Hyperreal Delta Functions as a New General Tool for Modeling Systems with Infinitely High Densities

In general, the state of a system in which a physical quantity such as mass is distributed over space can be modeled by a function that represents the density distribution. The purpose of this paper is to introduce special functions that can be applied when in the system to be modeled, where the quantity is distributed over isolated points. For that matter, the expanded real numbers are introduced as an ordered subring of the hyperreal number field that does not contain any infinitesimals, and hyperreal delta functions are defined as special functions from the real numbers to the expanded real numbers satisfying the condition that (i) the support is a singleton, and (ii) the integral over the reals is a nonzero real number. These newly defined hyperreal delta functions, and tensor products thereof, then provide a general tool, applicable for the mathematical modeling of physical systems in which infinitely high densities occur.

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Dimensional Analysis in Mathematical Modeling Systems: A Simple Numerical Method

10.21236/ada236904 ◽

1991 ◽

Cited By ~ 1

Author(s):

Hemant K. Bhargava

Keyword(s):

Mathematical Modeling ◽

Numerical Method ◽

Dimensional Analysis ◽

Modeling Systems

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Mathematical modeling of the current density distribution in a high-frequency electrosurgery

2015 16th International Conference on Computational Problems of Electrical Engineering (CPEE) ◽

10.1109/cpee.2015.7333379 ◽

2015 ◽

Cited By ~ 3

Author(s):

Volodymyr Sydorets ◽

Alexei Lebedev ◽

Andrey Dubko

Keyword(s):

Mathematical Modeling ◽

Density Distribution ◽

Current Density ◽

High Frequency ◽

Current Density Distribution

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Remarks on systems and differential operators on groups

Facta universitatis - series Electronics and Energetics ◽

10.2298/fuee0503531s ◽

2005 ◽

Vol 18 (3) ◽

pp. 531-545 ◽

Cited By ~ 3

Author(s):

Radomir Stankovic ◽

Miomir Stankovic ◽

Claudio Moraga

Keyword(s):

Mathematical Modeling ◽

System Theory ◽

Fourier Analysis ◽

Formal Specification ◽

Differential Operators ◽

Computer Systems ◽

Complex Numbers ◽

Real Numbers ◽

Large Size ◽

Design And Testing

Complexity and large size of contemporary control, communication, and computer systems impose strong challenges to methods of their mathematical modeling, design, and testing. Classical approaches to their formal specification by exploiting theory of systems on groups of real numbers R and complex numbers C, often do not fulfill requirements of practice. For that reason, theory of systems on groups different from R and C has been developed. Differential operators and spectral (Fourier) analysis on groups play the same important role in such systems as in the case of systems modeled by signals defined on R and C. This paper first briefly reviews some aspects of research in system theory on groups and then presents an extension of the notion of Gibbs differentiation to matrix- valued functions on finite non-Abelian groups.

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Mathematical Modeling of Cyber-Socially-Physical Systems in Transport

10.1007/978-3-030-81619-3_48 ◽

2021 ◽

pp. 419-427

Author(s):

Pavel Elugachev ◽

Elzarbek Esharov ◽

Boris Shumilov ◽

Altynbek Kuduev

Keyword(s):

Mathematical Modeling ◽

Physical Systems

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Mathematical Modeling in Materials Science and Engineering

MRS Bulletin ◽

10.1557/s0883769400038793 ◽

1994 ◽

Vol 19 (1) ◽

pp. 11-13

Author(s):

Julian Szekely

Keyword(s):

Mathematical Modeling ◽

Systems Analysis ◽

Solid Mechanics ◽

Materials Science ◽

Science And Engineering ◽

Physical Systems ◽

The Past ◽

Modeling Techniques ◽

Materials Science And Engineering ◽

The Common

During the past two decades, mathematical modeling has been gaining acceptance as a legitimate part of materials science and engineering. However, as common to all relatively new disciplines, we still lack a realistic perspective regarding the uses, limitations, and even the optimal methodologies of mathematical modeling techniques.The term “mathematical modeling” covers a broad range of activities, including molecular dynamics, other atomistic scale systems, continuum fluid and solid mechanics, deformation processing, systems analysis, input-output models, and lifecycle analyses. The common point is that we use algebraic expressions or differential equations to represent physical systems to varying degrees of approximation and then manipulate these equations, using computers, to obtain graphical output.While it is becoming an accepted fact that some kind of mathematical modeling will be needed to make most research programs complete, there is still considerable ambiguity as to what form this should take and what might be the actual usefulness of such an effort.Among the more seasoned and successful practitioners of this art, clear guidelines have emerged regarding the uses and limitations of the mathematical modeling approach. We seek to illustrate these uses through the successful modeling examples presented by some leading practitioners. Some general principles may be worth repeating as an introduction to this interesting collection of articles.

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Pair Correlations and Random Walks on the Integers

Uniform distribution theory ◽

10.1515/udt-2016-0008 ◽

2016 ◽

Vol 11 (1) ◽

pp. 159-164

Author(s):

Radhakrishnan Nair ◽

Entesar Nasr

Keyword(s):

Random Walk ◽

Random Walks ◽

Real Number ◽

Pair Correlation ◽

Real Numbers ◽

Pair Correlations ◽

Nonzero Real Number ◽

Correlation Estimate

AbstractThe paper gives conditions for a sequence of fractional parts of real numbers $\left( {\{ a_n x\} } \right)_{n = 1}^\infty $ to satisfy a pair correlation estimate. Here x is a fixed nonzero real number and $\left( {a_n } \right)_{n = 1}^\infty $ is a random walk on the integers.

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Novel Refinements via n –Polynomial Harmonically s –Type Convex Functions and Application in Special Functions

Journal of Function Spaces ◽

10.1155/2021/6615948 ◽

2021 ◽

Vol 2021 ◽

pp. 1-17

Author(s):

Saad Ihsan Butt ◽

Saima Rashid ◽

Muhammad Tariq ◽

Miao-Kun Wang

Keyword(s):

Convex Function ◽

Hypergeometric Function ◽

Convex Functions ◽

Special Functions ◽

Integral Inequalities ◽

Real Numbers ◽

Algebraic Properties ◽

Type Integral ◽

Harmonically Convex Functions ◽

Formula Type

In this work, we introduce the idea of n –polynomial harmonically s –type convex function. We elaborate the new introduced idea by examples and some interesting algebraic properties. As a result, new Hermite–Hadamard, some refinements of Hermite–Hadamard and Ostrowski type integral inequalities are established, which are the generalized variants of the previously known results for harmonically convex functions. Finally, we illustrate the applicability of this new investigation in special functions (hypergeometric function and special mean of real numbers).

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