Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations

This paper studies the existence of a positive solution to the second-order periodic boundary value problem

Relationships between characteristics in periodic Poisson queues

We study single server periodic queues in the day equilibrium conditions. The following characteristics of interest are considered at time of dayt: V^p(t)-the work load, L^p(t)-the number of customers and u^p(t)-the departure rate. We give relationships between E[V^p(t)], E[L^p(t)] and u^p(t). We also prove that E[V^p(t)] < and E[L^p(t)] < provided the second moment of the service time is finite.     

Singular spherical maximal operators on a class of two step nilpotent lie groups

LetH ⁿ ≅ℝ²ⁿ ⋉ℝ be the Heisenberg group and letμ _t be the normalized surface measure for the sphere of radiust in ℝ²ⁿ . Consider the maximal function defined byM f=sup _t>0|f*μ _t |. We prove forn≥2 thatM defines an operator bounded onL ^p (H ⁿ ) provided thatp>2n/(2n−1). This improves an earlier result by Nevo and Thangavelu, and the range forL ^p boundedness is optimal. We also extend the result to a more general class of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type. The second author was supported in part by the National Science Foundation.     

ON Sums of Products of Polynomials

Abstract

Given a polynomial P(t₁ ,…, t _n) = σ a_a t_a ^a1 …t_n ^an in several variables, we consider the p-norms |P|_p = (σ |a_a | ^p )^1/p (1≥ p < ∞) and |p|_∞ = max |a_a |. Our goal is to establish a generalization to the p-norms (1 ≥ p ≥ ∞) of a theorem originally obtained by P. Enflo for the l-norm.     

On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu _t=u_xx, u_t=(u^m)_xxand (m>1) forx>0,t>0 with nonlinear boundary conditions−u _x=u^p,−(u ^m)_x=u^pand forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp ₀,p_c(withp ₀<p_c)such that forp∃(0,p ₀],all solutions are global while forp∃(p₀,p_c] any solutionu≢0 blows up in a finite time and forp>p _csmall data solutions exist globally in time while large data solutions are nonglobal. We havep _c=2,p _c=m+1 andp _c=2m for each problem, whilep ₀=1,p ₀=1/2(m+1) andp ₀=2m/(m+1) respectively. This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.     

The Gaussian cotype of operators fromC(K)

We show that the canonical embeddingC(K) →L _Φ(μ) has Gaussian cotypep, where μ is a Radon probability measure onK, and Φ is an Orlicz function equivalent tot ^p(logt) ^p/2 for larget.     

Littlewood–Paley theorem on spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L ^p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L ^p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L ^p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L ^p(t) (ℝ ⁿ ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ ⁿ , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.     

p-Power Points and Modules of Constant p-Power Jordan Type

We study finitely generated modules over k[G] for a finite abelian p-group G, char(k) = p, through restrictions to certain subalgebras of k[G]. We define p-power points, shifted cyclic p-power order subgroups of k[G], and give characterizations of these. We define modules of constant p ^t -Jordan type, constant p ^t -power-Jordan type as generalizations of modules of constant Jordan type, and p ^t -support, nonmaximal p ^t -support spaces. We obtain a filtration of modules of constant Jordan type with modules of constant p-power Jordan type as the last term and give examples of non-isomorphic modules of constant p-power Jordan type having the same constant Jordan type.     

Elliptic operators with general Wentzell boundary conditions,analytic semigroups and the angle concavity theorem

We prove a very general form of the Angle Concavity Theorem, which says that if (T (t)) defines a one parameter semigroup acting over various L^p spaces (over a fixed measure space), which is analytic in a sector of opening angle θ_p, then the maximal choice for θ_p is a concave function of 1 – 1/p. This and related results are applied to give improved estimates on the optimal L^p angle of ellipticity for a parabolic equation of the form ?u /?t = Au, where A is a uniformly elliptic second order partial differential operator with Wentzell or dynamic boundary conditions. Similar results are obtained for the higher order equation ?u /?t = (–1)^{m +l}A^mu, for all positive integers m.     

Oscillation and Nonoscillation for Second Order Linear Differential Equations

New oscillation and nonoscillation theorems are obtained for the second order linear differential equationu″ + p(t)u = 0, wherep(t) ∈ C[0, ∞) andp(t) ≥ 0. Conditions only about the integrals ofp(t) on every interval [2ⁿt₀, 2^{n + 1}t₀] (n = 1, 2,…) for some fixedt₀ > 0 are used in the results.     

Thermal avalanche for blowup solutions of semilinear heat equations

We consider the semilinear heat equation u_t = Δu + u^p both in ?^N and in a bounded domain with homogeneous Dirichlet boundary conditions, with 1 < p < p_s where p_s is the Sobolev exponent. This problem has solutions with finite‐time blowup; that is, for large enough initial data there exists T < ∞ such that u is a classical solution for 0 < t < T, while it becomes unbounded as t ↗ T. In order to understand the situation for t > T, we consider a natural approximation by reaction problems with global solution and pass to the limit. As is well‐known, the limit solution undergoes complete blowup: after it blows up at t = T, the continuation is identically infinite for all t > T. We perform here a deeper analysis of how complete blowup occurs. Actually, the singularity set of a solution that blows up as t ↗ T propagates instantaneously at time t = T to cover the whole space, producing a catastrophic discontinuity between t = T^? and t = T⁺. This is called the “avalanche.” We describe its formation as a layer appearing in the limit of the natural approximate problems. After a suitable scaling, the initial structure of the layer is given by the solution of a limit problem, described by a simple ordinary differential equation. As t proceeds past T, the solutions of the approximate problems have a traveling wave behavior with a speed that we compute. The situation in the inner region depends on the type of approximation: a conical pattern is formed with time when we approximate the power u^p by a flat truncation at level n, while for tangent truncations we get an exponential increase in time and a diffusive spatial pattern. © 2003 Wiley Periodicals, Inc.     

Covering arrays of strength 3 and 4 from holey difference matrices

A covering array CA(N; t, k, v) is an N × k array with entries from a set X of v symbols such that every N × t sub-array contains all t-tuples over X at least once, where t is the strength of the array. The minimum size N for which a CA(N; t, k, v) exists is called the covering array number and denoted by CAN(t, k, v). Covering arrays are used in experiments to screen for interactions among t-subsets of k components. One of the main problems on covering arrays is to construct a CA(N; t, k, v) for given parameters (t, k, v) so that N is as small as possible. In this paper, we present some constructions of covering arrays of strengths 3 and 4 via holey difference matrices with prescribed properties. As a consequence, some of known bounds on covering array number are improved. In particular, it is proved that (1) CAN(3, 5, 2v) ≤ 2v ²(4v + 1) for any odd positive integer v with gcd(v, 9) ≠ 3; (2) CAN(3, 6, 6p) ≤ 216p ³ + 42p ² for any prime p > 5; and (3) CAN(4, 6, 2p) ≤ 16p ⁴ + 5p ³ for any prime p ≡ 1 (mod 4) greater than 5.     

Global asymptotic stability for damped half-linear oscillators

A necessary and sufficient condition is established for the equilibrium of the oscillator of half-linear type with a damping term, (?_p(x^′))^′+h(t)?_p(x^′)+?_p(x)=0 to be globally asymptotically stable. The obtained criterion is given by the form of a certain growth condition of the damping coefficient h(t) and it can be applied to not only the cases of large damping and small damping but also the case of fluctuating damping. The presented result is new even in the linear cases (p=2). It is also discussed whether a solution of the half-linear differential equation (r(t)?_p(x^′))^′+c(t)?_p(x)=0 that converges to a non-zero value exists or not. Some suitable examples are included to illustrate the results in the present paper.     

The mean value of the zeta-function on <Emphasis Type="Italic">σ</Emphasis>=1

Let R(T): = ò₁^T|z(1+it)|² dt - z(2)T+plogTR(T):= \int_{1}^{T}|\zeta (1+it)|^{2}\,\mathrm{d}t - \zeta (2)T+\pi\log T. We derive a precise explicit expression for R(t) which is used to derive asymptotic formulas for ò₁^TR(t) dt\int_{1}^{T}R(t)\,\mathrm{d}t and ò₁^TR²(t) dt\int_{1}^{T}R^{2}(t)\,\mathrm{d}t. These results improve on earlier upper bounds of Balasubramanian, Ramachandra and the author for the integrals in question.     

Rédei-Funktionen für Zweierpotenzen

L. Rédei has introduced in 1946 a class of rational functions over finite fields of orderp ^t –p being an odd prime — and has proved some interesting results on permutations which are induced by these functions. Recently, similar results have been found for factor rings of the integers of orderp ^t ; moreover, these functions have been applied to construct public key cryptosystems. In this paper we consider functions of Rédei type over finite fields and factor rings of the integers of order 2 ^t . We show that most of the results for oddp also hold in this case.

     

An approximation of a nonlinear integral equation driven by a function of boundedp-variation

The uniform distance between the solution of a nonlinear equation driven by a functionh with boundedp-variation and its Milstein-type approximation is estimated by δ _n v γ _p (n) v γ _p ² (n), where δ _n =max(t _k −t _k−1 ) is the maximum step size of the approximation on the interval [0,T], γ _p (n)=max υ _p ^1/p (h;[t _k-1,t _k ]), 1 <p < 2, and υ _p (h;[t _k-1,t _k ]) is thep-variation of the functionh on [t _k-1,t _k]. In particular, ifh is a Lipschitz function of order α, then the uniform distance has the bound δ _n ^α for δ_n <1. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius; Vilnius Technical University, Saulétekio 11, 2054 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 317–330, July–September, 1999.     

A characterization of Chebyshev curves inR n

Given a totally ordered setT containing at leastn+1 elements (say a subset ofR ¹), the graph of the functiona:TR ⁿ is called a Chebyshev curve (inR ⁿ) if the determinant of the matrix (a(t ₁),a(t ₂), ...,a(t _n)) is either positive whenevert ₁<t ₂<...<t _n or negative whenevert ₁<t ₂<...<t _n. For finiteT a characterization of these curves (sequences) has been given by the author.In this paper the result is extended to non-finiteT. The characterization proved here is an improved (reformulated) version of that given by the author for infiniteT.     

Capture in linear functional differential games of pursuit

The problem of capture in a pursuit game which is described by a linear retarded functional differential equation is considered. The initial function belongs to the Sobolev space W₂⁽¹⁾. The target is either a subset of W₂⁽¹⁾ a point in W₂⁽¹⁾, a subset of the Euclidean space Eⁿ or a point of Eⁿ. There is capture if the initial function can be forced to the target by the pursuer no matter what the quarry does. The concept of capture therefore formalizes the concepts of controllability under unpredictable disturbances. This is proved to be equivalent to the controllability of an associated linear retarded functional differential equation. There is nothing in (2) (6) or (7) below which restricts the control sets to be of the same dimension as the phase space. Our results can be applied in (2) for example, if the constraint sets Q′, P′ are subsets of E^m and Eⁱ respectively with q(t) = C(t) q′(t), − p(t) = B(t) p′(t), q′(t) ε E^mp′(t) ε E^r and B(t) is an n × r′-matrices and C(t) an n × m-matrix.     

Existence of periodic solutions for a fourth-order -Laplacian equation with a deviating argument

In this paper, we study the existence of periodic solutions for a fourth-order p-Laplacian differential equation with a deviating argument as follows:

[φ_p(u^″(t))]^″+f(u^″(t))+g(u(t−τ(t)))=e(t).

Controllability of stochastic systems with random delay

The object of the present paper is to study the stability behavior of a nonlinear stochastic differential system with random delay of the form ?(t; ω), ω; u(t)) + ?? (t, ω) ?(z(t — y(t; ω); ω) where ω ω, the supporting set of a probability measure space (ω, A, P), x(t, ω) in an n-dimensional random function; u(t) is an m-dimensional control vector, A(t, ω) in an n X p matrix function and ø in a p-dimensional random function defined on R^p X ω and y(t, ω) is a random delay with z(t, ω) being a p-dimensional observation vector defined a specific way. Conditions are given that guarantee the existence of an admissible control u, under the influence of which the sample paths of the stochastic system can be guided arbitrarily close to the origin with an assigned probability.