A new fast direct algorithm for implementing a finite element method (FEM) of order on rectangles as applied to boundary value problems for Poisson-type equations is described that extends a well-known algorithm for the case of difference schemes or bilinear finite elements (n = 1). Its core consists of fast direct and inverse algorithms for expansion in terms of eigenvectors of one-dimensional eigenvalue problems for an nth-order FEM based on the fast discrete Fourier transform. The amount of arithmetic operations is logarithmically optimal in the theory and is rather attractive in practice. The algorithm admits numerous further applications (including the multidimensional case).

The Dotsenko-Fateev integral, an analytic function of one complex variable arising in conformal field theory, is generalized in a natural way to an analytic function of two complex variables. A system of partial differential equations and a Pfaffian system of Fuchsian type are derived for this generalized Dotsenko- Fateev integral. The Fuchsian system permits to obtain local expansions of solutions in the neighborhoods of singularities of the system.

Рассматривается выборка X_1, . . . , X_n, состоящая из независимых одинаково рас- пределенных векторов в Rp, имеющих нулевое среднее и ковариационную матрицу \Sigma. Задача восстановления спектральных проекторов ковариационных матриц вы- сокой размерности по выборке наблюдений является одной из ключевых задач ста- тистики и возникает во многих приложениях. В недавней работе В. Колчинского и К. Луничи 2015 года были получены неасимптотические оценки нормы Фробениуса расстояния между выборочным и истинным проекторами \|\hat P_r − P_􏰀r \|_2^2, а также исследовано асимптотическое поведение для больших выборок. Эта работа позво- ляет строить асимптотические доверительные множества для истинного проектора P_r в предположении, что нам известны моментные характеристики наблюдаемых величин. В настоящей работе рассматривается бутстреп-метод построения довери- тельных множеств для спектрального проектора P_r ковариационной матрицы \Sigma с помощью имеющихся данных. Данный подход не использует асимптотическое распределение величины \|\hat P_r − P_􏰀r \|_2^2 и не требует вычисления моментных характеристик последней. В работе изучается вопрос качества бутстреп-аппроксимации.

The notion of a tolerance of an element of a combinatorial optimization problem is often used for stability analysis of an optimal solution and it is a base for design branch-and-bound algorithms solving such problems. In this paper we show that for the weighted independent set problem on trees with n vertices all upper and lower tolerances related to this solution can be computed with O(n) time.

Entropy balance in the one-dimensional hyperbolic quasi-gasdynamic (HQGD) system of equations is analyzed. In regular flow regimes, it is shown that the behavior of entropy in the HQGD system is determined by terms involving the natural viscosity and thermal conductivity coefficients. The total entropy production differs from the Navier–Stokes equations for viscous compressible heat-conducting gases by the second order terms with respect to a relaxation parameter. Additionally, a similar analysis of energy balance is performed for the simpler case of the barotropic HQGD system, which is of interest for some applications.

A novel approach to the fair division problem is proposed, which is based on the concept of a priori estimates and ideas of dynamical systems theory. For several problems on the division of a resource with discrete components, this approach leads to explicit constructive solutions in cases for which even the existence of solutions has not been previously known.