7,980 research outputs found
Basic calculus on time scale with mathematica
Mathematical modeling of time dependent systems are always interesting for applied mathematicians. First continuous and then discrete mathematical modeling are built during the mathematical development from ancient to the modern times. By the discovery of the time scales, the problem of irregular controlling of time dependent systems is solved in 1990's. In this paper, we explain the derivative of functions on time scales and the solutions of some basic calculus problems by using Mathematica. © Springer-Verlag Berlin Heidelberg 2003
Basics of Qualitative Theory of Linear Fractional Difference Equations
Tato doktorská práce se zabývá zlomkovým kalkulem na diskrétních množinách, přesněji v rámci takzvaného (q,h)-kalkulu a jeho speciálního případu h-kalkulu. Nejprve jsou položeny základy teorie lineárních zlomkových diferenčních rovnic v (q,h)-kalkulu. Jsou diskutovány některé jejich základní vlastnosti, jako např. existence, jednoznačnost a struktura řešení, a je zavedena diskrétní analogie Mittag-Lefflerovy funkce jako vlastní funkce operátoru zlomkové diference. Dále je v rámci h-kalkulu provedena kvalitativní analýza skalární a vektorové testovací zlomkové diferenční rovnice. Výsledky analýzy stability a asymptotických vlastností umožňují vymezit souvislosti s jinými matematickými disciplínami, např. spojitým zlomkovým kalkulem, Volterrovými diferenčními rovnicemi a numerickou analýzou. Nakonec je nastíněno možné rozšíření zlomkového kalkulu na obecnější časové škály.This doctoral thesis concerns with the fractional calculus on discrete settings, namely in the frame of the so-called (q,h)-calculus and its special case h-calculus. First, foundations of the theory of linear fractional difference equations in (q,h)-calculus are established. In particular, basic properties, such as existence, uniqueness and structure of solutions, are discussed and a discrete analogue of the Mittag-Leffler function is introduced via eigenfunctions of a fractional difference operator. Further, qualitative analysis of a scalar and vector test fractional difference equation is performed in the frame of h-calculus. The results of stability and asymptotic analysis enable us to specify the connection to other mathematical disciplines, such as continuous fractional calculus, Volterra difference equations and numerical analysis. Finally, a possible generalization of the fractional calculus to more general settings is outlined.
Reductionism and the Universal Calculus
In the seminal essay, "On the unreasonable effectiveness of mathematics in
the physical sciences," physicist Eugene Wigner poses a fundamental
philosophical question concerning the relationship between a physical system
and our capacity to model its behavior with the symbolic language of
mathematics. In this essay, I examine an ambitious 16th and 17th-century
intellectual agenda from the perspective of Wigner's question, namely, what
historian Paolo Rossi calls "the quest to create a universal language." While
many elite thinkers pursued related ideas, the most inspiring and forceful was
Gottfried Leibniz's effort to create a "universal calculus," a pictorial
language which would transparently represent the entirety of human knowledge,
as well as an associated symbolic calculus with which to model the behavior of
physical systems and derive new truths. I suggest that a deeper understanding
of why the efforts of Leibniz and others failed could shed light on Wigner's
original question. I argue that the notion of reductionism is crucial to
characterizing the failure of Leibniz's agenda, but that a decisive argument
for the why the promises of this effort did not materialize is still lacking.Comment: 11 pages, 1 figur
A distributed procedure for computing stochastic expansions with Mathematica
The solution of a (stochastic) differential equation can be locally approximated by a (stochastic) expansion. If the vector field of the differential equation is a polynomial, the corresponding expansion is a linear combination of iterated integrals of the drivers and can be calculated using Picard Iterations. However, such expansions grow exponentially fast in their number of terms, due to their specific algebra, rendering their practical use limited.
We present a Mathematica procedure that addresses this issue by reparametrizing the polynomials and distributing the load in as small as possible parts that can be processed and manipulated independently, thus alleviating large memory requirements and being perfectly suited for parallelized computation. We also present an iterative implementation of the shuffle product (as opposed to a recursive one, more usually implemented) as well as a fast way for calculating the expectation of iterated Stratonovich integrals for Brownian motion
3D Printing A Pendant with A Logo
The purpose of this short paper is to describe a project to manufacture a
3D-print of a pendant that includes a logo. The methods described in this paper
involve processing the image of the logo through a Mathematica script. These
methods can be applied to many logos and other images. With the Mathematica
script, a STereoLithography (.stl) file is created that can be used by a 3D
printer. Finally, the object is created on a 3D printer. We assume that the
reader is familiar with the basics of 3D printing.Comment: 8 pages, 7 figure
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