More on the Strength of Engeler’s Lemma

We introduce semilattices equipped with a partial n-ary operation ρ. A useful separation lemma for such nontrivial complete partial ρ-semilattices is proved equivalent to PIT, the prime ideal theorem. The relation of various versions of the Lemma to each other and to PIT is also explored. Supported by the Ministry of Education of the Czech Republic under the project MSM0021622409.     

Maximum principles for second order dynamic equations on time scales

This paper establishes some new maximum principles for second order dynamic equations on time scales, including: a strong maximum principle; a generalized maximum principle; and a boundary point lemma. The new results include, as special cases, well-known ideas for ordinary differential equations and difference equations.     

Integrability of Solutions of Nonlinear Elliptic Equations with Right-Hand Sides from Classes Close to L 1

We establish a number of results on the integrability of the entropy and the weak solutions of the Dirichlet problem for nonlinear elliptic equations of second order depending on the integrability properties of the right-hand sides of these equations.     

Spanning triangulations in graphs

We prove that every graph of sufficiently large order n and minimum degree at least 2n/3 contains a triangulation as a spanning subgraph. This is best possible: for all integers n, there are graphs of order n and minimum degree ?2n/3? ? 1 without a spanning triangulation. © 2005 Wiley Periodicals, Inc. J Graph Theory     

On variants of the Johnson–Lindenstrauss lemma

The Johnson–Lindenstrauss lemma asserts that an n‐point set in any Euclidean space can be mapped to a Euclidean space of dimension k = O(ε^‐2 log n) so that all distances are preserved up to a multiplicative factor between 1 ? ε and 1 + ε. Known proofs obtain such a mapping as a linear map R ⁿ → R ^k with a suitable random matrix. We give a simple and self‐contained proof of a version of the Johnson–Lindenstrauss lemma that subsumes a basic versions by Indyk and Motwani and a version more suitable for efficient computations due to Achlioptas. (Another proof of this result, slightly different but in a similar spirit, was given independently by Indyk and Naor.) An even more general result was established by Klartag and Mendelson using considerably heavier machinery. Recently, Ailon and Chazelle showed, roughly speaking, that a good mapping can also be obtained by composing a suitable Fourier transform with a linear mapping that has a sparse random matrix M; a mapping of this form can be evaluated very fast. In their result, the nonzero entries of M are normally distributed. We show that the nonzero entries can be chosen as random ± 1, which further speeds up the computation. We also discuss the case of embeddings into R ^k with the ?₁ norm. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

局部平方可积鞅Rosenthal}不等式的另一形式

本文研究局部平方可积鞅的一种Rosenthal不等式中常数的性态,证明了其系数与离散参数鞅情形有相同的增长阶.     

Impulsive synchronization of two coupled complex networks with time-delayed dynamical nodes

In this paper, we investigate the impulsive synchronization between two coupled complex networks with time-delayed dynamical nodes. Based on the Lyapunov stability, the linear feedback control and the impulsive control theories, the linear feedback and the impulsive controllers are designed separately. By using the generalized Barbalat's lemma, the global asymptotic impulsive synchronization of the drive—response complex networks is derived and some corresponding sufficient conditions are also obtained. Numerical examples are presented to verify the effectiveness and the correctness of the synchronization criteria.     

Multiscale cohesive failure modeling of heterogeneous adhesives

A novel multiscale cohesive approach that enables prediction of the macroscopic properties of heterogeneous thin layers is presented. The proposed multiscale model relies on the Hill's energy equivalence lemma, implemented in the computational homogenization scheme, to couple the micro- and macro-scales and allows to relate the homogenized cohesive law used to model the failure of the adhesive layer at the macro-scale to the complex damage evolution taking place at the micro-scale. A simple isotropic damage model is used to describe the failure processes at the micro-scale. We establish the upper and lower bounds on the multiscale model and solve several examples to demonstrate the ability of the method to extract physically based macroscopic properties.     

Noncommutative Differential Calculus and Its Application on Discrete Spaces

We present the noncommutative differential calculus on the function space of the infinite set and construct a homotopy operator to prove the analogue of the Poincare lemma for the difference complex. Then the horizontal and vertical complexes are introduced with the total differential map and vertical exterior derivative. As the application of the differential calculus, we derive the schemes with the conservation of symplecticity and energy for Hamiltonian system and a two-dimensional integral models with infinite sequence of conserved currents. Then an Euler-Lagrange cohomology with symplectic structure-preserving is given in the discrete classical mechanics.