maass operators and eisenstein series let g k be a hermitian symmetric space of the non compact type and let f c g be an arithmetic subgroup it is obvious that the enveloping algebra u g 0 of the complexified lie algebra of g gives rise to operators which transform automorphic forms on n with respect to f to f automorphic forms of a different type however it is in general not easy to write these differential operators explicitly in the coordinates of in the case g sp n ir when n is the siegel upper half plane of genus n maass constructed certain explicit operators of this type as a consequence of his construction he was able to show that his operators transform certain simple eisenstein series for f to other specific simple eisenstein series for f for a summary of his results and for references see 6 in this paper we generalize these results of maass to arbitrary hermitian symmetric spaces of the non compact type namely we show that the enveloping algebra u g 0 gives rise to differential operators a such that for certain pairs of canonical automorphy factors j j on g we have aj j cf theorem 2 4 and corollary 3 4 we do this abstractly using the transformation properties of the automorphy factors in particular we do not give explicit formulas for our differential operators which we call maass operators it would be interesting to do so in general we have also restricted our attention to cases in which one of the automorphy factors j j is scalar valued the same methods probably work more generally our interest in these operators arose from their use by shimura in his study of rationality properties for special values of zeta functions see 8 9 and subsequent papers katz 4 has also introduced these operators in another form the author has extended shimura s method to the case of siegel modular forms in 3 and intends to use the results of the present paper in future work of the same kind for this reason a brief and somewhat obtuse discussion of the rationality of maass operators has been included in sect 4 in any specific i e coordinatized example the results of sect 4 can be improved upon

endoscopic lifts to the siegel modular threefold related to klein s cubic threefold let cal a lev 11 be the moduli space of 1 11 polarized abelian surfaces with a canonical level structure let chi be a primitive character of order 5 with conductor 11 in this paper we construct five endoscopic lifts pi i 0 le i le 4 from two elliptic modular forms f otimes chi i of weight 2 and g otimes chi i of weight 4 with complex multiplication by bbb q sqrt 11 such that pi i infty gives a non holomorphic differential form on cal a lev 11 for each i 0 le i le 4 then their spinor l functions are of form l s 1 f otimes chi i l s g otimes chi i such that l s g otimes chi i does not appear in the l function of cal a lev 11 for any i 0 le i le 4 the existence of such lifts is motivated by the computation of the l function of klein s cubic threefold which is a birational smooth model of cal a lev 11

local and global strong solutions to continuous coagulation fragmentation equations with diffusion we consider the diffusive continuous coagulation fragmentation equations with and without scattering and show that they admit unique strong solutions for a large class of initial values if the latter values are small with respect to a suitable norm we provide sufficient conditions for global in time existence in the absence of fragmentation

on certain zeta functions attached to two hilbert modular forms ii the case of automorphic forms on a quaternion algebra as to the meaning of the symbols the reader is referred to the introduction of part i we recall here only that f is a totally real algebraic number field of degree n and w denotes the fourier coefficients of an elliptic modular form 52 z e wq a e2q iaz in part i we investigated d assuming that ex a n q s is an l function of a cm field with an algebraic valued hecke character in the present part ii we treat the case where x a can be obtained as eigenvalues of hecke operators t a on a quaternion algebra b over f thus we begin our study by recalling the theory of automorphic forms on the product o r of r copies of the upper half plane with respect to congruence subgroups of b x here r is the number of archimedean primes of f unramified in b then we introduce in section 2 the notion of arithmeticity or q rationality of such automorphic forms and show that the space of automorphic forms can be spanned by the arithmetic ones our main theorems will be stated in section 3 and proved in section 6 sections 4 and 5 are preliminaries to the proof the central result specialized to the case r 1 asserts that if g is a q rational eigenform such that gj t a x a g for all integral ideals a of f then d s0 for certain integers so are algebraic numbers times 7tdg g where d is an integer determined by the weight of g and is the petersson inner product

pullbacks of saito kurokawa lifts pullbacks of siegel eisenstein series have been studied by garrett 17 18 bocherer 6 7 heim 26 and play a key role in the proof of the algebraicity of critical values of certain automorphic l functions more generally one might consider pullbacks of siegel cusp forms for example ikeda 30 gave a conjectural formula for pullbacks of ikeda lifts 29 in terms of critical values of l functions for spn gl2 also the gross prasad conjecture 22 23 8 27 8 would relate pullbacks of siegel cusp forms of degree 2 to central critical values of l functions for gsp2 gl2 gl2 indeed bocherer furusawa and schulze pillot 8 gave an explicit formula for pullbacks of yoshida lifts 57 in this paper we give an explicit formula for pullbacks of saito kurokawa lifts and prove the algebraicity of central critical values of certain l functions for sp1 gl2 to be precise let be an odd positive integer let f s2 sl2 z be a normalized hecke eigenform and h s 1 2 0 4 a hecke eigenform associated to f by the shimura correspondence let f s 1 sp2 z be the saito kurokawa lift of h for each normalized hecke eigenform g s 1 sl2 z we consider the period integral f h h g g given by f h h g g

integrality of a ratio of petersson norms and level lowering congruences we prove integrality of the ratiof f g g outside an explicit finite set of primes where g is an arithmetically normalized holomorphic newform on a shimura curve f is a normalized hecke eigenform on gl 2 with the same hecke eigenvalues as g and denotes the petersson inner product the primes dividing this ratio are shown to be closely related to certain level lowering con gruences satisfied by f and to the central values of a family of rankin selberg l functions finally we give two applications the first to proving the integral ity of a certain triple product l value and the second to the computation of the faltings height of jacobians of shimura curves

irreducibility and exponential mixing of some stochastic hydrodynamical systems driven by pure jump noise in this paper we derive several results related to the long time behavior of a class of stochastic semilinear evolution equations in a separable hilbert space h d mathfrak u t rm a mathfrak u t rm b mathfrak u t mathfrak u t dt dl t quad mathfrak u 0 x in rm h here a is a positive self adjoint operator and b is a bilinear map and the driving noise l is basically a d rm a 1 2 valued levy process satisfying several technical assumptions by using a density transformation theorem type for levy measure we first prove a support theorem and an irreducibility property of the ornstein uhlenbeck processes associated to the nonlinear stochastic problem second by exploiting the previous results we establish the irreducibility of the nonlinear problem provided that for a certain gamma in 0 1 4 b is continuous on d rm a gamma times d rm a gamma with values in d rm a 1 2 using a coupling argument the exponential ergodicity is also proved under the stronger assumption that b is continuous on rm h times rm h while the latter condition is only satisfied by the nonlinearities of goy and sabra shell models the assumption under which the irreducibility property holds is verified by several hydrodynamical systems such as the 2d navier stokes magnetohydrodynamics equations the 3d leray varvec alpha model the goy and sabra shell models

bessel models for gsp 4 methods of theta correspondence are used to analyze local and global bessel models for gsp 4 proving a conjecture of gross and prasad which describes these models in terms of local epsilon factors in the local case and the non vanishing of central critical l value in the global case

fragmentation of random trees we study fragmentation of a random recursive tree into a forest by repeated removal of nodes the initial tree consists of n nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly selected existing node as nodes are removed from the tree one at a time the tree dissolves into an ensemble of separate trees namely a forest we study statistical properties of trees and nodes in this heterogeneous forest and find that the fraction of remaining nodes m characterizes the system in the limit we obtain analytically the size density of trees of size s the size density has power law tail with exponent therefore the tail becomes steeper as further nodes are removed and the fragmentation process is unusual in that exponent increases continuously with time we also extend our analysis to the case where nodes are added as well as removed and obtain the asymptotic size density for growing trees

new solutions to the fragmentation equation a new general class of exact explicit scaling solutions to the fragmentation equation is given this class is described by a breakage rate a x xlambda and the daughter distribution function yb x mod y a gamma xy gamma 2 1 a delta x y delta 2 and includes as special cases all previously known scaling solutions to the fragmentation equation for a subset of this class with gamma lambda and a delta delta lambda the complete time dependent solution for a monodisperse initial condition is also given

kinetics of fragmentation a general discussion of the kinetics of continuous irreversible fragmentation processes is presented for a linear process where particle breakup is driven by an external force the authors develop a scaling theory to describe the evolution of the cluster size distribution they treat the general case where the breakup rate of a cluster of mass x varies as xlambda when lambda 0 corresponding to larger clusters more likely to break up the scaled cluster size distribution phi x decays with the scaled mass x as x 2 exp axlambda as x for small mass phi x has the log normal form exp a ln2 x if the breakup kernel has a small size cutoff while phi x has a power law tail in the absence of a cutoff they also show that a conventional scaling picture applies only for the case lambda 0 for lambda 0 they develop an alternative formulation for the cluster size distribution in which the typical mass scale is determined by the initial condition in this regime they also investigate the nature of a shattering transition where mass is lost to a dust phase of zero mass particles they also study the kinetics of a nonlinear collision induced fragmentation process they analyse the asymptotic behaviour of a simple minded class of models in which a two particle collision results in either 1 both particles splitting into two equal pieces 2 only the larger particle splitting in two or 3 only the smaller particle splitting they map out the kinetics of these models by scaling arguments and by analytic and numerical solutions of the rate equations scaling is found to hold for different ranges of homogeneity index for the three models

convergence to equilibrium for the discrete coagulation fragmentation equations with detailed balance under the condition of detailed balance and some additional restrictions on the size of the coefficients we identify the equilibrium distribution to which solutions of the discrete coagulation fragmentation system of equations converge for large times thus showing that there is a critical mass which marks a change in the behavior of the solutions this was previously known only for particular cases as the generalized becker doring equations our proof is based on an inequality between the entropy and the entropy production which also gives some information on the rate of convergence to equilibrium for solutions under the critical mass

distribution of the number of fragmentations in continuous fragmentation the continuous fragmentation equation is generalized to include the number of fragmentations as a variable the fragmentation rate and the probability for the production of progeny size y from parent size x are both made dependent on the number n of fragmentations experienced by the fragment during its history an interpretation where n is the number of impacts experienced by the fragment is possible by also allowing a single progeny equalling the parent in size solutions for the cases where does not depend on x are studied the theory is adapted to model clustering in fragmentation generated cellular structures

correlations between fragment sizes in sequential fragmentation size correlations in a sequential fragmentation process originate from the correlations the progeny of a single fragmentation event necessarily have a multidimensional equation describing the binary fragmentation of a unit segment is formulated the fragments are assumed to keep their positions with respect to the original segment if the fragmentation rate is not dependent on fragment size the distribution for n successive adjacent fragments is shown to asymptotically approach an n dimensional log normal distribution explicit forms of the asymptotic distribution and the correlations are given in terms of the parameters of the probability that a fragmentation event with parent size y produces progeny size x

on a discrete dynamical model for local turbulence works towards an explanation of the observed experimental deviation of k41 exponent law are generally based on an energy cascade process an alternative approach based on the analysis of turbulent sequences allows us to construct a sequence model where this deviation has a dynamical origin we finally show why we were forced to go beyond the markovian model and incorporate into the model the frequent long flights an intermittent effect also present in the experimental data

distribution of droplets in a turbulent spray a statistical description of droplets in turbulent spray connected with the description of turbulent dissipation is presented the ideas of similarity the cascade processes and the infinitely divisible distributions are used in this description formulas for characteristic droplet sizes and corresponding probability distributions are obtained along with a simple formula for turbulent dissipation in flow near a ship copyright ital 1997 ital the american physical society

models of fragmentation with power law log normal distributions abstract two models of binary fragmentation are introduced in which a time dependent transition size produces two regions of fragment sizes above and below the transition size in the models we consider a fixed rate of fragmentation for the largest fragment and two different rates of fragmentation for the two regions of sizes above and below the transition size the models are solved exactly in the long time limit to reveal stable time invariant solutions for the fragment size distributions a rate of fragmentation proportional to the inverse of fragment size in the smaller size region produces a power law distribution in that region a rate of fragmentation combined of two terms one proportional to the inverse of the fragment size and the other proportional to a logarithmic function of the fragment size in the larger size region produces a log normal distribution in that region special cases of the models with no fragmentation for the smaller fragments are also considered the similarities between the stable distributions in our models and power law log normal distributions from experimental work on shock fragmentation of long thin glass rods and rupture of mercury droplets are investigated

a summation formula for triples of quadratic spaces abstract let v 1 v 2 v 3 be a triple of even dimensional vector spaces over a number field f equipped with nondegenerate quadratic forms q 1 q 2 q 3 respectively let y i 1 v i be the closed subscheme consisting of v 1 v 2 v 3 on which q 1 v 1 q 2 v 2 q 3 v 3 motivated by conjectures of braverman and kazhdan and related work of lafforgue ngo and sakellaridis we prove an analogue of the poisson summation formula for certain functions on this space

infinitely divisible distributions in turbulence the imbedding of the scale similarity of random fields into the theory of infinitely divisible probability distributions is considered the general probability distribution for the breakdown coefficients of turbulent energy dissipation is obtained along with corresponding similarity exponents related issues of self similarity and asymptotic behavior of statistical characteristics are also considered

multifractal cascades with interactions abstract we consider a nonlinear version of the random model where the fragmentation probability of an eddy depends directly on the activity of the neighbouring regimes we present an exact calculation for the complete multifractal spectrum inspired by concepts recently introduced in the theory of fractal growth we point out how this generalization might be important for developing a first principle theory based on the navier strokes equations

a new three parameter cloud aerosol particle size distribution based on the generalized inverse gaussian density function we describe a three parameter cloud aerosol size distribution based on the generalized inverse gaussian density function that provides a simple and convenient recipe for characterizing aerosol radiative properties the feature that makes this distribution attractive is that all of its moments can be calculated analytically providing simple formulas for the effective radius and variance and relating these to other important quantities such as the particle number and mass densities another advantage of this distribution is the smooth exponential cut off not only for large particles sizes but also in small particles regime which prevents the singularity that occurs at zero size for large effective variances for distributions like the popular gamma distribution unlike most size distributions in current use the proposed functional form has a third parameter order which provides greater flexibility in representing different distributional shapes analytical expressions for the moments of the generalized inverse gaussian distribution in the generic case contain ratios of modified bessel functions however for half integer orders these expressions are simple ratios of polynomials as a practical example we compare extinction efficiency factors and asymmetry parameters computed for several different orders of the proposed distribution function

on maass lifts and the central critical values of triple product l functions we express certain period integrals of maass lifts which appear in the gross prasad conjecture in terms of the central critical values of triple product l functions in the imbalanced case

a note on the self similar solutions to the spontaneous fragmentation equation we provide a method to compute self similar solutions for various fragmentation equations and use it to compute their asymptotic behaviours our procedure is applied to specific cases i the case of mitosis where fragmentation results into two identical fragments ii fragmentation limited to the formation of sufficiently large fragments and iii processes with fragmentation kernel presenting a power like behaviour

the discrete diffusive coagulation fragmentation equations with scattering we consider the discrete coagulation fragmentation equations with diffusion presupposing a maximal cluster size such a feature requires a new interaction mechanism in order to prevent occurrence of too large clusters being produced by coagulation existence of a unique solution for this model is proven and long time behaviour is studied in situations where equilibria are explicitly known

on the solution of the integro differential fragmentation equation with continuous mass loss in this paper we obtain a complete classification of all possible non trivial similarity solutions of the integro differential fragmentation equation with continuous mass loss rate these solutions include the effects of both continuous and discrete mass loss rates the similarity solutions are compared with the solutions found earlier the comparison shows that some previous solutions are obtained as special cases from our solutions the results reported here provide further evidence of the usefulness of the lie group method for obtaining similarity solutions for either differential or integro differential equations

meloy s law of morphological coefficients as a scaling condition for the fourier representation of particle shape a stochastic renormalization approach a new theoretical model for the description of particle shape distributions is suggested within its framework a complex fourier representation of particle shape is used a frequency distribution expressing the contributions of the different complex harmonics to the dispersion of a position vector is defined the author assumes that particles with different shapes are generated during a scale invariant fragmentation process starting from an arbitrary shape distribution the fragmentation leads to a succession of different shape distributions the asymptotic behaviour of this sequence is analysed by means of a stochastic renormalisation approach two different scaling laws are found one scaling law describes the behaviour of the moments of amplitudes attached to the fourier representation of particle shape whereas the other corresponds to the renormalized size distribution in particular the author gives a theoretical derivation of the empirical law of morphological coefficients discovered by meloy 1978 a physical interpretation of the scaling conditions is given based on a many particle branched chain approach the author suggests that the scaling behaviour is mainly due to the time homogeneous markovian structure of the many particle evolution equations

on the existence of moments of solutions to fragmentation equations abstract we consider the multiple fragmentation equations with polynomially bounded fragmentation rates both in the discrete and continuous cases the theory of semigroups of operators on frechet spaces is used to produce a simple proof that if moments of all non negative orders of solutions are initially finite then they remain finite for all future times moreover a class of fragmentation processes is identified in which the existence of the first moment of the initial distribution suffices for the existence of all other moments for positive times

an analytical solution of coagulation processes with collision induced fragmentation we propose a single species coagulation model with collision induced fragmentation in which a pair of clusters can coagulate into a larger one if their encounter is a complete inelastic collision otherwise one of them will fragment into two smaller clusters due to a destructive collision consider a simple system with the constant coagulation kernel i i j i and the size dependent fragmentation kernel j l i j 2jlu we then investigate analytically the kinetic behavior of such a system by means of the rate equation approach it is indicated that the results are crucially dependent on the value of index u in the case of u 1 and i 2j u here x is the riemann zeta function the cluster size distribution ak t approaches the conventional scaling form and the system will evolve permanently in time while in other cases ak t always takes the modified scaling form moreover a balance can be established between coagulation and collision induced breakage processes and thus the system can eventually evolve to a steady state

multiscaling in fragmentation abstract we review recent results on random fragmentation of multidimensional objects in addition to the obvious volume conservation these processes exhibit an infinite number of hidden conservation laws the fragment size distribution is characterized by an infinite number of scales and thus exhibits multiscaling nevertheless the volume distribution function shows ordinary scaling i e it is characterized by a single scale

similarity solutions for fragmenting systems with continuous mass loss a linear rate equation describing fragmentation processes which includes both continuous and discrete loss of mass is reduced to a partial differential equation different types of similarity solutions are obtained by applying lie s similarity method

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

similarity solutions in fragmentation kinetics for different homogeneous fragmentation models the authors derive a common integro differential equation they assume that the fragmentation rate is homogeneous of order gamma in both arguments a reduction of the integro differential equation to a partial differential equation delivers different types of similarity solutions by applying lie s similarity method calculation of the adjoint algebra performs a classification of the non trivial solutions

on the similarity solution of the fragmentation equation an analysis of the similarity solution of the linear homogeneous fragmentation equation is given for a general volume conserving daughter fragment distribution for the special case of a polynomial daughter distribution of degree p an exact basic similarity solution for the time evolution of the particle volume distribution is derived the solution is proportional to the meijer g function which may be represented as a linear combination of generalized hypergeometric functions the properties of generalized hypergeometric functions and g functions are given in the special function literature and include the continuation of the solution to large values of the similarity variable which is given here for special cases

transcending classical invariant theory let sp2 r sp2 sp be the real symplectic group of rank n let sp denote the two fold cover of sp there is a unitary representation constructed by shale sh and weil wa of sp that is of considerable interest in the theory of automorphic forms we denote this representation by co and call it the oscillator representation the purpose of this paper is to establish about c a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co we begin by formulating the result let e be a reductive subgroup of sp and let e denote the inverse image of e in sp denote by r e the set of infinitesimal equivalence classes of continuous irreducible admissible representations of e on locally convex spaces let co be the smooth representation of sp associated to co let co be realized on a hilbert space and let the subspace of smooth vectors on which space co is defined be written denote by r9 e co the set of elements of e which are realized as quotients by co e invariant closed subspaces of t g consider a reductive dual pair g g c sp h2 it is not hard to show that g and g commute with one another hence we may regard co i g g as a representation of gx g it is well known f that r gx g s9 g xr9 g the identification associates to p e r g and p e i g the tensor product p x p we note that p x p is not defined as a topological vector space nevertheless the infinitesimal equivalence class of p op is well defined select p x p e g x g suppose that in fact p p e 9 g g c then clearly p e r g c and p e 9 g c hence g co defines the graph of a correspondence between certain subsets of r9 g cv and j g co in fact the situation is quite precise

a theoretical explanation of grain size distributions in explosive rock fragmentation we have measured grain size distributions of the results of laboratory decompression explosions of volcanic rock the resulting distributions can be approximately represented by gamma distributions of weight per cent as a function of formula see text where d is the grain size in millimetres measured by sieving with a superimposed long tail associated with the production of fines we provide a description of the observations based on sequential fragmentation theory which we develop for the particular case of self similar fragmentation kernels and we show that the corresponding evolution equation for the distribution can be explicitly solved yielding the long time lognormal distribution associated with kolmogorov s fragmentation theory particular features of the experimental data notably time evolution advection truncation and fines production are described and predicted within the constraints of a generalized reductive fragmentation model and it is shown that the gamma distribution of coarse particles is a natural consequence of an assumed uniform fragmentation kernel we further show that an explicit model for fines production during fracturing can lead to a second gamma distribution and that the sum of the two provides a good fit to the observed data

mutually catalyzed birth of population and assets in exchange driven growth we propose an exchange driven aggregation growth model of population and assets with mutually catalyzed birth to study the interaction between the population and assets in their exchange driven processes in this model monomer or equivalently individual exchange occurs between any pair of aggregates of the same species population or assets the rate kernels of the exchanges of population and assets are k k l kkl and l k l lkl respectively at which one monomer migrates from an aggregate of size k to another of size l meanwhile an aggregate of one species can yield a new monomer by the catalysis of an arbitrary aggregate of the other species the rate kernel of asset catalyzed population birth is i k l iklmu and that of population catalyzed asset birth is j k l jklnu at which an aggregate of size k gains a monomer birth when it meets a catalyst aggregate of size l the kinetic behaviors of the population and asset aggregates are solved based on the rate equations the evolution of the aggregate size distributions of population and assets is found to fall into one of three categories for different parameters mu and nu i population asset aggregates evolve according to the conventional scaling form in the case of mu 0 mu 0 and nu 0 and iii both population and asset aggregates undergo gelation transitions at a finite time in the case of mu nu 0

microcanonical multifractal formalism a geometrical approach to multifractal systems part i singularity analysis multifractal formalism in the microcanonical framework has proved to be a valuable approach to understand and analyze complex signals typically associated with natural phenomena in scale invariant systems in this paper we discuss the multifractal microcanonical formalism in a comprehensive unified way including new theoretical proofs and validation tests on real signals so completing some known gaps in the foundations of this theory we also review the latest advances and describe the present perspectives in this field some technical details on the implementation of involved algorithms and relevant open issues are also discussed

origin of lognormal like distributions with a common width in a growth and division process lognormal statistical distributions are observed in a variety of scientific fields the widths of these distributions in the log scale are often similar but the underlying mechanism that maintains these widths within a small range has not been well explained we show that a stochastic process of halving followed by addition can yield a stationary distribution that resembles the universal lognormal distribution with a certain width the mechanism that we propose here would provide insight into the essence of why lognormal like distributions in many systems have a common width

automata over continuous time the principal objective of this paper is to lift basic concepts of the classical automata theory from discrete to continuous real time it is argued that the set of finite memory retrospective functions is the set of functions realized by finite state devices we show that the finite memory retrospective functions are speed independent i e they are invariant under stretchings of the time axis therefore such functions cannot deal with metrical aspects of the reals we classify and analyze phenomena which appear at continuous time and are invisible at discrete time

applications of nonclassical symmetries in this paper we discuss new applications of the nonclassical method of symmetry reduction the equations treated are well known in mathematical physics and are used to describe multisoliton solutions of the burger equation they are also useful in the study of kac moody algebras and in the examination of nonlinear waves in the theory of elasticity with the help of the nonclassical symmetry method the partial differential equations are reduced to ordinary differential equations which are solvable explicitly

generalized heegner cycles and p adic rankin l series this article studies a distinguished collection of so called generalized heegner cycles in the product of a kuga sato variety with a power of a cm elliptic curve its main result is a p adic analogue of the gross zagier formula which relates the images of generalized heegner cycles under the p adic abel jacobi map to the special values of certain p adic rankin l series at critical points that lie outside their range of classical interpolation

on the nonvanishing of the central value of the rankin selberg l functions first published in journal of the american mathematical society in volume 17 issue 3 published by the american mathematical society

on non scale invariant infinitely divisible cascades multiplicative processes multifractals and more recently also infinitely divisible cascades have seen increased popularity in a host of applications requiring versatile multiscale models ranging from hydrodynamic turbulence to computer network traffic from image processing to economics the methodologies prevalent as of today rely to a large extent on iterative schemes used to produce infinite detail and repetitive structure across scales while appealing due to their simplicity these constructions have limited applicability as they lead by default to power law progression of moments through scales to nonstationary increments and often to inherent log periodic scaling which favors an exponential set of scales this paper studies and develops a wide class of infinitely divisible cascades idc thereby establishing the first reported cases of controllable scaling of moments in non power law form embedded in the framework of idc these processes exhibit stationary increments and scaling over a continuous range of scales criteria for convergence further statistical properties as well as matlab routines are provided

some local global non vanishing results for theta lifts from orthogonal groups we firstly improve a theorem of b roberts which characterizes non vanishing of a global theta lift from o x to sp n in terms of non vanishing of local theta lifts in particular we will remove all the archimedean conditions imposed upon his theorem sec ondly following roberts we will apply our theorem to theta lifting of low rank similitude groups namely we characterize the non vanishing condition of a global theta lift from go 4 to gsp 2 in our improved setting also we consider non vanishing conditions of a global theta lift from go 4 to gsp 1 and explicitly compute the lift when it exists

the multifractal structure of contrast changes in natural images from sharp edges to textures we present a formalism that leads naturally to a hierarchical description of the different contrast structures in images providing precise definitions of sharp edges and other texture components within this formalism we achieve a decomposition of pixels of the image in sets the fractal components of the image such that each set contains only points characterized by a fixed strength of the singularity of the contrast gradient in its neighborhood a crucial role in this description of images is played by the behavior of contrast differences under changes in scale contrary to naive scaling ideas where the image is thought to have uniform transformation properties field 1987 each of these fractal components has its own transformation law and scaling exponents a conjecture on their biological relevance is also given

green forms and the arithmetic siegel weil formula we construct natural green forms for special cycles in orthogonal and unitary shimura varieties in all codimensions and for compact shimura varieties of type mathrm o p 2 and mathrm u p 1 we show that the resulting local archimedean height pairings are related to special values of derivatives of siegel eisentein series a conjecture put forward by kudla relates these derivatives to arithmetic intersections of special cycles and our results settle the part of his conjecture involving local archimedean heights

the nonlinear fragmentation equation we study the kinetics of nonlinear irreversible fragmentation here fragmentation is induced by interactions collisions between pairs of particles and modelled by general classes of interaction kernels for several types of breakage models we construct initial value and scaling solutions of the fragmentation equations and apply the non vanishing mass flux criterion for the occurrence of shattering transitions these properties enable us to determine the phase diagram for the occurrence of shattering states and of scaling states in the phase space of model parameters

new similarity solutions for fragmenting systems with continuous loss of mass in a recent article saied and el wakil discussed the similarity solutions of a fragmenting system their article is an extension of our own work on an equation including continuous loss of mass saied and el wakil restrict their considerations to similarity solutions on a special choice of parameters we demonstrate here that the symmetry group given in saied and el wakil s article is incomplete furthermore we are able to construct new similarity solutions derived from the missing subgroup

general solutions and scaling violation for fragmentation with mass loss explicit solutions of a linear rate equation lead to improved understanding of fragmenlatian with discrete and continuous mass loss discrete mass loss provides a general expression for the overall mass 10 rate which accounts for mass loss in the shattering regime where runaway fragmentation for small panicles producer a phase of zero mass panicles an explicit solution far the recession regime where small panicles lose large fractionsoftheir mass to surface recession shows that the total number ofpaniclesincreases with time until the typical panicles become small enough to lose all of their mass and disappear rather than to break a general series solution is presented for fragmentation with continuous and discrete mass loss a proof that continuous mass loss precludes dynamic scaling is presented

on certain zeta functions attached to two hilbert modular forms i the case of hecke characters to make our exposition smooth let us first introduce some notational conventions for an algebraic number field k of finite degree we denote by jk the set of all embeddings of k into c and by ik the free z module generated by the elements of jk we then put rik ik0 r and cik ik0 c if p eapau e ik with a e jk and po e z we put xp fl xa pfor 0 x e k this is meaningful for p e cik if xa are all real and positive throughout the paper we denote by d the complex upper half plane and by f a totally real algebraic number field of degree n now the zeta function to be studied in this paper when suitably specialized has the form

expressive completeness of duration calculus this paper compares the expressive power of first order monadic logic of order a fundamental formalism in mathematical logic and the theory of computation with that of the propositional version of duration calculus pdc a formalism for the specification of real time systems our results show that the propositional duration calculus is expressively complete for first order monadic logic of order our semantics for pdc conservatively extends the standard semantics to all positive including infinite length intervals hence in view of the expressive completeness liveness properties can be specified in pdc this observation refutes a widely believed misconception that the duration calculus cannot specify liveness properties

an explicit solution to a discrete fragmentation model the discrete binary fragmentation equation is solved explicitly far a model where tnenetrateoibreakupoiaparlicie oishek o equals k lj k l andthedaughter ri e distribution b lx equals 2 k 1 this system is closely related to a model of polymer degradation considered by simha in which ax i and b is as above in the continuum limit both of these models go over to a continuous fragmentation model in which all particles break with an equal rate a x i and the daughter size distribution is uniform b yl x z x which is at the borderline of the shattering transition

an operational calculus framework to characterize droplet size populations from turbulent breakup by a small number of parameters a systematic operational calculus framework that characterizes droplet bubble size distributions resulting from turbulent breakup of an immiscible fluid into a carrier one is presented the proposed formulation is derived from dynamical arguments a finite difference formulation of the integro differential continuous coagulation and fragmentation equation is shown to exhibit the same structure as a discrete sequence of mellin convolutions between the probability distribution of the evolving dispersed phase and a generic kernel this kernel may have its physical correspondence with the probability distribution resulting from a single breakup event e g a liquid ligament breakup in a ligament mediated spray formation the number of convolution steps in the sequence can be reduced to a single parameter as an illustration this procedure is applied to the exponential and the gamma distributions obtaining as a result the frechet distribution earlier used by rosin and rammler 1934 kolloid zeitschrift 67 16 26 and by nukiyama and tanasawa 1939 trans soc mech eng japan 5 62 7 thus the framework introduced in this work provides a physical foundation for the success of the frechet distribution in accurately fitting experimentally measured droplet size distributions in sprays and emulsions

decidable verification for reducible timed automata specified in a first order logic with time we consider one type of first order timed logic fotl with explicit continuous time fotl is sufficiently expressible from the user s point of view to rewrite directly requirements specifications often given in a language close to the natural one and it permits to represent the set of runs of timed programs thus fotl is apt to formalize the verification problem for timed systems our main goal is to describe in semantical terms interesting decidable classes of the verification problem within this setting we prove that under some finiteness properties of the requirements and algorithm specifications the verification problem represented in fotl becomes decidable the finiteness properties we introduce finite refutability and finite satisfiability are undecidable in the general case however finite refutability is often easy to verify on the other hand we give a sufficient condition namely reducibility which ensures the finite satisfiability for timed automata and we prove that the reducibility is decidable this is the main result of the paper as a consequence the verification of any finitely refutable requirements is decidable for reducible timed automata

refined scaling hypothesis for breakdown coefficients in turbulence we study the statistics of novikov s breakdown coefficients which represent ratios between energy dissipation rates at different length scales in turbulence the distribution of their logarithms is shown to be very closely reproduced by an analytic function that we obtain from a hierarchical stochastic process for the turbulent cascade correlations and deviation from gaussianity in the model are accounted for by two parameters one of which can be interpreted as a generalized dimension finally we illustrate the lack of power law scaling in the moments of the breakdown coefficients and propose an analytical approximation scheme for them

similarity solution for fragmentation with volume change the similarity form of the solution of the linear homogeneous equation for fragmentation with volume change is given the solution shows the time evolution of the particle volume distribution when a particle splits into a polynomial distribution of fragment volumes the limits for small and large values of the similarity variable are derived the long time limit for all values of the similarity variable is given and the characteristic time to approach the limit is identified the solution shows the effect of volume change on the particle volume distribution and on the time dependence of the moments of the distribution