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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 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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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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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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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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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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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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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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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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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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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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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 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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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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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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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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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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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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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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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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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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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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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 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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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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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 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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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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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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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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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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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 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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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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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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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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In recent years, a great deal of interest has focused on conducting inference on the parameters in a linear model in the highdimensional setting. In this paper, we consider a simple and very na\"{i}ve twostep procedure for this task, in which we (i) fit a lasso model in order to obtain a subset of the variables; and (ii) fit a least squares model on the lassoselected set. Conventional statistical wisdom tells us that we cannot make use of the standard statistical inference tools for the resulting least squares model (such as confidence intervals and $p$values), since we peeked at the data twice: once in running the lasso, and again in fitting the least squares model. However, in this paper, we show that under a certain set of assumptions, with high probability, the set of variables selected by the lasso is deterministic. Consequently, the na\"{i}ve twostep approach can yield confidence intervals that have asymptotically correct coverage, as well as pvalues with proper TypeI erro
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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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We show that every interval in the homomorphism order of finite undirected graphs is either universal or a gap. Together with density and universality this "fractal" property contributes to the spectacular properties of the homomorphism order. We first show the fractal property by using Sparse Incomparability Lemma and then by more involved elementary argument.
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We apply geometric techniques from representation theory to the study of homologically finite differential graded (DG) modules $M$ over a finite dimensional, positively graded, commutative DG algebra $U$. In particular, in this setting we prove a version of a theorem of Voigt by exhibiting an isomorphism between the Yoneda Ext group $\operatorname{YExt}^1_U(M,M)$ and a quotient of tangent spaces coming from an algebraic group action on an algebraic variety. As an application, we answer a question of Vasconcelos from 1974 by showing that a local ring has only finitely many semidualizing complexes up to shiftisomorphism in the derived category $\mathcal{D}(R)$.
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Let $\Lambda$ be a quasiprojective variety and assume that, either $\Lambda$ is a subvariety of the moduli space $\mathcal{M}_d$ of degree $d$ rational maps, or $\Lambda$ parametrizes an algebraic family $(f_\lambda)_{\lambda\in\Lambda}$ of degree $d$ rational maps on $\mathbb{P}^1$. We prove the equidistribution of parameters having $p$ distinct neutral cycles towards the $p$th bifurcation current letting the periods of the cycles go to $\infty$, with an exponential speed of convergence. We deduce several fundamental consequences of this result on equidistribution and counting of hyperbolic components. A key step of the proof is a locally uniform version of the quantitative approximation of the Lyapunov exponent of a rational map by the $\log^+$ of the modulus of the multipliers of periodic points.
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The class of connected LOG (Labelled Oriented Graph) groups coincides with the class of fundamental groups of complements of closed, orientable 2manifolds embedded in S^4, and so contains all knot groups. We investigate when Campbell and Robertson's generalized Fibonacci groups H(r,n,s) are connected LOG groups. In doing so, we use the theory of circulant matrices to calculate the Betti numbers of their abelianizations. We give an almost complete classification of the groups H(r,n,s) that are connected LOG groups. All torus knot groups and the infinite cyclic group arise and we conjecture that these are the only possibilities. As a corollary we show that H(r,n,s) is a 2generator knot group if and only if it is a torus knot group.
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We derive a family of highorder, structurepreserving approximations of the Riemannian exponential map on several matrix manifolds, including the group of unitary matrices, the Grassmannian manifold, and the Stiefel manifold. Our derivation is inspired by the observation that if $\Omega$ is a skewHermitian matrix and $t$ is a sufficiently small scalar, then there exists a polynomial of degree $n$ in $t\Omega$ (namely, a Bessel polynomial) whose polar decomposition delivers an approximation of $e^{t\Omega}$ with error $O(t^{2n+1})$. We prove this fact and then leverage it to derive highorder approximations of the Riemannian exponential map on the Grassmannian and Stiefel manifolds. Along the way, we derive related results concerning the supercloseness of the geometric and arithmetic means of unitary matrices.
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This paper presents an efficient method to perform Structured Matrix Approximation by Separation and Hierarchy (SMASH), when the original dense matrix is associated with a kernel function. Given points in a domain, a tree structure is first constructed based on an adaptive partitioning of the computational domain to facilitate subsequent approximation procedures. In contrast to existing schemes based on either analytic or purely algebraic approximations, SMASH takes advantage of both approaches and greatly improves the efficiency. The algorithm follows a bottomup traversal of the tree and is able to perform the operations associated with each node on the same level in parallel. A strong rankrevealing factorization is applied to the initial analytic approximation in the separation regime so that a special structure is incorporated into the final nested bases. As a consequence, the storage is significantly reduced on one hand and a hierarchy of the original grid is constructed on the o
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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 give an elementary construction of a $p\geq 1$singular GelfandTsetlin $\mathfrak{gl}_n(\mathbb C)$module in terms of local distributions. This is a generalization of the universal $1$singular GelfandTsetlin $\mathfrak{gl}_n(\mathbb C)$module obtained in [FGR1]. We expect that the family of new GelfandTsetlin modules that we obtained will lead to a classification of all irreducible $p>1$singular GelfandTsetlin modules. So far such a classification is known only for singularity $n=1$.
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A problem of Glasner, now known as Glasner's problem, asks whether every minimally almost periodic, monothetic, Polish groups is extremely amenable. The purpose of this short note is to observe that a positive answer is obtained under the additional assumption that the universal minimal flow is metrizable.
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We prove that if $M$ and $N$ are Riemannian, oriented $n$dimensional manifolds without boundary and additionally $N$ is compact, then Sobolev mappings $W^{1,n}(M,N)$ of finite distortion are continuous. In particular, $W^{1,n}(M,N)$ mappings with almost everywhere positive Jacobian are continuous. This result has been known since 1976 in the case of mappings $W^{1,n}(\Omega,\mathbb{R}^n)$, where $\Omega\subset\mathbb{R}^n$ is an open set. The case of mappings between manifolds is much more difficult.
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In the variational study of singular Lagrange systems, the zero energy solutions play an important role. Here for the planar anisotropic Kepler problem, we give a complete classification of the zero energy solutions under some nondegenerate condition. A method is also developed to compute the Morse index of a zero energy solution. In particular an interesting connecting between the Morse index and the oscillating behavior of these solutions is established.
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We classify finite dimensional $H_{m^2}(\zeta)$simple $H_{m^2}(\zeta)$module Lie algebras $L$ over an algebraically closed field of characteristic $0$ where $H_{m^2}(\zeta)$ is the $m$th Taft algebra. As an application, we show that despite the fact that $L$ can be nonsemisimple in ordinary sense, $\lim_{n\to\infty}\sqrt[n]{c_n^{H_{m^2}(\zeta)}(L)} = \dim L$ where $c_n^{H_{m^2}(\zeta)}(L)$ is the codimension sequence of polynomial $H_{m^2}(\zeta)$identities of $L$. In particular, the analog of Amitsur's conjecture holds for $c_n^{H_{m^2}(\zeta)}(L)$.
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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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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 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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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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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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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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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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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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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 present families of large undirected and directed Cayley graphs whose construction is related to butterfly networks. One approach yields, for every large $k$ and for values of $d$ taken from a large interval, the largest known Cayley graphs and digraphs of diameter $k$ and degree $d$. Another method yields, for sufficiently large $k$ and infinitely many values of $d$, Cayley graphs and digraphs of diameter $k$ and degree $d$ whose order is exponentially larger in $k$ than any previously constructed. In the directed case, these are within a linear factor in $k$ of the Moore bound.
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We construct a \emph{single} $\mathcal{L}_{\omega_1,\omega}$sentence $\psi$ that codes Kurepa trees to prove the consistency of the following: (1) The spectrum of $\psi$ is consistently equal to $[\aleph_0,\aleph_{\omega_1}]$ and also consistently equal to $[\aleph_0,2^{\aleph_1})$, where $2^{\aleph_1}$ is weakly inaccessible. (2) The amalgamation spectrum of $\psi$ is consistently equal to $[\aleph_1,\aleph_{\omega_1}]$ and $[\aleph_1,2^{\aleph_1})$, where again $2^{\aleph_1}$ is weakly inaccessible. This is the first example of an $\mathcal{L}_{\omega_1,\omega}$sentence whose spectrum and amalgamation spectrum are consistently both rightopen and rightclosed. It also provides a positive answer to a question in [14]. (3) Consistently, $\psi$ has maximal models in finite, countable, and uncountable many cardinalities. This complements the examples given in [1] and [2] of sentences with maximal models in countably many cardinalities. (4) $2^{\aleph_0}<\aleph_{\omega_1}<2^{\ale
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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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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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The paper explores various special functions which generalize the twoparametric MittagLeffler type function of two variables. Integral representations for these functions in different domains of variation of arguments for certain values of the parameters are obtained. The asymptotic expansions formulas and asymptotic properties of such functions are also established for large values of the variables. This provides statements of theorems for these formulas and their corresponding properties.
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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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These notes offer a lightening introduction to topological quantum field theory in its functorial axiomatisation, assuming no or little prior exposure. We lay some emphasis on the connection between the path integral motivation and the definition in terms symmetric monoidal categories, and we highlight the algebraic formulation emerging from a formal generatorsandrelations description. This allows one to understand (oriented, closed) 1 and 2dimensional TQFTs in terms of a finite amount of algebraic data, while already the 3dimensional case needs an infinite amount of data. We evade these complications by instead discussing some aspects of 3dimensional extended TQFTs, and their relation to braided monoidal categories.
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