scholarly journals On Ordinary, Linear q-Difference Equations, with Applications to q-Sato Theory

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Thomas Ernst
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Basis Function ◽  
Crucial Role ◽  
Appell Polynomials ◽  
Schur Polynomials ◽  
Homogeneous Case ◽  
The Matrix ◽  
Sato Theory

The purpose of this paper is to develop the theory of ordinary, linear q-difference equations, in particular the homogeneous case; we show that there are many similarities to differential equations. In the second part we study the applications to a q-analogue of Sato theory. The q-Schur polynomials act as basis function, similar to q-Appell polynomials. The Ward q-addition plays a crucial role as operation for the function argument in the matrix q-exponential and for the q-Schur polynomials.

scholarly journals Finite-difference equations of quasistatic motion of the shallow concrete shells in nonlinear setting

2020 ◽  
Vol 7 (1) ◽  
pp. 48-55 ◽  
Author(s):  
Bolat Duissenbekov ◽  
Abduhalyk Tokmuratov ◽  
Nurlan Zhangabay ◽  
Zhenis Orazbayev ◽  
Baisbay Yerimbetov ◽  
...  
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Difference Equations ◽  
Constant Load ◽  
Time Interval ◽  
Test Case ◽  
Algebraic Equations ◽  
Nonlinear Properties ◽  
Finite Difference Equations ◽  
Concrete Shells

AbstractThe study solves a system of finite difference equations for flexible shallow concrete shells while taking into account the nonlinear deformations. All stiffness properties of the shell are taken as variables, i.e., stiffness surface and through-thickness stiffness. Differential equations under consideration were evaluated in the form of algebraic equations with the finite element method. For a reinforced shell, a system of 98 equations on a 8×8 grid was established, which was next solved with the approximation method from the nonlinear plasticity theory. A test case involved computing a 1×1 shallow shell taking into account the nonlinear properties of concrete. With nonlinear equations for the concrete creep taken as constitutive, equations for the quasi-static shell motion under constant load were derived. The resultant equations were written in a differential form and the problem of solving these differential equations was then reduced to the solving of the Cauchy problem. The numerical solution to this problem allows describing the stress-strain state of the shell at each point of the shell grid within a specified time interval.

scholarly journals p-adic confluence of q-difference equations

2008 ◽  
Vol 144 (4) ◽  
pp. 867-919 ◽  
Author(s):  
Andrea Pulita
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Root Of Unity

AbstractWe develop the theory of p-adic confluence of q-difference equations. The main result is the fact that, in the p-adic framework, a function is a (Taylor) solution of a differential equation if and only if it is a solution of a q-difference equation. This fact implies an equivalence, called confluence, between the category of differential equations and those of q-difference equations. We develop this theory by introducing a category of sheaves on the disk D−(1,1), for which the stalk at 1 is a differential equation, the stalk at q isa q-difference equation if q is not a root of unity, and the stalk at a root of unity ξ is a mixed object, formed by a differential equation and an action of σξ.

Matrix-Mineral Relationships-A. Morphologist's Viewpoint

1979 ◽  
Vol 58 (2_suppl) ◽  
pp. 922-929 ◽  
Author(s):  
M.U. Nylen
Keyword(s):  
Organic Material ◽  
Crucial Role ◽  
Crystal Surface ◽  
Organic Matrix ◽  
The Other ◽  
Enamel Matrix ◽  
Initial Cell ◽  
Cell Product ◽  
Ultrastructural Morphology ◽  
The Matrix

The literature on the ultrastructural morphology of the enamel matrix and its relationship to the crystals is reviewed. Two morphological entities of the matrix are discussed: One is the so-called stippled material which may be the initial cell product; the other, variously described as fibrillar, lamellar, tubular or helical, is thought by many to play a crucial role in nucleation and orientation of the crystals. A number of observations, however, suggest that the latter structures form secondarily to the crystals and that in reality they represent organic material adsorbed to the crystal surface and maintained as independent structures upon removal of the mineral. The need for additional studies is stressed including systematic studies of interactions between constituents of the organic matrix and the apatite crystals.

Wavelet Multi-Scale Solution for Deflection of Thin Rectangular Plates under Temperature

2012 ◽  
Vol 166-169 ◽  
pp. 2871-2875
Author(s):  
Yan Chang Wang ◽  
Ke Liang Ren ◽  
Yan Dong ◽  
Ming Guang Wu
Keyword(s):  
Differential Equations ◽  
Rectangular Plate ◽  
Thermal Environment ◽  
Operational Matrix ◽  
Rectangular Plates ◽  
Thin Rectangular Plate ◽  
Multi Scale ◽  
Scale Method ◽  
The Matrix ◽  
Operational Matrix Of Integration

To consider the deformation of thin rectangular plate under temperature. In this paper, the wavelet multi-scale method was used to solve the thin plate governing differential equations with four different initial or boundary conditions. An operational matrix of integration based on the wavelet was established and the procedure for applying the matrix to solve the differential equations was formulated, and got the deflection of thin rectangular plates under temperature. The result provides a theoretical reference for solving thin rectangular plate deflection in thermal environment using multi-scale approach.

On obtaining limiting values of a class of infinite products

2018 ◽  
Vol 102 (555) ◽  
pp. 428-434
Author(s):  
Stephen Kaczkowski
Keyword(s):  
Difference Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Diffusion Theory ◽  
Numerical Technique ◽  
Applied Mathematics ◽  
Flow Analysis ◽  
Step Size ◽  
Infinite Products ◽  
Fluid Flow Analysis

Difference equations have a wide variety of applications, including fluid flow analysis, wave propagation, circuit theory, the study of traffic patterns, queueing analysis, diffusion theory, and many others. Besides these applications, studies into the analogy between ordinary differential equations (ODEs) and difference equations have been a favourite topic of mathematicians (e.g. see [1] and [2]). These applications and studies bring to light the similar character of the solutions of a difference equation with a fixed step size and a corresponding ODE.Also, an important numerical technique for solving both ordinary and partial differential equations (PDEs) is the method of finite differences [3], whereby a difference equation with a small step size is utilised to obtain a numerical solution of a differential equation. In this paper, elements of both of these ideas will be used to solve some intriguing problems in pure and applied mathematics.

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.

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