encoding tasks and renyi entropy a task is randomly drawn from a finite set of tasks and is described using a fixed number of bits all the tasks that share its description must be performed upper and lower bounds on the minimum pth moment of the number of performed tasks are derived the case where a sequence of tasks is produced by a source and n tasks are jointly described using nr bits is considered if r is larger than the renyi entropy rate of the source of order 1 1 provided it exists then the th moment of the ratio of performed tasks to n can be driven to one as n tends to infinity if r is smaller than the renyi entropy rate this moment tends to infinity the results are generalized to account for the presence of side information in this more general setting the key quantity is a conditional version of renyi entropy that was introduced by arimoto for iid sources two additional extensions are solved one of a rate distortion flavor and the other where different tasks may have different nonnegative costs finally a divergence that was identified by sundaresan as a mismatch penalty in the massey arikan guessing problem is shown to play a similar role here

uniform random number generation from markov chains non asymptotic and asymptotic analyses in this paper we derive non asymptotic achievability and converse bounds on the random number generation with without side information our bounds are efficiently computable in the sense that the computational complexity does not depend on the block length we also characterize the asymptotic behaviors of the large deviation regime and the moderate deviation regime by using our bounds which implies that our bounds are asymptotically tight in those regimes we also show the second order rates of those problems and derive single letter forms of the variances characterizing the second order rates furthermore we address the relative entropy rate and the modified mutual information rate for these problems

oscillatory critical amplitudes in hierarchical models we study the oscillatory critical amplitudes of theq states potts model on a diamond hierarchical lattice we consider an example of the generic case finite critical index as well as the degenerate case essential singularity in both cases we compare the magnitude of the oscillations with geometrical characteristics of the julia set of zeroes of the partition function

the spectrum of a quasiperiodic schrodinger operator the spectrum h of the tight binding fibonacci hamiltonian hmn m n 1 m 1 n m n v n v n chi omega 3 omega 2 n 1 1 is the golden number is shown to coincide with the dynamical spectrum the set on which an infinite subsequence of traces of transfer matrices is bounded the point spectrum is absent for any and h is a cantor set for 4 combining this with casdagli s earlier result one finds that the spectrum is singular continuous for 16

electronic properties of the tight binding fibonacci hamiltonian the analysis of the electronic properties of the tight binding fibonacci hamiltonian is carried out using dynamical systems techniques two classes of fibonacci sequences are considered corresponding to the cases when there are two types of building blocks and also when there are three types of building blocks the recursion relations for the traces of the transfer matrices are determined and studied for various extensions of the fibonacci case with the golden mean some differences are obtained between the various types of second order fibonacci sequences and are most likely due to long range order applications of this work to the transmission of light through a multilayered medium and the electrical resistance of a one dimensional quasicrystal as determined by the landauer formula are presented

the landauer conductance of generalised fibonacci superlattices numerical results abstract the quantum model for quasi one dimensional generalised fibonacci superlattices gfs is formulated the landauer electrical conductance l of gfs is studied numerically it is shown that l as a function of incident particle energy e oscillates strongly and exhibits resonant character we have verified numerically that l e reaches its local maximum for e corresponding to energy eigenenvalues of charge in superlattices

remarks on the spectral properties of tight binding and kronig penney models with substitution sequences we comment on some recent investigations on the electronic properties of models associated with the thue morse chain point out that their conclusions contradict rigorously proven theorems and indicate some of the sources of these misinterpretations we briefly review and explain the current status of mathematical results in this field and discuss some conjectures and open problems

on the hausdorff dimension of the spectrum of the thue morse hamiltonian we show that the hausdorff dimension of the spectrum of the thue morse hamiltonian has a common positive lower bound for all coupling

csisza spl acute r s cutoff rates for the general hypothesis testing problem in csisza spl acute r established the concept of forward spl beta cutoff rate for the error exponent hypothesis testing problem based on independent and identically distributed i i d observations given spl beta 0 spl alpha spl ne 1 similarly for 0 spl beta 1 csisza spl acute r also established the concept of reverse spl beta cutoff rate for the correct exponent hypothesis testing problem in this work we extend csisza spl acute r s results by investigating the forward and reverse spl beta cutoff rates for the hypothesis testing between two arbitrary sources with memory we demonstrate that the lim inf re spl acute nyi spl alpha divergence rate provides the expression for the forward spl beta cutoff rate we also show that if the log likelihood large deviation spectrum admits a limit then the reverse spl beta cutoff rate equals the liminf spl alpha divergence rate where spl alpha 1 1 spl beta and 0 spl beta spl beta sub max where spl beta sub max is the largest spl beta 1 for which the lim inf 1 1 spl beta divergence rate is finite for spl beta sub max spl les spl beta 1 we show that the reverse cutoff rate is in general only upper bounded by the lim inf re spl acute nyi divergence rate unlike in where the alphabet for the source coding cutoff rate problem was assumed to be finite we assume arbitrary countable or continuous source alphabet we also provide several examples to illustrate our forward and reverse spl beta cutoff rates results and the techniques employed to establish them

some properties of renyi entropy and renyi entropy rate in this paper we define the conditional renyi entropy and show that the so called chain rule holds for the renyi entropy then we introduce a relation for the rate of renyi entropy and use it to derive the rate of the renyi entropy for an irreducible aperiodic markov chain we also show that the bound for the renyi entropy rate is simply the shannon entropy rate

symmetric duality with invexity in variational problems abstract weak and strong duality results are established under invexity hypotheses for symmetric dual variational problems without positivity constraints self dual problems and static symmetric dual programs are included as special cases

anomalous thermostat and intraband discrete breathers abstract we investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice including quenched disorder the coupling between both parts is bilinear elimination of the harmonic degrees of freedom leads to a nonlinear langevin equation with memory kernels t and noise term t for the anharmonic coordinates q t q t for zero temperature i e for t 0 we prove that the support of the fourier transform of t and of the time averaged velocity velocity correlation functions k t of the anharmonic system cannot overlap as a consequence the asymptotic solutions can be constant periodic quasiperiodic or almost periodic and possibly weakly chaotic for a sinusoidal trajectory q t with frequency we find that the energy e t transferred to the harmonic system up to time t is proportional to t if equals one of the phonon frequencies it is 2 we prove that there is a zero measure set l such that for in its full measure complement r l it is 0 i e there is no energy dissipation under certain conditions l contains a subset l such that for l the dissipation rate is nonzero and may be subdissipative 0 1 or superdissipative 1 2 compared to ordinary dissipation 1 consequently the harmonic bath does act as an anomalous thermostat in variance with the common belief that elimination of a macroscopically large number of degrees of freedom always generates dissipation forcing convergence to equilibrium intraband discrete breathers are such solutions which do not relax we prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency exist for all in a cantor set c k of finite lebesgue measure this is achieved by estimating the contribution of small denominators appearing for g t related to t for c k the small denominators do not lead to divergencies such that g t is a smooth and bounded function in t

absence of localization in a class of schrodinger operators with quasiperiodic potential we prove that a class of discrete schrodinger operators with a quasiperiodic potential taking only a finite number of values exhibits purely continuous spectrum in particular they cannot have localized eigenvectors

spectral continuity for aperiodic quantum systems i general theory abstract how does the spectrum of a schrodinger operator vary if the corresponding geometry and dynamics change is it possible to define approximations of the spectrum of such operators by defining approximations of the underlying structures in this work a positive answer is provided using the rather general setting of groupoid c algebras a characterization of the convergence of the spectra by the convergence of the underlying structures is proved in order to do so the concept of continuous field of groupoids is slightly extended by adding continuous fields of cocycles with this at hand magnetic schrodinger operators on dynamical systems or delone systems fall into this unified setting various approximations used in computational physics like the periodic or the finite cluster approximations are expressed through the tautological groupoid which provides a universal model for fields of groupoids the use of the hausdorff topology turns out to be fundamental in understanding why and how these approximations work

frustration effects in quasicrystals an exactly soluble example in one dimension frustration effects due to competing interactions between magnetic atoms in quasicrystals are studied in an exactly soluble example the ising chain in a quasiperiodic two valued magnetic field at zero temperature the model exhibits an infinity of pure phases characterised by a modulation of the order parameter which reflects the aperiodic structure of the system up to a larger and larger length scale

escher like quasiperiodic heterostructures quasiperiodic heterostructures present unique structural electronic and vibrational properties connected to the existence of incommensurate periods we go beyond previous schemes such as fibonacci or thue morse based on substitutional sequences by introducing construction rules generated by tessellations of the unit disc by regular polygons we explore some of the properties exhibited by these systems

repeller structure in a hierarchical model i topological properties the repeller associated with the renormalization dynamics of the spectral problem of a hierarchical tight binding schrodinger equation is studied analysis of escaping regions and of stable and unstable manifolds provide complementary descriptions of the recurrent set whose structure undergoes relevant changes when the growth rater of the potential barriers is modified the minimal region containing the repeller is determined and the mechanism originating a cantor set structure along the unstable manifold is revealed the repeller is continuous along the stable manifold forr 2 finally we show the existence of a pointlike component of the spectrum located at its upper extremum forr 1 and we present the associated wavefunctions

critical properties of electron eigenstates in incommensurate systems this paper describes some properties of the eigenvalue equation n 1 n 1 2 cos 2 n n e n this is an example of the more general problem of a hermitian eigenvalue equation in the form of a difference equation with periodic coefficients these equations arise in solid state physics they occur in connection with tight binding models for electrons in one dimensional solids with an incommensurate modulation of the structure and in models for the energy bands of bloch electrons moving in a plane with a perpendicular magnetic field the model studied has a critical point when 1 following some earlier work by azbel azbel m ya phys rev lett 43 1954 1979 an approximate renormalization group transformation is derived this predicts that the spectrum and eigenstates have a remarkable recursive structure at the critical point which is dependent on the expansion of as a continued fraction also when is an irrational number there is a localization transition from extended states to localized states as increases through the critical point this localization transition which was previously discovered by aubry andre aubry s andre g ann israel phys soc 3 133 1979 using the thouless formula for the localization length is explained by the renormalization group transformation derived here

transmission fingerprints of quasi periodic chains abstract we propose a method to characterize quasi periodic chains based on the transmission probabilities we consider finite layers which are building up following the fibonaci thue morse and cantor sequences the spectra show an undoubtedly fractal behavior with a distinct appearence for each chain furthermore we construct return maps which present the same characteristic attractor for all peaks of energies within the same kind of chain

renyi divergence and kullback leibler divergence renyi divergence is related to renyi entropy much like kullback leibler divergence is related to shannon s entropy and comes up in many settings it was introduced by renyi as a measure of information that satisfies almost the same axioms as kullback leibler divergence and depends on a parameter that is called its order in particular the renyi divergence of order 1 equals the kullback leibler divergence we review and extend the most important properties of renyi divergence and kullback leibler divergence including convexity continuity limits of sigma algebras and the relation of the special order 0 to the gaussian dichotomy and contiguity we also show how to generalize the pythagorean inequality to orders different from 1 and we extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs for all orders and present several other minimax results

influence of the anisotropy parameter δ on the spectrum of the generalized q symmetrized harper equation the 1 generalization of the q symmetrized harper equation is discussed in terms of wavefunctions expressed by laurent series proceeding by recursion leads to a nontrivial dependent generalization of the characteristic energy polynomial with a special emphasis on a continuous dependence on the commensurability parameter the multiplicity parameter which is responsible for the amount of coprime realizations of the commensurability parameter is also accounted for the present energies have been derived so as to reproduce particular ones obtained before as limiting cases

some remarks on discrete aperiodic schrodinger operators we consider schrodinger operators onl2 with deterministic aperiodic potential and schrodinger operators on the l2 space of the set of vertices of penrose tilings and other aperiodic self similar tilings the operators onl2 fit into the formalism of ergodic random schrodinger operators hence their lyapunov exponent integrated density of states and spectrum are almost surely constant we show that they are actually constant the lyapunov exponent for one dimensional schrodinger operators with potential defined by a primitive substitution the integrated density of states and the spectrum in arbitrary dimension if the system is strictly ergodic we give examples of strictly ergodic schrodinger operators that include several kinds of almost periodic operators that have been studied in the literature for schrodinger operators on penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular penrose tiling under consideration

symmetric duality for a class of nondifferentiable multi objective fractional variational problems abstract we introduce a symmetric dual pair for a class of nondifferentiable multi objective fractional variational problems weak strong converse and self duality relations are established under certain invexity assumptions the paper includes extensions of previous symmetric duality results for multi objective fractional variational problems obtained by kim lee and schaible d s kim w j lee s schaible symmetric duality for invex multiobjective fractional variational problems j math anal appl 289 2004 505 521 and symmetric duality results for the static case obtained by yang wang and deng x m yang s y wang x t deng symmetric duality for a class of multiobjective fractional programming problems j math anal appl 274 2002 279 295 to the dynamic case

symbolic dynamics for the renormalization map of a quasiperiodic schrodinger equation a rigorous analysis is given of the dynamics of the renormalization map associated to a discrete schrodinger operatorh onl 2 defined byh n n 1 n 1 vf n n wherev is a real parameter f is a certain discontinuous period 1 function and sigma left 1 sqrt 5 right mathord left vphantom left 1 sqrt 5 right 2 right kern nulldelimiterspace 2 is the golden mean the renormalization map forh is a diffeomorphism t of 3 preserving a cubic surfaces v forv 8 we prove that the non wandering set of the restriction oft tos v is a hyperbolic set on whicht is conjugate to a subshift on six symbols it follows from results in dynamical systems theory that the optimally approximating periodic operators toh have spectra which obey a global scaling law we also define a set which we call the pseudospectrum of the operatorh we prove it to be a cantor set of measure zero and obtain bounds on its hausdorff dimension it is an open question whether the pseudospectrum coincides with the spectrum ofh

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

critical points for finite fibonacci chains of point delta interactions and orthogonal polynomials for a one dimensional schrodinger operator with a finite number n of point delta interactions with a common intensity the parameters are the intensity the n 1 intercenter distances and the mass critical points are points in the parameters space of the hamiltonian where one bound state appears or disappears the study of critical points for hamiltonians with point delta interactions arranged along a fibonacci chain is shown to be closely related to the study of the so called fibonacci operator a discrete one dimensional schrodinger type operator which occurs in the context of tight binding hamiltonians these critical points are the zeros of orthogonal polynomials previously studied in the context of special diatomic linear chains with elastic nearest neighbor interaction properties of the zeros location asymptotic behavior gaps are investigated the perturbation series from the solvable periodic case is determined the measure which yields orthogonality is investigated numerically from the zeros it is shown that the transmission coefficient at zero energy can be expressed in terms of the orthogonal polynomials and their associated polynomials in particular it is shown that when the number of point delta interactions is equal to a fibonacci number minus 1 i e when the intervals between point delta interactions form a palindrome all the fibonacci chains at critical points are completely transparent at zero energy

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

on kicked systems modulated along the thue morse sequence we give necessary conditions for the fourier transform of the time evolution operator of kicked systems modulated along the thue morse sequence to present dirac s delta functions the case of two level systems is analysed in detail and it is found based on the trace map and numerical investigations that generically its time evolution has no quasi periodic component

symmetric duality for the multiobjective fractional variational problem with partial invexity a symmetric dual problem for a class of multiobjective fractional variational problems is first formulated under partial invexity assumptions on the functions involved dual theorems are proved through a parametric approach to related properly efficient solutions of the primal and dual problems finally self dual and related problems are considered

block variables for deterministic aperiodic sequences we use the concept of block variables to obtain a measure of order disorder for some one dimensional deterministic aperiodic sequences for the thue morse sequence the rudin shapiro sequence and the period doubling sequence it is possible to obtain analytical expressions in the limit of infinite sequences for the fibonacci sequence we present some analytical results which can be supported by numerical arguments it turns out that the block variables show a wide range of different behaviour some of them indicating that some of the considered sequences are more random than other however the method does not give any definite answer to the question of which sequence is more disordered than the other and in this sense the results obtained are negative we compare this with some other ways of measuring the amount of order disorder in such systems and there seems to be no direct correspondence between the measures

a family of generalized fibonacci lattices self similarity and scaling of the wavefunction a family of generalized fibonacci lattices which are locally isomorphic and exhibit a peculiar type of self similarity have been found to exist by shifting the position of the strip initial phase in the projection method the necessary and sufficient conditions of generating such self similar quasiperiodic lattices are established the power law growth behaviour of the wavefunction at e 0 in the off diagonal model defined on some of these lattices has been analysed it is shown that although various structures resulting from different initial phases are in a local isomorphism class they lead to a variety of maximum exponents of power for the scaling of the wavefunction

the renyi redundancy of generalized huffman codes huffman s algorithm gives optimal codes as measured by average codeword length and the redundancy can be measured as the difference between the average codeword length and shannon s entropy if the objective function is replaced by an exponentially weighted average then a simple modification of huffman s algorithm gives optimal codes the redundancy can now be measured as the difference between this new average and a renyi s 1961 generalization of shannon s entropy by decreasing some of the codeword lengths in a shannon code the upper bound on the redundancy given in the standard proof of the noiseless source coding theorem is improved the lower bound is improved by randomizing between codeword lengths allowing linear programming techniques to be used on an integer programming problem these bounds are shown to be asymptotically equal the results are generalized to the renyi case and are related to r g gallager s 1978 bound on the redundancy of huffman codes

the kullback leibler divergence rate between markov sources in this work we provide a computable expression for the kullback leibler divergence rate lim sub n spl rarr spl infin 1 nd p sup n spl par q sup n between two time invariant finite alphabet markov sources of arbitrary order and arbitrary initial distributions described by the probability distributions p sup n and q sup n respectively we illustrate it numerically and examine its rate of convergence the main tools used to obtain the kullback leibler divergence rate and its rate of convergence are the theory of nonnegative matrices and perron frobenius theory similarly we provide a formula for the shannon entropy rate lim sub n spl rarr spl infin 1 nh p sup n of markov sources and examine its rate of convergence

wave propagation through cantor set media chaos scaling and fractal structures propagation of waves through cantor set media is investigated by renormalization group analysis for specific values of wave numbers transmission coefficients are shown to be governed by the logistic map and in the chaotic region they show sensitive dependence on small changes in parameters of the system such as the index of refraction for other values of wave numbers our numerical results suggest that light transmits completely or reflects completely by the cantor set media text c ensuremath infty it is also shown that transmission coefficients exhibit a local scaling behavior near complete transmission if the complete transmission is achieved at a wave number ensuremath kappa ensuremath kappa ensuremath ast with a rational ensuremath kappa ensuremath ast ensuremath pi the scaling function is obtained analytically by using the euler totient function and the local scaling behavior is confirmed numerically

beyond bragg mirrors the design of aperiodic omnidirectional multilayer reflectors a new method that enables omnidirectional reflectivity for all polarizations of incident light over a wide selectable range of wavelengths was used to design aperiodic multilayer mirrors choosing the materials and with the desired threshold value of the reflection coefficient also being given the resulting reflectors work for all polarizations in a predetermined range of incidence angles and wavelengths they are 1d photonic aperiodic crystals composed of a stack of layers arranged according to a deterministic aperiodic substitutive sequence appropriately determined in the calculation of the optical multilayer properties with the rayleigh abeles matrices the use of the computationally efficient building block recurrence or trace antitrace map techniques avoids numerical instability and strongly reduces computational time experimental measurements of a prototype satisfying predetermined requirements fabricated after the sequence generated by a b bba bbba show excellent agreement

wave propagation in quasiperiodic structures stop pass band distribution and prestress effects the band structure of dispersion diagrams for axial and flexural waves of quasiperiodic infinite beams is investigated every structure is composed of a repeated elementary cell generated adopting the fibonacci sequence different terms of this sequence are investigated the problem is then solved evaluating the transmission matrix of the cell and applying the floquet bloch technique in the case of axial vibrations a homogeneous rod where the quasiperiodicity is given by the particular distribution of external springs is considered while for the flexural problem the quasiperiodicity is defined in terms of relative distance between the simple supports that sustain the beam in both cases it is shown that for different fibonacci sequences the number of stop pass bands within a defined range of frequencies changes and follows the fibonacci recursion rule showing a self similar pattern in addition the overall dispersion characteristic can be interpreted in terms of an invariant function of the circular frequency independent of the sequence generating the elementary cell for flexural waves the effects of the axial prestress on dispersion diagrams is highlighted and the frequency shift of the stop pass band positions is quantified it is noted that a tensile axial prestress promotes length reduction of pass bands while leaves almost unchanged the length of stop band intervals

aperiodic and correlated disorder in xy chains exact results we study thermodynamic properties surface magnetization specific heat and susceptibility of xy quantum chains with coupling constants following arbitrary substitution sequences generalizing an exact renormalization group rg transformation originally formulated for ising quantum chains we obtain exact relevance criteria of harris luck type for this class of models for two letter substitution rules a detailed classification is given of sequences leading to irrelevant marginal or relevant aperiodic modulations we find that the relevance of the same aperiodic sequence of couplings in general will be different for xy and ising quantum chains by our method continuously varying critical exponents may be calculated exactly for arbitrary two letter substitution rules with marginal aperiodicity a number of examples are given including the period doubling three folding and precious mean chains we also discuss extensions of the renormalization approach to a special class of long range correlated random chains generated by random substitutions

electronic stability of disordered systems application to quasicrystals the author studies energetic stability of disordered binary alloys in which atoms are interacting via a tight binding hamiltonian in 1d for a weak potential the gain in energy compared to the periodic linear chain is derived as a function of the electronic filling factor and the disorder very precise results are given in the case of a quasiperiodic arrangement this kind of phase plays a very particular role since it is shown that for a given filling factor v or fermi level position the most stable structure is a quasiperiodic one associated to v moreover for this structure the fluctuations of the density are unbounded if v and the stoichiometric coefficients do not fulfil an arithmetical condition different features are observed for rational and irrational values of v

structure factor of 1d systems superlattices based on two letter substitution rules i delta bragg peaks the recent generalization to the case of arbitrary tile lengths and arbitrary scattering factors of the calculation of the structure factor of 1d substitutional systems is studied in detail this method makes a easy to find all the peaks in the diffraction spectrum of a system the well known periodic and quasiperiodic spectra with delta peaks at integer multiples of a single number and integer linear combinations of two incommensurate frequencies respectively were found to be the l 0 subsets of two more general types of spectra infinite periodic or limit periodic and infinite quasiperiodic or limit quasiperiodic characterized by rational numbers of the type m nl l 0 infinity in place of the above integers substitution rules that produce quasicrystalline quasiperiodic and infinite quasiperiodic spectra give the same type of spectrum for all values of the ratio rho rho a rho b of the two tile lengths rho a and rho b this is not the case for the other rules thus the same substitution rule such as the copper mean rule can give an infinite periodic spectrum for a single rational ratio rho rho a rho b of the two tile lengths rho a and rho b a periodic like spectrum for other rational rho and a spectrum in some aspects similar to that of a random system when rho is an irrational number on the other hand a thue morse system diffracts as a periodic crystal when p not 1 but has no non trivial delta peaks when rho 1 other thue morse like systems can have infinite periodic spectra for all rho

csiszar s cutoff rates for arbitrary discrete sources csiszar s 1995 forward spl beta cutoff rate given a fixed spl beta 0 for a discrete source is defined as the smallest number r sub 0 such that for every r r sub 0 there exists a sequence of fixed length codes of rate r with probability of error asymptotically vanishing as e sup n spl beta r r0 for a discrete memoryless source dms the forward spl beta cutoff rate is shown by csiszar to be equal to the source renyi 1961 entropy an analogous concept of reverse spl beta cutoff rate regarding the probability of correct decoding is also characterized by csiszar in terms of the renyi entropy in this work csiszar s results are generalized by investigating the spl beta cutoff rates for the class of arbitrary discrete sources with memory it is demonstrated that the limsup and liminf renyi entropy rates provide the formulas for the forward and reverse spl beta cutoff rates respectively consequently new fixed length source coding operational characterizations for the renyi entropy rates are established

generalized symmetric duality for multiobjective variational problems with invexity abstract we formulate a pair of multiobjective nonlinear generalized symmetric dual variational problems involving vector valued functions which unify the wolfe and mond weir models for a special case our problems become the symmetric dual pair of d s kim and w j lee 1998 j math anal appl 218 34 48 under invexity assumptions we establish weak strong and converse duality theorems for our variational problems by using the concept of efficiency also we give the static case of our multiobjective generalized symmetric duality results several known results are obtained as special cases

singular continuous spectrum for palindromic schrodinger operators we give new examples of discrete schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum if the hullx of the potential is strictly ergodic then the existence of just one potentialx inx for which the operator has no eigenvalues implies that there is a generic set inx for which the operator has purely singular continuous spectrum a sufficient condition for the existence of such anx is that there is az x that contains arbitrarily long palindromes thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inx the class includes well known substitutions like fibonacci thue morse period doubling binary non pisot and ternary non pisot we also show that the operator has no absolutely continuous spectrum for allx x ifx derives from a primitive substitution for potentials defined by circle maps x n 1 j 0 n we show that the operator has purely singular continuous spectrum for a generic subset inx for all irrational and every half open intervalj

localized wavefunctions in quantum systems with multiwell potentials one dimensional systems with the multiwell potential v x 2 1cos x 2 2cos x are considered localized wavefunctions in the case when the energy is greater than the maximum value of the potential are presented wigner and weyl functions corresponding to the wavefunctions of these systems are also studied it is shown that they are very sensitive to the value of the parameter

symmetric duality for a class of multiobjective fractional programming problems this paper is concerned with symmetric duality for a class of nondifferentiable multiobjective fractional programming problems two weak duality theorems and two strong duality theorems are proved discussion on some special cases shows that results in this paper extend previous work in this area

renormalisation procedure for the quasiperiodic schrodinger equation a renormalisation procedure is constructed for the one dimensional schrodinger equation with a quasiperiodic potential the renormalisation transformation has a trivial fixed point describing the behaviour of the system near the unperturbed free motion and a non trivial fixed point corresponding to the critical case a number system of an irrational base is introduced for the scaling of the spectra

renyi s divergence and entropy rates for finite alphabet markov sources in this work we examine the existence and the computation of the renyi divergence rate lim sub n spl rarr spl infin 1 n d sub spl alpha p sup n spl par q sup n between two time invariant finite alphabet markov sources of arbitrary order and arbitrary initial distributions described by the probability distributions p sup n and q sup n respectively this yields a generalization of a result of nemetz 1974 where he assumed that the initial probabilities under p sup n and q sup n are strictly positive the main tools used to obtain the renyi divergence rate are the theory of nonnegative matrices and perron frobenius theory we also provide numerical examples and investigate the limits of the renyi divergence rate as spl alpha spl rarr 1 and as spl alpha spl darr 0 similarly we provide a formula for the renyi entropy rate lim sub n spl rarr spl infin 1 n h sub spl alpha p sup n of markov sources and examine its limits as spl alpha spl rarr 1 and as spl alpha spl darr 0 finally we briefly provide an application to source coding

diffusion on a quasiperiodic chain an exact solution to the problem of diffusion on a quasiperiodic chain is provided by a decimation method involving scaling by an irrational length factor

dynamical system related to quasiperiodic schrodinger equations in one dimension a dynamical map is obtained from a class of quasiperiodic discrete schrodinger equations in one dimension which include the fibonacci system the potentials are constant except for steps at special points

franco american meeting on the mathematics of random and almost periodic potentials we present the abstracts of papers given at a conference held at the ecole polytechnique in june 1983 on the mathematics of random and almost periodic potentials

upper bounds in quantum dynamics of course the case of main interest is where h is given by l2 rd or 2 zd h is a schr dinger operator of the form a v and ip 0 is a localized wavepacket under the time evolution 1 the wavepacket i t will in general spread out with time finding quantitative bounds for this spreading in concrete cases and developing methods for proving such bounds has been the objective of a great number of papers in recent years one is often interested in the growth behavior of the moments of the position operator that is the function 11 x p 2 i 2 these questions are particularly complicated and interesting in situations where singular continuous spectra occur the discovery of the occurrence of such spectra for many models of physical relevance has triggered some activity on the quantum dynamical side for models with absolutely continuous spectra on the other hand quantum dynamical questions have been investigated much earlier in the context of scattering theory and have been answered to a great degree of satisfaction models with point spectrum or more restrictively models exhibiting anderson localization have also been investigated from a quantum dynamical point of view while it is not true in general that spectral localization implies dynamical localization 26 73 it has been shown that the three main approaches to spectral localization the fractional moment method 3 4 5 multi scale analysis 28 27 75 and the bourgain goldstein schlag method 12 37 yield dynamical localization as a by product 2 13 22 31 34 quantum dynamics for schr dinger operators with singular continuous spectra is much less understood the two primary examples motivated by physics are the harper model also known as the almost mathieu operator at critical cou pling which describes a bloch electron in a constant magnetic field with irrational flux through a unit cell and the fibonacci model which describes a standard one dimensional quasicrystal both models have been studied heavily in both the physics and mathematics communities

spectrum and extended states in a harmonic chain with controlled disorder effects of the thue morse symmetry along the lines of previous work we give the general framework together with a detailed and rigorous study of the spectrum and born von karman eigenstates of a 1d harmonic chain with controlled disorder determined by the thue morse sequence the spectrum is a cantor like set we prove numerically that its measure is zero and calculate its bouligand minkowski dimension box dimension we prove that the value of the ids on each of the gaps is 2k 1 3 2 p withk andp integers finally we also prove that points in a dense subset of the spectrum give rise to extended states an exceptional property due to the symmetry of the thue morse substitution which can have important applications to multilayered structures and we illustrate this situation

anomaly and singular continuous spectrum in a one dimensional incommensurate system the authors study a tight binding model given by psi n 1 psi n 1 v cos 2 pi qn theta psi n e psi n where q is a liouville number it is demonstrated that all eigenstates are critical with singular continuous spectra for all values of v these critical states appear to be composed of connected extended states they also find a band centre anomaly the resistance of the band centre displays coupled oscillations and is much larger than that of other states

scaling localization and bandwidths for equations with competing periods abstract the finite size scaling theory of the measure of the spectrum of harper s equation is reexamined for sequences of fractions tending to a rational limit a simple criterion is derived which determines whether the corrections to scaling behave as log p 2 or p 2 as the denominator p is increased the question of how special are the properties of the harper equation is studied it is shown that if the pure cosine term in the diagonal term is replaced by a distorted periodic function the different subbands undergo a transition from localized to extended at a value of the strength of the off diagonal term that depends on energy in contrast to the harper equation where the transition is energy independent this has a crucial effect on the measure of the spectrum

number of scaling factors in incommensurate systems the authors study the scaling properties of an incommensurate system which obeys simple clustering rules the one dimensional tight binding schrodinger equation with a fibonacci sequence potential they find that the clustering rules do not guarantee a finite number of scaling factors they find the energy spectrum by recursion the clustering rules and recursion relations suggest different descriptions of the same spectrum based on the thermodynamic formalism they compare these descriptions calculate their associated scaling functions and construct the mapping between them

symmetric duality for multiobjective variational problems with invexity abstract we formulate a pair of multiobjective nonlinear symmetric dual variational problems for the single objective problems our problems become the symmetric dual pair of i smart and b mond j math anal appl 152 1990 536 545 under invexity assumptions we establish the weak strong converse and self duality theorems for our variational problems by using the concept of efficiency also we give the static case of our symmetric duality results

a natural class of generalized fibonacci chains in this paper we propose a class of substitution rules that generate quasiperiodic chains sharing their typid properties with the quasiperiodic fibonacci chain for a subclass we explicitly con vct the atomic surface moreover scaling properties of the energy specmm are discussed in relation to the dynamics of trace maps

symmetric duality for invex multiobjective fractional variational problems we introduce a symmetric dual for multiobjective fractional variational problems under certain invexity assumptions we establish weak strong and converse duality theorems as well as self duality relations the paper includes an extension of previous symmetric duality results for the static case obtained by weir to the dynamic case

calculating dynamic structure factors with the real space renormalisation group the authors develop exact methods based on the real space renormalisation group for calculating dynamic structure factors of heisenberg ferromagnets the method is applied to a pure chain a sierpinski gasket and a quasi one dimensional model of a dilute chain the result for the sierpinski gasket exhibits a rich structure because of the requirement of discrete scale invariance by the use of the distribution scaling the method is generalised to treat a randomly diluted chain

almost periodic one dimensional systems the schrodinger equation with potential v x 2 1cos x 2 2cos x is considered its solution reduces to the problem of finding the eigenvectors of a matrix the eigenvalues of this matrix show a band structure which is very sensitive to the value of the parameter the solutions of the schrodinger equation are presented and their physical meaning is discussed the potential v x has a complex multiwell structure and quantum tunneling occurs the accuracy of all approximations is carefully studied

density of states and dynamic scaling on the vicsek snowflake fractal real space rescaling techniques have been employed to obtain recursion relations for the parameters characterising the dynamics of both the heisenberg ferromagnet and antiferromagnet on the infinite vicsek snowflake fractal the authors have calculated the associated dynamic exponents zf and za by linearising about the respective fixed points of the ferromagnetic antiferromagnetic recursion relations and find them to satisfy the relation za zf 2 in addition using a functional integral method both the ferromagnetic and antiferromagnetic density of states rho f omega and rho a omega have been calculated exactly and the resulting spectra were found to satisfy the scaling equation rho alpha omega approximately bz alpha df rho lambda alpha omega near their respective fixed points alpha a f df denoting the fractal dimension finally the amplitudes galpha omega appearing in the solution to the scaling equation rho alpha omega approximately galpha omega omega dfz alpha 1 for the density of states rho alpha omega were found to be periodic functions of ln omega with period ln lambda alpha where lambda alpha is the eigenvalue associated with ferromagnetic alpha f or antiferromagnetic alpha a fixed point

symmetric duality with pseudo invexity in variational problems abstract weak and strong duality results are established under pseudo invexity hypotheses for symmetric dual variational problems self dual problems and static symmetric dual programs are included as special cases

rescaling entropy and divergence rates based on rescaling by some suitable sequence instead of the number of time units the usual notion of divergence rate is here extended to define and determine meaningful generalized divergence rates rescaling entropy rates appears as a special case suitable rescaling is naturally induced by the asymptotic behavior of the marginal divergences closed form formulas are obtained as soon as the marginal divergences behave like powers of some analytical functions a wide class of countable markov chains is proved to satisfy this property most divergence and entropy functionals defined in the literature are concerned e g the classical shannon kullback leibler renyi and tsallis for illustration purposes ferreri or basu harris hjort jones among others are also considered

quantum diffusion in the generalized harper equation we study numerically the dynamic and spectral properties of a one dimensional quasi periodic system where site energies are given by with denoting the kth quasiperiodic lattice site when is given by the reciprocal lattice vector with n and m being successive fibonacci numbers the variance of the wavepacket is found to grow quadratically in time regardless of the potential strength v for other values of f there exists a critical value of v beyond which the growth of the wavepacket variance is bounded in particular an anomalous diffusion takes place for corresponding to with generic integers m and n the level spacing distribution is also examined and the corresponding exponent is observed to decrease with v

classical representation of the one dimensional anderson model a new approach is applied to the one dimensional anderson model by using a two dimensional hamiltonian map for a weak disorder this approach allows for a simple derivation of correct expressions for the localization length both at the centre and at the edge of the energy band where standard perturbation theory fails approximate analytical expressions for strong disorder are also obtained

error exponents of modulo additive noise channels with side information at the transmitter consider the optimum strategy for using channel state side information in transmission over a modulo additive noise channel with state dependent noise where the receiver does not have access to the side information si previous work showed that capacity wise the optimum transmitter shifts each code letter by a prediction of the noise sample based on the si we show that this structure achieves also the random coding error exponent and therefore is optimum at some range of rates below capacity specifically the optimum transmitter predictor minimizes the renyi entropy of the prediction error the renyi order depends on the rate and goes to one corresponding to shannon entropy for rates close to capacity in contrast it is shown that this prediction strategy may not be optimal at low transmission rates

statistics of resonances and delay times in random media beyond random matrix theory we review recent developments in quantum scattering from mesoscopic systems various spatial geometries whose closed analogues show diffusive localized or critical behaviour are considered these are the features that cannot be described by the universal random matrix theory results instead one has to go beyond this approximation and incorporate them in a non perturbative way here we pay particular attention to the traces of these non universal characteristics in the distribution of the wigner delay times and resonance widths the former quantity captures time dependent aspects of quantum scattering while the latter is associated with the poles of the scattering matrix

symmetric duality for multiobjective fractional variational problems with generalized invexity the concept of symmetric duality for multiobjective fractional problems has been extended to the class of multiobjective variational problems weak strong and converse duality theorems are proved under generalized invexity assumptions a close relationship between these problems and multiobjective fractional symmetric dual problems is also presented

sensitivity analysis for generalized nonlinear parametric a η m maximal monotone operator inclusion systems with relaxed cocoercive type operators abstract by using lim s inequalities nadler s results the new parametric resolvent operator technique associated with a m maximal monotone operators in this paper the existence theorem for a new class of generalized nonlinear parametric a m maximal monotone operator inclusion systems with relaxed cocoercive type operators in hilbert spaces is analyzed and established our results generalize sensitivity analysis results of other recent works on strongly monotone quasi variational inclusions nonlinear implicit quasi variational inclusions and nonlinear mixed quasi variational inclusion systems in hilbert spaces

trace maps for arbitrary substitution sequences the authors prove the existence and show how to construct trace maps for products of 2 2 matrices generated by arbitrary substitution sequences the dimension of the underlying space of the trace map is lower than the one suggested recently by other authors

maximum renyi entropy rate the supremum of the renyi entropy rate over the class of discrete time stationary stochastic processes whose marginals are supported by some given set and satisfy some given cost constraint is computed unlike the shannon entropy the renyi entropy of a random vector can exceed the sum of the renyi entropies of its components and the supremum is therefore typically not achieved by memoryless processes it is nonetheless related to shannon s entropy when the renyi parameter exceeds one the supremum is equal to the corresponding supremum of shannon s entropy and when it is smaller than one the supremum equals the logarithm of the volume of the support set a burg like supremum of the renyi entropy rate over the class of stochastic processes whose autocovariance function begins with some given values is also solved it is not achieved by gauss markov processes but it is nonetheless related to burg s supremum the two are equal when the renyi parameter exceeds one and the former is infinite otherwise

joint source channel coding error exponent for discrete communication systems with markovian memory we study the error exponent ej for reliably transmitting a discrete stationary ergodic markov sem source q over a discrete channel w with additive sem noise via a joint source channel jsc code we first establish an upper bound for ej in terms of the renyi entropy rates of the source and noise processes we next investigate the analytical computation of ej by comparing our bound with gallager s lower bound 1968 when the latter one is specialized to the sem source channel system we also note that both bounds can be represented in csiszar s form 1980 as the minimum of the sum of the source and channel error exponents our results provide us with the tools to systematically compare ej with the tandem separate coding exponent ej we show that as in the case of memoryless source channel pairs ej les 2er and we provide explicit conditions for which ej et numerical results indicate that ej ap 2et for many sem source channel pairs hence illustrating a substantial advantage of jsc coding over tandem coding for systems with markovian memory

symmetric duality for multiobjective fractional variational problems involving cones in this paper a pair of multiobjective fractional variational symmetric dual problems over cones is formulated weak strong and converse duality theorems are established under generalized f convexity assumptions moreover self duality theorem is also discussed

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

hierarchical band clustering and fractal spectra in incommensurate systems an analytic derivation is given of rules exactly describing the hierarchical band clustering in the spectrum of harper s equation for various incommensurate systems these rules involving transformations of the incommensurability parameter phi are shown to apply exactly for arbitrary coupling strength the approach also provides energy scale factors and hence the fractal dimension df 0 50 or 0 01 of the self similar spectrum occurring when phi takes a fixed point value phi g the golden mean

dynamical properties of quasi crystals fibonacci chain and penrose lattice the discrete scale invariance of quasi periodic systems leads to a scaling relationship omega l zf l between a characteristic dynamic variable omega and length l where f l is a periodic function of ln l the dynamic exponent z can be calculated for diffusion spin wave and phonon dynamics where omega is respectively gamma omega omega 2 where gamma and omega are characteristic rates or frequencies respectively by exploiting a crossover argument which results in z df t where df is the fractal dimension and t is the length scaling exponent for the conductance the latter exponents and hence z are calculated for a fibonacci chain model df t z 1 1 2 and via an exact bond moving technique for the penrose lattice df t z 2 0 2 the technique also provides the thermal exponent of the heisenberg spin model on the two quasi crystals nu 1 infinity respectively

a binary communication channel with memory based on a finite queue a model for a binary additive noise communication channel with memory is introduced the channel noise process which is generated according to a ball sampling mechanism involving a queue of finite length m is a stationary ergodic mth order markov source the channel properties are analyzed and several of its statistical and information theoretical quantities e g block transition distribution autocorrelation function acf capacity and error exponent are derived in either closed or easily computable form in terms of its four parameters the capacity of the queue based channel qbc is also analytically and numerically compared for a variety of channel conditions with the capacity of other binary models such as the well known gilbert elliott channel gec the fritchman channel and the finite memory contagion channel we also investigate the modeling of the traditional gec using this qbc model the qbc parameters are estimated by minimizing the kullback leibler divergence rate between the probability of noise sequences generated by the gec and the qbc while maintaining identical bit error rates ber and correlation coefficients the accuracy of fitting the gec via the qbc is evaluated in terms of acf channel capacity and error exponent numerical results indicate that the qbc provides a good approximation of the gec for various channel conditions it thus offers an interesting alternative to the gec while remaining mathematically tractable

schrodinger operators with rudin shapiro potentials are not palindromic we prove a conjecture of a hof o knill and b simon commun math phys 174 149 159 1995 by showing that the rudin shapiro sequence is not palindromic i e does not contain arbitrarily long palindromes we prove actually this property for all paperfolding sequences and all rudin shapiro sequences deduced from paperfolding sequences as a consequence and as guessed by the above authors their method cannot be used for establishing that discrete schrodinger operators with rudin shapiro potentials have a purely singular continuous spectrum

spectral properties of a tight binding hamiltonian with period doubling potential we study a one dimensional tight binding hamiltonian with a potential given by the period doubling sequence we prove that its spectrum is purely singular continuous and supported on a cantor set of zero lebesgue measure for all nonzero values of the potential strength moreover we obtain the exact labelling of all spectral gaps and compute their widths asymptotically for small potential strength

a variational characterization of renyi divergences atar chowdhary and dupuis have recently exhibited a variational formula for exponential integrals of bounded measurable functions in terms of renyi divergences we show that a variational characterization of the renyi divergences between two probability distributions on a measurable space in terms of relative entropies when combined with the elementary variational formula for exponential integrals of bounded measurable functions in terms of relative entropy yields the variational formula of atar chowdhary and dupuis as a corollary we then develop an analogous variational characterization of the renyi divergence rates between two stationary finite state markov chains in terms of relative entropy rates when combined with varadhan s variational characterization of the spectral radius of square matrices with nonnegative entries in terms of relative entropy this yields an analog of the variational formula of atar chowdary and dupuis in the framework of stationary finite state markov chains