r matrix approach to quantum superalgebras suq m n quantum superalgebras suq m n are studied in the framework of r matrix formalism explicit parametrization of l and l matrices in terms of suq m n generators are presented we also show that quantum deformation of nonsimple superalgebra su n n requires its extension to u n n

equivalent continuous and discrete realizations of levy flights a model of one dimensional motion of an inertial particle abstract the paper is devoted to the relationship between the continuous markovian description of levy flights developed previously see e g i a lubashevsky truncated levy flights and generalized cauchy processes eur phys j b 82 2011 189 195 and references therein and their equivalent representation in terms of discrete steps of a wandering particle a certain generalization of continuous time random walks to simplify understanding the key points of the technique to be created our consideration is confined to the one dimensional model for continuous random motion of a particle with inertia its dynamics governed by stochastic self acceleration is described as motion on the phase plane x v comprising the position x and velocity v d x d t of the given particle a notion of random walks inside a certain neighborhood l of the line v 0 the x axis and outside it is developed it enables us to represent a continuous trajectory of particle motion on the plane x v as a collection of the corresponding discrete steps each of these steps matches one complete fragment of the velocity fluctuations originating and terminating at the boundary of l as demonstrated the characteristic length of particle spatial displacement is mainly determined by velocity fluctuations with large amplitude which endows the derived random walks along the x axis with the characteristic properties of levy flights using the developed classification of random trajectories a certain parameter free core stochastic process is constructed its peculiarity is that all the characteristics of levy flights similar to the exponent of the levy scaling law are no more than the parameters of the corresponding transformation from the particle velocity v to the related variable of the core process in this way the previously found validity of the continuous markovian model for all the regimes of levy flights is explained based on the obtained results an efficient single peak approximation is constructed in particular it enables us to calculate the basic characteristics of levy flights using the probabilistic properties of extreme velocity fluctuations and the shape of the most probable trajectory of particle motion within such extreme fluctuations

finite dimensional irreducible representations of the quantum superalgebra u q gl n 1 it is shown that every finite dimensional irreducible module over the general linear lie superalgebragl n 1 can be deformed to an irreducible module ofu q gl n 1 aq analogue of the universal enveloping algebra ofgl n 1 the results are extended also to all kac modules which in the atypical cases remain indecomposible within each module expressions for the transformations of the gel fand zetlin basis under the action of the algebra generators are written down an analogoue of the poincare birkhoff witt theorem is formulated

q oscillator realizations of the quantum superalgebras slq m n andospq m 2n realizations of the quantum superalgebras corresponding to thea m n b m n c n 1 andd m n series are given in terms of the creation and annihilation operators ofq deformed bose and fermi oscillators

on the quantum symmetries associated with the two parameter free fermion model the quantum algebras associated with the r matrix obtained from the two parameter free fermion model are examined it is shown that in addition to uq s gl 1 1 there exists another algebra which is isomorphic to uq s gl 1 1 as an algebra but has a different hopf structure it does not have a classical limit and is related to up sl 2 c at a root of unity p i i2 1

induced representations of the multiparameter hopf superalgebras uuq gl m n and uuq sl m n we construct the induced holomorphic representations of the multiparameter hopf superalgebras uuq gl m n and uuq sl m n the first superalgebra we constructed earlier as the dual of the multiparameter quantum deformation of the supergroup gl m n the second superalgebra is a hopf subalgebra of the first for a special choice of the parameters the representations are labelled by m n integer numbers respectively m n 1 complex numbers and act in the space of formal power series of m n m n 1 2 non commuting variables of which mn are odd and the rest are even these variables generate a q deformation of a flag supermanifold of the supergroup gl m n respectively sl m n

finite dimensional representations of the quantum superalgebra uq gl 3 2 in a reduced uq gl 3 2 contains implies uq gl 3 1 contains implies uq gl 3 basis for generic q we give expressions for the transformations of all essentially typical finite dimensional modules of the hopf superalgebra uq ql 3 2 the latter is a deformation of the universal enveloping algebra of the lie superalgebra gl 3 2 the basis within each module is similar to the gel fand zetlin basis for gl 5 we write down expressions for the transformations of the basis under the action of the chevalley generators

a description of the quantum superalgebrauq sl n 1 m via creation and annihilation generators a description of the quantum superalgebra uq sl n 1 m and in particular of the special linear superalgebra sl n 1 m via creation and annihilation generators cags is given it provides an alternative to the canonical description of uq sl n 1 m in terms of chevalley generators a conjecture that the fock representations of the cags provide microscopic realizations of exclusion statistics is formulated

finite dimensional representations of the quantum superalgebra uq gl 2 2 ii nontypical representations at generic q the construction approach proposed in the previous paper n a ky j math phys 35 2583 1994 allows us there and in the present paper to construct at generic deformation parameter q all finite dimensional representations of the quantum lie superalgebra uq gl 2 2 the finite dimensional uq gl 2 2 modules wq constructed in the previous paper are either irreducible or indecomposable if a module wq is indecomposable i e when the condition 4 41 in the previous paper does not hold there exists an invariant maximal submodule of wq say iqk such that the factor representation in the factor module wq iqk is irreducible and called nontypical here in this paper indecomposable representations and nontypical finite dimensional representations of the quantum lie superalgebra uq gl 2 2 are considered and classified as their module structures are analyzed and the matrix elements of all nontypical representations are written down explicitly

generalized space time fractional diffusion equation with composite fractional time derivative abstract we investigate the solution of space time fractional diffusion equations with a generalized riemann liouville time fractional derivative and riesz feller space fractional derivative the laplace and fourier transform methods are applied to solve the proposed fractional diffusion equation the results are represented by using the mittag leffler functions and the fox h function special cases of the initial and boundary conditions are considered numerical scheme and grunwald letnikov approximation are also used to solve the space time fractional diffusion equation the fractional moments of the fundamental solution of the considered space time fractional diffusion equation are obtained many known results are special cases of those obtained in this paper we investigate also the solution of a space time fractional diffusion equations with a singular term of the form x t 1 0

levy walks in random scattering and growth of waves random spatial wave scattering and stochastic wave growth are studied where one or both of the random processes can be described by a levy walk this analysis extends previous work on randomly growing and scattering waves where both the random processes are modeled by gaussian diffusive statistics both random spatial scattering and stochastic wave growth modeled by levy walks are studied separately together and in combination with gaussian processes transmission coefficients lasing thresholds and energy densities in the medium are obtained for the different permutations

free fermions and the alexander conway polynomial we show how the conway alexander polynomial arises from theq deformation of z2 graded sl n n algebras in the simplestsl 1 1 case we then establish connection between classical knot theory and its modern versions based on quantum groups we first shown how the crystal and the fundamental group of the complement of a knot give rise naturally to the burau representation of the braid group the burau matrix is then transformed into theu q sl 1 1 r matrix by going to the exterior power algebra using a det str identity this allows us to recover the state model of k2 89 as well we also show how theu q sl 1 1 algebra describes free fermions propagating on the knot diagram we rewrite the conway alexander polynomial as a berezin integral and thus as an apparently new determinant

competition between levy jumps and continuous drift motivated by certain birth death processes with strongly fluctuating birth rates we consider a level crossing problem for a random process being a superposition of a continuous drift to the left and jumps to the right the lengths of the corresponding jumps follow a one sided extreme levy law of index we concentrate on the case 0 1 and discuss the probability of crossing a left boundary extinction we show that this probability decays exponentially as a function of the initial distance to the boundary such behavior is universal for all 1 and is exemplified by an exact solution for the special case 12 the splitting probabilities are also discussed

canonical q deformations of noncompact lie super algebras the author proposes a procedure for q deformations of the real forms g of complex lie super algebras associated with generalized cartan matrices the procedure gives different q deformations for the nonconjugate cartan subalgebras of g the author gives several illustrations e g q deformed lorentz and conformal super algebras the q deformed conformal algebra contains as a subalgebra a q deformed poincare algebra and as hopf subalgebras two conjugate 11 generator q deformed weyl algebras the q deformed lorentz algebra is a hopf subalgebra of both weyl algebras

optimization of random searches on defective lattice networks we study the general problem of how to search efficiently for targets randomly located on defective lattice networks i e regular lattices which have some fraction of its nodes randomly removed we consider large but finite triangular lattices and assume for the search dynamics that the walker chooses steps lengths lj from the power law distribution p lj approximately lj mu with the exponent mu regulating the strategy of the search process at each step lj the searcher moves in straight lines and constantly looks within a detection radius of vision rv for the targets along the way if there is contact with a defect the movement stops and a new step length is chosen hence the presence of defects decreases the efficiency of the overall process we study numerically how three different aspects of the lattice influence the optimization of the search efficiency i the type of boundary conditions ii the concentration of targets and defects and iii the category or class of search destructive nondestructive or regenerative motivated by the results we develop a type of mean field model for the problem and obtain an analytical approximation for the search efficiency function finally we discuss in the context of searches how defective lattices compare with perfect lattices and with continuous environments

texture recognition based on diffusion in networks a texture recognition method is proposed the image is modeled into networks the texture is characterized by the diffusion over the networks it is demonstrated that directed networks are better to texture recognition the proposed method outperform the state of the art much work has been done in the field of texture analysis and classification while promising classification methods have been proposed most of them rely on classical image analysis approaches this paper presents a texture classification method based on diffusion in directed networks first an image is modeled as a directed network by mapping each pixel as a node and connecting two nodes up to a maximum distance r to reveal texture properties links between two nodes are removed based on the pixel intensity difference once such a network is obtained the activity of each node is estimated by random walks and combined into a histogram to describe the image the main contribution of this paper is the use of directed networks which tends to provide better performance than in undirected cases also we have shown that the activity induced on these networks can be effectively used as texture descriptor experimental results show that the proposed method is favorably compared to traditional texture methods on widely used texture datasets the proposed method is also found to be promising for plant species classification using samples of leaf texture

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

r matrix and covariant q superoscillators for uq gl 1 mod 1 the author presents a construction based on an r matrix formalism of covariant quantum superoscillator algebras for uq gl 1 mod 1 he shows that the complete structure of the oscillator algebras stems from a suitable combination of properties of the r matrix covariance of the oscillators at the deformed level and associativity

the massless representations of the conformal quantum algebra the massless representations of the conformal quantum algebra uq su 2 2 for complex q such that mod q mod 1 are studied in detail by factorizing out all singular vectors of the corresponding verma modules a simple basis for these representations is explicitly constructed this basis is new also for the usual conformal algebra su 2 2 this construction allows a straightforward treatment of the case q a root of unity when the representations are unitary and finite dimensional

subsingular vectors and conditionally invariant q deformed equations we give a systematic discussion of the relation between subsingular vectors of verma modules over semisimple lie algebras g and differential equations which are conditionally g invariant this is extended to the drinfeld jimbo q deformation uq g of g we treat in detail the conformal algebra su 2 2 its complexification sl 4 and their q deformations the conditionally invariant equations are the d alembert equation and a new equation arising from a subsingular vector proposed by bernstein gel fand gel fand 1971 we also give the q difference analogues of these equations

scaling the fractional advective dispersive equation for numerical evaluation of microbial dynamics in confined geometries with sticky boundaries microbial motility is often characterized by run and tumble behavior which consists of bacteria making sequences of runs followed by tumbles random changes in direction as a superset of brownian motion levy motion seems to describe such a motility pattern the eulerian fokker planck equation describing these motions is similar to the classical advection diffusion equation except that the order of highest derivative is fractional email protected 0 2 the lagrangian equation driven by a levy measure with drift is stochastic and employed to numerically explore the dynamics of microbes in a flow cell with sticky boundaries the eulerian equation is used to non dimensionalize parameters the amount of sorbed time on the boundaries is modeled as a random variable that can vary over a wide range of values salient features of first passage time are studied with respect to scaled parameters

on the two parameter quantum supergroups and quantum superplanes starting from the assumption that the generators of quantum supergroup are not supercommutatives with the coordinates of quantum superplane instead of usual supercommutativity in the manin approach we construct in a natural way the two parameter quantum supergroup the virtue of this formulation is that it leads to many probable quantum superplanes associated with a given quantum supergroup some interesting examples are presented as a by product of this formulation a correspondence between the one and two parameter deformed differential structures on the quantum superplane is presented

multiparametric quantization of the special linear superalgebra it is shown how using the graded faddeev reshetikhin takhtajan frt formalism one can multiparametrize the quantum superalgebra slq n m since the naive serre chevalley relations are not adequate for the presentation of this algebra the quantization is carried out in the cartan weyl basis the final results can be generalized to any finite dimensional superalgebra with a symmetrizable cartan matrix

the influence of the environment on levy random search efficiency fractality and memory effects an open problem in the field of random searches relates to optimizing the search efficiency in fractal environments here we address this issue through a systematic study of levy searches in landscapes encompassing several degrees of target aggregation and fractality for scarce resources non destructive searches with unrestricted revisits to targets are shown to present universal optimal behavior irrespective of the general scaling properties of the spatial distribution of targets in contrast no such universal behavior occurs in the destructive case with forbidden revisits in which the optimal strategy strongly depends on the degree of target aggregation we also investigate how the presence of memory and learning skills of the searcher affect the search efficiency by considering a limiting model in which the searcher learns through recent experience to recognize food rich areas we find that a statistical memory of previous encounters does not necessarily increase the rate of target findings in random searches instead there is an optimal extent of memory dependent on specific details of the search space and stochastic dynamics which maximizes the search efficiency this finding suggests a more general result namely that in some instances there are actual advantages to ignoring certain pieces of partial information while searching for objects

first passage time moments of asymmetric levy flights we investigate the first passage dynamics of symmetric and asymmetric l evy flights in a semi infinite and bounded intervals by solving the space fractional diffusion equation we analyse the fractional order moments of the first passage time probability density function for different values of the index of stability and the skewness parameter a comparison with results using the langevin approach to l evy flights is presented for the semi infinite domain in certain special cases analytic results are derived explicitly and in bounded intervals a general analytical expression for the mean first passage time of l evy flights with arbitrary skewness is presented these results are complemented with extensive numerical analyses

induced representations of the two parametric quantum deformation upq gl 2 2 the two parametric quantum superalgebra up q gl 2 2 and its induced representations are considered a method for constructing all finite dimensional irreducible representations of this quantum superalgebra is also described in detail it turns out that finite dimensional representations of the two parametric up q gl 2 2 even at generic deformation parameters are not simply trivial deformations from those of the classical superalgebra gl 2 2 unlike the one parametric cases

levy flights in an infinite potential well as a hypersingular fredholm problem we study levy flights with arbitrary index 0 2 inside a potential well of infinite depth such a problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems the major technical tool is a transformation of the eigenvalue problem for initial fractional schrodinger equation into that for fredholm integral equation with hypersingular kernel the latter equation is then solved by means of expansion over the complete set of orthogonal functions in the domain d reducing the problem to the spectrum of a matrix of infinite dimensions the eigenvalues and eigenfunctions are then obtained numerically with some analytical results regarding the structure of the spectrum

optimization of random searches on regular lattices we investigate random searches on isotropic and topologically regular square and triangular lattices with periodic boundary conditions and study the efficiency of search strategies based on a power law distribution p ensuremath ell ensuremath sim ensuremath ell ensuremath ensuremath mu of step lengths ensuremath ell we consider both destructive searches in which a target can be visited only once and nondestructive searches when a target site is always available for future visits we discuss i the dependence of the search efficiency on the choice of the lattice topology ii the relevance of the periodic boundary conditions iii the behavior of the optimal power law exponent ensuremath mu mathrm opt as a function of target site density iv the differences between destructive and nondestructive environments and finally v how the results for the discrete searches differ from the continuous cases previously studied

first passage leapovers of levy flights and the proper formulation of absorbing boundary conditions an important open problem in the theory of levy flights concerns the analytically tractable formulation of absorbing boundary conditions although the nonlocal approach where the dynamical operators are given restrictions on their domain properties has yielded substantial insights regarding the statistics of first passage the resultant modifications to the dynamical equations hinder the detailed analysis possible in the absence of these nonlocal restrictions in this study it is demonstrated that using the first hit distribution related to the first passage leapover as the absorbing sink preserves the tractability of the dynamical equations for a particle undergoing levy flight in particular knowledge of the first hit distribution is sufficient to fully determine the first passage time and position density of the particle without requiring operator domain restrictions or numerical simulations in addition we report on the first hit and leapover properties of first passages and arrivals for levy flights of arbitrary skew parameter and extend these results to levy flights in a certain ubiquitous class of potentials satisfying an integral condition

fractional laplacians in bounded domains killed reflected censored and taboo levy flights the fractional laplacian 2 0 2 has many equivalent albeit formally different realizations as a nonlocal generator of a family of stable stochastic processes in r n on the other hand if the process is to be restricted to a bounded domain there are many inequivalent proposals for what a boundary data respecting fractional laplacian should actually be this ambiguity not only holds true for each specific choice of the process behavior at the boundary e g absorbtion reflection conditioning or boundary taboos but extends as well to its particular technical implementation dirichlet neumann etc problems the inferred jump type processes are inequivalent as well differing in their spectral and statistical characteristics which may strongly influence the ability of the formalism if uncritically adopted to provide an unambiguous description of real geometrically confined physical systems with disorder specifically that refers to their relaxation properties and the near equilibrium asymptotic behavior in the present paper we focus on levy flight induced jump type processes which are constrained to stay forever inside a finite domain this refers to a concept of taboo processes imported from brownian to levy stable contexts to so called censored processes and to reflected levy flights whose status still remains to be unequivocally settled as a by product of our fractional spectral analysis with reference to neumann boundary conditions we discuss disordered semiconducting heterojunctions as the bounded domain problem

levy flight random searches in biological phenomena abstract there has been growing interest in the study of levy flights observed in the movements of biological organisms performing random walks while searching for other organisms here we approach the problem of what is the best statistical strategy for optimizing the encounter rate between searcher and target organisms either of the same or of different species in terms of a limiting generalized searcher target model e g predator prey mating partner pollinator flower in this context we discuss known results showing that for fixed targets an inverse square density distribution of step lengths can optimize the encounter rate for moving targets we review how the encounter rate depends on whether organisms move in levy or brownian random walks we discuss recent findings indicating that levy walks confer a significant advantage for increasing encounter rates only when the searcher is larger or moves rapidly relative to the target and when the target density is low

finite dimensional representations of the quantum superalgebra u q gl n m and related q identities explicit expressions for the generators of the quantum superalgebrau q gl n m acting on a class of irreducible representations are given the class under consideration consists of all essentially typical representations for these a gel fand zetlin basis is known the verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set ofq number identities

levy flights and random searches in this work we discuss some recent contributions to the random search problem our analysis includes superdiffusive levy processes and correlated random walks in several regimes of target site density mobility and revisitability we present results in the context of mean field like and closed form average calculations as well as numerical simulations we then consider random searches performed in regular lattices and lattices with defects and we discuss a necessary criterion for distinguishing true superdiffusion from correlated random walk processes we invoke energy considerations in relation to critical survival states on the edge of extinction and we analyze the emergence of levy behavior in deterministic search walks finally we comment on the random search problem in the context of biological foraging

quantum flag manifolds quantum symmetric spaces and their associated universal k matrices abstract let u be a connected simply connected compact lie group with complexification g let u and g be the associated lie algebras let be the dynkin diagram of g with underlying set i and let u q u be the associated quantized universal enveloping algebra of u for some 0 q distinct from 1 let o q u be the coquasitriangular quantized function hopf algebra of u whose drinfeld double o q g r we view as the quantized function algebra of g considered as a real algebraic group we show how the datum of an involution of and a invariant function i r can be used to deform o q g r into a algebra o q id g r by a modification of the drinfeld double construction we then show how by a generalized theory of universal k matrices a specific subalgebra of o q id g r admits homomorphisms into both u q u and o q u the images being coideal subalgebras of respectively u q u and o q u we illustrate the theory by showing that two main classes of examples arise by such coideals namely quantum flag manifolds and quantum symmetric spaces except possibly for certain exceptional cases in the former case this connects to work of the first author and neshveyev while for the latter case we heavily rely on recent results of balagovic and kolb

average time spent by levy flights and walks on an interval with absorbing boundaries we consider a levy flyer of order alpha that starts from a point x 0 on an interval o l with absorbing boundaries we find a closed form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed these two quantities are equivalent to the mean first passage times for levy flights and levy walks respectively using fractional differential equations with a riesz kernel we find exact analytical expressions for both quantities in the continuous limit we show that numerical solutions for the discrete levy processes converge to the continuous approximations in all cases except the case of alpha 2 and the cases of x 0 0 and x 0 l for alpha 2 when the second moment of the flight length distribution exists our result is replaced by known results of classical diffusion we show that if x 0 is placed in the vicinity of absorbing boundaries the average total length has a minimum at alpha 1 corresponding to the cauchy distribution we discuss the relevance of this result to the problem of foraging which has received recent attention in the statistical physics literature

q difference intertwining operators for uq sl n general setting and the case n 3 we construct representations pi r of the quantum algebra uq sl n labelled by n 1 complex numbers ri and acting in the space of formal power series of n n 1 2 non commuting variables these variables generate a flag manifold of the matrix quantum group slq n which is dual to uq sl n the conditions for reducibility of pi r and the procedure for the construction of the q difference intertwining operators are given the representations and q difference intertwining operators are given in the most explicit form for n 3

the spectrum of large powers of the laplacian in bounded domains we present exact results for the spectrum of the nth power of the laplacian in a bounded domain we begin with the one dimensional case and show that the whole spectrum can be obtained in the limit of large n we also show that it is a useful numerical approach valid for any n finally we discuss implications of this work and present its possible extensions for non integer n and for 3d laplacian problems

continuous time multidimensional markovian description of levy walks the paper presents a multidimensional model for nonlinear markovian random walks that generalizes the one we developed previously i lubashevsky r friedrich and a heuer phys rev e 79 011110 2009 in order to describe the levy type stochastic processes in terms of continuous trajectories of walker motion this approach may open a way to treat levy flights or levy random walks in inhomogeneous media or systems with boundaries in the future the proposed model assumes the velocity of a wandering particle to be affected by a linear friction and a nonlinear langevin force whose intensity is proportional to the magnitude of the velocity for its large values based on the singular perturbation technique the corresponding fokker planck equation is analyzed and the relationship between the system parameters and the levy exponent is found following actually the previous paper we demonstrate also that anomalously long displacements of the wandering particle are caused by extremely large fluctuations in the particle velocity whose duration is determined by the system parameters rather than the duration of the observation interval in this way we overcome the problem of ascribing to levy random walk non markov properties