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In this paper, we elucidate the key role played by the cosymplectic geometry in the theory of time dependent Hamiltonian systems. In particular, we generalize the cosymplectic structures to timedependent NambuPoisson Hamiltonian systems and corresponding Jacobi's last multiplier for 3D systems. We illustrate our constructions with various examples.
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We consider the random walk on the hypercube which moves by picking an ordered pair $(i,j)$ of distinct coordinates uniformly at random and adding the bit at location $i$ to the bit at location $j$, modulo $2$. We show that this Markov chain has cutoff at time $\frac{3}{2}n\log n$ with window of size $n$, solving a question posed by Chung and Graham (1997).
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This paper analyzes the iterationcomplexity of a generalized alternating direction method of multipliers (GADMM) for solving linearly constrained convex problems. This ADMM variant, which was first proposed by Bertsekas and Eckstein, introduces a relaxation parameter $\alpha \in (0,2)$ into the second ADMM subproblem. Our approach is to show that the GADMM is an instance of a hybrid proximal extragradient framework with some special properties, and, as a by product, we obtain ergodic iterationcomplexity for the GADMM with $\alpha\in (0,2]$, improving and complementing related results in the literature. Additionally, we also present pointwise iterationcomplexity for the GADMM.
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In the paper, we present a high order fast algorithm with almost optimum memory for the Caputo fractional derivative, which can be expressed as a convolution of $u'(t)$ with the kernel $(t_nt)^{\alpha}$. In the fast algorithm, the interval $[0,t_{n1}]$ is split into nonuniform subintervals. The number of the subintervals is in the order of $\log n$ at the $n$th time step. The fractional kernel function is approximated by a polynomial function of $K$th degree with a uniform absolute error on each subinterval. We save $K+1$ integrals on each subinterval, which can be written as a convolution of $u'(t)$ with a polynomial base function. As compared with the direct method, the proposed fast algorithm reduces the storage requirement and computational cost from $O(n)$ to $O((K+1)\log n)$ at the $n$th time step. We prove that the convergence rate of the fast algorithm is the same as the direct method even a high order direct method is considered. The convergence rate and efficiency of th
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In this article, a sixparameter family of highly connected 7manifolds which admit an SO(3)invariant metric of nonnegative sectional curvature is constructed and the EellsKuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)invariant metric of nonnegative curvature.
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In \cite{JS} Jensen and Su constructed 0Schur algebras on double flag varieties. The construction leads to a presentation of 0Schur algebras using quivers with relations and the quiver approach naturally gives rise to a new class of algebras. That is, the path algebras defined on the quivers of 0Schur algebras with relations modified from the defining relations of 0Schur algebras by a tuple of parameters $\ut$. In particular, when all the entries of $\ut$ are 1, we have 0Schur algerbas. When all the entries of $\ut$ are zero, we obtain a class of degenerate 0Schur algebras. We prove that the degenerate algebras are associated graded algebras and quotients of 0Schur algebras. Moreover, we give a geometric interpretation of the degenerate algebras using double flag varieties, in the same spirit as \cite{JS}, and show how the centralizer algebras are related to nilHecke algebras and nilTemperlyLieb algebras
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This paper deals with initial value problems for fractional functional differential equations with bounded delay. The fractional derivative is defined in the Caputo sense. By using the Schauder fixed point theorem and the properties of the MittagLeffler function, new existence and uniqueness results for global solutions of the initial value problems are established. In particular, the unique existence of global solution is proved under the condition close to the Nagumotype condition.
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In the paper, we study the minimization problem of a nonconvex sparsity promoting penalty function $$P_{a}(x)=\sum_{i=1}^{n}p_{a}(x_{i})=\sum_{i=1}^{n}\frac{ax_{i}}{1+ax_{i}}$$ in compressed sensing, which is called fraction function. Firstly, we discuss the equivalence of $\ell_{0}$ minimization and fraction function minimization. It is proved that there corresponds a constant $a^{**}>0$ such that, whenever $a>a^{**}$, every solution to $(FP_{a})$ also solves $(P_{0})$, that the uniqueness of global minimizer of $(FP_{a})$ and its equivalence to $(P_{0})$ if the sensing matrix $A$ satisfies a restricted isometry property (RIP) and, last but the most important, that the optimal solution to the regularization problem $(FP_{a}^\lambda)$ also solves $(FP_{a})$ if the certain condition is satisfied, which is similar to the regularization problem in convex optimal theory. Secondly, we study the properties of the optimal solution to the regularization problem $(FP^{\lambda}_{a})$
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We study ${\mathbb Z}$graded thread $W^+$modules $$V=\oplus_i V_i, \; \dim{V_i}=1, \infty \le k< i < N\le +\infty, \; \dim{V_i}=0, \; {\rm \; otherwise},$$ over the positive part $W^+$ of the Witt (Virasoro) algebra $W$. There is wellknown example of infinitedimensional ($k=\infty, N=\infty$) twoparametric family $V_{\lambda, \mu}$ of $W^+$modules induced by the twisted $W$action on tensor densities $P(x)x^{\mu}(dx)^{\lambda}, \mu, \lambda \in {\mathbb K}, P(x) \in {\mathbb K}[t]$. Another family $C_{\alpha, \beta}$ of $W^+$modules is defined by the action of two multiplicative generators $e_1, e_2$ of $W^+$ as $e_1f_i=\alpha f_{i+1}$ and $e_2f_j=\beta f_{j+2}$ for $i,j \in {\mathbb Z}$ and $\alpha, \beta$ are two arbitrary constants ($e_if_j=0, i \ge 3$). We classify $(n+1)$dimensional graded thread $W^+$modules for $n$ sufficiently large $n$ of three important types. New examples of graded thread $W^+$modules different from finitedimensional quotients of $V_{\lam
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Millimeter wave (mmWave) communications have recently attracted large research interest, since the huge available bandwidth can potentially lead to rates of multiple Gbps (gigabit per second) per user. Though mmWave can be readily used in stationary scenarios such as indoor hotspots or backhaul, it is challenging to use mmWave in mobile networks, where the transmitting/receiving nodes may be moving, channels may have a complicated structure, and the coordination among multiple nodes is difficult. To fully exploit the high potential rates of mmWave in mobile networks, lots of technical problems must be addressed. This paper presents a comprehensive survey of mmWave communications for future mobile networks (5G and beyond). We first summarize the recent channel measurement campaigns and modeling results. Then, we discuss in detail recent progresses in multiple input multiple output (MIMO) transceiver design for mmWave communications. After that, we provide an overview of the solution for
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We study minimal Lorentz surfaces in the pseudoEuclidean 4space with neutral metric whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the inequality $K^2\varkappa^2 >0$. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of this class we introduce geometrically determined canonical parameters and prove that the Gauss curvature and the normal curvature of the surface satisfy a system of two natural partial differential equations. Conversely, any solution to this system determines a unique (up to a rigid motion) minimal Lorentz surface of general type such that the given parameters are canonical.
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At first glance the notion of an algebra with a generalized $H$action may appear too general, however it enables to work with algebras endowed with various kinds of additional structures (e.g. Hopf (co)module algebras, graded algebras, algebras with an action of a (semi)group by (anti)endomorphisms). This approach proves to be especially fruitful in the theory of polynomial identities. We show that if $A$ is a finite dimensional (not necessarily associative) algebra simple with respect to a generalized $H$action over a field of characteristic $0$, then there exists $\lim_{n\to\infty}\sqrt[n]{c_n^H(A)} \in \mathbb R_+$ where $\left(c_n^H(A)\right)_{n=1}^\infty$ is the sequence of codimensions of polynomial $H$identities of $A$. In particular, if $A$ is a finite dimensional (not necessarily group graded) gradedsimple algebra, then there exists $\lim_{n\to\infty}\sqrt[n]{c_n^{\mathrm{gr}}(A)} \in \mathbb R_+$ where $\left(c_n^{\mathrm{gr}}(A)\right)_{n=1}^\infty$ is the sequence of co
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The dramatic increase of observational data across industries provides unparalleled opportunities for datadriven decision making and management, including the manufacturing industry. In the context of production, datadriven approaches can exploit observational data to model, control and improve the process performance. When supplied by observational data with adequate coverage to inform the true process performance dynamics, they can overcome the cost associated with intrusive controlled designed experiments and can be applied for both monitoring and improving process quality. We propose a novel integrated approach that uses observational data for process parameter design while simultaneously identifying the significant control variables. We evaluate our method using simulated experiments and also apply it to a realworld case setting from a tire manufacturing company.
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In this paper, based on the block operator technique and operator spectral theory, the general explicit expressions for intertwining operators and direct rotations of two orthogonal projections have been established. As a consequence, it is an improvement of Kato's result (Perturbation Theory of Linear operators, SpringerVerlag, Berlin/Heidelberg, 1996); J. Avron, R. Seiler and B. Simon's Theorem 2.3 (The index of a pair of projections, J. Funct. Anal. 120(1994) 220237) and C. Davis, W.M. Kahan, (The rotation of eigenvectors by a perturbation, III. SIAM J. Numer. Anal. 7(1970) 146).
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Due to complexity and invisibility of human organs, diagnosticians need to analyze medical images to determine where the lesion region is, and which kind of disease is, in order to make precise diagnoses. For satisfying clinical purposes through analyzing medical images, registration plays an essential role. For instance, in ImageGuided Interventions (IGI) and computeraided surgeries, patient anatomy is registered to preoperative images to guide surgeons complete procedures. Medical image registration is also very useful in surgical planning, monitoring disease progression and for atlas construction. Due to the significance, the theories, methods, and implementation method of image registration constitute fundamental knowledge in educational training for medical specialists. In this chapter, we focus on image registration of a specific human organ, i.e. the lung, which is prone to be lesioned. For pulmonary image registration, the improvement of the accuracy and how to obtain it in o
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Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a connected reductive algebraic group over $k$. Under some standard hypothesis on $G$, we give a direct approach to the finite $W$algebra $U(\mathfrak g,e)$ associated to a nilpotent element $e \in \mathfrak g = \operatorname{Lie} G$. We prove a PBW theorem and deduce a number of consequences, then move on to define and study the $p$centre of $U(\mathfrak g,e)$, which allows us to define reduced finite $W$algebras $U_\eta(\mathfrak g,e)$ and we verify that they coincide with those previously appearing in the work of Premet. Finally, we prove a modular version of Skryabin's equivalence of categories, generalizing recent work of the second author.
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In this work we deal with the recently introduced concept of weaving frames. We extend the concept to include multiwindow frames and present the first sufficient criteria for a family of multiwindow Gabor frames to be woven. We give a Hilbert space norm criterion and a pointwise criterion in phase space. The key ingredient are localization operators in phase space and we give examples of woven multiwindow Gabor frames consisting of Hermite functions.
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A rectangular parallelepiped is called a cuboid (standing box). It is called perfect if its edges, face diagonals and body diagonal all have integer length. Euler gave an example where only the body diagonal failed to be an integer (Euler brick). Are there perfect cuboids? We prove that there is no perfect cuboid.
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For $q,n,d \in \mathbb{N}$, let $A_q(n,d)$ be the maximum size of a code $C \subseteq [q]^n$ with minimum distance at least $d$. We give a divisibility argument resulting in the new upper bounds $A_5(8,6) \leq 65$, $A_4(11,8)\leq 60$ and $A_3(16,11) \leq 29$. These in turn imply the new upper bounds $A_5(9,6) \leq 325$, $A_5(10,6) \leq 1625$, $A_5(11,6) \leq 8125$ and $A_4(12,8) \leq 240$. Furthermore, we prove that for $\mu,q \in \mathbb{N}$, there is a 11correspondence between symmetric $(\mu,q)$nets (which are certain designs) and codes $C \subseteq [q]^{\mu q}$ of size $\mu q^2$ with minimum distance at least $\mu q  \mu$. We derive the new upper bounds $A_4(9,6) \leq 120$ and $A_4(10,6) \leq 480$ from these `symmetric net' codes.
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We consider a network design problem with random arc capacities and give a formulation with a probabilistic capacity constraint on each cut of the network. To handle the exponentiallymany probabilistic constraints a separation procedure that solves a nonlinear minimum cut problem is introduced. For the case with independent arc capacities, we exploit the supermodularity of the set function defining the constraints and generate cutting planes based on the supermodular covering knapsack polytope. For the general correlated case, we give a reformulation of the constraints that allows to uncover and utilize the submodularity of a related function. The computational results indicate that exploiting the underlying submodularity and supermodularity arising with the probabilistic constraints provides significant advantages over the classical approaches.
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We establish upper bounds for the convolution operator acting between interpolation spaces. This will provide several examples of Young Inequalities in different families of function spaces. We use this result to prove a bilinear interpolation theorem and we show applications to the study of bilinear multipliers.
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We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong onesheeted graphical subsequential convergence, discuss the limit behaviour when multisheeted convergence happens and derive various consequences in terms of finiteness and topological control.
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We have undertaken an algorithmic search for new integrable semidiscretizations of physically relevant nonlinear partial differential equations. The search is performed by using a compatibility condition for the discrete Lax operators and symbolic computations. We have discovered a new integrable system of coupled nonlinear Schrodinger equations which combines elements of the AblowitzLadik lattice and the triangularlattice ribbon studied by Vakhnenko. We show that the continuum limit of the new integrable system is given by uncoupled complex modified Kortewegde Vries equations and uncoupled nonlinear Schrodinger equations.
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We determine barycentric coordinates of triangle centers in the elliptic plane. The main focus is put on centers that lie on lines whose euclidean limit (triangle excess $\rightarrow 0$) is the Euler line or the Brocard line. We also investigate curves which can serve in elliptic geometry as substitutes for the euclidean ninepointcircle, the first Lemoine circle or the apollonian circles.
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We propose new iterative methods for computing nontrivial extremal generalized singular values and vectors. The first method is a generalized Davidsontype algorithm and the second method employs a multidirectional subspace expansion technique. Essential to the latter is a fast truncation step designed to remove a low quality search direction and to ensure moderate growth of the search space. Both methods rely on thick restarts and may be combined with two different deflation approaches. We argue that the methods have monotonic and (asymptotic) linear convergence, derive and discuss locally optimal expansion vectors, and explain why the fast truncation step ideally removes search directions orthogonal to the desired generalized singular vector. Furthermore, we identify the relation between our generalized Davidsontype algorithm and the JacobiDavidson algorithm for the generalized singular value decomposition. Finally, we generalize several known convergence results for the Hermitian
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In the setting of symplectic manifolds which are convex at infinity, we use a version of the Aleksandrov maximum principle to derive uniform estimates for Floer solutions that are valid for a wider class of Hamiltonians and almost complex structures than is usually considered. This allows us to extend the class of Hamiltonians which one can use in the direct limit when constructing symplectic homology. As an application, we detect elements of infinite order in the symplectic mapping class group of a Liouville domain, and prove existence results for translated points.
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This paper deals with the Orthogonal Procrustes Problem in R^D by considering either two distinct point configurations or the distribution of distances of two point configurations. The objective is to align two distinct point configurations by first finding a correspondence between the points and then constructing the map which aligns the configurations.This idea is also extended to epsilondistorted diffeomorphisms which were introduced in [30] by Fefferman and Damelin. Examples are given to show when distributions of distances do not allow alignment if the distributions match, and when we can partition our configurations into polygons in order to construct the maximum possible correspondences between the configurations, considering their areas. Included is also a brief overview of reconstructing configurations, given their distance distributions. Finally, some algorithms are described for configurations with matching points along with examples, where we find a permutation which will
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A linear automorphism of Euclidean space is called bicircular its eigenvalues lie in the disjoint union of two circles $C_1$ and $C_2$ in the complex plane where the radius of $C_1$ is $r_1$, the radius of $C_2$ is $r_2$, and $0 < r_1 < 1 < r_2$. A wellknown theorem of Philip Hartman states that a local $C^{1,1}$ diffeomorphism $T$ of Euclidean space with a fixed point $p$ whose derivative $DT_p$ is bicircular is $C^{1,\beta}$ linearizable near $p$. We generalize this result to $C^{1,\alpha}$ diffeomorphisms $T$ where $0 < \alpha < 1$. We also extend the result to local diffeomorphisms in Banach spaces with $C^{1,\alpha}$ bump functions. The results apply to give simpler proofs under weaker regularity conditions of classical results of L. P. Shilnikov on the existence of horseshoe dynamics near socalled saddlefocus critical points of vector fields in $R^3$.
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Certain systems of inviscid fluid dynamics have the property that for solutions with just a modest amount of regularity in Eulerian variables, the corresponding Lagrangian trajectories are analytic in time. We elucidate the mechanisms in fluid dynamics systems that give rise to this automatic Lagrangian analyticity, as well as mechanisms in some particular fluids systems which prevent it from occurring. We give a conceptual argument for a general fluids model which shows that the fulfillment of a basic set of criteria results in the analyticity of the trajectory maps in time. We then apply this to the incompressible Euler equations, obtaining analyticity for vortex patch solutions in particular. We also use the method to prove the Lagrangian trajectories are analytic for solutions to the pressureless EulerPoisson equations, for initial data with moderate regularity. We then examine the compressible Euler equations, and find that the finite speed of propagation in the system is incompa
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Topos quantum theory provides representations of quantum states as direct generalizations of the probability distribution, namely probability valuation. In this article, we consider extensions of a known bijective correspondence between quantum states and probability valuations to composite systems and to state transformations. We show that multipartite probability valuations on composite systems have a bijective correspondence to positive over pure tensor states, according to a candidate definition of the composite systems in topos quantum theory. Among the multipartite probability valuations, a special attention is placed to Markov chains which are defined by generalizing classical Markov chains from probability theory. We find an incompatibility between the multipartite probability valuations and a monogamy property of quantum states, which trivializes the Markov chains to product probability valuations. Several observations on the transformations of probability valuations are deduc
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Let $k,N \in \mathbb{N}$ with $N$ squarefree and $k>1$. Let $f(z) \in M_{2k}(\Gamma_0(N))$ be a modular form. We prove an orthogonal relation, and use this to compute the coefficients of Eisenstein part of $f(z)$ in terms of sum of divisors function. In particular, if $f(z) \in E_{2k}(\Gamma_0(N))$, then the computation will to yield to an expression for Fourier coefficients of $f(z)$. We give three applications of the results. First, we give formulas for convolution sums of the divisor function to extend the result by Ramanujan. Second, we give formulas for number of representations of integers by certain infinite families of quadratic forms. And at last, we determine a formula for Fourier coefficients of $f(z)\in E_{2k}(\Gamma_0(N))$, where $f(z)$ is an eta quotient, and then we show that the set $\{ f(z) \in E_{2k}(\Gamma_0(N)), k \geq 1 \}$ is finite for all $N \in \mathbb{N}$ squarefree.
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The capacity of symmetric, neighboring and consecutive sideinformation single unicast index coding problems (SNCSUICP) with number of messages equal to the number of receivers was given by Maleki, Cadambe and Jafar. For these index coding problems, an optimal index code construction by using Vandermonde matrices was proposed. This construction requires all the sideinformation at the receivers to decode their wanted messages and also requires large field size. In an earlier work, we constructed binary matrices of size $m \times n (m\geq n)$ such that any $n$ adjacent rows of the matrix are linearly independent over every field. Calling these matrices as Adjacent Independent Row (AIR) matrices using which we gave an optimal scalar linear index code for the onesided SNCSUICP for any given number of messages and onesided sideinformation. By using Vandermonde matrices or AIR matrices, every receiver needs to solve $KD$ equations with $KD$ unknowns to obtain its wanted message, wher
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We present a covariant multisymplectic formulation for the EinsteinHilbert model of General Relativity. As it is described by a secondorder singular Lagrangian, this is a gauge field theory with constraints. The use of the unified LagrangianHamiltonian formalism is particularly interesting when it is applied to these kinds of theories, since it simplifies the treatment of them; in particular, the implementation of the constraint algorithm, the retrieval of the Lagrangian description, and the construction of the covariant Hamiltonian formalism. In order to apply this algorithm to the covariant field equations, they must be written in a suitable geometrical way which consists of using integrable distributions, represented by multivector fields of a certain type. We apply all these tools to the EinsteinHilbert model without and with energymatter sources. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomentum (covari
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We consider the flow in direction $\theta$ on a translation surface and we study the asymptotic behavior for $r\to 0$ of the time needed by orbits to hit the $r$neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as \emph{hitting time}. For flat tori the limsup of hitting time is equal to the diophantine type of the direction $\theta$. In higher genus, we consider an extended geometric notion of diophantine type of a direction $\theta$ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the diophantine type. For any squaretiled surface with the same topology, the diophantine type itself is a lower bound. Moreover both bounds are sharp for big sets of directions. Our results apply to Lshaped billiards.
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Conjectural results for cohomological invariants of wild character varieties are obtained by counting curves in degenerate CalabiYau threefolds. A conjectural formula for Epolynomials is derived from the GromovWitten theory of local CalabiYau threefolds with normal crossing singularities. A refinement is also conjectured, generalizing existing results of Hausel, Mereb and Wong as well as recent joint work of Donagi, Pantev and the author for weighted Poincar\'e polynomials of wild character varieties.
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In this paper additive bifree convolution is defined for general Borel probability measures, and the limiting distributions for sums of bifree pairs of selfadjoint commuting random variables in an infinitesimal triangular array are determined. These distributions are characterized by their bifreely infinite divisibility, and moreover, a transfer principle is established for limit theorems in classical probability theory and Voiculescu's bifree probability theory. Complete descriptions of bifree stability and fullness of planar probability distributions are also set down. All these results reveal one important feature about the theory of bifree probability that it parallels the classical theory perfectly well. The emphasis in the whole work is not on the tool of bifree combinatorics but only on the analytic machinery.
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In this note we construct conifold transitions between several CalabiYau threefolds given by Pfaffians in weighted projective spaces and CalabiYau threefolds appearing as complete intersections in toric varieties. We use the obtained results to predict mirrors following ideas of \cite{BCKS, Batsmalltoricdegen}. In particular we consider the family of CalabiYau threefolds of degree 25 in $\mathbb{P}^9$ obtained as a transverse intersection of two Grassmannians in their Plucker embeddings.
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Multiple hypothesis testing is a central topic in statistics, but despite abundant work on the false discovery rate (FDR) and the corresponding TypeII error concept known as the false nondiscovery rate (FNR), a finegrained understanding of the fundamental limits of multiple testing has not been developed. Our main contribution is to derive a precise nonasymptotic tradeoff between FNR and FDR for a variant of the generalized Gaussian sequence model. Our analysis is flexible enough to permit analyses of settings where the problem parameters vary with the number of hypotheses $n$, including various sparse and dense regimes (with $o(n)$ and $\mathcal{O}(n)$ signals). Moreover, we prove that the BenjaminiHochberg algorithm as well as the BarberCand\`{e}s algorithm are both rateoptimal up to constants across these regimes.
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We study the rate of mixing of observables of Z^dextensions of probability preserving dynamical systems. We explain how this question is directly linked to the local limit theorem and establish a rate of mixing for general classes of observables of the Z^2periodic Sinai billiard. We compare our approach with the induction method.
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Given a reductive representation $\rho: \pi_1(S)\rightarrow G$, there exists a $\rho$equivariant harmonic map $f$ from the universal cover of a fixed Riemann surface $\Sigma$ to the symmetric space $G/K$ associated to $G$. If the Hopf differential of $f$ vanishes, the harmonic map is then minimal. In this paper, we investigate the properties of immersed minimal surfaces inside symmetric space associated to a subloci of Hitchin component: $q_n$ and $q_{n1}$ case. First, we show that the pullback metric of the minimal surface dominates a constant multiple of the hyperbolic metric in the same conformal class and has a strong rigidity property. Secondly, we show that the immersed minimal surface is never tangential to any flat inside the symmetric space. As a direct corollary, the pullback metric of the minimal surface is always strictly negatively curved. In the end, we find a fully decoupled system to approximate the coupled Hitchin system.
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The study of higherdimensional black holes is a subject which has recently attracted a vast interest. Perhaps one of the most surprising discoveries is a realization that the properties of higherdimensional black holes with the spherical horizon topology are very similar to the properties of the well known fourdimensional Kerr metric. This remarkable result stems from the existence of a single object called the principal tensor. In our review we discuss explicit and hidden symmetries of higherdimensional black holes. We start with the overview of the Liouville theory of completely integrable systems and introduce Killing and KillingYano objects representing explicit and hidden symmetries. We demonstrate that the principal tensor can be used as a 'seed object' which generates all these symmetries. It determines the form of the black hole geometry, as well as guarantees its remarkable properties, such as special algebraic type of the spacetime, complete integrability of geodesic mot
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We establish the monotonicity property for the mass of nonpluripolar products on compact K\"ahler manifolds in the spirit of recent results due to Witt Nystr\"om. Building on this, we initiate the variational study of complex MongeAmp\`ere equations with prescribed singularity. As applications, we prove existence and uniqueness of K\"ahlerEinstein metrics with prescribed singularity, and we also provide the log concavity property of nonpluripolar products with small unbounded locus.
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In this paper, we derive the pointwise upper bounds and lower bounds on the gradients of solutions to the Lam\'{e} systems with partially infinite coefficients as the surface of discontinuity of the coefficients of the system is located very close to the boundary. When the distance tends to zero, the optimal blowup rates of the gradients are established for inclusions with arbitrary shapes and in all dimensions.
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Motivated by massive deployment of low data rate Internet of things (IoT) and ehealth devices with requirement for highly reliable communications, this paper proposes receive beamforming techniques for the uplink of a singleinput multipleoutput (SIMO) multiple access channel (MAC), based on a peruser probability of error metric and onedimensional signalling. Although beamforming by directly minimizing probability of error (MPE) has potential advantages over classical beamforming methods such as zeroforcing and minimum mean square error beamforming, MPE beamforming results in a nonconvex and a highly nonlinear optimization problem. In this paper, by adding a set of modulationbased constraints, the MPE beamforming problem is transformed into a convex programming problem. Then, a simplified version of the MPE beamforming is proposed which reduces the exponential number of constraints in the MPE beamforming problem. The simplified problem is also shown to be a convex programming pro
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In this paper we obtain at least 61 new singly even (Type I) binary [72,36,12] selfdual codes as a quasicyclic codes with m=2 (tailbitting convolutional codes) and at least 13 new doubly even (Type II) binary [72,36,12] selfdual codes by replacing the first row in each circulant in a double circulant code by "all ones" and "all zeros" vectors respectively.
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We establish new estimates for the constant $J_a(k,\alpha)$ in the BrudnyiJackson inequality for approximation of $f \in C[1,1]$ by algebraic polynomials: $$ E_{n}^a (f) \le J_a(k, \alpha) \ \omega_k (f, \alpha \pi /n ), \quad \alpha >0 $$ The main result of the paper implies the following inequalities $$ 1/2< J_a (2k, \alpha) < 10, \quad n \ge 2k(2k1), \quad \alpha \ge 2 $$
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On the basis of loop group decompositions (Birkhoff decompositions), we give a discrete version of the nonlinear d'Alembert formula, a method of separation of variables of difference equations, for discrete constant negative Gauss curvature (pseudospherical) surfaces in Euclidean three space. We also compute two examples by this formula in detail.
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We study the equation \begin{equation} (\Delta)^{s}u+V(x)u= (I_{\alpha}*u^{p})u^{p2}u+\lambda(I_{\beta}*u^{q})u^{q2}u \quad\mbox{ in } \R^{N}, \end{equation} where $I_\gamma(x)=x^{\gamma}$ for any $\gamma\in (0,N)$, $p, q >0$, $\alpha,\beta\in (0,N)$, $N\geq 3$ and $ \lambda \in R$. First, the existence of a groundstate solutions using minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy signchanging solutions is investigated by considering the Nehari nodal set.
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Let $p$ be a prime number and let $\mathrm{Cl}_{\mathbb{Q}(\sqrt{p})}$ denote the class group of the imaginary quadratic number field $\mathbb{Q}(\sqrt{p})$. We use Vinogradov's method to show that a spin symbol governing the $16$rank of $\mathrm{Cl}_{\mathbb{Q}(\sqrt{p})}$ is equidistributed, conditional on a standard conjecture about short character sums. This proves that the density of the set of prime numbers $p$ for which $\mathrm{Cl}_{\mathbb{Q}(\sqrt{p})}$ has an element of order $16$ is equal to $\frac{1}{16}$.
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We prove that every function $f:\mathbb{R}^n\to \mathbb{R}$ satisfies that the image of the set of critical points at which the function $f$ has Taylor expansions of order $n1$ and nonempty subdifferentials of order $n$ is a Lebesguenull set. As a byproduct of our proof, for the proximal subdifferential $\partial_{P}$, we see that for every lower semicontinuous function $f:\mathbb{R}^2\to\mathbb{R}$ the set $f(\{x\in\mathbb{R}^2 : 0\in\partial_{P}f(x)\})$ is $\mathcal{L}^{1}$null.
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Let X be a smooth projective curve over a field of characteristic zero. We calculate the motivic class of the moduli stack of semistable Higgs bundles on X. We also calculate the motivic class of the moduli stack of vector bundles with connections by showing that it is equal to the class of the stack of semistable Higgs bundles of the same rank and degree zero. We follow the strategy of Mozgovoy and Schiffmann for counting Higgs bundles over finite fields. The main new ingredient is a motivic version of a theorem of Harder about Eisenstein series claiming that all vector bundles have approximately the same motivic class of Borel reductions as the degree of Borel reduction tends to $\infty$.
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We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or pvalued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in CODF, we establish a relative Galois correspondence for relatively definable subgroups of the group of differential order automorphisms.
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We establish an asymptotic profile that sharply describes the behavior as $t\to\infty$ for solutions to a nonsolenoidal approximation of the incompressible NavierStokes equations introduced by Temam. The solutions of Temam's model are known to converge to the corresponding solutions of the classical NavierStokes, e.g., in $L^3\_{\rm loc} (R^+ \times R^3)$, provided $\epsilon\to0$, where $\epsilon>0$ is the physical parameter related to the artificial compressibility term. However, we show that such model is no longer a good approximation of NavierStokes for large times: indeed, its solutions can decay much slower as $t\to+\infty$ than the corresponding solutions of NavierStokes.
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We study locally univalent functions $f$ analytic in the unit disc $\mathbb{D}$ of the complex plane such that ${f"(z)/f'(z)}(1z^2)\leq 1+C(1z)$ holds for all $z\in\mathbb{D}$, for some $0<C<\infty$. If $C\leq 1$, then $f$ is univalent by Becker's univalence criterion. We discover that for $1<C<\infty$ the function $f$ remains to be univalent in certain horodiscs. Sufficient conditions which imply that $f$ is bounded, belongs to the Bloch space or belongs to the class of normal functions, are discussed. Moreover, we consider generalizations for locally univalent harmonic functions.
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Channel state feedback plays an important role to the improvement of link performance in current wireless communication systems, and even more in the next generation. The feedback information, however, consumes the uplink bandwidth and thus generates overhead. In this paper, we investigate the impact of channel state feedback and propose an improved scheme to reduce the overhead in practical communication systems. Compared with existing schemes, we introduce a more accurate channel model to describe practical wireless channels and obtain the theoretical lower bounds of overhead for the periodical and aperiodical feedback schemes. The obtained theoretical results provide us the guidance to optimise the design of feedback systems, such as the number of bits used for quantizing channel states. We thus propose a practical feedback scheme that achieves low overhead and improved performance over currently widely used schemes such as zero holding. Simulation experiments confirm its advantages
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In this paper the model for a highly viscous droplet sliding down an inclined plane is analyzed. It is shown that, provided the slope is not too steep, the corresponding moving boundary problem possesses classical solutions. Wellposedness is lost when the relevant linearization ceases to be parabolic. This occurs above a critical incline which depends on the shape of the initial wetted region as well as on the liquid's mass. It is also shown that translating circular solutions are asymptotically stable if the kinematic boundary condition is given by an affine functionand provided the incline is small.
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A theorem of Tverberg from 1966 asserts that every set $X\subset\mathbb{R}^d$ of $n=T(d,r)=(d+1)(r1)+1$ points can be partitioned into $r$ pairwise disjoint subsets, whose convex hulls have a point in common. Thus every such partition induces an integer partition of $n$ into $r$ parts (that is, $r$ integers $a_1,\ldots,a_r$ satisfying $n=a_1+\cdots+a_r$), where the parts $a_i$ correspond to the number of points in every subset. In this paper, we prove that for any partition $a_i\le d+1$, $i=1,\ldots,r$, there exists a set $X\subset\mathbb{R}^d$ of $n$ points, such that every Tverberg partition of $X$ induces the same partition on $n$, given by the parts $a_1,\ldots,a_r$.
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We extend Noether's theorem to the setting of multisymplectic geometry by exhibiting a correspondence between conserved quantities and continuous symmetries on a multiHamiltonian system. We show that a homotopy comomentum map interacts with this correspondence in a way analogous to the moment map in symplectic geometry. We apply our results to generalize the theory of the classical momentum and position functions from the phase space of a given physical system to the multisymplectic phase space. We also apply our results to manifolds with a torsionfree $G_2$ structure.
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The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of complexes (algebras) are introduced. Cohomological range leads to the concepts of derived bounded algebras and strongly derived unbounded algebras naturally. The first and second BrauerThrall type theorems for the bounded derived category of a finitedimensional algebra over an algebraically closed field are obtained. The first BrauerThrall type theorem says that derived bounded algebras are just derived finite algebras. The second BrauerThrall type theorem says that an algebra is either derived discrete or strongly derived unbounded, but not both. Moreover, piecewise hereditary algebras and derived discrete algebras are characterized as the algebras of finite global cohomological width and finite global cohomological length respectively.
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When the copula of the conditional distribution of two random variables given a covariate does not depend on the value of the covariate, two conflicting intuitions arise about the best possible rate of convergence attainable by nonparametric estimators of that copula. In the end, any such estimator must be based on the marginal conditional distribution functions of the two dependent variables given the covariate, and the best possible rates for estimating such localized objects is slower than the parametric one. However, the invariance of the conditional copula given the value of the covariate suggests the possibility of parametric convergence rates. The more optimistic intuition is shown to be correct, confirming a conjecture supported by extensive Monte Carlo simulations by I. Hobaek Haff and J. Segers [Computational Statistics and Data Analysis 84:113, 2015] and improving upon the nonparametric rate obtained theoretically by I. Gijbels, M. Omelka and N. Veraverbeke [Scandinavian J
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Through the Selberg zeta approach, we reduce the exponent in the error term of the prime geodesic theorem for cocompact Kleinian groups or Bianchi groups from Sarnak's $\frac{5}{3}$ to $\frac{3}{2}$. At the cost of excluding a set of finite logarithmic measure, the bound is further improved to $\frac{13}{9}$.
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In how many ways can $n$ queens be placed on an $n \times n$ chessboard so that no two queens attack each other? This is the famous $n$queens problem. Let $Q(n)$ denote the number of such configurations, and let $T(n)$ be the number of configurations on a toroidal chessboard. We show that for every $n$ of the form $4^k+1$, $T(n)$ and $Q(n)$ are both at least $n^{\Omega(n)}$. This result confirms a conjecture of Rivin, Vardi and Zimmerman for these values of $n$. We also present new upper bounds on $T(n)$ and $Q(n)$ using the entropy method, and conjecture that in the case of $T(n)$ the bound is asymptotically tight. Along the way, we prove an upper bound on the number of perfect matchings in regular hypergraphs, which may be of independent interest.
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Sequences with low autocorrelation property have been applied in codedivision multiple access communication systems, radar and cryptography. Using the inverse Gray mapping, a quaternary sequence of even length $N$ can be obtained from two binary sequences of the same length, which are called component sequences. In this paper, using interleaving method, we present several classes of component sequences from twinprime sequences pairs or GMW sequences pairs given by Tang and Ding in 2010; two, three or four binary sequences defined by cyclotomic classes of order $4$. Hence we can obtain new classes of quaternary sequences, which are different from known ones, since known component sequences are constructed from a pair of binary sequences with optimal autocorrelation or Sidel'nikov sequences.
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In this paper, we develop an elementary proof of the change of variables in multiple integrals. Our proof is based on an induction argument. Assuming the formula for (m1)integrals, we define the integral over hypersurface in Rm, establish the divergent theorem and then use the divergent theorem to prove the formula for mintegrals. In addition to its simplicity, an advantage of our approach is that it yields the Brouwer Fixed Point Theorem as a corollary.
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The present article is devoted to functions from a certain subclass of nondifferentiable functions. The arguments and values of considered functions represented by the sadic representation or the negasadic representation of real numbers. The technique of modeling such functions is the simplest as compared with wellknown techniques of modeling nondifferentiable functions. In other words, values of these functions are obtained from the sadic or negasadic representation of the argument by a certain change of digits or combinations of digits.
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Hausdorff dimension results are a classical topic in the study of path properties of random fields. This article presents an alternative approach to Hausdorff dimension results for the sample functions of a large class of selfaffine random fields. We present a close relationship between the carrying dimension of the corresponding selfaffine random occupation measure introduced by U. Z\"ahle and the Hausdorff dimension of the graph of selfaffine fields. In the case of exponential scaling operators, the dimension formula can be explicitly calculated by means of the singular value function. This also enables to get a lower bound for the Hausdorff dimension of the range of general selfaffine random fields under mild regularity assumptions.
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