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We show a method to build new examples of Lie algebras admitting LCS or LCK structures starting with a smaller dimensional Lie algebra endowed with a LCS or LCK structure respectively, and a suitable representation. We also study the existence of lattices in the associated simply connected Lie groups in order to obtain compact examples of manifolds admitting these kind of structures. Finally we show that the Lie algebra underlying of the well known OeljesklausToma solvmanifold can me reobtained using our construction.
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In this article we investigate partially truncated correlation functions (PTCF) of infinite continuous systems of classical point particles with pair interaction. We derive KirkwoodSalsburg (KS)type equations for the PTCF and write the solutions of these equations as a sum of contributions labelled by certain special graphs (forests), the connected components of which are tree graphs. We generalize the method introduced by Minlos and Pogosyan in the case of truncated correlations. These solutions make it possible to derive strong cluster properties for PTCF which were obtained earlier for lattice spin systems.
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In this article we give normal forms in a neighbourhood of a compact orbit of a Poisson Lie group action on a $b$symplectic manifold. In particular we establish cotangent models for Poisson group actions on $b$Poisson manifolds and a $b$symplectic slice theorem. We examine interesting particular instances of PoissonLie group actions on $b$symplectic manifolds. Also, we revise the notion of cotangent lift and twisted $b$cotangent lift introduced in \cite{km} and provide a generalization of the twisted $b$cotangent lift to higher dimensional torus actions. We introduce the notion of $b$Lie group and the associated $b$symplectic structures in its $b$cotangent bundle together with their reduction theory.
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We explore combinatorial formulas for deformations of highest weight characters of the odd orthogonal group $SO(2n+1)$. Our goal is to represent these deformations of characters as partition functions of statistical mechanical models  in particular, twodimensional solvable lattice models. In Cartan type $A$, Hamel and King [8] and Brubaker, Bump, and Friedberg [3] gave square ice models on a rectangular lattice which produced such a deformation. Outside of type $A$, icetype models were found using rectangular lattices with additional boundary conditions that split into two classes  those with `nested' and `nonnested bends.' Our results fill a gap in the literature, providing the first such formulas for type $B$ with nonnested bends. In type $B$, there are many known combinatorial parameterizations of highest weight representation basis vectors as catalogued by Proctor [19]. We show that some of these permit icetype models via appropriate bijections (those of Sundaram [21] and
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Despite CRISPR baby controversy, Harvard University will begin geneediting
1130 MIT Technology 4437 
Let $f(X)=X(1+aX^{q(q1)}+bX^{2(q1)})\in\Bbb F_{q^2}[X]$, where $a,b\in\Bbb F_{q^2}^*$. In a series of recent papers by several authors, sufficient conditions on $a$ and $b$ were found for $f$ to be a permutation polynomial (PP) of $\Bbb F_{q^2}$ and, in characteristic $2$, the sufficient conditions were shown to be necessary. In the present paper, we confirm that in characteristic 3, the sufficient conditions are also necessary. More precisely, we show that when $\text{char}\,\Bbb F_q=3$, $f$ is a PP of $\Bbb F_{q^2}$ if and only if $(ab)^q=a(b^{q+1}a^{q+1})$ and $1(b/a)^{q+1}$ is a square in $\Bbb F_q^*$.
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In this paper, we first present a GearhardtPr\"uss type theorem with a sharp bound for maccretive operators. Then we give two applications: (1) give a simple proof of the result proved by Constantin et al. on relaxation enhancement induced by incompressible flows; (2) show that shear flows with a class of Weierstrass functions obey logarithmically fast dissipation timescales.
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Strong invariants of evendimensional topological insulators of independent Fermions are expressed in terms of an invertible operator on the Hilbert space over the boundary. It is given by the Cayley transform of the boundary restriction of the halfspace resolvent. This dimensional reduction is routed in new representation for the $K$theoretic exponential map. It allows to express the invariants via the reflection matrix at the Fermi energy, for the scattering setup of a wire coupled to the halfspace insulator.
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In the framework of Density Functional Theory with Strongly Correlated Electrons we consider the so called bond dissociating limit for the energy of an aggregate of atoms. We show that the multimarginals optimal transport cost with Coulombian electronelectron repulsion may correctly describe the dissociation effect. The variational limit is completely calculated in the case of N=2 electrons. The theme of fractional number of electrons appears naturally and brings into play the question of optimal partial transport cost. A plan is outlined to complete the analysis which involves the study of the relaxation of optimal transport cost with respect to the weak* convergence of measures.
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The millimeter wave (mmWave) bands and other high frequencies above 6~GHz have emerged as a central component of FifthGeneration (5G) cellular standards to deliver high data rates and ultralow latency. A key challenge in these bands is blockage from obstacles, including the human body. In addition to the reduced coverage, blockage can result in highly intermittent links where the signal quality varies significantly with motion of obstacles in the environment. The blockages have widespread consequences throughout the protocol stack including beam tracking, link adaptation, cell selection, handover and congestion control. Accurately modeling these blockage dynamics is therefore critical for the development and evaluation of potential mmWave systems. In this work, we present a novel spatial dynamic channel sounding system based on phased array transmitters and receivers operating at 60 GHz. Importantly, the sounder can measure multiple directions rapidly at high speed to provide detaile
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We study the extremal particles of the twodimensional Coulomb gas with confinement generated by a radially symmetric positive background in the determinantal case and the zeros of the corresponding random polynomials. We show that when the background is supported on the unit disk, the point process of the particles outside of the disk converges towards a universal point process, i.e. that does not depend on the background. This limiting point process may be seen as the determinantal point process governed by the Bergman kernel on the complement of the unit disk. It has an infinite number of particles and its maximum is a heavy tailed random variable. To prove this convergence we study the case where the confinement is generated by a positive background outside of the unit disk. For this model we show that the point process of the particles inside the disk converges towards the determinantal point process governed by the Bergman kernel on the unit disk. In the case where the background
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We generalize DengDu's folding argument, for the bounded derived category D(Q) of an acyclic quiver Q, to the finite dimensional derived category D(Gamma Q) of the Ginzburg algebra Gamma Q associated to Q. We show that the Fstable category of D(Gamma Q) is equivalent to the finite dimensional derived category D(Gamma\SS) of the Ginzburg algebra Gamma\SS associated to the specie \SS, which is folded from Q. Then we show that, if (Q,\SS) is of Dynkin type, the principal component Stab_0 D(Gamma\SS) of the space of the stability conditions of D(Gamma\SS) is canonically isomorphic to the principal component Stab_0^F D(Gamma Q) of the space of Fstable stability conditions of D(Gamma Q). As an application, we show that, if (Q,\SS) is of type (A_3, B_2) or (D_4, G_2), the space Stab^N D(Gamma Q) of numerical stability conditions in Stab^0 D(Gamma Q), consists of Br Gamma Q/Br Gamma\SS many connected components, each of which is isomorphic to Stab^0 D(Gamma\SS) \cong Stab^F D(Gamma Q).
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We give an overview of the general framework of forms of Bak, Tits and Wall, when restricting to vector spaces over fields, including its relationship to the classical notions of Hermitian, alternating and quadratic forms. We then prove a version of Witt's lemma in this context, showing in particular that the action of the group of isometries of a space equipped with a form is transitive on isometric subspaces.
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We analyze the forward performance process in a general semimartingale market accounting for portfolio constraints, when investor's preferences are homothetic. We provide necessary and sufficient conditions for the construction of such a performance process, and establish its connection to the solution of an infinitehorizon quadratic backward stochastic differential equation (BSDE) driven by a semimartingale. We prove the existence and uniqueness of a solution to our infinitehorizon BSDE using techniques based on Jacod's decomposition and an extended argument of the comparison principle for finitehorizon BSDEs. We show the equivalence between the factor representation of the BSDE solution and the smooth solution to the illposed partial integraldifferential HamiltonJacobiBellman (HJB) equation arising in the extended semimartingale factor framework. Our study generalizes existing results on forward performance in Brownian settings, and shows that timemonotone processes are prese
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We investigate fractional colorings of graphs in which the amount of color given to a vertex depends on local parameters, such as its degree or the clique number of its neighborhood; in a \textit{fractional $f$coloring}, vertices are given color from the $[0, 1]$interval and each vertex $v$ receives at least $f(v)$ color. By Linear Programming Duality, all of the problems we study have an equivalent formulation as a problem concerning weighted independence numbers. However, these problems are most natural in the framework of fractional coloring, and the concept of coloring is crucial to most of our proofs. Our results and conjectures considerably generalize many wellknown fractional coloring results, such as the fractional relaxation of Reed's Conjecture, Brooks' Theorem, and Vizing's Theorem. Our results also imply previously unknown bounds on the independence number of graphs. We prove that if $G$ is a graph and $f(v) \leq 1/(d(v) + 1/2)$ for each $v\in V(G)$, then either $G$ has
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This paper studies both the conductance and charge transport on 2D orbifolds in a strong magnetic field. We consider a family of Landau Hamiltonians on a complex, compact 2D orbifold $Y$ that are parametrised by the Jacobian torus $J(Y)$ of $Y$. We calculate the degree of the associated stable holomorphic spectral orbibundles when the magnetic field $B$ is large, and obtain fractional quantum numbers as the conductance and a refined analysis also gives the charge transport. A key tool studied here is a nontrivial generalisation of the Nahm transform to 2D orbifolds.
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In this article we study a class of delay differential equations with infinite delay in weighted spaces of uniformly continuous functions. We focus on the integrated semigroup formulation of the problem and so doing we provide an spectral theory. As a consequence we obtain a local stability result and a Hopf bifurcation theorem for the semiflow generated by such a problem
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The search for generating compatibility conditions (CC) for a given operator is a very recent problem met in General Relativity in order to study the Killing operator for various standard useful metrics (Minkowski, Schwarschild and Kerr). In this paper, we prove that the link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the socalled snake lemma. It is therefore quite difficult to grasp it in general and even more difficult to use it on explicit examples. It does not seem that any one of the results presented in this paper is known as most of the other authors who studied the above problem of computing the total number of generating CC are confusing this number with a kind of differential transcendence degree, also c
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We investigate a subalgebra of the TemperleyLieb algebra called the JonesWenzl algebra, which is obtained by action of certain JonesWenzl projectors. This algebra arises naturally in applications to conformal field theory and statistical physics. It is also the commutant (centralizer) algebra of the Hopf algebra $U_q(\mathfrak{sl}_2)$ on its typeone modules  this fact is a generalization of the $q$SchurWeyl duality of Jimbo. In this article, we find two minimal generating sets for the JonesWenzl algebra. In special cases, we also find all of the independent relations satisfied by these generators.
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We geometrically describe a canonical sequence of modular blowups of the relative Picard stack of the Artin stack of prestable genus two curves. The final blowup stack locally diagonalizes certain tautological derived objects. This implies a resolution of the primary component of the moduli space of genus two stable maps to projective space and meanwhile makes the whole moduli space admit only normal crossing singularities. Our approach should extend to higher genera.
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We construct geometric models for the $\mathbb P^1$spectrum $M_{\mathbb P^1}(Y)$, which computes in GarkushaPanin's theory of framed motives \cite{GP14} a positively motivically fibrant $\Omega_{\mathbb P^1}$ replacement of $\Sigma_{\mathbb P^1}^\infty Y$ for a smooth scheme $Y\in \Sm_k$ over a perfect field $k$. Namely, we get the $T$spectrum in the category of pairs of smooth indschemes that defines $\mathbb P^1$spectrum of pointed sheaves termwise motivically equivalent to $M_{\mathbb P^1}(Y)$.
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We present a new penalty term approximating the CiarletNe\v{c}as condition (global invertibility of deformations) as a soft constraint for hyperelastic materials. For nonsimple materials including a suitable higher order term in the elastic energy, we prove that the penalized functionals converge to the original functional subject to the CiarletNe\v{c}as condition. Moreover, the penalization can be chosen in such a way that all low energy deformations, selfinterpenetration is completely avoided even for sufficiently small finite values of the penalization parameter. We also present numerical experiments in 2d illustrating our theoretical results.
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The Go Blog looks forward to version 2 of the Go language. "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."
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We consider the Graph Isomorphism problem for classes of graphs characterized by two forbidden induced subgraphs $H_1$ and $H_2$. By combining old and new results, Schweitzer settled the computational complexity of this problem restricted to $(H_1,H_2)$free graphs for all but a finite number of pairs $(H_1,H_2)$, but without explicitly giving the number of open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomialtime solvable on graph classes of bounded cliquewidth. By combining previously known results for Graph Isomorphism with known results for boundedness of cliquewidth, we reduce the number of open cases to 14. By proving a number of new results we then further reduce this number to seven. By exploiting the strong relationship between Graph Isomorphism and cliquewidth, we also prove that the class of $(\mbox{gem},P_1+2P_2)$free graphs has unbounded cliquewidth. This reduces the number of open cases for boundedness of cliquewidth for $(H_1,H_2)$free grap
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To every topologically transitive Cantor dynamical system $(X, \varphi)$ we associate a group $T(\varphi)$ acting faithfully by homeomorphism on the real line. It is defined as the group of homeomorphisms of the suspension flow of $(X, \varphi)$ which preserve every leaf and acts by dyadic piecewise linear homeomorphisms in the flow direction. We show that if $(X, \varphi)$ is minimal, the group $T(\varphi)$ is simple, and if $(X, \varphi)$ is a subshift the group $T(\varphi)$ is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple leftorderable groups. We show that if the system $(X, \varphi)$ is minimal, every action of the group $T(\varphi)$ on the circle has a fixed point, providing examples of so called "orderable monsters". We additionally have the following: for every subshift $(X, \varphi)$ the group $T(\varphi)$ does not have nontrivial subgroups with Kazhdan's property (T); for every
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Let $G$ be a nontrivial connected, edgecolored graph. An edgecut $S$ of $G$ is called a rainbow cut if no two edges in $S$ are colored with a same color. An edgecoloring of $G$ is a rainbow disconnection coloring if for every two distinct vertices $s$ and $t$ of $G$, there exists a rainbow cut $S$ in $G$ such that $s$ and $t$ belong to different components of $G\setminus S$. For a connected graph $G$, the {\it rainbow disconnection number} of $G$, denoted by $rd(G)$, is defined as the smallest number of colors such that $G$ has a rainbow disconnection coloring by using this number of colors. In this paper, we show that for a connected graph $G$, computing $rd(G)$ is NPhard. In particular, it is already NPcomplete to decide if $rd(G)=3$ for a connected cubic graph. Moreover, we prove that for a given edgecolored (with an unbounded number of colors) connected graph $G$ it is NPcomplete to decide whether $G$ is rainbow disconnected.
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We give a concrete presentation for the general linear group defined over a ring which is a finitely generated free $\mathbb{Z}$module or the integral Clifford group $\Gamma_n(\mathbb{Z})$ of invertible elements in the Clifford algebra with integral coefficients. We then use this presentation to prove that the elementary linear group over $\Gamma_n(\mathbb{Z})$ has a nontrivial decomposition as a free product with amalgamated subgroup the elementary linear group over $\Gamma_{n1}(\mathbb{Z})$. This allows to obtain applications to the unit group $\mathcal{U}(\mathbb{Z} G)$ of an integral group ring $\mathbb{Z} G$ of a finite group $G$. In particular, we prove that $\mathcal{U} (\mathbb{Z} G)$ is hereditary (FA), i.e. every subgroup of finite index has property (FA), or is commensurable with a nontrivial amalgamated product. In the case $\mathcal{U}(\mathbb{Z} G)$ is not hereditary (FA), we investigate subgroups of finite index in $\mathcal{U}(\mathbb{Z} G)$ that have a nontrivial
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We compute the Dolbeault cohomology of certain domains contained in Cousin groups defined by lattices which satisfy a strong dispersiveness condition. As a consequence we obtain a description of the Dolbeault cohomology of OeljeklausToma manifolds and in particular the fact that the Hodge decomposition holds for their cohomology.
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In this paper we construct families of homology spheres which bound 4manifolds with intersection forms isomorphic to $E_8$. We show that these families have arbitrary large correction terms. This result says that among homology spheres, the difference of the maximal rank of minimal sublattice of definite filling and the maximal rank of even definite filling is arbitrarily large.
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A new mathematical method for the dynamic analysis of nonlinear ecological systems has recently been developed by the author and was presented in a separate article. Based on this methodology, multiple new dynamic ecological system measures and indices of matrix, vector, and scalar types are systematically introduced in the present paper. These mathematical system analysis tools are quantitative ecological indicators that monitor the flow distribution and storage organization, quantify the effect and utility of one compartment directly or indirectly on another, identify the system efficiency and stress, measure the compartmental exposure to system flows, determine the residence time and compartmental activity levels, and ascertain the restoration time and resilience in the case of disturbances. Major flow and stockrelated concepts and quantities of the current static network analyses are also extended to nonlinear dynamic settings and integrated with the proposed dynamic measures and
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In this article, we study EinsteinWeyl structures on almost cosymplectic manifolds. First we prove that an almost cosymplectic $(\kappa,\mu)$manifold is Einstein or cosymplectic if it admits a closed EinsteinWeyl structure or two EinsteinWeyl structures. Next for a three dimensional compact almost $\alpha$cosymplectic manifold admitting closed EinsteinWeyl structures, we prove that it is Riccflat. Further, we show that an almost $\alpha$cosymplectic admitting two EinsteinWeyl structures is either Einstein or $\alpha$cosymplectic, provided that its Ricci tensor is commuting. Finally, we prove that a compact $K$cosymplectic manifold with a closed EinsteinWeyl structure or two special EinsteinWeyl structures is cosymplectic.
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We construct stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties. In particular, this gives an elliptic generalization of the results of arXiv:1211.1287. We apply them to the computation of the monodromy of $q$difference equations arising the enumerative Ktheory of rational curves in Nakajima varieties, including the quantum KnizhnikZamolodchikov equations.
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Let $H := \begin{pmatrix} 1 & {\mathbf R} & {\mathbf R} \\ 0 & 1 &{\mathbf R} \\ 0 & 0 & 1 \end{pmatrix}$ denote the Heisenberg group with the usual CarnotCarath\'eodory metric $d$. It is known (since the work of Pansu and Semmes) that the metric space $(H,d)$ cannot be embedded in a bilipchitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any $0 < \varepsilon < 1$, the snowflaked metric space $(H,d^{1\varepsilon})$ embeds into an infinitedimensional Hilbert space with distortion $O( \varepsilon^{1/2} )$. This distortion bound was shown by Austin, Naor, and Tessera to be sharp for the Heisenberg group $H$. Assouad's argument allows $\ell^2$ to be replaced by ${\mathbf R}^{D(\varepsilon)}$ for some dimension $D(\varepsilon)$ dependent on $\varepsilon$. Naor and Neiman showed that $D$ could be taken independent of $\varepsilon$, at the cost of worsening the bound on the distortion to $O( \varepsilon^{1+c_D} )$, where $c_D
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In early study of Engel manifolds from R. Montgomery, the Cartan prolongation and the development map are central figures. However, the development map can be globally defined only if the characteristic foliation is "nice". In this paper, we introduce the Cartan prolongation of a contact 3orbifold and the development map associated to a more general Engel manifold, and we give necessary and sufficient condition for the Cartan prolongation to be a manifold. Moreover, we explain the Cartan prolongation of a 3dimensional contact \'etale Lie groupoid and the development map associated to any Engel manifold, and we proof that all Engel manifolds obtained as the Cartan prolongation of a "space" with contact structure are obtained from a contact 3orbifold.
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We prove that, under a well known conjecture in the finitely generated case, Galois cohomology satisfies several surprisingly strong versions of Koszul properties. We point out several unconditional results which follow from our work. We show how these enhanced versions are preserved under certain natural operations on algebras, generalising several results that were previously established only in the commutative case.
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Shearer's inequality bounds the sum of joint entropies of random variables in terms of the total joint entropy. We give another lower bound for the same sum in terms of the individual entropies when the variables are functions of independent random seeds. The inequality involves a constant characterizing the expansion properties of the system. Our results generalize to entropy inequalities used in recent work in invariant settings, including the edgevertex inequality for factorofIID processes, Bowen's entropy inequalities, and Bollob\'as's entropy bounds in random regular graphs. The proof method yields inequalities for other measures of randomness, including covariance. As an application, we give upper bounds for independent sets in both finite and infinite graphs.
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Delays are an important phenomenon arising in a wide variety of real world systems. They occur in biological models because of diffusion effects or as simplifying modeling elements. We propose here to consider delayed stochastic reaction networks. The difficulty here lies in the fact that the statespace of a delayed reaction network is infinitedimensional, which makes their analysis more involved. We demonstrate here that a particular class of stochastic timevarying delays, namely those that follow a phasetype distribution, can be exactly implemented in terms of a chemical reaction network. Hence, any delayfree network can be augmented to incorporate those delays through the addition of delayspecies and delayreactions. Hence, for this class of stochastic delays, which can be used to approximate any delay distribution arbitrarily accurately, the statespace remains finitedimensional and, therefore, standard tools developed for standard reaction network still apply. In particular
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We study the ergodicity of nonautonomous discrete dynamical systems with nonuniform expansion. As an application we get that any uniformly expanding finitely generated semigroup action of $C^{1+\alpha}$ local diffeomorphisms of a compact manifold is ergodic with respect to the Lebesgue measure. Moreover, we will also prove that every exact nonuniform expandable finitely generated semigroup action of conformal $C^{1+\alpha}$ local diffeomorphisms of a compact manifold is Lebesgue ergodic.
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This paper presents a new algorithm based on interval methods for rigorously constructing inner estimates of feasible parameter regions together with enclosures of the solution set for parameterdependent systems of nonlinear equations in low (parameter) dimensions. The proposed method allows to explicitly construct feasible parameter sets around a regular parameter value, and to rigorously enclose a particular solution curve (resp. manifold) by a union of inclusion regions, simultaneously. The method is based on the calculation of inclusion and exclusion regions for zeros of square nonlinear systems of equations. Starting from an approximate solution at a fixed set $p$ of parameters, the new method provides an algorithmic concept on how to construct a box $\mathbf{s}$ around $p$ such that for each element $s\in \mathbf{s}$ in the box the existence of a solution can be proved within certain error bounds.
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Here we extend the notion of targetlocal Gromov convergence of pseudoholomorphic curves to the case in which the target manifold is not compact, but rather is exhausted by compact neighborhoods. Under the assumption that the curves in question have uniformly bounded area and genus on each of the compact regions (but not necessarily global bounds), we prove a subsequence converges in an exhaustive Gromov sense.
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An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as a variable. The algorithm, in principle, utilizes the minimal annihilating polynomials for eliminating redundant calculations. Furthermore, in the actual computation, the algorithm computes candidates of eigenvectors by utilizing pseudo annihilating polynomials and verifies their correctness. The experimental results show that our algorithms have better performance compared to conventional methods.
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We give an efficient algorithm that, given a graph $G$ and a partition $V_1,\ldots,V_m$ of its vertex set, finds either an independent transversal (an independent set $\{v_1,\ldots,v_m\}$ in $G$ such that $v_i\in V_i$ for each $i$), or a subset $\mathcal B$ of vertex classes such that the subgraph of $G$ induced by $\bigcup\mathcal B$ has a small dominating set. A nonalgorithmic proof of this result has been known for a number of years and has been applied to solve many other problems. Thus we are able to give algorithmic versions of many of these applications, a few of which we describe explicitly here.
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In this paper, we present finite element approximations of a class of Generalized random fields defined over a bounded domain of R d or a smooth ddimensional Riemannian manifold (d $\ge$ 1). An explicit expression for the covariance matrix of the weights of the finite element representation of these fields is provided and an analysis of the approximation error is carried out. Finally, a method to generate simulations of these weights while limiting computational and storage costs is presented.
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The relationship between the sparsest cut and the maximum concurrent multiflow in graphs has been studied extensively. For general graphs with $k$ terminal pairs, the flowcut gap is $O(\log k)$, and this is tight. But when topological restrictions are placed on the flow network, the situation is far less clear. In particular, it has been conjectured that the flowcut gap in planar networks is $O(1)$, while the known bounds place the gap somewhere between $2$ (Lee and Raghavendra, 2003) and $O(\sqrt{\log k})$ (Rao, 1999). A seminal result of Okamura and Seymour (1981) shows that when all the terminals of a planar network lie on a single face, the flowcut gap is exactly $1$. This setting can be generalized by considering planar networks where the terminals lie on $\gamma>1$ faces in some fixed planar drawing. Lee and Sidiropoulos (2009) proved that the flowcut gap is bounded by a function of $\gamma$, and Chekuri, Shepherd, and Weibel (2013) showed that the gap is at most $3\gamma
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What Google did not make public was that an employee had accused Mr. Rubin of sexual misconduct. The woman, with whom Mr. Rubin had been having an extramarital relationship, said he coerced her into performing oral sex in a hotel room in 2013, according to two company executives with knowledge of the episode. Google investigated and concluded her claim was credible, said the people, who spoke on the condition that they not be named, citing confidentiality agreements. Mr. Rubin was notified, they said, and Mr. Page asked for his resignation. Google could have fired Mr. Rubin and paid him little to nothing on the way out. Instead, the company handed him a $90 million exit package, paid in installments of about $2 million a month for four years, said two people with knowledge of the terms. The last payment is scheduled for next month. Mr. Rubin was one of three executives that Google protected over the past decade after they were accused of sexual misconduct. In two instances, it
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harrymcc writes: The better AI gets at teaching itself to perform tasks in ways beyond the skills of mere humans, the more likely it is that it may unwittingly behave in ways a human would consider unethical. To explore ways to prevent this from happening, IBM researchers taught AI to play PacMan without ever gobbling up the ghosts. And it did so without ever explicitly telling the software that this was the goal. Over at Fast Company, I wrote about this project and what IBM learned from conducting it. The researchers built a piece of software that could balance the AI's ratio of selfdevised, aggressive game play to humaninfluenced ghost avoidance, and tried different settings to see how they affected its overall approach to the game. By doing so, they found a tipping point  the setting at which PacMan went from seriously chowing down on ghosts to largely avoiding them.
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We introduce a connection between a nearterm quantum computing device, specifically a Gaussian boson sampler, and the graph isomorphism problem. We propose a scheme where graphs are encoded into quantum states of light, whose properties are then probed with photonnumberresolving detectors. We prove that the probabilities of different photondetection events in this setup can be combined to give a complete set of graph invariants. Two graphs are isomorphic if and only if their detection probabilities are equivalent. We present additional ways that the measurement probabilities can be combined or coarsegrained to make experimental tests more amenable. We benchmark these methods with numerical simulations on the Titan supercomputer for several graph families: pairs of isospectral nonisomorphic graphs, isospectral regular graphs, and strongly regular graphs.
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In this paper, we define a class of slice mappings of several Clifford variables, and the corresponding slice regular mappings. Furthermore, we establish the growth theorem for slice regular starlike or convex mappings on the unit ball of several slice Clifford variables, as well as on the bounded slice domain which is slice starlike and slice circular.
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We consider hamiltonian models representing an arbitrary number of spin $1/2$ fermion quantum fields interacting through arbitrary processes of creation or annihilation of particles. The fields may be massive or massless. The interaction form factors are supposed to satisfy some regularity conditions in both position and momentum space. Without any restriction on the strength of the interaction, we prove that the Hamiltonian identifies to a selfadjoint operator on a tensor product of antisymmetric Fock spaces and we establish the existence of a ground state. Our results rely on new interpolated $N_\tau$ estimates. They apply to models arising from the Fermi theory of weak interactions, with ultraviolet and spatial cutoffs.
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In the last two decades, significant effort has been put in understanding and designing socalled structurepreserving numerical methods for the simulation of mechanical systems. Geometric integrators attempt to preserve the geometry associated to the original system as much as possible, such as the structure of the configuration space, the energy behaviour, preservation of constants of the motion and of constraints or other structures associated to the continuous system (symplecticity, Poisson structure...). In this article, we develop highorder geometric (or pseudovariational) integrators for nonholonomic systems, i.e., mechanical systems subjected to constraint functions which are, roughly speaking, functions on velocities that are not derivable from position constraints. These systems realize rolling or certain kinds of sliding contact and are important for describing different classes of vehicles.
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Let $(X,H)$ be a polarized K3 surface with $\mathrm{Pic}(X) = \mathbb Z H$, and let $C\in H$ be a smooth curve of genus $g$. We give an upper bound on the dimension of global sections of a semistable vector bundle on $C$. This allows us to compute the higher rank Clifford indices of $C$ with high genus. In particular, when $g\geq r^2\geq 4$, the rank $r$ Clifford index of $C$ can be computed by the restriction of LazarsfeldMukai bundles on $X$ corresponding to line bundles on the curve $C$. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank $r$ Clifford index of a degree $d(\geq 5)$ smooth plane curve is $d4$, which is the same as the Clifford index of the curve.
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We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal of an ample groupoid algebra over a field is an intersection of kernels of induced representations from isotropy groups, with a much shorter proof, by using the author's Disintegration Theorem for groupoid representations. We also prove that every primitive ideal is the kernel of an induced representation from an isotropy group; however, we are unable to show, in general, that it is the kernel of an irreducible induced representation. If each isotropy group is finite (e.g., if the groupoid is principal) and if the base ring is Artinian (e.g., a field), then we can show that every primitive ideal is the kernel of an irreducible representation induced from isotropy.
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High dimensional error covariance matrices are used to weight the contribution of observation and background terms in data assimilation procedures. As error covariance matrices are often obtained by sampling methods, the resulting matrices are often degenerate or illconditioned, making them too expensive to use in practice. In order to combat these problems, reconditioning methods are used. In this paper we present new theory for two existing methods that can be used to reduce the condition number of (or 'recondition') any covariance matrix: ridge regression, and the minimum eigenvalue method. These methods are used in practice at numerical weather prediction centres, but their theoretical impact on the covariance matrix itself is not well understood. Here we address this by investigating the impact of reconditioning on variances and covariances of a general covariance matrix in both a theoretical and practical setting. Improved theoretical understanding provides guidance to users wit
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The Defense Advanced Research Projects Agency (DARPA), a division of the U.S. Department of Defense responsible for the development of emerging technologies, is one of the birthplaces of machine learning, a kind of artificial intelligence (AI) that mimics the behavior of neurons in the brain. Dr. Brian Pierce, director of DARPA's Innovation Office, spoke about the agency's recent efforts at a VentureBeat summit. From the report: One area of study is socalled "common sense" AI  AI that can draw on environmental cues and an understanding of the world to reason like a human. Concretely, DARPA's Machine Common Sense Program seeks to design computational models that mimic core domains of cognition: objects (intuitive physics), places (spatial navigation), and agents (intentional actors). "You could develop a classifier that could identify a number of objects in an image, but if you ask a question, you're not going to get an answer," Pierce said. "We'd like to get away from having an enor
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