Chaotic and bifurcation dynamic behavior of a simply supported rectangular orthotropic plate with thermo-mechanical coupling

This paper presents a novel analytical approach utilizing fractal dimension criteria and the maximum Lyapunov exponent to characterize the conditions which can potentially lead to the chaotic motion of a simply supported thermo-mechanically coupled orthotropic rectangular plate undergoing large deflections. The study commences by deriving the governing partial differential equations of the rectangular plate, and then applies the Galerkin method to simplify these equations to a set of three ordinary differential equations. The associated power spectra, phase plots, Poincaré map, maximum Lyapunov exponents, and fractal and bifurcation diagrams are computed numerically. These features are used to characterize the dynamic behavior of the orthotropic rectangular plate under various excitation conditions. The maximum Lyapunov exponents and the correlation dimensions method indicate that chaotic motion of the orthotropic plate occurs at η₁ = 1.0, , and for an external force of . The application of an external in-plane force of magnitude causes the orthotropic plate to perform bifurcation motion. Furthermore, when , aperiodic motion of the plate is observed. Hence, the dynamic motion of a thermo-mechanically coupled orthotropic rectangular plate undergoing large deflections can be controlled and manipulated to achieve periodic motion through an appropriate specification of the system parameters and loads.     

Space-efficient geometric divide-and-conquer algorithms

We develop a number of space-efficient tools including an approach to simulate divide-and-conquer space-efficiently, stably selecting and unselecting a subset from a sorted set, and computing the kth smallest element in one dimension from a multi-dimensional set that is sorted in another dimension. We then apply these tools to solve several geometric problems that have solutions using some form of divide-and-conquer. Specifically, we present a deterministic algorithm running in time using extra memory given inputs of size n for the closest pair problem and a randomized solution running in expected time and using extra space for the bichromatic closest pair problem. For the orthogonal line segment intersection problem, we solve the problem in time using extra space where n is the number of horizontal and vertical line segments and k is the number of intersections.     

Fractal sets determined by dilation-stable processes

Let be a dilation-stable process on . We determine a Hausdorff measure function (a) such that the fractal set X[0,1]={X(t):0t1} has positive finite -measure. We also investigate the packing measure of X[0,1].     

The number of translates of a closed nowhere dense set required to cover a Polish group

For a Polish group let be the minimal number of translates of a fixed closed nowhere dense subset of required to cover . For many locally compact this cardinal is known to be consistently larger than which is the smallest cardinality of a covering of the real line by meagre sets. It is shown that for several non-locally compact groups . For example the equality holds for the group of permutations of the integers, the additive group of a separable Banach space with an unconditional basis and the group of homeomorphisms of various compact spaces.     

Minimum weight pseudo-triangulations

We consider the problem of computing a minimum weight pseudo-triangulation of a set of n points in the plane. We first present an -time algorithm that produces a pseudo-triangulation of weight which is shown to be asymptotically worst-case optimal, i.e., there exists a point set for which every pseudo-triangulation has weight , where is the weight of a minimum weight spanning tree of . We also present a constant factor approximation algorithm running in cubic time. In the process we give an algorithm that produces a minimum weight pseudo-triangulation of a simple polygon.     

Nonlinear maps preserving Lie products on factor von Neumann algebras

In this paper, we prove that every bijective map preserving Lie products from a factor von Neumann algebra into another factor von Neumann algebra is of the form A→ψ(A)+ξ(A), where is an additive isomorphism or the negative of an additive anti-isomorphism and is a map with ξ(AB-BA)=0 for all .     

Augmenting the connectivity of geometric graphs

Let G be a connected plane geometric graph with n vertices. In this paper, we study bounds on the number of edges required to be added to G to obtain 2-vertex or 2-edge connected plane geometric graphs. In particular, we show that for G to become 2-edge connected, additional edges are required in some cases and that additional edges are always sufficient. For the special case of plane geometric trees, these bounds decrease to and , respectively.     

Linear maps preserving idempotency of products or triple jordan products of operators

Let be the algebra of all bounded linear operators on a complex Banach space . We give the concrete forms of linear surjective maps on which preserve the nonzero idempotency of either products of two operators or triple Jordan products of two operators.     

Uniform asymptotic expansions of the Pollaczek polynomials

Two uniform asymptotic expansions are obtained for the Pollaczek polynomials P_n(cosθ;a,b). One is for , , in terms of elementary functions and in descending powers of . The other is for , in terms of a special function closely related to the modified parabolic cylinder functions, in descending powers of n. This interval contains a turning point and all possible zeros of P_n(cosθ) in θ(0,π/2].     

The minimum-area spanning tree problem

Motivated by optimization problems in sensor coverage, we formulate and study the Minimum-Area Spanning Tree (mast) problem: Given a set of n points in the plane, find a spanning tree of of minimum “area”, where the area of a spanning tree is the area of the union of the n−1 disks whose diameters are the edges in . We prove that the Euclidean minimum spanning tree of is a constant-factor approximation for mast. We then apply this result to obtain constant-factor approximations for the Minimum-Area Range Assignment (mara) problem, for the Minimum-Area Connected Disk Graph (macdg) problem, and for the Minimum-Area Tour (mat) problem. The first problem is a variant of the power assignment problem in radio networks, the second problem is a related natural problem, and the third problem is a variant of the traveling salesman problem.     

Two-to-one quantum teleportation protocol and its application

In this paper, we propose a two-to-one quantum teleportation protocol with three-particle entangled states. Two senders teleport the product of their quantum states to one receiver. The quantum product is of two types. The receiver will recover one type with probability . We further apply our protocol to send the message secretly. The receiver can recover the original message with probability after performing our protocol k times.     

Sign balance for finite groups of Lie type

A product formula for the parity generating function of the number of 1’s in invertible matrices over is given. The computation is based on algebraic tools such as the Bruhat decomposition. It is somewhat surprising that the number of such matrices with odd number of 1’s is greater than the number of those with even number of 1’s. The same technique can be used to obtain a parity generating function also for symplectic matrices over . We present also a generating function for the sum of entries of matrices over an arbitrary finite field calculated in . The Mahonian distribution appears in these formulas.     

Strange distributionally chaotic triangular maps II

The notion of distributional chaos was introduced by Schweizer and Smítal [Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans Am Math Soc 1994;344:737–854] for continuous maps of the interval. For continuous maps of a compact metric space three mutually non-equivalent versions of distributional chaos, DC1–DC3, can be considered. In this paper we study distributional chaos in the class of triangular maps of the square which are monotone on the fibres. The main results: (i) If has positive topological entropy then F is DC1, and hence, DC2 and DC3. This result is interesting since similar statement is not true for general triangular maps of the square [Smítal and Štefánková, Distributional chaos for triangular maps, Chaos, Solitons & Fractals 2004;21:1125–8]. (ii) There are which are not DC3, and such that not every recurrent point of F₁ is uniformly recurrent, while F₂ is Li and Yorke chaotic on the set of uniformly recurrent points. This, along with recent results by Forti et al. [Dynamics of homeomorphisms on minimal sets generated by triangular mappings, Bull Austral Math Soc 1999;59:1–20], among others, make possible to compile complete list of the implications between dynamical properties of maps in , solving a long-standing open problem by Sharkovsky.     

A new discrete integrable system and its discrete integrable coupling system

In terms of the properties of the known loop algebra and difference operators, a new algebraic system X is constructed, which is devote to working out the well-known generalized cubic Volterra lattice equations hierarchy. Then an extended algebraic system of X is presented, from which the integrable coupling system of generalized cubic Volterra lattice equations are obtained.     

Limit behavior of global attractors for the complex Ginzburg–Landau equation on infinite lattices

In this work, the authors first show the existence of global attractors for the following lattice complex Ginzburg–Landau equation:

and for the following lattice Schrödinger equation:

Then they prove that the solutions of the lattice complex Ginzburg–Landau equation converge to that of the lattice Schrödinger equation as ε→0+. Also they prove the upper semicontinuity of as ε→0+ in the sense that .     

An atlas for tridiagonal isospectral manifolds

Let be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix Λ with simple spectrum. We introduce bidiagonal coordinates, charts defined on open dense domains forming an explicit atlas for . In contrast to the standard inverse variables, consisting of eigenvalues and norming constants, every matrix in now lies in the interior of some chart domain. We provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics, both for continuous and discrete time.     

Limit shape of convex lattice polygons with minimal perimeter

Let n be a positive integer and · any norm in . Denote by B the unit ball of · and the class of convex lattice polygons with n vertices and least ·-perimeter. We prove that after suitable normalization, all members of tend to a fixed convex body, as n→∞.     

A result about the density of iterated line intersections in the plane

Let S be a finite set of points in the plane and let be the set of intersection points between pairs of lines passing through any two points in S. We characterize all configurations of points S such that iteration of the above operation produces a dense set. We also discuss partial results on the characterization of those finite point-sets with rational coordinates that generate all of through iteration of .     

Non-associative algebras containing the matrix ring

The Lie admissible non-associative algebra is defined in the papers [Seul Hee Choi, Ki-Bong Nam, Derivations of a restricted Weyl type algebra I, Rocky Mountain J. Math. 37 (6) (2007) 1813–1830; Seul Hee Choi, Ki-Bong Nam, Weyl type non-associative algebra using additive groups I, Algebra Colloq. 14 (3) (2007) 479–488; Ki-Bong Nam, On Some Non-associative Algebras using Additive Groups, Southeast Asian Bull. Math., vol. 27, Springer-Verlag, 2003, 493–500]. We define in this work the algebra which generalizes the previous one and is not Lie admissible. We prove that the antisymmetrized Lie algebra is simple and contains the simple Lie algebra . We also prove that the matrix ring is embedded in .     

A non-normal topology generated by a two-point selection

We construct a two-point selection , where is the set of the irrational numbers, such that the space is not normal and it is not collectionwise Hausdorff either. Here, τ_f denotes the topology generated by the two-point selection f. This example answers a question posed by V. Gutev and T. Nogura. We also show that if is a two-point selection such that the topology τ_f has countable pseudocharacter, then τ_f is a Tychonoff topology.