Discontinuity Waves, Shock Formation and Critical Temperature in Crystals

The propagation of discontinuity waves in a rigid heat conductor at low temperatures is studied by using a generalized non-linear Maxwell–Cattaneo equation developed in the framework of extended thermodynamics. The critical time (i.e., the instant in which a shock wave formation occurs) is evaluated in both cases of infinite and finite heat conductivity. The critical temperature θ̃, pointed out in our previous papers concerning the propagation of shock and simple waves, once more plays an important role: in fact, now it determines two different regimes for the wave propagation and this phenomenon, from a mathematical point of view, is related to the loss of the genuine non-linearity when θ = θ̃. In the last sections some numerical results are given and a brief analysis about the evolution of a possible initial wave profile is performed.     

Exp-function method for constructing exact solutions of Sharma–Tasso–Olver equation

In this paper we use the Exp-function method for the analytic treatment of Sharma–Tasso–Olver equation. New solitonary solutions are formally derived. Change of parameters, which drastically changes the characteristics of the equations, is examined. It is shown that the Exp-function method provides a powerful mathematical tool for solving high-dimensional nonlinear evolutions in mathematical physics. The proposed schemes are reliable and manageable.     

Solitons and other solutions to a new coupled nonlinear Schrodinger type equation

In this paper, the first integral method combined with Liu's theorem is applied to integrate a new coupled nonlinear Schrodinger type equation. Using this combination, more new exact traveling wave solutions are obtained for the considered equation using ideas from the theory of commutative algebra. In addition, more solutions are also obtained via the application of semi-inverse variational principle due to Ji-Huan He. The used approaches with the help of symbolic computations via Mathematica 9, may provide a straightforward effective and powerful mathematical tools for solving nonlinear partial differential equations in mathematical physics.     

Various definitions of the derivative in mathematical programming

This paper introduces seven derivatives in mathematical programming in locally convex topological vector spaces. All these derivatives have been known in various fields of mathematical sciences but they have never been used before in mathematical programming. The weakest of the seven derivatives is the compact derivative of Gil de Lamadrid and Sova. The derivative used by Neustadt in optimization theory is stronger than the compact derivative and it is equivalent to the derivative introduced by Michal and Bastiani. The main results of the paper show that the optimality conditions of both Lagrange—Kuhn—Tucker type and Caratheodory—John type hold for compactly differentiable functions. In the case of finite-dimensional spaces all these seven derivatives are equivalent to the Fréchet derivative.Research partly supported by the National Research Council of Canada. This paper was presented at the VIIIth International Symposium on Mathematical Programming held at Stanford University, August 27–31, 1973.     

How powerful is continuous nonlinear information for linear problems?

There are many papers dealing with the approximate solution of linear problems where only partial information is available. Two types of information have been considered: linear and discontinuous nonlinear. In particular, we know that discontinuous nonlinear information is far more powerful than linear information. In this paper we study continuous nonlinear information for linear problems, and we prove that:

• -it is no more powerful than linear information in the worst case setting,

• -it is much more powerful than linear information in the average case setting.

     

Exact travelling wave solutions for the generalized nonlinear Schrödinger (GNLS) equation with a source by Extended tanh–coth, sine–cosine and Exp-Function methods

The capability of Extended tanh–coth, sine–cosine and Exp-Function methods as alternative approaches to obtain the analytic solution of different types of applied differential equations in engineering mathematics has been revealed. In this study, the generalized nonlinear Schrödinger (GNLS) equation is solved by three different methods. To obtain the single-soliton solutions for the equation, the Extended tanh–coth and sine–cosine methods are used. Furthermore, for this nonlinear evolution equation the Exp-Function method is applied to derive various travelling wave solution. Results show that while the first two procedures easily provide a concise solution, the Exp-Function method provides a powerful mathematical means for solving nonlinear evolution equations in mathematical physics.     

Peano's concept of number

Giuseppe Peano's development of the real number system from his postulates for the natural numbers and some of his views on definitions in mathematics are presented in order to clarify his concept of number. They show that his use of the axiomatic method was intended to make mathematical theory clearer, more precise, and easier to learn. They further reveal some of his reasons for not accepting the contemporary “philosophies” of logicism and formalism, thus showing that he never tried to found mathematics on anything beyond our experience of the material world.     

New exact Jacobi elliptic functions solutions for the generalized coupled Hirota-Satsuma KdV system

In this paper, an generalized Jacobi elliptic functions expansion method with computerized symbolic computation is used for constructing more new exact Jacobi elliptic functions solutions of the generalized coupled Hirota-Satsuma KdV system. As a result, eight families of new doubly periodic solutions are obtained by using this method, some of these solutions are degenerated to solitary wave solutions and triangular functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the applied method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.     

Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine-cosine method

In this paper, we establish exact solutions for (2 + 1)-dimensional nonlinear evolution equations. The sine-cosine method is used to construct exact periodic and soliton solutions of (2 + 1)-dimensional nonlinear evolution equations. Many new families of exact traveling wave solutions of the (2 + 1)-dimensional Boussinesq, breaking soliton and BKP equations are successfully obtained. These solutions may be important of significance for the explanation of some practical physical problems. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.     

Unique solvability of the inverse problem of impulse electromagnetic sounding

The article considers the inverse problem of impulse electromagnetic sounding with a field source in the form of a powerful one-time pulse. Unique solvability is proved for the inverse problem involving simultaneous determination of both the equation coefficient and the boundary condition in the boundary-value problem for an equation of parabolic type.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 18–21, 1993.     

Mathematical model for the novel coronavirus (2019-nCOV) with clinical data using fractional operator

Coronavirus infection (COVID-19) is a considerably dangerous disease with a high demise rate around the world. There is no known vaccination or medicine until our time because the unknown aspects of the virus are more significant than our theoretical and experimental knowledge. One of the most effective strategies for comprehending and controlling the spread of this epidemic is to model it using a powerful mathematical model. However, mathematical modeling with a fractional operator can provide explanations for the disease's possibility and severity. Accordingly, basic information will be provided to identify the kind of measure and intrusion that will be required to control the disease's progress. In this study, we propose using a fractional-order SEIARPQ model with the Caputo sense to model the coronavirus (COVID-19) pandemic, which has never been done before in the literature. The stability analysis, existence, uniqueness theorems, and numerical solutions of such a model are displayed. All results were numerically simulated using MATLAB programming. The current study supports the applicability and influence of fractional operators on real-world problems.     

Explicit solutions of (2 + 1)-dimensional nonlinear KP-BBM equation by using Exp-function method

This paper is devoted to studying the (2 + 1)-dimensional KP-BBM wave equation. Exp-function method is used to conduct the analysis. The generalized solitary solutions, periodic solutions and other exact solutions for the (2 + 1)-dimensional KP-BBM wave equation are obtained via this method with the aid of symbolic computational system. It is also shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.     

Scheduling jobs with time-resource tradeoff via nonlinear programming

We consider a scheduling problem where the processing time of any job is dependent on the usage of a discrete renewable resource, e.g. personnel. An amount of k units of that resource can be allocated to the jobs at any time, and the more of that resource is allocated to a job, the smaller its processing time. The objective is to find a resource allocation and a schedule that minimizes the makespan. We explicitly allow for succinctly encodable time-resource tradeoff functions, which calls for mathematical programming techniques other than those that have been used before. Utilizing a (nonlinear) integer mathematical program, we obtain the first polynomial time approximation algorithm for the scheduling problem, with performance bound (3+ε) for any ε>0. Our approach relies on a fully polynomial time approximation scheme to solve the nonlinear mathematical programming relaxation. We also derive lower bounds for the approximation.     

A note on the uniformly most powerful tests in the presence of nuisance parameters

When a testing problem has nuisance parameters, the uniformly most powerful (UMP) tests do not generally exist. Exceptional examples were given by Dubey (1962, Skand. Aktuarietidskr., 45, 25–38; 1963, Skand. Aktuarietidskr., 46, 1–24) and Takeuchi (1968, Ann. Math. Statist., 40, 1838–1839) for the exponential distributions. What is essential for proving the existence of UMP tests lies in a special relationship between null hypothesis and the alternative. Assuming a similar relationship between them, a similar kind of result can be shown under more general situation.     

Coderivatives in parametric optimization

We consider parametric families of constrained problems in mathematical programming and conduct a local sensitivity analysis for multivalued solution maps. Coderivatives of set-valued mappings are our basic tool to analyze the parametric sensitivity of either stationary points or stationary point-multiplier pairs associated with parameterized optimization problems. An implicit mapping theorem for coderivatives is one key to this analysis for either of these objects, and in addition, a partial coderivative rule is essential for the analysis of stationary points. We develop general results along both of these lines and apply them to study the parametric sensitivity of stationary points alone, as well as stationary point-multiplier pairs. Estimates are computed for the coderivative of the stationary point multifunction associated with a general parametric optimization model, and these estimates are refined and augmented by estimates for the coderivative of the stationary point-multiplier multifunction in the case when the constraints are representable in a special composite form. When combined with existing coderivative formulas, our estimates are entirely computable in terms of the original data of the problem. Key words.parametric optimization – variational analysis – sensitivity – Lipschitzian stability – generalized differentiation – coderivativesThis research was partly supported by the National Science Foundation under grant DMS-0072179.     

Formal and precise derivation of the Green functions for a simple potential

In formal scattering theory, the Green functions are obtained as solutions of a distributional equation. In this paper, we use the Sturm–Liouville theory to compute the Green functions within a rigorous mathematical theory. We shall show that both the Sturm–Liouville theory and the formal treatment yield the same Green functions. We shall also show how the analyticity of the Green functions as functions of the energy keeps track of the so-called “incoming” and “outgoing” boundary conditions.     

On the joint distribution of studentized order statistics

In this paper, the joint distribution of some special linear combinations of the (internally) studentized order statistics are derived for both normal and exponential populations; the exact relationship between their pdf's is also obtained. The exact sampling distributions of studentized extreme deviation statistic, which has been proposed by Pearson and Chandra Sekar (1936,Biometrika,28, 308–320), are derived for these two populations. An application to the most powerful location and scale invariant test is discussed briefly.     

Toward the problem-centered classroom: trends in mathematical problem solving in Japan

In this paper, I summarize the influence of mathematical problem solving on mathematics education in Japan. During the 1980–1990s, many studies had been conducted under the title of problem solving, and, therefore, even until now, the curriculum, textbook, evaluation and teaching have been changing. Considering these, it is possible to identify several influences. They include that mathematical problem solving helped to (1) enable the deepening and widening of our knowledge of the students’ processes of thinking and learning mathematics, (2) stimulate our efforts to develop materials and effective ways of organizing lessons with problem solving, and (3) provide a powerful means of assessing students’ thinking and attitude. Before 1980, we had a history of both research and practice, based on the importance of mathematical thinking. This culture of mathematical thinking in Japanese mathematics education is the foundation of these influences.     

Mathematical model of the solid-phase extrusion of polymeric composite materials

A mathematical model of the deformation of a structurally inhomogeneous material in a conical die is developed. The model accounts for the possibility of the material's compaction and loosening during extrusion. It is demonstrated that both a monotonic decrease, and extremal variation in microporosity are possible with increasing degree of drawing, depending on the character of the composite and deforming tool. Computed results are compared with experimental data obtained for superhigh-molecular polyethylene-based composites filled by polymerization. Good correspondence is established between theory and experiment.Translated from Mekhanika Kompozitnyk Materialov. Vol. 31. No. 6. pp. 834–839. November–December, 1995.