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Derivatives of rational expressions with multiplicity

Sylvain Lombardy, Jacques Sakarovitch
2005 Theoretical Computer Science  
We formalize and generalize the idea of "partial derivatives" introduced in 1995 by Antimirov, in order to obtain a construction of an automaton with multiplicity from a rational expression describing  ...  set of derived terms, that yields a finite automaton with multiplicity whose behaviour is the series denoted by the expression.  ...  Rutten and several anonymous referees for their corrections and their comments that have helped them to improve the content and the presentation of this paper.  ... 
doi:10.1016/j.tcs.2004.10.016 fatcat:leqx2j4kp5ddbgmwdropsjxk64

A simple method for computing the values of the derivatives of a rational function

Jon Rokne
1986 Journal of Computational and Applied Mathematics  
as required thus obtaining a rational expression for the derivative bf the order required.  ...  The procedure normally employed for the calculation of the derivatives of a rational function at a given point consists of applying the rules for forming derivatives to the rational function as many times  ...  as required thus obtaining a rational expression for the derivative bf the order required.  ... 
doi:10.1016/0377-0427(86)90078-6 fatcat:vhv3ezkzznh3lhqk7bessy4iym

Derived-Term Automata of Multitape Rational Expressions (Long version) [article]

Akim Demaille
2016 arXiv   pre-print
We define expansions, which generalize the concept of derivative of a rational expression, but relieved from the need of a free monoid.  ...  We introduce (weighted) rational expressions to denote series over Cartesian products of monoids. To this end, we propose the operator | to build multitape expressions such as (a^+|x + b^+|y)^*.  ...  This algorithm relies on rational expansions. They are to the derivatives of rational expressions what differential forms are to the derivatives of functions.  ... 
arXiv:1608.00749v1 fatcat:44gt5g6j7vaj7aolirq74cie4m

Solution of Mumford's second problem [article]

Julia Bernatska, Yaacov Kopeliovich
2022 arXiv   pre-print
A complete solution of Mumford's second problem about representation of theta derivatives with rational characteristics in terms of theta constants with rational characteristics is found.  ...  An explicit formula for computing such an expression for theta derivative with an arbitrary rational characteristic is derived, and illustrated with examples.  ...  Following Jacobi's theta derivative identity Mumford posed the problem about expressing a theta derivative with rational characteristic as a cubic polynomial of theta constants with rational characteristics  ... 
arXiv:2108.00130v2 fatcat:pkqvdjuzsnazxcvy2fxnkfnlkm

A complete axiom system for rational sets with multiplicity

Masato Morisaki, Ko Sakai
1980 Theoretical Computer Science  
automata with multiplicity in semlilr , K).  ...  is complete; that is, all and only equalities between rational K-subsets are derivable in the system.  ...  Acknowledgment We thank Professor Masako Takahashi at Tokyo Institute of Technology for her valuable advice and encouragement.  ... 
doi:10.1016/0304-3975(80)90038-9 fatcat:ntx6etbj65epjambg3qp5nzl4m

Derived-Term Automata for Extended Weighted Rational Expressions [chapter]

Akim Demaille
2016 Lecture Notes in Computer Science  
Inspired by derived-term based algorithms, its core relies on a different construct, rational expansions.  ...  We present an algorithm to build an automaton from a rational expression. This approach introduces support for extended weighted expressions.  ...  In particular, an anonymous reviewer of ICALP 2016 contributed the proof of Theorem 1, much simpler than the original one (which was still based on derivatives), and proposed the benchmark of Sect. 4.4  ... 
doi:10.1007/978-3-319-46750-4_20 fatcat:54s3we3xejdbzo3hyzof4cweay

Derived-term Automata for Extended Weighted Rational Expressions [article]

Akim Demaille
2016 arXiv   pre-print
Inspired by derived-term based algorithms, its core relies on a different construct, rational expansions.  ...  We present an algorithm to build an automaton from a rational expression. This approach introduces support for extended weighted expressions.  ...  Acknowledgments Interactions with A. Duret-Lutz, S. Lombardy, L. Saiu and J. Sakarovitch resulted in this work. Anonymous reviewers made very helpful comments.  ... 
arXiv:1605.01530v1 fatcat:t7ga46j4lrhdph5h3oradqdayy

Computation of lower derivatives of rational triangular Bézier surfaces and their bounds estimation

Lei Zhang, Guo-jin Wang
2005 Journal of Zhejiang University: Science A  
presents some explicit formulas of the first and second derivatives of rational triangular Bézier surface with respect to each variable (including the mixed derivative) and derives some estimations of  ...  Thus, we have represented the second mixed derivatives of the rational triangular B-B surface  ...  Section 2 in a different expression, and then estimate the bound of each derivative based on the new expression.  ... 
doi:10.1631/jzus.2005.as0108 fatcat:yjjpqzfjhrd7jpfth4zravgq5y

Derivatives of the Hurwitz Zeta function for rational arguments

Jeff Miller, Victor S. Adamchik
1998 Journal of Computational and Applied Mathematics  
The functional equation for the Hurwitz Zeta function ((s,a) is used to obtain formulas for derivatives of ((s,a) at negative odd s and rational a.  ...  For several of these rational arguments, closed-form expressions are given in terms of simpler transcendental functions, like the logarithm, the polygamma fimction, and the Riemann Zeta function.  ...  Summed trigonometric terms give rise to sums of Hurwitz Zeta derivatives with alternating signs that cannot be removed with the multiplication formula or other known identities.  ... 
doi:10.1016/s0377-0427(98)00193-9 fatcat:iflowdfipbhb5lnfzl5z5y37pe

Symbolic Synthesis of Mealy Machines from Arithmetic Bitstream Functions

H.H. Hansen, J. Rutten
2010 Scientific Annals of Computer Science  
The synthesis algorithm can be seen as an analogue of Brzozowski's construction of a finite deterministic automaton from a regular expression.  ...  It is based on a coinductive characterisation of the operators of 2-adic arithmetic in terms of stream differential equations.  ...  p) × Bin(q) (with on the left multiplication of rationals and on the right multiplication of bitstreams), and similarly for the other operators.  ... 
doaj:4b357923a7e04c029d14e6197848b0d6 fatcat:gq6audzktzajviwltymsrns2gm

Derivatives of rational expressions and related theorems

Jean-Marc Champarnaud, Gérard Duchamp
2004 Theoretical Computer Science  
Our aim is to study the set of K-rational expressions describing rational series.  ...  More precisely we are concerned with the deÿnition of quotients of this set by coarser and coarser congruences which lead to an extension-in the case of multiplicities-of some classical results stated  ...  The set of partial derivatives of every regular expression in R( ) is ÿnite. Series and rational expressions 3.1.  ... 
doi:10.1016/j.tcs.2003.10.005 fatcat:cpekftggdbguxjrk57wxhxsy7i

Closed-form formulae for the derivatives of trigonometric functions at rational multiples of π

Djurdje Cvijović
2009 Applied Mathematics Letters  
chi function, that the values of all derivatives of four trigonometric functions at rational multiples of π can be expressed in closed form as simple finite sums involving the Bernoulli and Euler polynomials  ...  In this sequel to our recent note it is shown, in a unified manner, by making use of some basic properties of certain special functions, such as the Hurwitz zeta function, Lerch zeta function and Legendre  ...  This work was financially supported by the Ministry of Science of the Republic of Serbia under research project number 144004.  ... 
doi:10.1016/j.aml.2008.07.019 fatcat:tbthrh72v5c43gwu4e5egda32u

Gnoseology of fractality: fractal optics

Gennady S. Melnikov, Ivan Cindrich, Sing H. Lee
1997 Diffractive and Holographic Device Technologies and Applications IV  
The paper deals with the theory and experiment of fractal optics.  ...  Basing upon analysis of these properties and attracting results of development of the theory of numbers (integers, rational, irrational) the author suggests his interpretation of gnoseology of fractality  ...  The author wishes to express deep thanks to colleagues and co-authors of some publications Larionov S.A., Mikheev P.A., Tsvetkov E.A. for conformity of ideas in theoretical and experimental interpretation  ... 
doi:10.1117/12.274402 fatcat:joapnw6rzzhuvi2hck5cwnblxi

On the asymptotic structure of the polynomials of minimal diophantic deviation from zero

Emiliano Aparicio Bernardo
1988 Journal of Approximation Theory  
PC)(x) has integer coefficients, by applying the above argument to the expression k=l p = 0, 1, . . . . n -1, (25) we derive that they are rational integers.  ...  Zf (24) then each root of the polynomial f (x) is a zero of the polynomials PN(x) of the above theorem with multiplicity order greater than or equal to r = NA. Proof: As (l/r!)  ... 
doi:10.1016/0021-9045(88)90093-7 fatcat:2uq5egeryjeftegegusqofh2yq

How expressions can code for automata

Sylvain Lombardy, Jacques Sakarovitch
2005 RAIRO - Theoretical Informatics and Applications  
The notion of derived term of an expression, introduced by Antimirov, appears to be instrumental in this problem.  ...  In this paper we investigate how it is possible to recover an automaton from a rational expression that has been computed from that automaton.  ...  The authors are pleased to thanks the careful referees who pointed out several inaccuracies in the definitions, and whose remarks were of precious help for improving the presentation of the paper.  ... 
doi:10.1051/ita:2005013 fatcat:zzss6zvqlbd4lnhpjncvbielxi
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