Closed Forms: What They Are and Why We Care

Borwein, Jonathan M.; Crandall, Richard E.

doi:10.1090/noti936

Cited by 80 publications

(55 citation statements)

References 49 publications

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“…Many of approaches are applied to develop mobile channel models by a Let us define the key definitions that are the initial formulation of the mathematical descriptions of the process and systems. An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set [2]. On the other side, the closed-form expression represents an implicit solution that is contained in a mathematical expression.…”

Section: Related Workmentioning

confidence: 99%

An Iteration-Based Simulation Method for Getting Semi- Symbolic Solution of Non-coherent FSK/ASK System by Using Computer Algebra Systems

Mladenović¹,

MAKOV²,

VORONIN³

et al. 2016

SIC

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Abstract:The paper presents a development of an IBSM (Iteration-Based Simulation Method) to obtain semi-symbolic solutions of digital telecommunication non-coherent FSK/ASK system by using computer algebra systems. These solutions are applied for derivation of the symbol error probability (SEP) needed for quantitative and qualitative analysis of performances of telecommunication systems in optimization and design of low-complexity implementation into highcomplexity structures. Various software tools, used in telecommunications for calculating, designing and analysing, have been developed from the viewpoint of numeric-only algorithms. But, many shortcomings cannot be neglected, such as generating of a great amount of numeric data, losing insight into the phenomenon of investigation, and numerical computation manipulates with numerical values. For these reasons, IBSM provides design, optimization, manipulation, and simplification by introducing a new parameter of iteration. The IBSM is applied to aforementioned system when the uncorrelated and correlated noise and the interferences are present in both upper and lower frequencies, respectively. The impact of number of iterations, number of bits, correlation coefficient and different values of interferences to symbol error probability are derived on semi-symbolic forms. Making use of IBSM gives an opportunity to solve very complex analyses and derive many sophisticated conclusions.

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Section: Related Workmentioning

confidence: 99%

An Iteration-Based Simulation Method for Getting Semi- Symbolic Solution of Non-coherent FSK/ASK System by Using Computer Algebra Systems

Mladenović¹,

MAKOV²,

VORONIN³

et al. 2016

SIC

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show abstract

“…Upon completion, our PSLQ program reported an exclusion bound of 2.351 × 10 19 . This means that in any integer linear relation among the set of 149 constants that is not listed above, the Euclidean norm of the corresponding vector of coefficients must exceed 2.351 × 10…”

Section: Relations Amongst U Constantsmentioning

confidence: 99%

“…This paper is, in consequence, in part in homage to our friend and collaborator Richard Crandall who died after a short illness on December 20, 2012. It is also a fitting tribute to Dick Askey, whose role in the development of modern special function theory has been prodigious (see [1] and the discussion in [19]). In Indiscrete Thoughts [34, p. 216] Gian-Carlo Rota writes "special functions [is] a Wisconsin subject," but in this setting 'Askey' is a synonym for 'Wisconsin'.…”

Section: Introductionmentioning

confidence: 99%

Computation and theory of Mordell–Tornheim–Witten sums II

Bailey

Borwein

2015

Journal of Approximation Theory

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In [8] the current authors, along with the late and much-missed Richard Crandall , considered generalized Mordell-Tornheim-Witten (MTW) zeta-function values along with their derivatives, and explored connections with multiple-zeta values (MZVs). This entailed use of symbolic integration, high precision numerical integration, and some interesting combinatorics and special-function theory. The original motivation was to represent objects such as Eulerian log-gamma integrals; and all such integrals were expressed in terms of a MTW basis. Herein, we extend the research envisaged in [8] by analyzing the relations between a significantly more general class of MTW sums. This has required significantly more subtle scientific computation and concomitant special function theory.

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“…On the one hand, there are numerical and graphical solutions; their main advantage is that almost always they may be determined; they are not considered here. On the other hand, there are so-called exact or closed form solutions, a detailed discussion of this concept may be found in [1]. Thereby the goal is to obtain finite expressions in terms of known functions like elementary, Liouvillian or special functions that annihilate the given differential equation upon substitution; they exist only in exceptional cases.…”

Section: Introduction: Description Of the Problemmentioning

confidence: 99%

“…Drach [3] has described this situation as follows: "Lie's théorie des groupes à l'intégration des équations n'est pas la véritable généralization de la méthode employée par Galois pour les équations algébriques". 1 This judgement is based on the following observation. There are equations with a large group of Lie symmetries that may not be utilized for solving it as may be seen in several examples given below.…”

Section: Introduction: Description Of the Problemmentioning

confidence: 99%

Decomposition of ordinary differential equations

Schwarz

2017

Bull. Math. Sci.

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Decompositions of linear ordinary differential equations (ode's) into components of lower order have successfully been employed for determining their solutions. Here this approach is generalized to nonlinear ode's. It is not based on the existence of Lie symmetries, in that it is a genuine extension of the usual solution algorithms. If an equation allows a Lie symmetry, the proposed decompositions are usually more efficient and often lead to simpler expressions for the solution. For the vast majority of equations without a Lie symmetry decomposition is the only available systematic solution procedure. Criteria for the existence of diverse decomposition types and algorithms for applying them are discussed in detail and many examples are given. The collection of Kamke of solved equations, and a tremendeous compilation of random equations are applied as a benchmark test for comparison of various solution procedures. Extensions of these proceedings for more general types of ode's and also partial differential equations are suggested.

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Closed Forms: What They Are and Why We Care

Cited by 80 publications

References 49 publications

An Iteration-Based Simulation Method for Getting Semi- Symbolic Solution of Non-coherent FSK/ASK System by Using Computer Algebra Systems

An Iteration-Based Simulation Method for Getting Semi- Symbolic Solution of Non-coherent FSK/ASK System by Using Computer Algebra Systems

Computation and theory of Mordell–Tornheim–Witten sums II

Decomposition of ordinary differential equations

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