ArXiv math

Authors: Chris Aholt, Luke Oeding

Techniques from representation theory, symbolic computational algebra, and numerical algebraic geometry are used to find the minimal generators of the ideal of the trifocal variety. An effective test for determining whether a given tensor is a trifocal tensor is also given.

Authors: Hasan Almanasreh

We apply hp-cloud method to the radial Dirac eigenvalue problem. The difficulty of occurrence of spurious values among the genuine eigenvalues is treated. The method of treatment is based on assuming hp-cloud Petrov-Galerkin scheme to construct the weak formulation of the problem which adds a consistent diffusivity to the variational formulation without deforming the equation. The size of the artificially added diffusion term is controlled by a derived stability parameter ($\tau$). The derivation of $\tau$ considers the limit behavior of the eigenvalues at infinity. The importance of $\tau$ is of being applicable for generic basis functions. This is together with choosing appropriate intrinsic enrichments in the construction of the cloud shape functions.

Authors: André LeClair, Denis Bernard

Using generic properties of Clifford algebras in any spatial dimension, we explicitly classify Dirac hamiltonians with zero modes protected by the discrete symmetries of time-reversal, particle-hole symmetry, and chirality. Assuming the boundary states of topological insulators are Dirac fermions, we thereby holographically reproduce the Periodic Table of topological insulators found by Kitaev and Ryu. et. al, without using topological invariants nor K-theory. In addition we find candidate Z_2 topological insulators in classes AI, AII in dimensions 0,4 mod 8 and in classes C, D in dimensions 2,6 mod 8.

Authors: Herbert R Stahl

Let f be a function meromorphic in a neighborhood of infinity. The central problem in the present investigation is to find the largest domain D \subset C to which the function f can be extended in a meromorphic and singlevalued manner. 'Large' means here that the complement C\D is minimal with respect to (logarithmic) capacity. Such extremal domains play an important role in Pad'e approximation. In the paper a unique existence theorem for extremal domains and their complementary sets of minimal capacity is proved. The topological structure of sets of minimal capacity is studied, and analytic tools for their characterization are presented; most notable are here quadratic differentials and a specific symmetry property of the Green function in the extremal domain. A local condition for the minimality of the capacity is formulated and studied. Geometric estimates for sets of minimal capacity are given. Basic ideas are illustrated by several concrete examples, which are also used in a discussion of the principal differences between the extremality problem under investigation and some classical problems from geometric function theory that possess many similarities, which for instance is the case for Chebotarev's Problem.

Authors: Kevin P. Thompson

The set of trigonometric functions in taxicab geometry is completed and derivatives of all of the taxicab trigonometric functions are explored.

Authors: Alvaro Bustinduy, Luis Giraldo

It is proved that any polynomial vector field in two complex variables which is complete on a non-algebraic trajectory is complete.

Authors: Nanao Kita

This paper is concerned with the structures of general graphs with perfect matchings. We first reveal a partially ordered structure among elementary components of general graphs with perfect matchings. Our second result is a generalization of the canonical partition to a decomposition theorem for each elementary components of general graphswith perfect matchings. It includes a short proof for the canonical partition by A. Kotzig [1959--1960]. These results gives decompositions of a graph which are canonical, that is, unique to it.

Authors: Thomas Bauer, David Schmitz

Zariski chambers are natural pieces into which the big cone of an algebraic surface decomposes. They have so far been studied both from a geometric and from a combinatorial perspective. In the present paper we complement the picture with a metric point of view by studying a suitable notion of chamber sizes. Our first result gives a precise condition for the nef cone volume to be finite and provides a method for computing it inductively. Our second result determines the volumes of arbitrary Zariski chambers from nef cone volumes of blow-downs. We illustrate the applicability of this method by explicitly determining the chamber volumes on Del Pezzo and other anti-canonical surfaces.

Authors: Alexandre Mauroy, Rodolphe Sepulchre

This paper establishes a global contraction property for networks of phase-coupled oscillators characterized by a monotone coupling function. The contraction measure is a total variation distance. The contraction property determines the asymptotic behavior of the network, which is either finite-time synchronization or asymptotic convergence to a splay state.

Authors: Daniel Hernández-Hernández, Leonel Pérez-Hernández

The minimality of the penalization function associated with a convex risk measure is analyzed in this paper. First, in a general static framework, we provide necessary and sufficient conditions for a penalty function defined in a convex and closed subset of the absolutely continuous measures with respect to some reference measure $\mathbb{P}$ to be minimal. When the probability space supports a L\'{e}vy process, we establish results that guarantee the minimality property of a penalty function described in terms of the coefficients associated with the density processes. The set of densities processes is described and the convergence of its quadratic variation is analyzed.

Authors: Albert Fannjiang

This note extends Candes, Romberg and Tao's proof of exact recovery of discrete piecewise constant objects by discrete total variation (TV) minimization with noiseless incomplete Fourier data to the case of noisy data. Specifically, the inverse scattering by piecewise constant objects is formulated as discrete TV-min with proposed measurement schemes.

A TV-error bound, independent of pixel size, is derived for the TV-minimizer with the number of data roughly proportional to the sparsity of the object gradient. TV and maximum error bounds, independent of pixel size, are obtained for the greedy recovery by Orthogonal Matching Pursuit with the number of data roughly proportional to the square of the gradient sparsity.

Authors: Nasir Ganikhodjaev, Dmitriy Zanin

In the paper a Volterra quadratic stochastic operators of three dimensional simplex into itself is considered.The full description of ergodic properties such operators is given.

Authors: Xia Liao

Let $X$ be a nonsingular complex projective variety and $D$ a locally quasi-homogeneous free divisor in $X$. In this paper we prove that the classes $c_{SM}(1_U)$ and $c(\textup{Der}_X(-\log D))\cap [X]$ agree after push-forward to the ambient projective space.

Authors: Weiping Yan, Yong Li

This paper is devoted to the study of a class of singular perturbation elliptic type problems on compact Lie groups or Homogeneous spaces $\mathcal{M}$. By constructing a suitable Nash-Moser-type iteration scheme on compact Lie groups and Homogeneous spaces, we overcome the clusters of "small divisor" problem, then the existence of solutions for nonlinear elliptic equations with a singular perturbation is established. Especially, if $\mathcal{M}$ is the standard torus $\textbf{T}^n$ or the spheres $\textbf{S}^n$, our result shows that there is a local uniqueness of spatially periodic solutions for nonlinear elliptic equations with a singular perturbation.

Authors: Weiping Yan, Yong Li

This paper is devoted to the study of the dynamical behavior of the critically dissipative quasi-geostrophic equation in $\textbf{T}^2$. We prove that this system possesses time-dependent periodic solutions, bifurcating from a smooth steady solution, i.e. a Hopf-Bifurcation theorem.

Authors: M.S. Moslehian, M. Chakoshi

Suppose that $\mathscr{X}$ and $\mathscr{Y}$ are Hilbert $C^*$-modules. Let $t:{\rm Dom}(t) \subseteq \mathscr{X}\to \mathscr{Y}$ be a regular operator such that $\mathscr{X}={\rm Ker}(t) \oplus \bar{{\rm Ran}(t^*)}$ and $\mathscr{Y} ={\rm Ker}(t^*) \oplus \bar{{\rm Ran}(t)}$. We extend two significant results involving the Moore--Penrose inverse of Gram operators on Hilbert spaces to Hilbert $C^*$-modules by establishing that (i) $(t^*t)^\dagger=t^\dagger t^*{^\dagger}$ and (ii) $t^\dagger=t^*(tt^*)^\dagger=(t^*t)^\dagger t^*$. Furthermore, (i) holds for all such operators $t$ if and only if so does (ii). We present some applications. In particular, we show that a regular operator $t:{\rm Dom}(t) \subseteq \mathscr{X}\to \mathscr{X}$ is normal if and only if $t^\dagger$ commutes with $t$ and $t^*$.

Authors: Curt Schieler, Paul Cuff

A secret key can be used to conceal information from an eavesdropper during communication, as in Shannon's cipher system. Most theoretical guarantees of secrecy require the secret key space to grow exponentially with the length of communication. Here we show that when an eavesdropper attempts to reconstruct an information sequence, as posed in the literature by Yamamoto, very little secret key is required to effect unconditionally maximal distortion; specifically, we only need the secret key space to increase unboundedly, growing arbitrarily slowly with the blocklength. As a corollary, even with a secret key of constant size we can still cause the adversary arbitrarily close to maximal distortion, regardless of the length of the information sequence.

Authors: Laura Ciobanu, Susan Hermiller

In this paper we introduce the geodesic conjugacy language and geodesic conjugacy growth series for a finitely generated group. We study the effects of various group constructions on rationality of both the geodesic conjugacy growth series and spherical conjugacy growth series, as well as on regularity of the geodesic conjugacy language and spherical conjugacy language. In particular, we show that regularity of the geodesic conjugacy language is preserved by the graph product construction, and rationality of the geodesic conjugacy growth series is preserved by both direct and free products.

Authors: Saeed Hajizadeh, Ghosheh Abed Hodtani

Three-receiver broadcast channel (BC) is of interest due to its information theoretical differences with two receiver one. In this paper, we derive achievable rate regions for two classes of 3-receiver BC with side information available at the transmitter, Multilevel BC and 3-receiver less noisy BC, by using superposition coding, Gel'fand-Pinsker binning scheme and Nair-El Gamal indirect decoding. Our rate region for multilevel BC subsumes the Steinberg rate region for 2-receiver degraded BC with side information as its special case. We also find the capacity region of 3-receiver less noisy BC when side information is available both at the transmitter and at the receivers.

Authors: Raziuddin Siddiqui

In this paper we consider the Grassmannian complex of projective configurations in weight 2 and 3, and Cathelineau's infinitesimal polylogarithmic complexes as well as a tangential complex to the famous Bloch-Suslin complex (in weight 2) and to Goncharov's motivic complex (in weight 3), respectively, as proposed by Cathelineau \cite{Cath3}. Our main result is a morphism of complexes between the Grassmannian complexes and the associated infinitesimal polylogarithmic complexes. In order to establish this connection we introduce an $F$-vector space $\beta^D_2(F)$, which is an intermediate structure between a $\varmathbb{Z}$-module $\mathcal{B}_2(F)$ (scissors congruence group for $F$) and Cathelineau's $F$-vector space $\beta_2(F)$ which is an infinitesimal version of it. The structure of $\beta^D_2(F)$ is also infinitesimal but it has the advantage of satisfying similar functional equations as the group $\mathcal{B}_2(F)$. We put this in a complex to form a variant of Cathelineau's infinitesimal complex for weight 2.