Form Of Linear Operators Research Articles

Effective Hamiltonians and Lagrangians for Conditioned Markov Processes at Large Volume

When analysing statistical systems or stochastic processes, it is often interesting to ask how they behave given that some observable takes some prescribed value. This conditioning problem is well understood within the linear operator formalism based on rate matrices or Fokker-Planck operators, which describes the dynamics of many independent random walkers. Relying on certain spectral properties of the biased linear operators, guaranteed by the Perron-Frobenius theorem, an effective process can be found such that its path probability is equivalent to the conditional path probability. In this paper, we extend those results for nonlinear Markov processes that appear when the many random walkers are no longer independent, and which can be described naturally through a Lagrangian-Hamiltonian formalism within the theory of large deviations at large volume. We identify the appropriate spectral problem as being a Hamilton-Jacobi equation for a biased Hamiltonian, for which we conjecture that two special global solutions exist, replacing the Perron-Frobenius theorem concerning the positivity of the dominant eigenvector. We then devise a rectification procedure based on a canonical gauge transformation of the biased Hamiltonian, yielding an effective dynamics in agreement with the original conditioning. Along the way, we present simple examples in support of our conjecture, we examine its consequences on important physical objects such as the fluctuation symmetries of the biased and rectified processes as well as the dual dynamics obtained through time-reversal. We apply all those results to simple independent and interacting models, including a stochastic chemical reaction network and a population process called the Brownian Donkey.

Perturbation Method in Problems of Linear Matrix Regression

Within the frameworks of linear regression this work studied estimates linear by observations, in particular the unbiased ones that leads to unbiased equations, among solutions of which the ones minimum in norm are distinguished, that allows us to minimize the root mean square error at uncorrelated observation errors with equal variances. Preliminary the problem of linear regression analysis is presented in the form of linear operator in a space of independent rectangular matrices connected with the equation of linear functions unbiasedness from matrix parameters. It is assumed that for this operator in the unperturbed version its SVD representation is known as well as SVD representation of the pseudo inverse to it operator. Since it is necessary to determine the singular set of the perturbed operator for determining eigenvalues and eigenvectors of the special symmetric matrix we employ the perturbation method, according to the general theory of operators in Euclidean space we determine the eigenmatrices of conjugate perturbed operator. Assuming that the problem of the linear regression analysis at the presence of perturbation of observation matrices can be solved in real time conditions, the resulting formulas are given in the first approximation of the small parameter. We present the test example which besides a small parameter includes parameters of random perturbations.

On facilitated computation of mesoscopic behavior of reaction‐diffusion systems

Various cellular and subcellular biological systems occur in the conditions where both reactions and diffusion take place. Since the concentration of species varies spatially, application of reaction‐diffusion master equation has served as an effective method to handle these complicated systems; yet solving these equation incurs a large CPU time penalty. Counter to the traditional technique of generating many sample paths, this article introduces a method which combines Grima's effective rate equation approach (Grima, J Chem Phys. 2010;133:3) with a linear operator formalism for diffusion to capture averaged species behaviors. The formulation also shows correct results at various choices of compartment sizes, which have been found to be an important factor that can affect accuracy of the final predictions (Erban, Chapman, Phys Biol. 2009;6:4). It is shown that the method presented allows the computation of the mesoscopic average of reaction‐diffusion systems at considerably accelerated rates (exceeding a thousand fold) over those based on sample path averages. © 2017 American Institute of Chemical Engineers AIChE J, 63: 5258–5266, 2017

Zero-term rank and zero-star cover number of symmetric matrices and their linear preservers

This paper concerns two notions of matrix functions over semirings; zero-term rank and zero-star cover number. We study linear operators that preserve these two matrix functions of symmetric matrices. Consequently, we determine the form of linear operators T on symmetric matrices that preserves zero-term rank (zero-star cover number). We also show that a linear operator T on symmetric matrices preserves zero-term rank (zero-star cover number) if and only if T preserves zero-term ranks (zero-star cover numbers, respectively) 1 and 2. Other characterizations of zero-term rank (zero-star cover number, respectively) preservers are also given.

Heat Transfer Analysis on the Hiemenz Flow of a Non-Newtonian Fluid: A Homotopy Method Solution

The mathematical model for the incompressible two-dimensional/axisymmetric non-Newtonian fluid flows and heat transfer analysis in the region of stagnation point over a stretching/shrinking sheet and axisymmetric shrinking sheet is presented. The governing equations are transformed into dimensionless nonlinear ordinary differential equations by similarity transformation. Analytical technique, namely, the homotopy perturbation method (HPM) with general form of linear operator is used to solve dimensionless nonlinear ordinary differential equations. The series solution is obtained without using the diagonal Padé approximants to handle the boundary condition at infinity which can be considered as a clear advantage of homotopy perturbation technique over the decomposition method. The effects of the pertinent parameters on the velocity and temperature field are discussed through graphs. To the best of authors’ knowledge, HPM solution with general form of linear operator for two-dimensional/axisymmetric non-Newtonian fluid flows and heat transfer analysis in the region of stagnation point is presented for the first time in the literature.

A direct proof of theorem on generalized Jordan form of linear operators

The present paper gives a direct proof of the following result: for any linear operator over arbitrary field there exists a basis in which it has a polyquasicyclic matrix, i.e. a generalized Jordan form of second kind. The polyquasicyclic form of a linear operator is uniquely determined up to the order of direct summands on the diagonal, and it is shown that the generalized Jordan form of second kind is a link that connects the classical Jordan form and the rational canonical form.

Chromotomography for a rotating-prism instrument using backprojection, then filtering

A simple closed-form solution is derived for reconstructing a 3D spatial-chromatic image cube from a set of chromatically dispersed 2D image frames. The algorithm is tailored for a particular instrument in which the dispersion element is a matching set of mechanically rotated direct vision prisms positioned between a lens and a focal plane array. By using a linear operator formalism to derive the Tikhonov-regularized pseudoinverse operator, it is found that the unique minimum-norm solution is obtained by applying the adjoint operator, followed by 1D filtering with respect to the chromatic variable. Thus the filtering and backprojection (adjoint) steps are applied in reverse order relative to an existing method. Computational efficiency is provided by use of the fast Fourier transform in the filtering step.

Statistical signal analysis for the inverse source problem of electromagnetics

A statistical signal analysis for the inverse source problem of electromagnetics is given. This correspondence considers the problem of estimating either the near field or the radiating current distribution from a measurement of the far field. The solution is derived via a linear operator formalism, and the ill-posedness of the reconstruction is quantified by using the Cramer-Rao lower bound, which is explicitly given in terms of the multipole expansion of the electromagnetic field. A numerical study is included to illustrate the theoretical results.

The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution

We address the inverse source problem of finding the time-harmonic current distribution (source) with minimum L/sup 2/ norm (minimum energy) that generates a prescribed electromagnetic field outside the source's region of support. Using the well-known multipole expansion of the electromagnetic field we compute (via a linear operator formalism) the sought-after minimum L/sup 2/ norm-current distribution consistent with the data.

Complete Spectral Representation for the Electromagnetic Field of Planar Multi-Layered Waveguides Containing Pseudochiral Ω-Media

A complete spectral representation for the electromagnetic field of planar multilayered waveguides inhomogeneously filled with pseudochiral Ω -media is presented. The problem of guided electromagnetic wave propagation is reduced to an eigenvalue equation related to a 2x2 matrix differential operator. Using the concept of adjoint waveguide, general bi-orthogonality relations for the hybrid modes (either from the discrete or from the continuous spectrum) are derived. For the special case of homogeneous layers the linear operator formalism is reduced to a simple 2 x 2 coupling matrix eigenvalue problem. Finally, as an example of application, the surface and the radiation modes of a grounded pseudochiral Ω -slab waveguide are analyzed.

A linear-operator formalism for bianisotropic waveguides

A bianisotropic waveguide can be defined as a cylindrical waveguide filled with bianisotropic materials, and all the conventional waveguides are special cases of the bianisotropic waveguide. In this paper, guided wave propagation in bianisotropic waveguide is analyzed by the theory of linear operators, and two types of adjoint waveguides and inner products are introduced respectively. Based on the concept of adjoint waveguides, the functional expressions of the field equations can be obtained, and from which the eigenvalue problem of the bianisotropic waveguide can be solved. Also, bi-orthogonality relations of guided modes are derived. These biorthogonality relations reported here can be used to expand electromagnetic fields in terms of a complete set of modes in straight bianisotropic waveguide. As an example of application, mode matching formulae for a discontinuity problem are given.

New biorthogonality relations for inhomogeneous biisotropic planar waveguides

Using a linear operator formalism this paper presents new biorthogonality relations for the hybrid modes supported by planar waveguides inhomogeneously filled with general biisotropic media. In the special case of lossless biisotropic media, the linear operator is self-adjoint, the original and adjoint waveguides are identical and new orthogonality relations can be derived. As an example of application, the radiation modes of a grounded nonreciprocal and lossless biisotropic slab waveguide are analyzed in terms of a pair of incident transverse electric (ITE) and incident transverse magnetic (ITM) continuous modes, which have the advantage of being mutually orthogonal and of having a clear physical interpretation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A linear-operator formalism for the analysis of inhomogeneous biisotropic planar waveguides

Using the theory of linear operators, guided electromagnetic wave propagation in inhomogeneous (nonreciprocal) biisotropic planar structures is analyzed in terms of a 2*2 matrix differential operator. Based on the concept of adjoint waveguide, a new biorthogonality relation for the guided hybrid modes is derived. For the special case of reciprocal biisotropic media or chiral media, the linear-operator formalism leads to a self-adjoint problem. As an application example, a general analysis of the radiation modes of a grounded chiroslabguide is also presented. >

Spectral representation of self-adjoint problems for layered anisotropic waveguides

Layered waveguides with lossless anisotropic layers in the polar configuration are analyzed through the unifying concept of a real self-adjoint operator. For a suitable definition of two-vector transverse eigenfunctions, general properties such as orthogonality and completeness relations are derived. The linear operator formalism is applied to closed waveguides inhomogeneously filled with anisotropic materials, including crystals and gyrotropic media. As an extension of the former theory to open waveguides, a grounded uniaxial dielectric slab with a coplanar optic axis is also analyzed: as for open isotropic waveguides, a complete spectral representation including the surface (proper eigenfunctions) and the pseudosurface modes (improper eigenfunctions) is presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Multicomponent reaction and diffusion in a tubular reactor: an operator-theoretic solution

Mathematical models describing isothermal tubular reactors in which first-order chemical reactions and uncoupled multicomponent diffusion occur are solved by a linear operator formalism. A pseudohomogeneous axial dispersion model accounting for both bulk-fluid-phase dispersion and intraparticle diffusion is solved to examine the effect of widely disparate configuration diffusion coefficients on selectivity in the reactor. The operator formalism is shown to obtain also solutions for the mathematically more complicated situation of a wall-catalyzed reactor with a nonuniform, fully developed flow profile. Computations presented for the axial dispersion model demonstrate that intraparticle diffusion hindrances in molecular sieve catalysts can drastically affect the selectivity of the xylene isomerization system. The influence of reactor parameters (Peclet and Damkohler numbers) and reactor feed conditions on product selectivity is investigated in order to elucidate the conditions under which optimum selectivity can be expected.

Orthogonality and normalization of radiation modes in dielectric waveguides

By expressing Maxwell’s equations in a linear-operator formalism, it is shown that the orthogonality and normalization properties of the continuous spectrum of radiation modes in a dielectric waveguide with arbitrary refractive-index profile and cross-sectional shape can be established directly from the properties of much simpler free-space fields.

General form of linear operators which commute with the operation of differentiation

In this paper we indicate the general form of linear operators which commute with the operation of differentiation and which act in a space of functions which are analytic in a bounded simply connected domain and are continuous in a very weak sense; the properties of such operators are studied.