Journal of the Institute of Mathematics of Jussieu

This paper presents a novel approach to interpreting logical formulas for synthesizing algorithms and programs. The proposed method combines features of Kleene realizability and Gödel's “dialectica” interpretation but does not rely on them directly. A simple version of positive predicate logic without functions is considered, including conjunction, disjunction, implication, and universal and existential quantifiers. A new realizability semantics for formulas and sequents is described, which considers not just a realization of a formula, but a realization with additional support. The realization roughly corresponds to Kleene realizability. The support provides additional data in favor of the correctness of the realization. The support must confirm that the realization works correctly for the formula under any valid conditions of application. A proof language is presented for which a correctness theorem is proved showing that any derivable sequent has a realization and support confirming that this realization works correctly for this formula under any valid conditions with a suitable interpreter for the programs used.

One of the main issues in statistical mechanics is the phase transition phenomenon. It happens when there are at least two distinct Gibbs measures in the model. It is known that the ferromagnetic Potts model with $q$ states possesses, at sufficiently low temperatures, at most $2^{q}-1$ translation-invariant splitting Gibbs measures. For continuous Hamiltonians, in the space of probability measures, the Gibbs measures form a non-empty, convex, compact set. Extremal measures, which corresponds to the extreme points of this set, determines pure phases. We study the extremality of the translation-invariant splitting Gibbs measures for the ferromagnetic $q$-state Potts model on the Cayley tree of order three. We define the regions where the translation-invariant Gibbs measures for this model are extreme or not. We reduce description of Gibbs measures to solving a non-linear functional equation, each solution of which corresponds to one Gibbs measure.

The problem of coordinating the semantics of data presented within different models has remained relevant for a significant period of time. First of all, this is related to the convenience of work for users who are accustomed to certain tools, for example, spreadsheets. The data prepared in these environments needs to be loaded into a centralized database, which makes it possible to get rid of duplication and inconsistency of data. An obstacle to this path is the problem of data reconciliation. Editing data directly in the database is a difficult task for non-programmer users. The traditional way to solve this problem is to develop special applications that have limited functionality. This paper proposes a technology that allows editing data in a database using spreadsheets, making their rich functionality available. The main difference from similar approaches is the use of the “Transformation” model, which makes the presentation of data convenient for human perception. Since the “Transformation” data model differs significantly from the relational model, there is a need to reconcile data between the database and spreadsheets. To solve similar problems, L.A. Kalinichenko proposed a method of commutative transformations in databases. In this paper, this technique, with some modifications, is used in algorithms for transferring data from a database to “Transformation” and back. The article presents an overview of works on the problem of data matching in various sources, a description of the data model “Transformation”, including: a description of the table schema, conditions for the existence of a table instance and data editing operations. The paper describes an algorithm for loading data into a table from a database and the algorithm for transforming data in a database in accordance with changes in the table, defines the conditions for the commutativity of the transformations, and presents a proof of the correctness of the transformations.

Issues related to solving the additive problem of sequential traversal of sets with precedence restrictions and cost functions that allow dependence on the list of tasks are considered. The basic method is a broadly understood dynamic programming (DP), supplemented in the case of problems of appreciable dimension by decompositions of the family of tasks and transformation of the parameters of the original problem. Possible applications are related, in particular, to the problem of tool control in figured sheet cutting of parts on CNC machines. In this problem, an important circumstance is taking into account the precedence conditions, which have, in particular, the following meaning: in the case of a part with holes, cutting of each of the internal contours (corresponding to the holes) should precede cutting of the external contour. The quality criterion itself in this problem, as a rule, is additive. Another type of constraints concerns avoiding thermal deformations of parts. When using the approach with penalties for violating the conditions associated with effective heat dissipation during cutting, cost functions arise that allow dependence on the list of tasks completed to date. Note that in another applied problem, namely, in the problem of dismantling radiation hazardous objects, cost functions arise with dependence on the list of tasks that have not been completed at the moment (and, consequently, concern the objects that have not been dismantled). As a result, we arrive at a very general problem with precedence constraints and cost functions with dependence on the list of tasks. The decomposition applied in the case of a noticeable dimensionality with subsequent implementation of the DP requires, on the one hand, the development of clustering methods, and, on the other, the construction of an adequate structure for distributing global precedence conditions among clusters. In the theoretical part of the work, the case of two clusters is discussed, which makes it possible to cover with a single scheme a number of practically interesting problems of a range (in terms of dimensionality) type. An algorithm for constructing a composite solution is indicated, including a stage of clustering training based on a greedy algorithm. This “composite” algorithm is implemented on a PC; a computational experiment was carried out.

Let $U$ be the set of admissible controls, $T>0$, and let $W[0;\tau]$, $\tau\in(0;T]$, be a scale of Banach spaces such that the set of restrictions of functions from $W=W[0;T]$ to $[0;\tau]$ coincides with $W[0;\tau]$; let $F[.;u]\colon W\to W$ be a controlled Volterra operator, $u\in U$. Earlier, for the operator equation $x=F[x;u]$, $x\in W$, the author introduced a comparison system in the form of a functional integral equation in the space $\mathbf{C}[0;T]$. It was established that to preserve (under small perturbations of the right-hand side) the global solvability of the operator equation, it is sufficient to preserve the global solvability of the specified comparison system, and the corresponding sufficient conditions were established. In this paper, further examples of application of this theory are considered: nonlinear wave equation, strongly nonlinear wave equation, nonlinear heat equation, strongly nonlinear parabolic equation.

In the present article, we set forth with the new notion of quasi statistically rough convergence in the gradual normed linear spaces. We establish significant results that present several fundamental properties of this new notion. We also introduce the notion of $st_{q}^{r}(\mathcal{G})$-limit set and prove that it is gradually closed, convex, and plays an important role for the quasi statistically boundedness of a sequence in a gradual normed linear space.

The paper studies the motions of a dynamically symmetric satellite (rigid body) relative to the center of mass in the central Newtonian gravitational field on a weakly elliptical orbit in the neighborhood of its stationary rotation (cylindrical precession). We consider the values of the parameters for which, in the limiting case of a circular orbit, one of the frequencies of small linear oscillations is equal to unity and the other is equal to zero, and the rank of the coefficient matrix of the linearized equations of the perturbed motion is equal to two, as well as a small neighborhood of this resonant point in the three-dimensional space of parameters. The resonant periodic motions of the satellite, analytical in fractional powers of a small parameter (the eccentricity of the orbit of the satellite's center of mass), are constructed. A rigorous nonlinear analysis of their stability is carried out. The methods of KAM theory are used to describe two- and three-frequency conditionally periodic motions of a satellite, with frequencies of different orders in a small parameter. A number of general theoretical issues concerning the considered multiple parametric resonance in Hamiltonian systems with two degrees of freedom that are close to autonomous and periodic in time are discussed. Several qualitatively different variants of parametric resonance regions are constructed. It is shown that in the general case the nature of nonlinear resonant oscillations of the system is determined by the first approximation system in a small parameter.

At the river bed curves, secondary flow normal to the main flow direction are formed. Depending on the channel geometry, there may be several secondary flows in the cross-section, and they may have different scales. Even a small secondary cross-section flow affects the parameters of the hydrodynamic flow and this influence must be taken into account when modeling riverbed processes and researching coast deformations of the channel. Three-dimensional modeling of such multi-scale processes requires large computational costs and is currently possible only for small model channels. Therefore, a reduced-dimensional model is proposed in this paper to study coastal processes. The performed reduction of the problem from a three-dimensional model of river flow motion to a two-dimensional one in the plane of the channel cross-section assumes that the hydrodynamic flow is quasi-stationary and the hypotheses on the asymptotic behavior of the flow along the flow coordinate are fulfilled for it. Taking into account these limitations, a mathematical model of the problem of a stationary turbulent calm river flow in a channel cross-section is formulated in this work. The problem is formulated in a mixed velocity–vortex–stream function formulation. Specifying of the boundary conditions on the flow free surface for the velocity field determined in the normal and tangential directions to the cross-section axis is required as additional conditions for the problem reduction. It is assumed that the values of this velocity field should be determined from the solution of auxiliary problems or obtained from data of natural or experimental measurements. The finite element method in the Petrov–Galerkin formulation is used for the numerical solution of the formulated problem. A discrete analog of the problem is obtained and an algorithm for its solution is proposed. The performed numerical studies showed generally good agreement between the obtained solutions and the known experimental data. The authors associate the errors in the numerical results with the need for a more accurate determination of the radial component of the velocity field in the cross-section by selecting and calibrating a more suitable model for turbulent viscosity calculating and a more accurate determination of the boundary conditions on the cross-section free boundary.

In this paper we study specific features of the relations between topological and algebraic structures of quasigroups and loops. We study the measurability of subsets of topological quasigroups and loops with respect to invariant measures. We study the family of non-measurable subsets in locally compact non-discrete loops. We find out the existence of locally $\mu$-zero subsets that are not $\mu$-zero in a locally compact left quasigroup that is not $\sigma$-compact. We study quotient spaces of measurable spaces on quasigroups. Moreover, we study homogeneous spaces of quasigroups and countable separability of subsets in them.

The problem of bringing a trajectory to a neighborhood of zero under disturbance is considered in terms of a differential pursuit game. The dynamics are described by a nonlinear autonomous system of second-order differential equations. The set of values of the pursuer's controls is finite, and that of the evader (disturbance) is compact. The goal of the control, that is, the goal of the pursuer, is to bring, within a finite time, the trajectory to any predetermined neighborhood of zero, regardless of the actions of the disturbance. To construct the control, the pursuer knows only the phase coordinates and the value of the velocity at some discrete moments of time and the choice of the disturbance control is unknown. Conditions are obtained for the existence of a set of initial positions, from each point of which a capture occurs in the specified sense. Moreover, this set contains a certain neighborhood of zero. The winning control is constructed constructively and has an additional property specified in the theorem. In addition, an estimate of the time required to bring the speed from one given point to the neighborhood of another given point under disturbance conditions was obtained.

The paper considers the problem of reducing the complexity of $NP$-hard problems by using related problems for which an optimal or acceptable solution is already known. For multi-agent routing tasks, a technique is used based on network clustering consistent with traveling salesman routes on each cluster and constructing routes that take into account the limitation of delivery time windows. A mathematical model is presented that corresponds to a block of pseudo-Boolean conditional optimization with constraints in the form of disjunctive normal forms that allows polynomial solvability and a block of time constraints. The results of choosing metaheuristics based on related problems are used in a program for the delivery of goods by many agents to consumers located at the vertices of the regional infrastructure road network.

For block matrix linear control systems, we study the property of arbitrary matrix coefficient assignability for the characteristic matrix polynomial. This property is a generalization of the property of eigenvalue spectrum assignability or arbitrary coefficient assignability for the characteristic polynomial from system with scalar $(s=1)$ block matrices to systems with block matrices of higher dimensions $(s>1)$. Compared to the scalar case $(s=1)$, new features appear in the block cases of higher dimensions $(s>1)$ that are absent in the scalar case. New properties of arbitrary (upper triangular, lower triangular, diagonal) matrix coefficient assignability for the characteristic matrix polynomial are introduced. In the scalar case, all the described properties are equivalent to each other, but in block matrix cases of higher dimensions this is not the case. Implications between these properties are established.

This work studies direct initial boundary value and inverse coefficient determination problems for a one-dimensional partial differential equation with multi-term orders fractional Riemann–Liouville derivatives. The unique solvability of the direct problem is investigated and a priori estimates for its solution are obtained in weighted spaces, which will be used for studying the inverse problem. Then, the inverse problem is equivalently reduced to a nonlinear integral equation. The fixed-point principle is used to prove the unique solvability of this equation.

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f\colon E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we characterize $s$-bridge graphs with local antimagic chromatic number 2.

et sets of functions $Z$ and $\Omega$ on the time interval $T$ be given, let there also be a multifunction (m/f) $\alpha$ acting from $\Omega$ to $Z$ and a finite set $\Delta$ of moments from $T$. The work deals with the following questions: the first one is the connection between the possibility of stepwise construction (specified by $\Delta$) of a selector $z$ of $\alpha(\omega)$ for an unknown step-by-step implemented argument $\omega\in\Omega$ and the existence of a multiselector (m/s) $\beta$ of the m/f $\alpha$ with a non-anticipatory property of special kind (we call it partially or $\Delta$-non-anticipated); the second question is when and how non-anticipated m/s could be expressed by means of partially non-anticipated one; and the last question is how to build the above $\Delta$-non-anticipated m/s $\beta$ for a given pair $(\alpha,\Delta)$. The consideration of these questions is motivated by the presence of such step-by-step procedures in the differential game theory, for example, in the alternating integral method, in pursuit--evasion problems posed with use of counter-strategies, and in the method of guide control. It is shown that the step-by-step construction of the value $z\in\alpha(\omega)$ can be carried out for any steps-implemented argument $\omega$ if and only if the above m/s $\beta$ is non-empty-valued. The key point of the work is the description of finite-step procedure for calculation of this $\Delta$-non-anticipated m/s $\beta$. Conditions are given that guarantee the m/s $\beta$ be a non-anticipative one. Illustrative examples are considered that include, in particular, control problems with disturbance.