Semi-orthogonal spline-wavelets with derivatives and the algorithm with splitting

This paper deals with the use of a scalar product with derivatives for constructing semi-orthogonal spline-wavelets. The reduction of supports of such wavelets in comparison with the classical semi-orthogonal wavelets is shown. For splines of the third degree, the algorithm of wavelet-transformation in the formof the solution to a three-diagonal systemof linear equations with strict diagonal prevalence has been obtained. The results of numerical experiments on the calculation of derivatives of a discretely given function are presented.     

The motion of a body with a cavity partly filled with a viscous liquid

Many papers are concerned with the dynamics of a rigid body with a cavity filled with liquid (see the bibliography in [1]). The present paper deals with the motion of a rigid body having a cavity partly filled with a viscous incompressible liquid, and having a free surface. The shape of the cavity is arbitrary. The problem is considered in a linear formulation. The oscillations of the body with respect to its center of inertia and the motion of the liquid in the cavity are assumed small. The viscosity of the liquid is considered low. The solution of the problem of the oscillations of a body with a cavity partly filled with an ideal liquid is used as an initial approximation [1 to 6]. The viscosity is taken into consideration by the boundary layer method used before in similar problems [1 and 7 to 10). General equations are derived for the dynamics of a body filled with a liquid, for an arbitrary form of cavity. The coefficients of those integro-differential equations depend only on the solution of the problem of the oscillations of a body with a cavity of the given form filled with an ideal liquid. Since the corresponding problem has been solved for cavities of many forms [1 to 6, 11 and 12] in the case of an ideal liquid, the determination of the characteristic coefficients is reduced to the evaluation of quadratures. Several particular cases of motion are considered.     

Simulation of flows with strong shocks with an adaptive conservative scheme

In the present work, numerical simulations of unsteady flows with moving shocks are presented. An unsteady mesh adaptation method, based on error equidistribution criteria, is adopted to capture the most important flow features. The modifications to the topology of the grid are locally interpreted in terms of continuous deformation of the finite volumes built around the nodes. The arbitrary Lagrangian–Eulerian formulation of the Euler equations is then applied to compute the flow variable over the new grid without resorting to any explicit interpolation step. The numerical results show an increase in the accuracy of the solution, together with a strong reduction of the computational costs, with respect to computations with a uniform grid using a larger number of nodes.     

Stokeswaves with vorticity

The existence of periodic waves propagating downstream on the surface of a two-dimensional infinitely deep body of water under the force of gravity is established for a general class of vorticities. When reformulated as an elliptic boundary value problem in a fixed semi-infinite cylinder with a parameter, the operator describing the problem is nonlinear and non-Fredholm. A global connected set of nontrivial solutions is obtained via singular theory of bifurcation. The proof combines a generalized degree theory, global bifurcation theory, and Whyburn’s lemma in topology with the Schauder theory for elliptic problems and the maximum principle.     

Martingales with sufficient statistics

The property of countable convexity of the set which contains all dominated martingale laws, for a family of measurable functions with values in a separable Borel space, is proved to be equivalent to the existence of a sufficient statistic. The result is then used to derive a representation of such laws in terms of their extreme points, which are laws of uniformly integrable martingales. Finally, we show that martingales with sufficient statistics converge with probability 1.     

Knapsack problems with setups

Knapsack problems with setups find their application in many concrete industrial and financial problems. Moreover, they also arise as subproblems in a Dantzig–Wolfe decomposition approach to more complex combinatorial optimization problems, where they need to be solved repeatedly and therefore efficiently. Here, we consider the multiple-class integer knapsack problem with setups. Items are partitioned into classes whose use implies a setup cost and associated capacity consumption. Item weights are assumed to be a multiple of their class weight. The total weight of selected items and setups is bounded. The objective is to maximize the difference between the profits of selected items and the fixed costs incurred for setting-up classes. A special case is the bounded integer knapsack problem with setups where each class holds a single item and its continuous version where a fraction of an item can be selected while incurring a full setup. The paper shows the extent to which classical results for the knapsack problem can be generalized to these variants with setups. In particular, an extension of the branch-and-bound algorithm of Horowitz and Sahni is developed for problems with positive setup costs. Our direct approach is compared experimentally with the approach proposed in the literature consisting in converting the problem into a multiple choice knapsack with pseudo-polynomial size.     

Bogolyubov-type theorem with constraints induced by a control system with hysteresis effect

We consider the problem of minimization of an integral functional with nonconvex with respect to the control integrand. We minimize our functional over the solution set of a control system described by two ordinary differential equations subject to a control constraint given by a multivalued mapping with closed nonconvex values. The coefficients of the equations and the constraint depend on the phase variables. One of the equations contains the subdifferential of the indicator function of a closed convex set depending on the unknown phase variable. The equation containing the subdifferential describes an input–output relation of hysteresis type.     

Tuberculosis model with relapse via fractional conformable derivative with power law

The present study explores the tuberculosis dynamics with relapse with a nonlocal conformable derivative in the Caputo sense. The real data of tuberculosis cases since 2002 to 2017 are used to the set the parameters. The numerical results with the realistic parameters are chosen and present the graphical results. Further, we assign different values to the fractional parameters α and β and discuss its effect on the system variables. The use of these two fractional operators on the model simultaneously with realistic data gives reliable results.     

Categories with families and first-order logic with dependent sorts

First-order logic with dependent sorts, such as Makkai's first-order logic with dependent sorts (FOLDS), or Aczel's and Belo's dependently typed (intuitionistic) first-order logic (DFOL), may be regarded as logic enriched dependent type theories. Categories with families (cwfs) is an established semantical structure for dependent type theories, such as Martin-Löf type theory. We introduce in this article a notion of hyperdoctrine over a cwf, and show how FOLDS and DFOL fit in this semantical framework. A soundness and completeness theorem is proved for DFOL. The semantics is functorial in the sense of Lawvere, and uses a dependent version of the Lindenbaum-Tarski algebra for a DFOL theory. Agreement with standard first-order semantics is established. Applications of DFOL to constructive mathematics and categorical foundations are given. A key feature is a local propositions-as-types principle.     

Children with mathematical difficulties cope with modelling tasks: what develops?

This study investigates the participation and knowledge growth of children with mathematical difficulties as they work in groups with their classmates on a year-long sequence of modelling tasks. It involves observations of a class of 23 fifth graders, 9 of whom were identified as having difficulties in mathematics. All the students worked for 8 months on a sequence of 12 modelling task in heterogeneous groups. The findings show a gradual growth in modelling competencies and mathematical knowledge of the students with mathematical difficulties together with an increase in their contribution to the group. The growth in modelling competencies involved their ability to analyse situations and the growth of mathematical knowledge was evident in offering mathematical ideas during group work and in a better posttest performance. Student reflections indicated their awareness of these changes and of the appreciation of their ideas by their peers.     

Boundary Value Problems with a Parameter of Differential Equations with Deviated Arguments

The aim of the present paper is to study boundary value problems with a parameter of differential equations with deviated arguments. The existence, uniqueness and continuous dependence of the solution on the right-hand side of our problem are considered.     

Sets with two associative operations

In this paper we consider duplexes, which are sets with two associative binary operations. Dimonoids in the sense of Loday are examples of duplexes. The set of all permutations carries a structure of a duplex. Our main result asserts that it is a free duplex with an explicitly described set of generators. The proof uses a construction of the free duplex with one generator by planary trees.     

Homology with Trivial Coefficients and Universal Central Extensions of Algebras with Bracket

A 5-term exact sequence and the interpretation of low dimensional groups of homology with trivial coefficients of algebras with bracket are obtained. The endofunctor in the category of algebras with bracket which assigns to a perfect algebra with bracket its universal central extension is constructed and the characterization of the universal central extension by means of the low-dimensional homology groups is done. The conditions required to lift an automorphism or a derivation of A to A′ in a covering f:A′ ? A are analyzed.     

An inventory model with backorders with fuzzy parameters and decision variables

The paper considers an inventory model with backorders in a fuzzy situation by employing two types of fuzzy numbers, which are trapezoidal and triangular. A full-fuzzy model is developed where the input parameters and the decision variables are fuzzified. The optimal policy for the developed model is determined using the Kuhn-Tucker conditions after the defuzzification of the cost function with the graded mean integration (GMI) method. Numerical examples and a sensitivity analysis study are provided to highlight the differences between crisp and the fuzzy cases.     

Preconditioning of two-by-two block matrix systems with square matrix blocks,with applications

Two-by-two block matrices of special form with square matrix blocks arise in important applications, such as in optimal control of partial differential equations and in high order time integration methods.Two solution methods involving very efficient preconditioned matrices, one based on a Schur complement reduction of the given system and one based on a transformation matrix with a perturbation of one of the given matrix blocks are presented. The first method involves an additional inner solution with the pivot matrix block but gives a very tight condition number bound when applied for a time integration method. The second method does not involve this matrix block but only inner solutions with a linear combination of the pivot block and the off-diagonal matrix blocks.Both the methods give small condition number bounds that hold uniformly in all parameters involved in the problem, i.e. are fully robust. The paper presents shorter proofs, extended and new results compared to earlier publications.     

The Cauchy-Nicoletti problem with poles

The Cauchy-Nicoletti boundary value problem for a system of ordinary differential equations with pole-type singularities is investigated. The conditions of the existence, uniqueness, and nonuniqueness of a solution in the class of continuously differentiable functions are given. The classical Banach contraction principle is combined with a special transformation of the original problem.     

Games with informants

The purpose of this paper is to investigate the influence of information in non-cooperative games. We shall use methods being provided by information theory. The asymptotic value of a class of two-person zero-sum games is investigated in which one of the players is supplied with some knowledge of the controls chosen by the other. According to this information he then chooses his own controls. Due to this asymmetry a higher payoff arises compared to the outcome of games being played under the usual assumptions. The main result of this chapter is a computable formula for the value of these games.     

Boundary-Value Problem with Impulsive Action for a Parabolic Equation with Degeneration

Ukrainian Mathematical Journal - For a second-order parabolic equation, we consider a problem with oblique derivative and impulsive action. The coefficients of the equation and the boundary...     

Regular spaces with symmetries

Regular spaces with symmetries are manifolds with a multiplication that satisfies certain conditions. The concepts of translation group and general triple system of a regular space with symmetries are introduced and studied.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 113–120, July, 1973.     

Subharmonic Solutions with Prescribed Minimal Period of a Forced Pendulum Equation with Impulses

We consider a mathematical model which describes the sliding frictional contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic, the material’s behavior is described with a viscoplastic constitutive law with internal state variable and the contact is modelled with normal compliance and unilateral constraint. The wear of the contact surfaces is taken into account, and is modelled with a version of Archard’s law. We derive a mixed variational formulation of the problem which involve implicit history-dependent operators. Then, we prove the unique weak solvability of the contact model. The proof is based on a fixed point argument proved in Sofonea et al. (Commun. Pure Appl. Anal. 7:645–658, 2008), combined with a recent abstract existence and uniqueness result for mixed variational problems, obtained in Sofonea and Matei (J. Glob. Optim. 61:591–614, 2014).