on metaplectic modular categories and their applications for non abelian simple objects in a unitary modular category the density of their braid group representations the p hard evaluation of their associated link invariants and the bqp completeness of their anyonic quantum computing models are closely related we systematically study such properties of the non abelian simple objects in the metaplectic modular categories so m 2 for an odd integer m 3 the simple objects with quantum dimensions sqrt m have finite image braid group representations and their link invariants are classically efficient to evaluate we also provide classically efficient simulations of their braid group representations these simulations of the braid group representations can be regarded as qudit generalizations of the knill gottesmann theorem for the qubit case the simple objects of dimension 2 give us a surprising result while their braid group representations have finite images and are efficiently simulable classically after a generalized localization their link invariants are p hard to evaluate exactly we sharpen the p hardness by showing that any sufficiently accurate approximation of their associated link invariants is already p hard

a modular functor which is universal for quantum computation we show that the topological modular functor from witten chern simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor s state space a computational model based on chern simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model the chief technical advance the density of the irreducible sectors of the jones representation has topological implications which will be considered elsewhere

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

on classification of modular tensor categories we classify all unitary modular tensor categories umtcs of rank 4 there are a total of 35 umtcs of rank 4 up to ribbon tensor equivalence since the distinction between the modular s matrix s and s has both topological and physical significance so in our convention there are a total of 70 umtcs of rank 4 in particular there are two trivial umtcs with s 1 each such umtc can be obtained from 10 non trivial prime umtcs by direct product and some symmetry operations explicit data of the 10 non trivial prime umtcs are given in sect 5 relevance of umtcs to topological quantum computation and various conjectures are given in sect 6

low dimensional representations of the three component loop braid group motivated by physical and topological applications we study representations of the group lb3 of motions of 3 unlinked oriented circles in 3 our point of view is to regard the three strand braid group b3 as a subgroup of lb3 and study the problem of extending b3 representations we introduce the notion of a standard extension and characterize b3 representations admitting such an extension in particular we show using a classification result of tuba and wenzl pacific j math 197 491 510 2001 that every irreducible b3 representation of dimension at most 5 has a standard extension we show that this result is sharp by exhibiting an irreducible 6 dimensional b3 representation that has no extensions standard or otherwise we obtain complete classifications of 1 irreducible 2 dimensional lb3 representations 2 extensions of irreducible 3 dimensional b3 representations and 3 irreducible lb3 representations whose restriction to b3 has abelian image

protected gates for topological quantum field theories we study restrictions on locality preserving unitary logical gates for topological quantum codes in two spatial dimensions a locality preserving operation is one which maps local operators to local operators for example a constant depth quantum circuit of geometrically local gates or evolution for a constant time governed by a geometrically local bounded strength hamiltonian locality preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized and hence sufficiently dilute errors remain correctable by invoking general properties of two dimensional topological field theories we find that the locality preserving logical gates are severely limited for codes which admit non abelian anyons in particular there are no locality preserving logical gates on the torus or the sphere with m punctures if the braiding of anyons is computationally universal furthermore for ising anyons on the m punctured sphere locality preserving gates must be elements of the logical pauli group we derive these results by relating logical gates of a topological code to automorphisms of the verlinde algebra of the corresponding anyon model and by requiring the logical gates to be compatible with basis changes in the logical hilbert space arising from local f moves and the mapping class group

density of the so 3 tqft representation of mapping class groups we show that for an odd prime r 3 and an integer g 1 in the projective representation given by the so 3 witten reshitikhin turaev theory at an rth root of unity the image of the mapping class group of a surface of genus g is dense

a combinatorial approach to topological quantum field theories and invariants of graphs the combinatorial state sum of turaev and viro for a compact 3 manifold in terms of quantum 6j symbols is generalized by introducing observables in the form of coloured graphs they satisfy braiding relations and allow for surgeries and a discussion of cobordism theory application of these techniques give the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any riemann surface with arbitrary coloured punctures

experimental approximation of the jones polynomial with one quantum bit we present experimental results approximating the jones polynomial using 4 qubits in a liquid state nuclear magnetic resonance quantum information processor this is the first experimental implementation of a complete problem for the deterministic quantum computation with one quantum bit model of quantum computation which uses a single qubit accompanied by a register of completely random states the jones polynomial is a knot invariant that is important not only to knot theory but also to statistical mechanics and quantum field theory the implemented algorithm is a modification of the algorithm developed by shor and jordan suitable for implementation in nmr these experimental results show that for the restricted case of knots whose braid representations have four strands and exactly three crossings identifying distinct knots is possible 91 of the time

homological error correction classical and quantum codes we prove several theorems characterizing the existence of homological error correction codes both classically and quantumly not every classical code is homological but we find a family of classical homological codes saturating the hamming bound in the quantum case we show that for nonorientable surfaces it is impossible to construct homological codes based on qudits of dimension d 2 while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension d we give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries we show how the original shor s 9 qubit code can be visualized as a homological quantum code we study the problem of constructing quantum codes with optimal encoding rate in the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality for homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate we provide the tools of homology group theory for graphs embedded on surfaces in a self contained manner

tempered stable levy motion and transient super diffusion the space fractional diffusion equation models anomalous super diffusion its solutions are transition densities of a stable levy motion representing the accumulation of power law jumps the tempered stable levy motion uses exponential tempering to cool these jumps a tempered fractional diffusion equation governs the transition densities which progress from super diffusive early time to diffusive late time behavior this article provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation with drift a temporal and spatial second order crank nicolson method is developed based on a finite difference formula for tempered fractional derivatives a new exponential rejection method for simulating tempered levy stables is presented to facilitate particle tracking codes

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

on quantum gauge theories in two dimensions two dimensional quantum yang mills theory is studied from three points of view i by standard physical methods ii by relating it to the largek limit of three dimensional chern simons theory and two dimensional conformal field theory iii by relating its weak coupling limit to the theory of reidemeister ray singer torsion the results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces

applications of inverse tempered stable subordinators abstract the inverse tempered stable subordinator is a stochastic process that models power law waiting times between particle movements with an exponential tempering that allows all moments to exist this paper shows that the probability density function of an inverse tempered stable subordinator solves a tempered time fractional diffusion equation and its folded density solves a tempered time fractional telegraph equation two explicit formulae for the density function are developed and applied to compute explicit solutions to tempered fractional cauchy problems where a tempered fractional derivative replaces the first derivative in time several examples are given including tempered fractional diffusion equations on bounded or unbounded domains and the probability distribution of a tempered fractional poisson process it is shown that solutions to the tempered fractional diffusion equation have a cusp at the origin

computer calculation of witten s 3 manifold invariant witten s 2 1 dimensional chern simons theory is exactly solvable we compute the partition function a topological invariant of 3 manifolds on generalized seifert spaces thus we test the path integral using the theory of 3 manifolds in particular we compare the exact solution with the asymptotic formula predicted by perturbation theory we conclude that this path integral works as advertised and gives an effective topological invariant

on the classification of the grothendieck rings of non self dual modular categories abstract we develop a symbolic computational approach to classifying low rank modular fusion categories up to finite ambiguity by a generalized form of ocneanu rigidity due to etingof ostrik and nikshych it is enough to classify modular fusion algebras of a given rank that is to determine the possible grothendieck rings with modular realizations we use this technique to classify modular categories of rank at most 5 that are non self dual i e those for which some object is not isomorphic to its dual object

quantum computation and the evaluation of tensor networks we present a quantum algorithm that additively approximates the value of a tensor network to a certain scale when combined with existing results this provides a complete problem for quantum computation the result is a simple new way of looking at quantum computation in which unitary gates are replaced by tensors and time is replaced by the order in which the tensor network is swallowed we use this result to derive new quantum algorithms that approximate the partition function of a variety of classical statistical mechanical models including the potts model

subordinated continuous time ar processes and their application to modeling behavior of mechanical system abstract many real data exhibit behavior adequate to subdiffusion processes very often it is manifested by so called trapping events the visible evidence of subdiffusion we observe not only in financial time series but also in technical data in this paper we propose a model which can be used for description of such kind of data the model is based on the continuous time autoregressive time series with stable noise delayed by the infinitely divisible inverse subordinator the proposed system can be applied to real datasets with short time dependence visible jumps and mentioned periods of stagnation in this paper we extend the theoretical considerations in analysis of subordinated processes and propose a new model that exhibits mentioned properties we concentrate on the main characteristics of the examined subordinated process expressed mainly in the language of the measures of dependence which are main tools used in statistical investigation of real data we present also the simulation procedure of the considered system and indicate how to estimate its parameters the theoretical results we illustrate by the analysis of real technical data

state sum invariants of compact 3 manifolds with boundary and 6j symbols the authors extend the combinatorial construction of invariants of smooth compact closed 3 manifolds as given by turaev and viro 1991 to obtain invariants of 3 manifolds and with a boundary the techniques uses quantum 6j symbols associated to the quantized universal enveloping algebra uq sl 2 c q a root of unity to give a construction of a state sum for given triangulation this state sum is then invariant under subdivisions and isotopes of the boundary their methods also lead to a simplified proof of the main result of turaev and viro that the state sum is independent of the triangulation and hence gives rise to an invariant of the manifold they use surgery to calculate the state sum for some closed manifolds their results also confirm a finding of turaev 1990 which relates this theory to the topological quantum field theory with a chern simons action in the sense of witten 1989

quantum computational networks the theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines quantum gates are the generalization of classical logic gates a single type of gate the universal quantum gate together with quantum unit wires is adequate for constructing networks with any possible quantum computational property

geometric brownian motion with tempered stable waiting times one of the earliest system that was used to asset prices description is black scholes model it is based on geometric brownian motion and was used as a tool for pricing various financial instruments however when it comes to data description geometric brownian motion is not capable to capture many properties of present financial markets one can name here for instance periods of constant values therefore we propose an alternative approach based on subordinated tempered stable geometric brownian motion which is a combination of the popular geometric brownian motion and inverse tempered stable subordinator in this paper we introduce the mentioned process and present its main properties we propose also the estimation procedure and calibrate the analyzed system to real data

the maz ya shaposhnikova limit in the magnetic setting abstract we prove a magnetic version of the maz ya shaposhnikova singular limit of nonlocal norms with vanishing fractional parameter this complements a general convergence result recently obtained by authors when the parameter approaches one

multiplicity results for magnetic fractional problems abstract the paper deals with the existence of multiple solutions for a boundary value problem driven by the magnetic fractional laplacian a s that is a s u f u u in u 0 in r n where is a real parameter f is a continuous function and is an open bounded subset of r n with lipschitz boundary we prove that the problem admits at least two nontrivial weak solutions under two different sets of conditions on the nonlinear term f which are dual in a suitable sense

kramers escape problem for fractional klein kramers equation with tempered α stable waiting times in this paper we extend the subdiffusive klein kramers model in which the waiting times are modeled by the stable laws to the case of waiting times belonging to the class of tempered stable distributions we introduce a generalized version of the klein kramers equation in which the fractional riemman liouville derivative is replaced with a more general integro differential operator this allows a transition from the initial subdiffusive character of motion to the standard diffusion for long times to be modeled taking advantage of the corresponding langevin equation we study some properties of the tempered dynamics in particular we approximate solutions of the tempered klein kramers equation via monte carlo methods also we study the distribution of the escape time from the potential well and compare it to the classical results in the kramers escape theory finally we derive the analytical formula for the first passage time distribution for the case of free particles we show that the well known sparre andersen scaling holds also for the tempered subdiffusion

nonlinear fractional magnetic schrodinger equation existence and multiplicity abstract in this paper we focus our attention on the following nonlinear fractional schrodinger equation with magnetic field e 2 s a e s u v x u f u 2 u in r n where e 0 is a parameter s 0 1 n 3 a s is the fractional magnetic laplacian v r n r and a r n r n are continuous potentials and f r n r is a subcritical nonlinearity by applying variational methods and ljusternick schnirelmann theory we prove existence and multiplicity of solutions for e small

chern simons theory with finite gauge group we construct in detail a 2 1 dimensional gauge field theory with finite gauge group in this case the path integral reduces to a finite sum so there are no analytic problems with the quantization the theory was originally introduced by dijkgraaf and witten without details the point of working it out carefully is to focus on the algebraic structure and particularly the construction of quantum hilbert spaces on closed surfaces by cutting and pasting this includes the verlinde formula the careful development may serve as a model for dealing with similar issues in more complicated cases

on the fractional schrodinger kirchhoff equations with electromagnetic fields and critical nonlinearity abstract in this paper we consider the fractional schrodinger kirchhoff equations with electromagnetic fields and critical nonlinearity e 2 s m u s a e 2 a e s u v x u u 2 s 2 u h x u 2 u x r n u x 0 as x where a e s is the fractional magnetic operator with 0 s 1 2 s 2 n n 2 s m r 0 r is a continuous nondecreasing function v r n r 0 and a r n r n are the electric and magnetic potentials respectively by using the fractional version of the concentration compactness principle and variational methods we show that the above problem i has at least one solution provided that e e and ii for any m n has m pairs of solutions if e e m where e and e m are sufficiently small positive numbers moreover these solutions u e 0 as e 0

numerical approach to the fractional klein kramers equation subdiffusion in the presence of an external force field can be described in phase space by the fractional klein kramers equation in this paper we explore the stochastic structure of this equation using a subordination method we define a random process whose probability density function is a solution of the fractional klein kramers equation the structure of the introduced process agrees with the two stage scenario underlying the anomalous diffusion mechanism in which trapping events are superimposed on the langevin dynamics we develop an efficient computer algorithm for visualization of fractional klein kramers dynamics and present some simulation results based on monte carlo techniques

quantum morphing and the jones polynomial we will explore the experimental observation that on the set of knots with bounded crossing number algebraically independent vassiliev invariants become correlated as noticed first by s willerton we will see this through the value distribution of the jones polynomial at roots of unit as the degree of the roots of unit is getting larger the higher order fluctuation is diminishing and a more organized shape will emerge from a rather random value distribution of the jones polynomial we call such a phenomenon quantum morphing evaluations of the jones polynomial at roots of unity play a crucial role for example in the volume conjecture

arithmetic brownian motion subordinated by tempered stable and inverse tempered stable processes abstract in the last decade the subordinated processes have become popular and have found many practical applications therefore in this paper we examine two processes related to time changed subordinated classical brownian motion with drift called arithmetic brownian motion the first one so called normal tempered stable is related to the tempered stable subordinator while the second one to the inverse tempered stable process we compare the main properties such as probability density functions laplace transforms ensemble averaged mean squared displacements of such two subordinated processes and propose the parameters estimation procedures moreover we calibrate the analyzed systems to real data related to indoor air quality

characterization of the traces on the boundary of functions in magnetic sobolev spaces abstract we characterize the trace of magnetic sobolev spaces defined in a half space or in a smooth bounded domain in which the magnetic field a is differentiable and its exterior derivative corresponding to the magnetic field da is bounded in particular we prove that for d 1 and p 1 the trace of the magnetic sobolev space w a 1 p r d 1 is exactly w a 1 1 p p r d where a x a 1 a d x 0 for x r d with the convention a a 1 a d 1 when a c 1 r d 1 r d 1 we also characterize fractional magnetic sobolev spaces as interpolation spaces and give extension theorems from a half space to the entire space

diffusion and relaxation controlled by tempered α stable processes we derive general properties of anomalous diffusion and nonexponential relaxation from the theory of tempered alpha stable processes the tempering results in the existence of all moments of operational time the subordination by the inverse tempered alpha stable process provides diffusion relaxation that occupies an intermediate place between subdiffusion cole cole law and normal diffusion exponential law here we obtain explicitly the fokker planck equation and the cole davidson relaxation function this model includes subdiffusion as a particular case

bivariate sub gaussian model for stock index returns abstract financial time series are commonly modeled with methods assuming data normality however the real distribution can be nontrivial also not having an explicitly formulated probability density function in this work we introduce novel parameter estimation and high powered distribution testing methods which do not rely on closed form densities but use the characteristic functions for comparison the approach applied to a pair of stock index returns demonstrates that such a bivariate vector can be a sample coming from a bivariate sub gaussian distribution the methods presented here can be applied to any nontrivially distributed financial data among others

american option valuation under time changed tempered stable levy processes abstract given that the underlying assets in financial markets exhibit stylized facts such as leptokurtosis asymmetry clustering properties and heteroskedasticity effect this paper presents a novel model for pricing american option under the assumptions that the stock price processes are governed by time changed tempered stable levy process as this model is constructed by introducing random time changes into tempered stable ts processes which specially refer to normal tempered stable nts distribution as well as classical tempered stable cts distribution it permits infinite jumps as well as capturing random varying time in stochastic volatility consequently taking into account the empirical facts such as leptokurtosis skewness and volatility clustering behaviors we employ the fourier cosine technique to calculate american option and propose the improved particle swarm optimization ipso intelligent algorithm for model calibration to demonstrate the advantage of the constructed model we carry out empirical research on american index option in financial markets across wide ranges of models with the time changing normal tempered stable distribution model yielding a superior performance than others

overshooting and undershooting subordination scenario for fractional two power law relaxation responses in this paper we propose a transparent subordination approach to anomalous diffusion processes underlying the nonexponential relaxation we investigate properties of a coupled continuous time random walk that follows from modeling the occurrence of jumps with compound counting processes as a result two different diffusion processes corresponding to over and undershooting operational times respectively have been found we show that within the proposed framework all empirical two power law relaxation patterns may be derived this work is motivated by the so called less typical relaxation behavior observed e g for gallium doped cd0 99mn0 01te mixed crystals

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

the two eigenvalue problem and density of jones representation of braid groups 1 the two eigenvalue problem 179 2 hecke algebra representations of braid groups 183 3 duality of jones wenzl representations 186 4 closed images of jones wenzl sectors 188 5 distribution of evaluations of jones polynomials 192 6 fibonacci representations 196

analytic approach to the problem of convergence of truncated levy flights towards the gaussian stochastic process an analytic expression for characteristic function defining a truncated l evy flight is derived it is shown that the characteristic function yields results in agreement with recent simulations of truncated l evy flights by mantegna and stanley phys rev lett 73 2946 1994 with the analytic expression for the characteristic function the convergence of the l evy process towards the gaussian is demonstrated without simulations in the calculation of first return probability the simulations are replaced by numerical integration using simple quadratures

tempered stable levy motion driven by stable subordinator a b s t r a c t in this article we propose a new model for financial data description combining two independent mechanisms namely the tempered stable process and inverse stable subordinator we obtain a new model which captures not only the tempered stable character of the underlying data but also such a property as periods in which the values of an asset stay on the same level moreover we classify our system to the family of subdiffusive processes and investigate its tail behavior we describe in detail testing and estimation procedures for the proposed model in the last step we calibrate our model to the real data

calibration of the subdiffusive arithmetic brownian motion with tempered stable waiting times in the classical analysis many models used to real data description are based on the standard brownian diffusion type processes however some real data exhibit characteristic periods of constant values in such cases the popular systems seem not to be applicable therefore we propose an alternative approach based on the combination of the popular brownian motion with drift called also the arithmetic brownian motion and tempered stable subordinator the probability density function of the proposed model can be described by a fokker planck type equation and therefore it has many similar properties as the popular brownian motion with drift in this paper we propose the estimation procedure for the considered tempered stable subdiffusive arithmetic brownian motion and calibrate the analyzed process to the real financial data

fokker planck type equations associated with fractional brownian motion controlled by infinitely divisible processes abstract in this paper we study the anomalous diffusion process driven by fractional brownian motion delayed by general infinitely divisible subordinator we show the analyzed process is the stochastic representation of the fokker planck type equation that describes the probability density function of an introduced model moreover we study main characteristics of the examined process the first two moments that allow us in special cases for classification of it as a system with accelerating subdiffusion property

quasi quantum groups knots three manifolds and topological field theory we show how to construct starting from a quasi hopf algebra or quasiquantum group invariants of knots and links in some cases these invariants give rise to invariants of the three manifolds obtained by surgery along these links this happens for a finite dimensional quasi quantum group whose definition involves a finite groupg and a 3 cocycle which was first studied by dijkgraaf pasquier and roche we treat this example in more detail and argue that in this case the invariants agree with the partition function of the topological field theory of dijkgraaf and witten depending on the same datag

subordinated α stable ornstein uhlenbeck process as a tool for financial data description abstract the classical financial models are based on the standard brownian diffusion type processes however in the exhibition of some real market data like interest or exchange rates we observe characteristic periods of constant values moreover in the case of financial data the assumption of normality is often unsatisfied in such cases the popular vasicek model that is a mathematical system describing the evolution of interest rates based on the ornstein uhlenbeck process seems not to be applicable therefore we propose an alternative approach based on a combination of the popular ornstein uhlenbeck process with a stable distribution and subdiffusion systems that demonstrate such characteristic behavior the probability density function of the proposed process can be described by a fokker planck type equation and therefore it can be examined as an extension of the basic ornstein uhlenbeck model in this paper we propose the parameters estimation method and calibrate the subordinated vasicek model to the interest rate data

fractional fokker planck equation with tempered α stable waiting times langevin picture and computer simulation in this paper we introduce a langevin type model of subdiffusion with tempered stable waiting times we consider the case of space dependent external force fields the model displays subdiffusive behavior for small times and it converges to standard gaussian diffusion for large time scales we derive general properties of tempered anomalous diffusion from the theory of tempered stable processes in particular we find the form of the fractional fokker planck equation corresponding to the tempered subdiffusion we also construct an algorithm of simulation of sample paths of the introduced process we apply the algorithm to approximate solutions of the fractional fokker planck equation and to study statistical properties of the tempered subdiffusion via monte carlo methods

a semilinear schrodinger equation in the presence of a magnetic field we consider the semilinear stationary schrodinger equation in a magnetic field i a 2u v x u g x u u in n where v is the scalar or electric potential and a is the vector or magnetic potential we study the existence of nontrivial solutions both in the critical and in the subcritical case respectively g x u u 2 2 and g x u c 1 u p 2 where 2 p 2 the results are obtained by variational methods for g critical we use constrained minimization and for subcritical g we employ a minimax type argument in the latter case we also study the existence of infinitely many geometrically distinct solutions

non linear ground state representations and sharp hardy inequalities we determine the sharp constant in the hardy inequality for fractional sobolev spaces to do so we develop a non linear and non local version of the ground state representation which even yields a remainder term from the sharp hardy inequality we deduce the sharp constant in a sobolev embedding which is optimal in the lorentz scale in the appendix we characterize the cases of equality in the rearrangement inequality in fractional sobolev spaces

recognition of stable distribution with levy index α close to 2 we address the problem of recognizing stable levy distribution with levy index close to 2 from experimental data we are interested in the case when the sample size of available data is not large thus the power law asymptotics of the distribution is not clearly detectable and the shape of the empirical probability density function is close to a gaussian we propose a testing procedure combining a simple visual test based on empirical fourth moment with the anderson darling and jarque bera statistical tests and we check the efficiency of the method on simulated data furthermore we apply our method to the analysis of turbulent plasma density and potential fluctuations measured in the stellarator type fusion device and demonstrate that the phenomenon of the l h transition from low confinement l mode to a high confinement h mode which occurs in this device is accompanied by the transition from levy to gaussian fluctuation statistics

on the bourgain brezis and mironescu theorem concerning limiting embeddings of fractional sobolev spaces the article is concerned with the bourgain brezis and mironescu theorem on the asymptotic behaviour of the norm of the sobolev type embedding operator ws p lpn n sp as s 1 and s n p thei

regularity properties of nonlocal minimal surfaces via limiting arguments abstract we prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter s when s 1 as a consequence we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is less or equal than 7 and s is close to 1

quantum algorithm for linear systems of equations solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems given a matrix a and a vector b find a vector x such that ax b we consider the case where one does not need to know the solution x itself but rather an approximation of the expectation value of some operator associated with x e g x dagger mx for some matrix m in this case when a is sparse n x n and has condition number kappa the fastest known classical algorithms can find x and estimate x dagger mx in time scaling roughly as n square root kappa here we exhibit a quantum algorithm for estimating x dagger mx whose runtime is a polynomial of log n and kappa indeed for small values of kappa i e poly log n we prove using some common complexity theoretic assumptions that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm

why should anyone care about computing with anyons in this article we present a pedagogical introduction of the main ideas and recent advances in the area of topological quantum computation we give an overview of the concept of anyons and their exotic statistics present various models that exhibit topological behaviour and establish their relation to quantum computation possible directions for the physical realization of topological systems and the detection of anyonic behaviour are elaborated

representations of the loop braid groups from braided tensor categories the loop braid group is the motion group of unknotted oriented circles in r3 in this paper we study their representations through the approach inspired by two dimensional topological phases of matter in principle the motion of loops in r3 reduces to the motions of points in a two dimensional sliced plane we realize this physical picture in terms of braided tensor categories and their braid group representations

models of anomalous diffusion the subdiffusive case abstract the paper discusses a model for anomalous diffusion processes their one point probability density functions p d f are exact solutions of fractional diffusion equations the model reflects the asymptotic behavior of a jump anomalous random walk process with random jump sizes and random inter jump time intervals with infinite means and variances which do not satisfy the law of large numbers in the case when these intervals have a fractional exponential p d f the fractional komogorov feller equation for the corresponding anomalous diffusion is provided and methods of finding its solutions are discussed finally some statistical properties of solutions of the related langevin equation are studied the subdiffusive case is explored in detail the emphasis is on a rigorous presentation which however would be accessible to the physical sciences audience

conformal field theory and the cohomology of the moduli space of stable bundles let g be a compact riemann surface of genus g 2 and let be a line bundle over g of degree 1 then the moduli space of rank 2 stable bundles v over g such that v was shown by seshadri 19 to be a nonsingular projective variety ng its rational cohomology ring h ng is exceedingly rich and despite years of study has never quite been computed in full the object of this paper is to give an essentially complete characterization of this ring or at least to reduce the problem to a matter of linear algebra in particular we find a proof of newstead s conjecture that pg1 ng 0 we also obtain a formula for the volume of ng which can be regarded as a twisted version of the formula for the degree 0 moduli space recently announced by witten the approach we shall take is not from algebraic geometry but from mathematical physics it relies on the su 2 wess zumino witten model which is a functor zk associating a finite dimensional vector space to each riemann surface with marked points the relationship with the moduli space is that when the level k of the functor is even the vector space associated to g with no marked points can be identified with h ng l where l is a fixed line bundle over ng now the work of verlinde provides us with a means of calculating the dimension of any vector space arising from our functor and in particular dimh ng l which we shall denote d g k on the other hand newstead 14 found explicit generators for h ng and we can also express d g k in terms of them using a riemann roch theorem equating the two formulas enables us to evaluate any monomial in the generators on the fundamental class of ng and by poincare duality this is sufficient at least in principle to determine the ring structure of h ng in the discussion above one important point has been skated over the su 2 wzw model is of course associated to bundles of degree 0 not degree 1 so in order to make use of verlinde s work it is necessary to formulate a twisted version of the field theory this is carried out in 2 but the crucial properties of the twisted theory analogous to those which make the ordinary theory a modular functor are not proved in this paper rather we will confine ourselves to exploring the consequences of these claims and hope to return to justify them in a later paper an outline of the remaining sections goes as follows in 3 we review those parts of verlinde s work we shall need show how they must be modified in the twisted case and work out some explicit formulas for d g k in 4 we study the cohomology of ng which we regard throughout as a space of representations via the theorem of narasimhan and seshadri we define newstead s generators and i of h ng

interpolations and fractional sobolev spaces in carnot groups abstract in this paper we present an interpolation approach to the fractional sobolev spaces in carnot groups using the k method this approach provides us with a different characterization of these sobolev spaces moreover it provides us with the limiting behavior of the fractional sobolev norms at the end points this allows us to deduce results similar to the bourgain brezis mironescu and maz ya shaposhnikova in the case p 1 and davila s result in the case p 1 also this allows us to deduce the limiting behavior of the fractional perimeter in carnot groups

universal quantum computation with metaplectic anyons we show that braidings of the metaplectic anyons x in so 3 2 su 2 4 with their total charge equal to the metaplectic mode y supplemented with projective measurements of the total charge of two metaplectic anyons are universal for quantum computation we conjecture that similar universal anyonic computing models can be constructed for all metaplectic anyon systems so p 2 for any odd prime p 5 in order to prove universality we find new conceptually appealing universal gate sets for qutrits and qupits