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

partial compact quantum groups abstract compact quantum groups of face type as introduced by hayashi form a class of quantum groupoids with a classical finite set of objects using the notions of weak multiplier bialgebras and weak multiplier hopf algebras resp due to bohm gomez torrecillas lopez centella and van daele wang we generalize hayashi s definition to allow for an infinite set of objects and call the resulting objects partial compact quantum groups we prove a tannaka kre n woronowicz reconstruction result for such partial compact quantum groups using the notion of partial fusion c categories as examples we consider the dynamical quantum su 2 groups from the point of view of partial compact quantum groups

minimizing total tardiness on one machine is np hard the problem of minimizing the total tardiness for a set of independent jobs on one machine is considered lawler has given a pseudo polynomial time algorithm to solve this problem in spite of extensive research efforts for more than a decade the question of whether it can be solved in polynomial time or it is np hard in the ordinary sense remained open in this paper the problem is shown to be np hard in the ordinary sense

algebraic quantum hypergroups abstract an algebraic quantum group is a regular multiplier hopf algebra with integrals in this paper we will develop a theory of algebraic quantum hypergroups it is very similar to the theory of algebraic quantum groups except that the comultiplication is no longer assumed to be a homomorphism we still require the existence of a left and of a right integral there is also an antipode but it is characterized in terms of these integrals we construct the dual just as in the case of algebraic quantum groups and we show that the dual of the dual is the original quantum hypergroup we define algebraic quantum hypergroups of compact type and discrete type and we show that these types are dual to each other the algebraic quantum hypergroups of compact type are essentially the algebraic ingredients of the compact quantum hypergroups as introduced and studied in an operator algebraic context by chapovsky and vainerman we will give some basic examples in order to illustrate different aspects of the theory in a separate note we will consider more special cases and more complicated examples in particular in that note we will give a general construction procedure and show how known examples of these algebraic quantum hypergroups fit into this framework

attractor reliability reveals deterministic structure in neuronal spike trains when periodic current is injected into an integrate and fire model neuron the voltage as a function of time converges from different initial conditions to an attractor that produces reproducible sequences of spikes the attractor reliability is a measure of the stability of spike trains against intrinsic noise and is quantified here as the inverse of the number of distinct spike trains obtained in response to repeated presentations of the same stimulus high reliability characterizes neurons that can support a spike time code unlike neurons with discharges forming a renewal process such as a poisson process these two classes of responses cannot be distinguished using measures based on the spike time histogram but they can be identified by the attractor dynamics of spike trains as shown here using a new method for calculating the attractor reliability we applied these methods to spike trains obtained from current injection into cortical neurons recorded in vitro these spike trains did not form a renewal process and had a higher reliability compared to renewal like processes with the same spike time histogram

firing variability is higher than deduced from the empirical coefficient of variation a convenient and often used summary measure to quantify the firing variability in neurons is the coefficient of variation cv defined as the standard deviation divided by the mean it is therefore important to find an estimator that gives reliable results from experimental data that is the estimator should be unbiased and have low estimation variance when the cv is evaluated in the standard way empirical standard deviation of interspike intervals divided by their average then the estimator is biased underestimating the true cv especially if the distribution of the interspike intervals is positively skewed moreover the estimator has a large variance for commonly used distributions the aim of this letter is to quantify the bias and propose alternative estimation methods if the distribution is assumed known or can be determined from data parametric estimators are proposed which not only remove the bias but also decrease the estimation errors if no distribution is assumed and the data are very positively skewed we propose to correct the standard estimator when defining the corrected estimator we simply use that it is more stable to work on the log scale for positively skewed distributions the estimators are evaluated through simulations and applied to experimental data from olfactory receptor neurons in rats

model based gaussian and non gaussian clustering abstract the classification maximum likelihood approach is sufficiently general to encompass many current clustering algorithms including those based on the sum of squares criterion and on the criterion of friedman and rubin 1967 however as currently implemented it does not allow the specification of which features orientation size and shape are to be common to all clusters and which may differ between clusters also it is restricted to gaussian distributions and it does not allow for noise we propose ways of overcoming these limitations a reparameterization of the covariance matrix allows us to specify that some features but not all be the same for all clusters a practical framework for non gaussian clustering is outlined and a means of incorporating noise in the form of a poisson process is described an approximate bayesian method for choosing the number of clusters is given the performance of the proposed methods is studied by simulation with encouraging results the methods are applied to the analysis of a data set arising in the study of diabetes and the results seem better than those of previous analyses rh

a note on the equivalence of two heuristics to minimize total tardiness abstract over the last thirty years many researchers have studied single machine static and deterministic scheduling with the objective of minimizing total tardiness it has been established that the tardiness problem is np hard so it is unlikely that a polynomial time algorithm can be found for developing optimal solutions to this problem the modified due date rule mdd is generally considered to be an efficient heuristic that deals with the tardiness problem recently panwalkar et al have proposed the psk rule as effective in dealing with tardiness the purpose of this paper is to show that the psk rule is an implementation of the mdd rule furthermore the relationship between the mdd rule and the wi wilkeson and irwin rule is clarified

information geometry of interspike intervals in spiking neurons an information geometrical method is developed for characterizing or classifying neurons in cortical areas whose spike rates fluctuate in time under the assumption that the interspike intervals of a spike sequence of a neuron obey a gamma process with a time variant spike rate and a fixed shape parameter we formulate the problem of characterization as a semiparametric statistical estimation where the spike rate is a nuisance parameter we derive optimal criteria from the information geometrical viewpoint when certain assumptions are added to the formulation and we show that some existing measures such as the coefficient of variation and the local variation are expressed as estimators of certain functions under the same assumptions

2 cocycles and twisting of kac algebras we describe the twisting construction with the help of 2 cocycles on hopf von neumann and george kac algebras we show that twisted kac algebras are again kac algebras using this construction we give a wide class of new quantizations of the heisenberg group and describe several series of non trivial finite dimensional hopf algebras kac algebras of dimensions 4n and as twisting of finite groups

schur weyl quasi duality and co triangular hopf quasigroups schur weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic and plays a central role in many parts of algebraic lie theory in this paper we will introduce the notion of schur weyl quasi duality and study it for this generally we consider a braided vector space v c and its braided lie algebra endk v then we can construct its braided enveloping algebra u endk v which is a connected braided c cocommutative hopf algebra let h be a triangular hopf quasigroup with bijective antipode and b be a cotriangular hopf quasigroup with bijective antipode let v be any finite dimensional vector space in the category lq h r b of generalized long quasimodules we show that u endkv h b ksn v n is a schur weyl quasi duality under suitable conditions

quantum group of automorphisms of a finite quantum group abstract a notion of a quantum automorphism group of a finite quantum group generalising that of a classical automorphism group of a finite group is proposed and a corresponding existence result proved

the ornstein uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex cortical neurons of behaving animals generate irregular spike sequences recently there has been a heated discussion about the origin of this irregularity softky and koch 1993 pointed out the inability of standard single neuron models to reproduce the irregularity of the observed spike sequences when the model parameters are chosen within a certain range that they consider to be plausible shadlen and newsome 1994 on the other hand demonstrated that a standard leaky integrate and fire model can reproduce the irregularity if the inhibition is balanced with the excitation motivated by this discussion we attempted to determine whether the ornstein uhlenbeck process which is naturally derived from the leaky integration assumption can in fact reproduce higher order statistics of biological data for this purpose we consider actual neuronal spike sequences recorded from the monkey prefrontal cortex to calculate the higher order statistics of the interspike intervals consistency of the data with the model is examined on the basis of the coefficient of variation and the skewness coefficient which are respectively a measure of the spiking irregularity and a measure of the asymmetry of the interval distribution it is found that the biological data are not consistent with the model if the model time constant assumes a value within a certain range believed to cover all reasonable values this fact suggests that the leaky integrate and fire model with the assumption of uncorrelated inputs is not adequate to account for the spiking in at least some cortical neurons

multiplier hopf algebras of discrete type abstract in this paper we study regular multiplier hopf algebras with cointegrals they are a certain class of multiplier hopf algebras still sharing many nice properties with the much smaller class of finite dimensional hopf algebras recall that a multiplier hopf algebra is a pair a whereais an algebra over c possibly without identity and is a comultiplication ona a homomorphism ofainto the multiplier algebram a a ofa a satisfying certain properties the typical example is the algebraaof complex functions with finite support in a groupg with pointwise multiplication and where the comultiplication is defined by f p q f pq wheneverf aandp q g a left cointegral in a multiplier hopf algebra is an elementh asuch thatah a hfor alla awhere is the counit ofa in the group example this is the function that is 1 on the identity of the group and 0 everywhere else in this paper we show that cointegrals are unique up to a constant if they exist and that they are faithful we also show that on a regular multiplier hopf algebra with a left cointegral there exists also a left integral recall that a left integral is a linear functional onasuch that a a 1 where is the identity map and where the equation is to be considered inm a a multiplier hopf algebra with cointegrals is therefore an algebraic quantum group of discrete type we will also obtain different necessary and sufficient conditions on the algebraafor a multiplier hopf algebra a to have cointegrals i e to be of discrete type the algebras turn out to be frobenius quasi frobenius and kasch

finiteness conditions co frobenius hopf algebras and quantum groups abstract some finiteness conditions for infinite dimensional coalgebras particularly right or left semiperfect coalgebras or co frobenius hopf algebras are studied as well examples of co frobenius hopf algebras are constructed via a hopf algebra structure on an ore extension of a group algebra and it is shown that bicrossproducts of co frobenius hopf algebras are themselves co frobenius

information capacity in the weak signal approximation we derive an approximate expression for mutual information in a broad class of discrete time stationary channels with continuous input under the constraint of vanishing input amplitude or power the approximation describes the input by its covariance matrix while the channel properties are described by the fisher information matrix this separation of input and channel properties allows us to analyze the optimality conditions in a convenient way we show that input correlations in memoryless channels do not affect channel capacity since their effect decreases fast with vanishing input amplitude or power on the other hand for channels with memory properly matching the input covariances to the dependence structure of the noise may lead to almost noiseless information transfer even for intermediate values of the noise correlations since many model systems described in mathematical neuroscience and biophysics operate in the high noise regime and weak signal conditions we believe that the described results are of potential interest also to researchers in these areas

a mathematical theory of energy efficient neural computation and communication a neuroscience based mathematical model of how a neuron stochastically processes data and communicates information is introduced and analyzed call the neuron in question neuron j or just j the information j transmits approximately describes the time varying intensity of the excitation j is continuously experiencing from neural spike trains delivered to its synapses by thousands of other neurons neuron j encodes this excitation history into a sequence of time instants at which it generates neural spikes of its own by propagating these spikes along its axon j acts as a multiaccess partially degraded broadcast channel with thousands of input and output terminals that employs a time continuous version of pulse position modulation the mathematical model features three parameters m and b which largely characterize j as an engine of computation and communication each set of values of these parameters corresponds to a long term maximization of the bits j conveys to its targets per joule it expends doing so which is achieved by distributing the random duration between successive spikes j generates according to a gamma pdf with parameters and b and distributing b a according to a beta probability density with parameters and m where a is the random intensity of the effectively poisson process of spikes that arrive to the union of all of j s synapses at a randomly chosen time instant

the single machine total tardiness scheduling problem review and extensions abstract we review the latest theoretical developments for the single machine total tardiness 1 t scheduling problem and propose extensions to some of them we also review and in some cases extend exact algorithms fully polynomial time approximation schemes heuristic algorithms special cases and generalizations of the 1 t problem our findings indicate that the 1 t problem continues to attract significant research interest from both a theoretical and a practical perspective even though the problem is ordinary np hard the current state of the art algorithms are capable of solving problems with up to 500 jobs

quasitriangular g cograded multiplier hopf algebras we put the known results on the antipode of a usual quasitriangular hopf algebra into the framework of multiplier hopf algebras we illustrate with examples which cannot be obtained by using classical hopf algebras the focus of the present paper lies on the class of the so called g cograded multiplier hopf algebras by doing so we bring the results of quasitriangular hopf group coalgebras as introduced by turaev to the more general framework of multiplier hopf algebras

coribbon hopf face algebras generated by lattice models abstract by studying points of the underlying quantum groups of coquasitriangular hopf face algebras we construct ribbon categories for each lattice models without spectral parameter of both vertex and face type also we give a classification of the braiding and the ribbon structure on quantized classical groups and modular tensor categories closely related to quantum su n l invariants of 3 manifolds

bicrossproducts of multiplier hopf algebras abstract in this paper we generalize majid s bicrossproduct construction we start with a pair a b of two regular multiplier hopf algebras we assume that b is a right a module algebra and that a is a left b comodule coalgebra the right action of a on b gives rise to the smash product a b the left coaction of b on a gives a possible coproduct on a b we discuss in detail the necessary compatibility conditions between the action and the coaction for to be a proper coproduct on a b the result is again a regular multiplier hopf algebra majid s construction is obtained when we have hopf algebras we also look at the dual case constructed from a pair c d of regular multiplier hopf algebras where now c is a left d module algebra while d is a right c comodule coalgebra we show that indeed these two constructions are dual to each other in the sense that a natural pairing of a with c and of b with d yields a duality between a b and the smash product c d we show that the bicrossproduct of an algebraic quantum group is again an algebraic quantum group i e a regular multiplier hopf algebra with integrals the algebra case is also considered some special cases are treated and they are related with other constructions available in the literature the basic example coming from a not necessarily finite group g with two subgroups h and k such that g k h and h k e where e is the identity of g is used to illustrate our theory more examples will be considered in forthcoming papers on the subject

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

on idempotent states on quantum groups idempotent states on a compact quantum group are shown to yield group like projections in the multiplier algebra of the dual discrete quantum group this allows to deduce that every idempotent state on a finite quantum group arises in a canonical way as the haar state on a finite quantum hypergroup a natural order structure on the set of idempotent states is also studied and some examples discussed

tensor product of principal unitary representations of quantum lorentz group and askey wilson polynomials we study the tensor product of principal unitary representations of the quantum lorentz group prove a decomposition theorem and compute the associated intertwiners we show that these intertwiners can be expressed in terms of complex continuations of 6j symbols of uq su 2 these intertwiners are expressed in terms of q racah polynomials and askey wilson polynomials the orthogonality of these intertwiners imply some relation mixing these two families of polynomials the simplest of these relations is the orthogonality of askey wilson polynomials

differences in spiking patterns among cortical neurons spike sequences recorded from four cortical areas of an awake behaving monkey were examined to explore characteristics that vary among neurons we found that a measure of the local variation of interspike intervals lv is nearly the same for every spike sequence for any given neuron while it varies significantly among neurons the distributions of lv values for neuron ensembles in three of the four areas were found to be distinctly bimodal two groups of neurons classified according to the spiking irregularity exhibit different responses to the same stimulus this suggests that neurons in each area can be classified into different groups possessing unique spiking statistics and corresponding functional properties

modeling spiking behavior of neurons with time dependent poisson processes three kinds of interval statistics as represented by the coefficient of variation the skewness coefficient and the correlation coefficient of consecutive intervals are evaluated for three kinds of time dependent poisson processes pulse regulated sinusoidally regulated and doubly stochastic among these three processes the sinusoidally regulated and doubly stochastic poisson processes in the case when the spike rate varies slowly compared with the mean interval between spikes are found to be consistent with the three statistical coefficients exhibited by data recorded from neurons in the prefrontal cortex of monkeys

hopf c algebras in this paper we define and study hopf c algebras roughly speaking a hopf c algebra is a c algebra a with a comultiplication a m a a such that the maps a b 7 a 1 b and a b 7 a 1 b have their range in a a and are injective after being extended to a larger natural domain the haagerup tensor product a h a in a purely algebraic setting these conditions onare closely related with the existence of a counit and antipode in this topological context things turn out to be much more subtle but nevertheless one can show the existence of a suitable counit and antipode under these conditions the basic example is the c algebra c0 g of continuous complex functions tending to zero at infinity on a locally compact group where the comultiplication is obtained by dualizing the group multiplication but also the reduced group c algebra c r g of a locally compact group with the well known comultiplication falls in this category in fact all locally compact quantum groups in the sense of 13 and 14 such as the compact and discrete ones as well as most of the known examples are included

capacity of a single spiking neuron channel information transfer through a single neuron is a fundamental component of information processing in the brain and computing the information channel capacity is important to understand this information processing the problem is difficult since the capacity depends on coding characteristics of the communication channel and optimization over input distributions among other issues in this letter we consider two models the temporal coding model of a neuron as a communication channel assumes the output is where is a gamma distributed random variable corresponding to the interspike interval that is the time it takes for the neuron to fire once the rate coding model is similar the output is the actual rate of firing over a fixed period of time theoretical studies prove that the distribution of inputs which achieves channel capacity is a discrete distribution with finite mass points for temporal and rate coding under a reasonable assumption this allows us to compute numerically the capacity of a neuron numerical results are in a plausible range based on biological evidence to date

the larson sweedler theorem for multiplier hopf algebras abstract any finite dimensional hopf algebra has a left and a right integral conversely larsen and sweedler showed that if a finite dimensional algebra with identity and a comultiplication with counit has a faithful left integral it has to be a hopf algebra in this paper we generalize this result to possibly infinite dimensional algebras with or without identity we have to leave the setting of hopf algebras and work with multiplier hopf algebras moreover whereas in the finite dimensional case there is a complete symmetry between the bialgebra and its dual this is no longer the case in infinite dimensions therefore we consider a direct version with integrals and a dual version with cointegrals of the larson sweedler theorem we also add some results about the antipode furthermore in the process of this paper we obtain a new approach to multiplier hopf algebras with integrals

the drinfel d double of multiplier hopf algebras abstract let a b be a pairing of two regular multiplier hopf algebras a and b the drinfel d double associated to this pairing is constructed by using appropriate representations of a and b on the same vector space b a we realize the drinfel d double denoted by d as an algebra of operators on the vector space b a in the case that a b is a multiplier hopf algebra pairing we prove that d is again a multiplier hopf algebra if a and b carry positive integrals we prove that d also has a positive integral this proof is not given before

twisted tensor product of multiplier hopf algebras abstract we combine two not necessarily commutative or co commutative multiplier hopf algebras to a new multiplier hopf algebra we use the construction of a twisted tensor product for algebras as introduced by a van daele we then proceed to find sufficient conditions on the twist map for this twisted tensor product to be a multiplier hopf algebra with the natural comultiplication for usual hopf algebras we find that majid s double crossed product by a matched pair of hopf algebras is exactly the twisted tensor product hopf algebra according to an appropriate twist map starting from two dually paired multiplier hopf algebras we construct the drinfel d double multiplier hopf algebra in the framework of twisted tensor products

hopf galois systems abstract we introduce the concept of hopf galois system a reformulation of the notion of galois extension of the base field for a hopf algebra the main feature of our definition is a generalization of the antipode of an ordinary hopf algebra we present several examples which indicate that although our axiomatic is slightly more complicated than the classical one it is also more natural and easier to handle with the main application of hopf galois systems is the construction of monoidal equivalences between comodule categories

on the central charge of a factorizable hopf algebra abstract for a semisimple factorizable hopf algebra over a field of characteristic zero we show that the value that an integral takes on the inverse drinfel d element differs from the value that it takes on the drinfel d element itself by at most a fourth root of unity this can be reformulated by saying that the central charge of the hopf algebra is an integer if the dimension of the hopf algebra is odd we show that these two values differ by at most a sign which can be reformulated by saying that the central charge is even we give a precise condition on the dimension that determines whether the plus sign or the minus sign occurs to formulate our results we use the language of modular data

the martin boundary of a discrete quantum group we consider the markov operator p on a discrete quantum group given by convolution with a q tracial state in the study of harmonic elements x p x x we define the martin boundary a it is a separable c algebra carrying canonical actions of the quantum group and its dual we establish a representation theorem to the effect that positive harmonic elements correspond to positive linear functionals on a the c algebraa has a natural time evolution and the unit can always be represented by a kms state any such state gives rise to a u c p map from the von neumann closure of a in its gns representation to the von neumann algebra of bounded harmonic elements which is an analogue of the poisson integral under additional assumptions this map is an isomorphism which respects the actions of the quantum group and its dual next we apply these results to identify the martin boundary of the dual of suq 2 with the quantum homogeneous sphere of podle this result extends and unifies previous results by ph biane and m izumi

twisted tensor coproduct of multiplier hopf algebras we put a non trivial comultiplication on the natural tensor product algebra of two multiplier hopf algebras by means of a cotwisting map as a special case we characterize the dual of the drinfel d double of an algebraic quantum group because any finite dimensional hopf algebra is an algebraic quantum group our characterization applies to the dual of the drinfel d double of a finite dimensional hopf algebra then it coincides with a result of lu

the analytic structure of algebraic quantum groups abstract in adv math 140 1998 323 366 van daele introduced the notion of an algebraic quantum group we proved in internat j math 8 8 1997 1067 1139 that such algebraic quantum groups give rise to reduced c algebraic quantum groups in the sense of j kustermans s vaes ann sci ecole norm sup 4 33 2000 837 934 after introducing the universal c algebraic quantum group associated to an algebraic one we will pull down the analytic structure of this c algebraic quantum group to the algebraic quantum group the multiplier algebra of the dual quantum group will be realized as a space of linear functionals on the original algebra we also identify the analytic structure of the dual quantum group in terms of the analytic structure of the original algebraic quantum group

a method for selecting the bin size of a time histogram the time histogram method is the most basic tool for capturing a time dependent rate of neuronal spikes generally in the neurophysiological literature the bin size that critically determines the goodness of the fit of the time histogram to the underlying spike rate has been subjectively selected by individual researchers here we propose a method for objectively selecting the bin size from the spike count statistics alone so that the resulting bar or line graph time histogram best represents the unknown underlying spike rate for a small number of spike sequences generated from a modestly fluctuating rate the optimal bin size may diverge indicating that any time histogram is likely to capture a spurious rate given a paucity of data the method presented here can nevertheless suggest how many experimental trials should be added in order to obtain a meaningful time dependent histogram with the required accuracy

kirby elements and quantum invariants we define the notion of a kirby element of a ribbon category mathcal c not necessarily semisimple kirby elements lead to 3 manifold invariants we characterize a class of kirby elements the algebraic kirby elements in terms of the structure maps of a hopf algebra in mathcal c this class is sufficiently large to recover the quantum invariants of 3 manifolds of reshetikhin and turaev of hennings kauffman and radford and of lyubashenko when these are well defined the cases of a semisimple ribbon category and of a category of representations are explored in detail

on fusion categories using a variety of methods developed in the literature in particular the theory of weak hopf algebras we prove a number of general results about fusion categories in characteristic zero we show that the global dimension of a fusion category is always positive and that the s matrix of any not necessarily hermitian modular category is unitary we also show that the category of module functors between two module categories over a fusion category is semisimple and that fusion categories and tensor functors between them are undeformable generalized ocneanu rigidity in particular the number of such categories functors realizing a given fusion datum is finite finally we develop the theory of frobenius perron dimensions in an arbitrary fusion category at the end of the paper we generalize some of these results to positive characteristic

information transmission using non poisson regular firing in many cortical areas neural spike trains do not follow a poisson process in this study we investigate a possible benefit of non poisson spiking for information transmission by studying the minimal rate fluctuation that can be detected by a bayesian estimator the idea is that an inhomogeneous poisson process may make it difficult for downstream decoders to resolve subtle changes in rate fluctuation but by using a more regular non poisson process the nervous system can make rate fluctuations easier to detect we evaluate the degree to which regular firing reduces the rate fluctuation detection threshold we find that the threshold for detection is reduced in proportion to the coefficient of variation of interspike intervals

analytically solvable autocorrelation function for weakly correlated interevent times long term temporal correlations observed in event sequences of natural and social phenomena have been characterized by algebraically decaying autocorrelation functions such temporal correlations can be understood not only by heterogeneous interevent times iets but also by correlations between iets in contrast to the role of heterogeneous iets on the autocorrelation function little is known about the effects due to the correlations between iets to rigorously study these effects we derive an analytical form of the autocorrelation function for the arbitrary iet distribution in the case with weakly correlated iets where the farlie gumbel morgenstern copula is adopted for modeling the joint probability distribution function of two consecutive iets our analytical results are confirmed by numerical simulations for exponential and power law iet distributions for the power law case we find a tendency of the steeper decay of the autocorrelation function for the stronger correlation between iets our analytical approach enables us to better understand long term temporal correlations induced by the correlations between iets

amenability and co amenability of algebraic quantum groups ii abstract we continue our study of the concepts of amenability and co amenability for algebraic quantum groups in the sense of a van daele and our investigation of their relationship with nuclearity and injectivity one major tool for our analysis is that every non degenerate representation of the universal c algebra associated to an algebraic quantum group has a unitary generator which may be described in a concrete way

double construction for crossed hopf coalgebras abstract we provide an analog of the drinfeld quantum double construction in the context of crossed hopf group coalgebras introduced by turaev we prove that provided the base group is finite the double of a semisimple crossed hopf group coalgebra is both modular and unimodular

minimizing total tardiness on a single machine with unequal release dates this study addresses the problem of minimizing total tardiness on a single machine with unequal release dates dominance properties established in previous literatures and herein are adopted to develop branch and bound and heuristic procedures computational experiments were conducted to evaluate the approaches the results revealed that the branch and bound algorithm is efficient in solving hard problems and easy problems that involve up to 50 and 500 jobs respectively the computational effectiveness of the heuristic is also reported

quantum quasigroups and loops abstract quantum quasigroups and quantum loops are self dual objects providing a general framework for the nonassociative extension of quantum group techniques bialgebra reducts of hopf algebras are quantum loops while sufficient conditions are given for quantum loop structure to augment to a hopf algebra the moufang hopf algebras of benkart et al the hopf quasigroups and coquasigroups of klim majid and the coassociative h bialgebras of perez izquierdo for instance the universal enveloping algebras of sabinin algebras all form quantum loops other quantum quasigroups offer natural nonassociative extensions of hopf algebra constructions quasigroup algebras dual quasigroup algebras and quantum couples of groups with quasigroups further examples include an algebra of rooted binary trees and an algebra of skein polynomials

estimating instantaneous irregularity of neuronal firing cortical neurons in vivo had been regarded as poisson spike generators that convey no information other than the rate of random firing recently using a metric for analyzing local variation of interspike intervals researchers have found that individual neurons express specific patterns in generating spikes which may symbolically be termed regular random or bursty rather invariantly in time in order to study the dynamics of firing patterns in greater detail we propose here a bayesian method for estimating firing irregularity and the firing rate simultaneously for a given spike sequence and we implement an algorithm that may render the empirical bayesian estimation practicable for data comprising a large number of spikes application of this method to electrophysiological data revealed a subtle correlation between the degree of firing irregularity and the firing rate for individual neurons irregularity of firing did not deviate greatly around the low degree of dependence on the firing rate and remained practically unchanged for individual neurons in the cortical areas v1 and mt whereas it fluctuated greatly in the lateral geniculate nucleus of the thalamus this indicates the presence and absence of autocontrolling mechanisms for maintaining patterns of firing in the cortex and thalamus respectively

empirical bayes interpretations of random point events given a sequence of apparently random point events such as neuronal spikes one may interpret them as being derived either irregularly in time from a constant rate or regularly from a fluctuating rate to determine which interpretation is more plausible in any given case we employ the empirical bayes method for a sequence of point events derived from a rate fluctuating gamma process the marginal likelihood function can possess two local maxima and the system exhibits a first order phase transition representing the switch of the most plausible interpretation from one to the other

modular invariant frobenius algebras from ribbon hopf algebra automorphisms abstract for any finite dimensional factorizable ribbon hopf algebra h and any ribbon automorphism of h we establish the existence of the following structure an h bimodule f and a bimodule morphism z from lyubashenko s hopf algebra object k for the bimodule category to f this morphism is invariant under the natural action of the mapping class group of the one punctured torus on the space of bimodule morphisms from k to f we further show that the bimodule f can be endowed with a natural structure of a commutative symmetric frobenius algebra in the monoidal category of h bimodules and that it is a special frobenius algebra iff h is semisimple the bimodules k and f can both be characterized as coends of suitable bifunctors the morphism z is obtained by applying a monodromy operation to the coproduct of f a similar construction for the product of f exists as well our results are motivated by the quest to understand the bulk state space and the bulk partition function in two dimensional conformal field theories with chiral algebras that are not necessarily semisimple

discrete quantum groups letgbe any discrete group consider the algebraaof all complex functions with finite support ongwith pointwise operations the multiplication onginduces a comultiplication onaby f p q f pq wheneverf aandp q g ifgis finite one can identify the algebra of complex functions ong gwitha aso that a a a then a is a hopf algebra ifgis infinite we still have f g 1 and f 1 g ina afor allfandg in this case a is a multiplier hopf algebra in fact it is a multiplier hopf algebra whenais given the natural involution defined by formula for allf aandp g in this paper we call a multiplier hopf algebra a adiscrete quantum group if the underlying algebraais a direct sum of full matrix algebras we study these discrete quantum groups and we give a simple proof of the existence and uniqueness of a left and a right invariant haar measure

measures of statistical dispersion based on shannon and fisher information concepts we propose and discuss two information based measures of statistical dispersion of positive continuous random variables the entropy based dispersion and fisher information based dispersion although standard deviation is the most frequently employed dispersion measure we show that it is not well suited to quantify some aspects that are often expected intuitively such as the degree of randomness the proposed dispersion measures are not entirely independent though each describes the quality of probability distribution from a different point of view we discuss relationships between the measures describe their extremal values and illustrate their properties on the pareto the lognormal and the lognormal mixture distributions application possibilities are also mentioned

radford s s 4 formula for co frobenius hopf algebras abstract this note extends radford s formula for the fourth power of the antipode of a finite dimensional hopf algebra to co frobenius hopf algebras and studies equivalent conditions to a hopf algebra being involutory for finite dimensional and co frobenius hopf algebras

estimating spiking irregularities under changing environments we considered a gammadistribution of interspike intervals as a statistical model for neuronal spike generation a gamma distribution is a natural extension of the poisson process taking the effect of a refractory period into account the model is specified by two parameters a time dependent firing rate and a shape parameter that characterizes spiking irregularities of individual neurons because the environment changes over time observed data are generated from a model with a time dependent firing rate which is an unknown function a statistical model with an unknown function is called a semiparametric model and is generally very difficult to solve we used a novel method of estimating functions in information geometry to estimate the shape parameter without estimating the unknown function we obtained an optimal estimating function analytically for the shape parameter independent of the functional form of the firing rate this estimation is efficient without fisher information loss and better than maximum likelihood estimation we suggest a measure of spiking irregularity based on the estimating function which may be useful for characterizing individual neurons in changing environments

optimizing time histograms for non poissonian spike trains the time histogram is a fundamental tool for representing the inhomogeneous density of event occurrences such as neuronal firings the shape of a histogram critically depends on the size of the bins that partition the time axis in most neurophysiological studies however researchers have arbitrarily selected the bin size when analyzing fluctuations in neuronal activity a rigorous method for selecting the appropriate bin size was recently derived so that the mean integrated squared error between the time histogram and the unknown underlying rate is minimized shimazaki shinomoto 2007 this derivation assumes that spikes are independently drawn from a given rate however in practice biological neurons express non poissonian features in their firing patterns such that the spike occurrence depends on the preceding spikes which inevitably deteriorate the optimization in this letter we revise the method for selecting the bin size by considering the possible non poissonian features improvement in the goodness of fit of the time histogram is assessed and confirmed by numerically simulated non poissonian spike trains derived from the given fluctuating rate for some experimental data the revised algorithm transforms the shape of the time histogram from the poissonian optimization method

non compact quantum groups associated with abelian subgroups letg be a lie group for any abelian subalgebra mathfrak h of the lie algebra g ofg and any r in mathfrak h lambda mathfrak h the difference of the left and right translates ofr gives a compatible poisson bracket ong we show how to construct the corresponding quantum group in thec algebra setting the main tool used is the general deformation quantization construction developed earlier by the author for actions of vector groups onc algebras

quantum groups with invariant integrals quantum groups have been studied intensively for the last two decades from various points of view the underlying mathematical structure is that of an algebra with a coproduct compact quantum groups admit haar measures however if we want to have a haar measure also in the noncompact case we are forced to work with algebras without identity and the notion of a coproduct has to be adapted these considerations lead to the theory of multiplier hopf algebras which provides the mathematical tool for studying noncompact quantum groups with haar measures i will concentrate on the algebra case and assume positivity of the invariant integral doing so i create an algebraic framework that serves as a model for the operator algebra approach to quantum groups indeed the theory of locally compact quantum groups can be seen as the topological version of the theory of quantum groups as they are developed here in a purely algebraic context

an algebraic framework for group duality abstract a hopf algebra is a pair a whereais an associative algebra with identity and a homomorphism formatoa asatisfying certain conditions if we drop the assumption thatahas an identity and if we allow to have values in the so called multiplier algebram a a we get a natural extension of the notion of a hopf algebra we call this a multiplier hopf algebra the motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group also for these multiplier hopf algebras there is a natural notion of left and right invariance for linear functionals called integrals in hopf algebra theory we show that if such invariant functionals exist they are unique up to a scalar and faithful for a regular multiplier hopf algebra a i e with invertible antipode with invariant functionals we construct in a canonical way the dual it is again a regular multiplier hopf algebra with invariant functionals it is also shown that the dual of is canonically isomorphic with the original multiplier hopf algebra a it is possible to generalize many aspects of abstract harmonic analysis here one can define the fourier transform one can prove plancherel s formula because any finite dimensional hopf algebra is a regular multiplier hopf algebra and has invariant functionals our duality theorem applies to all finite dimensional hopf algebras then it coincides with the usual duality for such hopf algebras but our category of multiplier hopf algebras also includes in a certain way the discrete quantum groups and the compact quantum groups our duality includes the duality between discrete quantum groups and compact quantum groups in particular it includes the duality between compact abelian groups and discrete abelian groups one of the nice features of our theory is that we have an extension of this duality to the non abelian case but within one category this is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type we prove that also these are dual to each other we treat an example that is sufficiently general to illustrate most of the different features of our theory it is also possible to construct the quantum double of drinfel d within this category this provides a still wider class of examples so we obtain many more than just the compact and discrete quantum within this setting

duals of quasitriangular hopf superalgebras and the classical limit the duality of quasitriangular hopf superalgebras and superbialgebras with so called main commutation relations mcr which is established by means of bilinear forms with values in a finite dimensional representation of the universal r matrix r is used to calculate relations of quantum supercoordinates for several examples the tensor product of representations of r with respect to the comultiplication then leads to the construction of an accordingly deformed tensor product of mcr type superbialgebras the correspondingly deformed tensor product of quantum vector superspaces is then established in the classical limit of lie superbialgebras one similarly obtains supercommutation relations which may be understood in case of quantum groups with deformation parameter q 1 as mcr in the first order of ln q again the deformed tensor product of such semiclassical mcr type lie superbialgebras is constructed

on the duality of generalized lie and hopf algebras abstract we show how under certain conditions an adjoint pair of braided monoidal functors can be lifted to an adjoint pair between categories of hopf algebras this leads us to an abstract version of michaelis theorem stating that given a hopf algebra h there is a natural isomorphism of lie algebras q h p h where q h is the dual lie algebra of the lie coalgebra of indecomposables of h and p h is the lie algebra of primitive elements of the sweedler dual of h we apply our theory to turaev s hopf group co algebras

on the antipode of a quasitriangular hopf algebra let a r be a quasitriangular hopf algebra with antipode s in the category of vector spaces over a field k in this paper we show that s2 is inner which means s is bijective and when a is finite dimensional there is a grouplike element h e a such that s4 a h r for all a e a suppose that a is any hopf algebra with antipode s over the field k proposition 1 of section 1 gives a sufficient condition for 3 to be inner which we apply to quasitriangular hopf algebras in section 2 its statement and proof are based on calculations found in 2 pp 66671 generally s2 need not be inner suppose that a is finite dimensional if s2 were inner then all ideals of a would be invariant under s2 and hence all subcoalgebras of the dual hopf algebra a would be invariant under s where s s is the antipode of a there are examples 5 over algebraically closed lields in which the latter is not the case section 2 begins with a discussion of properties of quasit angular hopf algebras a r over the held k used in this paper write r cr c r ea a and set u cs r r the main result of section 2 is that u is invertible and s2 a uau for all a e a this result was established in z exercise 7 3 61 under the hypothesis that s is bijective if a is any finite dimensional hopf algebra with antipode s over the field k there are distinguished grouplike elements g e a and tl e a which relate left and right integrals and which together describe s4 in section 3 we consider s4 when a r is a unite dimensional quasit angular hopf algebra let h iiu where u c s r r l is defined as above and u s u i by relating h to g and 1 we are able to show that h is a grouplike element and that s4 a hah for all a e a when a is unimodular we prove that h g thus the product vu does not depend on r in the unimodular case we assume that the reader has some familiarity with the elementary aspects of the theory of hopf algebras a recommended reference is 6

harmonic analysis on the quantum lorentz group using a continuation of 6j symbols of su q 2 with complex spins we give a new description of the unitary representations of sl q 2 and find explicit expressions for the characters of unitary representations of sl q 2 we prove a plancherel theorem for the quantum lorentz group

multiplier bialgebras in braided monoidal categories abstract multiplier bimonoids or bialgebras in arbitrary braided monoidal categories are defined they are shown to possess monoidal categories of comodules and modules these facts are explained by the structures carried by their induced functors

paths in quantum cayley trees and l 2 cohomology abstract we study existence uniqueness and triviality of path cocycles in the quantum cayley graph of universal discrete quantum groups in the orthogonal case we find that the unique path cocycle is trivial in contrast with the case of free groups where it is proper in the unitary case it is neither bounded nor proper from this geometrical result we deduce the vanishing of the first l 2 betti number of a o i n

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h