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D V Chudnovsky

Publications and source records attributed to D V Chudnovsky.

8 recordsLinked to original sources
PubMed1999

Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations.

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent "accurately" harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in "accurate" reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Pade approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.

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PubMed1998

Classification of hypergeometric identities for pi and other logarithms of algebraic numbers.

This paper provides transcendental and algebraic framework for the classification of identities expressing pi and other logarithms of algebraic numbers as rapidly convergent generalized hypergeometric series in rational parameters. Algebraic and arithmetic relations between values of p+1Fp hypergeometric functions and their values are analyzed. The existing identities are explained, and new exhaustive classes of new ones are presented.

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PubMed1989

The computation of classical constants.

Hypergeometric representations of classical constants and efficient algorithms for their calculation are discussed. Particular attention is devoted to algorithms for computing pi.

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PubMed1987

Algebraic complexities and algebraic curves over finite fields.

We consider the problem of minimal (multiplicative) complexity of polynomial multiplication and multiplication in finite extensions of fields. For infinite fields minimal complexities are known [Winograd, S. (1977) Math. Syst. Theory 10, 169-180]. We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear coding theory and algebraic curves over finite fields.

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PubMed1985

Padé approximations and diophantine geometry.

Using methods of Padé approximations we prove a converse to Eisenstein's theorem on the boundedness of denominators of coefficients in the expansion of an algebraic function, for classes of functions, parametrized by meromorphic functions. This result is applied to the Tate conjecture on the effective description of isogenies for elliptic curves.

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PubMed1983

Rational approximations to solutions of linear differential equations.

Rational approximations of Padé and Padé type to solutions of differential equations are considered. One of the main results is a theorem stating that a simultaneous approximation to arbitrary solutions of linear differential equations over C(x) cannot be "better" than trivial ones implied by the Dirichlet box principle. This constitutes, in particular, the solution in the linear case of Kolchin's problem that the "Roth's theorem" holds for arbitrary solutions of algebraic differential equations. Complete effective proofs for several valuations are presented based on the Wronskian methods and graded subrings of Picard-Vessiot extensions.

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PubMed1983

Laws of composition of Bäcklund transformations and the universal form of completely integrable systems in dimensions two and three.

Bäcklund transformations are defined as operations on solutions of a Riemann boundary value problem (vector bundles over P(1)) that add apparent singularities. For solutions of difference and differential linear spectral problems, Bäcklund transformations are presented in explicit form through the Christoffel formula and its generalizations. Identities satisfied by iterations of elementary Bäcklund transformations are represented in the form of the law of addition or as the three-dimensional difference equation of Hirota's type. Matrix two-dimensional isospectral deformation equations are imbedded into three-dimensional scalar systems of Kadomtzev-Petviashvili (law of addition) form. Two-dimensional matrix systems correspond to reductions of Kadomtzev-Petviashvili equations with pseudodifferential operators satisfying algebraic equations.

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