Leray-Schauder Degree Method in Functional Boundary Value Problems Depending on the Parameter

Using the Leray-Schauder degree method sufficient conditions for the one-parameter boundary value problem x″ = f(t, x, x′, λ), α(x) = A, x(0) ? x(1) = B, x′(0) ? x′(1) = C, are stated. The application is given for a class of functional boundary value problems for nonlinear third-order functional differential equations depending on the parameter.     

Quasi‐symmetric designs with fixed difference of block intersection numbers

The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x,y and λ > 1 are proved.

• (1) Let D be a quasi‐symmetric design with z = y ? x and v ≥ 2k. If x ≥ 1 + z + z³ then λ < x + 1 + z + z³.
• (2) Let D be a quasi‐symmetric design with intersection numbers x, y and y ? x = 1. Then D is a design with parameters v = (1 + m) (2 + m)/2, b = (2 + m) (3 + m)/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2.
• (3) Let D be a triangle free quasi‐symmetric design with z = y ? x and v ≥ 2k, then x ≤ z + z².
• (4) For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x, y = x + z.
• (5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x, y = x + 2.

© 2006 Wiley Periodicals, Inc. J Combin Designs 15: 49–60, 2007     

Quasi-periodic solutions of a semilinear Liénard equation at resonance

We are concerned with the existence of quasi-periodic solutions for the following equation

A generalization of Fermat's little theorem

Fermat's Little Theorem states that if p is a prime number and gcd (x,p) = 1, then x^p?1 ≡ 1 (modp) If the requirement that gcd (x,p) = 1 is dropped, we can say x^p ≡ x(modp)for any integer x. Euler generalized Fermat's Theorem in the following way: if gcd (x,n) = 1 then x^φ(n) ≡ 1(modn), where φ is the Euler phi-function. It is clear that Euler's result cannot be extended to all integers x in the same way Fermat's Theorem can; that is, the congruence x^φ(n)+1 ≡ x(modn)is not always valid. In this note we show exactly when the congruence x^φ(n)+1 ≡ x(modn) is valid.     

Classification of all noncommutative polynomials whose Hessian has negative signature one and a noncommutative second fundamental form

Every symmetric polynomial p = p(x) = p(x ₁,..., x _g ) (with real coefficients) in g noncommuting variables x ₁,..., x _g can be written as a sum and difference of squares of noncommutative polynomials:

On integral equations of stationary distributions for biological systems

In this paper, properties of solutions of the convolution-type integral equation

Positive periodic solutions and eigenvalue intervals for systems of second order differential equations

In this paper, we employ a well‐known fixed point theorem for cones to study the existence of positive periodic solutions to the n ‐dimensional system x ″ + A (t)x = H (t)G (x). Moreover, the eigenvalue intervals for x ″ + A (t)x = λH (t)G (x) are easily characterized. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of Initial Value Problems with Monogenic Initial Functions in Banach Spaces with L p -Norm

This paper deals with the initial value problem of the type

Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

For x = (x ₁, x ₂, …, x _n ) ∈ (0, 1 ] ⁿ and r ∈ { 1, 2, … , n}, a symmetric function F _n (x, r) is defined by the relation

On complete recovery of geophysical data

One considers an elastic halfspace with depth (x₁) dependent density Q and Lamé moduli (λ,m?). Impulsive stresses τli = δ(x₂, x₃)δ(t) for x₁ = 0 are applied with displacement responses u_i = g_i(t, x₂, x₃) at x₁ = 0 (i = 1, 2, 3). Let v_i(t, x₁) = ∫∫u_idx₂dx₃ (i = 1, 2 is enough) and set w(t, x₁) = ∫∫x₂u₁dx₂dx₃. One obtains a system of 3 differential equations for v₁, v₂, and w to which the spectral techniques of inverse scattering theory are applied as in [25]. The inverse problems for the uncoupled v_i can then be solved to produce 2 functions A₁ and A₂ involving (Q, λ, μ) as functions of “bound” variables y₁ and y₂ containing (Q, λ, μ), between which a relation then is determined. Analysis of the coupled equation for w then leads to a Fredholm integral equation whose solution provides an additional relation between (Q, λ, μ) from which (Q, λ, μ) can be determined as functions of x₁. The integral equation is reduced to a Volterra type equation by results and techniques of transmutation and then solved by a modification of standard techniques. A number of features and results of independent mathematical interest arise from the transmutation theory.     

A single groupoid identity for Steiner loops

A loop which satisfies the identitiesx ² =e, xe = ex = x, andx(yx) = (xy) x = y is called a generalized Steiner loop. In this paper it is shown that a generalized Steiner loop is a groupoid with a single lawx(((yy) z) x) = z.     

From Chaos to Global Convergence

The main purpose of this paper is to investigate dynamical systems F : \mathbbR² ? \mathbbR²{F : \mathbb{R}^2 \rightarrow \mathbb{R}^2} of the form F(x, y) = (f(x, y), x). We assume that f : \mathbbR² ? \mathbbR{f : \mathbb{R}^2 \rightarrow \mathbb{R}} is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x₀ = (x ₀, y ₀), such that the orbit

Positive solutions of non‐positone Dirichlet boundary value problems with singularities in the phase variables

The paper presents existence results for positive solutions of the differential equations x ″ + μh (x) = 0 and x ″ + μf (t, x) = 0 satisfying the Dirichlet boundary conditions. Here μ is a positive parameter and h and f are singular functions of non‐positone type. Examples are given to illustrate the main results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Diophantine Equations (2 n − 1)(6 n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2

In this paper we prove that the equation (2 ⁿ – 1)(6 ⁿ – 1) = x ² has no solutions in positive integers n and x. Furthermore, the equation (a ⁿ – 1) (a ^kn – 1) = x ² in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).     

Periodic and Homoclinic Solutions of Extended Fisher–Kolmogorov Equations

In this paper we study the existence of periodic solutions of the fourth-order equations u^iv − pu″ − a(x)u + b(x)u³ = 0 and u^iv − pu″ + a(x)u − b(x)u³ = 0, where p is a positive constant, and a(x) and b(x) are continuous positive 2L-periodic functions. The boundary value problems (P₁) and (P₂) for these equations are considered respectively with the boundary conditions u(0) = u(L) = u″(0) = u″(L) = 0. Existence of nontrivial solutions for (P₁) is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for (P₂) is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation u^iv + pu″ + a(x)u − b(x)u² − c(x)u³ = 0, where p is a constant, and a(x), b(x), and c(x) are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentration-compactness arguments are used.     

On an infinite class of non-bipartite and non-cayley graphs having 2-arc transitive automorphism groups

Letq be a prime of the formq = 40x + 13,q = 40x + 27,q = 40x + 37, orq = 40x + 43. Then a connected, undirected, 4-valent, non-bipartite graph on whichPSL ₂ (q) acts 2-arc transitively is non-Cayley.     

The conformal plate buckling equation

The linear equation Δ²u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?² has the particularly interesting nonlinear generalization Δ_g²u = 1, where Δ_g = e^?2u Δ is the Laplace‐Beltrami operator for the metric g = e^2ug₀, with g₀ the standard Euclidean metric on ?². This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+K_g(x) exp(2u(x)) = 0 and Δ K_g(x) + exp(2u(x)) = 0, with x ∈ ?², describing a conformally flat surface with a Gauss curvature function K_g that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ K_g exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K₊ ∈ L¹(B₁(y)), uniformly w.r.t. y ∈ ?². We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K^* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K^*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc.     

Exchange of Stabilities in Autonomous Systems

Equilibrium solutions y = ?(x) of an autonomous system of differential equations, depending on a parameter x, are considered. Bifurcation of a second family of solutions y = ψ(x) and exchange of stabilities is supposed to occur at (x,y) = (0,0). Considering x as slowly varying leads to a singularly perturbed initial-value problem whose reduced path encounters a point of bifurcation. Rigorous asymptotic estimates are found for the difference between the (unique) solution of the full problem and that solution of the reduced problem which proceeds along stable segments of the reduced path.     

Self‐converse Mendelsohn designs with block size 4 and 5

A (v, k, λ)‐Mendelsohn design(X, ℬ︁) is called self‐converse if there is an isomorphic mapping ƒ from (X, ℬ︁) to (X, ℬ︁⁻¹), where ℬ︁⁻¹ = {B⁻¹ = 〈x_k, x_k−1,…,x₂, x₁〉: B = 〈x₁, x₂,…,x_k−1, x_k〉 ϵ ℬ︁}. In this paper, we give the existence spectrum for self‐converse (v, 4, 1)– and (v, 5, 1)– Mendelsohn designs. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 411–418, 2000