Interior transition layers of solutions to the perturbed elliptic Sine-Gordon equation on an interval

We consider the perturbed elliptic Sine-Gordon equation on an interval-u″t+γsinu(t)=μf(u(t)),t ∈I := (-T, T),u(t) > 0,t ∈I,u(±T)=0 where λ, μ>0 are parameters andT>0 is a constant. By applying variational methods subject to the constraint depending on λ, we obtain eigenpairs (μ,u)=(μ(λ),u _λ) which solve this eigenvalue problem for a given λ>0. Then we study the asymptotic behavior ofu _λ and μ(λ) as λ→∞. Especially, we study the location of interior transition layers ofu _λ as λ→∞. This research has been supported by the Japan Society for the Promotion of Science.     

Asymptotic relation for the density of a multidimensional random evolution with rare poisson switchings

A symmetric random evolution X(t) = (X ₁ (t), …, X _m (t)) controlled by a homogeneous Poisson process with parameter λ > 0 is considered in the Euclidean space ℝ ^m , m ≥ 2. We obtain an asymptotic relation for the transition density p(x, t), t > 0, of the process X(t) as λ → 0 and describe the behavior of p(x, t) near the boundary of the diffusion domain in spaces of different dimensions. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1631 – 1641, December, 2008.     

Stability Theory of General Contractions for Delay Equations

We study the persistence of the asymptotic stability of delay equations both under linear and nonlinear perturbations. Namely, we consider nonautonomous linear delay equations v′ = L(t)v _t with a nonuniform exponential contraction. Our main objective is to establish the persistence of the nonuniform exponential stability of the zero solution both under nonautonomous linear perturbations, i.e., for the equation v′ = (L(t) + M(t))v _t , thus discussing the so-called robustness problem, and under a large class of nonlinear perturbations, namely for the equation v′ = L(t)v _t + f(t, v _t ). In addition, we consider general contractions e ^−λρ(t) determined by an increasing function ρ that includes the usual exponential behavior with ρ(t) = t as a very special case. We also obtain corresponding results in the case of discrete time.     

Selection from poisson processes

Summary There are givenk Poisson processes with mean arrival times 1/λ₁,...1/λ _k . Let λ_[1]≦λ_[2]≦...≦λ_[k] denote the ordered set of values λ₁...,λ_[k]. We consider three procedures for selecting the process corresponding to λ_[k]. The processes are observed until there areN arrivals from any of the given processes, when the processes are observed continuously, or until there are at leastN arrivals, when the processes are observed at successive intervals of time whereN is a pre-determined positive integer. In the continuous case, the process for which theNth arrival time is shortest, is selected. In the discrete case, the selection involves certain randomization. Given (λ_[k]/λ_[k-1])≧0>1, it is shown that the probability of a correct selection (Pcs) is minimized whenθλ_[1]=θλ_[2]=...=θλ_[k-1]=θλ_[k]=θλ, say. The Pcs for this configuration is independent of λ for two of the given procedures, and monotone increasing in λ for the third. The value ofN is determined by a lower bound placed on the value of the Pcs. The problem of selecting from given Poisson processes for the discrete case is related to the problem of selecting from given Poisson populations. An application of the given procedures to a problem of selecting the “most probable event” from a multinomial population, is considered.     

Transient behavior ofM/M ij/1 queues

We obtain the time dependent probabilities for the joint distribution of the number of arrivals and departures in [0,t] for theM/M ^ij/1 queue. This queue has the exponential service with parametersμ _ij, depending on the types of the successive customers attended. We provide an intuitive interpretation of the solution and also present some numerical results, including time dependent event probabilities and queue length.     

Convergence analysis of the projective scaling algorithm based on a long-step homogeneous affine scaling algorithm

In this paper we give a new convergence analysis of a projective scaling algorithm. We consider a long-step affine scaling algorithm applied to a homogeneous linear programming problem obtained from the original linear programming problem. This algorithm takes a fixed fraction λ≤2/3 of the way towards the boundary of the nonnegative orthant at each iteration. The iteration sequence for the original problem is obtained by pulling back the homogeneous iterates onto the original feasible region with a conical projection, which generates the same search direction as the original projective scaling algorithm at each iterate. The recent convergence results for the long-step affine scaling algorithm by the authors are applied to this algorithm to obtain some convergence results on the projective scaling algorithm. Specifically, we will show (i) polynomiality of the algorithm with complexities of O(nL) and O(n ² L) iterations for λ<2/3 and λ=2/3, respectively; (ii) global covnergence of the algorithm when the optimal face is unbounded; (iii) convergence of the primal iterates to a relative interior point of the optimal face; (iv) convergence of the dual estimates to the analytic center of the dual optimal face; and (v) convergence of the reduction rate of the objective function value to 1−λ.     

On the multiplicity of the eigenvalues of a graph

Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q ₁(t)q ₂(t)² ... q _m (t)^m, where each q _i (t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of q _j (t). We give an algorithm to construct the polynomials q _i (t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) = |λ_i| of G, where λ₁, λ₂, ..., λ_n are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. This work was done during a visit of the second named author to UNAM.     

Asymptotic analysis of the M/G/1 queue with a time‐dependent arrival rate

We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(εt) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p _n(t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= εt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching. This revised version was published online in June 2006 with corrections to the Cover Date.     

Growth and Hölder conditions for the sample paths of Feller processes

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C _c ^∞(ℝ ⁿ )⊂D(A) and A|C _c ^∞(ℝ ⁿ ) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|_∞≤c(1+|ξ|²) and |Imp(x,ξ)|≤c ₀Rep(x,ξ). We show that the associated Feller process {X _t } _{t ≥0} on ℝ ⁿ is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β_∞ ^x :={λ>0:lim _{|ξ|→∞ | x − y |≤2/|ξ|}|p(y,ξ)|/|ξ|^λ=0} or δ_∞ ^x :={λ>0:liminf _{|ξ|→∞ | x − y |≤2/|ξ| |ε|≤1}|p(y,|ξ|ε)|/|ξ|^λ=0}, and obtain a.s. (ℙ ^x ) that lim _{t →0} t ^−1/λ _{s ≤ t} |X _s −x|=0 or ∞ according to λ>β_∞ ^x or λ<δ_∞ ^x . Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27]. Received: 21 July 1997 / Revised version: 26 January 1998     

On the replications of certain multiplicative designs

Bounds on the number of row sums in ann×n, non-singular (0,1)-matrixA sarisfyingA ^tA=diag (k ₁-λ₁,…,k _n-λ_n),k _j>λ_j>0,λ₁=…=λ_e,λ_e+1=…=λ_n are obtained which extend previous results for such matrices.     

On a class of semilinear weakly hyperbolic equations

Abstract  In this paper, we deal with some global existence results for the large data smooth solutions of the Cauchy Problem associated with the semilinear weakly hyperbolic equations

Hölder continuity for spatial and path processes via spectral analysis

For ν(dθ), a σ-finite Borel measure on R ^d , we consider L ²(ν(dθ))-valued stochastic processes Y(t) with te property that Y(t)=y(t,·) where y(t,θ)=∫ ^t ₀ e ^{−λ(θ)( t − s )} dm(s,θ) and m(t,θ) is a continuous martingale with quadratic variation [m](t)=∫ ^t ₀ g(s,θ)ds. We prove timewise H?lder continuity and maximal inequalities for Y and use these results to obtain Hilbert space regularity for a class of superrocesses as well as a class of stochastic evolutions of the form dX=AXdt+GdW with W a cylindrical Brownian motion. Maximal inequalities and H?lder continuity results are also provenfor the path process _t (τ)≗Y(τt∧t). Received: 25 June 1999 / Revised version: 28 August 2000 /?Published online: 9 March 2001     

Generalization of the perron effect whereby the characteristic exponents of all solutions of two differential systems change their sign from negative to positive

We obtain a generalization of the complete Perron effect whereby the characteristic exponents of all solutions change their sign from negative for the linear approximation system to positive for a nonlinear system with perturbations of higher-order smallness [Differ. Uravn., 2010, vol. 46, no. 10, pp. 1388–1402]. Namely, for arbitrary parameters λ ₁ ≤ λ ₂ < 0 and m > 1 and for arbitrary intervals [b _i , d _i ) ⊂ [λ _i ,+∞), i = 1, 2, with boundaries d ₁ ≤ b ₂, we prove the existence of (i) a two-dimensional linear differential system with bounded coefficient matrix A(t) infinitely differentiable on the half-line t ≥ 1 and with characteristic exponents λ ₁(A) = λ ₁ ≤ λ ₂(A) = λ ₂ < 0; (ii) a perturbation f(t, y) of smallness order m > 1 infinitely differentiable with respect to time t > 1 and continuously differentiable with respect to y ₁ and y ₂, y = (y ₁, y ₂) ∈ R ² such that all nontrivial solutions y(t, c), c ∈ R ², of the nonlinear system .y = A(t)y + f(t, y), y ∈ R ², t ≥ 1, are infinitely extendible to the right and have characteristic exponents λ[y] ∈ [b ₁, d ₁) for c ₂ = 0 and λ[y] ∈ [b ₂, d ₂) for c ₂ ≠ 0.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

Busy period,virtual waiting time and number of customers in G<Superscript><Emphasis Type="Italic">δ</Emphasis></Superscript>|M<Superscript><Emphasis Type="Italic">ϰ</Emphasis></Superscript>|1|B system

In this article, we determine the integral transforms of several two-boundary functionals for a difference of a compound Poisson process and a compound renewal process. Another part of the article is devoted to studying the above-mentioned process reflected at its infimum. We use the results obtained to study a G ^δ |M ^ϰ |1|B system with batch arrivals and finite buffer in the case when δ∼ge(λ). We derive the distributions of the main characteristics of the queuing system, such as the busy period, the time of the first loss of a customer, the number of customers in the system, the virtual waiting time in transient and stationary regimes. The advantage is that these results are given in a closed form, namely, in terms of the resolvent sequences of the process.     

Riesz transforms for a non-symmetric Ornstein-Uhlenbeck semigroup

Let (ℋ _t ) _t≥0 be the Ornstein-Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix −λ(I+R), where λ>0 and R is a skew-adjoint matrix and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L ²(γ _∞). We investigate the weak type 1 estimate of the Riesz transforms for (ℋ _t ) _t≥0. We prove that if the matrix R generates a one-parameter group of periodic rotations then the first order Riesz transforms are of weak type 1 with respect to the invariant measure γ _∞. We also prove that the Riesz transforms of any order associated to a general Ornstein-Uhlenbeck semigroup are bounded on L ^p (γ _∞) if 1<p<∞. The authors have received support by the Italian MIUR-PRIN 2005 project “Harmonic Analysis” and by the EU IHP 2002-2006 project “HARP”.     

The convolution structure for Jacobi function expansions

The product ϕ _λ ^(α,β) (t₁)ϕ _λ ^(α,β) (t₂) of two Jacobi functions is expressed as an integral in terms of ϕ _λ ^(α,β) (t₃) with explicit non-negative kernel, when α≧β≧−1/2. The resulting convolution structure for Jacobi function expansions is studied. For special values of α and β the results are known from the theory of symmetric spaces.     

A new class of 3-fold perfect splitting authentication codes

Restricted strong partially balanced t-designs were first formulated by Pei, Li, Wang and Safavi-Naini in investigation of authentication codes with arbitration. In this article, we will prove that splitting authentication codes that are multi-fold perfect against spoofing can be characterized in terms of restricted strong partially balanced t-designs. We will also investigate the existence of restricted strong partially balanced 3-designs RSPBD 3-(v, b, 3 × 2; λ₁, λ₂, 1, 0)s, and show that there exists an RSPBD 3-(v, b, 3 × 2; λ₁, λ₂, 1, 0) for any v o 9 (mod 16){v\equiv 9\ (\mbox{{\rm mod}}\ 16)} . As its application, we obtain a new infinite class of 3-fold perfect splitting authentication codes.     

Constrained energy problems with applications to orthogonal polynomials of a discrete variable

Given a positive measure Σ with gs > 1, we write Με ℳ^Σ if Μ is a probability measure and Σ—Μ is a positive measure. Under some general assumptions on the constraining measure Σ and a weight functionw, we prove existence and uniqueness of a measure λ^Σ _w that minimizes the weighted logarithmic energy over the class ℳ^Σ. We also obtain a characterization theorem, a saturation result and a balayage representation for the measure λ^Σ _w As applications of our results, we determine the (normalized) limiting zero distribution for ray sequences of a class of orthogonal polynomials of a discrete variable. Explicit results are given for the class of Krawtchouk polynomials. The research done by this author is in partial fulfillment of the Ph.D. requirements at the University of South Florida. The research done by this author was supported, in part, by U.S. National Science Foundation under grant DMS-9501130.