topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

a simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension it has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplex type algorithms for linear programming in this paper the average number of steps performed by a simplex algorithm the so called self dual method is analyzed the algorithm is not started at the traditional point 1 l t but points of the form 1 e e 2 t with e sufficiently small are used the result is better in two respects than those of the previous analyses first it is shown that the expected number of steps is bounded between two quadratic functions c 1 min m n 2 and c 2 min m n 2 of the smaller dimension of the problem this should be compared with the previous two major results in the field borgwardt proves an upper bound of o n 4 m 1 n 1 under a model that implies that the zero vector satisfies all the constraints and also the algorithm under his consideration solves only problems from that particular subclass smale analyzes the self dual algorithm starting at 1 1 t he shows that for any fixed m there is a constant c m such the expected number of steps is less than c m ln n m m 1 megiddo has shown that under smale s model an upper bound c m exists thus for the first time a polynomial upper bound with no restrictions except for nondegeneracy on the problem is proved and for the first time a nontrivial lower bound of precisely the same order of magnitude is established both borgwardt and smale require the input vectors to be drawn from spherically symmetric distributions in the model in this paper invariance is required only under certain

on the distributions of scan statistics abstract this article considers the distribution of the scan statistic over some interval of the real line both for a fixed number of independent uniform points and for a poisson process it provides a new series method of calculating this distribution in the poisson case and a simple self contained recursive approximation in the fixed case these results work best at low densities which is precisely the case when the closed formulas are computationally impractical the results are compared with values tabulated by neff and naus 1980 and with a recent approximation due to naus 1982 there is also an explanation of why naus s approximation is good in the poisson case and a derivation of recursive bounds in the fixed case

on the average number of steps of the simplex method of linear programming the goal is to give some theoretical explanation for the efficiency of the simplex method of george dantzig fixing the number of constraints and using dantzig s self dual parametric algorithm we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average

a useful upper bound for the tail probabilities of the scan statistic when the sample size is large abstract although a number of approximations to the distribution under uniformity of the scan statistic are known they are when the sample size is large either difficult to compute or numerically unstable a simple upper bound for the upper tail of the distribution based on a result of hunter 1976 is derived it is found that for large sample sizes and a reasonably wide range of window widths this bound gives a numerically stable approximation that is sufficiently accurate for hypothesis testing purposes

detection of minimal epidemics epidemiologists seeking epidemics of events in time and attempting to distinguish real epidemics from random clusters need to calculate the random probability of n or more events occurring within a time window of duration w and for any position of the window as it scans continuously across the period of observation exact algebraic solutions have been proposed by other workers but over a wide range of important values systematic arithmetic evaluation is beyond the capacity of existing computers this paper proposes an approximate test which is quick and simple and which achieves the same purpose an example of its application is given

some distribution independent results about the asymptotic order of the average number of pivot steps of the simplex method this paper is concerned with the average number of pivot steps of the simplex method which are required to solve linear programming problems of the following kind mbox max v t x quad mbox subject to a t i x leq 1 quad mbox for i 1 ldots m mbox where a i v x in mathbb r n we investigate a certain variant of the simplex method which is particularly suited for theoretical considerations for every pair m nm n we introduce probability spaces whose random samples are the linear programming problems of the above mentioned type the vectors ai v are supposed to be distributed independently identically and symmetrically under rotations on a bounded subset of rn we deduce upper and lower bounds for the expectation values of the number of steps required in phase ii of the simplex method the evaluation of these bounds yields the following result for m and for fixed n 1 the expectation values become om1 n 1 2 there exists a distribution whose expectation values increase like m1 n 1 3 for any distribution which satisfies our conditions the expectation values tend to infinity 4 for every e 0 there are distributions whose expectation values become ome

on smith s determinant abstract we give a brief review of papers relating to smith s determinant and point out a common structure that can be found in many extensions and analogues of smith s determinant we present the common structure in the language of posets we also make an investigation on a conjecture of beslin and ligh on greatest common divisor gcd matrices in the sense of meet matrices and give characterizations of the posets satisfying the conjecture further we give a counterexample for the conjecture of bourque and ligh that the least common multiple matrix on any gcd closed set is invertible

the efficiency of the simplex method a survey the linear programming problem is by far the most widely used optimization model its impact on economic and government modeling is immense the simplex method for solving the linear programming lp problem due to george dantzig has been an extremely efficient computational tool for almost four decades the method has been the subject of intense investigations for many years but some major aspects of its behavior are not fully understood yet the purpose of this paper is to survey the body of knowledge on the efficiency of the simplex method from both practical and theoretical points of view adopting the number of iterations pivot steps as the yardstick for efficiency we survey four aspects of the issue 1 reports on practical experience of the method s performance on real life lp problems 2 results on controlled monte carlo experiments solving lp problems which were randomly generated according to some predetermined distributions 3 complexity results including theoretical analyses on both upper and lower bounds for the performance of the simplex as well as non simplex algorithms for lp 4 results of recent theoretical studies using probabilistic analysis to derive bounds on the average behavior of the simplex method we discuss the consequences and limitations of the various studies special emphasis is given to open questions

a test for spatial disease clustering adjusted for multiple testing tango 1995 proposed a test statistic c for detecting spatial disease clusters however like most other methods it requires a value of the scale parameter which adjusts for the size of cluster to be detected in advance of its use when an appropriate size of cluster cannot be predicted and many clustered areas are expected we usually repeat the procedure by changing the size parameter but this practice clearly faces a multiple testing problem in this paper to ameliorate this problem we propose an extended test statistic which searches the minimum of the profile p value of c for the parameter on cluster size which varies continuously from a small value near zero upwards until it reaches to about half the size of the whole study area monte carlo simulation study shows that the power of the proposed method is shown to be reasonably high compared with the best power attained by c unadjusted for multiple testing in all the cluster models considered the proposed procedure is illustrated with some disease maps simulated in the tokyo metropolitan area

quasi quantum groups knots three manifolds and topological field theory we show how to construct starting from a quasi hopf algebra or quasiquantum group invariants of knots and links in some cases these invariants give rise to invariants of the three manifolds obtained by surgery along these links this happens for a finite dimensional quasi quantum group whose definition involves a finite groupg and a 3 cocycle which was first studied by dijkgraaf pasquier and roche we treat this example in more detail and argue that in this case the invariants agree with the partition function of the topological field theory of dijkgraaf and witten depending on the same datag

an approximation for the distribution of the scan statistic the scan statistic evaluates whether an apparent cluster of disease in time is due to chance the statistic employs a moving window of length w and finds the maximum number of cases revealed through the window as it scans or slides over the entire time period t computation of the probability of observing a certain size cluster under the hypothesis of a uniform distribution is infeasible when n the total number of events is large and w is of moderate or small size relative to t we give an approximation that is an asymptotic upper bound easy to compute and for the purposes of hypothesis testing more accurate than other approximations presented in the literature the approximation applies both when n is fixed and when n has a poisson distribution we illustrate the procedure on a data set of trisomic spontaneous abortions observed in a two year period in new york city

approximations for distributions of scan statistics abstract certain statistical applications deal with the extremal distributions of the number of points in a moving interval or window of fixed length this article gives an approximation that is highly accurate for several of these distributions applications include the maximum cluster of points on a line or circle multiple coverage by subintervals or subarcs of fixed size the length of the longest success run in bernoulli trials and the generalized birthday problem

semisimple hopf algebras via geometric invariant theory abstract we study hopf algebras via tools from geometric invariant theory we show that all the invariants we get can be constructed using the integrals of the hopf algebra and its dual together with the multiplication and the comultiplication and that these invariants determine the isomorphism class of the hopf algebra we then define certain canonical subspaces i n v i j of tensor powers of h and h and use the invariant theory to prove that these subspaces satisfy a certain non degeneracy condition using this non degeneracy condition together with results on symmetric monoidal categories we prove that the spaces i n v i j can also be described as h i h j a where a is the group of hopf automorphisms of h as a result we prove that the number of possible hopf orders of any semisimple hopf algebra over a given number ring is finite we give some examples of these invariants arising from the theory of frobenius schur indicators and from reshetikhin turaev invariants of three manifolds we give a complete description of the invariants for a group algebra proving that they all encode the number of homomorphisms from some finitely presented group to the group we also show that if all the invariants are algebraic integers then the hopf algebra satisfies kaplansky s sixth conjecture the dimensions of the irreducible representations of h divide the dimension of h

a sharp upper bound for the expected number of shadow vertices in lp polyhedra under orthogonal projection on two dimensional planes consider a set a1 am u v n satisfying a mild nondegeneracy condition when the polyhedron x x a1tx 1 amtx 1 n is orthogonally projected on the two dimensional plane linu v then some of its vertices may be mapped on vertices of the two dimensional image of x these vertices of x will be called shadow vertices under that projection their number will be denoted by s this figure s has an outstanding relevance for the complexity of a special simplex method the dimension by dimension method for solving lps of the type maximize vtx s t a1tx 1 amtx 1 where x v a1 am n and m n this method applies the shadow vertex algorithm in n 1 stages of increasing dimension by constructing a simplex path that visits only shadow vertices in each stage specific versions of the vectors u and v are used to define the projection plane and to control the application of the shadow vertex algorithm so s delivers an upper bound for the length of such a path in one stage we want to study the average case complexity of these algorithms and are therefore interested in em ns the expected value of s for that purpose we assume a distribution of the linear programming problems corresponding to the rotation symmetry model rsm the vectors a1 am u v are distributed on n 0 independently identically and symmetrically under rotations in this paper we improve the upper bounds for em ns from our polynomiality proofs of 1982 and 1987 there we had found an upper bound of m1 n 1 i n3 i const valid for all rsm distributions and all pairs m n based on a refined evaluation technique for space angles on a reorganization and on substantial modifications of the old proof we are now able to confirm a bound em ns m1 n 1 i n2 i const for all pairs m n with m n this result had been desired since 1982 because then a bound m1 n 1 i n2 i const had been derived exclusively for the situation where n is fixed and m tends to infinity all the time the question had been open whether the asymptotical behaviour would be better than the behaviour in moderate dimensions or whether the bound for the moderate case was crude in addition we know from an asymptotic analysis in 1987 that for a special rsm distribution namely the uniform distribution on the unit sphere there is a lower bound for em ns of type m1 n 1 i n2 i const the combination of this result and the new upper bound shows that the new bound is sharp

monoidal morita invariants for finite group algebras abstract two hopf algebras are called monoidally morita equivalent if module categories over them are equivalent as linear monoidal categories we introduce monoidal morita invariants for finite dimensional hopf algebras based on certain braid group representations arising from the drinfeld double construction as an application we show for any integer n the number of elements of order n is a monoidal morita invariant for finite group algebras we also describe relations between our construction and invariants of closed 3 manifolds due to reshetikhin and turaev

polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems we show that a particular pivoting algorithm which we call the lexicographic lemke algorithm takes an expected number of steps that is bounded by a quadratic inn when applied to a random linear complementarity problem of dimensionn we present two probabilistic models both requiring some nondegeneracy and sign invariance properties the second distribution is concerned with linear complementarity problems that arise from linear programming in this case we give bounds that are quadratic in the smaller of the two dimensions of the linear programming problem and independent of the larger similar results have been obtained by adler and megiddo

power of the scan statistic for detection of clustering the scan statistic is the maximum number of events in an interval of fixed length w as the subinterval moves over the entire time frame previous research derived the null distribution of the scan statistic under the conditional model which assumed that the total number of events was fixed and under the unconditional model which let the total number of events be a random variable this paper derives approximations for the power of the scan test for a pulse alternative under this alternative the relative risk of disease on a subinterval tau tau w tau unknown is theta fold as high as it is for other subintervals of length w two sets of approximations are given for each model the first approximation gives highly accurate results but requires use of a personal computer the second procedure can be performed on a hand held calculator and appears very accurate for the cases examined

the many facets of linear programming we examine the history of linear programming from computational geometric and complexity points of view looking at simplex ellipsoid interior point and other methods

approximations and bounds for the distribution of the scan statistic abstract the scan statistic is used in many areas of science to test the null hypothesis of uniformity against a clustering alternative this article derives approximations and bounds for the distribution of scan statistics an extensive simulation study is carried out to compare the approximations and the bounds derived in this article with other known approximations and bounds the new lower and upper bounds that have been derived are very useful in evaluating the approximations and the simulation a quantity of average number of points in the scanning interval of length d d is introduced to study the approximations for the distribution of the scan statistic for low values of d the approximation derived in this article is the most accurate one the importance of the scan statistics arises from their applications in many areas including nuclear physics geology radio optics photography and epidemiology

central invariants and higher indicators for semisimple quasi hopf algebras in this paper we define the higher frobenius schur fs indicators for finite dimensional modules of a semisimple quasi hopf algebra h via the categorical counterpart developed in a 2005 preprint when h is an ordinary hopf algebra we show that our definition coincides with that introduced by kashina sommerhauser and zhu we find a sequence of gauge invariant central elements of h such that the higher fs indicators of a module v are obtained by applying its character to these elements as an application we show that fs indicators are sufficient to distinguish the four gauge equivalence classes of semisimple quasi hopf algebras of dimension eight corresponding to the four fusion categories with certain fusion rules classified by tambara and yamagami three of these categories correspond to well known hopf algebras and we explicitly construct a quasi hopf algebra corresponding to the fourth one using the kac algebra we also derive explicit formulae for fs indicators for some quasi hopf algebras associated to group cocycles

testing for elevated disease rates using smoothed estimates a method of testing for elevated disease rates using smoothed estimates is introduced for counts aggregated to a specific administrative level a test statistic is developed based on an approximation to the smoothed estimate from a penalized multinomial likelihood and an approximate critical value is derived empirical investigation reveals that smoothing can increase the ability to detect elevated rates the methods are applied to data on gastric cancer in nova scotia canada