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We describe an algorithm to compute bases of antisymmetric vectorvalued cusp forms with rational Fourier coefficients for the Weil representation associated to a finite quadratic module. The forms we construct always span all cusp forms in weight at least three. These formulas are useful for computing explicitly with theta lifts.
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The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of $k$cosymplectic geometry. Since these field theories are singular, we need to introduce the notion of $k$precosymplectic structure, which is a generalization of the $k$cosymplectic structure. Next $k$precosymplectic Hamiltonian systems are introduced in order to describe singular field theories, both in Lagrangian and Hamiltonian formalisms. Finally, we develop a constraint algorithm in order to find a submanifold where the existence of solutions of the field equations is ensured. The case of affine Lagrangians is studied as a relevant example.
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We introduce the notion of a continuous Schauder frame for a Banach space. This is both a generalization of continuous frames and coherent states for Hilbert spaces and a generalization of unconditional Schauder frames for Banach spaces. As a natural example, we prove that any wavelet for $L_p(\R)$ with $1<p<\infty$ generates a continuous wavelet Schauder frame. Furthermore, we generalize the properties shrinking and boundedly complete to the continuous Schauder frame setting, and prove that many of the fundamental James theorems still hold in this general context.
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In Eikonal equations, rarefaction is a common phenomenon known to degrade the rate of convergence of numerical methods. The `factoring' approach alleviates this difficulty by deriving a PDE for a new (locally smooth) variable while capturing the rarefactionrelated singularity in a known (nonsmooth) `factor'. Previously this technique was successfully used to address rarefaction fans arising at point sources. In this paper we show how similar ideas can be used to factor the 2D rarefactions arising due to nonsmoothness of domain boundaries or discontinuities in PDE coefficients. Locations and orientations of such rarefaction fans are not known in advance and we construct a `justintime factoring' method that identifies them dynamically. The resulting algorithm is a generalization of the Fast Marching Method originally introduced for the regular (unfactored) Eikonal equations. We show that our approach restores the firstorder convergence and illustrate it using a range of maze navigat
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We develop an approach to study Coulomb branch operators in 3D $\mathcal{N}=4$ gauge theories and the associated quantization structure of their Coulomb branches. This structure is encoded in a onedimensional TQFT subsector of the full 3D theory, which we describe by combining several techniques and ideas. The answer takes the form of an associative noncommutative starproduct algebra on the Coulomb branch. For `good' and `ugly' theories (according to GaiottoWitten classification), we also have a trace map on this algebra, which allows to compute correlation functions and, in particular, guarantees that the starproduct satisfies a truncation condition. This work extends previous work on Abelian theories to the nonAbelian case by quantifying the monopole bubbling that describes screening of GNO boundary conditions. In our approach, the monopole bubbling is determined from the algebraic consistency of the OPE. This also yields a physical proof of the BullimoreDimofteGaiotto abelia
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This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence $\{P_i\}_{i=1}^{\infty}$ generated by a threeterm recurrence relation $P_i(x)+ Q_1(x)P_{i1}(x) +Q_2(x) P_{i2}(x)=0$ with the standard initial conditions $P_{0}(x)=1, P_{1}(x)=0,$ where $Q_1(x)$ and $Q_2(x)$ are arbitrary real polynomials.
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Twodimensional rational CFT are characterised by an integer $\ell$, the number of zeroes of the Wronskian of the characters. For $\ell<6$ there is a finite number of theories and most of these are classified. Recently it has been shown that for $\ell\ge 6$ there are infinitely many admissible characters that could potentially describe CFT's. In this note we examine the $\ell=6$ case, whose central charges lie between 24 and 32, and propose a classification method based on cosets of meromorphic CFT's. We illustrate the method using theories on Kervaire lattices with complete root systems.
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We study derivativefree methods for policy optimization over the class of linear policies. We focus on characterizing the convergence rate of these methods when applied to linearquadratic systems, and study various settings of driving noise and reward feedback. We show that these methods provably converge to within any prespecified tolerance of the optimal policy with a number of zeroorder evaluations that is an explicit polynomial of the error tolerance, dimension, and curvature properties of the problem. Our analysis reveals some interesting differences between the settings of additive driving noise and random initialization, as well as the settings of onepoint and twopoint reward feedback. Our theory is corroborated by extensive simulations of derivativefree methods on these systems. Along the way, we derive convergence rates for stochastic zeroorder optimization algorithms when applied to a certain class of nonconvex problems.
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An amoeba is the image of a subvariety of an algebraic torus under the logarithmic moment map. We consider some qualitative aspects of amoebas, establishing some results and posing problems for further study. These problems include determining the dimension of an amoeba, describing an amoeba as a semialgebraic set, and identifying varieties whose amoebas are a finite intersection of amoebas of hypersurfaces. We show that an amoeba that is not of full dimension is not such a finite intersection if its variety is nondegenerate and we describe amoebas of lines as explicit semialgebraic sets.
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We consider the highly nonlinear and ill posed inverse problem of determining some general expression $F(x,t,u,\nabla_xu)$ appearing in the diffusion equation $\partial_tu\Delta_x u+F(x,t,u,\nabla_xu)=0$ on $\Omega\times(0,T)$, with $T>0$ and $\Omega$ a bounded open subset of $\mathbb R^n$, $n\geq2$, from measurements of solutions on the lateral boundary $\partial\Omega\times(0,T)$. We consider both linear and nonlinear expression of $F(x,t,\nabla_xu,u)$. In the linear case, the equation is a convectiondiffusion equation and our inverse problem corresponds to the unique recovery, in some suitable sense, of a time evolving velocity field associated with the moving quantity as well as the density of the medium in some rough setting described by nonsmooth coefficients on a Lipschitz domain. In the nonlinear case, we prove the recovery of more general quasilinear expression appearing in a nonlinear parabolic equation. Our result give a positive answer to the unique recovery of a gen
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We solve the differentiablity problem for the evolution map in Milnor's infinite dimensional setting. We first show that the evolution map of each $C^k$semiregular Lie group admits a particular kind of sequentially continuity $$ called Mackey continuity $$ and then prove that this continuity property is strong enough to ensure differentiability of the evolution map. In particular, this drops any continuity presumptions made in this context so far. Remarkably, Mackey continuity rises directly from the regularity problem itself $$ which makes it particular among the continuity conditions traditionally considered. As a further application of the introduced notions, we discuss the strong Trotter property in the sequentially, and the Mackey continuous context.
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Let $A$ be a finite dimensional algebra over a field $F$ of characteristic zero. If $L$ is a Lie algebra acting on $A$ by derivations, then such an action determines an action of its universal enveloping algebra $U(L)$. In this case we say that $A$ is an algebra with derivation or an $L$algebra. Here we study the differential $L$identities of $A$ and the corresponding differential codimensions, $c_n^L (A)$, when $L$ is a finite dimensional semisimple Lie algebra. We give a complete characterization of the corresponding ideal of differential identities in case the sequence $c_n^L (A)$, $n=1,2,\dots$, is polynomially bounded. Along the way we determine up to PIequivalence the only finite dimensional $L$algebra of almost polynomial growth.
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It has been known for more than 40 years that there are posets with planar cover graphs and arbitrarily large dimension. Recently, Streib and Trotter proved that such posets must have large height. In fact, all known constructions of such posets have two large disjoint chains with all points in one chain incomparable with all points in the other. Gutowski and Krawczyk conjectured that this feature is necessary. More formally, they conjectured that for every $k\geq 1$, there is a constant $d$ such that if $P$ is a poset with a planar cover graph and $P$ excludes $\mathbf{k}+\mathbf{k}$, then $\dim(P)\leq d$. We settle their conjecture in the affirmative. We also discuss possibilities of generalizing the result by relaxing the condition that the cover graph is planar.
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In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate, in particular, conditions for them to constitute a "continuous basis" for the smallest space $\mathcal D$ of a rigged Hilbert space. This analysis requires suitable extensions of familiar notions as those of frame, Riesz basis and orthonormal basis. A motivation for this study comes from the Gel'fandMaurin theorem which states, under certain conditions, the existence of a family of generalized eigenvectors of an essentially selfadjoint operator on a domain $\mathcal D$ which acts like an orthonormal basis of the Hilbert space $\mathcal H$. The corresponding object will be called here a {\em Gel'fand distribution basis}. The main results are obtained in terms of properties of a conveniently defined {\em synthesis operator}.
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Dynamical transitions of the Acetabularia whorl formation caused by outside calcium concentration is carefully analyzed using a chemical reaction diffusion model on a thin annulus. Restricting ourselves with Turing instabilities, we found all three types of transition, continuous, catastrophic and random can occur under different parameter regimes. Detailed linear analysis and numerical investigations are also provided. The main tool used in the transition analysis is Ma \& Wang's dynamical transition theory including the center manifold reduction.
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The design of block codes for short information blocks (e.g., a thousand or less information bits) is an open research problem that is gaining relevance thanks to emerging applications in wireless communication networks. In this paper, we review some of the most promising code constructions targeting the short block regime, and we compare them with both finitelength performance bounds and classical errorcorrection coding schemes. The work addresses the use of both binary and highorder modulations over the additive white Gaussian noise channel. We will illustrate how to effectively approach the theoretical bounds with various performance versus decoding complexity tradeoffs.
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Modeling via fractional partial differential equations or a L\'evy process has been an active area of research and has many applications. However, the lack of efficient numerical computation methods for general nonlocal operators impedes people from adopting such modeling tools. We proposed an efficient solver for the convectiondiffusion equation whose operator is the infinitesimal generator of a L\'evy process based on $\mathcal{H}$matrix technique. The proposed Crank Nicolson scheme is unconditionally stable and has a theoretical $\mathcal{O}(h^2+\Delta t^2)$ convergence rate. The $\mathcal{H}$matrix technique has theoretical $\mathcal{O}(N)$ space and computational complexity compared to $\mathcal{O}(N^2)$ and $\mathcal{O}(N^3)$ respectively for the direct method. Numerical experiments demonstrate the efficiency of the new algorithm.
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The cost and memoryefficient numerical simulation of coupled volumebased multiphysics problems like flow, transport, wave propagation and others remains a challenging task with finite element method (FEM) approaches. Goaloriented space and time adaptive methods derived from the dual weighted residual (DWR) method appear to be a shiny key technology to generate optimal spacetime meshes to minimise costs. Current implementations for challenging problems of numerical screening tools including the DWR technology broadly suffer in their extensibility to other problems, in high memory consumption or in missing system solver technologies. This work contributes to the efficient embedding of DWR spacetime adaptive methods into numerical screening tools for challenging problems of physically relevance with a new approach of flexible data structures and algorithms on them, a modularised and complete implementation as well as illustrative examples to show the performance and efficiency.
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The dynamics of the fourbody problem have attracted increasing attention in recent years. In this paper, we extend the basic equilateral fourbody problem by introducing the effect of radiation pressure, PoyntingRobertson drag, and solar wind drag. In our setup, three primaries lay at the vertices of an equilateral triangle and move in circular orbits around their common center of mass. Here, one of the primaries is a radiating body and the fourth body (whose mass is negligible) does not affect the motion of the primaries. We show that the existence and the number of equilibrium points of the problem depend on the mass parameters and radiation factor. Consequently, the allowed regions of motion, the regions of the basins of convergence for the equilibrium points, and the basin entropy will also depend on these parameters. The present dynamical model is analyzed for three combinations of mass for the primaries: equal masses, two equal masses, different masses. As the main results, we
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We study the superlinear oscillator equation $\ddot{x}+ \lvert x \rvert^{\alpha1}x = p(t)$ for $\alpha\geq 3$, where $p$ is a quasiperiodic forcing with no Diophantine condition on the frequencies and show that typically the set of initial values leading to solutions $x$ such that $\lim_{t\to\infty} (\lvert x(t) \rvert + \lvert \dot{x}(t) \rvert) = \infty$ has Lebesgue measure zero, provided the starting energy $\lvert x(t_0) \rvert + \lvert \dot{x}(t_0) \rvert$ is sufficiently large.
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We prove existence of all possible biaxisymmetric nearhorizon geometries of 5dimensional minimal supergravity. These solutions possess the crosssectional horizon topology $S^3$, $S^1\times S^2$, or $L(p,q)$ and come with prescribed electric charge, two angular momenta, and a dipole charge (in the ring case). Moreover, we establish uniqueness of these solutions up to an isometry of the symmetric space $G_{2(2)}/SO(4)$.
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We prove existence and uniqueness of solutions to FokkerPlanck equations associated to Markov operators multiplicatively perturbed by degenerate timeinhomogeneous coefficients. Precise conditions on the timeinhomogeneous coefficients are given. In particular, we do not necessarily require the coefficients to be neither globally bounded nor bounded away from zero. The approach is based on constructing random timechanges and studying related martingale problems for Markov processes with values in locally compact, complete and separable metric spaces.
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We characterise the existentially closed models of the theory of exponential fields. We find the amalgamation bases and characterise the types over them. We define a notion of independence and show that independent systems of higher dimension can also be amalgamated. Using these results we position the category of existentially closed exponential fields in the stability hierarchy as NSOP$_1$ but TP$_2$.
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We consider dynamical systems $T: X \to X$ that are extensions of a factor $S: Y \to Y$ through a projection $\pi: X \to Y$ with shrinking fibers, i.e. such that $T$ is uniformly continuous along fibers $\pi^{1}(y)$ and the diameter of iterate images of fibers $T^n(\pi^{1}(y))$ uniformly go to zero as $n \to \infty$. We prove that every $S$invariant measure has a unique $T$invariant lift, and prove that many properties of the original measure lift: ergodicity, weak and strong mixing, decay of correlations and statistical properties (possibly with weakening in the rates).The basic tool is a variation of the Wasserstein distance, obtained by constraining the optimal transportation paradigm to displacements along the fibers. We extend to a general setting classical arguments, enabling to translate potentials and observables back and forth between $X$ and $Y$.
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The problem of sequential probability forecasting is considered in the most general setting: a model set C is given, and it is required to predict as well as possible if any of the measures (environments) in C is chosen to generate the data. No assumptions whatsoever are made on the model class C, in particular, no independence or mixing assumptions; C may not be measurable; there may be no predictor whose loss is sublinear, etc. It is shown that the cumulative loss of any possible predictor can be matched by that of a Bayesian predictor whose prior is discrete and is concentrated on C, up to an additive term of order $\log n$, where $n$ is the time step. The bound holds for every $n$ and every measure in C. This is the first nonasymptotic result of this kind. In addition, a nonmatching lower bound is established: it goes to infinity with $n$ but may do so arbitrarily slow.
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Given two endomorphisms $\tau_1,\tau_2$ of $\mathbb{C}^m$ with $m \ge 2n$ and a general $n$dimensional subspace $\mathcal{V} \subset \mathbb{C}^m$, we provide eigenspace conditions under which $\tau_1(v_1)=\tau_2(v_2)$ for $v_1,v_2 \in \mathcal{V}$ can only be true if $v_1=v_2$. As a special case, we recover the result of Unnikrishnan et al. in which $\tau_1,\tau_2$ are permutations composed with coordinate projections.
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In electricity markets with a dualpricing scheme for balancing energy, controllable production units typically participate in the balancing market as "active" actors by offering regulating energy to the system, while renewable stochastic units are treated as "passive" participants that create imbalances and are subject to less competitive prices. Against this background, we propose an innovative market framework whereby the participant in the balancing market is allowed to act as an active agent (i.e., a provider of regulating energy) in some trading intervals and as a passive agent (i.e., a user of regulating energy) in some others. To illustrate and evaluate the proposed market framework, we consider the case of a virtual power plant (VPP) that trades in a twosettlement electricity market composed of a dayahead and a dualprice balancing market. We formulate the optimal market offering problem of the VPP as a threestage stochastic program, where uncertainty is in the dayahead el
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In the developing theory of infinitedimensional quantum channels the relevance of the energyconstrained diamond norms was recently corroborated both from physical and informationtheoretic points of view. In this paper we study necessary and sufficient conditions for differentiability with respect to these norms of the strongly continuous semigroups of quantum channels (quantum dynamical semigroups). We show that these conditions can be expressed in terms of the generator of the semigroup. We also analyze conditions for representation of a strongly continuous semigroup of quantum channels as an exponential series converging w.r.t. the energyconstrained diamond norm. Examples of semigroups having such a representation are presented.
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We construct a nontrivial type of 1step exceptional BannaiIto polynomials which satisfy discrete orthogonality by using a generalized Darboux transformation. In this generalization, the Darboux transformed BannaiIto operator is directly obtained through an intertwining relation. Moreover, the seed solution, which consists of a gauge factor and a polynomial part, plays an important role in the construction of these 1step exceptional BannaiIto polynomials. And we show that there are 8 classes of gauge factors. We also provide the eigenfunctions of the corresponding multiplestep exceptional BannaiIto operator which can be expressed as a 3 x 3 determinant.
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Optimal control problems are inherently hard to solve as the optimization must be performed simultaneously with updating the underlying system. Starting from an initial guess, Howard's policy improvement algorithm separates the step of updating the trajectory of the dynamical system from the optimization and iterations of this should converge to the optimal control. In the discrete spacetime setting this is often the case and even rates of convergence are known. In the continuous spacetime setting of controlled diffusion the algorithm consists of solving a linear PDE followed by maximization problem. This has been shown to converge, in some situations, however no global rate of is known. The first main contribution of this paper is to establish global rate of convergence for the policy improvement algorithm and a variant, called here the gradient iteration algorithm. The second main contribution is the proof of stability of the algorithms under perturbations to both the accuracy of t
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A graph $G$ is said to be chordal if it has no induced cycles of length four or more. In a recent preprint Culbertson, Guralnik, and Stiller give a new characterization of chordal graphs in terms of sequences of what they call `edgeerasures'. In this note we show that these moves are in fact equivalent to a linear quotient ordering on $I_{\overline{G}}$, the edge ideal of the complement graph $\overline G$. Known results imply that $I_{\overline G}$ has linear quotients if and only if $G$ is chordal, and hence this recovers an algebraic proof of their characterization. We investigate higherdimensional analogues of this result, and show that in fact linear quotients for more general circuit ideals of $d$clutters can be characterized in terms of removing exposed circuits in the complement clutter. Restricting to properly exposed circuits can be characterized by a homological condition. This leads to a notion of higher dimensional chordal clutters which borrows from commutative algebra
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We recall Charles Babbage's 1819 criterion for primality, based on simultaneous congruences for binomial coefficients, and extend it to a leastprimefactor test. We also prove a partial converse of his nonprimality test, based on a single congruence. Two problems are posed. Along the way we encounter Bachet, Bernoulli, Bezout, Euler, Fermat, Kummer, Lagrange, Lucas, Vandermonde, Waring, Wilson, Wolstenholme, and several contemporary mathematicians.
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For each $p>1$ and each positive integer $m$ we give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L^m_p(R)$ to an arbitrary closed subset $E$ of the real line. We show that the classical one dimensional Whitney extension operator is "universal" for the scale of $L^m_p(R)$ spaces in the following sense: for every $p\in(1,\infty]$ it provides almost optimal $L^m_p$extensions of functions defined on $E$. The operator norm of this extension operator is bounded by a constant depending only on $m$. This enables us to prove several constructive $L^m_p$extension criteria expressed in terms of $m^{th}$ order divided differences of functions.
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This is a supplement for "Pearls in graph theory"  a textbook written by Nora Hartsfield and Gerhard Ringel. We discuss bounds on Ramsey numbers, the probabilistic method, deletioncontraction formulas, the matrix theorem, chromatic polynomials, the marriage theorem and its relatives, the Rado graph, and generating functions.
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In this paper we are concerned with the analysis of heavytailed data when a portion of the extreme values is unavailable. This research was motivated by an analysis of the degree distributions in a large social network. The degree distributions of such networks tend to have power law behavior in the tails. We focus on the Hill estimator, which plays a starring role in heavytailed modeling. The Hill estimator for this data exhibited a smooth and increasing "sample path" as a function of the number of upper order statistics used in constructing the estimator. This behavior became more apparent as we artificially removed more of the upper order statistics. Building on this observation we introduce a new version of the Hill estimator. It is a function of the number of the upper order statistics used in the estimation, but also depends on the number of unavailable extreme values. We establish functional convergence of the normalized Hill estimator to a Gaussian process. An estimation proc
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Dyson's celebrated constant term conjecture ({\em J. Math. Phys.}, 3 (1962): 140156) states that the constant term in the expansion of $\prod_{1\leqq i\neq j\leqq n} (1x_i/x_j)^{a_j}$ is the multinomial coefficient $(a_1 + a_2 + \cdots + a_n)!/ (a_1! a_2! \cdots a_n!)$. The definitive proof was given by I. J. Good ({\em J. Math. Phys.}, 11 (1970) 1884). Later, Andrews extended Dyson's conjecture to a $q$analog ({\em The Theory and Application of Special Functions}, (R. Askey, ed.), New York: Academic Press, 191224, 1975.) In this paper, closed form expressions are given for the coefficients of several other terms in the Dyson product, and are proved using an extension of Good's idea. Also, conjectures for the corresponding $q$analogs are supplied. Finally, perturbed versions of the $q$Dixon summation formula are presented.
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Beck's distributive laws provide sufficient conditions under which two monads can be composed, and monads arising from distributive laws have many desirable theoretical properties. Unfortunately, finding and verifying distributive laws, or establishing if one even exists, can be extremely difficult and errorprone. We develop generalpurpose techniques for showing when there can be no distributive law between two monads. Two approaches are presented. The first widely generalizes ideas from a counterexample attributed to Plotkin, yielding generalpurpose theorems that recover the previously known situations in which no distributive law can exist. Our second approach is entirely novel, encompassing new practical situations beyond our generalization of Plotkin's approach. It negatively resolves the open question of whether the list monad distributes over itself. Our approach adopts an algebraic perspective throughout, exploiting a syntactic characterization of distributive laws. This appr
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If spectrum of a Schr\"{o}dinger oparator with a nonHermitian potential contains a spectral singularity (SS), the latter requires exact matching of the parameters characterizing the potential. We provide a necessary and sufficient condition for a potential to have a SS at a given wavelength. It is shown that potentials with SS at prescribed wavelengths can be obtained by a simple and effective procedure. In particular, the developed approach allows one to obtain potentials with several SSs or with SSs of the second order and potentials obeying a given symmetry, say, PTsymmetric potentials. Also, the problem can be solved when it is required to obtain a potential obeying a given symmetry, say, $\mathcal{PT}$symmetric potential. We illustrate all opportunities with examples. We also describe splitting of the SSs of the second order, under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.
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We prove that the known formulae for computing the optimal number of maximally entangled pairs required for entanglementassisted quantum errorcorrecting codes (EAQECCs) over the binary field holds for codes over arbitrary finite fields as well. We also give a GilbertVarshamov bound for EAQECCs and constructions of EAQECCs coming from punctured selforthogonal linear codes which are valid for any finite field.
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We establish existence of the etainvariant as well as of the AtiyahPatodiSinger and the CheegerGromov rhoinvariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature, the GaussBonnet and the spin Dirac operator. We derive an analogue of the AtiyahPatodiSinger index theorem for incomplete edge spaces and their noncompact infinite Galois coverings with edge singular boundary. Our arguments employ microlocal analysis of the heat kernel asymptotics on incomplete edge spaces and the classical argument of AtiyahPatodiSinger. As an application, we discuss stability results for the two rhoinvariants we have defined.
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In this paper we study the $(2+1)$dimensional DiracDunkl oscillator coupled to an external magnetic field. Our Hamiltonian is obtained from the standard Dirac oscillator coupled to an external magnetic field by changing the partial derivatives by the Dunkl derivatives. We solve the DunklKleinGordontype equations in polar coordinates in a closed form. The angular part eigenfunctions are given in terms of the JacobiDunkl polynomials and the radial functions in terms of the Laguerre functions. Also, we compute the energy spectrum of this problem and show that, in the nonrelativistic limit, it properly reduces to the Hamiltonian of the two dimensional harmonic oscillator.
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We discuss generalized partition function of 2d CFTs decorated by higher qKdV charges on thermal cylinder. We propose that in the large central charge limit qKdV charges factorize such that generalized partition function can be rewritten in terms of auxiliary noninteracting bosons. The explicit expression for the generalized free energy is readily available in terms of the boson spectrum, which can be deduced from the conventional thermal expectation values of qKdV charges. In other words, the picture of the auxiliary noninteracting bosons allows extending thermal onepoint functions to the full nonperturbative generalized partition function. We verify this conjecture for the first seven qKdV charges using recently obtained pertrubative results and find corresponding contributions to the auxiliary boson masses. We further extend these results by conjecturing the full spectrum of bosons and find an exact expression for the generalized partition function as a function of infinite towe
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The class of split matroids arises by putting conditions on the system of split hyperplanes of the matroid base polytope. It can alternatively be defined in terms of structural properties of the matroid. We use this structural description to give an excluded minor characterisation of the class.
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In this paper, sufficient conditions are given for the existence of limiting distribution of a conservative affine process on the canonical state space $\mathbb{R}_{\geqslant0}^{m}\times\mathbb{R}^{n}$, where $m,\thinspace n\in\mathbb{Z}_{\geqslant0}$ with $m+n>0$. Our main theorem extends and unifies some known results for OUtype processes on $\mathbb{R}^{n}$ and onedimensional CBI processes (with state space $\mathbb{R}_{\geqslant0}$). To prove our result, we combine analytical and probabilistic techniques; in particular, the stability theory for ODEs plays an important role.
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Facial recognition has to be regulated to protect the public, says AI repor
1207 MIT Technology 6337 
In this paper we present novel $ADE$ correspondences by combining an earlier induction theorem of ours with one of Arnold's observations concerning Trinities, and the McKay correspondence. We first extend Arnold's indirect link between the Trinity of symmetries of the Platonic solids $(A_3, B_3, H_3)$ and the Trinity of exceptional 4D root systems $(D_4, F_4, H_4)$ to an explicit Clifford algebraic construction linking the two ADE sets of root systems $(I_2(n), A_1\times I_2(n), A_3, B_3, H_3)$ and $(I_2(n), I_2(n)\times I_2(n), D_4, F_4, H_4)$. The latter are connected through the McKay correspondence with the ADE Lie algebras $(A_n, D_n, E_6, E_7, E_8)$. We show that there are also novel indirect as well as direct connections between these ADE root systems and the new ADE set of root systems $(I_2(n), A_1\times I_2(n), A_3, B_3, H_3)$, resulting in a web of threeway ADE correspondences between three ADE sets of root systems.
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For a domain $D \subset \mathbb C^n$, $n \ge 2$, let $F^k_D(z)=K_D(z)\lambda\big(I^k_D(z)\big)$, where $K_D(z)$ is the Bergman kernel of $D$ along the diagonal and $\lambda\big(I^k_D(z)\big)$ is the Lebesgue measure of the Kobayashi indicatrix at the point $z$. This biholomorphic invariant was introduced by B\l ocki and in this note, we study the boundary behaviour of $F^k_D(z)$ near a finite type boundary point where the boundary is smooth, pseudoconvex with the corank of its Levi form being at most $1$. We also compute its limiting behaviour near the boundary of certain other basic classes of domains.
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The classical Eulerian polynomials $A_n(t)$ are known to be gamma positive. Define the positive Eulerian polynomial $\mathsf{AExc^{+}}_n(t)$ as the polynomial obtained when we sum excedances over the alternating group. We show that $\mathsf{AExc^{+}}_n(t)$ is gamma positive iff $n \geq 5$ and $n \equiv 1$ (mod 2). When $n \geq 4$, and $n \equiv 0$ (mod 2) we show that $\mathsf{AExc^{+}}_n(t)$ can be written as a sum of two gamma positive polynomials. Similar results are shown when we consider the positive typeD and typeD Eulerian polynomials. Finally, we show gamma positivity results when we sum excedances over derangements with positive and negative sign. Our main resuls is that the polynomial obtained by summing excedance over a conjugacy class indexed by $\lambda$ is gamma positive.
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We shall generalize the notion of a Laver table to algebras which may have many generators, several fundamental operations, fundamental operations of arity higher than 2, and to algebras where only some of the operations are selfdistributive or where the operations satisfy a generalized version of selfdistributivity. These algebras mimic the algebras of rankintorank embeddings $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ in the sense that composition and the notion of a critical point make sense for these sorts of algebras.
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In Kac's classification of finitedimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finitedimensional Lie algebras, but with additional types of nodes. For example, $A(n1,0) = \mathfrak{sl}(1n)$ can be constructed by adding a "gray" node to the Dynkin diagram of $A_{n1} = \mathfrak{sl}(n)$, corresponding to an odd null root. The Cartan superalgebras constitute a different class, where the simplest example is $W(n)$, the derivation algebra of the Grassmann algebra on $n$ generators. Here we present a novel construction of $W(n)$, from the same Dynkin diagram as $A(n1,0)$, but with additional generators and relations.
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We consider links that are alternating on surfaces embedded in a compact 3manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating links of Menasco. We use this to prove various facts about the hyperbolic geometry of generalisations of alternating links, including weakly generalised alternating links described by the first author. We give diagrammatical properties that determine when such links are hyperbolic, find the geometry of their checkerboard surfaces, bound volume, and exclude exceptional Dehn fillings.
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On a reduced analytic space $X$ we introduce the concept of a generalized cycle, which extends the notion of a formal sum of analytic subspaces to include also a form part. We then consider a suitable equivalence relation and corresponding quotient $\mathcal{B}(X)$ that we think of as an analogue of the Chow group and a refinement of de Rham cohomology. This group allows us to study both global and local intersection theoretic properties. We provide many $\mathcal{B}$analogues of classical intersection theoretic constructions: For an analytic subspace $V\subset X$ we define a $\mathcal{B}$Segre class, which is an element of $\mathcal{B}(X)$ with support in $V$. It satisfies a global King formula and, in particular, its multiplicities at each point coincide with the Segre numbers of $V$. When $V$ is cut out by a section of a vector bundle we interpret this class as a MongeAmp\`eretype product. For regular embeddings we construct a $\mathcal{B}$analogue of the Gysin morphism.
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Seeking compliance with Linux's new Code of Conduct, Intel software engineer Jarkko Sakkinen recently requested comments on a set of changes to kernel code comments which Neowin described as "replacing the Fword with 'hug'. " 80 comments quickly followed on the Linux Kernel Maintainer's List: Several contributors responded to the alterations calling them insane. One wondered if Sakkinen was just trying to make a joke, and another called it censorship and said he'd refuse to apply any sort of patches like this to the code he's in charge of... Some of the postchange comments read "Some Athlon laptops have really hugged PST tables", "If you don't see why, please stay the hug away from my code", and "Only Sun can take such nice parts and hug up the programming interface". Eventually LWN.net publisher Jonathan Corbet deflated most of the controversy by pointing out that Linux's new Code of Conduct applies to future comments but clearly indicates that it does not apply explicitly to pas
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A logistics division of DHL announced today that it will invest $300 million to modernize 60 percent of its warehouses in North America with more IoT sensors and robots. Robotic process automation and software made to reduce workflow interruptions will also play a role. VentureBeat reports: Such technology is already in operation in 85 DHL facilities, or roughly 20 percent of warehouses across North America. Funding announced today will bring emerging technology to 350 of DHL Supply Chain's 430 operating sites. The company has more than 35,000 employees in North America. Conversations are ongoing with more than 25 robotics and process automation industry leaders, DHL Supply Chain president of retail Jim Gehr said. DHL Supply Chain warehouse robots will work primarily with unitpicking operations and will be able to complete a range of tasks, from collaborative piece picking to shuttling items across a factory to following human packers.
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Internal Facebook documents seized by British lawmakers suggest that the social media giant once considered selling access to user data, according to extracts obtained by the Wall Street Journal. Back in April, Facebook CEO Mark Zuckerberg told congress unequivocally that, "We do not sell data." But these documents suggest that it was something that the company internally considered doing between 2012 and 2014, while the company struggled to generate revenue after its IPO. This just goes to show that no matter what promises a company makes, once the shareholders come knocking, they'll disregard all promises, morals, and values they claim to have.
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As of today, Fedora 27 will not be getting any more updates, including security updates. Users should be planning to upgrade more or less immediately. "Fedora 28 will continue to receive updates until 4 weeks after the release of Fedora 30. The maintenance schedule of Fedora releases is documented on the Fedora Project wiki. The Fedora Project wiki also contains instructions on how to upgrade from a previous release of Fedora to a version receiving updates."
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George H.W. Bush, the 41st president of the United States, has passed away tonight at the age of 94. As The Washington Post reports, he was "the last veteran of World War II to serve as president, he was a consummate public servant and a statesman who helped guide the nation and the world out of a fourdecade Cold War that had carried the threat of nuclear annihilation." From the report: Although Mr. Bush served as president three decades ago, his values and ethic seem centuries removed from today's acrid political culture. His currency of personal connection was the handwritten letter  not the social media blast. He had a competitive nature and considerable ambition that were not easy to discern under the sheen of his New England politesse and his earnest generosity. He was capable of running hardedge political campaigns, and took the nation to war. But his principal achievements were produced at negotiating tables. Despite his grace, Mr. Bush was an easy subject for caricature. He
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A major difference between Go 1 and Go 2 is who is going to influence the design and how decisions are made. Go 1 was a small team effort with modest outside influence; Go 2 will be much more communitydriven. After almost 10 years of exposure, we have learned a lot about the language and libraries that we didn't know in the beginning, and that was only possible through feedback from the Go community. The Go team s revealing some things about the future of the programming language.
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We explore the representation theory of Renner monoids associated to classical groups and their Hecke algebras. In Cartan type $A_n$, the Hecke algebra is a natural deformation of the rook monoid algebra, and its representation theory has been studied extensively by Solomon and Halverson, among others. It is known that the character tables are block upper triangular, i.e. $M=AY=YB$ for some matrices $A$ and $B$. We compute the $A$ and $B$ matrices in Cartan type $B_n$ by using the results of Li, Li, and Cao to pursue analogous results to those of Solomon. We then compute some type $B_n$ Hecke algebra character values by using the same $B$ matrix as in the monoid case.
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