Spectral Estimations for Can nical Systems

Two–dimensional canonical systems are boundary value problems of the form with y₁(0) = 0 and Weyl's limit point case at L. The 2 × 2 matrix valued function H is real, symmetric and nonnegative, . The correspondence between canonical systems and their Titchmarsh–Weyl coefficients Q is a bijection between the class of all matrix functions H with tr H(x) = 1 a.e. on [0, L) and the class of the Nevanlinna functions ℕ augmented by the function Q ≡ 8. Each Titchmarsh–Weyl coefficient Q ∈ ℕ can be represented by means of a measure σ, the so–called spectral measure of the canonical system. In this note matrix functions H are specified whose corresponding spectral measures σ satisfy conditions of the form or . Herewith we generalize corresponding results of M.G. Krein and I. S. Kac for so–called vibrating strings.     

On the Harnack principle for strongly elliptic systems with nonsmooth coefficients

In this paper we prove the following theorem (for notation and definitions, see the paragraphs below): “Let Ω ⊆ ℝⁿ be a domain, m ∈ ℕ, and λ, q > 0. Then, there exists r (= r(λ, q)) > 1 such that for every 0 < p < q, whenever are weak solutions of a strongly elliptic system with m equations of ellipticity λ satisfying ∈ 𝒫_r a.e. and Ω′ ⊆ Ω subdomain, the following inequalities hold: where C (= C(n,m,λ,q,p,Ω,Ω′)) is a positive constant.” © 1999 John Wiley & Sons, Inc.     

Decay estimates for the wave and Dirac equations with a magnetic potential

We study the electromagnetic wave equation and the perturbed massless Dirac equation on ℝ_t × ℝ³: where the potentials A(x), B(x), and V(x) are assumed to be small but may be rough. For both equations, we prove the expected time decay rate of the solution where the norm ‖f‖_X can be expressed as the weighted L²-norm of a few derivatives of the data f. © 2006 Wiley Periodicals, Inc.     

Necessary and Sufficient Conditions for the Oscillation of a Second Order Linear Differential Equation

In this paper we give a necessary and sufficient condition for the oscillation of the second order linear differential equation where p is a locally integrable function and either or where We give some applications which show how these results unify and imply some classical results in oscillation theory.     

The ground state eigenvalue of Hill's equation with white noise potential

The distribution of the ground state eigenvalue λ₀(Q) of Hill's operator Q = −d²/dx² + q(x) on the circle of perimeter 1 is expressed in two different ways in case the potential q is standard white noise. Let WN be the associated white noise measure, and let CBM be the measure for circular Brownian motion p(x), 0 ≤ x < 1, formed from the standard Brownian motion b(x), 0 ≤ x ≤ 1, starting at b(0) = a, by conditioning so that b(1) = a, and distributing this common level over the line according to the measure da. The connection is based upon the Ricatti correspondence q(x) = λ + p′ (x) + p²(x). The two versions of the distribution are (1) in which $\overline{p}$ is the mean value ∫ pdx, and (2) the left‐hand side of (2) being the density for (1) and CBM₀ the conditional circular Brownian measure on $\overline{p}$ = 0. (1) and (2) are related by the divergence theorem in function space as suggested by the recognition of the Jacobian factor the outward‐pointing normal component of the vector field v(x) = ∂Δ(λ)/∂q(x), 0 ≤ x < 1, Δ being the Hill's discriminant for Q. The Ricatti correspondence prompts the idea that (1) and (2) are instances of the Cameron‐Martin formula, but it is not so: The latter has to do with the initial value problem for Ricatti, but it is the periodic problem that figures here, so the proof must be done by hand, by finite‐dimensional approximation. The adaptation of 1 and 2 to potentials of Ornstein‐Uhlenbeck type is reported without details. © 1999 John Wiley & Sons, Inc.     

Existence of multiple positive solutions for −Δu−μ(u/∣x∣2)=u2*−1+σf(x)

In this paper, we consider the semilinear elliptic problem where Ω??^N (N?3) is a bounded smooth domain such that 0∈Ω, σ>0 is a real parameter, and f(x) is some given function in L^∞(Ω) such that f(x)?0, f(x)?0 in Ω. Some existence results of multiple solutions have been obtained by implicit function theorem, monotone iteration method and Mountain Pass Lemma. Copyright © 2002 John Wiley & Sons, Ltd.     

On the Range of Two Convolution Operators

Let _(ω)(ℝ) denote the non–quasianalytic class of Beurling type on ℝ. For μ, ν ∈ ′_(ω)(ℝ) we give necessary conditions for the inclusion T_ν( _(ω)(ℝ)) ⊂ T_μ( _(ω)(ℝ)), thus extending previous work of Malgrange and Ehrenpreis .     

Strengths and Weaknesses of LH Arithmetic

In this paper we provide a new arithmetic characterization of the levels of the og‐time hierarchy (LH). We define arithmetic classes and that correspond to ‐LOGTIME and ‐LOGTIME, respectively. We break and into natural hierarchies of subclasses and . We then define bounded arithmetic deduction systems ′ whose ‐definable functions are precisely B( ‐LOGTIME). We show these theories are quite strong in that (1) LIOpen proves for any fixed m that , (2) TAC, a theory that is slightly stronger than ′ whose (LH)‐definable functions are LH, proves LH is not equal to ‐TIME(s) for any m> 0, where 2^s ∈L, s(n) ∈ ω(log n), and (3) TAC proves LH ≠ for all k and m. We then show that the theory TAC cannot prove the collapse of the polynomial hierarchy. Thus any such proof, if it exists, must be argued in a stronger systems than ours.     

Integrability Properties of the Maximal Operator on Partial Sums of Fourier Series in Orlicz Spaces

Let S* (f be the majorant function of the partial sums of the trigonometric Fourier series of f. In this paper we consider the Orlicz space Lπ and give a generalization of Soria's result [S1]. Let π (t) be a concave function with some nice properties and . If there exists a positive constant a₀ < 1 such that then we have .     

Hadamard matrices constructed from supplementary difference sets in the class ℋ︁1

In this article we give the definition of the class ??₁ and prove: (1) ??₁(v) ≠ ? for v ∈ ?? = ??₁ ∪ ??₂ ∪ ??₃ where (2) there exists 2 ? {2q²; q² ± q, q²;q² ± q} supplementary difference sets for q² ∈ ??; (3) there exists an Hadamard matrix of order 4v for v ∈ ??; (4) if t is an order of T-matrices, there exists an Hadamard matrix of order 4tv for v ∈ ??. © 1994 John Wiley & Sons, Inc.     

Supercritical Branching Brownian Motion and K-P-P Equation In the Critical Speed-Area

If R_t is the position of the rightmost particle at time t in a one dimensional branching brownian motion, whore α is the inverse of the mean life time and m is the mean of the reproduction law. If Z_t denotes the random point measure of particles living at time t, we get in the critical area {c = c₀} The function u(t, x) = P(R_t > x) is studied as a solution of the K-P-P equation for some function f. Conditioned on non-extinction of the spatial tree in the c₀-direction, a limit distribution is obtained and characterized.     

Radially symmetric Poisson–Boltzmann equation in a domain expanding to infinity

The electric potential u in a solution of an electrolyte around a linear polyelectrolyte of the form of a cylinder satisfies We study the problem when R → ∞.     

Data Based Regularization Matrices for the Tikhonov-Phillips Regularization

In Tikhonov-Phillips regularization of general form the given ill-posed linear system is replaced by a Least Squares problem including a minimization of the solution vector x, relative to a seminorm with some regularization matrix L. Based on the finite difference matrix L_k, given by a discretization of the first or second derivative, we introduce the seminorm where the diagonal matrix and is the best available approximate solution to x. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inversion of the Radon transform with incomplete data

The Radon transform R(p, θ), θ∈S^n?1, p∈?¹, of a compactly supported function f(x) with support in a ball B_a of radius a centred at the origin is given for all $ \theta \in \mathop {S^{n - 1} }\limits^\tilde $, where $ \mathop {S^{n - 1} }\limits^\tilde $ is an open set on S^n?1, and all p∈(? ∞, ∞), n≥2. An approximate formula is given to calculate f(x) from the given data.     

Analytic aspects of the Toda system: I. A Moser‐Trudinger inequality

In this paper, we analyze solutions of the open Toda system and establish an optimal Moser‐Trudinger type inequality for this system. Let Σ be a closed surface with area 1 and K = (a_ij)_{N × N} the Cartan matrix for SU(N + 1), i.e., We show that has a lower bound in (H¹(Σ))^N if and only if This inequality is optimal. As a direct consequence, if M_j < for 4π for j = 1, 2, …, N, Φ_M has a minimizer u that satisfies © 2001 John Wiley & Sons, Inc.     

Comparing Packing Measures to Hausdorff Measures on the Line

For each 0 < s < 1, define where , denote respectively the s‐dimensional packing measure and Hausdorff measure, and the infimum is taken over all the sets E ⊂ R with . In this paper we give a nontrivial estimation of c(s), namely, for each 0 < s < 1, where . As an application, we obtain a lower density theorem for Hausdorff measures.     

On directed diffusion with measurable background

We show that the ‘directed diffusion equation’ with periodic boundary conditions has a unique weak solution u whenever b is measurable and bounded above and below by positive constants. Also, lim_t→∞u(t,x) in L^p, 1?p≤∞.     

Nontrivial periodic solutions of a nonlinear beam equation

We consider the existence of a nontrivial solution of the following equation: where g is a nondecreasing function defined on R¹, satisfies g(O) = O, and some other additional conditions. Our results and methods are quite similar to those associated with recent work on the nonlinear wave equation [1]-[8]: .     

Almost Sure Convergence in Extreme Value Theory

Let X₁, …, X_n be independent random variables with common distribution function F. Define and let G(x) be one of the extreme-value distributions. Assume F ∈ D(G), i.e., there exist a_n> 0 and b_n ∈ ? such that . Let l(?∞,x)(·) denote the indicator function of the set (?∞,x) and S(G) =: {x : 0 < G(x) < 1}. Obviously, 1(?∞,x)((M_n?b_n)/a_n) does not converge almost surely for any x ∈ S(G). But we shall prove .     

Quasilinear degenerate parabolic equations of Kirchhoff type

We investigate the evolution problem where H is a Hilbert space, A is a self‐adjoint linear non‐negative operator on H with domain D(A), and is a continuous function. We prove that if , and , then there exists at least one global solution, which is unique if either m never vanishes, or m is locally Lipschitz continuous. Moreover, we prove that if for all , then this problem is well posed in H. On the contrary, if for some it happens that for all , then this problem has no solution if with β small enough. We apply these results to degenerate parabolic PDEs with non‐local non‐linearities. Copyright © 1999 John Wiley & Sons, Ltd.