bounds for fourth order 0 1 difference equations this note examines bounds for fourth order linear difference equations with coefficients restricted to the unit interval it is shown that all solutions are of order strictly less than 3 2 n the bound is shown to be nearly best possible applications to zero one banded matrices are also discussed

p adic quantum mechanics with p adic valued functions an extension of the formalism of quantum mechanics to the case where the canonical variables and functions are valued in a field of p adic numbers is considered a new p adic integral calculus is used for the realization of the gauss representation in p adic quantum mechanics

oscillation theorems for higher order nonlinear delay difference equations abstract some new oscillation results for difference equations of the form 1 1 and forced difference equations of the form 1 2 are established the results obtained are discrete analogues of some known results for delay differential equations

p adic superanalysis ii supermanifolds and differential operators this article is the second part of a work in which p adic supermanifold theory is studied by using the algebraic approach introduced in the first part of this work p adic superdifferential maps are introduced and by restricting attention to the class of strictly differential maps the foundation of p adic supermanifold theory is developed herein in particular it is shown that the superfield expansion theorem is no longer true a superdifferential odd variables map which is not a polynomial is constructed finally tangent space and lie derivatives are constructed and it is shown that no complex valued fermion field of the p adic argument could exist

explicit bounds for multidimensional linear recurrences with restricted coefficients this note employs path counting techniques to extend recent results on bounds for odd order linear recurrences to higher dimensions the results imply optimal zero free polydisks for multivariable power series with 0 1 coefficients among the applications is a result that states that the optimal zero free polydisk has radius approximately 1 v 1 for large dimensions v

growing random sequences we consider the random sequence xn xn 1 xq with 0 where q 0 1 n 1 is chosen randomly from a probability distribution pn q when all q are chosen with equal probability i e pn q 1 n we obtain an exact solution for the mean xn and the divergence of the second moment x2n as functions of n and for 1 we examine the divergence of the mean value of xn as a function of n for the random sequences generated by power law and exponential pn q and for the non random sequence pn q q n 1

poisson brackets as the adelic limit of quantum commutators we show that the adelic limit of the quantum commutator between operators gives a suitable generalization of the poisson bracket on p adic phase space

how do random fibonacci sequences grow we study the random fibonacci sequences defined by f 1 f 2 widetilde f 1 widetilde f 2 1 and for n 1 f n 2 f n 1 f n linear case and widetilde f n 2 widetilde f n 1 pm widetilde f n non linear case where each sign is independent and either with probability p or with probability 1 p 0 p 1 our main result is that the exponential growth of f n for 0 p 1 and of widetilde f n for 1 3 p 1 is almost surely given by int limits 0 infty log x d nu alpha x where is an explicit function of p depending on the case we consider and is an explicit probability distribution on mathbb r defined inductively on stern brocot intervals in the non linear case the largest lyapunov exponent is not an analytic function of p since we prove that it is equal to zero for 0 p 1 3 we also give some results about the variations of the largest lyapunov exponent and provide a formula for its derivative

random geometric series integer sequences where each element is determined by a previous randomly chosen element are investigated analytically in particular the random geometric series xn 2xp with 0 p n 1 is studied at large n the moments grow algebraically xsn n s with s 2s 1 while the typical behaviour is xn nln 2 the probability distribution is obtained explicitly in terms of the stirling numbers of the first kind and it approaches a log normal distribution asymptotically

second order bounds for linear recurrences with negative coefficients this paper introduces a generalization of fibonacci and pell polynomials in order to obtain optimal second order bounds for general linear recurrences with negative coefficients an important aspect of the derived bounds is that they are applicable and easily computable the results imply bounds on all entries in inverses of triangular matrices as well as on coefficients of reciprocals of power series

monotone convex sequences and cholesky decomposition of symmetric toeplitz matrices this paper studies off diagonal decay in symmetric toeplitz matrices it is shown that if the generating sequence of the matrix is monotone positive and convex then the monotonicity and positivity are maintained through triangular decomposition the work is motivated by recent results on explicit bounds for inverses of triangular matrices

on the fourier transform and the spectral properties of the p adic momentum and schrodinger operators a momentum representation for p adic quantum mechanics is constructed it uses a fourier transform in the space with respect to the p adic lebesgue distribution and a study of its properties the momentum representation is used to investigate the spectral properties of the p adic momentum operator and to study the p adic valued schrodinger equation for the p adic harmonic oscillator

random fibonacci sequences solutions to the random fibonacci recurrence xn 1 xn xn 1 decrease increase exponentially xn exp n for sufficiently small large in the limits 0 and we expand the lyapunov exponent in powers of and 1 respectively for the classical case of 1 we obtain exact non perturbative results in particular an invariant measure associated with ricatti variable rn xn 1 xn is shown to exhibit plateaux around all rational r

oscillatory property of higher order nonlinear difference equations abstract in this paper we study the oscillatory and asymptotic behaviour of solutions of higher order nonlinear difference equations of the form r n d 1 x n f n n 0 we obtain some necessary and sufficient conditions for all bounded solutions of to be oscillatory and for to have a nonoscillatory solution of a special form

a bound for linear recurrence relations with unbounded order this paper considers linear recurrence relations with unbounded order where the coefficients are restricted to intervals of negative real numbers the optimal inequalities under the given constraints exhibit a second order structure

computation of growth rates of random sequences with multi step memory we extend the generating function approach to the computation of growth rate of random fibonacci sequences with long memory functional iteration equations are obtained and its general form is conjectured and proved based on which an asymptotic representation of the growth rate is obtained the validity of both the derived and the conjectured formula are verified upon comparison with monte carlo simulation a numerical scheme of the functional iteration is designed and implemented successfully

computing yapunov constants for random recurrences with smooth coefficients in recent years there has been much interest in the growth and decay rates lyapunov constants of solutions to random recurrences such as the random fibonacci sequence x n 1 x n x n 1 many of these problems involve nonsmooth dynamics nondi1erentiable invariant measures making computations hard here however we consider recurrences with smooth random coe cients and smooth invariant measures by computing discretised invariant measures and applying richardson extrapolation we can compute lyapunov constants to 10 digits of accuracy in particular solutions to the recurrence x n 1 x n cn 1x n 1 where the cn are independent standard normal variables increase exponentially almost surely at the asymptotic rate 1 0574735537 n solutions to the related recurrences x n 1 cn 1x n x n 1 and x n 1 cn 1x n dn 1x n 1 where the dn are also independent standard normal variables increase decrease at the rates 1 1149200917 n and 0 9949018837 n respectively c 2001 elsevier science b v all rights reserved

novel computation of the growth rate of generalized random fibonacci sequences a new formulation is proposed for the computation of average growth rates of generalized random fibonacci sequences based on the new formula a novel numerical scheme is designed and successfully implemented and interesting analytic asymptotic expansions are obtained for several examples

numerical results on some generalized random fibonacci sequences random fibonacci sequences are stochastic versions of the classical fibonacci sequence f n 1 f n f n 1 for n 0 and f 0 f 1 1 obtained by randomizing one or both signs on the right side of the defining equation and or adding a growth parameter these sequences may be viewed as coming from a sequence of products of i i d random matrices and their rate of growth measured by the associated lyapunov exponent following the techniques presented by embree and trefethen in their numerical paper embree and trefethen 1999 2 we study the behavior of the lyapunov exponents as a function of the probability p of choosing in the sign randomization

the combinatorial power of the companion matrix abstract using combinatorial methods we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to waring s formula on symmetric functions the general solution to homogeneous linear recurrence relations the multiplicative inverse of formal power series the generating function of compositions of numbers a unified approach to chebyshev polynomials including two recently discovered classes that satisfy analogous smallest norm and orthogonality properties subject to different weight functions dickson polynomials of various kinds arising from the theory of finite fields combinatorial expansions of toeplitz matrices and the recent notion of cycle dissections involving a bijective study of waring s formula

growth rate for the expected value of a generalized random fibonacci sequence we study the behaviour of generalized random fibonacci sequences defined by the relation gn gn 1 gn 2 where the sign is given by tossing an unbalanced coin giving probability p to the sign we prove that the expected value of gn grows exponentially fast for any 0 2 4 when is of the form 2cos k for some fixed integer k 3 in both cases we give an algebraic expression for the growth rate

growth and structure of stochastic sequences we introduce a class of stochastic integer sequences in these sequences every element is a sum of two previous elements at least one of which is chosen randomly the interplay between randomness and memory underlying these sequences leads to a wide variety of behaviours ranging from stretched exponential to log normal to algebraic growth interestingly the set of all possible sequence values has an intricate structure

numerical differentiation for use in integrating unified constitutive equations the nonss method for numerically integrating time dependent constitutive equations requires evaluation of the partial derivatives of the constitutive equations a numerical method for evaluating these partial derivatives has been developed thus avoiding the cumbersome task of analytically re deriving the partial derivatives when improvements to the constitutive model are evaluated this method is based upon the adaptive finite difference algorithm published by stepleman and winarsky modifications of their algorithm are described that allow its efficient use within an extended version of nonss called nonss nd using the matmod constitutive equations the capabilities of nonss nd are demonstrated by comparing the results of a wide variety of deformation simulations with those of nonss for which the derivatives are computed from analytical solutions the numerical differentiation within nonss nd is shown to be sufficiently accurate for use in integrating time dependent constitutive equations but results in a 30 100 per cent increase in computation time multi element structural calculations are therefore found to be uneconomical but nonss nd is well suited for testing alternative constitutive models owing to its accuracy and its general applicability to highly non linear systems of differential equations

oscillation theorems for neutral difference equations abstract oscillation criteria for first order difference equations of the form x n q n x n 0 and x n px n k q n x n f n are established here xn xn 1 xn is the forward difference operator 1 p is a real number and k are nonnegative integers qn fn are sequences of nonnegative real numbers and r r is continuous oscillation criteria for nonlinear difference equations of sublinear as well as superlinear type are also established

exact lyapunov exponent for infinite products of random matrices despite significant work since the original paper by h furstenberg 1963 explicit formulae for lyapunov exponents of infinite products of random matrices are available only in a very few cases in this work we give a rigorous explicit formula for the lyapunov exponent for some binary infinite products of random 2 2 real matrices all these products are constructed using only two types of matrices a and b which are chosen according to a stochastic process the matrix a is singular namely its determinant is zero this formula is derived by using a particular decomposition for the matrix b which allows us to write the lyapunov exponent as a sum of convergent series the key point is the computation of all the integer powers of b which is achieved by a suitable change of frame the computation then follows by looking at each of the special types of b hyperbolic parabolic and elliptic finally we show with an example that the lyapunov exponent is a discontinuous function of the given parameter

weak disorder in fibonacci sequences we study how weak disorder affects the growth of the fibonacci series we introduce a family of stochastic sequences that grow by the normal fibonacci recursion with probability 1 but follow a different recursion rule with a small probability we focus on the weak disorder limit and obtain the lyapunov exponent that characterizes the typical growth of the sequence elements using perturbation theory the limiting distribution for the ratio of consecutive sequence elements is obtained as well a number of variations to the basic fibonacci recursion including shift doubling and copying are considered

on the closed representation for the inverses of hessenberg matrices the general representation for the elements of the inverse of any hessenberg matrix of finite order is here extended to the reduced case with a new proof those entries are given with proper hessenbergians from the original matrix it justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order and some elementary properties of the inverse these results are applied on the resolvent matrix associated to a finite hessenberg matrix in standard form two examples on the unit disk are given

random fibonacci sequences and the number 1 13198824 begin abstract for the familiar fibonacci sequence defined by f 1 f 2 1 and f n f n 1 f n 2 for n 2 f n increases exponentially with n at a rate given by the golden ratio 1 sqrt 5 2 1 61803398 ldots but for a simple modification with both additions and subtractions the it random fibonacci sequences defined by t 1 t 2 1 and for n 2 t n pm t n 1 pm t n 2 where each pm sign is independent and either or with probability 1 2 it is not even obvious if abs t n should increase with n our main result is that begin equation sqrt n abs t n rightarrow 1 13198824 ldots text as n rightarrow infty end equation with probability 1 finding the number 1 13198824 ldots involves the theory of random matrix products stern brocot division of the real line a fractal like measure a computer calculation and a rounding error analysis to validate the computer calculation end abstract

on explicit formulas for the principal matrix logarithm we describe a method for evaluating both the fibonacci horner and the polynomial decomposition of the principal matrix logarithm with a view to solve the lifting problem of its explicit computation the binet formula for linear recursive sequences serves as a triggering factor for giving the exact formula we supply some illustrative examples

solutions of linear difference equations with variable coefficients the explicit solution of a linear difference equation of unbounded order with variable coefficients is presented as special cases the solutions of nonhomogeneous and homogeneous linear difference equations of order n with variable coefficients are obtained from these solutions we also get expressions for the product of companion matrices and the power of a companion matrix

a transfer operator approach to random fibonacci sequences we consider two stochastic variants of a fibonacci sequence generated by xn 1 xn xn 1 and xn 1 xn xn 1 respectively both the sequences begin with x0 x1 1 and the signs in each iteration are independently chosen with probability p for we focus on the generalized lyapunov exponent q of xn which gives the growth when q 0 or decay when q 0 rate of the ensemble average of xn q when n by using a technique that reduces the non increasing segments in the sequences we map the random sequences to a product of random 2 2 nonnegative matrices which is closely related to the farey fraction spin chain model we show that there exists a critical value q below which the ensemble average of xn q is dominated by the non increasing or linearly increasing samples and hence q is a constant both q and q and its derivatives can be precisely calculated with the transfer operator method moreover when q is a positive integer q is exactly determined by a system of polynomial equations the analysis can be readily generalized to the random sequence generated by xn 1 2cos k xn xn 1 or xn 1 2cos k xn xn 1 for any integer k 3

products of random matrices and derivatives on p c f fractals abstract we define and study intrinsic first order derivatives on post critically finite fractals and prove differentiability almost everywhere with respect to self similar measures for certain classes of fractals and functions we apply our results to extend the geography is destiny principle to these cases and also obtain results on the pointwise behavior of local eccentricities on the sierpinski gasket previously studied by oberg strichartz and yingst and the authors we also establish the relation of the derivatives to the tangents and gradients previously studied by strichartz and the authors our main tool is the furstenberg kesten theory of products of random matrices

on compact representations for the solutions of linear difference equations with variable coefficients a comprehensive treatment on compact representations for the solutions of linear difference equations with variable coefficients of both n th and unbounded order is presented the equivalence between their celebrated combinatorial and determinantal representations is considered a corresponding representation is proposed using determined nested sums of their variable coefficients it makes explicit all the sum of products involved in the previous representations of such solutions some basic applications are also illustrated

on the approximation of invariant measures given a discrete dynamical system defined by the map x x the density of the absolutely continuous a c invariant measure if it exists is the fixed point of the frobenius perron operator defined on l1 x ulam proposed a numerical method for approximating such densities based on the computation of a fixed point of a matrix approximation of the operator t y li proved the convergence of the scheme for expanding maps of the interval g keller and m blank extended this result to piecewise expanding maps of the cube in n we show convergence of a variation of ulam s scheme for maps of the cube for which the frobenius perron operator is quasicompact we also give sufficient conditions on for the existence of a unique fixed point of the matrix approximation and if the fixed point of the operator is a function of bounded variation we estimate the convergence rate

semicycles of solutions of nonlinear difference equations with several delays abstract in this paper we study the semicycles of oscillatory solutions of nonlinear difference equation with several delays where pi n is a real sequence with pi n 0 for all large n l mi kij are positive integers ij i 1 2 l j 1 2 mi are nonnegative real numbers with mij 1 ij 1 i 1 2 l upper bound of numbers of terms of semicycles are determined

existence for nonoscillatory solutions of higher order nonlinear neutral difference equations abstract in this paper we consider the following forced higher order nonlinear neutral difference equation m x n c n x n k s 1 u p s n f s x n r s q n n n 0 where m u 1 k 0 and rs 0 are integers cn p s n s 1 2 u and qn are sequence of real numbers and f s c r r s 1 2 u by using krasnoselskii s fixed point theorem and some new techniques we obtain sufficient conditions for the existence of nonoscillatory solutions for general p s n s 1 2 u and qn which means that we allow oscillatory p s n s 1 2 u and qn

chaos in the lorenz equators a computer assisted proof part ii details details of a new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described the technique combines abstract existence results based on the conley index theory with rigorous computer assisted computations as an application of these methods it is proven that for some explicit parameter values the lorenz equations exhibit chaotic dynamics

on the cycle expansion for the lyapunov exponent of a product of random matrices the cycle expansion of the thermodynamical zeta function for the lyapunov exponent of a product of random matrices typically converges exponentially with the maximal cycle length mainieri 1992 phys rev lett 68 1965 in this paper we show that the convergent exponents are given by the spectrum of a properly defined evolution operator which describes how a steady distribution of vector direction is established under the action of random matrices the exponential decay terms are automatically eliminated in the cycle expansion of the spectral determinant which greatly accelerates the convergence provided all matrix elements are positive numbers as a marginal case the random fibonacci series is studied in detail and it is shown that this method is helpful

oscillation for even order delay difference equations with unstable type in this paper we first prove the even odd delay difference equation with unstable type d mx email protected i 1 np i n x n k i n 0 1 2 has always a positive solution which tends infinity as n then we obtain some oscillation criteria for all bounded solutions of eq

asymptotic behaviour of solutions of higher order difference and partial difference equations with distributed deviating arguments we shall study the asymptotic behaviour of solutions of the difference equation m y j g j y j s a 1 d m y j y j s b x at j x f y h j x 0 j j 0 where a 0 as an application of the results obtained oscillation theorems are established for the partial difference equation m y j g j y j s b x a i j x f y h j x d j ly i j m l 1d l j ly i t l j i e w j

growth and decay of random fibonacci sequences for 0 they grow exponentially by formulating the problem as a markov chain involving random matrix products and computing its invariant measure a fractal the lyapunov constant limn x n 1 n is determined numerically for a wide range of values and its dependence on is observed to be non smooth the limit is defined in the almost sure sense this generalizes recent work of viswanath who proved 1 1 131 988 24 by a simple rescaling these results also apply to the more general random recurrence x n 1 x n x n 1 for fixed and these random recurrence relations have links with many fields including ergodic theory dynamical systems heavy tailed statistics spectral theory continued fractions and condensed matter physics

infinitesimal perturbation analysis for general discrete event systems a rigorous extension of the recent perturbation analysis approach to more general discrete event systems is given first a general class of systems and performance measures is defined and some basic reprsentational and linearity properties are derived next a sample gradient of performance with respect to a parameter of the system is defined then for certain parameters of such systems an infinitesimal perturbation analysis algorithm is derived it is proved that this algorithm gives exact values for the sample gradients of performance with respect to the parameters by observing only one sample path of the deds however the sample gradient may or may not be a good estimate of the gradient of the performance measure this point is elaborated in the body of the paper the computational complexity of this algorithm is bound to be linear in the number of events these results offer the potential for very efficient calculation of the gradients a fact that can be used for design operation of computer systems communication networks manufacturing systems and many other real world systems particularly since restrictive assumptions e g exponential distributions are not required of the system

power series with restricted coefficients and a root on a given ray we consider bounds on the smallest possible root with a specified argument o of a power series f z 1 n 1 a i z i with coefficients a i in the interval g g we describe the form that the extremal power series must take and hence give an algorithm for computing the optimal root when o 2 is rational when g 2 2 3 we show that the smallest disc containing two roots has radius g 1 1 coinciding with the smallest double real root possible for such a series it is clear from our computations that the behaviour is more complicated for smaller g we give a similar procedure for computing the smallest circle with a real root and a pair of conjugate roots of a given argument we conclude by briefly discussing variants of the beta numbers where the defining integer sequence is generated by taking the nearest integer rather than the integer part we show that the conjugates a of these pseudobeta numbers either lie inside the unit circle or their reciprocals must be roots of 1 2 1 2 power series in particular we obtain the sharp inequality 3 2

the resistance of randomly grown trees copyright 2011 iop publishing ltd this is a preprint version of the published article which can be accessed from the link below

a representation of the solution of the nth order linear difference equation with variable coefficients abstract the linear difference equation of the n th order with variable coefficients and a related difference equation are considered it is shown that for each problem the determinants of submatrices of a single solution matrix provides the solution sequence applications to sequences and series are shown results on toeplitz matrices involving the roots of associated polynomial equations are obtained

the random fibonacci recurrence and the visible points of the plane in this letter we show a connection between the random fibonacci recurrence and the visible points of the plane in particular we show that by suitably modifying the rules of the random fibonacci map there is a unique correspondence between the visible points points with relative prime coordinates of the first quadrant and the vertices of a self similar graph what we call the fibonacci graph the proposed random recurrence can then be interpreted as a random walk on this graph

oscillations of higher order linear difference equations abstract consider the higher order linear difference equation where m 1 is an odd integer and p n is a sequence of nonnegative real numbers we obtain several new sufficient conditions for the oscillation of all solutions of equation examples which dwell upon the importance of our results are also included

oscillation and nonoscillation for second order linear difference equations new oscillation and nonoscillation theorems are obtained for the second order linear difference equation 2xn 1 pnxn 0 where if pn n 1 is a real sequence with pn 0

anti hadamard matrices abstract an anti hadamard matrix may be loosely defined as a real 0 1 matrix which is invertible but only just let a be an invertible 0 1 matrix with eigenvalues i singular values i and inverse b bij we are interested in the four closely related problems of finding n mina i i n mina i i n maxa i j bij and n maxa ijb2ij then a is an anti hadamard matrix if it attains n we show that n n are between 2n 1 n 4 n 2 and c n 2 274 n where c is a constant c 2 274 n n 2 n 4 n 2 and c 5 172 n n 4n 2 n 4 n we also consider these problems when a is restricted to be a toeplitz triangular circulant or 1 1 matrix besides the obvious application to finding the most ill conditioned 0 1 matrices there are connections with weighing designs number theory and geometry

double roots of 1 1 power series and related matters for a given collection of distinct arguments vec theta theta 1 ldots theta t multiplicities vec k k 1 ldots k t and a real interval i u v containing zero we are interested in determining the smallest r for which there is a power series f x 1 sum n 1 infty a i x i with coefficients a i in i and roots alpha 1 re 2 pi i theta 1 ldots alpha t re 2 pi i theta t of order k 1 ldots k t respectively we denote this by r vec theta vec k i we describe the usual form of the extremal series we give a sufficient condition which is also necessary when the extremal series possesses at least left sum i 1 t delta theta i k i right 1 coefficients strictly inside i where delta theta i is 1 or 2 as alpha i is real or complex we focus particularly on r theta 2 1 1 the size of the smallest double root of a 1 1 power series lying on a given ray of interest in connection with the complex analogue of work of boris solomyak on the distribution of the random series sum pm lambda n we computed the value of r theta 2 1 1 for the rationals theta in 0 1 2 of denominator less than fifty the smallest value we encountered was r 4 29 2 1 1 0 7536065594 for the one sided intervals i 0 1 and 1 0 the corresponding smallest values were r 11 30 2 0 1 8237251991 and r 1 3 2 1 0 8656332072