Synchronization of complex dynamical networks with time delays

In the present paper, two kinds of dynamical complex networks are considered. The first is that elements of every node have different time delays but all nodes in such networks have the same time-delay vector. The second is that different nodes have different time-delay vectors, and the elements of each node also have different time delays. Corresponding synchronization theorems are established. Numerical examples show the efficiency of the derived theorems.     

Eigenvalue ratios of vibrating strings

Summary New results for the eigenvalue ratios of vibrating strings are presented partially in connection with previous results concerning Schr?dinger operators.     

Existence of energy minimizing vortices attached to a flat-top seamount

The existence of an energy minimizer relative to a class of rearrangements of a given function is proved. The minimizers are stationary and stable solutions of the two-dimensional barotropic vorticity equation, governing the evolution of geophysical flow over a surface of variable height. The theorem proved implies the existence of a family of stable anticyclonic vortices with cyclonic potential vorticity over a seamount, and a corresponding family of cyclonic vortices with anticyclonic potential vorticity over a localized depression. The seamount is described by a characteristic function (corresponding to a flat top) with arbitrary shape.     

A regression study of the number of efficient extreme points in multiple objective linear programming

In this paper we employ regression analysis to construct relationships for predicting the number of efficient extreme points in MOLPs (multiple objective linear programs) with up to 120,000 efficient extreme points, and the CPU time to compute them. Principal among the factors affecting the number of efficient extreme points and CPU time are the number of objectives, criterion cone size, number of constraints, number of variables, and the nonzero density of the constraint matrix. The regression equations show the degree to which interactions are present among the factors and provide a more formal basis for understanding how the complexity of the efficient set, an indicator of the difficulty involved in solving a multiple criteria problem, increases with problem size.     

Structural manipulation of pyrochlores: Thermal evolution of metastable Gd2(Ti1−yZry)2O7 powders prepared by mechanical milling

The structural and microstructural characteristics of metastable Gd₂(Ti_1−yZr_y)₂O₇ powders prepared by mechanical milling have been studied by a combination of XRD and Raman spectroscopy. Irrespective of their Zr content, as-prepared powder phases present an anion-deficient fluorite-type of structure as opposed to the pyrochlore equilibrium configuration obtained for the same solid solution by other synthetic routes. These fluorites are stable versus thermal activation, at least up to temperatures of 800 °C. For the Ti-rich compositions, thermal treatments at higher temperatures facilitate the rearrangement of the cation and anion substructures and the relaxation of mechanochemically induced defects whereas for compositions with high Zr content, the fluorite crystal structure is retained even at temperatures as high as 1200 °C. Interestingly enough, transient pyrochlores showing a very unusual cation distribution were observed during the thermally induced defect-recovery process.     

Note on a conjecture of toft

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision ofK ₄. We show that if a graphG has a degree three nodev such thatG-v is 3-colourable, then eitherG is 3-colourable or it contains a fully oddK ₄. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision ofK ₄. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)     

Characteristics and structures of weak efficient surfaces of production possibility sets

This paper studies the characteristics and structure of the weak surface of the production possibility set. We apply techniques and methods of transferring a polyhedral cone from its intersection form to its sum form, identify an intersection representation of the production possibility set. We give the structure theorem of weak surface of the production possibility set, which includes three complementary slackness conditions. We define the input weak efficiency and output weak efficiency for different DEA models according to the representation of the intersection form. It investigates the characteristics of the weak surfaces, and proves the structure theorems of input weak DEA efficiency and output weak DEA efficiency. The structure theorems establish weighted combination of inputs and outputs that are weak DEA efficient. Numerical examples are provided for illustration.     

Testing exponentiality against a class of alternatives which includes the RNBUE distributions based on the empirical laplace transform

A scale-free test for exponentiality is proposed which is consistent within an extended set of models, including, but not limited to, the “renewal new better than used in expectation” (RNBUE) class of life distributions. The limiting null distribution of the test statistics is derived, and the approximate local Bahadur efficiency is calculated for several families of alternatives. Finite-sample properties of the proposed procedures are investigated via simulation. Bibliography: 24 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2006, pp. 63–77.