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Academician Ol'ga Aleksandrovna Ladyzhenskaya passed away on 12 January 2004. She was a distinguished mathematician and a member of the editorial board of this journal from 1961. The board and editorial staff express their deep condolences to her family and friends.

Let be an elliptic curve defined over the rationals, with rational 2-torsion. We prove a uniform bound for the number of rational points on  of height of the form , valid for every fixed and a suitable positive computable constant . We give an application of this result to the counting of quadruples of distinct primes that do not exceed  and satisfy for all , where are given integers. This is applied by Konyagin (in the paper [3], which is published simultaneously with the present one) to a problem on the large sieve by squares.

Necessary and (or) sufficient conditions on a closed set are given for any function , continuous on  and -analytic on , to be the uniform limit on  of a sequence of -analytic entire or -analytic meromorphic functions.

We study the large-time asymptotic behaviour of solutions of the Cauchy problem for a system of non-linear evolution equations with dissipation. In the case when the initial data are small, we construct solutions using the contraction-mapping principle. When the initial data are large, we obtain the large-time asymptotics of solutions by taking into account a certain symmetry of the non-linear terms.

Let be a positive integer and the maximal cardinality of a subset such that is squarefree for all , . For large  we obtain new upper and lower bounds for .

We prove that any two irreducible cuspidal Hurwitz curves and  (or, more generally, two curves with -type singularities) in the Hirzebruch surface with the same homology classes and sets of singularities are regular homotopic. Moreover, they are symplectically regular homotopic if and  are symplectic with respect to a compatible symplectic form.

We describe a solution of the problem of finding rational trigonometric functions with fixed denominator that deviate least from zero on several subintervals of the period. The resulting representation is used to prove inequalities that estimate the derivatives of rational trigonometric and algebraic functions with fixed denominator in terms of their values on several intervals. Particular cases of these inequalities include the well-known inequalities of Videnskii, Rusak, Totik and others.

De Rham curves are obtained from a polygonal arc by passing to the limit in repeatedly cutting off the corners: at each step, the segments of the arc are divided into three pieces in the ratio , where is a given parameter. We find explicitly the sharp exponent of regularity of such a curve for any . Regularity is understood in the natural parametrization using the arclength as a parameter. We also obtain a formula for the local regularity of a de Rham curve at each point and describe the sets of points with given local regularity. In particular, we characterize the sets of points with the largest and the smallest local regularity. The average regularity, which is attained almost everywhere in the Lebesgue measure, is computed in terms of the Lyapunov exponent of certain linear operators. We obtain an integral formula for the average regularity and derive upper and lower bounds.

We prove the rationality of a non-Gorenstein Fano threefold of Fano index one and degree eight having terminal cyclic quotient singularities and Picard group . This threefold can be described as the quotient of a double covering of  ramified in a smooth quartic surface by an involution fixing eight different points.

Let be a domain such that is strictly pseudoconvex and let be an open subset. We define the hull with respect to the algebra and study its properties. It is proved that every continuous function on  can be extended to a continuous function on  whose graph is locally foliated by holomorphic curves.

Errata to the paper by N.A. Tyurin "The dynamical correspondence in algebraic Lagrangian geometry" Izv. Ross. Akad. Nauk Ser. Mat. 66:3, 2002, 175-196. English transl.: Izv. Math., 66, 2002, 611-629.