Linear differential equations with entire coefficients of small growth

We prove that if n \geqq 3 n \geqq 3 and A₀, ?, A_n-2 A_0, \ldots, A_{n-2} are entire functions of small growth, not all polynomials, then the linear differential equation¶¶ w⁽ⁿ⁾ + ?_j=0^n-2 A_j w^(j) = 0 w^{(n)} + \sum\limits_{j=0}^{n-2} A_j w^{(j)} = 0 ¶¶ cannot have a fundamental set of solutions each with few zeros.     

LARGE TIME BEHAVIOR OF SOLUTIONS TO SOME DEGENERATE PARABOLIC EQUATIONS

The purpose of this paper is to study the limit in L ¹(Ω), as t → ∞, of solutions of initial-boundary-value problems of the form u_t ? Δw = 0 and u ∈ β(w) in a bounded domain Ω with general boundary conditions ?w/?η + γ(w) ? 0. We prove that a solution stabilizes by converging as t → ∞ to a solution of the associated stationary problem. On the other hand, since in general these solutions are not unique, we characterize the true value of the limit and comment the results on the related concrete situations like the Stefan problem and the filtration equation.     

Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part II: Clenshaw-Lord type approximants

Laurent-Padé (Chebyshev) rational approximantsP _m (w, w ⁻¹)/Q _n (w, w ⁻¹) of Clenshaw-Lord type [2,1] are defined, such that the Laurent series ofP _m /Q _n matches that of a given functionf(w, w ⁻¹) up to terms of orderw ^±(m+n) , based only on knowledge of the Laurent series coefficients off up to terms inw ^±(m+n) . This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent series ofP _m matches that ofQ _n f up to terms of orderw ^±(m+n ), but based on knowledge of the series coefficients off up to terms inw ^±(m+2n). The Clenshaw-Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of the Padé-Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations are obtainable for allm≥0,n≥0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé-Chebyshev approximants. Remarkably similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw-Lord type methods, thus validating the use of either.     

Gâteaux derivatives and their applications to approximation in Lorentz spaces Γp,w

We establish the formulas of the left‐ and right‐hand Gâteaux derivatives in the Lorentz spaces Γ_p,w = {f: ∫₀^α (f **)^p w < ∞}, where 1 ≤ p < ∞, w is a nonnegative locally integrable weight function and f ** is a maximal function of the decreasing rearrangement f * of a measurable function f on (0, α), 0 < α ≤ ∞. We also find a general form of any supporting functional for each function from Γ_p,w , and the necessary and sufficient conditions for which a spherical element of Γ_p,w is a smooth point of the unit ball in Γ_p,w . We show that strict convexity of the Lorentz spaces Γ_p,w is equivalent to 1 < p < ∞ and to the condition ∫₀^∞ w = ∞. Finally we apply the obtained characterizations to studies the best approximation elements for each function f ∈ Γ_p,w from any convex set K ? Γ_p,w (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On properties of entire solutions of linear differential equations with polynomial coefficients

We investigate properties of entire solutions of differential equations of the form

A second-order,chaos-free,explicit method for the numerical solution of a cubic reaction problem in neurophysiology

A second order explicit method is developed for the numerical solution of the initialvalue problem w′(t) ≡ dw(t)/dt = ?(w), t > 0, w(0) = W⁰, in which the function ?(w) = αw(1 ? w) (w ? a), with α and a real parameters, is the reaction term in a mathematical model of the conduction of electrical impulses along a nerve axon. The method is based on four first-order methods that appeared in an earlier paper by Twizell, Wang, and Price [Proc. R. Soc. (London) A 430 , 541–576 (1990)]. In addition to being chaos free and of higher order, the method is seen to converge to one of the correct steady-state solutions at w = 0 or w = 1 for any positive value of α. Convergence is monotonic or oscillatory depending on W⁰, α, a, and l, the parameter in the discretization of the independent variable t. The approach adopted is extended to obtain a numerical method that is second order in both space and time for solving the initial-value boundary-value problem ?u/?t = κ?²u/?x² + αu(1 ? u)(u ? a) in which u = u(x,t). The numerical method so developed obtained the solution by solving a single linear algebraic system at each time step. © 1993 John Wiley & Sons, Inc.     

Life span of solutions of a weakly coupled parabolic system

We consider the Cauchy problem for the weakly coupled parabolic system ∂ _t w _λ−Δ w _λ = F(w _λ) in R ^N , where λ > 0, w _λ = (u _λ, v _λ), F(w _λ) = (v _λ ^p , u _λ ^q ) for some p, q ≥ 1, pq > 1, and w_l(0) = (lj₁, l^\fracq+1p+1j₂)w_{\lambda}(0) = ({\lambda}{\varphi}_1, {\lambda}^{\frac{q+1}{p+1}}{\varphi}_2), for some nonnegative functions φ₁, φ₂ ?\in C ₀(R ^N ). If (p, q) is sub-critical or either φ₁ or φ₂ has slow decay at ∞, w _λ blows up for all λ > 0. Under these conditions, we study the blowup of w _λ for λ small.     

Volume mean densities for the heat equation

It is shown that, for solid caps D of heat balls in ? ^{d + 1} with center z ₀ = (0, 0), there exist Borel measurable functions w on D such that inf w(D) > 0 and ∫ v(z)w(z) dz ≤ v(z ₀), for every supertemperature v on a neighborhood of D?. This disproves a conjecture by N. Suzuki and N.A. Watson. On the other hand, it turns out that there is no such volume mean density, if the bounded domain D in ? ^d × (?∞, 0) is only slightly wider at z ₀ than a heat ball.     

On a nonlinear stochastic integral equation with application to control systems

The aim of the present paper is to study a nonlinear stochastic integral equation of the form

Dynamical behavior of endomorphisms on certain invariant sets

We study the Hausdorff dimension of the intersection between local stable manifolds and the respective basic sets of a class of hyperbolic polynomial endomorphisms on the complex projective space ?². We consider the perturbation (z ² +?z +b?w ², w ²) of (z ², w ²) and we prove that, for b sufficiently small, it is injective on its basic set Λ_? close to Λ:= {0} × S ¹. Moreover we give very precise upper and lower estimates for the Hausdorff dimension of the intersection between local stable manifolds and Λ _? , in the case of these maps.     

Constructive existence theorems for problems of Thomas-Fermi Type

A minimal positive solution of the Thomas-Fermi problem ? = λt^?1/2 w^3/2, w(0) = 1, w(1) = w(1) is shown to exist for each λ > 0. It is proved that all positive solutions, for a given value of λ, are strictly ordered and that the minimal positive solution w_λ is a decreasing function of λ. Upper and lower analytic bounds for w _λ are given and these bounds are shown to initiate sequences of Picard and Newton iterates which converge monotonically to w _λ. A comparative analysis of the efficiency of the iteration schemes is presented. The methods used are of a general nature and can be applied to a variety of nonlinear boundary value problems of convex type [14].     

On tightly κ-filtered Boolean algebras

In this article we study the notion of tight ?-filteredness of a Boolean algebra for infinite regular cardinals ?. Tight à₀ \aleph_0 -filteredness is projectivity. We give characterizations of tightly ?-filtered Boolean algebras which generalize the internal characterizations of projectivity given by Haydon, Šcepin, and Koppelberg (see [15] or [17]). We show that for each ? there is an rc-filtered Boolean algebra which is not tightly ?-filtered. This generalizes a result of Šcepin (see [15]). We prove that no complete Boolean algebra of size larger than à₂ \aleph_2 is tightly à₁ \aleph_1 -filtered. We give a new example of a model of set theory where \frak P(w) \frak P(\omega) is tightly s-filtered. We study the effect of the tight s-filteredness of \frak P(w) \frak P(\omega) on the automorphism group of \frak P(w)/fin \frak P(\omega)/fin .     

The slow wave solution to the FitzHugh-Nagumo equations

This paper gives a new existence proof for a travelling wave solution to the FitzHugh-Nagumo equations, u_t = u_xx +f(u)?w, w _t = ? (u?γw). The proof uses a contraction mapping argument, and also shows that the solution (u, c, w) to the travelling wave equations, where c is the wave speed, converges as ? → 0⁺ to the solution to the equations having ?=0, c=0, and w=0.     

Approximation of the vth Root of N

We present a class of functions g_K(w), K ≥ 2, for which the recursive sequences w_{n + 1} = g_K(w_n) converge to N^1/v with relative error . Newton's method results when K = 2. The coefficients of the g_K(w) form a triangle, which is Pascal's for v = 2. In this case, if w₁ = x₁/y₁, where x₁, y₁ is the first positive solution of Pell's equation x² ? Ny² = 1, then w_{n + 1} = x_{n + 1}/y_{n + 1} is the Kⁿpth or 2Kⁿpth convergent of the continued fraction for , its period p being even or odd.     

On stable range conditions

In this paper, we show that a ring R satisfies unit 1-stable range if and only if a₁R + ? + a_mR = dR with m ≥ 2,a ₁, ?a_m ?R implies that there exist u₁ , ?u_m ? U(R) such that a₁u₁ +?+a_mu_m = d and an exchange ring R has stable range one if and only if a₁R+?+a_mR = dR with m ≥ 2,a ₁,?,a_m ? R implies that there exist unit-regular w ₁,?,w_m ? R such that a₁w₁ +?+ a_mw_m = d. Also we show that an exchange ring R satisfies the n-stable range condition if and only if a( ⁿR)+bR = dR with a ? Rⁿ,b,d ? R implies that there exist unimodular regular w ? ⁿ R and: y ? R such that aw+by = d.     

Lagrangian barriers and symplectic embeddings

We prove that every symplectic Kähler manifold (M;W) (M;\Omega) with integral [W] [\Omega] decomposes into a disjoint union (M,W) = (E,w₀) \coprod D (M,\Omega) = (E,\omega_0) \coprod \Delta , where (E,w₀) (E,\omega_0) is a disc bundle endowed with a standard symplectic form w₀ \omega_0 and D \Delta is an isotropic CW-complex. We perform explicit computations of this decomposition on several examples.¶As an application we establish the following symplectic intersection phenomenon: There exist symplectically irremovable intersections between contractible domains and Lagrangian submanifolds. For example, we prove that every symplectic embedding j:B²ⁿ(l) ? \Bbb CPⁿ \varphi:B^{2n}(\lambda) \to {\Bbb C}P^n of a ball of radius l² 3 1/2 \lambda^2 \ge 1/2 must intersect the standard Lagrangian real projective space \Bbb RPⁿ ì \Bbb CPⁿ {\Bbb R}P^n \subset {\Bbb C}P^n .     

Boundedness in nonlinear oscillations

In this paper, the boundedness of all solutions of the nonlinear differential equation (φ_p(x′))′ + αφ_p(x⁺) – βφ_p(x^–) + f(x) = e(t) is studied, where φ_p(u) = |u|^p–2 u, p ≥ 2, α, β are positive constants such that = 2w^–1 with w ∈ ?⁺\?, f is a bounded C⁵ function, e(t) ∈ C⁶ is 2π_p‐periodic, x⁺ = max{x, 0}, x^– = max{–x, 0}. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The asymptotic number of labeled connected graphs with a given number of vertices and edges

Let c(n, q) be the number of connected labeled graphs with n vertices and q ≤ N = (2n ) edges. Let x = q/n and k = q ? n. We determine functions w_k ? 1. a(x) and φ(x) such that c(n, q) ? w_k(q^N)eⁿφ(x)+a(x) uniformly for all n and q ≥ n. If ? > 0 is fixed, n→ ∞ and 4q > (1 + ?)n log n, this formula simplifies to c(n, q) ? (N^q) exp(–ne^?2q/n). on the other hand, if k = o(n^1/2), this formula simplifies to c(n, n + k) ? 1/2 w_k (3/π)^1/2 (e/12k)^k/2nⁿ?^(3k?1)/2.     

Characterization of polyanalytic functions by meromorphic extensions from chains of circles

Let C _t = {z ∈ ℂ: |z − c(t)| = r(t), t ∈ (0, 1)} be a C ¹-family of circles in the plane such that lim _t→0+ C _t = {a}, lim _t→1− C _t = {b}, a ≠ b, and |c′(t)|² + |r′(t)|² ≠ 0. The discriminant set S of the family is defined as the closure of the set {c(t) + r(t)w(t), t ∈ [0, 1]}, where w = w(t) is the root of the quadratic equation ̅c′(t)w ² + 2r′(t)w + c′(t) = 0 with |w| < 1, if such a root exists.