mckay correspondence for symplectic quotient singularities let v be a finite dimensional complex vector space and g sl v a finite subgroup the quotient variety x v g is usually singular if dim v 2 then there exists a canonical smooth projective resolution y x of the singular variety x this resolution is crepant which in this case means that the manifold y has trivial canonical bundle we recall the precise definitions in definition 1 2 the fiber 1 0 y over the singular point 0 x v g is a rational curve whose components are numbered by the nontrivial conjugacy classes in g the homology classes of these components freely generate h2 y q all the other homology groups of y are trivial except for h0 y q q this situation was described by j mckay in mck it is known as the mckay correspondence in higher dimensions the picture splits into two parts one can consider two separate questions 1 when does the quotient x v g admit a smooth crepant resolution y x 2 assuming there exists a smooth crepant resolution y x what can one say about the homology h y q

normality of marsden weinstein reductions for representations of quivers we prove that the marsden weinstein reductions for the moment map associated to representations of a quiver are normal varieties we give an application to conjugacy classes of matrices

reflection functors and symplectic reflection algebras for wreath products we construct reflection functors on categories of modules over deformed wreath products of the preprojective algebra of a quiver these functors give equivalences of categories associated to generic parameters which are in the same orbit under the weyl group action we give applications to the representation theory of symplectic reflection algebras of wreath product groups

recursive prediction error identification using the nonlinear wiener model the nonlinear wiener model consisting of a linear dynamic block in cascade with a static nonlinearity is considered a recursive prediction error identification algorithm based on the wiener model is derived the linear dynamic block is modelled as a siso transfer function operator and the static nonlinearity is approximated with a piecewise linear function a theoretical analysis of the method is carried out and conditions for local convergence to the true parameter vector are given in particular the analysis shows that the input signal should be such that there is signal energy in the whole range of the piecewise linear approximation a numerical example illustrates the performance of the algorithm further practical guidelines on how to apply the algorithm are also included in the paper

sheaves on fibered threefolds and quiver sheaves this paper classifies a class of holomorphic d branes closely related to framed torsion free sheaves on threefolds fibered in resolved ade surfaces over a general curve c in terms of representations with relations of a twisted kronheimer nakajima type quiver in the category coh c of coherent sheaves on c for the local calabi yau case c cong bf a 1 and special choice of framing one recovers the n 1 ade quiver studied by cachazo katz vafa

noncommutative resolutions of ade fibered calabi yau threefolds in this paper we construct noncommutative resolutions of a certain class of calabi yau threefolds studied by cachazo et al geometric transitions and n 1 quiver theories http arxiv org abs hep th 0108120v2 2001 the threefolds under consideration are fibered over a complex plane with the fibers being deformed kleinian singularities the construction is in terms of a noncommutative algebra introduced by ginzburg calabi yau algebras http arxiv org abs math 0612139v2 2006 which we call the n 1 ade quiver algebra

poisson deformations of symplectic quotient singularities abstract we establish a connection between smooth symplectic resolutions and symplectic deformations of a possibly singular affine poisson variety in particular let v be a finite dimensional complex symplectic vector space and g sp v a finite subgroup our main result says that the so called calogero moser deformation of the orbifold v g is in an appropriate sense a versal poisson deformation that enables us to determine the algebra structure on the cohomology h x c of any smooth symplectic resolution x v g multiplicative mckay correspondence we prove further that if g gl h is an irreducible weyl group and v h h then no smooth symplectic resolution of v g exists unless g is of types a b c

parameter identification of wiener systems with discontinuous nonlinearities abstract the paper deals with modeling and parameter identification of wiener type nonlinear dynamic systems having discontinuous asymmetric piecewise linear characteristics with preloads and dead zones multiple application of a decomposition technique provides a special form of static nonlinearity description its incorporation into the wiener model leads to an input output equation where all the model parameters are separated and can be estimated iteratively using input output data and internal variable estimates more illustrative examples are included to demonstrate the feasibility of the proposed approach

identification of fir wiener systems with unknown non invertible polynomial non linearities wiener systems consist of a linear dynamic system whose output is measured through a static non linearity in this paper we study the identification of single input single output wiener systems with finite impulse response dynamics and polynomial output non linearities using multi index notation we solve a least squares problem to estimate products of the coefficients of the non linearity and the impulse response of the linear system we then consider four methods for extracting the coefficients of the non linearity and impulse response direct algebraic solution singular value decomposition multi dimensional singular value decomposition and prediction error optimization

symplectic resolutions for nilpotent orbits in this paper firstly we calculate picard groups of a nilpotent orbit in a classical complex simple lie algebra and discuss the properties of being factorial and factorial for the normalization tilde of the closure of then we consider the problem of symplectic resolutions for tilde our main theorem says that for any nilpotent orbit in a semi simple complex lie algebra equipped with the kostant kirillov symplectic form if for a resolution z tilde the 2 form defined on 1 extends to a symplectic 2 form on z then z is isomorphic to the cotangent bundle t g p of a projective homogeneous space and is the collapsing of the zero section it proves a conjecture of cho miyaoka shepherd barron in this special case using this theorem we determine all varieties tilde which admit such a resolution

adaptive control of weiner type nonlinear systems abstract this study is concerned with the problem of adaptive control of a wiener type nonlinear process it is assumed that the linear dynamic part of the process can be represented by a pulse transfer function of known order with stable inverse followed by a known time delay the memoryless nonlinearity is one to one on a compact set and belongs to one of two classes 1 piecewise linear with known breakpoints or 2 smooth function in which case an approximation is made it is represented by linear splines the output signal from the linear part is not available for measurement the global stability of the proposed model reference adaptive control scheme is established subject to the assumption that the nonlinearity can be represented exactly by the linear spline function with a given set of breakpoints otherwise there will be a residual tracking error which depends on the nonlinearity approximation error but the boundedness of all signals can be assured the proposed scheme is used for adaptive control of the ph process in a continuous flow reactor

representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities we give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products with parameter t 0 and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras

calogero moser space and kostka polynomials abstract we consider the canonical map from the calogero moser space to symmetric powers of the affine line sending conjugacy classes of pairs of n n matrices to their eigenvalues we show that the character of a natural c action on the scheme theoretic zero fiber of this map is given by kostka polynomials a similar result is proved for a cyclic version of the calogero moser space

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

self adjusting binary search trees the splay tree a self adjusting form of binary search tree is developed and analyzed the binary search tree is a data structure for representing tables and lists so that accessing inserting and deleting items is easy on an n node splay tree all the standard search tree operations have an amortized time bound of o log n per operation where by amortized time is meant the time per operation averaged over a worst case sequence of operations thus splay trees are as efficient as balanced trees when total running time is the measure of interest in addition for sufficiently long access sequences splay trees are as efficient to within a constant factor as static optimum search trees the efficiency of splay trees comes not from an explicit structural constraint as with balanced trees but from applying a simple restructuring heuristic called splaying whenever the tree is accessed extensions of splaying give simplified forms of two other data structures lexicographic or multidimensional search trees and link cut trees

self organizing heuristics for implicit data structures self organizing heuristics are presented for data structures containing elements with different weights access probabilities the structures use just a constant number of locations in addition to those required for the values of the elements and support average access times that are within a constant multiplicative factor of optimal data structures and corresponding heuristics with essentially optimal average unsuccessful search times are also given for the case in which there are probabilities of access associated with the intervals between consecutive element values

germs of arcs on singular algebraic varieties and motivic integration we study the scheme of formal arcs on a singular algebraic variety and its images under truncations we prove a rationality result for the poincare series of these images which is an analogue of the rationality of the poincare series associated to p adic points on a p adic variety the main tools which are used are semi algebraic geometry in spaces of power series and motivic integration a notion introduced by m kontsevich in particular we develop the theory of motivic integration for semi algebraic sets of formal arcs on singular algebraic varieties we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of oesterle

rational cherednik algebras and hilbert schemes let h c be the rational cherednik algebra of type a n 1 with spherical subalgebra u c e h c e then u c is filtered by order of differential operators with associated graded ring mbox gr u c mathbb c mathfrak h oplus mathfrak h where w is the n th symmetric group we construct a filtered mathbb z algebra b such that under mild conditions on c the category b qgr of graded noetherian b modules modulo torsion is equivalent to u c mod the associated graded mathbb z algebra mbox gr b has mbox gr b lqgr simeq coh hilb n this can be regarded as saying that u c simultaneously gives a non commutative deformation of mathfrak h oplus mathfrak h w and of its resolution of singularities hilb n rightarrow mathfrak h oplus mathfrak h as we show elsewhere this result is a powerful tool for studying the representation theory of h c and its relationship to hilb n

homological aspects of noetherian pi hopf algebras and irreducible modules of maximal dimension abstract we prove that a noetherian hopf algebra of finite global dimension possesses further attractive homological properties at least when it satisfies a polynomial identity this applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity as well as to classical enveloping algebras in positive characteristic in all three cases we show that these algebras are auslander regular and macaulay we derive representation theoretic consequences concerning the coincidence of the non azumaya and singular loci for each of the above three classes of algebras

piecewise linear identification of non linear systems methods of identifying a series of piecewise linear models which approximate to a non linear system over some operating range are investigated spatial linear models and models with signal dependent parameters are considered

studies of self organizing placement algorithms for linear and finger lists abstract various algorithms that use self organization are presented for an np complete placement problem the problem is to find the optimal placement of records in a linear finger list where each access sequence begins at the previously accessed record the only goal is to produce a good final list the algorithms use a cooling schedule to produce the final static list where reorganizations are progressively restricted empirical results show that the simplest such scheme produces the best results only constant extra storage was allowed however the results are very robust

finite groups acting linearly hochschild cohomology and the cup product abstract when a finite group acts linearly on a complex vector space the natural semi direct product of the group and the polynomial ring over the space forms a skew group algebra this algebra plays the role of the coordinate ring of the resulting orbifold and serves as a substitute for the ring of invariant polynomials from the viewpoint of geometry and physics its hochschild cohomology predicts various hecke algebras and deformations of the orbifold in this article we investigate the ring structure of the hochschild cohomology of the skew group algebra we show that the cup product coincides with a natural smash product transferring the cohomology of a group action into a group action on cohomology we express the algebraic structure of hochschild cohomology in terms of a partial order on the group modulo the kernel of the action this partial order arises after assigning to each group element the codimension of its fixed point space we describe the algebraic structure for coxeter groups where this partial order is given by the reflection length function a similar combinatorial description holds for an infinite family of complex reflection groups

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

deformations of algebras defined by tilting bundles abstract in this paper we produce noncommutative algebras derived equivalent to deformations of schemes with tilting bundles we do this in two settings first proving that a tilting bundle on a scheme lifts to a tilting bundle on an infinitesimal deformation of that scheme and then expanding this result to c equivariant deformations over schemes with a good c action in both these situations the endomorphism algebra of the lifted tilting bundle produces a deformation of the original endomorphism algebra and this is a graded deformation in the c equivariant case we apply our results to rational surface singularities generalising the deformed preprojective algebras of crawley boevey and holland and also to symplectic situations where the deformations produced are related to symplectic reflection algebras

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

twisted graded hecke algebras abstract we generalize graded hecke algebras to include a twisting two cocycle for the associated finite group we give examples where the parameter spaces of the resulting twisted graded hecke algebras are larger than that of the graded hecke algebras we prove that twisted graded hecke algebras are particular types of deformations of a crossed product

brief wiener models of direction dependent dynamic systems analysis of systems with direction dependent dynamics is currently limited to cases in which the dynamics in the two directions of the output are first order results for such systems have been published for both pseudo random maximum length binary mlb and inverse repeat maximum length binary ir mlb inputs these relatively limited analytical results make it useful to examine alternative ways of modelling such systems and in this paper wiener models are considered for this purpose methods for optimising the wiener model parameters by matching the system and model cross correlation functions outputs and discrete fourier transforms of the outputs are considered and the results are compared these methods are also applied to a first order direction dependent system with a maximum length ternary mlt input for which no analytical results are currently available and to a second order system with an ir mlb input

deformed preprojective algebras and symplectic reflection algebras for wreath products abstract we determine the pbw deformations of the wreath product of a symmetric group with a deformed preprojective algebra of an affine dynkin quiver in particular we show that there is precisely one parameter which does not come from deformation of the preprojective algebra we prove that the pbw deformation is morita equivalent to a corresponding symplectic reflection algebra for wreath product

parameter identification of wiener systems with multisegment piecewise linear nonlinearities the paper deals with the identification of wiener systems having multisegment piecewise linear nonlinearities a special form of nonlinearity representation is used in the wiener system description the resulting output equation contains the least possible number of parameters required for the blocks of given system the proposed iterative method enables simultaneous estimation of both the linear block parameters and all the parameters characterizing the nonlinearity i e the slopes of linear segments and the constants determining the partition of domain limits for the constants are the only a priori knowledge required