relaxation rate $\Omega (q)$ vs. $q^2$ for temperatures between $40$ and $49 { {\,}^{\circ} \mbox{C}}$ . The data fit well to a diffusion law (although there is a slight indication of super-diffusive behaviour). Since the scattered intensity is related to the variations in refractive index produced by concentration fluctuations,
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
author:
- 'Yuta Tsuchimoto, Patrick Knüppel, Aymeric Delteil, Zhe Sun, Martin Kroner,'
- Ataç Imamoğlu
title: |
SUPPLEMENTAL MATERIAL\
Quantum interface between photonic and superconducting qubits
---
Quantum Monte Carlo method
--------------------------
We consider optical to microwave conversion. In the quantum trajectory formalism, the time evolution of the system is given by the Schrödinger equation: $$\frac{d}{dt}\ket{\psi(t)} = \frac{1}{i\hbar} H_{\mathrm{eff}} \ket{\psi(t)},$$ where $\psi(t)$ is a stochastic wavefunction and $H_{\mathrm{eff}}$ is the non-Hermitian Hamiltonian given by $$\label{hamiltonian}
H_{\mathrm{eff}} = H_{\mathrm{s}} + H_{\mathrm{t}} + H_{\mathrm{st}} - \frac{i \hbar}{2} \sum_k \hat{C}^\dagger_k \hat{C}_k.$$ where $H_{\mathrm{s}}$ ($H_{\mathrm{t}}$) is the source (target) Hamiltonian and $H_{\mathrm{st}}$ the interaction Hamiltonian. The collapse operators $\hat{C}_k$ of this system are $$\begin{aligned}
\hat{C}_1 &= \sqrt{\Gamma_{\mathrm{FE}}^{({\mathrm{s}})}} \sigma_{\mathrm{EF}}^{({\mathrm{s}})} + \sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})} \eta} \sigma_{\mathrm{GF}}^{({\mathrm{t}})}, \\
\hat{C}_2 &= \sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})} (1-\eta)} \sigma_{\mathrm{GF}}^{({\mathrm{t}})}, \\
\hat{C}_3 &= \sqrt{\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}} \sigma_{\mathrm{GE}}^{({\mathrm{t}})}, \\
\hat{C}_4 &= \sqrt{\kappa_{\mathrm{c}}} \hat{a}_{\mathrm{c}},
\end{aligned}$$ where $\sigma_{ij} = \ket{i}\bra{j}$ express the projection ($i=j$) and lowering or rising operator ($i \neq j$) respectively. (s) and (t) stand for source and target (interface). $\hat{C}_1$ corresponds to a detection event where photons are emitted either in the $\ket{F}_{\mathrm{s}} \rightarrow \ket{E}_{\mathrm{s}}$ or $\ket{F,0_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transitions. The latter transition is coupled to the incident light with a coupling efficiency $\eta$. The operator $\hat{C}_2$ denotes an event originating from the $\ket{F,0_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transition which does not couple to the incident mode. $\hat{C}_3$ describes an event associated with the $\ket{E,1_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,1_{\mathrm{c}}}_{\mathrm{t}}$ transition. Finally, $\hat{C}_4$ accounts for the cavity decay event. By substituting these collapse operators, equation \[hamiltonian\] becomes: $$\begin{aligned}
H_{\mathrm{eff}} = && \hbar \Omega_{\mathrm{L}} (\sigma_{\mathrm{GF}}^{({\mathrm{s}})} + \sigma_{\mathrm{FG}}^{({\mathrm{s}})}) + \hbar g_{\mathrm{c}} (\hat{a}_{\mathrm{c}} \sigma_{\mathrm{FE}}^{({\mathrm{t}})} + \hat{a}_{\mathrm{c}}^\dagger \sigma_{\mathrm{EF}}^{({\mathrm{t}})})\nonumber\\ &&- i\hbar \frac{\Gamma_{\mathrm{FE}}^{({\mathrm{s}})}}{2}\sigma_{\mathrm{FF}}^{({\mathrm{s}})} - i\hbar \frac{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}}{2}\sigma_{\mathrm{FF}}^{({\mathrm{t}})} - i \hbar \frac{\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}}{2}\sigma_{\mathrm{EE}}^{({\mathrm{t}})}\nonumber\\ && -i\hbar \sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}\Gamma_{\mathrm{FE}}^{({\mathrm{s}})}\eta} \sigma_{\mathrm{EF}}^{({\mathrm{s}})} \sigma_{\mathrm{FG}}^{({\mathrm{t}})} -i \hbar \frac{\kappa_c}{2}\hat{a}_{\mathrm{c}}^\dagger \hat{a}_{\mathrm{c}}.\end{aligned}$$ Here, we neglected the term $\sigma_{\mathrm{FE}}^{({\mathrm{s}})}\sigma_{\mathrm{GF}}^{({\mathrm{t}})}$ describing the reverse process where an emitted photon from the $\ket{F,0_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transition drives the $\ket{E}_{\mathrm{s}} \rightarrow \ket{F}_{\mathrm{s}}$ transition because we assume unidirectional coupling realized by a Faraday rotator or a chiral waveguide. We calculated the time evolution of the stochastic wave function using the quantum Monte Carlo wave function approach. We set the initial state as the ground states for both the source and interface and assumed that the cavity does not have microwave photons, i.e. $\ket{\psi_{\mathrm{initial}}} = \ket{G}_{\mathrm{s}}\ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$. As the excitation laser pulse, we chose a Gaussian pulse with a peak Rabi frequency $\Omega_0$ which is of the same order as $\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}$. The bandwidth of the pulse was set to be smaller than $\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}$. The generated single-photon pulse shape from the $\ket{F}_{\mathrm{s}} \rightarrow \ket{E}_{\mathrm{s}}$ transition was ensured to be Gaussian by keeping $\Omega_0/\Gamma_{\mathrm{FE}}^{({\mathrm{s}})}$ small. $\eta$ was assumed to be 1.0. By assuming a realistic CQD and SC cavity, we set each parameter as follows: $\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}/2\pi = 300{\mskip3mu}{\mathrm{MHz}}$, $\kappa_c/2\pi = 3 {\mskip3mu}{\mathrm{MHz}}$, $g_c/2\pi = 50-400{\mskip3mu}{\mathrm{MHz}}$, $\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}/2\pi = 0-750 {\mskip3mu}{\mathrm{MHz}}$. Since the $\ket{E,1_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,1_{\mathrm{c}}}_{\mathrm{t}}$ transition heralds a successful optical-to-microwave photon conversion, we counted this event to estimate the conversion efficiency and rate.
Next, we consider microwave to optical conversion. Here, the collapse operators of this scheme are as follows: $$\begin{aligned}
\hat{C'}_1 &= \sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}} \sigma_{\mathrm{GF}}^{({\mathrm{t}})}, \\
\hat{C'}_2 &= \sqrt{\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}} \sigma_{\mathrm{GE}}^{({\mathrm{t}})}, \\
\hat{C'}_3 &= \sqrt{\kappa_{\mathrm{c}}} \hat{a}_{\mathrm{c}}.
\end{aligned}$$ The effective Hamiltonian is then given by $$\begin{aligned}
H_{\mathrm{eff}} =&& \hbar \Omega_{0} (\sigma_{\mathrm{GE}}^{({\mathrm{t}})} + \sigma_{\mathrm{EG}}^{({\mathrm{t}})}) + \hbar g_{\mathrm{c}} (\hat{a}_{\mathrm{c}} \sigma_{\mathrm{FE}}^{({\mathrm{t}})} + \hat{a}_{\mathrm{c}}^\dagger \sigma_{\mathrm{EF}}^{({\mathrm{t}})})\nonumber\\ && - i\hbar \frac{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}}{2}\sigma_{\mathrm{FF}}^{({\mathrm{t}})}
- i \hbar \frac{\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}}{2}\sigma_{\mathrm{EE}}^{({\mathrm{t}})} -i \hbar \frac{\kappa_{\mathrm{c}}}{2} \hat{a}_{\mathrm{c}}^\dagger \hat{a}_{\mathrm{c}}\end{aligned}$$ The initial state is $\ket{\psi_{\mathrm{initial}}} = \ket{G,1_{\mathrm{c}}}_{\mathrm{t}}$. The parameters used for this simulation are the same as those of the optical to microwave conversion except for $\Omega_0 = \Gamma_{\mathrm{FG}}^{({\mathrm{t}})}/3$. We counted the decay event $\hat{C'}_1$ to estimate the conversion efficiency and rate.
Analytical formula for the optical-to-microwave conversion efficiency
---------------------------------------------------------------------
We consider a simple case where weak coherent field
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: |
We have used the event generator LUCIAE to analyse NA35 data of the $\bar p$ and $\bar{\Lambda}$ yields, the ratio $\bar{\Lambda}$/$\bar p$, and the transverse mass distributions of $\bar p$ and $\bar{\Lambda}$ in pp and central sulphur-nucleus collisions at 200A GeV. The NA35 data could be reproduced reasonably if one assumes that the s quark pair suppression factor and concerns in nucleus-nucleus collisions are larger than the nucleon-nucleon collision. It seems to indicate that NA35 data might imply the reduction of strangeness suppression in ultrarelativistic nucleus-nucleus collisions comparing to the nucleon-nucleon collision at the same energy. However, the ratio $\bar{\Lambda}$/$\bar p$ approaching to unity in AA collisions comparing to the pp collision does not necessarily mean a flavor symmetry since hadronic rescattering plays a role as well.
PACS number: 25.75.+r
---
**[Strange Antibaryon Production Data and Reduction of Strangeness Suppression in Sulphur-Nucleus Collisions at 200A GeV]{}\
Sa Ben-Hao$^{1,2,3,5}$ and Tai An$^4$\
**
ttttt = tt = 1. CCAST (World Lab.), P. O. Box 8730 Beijing, China.\
2. China Institute of Atomic Energy, P. O. Box 275 (18),\
Beijing, 102413 China.\
3. Department of Physics, University of Hiroshima,\
Higashi-Hiroshima, 739 Japan.\
4. Institute of High Energy Physics, Academia Sinica,\
P. O. Box 918, Beijing, 100039 China.\
5. Institute of Theoretical Physics, Academia Sinica,\
Beijing China.\
0.8cm =0.3cm =0.3cm Strangeness production is expected to be a powerful probe for the mechanism of nucleus-nucleus collisions. Strangeness enhancement, the increased strangeness particle production in nucleus-nucleus collisions comparing to the nucleon-nucleon collision, is predicted to be a sensitive signature of the QGP formation in the ultrarelativistic nucleus-nucleus collisions \[1\].
The first experimental results of the enhanced production of strange particles in nucleus-nucleus collisions at 200A GeV incident energy was reported six years ago \[2\]. Later on this enhancement was confirmed by more and more experiments \[3-7\].
Strange antibaryon production, in particular, might bring messages of the equilibrium and flavour symmetry of quarks in ultrarelativistic nucleus-nucleus collisions. It has been estimated \[8\] that if there is QGP formed in the ultrarelativistic nucleus-nucleus collisions, strange (antistrange) quark pair might be copiously reproduced, resulting in an approximate flavour symmetry among u, d and s quarks \[9\]. Thus the ratio of $\bar{\Lambda}$/$\bar p$ should approach unity, since $\bar{\Lambda}$ is composed of $\bar u\bar d\bar s$ and $\bar p$ is $\bar u\bar d\bar d$.
Recently published NA35 data of antibaryon production in sulphur-nucleus collisions at 200A GeV \[7\] did really observe that the ratios $\bar{\Lambda}$ /$\bar p$ are all approaching, even over, unity and far exceeding the corresponding value of $\sim$0.25 in the nucleon-nucleon collision at the same energy. Although the NA35 data of the $\bar{\Lambda}$/$\bar p$ has been fairly reproduced by RQMD 1.08 \[10,7\] with fusion of overlpping strings into a color rope and with hadronic rescattering, the data of transverse mass distributions of $\bar p$ and $\bar{\Lambda}$ have not been explained yet, the physics behind the data have not been exposed especially.
Based on the idea that the strangeness enhancement in ultrarelativistic nucleus-nucleus collision compared to the nucleon-nucleon collision at the same energy could be investigated via the reduction of s quark suppression \[11-14\], in this letter the event generator LUCIAE \[15\] is used to analyse the NA35 data and to explore the physics behind the data.
LUCIAE was updated based on FRITIOF 7.02 \[16\] by taking into account the collective string interaction \[17-18\] during the emission of gluon bremsstrahlung and the rescattering of produced particles \[19\]. One knows that in FRITIOF 7.02 \[16\], two colliding hadrons are excited due to the longitudinal momentum transfer and/or a Rutherford parton scattering. The highly excited states will emit bremsstrahlung gluons according to the soft radiation model. The deexcited states are then treated as Lund strings allowing to decay into the final hadronic state due to the Lund fragmentation scheme. However, in the ultrarelativistic nucleus-nucleus collisions there are generally many excited strings formed close by each other. These strings would behave like vortex lines in a color superconducting QCD vacuum and will interact with each other (as the repulsive interaction suffered by the “ordinary” vortex lines in a type II superconductor). This kind of collective interaction among strings, not included in FRITIOF 7.02, is depicted by the firecracker model \[15,18\] dealing with the large p$_t$ gluon (firecracker gluon) production from the collective interaction among strings. The firecracker gluon will work as a gluon-kink excitation on a string, which tends to shorten the longitudinal size of the string and then brings about the increasing of the string tension (thus it should be regarded as an effective string tension). In firecracker model it is assumed that the groups of neighboring strings might form interacting quantum states, the large common energy density (corresponding to the collective interaction among strings) might then affect the emission of gluonic bremsstrahlung \[17-18\].
A rescattering model has been developed to describe the reinteraction of produced particles, from FRITIOF event generator, with each other and with the participant and the spectator nucleons \[19\]. In the model, the produced particles and the participant (wounded) nucleons, from FRITIOF event generator, are distributed randomly in the geometrical overlapping region between the projectile and the target nuclei under a given impact parameter. The target (projectile) spectator nucleons are distributed randomly outside the overlapping region and inside the target (projectile) sphere. A rescattering cascade process has evolved since then, cf. Ref. \[15,19\] for the detail. The considered inelastic reactions, concerning strangeness, are here cataloged into:
tttttttttttttttttttttttttttttttttttttttttttttttttttt= $\pi\pi \rightleftharpoons k\bar{k}$;\
$\pi N \rightleftharpoons kY$, $\pi\bar{N} \rightleftharpoons \bar{k}\bar{Y}$;\
$\pi Y \rightleftharpoons k\Xi$, $\pi\bar{Y} \rightleftharpoons \bar{k}\bar{\Xi}$;\
$\bar{k}N \rightleftharpoons \pi Y$ , $k\bar{N} \rightleftharpoons \pi\bar{Y}$;\
$\bar{k}Y \rightleftharpoons \pi\Xi$, $k\bar{Y} \rightleftharpoons \pi\bar{\Xi}$;\
$\bar{k}N \rightleftharpoons k\Xi$, $k\bar{N} \rightleftharpoons \bar{k}\bar{\Xi}$;\
$\pi\Xi \rightleftharpoons k\Omega^- $, $\pi\bar{\Xi} \rightleftharpoons \bar{k}\overline{\Omega^-}$;\
$k\bar{\Xi} \rightleftharpoons \pi\overline{\Omega^-}$, $\bar{k}\Xi \rightleftharpoons \pi\Omega^-$;\
$\bar{N}N$ annihilation;\
$\bar{Y}N$ annihilation;\
where $Y$ refers to the $\Lambda$ or $\Sigma$ and $\Xi$ refers to the $\Xi^-$ or $\Xi^0$. There are 299 inelastic reactions involved altogether. As the reactions introduced above do not make up the full inelastic cross section, the remainder is again treated as elastic scattering \[19\]. The cross section of $\pi\pi \rightarrow k\bar{k}$ is taken to be 2.0 mb as usual \[10\]. The isospin averaged parametrization of Ref. \[9\] is adopted for the cross sections of the reactions $\pi N \rightarrow kY$ and for the other strange quark production reactions. Of course, the difference in threshold energy among reactions is taken into account. Following Ref. \[9\], the cross section of strange quark exchange reaction, $\bar{k}N\rightarrow\pi Y
$ for instance, is assumed to be equal to ten times the value of the cross section of the strangeness production reaction. As for the cross section of the reverse reaction, the detailed balance assumption \[20\] is required.
The cross sections of the inelastic reactions given by the isospin averaged parameterization formulas of \[9\] for the $\pi N \rightarrow kY$ decrease exponentially with the CMS
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
author:
- 'Alexander Grigor’yan and Meng Yang'
title: 'Local and Non-Local Dirichlet Forms on the Sierpiński Carpet'
---
[^1] [^2] [^3] [^4]
Introduction
============
Sierpiński carpet (SC) is a typical example of non p.c.f. (post critically finite) self-similar sets. It was first introduced by Wacław Sierpiński in 1916 which is a generalization of Cantor set in two dimensions, see Figure \[fig\_SC\].
![Sierpiński Carpet[]{data-label="fig_SC"}](carpet){width="50.00000%"}
SC can be obtained as follows. Divide the unit square into nine congruent small squares, each with sides of length $1/3$, remove the central one. Divide each of the eight remaining small squares into nine congruent squares, each with sides of length $1/9$, remove the central ones, see Figure \[fig\_construction\]. Repeat above procedure infinitely many times, SC is the compact connected set $K$ that remains.
(0,0)–(9,0)–(9,9)–(0,9)–cycle; (3,3)–(6,3)–(6,6)–(3,6)–cycle;
(11,0)–(20,0)–(20,9)–(11,9)–cycle; (14,3)–(17,3)–(17,6)–(14,6)–cycle; (12,1)–(13,1)–(13,2)–(12,2)–cycle; (15,1)–(16,1)–(16,2)–(15,2)–cycle; (18,1)–(19,1)–(19,2)–(18,2)–cycle; (12,4)–(13,4)–(13,5)–(12,5)–cycle; (18,4)–(19,4)–(19,5)–(18,5)–cycle; (12,7)–(13,7)–(13,8)–(12,8)–cycle; (15,7)–(16,7)–(16,8)–(15,8)–cycle; (18,7)–(19,7)–(19,8)–(18,8)–cycle;
In recent decades, self-similar sets have been regarded as underlying spaces for analysis and probability. Apart from classical Hausdorff measures, this approach requires the introduction of Dirichlet forms. Local regular Dirichlet forms or associated diffusions (also called Brownian motion (BM)) have been constructed in many fractals, see [@BP88; @BB89; @Lin90; @KZ92; @Kig93; @Bar98; @Kig01]. In p.c.f. self-similar sets including Sierpiński gasket, this construction is relatively transparent, while similar construction on SC is much more involved.
For the first time, BM on SC was constructed by Barlow and Bass [@BB89] using *extrinsic* approximation domains in ${\mathbb{R}}^2$ (see black domains in Figure \[fig\_construction\]) and time-changed reflected BMs in those domains. Technically, [@BB89] is based on the following two ingredients in approximation domains:
(a) \[enum\_a\] Certain resistance estimates.
(b) \[enum\_b\] Uniform Harnack inequality for harmonic functions with Neumann boundary condition.
For the proof of the uniform Harnack inequality, Barlow and Bass used certain probabilistic techniques based on Knight move argument (this argument was generalized later in [@BB99a] to deal also with similar problems in higher dimensions).
Subsequently, Kusuoka and Zhou [@KZ92] gave an alternative construction of BM on SC using *intrinsic* approximation graphs and Markov chains in those graphs. However, in order to prove the convergence of Markov chains to a diffusion, they used the two aforementioned ingredients of [@BB89], reformulated in terms of approximation graphs.
However, the problem of a purely analytic construction of a local regular Dirichlet form on SC (similar to that on p.c.f. self-similar sets) has been open until now and was explicitly raised by Hu [@Hu13]. The main result of this paper is a direct purely *analytic* construction of a local regular Dirichlet form on SC.
The most essential ingredient of our construction is a certain resistance estimate in approximation graphs which is similar to the ingredient (\[enum\_a\]). We obtain the second ingredient—the uniform Harnack inequality in approximation graphs as a consequence of (\[enum\_a\]). A possibility of such an approach was mentioned in [@BCK05]. In fact, in order to prove a uniform Harnack inequality in approximation graphs, we extend resistance estimates from finite graphs to the infinite graphical SC (see Figure \[fig\_graphSC\]) and then deduce from them a uniform Harnack inequality-first on the infinite graph and then also on finite graphs. By this argument, we avoid the most difficult part of the proof in [@BB89].
in [0,1,...,27]{} (,0)–(,28);
in [0,1,...,27]{} (0,)–(28,);
in [0,1,2]{} in [0,1,2]{} (9\*+3,9\*+3)–(9\*+6,9\*+3)–(9\*+6,9\*+6)–(9\*+3,9\*+6)–cycle;
(9,9)–(18,9)–(18,18)–(9,18)–cycle;
in [0,1,...,27]{} in [0,0.5,1,...,27.5]{} (,) circle (0.08);
in [0,1,...,27]{} in [0,0.5,1,...,27.5]{} (,) circle (0.08);
(9.25,9.25)–(17.75,9.25)–(17.75,17.75)–(9.25,17.75)–cycle;
in [0,1,2]{} in [0,1,2]{} (9\*+3.25,9\*+3.25)–(9\*+5.75,9\*+3.25)–(9\*+5.75,9\*+5.75)–(9\*+3.25,9\*+5.75)–cycle;
The self-similar local regular Dirichlet form ${\mathcal{E}}_{{\mathrm{loc}}}$ on SC has the following self-similarity property. Let $f_0,\ldots,f_7$ be the contraction mappings generating SC. For all function $u$ in the domain ${\mathcal{F}}_{{\mathrm{loc}}}$ of ${\mathcal{E}}_{{\mathrm{loc}}}$ and for all $i=0,\ldots,7$, we have $u\circ f_i\in{\mathcal{F}}_{{\mathrm{loc}}}$ and $${\mathcal{E}}_{{\mathrm{loc}}}(u,u)=\rho\sum_{i=0}^7{\mathcal{E}}_{{\mathrm{loc}}}(u\circ f_i,u\circ f_i).$$ Here $\rho>1$ is a parameter from the aforementioned resistance estimates, whose exact value remains still unknown. Barlow, Bass and Sherwood [@BB90; @BBS90] gave two bounds as follows:
- $\rho\in[7/6,3/2]$ based on shorting and cutting technique.
- $\rho\in[1.25147,1.25149]$ based on numerical calculation.
McGillivray [@McG02] generalized above estimates to higher dimensions.
The heat semigroup associated with ${\mathcal{E}}_{{\mathrm{loc}}}$ has a heat kernel $p_t(x,y)$ satisfying the following estimates: for all $x,y\in K,t\in(0,1)$ $$\label{eqn_hk}
p_t(x,y)\asymp\frac{C}{t^{\alpha/\beta^*}}\exp\left(-c\left(\frac{|x-y|}{t^{1/\beta^*}}\right)^{\frac{\beta^*}{\beta^*-1}}\right),$$ where $\alpha=\log8/\log3$ is the Hausdorff dimension of SC and $$\label{eqn_beta_up}
\beta^*:=\frac{\log(8\rho)}{\log3}.$$ The parameter $\beta^*$ is called the *walk dimension of BM* and is frequently denoted also by $d_w$. The estimates (\[eqn\_hk\]) were obtained by Barlow and Bass [@BB92; @BB99a] and by Hambly, Kumagai, Kusuoka and Zhou [@HKKZ00]. Equivalent conditions of sub-Gaussian heat kernel estimates for local regular Dirichlet forms on metric measure spaces were explored by many authors, see Andres and Barlow [@AB15], Grigor’yan and Hu [@GH14a; @GH14b], Grigor’yan, Hu and Lau [@GHL10; @GHL15], Grigor’yan and Telcs [@GT12]. We give an alternative proof of the estimates (\[eqn\_hk\]) based on the approach developed by the first author and others.
Consider the following stable-like non-local quadratic form $$\begin{aligned}
&{\mathcal{E}}_\beta(u,u)=\int_K\int_K\frac{(u(x)-u(y))^2}{|x-y|^{\alpha+\beta}}\nu({\mathrm{d}}x)\nu({\mathrm{d}}y),\\
&{\mathcal{F}}_\beta={\left\{
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: |
A new measure of the crystal-field strength, complementary to the conventional one, is defined. It is based on the rotational invariants $\left|B_{k0}\right|_{\rm av}$ or $\left|\sum_{k}B_{k0}\right|_{\rm av}$, $k=2,4,6$, of the crystal-(ligand)-field (CF) Hamiltonian ${\cal H}_{\rm CF}$ parametrizations, i.e. on the axial CF parameters modules averaged over all reference frame orientations. They turn out to be equal to $\left|{\cal
H}_{\rm CF}^{(k)}\right|_{\rm av}$ and $\left|{\cal H}_{\rm CF}\right|_{\rm av}$, respectively. While the traditional measure is established on the parametrization modules or on the second moment of the CF energy levels, the introduced scale employs rather the first moment of the energy modules and has better resolving power. The new scale is able to differentiate the strength of various iso-modular parametrizations according to the classes of rotationally equivalent parametrizations. Using both the compatible CF strength measures one may draw more accurate conclusions about the Stark levels arrays and particularly their total splitting magnitudes.
author:
- '**J. Mulak$^{1}$ and M. Mulak$^{2}$**'
date: |
[*$^{1}$ W Trzebiatowski Institute of Low Temperature and Structure Research,\
Polish Academy of Sciences, 50–950, PO Box 1410, Wroclaw, Poland\
$^{2}$ Institute of Physics, Wroclaw University of Technology,\
Wyb. Wyspianskiego 27, 50–370 Wroclaw, Poland*]{}
title: '**On a complementary scale of crystal-field strength**'
---
[*PACS*]{}: 71.70.Ch\
[*Key words*]{}: crystal-field strength, crystal-field splitting\
1. Introduction {#introduction .unnumbered}
===============
Solid state experimentalists, especially spectroscopists, still need a reliable scale quantitatively characterizing the effect of crystal-field interaction, i.e. defining the so-called crystal-field strength. Such a parameter could directly verify and compare various parametrizations of the crystal-field Hamiltonian ${\cal
H}_{\rm CF}$, which may come from different fittings experimental data when the orientations of reference frames associated with these parametrizations are unknown in the majority of cases.
Although such a conventional scale for measuring the strength of the crystal-field has been already introduced over twenty years ago \[1,2\], in some cases it seems to be insufficiently precise. It employs the basic rotational invariants of the ${\cal H}_{\rm CF}$, i.e. the modules of its $2^{k}$-pole components ${\cal H}_{\rm
CF}^{(k)}$, defined as $M_{k}=\left(\sum_{q}|B_{kq}|^{2}\right)^{1/2}$, as well as uses the global ${\cal
H}_{\rm CF}$ modulus $M=\left(\sum_{k}\sum_{q}|B_{kq}|^{2}\right)^{1/2}$. In the first case the partial crystal-field strength is defined as $S_{k}\!=\!\left(\frac{1}{2k+1}\right)^{1/2}M_{k}$, while in the second case the global crystal-field strength is given by $S=\left(\sum_{k}S_{k}^{2}\right)^{1/2}$. Throughout the paper the tensor (Wybourne) notation for the crystal-field Hamiltonian and the crystal-field parameters (CFPs), ${\cal H}_{\rm CF}=\sum_{k}\sum_{q}B_{kq}C_{q}^{(k)}$, is consistently used \[3\]. The summations over $k$ and $q$ indices run, in each individual case, over strictly specified values according to the kind of central ion and its point symmetry.
Both the parameters $S_{k}$ and $S$ themselves are not a direct measure of the real magnitude of the initial state splitting, since the crystal-field effect depends also on the properties of an object (a paramagnetic ion) upon which the ${\cal H}_{\rm CF}$ acts. Namely, the response of the system to the ${\cal H}_{\rm CF}$ perturbation reflects the symmetry of the electron density distribution of the central ion open-shell. For instance, an $S$-type ion like Gd$^{3+}$ feels no crystal field (in the first order of perturbation) no matter how strong is the surrounding field.
The effect of splitting can be most simply expressed by the so-called second moments $\sigma_{k}^{2}$ or $\sigma^{2}$ of the CF sublevels within the initial state upon switching on the ${\cal H}^{(k)}_{\rm CF}$ (or ${\cal H}_{\rm CF}$) perturbation. In fact, the second moment is easily represented by the scalar crystal-field strength parameters, either $S_{k}$ or $S$ (section 2). However, although the effective ${\cal H}^{(k)}_{\rm
CF}$ multipoles (for $k=2,4,6$) contribute to the energy of individual Stark levels independently (as an algebraic sum), the simple linear relations between $\sigma_{k}^{2}$ (or $\sigma^{2}$), and $S_{k}^{2}$ (or $S^{2}$) are always fulfilled. As it is proved these relations strongly confine both the maximal $(\Delta {\cal
E}_{\rm max})$ and minimal $(\Delta {\cal E}_{\rm min})$ nominally allowed splittings of the initial state (section 3). Moreover the actual crystal field splittings $\Delta E$ can be additionally restricted (section 5). Naturally, all the iso-modular ${\cal H}_{\rm CF}$ parametrizations correspond to the same crystal-field strengths $S_{k}$ and $S$. However, apart from the modules $M_{k}$ and $M$, there exist also other rotational invariants of the ${\cal H}^{(k)}_{\rm CF}$ or ${\cal H}_{\rm CF}$ which distinguish the whole classes of the rotationally equivalent ${\cal H}_{\rm CF}$ parameterizations, in other words the parameterizations referring to the same real crystal-field potential, but expressed in variously oriented reference frame. Interestingly, the new invariants turn out to be the average values of the axial parameter modulus $|B_{k0}|_{\rm av}$, $k=2,4,6$, in the case of ${\cal H}^{(k)}_{\rm CF}$, or $|\sum_{k}B_{k0}|_{\rm av}$ for the global ${\cal H}_{\rm CF}$ obtained after the averaging over all orientations of the reference frame, i.e. over the solid angle $4\pi$. As it is shown in the paper the average value of the axial parameter modulus or the average of the modulus of their sum are just equal to $|{\cal H}^{(k)}_{\rm CF}|_{\rm av}$ and $|{\cal H}_{\rm CF}|_{\rm av}$, respectively (section 4).
The new scale of the crystal-field strength based on the above invariants is in principle consistent with the conventional one but it reveals more resolving power. Applying the new measure to the iso-modular parametrizations may lead to different strength parameters what is exemplified below for several cases (section 5). The introduced more subtle strength gradation established rather on the first moment of the sublevel energy modules gives, comparing to the second moment, additional information about the Stark levels array for various iso-modular ${\cal H}_{\rm CF}$s, including the magnitude of the total splitting gap of the states. In this paper we confine ourselves to the pure model states of the zero-order approximation with a well defined angular momentum quantum number and the corresponding degeneration. These could be for instance Russell-Saunders coupled states $|\alpha L S J \rangle$ coming from the $^{2S+1}$L terms, where $\alpha$ stands for the remaining quantum numbers needed for their complete determination. Such states have a well defined quantum number $J$ and the degeneration $2J+1$. The derivation of the analogical expressions including $J$-mixing effects \[4\] or a transformation to other functional bases of the zero-order approximation can be accomplished by using standard angular momentum re-coupling techniques \[4-8\]. In section 5 we analyse by way of example the crystal-field splitting of $p^{1}$, $d^{1}$ and $f^{1}$ one-electron configurations and a typical complex state $^{3}H_{4}$ for various iso-modular ${\cal H}_{\rm CF}^{(k)}$, $k=2,4,6$. In the first three cases we avoid complex states re-coupling procedure which is a side issue to the problem under consideration. Since we study the differentiation of the effects due to various iso-modular Hamiltonians ${\cal H}_{\rm CF}^{(k)}$, all CFPs values along with the Stark levels energies are given in $M_{k}$ units.
2. Conventional definition of the crystal-field strength parameter {#conventional-definition-of-the-crystal-field-strength-parameter .unnumbered}
------------------------------------------------------------------
The comparison and scaling of the crystal-field impact can be based upon the two types of scalar quantities, $M_{k}$ and/or $M$, since both of them are rotationally invariant. A scalar crystal-field strength parameter of this kind was given firstly by Auzel and Malta \[1,2\] as (in original notation): $$\begin{aligned}
N_{v} &=& \left[\sum\limits_{k,q}|B_{q}^{k}|^2\left(\frac{2\pi}{2
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: |
Clinical applicability of automated decision support systems depends on a robust, well-understood classification interpretation. Artificial neural networks while achieving class-leading scores fall short in this regard. Therefore, numerous approaches have been proposed that map a salient region of an image to a diagnostic classification. Utilizing heuristic methodology, like blurring and noise, they tend to produce diffuse, sometimes misleading results, hindering their general adoption.
In this work we overcome these issues by presenting a model agnostic saliency mapping framework tailored to medical imaging. We replace heuristic techniques with a strong neighborhood conditioned inpainting approach, which avoids anatomically implausible artefacts. We formulate saliency attribution as a map-quality optimization task, enforcing constrained and focused attributions. Experiments on public mammography data show quantitatively and qualitatively more precise localization and clearer conveying results than existing state-of-the-art methods.
address: 'VRVis Zentrum für Virtual Reality und Visualisierung Forschungs-GmbH, Vienna, Austria'
bibliography:
- 'refs.bib'
title: Interpreting medical image classifiers by optimization based counterfactual impact analysis
---
at (current page.north) ;
Classifier Decision Visualization, Image Inpainting, Mammography, Explainable AI
Introduction {#sec:intro}
============
Consulting radiologists will routinely back their findings by pinpointing and describing a specific region on a radiograph. Contrary, acting as a highly efficient black box, artificial neural networks (ANN) [@litjens2017] fall short of this form of explanation for their predictions. ANNs’ high dimensional, nonlinear nature, does not induce a canonical map between derived prediction and input image. Understandably, a plethora of approaches have been presented that try to derive a so called saliency map, that is, a robust mapping between pixel space and prediction class [@zintgraf2017; @fong2017; @dabkowski2017; @simonyan2013; @zhou2016; @shrikumar17; @chang2018; @petsiuk2018; @uzunova2019].
![Our saliency mapping framework: (i) classification score of an image is obtained, (ii) a hole mask is generated, inpainted and classified, (iii) saliency loss is computed based on the score difference of original/inpainted images and map quality, (iv) optimization is continued for a fixed number of steps and a result map/mask is derived.[]{data-label="fig:overview"}](architecture_2.pdf){width=".48\textwidth"}
Most frequently this form of reasoning is based on *local explanations* (LE), i.e. on concrete maps for image-prediction pairs [@fong2017; @lipton2016]. A clinically applicable LE needs to be *informative* for the radiologists, that is, focusing on regions coinciding with medical knowledge [@lombrozo2006]. Moreover, a methodologically sound LE is *faithful* to the classifier, i.e. dependent on architecture, parametrization, and its preconditions like training-set distribution [@adebayo2018].
*Direct approaches* efficiently utilize the assumed analytic nature or the layered architecture of an ANN classifier to derive the desired saliency map for a LE [@simonyan2013; @zhou2016]. While frequently applied, the obtained results of this class are possibly incomplete, diffuse, hard to interpret, and as recent work shows misleading [@zintgraf2017; @fong2017; @dabkowski2017; @shrikumar17; @adebayo2018]. Thereby they violate both criteria, informativeness and faithfulness, hindering their general application in medical imaging.
Contrary, *reference based* LE approaches [@fong2017] try to mitigate these issues by studying how the given classifier reacts to perturbations of the input image. Using the original input as a reference and marginalizing a dedicated image region’s contribution, they estimate this region’s effect on the classification score. Solutions mainly vary in the ways this marginalization is achieved. They range from heuristic approaches, e.g. blurring, noise, or graying out [@dabkowski2017; @fong2017], over local neighbourhood conditioning [@zintgraf2017], to utilizing strong conditional generative models [@chang2018; @uzunova2019]. These methods address *informativeness*, however, applied to medical images, they introduce noise, possibly pathological indications, anatomical implausible tissue or other adversarial artefacts. By this, they amplify the out-of-distribution problem, similar to an adversarial attack: they expect a meaningful classification result for an image, that is not within the training-set distribution. Hence, they fall short of *faithfulness* for clinical applications.
Marginalization for medical imaging, i.e. the replacement of pathological regions with counterfactual healthy tissue, is being actively explored and addressed by generative adversarial network setups (GAN). Besides promising results, authors report resolution limitations, and the same underlying out-of-distribution issue[@bermudez2018; @baumgartner2017; @becker2019; @andermatt2019].
**Contribution:** We address the open challenge of *faithful and informative* medical black-box classifier interpretation by expanding natural image classifier visualization approaches [@zintgraf2017; @dabkowski2017; @fong2017]. We propose a reference based optimization framework tailored to medical images, focusing on the interactions between original and marginalized image classification-scores, and map quality. To tackle anatomical correctness of marginalization in medical images, partial convolution inpainting [@liu2018] is adapted. Hence, instead of a globally acting GAN, we utilize local per-pixel reconstruction without sacrificing global image composition. We validate our approach on publicly available mammography data, and show quantitatively and qualitatively more precise localization, and clearer conveying results than existing state-of-the-art methods.
Methods {#sec:methods}
=======
Our goal is to estimate a *faithful* and *informative* saliency map between a medical image and its classification score: given an image, we search for and visually attribute the *specific* pixel-set that contributes towards a confident classification for a fixed class (see Fig. \[fig:overview\]). Following [@dabkowski2017; @zintgraf2017] we formulate the general problem as finding the *smallest deletion region* (SDR) of a class $c$, i.e. the pixel-set whose marginalization w.r.t. the classifier lowers the classification score for $c$.
**Image-wise Saliency Mapping**: Informally, we search for the smallest smooth map, that indicates the regions we need to change (inpaint) such that we get a *sufficiently healthy* image able to *fool the classifier*. We formalize the problem as follows: Let $I$ denote an image of a domain $\mathcal{I}$ with pixels on a discrete grid $m_1 \times m_2$, $c$ a fixed class, and $f$ a classifier capable of estimating $p(c|I)$, the probability of $c$ for $I$. Also let $M$ denote the saliency mask for image $I$ and class $c$, hence $M \in M^{m_1 \times m_2}(\{0,1\})$. We use *total variation* $tv(M)$ [@dabkowski2017], and *size* $ar(M)$, to measure the mask’s shape. Note that *size* here is ambiguous. Experimentally we found dice overlap with regions-of-interest like organ masks to be favourable over the map’s average pixel value[@dabkowski2017]. With $\odot$ denoting elementwise multiplication, and $ \pi(M)$ the inpainting result of a hole image $I \odot M$, we can define $ \phi(M) := -1 \cdot \log ( p(c| \pi(M)) ) $ and $\psi(M) := \log (\text{odds}(I)) - \log (\text{odds}(\pi(M))) $, where $\text{odds}(I) = \frac{ p(c|I) }{ 1 - p(c|I)}$. Both, $\phi$ and $\psi$, weigh the new probability of the inpainted image. If we assume class $c$ to denote *pathological*, then healthy images, and large score differences will be favoured. With this preparation we define our desired optimization function as $$\mathcal{L}(M) := \lambda_1 \cdot (\phi(M) + \psi(M)) + \lambda_2 \cdot tv(M) + \lambda_3 \cdot ar(M)$$ where $\lambda_i \in {\rm I\!R}$ are regularization parameters, and search for $\arg\min_{M} \; \mathcal{L}(M)$. There are two collaborating parts in $\mathcal{L}$. The first term enforces the class probability to drop, the latter two emphasize an informative mask. Focusing on medical images, $\mathcal{L}$ directly solves the SDR task, thereby minimizing medically implausible and adversarial artefacts caused by inpainting of large classifier-neutral image regions, as observable in [@liu2018; @dabkowski2017; @fong2017]. The optimization problem is solved by local search through stochastic gradient descent, starting from a regular grid initialization. By design, no restrictions are applied on the classifier $f$. For optimization we relax the mask’s domain to $M^{m_1 \times m_2}([0,
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'In this paper, we consider finitely many interval maps simultaneously acting on the unit interval $I = [0,\, 1]$ in the real line $\mathbb{R}$; each with utmost finitely many jump discontinuities and study certain important statistical properties. Even though we use the symbolic space on $N$ letters to reduce the case of simultaneous dynamics to maps on an appropriate space, our aim in this paper remains to resolve ergodicity, rates of recurrence, decay of correlations and invariance principles leading upto the central limit theorem for the dynamics that evolves through simultaneous action. In order to achieve our ends, we define various Ruelle operators, normalise them by various means and exploit their spectra.'
author:
- |
Aswin Gopakumar\
[aswin15@iisertvm.ac.in]{}\
Kirthana Rajasekar\
[kirthanarajasekar15@iisertvm.ac.in]{}\
Shrihari Sridharan\
[shrihari@iisertvm.ac.in]{}\
[*Indian Institute of Science Education and Research*]{}\
[*Thiruvananthapuram (IISER-TVM), India.*]{}
title: '<span style="font-variant:small-caps;">Simultaneous Action of Finitely Many Interval Maps: Some Dynamical and Statistical Properties</span>'
---
--------------------- --- -----------------------------------------------------
**Keywords** : Growth of typical trajectories;
Invariance principles;
Ruelle operator and the pressure function;
Simultaneous action of finitely many interval maps.
**AMS Subject** : 37E05, 37C35, 37D35, 37B10.
**Classifications**
--------------------- --- -----------------------------------------------------
Introduction
============
Various dynamical properties and statistical properties help us understand the behaviour of dynamics caused by the action of a transformation $T$ on some phase space $X$. Important among such properties include the Birkhoff’s pointwise ergodic theorem, asymptotic estimates on rates of recurrence of typical orbits, decay of correlations, invariance principles, central limit theorem, law of iterated logarithms [*etc*]{}. Each of these theorems provide us a deeper glimpse into the structure, the dynamics builds in its phase space of action or an invariant subset, thereof.
Birkhoff’s pointwise ergodic theorem observes a considered dynamical system through a real-valued continuous function and states that the sequence of local time averages along the orbit of any typical point converges to the global space average, whenever the transformation $T$ acting on $X$ is ergodic. Though the result is extremely strong, it does allow some points (though negligible, meaning with collective measure zero) to fluctuate from this mean behaviour. Thus, an interesting study in the dynamics of such ergodic systems is to obtain a good understanding of the set of points that violate the Birkhoff’s ergodic conditions. An easy way to approach this subject locally is to work out the ergodic sums of the observables and consider the cardinality of the set of points whose ergodic sum calculated at various times remain inside some chosen interval $[a,\, b] \subset \mathbb{R}$. However, on a global scale, an alternate way to understand the deviation from the average behaviour of typical orbits is achieved by formalising the central limit theorem.
The central limit theorem is an important tool in mathematics that distributes the random variables along a bell-curve (normal distribution), as more and more independent random variables are appropriately included under the ambit of study. This is a central object of investigation in understanding deterministic dynamical systems, owing to its natural appeal, when we consider the various orbits in the phase space. However, as important as the central limit theorem is, we see that they are subsumed by more general invariance principles. A sequence of random variables $\big\{ X_{n} \big\}_{n\, \ge\, 1}$ is said to satisfy an almost sure invariance principle if the sequence can be approximated almost surely by another sequence, preferably with certain desired properties and with a relatively small margin of error.
Several mathematicians have studied these properties in many deterministic dynamical systems, where the phase space is a compact interval of the real line, [@md:86; @ci:96; @lsv:99; @ps:02; @cr:07; @cm:15], the Julia set of some rational map that occurs as a compact subset of the Riemann sphere, [@du:91; @dpu:96; @ss:07; @ss:09], [*etc.*]{}, however, with a single transformation acting on the appropriate space, based on which the dynamics evolves. Examples of continuous time dynamical systems that has remained in the focus of the dynamics community include expanding flows restricted on a compact subset of the Riemannian manifold, [@dp:84; @spl:89; @mp:91] or some mixing Axiom $A$ diffeomorphism restricted on a basic set, [@mr:73; @ks:90; @rs:93; @na:00; @ps:01]. A particularly desirable feature of all the above-mentioned maps restricted on their respective sets is that they can be studied through an associated symbolic model [@rb:73; @mr:73]. There are also various studies carried out by several mathematicians that analyse statistical results in various settings of dynamical systems. Prominent among them include [@ps:75; @hh:80; @cp:90; @pp:90; @ps:94; @lsy:99; @si:99; @fmt:03; @mtk:05; @mn:05; @hntv:17].
What we intend to investigate in this paper is slightly richer in dynamics, than what is explained so far. In this paper, we consider the compact unit interval $[0,\, 1] \subset \mathbb{R}$; however with finitely many maps acting on the space simultaneously. Thus, the dynamics evolves along the multiple branches provided by each of these maps. In fact, we work with finitely many interval maps defined on $[0,\, 1]$; each of which has a discrete set of utmost finitely many discontinuities. As an expert reader may realise, these results are readily transferable to various settings including the simultaneous action of finitely many rational maps restricted on the appropriate Julia set, as defined in [@hs:00] or to the action of a holomorphic correspondence restricted on the support of its Dinh-Sibony measure, as defined in [@bs:16]. We shall explain our claim of transferability of the main theorems of this paper, as written above, in the final section, .
This paper is structured as follows: In the next two introductory sections and , we narrate the basic settings of this paper and develop certain notations and dynamical notions, however only as skeletal to enable us to state the main theorems of this paper in section . The main results of this paper describe the ergodicity of the system in theorems and , the rates of recurrence in theorems and , the exponential decay of correlations in theorems and , almost sure invariance principles in theorems and and a few more statistical properties such as the central limit theorems and the laws of iterated logarithms in theorems and . The reason why each theorem appears twice in the list above will be clear, by the time we reach section . In section , we recall the setting of symbolic dynamics that comes in handy as a book-keeping mechanism in our study. In sections , and , we define three kinds of Ruelle operators on the appropriate Banach space of continuous functions and Hölder continuous functions defined on the phase spaces that interest us, compare their spectra and normalise them in different ways in order that they help us in proving our main theorems. Having achieved these, we embark on writing the proofs of the main theorems in sections , , , and . We conclude the paper with a few remarks in section .
Preliminaries and the pressure function {#prelims}
=======================================
In this section, we explain the setting of our paper and define certain basic terminologies that help us in constructing the necessary notions to state our main results.
Let $I$ denote the unit interval on the real line, [*i.e.*]{}, $I = [0,\, 1]$. We are interested in studying the dynamics of finitely many interval maps acting simultaneously on $I$, *i.e.*, given $N \in \mathbb{N}$ and $1 \leq d \leq N$, we consider the interval maps $T_{d} : I \longrightarrow I$ of degree $(d + 1)$ given by $$T_{d}\, (x)\ \ :=\ \ (d + 1)\; x \pmod 1.$$ The simultaneous action is explained as follows: For any $x_{0} \in I$, its forward orbit at times $t = 0,\, 1,\, 2,\, \cdots,\, n,\, \cdots$ is defined as $$\label{1storbit}
\left\{ x_{0},\, x_{1} \in \bigcup_{d\, =\, 1}^{N} T_{d} (x_{0}),\, x_{2} \in \bigcup_{d\, =\, 1}^{N} T_{d} (x_{1}),\, \cdots,\, x_{n} \in \bigcup_{d\, =\, 1}^{N} T_{d} (x_{n - 1}), \cdots \right\}.$$ Thus, at every stage, we have $N$ many maps to choose from to move forward and the totality of all these branches describe the forward orbit. Observe that the dynamics that arises out of such a process can also be described by the action of a semigroup generated by the same interval maps, $\mathscr{S} = \big\
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: |
We consider the problem of viscosity solution of integro-partial differential equation(IPDE in short) with one obstacle via the solution of reflected backward stochastic differential equations(RBSDE in short) with jumps. We show existence and uniqueness of a continuous viscosity solution of equation with non local terms, in case the generator is not monotonous and Levy’s measure is infinite.\
author:
- '<span style="font-variant:small-caps;">L. SYLLA</span>'
date: 'December, 21, 2017'
title: ': '
---
**Keywords**: Integro-partial differential equation; Reflected stochastic differential equations with jumps; Viscosity solution; Non-local operator.\
\
**MSC 2010 subject classifications**: 35D40, 35R09, 60H30.
Introduction
============
We consider the following system of integro-partial differential equation with one-obstacle $\ell$, which is a function of $(t,x)$: $\forall i\in\{1,\ldots,m\}$, $$\label{eq1}
\left
\{\begin{array}{ll}
\min\Big\{ u^{i}(t,x)-\ell(t,x);-\partial_{t}u^{i}(t,x)-b(t,x)^{\top}\mathrm{D}_{x}u^{i}(t,x)-\frac{1}{2}\mathrm{Tr}(\sigma\sigma^{\top}(t,x)\mathrm{D}^{2}_{xx}u^{i}(t,x))\\
\quad\quad-\mathrm{K}_{i}u^{i}(t,x)-\mathit{h}^{(i)}(t,x,u^{i}(t,x),(\sigma^{\top}\mathrm{D}_{x}u^{i})(t,x),\mathrm{B}_{i}u^{i}(t,x))\Big\}=0,\quad (t,x)\in\left[ 0,T\right] \times\mathbb{R}^{k};\\
u^{i}(T,x)=g^{i}(x);
\end{array}
\right.$$ where the operators $\mathrm{B}_{i}$ and $\mathrm{K}_{i}$ are defined as follows: $$\begin{aligned}
\mathrm{B}_{i}u^{i}(t,x) & = & \displaystyle\int_{\mathrm{E}}\gamma^{i}(t,x,e)(u^{i}(t,x+\beta(t,x,e))-u^{i}(t,x))\lambda(\mathrm{d} e);\label{2.2}\\
\mathrm{K}_{i}u^{i}(t,x) & = & \displaystyle\int_{\mathrm{E}}(u^{i}(t,x+\beta(t,x,e))-u^{i}(t,x)-\beta(t,x,e)^{\top}\mathrm{D}_{x}u^{i}(t,x))\lambda(de).\nonumber\end{aligned}$$ The resolution of (\[eq1\]) is in connection with the following system of backward stochastic differential equations with jumps and one-obstacle $\ell$:
$$\label{eq2}
\left
\{\begin{array}{ll}
(i)~dY^{i;t,x}_{s}=-f^{(i)}(s,X^{t,x}_{s},(Y^{i;t,x}_{s})_{i=1,m},Z^{i;t,x}_{s},U^{i;t,x}_{s})ds-
\mathrm{d}\mathrm{K}^{i;t,x}_{s}\\
\quad\quad\quad\quad\quad\quad
\quad\quad+Z^{i;t,x}_{s}\mathrm{d}
\mathrm{B}_{s}+\displaystyle\int_{\mathrm{E}}\mathrm{U}^{i;t,x}
_{s}(e)\tilde{\mu}(\mathrm{d}s,\mathrm{d}e),\quad s\leq T;\\
(ii)~Y^{i;t,x}_{s}\geq \ell(s,X^{t,x}_{s})~\textrm{and}~ \displaystyle\int^{T}_{0}(Y^{i;t,x}_{s}- \ell(s,X^{t,x}_{s}))\mathrm{d}\mathrm{K}^{i;t,x}_{s}=0;
\end{array}
\right.$$
and\
the following standard stochastic differential equation of diffusion-jump type: $$\label{2.4}
X^{t,x}_{s}=x+\displaystyle\int^{s}_{t}b(r,X^{t,x}_{r})\, \mathrm{d}r+\displaystyle\int^{s}_{t}\sigma(r,X^{t,x}_{r})\, \mathrm{d}B_{r}+\displaystyle\int^{s}_{t}
\displaystyle\int_{E}\beta(r,X^{t,x}_{r-},e)\tilde{\mu}(dr,de),$$ for $s\in[t,T]$ and $X^{t,x}_{s}=x$ if $s\leq t$.\
It is recalled that pioneering work was done for the resolution of (\[eq1\]), among these works we can mention those of Barles and al. [@bar] in case without obstacle, Harraj and al. [@har] in the case with two obstacles; with as common point the hypothesis of monotony on the generator and $\gamma \geq 0$. But recently Hamadène and Morlais relaxed these conditions with $\lambda(.)$ finite [@hamaMor].\
In this work we propose to solve (\[eq1\]) by relaxing the monotonicity of the generator and the positivity of $\gamma$ and assuming that $\lambda=\infty$.\
Our paper is organized as follows: in the next section we give the notations and the assumptions of our objects; in section $3$ we recall a number of existing results; in section $4$ we build estimates and properties for a good resolution of our problem; section $5$ is reserved to give our main result and the section $6$ for doing an extension of our result.\
And in the end, classical definition of the concept of viscosity solution is put in appendix.
Notations and assumptions
=========================
Let $\left(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\leq T},\mathbb{P}\right)$ be a stochastic basis such that $\mathcal{F}_{0}$ contains all $\mathbb{P}-$null sets of $\mathcal{F}$, and $\mathcal{F}_{t}=\mathcal{F}_{t+}:=\bigcap_{\epsilon>0}
\mathcal{F}_{t+\epsilon},~t\geq 0$, and we suppose that the filtration is generated by the two mutually independents processes:\
(i) $B:=(B_{t})_{t\geq 0}$ a $d$-dimensional Brownian motion and,\
(ii) a Poisson random measure $\mu$ on $\mathbb{R}^{+}\times\mathrm{E}$ where $\mathrm{E}:=\mathbb{R}^{\ell}-\{0\}$ is equipped with its Borel field $\mathcal{E}$ $(\ell\geq 1)$. The compensator $\nu(\mathrm{d}t,\mathrm{d}e)=\mathrm{d}t\lambda(\mathrm{d}e)$ is such that $\{\tilde{\mu}(\left[0,t\right]\times A)=(\mu-\lambda)(\left[ 0,t\right]\times A)\}_{t\geq 0}$ is a martingale for all $A\in\mathcal{E}$ satisfying $\lambda(A)<\infty$. We also assume that $\lambda$ is a $\sigma$-finite measure on $(E,\mathcal{E})$, integrates the function $(1\wedge\mid e\mid ^{2})$ and $\lambda(E)=\infty$.\
Let’s now introduce the following spaces:\
(iii) $\mathcal{P}~(resp.~\mathbf{P})$ the field on $\left[0,T\right]\times \Omega$ of $\mathcal{F}_{t\leq T}$-progressively measurable (resp. predictable) sets.\
(iv) For $\kappa\geq 1$, $\mathbb{L}^{2}_{\kappa}(\lambda)$ the space of Borel measurable functions $\varphi:=(\varphi(e))_{e\in E}$ from $E$ into $\mathbb{R}^{\kappa}$ such that $\|\varphi\|^{2}_{\mathbb{L}^{2}_{\kappa}(\lambda)}=\displaystyle\int_{E}\left|\varphi(e)\right|^{2}_{\kappa}\lambda(\mathrm{d}e)<\infty$; $\mathbb{L}^{2}_{1}(\lambda)$ will be simply denoted by $\mathbb{L}^{2}(\lambda)$;\
(v) $\mathcal{S}^{2}(\mathbb{R}^{\kappa})$ the space of RCLL (for right continuous with left limits) $\mathcal{P}$-measurable and $\mathbb{R}^{\kappa}$-valued processes such that $\mathbb{E}[\sup_{s\leq T} \left|Y_{s}\right|^{2}]<\infty$; $\mathcal{A}^{2}_{c}$ is its subspace of continuous non-decreasing processes $(\mathrm{K}_{t})_{t\leq T}$ such that $\mathrm{K}_{0}=0$ and $\mathbb{E}\left[(\mathrm{K}_{T})^{2} \right]<\infty$;\
(vi) $\mathbb
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We construct a fundamental region for the action on the $2d+1$-dimensional affine space of some free, discrete, properly discontinuous groups of affine transformations preserving a quadratic form of signature $(d+1, d)$, where $d$ is any odd positive integer.'
author:
- Ilia Smilga
date: 'Received: date / Accepted: date'
title: Fundamental domains for properly discontinuous affine groups
---
Introduction {#sec:intro}
============
Background and motivation {#sec:background}
-------------------------
The present paper is part of a larger effort to understand discrete groups $\Gamma$ of affine transformations (subgroups of the affine group $GL_n(\mathbb{R}) \rtimes \mathbb{R}^n$) acting properly discontinuously on the affine space $\mathbb{R}^n$. The case where $\Gamma$ consists of isometries (in other words, $\Gamma \subset O_n(\mathbb{R}) \rtimes \mathbb{R}^n$) is well-understood: a classical theorem by Bieberbach says that such a group always has an abelian subgroup of finite index.
Define a *crystallographic* group to be a discrete group $\Gamma \subset GL_n(\mathbb{R}) \rtimes \mathbb{R}^n$ acting properly discontinuously and such that the quotient space $\mathbb{R}^n / \Gamma$ is compact. In [@Aus64], Auslander conjectured that any crystallographic group is virtually solvable, that is, contains a solvable subgroup of finite index. Later, Milnor [@Mil77] asked whether this statement is actually true for any affine group acting properly discontinuously. The answer turned out to be negative: Margulis [@Mar83; @Mar87] gave a counterexample in dimension 3. On the other hand, Fried and Goldman [@FG83] proved the Auslander conjecture in dimension 3 (the cases $n=1$ and $2$ are easy). Later, Abels, Margulis and Soifer proved it in dimension $n \leq 6$. See [@AbSur] for a survey of already known results.
In his PhD thesis and subsequent papers [@Dru92; @Dru93], Drumm elaborated on Margulis’s result by explicitly describing fundamental domains for the groups $\Gamma$ introduced by Margulis, which allowed him in particular to deduce the topology of the quotient $\mathbb{R}^3 / \Gamma$. On the other hand, Abels, Margulis and Soifer [@AMS02] constructed a family of counterexamples to Milnor’s conjecture in dimension $4n+3$, preserving a quadratic form of signature $(2n+2,2n+1)$. The purpose of this paper is to adapt Drumm’s construction to Abels-Margulis-Soifer groups: describe a fundamental domain and deduce the topology of the quotient space. Here is the main result:
Let $d$ be an odd positive integer. Then any generalized Schottky subgroup of $SO(d+1, d)$ with sufficiently contracting generators has a nonempty open set of affine deformations $\Gamma$ that act properly discontinuously on $\mathbb{R}^{d+1, d}$, with the quotient $\mathbb{R}^{d+1, d}/\Gamma$ homeomorphic to a solid $(2d+1)$-dimensional handlebody.
To do this, we use mainly two sources of inspiration. The first one is of course [@AMS02], the original work of Abels, Margulis and Soifer. The second one is an article by Charette and Goldman [@CG00] presenting Drumm’s results.
Plan of the paper {#sec:plan}
-----------------
We start, in section \[sec:basic\], by giving some elementary geometrical properties of a space equipped with a form of signature $(d+1, d)$ where $d$ is odd. We describe, in subsection \[sec:MTIS’es\], its maximal totally isotropic subspaces; in subsection \[sec:frames\], its pseudohyperbolic maps (roughly maps whose space of fixed points has the smallest possible dimension); in subsection \[sec:orientation\], an orientation trick (taken from [@AMS02]) that allows to extend any two transversal maximal totally isotropic subspaces into half-$d+1$-dimensional spaces that still have zero intersection. Finally, in subsection \[sec:metric\], we introduce metrics on various spaces (in particular projective spaces) we need to work with, and we define the strength of contraction of a pseudohyperbolic map.
In the next two sections, we consider subgroups of $SO(d+1,d)$ generated by pseudohyperbolic maps. In section \[sec:group\], we study their action on $\mathbb{P}(\Lambda^d \mathbb{R}^{d+1,d})$. We show that, provided the generators are sufficiently contracting, such a group is free and every element is pseudohyperbolic. We also control the geometry and strength of contraction of all cyclically reduced words on the generators. This result is very similar to Lemma 5.24 from [@AMS02], and we follow closely its proof. (For a more concise proof of a similar result, see also section 6 of [@Ben96].)
In section \[sec:tennis\_ball\], we study the action of these subgroups directly on $\mathbb{P}(\mathbb{R}^{d+1,d})$. We show that, supposing again that the generators are sufficiently contracting, this action is similar to the action of a Schottky group (which shows again that such a group is free). The way we construct the fundamental domain was partly inspired by Drumm’s ideas, but his “crooked planes” do not directly generalize to higher dimensions. Instead, we have used “angular” neighborhoods of some half-spaces (namely of the “positive wings” defined in section \[sec:orientation\]).
Finally, in section \[sec:affine\], we study affine groups $\Gamma$ whose linear parts satisfy the conditions of the previous two sections. We prove the Main Theorem (after stating it more precisely: see Theorem \[main\_theorem\]). Here we closely follow section 4 of [@CG00]. First, we describe a set $\mathcal{H}^0$ as the complement to $2n$ “sources” and “sinks” corresponding to the $n$ generators of $\Gamma$. We show (Proposition \[fundamental\_region\]) that under some conditions, $\mathcal{H}^0$ is a fundamental domain for $\Gamma$. Indeed, we see immediately that its images under elements of the group “fit together nicely”. To prove that they cover the whole space, by contradiction, we turn our attention to a hypothetical point not covered by any “tile”. We include it in a nested sequence of domains, then show (by methods adapted from [@CG00]) that these domains must, in a sense, run away to infinity.
Conventions, definitions and basic properties {#sec:basic}
=============================================
Let $p$ and $q$ be two positive integers. We write $\mathbb{R}^{p, q}$ as shorthand for the space $\mathbb{R}^{p+q}$ equipped with a quadratic form $Q$ of signature $(p, q)$. The group of automorphisms of $\mathbb{R}^{p, q}$ (that is, automorphisms of $\mathbb{R}^{p+q}$ that preserve the quadratic form) is $O(p, q)$. This group has four connected components; we call $SO^+(p,q)$ the connected component of the identity.
We equip $\mathbb{R}^{p, q}$ with some additional structure. We choose a maximal positive definite subspace $S$ of $\mathbb{R}^{p, q}$, and we set $T = S^\perp$ the corresponding maximal negative definite subspace. We may then define orthogonal projections $\pi_S: \mathbb{R}^{p, q} \to S$ and $\pi_T: \mathbb{R}^{p, q} \to T$, and positive definite forms $N_S := {{\left. Q \right|}_{S}}$ and $N_T := -{{\left. Q \right|}_{T}}$, so that $$\label{eq:form_decomposition}
\forall x \in \mathbb{R}^{p,q},\quad Q(x) = N_S(\pi_S(x)) - N_T(\pi_T(x)).$$
Maximal totally isotropic subspaces {#sec:MTIS'es}
-----------------------------------
From now on, the acronym MTIS stands for a maximal totally isotropic subspace. If $V$ is a MTIS of $\mathbb{R}^{p, q}$, then (supposing that $p \geq q$) we have $\dim V = q$, $V \subset V^\perp$ and $\dim V^\perp = p$. We write $\mathscr{L}$ the set of all MTIS’es.
A very useful tool for the study of MTIS’es is the following bijection between $\mathscr{L}$ and the space $O(T, S)$ of orthogonal linear maps from $T$ to $S$ (seen as Euclidean spaces via the forms $N_S$ and $N_T$): $$\label{eq:MTIS_bijection}
\xymatrix@R=3pt{
\mathscr{L} \ar@{<->}[r]^{\sim} & O(T, S) \\
V \ar@{|->}[r] & f_V := \pi_S \
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'LOFT-P is a mission concept for a NASA Astrophysics Probe-Class ($<$\$1B) X-ray timing mission, based on the LOFT M-class concept originally proposed to ESA’s M3 and M4 calls. LOFT-P requires very large collecting area, high time resolution, good spectral resolution, broad-band spectral coverage (2–30 keV), highly flexible scheduling, and an ability to detect and respond promptly to time-critical targets of opportunity. It addresses science questions such as: What is the equation of state of ultra dense matter? What are the effects of strong gravity on matter spiraling into black holes? It would be optimized for sub-millisecond timing of bright Galactic X-ray sources including X-ray bursters, black hole binaries, and magnetars to study phenomena at the natural timescales of neutron star surfaces and black hole event horizons and to measure mass and spin of black holes. These measurements are synergistic to imaging and high-resolution spectroscopy instruments, addressing much smaller distance scales than are possible without very long baseline X-ray interferometry, and using complementary techniques to address the geometry and dynamics of emission regions. LOFT-P would have an effective area of $>$6 m$^2$, $>10\times$ that of the highly successful Rossi X-ray Timing Explorer (RXTE). A sky monitor (2–50 keV) acts as a trigger for pointed observations, providing high duty cycle, high time resolution monitoring of the X-ray sky with $\sim$20 times the sensitivity of the RXTE All-Sky Monitor, enabling multi-wavelength and multi-messenger studies. A probe-class mission concept would employ lightweight collimator technology and large-area solid-state detectors, segmented into pixels or strips, technologies which have been recently greatly advanced during the ESA M3 Phase A study of LOFT. Given the large community interested in LOFT ($>$800 supporters[^1], the scientific productivity of this mission is expected to be very high, similar to or greater than RXTE ($\sim 2000$ refereed publications). We describe the results of a study, recently completed by the MSFC Advanced Concepts Office, that demonstrates that such a mission is feasible within a NASA probe-class mission budget.'
author:
- 'Colleen A. Wilson-Hodge'
- 'Paul S. Ray'
- Deepto Chakrabarty
- Marco Feroci
- Laura Alvarez
- Michael Baysinger
- Chris Becker
- Enrico Bozzo
- Soren Brandt
- Billy Carson
- Jack Chapman
- Alexandra Dominguez
- Leo Fabisinski
- Bert Gangl
- Jay Garcia
- Christopher Griffith
- Margarita Hernanz
- Robert Hickman
- Randall Hopkins
- Michelle Hui
- Luster Ingram
- Peter Jenke
- Seppo Korpela
- Tom Maccarone
- Malgorzata Michalska
- Martin Pohl
- Andrea Santangelo
- Stephane Schanne
- Andrew Schnell
- Luigi Stella
- Michiel van der Klis
- Anna Watts
- Berend Winter
- Silvia Zane
- 'on behalf of the LOFT Consortium, the US-LOFT SWG, and the LOFT-P collaboration'
bibliography:
- 'report.bib'
title: 'Large Observatory for x-ray Timing (LOFT-P): A Probe-class Mission Concept Study'
---
INTRODUCTION {#sec:intro}
============
*LOFT-P* is a probe-class X-ray observatory designed to work in the 2–30 keV band with huge collecting area ($>10 \times$ NASA’s highly successful *Rossi X-ray Timing Explorer (RXTE)*) and good spectral resolution ($<$260 eV). It is optimized for the study of matter in the most extreme conditions found in the Universe and addresses several key science areas including:
- [Probing the behavior of matter spiraling into black holes (BHs) to explore the effects of strong gravity and measure the masses and spins of BHs.]{}
- [Using multiple neutron stars (NSs) to measure the ultradense matter equation of state over an extended range.]{}
- [Continuously surveying the dynamic X-ray sky with a large duty cycle and high time-resolution to characterize the behavior of X-ray sources over a vast range of time scales.]{}
- [Enabling multiwavelength and multi-messenger study of the dynamic sky through cross-correlation with high-cadence time-domain surveys in the optical and radio (LSST, LOFAR, SKA pathfinders) and with gravitational wave interferometers like LIGO and VIRGO.]{}
Detailed simulations[@yellowbook; @Watts2016] have demonstrated that an order of magnitude larger collecting area than *RXTE* (i.e., $>$6 m$^2$) is required to meet these BH and NS objectives, and a previous engineering study[@Ray2011] has shown that such an instrument is too large for the Explorer (EX) class and requires a probe-class mission.
The *LOFT-P* mission concept, which has been under study in both the Europe and the US since 2010[@yellowbook; @Feroci2012; @Feroci2014; @Feroci2016], comprises two instruments. The Large Area Detector (LAD) consists of collimated arrays of silicon drift detectors (SDDs) with a 1-degree field of view and a baseline peak effective area of 10 m$^2$ at 8 keV (Fig. \[fig:loft-p\_effarea\]), optimized for submillisecond timing and spectroscopy of NSs and BHs. The sensitive Wide Field Monitor (WFM) is a 2–50 keV coded-mask imager (also using SDDs) that acts as a trigger for pointed LAD observations of X-ray transients and also provides nearly continuous imaging of the X-ray sky with a large instantaneous field of view.
\[ht\]
![ \[fig:loft-p\_effarea\] Effective area as a function of area shown for the LOFT-P LAD baseline concept. Several existing and planned missions are shown for comparison.](LOFT-P_effarea.pdf){height="5cm"}
We first presented LOFT-P as a concept, based on the ESA M3 studies of LOFT[@yellowbook], at the American Astronomical Society (AAS) High Energy Astrophysics Division (HEAD): High-Energy Large- and Medium-class Space Missions in the 2020s meeting in 2015[^2], where it was well received. It was later presented as an example probe-class mission in the NASA Physics of the Cosmos Program Analysis Group (PhysPAG) final presentation to the head of NASA’s Astrophysics Division, to demonstrate the strong community support for creation of a “probe class,” for NASA astrophysics missions that cost between \$500M and \$1B. We submitted a white paper[@Wilson2016] describing LOFT-P science and this simple assessment to NASA’s PhysPAG’s Call for White Papers: Probe-class Mission Concepts, for which 14 white papers were received[^3]. At the April 2016 HEAD meeting, NASA’s PhysPAG endorsed the option that NASA issue a ROSES solicitation for Astrophysics Probe mission concept study proposals for input to the 2020 Astrophysics Decadal Survey[^4]. In May 2016 the Advanced Concepts Office at NASA MSFC performed a preliminary study (Fig. \[fig:loft-p\_pic\]) to verify the cost of *LOFT-P* as a US-led probe-class mission and to investigate a US-led design on a US launcher, in preparation.
\[ht\]
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ \[fig:loft-p\_pic\] LOFT-P spacecraft configuration with 122 LAD modules and 10 WFM cameras (left). This configuration fits within the volume of a Falcon 9 fairing (right). A Falcon Heavy is required to deliver LOFT-P to a 0 $\deg$ orbit from Cape Canaveral. An astronaut is added to both figures to give a sense of scale.](loft-p_pic_v2.pdf "fig:"){height="7cm"} ![ \[fig:loft-p\_pic\] LOFT-P spacecraft configuration with 122 LAD modules and 10 WFM cameras (left). This configuration fits within the volume of a Falcon 9 fairing (right). A Falcon Heavy is required to deliver LOFT-P to a 0 $\deg$ orbit from Cape Canaveral. An astronaut is added to both figures to give a sense of scale.](LOFT-P_in_Falcon9_v2.png "fig:"){height="7cm"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
SCIENCE GOALS AND MISSION REQUIREMENTS
======================================
Science Goals
-------------
[**Strong gravity and black hole spin.**]{} Unlike the small perturbations of Newtonian gravity found in the weak-field regime of general relativity (GR), strong-field gravity results in gross deviations from Newtonian physics and qualitatively new behavior for motion near compact objects, including the existence of event horizons and an innermost stable circular orbit (ISCO). *LOFT-P* observations will probe strong gravitational fields of NSs and BHs in a way that is complementary to gravitational wave interferometers like LIGO and VIRGO. Accretion flows and the X-ray photons they emit are
|
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}
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Cuprate high-temperature superconductors (HTS) are layered anisotropic materials. Therefore the electrodynamic problem of the magnetic field penetration depth in HTS in the low-field limit is characterized by two length parameters, namely, $\lambda_{ab}$ controlled by screening currents running in the CuO$_2$ planes (in-plane penetration depth) and $\lambda_c$ due to currents running in the direction perpendicular to these planes (out-of-plane or $c$-axis penetration depth). The temperature dependence of the penetration depth in HTS is largely determined by the superconductivity mechanism. It is known (see, e.g., Ref. [@Tru1] and references therein) that $\Delta\lambda_{ab}(T)\propto T$ in the range $T<T_c/3$ in high-quality HTS samples at the optimal level of doping, and this observation has found the most simple interpretation in the $d$-wave model of the high-frequency response in HTS [@Scal]. Measurements of $\lambda_c(T)$ are quoted less frequently than those of $\lambda_{ab}(T)$. Most of such data published by far were derived from microwave measurements of the surface impedance of HTS crystals [@Shib1; @Mao; @Kit1; @Bon1; @Jac1; @Shib2; @Srik; @Kit2; @Hos]. There is no consensus in literature about $\Delta\lambda_c(T)$ at low temperatures. Even in reports on low-temperature properties of high-quality YBCO crystals, which are the most studied objects, one can find both linear, $\Delta\lambda_c(T)\propto T$ [@Mao; @Srik], and quadratic dependences [@Hos] in the range $T<T_c/3$. In BSCCO materials, the shape of $\Delta\lambda_c(T)$ depends on the level of oxygen doping: in samples with maximal $T_c\simeq 90$ K $\Delta\lambda_c(T)\propto T$ at low temperatures [@Jac1; @Shib2]; at higher oxygen contents (overdoped samples) $T_c$ is lower and the linear function $\Delta\lambda_c(T)$ transforms to a quadratic one [@Shib2]. The common feature of all microwave experiments is that the change in the ratio $\Delta\lambda_c(T)/\lambda_c(0)$ is smaller than in $\Delta\lambda_{ab}(T)/\lambda_{ab}(0)$ because in all HTS $\lambda_c(0)\gg\lambda_{ab}(0)$. The length $\lambda_c(0)$ is especially large in BSCCO crystals, $\lambda_c(0)>10$ $\mu$m and, according to some estimates, it ranges up to $\sim 500$ $\mu$m. The large spread of $\lambda_c(0)$ is caused by two factors, namely, the poor accuracy of the techniques used in determination of $\lambda_c(0)$ and effects of local and extended defects in tested samples, whose range is of order of 1 mm and comparable to both $\lambda_c$ and total sample dimensions.
Recently we suggested [@Nic] a new technique for determination of $\lambda_c(0)$ based on the measurements of the surface barrier field $H_J(T)\propto 1/\lambda_c(T)$ at which Josephson vortices penetrate into the sample. The field $H_J$ corresponds to the onset of microwave absorption in the locked state of BSCCO single crystals. This paper suggests an alternative technique based on comparison between microwave measurements of BSCCO crystals aligned differently with respect to ac magnetic field and a numerical solution of the electrodynamic problem of the magnetic field distribution in an anisotropic plate at an arbitrary temperature. Moreover, since $\lambda_c(0)$ in BSCCO single crystals is relatively large, we managed to determine $\lambda_c(T)$ from the temperature dependences of ac-susceptibility and compare these measurements to results of microwave experiments.
Single crystals of BSCCO were grown by the floating-zone method [@Tam] and shaped as rectangular platelets. This paper presents measurements of two BSCCO samples with various levels of oxygen doping. The first sample (\#1), characterized by a higher critical temperature, $T_c\approx 90$ K (optimally doped), has dimensions $a\times b\times c\simeq 1.5\times 1.5\times 0.1$ mm$^3$ ($a\approx b$). The second (\#2, $a\times b\times c\simeq 0.8\times 1.8\times 0.03$ mm$^3$) is slightly overdoped ($T_c\approx 84$ K).
When measuring the ac-susceptibility $\chi=\chi'-i\chi''$, we placed a sample inside one of two identical induction coils. The coils were connected to one another, and the out-of-phase and in-phase components of the imbalance signal were measured at a frequency of $10^5$ Hz. These components are proportional to the real and imaginary parts of the sample magnetic moment $M$, respectively: $M=\chi vH_0$, where $v$ is the sample volume and $H_0$ is the ac magnetic field amplitude, which was within 0.1 Oe in our experiments.
Figure 1 shows temperature dependences $\chi'(T)/|\chi'(0)|$ in
sample \#1 for three different sample alignments with respect to the ac magnetic field: the transverse (T) orientation, ${\bf H}_{\omega}\parallel {\bf c}$, (the inset on the left of Fig. 1), when the screening current flows in the $ab$-plane (full circles); in the longitudinal (L) orientation, ${\bf H}_{\omega}\perp {\bf c}$, (the inset on the right of Fig. 1, ${\bf H}_{\omega}$ is parallel to the $b$-edge of the crystal), when currents running in the directions of both CuO$_2$ planes and the $c$-axis are present (up triangles); in the L-orientation, ${\bf H}_{\omega}\perp {\bf c}$, whose difference from the previous configuration is that the sample is turned around the $c$-axis through $90^\circ$ (down triangles). Fig. 1 clearly shows that at $T<T_c$ $\chi'_{ab}(T)$ is notably smaller in the T-orientation than $\chi'_{ab+c}(T)$ in the L-orientation (the subscripts of $\chi'$ denote the direction of the screening current). The coincidence of $\chi'_{ab+c}(T)$ curves at ${\bf H}_{\omega}\perp {\bf c}$ and the small width of the superconducting transition at ${\bf H}_{\omega}\parallel {\bf c}$ ($\Delta T_c< 1$ K) indicate that the quality of the tested sample \#1 is fairly high. This is supported by precision measurements of surface impedance $Z_s(T)=R_s(T)+iX_s(T)$ of sample \#1 at frequency $f=9.4$ GHz in the T-orientation, which are plotted in Fig. 2. The measurement technique was described in detail elsewhere [@Tru1]. It applies to both surface impedance components $R_s(T)$ and $X_s(T)$:
$$R_s=\Gamma_s\,\Delta(1/Q),\qquad X_s=-2\,\Gamma_s\,\delta f/f,\label{RX}$$
Here $\Gamma_s=\omega \mu_0\int_V H^2_{\omega}dV /[\int_S
H_s^2 dS]$ is the sample geometrical factor ($\omega=2\pi f$, $\mu_0=4\pi\cdot10^{-7}$ H/m, $V$ is the volume of the cavity, $H_{\omega}$ is the magnetic field generated in the cavity, $S$ is the total sample surface area, and $H_s$ is the tangential component of the microwave magnetic field on the sample surface); $\Delta (1/Q)$ is the difference between the values 1/Q of the cavity with the sample inside and empty cavity; $\delta f$ is the frequency shift relative to that which would be measured for a sample with perfect screening, i.e., no penetration of the microwave fields. In the experiment we measure the difference $\Delta f(T)$ between resonant frequency shifts with temperature of the loaded and empty cavity, which is equal to $\Delta f(T)=\delta f(T)+f_0$, where $f_0$ is a constant [@exp]. The constant $f_0$ includes both the perfect-conductor shift and the uncontrolled contribution caused by opening and closing the cavity. In HTS single crystals, the constant $f_0$ can be directly derived from measurements of the surface impedance in the normal state; in particular, in the T-orientation $f_0$ can be derived from the condition that the real and imaginary parts of the impedance should be equal above $T_c$ (normal skin-effect). In Fig. 2 $R_s(T)=X_s(T)$ at $T\ge T_c$, and its temperature dependence is adequately described by the expression $2R_s^2(T)/\omega
\mu_0=\rho(T)=\rho_0+bT$ with $\rho_0\approx 13$ $\mu\Omega\cdot$cm and $b\approx 0.3
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We computed the power spectrum of weak cosmic shear in models with non-Gaussian primordial density fluctuations. Cosmological initial conditions deviating from Gaussianity have recently attracted much attention in the literature, especially with respect to their effect on the formation of non-linear structures and because of the bounds that they can put on the inflationary epoch. The fully non-linear matter power spectrum was evaluated with the use of the physically motivated, semi-analytic halo model, where the mass function and linear halo bias were suitably corrected for non-Gaussian cosmologies. In agreement with previous work, we found that a level of non-Gaussianity compatible with CMB bounds and with positive skewness produces an increase in power of the order of a few percent at intermediate scales. We then used the matter power spectrum, together with observationally motivated background source redshift distributions in order to compute the cosmological weak lensing power spectrum. We found that the degree of deviation from the power spectrum of the reference Gaussian model is small compared to the statistical error expected from even future weak lensing surveys. However, summing the signal over a large range of multipoles can beat down the noise, bringing to a significant detection of non-Gaussianity at the level of $|f_\mathrm{NL}| \simeq $ few tens, when all other cosmological parameters are held fixed. Finally, we have shown that the constraints on the level of non-Gaussianity can be improved by $\sim 20\%$ with the use of weak lensing tomography.'
author:
- |
C. Fedeli$^{1,2,3}$ and L.Moscardini$^{1,2,3}$\
$^1$ Dipartimento di Astronomia, Università di Bologna, Via Ranzani 1, I-40127 Bologna, Italy (cosimo.fedeli@unibo.it)\
$^2$ INAF-Osservatorio Astronomico di Bologna, Via Ranzani 1, I-40127 Bologna, Italy\
$^3$ INFN, Sezione di Bologna, Viale Berti Pichat 6/2, I-40127 Bologna, Italy\
bibliography:
- './master.bib'
title: 'Cosmic shear statistics in cosmologies with non-Gaussian initial conditions'
---
Introduction {#sct:introduction}
============
One of the major success of the inflationary scenario for the early Universe is that it explains the formation of the seed fluctuations in the dark matter density field that, due to gravitational instability, eventually formed non-linear structures such as galaxies, galaxy clusters and voids (@GU81.1 [@BR84.1; @KO87.1]). In the simplest model of inflation, the early accelerated expansion phase of the Universe was driven by a single, minimally coupled scalar field. In this case, density fluctuations are predicted to follow an almost Gaussian probability distribution. Significant deviations from Gaussianity are however predicted in many of the more elaborated models of inflation that have been developed up to date (@MA86.1 [@AL87.1]).
The most recent analysis of the Cosmic Microwave Background radiation power spectrum of temperature fluctuations (CMB henceforth, @DU09.1 [@KO09.1], see also @SM09.1) is consistent with Gaussian primordial density perturbations, although a significant level of non-Gaussianity is still allowed. Inflationary models exist predicting a scale dependent behavior for the non-Gaussian amplitude [@LO08.1], implying that the amount of deviation from Gaussianity might be different between the large scales probed by the CMB and the small scales probed by galaxies and galaxy clusters.
In light of this, it is important to understand the effect on structure formation in the Universe of non-Gaussianity levels compatible with CMB bounds and/or with a significant scale evolution. This kind of problem has recently attracted much attention in the literature, with efforts directed towards the abundance of nonlinear structures (@MA00.2; @VE00.1; @MA04.1 [@KA07.1]; @GR07.1 [@GR09.1; @MA09.2]), halo biasing (@DA08.1 [@MC08.1]; @FE09.1), galaxy bispectrum [@SE07.2; @JE09.1], mass density distribution (@GR08.2) and topology [@MA03.2; @HI08.2], integrated Sachs-Wolfe effect [@AF08.1; @CA08.1], Ly-$\alpha$ flux from low-density intergalactic medium [@VI09.1], $21$ cm fluctuations [@CO06.2; @PI07.1] and reionization (@CR09.1).
One particular observable quantity that should be affected in a non-trivial way by non-Gaussianity is the fully non-linear power spectrum of the large-scale dark matter distribution. Studies of the effect of non-Gaussian initial conditions on this observable have been recently put forward with numerical $n$-body simulations [@GR08.2], with renormalized perturbation theory [@TA08.1] and by using both [@GI09.1]. Although differences exist between different works, they all agree in setting the effect of non-Gaussianity to a few percent at most on mildly non-linear scales.
Observationally, the matter power spectrum on scales smaller than CMB scales is usually measured by looking at the distribution of pairs of galaxies, that are known to be biased tracers of the underlying matter density field. More recently however the gravitational deflection of light has also been shown to be usable in order to map the large scale distribution of dark matter, having the additional advantage of being insensitive to the problems related with the bias of luminous tracers. The tradeoff for this advantage is that cosmic shear can measure only a projected version of the matter power spectrum, that depends on the assumed redshift distribution of the background source galaxies.
In this paper we focused on this approach, namely we aimed at understanding what kind of constraints can be put on deviations from primordial Gaussianity using the weak lensing power spectrum. As an example, attention was devoted to planned wide field optical surveys, such as the ESA Cosmic Vision project EUCLID [@LA09.1]. The rest of this work is organized as follows. In Section \[sct:ng\] we describe the non-Gaussian cosmologies considered in this work and how deviations from Gaussianity alter the mass function and the halo bias, both required for computing the non-linear power spectrum. In Section \[sct:modeling\] we discuss in detail the way in which we modeled the fully non-linear matter power spectrum, with particular attention to the assumptions, advantages and drawbacks underlying the method. In Section \[sct:results\] we describe our results concerning the weak lensing power spectrum, and in Section \[sct:discussion\] we draw our conclusions.
For the relevant calculations we adopted as a reference cosmology the one resulting from the best fit WMAP-$5$ parameters together with type-Ia supernovae and the observed Baryon Acoustic Oscillation (BAO). The present values of the density parameters for matter, dark energy and baryons are $\Omega_{\mathrm{m},0} = 0.279$, $\Omega_{\Lambda,0} = 0.721$ and $\Omega_{\mathrm{b},0} = 0.046$, respectively. The Hubble constant reads $H_0 = h$100 km s$^{-1}$ Mpc$^{-1}$, with $h = 0.701$. The slope of the primordial power spectrum of density fluctuations is $n = 0.96$, while the normalization is set by the *rms* of the density field on a comoving scale of $8 h^{-1}$ Mpc, $\sigma_8 = 0.817$. To construct the linear power spectrum we used the matter transfer function of @BA86.1, modified according to the shape factor of @SU95.1. The more sophisticated prescription of @EI98.1 is almost coincident with the former, except for the presence of the BAO, that is not of interest here.
Non-Gaussian cosmologies {#sct:ng}
========================
Simple generalizations of the most standard model of inflation give rise to seed primordial density fluctuations that follow a non-Gaussian probability distribution. A particularly simple way to parametrize the deviation of this distribution from a Gaussian consists of writing the Bardeen’s gauge invariant potential $\Phi$ as the sum of a linear Gaussian term and a quadratic correction [@SA90.1; @GA94.1; @VE00.1; @KO01.1],
$$\label{eqn:ng}
\Phi = \Phi_\mathrm{G} + f_\mathrm{NL}*\left( \Phi_\mathrm{G}^2 - \langle \Phi_\mathrm{G}^2 \rangle \right).$$
In Eq. (\[eqn:ng\]) the symbol $*$ denotes convolution between functions, and reduces to simple multiplication only in the particular case in which $f_\mathrm{NL}$ is a constant, while in general it is a function of the scale. Note that on scales smaller than the Hubble radius $\Phi$ equals minus the Newtonian peculiar gravitational potential.
In the following, we adopted the Large Scale Structure convention (as opposed to the CMB convention, see @AF08.1 [@PI09.1; @CA08.1]; @GR09.1) for defining the fundamental parameter $f_\mathrm{NL}$. According to this, the primordial value of $\Phi$ has to be linearly extrapolated at $z = 0$, and as a consequence the constraints given on $f_\mathrm{NL}$ by the CMB have to be raised of $\sim 30\%$ to comply with this paper
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We exploit the intrinsic difference between disordered and crystalline solids to create systems with unusual and exquisitely tuned mechanical properties. To demonstrate the power of this approach, we design materials that are either virtually incompressible or completely auxetic. Disordered networks can be efficiently driven to these extreme limits by removing a very small fraction of bonds via a selected-bond removal procedure that is both simple and experimentally relevant. The procedure relies on the nearly complete absence of any correlation between the contributions of an individual bond to different elastic moduli. A new principle unique to disordered solids underlies this lack of correlation: independence of bond-level response.'
author:
- 'Carl P. Goodrich'
- 'Andrea J. Liu'
- 'Sidney R. Nagel'
title: 'Tuning by pruning: exploiting disorder for global response and the principle of bond-level independence'
---
The properties of amorphous solids are essentially and qualitatively different from those of simple crystals [@Goodrich:2014fl]. In a crystal, identical unit cells are interminably and symmetrically repeated, ensuring that all cells make identical contributions to the solid’s global response to an external perturbation [@Ashcroft:1976ud; @kittel2004introduction]. Unless a crystal’s unit cell is very complicated, all particles or inter-particle bonds contribute nearly equally to any global quantity, so that each bond plays a similar role in determining the physical properties of the solid. For example, removing a bond in an ordered array or network decreases the overall elastic strength of the system, but in such a way that the resistance to shear and the resistance to compression drop in tandem [@Feng:1985vr] so that their ratio is nearly unaffected. Disordered materials are not similarly constrained. We will show that as a consequence, one can exploit disorder to achieve a unique, varied, textured and tunable global response.
A tunable global response is a corollary to a new principle that emerges for disordered matter: independence of bond-level response. This independence refers not only to the dearth of strong correlations between the response of different bonds, but also, and more importantly, to the response of any specific bond to different external perturbations. We will demonstrate this by constructing selected-bond-removal networks, where individual bonds, or springs, are successively removed to drive the overall system into different regimes of behavior, characterized by ratios of different mechanical responses. Starting from the same initial network, we can remove as few as 2% of the bonds to produce a network with a ratio of the shear to bulk modulus, $G/B$, that is either nearly zero (incompressible limit) or nearly infinite (maximally auxetic [@Greaves:2011ku]) merely by removing different sets of bonds. Moreover, by using different algorithms or starting with different configurations, we find that the region within which the bonds are removed can be confined to strips of controllable size, ranging from a few bond lengths to the size of the entire sample. This has the practical consequence that one can achieve precise spatial control in tuning properties of the material from region to region within the network–as is needed for creating origami [@Witten:2007cq; @Mahadevan:2005hr] or kirigami [@Castle:2014jg] materials.
We construct networks numerically by starting with a configuration of particles produced by a standard jamming algorithm [@OHern:2003vq; @Liu:2010jx]. We place $N$ soft repulsive particles at random in a box of linear size $L$ and minimize the total energy until there is force balance on each particle. We work in either two or three dimensions and start with a packing fraction, $\phi$, that is above the jamming density. After minimizing the energy of a configuration, we create a network by replacing each pair of interacting particles with an unstretched spring of unit stiffness between nodes at the particle centers [@Wyart:2005jna]. We characterize the network by the excess coordination number ${\Delta Z}\equiv Z - Z_\text{iso}$, where $Z$ is the average number of bonds at each node and $Z_\text{iso} \equiv 2d - 2d/N$ is the minimum for a system to maintain rigidity in $d$ dimensions [@Goodrich:2012ck].
For each network, we use linear response to calculate the contribution $B_i$ of each bond $i$ to the bulk modulus, $B=\sum_i B_i$ (see Appendix for details). The distribution of $B_i$ in three dimensions is shown in blue in Fig. \[fig:Ri\_distributions\_3d\], where data are averaged over 500 configurations, each with approximately 4000 nodes and an initial excess coordination number ${\Delta Z_\text{initial}}\approx 0.127$ (corresponding to a total number of bonds that is about 2% above the minimum needed for rigidity).
Similarly, we can start with the same initial network and calculate $G_i$, the contribution of each bond to the angle-averaged shear modulus, $G=\sum_i G_i$. (A finite system is not completely isotropic, so the shear modulus varies with direction [@DagoisBohy:2012dh]; we calculate the angle-averaged shear modulus, which approaches the isotropic shear modulus in the infinite system size limit [@Goodrich:2014iu].) The resulting distribution for $G_i$ is shown in purple in Fig. \[fig:Ri\_distributions\_3d\]. Note that the distributions of the bond contributions to $B$ and $G$ are continuous, very broad, and non-zero in the limit $B_i,G_i \rightarrow 0$. That is, some bonds have nearly zero contribution to the bulk or shear modulus while others contribute disproportionately. For both $B$ and $G$, the distribution decays as a power law at low values of $B_i$ or $G_i$. These power laws are terminated above ${\left< B_i \right>}$ and ${\left< G_i \right>}$ by approximately exponential cut-offs. In comparison, the distributions for a perfect crystal would be composed of discrete delta functions.
![\[fig:Ri\_distributions\_3d\]Bond-level response. Distribution on a log-log scale (inset: log-linear scale) of the contribution of each bond to the macroscopic bulk and shear moduli, $B_i$ and $G_i$, for $3d$ networks with ${\Delta Z_\text{initial}}\approx 0.127$. Here $i$ indexes bonds. At low $B_i$ or $G_i$, the distributions follow power-laws with exponents $-0.51$ and $-0.38$, respectively. At high values, the distributions decay over a range that is broad compared to their means, ${\left< B_i \right>}$ and ${\left< G_i \right>}$.](fig1.pdf){width="\linewidth"}
![\[fig:bond\_level\_independence\]Independence of bond-level response. ([**A**]{}) Joint probability distribution of $B_i$ and $G_i$ for $3d$ networks with ${\Delta Z_\text{initial}}\approx 0.127$. There is little apparent correlation between the response to compression ($B_i$) and to shear ($G_i$) for a given bond $i$. ([**B**]{}) The value of $G$ when bonds with the largest (purple squares) and smallest (purple circles) $B_i$ are removed is nearly indistinguishable from $G$ when bonds are removed at random (purple crosses). Similarly, $B$ is very similar whether bonds with the largest $G_i$ (blue triangles) are removed or bonds are removed at random (blue pluses).](fig2_combined2.pdf){width="0.9\linewidth"}
![\[fig:Global\_response\]Tuning global response in three dimensions. The ratio of shear to bulk modulus, $G/B$, for four pruning algorithms. Error bars (included) are smaller than the symbols. Lines are fits to the data over the indicated range and have slopes, from top to bottom, of -7.96, -0.01, 1.01, and 1.82. Starting with the same initial conditions, we can tune global response by 16 orders of magnitude by pruning of order 2% of the bonds.](fig3.pdf){width="0.9\linewidth"}
![\[fig:Global\_response\_2d\]Tuning global response in two dimensions. The ratio of shear to bulk modulus, $G/B$, for four pruning algorithms. Error bars (included) are smaller than the symbols. Lines are fits to the data over the indicated range and have slopes, from top to bottom, of -5.36, -0.26, 1.27, and 3.05. Starting with the same initial conditions, we can tune global response by 17 orders of magnitude by pruning of order 1% of the bonds.](fig3_2d.pdf){width="0.9\linewidth"}
We next ask if there is a correlation between how an individual bond responds to shear and how it responds to compression. Do bonds with a large contribution to the bulk modulus also have a proportionately large contribution to the shear modulus? Fig. \[fig:bond\_level\_independence\]a shows the joint probability distribution $P(B_i,G_i)$. There is a nearly vanishing (but not identically zero) correlation between how individual bonds respond to shear and how they respond to compression. This is qualitatively different from what one would find for a simple crystal. Thus,
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{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We show that a star orbiting close enough to an adiabatically grown supermassive black hole (SMBH) can capture weakly interacting massive particles (WIMPs) at an extremely high rate. The stellar luminosity due to annihilation of captured WIMPs in the stellar core may be comparable to or even exceed the luminosity of the star due to thermonuclear burning. The model thus predicts the existence of unusual stars, essentially WIMP burners, in the vicinity of a SMBH. We find that the most efficient WIMP burners are stars with degenerate electron cores, e.g. white dwarfs (WDs); such WDs may have a very high surface temperature. If found, such stars would provide evidence for the existence of particle dark matter and can possibly be used to establish its density profile. On the other hand, the lack of such unusual stars may provide constraints on the WIMP density near the SMBH, as well as the WIMP-nucleus scattering and pair annihilation cross-sections.'
author:
- 'Igor V. Moskalenko'
- 'Lawrence L. Wai'
title: Dark matter burners
---
Introduction
============
The nature of the non-baryonic dark matter, which dominates the visible matter by about 4:1, is perhaps the most interesting experimental challenge for contemporary particle astrophysics. A hint for a solution has been found in particle physics where the WIMPs arise naturally in supersymmetric extensions of the Standard Model [e.g., @haber-kane], among other possibilities. The WIMP is typically defined as a stable, electrically neutral, massive particle. Assuming that non-baryonic dark matter is dominated by WIMPs, the pair annihilation cross-section is related to the observed relic density [@jkg96; @bergstrom00]. A pair of WIMPs can annihilate producing ordinary particles and [$\gamma$-rays]{}.
WIMPs are expected to form high density clumps according to N-body simulations of test particles with only gravitational interactions [@Navarro97; @Moore99]. The highest density “free space” dark matter regions occur for dark matter particles captured within the gravitational potential of adiabatically grown SMBHs [@gs99; @gp04; @bm05]. Higher dark matter densities are possible for dark matter particles captured inside of stars or planets. Any star close enough to a SMBH can capture a large number of WIMPs during a short period of time. Annihilation of captured WIMPs may lead to considerable energy release in stellar cores thus affecting the evolution and appearance of such stars.
Such an idea has been first proposed by @salati89 and further developed by @bouquet89 who applied it to main-sequence stars. The model led to the conclusion of suppression of stellar core convection, thus predicting a concentration of stars in the Galactic Center masquerading as cold red giants.
An order-of-magnitude estimate of the WIMP capture rates for stars of various masses and evolution stages [@MW06] lead us to the conclusion that WDs, fully burned stars without their own energy supply, are the most promising candidates to look for. In this paper we calculate the WIMP capture by WDs located in a high density dark matter region, and discuss their observational features. We use current limits on WIMP-nucleus interaction and WIMP annihilation cross sections, as well as recent estimates of WIMP energy density near an adiabatically grown SMBH.
{width="98.00000%"}
WIMP accumulation in stars
==========================
In a steady state the WIMP capture rate $C$ is balanced by the annihilation rate [@gs87] $$C=A N_\chi^2,
\label{balance}$$where $$A=\frac{{\langle\sigma_a v\rangle}}{\pi^{3/2} r_\chi^3},
\label{A}$$${\langle\sigma_a v\rangle}$ is the velocity averaged WIMP pair annihilation cross-section, the effective radius $$r_\chi=c \left(\frac{3T_c}{2\pi G\rho_c m_\chi}\right)^{1/2}$$is determined by matching the star core temperature $T_c$ with the gravitational potential energy (assuming thermal equilibrium), $c$ is the speed of light, $G$ is the gravitational constant, $\rho_c$ is the star core density, and $m_\chi$ is the WIMP mass. The total number of WIMPs captured by a star is $$N_\chi=C \tau_{eq} \tanh(\tau_*/\tau_{eq}),
\label{Nchi}$$where $\tau_*$ is the star’s age, and the equilibrium time scale is given by $$\tau_{eq}=(CA)^{-1/2}.
\label{taueq}$$ The number density distribution of WIMPs can be estimated as [@ps85; @gs87; @bottino02]: $$n_\chi(r)=n_\chi^c \exp(-r^2/r_\chi^2),
\label{chidensity}$$where $n_\chi^c=N_\chi/V_{\rm eff}$ is the central WIMP number density. In thermal equilibrium, the effective radius $r_\chi=r_T$ is determined by the core temperature $T_c$ and density $\rho_c$ $$r_T=c \left(\frac{3T_c}{2\pi G\rho_c m_\chi}\right)^{1/2},\nonumber
\label{rT}$$where $c$ is the speed of light, $G$ is the gravitational constant, and $m_\chi$ is the WIMP mass. Limits from direct detection of dark matter on the WIMP-nucleon cross-section imply that only a fraction of the WIMPs crossing the star will scatter and be captured. The capture rate for a Maxwellian WIMP velocity distribution (in the observer’s frame) by a star moving with an arbitrary velocity $v_*$ relative to the observer is given by [@gould87]: $$C=4\pi \int_0^{R_*} dr\, r^2\, \frac{dC(r)}{dV},
\label{gould2.27}$$where $$\begin{aligned}
\frac{dC(r)}{dV}&=&
\left(
\frac{6}{\pi}
\right)^{1/2}
\sigma_0 A_n^4 \frac{\rho_*}{M_n}\frac{\rho_\chi}{m_\chi}
\frac{v^2(r)}{\bar{v}^2} \frac{\bar{v}}{2\eta A^2}
\label{gould2.24}\\
&\times&
\left\{
\left(
A_+A_- -\frac12
\right)
\left[
\chi(-\eta,\eta) -\chi(A_-,A_+)
\right]
\right.\nonumber\\
&+&\left.
\frac12A_+e^{-A_-^2} -\frac12A_- e^{-A_+^2}-\eta e^{-\eta^2}
\right\},\nonumber\\
A^2&=& \frac{3v^2(r)\mu}{2\bar{v}^2\mu_-^2},\nonumber\\
A_\pm&=&A\pm\eta,\nonumber\\
\eta&=&\frac{3v_*^2}{2\bar{v}^2}, \nonumber\\
\chi(a,b)&=&\int_a^b dy\, e^{-y^2}=
\frac{\sqrt\pi}{2}[{\rm erf}(b)-{\rm erf}(a)],
\nonumber\end{aligned}$$$\rho_\chi$ is the ambient WIMP energy density, $A_n$ is the atomic number of the star’s nuclei, $M_n$ is the nucleus mass, $\bar{v}$ is the WIMP velocity dispersion, and $\mu=m_\chi/M_n$, $\mu_-=(\mu-1)/2$. The escape velocity at a given radius $r$ *inside* of a star is given by $$v(r)=
\left[
2G\int_{V_*} dV\, \frac{\rho_*(r)}{r}
\right]^{1/2}
=
\left[
\frac{GM_*}{R_*}
\left(
3 -\frac{r^2}{R_*^2}\right)
\right]^{1/2},
\label{vesc}$$where we assumed the same mass density $\rho_*=M_*/V_*$ and the same chemical composition over the entire scattering volume $V_*$. This is a reasonable assumption for a degenerate electron core. Near a SMBH, where orbital motion around a single mass dominates, the test particle (WIMP or star) velocities are Keplerian $v_*=\bar{v}$; in this case $\eta=3/2$, although the exact value does not significantly change the result. The value of the spin-independent WIMP-nucleon scattering cross-section $\sigma_0$ is limited by direct detection experiments, i.e. less than $10^{-43}$ cm$^2$ [@cdms06]. If the star is composed of nuclei with atomic number $A_n$, the cross section increases by a coherent factor of $A_n^4$.
If a WD is heavy ($M\ga M_\odot$) and/or $A_n\gg1$,
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We present a search for Trojan companions to 25 transiting exoplanets. We use the technique of Ford & Gaudi, in which a difference is sought between the observed transit time and the transit time that is calculated by fitting a two-body Keplerian orbit to the radial-velocity data. This technique is sensitive to the imbalance of mass at the L4/L5 points of the planet-star orbit. No companions were detected above 2$\sigma$ confidence. The median 2$\sigma$ upper limit is 56 $M_\earth$, and the most constraining limit is 2.8 $M_\earth$ for the case of GJ 436. A similar survey using forthcoming data from the [ *Kepler*]{} satellite mission, along with the radial-velocity data that will be needed to confirm transit candidates, will be sensitive to 10–50 $M_\earth$ Trojan companions in the habitable zones of their parent stars. As a by-product of this study, we present empirical constraints on the eccentricities of the planetary orbits, including those which have previously been assumed to be circular. The limits on eccentricity are of interest for investigations of tidal circularization and for bounding possible systematic errors in the measured planetary radii and the predicted times of secondary eclipses.'
author:
- 'N. Madhusudhan & Joshua N. Winn'
title: Empirical Constraints on Trojan Companions and Orbital Eccentricities in 25 Transiting Exoplanetary Systems
---
Introduction
============
Trojan companions are bodies in a 1:1 mean-motion resonance with a planet, librating around one of the two triangular Lagrange points (L4 and L5) of the planet’s orbit around the star. The archetypal example is the population of Trojan asteroids in resonance with Jupiter. Trojan companions to Neptune and Mars have also been detected (Sheppard and Trujillo 2006, Rivkin et al. 2007). Another interesting example is the pair of Saturnian satellites Calypso and Telesto, which are in 1:1 resonance with their fellow satellite Tethys (Reitsema 1981). The presence of Trojan companions and their orbital and physical characteristics have been considered as clues to processes in planet formation and migration. Several recent studies have examined the capture and survival of Trojans in the context of suspected changes in the orbital architecture of the Solar system (Morbidelli et al. 2005, Chiang and Lithwick 2005, Kortenkamp et al. 2004).
Although the Trojan-to-planet mass ratios in the Solar system are very small ($m_T/m_P \sim 10^{-7}$ for Jupiter), it is conceivable that Trojans with much higher mass ratios exist in exoplanetary systems. For circular orbits, even very massive Trojans can be dynamically stable. Laughlin & Chambers (2002) explored the viability of Trojans with mass ratios of unity (i.e., co-orbital planets of equal mass), finding that such configurations can be dynamically stable over time scales comparable to or longer than stellar lifetimes. More generally, the stability of the L4/L5 points depends on the orbital eccentricity and the relative masses of the Trojan, planet, and star (see, e.g., Nauenberg 2002, Dvorak et al. 2004). For many of the known exoplanets, considerations of dynamical stability allow for massive Trojan companions. For example, at least 7 of the known gas giant planets that are within the habitable zones of their parent stars could have dynamically-stable, terrestrial-mass Trojan companions (Schwarz et al. 2007).
Several methods have been proposed to detect Trojan companions to exoplanets. A Trojan may be massive enough to perturb the stellar motion by an amount that is detectable in the radial-velocity (RV) orbit of the star (Laughlin & Chambers 2002). A Trojan in a nearly edge-on orbit may be large enough for its transit to be detected photometrically (see, e.g., Croll et al. 2007). For a transiting planet, the gravitational perturbations from a Trojan companion may cause a detectable pattern in the recorded transit times (Ford & Holman 2007). Alternatively, Ford & Gaudi (2006) proposed comparing the measured transit times with the times that would be expected based only on the RV data and the assumption of a two-body orbit.
An important virtue of the latter technique is that a sensitive search for Trojans can be performed using only the RV and photometric data that are routinely obtained while confirming transit candidates and characterizing the planets. This is in contrast to the first three methods, for which it is generally necessary to gather new and highly specialized data (very precise RVs, continuous space-borne photometry, and a long sequence of precisely-measured transit times, respectively). For example, Ford & Gaudi (2006) and Narita et al. (2007) placed upper limits on Trojan companions of approximately Neptune mass to the transiting planets HD 209458b, HD 149026b and TrES-1b, using data gathered for other purposes.
In this paper, we present a search for Trojan companions to 25 known transiting exoplanetary systems for which suitable data are available, using the method of Ford & Gaudi (2006, hereafter, “FG”). This paper is organized as follows. The method is described in § 2. The compilation and analysis of the data is described in § 3. The results are given in § 4. These results are summarized and discussed in § 5, which also looks ahead to the prospects for a similar search using data from the [*Kepler*]{} mission (Borucki et al. 2008).
As will be explained in § 2, the orbital eccentricity of the planet-star orbit affects the interpretation of the data. Hence, a necessary part of our analysis was the determination of the orbital eccentricity for each system, or the justification of the common assumption that the orbit is circular due to tidal effects. These issues are investigated systematically in § 3. Our findings may be of interest independently of our results on Trojan companions, not only because of the connection to the theory of tidal circularization, but also because the orbital eccentricity affects estimates of the planetary radius via transit photometry, as well as the predicted times of planetary occultations (secondary eclipses). We discuss these points in § 5.
Method {#sec:method}
======
The basic idea of the FG method is to compare the measured transit time with the expected transit time that is calculated by fitting a two-body Keplerian orbit to the RV data. We will denote by $t_O$ the observed transit time, and by $t_C$ the calculated transit time, in which the calculation is based on fitting a two-body Keplerian orbit to the RV data. The presence of a Trojan companion as a third body would cause a timing offset $\Delta t = t_O - t_C$.
This is most easily understood for the case of a planet on a circular orbit. In such a case, if there is no Trojan companion, the force vector on the star points directly at the planet, and the observed transit time $t_O$ coincides with the time $t_V$ when the orbital velocity of the star is in the plane of the sky (i.e., the time corresponding to the null in the RV variation). If instead there is a single Trojan located at L4 or L5 (or librating with a small amplitude), then the force vector on the star does not point directly at the planet; it is displaced in angle toward the Trojan companion, given by $\tan(\phi) \simeq \sqrt{3}
\epsilon/(2-\epsilon)$ where, $\epsilon = m_T/(m_P + m_T)$ for a Trojan mass $m_T$ and a planet mass $m_P$ (Ford & Gaudi 2006). As a result, $t_O$ occurs earlier or later than $t_V$, and the time difference is given by $\Delta t = \pm \phi P/2 \pi$. For small values of the Trojan-to-planet mass ratio, the magnitude of $t_O - t_V$ is proportional to $m_T$, (Ford & Gaudi 2006): $$\label{eq:deltat-mt-circular}
\Delta t \simeq \pm 37.5~\text{min}~\bigg( \frac{P}{3 \, \textrm{days}} \bigg)
\bigg( \frac{m_T}{10 \, M_\earth} \bigg)
\bigg( \frac{0.5\,M_{\rm Jup}}{m_P + m_T} \bigg) .$$ The positive sign corresponds to a mass excess at the L4 point (leading the planet) while the negative sign corresponds to a mass excess at the L5 point (lagging the planet). Thus, given a $\Delta t$, the mass excess can be estimated using Eq. (\[eq:deltat-mt-circular\]), assuming small Trojan-to-planet mass ratio.
More generally, the mass excess is given by: $$\label{eq:deltat-mt-x}
m_T = m_P \bigg(\frac{2\, \tan(2\pi \Delta t/P)}{\sqrt{3} - |\tan(2\pi \Delta t/P)|}\bigg).$$
For an eccentric two-body orbit, the transit time does not generally coincide with
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: |
Understanding the difference between group orbits and their closures is a key difficulty in geometric complexity theory (GCT): While the GCT program is set up to separate certain orbit closures, many beautiful mathematical properties are only known for the group orbits, in particular close relations with symmetry groups and invariant spaces, while the orbit closures seem much more difficult to understand. However, in order to prove lower bounds in algebraic complexity theory, considering group orbits is not enough.
In this paper we tighten the relationship between the orbit of the power sum polynomial and its closure, so that we can separate this orbit closure from the orbit closure of the product of variables by just considering the symmetry groups of both polynomials and their representation theoretic decomposition coefficients. In a natural way our construction yields a multiplicity obstruction that is neither an occurrence obstruction, nor a so-called vanishing ideal occurrence obstruction. All multiplicity obstructions so far have been of one of these two types. Our paper is the first implementation of the ambitious approach that was originally suggested in the first papers on geometric complexity theory by Mulmuley and Sohoni (SIAM J Comput 2001, 2008): Before our paper, all existence proofs of obstructions only took into account the symmetry group of one of the two polynomials (or tensors) that were to be separated. In our paper the multiplicity obstruction is obtained by comparing the representation theoretic decomposition coefficients of both symmetry groups. Our proof uses a semi-explicit description of the coordinate ring of the orbit closure of the power sum polynomial in terms of Young tableaux, which enables its comparison to the coordinate ring of the orbit.
author:
- 'Christian Ikenmeyer[^1] and Umangathan Kandasamy[^2]'
date: 'November 10, 2019'
title: 'Implementing geometric complexity theory: On the separation of orbit closures via symmetries'
---
[0.9]{} **Acknowledgements:** This paper was written partially when CI was at the Max Planck Institute for Informatics, at the Max Planck Institute for Software Systems, and at the University of Liverpool. This paper contains results that are present in UK’s master’s thesis at the Universität des Saarlandes.
[0.9]{} **2012 ACM CCS:** Theory of computation $\rightarrow$ Algebraic complexity theory
[0.9]{} **Keywords:** Geometric complexity theory, group orbit, orbit closure, multiplicity obstruction
Motivation: Geometric complexity theory
=======================================
#### Symmetries
The idea of using the symmetries of the determinant ${\textup{det}}_n := \sum_{\pi \in {\mathfrak{S}}_n}{\textup{sgn}}(\pi)\prod_{i=1}^n x_{i,\pi(i)}$ and the permanent ${\textup{per}}_m := \sum_{\pi \in {\mathfrak{S}}_m} \prod_{i=1}^m x_{i,\pi(i)}$ to separate algebraic complexity classes was pioneered by Mulmuley and Sohoni in 2001 [@MS:01]. This approach is based on the observation that ${\textup{det}}_n$ and ${\textup{per}}_m$ are both characterized by their respective symmetry groups. For example, consider homogeneous degree $n$ polynomials in $n^2$ variables $x_{1,1},\ldots,x_{n,n}$. Let $X$ denote the $n \times n$ matrix whose entry in row $i$ and column $j$ is $x_{i,j}$. Then ${\textup{det}}(X)={\textup{det}}_n$. Now, for matrices $A,B \in {\textup{SL}}_n({\mathbb{C}})$ the entries of the matrix $AXB$ are homogeneous linear polynomials in the $n^2$ variables. The crucial fact is that every homogeneous degree $n$ polynomial $q$ in $n^2$ variables that satisfies $q(AXB)=q(X)$ equals $\alpha \cdot {\textup{det}}_n$ for some scalar $\alpha \in {\mathbb{C}}$. This means that ${\textup{det}}_n$ is *characterized by its symmetries*. For the permanent polynomial, an analogous statement holds, and also for many other structurally simpler polynomials, for example for the power sum polynomial $x_1^D+\cdots+x_m^D$ and for the product of variables $x_1 x_2 \cdots x_D$, see [@Ike:19].
#### Algebraic complexity theory
An *affine projection* of a polynomial is its evaluation at a point whose coordinates are given by affine linear polynomials, e.g., $(x_1+x_2+1)^2 = x_1^2 + 2 x_1 x_2 + x_2^2 + 2 x_1 + 2 x_2 + 1$ is an affine projection of $x_1^2$. Kayal proved that it is ${\textup{\textsf{NP}}}$-hard to decide whether a polynomial is an affine projection of another polynomial [@Kay:12]. Valiant proved [@Val:79b] that every polynomial $p$ is an affine projection of some ${\textup{det}}_n$ for $n$ large enough. The smallest $n$ for which this is possible is called the *determinantal complexity* ${\textup{\textsf{dc}}}(p)$. The class of sequences of polynomials $(p_m)$ whose sequence of natural numbers ${\textup{\textsf{dc}}}(p_m)$ is polynomially bounded is called ${\ensuremath{\textup{\textsf{VP}}_{\textsf{s}}}}$. For the permanent we can define the *permanental complexity* ${\textup{\textsf{pc}}}(p)$ in a completely analogous manner: ${\textup{\textsf{pc}}}(p)$ is the smallest $n$ such that $p$ is an affine projection of ${\textup{per}}_n$. The class of sequences of polynomials $(p_m)$ whose ${\textup{\textsf{pc}}}(p_m)$ is polynomially bounded is called ${\textup{\textsf{VNP}}}$. Since ${\textup{\textsf{pc}}}({\textup{det}}_n)$ is polynomially bounded, ${\ensuremath{\textup{\textsf{VP}}_{\textsf{s}}}}\subseteq {\textup{\textsf{VNP}}}$. Valiant’s flagship conjecture in algebraic complexity theory, which is also known as the *determinant versus permament* conjecture can be succinctly phrased as ${\ensuremath{\textup{\textsf{VP}}_{\textsf{s}}}}\neq {\textup{\textsf{VNP}}}$. This is equivalent to conjecturing that ${\textup{\textsf{dc}}}({\textup{per}}_m)$ grows superpolynomially fast.
#### Homogeneous projections and endomorphism orbits
It will be beneficial to phrase Valiant’s conjecture in a homogeneous setting: A *homogeneous projection* of a homogeneous polynomial (i.e., all monomials have the same total degree) is its evaluation at a point whose coordinates are given by homogeneous linear polynomials. The set of all homogeneous projections of ${\textup{det}}_n$ to polynomials in the variables $x_1,\ldots,x_N$ can then be written as $\{{\textup{det}}_n(\ell_1,\ldots,\ell_{n^2}) \mid \text{$\ell_i$ is a homogeneous linear polynomial in $x_1,\ldots,x_N$}\}$. Note that we put the $n \times n = n^2$ inputs of the determinant in a linear order. The polynomial function $(x_{1,1},\ldots,x_{n,n}) \mapsto {\textup{det}}_n(\ell_1,\ldots,\ell_{n^2})$ equals the composition ${\textup{det}}_n \circ A$, where $A$ is the linear map $(x_{1,1},\ldots,x_{n,n}) \mapsto (\ell_1,\ldots,\ell_{n^2})$. As it is common in representation theory, we write $A \cdot {\textup{det}}_n$ or just $A {\textup{det}}_n$ for ${\textup{det}}_n \circ A$. The *endomorphism orbit* ${\textup{End}}_{n^2}{\textup{det}}_n$ is defined as $\{A {\textup{det}}_n \mid A \in {\mathbb{C}}^{n^2 \times n^2}\}$, which is the set of all homogeneous projections of ${\textup{det}}_n$ to polynomials in at most $n^2$ variables. Since all polynomials in $A {\textup{det}}_n$ are homogeneous of degree $n$, we have ${\textup{per}}_m \notin {\textup{End}}_{n^2} {\textup{det}}_n$ for any $m \neq n$. This slight technicality is treated by a procedure called *padding*: For fixed $m$, $n$ with $m < n$, define the *padded permanent* ${\textup{per}}_{m,n} := (x_{n,n})^{n-m}\cdot {\textup{per}}_m$. Let ${\textup{\textsf{dc}}}'({\textup{per}}_{m,n})$ denote the smallest $n$ such that ${\textup{per}}_{m,n} \in {\textup{End}}_{n^2} {\textup{det}}_n$. A short calculation shows that Valiant’s conjecture is equivalent to the conjecture that ${\textup{\textsf{dc}}}'({\textup{per}}_m)$ grows superpolynomially.
#### Group orbits
It turns out that if we restrict ${\textup{End}}_{n^2} {\textup{det}}_n$ to only the points $A{\textup{det}}_n$ for which $A$ is invertible, we get the much simpler *group orbit* ${\
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
author:
- 'Adam Back\'
- 'Iddo Bentov\'
title: Note on fair coin toss via Bitcoin
---
Introduction
============
In this short note we show that the Bitcoin network can allow remote parties to gamble with their bitcoins by tossing a fair or biased coin, with no need for a trusted party, and without the possibility of extortion by dishonest parties who try to abort. The superfluousness of having a trusted party implies that there is no house edge, as is the case with centralized services that are supposed to generate a profit.
One simple way to accomplish a coin toss protocol with Bitcoin is via a protocol fork that adds to the Bitcoin scripting language an opcode that puts on the stack the hash of the block in which the transaction resides. However, this implies that the parties have to wait for 10 minutes on average until the result of the bet becomes known. Worse still, such an opcode should have a maturity time of e.g. 100 blocks due to possible reorgs, thus the winning party will have to wait for more than 16 hours before being able to spend the coins that she won.
We propose an alternative coin toss protocol that utilizes the current Bitcoin implementation, i.e. with no need for a protocol fork. Further, with our protocol it is not necessary to wait for the next solved block, and instead the amount of coins of the bet can dictate the appropriate confidence level that the parties require. This means that 0-confirmations security for low value bets does not use the PoW irreversibility property, and instead the mining race degrades into a network race. Hence this is similar to Point of Sale for low value transactions with Bitcoin, as merchants can take a small risk by accepting unconfirmed transactions, while listening on the network to detect double-spending attempts.
Protocol
========
The reason why we can resist malicious adversaries who abort upon discovering that they lost the bet is that the Bitcoin scripting language allows us to have a primitive with which Alice locks a certain amount of her coins until a specified time in the future, and Bob can spend these coins to an arbitrary address at any time upon meeting certain arbitrary conditions that were specified in advance via a Bitcoin script, otherwise the locked coins are returned to Alice.
This primitive can be implemented in Bitcoin as follows: Alice creates a “principle” transaction that takes inputs that she controls, and can be spent according to “(Alice’s signature AND Bob’s signature) OR (arbitrary conditions)”. Alice keeps the “principle” transaction private, and creates another “refund” transaction that spends the ”principle” transaction to an output address that she controls, but has locktime set in the future. Alice then signs the “refund” transaction, and sends Bob a private message with the the “refund” transaction, asking Bob to sign it. Notice that since Bob only sees the hash of the “principle” transaction, he can protect himself from being tricked into signing a malevolent transaction that steals his other coins by generating a fresh secret key and asking Alice to create the “principle” transaction with the corresponding public address of this key. Hence, Bob sends Alice a private message with his signature for the “refund” transaction. Now Alice broadcasts the “principle” transaction to the Bitcoin network. If Bob (or anyone) cannot meet the conditions that the “principle” transaction specified, Alice will recover her coins after the locktime expires. [@g01]
Suppose that Alice and Bob wish to do a fair coin toss where each of them inputs X coins and the winner gets the 2X coins. This can be done by selecting the winner according to the least significant bit of two committed secrets, with the following protocol:
1. Alice picks some random secret $A1$ and sends a private message to Bob with the value $A2=\texttt{SHA256}(A1)$.
2. Bob picks some random secret $B1$ and sends a private message to Alice with the value $B2=\texttt{SHA256}(B1)$.
3. Bob creates a “bet”transaction that takes as input $2X$ of his own coins, and can be spent by: \[Alice’s signature AND Bob’s signature\] OR \[$\texttt{SHA256}(A)==A2$ AND $\texttt{SHA256}(B)==B2$ AND (($(A\ \texttt{xor}\ B)\ \texttt{mod}\ 2 == 0$ AND Alice’s signature) OR ($(A\ \texttt{xor}\ B)\ \texttt{mod}\ 2 == 1$ AND Bob’s signature))\]
4. Bob asks Alice to sign a “refund\_bet” transaction which spends his $2X$ coins to an address that he controls, and has locktime of (say) 20 blocks into the future.
5. Bob broadcasts the “bet” transaction to the Bitcoin network.
6. Alice creates a “reveal” transaction that takes as input $X$ of her own coins, and can be spent by: \[Alice’s signature AND Bob’s signature\] OR \[$\texttt{SHA256}(B)==B2$ AND Bob’s signature\]
7. Alice asks Bob to sign a “refund\_reveal” transaction which spends her $X$ coins to an address that she controls, and has locktime of (say) 10 blocks into the future.
8. Alice broadcasts the “reveal” transaction to the Bitcoin network (when she is confident enough that the “bet” transaction will not be reversed).
9. Bob redeems the “reveal” transaction by revealing B1 (when he is confident enough that the “reveal” transaction will not be reversed).
10. Alice redeems the “bet” transaction if she won, otherwise she sends $A1$ to Bob so that he could redeem the “bet” transaction without waiting for the locktime to expire.
This protocol is sound because the locktime in step (7) is shorter than the locktime of step (4), therefore Bob cannot cheat by broadcasting the transaction that reveals $B1$ in step (9) right before the locktime of step (4) expires.
By using more bits from the two committed secrets, Alice and Bob can bet with a biased coin so that the party with the worse odds wins the larger amount.
Currently `OP_MOD` is considered nonstandard in the Bitcoin protocol, but we can for example combine `OP_SHA1` and `OP_GREATERTHAN` to gain a similar effect.
[1]{} A. Back, I. Bentov, et al. , [*Bitcoin forum thread*]{}, August 2013. <https://bitcointalk.org/index.php?topic=277048.0>. `Last accessed on December 5, 2013`. G. Maxwell. , [*Bitcoin wiki*]{}. <https://en.bitcoin.it/wiki/Zero_Knowledge_Contingent_Payment>. `Last accessed on December 5, 2013`.
[revision 7]{}
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'Starting from rotational invariance we derive sum rules for the single–spin asymmetries in inclusive production and binary processes. We also get sum rules for spin correlation parameters in elastic $pp$–scattering.'
author:
- |
S.M. Troshin, N.E. Tyurin\
*Institute for High Energy Physics,\
*Protvino, Moscow Region, 142280, Russia**
title: Sum rules for spin asymmetries
---
An important role of spin effects for analysis of hadron interaction dynamics is widely recognized nowadays. The space–time structure of the strong interactions provides a number of constraints for the spin observables (cf. [@bks]) and, as it will be shown further, allows us to get a useful sum rule for the single–spin asymmetries and spin correlation parameters.
Let us consider first single–spin asymmetry in hadron production $$h_1+h_2\to h_3+X,$$ where the beam or target hadron $h_{1,2}$ is transversely polarized. Let $\xi$ stands for the set of variables related to the hadron $h_3$. The definition of asymmetry $A_N$ is well known $$A_N(s,\xi)=
\left[\frac{d\sigma^\uparrow}{d\xi}(s,\xi)-\frac{d\sigma^\downarrow}{d\xi}(s,\xi)\right]/
\left[\frac{d\sigma^\uparrow}{d\xi}(s,\xi)+\frac{d\sigma^\downarrow}{d\xi}(s,\xi)\right].$$ The equality of the integrated inclusive cross-sections follows from rotational invariance in a straightforward way, i.e. $$\int
\frac{d\sigma^\uparrow}{d\xi}(s,\xi)d\xi=\int \frac{d\sigma^\downarrow}{d\xi}(s,\xi)d\xi.$$ Then from the definition of $A_N$ we have the following sum rule $$\label{srincl}
\int A_N(s,\xi)\frac{d\sigma}{d\xi}(s,\xi)d\xi=0,$$ where ${d\sigma}/{d\xi} $ is the unpolarized cross-section. Eq. (\[srincl\]) should be taken into account at the construction of models intended to explain the significant single–spin asymmetries observed in the inclusive processes. Similar sum rule (with replacement $A_N$ $\to$ $P$) takes place when the polarization of the final hadron $h_3$ can be measured ($\Lambda$–hyperon for example).
The above arguments can be applied to analyzing power in elastic and binary processes, e.g. from the equality $$\sigma_{el}^\uparrow(s)=\sigma_{el}^\downarrow(s)$$ we should have $$\label{srelas}
\int_{-s+4m^2}^0 A(s,t)\frac{d\sigma}{dt}(s,t)dt=0,$$ where $A(s,t)$ is the analyzing power in elastic scattering and ${d\sigma}/{dt}$ is the unpolarized cross–section for the elastic scattering of the particles with equal masses. Sum rule for the inelastic binary processes has the similar form with minor kinematical changes of the integration limits. [*From Eq. (\[srelas\]) we arrive to conclusion that $A(s,t)$ should have sign–changing $t$-dependence since ${d\sigma}/{dt}$ is positive*]{}. This conclusion on the $t$–dependence is useful for the planning of the future experiments on the analyzing power measurements in elastic scattering at higher values of $t$ [@adk]. It should be noted that change of sign of $A$ in elastic $pp$–scattering was revealed for the first time in the measurements at 40 $GeV/c$ [@kaz] and considered as a new experimental regularity in the analyzing power $t$–dependence that time.
Oscillating pattern of analyzing power $t$-dependence with amplitude of oscillations increasing with $t$ observed in various experiments in elastic and binary processes [@bin], is in conformity with the sum rule Eq. (\[srelas\]). Such oscillating dependence has obtained model explanation in [@osa]. It should be noted that the Eq. (\[srincl\]) does not imply similar $p_{\perp}$–dependence for the single–spin asymmetries in the inclusive processes and the corresponding experimental data have not revealed oscillations (cf. e.g. [@bks]).
Using rotational invariance combined with particle identity we can obtain similar sum rules for the spin correlation parameters in elastic and inclusive $pp$–scattering. Spin correlation parameters are the spin observables which describe dependence of the interaction on the relative orientations of the spins of the two particles (cf. [@bks]). We will consider scattering when both protons in [*the initial state*]{} are polarized. Definition of spin correlation parameter $A_{nn}$ is the following $$\label{ann}
A_{nn}=\frac{{\frac{d\sigma_{^{\uparrow\uparrow }} }{dt}} +
{\frac{d\sigma_{^{\downarrow\downarrow }} }{dt}} -
{\frac{d\sigma_{^{\uparrow\downarrow }}}{dt}} -
{\frac{d\sigma_{^{\downarrow\uparrow }}}{dt}}}
{{\frac{d\sigma_{^{\uparrow\uparrow }}}{dt}}
+{\frac{d\sigma_{^{\downarrow\downarrow }} }{dt}}
+ {\frac{d\sigma_{^{\uparrow\downarrow }}}{dt}}
+ {\frac{d\sigma_{^{\downarrow\uparrow }} }{dt}}},$$ where index $n$ means that spins polarized along a normal to the scattering plane. Other parameters $A_{ll}$, $A_{ss}$ and $A_{sl}$ have the similar to Eq. (\[ann\]) structure and differ by the orientation of spins in the initial state. Rotational invariance and particle identity leads to the following equality $$\label{srcor}
\Delta\sigma^{el}_T(s)=-4\int_{-s+4m^2}^0
A_{nn}(s,t)\frac{d\sigma}{dt}dt,$$ where $\Delta\sigma^{el}_T(s)$ is the cross section difference with protons polarized along normal to beam direction: $$\Delta\sigma^{el}_T(s)\equiv \sigma^{el}_{\uparrow\downarrow}(s)-
\sigma^{el}_{\uparrow\uparrow}(s)$$ Parity conservation combined with particle identity allows us to get another relation $$\label{srcoral}
\Delta\sigma^{el}_L(s)=-4\int_{-s+4m^2}^0
A_{ll}(s,t)\frac{d\sigma}{dt}dt,$$ where $\Delta\sigma^{el}_L(s)$ is the cross section difference for the protons polarized along beam direction: $$\Delta\sigma^{el}_L(s)\equiv
\sigma^{el}_{{^\rightarrow_\leftarrow}}(s)-
\sigma^{el}_{{^\rightarrow_\rightarrow}}(s)$$ We also have due to rotational invariance that $$\label{srcoreq}
\int_{-s+4m^2}^0 [A_{nn}(s,t)- A_{ss}(s,t)]\frac{d\sigma}{dt}dt=0,$$ And parity conservation and rotational invariance provide $$\label{srcoral1}
\int_{-s+4m^2}^0 A_{sl}(s,t)\frac{d\sigma}{dt}dt=0.$$
Similar relations can be written for the spin correlation parameters in the inclusive processes.
All above sum rules should be, of course, in agreement with the experimental data and therefore they can be used for the extrapolation to the region where data are absent at the moment. These sum rules are also interesting as a test ground for the models and must be obeyed under their construction.
[9]{} C. Bourrely, E. Leader, J. Soffer, Phys. Rep. 59, 95 1980;\
S.M. Troshin, N.E. Tyurin, [*Spin Phenomena in Particle Interactions*]{}, World Scientific Publishing Co., Singapore, 1994;\
E. Leader, [*Spin in Particle Physics*]{}, Cambridge University Press, UK, 2001. V.G. Luppov, et al., AIP Conf. Proc. 675, 538, 2003. Yu. M. Kazarinov, et al., Nucl. Phys. B 124, 391, 1977. I. Auer, et al., Phys. Lett. 70, 475, 1977;\
G. Fidecaro, et al., Phys. Lett. 76, 369, 1978;\
J. Antille, et al., Nucl. Phys. 185, 1, 1981;\
V.D. Apokin, et al. Sov. Journal of Nucl. Phys. 45, 1355, 1987;\
D.G. Crabb, et al., Phys. Rev. Lett. 41, 1350, 1978; Phys. Rev. Lett. 65, 3241, 1990. S.M. Troshin, N.E. Tyurin, Proceedings “Polarization Phenomena In Nuclear Physics”, Osaka 1985, 954.
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We develop a new abstract derivation of the observability inequalities at two points in time for Schrödinger type equations. Our approach consists of two steps. In the first step we prove a Nazarov type uncertainty principle associated with a non-negative self-adjoint operator $H$ on $L^2(\mathbb{R}^n)$. In the second step we use results on asymptotic behavior of $e^{-itH}$, in particular, minimal velocity estimates introduced by Sigal and Soffer. Such observability inequalities are closely related to unique continuation problems as well as controllability for the Schrödinger equation.'
address:
- ' Shanlin Huang, School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, P.R.China '
- ' Avy Soffer, Department of Mathematics, Rutgers University, Piscataway, 08854-8019, USA '
author:
- 'Shanlin Huang, Avy Soffer'
---
Introduction
============
In a recent paper by Wang, Wang and Zhang [@WWZ], they established a new type of observability inequality at two points in time for the free Schrödinger equation. More precisely, let $u(x, t)$ satisfy $$\begin{aligned}
\label{equ1.1}
\begin{cases}
i\partial_{t}u +\Delta u=0,\,\,\,(x, t)\in \mathbb{R}^n\times (0, \infty),\\
u(0,x)=u_{0}\in L^2(\mathbb{R}^n).
\end{cases}\end{aligned}$$ Then given any $r_1,\,r_2>0$, and $t_1>t_2\ge 0$, there exists a positive constant $C$ depending only on $n$ such that $$\begin{aligned}
\label{equ1.2}
\int_{\mathbb{R}^n}|u_0|^2\,dx\leq Ce^{C\frac{r_1r_2}{t_2-t_1}}\left(\int_{|x|\ge r_1}|u(x, t_1)|^2\,dx+\int_{|x|\ge r_2}|u(x, t_2)|^2\,dx\right), \,\,u_0\in L^2(\mathbb{R}^n).\end{aligned}$$ The proof in [@WWZ] is based on the fact that in the free case, one has the identity $$\begin{aligned}
\label{equ1.3}
(2it)^{\frac{n}{2}}e^{-i|x|^2/4t}u(x, t)=\widehat{e^{-i|\cdot|^2/4t}u_0}(x/2t),\,\,\, \text{for all}\,\,t>0,\end{aligned}$$ where $\widehat{\cdot}$ denotes the Fourier transform. After applying with a scaling argument, it’s easy to see that the estimate is equivalent to the following Nazarov’s uncertainty principle built up in [@Jam]: $$\begin{aligned}
\label{equ1.4}
\int_{\mathbb{R}^n}|f(x)|^2\,dx\leq Ce^{Cr_1r_2}(\int_{|x|\ge r_1}|f(x)|^2\,dx+\int_{|\xi|\ge r_2}|\hat{f}(\xi)|^2\,d\xi), \,\,\,\,f\in L^2(\mathbb{R}^n).\end{aligned}$$
A natural question is whether such kind of observability inequalities still hold for more general Hamiltonian. We mention that the approach in [@WWZ] is restricted to the free Laplacian, since the argument there is essentially relying on the formula , which in turn follows from the fundamental solution of $e^{it\Delta}$. For general $H$, no such explicit solutions are available, thus one needs to proceed differently.
The motivation of this paper is to develop an abstract approach to obtain observability inequalities at two points in time for $e^{-itH}$ under some general assumptions on $H$. Then we apply it to special cases including Schrödinger equation with potentials and fractional Schrödinger equations.
We first point out that may fail if $H$ has eigenvalues. Indeed if $H\phi=\lambda\phi$, for some $\lambda\in \mathbb{R}$ and $\phi\in L^2$. Then $u(x, t)=e^{-i\lambda t}\phi(x)$ is a solution of the following Cauchy problem $$i\partial_{t}u =H u, \qquad u(0,x)=u_{0}(x)\in L^2(\mathbb{R}^n).$$ After choosing $r_1=r_2=\sqrt{t_2-t_1}$ in , we find that the RHS of is equal to $C\int_{|x|\ge \sqrt{t_2-t_1}}{|\phi|^2\,dx}$ with some fixed constant $C$, which goes to zero as $t_2-t_1\rightarrow \infty$. Hence estimate can’t hold for such $\phi$. Therefore, we only expect to hold for vectors lying in the continuous subspace of $H$.
We proceed to illustrate the key idea and main tools used in our approach. To simplify matters, we change the uncertainty principle into a form concerning two projection operators on $L^2$, i.e., for any $r>0$ $$\begin{aligned}
\label{equ1.5}
\|f\|^2\leq C\left(\|\chi(|x|\ge r)f\|^2+\|\chi(H\ge r^{-2})f\|^2\right), \,\,\,\,f\in L^2(\mathbb{R}^n).\end{aligned}$$ where $H=-\Delta$ and $C$ is a constant depending only on the dimension. We mention that inequality indicates that if the initial data is localized in a ball, then its “energy” must have a positive lower bound. Actually, it’s easy to see is equivalent to the following $$\|\chi(|x|\le r)f\|\leq C\|\chi(H\ge r^{-2})f\|,\,\,\,\text{for any}\,\, r>0.$$
Having established this type of uncertainty principle for $H$, we can use propagation estimates, in particular [**minimal velocity estimates**]{} to further study the asymptotic behavior of $e^{-itH}f$. To provide intuition in understanding of this method, let us consider the simple case $H=-\Delta$, and assume $f$ is a Schwartz function such that $f\in Ran\, \chi(H\ge \delta )$ with some $\delta>0$, hence $\hat{f}$ is smooth and $\text{supp}\,\hat{f}\subset \{\xi\in \mathbb{R}^n,\,|\xi|\ge \sqrt{\delta}\}$. Then a integration by parts argument yields $$\int_{\frac{|x|}{t}<\sqrt{\delta}}{|e^{-itH}f|^2\,dx}=O(t^{-m}),\,\,\,\text{as}\,\, t\rightarrow\infty$$ for any $m>0$. In this sense, the evolution $e^{-itH}f$ is said to have a minimal velocity $v_{min}=\sqrt{\delta}>0$. Roughly speaking, the goal of minimal velocity estimates is to obtain similar results for general Hamiltonian via an abstract way. And it’s based on choosing observable (self-adjoint operator) $A$ so that the commutator $i[H, A]$ is positive definite, see section \[sec2.2\] for further discussion. Such estimates are crucial in our proof, since it provides quantitative information about the rate with which the wave $e^{-itH}f$ moves out to spacial infinity. As a comparison, we recall that the RAGE theorem (see e.g. [@RS] ) indicates that for certain Schrödinger operators $H=-\Delta+V$, $e^{-itH}f$ is escaping any fixed ball in a mean ergodic sense: $$\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T{\,dt}\int_{|x|\leq R}{|e^{-itH}f|^2\, dx}=0.$$ We mention that minimal velocity estimates were first appeared in the work of Sigal and Soffer [@SS], which turned out to be very useful in scattering theory and theory of resonances, we refer to [@Ski; @SS90; @SW; @HSS] and references therein for further extensions and applications. One of the novelties in our paper is that we establish the close relationship between observability inequalities and minimal velocity estimates for Schrödinger type equations.
Now we turn to some applications. As is pointed out in [@WWZ], estimate can be used to derive controllability for Schrödinger equations. It is also closely related to quantitative unique continuation problems for Schrödinger equations. In Sect. \[sec4\], we shall use observability inequalities built up in this paper to obtain results concerning unique continuation properties of Schrödinger equations with potentials as well as fractional Schrödinger equations. Such kind of results for certain linear and nonlinear Schrödinger equations were considered by Ionescu and Kenig [@IK] based on the use of Carleman estimates. For the uncertainty principle and unique continuation inequalities for Schrödinger equations, we would like to refer a series of paper by Escauriaza, et al. [@EKPV1; @EKPV2; @EKPV3; @EKPV4] and references therein.
The rest of the paper is organized as follows. Section
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We study an extended QCD model in $2D$ obtained from QCD in $4D$ by compactifying two spatial dimensions and projecting onto the zero-mode subspace. This system is found to induce a dynamical mass for transverse gluons – adjoint scalars in $QCD_2$, and to undergo a chiral symmetry breaking with the full quark propagators yielding non-tachyonic, dynamical quark masses, even in the chiral limit. We construct the hadronic color singlet bound-state scattering amplitudes and study quark-antiquark bound states which can be classified in this model by their properties under Lorentz transformations inherited from $4D$.'
title: 'Quark-Antiquark Bound States in an Extended $QCD_2$ Model[^1]'
---
[Pedro Labra[ñ]{}a, Jorge Alfaro]{}\
Facultad de Física, Pontificia Universidad Católica de Chile.\
plabrana@puc.cl, jalfaro@puc.cl\
\
St.Petersburg State University and INFN, Sezione di Bologna.\
andrianov@bo.infn.it
We study a QCD reduced model in $2D$ which can be formally obtained from QCD in $4D$ by means of a classical dimensional reduction from $4D$ to $2D$ and neglecting heavy K-K (Kaluza-Klein) states. Thus only zero-modes in the harmonic expansion in compactified dimensions are retained. As a consequence, we obtain a two dimensional model with some resemblances of the real theory in higher dimension, that is, in a natural way adding boson matter in the adjoint representation to $QCD_2$ [@light; @Alfaro:2003yy]. The latter fields being scalars in $2D$ reproduce transverse gluon effects [@adjoint]. Furthermore this model has a richer spinor structure than just $QCD_2$ giving a better resolution of scalar and vector states which can be classified by their properties inherited from $4D$ Lorentz transformations. The model is analyzed in the light cone gauge and using large $N_c$ limit. The contributions of the extra dimensions are controlled by the radiatively induced masses of the scalar gluons as they carry a piece of information of transverse degrees of freedom. We consider their masses as large parameters in our approximations yet being much less than the first massive K-K excitation. This model might give more insights into the chiral symmetry breaking regime of $QCD_4$. Namely, we are going to show that the inclusion of solely lightest K-K boson modes catalyze the generation of quark dynamical mass and allows us to overcome the problem of tachyonic quarks present in $QCD_2$.
We start with the $QCD$ action in $(3+1)$ dimensions for one flavor (extension to more flavors is straightforward): $$S_{QCD}= \int d^4x \left[-\frac{1}{2{\tilde g}^2}tr(G_{\mu\nu}^2)
+ {\bar \Psi}\,(i\gamma^\mu \, D_\mu - m)\,\Psi \right].$$
Follow the scheme of [@Alfaro:2003yy] we proceed to make a dimensional reduction of $QCD$, at the classical level, from $4D$ to $2D$. For this we consider the coordinates $x_{2,3}$ being compactified in a 2-Torus, respectively the fields being periodic on the intervals ($0\leq x_{2,3}\leq L=2\pi R$). Next we assume $L$ to be small enough in order to get an effective model in $2D$ dimensions. Then by keeping only the zero K-K modes, we get the following effective action in $2D$, after a suitable rescaling of the fields: $$\begin{aligned}
\label{Lmodelo2D}
S_2 &=& \int \!d^2x\,\,tr \!\left[-\frac{1}{2}F_{\mu\nu}^2+ (D_\mu
\phi_1)^2 + (D_\mu \phi_2)^2 \right] +
{\bar \psi}_1\,(i\gamma^\mu \, D_\mu - m)\,\psi_1 \nonumber\\
%
&+& {\bar \psi}_2\,(i\gamma^\mu \, D_\mu - m)\,\psi_2 -
i\frac{g}{\sqrt{N_c}}\left({\bar \psi}_1\,\gamma^5\,\phi_1\,\psi_2 +
{\bar \psi}_2\,\gamma^5\,\phi_1\,\psi_1\right) \\
%
&-& i\frac{g}{\sqrt{N_c}}\left({\bar \psi}_1\,\gamma^5\,\phi_2\,\psi_1 -
{\bar \psi}_2\,\gamma^5\,\phi_2\,\psi_2\right) +
\frac{g^2}{N_c}\,tr[\phi_1,\phi_2]^2\,,
\nonumber\end{aligned}$$ where we have defined the coupling constant of the model $g^2= N_c\,{\tilde g}^2/L^2$. We expect [@Coleman1] the infrared mass generation for the two-dimensional scalar gluons $\phi_i$. To estimate the masses of scalar gluons $\phi_i$ we use the Schwinger-Dyson equations as self-consistency conditions, we get: $$M^2=\frac{2 N_c {\tilde g}^2}{L^2} \int^\Lambda \!\!
\frac{d^2p}{(2\pi)^2} \,\frac{1}{p^2+M^2} \,
= \frac{N_c {\tilde g}^2\,\Lambda^2}{8\pi^3}\,\,
\log\frac{\Lambda^2+M^2}{M^2}\,, \label{glumass}$$
![Inhomogeneous Bethe-Salpeter equation for quark-antiquark scattering amplitude $T_5$.[]{data-label="F1"}](Diagramas3b.eps){width="90.00000%"}
thus $M^2$ brings an infrared cutoff as expected. We notice that the gluon mass remains finite in the large-$N_c$ limit if the QCD coupling constant decreases as $1/N_c$ in line with the perturbative law of $4D$ QCD. We adopt the approximation $M \ll \Lambda\simeq 1/R$ to protect the low-energy sector of the model and consider the momenta $|p_{0,1}| \sim M$. Thereby we retain only leading terms in the expansion in $p^2/\Lambda^2$ and $M^2/\Lambda^2$, and also neglect the effects of the heavy K-K modes in the low-energy Wilson action. We observe that the limit $M \ll \Lambda$ supports consistently both the fast decoupling of the heavy K-K modes and moderate decoupling of scalar gluons [@Alfaro:2003yy], the latter giving an effective four-fermion interaction different from [@Burkardt]. Also allow us to define the “heavy-scalar” expansion parameter $A=g^2/(2\pi\,M^2)=1/log \frac{\Lambda^2}{M^2} \ll 1$. Now we proceed to the study of bound states of quark-antiquark. In our reduction we have four possible combinations of quark bilinears to describe these states with valence quarks: $(\psi_1\,{\bar \psi}_1),(\psi_1\,{\bar \psi}_2),$ $
(\psi_2\,{\bar \psi}_1),(\psi_2\,{\bar \psi}_2 )$. We need to compute the full quark-antiquark scattering amplitude $T$ in the different channels. As an example we are going to show the computing of $T_5$ which correspond to the scattering ($q_1 + \bar{q}_2 \longrightarrow q_1 + \bar{q}_2$). It satisfies, the equation given graphically in , in the large $N_c$ limit and in ladder exchange approximation (non-ladder contribution are estimated to be of higher order in the $A$ expansion). Notice that in the equation for $T_5$ the amplitude $T_8$ appears, which correspond to the process $q_2 + \bar{q}_1 \longrightarrow q_1 + \bar{q}_2$. This means that the equations for $T_5$ and $T_8$ are coupled. In Figure \[F1\] all internal fermion lines correspond to dressed quark propagators which were determined in [@Alfaro:2003yy] by using the large $N_c$ limit and the one-boson exchange approximation. There are two kind of solutions for the dressed quark propagators. In one solution, the perturbative one, we have tachyonic quarks in the chiral limit as in $QCD_2$ and our model could be interpreted as a perturbation from the result of chiral QCD in $2D$. But we are not allowed to consider that possibility because the spectrum for the lowest $q {\bar q}$ bound states becomes imaginary if one takes into account the scalar field exchange. The second solution, the non-perturbative one, supports non-tachyonic quarks with masses going to zero, in the chiral limit and also yield real masses for the $q {\bar q}$ bound states.
![$q\bar{q}$
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We utilized the Very Large Array to make multifrequency polarization measurements of 20 radio sources viewed through the IC 1805 HII region and “Superbubble”. The measurements at frequencies between 4.33 and 7.76 GHz yield Faraday rotation measures (RMs) along 27 lines of sight to these sources. The RMs are used to probe the plasma structure of the IC 1805 HII region and to test the degree to which the Galactic magnetic field is heavily modified (amplified) by the dynamics of the HII region. We find that IC 1805 constitutes a “Faraday rotation anomaly”, or a region of increased RM relative to the general Galactic background value. The $|$RM$|$ due to the nebula is commonly 600 – 800 rad m$^{-2}$. However, the observed RMs are not as large as predicted by simplified analytic models that include substantial amplification of the Galactic magnetic field within the shell. The magnitudes of the observed RMs are consistent with shells in which the Galactic field is unmodified, or increased by a modest factor, such as due to magnetic flux conservation. We also find that with one exception, the sign of the RM is that expected for the polarity of the Galactic field in this direction. Finally, our results show intriguing indications that some of the largest values of $|$RM$|$ occur for lines of sight that pass outside the fully ionized shell of the IC 1805 HII region but pass through the Photodissociation Region associated with IC 1805.'
author:
- 'Allison H. Costa and Steven R. Spangler'
bibliography:
- 'FRbib.bib'
- 'LBVbib.bib'
- 'compbib.bib'
- 'FollowupBib.bib'
title: A Faraday Rotation Study of the Stellar Bubble and Region Associated with the W4 Complex
---
Introduction\[intro\]
=====================
Young massive stars in OB associations photoionize the surrounding gas, creating an region, and their powerful stellar winds can inflate a bubble around the star cluster. Magnetic fields are important to the dynamics of these structures [@Tomisaka:1990; @Ferriere:1991; @Vallee:1993; @Tomisaka:1998; @Haverkorn:2004; @Sun:2008; @Stil:2009], and they can elongate the cavity preferentially in the direction of the magnetic field and thicken the shell perpendicular to the field [@Ferriere:1991; @deAvillez:2005; @Stil:2009], causing deviations from the classical structure of the @Weaver:1977 wind-blown bubble. Knowledge of the magnitude and direction of the magnetic field within stellar bubbles and regions is important for simulations and for understanding how the magnetic field interacts with and modifies these structures.
In previous work (i.e., @Savage:2013 and @Costa:2016), we investigated whether the Galactic magnetic field is amplified in the shell of the Rosette Nebula, an region and stellar bubble associated with NGC 2244 ($\ell$ = 206.5, $b$ = –2.1). Other similar work investigating magnetic fields near massive star clusters has been done by @Harvey:2011 and @Purcell:2015. In this work, we continue our investigation of how regions and stellar bubbles modify the ambient Galactic magnetic field by considering another example of a young star cluster and an region that appears to be formed into a shell by the effect of stellar winds.
Faraday Rotation and Magnetic Fields in the Interstellar Medium
---------------------------------------------------------------
Faraday rotation measurements probe the line of sight (LOS) component of the magnetic field in ionized parts of the interstellar medium (ISM), provided there is an independent estimate of the electron density. Faraday rotation is the rotation in the plane of polarization of a wave as it passes through magnetized plasma and is described by the equation $$\chi=\chi_{0}+\left[\left(\frac{e^{3}}{2\pi m_{e}^{2}c^{4}}\right)\int{n_e\ \mathbf{B}\cdot \textrm{d}\mathbf{s}}\right]\lambda^{2},
\label{eq:rmorg}$$ where $\chi$ is the polarization position angle, $\chi_0$ is the intrinsic polarization position angle, the quantities in the parentheses are the usual standard physical constants in cgs units, *n$_{\textrm{e}}$* is the electron density, **B** is the vector magnetic field, d**s** is the incremental path length interval along the LOS, and $\lambda$ is the wavelength. We define the terms in the square bracket as the rotation measure, RM, and we can express the RM in mixed but convenient interstellar units as $$\textnormal{RM}=0.81\intn_{e} \ (\text{cm$^{-3}$}) \ \mathbf{B} \ (\mu\text{G})\cdot \textrm{d}\mathbf{s} \text{ (pc) rad m$^{-2}$.}
\label{eq:rmprat}$$
The Region and Stellar Bubble Associated with the W4 Complex\[sec:structure\]
-----------------------------------------------------------------------------
The region and stellar bubble of interest for the present study is IC 1805, which is located in the Perseus Arm. The star cluster responsible for the region and stellar bubble is OCl 352, which is a young cluster (1–3 Myr) [@Basu:1999]. OCl 352 has 60 OB stars [@Shi:1999]. Three of these are the O stars HD 15570, HD 15558, and HD 15629, and they have mass loss rates between 10$^{-6}$ and 10$^{-5}$ [@Massey:1995] and terminal wind velocities of 2200 – 3000 [@Garmany:1988; @Groenewegen:1989; @Bouret:2012]. We adopt the nominal center of the star cluster to be R.A.(J2000) = 02$^h$ 23$^m$ 42$^s$, decl.(J2000) = +6127$'$ 0$''$ ($\ell$ = 134.73, *b* = +0.92) [@Guetter:1989] and a distance of 2.2 kpc to IC 1805 to conform with previous studies of the region (e.g., @Normandeau:1996 [@Dennison:1997; @Reynolds:2001; @Terebey:2003; @Gao:2015]). In the literature, other distance values include: 2.35 kpc [@Massey:1995; @Basu:1999; @West:2007; @Lagrois:2012], 2 kpc [@Dickel:1980], 2.04 kpc [@Feigelson:2013; @Townsley:2014], and 2.4 $\pm$ 0.1 kpc [@Guetter:1989].
We refer to the region between –0.2 $<$ *b* $<$ 2 as IC 1805. This structure is also known as the Heart Nebula for its appearance at optical wavelengths. We differentiate this region from the northern latitudes that constitute the W4 Superbubble [@Normandeau:1996; @West:2007; @Gao:2015], and we use the nomenclature of W4 to describe the entire region, which includes IC 1805 and the W4 Superbubble. Below we summarize the structure of IC 1805 and Figure \[fig:cartoon\] is a cartoon diagram of the structure described here.
- *South.* On the southern portion of IC 1805, there is a loop structure of ionized material at 134$<$ $\ell$ $<$ 136, *b* $<$ 1, which we call the southern loop. @Terebey:2003 find that at far infrared and radio wavelengths, the shell structure is well defined and ionization bounded, since the ionized gas lies interior to the dust shell. However, they also find that there is warm dust that extends past the southern loop and a faint ionized halo (see their Figure 6). @Terebey:2003 argue that the shell is patchy and inhomogeneous in density, which allows ionizing photons to escape. @Gray:1999 discuss extended emission surrounding IC 1805 and suggest that it may be evidence of an extended region [@Anantharamaiah:1985]. Also surrounding IC 1805 are patchy regions of [@Braunsfurth:1983; @Hasegawa:1983; @Sato:1990] and CO [@Heyer:1998; @Lagrois:2009].
@Terebey:2003 model the structure of the southern loop using radio continuum data. They assume a spherical shell and place OCl 352 at the top edge of the bubble instead of at the center to accommodate spherical symmetry (see their Figures 4 and 5). The center of their shell model is at ($\ell$, $b$) = (135.02, 0.42). They find an inner radius of 30 arcmin (19 pc) and a shell thickness of 10 arcmin (6 pc) and 2.5 arcmin (2 pc) for a thick and thin shell model, respectively. @Terebey:2003 report electron densities of 10 and 20 for the thick and thin shell models, respectively (see Section 3.5 and Table 3 of @Terebey:2003). While we utilize and discuss these models in the following sections, the center position of the shell in @Terebey:2003 was selected to fit the ionized shell, and as such, the shell parameters should only be used to describe the bottom of IC 1805. For latitudes near the star cluster, the model fails, as the star cluster is at the top edge of the bubble instead of at the center.
- *East.* On the eastern edge of IC 18
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'The work on large-scale graph analytics to date has largely focused on the study of static properties of graph snapshots. However, a static view of interactions between entities is often an oversimplification of several complex phenomena like the *spread of epidemics*, *information diffusion*, *formation of online communities*, and so on. Being able to find temporal interaction patterns, visualize the evolution of graph properties, or even simply compare them across time, adds significant value in reasoning over graphs. However, because of lack of underlying data management support, an analyst today has to manually navigate the added temporal complexity of dealing with large evolving graphs. In this paper, we present a system, called *Historical Graph Store*, that enables users to store large volumes of historical graph data and to express and run complex temporal graph analytical tasks against that data. It consists of two key components: a [*Temporal Graph Index*]{} (TGI), that compactly stores large volumes of historical graph evolution data in a partitioned and distributed fashion; it provides support for retrieving snapshots of the graph as of any timepoint in the past or evolution histories of individual nodes or neighborhoods; and a Spark-based [*Temporal Graph Analysis Framework*]{} (TAF), for expressing complex temporal analytical tasks and for executing them in an efficient and scalable manner. Our experiments demonstrate our system’s efficient storage, retrieval and analytics across a wide variety of queries on large volumes of historical graph data.'
author:
- |
Udayan Khurana\
\
Amol Deshpande\
\
bibliography:
- 'TGIdraft-extended.bib'
title:
- Historical Graph Data Management
- Storing and Analyzing Historical Graph Data at Scale
---
Introduction
============
Graphs are useful in capturing behavior involving interactions between entities. Several processes are naturally represented as graphs – social interactions between people, financial transactions, biological interactions among proteins, geospatial proximity of infected livestock, and so on. Many problems based on such graph models can be solved using well-studied algorithms from graph theory or network science. Examples include finding driving routes by computing shortest paths on a network of roads, finding user communities through dense subgraph identification in a social network, and many others. Numerous graph data management systems have been developed over the last decade, including specialized graph database systems like Neo4j, Titan, etc., and large-scale graph processing frameworks such as Pregel [@pregel], Giraph, GraphLab [@distgraphlab], GraphX [@graphx], GraphChi [@kyrola2012graphchi], etc. However much of the work to date, especially on cloud-scale graph data management systems, focuses on managing and analyzing a single (typically, current) static snapshot of the data. In the real world, however, interactions are a dynamic affair and any graph that abstracts a real-world process changes over time. For instance, in online social media, the friendship network on Facebook or the “follows” network on Twitter change steadily over time, whereas the “mentions” or the “retweet” networks change much more rapidly. Dynamic cellular networks in biology, evolving citation networks in publications, dynamic financial transactional networks, are few other examples of such data. Lately, we have seen an increasing merit in dynamic modeling and analysis of network data to obtain crucial insights in several domains such as cancer prediction [@taylor2009dynamic], epidemiology [@gross2006epidemic], organizational sociology [@gulati1999interorganizational], molecular biology [@eisenberg2000protein], information spread on social networks [@lerman2010information] amongst others.
In this work, our focus is on providing the ability to analyze and to reason over the entire history of the changes to a graph. There are many different types of analyses of interest. For example, an analyst may wish to study the evolution of well-studied static graph properties such as centrality measures, density, conductance, etc., over time. Another approach is through the search and discovery of temporal patterns, where the events that constitute the pattern are spread out over time. Comparative analysis, such as juxtaposition of a statistic over time, or perhaps, computing aggregates such as *max* or *mean* over time, possibly gives another style of knowledge discovery into temporal graphs. Most of all, a primitive notion of just being able to access past states of the graphs and performing simple static graph analysis, empowers a data scientist with the capacity to perform analysis in arbitrary and unconventional patterns.
Supporting such a diverse set of temporal analytics and querying over large volumes of historical graph data requires addressing several data management challenges. Specifically, there is a want of techniques for storing the historical information in a compact manner, while allowing a user to retrieve graph snapshots as of any time point in the past or the evolution history of a specific node or a specific neighborhood. Further the data must be stored and queried in a distributed fashion to handle the increasing scale of the data. We must also develop an expressive, high-level, easy-to-use programming framework that will allow users to specify complex temporal graph analysis tasks, while ensuring that the specified tasks can be executed efficiently in a data-parallel fashion across a cluster.
In this paper, we present a graph data management system, called [*Historical Graph Store (HGS)*]{}, that provides an ecosystem for managing and analyzing large historical traces of graphs. HGS consists of two key distinct components. First, the [*Temporal Graph Index (TGI)*]{}, is an index that compactly stores the entire history of a graph by appropriately partitioning and encoding the differences over time (called [*deltas*]{}). These deltas are organized to optimize the retrieval of several temporal graph primitives such as neighborhood versions, node histories, and graph snapshots. TGI is designed to use a distributed key-value store to store the partitioned deltas, and can thus leverage the scalability afforded by those systems (our implementation uses Apache Cassandra[^1] key-value store). TGI is a tunable index structure, and we investigate the impact of tuning the different parameters through an extensive empirical evaluation. TGI builds upon our prior work on DeltaGraph [@icdepaper], where the focus was on retrieving individual snapshots efficiently; we discuss the differences between the two in more detail in Section \[sec:tgi\].
The second component of HGS is a *Temporal Graph Analysis Framework (TAF)*, which provides an expressive library to specify a wide range of temporal graph analysis tasks and to execute them at scale in a cluster environment. The library is based on a novel set of *temporal graph operators* that enable a user to analyze the history of a graph in a variety of manners. The execution engine itself is based on Apache Spark [@zaharia2010spark], a large-scale in-memory cluster computing framework.
[ ]{} The rest of the paper is organized as follows. In Section \[sec:related\], we survey the related work on graph data stores, temporal indexing, and other topics relevant to the scope of the paper. In Section \[sec:overview\], we provide a sketch of the overall system, including key aspects of the underlying components. We then present the Temporal Graph Index and the Temporal Graph Analytics Framework in detail in Section \[sec:tgi\] and Section \[sec:taf\], respectively. In Section \[sec:experiments\], we provide an empirical evaluation of the various system components such as the graph retrieval, scalability of temporal analytics, etc. We conclude with a summary and a list of future directions in Section \[sec:conclusion\].
Related Work {#sec:related}
============
In the recent years, there has been much work on graph storage and graph processing systems and numerous systems have been designed to address various aspects of graph data management. Some examples include Neo4J, AllegroGraph [@aasman2006allegro], Titan[^2], GBase [@kang2011gbase], Pregel [@pregel], Giraph, GraphChi [@kyrola2012graphchi], GraphX [@graphx], GraphLab [@distgraphlab], and Trinity [@shao2013trinity]. These systems use a variety of different models for representation, storage, and querying, and there is a lack of standardized or widely accepted models for the same. Most graph querying happens through programmatic access to graphs in languages such as Java, Python or C++. Graph libraries such as Blueprints[^3] provide a rich set of implementations for graph theoretic algorithms. SPARQL [@perez2006semantics] is a language used to search patterns in linked data. It works on an underlying RDF representation of graphs. T-SPARQL [@grandi2010t] is a temporal extension of SPARQL. He et al. [@he:sigmod08], provide a language for finding sub-graph patterns using a graph as a query primitive. Gremlin[^4] is a graph traversal language over the property graph data model, and has been adopted by several open-source systems. For large-scale graph analysis, perhaps the most popular framework is the vertex-centric programming framework, adopted by Giraph, GraphLab, GraphX, and several other systems; there have also been several proposals for richer and more expressive programming frameworks in recent years. However, most of these prior systems largely focus on analyzing a single snapshot of the graph data, with very little support for handling dynamic graphs, if any.
A few recent papers address the issues of storage and retrieval in dynamic graphs. In our prior work, we proposed DeltaGraph [@icdepaper], an index data structure that compactly stores the history of all changes in a dynamic graph and provides efficient snapshot reconstruction. G
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We study the nonequilibrium dynamics in the pump-probe spectroscopy of excitonic insulators using the spinless two-orbital model with phonon degrees of freedom in the time-dependent mean-field approximation. We introduce the pulse light as a time-dependent vector potential via the Peierls phase in the Hamiltonian. We find that, in the Bose-Einstein condensation regime where the normal state is semiconducting, the excitonic order is suppressed when the frequency of the pulse light is slightly larger than the band gap, while the order is enhanced when the frequency of the pulse is much larger than the band gap. We moreover find that the excitonic order is completely destroyed in the former situation if the intensity of the pulse is sufficiently strong. In the BCS regime where the normal state is semimetallic, we find that the excitonic order is always suppressed, irrespective of the frequency of the pulse light. The quasiparticle band structure and optical conductivity spectrum after the pumping are also calculated for the instantaneous states.'
author:
- Tetsuhiro Tanabe$^1$
- Koudai Sugimoto$^2$
- Yukinori Ohta$^1$
title: 'Nonequilibrium dynamics in the pump-probe spectroscopy of excitonic insulators'
---
Introduction
============
Nonequilibrium dynamics induced by applying the intense laser pulse have recently been a new way of investigating a variety of quantum condensed phases. Recent achievement of the time resolution of a femto-second order enables one to perform experiments studying the ultrafast dynamics of materials. Examples include a success of observing light-induced superconductivity [@Fausti2011Science; @Mitrano2016Nature] and a pump-probe measurement of melting of charge-density-wave orders [@Perfetti2006PRL; @Schmitt2008Science].
The pump-probe measurement is also applicable to the study of excitonic condensation. In the excitonic phase, holes in the valence band and electrons in the conduction band form pairs called excitons, just like Cooper pairs of electrons in superconductivity, and they undergo quantum condensation at low temperatures [@Jerome1967PR; @Halperin1968RMP]. The realization of such condensation has been suggested in transition-metal chalcogenides $1T$-TiSe$_2$ [@Cercellier2007PRL; @Kogar2017Science] and Ta$_2$NiSe$_5$ [@Wakisaka2009PRL; @Seki2014PRB]. Here, we should note that, since the spin-singlet excitonic state necessarily couples to the phonon degrees of freedom [@Phan2013PRB; @Zenker2014PRB; @Kaneko2013PRB; @Sugimoto2016PRB; @Kaneko2018PRB], it is difficult to single out the excitonic contributions at least in equilibrium state experiments. There are, however, some attempts to distinguish between the excitonic and phononic contributions using the nonequilibrium dynamics induced by laser pulse in 1$T$-TiSe$_2$ [@Rohwer2011Nature; @Mohr-Vorobeva2011PRL; @Hellmann2012NC; @Monney2016PRB]. In Ta$_2$NiSe$_5$, Mor *et al.* found that the band gap can be controlled by the excitation density [@Mor2017PRL] and argued that its nonequilibrium phenomena come from the exciton dynamics [@Mor2018PRB]. Coherent order parameter oscillations caused by the induced phonons were also observed [@Werdehausen2018JPC; @Werdehausen2018SA].
The pump-probe spectroscopy experiments in the excitonic phases have been interpreted from the theoretical point of view. While the GW calculations showed that the excitonic order vanishes after applying the laser pulse in the BCS regime where the normal state is semimetallic [@Golez2016PRB], Murakami *et al.* [@Murakami2017PRL] recently showed that the excitonic order can be enhanced by the laser pulse in the Bose-Einstein condensation (BEC) regime where the normal state is semiconducting. Tanaka *et al.* [@Tanaka2018PRB] also showed that the switching between the melting and enhancement of excitonic orders can occur when the order varies from the BCS regime to BEC regime.
Note that all of these calculations [@Golez2016PRB; @Murakami2017PRL; @Tanaka2018PRB] assumed that the laser pulse excites the electrons in the valence-band orbital directly to the conduction-band orbital via the dipole transition. However, as was discussed in Ref. \[\], the matrix elements of the dipole transition can be small in the case where the valence-band and conduction-band orbitals are well-localized and are spatially separated in distant positions, just as in Ta$_2$NiSe$_5$ [@Kaneko2013PRB]. The Peierls term, on the other hand, can survive even in such situations [@Wissgott2012PRB]. Thus, there is another way of treating the laser pulse, which is to introduce a time-dependent vector potential via the Peierls phase in a tight-binding Hamiltonian.
In this paper, we study the nonequilibrium dynamics of excitonic insulator states applying the time-dependent mean-field approximation to the spinless two-orbital model in one-dimension (1D) with phonon degrees of freedom, whereby we simulate the situation where an optical laser pulse is applied to the system as a pump light. Unlike preceding studies, we here introduce the pulse light as a time-dependent vector potential via the Peierls phase in the Hamiltonian of the external field, assuming the situations where the dipole matrix elements are small. We note that the spontaneous hybridization between the valence-band and conduction-band orbitals occurs in the symmetry-broken excitonic insulator state, so that the interband excitations by the Peierls mechanism can work in the present model. We thus investigate the time evolution of the excitonic order parameter in both the BEC and BCS regimes, paying particular attention to its dependence on the frequency and intensity of the laser light.
We will show that, in the BEC regime where the normal state is semiconducting, the excitonic order is suppressed when the frequency of the pulse light is slightly larger than the band gap, while the order is enhanced when the frequency of the pulse is much larger than the band gap. In the BCS regime where the normal state is semimetallic, we will show that the excitonic order is always suppressed, irrespective of the frequency of the pulse light. We will demonstrate that the excitonic order parameter oscillation occurs in agreement with experiment. We will also calculate the optical conductivity spectrum assuming a single-time, instantaneous response for a quasi-steady state after pumping and demonstrate the measurement is a useful way for probing the nonequilibrium dynamics of excitonic insulator states.
The rest of this paper is organized as follows. In Sec. II, we introduce the spinless two-orbital model, define the laser pulse light, and derive the equations of motions for the excitonic order parameters in the time-dependent mean-field approximation. In Sec. III, we present results for nonequilibrium dynamics induced by laser pulse in both BEC and BCS regimes. We also present results for the optical conductivity spectra in the nonequilibrium state. We summarize our results and discuss their experimental significance in Sec. IV.
Model and Method
================
Spinless two-orbital model
--------------------------
As a minimum model for describing the spin-singlet excitonic insulator state coupled with phonon degrees of freedom, we consider the spinless two-orbital model (or extended Falicov-Kimball model [@Ihle2008PRB; @Zenker2011PRB; @Seki2011PRB; @Ejima2014PRL]) defined on the 1D lattice \[see Fig. \[fig1\](a)\], interacting with Einstein phonons of frequency $\omega_{0}$ [@Murakami2017PRL]. This model may be relevnt to the electronic state of an excitonic insulator candidate Ta$_2$NiSe$_5$ with a quasi-1D crystal structure [@Seki2014PRB], although the method discussed below is applicable to higher dimensional systems as well. Our model is defined by the Hamiltonian $$H = H_\mathrm{e} + H_{\mathrm{e,int}} + H_{\mathrm{ph}} + H_{\mathrm{e-ph}}
\label{eq:Hamiltonian}$$ with $$\begin{aligned}
&H_\mathrm{e}
= -\sum_{i, \alpha} \big( J_{\alpha} c^\dagger_{i + 1, \alpha} c_{i, \alpha} + \mathrm{H.c.} \big)
+ \sum_{i, \alpha} \Delta_{\alpha} c^\dagger_{i, \alpha} c_{i, \alpha}
\label{eq:H_e}
\\
&H_{\mathrm{e,int}}
= U \sum_i c^\dagger_{i, 0} c_{i, 0} c^\dagger_{i, 1} c_{i, 1}
\\
&H_{\mathrm{ph}}
= \omega_{0} \sum_{i} b^\dagger_{i} b_{i}
\\
&H_{\mathrm{e-ph}}
= g \sum_{i} \big( b^\dagger_{i} + b_{i} \big) \big( c^\dagger_{i, 1} c_{i, 0} + \mathrm{H.c.} \big),\end{aligned}$$ where $c_{i, \alpha}$ ($c^\dagger
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'Gas accretion is necessary to maintain star formation, spiral and bar structure, and secular evolution in galaxies. This can occur through tidal interaction, or mass accretion from cosmic filaments. Different processes will be reviewed to drive gas towards galaxy centers and trigger starbursts and AGN. The efficiency of these dynamical processes can be estimated through simulations and checked by observations at different redshift, across the Hubble time. Large progress has been made on galaxies at moderate and high redshifts, allowing to interpret the star formation history and star formation efficiency as a function of gas content, dynamical state and galaxy evolution.'
author:
- Francoise Combes
title: Gas accretion in disk galaxies
---
Introduction
============
In the last decade, cosmological simulations have emphasized the importance of cold gas accretion onto galaxies in the mass assembly, in particular at high redshift (e.g. Keres et al. 2005, Dekel et al. 2009, Devriendt et al. 2010). This mode of accretion is thought to be about one order of magnitude more important than galaxy mergers in mass assembly, unlike what was assumed in the hierarchical scenario.
In parallel, simulations of isolated galaxies show how important is the gas accretion to maintain star formation at a constant level, as is observed in spiral galaxies, to explain abundance gradients, and also to maintain the spiral structure. In the following, the impact of gas accretion is first described, and then we will review the evidence of circumgalactic gas inflow.
Role of gas accretion: secular evolution, bars
==============================================
Secular evolution involves mainly the disk galaxies of the Hubble sequence: all spirals and irregulars. As for the ellipticals, their formation scenario is still heavily relying on mergers, either a few major mergers, or more likely a series of minor mergers, to heat the stellar component, destroy disks progressively, and cancel out any angular momentum, explaining its low values in this class.
In disk galaxies, the main motor of evolution is non-axisymmetries and bars. When the disk is abundant in gas, as are most high redshift disky objects, then the bars are not long-lived, but weakened and destroyed by the accumulation of mass in the centers, and by the exchange of angular momentum between the gas and the stars of the bar (Friedli & Benz 1993, Berentzen et al. 1998, Bournaud & Combes 2002). A weakened bar would transiently look like a lens inside its inner ring (e.g. Laurikainen et al. 2009).
The frequency of bars in disk galaxies as a function of mass shows an interesting bimodality (Nair & Abraham 2010), with two maxima in the blue cloud and in the red sequence. This was also found by Masters et al (2011) with the Galaxy Zoo, although not by the S4G consortium (see K. Sheth, this meeting), but this could be due to selection effects.
Bars can act in conjunction with spirals, to redistribute the angular-momentum across galaxies, and to modify significantly stellar radial profiles. The non-linear interactions at resonances overlap multiply the effects (Minchev et al 2011). In less then 3 Gyrs, the effective sizes of galaxy disks may be multiplied by 3, and the corresponding radial migration brings high-velocity dispersion stars in the outer parts. Disk thickening is also substantial (Minchev et al 2012). When gas accretion is considered, the strength of bars can be re-boosted, and new stars formed out of the accreted gas continuously re-shape the radial stellar profiles. It is then possible to obtain the three observed types: Type I as a single exponential disk, Type II as a truncated one, when star formation is newly forming at the break, and beyond the break the star formation threshold is not yet reached, and Type III, the anti-truncated profile can be obtained in case of strong gas accretion in the outer parts (cf Fig \[fig:minchev-12\]). The Type III morphology appears relatively transient, and able to evolve into Type II or Type I. Its presence could be a tracer of accretion events. There is some correlation of these Type II with weakened bars, as expected from strong gas accretion (Bournaud & Combes 2002, Combes 2011).
When the gas accretion occurs essentially from non-aligned cosmic filaments, characteristic signatures may occur, such as inclined and warped rings (Roskar et al 2010), or even polar rings, when the accretion is near polar. Brook et al (2008) have shown how a lenticular system is first formed from matter accretion, which suddenly stops when the birth filament is consumed out. The next filament in the perpendicular direction then fuels a relatively stable polar ring system. Gas accretion may mimick galaxy interactions, since it can produce asymmetries, lopsidedness, clumpiness, and sfatbursts. Even if the accretion is globally symmetric and isotropic, it may be temporarily on one side only.
Gas accretion replenishes the extended gas reservoirs present around most spiral galaxies. This gas slowly spirals in, when in quiescent state. However, the first tidal interaction may drive the gas violently towards the center, and strongly affects abundance gradients, that can even be reversed (Montuori et al 2010): low-metallicity gas flows into the center and dilutes the abundance of the central gas, in a time-scale shorter than the time required for this gas to re-enrich through the triggered nuclear starburst. Such gradient reversals have been observed at high redshift in the MASSIV survey (Queyrel et al 2012).
Inside-out disk formation, inflow/outflow, metallicity
======================================================
Gas accretion occurs in the outer parts of disks, and is a way to explain inside-out disk formation. There are now multiple evidence of this progressive mode of disk formation. Through observations of galaxies as a function of redshift, it is possible to track the evolution directly. However, finding the progenitors of today galaxies is not easy. Statistically, this is solved by matching the galaxies at a given cumulative number density, for instance 1.4 10$^{-4}$ Mpc$^{-3}$. Plotting the mass of these matched galaxies at a given redshift, Patel et al (2013) follow the mass increase of galaxies over z=3 to 0. They notice that this mass increase involves only the mass of the outer parts, while the mass inside 2kpc is stable in all the same galaxies. This fact is related to the observations that the normalised galaxy radius, at a given mass, increases by a factor 3 from z=3 to 0 (Newman et al 2012). It is possible that dry minor mergers explain this increase of size, without much mass accretion, in the case of quenched galaxies. For star forming disk galaxies, external gas accretion, accompanied by secular evolution, is necessary. In the GASS sample of local galaxies detected in HI-21cm, Wang et al (2011) find that galaxies richer in HI are bluer, and the more so in the outer parts: the radial gradient of color is a function of HI-mass fraction. They conclude that the gas must be accreted slowly, and relaxed, since there is no correlation between HI-fraction and lopsidedness in this sample.
More directly, looking at H$\alpha$ in the 3D-HST project in 57 galaxies at z$\sim$ 1, Nelson et al (2012) find that the effective radii of galaxies in ionised gas and new stars is about 1.3 larger than the effective radius of the rest-frame R-band stellar continuum, representing older stars. The effet is larger for massive galaxies, that certainly are the first to form inside-out.
In this inside-out scenario, with gas accretion, the radial migration will produce a typical reversal of stellar ages in the outer parts: the gas is accreted at the radius of the break, which is where the new stars are formed, accentuating the negative age gradient from the center. After the break, only old stars migrated from the center are expected, and there is now a positive gradient (Roskar et al 2008). This age gradient reversal has been observed in M33 by Williams et al (2009). Some other galaxies do not show any break, but a flat age gradient in the outer parts (Vlajic et al 2011).
Evidence of gas accretion: HVC, warps, QSO absorption
=====================================================
Most of the gas from cosmic filaments is accreted at large scales, settles down to the disk, and spirals in progressively. Since the alignment process occurs through precession and dissipation, with a time-scale of the order of a few dynamical times at these large radii, this can take some Gyrs, during which galaxies appear warped or perturbed in the outer parts. Warps and polar rings are therefore the best tracers of external accretion, and indeed most spiral galaxies are observed to be warped (e.g. Briggs 1990, Binney 1992, Reshetnikov & Combes 1998). Also the frequency of asymmetries and lopsidedness in spiral galaxies cannot be explained but with external accretion (Jog & Combes 2009).
Searches have been done of extra-planar gas in the halo of spiral galaxies (Fraternali et al. 2002, Heald et al 2011, Gentile et al. 2013) and the quantities found are relatively small, NGC 891 being the most remarkable for its gas entension. The origin of this gas is multiple. Some gas can be ejected into the halo by stellar feedback (fountain effect), or through tidal disruption of satellites. In the Milky Way, the High Velocity Clouds (HVC) and the Magellanic stream are good examples. This gas is accreted progressively, with an interface of multiphase gas, but at a rate lower than the star formation rate (0.
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We construct an invariant of certain open four-manifolds using the Heegaard Floer theory of Ozsvath and Szabo. We show that there is a manifold $X$ homeomorphic to ${\mathbb{R}}^4$ for which the invariant is non-trivial, showing that $X$ is an exotic ${\mathbb{R}}^4$. This is the first invariant that detects exotic ${\mathbb{R}}^4$’s.'
address: |
Stat Math Unit,\
Indian Statistical Institute,\
Bangalore 560059, India
author:
- Siddhartha Gadgil
title: 'Open manifolds, Ozsvath-Szabo invariants and Exotic ${\mathbb{R}}^4$’s'
---
Introduction
============
In this paper, we construct invariants of certain open $4$-manifolds using the Heegaard Floer theory of Ozsvath and Szabo, and show that our invariants can detect exotic ${\mathbb{R}}^4$s. Previous constructions of exotic ${\mathbb{R}}^4$’s used indirect arguments to establish exoticity.
Given an $(n+1)$-dimensional field theory, a direct limit construction can be used to construct an invariant of open $(n+1)$-dimensional manifolds (which we see in detail later). The subtlety in the case of Ozsvath-Szabo invariants is that they do not give a field theory, but satisfy a more complicated composition law. However if we restrict to a class of cobordisms, which we call *admissible cobordisms*, we do get a field theory. Using this, we construct our invariants.
Recall that the Ozsvath-Szabo invariants of a smooth, oriented $3$-manifold $M$ associate homology groups to $M$ equipped with a $Spin^c$ structure $t$. Further, given a smooth cobordism $W$ between $3$-manifolds $M_1$ and $M_2$ and a $Spin^c$ structure ${\mathfrak{s}}$ on $W$, we get an induced map on the groups associated to the restrictions of ${\mathfrak{s}}$ to $M_1$ and $M_2$. To make this into a field theory, one needs a composition rule for a cobordism $W_1$ from $M_1$ to $M_2$ equipped with a $Spin^c$ structure ${\mathfrak{s}}_1$ and a cobordism $W_2$ from $M_2$ to $M_3$ equipped with a $Spin^c$ structure ${\mathfrak{s}}_2$ with ${\mathfrak{s}}_1|_{M_2}={\mathfrak{s}}_2|_{M_2}$. However, such $Spin^c$ structures ${\mathfrak{s}}_1$ and ${\mathfrak{s}}_2$ do not in general uniquely determine a $Spin^c$ structure on the composition $W=W_1\coprod_{M_2} W_2$ of $W_1$ and $W_2$. We do have a weaker composition law, where we sum over $Spin^c$ structures on $W$ restricting to ${\mathfrak{s}}_1$ and ${\mathfrak{s}}_2$.
We now find sufficient conditions under which ${\mathfrak{s}}_1$ and ${\mathfrak{s}}_2$ uniquely determine a $Spin^c$ structure ${\mathfrak{s}}$ on $W$. The $Spin^c$ structures on a manifold $X$ are a torseur of $H^2(X,{\mathbb{Z}})$. Consider the Mayer-Vietoris sequence for $W=W_1\cup W_2$ $$\to H^1(W_1)\oplus H^1(W_2)\to H^1(M_2)\overset{\delta}\to H^2(W)\to
H^2(W_1)\oplus H^2(W_2)\to H^2(M_2)$$
From this sequence, it follows that, given ${\mathfrak{s}}_1$ and ${\mathfrak{s}}_2$ as above, there is a unique $Spin^c$ structure ${\mathfrak{s}}$ on $W$ which restricts to ${\mathfrak{s}}_1$ and ${\mathfrak{s}}_2$ if and only if the coboundary map $\delta:H^1(M_2)\to
H^2(W)$ is trivial. This is equivalent to the map induced by inclusions $H^1(W_1)\oplus H^1(W_2)\to H^1(M_2)$ being surjective. Motivated by this, we make the following definition.
A smooth $4$-dimensional cobordism $W$ from $M_1$ to $M_2$ is admissible if the map induced by inclusion $H^1(W)\to H^1(M_2)$ is surjective.
We shall see basic properties of such cobordisms in Section \[cnvx\]. We now turn to the corresponding notions for open manifolds. Let $X$ be an open $4$-manifold which we assume for simplicity has one end. Let $K_1\subset K_2\subset \dots$ be an exhaustion of $X$ by compact manifolds and let $M_i={\partial}K_i$. We assume here and henceforth (for all exhaustions) that $K_i\subset
int(K_{i+1})$. For $i<j$, let $W_{ij}=K_j-int(K_i)$ be cobordisms from $M_i$ to $M_j$.
The exhaustion $\{K_i\}$ of $X$ is said to be admissible if each cobordism $W_{ij}$, $i,j\in{\mathbb{N}}$, $i<j$, is admissible. The manifold $X$ is said to be admissible if it has an admissible exhaustion.
We shall need to consider the appropriate notion of $Spin^c$ structures for the ends of $4$-manifolds.
An asymptotic $Spin^c$ structure ${\mathfrak{s}}$ on $X$ is a $Spin^c$ structure on $X-K$ for a compact subset $K\subset X$. Two asymptotic $Spin^c$ structures ${\mathfrak{s}}_1$ and ${\mathfrak{s}}_2$, defined on $X-K_1$ and $X-K_2$, are said to be equal if there is a compact set $K_0\supset K_1,K_2$ with ${\mathfrak{s}}_1|_{M-K_0}={\mathfrak{s}}_2|_{M-K_0}$.
Given an admissible open $4$-manifold $X$ and an asymptotic $Spin^c$ structure ${\mathfrak{s}}$, we can define invariants of $X$, which we call the *End Floer Homology*, using direct limits. We shall see in Section \[inv\] that an admissible exhaustion gives a directed system.
There is an invariant $HE(X,{\mathfrak{s}})$ which is the direct limit of the reduced Heegaard Floer homology groups $HF^+_{red}(M_i,{\mathfrak{s}}|_{M_i})$ under morphisms induced by the cobordisms $W_{ij}$. Furthermore this is independent of the admissible exhaustion of $X$.
We shall also need a *twisted* version of these invariants. Let $K\subset X$ be a compact set, ${\mathfrak{s}}$ a $Spin^c$-structure on $X-K$ and $\omega$ a $2$-form on $X-K$. Then we consider the reduced Floer theory with $\omega$-twisted coefficients (as in [@OZ4]). Once more we get a directed system whose limit gives an invariant $\underline{HE}(X,{\mathfrak{s}})$.
By taking an exhaustion of ${\mathbb{R}}^4$ by balls, we have the following proposition.
For the unique asymptotic $Spin^c$ structure ${\mathfrak{s}}$ on ${\mathbb{R}}^4$ (and any $2$-form $\omega$ on ${\mathbb{R}}^4-K$ with $K$ compact), we have $\underline{HE}({\mathbb{R}}^4,{\mathfrak{s}})=0$.
Our main result is that there are manifolds homeomorphic to ${\mathbb{R}}^4$ but with non-vanishing end Floer homology.
\[exot\] There is a $4$-manifold $X$ homeomorphic to ${\mathbb{R}}^4$ such that there is a compact set $K\subset X$, a $Spin^{{\mathbb{C}}}$ structure ${\mathfrak{s}}$ on $X-K$ and a closed $2$-form $\omega$ on $X-K$ with $\underline{HE}(X,{\mathfrak{s}})\neq 0$ with $\omega$-twisted coefficients.
Thus, $X$ is an exotic ${\mathbb{R}}^4$. Previous constructions of exotic ${\mathbb{R}}^4$’s used indirect arguments to show that they are exotic. The *End Floer homology* is the first invariant that detects exotic ${\mathbb{R}}^4$’s.
Admissible cobordisms and admissible ends {#cnvx}
=========================================
We henceforth assume that all our manifolds are smooth and oriented and all cobordisms are compact and $4$-dimensional. By $W:M_1\to M_2$ we mean a smooth cobordism from the closed $3$-manifold $M_1$ to the closed $3$-manifold $M_2$. Given $W_1
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{
"pile_set_name": "ArXiv"
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---
abstract: 'We show that the usual sufficient criterion for a very general hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to quasi-smooth hypersurfaces in complete simplicial toric varieties. This sufficient condition always holds for very general K3 surfaces embedded in Fano toric 3-folds.'
address:
- |
$^\P$ Department of Mathematics, University of Pennsylvania,\
David Rittenhouse Laboratory, 209 S 33rd Street,\
Philadelphia, PA 19104, USA[^1][^2]
- '$^\S$ Istituto Nazionale di Fisica Nucleare, Sezione di Trieste'
author:
- 'Ugo Bruzzo$^{\P\S\dag}$ and Antonella Grassi$^\P$'
title: |
Picard group of hypersurfaces\
in toric 3-folds
---
SISSA Preprint 78/2010/fm\
[arXiv:1011.1003]{}
[^3]
Introduction
============
In this paper we study the Noether-Lefschetz problem for hypersurfaces in complete simplicial toric threefolds, namely, we prove that under a certain condition, a very general hypersurface in an ample linear system in such a toric threefold ${{{{\mathbb P}}_\Sigma}}$ has the same Picard number as ${{{{\mathbb P}}_\Sigma}}$. In particular, this holds for a very general K3 hypersurface in the anticanonical system of a simplicial toric Fano threefold. (A property is very general if it holds in the complement of countably many proper closed subvarieties [@Lazarsfeld].)
This result can be regarded on one hand as a first step towards the study of Noether-Lefschetz loci of the moduli space of $K3$ hypersurfaces in a toric Fano threefolds; see also the recent works of [@KMPS; @MPP; @Kloos07]. On the other hand, this completes the picture for computing the Picard number for certain hypersurfaces in the anticanonical system of a toric Fano variety, by handling the unknown case in dimension $3$.
Recall that the Picard number, $\rho(Y)$ of a variety $Y$ is the rank of the Néron-Severi group, that is of the image of the Picard group in the second cohomology group with integer coefficients. The Picard group and the Picard number of a toric variety ${{{{\mathbb P}}_\Sigma}}$ can be easily computed from the combinatorial data of $\Sigma$. Let $X$ be a nondegenerate hypersurface in the anticanonical system of a simplicial toric Fano variety ${{{{\mathbb P}}_\Sigma}}$, with $\dim {{{{\mathbb P}}_\Sigma}}\geq 4$ (note that a general hypersurface is also nondegenerate). In the 80s and 90s it was shown [@CoKa99; @Bat94; @DK; @AGM], that the Picard number of any such $X$ can be explicitly computed from combinatorial data. This result was a pivotal ingredient in describing the toric version of mirror symmetry (see for example [@CoKa99]). The argument in the above papers is essentially topological and computes the dimension of the second cohomology group of $X$, which happens to be equal to $\rho(X)$ if $\dim (X) \geq 3$, but not necessarily if $\dim (X)=2$. In addition, even the statement in the above papers does not hold when $\dim {{{{\mathbb P}}_\Sigma}}=3$, as we see from the case of Fermat’s quartic in ${{\mathbb P}}^3$, which is nondegenerate.
This type of result was generalized by Roan to the case of toric varieties (not necessarily Fano) also for the case when the ambient variety has dimension $d\ge 4$ [@Roan96], and by Ravindra and Srinivas to general normal varieties, still with the restriction $d\ge 4$ [@RaviSri06].
This paper then fills the gap for $\dim {{{{\mathbb P}}_\Sigma}}= 3$. It was already known that $\rho (X)= \rho({{{{\mathbb P}}_\Sigma}})$ for particular cases of toric Fano threefolds, namely certain weighted projective spaces [@CoxWeighted; @SJ; @deJongSteen], as in the higher dimensional case. The techniques used in the case of weighted projective spaces are very much tailored to that specific case [@Dolgy; @SJ; @deJongSteen]. On the other hand, the classical infinitesimal techniques introduced in the 70s by Griffiths, Steenbrink and collaborators to solve the Noether-Lefschetz problem in the smooth case (see for example [@CMP03]) cannot be used due to the presence of singularities. Our argument is partly inspired by Cox’s paper [@CoxWeighted]: it generalizes the classical infinitesimal techniques and combines them with more recent results about toric varieties, their Cox ring and their cohomology [@BaCox94]. In fact, $X$ and ${{{{\mathbb P}}_\Sigma}}$ are projective orbifolds, and a pure Hodge structure can be defined for them; this will be a key tool in the proof.
In Section \[background\] we mostly recall some relevant results from [@BaCox94], and adapt them to the set up of [@CMP03]. We start with basic properties of simplicial toric varieties and general hypersurfaces defined by ample divisors. Moreover we note that the exact sequence defining the primitive cohomology in middle dimension of such a hypersurface splits orthogonally with respect to the intersection pairing. The middle cohomology is the sum of the primitive cohomology and the “fixed” cohomology, i.e., the cohomology inherited from the ambient toric variety; the splitting is consistent with the Hodge decomposition. We then state some results of [@BaCox94] which express the primitive cohomology in middle degree in terms of the Jacobian ring of the hypersurface; here we assume that ambient space has odd dimension.
Section \[Pic\] contains the bulk of the argument: we proceed along the lines of the infinitesimal arguments of Griffiths for smooth varieties and adapt it to the toric case. We start from the moduli space of quasi-smooth hypersurfaces constructed in [@BaCox94], consider a natural Gauss-Manin connection, proceed to prove an infinitesimal Noether-Lefschetz theorem and then the needed global Noether-Lefschetz theorem. Finally, we focus on the case of $K3$ hypersurfaces in the anticanonical system of a simplicial toric Fano threefold.
The suggestion that a very general hypersurface in a toric Fano threefold ${{{{\mathbb P}}_\Sigma}}$ has the same Picard number as the ambient variety can be found, in a different language, in an unpublished paper by Rohsiepe [@Rohsiepe] (see the formula and Remark in the middle of page 3), based on some dimension counting arguments and trying to generalize to the case $\dim {{{{\mathbb P}}_\Sigma}}=3 $ a formula that Batyrev proved for $\dim{{{{\mathbb P}}_\Sigma}}= 4$ [@Bat94].
[**Acknowledgements.**]{} We thank Eduardo Cattani, Alberto Collino, David Cox, Igor Dolgachev, Luca Migliorini, Vittorio Perduca, Domingo Toledo and the referee for useful discussions and suggestions. We are grateful for the hospitality and support offered by the University of Pennsylvania and SISSA. The first author would also like to thank the staff and the scientists at Penn’s Department of Mathematics for providing an enjoyable and productive atmosphere.
Hypersurfaces in simplicial complete toric varieties {#background}
====================================================
In this section we recall some basic facts about hypersurfaces in toric varieties and their cohomology. We mainly follow the notation in [@BaCox94]. All schemes are schemes over the complex numbers.
Preliminaries and notation
--------------------------
Let $M$ be a free abelian group of rank $d$, let $N={\operatorname{Hom}}(M,{{\mathbb Z}})$, and $N_{{\mathbb R}}=N\otimes_{{\mathbb Z}}{{\mathbb R}}$.
[@BaCox94 Def. 1.1 and 1.3]
1. A convex subset $\sigma\subset N_{{\mathbb R}}$ is a rational $k$-dimensional simplicial cone if there exist $k$ linearly independent primitive elements $e_1,\dots,e_k\in N$ such that $\sigma = \{\mu_1e_1+\dots+\mu_ke_k\}$, with $\mu_i$ nonnegative real numbers. The generators $e_i$ are said to be [*integral*]{} if for every $i$ and any nonnegative rational number $\mu$, the product $\mu\,e_i$ is in $N$ only if $\mu$ is an integer.
2. Given two rational simplicial cones $\sigma$, $\sigma'$, one says that $\sigma'$ is a face of $\sigma$ (we then write $\sigma' < \sigma$) if the set of integral generators of $\sigma'$ is a subset of the set of integral generators of $\sigma$.
3. A finite set $\Sigma=\{\sigma_1,\dots,\sigma_s\}$ of rational simplicial cones is called a rational simplicial complete $d$-dimensional fan if
1. all faces of cones in $\Sigma$ are in $\Sigma$;
2. if $\sigma,\sigma'\in\Sigma$, then $\sigma\
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{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'From Crofton’s formula for Minkowski tensors we derive stereological estimators of translation invariant surface tensors of convex bodies in the $n$-dimensional Euclidean space. The estimators are based on one-dimensional linear sections. In a design based setting we suggest three types of estimators. These are based on isotropic uniform random lines, vertical sections, and non-isotropic random lines, respectively. Further, we derive estimators of the specific surface tensors associated with a stationary process of convex particles in the model based setting.'
author:
- 'Astrid Kousholt[^1]'
- Markus Kiderlen
- Daniel Hug
bibliography:
- 'litteratur.bib'
title: Surface tensor estimation from linear sections
---
**Keywords** Crofton formula, Minkowski tensor, stereology, isotropic random line, anisotropic random line, vertical section estimator, minimal variance estimator, stationary particle process, stereological estimator\
\
**MSC2010** 60D05, 52A22, 53C65, 62G05, 60G55
Introduction
============
In recent years, there has been an increasing interest in Minkowski tensors as descriptors of morphology and shape of spatial structures of physical systems. For instance, they have been established as robust and versatile measures of anisotropy in [@Beisbart2002; @SM10; @Schroder-Turk2013]. In addition to the applications in materials science, [@Beisbart] indicates that the Minkowski tensors lead to a putative taxonomy of neuronal cells. From a pure theoretical point of view, Minkowski tensors are, likewise, interesting. This is illustrated by Alesker’s characterization theorem [@Alesker1999], stating that the basic tensor valuations (products of the Minkowski tensors and powers of the metric tensor) span the space of tensor-valued valuations satisfying some natural conditions.
This paper presents estimators of certain Minkowski tensors from measurements in one-dimensional flat sections of the underlying geometric structure. We restrict attention to translation invariant Minkowski tensors of convex bodies, more precisely, to those that are derived from the top order surface area measure; see Section \[prelim\] for a definition. As usual, the estimators are derived from an integral formula, namely the Crofton formula for Minkowski tensors. We adopt the classical setting where the sectioning space is affine and integrated with respect to the motion invariant measure. Rotational Crofton formulae where the sectioning space is a linear subspace and the rotation invariant measure on the corresponding Grassmannian is used, are established in [@A-C2013]. The latter formulae were the basis for local stereological estimators of certain Minkowski tensors in [@Jensen2013] (for $j \in \{1, \dots, n-1\}, s,r \in \{0,1\}$ and $j=n, s=0, r \in {{\mathbb N}}$ in the notation of and , below).
Kanatani [@Kanatani1984; @Kanatani1984a] was apparently the first to use tensorial quantities to detect and analyse structural anisotropy via basic stereological principles. He expresses the expected number $N(m)$ of intersections per unit length of a probe with a test line of given direction $m$ as the cosine transform of the spherical distribution density $f$ of the surface of the given probe in ${{\mathbb R}}^n$ for $n=2,3$. The relation between $N$ and $f$ is studied by expanding $f$ into spherical harmonics and by using the fact that these are eigenfunctions of the cosine transform. In order to express his results independently of a particular coordinate system, Kanatani uses tensors. For a fixed $s$, he considers the vector space $V_s$ of all symmetric tensors spanned by the elementary tensor products $u^{\otimes s}$ of vectors $u$ from the unit sphere $S^{n-1}$. Let $\hat T$ denote the deviator part (or trace-free part) of some symmetric tensor $T$. The tensors $\widehat{(u^{\otimes k})}$, for $k\le s$ and $u\in S^{n-1}$, then span $V_s$ and the components of $\widehat{(u^{\otimes k})}$ with respect to an orthonormal basis of ${{\mathbb R}}^n$ are spherical harmonics of degree $k$, when considered as functions of $u$. Hence, $u\mapsto \widehat{(u^{\otimes k})}$ is an eigenfunction of the cosine transform (Kanatani calls it ‘Buffon transform’), which in fact is the underlying integral transform when considering Crofton integrals with lines, as we shall see below in . In [@Kanatani1984b; @Kanatani1985], he suggests to use these ‘fabric tensors’ to detect surface motions and the anisotropy of the crack distribution in rock.
General Crofton formulas in ${{\mathbb R}}^n$ with arbitrary dimensional flats and for general Minkowski tensors (defined in ) of arbitrary rank are given in [@Hug]. Theorem \[thm\] is a special case of one of these results, for translation invariant surface tensors and one-dimensional sections, that is, sections with lines. In comparison to [@Hug], we get simplified constants in the case considered and obtain this result by an elementary independent proof. In contrast to Kanatani’s approach, our proof does not rely on spherical harmonics. Here we focus on relative Crofton formulas in which the Minkowski tensors of the sections with lines are calculated relative to the section lines and not in the ambient space (Crofton formulas of the second type may be called extrinsic Crofton formulas). A quite general investigation of integral geometric formulas for translation invariant Minkowski tensors, including extrinsic Crofton formulas, is provided in [@BernigHug].
In Theorem \[thm\] we prove that the relative Crofton integral for tensors of arbitrary even rank $s$ of sections with lines is equal to a linear combination of surface tensors of rank at most $s$. From this we deduce by the inversion of a linear system that any translation invariant surface tensor of even rank $s$ can be expressed as a Crofton integral. The involved measurement functions then are linear combinations of relative tensors of rank at most $s$. This implies that the measurement functions only depend on the convex body through the Euler characteristic of the intersection of the convex body and the test line.
Our results do not allow to write surface tensors of odd rank as Crofton integrals based on sections with lines. This drawback is not a result of our method of proof. Indeed, apart from the trivial case of tensors of rank one, there does not exist a translation invariant or a bounded measurement function that expresses a surface tensor of odd rank as a Crofton integral; see Theorem \[s ulige\] for a precise statement of this fact.
In Section \[sec estimation\] the integral formula for surface tensors of even rank is transferred to stereological formulae in a design based setting. Three types of unbiased estimators are discussed. Section \[IUR\] describes an estimator based on isotropic uniform random lines. Due to the structure of the measurement function, it suffices to observe whether the test line hits or misses the convex body in order to estimate the surface tensors. However, the resulting estimators possess some unfortunate statistical properties. In contrast to the surface tensors of full dimensional convex bodies, the estimators are not positive definite. For convex bodies, which are not too eccentric (see ), this problem is solved by using $n$ orthogonal test lines in combination with a measurement of the projection function of order $n-1$ of the convex body.
In applications it might be inconvenient or even impossible to construct the isotropic uniform random lines, which are necessary for the use of the estimator described above. Instead, it might be a possibility to use vertical sections; see Definition \[VUR\]. A combination of Crofton’s formula and a result of Blaschke-Petkantschin type allows us to formulate a vertical section estimator. The estimator, which is discussed in Section \[sec VUR\], is based on two-dimensional vertical flats.
The third type of estimator presented in the design based setting is based on non-isotropic linear sections; see Section \[Sec noniso\]. For a fixed convex body in ${{\mathbb R}}^2$ there exists a density for the distribution of test line directions in an importance-sampling approach that leads to minimal variance of the non-isotropic estimator, when we consider one component of a rank 2 tensor, interpreted as a matrix. In practical applications, this density is not accessible, as it depends on the convex body, which is typically unknown. However, there does exist a density independent of the underlying convex body yielding an estimator with smaller variance than the estimator based on isotropic uniform random lines. If *all* components of the tensor are sought for, the non-isotropic approach requires three test lines, as two of the four components of a rank 2 Minkowski tensor coincide due to symmetry. It should be avoided to use a density suited for estimating one particular component of the tensor to estimate any other component, as this would increase variance of the estimator. In this situation, however, a smaller variance can be obtained by applying an estimator based on *three* isotropic random lines (each of which can be used for the estimation of *all* components of the tensor).
In Section \[SecModel\] we turn to a model-based setting. We discuss estimation of the *specific (translation invariant) surface tensors* associated with a stationary process of convex particles; see for a definition. In [@RSRS06] the problem of estimating the area moment tensor (rank $2$) associated with a stationary process of convex particles via planar sections is discussed. We consider estimators of the specific surface tensors of arbitrary even rank based on one-dimensional linear sections. Using the Crofton
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{
"pile_set_name": "ArXiv"
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---
author:
- Christopher Broadbent
- Arnaud Carayol
- 'Matthew Hague[^1]'
- Olivier Serre
bibliography:
- 'references.bib'
title: ' Emptiness of Stack Automata is NEXPTIME-complete: A Correction [^2] '
---
[^1]: Supported by EPSRC \[EP/K009907/1\].
[^2]: We thank an anonymous reviewer for pointing out the error in the original PSPACE algorithm.
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'Suppose that $E$ and $E''$ denote real Banach spaces with dimension at least $2$ and that $D\varsubsetneq E$ and $D''\varsubsetneq E''$ are uniform domains with homogeneously dense boundaries. We consider the class of all $\varphi$-FQC (freely $\varphi$-quasiconformal) maps of $D$ onto $D''$ with bilipschitz boundary values. We show that the maps of this class are $\eta$-quasisymmetric. As an application, we show that if $D$ is bounded, then maps of this class satisfy a two sided Hölder condition. Moreover, replacing the class $\varphi$-FQC by the smaller class of $M$-QH maps, we show that $M$-QH maps with bilipschitz boundary values are bilipschitz. Finally, we show that if $f$ is a $\varphi$-FQC map which maps $D$ onto itself with identity boundary values, then there is a constant $C\,,$ depending only on the function $\varphi\,,$ such that for all $x\in D$, the quasihyperbolic distance satisfies $k_D(x,f(x))\leq C$.'
address:
- 'Yaxiang. Li, Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China'
- 'Matti. Vuorinen, Department of Mathematics and Statistics, University of Turku, FIN-20014 Turku, Finland'
- 'Xiantao. Wang, Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China'
author:
- 'Y. Li'
- 'M. Vuorinen'
- 'X. Wang ${}^{~\mathbf{*}}$'
title: Quasiconformal maps with bilipschitz or identity boundary values in Banach spaces
---
Introduction and main results {#intro}
=============================
Many results of classical function theory have their counterparts in the context of quasiconformal maps in the Euclidean $n$-dimensional space $\mathbb{R}^n$. J. Väisälä [@Vai6-0; @Vai6; @Vai5] has developed a theory of quasiconformality in the Banach space case which differs from the finite dimensional theory in many respects because tools such as conformal invariants and measures of sets are no longer available. These classical tools are replaced by fundamental objects from metric space geometry such as curves, their lengths, and approximately length minimizing curves. Väisälä used these notions in the setup of several metric space structures on the same underlying Banach space and developed effective methods based on these basic notions. In addition to the norm metric he considered two hyperbolic type metric structures, the quasihyperbolic metric and the distance ratio metric. The quasihyperbolic metric $k_D$ of a domain $D$ has a key role as quasiconformality is defined in terms of it in the Banach space case. Only recently some basic properties of quasihyperbolic metric have been studied: the convexity of quasihyperbolic balls was studied by R. Klén [@k; @k2], A. Rasila and J. Talponen [@rt; @krt], Väisälä [@Vai6']. Rasila and Talponen also proved the smoothness of quasihyperbolic geodesics in [@rt2] applying now stochastic methods.
Given domains $D, D'$ in Banach spaces $E$ and $E'$, respectively, our basic problem is to study the class of homeomorphisms $f\in QC^L_{\varphi}(D,D')$, where $$\label{intro-eq-1}
QC^L_{\varphi}(D,D') = \{f: \overline{D}\to \overline{D}'\; { \rm homeo}\;\Big| f|_{D} \;{\rm is}\; {\rm a}\;\varphi{\rm -FQC}\;{\rm map}\; {\rm and}\; f|_{\partial D}\; {\rm is}\; L{\rm -bilipschitz}\}\, .$$ For the definition of $\varphi$-FQC and $L$-bilipschitz maps see Section \[sec-2\]. The class $QC^L_{\varphi}(D,D')$ is very wide and many particular cases of interest are obtained by choosing $D, D', \varphi, L$ in a suitable way as we will see below.
Our first result deals with the case when both $D$ and $D'$ are uniform domains. In this case we prove that the class consists of quasisymmetric maps. More precisely, we prove the following theorem.
\[thm1.2\] Let $D\subsetneq E$, $D'\subsetneq E'$ be $c$-uniform domains. If $f\in QC^L_{\varphi}(D,D')$, then $f$ is $\eta$-QS in $\overline{D}$ with $\eta$ depending on $c$, $L$ and $\varphi$ only.
Applying this result to the case of a bounded domain $D$ we obtain the second result. Recall that in the case of ${\mathbb R}^n$ results of this type have been proved by R. Näkki and B. Palka [@np]. For the definitions, see Section \[sec-2\].
\[thm1.3\] Let $D\subsetneq E$, $D'\subsetneq E'$ be $c$-uniform domains. If $f\in QC^L_{\varphi}(D,D')$ and $D$ is bounded, then for all $x,y\in D$, $$\frac{|x-y|^{1/{\alpha}}}{C}\leq|f(x)-f(y)|\leq C|x-y|^{\alpha},$$ where $C\geq 1$ and $\alpha\in(0,1)$ depend on $c$, $L$, $\varphi$ and $\operatorname{diam}(D)$.
Our third result concerns the case when both $D$ and $D'$ are uniform domains and $\varphi(t)=Mt$ for some fixed $M \ge 1\,.$ We also require a density condition of the boundary of a domain. This $(r_1,r_2)$-HD condition will be defined in Section 2.
\[thm1.4\] Let $D\subsetneq E$, $D'\subsetneq E'$ be $c$-uniform domains and the boundary of $D$ be $(r_1,r_2)$-HD. If $f\in QC^L_{\varphi}(D,D')$ with $\varphi(t)=Mt$, then $f$ is $M'$-bilipschitz in $\overline{D}$, where $M'$ depends only on $c$, $r_1$, $r_2$, $L$ and $M$.
Our fourth result deals with the case when $D=D'$, $L=1$ and, moreover, the boundary mapping $f|_{\partial D}:\partial D\to\partial D$ is the identity. This problem has been studied very recently in [@MV; @M2; @VZ]. Originally, the problem was motivated by Teichmüller’s work on plane quasiconformal maps [@K; @T] and then extended to the higher dimensional case by several authors: [@AV], [@M2], [@MV; @VZ]. Our result is as follows.
\[thm1.1\] Let $D\subsetneq E$ be a $c$-uniform domain with $(r_1,r_2)$-HD boundary. If $f$ is a $\varphi$-FQC map which maps $D$ onto itself with identity boundary values, then for all $x\in D$, $$k_D(x, f(x))\leq C,$$ where $C$ is a constant depending on $r_1$, $r_2$, $c$ and $\varphi$ only.
For the case $n=2$, when $D$ is the unit disk, the sharp bound is due to Teichmüller [@K; @T]. For the case of unit ball in $\mathbb{R}^n, n\ge 2,$ nearly sharp results appear in [@MV; @VZ]. In both of these cases one uses the hyperbolic metric in place of the quasihyperbolic metric.
We do not know whether there are sharp results for the Banach spaces, too. For instance, it is an open problem whether Theorem \[thm1.1\] could be refined for the case $D=\mathbb{B}$, the unit ball, to the effect that $C\rightarrow 0$ when $\varphi$ approaches the identity map.
The organization of this paper is as follows. In Section \[sec-4\], we will prove Theorems \[thm1.2\], \[thm1.3\], \[thm1.4\] and \[thm1.1\]. In Section \[sec-2\], some preliminaries are stated.
Preliminaries {#sec-2}
=============
We adopt mostly the standard notation and terminology from Väisälä [@Vai6-0; @Vai5]. We always use $E$ and $E'$ to denote real Banach spaces with dimension at least $2$. The norm of a vector $z$ in $E$ is written as $|z|$, and for every pair of points $z_1$, $z_2$ in $E$, the distance between them is denoted by $|z_1-z_2|$, the closed line segment with endpoints $z_1$ and $
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'This paper considers the problem of acquiring an unknown target location (among a finite number of locations) via a sequence of measurements, where each measurement consists of simultaneously probing a group of locations. The resulting observation consists of a sum of an indicator of the target’s presence in the probed region, and a zero mean Gaussian noise term whose variance is a function of the measurement vector. An equivalence between the target acquisition problem and channel coding over a binary input additive white Gaussian noise (BAWGN) channel with state and feedback is established. Utilizing this information theoretic perspective, a two-stage adaptive target search strategy based on the sorted Posterior Matching channel coding strategy is proposed. Furthermore, using information theoretic converses, the fundamental limits on the target acquisition rate for adaptive and non-adaptive strategies are characterized. As a corollary to the non-asymptotic upper bound of the expected number of measurements under the proposed two-stage strategy, and to non-asymptotic lower bound of the expected number of measurements for optimal non-adaptive search strategy, a lower bound on the adaptivity gain is obtained. The adaptivity gain is further investigated in different asymptotic regimes of interest.'
author:
- 'Anusha Lalitha, Nancy Ronquillo, and Tara Javidi, [^1] [^2]'
bibliography:
- 'HypTest.bib'
title: Improved Target Acquisition Rates with Feedback Codes
---
Introduction {#sec:intro}
============
Consider a single target acquisition over a search region of width $B$ and resolution up to width $\delta$. Mathematically, this is the problem of estimating a unit vector $\mbf{W} \in \{0,1\}^{\frac{B}{\delta}}$ via a sequence of noisy linear measurements $$\label{eq:linear1}
Y_n = \langle {\mbf{S}_n},\mbf{W} + \mbs{\Xi}_n\rangle, \quad n = 1,2, \ldots, \tau,$$ where a binary measurement vector $\mbf{S}_n \in \{0,1\}^{\frac{B}{\delta}}$ denotes the locations inspected and the vector $\mbs{\Xi}_n \in \mathbb{R}^{\frac{B}{\delta}}$ denotes the additive measurement noise per location. More generally, the observation $Y_n$ at time $n$ can be written as $$\label{eq:linear2}
Y_n = \langle\mbf{S}_n, \mbf{W}\rangle + Z_n(\mbf{S}_n),$$ where $Z_n(\mbf{S}_n)$ is a noise term whose statistics are a function of the measurement vector $\mbf{S}_n$. The goal is to design the sequence of measurement vectors $\{\mbf{S}_n\}_{n = 1}^{\tau}$, such that the target location $\textbf{W}$ is estimated with high reliability, while keeping the (expected) number of measurements $\tau$ as low as possible.
In this paper, we first consider the linear model when the elements of $\mbf{\Xi}_n$ are i.i.d Gaussian with zero mean and variance $\delta \sigma^2$. This means that $Z_n(\mbf{S}_n)$ in are distributed as $\mathcal{N}(0, |\mbf{S}_n| \delta \sigma^2)$, and show that the problem of searching for a target under measurement dependent Gaussian noise $Z_n(\mbf{S}_n)$ is equivalent to channel coding over a binary additive white Gaussian noise (BAWGN) channel with state and feedback (in Section 4.6 [@Gallager]). This allows us not only to retrofit the known channel coding schemes based on sorted Posterior Matching (sort PM) [@SungEnChiu] as adaptive search strategies, but also to obtain information theoretic converses to characterize fundamental limits on the target acquisition rate under both adaptive and non-adaptive strategies. As a corollary to the non-asymptotic analysis of our sorted Posterior-Matching-based adaptive strategy and our converse for non-adaptive strategy, we obtain a lower bound on the adaptivity gain.
Our Contributions
-----------------
Our main results are inspired by the analogy between target acquisition under measurement dependent noise and channel coding with state and feedback. This connection was utilized in [@DBLP:journals/corr/KaspiSJ16] under a Bernoulli noise model. In this paper, in Proposition \[prop:connection\], we formalize the connection between our target acquisition problem with Gaussian measurement dependent noise and channel coding over a BAWGN channel with state. Here, the channel state denotes the variance of the measurement dependent noise $ |\mbf{S}_n| \delta \sigma^2$. Since feedback codes i.e., adapting the codeword to the past channel outputs, are known to increase the capacity of a channel with state and feedback. This motivates us to use adaptivity when searching, i.e., to utilize past observations $\{Y_1, Y_2, \ldots, Y_{n-1}\}$ when selecting the next measurement vector $\mbf{S}_n$. Furthermore, this information theoretic perspective allows us to quantify the increase in the adaptive target acquisition rate. Our analysis of improvement in the target acquisition rate as well as the adaptivity gain, measured as the reduction in expected number of measurements, while using an adaptive strategy over a non-adaptive strategy has two components. Firstly, we utilize information theoretic converse for an optimal non-adaptive search strategy to obtain a non-asymptotic lower bound on the minimum expected number of measurements required while maintaining a desired reliability. As a consequence, this provides the best non-adaptive target acquisition rate. Secondly, we utilize a feedback code based on Posterior Matching as a two-stage adaptive search strategy and obtain a non-asymptotic upper bound on the expected number of measurements while maintaining a desired reliability. These two components of our analysis allow us to characterize a lower bound on the increased target acquisition rate due to adaptivity.
Our non-asymptotic analysis of adaptivity gain reveals two qualitatively different asymptotic regimes. In particular, we show that adaptivity gain depends on the manner in which the number of locations grow. We show that the adaptivity grows logarithmically in the number of locations, i.e., $O\left(\log \frac{B}{\delta} \right)$ when refining the search resolution $\delta$ ($\delta$ going to zero) and while keeping total search width $B$ fixed. On the other hand, we show that as the search width $B$ expands while keeping search resolution $\delta$ fixed, the adaptivity gain grows in the number of locations as $O\left(\frac{B}{\delta} \log \frac{B}{\delta} \right)$.
The problem of searching for a target under a binary measurement dependent noise, whose crossover probability increases with the weight of the measurement vector was studied by [@DBLP:journals/corr/KaspiSJ16] and analyzed under sort PM strategy in [@SungEnChiu]. In particular, [@DBLP:journals/corr/KaspiSJ16] and [@SungEnChiu] provide asymptotic analysis of the adaptivity gain for the case where $B = 1$ and $\delta $ approaches zero. Our prior work [@8007098] by utilizing a (suboptimal) hard decoding of Gaussian observation $Y_n$, strengthens [@DBLP:journals/corr/KaspiSJ16] and [@SungEnChiu] by also accounting for the regime in which $B$ grows. While the analysis in [@8007098] strengthens the non-asymptotic bounds in [@SungEnChiu] with Bernoulli noise it failed to provide tight analysis for our problem with Gaussian observations. In this paper, by strengthening our analysis in [@8007098] we extend the prior work in three ways: (i) we consider the soft Gaussian observation $Y_n$, (ii) we obtain non-asymptotic achievability and converse analysis, and (iii) we characterize tight non-asymptotic adaptivity gain in the two asymptotically distinct regimes of $B \to \infty$ and $\delta \to 0$.
Applications
------------
Our problem formulation addresses two challenging engineering problems which arise in the context of modern communication systems. We will discuss the two problems in the following examples and then provide the details of the state of art.
Consider the problem of detecting the direction of arrival for initial access in millimeter wave (mmWave) Communications. In mmWave communication, prior to data transmission the base station is tasked with aligning the transmitter and receiver antennas in the angular space. In other words, the base station’s antenna pattern can be viewed as a measurement vector $\mbf{S}_n$ searching the angular space $B \subset (0, 360^{\circ})$. At each time $n$, the noise intensity depends on the base station’s antenna pattern $\mbf{S}_n$ and the noisy observation $Y_n$ is a function of measurement dependent noise $Z_n(\mbf{S}_n)$. Here it is natural to characterize the fundamental limit on the measurement time as a function of asymptotically small $\delta$.
Consider the problem of opportunistically searching for a vacant subband of bandwidth $\delta$ over a total bandwidth of $B$. In this problem secondary user desires to locate the single stationary vacant subband quickly and reliably, by making measurements $\mbf{
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We present a new approach to equivariant version of the topological complexity, called a symmetric topological complexity. It seems that the presented approach is more adequate for the analysis of an impact of symmetry on the motion planning algorithm than the one introduced and studied by Colman and Grant. We show many bounds for the symmetric topological complexity comparing it with already known invariants and prove that in the case of a free action it is equal to the Farber’s topological complexity of the orbit space. We define the Whitehead version of it.'
address:
- |
Theoretical Computer Science Department\
Faculty of Mathematics and Computer Science\
Jagiellonian University\
30-348 Kraków, Poland
- 'Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Poznań, ul. Umultowska 87, 61-614 Poznań, Poland'
author:
- Wojciech Lubawski
- 'Wacław Marzantowicz$^{1}$'
title: A new approach to the equivariant topological complexity
---
[^1]
Introduction
============
A topological invariant introduced by Farber in [@farber1; @farber2], and called the topological complexity, was the first to estimate a complexity of motion planning algorithm. With the configuration space $X$ of a mechanical robot he associated a natural number $TC(X)$ called topological complexity of $X$. To be more precise he considered the natural fibration $$\label{eq1}\pi\colon PX \rightarrow X\times X$$ from the free path space in $X$ which assigns to a path $\gamma$ defined on the unit interval its ends $(\gamma(0), \gamma(1))$. The topological complexity is the least $n$ such that $X\times X$ can be covered by $n$ open sets $U_1 , \ldots , U_n$ such that for each $i$ there is a homotopical section $s_i\colon U_i\rightarrow PX$ to $\pi$. This invariant is a special case of the well known Lusternik-Schnirelmann (or LS for short) category of $X\times X$ (cf [@colman-grant] for more detailed exposition of this notion and other references).
In this paper we discuss the following question: If the mechanical robot admits a symmetry with respect to a compact Lie group (and therefore the configuration space $X$ admits it too) what is an appropriate definition of the topological complexity that takes into account that symmetry? An answer is not that simple as it may look like and it is not unique. We define an invariant, different than the equivariant topological complexity introduced by H. Colman and A. Grant in [@colman-grant], called the symmetric topological complexity. By showing its properties we would like to demonstrate that in many situations it is better than that of [@colman-grant].
Let $G$ be a compact Lie group. Let us assume that $X$ is a $G$ space, i.e. $G$ acts continuously on $X$ (therefore we assume that $G$ is the “symmetry group” that appears in $X$). The formulae for topological complexity uses the natural fibration \[eq1\]. If the space $X$ is a $G$ space then $PX$ is a $G$ space in a natural way, and so does $X\times X$ by the diagonal action. It would be natural to define the equivariant complexity by assuming that all maps are $G$ maps. Actually this approach has been studied in [@colman-grant]. We will use the notation introduced there $TC_G(X)$ to denote this invariant. In spite of its mathematical naturalness this approach has some disadvantages that we present below.
![A symmetric robot arm with an action of $\tau$[]{data-label="fig:1"}](robotarm.png)
Let us consider a mechanical robot arm that admits a symmetry. For simplicity let us assume that $G = \mathbb Z / 2=\{ 1,\tau \}$ as showed in the picture \[fig:1\]. The element $\tau$ acts by interchanging the part $A$ of the arm with $B$. Assume we are given a path $\xi$ between points $x$ and $y$ in the configuration space $X$, as noted in picture \[fig:2\].
![A path in configuration space[]{data-label="fig:2"}](robotarmMove.png)
Note that although points $x$ and $\tau x\in X$ are distinct in the configuration space there is no physical difference between these two states of a mechanical arm as can be observed from picture \[fig:1\]. Therefore it is natural to require that the path $\xi$ determines a path between $\tau x$ and $\tau y$ – namely $\tau \xi$. This natural requirement leads us to a definition of equivariant topological complexity $TC_G(X)$. On the other hand if the task the mechanical arm is supposed to perform is symmetric we would like the path $\xi$ to determine the following four paths – between $x$ and $y$, $\tau x$ and $\tau y$ as well as between $x$ and $\tau y$, $\tau x$ and $y$. In other words we would like to exploit the $G\times G$ structure of the space $X\times X$. The main problem is that usually $PX$ is not a $G\times G$ space. We will show in section \[2.\] how to deal with this problem by defining so called symmetric topological complexity, $STC_G(X)$.
Unlike many other equivariant versions of numerical invariants the equivariant topological complexity does not have the required mathematical properties – for example when the group $G$ acts freely on $X$ then in general $TC_G(X)\neq TC(X/G)$ where $X / G$ is the orbit space and $TC(X/G)$ is the topological complexity of $X/G$. We will show that in our case $STC_G(X) = TC(X/G)$. A bridge to apply advanced homotopy theory in the theory of Lusternik-Schnirelmann category is the Whitehead version of it (cf. [@whitehead1]). We will show that for the symmetric topological complexity we can define a Whitehead version of it and for a finite group $G$ it gives the original symmetric topological complexity. We conjecture that the same holds for any compact Lie group. Finally we provide examples which distinguish the equivariant topological complexity and the symmetric topological complexity and calculate the latter in several cases.
Lusternik-Schnirelmann category
===============================
Basic definitions
-----------------
In this section we define and give some basic properties of a version of an equivariant Lusternik-Schnirelmann category for topological spaces that we will use later on in our considerations. We shall the standard notations of the theory of compact Lie group transformations of [@bredon].
Let $G$ be a Lie group and let $A$ be a closed $G$ subset of a $G$ space $X$.
A metrizable $G$-space $X$ we call a $G$-ANR if for every equivariant imbedding $\iota: X \to Y $ as a closed $G$-subset of a metrizable $G$-space $ Y$ there exists a $G$ neighborhood $U$ of $\iota(X)$ in $Y$ and a continuous equivariant retraction $U\to
\iota(X)$.
Throughout this paper we assume that $X$ is a compact $G$-ANR (see [@murayama] for the properties of $G$-ANRs). The class of $G$-ANRs includes $G$-ENRs (cf. [@jaworowski] for the definition), countable $G$-CW complexes thus smooth $G$-manifolds with smooth action of $G$.
\[1:1\] We call on open $G$ set $U\subseteq X$ *$G$-compressable* into $A$ whenever the inclusion map $\iota_U\colon U\subseteq X$ is $G$ homotopic to $c\colon U\rightarrow X$ such that $c(U)\subseteq A$.
This allows us to define our main tool
\[1:2\]\[Adfi3\] An $A$-Lusternik-Schnirelmann $G$-category of a $G$ space $X$ is the least $n$ such that $X$ can be covered by $U_1,\ldots , U_n$ open $G$ subsets of $X$ each $G$-compressable into $A$. We denote in by $_Acat_G(X)$.
Remind, we say that a $G$-space $X$ is $G$-path-connected if for every closed subgroup $H\subset G$ the space $X^H$ is path-connected.
Note that we have a relation to the standard Lusternik-Schnirelmann category . By $\ast $ we denote a fixed one point subset of $X$ provided it is invariant, i.e $\ast
\in X^G$.
\[1:3\] If $X$ is path connected and $G$ is the trivial group then we have $$_\ast cat_G(X) = cat(X).$$ If $\ast\in X^G$ and $X^H$ is path connected for all closed subgroups $
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'Discovery of high-redshift ($z > 6$) supermassive black holes (BHs) may indicate that the rapid (or super-Eddington) gas accretion has aided their quick growth. Here, we study such rapid accretion of the primordial gas on to intermediate-mass ($10^2 - 10^5~M_\odot$) BHs under anisotropic radiation feedback. We perform two-dimensional radiation hydrodynamics simulations that solve the flow structure across the Bondi radius, from far outside of the Bondi radius down to a central part which is larger than a circum-BH accretion disc. The radiation from the unresolved circum-BH disc is analytically modeled considering self-shadowing effect. We show that the flow settles into a steady state, where the flow structure consists of two distinct parts: (1) bipolar ionized outflowing regions, where the gas is pushed outward by thermal gas pressure and super-Eddington radiation pressure, and (2) an equatorial neutral inflowing region, where the gas falls toward the central BH without affected by radiation feedback. The resulting accretion rate is much higher than that in the case of isotropic radiation, far exceeding the Eddington-limited rate to reach a value slightly lower than the Bondi one. The opening angle of the equatorial inflowing region is determined by the luminosity and directional dependence of the central radiation. We find that photoevaporation from its surfaces set the critical opening angle of about ten degrees below which the accretion to the BH is quenched. We suggest that the shadowing effect allows even stellar-remnant BHs to grow rapidly enough to become high-redshift supermassive BHs.'
author:
- |
Kazuyuki Sugimura,$^1$[^1] Takashi Hosokawa,$^{2,3,4}$ Hidenobu Yajima$^{1,5}$ and Kazuyuki Omukai$^{1,3}$\
$^1$Astronomical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan\
$^2$Department of Physics, Kyoto University, Sakyo, Kyoto 606-8502, Japan\
$^3$Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA\
$^4$Department of Physics and Research Center for the Early Universe, the University of Tokyo, Bunkyo, Tokyo 113-0033, Japan\
$^5$Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Aoba, Sendai 980-8578, Japan
title: Rapid Black Hole Growth under Anisotropic Radiation Feedback
---
-1cm
quasars: supermassive black holes-cosmology: theory.
Introduction {#sec:intro}
============
Discovery of high-$z$ ($z\gtrsim6$) quasars suggests that supermassive black holes (SMBHs) already exist when the age the Universe is less than $1{\,\mathrm{Gyr}}$ [see, e.g., @Fan:2001aa; @Willott:2010aa; @Mortlock:2011aa; @Venemans:2013aa; @Wu:2015aa]. This poses a question about the formation mechanism of SMBHs in such a short interval. Among the scenarios for the SMBH seed formation [see, e.g., @Volonteri:2012ab; @Haiman:2013aa for a review], including the dense stellar cluster scenario [see, e.g., @Omukai:2008aa; @Devecchi:2009aa; @Katz:2015ab; @Yajima:2016aa and reference therein], following two are the most studied: the direct collapse BH (DCBH) and the population III (Pop III) remnant BH scenarios.
In the former scenario, supermassive stars of $\sim 10^5\,M_\odot$ collapse to form seed BHs with approximately the same mass. Specifically, supermassive stars are envisaged to form in exceptional environments in the high-$z$ Universe, for example, in atomic-cooling halos where the ${\mathrm{H_2}}$ cooling is totally suppressed by very strong far ultraviolet (FUV) irradiation [e.g., @Sugimura:2014aa]. While the seed BHs in this case are rather massive with $\sim
10^5\,M_\odot$, their number density might be too small to explain all the observed high-$z$ SMBHs due to the stringent necessary conditions [@Dijkstra:2008aa; @Dijkstra:2014aa; @Agarwal:2012aa; @Sugimura:2014aa; @Sugimura:2016aa; @Inayoshi:2015ab; @Chon:2016aa].
In the latter scenario, the remnant BHs of Pop III stars [@Yoshida:2008aa; @Hosokawa:2011aa; @Hosokawa:2016aa] are thought as SMBH seeds [@Alvarez:2009aa; @Jeon:2012aa]. Contrary to the DCBH scenario, they are abundant but the problem is whether they can actually grow to the SMBHs from smaller initial mass of $\lesssim 10^3\,M_\odot$ [@Susa:2014aa; @Hirano:2015aa] within the available time. Although BHs can acquire the mass by collisions with other BHs [@Tanikawa:2011aa], the BH collisions often result in ejection of the merged BHs from the host halo due to the recoil of gravitational wave emission [e.g., @Baker:2006aa; @Koppitz:2007aa]. Thus, the feasibility of this scenario relies on whether the rapid accretion on to seed BHs is possible or not [@Madau:2014aa; @Alexander:2014aa; @Volonteri:2015aa].
Recently, a number of authors have studied the BH accretion under radiation feedback [e.g., @Milosavljevic:2009aa; @Milosavljevic:2009ab; @Park:2011aa; @Park:2012aa; @Park:2013aa]. They solve the gas dynamics over the scale of the Bondi radius, where the accretion rate on to the circum-BH disc is physically determined. Although the central circum-BH disc is not spatially resolved, subgrid models that provide analytic prescriptions of its emissivity have been used. They have shown that the accretion rate is significantly reduced to $\lesssim1\%$ of that without radiation feedback (i.e., the Bondi rate) in case with modest BH mass and ambient density (e.g., $10^2\,M_\odot$ and $10^5{\,\mathrm{cm^{-3}}}$). Only in case with very high BH mass and/or ambient density (e.g., $10^4\,M_\odot$ and $10^5{\,\mathrm{cm^{-3}}}$), the accretion rate reaches to the Bondi value because of inefficient radiation feedback, as recently shown by [@Inayoshi:2016ac] [see also @Li:2011aa; @Pacucci:2015ab; @Park:2016ab for other mechanisms of efficient accretion]. However, all those calculations assume isotropic radiation (in either one- or two-dimensional simulations), whereas in reality the radiation from the BH accretion disc should be anisotropic. The flow structure will be significantly altered in such anisotropic radiation field. Although the BH accretion under anisotropic radiation has been studied in the context of active galactic nuclei (AGN) with the BH mass $\ga 10^6\,M_\odot$ [@Proga:2007aa; @Kurosawa:2009aa; @Novak:2011aa; @Barai:2012aa], the nature of accretion on to stellar-mass BHs would be quite different.
The anisotropic BH irradiation has been examined with different models of the BH accretion discs, including the “standard disc” for moderate accretion rates [@Shakura:1973aa], and “slim disc” for the higher rates [@Abramowicz:1988aa]. In particular, recent multi-dimensional simulations have investigated inner structure of the slim disc within roughly a hundred Schwarzschild radii, showing that the accretion rates can indeed exceed the Eddington-limited rate [e.g., @Ohsuga:2005aa; @Jiang:2014aa; @McKinney:2014aa; @Fragile:2014aa; @Takahashi:2015aa; @Sc-adowski:2016aa]. These studies show that the high-energy photons are predominantly emitted in polar directions from the inner part of the disc. However, the outer structure of the disc, which is not solved in the above simulations, should also modify the anisotropic radiation field. For instance, disc winds such as the line-driven AGN winds launched from the outer region will absorb a part of photons coming from the inner region [e.g., @Proga:2000aa; @Proga:2004aa; @Nomura:2016aa]. Since numerical simulations solving the whole structure of the disc are still infeasible, it is very uncertain how much anisotropy the BH accretion discs actually create.
In this paper, we will investigate accretion of the primordial gas on to BHs under the anisotropic radiation feedback from the central circum-BH accretion discs, considering the shadowing effect by the outer part of the discs. We perform a set of proof-of-concept two-dimensional (2D) radiation hydrodynamics (RHD) simulations, assuming that BHs are initially embedded in homogeneous and static media. We do not attempt to simulate the realistic directional dependence of BH irradiation in consideration of its high uncertainties; instead, we model it in a simple fashion to study
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
address: |
Jefferson Physical Laboratory, Harvard University,\
Cambridge, MA 02138 USA
author:
- Minjae Cho
bibliography:
- 'open\_closed.bib'
title: 'Open-closed Hyperbolic String Vertices'
---
= .8mm
\
\
Introduction
============
In recent years, it has become clear that a well defined and consistent perturbative formulation of string theory requires the framework of string field theory (we refer readers to [@Zwiebach:1992ie; @Sen:2015uaa; @deLacroix:2017lif] and references therein for an overview of the subject). For example, traditional analytic continuation involved in the computation of string amplitudes to cure the divergences in moduli integration naturally arises in string field theory as explained in [@Sen:2019jpm]. Furthermore, prescriptions given by string field theory provide not only unambiguous recipe to compute physical quantities in a given background, but also descriptions of more general backgrounds arising as solutions to string field equations of motions. For example, this idea was used to study strings in Ramond-Ramond flux backgrounds [@Cho:2018nfn].
In practice, computations that appear in string field theory are those of worldsheet conformal field theories. Therefore, we in principle can compute relevant quantities in a rather strightforward manner. Being a field theory, string field theory carries vertices which are roughly speaking integration of worldsheet correlators of off-shell string fields over specific parts of the moduli spaces. However, such off-shell objects in general depend on how one coordinatizes Riemann surfaces and this ambiguity exactly amounts to string field redefinitions [@Hata:1993gf; @Sen:1993ic; @Sen:2014dqa; @Sen:2015hha]. Thus, the choice of vertices amounts to which coordinatization of Riemann surfaces to use and which parts of the moduli spaces to cover.
Of course, not all such arbitrary choices of vertices are consistent. There is a very natural requirement on string vertices when the homomorphism between Batalin-Vilkoviski (BV)-algebras of surfaces and string fields are considered [@Sen:1994kx; @Sen:1993kb]. The requirement is called geometric master equation and the job of finding the solutions is of fundamental interest in the framework of string field theory. In the past, such solutions were found using various metrics, an example being minimal area metrics [@Zwiebach:1990nh; @Zwiebach:1990qj; @Zwiebach:1992bw; @Headrick:2018ncs; @Headrick:2018dlw]. There were also approximate constructions using the hyperbolic metrics [@Moosavian:2017fta; @Moosavian:2017qsp; @Moosavian:2017sev; @Pius:2018pqr].
Recently, a nice explicit construction of closed string vertices using hyperbolic metric was achieved in [@Costello:2019fuh]. One starts with a bordered hyperbolic Riemann surface with specified border lengths and systolic constraints, and grafts flat semi-infinite cylinders to the borders to make them into punctures. Upon connecting such vertices using closed string propagator which is represented as a flat finite cylinder, the resulting metric is the Thurston metric (for an overview, see [@tanigawa1995grafting]). As string theory requires us to integrate over the moduli space, one possible advantage of hyperbolic vertices from string theory perspective is that there is a better understanding of moduli integration in such metrics [@1998InMat.132..607M; @Mirzakhani:2006fta; @Mirzakhani:2006eta].
In this work, we generalize the construction of hyperbolic string vertices to oriented open-closed string field theory [@Zwiebach:1990qj; @Zwiebach:1997fe] (we will omit the term “oriented” from now on and it is always assumed). Hyperbolic surfaces to be considered are bordered hyperbolic surfaces (we refer readers to Chapter 1 of [@Buser1992GeometryAS] for a gentle introduction to these surfaces). Their boundaries are piecewise geodesic and some of geodesic sides will correspond to open string punctures, while the other sides belong to boundaries. Some of the borders which are closed geodesics will correspond to closed string punctures as in [@Costello:2019fuh], while the others will correspond to boundaries. We will define a family of subsets of such bordered hyperbolic surfaces, and show that it solves the geometric master equation. The essential ingredients of the proof are collar theorems. Such theorems are well-known for hyperbolic bordered Riemann surfaces where boundaries are all smooth closed geodesics. We will extend them to the case of bordered hyperbolic surfaces under consideration. We will also give explicit description of all zero and one-dimensional open-closed hyperbolic string vertices. This description will show that the family of hyperbolic vertices we constructed do not include a point where the theory becomes Witten’s cubic theory [@Witten:1985cc; @Zwiebach:1992bw], as already discussed in [@Costello:2019fuh].
The paper is organized as follows. In section \[geometricmastereq\], we review the geometric master equation for open-closed string vertices [@Zwiebach:1997fe] and provide a proof that Feynman diagrams built out of the solutions to the geometric master equation represent fundamental classes in relative homologies of interest. Then, we proceed to discuss relevant geometric objects and theorems in section \[hyperbolicsurfaces\]. Using these objects, in section \[hyperbolicvertices\] we will define hyperbolic open-closed string vertices and prove that they solve the geometric master equation. We describe zero and one-dimensional hyperbolic vertices in section \[lowdimvertices\]. We conclude with remarks and discussions in section \[discussions\].
Open-closed geometric master equation and Feynman diagrams {#geometricmastereq}
==========================================================
In this section, we review the general framework of open-closed string field theory and the corresponding geometric master equation, which we will solve in later sections. All discussions are standard and we will closely follow [@Zwiebach:1997fe]. Then, we will show that the Feynman diagrams built out of the solutions cover the moduli space exactly once, following the ideas presented in [@Costello:2019fuh].
Moduli spaces, total spaces, and singular chains
------------------------------------------------
In open-closed string field theory, the geometric objects under consideration are bordered Riemann surfaces with marked bulk and boundary punctures. In order to specify the moduli space, one specifies genus $g$, number of bulk punctures $n$, number of boundary components $b$, and number $m_i$ of boundary punctures on the $i$-th boundary, with $i=1,2,...,b$. We will denote the corresponding moduli space as ${\cal M}^{g,n}_{b,\{m_i\}}$. Open-closed string vertices take all possible values of these parameters satisfying \[moduliconditions\] &i) n3 g=b=0,\
&ii) n1 g=1, b=0,\
&iii) m\_13 g=0, b=1.
Off-shell amplitudes of string fields are integration of worldsheet correlators over a given moduli space. As already mentioned, the result depends on the coordinatization of Riemann surfaces. We first introduce local coordinates around bulk and boundary punctures. For bulk punctures, we will take flat unit disk $\{z\in\mathbb{C}|~|z|\leq1\}$ whose origin is the location of the puncture. For boundary punctures, we will take semi-disk $\{z\in\mathbb{C}|~|z|\leq1~\text{and}~\text{Im}(z)\geq0\}$ on flat upper-half plane, where the origin is the location of the puncture and the real axis is the boundary. Then, the choice of embedding of disks and semi-disks corresponds to coordinatization of Riemann surfaces. Over the moduli space ${\cal M}^{g,n}_{b,\{m_i\}}$, we will take the fiber to be such embeddings modulo phase rotations for disk coordinates around bulk punctures. The resulting total space is denoted as $\hat{\cal P}^{g,n}_{b,\{m_i\}}$. By forgetting about coordinates, one can naturally project down to the moduli space, $\pi:\hat{\cal P}^{g,n}_{b,\{m_i\}}\rightarrow{\cal M}^{g,n}_{b,\{m_i\}}$.
Typically, one would choose a section in $\hat{\cal P}^{g,n}_{b,\{m_i\}}$ over ${\cal M}^{g,n}_{b,\{m_i\}}$ and compute off-shell amplitudes by integrating along the section. However, as pointed out in [@Costello:2019fuh], one in general can allow for singular chains with real coefficients, as chains are natural objects to integrate over. We also assume that chains are symmetrized over the punctures.
Vertices and Feynman diagrams
-----------------------------
Say we made a choice of chains for all zero dimensional moduli spaces, where chains are in the fundamental homology class of the corresponding moduli spaces when pushed forward to it. Roughly speaking, it means that chains cover the moduli space (which is a point here) exactly once taking into account the multiplicities. Then, we can construct Feynman diagrams by combining these vertices using either open string or closed string propagators. Explicitly, closed string propagator plumbs two bulk punctures with local disk coordinates $z$ and $w$ via $zw=e^{-s+i\theta}$ for all $s\geq0$ and $0<\
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We use an effective one-dimensional Gross–Pitaevskii equation to study bright matter-wave solitons held in a tightly confining toroidal trapping potential, in a rotating frame of reference, as they are split and recombined on narrow barrier potentials. In particular, we present an analytical and numerical analysis of the phase evolution of the solitons and delimit a velocity regime in which soliton Sagnac interferometry is possible, taking account of the effect of quantum uncertainty.'
author:
- 'J. L. Helm'
- 'S. L. Cornish'
- 'S. A. Gardiner'
title: 'Sagnac Interferometry Using Bright Matter-Wave Solitons'
---
A Bose–Einstein condensate (BEC) with attractive inter-atomic interactions can support soliton-like structures referred to as bright solitary matter-waves [@khaykovich_etal_science_2002; @strecker_etal_nature_2002; @cornish_etal_prl_2006; @Marchant_etal_2013; @Nguyen2014]. These propagate without dispersion [@morgan_etal_pra_1997], are robust to collisions with other bright solitary matter-waves and with slowly varying external potentials [@parker_etal_physicad_2008; @billam_etal_pra_2011], and have center-of-mass trajectories well-described by effective particle models [@martin_etal_prl_2007; @martin_etal_pra_2008; @poletti_etal_prl_2008]. Such soliton-like properties are due to the mean-field description of an atomic BEC reducing to the nonlinear Schrödinger equation in a homogeneous, quasi-one-dimensional (quasi-1D) limit, which for the case of attractive interactions supports the bright soliton solutions well-known in the context of nonlinear optics [@zakharov_shabat_1972_russian; @satsuma_yajima_1974; @gordon_ol_1983; @haus_wong_rmp_1996; @Helczynski_ps_2000]. The quasi-1D limit is experimentally challenging for attractive condensates [@billam_etal_variational_2011], but solitary wave dynamics remain highly soliton-like outside this limit [@cornish_etal_prl_2006; @billam_etal_pra_2011].
A bright solitary wave colliding with a narrow potential barrier is a good candidate mechanism to create two mutually coherent localised condensates, much as a beam-splitter splits the light of an optical interferometer. This has been extensively investigated in the quasi-1D, mean-field description of an atomic BEC [@HELM_PRA_2012; @kivshar_malomed_rmp_1989; @ernst_brand_pra_2010; @lee_brand_2006; @cao_malomed_pla_1995; @holmer_etal_cmp_2007; @holmer_etal_jns_2007; @POLO_etal_PRA_2013; @Molmer_arxiv_2012; @Minmar_thesis_2012; @abdullaev_brazhnyi_2012], and sufficiently fast collisions do lead to the desired beam-splitting effect [@holmer_etal_cmp_2007; @holmer_etal_jns_2007]. Consequently, bright solitary matter-waves, with their dispersion-free propagation, present an intriguing candidate system for future interferometric devices [@HELM_PRA_2012; @strecker_etal_nature_2002; @cornish_etal_physicad_2009; @weiss_castin_prl_2009; @streltsov_etal_pra_2009; @billam_etal_pra_2011; @al_khawaja_stoof_njp_2011; @martin_ruostekoski_njp_2010; @mcdonald_prl_2014]. Previous work [@martin_ruostekoski_njp_2010; @POLO_etal_PRA_2013; @HELM_PRA_2014] considered a Mach–Zehnder interferometer using a narrow potential barrier to split harmonically trapped solitary waves, based on the configuration of a recent experiment [@Nguyen2014]. These demonstrated one can also recombine solitary waves if they collide at the barrier; the collision dynamics are explained more fully in [@HELM_PRA_2012]. In these collisions, the relative atomic populations within the two outgoing solitary waves are governed by the relative phase $\Delta$ between the incoming ones. The mean-field nonlinearity can lead to the relative populations of the outgoing waves exhibiting greater sensitivity to small variations in the phase $\Delta$; however, simulations including quantum noise in the initial condition [@HELM_PRA_2014] or via the truncated Wigner method [@blakie_etal_ap_2008], demonstrated that enhanced number fluctuations counteract this improvement [@martin_ruostekoski_njp_2010].
![Stages of Sagnac interferometry. An incoming soliton splits at time $T_{\mathrm{s}}$ on a barrier into two solitons of equal amplitude and opposite velocity. After circumnavigating the ring trap, at time $T_{\mathrm{c}}$ the solitons recombine either at the same barrier (a), or a second barrier (b) antipodal to the first, illustrated in both cases with angular rotation $\Omega=1.875 \times 10^{-3}$, and ring circumference $L=40\pi$. The resulting phase difference, incorporating the Sagnac phase due to the rotating reference frame, is read out via the population difference in the final output products within the positive (shaded) and negative domains. (c) Final population in the positive domain $I_{+}$ as a function of $\Omega$, with $L=40\pi$ and initial soliton velocity $v=4$. The sensitivity of the single barrier case (dashed line) is twice that of the double barrier case (solid line) because the interrogation time $T_{\mathrm{c}}-T_{\mathrm{s}}$ is doubled.[]{data-label="fig:overview"}](fig_diagram.png){width="\columnwidth"}
We extend the framework of soliton interferometry to measurement of the Sagnac effect, first observed in an atom interferometer by Riehle [*et al.* ]{}[@Riehle_etal_prl_1991]. In this experiment the observation manifested as a shift in the Ramsey fringes produced by passing an atomic beam of $^{40}$Ca through four travelling waves in a Ramsey geometry, producing an atomic beam interferometer. What we present differs from the Riehle setup in two ways. Firstly, in [@Riehle_etal_prl_1991] some phase information is transported optically. In our system atom-light interactions serve only to coherently split the condensate; any resulting phase dynamics are incidental. Secondly, our system results, not in an interference fringe shift, but a population shift between the positive and negative domains of the interferometer. The Sagnac effect is inferred from measurements of particle numbers [@halkyard_etal_pra_2010] in the spatially distinct condensates on either side of the barrier, and not the structure of those condensates. (which are expected to remain soliton-like). We consider an experimental configuration, contained entirely within a rotating frame, where there is a smooth ring-shaped trapping potential (implemented by, e.g., using a spatial light modulator [@moulder_etal_pra_2012], time-averaging with acousto-optic deflectors [@henderson_etal_njp_2009], or imaging an intensity mask [@corman_etal_prl_2014]) and narrow barriers realised with optical light sheets, focussed using high numerical aperture lenses. Solitons, initially produced in an optical trap, can be adiabatically transferred into the ring, with the initial velocity set by moving the optical potential during the transfer [@rakonjac_etal_ol_2012]. Key sources of error include: uncertainty in the value of the soliton velocity relative to the barrier strength and, in turn, the barrier transmission level [@HELM_PRA_2012]; initial particle number, which determines the ground-state energy and so sets the low-energy splitting threshold, close to which the system becomes sensitive to otherwise small fluctuations in the velocity [@martin_ruostekoski_njp_2010]; and measurement of final particle number.
Within the Gross–Pitaevskii equation (GPE) framework, we consider $N$ bosonic atoms of mass $m$ and scattering length $a_{s}$, in an effective 1D configuration due to a tightly confining (frequency $\omega_{r}$) harmonic trapping potential in the degrees of freedom perpendicular to the direction of free motion, implying an interaction strength of ${g_{\mathrm{1D}}}= 2\hbar\omega_{r}a_{s}$ per particle. We use “soliton units” [@martin_etal_pra_2008] (equivalent to $\hbar = m = |{g_{\mathrm{1D}}}|N =1$), where position, time and energy are in units of $\hbar^{2}/m|{g_{\mathrm{1D}}}|N$, $\hbar^{3}/m{g_{\mathrm{1D}}}^{2}N^{2}$, and $m{g_{\mathrm{1D}}}^{2}N^{2}/\hbar^{2}$ [@
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: |
The lack of detailed balance in active colloidal suspensions allows dissipation to determine stationary states. Here we show that slow viscous flow produced by polar or apolar active colloids near plane walls mediates attractive hydrodynamic forces that drive crystallization. Hydrodynamically mediated torques tend to destabilize the crystal but stability can be regained through critical amounts of bottom-heaviness or chiral activity. Numerical simulations show that crystallization is not nucleational, as in equilibrium, but is preceded by a spinodal-like instability. Harmonic excitations of the active crystal relax diffusively but the normal modes are distinct from an equilibrium colloidal crystal. The hydrodynamic mechanisms presented here are universal and rationalize recent experiments on the crystallization of active colloids.\
\
DOI: [10.1103/PhysRevLett.117.228002](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.117.228002)\
author:
- Rajesh Singh
- 'R. Adhikari'
title: Universal hydrodynamic mechanisms for crystallization in active colloidal suspensions
---
In active colloidal suspensions [@palacci2013living; @petroff2015fast], energy is continuously dissipated into the ambient viscous fluid. The balance between dissipation and fluctuation that prevails in equilibrium colloidal suspensions [@einstein1905theory; @kubo1966fluctuation] is, therefore, absent. Nonequilibrium stationary states in active suspensions, then, are determined by both dissipative and conservative forces, quite unlike passive suspensions where detailed balance prevents dissipative forces from determining phases of thermodynamic equilibrium. In this context, it is of great interest to enquire how thermodynamic phase transitions driven by changes in free energy are modified in the presence of sustained dissipation.
In two recent experiments disordered suspensions of active colloids have been observed to spontaneously order into two-dimensional hexagonal crystals when confined at a plane wall. Bottom-heavy synthetic active colloids which catalyze hydrogen peroxide when optically illuminated are used in the first experiment [@palacci2013living] while chiral fast-swimming bacteria of the species *Thiovulum majus* are used in the second experiment [@petroff2015fast]. Given this remarkably similar crystallization in two disparate active suspensions it is natural to ask if the phenomenon is universal and to search for mechanisms, necessarily involving dissipation, that drive it.
Our current understanding of phase separation in particulate active systems is derived from the coarse-grained theory of motility-induced phase separation (MIPS) where active particles are advected by a density-dependent velocity [@tailleur2008statistical; @cates2010arrested; @cates2013active; @cates2015]. Microscopic models with kinematics consistent with MIPS also show phase separation and crystallization of hard active disks have been reported in two dimensions [@henkes2011active; @fily2012athermal; @bialke2012crystallization; @redner2013structure]. However, these models ignore exchange of the locally conserved momentum of the ambient fluid with that of the active particles and are, thus, best applied to systems where such exchanges can be ignored. Fluid flow is an integral part of the physics in [@palacci2013living; @petroff2015fast] and a momentum-conserving theory, currently lacking, is essential to identify the dissipative forces and torques that drive crystallization.
In this Letter we present a microscopic theory of active crystallization that connects directly to the experiments described above. Specifically, we account for the *three-dimensional* active flow in the fluid and the effect of a plane wall on this flow. Representing activity by slip in a thin boundary layer at the colloid surface [@ghose2014irreducible; @singh2014many; @singh2016traction] we rigorously compute the long-ranged many-body hydrodynamic forces and torques on the colloids. Thus we estimate Brownian forces and torques to be smaller than their active counterparts by factors of order $10^{2}$ (for synthetic colloids in [@palacci2013living]) to $10^{4}$ (for bacteria in [@petroff2015fast]) making them largely irrelevant for active crystallization. We integrate the resulting deterministic balance equations numerically to obtain dynamical trajectories.
Our main numerical results are summarized in Fig. (\[fig:Dynamics-of-crystallization\]). Panels (a)-(c) show the spontaneous destabilization of the uniform state by attractive active hydrodynamic forces, the formation of multiple crystallites, and their coalescence into a single hexagonal crystal at late times. Panels (d)-(f) show the structure factor at corresponding times. The route to crystallization is not through activated processes that produce critical nuclei, but through a spinodal-like instability produced by the unbalanced long-ranged active attraction. The uniform state is, therefore, always unstable and crystallization occurs for all values of density, in contrast to the finite density necessary for crystallization in MIPS models [@cates2015]. Active hydrodynamic torques tend to destabilize the ordered state but stability is regained when these are balanced by external torques (from bottom-heaviness in [@palacci2013living]) or by chiral activity (from bacterial spin in [@petroff2015fast]). Crystallites of chiral colloids rotate at an angular velocity that is inversely proportional to the number of colloids contained in them, as shown in panel (g). This is in excellent agreement with the experiment [@petroff2015fast]. The critical values of bottom-heaviness and chirality above which orientational stability, and, hence, positional order, is ensured is shown in panel (h). We now present our model and detail the derivation of our results.
![Panels (a)-(c) are instantaneous configurations during the crystallization of $1024$ active colloids of radius $b$ at a plane wall. The colloids are colored by their initial positions. Panels (d)-(f) show the structure factor $S(\mathbf{k})$ at corresponding instants. Wavenumbers are scaled by the modulus of the reciprocal lattice vector $k_{0}$ and the contribution from $\mathbf{k}=0$ is discarded. Panel (g) shows the variation of the angular velocity $\mathbf{\Omega}_{c}$ of a crystallite with the number $N$ of colloids in it. A typical configuration is shown in the inset. Panel (h) is the state diagram for orientational stability in terms of the measure of chirality $V_{0}^{(3a)}$ and bottom-heaviness $T_{0}$ (see text). Each dot represents one simulation. Here $v_{s}$ is the self-propulsion speed of an isolated colloid, $\tau=b/v_{s}$, and $\epsilon$ is the scale of the repulsive steric potential.\[fig:Dynamics-of-crystallization\]](Figure1){width="45.00000%"}
*Model:* We consider $N$ spherical active colloids of radius $b$ near a plane wall with center-of-mass coordinates $\mathbf{R}_{i},$ orientation $\mathbf{p}_{i}$, linear velocity $\mathbf{V}_{i}$, and angular velocity $\boldsymbol{\Omega}_{i}$, where $i=1\ldots N$. Activity is imposed through a slip velocity $\mathbf{v}_{i}^{\mathcal{A}}$ which is a general vector field on the surface $S_{i}$ of the $i$-th colloid satisfying $\int{\bm{\hat{\rho}}_{i}\cdot}\mathbf{v}_{i}^{\mathcal{A}}\,d\text{S}_{i}=0$ [@slipConstraint], where $\boldsymbol{\rho}_{i}$ is the vector from the center of the colloid to a point on its surface. The fluid velocity $\mathbf{v}$ is subject to slip boundary conditions $$\mathbf{v}(\mathbf{R}_{i}+\boldsymbol{\rho}_{i})=\mathbf{V}_{i}+\bm{\Omega}_{i}\times\bm{\rho}_{i}+\mathbf{v}_{i}^{\mathcal{A}}(\bm{\rho}_{i}).\label{eq:slip-RBM-BC}$$ on the colloid surfaces, to a no-slip boundary condition $\mathbf{v}=0$ at the plane wall located at $z=0$, and to a quiescent boundary condition at large distances from the wall. The slip is conveniently parametrized by an expansion $\mathbf{v}^{\mathcal{A}}(\mathbf{R}_{i}+\boldsymbol{\rho}_{i})=\sum_{l=1}^{\infty}\tfrac{1}{(l-1)!(2l-3)!!}\,\mathbf{V}_{i}^{(l)}\cdot\mathbf{Y}^{(l-1)}(\bm{\hat{\rho}}_{i})$ in irreducible tensorial spherical harmonics $\mathbf{Y}^{(l)}(\bm{\hat{\rho}})=(-1)^{l}\rho^{l+1}\bm{\nabla}{}^{(l)}\rho^{-1}$, where $\bm{\nabla}^{(l)}=\bm{\nabla}_{\alpha_{1}}\dots\bm{\nabla}_{\alpha_{l}}$. The expansion coefficients $\mathbf{V}_{i}^{(l)}$ are $l$-th rank reducible Cartesian tensors with three irreducible parts of ranks $l,$ $l-1,$ and $l-2$, corresponding to symmetric traceless, antisymmetric and pure trace combinations of the reducible indices. We denote these by $\mathbf{V}_{i}^{(ls)}$, $\mathbf{V}_{i}^{(la)}$ and $\mathbf{V}_{i}^{(lt)}$ respectively. The leading contributions from the slip, $$\begin{aligned}
\mathbf{v}_{i}^{\mathcal{A}}(\bm{\rho}_{i}) & = & \underbrace{-\mathbf{V}_{i}^{\mathcal{A}}+\tfrac{1}{15}\mathbf{V}_{i}^{(3t)}\cdot\mathbf{Y}^{(2)}}_{\mathrm
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'The IceCube Neutrino Observatory detects high energy astrophysical neutrinos in two event topologies: tracks and cascades. Since the flavor composition of each event topology differs, tracks and cascades can be used to test the neutrino properties and the mechanisms behind the neutrino production in astrophysical sources. Assuming a conventional model for the neutrino production, the IceCube data sets related to the two channels are in $>3\sigma$ tension with each other. Invisible neutrino decay with lifetime $\tau/m=10^2$ s/eV solves this tension. Noticeably, it leads to an improvement over the standard non-decay scenario of more than $3\sigma$ while remaining consistent with all other multi-messenger observations. In addition, our invisible neutrino decay model predicts a reduction of $59\%$ in the number of observed $\nu_\tau$ events which is consistent with the current observational deficit.'
author:
- 'Peter B. Denton'
- Irene Tamborra
bibliography:
- 'Tracks\_Cascades.bib'
title: 'Invisible Neutrino Decay Resolves IceCube’s Track and Cascade Tension'
---
[*Introduction.*—]{}The IceCube Neutrino Observatory measures high energy astrophysical neutrinos with energies reaching up to few PeVs [@Aartsen:2013bka; @Aartsen:2013jdh; @Aartsen:2016xlq]. While numerous source candidates have been proposed to interpret the observed data, no clear picture has yet emerged [@Anchordoqui:2013dnh; @Meszaros:2015krr; @Waxman:2015ues; @Murase:2015ndr].
According to the conventional framework, adopted in this work, high energy astrophysical neutrinos are produced primarily by charged pion decay. Charged pions decay to a muon and a muon neutrino, and the muon in turn decays on to a positron, electron neutrino, and a muon antineutrino resulting in a neutrino flavor ratio at the source of $\nu_e:\nu_\mu:\nu_\tau = 1:2:0$, each with approximately the same energy. After neutrino oscillations, the flavor ratio at the Earth is roughly $1:1:1$ leading to the expectation that the spectral distributions of neutrinos will be the same for any flavor, see e.g. [@Anchordoqui:2013dnh; @Farzan:2008eg]. This is independent of the source class since any mechanism that produces high energy neutrinos will do so dominantly as a result of charged pion decays. Hence, within this picture, the only possible result is equal fluxes for each flavor.
Single power law (SPL) and broken power law (BPL) fits have been considered to interpret the neutrino data [@Aartsen:2015knd; @Chen:2014gxa; @Palladino:2016zoe; @Anchordoqui:2016ewn; @Palladino:2018evm; @Chianese:2017jfa; @Vincent:2016nut; @Palomares-Ruiz:2015mka; @Sui:2018bbh]. They favor a SPL, with a possible break to explain the excess of events below 100 TeV [@Chianese:2017jfa; @Denton:2018tdj].
IceCube is partially sensitive to the flavor state of the neutrino through two distinct event topologies: track events resulting dominantly from $\nu_\mu$ interactions [@Aartsen:2016xlq], and nearly spherical cascade events resulting dominantly from $\nu_e$ and $\nu_\tau$ interactions [@Niederhausen:2015svt]. The IceCube Collaboration has interpreted each of these data sets in terms of the true per-flavor neutrino flux at the Earth under the assumption that the flavor ratio remains constant at $1:1:1$ for all energies and that the flux follows a SPL [@Aartsen:2015knd]. It is found that those two different channels produce results in tension with each other [@Aartsen:2016xlq], as shown in Fig. \[fig:ICdata\].
![IceCube track [@Aartsen:2016xlq] and cascade [@Niederhausen:2015svt] data samples. The tension between the two data samples is driven on the high energy end by the observation of six tracks with energies $E_\nu>1$ PeV. On the low energy side there is an apparent excess of events in the cascade channel [@Denton:2018tdj].[]{data-label="fig:ICdata"}](Inu_Both){width="\columnwidth"}
IceCube finds that the best fit per-flavor astrophysical spectral index and normalization from the track analysis over $E_\nu \in[194$ TeV, $7.8$ PeV$]$ is $\gamma_{t,{\rm IC}}=2.13\pm0.13$, $\Phi_{t,{\rm IC}}=0.90^{+0.30}_{-0.27}$ [@Aartsen:2016xlq] and the best fit from the cascade analysis over $E_\nu \in[13$ TeV, $7.9$ PeV$]$ is $\gamma_{c,{\rm IC}}=2.67^{+0.12}_{-0.13}$, $\Phi_{c,{\rm IC}}=2.3^{+0.7}_{-0.6}$ [@Niederhausen:2015svt] where $\gamma_i$ is the spectral index and $\Phi_i$ is the flux normalization at $E_\nu = 100$ TeV in units of $10^{-18}$ GeV$^{-1}$ cm$^{-2}$ sr$^{-1}$ s$^{-1}$.
In this Letter, we combine spectral and flavor information simultaneously to investigate the tension between the data sets associated to the two event topologies. We explore several modifications to the standard picture of the high energy astrophysical neutrino flux beyond what is foreseen within the Standard Model [^1].
We determine the diffuse intensity at the Earth after oscillations, convert this into the per-flavor intensity from each of the track and cascade channels, and fit a power law to each assuming a $1:1:1$ flavor ratio to compare a model to IceCube’s observations. We then compare the normalizations and spectral indices to the measured ones by combining both tracks and cascades under the assumption that the correlation between the normalizations and spectral indexes are small. Invisible neutrino decay provides a good fit to the data and is preferred over the Standard Model at more than $3\sigma$ removing the tension. Our proposed solution is not in contradiction with existing multi-messenger constraints and also explains the current deficit in the observation of $\nu_\tau$ events.
[*Standard Neutrino Source Model.*—]{} For the sake of generality, we model the neutrino spectral distribution in such a way to be agnostic about the mechanism of the neutrino production, i.e. $p\gamma$ or $pp$ interactions. We consider a general BPL model at the source parameterized by the break energy in the source frame $\tilde E_{\nu,b}$ and the change in the spectral index $\Delta$, such that the spectral index below the break energy is $\gamma$ and it is $\gamma+\Delta$ above it [@Anchordoqui:2013dnh; @Meszaros:2015krr; @Waxman:2015ues; @Murase:2015ndr; @Meszaros:2001vi]. The SPL case is then recovered for $\Delta = 0$.
This model is further generalized to the case where the break energy for neutrinos coming from muon decay ($\nu_e$ and $\nu_\mu$) is different than that from pion decay ($\nu_\mu$). Pions and muons lose energy in $p\gamma$ sources, e.g. in the presence of magnetic fields due to synchrotron losses and they may have separate break energies, $\tilde E_{\nu,b,\mu}$ and $\tilde E_{\nu,b,\pi}$. For example, for synchrotron losses, the neutrino break energy scales like $m_i^{5/2}\tau_i^{-1/2}$ for $i\in\{\pi,\mu\}$ where $m$ ($\tau$) is the mass (lifetime) of the particle, so the ratio of the neutrino break energies is $R_{\pi,\mu} \equiv \tilde E_{\nu,b,\pi}/\tilde E_{\nu,b,\mu} \simeq 18.4$ when synchrotron cooling dominates. The simpler BPL model introduced above is recovered when $R_{\pi,\mu}=1$. Thus there are at most five free parameters in the BPL model: $\gamma$, $\Delta$, $\tilde E_{\nu,b}$, $R_{\pi,\mu}$, and the neutrino flux normalization $\Phi_\nu$.
The IceCube neutrino flux is considered to be dominantly extragalactic and compatible with a diffuse origin [@Denton:2017csz; @Aartsen:2017ujz; @Ando:2015bva; @Aartsen:2015knd]. Hence, the expected diffuse neutrino intensity at the Earth for the flavor $\nu_\beta$ ($\beta=e,\mu,\tau$) is $$\mathcal{I}_{\nu_\beta}=\sum_{\nu_\alpha} d_H\int_0^{z_{\max}}dz\frac{F_{\nu_\alpha}((1+z)E_\nu)\rho(z)}{h(z)}\bar P(\nu_\alpha \
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
author:
- 'E. Ding'
- 'H. N. Chan'
- 'K. W. Chow'
- 'K. Nakkeeran'
- 'B. A. Malomed'
bibliography:
- 'ref.bib'
title: Exact states in waveguides with periodically modulated nonlinearity
---
Introduction
============
It is commonly known that optical spatial solitons arise in planar and bulk waveguides through the balance of the Kerr nonlinearity and transverse diffraction [@agrawal]. Modern fabrication technologies make it possible to create waveguides featuring spatially inhomogeneous nonlinearities that support novel classes of propagation patterns [@boris1]. In particular, spatially inhomogeneous waveguides with a defocusing nonlinearity, whose local strength grows toward the periphery, can support diverse species of fundamental and higher-order solitons, including vortices, necklace rings, vortex gyroscopes, *hopfions*, and complex hybrid modes [boris2,Lei\_Wu,wu, zhong,Radik,Yasha,hybrids]{}, as well as *localized dark solitons* [@Zeng]. Similar nonlinearity landscapes, featuring different growth rates of the local nonlinearity in opposite transverse directions, support strongly asymmetric bright solitons [@boris4]. Asymmetric solitons also appear spontaneously if the nonlinearity profile features a dual-well structure [@dual; @Nir]. Furthermore, a combination of the fast growing local strength of the defocusing nonlinearity with the usual $\mathcal{PT}$-symmetric gain-loss profile makes it possible to produce solitons that exhibit *unbreakable* $\mathcal{PT}$ symmetry [@unbreakable; @raju; @2D], which is essential for constructing robust solitons in such systems [@Demetri; @PTreview1; @PTreview2]. It is also relevant to mention that a combination of $\mathcal{PT}$-symmetric with competing nonlinearities supports spatiotemporal solitons [@ref1].
Considerable interest has also been drawn to models with uniform nonlinearity, either self-defocusing or focusing, and specially designed periodic potentials that support exact periodic wave solutions [Carr1,Carr2]{}. Although both the particular potentials and the corresponding exact periodic solutions are not generic, and the analysis of their stability can only be performed numerically, these models provide direct insight into the possibility to support periodic wave patterns by utilizing the interplay of periodic potentials and the ubiquitous cubic nonlinearity. Furthermore, nontrivial exact solutions serve as benchmarks which suggest the shape of generic solutions. The inverse problem, aimed at engineering waveguiding potentials adjusted to maintaining periodic waves with prescribed properties, is a physically relevant issue too [@Spain].
In this work, we introduce a model with a class of periodic modulations that represent spatially periodic *pseudopotentials* [@pseudo] induced by the local nonlinearity. This model admits exact solutions in the form of periodic wave patterns which, in the limiting case of an infinite modulation period, become bright solitons. Stability of these patterns is studied numerically. The same model can also be used to solve the inverse problem of engineering a nonlinearity-modulation profile needed to support a wave pattern with prescribed period and amplitude.
The Mathematical Model
======================
The light propagation in a planar waveguide with spatially modulated nonlinearity is described by the model that is based on the scaled nonlinear Schrödinger equation for the electromagnetic wave amplitude $%
\Psi (x,z)$, $$i\Psi _{z}+\Psi _{xx}+g(x)|\Psi |^{2}\Psi =0, \label{eq:nls}$$where $x$ and $z$ are the transverse and longitudinal coordinates, respectively. The periodically-modulated nonlinearity profile is defined by $%
g(x)$, chosen as $$g(x)=\frac{\alpha }{\mathrm{dn}^{2}(x)}+\beta +\gamma ~\mathrm{dn}^{2}(x),
\label{eq:g}$$where $\alpha $, $\beta $, and $\gamma $ are real constants, and $\mathrm{dn}%
(x)$ is the standard Jacobi elliptic function with modulus $\sqrt{m}$ and period $2K$ ($K$ being the complete elliptic integral of the first kind). It is relevant to mention that, in the general case, the periodic inhomogeneity affects not only the local nonlinearity, but also the local refractive index, which would generate an additional term $U(x)\Psi $ in Eq. (\[eq:nls\]), with an effective spatially-periodic potential, $U(x)$. Nevertheless, specific experimental methods, such as resonant doping, make it possible to create waveguides in which the nonlinearity is affected by the periodic modulation, while the refractive index remains nearly constant [@boris1].
The same model, with propagation distance $z$ replaced by time $t$, represents the scaled Gross-Pitaevskii equation for the mean-field wave function of an atomic Bose-Einstein condensate (BEC), for which the periodic nonlinearity modulation can be induced by means of the Feshbach resonance in a spatially non-uniform magnetic or optical field. In particular, the necessary periodic profile of the magnetic field can be accurately implemented by means of the known technique based on the use of appropriately designed magnetic lattices [@magnetic]. Furthermore, the spatially periodic distribution of the local nonlinearity coefficient in BEC has been experimentally realized by means of an optical lattice [Feshbach]{}. A particular anharmonic profile corresponding to Eq. (\[eq:g\]) can be effectively approximated by a superposition of several harmonics of its Fourier decomposition, represented by the respective optical lattices. In a real experiment, the setting also includes an overall parabolic trapping potential. However, in many situations the characteristic scale of this potential is much larger than the period of the spatial modulation, which makes it possible to neglect the trapping potential while analyzing the effects of periodic lattices [@Heidelberg].
We look for a stationary solution to Eq. (\[eq:nls\]) in the form of a $\mathrm{dn}$-wave": $$\Psi (x,z)\equiv \psi (x)e^{-i\Omega z}=\frac{A_{0}\;\mathrm{dn}(x)}{1+b\;%
\mathrm{dn}^{2}(x)}e^{-i\Omega z}, \label{eq:ansatz}$$where $A_{0}$ and $b$ are real constants, and $-\Omega $ is the propagation constant (or the chemical potential, in the BEC model). Note that this ansatz is nonsingular under the conditions $b>-1$ or $b<-1/(1-m)$. Substituting it into Eq. (\[eq:nls\]) gives rise to the following system of equations for the three ansatz parameters $A_{0}^{2}$, $b$, and $\Omega $: $$\left\{ \begin{aligned} 2+A_{0}^{2} \; \alpha -6b(m-1)-m+\Omega =0 \; , \\
A_{0}^{2} \; \beta +2b^{2}(m-1)+2b(\Omega +3m-6)-2=0 \; , \\ 6b+A_{0}^{2} \;
\gamma +b^{2}(2-m+\Omega )=0 \; . \end{aligned}\right. \label{eq:para}$$One can calculate any three parameters from this system for given values of the others. In particular, this allows one to address the above-mentioned inverse problem, aimed at determining the nonlinearity modulation profile (see Eq. (\[eq:g\])) needed for maintaining a particular wave pattern.
Stability Analysis
==================
The stability of the $\mathrm{dn}$-wave denoted by Eq. (\[eq:ansatz\]) is investigated by means of the standard linearization procedure [linstab1,linstab2]{}. Substituting $\Psi (x,z)=\left[ \psi (x)+u(x,z)\right]
e^{-i\Omega z}$ into Eq. (\[eq:nls\]), with the small complex perturbation defined as $u\left( x,z\right) \equiv R(x,z)+iI(x,z)$, we arrive at the linearized system, $$\left\{ \begin{aligned} \partial _{z}R &=&\left( -\Omega -g(x)\psi
^{2}(x)-\partial _{x}^{2}\right) I \; ,\\ \partial _{z}I &=&\left( \Omega
+3g(x)\psi ^{2}(x)+\partial _{x}^{2}\right) R \; . \end{aligned}\right.
\label{eq:linear}$$The stability of the $\mathrm{dn}$-wave is determined by substituting $%
\left\{ R(x,z),I(x,z)\right\} =\left\{ P(x),Q(x)\right\} \exp (\lambda z)$ into the above equations. The resulting problem for stability eigenvalue $%
\lambda $ is solved numerically using the finite-difference method. In particular, *modulational instability* of periodic states [agrawal]{} is accounted for by eigenvalues with $\mathrm{Re}(\lambda )>0$. Generic examples of stable and unstable $\mathrm{dn}$-waves are presented in Fig. \[fig:stab\]. The stability, as predicted by the calculation of the eigenvalue spectra, is corroborated by direct simulations of Eq. ([eq:nls]{}), using the Fourier transform in $x$ and a fourth-order Runge-Kutta algorithm in $z$.
The dependence of the stability of the $\mathrm{dn}$-waves on the system’s parameters can be explored with the help
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: |
A famous result due to Grothendieck asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{2}$ is absolutely $(1,1)$-summing. If $n\geq2,$ however, it is very simple to prove that every continuous $n$-linear operator from $\ell_{1}\times\cdots\times\ell_{1}$ to $\ell_{2}$ is absolutely $\left( 1;1,...,1\right) $-summing, and even absolutely $\left( \frac{2}{n};1,...,1\right) $-summing$.$ In this note we deal with the following problem:
Given a positive integer $n\geq2$, what is the best constant $g_{n}>0$ so that every $n$-linear operator from $\ell_{1}\times\cdots\times\ell_{1}$ to $\ell_{2}$ is absolutely $\left( g_{n};1,...,1\right) $-summing?
We prove that $g_{n}\leq\frac{2}{n+1}$ and also obtain an optimal improvement of previous recent results (due to Heinz Juenk $\mathit{et}$ $\mathit{al}$, Geraldo Botelho $\mathit{et}$ $\mathit{al}$ and Dumitru Popa) on inclusion theorems for absolutely summing multilinear operators.
address: 'UFRN/CERES - Centro de Ensino Superior do Seridó, Rua Joaquim Gregório, S/N, 59300-000, Caicó- RN, Brazil'
author:
- 'A. T. Bernardino'
title: 'On cotype and a Grothendieck-type theorem for absolutely summing multilinear operators'
---
Introduction
============
Grothendieck’s theorem for absolutely summing operators asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{2}$ is absolutely $(1;1)$-summing (and hence absolutely $(p;p)$-summing for every $p\geq1$). For the linear theory of absolutely summing operators we refer to [@df; @djt] (see also [@bpr; @ku2; @seo] for recent developments).
In the multilinear setting, D. Pérez-García, in his PhD thesis [@dav] (see also [@bom] and [@botp] for a different proof), proved that every continuous $n$-linear operator from $\ell_{1}\times\cdots\times
\ell_{1}$ to $\ell_{2}$ is multiple $(1;1,...,1)$-summing (in fact, multiple $(p;p,...,p)$-summing for every $1\leq p\leq2$)$.$ This result can be regarded as the multilinear version of Grothendieck’s theorem.
Let us recall the notions.
The letters $X_{1},...,X_{n},X,Y$ will always denote Banach spaces over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and $X^{\ast}$ represents the topological dual of $X$.
For any $s>0,$ we denote the conjugate of $s$ by $s^{\ast}$. Given a positive integer $n$, the space of all continuous $n$-linear operators from $X_{1}\times\cdots\times X_{n}$ to $Y$ endowed with the $\sup$ norm is denoted by $\mathcal{L}(X_{1},...,X_{n};Y).$ For $p>0$, the vector space of all sequences $\left( x_{j}\right) _{j=1}^{\infty}$ in $X$ such that$$\left\Vert \left( x_{j}\right) _{j=1}^{\infty}\right\Vert _{p}=\left(
\sum_{j=1}^{\infty}\left\Vert x_{j}\right\Vert ^{p}\right) ^{\frac{1}{p}}<\infty$$ is denoted by $\ell_{p}\left( X\right) .$ We represent by $\ell_{p}^{w}\left( X\right) $ the linear space of the sequences $\left(
x_{j}\right) _{j=1}^{\infty}$ in $X$ such that $\left( \varphi\left(
x_{j}\right) \right) _{j=1}^{\infty}\in\ell_{p}\left( \mathbb{K}\right) $ for every $\varphi\in X^{\ast}$.
If $0<p,q_{1},...,q_{n}<\infty$ and $\frac{1}{p}\leq\frac{1}{q_{1}}+\cdots+\frac{1}{q_{n}},$ a multilinear operator $T\in\mathcal{L}(X_{1},...,X_{n};Y)$ is absolutely ** $(p;q_{1},...,q_{n})$-summing if $(T(x_{j}^{(1)},...,x_{j}^{(n)}))_{j=1}^{\infty}\in\ell_{p}(Y)$ for every $(x_{j}^{(k)})_{j=1}^{\infty}\in\ell_{q_{k}}^{w}(X_{k}),k=1,...,n.$ In this case we write $T\in\Pi_{\left( p;q_{1},...,q_{n}\right) }^{n}\left(
X_{1},...,X_{n};Y\right) $. For details we refer to [@am].
When $1\leq q_{1},...,q_{n}\leq p<\infty$ a multilinear operator $T\in\mathcal{L}(X_{1},...,X_{n};Y)$ is multiple ** $(p;q_{1},...,q_{n})$-summing if $(T(x_{j_{1}}^{(1)},...,x_{j_{n}}^{(n)}))_{j_{1},..,j_{n}=1}^{\infty}\in\ell_{p}(Y)$ for every $(x_{j}^{(k)})_{j=1}^{\infty}\in
\ell_{q_{k}}^{w}(X_{k}),k=1,...,n.$ In this case we write $T\in\Pi_{m\left(
p;q_{1},...,q_{n}\right) }^{n}\left( X_{1},...,X_{n};Y\right) $. For details we mention [@bom; @collec] and for recent developments and applications related to the multilinear and polynomial theory we refer to [@ag; @bbb; @bh; @an; @dgm; @se; @lit; @mat; @ppp] and references therein. For $n=1$ we write $\Pi$ instead of $\Pi^{1}$ and we recover the classical theory of absolutely summing linear operators.
For $1\leq q_{1},...,q_{n}\leq p<\infty,$ the inclusion $$\Pi_{m\left( p;q_{1},...,q_{n}\right) }^{n}\left( X_{1},...,X_{n};Y\right)
\subseteqq\Pi_{\left( p;q_{1},...,q_{n}\right) }^{n}\left( X_{1},...,X_{n};Y\right)$$ is obvious. So, the following coincidence result is an immediate consequence of Pérez-García multilinear version of Grothendieck’s theorem:
For every positive integer $n$, $$\Pi_{\left( 1;1,...,1\right) }^{n}\left( \ell_{1},...,\ell_{1};\ell
_{2}\right) =\mathcal{L}\left( \ell_{1},...,\ell_{1};\ell_{2}\right) .$$
However, using that $\ell_{1}$ has cotype $2$ it is easy to prove that the above result is far from being optimal. In fact, we have the following improvement (see [@irish; @irishd]):
For every positive integer $n\geq2$, $$\Pi_{\left( \frac{2}{n};1,...,1\right) }^{n}\left( \ell_{1},...,\ell
_{1};\ell_{2}\right) =\mathcal{L}\left( \ell_{1},...,\ell_{1};\ell
_{2}\right) . \label{dda}$$
So, the following problem is quite natural:
\[xc\]Given a positive integer $n\geq2$, what is the best constant $g_{n}>0$ so that$$\Pi_{\left( g_{n};1,...,1\right) }^{n}\left( \ell_{1},...,\ell_{1};\ell
_{2}\right) =\mathcal{L}\left( \ell_{1},...,\ell_{1};\ell_{2}\right) ?$$
If we test $n=1$ in (\[dda\]) we obtain $$\Pi_{(2;1)}\left( \ell_{1};\ell_{2}\right) =\mathcal{L}\left( \ell_{1};\ell_{2}\right)$$ which is not surprising at all, in view of Grothendieck
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We study two properties of two modules over a local hypersurface $R$: decency, which is close to proper intersection of the supports, and $\operatorname{Tor}$-rigidity. We show that the vanishing of Hochster’s function $\theta^R(M,N)$, known to imply decent intersection, also implies rigidity. We investigate the vanishing of $\theta^R(M,N)$ to obtain new results about decency and rigidity over hypersurfaces. Many applications are given.'
address: |
Department of Mathematics\
University of Kansas\
Lawrence, KS 66045-7523 USA
author:
- Hailong Dao
title: 'Decency and Tor-rigidity for modules over hypersurfaces'
---
[^1]
Introduction
============
Throughout this paper we will deal exclusively with a local, Noetherian, commutative ring $R$ and finitely generated modules over $R$.
Two $R$-modules $M$ and $N$ such that $l(M{\otimes}_RN)<\infty $ are said to *intersect decently* if $\dim M + \dim N \leq \dim
R$. We say that $M$ is *decent* if for all $N$ such that $l(M{\otimes}_RN)<\infty $, $M$ and $N$ intersect decently. This property arises naturally from Serre’s work on intersection multiplicity ([@Se]), which shows that over a regular local ring, any two modules intersect decently. In fact, to have a satisfying local intersection theory, one needs modules to intersect decently as a minimum requirement. However, sufficient conditions for decent intersection become much more elusive in general, even when $R$ is a hypersurface. For example, one can look at a famous open question in Commutative Algebra:
Let $A$ be a regular local ring, and suppose $R$ is a local ring such that $A\subset R$ and $R$ is module-finite over $A$. Then $A$ is a direct summand of $R$ as an $A$-module.
The Direct Summand Conjecture could be proved if one could show a certain module over a local hypersurface is decent (see [@Ho1]). As another example, it was conjectured by Peskine and Szpiro (cf. [@PS]) that in general, a module of finite projective dimension is decent. This question is open even for hypersurfaces of ramified regular local rings.
A pair of modules $(M,N)$ is called *rigid* if for any integer $i\geq0$, $\operatorname{Tor}_i^R(M,N)=0$ implies $\operatorname{Tor}_j^R(M,N)=0$ for all $j\geq i$. Moreover, $M$ is *rigid* if for all $N$, the pair $(M,N)$ is rigid. Auslander studied rigidity in order to understand torsion on tensor products ([@Au]). He also observed that rigidity of $M$ implies other nice properties, such as any $M$-sequence must be an $R$-sequence. To further demonstrate the usefulness of rigidity, let us recall the:
([@HW1], 2.5) Let $R$ be a complete intersection. Let $M,N$ be non-zero finitely generated modules over $R$ such that $\operatorname{Tor}_i^R(M,N)=0$ for all $i\geq 1$. Then: $$\operatorname{depth}(M) + \operatorname{depth}(N) = \operatorname{depth}(R)+\operatorname{depth}(M{\otimes}_RN)$$
Thus, rigidity allows us to force a very strong condition on the depths of the modules by proving the vanishing of one single $\operatorname{Tor}$ module. Auslander’s work, combined with that of Lichtenbaum ([@Li]), showed that modules over regular local rings are rigid. Huneke and Wiegand ([@HW1],[@HW2]) continued this line to study rigidity over hypersurfaces.
The classical condition conjectured to be sufficient for both rigidity and decency was that one of the modules must have finite projective dimension. In general, this is open for decency and false for rigidity (see [@He]). In any case, having finite projective dimension is too restrictive for the most interesting applications, so a key question arises: *Are there more flexible sufficient conditions for rigidity and decency?* A good answer here would not only make applications easier, it would also shed some light on the behaviour of modules of finite projective dimension with respect to the two conditions.In this paper, we try to answer this question for modules over hypersurfaces using a mixture of results and techniques from Commutative Algebra and Intersection Theory. Perhaps not surprisingly, our answers often involve some conditions about the classes of the modules in the Grothendieck group of finitely generated modules over $R$.
Throughout this note we will assume that our hypersurface $R$ comes from an unramified or equicharacteristic regular local ring (we call such hypersurfaces “admissible", following Hochster). Since we need to apply results such as Serre’s Positivity and Non-negativity of $\chi_i$, which are open in general for the ramified case, this is necessary. In some particular instances, such as in low dimensions, this restriction can be relaxed, however we feel it may disrupt the flow of the paper to comment on every such case. The reader will lose very little by thinking of the equicharacteristic (containing a field) case.
Section \[2\] is a review of basic notation and some preliminary results. Of particular interest is Hochster’s theta function. For a local hypersurface $R$ and a pair of finitely generated $R$ modules $M,N$ such that $l(\operatorname{Tor}_i^R(M,N))<\infty$ for all $i\gg0$, we can define: $$\theta^R(M,N) = l(\operatorname{Tor}_{2e+2}^R(M,N)) - l(\operatorname{Tor}_{2e+1}^R(M,N))$$ Here $e$ is any sufficiently large integer. The theta function was first introduced by Hochster \[Ho1\] in his study of the direct summand conjecture. We recall the basic properties of $\theta(M,N)$ and prove a key technical result: [the vanishing of $\theta^R(M,N)$ implies rigidity of (M,N)]{}(proposition \[rg1\]).
In section \[3\] we study the vanishing of $\theta^R(M,N)$ when $R$ is a hypersurface with isolated singularity. The key advantage with such rings is that $\theta^R(M,N)$ is always defined for a pair $M,N$, so we can “move" the modules easily into favorable positions where vanishing of $\theta$ is more evident. We give a fairly complete picture when the dimension of the ring is at most $4$ and obtain many results in higher dimension (see \[dim1\] to \[dim2,3\]). One of them (\[moving\]) states that when $R$ contains a field and $\dim M + \dim N \leq \dim R$, then $\theta^R(M,N)=0$. This result partly answers a question raised to the author by Roger Wiegand. Our results also point to a conjecture that $\theta^R(M,N)$ should always vanish if $\dim R$ is [*even*]{}.
In section \[4\] we study rigidity over hypersurfaces in general. We prove new criteria for rigidity (theorem \[rig1.1\]), as well as a connection to decency when one of the modules is Cohen-Macaulay(see theorem \[rigidandproper\]).
In section \[5\] we apply our results from previous sections to give new proofs of seemingly different results in the literature, as well as new results. First, we show that if $char(R)=p>0$, then $^eR$ is rigid, where $^eR$ is $R$ with the module structure defined by the $e$-th iteration of the Frobenius homeomorphism (theorem \[AM\]). This is the hypersurface case of a theorem by Avramov and Miller ([@AM]). We also apply our rigidity results to gain understanding of depth of tensor products, following the same line of investigation done by Auslander, Huneke and Wiegand(see \[vanishingiso\], \[HWmain\]). Finally, we switch our attention to projective hypersurfaces and prove a result on intersection of subvarieties(theorem \[projhyper\]) using what we know about decency.
Finally, section \[lastsection\] discusses our attempts to generalize the theta function and some additional results we gain on the way which improve on previous works by Murthy and Jorgensen. We also give numerous examples to illustrate our results throughout the paper and show that they are optimal in certain senses.
After this preprint appeared online, a few works by the authors and other researchers have been focused on extensions and applications of the ideas and results in here. For example, [@Da2] uses the $\operatorname{Tor}$-rigidity results here to extend Auslander’s theorems on $\operatorname{Hom}(M,M)$ over regular local rings to hypersurfaces, and [@Da3] applies such results on understanding Van den Bergh’s definition of non-commutative crepant resolutions. Papers [@CeDa; @Ce; @Da] deal with various generalizations to complete intersections as well as analogous results for vanishing of $\
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'The aim of this paper is to study the Harnack type logarithmic submajorisation and Fuglede-Kadison determinant inequalities for operators in a finite von Neumann algebra. In particular, the Harnack type determinant inequalities due to Lin-Zhang[@LZ2017] and Yang-Zhang[@YZ2020] are extended to the case of operators in a finite von Neumann algebra.'
address:
- 'College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China'
- 'College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China'
author:
- Yazhou Han
- Cheng Yan
title: Harnack type inequalities for operators in logarithmic submajorisation
---
Introduction
============
The classical Harnack inequality, named after Carl Gustav Axel von Harnack, gives an estimate from above and an estimate from below for a positive harmonic function in a domain. Even though the classical Harnack inequality is almost trivially derived from the Poisson formula, the consequences that may be deduced from Harnack inequality are particularly of great importance. Later, these inequalities became an important tool in the general theory of harmonic functions and partial differential equations. There exist as yet extensive works on generalized Harnack inequalities in various forms, see [@M2007; @W2013; @YZ2020] for a nice introduction about the inequality. The purpose of this paper is to investigate the Harnack type determinant inequality for operators and matrices.
With the help of Language multiplier method, the following Harnack type determinant inequality was established by Tung[@T1964], as a tool to study Harnack inequality: If $Z\in \mathbb{M}_n$ is a complex matrix with singular values $r_k$ with $0\leq r_k<1, k=1, 2, ..., n$, then $$\label{inequ. T1}
\prod_{k=1}^n\frac{1-r_k}{1+r_k}\leq\frac{\det(\mathbb{I}-Z^*Z)}{|\det(\mathbb{I}-UZ)|^2}\leq
\prod_{k=1}^n\frac{1+r_k}{1-r_k}, U\in \mathbb{U}_n,$$ where $\mathbb{U}_n$ denote the set of all $n\times n$ unitary matrices $U$. From these bounds Tung obtained upper and lower bounds of a Poisson kernel on $\mathbb{U}$(see [@T1964]), hence that the so-called Harnack’s first and second theorems are established. Tung’s work drew immediate attention of Hua and Marcus. Using majorisation theory and singular value (eigenvalue) inequalities of Weyl, Marcus [@M1965] gave another proof of (\[inequ. T1\]) and gave an equivalent form of (\[inequ. T1\]). Almost at the same time, a proof of (\[inequ. T1\]) was also given by Hua[@H1965] based on the determinantal inequality he had previously obtained in[@H1955]. In the past several ten years, Tung’s work has attracted attentions of mathematicians and been extended to various cases (see [@M2007; @W2013; @Z2005; @JL2020; @L2015; @YZ2020] and the references therein for more details). Among these outstanding works we will be interested in Lin-Zhang’s and Yang-Zhang’s work. Specifically, with $A=UZ$, (\[inequ. T1\]) is equivalently rewritten in terms of eigenvalues ([@JL2020; @YZ2020]) as $$\label{inequ. YZ1}
\prod_{k=1}^n\frac{1-r_k}{1+r_k}\leq\prod_{k=1}^n
\lambda_k((\mathbb{I}-A^*)^{-1}(\mathbb{I}-A^*A)(\mathbb{I}-A)^{-1})
\leq\prod_{k=1}^n\frac{1+r_k}{1-r_k}.$$ (\[inequ. YZ1\]) leads to the study of inequalities of logarithmic submajorisation of eigenvalues and singular values. Following this line, an interesting generalization of (\[inequ. YZ1\]) is presented by Yang-Zhang[@YZ2020] and Jing-Lin[@JL2020] as follows: $$\label{FK det 1}
\prod_{k\in K}
\lambda_k((\mathbb{I}-A^*)^{-1}(\mathbb{I}-A^*A)(\mathbb{I}-A)^{-1})
\leq\prod_{k\in K}\frac{1+r_k}{1-r_k},$$ $$\label{FK det 2}
\prod_{i\in K}
\lambda_{n-k+1}((\mathbb{I}-A^*)^{-1}(\mathbb{I}-A^*A)(\mathbb{I}-A)^{-1})
\geq\prod_{k\in K}(1-r_k^2)\prod_{i=1}^{|K|}\frac{1}{(1+r_i)^2},$$ where $K$ is a subset of $\{r_1, r_2, \cdots, r_n\}$ and $|K|$ denote the number of terms in $K$. The main theme of the paper is to continue with Jiang-Lin and Yang-Zhang’s work and to show their results hold in the case of operators in finite von Neumann algebras.
We are concerned with the Harnack type logarithmic submajorisation inequality and Fuglede-Kadison determinant inequality for operators in a finite von Neumann algebra. The properties of the logarithmic submajorisation and Fuglede-Kadison determinant for operators in a finite von Neumann algebra was investigated by many authors, see for example [@B1983; @BL2008; @HSZ2020]. Those properties are important, for example, in investigation of noncommutative Hardy spaces and invariant subspaces for operators in von Neumann algebra. By adapting the techniques in [@YZ2020; @FK1986; @N1987], we obtain some inequalities which is related to the Harnack type logarithmic submajorisation inequality and Fuglede-Kadison determinant inequality. In particular, we show that the inequalities (\[FK det 1\]) and (\[FK det 2\]) hold for operators in a finite von Neumann algebra. We will conclude this paper with a series of logarithmic submajorisation submajorization inequalities which is related to Cayley transform.
Preliminaries
=============
von Neumann algebras
--------------------
Suppose that $\mathcal{H}$ is a separable Hilbert space over the field $\mathbb{C}$ and $\mathbb{I}$ is the identity operator in $\mathcal{H}$. We will denote by $\mathcal{B}(\mathcal{H})$ the $*$-algebra of all linear bounded operators in $\mathcal{H}$. Let $\mathcal{M}$ be a $*$-subalgebra of $\mathcal{B}(\mathcal{H})$ containing the identity operator $\mathbb{I}$. Then $\mathcal{M}$ is called a von Neumann algebra if $\mathcal{M}$ is weak operator closed. Let $\mathcal{M}^+$ denote the positive part of $\mathcal{M}$. We recall that a weight on $\mathcal{M}$ is a map $\tau: \mathcal{M}^+\rightarrow [0, \infty]$ satisfying
1. $\tau(x+y)=\tau(x)+\tau(y),$ for all $x, y\in \mathcal{M}^+$;
2. $\tau(\alpha x)=\alpha\tau(x)$ for all $x\in \mathcal{M}^+$ and $\alpha\in [0, \infty)$, with the convention $0\cdot\infty=0.$
The weight $\tau$ is called faithful if $\tau(x^*x)=0$ implies $x=0$, normal if $x_i\uparrow_i x$ in $\mathcal{M}^+$ implies that $0\leq\tau(x_i)\uparrow_i\tau(x)$, tracial if $\tau(x^*x)=\tau(xx^*)$ for all $x\in\mathcal{M}$. Note that since $(x_i)$ is bounded there is $x$ in $\mathcal{M}^+$ such that, for any $h$ in $\mathcal{H}$, $\langle x_ih, h\rangle\uparrow\langle xh, h\rangle$, which implies that $x_i$ tends to $x$ weak\* and hence $x\in \mathcal{M}^+$. The operator $x$ is obviously the least upper bound of $(x_i)$, it is natural to denote it by $\sup_i x_i$. The space $\mathcal{M}$ is a partially ordered vector space under the ordering $x\geq0$ defined by $\langle x\xi, \xi\rangle\geq0, \xi\in \mathcal{H}$. Recall that $x\in\mathcal{M}$ is contractive if $\|x\|\leq1$ and strict contractive if $\|x\|<1$. Moreover, if $x$ is strict contractive, then $\mathbb{I}-x^*x$ is invertible and $\mathbb{I}-x^*x\geq0$.
It is also customary to say trace instead of tracial weight. A trace $\tau$ is called finite if $\tau(\mathbb{I})<\infty.$ A finite trace $\tau$ is extended uniquely to a positive linear functional on $\mathcal{M}$ which will also
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-selfadjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trace class case, the generic existence of transmission eigenvalues is established.'
address:
- |
M. Hitrik, Department of Mathematics\
UCLA\
Los Angeles\
CA 90095-1555\
USA
- |
K. Krupchyk, Department of Mathematics and Statistics\
University of Helsinki\
P.O. Box 68\
FI-00014 Helsinki\
Finland
- |
P. Ola, Department of Mathematics and Statistics\
University of Helsinki\
P.O. Box 68\
FI-00014 Helsinki\
Finland
- |
L. Päivärinta, Department of Mathematics and Statistics\
University of Helsinki\
P.O. Box 68\
FI-00014 Helsinki\
Finland
author:
- Michael Hitrik
- Katsiaryna Krupchyk
- Petri Ola
- Lassi Päivärinta
title: Transmission eigenvalues for elliptic operators
---
Introduction
============
Let $P_0(D)$ be an elliptic partial differential operator on ${\mathbb{R}}^n$, $n\ge 2$, of order $m\ge 2$ with constant real coefficients, $$P_0(D)=\sum_{|\alpha|\le m} a_{\alpha}D^\alpha, \quad a_\alpha\in{\mathbb{R}}, \quad D_j=-i\frac{\partial}{\partial x_j},\quad j=1,\dots,n.$$ Let $\Omega\subset {\mathbb{R}}^n$ be a bounded domain with a $C^\infty$-boundary and assume that $V\in C^\infty(\overline{\Omega}, {\mathbb{R}})$ with $V>0$ in $\overline{\Omega}$. The interior transmission problem associated to $P_0$ and $V$ is the following degenerate boundary value problem, $$\label{eq_TE_acoustic}
\begin{aligned}
(P_0-\lambda)v=0 \quad &\text{in} \quad \Omega,\\
(P_0-\lambda(1+ V))w=0 \quad &\text{in} \quad \Omega,\\
v-w \in H^{m}_0(\Omega).
\end{aligned}$$ Here $H^m_0(\Omega)$ is the standard Sobolev space, defined as the closure of $C^\infty_0(\Omega)$ in the Sobolev space $H^m(\Omega)$. We say that $\lambda\in {\mathbb{C}}$ is a transmission eigenvalue if the problem has non-trivial solutions $0\ne v\in L^2_{\textrm{loc}}(\Omega)$ and $0\ne w\in L^2_{\textrm{loc}}(\Omega)$.
In the recent paper [@HitKruOlaPai], we have studied the interior transmission problem and transmission eigenvalues for multiplicative sign-definite perturbations of linear partial differential operators with constant real coefficients. Sufficient conditions for the discreteness of the set of transmission eigenvalues and for the existence of real transmission eigenvalues were obtained. In particular, in the elliptic case, the set of transmission eigenvalues is discrete and in [@HitKruOlaPai], the existence of real transmission eigenvalues was obtained for certain elliptic operators such as the biharmonic operator and the Dirac system in ${\mathbb{R}}^3$.
The purpose of the present note is to point out an approach to the study of the transmission eigenvalues in the elliptic case, based on a reduction to the eigenvalue problem for a compact non-selfadjoint operator. By an application of Lidskii’s theorem, we obtain sufficient conditions for the existence of (possibly complex) transmission eigenvalues, and the completeness of the set of the generalized eigenvectors, as well as demonstrate the generic existence of transmission eigenvalues. Let us mention explicitly that in this approach, we were directly inspired by the recent works [@AboRob; @ChaHelLap04; @HelRobWang; @Rob2004], where similar ideas in dealing with quadratic eigenvalue problems have been used to study hypoelliptic partial differential operators which are not analytic hypoelliptic.
The significance of transmission eigenvalues and of the interior transmission eigenvalue problem comes from inverse scattering theory, and originally, this problem was introduced in [@ColMonk88] in this context. The real transmission eigenvalues can be characterized as those values for which the scattering amplitude is not injective, see [@ColPaiSyl; @HitKruOlaPai]. Furthermore, in reconstruction algorithms of inverse scattering theory [@CakColbook; @ColKir96; @KirGribook], transmission eigenvalues correspond to frequencies that one needs to avoid in the reconstruction procedure.
Recently there has been a large number of works devoted to the interior transmission eigenvalue problem [@CakColGint_complex; @CakColHous10; @CakDroHou; @ColKirPai; @kir07; @paisyl08], with the major part being concerned with the case $P_0=-\Delta$. The existing results establish the discreteness of the set of transmission eigenvalues, [@ColKirPai], and give sufficient conditions for the existence of an infinite set of real transmission eigenvalues, [@CakDroHou; @paisyl08]. We would particularly like to mention the recent paper [@CakColGint_complex], where the existence of complex transmission eigenvalues was shown, assuming that the perturbation $V$ in is constant and sufficiently small.
In this note, we have chosen to base our presentation on the generalized acoustic wave equation $(P_0-\lambda(1+V))u=0$. Under the assumption that the full symbol of $P_0$ is non-negative, all the results could equally well have been derived for the following interior transmission problem associated to the Schrödinger equation $(P_0+V-\lambda)u=0$, $$\begin{aligned}
(P_0-\lambda)v=0 \quad &\text{in} \quad \Omega,\\
(P_0+V-\lambda)w=0 \quad &\text{in} \quad \Omega,\\
v-w \in H^{m}_0(\Omega).\end{aligned}$$
The structure of this note is as follows. In Section 2 we reduce the interior transmission problem to an eigenvalue problem for a compact non-selfadjoint operator in a suitable Schatten class. As a consequence of this reduction, in Section 3, we derive sufficient conditions for the existence of transmission eigenvalues and completeness of the generalized eigenstates. Finally, in Section 4, we show the generic existence of transmission eigenvalues in the trace class case.
Reduction to an eigenvalue problem for a non-selfadjoint compact operator
=========================================================================
From [@HitKruOlaPai], let us recall the following characterization of transmission eigenvalues.
\[thm\_equivalence\] Assume that $V\in C^\infty(\overline{\Omega}, {\mathbb{R}})$ with $V>0$ in $\overline{\Omega}$. A complex number $\lambda\ne 0$ is a transmission eigenvalue if and only if there exists $0\ne u\in H^{m}_0(\Omega)$ satisfying $$T_\lambda u:=(P_0-\lambda(1+V))\frac{1}{V}(P_0-\lambda)u=0\quad \text{in}\quad \mathcal{D}'(\Omega).$$
The question of deciding whether $0\ne \lambda\in {\mathbb{C}}$ is a transmission eigenvalue is therefore equivalent to finding a non-trivial solution $u\in H^m_0(\Omega)$ of the following quadratic eigenvalue problem $$\label{eq_quadratic}
T_\lambda u=(A-\lambda B +\lambda^2 C)u=0,$$ where $$A=P_0\frac{1}{V}P_0,\quad B=\frac{1}{V}P_0+P_0\frac{1}{V}+P_0,\quad C=1+\frac{1}{V}.$$ Consider the following factorization $$\begin{aligned}
T_\lambda=C^{1/2}L_\lambda C^{1/2}, \quad L_\lambda&=\tilde A-\lambda \tilde B+\lambda^2,\\
\tilde A&=C^{-1/2}AC^{-1/2}, \quad \tilde B=C^{-1/2}BC^{-1/2}.\end{aligned}$$ In [@HitKruOlaPai] it was proved that the operator $\tilde A$, equipped with the domain $$\mathcal{D}(\tilde A)=H^{2m}(\Omega)\cap H^m_0(\Omega),$$ is a self-adjoint operator on $L^2(\Omega)$ with a discrete spectrum. Here the regularity assumption on $V$ can be relaxed to $V\in C^N(\overline{\Omega})$, with $N$ being large enough but finite.
\[prop\_properties\]
- The operator $\tilde A$ is positive, and $ \mathcal{D}(\tilde A^{1/2})=H_0^m(\Omega)$.
- The operators $\tilde B\tilde A^{-1/2}$ and $\tilde A^{-1/2}\tilde B$ are bounded in
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We reanalyze the prompt muon neutrino flux from gamma-ray bursts (GRBs), at the example of the often used reference Waxman-Bahcall GRB flux, in terms of the particle physics involved. We first reproduce this reference flux treating synchrotron energy losses of the secondary pions explicitly. Then we include additional neutrino production modes, the neutrinos from muon decays, the magnetic field effects on all secondary species, and flavor mixing with the current parameter uncertainties. We demonstrate that the combination of these effects modifies the shape of the original Waxman-Bahcall GRB flux significantly, and changes the normalization by a factor of three to four. As a consequence, the gamma-ray burst search strategy of neutrino telescopes may be based on the wrong flux shape, and the constraints derived for the GRB neutrino flux, such as the baryonic loading, may in fact be already much stronger than anticipated.'
author:
- Philipp Baerwald
- 'Svenja H[ü]{}mmer'
- Walter Winter
title: 'Magnetic Field and Flavor Effects on the Gamma-Ray Burst Neutrino Flux'
---
Neutrino telescopes, such as IceCube [@Ahrens:2003ix] or ANTARES [@Aslanides:1999vq], are designed to detect neutrinos from astrophysical sources. There are numerous candidate sources, see [Ref.]{} [@Becker:2007sv] for a review and [Ref.]{} [@Rachen:1998fd] for the general theory. We focus on the prompt emission of gamma-ray bursts (GRBs) in this letter, where photohadronic interactions are expected to lead to a significant flux of neutrinos [@Waxman:1997ti]. So far, no extraterrestrial high energy neutrino flux has been detected yet. That is, for sources optically thin to neutrons, consistent with generic bounds [@Waxman:1998yy; @Mannheim:1998wp] which are just being touched by IceCube. The search for GRB neutrinos has been driven by analytical estimates for the shape and normalization, the simplest one being the Waxman-Bahcall (WB) flux [@Waxman:1998yy]. More recent analyses, such as the stacking analysis in [Ref.]{} [@Abbasi:2009ig], relating the neutrino flux to the observed gamma-ray flux, are based on the analytical generalization of this flux for arbitrary input parameters following [Ref.]{} [@Guetta:2003wi]. These calculations typically approximate the $\Delta(1232)$ resonance for the charged pion production $$p + \gamma \rightarrow \Delta^+ \rightarrow \left\{\begin{array}{lc} n + \pi^+ & \text{1/3 of all cases} \\ p + \pi^0 & \text{2/3 of all cases} \end{array} \right. \label{equ:Delta}$$ in some form. However, the GRB neutrino flux computation has been updated over the last ten years from the particle physics point of view by improving the description of the photo-meson production processes, and it has been obvious there is a substantial impact from magnetic field effects and flavor mixing on the neutrino flux as well; see, [[*e.g.*]{}]{}, [Refs.]{} [@Mucke:1999yb; @Murase:2005hy; @Kashti:2005qa; @Lipari:2007su; @Hummer:2010vx]. In this letter, we make the impact of these effects very explicit by revising the often used WB reference flux from [Ref.]{} [@Waxman:1998yy]. We include the relevant pion production modes and neutrinos from kaon and neutron decays. We treat the magnetic field effects on each charged particle species explicitly, and we include flavor effects/flavor mixing. Note that we keep our considerations as independent of the astrophysical source model as possible to factor out the particle physics effects, which are much better known than the details of the astrophysical model. The purpose of this letter is to demonstrate how the original WB flux changes in both shape and normalization effect by effect, and where the main impact comes from. We also discuss the impact on data analyses. The technology used in this letter is based on [Refs.]{} [@Hummer:2010vx; @Hummer:2010ai], where details can be found.
![\[fig:photo\] The WB flux from [[Eq.]{} (\[equ:WB\])]{} (thin dashed curve), the numerically reproduced flux using the $\Delta^+$ resonance only (lower solid curve), and the WB flux including higher resonances, direct production/$t$-channel processes, and multi pion production (high energy processes), which are successively switched on, leading to the final upper solid curve. Here the $\nu_\mu$ flux from $\pi^+$ and $\pi^-$ decays is considered. The normalization of our result to the numerically reproduced WB flux (gray dashed curve) is described in the main text.](Neutrinocontributionsplusminusz2){width="0.85\columnwidth"}
In the standard picture, protons collide with photons, possibly from synchrotron emission of co-accelerated electrons or positrons (see, [[*e.g.*]{}]{}, [Ref.]{} [@Dermer:2003zv]), leading to pion production by processes as, for instance, [[Eq.]{} (\[equ:Delta\])]{}. The charged pions then decay further into neutrinos, such as by $\pi^+ \rightarrow \mu^+ + \nu_\mu$, $\mu^+ \rightarrow e^+ + \nu_e + \bar{\nu}_\mu$. For the shape of the WB flux, consider only the $\nu_\mu$ from pion decays for the moment. It is often assumed that the target photon field corresponds to the observed prompt GRB flux, which is typically parameterized by $dN_{\gamma}(E)/dE \propto E^{\alpha_{\gamma}}$ for $E < \varepsilon_{\gamma,\text{break}}$ and $dN_{\gamma}(E)/dE \propto E^{\beta_{\gamma}}$ for $E > \varepsilon_{\gamma,\text{break}}$ in the observer’s frame, where $\alpha_\gamma \simeq -1$, $\beta_\gamma \simeq -2$, and the break $\varepsilon_{\gamma,\text{break}}$ at a few hundred keV. If the protons are injected with a power law with injection index two, one obtains for the prompt GRB neutrino flux, referred to as “WB flux”, $$E^2_{\nu} \frac{\mathrm{d}N_{\nu}}{\mathrm{d}E_{\nu}} \propto \left\{ \begin{array}{ll} (E_{\nu} / \varepsilon^b_{\nu})^{\alpha_{\nu}} & \text{for} \; E_{\nu} < \varepsilon^b_{\nu} \\ (E_{\nu} / \varepsilon^b_{\nu})^{\beta_{\nu}} & \text{for} \; \varepsilon^b_{\nu} \leq E_{\nu} < \varepsilon^s_{\nu} \\ (E_{\nu} / \varepsilon^b_{\nu})^{\beta_{\nu}} (E_{\nu} / \varepsilon^s_{\nu})^{-2} & \text{for} \; E_{\nu} \geq \varepsilon^s_{\nu} \end{array} \right. \label{equ:WB}$$ with $\alpha_\nu = -\beta_\gamma - 1 \simeq +1$, $\beta_\nu = - \alpha_\gamma - 1 \simeq 0$, $\varepsilon^b_{\nu} \simeq 10^5 \, \giga\electronvolt$ and $\varepsilon^s_{\nu} \simeq 10^7 \, \giga\electronvolt$. For the analytical estimates of the break energies, we follow the treatment in [Ref.]{} [@Guetta:2003wi], assuming that $\Gamma = 10^{2.5}$ and $\textit{z} = 2$; see, [[*e.g.*]{}]{}, [Refs.]{} [@Guetta:2003wi; @Wanderman:2009es]. The first break energy $\varepsilon^b_{\nu}$ can be related to $\varepsilon_{\gamma,\text{break}}$ from the threshold of the photohadronic interactions at the source. As a minor difference to [Ref.]{} [@Guetta:2003wi], where heads-on collisions between photons and protons are assumed for the threshold, we include the effect that the pion production efficiency peaks at higher center-of-mass energies (see Fig. 4 in [Ref.]{} [@Hummer:2010vx]) to match our numerical results. This leads to a factor of two higher photon energy break ($14.8 \, \mathrm{keV}$) in the source frame to match the $\varepsilon^b_{\nu} \simeq 10^5 \, \giga\electronvolt$ for the chosen parameter set. The second break comes from pion cooling in the magnetic field. It can be computed from the energy where the pion decay rate equals the synchrotron loss rate. In order to reproduce $\varepsilon^s_{\nu} \simeq 10^7 \, \giga\electronvolt$, one has $B \simeq 3 \cdot 10^5 \, \mathrm{G}$. Note that, in the light of recent Fermi data, it is not clear how “typical” this parameter set is, which, however, does not affect the logic of this letter. As another relevant parameter, we choose the maximum proton energy by balancing synchrotron loss and acceleration rates with an acceleration efficiency of 10% [@Hillas:1985is]. For the expected normalization of the flux in [[Eq.]{} (\[equ:WB\])]{}, we use [@Waxman:1998yy] (updated in [Ref
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'This is a survey on the equivariant cohomology of Lie group actions on manifolds, from the point of view of de Rham theory. Emphasis is put on the notion of equivariant formality, as well as on applications to ordinary cohomology and to fixed points.'
address: |
Philipps Universität Marburg\
Fachbereich Mathematik und Informatik\
Hans-Meerwein-Straße\
35032 Marburg\
Germany
author:
- Oliver Goertsches
- Leopold Zoller
title: |
Equivariant de Rham cohomology:\
Theory and applications
---
Introduction
============
Equivariant cohomology is a topological invariant, not of spaces, but of group actions. It encodes in a subtle way information on the topology of the space, the isotropy groups of the action, and the orbit stratification, in particular on the fixed points of the action. In was introduced by Borel [@Borel] and H. Cartan [@Cartan1], [@Cartan2] in the 1950s and has found numerous applications wherever symmetries of geometric objects play a role. These purpose of these notes is twofold: they try to give a gentle introduction to this beautiful theory from the point of view of de Rham theory, and to survey both classical and more recent applications.
In the first few sections we introduce three different types of cohomology one can associate to a Lie group action on a manifold: cohomology of invariant forms, basic cohomology, and our main player, equivariant cohomology. After comparing them to each other and to ordinary (de Rham) cohomology we prove some basic results on equivariant cohomology like the homotopy axiom and the Mayer-Vietoris sequence.
We explain how equivariant cohomology can be used to gain information on both the ordinary cohomology of the manifold $M$ acted on, as well as on the fixed point set of the action. The main tool to relate equivariant cohomology to the fixed point set is the Borel localization theorem, which is the topic of Section \[sec:borellocalization\]. We explain how one uses it to show the equalities of the Euler characteristics of $M$ and the fixed point set $M^T$, as well as the inequality of total Betti numbers $\dim H^*(M^T)\leq \dim H^*(M)$, in Section \[sec:fixedpoints\].
Starting with Section \[sec:equivariantformality\] we make use of the spectral sequence of the Cartan model, as there we introduce another main topic of this survey, the notion of equivariant formality. All necessary knowledge on spectral sequences is contained in the appendix; in particular, there one can find details on the relation between the equivariant cohomology and the final page of the spectral sequence that are usually glossed over in the literature. Equivariant formality of an action is the condition that the spectral sequence of the Cartan model degenerates at $E_1$. In Theorem \[thm:bigthmequivformal\] we prove some equivalent formulations of this condition, one of which enables one to compute ordinary from equivariant cohomology. We apply this to obtain information on the cohomology of homogeneous spaces in Section \[sec:cohomhomspaces\], and of GKM manifolds in Section \[sec:HMHTM\].
In the last sections we give a short overview on some recent developments. The choice of material is rather biased and not meant to be exhaustive. We will explain some results surrounding the notions of Cohen-Macaulay actions and equivariant basic cohomology.
Throughout the paper we try to present the material in an easily accessible way, sometimes sacrificing greater generality for simplicity of the arguments. We do not give proofs for every result, but do so whenever we were not able to find a good reference in the literature; sometimes we provide a different proof. We will assume that the reader is familiar with the theory of actions of compact Lie groups on differentiable manifolds.
In preparation of this paper a wealth of literature was helpful, such as the monographs [@AlldayPuppe; @BeGeVe; @GuilleminSternberg; @Hsiang], as well as [@GGK Appendix C] and [@Bott].\
[*Acknowledgements.*]{} Parts of this paper stem from the first named author’s lectures at the University of Hamburg in 2012, and at the Philipps University of Marburg in 2018. We would like to thank the participants of these courses for their interest in the topic and their valuable comments. We are grateful to Michèle Vergne for several remarks on a previous version of this paper. We are especially indebted to Jeffrey Carlson for several enlightening discussions, as well as for a very thorough reading of a previous version and numerous suggestions that improved the presentation of this paper. The second named author is supported by the German Academic Scholarship foundation.
Invariant and basic differential forms
======================================
Let $G$ be a Lie group acting on a differentiable manifold $M$, with Lie algebra ${\mathfrak{g}}$. We denote, for $X\in {\mathfrak{g}}$, the induced fundamental vector field by $$\overline{X}_p:= \left.\frac{d}{dt}\right|_{t=0} \exp(tX)\cdot p.$$
A differential form $\omega\in \Omega(M)$ is called *$G$-invariant* if $g^*\omega=\omega$ for all $g\in G$. The space of $G$-invariant differential forms is denoted $\Omega(M)^G$.
The space $\Omega(M)^G$ is clearly invariant under the differential $d\colon\Omega(M)\to \Omega(M)$, i.e., $(\Omega(M)^G,d)$ is a subcomplex of $(\Omega(M),d)$ and we can consider its cohomology. However, if $G$ is connected and compact, this cohomology does not contain more information than the usual de Rham cohomology because of the following theorem due to É. Cartan [@ECartan]:
\[thm:cohomofinvariantforms\] If $G$ is a compact and connected Lie group acting on a differentiable manifold $M$, then the inclusion map $\Omega(M)^G\to \Omega(M)$ induces an isomorphism $H^*(\Omega(M)^G)\to H^*(M)$ in cohomology.
The proof of this result can be found e.g. in [@Onishchik §9]. One shows that the averaging operator $\mu\colon\Omega(M)\to \Omega(M)$ given by $$\mu(\omega)(v_1,\ldots,v_n):= \int_G (g^*\omega) (v_1,\ldots,v_n);$$ is chain homotopic to the identity. Of course, if $G$ is not connected, then this inclusion does not induce an isomorphism, see Examples \[ex:cohomfinitequotient\] and \[ex:cohomRPn\] below.
A different type of topological information is encoded in the complex of $G$-basic differential forms.
\[defn:basicforms\] Given an action of a Lie group $G$ on a smooth manifold $M$, a differential form $\omega\in \Omega(M)$ is called *($G$-)horizontal* if $i_{\overline{X}}\omega=0$ for all $X\in {\mathfrak{g}}$. It is called *$G$-basic* if it is both $G$-invariant and horizontal. The space of such differential forms is denoted $\Omega_{{\operatorname{bas}}G}(M)$.
Just like the $G$-invariant differential forms, also the basic differential forms comprise a subcomplex of the de Rham complex. In fact, for $\omega\in \Omega_{{\operatorname{bas}}G}(M)$, the form $d\omega$ is again (invariant and) horizontal because by the Cartan formula $i_{\overline{X}} d\omega = {{\mathcal L}}_{\overline{X}}\omega - di_{\overline{X}}\omega = 0$. Here, ${{\mathcal L}}$ denotes the Lie derivative.
We define the *basic cohomology* of the $G$-action to be $$H^*_{{\operatorname{bas}}G}(M):=H^*(\Omega_{{\operatorname{bas}}G}(M),d).$$
Recall that if the $G$-action on $M$ is free, then the orbit space $M/G$ is a smooth manifold, and the projection $\pi\colon M\to M/G$ is smooth. In general, for an arbitrary action of a compact Lie group, $M/G$ is just a topological Hausdorff space.
\[prop:basiccohomfreeaction\] Consider a free action of a (not necessarily connected) compact Lie group $G$ on a smooth manifold $M$, and consider the projection $\pi\colon M\to M/G$. Then $\pi^*$ defines an isomorphism of complexes $\pi^*\colon \Omega(M/G)\to \Omega_{{\operatorname{bas}}G}(M)$. In particular, $$H^*_{{\operatorname{bas}}G}(M)\cong H^*(M/G).$$
If $\omega\in \Omega(M/G)$, then $\pi^*\omega$ is $G$-invariant because for any $g\in G$ we have $$g^*\pi^*\omega = (\pi\circ g)^*\omega = \pi^*\omega.$$ At each $p\in M$, we have $\ker d\pi_p = T_pG\cdot p$. Thus, $\pi^*\omega$ is horizontal as well.
If conversely $\eta$ is a $G$-basic $k$-form on $M$, then we
|
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"pile_set_name": "ArXiv"
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0.3cm
[ROME prep.1249/99\
hep-th/9904200]{}
[**THE ABELIAN PROJECTION VERSUS THE HITCHIN FIBRATION OF $K(D)$ PAIRS IN FOUR-DIMENSIONAL $QCD$**]{}\
\
INFN Sezione di Roma\
Dipartimento di Fisica, Universita’ di Roma ‘La Sapienza’\
Piazzale Aldo Moro 2 , 00185 Roma\
\
We point out that the concept of Abelian projection gives us a physical interpretation of the role that the Hitchin fibration of parabolic $K(D)$ pairs plays in the large-$N$ limit of four-dimensional $QCD$.\
This physical interpretation furnishes also a simple criterium for the confinement of electric fluxes in the large-$N$ limit of $QCD$.\
There is also an alternative, compatible interpretation, based on the $QCD$ string.
April 1999
Introduction
============
Some years ago ’t Hooft introduced the concept of Abelian projection [@H1] into non-Abelian gauge theories, in order to explain the confinement of quarks in four-dimensional $QCD$ as a dual Meissner effect in a dual superconductor [@H02; @M].\
The Abelian projection allows us, by a careful choice of the gauge, to describe the physical variables of a non-Abelian $SU(N)$ gauge theory, without scalar matter fields, as a set of electric charges and magnetic monopoles interacting via a residual $U(1)^{N-1}$ Abelian gauge coupling.\
The occurrence of magnetic monopoles into a non-Abelian gauge theory without matter fields is perhaps the most crucial feature of the Abelian projection, that furnishes a precise understanding of the structure of the phases of non-Abelian gauge theories, according to the following alternatives [@H3].\
If there is a mass gap, either the electric charge condenses in the vacuum (Higgs phase) or the magnetic charge does (confinement phase). If there is no mass gap, the electric and magnetic fluxes coexist (Coulomb phase).\
Recently, in an apparently unrelated development [@MB], some mathematical control was gained over the large-$N$ limit of four-dimensional $QCD$, mapping, by means of a chain of changes of variables, the function space of the $QCD$ functional integral into an elliptic fibration of Hitchin bundles.\
Hitchin bundles [@Hi] are themselves a fibration of $U(1)$ bundles over spectral branched covers of a Riemann surface, that, in the case of [@MB], is a torus.\
In this paper, we point out that the map in [@MB] is a version, in a perhaps global algebraic-geometric setting, of the concept of Abelian projection [@H1].\
In fact, the branching points of the spectral cover are identified with the magnetic monopoles of the Abelian projection, the parabolic points of the cover with (topological) electric charges and the $U(1)$ gauge group on the cover with a global version (on the cover) of the $U(1)^{N-1}$ gauge group of the Abelian projection.\
The identifications that we have just outlined provide a physical interpretation of the mathematical construction in [@MB]. Indeed it is precisely this physical interpretation that explains naturally why the functional integral, once it is expressed as a functional measure supported over the collective field of the Hitchin fibration, is dominated by a saddle-point condition in the large-$N$ limit.\
On the other side, we may think that the mathematical proof, that the variables of the Abelian projection really capture the physics of four-dimensional $QCD$ in the large-$N$ limit, relies on the fact that those variables may be employed to dominate the functional integral in the large-$N$ limit.\
The only qualitative feature, in the treatment in [@MB], that was not already present in the concept of the Abelian projection, is the occurrence of Riemann surfaces and it is due to the global algebraic-geometric nature of the methods in [@MB]. This, however, makes contact, at least qualitatively, with another long-standing conjecture about the $QCD$ confinement, the occurrence of string world sheets [@GT] and the string program [@Po].\
Our last concluding remark is that the electric/magnetic alternative [@H3] and the physical interpretation based on the Abelian projection, applied in the mathematical framework of [@MB], give us a simple qualitative criterium to characterize the confinement phase of $QCD$ in the large-$N$ limit: confinement is equivalent to magnetic condensation, in absence of electric (parabolic) singularities of the spectral covers.\
An alternative, compatible interpretation, based on the idea that $QCD$ is equivalent, in the large-$N$ limit, to a theory of strings [@GT; @Po] is outlined in the following section. The rest of the paper is devoted to a technical explanation of the correspondence between the Abelian projection and the Hitchin fibration in four-dimensional $QCD$.
The Hitchin fibration as the Abelian projection in the gauge in which the Higgs current is a triangular matrix
==============================================================================================================
The Abelian projection, according to [@H1], is really the choice of a gauge-fixing in such a way that, after the gauge-fixing, the theory is no longer locally invariant under $SU(N)$ but only under its Cartan subgroup $U(1)^{N-1}$. The important point about this projection is that it is defined strictly locally, that is, the gauge rotation $\Omega$ performed at each point in space-time to implement the gauge-fixing condition, does not depend on the values of the physical fields in other points of space-time. This then guarantees that all observables in the new gauge frame are still locally observable. There are no propagating ghosts. But $\Omega$ is not completely defined. There is a subgroup, $U(1)^{N-1}$, of gauge rotations that may still be performed. And this is why the theory, after the Abelian projection, looks like a local $U(1)^{N-1}$ gauge theory.\
If one now tries to gauge-fix this remaining gauge freedom, one discovers that it cannot be done locally, without encountering apparent difficulties. But local gauge-fixing is not needed, since the residual gauge symmetry is the one of a familiar Abelian theory.\
There may be, however, isolated points, where the local gauge-fixing condition has coinciding eigenvalues, where the gauge symmetry is not $U(1)^{N-1}$ but a larger group. Here singularities appear, the magnetic monopoles. So we see that, topologically, the full theory can only be topologically equivalent to the $U(1)^{N-1}$ gauge theory if the latter is augmented with monopole singularities where the $U(1)$ conservation laws for the vortices are broken down into the (less restrictive) conservation laws of the $SU(N)$ vortices.\
When we try to gauge-fix completely, we hit upon the Dirac strings, whose end points are the magnetic monopoles.\
In addition to the magnetic monopoles, in the $QCD$ case, the gauge-fixed theory contains also gluon and quark fields, that are charged with respect to the residual $U(1)^{N-1}$.\
Therefore we have a set of electric charges and magnetic monopoles interacting via a residual $U(1)^{N-1}$ Abelian gauge coupling.\
We now compare this description with the one that arises in [@MB], for the pure gauge theory without quark matter fields.\
The functional integral for $QCD$ in [@MB] is defined in terms of the variables $(A_z, A_{\bar z}, \Psi_z, \Psi_{\bar z})$, obtained by means of a partial duality transformation from $(A_z, A_{\bar z}, A_u, A_{\bar u})$, where $(z, \bar z, u, \bar u)$ are the complex coordinates on the product of two two-dimensional tori, over which the theory is defined.\
$(A_z, A_{\bar z}, \Psi_z, \Psi_{\bar z})$ define the coordinates of an elliptic fibration of $T^* {\cal A}$, the cotangent bundle of unitary connections on the $(z, \bar z)$ torus, whose base is the $(u, \bar u)$ torus.\
$\Psi_z$ transforms as a field strength by gauge transformations and it is a non-hermitian matrix.\
Following Hitchin [@Hi], the gauge is chosen in which $\Psi_z$ is a triangular matrix, for example lower triangular, that leaves a $U(1)^{N-1}$ residual gauge freedom as in the Abelian projection.\
The points in space-time where $\Psi_z$ has a pair of coinciding eigenvalues, correspond to monopoles. In addition there are the charged components of $(A_z, A_{\bar z}, \Psi_z, \Psi_{\bar z})$. We have thus a set of charges and monopoles with a residual $U(1)^{N-1}$, according to the Abelian projection.\
In [@MB], however, it is found a dense set in the functional integral over (the elliptic fibration of) $T^* {\cal A}$, with the property that the quotient by the action of the gauge group exists as a Hausdorff (separable) manifold.\
This dense set is defined in [@MB] as the set of pairs $(A, \Psi)$ that are solutions of the following differential equations (elliptically fibered over the $(u
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'Equivalence of convex optimization and variational inequality is well established in the literature such that the latter is formally recognized as a fixed point problem of the former. Such equivalence is also known to exist between a saddle-point problem and the variational inequality. The variational inequality is a static problem which can be further studied within the dynamical settings using a framework called the projected dynamical system whose stationary points coincide with the static solutions of the associated variational inequality. Variational inequalities have rich properties concerning the monotonicity of its vector-valued map and the uniqueness of its solution, which can be extended to the convex optimization and saddle-point problems. Moreover, these properties also extend to the representative projected dynamical system. The objective of this paper is to harness rich monotonicity properties of the representative projected dynamical system to develop the solution concepts of the convex optimization problem and the associated saddle-point problem. To this end, this paper studies a linear inequality constrained convex optimization problem and models its equivalent saddle-point problem as a variational inequality. Further, the variational inequality is studied as a projected dynamical system[@friesz1994day] which is shown to converge to the saddle-point solution. By considering the monotonicity of the gradient of Lagrangian function as a key factor, this paper establishes exponential convergence and stability results concerning the saddle-points. Our results show that the gradient of the Lagrangian function is just monotone on the Euclidean space, leading to only Lyapunov stability of stationary points of the projected dynamical system. To remedy the situation, the underlying projected dynamical system is formulated on a Riemannian manifold whose Riemannian metric is chosen such that the gradient of the Lagrangian function becomes strongly monotone. Using a suitable Lyapunov function, the stationary points of the projected dynamical system are proved to be globally exponentially stable and convergent to the unique saddle-point.'
author:
- |
P. A. Bansode[^1]\
V. Chinde[^2]\
S. R. Wagh[^3]\
R. Pasumarthy[^4]\
N. M. Singh[^5]
bibliography:
- 'dsvm\_pd.bib'
title: 'On the Exponential Stability of Projected Primal-Dual Dynamics on a Riemannian Manifold'
---
Introduction
============
The convex optimization methods have remained as the subject of substantial research for many decades. The primal-dual gradient-based method is one of such methods which dates back to late $1950s$[@arrow1958studies]. Lately, these methods (also referred to as primal-dual dynamics as its dynamical system equivalent) have found many applications in the networked systems (viz, the power networks [@zhao2014design; @mallada2017optimal; @yi2015distributed] and the wireless networks [@feijer2010stability; @chen2012convergence; @ferragut2014network]), and building automation systems [@kosaraju2018stability]. The primal-dual dynamics (PD dynamics) seek a solution to the saddle-point problem representing the original constrained convex optimization problem by taking gradient descent along the primal variable and gradient ascent along the dual variable. From the perspectives of systems and control theory, the PD dynamics have much to offer in terms of stability and convergence with respect to the saddle point solution.
During recent years the notions of stability of PD dynamics have evolved. The asymptotic stability of PD dynamics has been established as one of the most fundamental notions. Feijer [*et al.*]{}[@feijer2010stability] explores the PD dynamics with applications to network optimization problems and prove its asymptotic stability. The dual dynamics pertaining to the inequality constraints have been shown to include switching projections that restrict the dual variables to the set of nonnegative real numbers. Due to the switching projections, the PD dynamics becomes discontinuous, which is further modeled as a hybrid dynamical system. A Krasovskii-type Lyapunov function along with LaSalle invariance principle of hybrid systems [@lygeros2003dynamical] have been utilized to prove the asymptotic stability of the PD dynamics. In [@cherukuri2016asymptotic] it is proved that the PD dynamics is a special case of the projected dynamical systems. It uses the invariance principle of Carathéodory solutions to show that the saddle-point solution of PD dynamics is unique and globally asymptotically stable. Although widely established, the notion of asymptotic stability does not offer explicit convergence bounds of the PD dynamics which is an essential factor in case of the on-line optimization techniques. One must ensure that the trajectories converge to the saddle point solution in finite time. To explicitly obtain stronger convergence rates, research interests have shifted towards the notions of global exponential convergence and stability.
The pathway leading to the exponential stability of the PD dynamics is not as straightforward as it is for its asymptotic stability. The existence of right-hand side discontinuities and non-strongly monotone gradient of the associated Lagrangian function seem to prevent the saddle-point solution from being exponentially stable. The globally exponential stability has been the most desirable yet often formidable aspect of PD dynamics, which guarantees a minimum rate of convergence to the saddle point. While exhaustive literature on asymptotic stability of the PD dynamic can be encountered, its exponential stability has not been explored except for the recent studies [@cortes2018distributed; @nguyen2018contraction; @qu2019exponential; @dhingra2018proximal]. The optimization problem considered in [@cortes2018distributed] proves the exponential stability of the PD dynamic for an equality constrained optimization problem. Robustness and contraction analysis of the primal-dual dynamics establishing exponential convergence to the saddle-point solution is presented in [@nguyen2018contraction]. In [@qu2019exponential], the PD dynamics is proved to be globally exponentially stable for linear equality and inequality constrained convex optimization problem. Under assumptions on strong convexity and smoothness of the objective function and full row rank conditions of the constraint matrices, the PD dynamics is shown to have global exponential convergence to the saddle point solution. It mainly proposes the augmented Lagrangian function that results in a PD dynamics which does not have right-hand side discontinuities. By employing a quadratic Lyapunov function that has non-zero off-diagonal block matrices, it shows that the PD dynamics is globally exponentially stable. In [@dhingra2018proximal] a composite optimization problem is considered in which the objective function is represented as a sum of differentiable non-convex component and convex non-differentiable regularization component. A continuously differentiable proximal augmented Lagrangian is obtained by using a Moreau envelope of the regularization component. This results in a continuous-time PD dynamics which under the assumption of strong convexity of the objective function, is shown to be exponentially stable by employing a framework of the integral quadratic constraints (IQCs) [@587335]. By using a well-known result pertaining to linear systems with nonlinearities in feedback connection that satisfy IQCs [@hu2016exponential], it proves the global exponential stability of the PD dynamics.
Motivation and contribution
---------------------------
For a sufficiently small step size, a Euler discretized globally exponentially stable PD dynamics leads to geometric convergence to the saddle-point solution[@stuart1994numerical]. This property has been widely appreciated in recent articles such as [@dhingra2018proximal; @qu2019exponential]. The existing methods have considered the augmented Lagrangian techniques at a pivotal position for proving the globally exponential stability of the PD dynamics. This paper does not rely on augmented Lagrangian techniques to arrive at exponentially stable saddle-point solution. It presents a complementary approach that uses a combined framework of variational inequalities, projected dynamical systems [@nagurney2012projected], and the theory of Riemannian manifolds[@udriste1994convex; @da2002contributions] to derive conditions that lead to the global exponential stability of the saddle-point solution.
This paper exploits an equivalence between a constrained optimization problem and a variational inequality problem as discussed in [@kinderlehrer1980introduction; @facchinei2007finite; @nagurney2012projected]. They are shown to be equivalent when the vector-valued map of the variational inequality is the gradient of the objective function of the underlying optimization problem. Besides that, when the objective function is convex the Karush-Kuhn-Tucker (KKT) conditions of both problems reveal that the Lagrangian function associated with the variational inequality and the Lagrangian of the optimization problem have the exactly same saddle-point[@facchinei2007finite]. This further hints at formulating the saddle-point problem (of the corresponding optimization problem) as a variational inequality when both primal, as well as dual variables, are of interest.
Variational inequalities are equivalent to fixed point problems [@nagurney2012projected] in the sense that they yield only the static solutions. Thus the variational inequality formulation of the saddle-point problem would only result in the static description of the saddle-point. To understand the dynamic behavior of such variational inequality, this paper brings in the framework of projected dynamical systems[@friesz1994day; @nagurney2012projected]. The projected dynamical system combines essential features of both variational inequalities and dynamical systems such that its solution coincides with the static equilibrium of the variational inequality problem. These dynamical systems have interesting features which they derive from the underlying variational inequality problem. In [@gao2003exponential] a globally projected dynamical system [@friesz1994day] is proved to be exponential stable when the vector-valued mapping concerning the variational inequality problem is strongly monotone. This motivates to represent the saddle-point problem as a variational inequality and use the framework of the projected dynamical system for proposing a new dynamical system that is equivalent to the PD dynamics. Aiming at exponential stability of the saddle-point solution, this paper indirectly poses the saddle-point problem as a projected dynamical system (regarded hereafter as the projected primal-dual dynamics). While deriving the stability results of the proposed dynamics, our analysis reveals that the gradient of the Lagrangian function is
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We show that within the inverse seesaw mechanism for generating neutrino masses minimal supergravity is more likely to have a sneutrino as the lightest superparticle than the conventional neutralino. We also demonstrate that such schemes naturally reconcile the small neutrino masses with the correct relic sneutrino dark matter abundance and accessible direct detection rates in nuclear recoil experiments.'
author:
- 'C. Arina'
- 'F. Bazzocchi'
- 'N. Fornengo'
- 'J. C. Romao'
- 'J. W. F. Valle'
title: Minimal supergravity sneutrino dark matter and inverse seesaw neutrino masses
---
Introduction
============
Over the last fifteen years we have had solid experimental evidence for neutrino masses and oscillations [@Maltoni:2004ei], providing the first evidence for physics beyond the Standard Model. On the other hand, cosmological studies clearly show that a large fraction of the mass of the Universe in dark and must be non–baryonic.
The generation of neutrino masses may provide new insight on the nature of the dark matter [@Berezinsky:1993fm]. In this Letter we show that in a minimal supergravity (mSUGRA) scheme where the smallness of neutrino masses is accounted for within the inverse seesaw mechanism the lightest supersymmetric particle is likely to be represented by the corresponding neutrino superpartner (sneutrino), instead of the lightest neutralino. This opens a new window for the mSUGRA scenario. Here we consider the implications of the model for the dark matter issue. We demonstrate that such a model naturally reconciles the small neutrino masses with the correct relic abundance of sneutrino dark matter and experimentally accessible direct detection rates.
Minimal SUGRA inverse seesaw model {#Model}
==================================
Let us add to the Minimal Supersymmetric Standard Model (MSSM) three sequential pairs of 1 singlet neutrino superfields ${\widehat
\nu^c}_i$ and $\widehat{S}_i$ ($i$ is the generation index), with the following superpotential terms [@mohapatra:1986bd; @Deppisch:2004fa], $${\cal W} = {\cal W_{\rm MSSM}} +\varepsilon_{ab}\,
h_{\nu}^{ij}\widehat L_i^a\widehat \nu^c_j\widehat H_u^b
+ M_{R}^{ij}\widehat \nu^c_i\widehat S_j
+\frac{1}{2}\mu_S^{ij} \widehat S_i \widehat S_j
\label{eq:Wsuppot}$$ where ${\cal W_{\rm MSSM}}$ is the usual MSSM superpotential. In the limit $\mu_S^{ij} \to 0$ there are exactly conserved lepton numbers assigned as $(1,-1,1)$ [@mohapatra:1986bd; @Deppisch:2004fa] for $\nu$, $\nu^{c}$ and $S$, respectively.
The extra singlet superfields induce new terms in the soft–breaking Lagrangian: $$\begin{aligned}
\mathcal{-L}_{\rm soft} &=& \mathcal{-L}_{\rm soft}^{\rm MSSM} +
\tilde{\nu}^c_i\ \mathbf{M^2_{\nu^c}}_{ij} \tilde{\nu}^c_j + \tilde{S}_i\,
\mathbf{M^2_{S}}_{ij} \tilde{S}_j \\
& & + \varepsilon_{ab}\,
A_{h_{\nu}}^{ij} \tilde{L}_i^a \tilde{\nu}^c_j H_u^b
+ B_{M_{R}}^{ij} \tilde{\nu}^c_i \tilde{S}_j
+\frac{1}{2} B_{\hat{\mu_S}}^{ij} \tilde{S}_i \tilde{S}_j \nonumber
\label{eq:soft}\end{aligned}$$ where $\mathcal{L}_{\rm soft}^{\rm MSSM}$ is the MSSM SUSY–breaking Lagrangian.
Small neutrino masses are generated through the inverse seesaw mechanism [@mohapatra:1986bd; @Deppisch:2004fa; @Nunokawa:2007qh]: the effective neutrino mass matrix $m^{\rm eff}_{\nu}$ is obtained by the following relation: $$\label{eq:1}
m^{\rm eff}_{\nu}= -v_u^2 h_{\nu} \left(M_R^T\right)^{-1} {\mu_S}
M_R^{-1} h_{\nu}^T = \left(U^T\right)^{-1} m_{\mu}^{\rm diag}\ U^{-1}$$ where $h_\nu$ defines the Yukawa matrix and $v_u$ is the $H_u$ vacuum expectation value. The smallness of the neutrino mass is ascribed to the smallness of the $\mu_S$ parameter, rather than the largeness of the Majorana–type mass matrix $M_R$, as required in the standard seesaw mechanism [@Nunokawa:2007qh]. In this way light (eV scale or smaller) neutrino masses allow for a sizeable magnitude for the Dirac–type mass $m_D=v_u h_\nu$ and a TeV–scale mass for the right-handed neutrinos, features which have been shown to produce an interesting sneutrino dark matter phenomenology [@Arina:2007tm].
The main feature of our model is that the nature of the dark matter candidate, its mass and couplings all arise from the same sector responsible for the generation of neutrino masses. In order to illustrate the mechanism we consider the simplest one-generation case, for simplicity. In this case where the sneutrino mass matrix reads $$\begin{aligned}
\mathcal{M}^2 =
\begin{pmatrix}
\mathcal{M}^2_+ & \mathbf{0}\cr
\mathbf{0} & \mathcal{M}^2_-\cr
\end{pmatrix}
$$ where the two sub–matrices $\mathcal{M_\pm}^2$ are:
$$\begin{aligned}
\label{eq:snumatrix}
\mathcal{M_{\pm}}^2 =
\begin{pmatrix}
m^2_L+\frac{1}{2} m^2_Z \cos 2\beta+m^2_D & \pm (A_{h_{\nu}}v_u-\mu m_D {\rm cotg} \beta) & m_D M_R\cr
\pm (A_{h_{\nu}}v_u-\mu m_D {\rm cotg}\beta) & m^2_{\nu^c}+M_R^2+m^2_D & \mu_S M_R \pm B_{M_R}\cr
m_D M_R & \mu_S M_R \pm B_{M_R} & m^2_S+\mu^2_S+M^2_R\pm B_{\mu_S}
\end{pmatrix}
$$
in the CP eigenstates basis: $\Phi^{\dag} = (\snu_{+}^\ast
\,\tilde{\nu}_{+}^{c\ast} \, \tilde{S}_+^\ast \,\, \snu_-^\ast \,
\tilde{\nu}_-^{c\ast} \, \tilde{S}_-^\ast)$. Once diagonalized, the lightest of the six mass eigenstates is our dark matter candidate and it is stable by $R$–parity conservation.
A novel supersymmetric spectrum
===============================
![Supersymmetric particle spectrum in the standard MSUGRA scheme \[panel (a)\] and in the inverse seesaw mSUGRA model \[panel (b)\] with parameters chosen as: $m_0= 358$ GeV, $m_{1/2}= 692$ GeV, $A_0
= 0$, $\tan\beta=35$ and sign $\mu >0$. The sneutrino sector has the additional parameter $B_{\mu_S}$, fixed at 10 GeV$^2$. The squark sector is not shown. []{data-label="fig:spectrum"}](tower2_new.eps){width="\columnwidth"}
![The $m_{0}-m_{1/2}$ plane for $\tan\beta=35$, $A_0=0$ and $\mu>0$. The red and yellow areas denote the set of supersymmetric parameters where the sneutrino is the LSP in inverse seesaw models (notice that it includes all the yellow region where the $\tilde{\tau}$ is the LSP in the standard mSUGRA case). The white region has the neutralino as LSP in both standard and modified mSUGRA. For the sneutrino LSP region, the additional parameters are: $B_{\mu_S}= 10 {\, {\rm GeV}}^2$, $M_R=500 {\, {\rm GeV}}$, $m_D=50 {\, {\rm GeV}}$ and $\mu_S=1$ eV. The blue region is excluded (see text). []{data-label="fig:m0mh12"}](mhalfm0_new.eps){width="\columnwidth"}
Let us now consider the model within a minimal SUGRA scenario. In the absence of the singlet neutrino superfields, the mSU
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
author:
- Tianming Liu
- Haoyu Wang
- Li Li
- Xiapu Luo
- Feng Dong
- Yao Guo
- Liu Wang
- 'Tegawend[é]{} F. Bissyand[é]{}'
- Jacques Klein
bibliography:
- 'cite.bib'
title: 'MadDroid: Characterizing and Detecting Devious Ad Contents for Android Apps'
---
<ccs2012> <concept> <concept\_id>10002978.10003022</concept\_id> <concept\_desc>Security and privacy Software and application security</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003260.10003272</concept\_id> <concept\_desc>Information systems Online advertising</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003120.10003138</concept\_id> <concept\_desc>Human-centered computing Ubiquitous and mobile computing</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Conclusion
==========
In this paper, we perform a large-scale characterization study of mobile ad content, which has been largely overlooked by the research community. We first create a comprehensive categorization of devious mobile ad contents, then we build [*MadDroid*]{}, a framework for automated detection of devious mobile ad contents. By applying [*MadDroid*]{} to 40,000 Android apps, we find that devious ad contents are prevalent: 6% of apps in our study are identified as delivering devious ad contents. To the best of our knowledge, [*MadDroid*]{} is the first attempt towards mitigating threats from both ad-load and ad-click introduced by mobile ad contents.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was partly supported by the National Natural Science Foundation of China (No.61702045 and No.61772042), by the Hong Kong RGC Projects (No.152223/17E, CityU C1008-16G), by the Australian Research Council (ARC) under projects DE200100016 and DP200100020, by the Fonds National de la Recherche (FNR), Luxembourg, under project CHARACTERIZE C17/IS/11693861, by the SPARTA project which has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 830892.
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'A short summary of constraints on the parameter space of supersymmetric models is given. Experimental limits from high energy colliders, electroweak precision data, flavor and Higgs physics, and cosmology are considered. The main focus is on the MSSM with conserved R- and CP-parity and minimal flavor violation, but more general scenarios and extended models will also be discussed briefly.'
author:
- 'A. Freitas'
title: |
\
Status of Constraints on Supersymmetry
---
INTRODUCTION
============
The purpose of this contribution is to summarize the constraints on supersymmetric models from various experimental results. Due to the large wealth of experimental searches for physics beyond the standard model (SM) and phenomenological studies on supersymmetry (SUSY) it is impossible to cover all of them in this short review. Thus the author apologizes that many valuable studies are not mentioned or cited in this report.
To set the scene, a short review of the most widely studied SUSY models is given in the next section. The following sections discuss constraints on the parameter space of these models from high energy colliders, electroweak precision data, flavor and Higgs physics, and cosmology, respectively. Finally, some qualitative comments on more general SUSY models are presented before the summary.
SUSY MODELS
===========
The most extensively studied SUSY model is the Minimal Supersymmetric Standard Model (MSSM), with the particle content listed in Table \[mssm\]. In addition the MSSM imposes R-parity, assigning $R_p = +1$ for the Higgs boson, gauge bosons, leptons, and quarks, and $R_p = -1$ for their supersymmetric partners (neutralinos, charginos, gluino, sleptons, and squarks). As a result the superpotential has the form $$W_{\rm MSSM} = y_{\rm u} \hat{Q} \cdot \hat{H}_2 \, \hat{U}^c
+ y_{\rm d} \hat{Q} \cdot \hat{H}_1 \, \hat{D}^c
+ y_{\rm e} \hat{L} \cdot \hat{H}_1 \, \hat{E}^c
- \mu \hat{H}_1 \cdot \hat{H}_2 \,.
\label{wmssm}$$ For a general introduction to the MSSM and notational definitions, see [*e.$\,$g.*]{} Ref. [@martin].
In the major part of this work, an even more minimal version of the MSSM is assumed where the CKM matrix is the only source of CP violation and flavor violation. In other words, the SUSY breaking parameters are assumed to be real and flavor blind.
**Spin 0** **Spin 1/2** **Spin 1**
----------------------------------------------------------------- ------------------------------------------------------- -----------------
Neutral Higgses Neutralinos Photon $\gamma$
$h_0,\,H_0,\,A_0$ ${\tilde{\chi}^0}_1 \dots {\tilde{\chi}^0}_4$ $Z$ boson
\[.5ex\] Charged Higgs $H^\pm$ Chargino ${\tilde{\chi}}^\pm_1, {\tilde{\chi}}^\pm_2$ $W^\pm$ bosons
\[.5ex\] Gluino $\tilde{g}$ gluon $g$
\[.5ex\] sleptons $\tilde{e}$, $\tilde{\mu}$, $\tilde{\nu}$,... leptons $e$, $\mu$, $\nu$, ...
squarks $\tilde{u}$, $\tilde{d}$, ... quarks $u$, $d$, ...
: Particle content of the MSSM
\[mssm\]
This still leaves more than one dozen [*a priori*]{} unknown SUSY breaking parameters. Many experimental searches and phenomenological analyses thus consider specific SUSY breaking scenarios:
- *mSUGRA/CMSSM:* In *minimal Supergravity* (mSUGRA) or the *constrained MSSM* (CMSSM) the scale of SUSY breaking is situated near the scale of gauge coupling unification, $M_{\rm GUT} \approx 2\times 10^{16}{{\rm \ GeV}}$. At this scale, there is one common mass parameter each for the gauginos, scalars and triple-scalar couplings ($A$-terms), respectively. At lower energies, a more complex SUSY mass spectrum emerges due to renormalization group running. As a result, the colored SUSY partners (squarks and gluino) are substantially heavier than the weakly coupled SUSY particles. The lightest SUSY particle (LSP) is typically the lightest neutralino ${\tilde{\chi}^0}_1$, with ${m_{\tilde{\chi}^0_{1}}} \sim {\cal O}(100 {{\rm \ GeV}})$.
- *GMSB:* In *gauge mediated SUSY breaking* (GMSB) the breaking of supersymmetry is transmitted by gauge interactions. The minimal version, which introduces messengers in the fundamental representation of SU(5), produces ${\cal O}(100 {{\rm \ GeV}})$ SUSY masses for a messenger scale $\Lambda_{\rm mess} \sim 100 {{\rm \ TeV}}$. Similar to mSUGRA, the gauge couplings and gaugino masses unify at $M_{\rm GUT}$, but the sfermion masses do not unify at any scale. The triple-scalar couplings ($A$-terms) are almost zero at the messenger scale $\Lambda_{\rm mess} \sim 100 {{\rm \ TeV}}$ and remain relatively small at the electroweak scale. In GSMB, the LSP is typically the gravitino, with $m_{\tilde{G}} \sim 100 {\rm \ eV} \dots 1 {{\rm \ GeV}}$.
- *AMSB:* In general, soft supersymmetry breaking terms receive contributions from the super-Weyl anomaly via loop effects. *Anomaly mediated supersymmetry breaking* (AMSB) becomes relevant only if other SUSY breaking mechanisms are suppressed or absent. AMSB predicts the gaugino mass ratios $|M_1| : |M_2| : |M_3| \approx 2.8 : 1 : 7.1$, so that the LSP is typically the lightest neutralino ${\tilde{\chi}^0}_1$ with a dominant wino component. The chargino ${\tilde{\chi}}^\pm_1$ is a almost pure wino and very close in mass to the LSP.
A shortcoming of the MSSM is the appearance of the $\mu$-term (the last term in eq. (\[wmssm\])) which must be of the order of the electroweak scale for successful electroweak symmetry breaking, leading to the unnatural hierarchy $\mu \ll M_{\rm GUT}$. One solution to this puzzle is the introduction of an additional singlet chiral superfield so that the general superpotential becomes $$W_{\rm MSSM+S} = \lambda \hat{S} \hat{H}_1 \cdot \hat{H}_2 + \kappa \hat{S}^3 +
m_{\rm S} \hat{S}^2 +
t_{\rm S} \hat{S} +
\mbox{Yukawa terms}.
\label{smssm}$$ In this general form the superpotential again has several dimensionful parameters which have to be much smaller than the GUT scale. However, the unwanted terms can be set to zero by introducing new symmetries, for example
- *Next-to-minimal MSSM (NMSSM):* A global $\mathbb{Z}_3$ symmetry mandates $m_{\rm S}=t_{\rm S}=0$, but could lead to cosmological domain walls [@domain].
- *Nearly minimal MSSM (nMSSM):* Imposing a global $\mathbb{Z}_5$ or $\mathbb{Z}_7$ symmetry forbids all singlet self-couplings at tree-level, $m_{\rm S}=t_{\rm S}=\kappa=0$. However, supergravity effects combined with SUSY breaking allow a contribution to $t_{\rm S}$ at the six- or seven-loop level, naturally generating a value $t_{\rm S} \sim {\cal O}$(TeV) as required for successful electroweak symmetry breaking [@nMSSM].
- *U(1)-extended MSSM (UMSSM):* This model introduces a U(1) gauge symmetry under which the Higgs and singlet field are charged. As a result, $m_{\rm S}=t_{\rm S}=\kappa=0$, but new D-term contributions to the Higgs potential appear which play an important role in achieving realistic electroweak symmetry breaking [@UMSSM].
HIGH ENERGY COLLIDERS
=====================
Searches for SUSY particles at $e^+e^-$ colliders are largely independent on the details of the model or scenario. Roughly speaking, results from LEP exclude sparticles up to the beam energy $E_{\rm beam} \sim 100 {{\rm \ GeV}}$. The actual exclusion bounds [@lepsusy; @pdg] are listed in Table \[lepsusy\]. The exact limits vary as a result of the different pair-production cross sections for different particles types. Furthermore, some of the searches fail if the mass difference between the pair-produced sparticle $\tilde{X}$ and the LSP becomes too small, $m_{\tilde{X}} - m_{\rm
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'The bosonic $su(n)$ Hubbard model was recently introduced. The model was shown to be integrable in one dimension by exhibiting the infinite set of conserved quantities. I derive the $R$-matrix and use it to show that the conserved charges commute among themselves. This new matrix is a non-additive solution of the Yang-Baxter equation. Some properties of this matrix are derived.'
author:
- |
[**Z. Maassarani**]{}[^1]\
\
[Département de Physique, Pav. A-Vachon]{}\
[Université Laval, Ste Foy, Qc, G1K 7P4 Canada]{}[^2]\
title: 'Exact integrability of the $su(n)$ Hubbard model'
---
23.50cm -1.7cm 0.6cm
\#1[\#1 ]{} \#1[[1 \#1]{}]{}
=msbm10 =msbm7 =msbm5 = == \#1[[\#1]{}]{}
PACS numbers: 75.10.-b, 75.10.Jm, 75.10.Lp\
Key words: Hubbard model, $su(n)$ spin-chain, integrability
October $6^{\rm th}$ 1997\
LAVAL-PHY-25/97\
Introduction
============
The two-dimensional Hubbard model was introduced to describe the effects of correlation for $d$-electrons in transition metals [@guhu]. It was then shown to be relevant to the study of high-$T_c$ superconductivity of cuprate compounds.
In one dimension the model is integrable [@liwu; @sh12; @woa]. The integrability framework of the model is the quantum inverse scattering method [@qism]. However, despite sharing many properties with discrete quantum integrable models, the model has a peculiar integrable structure which defines a class of its own.
In seeking to generalize the Hubbard model in any dimension, it was therefore natural to look for a one-dimensional generalization which is integrable. An $n$-state generalized model which contains the usual $su(2)$ model was recently introduced in [@hn]. This $su(n)$ Hubbard model was shown to possess an infinite set of conserved charges and to have an extended $su(n)$ symmetry. The model is built by coupling two copies of the recently discovered $su(n)$ XX ‘free-fermions’ model [@mm]. For $n=2$ a fermionic formulation exists, but for $n > 2$ finding an analogous framework is a tantalizing problem.
In this work I derive the $R$-matrix of the model; this provides a direct proof of the commutation of the conserved charges among themselves. Section two gives the definition of the bosonic Hamiltonian and the transfer matrix. The $R$-matrix intertwining the monodromy matrices is derived in section three. In section four some properties of this new matrix are given. I conclude with some remarks and outline some outstanding issues.
The model
=========
Let $E^{\af\be}$ be the $n\times n$ matrix with a one at row $\af$ and column $\be$ and zeros otherwise. The $su(n)$ Hubbard Hamiltonian on a ring then reads [@hn]: H\_2 &=&\_i h\_[ii+1]{} +\_i h\^[’]{}\_[ii+1]{} + U\_i h\^c\_i\[h2\]\
&=& \_i \_[< n]{} (x E\_i\^[n]{} E\_[i+1]{}\^[n]{} + x\^[-1]{} E\_i\^[n]{} E\_[i+1]{}\^[n]{} + (EE\^[’]{})) + U \_i (\_i +) (\^[’]{}\_i +)where $\rho = \sum_{\af < n} E^{\af\af} -(n-1) E^{nn}$, and primed and unprimed quantities correspond to two commuting copies of the $E$ matrices. The Hamiltonians $h$ and $h^{'}$ are $su(n)$ XX Hamiltonians [@mm]. The complex free parameter $x$ is a deformation inherited from the XX model. The Hamiltonian $H_2$ is defined in one dimension but can be evidently defined on any lattice; integrability is lost however.
For $n=2$ and $x=1$, and using Pauli matrices, the Hamiltonian is just the integrable bosonic version of the usual Hubbard Hamiltonian [@sh12]: H\_2\^[(2)]{}= \_i (\^x\_i \^x\_[i+1]{} + \^y\_i \^y\_[i+1]{}) + (\^[’]{} ) + U\_i \^z\_i\^[’z]{}\_i The Hamiltonians can be written simply in terms of $su(n)$ hermitian traceless matrices. For $|x|=1$ the Hamiltonians are hermitian.
The transfer matrix is the generator of the infinite set of conserved quantities. Its construction was given in [@hn]. We recall it here. Consider first the $R$-matrix of the $su(n)$ XX model [@mm]: R() &=& a() \[E\^[nn]{}E\^[nn]{}+\_[[, <n]{}]{} E\^E\^\]\
& & + b()\_[[<n]{}]{}(x E\^[nn]{}E\^ + x\^[-1]{} E\^E\^[nn]{})\
& & + c() \_[[<n]{}]{}(E\^[n ]{}E\^[n]{} + E\^[n]{}E\^[n]{}) where $a(\la)=\cos(\la)$, $b=\sin(\la)$ and $c(\la)=1$. The functions $a$, $b$ and $c$ satisfy the ‘free-fermion’ condition: $a^2 +b^2 = c^2$. For this set of parameters, a Jordan-Wigner transformation turns the $U=0$ Hamiltonian density for $su(2)$ into a fermionic expression for free fermions hopping on the lattice.
Consider also the matrix I\_0 (h) =() + () C\_0 C\^[’]{}\_0 =( C\_0 C\^[’]{}\_0) where $C=\sum_{\af < n} E^{\af\af}-E^{nn}$. We stress that $C$ turns out to be the fundamental matrix, not the $su(n)$ generator $\rho$. We have $\rho +\frac{n-2}{2}{\rm Id} = \frac{n}{2} C$, for $n \geq 2$. The parameter $h$ is related to the spectral parameter $\lambda$ by (2h) = (2) \[rela\] One chooses for $h(\la)$ the principal branch which vanishes for vanishing $\la$ or $U$. Then for $U=0$ the monodromy matrix becomes a tensor product of two uncoupled XX models. The Lax operator at site $i$ is given by: L\_[0i]{} () = I\_0(h) R\_[0i]{}() R\^[’]{}\_[0i]{}() I\_0(h) and the monodromy matrix is a product of Lax operators, $T(\lambda)= L_{0M}(\la)...L_{01}(\la)$, where $M$ is the number of sites on the chain. The transfer matrix is the trace of the monodromy matrix over the auxiliary space 0: $\tau (\la)= {\rm Tr}_0 \;\left[\left( L_{0M}...L_{01}\right)(\la)\right]$. One possible set of conserved quantities is given by H\_[p+1]{} = ([d\^p ()d\^p]{})\_[=0]{} The proof that $H_2$ commutes with $\tau(\lambda)$ was given in [@hn]. The derivative of the matrix $I$ gives the coupling term appearing in (\[h2\]). Note that the definition involving a logarithm has two benefits. Besides giving the most local operators, it further disentangles the two copies.
The $R$-matrix
==============
We derive the $R$-matrix intertwining two monodromy matrices at different spectral parameters. To this end we generalize the algebraic method of the Decorated Star Triangle Equation introduced by Shastry [@sh3].
The XX $R$-matrix satisfies the regularity property $\check R (0) = {\rm Id}$, the unitarity condition $\check R (\lambda) \check R (-\lambda) = {\rm Id} \;\cos^2\lambda$ and the Yang-Baxter equation \_[12]{}(-) R\_[13]{}()R\_[23]{}() = R\_[13]{}() R\_[23]{}()\_[12]{}(-)\[ybec\] where $R=P \check{R}$ and $P$ is the permutation operator on the tensor product of two $n$-dimensional spaces. It is easy to verify that it also satisfies a decorated Yang-Baxter equation \_[12]{}(+) C\_1 R\_[13]{}() R\_[23]{}() = R\_[13]{}() R\_[23]{}()C\_2\_[12]{}(+)\[dybec\]
We now look for the $R$-matrix intertwining two $L$-matrices: (\_1,\_2) L(\_1)L(\_2) = L(\_2)L(\_1) (\_1,\_2)\[rll\] The $su(2)$ case lead us to consider the following Ansatz [@sh3]: (\_1,\_2) &=& I\_[12]{}(h\_2) I\_[34]{}(h\_1) ( \_[13
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We calculate the time independent four-point function in high temperature ($T$) QCD and obtain the leading momentum dependent terms. Furthermore, we relate these derivative interactions to derivative terms in a recently proposed finite $T$ effective action based on the SU(3) Wilson Line and its trace, the Polyakov Loop. By this procedure we thus obtain a perturbative matching at finite $T$ between QCD and the effective model. In particular, we calculate the leading perturbative QCD-correction to the kinetic term for the Polyakov Loop.'
address: |
Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA.\
email: wirstam@bnl.gov
author:
- 'J. Wirstam'
title: |
[BNL-NT-01/11]{}
One-Loop QCD Corrections to the Thermal Wilson Line Model
---
Introduction {#I}
============
At high temperatures QCD is expected to be found in a new phase, the quark-gluon plasma. While the thermal excitations are hadrons and glueballs at low $T$, the degrees of freedom in the plasma phase are the quarks and gluons. This new state of matter is believed to have existed during the first microseconds after the Big Bang, and much of the recent interest stems from the fact such conditions may be produced in heavy-ion collisions. Already the results from CERN-SPS seem to hint in that direction, and the experiments at higher energies at BNL-RHIC have provided a wealth of new interesting results after its first year of running [@rhicwebpage]. To understand and interpret the experimental signatures in terms of the evolution of the initial stage after a heavy-ion collision is clearly a very challenging theoretical task.
The most convincing theoretical results that a drastic change in the degrees of freedom takes place at a certain $T$ come from lattice studies. In the pure glue theory, there is a phase transition between the confined and deconfined phases at a critical temperature $T_c \simeq 270$ MeV [@latticesu3]. When massless quarks are added, there is similarly a phase transition to a chirally symmetric phase at $T_c \simeq 155-175$ MeV [@chiral], where the precise value depends on the number of flavors. Recent lattice simulations also suggest that the chiral transition is simultaneous with the deconfining one [@digal]. At physical quark masses the situation is not completely clear, in the sense that there may not be a true phase transition but only a rapid cross-over [@transcross]. Nevertheless, lattice simulations have shown that the pressure, divided by the ideal gas result and plotted against $T/T_c$, is almost independent of the number of flavors [@indepofnf].
Due to asymptotic freedom, the quark-gluon plasma behaves as an ideal gas at asymptotically high temperatures. Up to corrections of the order of 20 percent, this behavior holds down to temperatures $T\simeq 3T_c$, even though each higher order term in a straightforward perturbative expansion gives widely different contributions in this temperature regime [@pertpressure; @nieto]. Instead, at $T\leq 5T_c$ one needs a resummed effective theory in terms of quasi particles, the HTL effective action [@htl]. Such an effective description correctly reproduces thermodynamic quantities like the pressure, as measured by the lattice, down to approximately $2T_c$ [@blaizotetal; @peshier].
Despite this progress, it is of course highly desirable to actually have an analytical description at $T\simeq T_c$, close to the critical temperature. Since the QCD coupling constant $g
\simeq 2.5$ at $T_c$ (using a renormalization scale $\mu =2\pi T$), one is presumably forced to consider effective models that go beyond the fundamental QCD Lagrangian. In a recent paper [@robmodel], such an effective theory was constructed in terms of the thermal Wilson Line ${\bf L}$, $$\begin{aligned}
{\bf L} = {\cal P} \exp \left [ ig\int_0^{\beta} d\tau A_0 (\vec{x}, \tau ) \right ] , \label{wilsonline}\end{aligned}$$ where ${\cal P}$ denotes path ordering, $\beta$ is the inverse temperature and $A_0 = A_0^a T^a$ the time component of the gluon field, with $T^a$ the generators of the fundamental SU(3) representation, $a=1,\ldots ,8$. The trace of the Wilson Line is proportional to the Polyakov Loop $l$, $l=(1/3){\rm Tr}\,{\bf L}$. In the pure Yang-Mills theory, the Polyakov Loop is an order parameter for a global Z(3) symmetry separating the confined and deconfined phases, with $\langle l\rangle \neq 0$ ($\langle l\rangle =0$) above (below) the phase transition [@refsforl]. When dynamical quarks are introduced, $l$ ceases to be an order parameter in the strict sense, but the susceptibility of $l$ still peaks strongly at $T_c$ [@digal; @lwithquarks].
In the effective theory [@robmodel], the pressure of the quark-gluon plasma at $T>T_c$ is completely due to the condensate of $l$, and below $T_c$, where $\langle l\rangle =0$, the pressure vanishes. Moreover, the effective potential $V(l)$ changes extremely rapidly around $T_c$. Hence, as the system cools it may find itself trapped at the wrong value of $\langle l\rangle$. By coupling the effective field $l$ to hadronic degrees of freedom, e.g. the pions, hadrons can be produced as $l$ evolves from $\langle l\rangle \neq 0$ and subsequently oscillates around $\langle l\rangle =0$. This scenario is somewhat reminiscent of reheating after inflation [@linde], and much attention has lately been paid to that aspect of the model [@robadrian; @particleprod]. Remarkably, many qualitative features observed at RHIC are in accordance with the model predictions.
However, when it comes to questions related to the change of the expectation value of $l$ and particle production around $T_c$, one has to take into account the variation of $l$ in space-time. In this paper we address the question of radiative corrections to the spatial variation, by considering the leading one-loop QCD contribution to the spatial derivatives of $l$. Previous work [@robadrian; @particleprod] took into account only the classical kinetic term in the (Euclidean) effective action, $\Gamma (l) = (1/2)\{ |{\partial}_t l|^2 + |{\partial}_i l|^2\} + V(l)$. While such an approach certainly is justified at these preliminary stages, it is important to estimate how much the radiative QCD-effects can affect $\Gamma (l)$ around $T_c$, where the QCD coupling constant becomes large. As for the parameters in the potential $V(l)$, they can be fitted by comparing to QCD lattice results and so are well defined at all $T$. The kinetic term, on the other hand, has to be matched to perturbatively calculated terms in QCD, and could therefore receive large radiative corrections. The magnitude of the first one-loop QCD correction can then hopefully serve as a guideline to the importance of loop effects, and indicate how reliable the above form of $\Gamma (l)$ is at $T_c$. We want to stress that the radiative corrections to be discussed come from QCD, and not from fluctuations in $\Gamma (l)$.
To study the correction to the kinetic term $|{\partial}_i l|^2$, we first consider the one-loop induced quartic terms in QCD, that contain four powers of the external field $A_0$ and two powers of the external momenta. These terms contribute to the high $T$, dimensionally reduced QCD effective action $\Gamma (A_0)$ [@nadkarni; @landsman], and apart from providing a correction to $\Gamma (A_0)$ they can also be related to the kinetic term in $\Gamma (l)$. As a byproduct we obtain some additional derivative interactions in $\Gamma (l)$.
The paper is organized as follows. In the next section, we give the perturbative QCD calculation that corresponds to the leading derivative interactions in $\Gamma (A_0)$. In Sec. III we make the actual matching from an effective theory in terms of $A_0$ to the one in $l$, and discuss the validity of the results. We end with our conclusions and an outlook. Our conventions and some technical details can be found in the appendix.
Perturbative calculation of the four-point function {#II}
===================================================
In the high temperature regime, long distance phenomena (i.e. $|\vec{x}|\gg\beta$) are dominated by the static sector of QCD. At high $T$ it therefore makes sense to use dimensional reduction and integrate out all the nonstatic modes in the theory [@dimred]. With only the static modes left, the full QCD Lagrangian is reduced to a three-dimensional theory. In principle the integrating-out procedure gives rise to an infinite number of interaction terms, but higher dimensional operators become more suppressed by powers of the QCD coupling constant $g$ and/or $T$. In full QCD, the following terms in the resulting effective action $\Gamma (A_0)$ have been calculated [@nadkarni; @landsman], $$\begin{aligned}
\Gamma (A_0) = \beta \!\int \! d^3x
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We derive new integral representations for objects arising in the classical theory of elliptic functions: the Eisenstein series $E_s$, and Weierstrass’ $\wp$ and $\zeta$ functions. The derivations proceed from the Laplace-Mellin transformation for multipoles, and an elementary lemma on the summation of 2D geometric series. In addition, we present new results concerning the analytic continuation of the Eisenstein series as an entire function in $s$, and the value of the conditionally convergent series, denoted by $\widetilde{E}_2$ below, as a function of summation over increasingly large rectangles with arbitrary fixed aspect ratio[^1].'
author:
- 'Andrew Dienstfrey[^2] and Jingfang Huang[^3]'
bibliography:
- '/home/andrewd/Bibliography/master.bib'
title: Integral Representations for Elliptic Functions
---
Introduction
============
In this paper we revisit the classical theory of elliptic functions as developed by Eisenstein and Weierstrass. Both of these researchers represented the meromorphic functions appearing in their theories as summations over a given lattice of elementary pole functions of a prescribed order. Our fundamental observation is that pole functions may be represented by exponentially-damped, oscillatory integrals. These representations depend on the complex half-planes in which the singularities lie, and are natural variants of the classical Mellin, or Laplace-Mellin, formulas which are valid for isolated poles lying in the right half plane, where $\Re(\w)>0$. In more recent times such integral representations have resurfaced in the development of fast multipole methods for the solution of Poisson problems; in this context they often are referred to as “plane-wave” representations ([@hr98] and, more recently, [@cgr99]). A key feature of these integral representations is that the pole centers appear in the exponents of the integrands. As a consequence the lattice summations are transformed into geometric series which may be summed explicitly underneath the integral. The result is a new class of integral representations for the Eisenstein series and other meromorphic functions of Weierstrass’ theory.
A brief summary of the paper follows. In the first section we review the definitions of the Eisenstein series $E_n$ and the Weierstrass functions ¶ and . We will analyze a generalization of Eisenstein’s series which we denote by $\tEs$, the differences being: first, we consider $s=\sig+\i t\in\C$, and second, we define $\tEs$ as a limit over lattice squares of increasing size, a significant point when $\Re(s)\le2$ and the sums are not absolutely convergent. In addition, in this preliminary section we provide elementary derivations of the requisite plane-wave formulas for general pole functions of the form $f(\w)=\w^{-s}$, and a summation identity for a two-dimensional geometric series.
In the next section we employ the plane-wave representations and the summation identity to derive an integral representation for $\tEs$ for the case $\ReSTwo$. Integral representations for Eisenstein’s $E_n$ naturally follow for $s=n\ge 3$. Subsequently, we derive an alternative representation for $\tEs$ as a contour integral from which we deduce that the sums $\tEs$, defined unambiguously for , admit an analytic continuation as an [**entire**]{} function to the whole of the complex plane. As a corollary, we prove the existence of a finite limit for $\tE_2$. We discuss $\tE_2$ and its relation to Eisenstein’s, $E_2$. As the summation processes defining these two conditionally convergent series are distinct, one expects different limiting values. We derive a closed form correction term for this difference. (Note that both Eisenstein’s convention and ours give $E_1=\tE_1=0$.) In the following section we derive analogous integral formulas for Weierstrass’ ¶ and functions. We conclude the paper with a brief discussion of these integral representations in relation to previous research in the theories of lattice sums, and elliptic functions.
In this last regard we take a moment to mention here that our formulas for $\tEs$ are the natural lattice analogues to the well-known representation for Riemann’s zeta function (which we denote with the subscript $\z_R$ so as to distinguish it from Weierstrass’ function of the same name) \[RiemannZ\] \_R(s) = \_[n=1]{}\^ = ł\^[s-1]{} \^[-ł]{} , (s)>1. For example, in the case of a square lattice we derive the following integral expression for the classical Eisenstein series E\_k(i) &=& \_[n=-]{}\^\_[m=-]{}\^\
\[EkSquare\] &=& ł\^[k-1]{} \^[-ł]{} , where $k$ is a positive integer divisible by four ($E_k(i)=0$ otherwise). The similarity between (\[RiemannZ\]) and (\[EkSquare\]) is clear. For more general lattices, we replace $\i$ by $\t$, $k\in\N$ by $s\in\C$, and the single trigonometric ratio in (\[EkSquare\]) by a sum of analogous ratios denoted by and defined in (\[fOne:def\]) and (\[fTwo:def\]). The general expression is given in theorem \[Eisenstein:thm\].
We note that a subset of the results presented below appeared previously in a slightly different form [@huang99].
Preliminaries
=============
In this section we review the definitions of the Eisenstein series and the Weierstrass ¶ and functions. Furthermore, we derive elementary lemmas concerning plane-wave representations and a geometric series identity, both of which we will use repeatedly in the subsequent sections.
The Eisenstein series and Elliptic functions
--------------------------------------------
We are given a general lattice $\Lambda\subset \C$ which we describe by generators $\mu, \nu$: $$\Lambda=\{m\cdot\mu + n\cdot\nu| m,n\in\Z\}$$ where $\mu,\nu$ are complex numbers such that the lattice ratio, $\t=\nu/\mu$, is not real. The fundamental parallelogram is the set, $\L_0=\{\al+\beta\t| |\al|\le 1/2, |\beta|\le 1/2\}$. Eisenstein began his investigations of doubly periodic, meromorphic functions with periods $\mu$ and $\nu$ through the study his eponymous series (see, for example, [@weil76] and [@schoeneberg74]) \[EisensteinEn\] E\_k = , k1; the elimination of the term $m=n=0$ is implicit here and below. For the magnitude of each term in the summand we have the bounds \[MagBound\] . Applying (\[MagBound\]) and elementary estimates one verifies that the series (\[EisensteinEn\]) are absolutely convergent for $n\ge
3$, and thus are well-defined functions of the lattice $E_n=E_n(\Lambda)$. The same estimates indicate that the series $E_n$ are absolutely divergent for $n=1$ or $2$, hence the limiting operation specified in (\[EisensteinEn\]) plays a non-trivial role in the definition of these sums. Eisenstein proved that the limiting procedure (\[EisensteinEn\]) yields finite values of $E_n$ even in these cases. We observe that, in particular, $E_1=0$. This is in keeping with the fact that, using a symmetry argument in the absolutely convergent case, one may prove that $E_n=0$ for all odd $n\ge3$. Therefore, from the point of view of convergence, the only “interesting” sum is $E_2$.
Eisenstein was cognizant that the value of $E_2$ depends on the choice of limiting procedures; and he derived many identities which connect his summation process for $E_2$ to others ([@weil76]). We choose yet a different summation convention and define $\tEs$ as the limit of partial sums over “lattice-squares” of increasing size. We generalize further in considering complex exponents. Specifically, we define $\tEs$ by \[HDEn\] = , which we consider, initially, for $s=1, s=2$ and $\Re(s)>2$. For non-integer $s$ we consider the branch of the function $\z^{s}$ with a cut along the “negative diagonal” of the lattice, $\{z=-t(\mu+\nu),\
t>0\}$. For all $z$ in the closure of this cut plane we have \[BranchW\] (z) +2,\
=(--). We further enforce the convention that points of the lattice lying along the diagonal are considered symmetrically, $$\fr{1}{(-m\mu-m\nu)^s} = \fr{1}{2(m|\mu+\nu|)^s}
\lp\fr{1}{\e^{\i\th s}}
+ \fr{1}{\e^{\i(\th+2\pi)s}}\rp$$ We will return to this point later. Finally, we note that the large $K$ limiting convention defined in (\[HDEn\]) is relevant only in the cases $s=1$ or $s=2$.
We will derive integral representations for $\tEs$. Naturally, restricting $s$ to the positive integers, our formula yields an integral representation for
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
author:
- 'B. Tuguldur and Ts. Gantsog'
title: 'Some features of the Driven Jaynes-Cummings system'
---
[School of Physics and Electronics, National University of Mongolia]{}
Introduction
============
The Jaynes-Cummings model (JCM) [@jaynes] is one of the simplest models describing the interaction of light with matter, where a single two-level atom interacts with a single mode of quantized radiation field in the electric dipole and rotating wave approximations. This model is of fundamental importance to the field of quantum optics; many interesting features have been predicted for both the atomic variables and the statistical properties of the field through the years, beginning with the well-known phenomena of collapses and revivals of the atomic population inversion oscillation [@eberly]. Numerous extensions of the JCM have been considered [@alsing; @deb; @gerry] and many experiments reported [@rempe; @diedrich; @thompson; @childs; @brune].
One of the most interesting features of the JCM is that if the atom is initially prepared in its upper state and if the cavity field is initially in the coherent state, then the quantized field evolves into an almost pure state at half of the atomic-revival time [@eiselt; @phoenix; @gea]. This approximately pure state is equal to a superposition (Schr$\ddot{\text{o}}$dinger cat) state composed of two states of light having the same amplitude, but opposite phase. The amplitude of the component states is approximately equal to the amplitude of the initial coherent state of the field mode [@buzek].
Buzek et al. [@buzek1] showed that by driving the atom with the external classical field, superposition states of the quantized cavity mode with arbitrary amplitudes and phases of component states can be produced. Recently, Gea-Banacloche and coworkers [@gea1] have verified the existence of maximally entangled state in externally driven JCM. Optical cavities with atoms have been proposed for quantum information processing [@pellizzari; @pellizzari1].
In this paper we study the dynamics of the JCM when a quantized cavity mode is pumped continuously by an external classical field. In Sec.2 we derive the eigenstates and eigenenergies of the driven Jaynes-Cummings system. Using these results we find the time dependent state vector of the system for a given initial condition. In Sec.3 the dynamics of the atomic inversion, mean photon number and the phase space distribution function $Q(\alpha,\alpha^*)$ are examined. In Sec.4 we give approximate solutions that enable us to explain the features observed. Finally in Sec.5 we summarize our results.
Driven Jaynes-Cummings System
=============================
We consider the driven Jaynes-Cummings system with the external field driving the cavity mode. The interaction Hamiltonian of the system in the interaction picture is given by \[3\] $$\hat{H}_I=g(\hat{a}\hat{\sigma}_++\hat{a}^\dagger\hat{\sigma}_-)+\mathscr{E}(\hat{a}^\dagger e^{i\phi}+\hat{a} e^{-i\phi}),\label{hhh}$$ where $g$ is the coupling constant between the atom and the cavity mode; $\mathscr{E}$ is the amplitude of the driving field; $\phi$ is the phase of the classical field; $\hat{a}^\dagger$, $\hat{a}$ are creation and annihilation operators for the cavity mode; and $\hat{\sigma}_+$ and $\hat{\sigma}_-$ are atomic pseudospin operators. We assumed that the driving field is in resonance with the atom and the cavity mode. The steady state solution of the Schr$\ddot{\text{o}}$dinger equation for the Hamiltonian (1) is possible for $\mathscr{E}<g/2$. The quasieigenenergies and the corresponding eigenstates are given by [@alsing] $$\begin{array}{l}
E_0=0,\\
\displaystyle{E_{n1}=g\sqrt{n}\left[1-\left(\frac{2\mathscr{E}}{g}\right)^2\right]^{3/4},\quad n=1,2,3,...,}\\
\displaystyle{E_{n2}=-g\sqrt{n}\left[1-\left(\frac{2\mathscr{E}}{g}\right)^2\right]^{3/4},\quad n=1,2,3,...}
\end{array}$$ and
\[steady\] $$\begin{gathered}
\vert\psi_{En1}\rangle=c_p(S(\eta)D(\beta_{n+})\vert n-1\rangle+L_+ S(\eta)D(\beta_{n+})\vert n\rangle)\vert -\rangle\\
+c_p(L_+S(\eta)D(\beta_{n+})\vert n-1\rangle+e^{2i\phi} S(\eta)D(\beta_{n+})\vert n\rangle)\vert +\rangle,\end{gathered}$$ $$\begin{gathered}
\vert\psi_{En2}\rangle=c_p(S(\eta)D(\beta_{n-})\vert n-1\rangle-L_+ S(\eta)D(\beta_{n-})\vert n\rangle)\vert -\rangle\\
+c_p(L_+S(\eta)D(\beta_{n-})\vert n-1\rangle-e^{2i\phi} S(\eta)D(\beta_{n-})\vert n\rangle)\vert +\rangle,\end{gathered}$$
where if quasienergies are positive (negative) $E_{n1}>0$ ($E_{n2}<0$) then $\displaystyle{\beta_{n+}=-\frac{2\mathscr{E}}{g}\sqrt{n}e^{i\phi}}$ ($\displaystyle{\beta_{n-}=\frac{2\mathscr{E}}{g}\sqrt{n}e^{i\phi}}$), $\eta=re^{i\theta}$, $e^{2r}=\sqrt{1-(2\mathscr{E}/g)^2}$, $\theta=2\phi$, $c_p=(1/2)\sqrt{1-\sqrt{1-(2\mathscr{E}/g)^2}}$ and $L_{+}=-(ge^{i\phi}/2\mathscr{E})(1+\sqrt{1-(2\mathscr{E}/g)^2})$. For an external field amplitude larger than $\mathscr{E}=g/2$, no normalizable steady states exist. This value is the threshold condition for spontaneous dressed-state polarization.
If the atom is initially prepared in its upper state $\vert +\rangle$ and if the cavity field is prepared initially in the coherent state $\vert \alpha_0\rangle$, then the time evolution of the system is given by $$\vert\psi(t)\rangle=\vert\psi^+(t)\rangle+\vert\psi^-(t)\rangle,\label{time}$$ where
\[timeevolution\] $$\begin{gathered}
\vert\psi^+(t)\rangle=\vert c_p\vert^2
\sum_{n=0}^{\infty}(L_+^*\langle n-1;-\beta_{n+};-\eta\vert\alpha_0\rangle+e^{-2i\phi}\langle n;-\beta_{n+};-\eta\vert\alpha_0\rangle)e^{-iE_nt}\\
\left[(L_+\vert\eta;\beta_{n+};n-1\rangle+e^{2i\phi}\vert\eta;\beta_{n+};n\rangle)\vert +\rangle
+(\vert\eta;\beta_{n+};n-1\rangle+L_+\vert\eta;\beta_{n+};n\rangle)\vert -\rangle\right],\label{df1}\end{gathered}$$ $$\begin{gathered}
\vert\psi^-(t)\rangle=\vert c_p\vert^2
\sum_{n=0}^{\infty}(L_+^*\langle n-1;-\beta_{n-};-\eta\vert\alpha_0\rangle-e^{-2i\phi}\langle n;-\beta_{n-};-\eta\vert\alpha_0\rangle)e^{iE_nt}\\
\left[(L_+\vert \eta;\beta_{n-};n-1\rangle-e^{2i\phi}\vert \eta;\beta_{n-};n\rangle)\vert +\rangle
+(\vert \eta;\beta_{n-};n-1\rangle-L_+\vert\eta;\beta_{n-};n\rangle)\vert -\rangle\right],\label{df2}\end{gathered}$$
and $E_n=g\sqrt{n}(1-(2\mathscr{E}/g)^2)^{3/4}$ and squeezed and displaced Fock states are denoted as $S(\eta)D(\beta_{n\pm})\vert n\rangle=\vert\eta;\beta_{n\pm};n\rangle$. This result is exact and no approximations are made. We shall apply it to study quantum dynamics of the system in the following section.
Quantum dynamics of the driven JCM
==================================
Atomic inversion and the mean photon number of the cavity mode
--------------------------------------------------------------
Using explicit expressions for the state vector given by equations and we can study the dynamical properties of the system under consideration. Firstly
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: |
We study some basic properties of the variety of characters in ${\mathrm{PSL}_2 (\C)}$ of a finitely generated group. In particular we give an interpretation of its points as characters of representations. We construct 3-manifolds whose variety of characters has arbitrarily many components that do not lift to ${\mathrm{SL}_2 (\C)}$. We also study the singular locus of the variety of characters of a free group.\
[*MSC*: 57M50, 57M05, 20C15\
*Keywords*: Representation spaces; variety of characters; ${\mathrm{PSL}_2 (\C)}$]{}
author:
- Michael Heusener and Joan Porti
title: 'The variety of characters in ${\mathrm{PSL}_2 (\C)}$'
---
Introduction
============
The varieties of representations and characters have many applications in 3-dimensional topology and geometry. The variety of ${\mathrm{SL}_2 (\C)}$-characters has been intensively studied since the seminal paper of Culler and Shalen [@CS], but for many applications it is more convenient to work with ${\mathrm{PSL}_2 (\C)}$ instead of ${\mathrm{SL}_2 (\C)}$ (see [@BZ] and [@BMP] for instance). The purpose of this note is to study some basic properties of the variety of characters in ${\mathrm{PSL}_2 (\C)}$. Most of the results of invariant theory that we use can be found in any standard reference (e.g. [@KSS], [@Kraft], [@PV]).
Throughout this paper, $\Gamma$ will denote a finitely generated group.
The set of all representations of $\Gamma$ in ${\mathrm{PSL}_2 (\C)}$ is denoted by $
R(\Gamma)
$ and it is called the *variety of representations of $\Gamma$ in ${\mathrm{PSL}_2 (\C)}$*.
The variety of representations $R(\Gamma)$ has a natural structure as an affine algebraic set over the complex numbers given as follows: the group ${\mathrm{PSL}_2 (\C)}$ is algebraic (see Section \[sec:invariants\]). Given a presentation $\Gamma=\langle
\gamma_1,\ldots,\gamma_s\mid (r_i)_{i\in I} \rangle$ we have a natural embedding: $$\begin{array}{rcl}
R(\Gamma)&\to &{\mathrm{PSL}_2 (\C)}\times\cdots\times {\mathrm{PSL}_2 (\C)}\\
\rho & \mapsto & (\rho(\gamma_1),\ldots,\rho(\gamma_s))
\end{array}$$ and the defining equations are induced by the relations. This structure can be easily seen to be independent of the presentation. In fact using the isomorphism ${\mathrm{PSL}_2 (\C)}\cong {\mathrm{SO}_{3} (\C)}$, $R(\Gamma)$ has a structure of an affine set (see Lemma \[lem:SO3\]).
The action of ${\mathrm{PSL}_2 (\C)}$ on $R(\Gamma)$ by conjugation is algebraic. The quotient $R(\Gamma)/{\mathrm{PSL}_2 (\C)}$ may be not Hausdorff and it is more convenient to consider the algebraic quotient of invariant theory, because ${\mathrm{PSL}_2 (\C)}$ is reductive.
The *variety of ${\mathrm{PSL}_2 (\C)}$-characters* $X(\Gamma)$ is the quotient $R(\Gamma)/\!/{\mathrm{PSL}_2 (\C)}$ of invariant theory.
This definition means that $X(\Gamma)$ is an affine algebraic set together with a regular map $t\co R(\Gamma) \to X(\Gamma)$ which induces an isomorphism $$t^{*}\co \C[X(\Gamma)]\to\C[R(\Gamma)]^{{\mathrm{PSL}_2 (\C)}}$$ (i.e.the regular functions on $X(\Gamma)$ are precisely the regular functions on $R(\Gamma)$ invariant by conjugation). We will use the notation $R(M)=R(\pi_1M)$ and $X(M)=X(\pi_1M)$ if $M$ is a path-connected topological space.
In this paper we study the basic properties of $X(\Gamma)$.
First we explain the name “variety of characters": given a representation $\rho\co\Gamma\to {\mathrm{PSL}_2 (\C)}$, its character is the map $$\begin{array}{rcl}
\chi_{\rho}\co\Gamma&\to&\C\\
\gamma&\mapsto &{\operatorname{tr}}^2(\rho(\gamma))
\end{array}$$
\[thm:PSL2characters\] There is a natural bijection between $X(\Gamma)$ and the set of characters of representations $\rho\in
R(\Gamma)$. This bijection maps every $t(\rho)\in X(\Gamma)$ to the character $\chi_\rho$.
In many cases the representations of $R(\Gamma)$ lift to ${\mathrm{SL}_2 (\C)}$, for instance if $\Gamma$ is a free group. In such a case, $X(\Gamma)$ is just a quotient of the usual variety of characters in ${\mathrm{SL}_2 (\C)}$ (See Proposition \[prop:naturaliso\]). This quotient is the definition already used in [@Bur90], [@HLM1],[@HLM2] and [@Ril84] for 2-bridge knot exteriors. The explicit computation for the figure eight knot exterior is found in [@GM].
There are cases where representations do not lift to ${\mathrm{SL}_2 (\C)}$, for instance the holonomy representation of an orientable hyperbolic 3-orbifold with 2 torsion. The next result proves that there are manifolds with arbitrarily many components of characters that do not lift.
\[thm:nolifts\] For every $n$, there exist a compact irreducible 3-manifold $M$ with $\partial M$ a 2-torus such that $X(M)$ has at least $n$ irreducible one dimensional components whose characters do not lift to ${\mathrm{SL}_2 (\C)}$.
In Section \[sec:invariants\] we prove Theorem \[thm:PSL2characters\]. In Section \[sec:irreducibility\] we study the fiber of the projection $t\co R(\Gamma)\to X(\Gamma)$, introducing the different notions of irreducibility. Section \[sec:lifts\] is devoted to the study of lifts of representations and the proof of Theorem \[thm:nolifts\]. In the last section we determine the singular set of $X(\Gamma)$ when $\Gamma\cong F_n$ is the free group of rank $n\geq 3$.
Invariants of ${\mathrm{PSL}_2 (\C)}$ {#sec:invariants}
=====================================
Before proving Theorem \[thm:PSL2characters\] we quickly review some basic notions of algebraic geometry and invariant theory (that the reader may prefer to skip and go directly to the proof in Subsection \[ss:ProofTheorem\]). For details see [@KSS], [@Kraft] or [@PV].
Basic notions of invariant theory
---------------------------------
A closed algebraic subset $Z\subset \C^{N}$ is called *affine*. We denote by $\C[Z]$ the ring of regular functions on $Z$. An algebraic group $G$ that acts algebraically on $Z$ acts naturally on $\C[Z]$ via $g f (z) := f(g^{-1} z) $. We denote by $\C[Z]^{G} $ the ring of invariant functions, i.e. functions $f\in\C[Z]$ for which $g
f = f$ for all $g\in G$.
The group $G$ is called *reductive* if it has the following property: for each finite dimensional rational representation $\rho\co G\to \mathrm{GL}(V)$ and every $G$-invariant subspace $W\subset V$ there exist a complementary $G$-invariant subspace $W'\subset V$, i.e. $V= W'\oplus W$.
If $Z$ is affine and $G$ is reductive, then the ring $\C[Z]^{G} $ is finitely generated. The affine set $Y$ such that $\C[Y]\cong
\C[Z]^{G} $ is called the *algebraic quotient* and it is denoted by $Z /\!/\/ G$.
We shall use the following properties of reductive groups:
- By Maschke’s theorem, finite groups are reductive.
- More generally, let $G\subset \mathrm{GL}_{n}(\C)$ be a linear algebraic group. The group $G$ is reductive if there is a Zariski-dense subgroup $K\subset G$ which is compact in the classical topology. It follows that $\mathrm{GL}_{n}(\C)$, $\mathrm{SL}_{n}(\C)$, $\textrm{O}_{n}(\C)$, $\textrm{SO}_{n}(\C)$ and $\textrm{Sp}_{n}(\C)$ are reductive.
- Let $G$ be a reductive linear algebraic group. Let $Y$ and $Z$ be varieties on which $G$ acts and let $f\co X\to Y$ be a $G$-invariant regular map. If $f^{*}\co \C[Y]\to\C[
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=1
Introduction
============
The aim of this article is to study rational parallelisms of algebraic varieties by means of the transcendence of their symmetries. Our original motivation was to understand the possible obstructions to the third Lie theorem for algebraic Lie pseudogroups. This article is concerned with the simply transitive case. These obstructions should appear in the Galois group of certain connection associated to a Lie algebroid. However, we have written the article in the language of regular and rational parallelisms of algebraic varieties and their symmetries.
A theorem of P. Deligne says that any Lie algebra can be realized as a parallelism of an algebraic variety. This is a sort of algebraic version of the third Lie theorem. Notwithstanding, there is one main problem: given an algebraic variety with a parallelism, how far is it from being an algebraic group? Is it possible to conjugate this parallelism with the canonical parallelism of invariant vector fields on an algebraic group?
In the analytic context, from the Darboux–Cartan theorem [@sharpe p. 212], a $\mathfrak{g}$-parallelized complex manifold $M$ has a natural $(G,G)$ structure where $G$ is a Lie group with $\mathfrak{lie}(G) = \mathfrak{g}$. The obstruction to be a covering of $G$, as manifold with a $(G,G)$ structure, is contained in a monodromy group [@sharpe p. 130]. In [@Wang], Wang proved that parallelized compact complex manifolds are biholomorphic to quotients of complex Lie groups by discrete cocompact subgroups. This result has been extended by Winkelmann in [@Winkelmann1; @Winkelmann2] for some open complex manifolds.
In this article we address the problem of classification of rational parallelisms on algebraic varieties up to birational transformations. Such a classification seems impossible in the algebraic category but we prove a criterion to ensure that a parallelized algebraic variety is isogenous to an algebraic group. Summarizing, we pursue the following plan: We regard infinitesimal symmetries of a rational parallelism as horizontal sections of a connection that we call the reciprocal Lie connection. This connection has a Galois group which is represented as a group of internal automorphisms of a Lie algebra. The obstruction to the algebraic conjugation to an algebraic group, under some assumptions, appear in the Lie algebra of this Galois group.
In Section \[section\_parallelisms\] we introduce the basic definitions; several examples of parallelisms are given here. In Section \[section\_lie\] we study the properties of connections on the tangent bundle whose local analytic horizontal sections form a sheaf of Lie algebras of vector fields. We call them [*Lie connections*]{}. They always come by pairs, and they are characterized by having vanishing curvature and constant torsion (Proposition \[prop\_Lie\_char\]). We see that each rational parallelism has an attached pair of Lie connections, one of them with trivial Galois group. We compute the Galois groups of some parallelisms given in examples (Proposition \[prop\_example\]), and prove that any algebraic subgroup of ${\rm PSL}_2(\mathbf C)$ appears as the differential Galois group of a $\mathfrak{sl}_2(\mathbf C)$-parallelism (Theorem \[thm\_SL2\]). Section \[section\_DC\] is devoted to the construction of the isogeny between a $\mathfrak g$-parallelized variety and an algebraic group $G$ whose Lie algebra is $\mathfrak g$. In order to do this, we introduce the Darboux–Cartan connection, a $G$-connection whose horizontal sections are parallelism conjugations. We prove that if $\mathfrak g$ is centerless then the Darboux–Cartan connection and the reciprocal Lie connection have isogenous Galois groups. We prove that the only centerless Lie algebras $\mathfrak{g}$ such that there exists a $\mathfrak{g}$-parallelism with a trivial Galois group are algebraic Lie algebras, i.e., Lie algebras of algebraic groups. In particular this allows us to give a criterion for a parallelized variety to be isogenous to an algebraic group. The vanishing of the Lie algebra of the Galois group of the reciprocal connection is a necessary and sufficient condition for a parallelized variety to be isogenous to an algebraic group:
Let $\mathfrak g$ be a centerless Lie algebra. An algebraic variety $(M,\omega)$ with a rational parallelism of type $\mathfrak g$ is isogenous to an algebraic group if and only if $\mathfrak{gal}(\nabla^{\rm rec}) = \{0\}$.
The notion of [*isogeny*]{} can be extended beyond the simply-transitive case. Let us consider a complex Lie algebra $\mathfrak g$. An [*infinitesimally homogeneous variety*]{} of type $\mathfrak g$ is a pair $(M,\mathfrak s)$ consisting of a complex smooth irreducible variety $M$ and a finite-dimensional Lie algebra isomorphic to $\mathfrak g$ that spans the tangent bundle of $M$ on the generic point.
We are interested in conjugation by rational or by algebraic maps, so that, whenever necessary, we replace $M$ by a suitable Zariski open subset. In this context, we say that a dominant rational map $f\colon M_1 \dasharrow M_2$ between varieties of the same dimension conjugates the infinitesimally homogeneous varieties $(M_1,\mathfrak s_1)$ and $(M_2,\mathfrak s_2)$ if $f^*(\mathfrak s_2) = \mathfrak s_1$. We say that $(M_1,\mathfrak s_1)$ and $(M_2,\mathfrak s_2)$ are [*isogenous*]{} if they are conjugated to the same infinitesimally homogeneous space of type $\mathfrak g$.
Under some hypothesis on the Lie algebra $\mathfrak s\subset \mathfrak X(M)$ one can prove that $(M,\mathfrak s)$ is isogenous to a homogeneous space $(G/H,\mathfrak{lie}(G)^{\rm rec})$ with the action of right invariant vector fields. These hypothesis are satisfied by transitive actions of $\mathfrak{sl}_{n+1}(\mathbf C)$ on $n$-dimensional varieties. As a particular case of Theorem \[homogeneous\] one has
Let $(M,\mathfrak s)$ be an infinitesimally homogeneous variety of complex dimension $n$ such that $\mathfrak s$ is isomorphic to $\mathfrak{sl}_{n+1}(\mathbf C)$. Then there exists a dominant rational map $M \dasharrow \mathbf{CP}_n$ conjugating $\mathfrak s$ with the Lie algebra $\mathfrak{sl}_{n+1}(\mathbf C)$ of projective vector fields in $\mathbf{CP}_n$.
Appendix \[ApA\] is devoted to a geometrical presentation of Picard–Vessiot theory for linear and principal connections. Finally, Appendix \[apB\] contains a detailed proof of Deligne’s theorem of the realization of a regular parallelism modeled over any finite-dimensional Lie algebra. This includes also a computation of the Galois group that turns out to be, for this particular construction, an algebraic torus.
Parallelisms {#section_parallelisms}
============
Let $M$ be a smooth connected affine variety over $\mathbf C$ of dimension $r$. We denote by $\mathbf C[M]$ its ring of regular functions and by $\mathbf C(M)$ its field of rational functions. Analogously, we denote by $\mathfrak X[M]$ and $\mathfrak X(M)$ respectively the Lie algebras of regular and rational vector fields in $M$, and so on.
Let $\mathfrak g$ be a Lie algebra of dimension $r$. We fix a basis $A_1,\ldots,A_r$ of $\mathfrak g$, and the following notation for the associated structure constants $[A_i,A_j] = \sum_{k}\lambda_{ij}^kA_k$.
A parallelism of type $\mathfrak g$ of $M$ is a realization of the Lie algebra $\mathfrak g$ as a Lie algebra of pointwise linearly independent vector fields in $M$. More precisely:
A regular parallelism of type $\mathfrak g$ in $M$ is a Lie algebra morphism, $\rho\colon \mathfrak g \to \mathfrak X[M]$ such that $\rho A_1(x), \ldots, \rho A_r(x)$ form a basis of $T_xM$ for any point $x$ of $M$.
\[ex:AGP\]Let $G$ be an algebraic group and $\mathfrak g$ be its Lie algebra of left invariant vector fields. Then the natural inclusion $\mathfrak g\subset \mathfrak X[G]$ is a regular parallelism of $G$. The Lie algebra $\mathfrak g^{\rm rec}$ of right invariant vector fields is another regular parallelism of the same type. Let invariant and right invariant vector fields commute, hence, an algebraic group is naturally endowed with a pair of commuting parallelisms of the same type.
From Example \[ex:AGP\], it is clear that any *algebraic* Lie algebra is realized as a parallelism of some algebraic variety. On the other hand, Theorem \[TDeligne\] due to P. Deligne and published in [@Malgrange], ensures that any Lie algebra is realized as a regular parallelism of an algebraic variety. Analogously, we have the definitions of rational and local analytic parallelism. Note that a rational parallelism in $M$ is a regular parallelism in a Zariski open subset $M^\star \subseteq M$.
There is dual definition, equivalent to that of parallelism. This is more suitable for calculations.
A regular parallelism form (or coparallel
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---
author:
- 'Taiya <span style="font-variant:small-caps;">Munenaka</span> and Hirohiko <span style="font-variant:small-caps;">Sato</span>[^1]'
title: ' A Novel Pyrochlore Ruthenate: Ca$_{2}$Ru$_{2}$O$_{7}$ '
---
Frustration results in many types of unexpected phenomena. Among three-dimensional frustrated systems, the pyrochlore lattice is particularly interesting for its strong frustration originating from a purely geometric reason. A typcial pyrochlore oxide has the composition A$_{2}$B$_{2}$O$_{7}$. In this system, the B sites (and also the A sites) form a three-dimensional network based on the B$_{4}$ tetrahedron. From another point of view, we can regard the pyrochlore lattice as a three-dimensional version of a Kagomé lattice. Therefore, perfect geometric frustration is inherent in this structure, and many interesting phenomena emerge. For example, Y$_{2}$Mo$_{2}$O$_{7}$ exhibits spin-glass behavior,[@gingras97; @gardner99; @miyoshi00] demonstrating that the geometric frustration due to the antiferromagnetic pyrochlore lattice itself is responsible for the glassy state, even if there is no structural disorder. On the other hand, the nearest-neighbor interaction is ferromagnetic in Ho$_{2}$Ti$_{2}$O$_{7}$ and Dy$_{2}$Ti$_{2}$O$_{7}$. In this case, single-ion magnetic anisotropy causes another type of frustration and consequently, “spin ice” behavior appears.[@harris97; @ramirez99; @higashinaka05]
Conductive pyrochlores are also remarkable systems. For Nd$_{2}$Mo$_{2}$O$_{7}$, there was the epoch-making interpretation that the anomalous Hall effect detects the chirality of Nd magnetic moments.[@taguchi03] A theoretical study proved that the Berry phase plays an important role in systems with a chiral spin arrangement.[@onoda03] Tl$_{2}$Mn$_{2}$O$_{7}$ has metallic conductivity and undergoes a ferromagnetic transition. Near the transition temperature, a giant magnetoresistance appears[@shimakawa96]. In Cd$_{2}$Re$_{2}$O$_{7}$, a superconducting transition was discovered at 1.5 K.[@sakai01; @hanawa01] Furthermore, $\beta$-type pyrochlore osmates, AOs$_{2}$O$_{6}$ (A = K, Rb, Cs), also exhibit superconductivity with a relatively high $T_{c}$ (9.7 K for A = K).[@yonezawa04; @yonezawa04b; @hiroi04; @hiroi05]
While searching for new materials with exotic electronic states, we have become interested in pyrochlore ruthenates. Because ruthenium 4$d$-orbitals have a character intermediate between localized and itinerant orbitals, a variety of electronic phases appear. In particular, the discovery of spin-triplet superconductivity in Sr$_{2}$RuO$_{4}$[@maeno94; @ishida98] has aroused the interest of many material scientists.
Ruthenates with pyrochlore structures have also been actively investigated. Bi$_{2}$Ru$_{2}$O$_{7}$ and Pb$_{2}$Ru$_{2}$O$_{6.5}$ are metallic with Pauli paramagnetism,[@longo69; @cox83; @hsu88] whereas Ln$_{2}$Ru$_{2}$O$_{7}$ and Y$_{2}$Ru$_{2}$O$_{7}$ are insulators with localized magnetic moments.[@aleonard62; @subramanian83; @yoshii99; @ito00] Tl$_{2}$Ru$_{2}$O$_{7}$ undergoes a metal-insulator transition.[@takeda98] These observations reveal that pyrochlore ruthenates display a variety of electronic phases, widely distributed over the Mott boundary. Their electronic states are very sensitive to the Ru-O distance or the Ru-O-Ru bond angle, which is related to the radius of the cations on the A site. In addition to controlling the band width by changing the cation radius, filling control of the $4d$-band also seems important in searching for novel electronic phases. However, there have been few trials[@yoshii99] on controlling the band filling of pyrochlore ruthenates. This is probably because the cation on the A site is trivalent in most stable pyrochlore ruthenates. Apart from Cd$_{2}$Ru$_{2}$O$_{7}$[@wang98], there are no reports on stoichiometric pyrochlores composed of only Ru$^{5+}$. In the present study, we succeeded in synthesizing a new pyrochlore ruthenate with Ru$^{5+}$, Ca$_{2}$Ru$_{2}$O$_{7}$, by maintaining a high-oxidization atmosphere.
Single crystals of Ca$_{2}$Ru$_{2}$O$_{7}$ were synthesized by a hydrothermal method. A mixture of RuO$_{2}$ (40 mg), obtained by oxidizing Ru metal (Furuya Metals, 99.99% purity), CaO (34 mg, Soekawa Chemical, 99.99% purity), and 0.3 ml of 30% H$_{2}$O$_{2}$ solution was encapsulated in a gold tube. Then, it was kept in an autoclave under 150 MPa hydrostatic pressure at 600$^{\circ}$C for 3 days. The chemical composition was determined using an energy dispersive X-ray spectrometer (EDS) installed on a scanning electron microscope. The crystal structure was analyzed using a single crystal and an imaging-plate X-ray diffractometer (Rigaku, R-Axis RAPID), in which Mo-K$\alpha$ radiation was generated using an X-ray tube and monochromized using graphite. We also used a powder X-ray diffractometer to check whether there was any contamination due to impurity phases. The magnetic susceptibility between 2 and 400 K was measured using a superconducting quantum-interference-device magnetometer. In the measurement, approximately 10 mg of nonoriented single crystals were wrapped in a piece of aluminum foil. The resistivity was measured by a DC four-wire method on an array of single crystals, connected with each other, in a closed-cycle helium refrigerator whose temperature range was between 5.5 and 300 K. The array was composed of four single crystals, and we attached the voltage leads to the same crystal located at the center. Therefore, we consider that the observed resistivity approximately reflects the behavior of a single crystal.
The obtained materials were black crystals with an octahedral shape. Observations using a microscope did not detect any other type of crystal as shown in Fig. \[fig1\](a). An EDS analysis showed that the atomic compositional ratio of Ca and Ru is almost 1:1. The single-crystal X-ray diffraction revealed an F-type cubic unit cell with $a = 10.197$ Å, which is very close to 10.143 Å for Y$_{2}$Ru$_{2}$O$_{7}$[@kennedy95]. This strongly suggested that the our material has a pyrochlore structure. A further structural refinement was carried out by observing about 2000 reflections at room temperature. The results are summarized in Table \[table1\],
------- ---------- ----------- ------- ------- ----------
Atom Position $x$ $y$ $z$ $B_{eq}$
Ru(1) 16$c$ 0 0 0 0.602(4)
Ca(1) 16$d$ 0.5 0.5 0.5 2.587(9)
O(1) 48$f$ 0.3219(1) 0.125 0.125 1.48(2)
O(2) 8$b$ 0.375 0.375 0.375 3.65(3)
------- ---------- ----------- ------- ------- ----------
: Fractional atomic coordinates and equivalent isotropic displacement parameters (Å$^{2}$) for Ca$_{2}$Ru$_{2}$O$_{7}$. The lattice symmetry and the space group are *cubic* and $Fd\bar{3}m$ (\#227), respectively. The lattice parameters are $a = 10.197(2)$ Å, $V = 1060.4(3)$ Å$^3$ and $Z=8$. The final reliability factor is $R(F) = 2.7 \%$ for 1888 observed reflections.[]{data-label="table1"}
![(a) Micrograph of single crystals of Ca$_{2}$Ru$_{2}$O$_{7}$. The typical dimensions of the crystals are $0.1 \times 0.1 \times 0.1$ mm$^{3}$. (b) X-ray powder pattern of batch used for magnetic measurement. The whole batch was ground before measurement. The weak peaks at 28.1$^{\circ}$ and at 54.3$^{\circ}$ are from traces of RuO$_{2}$.[]{data-label="fig1"}](fig1.eps){width="1.0\linewidth"}
and they coincide with those for a pyrochlore structure. The analysis did not detect any clear evidence that the composition deviates from the ideal pyrochlore, although the temperature factors of Ca(1) and O(2) seem unexpectedly large. We also analyzed the powder X-ray diffraction
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LYCEN 2001-38\
****
First Results of the EDELWEISS WIMP Search using a 320 g Heat-and-Ionization Ge Detector
[The EDELWEISS Collaboration:]{}\
A. Benoit$^{1}$, L. Bergé$^{2}$, A. Broniatowski$^{2}$, B. Chambon$^{3}$, M. Chapellier$^{4}$, G. Chardin$^{5}$, P. Charvin$^{5,6}$, M. De Jésus$^{3}$, P. Di Stefano$^{5}$, D. Drain$^{3}$, L. Dumoulin$^{2}$, J. Gascon$^{3}$, G. Gerbier$^{5}$, C. Goldbach$^{7}$, M. Goyot$^{3}$, M. Gros$^{5}$, J.P. Hadjout$^{3}$, A. Juillard$^{2,5}$, A. de Lesquen$^{5}$, M. Loidl$^{5}$, J. Mallet$^{5}$, S. Marnieros$^{2}$, O. Martineau$^{3}$, N. Mirabolfathi$^{2}$, L. Mosca$^{5}$, L. Miramonti$^{5}$, X.-F. Navick$^{5}$, G. Nollez$^{7}$, P. Pari$^{4}$, M. Stern$^{3}$, L. Vagneron$^{3}$
[$^{1}$Centre de Recherche sur les Très Basses Températures, SPM-CNRS, BP 166, 38042 Grenoble, France\
$^{2}$Centre de Spectroscopie Nucléaire et de Spectroscopie de Masse, IN2P3-CNRS, Université Paris XI, bat 108, 91405 Orsay, France\
$^{3}$Institut de Physique Nucléaire de Lyon-UCBL, IN2P3-CNRS, 4 rue Enrico Fermi, 69622 Villeurbanne Cedex, France\
$^{4}$CEA, Centre d’Études Nucléaires de Saclay, DSM/DRECAM, 91191 Gif-sur-Yvette Cedex, France\
$^{5}$CEA, Centre d’Études Nucléaires de Saclay, DSM/DAPNIA, 91191 Gif-sur-Yvette Cedex, France\
$^{6}$Laboratoire Souterrain de Modane, CEA-CNRS, 90 rue Polset, 73500 Modane, France\
$^{7}$Institut d’Astrophysique de Paris, INSU-CNRS, 98 bis Bd Arago, 75014 Paris, France ]{}
[**Abstract**]{}
The EDELWEISS collaboration has performed a direct search for WIMP dark matter using a 320 g heat-and-ionization cryogenic Ge detector operated in a low-background environment in the Laboratoire Souterrain de Modane. No nuclear recoils are observed in the fiducial volume in the 30-200 keV energy range during an effective exposure of [4.53]{} kg$\cdot$days. Limits for the cross-section for the spin-independent interaction of WIMPs and nucleons are set in the framework of the Minimal Supersymmetric Standard Model (MSSM). The central value of the signal reported by the experiment DAMA is excluded at 90% CL.
[**Introduction**]{}
A general picture of matter and energy in the Universe is now emerging (see e.g. Ref. [@bib-review] for a review), suggesting that our Galaxy could be immersed in a halo of Dark Matter made of Weakly Interacting Massive Particles (WIMPs). The collision of a WIMP with an atomic nucleus would produce a nuclear recoil with a kinetic energy of the order of ten keV [@bib-sandl]. In the event that WIMPs are the neutralinos of the Minimal Supersymmetric extension of the Standard Model (MSSM), interaction rates per kilogram of matter would vary between 1 event per day to one per decade, depending on model parameters [@bib-mssm].
Experimental searches for these recoils in germanium ionization detectors [@bib-ge] and NaI scintillators [@bib-nai] have yielded upper limits on their rate per kilogram of detector material, which are interpreted in the framework of the MSSM in terms of limits on the WIMP-nucleon interaction cross-section. These searches are limited by the interaction rate due to natural radioactivity, which is at best limited to approximately 1 count/kg/day in the low energy range where recoils are expected.
Recently, the DAMA experiment has reported an annual modulation of the low-energy rate recorded in their $~100$ kg NaI detector array over a period of four years [@bib-dama]. This was attributed [@bib-dama] to the modulation of the WIMP flux impinging on the detector due to the Earth rotation around the Sun corresponding to a WIMP mass of 52$\pm^{10}_{8}$ GeV/c$^2$ and a WIMP-nucleon interaction cross-section of (7.2$\pm^{0.4}_{0.9}$) $\times$10$^{-6}$pb. In contrast, the CDMS collaboration [@bib-cdms] observed no excess of nuclear recoils above the rate expected from the scattering of cosmic-ray induced neutrons after accumulating an exposure of 10.6 kg$\cdot$days in the fiducial volume of their heat-and-ionization cryogenic germanium detectors. The two experimental results are not compatible if one applies the standard procedure to scale rates in different detectors described in ref. [@bib-sandl].
More data are needed to resolve definitely this discrepancy. The most exciting developments here are to be expected from the rapidly evolving domain of heat-and-ionization (or heat-and-scintillation [@bib-cresst]) cryogenic detector technology, which provides excellent event-by-event rejection of the dominating $\gamma$-ray background. The EDELWEISS collaboration has recently commissioned a massive (320 g) heat-and-ionization Ge detector [@bib-navick]. We report here on the first physics results obtained with this detector in a low-background environment in the underground site of the Laboratoire Souterrain de Modane (LSM). In this deep-underground experiment, the cosmic-ray induced neutron background that limited the recent CDMS results (one nuclear scattering per kg$\cdot$day above 10 keV recoil energy [@bib-cdms]) should be reduced by orders of magnitude. The CDMS and EDELWEISS detectors differ by their mass, geometry and electrode implantation scheme, and a comparison of their performance will benefit the development of this novel technology. The results presented in this letter represent a significant improvement relative to our previous results [@bib-distefano; @bib-benoit], obtained with a 70 g detector and with higher background levels.
[**Experimental Setup**]{} The experimental site is the Laboratoire Souterrain de Modane in the Fréjus Tunnel under the French-Italian Alps. The 1780 m rock overburden (4800 m water equivalent) results in a muon flux of about 4 m$^{-2}$day$^{-1}$ in the experimental hall and the flux of neutrons in the 2-10 MeV range is .
The detector is mounted in a dilution cryostat shielded from the radioactive environment by 10 cm of copper and 15 cm of lead [@bib-cryostat]. Pure nitrogen gas is circulated around the cryostat in order to reduce radon accumulation. The radioactivity of all material in the close vicinity of the detectors was measured using a dedicated low-background germanium $\gamma$-ray detector, also at the LSM. All electronic components were moved away from the detector and hidden behind a 7 cm thick archeological lead shield. The entire setup is surrounded by a 30 cm thick paraffin shielding against neutrons. According to Monte Carlo simulations of the various shields based on the measured neutron flux in the experimental hall, the rate of neutron scattering events producing nuclear recoils above 30 keV is expected to be of the order of 0.03 per kg and per day.
The detector [@bib-navick] is a cylindrical Ge single crystal with a diameter of 70 mm and a thickness of 20 mm. The edges have been beveled at an angle of 45$^o$. The plane surfaces and chamfers have been metalized for ionization measurement. The electrodes are made of 100 nm Al layers sputtered on the surfaces after etching. The top electrode is divided in a central part and a guard ring, electrically decoupled for radial localization of the charge deposition. During data taking, a voltage $V_o =$ 6.37 V was applied to the top electrodes. The electrodes were regularly shorted in order to prevent charge accumulation due to trapping of carriers in the detector volume.
The cross-talk between the centre and guard ring electrode signal is approximately 10%. This does not affect the ionization energy resolution since the cross-talk amplitude is a fixed fraction of the signal amplitude on the other electrode. Its shape is constant in time and this effect is easily taken into account by the simultaneous analysis of the signals recorded on both electrodes.
The thermal sensor consists of a Neutron Transmutation Doped germanium crystal (NTD) of 4 mm
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---
abstract: 'Traditional 3D convolutions are computationally expensive, memory intensive, and due to large number of parameters, they often tend to overfit. On the other hand, 2D CNNs are less computationally expensive and less memory intensive than 3D CNNs and have shown remarkable results in applications like image classification and object recognition. However, in previous works, it has been observed that they are inferior to 3D CNNs when applied on a spatio-temporal input. In this work, we propose a convolutional block which extracts the spatial information by performing a 2D convolution and extracts the temporal information by exploiting temporal differences, i.e., the change in the spatial information at different time instances, using simple operations of shift, subtract and add without utilizing any trainable parameters. The proposed convolutional block has same number of parameters as of a 2D convolution kernel of size $n\times n$, i.e. $n^2$, and has $n$ times lesser parameters than an $n\times n \times n$ 3D convolution kernel. We show that the 3D CNNs perform better when the 3D convolution kernels are replaced by the proposed convolutional blocks. We evaluate the proposed convolutional block on UCF101 and ModelNet datasets. All the codes and pretrained models will be publicly available at \_.'
author:
- Gagan Kanojia
- Sudhakar Kumawat
- Shanmuganathan Raman
title: Exploring Temporal Differences in 3D Convolutional Neural Networks
---
Introduction {#sec:intro}
============
Lately, 3D convolutional neural networks are gaining popularity over the 2D CNNs when the task is to deal with 3D data representations which could be videos, shapes or other formats [@hara2018can; @tran2017convnet]. This is because 2D CNN lack in exploiting the temporal information. 3D CNNs are more proficient than 2D CNNs in extracting temporal information and utilizing it to perform specific tasks. It has been shown that a 3D CNN of same depth as that of a 2D CNN performs better on tasks like action recognition [@hara2018can; @tran2018closer]. However, this proficiency comes with a cost in terms of the number of learnable parameters, memory requirements, and risks of overfitting. For example, 3D ResNet (18 layers) [@hara2018can] has around 3 times more parameters than the 2D ResNet (18 layers) [@he2016deep].\
In this work, our focus is on acquiring both spatial and temporal structure of the 3D data while reducing the cost in terms of trainable parameters. We propose a convolutional block which exploits both the spatial information and the temporal information by utilizing a 2D convolution and temporal differences, i.e., the change in the spatial information at different time instances, using simple operations of shift, subtract and add. We have also incorporated temporal max pooling in order to downsample the temporal depth of the feature maps along the depth of the network. None of the operations other than 2D convolution require trainable parameters which makes the number of trainable parameters of the proposed convolutional block equal to the 2D convolution kernel with same kernel size. The major contributions of the work are as follows. **(a)** We propose a novel convolutional block which captures spatial information by performing a 2D convolution and captures temporal information using simple operations of shift, subtract and add. **(b)** We reduce the number of parameters by $n$ times by replacing the 3D convolution kernel of size $n\times n\times n$ with the proposed convolution block comprising a 2D convolution kernel of size $1\times n\times n$. **(c)** We show that the proposed convolutional block helps the 3D CNNs to perform better while utilizing lesser parameters than the 3D convolution kernels.
Related work {#sec:related}
============
In recent years, 2D CNNs have been dominating several applications of computer vision like object detection [@he2016deep] and image classification[@he2016deep]. However, they lack in extracting the temporal information present in the spatio-temporal data [@tran2018closer]. There are works which extend the 2D CNNs on videos by processing the video frames individually and then combining the extracted information along the temporal dimension to obtain the output [@xu2015discriminative; @girdhar2017actionvlad]. Recently, 3D CNNs have shown great potential in dealing with the spatio-temporal data or 3D CAD models as inputs [@tran2015learning; @zhi2017lightnet; @maturana2015voxnet]. It has been observed that 3D CNNs are much better in exploiting the temporal information than 2D CNNs[@tran2018closer]. However, 3D CNNs are computationally expensive and they are prone to overfit due to their large number of parameters. Hence, the researchers moved on to find better and more efficient ways of mimicking 3D convolutions. There has been notable advances in the separable convolutions in 2D CNNs to reduce the space-time complexity [@sandler2018mobilenetv2; @chollet2017xception; @xie2017aggregated]. In many works, the idea of separable convolutions has been extended to 3D CNNs [@sun2015human; @xie2018rethinking; @qiu2017learning; @tran2018closer]. In [@qiu2017learning], the authors proposed the idea of replacing the 3D convolution kernel by a 2D convolution kernel to capture the spatial information followed by a 1D convolution kernel to convolve along the temporal direction. They showed that the proposed technique has several advantages, like parameter reduction and better performance, over the 3D convolutions, which has been further explored in [@tran2018closer]. Temporal differences has been explored in few recent works [@wang2016temporal; @lee2018motion]. Wang *et al.* [@wang2016temporal] use difference in two frames as the approximation of motion information. Similarly, Lee *et al.* [@lee2018motion] propose a motion block which extracts features using spatial and temporal shifts. In this work, we only rely on the temporal differences. Instead of relying on only the adjacent frames, we compute aggregated temporal differences over several frames. The proposed SSA Layer does not involve any trainable parameter to extract temporal information via temporal differences. Our focus is to propose an efficient alternative to the 3D convolution filters which utilizes lesser parameters without compromising the performance.
Proposed Approach {#sec:conv_block}
=================
In this section, we discuss the proposed convolutional block which extracts both spatial and temporal information. The proposed convolutional block has three parts: 2D convolution kernel, SSA layer, and temporal pooling layer. Here, SSA stands for Shift, Subtract and Add. Let the input to the proposed convolutional block be $\mathcal{X} \in \mathbb{R}^{c\times f \times h \times w}$. Here, $\mathcal{X}$ is the output feature maps of the previous convolutional block or layer, $c$ is number of channels, $f$ corresponds to the temporal depth, and $h$ and $w$ are the height and width of $\mathcal{X}$, respectively.\
**2D convolution.** In traditional 3D CNNs, the feature maps are convolved with a 3D filter $\hat{g} \in \mathbb{R}^{c\times k \times k \times k}$ with $c$ channels and kernel size $k \times k \times k$ [@hara2018can]. In the proposed framework, first we obtain $\mathcal{X}_c = \mathcal{X}\star g$. Here, $\star$ stands for convolution, and $g$ is a 2D filter of kernel size $1\times k \times k$ and $c$ channels. The purpose of the 2D convolution is to extract the spatial information present in the input feature maps[@zeiler2014visualizing]. We, then, pass $\mathcal{X}_c$ through the proposed SSA layer to obtain the temporal structure of the feature maps.\
**SSA Layer.** SSA stands for Shift, Subtract and Add operations performed in the SSA layer. The purpose of the SSA layer is to extract the temporal information present in the spatio-temporal data. For example, in action recognition, motion features extracted from the videos can hold important information. In order to capture the motion information, optical flow techniques can be used [@dosovitskiy2015flownet]. However, capturing optical flow is in itself a computationally expensive task which can require a dedicated network [@dosovitskiy2015flownet]. In the proposed SSA layer, we rely on temporal differences, i.e., the change in the spatial information at different time instances, to extract the necessary temporal information present in the spatio-temporal data.In the case of action recognition, temporal differences can provide the rough extimate of the location of moving objects or non-rigid bodies [@park2013exploring]. However, there is a possibility that there has not been enough change occurred in the adjacent frames. Hence, we take multiple frames into the consideration. The difference could be due to motion like in the case of action recognition or due to the structure of the input, like in the case of shapes. This makes the SSA layer to be used in a more general sense.\
Let the input to the SSA layer be $\mathcal{X}_c \in \mathbb{R}^{c\times f \times h \times w}$. Here, $c$ is the number of channels, $f$ is the temporal depth, and $h$ and $w$ are the height and width of $\mathcal{X}_c$, respectively. We obtain the temporal differences between the
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We report the results from a new, highly sensitive ($\Delta T_{mb} \sim 3 $mK) survey for thermal OH emission at 1665 and 1667 MHz over a dense, 9 x 9-pixel grid covering a $1\degr \times 1\degr $ patch of sky in the direction of $l = 105\fdg00, b = +2\fdg50$ towards the Perseus spiral arm of our Galaxy. We compare our Green Bank Telescope (GBT) 1667 MHz OH results with archival (1-0) observations from the Five College Radio Astronomy Observatory (FCRAO) Outer Galaxy Survey within the velocity range of the Perseus Arm at these galactic coordinates. Out of the 81 statistically-independent pointings in our survey area, 86% show detectable OH emission at 1667 MHz, and 19% of them show detectable CO emission. We explore the possible physical conditions of the observed features using a set of diffuse molecular cloud models. In the context of these models, both OH and CO disappear at current sensitivity limits below an A$_{\rm v}$ of 0.2, but the CO emission does not appear until the volume density exceeds 100-200 . These results demonstrate that a combination of low column density A$_{\rm v}$ and low volume density $n_{H}$ can explain the lack of CO emission along sight lines exhibiting OH emission. The 18-cm OH main lines, with their low critical density of $n^{*}$ $ \sim 1 $ , are collisionally excited over a large fraction of the quiescent galactic environment and, for observations of sufficient sensitivity, provide an optically-thin radio tracer for diffuse H$_2$.'
author:
- 'Michael P. Busch'
- 'Ronald J. Allen'
- 'Philip D. Engelke'
- 'David E. Hogg'
- 'David A. Neufeld'
- 'Mark G. Wolfire'
bibliography:
- 'Mendeley.bib'
title: |
The Structure of Dark Molecular Gas in the Galaxy - II.\
Physical State of ”CO-Dark” Gas in the Perseus Arm
---
[^1]
Introduction
============
The formation of molecular hydrogen gas from atomic hydrogen in galaxies is widely considered to be a critical step for the formation of new stars from the interstellar gas, and hence it is one of the most important processes that occurs in the interstellar medium (ISM). Unfortunately, as a symmetric molecule without a dipole moment, is practically invisible in emission at the temperature range of 10-100K expected for the bulk of the ISM in the Galaxy, and indirect estimates are required that make use of surrogate tracers. The most universally-accepted surrogate tracer for in the ISM is the lowest-energy rotational spectral line of (1-0) at $\lambda = 3 $mm. This line is relatively bright and easily observed, often with instrumentation designed specifically for that purpose. CO observations are commonly used jointly with an empirically-derived conversion factor, usually called the “X-factor” [see @Bolatto2013TheFactor for a review]. The strength of the 3-mm CO line emission is measured in units of K $\kmps$ and, by multiplying this line strength by the X-factor, one directly obtains an estimate of the column density.
A growing body of observational evidence points to an extra component of the ISM not traced by either the 21-cm line or (1-0) emission. This excess component is usually referred to as “dark gas” [@Grenier2005UnveilingNeighborhood; @Wolfire2010TheGas]. This dark gas may be a large fraction of the total molecular gas content in the Galaxy [@Pineda2013AComponents; @Li2015QuantifyingGas]. Quantifying how much dark gas exists is therefore of great interest in an effort to calibrate the X-factor and work towards a single prescription for scaling between a molecular tracer line emission and an accurate total molecular mass of . @Wolfire2010TheGas constructed models of molecular cloud surfaces and determined that dark gas in molecular cloud surfaces can amount to $\sim 30\%$ of the total molecular mass of the cloud. Here, the column density is large enough so that has sufficient column to remain self-shielded against the ambient UV flux, but CO is photodissociated and the carbon is the form of C or C$^{+}$. However, a large fraction of the faint CO gas might also arise in the diffuse ISM [@Papadopoulos2002MolecularDistances; @Liszt2010TheGas; @Allen2012Faint5circ; @Allen2015The+1deg; @Xu2016EvolutionTaurusb].
Initially discovered in absorption at centimeter radio wavelengths in the diffuse ISM[^2], OH has also recently been detected in the far-IR [@Wiesemeyer2016Far-infraredClouds], also in absorption. The work reported here mainly concerns new 18-cm emission observations in the outer Galaxy. [@Allen2012Faint5circ] reported faint and widespread 18-cm OH emission in a blind survey of a small region in the second quadrant of the Galactic plane using the 25-m radio telescope at Onsala, Sweden. However, owing to spectrometer limitations and radio interference in the spectra, the Onsala blind survey was limited to the 1667 MHz OH line and to within 2 kpc of the Sun. Similar results were presented by [@Dawson2014SPLASH:Region] with the SPLASH survey using the Parkes Telescope; however, their survey of OH encountered high levels of background synchrotron continuum emission from the inner Galaxy at levels approaching the typical excitation temperatures of the 18-cm OH lines, effectively suppressing the extended OH emission outside of the ‘CO-bright’ clouds. In the outer Galaxy, we can typically avoid this issue as the integrated background synchrotron emission is much weaker. More recent observations in the outer Galaxy by @Allen2015The+1deg [hereafter Paper 1] using the GBT have shown main-line (1665 and 1667 MHz) OH emission to be present in regions largely devoid of CO emission, strongly suggesting that OH may be a good tracer of the dark gas. The main lines of OH emission in these regions are observed to be in the 5:9 ratio characteristic of optically-thin emission lines with level populations in local thermodynamic equilibrium (LTE). This makes calculating a column density of OH relatively straightforward as long as a good estimate for the excitation temperatures can be found. With an OH column density known from emission observations, the column density can be directly estimated using the N()/N(OH) ratio of approximately $10^{-7}$ measured from UV absorption data [@Weselak2010TheMolecules; @Nguyen2018Dust-GasISM; @Engelke2018OHW5].
In this paper we present the results of a densely-sampled, blind, highly-sensitive GBT emission survey of the two main OH lines in a one-square-degree field of the Perseus Arm. The purpose of this survey is to further explore the apparent connection of extended OH emission to the so-called ’dark molecular gas’, and attempt to resolve structures of diffuse molecular gas on a large scale that are otherwise invisible to CO observations.
In Paper 1, we were able to compare sparse OH survey measurements taken with the GBT (FWHM $\sim$ 7.6’) with spectra provided directly from archives of the CFA CO survey [@Dame2001TheSurvey], since both data sets were observed at closely similar angular resolution ($\sim 7.6'$ for the GBT vs $\sim 8.4'$ for the CfA telescopes). However, the Five College Radio Astronomy Observatory (FCRAO) “Outer Galaxy Survey” [@Heyer1998TheGalaxy] has significantly higher angular resolution ($\sim 50\arcsec$) than the CFA survey, and hence significantly better sensitivity when smoothed to the GBT beam. We have therefore compared our OH data with a smoothed version of the FCRAO data. This allows for a direct, dense, observational comparison between these two important molecular tracers with comparable sensitivities for the first time.
It was observed in Paper 1 that OH emission was widespread, and that there was varying structure to the emission profiles on scales of $\sim$ 30 pc at the distance of the Perseus Arm feature. In an attempt to resolve the structure of this emission a new observing program was undertaken to increase the coverage of the original 2015 sparse survey. In this paper we therefore chose to restrict our analysis to the Perseus Arm feature in particular in an attempt to map the OH emission spatially and compare it to the CO emission at the $\sim$ 7 pc scale, as we show in Sect. \[morphology\]. Due to the absence of the kinematic distance ambiguity in the outer Galaxy, we are able to differentiate components of the OH spectra (see Fig. \[fig:exampleSpectrum\]). The distance of the Perseus Arm offered us the best consistently apparent option to measure this structure. The availability of accurate parallax distances in this direction (see Sect. \[distancetoarm\]) made this spatial comparison possible. Statistical and spatial comparisons of OH and CO emission between the local and inter-arm features would also be interesting and will be the subject of future papers.
Observations and Data {#Observations}
=====================
We carried out highly-sensitive observations of main-line OH emission at 18 cm with the Robert C. Byrd Green Bank Telescope (GBT) in project AGBT14B\_031 at the 81 locations indicated with small circles in Fig.\[fig:osdArea\]. For comparison with the OH data, we used the (1-0)
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'Cluster structure detection is a fundamental task for the analysis of graphs, in order to understand and to visualize their functional characteristics. Among the different cluster structure detection methods, spectral clustering is currently one of the most widely used due to its speed and simplicity. Yet, there are few theoretical guarantee to recover the underlying partitions of the graph for general models. This paper therefore presents a variant of spectral clustering, called $\ell_1$-spectral clustering, performed on a new random model closely related to stochastic block model. Its goal is to promote a sparse eigenbasis solution of a $\ell_1$ minimization problem revealing the natural structure of the graph. The effectiveness and the robustness to small noise perturbations of our technique is confirmed through a collection of simulated and real data examples.'
author:
- |
[Champion Camille$^{1}$, Blazère Mélanie$^1$, Burcelin Rémy$^2$, Loubes Jean-Michel$^1$, Risser Laurent$^3$]{}\
[$^1$ Toulouse Mathematics Institute (UMR 5219)]{}\
[University of Toulouse F-31062 Toulouse, France]{}\
[$^2$ Metabolic and Cardiovascular Diseases Institute (UMR 1048)]{}\
[University of Toulouse F-31432 Toulouse, France]{}\
[$^3$ Toulouse Mathematics Institute (UMR 5219)]{}\
[CNRS F-31062 Toulouse, France]{}
bibliography:
- 'example\_paper.bib'
title: Robust spectral clustering using LASSO regularization
---
[*Keywords:*]{} Spectral clustering, community detection, eigenvectors basis, $\ell_1$-penalty.
Introduction {#section1}
============
Graphs play a central role in complex systems as they can conveniently model interactions between the variables of a system. Finding variable sets with similar attributes can then help understanding the mechanisms underlying a complex system. Graphs are commonly used in a wide range of applications, ranging from Mathematics (graph theory) to Physics [@Hopfield82], Social Networks [@Handcock10], Informatics [@Pastor07] or Biology [@Jeong00; @Meunier09]. For instance, in genetics, groups of genes with high interactions are likely to be involved in a same function that drives a specific biological process.
One of the most relevant features when analyzing graphs is cluster structures. Clusters are generally defined as connected subsets of nodes that are more densely connected to each other than to the rest of the graph. Different strategies make it possible to define more specifically variable clusters depending on whether this property of vertices is considered locally (on a connected subset of vertices) or globally (on the whole network). First, cliques (subset of vertices such that every two distinct vertices in the clique are adjacent)[@Wasserman94], n-clique (maximal subgraph such that the distance of each pair of its vertices is not larger than n) [@Wasserman94] and k-core (maximal connected subgraph of G in which all vertices have degree at least k) [@Seidman83] characterize local cluster structure. Secondly, one of the global cluster structure definition is based on the notion of modularity [@Newman04; @Newman06] that quantifies the extent to which the fraction of the edges that fall within the given groups differs from the expected fraction if edges were distributed at random. The most popular random model is proposed by [@Newman04], where edges are reconnected randomly, under the constraint that the expected degree of each vertex corresponds to the degree of the vertex in the original graph. The last definition of cluster structure, and the most natural is related to similarity between each pair of vertices, that includes local or global definitions of a cluster structure. It is really natural to assume that cluster structures are groups of vertices that are close to each other. Similarity measures are the foundations of traditional methods as detailed below. These include traditional distance measures such as Manhattan or Euclidean distances or computing correlations between rows of the adjacency matrix or random walk based similarities [@Pons05].
Once the definition of cluster structure is fixed, it is crucial to build efficient procedures and algorithms for the identification of such structures in the network. The ability to find and to analyze such groups can provide an invaluable help in understanding and visualizing the functional components of the whole graph [@Girvan02; @Newman04]. Classical techniques for data clustering, like hierarchical clustering, partitioning clustering and spectral clustering, detailed below, are sometimes adopted for graph clustering too. Hierarchical clustering [@Hastie01] builds a hierarchy of nested clusters organized as a tree. partitioning clustering [@Pothen97] decomposes the graph into a set of disjoint clusters. Given $N$ variables/nodes, it builds $k$ partitions of the data by satisfying: (i) each group contains at least one point (ii) each point belongs to exactly one group. In recent years, spectral clustering has become one of the most widely used methods due to its speed and simplicity [@Luxburg07; @Chung97; @Ng02; @Fortunato10]. This method extracts the geometry and local information of the dataset by computing the top or bottom eigenvectors of specially constructed matrices. The observations are projected into this eigenspace to reduce the dimensionality of the problem and $k$-means procedure is then applied in an easier subspace to detect clusters.
$k$-means, that belongs to partitional clustering methods, aims to find a set of $k$ cluster centers of a dataset such that the sum squared of distances of each point to its closest cluster center is minimized. Lloyd’s 1957 procedure [@Lloyd82] remains one of the widely used because of its speed and simplicity. It has been studied for several decades [@Lloyd82; @Wu08] and many versions of this technique has recently been developed. [@Xu19] proposed alternatives to Lloyd’s algorithm that preserves its simplicity, makes it more robust to initialization and relieves its tendency to get trapped by local minima. [@Lattanzi19] developed a new variant of $k$-means++ seeding algorithm [@Arthur07] to achieve a constant approximation guarantee.
**Our contribution.** Observed real networks differ from random graphs from their edge distribution and from their underlying structures. Erdös Renyi random graphs models [@Erdos59], where all the pairs of nodes have equal probability of being connected by an edge, independently of all other pairs fail to model real observed graphs. Additionnally, stochastic block models are not always relevant to infer their structures. To remedy this problem, we developed a new random model, closely related to stochastic block model, but better suited to model graphs that have been inferred from the observations. In practice, graphs that are studied are not known beforehand but often estimated.To achieve a good clustering recovery, random graph models are often associated to their similarity matrix to maintain the clustering structure of the graph. [@Wang16] developed a model to learn a doubly stochastic matrix which encodes the probability of each pair of data points to be connected, used to normalize the affinity matrix such that the data graph is more suitable for clustering tasks. [@Peng15] has shown that for a wide class of graphs, spectral clustering gives a good approximation of the optimal cluster. In our model, we assume that a group does not emerge by chance but because there exists an underlying structure. This randomized version of the deterministic graph with exact cluster structure, is used to check whether it displays the original cluster structure. [@Sussman12; @Lei15; @Rohe11] proved consistency of spectral clustering applied to stochastic block models for some specific adjacency type matrices. Even if the consistency of spectral clustering has been proved for stochastic block models, there is no convergence guarantee for general models. Thus, $k$-means can fail to reach the true underlying partitions of the graph. Moreover, spectral clustering technique fails to recover the original clusters when it comes to a higher randomization coefficient. This is mostly due to the computed eigenbasis that is not equally informative.
In order to tackle this issue, we develop an alternative method to the spectral clustering that promote a sparse eigenvectors basis solution of an $\ell_0$ optimization problem, corresponding to the indicator vectors of each cluster. Since the natural constrained $\ell_0$ is a NP-hard problem, it was then replaced by its convex relaxation $\ell_1$ [@Ramirez13]. Actually we can show that the solution of the $\ell_0$ optimization problem is still the same when replacing the $\ell_0$-norm by the $\ell_1$-norm if we add a constraint on the maximum of the coefficients. Hence, the algorithm turns out to solve an $\ell_1$-penalty optimization problem that is feasible and easy to implement, even for very large graphs. In a wider scope, research papers have explored differently regularized spectral clustering to robustly identify clusters in large networks. Although [@Zhang18] and [@Joseph16] show the effect of regularization on spectral clustering through graph conductance and respectively through stochastic block models. Equally, [@Lara19], shows on a simple block model that the spectral regularization separates the underlying blocks of the graph.
In this paper, we introduce in Section \[section2\] a new random graph model, used to solve spectral clustering (Section \[section3\]) and its new variant (Section \[section4\]) objective function. We prove the efficiency and accuracy of the variant algorithm in Section \[section5\] through experiments on simulated and real medical dataset (Section \[section6\]).
New random graph model {#section2}
======================
Notations
---------
This work considers the framework of an unweighted undirected graph $G(V,E)$ with no self-loops consisting of vertices $V=\left\{ 1
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We theoretically investigate heat transport in temperature-biased Josephson tunnel junctions in the presence of an in-plane magnetic field. In full analogy with the Josephson critical current, the phase-dependent component of the heat flux through the junction displays *coherent diffraction*. Thermal transport is analyzed in three prototypical junction geometries highlighting their main differences. Notably, minimization of the Josephson coupling energy requires the quantum phase difference across the junction to undergo $\pi$ *slips* in suitable intervals of magnetic flux. An experimental setup suited to detect thermal diffraction is proposed and analyzed.'
author:
- 'F. Giazotto'
- 'M. J. Martínez-Pérez'
- 'P. Solinas'
title: Coherent diffraction of thermal currents in Josephson tunnel junctions
---
Introduction {#intro}
============
The impressive advances achieved in nanoscience and technology are nowadays enabling the understanding of one central topic in science, i.e., *thermal flow* in solid-state nanostructures [@Giazotto2006; @Dubi2011]. Control and manipulation [@heattransistor; @ser] of thermal currents in combination with the investigation of the origin of dissipative phenomena are of particular relevance at such scale where heat deeply affects the properties of the systems, for instance, from *coherent caloritronic* circuits, which allow enhanced operation thanks to the quantum phase [@Meschke2006; @Vinokur2003; @Eom1998; @Chandrasekhar2009; @Ryazanov1982; @Panaitov1984; @virtanen2007; @Martinez2013], to more developed research fields such as ultrasensitive radiation detectors [@Giazotto2006; @Giazotto2008] or cooling applications [@Giazotto2006; @Giazotto2002]. In this context it has been known for more than $40$ years that heat transport in Josephson junctions can be, in principle, phase-dependent [@Maki1965; @Guttman97; @Guttman98; @Zhao2003; @Zhao2004; @Golubev2013]. The first ever Josephson thermal interferometer has been, however, demonstrated only very recently [@giazotto2012; @martinez2012; @giazottoexp2012; @simmonds2012], therefore proving that phase coherence extends to thermal currents as well. The heat interferometer of Ref. [@giazottoexp2012] might represent a prototypical circuit to implement novel-concept coherent caloritronic devices such as heat transistors [@martinez2012], thermal splitters and rectifiers [@Martinez2013].
In the present work we theoretically analyze heat transport in temperature-biased extended Josephson tunnel junctions showing that the phase-dependent component of thermal flux through the weak-link interferes in the presence of an in-plane magnetic field leading to *heat diffraction*, in analogy to what occurs for the Josephson critical current. In particular, thermal transport is investigated in three prototypical *electrically-open* junctions geometries showing that the quantum phase difference across the junction undergoes $\pi$ *slips* in order to minimize the Josephson coupling energy. These phase slips have energetic origin and are not related to fluctuations as conventional phase slips in low-dimensional superconducting systems [@Langer1967; @Zaikin1997; @astafiev2012]. We finally propose how to demonstrate thermal diffraction in a realistic microstructure, and to prove such $\pi$ slips exploiting an uncommon observable such as the heat current.
The paper is organized as follows: In Sec. \[model\] we describe the general model used to derive the behavior of the heat current in a temperature-biased extended Josephson tunnel junction. In Sec. \[results\] we obtain the conditions for the quantum phase difference across an electrically-open short Josephson junction in the presence of an in-plane magnetic field, and the resulting behavior of the phase-dependent thermal current. In particular, we shall demonstrate the occurrence of phase-slips of $\pi$, independently of the junction geometry, in order to minimize the Josephson coupling energy. The phase-dependent heat current in three specific junction geometries is further analyzed in Sec. \[differentgeo\], where we highlight their main differences. In Sec. \[experiment\] we suggest and analyze a possible experimental setup suited to detect heat diffraction through electronic temperature measurements in a microstructure based on an extended Josephson junction, and to demonstrate the existence of $\pi$ slips. Finally, our results are summarized in Sec. \[summary\].
![(Color online) (a) Cross section of a temperature-biased extended S$_1$IS$_2$ Josephson tunnel junction in the presence of an in-plane magnetic field $H$. The heat current $J_{S_1\rightarrow S_2}$ flows along the $z$ direction whereas $H$ is applied in the $x$ direction, i.e., parallel to a symmetry axis of the junction. Dashed line indicates the closed integration contour, $T_i$, $t_i$ and $\lambda_i$ represent the temperature, thickness and London penetration depth of superconductor S$_i$, respectively, and $d$ is the insulator thickness. $\Phi$ denotes the magnetic flux piercing the junction. Prototypical junctions with rectangular, circular, and annular geometry are shown in panel (b), (c) and (d), respectively. $L$, $W$, $R$ and $r$ represent the junctions geometrical parameters. []{data-label="fig1"}](fig1.pdf){width="\columnwidth"}
Model
=====
Our system is schematized in Fig. \[fig1\](a), and consists of an extended Josephson tunnel junction composed of two superconducting electrodes S$_1$ and S$_2$ in thermal steady-state residing at different temperatures $T_1$ and $T_2$, respectively. We shall focus mainly on symmetric Josephson junctions in the *short* limit, i.e., with lateral dimensions much smaller than the Josephson penetration depth \[see Fig. \[fig1\](b,c,d)\], $L,W,R,r\ll \lambda_J=\sqrt{\frac{\pi \Phi_0}{\mu_0i_ct_H}}$, where $\Phi_0=2.067\times 10^{-15}$ Wb is the flux quantum, $\mu_0$ is vacuum permeability, $i_c$ is the critical current areal density of the junction, and $t_H$ is the junction effective magnetic thickness to be defined below. In such a case the self-field generated by the Josephson current in the weak-link can be neglected with respect to the externally applied magnetic field, and no traveling solitons can be originated. $t_i$ and $\lambda_i$ denote the thickness and London penetration depth of superconductor S$_i$, respectively, whereas $d$ labels the insulator thickness. We choose a coordinate system such that the applied magnetic field ($H$) lies parallel to a symmetry axis of the junction and along $x$, and that the junction electrodes planes are parallel to the $xy$ plane. Furthermore, the junction lateral dimensions are assumed to be much larger than $d$ so that we can neglect the effects of the edges, and each superconducting layer is assumed to be thicker than its London penetration depth (i.e., $t_i>\lambda_i$) so that $H$ will penetrate the junction in the $z$ direction within a thickness $t_H=\lambda_1+\lambda_2+d$ [@magneticlength]. For definiteness, we assume $T_1\geq T_2$ so that the Josephson junction is temperature biased only, and no electric current flows through it. If $T_1\neq T_2$ there is a finite electronic heat current $J_{S_1\rightarrow S_2}$ flowing through the junction from S$_1$ to S$_2$ \[see Fig. \[fig1\](a)\] which is given by [@Maki1965; @Guttman97; @Guttman98; @Zhao2003; @Zhao2004; @Golubev2013] $$J_{S_1\rightarrow S_2}(T_1,T_2,\varphi)=J_{qp}(T_1,T_2)-J_{int}(T_1,T_2)\textrm{cos}\varphi.
\label{heatcurrent}$$ Equation (\[heatcurrent\]) describes the oscillatory behavior of the thermal current flowing through a Josephson tunnel junction as a function of $\varphi$ predicted by Maki and Griffin [@Maki1965], and experimentally verified in Ref. [@giazottoexp2012]. In Eq. (\[heatcurrent\]), $J_{qp}$ is the usual heat flux carried by quasiparticles [@Giazotto2006; @Frank1997], $J_{int}$ is the phase-dependent part of the heat current which is peculiar of Josephson tunnel junctions, and $\varphi$ is the macroscopic quantum phase difference between the superconductors. By contrast, the Cooper pair condensate does not contribute to heat transport in a static situation [@Maki1965; @Golubev2013; @giazottoexp2012].
The two terms appearing in Eq. (\[heatcurrent\]) read [@Maki1965; @Guttman97; @Guttman98; @Zhao2003; @Zhao2004; @Golubev2013] $$J_{qp}=\frac{1}{e^2 R_J} \int^{\infty}_{0} d\varepsilon \varepsilon \mathcal{N}_1 (\varepsilon,T_1)\mathcal{N}_2 (\varepsilon,T_2)[f(\varepsilon,T_2)-f(\varepsilon,T_1)],$$ and $$J_{int}=\frac{
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We investigate the influence of the additional third level on the dynamic evolution of a Two-Level system interacting with a coherent field in the strong coupling regime where Rotating Wave Approximation is not valid. We find that the additional level has great influence on the evolution of the system population. Our results show that the Two-Level model is not a good approximation in this strong light-matter coupling regime. We further investigate the parameter space where the Two-Level model can still be justified.'
author:
- Faheel Ather Hashmi
- 'Shi-Yao Zhu'
title: 'The justification of the ‘Two-Level Approximation’ in strong light-matter interaction'
---
Introduction
============
‘Two-level approximation’ is a very convenient tool to study light-matter interaction because of its simplicity and because of its applicability to a large number of real light matter interaction scenarios. This approximation together with Rotating Wave Approximation (RWA) can be easily justified when two distinct levels of a system weakly interact with the light field at resonance or quasi-resonance. The other levels of the system are far apart in energy and can be ignored along with the ‘Counter Rotating Terms (CRT)’. However, RWA becomes invalid in strong light-matter coupling regime. This strong coupling regime is becoming more accessible for experiments [@niemczyk10; @forn10; @you11; @you13], and hence there is a surge of interest in going beyond RWA (taking into account the effects of CRT) in the description of the interaction [@casanova10; @albert12]. The breakdown of RWA and the role of CRT has also been reported [@larson12; @wang09]. In the strong coupling regime, the two-level system without RWA has been very extensively studied [@irish07; @larson07; @werlang08; @Braak11; @zubairy88; @Milonni83; @Spreeuw90]. In these studies the counter-rotating terms are taken into account, but the two-level approximation is still adopted. The counter-rotating terms involve the contribution proportional to $1/(\omega_{ab}+\omega_0)$ where $\omega_{ab}$ is the transition frequency of the system and $\omega_0$ is the frequency of the field. If there is an additional level of the system (say ${\left|c\right\rangle}$ with transition frequency $\omega_{ac}$), the rotating wave terms on this transition will result in the contribution proportional to $1/(\omega_{ac}-\omega_0)$, which can be larger than the contribution proportional to $1/(\omega_{ab}+\omega_0)$. Consequently, neglecting the additional levels is questionable. Therefore, in the strong coupling regime where CRT effects have to be taken into account, the question ‘whether the effects due to additional levels can be neglected?’ needs to be treated carefully. In this paper, we consider the effects of the third level on the population dynamics of a two-level system in strong coupling regime, and focus on when the third level can be neglected.
The three level system interacting with resonant or quasi-resonant fields beyond RWA has also been studied, mainly in the context of trapping dark state in $\Lambda$ configuration [@unanyan00; @ho85; @matisov95; @xiaohong12; @sanchez04]. However, these studies concerned with the effects of CRT terms on the three level system, and did not discuss the consequence of the third level on the evolution. In the present work we consider the three level ‘V-system’, where the third level is non-resonant and is far away in energy from the two-level transition, and study the effects of the third level on the population dynamics. Our focus is on when the effects of the third level will diminish in the strong coupling regime.
Model
=====
![A three level system in ‘V’ configuration. We are interested in the effects of the additional level ${\left|c\right\rangle}$ on the population dynamics of level ${\left|b\right\rangle}$.[]{data-label="fig:1"}](system)
Consider a three level system in ‘V’ configuration as shown in FIG. \[fig:1\] interacting with a single mode quantized field. The Hamiltonian of the system can be written as (with $\hbar=1$) $$\label{eqHamil}
\begin{aligned}
{\hat{H}}=& {\omega_{ab}}{{\left|b\right\rangle}{\left\langleb\right|}} + {\omega_{ac}}{{\left|c\right\rangle}{\left\langlec\right|}} + {\omega_{0}}{\hat{b}^{\dagger}}{\hat{b}}\\ & +{g_{ab}}{\left( {\hat{b}^{\dagger}}+ {\hat{b}}\right)} {\left[{{\left|a\right\rangle}{\left\langleb\right|}} + {{\left|b\right\rangle}{\left\langlea\right|}} \right]}
\\ & +{g_{ac}}{\left( {\hat{b}^{\dagger}}+ {\hat{b}}\right)} {\left[{{\left|a\right\rangle}{\left\langlec\right|}} + {{\left|c\right\rangle}{\left\langlea\right|}} \right]},
\end{aligned}$$ where ${\omega_{ab}}$ and ${\omega_{ac}}$ are the transition frequencies for the excited states and ${\hat{b}^{\dagger}}$, ${\hat{b}}$, and ${\omega_{0}}$ are the creation and annihilation operators and the frequency for the field. The coupling constants ${g_{ab}}$ and ${g_{ac}}$ are real. We choose ${\left|n,l\right\rangle}$ as the basis to write the system ket where $n$ is the number of photon and $l=\{a,b,c\}$ denotes the level of the system. The system ket can be written as $$\begin{aligned}
{\left|\Psi\right\rangle}= \sum_n^{\infty}\sum_{p=0}^{1} e^{-i{\hat{H_0}}t} a_{2n+p} {\left|2n+p,a\right\rangle} + b_{2n+p}{\left|2n+p,b\right\rangle} + c_{2n+p}{\left|2n+p,c\right\rangle}
\label{eqKet}\end{aligned}$$ In this expression ${\hat{H_0}}={\omega_{ab}}{{\left|b\right\rangle}{\left\langleb\right|}} + {\omega_{ac}}{{\left|c\right\rangle}{\left\langlec\right|}} + {\omega_{0}}{\hat{b}^{\dagger}}{\hat{b}}$ is the non-interacting part of the Hamiltonian in eq(\[eqHamil\]). The system ket eq(\[eqKet\]) actually consists of two independent kets that separately satisfy the Schrödinger equation. This is because of the fact that the Hamiltonian admits even and odd parity chains of evolution like the JCM Hamiltonian with CRT [@casanova10]. One such chain consists of the states with even number of photons in ${\left|a\right\rangle}$ and odd number of photons in ${\left|b\right\rangle}$ and ${\left|c\right\rangle}$, and the other chain has states with odd number of photons in ${\left|a\right\rangle}$ and even number of photons in ${\left|b\right\rangle}$ and ${\left|c\right\rangle}$. The time evolution of the amplitudes is given by $$\label{eqMaster}
\begin{aligned}
i \dot{a}_{2n+p} &= {g_{ab}}\left[\sqrt{2n+p} \, b_{2n+p-1} {e^{-i \Delta_{ab}t}}+ \sqrt{2n+p+1} \,b_{2n+p+1}{e^{-i \Delta_{ab}^{\prime}t}}\right]
\\ &+ {g_{ac}}\left[\sqrt{2n+p} \, c_{2n+p-1} {e^{-i \Delta_{ac}t}}+ \sqrt{2n+p+1} \,c_{2n+p+1}{e^{-i \Delta_{ac}^{\prime}t}}\right]
\\
i \dot{b}_{2n+p} &= {g_{ab}}\left[\sqrt{2n+p+1} \,a_{2n+p+1} {e^{i \Delta_{ab}t}}+ \sqrt{2n+p} \,a_{2n+p-1}{e^{i \Delta_{ab}^{\prime}t}}\right]
\\
i \dot{c}_{2n+p} &= {g_{ac}}\left[\sqrt{2n+p+1} \,a_{2n+p+1} {e^{i \Delta_{ac}t}}+ \sqrt{2n+p} \,a_{2n+p-1}{e^{i \Delta_{ac}^{\prime}t}}\right]
\end{aligned}$$ Here $\Delta_{ab}= \omega_{ab}-\omega_0$, $\Delta_{ab}^{\prime}=\omega_{ab}+\omega_0$, $\Delta_{ac}= \omega_{ac}-\omega_0$, and $\Delta_{ac}^{\prime}=\omega_{ac}+\omega_0$. $p$ in each equation is either $0$ for state ${\left|a\right\rangle}$ and $
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
author:
- 'Jan-e Alam'
title: In search of quark gluon plasma in nuclear collisions
---
[**Abstract**]{}\
At high temperatures and densities the nuclear matter undergoes a phase transition to a new state of matter called quark gluon plasma (QGP). This new state of matter which existed in the universe after a few microsecond of the big bang can be created in the laboratory by colliding two nuclei at relativistic energies. In this presentation we will discuss how the the properties of QGP can be extracted by analyzing the spectra of photons, dileptons and heavy flavours produced in nuclear collisions at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies.
Introduction
============
The theory of strong interaction - Quantum Chromodynamics (QCD) has a unique feature - it possess the property of asymptotic freedom which implies that at very high temperatures and/or densities nuclear matter will convert to a deconfined state of quarks and gluons [@collins]. Recent lattice QCD based calculations [@lqcd] indicate that the value of the temperature for the nuclear matter to QGP transition $\sim 175$ MeV. It is expected that such high temperature can be achieved in the laboratory by colliding nuclei at RHIC and LHC energies.
A high multiplicity system of deconfined quarks and gluons with power law type of momentum distributions can created just after the nuclear collisions at high energies. Interactions among these constituents may alter the momentum distribution of quarks and gluons from a power law to an exponential one - resulting in a thermalized state of quarks and gluons with initial temperature, $T_i$. This thermalized system with high internal pressure expands very fast as a consequence it cools and reverts to hadronic matter at a temperature, $T_c\sim 175$ MeV. The hadrons formed after the hadronization of quarks may maintain thermal equilibrium among themselves until the expanding system becomes too dilute to support collectivity at a temperature, $T_F (\sim 120$ MeV) called freeze-out temperature from where the hadrons fly freely from the interaction zone to the detector.
The electromagnetic (EM) probes [@mclerran] (see [@rapp; @alam1; @alam2] for review) [*i.e.*]{} real photons and dileptons can be used to follow the evolution of the system from the pristine partonic stage to the final hadronic stage through an intermediary phase transition or cross over. In the state of QGP some of the symmetries of the physical vacuum may either be restored or broken - albeit transiently. The electromagnetic probes, especially the lepton pairs can be used very effectively to investigate whether these symmetries in the system are restored/broken at any stage of the evolving matter. Results from theoretical calculations will be shown in the presentation to demonstrate this aspect of the electromagnetically interacting probes. We will demonstrate that lepton pairs can be used very effectively to probe the collective motion (radial and elliptic) of the system.
The other promising probe of the QGP that will be discussed here - is the depletion of the transverse momentum spectra of energetic quarks (and gluons) in QGP. The magnitude of the depletion can be used to estimate the transport coefficients of QGP which is turn can be used to understand the fluidity of the matter.
The transport coefficients of QGP and hot hadrons calculated by using perturbative QCD and effective field theory respectively have been applied to evaluate the nuclear suppression ($R_{\mathrm AA}$) of heavy flavours. Theoretical results on $R_{\mathrm AA}$ will be compared with the experimental data available from RHIC and LHC energies. The azimuthal asymmetry of the system estimated through the single leptons originating from the decays of open heavy flavours produced from the fragmentation of heavy quarks will also be discussed.
The electromagnetic probes
==========================
The dilepton production per unit four-volume from a thermal medium produced in heavy ion collisions is well known to be given by: &=&- L(M\^2)f\_[BE]{}(p\_0) g\^\
&&W\_(p\_0,[p]{}) \[eq1\] where the factor $L(M^2)=(1+{2m_l^2}/{M^2})~
(1-4m_l^2/M^2)^{1/2}~$ is of the order of unity for electrons, $M(=\sqrt{p^2})$ being the invariant mass of the pair and the hadronic tensor $W_{\mu\nu}$ is defined by W\_(p\_0,[p]{})=d\^4xe\^[ipx]{}\[eq2\] where $J^{em}_\mu(x)$ is the electromagnetic current and $\langle.\rangle$ indicates ensemble average. For a deconfined thermal medium such as the QGP, Eq. (\[eq1\]) leads to the standard rate for lepton pair productions from $q\bar q$ annihilation at lowest order.
The production of low mass dileptons from the decays of light vector mesons in the hadronic matter can be obtained as (see [@sabya] for details): &=&- f\_[BE]{}(p\_0) g\^\
&&\_[R=,,]{}K\_R \^R\_(p\_0,[p]{}) \[eq2\] where $f_{BE}$ is the thermal distributions for bosons, $\rho^R_{{\mu\nu}}(q_0,{\vec q})$ is the spectral function of the vector meson $R (=\rho,\omega,\phi)$ in the medium, $K_R=F_R^2 m_R^2$, $m_R$ is the mass of $R$ and $F_R$ is related to the decay of $R$ to lepton pairs.
The interaction of the vector mesons with the hadrons in the thermal bath will shift the location of both the pole and the branch cuts of the spectral function - resulting in mass modification or broadening - which can be detected through the dilepton measurements and may be connected with the restoration of chiral symmetry in the thermal bath. In the present work the interaction of $\rho$ with thermal $\pi$,$\omega$, $a_1$, $h_1$ [@sabya; @sabya2] and nucleons [@ellis] have been considered to evaluate the in-medium spectral function of $\rho$. The finite temperature width of the $\omega$ spectral function has been taken from [@Weise].
To evaluate the dilepton yield from a dynamically evolving system produced in heavy ion collisions (HIC) one needs to integrate the fixed temperature production rate given by Eq. \[eq2\] over the space time evolution of the system - from the initial QGP phase to the final hadronic freeze-out state through a phase transition in the intermediate stage. We assume that the matter is formed in QGP phase with zero net baryon density at temperature $T_i$ in HIC. Ideal relativistic hydrodynamics with boost invariance [@bjorken] has been applied to study the evolution of the system. The EoS required to close the hydrodynamic equations is constructed by taking results from lattice QCD for high $T$ [@lqcd] and hadron resonance gas comprising of all the hadronic resonances up to mass of $2.5$ GeV [@victor; @bmja] for lower $T$. The system is assumed to get out of chemical equilibrium at $T=T_{ch}=170$ MeV [@tsuda]. The kinetic freeze-out temperature $T_{F}=120$ MeV fixed from the $p_T$ spectra of the produced hadrons.
Invariant mass spectra of lepton pairs
--------------------------------------
The $M$ distribution of lepton pairs originating from quark matter (QM) and hadronic matter (HM) with and without medium effects on the spectral functions of $\rho$ and $\omega$ are displayed in Fig. \[fig1\]. We observe that for $M\,>\,M_{\phi}$ the QM contributions dominate. For $M_{\rho}\lesssim M\lesssim M_{\phi}$ the HM shines brighter than QM. For $M\,<M_{\rho}$, the HM (solid line) over shines the QM due to the enhanced contributions primarily from the medium induced broadening of $\rho$ spectral function. However, the contributions from QM and HM become comparable in this $M$ region if the medium effects on $\rho$ spectral function is ignored (dotted line). Therefore, the results depicted in Fig. \[fig1\] indicate that a suitable choice of $M$ window will enable us to unravel the contributions from a particular phase (QM or HM). An appropriate choice of $M$ window will also allow us to extract the medium induced effects.
To further quantify these points we evaluate the following [@payalv2]: $$\begin{aligned}
&&F=
\frac{\int^\prime
\left(\frac{dN}{d^4xd^2p_TdM^2dy}\right)dxdyd\eta\tau d\tau d^2p_TdM^2}
{\int\left(\frac{dN}{d^4xd^2p_TdM^2dy}\right)dxdyd\eta\tau d\tau d^2p_TdM^2}\nn
\label{eq3}\end{aligned}$$ where the $M$ integration in both the numerator and denominator are performed for selective windows from $M_1$ to $M_2$ with mean $M$ defined as $\langle M\rangle = (M_1+M_2)/2$. While in the denominator the integration is done over the entire lifetime
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'Vision-to-Language tasks aim to integrate computer vision and natural language processing together, which has attracted the attention of many researchers. For typical approaches, they encode image into feature representations and decode it into natural language sentences. While they neglect high-level semantic concepts and subtle relationships between image regions and natural language elements. To make full use of these information, this paper attempt to exploit the text-guided attention and semantic-guided attention to find the more correlated spatial information and reduce the semantic gap between vision and language. Our method includes two level attention networks. One is the text-guided attention network which is used to select the text-related regions. The other is semantic-guided attention network which is used to highlight the concept-related regions and the region-related concepts. At last, all these information are incorporated to generate captions or answers. Practically, image captioning and visual question answering experiments have been carried out, and the experimental results have shown the excellent performance of the proposed approach.'
author:
- 'Xuelong Li, Aihong Yuan, and Xiaoqiang Lu, [^1] [^2] [^3] [^4] [^5] [^6]'
bibliography:
- 'egbib.bib'
title: 'Vision-to-Language Tasks Based on Attributes and Attention Mechanism'
---
[Shell : 3G Structure for Image Caption Generation]{}
Image captioning, visual question answering, deep learning, multi-modal.
Introduction
============
ision-to-Language (V2L) tasks aim to integrate natural language processing and computer vision together. Typical V2L tasks are image captioning [@ATT2U; @DBLP:conf/aaai/LiuMSY17; @DBLP:conf/cits/QuLTL16; @Binqiang], visual question answering (VQA) [@DBLP:conf/icml/XiongMS16; @DBLP:conf/cvpr/ShihSH16; @TGIF-QA] and video description [@DBLP:conf/cvpr/DonahueHGRVDS15; @HVC; @TDVD; @DBLP:conf/ijcai/LiZL17; @DBLP:conf/mm/ZhaoLL17]. Recently, due to the advent of artificial intelligence (AI), V2L tasks have attracted extensive attention. Practically, V2L tasks enable many important applications, including early childhood education, human-robot interaction, visually impaired people assistance and so on.
Many recent approaches for V2L tasks have achieved a lot of gratifying results through combining Convolutional Neural Networks (CNNs) and Recurrent Neural networks (RNNs) for image encoding and text generating, respectively [@DBLP:conf/cvpr/KarpathyL15; @DBLP:conf/cvpr/VinyalsTBE15; @DBLP:journals/corr/MaoXYWY14a; @DBLP:conf/nips/RenKZ15]. Concretely, a CNN pre-trained on ImageNet [@DBLP:journals/ijcv/RussakovskyDSKS15] is used to extract global image feature while a RNN is used to encode the language information. Most of the recently approaches are belong to the “CNN-RNN” paradigm and these approaches have attained some promising results, further improvements should be got over some limitations.
Motivation and Overview
-----------------------
![Overview of our scheme for vision-to-language tasks. The proposed approach is composed of two level attention modules. The first attention module is text-guided attention (*i.e.*, word-guided attention for image captioning and question-guided attention for VQA) which is used to dig up the mapping relation between word and image region. Semantic-guided attention is the second attention module, which is used for explore the relationship between image and semantic concept. At last, the two parts of representations which output from the two attention modules are jointly embedded into multi-modal space to generate the corresponding captions or answers.[]{data-label="introduction"}](intro_1.pdf "fig:"){width="0.90\linewidth"}\
Image high-level semantic concepts (also called image attributes, *i.e.*, objects, actions, scenes and object’s attributes of images) are very important information for V2L tasks. In previous works, the spatial attention based methods (to distinguish the semantic-guided attention network, the spatial attention network is called text-guided attention network in this paper) are the most popular scheme for V2L tasks. Namely, the relationships between natural language elements with image regions via computing the attention weights between words/questions and image regions. The core idea of the spatial attention mechanism is that every word in captions or every question should only correspond to one or several regions of an image. Although the spatial attention based methods can dig up the subtle relationships between image and text elements, high-level semantic concepts have not been fully utilized for V2L tasks in most of the previous works, while these concepts are important for humans when understanding a scene [@MLAN; @DBLP:conf/cvpr/WuSLDH16]. Actually, the semantic concepts bridge images and natural language information together and can contribute significantly to eliminate the well-known semantic gap. That is because the semantic concepts are not only the important high-level visual information for images, but also the important component of captions. For example, image in Fig. \[introduction\] shows an brown cat. According to the semantic concepts in this paper, the words “*brown*” and “*cat*” are high-level information of the image, and they provide very important information for understanding the image. Meanwhile, the two words also the components of the corresponding captions. To eliminate the semantic gap between images and natural language, the high-level semantic information is an extra input of the proposed method.
Moreover, every high-level semantic concept should correlated to a specific image region. Some previous methods, such as [@DBLP:conf/cvpr/WuSLDH16; @TPAMI/Q], attempt to use the image semantic information to complete the V2L tasks, the semantic information is encoded into vectors and directly input into the language generating model. However, this process is not the optimal one because it cannot dig up the relationships between the semantic concepts and image regions. In this paper, a semantic-guided attention network is designed to explore the relationships between image semantic concepts and image regions. Namely, the image semantic concepts information is used to attend the corresponding regions. Other works like [@MLAN] use the attention mechanism for image semantic information, but the guidance information is natural language. In other words, the high-level semantic information is attended the by the corresponding caption or the question. Actually, every semantic concept is more correlated to a specific image regions. For instance, image in Fig. \[introduction\] shows a brown cat. Both the concept words “*cat*” and “*brown*” are important elements of the captions, while they correspond to the “cat” regions. So, exploring the relationships between image regions and image semantic concepts are more effective than the relationships between image semantic concepts and natural language elements.
Motivated by the aforementioned two reasons, we propose a methods with a semantic-guided attention network. The semantic-guided attention network contains two sub-parts which are used to highlight the concept-related regions and the region-related concepts, respectively. In addition, text-guided attention network is also reserved to explore the subtle relationships between image regions and natural language parts. For example, when describing the content of the image in Fig. \[introduction\], the phrase “*brown cat*” should map with the “*cat*” region of the image. For VQA task, the question is “*What is the cat on?*”. When answering this question, the “*cat*” and its surrounding regions should be focused on because these regions are most related to the question. So, to simultaneously learn the relationships between high-level semantic concepts and image regions and the correlations between natural language elements and image regions, we unify two sub-attention networks (semantic-guided attention network and tex-guided attention network) into a framework.
Fig. \[introduction\] shows the overall scheme of the proposed approach. The approach mainly includes two level attention networks. One is the text-guided attention network which is used to select text-related regions. The text-guided attention network has two variants for image captioning and VQA, respectively. In image captioning task, the text-guided attention network is called word-guided attention which is used to explore the relationships between words and image regions. In VQA task, the text-guided attention network is called question-guided attention which is used to select the image regions corresponding to the question. The other is the semantic-guided attention network which is used to dig up the relationship between image regions and high-level semantic concepts. The outputs of these two networks are projected into the same multi-modal space to generate captions or answers.
Contributions
-------------
The core contributions can be summarized as follows:
1\) An approach based on image high-level semantic attributes and local image features is proposed to address the challenges of V2L tasks. Specially, the high-level semantic attributes information is used to reduce the semantic gap between images and text.
2\) An novel semantic-guided attention network is designed to explore the mapping relationships between semantic attributes and image regions. The semantic-guided attention network highlights the concept-related regions and selects the region-related concepts.
3\) Two special V2L tasks (*i
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
author:
- 'Y. Shimajiri, Ph. Andr$\acute{\rm e}$, P. Palmeirim , D. Arzoumanian , A. Bracco , V. K$\ddot{\rm o}$nyves , E. Ntormousi , and B. Ladjelate'
bibliography:
- 'B211\_accretion.bib'
date: 'Received ; accepted '
title: |
Probing accretion of ambient cloud material\
into the Taurus B211/B213 filament
---
[$Herschel$ observations have emphasized the role of molecular filaments in star formation. However, the origin and evolution of these filaments are not yet well understood, partly because of the lack of kinematic information.]{} [We aim to confirm that Taurus B211/B213 filament is accreting background cloud material from a kinematic viewpoint and to investigate the potential influence of large-scale external effects on the formation of the filament. ]{} [To examine whether the B211/B213 filament is accreting background gas due to its gravitational potential, we produced a toy accretion model and compared its predictions to the velocity patterns observed in $^{12}$CO (1–0) and $^{13}$CO (1–0). We also examined the spatial distributions of H${\alpha}$, $Planck$ 857 GHz dust continuum, and HI emission to search for evidence of large-scale external effects.]{} [We estimated the depth of the Taurus cloud around the B211/B213 filament to be $\sim$0.3–0.7 pc under the assumption that the density of the gas is the same as the critical density of $^{13}$CO (1–0). Compared to a linear extent of > 10 pc in the plane of the sky, this suggests that the 3D morphology of the cloud surrounding the B211/B213 filament is sheet-like. Position-velocity ($PV$) diagrams observed in $^{12}$CO (1–0) and $^{13}$CO (1–0) perpendicular to the filament axis show that the emission from the gas surrounding B211/B213 is redshifted to the northeast of the filament and blueshifted to the southwest, respectively, and that the velocities of both components approach the velocity of the B211/B213 filament as the line of sight approaches the crest of the filament. The $PV$ diagrams predicted by our accretion model are in good agreement with the observed $^{12}$CO (1–0) and $^{13}$CO (1–0) $PV$ diagrams, supporting the scenario of mass accretion into the filament proposed by Palmeirim et al. Moreover, inspection of the spatial distribution of the H$\alpha$ and $Planck$ 857 GHz emission in the Taurus-California-Perseus region on scales up to >200 pc suggests that the B211/B213 filament as a result of an expanding supershell generated by the Per OB2 association. ]{} [Based on these results, we propose a scenario in which the B211/B213 filament was initially formed by large-scale compression of HI gas and then is now growing in mass due to the gravitational accretion of ambient cloud molecular gas.]{}
Introduction {#Sect1}
============
{width="170mm"}
\[fig:3color\]
![Map of $^{12}$CO (1–0) optical depth derived from the @Goldsmith08 $^{12}$CO (1–0) and $^{13}$CO (1–0) data.[]{data-label="fig:tau"}](tau.jpg){width="90mm"}
The observations of the [*Herschel*]{} Gould Belt survey (HGBS) have revealed an omnipresence of parsec-scale filaments in molecular clouds and emphasized their importance for solar-type star formation [e.g. @Andre10; @Menshchikov10; @Arzoumanian11; @Palmeirim13]. In particular, most $Herschel$ prestellar cores are found to lie in dense (thermally supercritical) filaments, suggesting that cores generally form by filament fragmentation [eg. @Konyves15; @Marsh16; @Benedettini18]. Molecular line observations of the velocity field around cores and filaments further support this view [@Tafalla15]. Based on the HGBS results, @Andre14 proposed a filament paradigm for star formation, whereby large-scale compression of interstellar material in supersonic flows generates a quasi-universal web of $\sim$0.1-pc wide filaments in the cold interstellar medium (ISM) and then the denser filaments fragment into prestellar cores by gravitational instability. Recently, @Shimajiri17 found that the star formation efficiency in dense molecular gas ($A_{\rm v}$ > 8), where filamentary structures dominate the mass budget, is remarkably uniform over a wide range of scales from 1-10 pc to >10 kpc [see also, @Gao04; @Lada10; @Lada12; @Chen15]. Furthermore, @Shimajiri17 proposed that this common star formation efficiency in dense gas results from the microphysics of star formation in filaments [see also @Andre14]. This result suggests the existence of a universal “star formation law” converting dense molecular gas into stars along filaments. Therefore, unveiling how molecular filaments grow in mass and fragment is crucial to understanding star formation in filaments.
The B211/B213 filament system is located in the Taurus molecular cloud, which is one of the nearest star-forming regions [$d$$\sim$140 pc, @Elias78]. Wide-field mapping observations in $^{12}$CO, $^{13}$CO, C$^{18}$O, N$_2$H$^+$, and SO emission revealed a whole network of filamentary structures in the B211/B213 area [@Goldsmith08; @Hacar13; @Panopoulou14; @Tafalla15]. @Goldsmith08 and @Palmeirim13 found that many low-density striations are elongated parallel to the magnetic field, and that blueshifted and redshifted components in both $^{12}$CO (1–0) and $^{13}$CO (1–0) emission are distributed to the southwest and the northeast of the B211/B213 filament, respectively, as shown in Fig. \[fig:3color\]. This morphology was suggestive of mass accretion along magnetic field lines into the B211/B213 filament. To quantify mass accretion, @Palmeirim13 assumed cylindrical geometry and used the observed mass per unit length $M_{\rm line}$ to estimate the gravitational acceleration $\phi$($R$) = 2$GM_{\rm line}/R$ of a piece of gas in free-fall toward the filament (where $R$ and $G$ denote radius from filament center and the gravitational constant, respectively). The free-fall velocity of gas initially at rest at a cylindrical radius $R_{\rm init}\sim$2 pc was estimated to reach 1.1 km s$^{-1}$ when the material reached the outer radius $R_{\rm out}\sim$0.4 pc of the B211/B213 filament. This estimation was consistent with the velocity observed in CO, suggesting that the background gas accretes into the B211/B213 filament owing to the gravitational potential of the B211/B213 filament. However, the velocity structure was not investigated in detail. Investigation of the velocity structure is crucial to confirm this suggested scenario from the kinematic viewpoint. This is the topic of the present paper.
The paper is organized as follows: in Sect. \[Sect2\], we describe the $^{12}$CO (1–0) and $^{13}$CO (1–0) data, as well as complementary H${\alpha}$, 857 GHz, and HI data. In Sect. \[Sect3\], we estimate the optical depth of the $^{12}$CO (1–0) line and present the $^{12}$CO (1–0) and $^{13}$CO (1–0) velocity structures observed in the B211/B213 cloud. In Sect. \[Sect4\], we discuss the cloud structure, whether the surrounding material accretes onto the B211/B213 filament from the kinematic viewpoint, and whether the filament is formed by large-scale compression. In Sect. \[Sect5\], we summarize our results.
{width="155mm"} {width="155mm"}
Observational data {#Sect2}
==================
In this paper, we used the $^{12}$CO (1–0) and $^{13}$CO (1–0) data obtained by @Goldsmith08 [@Narayanan08] with the 14 m diameter millimeter-wave telescope of the Five College Radio Astronomy Observatory (FCRAO). The half-power beam width of the telescope was 45$\arcsec$ for $^{12}$CO (1–0) and 47$\arcsec$ for $^{13}$CO (1–0). We applied Gaussian spatial smoothing to improve the signal to noise ratio, resulting in an effective beam resolution of $\sim$76$\arcsec$, corresponding to $\sim$0.05 pc at a distance of 140 pc. The velocity resolution of the data is 0.26 km s$^{-1}$ for $^{12}$CO (1–0) and 0.27 km s$^{-1}$ for $^{13}$CO (1–0). The rms noise level is 0.1 K ($T_{\rm A}^*$) for $^{12}$CO (1–0) and 0.05 K ($T
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"pile_set_name": "ArXiv"
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| null | null | null |
---
abstract: 'We formulate axisymmetric general relativity in terms of real Ashtekar–Barbero variables. We study the constraints and equations of motion and show how the Kerr, Schwarzschild and Minkowski solutions arise. We also discuss boundary conditions. This opens the possibility of a midisuperspace quantization using loop quantum gravity techniques for spacetimes with axial symmetry and time dependence.'
author:
- 'Rodolfo Gambini$^{1}$, Esteban Mato$^{1}$, Javier Olmedo$^{2,3}$, Jorge Pullin$^{3}$'
title: Classical axisymmetric gravity in real Ashtekar variables
---
Introduction
============
Due to the complexities of the quantized version of the Einstein equations in loop quantum gravity, the study of mini and midisuperspaces has proved a valuable tool to gain insights into the physics of the theory. The study first started with homogeneous cosmologies, giving rise to loop quantum cosmology (see [@Ashtekar:2011ni] and references therein). It was later expanded to include spherically symmetric spacetimes (see [@Gambini:2013hna] and references therein), including charged black holes [@Gambini:2014qta]. In both cases interesting physical insights, like the elimination of singularities due to quantum effects, were found. It is natural to try to extend these studies to situations with less symmetry, like the case of axisymmetric spacetimes, which include physically important situations, like the Kerr geometry. There is virtually no literature on the subject. An exception is the work on isolated horizons and black hole entropy [@krasnov:1998; @bojowald:2000; @perez:2011; @bianchi:2011; @frodden:2014; @achour:2016; @croken:2017]. An early study of spacetimes with one Killing vector field made some progress, partially addressing the situation of axial symmetry in complex connection variables [@Husain:1989mp]. Some progress was also made in planar space-times ([@Neville:2013xba] and references therein, [@Hinterleitner:2011rb]) and the case of two spatial Killing vector fields was also discussed for the Gowdy models [@Husain:1989qq], including the use of hybrid quantizations (see [@ElizagaNavascues:2016vqw] for a review). Some of these studies were in terms of the early form of the Ashtekar variables which were complex.
Here we would like to discuss the case of axisymmetric space-time using real Ashtekar–Barbero variables. We introduce a suitable Killing vector field and coordinates adapted to it. We will also show how the Kerr, Schwarzschild and Minkowski solutions arise. Besides, we will make some remarks on boundary conditions. This completes a classical setup suitable to perform a loop quantization, which we will discuss in a subsequent paper. This is the first example of a system with only one Killing vector field to be formulated with the real Ashtekar–Barbero variables.
The organization of this paper is as follows. In section 2 we discuss a set of symmetry adapted variables and set up the kinematics of the problem. In section 3 we introduce the constraints of general relativity in terms of the reduced axisymmetric variables introduced. In section 4 we work out the equations of motion. In section 5 we check that some particular solutions of interest including the Kerr, Schwarzschild and Minkowski space-times solve the equations we present. In section 6 we discuss boundary conditions. We end with a discussion.
Kinematics: symmetry adapted variables
======================================
Here we will impose a symmetry reduction due to a spatial Killing field with orbits tangent to $S^1$. Let us consider a choice of fiducial coordinates $\{x,y,\phi\}$, where $\phi\in S^1$ and $x,y\in \mathbb{R}$. The Killing field will be then $$K^a=\left(\partial_\phi\right)^a.$$
We will be following the typical reduction procedure adopted for connection variables. Namely, a connection $ A=A_a^i\tau_i \mathrm{d}x^a$ will be invariant under the Killing symmetries if it satisfies the condition $${\cal L}_{\tilde K}A^i_a=\epsilon_{ijk}\lambda^jA^k_a,$$ where $\tilde K = \lambda_i \partial^i=\lambda_3\partial_\phi$ and $\lambda_1=0=\lambda_2$. The previous equation amounts to $$\label{eq:axi-sym}
\partial_\phi A^i_a=\epsilon_{i3k}A^k_a.$$ Notice that we are imposing that the Lie derivative be proportional to a constant $O(2)$ gauge transformation [@cordero]. We have found this to be the simplest choice that is general enough to recover all solutions with axisymmetry. In other situations one may need to consider $\lambda^i$ that are more general, perhaps including spatial dependence.
The same equation is valid for the densitized triad $E^a_i$. The most general solution (see Appendix \[app:red\]) to these equations are $$\begin{aligned}
\label{eq:red-A}
A&=&A_a^i\tau_i \mathrm{d}x^a= \left((\cos(\phi)\tau_1 + \sin(\phi) \tau_2) \a_a^1 + (-\sin(\phi) \tau_1 + \cos(\phi) \tau_2) \a_a^2 + \a_a^3 \tau_3 \right)\mathrm{d}x^a \\ \label{eq:red-E}
E&=&E^a_i \tau^i \partial_a= \left((\cos(\phi)\tau^1 + \sin(\phi) \tau^2) \e^a_1 + (-\sin(\phi) \tau^1 + \cos(\phi) \tau^2) \e^a_2 + \e^a_3 \tau^3 \right)\partial_a,\end{aligned}$$ where the symmetry adapted variables $(\a_a^i,\e^b_j)$ do not depend on the angular coordinate $\phi$, i.e. only on $(x,y)$, and are canonically conjugate. In order to prove this, it is very easy to verify that $$\Omega=\frac{1}{8\pi G \beta}\int dx dy d\phi \;\delta E^a_i\wedge \delta A^i_a = \frac{1}{4 G \beta}\int dx dy \;\delta \e^a_i\wedge \delta\a^i_a,$$ with $\beta$ the Immirzi parameter. In other words, $$\{\a^i_a(\vec x),\e^b_j(\vec x')\}=4 G\beta\,\delta^i_j\delta^b_a\delta^{(2)}(\vec x-\vec x').$$ Another geometrical quantity that will be useful and can be computed now is the determinant of the symmetry-reduced densitized triad $$E = \det(E) =\frac{1}{3!}\varepsilon_{abc}\varepsilon^{ijk}E^a_iE^b_jE^c_k=\frac{1}{3!}\varepsilon_{abc}\varepsilon^{ijk}\e^a_i\e^b_j\e^c_k=\det (\e)=\e.$$ The inverse of the densitized triad, $E^i_a$, takes a similar form as $E^a_i$, but replacing $\e^a_i$ by $\e_a^i$. One can easily see that $\e^i_a$ fulfills $\e^i_a\e^a_j=\delta^i_j$ and $\e^i_a\e^b_i=\delta^a_b$, i.e. it is the inverse of $\e^a_i$ (and therefore it can be written in terms of $\e^a_i$). Then, the symmetry-reduced spatial metric can be written as $$q_{ab}=EE^i_aE^i_b=\e\,\e^i_a\e^i_b,$$ and it only depends on $\e^a_i$.
Similarly, the same reduction process can be applied to the extrinsic curvature $ K=K_a^i\tau_i \mathrm{d}x^a$, in its triadic form, and the spin connection $ \Gamma=\Gamma_a^i\tau_i \mathrm{d}x^a$, namely $$\begin{aligned}
K&&=\left((\cos(\phi)\tau_1 + \sin(\phi) \tau_2) \k_a^1 + (-\sin(\phi) \tau_1 + \cos(\phi) \tau_2) \k_a^2 + \k_a^3 \tau_3 \right)\mathrm{d}x^a,\\
\Gamma&&=\left((\cos(\phi)\tau_1 + \sin(\phi) \tau_2) {\gamma}_a^1 + (-\sin(\phi) \tau_1 + \cos(\phi) \tau_2) \gamma_a^2 + \gamma_a^3 \tau_3 \right)\mathrm{d}x^a.\end{aligned}$$ Actually, the components of the symmetry-
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'The past few years have seen remarkable progress in the theory and phenomenology of QCD, bringing perturbative and nonperturbative methods into closer contact with each other and with experiment.'
title:
---
ø Ł
\#1 [\[\#1\]]{}
ITP-SB-99-22
[^1]
George Sterman
*Institute for Theoretical Physics, SUNY at Stony Brook, Stony Brook, NY 11794-3840*
Introduction: QCD a Group Portrait
==================================
In this talk, I will summarize some recent developments in QCD, concentrating on, but not limited to, topics discussed at this meeting. Details, of course, can be found in the talks themselves, presented in lively sessions organized by Lance Dixon and Joey Huston for perturbative QCD, and by Paul Mackenzie and Claude Bernard for nonperturbative QCD. The future of the field is in the already-advanced convergence of these topics, sometimes thought of as nearly independent. Progress on heavy quark physics is described in the proceedings from separate parallel sessions and plenary talks. I must necessarily pass over some of the most interesting recent advances in other closely related fields as well for lack of space.
I have tried in the following to interleave perturbative and nonperturbative treatments of QCD dynamics. Let me begin with a few general comments on the place of QCD studies in high energy physics.
[*Why QCD?*]{} By now it is clear that QCD is a “correct", or phenomenologically relevant theory, at least the way classical Maxwell theory is “correct". Classical electromagnetism is an effective theory appropriate to the limit of many photons; quantum chromodynamics might be the long-distance limit of some more fundamental, underlying theory. Its self-consistency, however, leaves us to free to study QCD in its own terms.
This is a fascinating, and dauting, task, despite the fact that QCD is defined through a single dimensionless parameter, $\alpha_s$. The overall dimensional scale is determined by comparison with other interactions, for example, by measuring the strong coupling at the mass of the Z: $\alpha_s(M_{\rm Z})$. Its intrinsic interest aside, QCD has important “practical" applications, in the calculation of backgrounds to new physics and, for hadron colliders particularly, in predictions for new particle cross sections. From a theoretical point of view, however, what makes QCD so attractive is that it is a quantum field theory that requires all orders in perturbation theory and nonperturbative analysis to confront data at available energies.
[*QCD is the exemplary quantum field theory.*]{} QCD exhibits most of the classic quantum field-theoretic phenomena discovered in the sixties and seventies, including asymptotic freedom, confinement, spontaneous symmetry breaking and instantons. The problems of strong interactions that gave rise to QCD were also, in the same time period, the original inspiration for concepts of duality and strings. As we saw at this conference, the strong interactions, now understood as QCD, are once more the meeting ground for field theory, duality and string theory.
Again, QCD is evidently the correct theory of the strong interactions. Given its depth, however, “tests" of QCD should be thought of as tests, or perhaps better, “explorations", of quantum field theory itself. Complementary to the extraordinary accuracy of selected perturbative predictions in quantum electrodynamics are the broad predictions of QCD, interweaving nonperturbative and perturbative scales and phenomena.
QCD at the Shortest Distances
=============================
It was the asymptotic freedom of QCD that first drew attention to gauge field theory as the unique description of the strong interactions at short distances. This theme continues to unfold in current experiments, and to provide a basis for extrapolations between energy scales and the detection of signals for new physics. Let us begin with a run-through of the underlying methods [^2].
Methods
-------
[*Infrared safety.*]{} Infrared safe quantities are insensitive to long-distance effects, and may be calculated in perturbation theory [@tasi95; @glover]. An infrared safe quantity may or may not be directly observable. The classic examples include the total cross section for $\e^+\e^-$ annihilation to hadrons, and jet and event shape cross sections in $\e^+\e^-$. These can be written in the general form, Q\^2 \_[phys]{}(Q\^2) = \_n c\_n(Q\^2/\^2) \^n() , \[IRS\] with the $c_n$ dimensionless functions of the ratio of the hard scale to the renormalization scale $\mu$.
[*Factorization.*]{} Cross sections for deep-inelastic scattering (DIS) and for jet or heavy quark production in hadron-hadron scattering, are not purely perturbative, but appear as convolutions in parton momentum fractions of distributions $f_{a/h}(\xi,\m)$ of partons $a$ in hadrons $h$, with perturbative hard-scattering functions $\hat\sigma^{\rm PT}$, Q\^2\_[phys]{}(Q,x) = \_[[partons]{} a]{}\_a\^[PT]{}(Q/,())f\_[a/h]{}() = \_[[partons]{} a]{} \_x\^1 d \_a\^[PT]{}(x/,Q/,()) f\_[a/h]{}(,) , \[fact\] where in the second equality we have exhibited the convolution appropriate to deeply inelastic scattering (with $x=Q^2/2p\cdot q$). Corrections to Eq. (\[fact\]) are suppressed by ${\cal O}(1/Q^2)$ [@tasi95]. In this formula, there is a complementarity between the roles of parton distributions and the hard scattering. The distributions $f_{a/h}$ are universal among hard-scattering processes, but particular to hadron $h$, while the functions $\hat \sigma^{\rm PT}$ are particular to the process, but universal among external hadrons. This last feature enables us to calculate realistic $\hat\sigma^{\rm PT}$ in “unrealistic", but technically manageable (infrared regulated) scattering processes, in which the initial state hadrons are partons.
[*Evolution.*]{} The physical cross sections of Eqs. (\[IRS\]) and (\[fact\]) above must both be independent of the scales $\m$ that define the parton distributions: $\m d\sigma_{\rm phys}/d\m=0$. This self-consistency requirement is readily translated into the “DGLAP" equation for the evolution of parton densities, =\_b P\_[ab]{}(/,())f\_[b/h]{}(,) . \[evol\] Here, the dimensionless kernel $P$ depends only on variables that are in common between the hard scattering functions and the parton distributions: $\alpha_s$ and the momentum fractions. The scale-independence of physical quantities and their relations can be studied systematically [@commensurate].
The idealized pattern for determining and applying the distributions may be summarized as follows. We measure one cross section, $\sigma_{\rm phys}$, at momentum transfer $Q_0$. Given a “next-to-leading" order calculation of its hard scattering functions $\hat\sigma_a^{\rm (NLO)}$, we determine NLO parton distributions $f_{a/h}^{\rm (NLO)}$ at $\m=Q_0$. Using evolution, we can then predict $\sigma_{\rm phys}$ for any hard process at all $Q$.
The coefficients $c_n$ in Eq. (\[IRS\]) are known for many processes to NLO [@glover]. They have been determined at NNLO for inclusive DIS and Drell-Yan [@nnlo], and even to three loops [@3loop] for selected quantities. Generally, two loop corrections are available only for single-scale processes, and the calculation of two-loop corrections for genuine scattering diagrams is an as-yet unsolved, but actively studied, problem in QCD [@kaufmann; @bern].
Perturbative QCD is most successful for inclusive processes, and/or single-scale semi-inclusive. Evolution in DIS is perhaps still the best illustration (see below). The current experimental situation in hadronic hard-scattering is reviewed in Ref. [@huston]. In multiscale problems, logarithms of ratios of distinct but perturbative scales often require resummation to all orders. Formally beyond perturbative resummation, but not always less important numerically, are corrections suppressed by powers of the hard scale(s). In DIS, and a few other cases, these corrections can be described by the operator product expansion. We shall encounter below “generalizations" of this famous technique, usually in the form of effective field theories.
Prime Examples
--------------
[*Tevatron Jets.*]{} The most impressive success in orders of magnitude continues to be the Tevatron inclusive single-jet and dijet cross sections [@huston; @bhatta; @hauser; @krane], as illustrated in Fig. \[cteqSinglejet\], which shows a plot from Ref. [@cteq5]. According to Eq. (\[fact\]), these cross sections are of the form \_[p|pJ+X]{}=\_[ab]{} f\_[a/p]{}\^[(NLO)]{}\_[abJ+X]{}f\_[b/|p]{} . \[1jet\] We find a consistency between NLO theory and experiment at a few tens of percent, well within the overall systematic errors, over a range in which the cross section decreases by seven or so factors of ten. As the figure shows, reasonable choices of parton distributions (in this case CTEQ5HJ) can account even for the highest-momentum data, although a slight difference remains between the D0 and CDF
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'A resolvent-based reduced-order representation is used to capture time-averaged second-order statistics in turbulent channel flow. The recently-proposed decomposition of the resolvent operator into two distinct families related to the Orr-Sommerfeld and Squire operators \[K. Rosenberg and B. J. McKeon, Efficient representation of exact coherent states of the Navier-Stokes equations using resolvent analysis, Fluid Dynamics Research 51, 011401 (2019)\] results in dramatic improvement in the ability to match all components of the energy spectra and the $uv$ cospectrum. The success of the new representation relies on the ability of the Squire modes to compete with the vorticity generated by Orr-Sommerfeld modes, which is demonstrated by decomposing the statistics into contributions from each family. It is then shown that this competition can be used to infer a phase relationship between the two sets of modes. Additionally, the relative Reynolds number scalings for the two families of resolvent weights are derived for the universal classes of resolvent modes presented by Moarref *et al.* \[R. Moarref, A. S. Sharma, J. A. Tropp, and B. J. McKeon, Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels, Journal of Fluid Mechanics 734, 275 (2013)\]. [These developments can be viewed as a starting point for further modeling efforts to quantify nonlinear interactions in wall-bounded turbulence.]{}'
author:
- 'Ryan M. McMullen'
- Kevin Rosenberg
- 'Beverley J. McKeon'
bibliography:
- 'references.bib'
title: |
Interaction of forced Orr-Sommerfeld and Squire modes\
in a low-order representation of turbulent channel flow
---
\[sec:intro\]Introduction and background
========================================
Techniques from linear systems theory applied to wall-bounded turbulent shear flows have met with much success. For example, analyses of the Navier-Stokes equations (NSE) linearized about the turbulent mean velocity predict the spanwise length scales associated with the near-wall cycle and large-scale structures in the outer region of the flow from both a transient growth [@delAlamo2006; @cossu2009; @willis2010] and energy amplification of harmonic and stochastic forcing perspective [@hwang2010; @willis2010]. More recently, the linearized equations have been used to develop linear estimators [@illingworth2018; @madhusudanan2019; @towne2020] and compute impulse responses [@vadarevu2019] that qualitatively reproduce the coherence and self-similarity of large-scale motions.
As turbulence is an inherently nonlinear phenomenon, a complete model must account for nonlinear interactions. A common approach to incorporate the effects of nonlinearity into linear models is to augment the linearized equations with an eddy viscosity, such that the turbulent mean profile is fixed as an equilibrium solution of the modified mean momentum equation [@delAlamo2006; @cossu2009; @willis2010; @hwang2010; @illingworth2018; @madhusudanan2019; @vadarevu2019]. While this approach justifies linearization about the turbulent mean profile, it precludes the study of finite-amplitude fluctuations, since their nonlinear interactions would feed back on and further alter the mean. Instead of using an eddy viscosity, @zare2017 considered colored-in-time stochastic forcing of the linearized NSE in the problem of completing partially-known second-order statistics. Notably, they demonstrated that their approach can be equivalently represented as a low-rank modification of the original equations.
In a different approach to dealing with nonlinearity, the resolvent analysis framework introduced by @mckeon2010 retains the nonlinear term and interprets it as endogenous forcing of the linear dynamics through triadic interactions with the velocity fluctuations at other wavenumber-frequency combinations. This framework eliminates the need to incorporate an eddy viscosity for self-consistency, as no linearization is performed. [@landahl1967 arrived at a similar formulation, deriving a forced Orr-Sommerfeld equation in the study of wall-pressure fluctuations, but focused on obtaining approximate solutions in eigenfunction expansions.]{}
Closure of the loop requires determination of the forcing such that it yields the correct velocity Fourier modes, as well as the mean velocity profile, which is assumed known. The forcing can be expanded as a sum over a set of basis functions such that the unknowns are the complex amplitudes, called the resolvent weights. An exact equation for the weights can be formulated [@mckeon2013], though it is intractable to solve for most complex flows of interest. Consequently, there have been previous attempts to estimate the weights from data, e.g. by using either a single time series or power spectral density of the velocity fluctuations [@gomez2016; @beneddine2016]. Alternatively, @moarref2014 used convex optimization to compute the weights for a resolvent-based low-order representation of time-averaged velocity spectra that minimize the deviation from spectra obtained from a direct numerical simulation (DNS) of $\Rey=2003$ channel flow [@hoyas2006]. @towne2018 established a link between resolvent analysis and spectral proper orthogonal decomposition (SPOD) and showed that if the resolvent weights are treated as stochastic quantities, their covariance matrix can be calculated from the SPOD modes, which inherently rely on statistical data.
[In special cases where full information of the nonlinear forcing is available, such as for exact coherent states (ECS), the resolvent weights can be computed exactly by projecting the forcing onto the aforementioned set of basis functions [@sharma2016]. For ECS families in channel and pipe flow, which come in pairs of upper and lower branch solutions, the lower branch ones are typically well-represented by only a few resolvent modes, whereas many of the upper branch solutions are not captured as efficiently. Furthermore, the wall-normal and spanwise velocity components converge much more slowly than the streamwise velocity. However, an alternative decomposition of the resolvent operator recently proposed by @rosenberg2019a yields two families of modes related to the Orr-Sommerfeld and Squire operators from classical linear stability theory. By projecting the same channel ECS, they demonstrated that the new sets of basis functions enable a much more compact representation of both branches of solutions, and, notably, all three velocity components converge at roughly the same rate. Subsequent analysis attributed the improved efficacy of the alternative decomposition to the isolation of the wall-normal velocity response into the Orr-Sommerfeld modes, such that the Squire wall-normal vorticity is free to interact with that generated by the Orr-Sommerfeld modes [@rosenberg2018].]{}
[While the utility of the decomposition into Orr-Sommerfeld and Squire modes for highly simplified flows like ECS has been established, an open question is whether or not it remains relevant for high Reynolds number turbulence. In the present work, it is shown that the second-order statistics of turbulent channel flow can be accurately represented using a low-order approximation based on this framework. It is additionally shown that the vorticity produced by the Orr-Sommerfeld and Squire modes act to oppose each other, and this observation reveals information about how the resolvent weights for the two families scale relative to each other with Reynolds number. Altogether, these insights point to a mechanism in turbulent channel flow that is important for low-order modeling efforts.]{}
\[sec:form\]Formulation
=======================
\[sec:res\]Resolvent analysis of turbulent channel flow
-------------------------------------------------------
The approach is based on the resolvent analysis framework of @mckeon2010, in which the incompressible NSE,
\[eq:NSE\] $$\begin{aligned}
\partial_t \tilde{\vect{u}} + \left( \tilde{\vect{u}} \boldsymbol{\cdot} \bnab \right) \tilde{\vect{u}} = - \bnab \tilde{p} + \Rey^\inv\bnab^2 \tilde{\vect{u}}&,
\\
\bnab \boldsymbol{\cdot} \tilde{\vect{u}} = 0&,\end{aligned}$$
here nondimensionalized using the friction velocity $u_\tau$ and channel half-height $h$, are first Reynolds decomposed as $\tilde{\vect{u}} = \vect{U} + \vect{u}$, where $\vect{U} = \begin{pmatrix}U(y) & 0 & 0\end{pmatrix}^\trsp$ is the turbulent mean velocity profile and $\vect{u}$ are the fluctuations about the mean, and then Fourier transformed in the homogeneous wall-parallel and temporal directions $x$, $z$, and $t$, resulting in equations for the Fourier coefficients, denoted by $\hat{\blankop}\,$. For each wavenumber-frequency triplet $\begin{pmatrix}k_x & k_z & \omega\end{pmatrix}^\trsp \neq \mathbf{0}$, we have
$$\begin{aligned}
i\omega \hat{\vect{u}} + \left( \vect{U} \boldsymbol{\cdot} \bnab \right) \hat{\vect{u}} + \left( \hat{\vect{u}} \boldsymbol{\cdot} \bnab \right) \vect{U} + \bnab \hat{p} - \Rey^\inv\bnab^2 \hat{\vect{u}} =& \hat{\
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We consider the problems of error-correcting codes and image restoration with multiple stages of dynamics. Information extracted from the former stage can be used selectively to improve the performance of the latter one. Analytic results were derived for the mean-field systems using the cavity method. We find that it has the advantage of being tolerant to uncertainties in hyperparameter estimation, as confirmed by simulations.'
address:
- 'Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong'
- 'Department of Physics, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152-8551, Japan'
author:
- 'K. Y. Michael Wong'
- Hidetoshi Nishimori
title: |
Error-correcting codes and image restoration\
with multiple stages of dynamics
---
INTRODUCTION
============
The corruption of signals by noise is a common problem encountered in information processing. To retrieve signals from messages corrupted during the transmission through noisy channels, various error-correcting codes have been proposed [@eliece]. In particular, the error-correction mechanism of a class of parity-checking codes can be considered as the search for thermodynamically stable states of a Hamiltonian constructed in terms of the message bits [@sourlas]. These codes have been demonstrated to saturate the Shannon information bound in the limit that each encoded bit checks the parity of an infinitely large number of message bits [@sourlas; @kabasaad]. While in practice, each encoded bit can only check the parity of a finite number of message bits, these codes still maintain a very low bit error probability.
The need to retrieve signals from corrupted messages is also inherent in image restoration [@geman]. Although parity-checking bits may not be explicitly introduced for the task, prior knowledge about the images plays a similar role. For example, the smoothness of real-world images provides a mechanism for checking the pixel values in comparison with those of their neighbors. A corresponding Hamiltonian, consisting of a ferromagnetic bias to reflect the smoothening tendency, can be constructed in terms of the image pixels. Modern techniques of image restoration based on Markov random fields correspond to the search for thermodynamically stable states of the Hamiltonian system, using methods such as simulated annealing [@geman].
In a recent paper, we have shown that the problems of error-correcting codes and image restoration can be formulated in a unified framework [@nishiwong]. In both tasks, the choice of the so-called hyperparameters is an important factor in determining their performances. Hyperparameters refer to the coefficients of the various interactions appearing in the Hamiltonian of the tasks. In error correction, they determine the statistical significance given to the parity-checking terms and the received bits. Similarly in image restoration, they determine the statistical weights given to the prior knowledge and the received data. It was shown, by the use of inequalities, that the optimal choice of the hyperparameters correspond to the Maximum Posterior Marginal (MPM) method, where there is a match between the source and model priors. The choice of these values correspond to the Nishimori point in the space of hyperparameters [@nishimori]. It is equivalent to a thermodynamic process at finite temperature, and the task performance is better than the Maximum A Posteriori probability (MAP) method, where the values of the hyperparameters are taken to infinity, equivalent to a zero temperature process. Furthermore, from the analytic solution of the infinite-range model and the Monte Carlo simulation of finite-dimensional models, it was shown that an inappropriate choice of the hyperparameters can lead to a rapid degradation of the tasks.
In fact, hyperparameter estimation has been the subject of many previous studies [@zhou], a recently popular one using the “evidence framework” [@mackay]. However, if the prior models the source poorly, no hyperparameters can be reliable [@pryce]. Even if they can be estimated accurately through steady-state statistical measurements, they may fluctuate when interfered by bursty noise sources in communication channels. Hence it is important to devise decoding or restoration procedures which are robust against the uncertainties in hyperparameter estimation.
In this paper we propose the technique of selective freezing as a method to increase the tolerance to uncertainties in hyperparameter estimation. The technique has been studied for pattern reconstruction in neural networks, where it led to an improvement in the retrieval precision, a widening of the basin of attraction, and a boost in the storage capacity [@wong]. The idea is best illustrated for Ising bits or pixels with binary states $\pm 1$, though it can be easily generalized to other cases. In a finite temperature thermodynamic process, the Ising variables keep moving under thermal agitation. Some of them have smaller thermal fluctuations than the others, implying that they are more certain to stay in one state than the other. This stability implies that they have a higher probability to stay in the correct state for error-correction or image restoration tasks, even when the hyperparameters are not optimally tuned. It may thus be interesting to separate the thermodynamic process into two stages. In the first stage we select those relatively stable bits or pixels whose time-averaged states have a magnitude exceeding a certain threshold. In the second stage we subsequently fix (or freeze) them in the most probable thermodynamic states (for Ising variables this corresponds to the sign of the time-averaged state). Thus these selectively frozen bits or pixels are able to provide a more robust assistance to the less stable bits or pixels in their search for the most probable states. The selective freezing procedure reduces to the usual finite-temperature decoding or restoration process if all bits or pixels are frozen (since nothing happens in the second stage), or no bits or pixels are frozen (since the second stage is merely a continuation of the equilibration process of the first stage).
The two-stage thermodynamic process can be studied analytically in the mean-field model, which provides a qualitative guide to the behavior of more realistic cases of lower dimensions. However, it is necessary to give a remark about the theoretical approach. That is, as far as we have tried, the analytical solution has been inaccessible by the more conventional replica method. Rather, we have to use the cavity method to obtain the equations for the order parameters. In particular, the cavity method leads to the appearance of a term called the trans-susceptibility, which correctly describes the effects of the thermodynamics of the first stage on that of the second.
The paper is organized as follows. In Section II we briefly review the formulation of error-correcting codes and image restoration in a unified framework. In Sections III and IV, we consider the mean-field model for error-correcting codes and image restoration respectively. We derive the equations for the order parameters of the two-stage thermodynamics using the cavity method, and present numerical results illustrating the robustness of selective freezing against uncertainties in hyperparameter estimation. We further demonstrate that even when the noise model changes without the receiver/restoration agent realizing the change (i.e. it makes a wrong estimation of the prior), the task performance is still robust. For the more realistic cases of lower dimensions, simulation results illustrate the relevance of the infinite-range model in providing qualitative guidance. The conclusion is given in Section V.
FORMULATION
===========
Consider an information source which generates data represented by a set of Ising spins $\{\xi_i\}$, where $\xi_i=\pm 1$ and $i=1, \cdots, N$. The data is generated according to the source prior $P_s(\{\xi_i\})$. For error-correcting codes transmitting unbiased messages, all sequences are equally probable and $P_s(\{\xi\})=2^{-N}$. For images with smooth structures, the prior consists of ferromagnetic Boltzmann factors, which increase the tendencies of the neighboring spins to stay at the same spin states, that is, $$P_s(\{\xi\})=\frac{1}{Z(\beta_s)}
\exp \left({\beta_s\over z}\sum_{\langle ij\rangle}
\xi_i \xi_j \right).
\label{prior}$$ Here $\langle ij\rangle$ represents pairs of neighboring spins, $z$ is the valency of each site, and the partition function $Z(\beta_s)$ is given by $$Z(\beta_s)={\rm Tr}_\xi
\exp \left({\beta_s\over z}\sum_{\langle ij\rangle}
\xi_i \xi_j \right).
\label{partition}$$ The data is coded by constructing the codewords, which are the products of $p$ spins $J^0_{i_1 \cdots i_p}=\xi_{i_1}\cdots\xi_{i_p}$ for appropriately chosen sets of of indices $\{i_1,\cdots,i_p\}$, the choice of which determines the type of code. Each spin may appear in a number of $p$-spin codewords; the number of times of appearance is called the valency $z_p$. The Sourlas code [@sourlas] is equivalent to the infinite-range model in which all possible codewords of $p$ spins are chosen from $N$ spins. On the other hand, the Kabashima-Saad code [@kabasaad] consists of combinations in which each spin appears in a random pre-selection of $z_p$ codewords. For conventional image restoration, codewords with only $p=1$ are transmitted, corresponding to the pixels in the image; the inclusion of terms with $p>1$, and their positive effects on restoring the original image, have also been discussed in [@nishiwong]. For simplicity, we restrict ourselves to the case of a single non-vanishing value of $p
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ł c v § ¶ Ø Ł i
LPT-ENS 02/42\
Bernard Julia$^a$
0.5cm
$^a$Laboratoire de Physique Th[é]{}orique CNRS-ENS\
24 rue Lhomond, F-75231 Paris Cedex 05, France[^1]\
0.5cm
[**ABSTRACT**]{}
It seems to me at this time that two recent developments may permit fast progress on our way to understand the symmetry structure of toroidally (compactified and) reduced M-theory. The first indication of a (possibly) thin spot in the wall that prevents us from deriving a priori the U-duality symmetries of these models is to be found in the analysis of the hyperbolic billiards that control the chaotic time evolution of (quasi)homogeneous anisotropic String, Supergravity or Einstein cosmologies near a spacelike singularity. What happens is that U-duality symmetry controls chaos via negative constant curvature. On the other hand it was noticed in 1982 that (symmetrizable) ”hyperbolic” Kac-Moody algebras have maximal rank ten, exactly like superstring models and that two of these four rank ten algebras matched physical theories. My second reason for optimism actually predates also the previous breakthrough, it was the discovery in 1998 of surprising superalgebras extending U-dualities to all (p+1)-forms (associated to p-branes). They have a super-nonlinear sigma model structure similar to the symmetric space structure associated to 0-forms and they obey a universal self-duality field equation. As the set of forms is doubled to implement electric-magnetic duality, they obey a set of first order equations. More remains to be discovered but the beauty and subtlety of the structure cannot be a random emergence from chaos. In fact we shall explain how a third maximal rank hyperbolic algebra $BE_{10}$ controls heterotic cosmological chaos and how as predicted Einstein’s General Relativity fits into the general picture.
Classifications
===============
It is well known to conformal field theorists but a much more general and venerable fact that positive definite symmetric matrices with integer entries tend to appear in many classification problems. More precisely the ADE philosophy is to list such occurrences and try to relate them to each other. Let us be a bit more specific, this set of problems corresponds to the emergence in various contexts of symmetric matrices with diagonal entries equal to 2 and negative integral off-diagonal entries. The prototype of a successful correspondence is the work of Brieskorn (with some help from Grothendieck) realising the simple complex Kleinian singularities related to discrete subgroups of SU(2) as rational singularities in the set of unipotent elements of the corresponding complex Lie group of type ADE. The singularity is at the conjugacy class of subregular elements in other words the elements whose centraliser has 2 more dimensions than regular ones namely $r+2$ instead of $r$, the group rank. The resolution of the singularity introduces exceptional divisors whose intersection matrix is the opposite of the group Cartan matrix, there the ADE matrices arise in two different disguises but there is a relation to the Lie group in both cases.
Halfway (in time) between the Brieskorn results and the classification of modular invariant partition functions by Cappelli et al. $N=8$ supergravity was constructed in 4 dimensions as well as its toroidal decompactification family up to 11 spacetime dimensions. It was quickly remarked to me by Y. Manin that the $E_r$ internal symmetry groups appearing upon compactification on a r-torus suggested a role for (regular) del Pezzo surfaces, these are variants of $CP_2$ or $CP_1 \times CP_1$ which admit a canonical projective embedding . However the other pure supergravities in 4 dimensions with fewer supersymmetries also belonged to families which together formed a magic triangle of theories. They led also to ADE groups yet not all in split form, this meant that real geometry was to be tackled rather than the simpler complex algebraic one.
It also rapidly became clear if not rigorously established that the $E_r$ family included the infinite dimensional $E_9 \equiv E_8^{(1)}$ in 2 dimensions ie. after compactification on a 9-torus. Our partial understanding of these duality groups includes their $A_{r-1} \times \R$ subgroups related to the r ignorable coordinates. The infinite dimensional hidden symmetry is related to the existence of a Lax pair in 2 residual dimensions leading to a quasi integrable situation yet it was known that chaos remained prevalent in some particular flows.
In 1982 I noticed that a naive extrapolation to 1 dimension would suggest a role for $E_{10}$ as one would expect its subalgebra $A_9$ to appear there, however the implementation was problematic. Nevertheless a similar analysis of so-called type I supergravity in other words pure (without matter multiplet) supergravity in 10 dimensions led to the suggestion of a corresponding role for the other hyperbolic Kac-Moody algebra $DE_{10}$ the “overextended” $D_8$ in split form (namely $SO(8,8)$ affinized with one more $SL(2,\R)$ generating subgroup right next to the affinizing $SL(2,\R)$ in the Dynkin diagram as implied by the $A_{r-1}$ argument). Now according to Bourbaki (actually Chein) $E_{10}$ and $DE_{10}$ are exactly the two simply laced hyperbolic Kac-Moody algebras of maximal rank. There are only two more (non simply laced) hyperbolic Kac-Moody algebras of maximal rank: $BE_{10}$ (see section 3) and $CE_{10}$ (to be seen).
Let us recall that hyperbolicity here means that the Dynkin diagram becomes a product of finite or affine Dynkin diagrams after removal of any node (this gives nice arithmetic properties as well). The rank ten here is directly related to the rank eight of $E_8$, the largest exceptional simple Lie group which itself comes about also from some subtle classification analysis. On the other hand ten is the critical dimension of superstring models and as such it is related to quantum conformal invariance. How could the two derivations be related? They must be as the rank of the Lie group is closely related to the dimension of the compactification torus!
Still a puzzling fact emerged from the subsequent construction of heterotic strings namely the possibility of duality groups of rank higher than 8 for instance 16 in dimension 3 ($SO(8,24)$) corresponding putatively to rank 18 in one dimension. This paradoxical apparent violation of the bound ten on the rank will be clarified in section 3. Let us also remark that we are talking of superstrings and (bosonic) Kac-Moody algebras, hence fermionic structures should emerge in an a priori bosonic context. The upper limit 26 does not arise so simply yet, although it can be argued to be the sum of ten and sixteen the latter being the rank of the two even euclidean unimodular lattices dictated again by quantum anomaly considerations.
Let us add some examples of ADE objects. The basic one is the set of integral positive definite matrices occuring as Cartan matrices of simple simply laced complex Lie groups. The positive definiteness, resp positive semi-definiteness, resp hyperbolicity guarantees a relatively simple classification, for instance in the positive definite case (that of finite dimensional simply laced Lie algebras) there is a finite number of objects for each rank; on the other hand the three cases of $E_6, E_7, E_8$ look exceptional in this context. In the semi-definite case also called affine Kac-Moody situation very much the same is true; in the hyperbolic case however as we have seen the rank is bounded and there are finitely many instances of a given rank except when it is equal to two.
The finite dimensional (irreducible) Coxeter groups generated by reflections are closely related objects, their list encompasses that of the Weyl groups of the simple Lie groups but one gets 3 extra Coxeter diagrams with rotation angle $2\pi/5$ and an infinite family of dihedral groups of rotation angles $2\pi/k$ for non cristallographic k’s integers at least 7. These are all nonsimply laced cases. It is important to note that non simply laced Lie groups have a symmetrisable Cartan matrix: a basic assumption for most of Kac-Moody theory and Borcherds algebras. The reflections preserve a symmetric form that is a symmetrisation of the non symmetric Cartan matrix.
By definition Coxeter groups admit a finite presentation by involutions $S_i
\, , \, i=1,\dots,r$ satisfying $$( S_i S_j)^{m_{ij}} = I \, , \, i\neq j.$$ In the simply laced case we consider only exponents $ m_{ij} =2$ for commuting involutions or 3 for dihedral subgroups of order 6. The matrix is encoded by a Dynkin diagram with r vertices and simple bonds for exponent 3; it turns out that no loop is allowed, that at most three legs occur and finally that the sum $1/m+1/n+1/p$ of the inverses of the number of vertices (including the potentially trivalent vertex) on each leg must be strictly larger than one. One recognises two infinite families $A_k, D_k$ and three exceptions $E_6, E_7, E_8$ with numbers of vertices respectively $(m,n,p
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{
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---
abstract: 'Given a partial action $\alpha$ of a connected groupoid ${\mathcal{G}}$ on an associative ring $A$ we investigate under what conditions the partial skew groupoid ring $A\star_{\alpha}{\mathcal{G}}$ can be realized as a partial skew group ring. In such a case applications concerning to the separability, semisimplicity and Frobenius property of the ring extension $A\subset A\star_{\alpha}{\mathcal{G}}$ as well as to the artinianity of ${{A}}\star_{{\alpha}}{\mathcal{G}}$ are given.'
address:
- |
Departamento de Matemática, Universidade Federal de Santa Maria, 97105-900\
Santa Maria-RS, Brasil
- |
Instituto de Matemática e Estatística, Universidade federal de Porto Alegre, 91509-900\
Porto Alegre-RS, Brazil
- 'Escuela de Matematicas, Universidad Industrial de Santander, Cra. 27 Calle 9 ´ UIS Edificio 45, Bucaramanga, Colombia'
author:
- Dirceu Bagio
- Antonio Paques
- Héctor Pinedo
title: On partial skew groupoids rings
---
\#1[[**(\*\*\* \#1 \*\*\*)**]{} ]{}
\#1
[^1] [^2]
Introduction
============
In this work we will consider partial actions of groupoids on rings. We are interested in studying the structure of the corresponding partial skew groupoids rings.
Partial groupoids actions on rings were introduced in [@BP] and they are a natural generalization of partial group actions. It is well known that every groupoid is a direct sum of its connected component. A partial action of a groupoid on a ring $A$ is completely determined by the partial actions of its connected components on $A$. Thence, we can reduce the study of groupoid partial action on rings to the context of connected groupoids.
The structure of a connected groupoid is also well known. If ${\mathcal{G}}$ is a connected groupoid then ${\mathcal{G}}\simeq {\mathcal{G}}_0^2\times {\mathcal{G}}(x)$, where ${\mathcal{G}}_0^2$ is the coarse groupoid associated to the set ${\mathcal{G}}_0$ of the objects of ${\mathcal{G}}$ and ${\mathcal{G}}(x)$ is the isotropy group of an object $x$ of ${\mathcal{G}}$.
For a partial action $\alpha$ of a connected groupoid ${\mathcal{G}}$ on a ring $A$ we can construct the partial skew groupoid ring $A\star_{\alpha}{\mathcal{G}}$. If ${\mathcal{G}}_0$ is finite and $\alpha$ is unital then $A\star_{\alpha}{\mathcal{G}}$ is an associative and unital ring which is an extension of $A$.
The partial skew groupoid rings have an important role in the partial Galois theory for groupoids as it is explicit in Theorem 5.3 of [@BP]. They also are examples of Leavitt path algebras, which are important in the theory of $C^{\ast}$-algebras (see Theorem 3.11 of [@GY]). In the last years, algebraic properties to the extension $A\subset A\star_{\alpha}{\mathcal{G}}$ have been studied. For example, the separability and semisimplicity properties of the extension $A\subset A\star_{\alpha}{\mathcal{G}}$ were studied in [@BPi] whereas in [@NOP] the authors investigate chain conditions between $A$ and $A\star_{\alpha}{\mathcal{G}}$.
Our purpose in this work is to study the following problem. Let ${\mathcal{G}}$ be a connected groupoid such that ${\mathcal{G}}_0$ is finite and $\alpha$ a unital partial action of ${\mathcal{G}}$ on a ring $A$. Does the factorization ${\mathcal{G}}\simeq {\mathcal{G}}_0^2\times {\mathcal{G}}(x)$ induce a factorization of $A\star_{\alpha}{\mathcal{G}}$? Theorem 4.1 provides sufficient conditions for the answer to this question to be affirmative. Precisely, when $\alpha$ is a group-type partial action, we construct a groupoid action $\beta$ of ${\mathcal{G}}_0^2$ on $A$ and a partial group action $\gamma$ of ${\mathcal{G}}(x)$ on $A\star_{\beta} {\mathcal{G}}_0^2$ and we prove that $A\star_{\alpha}{\mathcal{G}}\simeq (A\star_{\beta} {\mathcal{G}}_0^2)\star_{\gamma}{\mathcal{G}}(x)$.
We organize our work as follows. The background about groupoids is presented in Section 2. The topics of partial groupoid actions that will be used are in Section 3. In Section 4, we construct the actions $\beta$ and $\gamma$ which allow us to prove the factorization of ${{A}}\star_{{\alpha}}{\mathcal{G}}$ mentioned in the previous paragraph. Applications of this result concerning to the separability, semisimplicity and Frobenius property of the extension $A\subset {{A}}\star_{{\alpha}}{\mathcal{G}}$ as well as to the artinianity of ${{A}}\star_{{\alpha}}{\mathcal{G}}$ are given in Section 5.
Conventions {#subsec:conv .unnumbered}
-----------
Throughout this work, by ring we mean an associative and not necessarily unital ring. The center of a ring $A$ will be denoted by $C(A)$. We will denote the cardinality of a finite set $X$ by $|X|$.
Groupoids
=========
We recall that a [*groupoid*]{} is a small category in which every morphism is an isomorphism. The set of the objects of a groupoid ${\mathcal{G}}$ will be denoted by ${\mathcal{G}}_0$. If $g:x\to y$ is a morphism of ${\mathcal{G}}$ then $s(g)=x$ and $t(g)=y$ are called the [*source*]{} and the [*target*]{} of $g$ respectively. We will identify any object $x$ of ${\mathcal{G}}$ with its identity morphism, that is, $x={{\rm id}}_x$. The [*isotropy group*]{} associated to an object $x$ of ${\mathcal{G}}$ is the group ${\mathcal{G}}(x)=\{g\in {\mathcal{G}}:\,s(g)=t(g)=x\}$.
The composition of morphisms of a groupoid ${\mathcal{G}}$ will be denoted via concatenation. Hence, for $g,h\in {\mathcal{G}}$, there exists $gh$ if and only if $t(h)=s(g)$. Notice that, if $g\in {\mathcal{G}}$ then $s(g)=g^{-1}g$ and $t(g)=gg{{}^{-1}}$. Also, $s(gh)=s(h)$ and $t(gh)=t(g)$ for all $g,h\in {\mathcal{G}}$ with $t(h)=s(g)$.
A groupoid ${\mathcal{G}}$ is said to be [*connected*]{} if for any $x,y\in {\mathcal{G}}_0$ there exists a morphism $g\in {\mathcal{G}}$ such that $s(g)=x$ and $t(g)=y$, that is, the morphism $g$ connects the objects $x$ and $y$. It is well-known that any groupoid is a disjoint union of connected subgroupoids. In order to justify this fact, we consider the following equivalence relation on ${\mathcal{G}}_0$: for any $x,y\in{\mathcal{G}}_0$, $x\sim y$ if and only if there exists $ g\in{\mathcal{G}}$ such that $s(g)=x$ and $t(g)=y$. Every equivalence class $X\in{\mathcal{G}}_0/\!\!\sim$ determines a full connected subgroupoid ${\mathcal{G}}_X$ of ${\mathcal{G}}$. The set of objects of ${\mathcal{G}}_X$ is $X$. The set ${{\mathcal{G}}_X}(x,y)$ of morphisms of ${\mathcal{G}}_X$ from $x$ to $y$ is equal to ${{\mathcal{G}}}(x,y)$, for all $x,y\in X$. By construction, ${\mathcal{G}}$ is the disjoint union of the subgroupoids ${\mathcal{G}}_X$, i. e. $$\label{direct-sum}
{\mathcal{G}}=\dot\cup_{X\in {\mathcal{G}}_0/\!\sim}{\mathcal{G}}_X.$$
For the convenience of the reader, we will prove a well-known result about the structure of connected groupoids. In order to do this, we need to introduce some extra notation. Let $X$ be a nonempty set and $X^2=X\times X$. Then $X^2$ is a groupoid. The source and target maps of $X^2$ are, respectively, $s(x,y)=x$ and $t(x,y)=y$, for all $x,y\in X$. The rule of composition is given by: $(y,z)(x,y)=(x,z)$, for all $x,y,z\in X$. The groupoid $X^2$ is called the [*coarse groupoid associated to
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---
abstract: 'A successful state transfer (or teleportation) experiment must perform better than the benchmark set by the ‘best’ measure and prepare procedure. We consider the benchmark problem for the following families of states: (i) displaced thermal equilibrium states of given temperature; (ii) independent identically prepared qubits with completely unknown state. For the first family we show that the optimal procedure is heterodyne measurement followed by the preparation of a coherent state. This procedure was known to be optimal for coherent states and for squeezed states with the ‘overlap fidelity’ as figure of merit. Here we prove its optimality with respect to the trace norm distance and supremum risk. For the second problem we consider $n$ i.i.d. spin-$\frac12$ systems in an arbitrary unknown state $\rho$ and look for the measurement-preparation pair $(M_{n},P_{n})$ for which the reconstructed state $\omega_{n}:=P_{n}\circ M_{n} (\rho^{\otimes n})$ is as close as possible to the input state, i.e. $\|\omega_{n}- \rho^{\otimes n}\|_{1}$ is small. The figure of merit is based on the trace norm distance between input and output states. We show that asymptotically with $n$ the this problem is equivalent to the first one. The proof and construction of $(M_{n},P_{n})$ uses the theory of [*local asymptotic normality*]{} developed for state estimation which shows that i.i.d. quantum models can be approximated in a strong sense by quantum Gaussian models. The measurement part is identical with ‘optimal estimation’, showing that ‘benchmarking’ and estimation are closely related problems in the asymptotic set-up.'
author:
- Mădălin Guţă
- Peter Bowles
- Gerardo Adesso
date: 'September 17, 2010'
title: |
Quantum teleportation benchmarks for independent and identically-distributed\
spin states and displaced thermal states
---
[^1]
Introduction
============
Quantum teleportation [@Telep] and quantum state storage [@Storage] are by now well-established protocols in quantum information science. In both cases the procedure amounts to mapping one quantum state onto another (at a remote location in the case of teleportation), by making use of quantum correlations in the form of entanglement or interaction between systems. However, approximate transformations could also be accomplished without any use of quantum correlations, by means of classical ‘measure and prepare’ (MAP) schemes. Whilst in the ideal case, the entanglement resource gives quantum teleportation a clear advantage in terms of performance, there exists inevitable degradation of the quantum channel in realistic implementations. This has led to a number of investigations into the existence of optimal MAP schemes to locate classical-quantum boundaries and assign precise benchmarks for proving the presence of quantum effects [@bfkjmo]. Any experimental demonstration of quantum teleportation and state storage has to perform better than the optimal MAP scheme, to be certified as a truly quantum demonstration. A review of the quantum benchmarks for completely unknown pure input states of $d$-dimensional systems can be found in [@BrussMac]. More recent research has largely focused on benchmarks originating in the context of teleportation and quantum memory for continuous variable (CV) systems [@COVAQIAL], with notable results obtained for transmission of pure and mixed coherent input states, and squeezed states [@Hammerer; @AdessoC; @Namiki; @Owari; @Calsamiglia]. Beautiful experiments [@Furusawa98; @memorypolzik; @telepolzik; @polzikalessio] involving light (Gaussian modes) and matter (coherent and spin-squeezed atomic ensembles) have demonstrated unambiguous quantum teleportation, storage and retrieval of these infinite-dimensional quantum states with a measured ‘fidelity’ between input and output exceeding the benchmark set by the optimal MAP strategy (see also [@noteluk; @luk]).
In each of the above cases, the benchmarks deal with the case of teleportation or storage of [*single*]{} input states drawn from a set, in a Bayesian or pointwise set-up. To date, there exist no nontrivial benchmarks for the transmission of multiple copies of quantum states – a ‘quantum register’ – in particular for an ensemble of $n$ independent and identically-distributed (i.i.d.) qubits. Such a task comes as a primitive in distributed quantum communication. Quantum registers can be locally initialised and then transferred to remote processing units where a quantum computation is going to take place. Also, in hybrid interfaces between light and matter [@qinternet], storage and retrieval of e.g. coherent states, involves mapping the state of $n$ i.i.d. atoms onto a light mode (back and forth). Therefore, strictly speaking, a quantum benchmark for this precise input ensemble would be needed to assess the success of the experiment. In the current practice [@memorypolzik; @telepolzik] the problem is circumvented by noting that the collective spin components of the atomic ensemble (with $n \sim 10^{12}$ [@PolzikNAT]) approximately satisfy canonical commutation relations, henceforth the atomic system is treated [*a priori*]{} as a CV system, and the corresponding benchmarks are used.
In this paper we put this procedure on firm grounds, by proving rigorously that the optimal MAP scheme for teleportation and storage of $n$ i.i.d. unknown [*mixed*]{} qubits converges when $n \rightarrow \infty$ to the optimal MAP scheme for a single-mode displaced thermal state. Additionally, we also prove that the heterodyne measurement followed by the preparation of a coherent state is optimal MAP scheme for displaced thermal states when the figure of merit is the trace norm distance. The same scheme is known to be optimal for thermal and squeezed states, but for a figure of merit based on overlap fidelity [@Owari].
The key tool in deriving our results is the theory of local asymptotic normality (LAN) for quantum states which is the quantum extension of a fundamental concept in mathematical statistics introduced by Le Cam [@LeCam]. In the classical context this roughly means that a large i.i.d. sample $X_{1},\dots , X_{n}$ from an unknown distribution contains approximately the same amount of [*statistical*]{} information as a single sample from a Gaussian (normal) distribution with unknown mean and known variance.
In the quantum case, LAN means that the joint (mixed) state $\rho_{\theta}^{\otimes n}$ of $n$ identically prepared (finite dimensional) quantum systems can be transferred by means of a quantum channel to a quantum-classical Gaussian state, with asymptotically vanishing loss of statistical information. More precisely, for any fixed $\theta_{0}$ there exist quantum channels $T_{n}, S_{n}$ such that $$\begin{aligned}
& & \quad \left\|T_{n} \left(\rho_{\theta_{0}+u/\sqrt{n}}^{\otimes n}\right) -
N_{{u}}\otimes \Phi_{{u}} \right \|_{1} \nonumber \\
&\mbox{and}& \quad
\left\|\rho_{\theta_{0}+u/\sqrt{n}}^{\otimes n} - S_{n} \left(
N_{{u}}\otimes \Phi_{{u}}\right) \right\|_{1} \nonumber\end{aligned}$$ converge to zero as $n\to\infty$, [*uniformly*]{} over a $n^{-1/2+\epsilon}$ local neighbourhood of the state $\rho_{\theta_{0}}$. Here $N_{{u}}$ is a classical normal distribution and $\Phi_{{u}}$ is a Gaussian state on an ensemble of oscillators whose means are linear transformations of ${u}$ and the covariance matrices depend only on $\theta_{0}$. The qubit case is described in detail in Section \[sec.lan\] and the precise result is formulated in Theorem \[th.qlan.qubits\]. The LAN theory has been used to find asymptotically optimal estimation procedures for qubits and qudits and to show that the Holevo bound for state estimation is achievable . Here we use it to solve the benchmark problem for qubits by casting it into the corresponding one for displaced thermal states. The following diagram illustrates the asymptotically optimal MAP scheme: the measurement $M_{n}$ consists in composing the channel $T_{n}$ with the heterodyne measurement $H$. The preparation procedure consists in creating the coherent state $|\alpha_{\hat{\vec{u}}}\rangle$ and mapping it back to the qubits space by the channel $S_{n}$. The optimality of the scheme is proved in Theorem \[th.main\]. $$\label{comm.diagram}
\begin{CD}
\rho^{n}_{\vec{u}} @>M_n>> X_n @>P_n>>\omega(X_{n}) \\
@V{T_n}VV @. @AA {S_n} A \\
\Phi_{\vec{u}} \otimes N_{\vec{u}}
@> H >> \hat{\vec{u}} @>P>> |\alpha_{\hat{\vec{u}}} \rangle\langle\alpha_{\hat{\vec{u}}} |\otimes \delta_{\hat{u}_{3}} \\
\end{CD}$$
The paper is organised as follows. In Section \[sec.stat.formulation\] we give a precise statistical formulation of the benchmark problem, and we explain in some detail the definition of the asymptotic risk. A brief overview of the necessary classical and quantum concepts from the LAN theory is given in Section \[sec.lan\]. In Section \[sec.thermal\] we then revisit the benchmark problem for displaced thermal states. When the figure of merit is the overlap fidelity, the solution was found in [@
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---
abstract: 'The aim of this paper is that of discussing Closed Graph Theorems for bornological vector spaces in a way which is accessible to non-experts. We will see how to easily adapt classical arguments of functional analysis over $\R$ and $\C$ to deduce Closed Graph Theorems for bornological vector spaces over any complete, non-trivially valued field, hence encompassing the non-Archimedean case too. We will end this survey by discussing some applications. In particular, we will prove de Wilde’s Theorem for non-Archimedean locally convex spaces and then deduce some results about the automatic boundedness of algebra morphisms for a class of bornological algebras of interest in analytic geometry, both Archimedean (complex analytic geometry) and non-Archimedean.'
author:
- Federico Bambozzi
title: Closed graph theorems for bornological spaces
---
Introduction {#introduction .unnumbered}
============
This paper aims to discuss the Closed Graph Theorems for bornological vector spaces in a way which is accessible to non-specialists and to fill a gap in literature about the non-Archimedean side of the theory at the same time. In functional analysis over $\R$ or $\C$ bornological vector spaces have been used since long time ago, without becoming a mainstream tool. It is probably for this reason that bornological vector spaces over non-Archimedean valued fields were rarely considered. Over the last years, for several reasons, bornological vector spaces have drawn new attentions: see for example [@Bam], [@BaBe], [@BaBeKr], [@CAM] and [@M]. These new developments involve the non-Archimedean side of the theory too and that is why it needs adequate foundations. Among the classical books on the theory of bornological vector spaces, the only one considering also non-Archimedean base fields, in a unified fashion with the Archimedean case, is [@H2]. But that book cannot cover all the theory, and in particular it lacks of a unified treatment of the Closed Graph Theorems. This work comes out from the author’s need for a reference for these theorems and also in the hope that in the future bornological vector spaces will gain more popularity and that this work may be useful for others.
The Closed Graph Theorem for Banach spaces over $\R$ and $\C$ is one of the most celebrated classical theorems of functional analysis. Over the years it has been generalized in many ways, for example to Fréchet and LF spaces as a consequence of the Open Mapping Theorems. Although it is a classical argument that the Closed Graph Theorem can be deduced from the Open Mapping Theorem, people have understood that the Closed Graph Theorem can be proved in an independent way, with argumentations that also work when the Open Mapping Theorem fails. The two most famous examples of this kind of proofs are given in [@POPA] by Popa and [@DW] by de Wilde. In particular, de Wilde’s Theorem is probably the most general Closed Graph Theorem for locally convex spaces, and states the following:
(De Wilde’s Closed Graph Theorem)\
If $E$ is an ultrabornological locally convex space and $F$ is a webbed locally convex space over $\R$ or $\C$, then every linear map $f: E \to F$ which has bornologically closed graph with respect to the convex bornologies on $E$ and $F$ that are generated by all bounded Banach disks in $E$ and in $F$, respectively, is continuous even if regarded as a mapping into the ultrabornologification $F_\uborn$ of $F$.
The terminology of the theorem will be explained in the course of this work. What is interesting to notice is that, although we would like to prove a theorem for locally convex spaces we are naturally led to talk about bornologies and bounded maps. Popa’s Theorem, on the contrary, is an explicit bornological statement which is the Archimedean case of our Theorem \[thm:net\].
The content of the paper is the following: in the first section we give an overview of the theory of bornological vector spaces. In particular, since we adopt the unusual attitude of discussing the Archimedean and non-Archimedean case of the theory at the same time, we spend some time in recalling basic definitions and discuss in details the notions from the theory bornological vector spaces that will be used. In the second section we will introduce the notion of bornological nets and then give the main examples of bornological vector spaces endowed with nets. We will then deduce our first Closed Graph Theorem, which is the unified version of Popa’s Theorem (cf. [@POPA]), stated as follows:
Let $E$ and $F$ be separated convex bornological vector spaces, where $E$ is complete and $F$ has a net compatible with its bornology. Then, every linear map $f: E \to F$ with bornologically closed graph is bounded.
In the subsequent section the notion of bornological net is generalized by the notion of bornological web and the analogous Closed Graph Theorem for webbed bornological vector spaces is proved quite easily as a consequence of the previous discussion. In this case our theorem is the direct generalization, for all base fields, of the Closed Graph Theorem proved by Gach in [@G], Theorem 4.3. In the last section we discuss some applications of the theorems we proved. We will see how one can deduce Isomorphism Theorems from Closed Graph Theorems and following [@G] we will see how de Wilde’s Closed Graph Theorem can be deduced. We would like to remark that for non-Archimedean base fields we need to add some restrictions, that do not affect the Archimedean side of the theory. Our generalization of de Wilde’s Theorem is the following:
If $E$ is an ultrabornological locally convex space and $F$ is a polar webbed locally convex space defined over a spherically complete field $K$, then every linear map $f: E \to F$ which has bornologically closed graph with respect to the convex bornologies on $E$ and $F$ that are generated by all bounded Banach disks in $E$ and in $F$, respectively, is continuous even if regarded as a mapping into the ultrabornologification $F_\uborn$ of $F$.
Therefore, in order to deduce de Wilde’s Theorem for non-Archimedean base fields we needed to suppose that the base field $K$ is spherically complete and that $F$ is a polar locally convex space (cf. Definition \[defn:polar\]), conditions which are always satisfied when $K$ is Archimedean. We remark that these hypothesis on $K$ and $F$ are only used in Lemma \[lemma:polar\] and in Lemma \[lemma:web\_ultraborn\]; one might ask if it is possible to prove that lemmata without these restrictions. We do not address this problem in this work. Finally, in the last part of the paper we show how to use the Closed Graph Theorems for bornological spaces to deduce that all algebra morphisms between dagger affinoid algebras, as defined in [@Bam], are bounded. This application, and others coming in [@BaBeKr] and planned in future works, are our main motivations for this study.
Bornological spaces and Closed Graph Theorems
=============================================
Closed graph {#closed-graph .unnumbered}
------------
Let $u: E \to F$ be a map of sets, then the set $$\Gamma(u) = \{ (x, y) \in E \times F \ | y = u(x) \ \}$$ is called the graph of $u$. If $E$ and $F$ are Hausdorff topological spaces and $u$ a continuous map, then $\Gamma(u)$ is a closed subspace of $E \times F$ endowed with the product topology. This is a basic property of Hausdorff topological spaces. If $E$ and $F$ are vector spaces over a field $K$ and $u$ is a linear map, then $\Gamma(u)$ is a vector subspace of $E \times F$. If $E$ and $F$ are separated bornological vector spaces over a complete, non-trivially valued field $K$ and $u$ is linear and bounded, then $\Gamma(u)$ is bornologically closed in $E \times F$ endowed with the product bornology (see below for what it means for a subset of a bornological vector space to be bornologically closed). This assertion is pretty easy to check. Let $(x_n, u(x_n))$ be a sequence of elements of $\Gamma(u)$ which converges bornologically to $( x, y)$ in $E \times F$. Then, by definition of product bornology, $x_n \to x$ in $E$ and $u(x_n) \to y$ in $Y$. Since $u$ is bounded, the sequence $(u(x_n))$ converges bornologically to $u(x)$ and since $Y$ is separated, we must have $y = u(x)$. Therefore $(x, y) \in \Gamma(u)$ and $\Gamma(u)$ is bornologically closed in $E \times F$.
The Closed Graph Theorems are converses of the above statements for some special class of bornological or topological $K$-vector spaces. Here we pursue the main ideas of [@G] for which the born
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: |
We develop a new approach for building cryptographic implementations. Our approach goes the last mile and delivers assembly code that is provably functionally correct, protected against side-channels, and as efficient as hand-written assembly. We illustrate our approach using -, one of the mandatory ciphersuites in TLS 1.3, and deliver formally verified vectorized implementations which outperform the fastest non-verified code.
We realize our approach by combining the framework, which offers in a single language features of high-level and low-level programming, and the proof assistant, which offers a versatile verification infrastructure that supports proofs of functional correctness and equivalence checking. Neither of these tools had been used for functional correctness before. Taken together, these infrastructures empower programmers to develop efficient and verified implementations by game hopping, starting from reference implementations that are proved functionally correct against a specification, and gradually introducing program optimizations that are proved correct by equivalence checking.
We also make several contributions of independent interest, including a new and extensible verified compiler for , with a richer memory model and support for vectorized instructions, and a new embedding of in .
author:
-
bibliography:
- 'dblp.bib'
- 'abbrev3.bib'
- 'crypto.bib'
title: 'The Last Mile: High-Assurance and High-Speed Cryptographic Implementations'
---
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'Given a rational homology 3–sphere $M$ with $|H_1(M,\Z)|=b$ and a link $L$ inside $M$, colored by odd numbers, we construct a unified invariant $I_{M,L}$ belonging to a modification of the Habiro ring where $b$ is inverted. Our unified invariant dominates the whole set of the $SO(3)$ Witten–Reshetikhin–Turaev invariants of the pair $(M,L)$. If $b=1$ and $L=\emptyset$, $I_M$ coincides with Habiro’s invariant of integral homology 3–spheres. For $b>1$, the unified invariant defined by the third author is determined by $I_M$. One of the applications are the new Ohtsuki series (perturbative expansions of $I_M$ at roots of unity) dominating all quantum $SO(3)$ invariants.'
address:
- 'Institut für Mathematik, Universität Zurich, Winterthurerstrasse 190, 8057 Zürich, Switzerland'
- 'Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332–0160, USA '
author:
- Anna Beliakova
- Irmgard Bühler
- Thang Le
title: |
A unified quantum SO(3) invariant\
for rational homology 3–spheres
---
Introduction {#introduction .unnumbered}
============
Background {#background .unnumbered}
----------
The $SU(2)$ Witten–Reshetikhin–Turaev (WRT) invariant is defined for any closed oriented 3–manifold $M$ and any root of unity $\xi$ [@Tu]. Kirby and Melvin [@KM] introduced the $SO(3)$ version of the invariant $\tau_M(\xi)\in \Q(\xi)$ for roots of unity $\xi$ of odd order. If the order of $\xi$ is prime, then by the results of Murakami [@Mu] (also Masbaum–Roberts [@MR]), $\tau_M(\xi)$ is an algebraic integer. This integrality result was the starting point for the construction of finite type 3–manifold invariants, Ohtsuki series [@Ohtsukibook], integral TQFTs, representations of the mapping class group over $\Z[\xi]$ [@GM], and categorification of quantum 3–manifold invariants [@Kho]. The proofs in [@Mu] and [@MR] depend heavily on the arithmetic of $\Z[\xi]$ for a root of unity $\xi$ of [*prime*]{} order and do not extend to other roots of unity.
Is it true that $\tau_M(\xi)$ is always an algebraic integer (belongs to $\Z[\xi]$), even when the order of $\xi$ is not a prime? The positive answer to this question was given first for [*integral homology spheres*]{} by Habiro [@Ha], and then for arbitrary 3–manifolds by the first and third author [@BL], in connection with the study of “strong integrality”.
What Habiro proved for integral homology 3–spheres is actually much stronger than integrality. For any integral homology 3–sphere $M$, Habiro [@Ha] constructed a [*unified invariant*]{} $J_M$ whose evaluation at any root of unity coincides with the value of the Witten–Reshetikhin–Turaev invariant at that root. Habiro’s unified invariant $J_M$ is an element of the following ring (Habiro’s ring) $$\Habiro:=\lim_{\overleftarrow{\hspace{2mm}k\hspace{2mm}}}
\frac{\Z[q]}{
((q;q)_k)}, \qquad \text{ where} \quad (q;q)_k = \prod_{j=1}^k (1-q^j).$$ Every element $f(q)\in \Habiro$ can be written as an infinite sum $$f(q)= \sum_{k\ge 0} f_k(q)\, (1-q)(1-q^2)...(1-q^k),$$ with $f_k(q)\in \Z[q]$. When $q=\xi$, a root of unity, only a finite number of terms on the right hand side are not zero, hence the right hand side gives a well–defined value, called the evaluation $\ev_\xi(f(q))$. Since $f_k(q)\in \Z[q]$, $\ev_\xi(f(q))\in \Z[\xi]$ is an algebraic integer. The fact that the unified invariant belongs to $\Habiro$ is stronger than just integrality of $\tau_M(\xi)$. We will refer to it as “strong” integrality.
The Habiro ring has beautiful arithmetic properties. Every element $f(q) \in \Habiro$ can be considered as a function whose domain is the set of roots of unity. Moreover, there is a natural Taylor series for $f$ at every root of unity. Two elements $f,g \in \Habiro$ are the same if and only if their Taylor series at a root of unity coincide. In addition, each function $f(q) \in \Habiro$ is totally determined by its values at, say, infinitely many roots of order $3^n,\, n\in \N$. Due to these properties the Habiro ring is also called a ring of “analytic functions at roots of unity” [@Ha]. Thus belonging to $\Habiro$ means that the collection of the $SO(3)$ WRT invariants is far from a random collection of algebraic integers; together they form a nice function.
Perturbative expansion at 1 of WRT invariants for rational homology 3–spheres was first constructed by Ohtsuki in the case when the order of the quantum parameter $\xi$ is prime [@Oh]. General properties of the Habiro ring imply that for any integral homology 3–sphere $M$, the Taylor expansion of the unified invariant $J_M$ at $q= 1$ coincides with the Ohtsuki series and dominates WRT invariants of $M$ at all roots of unity (not only of prime order).
To generalize Habiro’s results to rational homology 3–spheres, new ideas and techniques are required. Strong integrality of quantum invariants for rational homology 3–spheres was studied in [@Le] and [@BL]. Among other things, in [@Le] a unified invariant was constructed for the case when the order $r$ of the quantum parameter $\xi$ is coprime with $b$. In [@BL], it was proved that for any 3–manifold $M$ (not necessary a rational homology 3–sphere), the $SO(3)$ WRT invariant $\tau_M(\xi)$ is always an algebraic integer, i.e. $\tau_M(\xi)\in \Z[\xi]$ with no restriction on the order of $\xi$ at all. There we used a (2nd order) Laplace transform method [@BBL] and a difficult identity of Andrews [@And] in $q$–calculus, generalizing those of Rogers–Ramanujan.
Thus, although we have had integrality of all SO(3) WRT invariants, we still lacked a “strong integrality” for the case when $(r,b) \neq 1$. This is the main object of this paper.
In this paper we will generalize Habiro’s construction of the unified invariant to all rational homology 3–spheres. Our new unified invariant $I_M$ dominates SO(3) WRT invariants also in the case when the order $r$ of the quantum parameter is not coprime with $b=|H_1(M,\Z)|$. Although this includes the case $(r,b)=1$ of [@Le], the ring our invariant belongs to is simpler than the one obtained in [@Le] and [@BL]. In particular, we don’t need any fractional power of $q$. We show that the Taylor expansion of our unified invariant at a root of unity of order $c$ (new Ohtsuki series) dominates all WRT invariants with $r=cl$ and $(l,b)=1$.
For rational homology 3–spheres the universal finite type invariant was constructed by Le, Murakami and Ohtsuki [@LMO]. It determines Ohtsuki series and, hence, $\{\tau_M(\xi)\,|\,({\operatorname{ord}}(\xi),b)=1\}$ [@Le]. An interesting open question is whether the Le–Murakami–Ohtsuki invariant determines $I_M$.
Results {#results .unnumbered}
-------
The WRT or quantum $SO(3)$ invariant $\tau_{M,L}(\xi)$ is defined for a pair of a closed 3–manifold $M$ and a link $L$ in it, with link components colored by integers. Here $\xi$ is a root of unity of odd order. We will recall the definitions in Section \[defs\].
Suppose $M$ is a rational homology 3–sphere, i.e. $|H_1(M,\Z)|:={\rm card}\,H_1 (M,\Z) < \infty$. There is a unique decomposition $ H_1(M,\Z)=\bigoplus_{i}
\Z/{b_{i}\Z}$, where each $b_i$ is a prime power. We renormalize the $SO(3)$ WRT invariant of the pair $(M, L)$ as follows: $$\tau'_{M, L}(\xi)=\frac{\tau_{M,
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'We use resistive magnetohydrodynamical simulations with the nested grid technique to study the formation of protoplanetary disks around protostars from molecular cloud cores that provide the realistic environments for planet formation. We find that gaseous planetary-mass objects are formed in the early evolutionary phase by gravitational instability in regions that are decoupled from the magnetic field and surrounded by the injection points of the magnetohydrodynamical outflows during the formation phase of protoplanetary disks. Magnetic decoupling enables massive disks to form and these are subject to gravitational instability, even at $\sim$ 10 AU. The frequent formation of planetary-mass objects in the disk suggests the possibility of constructing a hybrid planet formation scenario, where the rocky planets form later under the influence of the giant planets in the protoplanetary disk.'
author:
- 'Shu-ichiro Inutsuka, Masahiro N. Machida, and Tomoaki Matsumoto'
title: Emergence of Protoplanetary Disks and Successive Formation of Gaseous Planets by Gravitational Instability
---
Introduction
============
Recent direct imaging of outer planets in extra-solar planetary systems [@Kalas+2008; @Marois+2008; @Lagrange+2008; @Greaves+2008; @Thalmann+2009] provides a challenging question to the theory of planet formation: how are giant planets formed in the distant regions far from the central star? Almost all the dynamical timescales for the various important processes in planet formation essentially scale with the Kepler rotation timescale [e.g., @KokuboIda2002]. In the standard core-accretion scenario, there are severe timescale constraints on the in situ formation of Jovian planets, even at 5 AU from the central star [e.g., @Pollack+1996; @IkomaNakazawaEmori2000; @HubickyjBodenheimerLissauer2005; @HoriIkoma2008; @Lissauer+2009; @Machida+2010]. Consequently, the formation of planets at greater distances (up to $\sim 10^2$ AU) seems to be impossible within an observationally reasonable timescale in the core-accretion scenario of planet formation [e.g., @IdaLin2004]. On the other hand, the formation of giant planets due to gravitational instability of massive protoplanetary disks has been extensively investigated by various authors [e.g., see review by @Durisen+2007]. However, most analyses focused only on the evolution of hypothetical disks and cannot be applied directly to actual systems. The result of gravitational instability or absence of it does depend on how the protoplanetary disks are formed, but the formation of the protoplanetary disk is closely related to the formation of the central star. In other words, the initial conditions of planet formation by gravitational instability should be provided by the star formation process.
The last decade has seen dramatic progress in our understanding of protostar formation [e.g., @AndreBasuInutsuka2008] and now provides us with the opportunity to study the formation phase of protoplanetary disks. A highlight of the recent non-ideal magnetohydrodynamical (MHD) calculations of protostellar collapse from molecular cloud cores is the driving of outflows from the first cores and well-collimated fast jets from the protostars; these may be regarded as proof of the importance of various physical processes such as the ohmic dissipation of magnetic fields due to the low degree of ionization at higher density phases. In this Letter, we study the formation phase of protoplanetary disks in a self-consistent, non-ideal MHD system. Our protostellar collapse calculations start from a molecular cloud core, include (practically) all the realistic physical processes, and show how knowledge of the protostar formation process provides convincing evidence for the formation of giant planets in the early phase.
Non-Ideal MHD Simulations
=========================
The initial condition of our resistive MHD simulation is the critical Bonner-Ebert Sphere (an isothermal sphere at gravitational equilibrium) with a temperature of 10 K and a radius of 4750 AU. In order to initiate gravitational collapse, we increase the density uniformly by a factor of 3 to an initial central density of $3~\times~10^6~{\rm cm}^3$. The mass is 1.6 times the solar mass. In a typical simulation, we use a uniform rotation of the angular velocity of $\Omega_{\rm init}=1.1\times10^{-13}{\rm s}^{-1}$ and a uniform magnetic field strength of $B_{\rm init}=37~\mu$ G in the initial state. The initial ratios of thermal, rotational, and magnetic energy to the negated gravitational energy are 0.3, 0.005, 0.014, respectively [cf., @Machida+2008c]. We adopt the nested-grid scheme in order to increase the spatial resolution of the central region [@Machida+2005]; the number of nest-grids is typically 12 and each grid has $n_x~\times~n_y~\times n_z = 128~\times~128\times~32$ cells [@Machida+2006c]. As a result, spatial resolution of the innermost grid is 0.58AU and that of the outermost grid is 1200AU.
To describe a realistic evolution of the magnetic field in protostar formation, we should take into account the non-ideal MHD effects of weakly ionized molecular gas. In general, the ambipolar diffusion is important in the low-density phase, but it is slow and not critical in the dynamically collapsing state, In the intermediate-density phase, the Hall term effect can produce a modest effect depending on the size distribution of dust grains [@WardleNg1999]. In contrast, ohmic dissipation dominates in the high-density phase, and is shown to be the most efficient mechanism for the dissipation of magnetic field in the magnetically supercritical cloud core [e.g., @NakanoNishiUmebayashi2002]. Therefore, we model the non-ideal effects of the magnetic field by the effective resistivity in the induction equation. We adopt the resistivity evolution of the fiducial model of @MachidaInutsukaMatsumoto2007 that corresponds to the ionization equilibrium in standard molecular clouds. As the cloud core collapses, the degree of ionization decreases with increasing density. The gas becomes magnetically decoupled at densities of around $10^{10} {\rm cm}^{-3}$ but couples again when the temperature exceeds about 1000 K. The adopted equation of state is the same as that used in @Machida+2009 that follows the radiation hydrodynamical calculations of protostellar collapse [@MasunagaInutsuka2000].
Results
=======
![ Bird’s eye-view of the result of non-ideal MHD simulation with nested grid technique, covering the evolution of the molecular cloud core to the protostar. The left panel shows the structure in the grid, level $l=8$, where the high-density region ($n=10^{10}{\rm cm}^{-3}$; blue isodensity surface) and magnetic field lines are plotted. Two cocoon-like structures (brown) above and below the flattened core denote the zero-velocity surface inside of which the gas is outflowing from the center. The density contours (color and contour lines) and velocity vectors (thin white arrows) are projected in each wall surface. The right upper panel shows the structure in part of the 10th grid, where we can clearly see the central cavity in the outflowing region. The right lower panel (12th grid) shows the protoplanetary disk in the formation phase, and two newly formed planetary-mass objects in the disk. []{data-label="fig:1"}](inutsuka_fig1.eps){width="170mm"}
Figure 1 shows a typical “bird’s eye-view” snapshot of our simulations. The timescale of the gravitational collapse (i.e., free-fall time) is a decreasing function of density; therefore, the dense central region shrinks faster than the less-dense surrounding regions. This property of gravitational collapse almost always leads to the successive decrease of mass inside the faster shrinking region in a run-away manner, This gravitational “run-away collapse” is decelerated by a gradual increase in the temperature of the central region, and eventually a quasi-steady object called “the first core” is formed [@Larson1969; @WinklerNewman1980a; @WinklerNewman1980b; @MasunagaMiyamaInutsuka1998]. It consists mainly of hydrogen molecules and has a radial extent exceeding 10 AU. The molecular gas surrounding the first core continues to accrete onto it, resulting in a slow but monotonic increase in density and temperature at the center of the first core. If the initial angular momentum of the molecular cloud core is on the order of the value suggested by observation, the resultant first core rotates significantly fast, and its formation corresponds to the onset of bipolar outflows driven by magnetic fields . In Figure 1, the outflow regions are shown by the two brown cocoon-like structure that correspond to the zero vertical velocity surfaces ($v_z=0$): gas inside the cocoons has a significant vertical velocity. Most of the angular momenta in gravitationally collapsing objects are removed by the Maxwell stress of the field, which is called magnetic braking [@MachidaInutsukaMatsumoto2007]. In addition the outflowing gas carries away angular momentum during this phase. When the central temperature becomes sufficiently high ($\sim 2\times 10^3$K), the dissociation of hydrogen molecules becomes significant, providing effective cooling that makes the core gravitationally unstable, triggering “
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'Vehicular networks, an enabling technology for Intelligent Transportation System (ITS), smart cities, and autonomous driving, can deliver numerous on-board data services, e.g., road-safety, easy navigation, traffic efficiency, comfort driving, infotainment, etc. Providing satisfactory quality of service (QoS) in vehicular networks, however, is a challenging task due to a number of limiting factors such as hostile wireless channels (e.g., high mobility or asynchronous transmissions), increasingly fragmented and congested spectrum, hardware imperfections, and explosive growth of vehicular communication devices. Therefore, it is highly desirable to allocate and utilize the available wireless network resources in an ultra-efficient manner. In this paper, we present a comprehensive survey on resource allocation (RA) schemes for a range of vehicular network technologies including dedicated short range communications (DSRC) and cellular based vehicular networks. We discuss the challenges and opportunities for resource allocations in modern vehicular networks and outline a number of promising future research directions.'
author:
- 'Md. Noor-A-Rahim, Zilong Liu, Haeyoung Lee, G. G. Md. Nawaz Ali, Dirk Pesch, Pei Xiao [^1]'
bibliography:
- 'refs.bib'
title: A Survey on Resource Allocation in Vehicular Networks
---
Intelligent Transportation System, Vehicular network, Autonomous Driving, DSRC V2X, Cellular V2X, Resource Allocation, Network Slicing, Machine Learning.
Introduction
============
The prevalent vision is that vehicles (e.g., cars, trucks, trains, etc.) will in the future be highly connected with the aid of ubiquitous wireless networks, anytime and anywhere, to provide unprecedented travel experiences and offer a series of far-reaching benefits such as significantly improved road safety, enhanced situational awareness, less traffic congestion, reduced pollution emission, and lower capital expenditure. Central to this vision is a scalable and intelligent vehicular network which is responsible for efficient information exchange among vehicles and/or between vehicles and infrastructure. As an instrumental enabler for ITS, smart cities, and autonomous driving, vehicular networks have attracted significant research interests in recent years both from the academic and industrial communities [@Liu2017CodingAssisted; @Wang2017Centrality; @Nguyen2018; @Cheng2015; @Ali2018_tvt]. In particular, the concept of connected vehicles, also known as vehicle-to-everything (V2X) communications, has gained substantial momentum by bringing in increased data throughput and enhanced road safety along with novel onboard computing and sensing technologies. So far, there are two major approaches for V2X communications: DSRC and cellular based vehicular communication [@Seo2016; @Bazzi2017]. DSRC is supported by a family of standards including the IEEE 802.11p amendment for Wireless Access in Vehicular Environments (WAVE), the IEEE 1609.1$\sim$0.4 standards for resource management, security, network service, and multi-channel operation [@Kenney2011DSRC]. On the other hand, cellular based vehicular communication, also called C-V2X, designed over cellular networks such as Long-Term Evolution (LTE) and 5G new radio (5G NR), allows every vehicle to communicate with different types of communication entities, such as pedestrians, roadside units (RSU), satellites, internet/cloud, and other vehicles. Both V2X techniques have their respective advantages and limitations when they are adopted in vehicular environments. As a result, an integration of such heterogeneous vehicular networks has been suggested to exploit their unique benefits, while addressing their individual drawbacks.
Wireless networks suffer from a wide range of impairments like shadowing, path loss, time- and/or frequency- selectivity of wireless channels, jamming and/or multi-user interference, etc. To deal with these impairments, radio resources (such as time slots, frequency bands, transmit power levels, etc.) should be allocated in an optimized manner to cater for instantaneous channels and network conditions. Dynamic Resource Allocation (RA) schemes are preferred as they give rise to significantly improved performance (compared to the Static RA schemes) by efficiently exploiting diversities from various dimensions [@Georgiadis2006; @Zhang2010_cog; @Wang2011_mul]. For instance, authors in [@Botsov2014; @Ren2015; @Sun2016; @Sun2016b; @Cheng2017] studied RA schemes for device-to-device (D2D) V2X networks by taking into account fast vehicular channel variations. Nevertheless, RA in vehicular networks are far more challenging due to the following reasons:
1. Highly dynamic mobility from low-speed vehicles (e.g., less than 60 km/h) to high-speed cars/trains (e.g., 500 km/h or higher) [@Zhang2011; @ZijunZhao2013]. The air interface design for high mobility communication, for instance, may require more time-frequency resources in order to combat the impairments incurred by Doppler spread/shifts and multi-path channels.
2. Vast range of data services (e.g., in-car multimedia entertaining, video gaming/conferencing, ultra-reliable and low-latency delivery of safety messages, high-precision map downloading, etc) with different QoS requirements in terms of reliability, latency, and data rates. In particular, some requirements (e.g., high data throughput against ultra-reliability) may be conflicting and hence it may be difficult to support them simultaneously.
3. Explosive growth of vehicular communication devices in the midst of increasingly fragmented and congested spectrum. Moreover, these devices usually have different hardware parameters and therefore may display a wide variation in their communication capabilities under different channel and network conditions. For example, a vehicular sensor device aiming for long battery life (e.g., more than 10 years) is unlikely to use sophisticated signal processing algorithms for power saving purposes whereas more system resources and more signal processing capabilities may be required for ultra-reliable transmission of safety messages.
Driven by these challenges, over the past decade, numerous disruptive ideas and techniques have been emerging aiming for optimizing/addressing various aspects/challenges of vehicular networks. In the existing literature, however, a survey with an extensive high-level overview as well as detailed up-to-date advances on RA in vehicular networks is still lacking to the best of our knowledge. To fill this gap and to stimulate more innovations in this area, we provide a comprehensive survey on the state-of-the-art of RA in vehicular networks and suggest a number of promising research directions.
This article is organized as follows. We start our discourse in Section II by a high-level overview of vehicular networks which include DSRC network, C-V2X network and heterogeneous network. Detailed literature surveys on these three types of vehicular networks are presented in Sections III-V, respectively. As machine learning is gaining ever-increasing research attention in numerous areas such as data-driven decision making, we provide a dedicated survey in Section VI on applications of machine learning for RA in vehicular networks. In Section VII, we summarize three important future directions of the RA research by taking advantage of network slicing, machine learning, and context awareness. Finally, this article is concluded in Section VIII.
Overview of Vehicular Networks
==============================
\[h\]
[0.5]{} {width="\textwidth"}
[0.5]{} {width="\textwidth"}
[0.6]{} {width="\figwidthh"}
DSRC Vehicular Network
----------------------
DSRC is a wide-consensus wireless technology that is designed to support ITS applications in vehicular networks. The underlying standard for DSRC is 802.11p, which is derived from IEEE 802.11e with small modifications in the QoS aspects. DSRC supports communications between vehicles and RSUs. The US Department of Transportation estimates that vehicle-to-vehicle (V2V) communications based on DSRC can eliminate up to 82% of all crashes involving unimpaired drivers in the US, and about 40% of all crashes occurred at intersections [@Kenney2011DSRC]. These statistics imply a significant potential for DSRC technology to reduce crashes and to improve road safety.
DSRC technology supports two classes of devices [@Morgan2010; @Hartenstein2010]: on-board unit (OBU) and road side unit (RSU), which are equivalent to the mobile station (MS) and base station (BS) in traditional cellular systems, respectively. An overview of a typical DSRC vehicular network in shown in Fig. \[fig:DSRC\]. The Federal Communications Commission in the United States has allocated 75 MHz licensed spectrum for DSRC communications in the 5.9 GHz frequency band [@Noor-A-Rahim2018_acc]. Out of the 75 MHz spectrum, 5 MHz is reserved as the guard band and seven 10-MHz channels are defined for DSRC communications. The available spectrum is configured into one control channel (CCH) and six service channels (SCHs). The CCH is reserved for carrying high-priority short messages or control data, while other data are transmitted over the SCHs. Several modulation and coding schemes (MCS) are supported by DSRC with the transmitter (TX) power ranging from 0 dBm to 28.8 dBm. Based on the communication environments, the coverage distance may range from 10m to 1km.
A fundamental mechanism for medium/channel access in DSRC technology
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
abstract: 'Shi and Xie \[8\] predicted the N-shape current-voltage characteristic $(CVC)$ for a 2DEG in a magnetic field where the zero-resistance state has been observed in recent experiments. However it is known that in the absence of a magnetic field the zero resistance state (zero differential resistance state) is achieved in a system with the S-shape $CVC$. The difference in the behaviour of systems with N- and S-shape $CVC$ was studied more than three decades ago (see for example the review \[9\] and references therein) and is briefly explained in this Comment. At present it is not clear whether the N-shape $CVC$ may lead to the zero resistance state in a 2DEG in a magnetic field.'
address: |
$^{1}$ Theoretische Physik III,\
Ruhr-Universität Bochum, D-44780 Bochum, Germany\
$^{2}$Institute of Radioengineering and Electronics of the Russian\
Academy of Sciencies, Moscow 103907, Russia.
author:
- 'A.F. Volkov$^{1,2}$'
title: 'Comment on: ”Radiation-Induced ”Zero-Resistance State” and the Photon Assisted Transport”'
---
An interesting effect has been observed in recent papers [@Klitzing; @Zudov]: the resistance of a 2-DEG subjected to microwave irradiation drops to zero in some interval of an applied magnetic field $B.$ Possible mechanisms for this phenomenon are discussed in a number of papers [@Philips; @Yale; @Andreev; @Anderson; @Shrivastava; @Shi; @Raikh; @Mikhailov]. It was shown in Refs. [@Yale; @Shi; @Raikh] that in the presence of irradiation and a sufficiently strong magnetic field the conductivity $\sigma _{xx}$ may become negative in a weak electric field $E_{x}$. In addition, using a simple model, the authors of Ref. [@Shi] have calculated the current-voltage characteristic $I(V)$ of an irradiated system with a density-of-states periodic in energy $\varepsilon $. They have demonstrated that not only regions with negative conductance $G$, but also regions with positive $G$, may have negative differential conductance $G_{d}=dI/dV$ on the current-voltage characteristic ($CVC$) curve.
In the present Comment I would like to note that effects in systems with negative differential conductance (or resistance) depend in a crucial way on the type of $CVC$, i.e. whether it has the N- or S-shape form (see, for example, the review [@KoganVolkovUs] and references therein). It was established long ago that the states corresponding to regions with negative $%
G_{d}$ on $CVC$ are unstable with respect to nonhomogeneous fluctuations. However the type of instability and final nonhomogeneous state which is formed as a result of the instability depend on the type of $CVC.$
In the case of the N-shape $CVC$ (three voltages $V_{i}$ correspond to one current $I$) a domain with a strong electric field (as happens, for example, in the Gunn effect) arises as a result of an instability in a homogeneous initial state (it is assumed that the total voltage is fixed). This domain moves in the direction of the applied electric field $E=V/L$ ($L$ is the length of the sample). In the presence of the high field domain the $CVC$ changes drastically: an almost flat region (plateau) appears on the $I(V)$ curve. The differential conductance $G_{d}$ (but not the resistance $%
R_{d}=dV/dI$) is zero on this plateau. The current $I_{pl}$ corresponding to this flat part of $CVC$ depends on the particular form of the $CVC$. In the case of a ”symmetric” $I(V)$ curve with absolute negative conductance (i.e. min{$dI/dV$} corresponds to $V=0$ ) this current is equal to zero (that is the total conductance is zero). This means that when the bias voltage varies in some limits, the current remains equal to zero. The $CVC$ obtained in Ref. [@Shi] on the basis of a toy model can be regarded as a chain of N-shape $I(V)$ curves.
In the case of the S-shape $CVC$ (three currents $I_{i}$ correspond to one voltage $V$) small perturbations increase in the direction transverse to the current $I$ if this current corresponds to the part of $CVC$ with negative $%
R_{d}$. As a result of this instability, a domain of a strong current density arises in the sample (it is assumed that total current is fixed, otherwise the system goes over to a state on the stable part of the $CVC$). In the presence of this current domain (or a filament with a higher current density) the $CVC$ is modified so that an almost vertical part appears on the $I(V)$ curve ($R_{d}$ is close to zero). The electric field $E_{ver}$ corresponding to this part of the $CVC$ is again determined by the shape of $%
CVC$. All these statements concerning the S-type of $CVC$ are based on the study of the so called superheating mechanism of the S-shape $I(V)$ curve [@KoganVolkovZh] . In the case of a ”symmetric” S-shape $CVC$ with negative $G$ (i.e. min{$dV/dI$} corresponds to $I=0$) the field $E_{ver}$ is zero (when the bias current varies in some limits, the voltage across the sample is zero ). Similar conclusions were made in a recent paper on a simple phenomenological model [@Andreev].
Of course the behaviour of a system with the N- or S-shape $CVC$ in the absence or presence of a magnetic field $B$ is different. In the latter case the Hall field $E_{y}$ arises for example if the transverse current $%
I_{y}$ is zero. The form of the $CVC$ of a homogeneous 2DEG (in the absence of domains of the electric field or current) depends on whether the Hall current $I_{y}$ or the Hall field $E_{y}$ is zero. For instance, if the $%
I_{x}(V_{x})$ dependence corresponds to the N-shape $CVC$ in the absence of the Hall field $E_{y}$ $(I_{y}\neq 0)$, this correspondence changes completely in the absence of the Hall current $I_{y}$ ($E_{y}\neq 0$). It acquires a complicate form and can not be assigned to the N- or S-shape type of $CVC$ (several values of $I_{x}$ correspond to one $E_{x}$ and several values of $E_{x}$ correspond to one $I_{x}$). Therefore one has to analyse the type of instability and the form of the final state of the system separately for these two cases ($I_{y}=0$ or $E_{y}=0$).
To summarize, one can conclude that the absolute negative conductance obtained in Refs. [@Yale; @Shi; @Raikh] is not sufficient to explain the observed zero resistance (or differential resistance) states. In a system with negative conductance at low dc fields $E,$ the $CVC$ may have either the N- or S-shape form, and the behaviour of the system will be different in these both cases. I also would like draw attention to the fact that negative conductance was predicted long ago in Refs.[@Ryzhii; @Suris; @Elesin; @V'yurkov] (2DEG and 3DEG in a quantizing magnetic field in the presence of microwave irradiation) and in Refs.[@Epstein] (superlattices under ac irradiation).
I would like to thank Sh.M.Kogan as well as B.Huckestein, N.Rick and J.Shi for useful discussions and comments.
R.G.Mani, J.H. Smet, K. von Klitzing, V. Narayanmurti, W.B. Jonson, V. Umansky, Nature [**420**]{}, 646 (2002).
M.A. Zudov, R.R. Du, L.N. Pfeiffer, and K.W. West, Phys.Rev.Lett. [**90**]{}, 046807 (2003); M.A. Zudov, R.R. Du, J.A. Simmons, and L. Reno, Phys.Rev. [**B 64**]{}, R201311 (2001).
J.C.Phillips, cond-mat/0212416, 0303181; 0303184.
A.C. Durst, S. Sachdev, N. Read, and S.M. Girvin, cond-mat/0301569.
A.V.Andreev, I.L. Aleiner, and A.J.Millis, cond-mat/0302063.
P.W. Anderson and W.F. Brinkman, cond-mat/0302129.
K. Shrivastava, cond-mat/0302320.
J.Shi and X.C. Xie, cond-
|
{
"pile_set_name": "ArXiv"
}
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---
author:
- |
M. Vivekanand[^1]\
National Center for Radio Astrophysics, TIFR,\
Pune University Campus, P. O. Box 3,\
Ganeshkhind, Pune 411007, India.
title: |
The issue of aliasing in 0943\
II Signal processing arguments
---
[**Abstract**]{}: [@DR1999; @DR2001] claim that the frequency of the very narrow feature, in the spectrum of radio flux variations of 0943, is an alias of its actual value. They also claim to have detected an amplitude modulation on the above phase modulation. This paper argues that both these claims are unjustified. [**Keywords**]{}: pulsars: general – pulsars: individual (0943, 0031): stars – neutron — fluctuation spectrum — aliased feature — signal processing — drifting sub pulses.
Introduction
============
The rotation powered radio pulsar 0943 exhibits sub pulses that are drifting systematically from period to period within the observable pulse window. This pulsar has a very narrow feature in the longitude resolved spectrum of intensity fluctuations; its $Q$, defined as its central frequency divided by its width, is relatively high (@TH1971; @BRC1975; @SO1975). Recently @DR1999 ([-@DR1999], [-@DR2001]; henceforth DR1999 and DR2001) put its $Q$ at $\ge$ 500. It has been speculated that this very narrow spectral feature, occurring at 0.465 cycles per pulsar period \[cpp\], could be an alias of the actual value 0.535 \[cpp\] (@SO1975). [@DB1973] noted such a general possibility in radio pulsars and claimed, based on the phase information in the fluctuation spectrum, that PSR B2303+30 has an aliased spectral feature. This was contested by [@SO1975], who state on their page 326 that “even a phase analysis, contrary to Backer’s statements (@DB1973), is unable to decide between the two possibilities”, viz., whether the spectral feature is aliased or not. Indeed, in a later paper Backer did not repeat such a claim for 0943, whose fluctuation spectrum is similar to that of PSR B2303+30 (@BRC1975); he quoted the the true value as 0.465 \[cpp\]. However, DR1999 and DR2001 recently claimed that the spectral feature is indeed an alias; that the actual value is 0.535 \[cpp\]. In their view [@SO1975] “came to the wrong conclusion” (DR2001). They also claim that the weak, symmetrically spaced sidebands, at 0.027 \[cpp\] away from the above spectral feature “strongly suggest ... a regular, highly periodic amplitude modulation of the ... drifting sub pulse sequences”.
This paper argues that DR1999 and DR2001 are unjustified in drawing these two conclusions. The question, whether 0943 has an aliased spectral feature or not, is still unresolved; and the latter observation is as likely, if not more likely, due to an additional phase modulation of the drifting sub pulses.
A review of the relevant signal processing
==========================================
The signal from an ideal pulsar with drifting sub pulses falls under the topic “pulse position modulation” (PPM); it consists of periodically occurring narrow pulses whose positions are modulated by another periodicity. Its general principles can be found in books on electronic communication engineering (see @BPL1998). The spectrum of a PPM signal when the position modulation is due to a pure “tone” (a single frequency) is given by [@PP1965] on his page 541 and by @SBS1966 ([-@SBS1966]; henceforth SBS1966) on their pages 252 – 253.
Frequency domain discussion
---------------------------
To begin with let us assume that the drifting sub pulse pattern is PPM due to a pure tone.
The abscissa in fig. 6-2-3 on page 251 of SBS1966 represents the phase of the sampling signal at which a pulse is observed; in our case the sampling signal has the pulsar period $P$ (sec), or frequency $1 / P$ Hz or 1 \[cpp\]. The ordinate represents the corresponding phase of the modulating signal; in our case it has period $P3$ pulsar periods (frequency $1 / P3$ \[cpp\]) which is the repetition time of the drifting sub pulse pattern. The motion in time of the drifting sub pulses in this figure is along a straight line of slope $1 / P3$. In the last para on their page 254, SBS1966 state “the average pulse repetition frequency should be at least twice the highest signal frequency in order to obtain the minimum number of samples necessary for satisfactory signal recovery”; i.e., $1 > 2 / P3$ ($\Rightarrow 1 / P3 < 0.5$) to avoid aliasing. This can be verified by considering two straight lines of slopes $< 0.5$ and $> 0.5$ in fig. 6-2-3 of SBS1966.
Thus, the Nyquist sampling criterion is the same for a PPM signal and a canonical pulse amplitude modulation (PAM) signal (amplitude modulation of periodic pulses). DR2001 are wrong when they claim in their conclusion that “A harmonic resolved fluctuation spectrum uses the information within the finite width of the pulse to achieve a Nyquist frequency of 1 \[cpp\], showing clearly that the primary feature is aliased”. Their average pulse repetition frequency is obviously $P$ (sec); so their Nyquist frequency is only 0.5 \[cpp\]. Consequently, they are also wrong in concluding that “the primary feature is aliased” based merely upon the Fourier technique; they require [ **additional and independent information**]{}, as discussed ahead.
Consider a PAM signal in which the amplitude modulation is due to a pure tone of frequency $\nu$ \[cpp\]. The amplitudes occur at frequencies $\nu$ \[cpp\], $1 \pm \nu$ \[cpp\], $2 \pm \nu$ \[cpp\], etc; and there is no difference in the amplitude spectra of the original ($\nu < 0.5$) and the aliased ($\nu > 0.5$) signal. However the phase spectra differ. Therefore one can distinguish between the original and aliased PAM signals only by using additional information such as the phase of the modulating signal. In the PPM case, the amplitudes in the spectra occur at frequencies $\nu$ \[cpp\], $1 \pm
m \times \nu$ \[cpp\], $2 \pm m \times \nu$ \[cpp\], etc., where $m = 1, 2, 3, ...$ is the order of the harmonic (see eq. 6-2-13 on page 252 of SBS1966). Now, both amplitude and phase spectra differ for the original and aliased signals. Therefore once again, one can distinguish between the original and aliased PPM signals only by using additional information such as the exact shape of the two amplitude spectra. In practice it is impossible to predict this exactly in the current pulsar context.
![ Simulated amplitude spectra of a pulsar signal showing the drifting sub pulse phenomenon. Only a small range of frequency has been shown for better visual comparison. The time series consists of $4 \times 1024 \times 1024$ samples, each of duration 0.25 milli seconds (ms). The pulsar parameters are those of 0943, taken from DR2001: period $P = 1.0977$ (sec), Gaussian integrated profile of width $= 0.031 \times P$ (sec), maximum time departure of drifting sub pulses $= 0.021 \times P$ (sec). The sub pulses are assumed to be Gaussian in shape of very narrow intrinsic width, of $\approx$ one sampling interval; making the width a more realistic value merely suppresses the spectra at higher frequencies. The drifting sub pulse phenomenon is modeled as a PPM signal with a saw-tooth modulation in time. [**Top panel**]{}: Original modulating frequency $= 0.465$ \[cpp\], or $P3 = 2.1505376$ periods; [**Bottom panel**]{}: aliased modulating frequency $= 0.535$ \[cpp\], or $P3 = 1.8691589$ periods, and with opposite drift direction. []{data-label="fig1"}](vivek2-fig1.eps){width="12.5cm"}
Fig. \[fig1\] simulates the amplitude spectra of a pulsar signal showing the drifting sub pulse phenomenon, modeled as a PPM signal with a saw-tooth modulation in time, for both the original modulating frequency (top panel), and its alias but with the opposite drift direction (bottom panel). The pulsar parameters are taken from DR2001 for 0943; however the figure is insensitive to small variations in these parameters. The algorithm was checked by reproducing the spectrum in eq. 6-2-13 of SBS1966. The difference between the two amplitude spectra in fig. \[fig1\] is generally $\le$ 5% of the maximum amplitude of pulsar harmonics, upto a frequency of 100 \[cpp\], for the harmonics $m = 1$ to $3$. For example, at $59.93$ \[cpp\], the second harmonics differ by $\approx$ 12.5% in the two panels; but their average amplitude is $\le 33$% of the maximum amplitude. The third harmonics, aliased to $59.605$ \[cpp\], differ by $\approx$ 9%; but their average amplitude is only about 4.5% of the maximum amplitude. At $110.395$ \[cpp\], the harmonics differ by $\approx$ 20%; but their average amplitude is now about 37.5% of the peak amplitude of the pulsar harmonics;
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{
"pile_set_name": "ArXiv"
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---
abstract: |
The conjecture of Bollobás and Komlós, recently proved by Böttcher, Schacht, and Taraz \[Math. Ann. 343(1), 175–205, 2009\], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and sublinear bandwidth appears as a subgraph of any $2n$-vertex graph $G$ with minimum degree $(1+\gamma)n$, provided that $n$ is sufficiently large. We show that this threshold can be cut in half to an essentially best-possible minimum degree of $(\frac12+\gamma)n$ when we have the additional structural information of the host graph $G$ being balanced bipartite.
This complements results of Zhao \[to appear in SIAM J. Discrete Math.\], as well as Hladký and Schacht \[to appear in SIAM J. Discrete Math.\], who determined a corresponding minimum degree threshold for $K_{r,s}$-factors, with $r$ and $s$ fixed. Moreover, it implies that the set of Hamilton cycles of $G$ is a generating system for its cycle space.
[*Keywords:*]{} Graph theory (05Cxx), Extremal combinatorics (05Dxx), Graph embedding
address:
- 'Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, D-85747 Garching bei München, Germany'
- 'Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, D-85747 Garching bei München, Germany'
- 'Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, D-85747 Garching bei München, Germany'
author:
- Julia Böttcher
- Peter Heinig
- Anusch Taraz
bibliography:
- 'bipartite.bib'
title: Embedding into bipartite graphs
---
Introduction
============
The Bollobás–Komlós conjecture, recently proved in [@BST09], provides a sufficient and essentially best possible minimum degree condition for the containment of $r$-chromatic spanning graphs $H$ of bounded maximum degree and small bandwidth. Here, a graph is said to have bandwidth at most $b$, if there exists a labelling of the vertices by numbers $1,\dots,n$, such that for every edge $\{i,j\}$ of the graph we have $|i-j| \le b$.
\[thm:bk\] For all $r,\Delta\in\mathbb{N}$ and $\gamma>0$, there exist constants $\beta>0$ and $n_0\in\mathbb{N}$ such that for every $n\geq n_0$ the following holds. If $H$ is an $r$-chromatic graph on $n$ vertices with $\Delta(H) \leq \Delta$ and bandwidth at most $\beta n$ and if $G$ is a graph on $n$ vertices with minimum degree $\delta(G) \geq (\frac{r-1}{r}+\gamma)n$, then $G$ contains a copy of $H$.
This theorem in particular implies that for any $\gamma>0$, every [bipartite]{} graph $H$ on $2n$ vertices with bounded degree and sublinear bandwidth appears as a subgraph of any $2n$-vertex graph $G$ with minimum degree $(1+\gamma)n$, provided that $n$ is sufficiently large. This bound is essentially best possible for an almost trivial reason: there are graphs $G$ with minimum degree just slightly below $n$ that are not connected. Such $G$ clearly do not contain a connected $H$ as a subgraph. These graphs are simply too different in structure from $H$.
One may ask, however, whether it is possible to lower the minimum degree threshold in Theorem \[thm:bk\] for graphs $G$ and $H$ that are structurally more similar and, in particular, have the same chromatic number. In this paper we will pursue this question for the case of balanced bipartite graphs, i.e., bipartite graphs on $2n$ vertices with $n$ vertices in each colour class.
Dirac’s theorem [@Dir] implies that a $2n$-vertex graph $G$ with minimum degree at least $n$ contains a Hamilton cycle. If $G$ is balanced bipartite, it follows from a theorem of Moon and Moser [@MooMos] that this minimum degree threshold can be cut almost in half.
\[thm:moon\] Let $G$ be a balanced bipartite graph on $2n$ vertices. If $\delta(G)\geq
\frac{n}{2}+1$, then $G$ contains a Hamilton cycle.
We prove that slightly increasing this minimum degree bound suffices to obtain all balanced bipartite graphs with bounded maximum degree and sublinear bandwidth as subgraphs, and thereby establishing the following bipartite analogue of Theorem \[thm:bk\], halving the minimum degree threshold in that result.
\[thm:bipbw\] For all $\gamma$ and $\Delta$ there is a positive constant $\beta$ and an integer $n_0$ such that for all $n\ge n_0$ the following holds. Let $G$ and $H$ be balanced bipartite graphs on $2n$ vertices such that $G$ has minimum degree $\delta(G)\ge(\frac12+\gamma)n$ and $H$ has maximum degree $\Delta$ and bandwidth at most $\beta n$. Then $G$ contains a copy of $H$.
Results of a similar nature have recently been established by Zhao [@Zhao_bip], and by Hladký and Schacht [@HlaSch] who considered the special case of coverings of $G$ with disjoint copies of complete bipartite graphs. Moreover, as a first step towards Theorem \[thm:bipbw\], in [@heinigbsc] this result was proved for a special balanced bipartite connected graph (the so-called Möbius ladder).
We remark that the bandwidth condition in Theorem \[thm:bipbw\] cannot be omitted. Indeed, Abbasi [@Abbasi] proved that the assertion of Theorem \[thm:bk\] gets false if $\beta>4\gamma$. The graph $H$ he constructs for this purpose is a balanced bipartite graph and it is not difficult to see that Abbasi’s host graph contains a bipartite subgraph meeting our conditions but not containing $H$. However, the bound on $\beta$ coming from our proof is very small, having a tower-type dependence on $1/\gamma$.
The proof of Theorem \[thm:bipbw\] is given in Section \[sec:proof\]. It is based on Szemerédi’s regularity lemma which we introduce in the following section. In Sections \[sec:G\] and \[sec:H\] we provide the proofs of the remaining lemmas that are used in the proof of Theorem \[thm:bipbw\].
The regularity method {#sec:reg}
=====================
In this section we formulate a version of Szemerédi’s regularity lemma [@Szemeredi76] that is convenient for our application (Lemma \[lem:reg\]), introduce all necessary definitions, and formulate an embedding lemma for spanning subgraphs (Lemma \[lem:gel\]).
The regularity lemma relies on the concept of a regular pair. To define this, let $G=(V,E)$ be a graph and $0\le \eps,d\le 1$. For disjoint nonempty vertex sets $U,W\subset V$ the *density* $d(U,W)$ of the pair $(U,W)$ is the number of edges that run between $U$ and $W$ divided by $|U||W|$. A pair $(U,W)$ with density at least $d$ is *$(\eps,d)$-regular* if $|d(U',W')-d(U,W)|\le\eps$ for all $U'\subset U$ and $W'\subset W$ with $|U'|\ge\eps|U|$ and $|W'|\ge\eps|W|$. The following useful property of regular pairs follows immediately from the definition.
\[lem:typical\] Let $G=(A,B)$ be an $(\eps, d)$-regular pair. Let $B'$ be a subset of $B$ with $|B'|\ge\eps|B|$. Then there are at most $\eps|A|$ vertices in $A$ with less than $(d-\eps)|B'|$ neighbours in $B'$.
The regularity lemma asserts that each graph admits a partition into relatively few vertex classes of equal size such that most pairs of these classes form an $\eps$-regular pair. The following definition makes this precise. A partition $V_0\dcup V_1 \dcup\dotsm\dcup V_k$ of $V$ with $|V_0|\le\eps|V|$ is *$(\eps,d)$-regular on* a graph $R=([k],E_R)$ if $ij \in E_R$ implies that $(V_i,V_j)$ is an $(\eps,d)$-regular pair in $G$. If such a partition exists, we also say that
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Manipulating the orbital occupation of valence electrons via epitaxial strain in an effort to induce new functional properties requires considerations of how changes in the local bonding environment affect the band structure at the Fermi level. Using synchrotron radiation to measure the x-ray linear dichroism of epitaxially strained films of the correlated oxide CaFeO~3~, we demonstrate that the orbital polarization of the Fe valence electrons is opposite from conventional understanding. Although the energetic ordering of the Fe $3d$ orbitals is confirmed by multiplet ligand field theory analysis to be consistent with previously reported strain-induced behavior, we find that the nominally higher energy orbital is more populated than the lower. We ascribe this inverted orbital polarization to an anisotropic bandwidth response to strain in a compound with nearly filled bands. These findings provide an important counterexample to the traditional understanding of strain-induced orbital polarization and reveal a new method to engineer otherwise unachievable orbital occupations in correlated oxides.'
author:
- 'Paul C. Rogge'
- 'Robert J. Green'
- Padraic Shafer
- Gilberto Fabbris
- 'Andi M. Barbour'
- 'Benjamin M. Lefler'
- Elke Arenholz
- 'Mark P. M. Dean'
- 'Steven J. May'
title: Inverted orbital polarization in strained correlated oxide films
---
The use of epitaxial strain to induce occupation of specific electron orbitals by removing orbital degeneracies has been pursued in transition metal oxides in an effort to engineer new electronic and magnetic properties [@Tokura_Manganites_XLD; @Aruta_LSMO_XLD; @Strain_OP_Csiszar; @Hansmann_Held_theory_nickelate_SL_OP; @Nickelate_holes_reduced_OP; @Chak_asymmetric_XLD; @Freeland_LNO_XLD; @Wu_nickelate_SL_XLD; @Wu_PrNiO3_orbital_polarization; @Bruno_nickelate_XLD; @Pesquera_LSMO_XLD]. Such strain-induced orbital polarization has been very successfully described by ligand field theory, which considers the overlap of electron orbitals between a central cation and its surrounding anions [@Cox_oxide_book; @Khomskii_book]. For transition metal perovskite oxides, the metal cation is octahedrally coordinated by six oxygen anions, or ligands. This $O_h$ symmetry splits the five degenerate $d$-levels into two groups: a lower, triply degenerate group ($t_{2g}$) and a doubly degenerate group ($e_g$) higher in energy by an amount $10Dq$. Whereas the lobes of the O $p$ orbitals point in between the $t_{2g}$ lobes, they directly overlap with the $e_g$ lobes, which comes at a coulombic energy cost that raises the $e_g$ orbitals in energy. Epitaxial strain alters the local crystal field and lifts the $t_{2g}$ and $e_g$ degeneracies. For example, tensile strain reduces the overlap between the $e_g$ orbital of $d_{x^2-y^2}$ symmetry and its ligands, thus lowering its energy relative to the other $e_g$ orbital, $d_{3z^2-r^2}$, by an amount $\Delta e_g$ \[see Fig. 1(a) inset\]. Unless the $e_g$ orbitals are fully filled, one subsequently expects $d_{x^2-y^2}$ to be preferentially occupied; the converse applies for compressive strain. This simple picture has been used to explain strain-induced orbital polarization in many systems, particularly *AB*O~3~ perovskite oxides [@Tokura_Manganites_XLD; @Aruta_LSMO_XLD; @Hansmann_Held_theory_nickelate_SL_OP; @Nickelate_holes_reduced_OP; @Chak_asymmetric_XLD; @Freeland_LNO_XLD; @Wu_nickelate_SL_XLD; @Wu_PrNiO3_orbital_polarization; @Bruno_nickelate_XLD; @Pesquera_LSMO_XLD]. In this Letter, we find that this model fails to explain orbital polarization in strained films of CaFeO~3~, which exhibit orbital polarization opposite to that described above.
To quantify the electron occupation of specific $e_g$ orbitals, we measure x-ray absorption across the Fe $L$- and O $K$-edge resonance energies using linearly polarized photons, which allows us to differentiate between $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ occupations. Analyzing the x-ray linear dichroism using multiplet ligand field theory reveals that the effect of epitaxial strain on the energetic ordering of the $e_g$ orbitals is consistent with the aforementioned considerations–stretched bonds are lower in energy than unstretched. Given this energetic landscape, however, the expected orbital *occupations* do not follow: The out-of-plane ($d_{3z^2-r^2}$) orbitals are more populated under tensile strain (and *vice versa* for compressive strain). We propose that this inverted orbital polarization arises from strain-induced anisotropic changes in the Fe-O-Fe bond angles and the resulting anisotropic bandwidths in bands that are more-than-half-filled. Such conditions are not limited to ferrates but could arise in other strongly hybridized systems, such as the rare-earth nickelates [@Sawatzky_bond_disproportionation].
![Polarization-dependent x-ray absorption measured by total electron yield (TEY) across the Fe $L$-edge for CaFeO~3~ under (a) tensile strain and (b) compressive strain. Inset: Octahedral crystal field splitting of transition metal $d$ levels for a (001)-oriented film under no strain ($c=a$) and under biaxial tensile strain ($c<a$). \[Fig1\_Fe\_XLD\]](Fig1.eps)
CaFeO~3~ films of 40 pseudocubic unit cells (${\sim}$15 nm thick) were deposited by oxygen plasma-assisted molecular beam epitaxy. Epitaxial strain was achieved by deposition on single crystal, (001)-oriented substrates: YAlO$_3$ (YAO, -2.0% strain), SrLaAlO$_4$ (SLAO, -0.7%), LaAlO$_3$ (LAO, 0.2%), (La$_{0.18}$Sr$_{0.82}$)(Al$_{0.59}$Ta$_{0.41}$)O$_3$ (LSAT, 2.3%), and SrTiO$_3$ (STO, 3.3%). As previously reported, the films are coherently strained and exhibit bulk-like electrical transport, indicating high-quality, stoichiometric films [@Rogge_PRM]. Prior to all measurements, the films were reoxidized by heating to ${\sim}$600 $^\circ$C in oxygen plasma (200 Watts, 1x$10^{-5}$ Torr chamber pressure) and then slowly cooled to room temperature in oxygen plasma. X-ray absorption spectroscopy was performed at the Advanced Light Source, Beamline 4.0.2 and at the National Synchrotron Light Source-II, Beamline 23-ID-1. The spectra were recorded at 290 K, where CaFeO~3~ is paramagnetic with metallic conductivity [@Woodward_CFO]. The x-ray incident angle was $20^{\circ}$ from the film plane, and a geometric correction was applied to the absorption measured with photons polarized out of the film plane [@SI_XLD].
Although CaFeO~3~ has an unusually high formal oxidation state of Fe^4+^, its ground state exhibits a significant self-doped ligand hole density due to its negative charge transfer energy, $\Delta$ [@Kawasaki_CFO_first_transport; @Bocquet_SFO_ligand_holes; @Woodward_CFO; @Matsuno_CFO_dispro; @Takeda_CFO; @Rogge_PRM]. In this regime, the transition metal cation does not adopt its formal oxidation state but instead keeps an extra electron that results in a hole ($\underline{L}^1$) on the oxygen ligand [@ZSA; @Sawatzky_neg_charge_trans_1; @Matsuno_CFO_dispro]. So while CaFeO~3~ has a nominal Fe configuration of $d^4$ ($e_g^1$), its ground state has a strong $d^5\underline{L}^1$ contribution. Because of the half-filled $d$-shell, this $d^5\underline{L}^1$ ($e_g^2$) state has no significant orbital polarization and is expected to decrease the degree of orbital polarization achievable in the Fe states.
X-ray absorption across the Fe $L$-edge for a CaFeO~3~ film under tensile strain is shown in Fig. \[Fig1\_Fe\_XLD\](a). The $L_3$ peak exhibits primarily a single, broad peak (with a small shoulder) that is consistent with nominal Fe^4+^ [@Abbate_SFO_XAS; @Reduced
|
{
"pile_set_name": "ArXiv"
}
| null | null | null |
---
author:
- 'S. Théado'
- 'G. Alecian'
- 'F. LeBlanc'
- 'S. Vauclair'
date: 'Received September 15, 2011; accepted'
title: 'The new Toulouse-Geneva Stellar Evolution Code including radiative accelerations of heavy elements'
---
[Atomic diffusion has been recognized as an important process that has to be considered in any computations of stellar models. In solar-type and cooler stars, this process is dominated by gravitational settling, which is now included in most stellar evolution codes. In hotter stars, radiative accelerations compete with gravity and become the dominant ingredient in the diffusion flux for most heavy elements. Introducing radiative accelerations into the computations of stellar models modifies the internal element distribution and may have major consequences on the stellar structure. Coupling these processes with hydrodynamical stellar motions has important consequences that need to be investigated in detail.]{} [We aim to include the computations of radiative accelerations in a stellar evolution code (here the TGEC code) using a simplified method (SVP) so that it may be coupled with sophisticated macroscopic motions. We also compare the results with those of the Montreal code in specific cases for validation and study the consequences of these coupled processes on accurate models of A- and early-type stars.]{} [We implemented radiative accelerations computations into the Toulouse-Geneva stellar evolution code following the semi-analytical prescription proposed by Alecian and LeBlanc. This allows more rapid computations than the full description used in the Montreal code.]{} [We present results for A-type stellar models computed with this updated version of TGEC and compare them with similar published models obtained with the Montreal evolution code. We discuss the consequences for the coupling with macroscopic motions, including thermohaline convection.]{}
Introduction
============
Accurate stellar modeling has recently been given a new boost with the advent of asteroseismology. The observations of oscillating stars and the analysis of the stellar oscillation properties brought new and powerful constraints on stellar models and allowed major progress in our understanding of stellar internal structure. Furthermore, since the discovery of the first extrasolar planets [@Wolszczan92; @Mayor95], the spectacular development of the exoplanet research field has also sparked renewed interest in stellar physics, the accurate knowledge of the host star being a necessary condition for characterizing the surrounding planets.
One of the most important successes of astrophysics was the understanding of the basics of stellar internal structure and evolution. This progress was supported by the computations of numerical models. Growing computational resources contributed to refining the description of the physics of the stellar medium (equation of state, opacities, nuclear reactions, etc) and allowed building a simplified but efficient and widely used “standard model”. However, this standard model does not take the effects of rotation and magnetic fields or the occurrence of accretion or mass loss into account, and it considers convection as the only chemical transport process.
Observations of chemical abundance anomalies in stars and unexpected stellar seismic behaviors have proved the necessity of including “non standard processes” in stellar evolutionary computations. In this framework, the main challenges encountered today by stellar physicists are to better determine the effects of rotation and magnetic fields and understand the chemical transport processes better. This last point, and more specifically modeling of atomic diffusion including the radiative accelerations on individual elements, represents a key ingredient for accurate stellar modeling.
The importance of atomic diffusion inside stars is well established: not only can it modify the atmospheric abundances [@Michaud70], as observed in the so-called “chemically peculiar stars”, but it can also have strong implications for the stellar internal structure [e.g. @Richard01]. In main-sequence solar-type and cooler stars (below about 1.2 M$_{\odot}$), the radiative accelerations on the heavy elements are generally slower than gravity in absolute value, owing to the small radiation flux compared to hotter stars [@MichaudMiChVaetal1976]. The elements heavier than hydrogen sink, even if the efficiency of this sinking may be modulated by the radiative accelerations [@Turcotte98].
One of the biggest success of helioseismology was to prove the importance of atomic diffusion in the Sun. Its introduction in solar evolutionary models has significantly improved the agreement between the sound speed profile inside solar models and that deduced from helioseismic inversion techniques [e.g. @JCD96; @Richard96; @Brun98; @Turcotte98; @Schlattl02]. (This agreement has however been spoiled by the new abundances proposed by Asplund et al. 2005, 2009.) Gravitational settling is now introduced in most stellar evolution codes.
The effects of atomic diffusion on the seismic frequency of main-sequence, solar-type star models have been studied by [@Theado05]. They have shown that element segregation significantly alters the internal structure of the models and their oscillations frequencies. The frequency differences between models with and without diffusion reach several microHertz for stars with masses greater than 1.3M$_{\odot}$.
In hotter stars, the radiative accelerations may become significantly stronger than gravity for many metals, which are pushed up. The variations with depth of the radiative accelerations of specific elements combined with the selective effects of gravitational settling can lead to element accumulation or depletion in various stellar layers. In A and B-type stars, iron-peak element accumulation appear in the Z-opacity bump (located at $\simeq$200’000K). The induced opacity increase may lead to local convection [@Richer00; @Richard01].
The iron-peak element accumulation in the opacity bump region can help trigger stellar pulsations, therefore improving the agreement between seismic observations and theoretical frequency spectra in many stars: e.g. in $\gamma$ Doradus, Am, SPB, $\beta$ Cephei or sdB stars [see @Theado09 for a detailed discussion of this subject].
The effects of the radiative accelerations on the oscillation frequencies have been tested by @Escobar12. Very weak effects are observed for models with masses up to 1.28M$_{\odot}$, but significant effects are expected for more massive stars.
Progress in the understanding of the physics of early-type stars is limited by the radiative accelerations not being computed in most stellar evolutionary codes. Up to now, the Montreal stellar evolution code is the only one in which a complete, accurate, and consistent treatment of radiative accelerations has been introduced [@Richer00; @Richard01; @Turcotte00]. The price to pay for this accuracy is that the heavy and CPU-time consuming computations of atomic processes do not allow additional sophisticated treatments of macroscopic motions. In the Montreal code, turbulence is only treated as an extremely simplified process with a turbulent diffusion coefficient proportional to a parameterized power of the density.
In this context, we introduced into the Toulouse-Geneva Evolution Code (hereafter TGEC) new computations of the radiative accelerations on heavy elements, following the semi-analytical prescription proposed by @Alecian02 and @LeBlanc04. This prescription leads to fast but reasonably accurate computations, which represent a good compromise between accuracy and CPU-time consumption and allows coupling with macroscopic motions. Such a treatment of abundance variations inside early-type stars is necessary for a good understanding of these stars.
In the present paper, we do not introduce thermohaline convection as described in @Theado09 because we want to present a detailed comparison with the Montreal code in which this specific physical process is not included. Very precise tests of the results obtained by the two codes for iron accumulation inside nearly identical models with similar physics are still underway. In particular, the computations of iron fluxes seem to lead to differences in some cases, which still have to be understood. These tests are beyond the scope of the present paper and will be presented elsewhere in the near future. Here we present a first step in the comparisons, namely the detailed study of the radiative accelerations obtained with the two methods. Some abundance profiles are only shown as indicators of the results that the TGEC code is presently able to obtain.
In the following section, we explain in detail the major improvements implemented in the TGEC code. In Sect. 3, we present a comparison between the Montreal and TGEC computations for two similar stellar models. In Sect. 4, additional models are presented to illustrate the capabilities of the updated TGEC version and its application fields. Our conclusions are given in Sect. 5.
New opacities and atomic diffusion computations in TGEC {#implementation}
=======================================================
The Toulouse-Geneva stellar evolution code is described in detail in @Hui08; however, the code has recently undergone major improvements that are reported in the following sections. The code can follow the time-dependent abundance variations of 21 species (12 elements and their main isotopes: H, $^3$He, $^4$He, $^6$Li, $^7$Li, $^9$Be, $^{10}$B, $^{12}$C, $^{13}$C, $^{14}$N, $^{15}$N, $^{16}$O, $^{17}$O, $^{18}$O, $^{20}$Ne, $^{22}$Ne, $^{24}$Mg, $^{25}$Mg, $^{26}$Mg, $^{40}$Ca, and $^{56}$Fe) in detail. The remaining metals are collected into an average species Z.
Opacities {#opacity}
---------
In the TGEC code, the opacities are computed using the OPCD v3.3 codes and data [@Seaton05]. They allow computating self-consistent Rosseland opacities taking the detailed composition of the chemical mixture into account. The opacities are recalculated by considering the abundance variations at each time step and at each mesh point.
Atomic
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{
"pile_set_name": "ArXiv"
}
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