Potential constants and mean amplitudes of vibrations of N-methyl acetamide

A normal co-ordinate treatment based on general quadratic forcefield has been applied to N-methyl acetamide molecule. In these calculations, the structure parameters of N-methyl acetamide, as given recently by Katzet al., and the Raman and infra-red data obtained by the authors are used and the potential constants have been obtained. The potential energy distribution among the various symmetry co-ordinates of each normal vibration and the mean amplitudes of vibrations have been calculated.     

Successive Minimization of the State Complexity of the Self-dual Lattices Using Korkin-Zolotarev Reduced Basis

This work presents a systematic method to successively minimize the state complexity of the self-dual lattices (in the sense that each section of the trellis has the minimum possible number of states fixing its preceding co-ordinates). This is based on representing the lattice on an orthogonal co-ordinate system corresponding to the Gram-Schmidt (GS) vectors of a Korkin-Zolotarev (KZ) reduced basis. As part of the computations, we give expressions for the GS vectors of a KZ basis of the K ₁₂, ₂₄, and BW _n lattices. It is also shown that for the complex representation of the ₂₄ and the BW _n lattices over the set of the Gaussian integers, we have: (i) the corresponding GS vectors are along the standard co-ordinate system, and (ii) the branch complexity at each section of the resulting trellis meets a certain lower bound. This results in a very efficient trellis representation for these lattices over the standard co-ordinate system.     

The Dirichlet problem for the Laplace equation in a starlike domain of a Riemann surface

We consider the Dirichlet problem for the Laplace equation in a starlike domain, i.e. a domain which is normal with respect to a suitable polar co-ordinates system. Such a domain can be interpreted as a non-isotropically stretched unit circle. We write down the explicit solution in terms of a Fourier series whose coefficients are determined by solving an infinite system of linear equations depending on the boundary data. Numerical experiments show that the same method works even if the considered starlike domain belongs to a two-fold Riemann surface.     

N-S方程组的通用形式及近似因式分解

基于张量分析,本文在任意曲线坐标系中导出了用原始变量表达的Navier-Stokes(以下简称N-S)方程组弱守恒型通用形式,其中速度采用了逆变或协变分量;与将复杂的坐标变换嵌入该方程组的流行做法相比,本文所得方程组的形式简捷、直观、更适于在贴体曲线坐标系中直接求解.文中详细讨论了这个方程的因式分解过程即将一个n维流动化为n步一维问题来求解,每一步只需解一个块三对角矩阵,从而避开了大型矩阵求逆,提高了解题速度,进一步推广和发展了Beam-Warming的因式分解法.     

Stream-function Co-ordinates in Rotational Flow and an Analysis of the Flow in a Shock Layer: II. Axisymmetric flow

The stream-function co-ordinates system proposed for the analysisof inviscid rotational flow in Part I, is applied to the analysisof the flow in a shock layer around an axisymmetric blunt-nosedbody in a hypersonic stream. The problem treated in this paperis the inverse problem to be solved with a prescribed shockshape. Numerical computations after the method of characteristicsor the forward integration technique are carried out for a sphericalshock and a power-law shock, by making use of an electroniccomputer. Some remarks are then made upon the singular characterof the solution in the supersonic region of the shock layer.     

Resultant forces and moments in mixed-mixed problems of the theory of elasticity

A problem is called mixed-mixed, when both normal and tangential displacements are prescribed on a part of the boundary, while the normal and tangential stresses are prescribed at the rest of the boundary. Exact closed form expressions have been derived for the resultant normal and tangential forces, tilting moment and torque, directly through the prescribed displacements, thus eliminating the need for determination of stresses. The problem solved treats a transversely isotropic elastic half-space, with arbitrary normal and tangential displacements prescribed inside a circle, and the rest of the boundary being stress-free. The interaction between an arbitrary force inside the half-space and a bonded punch is considered as an example. No similar result has ever been reported, even in the case of isotropy.     

The Initial Stages of Cooling of a Non-homogeneous Sphere

An asymptotic representation is obtained, valid for small valuesof the time variable, of the temperature distribution in a solidsphere cooling from a prescribed (non-symmetrical) initial distributionof temperature by radiative exchange with its surroundings.The thermal properties of the solid are assumed to be known(sufficiently smooth) functions of the radial co-ordinate r.The method of analysis applied to this problem can be used toinvestigate the initial stages of other time-dependent processesin non-homogeneous media.     

Affine mappings in the geometries of algebras

Modules and algebras over a commutative ring R and the geometry associated with an algebra are studied. The non-homogeneous members of a module, for which no maximal ideal of R contains every co-ordinate, the generators of a module, for which polynomials in the co-ordinates take arbitrary values and the non-singular members of an algebra are investigated in relation to the group G of non-singular R-linear transformations. Particular attention is given to bi-quadratic extensions of fields. Geometrical isomorphisms are proved to be exactly those transformations that can be written as the composition of a translation, a member of G and a co-ordinate-wise extension to the algebra of an automorphism of R.The author gratefully acknowledges the help and advice of Prof. Walter Benz of the University of Waterloo and the financial support of the University of Waterloo and the National Research Council of Canada.     

New closed-form Green function and integral formula for a thermoelastic quadrant

In this study a new Green’s function and a new Green-type integral formula for a boundary value problem (BVP) in thermoelastostatics for a quadrant are derived in closed form. On the boundary semi-straight-lines twice mixed homogeneous mechanical boundary conditions (one boundary semi-straight-line is free of loadings and normal displacements and tangential stresses are prescribed on the other one) are prescribed. The thermoelastic displacements are subject by a heat source applied in the inner points of the quadrant and by mixed non-homogeneous boundary heat conditions (on one boundary semi-straight-line the temperature is prescribed and the heat flux is given on the other one). When thermoelastic Green’s function is derived the thermoelastic displacements are generated by an inner unit point heat source, described by δ Dirac’s function. All results are obtained in elementary functions that are formulated in a special theorem. A closed-form solution for a particular BVP of thermoelastostatics for a quadrant also is included. Using the proposed approach it is possible to extend the obtained for quadrant results to any other canonical Cartesian domain.     

Local Wiener–Hopf factorization and indices over arbitrary fields

In this paper, a generalization to arbitrary fields of the usual Wiener–Hopf equivalence of complex valued rational matrix functions is given and the left local Wiener–Hopf factorization indices defined in our previous work [A. Amparan, S. Marcaida, I. Zaballa, Local realizations and local polynomial matrix representations of systems, Linear Algebra Appl. 425 (2007) 757–775] are proved to form a complete system of invariants for this equivalence relation. For the case when the field is algebraically closed a reduced form of a controllable matrix pair under the feedback equivalence is presented for which the controllability indices can be written as sums of the local controllability indices [A. Amparan, S. Marcaida, I. Zaballa, On the existence of linear systems with prescribed invariants for system similarity, Linear Algebra Appl. 413 (2006) 510–533].     

Characteristics of wave propagation in piezoelectric bent rods with arbitrary curvature

The wave propagation in the piezoelectric bend rods with arbitrary curvature is studied in this paper. Basic three-dimensional equations in an orthogonal curvilinear coordinate system (r, θ, s) are established. The Bessel functions in radial co-ordinate and triangle series in the angular co-ordinate are used to describe the displacements and electrical potential. Characteristics of dispersion, distributions of displacements and electrical potential over the cross section are calculated, respectively. In the numerical examples, the effects of the ratio of the two ellipse axes on the dispersion relations of the first three modes are observed. The characteristics of the distribution of displacements and electric potential in the cross section, along the radial and s direction are investigated.     

On the Bernstein Inequality for Rational Functions with a Prescribed Zero

We extend some results of Giroux and Rahman (Trans. Amer. Math. Soc.193(1974), 67–98) for Bernstein-type inequalities on the unit circle for polynomials with a prescribed zero atz=1 to those for rational functions. These results improve the Bernstein-type inequalities for rational functions. The sharpness of these inequalities is also established. Our approach makes use of the Malmquist–Walsh system of orthogonal rational functions on the unit circle associated with the Lebesgue measure.     

Inventory management with periodic ordering and minimum order quantities

Periodic review systems are commonly employed by distributors and retailers to replenish their inventories (for example, to co-ordinate in-bound transportation). It is also often the case that vendors specify minimum purchase quantities for physical (for example, packaging) or strategic reasons. When inventory systems recommend order quantities below the prescribed minimum, a decision must be made on whether or not to order. We describe how this environment can be modelled, and employ an extensive factorial experiment involving a simulation to demonstrate situations in which a common simple ‘rounding up’ decision rule performs poorly. In this paper, we discuss financial implications and implementation issues.     

韧性金属大变形拟流动角点理论及应用

本文提出韧性金属弹塑性大变形拟流动角点理论（ｑｕａｓｉ－ｆｌｏｗｃｏｒｎｅｒｔｈｅｏｒｙ）．该理论从塑性变形正交法则出发，将”模量衰减函数”及屈服面的尖点效应引入本构模型，从而实现了由正交法则本构模型向非正交法则本构模型以及从塑性加载向物理弹性却载的光滑过渡，使一般无角点各向异性硬化屈服函数与有角点硬化情形相结合成为可能．用于数值模拟各向异性金属薄板单向拉伸失稳与剪切带分析并与实验结果作比较，表明本文理论的有效性．     

Penalized Orthogonal Iteration for Sparse Estimation of Generalized Eigenvalue Problem

We propose a new algorithm for sparse estimation of eigenvectors in generalized eigenvalue problems (GEPs). The GEP arises in a number of modern data-analytic situations and statistical methods, including principal component analysis (PCA), multiclass linear discriminant analysis (LDA), canonical correlation analysis (CCA), sufficient dimension reduction (SDR), and invariant co-ordinate selection. We propose to modify the standard generalized orthogonal iteration with a sparsity-inducing penalty for the eigenvectors. To achieve this goal, we generalize the equation-solving step of orthogonal iteration to a penalized convex optimization problem. The resulting algorithm, called penalized orthogonal iteration, provides accurate estimation of the true eigenspace, when it is sparse. Also proposed is a computationally more efficient alternative, which works well for PCA and LDA problems. Numerical studies reveal that the proposed algorithms are competitive, and that our tuning procedure works well. We demonstrate applications of the proposed algorithm to obtain sparse estimates for PCA, multiclass LDA, CCA, and SDR. Supplementary materials for this article are available online.     

Discriminant and Clifford algebras

The centralizer of a square-central skew-symmetric unit in a central simple algebra with orthogonal involution carries a unitary involution. The discriminant algebra of this unitary involution is shown to be an orthogonal summand in one of the components of the Clifford algebra of the orthogonal involution. As an application, structure theorems for orthogonal involutions on central simple algebras of degree 8 are obtained. Received: 30 January 2001; in final form: 28 May 2001 / Published online: 1 February 2002     

Stream-function Co-ordinates in Rotational Flow and an Analysis of the Flow in a Shock Layer: I. Two-dimensional flow

For the analysis of two-dimensional rotational flow of inviscidfluids a new set of co-ordinates are proposed. The co-ordinatesare composed of a pair of functions, the stream function anda function to be constant on each trajectory perpendicular tothe stream-lines in the flow field. The theory is then appliedto the analysis of the flow in a shock layer around a blunt-nosedbody in a hypersonic stream. The problem treated in this paperis the inverse problem to be solved with a prescribed shockshape. Some numerical computations are carried out for a parabolicshape of shock, by making use of an electronic computer.     

Laminar flow separation in an axi-symmetric sudden smooth expanded circular tube

This paper concerns with the investigation of laminar flow separation and its consequences in a tube over a smooth expansion under the axi-symmetric approximations. A co-ordinate stretching has been made to map the expanded tube into a straight tube. The two-dimensional unsteady Navier-Stokes equations are solved approximately by using primitive variables in staggered grid. A thorough quantitative analysis is performed through numerical simulations of the desired quantities such as wall shear stress, axial velocity, pressure distribution etc. These quantities are presented graphically and their consequences in the flow field are analysed in details. The dependence of the flow field on the physical parameter like expansion height d and on the Reynolds number has been investigated in details. It is interesting to note that the peak value of wall shear stress decreases with increasing height of expansion and also with the increasing Reynolds number.     

The Design of axisymmetric ducts for incompressible flow with vorticity.

In this paper a numerical algorithm is described for solving the boundary value problem associated with axisymmetric, inviscid, incompressible, rotational (and irrotational) flow in order to obtain duct wall shapes from prescribed wall velocity distributions. The governing equations are formulated in terms of the stream function and the function as independent variables where for irrotational flow can be recognized as the velocity potential function, for rotational flow ceases being the velocity potential function but does remain orthogonal to the stream lines. A numerical method based on finite differences on a uniform mesh is employed. The technique described is capable of tackling the so–called inverse problem where the velocity wall distributions are prescribed from which the duct wall shape is calculated, as well as the direct problem where the velocity distribution on the duct walls are calculated from prescribed duct wall shapes. The two different cases as outlined in this paper are in fact boundary value problems with Neumann and Dirichlet boundary conditions respectively. Even though both approaches are discussed, only numerical results for the case of the Dirichlet boundary conditions are given. A downstream condition is prescribed such that cylindrical flow, that is flow which is independent of the axial coordinate, exists. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strip electro-mechanical yielding model for piezoelectric plate cut along two equal collinear cracks

The multiple-crack problems for piezoelectric ceramics till now have not yet address the crack opening arrest problem. The present work addresses this paucity. A 2-D strip-electro-mechanical yielding model is proposed for a transversely isotropic piezoelectric media weakened by two internal equal collinear straight cracks. The infinite boundary is prescribed with combined uniform constant in-plane mechanical and electrical loads. Developed mechanical and electric strip zones are arrested by prescribing over their rims uniform, normal, cohesive yield point stress and saturation limit electric displacement. Two cases are considered when saturation zone is bigger than developed yield zone and vice versa. Stroh formulation together with complex variable technique is employed to obtain the solution. Closed form expressions are derived for saturation zone length, yield zone length, crack opening displacement (COD), crack opening potential jump (COP) and energy release rate (ERR). An illustrative numerical study is prescribed to determine the effect of various parameters on the crack growth arrest and presented graphically. The results reveal that the model is capable of crack arrest under small-scale mechanical and electric yielding.