ranking and formal power series abstract we prove that the ranking problem for unambiguous context free languages is nc 1 reducible to the value problem for algebraic formal power series in noncommuting variables in the particular case of regular languages we show that the problem of ranking is nc 1 reducible to the problem of counting the number of strings of given length in suitable regular languages as a consequence ranking problems for regular languages are nc 1 reducible to integer division and hence computable by log space uniform boolean circuits of polynomial size and depth o log n log log n or by p uniform boolean circuits of polynomial size and depth o log n

excision in algebraic k theory introduction 1 excision and the homology of affine groups 2 volodin s spaces 3 excision for rings with the triple factorization property 4 lie analogue of volodin s construction 5 the homology of nilpotent groups and nilpotent lie algebras 6 a proof of the main theorems 7 the tensor product of h unital algebras 8 the matrix ring of a small category 9 the acyclicity of triangular complexes 10 a proof of karoubi s conjecture

numeration systems on a regular language arithmetic operations recognizability and formal power series abstract generalizations of numeration systems in which n is recognizable by a finite automaton are obtained by describing a lexicographically ordered infinite regular language l for these systems we obtain a characterization of recognizable sets of integers in terms of n rational formal series after a study of the polynomial regular languages we show that if the complexity of l is n l resp if l is the complement of a polynomial language then multiplication by n preserves recognizability only if l 1 resp if for some n finally we obtain sufficient conditions for the notions of recognizability for abstract systems and some positional number systems to be equivalent

quadratic functors on pointed categories we study polynomial functors of degree 2 called quadratic with values in the category of abelian groups ab and whose source category is an arbitrary category c with null object such that all objects are colimits of copies of a generating object e which is small and regular projective this includes all categories of models v of a pointed theory t more specifically we are interested in such quadratic functors f from c to ab which preserve filtered colimits and suitable coequalizers a functorial equivalence is established between such functors f c ab and certain minimal algebraic data which we call quadratic c modules these involve the values on e of the cross effects of f and certain structure maps generalizing the second hopf invariant and the whitehead product applying this general result to the case where e is a cogroup these data take a particularly simple form this application extends results of baues and pirashvili obtained for c being the category of groups or of modules over some ring here quadratic c modules are equivalent with abelian square groups or quadratic r modules respectively

homological stability for automorphism groups abstract given a family of groups admitting a braided monoidal structure satisfying mild assumptions we construct a family of spaces on which the groups act and whose connectivity yields via a classical argument of quillen homological stability for the family of groups we show that stability also holds with both polynomial and abelian twisted coefficients with no further assumptions this new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups such as the symmetric groups general linear groups and mapping class groups by making systematic the proofs of classical stability results we show that they all hold with the same type of coefficient systems obtaining in particular without any further work new stability theorems with twisted coefficients for the symmetric groups braid groups automorphisms of free groups unitary groups mapping class groups of non orientable surfaces and mapping class groups of 3 manifolds our construction can also be applied to families of groups not considered before in the context of homological stability as a byproduct of our work we construct the braided analogue of the category fi of finite sets and injections relevant to the present context and define polynomiality for functors in the context of pre braided monoidal categories

a taxonomy of problems with fast parallel algorithms the class nc consists of problems solvable very fast in time polynomial in log n in parallel with a feasible polynomial number of processors many natural problems in nc are known in this paper an attempt is made to identify important subclasses of nc and give interesting examples in each subclass the notion of nc 1 reducibility is introduced and used throughout problem r is nc 1 reducible to problem s if r can be solved with uniform log depth circuits using oracles for s problems complete with respect to this reducibility are given for many of the subclasses of nc a general technique the parallel greedy algorithm is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in nc 2 solvable by uniform boolean circuits of depth o log 2 n and polynomial size the class logcfl is given a new characterization in terms of circuit families the class det of problems reducible to integer determinants is defined and many examples given a new problem complete for deterministic polynomial time is given namely finding the lexicographically first maximal clique in a graph this paper is a revised version of s a cook 1983 in proceedings 1983 intl found comut sci conf lecture notes in computer science vol 158 pp 78 93 springer verlag berlin new york

logarithmic depth circuits for algebraic functions this paper describes circuits for computation of a large class of algebraic functions on polynomials power series and integers for which it has been a long standing open problem to compute in depth less than omega log n 2 algebraic circuits assume unit cost for elemental addition and multiplication this paper describes o log n depth algebraic circuits which given as input the coefficients of n degree polynomials over an appropriate ring compute the product of n o 1 polynomials the symmetric functions as well as division and interpolation of real polynomials also described are o log n depth algebraic circuits which are given as input the first n coefficients of a power series over an appropriate ring compute the product of n o 1 power series as well as division reciprocal and reversion of real power series furthermore this paper describes boolean circuits of depth o log n log log n which given n bit binary numbers compute the product of n numbers and integ

dimension of the torelli group for out fn let mathcal t n be the kernel of the natural map out fn gln we use combinatorial morse theory to prove that mathcal t n has an eilenberg maclane space which is 2n 4 dimensional and that h 2n 4 mathcal t n mathbb z is not finitely generated n 3 in particular this shows that the cohomological dimension of mathcal t n is equal to 2n 4 and recovers the result of krstic mccool that mathcal t 3 is not finitely presented we also give a new proof of the fact due to magnus that mathcal t n is finitely generated

the structure of the torelli group i a finite set of generators for j this is the first of three papers concerning the so called torelli group let m mg be a compact orientable surface of genus g having n boundary components and let 9 xg be its mapping class group that is the group of orientation preserving diffeomorphisms of m which are 1 on the boundary am modulo isotopies which fix 3m pointwise this group is also known to the complex analysts as the teichmuller group or modular group if n 0 or 1 let further 4 jg be the subgroup of d1 which acts trivially on h1 m z the topologists have no traditional name for a but the analysts tell me it was known classically and is called the torelli group several interesting problems and conjectures exist concerning f the principal one can be found in kirby s problem list k and asks if gg is finitely generated in this first paper we shall answer the question affirmatively for both gg 0 and fgg when g 3 and shall give a fairly simple set of generators two other conjectures were made by the author the first involves the subgroup j of f which is generated by twists on nulhomologous simple closed curves ji produces a surjective homomorphism t fgg 1 a3h1 m z which kills c and it is conjectured there that in fact 5y ker t the proof of this is the content of the second paper in the third paper we use the results of the second to compute the abelianization f f explicitly thereby verifying another conjecture in ji the first reasonably simple but infinite set of generators for fg0 was produced by powell in p his generators consist of two types a twists on bounding simple closed curves b opposite twists on a bounding pair of disjoint homologous simple closed curves each of which are nonbounding using his result the author showed in j2 that the maps of the second type which we call bp maps for bounding pair are actually sufficient to generate both kg 0 and fg 1 for g 3 and in fact that we need only those whose two curves bound a genus one subsurface of m note that for g 2 all bp maps are trivial and hence the result fails in this case in the finite set of generators produced in this paper only

portfolio selection problems with random fuzzy variable returns this paper considers several portfolio selection problems including probabilistic future returns with ambiguous expected returns assumed as random fuzzy variables random fuzzy portfolio selection problems are formulated as nonlinear programming problems based on both stochastic and fuzzy programming approaches since there is no efficient solution method to solve these problems directly main problems are transformed into equivalent deterministic quadratic programming problems using probabilistic chance constraints possibility measure and fuzzy goals and their efficient solution methods to find a global optimal solution of each problem is constructed furthermore numerical examples of portfolio selection problems are provided to illustrate our proposed models and solution methods compared with several previous basic models and to show that our proposed model is a versatile model to be applicable to various unexpected conditions

stabilization for mapping class groups of 3 manifolds we prove that the homology of the mapping class group of any 3manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3 manifold when both manifolds are compact and orientable the stabilization also holds for the quotient group by twists along spheres and disks and includes as particular cases homological stability for symmetric automorphisms of free groups automorphisms of certain free products and handlebody mapping class groups our methods also apply to manifolds of other dimensions in the case of stabilization by punctures the main result of this paper is a homological stability theorem for mapping class groups of 3 manifolds where the stabilization is by connected sum with an arbitrary 3 manifold more precisely we show that given any two compact connected oriented 3 manifolds n and p with n 6 the homology group hi 0diff n p p rel n z is independent of the number n of copies of p in the connected sum as long as n 2i 2 i e each homology group stabilizes with p we also prove an analogous result for boundary connected sum and a version for the quotient group of the mapping class group by twists along spheres and disks a group closely related to the automorphism group of the fundamental group of the manifold homological stability theorems were first found in the sixties for symmetric groups by nakaoka 36 and linear groups by quillen and now form the foundation of modern algebraic k theory see for example 28 part iv and 42 stability theorems for mapping class groups of surfaces were obtained in the eighties by harer and ivanov 14 25 and recently turned out to be a key ingredient to a solution of the mumford conjecture about the homology of the riemann moduli space 30 the other main examples of families of groups for which stability has been known are braid groups 1 and automorphism groups of free groups 17 18 the present paper extends significantly the class of groups for which homological stability is known to hold it suggests that it is a widespread phenomenon among families of groups containing enough symmetries in addition to the already mentioned stability theorems for mapping class groups of 3 manifolds corollaries of our main result include stability for handlebody subgroups of surface mapping class groups symmetric automorphism groups of free groups and automorphism groups of free products ng for many groups g using similar techniques we obtain stability results also for mapping class groups 0diff m n points rel m for m any m dimensional manifold with boundary m 2 even the case m 2 is new here as well as for the fundamental group 1conf m n of the configuration space of n unordered points in m our paper thus also unifies previous known results as we recover stability for braid groups as 1conf d n symmetric groups as 1conf d n and automorphism groups of free groups as aut nz

the birman craggs johnson homomorphism and abelian cycles in the torelli group in the 1970s birman craggs johnson bcj trans ams 237 283 309 1978 trans ams 261 1 423 422 1980 used rochlin s invariant for homology 3 spheres to construct a remarkable surjective homomorphism sigma mathcal i g 1 to b 3 where mathcal i g 1 is the torelli group and b 3 is a certain bf f 2 vector space of boolean square free polynomials by pulling back cohomology classes and evaluating them on abelian cycles we construct 2g 4 o g 3 dimensions worth of nontrivial elements of h 2 mathcal i g 1 bf f 2 which cannot be detected rationally these classes in fact restrict to nontrivial classes in the cohomology of the subgroup mathcal k g 1 mathcal i g 1 generated by dehn twists about separating curves we also use the casson morita algebra and morita s integral lift of the bcj map restricted to mathcal k g 1 to give the same lower bound on h 2 mathcal k g 1 bf z

representation theory and homological stability abstract we introduce the idea of representation stability and several variations for a sequence of representations v n of groups g n a central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability this makes it possible to extend classical theorems of homological stability to a much broader variety of examples representation stability also provides a framework in which to find and to predict patterns from classical representation theory littlewood richardson and murnaghan rules stability of schur functors to cohomology of groups pure braid torelli and congruence groups to lie algebras and their homology to the equivariant cohomology of flag and schubert varieties to combinatorics the n 1 n 1 conjecture the majority of this paper is devoted to exposing this phenomenon through examples in doing this we obtain applications theorems and conjectures beyond the discovery of new phenomena the viewpoint of representation stability can be useful in solving problems outside the theory in addition to the applications given in this paper it is applied by church ellenberg farb in preparation 20 to counting problems in number theory and finite group theory representation stability is also used by church 2012 19 to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds

homological stability for completed homology we prove that the completed homology groups of mathrm gl n mathbf z in fixed degree stabilize as n goes to infinity we also prove that the action of hecke operators on stable cohomology is trivial in a precisely defined sense

log depth circuits for division and related problems we present optimal depth boolean circuits depth o log n for integer division powering and multiple products we also show that these three problems are of equivalent uniform depth and space complexity in addition we describe an algorithm for testing divisibility that is optimal for both depth and space

central stability for the homology of congruence subgroups and the second homology of torelli groups abstract we prove a representation stability result for the second homology groups of torelli subgroups of mapping class groups and automorphism groups of free groups this strengthens the results of boldsen hauge dollerup and day putman we also prove a new representation stability result for the homology of certain congruence subgroups partially improving upon the work of putman sam these results follow from a general theorem on syzygies of certain modules with finite polynomial degree

the complexity of computing the number of strings of given length in context free languages abstract computing the number of strings of given length contained in a language is related to classical problems of combinatorics formal languages and computational complexity here we study the complexity of this problem in the case of context free languages it is shown that for unambiguous context free languages such a computation is easy and can be carried out by efficient parallel algorithms on the contrary for some context free languages of ambiguity degree two the problem becomes intractable these results are related to other classical subjects concerning counting problems exponential time recognizable languages and sparse sets