results for au:Wu_C in:math

- We study the redundancy of universally compressing strings $X_1,\dots, X_n$ generated by a binary Markov source $p$ without any bound on the memory. To better understand the connection between compression and estimation in the Markov regime, we consider a class of Markov sources restricted by a continuity condition. In the absence of an upper bound on memory, the continuity condition implies that $p(X_0|X^{-1}_{-m})$ gets closer to the true probability $p(X_0|X_{-\infty}^{-1})$ as $m$ increases, rather than vary around arbitrarily. For such sources, we prove asymptotically matching upper and lower bounds on the redundancy. In the process, we identify what sources in the class matter the most from a redundancy perspective.
- Jan 31 2018 math.NA arXiv:1801.09867v1We present an iterative support shrinking algorithm for $\ell_{p}$-$\ell_{q}$ minimization~($0 <p < 1 \leq q < \infty $). This algorithm guarantees the nonexpensiveness of the signal support set and can be easily implemented after being proximally linearized. The subproblem can be very efficiently solved due to its convexity and reducing size along iteration. We prove that the iterates of the algorithm globally converge to a stationary point of the $\ell_{p}$-$\ell_{q}$ objective function. In addition, we show a lower bound theory for the iteration sequence, which is more practical than the lower bound results for local minimizers in the literature.
- We study the asymptotic behavior of the asymptotic translation lengths on the curve complexes of pseudo-Anosov monodromies in a fibered cone of a fibered hyperbolic 3-manifold $M$ with $b_1(M) \geq 2$. For a sequence $(\Sigma_n, \psi_n)$ of fibers and monodromies in the fibered cone, we show that the asymptotic translation length on the curve complex is bounded above by $1/\chi(\Sigma_n)^{1+1/r}$ as long as their projections to the fibered face converge to a point in the interior, where $r$ is the dimension of the $\psi_n$-invariant homology of $\Sigma_n$ (which is independent of $n$). As a corollary, if $b_1(M) = 2$, the asymptotic translation length on the curve complex of such a sequence of primitive elements behaves like $1/\chi(\Sigma_n)^{2}$. Furthermore, together with a work of E. Hironaka, our theorem can be used to determine the asymptotic behavior of the minimal translation lengths of handlebody mapping class groups and the set of mapping classes with homological dilatation one.
- Recently, a conjecture on the linear complexity of a new class of generalized cyclotomic binary sequences of period $p^n$ were proposed by Z. Xiao et al. (Des. Codes Cryptogr. DOI 10.1007/s10623-017-0408-7). Later, for the case $f$ being the form $2^r$ with $r\ge 1$, Vladimir Edemskiy proved the conjecture (arXiv:1712.03947). In this paper, we first introduce a generic construction of $p^n$-periodic binary sequence based on the generalized cyclotomy, which admits a flexible support set and includes Xiao's construction as a special case. Then, under the assumption of $2$ being a primitive root modulo $p^2$, the linear complexity of the new proposed sequence over $\mathrm {GF} (2) $ is determined by using the Euler quotient. As a byproduct, in the case of $2$ being a primitive root modulo $p^2$, the conjecture given by Z. Xiao et al. is proved to be correct for a general $f$.
- Dec 12 2017 math.PR arXiv:1712.03808v1Markov chain Monte Carlo(MCMC) is a popular approach to sample from high dimensional distributions, and the asymptotic variance is a commonly used criterion to evaluate the performance. While most popular MCMC algorithms are reversible, there is a growing literature on the development and analyses of nonreversible MCMC. Chen and Hwang(2013) showed that a reversible MCMC can be improved by adding an antisymmetric perturbation. They also raised a conjecture that it can not be improved if there is no cycle in the corresponding graph. In this paper, we present a rigorous proof of this conjecture. The proof is based on the fact that the transition matrix with an acyclic structure will produce minimum commute time between vertices.
- Random hypergraphs and random simplicial complexes have potential applications in computer science and engineering. Various models of random hypergraphs and random simplicial complexes on n-points have been studied. Let L be a simplicial complex. In this paper, we study random sub-hypergraphs and random sub-complexes of L. By considering the minimal complex that a sub-hypergraph can be embedded in and the maximal complex that can be embedded in a sub-hypergraph, we define some operators on the space of probability functions on sub-hypergraphs of L. We study the compositions of these operators as well as their actions on the space of probability functions. As applications in computer science, we give algorithms generating large sparse random hypergraphs and large sparse random simplicial complexes.
- Nov 28 2017 math.AT arXiv:1711.09211v1In this paper, we study further properties and applications of weighted homology and persistent homology. We introduce the Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology. For applications, we show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. We also prove a theorem that allows us to calculate the mod $p^2$ weighted persistent homology given some information on the mod $p$ weighted persistent homology. In the paper, we include many examples to illustrate the concepts.
- We consider the $k$-error linear complexity of a new binary sequence of period $p^2$, proposed in the recent paper "New generalized cyclotomic binary sequences of period $p^2$", by Z. Xiao et al., who calculated the linear complexity of the sequences (Designs, Codes and Cryptography, 2017, https://doi.org/10.1007/s10623-017-0408-7). More exactly, we determine the values of $k$-error linear complexity over $\mathbb{F}_2$ for almost $k>0$ in terms of the theory of Fermat quotients. Results indicate that such sequences have good stability.
- We extend Zhang's notion of canonical measures to all (possibly non-compact) metric graphs. This will allow us to introduce a notion of "hyperbolic measures" on universal covers of metric graphs. Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. We will prove a generalized version of this theorem, allowing any infinite Galois cover to replace the universal cover. We will show all such limiting measures satisfy a version of Gauss-Bonnet formula which, using the theory of von Neumann dimensions, can be interpreted as a "trace formula". In the special case where the infinite cover is the universal cover, we will provide explicit methods to compute the corresponding limiting (hyperbolic) measure. Our ideas are motivated by non-Archimedean analytic and tropical geometry.
- Oct 25 2017 math.CV arXiv:1710.08593v1In this paper, we introduce certain $n$-th order nonlinear Loewy factorizable algebraic ordinary differential equations for the first time and study the growth of their meromorphic solutions in terms of the Nevanlinna characteristic function. It is shown that for generic cases all their meromorphic solutions are elliptic functions or their degenerations and hence their order of growth are at most two. Moreover, for the second order factorizable algebraic ODEs, all the meromorphic solutions of them (except for one case) are found explicitly. This allows us to show that a conjecture proposed by Hayman in 1996 holds for these second order ODEs.
- Kriging based on Gaussian random fields is widely used in reconstructing unknown functions. The kriging method has pointwise predictive distributions which are computationally simple. However, in many applications one would like to predict for a range of untried points simultaneously. In this work we obtain some error bounds for the (simple) kriging predictor under the uniform metric. It works for a scattered set of input points in an arbitrary dimension, and also covers the case where the covariance function of the Gaussian process is misspecified. These results lead to a better understanding of the rate of convergence of kriging under the Gaussian or the Matérn correlation functions, the relationship between space-filling designs and kriging models, and the robustness of the Matérn correlation functions.
- Oct 17 2017 math.AP arXiv:1710.05482v1We investigate the spreading behavior of two invasive species modeled by a Lotka-Volterra diffusive competition system with two free boundaries in a spherically symmetric setting. We show that, for the weak-strong competition case, under suitable assumptions, both species in the system can successfully spread into the available environment, but their spreading speeds are different, and their population masses tend to segregate, with the slower spreading competitor having its population concentrating on an expanding ball, say Bt, and the faster spreading competitor concentrating on a spherical shell outside Bt that disappears to infinity as time goes to infinity.
- Oct 04 2017 math.DG arXiv:1710.01242v2In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of $\mathbb{R}^{n+1}$ ($n\geqslant2$) is mean convex and star-shaped. Several interesting examples and some hyperbolic evolution equations for geometric quantities of the evolving hypersurfaces have been shown. Besides, under different assumptions for the initial velocity, we can get the expansion and the convergence results of a hyperbolic inverse mean curvature flow in the plane $\mathbb{R}^2$, whose evolving curves move normally.
- Oct 02 2017 math.OC arXiv:1709.10345v1The optimal control of epidemic-like stochastic processes is important both historically and for emerging applications today, where it can be especially important to include time-varying parameters that impact viral epidemic-like propagation. We connect the control of such stochastic processes with time-varying behavior to the stochastic shortest path problem and obtain solutions for various cost functions. Then, under a mean-field scaling, this general class of stochastic processes is shown to converge to a corresponding dynamical system. We analogously establish that the optimal control of this class of processes converges to the optimal control of the limiting dynamical system. Consequently, we study the optimal control of the dynamical system where the comparison of both controlled systems renders various important mathematical properties of interest.
- Sep 26 2017 math.OC arXiv:1709.07988v1The study of density-dependent stochastic population processes is important from a historical perspective as well as from the perspective of a number of existing and emerging applications today. In more recent applications of these processes, it can be especially important to include time-varying parameters for the rates that impact the density-dependent population structures and behaviors. Under a mean-field scaling, we show that such density-dependent stochastic population processes with time-varying behavior converge to a corresponding dynamical system. We analogously establish that the optimal control of such density-dependent stochastic population processes converges to the optimal control of the limiting dynamical system. An analysis of both the dynamical system and its optimal control renders various important mathematical properties of interest.
- Aug 23 2017 math.AT arXiv:1708.06722v4In this paper we develop the theory of weighted persistent homology. In 1990, Robert J. Dawson was the first to study in depth the homology of weighted simplicial complexes. We generalize the definitions of weighted simplicial complex and the homology of weighted simplicial complex to allow weights in an integral domain $R$. Then we study the resulting weighted persistent homology. We show that weighted persistent homology can tell apart filtrations that ordinary persistent homology does not distinguish. For example, if there is a point considered as special, weighted persistent homology can tell when a cycle containing the point is formed or has disappeared.
- May 24 2017 math.PR arXiv:1705.08046v1Let $F: \mathbb{L}^2(\Omega, \mathbb{R}) \to \mathbb{R}$ be a law invariant and continuously Fréchet differentiable mapping. Based on Lions \citeLions, Cardaliaguet \citeCardaliaguet (Theorem 6.2 and 6.5) proved that: \bea \labelDerivative D F (\xi) = g(\xi), \eea where $g: \mathbb{R} \to \mathbb{R}$ is a deterministic function which depends only on the law of $\xi$. See also Carmona \& Delarue \citeCD Section 5.2. In this short note we provide an elementary proof for this well known result. This note is part of our accompanying paper \citeWZ, which deals with a more general situation.
- May 02 2017 math.AT arXiv:1705.00151v2Hypergraph is a topological model for networks. In order to study the topology of hypergraphs, the homology of the associated simplicial complexes and the embedded homology have been invented. In this paper, we give some algorithms to compute the homology of the associated simplicial complexes and the embedded homology of hypergraphs as well as some heuristics for efficient computations.
- In [2], the authors constructed closed oriented hyperbolic surfaces with pseudo-Anosov homeomorphisms from certain class of integral matrices. In this paper, we present a very simple algorithm to compute the Teichmueller polynomial corresponding to those surface homeomorphisms.
- Kennedy and O'Hagan (2001) propose a model for calibrating some unknown parameters in a computer model and estimating the discrepancy between the computer output and physical response. This model is known to have certain identifiability issues. Tuo and Wu (2016) show that there are examples for which the Kennedy-O'Hagan method renders unreasonable results in calibration. In spite of its unstable performance in calibration, the Kennedy-O'Hagan approach has a more robust behavior in predicting the physical response. In this work, we present some theoretical analysis to show the consistency of predictor based on their calibration model in the context of radial basis functions.
- Feb 28 2017 math.OC arXiv:1702.07781v1We consider the optimal allocation of generic resources among multiple generic entities of interest over a finite planning horizon, where each entity generates stochastic returns as a function of its resource allocation during each period. The main objective is to maximize the expected return while at the same time managing risk to an acceptable level for each period. We devise a general solution framework and establish how to obtain the optimal dynamic resource allocation.
- Feb 21 2017 math.CO arXiv:1702.05750v1A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. (Electronic J. Combin. 18, \#P233, 2011) and Pan et al. (Electronic J. Combin. 20, \#P36, 2013) determined all pentavalent symmetric graphs of order $4pq$. In this paper, we shall generalize this result by determining all connected pentavalent symmetric graphs of order four times an odd square-free integer. It is shown in this paper that, for each of such graphs $\it\Gamma$, either the full automorphism group ${\sf Aut}\it\Gamma$ is isomorphic to ${\sf PSL}(2,p)$, ${\sf PGL}(2,p)$, ${\sf PSL}(2,p){\times}\mathbb{Z}_2$ or ${\sf PGL}(2,p){\times}\mathbb{Z}_2$, or $\it\Gamma$ is isomorphic to one of 8 graphs.
- Although the low energy fractional excitations of one dimensional integrable models are often well-understood, exploring quantum dynamics in these systems remains challenging in the gapless regime, especially at intermediate and high energies. Based on the algebraic Bethe ansatz formalism, we study spin dynamics in the antiferromagnetic spin-$\frac{1}{2}$ XXZ chain with the Ising anisotropy via the form-factor formulae. Various excitations at different energy scales are identified crucial to the dynamic spin structure factors under the guidance of sum rules. At small magnetic polarizations, gapless excitations dominate the low energy spin dynamics arising from the magnetic-field-induced incommensurability. In contrast, spin dynamics at intermediate and high energies is characterized by the two- and three-string states, which are multi-particle excitations based on the commensurate Néel ordered background. Our work is helpful for experimental studies on spin dynamics in both condensed matter and cold atom systems beyond the low energy effective Luttinger liquid theory.
- This paper is concerned with a constrained optimization problem over a directed graph (digraph) of nodes, in which the cost function is a sum of local objectives, and each node only knows its local objective and constraints. To collaboratively solve the optimization, most of the existing works require the interaction graph to be balanced or "doubly-stochastic", which is quite restrictive and not necessary as shown in this paper. We focus on an epigraph form of the original optimization to resolve the "unbalanced" problem, and design a novel two-step recursive algorithm with a simple structure. Under strongly connected digraphs, we prove that each node asymptotically converges to some common optimal solution. Finally, simulations are performed to illustrate the effectiveness of the proposed algorithms.
- This paper considers a distributed convex optimization problem with inequality constraints over time-varying unbalanced digraphs, where the cost function is a sum of local objectives, and each node of the graph only knows its local objective and inequality constraints. Although there is a vast literature on distributed optimization, most of them require the graph to be balanced, which is quite restrictive and not necessary. Very recently, the unbalanced problem has been resolved only for either time-invariant graphs or unconstrained optimization. This work addresses the unbalancedness by focusing on an epigraph form of the constrained optimization. A striking feature is that this novel idea can be easily used to study time-varying unbalanced digraphs. Under local communications, a simple iterative algorithm is then designed for each node. We prove that if the graph is uniformly jointly strongly connected, each node asymptotically converges to some common optimal solution.
- Dec 12 2016 math.GT arXiv:1612.03084v3In this note, we deduce a partial answer to the question in the title. In particular, we show that asymptotically almost all bi-Perron algebraic unit whose characteristic polynomial has degree at most $2n$ do not correspond to dilatations of pseudo-Anosov maps on a closed orientable surface of genus $n$ for $n\geq 10$. As an application of the argument, we also obtain a statement on the number of closed geodesics of the same length in the moduli space of area one abelian differentials for low genus cases.
- Dec 09 2016 math.OC arXiv:1612.02538v1We study the problem of recovering the underlining sparse signals from clean or noisy phaseless measurements. Due to the sparse prior of signals, we adopt an L0regularized variational model to ensure only a small number of nonzero elements being recovered in the signal and two different formulations are established in the modeling based on the choices of data fidelity, i.e., L2and L1norms. We also propose efficient algorithms based on the Alternating Direction Method of Multipliers (ADMM) with convergence guarantee and nearly optimal computational complexity. Thanks to the existence of closed-form solutions to all subproblems, the proposed algorithm is very efficient with low computational cost in each iteration. Numerous experiments show that our proposed methods can recover sparse signals from phaseless measurements with higher successful recovery rates and lower computation cost compared with the state-of-art methods.
- Nov 29 2016 math.NA arXiv:1611.08817v3Variational methods have become an important kind of methods in signal and image restoration - a typical inverse problem. One important minimization model consists of the squared $\ell_2$ data fidelity (corresponding to Gaussian noise) and a regularization term constructed by a potential function composed of first order difference operators. It is well known that total variation (TV) regularization, although achieved great successes, suffers from a contrast reduction effect. Using a typical signal, we show that, actually all convex regularizers and most nonconvex regularizers have this effect. With this motivation, we present a general truncated regularization framework. The potential function is a truncation of existing nonsmooth potential functions and thus flat on $(\tau,+\infty)$ for some positive $\tau$. Some analysis in 1D theoretically demonstrate the good contrast-preserving ability of the framework. We also give optimization algorithms with convergence verification in 2D, where global minimizers of each subproblem (either convex or nonconvenx) are calculated. Experiments numerically show the advantages of the framework.
- Nov 24 2016 math.OC arXiv:1611.07708v2We study an optimal control problem in which both the objective function and the dynamic constraint contain an uncertain parameter. Since the distribution of this uncertain parameter is not exactly known, the objective function is taken as the worst-case expectation over a set of possible distributions of the uncertain parameter. This ambiguity set of distributions is, in turn, defined by the first two moments of the random variables involved. The optimal control is found by minimizing the worst-case expectation over all possible distributions in this set. If the distributions are discrete, the stochastic min-max optimal control problem can be converted into a convensional optimal control problem via duality, which is then approximated as a finite-dimensional optimization problem via the control parametrization. We derive necessary conditions of optimality and propose an algorithm to solve the approximation optimization problem. The results of discrete probability distribution are then extended to the case with one dimensional continuous stochastic variable by applying the control parametrization methodology on the continuous stochastic variable, and the convergence results are derived. A numerical example is present to illustrate the potential application of the proposed model and the effectiveness of the algorithm.
- The Expectation-Maximization (EM) algorithm is an iterative method to maximize the log-likelihood function for parameter estimation. Previous works on the convergence analysis of the EM algorithm have established results on the asymptotic (population level) convergence rate of the algorithm. In this paper, we give a data-adaptive analysis of the sample level local convergence rate of the EM algorithm. In particular, we show that the local convergence rate of the EM algorithm is a random variable $\overline{K}_{n}$ derived from the data generating distribution, which adaptively yields the convergence rate of the EM algorithm on each finite sample data set from the same population distribution. We then give a non-asymptotic concentration bound of $\overline{K}_{n}$ on the population level optimal convergence rate $\overline{\kappa}$ of the EM algorithm, which implies that $\overline{K}_{n}\to\overline{\kappa}$ in probability as the sample size $n\to\infty$. Our theory identifies the effect of sample size on the convergence behavior of sample EM sequence, and explains a surprising phenomenon in applications of the EM algorithm, i.e. the finite sample version of the algorithm sometimes converges faster even than the population version. We apply our theory to the EM algorithm on three canonical models and obtain specific forms of the adaptive convergence theorem for each model.
- Oct 20 2016 math.CO arXiv:1610.06166v2We look at some properties of functions of binomial coefficients mod 2. In particular, we derive a set of recurrence relations for sums of products of binomial coefficients mod 2 and show that they result in sequences that are the run length transforms of well known basic sequences. In particular, we show that the sequence $a(n) = \sum_{k=0}^{n} \left[\left(\begin{array}{c}n-k2k\end{array}\right) \left(\begin{array}{c}n k\end{array}\right) \mod 2\right]$ is the run length transform of the Fibonacci numbers, that the sequence $a(n) = \sum_{k=0}^{n} \left[\left(\begin{array}{c}n+k n-k\end{array}\right) \left(\begin{array}{c}nk\end{array}\right) \mod 2\right]$ is the run length transform of the positive integers, and that the sequence $$a(n) = \sum_k=0^n \left[\left(\beginarraycn-k6k\endarray\right) \left(\beginarraycn k\endarray\right) \mod 2\right]$$ is the run length transform of Narayana's cows sequence.
- Aug 26 2016 math.CO arXiv:1608.07247v2We consider the minimal number of points on a regular grid on the plane that generates $n$ blocks of points of exactly length $k$. We illustrate how this is related to the $n$-queens problem on the toroidal chessboard and show that this number is upper bounded by $kn/3$ and approaches $kn/4$ as $n\rightarrow\infty$ when $k+1$ is coprime with $6$ or when $k$ is large.
- May 03 2016 math.CO arXiv:1605.00180v5In general graph theory, the only relationship between vertices are expressed via the edges. When the vertices are embedded in an Euclidean space, the geometric relationships between vertices and edges can be interesting objects of study. We look at the number of isosceles triangles where the vertices are points on a regular grid and show that they satisfy a recurrence relation when the grid is large enough. We also derive recurrence relations for the number of acute, obtuse and right isosceles triangles.
- Mar 29 2016 math.NT arXiv:1603.08493v1In 1998, Bird introduced Meertens numbers as numbers that are invariant under a map similar to the Godel encoding. In base 10, the only known Meertens number is 81312000. We look at some properties of Meertens numbers and consider variations of this concept. In particular, we consider variations where there is a finite time algorithm to decide whether such numbers exist.
- Mar 15 2016 math.NT arXiv:1603.04123v2Let $\pi$ be a genuine cuspidal representation of the metaplectic group of rank $n$. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension $2n+1$. We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands $L$-function of $\pi$ twisted by a character. The bulk of this article focuses on proving a case of regularised Siegel-Weil formula, on which the Rallis inner product formula is based and whose proof is missing in the literature.
- Feb 18 2016 math.DG arXiv:1602.05290v1By studying the monotonicity of the first nonzero eigenvalues of Laplace and p-Laplace operators on a closed convex hypersurface $M^n$ which evolves under inverse mean curvature flow in $\mathbb{R}^{n+1}$, the isoperimetric lower bounds for both eigenvalues were founded.
- Dec 04 2015 math.DS arXiv:1512.00908v3We calculate the area of the smallest triangle and the area of the smallest virtual triangle for many known lattice surfaces. We show that our list of the lattice surfaces for which the area of the smallest virtual triangle greater than .05 is complete. In particular, this means that there are no new lattice surfaces for which the area of the smallest virtual triangle is greater than .05. Our method follows an algorithm described by Smillie and Weiss and improves on it in certain respects.
- Nov 17 2015 math.DG arXiv:1511.04696v1In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent $n$ $(n\geq 2)$, then it has exactly the $n$-dimensional volume growth. Besides, two interesting applications have also been given. The one is that we show that if a complete $n$-dimensional Finsler manifold of nonnegative $n$-Ricci curvature satisfies the Gagliardo-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. The other one is that we give an alternative proof to Mao's main result in [23] for smooth metric measure spaces with nonnegative weighted Ricci curvature.
- Sep 07 2015 math.GM arXiv:1509.01456v2In this paper we consider how to use the convolution method to construct approximations, which consist of fuzzy numbers sequences with good properties, for a general fuzzy number. It shows that this convolution method can generate differentiable approximations in finite steps for fuzzy numbers which have finite non-differentiable points. In the previous work, this convolution method only can be used to construct differentiable approximations for continuous fuzzy numbers whose possible non-differentiable points are the two endpoints of 1-cut. The constructing of smoothers is a key step in the construction process of approximations. It further points out that, if appropriately choose the smoothers, then one can use the convolution method to provide approximations which are differentiable, Lipschitz and preserve the core at the same time.
- Sep 02 2015 math.GM arXiv:1509.00447v3In this paper, we present characterizations of totally bounded sets, relatively compact sets and compact sets in the fuzzy sets spaces $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ equipped with $L_p$ metric, where $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ are two kinds of general fuzzy sets on $\mathbb{R}^m$ which do not have any assumptions of convexity or star-shapedness. Subsets of $F_B(\mathbb{R}^m)^p$ include common fuzzy sets such as fuzzy numbers, fuzzy star-shaped numbers with respect to the origin, fuzzy star-shaped numbers, and the general fuzzy star-shaped numbers introduced by Qiu et al. The existed compactness criteria are stated for three kinds of fuzzy sets spaces endowed with $L_p$ metric whose universe sets are the former three kinds of common fuzzy sets respectively. Constructing completions of fuzzy sets spaces with respect to $L_p$ metric is a problem which is closely dependent on characterizing totally bounded sets. Based on preceding characterizations of totally boundedness and relatively compactness and some discussions on convexity and star-shapedness of fuzzy sets, we show that the completions of fuzzy sets spaces mentioned in this paper can be obtained by using the $L_p$-extension. We also clarify relation among all the ten fuzzy sets spaces discussed in this paper, which consist of five pairs of original spaces and the corresponding completions. Then, we show that the subspaces of $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ mentioned in this paper have parallel characterizations of totally bounded sets, relatively compact sets and compact sets. At last, as applications of our results, we discuss properties of $L_p$ metric on fuzzy sets space and relook compactness criteria proposed in previous work.
- Aug 13 2015 math.CO arXiv:1508.02934v1Tabulation of the set of permutations $r_j$ of $\{1,\cdots , n\}$ such that $\sum_{i=1}^n \prod_{j=1}^k r_j(i)$ is maximized or minimized.
- Jul 14 2015 math.AG arXiv:1507.03069v2This paper studies a class of Abelian varieties that are of $\mathrm{GL}_2$-type and with isogenous classes defined over a number field $k$ (i.e., $k$-virtual). We treat both cases when their endomorphism algebras are (1) a totally real field $K$ or (2) a totally indefinite quaternion algebra over a totally real field $K$. Among the isogenous class of such an Abelian variety, we identify one whose Galois conjugates can be described in terms of Atkin-Lehner operators and certain action of the class group of $K$. We deduce that such Abelian varieties are parametrised by finite quotients of certain PEL Shimura varieties. These new families of moduli spaces are further analysed when they are of dimension $2$. We provide explicit numerical bounds for when they are surfaces of general type. In addition, for two particular examples, we calculate precisely the coordinates of inequivalent elliptic points, study intersections of certain Hirzebruch cycles with exceptional divisors. We are able to show that they are both rational surfaces.
- Jul 14 2015 math.NT arXiv:1507.03297v2In this paper, we introduce a new family of period integrals attached to irreducible cuspidal automorphic representations $\sigma$ of symplectic groups $\mathrm{Sp}_{2n}(\mathbb{A})$, which detects the right-most pole of the $L$-function $L(s,\sigma\times\chi)$ for some character $\chi$ of $F^\times\backslash\mathbb{A}^\times$ of order at most $2$, and hence the occurrence of a simple global Arthur parameter $(\chi,b)$ in the global Arthur parameter $\psi$ attached to $\sigma$. We also give a characterisation of first occurrences of theta correspondence by (regularised) period integrals of residues of certain Eisenstein series.
- We give a general method to compute the expansion of topological tau functions for Drinfeld-Sokolov hierarchies associated to an arbitrary untwisted affine Kac-Moody algebra. Our method consists of two main steps: first these tau functions are expressed as (formal) Fredholm determinants of the type appearing in the Borodin-Okounkov formula, then the kernels for these determinants are found using a reduced form of the string equation. A number of explicit examples are given.
- Apr 01 2015 math.NT arXiv:1503.08883v2Consider the operation of adding the same number of identical digits to the left and to the right of a number n. In OEIS sequence A090287, it was conjectured that this operation will not produce a prime if and only if n is a palindrome with an even number of digits. We show that this conjecture is false by showing that this property also holds for n=231, n=420, and an infinite number of other values of n.
- Mar 27 2015 math.DG arXiv:1503.07639v2In this paper, we derive the CR Reilly's formula and its applications to studying of the first eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded p-minimal hypersurfaces. In particular, we obtain the first Dirichlet eigenvalue estimate in a compact pseudohermitian (2n+1)-manifold with boundary and the first eigenvalue estimate of the tangential sublaplacian on closed oriented embedded p-minimal hypersurfaces in a closed pseudohermitian (2n+1)-manifold of vanishing torsion.
- In this paper, we study the growth, in terms of the Nevanlinna characteristic function, of meromorphic solutions of three types of second order nonlinear algebraic ordinary differential equations. We give all their meromorphic solutions explicitly, and hence show that all of these ODEs satisfy the \it classical conjecture proposed by Hayman in 1996.
- Feb 09 2015 math.OC arXiv:1502.01840v1In this work we study the asymptotic behavior of the solutions of a class of abstract parabolic time optimal control problems when the generators converge, in an appropriate sense, to a given strictly negative operator. Our main application to PDEs systems concerns the behavior of optimal time and of the associated optimal controls for parabolic equations with highly oscillating coefficients, as we encounter in homogenization theory. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the solutions of the time optimal control problems for the systems with oscillating coefficients converge, in the usual norms, to the solution of the corresponding problem for the homogenized system. In order to prove our main theorem, we provide several new results, which could be of a broader interest, on time and norm optimal control problems.
- Dec 02 2014 math.GT arXiv:1412.0633v3Finite translation surfaces can be classified by the order of their singularities. When generalizing to infinite translation surfaces, however, the notion of order of a singularity is no longer well-defined and has to be replaced by new concepts. This article discusses the nature of two such concepts, recently introduced by Bowman and Valdez: linear approaches and rotational components. We show that there is a large flexibility in the spaces of rotational components and even more in the spaces of linear approaches. In particular, we prove that every finite topological space arises as space of rotational components. However, this space will still not contain enough information to describe an infinite translation surface. We showcase this through an uncountable family with the same space of rotational components but different spaces of linear approaches. Additionally, we study several known and new examples to illustrate the concept of linear approaches and rotational components.
- In this paper, we give sufficient conditions for a Perron number, given as the leading eigenvalue of an aperiodic matrix, to be a pseudo-Anosov dilatation of a compact surface. We give an explicit construction of the surface and the map when the sufficient condition is met.
- Sep 19 2014 math.OC arXiv:1409.5173v2The expected increase in the penetration of renewables in the approaching decade urges the electricity market to introduce new products - in particular, flexible ramping products - to accommodate the renewables' variability and intermittency. CAISO and MISO are leading the design of the new products. However, it is not clear how such products may affect the electricity market. In this paper, we are specifically interested in assessing how the new products distort the optimal energy dispatch by comparing with the case without such products. The distortion may impose additional cost, which we term as the "distortion cost". Using a functional approach, we establish the relationship between the distortion cost and the key parameters of the new products, i.e., the up and down flexible ramping requirements. Such relationship yields a novel routine to efficiently construct the functions, which makes it possible to efficiently perform the minimal distortion cost energy dispatch while guaranteeing a given supply reliability level. Both theoretical analysis and simulation results suggest that smartly selecting the parameters may substantially reduce the distortion cost. We believe our approach can assist the ISOs with utilizing the ramping capacities in the system at the minimal distortion cost.
- Sep 03 2014 math.NT arXiv:1409.0767v3Following the idea of [GJS09] for orthogonal groups, we introduce a new family of period integrals for cuspidal automorphic representations $\sigma$ of unitary groups and investigate their relation with the occurrence of a simple global Arthur parameter $(\chi,b)$ in the global Arthur parameter $\psi_\sigma$ associated to $\sigma$, by the endoscopic classification of Arthur ([Art13], [Mok13], [KMSW14]). The argument uses the theory of theta correspondence. This can be viewed as a part of the $(\chi,b)$-theory outlined in [Jia14] and can be regarded as a refinement of the theory of theta correspondences and poles of certain $L$-functions, which was outlined in [Ral91].
- Sep 03 2014 math.NT arXiv:1409.0770v1This article shows that for unitary dual reductive pairs the first occurrence of theta lift of an irreducible cuspidal automorphic representation is irreducible. It also proves a refined tower property for theta lifts and the involutive property for twisted theta lifts.
- Aug 22 2014 math.DS arXiv:1408.4828v3We answered a question by Barak Weiss on the uniform discreteness of the holonomy vectors of translation surfaces.
- We show that for any integers k and g, with g at least two, there are infinitely many closed hyperbolic 3-manifolds which are integral homology spheres with Casson invariant k, and Heegaard genus equal to g. This existence result is shown using random methods, using a model of random 3-manifolds arising from random walks on the mapping class group of a closed orientable surface.
- May 14 2014 math.DG arXiv:1405.3038v1In this paper, we study a general almost Schur Lemma on pseudo-Hermitian (2n+1)-manifolds $(M,J,\theta)$ for $n\geq2$. When the equality of almost Schur inequality holds, we derive the contact form $\theta$ is pseudo-Einstein and the pseudo-Hermitian scalar curvature is constant.
- A classical way to introduce tau functions for integrable hierarchies of solitonic equations is by means of the Sato-Segal-Wilson infinite-dimensional Grassmannian. Every point in the Grassmannian is naturally related to a Riemann-Hilbert problem on the unit circle, for which Bertola proposed a tau function that generalizes the Jimbo-Miwa-Ueno tau function for isomonodromic deformation problems. In this paper, we prove that the Sato-Segal-Wilson tau function and the (generalized) Jimbo-Miwa-Ueno isomonodromy tau function coincide under a very general setting, by identifying each of them to the large-size limit of a block Toeplitz determinant. As an application, we give a new definition of tau function for Drinfeld-Sokolov hierarchies (and their generalizations) by means of infinite-dimensional Grassmannians, and clarify their relation with other tau functions given in the literature.
- Response of initial elastic field to stiffness perturbation and its possible application is investigated. Virtual thermal softening is used to produce the stiffness reduction for demonstration. It is interpreted that the redistribution of the initial strain will be developed by the non-uniform temperature elevation, as which leads to the non-uniform reduction of the material stiffness. Therefore, the initial filed is related to the stiffness perturbation and incremental field in a matrix form after eliminating the thermal expansion effect.
- Mar 10 2014 math.OC arXiv:1403.1868v2Distributed generation resources have become significantly more prevalent in the electric power system over the past few years. This warrants reconsideration on how the coordination of generation resources is achieved. In this paper, we particularly focus on secondary frequency control and how to enhance it by exploiting peer-to-peer communication among the resources. We design a control framework based on a consensus-plus-global-innovation approach, which guarantees bringing the frequency back to its nominal value. The control signals of the distributed resources are updated in response to a global innovation corresponding to the ACE signal, and additional information exchanged via communication among neighboring resources. We show that such a distributed control scheme can be very well approximated by a PI controller and can stabilize the system. Moreover, since our control scheme takes advantage of both the ACE signal and peer-to-peer communication, simulation results demonstrate that our control scheme can stabilize the system significantly faster than the AGC framework. Also, an important feature of our scheme is that it performs $c\epsilon$-close to the centralized optimal economic dispatch, where $c$ is a positive constant depending only on the cost parameters and the communication topology and $\epsilon$ denotes the maximum rate of change of overall system.
- Feb 27 2014 math.DS arXiv:1402.6667v2We calculate the action of the group of affine diffeomorphisms on the relative cohomology of square-tiled surfaces that are normal abelian covers of the flat pillowcase, and as an application, answer a question raised by Smillie and Weiss.
- Transmission imaging, as an important imaging technique widely used in astronomy, medical diagnosis, and biology science, has been shown in [49] quite different from reflection imaging used in our everyday life. Understanding the structures of images (the prior information) is important for designing, testing, and choosing image processing methods, and good image processing methods are helpful for further uses of the image data, e.g., increasing the accuracy of the object reconstruction methods in transmission imaging applications. In reflection imaging, the images are usually modeled as discontinuous functions and even piecewise constant functions. In transmission imaging, it was shown very recently in [49] that almost all images are continuous functions. However, the author in [49] considered only the case of parallel beam geometry and used some too strong assumptions in the proof, which exclude some common cases such as cylindrical objects. In this paper, we consider more general beam geometries and simplify the assumptions by using totally different techniques. In particular, we will prove that almost all images in transmission imaging with both parallel and divergent beam geometries (two most typical beam geometries) are continuous functions, under much weaker assumptions than those in [49], which admit almost all practical cases. Besides, taking into accounts our analysis, we compare two image processing methods for Poisson noise (which is the most significant noise in transmission imaging) removal. Numerical experiments will be provided to demonstrate our analysis.
- We study itinerant ferromagnetism in multi-orbital Hubbard models in certain two-dimensional square and three-dimensional cubic lattices. In the strong coupling limit where doubly occupied orbitals are not allowed, we prove that the fully spin-polarized states are the unique ground states, apart from the trivial spin degeneracies, for a large region of fillings factors. Possible applications to p-orbital bands with ultra-cold fermions in optical lattices, and electronic 3d-orbital bands in transition-metal oxides, are discussed.
- Jun 20 2013 math.DG arXiv:1306.4539v1This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature $\kappa$ evolving in direction of its normal vector, where the speed is given by a power $\beta (\geq 1/m)$ of the $m$th mean curvature plus a volume preserving term, including the case of powers of the mean curvature and of the $\mbox{Gau\ss}$ curvature. The main result is that if the initial hypersurface satisfies that the ratio of the biggest and smallest principal curvature is close enough to 1 everywhere, depending only on $n$, $m$, $\beta$ and $\kappa$, then under the flow this is maintained, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces exponentially converge to a geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$, enclosing the same volume as the initial hypersurface.
- In this paper we define the torsion flow, a CR analogue of the Ricci flow. For homogeneous CR manifolds we give explicit solutions to the torsion flow illustrating various kinds of behavior. We also derive monotonicity formulas for CR entropy functionals. As an application, we classify torsion breathers.
- This paper studies steady-state traffic flow on a ring road with up- and down- slopes using a semi-discrete model. By exploiting the relations between the semi-discrete and the continuum models, a steady-state solution is uniquely determined for a given total number of vehicles on the ring road. The solution is exact and always stable with respect to the first-order continuum model, whereas it is a good approximation with respect to the semi-discrete model provided that the involved equilibrium constant states are linearly stable. In an otherwise case, the instability of one or more equilibria could trigger stop-and-go waves propagating in certain road sections or throughout the ring road. The indicated results are reasonable and thus physically significant for a better understanding of real traffic flow on an inhomogeneous road.
- Following the approach of Carlet et al.(2011)\citeCDM, we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection between these infinite-dimensional Frobenius manifolds and the finite-dimensional Frobenius manifolds on the orbit space of extended affine Weyl groups of type $A$ defined by Dubrovin and Zhang.
- In the free group $F_k$, an element is said to be primitive if it belongs to a free generating set. In this paper, we describe what a generic primitive element looks like. We prove that up to conjugation, a random primitive word of length $N$ contains one of the letters exactly once asymptotically almost surely (as $N \to \infty$). This also solves a question from the list `Open problems in combinatorial group theory' [Baumslag-Myasnikov-Shpilrain 02']. Let $p_{k,N}$ be the number of primitive words of length $N$ in $F_k$. We show that for $k \ge 3$, the exponential growth rate of $p_{k,N}$ is $2k-3$. Our proof also works for giving the exact growth rate of the larger class of elements belonging to a proper free factor.
- The optimality of the conventional maximum likelihood sequence estimation (MLSE), also known as the Viterbi Algorithm (VA), relies on the assumption that the receiver has perfect knowledge of the channel coefficients or channel state information (CSI). However, in practical situations that fail the assumption, the MLSE method becomes suboptimal and then exhaustive checking is the only way to obtain the ML sequence. At this background, considering directly the ML criterion for partial CSI, we propose a two-phase low-complexity MLSE algorithm, in which the first phase performs the conventional MLSE algorithm in order to retain necessary information for the backward VA performed in the second phase. Simulations show that when the training sequence is moderately long in comparison with the entire data block such as 1/3 of the block, the proposed two-phase MLSE can approach the performance of the optimal exhaustive checking. In a normal case, where the training sequence consumes only 0.14 of the bandwidth, our proposed method still outperforms evidently the conventional MLSE.
- A new primal-dual algorithm is presented for solving a class of non-convex minimization problems. This algorithm is based on canonical duality theory such that the original non-convex minimization problem is first reformulated as a convex-concave saddle point optimization problem, which is then solved by a quadratically perturbed primal-dual method. %It is proved that the popular SDP method is indeed a special case of the canonical duality theory. Numerical examples are illustrated. Comparing with the existing results, the proposed algorithm can achieve better performance.
- Dec 11 2012 math.GM arXiv:1212.2597v1In this paper, we presents a characterization of compact subsets of the fuzzy number space equipped with the level convergence topology. Based on this, it is shown that compactness is equivalent to sequential compactness on the fuzzy number space equipped with the level convergence topology. Diamond and Kloeden gave a characterization of compact sets in fuzzy number spaces equipped with the supremum metric, Fang and Xue also gave a characterization of compact sets in one-dimensional fuzzy number spaces equipped with supremum metric. The latter characterization is just the one-dimensional case of the former characterization. There exists conflict between the characterization given by us and the characterizations given by the above mentioned authors. We point out the characterizations gave by them is incorrect by a counterexample.