scholarly journals Asymptotics and Quotient Spaces of Solutions of Operator-Difference Equations and Differential Equations with Small Delay

2021 ◽  
Vol 12 (6) ◽  
pp. 1628-1637
Author(s):  
Zhumagul Zheentaeva
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Dynamic Systems ◽  
Dimensional Space ◽  
Single Point ◽  
Solution Space ◽  
Retarded Argument ◽  
Monotonic Transformation ◽  
Quotient Spaces ◽  
Point Representation

Formerly, in order to conduct the in-depth study of differential equations with delay, the author proposed the method of splitting the solution space reducing such equations to the systems of operator-difference equations. Using this method, the author assumed new conditions, i.e. the absolute domains for coefficients sufficient for the existence of special (slowly changing) solutions, and proved the presence of approximating and asymptotically approximating properties in them, as well as the asymptotic one-dimensional space of solutions of the initial problems for linear scalar differential equations with insignificantly retarded argument and the corresponding operator-difference equation systems (special solutions correspond, to the solutions with a slowly changing first component and a relatively small second component). For the purposes of the single-point representation of the obtained results and other data related to the theory of dynamic systems (the distance between the solution values tends to zero alongside the unlimited increase in argument), throughout this research paper the author uses the concept of the asymptotic equivalence of solutions for dynamic systems, as it was introduced by the author in their previous research. In order to shape the new mathematical objects, the concept of asymptotic Hausdorff equivalence of solutions for dynamic systems is introduced (the distance between solution values tends to zero with unlimited increase in argument of one solution and monotonic transformation of argument of another solution).

Translation-, Rotation-, Scale-, and Distortion-Invariant Object Recognition Through Self-Organization

1997 ◽  
Vol 08 (02) ◽  
pp. 173-179 ◽  
Author(s):  
Soheil Shams
Keyword(s):  
Object Recognition ◽  
Dimensional Space ◽  
Single Point ◽  
General Pattern ◽  
Solution Space ◽  
Spatial Location ◽  
Search Space ◽  
Recognition System ◽  
Visual Object ◽  
Visual Object Recognition

The task of visual object recognition is often complicated by the fact that a single 3-D object can undergo a number of transformations which can substantially alter its projection onto a 2-D surface, such as the retina. Such transformations include translation of the object in the visual field, changes in the size of the object, its orientation in the 2-D plane and the viewing perspective. For a general pattern recognition system to detect and recognize an object after such transformations, it must be able to associate widely differing patterns with the same object label. In this paper, a novel self-organizing model, called the Multiple Elastic Modules (MEM), is presented which attempts to solve this problem by searching a multi-dimensional space, where each axis is defined by one of the transformations (e.g scale, translation, rotation, etc.). A particular object of a specific size, orientation and spatial location is mapped onto a single point in this space. Of course, distortions and minor variations in an object's image will expand this point to a small localized area in this multi-dimensional space. Such a powerful representation scheme comes at a cost of high computational demand due to the combinatorially large search space. The MEM approach to solving this problem efficiently partitions the solution space to search the most promising areas for the correct match. Simulation results are presented on detecting a stick-figure object under translation, distortion, scale, and rotation transformations in a cluttered background.

Celestial Tapestry

2020 ◽  
Author(s):  
Nicholas Mee
Keyword(s):  
Mathematics Curriculum ◽  
Dimensional Space ◽  
Golden Section ◽  
Single Point ◽  
Historical Context ◽  
Computer Art ◽  
Lightning Strike ◽  
Secret Message ◽  
Axiomatic Method ◽  
Higher Dimensional

Celestial Tapestry places mathematics within a vibrant cultural and historical context, highlighting links to the visual arts and design, and broader areas of artistic creativity. Threads are woven together telling of surprising influences that have passed between the arts and mathematics. The story involves many intriguing characters: Gaston Julia, who laid the foundations for fractals and computer art while recovering in hospital after suffering serious injury in the First World War; Charles Howard, Hinton who was imprisoned for bigamy but whose books had a huge influence on twentieth-century art; Michael Scott, the Scottish necromancer who was the dedicatee of Fibonacci’s Book of Calculation, the most important medieval book of mathematics; Richard of Wallingford, the pioneer clockmaker who suffered from leprosy and who never recovered from a lightning strike on his bedchamber; Alicia Stott Boole, the Victorian housewife who amazed mathematicians with her intuition for higher-dimensional space. The book includes more than 200 colour illustrations, puzzles to engage the reader, and many remarkable tales: the secret message in Hans Holbein’s The Ambassadors; the link between Viking runes, a Milanese banking dynasty, and modern sculpture; the connection between astrology, religion, and the Apocalypse; binary numbers and the I Ching. It also explains topics on the school mathematics curriculum: algorithms; arithmetic progressions; combinations and permutations; number sequences; the axiomatic method; geometrical proof; tessellations and polyhedra, as well as many essential topics for arts and humanities students: single-point perspective; fractals; computer art; the golden section; the higher-dimensional inspiration behind modern art.

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.

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 On Global Dynamics in Duffing Equation with Quasiperiodic Perturbation

2020 ◽  
Vol 22 (2) ◽  
pp. 164-176
Author(s):  
Timofey N. Dragunov ◽  
Kirill E. Morozov ◽  
Albert D. Morozov
Keyword(s):  
Differential Equations ◽  
Dimensional Space ◽  
Nonlinear Differential Equations ◽  
Uniform Mesh ◽  
Global Dynamics ◽  
Nonlinear Operator Equations ◽  
Wide Range ◽  
Perturbations Of Initial Data ◽  
Analysis Of Stability ◽  
Stated Problem

An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its successive solution via construction of an auxiliary system of nonlinear differential equations that can be solved with the help of different numerical methods. The result is presented as a mesh function that consists of approximate values of the solution of stated problem and is constructed on a uniform mesh in a bounded domain of two-dimensional space. The advantages of the method are its simplicity and also its universality in the sense that the method can be applied for solving problems with a wide range of nonlinearities. Finally it should be mentioned that one of the important advantages of the proposed method is its stability to perturbations of initial data that is substantiated by methods for analysis of stability of solutions of systems of ordinary differential equations. Solving several model problems shows effectiveness of the proposed method.

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