the set of types ofn dimensional semisimple and cosemisimple hopf algebras is finite in this paper we give a partial answer to the above statement proving that the set of types of semisimple and cosemisimple hopf algebras of a given dimension is finite in any characteristic the proof uses the formalism of deformation theory of bialgebras which is concerned from a geometrical point of view with the study of the affine algebraic set b v of all bialgebra structures which can be defined on a vector space v there are two basic facts regarding b v first of all the transport of structures defines an action of gl v on b v such that the orbit of a bialgebra a contains all structures isomorphic to a so the set of types of bialgebras on v is in a one to one correspondence with the set of orbits second we are led in a natural way to a cohomology theory for bialgebras studying the tangent spaces t o and t b v where o is the closure a a a a of the orbit through a bialgebra a one of the most important results of the deformation theory of bialgebras is that the orbit of a bialgebra is open if the cohomology vanishes therefore the number of orbits which correspond to bialgebras having trivial cohomology is finite we obtain our result by proving that the cohomology of a semisimple hopf algebra a vanishes if the characteristic of k is 0 or more generally if a is

spiralling self avoiding walks an exact solution an exact solution is presented to a problem of spiralling self avoiding walks on the square lattice recently proposed by privman 1983 for n to infinity the number of n step spiral walks increases as cn approximately 2 23 5 4 pi n 7 4 exp 2 pi n 3 1 2 and their root mean square end to end distance behaves as rn approximately 1 2 square root pi 1n1 2 log n

a monoidal approach to splitting morphisms of bialgebras the main goal of this paper is to investigate the structure of hopf algebras with the property that either its jacobson radical is a hopf ideal or its coradical is a subalgebra let us consider a hopf algebra a such that its jacobson radical j is a nilpotent hopf ideal and h a j is a semisimple algebra we prove that the canonical projection of a on h has a section which is an h colinear algebra map furthermore if h is cosemisimple too then we can choose this section to be an h h bicolinear algebra morphism this fact allows us to describe a as a generalized bosonization of a certain algebra r in the category of yetter drinfeld modules over h as an application we give a categorical proof of radford s result about hopf algebras with projections we also consider the dual situation let a be a bialgebra such that its coradical is a hopf sub bialgebra with antipode then there is a retraction of the canonical injection of h into a which is an h linear coalgebra morphism furthermore if h is semisimple too then we can choose this retraction to be an h h bilinear coalgebra morphism then also in this case we can describe a as a generalized bosonization of a certain coalgebra r in the category of yetter drinfeld modules over h

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

on the number of spiral self avoiding walks on a triangular lattice an elementary derivation of the complete asymptotic expansion for the number sn of spiral self avoiding walks with n steps on a triangular lattice is given

spiral self avoiding walks self avoiding walks with a spiral constraint on the square lattice are enumerated up to 40 steps numerical evidence suggests that they belong to a new universality class

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

an exact formula for the number of spiral self avoiding walks an exact closed form expression is obtained for the number sn of spiral self avoiding walks with n steps on the square lattice this result is used to derive a complete asymptotic expansion for sn which is valid as n to infinity

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

spiral self avoiding walks on a triangular lattice end to end distance an exact solution is obtained for the root mean square end to end distance rn of spiral self avoiding walks with n steps on a triangular lattice for n to infinity rn increases as 2 pi 1 6n 1 2 log n

spiral self avoiding walks on a triangular lattice an exact result is obtained for the number sn of spiral self avoiding walks with n steps on a triangular lattice for n to infinity sn increases as 214 3 7 4 pi n 5 4 exp pi 2n 3 1 2

eigenvalue spread criteria in the particle swarm optimization algorithm for solving of constraint parametric problems this paper presents an alternative and efficient method for solving a class of constraint parametric optimization problems using particle swarm optimization algorithm pso in this paper for the first time pso is used for solving convex parametric programming but pso must be adaptive for doing it so for obtaining particles velocities adaptation weight and velocity boundaries in the updating of velocities are calculated in the recursive form computational complexity of the pso algorithm is decreased based on uniformity of population of particles the uniformity of which is obtained using eigenvalue spread of covariance of population two simple examples are provided for showing the efficiency of the proposed method for solving these examples eigenvalue spread criteria applied in the pso algorithm decrease 78 of computations

on the number of spiral self avoiding walks the authors consider the problem of spiral self avoiding walks as recently introduced by privman 1983 they prove that the number of n step spiral self avoiding walks is given by sn exp 2 pi n 3 12 n74 c 1 o 1 square root n where c pi 4 354 similar results for various subsets of these walks are also obtained

number of spiral self avoiding loops on a triangular lattice exact results are derived for the number cn of spiral self avoiding loops with n steps on a triangular lattice a closed form expression for the generating function g x sigma n 1infinity cnxn is obtained for n to infinity cn increases as n6 31 104

on the number of types of finite dimensional hopf algebras for an odd prime p we construct an infinite class of non isomorphic hopf algebras of dimension p 4 over an infinite field containing primitive p th roots of unity answering in the negative a long standing conjecture of kaplansky

generalized hopf ore extensions abstract we investigate some hopf algebra structures over an ore extension of a hopf algebra necessary and sufficient conditions for an ore extension of a hopf algebra to have a hopf algebra structure of a certain type are given this construction called a generalized hopf ore extension generalizes hopf ore extensions we describe the generalized hopf ore extensions of the enveloping algebras of lie algebras for some lie algebras g the generalized hopf ore extensions of u g are classified

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 exactly solvable self avoiding walks model ii triangular and honeycomb lattices a modified self avoiding walks model previously proposed for the square lattice is applied to the triangular and honeycomb lattices in the model the walker is restricted not to take any turn which will put the walker in a direction rotated by more than a certain angle phi max from any of the directions previously taken the generating functions are obtained and various quantities are evaluated exactly for all three square triangular and honeycomb lattices it is found that the model exhibits the characteristics reminiscent of one dimensional self avoiding walks in all directions while strongly retaining the anisotropic effect of the direction of the first step for the honeycomb lattice the authors have studied the model for phi max pi a model and phi max 3 2 pi b model unlike other walks the b model exhibits a special property the oscillations between even and odd steps in various quantities such as the number of n step walks do not decay as the number of steps increases