decomposition of matrices into three involutions abstract in this paper we generalize some results of c s ballantine concerning products of three n n complex involutions we prove that each n n complex matrix a with determinant 1 and dimker a n 2 for all c is the product of three involutions on the other hand we show that if an n n complex matrix a is the product of three involutions then m 2n r 3 and m 3n 4 where m dimker a and r dimker a 3 for any 0 and 4 1 we also completely characterize products of three 5 5 complex involutions

finite size effects and the supersymmetric sine gordon models we propose nonlinear integral equations to describe the groundstate energy of the fractional supersymmetric sine gordon models the equations encompass the n 1 supersymmetric sine gordon model as well as the id id adj perturbation of the su 2 l su 2 k su 2 l k models at rational level k a second set of equations is proposed for the groundstate energy of the n 2 supersymmetric sine gordon model

strong periodic solutions for a class of abstract evolution equations abstract we study a class of abstract nonlinear evolution equations in a separable hilbert space for which we prove existence of strong time periodic solutions provided the right hand side is periodic and c1 in time and small enough in the norm of the considered space we prove also uniqueness and stability of the solutions the results apply in particular in several models of hydrodynamics such as magneto micropolar and micropolar models and classical magnetohydrodynamics and navier stokes models with non homogeneous boundary conditions the existence part of the proof is based on a set of estimates for the family of finite dimensional approximate solutions

conformal spectrum of the six vertex model by way of a generalization of analytical methods recently introduced the conformal spectrum of the six vertex model with twisted boundary conditions is analysed nonlinear integral equations are derived from which all scaling dimensions and the spins can be extracted analytically in terms of rogers dilogarithms the scaling dimensions of low lying excitations are given by the so called gaussian dimensions

finite size corrections and scaling dimensions of solvable lattice models an analytic method a general analytic method for calculating finite size corrections central charges and scaling dimensions of solvable lattice models is presented the approach is to solve the special functional equations or inversion identities satisfied by the commuting row transfer matrices of these lattice models at criticality for purposes of illustration the method is applied to calculate the central charge c 4 5 and leading magnetic scaling dimension x 2 15 of hard hexagons these numbers are rational due to special values of rogers dilogarithms

expressing infinite matrices as products of involutions abstract we take into consideration ut k the group of upper triangular infinite matrices whose entries lying on the main diagonal are equal to either 1 or 1 we show that for any field k every element of this group can be expressed as a product of at most five involutions moreover we prove that if k is a field of characteristic different from 2 then four involutions suffice

on condiagonalizable matrices abstract we call a m n c a condiagonalizable matrix if a r a a or which is the same a l a a is diagonalizable by a conventional similarity transformation our main result is that any condiagonalizable matrix can be brought by a consimilarity transformation to a special block diagonal form with the diagonal blocks of orders one and two

the time periodic solution for a 2d dissipative klein gordon equation abstract in this paper we study a 2d dissipative klein gordon equation with periodic boundary condition the existence and uniqueness of a time periodic solution is proved by the galerkin method and leray schauder fixed point theorem

critical properties of the izergin korepin and solvable o n models and their related quantum spin chains the central charge of the izergin korepin model the corresponding quantum spin chain and the o n model is calculated analytically via the bethe ansatz the calculation extends a technique recently developed for the zamolodchikov fateev model in addition critical exponents and the central charge for these models are obtained from numerical solutions of bethe ansatz equations for finite systems as a physical application the authors find the exponents v 12 23 and gamma 53 46 for the theta transition of polymers in two dimensions

six vertex model with rotated boundary conditions we investigate the six vertex model on a square lattice rotated through an arbitrary angle with respect to the coordinate axes a model recently introduced by litvin and priezzhev auxiliary vertices are used to define an inhomogeneous system which leads to a one parameter family of commuting transfer matrices a product of commuting transfer matrices can be interpreted as a transfer matrix acting on zigzag walls in the rotated system using an equation for commuting transfer matrices we calculate their eigenvalues finite size properties of the model are discussed from the viewpoint of the conformal field theory

new results for exactly solvable critical rsos models and vertex models in two dimensions a method for analytically calculating critical properties of two dimensional exactly solvable models is presented it is based on the calculation of finite size corrections of the critical system and on predictions by conformal field theory it is applied to the restricted solid on solid models found by andrews baxter and forrester the analysis of the eigenvalues of the finite size transfer matrices is performed using special functional equations of inversion identity type the results are given in terms of rogers dilogarithms an outlook is given on related problems e g the application to vertex models and the calculation of free energy and correlation lengths of integrable quantum chains at finite temperature

expressing infinite matrices over rings as products of involutions abstract let k be an arbitrary field and r be an arbitrary associative ring with identity 1 slowik in 12 proved that each matrix of ut k the group of upper triangular infinite matrices whose entries lying on the main diagonal are equal to either 1 or 1 can be expressed as a product of at most five involutions in this article we extend the investigate to an arbitrary associative ring r with identity 1 our conclusion is that every element of ut r can be expressed as a product of at most four involutions we also prove that for the complex field every element of t c the group of upper triangular infinite matrices whose entries lying on the main diagonal satisfy a a 1 can be expressed as a product of at most three coninvolutions matrices satisfying a a i

products of two involutions with prescribed eigenvalues and some applications abstract let f be a field in djokovic product of two involutions arch math 18 1967 582 584 it was proved that a matrix a f n n can be written as a bc for some involutions b c f n n if and only if a is similar to a 1 in this paper we describe the possible eigenvalues of the matrices b and c as a consequence in case char f 2 we describe the possible similarity classes of p 11 p 22 p 1 when the nonsingular matrix p p ij f n n i j 1 2 and p 11 f s s varies when f is an algebraically closed field and char f 2 we also describe the possible similarity classes of a ij f n n i j 1 2 when a 11 and a 22 are square zero matrices and a 12 and a 21 vary

on time periodic solutions asymptotic stability and bifurcations of navier stokes equations in this article we investigate the time periodic solutions for two dimensional navier stokes equations with nontrivial time periodic force terms under the time periodic assumption of the force term the existence of time periodic solutions for two dimensional navier stokes equations has received extensive attention from many authors with the smallness assumption of the time periodic force we show that there exists only one time periodic solution and this time periodic solution is globally asymptotically stable in the h 1 h1 sense without smallness assumption of the force term there is no stability analysis theory addressed it is expected that when the amplitude of the force term is increasing the time periodic solution is no longer asymptotically stable in the last part of the article we use numerical experiments to study the bifurcation of the time periodic solutions when the amplitude of the force is increasing extrapolating to the heating of the earth by the sun the bifurcation diagram hints that when the earth receives a relatively small amount of solar energy regularly the time periodic fluid patterns are asymptotically stable while when the earth receives too much solar energy even though in a time periodic way the time periodic pattern of the fluid motions will lose its stability

excited states in the twisted xxz spin chain we compute the finite size spectrum for the spin chain with twisted boundary conditions for anisotropy in the regime and arbitrary twist the string hypothesis is employed for treating complex excitations the bethe ansatz equtions are solved within a coupled nonlinear integral equation approach with one equation for each type of string the root of unity quantum group invariant periodic chain reduces to the chain with a set of twist boundary conditions an integer multiple of for this model the restricted hilbert space corresponds to a unitary conformal field theory and we recover all primary states in the kac table in terms of states with specific twist and strings

decomposition of matrices into commutators of involutions abstract we consider the group sl n f of all n n matrices with determinant 1 over a field f we prove that if f is the complex number field or the real number field every matrix a in sl n f is a product of at most two commutators of involutions moreover two is the smallest such number

time periodic solution of the viscous camassa holm equation abstract this paper discusses the viscous camassa holm equation with a periodic boundary condition the existence and uniqueness of a time periodic solution are investigated by using the galerkin method and leray schauder fixed point theorem

a real coninvolutory analog of the polar decomposition abstract we study properties of coninvolutory matrices e i and derive a canonical form under similarity as well as a canonical form under unitary consimilarity for them we show that any complex matrix has a coninvolutory dilation and we characterize the minimum size of a coninvolutory dilation of a square matrix we characterize the m by n complex matrices a that can be factored as a re with r real and e coninvolutory and we discuss the uniqueness of this factorization when a is square and nonsingular

involutions and characters of upper triangular matrix groups we study the realizability over r of representations of the group u n of upper triangular n x n matrices over f 2 we prove that all the representations of u n are realizable over r if n 12 but that if n 13 u n has representations not realizable over r this theorem is a variation on a result that can be obtained by combining work of j arregi and a vera lopez and of the authors but the proof of the theorem in this paper is much more natural

spatial anisotropy and rotational invariance of critical hard squares we develop a new method to analyse solvable models defined on a square lattice rotated through an arbitrary angle phi with respect to the coordinate axes in the method we introduce auxiliary faces into a rotated system to relate it with an inhomogeneous one in the natural orientation phi 0 the inhomogeneous system is investigated by commuting a transfer matrix argument as an application of the method we consider interacting hard squares at the critical point we discuss relations between spatial anisotropy and rotational or conformal invariance of the model in calculations of finite size properties