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Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time., 50 pages

Multimineral log analysis is a quantitative formation evaluation tool for geologic and petrophysical reservoir characterization. Rock composition can be estimated by solving equations that relate log measurements to the petrophysical endpoints of minerals and fluids. Due to errors in log data and uncertainties in petrophysical endpoints of constituents, we have used effective medium models from rock physics as additional independent information to validate or constrain the results. We examine the Voigt-Reuss (VR) bound model, self-consistent approximation (SCA), and differential effective medium (DEM). The VR bound model provides the first-order quality control of multimineral results. We first show a conventional carbonate reservoir study with intervals in which the predicted effective medium models from multimineral results are inconsistent with measured elastic properties. We use the VR bound model as an inequality constraint in multimineral analysis for plausible alternative solutions. The SCA and DEM models provide good estimates in low-porosity intervals and imply geologic information for porous intervals. Then, we present a field case of the Bakken and Three Forks formations. A linear interpolation of the VR bound model helps validate multimineral results and approximate the elastic moduli of clay. There are two major advantages to using our new method: (1) Rock-physics effective medium models provide independent quality control of petrophysical multimineral results and (2) multimineral information leads to realistic rock-physics models.

The applicability of an efficient sixth convergence order scheme is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the scheme. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided (not given in earlier works) based on ω–continuity conditions. Numerical examples complete this article.

When the space shuttle was launched from the Kennedy Space Center, its escape velocity could be determined by solving a first-order ordinary differential equation. The same can be said for finding the flow of electromagnetic forces, the temperature of a cup of coffee, the population of a species as well as numerous other applications. In this chapter, we show how these problems can be expressed as first-order equations. We will focus our attention on setting up the problems and explaining the meaning of the subsequent solutions because the techniques for solving these problems were discussed in Chapter 2.

Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. In numerical analysis of DAEs, consistent initialization and index reduction are important preprocessing prior to numerical integration. Existing DAE solvers commonly adopt structural preprocessing methods based on combinatorial optimization. Unfortunately, the structural methods fail if the DAE has numerical or symbolic cancellations. For such DAEs, methods have been proposed to modify them to other DAEs to which the structural methods are applicable, based on the combinatorial relaxation technique. Existing modification methods, however, work only for a class of DAEs that are linear or close to linear. This paper presents two new modification methods for nonlinear DAEs: the substitution method and the augmentation method. Both methods are based on the combinatorial relaxation approach and are applicable to a large class of nonlinear DAEs. The substitution method symbolically solves equations for some derivatives based on the implicit function theorem and substitutes the solution back into the system. Instead of solving equations, the augmentation method modifies DAEs by appending new variables and equations. The augmentation method has advantages that the equation solving is not needed and the sparsity of DAEs is retained. It is shown in numerical experiments that both methods, especially the augmentation method, successfully modify high-index DAEs that the DAE solver in MATLAB cannot handle., Comment: A preliminary version of this paper is to appear in Proceedings of the 44th International Symposium on Symbolic and Algebraic Computation (ISSAC 2019), Beijing, China, July 2019

Problem and goal. In modern conditions, specialists in various subject areas who have an information culture and are able to solve complex professional problems using modern information and communication technologies are in demand. Currently, specialists in the field of applied mathematics are required, which plays an important role in the development of human civilization. Therefore, in the process of teaching various academic disciplines of applied mathematics at the university, including the discipline Equations of mathematical physics, attention should be paid to the development of students information culture. Methodology. When teaching students the discipline Equations of mathematical physics, it is extremely important that the teacher knows not only the content of teaching this discipline of applied mathematics, but also has practical experience in solving equations of mathematical physics by computer means. Such qualities of the teacher will allow him to successfully conduct training sessions in the conditions of informatization of teaching mentioned discipline. At the same time, it ought to be clearly understood that the use of computer technologies in teaching the discipline Equations of mathematical physics must be correct. The necessity to develop and implement in practice a variety of methodological approaches that allow students to develop an information culture in training sessions on that discipline is obvious. Results. The use of advanced pedagogical technologies in training sessions on the discipline Equations of mathematical physics, where computer technologies are used, will allow students to develop an information culture. Conclusion. Computer technologies that students use in the process of solving educational problems require them to have certain skills and abilities to identify their broad capabilities. Students are aware of the role of computer technologies in conducting applied scientific research, understand the role of computer modeling methodology and computational experiment in studying the world around them.

The purpose of this study was to explore the paths by which word-problem intervention, with versus without embedded prealgebraic reasoning instruction, improved word-problem performance. Students with mathematics difficulty (MD; n = 304) were randomly assigned to a business-as-usual condition or 1 of 2 variants of word-problem intervention. The prealgebraic reasoning component targeted relational understanding of the equal sign as well as standard and nonstandard equation solving. Intervention occurred for 16 weeks, 3 times per week, 30 min per session. Sequential mediation models revealed main effects, in which each intervention condition significantly and substantially outperformed the business-as-usual condition, corroborating prior research on the efficacy of schema word-problem intervention. Yet despite comparable effects on word-problem outcomes between the two word-problem conditions, the process by which effects accrued differed: An indirect path via equal-sign understanding and then equation solving was significant only for the word-problem intervention condition with embedded prealgebraic reasoning instruction. Additionally, the effect of this condition on equal-sign reasoning was strong. Given the link between equal-sign reasoning for success with algebra and the importance of algebra for success with advanced mathematics, results suggest an advantage for embedding prealgebraic reasoning instruction within word-problem intervention. (PsycInfo Database Record (c) 2021 APA, all rights reserved)

In this paper, an attempt is made to extend a linear two-dimensional model for stability analysis of the laminated annular microplate subject to external excitation. A new approach called hybrid optimization is introduced to solve optimization problems with a high sensitive objective function to decline computational costs and increase the predicted optimum results accuracy. Regarding this issue, generalized differential quadrature element method (GDQEM), particle swarm optimization (PSO), as well as genetic algorithm (GA) methods are coupled to improve the dynamic stability of the annular microsystems via finding an optimum frequency and fiber angle of layers simultaneously. Higher-order shear deformation theory (HSDT) and Hamilton’s principle are taken into consideration for the exact derivation of the general linear governing equations and boundary conditions of the axisymmetric laminated annular plate. Also, modified couple stress theory (MCST) is presented for presenting the size-dependency of the current microsystem. The GDQEM is used to solve the governing equations of the microsystem via its boundary domains. To enhance the genetic algorithms’ performance for solving equations, the optimizer approach of particle swarm has been employed as a GA’s operator. Precise convergence and practicality of the suggested mixed-method have been disclosed. Moreover, we would have proven that for achieving the convergence PSO’s and GA’s outcomes, we have to apply higher than fifteen iterations.

We present the semi-local convergence analysis of atwo-step derivative-free method for solving Banach space valuedequations. The convergence criteria are based only on the firstderivative and our idea of recurrent functions.

Investigation of factors (teacher knowledge and arithmetic practice as presented in textbooks) influencing students' understanding of equivalence, across samples from China, England, New Zealand, South Korea, Turkey, and the US.

We offer an alternative proof of a result of Conlon, Fox, Sudakov and Zhao on solving translation-invariant linear equations in dense Sidon sets. Our proof generalises to equations in more than five variables and yields effective bounds., Comment: v2: Typos corrected and suggestions from correspondents incorporated

We determine a radius of convergence for an efficient iterative method with frozen derivatives to solve Banach space defined equations. Our convergence analysis use \(\omega-\) continuity conditions only on the first derivative. Earlier studies have used hypotheses up to the seventh derivative, limiting the applicability of the method. Numerical examples complete the article.

It is possible to solve equations of degree $${\leqslant}4$$ in some bases of the field $$GF(p^{s})$$ , where $$p>3$$ , $$s=2^{k}r$$ , $$k\rightarrow\infty$$ , $$r=\pm 1(\textrm{mod}\ 6)$$ , and $$p,r=O(1)$$ with the bit complexity $$O_{r}(M(2^{k})kM(r)M(\lceil\log_{2}p)\rceil)=O_{r,p}(M(s)\log_{2}s),$$ where $$M(n)$$ is the complexity of polynomial multiplication. In a normal basis of the fields $$GF(3^{s})$$ , $$s=\pm 1(\textrm{mod}\ 6)$$ , all roots may be found with the bit complexity $$O(M(GF(3^{s}))\log_{2}s)$$ , where $$M(GF(q))$$ is the complexity of multiplication in the field $$GF(q)$$ . For normal bases in the fields $$GF(2^{s})$$ , where $$s=2r$$ , $$r\neq 0(\textrm{mod}\ 3)$$ , the bit complexity is $$O(M(GF(2^{s}))\log_{2}s)$$ .

Be it in the world of mathematics or real life, it is often rewarding to think out-of-the box while solving a problem. Accordingly, in this paper, our aim is to explore the various alternative approaches for solving algebraic equations and finding the number of real zeros. We will further delve deeper into the conceptual part of mathematics and understand how implementation of simple ideas can lead to an acceptable solution, which otherwise would have been tedious by considering the conventional approaches. In the pursuit of achieving the objective of this paper, we will consider few examples with full solutions coupled with precise explanation. It is also intended to leave something meaningful for the readers to explore further on their own. The fundamental objective of this paper is to emphasize on the importance of application of basic mathematical logic, concept of inequality, concept of domain and range of functions, concept of calculus and last but not the least the graphical approach in solving mathematical equations. As a further clarification on the scope of this paper, it is highly pertinent to bring to the understanding of the readers two important aspects - firstly, we will only deal with equations involving real variables; and secondly, this paper does not include topics related to number theory.

The nonlinear reaction-diffusion cycle in the thin membrane that describes the chemical reactions involving three species is studied. The model consists of the system of on nonlinear reaction-diffusion equations. The closed type of analytical expression of concentrations for the enzyme was developed by solving equations using the Taylor series formula. This results in the mixed Dirichlet and Neumann boundary conditions. Taylor series method similar to exponential function method. This technique provides approximate and simple solutions that are quick, easy to compute, and efficiently correct. These estimated findings are compared to the nuxmerical results. There is a good agreement with the simulation results.

This review/research paper deals with the reduction of nonlinear partial differential equations governing the dynamic behavior of structural mechanical members with emphasis put on theoretical aspects of the applied methods and signal processing. Owing to the rapid development of technology, materials science and in particular micro/nano mechanical systems, there is a need not only to revise approaches to mathematical modeling of structural nonlinear vibrations, but also to choose/propose novel (extended) theoretically based methods and hence, motivating development of numerical algorithms, to get the authentic, reliable, validated and accurate solutions to complex mathematical models derived (nonlinear PDEs). The review introduces the reader to traditional approaches with a broad spectrum of the Fourier-type methods, Galerkin-type methods, Kantorovich–Vlasov methods, variational methods, variational iteration methods, as well as the methods of Vaindiner and Agranovskii–Baglai–Smirnov. While some of them are well known and applied by computational and engineering-oriented community, attention is paid to important (from our point of view) but not widely known and used classical approaches. In addition, the considerations are supported by the most popular and frequently employed algorithms and direct numerical schemes based on the finite element method (FEM) and finite difference method (FDM) to validate results obtained. In spite of a general aspect of the review paper, the traditional theoretical methods mentioned so far are quantified and compared with respect to applications to the novel branch of mechanics, i.e. vibrational behavior of nanostructures, which includes results of our own research presented throughout the paper. Namely, considerable effort has been devoted to investigate dynamic features of the Germain–Lagrange nanoplate (including physical nonlinearity and inhomogeneity of materials). Modified Germain–Lagrange equations are obtained using Kirchhoff’s hypothesis and relations based on the modified couple stress theory as well as Hamilton’s principle. A comparative analysis is carried out to identify the most effective methods for solving equations of mathematical physics taking as an example the modified Germain–Lagrange equation for a nanoplate. In numerical experiments with reducing the problem of PDEs to ODEs based on Fourier’s ideas (separation of variables), the Bubnov–Galerkin method of static problems and Faedo–Galerkin method of dynamic problems are employed and quantified. An exact solution governing the behavior of nanoplates served to quantify the efficiency of various reduction methods, including the Bubnov–Galerkin method, Kantorovich–Vlasov method, variational iterations and Vaindiner’s method (the last three methods include theorems regarding their numerical convergence). The numerical solutions have been compared with the solutions obtained by various combinations of the mentioned methods and with solutions obtained by FDM of the second order of accuracy and FEM for triangular and quadrangular finite elements. The studied methods of reduction to ordinary differential equations show high accuracy and feasibility to solve numerous problems of mathematical physics and mechanical systems with emphasis put on signal processing.

Structure sense can be interpreted as an intuitive ability towards symbolic expressions, including skills to perceive, to interpret, and to manipulate symbols in different roles. This ability shows student algebraic proficiency in dealing with various symbolic expressions and is considered important to be mastered by secondary school students for advanced study or professional work. This study, therefore, aims to investigate students’ algebraic proficiency in terms of structure sense. To reach this aim, we set up a qualitative case study with the following three steps. First, after conducting a literature study, we designed structure sense tasks according to structure sense characteristics for the topic of equations. Second, we administered an individual written test involving 28 grade XI students (16-17 year-old). Third, we analyzed students’ written work using a structure sense perspective. The results showed that about two-thirds of the participated students lack of structure sense in which they tend to use more procedural strategies than structure sense strategies in solving equations. We conclude that the perspective of structure sense provides a fruitful lens for assessing students’ algebraic proficiency.

The article discusses the ways of developing skills and abilities to effectively solve problems when describing methods for solving equations and inequalities, clarifying theoretical knowledge, the basics of forming skills for practical application. The formation of mathematical concepts through solving problems in teaching mathematics opens the way to the development of mathematical thinking, the application of knowledge in practice, and the development of search skills. To master a mathematical concept, along with its definition, it is necessary to know its features and properties. This can be achieved primarily through problem solving and exercise. Problem solving is based on the development of new methods, the creation of algorithms, ways of developing practical skills in the methods and techniques mastered with the help of tasks.In addition, transforming equations and inequalities through the development of thinking skills helps to identify common or special properties in order to draw correct conclusions. Solving various problems, it shows what operations should be used to determine the situation in which a solution was found, and what features of the solution allow choosing the most effective methods. Thanks to the theoretical substantiation of the general article, it is possible to master convenient methods for solving equations and inequalities of various structures.

Firstly it will be shown that, using the concept of monomial orderings, the classical 1D theory for linear systems generalizes in a very natural way to (autonomous) Attasi nD-systems, giving the Attasi-Hankel matrix, the Attasi transfer function and Attasi state-space realizations. Secondly we will explain how one can associate an Attasi system to any set of polynomial equations having a finite number of solutions and how Attasi realization theory can be used to (1) describe a commutative matrix solution to the set of polynomial equations (this generalizes the Cayley-Hamilton theorem) and (2) find the (scalar) solutions to the set of polynomial equations by computing the joint eigenvalues and eigenvectors of the commuting matrices. Some remarks will be made about how the (discrete time) Attasi system equations can be solved recursively and how that can be used in large-scale eigensolvers. Using the associated Attasi system to a set of polynomial equations also helps to understand various different approaches to polynomial equation solving and multivariate polynomial and rational function minimization.

In this paper, we consider several ways to solve problems of high complexity that occur in the school course of mathematics, test tasks of mathematical literacy, and Olympic problems. Problem solving is the most productive form of learning mathematics, and this process should be a necessary component of all extracurricular activities conducted in mathematics. Mathematical Olympic problems, as a rule,will help to a large extent to learn, to find independent nonstandard methods of solving problems. The main goal of solving problems of high complexity plays a special role in the development of students. Helps them learn the material more strongly and consciously. The ability to analyze a given situation, compare data and search for data, determine the hidden properties of this case, synthesize useful information for solving problems, and form the necessary skills not only for solving problems. The high school program considered solving equations that are not defined using Euclid's rules for catching ordinary multipliers,classifying complex expressions by the properties of Prime numbers, and solving economic problems

Functional and graphic lines are one of the foundations of mathematics teaching methods. The advantage of this line is that the study of other important lines of mathematics is carried out through the prism of the concept of function. Based on the experience of teaching mathematics, we know that the concept of function is abstract and very difficult for students to understand, so in order to enhance the visualization of the researching objects and concepts when implementing functional and graphic lines, students need to increase the system of physical content tasks for studying and understanding functions. In school course of algebra, the functional-graphical method is rarely used for solving a system of equations with two unknowns, as well as for solving equations with two unknowns. The article deals with the problems of solving problems of physical content when studying a system of linear equations with two variables in school course of algebra. The emphasis is on the fact that the considered problems with physical content are interconnected with functionalgraphic lines in algebra and allow deepening the topic, revealing the practical content. The problems with physical content presented in the article are intended for studying linear functions of algebra and their graphs, studying functions, constructing and solving equations and a system of linear equations associated with these functions.

Performance-based design for a constructional steel frame in nonlinear-plastic region requires an improvement in order to achieve a reliable structural analysis. The need to explicitly consider the nonlinear behaviour of structures makes the numerical modelling approach much more favourable than expensive and potentially dangerous experimental work. The parameters considered in the analysis are not limited to the linear change of geometry and material yielding, but also include the effect of large deformations, geometrical imperfections, load eccentricities, residual stresses, strain-unloading, and the nonlinear boundary conditions. Such analysis requires the use of accurate mathematical modelling and effective numerical procedures for solving equations of equilibrium. With that in mind, this paper presents the mathematical formulations and finite element procedures of nonlinear inelastic steel frame analysis with quasi-static semi-rigid connections. Verification and validation of the developed analytical procedures are conducted and good agreements are obtained. It is an approach that enables the structural behaviour of constructional steel frames to be traced throughout the entire range of loading until failure. It also provides information on the derivation of the structural analysis by using finite element method.

An approximate analytical solution to the equations of the physical theory of meteors is obtained for a meteoroid moving as a single body under the assumption of constant ablation parameter. The approach allows predicting the velocity and mass of the meteoroid at any point of its trajectory. The accuracy of the solution is estimated. A numerical-analytical method for solving the equations is proposed for the case when the ablation parameter is variable. The soluitons obtained by different methods with constant and variable ablation parameters are compared.

Despite their widespread use for performing advanced computational tasks, digital signal processors suffer from several restrictions, including low speed, high power consumption and complexity, caused by costly analogue-to-digital converters. For this reason, there has recently been a surge of interest in performing wave-based analogue computations that avoid analogue-to-digital conversion and allow massively parallel operation. In particular, novel schemes for wave-based analogue computing have been proposed based on artificially engineered photonic structures, that is, metamaterials. Such kinds of computing systems, referred to as computational metamaterials, can be as fast as the speed of light and as small as its wavelength, yet, impart complex mathematical operations on an incoming wave packet or even provide solutions to integro-differential equations. These much-sought features promise to enable a new generation of ultra-fast, compact and efficient processing and computing hardware based on light-wave propagation. In this Review, we discuss recent advances in the field of computational metamaterials, surveying the state-of-the-art metastructures proposed to perform analogue computation. We further describe some of the most exciting applications suggested for these computing systems, including image processing, edge detection, equation solving and machine learning. Finally, we provide an outlook for the possible directions and the key problems for future research. Metamaterials provide a platform to leverage optical signals for performing specific-purpose computational tasks with ultra-fast speeds. This Review surveys the basic principles, recent advances and promising future directions for wave-based-metamaterial analogue computing systems.

In this paper, we present a new approach for solving equations of motion for the trapped motion of the infinitesimal mass m in case of the elliptic restricted problem of three bodies (ER3BP) (primaries $$M_\mathrm{Sun}$$ and $$m_\mathrm{planet}$$ are rotating around their common centre of masses on elliptic orbit): a new type of the solving procedure is implemented here for solving equations of motion of the infinitesimal mass m in the vicinity of the barycenter of masses $$M_\mathrm{Sun}$$ and $$m_\mathrm{planet}$$ . Meanwhile, the system of equations of motion has been successfully explored with respect to the existence of analytical way for presentation of the approximated solution. As the main result, equations of motion are reduced to the system of three nonlinear ordinary differential equations: (1) equation for coordinate x is proved to be a kind of appropriate equation for the forced oscillations during a long-time period of quasi-oscillations (with a proper restriction to the mass $$m_\mathrm{planet}$$ ), (2) equation for coordinate y reveals that motion is not stable with respect to this coordinate and condition $$y \sim 0$$ would be valid if only we choose zero initial conditions, and (3) equation for coordinate z is proved to be Riccati ODE of the first kind. Thus, infinitesimal mass m should escape from vicinity of common centre of masses $$M_\mathrm{Sun}$$ and $$m_\mathrm{planet}$$ as soon as the true anomaly f increases insofar. The main aim of the current research is to point out a clear formulation of solving algorithm or semi-analytical procedure with partial cases of solutions to the system of equations under consideration. Here, semi-analytical solution should be treated as numerical algorithm for a system of ordinary differential equations (ER3BP) with well-known code for solving to be presented in the final form.

A dedicated programmable calculator serves to find values of Lambert W0(х) function and related functions. Its software implementation is described and calculation examples specified. The calculator enables finding of values for top branch of Lambert function, reciprocal of Lambert function, logarithm of Lambert function, exponent whose index of power equals Lambert function, values of the first and second derivatives, values of generalized logarithm, generalized tangent, generalized cotangent, generalized sine, generalized cosine, hyperbolic generalized tangent, hyperbolic generalized cotangent, hyperbolic generalized sine, hyperbolic generalized cosine. The calculator permits to compute values of two most widely used types of certain integrals, whose integrand contains Lambert function. Various methods of Lambert function value calculation were compared and their applicability advised. Lambert function expansion by powers of variable х was shown to be valid only at the vicinity of zero. Relative error variation plots are specified which emerge at application of these procedures. The most feasible calculation of Lambert function values was shown to be executed either by approximation formulas or by solving equations which determine this function and are related to it. Methods of power series inversion and inverse function integration are discussed in much detail. These methods were used in our work, and the procedure of Lambert function integration as implicit function has been implemented in proposed calculator. The necessity to have a program for calculation of Lambert function values whose text is accessible to its authors is shown at an example of comparison between two programs intended for solution of algebraic equations. One of these programs made in one case a basic error in result, though running of its working module remained unchanged.