Rational Invariants of the Generalized Classical Groups

In this paper,we give transcendence bases of the rational invariants fields of the generalized classical groups and their subgroups B,N and T,and we also compute the orders of them.Furthermore,we give explicit generators for the rational invariants fields of the Borel subgroup and the Neron-Severi subgroup of the general linear group.

On the Construction and Classification of the Common Invariant Solutions for Some P(1,4) -Invariant Partial Differential Equations

We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.

Symmetries and Group-Invariant Solutions for Transonic Pressure-Gradient Equations

Lie symmetry group method is applied to study the transonic pressure-gradient equations in two-dimensionalspace.Its symmetry groups and corresponding optimal systems are determined,and several classes of irrotational groupinvariantsolutions associated to the symmetries are obtained and special case of one-dimensional rarefaction wave isfound.

Symmetry Reductions and Group-Invariant Solutions of (2+1)-Dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

With the aid of symbolic computation, we present the symmetry transformations of the (2+1)-dimensionalCaudrey-Dodd Gibbon-Kotera-Sawada equation with Lou's direct method that is based on Lax pairs. Moreover, withthe symmetry transformations we obtain the Lie point symmetries of the CDGKS equation, and reduce the equation withthe obtained symmetries. As a result, three independent reductions are presented and some group-invariant solutions ofthe equation are given.

The Invariant Rings of the Generalized Transvection Groups in the Modular Case

In this paper, first we investigate the invariant rings of the finite groups G ≤ GL（n, F;） generated by i-transvections and i-reflections with given invariant subspaces H over a finite field F;in the modular case. Then we are concerned with general groups G;（ω） and G;（ω）;named generalized transvection groups where ωis a k-th root of unity. By constructing quotient group and tensor, we calculate their invariant rings. In the end, we determine the properties of Cohen-Macaulay,Gorenstein, complete intersection, polynomial and Poincare series of these rings.

ON THE LOCAL SOLVABILITY OF DIFFERENTIAL EQUATIONS ASSOCIATED TO UNSOLVABLE INHOMOGENEOUS LEFT INVARIANT DIFFERENTIAL OPERATORS ON THE HEISENBERG GROUP

This paper studies the local solvability of the differental equations associated to unsolvable inhomogeneous left invariant differential operators on the Heisenberg group.It is provedthat for a class of inhomogeneous left invariant differental operators on the Heisenberg group,the local solvability of the corresponding equations is equivalent to the local sovability of the equations associated to their highest order terms.Then,under certain conditions on the highest order term,we obtain the necessary and sufficient conditions for the functon f to satisfy ill order for the differential equation Lu=f to be locally solvable

The Lattice of Fully Invariant Subgroups of the Cotorsion Hull

The paper considers the lattice of fully invariant subgroups of the cotorsion hull ?when a separable primary group T?is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation ?difined on the set ?of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice. The use of the relation ?essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group is isomorphic to the lattice of filters of the lower semilattice.

APPLICATION OF ONE-PARAMETER GROUPS OF TRANSFORMATION IN MECHANICS

In this paper,including some partial differential equations with a number of independent variables, which can he reduced by the infinitesimal form of the group, we obtain the theory of similarity transformation and its application of the second order nonlinear partial differential equations which have two independent variables and two dependent variables in mechanics.

Block Basis for Modular Coinvariants of Pseudo-Reflection Groups

We investigate the block basis for modular coinvariants of finite pseudo-reflection groups.We are particularly interested in the case of a subgroup G of the parabolic subgroups of GLn(q)which generalizes the Weyl groups of restricted Cartan typeLiealgebra.

Exact Solutions and Invariant Sets to General Reaction-Diffusion Equation

在这篇论文，我们介绍新不变的集合，并且到一般反应散开方程的不变的集合和准确答案被讨论。在那里存在，这被显示出属于不变的集合的方程的准确答案的一个班。

Continuous Symmetry Analysis of the Effects of City Infrastructures on Invariant Metrics for House Market Volatilities

The invariant metrics of the effects of park size and distance to public transportation on housing value volatilities in Boston,Milwaukee,Taipei and Tokyo are investigated.They reveal a Cobb-Douglas-like behavior.The scaleinvariant exponents corresponding to the percentage of a green area(a)are 7.4,8.41,14.1 and 15.5 for Boston,Milwaukee,Taipei and Tokyo,respectively,while the corresponding direct distances to the nearest metro station(d)are−5,−5.88,−10 and−10,for Boston,Milwaukee,Taipei and Tokyo,respectively.The multiphysics-based analysis provides a powerful approach for the symmetry characterization of market engineering.The scaling exponent ratio between park area percentages and distances to metro stations is approximately 3/2.The scaling exponent ratio expressed in the perceptual stimuli will remain invariant under group transformation.According to Stevens’power law,the perception-dependent feature spaces for parks and public transportation can be described as two-and three-dimensional conceptual spaces.Based on the prolongation structure of the Schroinger equation,the SL(2,R)models are used to analyze the house-price volatilities.Consistent with Shepard’s law,the rotational group leads to a Gaussian pattern,exhibiting an extension of the special linear group structure by embedding SO(3)■R(3)in SL(2,R).The influencing factors related to cognitive functioning exhibit substantially different scaleinvariant characteristics corresponding to the complexity of the socio-economic features.Accordingly,the contour shapes of the price volatilities obtained from the group-theoretical analysis not only corroborate the impact of the housing pricing estimation in these cities but also reveal the invariant features of their housing markets are faced with the forthcoming sustainable development of big data technologies and computational urban science research.

Relativistic Gauge Invariant Wave Equation of the Electron-Neutrino

With the right and the left waves of an electron, plus the left wave of its neutrino, we write the tensorial densities coming from all associations of these three spinors. We recover the wave equation of the electro-weak theory. A new non linear mass term comes out. The wave equation is form invariant, then relativistic invariant, and it is gauge invariant under the U(1)×SU(2), Lie group of electro-weak interactions. The invariant form of the wave equation has the Lagrangian density as real scalar part. One of the real equations equivalent to the invariant form is the law of conservation of the total current.

Including Space-Time in the Extended Group Cl3* of Relativistic Form-Invariance

The inclusion of space-time in the extended group of relativistic form-invariance, Cl3*, is specified as the inclusion of the whole space-time manifold in this multiplicative Lie group. First physical results presented here are: the geometric origin of the time arrow, a better understanding of the non-simultaneity in optics and a mainly geometric origin for the universe expansion, and its recent acceleration.

Invariant Sets and Exact Solutions to Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

<正> In this paper,we introduce a new invariant set _0={u:u_x=f'(x)F(u)+ε[g'(x)-f'(x)g(x)]F(u)×exp(-i∫~u 1/F(z) dz)},where f and g are some smooth functions of x,ε is a constant,and F is a smooth function to bedetermined.The invariant sets and exact solutions to nonlinear diffusion equation u_t=(D(u)u_x)_x+Q(x,u)u_x+P(x,u),are discussed.It is shown that there exist several classes of solutions to the equation that belong to the invariant set _0.

Kac-Moody-Virasoro Symmetry Algebra of (2+1)-Dimensional Dispersive Long-Wave Equation with Arbitrary Order Invariant

By Lie symmetry method, the Lie point symmetries and its Kac-Moody-Virasoro (KMV) symmetry algebra of (2+1)-dimensional dispersive long-wave equation (DLWE) are obtained, and the finite transformation of DLWE is given by symmetry group direct method, which can recover Lie point symmetries. Then KMV symmetry algebra of DLWE with arbitrary order invariant is also obtained. On basis of this algebra the group invariant solutions and similarity reductions are also derived.

Two Concepts in Optics of Anisotropic Dispersive Media and Polariton Case in Coordinate-Invariant Way

Two concepts of phenomenological optics of homogeneous, anisotropic and dispersive media are compared, the younger and more general concept of media with spatial dispersion and the older concept of (bi)-anisotropic media with material tensors for electric and magnetic induction which only depend on the frequency. The general algebraic form of the polarization vectors for the electric field and their one-dimensional projection operators is discussed without the degenerate cases of optic axis for which they become two-dimensional projection operators. Group velocity and diffraction coefficients in an approximate equation for the slowly varying amplitudes of beam solutions are calculated. As special case a polariton permittivity for isotropic media with frequency dispersion but without losses is discussed for the usual passive case and for the active case (occupation inversion of two energy levels that goes in direction of laser theory) and the group velocity is calculated. For this active case, regions of frequency and wave vector with group velocities greater than that of light in vacuum were found. This is not fully understood and due to large diffraction is likely only to realize in guided resonator form. The notion of “negative refraction” is shortly discussed but we did not find agreement with its assessment in the original paper.

基于数据表示不变性的域泛化研究

域泛化是人工智能近几年非常热门的一个研究方向,希望在不同的数据分布中学习到与任务相关的不变表征,即移除不同域在学习任务中的影响,从而提升模型的域泛化能力。为提升模型域泛化能力,利用基于不变性风险最小化的思想,具体将神经网络分为特征提取器和不变性分类器进行分别探究。在特征提取器上,采用了基于牛顿迭代的组白化方法来控制激活值的分布,从而使得不同的图像经过神经网络后能够去除部分域信息,以求达到域泛化的目的;在不变性分类器上,探究了特征和权重的规范化方法对模型域泛化效果的影响,并提出了基于余弦相似度损失函数的雪花算法,该算法提升了模型域泛化的准确率。此外,提供了关于雪花算法的理论推导并做了深入分析,为实验提供了理论支撑。

紧致齐性空间G/T上复结构的若干性质

设G为连通单连通s阶实幂零李群,且其上有左不变可积近复结构J,T为李群G的最大秩的格;考虑紧致齐性空间G/T上是否有幂零复结构;运用归纳法证明了复结构J可诱导出紧致齐性空间G/T的可积复结构J且是幂零的.

A CHARACTERIZATION OF LOCALLY COMPACT INNER AMENABLE GROUPS

For locally compact groups G, Kuan Yuan studied a notion of inner amenability groups, that is, if there exists an inner invariant mean on G. In this article, among other things, the author investigates the inner amenability on a locally compact group G. The author gives sufficient conditions and some necessary conditions about G to have an inner invariant mean.

Conditional Symmetry Groups of Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

The generalized conditional symmetry and sign-invarinat approaches are developed to study the nonlinear diffusion equations with x-dependent convection and source terms.We obtain conditions under which the equations admit the second-order generalized conditional symmetries and the first-order sign-invariants on the solutions.Several types of different generalized conditional symmetries and first-order sign-invariants for the equations with diffusion of power law are obtained.Exact solutions to the resulting equations are constructed.