On the existence of (3,λ)‐semiframes of type 3u

A (k,λ)‐semiframe of type g^u is a (k,λ)‐group‐divisible design of type g^u (??, ??, ??), in which the collection of blocks ?? can be written as a disjoint union ??=??∪?? where ?? is partitioned into parallel classes of ?? and ?? is partitioned into holey parallel classes, each holey parallel class being a partition of ??\G_j for some G_j∈??. In this paper, we shall prove that the necessary conditions for (3,λ)‐semiframes of type 3^u are also sufficient with one exception. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 253–265, 2009     

On –δu = K(x)u5 in ℝ3

In this paper, we study the equation –Δu = K(x)u⁵ in ?³ and provide a large class of positive functions K(x) for which we obtain infinitely many positive solutions which decay at infinity at the rate of |x|^?1. © 1993 John Wiley & Sons, Inc.     

Embeddings of Sλ(2, 4, v)

It is shown that the necessary condition for a given S_λ(2,4,u) to be embedded in some S_λ(2,4,v) as a subdesign, namely v ≥ 3u + 1, is also sufficient for the case of λ = 6. Combining this with the previously known results gives the same sufficiency for any positive integer λ. © 1994 John Wiley & Sons, Inc.     

Asymptotic analysis and estimates of blow‐up time for the radial symmetric semilinear heat equation in the open‐spectrum case

We estimate the blow‐up time for the reaction diffusion equation u_t=Δu+ λf(u), for the radial symmetric case, where f is a positive, increasing and convex function growing fast enough at infinity. Here λ>λ^*, where λ^* is the ‘extremal’ (critical) value for λ, such that there exists an ‘extremal’ weak but not a classical steady‐state solution at λ=λ^* with ∥w(?, λ)∥_∞→∞ as 0<λ→λ^*?. Estimates of the blow‐up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f(s)=e^s, for λ?λ^*?1, regarding the form of the solution during blow‐up and an asymptotic estimate of blow‐up time is obtained. Finally, some numerical results are also presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Family of Pearson discrete distributions generated by the univariate hypergeometric function 3F2(α1, α2, α3; γ1, γ2; λ)

In this work we present a study of the Pearson discrete distributions generated by the hypergeometric function ₃F₂(α₁, α₂, α₃;γ₁, γ₂; λ), a univariate extension of the Gaussian hypergeometric function, through a constructive methodology. We start from the polynomial coefficients of the difference equation that lead to such a function as a solution. Immediately after, we obtain the generating probability function and the differential equation that it satisfies, valid for any admissible values of the parameters. We also obtain the differential equations that satisfy the cumulants generating function, moments generating function and characteristic function, From this point on, we obtain a relation in recurrences between the moments about the origin, allowing us to create an equation system for estimating the parameters by the moment method. We also establish a classification of all possible distributions of such type and conclude with a summation theorem that allows us study some distributions belonging to this family. © 1997 by John Wiley & Sons, Ltd.     

Single‐cell discretization of O(kh2 + h4) for ∂u/∂n for three‐space dimensional mildly quasi‐linear parabolic equation

In this article, using a single computational cell, we report some stable two‐level explicit finite difference approximations of O(kh² + h⁴) for ?u/?n for three‐space dimensional quasi‐linear parabolic equation, where h > 0 and k > 0 are mesh sizes in space and time directions, respectively. When grid lines are parallel to x‐, y‐, and z‐coordinate axes, then ?u/?n at an internal grid point becomes ?u/?x, ?u/?y, and ?u/?z, respectively. The proposed methods are also applicable to the polar coordinates problems. The proposed methods have the simplicity in nature and use the same marching type of technique of solution. Stability analysis of a linear difference equation and computational efficiency of the methods are discussed. The results of numerical experiments are compared with exact solutions. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 327–342, 2003.     

A probabilistic Poisson representation for positive solutions of Δu = u2 in a planar domain

We use the stochastic process called the Brownian snake to investigate solutions of the partial differential equation Δu = u² in a domain D of class C² of the plane. We prove that nonnegative solutions are in one-to-one correspondence with pairs (K, v) where K is a closed subset of ∂D and v is a Radon measure on ∂D\K. Both Kand v are determined from the boundary behavior of the solution u. On the other hand, u can be expressed in terms of the pair (K, v) by an explicit probabilistic representation formula involving the Brownian snake. © 1997 John Wiley & Sons, Inc.     

Global continuation for first order systems over the half‐line involving parameters

Let X be one of the functional spaces W^1,p ((0, ∞), ?^N ) or C₀¹ ([0, ∞), ?^N ), we study the global continuation in λ for solutions (λ, u, ξ) ∈ ? × X × ?^k of the following system of ordinary differential equations: where ?^N = X₁ ⊕ X₂ is a given decomposition, with associated projection P: ?^N → X₁. Under appropriate conditions upon the given functions F and φ, this problem gives rise to a nonlinear Fredholm operator which is proper on the closed bounded subsets of ? × X × ?^k and whose zeros correspond to the solutions of the original problem. Using a new abstract continuation result, based on a recent degree theory for proper Fredholm mappings of index zero, we reduce the continuation problem to that of finding a priori estimates for the possible solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the decay and stability of global solutions to the 3‐D inhomogeneous Navier‐Stokes equations

In this paper, we investigate the large‐time decay and stability to any given global smooth solutions of the 3‐D incompressible inhomogeneous Navier‐Stokes equations. In particular, we prove that given any global smooth solution (a,u) of (1.2), the velocity field u decays to 0 with an explicit rate, which coincides with the L² norm decay for the weak solutions of the 3‐D classical Navier‐Stokes system [26,29] as t goes to ∞. Moreover, a small perturbation to the initial data of (a,u) still generates a unique global smooth solution to (1.2), and this solution keeps close to the reference solution (a,u) for t > 0. We should point out that the main results in this paper work for large solutions of (1.2). © 2010 Wiley Periodicals, Inc.     

Singly Periodic Solutions of the Allen-Cahn Equation and the Toda Lattice

The Allen-Cahn equation ? Δu = u ? u ³ in ?² has family of trivial singly periodic solutions that come from the one dimensional periodic solutions of the problem ?u″ =u ? u ³. In this paper we construct a non-trivial family of singly periodic solutions to the Allen-Cahn equation. Our construction relies on the connection between this equation and the infinite Toda lattice. We show that for each one-soliton solution to the infinite Toda lattice we can find a singly periodic solution to the Allen-Cahn equation, such that its level set is close to the scaled one-soliton. The solutions we construct are analogues of the family of Riemann minimal surfaces in ?³.     

A Tukey decomposition of 𝒫κλ and the tree property for directed sets

Generalizing a result of Todor?evi?, we prove the existence of directed sets D, E such that D ? ??_κλ and E ? ??_κλ but D × E ≥ ??_κλ in the Tukey ordering. As an application, we show that the tree property for directed sets introduced by Hinnion is not preserved under products. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence to a reaction–diffusion equation through a self‐similar‐like upper solution

A demonstration method is presented, which will ensure the existence of positive global solutions in time to the reaction–diffusion equation ?u_t+Δu+u^p=0 in ?ⁿ×[0, ∞), for exponents p?3 and space dimensions n?3. This method does not require the initial value to have a specific uniform smallness condition, but rather to satisfy a bell‐like form. The method is based on a specific upper solution, which models the diffusion process of the heat equation. The upper solution is not self‐similar, but does have a self‐similar‐like form. After transforming the reaction–diffusion problem into an equivalent one, whose initial value is uniformly very small, a local solution is obtained in the time interval [0, 1] by the use of this upper solution. This local solution is then extended to [0, ∞) through an infinite sequence of extensions. At each step, an appropriate change of variables will transform the extension into a problem nearly identical to the local problem in [0, 1]. These transformations exploit the diffusive and self‐similar‐like nature of the upper solution. Copyright © 2009 John Wiley & Sons, Ltd.     

Behaviour of (−Δ−k2−i0+)−1 outside fading obstacles,independant scattering hypothesis and applications

This paper deals with the behaviour of k‐outgoing solutions of ?Δu?k²u=f outside a fading soft obstacle. We extend an approach using the so‐called Lax–Phillips construction and the well‐known properties of the capacity of smooth obstacles. So, classical results are recovered in a straightforward manner. The previous approach enables us to consider the case of obstacles composed of many tiny spheres. Roughly speaking, we prove that the scattering amplitude is approximately the sum of the scattering amplitudes scattered by each isolated sphere, which is an alternative form of the first Born approximation. As a consequence, two inverse problems are solved. Copyright © 2003 John Wiley & Sons, Ltd.     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

Refined asymptotics for the blowup of ut — δu = up

This work is concerned with positive, blowing-up solutions of the semilinear heat equation u_t — δu = u^p in Rⁿ. Our main contribution is a sort of center manifold analysis for the equation in similarity variables, leading to refined asymptotics for u in a backward space-time parabola near any blowup point. We also explore a connection between the asymptotics of u and the local geometry of the blowup set.     

Certain classes of potentials for p ‐Laplacian to be non‐degenerate

Given a positive integer n and an exponent 1 ≤ α ≤ ∞. We will find explicitly the optimal bound r_n such that if the L^α norm of a potential q (t ) satisfies ‖q ‖equation/tex2gif-inf-2.gif < r_n then the n ^th Dirichlet eigenvalue of the onedimensional p ‐Laplacian with the potential q (t ): (|u ′|^{p –2} u ′)′ + (λ + q (t )) |u |^{p –2}u = 0 (1 < p < ∞) will be positive. Using these bounds, we will construct, for the Dirichlet, the Neumann, the periodic or the antiperiodic boundary conditions, certain classes of potentials q (t ) so that the p ‐Laplacian with the potential q (t ) is non‐degenerate, which means that the above equation with λ = 0 has only the trivial solution verifying the corresponding boundary condition. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Morse homology for the heat flow – Linear theory

Consider the linear parabolic partial differential equation ${\mathcal {D}}_u\xi =0$ which arises by linearizing the heat flow on the loop space of a Riemannian manifold M. The solutions are vector fields along infinite cylinders u in M. For these solutions we establish regularity and a priori estimates. We show that for nondegenerate asymptotic boundary conditions the solutions decay exponentially in L² in forward and backward time. In this case ${\mathcal {D}}_u$ viewed as linear operator from the parabolic Sobolev space ${\mathcal {W}}^{1,p}$ to L^p is Fredholm whenever p > 1. We close with an L^p estimate for products of first order terms which is a crucial ingredient in the sequel 13 to prove regularity and the implicit function theorem. The results of the present text are the base to construct in 13 an algebraic chain complex whose homology represents the homology of the loop space.     

On the Logistic Equation in the Complex Plane

The famous logistic differential equation is studied in the complex plane. The method used is based on a functional analytic technique which provides a unique solution of the ordinary differential equation (ODE) under consideration in H ₂(𝔻) or H ₁(𝔻) and gives rise to an equivalent difference equation for which a unique solution is established in ?₂ or ?₁. For the derivation of the solution of the logistic differential equation this discrete equivalent equation is used. The obtained solution is analytic in {z ∈ ?: |z| <T}, T > 0. Numerical experiments were also performed using the classical 4th order Runge–Kutta method. The obtained results were compared for real solutions as well as for solutions of the form y(t) = u(t) + iv(t), t ∈ ?. For t ∈ ? the solution derived by the present method, seems to have singularities, that is, points where it ceases to be analytic, in certain sectors of the complex plane. These sectors, depending on the values of the involved parameters, can move at different directions, join forming common sectors, or pass through each other and continue moving independently. Moreover, the real and imaginary part of the solution seem to exhibit oscillatory behavior near these sectors.     

Generalized Hopf bifurcation in Hilbert space

We consider a family of semilinear evolution equations in Hilbert space of the form with, in general, unbounded operators *A(λ), F(λ·) depending analytically on a real parameter λ. We assume that the origin is a stationary solution, i.e. F(λ,0) = 0, for all λ ε R and that the linearization (with respect to u) at the origin is given by du/dt + A(λ)u = 0. Our essential assumption is the following: A(λ) possesses one pair of simple complex conjugate eigenvalues μ(λ) = Re μ(λ) ± i Im μ(λ) such that Im μ(0) > 0 and for some m ε N or If m = 1 the curves of eigenvalues μ(λ) cross the imaginary axis transversally at ±i Im μ(0). In this case a unique branch of periodic solutions emanates from the origin at λ = 0 which is commonly called Hopf bifurcation. If μ(λ) and the imaginary axis are no longer transversal, i.e. m > 1, we call a bifurcation of periodic solutions, if it occurs, a generalized Hopf bifurcation. It is remarkable that up to m such branches may exist. Our approach gives the number of bifurcating solutions, their direction of bifurcation, and its asymptotic expansion. We regain the results of D. Flockerzi who established them in a completely different way for ordinary differential equations.     

Sufficient conditions for λ′‐optimality in graphs with girth g

For a connected graph the restricted edge‐connectivity λ′(G) is defined as the minimum cardinality of an edge‐cut over all edge‐cuts S such that there are no isolated vertices in G–S. A graph G is said to be λ′‐optimal if λ′(G) = ξ(G), where ξ(G) is the minimum edge‐degree in G defined as ξ(G) = min{d(u) + d(v) ? 2:uv ∈ E(G)}, d(u) denoting the degree of a vertex u. A. Hellwig and L. Volkmann [Sufficient conditions for λ′‐optimality in graphs of diameter 2, Discrete Math 283 (2004), 113–120] gave a sufficient condition for λ′‐optimality in graphs of diameter 2. In this paper, we generalize this condition in graphs of diameter g ? 1, g being the girth of the graph, and show that a graph G with diameter at most g ? 2 is λ′‐optimal. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 73–86, 2006