二阶线性时滞微分方程的广义振动性

本文主要研究了二阶线性时滞微分方程的广义振动性和广义非振动性,给出了方程广义振动和广义非振动的一些判定定理,同时给出了一类方程广义非振动的充要条件.     

脉冲中立型时滞抛物方程的振动性

本文研究了一类脉冲中立型时滞抛物方程解的振动性及强振动性,获得了此类脉冲中立型时滞抛物方程解振动和强振动的代数判据.     

一类二阶非线性差分方程的振动性与渐近性

研究了一类二阶非线性差分方程解的振动性与渐近性,建立了三个新的振动性与渐近性定理,推广和改进了已知的一些结果.     

非线性中立型泛函微分方程组的非振动解的存在性准则

本文利用 Banach 锥理论首次建立了非线性中立型泛函微分方程组的非振动解的存在性准则;同时,还给出了非线性高阶中立型方程的振动性和非振动性定理.     

二阶非线性摄动微分方程解的振动性与渐近性

研究了一类二阶非线性摄动微分方程解的振动性与渐近性,建立了四个新的振动性与渐近性定理,推广和改进了已知的一些结果.     

n阶线性脉冲微分方程解的振动性

脉冲微分方程解的振动性由于在物理、生态、工程等领域有其应用背景,引起了人们的广泛兴趣,正在成为研究的热点．本文讨论了一类n阶线性脉冲微分方程解的振动性和非振动性,分别给出了n为奇数和偶数时该方程振动和非振动的充分条件,并通过3个例题说明了文中定理的应用,所得结论改进并推广了部分文献的相关结果．     

高阶线性时滞微分方程的广义振动性

该文考虑了一类高阶线性常系数时滞微分方程 y( n) (t) py′(t) qy(t-τ) =0的广义振动性和广义非振动性 ,给出了一些该类方程广义振动和广义非振动的判定定理 .文中的定理 4还给出了一类非振动但广义振动的方程的判别法则     

二阶非线性泛函微分方程解的振动性质

研究了一类二阶非线性泛函微分方程解的振动性以及非振动解的有界性.在一定条件下,建立了几个新的振动性和有界性定理,其结果推广和改进了已有的一些结果.     

三阶非线性泛函微分方程振动的比较准则

本文利用比较法,与一阶时滞微分方程的振动性相比较,研究一类三阶非线性泛函微分方程的振动性,建立该类方程振动的新比较振动准则.本文结果推广和改进了最近文献中的一些新结果.     

二阶非线性摄动微分方程的振动性定理

研究了一类二阶非线性摄动微分方程解的振动性,建立了两个新的振动性定理,推广和改进了已有的结果.     

The generalized hierarchical product of graphs

A generalization of both the hierarchical product and the Cartesian product of graphs is introduced and some of its properties are studied. We call it the generalized hierarchical product. In fact, the obtained graphs turn out to be subgraphs of the Cartesian product of the corresponding factors. Thus, some well-known properties of this product, such as a good connectivity, reduced mean distance, radius and diameter, simple routing algorithms and some optimal communication protocols, are inherited by the generalized hierarchical product. Besides some of these properties, in this paper we study the spectrum, the existence of Hamiltonian cycles, the chromatic number and index, and the connectivity of the generalized hierarchical product.     

城市老旧居住小区交通环境评价指标与评价方法

定量的评判城市老旧居住小区整体交通环境质量并进而科学有效的提出老旧小区交通环境更新与改善措施,对城市老旧居住小区交通环境进行了调查,明确了老旧居住小区存在的突出交通问题,并提出了城市老旧居住小区交通环境评价的指标体系,包括行人跨路出行指数、无人行道路段长度占比、停车泊位供需比、路内非法停车占比、平均停车步行距离、泊位数量户数比、人均步行空间、无障碍通道、盲道设置比例、非机动车泊位供需比、地面停放车辆数小区面积比、外部穿行交通周转量小区面积比等。并给出了具体指标的计算方法与获得办法。以长沙市四个典型老旧居住小区为例,对评价方法进行了论证,并得到城市老旧居住小区存在的一般问题,包括人车混杂、人车交织现象严重,停车泊位短缺、车辆乱停乱放,交通设施、特别是弱势群体交通设施缺乏。给出的评价指标、评价方法与结论可为城市老旧小区改造更新,特别是其中很重要的交通环境更新提供决策依据。     

Rolling-ball method for estimating the boundary of the support of a point-process intensity

We suggest a generalisation of the convex-hull method, or ‘DEA’ approach, for estimating the boundary or frontier of the support of a point cloud. Figuratively, our method involves rolling a ball around the cloud, and using the equilibrium positions of the ball to define an estimator of the envelope of the point cloud. Constructively, we use these ideas to remove lines from a triangulation of the points, and thereby compute a generalised form of a convex hull. The radius of the ball acts as a smoothing parameter, with the convex-hull estimator being obtained by taking the radius to be infinite. Unlike the convex-hull approach, however, our method applies to quite general frontiers, which may be neither convex nor concave. It brings to these contexts the attractive features of the convex hull: simplicity of concept, rotation-invariance, and ready extension to higher dimensions. It admits bias corrections, which we describe and illustrate through implementation.     

Analysis of the Stress State of the Surface Layer of a Polymeric Mass upon Filling of Bulky Compression Molds

The stress state of the surface layer of a polymeric mass during filling of bulky compression molds is analyzed. It is shown that, at particular rheological characteristics of the mass, temperature, and filling rates, cracking of the surface layer occurs, which leads to defects in the finished products. A physical analysis of this process makes it possible to conclude that the cracks arise due to the normal stresses operating in the front region of the moving polymeric mass. It is found that, under certain flow conditions, areas with a pressure lower than the atmospheric one appear on the surface of the polymer. If the tensile stresses arising in these local regions are higher than the tensile strength of the mass, the continuity of the composition is broken in the direction determined by the greatest rate of the normal deformation. To confirm the reliability of the crack-formation mechanism proposed, the distribution of the pressure and normal stresses over the free surface is calculated based on a numerical method. These calculations show that, by comparing the stress level achieved in the front region with the tensile-strength characteristics of the polymeric composition, it is possible to predict, with a sufficient accuracy, the possibility of crack formation in the surface layer of such a mass under given flow conditions and thus to solve the question on flawless manufacturing of products.     

Quality rating of a metal matrix-diamondcomposite from its thermal conductivity and electric resistance

The efficiency of hot-pressed diamond-containing composite materials (DCM) for various tool applications is greatly affected by microdefects, namely, the residual porosity of the metal matrix, damaged diamond grains, and imperfect diamond-matrix interfaces. An instrumental evaluation of these microdefects, predetermining the quality of a tool equipped with DCM, is rather difficult due to the small size, the nonstandard shape, and the strong heterogeneity of specimens. Proposed here is an alternative, nondestructive technique of DCM quality rating, which includes the measurement of electric resistance and thermal conductivity of diamond-containing composites and processing the obtained data by the methods of composite mechanics. It exploits the fact that diamond, being a dielectric, possesses an extremely high thermal conductivity, which allows estimating the residual porosity of a sintered metal matrix from the ratio of specific electric resistances, one being measured and another predicted by a theory. These data, in turn, are utilized to predict the thermal conductivity ofDCMwith an imperfect matrix. Matching with experiments, after solving the inverse problem gives the thermal resistance of diamond-matrix interface, which, within the frame work of the given model, simulates the damage of both the diamond grains and their bonds with the matrix. Thus, the numerical rating of quality is given in terms of two dimensionless parameters. The first one, 0 < K < 1, reflects the quality of the sintered metal matrix, whereas the second one, 0 < R <1, is an aggregate measure of the integrity of diamond grains and the perfection degree of composite interfaces. The quite satisfactory agreement observed between the theory and experiment confirms the efficiency of the technique and the reliability of the data obtained. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 361–374, May–June, 2006.     

Calculation of damage and of force relaxation in amorpho-crystalline polymers

The dynamics of damage and of the relaxing force in amorpho-crystalline polymers under constant strain are calculated using the formulas for the probability of rupture of a deformed polymer molecule and a model representation of amorphous interlayers. The main parameters of the model are the maximum and minimum possible deformations of molecular chains, the energy of rupture activation, the function of the chain length distribution, the temperature, the macroscopic strain, and the relative dimensions of the amorphous interlayer. The conformity of the theoretical model and the association of the relaxation spectrum with the internal molecular and structural characteristics of the material are established.Zhambyl Technical Institute of Light and Food Industry, Taraz, Kazakhstan. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 499–508, July–August, 1999.     

Gaussian Versus Optimal Integration of Analytic Functions

We consider error estimates for optimal and Gaussian quadrature formulas if the integrand is analytic and bounded in a certain complex region. First, a simple technique for the derivation of lower bounds for the optimal error constants is presented. This method is applied to Szeg?-type weight functions and ellipses as regions of analyticity. In this situation, the error constants for the Gaussian formulas are close to the obtained lower bounds, which proves the quality of the Gaussian formulas and also of the lower bounds. In the sequel, different regions of analyticity are investigated. It turns out that almost exclusively for ellipses, the Gaussian formulas are near-optimal. For classes of simply connected regions of analyticity, which are additionally symmetric to the real axis, the asymptotic of the worst ratio between the error constants of the Gaussian formulas and the optimal error constants is calculated. As a by-product, we prove explicit lower bounds for the Christoffel-function for the constant weight function and arguments outside the interval of integration. September 7, 1995. Date revised: October 25, 1996.     

Development of the Tangent Modulus from the Euler-Problem to Nonsmooth Materials

History and development of the tangent modulus from the origins to the recent nonsmooth damaging versions are presented. Load history and stability analyses of structures of nonlinear reversible or irreversible materials are based on the concept of tangent modulus. Generally, instantaneously changing tangent modulus is needed and the solution yields iteration process. In the case of inelastic problems, the switch from loading to unloading of the material behaviour results in nonsmooth material functions. Nonsmooth, generally saw-tooth like behaviour happens in composite, laminated or rock type materials, or in the interaction of concrete and the reinforcement, too. Recently, damage and localization are in the focus of structural analyses, extending the tangent modulus to the negative cases, as well. Consequently, an overview of the history and development of the tangent modulus containing the recent modifications seems to be necessary. On the other hand, the more than a century long history of the tangent modulus is a marvellous study of the parallel development of mechanics and mathematics, by following the mutual inspiring effect of them through the activity of such pioneers like P.D. Panagiotopoulos in creating Nonsmooth Mechanics.     

The limiting state of a rigidly fixed nonlinearly elastic multilayer rod

The loss of the load-carrying capacity of a nonlinearly elastic multilayer rod is investigated. The rod, whose layers have various thickness and are made of different materials, is rigidly fixed at both its ends. Rigid contact conditions between the layers are assumed. The problem posed is solved by using the variational method of mixed type in combination with the Rayleigh-Ritz method. The initial analysis is reduced to the solution of the Cauchy problem for a nonlinear ordinary differential equation solved for the first derivative. As the initial condition, the maximum initial eccentricity of the rod is assumed. In the case of zero eccentricity, the Shanley critical force for an axially compressed rod is determined. For a three-layer rod whose outer layers have equal thickness and are made of the same material, numerically, for various degrees of nonlinearity, the effect of physicomechanical and geometric parameters on the critical load of buckling instability is determined. It is found that, by matching the heterogeneity of the rod, it is possible to raise its load-carrying capacity. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 347–360, May–June, 2006.     

Some curious properties and loci problems associated with cubics and other polynomials

The article begins with a well-known property regarding tangent lines to a cubic polynomial that has distinct, real zeros. We were then able to generalize this property to any polynomial with distinct, real zeros. We also considered a certain family of cubics with two fixed zeros and one variable zero, and explored the loci of centroids of triangles associated with the family. Some fascinating connections were observed between the original family of the cubics and the loci of the centroids of these triangles. For example, we were able to prove that the locus of the centroid of certain triangles associated with the family of cubics is another cubic whose zeros are in arithmetic progression. Motivated by this, in the last section of the article, we considered families of cubic polynomials whose zeros are in arithmetic progression, along with the loci of the special points of certain triangles arising from such families. Special points include the centroid, circumcentre, orthocentre, and nine-point centre of the triangles. Throughout the article, we used the computer algebra system, Mathematica®, to form conjectures and facilitate calculations. Mathematica® was also used to create various animations to explore and illustrate many of the results.