Analytic Coefficients Research Articles

Simultaneous determination of the diffusion and absorption coefficient from boundary data

We consider the inverse problem of determining both an unknown diffusion and an unknown absorption coefficient from knowledge of (partial) Cauchy data in an elliptic boundary value problem. For piecewise analytic coefficients, we prove a complete characterization of the reconstructible information. It is shown to consist of a combination of both coefficients together with the jumps in the leading order diffusion coefficient and its derivative.

Uniqueness properties of solutions to Schrödinger equations

nX j=1 ∂ 2 u ∂x 2 (x) = 0, (harmonic functions) in the unit ball {x ∈ R n : |x| 2, since harmonic functions are still real analytic in {x ∈ R n : |x| < 1}. In fact, it is well-known that if P(x,D) is a linear elliptic differential operator with real analytic coefficients, andP(x,D)u = 0 in a open set ⊂ R n , then u is real analytic in . Hence, the (s.u.c.p.) also holds for such solutions. Through the work of Hadamard [28] on the uniqueness of the Cauchy problem (which is closely related to the strong unique continuation property discussed earlier) it became clear (for applications in nonlinear problems) that it would be desirable to establish the strong unique continuation property for operators whose coefficients are not necessarily real analytic, oreven C ∞ . The first results in this direction were found in the pioneering work of Carleman [9] (when ,

Regularization of linear difference equations with analytic coefficients and their applications

We propose a regularization method for four-element linear difference equations with analytic coefficients. We study these equations in the class of functions which are holomorphic in the complex plane with a cruciform cut and vanish at infinity. We give several examples illustrating the dependence of the solvability properties of equations on the choice of periodic coefficients. We describe various applications.

Global solvability of real analytic complex vector fields in two variables

This paper deals with the global solvability of a complex vector field with real analytic coefficients in two real variables. The vector field is assumed to satisfy the Nirenberg–Treves condition ( P ) for local solvability. Normal forms for the vector field near the one-dimensional orbits are obtained and a generalization of the Riemann–Hilbert problem is considered.

Homogeneous weakly hyperbolic equations with time dependent analytic coefficients

This paper concerns the Cauchy problem for homogeneous weakly hyperbolic equations with time depending analytic coefficients. We give a sufficient condition for the C ∞ -well-posedness which is also necessary if the space dimension is equal to one. The main point of the paper consists in expressing our condition only in terms of the coefficients of the operator, without needing to know the behavior of the characteristic roots. This is made possible by using the so-called standard symmetrizer of a companion hyperbolic matrix.

New applications of Picardʼs successive approximations

The iterative method of successive approximations, originally introduced by Émile Picard in 1890, is a basic tool for proving the existence of solutions of initial value problems regarding ordinary first order differential equations. In the present paper, it is shown that this method can be modified to get estimates for the growth of solutions of linear differential equations of the type f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = 0 with analytic coefficients. A short comparison to the growth results in the literature, obtained by means of different methods, is also given. It turns out that many known results can be proved by applying Picardʼs successive approximations in an effective way. Self-contained considerations are carried out in the complex plane and in the unit disc, and some remarks about solutions of real linear differential equations are made.

On the non-analyticity locus of an arc-analytic function

Let X X be a real analytic manifold. A function f : X → R f:X\to \mathbb {R} is called arc-analytic if it is real analytic on each real analytic arc. In real analytic geometry there are many examples of arc-analytic functions that are not real analytic. They appear while studying the arc-symmetric sets and the blow-analytic equivalence. In this paper we show that the non-analyticity locus of an arc-analytic function is arc-symmetric. We also discuss the behavior of the non-analyticity locus under blowings-up. By a result of Bierstone and Milman, an arc-analytic function f : X → R f:X\to \mathbb {R} that satisfies a polynomial equation with real analytic coefficients, can be made analytic, over any relatively compact subset of X X , by a sequence of blowings-up with smooth centers. We show that these centers can be chosen, at each stage of the resolution, inside the non-analyticity loci.

Linear differential equations with solutions in the [formula omitted] spaces

For the complex differential equation f ( n ) + A n − 1 ( z ) f ( n − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = 0 with analytic coefficients A j in the unit disk, our contribution is to build up a relationship between the coefficients and solutions of the equation. We provide conditions for coefficients A j such that all solutions of the equation belong to the Q K spaces.

To the question of the minimal number of inputs for linear differential algebraic systems

We examine a control linear system of ordinary differential equations with an identically degenerate matrix coefficient of the derivative of the unknown vector function. We study the question of the minimal dimension of the control vector when the system could be fully controllable on any segment in the domain of definition. The problem is investigated in the cases of stationary systems and the systems with real analytic and smooth coefficients for which some structural forms can be defined.

ODEs and Wiman–Valiron theory in the unit disc

An asymptotic equality of Wiman–Valiron type is proved for the derivatives of analytic functions in the unit disc and applied to ODEs with analytic coefficients.

Analytic Solutions of Matrix Riccati Equations with Analytic Coefficients

For matrix Riccati equations of platoon-type systems and of systems arising from PDEs, assuming the coefficients are analytic or rational functions in a suitable domain, analyticity of the stabilizing solution is proved under various hypotheses. General results on analytic behavior of stabilizing solutions are developed as well.

Potential Coupling Through a Slot Backed by an Elliptical Cavity: Neumann and Dirichlet Boundary Conditions

Exact, closed-form expressions are presented for the solution to Laplace's equation everywhere in a specific cavity-backed aperture problem. A symmetrically placed slot enables the static field in a half-space above a hard or soft ground plane to penetrate into the interior of an elliptically shaped cavity. Separation-of-variables in elliptic coordinates yields a summable series, whereupon the aperture field or its normal derivative appears naturally as a series of edge-condition weighted Chebyshev polynomials in the Cartesian coordinates. Analytic coefficients explicitly display simple dependence upon the cavity and point-source geometry.

Elastodynamic Equations: Characteristics, Wavefronts and Rays

Summary We derive the characteristic equations, and the so-called Christoffel equation, for the vector elastodynamic equations in terms of both hypersurfaces of nonuniqueness and as wavefronts based on a physical definition. We follow Courant and Hilbert in defining a wavefront as a surface for which a solution may be zero on one side but nonzero on the other. We show that wavefronts defined in this way must be characteristics. Moreover, we give a partial converse showing that in the case of analytic coefficients, characteristics must (locally) be wavefronts. Furthermore, we show how the equations defining wavefronts, which are characteristics, can be obtained by letting frequency tend to infinity in a certain trial solution expressed in the frequency domain. The last approach might suggest that ray theory, which results from the Christoffel equation, is an asymptotic theory. Taking the limit, however, is just a technicality and is unnecessary to obtain the Christoffel equation, which is contained in the elastodynamic equations and their characteristic equations.

Some results on the well-posedness for second order linear equations

We investigate the Cauchy problem for second order hyperbolic equations of complete form, and we prove an extension of a classical result of Oleĭnik [10] concerning the well-posedness for equations in which are absent the terms with mixed time-space derivatives. Then, in space dimension $n=1$, we compare our results with those in [8] for equations with analytic coefficients, and those of [7] and [11] for homogeneous equations with coefficients depending only either on $t$ or on $x$. Moreover we exhibit, in space dimension $n\ge 2$, an equation of the form \begin{equation*} u_{tt} - \sum_{i,j=1}^{n} (a_{ij}(t,x)u_{x_{j}})_{x_{i}} = 0{,} \quad\text{with}\quad \sum a_{ij} \xi_{i}\xi_{j} \ge 0, \end{equation*} where the coefficients are analytic functions, for which the Cauchy problem is ill-posed. Finally, we present a sufficient condition for the well-posedness of $2 \times 2$ systems.

Logarithmic conversion of absorption detection in wavelength modulation spectroscopy with a current-modulated diode laser

Logarithmic-conversion data processing used in wavelength modulation spectroscopy (WMS) with a current-modulated diode laser as its source is analyzed and compared with second-to-first ratio detection. Analytic Fourier coefficients of logarithmic-converted residual amplitude modulation (RAM) of a light source are given. An experimental setup for methane absorption detection at 1650 nm is described. It is shown theoretically and experimentally that logarithmic-converted WMS cannot only eliminate the fluctuation of received light power, but also improve the signal-to-noise ratio significantly.

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.

Lower estimates on microstates free entropy dimension

By proving that certain free stochastic differential equations have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain $n$-tuples $X_{1},...,X_{n}$: we show that Abstract. By proving that certain free stochastic differential equations with analytic coefficients have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain n-tuples X_{1},...,X_{n}. In particular, we show that \delta_{0}(X_{1},...,X_{n})\geq\dim_{M\bar{\otimes}M^{o}}V where M=W^{*}(X_{1},...,X_{n}) and V=\{(\partial(X_{1}),...,\partial(X_{n})):\partial\in\mathcal{C}\} is the set of values of derivations A=\mathbb{C}[X_{1},... X_{n}]\to A\otimes A with the property that \partial^{*}\partial(A)\subset A. We show that for q sufficiently small (depending on n) and X_{1},...,X_{n} a q-semicircular family, \delta_{0}(X_{1},...,X_{n})>1. In particular, for small q, q-deformed free group factors have no Cartan subalgebras. An essential tool in our analysis is a free analog of an inequality between Wasserstein distance and Fisher information introduced by Otto and Villani (and also studied in the free case by Biane and Voiculescu).

On uniqueness in diffuse optical tomography

A prominent result of Arridge and Lionheart (1998 Opt. Lett. 23 882–4) demonstrates that it is in general not possible to simultaneously recover both the diffusion (aka scattering) and the absorption coefficient in steady-state (dc) diffusion-based optical tomography. In this work we show that it suffices to restrict ourselves to piecewise constant diffusion and piecewise analytic absorption coefficients to regain uniqueness. Under this condition both parameters can simultaneously be determined from complete measurement data on an arbitrarily small part of the boundary.

Holomorphic extension of fundamental solutions of elliptic linear partial differential operators of the second order with analytic coefficients

We prove that every fundamental solution of an elliptic linear partial differential operator of the second order with analytic coefficients and simple complex characteristics in an open set Ω ⊂ R n can be continued at least locally as a multi-valued analytic function in C n up to the complex bicharacteristic conoid. This extension ramifies or not along its singular set the bicharacteristic conoid and belongs to the Nilsson class. To cite this article: S. Lukasiewicz, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Minimal Microlocal Gevrey Regularity for "Sums of Squares"

A theorem of minimal microlocal Gevrey regularity is proved for operators that are sums of squares of vector fields with real analytic coefficients, thus providing a microlocal version of a well-known theorem of Derridj and Zuily (“Regularite analytique et Gevrey d'operateurs elliptiques degeneres,” Journal de Mathematiques Pures et Appliquees 52 (1973): 65-80).