Splines with maximal zero sets

Consider a spline s(x) of degree n with L knots of specified multiplicities R₁, …, R_L, which satisfies r sign consistent mixed boundary conditions in addition to s⁽ⁿ⁾(a) = 1. Such a spline has at most n + 1 ?r + ∑_{j = 1}^LR_j zeros in (a, b) which fulfill an interlacing condition with the knots if s(x) ? = 0 everywhere. Conversely, given a set of n ?r + ∑_{j = 1}^LR_j zeros then for any choice η₁ < ··· < η_L of the knot locations which fulfills the interlacing condition with the zeros, the unique spline s(x) possessing these knots and zeros and satisfying the boundary conditions is such that s⁽ⁿ⁾(x) vanishes nowhere and changes sign at η_j if and only if R_j is odd. Moreover there exists a choice of the knot locations, not necessarily unique, which makes $\text{¦s}{}^{\mathrm{(n)}}\text{(x)¦ ≡ 1}$. In particular, this establishes the existence of monosplines and perfect splines with knots of given multiplicities, satisfying the mixed boundary conditions and possessing a prescribed maximal zero set. An application is given to double-precision quadrature formulas with mixed boundary terms and a certain polynomial extremal problem connected with it.     

A constructive procedure to recover asymptotics of short-range or long-range potentials

We study an inverse scattering problem for a pair of Hamiltonians (H,H₀) on L²(Rⁿ), where H₀=-Δ and H=H₀+V, V being a short- or long-range potential. By an elementary constructive method, we show that the scattering operator S, which is localized near a fixed energy λ>0, determines the asymptotics of the potential V at infinity, in dimension n?3. This is done by studying the action of the scattering operator on suitable wave packets.     

Ruzsa’s constant on additive functions

A function f : N → R is called additive if f(mn)= f(m)+f(n)for all m, n with(m, n)= 1. Let μ(x)= max n≤x(f(n)f(n + 1))and ν(x)= max n≤x(f(n + 1)f(n)). In 1979, Ruzsa proved that there exists a constant c such that for any additive function f , μ(x)≤ cν(x 2 )+ c f , where c f is a constant depending only on f . Denote by R af the least such constant c. We call R af Ruzsa's constant on additive functions. In this paper, we prove that R af ≤ 20.     

A constructive description of Hölder classes on closed Jordan curves

Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, leta ε int Γ, and let R_n(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the pointa. Let Λ^α(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γ_n(z) > 0 on Γ such that f∈Λ^α(Γ) if and only if for any nonnegative integer n there exists a function R_n, R_n ε R_n(a) such that ¦f(z) ? R_n(z)¦ ≤ c_f·γ_n(z), z ε Γ. It is proved that the weights γ_n cannot be expressed simply in terms of ρ ₁ ⁺ /n^(z) and ρ ₁ ^- /n^(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on \(\mathbb{C}\backslash \mathbb{D}\) .     

On two-sided and asymptotic estimates for the norms of embedding operators of into L q (dμ)

Explicit upper and lower estimates are given for the norms of the operators of embedding of , n ∈ ?, in L _q (dµ), 0 < q < ∞. Conditions on the measure µ are obtained under which the ratio of the above estimates tends to 1 as n → ∞, and asymptotic formulas are presented for these norms in regular cases. As a corollary, an asymptotic formula (as n → ∞) is established for the minimum eigenvalues λ_{1, n, β} , β > 0, of the boundary value problems (?d ²/dx ²) ⁿ u(x) = λ|x| ^β?1, x ∈ (?1, 1), u ^(k)(±1) = 0, k ∈ {0, 1, ..., n ? 1}.     

On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries

We consider the set Σ(R,C) of all m×n matrices having 0-1 entries and prescribed row sums R=(r₁,…,rm) and column sums C=(c₁,…,cn). We prove an asymptotic estimate for the cardinality |Σ(R,C)| via the solution to a convex optimization problem. We show that if Σ(R,C) is sufficiently large, then a random matrix D∈Σ(R,C) sampled from the uniform probability measure in Σ(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions.     

Homoclinic solutions for a class of the second order Hamiltonian systems

We study the existence of homoclinic orbits for the second order Hamiltonian system , where q∈Rⁿ and V∈C¹(R×Rⁿ,R), V(t,q)=-K(t,q)+W(t,q) is T-periodic in t. A map K satisfies the “pinching” condition b₁|q|²?K(t,q)?b₂|q|², W is superlinear at the infinity and f is sufficiently small in L²(R,Rⁿ). A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second order differential equations.     

A boundary Schwarz lemma on the classical domain of type <Emphasis Type="Italic">I</Emphasis>

Let R _I (m, n) be the classical domain of type I in ? ^m×n with 1 ≤ m ≤ n. We obtain the optimal estimates of the eigenvalues of the Fréchet derivative Df(\(\mathop Z\limits^ \circ \)) at a smooth boundary fixed point \(\mathop Z\limits^ \circ \)of R _I (m, n) for a holomorphic self-mapping f of R _I (m, n). We provide a necessary and sufficient condition such that the boundary points of R _I (m, n) are smooth, and give some properties of the smooth boundary points of R _I (m, n). Our results extend the classical Schwarz lemma at the boundary of the unit disk Δ to R _I (m, n), which may be applied to get some optimal estimates in several complex variables.     

Kronecker multiplicities in the (k, ℓ) hook are polynomially bounded

The problem of decomposing the Kronecker product of S _n characters is one of the last major open problems in the ordinary representation theory of the symmetric group S _n . In this note λ and µ are partitions of n, n goes to infinity, and we prove upper and lower polynomial bounds for the multiplicities of the Kronecker product χ^λ ? χ^µ, where for some fixed k and ? both partitions λ and µ are in the (k, ?) hook.     

A construction for equidistant permutation arrays of index one

An equidistant permutation array (EPA) which we denote by A(r, λ; ν) is a ν × r array such that every row is a permutation of the integers 1, 2,…, r and such that every pair of distinct rows has precisely λ columns in common. R(r, λ) is the maximum ν such that there exists an A(r, λ; ν). In this paper we show that R(n² + n + 2, 1) ? 2n² + n where n is a prime power.     

Completeness of Averaged Scattering Solutions and Inverse Scattering at a Fixed Energy

We prove that the averaged scattering solutions to the Schrödinger equation with short-range electromagnetic potentials (V, A) where V(x) = O(|x|^?ρ), A(x) = O(|x|^?ρ), |x| → ∞, ρ > 1, are dense in the set of all solutions to the Schrödinger equation that are in L ²(K) where K is any connected bounded open set in ? ⁿ ,n ≥ 2, with smooth boundary. We use this result to prove that if two short-range electromagnetic potentials (V ₁, A ₁) and (V ₂, A ₂) in ? ⁿ , n ≥ 3, have the same scattering matrix at a fixed positive energy and if the electric potentials V _j and the magnetic fields F _j : = curl A _j , j = 1, 2, coincide outside of some ball they necessarily coincide everywhere. In a previous paper of Weder and Yafaev the case of electric potentials and magnetic fields that are asymptotic sums of homogeneous terms at infinity was studied. It was proven that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at a fixed positive energy. The combination of the new uniqueness result of this paper and the result of Weder and Yafaev implies that the scattering matrix at a fixed positive energy uniquely determines electric potentials and magnetic fields that are a finite sum of homogeneous terms at infinity, or more generally, that are asymptotic sums of homogeneous terms that actually converge, respectively, to the electric potential and to the magnetic field.     

Stochastic gradient algorithm with random truncations

Let f : R^d × R^d → R be a Borel-measurable function which satisfies ∫_R^d′|f(θ, x) < ∞, ∨θ ϵ R^d, where q₀(·) is a probability measure on (R^d′, B^d′). The problem of minimization of the function f₀(θ) = ∫_R^d′(θ, x)q₀(d), θ ϵ R^d, is considered for the case when the probability measure q₀(·) is unknown, but a realization of a non-stationary random process {X_n}_n⩾1 whose single probability measures in a certain sense tend to q₀(·), is available. The random process {X_n}_n⩾1 is defined on a common probability space, R^′-valued, correlated and satisfies certain uniform mix conditions. The function f(·, ·) is completely known. A stochastic gradient algorithm with random truncations is used for the minimization of f₀(·), and its almost sure convergence is proved.     

The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Let L be a non-negative self-adjoint operator acting on L²(R ⁿ ) satisfying a pointwise Gaussian estimate for its heat kernel. Let w be an A _r weight on R ⁿ × R ⁿ , 1 < r < ∞. In this article we obtain a weighted atomic decomposition for the weighted Hardy space H _L,w ^p (R ⁿ ×R ⁿ ), 0 < p ≤ 1 associated to L. Based on the atomic decomposition, we show the dual relationship between H _L,w ¹ (R ⁿ × R ⁿ ) and BMO_L,w(R ⁿ × R ⁿ ).     

The semiclassical resolvent and the propagator for non-trapping scattering metrics

Consider a compact manifold with boundary M with a scattering metric g or, equivalently, an asymptotically conic manifold (M^○,g). (Euclidean Rn, with a compactly supported metric perturbation, is an example of such a space.) Let Δ be the positive Laplacian on (M,g), and V a smooth potential on M which decays to second order at infinity. In this paper we construct the kernel of the operator ⁻¹(h²Δ+V−²(λ₀±i0)), at a non-trapping energy λ₀>0, uniformly for h∈(0,h₀), h₀>0 small, within a class of Legendre distributions on manifolds with codimension three corners. Using this we construct the kernel of the propagator, e−it(Δ/2+V), t∈(0,t₀) as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.     

Periodic boundary value problem of functional differential equations with perturbation

By coincidence degree, the existence of solution to the periodic boundary value problem of functional differential equations with perturbation     

Stability of nonlocal difference schemes in a subspace

We consider a family of two-layer difference schemes for the heat equation with nonlocal boundary conditions containing the parameter γ. In some interval γ ∈ (1, γ ₊), the spectrum of the main difference operator contains a unique eigenvalue λ ₀ in the left complex half-plane, while the remaining eigenvalues λ ₁, λ ₂, …, λ _N?1 lie in the right half-plane. The corresponding grid space H _N is represented as the direct sum H _N = H ₀⊕H _N?1 of a one-dimensional subspace and the subspace H _N?1 that is the linear span of eigenvectors µ⁽¹⁾, µ⁽²⁾, …, µ^(N?1). We introduce the notion of stability in the subspace H _N?1 and derive a stability criterion.     

Θ-type Calderón-Zygmund operators with non-doubling measures

Let µ be a Radon measure on ? ^d which may be non-doubling. The only condition that µ must satisfy is µ(B(x, r)) ≤ Cr ⁿ for all x∈? ^d , r > 0 and for some fixed 0 < n ≤ d. In this paper, under this assumption, we prove that θ-type Calderón-Zygmund operator which is bounded on L ²(µ) is also bounded from L ^∞(µ) into RBMO(µ) and from H _atb ^1,∞ (µ) into L ¹(µ). According to the interpolation theorem introduced by Tolsa, the L ^p (µ)-boundedness (1 < p < ∞) is established for θ-type Calderón-Zygmund operators. Via a sharp maximal operator, it is shown that commutators and multilinear commutators of θ-type Calderón-Zygmund operator with RBMO(µ) function are bounded on L ^p (µ) (1 < p < ∞).     

On a characterization of irreducibility of a non-negative matrix

Let A denote an n×n matrix with all its elements real and non-negative, and let r_i be the sum of the elements in the ith row of A, i=1,…,n. Let B=A?D(r₁,…,r_n), where D(r₁,…,r_n) is the diagonal matrix with r_i at the position (i,i). Then it is proved that A is irreducible if and only if rank B=n?1 and the null space of B^T contains a vector d whose entries are all non-null.     

Partitions of natural numbers with the same representation functions

For a set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A₀ be the set of all nonnegative integers a with even D(a) and A₁ be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R₂(A,n)=R₂(N?A,n) for all n?2N−1, then R₂(A,n)=R₂(N?A,n)?1 for all n?12N²−10N−2 except for A=A₀ or A=A₁; (b) if R₃(A,n)=R₃(N?A,n) for all n?2N−1, then R₃(A,n)=R₃(N?A,n)?1 for all n?12N²+2N. Several problems are posed in this paper.     

Rectangular arrays

In this paper we studied m×n arrays with row sums $\text{nr}\text{(n,m)}$ and column sums $\text{mr}\text{(n,m)}$ where (n,m) denotes the greatest common divisor of m and n. We were able to show that the function H_m,n(r), which enumerates m×n arrays with row sums and column sums $\text{nr}\text{(m,n)}$ and $\text{mr}\text{(n,m)}$ respectively, is a polynomial in r of degree (m?1)(n?1). We found simple formulas to evaluate these polynomials for negative values, ?r, and we show that certain small negative integers are roots of these polynomials. When we considered the generating function G_m,n(y) = Σ_r?0H_m,n(r)y^r, it was found to be rational of degree less than zero. The denominator of G_m,n(y) is of the form (1?y)^(m?1)(n?1)+3, and the coefficients of the numerator are non-negative integers which enjoy a certain symmetric relation.