Using methods of Pade 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. meromorphic functions of finite order of growth -p. This means that there exist entire functions H(u), Hj(u), ..., HJ(R) of 17 in C9 such that Uj(7) = Hj(77)/H(g) (i = 1, ..., n) and maxIIH(u9)1, IH,(Ii)l, ..., IHn(u9)11

Rational approximations of Pade and Pade 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.The purpose of this paper is to present simple and complete proofs.of several results on best rational approximations to solutions of differential equations. Our methods are based on studies of particular differential invariants associated with graded subrings of Picard-Vessiot extensions of differential fields (1). The origin of these studies is the famous Kolchin's problem (2) on the extension of "Roth's theorem" to approximation of solutions of arbitrary algebraic differential equations by rational functions. The Roth's theorem is used here in the broad sense as the property of numbers or functions that was proved by Roth (3) for algebraic numbers. For numbers 6, the Roth's theorem means that for any E > 0 and arbitrary rational integers p and q, 10 -(p/q)l > IqL 2-8 for q 2 qo(E). For functions y(x), the Roth's theorem means that for any E > 0 and arbitrary poly-P(x) nomials P(x) and In this paper, we prove functional solutions of linear differential equations over k(x) and also prove Schmidt's theorem and the generalization of Roth's theorem for the simultaneous approximations in several normings. The methods of this paper are also useful in our complete solution of the Kolehin problem for arbitrary algebraic differential equations. We want to acknowledge that C. Osgood announced a year ago an effectivization of Roth's theorem for algebraic functions and recently announced the Roth's and Schmidt's theorems for solutions of linear differential equations. After the Roth theorem (3) in 1955, its p-adic and g-adic generalizations appeared (11)(12)(13). This development was summarized by Mahler in his book (11), where he presented a general "approximation theorem" on rational approximations of algebraic numbers in archimedian and nonarchimedian metrics.Let Pi, P2, . Pr+r'+ r be a fixed system of r + r' + r" distinct primes. Also, for a real number and for a p-adic number

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 of 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 Padé 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.

Bacldund transformations are defined as operations on solutions of a Riemann boundary value problem (vector bundles over P1) that add apparent singularities. For solutions of difference and differential linear spectral problems, BAcldund transformations are presented in explicit form through the Christoffel formula and its generalizations. Identities satisfied by iterations of elementary BAcldund 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.The purpose of this article is to demonstrate how various completely integrable systems in dimensions two and three can be represented in a transparent form, in terms of identities implied by BAcklund transformations (BTs) algebra. Investigations of BTs (1-3) have shown how a successive application of BTs replaces continuous variables by discrete ones. In this way one constructs, starting from a two-dimensional completely integrable system (say, the operator nonlinear Schrodinger equation), lattice systems in one or two dimensions equivalent to and approximating the initial one. Here we classify lattice models through identities satisfied by BTs. In various infinitesimal limits and reductions, one recovers a variety of two-and three-dimensional difference, difference-differential, and differential, completely integrable systems of isospectral deformation origin. In this there is an analogy with Hirota's bilinear method (4). (7) and its generalizations. Christoffel's formula for the iteration of BTs corresponds to the equivalent situation when the weight function w(A) is multiplied by a function rational in A and a finite sum of 8 functions is added; see treatments by Szego (6) and Uvarov (8). In Uvarov's formula (8) In analogy to Eq. 2, if poles instead of zeros are added tof(A), then in Eq. 3 the corresponding polynomials of the first kind are substituted by polynomials of the second kind.Alternatively, the effect of an elementary BT can be described in terms of a single difference operator X(A). This operator X(A) is a difference analog of the vertex operator that represents the generating function for infinite dimensional affine algebras described by Lepowsky and Wilson (9). An operator X(A) acts on functions of infinitely many variables {ai} that are

This paper provides transcendental and algebraic framework for the classification of identities expressing and other logarithms of algebraic numbers as rapidly convergent generalized hypergeometric series in rational parameters. Algebraic and arithmetic relations between values of p؉1 F p 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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