A GROUP REPRESENTATION OF CANONICAL TRANSFORMATION

The mutual relationships between four generating functions F-1(q, Q), F-2(q, P), F-3(p, P), F-4(p, Q) and four kinds of canonical variables q, p, Q, P concerned in Hamilton's canonical transformations, can be got with linear transformations from seven basic formulae. All of them are Legendre's transformation, which are implemented by 32 matrices of 8 x 8 which are homomorphic to D-4 point group of 8 elements with correspondence of 4:1. Transformations and relationships of four state functions G(P, T), H(P, S), U(V, S), F(V, T) and four variables P, V, T, S in thermodynamics, are just the same Lagendre's transformations with the relationships of canonical transformations. The state functions of thermodynamics are summarily founded on experimental results of macroscope measurements, and Hamilton's canonical transformations are theoretical generalization of classical mechanics. Both group represents are the same, and it is to say, their mathematical frames are the same. This generality indicates the thermodynamical transformation is an example of one-dimensional Hamilton's canonical transformation.

A Disturbance Localization Method for Power System Based on Group Sparse Representation and Entropy Weight Method

This paper addresses the problem of complex and challenging disturbance localization in the current power system operation environment by proposing a disturbance localization method for power systems based on group sparse representation and entropy weight method.Three different electrical quantities are selected as observations in the compressed sensing algorithm.The entropy weighting method is employed to calculate the weights of different observations based on their relative disturbance levels.Subsequently,by leveraging the topological information of the power system and pre-designing an overcomplete dictionary of disturbances based on the corresponding system parameter variations caused by disturbances,an improved Joint Generalized Orthogonal Matching Pursuit(J-GOMP)algorithm is utilized for reconstruction.The reconstructed sparse vectors are divided into three parts.If at least two parts have consistent node identifiers,the node is identified as the disturbance node.If the node identifiers in all three parts are inconsistent,further analysis is conducted considering the weights to determine the disturbance node.Simulation results based on the IEEE 39-bus system model demonstrate that the proposed method,utilizing electrical quantity information from only 8 measurement points,effectively locates disturbance positions and is applicable to various disturbance types with strong noise resistance.

Representation of Spin Group Spin（p, q）

The representation η（P, q） of spin group Spin（p, q） in any dimensional space is given by induction, and the relation between two representations, which are obtained in two kinds of inductions from Spin（p, q） to Spin（p ＋ 1, q ＋ 1） are studied.

Special Unipotent Representations of Simple Linear Lie Groups of Type A

Let G be a special linear group over the real,the complex or the quaternion,or a special unitary group.In this note,we determine all special unipotent representations of G in the sense of Arthur and Barbasch-Vogan,and show in particular that all of them are unitarizable.

Coset Structure of Spin Group

This article considers One example is also given to take a the coset structure closer look at what of spin group via analyzing the expression of its representation. the coset and the subgroup are.

Reduction of truss topology optimization

A reduction of truss topology design problem formulated by semidefinite optimization （SDO） is considered. The finite groups and their representations are introduced to reduce the stiffness and mass matrices of truss in size. Numerical results are given for both the original problem and the reduced problem to make a comparison.

Everything Is Timelessly Connected: Metaphysics, Physics, Consciousness, Spacetime Travel, DNA, and the Dry Bones Prophesy (Ezekiel Ch. 37)

This paper provides a perception that all things are connected. Starting with the perception of Metaphysics and how matter exists out of total void, complementary matter, or dark matter, the incompleteness of Einstein relativistic theory. Multi spacetime universes and the jump drive for jumping within and between spacetime universes. Warp drive for space travel. DNA as sequence of momentary frequencies and how it related to Ezekiel’s dry bones prophecy. The outcome of neural patterns as cords of consciousness, consciousness as the collection of all cords of consciousness and the lack of uniqueness of individual consciousness. Finally, all things are cords of consciousness.

THE BANACH ALGEBRAS GENERATED BY OPERATORS WITH ONE-POINT SPECTRUM

In this article, we present some results related to the structure of Banach algebras generated by operators with one-point spectrum.

The calculation for C-G coefficients

The eigenfunction method put forward by Chen Jin-quan is illustrated. We apply this theory to the space group D1_ 6h. The selection rules of this space are worked out in the points of higher symmetry A,K,H in the first Brillion Zone. The C-G coefficients are calculated for K.HA.

Group Sparsity Residual Constraint Image Denoising Model with l_(1)/l_(2)Regularization

Group sparse residual constraint with non-local priors(GSRC)has achieved great success in image restoration producing stateof-the-art performance.In the GSRC model,the l_(1)norm minimization is employed to reduce the group sparse residual.In recent years,nonconvex regularization terms have been widely used in image denoising problems,which have achieved better results in denoising than convex regularization terms.In this paper,we use the ratio of the l_(1)and l_(2)norm instead of the l_(1)norm to propose a new image denoising model,i.e.,a group sparse residual constraint model with l_(1)/l_(2)minimization(GSRC-l_(1)/l_(2)).Due to the computational difficulties arisen from the non-convexity and non-linearity,we focus on a constrained optimization problem that can be solved by alternative direction method of multipliers(ADMM).Experimental results of image denoising show that the pro-posed model outperforms several state-of-the-art image denoising methods both visually and quantitatively.

Variables separated equations:Strikingly different roles for the Branch Cycle Lemma and the finite simple group classification

Davenport's Problem asks:What can we expect of two polynomials,over Z,with the same ranges on almost all residue class fields? This stood out among many separated variable problems posed by Davenport,Lewis and Schinzel.By bounding the degrees,but expanding the maps and variables in Davenport's Problem,Galois stratification enhanced the separated variable theme,solving an Ax and Kochen problem from their Artin Conjecture work.Denef and Loeser applied this to add Chow motive coefficients to previously introduced zeta functions on a diophantine statement.By restricting the variables,but leaving the degrees unbounded,we found the striking distinction between Davenport's problem over Q,solved by applying the Branch Cycle Lemma,and its generalization over any number field,solved by using the simple group classification.This encouraged Thompson to formulate the genus 0 problem on rational function monodromy groups.Guralnick and Thompson led its solution in stages.We look at two developments since the solution of Davenport's problem.Stemming from MacCluer's 1967 thesis,identifying a general class of problems,including Davenport's,as monodromy precise.R(iemann)E(xistence)T(heorem)'s role as a converse to problems generalizing Davenport's,and Schinzel's (on reducibility).We use these to consider:Going beyond the simple group classification to handle imprimitive groups,and what is the role of covers and correspondences in going from algebraic equations to zeta functions with Chow motive coefficients.

A Decomposition Theorem for G-Groups

Some classical results about linear representations of a finite group G have been also proved for representations of G on non-abelian groups （G-groups）. In this paper we establish a decomposition theorem for irreducible G-groups which expresses a suitable irreducible G-group as a tensor product of two projective G-groups in a similar way to the celebrated theorem of Clifford for linear representations. Moreover, we study the non-abelian minimal normal subgroups of G in which this decomposition is possible.

New method of applying conformal group to quantum fields

Most of previous work on applying the conformal group to quantum fields has emphasized its invariant aspects, whereas in this paper we find that the conformal group can give us running quantum fields, with some constants, vertex and Green functions running, compatible with the scaling properties of renormalization group method （RGM）. We start with the renormalization group equation （RGE）, in which the differential operator happens to be a generator of the conformal group, named dilatation operator. In addition we link the operator/spatial representation and unitary/spinor representation of the conformal group by inquiring a conformal-invariant interaction vertex mimicking the similar process of Lorentz transformation applied to Dirac equation. By this kind of application, we find out that quite a few interaction vertices are separately invaxiant under certain transformations （generators） of the conformal group. The significance of these transformations and vertices is explained. Using a particular generator of the conformal group, we suggest a new equation analogous to RGE which may lead a system to evolve from asymptotic regime to nonperturbative regime, in contrast to the effect of the conventional RGE from nonperturbative regime to asymptotic regime.

Some Special Families of Rank-2 Representations of π_(1) of Compact Riemann Surfaces

In this article,we give an explicit way to construct representations of thefundamental groupπ_(1)(X),where X is a hyperbolic curve over C.Our motivation isto study a special space in MDR(X,SL_(2)(C))which is called the space of permissibleconnections in Faltings(Compos Math 48(2):223-269,1983),or indigenous bundlesin Gunning(Math Ann 170:67-86,1967).We get representations by constructingHiggs bundles,and we show that the family we get intersects the space of permissibleconnections PC in a positive dimension.In this way,we actually get a deformation ofthe canonical representation in PC,and all these deformations are given by explicitconstructed Higgs bundles.We also estimate the dimension of this deformation space.

Tensor products of complementary series of rank one Lie groups

We consider the tensor product π_α ? π_βof complementary series representations π_α and π_β of classical rank one groups SO_0(n, 1), SU(n, 1) and Sp(n, 1). We prove that there is a discrete component π_(α+β)for small parameters α and β(in our parametrization). We prove further that for SO_0(n, 1) there are finitely many complementary series of the form π_(α+β+2j,)j = 0, 1,..., k, appearing in the tensor product π_α ? π_βof two complementary series π_α and π_β, where k = k(α, β, n) depends on α, β and n.

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp（n）/U（n）. We prove that an eigenvalue λs（f1,f2,...,fn） of the Lie group Sp（n）, corresponding to the representation with label （f1, f2,..., fn）, is an eigenvalue of the manifold Sp（n）/U（n）, if and only if f1, f2,..., fn are all even.

Orbifold Stiefel-Whitney Classes of Real Orbifold Vector Bundles over Right-Angled Coxeter Complexes

The author gives a definition of orbifold Stiefel-Whitney classes of real orbifold vector bundles over special q-CW complexes(i.e.,right-angled Coxeter complexes).Simi-larly to ordinary Stiefel-Whitney classes,orbifold Stiefel-Whitney classes here also satisfy the associated axiomatic properties.

A note on Higgs-de Rham flows of level zero

The notion of Higgs-de Rham flows was introduced by Lan et al.(2019),as an analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory.In this paper we investigate a small part of this theory,and study those Higgs-de Rham flows which are of level zero.We improve the original definition of level-zero Higgs-de Rham flows(which works for general levels),and establish a Hitchin-Simpson type correspondence between such objects and certain representations of fundamental groups in positive characteristic,which generalizes a classical results of Katz(1973).We compare the deformation theories of two sides in the correspondence,and translate the Galois action on the geometric fundamental groups of algebraic varieties defined over finite fields into the Higgs side.