on bounded weight codes the maximum size of a binary code is studied as a function of its length n minimum distance d and minimum codeword weight ssi w this function b n d w is first characterized in terms of its exponential growth rate in the limit n for fixed d n and w n the exponential growth rate of b n d w is shown to be equal to the exponential growth rate of a n d for 0 1 2 and equal to the exponential growth rate of a n d w for 1 2 1 second analytic and numerical upper bounds on b n d w are derived using the semidefinite programming sdp method these bounds yield a nonasymptotic improvement of the second johnson bound and are tight for certain values of the parameters

new code upper bounds from the terwilliger algebra and semidefinite programming we give a new upper bound on the maximum size a n d of a binary code of word length n and minimum distance at least d it is based on block diagonalizing the terwilliger algebra of the hamming cube the bound strengthens the delsarte bound and can be calculated with semidefinite programming in time bounded by a polynomial in n we show that it improves a number of known upper bounds for concrete values of n and d from this we also derive a new upper bound on the maximum size a n d w of a binary code of word length n minimum distance at least d and constant weight w again strengthening the delsarte bound and yielding several improved upper bounds for concrete values of n d and w

from quantum cellular automata to quantum lattice gases a natural architecture for nanoscale quantum computation is that of a quantum cellular automaton motivated by this observation we begin an investigation of exactly unitary cellular automata after proving that there can be no nontrivial homogeneous local unitary scalar cellular automaton in one dimension we weaken the homogeneity condition and show that there are nontrivial exactly unitary partitioning cellular automata we find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton this model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the dirac equation we describe two generalizations of this automaton the second of which to multiple interacting particles is the correct definition of a quantum lattice gas

classification of binary constant weight codes a binary code c f2n with minimum distance at least d and codewords of hamming weight w is called an n d w constant weight code the maximum size of an n d w constant weight code is denoted by a n d w and codes of this size are said to be optimal in a computer aided approach optimal n d w constant weight codes are here classified up to equivalence for d 4 n 12 d 6 n 14 d 8 n 17 d 10 n 20 with one exception d 12 n 23 d 14 n 26 d 16 n 28 and d 18 n 28 moreover several new upper bounds on a n d w are obtained leading among other things to the exact values a 12 4 5 80 a 15 6 7 69 a 18 8 7 33 a 19 8 7 52 a 19 8 8 78 and a 20 8 8 130 since a 15 6 6 70 this gives the first known example of parameters for which a n d w 1 a n d w with w n 2 a scheme based on double counting is developed for validating the classification results

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

estimates on the size of symbol weight codes the study of codes for powerline communications has garnered much interest over the past decade various types of codes such as permutation codes frequency permutation arrays and constant composition codes have been proposed over the years in this paper we study a type of code called bounded symbol weight codes which was first introduced by versfeld in 2005 and a related family of codes that we term constant symbol weight codes we provide new upper and lower bounds on the size of bounded symbol weight and constant symbol weight codes we also give direct and recursive constructions of codes for certain parameters

purely singular continuous spectrum for limit periodic cmv operators with applications to quantum walks abstract we show that a generic element of a space of limit periodic cmv operators has zero measure cantor spectrum we also prove a craig simon type theorem for the density of states measure associated with a stochastic family of cmv matrices and use our construction from the first part to prove that the craig simon result is optimal in general we discuss applications of these results to a quantum walk model where the coins are arranged according to a limit periodic sequence the key ingredient in these results is a new formula which may be viewed as a relationship between the density of states measure of a cmv matrix and its schur function

minimal representations of unitary operators and orthogonal polynomials on the unit circle abstract in this paper we prove that the simplest band representations of unitary operators on a hilbert space are five diagonal orthogonal polynomials on the unit circle play an essential role in the development of this result and also provide a parameterization of such five diagonal representations which shows specially simple and interesting decomposition and factorization properties as an application we get the reduction of the spectral problem of any unitary hessenberg matrix to the spectral problem of a five diagonal one two applications of these results to the study of orthogonal polynomials on the unit circle are presented the first one concerns krein s theorem the second one deals with the movement of mass points of the orthogonality measure under mono parametric perturbations of the schur parameters

a quantum dynamical approach to matrix khrushchev s formulas khrushchev s formula is the cornerstone of the so called khrushchev theory a body of results which has revolutionized the theory of orthogonal polynomials on the unit circle this formula can be understood as a factorization of the schur function for an orthogonal polynomial modification of a measure on the unit circle no such formula is known in the case of matrix valued measures this constitutes the main obstacle to generalize khrushchev theory to the matrix valued setting which we overcome in this paper it was recently discovered that orthogonal polynomials on the unit circle and their matrix valued versions play a significant role in the study of quantum walks the quantum mechanical analogue of random walks in particular schur functions turn out to be the mathematical tool which best codify the return properties of a discrete time quantum system a topic in which khrushchev s formula has profound and surprising implications we will show that this connection between schur functions and quantum walks is behind a simple proof of khrushchev s formula via quantum diagrammatic techniques for cmv matrices this does not merely give a quantum meaning to a known mathematical result since the diagrammatic proof also works for matrix valued measures actually this path counting approach is so fruitful that it provides different matrix generalizations of khrushchev s formula some of them new even in the case of scalar measures furthermore the path counting approach allows us to identify the properties of cmv matrices which are responsible for khrushchev s formula on the one hand this helps to formalize and unify the diagrammatic proofs using simple operator theory tools on the other hand this is the origin of our main result which extends khrushchev s formula beyond the cmv case as a factorization rule for schur functions related to general unitary operators 2015 wiley periodicals inc

a deterministic control based approach to fully nonlinear parabolic and elliptic equations we show that a broad class of fully nonlinear second order parabolic or elliptic pde s can be realized as the hamilton jacobi bellman equations of deterministic twoperson games more precisely given the pde we identify a deterministic discretetime two person game whose value function converges in the continuous time limit to the viscosity solution of the desired equation our game is roughly speaking a deterministic analogue of the stochastic representation recently introduced by cheridito soner touzi and victoir comm pure appl math 60 2006 1081 1110 in the parabolic setting with no u dependence it amounts to a semidiscrete numerical scheme whose timestep is a min max our result is interesting because the usual control based interpretations of second order pde s involve stochastic rather than deterministic control

anderson localization for quasi periodic cmv matrices and quantum walks abstract we consider cmv matrices both standard and extended with analytic quasi periodic verblunsky coefficients and prove anderson localization in the regime of positive lyapunov exponents this establishes the cmv analog of a result bourgain and goldstein proved for discrete one dimensional schrodinger operators we also prove a similar result for quantum walks on the integer lattice with suitable analytic quasi periodic coins

matrix valued szegő polynomials and quantum random walks we consider quantum random walks qrw on the integers a subject that has been considered in the last few years in the framework of quantum computation we show how the theory of cmv matrices gives a natural tool to study these processes and to give results that are analogous to those that karlin and mcgregor developed to study classical birth and death processes using orthogonal polynomials on the real line in perfect analogy with the classical case the study of qrws on the set of nonnegative integers can be handled using scalar valued laurent polynomials and a scalar valued measure on the circle in the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions we show how our tools yield results in the well known case of the hadamard walk but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods more precisely we consider qrws on the set of nonnegative integers the analysis of these cases leads to phenomena that are absent in the case of qrws on the integers even if one restricts oneself to a constant coin this is illustrated here by studying recurrence properties of the walk but the same method can be used for other purposes the presentation here aims at being self contained but we refrain from trying to give an introduction to quantum random walks a subject well surveyed in the literature we quote for two excellent reviews see 1 9 see also the recent notes 20 2009 wiley periodicals inc

bulk edge correspondence of one dimensional quantum walks we outline a theory of symmetry protected topological phases of one dimensional quantum walks we assume spectral gaps around the symmetry distinguished points 1 and 1 in which only discrete eigenvalues are allowed the phase classification by integer or binary indices extends the classification known for translation invariant systems in terms of their band structure however our theory requires no translation invariance whatsoever and the indices we define in this general setting are invariant under arbitrary symmetric local perturbations even those that cannot be continuously contracted to the identity more precisely we define two indices for every walk characterizing the behavior far to the right and far to the left respectively their sum is a lower bound on the number of eigenstates at 1 and 1 for a translation invariant system the indices add up to zero so one of them already characterizes the phase by joining two bulk phases with different indices we get a walk in which the right and left indices no longer cancel so the theory predicts bound states at 1 or 1 this is a rigorous statement of bulk edge correspondence the results also apply to the hamiltonian case with a single gap at zero

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

a robust control framework for option pricing a new approach is taken to the problem of option pricing in the standard framework the option pricing problem involves determining a price such that the option writer can guarantee a certain bound on the cost almost surely due to this form the problem may be reformulated in terms of deterministic differential games of the type employed in robust and h control different models yield different prices the standard model yields the black and scholes price both a deterministic model and the standard model with the ito integral replaced by the stratonovich integral yield the price corresponding to a stop loss hedging technique with these methods it can also easily be shown that for the standard model with a bounded stochastic volatility the black and scholes price corresponding to the upper bound for volatility is sufficient to hedge the option

three routes to the exact asymptotics for the one dimensional quantum walk we demonstrate an alternative method for calculating the asymptotic behaviour of the discrete one coin quantum walk on the infinite line via the jacobi polynomials that arise in the path integral representation we calculate the asymptotics using a method that is significantly easier to use than the darboux method it also provides a single integral representation for the wavefunction that works over the full range of positions n including throughout the transitional range where the behaviour changes from oscillatory to exponential previous analyses of this system have run into difficulties in the transitional range because the approximations on which they were based break down here the fact that there are two different kinds of approach to this problem path integral versus schrodinger wave mechanics is ultimately a manifestation of the equivalence between the path integral formulation of quantum mechanics and the original formulation developed in the 1920s we also discuss how and why our approach is related to the two methods that have already been used to analyse these systems

robust control approach to option pricing a representation theorem and fast algorithm the so called interval model for security prices together with a robust control approach allows one to construct a consistent theory of option pricing including discrete time trading and arbitrary transaction costs in this context a new representation theorem for the viscosity solution of the relevant isaacs differential quasi variational inequality leads to simple formulas and fast numerical algorithms to compute a hedging portfolio we argue that in spite of a less satisfactory market model the overall theory is not much less realistic than the classical black and scholes theory but rather only that it shifts from the portfolio model to the market model the place where the model is violated when sudden large price changes occur on the market as such and subject to a more detailed validation the new theory might be the basis of a possible alternative as a normative theory whenever transaction costs or discrete time trading are the main concerns

a new table of constant weight codes a table of binary constant weight codes of length nl28 is presented explicit constructions are given for most of the 600 codes in the table the majority of these codes are new the known techniques for constructing constant weight codes are surveyed and a table of unrestricted binary codes of length nl28 is given

symmetry in semidefinite programs this paper is a tutorial in a general and explicit procedure to simplify semidefinite programs which are invariant under the action of a symmetry group the procedure is based on basic notions of representation theory of finite groups as an example we derive the block diagonalization of the terwilliger algebra of the binary hamming scheme in this framework here its connection to the orthogonal hahn and krawtchouk polynomials becomes visible