Real Half-line Research Articles

The zig-zag walk with scattering and absorption on the real half line and in a lattice model

The Darwin–Hamilton equations, describing one-dimensional transport with scattering and absorption, are expanded into a recursion. The solution involves ballot numbers. The recurrence probability as function of scattering order is given by Catalan numbers. To reproduce this analytical result in a lattice model, a novel relation between Narayana and Catalan numbers is derived.

Application of Chebyshev Polynomials to the Approximate Solution of Singular Integral Equations of the First Kind with Cauchy Kernel on the Real Half-line

In this paper, exact solution of the characteristic equation with Cauchy kernel on the real half-line is presented. Next, the Chebyshev polynomials of the second kind, $U_{n}(x)$, and fourth kind, $W_{n}(x)$, are used to derive numerical solutions of Cauchy -type singular integral equations of the first kind on the real half-line. The collocation points are chosen as the zeros of the Chebyshev polynomials of the first kind, $T_{n+2}(x)$, and third kind, $V_{n+1}(x)$. Moreover, estimations of errors of the approximated solutions are presented. The numerical results are given to show the accuracy of the methods presented.

Mild solutions for perturbed evolution equations with infinite state-dependent delay

In this paper, we give sucient conditions to get the existence of mild solutions for two classes of rst order partial and neutral of perturbed evolution equations by using the nonlinear alternative of Avramescu for contractions operators in Fr echet spaces, combined with semigroup theory. The solution here is depending on an innite delay and is giving on the real half-line.

Complex form of the Hardy-Titchmarsh-Watson characterization of Fourier kernels and some of its consequences

Kernels of integral transforms of the form k(xy) are called Fourier kernels. Hardy and Titchmarsh (6) and Watson (15) studied self- reciprocal transforms with Fourier kernels on the positive half-real line R+ and obtained nice characterizing conditions and other properties. Neverthe- less, the Fourier transform has an inherent complex, unitary structure and by treating it on the whole real line more interesting properties can be ex- plored. In this paper, we characterize all unitary integral transforms on L 2 (R) with Fourier kernels. Characterizing conditions are obtained through a natu- ral splitting of the kernel on R+ and R−, and the conversion to convolution integrals. A collection of concrete examples are obtained, which also include discrete transforms (that are not integral transforms). Furthermore, we ex- plore algebraic structures for operator-composing integral kernels and identify a certain Abelian group associated with one of such operations. Fractional powers can also be assigned a new interpretation and obtained.

Quadratures associated with pseudo-orthogonal rational functions on the real half line with poles in [formula omitted

We consider a positive measure on [0,∞) and a sequence of nested spaces ℒ0⊂ℒ1⊂ℒ2⋯ of rational functions with prescribed poles in [−∞,0]. Let {φk}k=0∞, with φ0∈ℒ0 and φk∈ℒk∖ℒk−1, k=1,2,… be the associated sequence of orthogonal rational functions. The zeros of φn can be used as the nodes of a rational Gauss quadrature formula that is exact for all functions in ℒn⋅ℒn−1, a space of dimension 2n. Quasi- and pseudo-orthogonal functions are functions in ℒn that are orthogonal to some subspace of ℒn−1. Both of them are generated from φn and φn−1 and depend on a real parameter τ. Their zeros can be used as the nodes of a rational Gauss–Radau quadrature formula where one node is fixed in advance and the others are chosen to maximize the subspace of ℒn⋅ℒn−1 where the quadrature is exact. The parameter τ is used to fix a node at a preassigned point. The space where the quadratures are exact has dimension 2n−1 in both cases but it is in ℒn−1⋅ℒn−1 in the quasi-orthogonal case and it is in ℒn⋅ℒn−2 in the pseudo-orthogonal case.

Erratum to: The hitting distributions of a half real line for two-dimensional random walks

In Theorem 1.1 of my paper [1] (p. 373) there is an erroneous statement. The formula (3) of the theorem expresses the asymptotic form of H x (s), the hitting distribution of the non-positive half line for a random walk on Z started at x. Its first statement, which is true, asserts that (3) holds for x>0. The error is included in the second one, in which the validity of (3) is asserted also for x<0 under the additional moment condition E [|S| log |S|]<∞. For x<0 however, the righthand side of (3) must be multiplied by the ratio (|x|+|s|)/2|x−s|, namely the true statement must read: if E [|S| log |S|]<∞ in addition, then for x<0,

On the exact [formula omitted] Markov inequality on some unbounded domains in [formula omitted

We consider the Markov problem of finding the so-called Markov factor M ( U , K ) : = sup u ∈ U ‖ D u ‖ ‖ u ‖ , of the set of differentiable functions U , where D u : = | ∂ u | ℓ 2 stands for the ℓ 2 -norm of the gradient vector of u , and ‖ ⋅ ‖ is the weighted L 2 norm on the set K ⊂ R d . In the univariate case exact L 2 -Markov inequalities are known for algebraic polynomials on the real line, half line and intervals. We outline a variational approach to the above problem and show how this leads either to certain partial differential equations, or to a system of homogeneous linear equations. This method will be illustrated by using it to solve the L 2 Markov problem for the cases of d -dimensional spaces and d -dimensional hyperquadrants. In the case of d -spaces the solution is given for homogeneous polynomials, as well.

Competing particle systems evolving by interacting Lévy processes

We consider finite and infinite systems of particles on the real line and half-line evolving in continuous time. Hereby, the particles are driven by i.i.d. Lévy processes endowed with rank-dependent drift and diffusion coefficients. In the finite systems we show that the processes of gaps in the respective particle configurations possess unique invariant distributions and prove the convergence of the gap processes to the latter in the total variation distance, assuming a bound on the jumps of the Lévy processes. In the infinite case we show that the gap process of the particle system on the half-line is tight for appropriate initial conditions and same drift and diffusion coefficients for all particles. Applications of such processes include the modeling of capital distributions among the ranked participants in a financial market, the stability of certain stochastic queueing and storage networks and the study of the Sherrington–Kirkpatrick model of spin glasses.

On a class of singular Sturm–Liouville periodic boundary value problems

Keeping in mind the singular model for the periodic oscillations of the axis of a satellite in the plane of the elliptic orbit around its center of mass, we give sufficient conditions for the solvability of a class of singular Sturm–Liouville equations with periodic boundary value conditions. To this end, under a suitable change of variables, we present a new existence result for problems defined in the real half-line.

The equivalence of two functional equations involving the arithmetic mean, the geometric mean and their Gauss composition

In this paper, the equivalence of the two functional equations $$f\left(\frac{x+y}{2} \right)+f\left(\sqrt{xy} \right)=f(x)+f(y)$$ and $$2f\left(\mathcal{G}(x,y)\right)=f(x)+f(y)$$ will be proved by showing that the solutions of either of these equations are constant functions. Here I is a nonvoid open interval of the positive real half-line and \({\mathcal{G}}\) is the Gauss composition of the arithmetic and geometric means.

The hitting distributions of a half real line for two-dimensional random walks

For every two-dimensional random walk on the square lattice Z2 having zero mean and finite variance we obtain fine asymptotic estimates of the probability that the walk hits the negative real line for the first time at a site (s,0), when it is started at a site far from both (0, s) and the origin.

Properties of the solutions of the fourth-order Bessel-type differential equation

The structured Bessel-type functions of arbitrary even-order were introduced by Everitt and Markett in 1994; these functions satisfy linear ordinary differential equations of the same even-order. The differential equations have analytic coefficients and are defined on the whole complex plane with a regular singularity at the origin and an irregular singularity at the point of infinity. They are all natural extensions of the classical second-order Bessel differential equation. Further these differential equations have real-valued coefficients on the positive real half-line of the plane, and can be written in Lagrange symmetric (formally self-adjoint) form. In the fourth-order case, the Lagrange symmetric differential expression generates self-adjoint unbounded operators in certain Hilbert function spaces. These results are recorded in many of the papers here given as references. It is shown in the original paper of 1994 that in this fourth-order case one solution exists which can be represented in terms of the classical Bessel functions of order 0 and 1. The existence of this solution, further aided by computer programs in Maple, led to the existence of a linearly independent basis of solutions of the differential equation. In this paper a new proof of the existence of this solution base is given, on using the advanced theory of special functions in the complex plane. The methods lead to the development of analytical properties of these solutions, in particular the series expansions of all solutions at the regular singularity at the origin of the complex plane.

Semiclassical resonances for a two-level Schrödinger operator with a conical intersection

We study the resonant set of a two-level Schrodinger operator with a linear conical intersection. This model operator can be decomposed into a direct sum of first order systems on the real half-line. For these ordinary differential systems we locally construct exact WKB solutions, which are connected to global solutions, amongst which are resonant states. The main results are a generalized Bohr-Sommerfeld quantization condition and an asymptotic description of the set of resonances as a distorted lattice.

Periodic solutions for some nonautonomous semilinear boundary evolution equations

In this work we study the Massera problem for the existence of a periodic mild solution of a class of nonautonomous semilinear boundary evolution equations (0.1) { x ′ ( t ) = A m ( t ) x ( t ) + f ( t , x ( t ) ) , t ≥ 0 , L ( t ) x ( t ) = Φ ( t ) x ( t ) + g ( t , x ( t ) ) , t ≥ 0 , x ( 0 ) = x 0 . First, we prove the existence of a periodic solution for nonhomogeneous boundary evolution equations under the existence of a bounded solution on the right half real line. Next, by using a fixed point theorem, we investigate the existence of periodic solutions in the semilinear case. We end with an application to a periodic heat equation with semilinear boundary conditions.

Feynman and Feynman-Kac formulas for evolution equations with Vladimirov operator

A Feynman formula is a representation of a solution to the Cauchy problem for an evolution differential or pseudodifferential equation in terms of a limit of integrals over the Cartesian degrees of some space E . A Feynman‐Kac formula is a representation of a solution to the same problem in terms of a path integral. We assume that, on the path space, a countably additive measure or a pseudomeasure (of the type of the Feynman measure; see [2, 3]) is defined, and the multiple integrals in the Feynman formulas coincide with integrals of finite multiplicity approximating integrals with respect to this measure or pseudomeasure. In this paper, we obtain Feynman and Feynman‐ Kac formulas for solutions to the Cauchy problems for the heat equation with respect to complex-valued functions on the product of the real half-line and the p -adic line � � ; the role of the Laplace operator in these equations is played by the Vladimirov operator. Similar formulas can be obtained for Schrodinger-type equations and for the case of a multidimensional space over � � . Such equations may be useful in constructing mathematical models of processes on scales characterized by Planck length and time and phenomenological models in chemistry, continuum mechanics, and psychology (see [1, 4‐6] and the references therein).

Almost Automorphic Solutions for Partial Functional Differential Equations with Infinite Delay

In this work, we study the existence of almost automorphic solutions for partial functional differential equations with infinite delay. We assume that the undelayed part is not necessarily densely defined and satisfies the Hille-Yosida condition. We use the so-called reduction principle developed recently in [3], to show the existence of an almost automorphic solution under minimal condition. More precisely, the existence of an almost automorphic solution is proved when there is at least one bounded solution in the positive real half line. We give an application to the Lotka-Volterra model with diffusion.

On the spectrum of a Stokes-type operator arising from flow around a rotating body

We present the description of the spectrum of a linear perturbed Stokes-type operator which arises from equations of motion of a viscous incompressible fluid in the exterior of a rotating compact body. Considering the operator in the function space \(L^2_{\sigma}(\Omega)\) we prove that the essential spectrum consists of a set of equally spaced half lines parallel to the negative real half line in the complex plane. Our approach is based on a reduction to invariant closed subspaces of \(L^2_{\sigma}(\Omega)\) and on a Fourier series expansion with respect to an angular variable in a cylindrical coordinate system attached to the axis of rotation. Moreover, we show that the leading part of the operator is normal if and only if the body is axially symmetric about this axis.

SQUARES OF WHITE NOISE, SL(2, ℂ) AND KUBO–MARTIN–SCHWINGER STATES

We investigate the structure of Kubo–Martin–Schwinger (KMS) states on some extension of the universal enveloping algebra of SL (2, ℂ). We find that there exists a one-to-one correspondence between the set of all covariant KMS states on this algebra and the set of all probability measures dμ on the real half-line [0, +∞), which decrease faster than any inverse polynomial. This problem is connected to the problem of KMS states on square of white noise algebra.

On existence and asymptotic behaviour of solutions of a functional integral equation

The paper contains a result on the existence and asymptotic behaviour of solutions of a functional integral equation. That result is proved under rather general hypotheses. The main tools used in our considerations are the concept of a measure of noncompactness and the classical Schauder fixed point principle. The investigations of the paper are placed in the space of continuous and tempered functions on the real half-line. We prove an existence result which generalizes several ones concerning functional integral equations and obtained earlier by other authors. The applicability of our result is illustrated by some examples.

Product-limit estimators of the survival function with twice censored data

A model for competing (resp. complementary) risks survival data where the failure time can be left (resp. right) censored is proposed. Product-limit estimators for the survival functions of the individual risks are derived. We deduce the strong convergence of our estimators on the whole real half-line without any additional assumptions and their asymptotic normality under conditions concerning only the observed distribution. When the observations are generated according to the double censoring model introduced by Turnbull, the product-limit estimators represent upper and lower bounds for Turnbull's estimator.