statistical complexity measure of pseudorandom bit generators abstract pseudorandom number generators prng are extensively used in monte carlo simulations gambling machines and cryptography as substitutes of ideal random number generators rng each application imposes different statistical requirements to prngs as l ecuyer clearly states the main goal for monte carlo methods is to reproduce the statistical properties on which these methods are based whereas for gambling machines and cryptology observing the sequence of output values for some time should provide no practical advantage for predicting the forthcoming numbers better than by just guessing at random in accordance with different applications several statistical test suites have been developed to analyze the sequences generated by prngs in a recent paper a new statistical complexity measure phys lett a 311 2003 126 has been defined here we propose this measure as a randomness quantifier of a prngs the test is applied to three very well known and widely tested prngs available in the literature all of them are based on mathematical algorithms another prngs based on lorenz 3d chaotic dynamical system is also analyzed prngs based on chaos may be considered as a model for physical noise sources and important new results are recently reported all the design steps of this prng are described and each stage increase the prng randomness using different strategies it is shown that the mpr statistical complexity measure is capable to quantify this randomness improvement the prng based on the chaotic 3d lorenz dynamical system is also evaluated using traditional digital signal processing tools for comparison

powers of the euler class abstract let g be the topological group of orientation preserving homeomorphisms of the circle s 1 the classical euler class and its powers generate the cohomology of the classifying space b g in the transition from continuous g to discrete g the behavior of the powers of the euler class is altered significantly there are subgroups k of g for which the powers of the discrete euler class e n h 2 n b g z are non vanishing only up to a certain dimension in this work we determine the threshold dimension for certain groups of homeomorphisms of the circle our results are applied to the based mapping class groups m g and we show e n m g 0 n g but e n m g 0 n g at the threshold dimension n g the discrete euler class has torsion of order 2 g 2 g 1

on azema yor processes their optimal properties and the bachelier drawdown equation we study the class of azema yor processes defined from a gene ral semi martingale with a continuous running supremum process we show that they arise as unique strong solutions of the bachelier stochastic diff erential equation which we prove is equivalent to the drawdown equation solutions of the latter have the drawdown property they always stay above a given function of their past supre mum we then show that any process which satisfies the drawdow n property is in fact an azema yor process the proofs exploit group structure of the set of azema yor processes indexed by functions which we introduce secondly we study in detail azema yor martingales defined f rom a non negative local martingale converging to zero at infinity we establis h relations between

observability quadratic normal form for discrete time systems this note deals with quadratic observability normal form for nonlinear discrete time single input single output siso system first of all the main concept of quadratic equivalence with respect to the observability property is introduced for discrete time systems subsequently normal form structure for discrete time system is developed for system with unobservable linear approximation in one direction finally the effect of the so called resonant terms on the observer design and synchronization of chaotic systems is pointed out in an illustrative example

the smale conjecture for hyperbolic 3 manifolds isom m 3 diff m 3 the main result of this paper is if m is a closed hyperbolic 3 manifold then the inclusion of the isometry group isom m into the diffeomorphism group diff m is a homotopy equivalence

topological transversality holds in all dimensions on suppose m et x des sous varietes propres de y x a un microfibre normal il y a des sous ensembles fermes c d y et m est transversable a pres de c alors il y a une isotopie de m a support dans un voisinage donne de d c m x vers une sous variete transversale a pres de d

an iterated azema yor type embedding for finitely many marginals we solve the nn marginal skorokhod embedding problem for a continuous local martingale and a sequence of probability measures 1 n 1 n which are in convex order and satisfy an additional technical assumption our construction is explicit and is a multiple marginal generalization of the azema and yor in seminaire de probabilites xiii univ strasbourg strasbourg 1977 78 1979 90 115 springer solution in particular we recover the stopping boundaries obtained by brown hobson and rogers probab theory related fields 119 2001 558 578 and madan and yor bernoulli 8 2002 509 536 our technical assumption is necessary for the explicit embedding as demonstrated with a counterexample we discuss extensions to the general case giving details when n 3n 3 in our analysis we compute the law of the maximum at each of the nn stopping times this is used in henry labordere et al ann appl probab 26 2016 1 44 to show that the construction maximizes the distribution of the maximum among all solutions to the nn marginal skorokhod embedding problem the result has direct implications for robust pricing and hedging of lookback options

a stochastic representation theorem with applications to optimization and obstacle problems we study a new type of representation problem for optional processes with connections to singular control optimal stopping and dynamic allocation problems as an application we show how to solve a variant of skorohod s obstacle problem in the context of backward stochastic differential equations

a novel dynamic model of pseudo random number generator an interesting hierarchy of random number generators is introduced in this paper based on the review of random numbers characteristics and chaotic functions theory the main objective of this paper is to produce an ergodic dynamical system which can be implemented in random number generators in order to check the efficacy of pseudo random number generators based on this map we have carried out certain statistical tests on a series of numbers obtained from the introduced hierarchy the results of the tests were promising as the hierarchy passed the tests satisfactorily and offers a great capability to be employed in a pseudo random number generator

max plus decomposition of supermartingales and convex order application to american options and portfolio insurance we are concerned with a new type of supermartingale decomposition in the max plus algebra which essentially consists in expressing any supermartingale of class d as a conditional expectation of some running supremum process as an application we show how the max plus supermartingale decomposition allows in particular to solve the american optimal stopping problem without having to compute the option price some illustrative examples based on one dimensional diffusion processes are then provided another interesting application concerns the portfolio insurance hence based on the max plus martingale we solve in the paper an optimization problem whose aim is to find the best martingale dominating a given floor process on every intermediate date w r t the convex order on terminal values

homological stability for moduli spaces of high dimensional manifolds i s galatius was partially supported by nsf grants dms 1105058 and dms 1405001 the european research council erc under the european union s horizon 2020 research and innovation pro gramme grant agreement no 682922 as well as the danish national research foundation through the centre for symmetry and deformation dnrf92 and erc 682992 o randal williams was partially supported by the herchel smith fund and epsrc grant ep m027783 1

properties making a chaotic system a good pseudo random number generator we discuss the properties making a deterministic algorithm suitable to generate a pseudo random sequence of numbers high value of kolmogorov sinai entropy high dimensionality of the parent dynamical system and very large period of the generated sequence we propose the multidimensional anosov symplectic cat map as a pseudo random number generator we show what chaotic features of this map are useful for generating pseudo random numbers and investigate numerically which of them survive in the discrete state version of the map testing and comparisons with other generators are performed

algorithm for normal random numbers we propose a simple algorithm for generating normally distributed pseudorandom numbers the algorithm simulates n molecules that exchange energy among themselves following a simple stochastic rule we prove that the system is ergodic and that a maxwell like distribution that may be used as a source of normally distributed random deviates follows in the n limit the algorithm passes various performance tests including monte carlo simulation of a finite two dimensional ising model using wolff s algorithm it only requires four simple lines of computer code and is approximately ten times faster than the box muller algorithm s1063 651x 99 04409 8 pacs number s 02 70 lq 02 90 1p

logistic map a possible random number generator the logistic map is one of the simple systems exhibiting order to chaos transition in this work we have investigated the possibility of using the logistic map in the chaotic regime logmap for a pseudorandom number generator to this end we have performed certain statistical tests on the series of numbers obtained from the logmap we find that the logmap passes these tests satisfactorily and therefore it possesses many properties required of a pseudorandom number generator

homological stability and stable moduli of flat manifold bundles abstract we prove that the group homology of the diffeomorphism group of g s n s n int d 2 n as a discrete group is independent of g in a range provided that n 2 this answers the high dimensional version of a question posed by morita about surface diffeomorphism groups made discrete the stable homology is isomorphic to the homology of a certain infinite loop space related to the haefliger s classifying space of foliations one geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized mumford morita miller classes lift to a secondary r z invariants for flat g s n s n bundles provided g 0

a proof of the smale conjecture diff s3 0 4 the smale conjecture 9 is the assertion that the inclusion of the orthogonal group 0 4 into diff s3 the diffeomorphism group of the 3 sphere with the cw topology is a homotopy equivalence there are many equivalent forms of this conjecture some of which are listed in the appendix to this paper we shall prove

tendency towards maximum complexity in a nonequilibrium isolated system the time evolution equations of a simplified isolated ideal gas the open quotes tetrahedral close quotes gas are derived the dynamical behavior of the lopez ruiz endash mancini endash calbet complexity r lopez ruiz h l mancini and x calbet phys lett a 209 321 1995 is studied in this system in general it is shown that the complexity remains within the bounds of minimum and maximum complexity we find that there are certain restrictions when the isolated open quotes tetrahedral close quotes gas evolves towards equilibrium in addition to the well known increase in entropy the quantity called disequilibrium decreases monotonically with time furthermore the trajectories of the system in phase space approach the maximum complexity path as it evolves toward equilibrium

features of the extension of a statistical measure of complexity to continuous systems we discuss some aspects of the extension to continuous systems of a statistical measure of complexity introduced by l opez ruiz mancini and calbet phys lett a 209 321 1995 in general the extension of a magnitude from the discrete to the continuous case is not a trivial process and requires some kind of choice in the present study several possibilities appear available one of them is examined in detail some interesting properties desirable for any magnitude of complexity are discovered on this particular extension

hierarchy of chaotic maps with an invariant measure we give hierarchy of one parameter family x of maps at the interval 0 1 with an invariant measure using the measure we calculate kolmogorov sinai entropy or equivalently lyapunov characteristic exponent of these maps analytically where the results thus obtained have been approved with the numerical simulation in contrary to the usual one parameter family of maps such as logistic and tent maps these maps do not possess period doubling or period n tupling cascade bifurcation to chaos but they have single fixed point attractor for certain values of the parameter where they bifurcate directly to chaos without having period n tupling scenario exactly at those values of the parameter whose lyapunov characteristic exponent begins to be positive