using extension sets to aggregate partial rankings in a flexible setting this paper deals with the rank aggregation problem in a general setting in particular we approach the problem for any kind of ranking complete or incomplete and with or without ties the underlying idea behind our approach is to take into account the so called extension set of a ranking that is the set of permutations that are compatible with the given ranking in this way we aim to manage the uncertainty inherent to this problem when not all the items are ranked and or when some items are equally preferred we develop two approaches a general one based on this idea and a constrained version in which the extension set is limited by not allowing non ranked items to be placed in an existent bucket of tied items we test our proposal by coupling it with two different algorithms we formalize our approaches mathematically and also carry out an extensive experimental evaluation by using 22 datasets the results show that the use of our extension sets based approaches to compute the precedence matrix clearly outperforms the standard way of computing the preference matrix by only using the information explicitly provided by the rankings in the sample

poisson gerstenhaber brackets in representation algebras abstract we introduce cyclic bilinear forms on coalgebras and use them to generalize van den bergh s poisson brackets in representation algebras

azumaya loci and discriminant ideals of pi algebras abstract we prove that under mild assumptions for all positive integers l the zero set of the discriminant ideal d l r z r tr of a prime affine polynomial identity pi algebra r coincides with the zero set of the modified discriminant ideal m d l r z r tr of r and give an explicit description of this set in terms of the dimensions of the irreducible representations of r furthermore we prove that when l is the square of the pi degree of r this zero set is precisely the complement of the azumaya locus of r this description is used to classify the azumaya loci of the multiparameter quantized weyl algebras at roots of unity as another application we prove that the zero set of the top discriminant ideal of a prime affine pi algebra r coincides with the singular locus of the center of r provided that the discriminant ideal has height at least 2 r has finite global dimension and r is a cohen macaulay module over its center

nonnuclear subalgebras of af algebras we show that any non type i separable unital af algebra b can be modeled from inside by a nonnuclear c algebra and from outside by a nonexact c algebra more precisely there exist unital separable quasidiagonal c algebras a b c of real rank zero stable rank one such that a is nonnuclear c is nonexact and both a and c are asymptotically homotopy equivalent to b in particular a b and c have the same ordered k theory groups hence isomorphic ideal lattices and a and b have affinely homeomorphic trace spaces

calabi yau algebras and weighted quiver polyhedra dimer models have been used in string theory to construct path algebras with relations that are 3 dimensional calabi yau algebras these constructions result in algebras that share some specific properties they are finitely generated modules over their centers and their representation spaces are toric varieties in order to describe these algebras we introduce the notion of a toric order and show that all toric orders which are 3 dimensional calabi yau algebras can be constructed from dimer models on a torus toric orders are examples of a much broader class of algebras positively graded cancellation algebras for these algebras the cy 3 condition implies the existence of a weighted quiver polyhedron which is an extension of dimer models obtained by replacing the torus with any two dimensional compact orientable orbifold

the flat locus of brauer severi fibrations of smooth orders given a cayley hamilton smooth order a in a central simple algebra we determine the flat locus of the brauer severi fibration of a moreover we give a classification of all reduced central singularities where the flat locus differs from the azumaya locus and show that the fibers over the flat non azumaya points near these central singularities can be described as fibered products of graphs of projection maps this generalizes an old result of artin on the fibers of the brauer severi fibration of a maximal order over a ramified point finally we show these fibers are also toric quiver varieties and use this fact to compute their cohomology

decomposition rank and z stability we show that separable simple nonelementary unital c algebras with finite decomposition rank absorb the jiang su algebra mathcal z tensorially this has a number of consequences for elliott s program to classify nuclear c algebras by their k theory data in particular it completes the classification of c algebras associated to uniquely ergodic smooth minimal dynamical systems by their ordered k groups

vector bundles whose restriction to a linear section is ulrich an ulrich sheaf on an n dimensional projective variety x subseteq mathbb p n is an initialized acm sheaf which has the maximum possible number of global sections using a construction based on the representation theory of roby clifford algebras we prove that every normal acm variety admits a reflexive sheaf whose restriction to a general 1 dimensional linear section is ulrich we call such sheaves delta ulrich in the case n 2 where delta ulrich sheaves satisfy the property that their direct image under a general finite linear projection to mathbb p 2 is a semistable instanton bundle on mathbb p 2 we show that some high veronese embedding of x admits a delta ulrich sheaf with a global section

aggregating inconsistent information ranking and clustering we address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the extent of disagreement with the respective inputs specifically the problems we address are rank aggregation the feedback arc set problem on tournaments and correlation and consensus clustering we show that for all these problems and various weighted versions of them we can obtain improved approximation factors using essentially the same remarkably simple algorithm additionally we almost settle a long standing conjecture of bang jensen and thomassen and show that unless np bpp there is no polynomial time algorithm for the problem of minimum feedback arc set in tournaments

polynomial identity rings as rings of functions ii abstract in characteristic zero zinovy reichstein and the author generalized the usual relationship between irreducible zariski closed subsets of the affine space their defining ideals coordinate rings and function fields to a non commutative setting where varieties carry a pgl n action regular and rational functions on them are matrix valued coordinate rings are prime polynomial identity algebras and function fields are central simple algebras of degree n in the present paper much of this is extended to prime characteristic in addition a mistake in the earlier paper is corrected one of the results is that the finitely generated prime pi algebras of degree n are precisely the rings that arise as coordinate rings of n varieties in this setting for n 1 the definitions and results reduce to those of classical affine algebraic geometry

footnotes to a paper of domokos i abstract we use regev s double centralizer theorem from 7 to study invariants of matrices with group actions this theory then has applications to trace identities and poincare series

local structure of schelter procesi smooth orders in this paper we give the etale local classification of schelterprocesi smooth orders in central simple algebras in particular we prove that if is a central simple k algebra of dimension n2 where k is a field of trancendence degree d then there are only finitely many etale local classes of smooth orders in this result is a non commutative generalization of the fact that a smooth variety is analytically a manifold and so has only one type of etale local behaviour

a formal inverse to the cayley hamilton theorem abstract we prove that a q algebra r with formal trace can be realized as n n matrices if and only if it satisfies the cayley hamilton identity of degree n

necklace lie algebras and noncommutative symplectic geometry recently v ginzburg proved that calogero phase space is a coadjoint orbit for some infinite dimensional lie algebra coming from noncommutative symplectic geometry 12 in this note we generalize his argument to specific quotient varieties of representations of deformed preprojective algebras this result was also obtained independently by v ginzburg 13 using results of w crawley boevey and m holland 10 8 and 9 we give a combinatorial description of all the relevant couples alpha lambda which are coadjoint orbits we give a conjectural explanation for this coadjoint orbit result and relate it to different noncommutative notions of smoothness

the complexity of kemeny elections kemeny proposed a voting scheme which is distinguished by the fact that it is the unique voting scheme that is neutral consistent and condorcet bartholdi tovey and trick showed that determining the winner in kemeny s system is np hard we provide a stronger lower bound and an upper bound matching the lower bound namely we show that determining the winner in kemeny s system is complete for p np the class of sets solvable via parallel access to np

a description of the brauer severi scheme of trace rings of generic matrices abstract here we use invariant theory to describe the brauer severi scheme of the fibers of trace rings of generic matrices with an algebraically closed base field of characteristic zero when the trace ring is viewed as a sheaf of algebras over the variety of matrix invariants using this approach we first prove that the brauer severi scheme of a trace ring is isomorphic to proj q for a graded ring q whose generators we describe in the first section this description also has a relevant interpretation over base fields of arbitrary characteristic in the second section of this paper we show that the brauer severi scheme of the fiber of a trace ring over a point that is not too degenerate will have smooth irreducible components meeting transversally and describe these irreducible components as brauer severi schemes of certain algebras

embeddings for the jordan algebra of a bilinear form abstract let k be a field of characteristic zero and let j be a jordan algebra with a formal trace we prove that the algebra j can be embedded into a jordan algebra of a non degenerate symmetric bilinear form over some associative and commutative k algebra c if and only if j satisfies all trace identities of the jordan algebra of a non degenerate symmetric bilinear form over the field k this is an extension of results of procesi and berele concerning the analogous problem for the associative matrix algebras and the matrix algebras with involution as a consequence of these results we also prove that the ideal of all trace identities of the jordan algebra of a non degenerate symmetric bilinear form over k satisfies the specht property

generalized inductive limits of quasidiagonal c algebras if a is a unital quasidiagonal c algebra we construct a generalized inductive limit ba which is simple unital and inherits many structural properties from a if a is the unitization of a non simple purely infinite algebra e g the cone over a cuntz algebra then ba is tracially af which among other things lends support to a conjecture of toms

the nuclear dimension of c algebras abstract we introduce the nuclear dimension of a c algebra this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank our notion behaves well with respect to inductive limits tensor products hereditary subalgebras hence ideals quotients and even extensions it can be computed for many examples in particular it is finite for all uct kirchberg algebras in fact all classes of nuclear c algebras which have so far been successfully classified consist of examples with finite nuclear dimension and it turns out that finite nuclear dimension implies many properties relevant for the classification program surprisingly the concept is also linked to coarse geometry since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform roe algebra is dominated by the asymptotic dimension of the underlying space

non levi closed conjugacy classes of sp q 2 n we construct an explicit quantization of semi simple conjugacy classes of the complex symplectic group sp 2n with non levi isotropy subgroups through an operator realization on highest weight modules over the quantum group u q bigl mathfrak sp 2n bigr

quasi identities on matrices and the cayley hamilton polynomial abstract we consider certain functional identities on the matrix algebra m n that are defined similarly as the trace identities except that the coefficients are arbitrary polynomials not necessarily those expressible by the traces the main issue is the question of whether such an identity is a consequence of the cayley hamilton identity we show that the answer is affirmative in several special cases and moreover for every such an identity p and every central polynomial c with zero constant term there exists m n such that the affirmative answer holds for c m p in general however the answer is negative we prove that there exist antisymmetric identities that do not follow from the cayley hamilton identity and give a complete description of a certain family of such identities

non commutative resolutions of quotient singularities for reductive groups in this paper we generalize standard results about non commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups we show in particular that quotient singularities for reductive groups always have non commutative resolutions in an appropriate sense moreover we exhibit a large class of such singularities which have twisted non commutative crepant resolutions we discuss a number of examples both new and old that can be treated using our methods notably we prove that twisted non commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew symmetric matrices in contrast to almost all prior results in this area our techniques are algebraic and do not depend on knowing a commutative resolution of the singularity

the characteristic polynomial of an algebra and representations we pose two new linear preserver problems first given a field k and a finite dimensional k algebra b we show that the only linear maps k d b k d b which preserve the unit and the n th roots of unity for some n 2n 2 coprime to char k char k are the algebra homomorphisms second we consider linear maps a b a b between finite dimensional k algebras which preserve the cayley hamilton relations we show that if preservation of the cayley hamilton property is understood in a certain non commutative sense and a k da k d then the only such linear mappings are the algebra homomorphisms

clustering algorithms for chains we consider the problem of clustering a set of chains to k clusters a chain is a totally ordered subset of a finite set of items chains are an intuitive way to express preferences over a set of alternatives as well as a useful representation of ratings in situations where the item specific scores are either difficult to obtain too noisy due to measurement error or simply not as relevant as the order that they induce over the items first we adapt the classical k means for chains by proposing a suitable distance function and a centroid structure we also present two different approaches for mapping chains to a vector space the first one is related to the planted partition model while the second one has an intuitive geometrical interpretation finally we discuss a randomization test for assessing the significance of a clustering to this end we present an mcmc algorithm for sampling random sets of chains that share certain properties with the original data the methods are studied in a series of experiments using real and artificial data results indicate that the methods produce interesting clusterings and for certain types of inputs improve upon previous work on clustering algorithms for orders

an extension on statistical comparisons of classifiers over multiple data sets for all pairwise comparisons in a recently published paper in jmlr dem sar 2006 recommends a set of non parametric statistical tests and procedures which can be safely used for comparing the performance of classifiers over multiple data sets after studying the paper we realize that the paper correctly introduces the basic procedures and some of the most advanced ones when comparing a control method however it does not deal with some advanced topics in depth regarding these topics we focus on more powerful proposals of statistical procedures for comparing n n classifiers moreover we illustrate an easy way of obtaining adjusted and comparable p values in multiple comparison procedures

non simple purely infinite c algebras a c algebra a is defined to be purely infiniteif there are no characters on a a nd if for every pair of positive elements a b in a such that b lies in the closed two sided ideal generated by a there exists a sequence rn in a such that rn arn b this definition agrees with the usual definition by j cuntz when a is simple it is shown that the property of being purely infinite is preserved under extensions morita equivalence inductive limits and it passes to quotients and to hereditary sub c algebras it is shown that a is purely infinite for every c algebra a purely infinite c algebras admit no traces and conversely an approximately divisible exact c algebra is purely infinite if it admits no nonzero trace

some remarkable degenerations of quantum groups we show that an irreducible representation of a quantized enveloping algebraue at a lth root of 1 has maximal dimension l n if the corresponding symplectic leaf has maximal dimension 2n the method of the proof consists of a construction of a sequence of degenerations ofue the last one being aq commutative algebrau e 2n this allows us to reduce many problems concerningue to that concerningu e 2n

tackling the rank aggregation problem with evolutionary algorithms probabilistic reasoning and learning with permutation data has gained interest in recent years because its use in different ranking based real world applications therefore constructing a model from a given set of permutations or rankings has become a target problem in the machine learning community in this paper we focus on probabilistic modelling and concretely in the use of a well known permutation based distribution as it is the mallows model learning a mallows model from data requires the estimation of two parameters a consensus permutation p 0 and a dispersion parameter q since the exact computation of these parameters is an np hard problem it is natural to consider heuristics to tackle this problem an interesting approach consists in the use of a two step procedure first estimating p 0 and then computing q for a given p 0 this is possible because the optimal p 0 does not depend on q when following this approach computation of p 0 reduces to the rank aggregation problem which consists in finding the ranking which best represents such dataset in this paper we propose to use genetic algorithms to tackle this problem studying its performance with respect to state of the art algorithms specially in complex cases that is when the number of items to rank is large and there is few consensus between the available rankings which traduces in a low value for q after a series of experiments involving data of different type we conclude that our evolutionary approach clearly outperforms the remaining tested algorithms

noncommutative smoothness and coadjoint orbits abstract in math ag 0010030 math z to appear r bocklandt and the author proved that certain quotient varieties of representations of deformed preprojective algebras are coadjoint orbits for the necklace lie algebra n q of the corresponding quiver q a conjectural ring theoretical explanation of these results was given in terms of noncommutative smoothness in the sense of c procesi j algebra 107 1987 63 74 in this paper we prove these conjectures the main tool in the proof is the etale local description due to w crawley boevey math ag 0105247 along the way we determine the smooth locus of the marsden weinstein reductions for quiver representations

distance based ranking models on examine le modele de type mallows pour une distance generale et on etudie ensuite les distances pour lesquelles le modele peut etre decompose en facteurs representant des etapes independantes dans le processus de classement

algebraic families of galois representations and potentially semi stable pseudodeformation rings we construct and study the moduli of continuous representations of a profinite group with integral p adic coefficients we present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable when this profinite group is the absolute galois group of a p adic local field we show that these moduli spaces admit zariski closed loci cutting out galois representations that are potentially semi stable with bounded hodge tate weights and a given hodge and galois type as a consequence we show that these loci descend to the universal deformation ring of the corresponding pseudorepresentation

hopf algebras with trace and representations we study the restriction of representations of cayley hamilton algebras to subalgebras this theory is applied to determine tensor products and branching rules for representations of quantum groups at roots of 1