on the significance of absolutely continuous invariant measures abstract let be a mapping from an interval i into itself for a large class of such maps there exist ergodic measures invariant under which are absolutely continuous with respect to lebesgue measure and others which are continuous but not absolutely continuous the aim of this paper is to deal with the question which of these ergodic measures best describes the real dynamics of to do this we shall model the dynamics of by a markov chain which reflects the perturbations inherent in experimental work or the truncation error in computations when is expanding it is the absolutely continuous invariant measure that acts as a global attractor this explains the observed effect that the absolutely continuous invariant measure is the one that appears in experimental and numerical work

deformed oscillator algebra approach of some quantum superintegrable lissajous systems on the sphere and of their rational extensions we extend the construction of 2d superintegrable hamiltonians with separation of variables in spherical coordinates using combinations of shift ladder and supercharge operators to models involving rational extensions of the two parameter lissajous systems on the sphere these new families of superintegrable systems with integrals of arbitrary order are connected with jacobi exceptional orthogonal polynomials of type i or ii and supersymmetric quantum mechanics moreover we present an algebraic derivation of the degenerate energy spectrum for the one and two parameter lissajous systems and the rationally extended models these results are based on finitely generated polynomial algebras casimir operators realizations as deformed oscillator algebras and finite dimensional unitary representations such results have only been established so far for 2d superintegrable systems separable in cartesian coordinates which are related to a class of polynomial algebras that display a simpler structure we also point out how the structure function of these deformed oscillator algebras is directly related with the generalized heisenberg algebras spanned by the nonpolynomial integrals

parity effect of the initial capital based on parrondo s games and the quantum interpretation abstract in our previous study zhu et al quantum game interpretation for a special case of parrondo s paradox physica a 390 2011 579 the capital dependent parrondo s game where one game depends on the capital modulus m 4 was shown not to have a definite stationary probability distribution and that payoffs of the game depended on the parity of the initial capital this paper presents a generalization of these results to even m greater than 4 an intuitive explanation for producing this phenomenon is that the discrete time markov chain of the game is divided into two completely unrelated inner and outer rings the process taking the inner ring or outer ring of the game is determined by the initial capital of parity and then a win or loss of the game is determined quantum game theory is used to further analyze the phenomenon the results show that the explanation of the game corresponding to a stationary probability distribution is that the probability of the initial capital has reached parity

a revision of the lyapunov exponent in 1d quadratic maps abstract only misiurewicz points strictly have a lyapunov exponent positive measure in one dimensional 1d quadratic maps hence we represent here the lyapunov exponent of all the misiurewicz points which were found and inventoried by us in a former work as a result of this representation an infinity of alignments of misiurewicz points are obtained and two types of jumps in the values of are found

rigorous approximation of stationary measures and convergence to equilibrium for iterated function systems we study the problem of the rigorous computation of the stationary measure and of the rate of convergence to equilibrium of an iterated function system described by a stochastic mixture of two or more dynamical systems that are either all uniformly expanding on the interval either all contracting in the expanding case the associated transfer operators satisfy a lasota yorke inequality we show how to compute a rigorous approximations of the stationary measure in the l 1 norm and an estimate for the rate of convergence the rigorous computation requires a computer aided proof of the contraction of the transfer operators for the maps and we show that this property propagates to the transfer operators of the ifs in the contracting case we perform a rigorous approximation of the stationary measure in the wasserstein kantorovich distance and rate of convergence using the same functional analytic approach we show that a finite computation can produce a realistic computation of all contraction rates for the whole parameter space we conclude with a description of the implementation and numerical experiments

generalized deformed algebras f a1 and their applications generalized deformation of a1 is proposed some generalized deformed algebras f a1 and their unitary representations are discussed in detail f a1 involves not only lie algebras su 2 su 1 1 e2 and boson algebra as special cases but also quantum algebras suq 2 suq 1 1 q boson algebra and a large class of other nonlinear algebras it is found that these algebras can be mapped from one type to another by generalized deforming functionals in some cases possible applications in physical problems are outlined some examples are also given as illustrations

a direct formulation for sparse pca using semidefinite programming given a covariance matrix we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination this problem arises in the decomposition of a covariance matrix into sparse factors or sparse principal component analysis pca and has wide applications ranging from biology to finance we use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix where cardinality is constrained and derive a semidefinite programming based relaxation for our problem we also discuss nesterov s smooth minimization technique applied to the semidefinite program arising in the semidefinite relaxation of the sparse pca problem the method has complexity o n 4 sqrt log n epsilon where n is the size of the underlying covariance matrix and epsilon is the desired absolute accuracy on the optimal value of the problem

why computers like lebesgue measure let 0 1 0 1 be a transformation which has an absolutely continuous invariant measure let be a realistic deterministic model for we prove that if has long computer trajectories either periodic or non periodic then these computer trajectories have histograms which approach the density of for a large class of piecewise linear transformations we prove the existence of long periodic trajectories

quantum algebra deforming maps clebsch gordan coefficients coproducts r and u matrices quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts map induced coproducts are connected to the usual ones by similarity transformations u that may be expressed either in terms of clebsch gordan coefficients or in a universal operator form the product of two such u matrices yields the r matrix for a fixed value of the spectral parameter which bears on the yang baxterization of u as well as r all this is explicitly illustrated for the tensor product 1 2 j of su 2 q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a two parameter manifold some observations are made on the general structure of u and r matrices and of coproduct manifolds based on the solutions of hierarchies of partial difference equations applications of deforming maps and u matrices to the physics of spin chains are outlined

an observation of quadratic algebra dual family of nonlinear coherent states and their non classical properties in the generalized isotonic oscillator in this paper we construct nonlinear coherent states for the generalized isotonic oscillator and study their non classical properties in detail by transforming the deformed ladder operators suitably which generate the quadratic algebra we obtain heisenberg algebra from the algebra we define two non unitary and one unitary displacement type operators while the action of one of the non unitary type operators reproduces the original nonlinear coherent states the other one fails to produce a new set of nonlinear coherent states dual pair we show that these dual states are not normalizable for the nonlinear coherent states we evaluate the parameter a3 and examine the non classical nature of the states through quadratic and amplitude squared squeezing effect further we derive analytical formula for the p function q function and the wigner function for the nonlinear coherent states all of them confirm the non classicality of the nonlinear coherent states in addition to the above we obtain the

phase retrieval via wirtinger flow theory and algorithms we study the problem of recovering the phase from magnitude measurements specifically we wish to reconstruct a complex valued signal boldsymbol x in mathbb c n about which we have phaseless samples of the form y r left langle boldsymbol a r boldsymbol x rangle right 2 r 1 ldots m knowledge of the phase of these samples would yield a linear system this paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm in a nutshell this algorithm starts with a careful initialization obtained by means of a spectral method and then refines this initial estimate by iteratively applying novel update rules which have low computational complexity much like in a gradient descent scheme the main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements indeed the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources in theory a variation on this scheme leads to a near linear time algorithm for a physically realizable model based on coded diffraction patterns we illustrate the effectiveness of our methods with various experiments on image data underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval

phaseless near field far field transformation utilizing combinations of probe signals phaseless near field far field transformations based on nonconvex optimization suffer from local stationary points in order to avoid the corresponding suboptimal solutions the incorporation of combinations of probe signals into the cost functional is proposed and investigated the nonconvex optimization based on the wirtinger calculus relies on forward and adjoint radiation operator evaluations via the fast irregular antenna field transformation algorithm fiafta and is performed by a memory limited quasi newton method this allows the solution of large scale problems with utilization of the full flexibility of the fiafta in terms of measurement probe correction and arbitrary sample locations results for synthetic and measured near field data reveal significant improvements when the knowledge of probe signal combinations is utilized especially in applications with a large number of unknowns

randomly chosen chaotic maps can give rise to nearly ordered behavior abstract parrondo s paradox j m r parrondo g p harmer d abbott new paradoxical games based on brownian ratchets phys rev lett 85 2000 5226 5229 see also o e percus j k percus can two wrongs make a right coin tossing games and parrondo s paradox math intelligencer 24 3 2002 68 72 states that two losing gambling games when combined one after the other either deterministically or randomly can result in a winning game that is a losing game followed by a losing game a winning game inspired by this paradox a recent study j almeida d peralta salas m romera can two chaotic systems give rise to order physica d 200 2005 124 132 asked an analogous question in discrete time dynamical system can two chaotic systems give rise to order namely can they be combined into another dynamical system which does not behave chaotically numerical evidence is provided in j almeida d peralta salas m romera can two chaotic systems give rise to order physica d 200 2005 124 132 that two chaotic quadratic maps when composed with each other create a new dynamical system which has a stable period orbit the question of what happens in the case of random composition of maps is posed in j almeida d peralta salas m romera can two chaotic systems give rise to order physica d 200 2005 124 132 but left unanswered in this note we present an example of a dynamical system where at each iteration a map is chosen in a probabilistic manner from a collection of chaotic maps the resulting random map is proved to have an infinite absolutely continuous invariant measure acim with spikes at two points from this we show that the dynamics behaves in a nearly ordered manner when the foregoing maps are applied one after the other deterministically as in o e percus j k percus can two wrongs make a right coin tossing games and parrondo s paradox math intelligencer 24 3 2002 68 72 the resulting composed map has a periodic orbit which is stable

adaptive estimation of a quadratic functional by model selection we consider the problem of estimating s 2 when s belongs to some separable hilbert space and one observes the gaussian process y t langles t rangle sigmal t for all t epsilon mathbb h where l is some gaussian isonormal process this framework allows us in particular to consider the classical gaussian sequence model for which mathbb h l 2 mathbb n and l t sum lambda geq1 t lambda varepsilon lambda where varepsilon lambda lambda geq1 is a sequence of i i d standard normal variables our approach consists in considering some at most countable families of finite dimensional linear subspaces of mathbb h the models and then using model selection via some conveniently penalized least squares criterion to build new estimators of s 2 we prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s depending on the family of models from which they are built in particular in the context of the gaussian sequence model a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles ellipsoids l p bodies or besov bodies we take special care to describe the conditions under which the penalized estimator is efficient when the level of noise sigma tends to zero our construction is an alternative to the one by efroimovich and low for hyperrectangles and provides new results otherwise

parrondo effect in quantum coin toss simulations game a game b game c parrondo s games follow this basic structure where a and b are losing games and c is a winning game a phenomenon called parrondo s paradox these games can take on a wider class of definitions and exhibit these paradoxical results in this paper we show three paradoxical cases i the successive tossing of a single fair quantum coin gives a biased result a previously known result ii the random tossing of two quantum coins each with successive biased expectations gives an average random walk position of approximately zero iii the sequential periodic tossing of two quantum coins each with successive negative biased expectations gives an average random walk with positive expectation using these results we then propose a protocol for identifying and classifying quantum operations that span the same hilbert space for a two level quantum system

alternative hamiltonian quantization the authors take a standard hamiltonian quadratic in a set of creation and annihilation operators and demonstrate an alternative quantization by quommutators i e a set of rules for transposition of these operators different from the those of the usual canonical commutation relations which yields a new spectrum of a type reminiscent of logarithmic trajectories

sparse phase retrieval via truncated amplitude flow this paper develops a novel algorithm termed sparse truncated amplitude flow sparta to reconstruct a sparse signal from a small number of magnitude only measurements it deals with what is also known as sparse phase retrieval pr which is np hard in general and emerges in many science and engineering applications upon formulating sparse pr as an amplitude based nonconvex optimization task sparta works iteratively in two stages in stage one the support of the underlying sparse signal is recovered using an analytically well justified rule and subsequently a sparse orthogonality promoting initialization is obtained via power iterations restricted on the support and in the second stage the initialization is successively refined by means of hard thresholding based gradient type iterations sparta is a simple yet effective scalable and fast sparse pr solver on the theoretical side for any n dimensional k sparse k ll n signal boldsymbol x with minimum in modulus nonzero entries on the order of 1 sqrt k vert boldsymbol x vert 2 sparta recovers the signal exactly up to a global unimodular constant from about k 2 log n random gaussian measurements with high probability furthermore sparta incurs computational complexity on the order of k 2n log n with total runtime proportional to the time required to read the data which improves upon the state of the art by at least a factor of k finally sparta is robust against additive noise of bounded support extensive numerical tests corroborate markedly improved recovery performance and speedups of sparta relative to existing alternatives

dynamic parrondo s paradox abstract this paper is focused on studying parrondo s paradox in non linear dynamics specifically how the periodic combination of the individual maps f and g can give rise to chaos or order we construct dynamical systems exhibiting the paradox for several notions of chaos derived from topological dynamics the effect of altering the order of the combination i e considering f g or g f is analyzed as well as the robustness of the parrondo effect under small perturbations conditions for avoiding parrondian dynamics are also obtained placing special emphasis on the notion of chaos given by turbulence

high dimensional analysis of semidefinite relaxations for sparse principal components principal component analysis pca is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix however pca is well known to behave poorly in the large p small n setting in which the problem dimension p is comparable to or larger than the sample size n this paper studies pca in this high dimensional regime but under the additional assumption that the maximal eigenvector is sparse say with at most k nonzero components we consider a spiked covariance model in which a base matrix is perturbed by adding a k sparse maximal eigenvector and we analyze two computationally tractable methods for recovering the support set of this maximal eigenvector as follows a a simple diagonal thresholding method which transitions from success to failure as a function of the rescaled sample size dia n p k n k 2 log p k and b a more sophisticated semidefinite programming sdp relaxation which succeeds once the rescaled sample size sdp n p k n klog p k is larger than a critical threshold in addition we prove that no method including the best method which has exponential time complexity can succeed in recovering the support if the order parameter sdp n p k is below a threshold our results thus highlight an interesting trade off between computational and statistical efficiency in high dimensional inference

a report on the nonlinear squeezed states and their non classical properties of a generalized isotonic oscillator we construct nonlinear squeezed states of a generalized isotonic oscillator potential we demonstrate the non existence of a dual counterpart of nonlinear squeezed states in this system we investigate the statistical properties exhibited by the squeezed states in particular mandel s parameter second order correlation function photon number distributions and parameter a3 in detail we also examine the quadrature and amplitude squared squeezing effects finally we derive the expression for the s parameterized quasi probability distribution function of these states all this information about the system is new to the literature

generalized deformed su 2 algebra a generalized deformed algebra suphi 2 characterized by a structure function phi is obtained the usual su 2 and suq 2 algebras correspond to specific choices of the structure function phi the action of the generators of the algebra on the relevant basis vectors as well as the eigenvalues of the casimir operator are easily obtained possible applications in improving phenomenological models are discussed

nonlinear lie algebra and ladder operators for orbital angular momentum based on the nonlinear lie algebra a new framework is put forward to treat the problem of angular momentum from this framework the ladder operators for the orbital angular momentum can be solved as simultaneous eigenvectors of the lie derivations ad l2 and ad l2 in an operator space the ladder operators for solvable central potentials can also be found in a united way

phaseequal convex phase retrieval via alternating direction method of multipliers the problem of estimating an unknown complex signal from its magnitude only measurements is a classical problem known as phase retrieval this problem arises naturally because in some applications it is difficult or costly to measure the phase of the measurements in this paper we propose a new convex formulation referred to as phaseequal for solving the phase retrieval problem similar to phasemax t goldstein and c studer 2018 by goldstein and studer iee tit 2018 phaseequal works in the natural signal space and thus is computationally efficient we then extend phaseequal to the sparse signal case with the proposed convex formulation termed as compressed phaseequal different from existing convex compressed phase retrieval methods the proposed compressed phaseequal formulation does not involve any regularization parameter and thus is free of the parameter tuning issue which is always tricky in practice order wise recovery conditions for phaseequal and its sparse version i e compressed phaseequal are analyzed our theoretical results show that phaseequal resp compressed phaseequal achieves perfect recovery with mathcal o n resp mathcal o k text log frac n k magnitude measurements provided that a well correlated reference vector is available where k and n denote the number of nonzero entries in the complex sparse signal and the dimension of the signal respectively simulation results are provided to illustrate the effectiveness of the proposed methods

occurrence of complementary processes in parrondo s paradox abstract parrondo s paradox involves two losing processes producing a winning outcome we analyze the paradox with an original and novel method in which we start with one process and seek to construct a complementary process to achieve the paradox we then derive a general condition for the classical parrondo game to have a complementary process numerical simulation predicts that approximately two thirds of such losing games satisfy the required condition this suggests the common occurrence of the paradox indicative of many potentially undiscovered applications in real life scenarios involving stochastic processes

constructive approximations to densities invariant under nonexpanding transformations let 0 1 0 1 be a piecewise monotonic nonexpanding map which has an invariant densityg and is topologically conjugate to a piecewise monotonic expanding map where the conjugacy is absolutely continuous an effective computable method is presented for approximatingg

nonlinear deformations of su 2 and su 1 1 generalizing witten s algebra nonlinear deformations of su 2 and su 1 1 involving two deforming functions f j0 and g j0 are considered for g j0 1 they reduce to some algebras first studied by polychronakos 1990 and rocek 1991 spatial emphasis is laid on the case where g j0 is a linear function of j0 it is shown that for any lambda 2 3 there exist lambda 1 parameter algebras that are deformations of su 2 or su 1 1 respectively and for which f j0 is a polynomial of degree lambda for lambda 2 such algebras are equivalent to witten s 1990 first deformation of su 2 or su 1 1 for any lambda the spectrum of j0 is exponential instead of linear as in the case where g j0 1

annealed and quenched limit theorems for random expanding dynamical systems in this paper we investigate annealed and quenched limit theorems for random expanding dynamical systems making use of functional analytic techniques and more probabilistic arguments with martingales we prove annealed versions of a central limit theorem a large deviation principle a local limit theorem and an almost sure invariance principle we also discuss the quenched central limit theorem dynamical borel cantelli lemmas erdos renyi laws and concentration inequalities

on the number of invariant measures for higher dimensional chaotic transformations lets be a bounded region inr n and letp s l i 1 m be a partition ofs into a finite number of closed subsets having piecewisec2 boundaries of finite n 1 dimensional measure let s s be piecewisec2 onp and expanding in the sense that there exists 0 1 such that for anyi 1 2 m dt i 1 wheredt i 1 is the derivative matrix oft i 1 and is the euclidean matrix norm we prove that for some classes of such mappings for example jabtonski transformations or convexity preserving transformations the number of crossing points constitutes a bound for the number of ergodic absolutely continuous invariant measures we give examples showing that in general the simple bound of one dimensional dynamics cannot be generalized to higher dimensions in fact we show that it is possible to construct piecewise expandingc2 transformations on a fixed partition with a finite number of elements but which have an arbitrarily large number of ergodic absolutely continuous invariant measures

the coupling effect of the process sequence and the parity of the initial capital on parrondo s games abstract based on the original parrondo s game and on the case where game a and game b are played randomly with modulo m 4 the processes of the game are divided into odd and even numbered plays where the probability of playing game a in odd numbers is 1 and the probability of playing game a in even numbers is 2 by using the discrete time markov chain we find that the stationary probability distribution and mathematical expectation are not definite when 1 2 while they are definite when 1 2 meanwhile we perform a more in depth analysis according to the residue values divided by an integer n that is 1 2 3 n 1 0 we divide the process of the game into 1 2 3 n 1 n times where the probability of playing game a at each time is i i 1 2 n 1 n the general conclusions we obtain through analysis are 1 when the modulo m is odd whatever odd or even number n is and whatever value i is the stationary probability distribution is definite and the profit of the game does not depend on the initial value and 2 when the modulo m is even if n is odd then whatever value i is the stationary probability distribution is definite if n is even 1 2 n 1 n must be satisfied and then the stationary probability distribution is definite otherwise the stationary probability distribution has infinite solutions and the profit of the game depends on the initial value

higher dimensional point transformations and asymptotic measures for cellular automata abstract the evolution of a cellular automata is modelled by a higher dimensional point transformation on configuration space the transformation reflects the interaction of neighbours and has a probabilistic interpretation for certain point transformations the asymptotic measure of sets of configurations can be computed by using computer orbits

collapsing effects in numerical simulation of a class of chaotic dynamical systems and random mappings with a single attracting centre computer simulations of dynamical systems contain discretizations where finite machine arithmetic replaces continuum state spaces in some circumstances complicated theoretical behaviour has a tendency to collapse to trivial and degenerate behaviour as a result of discretizations characteristics of such collapsing effects often seem to depend on the corresponding discretization only in a random way we describe a procedure to construct a stochastic process with similar statistical characteristics results of computer modelling are discussed

periodic sequences of simple maps can support chaos abstract in this paper we explore the parrondo s paradox when several dynamically simple maps are combined in a periodic way producing chaotic dynamics we show that the paradox is not commutative that is it depends on the way that the maps are iterated we also see that the paradox happens more frequently when the number of maps that we iterate increases

toward a mathematical holographic principle in work started in 17 and continued in this paper our objective is to study selectors of multivalued functions which have interesting dynamical properties such as possessing absolutely continuous invariant measures we specify the graph of a multivalued function by means of lower and upper boundary maps tau 1 and tau 2 on these boundary maps we define a position dependent random map r p tau 1 tau 2 p 1 p which at each time step moves the point x to tau 1 x with probability p x and to tau 2 x with probability 1 p x under general conditions for each choice of p r p possesses an absolutely continuous invariant measure with invariant density f p let varvec tau be a selector which has invariant density function f one of our objectives is to study conditions under which p x exists such that r p has f as its invariant density function when this is the case the long term statistical dynamical behavior of a selector can be represented by the long term statistical behavior of a random map on the boundaries of g we refer to such a result as a mathematical holographic principle we present examples and study the relationship between the invariant densities attainable by classes of selectors and the random maps based on the boundaries and show that under certain conditions the extreme points of the invariant densities for selectors are achieved by bang bang random maps that is random maps for which p x in 0 1

absolutely continuous invariant measures for random maps with position dependent probabilities abstract a random map is discrete time dynamical system in which one of a number of transformations is randomly selected and applied at each iteration of the process usually the map k is chosen from a finite collection of maps with constant probability p k in this note we allow the p k s to be functions of position in this case the random map cannot be considered to be a skew product the main result provides a sufficient condition for the existence of an absolutely continuous invariant measure for position dependent random maps on 0 1 geometrical and topological properties of sets of absolutely continuous invariant measures attainable by means of position dependent random maps are studied theoretically and numerically

deformation of the askey wilson algebra with three generators a most general algebra with three generators and with ladder and duality properties gaw 3 is considered it is shown that the obtained algebra corresponds to p q deformation of the askey wilson algebra for the new algebra gaw 3 the ladder representation is constructed and the overlap problem is discussed

effects of phase space discretization on the long time behavior of dynamical systems abstract we investigate the effects that roundoff errors have on discrete time dynamical systems computer generated orbits terminate in loops whose lengths scale with the machine precision in a universal way we present arguments to explain this fact

collapsing of chaos in one dimensional maps in their numerical investigation of the family of one dimensional maps f x d 1 2jxj where 2 diamond et al p diamond et al physica d 86 1999 559 571 have observed the surprising numerical phenomenon that a large fraction of initial conditions chosen at random eventually wind up at 1 a repelling fixed point this is a numerical artifact because the continuous maps are chaotic and almost every true trajectory can be shown to be dense in 1 1 the goal of this paper is to extend and resolve this obvious contradiction we model the numerical simulation with a randomly selected map while they used 27 bit precision in computing f we prove for our model that this numerical artifact persists for an arbitrary high numerical prevision the fraction of initial points eventually winding up at 1 remains bounded away from 0 for every numerical precision 2000 elsevier science b v all rights reserved

attainable densities for random maps abstract we consider a random map t t where 1 2 k is a collection of maps of an interval and p 1 p 2 p k is a collection of the corresponding position dependent probabilities that is p k x 0 for k 1 2 k and k 1 k p k x 1 at each step the random map t moves the point x to k x with probability p k x for a fixed collection of maps t can have many different invariant probability density functions depending on the choice of the weighting probabilities most of the results in this paper concern random maps where is a family of piecewise linear semi markov maps we investigate properties of the set of invariant probability density functions of t that are attainable by allowing the probabilities in to vary in a certain class of functions we prove that the set of all attainable densities can be determined algorithmically we also study the duality between random maps generated by transformations and random maps constructed from a collection of their inverse branches such representation may be of greater interest in view of new methods of computing entropy w slomczynski j kwapien k yczkowski entropy computing via integration over fractal measures chaos 10 2000 180 188

approach of spherical harmonics to the representation of the deformed su 1 1 algebra the m shifting generators of su 2 algebra together with a pair of l shifting ladder symmetry operators have been used in the space of all spherical harmonics ylm in order to introduce a new set of operators expressing the transitions between them it is shown that the space of spherical harmonics whose l 2m or l 2m is given presents negative and positive irreducible representations of a deformed su 1 1 algebra respectively these internal symmetries also suggest new algebraic methods to construct the spherical harmonics in the framework of the spectrum generating algebras

synchronization of random linear maps abstract we study synchronization of random one dimensional linear maps for which the lyapunov exponent can be calculated exactly certain aspects of the dynamics of these maps are explained using their relation with a random walk we confirm that the lyapunov exponent changes sign at the complete synchronization transition we also consider partial synchronization of nonidentical systems it turns out that the way partial synchronization manifests depends on the type of differences in lyapunov exponent or in contraction points between the systems the crossover from partial to complete synchronization is also examined

exponential decay of correlations for random lasota yorke maps we consider random piecewise smooth piecewise invertible maps mainly on the interval but also in higher dimensions we assume that on the average and possibly without any stochastic uniformity i the maps expand distances ii do not have too many pieces iii do not have too large a distortion and iv are strongly mixing we assume it no markov property we prove that as in the classical case of the iteration of a fixed piecewise expanding map of the interval we have it exponential decay of random correlations our proof builds on the one given by c liverani for deterministic mixing and piecewise expanding interval maps

quantum game interpretation for a special case of parrondo s paradox abstract by using the discrete markov chain method parrondo s paradox is studied by means of theoretical analysis and computer simulation built on the case of game ab played in alternation with modulus m 4 we find that such a case does not have a definite stationary probability distribution and that payoffs of the game depend on the parity of the initial capital besides this paper reveals the phenomenon that processing in order produces non deterministic results while a random process produces deterministic results the quantum game method is used in a further study the results show that the explanation of the game corresponding to a stationary probability distribution is that the probability of the initial capital has reached parity

compactness of eigenvectors for certain families of non negative matrices conditions are given which guarantee that the normalized left eigenvectors of certain classes of non negative matrices of unbounded dimension interpreted as functions of 0 1 form precompact sets in l1 these eigenvectors are uniformly bounded above and in some cases uniformly bounded below

strange attractors that are not chaotic abstract it is shown that in certain types of dynamical systems it is possible to have attractors which are strange but not chaotic here we use the word strange to refer to the geometry or shape of the attracting set while the word chaotic refers to the dynamics of orbits on the attractor in particular the exponential divergence of nearby trajectories we first give examples for which it can be demonstrated that there is a strange nonchaotic attractor these examples apply to a class of maps which model nonlinear oscillators continuous time which are externally driven at two incommensurate frequencies it is then shown that such attractore are persistent under perturbations which preserve the original system type i e there are two incommensurate external driving frequencies this suggests that for systems of the typw which we have considered nonchaotic strange attractors may be expected to occur for a finite interval of parameter values on the other hand when small perturbations which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observed to be converted to a periodic or chaotic orbit thus we conjecture that in general continuous time systems flows which are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space

discrete time ratchets the fokker planck equation and parrondo s paradox parrondo s games manifest the apparent paradox where losing strategies can be combined to win and have generated significant multidisciplinary interest in the literature here we review two recent approaches based on the fokker planck equation that rigorously establish the connection between parrondo s games and a physical model known as the flashing brownian ratchet this gives rise to a new set of parrondo s games of which the original games are a special case for the first time we perform a complete analysis of the new games via a discrete time markov chain analysis producing winning rate equations and an exploration of the parameter space where the paradoxical behaviour occurs

phase retrieval using alternating minimization phase retrieval problems involve solving linear equations but with missing sign or phase for complex numbers information more than four decades after it was first proposed the seminal error reduction algorithm of gerchberg and saxton and fienup is still the popular choice for solving many variants of this problem the algorithm is based on alternating minimization i e it alternates between estimating the missing phase information and the candidate solution despite its wide usage in practice no global convergence guarantees for this algorithm are known in this paper we show that a resampling variant of this approach converges geometrically to the solution of one such problem finding a vector bf x from bf y bf a where bf y vert bf a t bf x vert and vert bf z vert denotes a vector of element wise magnitudes of bf z under the assumption that bf a is gaussian empirically we demonstrate that alternating minimization performs similar to recently proposed convex techniques for this problem which are based on lifting to a convex matrix problem in sample complexity and robustness to noise however it is much more efficient and can scale to large problems analytically for a resampling version of alternating minimization we show geometric convergence to the solution and sample complexity that is off by log factors from obvious lower bounds we also establish close to optimal scaling for the case when the unknown vector is sparse our work represents the first theoretical guarantee for alternating minimization albeit with resampling for any variant of phase retrieval problems in the non convex setting

absolutely continuous invariant measures for random non uniformly expanding maps we prove existence of at most denumerable many absolutely continuous invariant probability measures for random one dimensional dynamical systems with asymptotic expansion if the rate of expansion lyapunov exponents is bounded away from zero we obtain finitely many ergodic absolutely continuous invariant probability measures describing the asymptotics of almost every point we also prove a similar result for higher dimensional random non uniformly expanding dynamical systems the results are consequences of the construction of such measures for skew products with essentially arbitrary base dynamics and asymptotic expansion along the fibers in both cases our method deals with either critical o singular points for the random maps

optimal rates of convergence for noisy sparse phase retrieval via thresholded wirtinger flow this paper considers the noisy sparse phase retrieval problem recovering a sparse signal mathbf x in mathbb r p from noisy quadratic measurements y j mathbf a j mathbf x 2 varepsilon j j 1 ldots m with independent sub exponential noise varepsilon j the goals are to understand the effect of the sparsity of mathbf x on the estimation precision and to construct a computationally feasible estimator to achieve the optimal rates adaptively inspired by the wirtinger flow ieee trans inform theory61 2015 1985 2007 proposed for non sparse and noiseless phase retrieval a novel thresholded gradient descent algorithm is proposed and it is shown to adaptively achieve the minimax optimal rates of convergence over a wide range of sparsity levels when the mathbf a j s are independent standard gaussian random vectors provided that the sample size is sufficiently large compared to the sparsity of mathbf x

pseudo orbit shadowing in the family of tent maps on montre que les applications en tente ont la propriete d ombrage pour presque tous les parametres s mais elles n ont pas cette propriete pour un ensemble dense non denombrable des parametres on montre que pour une application tente chaque pseudo orbite peut etre approchee par une orbite actuelle d une application en tente avec une boucle peut etre legerement plus grande

statistical properties of discretizations of a class of chaotic dynamical systems computer simulations of dynamical systems contain discretizations where finite machine arithmetic replaces continuum state spaces in some circumstances complicated theoretical behavior has a tendency to collapse to trivial and degenerate behavior as a result of discretizations various statistical estimators associated with such collapsing effects often seem to depend on the corresponding discretization in a random way behavior of some statistical properties of collapse is discussed results of computer modeling of mappings x 1 1 2x l x 0 1 l 2 are presented

singular perturbations of piecewise monotonic maps of the interval let t 0 1 0 1 be a piecewise monotonic c2 and expanding map in computing an orbit i x0 i 0 we model the roundoff error at each iteration by a singular perturbation i e xn 1 xn w wherew is a random variable taking on discrete values in an interval e e the main result proves that this process admits an absolutely continuous invariant measure which approaches the absolutely continuous measure invariant under the deterministic map t as the precision of computation e 0

a switching scheme for synthesizing attractors of dissipative chaotic systems in this paper a periodic parameter switching scheme is proposed for synthesizing a large class of hyperbolic attractors of continuous time and autonomous dissipative chaotic systems depending linearly on a single real bifurcation parameter it is illustrated by numerical simulations that a wide range of hyperbolic attractors can be obtained by this new scheme the scheme can also be considered as an effective way for control and anticontrol of chaos

generalized deformed oscillator and nonlinear algebras the harmonic oscillator is deformed arbitrarily and the properties of the deformed oscillator algebra are studied the eigenstates and eigenvalues of the deformed oscillator are calculated this generalized deformed oscillator algebra is used to create nonlinear deformation of the classical su 2 algebra the deformed boson realizations of an arbitrary nonlinear su 2 algebra are calculated the algebra of the q deformed oscillator is a special case of the generalized deformed oscillator algebra

undersampled sparse phase retrieval via majorization minimization in the undersampled phase retrieval problem the goal is to recover an n dimensional complex valued signal from only m intensity measurements without phase information this inverse system is not only nonconvex but also underdetermined in this paper we propose to exploit the sparsity in the original signal and develop two low complexity algorithms with superior performance based on the majorization minimization framework the proposed algorithms are preferred to existing benchmark methods since at each iteration a simple convex surrogate problem is solved with a closed form solution that monotonically decreases the objective function value when the unknown signal is sparse in the standard basis the first algorithm c prime can produce a stationary point of the corresponding nonconvex phase retrieval problem when the unknown signal is not sparse in the standard basis the second algorithm sc prime can find a coordinatewise stationary point of the more challenging phase retrieval problem through sparse coding experimental results validate that the proposed algorithms have higher successful recovery rate and less normalized mean square error than existing up to date methods under the same setting

phase retrieval via reweighted amplitude flow this paper deals with finding an n dimensional solution boldsymbol x to a system of quadratic equations of the form y i langle boldsymbol a i boldsymbol x rangle 2 for 1 leq i leq m which is also known as the generalized phase retrieval problem for this np hard problem a novel approach is developed for minimizing the amplitude based least squares empirical loss which starts with a weighted maximal correlation initialization obtainable through a few power or lanczos iterations followed by successive refinements based on a sequence of iteratively reweighted gradient iterations the two stages initialization and gradient flow distinguish themselves from prior contributions by the inclusion of a fresh re weighting regularization procedure for certain random measurement models the novel scheme is shown to be able to recover the true solution boldsymbol x in time proportional to reading the data lbrace boldsymbol a i y i rbrace 1 leq i leq m this holds with high probability and without extra assumption on the signal vector boldsymbol x to be recovered provided that the number m of equations is some constant c 0 times the number n of unknowns in the signal vector namely m cn empirically the upshots of this contribution are first almost text 100 perfect signal recovery in the high dimensional say n geq 2000 regime given only an information theoretic limit number of noiseless equations namely m 2n 1 in the real gaussian case and second nearly optimal statistical accuracy in the presence of additive noise of bounded support finally substantial numerical tests using both synthetic data and real images corroborate markedly improved recovery performance and computational efficiency of the novel scheme relative to the state of the art approaches

a nonconvex splitting method for symmetric nonnegative matrix factorization convergence analysis and optimality symmetric nonnegative matrix factorization symnmf has important applications in data analytics problems such as document clustering community detection and image segmentation in this paper we propose a novel nonconvex variable splitting method for solving symnmf the proposed algorithm is guaranteed to converge to the set of karush kuhn tucker kkt points of the nonconvex symnmf problem furthermore it achieves a global sublinear convergence rate we also show that the algorithm can be efficiently implemented in parallel further sufficient conditions are provided that guarantee the global and local optimality of the obtained solutions extensive numerical results performed on both synthetic and real datasets suggest that the proposed algorithm converges quickly to a local minimum solution

the exact hausdorff dimension for a class of fractal functions in recent years a number of authors an incomplete list is 1 2 3 6 7 9 11 12 131 have considered a class of real functions whose graphs are in general fractal sets in r2 in this paper we give sufficient conditions for the fractal and hausdorff dimensions to be equal for a certain subclass of fractal functions the sets we consider are examples of self affine fractals generated using iterated function systems i f s falconer s has shown that for almost all such sets the fractal and hausdorff dimensions are equal and he gives a formula for the common dimension due originally to moran s these results however give no information about individual fractal functions in this paper we extend moran s original method and show that if certain conditions on the i f s are satisfied then the two dimensions are equal kono 111 and bedford 121 considered special cases of the subclass of fractal functions that we will introduce bedford and urbanski 13 use a nonlinear setting to present conditions for the equality of hausdorff and fractal dimension however their criteria are based on measure theoretic characterizations and the use of the concept of generalized pressure our criterion on the other hand is based on the underlying geometry of the attractor and is easier to verify we will show this on two specific examples which are more general than the self ahine functions presented in 13

what good are numerical simulations of chaotic dynamical systems abstract numerical simulations of mathematical models can suggest that the models are chaotic for example one can compute an orbit and its associated finite time lyapunov exponents and these computed exponents can be positive it is not clear how far these suggestions can be trusted because as is well known numerical methods can introduce spurious chaos or even suppress actual chaos this focused review examines the fidelity of numerical methods we look at the didactic example of the gauss map from the theory of continued fractions which allows a simple examination of backward error analysis for discrete dynamical systems and gives a clear picture of the effects of floating point arithmetic a similar use of backward error analysis in the form of defect control gives a useful understanding in the case of continuous dynamical systems finally we discuss limitations of this backward point of view

phase recovery maxcut and complex semidefinite programming phase retrieval seeks to recover a signal x in mathbb c p x c p from the amplitude a x a x of linear measurements ax in mathbb c n a x c n we cast the phase retrieval problem as a non convex quadratic program over a complex phase vector and formulate a tractable relaxation called phasecut similar to the classical maxcut semidefinite program we solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in gerchberg and saxton optik 35 237 246 1972 where each iteration is a matrix vector product numerical results show the performance of this approach over three different phase retrieval problems in comparison with greedy phase retrieval algorithms and matrix completion formulations

can two chaotic systems give rise to order abstract the recently discovered parrondo s paradox claims that two losing games can result under random or periodic alternation of their dynamics in a winning game losing losing winning in this paper we follow parrondo s philosophy of combining different dynamics and we apply it to the case of one dimensional quadratic maps we prove that the periodic mixing of two chaotic dynamics originates an ordered dynamics in certain cases this provides an explicit example theoretically and numerically tested of a different parrondian paradoxical phenomenon chaos chaos order

lyapunov exponents noise induced synchronization and parrondo s paradox we show that lyapunov exponents of a stochastic system when computed for a specific realization of the noise process are related to conditional lyapunov exponents in deterministic systems we propose to use the term stochastically induced regularity instead of noise induced synchronization and explain the reason why the nature of stochastically induced regularity is discussed in some instances it is a dynamical analog of parrondo s paradox

application of nonlinear deformation algebra to a physical system with poschl teller potential nonlinear deformation algebra are realized in a physical system with p schl teller potential the raising and lowering operators satisfying this algebra are constructed from which the eigenproblem of the system can be exactly solved by the operator method the physical meaning of two deforming functions involved in this algebra is also found in addition algebra is obtained naturally and discussions on the coherent state are also made

deterministic and random synthesis of discrete chaos in this paper two anticontrol algorithms for synthesis of discrete chaos are introduced in these algorithms the control parameter of a discrete dynamical system is switched either randomly or in a deterministic way between two or more values corresponding to periodic motions the result being chaotic behavior these algorithms require no knowledge of specific mathematical properties of the underlying map modeling the system the existence of chaos is demonstrated using various tools including graphical iteration histogram lyapunov exponent and surrogate tests in this paper these very simple and implementable chaotifiers are applied to the logistic map

invariant densities for random maps of the interval on etudie des applications aleatoires de 0 1 qui representent des systemes dynamiques sur le carre 0 1 0 1 on donne des conditions suffisantes pour qu une application aleatoire ait une mesure invariante absolument continue

solving systems of random quadratic equations via truncated amplitude flow this paper presents a new algorithm termed truncated amplitude flow taf to recover an unknown vector x from a system of quadratic equations of the form y i langle a i x rangle 2 where a i s are given random measurement vectors this problem is known to be np hard in general we prove that as soon as the number of equations is on the order of the number of unknowns taf recovers the solution exactly up to a global unimodular constant with high probability and complexity growing linearly with both the number of unknowns and the number of equations our taf approach adapts the amplitude based empirical loss function and proceeds in two stages in the first stage we introduce an orthogonality promoting initialization that can be obtained with a few power iterations stage two refines the initial estimate by successive updates of scalable truncated generalized gradient iterations which are able to handle the rather challenging nonconvex and nonsmooth amplitude based objective function in particular when vectors x and a i s are real valued our gradient truncation rule provably eliminates erroneously estimated signs with high probability to markedly improve upon its untruncated version numerical tests using synthetic data and real images demonstrate that our initialization returns more accurate and robust estimates relative to spectral initializations furthermore even under the same initialization the proposed amplitude based refinement outperforms existing wirtinger flow variants corroborating the superior performance of taf over state of the art algorithms

misiurewicz points in one dimensional quadratic maps abstract misiurewicz points are constituted by the set of unstable or repellent points sometimes called the set of exceptional points these points which are preperiodic and eventually periodic play an important role in the ordering of hyperbolic components of one dimensional quadratic maps in this work we use graphic tools to analyse these points by measuring their preperiods and periods and by ordering and classifying them

rigorous shadowing of numerical solutions of ordinary differential equations by containment an exact trajectory of a dynamical system lying close to a numerical trajectory is called a shadow we present a general purpose method for proving the existence of finite time shadows of numerical ode integrations of arbitrary dimension in which some measure of hyperbolicity is present and there are either 0 or 1 expanding modes or 0 or 1 contracting modes much of the rigor is provided automatically by interval arithmetic and validated ode integration software that is freely available the method is a generalization of a previously published containment process that was applicable only to two dimensional maps we extend it to handle maps of arbitrary dimension with the above restrictions and finally to odes the method involves building n cubes around each point of the discrete numerical trajectory through which the shadow is guaranteed to pass at appropriate times the proof consists of two steps first the rigorous computational verification of a simple geometric property which we call the inductive containment property and second a simple geometric argument showing that this property implies the existence of a shadow the computational step is almost entirely automated and easily adaptable to any ode problem the method allows for the rescaling of time which is a necessary ingredient for successfully shadowing odes finally the method is local in the sense that it builds the shadow inductively requiring information only from the most recent integration step rather than more global information typical of several other methods the method produces shadows of comparable length and distance to all currently published results finally we conjecture that the inductive containment property implies the existence of a shadow without restriction on the number of expanding and contracting modes although proof currently eludes us

low rank positive semidefinite matrix recovery from corrupted rank one measurements we study the problem of estimating a low rank positive semidefinite psd matrix from a set of rank one measurements using sensing vectors composed of i i d standard gaussian entries which are possibly corrupted by arbitrary outliers this problem arises from applications such as phase retrieval covariance sketching quantum space tomography and power spectrum estimation we first propose a convex optimization algorithm that seeks the psd matrix with the minimum ell 1 norm of the observation residual the advantage of our algorithm is that it is free of parameters therefore eliminating the need for tuning parameters and allowing easy implementations we establish that with high probability a low rank psd matrix can be exactly recovered as soon as the number of measurements is large enough even when a fraction of the measurements are corrupted by outliers with arbitrary magnitudes moreover the recovery is also stable against bounded noise with the additional information of an upper bound of the rank of the psd matrix we propose another nonconvex algorithm based on subgradient descent that demonstrates excellent empirical performance in terms of computational efficiency and accuracy

scalable solvers of random quadratic equations via stochastic truncated amplitude flow a novel approach termed stochastic truncated amplitude flow staf is developed to reconstruct an unknown n dimensional real complex valued signal boldsymbol x from m phaseless quadratic equations of the form psi i langle boldsymbol a i boldsymbol x rangle this problem also known as phase retrieval from magnitude only information is np hard in general adopting an amplitude based nonconvex formulation staf leads to an iterative solver comprising two stages s1 orthogonality promoting initialization through a stochastic variance reduced gradient algorithm and s2 a series of iterative refinements of the initialization using stochastic truncated gradient iterations both stages involve a single equation per iteration thus rendering staf a simple scalable and fast approach amenable to large scale implementations that are useful when n is large when lbrace boldsymbol a i rbrace i 1 m are independent gaussian staf provably recovers exactly any boldsymbol x in mathbb r n exponentially fast based on order of n quadratic equations staf is also robust in the presence of additive noise of bounded support simulated tests involving real gaussian lbrace boldsymbol a i rbrace vectors demonstrate that staf empirically reconstructs any boldsymbol x in mathbb r n exactly from about 2 3n magnitude only measurements outperforming state of the art approaches and narrowing the gap from the information theoretic number of equations m 2n 1 extensive experiments using synthetic data and real images corroborate markedly improved performance of staf over existing alternatives

the paradox of parrondo s games we introduce parrondo s paradox that involves games of chance we consider two fair games a and b both of which can be made to lose by changing a biasing parameter an apparently paradoxical situ

application of polynomial su 1 1 algebra to poschl teller potentials two novel polynomial su 1 1 algebras for the physical systems with the first and second poschl teller pt potentials are constructed and their specific representations are presented meanwhile these polynomial su 1 1 algebras are used as an algebraic technique to solve eigenvalues and eigenfunctions of the hamiltonians associated with the first and second pt potentials the algebraic approach explores an appropriate new pair of raising and lowing operators k of polynomial su 1 1 algebra as a pair of shift operators of our hamiltonians in addition two usual su 1 1 algebras associated with the first and second pt potentials are derived naturally from the polynomial su 1 1 algebras built by us

computing the topological entropy of general one dimensional maps a matrix theoretic method for computing the topological entropy of continuous piecewise monotonic maps of the interval is presented the method results in a constructive procedure which is easily implemented on the computer examples for families of unimodal nonunimodal and discontinuous maps are presented

phaselift exact and stable signal recovery from magnitude measurements via convex programming suppose we wish to recover a signal input amssym font abc cmmib10 def bi 1 hbox abc 1 bi x in bbb c n from m intensity measurements of the form that is from data in which phase information is missing we prove that if the vectors are sampled independently and uniformly at random on the unit sphere then the signal x can be recovered exactly up to a global phase factor by solving a convenient semidefinite program a trace norm minimization problem this holds with large probability provided that m is on the order of and without any assumption about the signal whatsoever this novel result demonstrates that in some instances the combinatorial phase retrieval problem can be solved by convex programming techniques finally we also prove that our methodology is robust vis a vis additive noise 2012 wiley periodicals inc

parrondo s paradox and complementary parrondo processes abstract parrondo s paradox has gained a fair amount of attention due to it being counter intuitive given two stochastic processes both of which are losing in nature it is possible to have an overall net increase in capital by periodically or randomly alternating between the two processes in this paper we analyze the paradox with a different approach in which we start with one process and seek to derive its complementary process we will also state the conditions required for this to occur possible applications of our results include the development of future models based on the paradox