In the article, the asymptotics of
the coefficients of the generating functions are found with a certain accuracy, which can be used to calculate the powers of layers of certain types of partially
ordered sets, as well as to calculate the values of the sums of boundary functionals when estimating
the number of antichains in such sets. In addition, applications of the obtained
results are considered using examples.
Keywords:
generating function; partially ordered set
The author considers an inverse coefficient problem for a model of sorption dynamics. The inverse problem is reduced to a nonlinear operator equation for an unknown coefficient. The differentiability of the nonlinear operator is proved. The Newton–Kantorovich method and the modified Newton–Kantorovich method are constructed for the numerical solution of the inverse problem. The results of numerical calculations are presented.
Keywords:
mathematical model of sorption dynamics, inverse problem, nonlinear operator equation, operator derivative, Newton–Kantorovich method
When describing the group behavior of high-frequency traders, a boundary value problem arises based on the concept of mean field games. The system consists of two coupled partial differential equations: the Hamilton–Jacobi–Bellman equation, which describes the evolution of the average payoff function in backward time, and the Kolmogorov–Fokker–Planck equation, which describes the evolution of the probability density distribution of traders in forward time. The system is ill-conditioned due to the turnpike effect. Under certain assumptions, it is possible to reduce the system to a set of Riccati equations; however, the question of the well-posedness of the reduced problem remains open. This work investigates this question, specifically the conditions for the existence and uniqueness of the solution to the boundary value problem depending on the model parameters.
Keywords:
mean field games, a system of Riccati equations, a boundary value problem for a system of ODEs
Statistical inference often assumes that the distribution being sampled is normal. As observed, following the normal distribution assumption blindly may affect the accuracy of inference and estimation procedures. For this reason, many tests for normality have been proposed in the literature. This paper deals with the problem of testing normality in the case when data consists of a number of small independent samples such that in each small sample observations are independent and identically distributed while from sample to sample they have different parameters but the same type of distribution (call this multi-sample data). In this case it is necessary to use test statistics which do not depent on the parameters. A natural way to exclude the nuisance location parameter is to replace the observations within each small group by diferences. We obtain some estimates of stability of such a decomposition and study and compare the power of eight selected normality tests in the case of multi-sample data. The following tests are considered: the Pearson chi-square test, the Kolmogorov–Smirnov, the Cramer–von Mises, the Anderson-Darling, the Shapiro–Wilk, the Shapiro–Francia, the Jarque–Bera, and the adjusted Jarque–Bera tests. Power comparisons of these eight tests were obtained via the Monte Carlo simulation of sample data generated from several alternative distributions.
Keywords:
normal distribution; test for normality; multi-sample data; decomposition stability; the Levy– Cramer theorem; Monte-Carlo simulation
This article presents a new algorithm for calculating singularity (boundary) type of ribbon surface of generalized pseudo-Anosov homeomorphism using the surface’s combinatorial description provided with the so-called configuration. As an additional output the fundamental group relators of the ribbon surface are calculated for its co-presentation associated with a given ribbon partition. In comparison to a known algorithm, the one which is presented in this article does not involve any auxiliary sets nor recurrent functions.
Keywords:
pseudo-Anosov homeomorphism, ribbon surface, singularity type, adjacency matrix, fundamental group
The creation of cryptographic systems based on lattice theory is a promising direction in the field of post-quantum cryptography. The aim of this work is to obtain new properties of lattices through related objects — dense packings of equal spheres. The article proposes a method for constructing lattice packings of equal spheres corresponding to the packing density of the “Lambda” series in dimensions 1–24, using a series of coefficients to the height of a fundamental parallelepiped of dimension (n−1): 1/2, 1/3, 1/2, 0, 1/2, 1/3, 1/2, √−1, 1/2, 1/3, 1/2, 0, 1/2, 1/3, 1/2, √−1, 1/2, 1/3, 1/2, 0, 1/2, 1/3, 1/2. The construction of lattice packings of equal spheres using this method was carried out up to dimension 11 inclusive.
Keywords:
post-quantum cryptography, geometry of numbers, lattice theory, arithmetic minima of positive quadratic forms, lattice packings of equal spheres, Hermite constant
This paper solves the construction problem of embedding the complete rooted binary and ternary trees with k, k =1, 2, . . . , levels in rectangular lattices (RL) with minimum length and near minimum height. It is assumed that different vertices of the tree go to different (main) vertices of the RL, with the leaves of the tree going to the vertices of the RL located on its horizontal sides. It is also assumed that the edges of the tree go to simple (transit) chains of the RL, which connect the images of their end vertices and do not pass through other main vertices, with no more than 1 (correspondingly 2) transit chains passing through the same edge (the same vertex) of the RL.
Keywords:
tree embedding, rectangular lattices, minimum length
Time optimal control problem with state constraint investigated in this article. The behavior of an object is described by a system of second-order linear differential equations. The coefficient matrix for state variables has various positive eigenvalues. The state constraint is linear. A admissible control is a piecewise continuous function that takes values from a given compact set. Sets of controllability to the origin are constructed for time intervals of various lengths. A study of the dependence of the solution of the problem on the parameter determining the state constraint was carried out.
Keywords:
time optimal control, state constraint, linear system, controllability set
In the work on real data, the modeling and assessment of the infection rate of a population of ixodid ticks with tick-borne encephalitis virus and Borrelia burgdorferi sensu lato is carried out using the maximum likelihood and moment methods, and their comparative analysis is given. A review of methods for solving direct and inverse problems of binary object propagation based on individual and group observations is made.
Keywords:
infection rate model, grouped observations, Bernoulli tests, maximum likelihood method, method of moments, ixodid ticks
The paper considers the algebraic properties of the Hadamard product (Shur product, componentwise product) of error-correcting linear codes. We discuss the question of the complexity of constructing the basis of Hadamard’s product by known bases of factors. Also, we introduce the concept of quotients, quasi quotients, and maximal (concerning inclusion) quasi quotient from the Hadamard division of one linear code to another. We establish an explicit form of the maximal quasi quotients of the Hadamard division. It proved the criterion of existence for a given code of an inverse code in a semiring formed by linear codes of length n with operations of sum and Hadamard product of codes. We also describe the explicit form of codes with an inverse code in
this semiring.
Keywords:
Hadamard product, Shur product, Component-wise product, McEliece public-key cryptosystem, algorithm, Hadamard quotient, Hadamard quasi quotient, Hadamard maximum quasi quotient
The problem of maximizing the horizontal coordinate of a point mass moving in a vertical plane under the influence of gravity, viscous friction, curve reaction force and thrust is considered. It is assumed that state constraints are imposed on the trajectory inclination angle. The system of equations belongs to a certain type, which allows us to reduce the optimal control problem with constraints on the phase variable to the optimal control problem with control constraints. The sequence and number of inclusions of phase constraints in the optimal trajectory and synthesis of optimal control are determined.
Keywords:
brachistochrone, phase constraints, Pontryagin’s maximum principle, boundary value problem, optimal trajectory
The article deals with the task of testing drugs for bioequivalence. Bioequivalence studies underlie the reproduction of drugs that have confirmed their efficacy and safety. The main method for testing the bioequivalence hypothesis is the procedure of two one-way Schuirmann tests. Due to inaccurate data, as well as omitted data at the drug trial stage, the Schuirmann criterion allows for errors that exceed a given probability of error of the first kind. Such situations are dangerous for patients who may receive a drug that is not equivalent to the original drug. The authors presented a new criterion that is more sensitive to differences in characteristics affecting the bioavailability of drugs, which reduces patient risk. Note that the new criterion generalizes the
classical Schuirmann criterion, preserving its useful properties.
The task of urgent detection of pathogen microorganisms in the human body is a relevant problem in the field of medicine. An approach based on seeding the biomaterial into the media nutritious and monitoring the bacterial colony growth is on the popular side by today’s standards. At the same time it possesses a set of downsides, caused generally by the human factor, introducing possible mistakes to medical verdicts. This work is dedicated to the development of intelligent data-driven technologies for processing microbiological analysis data in the form of photographic images of Petri dishes. Such technologies must allow reducing the dependence on the human factor as well as increasing the key factor quality of the analysis. The results of the conducted experiments allow making a conclusion that developed heuristic and neural network methods for detection and classification of the microorganism colonies surpass existing approaches and allow automating the key stages of the microbiological analysis as well as introducing proposed methods in practice.
It was proved earlier that product xy is a universal function for the class of linear functions depending on two arguments if k = 6l ± 1. Also it was shown that there is no universal polynomials for the class of linear functions depending on two arguments for any even k. In this paper we prove that polynomial xy + xz + yz is universal for classes of linear functions depending on three arguments for arbitrary odd k and polynomial
xy + zw is universal for classes of linear functions depending on four arguments for any k.
Keywords:
generation, universal function, sum modulo, polynimial
A one-line queueing system with two priority classes, relative priority, Poissonian input flow with random intensity and infinite number of places in queue for waiting is studied. The current intensity value is taken at the beginning of the time reckoned for the arrival of the next requirement. Successive values of the flow intensity form a Markov chain of a special kind. The heavy traffic limiting distribution of the queue length for the least
priority class is obtained.
Keywords:
Poissonian flow, random intensity, relative priority, queue length, heavy traffic
The paper describes algorithms for converting linear control systems with interval parameters from the
state space representations to the form of transfer functions and vice versa. The results are accompanied by
computational examples demonstrating the possibilities and limitations of the proposed approach.
Keywords:
interval systems, transfer functions, uncertainty
The article discusses the development of a system for identifying the channel of information leakage from a document photo taken from a computer screen via digital watermarks integrated into the image on the monitor. The article also presents the results of an experimental study to prove the properties of the algorithm and its resistance to deliberate attacks.
Recurrent sequences over a set of integers are considered, in which arbitrary superpositions of polynomial functions and functions close to polynomial ones are used as generating functions, — almost polynomial recurrent sequences. A series of functions of the form b · ji(x) is distinguished. Each of these functions, together with polynomial functions, allows us to construct generating functions that make it possible to determine almost polynomial recurrent sequences that simulate calculations on Minsky machines. Based on this result, algorithmically unsolvable problems related to these almost polynomial recurrent sequences are formulated. Consequences are obtained that significantly expand the range of functions capable of generating recurrent sequences with algorithmically unsolvable problems.
This paper is devoted to the problem of separation of mixtures of probability distributions. An optimization method is proposed as an alternative to the EM-algorithm (Expectation-Maximization) for statistical estimation of mixture parameters. The idea of approximating the distribution of increments (logarithms) of financial data by a mixture of normal laws is considered. The practical application of such an approximation to the problems of calculating and predicting volatility, as well as to the problem of calculating the risk measure (Value at Risk), is presented. The results obtained allow us to conclude that the application of mixtures of normal distributions to the description of financial data is adequate.
Keywords:
stochastic differential equations, finite mixtures of normal distributions, optimization method for separating mixtures of probability distributions, volatility, Value at Risk estimation
The article deals with localization methods of inverse magnetoencephalography tasks. Localization methods are important in real clinical practice. During neurosurgical interventions, various areas of the brain can be damaged, including non-recoverable areas. Since the location of functional areas in the human brain is individual, the doctor must be able to localize these areas in the preoperative period with high accuracy. The developed methods serve the solution of such an important problem.
Keywords:
independent component analysis, magnetoencephalography, current dipole
Similarity transformations are the main part of matrix theory, which studies numerous classes of special matrices. Accordingly, there are many ways of describing such classes. In most cases, one can verify whether a matrix belongs to the required class by a rational calculation, that is, by a finite algorithm using only arithmetical operations.
Keywords:
unitoid, cosquare, canonical form with respect to congruences, involution, Toeplitz decomposition
The use of a large number of drones set us the tasks of correct, transparent, and high-performance modeling of their collective activity within the framework of swarm behavior. For this, an example of transitioning from the description of the control system of a single drone to the behavior of a swarm of identical drones based on Hadamard algebra (product) was developed and given. It was found that the use of such an approach increases the performance of the model by dozens of times and allows modeling of swarm systems with a very large number of agents.
We study the existence of an implicit function, defined by an equation G(x, σ) = 0, in a neighborhood of an abnormal point (x0, σ0). We prove that if some λ-truncation of the mapping F(x) = G(x, σ0) is regular in
a certain direction, then the sought implicit function exists.
Keywords:
implicit function, abnormal point, nonlinear equation, real solution, truncation, regularity in direction
The paper considers the problem of feedback control for a group of several quadrocopters. The main goal is to transfer this group from a given initial position to a given target position for a fixed period of time, provided that at each intermediate time instant the quadrocopters must be in a small neighborhood of a fixed smooth curve in space. A centralized group control scheme is used. The main difficulty here is the complex, nonlinear mathematical model, describing the movement of each individual aircraft. It is necessary to develop computationally efficient algorithms for approximate search for feedback control, which allow to cope simultaneously with nonlinear dynamics, point-to-point constraints on control parameters, as well as state constraints arising in connection with group motion (in particular, due to the requirements of pairwise collision avoidance). Modifications of ellipsoidal calculus methods are used here to construct such algorithms.
Keywords:
group control, nonlinear dynamics, feedback control, ellipsoidal methods
It is considered the question about continuity of optimal time, as the function of initial state, for linear control objects. We obtained some generalization of the Theorems 21, 22 from the second Chapter of monograph E.B.Lee, L.Markus “Foundations of Optimal Control Theory,” J.Wiley and Sons, 1967, connected with continuity of optimal time, as the function of initial state, for linear controlled objects. In the article we broadly use the
apparatus of support functions from Convex Analysis. Our results are obtained in more constructive form than another general results In this direction. In the first part we consider stationary case and in the second part it is considered non-stationary case.
Keywords:
optimal time control, measured control functions, Cauchy formula, set-valued maps, integration
Approximate bilinear algorithms for multiplication of 2 × 3 and 3 × 4 matrices (complexity 18), 2 × 4 and 4×4 matrices (complexity 24), 2×5 and 5×4 matrices (complexity 30) are presented, and approximate bilinear algorithms for multiplication of 2×n and n×4 matrices (complexity 6n) over field of characteristic zero gotten via them.
Keywords:
approximate bilinear complexity, approximate matrix multiplication algorithms, objective function
Dirichlet boundary value problem for the elliptic equation in a rectangle with a noninteger degeneration and analytic coefficients is considered. Applying the method of spectral isolation of singularities, the formal solution to the problem is constructed as a series which exhibits its non-analytic dependence with respect to y at the origin. Green’s function method verifies the convergence of the series to the classical solution of the considered problem.
Keywords:
noninteger order degeneration, analytic coefficients, elliptic equation, equation with a small parameter
The time optimal control problem with a state constraint investigates in the article. The object’s behavior is described by a system of second-order differential equations. The coefficient matrix for state variables has zero eigenvalues. The state constraint is linear. An admissible control is a piecewise continuous function that takes values from a given compact set. Sets of controllability to the origin are constructed for time intervals of various lengths.
Keywords:
time optimal control, state constraint, linear system, controllability set
It was proved earlier that product xy is a universal function for the class of linear functions depending on two arguments if k = 6l±1. In the paper we prove that there is no universal polynomials for the class of linear
functions depending on two arguments for any even k.
Keywords:
generation, universal function, sum modulo, polynimial
Multiple structure protein alignments are essential to protein function analysis. Most methods of multiple protein structure alignment are based on pairwise alignment methods, where a resulting alignment is built up by merging pairwise alignments into the resulting alignment according to a guide tree. In this paper we propose the genetic algorithm for guide tree optimisation and provide the theoretical proof of convergence and experimental study of the proposed algorithm.
Is proposed the description of the implementation stabilizer construction algorithm for a switched linear system operating under parametric uncertainty in the Matlab package of applied mathematics is proposed.
In this paper asymptotic theorems are proved for estimates of the characteristic index, the scale parameter, and the shape and scale parameters for the remaining fixed parameters of the digamma distribution with a random sample size. Particular cases of limit distributions are given in the case when the sample size has a mixed Poisson distribution.
Keywords:
parameter estimation, digamma distribution, mixed distributions, random sample size
We discuss how to get a normal matrix from a binormal one and, conversely, how to get a binormal matrix from a normal one via the right multiplication on a suitable unitary matrix. Let N be a normal matrix badly conditioned with respect to inversion, that is, having a large condition number cond2N. We show that, among the binormal matrices B that can be obtained from N, there is a matrix whose eigenvalues have individual
condition numbers of order (cond2N)1/2.
Keywords:
normal matrix, binormal matrix, unitary matrix, condition number
The paper formulates and studies optimization problems of energy consumption and storage control for a small consumer, who does not make any impact on the market prices. The models take into account new
technical and economic tools: renewable energy sources and energy storages.
Keywords:
energy storages, optimal control, Lagrange theorem, stochastic optimization
A single server nonpreemptive priority queueing system with renewal type input stream and multiple vacations is studied. The distributions of intervals between arrivals, service time for each priority class and vacation periods are arbitrary and absolutely continuous. The extra component method, the Laplace transform and the special integral transform are used to obtain the non-stationary joint distribution of queue length for all priority classes.
Keywords:
nonpreemprive priority queue, single server, working vacations, queue length
Initial problems for the equations of motion of a viscous incompressible fluid and gas in Lagrangian variables are considered. It is shown that the motion of an incompressible fluid is not related to pressure. The pressure in the absence of external forces is constant, what allows the fluid to move freely. This motion is purely vortex in nature and is described by quasi-linear equations of the parabolic type. The existence and uniqueness of the classical periodic solution of the initial problem in Rn at n > 2 is proved. The equations of motion of liquid and gas in steady-state mode are obtained. The problem of turbulent flow of partially compressible liquid and gas is solved. It is established that there is no turbulent flow in an incompressible fluid. It is shown that as a result of frequency synchronization, spatially stable periodic structures arise.
Keywords:
Lagrange variables, steady-state, turbulence, frequency synchronization
Type I diabetes, also known as insulin-dependent diabetes, is a serious disease. There are two main approaches to the management of patients with this disease: the use of insulin pens and various types of insulin (such as ultrashort, short-acting and long-acting insulins) or the use of an insulin pump with ultrashort insulin. An important task remains the development of a reliable system for the accurate calculation of insulin doses using insulin pens in the treatment of type I diabetes. In this study, we investigate the accuracy of calculating insulin doses based on historical data. To solve this problem, we propose using various artificial intelligence (AI) methods, including decision trees, gradient acceleration, support vector machine (SVM) methods and various neural network architectures. We are conducting a comparative analysis of these approaches to determine their
effectiveness.
Keywords:
type I diabetes, machine learning, neural networks
The traffic load balancing problem is relevant in modern networks that have many alternative routes between any pair of nodes. Balancing provides a uniform load of network resources. The paper proposes a method for
adaptive queueing policy configuration on a switch to get a uniform queue load on the switch output ports. Because modern applications require the data transmission delay to be around milliseconds, reinforcement
learning method DQN was applied to solve the problem. An experimental study demonstrated the convergence of the proposed method during the training to uniform queue load at output ports.
One of the main tools for describing and predicting the occurrence of events is survival analysis, which allows you to predict not only the probability and time of events but also the changing of probability over time. This article presents Survivors, an open-source Python library that helps solve problems of survival analysis, build individual forecasts of survival and risk functions, investigate data dependencies, evaluate the quality of forecasts, and conduct experimental studies. The library uses new methods of constructing tree-based models of survival analysis with high sensitivity to real datasets. In particular, the paper presents a new histogram approach for searching the best split of censored data. The models can handle categorical and missing values, cases of informative censorship, and multimodal time distribution. The paper describes the architecture and components of the library, the features of the software implementation, and an experimental comparison with existing libraries of survival analysis.
A one-line queueing system with three incoming Poissonian flows is studied. Service time distributions are general and absolutely continuous for each flow. The first class requirements have nonpreemptive priority over
the second class requirements and preemptive-repeat-different priority over the third class requirements. The second class requirements have nonpreemptive priority over the third class requirements. The heavy traffic limiting distribution of the queue length for the third class is obtained while the system load tends to 1 and time tends to infinity.
The paper considers the asymptotic behavior of the reserve of an organization subjected to risk in the case when the number of factors leading to loss is random. Burr distribution is considered as loss distribution. An asymptotic comparison of the activities of such organizations is carried out in terms of the necessary additional number of such factors (asymptotic deficiency). Two examples illustrating tne obtained results are presented. The first example concerns extreme order statistics, and the second one deals with the truncated Poisson and binomial distributions.
Keywords:
Reserve of insurance company, sample of random size, Burr distribution, asymptotic expansions, truncated Poisson and binomial distributions, extreme order statistics, asymptotic deficiency
The concept of steady-state solutions of the Navier–Stokes equation is defined. Such solutions expand the concept of stationary, exponentially decrease in time, have a constant spatial velocity field and constant pressure in the absence of external fields. The method of their construction is considered, and the problem of Taylor vortices is solved. A mathematical model of a tornado is proposed. Within the framework of this model, a steady-state solution is obtained as an eigenfunction of the problem in the form of a vortex. Based on the Navier–Stokes equation, a model of the formation of the structure of a gas cloud is proposed. It is shown that due to the Coriolis force, spiral arms arise from flows of gas moving outward. It is proved that the number of arms m is even, and their structure does not depend on the angular velocity of rotation. A formula is obtained for the spiral twist angle depending on the cloud parameters for the case of m=2.
Keywords:
Taylor vortices, tornado, gas cloud, Coriolis force, galaxy
The paper considers inverse problems for hyperbolic equation with singular perturbation in which the function entering the source is unknown. The existence of inverse problems solutions is proved, numerical methods for inverse problems solving are developed and the results of computational experiments illustrating effectiveness of numerical methods are presented.
Technique of data preparation for hemodynamic calculations in a quasi-one-dimensional approximation on complex spatial graphs of elastic vessels is considered. A method is described for the automatic construction of volumetric (3D) models of graphs of the vascular system according to their frame models. The main approaches to carrying out calculations in parallel with the visualization of their results on volumetric models are discussed.
Keywords:
hemodynamic calculations, 3D models of the vascular system, visualization of calculated data on graphs
We consider representations of regular languages over symmetric groups such as finite automata and regular expressions. We show that the problem of computing cardinality of such representations is NP-hard.
Polynomiality characterizations are proposed for k-valued logic functions of one variable for a composite modulo k which is a power of a prime number. Based on these characterizations, for each prime number p, polynomiality checking algorithms are obtained for pm-valued logic functions of one variable, m > 1. In these algorithms, the calculations are performed in the residue ring modulo pm. In the case of the positive answer, the algorithms find the canonical polynomial of a input function. The complexity of the obtained algorithms are evaluated (with respect to number of operations from the field of p elements with possible constants).
The paper investigates a two-server system with a Poisson input, in which the service times on the servers have a Pareto distribution with the parameter a > 1. Using known asymptotics for the distribution of stationary waiting time, the distribution of the maximum stationary delay is obtained for two cases: p < 1 and 1 < p < 2. Estimates of the extremal index of stationary waiting time were obtained in numerical experiments using regenerative modeling.
Keywords:
stationary delay, two-server queueing system, extremal index
A method of intelligent forecasting of random processes is described. This method is based on a more comprehensive use of statistical regularities inherent in the evolution of the process under consideration. Within the framework of the proposed approach, on the stage of training, the feature space is complemented by the parameters of mixed probability models that make it possible to reconstruct the coefficients of the stochastic differential equation describing the process under consideration. The additional statistical information imposes supplementary restrictions on the research area and thus, narrows the set of possible solutions and directs learning by preliminary rejection of improbable or highly unlikely decisions, and hence, makes the training more efficient and the forecasts more accurate.
Keywords:
time series, stochastic differential equation, mixture of probability distributions, statistical separation of mixtures, forecasting
The matrix equation XAX = AXA is called the equation of the Young–Baxter type. We examine this equation for matrices of order 2 under the assumption that A is a nonsingular matrix, and we are only interested in the nonsingular solutions. Using a unified rule, one can associate with every such solution a matrix commuting with A. In other words, this matrix is an element of the centralizer MA of A. No obvious reasons exist for two distinct solutions X1 and X2 to generate the same element of MA. Nevertheless, all the solutions (and there are infinitely many of them) generate one and the same matrix in the centralizer. We give an explanation of
this surprising fact.
Keywords:
equation of the Young–Baxter type, similarity of matrices, centralizer of a matrix, diagonalizable matrix, Jordan block
The paper considers an approach to solving the problem of removing noise in a large data set from the sparsity class mp under condition of weak dependence. The approach is based on the method of controlling the false discovery rate (FDR). An upper asymptotic bound for the mean-square risk is obtained.
A method for approximate construction of the boundary of attainability domain is developed for a class of nonsmooth controlled dynamical systems on the plane that arise in economics. The method is based on an explicit procedure for smoothing the system and using the apparatus of the Pontryagin maximum principle. As an example, we consider the problem of constructing the boundary of the attainability domain for a controlled version of the well-known Kaldor’s model of business cycle.
Keywords:
control system, attainability domain, Pontryagin’s maximum principle, numerical methods, Kaldor’s model of business cycle
This paper discusses generative intelligence systems for image synthesis, provides a detailed description of one of them (DALL·E 2), and presents known examples of such systems in use. The rationale for preparing such a review is due to the current situation in the field of generative intelligence, with many inflated expectations and even fears, and a practical lack of description and analysis of their business use cases. The article will be useful to anyone who would like to understand the real possibilities and limitations of such systems.
Keywords:
generative intelligence, image synthesis, use cases of generative intelligence
A diffusion logistic model of information dissemination in a social network in the form of a one-dimensional unsteady parabolic equation is considered. The problem of parametric identification is posed as an extreme problem for finding a parameter in the form of a spatially distributed function of network bandwidth. Gradient optimization methods are applied. The results obtained demonstrated uniform convergence to the exact solution in the method with an adjustable descent direction.
Keywords:
social networks, mathematical modeling, identification, optimization
The problem of a non-publicized agreement between buyers of unlicensed spectrum in a spectrum auction for the coordinated formation of their price bids is considered in terms of game theory and operations research. Such auction participants are potential free riders, they hope on free access to the frequencies being sold, which causes their non-standard behavior. We propose two options for organizing an agreement for Vickrey auction in the case of complete information about the amounts of participants’ income from the use of the frequency purchased with shared expenses. It is shown that the lack of information leads to an equalizing distribution of payment between contracting buyers, and this significantly reduces their competitive advantage in the spectrum auction. Being a mechanism stimulating the true preferences uncovering, the Clark–Groves mechanism is analyzed, and its modified version is developed. Unfortunately, according to the results of the study, its application to the considered problem seems inappropriate. An alternative possibility of choosing a joint decision based on the Germeier–Vatel model is discussed.
Keywords:
agreement between free riders, game model for spectrum auction, Vickrey rule, Nash equilibrium, discovering preferences, Clarke–Groves mechanism, Germeier–Vatel model
The inverse problem of reconstructing the coefficient in the nonlinear equation of the model of the development of a homogeneous biological age-structured population is considered. The model takes into account the dependence of the parameters of the vital activity of individuals on the size of the population. The coefficients of the model are non-local and have an integral structure. Conditions are established to ensure the uniqueness of the solution of the inverse problem.
Keywords:
the inverse problem for the population model, age structuring, overpopulation, Volterra integral equation type II, the method of successive approximations, the method of compressive maps
The implicatively implicit extensions of all 27 single functions of three-valued logic are characterized. It is established that among them there are both extensions that coincide with the known implicatively closed classes, and extensions that are not closed with respect to the superposition operation. In addition, it is shown that for any k > 3, any implicatively implicit extension in Pk contains the class Hk of all homogeneous functions from Pk.
The article considers an attack-defense model, in which the attack has the ability to strike several blows on defense points in order to inflict the most damage. A method for constructing a solution in mixed strategies of the zero-sum game has been developed.
In this paper, we construct a complete classification of linear codes, which are obtained from codimension
1 subcodes of Reed-Muller codes using Hadamard’s product.
The additive schemes (splitting schemes) are the basis for the construction of efficient computational algorithms for the approximate solution of initial boundary value problems for non-stationary partial differential equations. Usually, splitting schemes are developed for the additive representation of the basic operator of the problem. Also of interest are problems where the operator splits at the time derivative of the solution. We propose splitting schemes for first order evolution equations. These schemes are based on the transformation of the original equation into an equivalent system of equations.
For a fixed period of time, a mathematical model for the treatment of psoriasis is considered, consisting of three differential equations. These equations describe the relationships between the major cell populations that are responsible for the occurrence, progression, and treatment of this disease. The model also contains a bounded control function reflecting the effects of a drug aimed at inhibiting interactions between specific cell populations. The task is to reduce the population of cells directly affecting the disease at the end of a given period of time. The analysis of such an optimal control problem is carried out using the Pontryagin maximum principle. The main attention is paid to clarifying the bang-bang property of the corresponding optimal control, as well as the possibility of its having singular regimens of various orders depending on the relationships between the parameters of the original model.
Keywords:
psoriasis, nonlinear control system, optimal control, Pontryagin maximum principle, switching function, bang-bang function, singular regimen
We study conditions under which components of the distribution of the difference of two independent and identically distributed random variables are determined uniquely up to a shift and reflection. This uniqueness is essential to some characterization problems. An algorithm is presented for estimation of the components when data are given in a symmetrized form.
Keywords:
decomposition of probability laws, characteristic function, convolution, symmetrization
The paper is devoted to the development of a new method for the approximate construction of the reachability set for a nonlinear control system with discrete time. Pointwise restrictions are imposed on the control parameters. To solve this problem, a technique previously developed and applied for the case of continuous time and differential equations is used. The estimate of the reachability set can be obtained as the level set of a special piecewise affine value function constructed on a grid of simplices in the state space. The paper presents formulas for calculating the coefficients of such a function, which make it possible to analyze the difference between the case with discrete time and the case with continuous time. An example of calculation of piecewise affine value functions and corresponding internal and external estimates of the reachability set is considered.
Keywords:
nonlinear dynamics, reachability set, value function, piecewise affine estimates
Let A and B be matrices of order n that are direct sums of nilpotent Jordan blocks. Suppose that A and B are not just different arrangements of the same blocks but, rather, they differ in the sizes of the blocks. It is shown that, in this case, A and B cannot be congruent. This result can be regarded as a new proof of the uniqueness of the singular part in the Horn–Sergeichuk canonical form of a singular matrix.
Keywords:
direct sum, Jordan block, congruence transformation, span of a system of vectors
The paper studies unit max switch constraints in unit commitment problem. The unit commitment is a mixed integer problem widely used in short term energy system scheduling. Its computational complexity strongly depends on its dimension. According to Russian power energy market regulations max switch constraint is submitted by a participant and is active for arbitrary seven day time period. In the paper it is shown however
that it is sufficient to set this constraint in the model for only certain time periods. These are determined by times in seven day prehistory of the planning horizon where the unit changed state. Hence, vast majority of constraints of this type are redundant and could be safely removed from the model. This increases efficiency of the methods used to solve the resulting problem.
Keywords:
wholesale electricity market, energy system scheduling, unit commitment, nonlinear optimization, mixed-integer programming
The paper describes a modified scenario method for risk estimation of financial instruments. It is based on the scenario method proposed by Jamshidian F. and Zhu Y. allowing to significantly reduce computation time of risk indicators on large portfolios compared to the Monte Carlo method. It is proposed to change the way of choosing scenarios and points of approximating distribution. It allows to improve the quality of approximation. More than that, the new method makes possible to remove restrictions on the type of distribution of portfolio price factors allowing to expand the scope of its application. The comparison of VaR (value at risk) estimation using original and modified method was performed on a financial portfolio consisting of an interest rate swap for cases when the values of the price factors have normal distribution, gamma distribution and Student’s t-distribution.
Keywords:
Monte Carlo method, scenario simulation, interest rate swap, Value-at-Risk
It was proved earlier that product xy is a universal function for the class of linear functions depending on two arguments if k = 6l ± 1. In the paper we prove that there is no universal polynomials for the class of linear functions depending on two arguments for any k dividing three. Also we show that there is no universal
polynomials for the class of linear functions depending on three arguments for any even k. Thus the criterion of existence of universal polynomials for the class of linear functions is obtained.
Keywords:
generation, universal function, sum modulo, polynomial
The problem of applying erasure coding methods at the transport level to restore lost packets is considered. This will allow to avoid multiple retransmissions of the same packet, reduce data transmission delays, and waste of network resources. The basic idea behind erasure coding methods is to introduce redundancy into the transmitted data, which will allow the lost data to be recovered on the receiver side. The paper considers various erasure coding methods at the transport level, selects the most promising ones based on the computational complexity of the encoding and decoding algorithms, as well as the effect of redundancy on the data transmission delay and the transport connection loss level. The required level of redundancy in the selected error-correcting coding methods is given, depending on the requirements for the loss level in the transport connection and its quality characteristics.
Keywords:
quality of service, erasure code, transport level
For an isotropic stratified elastic strip we consider the Poincaré–Steklov operator that maps normal stresses into normal displacements on part of the boundary. A new variant of the algorithm is proposed for computing a transfer function of this operator. This variant is based on preconditioned conjugate gradient method.
