fast algorithms for n dimensional restrictions of hard problems let m be a parallel ram with p processors and arithmetic operations addition and subtraction recognizing l n n in t steps inputs for m are given integer by integer not bit by bit then l can be recognized by a sequential linear search algorithm lsa in o n 4 log n t log p steps thus many n dimensional restrictions of np complete problems binary programming traveling salesman problem etc and even that of the uniquely optimum traveling salesman problem which is p 2 complete can be solved in polynomial time by an lsa this result generalizes the construction of a polynomial lsa for the n dimensional restriction of the knapsack problem previously shown by the author and destroys the hope of proving nonpolynomial lower bounds on lsas for any problem that can be recognized by a pram as above with 2 poly n processors in poly n time

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

topological lower bounds on algebraic random access machines we prove general lower bounds for set recognition on random access machines rams that operate on real numbers with algebraic operations times as well as rams that use the operations times lfloor rfloor we do it by extending a technique formerly used with respect to algebraic computation trees in the case of algebraic computation trees the complexity was related to the number of connected components of the set w to be recognized for rams four similar results apply to the number of connected components of w circ the topological interior of w two results use overline w circ the interior of the topological closure of w we present theorems that can be applied to a variety of problems and obtain lower bounds many of them tight for the following models 1 a ram which operates on real numbers using integers to address memory and either the operations times or times lfloor rfloor 2 a ram of each of the above instruction sets extended by allowing arbitrary real numbers to be used as memory addresses and adding a test for integer instruction 3 a ram of each of the above instruction sets which can compute with arbitrary real numbers as well as use them for memory addressing while the input is restricted to the integers for one result on this model we require that all program constants be rational

lower bounds for solving linear diophantine equations on random access machines the problem of recognizing the language l n l n k of solvable diophantine linear equations with n variables and solutions from o k n is considered the languages nen l n nen l n l the knapsack problem are np complete the o n 2 lower bound for l n 1 on linear search algorithms due to dobkin and lipton is generalized to an o n 2 log k 1 lower bound for l n k the method of klein and meyer auf der heide is further improved to carry over the o n 2 lower bound for l n 1 to random access machines rams in such a way that it holds for a large class of problems and for very small input sets by this method lower bounds that depend on the input size as is necessary for l n are proved thereby an o n 2 log k 1 lower bound is obtained for rams recognizing l n or l n k for inputs from 0 nk 0 n 2 n

a polynomial linear search algorithm for the n dimensional knapsack problem presentation d un algorithme de recherche lineaire qui reconnait le probleme de sac a dos a n dimensions par etapes de 2n 4 logn o n 3

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

on the limits of computations with the floor function abstract up to now few models of computation with the power of evaluating discontinuous functions have been analyzed and few of their lower bounds or results on the decidability of languages are known in this paper we present a model of an analytic computation tree act these trees operate on real numbers and are able to compare real numbers to evaluate functions on real numbers and to evaluate certain discontinuous functions like the floor function this model generalizes the model of algebraic computation trees introduced by ben or we show by topological arguments that by acts one cannot decide certain classes of languages examples of which are q n and the set of tuples x1 xn r n that have components which are z linearly or algebraically dependent

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

simulating probabilistic by deterministic algebraic computation trees a probabilistic algebraic computation tree probabilistic act which recognizes l rn in expected time t and which gives the wrong answer with probability 12 can be simulated by a deterministic act in o t2n steps the same result holds for linear search algorithms lsas the result for acts establishes a weaker version of results previously shown by the author for lsas namely that lsas can only be slightly sped up by their nondeterministic versions this paper shows that acts can only be slightly sped up by their probabilistic versions the result for lsas solves a problem posed by snir 1983 he found an example where probabilistic lsas are faster than deterministic ones and asked how large this gap can be

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

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

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

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

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

generalized finite automata over real and complex numbers in a recent work gandhi khoussainov and liu 7 introduced and studied a generalized model of finite automata able to work over arbitrary structures as one relevant area of research for this model the authors identify studying such automata over particular structures such as real and algebraically closed fields in this paper we start investigations into this direction we prove several structural results about sets accepted by such automata and analyze decidability as well as complexity of several classical questions about automata in the new framework our results show quite a diverse picture when compared to the well known results for finite automata over finite alphabets

lower time bounds for solving linear diophantine equations on several parallel computational models we consider parallel random access machines pram s with p processors and distributed systems of random access machines dram s with p processors being partially joint by wires according to a communication graph for these computational models we prove lower bounds for testing the solvability of linear diophantine equations and related problems including the knapsack problem these bounds are achieved by generalizing and simplifying a lower bound for parallel computation trees due to yao introducing a new type of computation trees which models computations of dram s and by generalizing a technique used by paul and simon and klein and meyer auf der heide to carry over lower bounds from computation trees to ram s thereby we prove that for many problems p processors cannot speed up a computation by a factor o p but only by a factor o log p 1 and in the case of dram s whose communication network has degree c by a factor o log c 1 only

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