spectrality of one dimensional self similar measures with consecutive digits abstract assume 0 1 and m is a prime let be the self similar measure defined by a 1 m j 0 m 1 1 a j a b we prove that l 2 contains an orthonormal basis of exponential functions if and only if 1 m k for some k n

on the spectra of a class of self affine measures on r2 abstract for i 1 2 let 0 i 1 for an expanding real matrix m d i a g 1 1 2 1 and a three element digit set d z 2 with cardinality d let m d be the self affine measure defined by m d 1 d d d m d m d let f 3 2 1 3 l 1 l 2 t l 1 l 2 n 0 l 1 l 2 2 and z m d x r 2 d d e 2 i d x 0 if a f 3 2 z m d a mod z 2 and x 0 1 2 d d e 2 i d x d 0 0 t then l 2 m d has an exponential orthonormal basis if and only if i 1 3 z 0 for i 1 2

spectral property of cantor measures with consecutive digits abstract we consider equally weighted cantor measures q b arising from iterated function systems of the form b 1 x i i 0 1 q 1 where q b we classify the q b so that they have infinitely many mutually orthogonal exponentials in l 2 q b in particular if q divides b the measures have a complete orthogonal exponential system and hence spectral measures improving the construction by dutkay et al 2009 3 we characterize all the maximal orthogonal sets when q divides b via a maximal mapping on the q adic tree in which all elements in are represented uniquely in finite b adic expansions and we can separate the maximal orthogonal sets into two types regular and irregular sets for a regular maximal orthogonal set we show that its completeness in l 2 q b is crucially determined by the certain growth rate of non zero digits in the tail of the b adic expansions of the elements furthermore we exhibit complete orthogonal exponentials with zero beurling dimensions these examples show that the technical condition in theorem 3 5 of dutkay et al 2011 4 cannot be removed for an irregular maximal orthogonal set we show that under some conditions its completeness is equivalent to that of the corresponding regularized mapping

maximum principle of state constrained optimal control governed by fluid dynamic systems this work deals with maximum principle of optimal control governed by some fluid dynamic systems which include three dimensional navier stokes equations some type of state constraint is involved

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

on the spectra of sierpinski type self affine measures abstract for a class of sierpinski type self affine measures on r 2 a characterization for all maximal orthogonal sets of exponentials is given by this characterization a sufficient condition for a maximal orthogonal set to be a basis of l 2 is given

on spectral n bernoulli measures abstract for 0 1 and n 1 an integer let be the self similar measure defined by i 0 n 1 1 n 1 i we prove that l 2 has an exponential orthonormal basis if and only if 1 q for some q 0 and n divides q the special case is the cantor measure with 1 2 k and n 2 16 which was proved recently to be the only spectral measure among the bernoulli convolutions with 0 1 4

the cardinality of orthogonal exponentials of planar self affine measures with three element digit sets abstract in this paper we consider the planar self affine measures m d generated by an expanding matrix m m 2 z and an integer digit set d 0 0 1 2 1 2 with 1 2 2 1 0 we show that if det m 3 z then the mutually orthogonal exponential functions in l 2 m d is finite and the exact maximal cardinality is given

on the orthogonal exponential functions of a class of planar self affine measures abstract for an expanding real matrix m 1 c 0 1 m 2 r and a digit set d 0 0 t 1 0 t 0 1 t let m d be the self affine measure generated by m and d in this paper we show that l 2 m d admits an infinite orthogonal set of exponential functions if and only if q p 1 r and c 1 for some positive integers p q r with p 3 z gcd p q 1 and q moreover if l 2 m d does not admit any infinite orthogonal set of exponential functions we estimate the number of orthogonal exponential functions in l 2 m d and give the exact maximal cardinality

on spectral cantor measures a probability measure in rd is called a spectral measure if it has an orthonormal basis consisting of exponentials in this paper we study spectral cantor measures we establish a large class of such measures and give a necessary and sufficient condition on the spectrum of a spectral cantor measure these results extend the studies by jorgensen and pedersen j anal math 75 1998 185 228 and strichartz j d analyse math 81 2000 209 238

some recent advances in theory and simulation of fractional diffusion processes to offer an insight into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of current interest i the well scaled passage to the limit from continuous time random walk under power law assumptions to space time fractional diffusion ii the asymptotic universality of the mittag leffler waiting time law in time fractional processes iii our method of parametric subordination for generating particle trajectories

uncoupled continuous time random walks solution and limiting behavior of the master equation a detailed study is presented for a large class of uncoupled continuous time random walks ctrws the master equation is solved for the mittag leffler survival probability the properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed finally some common objections found in the literature are thoroughly reviewed

tree structure of spectra of spectral self affine measures abstract for an integral self affine spectral measure if the zeros of its fourier transform are all integral vectors it is proven that any its spectrum has a tree structure for any subset with such tree structure a sufficient condition and a necessary condition for the subset to be a spectrum are given respectively applications are given to some known results as special cases

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

dynamical approach to levy processes we derive the diffusion process generated by a correlated dichotomous fluctuating variable y starting from a liouville like equation by means of a projection procedure this approach makes it possible to derive all statistical properties of the diffusion process from the correlation function of the dichotomous fluctuating variable f y t of special interest is that the distribution of the times of sojourn in the two states of the fluctuating process is proportional to d 2 f y t dt 2 furthermore in the special case where f y t has an inverse power law with the index b ranging from 0 to 1 thus making it nonintegrable we show analytically that the

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

spectral property of the bernoulli convolutions for 0 1 let be the bernoulli convolution associated with jorgensen and pedersen p jorgensen s pedersen dense analytic subspaces in fractal l2 spaces j anal math 75 1998 185 228 proved that if 1 q where q is an even integer then l2 has an exponential orthonormal basis we show that for any 0 1 l2 contains an infinite orthonormal set of exponential functions if and only if is the nth root of a fraction p q where p is an odd integer and q is an even integer

orthogonal exponential functions of self similar measures with consecutive digits in r abstract suppose that 0 1 and m 2 is an integer let m be the self similar measure defined by m 1 m j 0 m 1 m 1 j in this paper we prove that l 2 m contains an infinite orthonormal set of exponential functions if and only if q p 1 r for some p q r n with gcd p q 1 and gcd p m 1

spectral properties of a class of moran measures abstract let b b n n n and q q n n n be a sequence of positive integers respectively satisfying b n q n 1 let b q b 1 1 0 1 q 1 1 b 1 b 2 1 0 1 q 2 1 b q is called a moran measure which is a generalization of self affine measures in particular we call b q the n bernoulli convolution if b n b and q n n with 1 b r for all n n it is known 1 that b q is a spectral measure if q n divides b n for all n n in this paper we give a simpler proof to this result by developing the method given by dai 3 and simultaneously we construct a spectrum for the measure b q when q n divides b n for all n n moreover we give an equivalent condition for the n bernoulli convolution with 1 b r and a sufficient condition for the moran type measures b q with 1 b n z to admit an infinitely many orthogonal exponentials we also provide an equivalent characterization for the maximal orthogonal exponential sets of b q by defining a mixed tree mapping of the b q which is an important tool for us to characterize the maximality of orthogonal exponential set

short note on the emergence of fractional kinetics abstract in the present short note an idea is proposed to explain the emergence and the observation of processes in complex media that are driven by fractional non markovian master equations particle trajectories are assumed to be solely markovian and described by the continuous time random walk model but as a consequence of the complexity of the medium each trajectory is supposed to scale in time according to a particular random timescale the link from this framework to microscopic dynamics is discussed and the distribution of timescales is computed in particular when a stationary distribution is considered the timescale distribution is uniquely determined as a function related to the fundamental solution of the space time fractional diffusion equation in contrast when the non stationary case is considered the timescale distribution is no longer unique two distributions are here computed one related to the m wright mainardi function which is green s function of the time fractional diffusion equation and another related to the mittag leffler function which is the solution of the fractional relaxation equation

monte carlo simulation of uncoupled continuous time random walks yielding a stochastic solution of the space time fractional diffusion equation we present a numerical method for the monte carlo simulation of uncoupled continuous time random walks with a levy stable distribution of jumps in space and a mittag leffler distribution of waiting times and apply it to the stochastic solution of the cauchy problem for a partial differential equation with fractional derivatives both in space and in time the one parameter mittag leffler function is the natural survival probability leading to time fractional diffusion equations transformation methods for mittag leffler random variables were found later than the well known transformation method by chambers mallows and stuck for levy stable random variables and so far have not received as much attention nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the mittag leffler distribution combining the two methods we obtain an accurate approximation of space and time fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes

when does a bernoulli convolution admit a spectrum abstract in this paper we solve a long standing problem on bernoulli convolutions in particular we show that the bernoulli convolution with contraction rate 0 1 admits a spectrum if and only if is the reciprocal of an even integer

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

sierpinski type spectral self similar measures abstract we consider the sierpinski type self similar measure on r 2 with contraction ratio 0 1 we show that is a spectral measure if and only if 1 3 p for some integer p 0 a similar characterization for bernoulli convolution is due to dai 2 over which 1 2 p