invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

cohomology for frobenius kernels of sl2 abstract let sl 2 r be the r th frobenius kernels of the group scheme sl 2 defined over an algebraically closed field of characteristic p 0 in this paper we give for r 1 a complete description of the cohomology groups for sl 2 r we also prove that the reduced cohomology ring h sl 2 r k red is cohen macaulay geometrically we show for each r 1 that the maximal ideal spectrum of the cohomology ring for sl 2 r is homeomorphic to the fiber product g b u r finally we adapt our calculations to obtain analogous results for the cohomology of higher frobenius lusztig kernels of quantized enveloping algebras of type sl 2

strong integrality of quantum invariants of 3 manifolds we prove that the quantum so 3 invariant of an arbitrary 3 manifold m is always an algebraic integer if the order of the quantum parameter is co prime with the order of the torsion part of h 1 m z an even stronger integrality known as cyclotomic integrality was established by habiro for integral homology 3 spheres here we also generalize habiro s result to all rational homology 3 spheres

asymptotic expansions of witten reshetikhin turaev invariants for some simple 3 manifolds the link l this paper investigates the invariant z m 0 when g sk for a simple family of rational homology 3 spheres obtained by integer surgery around 2 n type torus knots in particular we find a closed formula for a formal power series z m eq h in h q 1 from which zf m 0 may be derived for all sufficiently large primes r we show that this formal power series may be viewed as the asymptotic expansion around 4 1 of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition for these particular manifolds most of which are not z homology spheres this extends work of ohtsuki and murakami in which the existence of power series with rational coefficients related to z m 0 was demonstrated for rational homology spheres the coefficients in the formal power series z m are expected to be identical to those obtained from a perturbative expansion of the witten chem simons path integral formula for z m 0 0 199 5 american institute of physics

cohomology of infinitesimal and discrete groups an infinitesimal algebraic group over a field k is a connected algebraic group scheme whose coordinate ring is a finite dimension al k algebra if one begins with a reductive algebraic group g defined over fp then one has a close relationship between the representation theory of g that of its infinitesimal subgroups g defined as the kernels of the frobenius endomorphisms rr g g and that of its finite chevalley subgroups g fq of fq rational points q pd a basic theorem of cline et al i 9 theorem 6 6 relates the cohomology of g in a rational representation to that of the finite groups g fq related work of cline et al 6 and of the authors 1 16 relates the cohomology of g to that of its infinitesimal subgroups a determination of the cohomology of the first infinitesimal subgroup g 1 of g with trivial coefficients has enabled the authors in 14 to make certain explicit cohomology calculations for the finite groups g fq with certain nontrivial coefficients in this paper we continue our study of infinitesimal subgroups of simple algebraic groups by proving various qualitative properties of their cohomology and by extending our computational knowledge in particular we provide a computation which applies in a range of degrees to chevalley groups over the prime field we also investigate the cohomology of infinitesimal subgroups of nilpotent groups enabling us for example to determine the cohomology of their lie algebras in certain cases we apply this lie algebra cohomology computation to determine the integral cohomology of various torsion free nilpotent groups modulo small prime torsion thereby extending a theorem of dwyer 12 in more detail the paper begins in sect 1 with a spectral sequence converging to the cohomology of the first infinitesimal subgroup g which was previously studied in 14 from this spectral sequence it easily follows that h g1 m is a noetherian h g1 k module for each finite dimensional rational gl module m we then apply the work of andersen and jantzen 1 on the gl cohomology of induced modules to obtain the analogous result for the second infinitesimal subgroup gz our analysis also permits a determination of the generic

extensions of simple modules for finite chevalley groups let g be an almost simple algebraic group over an algebraically closed field k of positive characteristic p we assume that g is defined and split over the prime field 5 for it 3 1 we denote by g n the finite group consisting of the points of g over the field with p elements our first result states that for a certain class of simple modules for g n the only extensions that can occur are those which are restrictions of g extensions see theorem 2 8 the full story about g extensions is not known yet except for a few rather small groups however much more is known about extensions for the algebraic group than about those for the finite group for instance the linkage principle l gives an upper bound for which g extensions may occur moreover a proof of the lusztig conjecture on the irreducible characters for g also implies strong statements about g extensions see 3 proposition 2 81 it is an open question whether all extensions between simple g n modules can be found via the extension theory for g we prove that in generic cases this is in fact so see theorem 3 2 moreover for the groups x 3 p and sp 4 p we are able to determine completely all extensions between simple modules from the corresponding known results for the algebraic groups x 3 k and sp 4 k our results extend previous results by cline parshall and scott s 91 who consider extensions of the trivial module by a minimal module and smith ls who considers extensions between certain minimal modules the method we use is a generalization of the one used by humphreys 131 who proves the existence of certain selfextensions for sp 4 p it should be pointed out that our methods do not work for p small on the other hand everything we do generalizes easily to the twisted chevalley groups see theorem 3 4 we demonstrate this by working out all extensions for su 3 p see section 4 3

integrality for tqfts we discuss ways that the ring of coefficients for a tqft can be reduced if one restricts somewhat the allowed cobordisms when we apply these methods to a tqft associated to so 3 at an odd prime p we obtain a functor from a somewhat restricted cobordism category to the category of free finitely generated modules over a ring of cyclotomic integers z 2p if p 1 mod 4 and z 4p if p 1 mod 4 where k is a primitive kth root of unity we study the quantum invariants of prime power order simple cyclic covers of 3 manifolds we define new invariants arising from strong shift equivalence and integrality similar results are obtained for some other tqfts but the modules are only guaranteed to be projective

group level spatial independent component analysis of fourier envelopes of resting state meg data article we developed a data driven method to spatiotemporally and spectrally characterize the dynamics of brain oscillations in resting state magnetoencephalography meg data the method called envelope spatial fourier independent component analysis esfica maximizes the spatial and spectral sparseness of fourier energies of a cortically constrained source current estimate we compared this method using a simulated data set against 5 other variants of independent component analysis and found that esfica performed on par with its temporal variant etfica and better than other ica variants in characterizing dynamics at time scales of the order of minutes

second cohomology groups for frobenius kernels and related structures abstract let g be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p let b be a borel subgroup of g and u be its unipotent radical in this paper we determine the second cohomology groups of b and its frobenius kernels for all simple b modules we also consider the standard induced modules obtained by inducing a simple b module to g and compute all second cohomology groups of the frobenius kernels of g for these induced modules also included is a calculation of the second ordinary lie algebra cohomology group of lie u with coefficients in k

the coloured jones function the invariantsjk k of a framed knotk coloured by the irreduciblesu 2 q module of dimensionk are studied as a function ofk by means of the universalr matrix it is shown that whenjk k is written as a power series inh withq eh the coefficient ofhd is an odd polynomial ink of degree at most 2d 1 this coefficient is a vassiliev invariant ofk in the second part of the paper it is shown that ask varies these invariants span ad dimensional subspace of the space of all vassiliev invariants of degreed for framed knots the analogous questions for unframed knots are also studied

extensions of modules for algebraic groups let g be a connected semi simple and simply connected algebraic group over an algebraically closed field k of prime characteristic p in this paper we show how one can obtain the extensions of any two simple g modules if one knows the socles of all the g modules extg l l where g1 denotes the kernel of the frobenius endomorphism on g and l as well as l are simple g1 modules alternatively we prove how all such g extensions can be found once one knows the extensions of any two simple g modules belonging to a specified finite set of simple modules with small highest weight such information is available in certain low rank cases it should also be pointed out that it is possible from lusztig s conjecture on irreducible characters 11 to derive the dimensions of ext l1 l2 when l1 and l2 are simple with such small highest weights to obtain our results on exth we need to know that there exists no g1 extension of a simple module by itself this turns out to be somewhat

model independent method for fmri analysis this paper presents a fast method for delineation of activated areas of the brain from functional magnetic resonance imaging fmri time series data the steps of the work accomplished are as follows 1 it is shown that the detection performance evaluated by the area under the receiver operating characteristic curve is directly related to the signal to noise ratio snr of the composite image generated in the detection process 2 detection and segmentation of activated areas are formulated in a vector space framework in this formulation a linear transformation image combination method is shown to be desirable to maximize the snr of the activated areas subject to the constraint of removing inactive areas 3 an analytical solution for the problem is found 4 image pixel vectors and expected time series pattern signature for inactive pixels are used to calculate weighting vector and identify activated regions 5 signatures of the activated regions are used to segment different activities 6 segmented images by the proposed method are compared with those generated by the conventional methods correlation t statistic and z statistic detection performance and snrs of the images are compared the proposed approach outperforms the conventional methods of fmri analysis in addition it is model independent and does not require a priori knowledge of the fmri response to the paradigm since the method is linear and most of the work is done analytically numerical implementation and execution of the method are much faster than the conventional methods

quantum invariants modular forms and lattice points ii we study the su 2 witten reshetikhin turaev wrt invariant for the seifert fibered homology spheres with m exceptional fibers we show that the wrt invariant can be written in terms of differential of the eichler integrals of modular forms with weight 1 2 and 3 2 by use of nearly modular property of the eichler integrals we shall obtain asymptotic expansions of the wrt invariant in the large n limit we further reveal that the number of the gauge equivalent classes of flat connections which dominate the asymptotics of the wrt invariant in n is related to the number of integral lattice points inside the m dimensional tetrahedron

blind source separation using temporal predictability a measure of temporal predictability is defined and used to separate linear mixtures of signals given any set of statistically independent source signals it is conjectured here that a linear mixture of those signals has the following property the temporal predictability of any signal mixture is less than or equal to that of any of its component source signals it is shown that this property can be used to recover source signals from a set of linear mixtures of those signals by finding an un mixing matrix that maximizes a measure of temporal predictability for each recovered signal this matrix is obtained as the solution to a generalized eigenvalue problem such problems have scaling characteristics of o n3 where n is the number of signal mixtures in contrast to independent component analysis the temporal predictability method requires minimal assumptions regarding the probability density functions of source signals it is demonstrated that the method can separate signal mixtures in which each mixture is a linear combination of source signals with supergaussian subgaussian and gaussian probability density functions and on mixtures of voices and music

extensions for finite chevalley groups i let g be a connected semisimple algebraic group defined and split over the field fp with p elements and k be the algebraic closure of fp assume further that g is almost simple and simply connected and let g fq be the finite chevalley group consisting of fq rational points of g where q pr for a non negative integer r in this paper formulas are found relating extensions between simple kg fq modules and extensions over g considered as an algebraic group over k one of these formulas which only holds for primes p 3 h 1 where h is the coxeter number of g is then used to show the vanishing of self extensions between simple kg fq modules except for certain simple modules when r 1 and the underlying root system is of type a1 or cn

a formula for casson s invariant suppose h is a homology 3 sphere obtained by dehn surgery on a link l in a homology 3 sphere m if every pair of components of l has zero linking number in m then we give a formula for the casson invariant x h in terms of a m the surgery coefficients of l and a certain coefficient from each of the conway polynomials of l and all its sublinks a few consequences of this formula are given introduction andrew casson has discovered an integral invariant of oriented homology 3 spheres which reduces mod 2 to the rochlin or n invariant 1 in this paper we give a formula for casson s invariant in the case where the homology sphere h is obtained by dehn surgery on a link l in a homology sphere m and furthermore the linking number between every pair of components of l is zero note that every homology 3 sphere can be obtained in this way with m s3 in this case the casson invariant x h can be expressed in terms of x m the surgery coefficients of l and a certain coefficient from each of the conway polynomials of l and all its sublinks to illustrate the utility of this formula we compute the casson invariant of homology spheres that arise as cyclic branched covers branched along untwisted double knots we also describe how the x invariant changes when a crossing of the framed link l between two strands of the same component is changed this provides at least in this setting an effective method for computing the invariant an interesting question is whether it is possible to show directly that the formula we give is preserved by the kirby rolfsen calculus and thus give an alternative proof of the invariance of x if l1 and l2 are two rationally framed links representing the same manifold then l1 can be transformed into l2 by a sequence of three types of moves homeomorphism twisting and trivial insertion or deletion 8 if the linking number between every pair of components of l1 is zero then the first two types of moves will preserve this property while the third move may not however if l1 can be transformed into l2 through a sequence of links all of whose linking numbers are all zero then we show that the formula for x is preserved we do not know if such a transformation is in general possible in order to state casson s theorem and the formula for x h we first establish some notation suppose l k1 kn is a framed link in a 3 manifold m with received by the editors october 22 1985 1980 mathematics subject classification 1985 revision primary 57m25 supported in part by the nsf under grant no dms 8502940 x 1986 american mathematical society 0002 9947 86 1 00 25 per page

extensions for frobenius kernels 1 1 let g be a simple algebraic group over an algebraically closed field k of characteristic p 0 for r 1 let gr be the rth frobenius kernel of g it is well known that the representations for g1 are equivalent to the restricted representations for lie g historically the cohomology for frobenius kernels has been best understood for large primes friedlander and parshall fp first computed the cohomology ring h g1 k for p 3 h 1 where h is the coxeter number of the underlying root system they proved that the cohomology ring can be identified with the coordinate algebra of the nullcone andersen and jantzen aj later verified this fact for p h furthermore they generalized this calculation by looking at h g1 h0 where h0 indb for p h their results had some restrictions on the type of root system involved kumar lauritzen and thomsen klt removed the restrictions on the root systems through the use of frobenius splittings the cohomology ring h g1 k modulo nilpotents can be identified in general with the coordinate algebra of the restricted nullcone n1 x lie g x p 0 for good primes nakano parshall and vella npv proved that this variety is irreducible and can be identified with the closure of some richardson orbit recently carlson lin nakano and parshall clnp have given an explicit description of n1 these recent results provide some indication that one can systematically study extensions of frobenius kernels for small primes by using general formulas which exhibit generic behavior for large primes

second cohomology groups for algebraic groups and their frobenius kernels let g be a simple simply connected algebraic group scheme defined over an algebraically closed field of characteristic p 0 let t be a maximal split torus in g b t be a borel subgroup of g and u its unipotent radical let f g g be the frobenius morphism for r 1 define the frobenius kernel gr to be the kernel of f iterated with itself r times define ur respectively br to be the kernel of the frobenius map restricted to u respectively b let x t be the integral weight lattice and x t be the dominant integral weights the computations of particular importance are h2 u1 k h2 br for x t h2 gr h0 for x t and h2 b for x t the above cohomology groups for the case when the field has characteristic 2 are computed in this paper these computations complete the picture started by bendel nakano and pillen 2007 5 for p 3 furthermore the computations show h2 gr h0 has a good filtration

frobenius splitting of cotangent bundles of flag varieties let g be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic p 0 let u be the unipotent radical of a borel subgroup b g and u the lie algebra of u springer 16 has shown for good primes that there is a b equivariant isomorphism u u where b acts through conjugation on u and through the adjoint action on u for g sln one has the well known equivariant isomorphism a 7 a i between unipotent and nilpotent upper triangular matrices let p be a good prime for g then there is an isomorphism of homogeneous bundles x g b u g b u where the latter can be identified with the cotangent bundle t g b of g b motivated in part by 12 we establish a link between the g invariant form on the steinberg module st h0 g b p 1 cf 1 8 and frobenius splittings 15 of the cotangent bundle of g b the representation h0 g b 2 p 1 is a quotient of the space of functions h x ox on x here h0 g b m denotes the g module induced from the b module m and half the sum of the roots r opposite to the roots

extensions for finite chevalley groups ii let g be a semisimple simply connected algebraic group defined and split over the field fp with p elements let g fq be the finite chevalley group consisting of the fq rational points of g where q p r and let g r be the rth frobenius kernel the purpose of this paper is to relate extensions between modules in mod g f q and mod g r with extensions between modules in mod g among the results obtained are the following for r 2 and p 3 h 1 the g f q extensions between two simple g f q modules are isomorphic to the g extensions between two simple p r restricted g modules with suitably twisted highest weights for p 3 h 1 we provide a complete characterization of h 1 g f q h0 where h 0 ind g b a and a is p r restricted furthermore for p 3 h 1 necessary and sufficient bounds on the size of the highest weight of a g module v are given to insure that the restriction map h 1 g v h 1 g f q v is an isomorphism finally it is shown that the extensions between two simple p r restricted g modules coincide in all three categories provided the highest weights are close together

on the cohomology of algebraic and related finite groups in the above theorem h gl m denotes the rational cohomology of the algebraic group gl with coefficients in m the rational gl module obtained from the adjoint module m m through twisting by the r th power of the frobenius endomorphism on gl the algebra structure on the cohomology is induced from the associative algebra structure on m on the other hand we provide the following systematic determination of certain rational cohomology groups h g v r where v r is the rational gmodule obtained from the rational g module v by applying the r th frobenius twist see 1 2 this is a somewhat sharpened version of theorem 4 of 36

a unified witten reshetikhin turaev invariant for integral homology spheres we construct an invariant j m of integral homology spheres m with values in a completion widehat mathbb z q of the polynomial ring q such that the evaluation at each root of unity gives the the su 2 witten reshetikhin turaev invariant m of m at thus j m unifies all the su 2 witten reshetikhin turaev invariants of m it also follows that m as a function on behaves like an analytic function defined on the set of roots of unity

effects of repeatability measures on results of fmri sica a study on simulated and real resting state effects spatial independent components analysis sica has become a widely applied data driven method for fmri data especially for resting state studies these sica approaches are often based on iterative estimation algorithms and there are concerns about accuracy due to noise repeatability measures such as icasso raicar and arabica have been introduced as remedies but information on their effects on estimates is limited the contribution of this study was to provide more of such information and test if the repeatability analyses are necessary we compared fastica based ordinary and repeatability approaches concerning mixing vector estimates comparisons included original fastica fsl4 melodic fastica and original and modified icasso the effects of bootstrapping and convergence threshold were evaluated the results show that there is only moderate improvement due to repeatability measures and only in the bootstrapping case bootstrapping attenuated power from time courses of resting state network related ics at frequencies higher than 0 1 hz and made subsets of low frequency oscillations more emphasized ic wise the convergence threshold did not have a significant role concerning the accuracy of estimates the performance results suggest that repeatability measures or strict converge criteria might not be needed in sica analyses of fmri data consequently the results in existing sica fmri literature are probably valid in this sense a decreased accuracy of original bootstrapping icasso was observed and corrected by using centrotype mixing estimates but the results warrant for thorough evaluations of data driven methods in general also given the fmri specific considerations further development of sica methods is strongly encouraged

on the melvin morton rozansky conjecture we prove a conjecture stated by melvin and morton and elucidated further by rozansky saying that the alexander conway polynomial of a knot can be read from some of the coefficients of the jones polynomials of cables of that knot i e coefficients of the colored jones polynomial we first reduce the problem to the level of weight systems using a general principle which may be of some independent interest and which sometimes allows to deduce equality of vassiliev invariants from the equality of their weight systems we then prove the conjecture combinatorially on the level of weight systems finally we prove a generalization of the melvin morton rozansky mmr conjecture to knot invariants coming from arbitrary semi simple lie algebras as side benefits we discuss a relation between the conway polynomial and immanants and a curious formula for the weight system of the colored jones polynomial

witten reshetikhin turaev invariants of seifert manifolds for seifert homology spheres we derive a holomorphic function of k whose value at integer k is the sl2 witten reshetikhin turaev invariant zk at q exp 2 i k this function is expressed as a sum of terms which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the witten chern simons path integral the trivial connection contribution is found to have an asymptotic expansion in powers of k 1 which for k an odd prime power converges k adically to the exact total value of the invariant zk at that root of unity evaluations at rational k are also discussed using similar techniques an expression for the coloured jones polynomial of a torus knot is obtained providing a trivial connection contribution which is an analytic function of the colour this demonstrates that the stationary phase expansion of the chern simons witten theory is exact for seifert manifolds and for torus knots in s3 the possibility of generalising such results is also discussed

non negative matrix factorization with sparseness constraints non negative matrix factorization nmf is a recently developed technique for finding parts based linear representations of non negative data although it has successfully been applied in several applications it does not always result in parts based representations in this paper we show how explicitly incorporating the notion of sparseness improves the found decompositions additionally we provide complete matlab code both for standard nmf and for our extension our hope is that this will further the application of these methods to solving novel data analysis problems

the gln q module structure of the symmetric algebra around the steinberg module abstract we determine the graded composition multiplicity in the symmetric algebra s v of the natural gl n q module v or equivalently in the coinvariant algebra of v for a large class of irreducible modules around the steinberg module this was built on a computation via connections to algebraic groups of the steinberg module multiplicity in a tensor product of s v with other tensor spaces of fundamental weight modules

probabilistic independent component analysis for functional magnetic resonance imaging we present an integrated approach to probabilistic independent component analysis ica for functional mri fmri data that allows for nonsquare mixing in the presence of gaussian noise in order to avoid overfitting we employ objective estimation of the amount of gaussian noise through bayesian analysis of the true dimensionality of the data i e the number of activation and non gaussian noise sources this enables us to carry out probabilistic modeling and achieves an asymptotically unique decomposition of the data it reduces problems of interpretation as each final independent component is now much more likely to be due to only one physical or physiological process we also describe other improvements to standard ica such as temporal prewhitening and variance normalization of timeseries the latter being particularly useful in the context of dimensionality reduction when weak activation is present we discuss the use of prior information about the spatiotemporal nature of the source processes and an alternative hypothesis testing approach for inference using gaussian mixture models the performance of our approach is illustrated and evaluated on real and artificial fmri data and compared to the spatio temporal accuracy of results obtained from classical ica and glm analyses

on extensions of simple modules over symmetric and algebraic groups here and below the cohomology of algebraic groups is the rational cohomology see 13 i 4 this result is quite useful for calculating some ext groups between simple g modules for example it immediately implies that extg l l 0 we would be interested in having a result similar to 1 for the symmetric group 6n for a partition of n we denote by s the corresponding specht module and if is p regular we write d for the corresponding irreducible module over f6n see 11 it is known that the specht modules play a role similar to that of weyl modules for algebraic groups so we are interested in results connecting ext6n d d and hom6n rad s d the first main result of the paper is

on blind separability based on the temporal predictability method this letter discusses blind separability based on temporal predictability stone 2001 xie he fu 2005 our results show that the sources are separable using the temporal predictability method if and only if they have different temporal structures i e autocorrelations consequently the applicability and limitations of the temporal predictability method are clarified in addition instead of using generalized eigendecomposition we suggest using joint approximate diagonalization algorithms to improve the robustness of the method a new criterion is presented to evaluate the separation results numerical simulations are performed to demonstrate the validity of the theoretical results

the universalr matrix burau representation and the melvin morton expansion of the colored jones polynomial abstract p melvin and h morton 9 studied the expansion of the colored jones polynomial of a knot in powers of q 1 and color they conjectured an upper bound on the power of color versus the power of 1 they also conjectured that the bounding line in their expansion generated the inverse alexander conway polynomial these conjectures were proved by d bar natan and s garoufalidis 1 we have conjectured 12 that other lines in the melvin morton expansion are generated by rational functions with integer coefficients whose denominators are powers of the alexander conway polynomial here we prove this conjecture by using the r matrix formula for the colored jones polynomial and presenting the universal r matrix as a perturbed burau matrix

ica based on asymmetry we build a new approach to ica which is based on the data asymmetry instead of densities with heavy tails we use asymmetric ones split gaussians we verified our approach on images sound and eeg data in the case of source signal reconstructing our approach gives better results independent component analysis ica one of the basic tools in data analysis aims to find a coordinate system in which the components of the data are independent most of existing methods are based on the minimization of the function of fourth order moment kurtosis skewness third order moment has received much less attention in this paper we present a competitive approach to ica based on the split gaussian distribution which is well adapted to asymmetric data consequently we obtain a method which works better than the classical approaches especially in the case when the underlying density is not symmetric which is a typical situation in the color distribution in images

bridged links and tangle presentations of cobordism categories abstract we develop a calculus of surgery data called bridged links which involves besides links also pairs of balls that describe one handle attachments as opposed to the usual link calculi of kirby and others this description uses only elementary local moves namely modifications and isolated cancellations and it is valid also on non simply connected and disconnected manifolds in particular it allows us to give a presentation of a 3 manifold by doing surgery on any other 3 manifold with the same boundary bridged link presentations on unions of handlebodies are used to give a cerf theoretical derivation of presentations of 2 1 dimensional cobordisms categories in terms of planar ribbon tangles and their composition rules as an application we give a different more natural proof of the matveev polyak presentations of the mapping class groups and furthermore find systematically surgery presentations of general mapping tori we discuss a natural extension of the reshetikhin turaev invariant to the calculus of bridged links invariance follows now similar as for knot invariants from simple identifications of the elementary moves with elementary categorical relations for invariances or cointegrals respectively hence we avoid the lengthy computations and the unnatural fenn rourke reduction of the original proofs moreover we are able to start from a much weaker modularity condition which implies the one of turaev generalizations of the presentation to cobordisms of surfaces with boundaries are outlined

detection of consistently task related activations in fmri data with hybrid independent component analysis fmri data are commonly analyzed by testing the time course from each voxel against specific hypothesized waveforms despite the fact that many components of fmri signals are difficult to specify explicitly in contrast purely data driven techniques by focusing on the intrinsic structure of the data lack a direct means to test hypotheses of interest to the examiner between these two extremes there is a role for hybrid methods that use powerful data driven techniques to fully characterize the data but also use some a priori hypotheses to guide the analysis here we describe such a hybrid technique hybica which uses the initial characterization of the fmri data from independent component analysis and allows the experimenter to sequentially combine assumed task related components so that one can gracefully navigate from a fully data derived approach to a fully hypothesis driven approach we describe the results of testing the method with two artificial and two real data sets a metric based on the diagnostic predicted sum of squares statistic was used to select the best number of spatially independent components to combine and utilize in a standard regressional framework the proposed metric provided an objective method to determine whether a more data driven or a more hypothesis driven approach was appropriate depending on the degree of mismatch between the hypothesized reference function and the features in the data hybica provides a robust way to combine the data derived independent components into a data derived activation waveform and suitable confounds so that standard statistical analysis can be performed

three manifold invariants associated with restricted quantum groups we show a simple relation between witten reshetikhin turaev su 2 invariant and the hennings invariant associated with the restricted quantum mathfrak sl 2 these invariants are defined in very different methods the former uses the representation theory of quantum mathfrak sl 2 while the latter uses the integral of the hopf algebra but they turn out to be the same for most rational homology 3 spheres up to a sign this relation can be used to prove the integrality of the former invariant

extensions for finite chevalley groups iii rational and generic cohomology abstract let g be a connected reductive algebraic group and b be a borel subgroup defined over an algebraically closed field of characteristic p 0 in this paper the authors study the existence of generic g cohomology and its stability with rational g cohomology groups via the use of methods from the authors earlier work new results on the vanishing of g and b cohomology groups are presented furthermore vanishing ranges for the associated finite group cohomology of g f q are established which generalize earlier work of hiller in addition to stability ranges for generic cohomology which improve on seminal work of cline parshall scott and van der kallen

second cohomology for finite groups of lie type let g be a simple simply connected algebraic group defined over mathbb f p given a power q p r of p let g mathbb f q subset g be the subgroup of mathbb f q rational points let l lambda be the simple rational g module of highest weight lambda in this paper we establish sufficient criteria for the restriction map in second cohomology h 2 g l lambda rightarrow h 2 g mathbb f q l lambda to be an isomorphism in particular the restriction map is an isomorphism under very mild conditions on p and q provided lambda is less than or equal to a fundamental dominant weight even when the restriction map is not an isomorphism we are often able to describe h 2 g mathbb f q l lambda in terms of rational cohomology for g we apply our techniques to compute h 2 g mathbb f q l lambda in a wide range of cases and obtain new examples of nonzero second cohomology for finite groups of lie type

spatiotemporal independent component analysis of event related fmri data using skewed probability density functions we introduce two independent component analysis ica methods spatiotemporal ica stica and skew ica and demonstrate the utility of these methods in analyzing synthetic and event related fmri data first stica simultaneously maximizes statistical independence over both time and space this contrasts with conventional ica methods which maximize independence either over time only or over space only these methods often yield physically improbable solutions second skew ica is based on the assumption that images have skewed probability density functions pdfs an assumption consistent with spatially localized regions of activity in contrast conventional ica is based on the physiologically unrealistic assumption that images have symmetric pdfs we combine stica and skew ica to form skew stica and use it to analyze synthetic data and data from an event related left right visual hemifield fmri experiment results obtained with skew stica are superior to those of principal component analysis spatial ica sica temporal ica stica and skew sica we argue that skew stica works because it is based on physically realistic assumptions and that the potential of ica can only be realized if such prior knowledge is incorporated into ica methods

the colored jones polynomials and the simplicial volume of a knot we show that the set of colored jones polynomials and the set of generalized alexander polynomials defined by akutsu deguchi and ohtsuki intersect non trivially moreover it is shown that the intersection is at least includes the set of kashaev s quantum dilogarithm invariants for links therefore kashaev s conjecture can be restated as follows the colored jones polynomials determine the hyperbolic volume for a hyperbolic knot modifying this we propose a stronger conjecture the colored jones polynomials determine the simplicial volume for any knot if our conjecture is true then we can prove that a knot is trivial if and only if all of its vassiliev invariants are trivial in 13 r m kashaev defined a family of complex valued link invariants indexed by integers n 2 using the quantum dilogarithm later he calculated the asymptotic behavior of his invariant and observed that for the three simplest hyperbolic knots it grows as exp vol k n 2 when n goes to the infinity where vol k is the hyperbolic volume of the complement of a knotk 14 this amazing result and his conjecture that the same also holds for any hyperbolic knot have been almost ignored by mathematicians since his definition of the invariant is too complicated though it uses only elementary tools the aim of this paper is to reveal his mysterious definition and to show that his invariant is nothing but a specialization of the colored jones polynomial the colored jones polynomial is defined for colored links each component is decorated with an irreducible representation of the lie algebra sl 2 c the original jones polynomial corresponds to the case that all the colors are identical to the 2 dimensional fundamental representation we show that kashaev s invariant with parameter n coincides with the colored jones polynomial in a certain normalization with every color the n dimensional representation evaluated at the primitive n th root of unity we have to normalize the colored jones polynomial so that the value for the trivial knot is one for otherwise it always vanishes on the other hand there are other colored polynomial invariants the generalized multivariable alexander polynomial defined by y akutsu t deguchi and t ohtsuki 1 they used the same lie algebra sl 2 c but a different hierarchy of representations their invariants are parameterized by c 1 parameters an integer date february 1 2008 1991 mathematics subject classification 57m25 57m50 17b37 81r50

quantum invariants at the sixth root of unity a general topological formula is given for thesu 2 quantum invariant of a 3 manifoldm at the sixth root of unity it is expressed in terms of the homology witt invariants and signature defects of the various 2 fold covers ofm and thus ties in with basic 4 dimensional invariants a discussion of the range of values of these quantum invariants is included and explicit evaluations are made for lens spaces

a unified quantum so 3 invariant for rational homology 3 spheres given a rational homology 3 sphere m with h1 m z b and a link l inside m colored by odd numbers we construct a unified invariant im l belonging to a modification of the habiro ring where b is inverted our unified invariant dominates the whole set of the so 3 witten reshetikhin turaev invariants of the pair m l i fb 1a ndl im coincides with habiro s invariant of integral homology 3 spheres for b 1 the unified in variant defined by the third author is determined by im important applica tions are the new ohtsuki series perturbative expansions of im dominating quantum so 3 invariants at roots of unity whose order is not a power of a prime these series are not known to be determined by the lmo invariant

an information maximization approach to blind separation and blind deconvolution we derive a new self organizing learning algorithm that maximizes the information transferred in a network of nonlinear units the algorithm does not assume any knowledge of the input distributions and is defined here for the zero noise limit under these conditions information maximization has extra properties not found in the linear case linsker 1989 the nonlinearities in the transfer function are able to pick up higher order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation this enables the network to separate statistically independent components in the inputs a higher order generalization of principal components analysis we apply the network to the source separation or cocktail party problem successfully separating unknown mixtures of up to 10 speakers we also show that a variant on the network architecture is able to perform blind deconvolution cancellation of unknown echoes and reverberation in a speech signal finally we derive dependencies of information transfer on time delays we suggest that information maximization provides a unifying framework for problems in blind signal processing

sheaf homology and complete reducibility abstract in the study of finite simple groups by means of the geometries provided by their local subgroups problems of structure often reduce to questions about modular representations in finite characteristic of particular interest are modules spanned by fixed points of sylow groups including high weight modules for chevalley groups for which the homology methods of ronan and smith 6 can be very useful this note presents an elementary sufficient condition for splitting of some reducible modules of this type among the applications we generalize a number of results appearing in recent work of timmesfeld 11 which provided the original inspiration for this analysis some of the ideas had appeared in a special case in 10

generic patterns for extensions of simple modules for finite chevalley groups abstract this paper describes generic patterns for the extensions between simple modules of a finite chevalley group a one to one correspondence between these extensions and the extensions between certain simple modules of the ambient algebraic group are established it is shown that an extension appears in a principal series representation of the finite chevalley group if and only if the corresponding extension for the algebraic group appears in a weyl module