On Marcel Riesz's Ultrahyperbolic Kernel

Let t = (t₁, …, t_n) be a point of ?ⁿ. We shall write . We put by definition R_α(u) = u^(α?n)/2/K_n(α); here α is a complex parameter, n the dimension of the space, and K_n(α) is a constant. First we evaluate □R_α(u) = R_α(u), where □ the ultrahyperbolic operator. Then we obtain the following results: R_?2k(u) = □^kδ; R₀(u) = δ; and □^kR_2k(u) = δ, k = 0, 1, …. The first result is the n-dimensional ultrahyperbolic correlative of the well-known one-dimensional formula . Equivalent formulas have been proved by Nozaki by a completely different method. The particular case µ = 1 was solved previously.     

On the Fourier Transform of Causal Distributions

We extend the distributional Bochner formula [1, p. 72, Theorem 26] to certain kinds of distributions. Theorem I.1 gives a formula [Eq. (I.1.14)] which makes it possible to obtain easily the Fourier transform of distributions of the form As applications of the formula (I.1.14) we evaluate the Fourier transforms of the distributions G_α(P±i0, m, n) [Eq. (I.4.1)] and H_α(P±i0,n) [Eq. (II.1.1)]. It follows from Theorem II.3 that H_zk(P±i0,n) is, for 2K≠n+2r, r=0,1..., an elementary solution of the n-dimensional ultrahyperbolic operator iterated k times.     

On the Fourier transforms of retarded Lorentz-invariant functions

In this article we evaluate the Fourier transforms of retarded Lorentz-invariant functions (and distributions) as limits of Laplace transforms. Our method works generally for any retarded Lorentz-invariant functions φ(t) (t?Rⁿ) which is, besides, a continuous function of slow growth. We give, among others, the Fourier transform of G_R(t, α, m², n) and G_A(t, α, m², n), which, in the particular case α = 1, are the characteristic functions of the volume bounded by the forward and the backward sheets of the hyperboloid u = m² and by putting α = ?k are the derivatives of k-order of the retarded and the advanced-delta on the hyperboloid u = m². We also obtain the Fourier transform of the function W(t, α, m², n) introduced by M. Riesz (Comm. Sem. Mat. Univ. Lund4 (1939)). We finish by evaluating the Fourier transforms of the distributional functions G_R(t, α, m², n), G_A(t, α, m², n) and W(t, α, m², n) in their singular points.     

A System of Schrödinger Equations in the Critical Case

A system of nonlinear Schrödinger equations u _k } / t=ia _k u _k+f _k (u,u ^*), t>0, k=1,... ,m; u _k (0,x)=u _k0 (x), where f _k are homogeneous functions of order 1+4/n, is considered. Sufficient conditions for the globality of the solution are obtained. The existence of the explicit blow-up solution is proved.     

Long time small solutions to nonlinear parabolic equations

A sharp result on global small solutions to the Cauchy problem $$u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $$ In Rⁿ is obtained under the the assumption thatf is C^1+r forr>2/n and ‖u ₀‖C²(R ⁿ ) +‖u ₀‖W ₁ ² (R ⁿ ) is small. This implies that the assumption thatf is smooth and ‖u ₀ ‖W ₁ ^k (R ⁿ )+‖u ₀‖W ₂ ^k (R ⁿ ) is small fork large enough, made in earlier work, is unnecessary.     

On the average genus of the random graph

The generalized Petersen graph GP (n, k), n ≤ 3, 1 ≥ k < n/2 is a cubic graph with vertex-set {u_j; i ? Z_n} ∪ {v_j; i ? Z_n}, and edge-set {u_iu_i, u_iv_i, v_iv_{i+k, i?}Z_n}. In the paper we prove that (i) GP(n, k) is a Cayley graph if and only if k² ? 1 (mod n); and (ii) GP(n, k) is a vertex-transitive graph that is not a Cayley graph if and only if k² ? -1 (mod n) or (n, k) = (10, 2), the exceptional graph being isomorphic to the 1-skeleton of the dodecahedon. The proof of (i) is based on the classification of orientable regular embeddings of the n-dipole, the graph consisting of two vertices and n parallel edges, while (ii) follows immediately from (i) and a result of R. Frucht, J.E. Graver, and M.E. Watkins [“The Groups of the Generalized Petersen Graphs,” Proceedings of the Cambridge Philosophical Society, Vol. 70 (1971), pp. 211-218]. © 1995 John Wiley & Sons, Inc.     

Graphs with unavoidable subgraphs with large degrees

Let ??(n, m) denote the class of simple graphs on n vertices and m edges and let G ∈ ?? (n, m). There are many results in graph theory giving conditions under which G contains certain types of subgraphs, such as cycles of given lengths, complete graphs, etc. For example, Turan's theorem gives a sufficient condition for G to contain a K_{k + 1} in terms of the number of edges in G. In this paper we prove that, for m = αn², α > (k - 1)/2k, G contains a K_{k + 1}, each vertex of which has degree at least f(α)n and determine the best possible f(α). For m = ?n²/4? + 1 we establish that G contains cycles whose vertices have certain minimum degrees. Further, for m = αn², α > 0 we establish that G contains a subgraph H with δ(H) ≥ f(α, n) and determine the best possible value of f(α, n).     

An asymptotic approximation of Wallis’ sequence

An asymptotic approximation of Wallis’ sequence W(n) = Π _k=1 ⁿ 4k ²/(4k ² − 1) obtained on the base of Stirling’s factorial formula is presented. As a consequence, several accurate new estimates of Wallis’ ratios w(n) = Π _k=1 ⁿ (2k−1)/(2k) are given. Also, an asymptotic approximation of π in terms of Wallis’ sequence W(n) is obtained, together with several double inequalities such as, for example,

Coincidence of Least Uniform Deviations of Functions from Polynomials and Rational Fractions

For a given nonincreasing vanishing sequence {a _n } _{n = 0} of nonnegative real numbers, we find necessary and sufficient conditions for a sequence {n _k } _{k = 0} to have the property that for this sequence there exists a function f continuous on the interval [0,1] and satisfying the condition that , k = 0,1,2,..., where E _n (f) and R _n,m (f) are the best uniform approximations to the function f by polynomials whose degree does not exceed n and by rational functions of the form r _n,m (x) = p _n (x)/q _m (x), respectively.     

Best Constants in the Miranda-Agmon Inequalities for Solutions of Elliptic Systems and the Classical Maximum Modulus Principle for Fluid and Elastic Half-spaces

The Dirichlet problem for elliptic systems of the second order with constant real and complex coefficients in the half-space  ^k ₊ = {x = (x ₁,…,x_k ): x_k > 0} is considered. It is assumed that the boundary values of a solution u = (u ₁,…,u _m) have the form ψ ₁ ξ ₁ + · · · + ψ _n ξ _n, 1 ≤ n ≤ m, where ξ ₁,· · ·,ξ _n is an orthogonal system of m-component normed vectors and ψ ₁,· · ·,ψ _n are continuous and bounded functions on ? ^k ₊. We study the mappings [C(? ^k ₊)] ⁿ ? (ψ ₁,…,ψ _n) → u(x) ?  ^m and [C(? ^k ₊)] ⁿ ? (ψ ₁,…,ψ _n) → u(x) ?  ^m generated by real and complex vector valued double layer potentials. We obtain representations for the sharp constants in inequalities between |u(x)| or |(z, u(x))| and ∥u| _xk =₀∥, where z is a fixed unit m-component vector, | · | is the length of a vector in a finite-dimensional unitary space or in Euclidean space, and (·,·) is the inner product in the same space. Explicit representations of these sharp constants for the Stokes and Lamé systems are given. We show, in particular, that if the velocity vector (the elastic displacement vector) is parallel to a constant vector at the boundary of a half-space and if the modulus of the boundary data does not exceed 1, then the velocity vector (the elastic displacement vector) is majorised by 1 at an arbitrary point of the half-space. An analogous classical maximum modulus principle is obtained for two components of the stress tensor of the planar deformed state as well as for the gradient of a biharmonic function in a half-plane.     

A proof of Freud's conjecture for exponential weights

LetW (x) be a function nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ²(x) are finite. Let {p _n (W ²;x)} ₀ denote the sequence of orthonormal polynomials with respect to the weightW ²(x), and let {A _n } ₁ and {B _n } ₁ denote the coefficients in the recurrence relation

On the existence of cycle frames and almost resolvable cycle systems

Suppose H is a complete m-partite graph K_m(n₁,n₂,…,n_m) with vertex set V and m independent sets G₁,G₂,…,G_m of n₁,n₂,…,n_m vertices respectively. Let G={G₁,G₂,…,G_m}. If the edges of λH can be partitioned into a set C of k-cycles, then (V,G,C) is called a k-cycle group divisible design with index λ, denoted by (k,λ)-CGDD. A (k,λ)-cycle frame is a (k,λ)-CGDD (V,G,C) in which C can be partitioned into holey 2-factors, each holey 2-factor being a partition of V?G_i for some G_i∈G. Stinson et al. have resolved the existence of (3,λ)-cycle frames of type g^u. In this paper, we show that there exists a (k,λ)-cycle frame of type g^u for k∈{4,5,6} if and only if , , u≥3 when k∈{4,6}, u≥4 when k=5, and (k,λ,g,u)≠(6,1,6,3). A k-cycle system of order n whose cycle set can be partitioned into (n−1)/2 almost parallel classes and a half-parallel class is called an almost resolvable k-cycle system, denoted by k-ARCS(n). Lindner et al. have considered the general existence problem of k-ARCS(n) from the commutative quasigroup for . In this paper, we give a recursive construction by using cycle frames which can also be applied to construct k-ARCS(n)s when . We also update the known results and prove that for k∈{3,4,5,6,7,8,9,10,14} there exists a k-ARCS(2kt+1) for each positive integer t with three known exceptions and four additional possible exceptions.     

Spectral Criteria of Abstract Sequences; Application to Difference Equations

We suppose that M is a closed subspace of l ^∞(J, X), the space of all bounded sequences {x(n)} _n?J ? X, where J ? {Z⁺,Z} and X is a complex Banach space. We define the M-spectrum σ_M (u) of a sequence u ? l ^∞(J,X). Certain conditions will be supposed on both M and σ_M (u) to insure the existence of u ? M. We prove that if u is ergodic, such that σ_M (u,) is at most countable and, for every λ ? σ_M (u), the sequence e^?iλnu(n) is ergodic, then u ? M. We apply this result to the operator difference equationu(n + 1) = Au(n) + ψ(n), n ? J,and to the infinite order difference equation Σ ^r _k=1 a_k (u(n + k) ? u(n)) + Σ _{s ? Z}?(n ? s)u(s) = h(n), n?J, where ψ?l ^∞(Z,X) such that ψ| _J ? M, A is the generator of a C ₀-semigroup of linear bounded operators {T(t)} _t>0 on X, h ? M, ? ? l ¹(Z) and a_k ?C. Certain conditions will be imposed to guarantee the existence of solutions in the class M.     

Second degree classical forms

A form (linear functional) u is called regular if there exists a sequence of polynomials {P_n}_n≥0, deg P_n = n which is orthogonal with respect to u. Such a form is said to be of second degree if there are polynomials B and C such that the Stieltjes function satisfies a relation of the form BS²(u) + CS(u) + D = 0.Classical forms correspond to classical orthogonal polynomials: sequences of polynomials whose derivatives also form an orthogonal sequence. In this paper, the authors determine all the classical forms which are of second degree. They show that Hermite, Laguerre and Bessel forms are not of second degree. Only Jacobi forms which satisfy a certain condition possess this property.     

A note on the oscillatory property for nonlinear difference equations and differential equations

Criteria are given to determine the oscillatory property of solutions of the nonlinear difference equation: Δ^du_n + ∑_{i = 1}^mp_inf_i(u_n, Δu_n,…,Δ^{d ? 1}u_n) = 0, n = 0, 1, 2,…, where d is an arbitrary integer, generalizing results that have been obtained by B. Szmanda (J. Math. Anal. Appl.79 (1981), 90–95) for d = 2. Analogous results are given for the differential equation: u^(d) + ∑_{i = 1}^mp_i(t)f_i(u, u′,…, u^{(d ? 1)}) = 0, t ? t₀, which coincide with the criteria given by 2., 3., 599–602) and 4., 5., 6., 715–719) for the case m = 1.     

A property of nonlinear elliptic equations when the right-hand side is a measure

LetA(u)=–diva(x, u, Du) be a Leray-Lions operator defined onW ₀ ^1,p () and be a bounded Radon measure. For anyu SOLA (Solution Obtained as Limit of Approximations) ofA(u)= in ,u=0 on , we prove that the truncationsT _k(u) at heightk satisfyA(T _k(u)) A(u) in the weak * topology of measures whenk + .

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Résumé SoitA(u)=–diva(x, u, Du) un opérateur de Leray-Lions défini surW ₀ ^1,p () et une mesure de Radon bornée. Pour toutu SOLA (Solution Obtenue comme Limite d'Approximations) deA(u)= dans ,u=0 sur , nous démontrons que les troncaturesT _k(u) à la hauteurk vérifientA(T _k(u)) A(u) dans la topologie faible * des mesures quandk + .

     

On the convex hull of the multiform numerical range

Let A ₁,…,A_m be nxn hermitian matrices. Definine

W(A ₁,…,A_m )={(xA₁x ^?,…xA_mx ^?):x?C ⁿ ,xx ^?=1}. We will show that every point in the convex hull of W(A ₁,…,A_m ) can be represented as a convex combination of not more than k(m,n) points in W(A ₁,…,A_m ) where k(m,n)=min{n,[√m]+δ _n ² _m+1}.     

Tracking and invariance problems for some classes of discrete delay systems

Consider the system with perturbation g _k ∈ ℝ ⁿ and output z _k = Cx _k . Here, A _k ,A _k ^(s) ∈ ℝ ^{n × n} , B _k ⁽¹⁾ ∈ ℝ ^{n × p} , B _k ⁽²⁾ ∈ ℝ ^{n × m} , C ∈ ℝ ^{p × n} . We construct a special Lyapunov-Krasovskii functional in order to synthesize controls u _k ⁽¹⁾ and u _k ⁽²⁾ for which the following properties are satisfied: