hopf algebras of low dimension the main aim of this paper is to classify all types of hopf algebras of dimension less thn or equal to 11 over an algebraically closed field of characteristic 0 if a is such a hopf algebra that is not semisimple then we shall prove that a or a is pointed this property will result from the fact that under some assumptions any hopf algebra that is generated as an algebra by a four dimensional simple subcoalgebra is a hopf quotient of the coordinate ring of quantum sl 2 k the first result allows us to reduce the classification to the case of pointed hopf algebras of dimension 8 we shall describe their types in the last part of the paper

an interior point cutting plane method for the convex feasibility problem with second order cone inequalities the convex feasibility problem in general is a problem of finding a point in a convex set that contains a full dimensional ball and is contained in a compact convex set we assume that the outer set is described by second order cone inequalities and propose an analytic center cutting plane technique to solve this problem we discuss primal and dual settings simultaneously two complexity results are reported the complexity of restoration procedure and complexity of the overall algorithm we prove that an approximate analytic center is updated after adding a second order cone cut socc ino 1 newton step and that the analytic center cutting plane method accpm with socc is a fully polynomial algorithm

stochastic boundary elements for two dimensional potential flow in homogeneous domains abstract a stochastic boundary element formulation is presented for the analysis of two dimensional steady state potential flow through homogeneous domains the operator of the governing differential equation is assumed to be random and is described by a set of correlated random variables the perturbation method in conjunction with the boundary element method is employed to derive the systems of equations for the unknown boundary variables and their respective first and second order derivatives with respect to the random variables these quantities are then used to calculate the desired response statistics a general procedure is developed which is next applied for the specific cases of random geometric configuration and random material parameter the random geometric configuration is modeled using a finite set of correlated random variables the random material parameter is modeled as a homogeneous random field which allows the use of deterministic fundamental solutions and integral representations for homogeneous domains the random field is first discretized into a set of correlated random variables and then the general procedure is applied a transformation of the correlated random variables into an uncorrelated set is performed to reduce the number of numerical operations the results for the boundary variables are used to calculate the response statistics of internal potentials these calculations require the modeling of the interior of the domain under consideration several models for representing the interior of the domain are presented for both random configuration and random material parameter and their influence on the response statistics is analysed distributed sources are considered in the present study using the particular integral approach a number of numerical examples are presented to demonstrate the validity of the present formulations the results obtained from the present analyses are compared with those obtained from monte carlo simulations with 5000 samples and a good agreement of results is observed

computation of heat conduction in materials with random variable thermophysical properties a method for the direct computation of mean values and variances of the temperature in conduction heated objects with random variable thermophysical properties is presented this method is based on a taylor expansion of the finite element formulation of the heat conduction equation and offers a powerful alternative to the computationally expensive monte carlo method both steady state and transient problems are considered some example problems are solved and the results discussed the simulations indicate that the variability of the thermophysical properties may cause a considerable variability of the temperature within the heated object this may have important consequences on the design of heating operations in food process engineering

random field finite elements the probabilistic finite element method pfem is formulated for linear and non linear continua with inhomogeneous random fields analogous to the discretization of the displacement field in finite element methods the random field is also discretized the formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization furthermore it is shown that a reduced set of the uncorrelated variables is sufficient for the second moment analysis based on the linear formulation of the pfem the method is then extended to transient analysis in non linear continua the accuracy and efficiency of the method is demonstrated by application to a one dimensional elastic plastic wave propagation problem and a two dimensional plane stress beam bending problem the moments calculated compare favourably with those obtained by monte carlo simulation also the procedure is amenable to implementation in deterministic fem based computer programs

stochastic boundary elements for two dimensional potential flow in non homogeneous media abstract a stochastic boundary element formulation is presented for the treatment of two dimensional problems of steady state potential flow in non homogeneous media that involve a random operator in the governing differential equation the randomness is introduced through the material parameter of the domain which is described as a non homogeneous random field the random field is discretized into a set of correlated random variables and a perturbation is applied to the differential equation of the problem this leads to differential equations for the unknown potential and its first and second order derivatives respectively evaluated at the mathematical expectations of the random variables resulting from the discretization of the random field an approximate method is applied for the solution of these equations by expressing the potential and its derivatives as a sum of functions of descending order these solutions are introduced into the differential equations and upon equating similar order terms a sequence of poisson s equations is obtained for each order a transformation of the correlated random variables into an uncorrelated set is performed to reduce the number of numerical operations by retaining a small number of transformed random variables the resulting equations are solved using the boundary element method to obtain the unknown boundary values of the potentials and their respective first and second order derivatives which are then used to compute the desired response statistics quadratic conforming boundary elements are used in the boundary integration and four node quadrilateral cells are used in the domain integration strongly singular terms of the boundary element matrices are obtained indirectly by applying a state of uniform unit potential over the entire contour of the object the singular domain integrals are calculated analytically direct solution techniques are used to calculate the response variables and their derivatives respectively a number of example problems are presented and the results are compared with those obtained from monte carlo simulations a good agreement of the results is observed

a nonlinear analytic center cutting plane method for a class of convex programming problems a cutting plane algorithm for minimizing a convex function subject to constraints defined by a separation oracle is presented the algorithm is based on approximate analytic centers the nonlinearity of the objective function is taken into account yet the feasible region is approximated by a containing polytope this containing polytope is regularly updated by adding a new cut through a test point each test point is an approximate analytic center of the intersection of a containing polytope and a level set of the nonlinear objective function we establish the complexity of the algorithm our complexity estimate is given in terms of the problem dimension the desired accuracy of an approximate solution and other parameters that depend on the geometry of a specific instance of the problem

abelian and non abelian second cohomologies of quantized enveloping algebras abstract for a class of pointed hopf algebras including the quantized enveloping algebras we discuss cleft extensions cocycle deformations and the second cohomology we present such a non standard method of computing the abelian second cohomology that derives information from the non abelian second cohomology classifying cleft extensions as a sample computation a quantum analogue of whitehead s second lemma for lie algebra cohomology is proved

primitive cohomology of hopf algebras abstract primitive cohomology of a hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra among many of its applications two classifications are presented firstly we classify all non locally pi pointed hopf algebra domains of gelfand kirillov dimension two and secondly we classify all pointed hopf algebras of rank one the first classification extends some results of brown goodearl and others in an ongoing project to understand all hopf algebras of low gelfand kirillov dimension the second generalizes results of krop radford and wang you chen which classified hopf algebras of rank one under extra hypothesis properties and algebraic structures of the primitive cohomology are discussed

conforming boundary elements in plane elasticity for shape design sensitivity the structural design sensitivity analysis of a two dimensional continuum using conforming continuous boundary elements is investigated implicit differentiation of the discretized boundary integral equations is performed to obtain design sensitivities in an efficient manner by avoiding the factorization of the perturbed matrices a singular formulation of the boundary element method is used implicit differentiation of the boundary integral equations produces terms that contain derivatives of the fundamental solutions employed in the analysis the behaviour of the singularity of these derivatives of the boundary element kernel functions with respect to the design variables is investigated a rigid body motion technique is presented to obtain the singular terms in the resulting sensitivity matrices thus avoiding the problems associated with their numerical integration a formulation for obtaining the design sensitivities of the continua under body forces of the gravitational and centrifugal types is also presented the design sensitivity results are seen to be of the same order of accuracy as the boundary element analysis results numerical data comparing the performance of conforming and non conforming formulations in the calculation of design sensitivities are also presented the accuracy of the present results is demonstrated through comparisons with existing analytical results

stochastic boundary elements in elastostatics abstract a stochastic boundary element formulation for the treatment of boundary value problems in two dimensional elastostatics that involve a random operator is presented a general perturbation procedure is formulated for the set of correlated random variables governing the response of the solid this procedure is then specialized for the cases of a random geometry and b random material properties the problems involving a random configuration are analyzed using the random variable model and those with a random material property are analyzed using the random field model the random field is first discretized into a set of correlated random variables which are then transformed into an uncorrelated set to simplify the analysis the derivatives of the boundary element matrices appearing in the systems of equations from the perturbation of the random variables are derived analytically the direct solution methods are used to obtain the response variables and their first and second order derivatives respectively quadratic conforming boundary elements are employed in the boundary element discretization and the strongly singular terms of the boundary element matrices and their first and second order derivatives are obtained using the conditions associated with rigid body motions of the solid the present formulation has been evaluated for a number of example problems through comparisons with the solutions obtained by monte carlo simulation a good agreement of the results is observed

fuzzy boundary element method for the analysis of imprecisely defined systems many engineering systems contain uncertainties that cannot be described by either deterministic or probabilistic approaches the uncertainties present may be associated with parameters that are vague imprecise or linguistic for example the geometry material properties external actions loads and boundary conditions may be imprecise in a practical system a fuzzy boundary element method for the analysis of imprecisely defined systems is developed starting from the basic concepts of fuzzy sets and fuzzy arithmetic the various steps of the boundary element method are redefined using fuzzy concepts the resulting fuzzy equations are solved using a fuzzified version of gaussian elimination procedure coupled with truncation the truncation method limits the growth of interval ranges of response parameters to obtain realistic and accurate solutions the procedure is illustrated by considering the analysis of a potential flow problem the methodology is applicable to uncertain systems that are described in linguistic terms as well as those that are described by incomplete information the approach represents a unique methodology that enables the analysis and design of many engineering systems more realistically

classification of connected hopf algebras of dimension p 3 i abstract let p be a prime and k be an algebraically closed field of characteristic p in this paper we provide the classification of connected hopf algebras of dimension p 3 except for the case when the primitive space of the hopf algebra is a two dimensional abelian restricted lie algebra each isomorphism class is presented by generators x y z with relations and hopf algebra structures let be the multiplicative group of p 2 p 1 th roots of unity when the primitive space is one dimensional and p is odd there is an infinite family of isomorphism classes which is naturally parameterized by a k 1

basis random variables in finite element analysis galerkin s method is applied to random operator equations appropriate hilbert spaces are defined for random functions and solutions are projected into these spaces allowing the first and second moment properties of the solution to be calculated an equivalent energy based approach similar to the rayleigh ritz method is developed from which a stochastic finite element technique is derived several one and two dimensional example problems are solved and the results discussed

an analytic center quadratic cut method for the convex quadratic feasibility problem we consider a quadratic cut method based on analytic centers for two cases of convex quadratic feasibility problems in the first case the convex set is defined by a finite yet large number n of convex quadratic inequalities we extend quadratic cut algorithm of luo and sun 3 for solving such problems by placing or translating the quadratic cuts directly through the current approximate center we show that in terms of total number of addition and translation of cuts our algorithm has the same polynomial worst case complexity as theirs 3 however the total number of steps where steps consist of damped newton steps function evaluations and arithmetic operations required to update from one approximate center to another is where e is the radius of the largest ball contained in the feasible set in the second case the convex set is defined by an infinite number of certain strongly convex quadratic inequalities we adapt the same quadratic cut method for the first case to the second one we show that in this case the quadratic cut algorithm is a fully polynomial approximation scheme furthermore we show that at each iteration k the total number steps as described above required to update from one approximate center to another is at most with e as defined above

lifting of quantum linear spaces and pointed hopf algebras of orderp3 abstract we propose the following principle to study pointed hopf algebras or more generally hopf algebras whose coradical is a hopf subalgebra given such a hopf algebraa consider its coradical filtration and the associated graded coalgebra gr a then gr ais a graded hopf algebra since the coradicala0ofais a hopf subalgebra in addition there is a projection gr a a0 letrbe the algebra of coinvariants of then by a result of radford and majid ris a braided hopf algebra and gr ais the bosonization or biproduct ofranda0 gr a r a0 the principle we propose to studyais first to studyr then to transfer the information to gr avia bosonization and finally to lift toa in this article we apply this principle to the situation whenris the simplest braided hopf algebra a quantum linear space as consequences of our technique we obtain the classification of pointed hopf algebras of orderp3 pan odd prime over an algebraically closed field of characteristic zero with the same hypothesis the characterization of the pointed hopf algebras whose coradical is abelian and has indexporp2 and an infinite family of pointed nonisomorphic hopf algebras of the same dimension this last result gives a negative answer to a conjecture of i kaplansky

an analytic center based column generation algorithm for convex quadratic feasibility problems we consider the analytic center based column generation algorithm for the problem of finding a feasible point in a set defined by a finite number of convex quadratic inequalities at each iteration the algorithm computes an approximate analytic center of the set defined by the intersection of quadratic inequalities generated in the previous iterations if this approximate analytic center is a solution then the algorithm terminates otherwise a quadratic inequality violated at the current center is selected and a new quadratic cut defined by a convex quadratic inequality is placed near the approximate center as the number of cuts increases the set defined by their intersection shrinks and the algorithm eventually finds a solution to the problem previously similar analytic center based column generation algorithms were studied first for the linear feasibility problem and later for the general convex feasibility problem our method differs from these early methods in that we use quadratic cuts in the computation instead of linear cuts moreover our method has a polynomial worst case complexity of o n ln frac 1 varepsilon on the total number of cuts to be used where n is the number of convex quadratic polynomial inequalities in the problem and varepsilon is the radius of the largest ball contained in the feasible set in contrast the early column generation methods using linear cuts can only solve the convex quadratic feasibility problem in pseudopolynomial time

hochschild cohomology and the coradical filtration of pointed coalgebras applications abstract in this paper we show that there is a close connection between the coradical filtration of a pointed coalgebra and the hochschild cohomology of that coalgebra with coefficients in some one dimensional bicomodules as an application for a given prime numberpand an algebraically closed fieldkof characteristic 0 we classify all pointed hopf algebras of dimensionp3overk

finite quantum groups and cartan matrices abstract we consider a class of hopf algebras having as an invariant a generalized cartan matrix for a hopf algebra of this kind we prove that it is finite dimensional if and only if its generalized cartan matrix is actually a finite cartan matrix under some mild hypothesis these results allow us to classify all the finite dimensional coradically graded pointed hopf algebras whose coradical has odd prime dimension p we also characterize coradically graded pointed hopf algebras of order p 4

an analytic center cutting plane approach for conic programming we analyze the problem of finding a point strictly interior to a bounded convex and fully dimensional set from a finite dimensional hilbert space we generalize the results obtained for the linear programming lp semidefinite programming sdp and second order core programming socp cases the cuts added by our algorithm are central and conic in our analysis we find an upper bound for the number of newton steps required to compute an approximate analytic center also we provide an upper bound for the total number of cuts added to solve the problem this bound depends on the quality of the cuts the dimensionality of the problem and the thickness of the set we are considering

connected hopf algebras of dimension p2 abstract let h be a finite dimensional connected hopf algebra over an algebraically closed field k of characteristic p 0 we provide the algebra structure of the associated graded hopf algebra gr h then we study the case when h is generated by a hopf subalgebra k and another element and the case when h is cocommutative when h is a restricted universal enveloping algebra we give a specific basis for the second term of the hochschild cohomology of the coalgebra h with coefficients in the trivial h bicomodule k finally we classify all connected hopf algebras of dimension p 2 over k

a new boundary element formulation for two and three dimensional thermoelasticity using particular integrals new two and three dimensional boundary element formulations are developed for steady state and transient uncoupled thermoelasticity these new procedures differ from previous work in that additional surface or volume integration is not required to incorporate thermal loads in the analysis instead thermal body forces are introduced in the boundary element system via particular integrals the present formulation is implemented in a general purpose multi region system and examples are presented to demonstrate the accuracy and versatility of the method

pointed hopf algebras of dimensionp3 abstract we give a structure theorem for pointed hopf algebras of dimension p 3 having coradical kc p where k is an algebraically closed field of characteristic zero combining this with previous results we obtain the complete classification of all pointed hopf algebras of dimension p 3

wavelet based model for stochastic analysis of beam structures a wavelet based model for stochastic analysis of beam structures is presented in this model the random processes representing the stochastic material and geometric properties are treated as stationary gaussian processes with specified mean and correlation functions using the karhunen loeve expansion the process is represented as a linear sum of orthonormal eigenfunctions with uncorrelated random coefficients the correlation and the eigenfunctions are approximated as truncated linear sums of compactly supported orthogonal wavelets and the integral eigenvalue problem is converted to a finite dimensional eigenvalue problem the energy principle based finite element approach is used to obtain the equilibrium and boundary conditions neumann expansion of the stiffness matrix is used to write the nodal displacement vector in terms of random coefficients the expectation operator is applied to the nodal displacements and their squares to obtain the mean and standard deviation of the displacements studies show that the results obtained using this method compare well with monte carlo and semianalytical techniques

the analytic center quadratic cut method for strongly monotone variational inequality problems convergence of an algorithm for strongly monotone variational inequality problems vips is investigated at each iteration the algorithm adds a quadratic cut through the analytic center of the consequently shrinking convex set it is shown that the sequence of analytic centers converges to the unique solution in cal o 1 sqrt k where k is the number of iterations

the analytic center cutting plane method with semidefinite cuts we analyze an analytic center cutting plane algorithm for convex feasibility problems with semidefinite cuts the problem of interest seeks a feasible point in a bounded convex set which contains a full dimensional ball with varepsilon 0 is a condition number of the field of cuts

constructing pointed hopf algebras by ore extensions abstract we present a general construction producing pointed co frobenius hopf algebras and give some classification results for the examples obtained

on hopf algebras of dimension 4p abstract in this paper we prove that a non semisimple hopf algebra h of dimension 4 p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided h contains more than two group like elements in particular we prove that non semisimple hopf algebras of dimensions 20 28 and 44 are pointed or their duals are pointed and this completes the classification of hopf algebras in these dimensions

finite dimensional cosemisimple hopf algebras in characteristic 0 are semisimple finite dimensional cosemisimple hopf algebras are a natural generalization of group algebras much of the representation theory of finite groups can be extended to such hopf algebras however there are still many unanswered basic questions for such hopf algebras one such question is a conjecture mentioned by kaplansky i the square of the antipode of a finite dimensional cosemisimple hopf algebra is the identity in this paper we prove that the fourth power of the antipode is the identity in a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 in the course of proving this we investigate two elements of the hopf algebra which are roughly analogous to the sum of the group elements in a group algebra we show that kaplansky s conjecture is equivalent to these two elements being equal for a finite dimensional hopf algebra a with antipode s over a field let l p q pq for p q e a denote the left module action of a on itself in this paper we study two elements of a defined by the trace function the element i defined by p a tr l p s for all pe a and the element x defined by p x tr l p for all p e a we show that 2 is a left integral and use it to prove theorem 3 3 the main result of this paper a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 is also semisimple hence its antipode has order 1 2 or 4 we extend the first corollary to 3 proposition 91 in theorems 4 3 and 4 4 by finding necessary and sufficient conditions for s to be a non zero left integral in which case x a and also study how x relates to the structure of a in particular when dim a 1 0 the right ideal xa may be thought of as a measure of the extent to which s i the elements ji and k are two examples of elements of a associated with an endomorphism of a in general for f end a we define a e a by p a tr l p cf for all pea thus i as2 and x a in section 2 267 0021 8693 88 33 00

iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems a new algorithm for the computation of the spectral expansion of the eigenvalues and eigenvectors of a random non symmetric matrix is proposed the algorithm extends the deterministic inverse power method using a spectral discretization approach the convergence and accuracy of the algorithm is studied for both symmetric and non symmetric matrices the method turns out to be efficient and robust compared to existing methods for the computation of the spectral expansion of random eigenvalues and eigenvectors