Vol 39, No 7 (2018)

A mathematical model for variably saturated flow in unconfined conditions is presented. The model is based on pseudo-unsaturated approach using Richards equation with piecewise linear dependencies between hydraulic head, water content and relative permeability. It is implemented in GeRa (Geomigration of Radionuclides) software package, which is designed for modeling groundwater flow and contaminant transport in porous media and uses finite volume methods on unstructured grids. We consider two nonlinear solvers for nonlinear equations arising from discretization of the Richards equation, namely Newton and Picard methods. A special method for correction of hydraulic head values within the iterations of nonlinear solvers is proposed. The developed numerical techniques are applied to two test cases: dam seepage and real-world groundwater flow problems.

The convergence and accuracy estimates are proved for Lagrange–Galerkin method, used for approximating the parabolic obstacle problem. The convergence analysis is based on the comparison of the solutions of Lagrange–Galerkin and backward Euler approximation schemes. First order in time step estimate for the difference of the solutions for above schemes in energy norm is proved under sufficiently weak requirements for the smoothness of the initial data. First order in time and space steps accuracy estimate for Lagrange–Galerkin method is derived in the case of discontinuous time derivative of the exact solution.

For the efficient error control of numerical solutions of the solid mechanics problems, the two requirements are important: an a posteriori error bound has sufficient accuracy and computation of the bound is cheap in respect to the arithmetic work. The first requirement can be formulated in a more specific form of consistency of an a posteriori bound, assuming that it is not improvable in the order and, at least, coincides in the order with the a priori error estimate. Several new a posteriori error bounds are presented, which improve accuracy and reduce the computational cost. Also for the first time a new consistent guaranteed a posteriori error bound is suggested.

This article considers the problem of bubbles existence while using the pool of renewable investment projects. The formulation of the Cantor–Lippman model for continuous time is described in this paper. The result allows to classify pools of investment projects into the arbitration, and the ineffective and the standard is proved. The estimation of the yield is found for each of the classes. The classification of pools and their yield calculation are based on the functions of the upper envelope of the Laplace transform of the investment projects cash flow functions. It is shown that for the case of a standard pool the yield can be obtained by computing the minimal positive root of the upper envelope. Also, it is shown that for the case of a standard pool the roots different from the minimum one refer to bubble strategies requiring permanent reinvesting to support growth and are not able to result in liquid final state for investors.

The problem of finding the minimal eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem for the ordinary differential equation with coefficients depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radiofrequency discharge at reduced pressures. A necessary and sufficient condition for the existence of a minimal eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem is established. The original differential eigenvalue problem is approximated by the finite element method on a uniform grid. The convergence of approximate eigenvalue and approximate positive eigenfunction to exact ones is proved. Investigations of this paper generalize well known results for eigenvalue problems with linear dependence on the spectral parameter.

In the paper we consider quantum (δ, ϵ)-hash functions in so called phase form (phase quantum (δ, ϵ)-hash function). It is known that ϵ-biased sets generate phase quantum (δ, ϵ)- hash function. We show that the construction is invertible, that is, phase quantum (δ, ϵ)-hash function defines ϵ-biased sets. Next, we present an efficient (in the sense of time and qubits needed) Branching program construction for phase quantum (δ, ϵ)-hash function.

As we proved earlier, for any triangulated category \(\underline C \) endowed with a weight structure w and a triangulated subcategory \(\underline D \) of \(\underline C \) (strongly) generated by cones of a set of morphism S in the heart \(\underline {Hw} \) of w there exists a weight structure w' on the Verdier quotient \(\underline {C'} = \underline C /\underline D \) such that the localization functor \(\underline C \to \underline {C'} \) is weight-exact (i.e., “respects weights”). The goal of this paper is to find conditions ensuring that for any object of \(\underline {C'} \) of non-negative (resp. non-positive) weights there exists its preimage in \(\underline C \) satisfying the same condition; we call a certain stronger version of the latter assumption the left (resp., right) weight lifting property. We prove that that these weight lifting properties are fulfilled whenever the set S satisfies the corresponding (left or right) Ore conditions. Moreover, if \(\underline D \) is generated by objects of \(\underline {Hw} \) then any object of \(\underline {Hw'} \) lifts to \(\underline {Hw} \). We apply these results to obtain some new results on Tate motives and finite spectra (in the stable homotopy category). Our results are also applied to the study of the so-called Chow-weight homology in another paper.

We consider an information reconciliation protocol for quantum key distribution (QKD). In order to correct down the error rate, we suggest a method, which is based on symmetric blind information reconciliation for the low-density parity-check (LDPC) codes. We develop a subsequent verification protocol with the use of ϵ-universal hash functions, which allows verifying the identity between the keys with a certain probability.

In our paper we give theoretical and practical estimations of the error probability in the well-known Miller–Rabin probabilistic primality test. We show that a theoretical probability of error 0.25 for a single round of the test is very overestimated and, in fact, error is diminishing with the growth of length of numbers involved by a rate limited with ln n/\(\sqrt n \).

A description of the monomial automorphisms group of an arbitrary linear cyclic code in term of polynomials is presented. This allows us to reduce a task of code’s monomial automorphisms calculation to a task of solving some system of equations (in general, nonlinear) over a finite field. The results are illustrated with examples of calculating the full monomial automorphisms groups for two codes.