the set of smooth quasi periodic schrodinger cocycles with positive lyapunov exponent is not open one knows that the set of quasi periodic schrodinger cocycles with positive lyapunov exponent is open and dense in analytic topology in this paper we construct cocycles with positive lyapunov exponent which can be arbitrarily approximated by ones with zero lyapunov exponent in the space of mathcal c l 1 le l le infty smooth quasi periodic cocycles which shows that the set of quasi periodic schrodinger cocycles with positive lyapunov exponent is not open in smooth topology

dct algorithms for composite sequence lengths this paper presents fast algorithms for the type ii and iii discrete cosine transforms of composite sequence length in particular a radix q algorithm where q is an odd integer is derived for uniform or mixed radix decomposition of the discrete cosine transform by combining the radix q and radix 2 algorithms a general decomposition method for any composite length is developed reduction of computational complexity can be achieved for many sequence lengths compared with that needed by the well known radix 2 algorithm furthermore both the proposed and chan and siu s 1993 mixed radix algorithms achieve the same computational complexity for n 3 2 sup p and 9 2 sup p however our algorithm uses a simpler decomposition approach and provides a wider range of choices of sequence lengths

large deviations of the lyapunov exponent and localization for the 1d anderson model the proof of anderson localization for the 1d anderson model with arbitrary e g bernoulli disorder originally given by carmona klein martinelli in 1987 is based in part on the multi scale analysis later in the 90s it was realized that for one dimensional models with positive lyapunov exponents some parts of multi scale analysis can be replaced by considerations involving subharmonicity and large deviation estimates for the corresponding cocycle leading to nonperturbative proofs for 1d quasiperiodic models in this paper we present a short proof along these lines for the anderson model to prove dynamical localization we also develop a uniform version of craig simon s bound that works in high generality and may be of independent interest

shifted fourier transform based tensor algorithms for the 2 d dct in this paper tensor algorithms for calculating the two dimensional 2 d discrete cosine transform dct are presented the tensor approach is based on the concept of the covering revealing the transforms which yields in particular the splitting of the shifted 2 sup r spl times 2 sup r point fourier and cosine transforms into 2 sup r 1 3 one dimensional 1 d incomplete 2 sup r point transforms the multiplicative complexity of the 2 d 2 sup r spl times 2 sup r point discrete cosine transforms in terms of the tensor representation is 4 sup r 3 2 sup r 2 r sup 2 7r 14 which is reduced to 4 sup r 8 3 2 sup r r sup 2 7r 10 20 3 when using the improved tensor algorithm the multiplicative complexity in the general l sup r spl times l sup r case with a prime l 2 as well as in the l sub 1 l sub 2 spl times l sub 1 l sub 2 case with arbitrary co prime l sub 1 l sub 2 1 is provided the examples of the tensor algorithms for calculating the 8 spl times 8 point dct through 104 88 and 84 multiplications are given in detail based on the proposed concept the fast algorithm for calculating the 1 d dct i is also developed the multiplicative complexity of the 2 sup r point dct i is 2 sup r 1 r 2 r 5 2 8 the comparative estimates with the known algorithms are given

positive lyapunov exponents for continuous quasiperiodic schrodinger equations we prove that the continuous one dimensional schrodinger equation with an analytic quasi periodic potential has positive lyapunov exponents in the bottom of the spectrum for large couplings

positive lyapunov exponents for schrodinger operators with quasi periodic potentials we present a new simple way to estimate the rate of exponential growth lyapunov exponent of solutions of the finite difference schrodinger equation h e psi n mathop limits def psi n 1 psi n 1 lambda f alpha n theta psi n heref is a non constant real analytic function of period 1 and is irrational for large we prove that the lyapunov exponent is positive for every energye in the spectrum ofh and a e in particular the absolutely continuous spectrum ofh is empty in the continuum we study the quasi periodic operator onl2 r h frac d 2 dx 2 k 2 cos x cos alpha x theta for largek and show that for wide intervals of low energies the lyapunov exponent is positive the main idea which originated from m herman s subharmonic argument 11 is to deform the phase to the complex plane this enables us to avoid small denominator problems by moving them off the axis making estimates much easier to perform we recover the information for real using an elementary extension of jensen s formula subharmonicity

purely absolutely continuous spectrum for almost mathieu operators using a recent result of sinai we prove that the almost mathieu operators acting onl2 ly n l 1 l cos n n have a purely absolutely continuous spectrum for almost all a provided that is a good irrational and is sufficiently small furthermore the generalized eigen functions are quasiperiodic

resonance tongues and instability pockets in the quasi periodic hill schrodinger equation this paper concerns hill s equation with a parametric forcing that is real analytic and quasi periodic with frequency vector d and a frequency or energy parameter a and a small parameter b the 1 dimensional schrodinger equation with quasi periodic potential occurs as a particular case in the parameter plane 2 a b for small values of b we show the following the resonance tongues with rotation number frac 1 2 langle bf k omega rangle bf k in mathbb z d have c boundary curves our arguments are based on reducibility and certain properties of the schrodinger operator with quasi periodic potential analogous to the case of hill s equation with periodic forcing i e d 1 several further results are obtained with respect to the geometry of the tongues one result regards transversality of the boundaries at b 0 another result concerns the generic occurrence of instability pockets in the tongues in a reversible near mathieu case that may depend on several deformation parameters these pockets describe the generic opening and closing behaviour of spectral gaps of the schrodinger operator in dependence of the parameter b this result uses a refined averaging technique also consequences are given for the behaviour of the lyapunov exponent and rotation number in dependence of a for fixed b

difference almost periodic schrodinger operators corollaries of localization we study the technique used in proving the exponential localization for one dimensional difference schrodinger operators with quasi periodic potential in this way we get some corollaries concerning the spectrum structure near the boundaries and the existence of bounded non exponentially decaying solutions of the equation on eigenvalues

a fast recursive algorithm for computing the discrete cosine transform the discrete cosine transform dct is widely applied in various fields including image data compression because it operates like the karhunen loeve transform for stationary random data this paper presents a recursive algorithm for dct with a structure that allows the generation of the next higher order dct from two identical lower order dct s as a result the method for implementing this recursive dct requires fewer multipliers and adders than other dct algorithms

on the multiplicative complexity of discrete cosine transforms the multiplicative complexity of discrete cosine transforms dcts of arbitrary dimensions on input sizes which are powers of two are obtained new upper bounds on the multiplicative complexity of scaled dcts on input sizes which are powers of two are also obtained

scaled dct s on input sizes that are composite given efficient scaled discrete cosine transforms dct s of size p and q where p and q are relatively prime we present methods for constructing fast algorithms to compute scaled dct s and scaled inverse dct s of size pq extensions to multidimensional transforms are also given

on resonances and the formation of gaps in the spectrum of quasi periodic schrodinger equations we consider one dimensional difference schrodinger equations h x n n 1 n 1 v x n n e n n z x 0 1 with real analytic potential function v x if l e w o is greater than 0 for all e e e and some diophantine 0 then the integrated density of states is absolutely continuous for almost every close to 0 as shown by the authors in earlier work in this paper we establish the formation of a dense set of gaps in spec h x w e e our approach is based on an induction on scales argument and is therefore both constructive as well as quantitative resonances between eigenfunctions of one scale lead to pre gaps at a larger scale to pass to actual gaps in the spectrum we show that these pre gaps cannot be filled more than a finite and uniformly bounded number of times to accomplish this one relates a pre gap to pairs of complex zeros of the dirichlet determinants off the unit circle amongst other things we establish a nonperturbative version of the co variant parametrization of the eigenvalues and eigenfunctions via the phases in the spirit of sinai s perturbative description of the spectrum via his function a this allows us to relate the gaps in the spectrum with the graphs of the eigenvalues parametrized by the phase our infinite volume theorems hold for all diophantine frequencies up to a set of hausdorff dimension zero

1d quasiperiodic operators latent symmetries a large class of discrete quasiperiodic operators is shown to be decomposed into orbits ofsl 2 z action with equal densities of states moreover under some natural assumptions all nontrivial representatives of the mentioned action transform operators with pure point spectrum into those with absolutely continuous spectrum some applications of these results are presented

on equitable resource allocation problems a lexicographic minimax approach in this expository paper we review a variety of resource allocation problems in which it is desirable to allocate limited resources equitably among competing activities applications for such problems are found in diverse areas including distribution planning production planning and scheduling and emergency services location each activity is associated with a performance function representing for example the weighted shortfall of the selected activity level from a specified target a resource allocation solution is called equitable if no performance function value can be improved without either violating a constraint or degrading an already equal or worse off i e larger performance function value that is associated with a different activity a lexicographic minimax solution determines this equitable solution that is it determines the lexicographically smallest vector whose elements the performance function values are sorted in nonincreasing order the problems reviewed include large scale allocation problems with multiple knapsack resource constraints multiperiod allocation problems for storable resources and problems with substitutable resources the solution of large scale problems necessitates the design of efficient algorithms that take advantage of special mathematical structures indeed efficient algorithms for many models will be described we expect that this paper will help practitioners to formulate and solve diverse resource allocation problems and motivate researchers to explore new models and algorithmic approaches

anderson localization for generic deterministic operators abstract we consider a class of ensembles of lattice schrodinger operators with deterministic potentials including quasi periodic potentials with diophantine frequencies depending upon an infinite number of parameters in an auxiliary measurable space using a variant of the multi scale analysis we prove anderson localization for generic ensembles in the strong disorder regime and establish an analog of minami type bounds for spectral spacings

singular continuous spectrum is generic in a variety of contexts we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators those with singular spectrum are a dense g delta

localization for a class of one dimensional quasi periodic schrodinger operators we prove for small and satisfying a certain diophantine condition the operator h varepsilon 2 delta frac 1 2 pi cos 2 pi j alpha theta j in mathbb z has pure point spectrum for almost all a similar result is established at low energy for h frac d 2 dx 2 k 2 cos 2 pi x cos 2 pi alpha x theta providedk is sufficiently large

positive lyapunov exponents for a class of ergodic schrodinger operators we present a simple method to estimate the lyapunov exponent e for the system psi j 1 psi j 1 v j omega psi j e psi j where v j is an ergodic family of potentials defined forj we assume that there is a constant 2 and large positive integersl l such that for almost every and everye there is an infinite sequence of disjoint intervalsj n with the following properties under these conditions we prove that meas e e o be l 6 where gb andb are positive constants and ldmeas rd refers to lebesgue measure

fast algorithms for type iii dct of composite sequence lengths this correspondence presents an odd factor algorithm for the type iii discrete cosine transform dct for uniform or mixed radix decomposition by jointly using the old factor and the existing radix 2 algorithms a general decomposition method for arbitrarily composite sequence length is developed a reduction of computational complexity can be achieved compared with that needed by other reported algorithms for m 2 sup m the decomposition approach has a simple computational structure and supports a wider range of choices for different sequence lengths

anderson localization for the almost mathieu equation a nonperturbative proof we prove that for any diophantine rotation angle and a e phase the almost mathieu operator h n n 1 n 1 cos 2 n n has pure point spectrum with exponentially decaying eigenfunctions for 15 we also prove the existence of some pure point spectrum for any 5 4

absence of localisation in the almost mathieu equation the author considers the discrete schrodinger operator on z with the potential lambda cos 2 pi alpha n theta this one dimensional model occurs in the study of an electron in a two dimensional periodic potential with a uniform magnetic field first it is proved that for every alpha and for lambda 2 this operator has no eigenvalue i e localised state furthermore at lambda 2 the eigenvalues if they exist belong to the set where the lyapunov exponent vanishes and the associated eigenvectors at in l2 z but are not summable

efficient block based frequency domain wavelet transform implementations subband decompositions for image coding have been explored extensively over the last few decades the condensed wavelet packet cwp transform is one such decomposition that was recently shown to have coding performance advantages over conventional decompositions a special feature of the cwp is that its design and implementation are performed in the cyclic frequency domain while performance gains have been reported efficient implementations of the cwp or more generally efficient implementations of cyclic filter banks have not yet been fully explored in this paper we present efficient block based implementations of cyclic filter banks along with an analysis of the arithmetic complexity block based cyclic filter bank implementations of the cwp coder are compared with conventional subband wavelet image coders whose filter banks are implemented in the time domain it is shown that block based cyclic filter bank implementations can result in cwp coding systems that outperform the popular image coding systems both in terms of arithmetic complexity and coding performance

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

dynamics of the quasi periodic schrodinger cocycle at the lowest energy in the spectrum in this paper we consider the quasi periodic schrodinger cocycle over mathbb t d d 1 and in particular its projectivization in the regime of large coupling constants and diophantine frequencies we give an affirmative answer to a question posed by m herman 21 p 482 concerning the geometric structure of certain strange non chaotic attractors which appear in the projective dynamical system we also show that for some phase the lowest energy in the spectrum of the associated schrodinger operator is an eigenvalue with an exponentially decaying eigenfunction this generalizes 39 to the multi frequency case d 1

kotani theory for one dimensional stochastic jacobi matrices we consider families of operators h on l2 given by h u n u n 1 u n 1 v n u n wherev is a stationary bounded ergodic sequence we prove analogs of kotani s results including that for a e ac h is the essential closure of the set ofe where e the lyaponov index vanishes and the result that ifv is non deterministic then ac is empty

dynamical localization ii with an application to the almost mathieu operator several recent works have established dynamical localization for schrodinger operators starting from control on the localization length of their eigenfunctions in terms of their centers of localization we provide an alternative way to obtain dynamical localization without resorting to such a strong condition on the exponential decay of the eigenfunctions furthermore we illustrate our purpose with the almost mathieu operator h cos 2 x 15 and with good diophantine properties more precisely for almost all for all q 0 and for all functions l2 mathbb z of compact support we show that mathop sup limits t left langle e ith theta lambda omega psi left x right q e ith theta lambda omega psi right rangle c psi the proof applies equally well to discrete and continuous random hamiltonians in all cases it uses as input a repulsion principle of singular boxes supplied in the random case by the multi scale analysis

proportional equity flow problem for terminal arcs the proportional equity flow problem extends a class of problems referred to as equity flow problems whose objective is to equitably distribute flow among the arcs in a flow circulation network the proportionally bounded flow circulation problem places lower and upper bounds on each arc flow that are nondecreasing continuous functions of the flow through one special arc and the objective is to maximize the flow through the special arc the proportional equity flow problem for terminal arcs problem ta is then defined as a special case where all the proportional arcs enter a sink vertex applications of both the general problem and problem ta are given two optimality conditions for problem ta are developed as well as an algorithm that is polynomially bounded for many types of nondecreasing continuous proportional bounding functions specifically the algorithm is shown to be polynomially bounded if the largest root of an equation can be found in polynomial time

taming of the wild group of magnetic translations we use a theorem of auslander and kostant on the representation theory of solvable lie groups for the study of some groups necessary for the description of certain quasi periodic systems of solid state physics we show that the magnetic translation group is tame type i if the magnetic field is not constant but fluctuating

lyapunov exponents of the schrodinger equation with quasi periodic potential on a strip we prove that all the non negative lyapunov exponents of difference schrodinger equation y n 1 q n y n y n 1 0 infty n infty are strictly positive herey n er m andq n is a symmetricm m matrix whose offdiagonal elements do not depend onn and the diagonal elements are quasiperiodic functions q nj theta lambda f j e 2 pi i theta n alpha e with allf i non constant analytic functions sufficiently large and any irrational number

anderson localization for the almost mathieu equation ii point spectrum for lambda 2 we prove that for any 2 and a e the pure point spectrum of the almost mathieu operator h n n 1 n 1 cos 2 n n contains the essential closure hat sigma of the spectrum corresponding eigenfunctions decay exponentially the singular continuous component if it exists is concentrated on a set of zero measure which is nowhere dense in hat sigma

floquet solutions for the 1 dimensional quasi periodic schrodinger equation we show that the 1 dimensional schrodinger equation with a quasiperiodic potential which is analytic on its hull admits a floquet representation for almost every energye in the upper part of the spectrum we prove that the upper part of the spectrum is purely absolutely continuous and that for a generic potential it is a cantor set we also show that for a small potential these results extend to the whole spectrum

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

spectral behavior of quasi periodic potentials spectral properties of schrodinger operators of the typeh v where is the laplacian v a quasiperiodic potential and a coupling constant are developed v is taken to be finite sum of exponentials with generic frequencies for small a strong stability is shown on the other hand examples in the finite diffeence case are given for which a transition in the type of spectrum occurs as is increased

the schrodinger equation with a quasi periodic potential we consider the schrodinger equation dx2 v e cosx cos cix 0 v eqi where e is small and a satisfies the diophantine inequality lp qcl c q2 forp qez q 0 we look for solutions of the form ikx ikx in x im ax 0 m x e q x e 2m ne inxne e if we try to solve for v v mn we are led to the schrodinger equation on the lattice z2 h k q ba v k q en where a is the discrete laplacian without diagonal terms and v k is some 2 potential on z we have two main results 4 for e sufficiently small h k has pure point spectrum for almost every k 2 for e sufficiently small the operator d dx 2 e cosx cos ax has no point spectrum to prove our results we must get decay estimates on the green s function e h 1 the decay of the eigenfunction follows from this in general we must keep track of small divisors which can make the green s function large this is accomplished by a kam kolmogorov arnold moser type of multiscale perturbation analysis

on the spectrum of multi frequency quasiperiodic schrodinger operators with large coupling we study multi frequency quasiperiodic schrodinger operators on mathbb z we prove that for a large real analytic potential satisfying certain restrictions the spectrum consists of a single interval the result is a consequence of a criterion for the spectrum to contain an interval at a given location that we establish non perturbatively in the regime of positive lyapunov exponent

stochastic schrodinger operators and jacobi matrices on the strip we discuss stochastic schrodinger operators and jacobi matrices with wave functions taking values in l so there are 2l lyaponov exponents 1 l 0 l 1 2l 1 our results include the fact that if 1 0 on a set positive measure thenv is deterministic and one that says that e exactly 2j s are zero is the essential support of the a c spectrum of multiplicity 2j

metal insulator transition for the almost mathieu operator we prove that for diophantine and almost every the almost mathieu operator h n n 1 ni 1 cos 2 n n exhibits localization for 2 and purely absolutely continuous spectrum for 2

localization for random schrodinger operators with correlated potentials we prove localization at high disorder or low energy for lattice schrodinger operators with random potentials whose values at different lattice sites are correlated over large distances the class of admissible random potentials for our multiscale analysis includes potentials with a stationary gaussian distribution whose covariance functionc x y decays as x y where 0 can be arbitrarily small and potentials whose probability distribution is a completely analytical gibbs measure the result for gaussian potentials depends on a multivariable form of nelson s best possible hypercontractive estimate

fast algorithms for the discrete cosine transform several fast algorithms for computing discrete cosine transforms dcts and their inverses on multidimensional inputs of sizes which are powers of 2 are introduced because the 1 d 8 point dct and the 2 d 8 8 point dct are so widely used they are discussed in detail algorithms for computing scaled dcts and their inverses are also presented these have applications in compression of continuous tone image data where the dct is generally followed by scaling and quantization

anderson localization for the 1 d discrete schrodinger operator with two frequency potential we prove the complete exponential localization of eigenfunctions for the 1 d discrete schrodinger operators with quasi periodic potentials having two basic frequencies it is shown also that for such operators there is no forbidden zones in the spectrum unlike the operators with one basic frequency

anderson localization for time quasi periodic random schrodinger and wave equations we prove that at large disorder with large probability and for a corresponding set of diophantine frequencies of large measure anderson localization in d is stable under localized time quasi periodic perturbations by proving that the associated quasi energy operator has pure point spectrum the main tools are the frohlich spencer mechanism for the random component and the bourgain goldstein schlag mechanism for the quasi periodic component this paper paves the way for the construction of time quasi periodic kam type of solutions of non linear random schrodinger equations in bw

towards equitable distributions via proportional equity constraints we propose a new approach to the formulation of models for solving problems of equitable distribution equity is achieved by the use of proportional equity constraints which require that each recipient must receive ashare of the total distribution which falls within a prespecified range the algorithmic implications of this new approach are illustrated using two mathematical models we consider the problem of maximizing total flow in a multiterminal network with proportional equity constraints an efficient algorithm for solving this problem is provided as a further application we show that the introduction of such constraints still permits easy solutions for the linear knapsack problem

sharp phase transitions for the almost mathieu operator it is known that the spectral type of the almost mathieu operator amo depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency we study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place which solves jitomirskaya s conjecture together with a previous work by avila this gives the sharp description of phase transitions for the amo for the a e phase

a metal insulator transition for the almost mathieu model we study the spectrum of the almost mathieu hamiltonian left h x psi right left n right psi left n 1 right psi left n 1 right 2 mu cos left x n theta right psi left n right n in mathbb z where is an irrational number andx is in the circle mathbb t for a small enough coupling constant and anyx there is a closed energy set of non zero measure in the absolutely continuous spectrum ofh for sufficiently high and almost allx we prove the existence of a set of eigenvalues whose closure has positive measure the two results are obtained for a subset of irrational numbers of full lebesgue measure

anderson localization for the almost mathieu equation iii semi uniform localization continuity of gaps and measure of the spectrum we show that the almost mathieu operator has semi uniform and thus dynamical localization for 15 and a e we also obtain a new estimate on gap continuity in for this operator with 29 or 4 29 and use it to prove that the measure of its spectrum is equal to for in this range and all irrational s

holder continuity of the integrated density of states for quasi periodic schrodinger equations and averages of shifts of subharmonic functions in this paper we consider various regularity results for discrete quasiperiodic schr6dinger equations n l pn 1 v 9 nw on eon with analytic potential v we prove that on intervals of positivity for the lyapunov exponent the integrated density of states is holder continuous in the energy provided w has a typical continued fraction expansion the proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the avalanche principle the latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks in the multi frequency case the integrated density of states is shown to have a modulus of continuity of the form exp log tl for some 0 a 1 but currently we do not obtain holder continuity in the case of more than one frequency we also present a mechanism for proving the positivity of the lyapunov exponent for large disorders for a general class of equations the only requirement for this approach is some weak form of a large deviation theorem for the lyapunov exponents in particular we obtain an independent proof of the herman sorets spencer theorem in the multi frequency case the approach in this paper is related to the recent nonperturbative proof of anderson localization in the quasi periodic case by j bourgain and m goldstein

operators with singular continuous spectrum iii almost periodic schrodinger operators we prove that one dimensional schrodinger operators with even almost periodic potential have no point spectrum for a dense g in the hull this implies purely singular continuous spectrum for the almost mathieu equation for coupling larger than 2 and a dense g in even if the frequency is an irrational with good diophantine properties

uniform anderson localization unimodal eigenstates and simple spectra in a class of haarsh deterministic potentials abstract we consider a particular class of families of multi dimensional lattice schrodinger operators with deterministic e g quasi periodic potentials generated by the hull given by an orthogonal series over the haar wavelet basis on the torus of arbitrary dimension with expansion coefficients considered as independent parameters in the strong disorder regime we prove anderson localization for generic operator families and show that all localized eigenfunctions are unimodal and feature uniform exponential decay away from their respective localization centers we also prove a variant of the minami estimate for deterministic potentials and simplicity of the spectrum

coarsening by ginzburg landau dynamics we study slowly moving solutions of the real ginzburg landau equation on the infinite line by a method inspired by the work of j carr and r l pego these solutions are functions taking alternately positive or negative values on large intervals we give a formula for the speed of the motion of the zeros of these functions and present a description of the collapse of two nearby zeros since we work on an infinite domain we can find initial conditions which lead to infinite sequences of collapses of kinks these results justify in part the approach to the coarsening problem by a simplified model proposed by a j bray b derrida and c godreche their model starts from a random distribution of domain lengths and evolves by successive elimination of the smallest domain our model is on one hand more realistic since it controls the original ginzburg landau equation in continuous time on the other hand we fail to gain sufficient control over the vanishing of pairs of kinks embedded within pairs of kinks at a greater distance

phase transition and semi global reducibility for 1d continuous schodinger operators with large analytic quasi periodic potentials of two frequencies one knows that the spectral measure is singular at the bottom of the spectrum and purely absolutely continuous in the upper part of the spectrum so there is a phase transition when energy increases in this paper we obtain the exact power law for the phase transition in energy by the semi global reducibility theory of analytic quasi periodic linear systems

absence of cantor spectrum for a class of schrodinger operators it is shown that the complete localization of eigenvectors for the almost mathieu operator entails the absence of cantor spectrum for this operator

a new algorithm to compute the discrete cosine transform a new algorithm is introduced for the 2m point discrete cosine transform this algorithm reduces the number of multiplications to about half of those required by the existing efficient algorithms and it makes the system simpler

the bipartite rationing problem in the bipartite rationing problem a set of agents share a single resource available in different types each agent has a claim over only a subset of the resource types and these claims overlap in arbitrary fashion the goal is to divide fairly the various types of resources between the claimants when resources are in short supply with a single type of resource this is the standard rationing problem o neill b 1982 a problem of rights arbitration from the talmud math soc sci 2 4 345 371 of which the three benchmark solutions are the proportional uniform gains and uniform losses methods we extend these methods to the bipartite context imposing the familiar consistency requirement the division is unchanged if we remove an agent respectively a resource and take away at the same time his share of the various resources respectively reduce the claims of the relevant agents the uniform gains and uniform losses methods have infinitely many consistent extensions but the proportional me

reducibility or nonuniform hyperbolicity for quasiperiodic schrodinger cocycles we show that for almost every frequency r q for every c o potential v r z r and for almost every energy e the corresponding quasiperiodic schr nodinger cocycle is either reducible or nonuniformly hyperbolic this result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic schr nodinger operator and allows us to complete the proof of the aubry andr le conjecture on the measure of the spectrum of the almost mathieu operator

universal lower bounds on eigenvalue splittings for one dimensional schrodinger operators we provide lower bounds on the eigenvalue splitting for d2 dx2 v x depending only on qualitative properties ofv for example ifv is c on a b ande n en 1 are two successive eigenvalues of d2 dx2 v withu a u b 0 boundary conditions and if lambda mathop max limits e in e n 1 e n x in a b e v x 1 2 then e n e n 1 geqq pi lambda 2 exp left lambda b a right the exponential factor in such bounds are saturated precisely in tunneling examples our results arenot restricted tov s of compact support but only require e n mathop lim limits overline x to infty v x

continuity of the measure of the spectrum for quasiperiodic schrodinger operators with rough potentials we study discrete quasiperiodic schrodinger operators on ell 2 mathbb z with potentials defined by holder functions we prove a general statement that for 1 2 and under the condition of positive lyapunov exponents measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants an important ingredient in our analysis is a general result on uniformity of convergence from above in the subadditive ergodic theorem for strictly ergodic cocycles

anderson transitions for a family of almost periodic schrodinger equations in the adiabatic case this work is devoted to the study of a family of almost periodic one dimensional schrodinger equations using results on the asymptotic behavior of a corresponding monodromy matrix in the adiabatic limit we prove the existence of an asymptotically sharp anderson transition in the low energy region more explicitly we prove the existence of energy intervals containing only singular spectrum and of other energy intervals containing absolutely continuous spectrum the zones containing singular spectrum and those containing absolutely continuous are separated by asymptotically sharp transitions the analysis may be viewed as utilizing a complex wkb method for adiabatic perturbations of periodic schrodinger equations the transition energies are interpreted in terms of phase space tunneling

fast algorithm for the 3 d dct ii recently many applications for three dimensional 3 d image and video compression have been proposed using 3 d discrete cosine transforms 3 d dcts among different types of dcts the type ii dct dct ii is the most used in order to use the 3 d dcts in practical applications fast 3 d algorithms are essential therefore in this paper the 3 d vector radix decimation in frequency 3 d vr dif algorithm that calculates the 3 d dct ii directly is introduced the mathematical analysis and the implementation of the developed algorithm are presented showing that this algorithm possesses a regular structure can be implemented in place for efficient use of memory and is faster than the conventional row column frame rcf approach furthermore an application of 3 d video compression based 3 d dct ii is implemented using the 3 d new algorithm this has led to a substantial speed improvement for 3 d dct ii based compression systems and proved the validity of the developed algorithm

discrete magnetic laplacian we consider a 2 dimensional discrete operator which we call the discrete magnetic laplacian dml it is an analogue of the magnetic schrodinger operator it follows from well known arguments that dml has the same spectrum as a subset inr as the almost mathieu operator am they also have the same integrated density of states ids which is known to be continuous we show that dml is an element in a ii1 factor and its ids can be expressed through the trace in the ii1 factor it follows that dml never has anyl2 eigenfunctions i e has no point spectrum then we formulate a natural algebraic conjecture which implies that the spectrum of dml hence the spectrum of am is a cantor set

an approach to wegner s estimate using subharmonicity we develop a strategy to establish a wegner estimate and localization in random lattice schrodinger operators on bbb z d which does not rely on the usual eigenvalue variation argument our assumption is that the potential v depends real analytically on and we use a distributional property of analytic functions in many variables an application is given to models where v n is a self adjoint matrix obtained by random unitary conjugation v n u n au of a fixed matrix a

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

uniform positivity and continuity of lyapunov exponents for a class of c 2 quasiperiodic schrodinger cocycles abstract we show that for a class of c 2 quasiperiodic potentials and for any fixed diophantine frequency the lyapunov exponent of the corresponding schrodinger cocycles as a function of energies are uniformly positive and weakly holder continuous as a corollary we obtain that the corresponding integrated density of states is weakly holder continuous as well our approach is of purely dynamical systems which depends on a detailed analysis of asymptotic stable and unstable directions we also apply it to more general sl 2 r cocycles which in turn can be applied to get uniform positivity and continuity of lyapunov exponents around unique nondegenerate extremal points of any smooth potential and to a certain class of c 2 szeg cocycles

the flow circulation sharing problem the flow circulation sharing problem is defined as a network flow circulation problem with a maximin objective function the arcs in the network are partitioned into regular arcs and tradeoff arcs where each tradeoff arc has a non decreasing tradeoff function associated with it all arcs have lower and upper bounds on their flow while the value of the smallest tradeoff function is maximized the model is useful in equitable resource allocation problems over time which is illustrated in a coal strike example and a submarine assignment example some properties including optimality conditions are developed each cut in the network defines a knapsack sharing problem which leads to an optimality condition similar to the max flow min cut theorem an efficient algorithm for both the continuous and integer versions of the flow circulation sharing problem is developed and computational experience given in addition efficient algorithms are developed for problems where some of the arcs have infinite flow upper bounds

growth of sobolev norms in linear schrodinger equations with quasi periodic potential in this paper we consider the following problem let iu t u v x t u 0 be a linear schrodinger equation periodic boundary conditions where v is a real bounded real analytic potential which is periodic in x and quasi periodic in t with diophantine frequency vector denote s t the corresponding flow map thus s t preserves the l 2 norm and our aim is to study its behaviour on h s t d s 0 our main result is the growth in time is at most logarithmic thus if h s then more precisely is proven in 1d and 2d when v is small we also exhibit examples showing that a growth of higher sobolev norms may occur in this context and is thus essentially best possible

methods of kam theory for long range quasi periodic operators on bf z nu pure point spectrum we consider the class of quasi periodic self adjoint operators x hat d x hat v x x s 1 1 1 on a multi dimensional lattice v with the matrix elements hat d mn x delta mn d x n omega hat v mn x v m n x n omega whered x 1 d x v n s 1 v n x v and v n x ee r n r 0 we prove that if e is small enough v n andd satisfy some conditions of smoothness andd is non degenerate then for a e and for a e x s 1 the operator x has pure point spectrum all its eigenfunctions belong tol 1 v

an algorithm for calculation of the discrete cosine transform by paired transform a new algorithm for splitting the one dimensional 1 d 2 sup r point discrete cosine transform dct into a set of short 2 sup k point type iv dcts k 1 r 1 is introduced the splitting is performed by means of paired transformation that is defined by the paired representation of signals with respect to the cosine transform a proposed method of calculating the 2 sup r point cosine transform requires 2 sup r 1 r multiplications and 2 sup r 1 3r 2 1 additions when r spl ges 2

a fast parametric maximum flow algorithm and applications the classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition finding the desired parameter value requires solving a sequence of related maximum flow problems in this paper it is shown that the recent maximum flow algorithm of goldberg and tarjan can be extended to solve an important class of such parametric maximum flow problems at the cost of only a constant factor in its worst case time bound faster algorithms for a variety of combinatorial optimization problems follow from the result

splitting of the low landau levels into a set of positive lebesgue measure under small periodic perturbations we study the spectral properties of a two dimensional schrodinger operator with a uniform magnetic field and a small external periodic field where and are small parameters representing as the direct integral of one dimensional quasi periodic difference operators with long range potential and employing recent results of e i dinaburg about anderson localization for such operators we assume to be typical irrational we construct the full set of generalised eigenfunctions for the low landau bands we also show that the lebesgue measure of the low bands is positive and proportional in the main order to