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In this paper, we study the behavior of solutions of a semilinear elliptic equation in the exterior of a compact set as $|x|\backslash to\; \backslash nomathbreak\; \backslash infty$. Such equations were considered by many authors (for example, Kondrat'ev, Landis, Oleinik, Veron, etc.). In the present paper, we study the case in which in the equation contains lower terms. The coefficients of the lower terms are arbitrary bounded measurable functions. It is shown that the solutions of the equation tend to zero as $|x|\backslash to\; \backslash nomathbreak\; \backslash infty$. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new system of generalized resolvent equations with corresponding system of variational inclusions in uniformly smooth Banach spaces. We establish an equivalence relation between system of generalized resolvent equations and system of variational inclusions. The iterative algorithms for finding the approximate solutions of system of generalized resolvent equations are proposed. The convergence of approximate solutions of system of generalized resolvent equations obtained by the proposed iterative algorithm is also studied. [ABSTRACT FROM AUTHOR]

The mode-I weight function proposed by Oore and Burns is an efficient and empirical mathematical relationship capable of estimating the Stress Intensity Factor in the presence of a generic planar flaw. In the case of penny shaped cracks or tunnel cracks the Oore-Burns equation provides the classic analytical expressions. In this paper, the authors prove analytically that the Oore-Burns equation is not the weight function for an ellipse in an infinite solid under traction loading. The absolute novelty is given by the analytic evaluation between the Oore-Burns integral and the Irwin solution in terms of an algebraic expression, which is independent from any numerical procedures for an ellipse close to a circle. [ABSTRACT FROM AUTHOR]

The optimal control of the solutions of the initial-finish problem for Sobolev-type equations is studied. The abstract results obtained in the paper are applied to the linear Hoff model on graphs. [ABSTRACT FROM AUTHOR]

Abstract  In this paper, we consider the action of (2, q) on the finite projective line $${\mathbb{F}}_q\cup\{\infty\}$$ for q ≡ 1 (mod 4) and construct several infinite families of simple 3-designs which admit PSL(2, q) as an automorphism group. Some of the designs are also minimal. We also indicate a general outline to obtain some other algebraic constructions of simple 3-designs. [ABSTRACT FROM AUTHOR]

The progress of Mathematics during the nineteenth century was characterised both by an enormous acquisition of new knowledge and by the attempts to introduce rigour in reasoning patterns and mathematical writing. Cauchy’s presentation of Mathematical Analysis was not immediately accepted, and many writers, though aware of that new style, did not use it in their own mathematical production. This paper is devoted to an episode of this sort that took place in Spain during the first half of the century: It deals with the presentation of a method for numerically solving algebraic equations by José Mariano Vallejo, a late Spanish follower of the Enlightenment ideas, politician, writer, and mathematician who published it in the fourth (1840) edition of his book Compendio de Matemáticas Puras y Mistas, claiming to have discovered it on his own. Vallejo’s main achievement was to write down the whole procedure in a very careful way taking into account the different types of roots, although he paid little attention to questions such as convergence checks and the fulfilment of the hypotheses of Rolle’s Theorem. For sure this lack of mathematical care prevented Vallejo to occupy a place among the forerunners of Computational Algebra. [ABSTRACT FROM AUTHOR]

The paper is concerned with the problem of inverting block matrices to which the well-known Frobenius— Schur formulas are not applicable. These can be square matrices with four noninvertible square or rectangular blocks as well as square or rectangular matrices with two blocks. With regard to rectangular matrices, the results obtained are a further step in the development of the canonization method, which is used for solving arbitrary matrix equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the Darboux transformation of the Darboux-Treibich-Verdier equation. On the basis of this transformation, we construct a generalization of the Darboux transformation to the case of the Heun equation and to other linear ordinary differential equations of second order. Examples are given. [ABSTRACT FROM AUTHOR]

This paper is devoted to the proof of the existence of a solution of the Cauchy problem for a system of equations of ultraparabolic type. [ABSTRACT FROM AUTHOR]

For the linear stationary dynamic systems, the necessary and sufficient conditions for output invariance to arbitrary external perturbations were substantiated by solving the matrix equations with the help of the canonization method. The conditions obtained were used to design controllers maintaining this invariance. The results were extended to the problem of parametric invariance. The paper was confined to the linear stationary systems without delays and constraints on control. An illustrating example was presented. [ABSTRACT FROM AUTHOR]

In this paper we show that all totally real minimal tori in correspond to doubly periodic finite gap solutions of the Tzitzéica equation Using the results on the Tzitzéica equation in integrable systems theory, we describe explicitly these tori by Prym-theta functions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the connection between the relative Dixmier property and the complete boundedness of bounded modular maps. For an inclusion of II 1-factors with the relative Dixmier property, we will show that is properly infinite where p is the GNS representation associated with a singular state on . We show that a bounded -multimodular bilinear map is completely bounded and that a bounded -bimodule map is completely bounded. Finally, we prove the row boundedness of a normal -bimodule map where is hyperfinite without the relative Dixmier property. [ABSTRACT FROM AUTHOR]

The paper deals with the three-dimensional dynamic problem for an elliptic crack interacting with a normally incident harmonic compression–expansion wave, considering the contact interaction of the crack faces. An asymmetric solution is obtained using an iteration algorithm developed earlier. Numerical results are presented [ABSTRACT FROM AUTHOR]

The paper studies the dependence of edge effects in a laminated composite of regular structure with a periodic system of symmetric interfacial cracks on the mechanical characteristics of the composite components. Loading the composite induces constant strain in the reinforcement direction. The problem is solved in exact formulation using the linear elastic equations for piecewise-homogeneous media and a criterion for quantitative evaluation of edge effects. An approximate solution of the problem is found by the mesh approach. A difference scheme for the mixed problem under consideration is derived. The edge-effect zones for normal stresses are constructed. The maximum values of the normal stress concentration factors are determined [ABSTRACT FROM AUTHOR]

In this paper, we study singular parabolic equations with “quadratic nonlinearity.” We consider questions of unique solvability, of stabilization of solutions at infinity, and of the presence of positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, we employ some new techniques to study the existence of positive periodic solution of n-species neutral delay system N_{i}^{\prime}(t)=N_i(t)\left[ a_i(t)-\sum_{j=1}^n \beta_{ij}(t)N_j(t)-\sum_{j=1}^n b_{ij}(t)N_j(t-\tau_{ij}(t))-\sum_{j=1}^n c_{ij}(t)N_{j}^\prime(t-\tau_{ij}(t))\r/math>As a corollary, we answer an open problem proposed by Y. Kuang. [ABSTRACT FROM AUTHOR]

In this paper, we study necessary and sufficient conditions for functions to be approximated uniformly on plane compact sets by polynomial solutions to second-order homogeneous elliptic equations with constant coefficients. Sufficient conditions for approximability are of reductive character, i.e., the possibility of approximating on some (simpler) parts of the compact set implies approximability on the entire compact set. [ABSTRACT FROM AUTHOR]

We consider the problem of determining an optimal driving strategy in a train control problem with a generalised equation of motion. We assume that the journey must be completed within a given time and seek a strategy that minimises fuel consumption. On the one hand we consider the case where continuous control can be used and on the other hand we consider the case where only discrete control is available. We pay particular attention to a unified development of the two cases. For the continuous control problem we use the Pontryagín principle to find necessary conditions on an optimal strategy and show that these conditions yield key equations that determine the optimal switching points. In the discrete control problem, which is the typical situation with diesel-electric locomotives, we show that for each fixed control sequence the cost of fuel can be minimised by finding the optimal switching times. The corresponding strategies are called strategies of optimal type and in this case we use the Kuhn-Tucker equations to find key equations that determine the optimal switching times. We note that the strategies of optimal type can be used to approximate as closely as we please the optimal strategy obtained using continuous control and we present two new derivations of the key equations. We illustrate our general remarks by reference to a typical train control problem. [ABSTRACT FROM AUTHOR]

Formal mathematics is a paragon of abstractness. It thus seems natural to assume that the mathematical expert should rely more on symbolic or conceptual processes, and less on perception and action. We argue instead that mathematical proficiency relies on perceptual systems that have been retrained to implement mathematical skills. Specifically, we investigated whether the visual system-in particular, object-based attention-is retrained so that parsing algebraic expressions and evaluating algebraic validity are accomplished by visual processing. Object-based attention occurs when the visual system organizes the world into discrete objects, which then guide the deployment of attention. One classic signature of object-based attention is better perceptual discrimination within, rather than between, visual objects. The current study reports that object-based attention occurs not only for simple shapes but also for symbolic mathematical elements within algebraic expressions-but only among individuals who have mastered the hierarchical syntax of algebra. Moreover, among these individuals, increased object-based attention within algebraic expressions is associated with a better ability to evaluate algebraic validity. These results suggest that, in mastering the rules of algebra, people retrain their visual system to represent and evaluate abstract mathematical structure. We thus argue that algebraic expertise involves the regimentation and reuse of evolutionarily ancient perceptual processes. Our findings implicate the visual system as central to learning and reasoning in mathematics, leading us to favor educational approaches to mathematics and related STEM fields that encourage students to adapt, not abandon, their use of perception. [ABSTRACT FROM AUTHOR]

For the problem of optimal control of the coefficients of the quasilinear elliptical equation, correctness of formulation was considered, and the necessary optimality conditions were established. [ABSTRACT FROM AUTHOR]

Every orthonomic system of partial differential equations is known to possess a finite number of integrability conditions sufficient to ensure the validity of them all. Here we show that a redundancy-free sufficient set of integrability conditions can be constructed in a time proportional to the number of equations cubed. [ABSTRACT FROM AUTHOR]

Abstract  We give a continuation-based implementation of rewriting for systems in the LCF tradition. These systems must construct explicit proofs of equations when rewriting, and currently do so in a way that can be very space-inefficient. An explicit representation of continuations improves performance on large terms, and on long-running computations. [ABSTRACT FROM AUTHOR]

The notions of (ϵ, ϵ∨ q)-fuzzy BCI-positive implicative (resp., BCI-implicative, BCI-commutative) ideals in BCI-algebras are introduced, and related properties are investigated. Some characterization theorems of these generalized fuzzy ideals are derived. The relationship among these generalized fuzzy ideals of BCI-algebras is considered. We also consider the concept of implication-based fuzzy implicative ideals in BCI-algebras. In particular, we discuss the implication operators in Lukasiewicz system of continuous-valued logic. [ABSTRACT FROM AUTHOR]

Abstract  Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein. [ABSTRACT FROM AUTHOR]

Consideration is given to the problem of synthesis of robust regulators of a desired structure that ensure the motion of a system on bounded neighborhoods of a set. The solution to the problem is based on the representation of motion in a connected phase plane and is reduced to studying the solvability of algebraic equations. [ABSTRACT FROM AUTHOR]

Abstract  A Kirchhoff-type equation describing the transversal vibrations of a beam is considered. The beam is clamped to a rigid base at one part of its edge and free at the remaining part. On the free part, it is subject to a feedback involving fractional derivatives instead of the classical velocity of the deflection and angular velocity. In presence of an external nonlinear source we prove that solutions blow up at a finite time. [ABSTRACT FROM AUTHOR]

The article examines the Poincaré inequality and the reverse Hölder inequality for the solution to the p-harmonic type system. The conjugate p-harmonic equation and the conjugate A-harmonic equation are the specific p-harmonic type system. A-harmonic equation is an important generalization of the p-harmonic equation and the p-harmonic equation is a natural extension of the usual Laplace equation. The reverse Hölder inequalities have been frequently used in analysis of partial differential equations and the theory of elasticity.

A finite-dimensional nondegenerate system of algebro-differential equations with discontinuances is considered. An equivalent system of discontinuous differential equations is constructed. Conditions of right-hand uniqueness of solution and of convergence of numerical methods are given. [ABSTRACT FROM AUTHOR]

The article presents a study on the mathematical equation of two-dimensional linear differential systems with deviating arguments using Kneser type solutions. It presents an equation with positive coefficient and two linearly independent solutions. The article also evaluates the existence and uniqueness of these types of solutions. It proves that Kneser type solution is equivalent to nonvanishing of the Wronskian fundamental system and positivity of Green's function of the problem.

The article presents a research on the iterative methods for generalized von Foerster equations with functional dependence. It examines the meeting of a natural iterative method to the exact solution of a differential-functional von Foerster-type equation that notes a single population depending on its total size, past time and states densities. It also assumes either Perron comparison conditions and some monotonicity. It also investigates on the natural conditions which ensures convergence of iterative methods.

The article presents a research on the general summability factor theorems. It aims to obtain ample and needed conditions for a series that is summable of an order by a triangular matrix method. It discusses the equation which starts with the substitution of a lower triangular matrix and the analysis of a series that is summable to a particular value. It also cites two theorems which proves the satisfaction of the condition of the equation with the substitution of constants in a sequence of positive constants.

A Simplified version of rate equations for enzyme-catalyzed reactions has been developed in which the rate equation is analyzed with two different mechanisms: ordered bi–ter and Ping-Pong bi–bi mechanisms. This procedure is able to develop the rate expressions accurately. Random sequential order mechanism can be effectively explained by this approach. [ABSTRACT FROM AUTHOR]

We study quasiperiodic (finite-gap) solutions of the Volterra chain satisfying an integrable boundary condition on the semiaxis. From the set of general finite-gap solutions, only those corresponding to the boundary-value problem are singled out, the relevant condition being expressed as a system of algebraic equations. [ABSTRACT FROM AUTHOR]

Using the structural joining method, we construct a uniformly valid explicit asymptotics of the solution of a perturbed model Lighthill equation with a regular singular point. [ABSTRACT FROM AUTHOR]

We consider model three-dimensional wave nonlinear equations of Sobolev type with cubic sources and, first of all, model three-dimensional equations of Benjamin-Bona-Mahony type and of Rosenau type with model cubic sources. We also study an essentially three-dimensional nonlinear equation of spin waves with cubic source. For these equations, we consider the first initial boundary-value problem in a bounded domain with smooth boundary. We prove local solvability in the strong generalized sense and, for an equation of Benjamin-Bona-Mahony type with source, we prove the unique solvability of the “weakened” solution. We obtain sufficient conditions for the “destruction” of the solutions to the problems under consideration. These conditions mean that the value of the initial perturbation in the norms of some Banach spaces is “large.” Finally, for an equation of Benjamin-Bona-Mahony type, we prove that the “weakened” solution fails in a finite time. [ABSTRACT FROM AUTHOR]

We construct an algorithm that tests a system of recurrence equations on a tree for the existence of a nontrivial solution and computes it. [ABSTRACT FROM AUTHOR]

A linear parabolic problem in a separable Hilbert space is solved approximately by the projection-difference method. The problem is discretized in space by the Galerkin method orientated towards finite-dimensional subspaces of finite-element type and in time by using the implicit Euler method and the modified Crank-Nicolson scheme. We establish uniform (with respect to the time grid) and mean-square (in space) error estimates for the approximate solutions. These estimates characterize the rate of convergence of errors to zero with respect to both the time and space variables. [ABSTRACT FROM AUTHOR]

We prove the following Whitney estimate. Given 0

d, and each function f \in Lp(O), Er-1(f,O)p \le C(d,r,p)?r(f, diam(O))p, where Er-1(f,O)p is the degree of approximation by polynomials of total degree, r - 1, and ?r(f,·)p is the modulus of smoothness of order r. Estimates like this can be found in the literature but with constants that depend in an essential way on the geometry of the domain, in particular, the domain is assumed to be a Lipschitz domain and the above constant C depends on the minimal head-angle of the cones associated with the boundary. The estimates we obtain allow us to extend to the multivariate case, the results on bivariate Skinny B-spaces of Karaivanov and Petrushev on characterizing nonlinear approximation from nested triangulations. In a sense, our results were anticipated by Karaivanov and Petrushev. [ABSTRACT FROM AUTHOR]

We study a Turán extremal problem on the largest mean value of a $1$-periodic even function with nonnegative Fourier coefficients, fixed value at zero, and support on a closed interval $[-h,h]$,

Sufficient conditions of technical stability are established for nonstationary automatic-control systems of variable structure. The case is considered where for appropriate controls and for all admissible initial perturbations from a predefined measurable set of initial states, the nonstationary parameters of a process under external actions vary within given ranges. The control process depends on the mismatch signal and its derivatives up to an order that is less by one than the order of the equations of motion. The associated system of differential equations contains discontinuous time-dependent coefficients. A relationship is established between the eigenvalues of the quadratic forms of the given Lyapunov functions and the technical-stability criteria. It is shown that the technical-stability conditions do not depend on the sliding-mode conditions for variable-structure control processes [ABSTRACT FROM AUTHOR]

An approach developed earlier to solve two-dimensional static problems for noncircular cylindrical shells is used to analyze the effect of the spatial frequency and amplitude of corrugation on the stress–strain state of shells. The results of the analysis are presented in the form of plots and tables [ABSTRACT FROM AUTHOR]

Two-dimensional magnetoelastic equations are derived. A technique is proposed to solve a boundary-value problem for an annular plate of variable thickness. A numerical example is considered [ABSTRACT FROM AUTHOR]

Galin's method is used to derive equations that relate the basic parameters of the problem on wedging an orthotropic space by a rigid rectangular wedge. To eliminate the stress singularity at the wedging crack tips, a Leonov–Panasyuk–Dugdale prefracture zone is assumed to exist at the crack front. The equations are derived using the COD criterion [ABSTRACT FROM AUTHOR]

We investigate tournaments that are projective in the variety that they generate, and free algebras over partial tournaments in that variety. We prove that the variety determined by three-variable equations of tournaments is not locally finite. We also construct infinitely many finite, pairwise incomparable simple tournaments. [ABSTRACT FROM AUTHOR]

The two‐dimensional transient problem of a floating body having a rectangular shape in a fluid layer of finite depth is considered. Vertical displacements of the body are specified. The problem is studied within the framework of the linear theory of potential ideal incompressible flow. The fluid flow equations reduce to an infinite system of Volterra integral equations of the second kind by the method of decomposition of the flow region. The system obtained is studied and solved numerically by the reduction method. A method of solving the problem for the flow velocity potential is proposed. The distribution of the hydrodynamic pressure and force acting on the body is determined. [ABSTRACT FROM AUTHOR]

A system of nonlinear equations proposed by E. Reissner for the problem of elastic pure bending of thin‐walled curvilinear tubes is considered. The formulation of the problem is refined, and the numerical solution of the equations obtained is compared with the finite‐element results. [ABSTRACT FROM AUTHOR]

For ten models of the isothermal behavior of materials, the solutions of boundary‐value problems are studied for five types of the experimentally reproducible uniform stress–strain state with unchanged directions of the principal axes. It is found that, for three models, the governing equations are similar to the relations of Hooke's law and valid within the same range of the ratio between the shear and bulk moduli. In these models, the specific strain energy can be represented as a sum of the energies due to changes in volume and shape. The ranges where the other three known models exhibit incorrect behavior are determined. [ABSTRACT FROM AUTHOR]

May [18] developed an algebraic choice model of pairwise preference comparison in which subjects respond precisely to ordinal information on attributes of comparison. This study considers a probabilistic choice model variation of May's model, in which subjects respond with various degrees of precision in comparison to May's model. This precision can be viewed as an indirect measure of the subject's level of perception of the attributes of comparison. The purpose of the study is to examine the expected likelihood with which subjects will have transitive responses, as the degree of precision is varied. Closed form representations are obtained for the expected likelihood of transitivity for three alternatives for each different level of precision. Results indicate that a relatively small change in this precision can lead to substantial changes in the expected likelihood of transitivity. [ABSTRACT FROM AUTHOR]

This note presents corrections to some misstated equations in Hespeler (Comput Econ, 31, 207–223pp). [ABSTRACT FROM AUTHOR]