Existence And Uniqueness Of A Traveling Wave Front Of A Model Equation In Synaptically Coupled Neuronal Networks

Consider the model equation in synaptically coupled neuronal networks@u@t+ m(u − n)= ( − au) Z 10(c) ZRK(x − y)H uy, t −1c|x − y| − dydc+ ( − bu) Z 10( ) ZRW(x − y)H(u(y, t − ) − )dyd.In this model equation, u = u(x, t) stands for the membrane potential of a neuron at position x andtime t. The kernel functions K 0 and W 0 represent synaptic couplings between neurons insynaptically coupled neuronal networks. The Heaviside step function H = H(u − ) represents thegain function and it is defined by H(u − ) = 0 for all u < , H(0) = 12 and H(u − ) = 1 for allu > . The functions and represent probability density functions. The function f(u) m(u − n)represents the sodium current, where m > 0 is a positive constant and n is a real constant. Theconstants a 0, b 0, 0, 0 and > 0 represent biological mechanisms. This model equationis motivated by previous models in synaptically coupled neuronal networks.We will couple together intermediate value theorem, mean value theorem and many techniquesin dynamical systems to prove the existence and uniqueness of a traveling wave front of this modelequation. One of the most interesting and difficult parts is the proof of the existence and uniquenessof the wave speed. We will introduce several auxiliary functions and speed index functions to provethe existence and uniqueness of the front and the wave speed.     

Hermitian geometry on the resolvent set(Ⅱ)

For an element A in a unital C*-algebra B, the operator-valued 1-form ω_A(z) =(z-A)~(-1) dz is analytic on the resolvent set ρ(A), which plays an important role in the functional calculus of A. This paper defines a class of Hermitian metrics on ρ(A) through the coupling of the operator-valued(1, 1)-form ?_A=-ω_A~*∧ω_A with tracial and vector states. Its main goal is to study the connection between A and the properties of the metric concerning curvature, arc length, completeness and singularity. A particular example is when A is quasi-nilpotent, in which case the metric lives on the punctured complex plane C \ {0}. The notion of the power set is defined to gauge the "blow-up" rate of the metric at 0, and examples are given to indicate a likely link with A's hyper-invariant subspaces.     

A Complement to the Valiron-Titchmarsh Theorem for Subharmonic Functions

The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros has been recently generalized onto subharmonic functions with the Riesz measure on a half-line in Rn, n ≥ 3. Here we extend the Drasin complement to the Valiron-Titchmarsh theorem and show that if u is a subharmonic function of this class and of order 0 ρ 1, then the existence of the limit limr →∞ logu(r)/N(r),where N(r) is the integrated counting function of the masses of u, implies the regular asymptotic behavior for both u and its associated measure.     

一类非稳态热耦合方程组解的存在性和正则性

In this paper,a coupled elliptic-parabolic system modeling a class of engineering problems with thermal effect is studied.Existence of a weak solution is first established through a result of Meyers' theorem and Schander fixed point theorem,where the coupled functionsσ(s),k(s) are assumed to be bounded in the C(IR×(0,T)).Ifσ(s),k(s) are Lipschitz continuous we prove that solution is unique under some restriction on integrability of solution.The regularity of the solution in dimension n≤2 is then analyzed...     

Traveling Wave Solutions of Nonlinear Scalar Integral Differential Equations Arising from Synaptically Coupled Neuronal Networks

Consider the following nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks. These model equations generalize many important nonlinear scalar integral differential equations aris ing from synaptically coupled neuronal networks. The kernel functions K and W represent synaptic couplings between neurons in synaptically coupled neuronal networks. The synaptic couplings can be very general, including not only pure excitations (modeled with nonnegative kernel functions), lateral inhibitions (modeled with Mexican hat kernel functions), lateral excitations (modeled with upside down Mexican hat kernel functions), but also synaptic couplings which may change sign for finitely many times or even infinitely many times. The function H = H(u − ) represents the Heaviside step function, which is defined by H(u − ) = 0 for all u < , H(0) = 1 2 and H(u − ) = 1 for all u > . The functions and represent probability density functions defined on (0,1). The parameter c > 0 represents the speed of an action potential and the parameter > 0 represents a constant delay. In these equations, u = u(x, t) stands for the membrane potential of a neuron at position x and time t. The positive constants > 0 and > 0 represent synaptic rates. The positive constants > 0 and > 0 represent thresholds for excitation of neurons. The function f = f(u) represents the sodium currents in neuronal networks. The positive constant w 0 > 0 is to be given. The authors will establish the existence and stability of traveling wave solutions of these nonlinear scalar integral differential equations by coupling together speed index functions, stability index func tions (often called Evans functions, that is, complex analytic functions), implicit function theorem, intermediate value theorem, mean value theorem, global strong maximum principle for Evans func tions, linearized stability criterion and many other important techniques in dynamical systems. They will find sufficient conditions satisfied by the synaptic couplings, by the probability density functions, by the synaptic rate constants and by the thresholds so that the traveling wave solutions and their wave speeds exist, and the stability of the traveling wave solutions is true. The main results obtained in this paper greatly improve many previous results.     

QUASI-REVERSIBLE MEASURES OF NEAREST NEIGHBOUR SPEED FUNCTIONS

Let S be a countable set with a graph structure. The process with state space \[X = {\{ 0,1\} ^s}\] is described in terms of a collection of nonnegative speed functions \[c(u, \cdot ),u \in S\]. In this paper we introduce the concept of qnasi-reversible measure for speed functions, and discuss some properties contained in the existence and uniqueness of quasi-reversible measures for the nearest neighbour speed functions, with the idea of field theory by Hou and chen[3]； In section 2, we show that qnasi-reyerisible measures are Markov random fields. A necessary and sufficient condition for the existence of quasi-reversible measures is presented. In seotion 3, a uniqueness theorem of quasi-reversible measures is given. The problem to determine the quasi-reversible measures in accordance with the speed function is discussed , for some particular cases, the quasi-reversible measures can be computed explicitly. In seotion 4, we show that if the speed functions are uniformly bounded, and each point of S has uniformly bounded boundary then the quasi-reyersible measures of the speed functions are reversible measures of the spin-flip process with the speed fnncfaons. Thus we obtain, the necessary and sufficient conditions for the existence and uniqueness of reversible measures for spin-flip process with nearest neighbour speed functions. Particularly if speed functions are defined by the nearest neighbour potential, then quasi-reversible measure exigt，thus our results can be applied to solve the uniqueness problem of Gibbs states with the nearest neighbour potential.     

Harmonic L~p-functions on Complete Riemannian Manifolds

This survey paper concerns some existence theorems of harmonic functions belonging to LP (M), M being a complete Riemannian manifold. It is well known that a function which is analytic and bounded on the whole complex plane must reduce to a constant.This classical result, known as Liouville's theorem, is also true on a higher-dimensional Euclidean spaces. The generalization of this theorem to other Riemannian manifolds is very interesting. Besides its beauty, the proof usally requires sharp estimates which provide deeper understanding of the Laplacian and hence give broad applications to problems in global analysis.The basic problem in this paper is to study how the geometric conditions of a complete Riemannian manifold affect the validity of the Liouville theorem. The paper consists of two parts. Part I describes the results systematically and Part I will be more technical and will contain the detailed proofs of the results given in the first part.     

Existence and Global Asymptotic Behavior of Positive Solutions for Sublinear and Superlinear Fractional Boundary Value Problems

In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form(P_(a,b)){D~αu(x) + f(x, u(x)) = 0, x ∈(0, 1),u(0) = u(1) = 0, D~(α-3)u(0) = a, u(1) =-b,where 3 α≤ 4, Dαis the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f(x, t) =-p(x)t~σ, with σ∈(-1, 1)and p being a nonnegative continuous function that may be singular at x = 0 or x = 1and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch¨auder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem(P_(0,0)).Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a + b 0 and f(x, t) = tφ(x, t), with φ(x, t) being a nonnegative continuous function in(0, 1)×[0, ∞) that is required to satisfy some suitable integrability condition. Using estimates on the Green's function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem(P_(a,b)), which behaves like the unique solution of the homogeneous problem corresponding to(P_(a,b)). Some examples are given to illustrate the existence results.     

一类新四阶边值问题的唯一性结果

In this paper, we investigate the existence and uniqueness of solutions for a new fourth-order differential equation boundary value problem:{u(4)(t) = f(t, u(t))-b, 0 t 1,u(0) = u′(0) = u′(1) = u(3)(1) = 0,where f ∈ C([0,1] ×(-∞,+∞),(-∞, +∞)),b ≥ 0 is a constant. The novelty of this paper is that the boundary value problem is a new type and the method is a new fixed point theorem ofφ-(h,e)-concave operators.     

KAM Tori for the Derivative Quintic Nonlinear Schrodinger Equation

This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation,iut—uxx+i(|u|4u)x=0,x eT.The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established.The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem.We mention that in the present paper the mean value of u does not need to be zero,but small enough,which is different from the assumption(1.7)in Geng-Wu[J.Math.Phys.、53,102702(2012)].     

ON THE EQUATION□Φ=|▽Φ|~2 IN FOUR SPACE DIMENSIONS

This paper considers the following Cauchy problem for semilinear wave equations in n space dimensionswhere A is the wave operator, F is quadratic in (?) with (?) = ( ).The minimal value of s is determined such that the above Cauchy problem is locally well-posed in H8. It turns out that for the general equation s must satisfyThis is due to Ponce and Sideris (when n = 3) and Tataru (when n≥5). The purpose of this paper is to supplement with a proof in the case n = 2,4.     

On transcendental meromorphic functions with radially distributed values

The growth of transcendental meromorphic functions in terms of their ordersis investigated in this paper when they and their derivatives have radially distributed valuesfollowing the discussion of the author. A simple and elementary way to study such subjectsis exhibited in this paper; that is, once an estimation of B(r, *) in terms of a few c(r, **) inthe Nevanlinna theory on angular domains is established, we can produce one result thatthe order of a mermorphic function with radially distributed values related to C(r, **) canbe estimated under the assumption of existence of suitable deficient value. The results ob-tained in this paper lead us to a new singular direction in terms of Nevanlinna charactersticinstead of the order of meromorphic functions.     

ON GROWTH OF MEROMORPHIC SOLUTIONS OF NONLINEAR DIFFERENCE EQUATIONS AND TWO CONJECTURES OF C.C. YANG

In this paper, we investigate the growth of the meromorphic solutions of the following nonlinear difference equationsf(z)n+ P_(n-1)(f) = 0,where n ≥ 2 and P_(n-1)(f) is a difference polynomial of degree at most n- 1 in f with small functions as coefficients. Moreover, we give two examples to show that one conjecture proposed by Yang and Laine [2] does not hold in general if the hyper-order of f(z) is no less than 1.     

On Laguerre Isopararmetric Hypersurfaces in R^7

Let x : M → R n be an umbilical free hypersurface with non-zero principal curvatures, then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. A classical theorem of Laguerre geometry states that M(n 3) is characterized by g and B up to Laguerre equivalence. A Laguerre isopararmetric hypersurface is defined by satisfying the conditions that C = 0 and all the eigenvalues of B with respect to g are constant. It is easy to see that all Laguerre isopararmetric hypersurfaces are Dupin hypersurfaces. In this paper, we established a complete classification for all Laguerre isopararmetric hypersurfaces with three distinct principal curvatures in R7.     

A METHOD FOR SOLVING THE INVERSE SCATTERING PROBLEM FOR SHAPE AND IMPEDANCE

The inverse problem considered in this paper is to determine the shape and the impedance of an obstacle from a knowledge of the time-harmonic incident field and the phase and amplitude of the far field pattern of the scattered wave in two-dimension. Single-layer potential is used to approach the scattered waves. An approximation method is presented and the convergence of the proposed method is established. Numerical examples are given to show that this method is both accurate and easy to use.     

Convergence of cascade algorithm for individual initial function and arbitrary refinement masks

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For any initial function φn, a cascade sequence (φn)n∞=1 constructed by the iteration φn=Cnφn-1=1,2.. where Cαis defined by g∈Lp(R) In this paper, we characterize the convergence of a cascade sequence in terms of a sequence of functions and in terms of joint spectral radius. As a consequence, it is proved that any convergent cascade sequence has a convergence rate of geometry, i.e., ||φ 1-φn||Lp(R)=O((？)n)for some (？)∈(0.1i). The condition of sum rules for the mask is not required. Finally, an example is presented to illustrate our theory.     

Complex and p-Adic Meromorphic Functions f′P′( f ),g′P′(g) Sharing a Small Function

Let K be a complete algebraically closed p-adic field of characteristic zero.We apply results in algebraic geometry and a new Nevanlinna theorem for p-adic meromorphic functions in order to prove results of uniqueness in value sharing problems, both on K and on C. Let P be a polynomial of uniqueness for meromorphic functions in K or C or in an open disk. Let f, g be two transcendental meromorphic functions in the whole field K or in C or meromorphic functions in an open disk of K that are not quotients of bounded analytic functions. We show that if f′P′( f) and g′P′(g) share a small function α counting multiplicity, then f = g, provided that the multiplicity order of zeros of P′satisfy certain inequalities. A breakthrough in this paper consists of replacing inequalities n ≥ k+2 or n ≥ k+3 used in previous papers by Hypothesis(G). In the p-adic context, another consists of giving a lower bound for a sum of q counting functions of zeros with(q-1) times the characteristic function of the considered meromorphic function.     

Resolvability in circulant graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d(u, s). A k-partition Π = {S 1 , S 2 , . . . , S k } of V (G) is called a resolving partition if for every two distinct vertices u, v ∈ V (G) there is a set S i in Π such that d(u, Si )≠ d(v, Si ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i-j (mod n) ∈ C , where CZn has the property that C =-C and 0 ■ C. The circulant graph is denoted by Xn, Δ where Δ = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn, 3 with connection set C = {1, n/2 , n-1} and prove that dim(Xn, 3 ) is independent of choice of n by showing that dim(Xn, 3 ) ={3 for all n ≡ 0 (mod 4), 4 for all n ≡ 2 (mod 4). We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C = {±1, ±2} and prove that pd(Xn, 4 ) is independent of choice of n and show that pd(X5,4 ) = 5 and pd(Xn,4 ) ={3 for all odd n ≥ 9, 4 for all even n ≥ 6 and n = 7.     

Catenoidal Layers for the Allen-Cahn Equation in Bounded Domains

This paper presents a new family of solutions to the singularly perturbed Allen-Cahn equation α~2Δu + u(1- u~2) = 0 in a smooth bounded domain Ω R~3, with Neumann boundary condition and α 0 a small parameter. These solutions have the property that as α→ 0, their level sets collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature intersecting ?Ω orthogonally and that is non-degenerate respect to ?Ω. The authors provide explicit examples of surfaces to which the result applies.     

A Generalized Lyapunov-Sylvester Computational Method for Numerical Solutions of NLS Equation with Singular Potential

In the present paper a numerical method is developed to approximate the solution of two-dimensional Nonlinear Schrdinger equation in the presence of a singular potential. The method leads to generalized Lyapunov-Sylvester algebraic operators that are shown to be invertible using original topological and differential calculus issued methods. The numerical scheme is proved to be consistent, convergent and stable using the Lyapunov criterion, lax equivalence theorem and the properties of the generalized Lyapunov-Sylvester operators.