Bifurcation and Stability Theory of Periodic Solutions For Integrodifferential Equations

This paper is concerned with a nonlinear integrodifferential equation (the delay logistic equation) governing the growth dynamics of a single species N(t) for time t > 0. This equation contains a positive parameter λ. Suppose that there exists a positive equilibrium solution N = c which is stable for all small values of λ. Assume also that this solution loses stability as λ is increased past a critical value λ*. This will correspond to a simple pure imaginary conjugate pair of roots of a characteristic equation associated with the linearized stability of N = c at λ = λ*. Then we will construct a unique bifurcating time periodic solution of the equation as a Taylor series in a parameter ε. Furthermore this solution exists either for supercritical values of the parameter (λ > λ*) or for subcritical values (λ < λ*). The stability behavior of this small periodic solution can be characterized according to whether the bifurcation is supercritical or subcritical-supercritical solutions are stable, but subcritical solutions are unstable. Therefore these results are analogous to Hoprs bifurcation theorem for autonomous systems of differential equations.     

Application of a Mickens finite‐difference scheme to the cylindrical Bratu‐Gelfand problem

The boundary value problem Δu + λe^u = 0 where u = 0 on the boundary is often referred to as “the Bratu problem.” The Bratu problem with cylindrical radial operators, also known as the cylindrical Bratu‐Gelfand problem, is considered here. It is a nonlinear eigenvalue problem with two known bifurcated solutions for λ < λ_c, no solutions for λ > λ_c and a unique solution when λ = λ_c. Numerical solutions to the Bratu‐Gelfand problem at the critical value of λ_c = 2 are computed using nonstandard finite‐difference schemes known as Mickens finite differences. Comparison of numerical results obtained by solving the Bratu‐Gelfand problem using a Mickens discretization with results obtained using standard finite differences for λ < 2 are given, which illustrate the superiority of the nonstandard scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 327–337, 2004     

Maximal Probability Inequalities for Stochastic Processes

Let X_a,b be nonnegative random variables with the property that X_a,b ≦ X_a,c + X_c.b for all 0≦ a < c < b ≦ T, where T > 0 is fixed. We define M_a,b = sup {X_a,c: a < c ≦ h} and establish bounds for P[M_a,b ≧ λ] in terms of given bounds for P[X_a,b ≧ λ], where λ runs through some interval (0, λ_o), 0 < λ_o ≦ ∞ fixed. These bounds explicitly involve a nonnegative function g(a, b) assumed to be quasi-superadditive with an index Q, i.e., g(a, c) + g(c, b) ≦ Qg(a, b) for all 0≦ a < c < b < T, where 1 ≦ Q < 2 is fixed. Maximal inequalities obtained in this way can be applied to stochastic processes exhibiting long-range dependence. Among others, these applications may include certain self-similar processes such as fractional Brownian motion, stochastic processes occurring in linear time series models, etc.     

On the limiting behaviour of solutions to boundary integral equations associated with time harmonic wave equations for small frequencies

The treatment of boundary value problems for Helmholtz equation and for the time harmonic Maxwell's equations by boundary integral equations leads to integral equations of the second kind which are uniquely solvable for small positive frequencies λ. However, the integral equations obtained in the limiting case λ = 0 which are related to boundary value problems of potential theory in general are not uniquely solvable since the corresponding boundary value problems are not. By first considering in a general setting of a Banach space X the limiting behaviour of solutions ?_λ to the equation ?_λ – K λ ? λ = fλ as λ → 0 where {K_λ: X → X, λ ∈ (0,α)}, α > 0, denotes a family of compact linear operators such that I - K_λ (I identity) is bijective for λ∈(0,α) whilst I - K₀ is not and ‖ K_λ – K₀‖ →, 0, ‖f_λ – f0‖ → 0, λ → 0, and then applying the results to the boundary integral operators, the limiting behaviour of the integral equations is considered. Thus, the results obtained by Mac Camey for the Helmholtz equation are extended to the case of non-connected boundaries and Werner's results on the integral equations for the Maxwell's equations are extended to the case of multiply connected boundaries.     

Further results on the maximum size of a hole in an incomplete t‐wise balanced design with specified minimum block size*

Kreher and Rees 3 proved that if h is the size of a hole in an incomplete balanced design of order υ and index λ having minimum block size , then, They showed that when t = 2 or 3, this bound is sharp infinitely often in that for each h ≥ t and each k ≥ t + 1, (t,h,k) ≠(3,3,4), there exists an ItBD meeting the bound. In this article, we show that this bound is sharp infinitely often for every t, viz., for each t ≥ 4 there exists a constant C_t > 0 such that whenever (h ? t)(k ? t ? 1) ≥ C_t there exists an ItBD meeting the bound for some λ = λ(t,h,k). We then describe an algorithm by which it appears that one can obtain a reasonable upper bound on C_t for any given value of t. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 256–281, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10014     

Perturbations with several independent parameters

In this paper we discuss the problem of determining a T-periodic solution $\text{x}{}^1\text{(·, λ)}$ of the differential equation x = A(t)x + f(t, x, λ) + b(t), where the perturbation parameter λ is a vector in a parameter-space R^k. The customary approach assumes that λ = λ(?), ??R. One then establishes the existence of an ?₀ > 0 such that the differential equation has a T-periodic solution $\text{x}{}^1\text{(·, λ(?))}$ for all ? satisfying 0 < ? < ?₀. More specifically it is usually assumed that λ(?) has the form λ(?) = ?λ₀ where λ₀ is a fixed vector in R^k. This means that attention is confined in the perturbation procedure to examining the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies along a line segment terminating at the origin in the parameter-space R^k. The results established here generalize this previous work by allowing one to study the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies through a “conical-horn” whose vertex rests at the origin in R^k. In the process an implicit-function formula is developed which is of some interest in its own right.     

Matrices and set intersections

The incidence matrix of a (υ, k, λ)-design is a (0, 1)-matrix A of order υ that satisfies the matrix equation AA^T=(k?λ)I+λJ, where A^T denotes the transpose of the matrix A, I is the identity matrix of order υ, J is the matrix of 1's of order υ, and υ, k, λ are integers such that 0<λ<k<υ?1. This matrix equation along with various modifications and generalizations has been extensively studied over many years. The theory presents an intriguing joining together of combinatorics, number theory, and matrix theory. We survey a portion of the recent literature. We discuss such varied topics as integral solutions, completion theorems, and λ-designs. We also discuss related topics such as Hadamard matrices and finite projective planes. Throughout the discussion we mention a number of basic problems that remain unsolved.     

Generalized Hilbert Functions

Let M be a finite module, and let I be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of I on M using the zeroth local cohomology functor. We show that our definition reconciliates with that of Ciuperc?. By generalizing Singh's formula (which holds in the case of λ(M/IM) < ∞), we prove that the generalized Hilbert coefficients 𝔧₀,…, 𝔧 _d?2 are preserved under a general hyperplane section, where d = dim M. We also keep track of the behavior of 𝔧 _d?1. Then we apply these results to study the generalized Hilbert function for ideals that have minimal j-multiplicity or almost minimal j-multiplicity. We provide counterexamples to show that the generalized Hilbert series of ideals having minimal or almost minimal j-multiplicity does not have the ‘expected’ shape described in the case where λ(M/IM) < ∞. Finally, we give a sufficient condition such that the generalized Hilbert series has the desired shape.     

On c‐Bhaskar Rao designs with block size 4

Several new families of c‐Bhaskar Rao designs with block size 4 are constructed. The necessary conditions for the existence of a c‐BRD (υ,4,λ) are that: (1)λ_min=?λ/3 ≤ c ≤ λ and (2a) c≡λ (mod 2), if υ > 4 or (2b) c≡ λ (mod 4), if υ = 4 or (2c) c≠ λ ? 2, if υ = 5. It is proved that these conditions are necessary, and are sufficient for most pairs of c and λ; in particular, they are sufficient whenever λ?c ≠ 2 for c > 0 and whenever c ? λ_min≠ 2 for c < 0. For c < 0, the necessary conditions are sufficient for υ> 101; for the classic Bhaskar Rao designs, i.e., c = 0, we show the necessary conditions are sufficient with the possible exception of 0‐BRD (υ,4,2)'s for υ≡ 4 (mod 6). © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 361–386, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10009     

Spin depolarization decay rates in α-symmetric stable fields on cubic lattices

We study the asymptotic, long-time behavior of the energy function where {X_s : 0 ≤ s < ∞} is the standard random walk on the d-dimensional lattice Z^d, 1 < α ≤ 2, and f:R⁺ → R⁺ is any nondecreasing concave function. In the special case f(x) = x, our setting represents a lattice model for the study of transverse magnetization of spins diffusing in a homogeneous, α-stable, i.i.d., random, longitudinal field {λV(x) : x ∈ Z^d} with common marginal distribution, the standard α-symmetric stable distribution; the parameter λ describes the intensity of the field. Using large-deviation techniques, we show that S_c(λ α f) = lim_t→∞ E(t; λ f) exists. Moreover, we obtain a variational formula for this decay rate S_c. Finally, we analyze the behavior S_c(λ α f) as λ → 0 when f(x) = x^β for all 1 ≥ β > 0. Consequently, several physical conjectures with respect to lattice models of transverse magnetization are resolved by setting β = 1 in our results. We show that S_c(λ, α, 1) ≈ λ^α for d ≥ 3, λ^agr;(ln 1/λ)^α−1 in d = 2, and in d = 1. © 1996 John Wiley & Sons, Inc.     

Combinatorial coverings from geometries over principal ideal rings

A t-(v, k, λ) covering is an incidence structure with v points, each block incident on exactly k points, such that every set of t distinct points is incident on at least λ blocks. By considering certain geometries over finite principal ideal rings, we construct infinite families of t-(v, k, λ) coverings having many interesting combinatorial properties. © 1999 John & Sons, Inc. J Combin Designs 7: 247–268, 1999     

Eigenvalues and the One-Dimensional p-Laplacian

We consider the boundary value problem (?_p(u′))′ + λF(t, u) = 0, with p > 1, t ∈ (0, 1), u(0) = u(1) = 0, and with λ > 0. The value of λ is chosen so that the boundary value problem has a positive solution. In addition, we derive an explicit interval for λ such that, for any λ in this interval, the existence of a positive solution to the boundary value problem is guaranteed. In addition, the existence of two positive solutions for λ in an appropriate interval is also discussed.     

Time-periodic solutions of wave equations on ℝ1 and ℝ3

The classical Paiey-Wiener theorem and Hilbert space methods are used to show the existence of time-periodic solutions of the wave equation w_tt?w_rr+λw=h, 0 < r < + ∞, which are radially symmetric and have exponential decay as r → + ∞. This problem is obtained when considering a one-dimensional or three-dimensional problem, and should be thought of as a linearization of a semilinear problem in which the associated linear operator has point spectrum (? ∞, λ). When λ ? 0 there is uniqueness, otherwise there is a non-trivial finite-dimensional null space. Estimates on w, w_t, w_r are obtained, which show that in either case there is a continuous correspondence h → w, where w is a uniquely characterized solution.     

On a parabolic equation in MEMS with fringing field

We consider the asymptotic behavior of the solutions to the equation ${u_{t}-u_{xx} = \lambda(1 + {\delta}u_{x}^{2})(1 - u)^{-2}}$ , which comes from Micro-Electromechanical Systems (MEMS) devices modeling. It is shown that when the fringing field exists (i.e., δ?> 0), there is a critical value λ _δ ^* > 0 such that if 0 < λ < λ _δ ^* , the equation has a global solution for some initial data; while for λ > λ _δ ^* , all solutions to the equation will quench at finite time. When the quenching happens, u has only finitely many quenching points for particular initial data. A one-side estimate is deduced for the quenching rate of u.     

A characterization of the tight 3‐sphere II

Recall that every closed, oriented, connected 3‐manifold M admits a contact form λ. We shall give conditions on a periodic orbit P₀ of the Reeb vector field defined by λ that allows the construction of an open book decomposition of M \ P₀ into embedded planes asymptotic to P₀ such that P₀ is the binding orbit of the decomposition. It follows that M is the tight 3‐sphere. As a by‐product, a new dynamical criterion for the tightness of a contact structure is derived in the proof. The results extend earlier results. © 1999 John Wiley & Sons, Inc.     

Slow mixing of Glauber dynamics for the hard‐core model on regular bipartite graphs

Let ∑ = (V,E) be a finite, d‐regular bipartite graph. For any λ > 0 let π_λ be the probability measure on the independent sets of ∑ in which the set I is chosen with probability proportional to λ^|I| (π_λ is the hard‐core measure with activity λ on ∑). We study the Glauber dynamics, or single‐site update Markov chain, whose stationary distribution is π_λ. We show that when λ is large enough (as a function of d and the expansion of subsets of single‐parity of V) then the convergence to stationarity is exponentially slow in |V(∑)|. In particular, if ∑ is the d‐dimensional hypercube {0,1}^d we show that for values of λ tending to 0 as d grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

The Stokes resolvent equations in 2d exterior domains

With methods of hydrodynamical potential theory we construct a solution of the Stokes resolvent equations in 2d exterior domains with C^1,α-boundaries, 0 < α ≤ 1. The resulting system of boundary integral equations is uniquely solvable for small resolvent parameter λ and allows the limiting procedure λ → 0. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A generalization of N-matrices

Ky Fan defines an N-matrix to be a matrix of the form A = tI ? B, B ? 0, λ < t < ?(B), where ?(B) is the spectral radius of B and λ is the maximum of the spectral radii of all principal submatrices of B of order n ? 1. In this paper, we define the closure (N₀-matrices) of N-matrices by letting λ ? t. It is shown that if A ∈ Z and A^-1 < 0, then A ∈ N. Certain inequalities of N-matrices are shown to hold for N₀-matrices, and a method for constructing an N-matrix from an M-matrix is given.     

Small ball probabilities for a Wiener process under weighted sup-norms,with an application to the supremum of bessel local times

LetR be the radial part of ad-dimensional Wiener process, starting from 0. In this paper, small ball probabilities are evaluated for sup_0<11(t ^–p R(t)) and sup _t 0(e ^–1 R(t)), withp[0, 1/2]. Chung's law of the iterated logarithm is established for the supremum of the local times of a two-dimensional Bessel process.     

A contact process with a semi-infected state on the complete graph

In this paper, we are concerned with a contact process with a semi-infected state on the complete graph C_n with n vertices. Our model is a special case of a general model introduced by Schinazi in 2003. In our model, each vertex is in one of three states, namely, “healthy,” “semi-infected,” or “fully-infected.” Only fully-infected vertices can infect others. A healthy vertex becomes semi-infected when being infected while a semi-infected vertex becomes fully-infected when being further infected. Each (semi- and fully-) infected vertex becomes healthy at constant rate. Our main result shows a phase transition for the waiting time until extinction of the fully-infected vertices. Conditioned on all the vertices are fully-infected when t = 0, we show that fully-infected vertices survive for exp?{O(n)} units of time when the infection rate λ > 4 while they die out in O(log?n) units of time when λ < 4.