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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.

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.

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.

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.

As applied to phase transformations and their formally analogous processes, mathematical descriptions suitable for engineering design are proposed for successive, bulk, and successive–bulk transformations with the structural rearrangement of the initial system within the proposed formal analogy of these processes. To specify the considered models, they are presented as applied to melt crystallization. Possible methods for solving equations of these models are discussed.

In the paper, an approach to numerical integration of equations governing motion of constrained mechanical systems is suggested. In the framework of this approach, unknown reaction forces acting on the system are treated as controls, and the algebraic equations that these reactions satisfy, as control goals. On the basis of the suggested approach, a technique for numerical solving equations of rolling is developed. The discussion is illustrated by the example of application of the algorithm to solving the problem of a heavy wheel with a pendulum (a prototype of a ball-shaped robot) rolling along a curvilinear profile without slippage.

The Delta robot, widely used in fast pick-and-place applications with pure position control, is a parallel kinematic chain with three rotational inputs resulting in three pure translations at the end-effector. This paper proposes a complete task-space impedance control with inverse dynamics to give this robot compliant behavior, enabling it to be used in tasks involving physical interaction. For that purpose, the well-known usage of dual quaternion algebra for kinematics modeling is novelly integrated with a neural network to compose a compact representation for the forward kinematics function, that is singularity-free and suitable for real-time calculation. This network computes the forward kinematics more than 150 times faster than a numeric equation solving algorithm, with an average estimation error of less than 0.5 mm. The proposed algorithm is implemented in a rigid body simulator, and the performance of the complete system is analyzed.

Algorithms of the optimal arrangement of heat sources with volumetric heat release within regions of a complex geometric shape are considered. The distribution found has the minimum total power and provides the temperature in the given temperature corridor. Finite-dimensional approximations of the original problem are constructed in the form of a linear programming problem. A method is given for constructing a finite-difference scheme for solving the heat equation, as well as a brief description of the developed software modules for constructing grids and solving equations. Several computer experiments have been carried out using the developed programs.

Numerical simulation of biofluids entails solving equations of fluid-structure interactions. At zero Reynolds number, solvers such as the Method of Regularized Stokeslets (MRS) give rise to large and dense matrices in practical applications where the number of structures immersed in the fluid is large. Building on previous work for an unbounded fluid domain, we first extend the Kernel-Independent Fast Multipole Method (KIFMM) for MRS to compute the matrix-vector products for the fluid flow induced by point forces above a stationary wall. In this case, the use of a regularized image system introduces additional terms to the solution which cause the matrix-vector multiplication to be quite challenging. In addition, we study the case where a linear system needs to be solved for the unknown forces that structures with known velocities exert on the fluid. Our main contribution is proposing several preconditioning techniques for the matrices associated with a few variants of MRS, including the case where a force-free, torque-free condition is imposed. They take advantage of the data-sparsity of FMM matrices as well as properties of Krylov subspaces. Our approach is memory efficient, capable of handling non-uniformly distributed structures and applicable to all FMM matrices. It enables efficient computation of the flow field surrounding a large group of dynamic micro-structures; in particular, we study the effects of fluid mixing caused by the periodic beating of a dense carpet of lung cilia.

The present study establishes an approximate optimal critic learning algorithm, based on the single-network integral reinforcement learning (IRL) algorithm and intends to solve the optimal control problem for an unknown nonlinear system with saturating actuators. The value function is formulated through building generalized nonquadratic functions. In order to solve the Hamilton–Jacobi–Bellman (HJB) equation, a novel optimal scheme for the control approximation, based on the off-policy iteration is presented. Moreover, the single-neural network implementation procedure is introduced to complete the iteration algorithm. The synchronous IRL policy iteration is proposed to update the weight of the critic neural network. Finally, reasonable simulation results are provided for confirming the effectiveness of the proposed optimal approximation control technique in solving equations for a linear and oscillating systems.

The previously developed volume-surface integral equation-based domain decomposition method (VSIE-DDM) generally requires huge matrix filling time. To alleviate this issue, the equivalent dipole-moment (EDM) method is applied to the VSIE-DDM for complex electricallylarge planar thin-layer composite objects. Therefore, an efficient hybrid VSIE-DDM-EDM method is developed, in which the EDM technique remarkably saves the calculation time of the VSIE-DDM matrix. In this hybrid method, a new simplified prism vector basis-based dipole model is proposed. Compared with the Schaubert–Wilton–Glisson basis-based dipole model, the proposed model significantly reduces the number of numerical integrations within the threshold for planar thin dielectrics. Furthermore, to ensure the flexibility of the solution for a variety of electromagnetic problems, two strategies for solving equations (inner–outer iteration scheme and direct-iterative hybrid method) are, respectively, adopted. Finally, the multilevel fast multipole algorithm (MLFMA) is utilized for iteration procedure. Numerical examples are shown to validate the accuracy and efficiency of the proposed method.

Gene Regulatory Networks are powerful models for describing the mechanisms and dynamics inside a cell. These networks are generally large in dimension and seldom yield analytical formulations. It was shown that studying the conditional expectations between dimensions (interactions or species) of a network could lead to drastic dimension reduction. These conditional expectations were classically given by solving equations of motions derived from the Chemical Master Equation. In this paper we deviate from this convention and take an Algebraic approach instead. That is, we explore the consequences of conditional expectations being described by a polynomial function. There are two main results in this work. Firstly, if the conditional expectation can be described by a polynomial function, then coefficients of this polynomial function can be reconstructed using the classical moments. And secondly, there are dimensions in Gene Regulatory Networks which inherently have conditional expectations with algebraic forms. We demonstrate through examples, that the theory derived in this work can be used to develop new and effective numerical schemes for forward simulation and parameter inference. The algebraic line of investigation of conditional expectations has considerable scope to be applied to many different aspects of Gene Regulatory Networks; this paper serves as a preliminary commentary in this direction.

This paper analyzes the local convergence of several iterative methods for approximating a locally unique solution of a nonlinear equation in a Banach space. It is shown that the local convergence of these methods depends of hypotheses requiring the first-order derivative and the Lipschitz condition. The new approach expands the applicability of previous methods and formulates their theoretical radius of convergence. Several numerical examples originated from real world problems illustrate the applicability of the technique in a wide range of nonlinear cases where previous methods can not be used.

This paper presents a triple acceleration method (TAM) for the topology optimization (TO), which consists of three parts: multilevel mesh, initial-value-based preconditioned conjugate-gradient (PCG) method, and local-update strategy. The TAM accelerates TO in three aspects including reducing mesh scale, accelerating solving equations, and decreasing the number of updated elements. Three benchmark examples are presented to evaluate proposed method, and the result shows that the proposed TAM successfully reduces 35–80% computational time with faster convergence compared to the conventional TO while the consistent optimization results are obtained. Furthermore, the TAM is able to achieve a higher speedup for large-scale problems, especially for the 3D TOs, which demonstrates that the TAM is an effective method for accelerating large-scale TO problems.

Given the growing concern for developing students’ algebraic ideas and thinking in earlier grades ( NCTM, 2000 ) it is important for students to have experiences that better prepare them for their formal introduction to algebra. Mobile puzzles seem to be an opportunity for exhibiting certain algebraic habits of mind as well as for demonstrating symbol-sense which might support students in their transition from arithmetic to algebra. These puzzles include multiple balanced collections of objects whose weights must be determined by the solver. The arms/beams must be perfectly balanced for it to hang properly. Therefore, they represent, in a pictorial way, systems of equations. Each arm/beam that balances two sets of objects (representing variables as unknown “weights”) represents an equation. The data derived from Grade-6 students who were asked to solve a collection of tasks reflect the presence of the “Puzzling and Persevering” and “Seeking and Using Structure” habits of mind. At the same time these data incorporate instances of some main components of symbol-sense such as “friendliness with symbols”, “manipulating and ‘reading through’ symbolic expressions”, and “choice of symbols”. Also discussed is the way this experience contributes to an intuitive application of the conventional rules for solving equations that will be later introduced to the students as the standard algebraic “moves”.

It has been found that there exists an essential similarity between solving equations and controlling dynamic systems: Both errors are expected to decrease to zero (or an acceptably tiny value) as soon as possible. By exploiting such a similarity, researchers have presented and investigated continuous-time recurrent neural network models for solving time-varying problems. To be compatible with digital computers, it is desirable to develop discrete-time neural algorithms from the control perspective for performance improvement. In this paper, a discrete-time zeroing neural algorithm is proposed for the solving system of linear equations with the aid of control techniques. To lay a basis for theoretical analyses, the proposed zeroing neural algorithm with nonlinearity is converted into a second-order linear system plus a residual term, and then, analyzed using the control theory. Theoretical results and numerical experiments are provided, which illustrate that the proposed neural algorithm possesses an improved performance compared to the existing solutions.

Currently, there are two non-contact measurement methods of the transmission line voltage based on field sensor: one is calculation with the inverse problem of electric field for the solution, and the other is to solve by the numerical integration algorithm. In general, the first one is confronted with data equation solving problems and difficulties in accurate calibration as well as low precision, while the second one is troubled by the complexity of algorithm equation and unsatisfactory integration node. In view of the above problems, this paper improves on the basis of the Gaussian integral algorithm to seek better nodes, in order to reduce the difficulty of solving and improve the measurement accuracy of the integral algorithm. First, starting from the Gaussian integral algorithm and making research to improve the Gaussian integral algorithm theory. Then, the finite element simulation of three-phase transmission line through Maxwell software is built, and the space electric field distribution data is acquired to calculate an integral node as well as its corresponding weights. Finally, the three-phase transmission line voltage measurement test platform with D-dot sensor voltage measurement system is built to test verification. The text results show that the voltage measurement method based on the improved Gaussian Integral has characteristics of a simple solution, better integral node, and higher precision, and all the measured voltage errors are less than 0.45%.

Examples of solving equations and inequalities, analyzing quadratic expressions, and reasoning with functions show three ways to engage students in this mathematical practice.

Many research studies have sought to explain why NCTM's vision for mathematics classrooms has not had greater impact on everyday instruction, with teacher beliefs often identified as an explanatory variable. Using instructional exchanges as a theoretical construct, this study explores the influence of teachers' institutional positions on the solving of equations in algebra classrooms. The experimental design uses surveys with embedded rich-media representations of classroom interaction to surface how teachers appraise correct solutions to linear equations where some solutions follow suggested textbook procedures for solving linear equations and others do not. This paper illustrates the feasibility of studying teaching with rich-media surveys and suggests new ways to support changes in everyday mathematics teaching.

A task frequently encountered in digital circuit design is the solution of a two-valued Boolean equation of the form h(X,Y,Z)=1, where h: B_2^(k+m+n)→ B_2 and X,Y, and Z are binary vectors of lengths k, m, and n, representing inputs, intermediary values, and outputs, respectively. The resultant of the suppression of the variables Y from this equation could be written in the form g(X,Z)=1 where g: B_2^(k+n)→ B_2. Typically, one needs to solve for Z in terms of X, and hence it is unavoidable to resort to ‘big’ Boolean algebras which are finite (atomic) Boolean algebras larger than the two-valued Boolean algebra. This is done by reinterpreting the aforementioned g(X,Z) as g(Z): B_(2^K)^n→ B_(2^K ), where B_(2^K ) is the free Boolean algebra FB(X_1,X_2…….X_k ), which has K= 2^k atoms, and 2^K elemnets. This paper describes how to unify many digital specifications into a single Boolean equation, suppress unwanted intermediary variables Y, and solve the equation g(Z)=1 for outputs Z (in terms of inputs X) in the absence of any information about Y. The paper uses a novel method for obtaining the parametric general solutions of the ‘big’ Boolean equation g(Z)=1. The parameters used do not belong to B_(2^K ) but they belong to the two-valued Boolean algebra B_2, also known as the switching algebra or propositional algebra. To achieve this, we have to use distinct independent parameters for each asserted atom in the Boole-Shannon expansion of g(Z). The concepts and methods introduced herein are demonsrated via several detailed examples, which cover the most prominent type among basic problems of digital circuit design.

The harmonic balance (HB) method is a very popular in software tools for simulation of radio-frequency (RF) circuits. In the yearly published papers, we have proposed the new method for solving HB equations. This method replaces the solution a large system of HB equations by two subsystems with very small dimensions, the solution of which is performed sequentially. In the first subsystem of equations, the number of harmonics is reduced to a certain minimum value (R), taking into account all the nodes of the circuit. In the second subsystem of equations, the number of circuit nodes is reduced to the value R, taking into account all harmonics. Previously unpublished relations and algorithms for solving equations by the Newton method are presented. The relations for calculating the Jacobian for both equations and the features of their implementation in the software are given. All algorithms are implemented in the circuit modeling software in the Matlab/Simulink system.

Merging the concept of metamaterials with the nanophotonics platform brings new functionalities to the science and technology of both fields. Since photons are usually controlled by materials, metamaterials have provided interesting platforms for engineering light-matter interaction. In recent years, a variety of features and exciting applications of meso-, micro-, and nanoscale metamaterials have been explored and developed. One of these applications is the development of material-based wave-based analog computing machines, which we are investigating in my group. We have designed material structures that can perform mathematical operations as waves propagate through them. Such materials have effectively become computing machines. For example, we have explored metastructures that can solve integral and differential equations and can invert matrices using waves as incoming waves interact with them. Moreover, we have also been investigating how collections of other photonic networks (such as the Mach-Zehnder interferometers), either by themselves or together with metamaterials, can provide another light-based method for solving equations and performing mathematical operations with light. In this talk, I will discuss the most recent results in some of our ongoing research programs on this topic, will present physical insights into these results, and will forecast future research directions in this area.

We have developed an automated processed calculator that works on the voice command of people and provides voice-generated desired output. It is specifically developed for the people who are partially or completely visually impaired. We have provided a way out for those people to do basic calculation like addition, subtraction, division, and multiplication by directly speaking the commands for calculation on microphone. These basic calculations are performed according to Bracket Of Division Multiplication Addition Subtraction (BODMAS) rule. Further, we have added various features for calculating quadratic equations F(X) = AX^2 + BX^1 + C linear equations F(X) = PX + B, trigonometric functions, and modern math working on the same procedure as mentioned above, thereby giving a medium to these people and helping them a way out to do these activities in more simplified manner. Our project is aimed to fully automate this process and add more features to it.

In order to improve the online teaching effect of equation solving, an online teaching method of equation solving based on transfer learning is designed. First of all, integrate the online teaching resources in the Internet, then the data conversion and classification processing, and finally build the learner behavior and knowledge resources relationship model, to recommend the relevant teaching content to learners, so as to achieve online teaching. The experimental results show that the online teaching method of equation solving based on transfer learning has high accuracy, short classification time and high student performance. It is proved that the online teaching method in this study can effectively improve the online teaching effect.

A two-step approach is proposed to accomplish high-fidelity whole-core resonance self-shielding calculation. Direct slowing-down equation solving based on the pin-cell scale is performed as the first step to simulate different operating conditions of the reactor. Resonance database is fitted using the results from the pin-cell calculation. Several techniques are used in the generation of the resonance database to estimate multiple types of resonance effects. The second step is the calculation of practical whole-core problem using the resonance database obtained from the first step. The transport solver is embedded both at the first step and the second step to establish the equivalence relationship between the fuel rod in the practical problem and the pin-cell at the first step. The numerical results show that the new approach have capability to perform high-fidelity resonance calculations for practical problem.

This chapter focuses on the numerical solution of the wave equation. Partial differential equations play a key role in many fields of science and engineering. Recently, however, the emergence of massively parallel computer systems has suggested an alternative computational approach. Many classical numerical methods are not intended for multiprocessor systems. We devote this paper to the investigation of new numerical methods for solving equations in partial derivatives with parallel computer systems using. The basic idea is founded on the assumption that the mesh function is monotonous and consistently bypasses the solution area in different directions. It is demonstrated that the application of the proposed method on large meshes is more preferable than the known methods.

Recurrent neural network (RNN), as a branch of artificial intelligence, shows powerful abilities to solve the complicated computational problems. Due to the similarity between solving equations and controlling dynamic systems, RNN-based approaches can also be analysed from the perspectives of control. Multi-linear systems, on the other hand, are a type of tensor equations with considerable complexity due to the special structure of tensors. In this paper, a power-sum function activated RNN model is proposed to find the solutions of the multi-linear systems with nonsingular ${\mathcal{M}}$-tensors. It is theoretically proved that the proposed RNN model is stable in the sense of Lyapunov stability theory and converges to the theoretical solution. In addition, computer simulations are provided to substantiate the effectiveness and superiority of the proposed RNN model.

It can be found that the exact solution of the equations has quite similarity to the pursuing of the minimum error in the field of control. Therefore, many control problems can be explained from the perspective of solving equations. As a powerful computing tool, neural dynamics model has been widely applied in addressing time-varying equations and optimization problems because of its remarkable convergence and robustness. For neural dynamics model, the selection of activation function plays a significant role in its performance. In this paper, a special nonlinearly activated (i.e. power-sum activation function) neural dynamics model is exploited to the experimental simulation of intention recognition of lower limb motion. Compared with the linear activation function, the corresponding analyses of convergence and robustness are given respectively, demonstrating the superior characteristics of the neural dynamics model with power-sum activation function.

This paper presents a design procedure for implementation of a matching network with spurious tones rejection using on-chip transformer for high speed DAC applications. Such circuits are complex to analyze and design, due to the higher order of the network. The presented design procedure involves matching the impedance at a desired center frequency based on the input impedance equation and with the aid of mathematical computational platform to avoid tedious mathematical derivations. A test case is considered with center frequency of 28GHz and detectable spurious tones at 30GHz and 26GHz to verify the proposed design procedure. The designed network matches a 50$\Omega$ load to a complex impedance of 12-j5.3 and simultaneously rejects the spurious tones by imposing notch filter at 30GHz and suppressing the other tone at 26GHz by 9. 5dB.

The solution of boundary value problems describing piecewise-homogeneous media on networks of radial basis functions is considered. The proposed algorithm is based on solving individual problems for each area with different properties of the medium and using a common error functional that takes into account errors at the interface between the media. This removes the restrictions on the use of radial basis functions and allows the use of radial basis functions with both unlimited and limited definition areas. We used the fast algorithm proposed by the authors for training networks of radial basis functions by the Levenberg-Marquardt method with analytical calculation of the Jacobi matrix. The algorithm makes it possible to reduce the number of iterations by several orders of magnitude compared to the gradient descent algorithm currently used and to obtain the accuracy of the solution, which is practically unattainable by the gradient descent algorithm. The results of solving the model problem showed the effectiveness of the proposed algorithm.

The use of live demonstrations in courses related to numerical problem solving combined with PowerPoint presentations is advocated in order to stimulate student interest and involvement and enhance learning. The proposed approach is demonstrated by solving a system of ordinary differential equations as a boundary value problem, utilizing finite difference approximations for solving partial differential equations, and investigating ill-conditioning in multiple linear regression. The educational benefits of this approach are discussed in detail. © 2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 17: 285–294, 2009; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/cae.20214

Solving equations in abstract spaces is important since many problems from diverse disciplines require it. The solutions of these equations cannot be obtained in a form closed. That difficulty forces us to develop ever improving iterative methods. In this paper we improve the applicability of such methods. Our technique is very general and can be used to expand the applicability of other methods. We use two methods of linear interpolation namely the Secant as well as the Kurchatov method. The investigation of Kurchatov&rsquo, s method is done under rather strict conditions. In this work, using the majorant principle of Kantorovich and our new idea of the restricted convergence domain, we present an improved semilocal convergence of these methods. We determine the quadratical order of convergence of the Kurchatov method and order 1 + 5 2 for the Secant method. We find improved a priori and a posteriori estimations of the method&rsquo, s error.

Physical models for equation solving typically lack feedback regarding their appropriate use. Such feedback is possible in virtual environments and could be implemented in hybrid models. Based on an epistemological analysis, this article presents a touch gesture as a way for users to signal they want to divide both sides of an equation and a design for feedback on the use of this so-called ‘division gesture’. The design is investigated by contrasting a case study, in which students used an app with the division gesture, with a preparatory study where students had to perform corresponding actions on physical manipulatives. This investigation revealed insight into feedback functions, steps of understanding dividing with this touch gesture and, furthermore, showed problems that students have with the boundary case where the dividend is 0. The study informs possible improvements of the design of the division gesture and of the overall learning environment. The results are reflected on, in order to illuminate known problems of learning how to solve linear equations, and theorized to contribute to the wider discussion around the design of digital and physical manipulatives, in particular the design of modes of interaction enabled by new technologies.

We present a transformation-system-based technique in the frame-work of string solving, by reformulating a classical combinatorics on words result, the Lemma of Levi. We further enrich the induced rules by simplification steps based on results from the combinatorial theory of word equations, as well as by the addition of linear length constraints. This transformation-system approach cannot solve all equations efficiently by itself. To improve the efficiency of our transformation-system approach we integrate existing successful string solvers, which are called based on several heuristics. The experimental evaluation we performed shows that integrating our technique as an inprocessing step improves in general the performance of existing solvers.

Proximity relations are binary fuzzy relations that satisfy reflexivity and symmetry properties, but are not transitive. They induce the notion of distance between function symbols, which is further extended to terms. Given two terms, we aim at bringing them “sufficiently close” to each other, by finding an appropriate substitution. We impose no extra restrictions on proximity relations, allowing a term in unification to be close to two terms that themselves are not close to each other. Our unification algorithm works in two phases: first reducing the equation solving problem to constraints over sets of function symbols, and then solving the obtained constraints. Termination, soundness and completeness of both algorithms are shown. The unification problem has finite minimal complete set of unifiers.

As industrial control technology continues to develop, modern industrial control is undergoing a transformation from manual control to automatic control. In this paper, we show how to evaluate and build machine learning models to predict the flow rate of the gas pipeline accurately. Compared with traditional practice by experts or rules, machine learning models rely little on the expertise of special fields and extensive physical mechanism analysis. Specifically, we devised a method that can automate the process of choosing suitable machine learning algorithms and their hyperparameters by automatically testing different machine learning algorithms on given data. Our proposed methods are used in choosing the appropriate learning algorithm and hyperparameters to build the model of the flow rate of the gas pipeline. Based on this, the model can be further used for control of the gas pipeline system. The experiments conducted on real industrial data show the feasibility of building accurate models with machine learning algorithms. The merits of our approach include (1) little dependence on the expertise of special fields and domain knowledge-based analysis; (2) easy to implement than physical models; (3) more robust to environment changes; (4) requiring much fewer computation resources when it is compared with physical models that call for complex equation solving. Moreover, our experiments also show that some simple yet powerful learning algorithms may outperform industrial control problems than those complex algorithms.

Boolean automata networks, genetic regulation networks, and metabolic networks are just a few examples of biological modeling by discrete dynamical systems (DDS). A major issue in modeling is the verification of the model against the experimental data or inducing the model under uncertainties in the data. Equipping finite discrete dynamical systems with an algebraic structure of commutative semiring provides a suitable context for hypothesis verification on the dynamics of DDS. Indeed, hypothesis on the systems can be translated into polynomial equations over DDS. Solutions to these equations provide the validation to the initial hypothesis. Unfortunately, finding solutions to general equations over DDS is undecidable. In this article, we want to push the envelop further by proposing a practical approach for some decidable cases in a suitable configuration that we call the Hypothesis Checking. We demonstrate that for many decidable equations all boils down to a "simpler" equation. However, the problem is not to decide if the simple equation has a solution, but to enumerate all the solutions in order to verify the hypothesis on the real and undecidable systems. We evaluate experimentally our approach and show that it has good scalability properties. 2012 ACM Subject Classification

It is known that for a stream cipher with state size less than 2.5 times the key size, it is possible to mount a Time-Memory-Data Trade-Off attack with an online complexity lower than the exhaustive key search. The search space is restricted by considering a fixed keystream prefix and deducing certain state bits by formulating equations. We show how by using SAT solving techniques one can automate this process of solving equations and obtain better parameters. This is demonstrated by mounting TMDTO attacks on ACORN v3 and Grain v1. We show that a TMDTO attack can be mounted on ACORN v3 with a preprocessing complexity 2171 and 2180 (without and with the help of a SAT solver) and the maximum of online time, memory and data complexities 2122 and 2120 respectively. For Grain v1, we show that it is possible to obtain parameters as T = 268.06, M = 264, D = 268 with a preprocessing complexity of 296. While our results do not refute any claim of the designers, these observations might be useful for further understanding of the ciphers.

This study examined the effects of students’ construction of computer-based educational games, using Scratch, on their mathematical equation-solving performance and their attitudes towards learning...

This study aims to identify the types of difficulties experienced by high school students in solving equations and trigonometric identities. The method used in this research is descriptive qualitative research method because researchers want to describe or describe the facts of students' difficulties in solving equations and trigonometric identities. The data collection technique in this study is by using respondents' ability tests and interviews. Based on the results of data analysis, there are three aspects of students 'difficulties in solving trigonometric equations and also there are three aspects of students' difficulties in solving trigonometric identity problems. The difficulties of students in solving trigonometric equations, namely the difficulty of students in deciphering the form of the problem, difficulty in factoring in the form of trigonometric quadratic equations, and difficulties using the basic trigonometric equations. Whereas, the difficulties of students in solving trigonometric identity problems, namely the difficulty of students applying general trigonometry formulas, difficulty describing each of the trigonometric comparison relationships, and difficulties in performing algebraic calculations/computation.

In this study, we aim to recover the interval hull of the set of feasible cost functions that can make uncertain observations optimal for a class of nonlinear constrained parametric optimization problems when all uncertainty and disturbances acting on observations or modeling are taken bounded but otherwise unknown. Fostering on inverse Karush–Kuhn–Tucker optimality conditions, we first state the solving equations as a constraint satisfaction problem, then show how to derive a safe overapproximation of the feasible solution set combining standard numerical tools and a posteriori validation with guaranteed methods based on interval analysis. The approach is evaluated on two well-tuned numerical examples: A discrete unicycle robot model and a planar elastica model, respectively.

The hygrothermal simulation of hygroscopic building materials is a real challenge, in terms of regulations and labelling, but also in decision-making. Today, we lack reference models for the hygrothermal behavior of a whole building. A scoping literature review is conducted to provide an overview of current state-of-the-art methods in order to address of these simulation methods. The most comprehensive studies are selected and examined in detail. These include physical models (White-box), which focus on solving equations that simulate the hygrothermal behavior of buildings, and data driven models, which involve implementing a prediction model using machine learning techniques (Black-box). On one hand, the white-box models are reviewed according to a two-category classification: CFD and Nodal approaches. On the other hand, the principal model used for black-box models is neural network models. The study highlights the need for a recognized method for hygrothermal simulations of hygroscopic envelopes. It provides a better understanding of the hygrothermal simulation, which helps to choose the most suitable tool or model. In addition, this review points out that there is limited application of data-driven methods to simulate the hygrothermal behavior of hygroscopic envelopes. This analysis study highlights future research gaps to overcome in order to stimulate data-driven building performance design.

A variety of strategies are used to construct algorithms for solving equations. However, higher order derivatives are usually assumed to calculate the convergence order. More importantly, bounds on error and uniqueness regions for the solution are also not derived. Therefore, the benefits of these algorithms are limited. We simply use the first derivative to tackle all these issues and study the ball analysis for two sixth order algorithms under the same set of conditions. In addition, we present a calculable ball comparison between these algorithms. In this manner, we enhance the utility of these algorithms. Our idea is very general. That is why it can also be used to extend other algorithms as well in the same way.