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We introduce the notion of $n$dimensional topological quantum field theory (TQFT) with defects as a symmetric monoidal functor on decorated stratified bordisms of dimension $n$. The familiar closed or openclosed TQFTs are special cases of defect TQFTs, and for $n=2$ and $n=3$ our general definition recovers what had previously been studied in the literature. Our main construction is that of "generalised orbifolds" for any $n$dimensional defect TQFT: Given a defect TQFT $\mathcal{Z}$, one obtains a new TQFT $\mathcal{Z}_{\mathcal{A}}$ by decorating the Poincar\'e duals of triangulated bordisms with certain algebraic data $\mathcal{A}$ and then evaluating with $\mathcal{Z}$. The orbifold datum $\mathcal{A}$ is constrained by demanding invariance under $n$dimensional Pachner moves. This procedure generalises both state sum models and gauging of finite symmetry groups, for any $n$. After developing the general theory, we focus on the case $n=3$.
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For a matrix A_2 weight W on R^p, we introduce a new notion of WCalder\'onZygmund matrix kernels, following earlier work in by Isralowitz. We state and prove a T1 theorem for such operators and give a representation theorem in terms of dyadic WHaar shifts and paraproducts, in the spirit of Hyt\"onen's Representation Theorem. Finally, by means of a Bellman function argument, we give sharp bounds for such operators in terms of bounds for weighted matrix martingale transforms and paraproducts.
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This paper considers multipleinput multipleoutput (MIMO) fullduplex (FD) twoway secrecy systems. Specifically, both multiantenna FD legitimate nodes exchange their own confidential message in the presence of an eavesdropper. Taking into account the imperfect channel state information (CSI) of the eavesdropper, we formulate a robust sum secrecy rate maximization (RSSRM) problem subject to the outage probability constraint of the achievable sum secrecy rate and the transmit power constraint. Unlike other existing channel uncertainty models, e.g., normbounded and Gaussiandistribution, we exploit a momentbased random distributed CSI channel uncertainty model to recast our formulate RSSRM problem into the convex optimization frameworks based on a Markov's inequality and robust conic reformulation, i.e., semidefinite programming (SDP). In addition, differenceofconcave (DC) approximation is employed to iteratively tackle the transmit covariance matrices of these legitimate nodes. Si
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We prove the existence of scaling limits for the projection on the backbone of the random walks on the Incipient Infinite Cluster and the Invasion Percolation Cluster on a regular tree. We treat these projected random walks as randomly trapped random walks (as defined in [BC\v{C}R15]) and thus describe these scaling limits as spatially subordinated Brownian motions
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Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil conjectures, which were proved by Weil for curves and later, by Deligne for varieties over finite fields. Much work was done in the search for a proof of these conjectures, including the development in algebraic geometry of a Weil cohomology theory for these varieties, which uses the Frobenius operator on a finite field. The zeta function is then expressed as a determinant, allowing the properties of the function to relate to those of the operator. The search for a suitable cohomology theory and associated operator to prove the Riemann hypothesis is still on. In this paper, we study the properties of the derivative operator $D = \frac{d}{dz}$ on a particular weighted Bergman space of entire functions. The operator $D$ can be naturally viewed as the `i
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The fundamental group of a smooth projective variety is fibered if it maps onto the fundamental group of smooth curve of genus 2 or more. The goal of this paper is to establish some strong restrictions on these groups, and in particular on the fundamental groups of Kodaira surfaces. In the specific case of a Kodaira surface, these results are in the form of restrictions on the monodromy representation into the mapping class group. When the monodromy is composed with certain standard representations, the images are Zariski dense in a semisimple group of Hermitian type.
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A probabilitymeasurepreserving action of a countable group is called stable if its transformationgroupoid absorbs the ergodic hyperfinite equivalence relation of type II_1 under direct product. We show that for a countable group G and its central subgroup C, if G/C has a stable action, then so does G. Combining a previous result of the author, we obtain a characterization of a central extension having a stable action.
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In this article we establish the validity of Prandtl layer expansions around Euler flows which are not shear. The presence of nonshear flows at the leading order creates a singularity of $o(\frac{1}{\sqrt{\epsilon}})$. A new $y$weighted positivity estimate is developed to control this leadingorder growth at the far field.
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This paper is devoted to strict $K$ monotonicity and $K$order continuity in symmetric spaces. Using the local approach to the geometric structure in a symmetric space $E$ we investigate a connection between strict $K$monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of $K$order continuity in a symmetric space $E$ on $[0,\infty)$ implies that the embedding $E\hookrightarrow{L^1}[0,\infty)$ does not hold. We finish this article with a complete characterization of $K$order continuity in a symmetric space $E$ that is written using a notion of order continuity under some assumptions on the fundamental function of $E$.
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In this paper, we study three asymptotic regimes that can be applied to ranking and selection (R&S) problems with general sample distributions. These asymptotic regimes are constructed by sending particular problem parameters (probability of incorrect selection, smallest difference in system performance that we deem worth detecting) to zero. We establish asymptotic validity and efficiency of the corresponding R&S procedures in each regime. We also analyze the connection among different regimes and compare the prelimit performances of corresponding algorithms.
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We consider the minimization of nonconvex functions that typically arise in machine learning. Specifically, we focus our attention on a variant of trust region methods known as cubic regularization. This approach is particularly attractive because it escapes strict saddle points and it provides stronger convergence guarantees than first and secondorder as well as classical trust region methods. However, it suffers from a high computational complexity that makes it impractical for largescale learning. Here, we propose a novel method that uses subsampling to lower this computational cost. By the use of concentration inequalities we provide a sampling scheme that gives sufficiently accurate gradient and Hessian approximations to retain the strong global and local convergence guarantees of cubically regularized methods. To the best of our knowledge this is the first work that gives global convergence guarantees for a subsampled variant of cubic regularization on nonconvex functions.
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In this article we consider the class $\mathcal{A}(p)$ which consists of functions that are meromorphic in the unit disc $\ID$ having a simple pole at $z=p\in (0,1)$ with the normalization $f(0)=0=f'(0)1 $. First we prove some sufficient conditions for univalence of such functions in $\ID$. One of these conditions enable us to consider the class $\mathcal{V}_{p}(\lambda)$ that consists of functions satisfying certain differential inequality which forces univalence of such functions. Next we establish that $\mathcal{U}_{p}(\lambda)\subsetneq \mathcal{V}_{p}(\lambda)$, where $\mathcal{U}_{p}(\lambda)$ was introduced and studied in \cite{BF1}. Finally, we discuss some coefficient problems for $\mathcal{V}_{p}(\lambda)$ and end the article with a coefficient conjecture.
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A number of recent works have proposed to solve the line spectral estimation problem by applying an offthegrid ex tension of sparse estimation techniques. These methods are more advantageous than classical line spectral estimation algorithms because they inherently estimate the model order. However, they all have computation times which grow at least cubically in the problem size, which limits their practical applicability for large problem sizes. To alleviate this issue, we propose a lowcomplexity method for line spectral estimation, which also draws on ideas from sparse estimation. Our method is based on a probabilistic view of the problem. The signal covariance matrix is shown to have Toeplitz structure, allowing superfast Toeplitz inversion to be used. We demonstrate that our method achieves estimation accuracy at least as good as current methods and that it does so while being orders of magnitudes faster.
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The {\it forwardbackward algorithm} is a powerful tool for solving optimization problems with a {\it additively separable} and {\it smooth} + {\it nonsmooth} structure. In the convex setting, a simple but ingenious acceleration scheme developed by Nesterov has been proved useful to improve the theoretical rate of convergence for the function values from the standard $\mathcal O(k^{1})$ down to $\mathcal O(k^{2})$. In this short paper, we prove that the rate of convergence of a slight variant of Nesterov's accelerated forwardbackward method, which produces {\it convergent} sequences, is actually $o(k^{2})$, rather than $\mathcal O(k^{2})$. Our arguments rely on the connection between this algorithm and a secondorder differential inclusion with vanishing damping.
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We are concerned with the existence and uniqueness issue for the inhomogeneous incompressible NavierStokes equations supplemented with H^1 initial velocity and only bounded nonnegative density. In contrast with all the previous works on that topics, we do not require regularity or positive lower bound for the initial density, or compatibility conditions for the initial velocity, and still obtain unique solutions. Those solutions are global in the twodimensional case for general data, and in the threedimensional case if the velocity satisfies a suitable scaling invariant smallness condition. As a straightforward application, we provide a complete answer to Lions' question in [25], page 34, concerning the evolution of a drop of incompressible viscous fluid in the vacuum.
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This study introduces a procedure to obtain general expressions, $y = f(x)$, subject to linear constraints on the function and its derivatives defined at specified values. These constrained expressions can be used describe functions with embedded specific constraints. The paper first shows how to express the most general explicit function passing through a single point in three distinct ways: linear, additive, and rational. Then, functions with constraints on single, two, or multiple points are introduced as well as those satisfying relative constraints. This capability allows to obtain general expressions to solve linear differential equations with no need to satisfy constraints (the "subject to:" conditions) as the constraints are already embedded in the constrained expression. In particular, for expressions passing through a set of points, a generalization of the Waring's interpolation form, is introduced. The general form of additive constrained expressions is introduced as well as
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A real reductive pair $(G,H)$ is called strongly spherical if the homogeneous space $(G\times H)/{\rm diag}(H)$ is real spherical. This geometric condition is equivalent to the representation theoretic property that ${\rm dim\,Hom}_H(\pi_H,\tau)<\infty$ for all smooth admissible representations $\pi$ of $G$ and $\tau$ of $H$. In this paper we explicitly construct for all strongly spherical pairs $(G,H)$ intertwining operators in ${\rm Hom}_H(\pi_H,\tau)$ for $\pi$ and $\tau$ spherical principal series representations of $G$ and $H$. These socalled \textit{symmetry breaking operators} depend holomorphically on the induction parameters and we further show that they generically span the space ${\rm Hom}_H(\pi_H,\tau)$. In the special case of multiplicity one pairs we extend our construction to vectorvalued principal series representations and obtain generic formulas for the multiplicities between arbitrary principal series. As an application, we prove the GrossPrasad conjecture f
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These are the notes of rather informal lectures given by the first coauthor in UPMC, Paris, in January 2017. Practical goal is to explain how to compute or estimate the Morse index of the second variation. Symplectic geometry allows to effectively do it even for very degenerate problems with complicated constraints. Main geometric and analytic tool is the appropriately rearranged Maslov index. In these lectures, we try to emphasize geometric structure and omit analytic routine. Proofs are often substituted by informal explanations but a welltrained mathematician will easily rewrite them in a conventional way.
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We relate some terms on the boundary of the Newton polygon of the Alexander polynomial $\Delta(x,y)$ of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize $\Delta(x,y)$ so that no $x^{1}$ or $y^{1}$ terms appear, but $x^{1}\Delta(x,y)$ and $y^{1}\Delta(x,y)$ have negative exponents, and so that terms of even total degree are positive and terms with odd total degree are negative. If the rational link has a reduced alternating diagram with no self crossings, then $\Delta(1, 0) = 1$. If the standard form of the rational link has $m$ monochromatic twist sites, and the $j^{\textrm{th}}$ monochromatic twist site has $\hat{q}_j$ crossings, then $\Delta(1, 0) = \prod_{j=1}^{m}(\hat{q}_j+1)$. Our proof employs Kauffman's clock moves and a lattice for the terms of $\Delta(x,y)$ in which the $y$power cannot decrease.
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This paper addresses the stability analysis of adaptive control systems with the tryoncediscard protocol. At every sampling time, an event trigger evaluates errors between the current value and the last released value of each measurement and determines whether to transmit the measurements and which measurements to transmit, based on the tryoncediscard protocol and given lower and upper thresholds. For both gainscheduling controllers and switching controllers that are adaptive to the maximum error of the measurements, we obtain sufficient conditions for the closedloop stability in terms of linear matrix inequalities.
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Let $S$ be a finite semigroup and let $A$ be a finite dimensional $S$graded algebra. We investigate the exponential rate of growth of the sequence of graded codimensions $c_n^S(A)$ of $A$, i.e $\lim\limits_{n \rightarrow \infty} \sqrt[n]{c_n^S(A)}$. For group gradings this is always an integer. Recently in [20] the first example of an algebra with a noninteger growth rate was found. We present a large class of algebras for which we prove that their growth rate can be equal to arbitrarily large nonintegers. An explicit formula is given. Surprisingly, this class consists of an infinite family of algebras simple as an $S$graded algebra. This is in strong contrast to the group graded case for which the growth rate of such algebras always equals $\dim (A)$. In light of the previous, we also handle the problem of classification of all $S$graded simple algebras, which is of independent interest. We achieve this goal for an important class of semigroups that is crucial for a solution of t
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We propose a continuoustime model of trading among riskneutral agents with heterogeneous beliefs. Agents face quadratic costsofcarry on their positions and as a consequence, their marginal valuation of the asset decreases when the magnitude of their position increases, as it would be the case for riskaverse agents. In the equilibrium models of investors with heterogeneous beliefs that followed the original work by Harrison and Kreps, investors are riskneutral, shortselling is prohibited and agents face a constant marginal cost of carrying positions. The resulting resale option guarantees that the equilibrium price exceeds the price of the asset in a static buyandhold model where speculation is ruled out. Our model features three main novelties. First, increasing marginal costs entail that the price depends on the exogenous supply. Second, in addition to the resale option, agents may also value an option to delay, and this may cause the market to equilibrate \emph{below} the st
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The Randi{\' c} index of a graph $G$, written $R(G)$, is the sum of $\frac 1{\sqrt{d(u)d(v)}}$ over all edges $uv$ in $E(G)$. %let $R(G)=\sum_{uv \in E(G)} \frac 1{\sqrt{d(u)d(v)}}$, which is called the Randi{\' c} index of it. Let $d$ and $D$ be positive integers $d < D$. In this paper, we prove that if $G$ is a graph with minimum degree $d$ and maximum degree $D$, then $R(G) \ge \frac{\sqrt{dD}}{d+D}n$; equality holds only when $G$ is an $n$vertex $(d,D)$biregular. Furthermore, we show that if $G$ is an $n$vertex connected graph with minimum degree $d$ and maximum degree $D$, then $R(G) \le \frac n2 \sum_{i=d}^{D1}\frac 12 \left( \frac 1{\sqrt{i}}  \frac 1{\sqrt{i+1}}\right)^2$; it is sharp for infinitely many $n$, and we characterize when equality holds in the bound.
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We introduce and study skew product Smale endomorphisms over finitely irreducible topological Markov shifts with countable alphabets. We prove that almost all conditional measures of equilibrium states of summable and locally Holder continuous potentials are dimensionally exact, and that their dimension is equal to the ratio of the (global) entropy and the Lyapunov exponent. We also prove for them a formula of Bowen type for the Hausdorff dimension of all fibers. We develop a version of thermodynamic formalism for finitely irreducible twosided topological Markov shifts with countable alphabets. We describe then the thermodynamic formalism for Smale skew products over countableto1 endomorphisms, and give several applications to measures on natural extensions of endomorphisms. We show that the exact dimensionality of conditional measures on fibers, implies the global exact dimensionality of the measure, in certain cases. We then study equilibrium states for skew products over endomor
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In this paper we study the structure of $\theta$cyclic codes over the ring $B_k$ including its connection to quasi$\tilde{\theta}$cyclic codes over finite field $\mathbb{F}_{p^r}$ and skew polynomial rings over $B_k.$ We also characterize Euclidean selfdual $\theta$cyclic codes over the rings. Finally, we give the generator polynomial for such codes and some examples of optimal Euclidean $\theta$cyclic codes.
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We classify all complex surfaces with quotient singularities that do not contain any smooth rational curves, under the assumption that the canonical divisor of the surface is not pseudoeffective. As a corollary we show that if $X$ is a log del Pezzo surface such that for every closed point $p\in X$, there is a smooth curve (locally analytically) passing through $p$, then $X$ contains at least one smooth rational curve.
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In this article, we discuss the rationality of the Betti Series of the universal module of nth order derivations of R_{m} where m is a maximal ideal of R. We proved that if R is a coordinate ring of an affine irreducible curve and if it has at most one singularity point, then the Betti series is rational.
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Qualcomm on Wednesday sued the manufacturers that make iPhones for Apple for failing to pay royalties on the chip maker's technology, widening its legal battle with the world's most valuable company. Qualcomm's lawsuit, filed Wednesday in a federal district court in San Diego, accuses Compal, Foxconn, Pegatron, and Wistron of breaching longstanding patentlicensing agreements with Qualcomm by halting royalty payments on Qualcomm technology used in iPhones and iPads. From a report: Apple sued Qualcomm in January, accusing it of overcharging for chips and refusing to pay some $1 billion in promised rebates. Qualcomm said in the complaint that Apple is trying to force the company to agree to a "unreasonable demand for a belowmarket direct license." Qualcomm said last month that Apple had decided to withhold royalty payments to its contract manufacturers that are owed to the chipmaker, for sales made in the first quarter of 2017, until the dispute is resolved in court. "While not disputin
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This note surveys  by means of 25 examples  the concept of varifold, as generalised submanifold, with emphasis on integral varifolds with mean curvature, while keeping prerequisites to a minimum. Integral varifolds are the natural language for studying the variational theory of the area integrand if one considers, for instance, existence or regularity of stationary (or, stable) surfaces of dimension at least three, or the limiting behaviour of sequences of smooth submanifolds under area and mean curvature bounds.
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We revisit Kesten's argument for the upper bound on the growth rate of DLA. We are able to make the argument robust enough so that it applies to many graphs, where only control of the heat kernel is required. We apply this to many examples including transitive graphs of polynomial growth, graphs of exponential growth, nonamenable graphs, supercritical percolation on Z^d, high dimensional preSierpinski carpets. We also observe that a careful analysis shows that Kesten's original bound on Z^3 can be improved from t^{2/3} to (t log t)^{1/2} .
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We show that every Carnot group G of step 2 admits a Hausdorff dimension one `universal differentiability set' N such that every realvalued Lipschitz map on G is Pansu differentiable at a point of N. This relies on the fact that existence of a maximal directional derivative of f at a point x implies Pansu differentiability at the same point x. We show that such an implication holds in Carnot groups of step 2 but fails in the Engel group which has step 3.
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P. Flajolet and B. Salvy [10] prove the famous theorem that a nonlinear Euler sum $S_{i_1i_2\cdots i_r,q}$ reduces to a combination of sums of lower orders whenever the weight $i_1+i_2+\cdots+i_r+q$ and the order $r$ are of the same parity. In this article, we develop an approach to evaluate the cubic sums $S_{1^2m,p}$ and $S_{1l_1l_2,l_3}$. By using the approach, we establish some relations involving cubic, quadratic and linear Euler sums. Specially, we prove the cubic sums $S_{1^2m,m}$ and $S_{1(2l+1)^2,2l+1}$ are reducible to zeta values, quadratic and linear sums. Finally, we evaluate the alternating cubic Euler sums ${S_{{{\bar 1}^3},2r + 1}}$ and show that it are reducible to alternating quadratic and linear Euler sums. The approach is based on Tornheim type series computations.
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Apollonian gaskets are formed by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We experimentally study the pair correlation, electrostatic energy, and nearest neighbor spacing of centers of circles from Apollonian gaskets. Even though the centers of these circles are not uniformly distributed in any `ambient' space, after proper normalization, all these statistics seem to exhibit some interesting limiting behaviors.
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In this note, for any two orthogonal projection $P,Q$ on a Hilbert space, the characterization of spectrum of anticommutator $PQ+QP$ has been obtained. As a corollary, the norm formula $$\parallel PQ+QP\parallel=\parallel PQ\parallel+\parallel PQ\parallel^2$$ has been got an alternative proof (see Sam Waltrs, Anticommutator norm formula for projection operators, arXiv:1604.00699vl [math.FA] 3 Apr 2016)
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We study the spectral properties of bounded and unbounded Jacobi matrices whose entries are bounded operators on possibly infinite dimensional complex Hilbert space. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous. Uniform asymptotics of generalized eigenvectors and conditions implying complete indeterminacy are also provided.
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The paper studies functions defined on continuous branching lines connected into a system. A notion of spectrum degeneracy for these systems of functions is introduced. This degeneracy is based on the properties of the Fourier transforms for processes representing functions on the branches that are deemed to be extended onto the real axis. This spectrum degeneracy ensures some opportunities for extrapolation and sampling. The topology of the system is taken into account via a restriction that these processes coincides on certain parts of real axis. It is shown that processes with this feature are everywhere dense in the set of processes equivalent to functions on the branching lines. Some applications to extrapolation and sampling are considered.
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Integer cuboids are rectangular Diophantine parallelepipeds. It has been discovered that these cuboids come in 3 varieties: Euler or body type, edge type, and face type. In all three cases, one edge or diagonal is irrational, all six others are rational. We discuss an exhaustive computer search procedure which uses the Pythagorean group Py(n) to locate all possible cuboids with a given edge n. Over the range of 44 to 155,000,000,000, for the smallest edge, 154,571 cuboids were discovered. They are listed in the Integer Cuboid Table.
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Efficient beam alignment is a crucial component in millimeter wave systems with analog beamforming, especially in fastchanging vehicular settings. This paper uses the vehicle's position (e.g., available via GPS) to query the multipath fingerprint database, which provides prior knowledge of potential pointing directions for reliable beam alignment. The approach is the inverse of fingerprinting localization, where the measured multipath signature is compared to the fingerprint database to retrieve the most likely position. The power loss probability is introduced as a metric to quantify misalignment accuracy and is used for optimizing candidate beam selection. Two candidate beam selection methods are derived, where one is a heuristic while the other minimizes the misalignment probability. The proposed beam alignment is evaluated using realistic channels generated from a commercial raytracing simulator. Using the generated channels, an extensive investigation is provided, which includes
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The Ricci iteration is a discrete analogue of the Ricci flow. According to Perelman, the Ricci flow converges to a KahlerEinstein metric whenever one exists, and it has been conjectured that the Ricci iteration should behave similarly. This article confirms this conjecture. As a special case, this gives a new method of uniformization of the Riemann sphere.
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We use invariant manifold results on Banach spaces to conclude the existence of spectral submanifolds (SSMs) in a class of nonlinear, externally forced beam oscillations. SSMs are the smoothest nonlinear extensions of spectral subspaces of the linearized beam equation. Reduction of the governing PDE to SSMs provides an explicit lowdimensional model which captures the correct asymptotics of the full, infinitedimensional dynamics. Our approach is general enough to admit extensions to other types of continuum vibrations. The modelreduction procedure we employ also gives guidelines for a mathematically selfconsistent modeling of damping in PDEs describing structural vibrations.
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We consider the Dirichlet Laplacian $H_\gamma$ on a 3D twisted waveguide with random Andersontype twisting $\gamma$. We introduce the integrated density of states $N_\gamma$ for the operator $H_\gamma$, and investigate the Lifshits tails of $N_\gamma$, i.e. the asymptotic behavior of $N_\gamma(E)$ as $E \downarrow \inf {\rm supp}\, dN_\gamma$. In particular, we study the dependence of the Lifshits exponent on the decay rate of the singlesite twisting at infinity.
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The goal of this paper is to give a short review of recent results of the authors concerning classical Hamiltonian many particle systems. We hope that these results support the new possible formulation of Boltzmann's ergodicity hypothesis which sounds as follows. For almost all potentials, the minimal contact with external world, through only one particle of $N$, is sufficient for ergodicity. But only if this contact has no memory. Also new results for quantum case are presented.
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We give sharp remainder terms of $L^{p}$ and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of HardyRellich type, all with sharp constants. A number of identities are derived including weighted and higher order types.
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We show that the following problems are NPcomplete. 1. Can the vertex set of a graph be partitioned into two sets such that each set induces a perfect graph? 2. Is the difference between the chromatic number and clique number at most $1$ for every induced subgraph of a graph? 3. Can the vertex set of every induced subgraph of a graph be partitioned into two sets such that the first set induces a perfect graph, and the clique number of the graph induced by the second set is smaller than that of the original induced subgraph? 4. Does a graph contain a stable set whose deletion results in a perfect graph? The proofs of the NPcompleteness of the four problems follow the same pattern: Showing that all the four problems are NPcomplete when restricted to trianglefree graphs by using results of Maffray and Preissmann on $3$colorability and $4$colorability of trianglefree graphs
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We consider two manifestations of nonpositive curvature: acylindrical actions (on hyperbolic spaces) and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for simultaneously studying many important families of groups, including mapping class groups, rightangled Coxeter groups, most 3manifold groups, rightangled Artin groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces. It is natural to try to develop an understanding of all such actions and to search for a "best" one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure, and in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindri
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We demonstrate the use of the Unified Transform Method or Method of Fokas for boundary value problems for systems of constantcoefficient linear partial differential equations. We discuss how the apparent branch singularities typically appearing in the global relation are removable, allowing the method to proceed, in essence, as for scalar problems. We illustrate the use of the method with boundary value problems for the KleinGordon equation and the linearized FitzhughNagumo system. The case of wave equations is treated separately in an appendix.
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Notions and results from quantum harmonic analysis, such as the convolution between functions and operators or between two operators, is identified as the appropriate setting for Berezin quantization and BerezinLieb inequalities. Based on this insight we provide a rigorous approach to generalized phasespace representation introduced by KlauderSkagerstam and their variants of BerezinLieb inequalities in this setting. Hence our presentation of the results of KlauderSkagerstam gives a more conceptual framework, which yields as a byproduct an interesting perspective on the connection between Berezin quantization and Weyl quantization.
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This paper concerns the continuous time meanvariance portfolio selection problem with a special nonlinear wealth equation. This nonlinear wealth equation has nonsmooth random coefficients and the dual method developed in [7] does not work. To apply the completion of squares technique, we introduce two Riccati equations to cope with the positive and negative part of the wealth process separately. We obtain the efficient portfolio strategy and efficient frontier for this problem. Finally, we find the appropriate subderivative claimed in [7] using convex duality method.
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We investigate a mixed $01$ conic quadratic optimization problem with indicator variables arising in meanrisk optimization. The indicator variables are often used to model nonconvexities such as fixed charges or cardinality constraints. Observing that the problem reduces to a submodular function minimization for its binary restriction, we derive three classes of strong convex valid inequalities by lifting the polymatroid inequalities on the binary variables. Computational experiments demonstrate the effectiveness of the inequalities in strengthening the convex relaxations and, thereby, improving the solution times for meanrisk problems with fixed charges and cardinality constraints significantly.
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Let $W$ be the ring of the Witt vectors of a perfect field of characteristic $p$, $\mathfrak{X}$ a smooth formal scheme over $W$, $\mathfrak{X}'$ the base change of $\mathfrak{X}$ by the Frobenius morphism of $W$, $\mathfrak{X}_{2}'$ the reduction modulo $p^{2}$ of $\mathfrak{X}'$ and $X$ the special fiber of $\mathfrak{X}$. We lift the Cartier transform of OgusVologodsky defined by $\mathfrak{X}_{2}'$ modulo $p^{n}$. More precisely, we construct a functor from the category of $p^{n}$torsion $\mathscr{O}_{\mathfrak{X}'}$modules with integrable $p$connection to the category of $p^{n}$torsion $\mathscr{O}_{\mathfrak{X}}$modules with integrable connection, each subject to suitable nilpotence conditions. Our construction is based on Oyama's reformulation of the Cartier transform of OgusVologodsky in characteristic $p$. If there exists a lifting $F:\mathfrak{X}\to \mathfrak{X}'$ of the relative Frobenius morphism of $X$, our functor is compatible with a functor constructed by Shiho f
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This paper presents a novel massconservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast. The proposed method solves the flow equation in a mixed formulation on a coarse grid by constructing multiscale basis functions. The resulting velocity field is mass conservative on the fine grid. Our main goal is to obtain firstorder convergence in terms of the mesh size which is independent of local contrast. This is achieved, first, by constructing some auxiliary spaces, which contain global information that can not be localized, in general. This is built on our previous work on the Generalized Multiscale Finite Element Method (GMsFEM). In the auxiliary space, multiscale basis functions corresponding to small (contrastdependent) eigenvalues are selected. These basis functions represent the highconductivity channels (which connect the boundaries of a coarse block). Next, we
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In topological quantum computing, information is encoded in "knotted" quantum states of topological phases of matter, thus being locked into topology to prevent decay. Topological precision has been confirmed in quantum Hall liquids by experiments to an accuracy of $10^{10}$, and harnessed to stabilize quantum memory. In this survey, we discuss the conceptual development of this interdisciplinary field at the juncture of mathematics, physics and computer science. Our focus is on computing and physical motivations, basic mathematical notions and results, open problems and future directions related to and/or inspired by topological quantum computing.
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In this paper, we define a general class of abstract aerial robotic systems named Laterally Bounded Force (LBF) vehicles, in which most of the control authority is expressed along a principal thrust direction, while in the lateral directions a (smaller and possibly null) force may be exploited to achieve fullpose tracking. This class approximates well platforms endowed with noncoplanar/noncollinear rotors that can use the tilted propellers to slightly change the orientation of the total thrust w.r.t. the body frame. For this broad class of systems, we introduce a new geometric control strategy in SE(3) to achieve, whenever made possible by the force constraints, the independent tracking of positionplusorientation trajectories. The exponential tracking of a feasible fullpose reference trajectory is proven using a Lyapunov technique in SE(3). The method can deal seamlessly with both under and fullyactuated LBF platforms. The controller guarantees the tracking of at least the posi
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We unveil the existence of a precise mapping between the ground state of noninteracting free fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: i) we consider the full family of admissible quantum boundary conditions (i.e., selfadjoint extensions) for the Laplacian on a bounded interval, and the corresponding projection correlation kernels; ii) we construct the grand canonical extensions at finite temperature of the projection kernels, interpolating from Poisson to random matrix eigenvalue statistics. The scaling limits in the bulk and at the edges are studied in a unified framework, and the question of universality is addressed. Whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is, a priori, not obvious. We complete the picture b
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Many realworld examples of fluidstructure interaction, including the transit of red blood cells through the narrow slits in the spleen and the intracellular trafficking of vesicles into dendritic spines, involve the nearcontact of elastic structures separated by thin layers of fluid. Motivated by such problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We demonstrate 2ndorder accurate convergence for simple twodimensional flows with known exact solutions to showcase the increased accuracy of this method compared to the standard immersed boundary method. Motivated by the phenomenon of wallinduced migration, we apply the lubricated immersed boundary method to simulate an elastic capsule near a wall in shear flow. We also simulate the dynamics of a deformable vesicle traveling through a narrow channel and observe that the ability of the lubricated method to capture the vesicle motion is
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We study Riemannian metrics on compact, torsionless, nongeometric $3$manifolds, i.e. whose interior does not support any of the eight model geometries. We prove a lower bound "\`a la Margulis" for the systole and a volume estimate for these manifolds, only in terms of an upper bound of entropy and diameter. We then deduce corresponding local topological rigidy results in the class $\mathscr{M}_{ngt}^\partial (E,D) $ of compact nongeometric 3manifolds with torsionless fundamental group (with possibly empty, nonspherical boundary) whose entropy and diameter are bounded respectively by $E, D$. For instance, this class locally contains only finitely many topological types; and closed, irreducible manifolds in this class which are close enough (with respect to $E,D$) are diffeomorphic. Several examples and counterexamples are produced to stress the differences with the geometric case.
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In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps, we identify YangMills connections obtained by approximations with respect to the YangMills {\alpha}energy. More specifically, we show that for the SU(2) Hopf fibration over the four sphere, for sufficiently small {\alpha} values the SO(4) invariant ADHM instanton is the unique {\alpha}critical point which has YangMills {\alpha}energy lower than a specific threshold.
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We show that the Lie algebra of polynomial vector fields on an irreducible affine variety X is simple if and only if X is a smooth variety. This completes the result of Jordan on the simplicity of the derivation algebra \cite{Jo}. Given proof is selfcontained and does not depend on the results of Jordan. Besides, the structure of the module of polynomial functions on an irreducible smooth affine variety over the Lie algebra of vector fields is studied. Examples of Lie algebras of polynomial vector fields on an Ndimensional sphere, nonsingular hyperelliptic curves and linear algebraic groups are considered.
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These notes present a first graduate course in harmonic analysis. The first part emphasizes Fourier series, since so many aspects of harmonic analysis arise already in that classical context. The Hilbert transform is treated on the circle, for example, where it is used to prove L^p convergence of Fourier series. Maximal functions and CalderonZygmund decompositions are treated in R^d, so that they can be applied again in the second part of the course, where the Fourier transform is studied. The final part of the course treats band limited functions, Poisson summation and uncertainty principles. Distribution functions and interpolation are covered in the Appendices. The references at the beginning of each chapter provide guidance to students who wish to delve more deeply, or roam more widely, in the subject.
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Let $X = \{1,1\}^\mathbb{N}$ be the symbolic space endowed with the product order. A Borel probability measure $\mu$ over $X$ is said to satisfy the FKG inequality if for any pair of continuous increasing functions $f$ and $g$ we have $\mu(fg)\mu(f)\mu(g)\geq 0$. In the first part of the paper we prove the validity of the FKG inequality on Thermodynamic Formalism setting for a class of eigenmeasures of the dual of the Ruelle operator, including several examples of interest in Statistical Mechanics. In addition to deducing this inequality in cases not covered by classical results about attractive specifications our proof has advantage of to be easily adapted for suitable subshifts. We review (and provide proofs in our setting) some classical results about the longrange Ising model on the lattice $\mathbb{N}$ and use them to deduce some monotonicity properties of the associated Ruelle operator and their relations with phase transitions. As is widely known, for some continuous potentia
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We study the class of groups having the property that every nonnilpotent subgroup is equal to its normalizer. These groups are either soluble or perfect. We completely describe the structure of soluble groups and finite perfect groups with the above property. Furthermore, we give some structural information in the infinite perfect case.
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Unbounded order convergence has lately been systematically studied as a generalization of almost everywhere convergence to the abstract setting of vector and Banach lattices. This paper presents a duality theory for unbounded order convergence. We define the unbounded order dual (or uodual) $X_{uo}^\sim$ of a Banach lattice $X$ and identify it as the order continuous part of the order continuous dual $X_n^\sim$. The result allows us to characterize the Banach lattices that have order continuous preduals and to show that an order continuous predual is unique when it exists. Applications to the FenchelMoreau duality theory of convex functionals are given. The applications are of interest in the theory of risk measures in Mathematical Finance.
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In optical transport networks, signal lightpaths between two terminal nodes can be different due to current network conditions. Thus the transmission distance and accumulated dispersion in the lightpath cannot be predicted. Therefore, the adaptive compensation of dynamic dispersion is necessary in such networks to enable a flexible routing and switching. In this paper, we present a detailed analysis on the adaptive dispersion compensation using the leastmeansquare (LMS) algorithms in coherent optical communication networks. It is found that the variablestepsize LMS equalizer can achieve the same performance with a lower complexity, compared to the traditional LMS algorithm.
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In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an $L^1$ function, extending the Lagrangian theory in \cite{BouchutCrippa13}. The proof is based on a combination of a stability estimate via optimal transport techniques developed in \cite{Seis16a} and some tools from harmonic analysis introduced in \cite{BouchutCrippa13}. In the second part of the paper, we address a question that arose in \cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data has low integrability. We show that this is the case even when the initial vorticity is only in~$L^1$, extending the proof for the $L^p$ case in \cite{CrippaSpirito15}.
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We study the chemotaxisfluid system \begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=\Delta n\nabla\!\cdot(\frac{n}{c}\nabla c),\ &x\in\Omega,& t>0, c_{t}&+&u\cdot\!\nabla c&=\Delta cnc,\ &x\in\Omega,& t>0, u_{t}&+&\nabla P&=\Delta u+n\nabla\phi,\ &x\in\Omega,& t>0, &&\nabla\cdot u&=0,\ &x\in\Omega,& t>0, \end{array}\right. \end{align*} under homogeneous Neumann boundary conditions for $n$ and $c$ and homogeneous Dirichlet boundary conditions for $u$, where $\Omega\subset\mathbb{R}^2$ is a bounded domain with smooth boundary and $\phi\in C^{2}\left(\bar{\Omega}\right)$. From recent results it is known that for suitable regular initial data, the corresponding initialboundary value problem possesses a global generalized solution. We will show that for small initial mass $\int_{\Omega}\!n_0$ these generalized solutions will eventually b
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This paper is devoted to study the sharp MoserTrudinger type inequalities in whole space $\mathbb R^N$, $N \geq 2$ in more general case. We first compute explicitly the \emph{normalized vanishing limit} and the \emph{normalized concentrating limit} of the MoserTrudinger type functional associated with our inequalities over all the \emph{normalized vanishing sequences} and the \emph{normalized concentrating sequences}, respectively. Exploiting these limits together with the concentrationcompactness principle of Lions type, we give a proof of the exitence of maximizers for these MoserTrudinger type inequalities. Our approach gives an alternative proof of the existence of maximizers for the MoserTrudinger inequality and singular MoserTrudinger inequality in whole space $\mathbb R^N$ due to Li and Ruf \cite{LiRuf2008} and Li and Yang \cite{LiYang}.
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We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from pointset topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.
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We consider random dynamical systems on manifolds modeled by a skew product which have certain geometric properties and whose measures satisfy quenched decay of correlations at a sufficient rate. We prove that the limiting distribution for the hitting and return times to geometric balls are both exponential for almost every realisation. We then apply this result to random $C^2$ maps of the interval and random parabolic maps on the unit interval.
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We study the 3D compressible barotropic radiation fluid dynamics system describing the motion of the compressible rotating viscous fluid with gravitation and radiation confined to a straight layer. We show that weak solutions in the 3D domain converge to the strong solution of the rotating 2D NavierStokesPoisson system with radiation for all times less than the maximal life time of the strong solution of the 2D system when the Froude number is small or to the strong solution of the rotating pure 2D Navier Stokes system with radiation.
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We consider the estimation of hidden Markovian process by using information geometry with respect to transition matrices. We consider the case when we use only the histogram of $k$memory data. Firstly, we focus on a partial observation model with Markovian process and we show that the asymptotic estimation error of this model is given as the inverse of projective Fisher information of transition matrices. Next, we apply this result to the estimation of hidden Markovian process. For this purpose, we define an exponential family of ${\cal Y}$valued transition matrices. We carefully discuss the equivalence problem for hidden Markovian process on the tangent space. Then, we propose a novel method to estimate hidden Markovian process.
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We give a description of the Hochschild cohomology for noncommutative planes (resp. quadrics) using the automorphism groups of the elliptic triples (resp. quadruples) that classify the ArtinSchelter regular $\mathbb{Z}$algebras used to define noncommutative planes and quadrics. For elliptic triples the description of the automorphism groups is due to BondalPolishchuk, for elliptic quadruples it is new.
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We prove that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$, and nonnegative integers $j$ and $r$ with $j\leqslant m$, the following two expressions \begin{align*} &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack n_1}^{1}\sum_{k=0}^{n_1} q^{j(k^2+k)(2r+1)k}[2k+1]^{2r+1}\prod_{i=1}^m {n_i+n_{i+1}+1\brack n_ik},\\[5pt] &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack n_1}^{1}\sum_{k=0}^{n_1}(1)^k q^{{k\choose 2}+j(k^2+k)2rk}[2k+1]^{2r+1}\prod_{i=1}^m {n_i+n_{i+1}+1\brack n_ik} \end{align*} are Laurent polynomials in $q$ with integer coefficients, where $[n]=1+q+\cdots+q^{n1}$ and ${n\brack k}=\prod_{i=1}^k(1q^{ni+1})/(1q^i)$. This gives a $q$analogue of some divisibility results of sums and alternating sums involving binomial coefficients and powers of integers obtained by Guo and Zeng. We also confirm some related conjectures of Guo and Zeng by establishing their $q$analogues. Several conjectural congruences for sums involving products of $q$ballot numbers $\left({2n\b
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We introduce LRT, a new Lagrangianbased ReachTube computation algorithm that conservatively approximates the set of reachable states of a nonlinear dynamical system. LRT makes use of the CauchyGreen stretching factor (SF), which is derived from an overapproximation of the gradient of the solution flows. The SF measures the discrepancy between two states propagated by the system solution from two initial states lying in a welldefined region, thereby allowing LRT to compute a reachtube with a balloverestimate in a metric where the computed enclosure is as tight as possible. To evaluate its performance, we implemented a prototype of LRT in C++/Matlab, and ran it on a set of wellestablished benchmarks. Our results show that LRT compares very favorably with respect to the CAPD and Flow* tools.
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We study the correlation functions of the Pfaffian Schur process using Macdonald difference operators. Sasamoto and Imamura \cite{SmIm04} introduced the Pfaffian Schur process for studying the polynuclear growth processes in halfspace. Later, Borodin and Rains \cite{BR05} derived the correlation functions of the Pfaffian Schur process using a Pfaffian analogue of the EynardMehta theorem. We present here an alternative derivation using Macdonald difference operators. One can find similar exposition for the Schur process in \cite{A14}.
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