Applications of interval arithmetic in solving polynomial equations by Wu''''s elimination method

Wu's elimination method is an important method for solving multivariate poly- nomial equations.In this paper,we apply interval arithmetic to Wu's method and convert the problem of solving polynomial equations into that of solving interval polynomial equa- tions.Parallel results such as zero-decomposition theorem are obtained for interval poly- nomial equations.The advantages of the new approach are two-folds:First,the problem of the numerical instability arisen from floating-point arithmetic is largely overcome.Second, the low efficiency of the algorithm caused by large intermediate coefficients introduced by exact compaction is dramatically improved.Some examples are provided to illustrate the effectiveness of the proposed algorithm.     

On solving systems of linear algebraic equations by Gibbs’s method

The paper presents an algorithm of approximate solution of a system of linear algebraic equations by the Monte Carlo method superimposed with ideas of simulating Gibbs and Metropolis fields. A solution in the form of a Neumann series is evaluated, the whole vector of solutions is obtained. The dimension of a system may be quite large. Formulas for evaluating the covariance matrix of a single simulation run are given. The method of solution is conceptually linked to the method put forward in a 2009 paper by Ermakov and Rukavishnikova. Examples of 3 × 3 and 100 × 100 systems are considered to compare the accuracy of approximation for the method proposed, for Ermakov and Rukavishnikova’s method and for the classical Monte Carlo method, which consists in consecutive estimation of the components of an unknown vector.     

On solving equations of algebraic sum of equal powers

It is well known that a system of equations of sum of equal powers can be converted to an algebraic equation of higher degree via Newton's identities. This is the Viete-Newton theorem. This work reports the generalizations of the Viete-Newton theorem to a system of equations of algebraic sum of equal powers. By exploiting some facts from algebra and combinatorics, it is shown that a system of equations of algebraic sum of equal powers can be converted in a closed form to two algebraic equations, whose degree sum equals the number of unknowns of the system of equations of algebraic sum of equal powers.     

Role of operator semirings in characterizing Γ-semirings in terms of fuzzy subsets

The operator semirings of a ??-semiring have been brought into use to study ??-semiring in terms of fuzzy subsets. This is accomplished by obtaining various relationships between the set of all fuzzy ideals of a ??-semiring and the set of all fuzzy ideals of its left operator semiring such as lattice isomorphism between the sets of fuzzy ideals of a ??-semiring and its operator semirings.     

Solvability of a system of equations for the Fourier–Chebyshev coefficients when solving ordinary differential equations by the Chebyshev series method

A solvability theorem for a system of equations with respect to approximate values of Fourier–Chebyshev coefficients is formulated. This theorem is a theoretical justification for numerical solution of ordinary differential equations using Chebyshev series.     

α-contractions and attractors for dissipative semilinear hyperbolic equations and systems

Summary In this paper we discuss the existence of compact attractor for the abstract semilinear evolution equation u=Au+f(t, u); the results are applied to damped partial differential equations of hyperbolic type. Our approach is a combination of Liapunov method with the theory of -eontractions.     

Optimization on class of operator equations in the probabilistic case setting

In this paper, we introduce a problem of the optimization of approximate solutions of operator equations in the probabilistic case setting, and prove a general result which connects the relation between the optimal approximation order of operator equations with the asymptotic order of the probabilistic width. Moreover, using this result, we determine the exact orders on the optimal approximate solutions of multivariate Preldholm integral equations of the second kind with the kernels belonging to the multivariate Sobolev class with the mixed derivative in the probabilistic case setting.     

On Γ-limit classes of perturbations defined by integral conditions

For the coefficients of linear differential systems, we consider classes of piecewise continuous perturbations that are infinitesimal in mean on the positive half-line with some positive piecewise continuous weight belonging to a given set. We obtain sufficient conditions for such a class to be Γ-limit, i.e., to admit the computation of a reachable upper bound of the exponents of linear differential systems with perturbations in that class by a formula similar to the well-known formulas for the central and exponential exponents.     

The nature and development of experts’ strategy flexibility for solving equations

Largely absent from the emerging literature on flexibility is a consideration of experts’ flexibility. Do experts exhibit strategy flexibility, as one might assume? If so, how do experts perceive that this capacity developed in themselves? Do experts feel that flexibility is an important instructional outcome in school mathematics? In this paper, we describe results from several interviews with experts to explore strategy flexibility for solving equations. We conducted interviews with eight content experts, where we asked a number of questions about flexibility and also engaged the experts in problem solving. Our analysis indicates that the experts that were interviewed did exhibit strategy flexibility in the domain of linear equation solving, but they did not consistently select the most efficient method for solving a given equation. However, regardless of whether these experts used the best method on a given problem, they nevertheless showed an awareness of and an appreciation of efficient and elegant problem solutions. The experts that we spoke to were capable of making subtle judgments about the most appropriate strategy for a given problem, based on factors including mental and rapid testing of strategies, the problem solver’s goals (e.g., efficiency, error-free execution, elegance) and familiarity with a given problem type. Implications for future research on flexibility and on mathematics instruction are discussed.     

Green–Samoilenko operator in the theory of invariant sets of nonlinear differential equations

We establish conditions for the existence of an invariant set of the system of differential equations

Inverse problem for equations of mixed type with Lavrent’ev-Bitsadze operator

For the equation of mixed elliptic-hyperbolic type $u_{xx} + (\operatorname{sgn} y)u_{yy} - b^2 u = f(x)$ in a rectangular domainD = {(x, y) | 0 < x < 1, ?α < y < β}, where α, β, and b are given positive numbers, we study the problem with boundary conditions $\begin{gathered} u(0,y) = u(1,y) = 0, - \alpha \leqslant y \leqslant \beta , \hfill \\ u(x,\beta ) = \phi (x),u(x,\alpha ) = \psi (x),u_y (x, - \alpha ) = g(x),0 \leqslant x \leqslant 1. \hfill \\ \end{gathered} $ . We establish a criterion for the uniqueness of the solution, which is constructed as the sum of the series in eigenfunctions of the corresponding eigenvalue problem and prove the stability of the solution.     

Construction of new generalizations of Wynn’s epsilon and rho algorithm by solving finite difference equations in the transformation order

We construct new sequence transformations based on Wynn’s epsilon and rho algorithms. The recursions of the new algorithms include the recursions of Wynn’s epsilon and rho algorithm and of Osada’s generalized rho algorithm as special cases. We demonstrate the performance of our algorithms numerically by applying them to some linearly and logarithmically convergent sequences as well as some divergent series.

     

The characterization of Γ-modules in terms of fuzzy soft Γ-submodules

In this paper, we focus on combining the theories of fuzzy soft sets with Γ-modules, and establishing a new framework for fuzzy soft Γ-submodules. The main contributions of the paper are 3-fold. First, we present the concepts of (R, S)-bi-Γ-submodules, quasi-Γ-submodules and regular Γ-modules. Meanwhile, some illustrative examples are given to show the rationality of the definitions introduced in this paper. Second, several new kinds of generalized fuzzy soft Γ-submodules are proposed, and related properties and mutual relationships are also investigated. Third, we discover some intrinsic connections between the generalized fuzzy soft Γ-submodules presented in this paper and crisp Γ-submodules, and describe the relationships between regular Γ-modules and the generalized fuzzy soft Γ-submodules presented in this paper.     

Monotone iterative methods for nonlinear operator equations ∗)

We give general affine invariant conditions for the monotone convergence of a class of iterative procedures for solving nonlinear operator equations. The theorems obtained in the paper generalize and unify many known results and provide a convenient framework for studying new iterative procedures     

Homological determination of Γ-modules

It is proved that if M is a profinitely generated -module which is free as a module over the ring of p-adic integers, then M is determined up to free direct factors by its homology. This result generalizes the theorem on homological determinacy of p-adic representations of a cyclic group [Ref. Zh. Mat., 3A, 318 (1971)].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeieniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 104–126, 1976.     

Legendre wavelet – quasilinearization technique for solving q-difference equations

Recently, various fixed point theorems have been used to prove the existence and uniqueness of the solutions for q-difference equations. In this paper, we obtain the existence and uniqueness theorems for a q-initial and a q-boundary value problem using the classical Newton’s method. Making use of the main theorems, a Legendre wavelet technique has been proposed to solve the q-difference equations numerically. The numerical simulation shows that the proposed scheme produces higher accuracy and is very straightforward to apply.