Intended Treatments of Arithmetic Average in U.S. and Asian School Mathematics Textbooks

This study examined how selected U.S. and Asian mathematics curricula are designed to facilitate students' understanding of the arithmetic average. There is a consistency regarding the learning goals among these curriculum series, but the focuses are different between the Asian series and the U.S. reform series. The Asian series and the U.S. commercial series focus the arithmetic average more on conceptual and procedural understanding of the concept as a computational algorithm than on understanding the concept as a representative of a data set; however, the two U.S. reform series focus the concept more on the latter. Because of the different focuses, the Asian and the U.S. curriculum series treat the concept differently. In the Asian series, the concept is first introduced in the context of “equal‐sharing” or “per‐unit‐quantity,” and the averaging formula is formally introduced at a very early stage. In the U.S. reform series, the concept is discussed as a measure of central tendency, and after students have some intuitive ideas of the statistical aspect of the concept, the averaging algorithm is briefly introduced.     

Positive definiteness of the gravitational radiation intensity in the theory of gravity with a nonzero graviton mass

A version of the theory of gravity is considered where the graviton mass is nonzero, and the gravitation radiation flux from an arbitrary spatially-bounded source is positive definite. The relation between energy losses by emission and the work of the source is established. It is shown that the total work includes the part produced by the interaction of the source with the radiation field and the part produced by the self-action of the field. The total work proves to be positive definite. The general form of the spectrum-angular distribution is obtained, accounting for the spin and polarization states. For spherically symmetric sources, the states with zero spin and zero projection of spin two on the momentum contribute to the emission.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 329–343, May, 1996.Translated by V. I. Serdobol'skii.     

Various methods of determining the natural frequencies and damping of composite cantilever plates. 1. Exact solution for the binomial model of deformation

On the basis of the classical theory of thin anisotropic laminated plates the article analyzes the free vibrations of rectangular cantilever plates made of fibrous composites. The application of Kantorovich's method for the binomial representation of the shape of the elastic surface of a plate yielded for two unknown functions a system of two connected differential equations and the corresponding boundary conditions at the place of constraint and at the free edge. The exact solution for the frequencies and forms of the free vibrations was found with the use of Laplace transformation with respect to the space variable. The magnitudes of several first dimensionless frequencies of the bending and torsional vibrations of the plate were calculated for a wide range of change of two dimensionless complexes, with the dimensions of the plate and the anisotropy of the elastic properties of the material taken into account. The article shows that with torsional vibrations the warping constraint at the fixed end explains the apparent dependence of the shear modulus of the composite on the length of the specimen that had been discovered earlier on in experiments with a torsional pendulum. It examines the interaction and transformation of the second bending mode and of the first torsional mode of the vibrations. It analyzes the asymptotics of the dimensionless frequencies when the length of the plate is increased, and it shows that taking into account the bending-torsion interaction in strongly anisotropic materials type unidirectional carbon reinforced plastic can reduce substantially the frequencies of the bending vibrations but has no effect (within the framework of the binomial model) on the frequencies of the torsional vibrations.Institute of Engineering Science Russian Academy of Sciences, St. Petersburg, Russia. St. Petersburg State University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 6, pp. 759–769, November–December, 1996.     

The Analysis and the Representation of Balanced Complex Polytopes in 2D

In this paper, we deepen the theoretical study of the geometric structure of a balanced complex polytope (b.c.p.), which is the generalization of a real centrally symmetric polytope to the complex space. We also propose a constructive algorithm for the representation of its facets in terms of their associated linear functionals. The b.c.p.s are used, for example, as a tool for the computation of the joint spectral radius of families of matrices. For the representation of real polytopes, there exist well-known algorithms such as, for example, the Beneath–Beyond method. Our purpose is to modify and adapt this method to the complex case by exploiting the geometric features of the b.c.p. However, due to the significant increase in the difficulty of the problem when passing from the real to the complex case, in this paper, we confine ourselves to examine the two-dimensional case. We also propose an algorithm for the computation of the norm the unit ball of which is a b.c.p. This work was supported by INdAM-GNCS.     

Scope for Exploitation of Consumers Through Speculation: A Dynamic- Programming Approach to the Case of the Indian Sugar Industry

The Government of India (G.O.I.) controls the quantity of sugar released into the domestic market on a monthly basis. This forms a part of the package of public policies affecting the development of the Indian sugar industry, which includes protection from foreign competition, simultaneous existence of an open market for sugar and distribution of sugar at subsidized prices to consumers through the public distribution system (P.D.S.), minimum support prices for the farmers producing sugar-cane, and differential prices paid to sugar factories for the sugar taken over by G.O.I. for the P.D.S.Using a dynamic-programming approach, this paper describes a worst case analysis of the scope for the traders to exploit the consumers under the current policy of controls over releases into the domestic market. The results show the marginal importance of such controls in saving the consumers from the exploitation by traders. Secondly, on the basis of wider research done by the authors, it is concluded that the scope provided to the traders for the exploitation of consumers through the other policies of the package is of such a magnitude as to make the controls over the releases into the market a relatively insignificant policy intervention for protecting the consumers.     

Availability optimization of a redundant system through dependency modeling

This paper presents a constructive approach to optimize the availability of a system through modeling the dependency of the components. Our goal is to minimize the system cost under the constraint that system availability must not be less than a given level. In particular, the components are dependent of each other. A function noted as the dependence function is introduced to model the dependency. It is demonstrated that, for a general form of the system cost, the dependence function guarantees a finite set of feasible solutions. An approach is then developed with the help of the dependence function to obtain the optimal solution. The resolution is illustrated by an interesting example, in which the system cost depends on the strength of the dependency. Our study reveals that the dependency is an essential and effective option to improve system reliability. Moreover, the modeling of dependency, i.e. the introduction of the dependence function is valuable for resolving the optimization problem.     

Existence of nonlinear boundary layer to the Boltzmann equation with reverse reflection boundary condition

Many physical models have boundaries. When the Boltzmann equation is used to study a physical problem with boundary, there usually exists a layer of width of the order of the Knudsen number along the boundary. Hence, the research on the boundary layer problem is important both in mathematics and physics. Based on the previous work, in this paper, we consider the existence of boundary layer solution to the Boltzmann equation for hard sphere model with positive Mach number. The boundary condition is imposed on incoming particles of reverse reflection type, and the solution is assumed to approach to a global Maxwellian in the far field. Similar to the problem with Dirichlet boundary condition studied in [S. Ukai, T. Yang, S.H. Yu, Nonlinear boundary layers of the Boltzmann equation: I. Existence, Comm. Math. Phys. 3 (2003) 373-393], the existence of a solution is shown to depend on the Mach number of the far field Maxwellian. Moreover, there is an implicit solvability condition on the boundary data. According to the solvability condition, the co-dimension of the boundary data related to the number of the positive characteristic speeds is obtained.     

Quasi-classical approach to the inverse scattering problem for the KdV equation,and solution of the Whitham modulation equations

We consider an initial value problem for the KdV equation in the limit of weak dispersion. This model describes the formation and evolution in time of a nondissipative shock wave in plasma. Using the perturbation theory in power series of a small dispersion parameter, we arrive at the Riemann simple wave equation. Once the simple wave is overturned, we arrive at the system of Whitham modulation equations that describes the evolution of the resulting nondissipative shock wave. The idea of the approach developed in this paper is to study the asymptotic behavior of the exact solution in the limit of weak dispersion, using the solution given by the inverse scattering problem technique. In the study of the problem, we use the WKB approach to the direct scattering problem and use the formulas for the exact multisoliton solution of the inverse scattering problem. By passing to the limit, we obtain a finite set of relations that connects the space-time parameters x, t and the modulation parameters of the nondissipative shock wave.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 44–61, January, 1996.     

On isoperimetric problems for domains with partly known boundaries

In the present paper, we discuss the isoperimetric problems for domains with partly known boundaries, i.e., the problem of determining a domain that minimizes the capacity functional in the class of plain double-connected domains having the same fixed area and outer boundary. The formulas for capacity variations obtained in this paper allows us to formulate necessary conditions.It is proved that the convexity of the fixed outer boundary implies the convexity of the inner boundary corresponding to an optimal domain. Then, we discuss the case where the fixed part of the boundary is a square.Further, we consider similar problems with more complicated functionals. We introduce the concept of a minimal function in the class of equimeasurable functions. This concept allows us to unify the approach to all of these problems. At the end, we produce a hypothesis that, if proved, would enable us to characterize the shape of the optimal domains in the isoperimetric problems mentioned above.The author wishes to express his appreciation to Dr. K. A. Lurie for his help and unceasing attention.     

Flow Structure in a Technical Scale Reactor with an Internal Reboiler

In this contribution, results of experimental flow investigations in a technical scale cylindrical reactor with centrally located internal reboiler are presented. The fluid motion in the kettle is induced by natural circulation. Two different geometrical configurations are considered. First, the liquid level is set below the evaporator's outlet. Thus, only the liquid phase penetrates the liquid surface. The second configuration characterizes the a direct injection of the two phase mixture into bulk liquid. In both cases the liquid leaves the evaporator radially and horizontally towards the cylindrical wall. Gathered results reveal the presence of the big, toroidal vortex in the kettle annulus. Independently on the boundary conditions, self similarity of velocity profiles in the near wall region is observed. In contrast, investigations of the momentum transfer between the jet and the bulk liquid reveal strong dependency on the geometrical configuration of the setup. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非Fourier温度场分布的奇摄动解

应用非Fourier热传导定律构建了单层材料中温度场模型，即一类在无界域上带小参数的奇摄动双曲方程，通过奇摄动展开方法，得到了该问题的渐近解.首先应用奇摄动方法得到了该问题的外解和边界层矫正项，通过对内解和外解的最大模估计和关于时间导数的最大模估计以及线性抛物方程理论，得到了内外解的存在唯一性，从而得到了解的形式渐近展开式.通过余项估计，给出了渐近解的L2估计，得到了渐近解的一致有效性，从而得到了无界域上温度场的分布.通过奇摄动分析，给出了非Fourier 温度场与Fourier 温度场的关系，描述了非Fourier温度场的具体形态.     

A two-level evolutionary algorithm for solving the facility location and design (1|1)-centroid problem on the plane with variable demand

In this work, the problem of a company or chain (the leader) that considers the reaction of a competitor chain (the follower) is studied. In particular, the leader wants to set up a single new facility in a planar market where similar facilities of the follower, and possibly of its own chain, are already present. The follower will react by locating another single facility after the leader locates its own facility. Both the location and the quality (representing design, quality of products, prices, etc.) of the new leader’s facility have to be found. The aim is to maximize the profit obtained by the leader considering the future follower’s entry. The demand is supposed to be concentrated at n demand points. Each demand point splits its buying power among the facilities proportionally to the attraction it feels for them. The attraction of a demand point for a facility depends on both the location and the quality of the facility. Usually, the demand is considered in the literature to be fixed or constant regardless the conditions of the market. In this paper, the demand varies depending on the attraction for the facilities. Taking variable demand into consideration makes the model more realistic. However, it increases the complexity of the problem and, therefore, the computational effort needed to solve it. Three heuristic methods are proposed to cope with this hard-to-solve global optimization problem, namely, a grid search procedure, a multistart algorithm and a two-level evolutionary algorithm. The computational studies show that the evolutionary algorithm is both the most robust algorithm and the one that provides the best results.     

道德风险影响下的保险最优投资研究

在考虑道德风险的情况下,以均值方差准则为目标研究保险人最优投资问题.假设保险盈余过程服从C-L模型,金融市场上存在一种无风险资产和一种风险资产可供投资,其中风险资产的价格过程服从几何布朗运动.在纯道德风险保险契约设计中,借鉴相关研究对努力水平和效用化努力成本的假设,量化道德风险对盈余过程的影响.在均值方差目标下,建立保险人最优投资问题的广义Hamilton-Jacobi-Bellman(HJB)方程,给出保险人时间一致的均衡投资策略和价值函数.结果显示累计索赔比例参数越大,公司对最优努力水平越敏感,采取措施降低道德风险有利于公司收益提升;努力成本参数越大,公司会降低努力水平减少支出,避免损失.     

On the conditions under which a satellite orbit intersects the surface of a central body of finite radius in the restricted double-averaged three-body problem

This paper is concerned with the applied problem of choosing long-living orbits of artificial Earth satellites whose evolution under the influence of gravitational perturbation from the Moon and the Sun may result in the collision of the satellite with the central body, as was shown by M.L. Lidov for the well-known example of “Vertical Moon.” We use solutions of the completely integrable system of evolution equations obtained by Lidov in 1961 by averaging twice the spatial circular restricted three-body problem in the Hill approximation. In order to apply the integrability of this problem in practice, we study the foliation of the manifold of levels of first integrals and the change of motion under crossing the bifurcation manifolds separating the foliated cells. As a result, we describe the manifold of initial conditions under which the orbit evolution leads to an inevitable collision of the satellite with the central body. We also find a lower bound for the practical applicability of the results, which is determined by the presence of gravitational perturbations caused by a polar flattening of the central body. Original Russian Text ? V.I. Prokhorenko, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 156–173.     

A refined model for three-point bending of sandwich panels 2. Transverse stresses

According to the relationships derived in [1], transverse normal and tangential stresses in a sandwich panel have been analyzed. Asymptotic formulas for the stress concentration area in the vicinity of point forces are derived. Analytical estimates of a normal stress at the central and end sections of the panel are deduced. The Saint-Venant effect of the degeneration of a panel of finite length into an infinite strip is studied. For the estimation of the concentration of the transverse tangential stress, the possibility of a superposition of the solution of the slippage problem of the face layers and the classical solution allowing for shear is substantiated. It is shown that the local Reissner-type effects are specified by reducing the concentration of the tangential stress in the face layers along the longitudinal coordinate and transition to the steady tangential stress state in the filler layer. The concentration coefficients of the tangential stress are derived as functions of the dimensional parameters of the panel section.Institute of Polymer Mechanics. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 1, pp. 66–93, January–February, 1998.     

Electron in the Aharonov-Bohm potential and in the Coulomb field in 2+1 dimensions

We obtain exact solutions of the Dirac equation in 2+1 dimensions and the electron energy spectrum in the superposition of the Aharonov-Bohm and Coulomb potentials, which are used to study the Aharonov-Bohm effect for states with continuous and discrete energy spectra. We represent the total scattering amplitude as the sum of amplitudes of scattering by the Aharonov-Bohm and Coulomb potentials. We show that the gauge-invariant phase of the wave function or the energy of the electron bound state can be observed. We obtain a formula for the scattering cross section of spin-polarized electrons scattered by the Aharonov-Bohm potential. We discuss the problem of the appearance of a bound state if the interaction between the electron spin and the magnetic field is taken into account in the form of the two-dimensional Dirac delta function. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 502–517, December, 2006. An erratum to this article is available at .     

On the singularities of the gradient of the solution to the Dirichlet-Neumann problem outside a plane cut

The problem considered in this paper deals with the Laplace equation outside a cut of sufficiently arbitrary form. The Dirichlet condition is given on one side of the cut and the Neumann condition on the other. Using the integral representation for the solution, we obtain explicit asymptotic formulas describing the singularity of the gradient of the solution at the edges of the cut. We discuss the effect of the disappearance of the singularity.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 364–377Original Russian Text Copyright © 2005 by P. A. Krutitskii, A. I. Sgibnev.This revised version was published online in April 2005 with a corrected issue number.