encounter distribution of two random walkers on a finite one dimensional interval we analyse the first passage properties of two random walkers confined to a finite one dimensional domain for the case of absorbing boundaries at the endpoints of the interval we derive the probability that the two particles meet before either one of them becomes absorbed at one of the boundaries for the case of reflecting boundaries we obtain the mean first encounter time of the two particles our approach leads to closed form expressions that are more easily tractable than a previously derived solution in terms of the weierstrass elliptic function

absorbing boundary in one dimensional anomalous transport in this paper we study the space time probability distribution q x t of a random walk subject to an absorbing boundary at the origin x 0 for motion controlled by l evy flights and l evy walks characterized by the exponent ensuremath gamma we find that the method of images usually applicable to brownian motion may break down for l evy processes we calculate the distribution q x t to be at xg0 after time tg0 having started at the origin assuming that the boundary is effective at time tg0 we show that q x t depends on the details of the underlying process q x t ensuremath sim mathit x ensuremath gamma 2 mathit t 1 1 ensuremath gamma 1 ensuremath le ensuremath gamma ensuremath le 2 for small x while total survival is independent of the spatial realization of motion and displays a universal behavior we also discuss the related smoluchowski boundary condition problem

exact partition functions for potts antiferromagnets on cyclic lattice strips abstract we present exact calculations of the zero temperature partition function of the q state potts antiferromagnet on arbitrarily long strips of the square triangular and kagome lattices with width l y 2 or 3 vertices and with periodic longitudinal boundary conditions from these in the limit of infinite length we obtain the exact ground state entropy s 0 k b ln w these results are of interest since this model exhibits nonzero ground state entropy s 0 0 for sufficiently large q and hence is an exception to the third law of thermodynamics we also include results for homeomorphic expansions of the square lattice strip the analytic properties of w q are determined and related to zeros of the chromatic polynomial for long finite strips

ground state entropy of the potts antiferromagnet on strips of the square lattice abstract we present exact solutions for the zero temperature partition function chromatic polynomial p and the ground state degeneracy per site w exponent of the ground state entropy for the q state potts antiferromagnet on strips of the square lattice of width ly vertices and arbitrarily great length lx vertices the specific solutions are for a ly 4 fbcy pbcx cyclic b ly 4 fbcy tpbcx mobius c ly 5 6 pbcy fbcx cylindrical and d l y 5 fbc y fbc x open where fbc pbc and tpbc denote free periodic and twisted periodic boundary conditions respectively in the lx limit of each strip we discuss the analytic structure of w in the complex q plane the respective w functions are evaluated numerically for various values of q several inferences are presented for the chromatic polynomials and analytic structure of w for lattice strips with arbitrarily great ly the absence of a nonpathological lx limit for real nonintegral q in the interval 0 0 for strips of the square triangular lattice is discussed

complex temperature phase diagrams for the q state potts model on self dual families of graphs and the nature of the q limit exact calculations of the potts model partition function z g q v have been presented for arbitrary q and temperature like variable v on self dual strip graphs g of the square lattice with fixed width l y and arbitrarily great length lx with two types of boundary conditions letting lx infinity the resultant free energy and complex temperature phase diagram have been computed including the locus b where the free energy is nonanalytic results are analyzed for widths l y 1 2 3 these results have been used to study the approach to the large q limit of b

asymptotic limits and zeros of chromatic polynomials and ground state entropy of potts antiferromagnets we study the asymptotic limiting function w g q lim n to infty p g q 1 n where p g q is the chromatic polynomial for a graph g with n vertices we first discuss a subtlety in the definition of w g q resulting from the fact that at certain special points q s the following limits do not commute lim n to infty lim q to q s p g q 1 n ne lim q to q s lim n to infty p g q 1 n we then present exact calculations of w g q and determine the corresponding analytic structure in the complex q plane for a number of families of graphs g including circuits wheels biwheels bipyramids and cyclic and twisted ladders we study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs all but a finite number of the zeros lie exactly on a unit circle whose position depends on the family using the connection of p g q with the zero temperature potts antiferromagnet we derive a theorem concerning the maximal finite real point of non analyticity in w g q denoted q c and apply this theorem to deduce that q c sq 3 and q c hc 3 sqrt 5 2 for the square and honeycomb lattices finally numerical calculations of w hc q and w sq q are presented and compared with series expansions and bounds

towards deterministic equations for levy walks the fractional material derivative levy walks are random processes with an underlying spatiotemporal coupling this coupling penalizes long jumps and therefore levy walks give a proper stochastic description for a particle s motion with broad jump length distribution we derive a generalized dynamical formulation for levy walks in which the fractional equivalent of the material derivative occurs our approach is expected to be useful for the dynamical formulation of levy walks in an external force field or in phase space for which the description in terms of the continuous time random walk or its corresponding generalized master equation are less well suited

langevin equations for continuous time levy flights we consider the combined effects of a power law l evy step distribution characterized by the step index f and a power law waiting time distribution characterized by the time index g on the long time behavior of a random walker the main point of our analysis is a formulation in terms of coupled langevin equations which allows in a natural way for the inclusion of external force fields in the anomalous case for f2 and g1 the dynamic exponent z locks onto the ratio f g drawing on recent results on l evy flights in the presence of a random force field we also find that this result is independent of the presence of weak quenched disorder for d below the critical dimension mathit d mathit c 2f 2 the disorder is relevant corresponding to a nontrivial fixed point for the force correlation function

statistics of residence time for levy flights in unstable parabolic potentials we analyze the residence time problem for an arbitrary markovian process describing nonlinear systems without a steady state we obtain exact analytical results for the statistical characteristics of the residence time for diffusion in a fully unstable potential profile in the presence of levy noise we get the conditional probability density of the particle position and the average residence time the noise enhanced stability phenomenon is observed in the system investigated results from numerical simulations are in very good agreement with analytical ones

families of graphs with chromatic zeros lying on circles we define an infinite set of families of graphs which we call p wheels and denote wh p n that generalize the wheel p 1 and biwheel p 2 graphs the chromatic polynomial for wh p n is calculated and remarkably simple properties of the chromatic zeros are found i the real zeros occur at q 0 1 p 1 for n p even and q 0 1 p 2 for n p odd and ii the complex zeros all lie equally spaced on the unit circle q p 1 1 in the complex q plane in the n to infty limit the zeros on this circle merge to form a boundary curve separating two regions where the limiting function w wh p q is analytic viz the exterior and interior of the above circle connections with statistical mechanics are noted

exact potts model partition functions on wider arbitrary length strips of the square lattice abstract we present exact calculations of the partition function of the q state potts model for general q and temperature on strips of the square lattice of width l y 3 vertices and arbitrary length l x with periodic longitudinal boundary conditions of the following types i fbc y pbc x cyclic ii fbc y tpbc x mobius iii pbc y pbc x toroidal and iv pbc y tpbc x klein bottle where fbc and t pbc refer to free and twisted periodic boundary conditions results for the l y 2 torus and klein bottle strips are also included in the infinite length limit the thermodynamic properties are discussed and some general results are given for low temperature behavior on strips of arbitrarily great width we determine the submanifold in the c 2 space of q and temperature where the free energy is singular for these strips our calculations are also used to compute certain quantities of graph theoretic interest

levy flights on the half line we study the probability distribution function pdf of the position of a levy flight of index 0 2 in the presence of an absorbing wall at the origin the solution of the associated fractional fokker planck equation can be constructed using a perturbation scheme around the brownian solution corresponding to 2 as an expansion in e 2 we obtain an explicit analytical solution exact at the first order in e which allows us to conjecture the precise asymptotic behavior of this pdf including the first subleading corrections for any careful numerical simulations as well as an exact computation for 1 confirm our conjecture

on the first passage time and leapover properties of levy motions we investigate two coupled properties of levy stable random motions the first passage times fpts and the first passage leapovers fpls while in general the fpt problem has been studied quite extensively the fpl problem has hardly attracted any attention considering a particle that starts at the origin and performs random jumps with independent increments chosen from a levy stable probability law x the fpt measures how long it takes the particle to arrive at or cross a target the fpl addresses a different question given that the first passage jump crosses the target then how far does it get beyond the target these two properties are investigated for three subclasses of levy stable motions i symmetric levy motions characterized by levy index 0 2 and skewness parameter 0 ii one sided levy motions with 0 1 1 and iii two sided skewed levy motions the extreme case 1 2 1

first passage and arrival time densities for levy flights and the failure of the method of images we discuss the first passage time problem in the semi infinite interval for homogeneous stochastic markov processes with levy stable jump length distributions x l x 1 x l namely levy flights lfs in particular we demonstrate that the method of images leads to a result which violates a theorem due to sparre andersen according to which an arbitrary continuous and symmetric jump length distribution produces a first passage time density fptd governed by the universal long time decay t 3 2 conversely we show that for lfs the direct definition known from gaussian processes in fact defines the probability density of first arrival which for lfs differs from the fptd our findings are corroborated by numerical results

families of graphs with w r g q functions that are nonanalytic at 1 q 0 denoting p g q as the chromatic polynomial for coloring an n vertex graph g with q colors and considering the limiting function w g q lim n to infty p g q 1 n a fundamental question in graph theory is the following is w r g q q 1 w g q analytic or not at the origin of the 1 q plane where the complex generalization of q is assumed this question is also relevant in statistical mechanics because w g q exp s 0 k b where s 0 is the ground state entropy of the q state potts antiferromagnet on the lattice graph g and the analyticity of w r g q at 1 q 0 is necessary for the large q series expansions of w r g q although w r is analytic at 1 q 0 for many g there are some g for which it is not for these w r has no large q series expansion it is important to understand the reason for this nonanalyticity here we give a general condition that determines whether or not a particular w r g q is analytic at 1 q 0 and explains the nonanalyticity where it occurs we also construct infinite families of graphs with w r functions that are non analytic at 1 q 0 and investigate the properties of these functions our results are consistent with the conjecture that a sufficient condition for w r g q to be analytic at 1 q 0 is that g is a regular lattice graph lambda this is known not to be a necessary condition

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

numerical methods for the fractional laplacian a finite difference quadrature approach the fractional laplacian delta alpha 2 is a nonlocal operator which depends on the parameter alpha and recovers the usual laplacian as alpha to 2 a numerical method for the fractional laplacian is proposed based on the singular integral representation for the operator the method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights the accuracy of the method is shown to be o h 3 alpha convergence of the method is proven the treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method numerical experiments on known exact solutions validate the predicted convergence rates computational examples include exponentially and algebraically decaying solutions with varying regularity the generalization to nonlinear equations involving the operator is discussed the obstacle problem for the fractional laplacian is computed

first passage time for subdiffusion the nonadditive entropy approach versus the fractional model we study the similarities and differences between different models concerning subdiffusion more particularly we calculate first passage time fpt distributions for subdiffusion derived from greens functions of nonlinear equations obtained from sharma mittal s tsallis s and gauss s nonadditive entropies then we compare these with fpt distributions calculated from a fractional model using a subdiffusion equation with a fractional time derivative all of greens functions give us exactly the same standard relation x 2 2d t which characterizes subdiffusion 0 1 but generally fpt distributions are not equivalent to one another we will show here that the fpt distribution for the fractional model is asymptotically equal to the sharma mittal model over the long time limit only if in the latter case one of the three parameters describing sharma mittal entropy r depends on and satisfies the specific equation derived in this paper whereas the other two models mentioned above give different fpt distributions with the fractional model greens functions obtained from the sharma mittal and fractional models for r obtained from this particular equation are very similar to each other we will also discuss the interpretation of subdiffusion models based on nonadditive entropies and the possibilities of the experimental measurement of subdiffusion models parameters

zeros of jones polynomials for families of knots and links abstract we calculate jones polynomials v l t for several families of alternating knots and links by computing the tutte polynomials t g x y for the associated graphs g and then obtaining v l t as a special case of the tutte polynomial for each of these families we determine the zeros of the jones polynomial including the accumulation set in the limit of infinitely many crossings a discussion is also given of the calculation of jones polynomials for non alternating links

improved lower bounds on the ground state entropy of the antiferromagnetic potts model we present generalized methods for calculating lower bounds on the ground state entropy per site s 0 or equivalently the ground state degeneracy per site w e s 0 k b of the antiferromagnetic potts model we use these methods to derive improved lower bounds on w for several lattices

chromatic polynomials for families of strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p gs m q and from these the asymptotic limiting functions w g s q lim n p g s q 1 n for families of n vertex graphs gs m comprised of m repeated subgraphs h adjoined to an initial graph i these calculations of w gs q for infinitely long strips of varying widths yield important insights into properties of w q for two dimensional lattices in turn these results connect with statistical mechanics since w q is the ground state degeneracy of the q state potts model on the lattice for our calculations we develop and use a generating function method which enables us to determine both the chromatic polynomials of finite strip graphs and the resultant w gs q function in the limit n from this we obtain the exact continuous locus of points b where w gs q is nonanalytic in the complex q plane this locus is shown to consist of arcs which do not separate the q plane into disconnected regions zeros of chromatic polynomials are computed for finite strips and compared with the exact locus of singularities b we find that as the width of the infinitely long strips is increased the arcs comprising b elongate and move toward each other which enables one to understand the origin of closed regions that result for the infinite 2d lattice

flow polynomials and their asymptotic limits for lattice strip graphs we present exact calculations of flow polynomials f g q for lattice strips of various fixed widths l y 4 and arbitrarily great lengths l x with several different boundary conditions square honeycomb and triangular lattice strips are considered we introduce the notion of flows per face fl in the infinite length limit we study the zeros of f g q in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros in the infinite length limit for the various families of strips the function fl is nonanalytic on this locus the loci are found to be noncompact for many strip graphs with periodic or twisted periodic longitudinal boundary conditions and compact for strips with free longitudinal boundary conditions we also find the interesting feature that aside from the trivial case l y 1 the maximal point q cf where crosses the real axis is universal on cyclic and mobius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value q cf 3 for the infinite square lattice

potts model on infinite graphs and the limit of chromatic polynomials given an infinite graph quasi transitive and amenable with maximum degree we show that reduced ground state degeneracy per site w r q of the q state antiferromagnetic potts model at zero temperature on is analytic in the variable 1 q whenever 2 e 3 q 1 this result proves in an even stronger formulation a conjecture originally sketched in 12 and explicitly formulated in 16 and 19 based on which a sufficient condition for w r q to be analytic at 1 q 0 is that is a regular lattice

ground state entropy of potts antiferromagnets on homeomorphic families of strip graphs abstract we present exact calculations of the zero temperature partition function and ground state degeneracy per site w for the q state potts antiferromagnet on a variety of homeomorphic families of planar strip graphs g ch k1 k2 k m where k1 k2 and k describe the homeomorphic structure and m denotes the length of the strip several different ways of taking the total number of vertices to infinity by sending i m with k1 k2 and k fixed ii k1 and or k2 with m and k fixed and iii k with m and p k1 k2 fixed are studied and the respective loci of points b where w is nonanalytic in the complex q plane are determined the b s for limit i are comprised of arcs which do not enclose regions in the q plane and for many values of p and k include support for re q b for limits ii and iii is the unit circle q 1 1

mean first passage time for anomalous diffusion when the random force acting on a particle diffusing in an interval 0 l and subjected to a constant external force is a gaussian white noise the brownian mean squared displacement is described by the seminal relation x 2 2 mathrm dt ensuremath gamma with ensuremath gamma 1 however for more complicated random forces the diffusion may be slower ensuremath gamma l1 subdiffusion or faster ensuremath gamma g1 superdiffusion than the normal diffusion for both these cases we calculated the mean free passage time mfpt the time needed to reach one of the traps at boundaries the simple formulas for the different diffusive regimes are compared quantitatively for the simplest case of the absence of an external field and for an initial position in the middle of the interval it turns out that the mfpt s for anomalous diffusion can be both larger or smaller than that for normal diffusion depending on the values of the length of the interval and the diffusion coefficient moreover the mfpt can show nonmonotonic changes with the degree of departure from normal diffusion

ground state entropy of the potts antiferromagnet on cyclic strip graphs we present exact calculations of the zero temperature partition function chromatic polynomial and the exponent of the ground state entropy s0 for the q state potts antiferromagnet on families of cyclic and twisted cyclic mobius strip graphs composed of p sided polygons our results suggest a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where s0 0 the chromatic zeros and their accumulation set exhibit the rather unusual property of including support for re q 0 and provide further evidence for a relevant conjecture

ground state entropy of potts antiferromagnets on cyclic polygon chain graphs we present exact calculations of chromatic polynomials for families of cyclic graphs consisting of linked polygons where the polygons may be adjacent or separated by a given number of bonds from these we calculate the exponential of the ground state entropy w for the q state potts model on these graphs in the limit of infinitely many vertices a number of properties are proved concerning the continuous locus of nonanalyticities in w our results provide further evidence for a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where s0 0

chromatic polynomials for lattice strips with cyclic boundary conditions abstract the zero temperature q state potts model partition function for a lattice strip of fixed width l y and arbitrary length l x has the form p g q j 1 n g c g j g j l x and is equivalent to the chromatic polynomial for this graph we present exact zero temperature partition functions for strips of several lattices with fbc y pbc x i e cyclic boundary conditions in particular the chromatic polynomial of a family of generalized dodecahedra graphs is calculated the coefficient c g j of degree d in q is c d u 2d q 2 where u n x is the chebyshev polynomial of the second kind we also present the chromatic polynomial for the strip of the square lattice with pbc y pbc x i e toroidal boundary conditions and width l y 4 with the property that each set of four vertical vertices forms a tetrahedron a number of interesting and novel features of the continuous accumulation set of the chromatic zeros b are found

ground state entropy in potts antiferromagnets abstract the q state potts antiferromagnet exhibits nonzero ground state entropy s0 g q 0 for sufficiently large q on a given n vertex lattice or graph g and its n limit g we present exact calculations of the zero temperature partition function z g q t 0 and w g q where s 0 k b ln w for this model on a number of families g these calculations have interesting connections with graph theory since z g q t 0 p g q where the chromatic polynomial p g q gives the number of ways of coloring the vertices of the graph g such that no adjacent vertices have the same color generalizing q from z to c we determine the accumulation set b of the zeros of p g q which constitute the continuous loci of points on which w is nonanalytic the potts antiferromagnet has a zero temperature critical point at the maximal value qc where b crosses the real q axis in particular exact solutions for w and b are given for infinitely long finite width strips of various lattices in addition to their intrinsic interest these yield insight into the approach to the 2d thermodynamic limit some corresponding results are presented for the exact finite temperature potts free energy on families of graphs finally we present rigorous upper and lower bounds on w for 2d lattices

exact potts model partition functions on ladder graphs abstract we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex ladder graphs i e strips of the square lattice with width ly 2 and arbitrary length lx with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of these ladder graphs and the thermodynamics is discussed by comparison with strip graphs of other widths we analyze how the singularities at the zero temperature critical point of the ferromagnet on infinite length finite width strips depend on the width we point out and study the following noncommutativity at certain special values q s lim n lim q q s z 1 n lim q q s lim n z 1 n it is shown that the potts antiferromagnet on both the infinite length line and ladder graphs with cyclic or mobius boundary conditions exhibits a phase transition at finite temperature if 0 b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros certain properties of the complex temperature phase diagrams are shown to exhibit close connections with those of the model on the square lattice showing that exact solutions on infinite length strips provide a way of gaining insight into these complex temperature phase diagrams

sw gr from the seiberg witten equations to pseudo holomorphic curves the purpose of this article is to explain how pseudo holomorphic curves in a symplectic 4 manifold can be constructed from solutions to the seiberg witten equations as such the main theorem proved here theorem 1 3 is an existence theorem for pseudo holomorphic curves this article thus provides a proof of roughly half of the main theorem in the announcement t1 that theorem theorem 4 1 asserts an equivalence between the seiberg witten invariants for a symplectic manifold and a certain gromov invariant which counts with signs the number of pseudoholomorphic curves in a given homology class the seiberg witten invariants were introduced to mathematicians by witten w based on his joint work with nat seiberg sw1 sw2 a description of these invariants is given in section 1 see also km1 t1 suffice it to say here that when x is a compact oriented 4 dimensional manifold with

potts model partition functions for self dual families of strip graphs abstract we consider the q state potts model on families of self dual strip graphs g d of the square lattice of width l y and arbitrarily great length l x with periodic longitudinal boundary conditions the general partition function z and the t 0 antiferromagnetic special case p chromatic polynomial have the respective forms j 1 n f l y c f l y j f l y j l x with f z p for arbitrary l y we determine i the general coefficient c f l y j in terms of chebyshev polynomials ii the number n f l y d of terms with each type of coefficient and iii the total number of terms n f l y we point out interesting connections between the n z l y d and temperley lieb algebras and between the n f l y and enumerations of directed lattice animals exact calculations of p are presented for 2 l y 4 in the limit of infinite length we calculate the ground state degeneracy per site exponent of the ground state entropy w q generalizing q from z to c we determine the continuous locus b in the complex q plane where w q is singular we find the interesting result that for all l y values considered the maximal point at which b crosses the real q axis denoted q c is the same and is equal to the value for the infinite square lattice q c 3 this is the first family of strip graphs of which we are aware that exhibits this type of universality of q c

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

ground state degeneracy of potts antiferromagnets homeomorphic classes with noncompact w boundaries abstract we present exact calculations of the zero temperature partition function z g q t 0 and ground state degeneracy w g q for the q state potts antiferromagnet on a number of families of graphs g for which generalizing q from z to c the boundary b of regions of analyticity of w in the complex q plane is noncompact passing through z 1 q 0 for these types of graphs since the reduced function w red q 1 w is nonanalytic at z 0 there is no large q taylor series expansion of w red the study of these graphs thus gives insight into the conditions for the validity of the large q expansions it is shown how such families of graphs can be generated from known families by homeomorphic expansion

chromatic polynomials and representations of the symmetric group abstract the chromatic polynomials considered in this paper are associated with graphs constructed in the following way take n copies of a complete graph k b and for i 1 2 n join each vertex in the ith copy to the same vertex in the i 1 th copy taking n 1 1 by convention previous calculations for b 2 and b 3 suggest that the chromatic polynomial contains terms that occur in levels in the present paper the levels are explained by using a version of the sieve principle and it is shown that the terms at level l correspond to the irreducible representations of the symmetric group sym l in the case of the two linear representations the terms can be calculated explicitly by methods based on the theory of distance regular graphs for the nonlinear representations the calculations are more complicated an illustration is given in section 10 where the complete chromatic polynomial for the case b 4 is obtained

fox function representation of non debye relaxation processes applying the liouville riemann fractional calculus we derive and solve a fractional operator relaxation equation we demonstrate how the exponent of the asymptotic power law decay t relates to the order of the fractional operatordv dtv 0 1 continuous time random walk ctrw models offer a physical interpretation of fractional order equations and thus we point out a connection between a special type of ctrw and our fractional relaxation model exact analytical solutions of the fractional relaxation equation are obtained in terms of fox functions by using laplace and mellin transforms apart from fractional relaxation fox functions are further used to calculate fourier integrals of kohlrausch williams watts type relaxation functions because of its close connection to integral transforms the rich class of fox functions forms a suitable framework for discussing slow relaxation phenomena

chromatic polynomials potts models and all that the q state potts model can be defined on an arbitrary finite graph and its partition function encodes much important information about that graph including its chromatic polynomial flow polynomial and reliability polynomial the complex zeros of the potts partition function are of interest both to statistical mechanicians and to combinatorists i give a pedagogical introduction to all these problems and then sketch two recent results a construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q plane except possibly for the disc q 1 1 b proof of a universal upper bound on the q plane zeros of the chromatic polynomial or antiferromagnetic potts model partition function in terms of the graph s maximum degree

lower bounds and series for the ground state entropy of the potts antiferromagnet on archimedean lattices and their duals we prove a general rigorous lower bound for w lambda q exp s 0 lambda q k b the exponent of the ground state entropy of the q state potts antiferromagnet on an arbitrary archimedean lattice lambda we calculate large q series expansions for the exact w r lambda q q 1 w lambda q and compare these with our lower bounds on this function on the various archimedean lattices it is shown that the lower bounds coincide with a number of terms in the large q expansions and hence serve not just as bounds but also as very good approximations to the respective exact functions w r lambda q for large q on the various lattices lambda plots of w r lambda q are given and the general dependence on lattice coordination number is noted lower bounds and series are also presented for the duals of archimedean lattices as part of the study the chromatic number is determined for all archimedean lattices and their duals finally we report calculations of chromatic zeros for several lattices these provide further support for our earlier conjecture that a sufficient condition for w r lambda q to be analytic at 1 q 0 is that lambda is a regular lattice

3 dimensional seiberg witten invariants and non kahlerian geometry the paper is concerned with seiberg witten invariants of closed 3 manifolds and their relation to the 4 dimensional invariants the main points are 1 the introduction of parameter dependent seiberg witten invariants for 3 manifolds with b1 0 these invariants depend on a complicated chamber structure defined by a differential geometric brill noether locus in the parameter space 2 a comparison theorem which shows that the seiberg witten invariants of 3 manifolds m with b 1 m 1 are simply the restrictions of the 4 dimensional invariants of the product s 1 m to the set of pull back spinc structures as an immediate consequence one gets the wall crossing formula for the invariants of 3 dimensional manifolds with b1 1 from the 4 dimensional wall crossing formula for manifolds with b 1 3 a novel method to compute the seiberg witten invariants of 3 manifolds m using non kahlerian complex geometry on s 1 m provided this product has such a structure which is compatible with a product metric and orientation induced from m this method relies essentially on a kobayashi hitchin correspondence which is new in the non kahler case 4 a complete description of those 3 manifolds to which our method applies these are riemannian manifolds which admit an oriented riemannian foliation by geodesics such manifolds are always diffeomorphic to seifert manifolds but the foliation has not to be seifert it can have non closed leaves 5 explicit computations of seiberg witten invariants for the 3 sphere s 3 using non seifert riemannian foliations we show that the chamber structure induced by the brill noether locus is very complicated there exist countably many dif ferent chambers and the seiberg witten invariant takes on all integer values

subdiffusive continuous time random walks with stochastic resetting we analyze two models of subdiffusion with stochastic resetting each of them consists of two parts subdiffusion based on the continuous time random walk scheme and independent resetting events generated uniformly in time according to the poisson point process in the first model the whole process is reset to the initial state whereas in the second model only the position is subject to resets the distinction between these two models arises from the non markovian character of the subdiffusive process we derive exact expressions for the two lowest moments of the full propagator stationary distributions and first hitting time statistics we also show with an example of a constant drift how these models can be generalized to include external forces possible applications to data analysis and modeling of biological systems are also discussed

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

bifurcation bimodality and finite variance in confined levy flights we investigate the statistical behavior of levy flights confined in a symmetric quartic potential well u x proportional variant x 4 at stationarity the probability density function features a distinct bimodal shape and decays with power law tails which are steep enough to give rise to a finite variance in contrast to free levy flights from a delta initial condition a bifurcation of the unimodal state is observed at t c 0 the nonlinear oscillator with potential u x ax 2 2 bx 4 4 a b 0 shows a crossover from unimodal to bimodal behavior at stationarity depending on the ratio a b

ground state entropy of potts antiferromagnets bounds series and monte carlo measurements we report several results concerning w ensuremath lambda q mathrm exp s 0 k b the exponent of the ground state entropy of the potts antiferromagnet on a lattice ensuremath lambda first we improve our previous rigorous lower bound on w mathrm hc q for the honeycomb hc lattice and find that it is extremely accurate it agrees to the first 11 terms with the large q series for w mathrm hc q second we investigate the heteropolygonal archimedean 4 ensuremath cdot 8 2 lattice derive a rigorous lower bound on w 4 ensuremath cdot 8 2 q and calculate the large q series for this function to o y 12 where y 1 q ensuremath 1 remarkably these agree exactly to all 13 terms calculated we also report monte carlo measurements and find that these are very close to our lower bound and series third we study the effect of non nearest neighbor couplings focusing on the square lattice with next nearest neighbor bonds

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

exact chromatic polynomials for toroidal chains of complete graphs abstract we present exact calculations of the partition function of the zero temperature potts antiferromagnet equivalently the chromatic polynomial for graphs of arbitrarily great length composed of repeated complete subgraphs kb with b 5 6 which have periodic or twisted periodic boundary condition in the longitudinal direction in the lx limit the continuous accumulation set of the chromatic zeros b is determined we give some results for arbitrary b including the extrema of the eigenvalues with coefficients of degree b 1 and the explicit forms of some classes of eigenvalues we prove that the maximal point where b crosses the real axis qc satisfies the inequality qc b for 2 b the minimum value of q at which b crosses the real q axis is q 0 and we make a conjecture concerning the structure of the chromatic polynomial for klein bottle strips

exact potts model partition function on strips of the triangular lattice abstract in this paper we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex strip graphs of width ly 2 and arbitrary length of the triangular lattice with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed considering the full generalization to arbitrary complex q and temperature we determine the singular locus b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros comparisons are made with our previous exact calculation of potts model partition functions for the corresponding strips of the square lattice our present calculations yield as special cases several quantities of graph theoretic interest

transfer matrices for the zero temperature potts antiferromagnet on cyclic and mobius lattice strips abstract we present transfer matrices for the zero temperature partition function of the q state potts antiferromagnet equivalently the chromatic polynomial on cyclic and mobius strips of the square triangular and honeycomb lattices of width l y and arbitrarily great length l x we relate these results to our earlier exact solutions for square lattice strips with l y 3 4 5 triangular lattice strips with l y 2 3 4 and honeycomb lattice strips with l y 2 3 and periodic or twisted periodic boundary conditions we give a general expression for the chromatic polynomial of a mobius strip of a lattice and exact results for a subset of honeycomb lattice transfer matrices both of which are valid for arbitrary strip width l y new results are presented for the l y 5 strip of the triangular lattice and the l y 4 and l y 5 strips of the honeycomb lattice using these results and taking the infinite length limit l x we determine the continuous accumulation locus of the zeros of the above partition function in the complex q plane including the maximal real point of nonanalyticity of the degeneracy per site w as a function of q

the phase transition in a bistable duffing system driven by levy noise in this paper the phase transition and mean first passage time mfpt induced by levy noise is investigated in a bistable duffing system numerically first we obtain the stationary probability density functions as the criterion of phase transition through the qualitative changes of the shapes of stationary probability distribution then the influences of levy noise parameters are discussed the results indicate that the stability index noise intensity and skewness parameter can lead to phase transition behaviors in bistable duffing system in addition the mfpt is calculated for various levy parameters different effects among stability index noise intensity and the skewness parameter on the phase transition and mfpt are observed furthermore distinctions between levy noise and gaussian noise are provided

upper and lower bounds for the ground state entropy of antiferromagnetic potts models we derive rigorous upper and lower bounds for the ground state entropy of the q state potts antiferromagnet on the honeycomb and triangular lattices these bounds are quite restrictive especially for large q copyright ital 1997 ital the american physical society

the structure of chromatic polynomials of planar triangulations and implications for chromatic zeros and asymptotic limiting quantities we present an analysis of the structure and properties of chromatic polynomials of one parameter and multi parameter families of planar triangulation graphs where is a vector of integer parameters we use these to study the ratio of to the tutte upper bound 1 n 5 where and n is the number of vertices in in particular we calculate limiting values of this ratio as n for various families of planar triangulations we also use our calculations to analyze zeros of these chromatic polynomials we study a large class of families with p 1 and p 2 and show that these have a structure of the form for p 1 where 1 q 2 2 q 3 and 3 1 and for p 2 we derive properties of the coefficients and show that has a real chromatic zero that approaches as one or more of the mi the generalization to p 3 is given further we present a one parameter family of planar triangulations with real zeros that approach 3 from below as m implications for the ground state entropy of the potts antiferromagnet are discussed

ground state degeneracy of potts antiferromagnets cases with noncompact w boundaries having multiple points at we present exact calculations of the zero temperature partition function and ground state degeneracy per site for the q state potts antiferromagnet on a number of families of graphs for which the boundary of regions of analyticity of w in the complex q plane is noncompact and has the properties that i in the plane the point z 0 is a multiple point on and ii includes support for these families are generated by the method of homeomorphic expansion our results give further insight into the conditions for the validity of large q series expansions for the reduced function

ground state entropy of potts antiferromagnets and the approach to the two dimensional thermodynamic limit we study the ground state degeneracy per site exponent of the ground state entropy w lambda l x infty times l y q for the q state potts antiferromagnet on infinitely long strips with width l y of 2d lattices lambda with free and periodic boundary conditions in the y direction denoted fbc y and pbc y we show that the approach of w to its 2d thermodynamic limit as l y increases is quite rapid for moderate values of q and l y simeq 4 w lambda l x infty times l y q is within about 5 and cal o 10 3 of the 2d value w lambda l x infty times l y infty q for fbc y and pbc y respectively the approach of w to the 2d thermodynamic limit is proved to be monotonic non monotonic for fbc y pbc y it is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states

a mesh free pseudospectral approach to estimating the fractional laplacian via radial basis functions abstract this paper investigates the use of radial basis function rbf interpolants to estimate a function s fractional laplacian of a given order through a mesh free pseudospectral method the mesh free approach yields an algorithm that can be implemented in high dimensional settings without adjustment moreover the fractional laplacian is defined in terms of the fourier transform and the symmetry of rbfs can be exploited to simplify the estimation problem convergence rates are established for rbfs when the function whose fractional laplacian to be estimated is compactly supported further results demonstrate convergence when a function is in the native space for a wendland rbf i e a sobolev space and satisfies a certain l 1 condition numerical experiments demonstrate the developed method by estimating the fractional laplacian of several functions and by solving a fractional poisson equation with extended dirichlet condition in one and two dimensions

transfer matrices and partition function zeros for antiferromagnetic potts models i general theory and square lattice chromatic polynomial we study the chromatic polynomials zero temperature antiferromagnetic potts model partition functions p g q for m n rectangular subsets of the square lattice with m 8 free or periodic transverse boundary conditions and n arbitrary free longitudinal boundary conditions using a transfer matrix in the fortuin kasteleyn representation in particular we extract the limiting curves of partition function zeros when n which arise from the crossing in modulus of dominant eigenvalues beraha kahane weiss theorem we also provide evidence that the beraha numbers b2 b3 b4 b5 are limiting points of partition function zeros as n whenever the strip width m is 7 periodic transverse b c or 8 free transverse b c along the way we prove that a noninteger beraha number except perhaps b10 cannot be a chromatic root of any graph

and did he search for you and could not find you we discuss stochastic search mechanisms for sparse targets these mechanisms may be normal diffusion in different dimensionalities as well as anomalous diffusion namely levy walks and subdiffusion we argue that under different boundary conditions either mechanism may be beneficial for the search as a concrete example we investigate facilitated diffusion of regulating proteins in gene regulation moreover we discuss potential pitfalls in acquiring diffusion behaviour from time series analysis of single particle trajectories

structural properties of potts model partition functions and chromatic polynomials for lattice strips abstract the q state potts model partition function equivalent to the tutte polynomial for a lattice strip of fixed width l y and arbitrary length l x has the form z g q v j 1 n z g c z g j z g j l x where v is a temperature dependent variable the special case of the zero temperature antiferromagnet v 1 is the chromatic polynomial p g q using coloring and transfer matrix methods we give general formulas for c x g j 1 n x g c x g j for x z p on cyclic and mobius strip graphs of the square and triangular lattice combining these with a general expression for the unique coefficient c z g j of degree d in q c d u 2d q 2 where u n x is the chebyshev polynomial of the second kind we determine the number of z g j s with coefficient c d in z g q v for these cyclic strips of width l y to be n z l y d 2d 1 l y d 1 1 2l y l y d for 0 d l y and zero otherwise for both cyclic and mobius strips of these lattices the total number of distinct eigenvalues z g j is calculated to be n z l y 2l y l y results are also presented for the analogous numbers n p l y d and n p l y for p g q we find that n p l y 0 n p l y 1 1 m l y 1 motzkin number n z l y 0 c l y the catalan number and give an exact expression for n p l y our results for n z l y and n p l y apply for both the cyclic and mobius strips of both the square and triangular lattices we also point out the interesting relations n z l y 2 n da tri l y and n p l y 2 n da sq l y where n da n denotes the number of directed lattice animals on the lattice we find the asymptotic growths n z l y l y 1 2 4 l y and n p l y l y 1 2 3 l y as l y some general geometric identities for potts model partition functions are also presented

large q series expansion for the ground state degeneracy of the q state potts antiferromagnet on the 3 12 2 lattice we calculate the large q series expansion for the ground state degeneracy exponent of the ground state entropy per site of the q state potts antiferromagnet on the 3 cdot 12 2 lattice to order o y 19 where y 1 q 1 we note a remarkable agreement to o y 18 between this series and a rigorous lower bound derived recently