Fundamental Mathematical Problems Research Articles

A New Sufficient & Necessary Condition for Testing Linear Separability Between Two Sets.

As a fundamental mathematical problem in the field of machine learning, the linear separability test still lacks a theoretically complete and computationally efficient method. This paper proposes and proves a sufficient and necessary condition for linear separability test based on a sphere model. The advantage of this test method is two-fold: (1) it provides not only a qualitative test of linear separability but also a quantitative analysis of the separability of linear separable instances; (2) it has low time cost and is more efficient than existing test methods. The proposed method is validated through a large number of experiments on benchmark datasets and artificial datasets, demonstrating both its correctness and efficiency.

Nested invariant tori foliating a vector field and its curl: Toward MHD equilibria and steady Euler flows in toroidal domains without continuous Euclidean isometries

This paper studies the problem of finding a three-dimensional solenoidal vector field such that both the vector field and its curl are tangential to a given family of toroidal surfaces. We show that this question can be translated into the problem of determining a periodic solution with periodic derivatives of a two-dimensional linear elliptic second-order partial differential equation on each toroidal surface and prove the existence of smooth solutions. Examples of smooth solutions foliated by toroidal surfaces that are not invariant under continuous Euclidean isometries are also constructed explicitly, and they are identified as equilibria of anisotropic magnetohydrodynamics. The problem examined here represents a weaker version of a fundamental mathematical problem that arises in the context of magnetohydrodynamics and fluid mechanics concerning the existence of regular equilibrium magnetic fields and steady Euler flows in bounded domains without continuous Euclidean isometries. The existence of such configurations represents a key theoretical issue for the design of the confining magnetic field in nuclear fusion reactors known as stellarators.

ПРОГНОЗУВАННЯ НЕСУЧОЇ СПРОМОЖНОСТІ ПЛИТНОГО ФУНДАМЕНТА ЗА ЧИСЛОВИМ МЕТОДОМ ГРАНИЧНИХ ЕЛЕМЕНТІВ

Construction is one of the leading branches of the national economy in the historical aspect of its development. The first design task is to determine the strength of building structures. Therefore, the study of the stress-strain state and related calculations are the most important in construction. The purpose of studying soil mechanics and foundation construction techniques is the calculation and construction of structures on or in soil. The main task is the construction of structures with a sufficient degree of reliability.The selection of an adequate theoretical model remains the main problem of soil mechanics. Indeed, the deformation of the dispersed granular material of the soil takes place during the mutual sliding of the grains, the rheology of the soil is complex, as evidenced by a large amount of experimental material. Today, the path of development of soil mechanics is related to the study of problems within the framework of the elastic-plastic dilatation model and the improvement of this model based on experiments. A mathematical model of a technical object at the micro level is a system of differential equations in partial derivatives, the exact solution of which can be obtained only in a few partial cases, therefore a discrete model is built using numerical methods that use the Poisson idea that the behavior of a complex model can be represented by the behavior of its individual component elements. The intensive development and widespread use of computers significantly brought fundamental mathematical problems closer to applied ones, and strengthened their mutual influence.The emergence of a new, powerful and general method of research - a numerical experiment, more than ever before closely connected the physical content of the problem, its mathematical formulation, numerical methods of calculation and modern computers. The work uses the numerical method of boundary elements. A promising way to develop foundations and foundation structures is to use the ratios of the theory of plastic flow, and the level of development of soil mechanics significantly affects the economy and reliability of the decisions made.

Kernel Approximation on Algebraic Varieties

Low-rank approximation of kernels is a fundamental mathematical problem with widespread algorithmic applications. Often the kernel is restricted to an algebraic variety, e.g., in problems involving sparse or low-rank data. We show that significantly better approximations are obtainable in this setting: the rank required to achieve a given error depends on the variety’s dimension rather than the ambient dimension, which is typically much larger. This is true in both high-precision and high-dimensional regimes. Our results are presented for smooth isotropic kernels, the predominant class of kernels used in applications. Our main technical insight is to approximate smooth kernels by polynomial kernels and to leverage two key properties of polynomial kernels that hold when they are restricted to a variety. First, their ranks decrease exponentially in the variety’s co-dimension. Second, their maximum values are governed by their values over a small set of points. Together, our results provide a general approach for exploiting (approximate) “algebraic structure” in datasets in order to efficiently solve large-scale data science problems.

Guaranteed Automatic Integration Library (GAIL): An Open-Source MATLAB Library for Function Approximation, Optimization, and Integration

Function approximation, integration, and optimization are three fundamental mathematical problems. They are especially challenging when the functions involved fluctuate wildly in certain parts of the domain, or if the domain is high dimensional. Ideally, algorithms to solve these problems should possess a rigorous mathematical framework, data-based (probabilistic) error bounds, and advanced sampling strategies for efficiency. The Guaranteed Automatic Integration Library (GAIL) is our multi-year research effort addressing these aforementioned challenges. GAIL is a free, open-source MATLAB software library with nine main algorithms undergirded by over a dozen peer-reviewed publications. GAIL solves problems in univariate and multivariate integration, and in univariate function approximation and optimization. GAIL algorithms adaptively sample data values of the input function and automatically stop when the error tolerance has been reached. In some cases, GAIL algorithms are proven to have asymptotically optimal computational cost. We consistently employ good software development practices for GAIL such as unit tests, searchable online documentation, and Git version control. GAIL is available at <a href="https://gailgithub.github.io/GAIL_Dev/" target="_blank">https://gailgithub.github.io/GAIL_Dev/</a>.

ПОШУК РЕЗЕРВІВ НЕСУЧОЇ ЗДАТНОСТІ ПІРАМІДАЛЬНИХ ПАЛЬ ЗА МГЕ

Vinnytsia National Technical University&#x0D; Construction – is one of the prominent branches of the national economy in the historical aspect of its development. The leading task of the designing is to determine the strength of building structures. Therefore, TDS research and related calculations are most responsible in construction.&#x0D; Mathematical model of a technical object at the micro level is a system of differential equations in partial derivatives, the exact solution of which can be obtained only in a few partial cases, so a discrete model is built using numerical methods that use Poisson's idea . Intensive development and widespread use of computers significantly brought fundamental mathematical problems closer to applied ones, strengthened their interaction. The emergence of a new, powerful and general method of research - numerical experiment, as never before closely linked the physical content of the problem, its mathematical formulation, numerical methods of calculation and modern computers. The numerical method of boundary elements is used in the work.&#x0D; A promising way to develop foundations and foundation structures is the usage of the relations of the theory of plastic flow, and the level of development of soil mechanics significantly affects the efficiency and reliability of decisions. One of the types of raising the bearing capacity of piles is, in addition to taking into account the nonlinear work of the soil base, also the arrangement by expanding the soil base of gravel additives. Compacted soil in the heel area allows to transfer higher loads to the pile.

РОБОТА ПІРАМІДАЛЬНИХ ПАЛЬ В ЗАЛЕЖНОСТІ ВІД КУТА КОНУСНОСТІ

One of the effective methods to ensure reliable operation of the structure is the use of pyramidal piles, which in comparison with prismatic piles have an increased bearing capacity due to the increase of the compacted area around the pyramidal pile in its upper part. In this paper, based on the consideration of the theoretical foundations of the rheology of clay soils and the use of the numerical method of boundary elements, the results of numerical studies of the behavior under the load of pyramidal piles are presented. Intensive development of soil mechanics, which significantly affects the cost-effectiveness of decisions, and the widespread use of computers have significantly brought fundamental mathematical problems in applications, strengthened their interaction. The use of a numerical experiment has more closely linked the physical content of a problem, its mathematical formulation, numerical calculation methods, and computers than ever before. Growing demands of construction practice lead to the complexity of applied tasks. Numerical experiment allows to allocate a rational grain, to describe the most important types of behavior of a base design, to carry out the mathematical substantiation, to write down the mathematical model which is checked by means of experiments. Due to the geological conditions of Ukraine in industrial and civil construction, it is advisable to use pyramidal piles, and the features of their interaction with the bases and theoretical calculation methods are insufficiently studied. The lack of reliable methods for estimating the bearing capacity of pyramidal piles leads to an increase in stock ratios, making imperfect decisions, and makes it difficult to widely implement them in construction practice. Therefore, the topic of VAT research of rational pyramidal piles is relevant today.

Dry Friction and Mechanical System Motion Implicit Equations

It is shown that forces acting on the mechanical system points could depend on accelerations of the system points. Differential equation system of the mechanical system motion appears to be implicit. It is not resolved with respect to senior derivatives. Fundamental mathematical problems appear associated with possibility and uniqueness of these equations' solution with respect to the senior derivatives. Such problems are common in mechanical systems with dry sliding friction and rolling friction. Such problems are missing in the point dynamics. However, such problems are rather typical in more complex mechanical systems appearing in the study of a rigid body motion, which entire mass is concentrated in a single point, as well as in systems with one degree of freedom. Four fairly simple examples of mechanical systems are considered, and their motion is described by implicit differential motion equations. Situations could appear in these systems, when motion equations are not solvable with respect to the senior derivatives (motion equations are missing), as well as situations, when there are several solutions with respect to senior derivatives (there are several different systems of the mechanical system motion equations). At the same time, one of the fundamental principles of mechanics is not fulfilled, i.e., the principle of determinism

Non-stationary Stochastic Multi-armed Bandit Problems with External Information on Stationarity

Multi-armed bandit problem is a fundamental mathematical problem in sequential optimization and reinforcementlearning that has a variety of application such as online recommendation system and clinical trial design. Multiarmedbandit problem can describe a situation in which a player tries to select a good choice sequentially from givencandidate choices to maximize the cumulative reward. In this paper, we consider the non-stationary multi-armed banditproblems. Non-stationary means the reward distribution of each arm varies with time. We point out that in somereal application, we can utilize information on the change of reward distribution. Especially we consider the type ofinformation that may restrict the rounds at which the reward distribution changes. Against such scenario, we proposea novel strategy called PM policy. The proposed policy is based on existing CUSUM-UCB policy and M-UCB policythat do not consider external information. Though such existing policies monitor all arms to detect the change ofreward distribution, our policy monitors only important arms and rounds. As a result, the ratio of unnecessary monitoringis reduced, and an efficient search can be performed. The regret bound of the proposed policy is described. Wealso show the effectiveness of the proposed method by numerical experiments.

The KR-Elliptic Curve Public Key Cryptosystem

The connection between some fundamental mathematical problems in number theory with cryptography has been done in this work. A new version of the elliptic curve public key cryptosystem (ECPKC) has been proposed by Karrar Taher and Ruma Ajeena. Their main idea is to design this cryptosystem focusses on generating the elliptic curve group, its order divides the Euler phi function that is computed securely based on two large primes. Faster and more secure computations to generate the keys, encryption and decryption processes, on the KR-ECPK algorithm are discussed. In comparing with other algorithms like RSA and ElGamal public key cryptosystem (EEPKC), the KR-ECPKC considers as an efficient algorithm. So, the KR-ECPKC is more appropriate for elliptic curve cryptographic applications.

On the implicit equations of a mechanical systems motion

It is shown that the forces acting on the points of the mechanical system may depend on their accelerations. The differential equations of a mechanical system motion prove to be implicit. It is not allowed with respect to higher derivatives. There are fundamental mathematical problems related to the possibility and the only solution of these equations with respect to higher derivatives. Implicit equations of motion are typical for mechanical systems with dry friction sliding and rolling. In the dynamics of material point such problems do not arise. But in more complex mechanical systems, including the study of the motion of a solid whole mass is concentrated at one point, as well as in systems with one degree of freedom, such a situation is very characteristic. The paper discusses four fairly simple examples of mechanical systems movement, which is described by implicit differential equations of motion.

Snow plow route optimization: A constraint programming approach

Many cities have to cope with annual snowfall, but are struggling to manage their snow plowing activities efficiently. Despite the fact that winter road maintenance has been a popular research subject for decades, very few papers propose scalable models that can incorporate side constraints encountered in real-life applications. In this work, we propose a Constraint Programming formulation for a Snow Plow Routing Problem (SPRP). The SPRP under consideration involves finding a set of vehicle routes to service a street network in a pre-defined service area, while accounting for various vehicle constraints and traffic restrictions. The fundamental mathematical problem underlying SPRP is the well-known Capacitated Arc Routing Problem (CARP). Common Mathematical Programming (MP) approaches for CARP are typically based on: (i) a graph transformation, thereby transforming CARP into an equivalent node routing problem, or (ii) a sparse network formulation. The CP formulation in this article is based on the former graph transformation. Using geospatial data from the city of Pittsburgh, we empirically show that our CP approach outperforms existing MP formulations for SPRP. For some of the larger instances, our CP model finds 26% shorter plowing schedules than alternative Integer Programming formulations. A test pilot held with actual vehicles proves the applicability of our approach in practice: our routes are 3–156% shorter than the routes the city of Pittsburgh generated with commercial routing software.

On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems

In this paper, we present some novel and intriguing findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviours of the eigenfunctions in two dimensions. We introduce a new notion of generalized singular lines of the Laplacian eigenfunctions, and carefully study these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We establish an accurate and comprehensive quantitative characterisation of the relationship. Roughly speaking, the vanishing order is generically infinite if the intersecting angle is irrational, and the vanishing order is finite if the intersecting angle is rational. In fact, in the latter case, the vanishing order is the degree of the rationality. The theoretical findings are original and of significant interest in spectral theory. Moreover, they are applied directly to some physical problems of great importance, including the inverse obstacle scattering problem and the inverse diffraction grating problem. It is shown in a certain polygonal setup that one can recover the support of the unknown scatterer as well as the surface impedance parameter by finitely many far-field patterns. Indeed, at most two far-field patterns are sufficient for some important applications. Unique identifiability by finitely many far-field patterns remains to be a highly challenging fundamental mathematical problem in the inverse scattering theory.

Generalized Sherman–Morrison–Woodbury formula based algorithm for the inverses of opposite-bordered tridiagonal matrices

Matrix inverse computation is one of the fundamental mathematical problems of linear algebra and has been widely used in many fields of science and engineering. In this paper, we consider the inverse computation of an opposite-bordered tridiagonal matrix which has attracted much attention in recent years. By exploiting the low-rank structure of the matrix, first we show that an explicit formula for the inverse of the opposite-bordered tridiagonal matrix can be obtained based on the combination of a specific matrix splitting and the generalized Sherman–Morrison–Woodbury formula. Accordingly, a numerical algorithm is outlined. Second, we present a breakdown-free symbolic algorithm of $$O(n^2)$$ for computing the inverse of an n-by-n opposite-bordered tridiagonal matrix, which is based on the use of GTINV algorithm and the generalized symbolic Thomas algorithm. Finally, we have provided the results of some numerical experiments for the sake of illustration.

Identifying the Riemann zeros by periodically driving a single qubit

The Riemann hypothesis, one of the most important open problems in pure mathematics, implies the most profound secret of prime numbers. One of the most interesting approaches to solve this hypothesis is to connect the problem with the spectrum of the physical Hamiltonian of a quantum system. However, none of the proposed quantum Hamiltonians have been experimentally feasible.Here, we report the first experiment to identify the first non-trivial zeros of the Riemann zeta function and the first two zeros of P\'olya's fake zeta function, using a novel Floquet method, through properly designed periodically driving functions. According to this method, the zeros of these functions are characterized by the occurrence of crossings of quasi-energies when the dynamics of the system are frozen. The experimentally obtained zeros are in excellent agreement with their exact values. Our study provides the first experimental realization of the Riemann zeros, which may provide new insights into this fundamental mathematical problem.

Hybrid Signature Scheme Integrating Elliptic Curve Cryptosystem with Ong, Schnorr, and Shamir Digital Signature Scheme

Transmitting data over unsecured Channels is one of the most indispensable issues which motivates researchers to study. In this paper, a new hybrid digital signature approach is going to be proposed which integrates the elliptic curve cryptosystem (ECC) into Ong, Schnorr, and Shamir signature (OSS). A self-invertible 8×8 key matrix will be utilized as a portion of the OSS digital signature. The strength of the proposed technique relies on combining couple of fundamental mathematical problems namely the discrete logarithm problem (DLP) and integer factorization problem (IFP). Elaborated results regarding security and time rendering analysis have been measured and confirmed its durability against diverse patterns of attacks. The submitted technique can be applied as a substitutional protocol if all the familiar systems are executed.

Properties of the bridge sampler with a focus on splitting the MCMC sample

Computation of normalizing constants is a fundamental mathematical problem in various disciplines, particularly in Bayesian model selection problems. A sampling-based technique known as bridge sampling (Meng and Wong in Stat Sin 6(4):831–860, 1996) has been found to produce accurate estimates of normalizing constants and is shown to possess good asymptotic properties. For small to moderate sample sizes (as in situations with limited computational resources), we demonstrate that the (optimal) bridge sampler produces biased estimates. Specifically, when one density (we denote as $$p_2$$) is constructed to be close to the target density (we denote as $$p_1$$) using method of moments, our simulation-based results indicate that the correlation-induced bias through the moment-matching procedure is non-negligible. More crucially, the bias amplifies as the dimensionality of the problem increases. Thus, a series of theoretical as well as empirical investigations is carried out to identify the nature and origin of the bias. We then examine the effect of sample size allocation on the accuracy of bridge sampling estimates and discovered that one possibility of reducing both the bias and standard error with a small increase in computational effort is by drawing extra samples from the moment-matched density $$p_2$$ (which we assume easy to sample from), provided that the evaluation of $$p_1$$ is not too expensive. We proceed to show how the simple adaptive approach we termed “splitting” manages to alleviate the correlation-induced bias at the expense of a higher standard error, irrespective of the dimensionality involved. We also slightly modified the strategy suggested by Wang et al. (Warp bridge sampling: the next generation, Preprint, 2019. arXiv:1609.07690) to address the issue of the increase in standard error due to splitting, which is later generalized to further improve the efficiency. We conclude the paper by offering our insights of the application of a combination of these adaptive methods to improve the accuracy of bridge sampling estimates in Bayesian applications (where posterior samples are typically expensive to generate) based on the preceding investigations, with an application to a practical example.

The Relationship of Arithmetic As Two Twin Peano Arithmetic(s) and Set Theory: A New Glance From the Theory of Information

The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat’s last theorem, four-color theorem as well as its new-formulated generalization as “four-letter theorem”, Poincare’s conjecture, “P vs NP” are considered over again, from and within the new-founding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested.

Waves in strongly nonlinear discrete systems.

The paper presents the main steps in the development of the strongly nonlinear wave dynamics of discrete systems. The initial motivation was prompted by the challenges in the design of barriers to mitigate high-amplitude compression pulses caused by impact or explosion. But this area poses a fundamental mathematical and physical problem and should be considered as a natural step in developing strongly nonlinear wave dynamics. Strong nonlinearity results in a highly tunable behaviour and allows design of systems with properties ranging from a weakly nonlinear regime, similar to the classical case of the Fermi-Pasta-Ulam lattice, or to a non-classical case of sonic vacuum. Strongly nonlinear systems support periodic waves and one of the fascinating results was a discovery of a strongly nonlinear solitary wave in sonic vacuum (a limiting case of a periodic wave) with properties very different from the Korteweg de Vries solitary wave. Shock-like oscillating and monotonous stationary stress waves can also be supported if the system is dissipative. The paper discusses the main theoretical and experimental results, focusing on travelling waves and possible future developments in the area of strongly nonlinear metamaterials.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

Parameterized Complexity of Sparse Linear Complementarity Problems

In this paper, we study the parameterized complexity of the linear complementarity problem (LCP), which is one of the most fundamental mathematical optimization problems. The parameters we focus on are the sparsities of the input and the output of the LCP: the maximum numbers of nonzero entries per row/column in the coefficient matrix and the number of nonzero entries in a solution. Our main result is to present a fixed-parameter algorithm for the LCP with the combined parameter. We also show that if we drop any of the three parameters, then the LCP is NP-hard or W[1]-hard. In addition, we show the nonexistence of a polynomial kernel for the LCP unless coNP $$\subseteq $$ NP/poly.