Linearization of nonautonomous models describing fluid filled elastic tubes and nonlinear elastic rods with variable cross-section

Summary The problem of reducing nonautonomous and nonlinear models of physical interest to canonical integrable forms is considered herein.First, in the theoretical framework of group analysis, we transform to autonomous form the systems describing fluid filled elastic tubes and nonlinear elastic rods with variable cross-section area. Thus, by means of a hodograph-like transformation or a combination of Bäcklund-like transformations, we reduce the autonomous models in point to linear form.To accomplish such a plan we characterize several model constitutive laws which can be used to approximate in suitable ranges of the concerned variables material laws based on experimental evidence.     

Canonical transformations of skew-normal variates

Conditions are given for linear functions of skew-normal random vectors to maximize skewness and kurtosis. As a direct implication, several measures of their multivariate skewness and kurtosis are shown to be equivalent. An estimator of the shape parameter with good statistical properties is also considered. These results are strictly related to canonical forms of skew-normal distributions and linear transformations to normality.     

Noether’s theorem for fractional variational problem from El-Nabulsi extended exponentially fractional integral in phase space

This paper focuses on studying Noether’s theorem in phase space for fractional variational problems from extended exponentially fractional integral introduced by El-Nabulsi. Both holonomic and nonholonomic systems are studied. First, the fractional variational problem from extended exponentially fractional integral, as well as El-Nabulsi–Hamilton’s canonical equations are established; second, the definitions and criteria of fractional Noether symmetric transformations and fractional Noether quasi-symmetric transformations are presented which are based on the invariance of El-Nabulsi–Hamilton action under the infinitesimal group transformations; finally, the fractional Noether’s theorem is established, which reveals the inner relationship between a fractional Noether symmetry and a fractional conserved quantity.     

Rotation and gyration of finite two-dimensional modes

Hermite-Gauss and Laguerre-Gauss modes of a continuous optical field in two dimensions can be obtained from each other through paraxial optical setups that produce rotations in (four-dimensional) phase space. These transformations build the SU(2) Fourier group that is represented by rigid rotations of the Poincaré sphere. In finite systems, where the emitters and the sensors are in NxN square pixellated arrays, one defines corresponding finite orthonormal and complete sets of two-dimensional Kravchuk modes. Through the importation of symmetry from the continuous case, the transformations of the Fourier group are applied on the finite modes.     

Generalized prolate spheroidal wave functions for optical finite fractional Fourier and linear canonical transforms

Prolate spheroidal wave functions (PSWFs) are known to be useful for analyzing the properties of the finite-extension Fourier transform (fi-FT). We extend the theory of PSWFs for the finite-extension fractional Fourier transform, the finite-extension linear canonical transform, and the finite-extension offset linear canonical transform. These finite transforms are more flexible than the fi-FT and can model much more generalized optical systems. We also illustrate how to use the generalized prolate spheroidal functions we derive to analyze the energy-preservation ratio, the self-imaging phenomenon, and the resonance phenomenon of the finite-sized one-stage or multiple-stage optical systems.     

Properties of the linear canonical integral transformation

We provide a general expression and different classification schemes for the general two-dimensional canonical integral transformations that describe the propagation of coherent light through lossless first-order optical systems. Main theorems for these transformations, such as shift, scaling, derivation, etc., together with the canonical integral transforms of selected functions, are derived.     

Canonical-covariant Wigner function in polar form

The two-dimensional Wigner function is examined in polar canonical coordinates, and covariance properties under the action of affine canonical transformations are derived.     

On the solution of hydrodynamical equations via Lie groups

Lie groups of homothetic transformations in the Euclidean space R₂ have been employed to determine and investigate certain classes of solutions of hydrodynamical equations of a perfect fluid. In particular, the conditions for a solution to be regular with respect to the one-parameter group of transformations have been determined. Furthermore, it has been shown that if the regularity conditions mentioned above are satisfied then the problem of obtaining regular solutions reduces to that of solving a system of equations not involving λ in the canonical coordinates (λ, μ) of the subgroup. Some special classes of flows have also been investigated.     

Binding transformation in a system of many bodies. Report 3. Group of binding transformations

It is shown that introduction into a set of bodies of the operations of mergence and separation generates a group of binding transformations. The action of this group on various binding states is illustrated by examples. Invariants of binding transformations are considered. Thus, in the method of finite elements binding transformations do not change the number of elements, however the number of nodal points is a variable quantity.Institute of Strength Problems, Kiev. Translated from Problemy Prochnosti, No. 9, pp. 89–93, September, 1989.     

Conservation laws in linear piezoelectricity

We use Noether's theorem on variational principles invariant under a group of infinitesimal transformations to obtain a class of conservation laws for linear piezoelectric materials and linear elastic dielectrics.     

Computing Cartan subalgebras of Lie algebras

We consider the algorithmic problem of computing Cartan subalgebras in Lie algebras over finite fields and algebraic number fields. We present a deterministic polynomial time algorithm for the case when the ground fieldk is sufficiently large. Our method is based on a solution of a linear algebra problem: the task of finding a locally regular element in a subspace of linear transformations. Also, we give a polynomial time algorithm for restricted Lie algebras over arbitrary finite fields. Both methods require an auxiliary procedure for finding non-nilpotent elements in subalgebras. This problem is also treated. Computational experiences are discussed at the end of the paper.Research supported in part by Hungarian National Foundation for Scientific Research grants T016503 and F4116     

线性正则域的Hartley变换及其不确定性原理

Hartley变换是傅里叶变换的推广, 它的一个非常好的性质就是把实信号变换成实信号, 从而减少计算量。近些年, 随着分数阶傅里叶变换在信号处理中被广泛的应用, 线性正则变换也逐渐被应用到信号处理, 所以把Hartley变换推广到正则域是一个有研究价值的问题。本文首先通过变化傅里叶变换域Hartley变换的核函数, 得到了一个具有共轭性的核函数, 之后, 通过把该核函数替换成线性正则变换的核函数, 从而得到了正则域的Hartley变换, 在这个定义的基础上, 得到了正则域Hartley变换满足实数性质和奇偶不变性, 之后再利用线性正则变换的Heisenberg不确定性原理, 得到了正则域Hartley变换的Heisenberg不确定性原理。     

线性正则正弦与余弦变换的卷积定理及其应用

针对奇、偶信号的去噪问题,提出了一种基于线性正则正(余)弦变换卷积定理的乘性滤波器设计方法。在现有线性正则变换域卷积理论的基础上,研究了两类线性正则正(余)弦变换卷积定理,利用所得卷积定理,通过合理选择滤波函数,设计了一类基于卷积定理的线性正则正(余)弦变换域带限信号的乘性滤波模型,并对算法的复杂度进行分析。研究表明,这种滤波模型特别适合处理奇、偶信号,并能有效降低乘积滤波的计算复杂度,提高运算效率。     

一类双壳结构振动特性的半解析法

以Hamilton正则方程的半解析法为基础,为一类双壳结构的振动特性提出了一种新的数学模型.基本步骤:(1)独立地建立内外壳和连接筋的线性方程组;(2)考虑到内外壳和连接筋的界面上的应力和位移的连续性,联立内外壳和连接筋的方程,从而得到全结构的方程组.主要优点是:采用了同一种Hamiltonian等参元离散壳和连接筋,结构的转动惯性、剪切变形等因素都得到了考虑,而且不限制壳的厚度和筋的高度;该方法象一般的有限元法一样适应复杂的边界条件和由多种材料构成的结构.本文的方法可推广用来研究加筋复合材料或加筋压电材料层合壳及相应的双壳结构的动力学问题.     

Hybrid and mixed-penalty finite elements for 3-D analysis of soft hydrated tissue

Solution of biomechanics problems involving three-dimensional (3-D) behaviour of soft tissue on geometries representative of such tissue in vivo will require the use of numerical methods. Toward this end, a pair of tetrahedral finite elements has been developed. The equations which are used to model the tissue behaviour for both elements are those commonly known as the linear biphasic equations. This model assumes that hydrated soft tissue is a mixture of two incompressible, immiscible phases, and employs mixture theory to derive governing equations for its mechanical behaviour. The finite element techniques applied to these equations for the two elements are the mixed-penalty method and the hybrid method. Both elements are described here, and the special requirements for 3-D analysis are discussed. Results obtained by solving canonical problems in two and three dimensions using both elements are presented and compared. Both elements are found to produce excellent results. The hybrid element is also noted to have advantages for non-linear analyses involving finite deformation which will require solution in the future.     

Micromechanics of Tension‐Compression Asymmetry of Polycrystalline Shape‐Memory‐Alloys

The asymmetry associated with martensitic transformations observed in tension/compression experiments of shape‐memory‐alloys (SMAs) is investigated on the basis of a recently suggested micromechanical model. The approach is based on crystallographic theory and utilizes a framework of energy minimization in a finite deformation context. Polycrystalline NiTi under tension demonstrates smaller phase‐transformation start‐strain, differe phase‐transformation stress‐levels and flatter phase‐transformation stress‐strain slopes than that under compression in our numerical simulation. The phase‐transformation start‐stress is followed to have a linear relationship with respect to the temperature within a certain range. These results agree well with experimental results reported in the literature.     

On conservation laws in viscoelastostatics

Summary Noether's theorem on variational principles in which the associated functionals are invariant under a group of infinitesimal transformations is used to establish conservation laws associated with linear viscoelastostatics. One of them may be regarded as a viscoelastic generalization of Rice'sJ-integral in elasticity. In addition, those conservation laws are discussed briefly in physics.     

A note on cyclic groups,finite fields,and the discrete logarithm problem

We show how the discrete logarithm problem in some finite cyclic groups can easily be reduced to the discrete logarithm problem in a finite field. The cyclic groups that we consider are the set of points on a singular elliptic curve over a finite field, the set of points on a genus 0 curve over a finite field given by the Pell equation, and certain subgroups of the general linear group.     

Recognition of Polynomial Plane Curves Under Affine Transformations

 We present a new method to decide if two algebraic plane curves are (or are not) affine equivalent. The method is based on describing the curves by means of local parametrizations around related points. To that end we introduce a new type of parametrizations, called Ancochea parametrizations, which are canonical under affine transformations. Received: February 1, 2001; revised version: February 8, 2002