spectral zeta functions of fractals and the complex dynamics of polynomials we obtain formulas for the spectral zeta function of the laplacian on symmetric finitely ramified fractals such as the sierpinski gasket and a fractal laplacian on the interval these formulas contain a new type of zeta function associated with a polynomial rational functions also can appear in this context it is proved that this zeta function has a meromorphic continuation to a half plane with poles contained in an arithmetic progression it is shown as an example that the riemann zeta function is the zeta function of a quadratic polynomial which is associated with the laplacian on an interval the spectral zeta function of the sierpinski gasket is a product of the zeta function of a polynomial and a geometric part the poles of the former are canceled by the zeros of the latter a similar product structure was discovered by m l lapidus for self similar fractal strings

the zeta function of the laplacian on certain fractals we prove that the zeta function of the laplacian a on self similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane we characterise the poles compute their residues and give expressions for some special values of the zeta function furthermore we discuss the presence of oscillations in the eigenvalue counting function thereby answering a question posed by j kigami and m lapidus for this class of fractals

categorification of a linear algebra identity and factorization of serre functors we provide a categorical interpretation of a well known identity from linear algebra as an isomorphism of certain functors between triangulated categories arising from finite dimensional algebras as a consequence we deduce that the serre functor of a finite dimensional triangular algebra a has always a lift up to shift to a product of suitably defined reflection functors in the category of perfect complexes over the trivial extension algebra of a

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

picard groups for derived module categories we introduce in this paper a generalization of picard groups to derived cat egories of algebras we study general properties of those groups and we construct braid group actions on these groups for particular classes of algebras

projective injective modules serre functors and symmetric algebras we describe serre functors for generalisations of the category o associated with a semisimple complex lie algebra in our approach projective injective modules that is modules which are both

a stochastic differential equation model with jumps for fractional advection and dispersion the path of a tracer particle through a porous medium is typically modeled by a stochastic differential equation sde driven by brownian noise this model may not be adequate for highly heterogeneous media this paper extends the model to a general sde driven by a levy noise particle paths follow a markov process with long jumps their transition probability density solves a forward equation derived here via pseudo differential operator theory and fourier analysis in particular the sde with stable driving noise has a space fractional advection dispersion equation fade with variable coefficients as the forward equation this result provides a stochastic solution to anomalous diffusion models and a solid mathematical foundation for particle tracking codes already in use for fractional advection equations

killing and the coxeter transformation of kac moody algebras on etudie le spectre de la transformation de coxeter associee a une algebre de kac moody arbitraire et plus particulierement a une algebre affine

dirac and magnetic schrodinger operators on fractals abstract in this paper we define local dirac operators and magnetic schrodinger hamiltonians on fractals and prove their essential self adjointness to do so we use the concept of 1 forms and derivations associated with dirichlet forms as introduced by cipriani and sauvageot and further studied by the authors jointly with rockner ionescu and rogers for simplicity our definitions and results are formulated for the sierpinski gasket with its standard self similar energy form we point out how they may be generalized to other spaces such as the classical sierpinski carpet

parameter estimation for fractional transport a particle tracking approach 1 space fractional advection dispersion models provide attractive alternatives to the classical advection dispersion equation for model applications that exhibit early arrivals and plume skewness this paper develops a flexible method for estimating the parameters of the fractional transport model on the basis of spatial plume snapshots or temporal breakthrough curve data a particle tracking approach provides error bars for the parameter estimates and a general method for model fitting and comparison via optimal weighted least squares a simple model of concentration variance based on the particle tracking approach identifies the optimal weights

quivers floer cohomology and braid group actions we consider the derived categories of modules over a certain family a m of graded rings and floer cohomology of lagrangian intersections in the symplectic manifolds which are the milnor fibres of simple singularities of type a m we show that each of these two rather different objects encodes the topology of curves on an m 1 punctured disc we prove that the braid group b m 1 acts faithfully on the derived category of a m modules and that it injects into the symplectic mapping class group of milnor fibers the philosophy behind our results is as follows using floer cohomology one should be able to associate to the milnor fibre a triangulated category its construction has not been carried out in detail this triangulated category should contain a full subcategory which is equivalent up to a slight difference in the grading to the derived category of a m modules the full embedding would connect the two occurrences of the braid group thus explaining the similarity between them

distance and tube zeta functions of fractals and arbitrary compact sets abstract recently the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets a of the n dimensional euclidean space r n for any integer n 1 it is defined by the lebesgue integral a s a d x a s n d x for all s c with re s sufficiently large and we call it the distance zeta function of a here d x a denotes the euclidean distance from x to a and a is the neighborhood of a where is a fixed positive real number we prove that the abscissa of absolute convergence of a is equal to dim b a the upper box or minkowski dimension of a particular attention is payed to the principal complex dimensions of a defined as the set of poles of a located on the critical line re s dim b a provided a possesses a meromorphic extension to a neighborhood of the critical line we also introduce a new closely related zeta function a s 0 t s n 1 a t d t called the tube zeta function of a assuming that a is minkowski measurable we show that under some mild conditions the residue of a computed at d dim b a the box dimension of a is equal to the minkowski content of a more generally without assuming that a is minkowski measurable we show that the residue is squeezed between the lower and upper minkowski contents of a we also introduce transcendentally quasiperiodic sets and construct a class of such sets using generalized cantor sets along with baker s theorem from the theory of transcendental numbers

fluid limit of the continuous time random walk with general levy jump distribution functions the continuous time random walk ctrw is a natural generalization of the brownian random walk that allows the incorporation of waiting time distributions psi t and general jump distribution functions eta x there are two well known fluid limits of this model in the uncoupled case for exponential decaying waiting times and gaussian jump distribution functions the fluid limit leads to the diffusion equation on the other hand for algebraic decaying waiting times psi approximately t 1 beta and algebraic decaying jump distributions eta approximately x 1 alpha corresponding to levy stable processes the fluid limit leads to the fractional diffusion equation of order alpha in space and order beta in time however these are two special cases of a wider class of models here we consider the ctrw for the most general levy stochastic processes in the levy khintchine representation for the jump distribution function and obtain an integrodifferential equation describing the dynamics in the fluid limit the resulting equation contains as special cases the regular and the fractional diffusion equations as an application we consider the case of ctrws with exponentially truncated levy jump distribution functions in this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory long jumps and truncation effects in the intermediate asymptotic regime the dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as tauc approximately lambda alpha beta where 1 lambda is the truncation length scale the asymptotic behavior of the propagator green s function of the truncated fractional equation exhibits a transition from algebraic decay for t tauc

complex dimensions of fractals and meromorphic extensions of fractal zeta functions abstract we study meromorphic extensions of distance and tube zeta functions as well as of geometric zeta functions of fractal strings the distance zeta function a s a d x a s n d x where 0 is fixed and d x a denotes the euclidean distance from x to a has been introduced by the first author in 2009 extending the definition of the zeta function l associated with bounded fractal strings l l j j 1 to arbitrary bounded subsets a of the n dimensional euclidean space the abscissa of lebesgue i e absolute convergence d a coincides with d dim b a the upper box or minkowski dimension of a the visible complex dimensions of a are the poles of the meromorphic continuation of the fractal zeta function i e the distance or tube zeta function of a to a suitable connected neighborhood of the critical line re s d we establish several meromorphic extension results assuming some suitable information about the second term of the asymptotic expansion of the tube function a t as t 0 where a t is the euclidean t neighborhood of a we pay particular attention to a class of minkowski measurable sets such that a t t n d m o t as t 0 with 0 and to a class of minkowski nonmeasurable sets such that a t t n d g log t 1 o t as t 0 where g is a nonconstant periodic function and 0 in both cases we show that a can be meromorphically extended at least to the open right half plane re s d and determine the corresponding visible complex dimensions furthermore up to a multiplicative constant the residue of a evaluated at s d is shown to be equal to m the minkowski content of a and to the mean value of g the average minkowski content of a respectively moreover we construct a class of fractal strings with principal complex dimensions of any prescribed order as well as with an infinite number of essential singularities on the critical line re s d finally using an appropriate quasiperiodic version of the above construction with infinitely many suitably chosen quasiperiods associated with a two parameter family of generalized cantor sets we construct maximally hyperfractal compact subsets of r n for n 1 arbitrary these are compact subsets of r n such that the corresponding fractal zeta functions have nonremovable singularities at every point of the critical line re s d

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 riemann zeta function and the one dimensional weyl berry conjecture for fractal drums based on his earlier work on the vibrations of drums with fractal boundary the first author has refined m v berry s conjecture that extended from the smooth to the fractal case h weyl s conjecture for the asymptotics of the eigenvalues of the laplacian on a bounded open subset of w see 16 we solve here in the one dimensional case that is when n 1 this modified weyl berry conjecture we discover in the process some unexpected and intriguing connections between spectral geometry fractal geometry and the riemann zeta function we therefore show that one can hear that is recover from the spectrum not only the minkowski fractal dimension of the boundary as was established previously by the first author but also under the stronger assumptions of the conjecture its minkowski content a fractal analogue of its length we also prove still in dimension one a related conjecture of the first author as well as its converse which characterizes the situation when the error estimates of the aforementioned paper are sharp

self similarity of volume measures for laplacians on p c f self similar fractals our main goal in this paper is to obtain a precise analogue of weyl s asymptotic formula for the eigenvalue distribution of laplacians on a certain class of finitely ramified or p c f self similar fractals building in particular on the work of 7 9 22 24 our main result consists in precisely identifying for the class of decimable fractals the volume measures constructed by the second author in 24 for general p c f fractals and showing that they are self similar

harmonic calculus on p c f self similar sets the main object of this paper is the laplace operator on a class of fractals first we establish the concept of the renormalization of difference operators on post critically finite p c f for short self similar sets which are large enough to include finitely ramified self similar sets and extend the results for sierpinski gasket given in 10 to this class under each invariant operator for renormalization the laplace operator green function dirichlet form and neumann derivatives are explicitly constructed as the natural limits of those on finite pre self similar sets which approximate the p c f self similar sets also harmonic functions are shown to be finite dimensional and they are characterized by the solution of an infinite system of finite difference equations

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

space fractional advection dispersion equations with variable parameters diverse formulas numerical solutions and application to the macrodispersion experiment site data 1 to model the observed local variation of transport speed an extension of the homogeneous space fractional advection dispersion equation fade to more general cases with space dependent coefficients drift velocity v and dispersion coefficient d has been suggested to provide a rigorous evaluation of this extension we explore the underlying physical meanings of two proposed and one other possible form of the fade by using the generalized mass balance law proposed by meerschaert et al 2006 when the classical fick s law is replaced by its generalized form in the first order mass conversation law the original fade with constant parameters extends to the advection dispersion equation ade with fractional flux denoted as ff ade when the net inflow of dispersive flux is from nonlocal concentration gradients following a fractional divergence we get the ade with fractional divergence fd ade when the total net inflow from both nonlocal advection and nonlocal concentration gradients follows a fractional form the fade contains fully fractional divergence ffd ade these three fades with constant parameters can also be obtained by proper choice of the two memory kernels in the nonlocal dispersive constitutive theory proposed by cushman et al 1994 while the space variable fades correspond to the conditional nonlocal theory proposed by neuman 1993 after specifying the general levy type functional form of the random distribution the corresponding langevin markov models can be found in many cases where the lagrangian stochastic processes can be conditioned directly on local aquifer properties at any practical measurable level and resolution the resulting lagrangian random walk particle tracking methods along with previous numerical solutions using implicit euler finite differences distinguish and elucidate the plume behavior described by these fades the fractional models are applied to fit the tritium plumes measured at the macrodispersion experiment test site when the local parameters gleaned from the hydraulic conductivity k distribution are varied even slightly i e a two zone model allowing the use of observed hydraulic conditions at the site the model fits are well within the variability of the data the extended fades describe the fast and space dependent leading edges of measured plumes in the regional scale alluvial system which was underestimated and could not be fully captured by the original fade with constant parameters applications also favor the fd ade model because of the ease of implementation and consistency with previous analysis of the k statistics

spectral analysis on infinite sierpinski gaskets we study the spectral properties of the laplacian on infinite sierpin ski gaskets we prove that the laplacian with the neumann boundary condition has pure point spectrum moreover the set of eigenfunctions with compact support is complete the same is true if the infinite sierpin ski gasket has no boundary but is false for the laplacian with the dirichlet boundary condition in all these cases we describe the spectrum of the laplacian and all the eigenfunctions with compact support to obtain these results first we prove certain new formulas for eigenprojectors of the laplacian on finite sierpin ski pre gaskets then we prove that the spectrum of the discrete laplacian on a sierpin ski lattice is pure point and the eigenfunctions are localized 1998 academic press

hyperfunctions and spectral zeta functions of laplacians on self similar fractals we investigate the spectral zeta function of fractal differential operators such as the laplacian on the unbounded i e infinite sierpinski gasket and a self similar sturm liouville operator associated with a fractal self similar measure on the half line in the latter case sabot discovered the relation between the spectrum of this operator and the iteration of a rational map of several complex variables called the renormalization map we obtain a factorization of the spectral zeta function of such an operator expressed in terms of the dirac delta hyperfunction a geometric zeta function and the zeta function associated with the dynamics of the corresponding renormalization map viewed either as a polynomial function on in the first case or in the second case as a polynomial on the complex projective plane our first main result extends to the case of the fractal laplacian on the unbounded sierpinski gasket a factorization formula obtained by the second author for the spectral zeta function of a fractal string and later extended by teplyaev to the bounded i e finite sierpinski gasket and some other decimable fractals furthermore our second main result generalizes these factorization formulas to the renormalization maps of several complex variables associated with fractal sturm liouville operators moreover as a corollary in the very special case when the underlying self similar measure is the lebesgue measure on 0 1 we obtain a representation of the riemann zeta function in terms of the dynamics of a certain polynomial in thereby extending to several variables an analogous result by teplyaev

on using random walks to solve the space fractional advection dispersion equations the solution of space fractional advection dispersion equations fade by random walks depends on the analogy between the fade and the forward equation for the associated markov process the forward equation which provides a lagrangian description of particles moving under specific markov processes is derived here by the adjoint method the fade however provides an eulerian description of solute fluxes there are two forms of the fade based on fractional flux ff ade and fractional divergence fd ade the ff ade is derived by taking the integer order mass conservation of non local diffusive flux while the fd ade is derived by taking the fractional order mass conservation of local diffusive flux the analogy between the fade and the forward equation depends on which form of the fade is used and on the spatial variability of the dispersion coefficient d in the fade if d does not vary in space then the fades can be solved by tracking particles following a markov process with a simple drift and an stable levy noise with index that corresponds to the fractional order of the fade if d varies smoothly in space and the solute concentration at the upstream boundary remains zero the fd ade can be solved by simulating a markov process with a simple drift an stable levy noise and an additional term with the dispersion gradient and an additional levy noise of order 1 however a non markov process might be needed to solve the ff ade with a space dependent d except for specific d such as a linear function of space

weyl s problem for the spectral distribution of laplacians on p c f self similar fractals we establish an analogue of weyl s classical theorem for the asymptotics of eigenvalues of laplacians on a finitely ramified i e p c f self similar fractalk such as for example the sierpinski gasket we consider both dirichlet and neumann boundary conditions as well as laplacians associated with bernoulli type multifractal measures onk from a physical point of view we study the density of states for diffusions or for wave propagation in fractal media more precisely let x be the number of eigenvalues less thanx then we show that x is of the order ofxds 2 asx where the spectral exponent ds is computed in terms of the geometric as well as analytic structures ofk further we give an effective condition that guarantees the existence of the limit ofx ds 2 x asx this condition is in some sense generic in addition we define in terms of the above spectral exponents and calculate explicitly the spectral dimension ofk

on the periodicity of coxeter transformations and the non negativity of their euler forms abstract we show that for piecewise hereditary algebras the periodicity of the coxeter transformation implies the non negativity of the euler form contrary to previous assumptions the condition of piecewise heredity cannot be omitted even for triangular algebras as demonstrated by incidence algebras of posets we also give a simple direct proof that certain products of reflections defined for any square matrix a with 2 on its main diagonal and in particular the coxeter transformation corresponding to a generalized cartan matrix can be expressed as a 1 a t where a a are closely associated with the upper and lower triangular parts of a

vibration modes of 3n gaskets and other fractals we rigorously study eigenvalues and eigenfunctions vibration modes on the class of self similar symmetric finitely ramified fractals which include the sierpinski gasket and other 3n gaskets we consider the classical laplacian on fractals which generalizes the usual one dimensional second derivative is the generator of the self similar diffusion process and has possible applications as the quantum hamiltonian we develop a theoretical matrix analysis including analysis of singularities which allows us to compute eigenvalues eigenfunctions and their multiplicities exactly we support our theoretical analysis by symbolic and numerical computations our analysis in particular allows the computation of the spectral zeta function on fractals and the limiting distribution of eigenvalues i e integrated density of states we consider such examples as the level 3 sierpinski gasket a fractal 3 tree and the diamond fractal