On the Geometry of the Nodal Lines of Eigenfunctions of the Two-Dimensional Torus

The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. We prove a variety of results on the width, some having stronger versions assuming a conjecture of Cilleruelo and Granville asserting a uniform bound for the number of lattice points on the circle lying in short arcs.     

On the variation of the spectra of matrices

Several “distances” between the spectra of two matrices are discussed and compared. Optimal bounds are given, which enable us to reduce certain bounds on the eigenvalue variation of matrices by a factor of about two. The results of Bhatia and Mukherjea and of Bhatia and Friedland on the eigenvalue variation are derived in an elementary way using results of Henrici on the spectral variation.     

On the dimensions of bivariate spline spaces and the stability of the dimensions

In this paper, the dimensions of bivariate spline spaces are studied using the Smoothing Cofactor-Conformality method. Based on the analysis on the conformality condition at one interior vertex, the stability (or singularity to the contrary) of the dimensions of general spline spaces is discussed in detail. By the aid of directed partition some new results on dimensions are obtained with the corresponding constraints depending on the degree, the smoothness order of the spline spaces and the structure of the partition as well.     

Effect of hyperviscosity on the geometry of the vorticity

We consider the two-dimensional modified Navier–Stokes equations by adding a p-Laplacian. We provide upper bounds on the Hausdorff length of the level sets of the vorticity. We express them in terms of the Landau–Lifschitz estimates of the number of degrees of freedom in turbulent flow. The proof, based on very standard methods, relies on a straightforward relation between the size of the nodal set of the vorticity and the growth of hyperdissipation.     

Effect of the incompressibility condition on the stability of the solutions of ill-posed problems of the theory of elasticity

One considers the energy functional with density depending only on the first invariant of the strain tensor. Under natural constraints on the density and in the absence of exterior actions, the zero displacements yield a minimum to the energy functional both for the compressible and the incompressible media. In the paper one proves the isolation of this minimum for incompressible media and its nonisolation for compressible media.Translated from Problemy Matemticheskogo Analiza, No. 10, pp. 62–71, 1986.     

On the Modelling of the Adoption of Process Innovations

An alternative view on the modelling of diffusion processesof process innovations has been introduced by Nooteboom (1989). Thebasic idea in that paper is to model the process of adoption ofprocess innovations as a gambling model with odds in favourdepending on expected returns and odds against depending on risk.In this paper it is investigated under which theoretical conditions posed on the expected return function the usual adopted assumptions Nooteboom (1989) with respect to the parameters of the gambling model are in fact reasonable. It is shown that only under certainconditions the assumptions made on the gambling model are true ontheoretical grounds. In all other cases a nonexample shows that theresults as mentioned in literature are in general not true.     

On the dimensions of bivariate spline spaces and the stability of the dimensions

In this paper, the dimensions of bivariate spline spaces are studied using the Smoothing Cofactor-Conformality method. Based on the analysis on the conformality condition at one interior vertex, the stability (or singularity to the contrary) of the dimensions of general spline spaces is discussed in detail. By the aid of directed partition some new results on dimensions are obtained with the corresponding constraints depending on the degree, the smoothness order of the spline spaces and the structure of the partition as well.     

Analysis of the steady motions of the tippe top

The motion of the tippe top on a horizontal plane is considered taking into account sliding friction within the Contensou model. The tippe top is modelled by two spherical segments rigidly joined by a rod directed along the common axis of symmetry of the segments. The dimensions of the spherical segments and the rod are chosen so that, as the axis of symmetry deviates from the upward vertical, the tippe top is supported on the plane at a point on one segment up to a certain critical value and at a point on the other segment at larger deviations (at points on both segments at the critical value). The motion of the tippe top is described by different equations in different regions of configuration space, and the motion is accompanied by impacts on the boundary of these regions. An effective potential of the system is constructed, and the type of its critical points is investigated. Poincaré–Chetayev bifurcation diagrams and generalized Smale diagrams are constructed for steady motions. Plots of the steady-state precessional motions have a discontinuity on the boundary between the regions indicated.     

Properties of the solitonic potentials of the heat operator

We study properties of the purely solitonic τ-function and potential of the heat equation in detail. We describe the asymptotic behavior of the potential and establish the ray structure of this asymptotic behavior on the plane (x₁, x₂) in dependence on the parameters of the potential.     

On the combinatorics of the Pfaff identity

Recently, there has been a revival of interest in the Pfaff identity on hypergeometric series because of the specialization of Simons and a generalization of Munarini. We present combinatorial settings and interpretations of the specialization and the generalization; one is based on free Dyck paths and free Schröder paths, and the other relies on a correspondence of Foata and Labelle between the Meixner endofunctions and bicolored permutations, and an extension of the technique developed by Labelle and Yeh for the Pfaff identity. Applying the involution on weighted Schröder paths, we derive a formula for the Narayana numbers as an alternating sum of the Catalan numbers.     

Regularization of the solution of the cauchy problem for the generalized Moisil-Theodoresco system

We consider the problem of analytic continuation of a solution of the generalized Moisil-Theodoresco system in a spatial domain on the basis of its values on part of the boundary of this domain, that is, the Cauchy problem. We construct an approximate solution of this problem on the basis of the Carleman matrix method.     

Acceleration of the convergence of the Laguerre series in the problem of inverting the Laplace transform

When the Laplace transform is inverted numerically, the original function is sought in the form of a series in the Laguerre polynomials. To accelerate the convergence of this series, the Euler-Knopp method is used. The techniques for selecting the optimal value of the parameter of the transform on the real axis and in the complex plane are proposed.     

Examples of the influence of the geometry on the propagation of progressive waves

In this paper, we give examples of the influence of the domain of propagation on progressive waves. More precisely, we numerically investigate the propagation of reaction diffusion waves in cylinders with variable radius. We show that, when the radius rapidly expands from a very small radius to a larger one, depending on the viscosity and the nonlinearity, the travelling wave may be blocked. The aim of this paper is to give numerical illustrations and quantifications of this effect, and to propose some conjectures which could be interesting subjects for further mathematical investigations.This work is linked to the study of spreading depression (SD), a propagative mechanism in brain and various tissues which has been observed in vivo and in vitro in many species since their discovery in 1944 by Leao. As a matter of fact, their direct observation in Man is still controversial. The complex structure of gray and white matter in humans may block the propagation of SD over large distances in brain and thus explain the difficulty of observing it. Medical consequences of the current numerical studies are detailed in [M.A. Dronne, et al., Influence of brain geometry on spreading depressions: A computationnal study, Progress in Biophysics and Molecular Biology 97 (1) (2008) 54–59] and a first mathematical approach given in [M.A. Dronne, E. Grenier, H. Gilquin, Modelization of spreading depressions following Nedergaard, preprint, 2003].     

Derivation of the Hydrodynamic Equations in the Framework of the Bogoliubov Functional Hypothesis

We generalize the Bogoliubov functional hypothesis to the case of multiparticle interaction depending on both the coordinates and momenta of particles. We illustrate this with the examples of two weakly relativistic models: the Darwin model in the theory of charged particles and the Fock model in the general theory of relativity. For these models, based on the chain of the BBGKY equations, we calculate weakly relativistic corrections to the classical transport coefficients and find the conditions under which there is no bijective relation between the parameters of the local equilibrium distribution and the hydrodynamic variables.     

Circulation hydrodynamics in the organs of the abdominal cavity in the presence of extracorporeal perfusion

The general hemodynamics and circulation in the organs of the abdominal cavity have been investigated in the presence of artificial circulation at a calculated rate of perfusion. The effect of vasoactive preparations (noradrenaline, Arfonad) on these characteristics is analyzed. The behavior of the microcirculation in the presence of artificial circulation created by an artificial circulation apparatus equipped with automatic control systems of original design has been studied. It has been found that disturbances of the microcirculation are responsible for the development of a state of hypoxia in organs and tissues in the presence of artificial circulation at a calculated rate of perfusion under conditions of controlled vasoconstriction or controlled hypotension. Stabilization of the arterial pressure and the functional state of the vascular bed in the presence of artificial circulation had a positive effect on the tissue blood flow and oxygen metabolism in the organs of the abdominal cavity.Paper presented at the First All-Union Conference on Engineering and Medical Biomechanics on the topic: Problems of Circulatory Hydrodynamics, Riga, October, 1975.D. I. Ulyanov Kuibyshev Medical Institute. Translated from Mekhanika Polimerov, No. 5, pp. 871–874, September–October, 1975.     

On the interrelation of the basic laws of continuum mechanics

It has been shown by the present authors in a recent paper [1] that if some conservation and balance laws of continuum mechanics are represented in a 4 × 4 form, balance of linear physical momentum (i. e., stress) and balance of mass become closely linked. This seemingly novel result was reached in a completely ad‐hoc fashion by treating time on the same level as the spatial coordinates, and not as parameter, as it is usually done. In order to place the above ad‐hoc result on a firmer foundation and since it is in the theory of relativity that space and time are considered on the same footing, an attempt is made to derive several tensors of continuum mechanics in a systematic manner as 4 × 4 invariant objects.     

The distribution of the eigenvalues of the discrete Laplacian

We estimate the distribution of the eigenvalues of the discrete Laplacian on a bounded set inR ⁿ . The proof is based on a variational technique similar to that used by Weyl for the Laplacian. As an application of our estimates we prove stability in the maximum norm for the Crank-Nicolson method for the heat equation on a bounded set.The research of the second author was supported in part by National Science Foundation grant GP-30735X.     

Study of the effect of the dressing of carbon fibers on the properties of carbon-fiber-reinforced plastics based on polyacetals

Studies are made of the mechanical and friction engineering properties of a copolymer of trioxane-1,3,5 with dioxolan-1,3 reinforced with dressed carbon fibers based on hydrated cellulose. Polyalkyloxysilanes are used as the finishes. The optimum concentrations of the finishes on the fibrous filler are determined. Relations are obtained to describe the dependence of the mechanical properties of carbon-fiber-reinforced plastics on the type of finish and its effect on the condition of the phase boundary between filler and the polymer matrix. The effect of adhesion of the polymer to the fibrous filler on the properties of a randomly reinforced thermoplastic composite is also determined. The dependences of the friction engineering properties of polyacetal-based carbon-plastics on the loading parameters of the friction-loading unit are examined. The material that is developed has high values of strength, elastic modulus, and notch toughness and low values of the friction coefficient and shrinkage. The material can be used as structural and anti-friction elements in rocketry, aviation, and the automobile industry.Ukrainian State Chemical Engineering University, Dnepropetrovsk. Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 90–97, January–February, 1997.     

On the stability of the polygonal isoperimetric inequality

We obtain a sharp lower bound on the isoperimetric deficit of a general polygon in terms of the variance of its side lengths, the variance of its radii, and its deviation from being convex. Our technique involves a functional minimization problem on a suitably constructed compact manifold and is based on the spectral theory for circulant matrices.     

农业现代化视阈下基于SPSS多元回归分析的全国农业总产值主要影响因素探究

随着我国城镇化的快速发展,大批农业劳动力向城市转移,农业总产值及其主要影响因素的研究已经成为热点问题.为研究我国农业总产值的主要影响因素,本文借助多元回归分析的相关理论抽取了2016年我国农业投入的31个省市的相关截面数据,采用最小二乘法,利用SPSS软件进行多元回归分析,得到回归模型,并基于该模型找出我国农业总产值的主要影响因素.