Integral representations of the Legendre chi function

We, by making use of elementary arguments, deduce integral representations of the Legendre chi function χs(z) valid for |z|<1 and Res>1. Our earlier established results on the integral representations for the Riemann zeta function ζ(2n+1) and the Dirichlet beta function β(2n), n∈N, are a direct consequence of these representations.     

Error bounds for the uniform asymptotic expansion of the incomplete gamma function

We derive simple, explicit error bounds for the uniform asymptotic expansion of the incomplete gamma function Γ(a,z) valid for complex values of a and z as |a|→∞. Their evaluation depends on numerically pre-computed bounds for the coefficients c_k(η) in the expansion of Γ(a,z) taken along rays in the complex η plane, where η is a variable related to z/a. The bounds are compared with numerical computations of the remainder in the truncated expansion.     

A Liapunov functional for a singular integral equation

We consider a scalar integral equation where a∈L²[0,∞), while C(t,s) has a significant singularity, but is convex when t−s>0. We construct a Liapunov functional and show that g(t,x(t))−a(t)∈L²[0,∞) and that x(t)−a(t)→0 pointwise as t→∞. Small perturbations are also added to the kernel. In addition, we consider both infinite and finite delay problems. This paper offers a first step toward treating discontinuous kernels with Liapunov functionals.     

Scalar conservation laws with general boundary condition and continuous flux function

We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0,⋅)=u₀ on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any Φ∈C(R;RN), u₀∈L^∞(Ω), f∈L^∞(Q), a∈L^∞((0,T)×∂Ω). In the L¹-setting, a corresponding result is proved for the more general notion of renormalised entropy solution.     

On normal families and differential polynomials for meromorphic functions

We consider the normality criterion for a families F meromorphic in the unit disc Δ, and show that if there exist functions a(z) holomorphic in Δ, a(z)≠1, for each z∈Δ, such that there not only exists a positive number ε₀ such that |an(a(z)−1)−1|?ε₀ for arbitrary sequence of integers an(n∈N) and for any z∈Δ, but also exists a positive number B>0 such that for every f(z)∈F, B|f^′(z)|?|f(z)| whenever f(z)f^″(z)−a(z)(f^′2(z))=0 in Δ. Then is normal in Δ.     

Fourier series for zeta function via Sinc

In this paper we derive some Fourier series and Fourier polynomial approximations to a function F which has the same zeros as the zeta function, ζ(z) on the strip {z∈C:0<Rz<1}. These approximations depend on an arbritrary positive parameter h, and which for arbitrary ε∈(0,1/2), converge uniformly to ζ(z) on the rectangle {z∈C:ε<Rz<1-ε,-π/h<Iz<π/h}.     

Inversion of analytically perturbed linear operators that are singular at the origin

Let H and K be Hilbert spaces and for each z∈C let A(z)∈L(H,K) be a bounded but not necessarily compact linear map with A(z) analytic on a region |z|<a. If A(0) is singular we find conditions under which A⁻¹(z) is well defined on some region 0<|z|<b by a convergent Laurent series with a finite order pole at the origin. We show that by changing to a standard Sobolev topology the method extends to closed unbounded linear operators and also that it can be used in Banach spaces where complementation of certain closed subspaces is possible. Our method is illustrated with several key examples.²     

Structure of boundary blow-up solutions for quasi-linear elliptic problems II: small and intermediate solutions

The structure of positive boundary blow-up solutions to quasi-linear elliptic problems of the form −Δ_pu=λf(u), u=∞ on ∂Ω, 1<p<∞, is studied in a bounded smooth domain , for a class of nonlinearities f∈C¹((0,∞)?{z₂})∩C⁰[0,∞) satisfying f(0)=f(z₁)=f(z₂)=0 with 0<z₁<z₂, f<0 in (0,z₁)∪(z₂,∞), f>0 in (z₁,z₂). Large, small and intermediate solutions are obtained for λ sufficiently large. It is known from Part I (see Structure of boundary blow-up solutions for quasilinear elliptic problems, part (I): large and small solutions, preprint), that the large solution is the unique large solution to the problem. We will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions. Our results are new even for the case p=2.     

Three-term asymptotic expansions for the moments of the random walk with triangular distributed interference of chance

In this study, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of ergodic distribution of the process X(t) are obtained when the random variable ζ₁, which is describing a discrete interference of chance, has a triangular distribution in the interval [s, S] with center (S + s)/2. Based on these results, the asymptotic expansions with three-term are obtained for the first four moments of the ergodic distribution of X(t), as a ≡ (S − s)/2 → ∞. Furthermore, the asymptotic expansions for the variance, skewness and kurtosis of the ergodic distribution of the process X(t) are established. Finally, by using Monte Carlo experiments it is shown that the given approximating formulas provide high accuracy even for small values of parameter a.     

A simple approach to asymptotic expansions for Fourier integrals of singular functions

In this work, we are concerned with the derivation of full asymptotic expansions for Fourier integrals as s → ∞, where s is real positive, [a, b] is a finite interval, and the functions f(x) may have different types of algebraic and logarithmic singularities at x = a and x = b. This problem has been treated in the literature by techniques involving neutralizers and Mellin transforms. Here, we derive the relevant asymptotic expansions by a method that employs simpler and less sophisticated tools.     

On a result of G. Brosch

We prove a uniqueness theorem for non-constant meromorphic functions f, g which share three values 0, 1, ∞ and f−a, g−b share the value 0 for a,b∉{0,1,∞}. Our theorem improves a result of G. Brosch.     

Generation of the integrated semigroups by superelliptic differential operators

Let A be a superelliptic differential operator of order 2m introduced by E.B. Davies [E.B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141-169]. In the case of 2m>N, he obtained the upper Gaussian bound of the integral kernel representing (e−zA)_z∈C+ and the estimates of the Lp-operator norm of the semigroup for all p∈[1,∞). The purpose of the present paper is to show that −i(A+k) (for some constant k>0) generates an integrated semigroup on Lα,p (weighted Lp space) and lp(Lα,q). To prove this we need norm estimates of (e−zA)_z∈C+ on each of these spaces. Also we get another norm estimate of (e−zA)_z∈C+ on Lp when 2m>N without using the integral kernel. This norm estimate is better than that in [E.B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141-169] and gives a better “times of the integration” of the integrated semigroup.     

Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions

Expansions in terms of Bessel functions are considered of the Kummer function ₁ F ₁(a; c, z) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials in the parameters of the Kummer function and the asymptotic behavior of these polynomials for large degree is given. Tables are given to show the rate of approximation of the asymptotic estimates. The numerical performance of the expansions is discussed together with the numerical stability of recurrence relations to compute the polynomials. The asymptotic character of the expansions is explained for large values of the parameter a of the Kummer function.     

Positive solutions of nonlinear three-point boundary-value problems

Let a∈C[0,1], b∈C([0,1],(−∞,0]). Let φ₁(t) be the unique solution of the linear boundary value problem

u″(t)+a(t)u′(t)+b(t)u(t)=0,t∈(0,1),u(0)=0,u(1)=1.     

Heteroclinic solutions for a class of the second order Hamiltonian systems

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system , where q∈Rn and V∈C¹(R×Rn,R), V?0. We will assume that V and a certain subset M⊂Rn satisfy the following conditions. M is a set of isolated points and #M?2. For every sufficiently small ε>0 there exists δ>0 such that for all (t,z)∈R×Rn, if d(z,M)?ε then −V(t,z)?δ. The integrals , z∈M, are equi-bounded and −V(t,z)→∞, as |t|→∞, uniformly on compact subsets of Rn?M. Our result states that each point in M is joined to another point in M by a solution of our system.     

Modified zeta functions as kernels of integral operators

The modified zeta functions _∑n∈Kn−s, where K⊂N, converge absolutely for . These generalise the Riemann zeta function which is known to have a meromorphic continuation to all of C with a single pole at s=1. Our main result is a characterisation of the modified zeta functions that have pole-like behaviour at this point. This behaviour is defined by considering the modified zeta functions as kernels of certain integral operators on the spaces L²(I) for symmetric and bounded intervals I⊂R. We also consider the special case when the set K⊂N is assumed to have arithmetic structure. In particular, we look at local Lp integrability properties of the modified zeta functions on the abscissa for p∈[1,∞].     

Existence Criterion for Estimates of Derivatives of Rational Functions

Suppose that K is a compact set in the open complex plane. In this paper, we prove an existence criterion for an estimate of Markov-Bernstein type for derivatives of a rational function R(z) at any fixed point z ₀ ∈ K. We prove that, for a fixed integer s, the estimate of the form |R ^(s) (z ₀)| ≤ C(K, z ₀, s)n‖R‖ _C(K), where R is an arbitrary rational function of degree n without poles on K and C is a bounded function depending on three arguments K, z ₀, and s, holds if and only if the supremum $$\omega (K,z_0 ,s) = \sup \left\{ {\frac{{\operatorname{dist} (z,K)}}{{\left| {z - z_0 } \right|^{s + 1} }}} \right\}$$ over z in the complement of K is finite. Under this assumption, C is less than or equal to const ·s!ω(K, z ₀, s).     

Global Strichartz estimates for the wave equation with time-periodic potentials

We obtain global Strichartz estimates for the solutions u of the wave equation for time-periodic potentials V(t,x) with compact support with respect to x. Our analysis is based on the analytic properties of the cut-off resolvent Rχ(z)=χ(U⁻¹(T)−zI)ψ₁, where U(T)=U(T,0) is the monodromy operator and T>0 the period of V(t,x). We show that if Rχ(z) has no poles z∈C, |z|?1, then for n?3, odd, we have a exponential decal of local energy. For n?2, even, we obtain also an uniform decay of local energy assuming that Rχ(z) has no poles z∈C, |z|?1, and Rχ(z) remains bounded for z in a small neighborhood of 0.     

On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry

We consider families (Yn) of degenerating hyperbolic surfaces. The surfaces are geometrically finite of fixed topological type. Let Zn be the Selberg Zeta function of Yn, and let zn be the contribution of the pinched geodesics to Zn. Extending a result of Wolpert's, we prove that Zn(s)/zn(s) converges to the Zeta function of the limit surface if Re(s)>1/2. The technique is an examination of resolvent of the Laplacian, which is composed from that for elementary surfaces via meromorphic Fredholm theory. The resolvent ⁻¹(Δn−t) is shown to converge for all t∉[1/4,∞). We also use this property to define approximate Eisenstein functions and scattering matrices.