efficient parallel rooting of complex polynomials on the unit circle an efficient method is presented for rooting complex polynomial equations of any order where the root space is restricted to the unit circle the method restates the evaluation of polynomials as a recursive algorithm involving only additions by means of the bilinear transformation the straight line uniformly spaced recursive polynomials evaluation method of a h nuttall 1987 is extended to the unit circle where the root positions are determined by thresholding general coefficient transformations are provided along with a comparison to the horner method

fokker planck equations for nonlinear dynamical systems driven by non gaussian levy processes the fokker planck equations describe time evolution of probability densities of stochastic dynamical systems and are thus widely used to quantify random phenomena such as uncertainty propagation for dynamical systems driven by non gaussian levy processes however it is difficult to obtain explicit forms of fokker planck equations because the adjoint operators of the associated infinitesimal generators usually do not have exact formulation in the present paper fokker planck equations are derived for nonlinear stochastic differential equations with non gaussian levy processes a few examples are presented to illustrate the method

fractional fokker planck equation for nonlinear stochastic differential equations driven by non gaussian levy stable noises the fokker planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by gaussian noises however there are both theoretical and empirical reasons to consider similar equations driven by strongly non gaussian noises in particular they yield strongly non gaussian anomalous diffusion which seems to be relevant in different domains of physics in this paper we therefore derive a fractional fokker planck equation for the probability distribution of particles whose motion is governed by a nonlinear langevin type equation which is driven by a levy stable noise rather than a gaussian we obtain in fact a general result for a markovian forcing we also discuss the existence and uniqueness of the solution of the fractional fokker planck equation

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

a fast parallel algorithm for determining all roots of a polynomial with real roots given a polynomial p z of degree n with m bit integer coefficients and an integer mu the problem of determining all its roots with error less than 2 mu is considered it is shown that this problem is in the class nc if p z has all real roots some very interesting properties of a sturm sequence of a polynomial with distinct real roots are proved and used in the design of a fast parallel algorithm for this problem using newton identities and a novel numerical integration scheme for evaluating a contour integral to high precision this algorithm determines good approximations to the linear factors of p z

root finding with threshold circuits we show that for any constant d complex roots of degree d univariate rational or gaussian rational polynomials given by a list of coefficients in binary can be computed to a given accuracy by a uniform tc 0 algorithm a uniform family of constant depth polynomial size threshold circuits the basic idea is to compute the inverse function of the polynomial by a power series we also discuss an application to the theory v tc 0 of bounded arithmetic

stochastic solution of fractional fokker planck equations with space time dependent coefficients abstract this paper develops solutions of fractional fokker planck equations describing subdiffusion of probability densities of stochastic dynamical systems driven by non gaussian levy processes with space time dependent drift diffusion and jump coefficients thus significantly extends magdziarz and zorawik s result in 14 fractional fokker planck equation describing subdiffusion is solved by our result in full generality from perspective of stochastic representation

stochastic foundations of fractional dynamics it is shown that fractional diffusion equations arise very naturally as the limiting dynamic equations of all continuous time random walks with decoupled temporal and spatial memories and with either temporal or spatial scale invariance fractal walks thus enlarging their stochastic foundations hitherto restricted to a particular case of fractal walk r hilfer and l anton phys rev e 51 r848 1995

optimal and nearly optimal algorithms for approximating polynomial zeros abstract we substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p x our new algorithms save both boolean and arithmetic sequential time versus the previous best algorithms of schonhage 1 pan 2 and neff and reif 3 in parallel nc implementation we dramatically decrease the number of processors versus the parallel algorithm of neff 4 which was the only nc algorithm known for this problem so far specifically under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p x to the unit disc x x 1 our algorithms which promise to be practically effective approximate all the zeros of p x within the absolute error bound 2 b by using order of n arithmetic operations and order of b n n 2 boolean bitwise operations in both cases up to within polylogarithmic factors the algorithms allow their optimal work preserving nc parallelization so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or b n n 2 boolean processors all the cited bounds on the computational complexity are within polylogarithmic factors from the optimum in terms of n and b under both arithmetic and boolean models of computation in the boolean case under the additional realistic assumption that n o b

levy anomalous diffusion and fractional fokker planck equation we demonstrate that the fokker planck equation can be generalized into a fractional fokker planck equation i e an equation which includes fractional space differentiations in order to encompass the wide class of anomalous diffusions due to a levy stable stochastic forcing a precise determination of this equation is obtained by substituting a levy stable source to the classical gaussian one in the langevin equation this yields not only the anomalous diffusion coefficient but a non trivial fractional operator which corresponds to the possible asymmetry of the levy stable source both of them cannot be obtained by scaling arguments the mono scaling behaviors of the fractional fokker planck equation and of its solutions are analysed and a generalization of the einstein relation for the anomalous diffusion coefficient is obtained this generalization yields a straightforward physical interpretation of the parameters of levy stable distributions furthermore with the help of important examples we show the applicability of the fractional fokker planck equation in physics

complexity of computations with matrices and polynomials this paper reviews the complexity of polynomial and matrix computations as well as their various correlations to each other and some major techniques for the design of algebraic and numerical algorithms

anomalous diffusion resulting from strongly asymmetric random walks for leftward and rightward flights pl t t 2m and pr t t 2h with mfih we calculate the dependence of the variance exponent g s 2 t g on the pdf exponents n m and h we find thatg is determined by the two smaller of the three pdf exponents and in some cases by only the smallest a pdf with decay exponent less

a storage efficient method for solving banded toeplitz systems a method is presented for solving the banded toeplitz system tx y by decomposing t into its asymptotic upper and lower triangular factors which are banded and toeplitz and a rank p correction matrix where p is the bandwidth of t this way of representing t requires only o p sup 2 words of storage and allows computation of x in o 2np operations a similar method is presented for the case in which t is bi infinite and y is zero outside a finite region

parameter estimation for the fractional schrodinger equation using bayesian method in this paper the fractional schrodinger equation is studied the bayesian method is put forward to estimate some relevant parameters of the equation results show that the estimated values can fit well with the exact solution the varying initial values and maximum iterations have little effect on the estimated results it indicates that the bayesian method is efficient for the multi parameter estimation for the fractional schrodinger equation this method can also be used to estimate parameters for the fractional schrodinger equation in other potential field

instability in the solution of banded toeplitz systems the author considers some algorithms for the solution of banded toeplitz systems and calculates certain elements of the solution first and then the remaining elements by forward or backward substitution it is shown that for symmetric matrices this method is almost always highly unstable a numerical example is given to support the argument

multidimensional advection and fractional dispersion extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second order equation this is revealed by the solutions of the equations unlike the gaussian the most general stable vector cannot be generated with an atomistic measure on the coordinate axes a random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle subsets are symmetric stable vectors that have previously appeared in the literature and the well known multidimensional brownian motion a multidimensional fractional differential operator is defined in the process

lower bounds in a parallel model without bit operations we define a natural and realistic model of parallel computation called the pram model without bit operations it is like the usual pram model the main difference being that no bit operations are provided it encompasses virtually all known parallel algorithms for weighted combinatorial optimization and algebraic problems in this model we prove that for some large enough constant b the mincost flow problem for graphs with n vertices cannot be solved deterministically or with randomization in sqrt n b expected time using 2 sqrt n b processors this is so even if we restrict every cost and capacity to be an integer nonnegative if it is a capacity of bitlength at most an for some large enough constant a a similar lower bound is also proved for the max flow problem it follows that these problems cannot be solved in our model deterministically or with randomization in omega n c expected time with 2 omega n c processors where c is an appropriate positive constant and n is the total bitlength of the input since these problems were known to be p complete this provides concrete support for the belief that p completeness implies high parallel complexity and for the p not nc conjecture our lower bounds also extend to the pram model with limited bit operations which provides instructions for parity and left or right shift by one bit our proof is based on basic algebraic geometry so we investigate if the algebrogeometric approach could also work for the p versus nc problem our results support this possibility and a close analysis of the limitation of our technique in this context suggests that such a proof of p not nc should somehow use geometric invariant theory in a deep way

comment on fokker planck equations for nonlinear dynamical systems driven by non gaussian levy processes j math phys 53 072701 2012 in the proof of theorem 1 in sun and duan j math phys 53 072701 2012 the authors use the taylor expansion to represent an arbitrary infinitely differentiable function with compact support which is incorrect we prove that although the derivation is incorrect the statement of theorem 1 remains valid if we add certain additional assumptions

local lipschitz continuity of the inverse of the fractional p laplacian holder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains let p 1 s 0 1 and rn an arbitrary bounded open set in the first part we consider the inverse s p p s 1 of the fractional p laplace operator p s with the dirichlet boundary condition we show that in the singular case p 1 2 the operator s p is locally lipschitz continuous on l and that global lipschitz continuity cannot be achieved we use this result to show that in the case n sp if 2nn 2s p 2 0 q npn sp 2 and are small constants then the nonlinear problem psu u qu u npn sp 2u hin u 0 on rn has at least one weak solution in the second part of the paper we prove that the operator p s generates a nonlinear submarkovian semigroup ss p t t 0 on l2 if p 2 and sp 0 such that for every t 0 and u v lq q 2 we have ss p t u ss p t v l c s t s u v lq s where s s and s are explicit constants depending only on n s p q

levy fights variations on a theme abstract levy s original ideas on limit distributions for sums of identically distributed random variables with infinite second moments are applied to a variety of topics including generating nondifferentiable lacunary series the riemann hypothesis and turbulent diffusion

computing matrix eigenvalues and polynomial zeros where the output is real surprisingly simple corollaries from the courant fischer minimax characterization theorem enable us to devise a very effective algorithm for the evaluation of a set s interleaving the set e of the eigenvalues of an n x n real symmetric tridiagonal rst matrix tn as well as a point that splits e into two subsets of comparable cardinalities furthermore we extend this algorithm so as to approximate all the n eigenvalues of tn at nearly optimal sequential and parallel cost that is at the cost of staying within polylogarithmic factors from the straightforward lower bounds the resulting improvement of the known processor bound in nc algorithms for the rst eigenproblem is roughly by factor n our approach extends the previous works m ben or and p tiwari j complexity 6 1990 pp 417 442 and m ben or et al siam j comput 17 1988 pp 1081 1092 for the approximation of the zeros of a polynomial having only real zeros and our algorithm leads to an alternative and simplified derivation of the known record parallel and sequential complexity estimates for the latter problem

path integral solutions of the governing equation of sdes excited by levy white noise abstract in this paper the probability density functions pdfs of scalar stochastic differential equations sdes subject to stable levy white noise are investigated the path integral pi method is extended to solve one dimensional space fractional fokker planck kolmogorov fpk equations which are the governing equations corresponded to scalar sdes excited by stable levy white noise first we derive a short time solution of the one dimensional space fractional fpk equation which is used in the chapman kolmogorov smoluchowski cks equation to obtain the pi solution then the accuracy of the pi solution is analyzed theoretically in terms of its characteristic function our results demonstrate that the pi method has a higher accuracy than the first order finite difference method for one step iteration in time finally several illustrative examples are carried out in detail to verify the feasibility and effectiveness of the pi method for solving one dimensional space fractional fpk equations we find that the pi solution agrees well with the exact solution or the monte carlo one

generalized local and nonlocal master equations for some stochastic processes in this paper we present a study on generalized local and nonlocal equations for some stochastic processes by considering the net flux change in a region determined by the transition probability we derive the master equation to describe the evolution of the probability density function some examples such as classical fokker planck equations models for levy process and stochastic coagulation equations are provided as illustrations a particular application is a consistent derivation of coupled dynamical systems for spatially inhomogeneous stochastic coagulation processes

optimal portfolio and consumption with habit formation in a jump diffusion market this paper studies optimal portfolio and consumption selection with habit formation in a jump diffusions incomplete market in continuous time the stochastic maximum principle for jump processes is applied to solve habit forming utility maximization problem we transform this problem into the case not involving habit formation in mechanically then the solution in the state feedback form is given the relationship between maximum principle and dynamic programming is employed to get the expression of the relative risk aversion rra coefficient and its distribution finally for a special case the stationary mean of the rra coefficient is obtained and the numerical experiment indicates our model with jump diffusions is better than the model in 1 to resolve the equity premium puzzle in a way

evidence for non gaussian scaling behavior in heterogeneous sedimentary formations vertical and horizontal fluctuations in permeability and porosity in sedimentary formations are analyzed and are found to be consistent with scaling models based on levy stable probability distributions the approach avoids the assumption of gaussian behavior and is supported by evidence from horizontal and vertical well logs and from permeability measurements on a sandstone outcrop and segments of core from a heterogeneous formation the incremental values in these measurement sequences are accurately modeled as having levy stable distributions the width of the distribution of increments depends on the spatial scale in a manner consistent with scaling behavior the width of the distribution is smaller for horizontal increments than for vertical increments reflecting the reduced variability in the horizontal direction the scaling parameters are in the range associated with antipersistence and are roughly the same magnitude in the vertical and horizontal directions the relationships between different physical properties are briefly studied and it is suggested that they be quantified through off diagonal terms in a multivariate levy width matrix simulations designed to reproduce the observed statistical features are also described these results have some fundamental implications as levy stable distributions require a different set of statistical tools and theoretical methods compared to finite variance distributions

computation of equilibriain noncooperative games this paper presents algorithms for finding equilibria of mixed strategy in multistage noncooperative games of incomplete information like probabilistic blindfold chess where at every opportunity a player can perform different moves with some probability these algorithms accept input games in extensive form our main result is an algorithm for computing aeguential equilibrium which is the most widely accepted notion of equilibrium for mixed strategies of noncooperative probabilistic games in mainstream economic game theory previously there were no known algorithms for computing sequential equilibria strategies except for the special case of single stage games the computational aspects of passage from a recursive presentation of a game to its extensive form are also discussed for nontrivial inputs the concatenation of this procedure with the equilibrium computation is time intensive but has low spatial requirements given a recursively represented game with a position space bound s n and a log space computable next move relation we can compute an example mixed strategy satisfying the sequential equilibria condition all in space bound o s n 2 furthermore in space o s n 3 we can compute the connected components of mixed strategies satisfying sequential equilibria

derivation of fokker planck equations for stochastic systems under excitation of multiplicative non gaussian white noise abstract fokker planck equations describe time evolution of probability densities of stochastic dynamical systems and play an important role in quantifying propagation and evolution of uncertainty although fokker planck equations can be written explicitly for systems excited by gaussian white noise they have remained unknown in general for systems excited by multiplicative non gaussian white noise in this paper we derive explicit forms of fokker planck equations for one dimensional systems modeled by marcus stochastic differential equations under multiplicative non gaussian white noise as examples to illustrate the theoretical results the derived formula is used to obtain fokker planck equations for nonlinear dynamical systems under excitation of i stable white noise ii combined gaussian and poisson white noise respectively

langevin equations for continuous time levy flights we consider the combined effects of a power law l evy step distribution characterized by the step index f and a power law waiting time distribution characterized by the time index g on the long time behavior of a random walker the main point of our analysis is a formulation in terms of coupled langevin equations which allows in a natural way for the inclusion of external force fields in the anomalous case for f2 and g1 the dynamic exponent z locks onto the ratio f g drawing on recent results on l evy flights in the presence of a random force field we also find that this result is independent of the presence of weak quenched disorder for d below the critical dimension mathit d mathit c 2f 2 the disorder is relevant corresponding to a nontrivial fixed point for the force correlation function