Entire Real Line Research Articles

Reversible Bifurcation of Homoclinic Solutions in Presence of an Essential Spectrum

We consider bifurcations of a class of infinite dimensional reversible dynamical systems which possess a family of symmetric equilibria near the origin. We also assume that the linearized operator at the origin Lɛ has an essential spectrum filling the entire real line, in addition to the simple eigenvalue at 0. Moreover, for parameter values ɛ < 0 there is a pair of imaginary eigenvalues which meet in 0 for ɛ = 0, and which disappear for ɛ > 0. The above situation occurs for example when one looks for travelling waves in a system of superposed perfect fluid layers, one being infinitely deep. We give quite general assumptions which apply in such physical examples, under which one obtains a family of bifurcating solutions homoclinic to every equilibrium near the origin. These homoclinics are symmetric and decay algebraically at infinity, being approximated at main order by the Benjamin–Ono homoclinic. For the water wave example, this corresponds to a family of solitary waves, such that at infinity the upper layer slides with a uniform velocity, over the bottom layer (at rest).

Approximation of Continuous Functions by de La Vallee-Poussin Operators

For the upper bounds of the deviations of a function defined on the entire real line from the corresponding values of the de la Vallee-Poussin operators, we find asymptotic equalities that give a solution of the well-known Kolmogorov–Nikol'skii problem.

The Lesser-Known Δ-Function in Number Theory

We actually use this definition for any nonnegative real number x and extend it to the entire real line by setting A(x, N) = A({x}, N). Here {x} = x [x] signifies the fractional part of x. As a function of x, A(x, N) is clearly periodic of period 1, with A(0, N) = 0. Large values of |A(x, N) I correspond to abnormal numbers of integers coprime to N in the interval [1, xN]. In particular, if N is the product of all primes up to some parameter y, such possible fluctuations of I A(x, N)I, as x varies, have a bearing on anticipated sieve bounds. These are modem versions of the classical sieve of Eratosthenes that produces primes, up to some fixed bound, from the list of positive integers

Exact solutions of a Schrödinger equation based on the Lambert function

An exactly solvable Schrödinger equation of the confluent Natanzon class is derived using the differential properties of the Lambert W function. This potential involves two constant parameters and is defined along the entire real line. Specific spatial forms demonstrating wells and deformed positive barriers are presented.

Homoclinic solutions of reversible systems possessing an essential spectrum

In this Note we consider bifurcations of a class of infinite dimensional reversible dynamical systems. These systems possess a family of equilibrium solutions near the origin. We also assume that the linearized operator at the origin L ɛ has an essential spectrum filling the entire real line, in addition to a simple eigenvalue at 0. Moreover, for parameter values ɛ < 0 there is a pair of imaginary eigenvalues which meet in 0 for ɛ = 0 , and which disappear for ɛ > 0 . We give assumptions on L ɛ and on the non-linear term which describe this situation. These assumptions are sufficient to prove the existence of a family of solutions homoclinic to the equilibrium solutions near the origin. The result of this Note applies when we look for solitary waves in superposed layers of perfect fluids, the bottom one being infinitely deep. To cite this article: M. Barrandon, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

An extension of the Euler Laplace transform inversion algorithm with applications in option pricing

We show that the Euler algorithm for Laplace transform inversion can be extended to functions defined on the entire real line, if they have specific decay features. Our objective is to apply the method to option pricing problems, specifically when inverting Laplace transforms of the option price in the logarithm of the strike.

Reconciling the Integral Equation and Markov Chain Approaches for Computing EWMA Average Run Lengths

The integral equation and Markov chain approaches for computing average run lengths for two-sided exponentially weighted moving average control charts are studied. For the integral equation approach, the choice of numerical method can greatly ease the burden of computation. Gaussian quadrature is recommended when the underlying process data arise from a distribution whose support is the entire real line; however, the Collocation method is to be preferred when the support is finite or semi-infinite. Results for EWMA average run length calculations are given for process data following normal, gamma, t, and uniform distributions. Ultimately, the Markov chain approach is shown to be equivalent to a special case of the integral equation method.

Fourier series and elliptic functions

Non-linear second-order differential equations whose solutions are the elliptic functions sn(t, k), cn(t, k) and dn(t, k) are investigated. Using Mathematica, high precision numerical solutions are generated. From these data, Fourier coefficients are determined yielding approximate formulas for these non-elementary functions that are correct to at least 11 decimal places. These formulas have the advantage over numerically generated data that they are computationally efficient over the entire real line. This approach is seen as further justification for the early introduction of Fourier series in the undergraduate curriculum, for by doing so, models previously considered hard or advanced, whose solution involves elliptic functions, can be solved and plotted as easily as those models whose solutions involve merely trigonometric or other elementary functions.

OPTIMAL CONTROL OF A SIMPLE IMMIGRATION PROCESS THROUGH THE INTRODUCTION OF A PREDATOR

This article is concerned with the problem of controlling a simple immigration process, which represents a pest population, by the introduction of a predator. It is assumed that the cost rate caused by the pests is an increasing function of their population size and that the cost rate of the controlling action is constant. The existence of a control-limit policy that minimizes the expected long-run average cost per unit time is established. The proof is based on the variation of a fictitious parameter over the entire real line.

Convergence for Fourier series solutions of the forced harmonic oscillator II

This paper compliments two recent articles by the author in this journal concerning solving the forced harmonic oscillator equation when the forcing is periodic. The idea is to replace the forcing function by its Fourier series and solve the differential equation term-by-term. Herein the convergence of such series solutions is investigated when the forcing function is bounded, piecewise continuous, and piecewise smooth. The series solution and its term-by-term derivative converge uniformly over the entire real line. The term-by-term differentiation produces a series for the second derivative that converges pointwise and uniformly over any interval not containing a jump discontinuity of the forcing function.

Pauli's theorem and quantum canonical pairs: the consistency of a bounded, self–adjoint time operator canonically conjugate to a Hamiltonian with non–empty point spectrum

In single Hilbert space, Pauli's well-known theorem implies that the existence of a self-adjoint time operator canonically conjugate to a given Hamiltonian signifies that the time operator and the Hamiltonian possess completely continuous spectra spanning the entire real line. Thus the conclusion that there exists no self-adjoint time operator conjugate to a semibounded or discrete Hamiltonian despite some well-known illustrative counterexamples. In this paper we evaluate Pauli's theorem against the single Hilbert space formulation of quantum mechanics, and consequently show the consistency of assuming a bounded, self-adjoint time operator canonically conjugate to a Hamiltonian with an unbounded, or semibounded, or finite point spectrum. We point out Pauli's implicit assumptions and show that they are not consistent in a single Hilbert space. We demonstrate our analysis by giving two explicit examples. Moreover, we clarify issues sorrounding the different solutions to the canonical commutation relations, and, consequently, expand the class of acceptable canonical pairs beyond the solutions required by Pauli's theorem.

The Best Extension Operators for Sobolev Spaces on the Half-Line

We describe the construction of extension operators with minimal possible norm τm from the half-line to the entire real line for the spaces \(W_2^m \) and derive the asymptotic estimate \(\tau _m \approx K_0 m\;\;({\text{as }}m \to \infty )\), where $$K_0 : = \tfrac{4}{\pi }\int_0^{x/4} {\ln (\cot x)} \;dx = 1.166243... = \ln 3.209912....$$

The Spectral Expansion on the Entire Real Line of Green's Function for a Two-Layer Medium in the Eigenfunctions of a Nonself-Adjoint Sturm–Liouville Operator

The Spectral Expansion on the Entire Real Line of Green's Function for a Two-Layer Medium in the Eigenfunctions of a Nonself-Adjoint Sturm–Liouville Operator

Flows equivalences

&lt;p&gt;Given a differential equation on an open set O of an n-manifold we can associate to it a pseudo-flow, that is, a flow whose trajectories may not be defined in the entire real line. In this paper we prove that this pseudo-flow is always equivalent to a flow with its trajectories defined in all R. This result extends a similar result of Vinograd stated in the n-dimensional Euclidean space.&lt;/p&gt;

Integrating the Korteweg–de Vries Equation with a Self-Consistent Source and “Steplike” Initial Data

It is shown that “steplike” solutions of the Korteweg–de Vries equation with a self-consistent source can be found by the inverse scattering method for the Sturm–Liouville operator on the entire real line.

Multiple layered solutions of the nonlocal bistable equation

The nonlocal bistable equation is a model proposed recently to study materials whose constitutive relations among the variables that describe their states are nonlocal. It resembles the local bistable equation (the Allen–Cahn equation) in some way, but contains a much richer set of solutions. In this paper we consider two types of solutions. The first are the periodic solutions on a finite interval. These solutions are observed in materials like elastic crystals undergoing martensitic phase transitions and diblock copolymers at low temperatures. They are constructed by a variational method known as the Γ-limit technique. The second are solutions on the entire real line with transition layers, which are found by the formal matched asymptotics argument. We construct them to compare with the single layer heteroclinic and traveling wave solutions of the local bistable equation. The existence of multiple layered solutions depends on a unique nonlocal feature: the presence of two properly balanced competing effects of the constitutive relation, the oscillation inhibiting effect and the oscillation forcing effect, which coexist at two different length scales.

Convergence for Fourier series solutions of the forced harmonic oscillator

Harmonic oscillator equations of the form ÿ + ⋋2y = h(t) where ⋋ is a real constant and h(t) is a continuous, piecewise smooth, periodic ‘forcing’ function are considered. The exact solution, obtained through the Laplace transform is cumbersome to handle over long t intervals, and thus solving ‘term-by-term’ by replacing h(t) by its Fourier series is an attractive and accurate alternative. But this solution is an infinite series involving sums of sine and cosine terms, and thus one should worry about convergence of a solution in this form. In the article, it is shown that such a series solution indeed converges uniformly over the entire real line and is twice continuously differentiable, the derivatives being calculated ‘term-by-term’. Only results commonly available in the undergraduate literature are used to verify this and in so doing, a non-trivial application of these results is given. Also included are some interesting problems suitable for undergraduate research.

On the uniform convergence of the empirical density of an ergodic diffusion

We investigate the uniform convergence of the density of the empirical measure of an ergodic diffusion. It is known that under certain conditions on the drift and diffusion coefficients of the diffusion, the empirical density ft converges in probability to the invariant density f, uniformly on the entire real line. We show that under the same conditions, uniform convergence of ft to f on compact intervals takes place almost surely. Moreover, we prove that under much milder conditions (the usual linear growth condition on the drift and diffusion coefficients and a finite second moment of the invariant measure suffice), we have the uniform convergence of ft to f on compacta in probability.

A method for the practical evaluation of the Hilbert transform on the real line

We investigate a method for the numerical evaluation of the weighted Hilbert transform over the entire real line, ⨍ −∞ ∞ exp(−x 2)f(x)(x−t) −1 dx , where the integral is taken in the Cauchy principal value sense. The method is based on a combination of two polynomial approximation processes. In this way, we obtain a procedure that is numerically stable and efficient. The algorithm can be implemented in a fast way, and existing well known software packages may be used. From the point of view of theoretical error bounds, the method is competitive. Generalizations to other approximation methods are also possible.

On the use of Coifman intervallic wavelets in the method of moments for fast construction of wavelet sparsified matrices

Orthonormal wavelets have been successfully used as basis and testing functions for the integral equations and extremely sparse impedance matrices have been obtained. However, in many practical problems, the solution domain is confined in a bounded interval, while the wavelets are originally defined on the entire real line. To overcome this problem, periodic wavelets have been described in the literature. Nonetheless, the unknown functions must take on equal values at the endpoints of the bounded interval in order to apply periodic wavelets as the basis functions. We present the intervallic Coifman wavelets (coiflets) for the method of moments (MoM). The intervallic wavelets release the endpoints restrictions imposed on the periodic wavelets. The intervallic wavelets form an orthonormal basis and preserve the same multiresolution analysis (MRA) of other usual unbounded wavelets. The coiflets possesses a special property that their scaling functions have many vanishing moments. As a result, the zero entries of the matrices are identified directly, without using a truncation scheme with an artificially established threshold. Further, the majority of matrix elements are evaluated directly without performing numerical integration procedures such as Gaussian quadrature. For an n/spl times/n matrix, the number of actual numerical integrations is reduced from n/sup 2/ to the order of 3n(2L-1), when the coiflets of order L is employed. The construction of intervallic wavelets is presented. Numerical examples of scattering problems are discussed and the relative error of this method is studied analytically.