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Subspace designs are a (large) collection of highdimensional subspaces $\{H_i\}$ of $\F_q^m$ such that for any lowdimensional subspace $W$, only a small number of subspaces from the collection have nontrivial intersection with $W$; more precisely, the sum of dimensions of $W \cap H_i$ is at most some parameter $L$. The notion was put forth by Guruswami and Xing (STOC'13) with applications to list decoding variants of ReedSolomon and algebraicgeometric codes, and later also used for explicit rankmetric codes with optimal list decoding radius. Guruswami and Kopparty (FOCS'13, Combinatorica'16) gave an explicit construction of subspace designs with nearoptimal parameters. This construction was based on polynomials and has close connections to folded ReedSolomon codes, and required large field size (specifically $q \ge m$). Forbes and Guruswami (RANDOM'15) used this construction to give explicit constant degree "dimension expanders" over large fields, and noted that subspace design
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The connective constant $\mu(G)$ of a quasitransitive graph $G$ is the asymptotic growth rate of the number of selfavoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph $G$. $\bullet$ We present upper and lower bounds for $\mu$ in terms of the vertexdegree and girth of a transitive graph. $\bullet$ We discuss the question of whether $\mu\ge\phi$ for transitive cubic graphs (where $\phi$ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). $\bullet$ We present strict inequalities for the connective constants $\mu(G)$ of transitive graphs $G$, as $G$ varies. $\bullet$ As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a nontrivial group element is declared to be a further gene
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Security updates have been issued by Arch Linux (chromium and nss), CentOS (bind and qemukvm), Debian (firefoxesr, ghostscript, hunspellenus, and uzbekwordlist), Fedora (phponeloginphpsaml), openSUSE (bind, gstreamerpluginsgood, and xen), Red Hat (bind, firefox, nss, nss and nssutil, and nssutil), and SUSE (ruby2.1).
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We relate composition and substitution in pre and postLie algebras to algebraic geometry. The ConnesKreimer Hopf algebras, and MKW Hopf algebras are then coordinate rings of the infinitedimensional affine varieties consisting of series of trees, resp.\ Lie series of ordered trees. Furthermore we describe the Hopf algebras which are coordinate rings of the automorphism groups of these varieties, which govern the substitution law in pre and postLie algebras.
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This article introduces proximal Cech complexes, restricted to the Euclidean plane. A Cechh complex of a set of points is the nerve of a collection of convex sets that are closed geometric balls all with the same radius, that have nonempty intersection. Both spatial and descriptive closed balls are considered. Cech complexes are proximal, provided the complexes are close to each other, either spatially or descriptively. A main result is that a descriptive Cech complex is homotopically equivalent to the description of the union of its closed balls.
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The objective of this paper is to introduce an artificial intelligence based optimization approach, which is inspired from Piagets theory on cognitive development. The approach has been designed according to essential processes that an individual may experience while learning something new or improving his / her knowledge. These processes are associated with the Piagets ideas on an individuals cognitive development. The approach expressed in this paper is a simple algorithm employing swarm intelligence oriented tasks in order to overcome singleobjective optimization problems. For evaluating effectiveness of this early version of the algorithm, test operations have been done via some benchmark functions. The obtained results show that the approach / algorithm can be an alternative to the literature in terms of singleobjective optimization. The authors have suggested the name: Cognitive Development Optimization Algorithm (CoDOA) for the related intelligent optimization approach.
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A $C^{*}$algebra $A$ has ideal property if any ideal $I$ of $A$ is generated as an ideal by the projections inside the ideal. Suppose that the limit $C^{*}$algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has ideal property. In this paper we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension drop interval algebras and matrix algebras over 2dimensional spaces with torsion $H^{2}$ groups. This result unifies such reduction theorems for real rank zero $AH$ algebras in [EGS] and [DG] and for simple $AH$ algebras in [Li4]. The result play essential role in the classification of $AH$ algebra with ideal property (see [GJL]).
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In this paper, vector optimization is considered in the framework of decision making and optimization in general spaces. Interdependencies between domination structures in decision making and domination sets in vector optimization are given. We prove some basic properties of efficient and of weakly efficient points in vector optimization. Sufficient conditions for solutions to vector optimization problems are shown using minimal solutions of functionals. We focus on the scalarization by functions with uniform sublevel sets, which also delivers necessary conditions for efficiency and weak efficiency. The functions with uniform sublevel sets may be, e.g., continuous or even Lipschitz continuous, convex, strictly quasiconcave or sublinear. They can coincide with an order unit norm on a subset of the space.
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Given a complex projective structure $\Sigma$ on a surface, Thurston associated a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms $\\phi_\Sigma\_\infty$ and $\\phi_\Sigma\_2$ of the quadratic differential $\phi_\Sigma$ of $\Sigma$ given by the Schwarzian derivative. We show that these give a unifying approach that generalizes a number of wellknown results for convex cocompact hyperbolic structures including bounds on the Lipschitz constant for the retract and the length of the bending lamination. We then use these bounds to begin a study of the WeilPetersson gradient flow of renormalized volume on the space $CC(N)$ of convex cocompact hyperbolic structures on a compact manifold $N$ with incompressible boundary. This leads to a proof of the conjecture that the renormalized volume has infimum given by onehalf the simplicial volume of $DN$, the double of $N$.
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We investigate small covers and quasitoric over the duals of neighborly simplicial polytopes with small number of vertices in dimensions $4$, $5$, $6$ and $7$. In the most of the considered cases we obtain the complete classification of small covers. The lifting conjecture in all cases is verified to be true. The problem of cohomological rigidity for small covers is also studied and we have found a whole new series of weakly cohomologically rigid simple polytopes. New examples of manifolds provide the first known examples of quasitoric manifolds in higher dimensions whose orbit polytopes have chromatic numbers $\chi (P^n)\geq 3n5$.
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In the present note we study Waldschmidt constants of StanleyReisner ideals of a hypergraph and a graph with vertices forming a bipyramid over a planar ngon. The case of the hypergraph has been studied by Bocci and Franci. We reprove their main result. The case of the graph is new. Interestingly, both cases provide series of ideals with Waldschmidt constants descending to 1. It would be interesting to known if there are bounded ascending sequences of Waldschmidt constants.
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Let $(R, \mathfrak{m}) $ be a Gorenstein local ring of dimension $d > 0$ and let $I$ be an ideal of $R$ such that $(0) \ne I \subsetneq R$ and $R/I$ is a CohenMacaulay ring of dimension $d$. There is given a complete answer to the question of when the idealization $A = R \ltimes I$ of $I$ over $R$ is an almost Gorenstein local ring.
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The scheduler is a topic of keen interest for the desktop user; the scheduling algorithm partially determines the responsiveness of the Linux desktop as a whole. Con Kolivas maintains a series of scheduler patch sets that he has tuned considerably over the years for his own use, focusing primarily on latency reduction for a better desktop experience. In early October 2016, Kolivas updated the design of his popular desktop scheduler patch set, which he renamed MuQSS. It is an update (and a name change) from his previous scheduler, BFS, and it is designed to address scalability concerns that BFS had with an increasing number of CPUs.
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Linux usage in networking hardware has been on the rise for some time. During the latest Netdev conference held in Montreal this April, people talked seriously about Linux running on high end, "top of rack" (TOR) networking equipment. Those devices have long been the realm of proprietary hardware and software companies like Cisco or Juniper, but Linux seems to be making some significant headway into the domain. Are we really seeing the rise of Linux in highend networking hardware?
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Let $ G $ be a simple graph of $ \ell $ vertices $ \{1, \dots, \ell \} $ with edge set $ E_{G} $. The graphical arrangement $ \mathcal{A}_{G} $ consists of hyperplanes $ \{x_{i}x_{j}=0\} $, where $ \{i, j \} \in E_{G} $. It is well known that three properties, chordality of $ G $, supersolvability of $ \mathcal{A}_{G} $, and freeness of $ \mathcal{A}_{G} $ are equivalent. Recently, Richard P. Stanley introduced $ \psi $graphical arrangement $ \mathcal{A}_{G, \psi} $ as a generalization of graphical arrangements. Lili Mu and Stanley characterized the supersolvability of the $ \psi $graphical arrangements and conjectured that the freeness and the supersolvability of $ \psi $graphical arrangements are equivalent. In this paper, we will prove the conjecture.
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In the present work, we extend our previous work with Gwilliam by realizing $\hat{A}(X)$ as the projective volume form associated to the BV operator in our quanitization of a onedimensional sigma model. We also discuss the associated integration/expectation map. We work in the formalism of $L_\infty$ spaces, objects of which are computationally convenient presentations for derived stacks. Both smooth and complex geometry embed into $L_\infty$ spaces and we specialize our results in both of these cases.
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In this note, we prove that the disk is a local maximum for the geometric probability that three points chosen uniformly at random in a bounded convex region of the plane form an acute triangle. This provides progress towards a conjecture by Glen Hall, which states that the probability is maximized by the disk. We prove a corresponding result in three dimensions as well. We also prove that if the isoperimetric ratio is sufficiently large, that the probability of picking an acute triangle is small, and so the maximum must occur in some GromovHausdorff compact set in the moduli space.
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At a campaign rally in Fort Dodge, Iowa, Donald Trump, speaking of the Islamic State, once told supporters, “I would bomb the shit out of ’em. I would just bomb those suckers.” During Trump’s nascent tenure as CommanderinChief, air strikes conducted by the U.S.led coalition in Iraq and Syria have increased. So have civilian casualties: coalition strikes killed more civilians in March than in any other month since our mostly aerial war against the Islamic State began, in late 2014. Last Thursday, the U.S. military also acknowledged that a strike had mistakenly killed eighteen of our local allies in Syria. The victims belonged to the Syrian Democratic Forces, or S.D.F., a coalition of Arab and Kurdish fighters that partners closely with U.S. Special Operations Forces and has been preparing for months to attack the Islamic State stronghold of Raqqa. The incident was widely presented as evidence that President Trump is following through on the promise he made in Fort Dodge, as well as a
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The quantum mechanical description of the chemical bond is generally given in terms of delocalized bonding orbitals, or, alternatively, in terms of correlations of occupations of localised orbitals. However, in the latter case, multiorbital correlations were treated only in terms of twoorbital correlations, although the structure of multiorbital correlations is far richer; and, in the case of bonds established by more than two electrons, multiorbital correlations represent a more natural point of view. Here, for the first time, we introduce the true multiorbital correlation theory, consisting of a framework for handling the structure of multiorbital correlations, a toolbox of true multiorbital correlation measures, and the formulation of the multiorbital correlation clustering, together with an algorithm for obtaining that. These make it possible to characterise quantitatively, how well a bonding picture describes the chemical system. As proof of concept, we apply the theory for the i
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Let $\mathcal{G}_{n,p,q}$ denote the set of strongly connected bipartite digraphs on $n$ vertices which contain a complete bipartite subdigraph $\overleftrightarrow{K_{p,q}}$, where $p, q, n$ are positive integers and $p+q \leq n$. In this paper, we study the Qindex (i.e. the signless Laplacian spectral radius) of $\mathcal{G}_{n,p,q}$, and determine the extremal digraph that has the minimum Qindex.
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We consider an intermediate category between the category of finite quivers and a certain category of pseudocompact associative algebras that contains all finite dimensional algebras. We define the completed path algebra and the Gabriel quiver as functors. We give an explicit quotient of the category of algebras on which these functors form an adjoint pair. We show that these functors respect ideals, obtaining in this way an equivalence between related categories.
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We consider the "searching for a trail in a maze" composite hypothesis testing problem, in which one attempts to detect an anomalous directed path in a lattice 2D box of side n based on observations on the nodes of the box. Under the signal hypothesis, one observes independent Gaussian variables of unit variance at all nodes, with zero, mean off the anomalous path and mean \mu_n on it. Under the null hypothesis, one observes i.i.d. standard Gaussians on all nodes. AriasCastro et al. (2008) showed that if the unknown directed path under the signal hypothesis has known the initial location, then detection is possible (in the minimax sense) if \mu_n >> 1/\sqrt log n, while it is not possible if \mu_n << 1/ log n\sqrt log log n. In this paper, we show that this result continues to hold even when the initial location of the unknown path is not known. As is the case with AriasCastro et al. (2008), the upper bound here also applies when the path is undirected. The improvement is
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I challenge anyone to receive a notification on Samsung s Galaxy S8 and not be charmed by the elegant blue pulse of light that traces the contours of the phone's gorgeous screen. This sort of subtlety, this sort of organic, emotive, instant appeal is not something I ever expected Samsung would be capable of. But the company once judged to have cynically copied Apple's iPhone design has exceeded all expectations this year: the 2017 version of Samsung's TouchWiz brings its software design right up to the high standard of its hardware. I have always hated TouchWiz. It was ugly, overbearing, complex, and annoying. Keyword here is was. As per my philosophy to never rot stuck in a single brand or platform, I replaced my Nexus 6P with a Samsung Galaxy S7 Edge a few weeks ago. I was assuming I'd have to root it and install a custom ROM on it within days, so I had the proper files and reading material ready to go the day the phone arrived. But as I was using the phone for a few days, it dawn
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Suppose $F$ is a nonarchimedean local field. The classical GodementJacquet theory is that one can use SchwartzBruhat functions on $n \times n$ matrices $M_n(F)$ to define the local standard $L$functions on $\mathrm{GL}_n$. The purpose of this partly expository note is to give evidence that there is an analogous and useful "approximate" GodementJacquet theory for the standard $L$functions on the special orthogonal groups $\mathrm{SO}(V)$: One replaces $\mathrm{GL}_n(F)$ with $\mathrm{GSpin}(V)(F)$ and $M_n(F)$ with $\mathrm{Clif}(V)(F)$, the Clifford algebra of $V$. More precisely, we explain how a few different local unramified calculations for standard $L$functions on $\mathrm{SO}(V)$ can be done easily using SchwartzBruhat functions on $\mathrm{Clif}(V)(F)$. We do not attempt any of the ramified or global theory of $L$functions on $\mathrm{SO}(V)$ using SchwartzBruhat functions on $\mathrm{Clif}(V)$.
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We study an interplay between operator algebras and geometry of rational elliptic curves. Namely, let $\mathcal{O}_B$ be the CuntzKrieger algebra given by square matrix $B=(b1, ~1, ~b2, ~1)$, where $b$ is an integer greater or equal to two. It is proved, that there exists a dense selfadjoint subalgebra of $\mathcal{O}_B$, which is isomorphic (modulo an ideal) to a twisted homogeneous coordinate ring of the rational elliptic curve $\mathcal{E}({\Bbb Q})=\{(x,y,z) \in {\Bbb P}^2({\Bbb C}) ~~ y^2z=x(xz)(x{b2\over b+2}z)\}$.
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We propose a piggybacking scheme for network coding where strong source inputs piggyback the weaker ones, a scheme necessary and sufficient to achieve the cutset upper bound at high/lowsnr regime, a new asymptotically optimal operational regime for the multihop Amplify and Forward (AF) networks.
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Let $G=(V, E)$ be a graph. A set $S\subseteq V(G)$ is a {\it dominating set} of $G$ if every vertex in $V\setminus S$ is adjacent to a vertex of $S$. The {\it domination number} of $G$, denoted by $\gamma(G)$, is the cardinality of a minimum dominating set of $G$. Furthermore, a dominating set $S$ is an {\it independent transversal dominating set} of $G$ if it intersects every maximum independent set of $G$. The {\it independent transversal domination number} of $G$, denoted by $\gamma_{it}(G)$, is the cardinality of a minimum independent transversal dominating set of $G$. In 2012, Hamid initiated the study of the independent transversal domination of graphs, and posed the following two conjectures: Conjecture 1. If $G$ is a noncomplete connected graph on $n$ vertices, then $\gamma_{it}(G)\leq\lceil\frac{n}{2}\rceil$. Conjecture 2. If G is a connected bipartite graph, then $\gamma_{it}(G)$ is either $\gamma(G)$ or $\gamma(G)+1$. We show that Conjecture 1 is not true in general. Very r
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In this work we study how a viral capsid can change conformation using techniques of Large Deviations Theory for stochastic differential equations. The viral capsid is a model of a complex system in which many units  the proteins forming the capsomers  interact by weak forces to form a structure with exceptional mechanical resistance. The destabilization of such a structure is interesting both per se, since it is related either to infection or maturation processes, and because it yields insights into the stability of complex structures in which the constitutive elements interact by weak attractive forces. We focus here on a simplified model of a dodecahederal viral capsid, and assume that the capsomers are rigid plaquettes with one degree of freedom each. We compute the most probable transition path from the closed capsid to the final configuration using minimum energy paths, and discuss the stability of intermediate states.
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We compute the mixed Hodge structure on the cohomology ring of complements of complex coordinate subspace arrangements. The mixed Hodge structure can be described in terms of the special bigrading on the cohomology ring of complements of complex coordinate subspace arrangements. Originally this bigrading was introduced in the setting of toric topology by V.M. Buchstaber and T.E. Panov.
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The adaptive classification of the interference covariance matrix structure for radar signal processing applications is addressed in this paper. This represents a key issue because many detection architectures are synthesized assuming a specific covariance structure which may not necessarily coincide with the actual one due to the joint action of the system and environment uncertainties. The considered classification problem is cast in terms of a multiple hypotheses test with some nested alternatives and the theory of Model Order Selection (MOS) is exploited to devise suitable decision rules. Several MOS techniques, such as the Akaike, Takeuchi, and Bayesian information criteria are adopted and the corresponding merits and drawbacks are discussed. At the analysis stage, illustrating examples for the probability of correct model selection are presented showing the effectiveness of the proposed rules.
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We introduce axial representations and modules over axial algebras as new tools to study axial algebras. All known interesting examples of axial algebras fall into this setting, in particular the Griess algebra whose automorphism group is the Monster group. Our results become especially interesting for Matsuo algebras. We vitalize the connection between Matsuo algebras and 3transposition groups by relating modules over Matsuo algebras with representations of 3transposition groups. As a byproduct, we define, given a Fischer space, a group that can fulfill the role of a universal 3transposition group.
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In this paper, we study contractions of the boundary of a Riemannian 2disc where the maximal length of the intermediate curves is minimized. We prove that with an arbitrarily small overhead in the lengths of the intermediate curves, there exists such an optimal contraction that is monotone, i.e., where the intermediate curves are simple closed curves which are pairwise disjoint. This proves a conjecture of Chambers and Rotman.
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Monte Carlo Tree Search (MCTS), most famously used in gameplay artificial intelligence (e.g., the game of Go), is a wellknown strategy for constructing approximate solutions to sequential decision problems. Its primary innovation is the use of a heuristic, known as a default policy, to obtain Monte Carlo estimates of downstream values for states in a decision tree. This information is used to iteratively expand the tree towards regions of states and actions that an optimal policy might visit. However, to guarantee convergence to the optimal action, MCTS requires the entire tree to be expanded asymptotically. In this paper, we propose a new technique called PrimalDual MCTS that utilizes sampled information relaxation upper bounds on potential actions, creating the possibility of "ignoring" parts of the tree that stem from highly suboptimal choices. This allows us to prove that despite converging to a partial decision tree in the limit, the recommended action from PrimalDual MCTS is
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Reflected random walk in higher dimension arises from an ordinary random walk (sum of i.i.d. random variables): whenever one of the reflecting coordinates becomes negative, its sign is changed, and the process continues from that modified position. Onedimensional reflected random walk is quite well understood from work in 7 decades, but the multidimensional model presents several new difficulties. Here we investigate recurrence questions.
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We formulate multiple SchrammLoewner evolutions (SLEs) for coset WessZuminoWitten (WZW) models. The resultant SLEs may describe the critical behavior of multiple interfaces for the 2D statistical mechanics models whose critical properties are classified by coset WZW models. The SLEs are essentially characterized by multiple Brownian motions on a Lie group manifold as well as those on the real axis. The drift terms of the Brownian motions, which come from interactions of interfaces, are explicitly determined by imposing a martingale condition on correlation functions among boundary condition changing operators. As a concrete example, we formulate multiple SLE on the $Z(n)$ parafermion model and calculate the crossing probability which is closely related to 3SLE drift terms.
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In the field of numerical algebraic geometry, positivedimensional solution sets of systems of polynomial equations are described by witness sets. In this paper, we define multiprojective witness sets which encode the multidegree information of an irreducible multiprojective variety. Our main results generalize the regeneration solving procedure, a trace test, and numerical irreducible decomposition to the multiprojective case. Examples are included to demonstrate this new approach.
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We show that if $X$ is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of $X$ as we wish. Furthermore, we show that, under the condensation open set condition, the lower dimension of an inhomogeneous selfsimilar set $E_C$ coincides with the lower dimension of the condensation set $C$, while the Assouad dimension of $E_C$ is the maximum of the Assouad dimensions of the corresponding selfsimilar set $E$ and the condensation set $C$. If the Assouad dimension of $C$ is strictly smaller than the Assouad dimension of $E$, then the upper regularity dimension of any measure supported on $E_C$ is strictly larger than the Assouad dimension of $E_C$. Surprisingly, the corresponding statement for the lower regularity dimension fails.
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We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. We call the resulting objects matroids over hyperfields. In fact, there are (at least) two natural notions of matroid in this context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, GrassmannPlucker functions, and dual pairs, and establish some basic duality theorems. We also show that if F is a doubly distributive hyperfield then the notions of weak and strong matroid over F coincide.
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In this paper we provide an integral representation of the fractional LaplaceBeltrami operator for general riemannian manifolds which has several interesting applications. We give two different proofs, in two different scenarios, of essentially the same result. One of them deals with compact manifolds with or without boundary, while the other approach treats the case of riemannian manifolds without boundary whose Ricci curvature is uniformly bounded below.
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Most people running Windows like having multiple apps running at the same time  and often, what's running in the background can drain your battery. In this latest Insider Preview build (Build 16176), we leveraged modern silicon capabilities to run background work in a powerefficient manner, thereby enhancing battery life significantly while still giving users access to powerful multitasking capabilities of Windows. With "Power Throttling", when background work is running, Windows places the CPU in its most energy efficient operating modes  work gets done, but the minimal possible battery is spent on that work. My biggest worry with technology like this is that it affects unsaved work. Luckily, you're supposed to be able to turn it on and off.
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Joint reconstruction has recently attracted a lot of attention, especially in the field of medical multimodality imaging such as PETMRI. Most of the developed methods rely on the comparison of image gradients, or more precisely their location, direction and magnitude, to make use of structural similarities between the images. A challenge and still an open issue for most of the methods is to handle images in entirely different scales, i.e. different magnitudes of gradients that cannot be dealt with by a global scaling of the data. We propose the use of generalized Bregman distances and infimal convolutions thereof with regard to the wellknown total variation functional. The use of a total variation subgradient respectively the involved vector field rather than an image gradient naturally excludes the magnitudes of gradients, which in particular solves the scaling behavior. Additionally, the presented method features a weighting that allows to control the amount of interaction between
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We present a construction of kinematic quantum states for theories of tensor fields of an arbitrary sort. The construction is based on projective techniques by Kijowski. Applying projective quantum states for Loop Quantum Gravity obtained by Lanery and Thiemann we construct quantum states for LQG coupled to tensor fields.
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Distributional approximations of (bi) linear functions of sample variancecovariance matrices play a critical role to analyze vector time series, as they are needed for various purposes, especially to draw inference on the dependence structure in terms of second moments and to analyze projections onto lower dimensional spaces as those generated by principal components. This particularly applies to the highdimensional case, where the dimension $d$ is allowed to grow with the sample size $n$ and may even be larger than $n$. We establish largesample approximations for such bilinear forms related to the sample variancecovariance matrix of a highdimensional vector time series in terms of strong approximations by Brownian motions. The results cover weakly dependent as well as many longrange dependent linear processes and are valid for uniformly $ \ell_1 $bounded projection vectors, which arise, either naturally or by construction, in many statistical problems extensively studied for
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We demonstrate that the five vortex equations recently introduced by Manton ariseas symmetry reductions of the antiselfdual YangMills equations in four dimensions. In particular the JackiwPi vortex and the Ambj\o rnOlesen vortex correspond to the gauge group $SU(1, 1)$, and respectively the Euclidean or the $SU(2)$ symmetry groups acting with twodimensional orbits. We show how to obtain vortices with higher vortex numbers, by superposing vortex equations of different types. Finally we use the kinetic energy of the YangMills theory in 4+1 dimensions to construct a metric on vortex moduli spaces. This metric is not positivedefinite in cases of noncompact gauge groups.
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In this note we discuss an interesting duality known to occur for certain complex reflection groups, we prove in particular that this duality has a concrete representation theoretic realisation. As an application, we construct matrix factorisations of the highest degree basic invariant which give free resolutions of the module of K\"{a}hler differentials of the coinvariant algebra $A$ associated to such a reflection group. From this one can read off the Hilbert series of ${\rm Der}_{\mathbb{C}}(A,A)$. This applies for instance when $A$ is the cohomology of any complete flag manifold, and hence has geometric consequences.
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In this paper we study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball. We show that among all such paths, the one that maximizes the covering probability is the monotonic increasing one that stays within distance 1 from the diagonal. As a result, we can obtain an upper bound on the exponent of covering probability of any such path when $d\ge 4$. This upper bound is asymptotically sharp for all monotonic paths.
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We examine connections between the gonality, treewidth, and orientable genus of a graph. Especially, we find that hyperelliptic graphs in the sense of Baker and Norine are planar. We give a notion of a bielliptic graph and show that each of these must embed into a closed orientable surface of genus one. We also find, for all $g\ge 0$, trigonal graphs of treewidth 3 and orientable genus $g$, and give analogues for graphs of higher gonality.
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Some comments on the Friedmann and Hagen's quantum mechanical derivation of the Wallis formula for $\pi$ are given. In particular, we demonstrate that Lorentz trial function, instead of the Gaussian one used by Friedmann and Hagen, also leads to the Wallis formula. The anatomy of the integrals, leading to the appearance of the Wallis ratio, are carefully revealed.
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Using properties of shuffles of copulas and tools from combinatorics we solve the open question about the exact region $\Omega$ determined by all possible values of Kendall's $\tau$ and Spearman's $\rho$. In particular, we prove that the wellknown inequality established by Durbin and Stuart in 1951 is only sharp on a countable set with sole accumulation point $(1,1)$, give a simple analytic characterization of $\Omega$ in terms of a continuous, strictly increasing piecewise concave function, and show that $\Omega$ is compact and simply connected but not convex. The results also show that for each $(x,y)\in \Omega$ there are mutually completely dependent random variables whose $\tau$ and $\rho$ values coincide with $x$ and $y$ respectively.
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We give a classification and complete algebraic description of groups allowing only finitely many (left multiplication invariant) circular orders. In particular, they are all solvable groups with a specific semidirect product decomposition. This allows us to also show that the space of circular orders of any group is either finite or uncountable. As a special case and first step, we show that the space of circular orderings of an infinite Abelian group has no isolated points, hence is homeomorphic to a cantor set.
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Let $G$ be a transitive permutation group of degree $n$. We say that $G$ is $2'$elusive if $n$ is divisible by an odd prime, but $G$ does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive and biquasiprimitive $2'$elusive permutation groups, extending earlier work of Giudici and Xu on elusive groups. As an application, we use our results to investigate automorphisms of finite arctransitive graphs of prime valency.
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When we have a proper action of a Lie group on a manifold, it is well known that we get a stratification by orbit types and it is known that this stratification satisfies the Whitney (b) condition. In a previous article we have seen that the stratification satisfies the strong Verdier condition. In this article we improve this result and obtain smooth local triviality.
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An edgecoloured path is \emph{rainbow} if all the edges have distinct colours. For a connected graph $G$, the \emph{rainbow connection number} $rc(G)$ is the minimum number of colours in an edgecolouring of $G$ such that, any two vertices are connected by a rainbow path. Similarly, the \emph{strong rainbow connection number} $src(G)$ is the minimum number of colours in an edgecolouring of $G$ such that, any two vertices are connected by a rainbow geodesic (i.e., a path of shortest length). These two concepts of connectivity in graphs were introduced by Chartrand et al.~in 2008. Subsequently, vertexcoloured versions of both parameters, $rvc(G)$ and $srvc(G)$, and a totalcoloured version of the rainbow connection number, $trc(G)$, were introduced. In this paper we introduce the strong total rainbow connection number $strc(G)$, which is the version of the strong rainbow connection number using totalcolourings. Among our results, we will determine the strong total rainbow connection
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We solve explicitly a twodimensional singular control problem of finite fuel type for infinite time horizon. The problem stems from the optimal liquidation of an asset position in a financial market with multiplicative and transient price impact. Liquidity is stochastic in that the volume effect process, which determines the intertemporal resilience of the market in spirit of Predoiu, Shaikhet and Shreve (2011), is taken to be stochastic, being driven by own random noise. The optimal control is obtained as the local time of a diffusion process reflected at a nonconstant free boundary. To solve the HJB variational inequality and prove optimality, we need a combination of probabilistic arguments and calculus of variations methods, involving Laplace transforms of inverse local times for diffusions reflected at elastic boundaries.
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Let $G$ be a complex semisimple algebraic group and $X$ be a complex symmetric homogeneous $G$variety. Assume that both $G$, $X$ as well as the $G$action on $X$ are defined over real numbers. Then $G(\mathbb{R})$ acts on $X(\mathbb{R})$ with finitely many orbits. We describe these orbits in combinatorial terms using Galois cohomology, thus providing a patch to a result of A.Borel and L.Ji.
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The analysis in Part I revealed interesting properties for subgradient learning algorithms in the context of stochastic optimization when gradient noise is present. These algorithms are used when the risk functions are nonsmooth and involve nondifferentiable components. They have been long recognized as being slow converging methods. However, it was revealed in Part I that the rate of convergence becomes linear for stochastic optimization problems, with the error iterate converging at an exponential rate $\alpha^i$ to within an $O(\mu)$neighborhood of the optimizer, for some $\alpha \in (0,1)$ and small stepsize $\mu$. The conclusion was established under weaker assumptions than the prior literature and, moreover, several important problems (such as LASSO, SVM, and Total Variation) were shown to satisfy these weaker assumptions automatically (but not the previously used conditions from the literature). These results revealed that subgradient learning methods have more favorable be
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