imprecise data envelopment analysis idea a review and improvement with an application abstract the standard data envelopment analysis dea method requires that the values for all inputs and outputs be known exactly when some outputs and inputs are unknown decision variables such as bounded data ordinal data and ratio bounded data the dea model becomes a non linear programming problem and is called imprecise dea idea there are two different approaches in dealing with imprecise outputs and inputs one uses scale transformations and variable alternations to convert the non linear idea model into a linear program the other converts imprecise data into exact data and then uses the standard linear dea model the current paper reviews and compares the two approaches through an efficiency analysis of a set of telephone offices a simplified approach is developed to reduce the computational burden if one uses the first approach the treatment of weight restrictions in idea is discussed it is shown that weight restrictions on imprecise data are redundant new developments and improvements to both approaches are provided

extracting computational entropy and learning noisy linear functions we study the task of deterministically extracting randomness from sources containing computational entropy the sources we consider have the form of a conditional distribution f x x for some function f and some distribution x and we say that such a source has computational min entropy k if any circuit of size 2k can only predict f x correctly with probability at most 2 k given input x sampled from x we first show that it is impossible to have a seedless extractor to extract from one single source of this kind then we show that it becomes possible if we are allowed a seed which is weakly random instead of perfectly random but contains some statistical min entropy or even a seed which is not random at all but contains some computational min entropy this can be seen as a step toward extending the study of multisource extractors from the traditional statistical setting to a computational setting we reduce the task of constructing such extractors to a problem in computational learning theory learning linear functions under arbitrary distribution with adversarial noise and we provide a learning algorithm for this problem in fact this problem is a well recognized one in computational learning theory and variants of this problem have been studied intensively before thus in addition to its application to extractors our learning algorithm also has independent interest of its own and it can be considered as the main technical contribution of this paper

qualitative factors in data envelopment analysis a fuzzy number approach abstract qualitative factors are difficult to mathematically manipulate when calculating the efficiency in data envelopment analysis dea the existing methods of representing the qualitative data by ordinal variables and assigning values to obtain efficiency measures only superficially reflect the precedence relationship of the ordinal data this paper treats the qualitative data as fuzzy numbers and uses the dea multipliers associated with the decision making units dmus being evaluated to construct the membership functions based on zadeh s extension principle a pair of two level mathematical programs is formulated to calculate the cuts of the fuzzy efficiencies fuzzy efficiencies contain more information for making better decisions a performance evaluation of the chemistry departments of 52 uk universities is used for illustration since the membership functions are constructed from the opinion of the dmus being evaluated the results are more representative and persuasive

a product theorem in free groups if a is a nite subset of a free group with at least two noncommuting elements thenja a aj jaj 2 logjaj o 1 more generally the same conclusion holds in an arbitrary virtually free group unless a generates a virtually cyclic subgroup the central part of the proof of this result is carried on by estimating the number of collisions in multiple products a1 ak we include a few simple observations showing that in this statistical context

improving the hadamard extractor in this paper we construct a strong randomness extractor with two independent bit input distributions with min entropies b x b y b x b y the probability of any particular output is upper bounded by 2 b x and 2 b y respectively for b x b y 1 our extractor produces one bit which is by the factor of 2 closer to the uniform distribution when compared to the hadamard extractor what is more this distance drops to zero if at least one of the min entropies raises to this is in sharp contrast to the hadamard extractor which fails to produce even a single unbiased bit even if one of the input distributions is uniform we also extend our construction to produce k bits of output with a bias that is by the factor of 3 2 smaller than that of the corresponding hadamard extractor and retains the ability to produce unbiased bits if one of the input distributions is uniform the strongness property of the extractor is maintained in both cases however in the multi bit scenario the bias is increased by the factor of 3 2

an algebro geometric realization of equivariant cohomology of some springer fibers we give an explicit affine algebraic variety whose coordinate ring is isomorphic as a w algebra with the equivariant cohomology of some springer fibers

extracting kolmogorov complexity with applications to dimension zero one laws we apply results on extracting randomness from independent sources to extract kolmogorov complexity for any a e 0 given a string x with k x a x we show how to use a constant number of advice bits to efficiently compute another string y y w x with k y 1 e y this result holds for both unbounded and space bounded kolmogorov complexity we use the extraction procedure for space bounded complexity to establish zero one laws for the strong dimensions of complexity classes within espace the unbounded extraction procedure yields a zero one law for the constructive strong dimensions of turing degrees

torus knots and the rational daha author s gorsky e oblomkov a rasmussen j shende v abstract 2014 we conjecturally extract the triply graded khovanov rozansky homology of the m n torus knot from the unique finite dimensional simple representation of the rational daha of type a rank n 1 and central character m n the conjectural differentials of gukov dunfield and the third author receive an explicit algebraic expression in this picture yielding a prescription for the doubly graded khovanov rozansky homologies we match our conjecture to previous conjectures of the first author relating knot homology to q t catalan numbers and to previous conjectures of the last three authors relating knot homology to hilbert schemes on singular curves

link homology theories from symplectic geometry for each positive integer n khovanov and rozansky constructed an invariant of links in the form of a doubly graded cohomology theory whose euler characteristic is the link polynomial we use lagrangian floer cohomology on some suitable affine varieties to build a similar series of link invariants and we conjecture them to be equal to those of khovanov and rozansky after a collapse of the bigrading our work is a generalization of that of seidel and smith who treated the case n 2

ideals of quasi symmetric functions and super covariant polynomials for sn the aim of this work is to study the quotient ring r n of the ring q x 1 x n over the ideal j n generated by non constant homogeneous quasi symmetric functions we prove here that the dimension of r n is given by c n the n th catalan number this is also the dimension of the space sh n of super covariant polynomials that is defined as the orthogonal complement of j n with respect to a given scalar product we construct a basis for r n whose elements are naturally indexed by dyck paths this allows us to understand the hilbert series of sh n in terms of number of dyck paths with a given number of factors

on the signature character of representations of p adic general linear groups in this article we calculate the signature character of certain hermitian representations of gln f for a p adic field f we further give a conjectural description for the signature character of unramified representations in terms of kostka numbers

rank order data in dea a general framework in data envelopment analysis dea performance evaluation is generally assumed to be based upon a set of quantitative data in many real world settings however it is essential to take into account the presence of qualitative factors when evaluating the performance of decision making units dmus very often rankings are provided from best to worst relative to particular attributes such rank positions might better be presented in an ordinal rather than numerical sense the paper develops a general frame work for modeling and treating qualitative data in dea and provides a unified structure for embedding rank order data into the dea framework the existing techniques are discussed and their equivalence is demonstrated both continuous and discrete projection models are provided it is shown that qualitative data can be treated in conventional dea methodology

a decision model for technology selection in the existence of both cardinal and ordinal data technology selection is an important part of management of technology to select the best technologies in the existence of both cardinal and ordinal data this paper proposes an innovative approach which is based on a mathematical programming a numerical example demonstrates the application of the proposed method

two block springer fibers of types c and d a diagrammatic approach to springer theory we explain an elementary topological construction of the springer representation on the homology of topological springer fibers of types c and d in the case of nilpotent endomorphisms with two jordan blocks the weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types b and d we determine the decomposition of the representations into irreducibles and relate our construction to classical springer theory as an application we obtain presentations of the cohomology rings of all two block springer fibers of types c and d moreover we deduce explicit isomorphisms between the kazhdan lusztig cell modules attached to the induced trivial module and the irreducible specht modules in types c and d

decomposition of green polynomials of type a and springer modules for hooks and rectangles abstract this paper deals with graded representations of the symmetric group on the cohomology ring of flags fixed by a unipotent matrix we consider a combinatorial property called the coincidence of dimension of the graded representations and give an interpretation in terms of representation theory of the symmetric group in the case where the corresponding partition of the unipotent matrix is a hook or a rectangle the interpretation is equivalent to a recursive formula of green polynomials at roots of unity

an efficient reduction from two source to nonmalleable extractors achieving near logarithmic min entropy the breakthrough result of chattopadhyay and zuckerman explicit two source extractors and resilient functions in proceedings of the 48th annual acm sigact symposium on theory of computing stoc

extracting randomness using few independent sources in this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1 2 specifically for every delta 0 we give an explicit construction for extracting randomness from a constant depending polynomially on 1 delta number of distributions over bits n each having min entropy delta n these extractors output n bits that are 2 n close to uniform this construction uses several results from additive number theory and in particular a recent result of bourgain et al we also consider the related problem of constructing randomness dispersers for any constant output length m our dispersers use a constant number of identical distributions each with requires min entropy omega log n and outputs every possible m bit string with positive probability the main tool we use is a variant of the stepping up lemma of erdods and hajnal used in establishing a lower bound on the ramsey number for hypergraphs

technologies ranking in the presence of both cardinal and ordinal data ranking of technologies is an important phase for technology transfer to rank the best technologies in the presence of both cardinal and ordinal data this paper proposes an innovative approach which is based on imprecise data envelopment analysis idea the objective of this paper is to propose a comprehensive reference that discusses the use of idea in technology ranking a numerical example demonstrates the application of the proposed method

affine springer fibers of type a and combinatorics of diagonal coinvariants abstract we calculate the borel moore homology of affine springer fibers of type a associated with some regular semisimple nil elliptic elements as a result we obtain bigraded s n modules whose bigraded frobenius series are a generalization of the symmetric functions introduced by haglund haiman loehr remmel and ulyanov

duality efficiency computations and interpretations in imprecise dea data envelopment analysis dea has proven to be a useful tool for assessing efficiency or productivity of organizations which is of vital practical importance in managerial decision making while dea assumes exact input and output data the development of imprecise dea idea broadens the scope of applications to efficiency evaluations involving imprecise information which implies various forms of ordinal and bounded data possibly or often occurring in practice the primary purpose of this article is to characterize the variable efficiency in idea since dea describes a pair of primal and dual models also called envelopment and multiplier models we can basically consider two idea models one incorporates imprecise data into envelopment model and the other includes the same imprecise data in multiplier model the issues of rising importance are thus the relationships between the two models and how to solve them the groundwork we will make includes a duality study which makes it possible to characterize the efficiency solutions from the two models and link with the efficiency bounds and classifications that some of the published idea studies have done the other purposes are to present computational aspects of the efficiency bounds and how to interpret the efficiency solutions the computational method developed here extends the previous idea method to effectively incorporate a more general form of strict ordinal data and partial orders in its framework which in turn overcomes some drawbacks of the previous approaches the interpretation of the resulting efficiency is also important but we have never seen it before

two source dispersers for polylogarithmic entropy and improved ramsey graphs in his 1947 paper that inaugurated the probabilistic method erdos proved the existence of 2 o 1 log n ramsey graphs on n vertices matching erdos s result with a constructive proof is cons

2 source dispersers for n o 1 entropy and ramsey graphs beating the frankl wilson construction the main result of this paper is an explicit disperser for two independent sources on n bits each of min entropy k 2 1 0 for some small absolute constant 0 0 put differently setting n 2 and k 2 we construct an explicit n n boolean matrix for which no k k sub matrix is monochromatic viewed as the adjacency matrix of a bipartite graph this gives an explicit construction of a bipartite k ramsey graph of 2n vertices this improves the previous the previous bound of k o n of barak kindler shaltiel sudakov and wigderson bks05 as a corollary we get a construction of a 2 log1 0 n non bipartite ramsey graph of 2 vertices significantly improving the previous bound of 2o n due to frankl and wilson fw81 we also give a construction of a new independent sources extractor that can extract from a constant number of sources of polynomially small min entropy with exponentially small error this improves independent sources extractor of rao rao06 which only achieved polynomially small error our dispersers combine ideas and constructions from several previous works in the area together with some new ideas in particular we rely on the extractors of raz raz05 and bourgain bou05 as well as an improved version of the extractor of rao rao06 a key ingredient that allows us to beat the barrier of k n is a new and more complicated variant of the challenge response mechanism of barak et al bks05 that allows us to locate the min entropy concentrations in a source of low min entropy

simulating independence new constructions of condensers ramsey graphs dispersers and extractors we present new explicit constructions of deterministic randomness extractors dispersers and related objects we say that a distribution x on binary strings of length n is a source if x assigns probability at most 2 n to any string of length n for every 0 we construct the following poly n time computable functions 2 source disperser d 0 1 n 2 0 1 such that for any two independent sources x1 x2 we have that the support of d x1 x2 is 0 1 bipartite ramsey graph let ne2n a corollary is that the function d is a 2 coloring of the edges of kn n the complete bipartite graph over two sets of n vertices such that any induced subgraph of size n by n is not monochromatic 3 source extractor e 0 1 n 3 0 1 such that for any three independent sources x1 x2 x3 we have that e x1 x2 x3 is o 1 close to being an unbiased random bit no previous explicit construction was known for either of these for any a component in these results is a new construction of condensers that may be of independent interest this is a function c 0 1 n 0 1 n c d where c and d are constants that depend only on such that for every source x one of the output blocks of c x is exponentially close to a 0 9 source this result was obtained independently by ran raz the constructions are quite involved and use as building blocks other new and known objects a recurring theme in these constructions is that objects that were designed to work with independent inputs sometimes perform well enough with correlated high entropy inputs the construction of the disperser is based on a new technique which we call the challenge response mechanism that in some sense allows identifying high entropy regions in a given pair of sources using only one sample from the two sources

les degres des varietes des matrices nilpotentes abstract nous calculons les degres des adherences des orbites de matrices nilpotentes et en particulier ceux des varietes a rang et indice donnes