Vibrational resonance in neuron populations with hybrid synapses

In this paper, vibrational resonance in excitable neuron populations with synapses is investigated by numerical simulation. In particular, the effect of the hybrid synapses on the signal detection and transmission in neural system is studied. Different topologies from regular and random networks to small-world networks are considered to analyze the dependence of vibrational resonance on the network structure and parameters. It is shown that there exists an optimal amplitude of high-frequency driving, enhancing the response of coupled neuron populations to a subthreshold signal. We find that chemical synaptic coupling is more efficient than the electrical coupling in signal detection and electrical synaptic coupling is better in signal transmission. Neuron populations with hybrid synapses compromise the merits of the two types of coupling and have an advantage in information communication.     

Conformal and semi-conformal biharmonic maps

We show that a conformal mapping between Riemannian manifolds of the same dimension n ≥ 3 is biharmonic if and only if the gradient of its dilation satisfies a certain second-order elliptic partial differential equation. On an Einstein manifold solutions can be generated from isoparametric functions. We characterise those semi-conformal submersions that are biharmonic in terms of their dilation and the fibre mean curvature vector field.      

Uniqueness of Fq-quadratic perfect nonlinear maps from Fq3 to Fq2

Let q be a power of an odd prime. We prove that all ${\mathbb{F}}_q$-quadratic perfect nonlinear maps from ${\mathbb{F}}_{q^3}$ to ${\mathbb{F}}_q^2$ are equivalent. We also give a geometric method to find the corresponding equivalence explicitly.     

Periodic orbits in two classes of piecewise smooth maps with positive nonlinear parts

In this paper we consider two classes of one dimensional piecewise smooth continuous maps that have been derived as normal forms for grazing bifurcations of piecewise smooth dynamical systems. These maps are linear on one side of the phase space and nonlinear on the other side. The case of nonlinear parts with negative coefficients has been studied previously and it is proved that period-adding scenarios are generic in this case. In contrast to this result, in our analytical and numerical results, the period-adding scenarios are not observed when the nonlinear parts have positive coefficients. Furthermore, our results suggest that the typical bifurcation scenario is period doubling cascade leading to chaos in this case, which is similar to that of the smooth logistic map.     

Biharmonic homogeneous hypersurfaces in compact symmetric spaces

In this paper, we study biharmonic hypersurfaces in Einstein manifolds. Then, we determine all the biharmonic hypersurfaces in irreducible symmetric spaces of compact type which are regular orbits of commutative Hermann actions of cohomogeneity one.     

Covering of nonlinear maps on a cone in neighborhoods of irregular points

Inverse function theorems for smooth nonlinear maps defined on convex cones in Banach spaces in a neighborhood of an irregular point are considered. The corresponding covering theorem is proved. The proofs are based on a Banach open mapping theorem for convex cones in Banach spaces, which is also proved in the paper. Sufficient conditions for tangency to the zero set of a nonlinear map without a priori regularity assumptions are obtained.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 483–497.Original Russian Text Copyright © 2005 by A. V. Arutyunov.This revised version was published online in April 2005 with a corrected issue number.     

On twistor Gauss maps of surfaces in 4-spheres

In this paper we study surfaces in S⁴ and their twistor Gauss maps. Some necessary and sufficient conditions that the twistor Gauss map is harmonic are given. We find many examples of nonisotropic harmonic maps from a surface to P ³.Supported by the National Natural Science Foundation of China and the Science Foundation of Zhejiang Province.     

Parametric resonance of ground states in the nonlinear Schrödinger equation

We study the global existence and long-time behavior of solutions of the initial-value problem for the cubic nonlinear Schrödinger equation with an attractive localized potential and a time-dependent nonlinearity coefficient. For small initial data, we show under some nondegeneracy assumptions that the solution approaches the profile of the ground state and decays in time like t^-1/4. The decay is due to resonant coupling between the ground state and the radiation field induced by the time-dependent nonlinearity coefficient.     

An algorithmic approach to the construction of homomorphisms induced by maps in homology

This paper is devoted to giving the theoretical background for an algorithm for computing homomorphisms induced by maps in homology. The principal idea is to insert the graph of a given continuous map into a graph of a multi-valued representable map . The multi-valued representable maps have well developed continuity properties and admit a finite coding that permits treating them by combinatorial methods. We provide the construction of the homomorphism induced by such that . The presented construction does not require subsequent barycentric subdivisions and simplicial approximations of . The main motivation for this paper comes from the project of computing the Conley Index for discrete dynamical systems.

     

Nonexistence of formal first integrals for general nonlinear systems under resonance

In this paper, we consider the nonintegrability for nonlinear systems under the simple resonant case, i.e., the Jacobian matrix of vector field at some fixed point has some single multiply zero eigenvalues, and some nonzero eigenvalues which are N-independent. By using the Poincaré-Dulac normal form theory, we give a necessary condition for the system under consideration to have formal first integral.     

Positive solution for nonlinear third-order multi-point boundary value problem at resonance

In this paper, positive solutions for a kind of third-order multi-point boundary value problem at resonance are investigated. By using the Leggett-Williams norm-type theorem due to O''Regan and Zima, existence result of at least one positive solution is established. An example is given to demonstrate the main results.     

Super efficiency in vector optimization with nearly convexlike set-valued maps

In this paper, we establish a scalarization theorem and a Lagrange multiplier theorem for super efficiency in vector optimization problem involving nearly convexlike set-valued maps. A dual is proposed and duality results are obtained in terms of super efficient solutions. A new type of saddle point, called super saddle point, of an appropriate set-valued Lagrangian map is introduced and is used to characterize super efficiency.     

Some useful set-valued maps in set optimization

In this article, we introduce several classes of set-valued maps which can be useful in set optimization due to their applications. Exactly, we present some set-valued maps defined by scalar and vector functions and study their properties such as continuity and convexity among others. In addition, we compute their asymptotic maps which can be employed to establish coercivity and existence results in the framework of set optimization problems. Finally, we expose some possible directions for further research.     

The mixed BVP for second order nonlinear ordinary differential equation at resonance

Efficient sufficient conditions are established for the solvability of the mixed problem where in the case where the homogeneous linear problem has nontrivial solutions.