fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london

beyond the poisson renewal process a tutorial survey after sketching the basic principles of renewal theory and recalling the classical poisson process we discuss two renewal processes characterized by waiting time laws with the same power asymptotics defined by special functions of mittag leffler and of wright type we compare these three processes with each other

monotonic continuous time random walks with drift and stochastic reset events in this paper we consider a stochastic process that may experience random reset events which suddenly bring the system to the starting value and analyze the relevant statistical magnitudes we focus our attention on monotonic continuous time random walks with a constant drift the process increases between the reset events either by the effect of the random jumps or by the action of the deterministic drift as a result of all these combined factors interesting properties emerge like the existence for any drift strength of a stationary transition probability density function or the faculty of the model to reproduce power law like behavior general formulas for two extreme statistics the survival probability and the mean exit time are also derived to corroborate in an independent way the results of the paper monte carlo methods were used these numerical estimations are in full agreement with the analytical predictions

density approach to ballistic anomalous diffusion an exact analytical treatment this paper addresses the problem of deriving the probability distribution density of a diffusion process generated by a nonergodic dichotomous fluctuation using the liouville equation density method the velocity of the diffusing particles fluctuates from the value of 1 to the value of 1 and back with the distribution density of time durations of the two states proportional to 1 in the asymptotic time limit the adopted density method allows us to establish an exact analytical expression for the probability distribution density of the diffusion process generated by these fluctuations contrary to intuitive expectations the central part of the diffusion distribution density is not left empty when moving from 2 ergodic condition to cr the monomodal distribution density with a minimum at the origin is turned into a bimodal one with a central bump whose intensity increases for 2 the

simultaneous encodings for range and next previous larger smaller value queries given an array of n elements from a total order we propose encodings that support various range queries range minimum range maximum and their variants and previous and next smaller larger value queries when query time is not of concern we obtain a 4 088n o n bit encoding that supports all these queries for the case when we need to support all these queries in constant time we give an encoding that takes 4 585n o n bits where n is the length of input array this improves the 5 08n o n bit encoding obtained by encoding the colored 2d min and max heaps proposed by fischer 11 we first extend the original dfuds 8 encoding of the colored 2d min max heap that supports the queries in constant time then we combine the extended dfuds of 2d min heap and 2d max heap using the minmax encoding of gawrychowski and nicholson 15 with some modifications we also obtain encodings that take lesser space and support a subset of these queries

membership in constant time and almost minimum space this paper deals with the problem of storing a subset of elements from the bounded universe mathcal m 0 ldots m 1 so that membership queries can be performed efficiently in particular we introduce a data structure to represent a subset of n elements of mathcal m in a number of bits close to the information theoretic minimum b left lceil lg m choose n right rceil and use the structure to answer membership queries in constant time

tree compression with top trees we introduce a new compression scheme for labeled trees based on top trees our compression scheme is the first to simultaneously take advantage of internal repeats in the tree as opposed to the classical dag compression that only exploits rooted subtree repeats while also supporting fast navigational queries directly on the compressed representation we show that the new compression scheme achieves close to optimal worst case compression can compress exponentially better than dag compression is never much worse than dag compression and supports navigational queries in logarithmic time

a simple optimal representation for balanced parentheses we consider succinct or highly space efficient representations of a static string consisting of n pairs of balanced parentheses which support natural operations such as finding the matching parenthesis for a given parenthesis or finding the pair of parentheses that most tightly enclose a given pair this problem was considered by jacobson space efficient static trees and graphs in proc of the 30th focs 1989 pp 549 554 and munro and raman succinct representation of balanced parentheses and static trees siam j comput 31 2001 762 776 who gave o n bit and 2n o n bit representations respectively that supported the above operations in o 1 time on the ram model of computation this data structure is a fundamental tool in succinct representations and has applications in representing suffix trees ordinal trees planar graphs and permutations we consider the practical performance of parenthesis representations first we give a new 2n o n bit representation that supports all the above operations in o 1 time this representation is conceptually simpler its space bound has a smaller o n term and it also has a simple and uniform o n time and space construction algorithm we implement our data structure and a variant of jacobson s and evaluate their practical performance speed and memory usage when used in a succinct representation of trees derived from xml documents as a baseline we compare our representations against a widely used implementation of the standard dom document object model representation of xml documents both succinct representations use orders of magnitude less space than dom and tree traversal operations are usually only slightly slower than in dom

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

succinct representations of graphs for a fixed graph property q the complexity of the problem given a graph g does g have property q is usually investigated as a function of v the number of vertices in g with the assumption that the input size is polynomial in v in this paper the complexity of these problems is investigated when the input graph is given by a succinct representation by a succinct representation it is meant that the input size is polylog in v it is shown that graph problems which are approached this way become intractable actually no nontrivial problem could be found which can be solved in polynomial time the main result is characterizing a large class of graph properties for which the respective succinct problem is np hard trying to locate these problems within the p time hierarchy shows that the succinct versions of polynomially equivalent problems may not be polynomially equivalent

large deviations for a reaction diffusion equation with non gaussian perturbations in this paper we establish a large deviations principle for the non gaussian stochastic reaction diffusion equation srde partial t nu varepsilon mathscr l nu varepsilon f x nu varepsilon varepsilon sigma x nu varepsilon ddot w tx as a random perturbation of the deterministic rde partial t nu 0 mathscr l nu 0 f x nu 0 here the space variable takes values on the unit circle s 1 and mathscr l is a strongly elliptic second order operator with constant coefficients the functions f and sigma are sufficiently regular so that there is a unique solution to the above srde for any continuous initial condition we also assume that there are positive constants m and m such that m leq sigma x y leq m for all x in s 1 and all y in mathbb r the perturbation ddot w tx is the formal derivative of a brownian sheet it is known that if the initial condition is continuous then the solution will also be continuous and moreover if the initial condition is assumed to be holder continuous of exponent kappa for some 0 kappa frac 1 2 then the solution will be holder continuous of exponent kappa 2 as a function of t x in this paper we establish the large deviations principle for nu varepsilon in the holder norm of exponent kappa 2 when the initial condition is holder continuous of exponent kappa for any 0 kappa frac 1 2 and when the initial condition is assumed only to be continuous we establish the large deviations principle for nu varepsilon in the supremum norm moreover we prove that these large deviations principles are uniform with respect to the initial condition

a fast general methodology for information theoretically optimal encodings of graphs we propose a fast methodology for encoding graphs with information theoretically minimum numbers of bits specifically a graph with property pi is called a em pi graph if pi satisfies certain properties then an n node m edge pi graph g can be encoded by a binary string x such that 1 g and x can be obtained from each other in o n log n time and 2 x has at most beta n o beta n bits for any continuous superadditive function beta n so that there are at most 2 beta n o beta n distinct n node pi graphs the methodology is applicable to general classes of graphs this paper focuses on planar graphs examples of such pi include all conjunctions over the following groups of properties 1 g is a planar graph or a plane graph 2 g is directed or undirected 3 g is triangulated triconnected biconnected merely connected or not required to be connected 4 the nodes of g are labeled with labels from 1 ldots ell 1 for ell 1 leq n 5 the edges of g are labeled with labels from 1 ldots ell 2 for ell 2 leq m and 6 each node respectively edge of g has at most ell 3 o 1 self loops respectively ell 4 o 1 multiple edges moreover ell 3 and ell 4 are not required to be o 1 for the cases of pi being a plane triangulation these examples are novel applications of small cycle separators of planar graphs and are the only nontrivial classes of graphs other than rooted trees with known polynomial time information theoretically optimal coding schemes

space efficient euler partition and bipartite edge coloring abstract we describe space efficient algorithms for two problems on undirected multigraphs euler partition partitioning the edges into a minimum number of trails and bipartite edge coloring coloring the edges of a bipartite multigraph with the minimum number of colors let n m and 1 be the numbers of vertices and of edges and the maximum degree respectively of the input multigraph for euler partition we reduce the amount of working memory needed by a logarithmic factor from n m log n m bits to o n m bits while preserving a running time of o n m for bipartite edge coloring still using o n m bits of working memory we achieve a running time of o n m min log log log m log this is o m log log if m n log n log log n log n to be compared with o m log for the fastest known algorithm instrumental in obtaining the latter result is a new data structure presented here the subgraph stack that allows graph algorithms centered around recursive calls on substantially smaller subgraphs to be implemented in a space efficient manner

stochastic volterra equations in banach spaces and stochastic partial differential equation abstract in this paper we study the existence uniqueness and large deviation estimate for stochastic volterra integral equations with singular kernels in 2 smooth banach spaces then we apply them to a large class of semilinear stochastic partial differential equations spde and obtain the existence of unique maximal strong solutions in the sense of sde and pde under local lipschitz conditions moreover stochastic navier stokes equations are also investigated

wright functions as scale invariant solutions of the diffusion wave equation the time fractional diffusion wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivative by a fractional derivative of order 0 2 using the similarity method and the method of the laplace transform it is shown that the scale invariant solutions of the mixed problem of signalling type for the time fractional diffusion wave equation are given in terms of the wright function in the case 0 1 and in terms of the generalized wright function in the case 1 2 the reduced equation for the scale invariant solutions is given in terms of the caputo type modification of the

on compact representations of all pairs shortest path distance matrices let g be an unweighted and undirected graph of n nodes and let d be the nxn matrix storing the all pairs shortest path distances in g since d contains integers in n its plain storage takes n 2log n 1 bits however a simple counting argument shows that n 2 2 bits are necessary to store d in this paper we investigate the question of finding a succinct representation of d that requires o n 2 bits of storage and still supports constant time access to each of its entries this is asymptotically optimal in the worst case and far from the information theoretic lower bound by a multiplicative factor log 23 1 585 as a result o 1 bits per pairs of nodes in g are enough to retain constant time access to their shortest path distance we achieve this result by reducing the storage of d to the succinct storage of labeled trees and ternary sequences for which we properly adapt and orchestrate the use of known compressed data structures this approach can be easily and optimally extended to graphs whose edge weights are positive integers bounded by a constant value

succinct representation of labeled trees we give a representation for labeled ordered trees that supports labeled queries such as finding the i th ancestor of a node with a given label our representation is succinct namely the redundancy is small o of the optimal space for storing the tree this improves the representation of he et al isaac 2012 8 which is succinct only when the entropy of the labels is 1

a variational representation for certain functionals of brownian motion in this paper we show that the variational representation log ee f w inf v e 1 2 int 0 1 parallel v s parallel 2 ds f w int 0 cdot v s ds holds where w is a standard d dimensional brownian motion f is any bounded measurable function that maps c 0 1 mathbb r d into mathbb r and the infimum is over all processes v that are progressively measurable with respect to the augmentation of the filtration generated by w an application is made to a problem concerned with large deviations and an extension to unbounded functions is given

succinct representation of balanced parentheses and static trees we consider the implementation of abstract data types for the static objects binary tree rooted ordered tree and a balanced sequence of parentheses our representations use an amount of space within a lower order term of the information theoretic minimum and support in constant time a richer set of navigational operations than has previously been considered in similar work in the case of binary trees for instance we can move from a node to its left or right child or to the parent in constant time while retaining knowledge of the size of the subtree at which we are positioned the approach is applied to produce a succinct representation of planar graphs in which one can test adjacency in constant time

on mittag leffler distributions and related stochastic processes random variables with mittag leffler distribution can take values either in the set of non negative integers or in the positive real line they can be of two different types one type 1 heavy tailed with index 0 1 the other type 2 possessing all its moments we investigate various stochastic processes where they play a key role among which the discrete space time neveu branching process the discrete space continuous time neveu branching process the continuous space time neveu branching process csbp and renewal processes with rare events its relation to discrete or continuous self decomposability and branching processes with immigration is emphasized special attention will be paid to the neveu csbp for its connection with the bolthausen sznitman coalescent in this context and following a recent work of mohle 2015 a type 2 mittag leffler process turns out to be the siegmund dual to neveu s csbp block counting process arising in sampling from p d e t 0 further combinatorial developments of this model are investigated

freidlin wentzell s large deviations for stochastic evolution equations abstract we prove a freidlin wentzell large deviation principle for general stochastic evolution equations with small perturbation multiplicative noises in particular our general result can be used to deal with a large class of quasi linear stochastic partial differential equations such as stochastic porous medium equations and stochastic reaction diffusion equations with polynomial growth zero order term and p laplacian second order term

large deviations for infinite dimensional stochastic dynamical systems the large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation the construction and justification of the approximations can be onerous especially in the case where the process state is infinite dimensional in this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of brownian noise the approach is based on a variational representation for functionals of brownian motion proofs of large deviations properties are reduced to demonstrating basic qualitative properties existence uniqueness and tightness of certain perturbations of the original process

succinct data structure for dynamic trees with faster queries abstract navarro and sadakane 19 gave a dynamic succinct data structure for storing an ordinal tree the structure supports tree queries in either o log n log log n or o log n time and insertion or deletion of a single node in o log n time in this paper we improve the result of navarro and sadakane by reducing the time complexities of some queries e g degree and level ancestor from o log n to o log n log log n

euler schemes and large deviations for stochastic volterra equations with singular kernels in this paper an euler type approximation is constructed for stochastic volterra equation with singular kernels which provides an algorithm for numerical calculation then the large deviation estimates of small perturbation to equations of this type are obtained we finally apply them to sdes with the kernel of fractional brownian motion with hurst parameter h 0 1

a computational method for solving stochastic ito volterra integral equations based on stochastic operational matrix for generalized hat basis functions abstract in this paper a new computational method based on the generalized hat basis functions is proposed for solving stochastic ito volterra integral equations in this way a new stochastic operational matrix for generalized hat functions on the finite interval 0 t is obtained by using these basis functions and their stochastic operational matrix such problems can be transformed into linear lower triangular systems of algebraic equations which can be directly solved by forward substitution also the rate of convergence of the proposed method is considered and it has been shown that it is o 1 n 2 further in order to show the accuracy and reliability of the proposed method the new approach is compared with the block pulse functions method by some examples the obtained results reveal that the proposed method is more accurate and efficient in comparison with the block pule functions method

regularities for semilinear stochastic partial differential equations abstract in this paper we study the regularities of solutions to semilinear stochastic partial differential equations in general settings and prove that the solution can be smooth arbitrarily when the data is sufficiently regular as applications we also study several classes of semilinear stochastic partial differential equations on abstract wiener space complete riemannian manifold as well as bounded domain in euclidean space

lp theory of semi linear spdes on general measure spaces and applications abstract in general settings applying evolutional semigroup arguments we prove the existence and uniqueness of l p solutions to semi linear spdes of the type d u t x l u t x f t x u t d t k g k t x u t d w t k u 0 x u 0 x x e where l is an unbounded linear negative operator on l p e b w t k t 0 k 1 2 is a sequence of independent brownian motions and e b is a general measure space we also discuss the regularities of solutions in sobolev spaces moreover a time discretized approximation for above equation is proved to convergence in holder spaces as applications we study several classes of solutions for different types spdes on abstract wiener space and riemannian manifold

succinct representations of permutations and functions we investigate the problem of succinctly representing an arbitrary permutation p on 0 n 1 so that p k i can be computed quickly for any i and any positive or negative integer power k a representation taking 1 e nlgn o 1 bits suffices to compute arbitrary powers in constant time for any positive constant e 1 a representation taking the optimal lgn o n bits can be used to compute arbitrary powers in o lgn lglgn time we then consider the more general problem of succinctly representing an arbitrary function f n n so that f k i can be computed quickly for any i and any integer power k we give a representation that takes 1 e nlgn o 1 bits for any positive constant e 1 and computes arbitrary positive powers in constant time it can also be used to compute f k i for any negative integer k in optimal o 1 f k i time we place emphasis on the redundancy or the space beyond the information theoretic lower bound that the data structure uses in order to support operations efficiently a number of lower bounds have recently been shown on the redundancy of data structures these lower bounds confirm the space time optimality of some of our solutions furthermore the redundancy of one of our structures surpasses a recent lower bound by golynski golynski soda 2009 thus demonstrating the limitations of this lower bound

a collocation technique for solving nonlinear stochastic ito volterra integral equations a numerical method for solving nonlinear stochastic ito volterra equations is proposed the method is based on delta function df approximations the properties of dfs and their operational matrix of integration together with the newton cotes nodes are presented and utilized to reduce the problem to the solution of a nonlinear system of algebraic equations the method is computationally attractive and some numerical examples are provided to illustrate its high accuracy

path queries on functions abstract let f 1 n 1 n be a function and l 1 n 1 indicate a label assigned to each element of the domain we design several compact data structures that answer various kinds of summary queries on the labels of paths in f for example we can find either the minimum label in f k i for a given i and any k 0 in a given range k 1 k 2 or the minimum label in f k i for a given i and k 0 using n lg n n lg o n lg n bits and time o n the inverse ackermann function within similar space we can count in time o lg n lg lg n the number of labels within a range and report each element with such labels in o lg n lg lg n additional time several other tradeoffs and possible queries are considered such as selection top r queries and majorities finally we consider queries that allow us navigate on the graph of the function such as the nearest common successor of two elements or the nearest successor or predecessor of an element within a range of labels

a numerical method for solving m dimensional stochastic ito volterra integral equations by stochastic operational matrix the multidimensional ito volterra integral equations arise in many problems such as an exponential population growth model with several independent white noise sources in this paper we obtain a stochastic operational matrix of block pulse functions on interval 0 1 to solve m dimensional stochastic ito volterra integral equations by using block pulse functions and their stochastic operational matrix of integration m dimensional stochastic ito volterra integral equations can be reduced to a linear lower triangular system which can be directly solved by forward substitution we prove that the rate of convergence is o h furthermore a 95 confidence interval of the errors mean is made the results shows that the approximate solutions have a credible degree of accuracy

symmetrical martingale solutions of backward doubly stochastic volterra integral equations abstract this paper aims to study a new class of integral equations called backward doubly stochastic volterra integral equations bdsvies for short the notion of symmetrical martingale solutions sm solutions for short is introduced for bdsvies and the existence and uniqueness theorem for bdsvies in the sense of sm solutions is established

the wright functions as solutions of the time fractional diffusion equation we revisit the cauchy problem for the time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order email protected 0 2 by using the fourier laplace transforms the fundamentals solutions green functions are shown to be high transcendental functions of the wright type that can be interpreted as spatial probability density functions evolving in time with similarity properties we provide a general representation of these functions in terms of mellin barnes integrals useful for numerical computation

random access to grammar compressed strings and trees grammar based compression where one replaces a long string by a small context free grammar that generates the string is a simple and powerful paradigm that captures sometimes with slight reduction in efficiency many of the popular compression schemes including the lempel ziv family run length encoding byte pair encoding sequitur and re pair in this paper we present a novel grammar representation that allows efficient random access to any character or substring without decompressing the string let s be a string of length n compressed into a context free grammar mathcal s of size n we present two representations of mathcal s achieving o log n random access time and either o n cdot alpha k n construction time and space on the pointer machine model or o n construction time and space on the ram here alpha k n is the inverse of the k th row of ackermann s function our representations also efficiently support decompression of any substring in s we can decompres

large deviation for navier stokes equations with small stochastic perturbation we consider a navier stokes evolution equation with a small stochastic perturbation the unperturbed navier stokes equation approximates incompressible viscous fluid flows in a two dimensional bounded region with an adhesive boundary condition the small stochastic perturbation accounts for often ignored quantum effects and other sources of fluctuations the weak solution process for the stochastic navier stokes equation is constructed via a girsanov type transformation large deviation results in terms of wentzel freidlin estimates are obtained for both the infinite dimensional brownian motion and the weak solution process for the stochastic navier stokes equation

analytically solvable autocorrelation function for weakly correlated interevent times long term temporal correlations observed in event sequences of natural and social phenomena have been characterized by algebraically decaying autocorrelation functions such temporal correlations can be understood not only by heterogeneous interevent times iets but also by correlations between iets in contrast to the role of heterogeneous iets on the autocorrelation function little is known about the effects due to the correlations between iets to rigorously study these effects we derive an analytical form of the autocorrelation function for the arbitrary iet distribution in the case with weakly correlated iets where the farlie gumbel morgenstern copula is adopted for modeling the joint probability distribution function of two consecutive iets our analytical results are confirmed by numerical simulations for exponential and power law iet distributions for the power law case we find a tendency of the steeper decay of the autocorrelation function for the stronger correlation between iets our analytical approach enables us to better understand long term temporal correlations induced by the correlations between iets

orderly spanning trees with applications we introduce and study orderly spanning trees of plane graphs this algorithmic tool generalizes canonical orderings which exist only for triconnected plane graphs although not every plane graph admits an orderly spanning tree we provide an algorithm to compute an orderly pair for any connected planar graph g consisting of an embedded planar graph h isomorphic to g and an orderly spanning tree of h we also present several applications of orderly spanning trees 1 a new constructive proof for schnyder s realizer theorem 2 the first algorithm for computing an area optimal 2 visibility drawing of a planar graph and 3 the most compact known encoding of a planar graph with o 1 time query support all algorithms in this paper run in linear time

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

schilder theorem for the brownian motion on the diffeomorphism group of the circle abstract we prove a large deviation principle for flows associated to stochastic differential equations with non lipschitz coefficients as an application we establish a schilder theorem for the brownian motion on the group of diffeomorphisms of the circle

combined data structure for previous and next smaller values let a be a static array storing n elements from a totally ordered set we present a data structure of optimal size at most nlog 2 3 22 o n bits that allows us to answer the following queries on a in constant time without accessing a 1 previous smaller value queries where given an index i we wish to find the first index to the left of i where a is strictly smaller than at i and 2 next smaller value queries which search to the right of i as an additional bonus our data structure also allows one to answer a third kind of query given indices i

large deviation principle for stochastic evolution equations the large deviation principle obtained by freidlin and wentzell for measures associated with finite dimensional diffusions is extended to measures given by stochastic evolution equations with non additive random perturbations the proof of the main result is adopted from the priouret paper concerning finite dimensional diffusions exponential tail estimates for infinite dimensional stochastic convolutions are used as main tools

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

l p estimates for the uniform norm of solutions of quasilinear spde s in this paper we prove l p estimates p 2 for the uniform norm of the paths of solutions of quasilinear stochastic partial differential equations spde of parabolic type our method is based on a version of moser s iteration scheme developed by aronson and serrin in the context of non linear parabolic pde

succinct data structures for searchable partial sums with optimal worst case performance abstract the notion of succinct indexes can be dated back from the debut of jacobson s thesis 1988 14 and has triggered many results in the last decade in traditional indexing some given data are preprocessed so as to support online queries and updates on the data as efficiently as possible when succinctness is involved we are restricted to index the data using only an information theoretically minimum number of bits this paper concerns the succinct indexing schemes for a well studied problem called searchable partial sums sps in sps an array a of n non negative k bit integers is preprocessed so as to support online sum and search queries and possibly update operation of individual entry a succinct indexing scheme would allow only k n o k n bits to represent the array a the only known result is that when k 1 in this case it is known as the dynamic bit array problem we can support both queries in o log b n time and update in o b amortized time for any b with lg n lg lg n b n this paper shows that even for k o lg lg n we can index a succinctly such that both query and update operations can be supported using the same time complexities moreover the time for update becomes the worst case time furthermore the tradeoff between the query times and the update time is optimal as implied by pa trascu and demaine s lower bound result 2006 24 in general when k o lg u we show a lower bound of lg n lg lg n time for the search query irrespective of the update time this gives a tighter lower bound as compared to that of pa trascu and demaine s which is a consequence of the requirement of succinctness on the other hand we give a succinct index that can support sum in o log b n time search in o log b n time and update in o b time where min lg lg n lg lg u lg lg lg u lg n lg lg n the query times are optimal when b n this paper also extends the searchable partial sums with insert and delete operations and provides a succinct data structure for some cases

numerical approach for solving stochastic volterra fredholm integral equations by stochastic operational matrix in this paper we obtain stochastic operational matrix of block pulse functions on interval 0 1 to solve stochastic volterra fredholm integral equations by using block pulse functions and their stochastic operational matrix of integration the stochastic volterra fredholm integral equation can be reduced to a linear lower triangular system which can be directly solved by forward substitution we prove that the rate of convergence is o h furthermore the results show that the approximate solutions have a good degree of accuracy

renewal theory with fat tailed distributed sojourn times typical versus rare renewal processes with heavy tailed power law distributed sojourn times are commonly encountered in physical modeling and so typical fluctuations of observables of interest have been investigated in detail to describe rare events the rate function approach from large deviation theory does not hold and new tools must be considered here we investigate the large deviations of the number of renewals the forward and backward recurrence times the occupation time and the time interval straddling the observation time we show how non normalized densities describe these rare fluctuations and how moments of certain observables are obtained from these limiting laws numerical simulations illustrate our results showing the deviations from arcsine dynkin darling kac l evy and lamperti laws

fractional levy motions and related processes this paper investigates some properties of a class of random motions called fractional levy motions flms and their fractal time extension flm identifies with fractional brownian motion fbm sampled in fractal levy time this two parameter class of processes borrows hyperbolic temporal dependence to fbm and heavy tailedness to levy flights or motions lm it is shown that there exists a family of flms which shares with standard brownian motion bm the strict diffusivity property that the dispersion measured in terms of quantiles grows as time raised to the power processes from this class are critical in that they separate both sub superdiffusive flms and finite infinite variance motions related ornstein uhlenbeck and multiplicative processes are also briefly investigated flms are self similar but not levy stable in sharp contrast to brownian whether fractional or not and standard or fractional stable in the taqqu wolpert sense levy motions fsms stable processes are self similar as a result of levy stability however the converse is false there are motions with stationary increments which are self similar but not stable the fractional levy motion is one of them in the pure jump process class it turns out that all processes discussed so far bm fbm lm flm fsm are self similar with stationary increments we finally introduce a natural one parameter family of fractional processes for which a weaker notion of self similarity seems to hold i e self similarity of the unidimensional distributions it is fractal time brownian motion ft bm such a process is obtained as a weak limit of fractal time random walk models with 0 1 the tail exponent of the waiting times ft bm identifies with brownian motion now sampled in fractal inverse levy time this construction extends to fbm lm and flm we therefore introduce and study ft fbm ft lm and ft flm which are the fractal time extensions of fbm lm and flm these processes are not strongly self similar nor stable nor do they have stationary increments

volterra equations driven by semimartingales existence and uniqueness of solutions is established for stochastic volterra integral equations driven by right continuous semimartingales this resolves in the affirmative a conjecture of m berger and v mizel

efficient and compact representations of prefix codes most of the attention in statistical compression is given to the space used by the compressed sequence a problem completely solved with optimal prefix codes however in many applications the storage space used to represent the prefix code itself can be an issue in this paper we introduce and compare several techniques to store prefix codes let n be the sequence length and n be the alphabet size then a naive storage of an optimal prefix code uses mathcal o hspace 5ex left n log n right bits our first technique shows how to use mathcal o hspace 5ex left n log log n n right bits to store the optimal prefix code then we introduce an approximate technique that for any 0 takes mathcal o hspace 5ex left n log log 1 epsilon right bits to store a prefix code with an average codeword length within an additive epsilon of the minimum finally a second approximation takes for any constant c 1 vphantom sum r r mathcal o n 1 c log n bits to store a prefix code with an average codeword length at most c times the minimum in all cases our data structures allow encoding and decoding of any symbol in mathcal o hspace 5ex left 1 right time we experimentally compare our new techniques with the state of the art showing that we achieve sixfold to eightfold space reductions at the price of a slower encoding 2 5 8 times slower and decoding 12 24 times slower the approximations further reduce this space and improve the time significantly up to recovering the speed of classical implementations for a moderate penalty in the average code length as a byproduct we compare various heuristic approximate and optimal algorithms to generate length restricted codes showing that the optimal ones are clearly superior and practical enough to be implemented

succinct representations of planar maps this paper addresses the problem of representing the connectivity information of geometric objects using as little memory as possible as opposed to raw compression issues the focus here is on designing data structures that preserve the possibility of answering incidence queries in constant time we propose in particular the first optimal representations for 3 connected planar graphs and triangulations which are the most standard classes of graphs underlying meshes with spherical topology optimal means that these representations asymptotically match the respective entropy of the two classes namely 2 bits per edge for 3 connected planar graphs and 1 62 bits per triangle or equivalently 3 24 bits per vertex for triangulations these representations support adjacency queries between vertices and faces in constant time

an efficient computational method for solving nonlinear stochastic ito integral equations because of the nonlinearity closed form solutions of many important stochastic functional equations are virtually impossible to obtain thus numerical solutions are a viable alternative in this paper a new computational method based on the generalized hat basis functions together with their stochastic operational matrix of ito integration is proposed for solving nonlinear stochastic ito integral equations in large intervals in the proposed method a new technique for computing nonlinear terms in such problems is presented the main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved error analysis of the proposed method is investigated and also the efficiency of this method is shown on some concrete examples the obtained results reveal that the proposed method is very accurate and efficient as two useful applications the proposed method is applied to obtain approximate solutions of the stochastic population growth models and stochastic pendulum problem

on mittag leffler type functions in fractional evolution processes abstract we review a variety of fractional evolution processes so defined being governed by equations of fractional order whose solutions turn out to be related to mittag leffler type functions the chosen equations are the simplest of the fractional calculus and include the abel integral equations of the second kind which are relevant in typical inverse problems and the fractional differential equations which govern generalized relaxation and oscillation phenomena

application of operational matrices for solving system of linear stratonovich volterra integral equation abstract in this paper we give a brief review on block pulse functions bpfs and obtain operational matrix and stochastic operational matrix of integration based on bpfs then these operational matrices are used to solve system of linear stratonovich volterra integral equations by applying proposed method the system of linear stratonovich volterra integral equations reduce to system of linear algebraic equations which can be solved by a convenient numerical method also the error analysis is proved under several mild conditions we show the rate of convergence is o h finally by applying this method on two examples we demonstrate accuracy and efficiency of the proposed method all of the numerical calculation are performed on computer using a program written in matlab