quantal problems with partial algebraization of the spectrum we discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly solvable problems e g harmonic oscillator and all others the problems belonging to this class are distinguished by the fact that a part of the eigenvalues and eigenfunctions can be found algebraically but not the whole spectrum the reason explaining the existence of the quasi exactly solvable problems is a hidden dynamical symmetry present in the hamiltonian for one dimensional motion this hidden symmetry issl 2 r it is shown that other groups lead to a partial algebraization in multidimensional quantal problems in particular sl 2 r sl 2 r so 3 andsl 3 r are relevant to two dimensional motion inducing a class of quasi exactly solvable two dimensional hamiltonians typically they correspond to systems in a curved space but sometimes the curvature turns out to be zero graded algebras open the possibility of constructing quasi exactlysolvable hamiltonians acting on multicomponent wave functions for example with a non minimal superextension ofsl 2 r we get a hamiltonian describing the motion of a spinor particle

subdiffusive master equation with space dependent anomalous exponent and structural instability we derive the fractional master equation with space dependent anomalous exponent we analyze the asymptotic behavior of the corresponding lattice model both analytically and by monte carlo simulation we show that the subdiffusive fractional equations with constant anomalous exponent in a bounded domain 0 l are not structurally stable with respect to the nonhomogeneous variations of parameter in particular the gibbs boltzmann distribution is no longer the stationary solution of the fractional fokker planck equation whatever the space variation of the exponent might be we analyze the random distribution of in space and find that in the long time limit the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events we show that subdiffusive fractional equations with the nonuniform random distribution of anomalous exponent is an illustration of a black swan the low probability event of the small value of the anomalous exponent that completely dominates the long time behavior of subdiffusive systems

mittag leffler analysis ii application to the fractional heat equation abstract mittag leffler analysis is an infinite dimensional analysis with respect to non gaussian measures of mittag leffler type which generalizes the powerful theory of gaussian analysis and in particular white noise analysis in this paper we further develop the mittag leffler analysis by characterizing the convergent sequences in the distribution space moreover we provide an approximation of donsker s delta by square integrable functions then we apply the structures and techniques from mittag leffler analysis in order to show that a green s function to the time fractional heat equation can be constructed using generalized grey brownian motion ggbm by extending the fractional feynman kac formula from schneider moreover we analyse ggbm show its differentiability in a distributional sense and the existence of corresponding local times

on semi markov processes and their kolmogorov s integro differential equations abstract semi markov processes are a generalization of markov processes since the exponential distribution of time intervals is replaced with an arbitrary distribution this paper provides an integro differential form of the kolmogorov s backward equations for a large class of homogeneous semi markov processes having the form of an abstract volterra integro differential equation an equivalent evolutionary differential form of the equations is also provided fractional equations in the time variable are a particular case of our analysis weak limits of semi markov processes are also considered and their corresponding integro differential kolmogorov s equations are identified

quasi exactly solvable lie algebras of differential operators in two complex variables the authors completely classify all quasi exactly solvable lie algebras of first order differential operators in two complex variables applications to quasi exactly solvable quantum problems are indicated

solvable non markovian dynamic network non markovian processes are widespread in natural and human made systems yet explicit modeling and analysis of such systems is underdeveloped we consider a non markovian dynamic network with random link activation and deletion rlad and heavy tailed mittag leffler distribution for the interevent times we derive an analytically and computationally tractable system of kolmogorov like forward equations utilizing the caputo derivative for the probability of having a given number of active links in the network and solve them simulations for the rlad are also studied for power law interevent times and we show excellent agreement with the mittag leffler model this agreement holds even when the rlad network dynamics is coupled with the susceptible infected susceptible spreading dynamics thus the analytically solvable mittag leffler model provides an excellent approximation to the case when the network dynamics is characterized by power law distributed interevent times we further discuss possible generalizations of our result

new quasi exactly solvable hamiltonians in two dimensions quasi exactly solvable schrodinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods such operators lie in the enveloping algebra of a finite dimensional lie algebra of first order differential operators the hidden symmetry algebra in this paper we develop some general techniques for constructing quasi exactly solvable operators our methods are applied to provide a wide variety of new explicit two dimensional examples on both flat and curved spaces of quasi exactly solvable hamiltonians corresponding to both semisimple and more general classes of lie algebras

realizations of real low dimensional lie algebras using a new powerful technique based on the notion of megaideal we construct a complete set of inequivalent realizations of real lie algebras of dimension no greater than four in vector fields on a space of an arbitrary finite number of variables our classification amends and essentially generalizes earlier works on the subject

fractional cauchy problems on bounded domains fractional cauchy problems replace the usual first order time derivative by a fractional derivative this paper develops classical solutions and stochastic analogues for fractional cauchy problems in a bounded domain dr d with dirichlet boundary conditions stochastic solutions are constructed via an inverse stable subordi nator whose scaling index corresponds to the order of the fractional time derivative dirichlet problems corresponding to iterated brow nian motion in a bounded domain are then solved by establishing a correspondence with the case of a half derivative in time 1 introduction in this paper we extend the approach of meerschaert and scheffler 23 and meerschaert et al 24 to fractional cauchy problems on bounded domains our methods involve eigenfunction expansions killed markov processes and inverse stable subordinators in a recent related paper 7 we establish a connection between fractional cauchy problems with index 1 2 on an unbounded domain and iterated brownian motion ibm defined as zt b yt where b is a brownian motion with values in r d and y is an independent one dimensional brownian motion since ibm is also the stochastic solution to a cauchy problem involving a fourth order derivative in space 2 14 that paper also establishes a connection between certain higher order cauchy problems and their time fractional analogues more generally baeumer meerschaert and nane 7 shows a connection between fractional cauchy problems with 1 2 and higher order cauchy problems that involve the square of the generator in the present paper we

the use of moment space bounds for evaluating the performance of a nonlinear digital communication system certain moments of the output of a bandpass nonlinear system whose input is a stationary gaussian random process will be used to evaluate the performance i e the average probability of error of the system the main application of this procedure is the analysis of a frequency translating saturating satellite link the basic analytical technique to be employed will be the use of moment space bounds it will be shown that by using at most a two dimensional gaussian quadrature formula to perform a numerical integration the bounding procedure presented in this paper can be used in a relatively straightforward manner with virtually any bounded bandpass nonlinearity numerical results corresponding to a hard limiter with only an am am characteristic will be presented as well as results corresponding to a soft limiter with both am am and am pm characteristics in the latter model the effects of intersymbol interference will also be included however computationally the procedure is only feasible for a small number of interfering terms

a construction of transitively differential groups of degree 3 some general results on lie algebras of first order differential operators in n real variables with polynomial degree 3 coefficients are demonstrated an infinite family of specific lie algebras and the corresponding lie groups are described

levy mixing related to distributed order calculus subordinators and slow diffusions abstract the study of distributed order calculus usually concerns fractional derivatives of the form 0 1 u m d for some measure m eventually a probability measure in this paper an approach based on levy mixing is proposed non decreasing levy processes associated with levy triplets of the form a y b y d s y are considered and the parameter y is randomized by means of a probability measure the related subordinators are studied from different points of view some distributional properties are obtained and the interplay with inverse local times of markov processes is explored distributed order integro differential operators are introduced and adopted in order to write explicitly the governing equations of such processes an application to slow diffusions delayed brownian motion is discussed

low traffic limit and first passage times for a simple model of the continuous double auction we consider a simplified model of the continuous double auction where prices are integers varying from 1 to n with limit orders and market orders but quantity per order limited to a single share for this model the order process is equivalent to two m m 1 queues we study the behaviour of the auction in the low traffic limit where limit orders are immediately matched by market orders in this limit the distribution of prices can be computed exactly and gives a reasonable approximation of the price distribution when the ratio between the rate of order arrivals and the rate of order executions is below 1 2 this is further confirmed by the analysis of the first passage time in 1 or n

semi markov models and motion in heterogeneous media in this paper we study continuous time random walks such that the holding time in each state has a distribution depending on the state itself for such processes we provide integro differential backward and forward equations of volterra type exhibiting a position dependent convolution kernel particular attention is devoted to the case where the holding times have a power law decaying density whose exponent depends on the state itself which leads to variable order fractional equations a suitable limit yields a variable order fractional heat equation which models anomalous diffusions in heterogeneous media

mittag leffler analysis i construction and characterization abstract we construct an infinite dimensional analysis with respect to non gaussian measures of mittag leffler type which we call mittag leffler measures it turns out that the well known wick ordered polynomials in gaussian analysis cannot be generalized to this non gaussian case instead of using wick ordered polynomials we prove that a system of biorthogonal polynomials called appell system is applicable to the mittag leffler measures therefore we are able to introduce a test function and a distribution space as an application we construct donsker s delta in a non gaussian setting as a weak integral in the distribution space

a class of graded lie algebras of vector fields and first order differential operators finite dimensional lie algebras of polynomial vector fields on rn that contain the elements xi and xi xi for i 1 n were studied to any lie algebra l of this class an n valued n n matrix a and a set of special elements s 1 n are associated it is proven that the pair a s necessarily satisfies two properties conversely to any pair a s satisfying those two properties is associated a lie algebra l a s such that l a s is maximal in the class of all l with matrix a and special elements s for the lie algebras l a s the possible extensions to first order differential operators and its modules of c functions are discussed 1994 american institute of physics

escape of a uniform random walk from an interval we study the first passage properties of a random walk in the unit interval in which the length of a single step is uniformly distributed over the finite range a a for a of the order of one the exit probabilities to each edge of the interval and the exit time from the interval exhibit anomalous properties stemming from the change in the minimum number of steps to escape the interval as a function of the starting point as a decreases first passage properties approach those of continuum diffusion but non diffusive effects remain because of residual discreteness effects

achievable rates for nonlinear volterra channels random coding theorems and achievable rates for nonlinear additive noise channels are presented modeling the channel s nonlinear behavior as a causal stationary volterra system upper bounds on the average error probability are obtained for maximum likelihood and weakly typical set decoding the proposed bounds are deduced by treating correct decoding regions as subspaces of high concentration measure and deploying exponential martingale inequalities due to the union bound effect and the i i d assumption imposed on the codewords components the deduced exponents constitute only lower bounds on the true random coding exponents of nonlinear channels cubic and fourth order nonlinearities are used as examples to illustrate the relation of the random coding exponents and achievable rates with respect to the channel s parameters

fractional diffusion in inhomogeneous media starting from the continuous time random walk ctrw scheme with the space dependent waiting time probability density function pdf we obtain the time fractional diffusion equation with varying in space fractional order of time derivative as an example we study the evolution of a composite system consisting of two separate regions with different subdiffusion exponents and demonstrate the effects of non trivial drift and subdiffusion whose laws are changed in the course of time

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

error rate estimates in digital communication over a nonlinear channel with memory the principal representations of krein are known to provide very tight upper and lower bounds on the error probability in digital communication over a linear channel with intersymbol interference these bounds depend on the signal to noise ratio the amount of intersymbol interference and the number of moments of the random interference variable the same technique can be applied to a nonlinear channel with memory however it is difficult to calculate these moments when the channel is nonlinear and has memory in this paper we describe a method for computing the moments for the case of an arbitrary nonlinear channel with finite memory the method of krein is then used to evaluate the error probability from the moments obtained we demonstrate the technique with two examples the model of the nonlinear channel consists of a linear filter in cascade with a memoryless nonlinearity followed by another linear filter if a sufficient number of moments are used we obtain accurate estimates of the true error probability

regenerative systems on the real line on considere les systemes regeneratifs sur la droite reelle et on etudie leur structure on etablit une correspondance entre les systemes regeneratifs stationnaires et les processus additifs markoviens qui sont stationnaires dans la composante spatiale

maximal subalgebras of vector fields for equivariant quantizations the elaboration of new quantization methods has recently developed the interest in the study of subalgebras of the lie algebra of polynomial vector fields over a euclidean space in this framework these subalgebras define maximal equivariance conditions that one can impose on a linear bijection between observables that are polynomial in the momenta and differential operators here we determine which finite dimensional graded lie subalgebras are maximal in order to characterize these we make use of results of guillemin singer and sternberg and kobayashi and nagano