Families of curve congruences on Lorentzian surfaces and pencils of quadratic forms

2011 ◽  
Vol 141 (3) ◽  
pp. 655-672 ◽  
Author(s):  
Ana Claudia Nabarro ◽  
Farid Tari
Keyword(s):  
Differential Equations ◽  
Projective Plane ◽  
Quadratic Forms ◽  
Adjoint Operator ◽  
Principal Curves ◽  
Conjugate Equation ◽  
Oriented Surface ◽  
Lorentzian Metric ◽  
Binary Differential Equations

We define and study families of conjugate and reflected curve congruences associated to a self-adjoint operator on a smooth and oriented surface M endowed with a Lorentzian metric g. These families trace parts of the pencil joining the equations of the -asymptotic and the -principal curves, and the pencil joining the -characteristic and the -principal curves, respectively. The binary differential equations (BDEs) of these curves can be viewed as points in the projective plane. Using the polar lines of various BDEs with respect to the conic of degenerate quadratic forms, we obtain geometric results on the above pencils and their relation with the metric g, on the type of solutions of a given BDE, of its -conjugate equation and on BDEs with orthogonal roots.

scholarly journals Normal modes of a waveguide as eigenvectors of a self-adjoint operator pencil

2021 ◽  
Vol 29 (1) ◽  
pp. 14-21
Author(s):  
Mikhail D. Malykh
Keyword(s):  
Electromagnetic Field ◽  
Differential Equations ◽  
Wave Equations ◽  
Adjoint Operator ◽  
Normal Modes ◽  
Operator Pencil ◽  
Homogeneous Case ◽  
Permittivity And Permeability ◽  
Simply Connected ◽  
Magnetic Potentials

A waveguide with a constant, simply connected section S is considered under the condition that the substance filling the waveguide is characterized by permittivity and permeability that vary smoothly over the section S, but are constant along the waveguide axis. Ideal conductivity conditions are assumed on the walls of the waveguide. On the basis of the previously found representation of the electromagnetic field in such a waveguide using 4 scalar functions, namely, two electric and two magnetic potentials, Maxwells equations are rewritten with respect to the potentials and longitudinal components of the field. It appears possible to exclude potentials from this system and arrive at a pair of integro-differential equations for longitudinal components alone that split into two uncoupled wave equations in the optically homogeneous case. In an optically inhomogeneous case, this approach reduces the problem of finding the normal modes of a waveguide to studying the spectrum of a quadratic self-adjoint operator pencil.

scholarly journals A convergence result for discreet steepest decent in weighted sobolev spaces

1997 ◽  
Vol 2 (1-2) ◽  
pp. 67-72 ◽  
Author(s):  
W. T. Mahavier
Keyword(s):  
Differential Equations ◽  
Sobolev Spaces ◽  
Quadratic Forms ◽  
Weighted Sobolev Spaces ◽  
Convergence Result ◽  
Sobolev Gradients

A convergence result is given for discrete descent based on Sobolev gradients arising from differential equations which may be expressed as quadratic forms. The argument is an extension of the result of David G. Luenberger on Euclidean descent and compliments the work of John W. Neuberger on Sobolev descent.

scholarly journals Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 586 ◽  
Author(s):  
Julio López-Saldívar ◽  
Margarita Man’ko ◽  
Vladimir Man’ko
Keyword(s):  
Differential Equations ◽  
Quadratic Forms ◽  
Frequency Converter ◽  
Continuous Variable ◽  
Variable Density ◽  
Gaussian State ◽  
Density Matrices ◽  
Mean Values ◽  
Invariant States ◽  
The Mean

In the differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and momentum operators or quadrature components. Specifically, we obtain in generic form the differential equations for the covariance matrix, the mean values, and the density matrix parameters of a multipartite Gaussian state, unitarily evolving according to a Hamiltonian H ^ . We also present the corresponding differential equations, which describe the nonunitary evolution of the subsystems. The resulting nonlinear equations are used to solve the dynamics of the system instead of the Schrödinger equation. The formalism elaborated allows us to define new specific invariant and quasi-invariant states, as well as states with invariant covariance matrices, i.e., states were only the mean values evolve according to the classical Hamilton equations. By using density matrices in the position and in the tomographic-probability representations, we study examples of these properties. As examples, we present novel invariant states for the two-mode frequency converter and quasi-invariant states for the bipartite parametric amplifier.

scholarly journals On Spectral Properties of Doubly Stochastic Matrices

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 369
Author(s):  
Mutti-Ur Rehman ◽  
Jehad Alzabut ◽  
Javed Hussain Brohi ◽  
Arfan Hyder
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Quadratic Forms ◽  
Spectral Properties ◽  
Singular Values ◽  
Low Rank ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices ◽  
The Relationship

The relationship among eigenvalues, singular values, and quadratic forms associated with linear transforms of doubly stochastic matrices has remained an important topic since 1949. The main objective of this article is to present some useful theorems, concerning the spectral properties of doubly stochastic matrices. The computation of the bounds of structured singular values for a family of doubly stochastic matrices is presented by using low-rank ordinary differential equations-based techniques. The numerical computations illustrating the behavior of the method and the spectrum of doubly stochastic matrices is then numerically analyzed.

Mutually conjugate solutions of formally self-adjoint differential equations

1979 ◽  
Vol 83 (1-2) ◽  
pp. 93-101 ◽  
Author(s):  
J. M. Hill ◽  
R. V. Nillsen
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Linear Differential Operator ◽  
Adjoint Operator ◽  
Linear Differential ◽  
Factorization Result

SynopsisLet L be a formally self-adjoint linear differential operator of order m with strictly positive leading coefficient and let m = 2n + 1 if m is odd, m = 2n if m is even. Let y1, y2,…, yn be n given mutually conjugate solutions of Ly = 0 on I, where I is some interval, whose Wronskian is non-zero on I. Then L = (−1)nQ*Q or L = (−1)nQ*DQ where Q is a differential operator of order n, Q* is the adjoint operator and D denotes differentiation. This fact is used to construct further solutions yn+1,−, ym of Ly = 0 so that y1,…, ym is a basis for the solutions of Ly = 0 and for which yi and yn+j are mutually conjugate if i ≠ j. If y1 ≠ 0 on I the degree of L may be lowered by 2 to obtain a formally self-adjoint operator L1 for which mutually conjugate solutions are constructed. If this process is continued a factorization result is obtained which is related to a result of Pólya.

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