A necessary and sufficient condition of existence of positive solutions for nonlinear singular differential systems

In this paper, by means of monotone iterative technique, a necessary and sufficient condition of the existence of positive solution for a class of nonlinear singular differential system is established, the results of the existence and uniqueness of the positive solution and the iterative sequence of solution are given. In the end, two classes extending boundary value differential systems are discussed and some further results are obtained.     

Dipole approximation in three-vortex dynamics

We solve the problem of the relative motion of two nearby vortices (a dipole pair) and a third vortex for different current functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 409–416, March, 2007.     

Chiral discrimination of a helically organized crown ether array parallel to the helix axis of polyisocyanate

The asymmetric polymerization of 4′‐isocyanatobenzo‐18‐crown‐6 with the lithium amide of (S)‐(2‐methoxymethyl)pyrrolidine successfully proceeded to afford end‐functionalized poly(4′‐isocyanatobenzo‐18‐crown‐6) with (S)‐(2‐methoxymethyl)pyrrolidine (polymer 2 ). In the circular dichroism (CD) spectrum of 2 , a clear positive Cotton effect was observed in the range of 240–350 nm corresponding to the absorption of the polymer backbone, indicating that 2 partially formed a one‐handed helical structure, which was preserved by the chirality of (S)‐(2‐methoxymethyl)pyrrolidine bonding to the terminal end in 2 . In the titration experiments for the CD intensity of 2 in the presence of D ‐ and L ‐Phe·HClO₄ (where Phe is phenylalanine), a small but remarkable difference was observed in the amount of the chiral guest needed for saturation of the CD intensity and in the saturated CD intensity, indicating that the extremely stable, one‐handed helical part should exist in the main chain of 2 , which was not inverted even when the unfavorable chiral guest for the predominant helical sense, L ‐Phe·HClO₄, was added. In addition, helical polymer 2 exhibited a chiral discrimination ability toward racemic guests; that is, the guests were extracted from the aqueous phase into the organic phase with enantiomeric excess. The driving force of the chiral discrimination ability of 2 should certainly be attributed to the one‐handed helical structure in 2 . © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 325–334, 2006     

Innately transitive subgroups of wreath products in product action

A permutation group is innately transitive if it has a transitive minimal normal subgroup, which is referred to as a plinth. We study the class of finite, innately transitive permutation groups that can be embedded into wreath products in product action. This investigation is carried out by observing that such a wreath product preserves a natural Cartesian decomposition of the underlying set. Previously we classified the possible embeddings in the case where the plinth is simple. Here we extend that classification and identify several different types of Cartesian decompositions that can be preserved by an innately transitive group with a non-abelian plinth. These different types of decompositions lead to different types of embeddings of the acting group into wreath products in product action. We also obtain a full characterisation of embeddings of innately transitive groups with diagonal type into such wreath products.

     

The application of Hölder exponent to traffic congestion warning

In this paper, we applied multifractal modeling techniques to analyze the traffic data collected from the Beijing Yuquanying. The results indicated that multifractal characteristics obviously exist in the traffic system; the degree of fractality of these traffic data tends to increase as the traffic system becomes congested; the Hölder exponent that measures the local rate of fractality may be used as indicators to predict the presence of the traffic congestion.     

Coupling leads to chaos

The main goal of this paper is to prove analytically the existence of strange attractors in a family of vector fields consisting of two Brusselators linearly coupled by diffusion. We will show that such a family contains a generic unfolding of a 4-dimensional nilpotent singularity of codimension 4. On the other hand, we will prove that in any generic unfolding Xμ of an n-dimensional nilpotent singularity of codimension n there are bifurcation curves of (n−1)-dimensional nilpotent singularities of codimension n−1 which are in turn generically unfolded by Xμ. Arguments conclude recalling that any generic unfolding of the 3-dimensional nilpotent singularity of codimension 3 exhibits strange attractors.     

Hidden structure in the randomness of the prime number sequence?

We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would eventually appear in applications. Finally, our theory allows us to link with three different but important topics: the Hardy–Littlewood conjecture, the statistical mechanics of spin systems, and the celebrated Sierpinski fractal.     

An explicit characterization of Calogero-Moser systems

Combining theorems of Halphen, Floquet, and Picard and a Frobenius type analysis, we characterize rational, meromorphic simply periodic, and elliptic KdV potentials. In particular, we explicitly describe the proper extension of the Airault-McKean-Moser locus associated with these three classes of algebro-geometric solutions of the KdV hierarchy with special emphasis on the case of multiple collisions between the poles of solutions. This solves a problem left open since the mid-1970s.

     

Dynamic behavior of discrete-time multiagent systems with general communication structures

In this paper, we discuss the dynamic behavior of networks of dynamic agents with general communication topologies. We first analyze the basic case: systems with communication topologies that have spanning trees, i.e., the systems that solve consensus problems. We establish an algebraic condition to characterize each agent's contributions to the final state. And we also study the influence of time-delays on each agent's contributions. Then, we investigate the general case: systems with weakly connected topologies. By using matrix theory, we prove that the states of internal agents will converge to a convex combination of boundary agents in the absence or presence of communication time-delays, and we also show that the coefficients of the convex combination are independent of time-delays even if the delays are time-varying. These results have broad applications in other areas, e.g., study of swarm behavior, formation control of vehicles, etc.     

Multi‐dimensional combustion waves for Lewis number close to one

This paper is devoted to the study of multi‐dimensional travelling wave solution for a thermo‐diffusive model, describing the propagation of curved flames in an infinite cylinder. The linear dependence of the components of the reaction rate together with the existence of an ignition temperature ensure that the corresponding linearized operator does not satisfy the Fredholm property. A direct consequence is that solvability conditions for the linearized operator are not known and classical methods of nonlinear analysis cannot be directly applied. We prove in this paper existence results of such travelling waves, by first introducing a suitable re‐formulation of the equations and then by choosing suitable weighted spaces that allows us to move the essential spectrum away from zero. Copyright © 2006 John Wiley & Sons, Ltd.