The Distributional Hankel Transform of Marcel Riesz's Ultrahyperbolic Kernel

In this article we give a sense to the distributional Hankel transform of Marcel Riesz's ultrahyperbolic kernel. First we evaluate Rα(u) in α = ?2k and α = 2k for the cases μ even and ν odd, μ even and ν even, and μ odd and ν odd, μ odd and ν even, where and Finally in Section 4 we obtain the distributional Hankel transform of Marcel Riesz's ultrahyperbolic kernel.     

On the Elementary Retarded,Ultrahyperbolic Solution of the Klein-Gordon Operator,Iterated k Times

Let t = (t₁,…,t_n) be a point of ?ⁿ. We shall write . We put, by the definition, W_α(u, m) = (m^?2u)^{(α ? n)/4}[π^{(n ? 2)/2}2^{(α + n ? 2)/2}Г(α/2)]J_{(α ? n)/2}(m²u)^1/2; here α is a complex parameter, m a real nonnegative number, and n the dimension of the space. W_α(u, m), which is an ordinary function if Re α ≥ n, is an entire distributional function of α. First we evaluate {□ + m²}W_{α + 2}(u, m) = W_α(u, m), where {□ + m²} is the ultrahyperbolic operator. Then we express W_α(u, m) as a linear combination of R_α(u) of differntial orders; R_α(u) is Marcel Riesz's ultrahyperbolic kernel. We also obtain the following results: W_?2k(u, m) = {□ + m²}^kδ, k = 0, 1,…; W₀(u, m) = δ; and {□ + m²}^kW_2k(u, m) = δ. Finally we prove that W_α(u, m = 0) = R_α(u). Several of these results, in the particular case µ = 1, were proved earlier by a completely different method.     

On Mean Convergence of Fourier-Bessel Series of Negative Order

The expansion of f ∈ L^p(0, 1) Fourier series of Bessel functions of order converges to f in L^p whenever Let be the space of p-integrable functions with respect to the measure t dt and where {s_n}, n = 1, 2, …, is the set of positive zeros of J_v. Then, the expansion of in a Fourier series of functions ψ_n, ?1 < ν < ?½, converges to in whenever      

Orthogonality via Transforms

Let e(x, t) = ∑p_n(x)tⁿ be the generating function of a polynomial sequence, and the transform of multiplication by x relative to e(x, t). We show that the sequence p_n(x) is orthogonal precisely when is a t-variable, i.e., maps K[t] into itself and increases degree by 1. We also show how transform techniques can shed light on the recursion relations and differential equations for p_n(x).     

The Asymptotic Solution of a Connection Problem of a Second Order Ordinary Differential Equation

The solutions of the equation are discussed in the limit ρ → 0. The solutions which oscillate about ? |t| as t → ∞ have asymptotic expansions whose leading terms are where Ã₊, , Ã_?, and are constants. The connection problem is to determine the asymptotic expansion at + ∞. In other words, we wish to find (Ã₊, ) as functions of Ã_? and The nonlinear solutions with Ã_± not small are analyzed by using the method of averaging. It is shown that this method breaks down for small amplitudes. In this case a solution can be obtained on [0, ∞) as a small amplitude perturbation about the exact nonoscillating solution W(t) whose asymptotic expansion is This is a solution of (1) which corresponds to Ã₊ ≡ 0 in (2). A quantity which determines the scale of the small amplitude response is ?W'(0). This quantity is found to be exponentially small. The determination of this constant is shown to reduce to a solution of the equation for the first Painlevé transcendent. The asymptotic behavior of the required solution is determined by solving an integral equation.     

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.     

Asymptotic Expansions for Second-Order Linear Difference Equations,II

Infinite asymptotic expansions are derived for the solutions to the second-order linear difference equation where p and q are integers, a(n) and b(n) have power series expansions of the form for large values of n, and a₀ ≠ 0, b₀ ≠ 0. Recurrence relations are also given for the coefficients in the asymptotic solutions. Our proof is based on the method of successive approximations. This paper is a continuation of an earlier one, in which only the special case p ≤ 0 and q = 0 is considered.     

A Solvable Nonlinear Wave Equation

It is pointed out that the nonlinear wave equation can be solved by quadratures. Here a and c are constants, A(y) and B(y) (arbitrary) functions; a t-dependence of all these quantities can also be accommodated. This wave equation can also be rewritten in the (purely differential) form via the substitutions .     

Eigenvalue Degeneracy and Existence and Non-existence of a Phase Transition in One Dimension

We consider a one dimensional Ising chain with interaction potential J(k) such that J(k) = 0 when k > n. By a perturbation argument we show that long range order exists at sufficiently low temperatures if and only if This is consistent with Dyson's recent theorems and in addition predicts that when J(k) = k^?2 there is no long range order.     

Oblique Interactions between Internal Solitary Waves

In this paper, we study the oblique interaction of weakly, nonlinear, long internal gravity waves in both shallow and deep fluids. The interaction is classified as weak when where Δ₁=|c_m/c_n?cosδ|, Δ₂=|c_n/c_m?cosδ|,c_m,n, are the linear, long wave speeds for waves with mode numbers m, n, δ is the angle between the respective propagation directions, and α measures the wave amplitude. In this case, each wave is governed by its own Kortweg-de Vries (KdV) equation for a shallow fluid, or intermediate long-wave (ILW) equation for a deep fluid, and the main effect of the interaction is an 0(α) phase shift. A strong interaction (I) occurs when Δ_1,2 are 0(α), and this case is governed by two coupled Kadomtsev-Petviashvili (KP) equations for a shallow fluid, or two coupled two-dimensional ILW equations for deep fluids. A strong interaction (II) occurs when Δ₁ is 0(α), and (or vice versa), and in this case, each wave is governed by its own KdV equation for a shallow fluid, or ILW equation for a deep fluid. The main effect of the interaction is that the phase shift associated with Δ₁ leads to a local distortion of the wave speed of the mode n. When the interacting waves belong to the same mode (i.e., m = n) the general results simplify and we show that for a weak interaction the phase shift for obliquely interacting waves is always negative (positive) for (1/2+cosδ)>0(<0), while the interaction term always has the same polarity as the interacting waves.     

On the Eigenvalue Problem for a Certain

This paper studies the spectral properties of the partial differential operator over a finite region Ω. This operator, which arises in the analysis of nonaxisymmetric, rapidly rotating compressible flows, is treated as a perturbation of the operator which is generated by the terms Using the fact that , when defined on a suitable domain, is closed and self-adjoint, it is shown that [when acting on elements of ] is an operator with compact resolvent whose generalized eigenvectors are complete in ?² (Ω).     

Asymptotic Estimates of Stirling Numbers

New asymptotic estimates are given of the Stirling numbers and , of first and second kind, respectively, as n tends to infinity. The approximations are uniformly valid with respect to the second parameter m.     

On the Generalizations of a Theorem of Beppo Levi

In this paper we give two generalizations of a theorem of Beppo Levi ([1], p. 347, Formula (12)). This theorem affirms that, under certain conditions, the following assertion is true: where φ(x) is a function that verifies φ(0) > 0; f(x) is defined and bounded in the interval (a, b) and continuous in the point 0 with f(0) ≠ 0; f(x) and φ(x) are integrable functions in the interval [a, b]; c >, 0 and υ > 1. This problem was studied by Laplace [2], Darboux [3], Stieltjes [4], Lebesgue [5], Romanovsky [6], and Fowler [7]. The first generalization (Section 1, Theorem 1.2, Formula (1.35)) says that, under certain conditions, the following formula is valid: where φ_n(x) is a sequence of functions and Bn(a) designates the n-dimentional ball of radius a and center in the origin. The extension follows by Romanovsky's method. The absolute maximum of φ(x) in the extremes of the interval of definition is treated in the second generalization of the Theorem of Beppo Levi (Section 2, Theorem 2.2, Formulas (2.1), (2.2)). We note that Beppo Levi proves this assertion in the interior of the interval.     

Asymptotics beyond All Orders in a Model of Crystal Growth

In its simplest form, the geometric model of crystal growth is a third-order, nonlinear, ordinary differential equation for θ(s, ε): A needle crystal is a solution that satisfies boundary conditions The geometric model admits a needle-crystal solution for ε = 0; for small ε, it admits an asymptotic expansion that is valid to all orders for such a solution. Even so, we prove that the geometric model in this form admits no needle crystal for any small, nonzero ε, a fact that lies beyond all orders of the asymptotic expansion. A more complicated version of the geometric model is where α represents crystalline anisitropy. We show that for 0 < α < 1, the geometric model admits needle crystals for a discrete set of values of α. The number of such values of α increases like ε^?1 as ε → 0.     

On weak and Moments Convergence of Randomly Indexed Sums

Let {X_n, n ? 1) be a sequence of independent random variables such that EX_n = a_n, E(X_n ? a_n)² = σ, n ? 1. Let {N_n, n ? 1} be a sequence of positive integer-valued random variables. Let us put In this paper we present necessary and sufficient conditions for weak and moments convergence of the sequence {(S-L_n)/s_n, n ? 1}, as n → ∞. Hermite polinomial type limit theorems are also considered. The obtained results extend the main theorem of M. Finkelstein and H. G. Tucker (1989).     

Asymptotic Behavior of the Pollaczek Polynomials and Their Zeros

In 1954, A. Novikoff studied the asymptotic behavior of the Pollaczek polynomials P_n(x; a, b) when , where t > 0 is fixed. He divided the positive t-axis into two regions, 0 < t < (a + b)^1/2 and t > (a + b)^1/2, and derived an asymptotic formula in each of the two regions. Furthermore, he found an asymptotic formula for the zeros of these polynomials. Recently M. E. H. Ismail (1994) reconsidered this problem in an attempt to prove a conjecture of R. A. Askey and obtained a two-term expansion for these zeros. Here we derive an infinite asymptotic expansion for , which holds uniformly for 0 < ε ≤ t ≤ M < ∞, and show that Ismail's result is incorrect.     

Inverse Spectral Transform for the Nonlinear Evolution Equation Generating the Davey-Stewartson and Ishimori Equations

A 2 + 1-dimensional nonlinear differential equation integrable by the inverse-spectral-transform method with the quartet operator representation is proposed. This GL(2, C)-valued chiral-field-type equation is the generating (prototype) equation for the Davey-Stewartson and Ishimori equations. It coincides with the nonlinear equation for the Davey-Stewartson eigenfunction ψ_DS. The initial-value problem for this equation is solved by the techniques for the and the nonlocal Riemann-Hilbert problem. The classes of exact solutions with the functional parameters and exponential-rational solutions are constructed by the method. The static lump solution in the case α = i and the exponentially localized solution at α = i are found. Other similar examples of nonlinear integrable equations in 2 + 1 and 1 + 1 dimensions are discussed.     

Hyper-rook Domain Inequalities

A hyper-rook domain of an element x in the space (words of length n over alphabets with k elements) is a sphere with center x and fixed radius j in Hamming distance. The number j determines the dimension of the hyper-rook domain. The classical (and far from solved) problem of covering by rook domains (here considered as the 1-dimensional case) is the problem of finding minimal coverings of by such spheres. Very few results are known in the literature for dimensions ≥ 2. We prove in this paper certain classes of inequalities based on coverings using matrices, which give upper and lower bounds for several cases of the problem for higher dimensions.     

Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source

We study a quasilinear parabolic–elliptic Keller–Segel system involving a source term of logistic type u_t = ? ? (?(u) ? u) ? χ ? ? (u ? v) + g(u), ? Δv = ? v + u in Ω × (0,T), subject to nonnegative initial data and the homogeneous Neumann boundary condition in a bounded domain with smooth boundary, n ≥ 1, χ > 0, ? ≥ c₁s^p for s ≥ s₀ > 1, and g(s) ≤ as ? μs² for s > 0 with a,g(0) ≥ 0, μ > 0. There are three nonlinear mechanisms included in the chemotaxis model: the nonlinear diffusion, aggregation and logistic absorption. The interaction among the triple nonlinearities shows that together with the nonlinear diffusion, the logistic absorption will dominate the aggregation such that the unique classical solution of the system has to be global in time and bounded, regardless of the initial data, whenever , or, equivalently, , which enlarge the parameter range , or , required by globally bounded solutions of the quasilinear K‐S system without the logistic source. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence and concentration of bound states for a Kirchhoff type problem with potentials vanishing or unbounded at infinity

In this paper, we study the existence and concentration behavior of positive solutions for the following Kirchhoff type equation: where ɛ is a positive parameter, a and b are positive constants, and 3<p<5. Let denotes the ground energy function associated with , , where is regard as a parameter. Suppose that the potential V(x) decays to zero at infinity like |x|^−α with 0<α≤2, we prove the existence of positive solutions u_ɛ belonging to for vanishing or unbounded K(x) when ɛ > 0 small. Furthermore, we show that the solution u_ɛ concentrates at the minimum points of as ɛ→0⁺.