fractal property of generalized m set with rational number exponent dynamic systems described by f c z z 2 c is called mandelbrot set m set which is important for fractal and chaos theories due to its simple expression and complex structure f c z z k c is called generalized m set k m set this paper proposes a new theory to compute the higher and lower bounds of generalized m set while exponent k is rational and proves relevant properties such as that generalized m set could cover whole complex number plane when k 1 and that k m set can be divided into p q isomorphic parts

jordan triple endomorphisms and isometries of unitary groups abstract in this paper we present the general form of all continuous endomorphisms of the group u n of n n complex unitary matrices with respect to the jordan triple product these are the continuous maps u n u n which satisfy v w v v w v v w u n the result is applied to determine the structure of certain isometries of u n these include the isometries relative to any metric given by a unitarily invariant norm on the space m n of all n n complex matrices and also the isometries relative to any member of a new class of metrics on u n recently introduced by chau li poon and sze 6

relation between quantum speed limits and metrics on u n recently chau 2011 quantum inform comp 11 721 found a family of metrics and pseudo metrics on n dimensional unitary operators that can be interpreted as the minimum resources given by certain tight quantum speed limit bounds needed to transform one unitary operator to another this result is closely related to the weighted l1 norm on here we generalize this finding by showing that every weighted lp norm on with 1 p 2 induces a metric and a pseudo metric on n dimensional unitary operators with quantum information theoretic meanings related to certain tight quantum speed limit bounds besides we investigate how far the correspondence between the existence of metrics and pseudo metrics of this type and the quantum speed limits can go

the numerical solution of linear multi term fractional differential equations systems of equations in this paper we show how the numerical approximation of the solution of a linear multi term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity we begin by showing how our method applies to a simple class of problems and we give a convergence result we solve the bagley torvik equation as an example we show how the method can be applied to a general linear multi term equation and give two further examples

technical communique a note on fractional order derivatives of periodic functions in this paper it is shown that the fractional order derivatives of a periodic function with a specific period cannot be a periodic function with the same period the fractional order derivative considered here can be obtained based on each of the well known definitions grunwald letnikov definition riemann liouville definition and caputo definition this concluded point confirms the result of a recently published work proving the non existence of periodic solutions in a class of fractional order models also based on this point it can be easily proved the absence of periodic responses in a wider class of fractional order models finally some examples are presented to show the applicability of the paper achievements in the solution analysis of fractional order systems

orbits connecting critical points of differential equations depending on a parameter a sufficient and necessary condition for a planar critical point to be isolated as an invariant set is given using the concept of an isolated invariant set some existence criteria of orbits connecting two critical points bifurcating from a single critical point for planar differential equations depending on a parameter are given

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

effective dimensions and relative frequencies consider the problem of calculating the fractal dimension of a set x consisting of all infinite sequences s over a finite alphabet s that satisfy some given condition p on the asymptotic frequencies with which various symbols from s appear in s solutions to this problem are known in cases where i the fractal dimension is classical hausdorff or packing dimension by work of volkmann and olsen or ii the fractal dimension is effective even finite state and the condition pcompletely specifies an empirical distribution p over s i e a limiting frequency of occurrence for every symbol in s in this paper we show how to calculate the finite state dimension equivalently the finite state compressibility of such a set x when the condition p only imposes partial constraints on the limiting frequencies of symbols our results automatically extend to less restrictive effective fractal dimensions e g polynomial time computable and constructive dimensions and they have the classical results i as immediate corollaries our methods are nevertheless elementary and in most cases simpler than those by which the classical results were obtained

computer arithmetic chaos and fractals abstract in this paper we explore aspects of computer arithmetic from the viewpoint of dynamical systems we demonstrate the effects of finite precision arithmetic in three uniformly hyperbolic chaotic dynamical systems bernoulli shifts cat maps and pseudorandom number generators we show that elementary floating point operations in binary computer arithmetic possess an inherently fractal structure each of these dynamical systems allows us to compare the exact results in integer arithmetic with those obtained by using floating point arithmetic

effective strong dimension in algorithmic information and computational complexity the two most important notions of fractal dimension are hausdorff dimension developed by hausdorff math ann 79 1919 pp 157 179 and packing dimension developed independently by tricot math proc cambridge philos soc 91 1982 pp 57 74 and sullivan acta math 153 1984 pp 259 277 both dimensions have the mathematical advantage of being defined from measures and both have yielded extensive applications in fractal geometry and dynamical systems lutz proceedings of the 15th ieee conference on computational complexity florence italy 2000 ieee computer society press piscataway nj 2000 pp 158 169 has recently proven a simple characterization of hausdorff dimension in terms of gales which are betting strategies that generalize martingales imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of hausdorff dimension including constructive computable polynomial space polynomial time and finite state dimensions work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science in this paper we show that packing dimension can also be characterized in terms of gales moreover even though the usual definition of packing dimension is considerably more complex than that of hausdorff dimension our gale characterization of packing dimension is an exact dual of and every bit as simple as the gale characterization of hausdorff dimension effectivizing our gale characterization of packing dimension produces a variety of effective strong dimensions which are exact duals of the effective dimensions mentioned above in general and in analogy with the classical fractal dimensions the effective strong dimension of a set or sequence is at least as great as its effective dimension with equality for sets or sequences that are sufficiently regular we develop the basic properties of effective strong dimensions and prove a number of results relating them to fundamental aspects of randomness kolmogorov complexity prediction boolean circuit size complexity polynomial time degrees and data compression aside from the above characterization of packing dimension our two main theorems are the following 1 if vec beta beta 0 beta 1 ldots is a computable sequence of biases that are bounded away from 0 and r is random with respect to vec beta then the dimension and strong dimension of r are the lower and upper average entropies respectively of vec beta 2 for each pair of delta 0 2 computable real numbers 0 alpha le beta le 1 there exists a in rm e such that the polynomial time many one degree of a has dimension alpha in e and strong dimension beta in e our proofs of these theorems use a new large deviation theorem for self information with respect to a bias sequence vec beta that need not be convergent

a fractional differential equation for a mems viscometer used in the oil industry a mathematical model is developed for a micro electro mechanical system mems instrument that has been designed primarily to measure the viscosity of fluids that are encountered during oil well exploration it is shown that in one mode of operation the displacement of the device satisfies a fractional differential equation fde the theory of fdes is used to solve the governing equation in closed form and numerical solutions are also determined using a simple but efficient central difference scheme it is shown how knowledge of the exact and numerical solutions enables the design of the device to be optimised it is also shown that the numerical scheme may be extended to encompass the case of a nonlinear spring where the resulting fde is nonlinear

non existence of finite time stable equilibria in fractional order nonlinear systems we note that in the literature it is often taken for granted that for fractional order system without delays whenever the system trajectory reaches the equilibrium it will stay there in fact this is the well known phenomenon of finite time stability however in this paper we will prove that for fractional order nonlinear system described by caputo s or riemann liouville s definition any equilibrium cannot be finite time stable as long as the continuous solution corresponding to the initial value problem globally exists in addition some examples of stability analysis are revisited and linear lyapunov function is used to prove the asymptotic stability of positive fractional order nonlinear systems

equivalence classes of second order ordinary differential equations with only a three dimensional lie algebra of point symmetries and linearisation there are seven equivalence classes of second order ordinary differential equations possessing only three lie point symmetries and hence not linearisible by means of a point transformation we examine the representatives of these classes for linearisibility by means of other types of transformation in particular we compare the potential for linearisibility and the possession of the painleve property the complete symmetry group is realised in the standard algebra for each of the equivalence classes

on the equivalence of linear equations under nonlocal transformation abstract linear n th order n 3 ordinary differential equations have been shown to possess n 1 n 2 or n 4 lie point symmetries each class contains equations which are equivalent under point transformation by taking the example of third order equations we show that all linear equations are equivalent if the class of transformation is broadened to include nonlocal transformations and hence the representative of this class of equations is y n 0

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

finite state dimension classical hausdorff dimension sometimes called fractal dimension was recently effectivized using gales betting strategies that generalize martingales thereby endowing various complexity classes with dimension structure and also defining the constructive dimensions of individual binary infinite sequences in this paper we use gales computed by multi account finite state gamblers to develop the finite state dimensions of sets of binary sequences and individual binary sequences the theorem of eggleston quart j math oxford ser 20 1949 31 36 relating hausdorff dimension to entropy is shown to hold for finite state dimension both in the space of all sequences and in the space of all rational sequences binary expansions of rational numbers every rational sequence has finite state dimension 0 but every rational number in 0 1 is the finite state dimension of a sequence in the low level complexity class ac0 our main theorem shows that the finite state dimension of a sequence is precisely the infimum of all compression ratios achievable on the sequence by information lossless finite state compressors

induced metric and matrix inequalities on unitary matrices recently chau 2011 quantum inform comput 11 721 showed that one can define certain metrics and pseudo metrics on u n the group of all n n unitary matrices based on the arguments of the eigenvalues of the unitary matrices more importantly these metrics and pseudo metrics have quantum information theoretical meanings so it is instructive to study this kind of metrics and pseudo metrics on u n here we show that any symmetric norm on induces a metric on u n furthermore using the same technique we prove an inequality concerning the eigenvalues of a product of two unitary matrices which generalizes a few inequalities obtained earlier by chau arxiv 1006 3614v1

stability and bifurcation of delayed fractional order dual congestion control algorithms in this technical note fractional order congestion control systems are introduced for the first time in comparison with the conventional integer order dual congestion control algorithms the fractional control algorithms are more accurate and versatile bifurcation theory in fractional order differential equations is still an outstanding problem sufficient conditions for the occurrence of hopf bifurcations are extended from integer order dynamical systems to fractional order cases then these conditions are used to establish the existence of hopf bifurcations for the delayed fractional order model of dual congestion control algorithms proposed in this note finally the onsets of bifurcations are identified where hopf bifurcations occur and a family of oscillations bifurcate from the equilibrium illustrative examples are also provided to demonstrate the theoretical results

unambiguous discrimination among oracle operators we address the problem of unambiguous discrimination among oracle operators the general theory of unambiguous discrimination among unitary operators is extended with this application in mind we prove that entanglement with an ancilla cannot assist any discrimination strategy for commuting unitary operators we also obtain a simple practical test for the unambiguous distinguishability of an arbitrary set of unitary operators on a given system using this result we prove that the unambiguous distinguishability criterion is the same for both standard and minimal oracle operators we then show that except in certain trivial cases unambiguous discrimination among all standard oracle operators corresponding to integer functions with fixed domain and range is impossible however we find that it is possible to unambiguously discriminate among the grover oracle operators corresponding to an arbitrarily large unsorted database the unambiguous distinguishability of standard oracle operators corresponding to totally indistinguishable functions which possess a strong form of classical indistinguishability is analysed we prove that these operators are not unambiguously distinguishable for any finite set of totally indistinguishable functions on a boolean domain and with arbitrary fixed range sets of such functions on a larger domain can have unambiguously distinguishable standard oracle operators and we provide a complete analysis of the simplest case that of four functions we also examine the possibility of unambiguous oracle operator discrimination with multiple parallel calls and investigate an intriguing unitary superoperator transformation between standard and entanglement assisted minimal oracle operators

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

brief paper synthesis of fractional laguerre basis for system approximation fractional differentiation systems are characterized by the presence of non exponential aperiodic multimodes although rational orthogonal bases can be used to model any l2 0 system they fail to quickly capture the aperiodic multimode behavior with a limited number of terms hence fractional orthogonal bases are expected to better approximate fractional models with fewer parameters intuitive reasoning could lead to simply extending the differentiation order of existing bases from integer to any positive real number however classical laguerre and by extension kautz and generalized orthogonal basis functions are divergent as soon as their differentiation order is non integer in this paper the first fractional orthogonal basis is synthesized extrapolating the definition of laguerre functions to any fractional order derivative completeness of the new basis is demonstrated hence a new class of fixed denominator models is provided for fractional system approximation and identification

undamped oscillations generated by hopf bifurcations in fractional order recurrent neural networks with caputo derivative in this paper a fractional order recurrent neural network is proposed and several topics related to the dynamics of such a network are investigated such as the stability hopf bifurcations and undamped oscillations the stability domain of the trivial steady state is completely characterized with respect to network parameters and orders of the commensurate order neural network based on the stability analysis the critical values of the fractional order are identified where hopf bifurcations occur and a family of oscillations bifurcate from the trivial steady state then the parametric range of undamped oscillations is also estimated and the frequency and amplitude of oscillations are determined analytically and numerically for such commensurate order networks meanwhile it is shown that the incommensurate order neural network can also exhibit a hopf bifurcation as the network parameter passes through a critical value which can be determined exactly the frequency and amplitude of bifurcated oscillations are determined

on the complete symmetry group of the classical kepler system a rather strong concept of symmetry is introduced in classical mechanics in the sense that some mechanical systems can be completely characterized by the symmetry laws they obey accordingly a complete symmetry group realization in mechanics must be endowed with the following two features 1 the group acts freely and transitively on the manifold of all allowed motions of the system 2 the given equations of motion are the only ordinary differential equations that remain invariant under the specified action of the group this program is applied successfully to the classical kepler problem since the complete symmetry group for this particular system is here obtained the importance of this result for the quantum kinematic theory of the kepler system is emphasized

entropy rates and finite state dimension the effective fractal dimensions at the polynomial space level and above can all be equivalently defined as the c entropy rate where c is the class of languages corresponding to the level of effectivization for example pspace dimension is equivalent to the pspace entropy rate at lower levels of complexity the equivalence proofs break down in the polynomial time case the p entropy rate is a lower bound on the p dimension equality seems unlikely but separating the p entropy rate from p dimension would require proving p np we show that at the finite state level the opposite of the polynomial time case happens the reg entropy rate is an upper bound on the finite state dimension we also use the finite state genericity of ambos spies and busse automatic forcing and genericity on the diagonalization strength of finit automata in proc fourth int conf on discrete mathematics and theoretical computer science 2003 springer berlin pp 97 108 to separate finite state dimension from the reg entropy rate however we point out that a block entropy rate characterization of finite state dimension follows from the work of ziv and lempel compression of individual sequences via variable rate coding ieee trans inform theory 24 1978 530 536 on finite state compressibility and the compressibility characterization of finite state dimension by dai et al finite state dimension theoret comput sci 310 1 3 2004 1 33 as applications of the reg entropy rate upper bound and the block entropy rate characterization we prove that every regular language has finite state dimension 0 and that normality is equivalent to finite state dimension 1

a note on dimensions of polynomial size circuits in this paper we use resource bounded dimension theory to investigate polynomial size circuits we show that for every i 0 p poly has ith order scaled p3 strong dimension 0 we also show that p polyi o has p3 dimension 1 2 and p3 strong dimension 1 our results improve previous measure results of lutz almost everywhere high nonuniform complexity j comput syst sci 44 2 1992 220 258 and dimension results of hitchcock and vinodchandran dimension entropy rates and compression in proc 19th ieee conf computational complexity 2004 pp 174 183 j comput syst sci to appear additionally we establish a supergale dilation theorem which extends the martingale dilation technique introduced implicitly by ambos spies terwijn and zheng resource bounded randomness and weakly complete problems theoret comput sci 172 1 2 1997 195 207 and made explicit by juedes and lutz weak completeness in e and e2 theoret comput sci 143 1 1995 149 158

dimension in complexity classes a theory of resource bounded dimension is developed using gales which are natural generalizations of martingales when the resource bound delta a parameter of the theory is unrestricted the resulting dimension is precisely the classical hausdorff dimension sometimes called fractal dimension other choices of the parameter delta yield internal dimension theories in e e2 espace and other complexity classes and in the class of all decidable problems in general if mathcal c is such a class then every set x of languages has a dimension in mathcal c which is a real number dim x mid mathcal c in 0 1 along with the elements of this theory two preliminary applications are presented for every real number 0 le alpha le frac 1 2 the set rm freq le alpha consisting of all languages that asymptotically contain at most alpha of all strings has dimension mathcal h alpha the binary entropy of alpha in e and in e2 for every real number 0 le alpha le 1 the set rm size alpha frac 2 n n consisting of all languages decidable by boolean circuits of at most alpha frac 2 n n gates has dimension alpha in espace

technical communique on type number concept in fractional order systems in this note the type number concept is defined for fractional order systems based on this definition it is shown that fractional order systems having type numbers more than 1 can not track some classes of reference inputs without any overshoot

pushdown dimension resource bounded dimension is a notion of computational information density of infinite sequences based on computationally bounded gamblers this paper develops the theory of pushdown dimension and explores its relationship with finite state dimension the pushdown dimension of any sequence is trivially bounded above by its finite state dimension since a pushdown gambler can simulate any finite state gambler we show that for every rational 0

quantum generalization of the horn conjecture recall that the lie algebra of the special unitary group su n is isomorphic to the real vector space of traceless hermitian matrices as representations of su n and hence the terminology additive eigenvalue problem for su n in 1962 horn 17 gave a conjectural solution to a problem equivalent to the above additive eigenvalue problem for su n by a recursively determined system of inequalities in 19 klyachko gave a solution to the additive eigenvalue problem for su n in terms of a certain system of inequalities to write down this system we need to know which structure constants in the cohomology of gr r n written in the schubert basis 0 2 be a positive integer characterize the possible eigenvalues a1 a2 as of matrices a a 2 a e su n which satisfy a a a 1

brief paper a proof for non existence of periodic solutions in time invariant fractional order systems the aim of this note is to highlight one of the basic differences between fractional order and integer order systems it is analytically shown that a time invariant fractional order system contrary to its integer order counterpart cannot generate exactly periodic signals as a result a limit cycle cannot be expected in the solution of these systems our investigation is based on caputo s definition of the fractional order derivative and includes both the commensurate or incommensurate fractional order systems

biased sinusoidal disturbance compensation with unknown frequency this note combines an adaptive frequency estimation scheme with a fractional order controller for unknown biased sinusoidal disturbance rejection the disturbance is estimated via an adaptive orthogonal signals generator based on a third order generalized integrator a fractional order controller is designed which guarantees the closed loop stability of the system if the location of the plant frequency response at the estimated frequencies lies in an half plane passing through the origin of the complex plane

modelling fractal patterns via the qualitative equivalence of a nonlinear ode abstract a model of fractal patterns using the qualitative equivalence behavior of the nonlinear equation x ax 4 bx 3 cx 2 dx e where a b c d e r set of real numbers a 0 is presented from the point of view of the critical points of the equation each distinct phase portrait of the equation including the directional arrow corresponds to a fractal pattern the phase portraits when the critical points are complex valued present the more interesting cases in all a total of twenty eight 28 distinct fractal patterns can be generated via the equation up to isomorphism eighteen 18 of these arise when all the critical points are real the relevant qualitative properties are existence and uniqueness of solutions

dimensions of copeland erdos sequences the base k copeland erdos sequence given by an infinite set a of positive integers is the infinite sequence cemk a formed by concatenating the base k representations of the elements of a in numerical order this paper concerns the following four quantities the finite state dimension dimfs cek a a finite state version of classical hausdorff dimension introduced in 2001 the finite state strong dimension dimfs cek a a finite state version of classical packing dimension introduced in 2004 this is a dual of dimfs cek a satisfying dimfs cek a dimfs ce k a the zeta dimension dim a a kind of discrete fractal dimension discovered many times over the past few decades the lower zeta dimension dim a a dual of dim a satisfying dim a dim a we prove the following dimfs cek a dim a this extends the 1946 proof by copeland and erdos that the sequence cek primes is borel normal dimfs cek a dim a these bounds are tight in the strong sense that these four quantities can have simultaneously any four values in 0 1 satisfying the four above mentioned inequalities

finite state dimension and real arithmetic we use entropy rates and schur concavity to prove that for every integer k 2 every nonzero rational number q and every real number a the base k expansions of a q a and q a all have the same finite state dimension and the same finite state strong dimension this extends and gives a new proof of wall s 1949 theorem stating that the sum or product of a nonzero rational number and a borel normal number is always borel normal

fractal dimension and logarithmic loss unpredictability we show that the hausdorff dimension equals the logarithmic loss unpredictability for any set of infinite sequences over a finite alphabet using computable feasible and finite state predictors this equivalence also holds for the computable feasible and finite state dimensions combining this with recent results of fortnow and lutz proc 15th ann conf on comput learning theory 2002 380 we have a tight relationship between prediction with respect to logarithmic loss and prediction with respect to absolute loss

attractors for differential equations with variable delays using the relatively new concept of a pullback attractor we present some results on the existence of attractors for differential equations with variable delay we give a variety of examples to which our result applies

hidden symmetries and linearization of the modified painleve ince equation the linearization and hidden symmetries of the modified painleve ince equation y yy y3 0 where and are constants are presented the linearization of this equation by a nonlocal transformation yields a damped stable or growing unstable harmonic oscillator equation for 0 hidden symmetries are analyzed by transforming the modified painleve ince equation to a third order ordinary differential equation ode which in general is invariant under a three parameter group by a riccati transformation a type i hidden symmetry is found of a second order ode found from the third order ode where a symmetry is lost in the reduction of order by the non normal subgroup invariants a type ii hidden symmetry occurs in the third order ode because the symmetries of a second order ode reduced from the third order ode by another set of normal subgroup invariants are increased

multi order fractional differential equations and their numerical solution we consider the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 and a b n 1 b j b j 1 1 0 b 1 1 combined with suitable initial conditions the derivatives are understood in the caputo sense we begin by discussing the analytical questions of existence and uniqueness of solutions and we investigate how the solutions depend on the given data moreover we propose convergent and stable numerical methods for such initial value problems

remarks on fractional derivatives in this paper we further discuss the properties of three kinds of fractional derivatives the grunwald letnikov derivative the riemann liouville derivative and the caputo derivative especially we compare the riemann liouville derivative with the caputo derivative and sequential property of the caputo derivative is also derived which is helpful in translating the higher fractional order differential systems into lower ones besides we also compare the riemann liouville derivative and the caputo derivative with the classical derivative

fractional ince equation with a riemann liouville fractional derivative we extend the classical treatment of the ince equation to include the effect of a fractional derivative term of order a 0 and amplitude c a fourier expansion is used to determine the eigenvalue curves a in function of the parameter the stability domains and the periodic stable solutions of the fractional ince equation two important observations are the detachment of the eigenvalue curves from the a axis in the a plane as well as the appearance of degenerate eigenvalues for suitable selections of the parameters the fractional solutions valid for the steady state of the system are not orthogonal and have no defined parity we also introduce a discrete numerical method to evaluate the riemann liouville fractional derivative with lower terminal at for a class of functions the case a 1 represents the ince equation with an additional constant damping

gambling using a finite state machine sequential gambling schemes in which the amount wagered on the future outcome is determined by a finite state fs machine are defined and analyzed it is assumed that the fs machine determines the fraction of the capital wagered at each time instance i on the outcome at the next time instance i 1 and that wagers are paid at even odds the maximal capital achieved by any fs machine is found and its dependence on an empirical entropy measure h sup fs x defined as the finite state complexity of x is shown a specific gambling scheme is then proposed based on the lempel ziv method for universal compression the capital gained by this method is found and it is observed that asymptotically its exponential growth rate dominates the experimental growth rate achieved by gambling using any fs machine furthermore this specific scheme suggests a class of gambling methods based on a class of variable to variable length lossless compression methods in which the capital is doubled for every bit compressed these results emphasize the relation between gambling and data compression

generalised entropies and asymptotic complexities of languages the paper explores connections between asymptotic complexity and generalised entropy asymptotic complexity of a language a language is a set of finite or infinite strings is a way of formalising the complexity of predicting the next element in a sequence it is the loss per element of a strategy asymptotically optimal for that language generalised entropy extends shannon entropy to arbitrary loss functions it is the optimal expected loss given a distribution on possible outcomes it turns out that the set of tuples of asymptotic complexities of a language w r t different loss functions can be described by means of the generalised entropies corresponding to the loss functions

universal prediction of individual sequences the problem of predicting the next outcome of an individual binary sequence using finite memory is considered the finite state predictability of an infinite sequence is defined as the minimum fraction of prediction errors that can be made by any finite state fs predictor it is proven that this fs predictability can be achieved by universal sequential prediction schemes an efficient prediction procedure based on the incremental parsing procedure of the lempel ziv data compression algorithm is shown to achieve asymptotically the fs predictability some relations between compressibility and predictability are discussed and the predictability is proposed as an additional measure of the complexity of a sequence

hausdorff dimension and oracle constructions bennett and gill relative to a random oracle a pa npa co npa with probability 1 siam j comput 10 1981 96 113 proved that pa npa relative to a random oracle a or in other words that the set o p np a pa npa has lebesgue measure 0 in contrast we show that o p np has hausdorff dimension 1 this follows from a much more general theorem if there is a relativizable and paddable oracle construction for a complexity theoretic statement then the set of oracles relative to which holds has hausdorff dimension 1 we give several other applications including proofs that the polynomial time hierarchy is infinite relative to a hausdorff dimension 1 set of oracles and that pa npa conpa relative to a hausdorff dimension 1 set of oracles

on quasi periodic properties of fractional sums and fractional differences of periodic functions this article is devoted to the study of discrete fractional calculus our goal is to investigate quasi periodic properties of fractional order sums and differences of periodic functions using riemann liouville and caputo type definitions we study concepts close to the well known idea of periodic function such as asymptotically periodicity or s asymptotically periodicity we use basic tools of discrete fractional calculus boundedness of sums and differences of fractional order of a given periodic function is also investigated