phase margins for multivariable control systems this paper presents a new definition for the phase margin of multivariable systems the properties of the phase margin are such that a stable closed loop multi input multi output system will remain stable for all unitary perturbations in the feedback path whose phases are less than the phase margin of the system it is also shown that the definition of phase margin is invariant under unitary transformations of the system

comments on stability margin evaluation for uncertain linear systems the bound derived in the above paper by gong thompson ibid vol 39 no 3 p 548 50 1994 is shown via two new stability criteria not to be the tightest possible for all unstructured perturbations a minor error of their theorem is also corrected

presentation functions fixed points and a theory of scaling function dynamics presentation functions provide the time ordered points of the forward dynamics of a system as successive inverse images they generally determine objects constructed on trees regular or otherwise and immediately determine a functional form of the transfer matrix of these systems presentation functions for regular binary trees determine the associated forward dynamics to be that of a period doubling fixed point they are generally parametrized by the trajectory scaling function of the dynamics in a natural way the requirement that the forward dynamics be smooth with a critical point determines a complete set of equations whose solution is the scaling function these equations are compatible with a dynamics in the space of scalings which is conjectured with numerical and intuitive support to possess its solution as a unique globally attracting fixed point it is argued that such dynamics is to be sought as a program for the solution of chaotic dynamics in the course of the exposition new information pertaining to universal mode locking is presented

maps of intervals with indifferent fixed points thermodynamic formalism and phase transitions we develop the thermodynamic formalism for a large class of maps of the interval with indifferent fixed points for such systems the formalism yields onedimensional systems with many body infinite range interactions for which the thermodynamics is well defined but gibbs states are not piecewise linear systems of this kind yield the soluble in a sense fisher models we prove that such systems exhibit phase transitions the order of which depends on the behavior at the indifferent fixed points we obtain the critical exponent describing the singularity of the pressure and analyze the decay of correlations of the equilibrium states at all temperatures our technique relies on establishing and exploiting a relation between the transfer operators of the original map and its suitable expanding induced version the technique allows one also to obtain a version of the bowen ruelle formula for the hausdorff dimension of repellers for maps with indifferent fixed points and to generalize fisher results to some nonsoluble models

polar codes for q ary channels q 2 r we study polarization for nonbinary channels with input alphabet of size q 2r r 2 3 using arikan s polarizing kernel h2 we prove that the virtual channels that arise in the process of channel evolution converge to q ary channels with capacity 0 1 2 r bits as a result of this analysis we show that polar codes support reliable transmission over discrete memoryless channels with q ary input for all rates below the symmetric capacity of the channel this leads to an explicit transmission scheme for q ary channels the block error probability of decoding using successive cancellation behaves as exp n where n is the code length and is any constant less than 0 5

linear and nonlinear binary kernels of polar codes of small dimensions with maximum exponents polar codes are constructed based on kernels with polarizing properties the performance of a polar code is characterized asymptotically in terms of the exponent of its kernel the pioneering work of arikan on polar codes is based on a linear kernel of dimension two and exponent 0 5 in this paper constructions of linear and nonlinear binary kernels of dimensions up to 16 are presented the kernels are obtained using computer search or by shortening longer kernels obtained by computer search and a computer program is used to determine their exponents it is proved that the constructed kernels have maximum exponents except in the case of nonlinear kernels of dimension 12 where it is demonstrated that the maximum exponent either equals that of the presented construction or assumes another specified value the results show that the minimum dimension for which there exists a linear kernel with exponent greater than 0 5 i e exceeds the exponent of the linear kernel proposed by arikan is 15 while this minimum dimension is 14 for nonlinear kernels furthermore it is shown that there is a linear kernel with maximum exponent up to dimension 11 for dimensions 13 14 15 and 16 there are nonlinear kernels with exponents larger than any of that of a linear kernel the kernels of these dimensions that have maximum exponent although nonlinear over textrm gf 2 are mathbb z 4 linear or mathbb z 2 mathbb z 4 linear

the minimum distance of codes in an array coming from telescopic semigroups the concept of an error correcting array gives a new bound on the minimum distance of linear codes and a decoding algorithm which decodes up to half this bound this gives a unified point of view which explains several improvements on the minimum distance of algebraic geometric codes moreover it is explained in terms of linear algebra and the theory of semigroups only

a generalization of strong kharitonov theorems to polytopes of polynomials the root clustering property of a certain class of polytopes of polynomials is examined it is shown that the polytopes of polynomials have only zeros within the left sector if and only if a subset of their edge polynomials has only zeros within the left sector certain conditions are derived under which the root clustering property of the polytopes of polynomials can be inferred from a subset of their vertices

number of first passage times as a measurement of information for weakly chaotic systems we consider a general class of maps of the interval having lyapunov subexponential instability xt x0 exp t x0 t where t grows sublinearly as t we outline here a scheme j stat phys 154 988 2014 whereby the choice of a characteristic function automatically defines the map equation and corresponding growth rate t this matching approach is based on the infinite measure property of such systems we show that the average information that is necessary to record without ambiguity a trajectory of the system tends to t suitably extending the kolmogorov sinai entropy and pesin s identity for such systems information behaves like a random variable for random initial conditions its statistics obeying a universal mittag leffler law we show that for individual trajectories information can be accurately inferred by the number of first passage times through a given turbulent phase space cell this enables us to calculate far more efficiently lyapunov exponents for such systems lastly we also show that the usual renewal description of jumps to the turbulent cell usually employed in the literature does not provide the real number of entrances there our results are supported by exhaustive numerical simulations

a counterexample of comments on stability margin evaluation for uncertain linear systems in the original paper c gong and s thompson ibid vol 39 p 548 50 1994 there is given what is claimed to be the tightest bound for all unstructured pertubations such that a specified system keeps its asymptotic stability j h su ibid vol 39 p 2523 4 1994 showed that it is not the tightest bound and gives another bound that bound is shown by a counterexample to be incorrect

a dual result to kharitonov s theorem a dual problem to kharitonov s problem involving a diamond instead of a rectangle is considered the results show that a family of polynomials with coefficients varying in the diamond is strictly hurwitz if and only if eight one dimensional exposed edges of the diamond are strictly hurwitz

exact invariant measures how the strength of measure settles the intensity of chaos the aim of this paper is to show how to extract dynamical behavior and ergodic properties from deterministic chaos with the assistance of exact invariant measures on the one hand we provide an approach to deal with the inverse problem of finding nonlinear interval maps from a given invariant measure then we show how to identify ergodic properties by means of transitions along the phase space via exact measures on the other hand we discuss quantitatively how infinite measures imply maps having subexponential lyapunov instability weakly chaotic as opposed to finite measure ergodic maps which are fully chaotic in addition we provide general solutions of maps for which infinite invariant measures are exactly known throughout the interval a demand from this field finally we give a simple proof that infinite measure implies universal mittag leffler statistics of observables rather than narrow distributions typically observed in finite measure ergodic maps

robustness analysis for linear dynamical systems with linearly correlated parametric uncertainties the authors propose an approach for robust pole location analysis of linear dynamical systems with parametric uncertainties linear control systems with characteristic polynomials whose coefficients are affine in a vector of uncertain physical parameters are considered a design region in complex plane for system pole placement and a nominal parameter vector generating a characteristic polynomial with roots in that region are given the proposed method allows the computation of maximal domains bounded by linear inequalities and centered at the nominal point in system parameter space preserving system poles in the given region the solution of this problem is shown to also solve the problem of testing robot location of a given polytope of polynomials in parameter space it is proved that for stability problems for continuous time systems with independent perturbations on polynomial coefficients this method generates the four extreme kharitonov polynomials

lyapunov statistics and mixing rates for intermittent systems we consider here a recent conjecture stating that correlation functions and tail probabilities of finite time lyapunov exponents would have the same power law decay in weakly chaotic systems we demonstrate that this conjecture fails for a generic class of maps of the pomeau manneville type we show further that typically the decay properties of such tail probabilities do not provide significant information on key aspects of weakly chaotic dynamics such as ergodicity and instability regimes our approaches are firmly based on rigorous results particularly the aaronson darling kac theorem and are also confirmed by exhaustive numerical simulations

a first order lyapunov robustness method for linear systems with uncertain parameters a method for stability robustness analysis based on a quadratic lyapunov function that varies linearly with uncertainty parameters is derived linear time invariant systems with structured uncertainties are discussed the lyapunov function is optimized numerically to maximize the robustness region in parameter space numerical results are given for four examples in which the first order method is compared to previous lyapunov methods for robustness analysis while the zero order method is slightly better than the first order method for one example the first order method is clearly superior in the other three more realistic examples the first order method is especially superior for applications to active control of flexible structures where robustness with respect to unmodeled coupling between modeled modes are important issues for such applications the first order method is much better at detecting the increased robustness associated with increased separation between frequencies

stability margin evaluation for uncertain linear systems a sufficient stability condition for parameter variation in a perturbed continuous time multivariable linear system represented by a state space model is presented starting with the existence of an algebraic riccati equation a stability bound is derived from the polar decomposition of the nominal system matrix unlike previous work the results are not dependent on the solution of the lyapunov equation and consequently not a function of an arbitrarily selected positive definite matrix in addition the bound would appear to be the tightest possible in that violation of the presented inequality can be shown to lead to instability

sporadicity between periodic and chaotic dynamical behaviors abstract we define the class of sporadic dynamical systems as the systems where the algorithmic complexity of kolmogorov kolmogorov a n 1983 russ math surv 38 29 40 and chaitin chaitin g j 1987 algorithmic information theory cambridge univ press cambridge u k as well as the logarithm of separation of initially nearby trajectories grow as nv0 log n v1 with 0 v0 1 or v0 1 and v1 0 as time n these systems present a behavior intermediate between the multiperiodic v0 0 v1 1 and the chaotic ones v0 1 v1 0 we show that intermittent systems of manneville manneville p 1980 j phys paris 41 1235 1243 as well as some countable markov chains may be sporadic and furthermore that the dynamical fluctuations of these systems may be of levy s type rather than gaussian

stability robustness bounds for linear state space models with structured uncertainty in this note we consider the robust stability analysis problem in linear state space models we consider systems with structured uncertainty some lower bounds on allowable perturbations which maintain the stability of a nominally stable system are derived these bounds are shown to be less conservative than the existing ones

thermodynamic phase transitions for pomeau manneville maps we study phase transitions in the thermodynamic description of pomeau manneville intermittent maps from the point of view of infinite ergodic theory which deals with diverging measure dynamical systems for such systems we use a distributional limit theorem to provide both a powerful tool for calculating thermodynamic potentials as also an understanding of the dynamic characteristics at each instability phase in particular topological pressure and renyi entropy are calculated exactly for such systems finally we show the connection of the distributional limit theorem with non gaussian fluctuations of the algorithmic complexity proposed by gaspard and wang proc natl acad sci usa 85 4591 1988

vardohus codes polar codes based on castle curves kernels in this paper we show some applications of algebraic curves to the construction of kernels of polar codes over a discrete memoryless channel which is symmetric w r t the field operations we will also study the minimum distance of the polar codes proposed their duals and the exponents of the matrices used for defining them all the restrictions that we make to our curves will be accomplished by the so called castle curves

polar codes characterization of exponent bounds and constructions polar codes were recently introduced by arikan they achieve the symmetric capacity of arbitrary binary input discrete memoryless channels under a low complexity successive cancellation decoding scheme the original polar code construction is closely related to the recursive construction of reed muller codes and is based on the 2 2 matrix 1 0 1 1 it was shown by arikan telatar that this construction achieves an error exponent of 1 2 i e that for sufficiently large blocklengths the error probability decays exponentially in the square root of the blocklength it was already mentioned by arikan that in principle larger matrices can be used to construct polar codes in this paper it is first shown that any l l matrix none of whose column permutations is upper triangular polarizes binary input memoryless channels the exponent of a given square matrix is characterized upper and lower bounds on achievable exponents are given using these bounds it is shown that there are no matrices of size smaller than 15 15 with exponents exceeding 1 2 further a general construction based on bch codes which for large i achieves exponents arbitrarily close to 1 is given at size 16 16 this construction yields an exponent greater than 1 2

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors the analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective liapunov exponents for hyperbolic attractors the latter determines the metric entropies and provides one constraint on the partial dimensions for nonhyperbolic attractors there are important modifications we discuss them for the examples of the logistic and henon map we show in particular that the generalized dimensions have singularities with noncontinuous derivative similar to first order phase transitions in statistical mechanics

alternative numerical computation of one sided levy and mittag leffler distributions we consider here the recently proposed closed form formula in terms of the meijer g functions for the probability density functions g ensuremath alpha x of one sided l evy stable distributions with rational index ensuremath alpha l k with 0l ensuremath alpha l1 since one sided l evy and mittag leffler distributions are known to be related this formula could also be useful for calculating the probability density functions ensuremath rho ensuremath alpha x of the latter we show however that the formula is computationally inviable for fractions with large denominators being unpractical even for some modest values of l and k we present a fast and accurate numerical scheme based on an early integral representation due to mikusinski for the evaluation of g ensuremath alpha x and ensuremath rho ensuremath alpha x their cumulative distribution function and their derivatives for any real index ensuremath alpha ensuremath in 0 1 as an application we explore some properties of these probability density functions in particular we determine the location and value of their maxima as functions of the index ensuremath alpha we show that ensuremath alpha ensuremath approx 0 567 and 0 605 correspond respectively to the one sided l evy and mittag leffler distributions with shortest maxima we close by discussing how our results can elucidate some recently described dynamical behavior of intermittent systems

codes from the suzuki function field we construct algebraic geometry ag codes from the function field f 2 sup 2n 1 x y f 2 sup 2n 1 defined by y 2 sup 2n 1 y x 2 sup 2n x where n is a positive integer these codes are supported by two places and many have parameters that are better than those of any comparable code supported by one place of the same function field to define such codes we determine and exploit the structure of the weierstrass gap set of an arbitrary pair of rational places of f 2 sup 2n 1 x y f 2 sup 2n 1 moreover we find some codes over f sub 8 with parameters that are better than any known code

source and channel polarization over finite fields and reed solomon matrices polarization phenomenon over any finite field fq with size q being a power of a prime is considered this problem is a generalization of the original proposal of channel polarization by arikan for the binary field as well as its extension to a prime field by sasoglu telatar and arikan in this paper a necessary and sufficient condition of a matrix over a finite field fq is shown under which any source and channel are polarized furthermore the result of the speed of polarization for the binary alphabet obtained by arikan and telatar is generalized to arbitrary finite field it is also shown that the asymptotic error probability of polar codes is improved by using the reed solomon matrices which can be regarded as a natural generalization of the 2 2 binary matrix used in the original proposal by arikan