Real Semiaxis Research Articles

The inverse laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation

When the Laplace transform F( p) of a function f( x) has no poles but is singular only on the real negative semiaxis because of a cut required to make it single-valued, the inverse transform f( x) can easily be computed by means of the integral of a real-valued function. This result is applied to the calculation of a class of exact eternal solutions of the Boltzmann equation, recently found by the authors. The new approach makes it easier to prove that these solutions are positive, as well as to study their asymptotics.

Approximation of the Hilbert Transform on the real semiaxis using Laguerre zeros

The authors propose two new algorithms for the computation of Cauchy principal value integrals on the real semiaxis. The proposed quadrature rules use zeros of Laguerre polynomials. Theoretical error estimates are proved and some numerical examples are showed.

Asymptotic Representations for Root Vectors of Nonselfadjoint Operators and Pencils Generated by an Aircraft Wing Model in Subsonic Air Flow

This paper is the second in a series of several works devoted to the asymptotic and spectral analysis of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works by A. V. Balakrishnan. The model is governed by a system of two coupled integrodifferential equations and a two parameter family of boundary conditions modeling the action of the self-straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite-meromorphic function on the complex plane having the branch cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. In the first paper and in the present one, our main object of interest is the dynamics generator of the differential parts of the system. It is a nonselfadjoint operator in the energy space with a purely discrete spectrum. In the first paper, we have shown that the spectrum consists of two branches and have derived their precise spectral asymptotics. In the present paper, we derive the asymptotical approximations for the mode shapes. Based on the asymptotical results of these first two papers, in the next paper, we will discuss the geometric properties of the mode shapes such as minimality, completeness, and the Riesz basis property in the energy space.

Improved conformal mapping of the Borel plane

The conformal mapping of the Borel plane can be utilized for the analytic continuation of the Borel transform to the entire positive real semi-axis and is thus helpful in the resummation of divergent perturbation series in quantum field theory. We observe that the rate of convergence can be improved by the application of Pad\'{e} approximants to the Borel transform expressed as a function of the conformal variable, i.e. by a combination of the analytic continuation via conformal mapping and a subsequent numerical approximation by rational approximants. The method is primarily useful in those cases where the leading (but not sub-leading) large-order asymptotics of the perturbative coefficients are known.

Approximation of Smooth Functions on the Semiaxis by Entire Functions of Bounded Half-Degree

In this paper, we study the pointwise approximation of functions defined on the real semiaxis and having an rth derivative bounded almost everywhere. The approximation is performed by means of entire functions of bounded half-degree, which were introduced by S. N. Bernstein. An asymptotically sharp estimate for pointwise approximation of this class of functions is obtained.

Algebraic Criteria of Absolute Stability

New algebraic criteria of absolute stability are obtained on the basis of a polynomial form of the frequency condition for absolute stability (Popov inequality). The analysis of the inequality is reduced to an analysis of the local properties of polynomials and a rational function on the negative real semiaxis. As a result, the stability criteria do not require the plotting of any sort of curves and can be expressed on the basis of algorithms of the Routh–Hurwitz type and calculations of the negative real roots of polynomials.

Sturmian basis functions for the harmonic oscillator

We point out that the discrete Sturmian set for the one-dimensional harmonic oscillator, constructed recently by Antonsen [Phys. Rev. A 60, 812 (1999)], is incomplete in ${L}_{{x}^{2}}^{2}(R)$ and thus does not form a basis in that Hilbert space. We show that for $Eg0,$ the spectrum of the Sturm-Liouville problem defining the Sturmian functions is mixed and consists of an infinite number of discrete positive eigenvalues, coinciding with those found by Antonsen, and the continuum of eigenvalues covering the negative real semiaxis. The discrete eigenvalues are simple while continuous eigenvalues are doubly degenerate. A basis in ${L}_{{x}^{2}}^{2}(R)$ comprises all Sturmian functions generated by that problem, i.e., those associated with discrete eigenvalues as well as those associated with continuum eigenvalues. We consider also the defining Sturm-Liouville problem in the case $El~0$ and find that then the discrete part of its spectrum is absent: the spectrum is purely continuous, doubly degenerate, and covers the negative real semiaxis; the associated Sturmian functions form a noncountable basis in ${L}_{{x}^{2}}^{2}(R).$

Riesz basis property of root vectors of non-self-adjoint operators generated by aircraft wing model in subsonic airflow

This paper is the third in a series of several works devoted to the asymptotic and spectral analysis of a model of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works by Balakrishnan. The model is governed by a system of two coupled integro-differential equations and a two-parameter family of boundary conditions modeling the action of the self-straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite-meromorphic function on the complex plane having the branch cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. In the first two papers (see [33, 34]) and in the present one, our main object of interest is the dynamics generator of the differential parts of the system. This generator is a non-self-adjoint operator in the energy space with a purely discrete spectrum. In the first paper, we have shown that the spectrum consists of two branches, and have derived their precise spectral asymptotics with respect to the eigenvalue number. In the second paper, we have derived the asymptotical approximations for the mode shapes. Based on the asymptotical results of the first two papers, in the present paper, we (a) prove that the set of the generalized eigenvectors of the aforementioned differential operator is complete in the energy space; (b) construct the set of vectors which is biorthogonal to the set of the generalized eigenvectors in the case when theremight be not only eigenvectors but associate vectors as well; and (c) prove that the set of the generalized eigenvectors forms a Riesz basis in the energy space. To prove the main result of the paper, we made use of the Nagy-Foias functional model for non-self-adjoint operators. The results of all three papers will be important for the reconstruction of the solution of the original initial-boundary-value problem from its Laplace transform in the forthcoming papers.

Phase optimized states via discrete coherent-state superpositions

It is shown that phase optimized quantum states of the electromagnetic field can be constructed by superpositions of a small number of coherent states along the positive real semiaxis in phase space. A systematic method is developed for finding optimal coherent-state superpositions.

ON ASYMPTOTIC PROPERTIES OF POLYNOMIALS ORTHOGONAL ON THE REAL AXIS

This paper contains a number of results on the logarithmic asymptotics and the asymptotic distribution of zeros of polynomials that are orthonormal on the real axis or semiaxis with respect to weight functions of the type .Bibliography: 10 titles.

Analytic properties of the Green's functions with respect to the coupling constant in an exactly solvable model of field theory

It is found in an exactly solvable model of field theory that Green's functions for boson fields are analytic in the coupling constant, except for a cut along the real semiaxis. In the case of fermion fields, Green's functions are analytic in the coupling constant near zero.

Collision of two plane free streams with division of one of them into two flows

The collision of two streams of ideal incompressible fluid with the division of one stream into two flows in opposite directions along the boundary of the other is considered. The second stream is shifted from its initial direction due to the collision. The ratio of the total pressure heads of the separated and shifted streams is e≤1. The flow regions corresponding to each of these streams map conformally onto the first and fourth quadrants of the complex plane so that the streamline separating the streams corresponds to the real semiaxis in both cases. By using the dynamic and kinematic conditions for this streamline, nonlinear relations are found linking the boundary values of the logarithms of the complex velocity in each of the two flow regions. These analytic functions are computed approximately by linearization at small values of e.