optimal binary index assignments for a class of equiprobable scalar and vector quantizers the problem of scalar and vector quantization in conjunction with a noisy binary symmetric channel is considered the issue is the assignment of the shortest possible distinct binary sequences to quantization levels or vectors so as to minimize the mean squared error caused by channel errors by formulating the assignment as a matrix or vector in the scalar case and showing that the mean squared error due to channel errors is determined by the projections of its columns onto the eigenspaces of the multidimensional channel transition matrix a class of source quantizer pairs is identified for which the optimal index assignment has a simple and natural form among other things this provides a simpler and more accessible proof of the result of crimmins et al 1969 that the natural binary code is an optimal index assignment for the uniform scalar quantizer and uniform source it also provides a potentially useful approach to further developments in source channel coding

length constrained map decoding of variable length encoded markov sequences in this paper we consider the problem of length constrained maximum a posteriori decoding of a markov sequence that is variable length encoded and transmitted over a binary symmetric channel we convert this problem into one of a maximum weight k link path in a weighted directed acyclic graph the induced graph optimization problem can be solved by a fast parameterized search algorithm that finds either the optimal solution with high probability or a good approximate solution otherwise the proposed algorithm has lower complexity and superior performance than the previous approximation algorithms

classification of matrix product unitaries with symmetries we prove that matrix product unitaries with on site unitary symmetries are completely classified by the chiral index and the cohomology class of the symmetry group g provided that we can add trivial and symmetric ancillas with arbitrary on site representations of g if the representations in both system and ancillas are fixed to be the same we can define symmetry protected indices spis which quantify the imbalance in the transport associated to each group element and greatly refines the classification these spis are stable against disorder and measurable in interferometric experiments our results lead to a systematic construction of two dimensional floquet symmetry protected topological phases beyond the standard classification and thus shed new light on understanding nonequilibrium phases of quantum matter

the stability in _ and _ of the projection onto finite element function spaces on etablit la stabilite de la l 2 projection sur quelques espaces a elements finis v h consideree comme une application dans l p et w p 1 1 p sous des hypotheses de regularite faibles

a generalization of yoshida nicolaescu theorem using partial signatures we give a simplified proof of the yoshida nicolaescu theorem in the product case using the theory of partial signatures as in giambo et al 2004 the theorem gives the equality of the spectral flow of a family of first order self adjoint differential operators defined on sections of a hermitian vector bundle over a partitioned manifold and the maslov index of the corresponding pair of cauchy data spaces no nondegeneracy assumption is made on the endpoints of the path of differential operators

on the joint source channel coding error exponent for discrete memoryless systems we investigate the computation of csisza spl acute r s bounds for the joint source channel coding jscc error exponent e sub j of a communication system consisting of a discrete memoryless source and a discrete memoryless channel we provide equivalent expressions for these bounds and derive explicit formulas for the rates where the bounds are attained these equivalent representations can be readily computed for arbitrary source channel pairs via arimoto s algorithm when the channel s distribution satisfies a symmetry property the bounds admit closed form parametric expressions we then use our results to provide a systematic comparison between the jscc error exponent e sub j and the tandem coding error exponent e sub t which applies if the source and channel are separately coded it is shown that e sub t spl les e sub j spl les 2e sub t we establish conditions for which e sub j e sub t and for which e sub j 2e sub t numerical examples indicate that e sub j is close to 2e sub t for many source channel pairs this gain translates into a power saving larger than 2 db for a binary source transmitted over additive white gaussian noise awgn channels and rayleigh fading channels with finite output quantization finally we study the computation of the lossy jscc error exponent under the hamming distortion measure

on factorization of m channel paraunitary filterbanks we systematically investigate the factorization of causal finite impulse response fir paraunitary filterbanks with given filter length based on the singular value decomposition of the coefficient matrices of the polyphase representation a fundamental order one factorization form is first proposed for general paraunitary systems then we develop a new structure for the design and implementation of paraunitary system based on the decomposition of hermitian unitary matrices within this framework the linear phase filterbank and pairwise mirror image symmetry filterbank are revisited their structures are special cases of the proposed general structures compared with the existing structures more efficient ones that only use approximately half the number of free parameters are derived the proposed structures are complete and minimal although the factorization theory with or without constraints is discussed in the framework of m channel filterbanks the results can be applied to wavelets and multiwavelet systems and could serve as a general theory for paraunitary systems

index assignment optimization for joint source channel map decoding channel optimized quantizer index assignment and maximum a posteriori map decoding have been extensively studied for error resilient communications an interesting and largely untreated problem is how to optimize the index assignment with respect to joint source channel map decoding in this paper we formulate the above problem as one of quadratic assignment and discuss its solutions from very general to some special cases for highly correlated gaussian markov sources and hamming distortion we can construct the optimal index assignment analytically for general cases simulated annealing algorithm is adopted to search for the optimal index assignment experimental results are presented to demonstrate the performance improvement of the index assignments optimized for map decoding over those designed for hard decision decoding e g gray code the reduction of symbol error rate and mean squared error can be as large as 40 and 50 respectively for highly correlated gaussian markov sources

spectral flow and level spacing of edge states for quantum hall hamiltonians we consider a non relativistic particle on the surface of a semi infinite cylinder of circumference l submitted to a perpendicular magnetic field of strength b and to the potential of impurities of maximal amplitude w this model is of importance in the context of the integer quantum hall effect in the regime of strong magnetic field or weak disorder b w it is known that there are chiral edge states which are localized within a few magnetic lengths close to and extended along the boundary of the cylinder and whose energy levels lie in the gaps of the bulk system these energy levels have a spectral flow uniform in l as a function of a magnetic flux which threads the cylinder along its axis through a detailed study of this spectral flow we prove that the spacing between two consecutive levels of edge states is bounded below by 2 l 1 with 0 independent of l and of the configuration of impurities this implies that the level repulsion of the chiral edge states is much stronger than that of extended states in the usual anderson model and their statistics cannot obey one of the gaussian ensembles our analysis uses the notion of relative index between two projections and indicates that the level repulsion is connected to topological aspects of quantum hall systems

on the accuracy of the finite element method plus time relaxation if u denotes a local spatial average of u then u u u is the associated fluctuation consider a time relaxation term added to the usual finite element method the simplest case for the model advection equation u t a u x t is u h t a u h v h u h v h f x t v h we analyze the error in this and more importantly higher order extensions and show that the added time relaxation term not only suppresses excess energy in marginally resolved scales but also increases the accuracy of the resulting finite element approximation

classification of quantum cellular automata there exists an index theory to classify strictly local quantum cellular automata in one dimension fidkowski et al in interacting invariants for floquet phases of fermions in two dimensions 2017 arxiv 1703 07360 gross et al in commun math phys 310 2 419 454 2012 po et al in phys rev b 96 245116 2017 we consider two classification questions first we study to what extent this index theory can be applied in higher dimensions via dimensional reduction finding a classification by the first homology group of the manifold modulo torsion second in two dimensions we show that an extension of this index theory including torsion fully classifies quantum cellular automata at least in the absence of fermionic degrees of freedom this complete classification in one and two dimensions by index theory is not expected to extend to higher dimensions due to recent evidence of a nontrivial automaton in three dimensions haah et al in nontrivial quantum cellular automata in higher dimensions 2018 arxiv 1812 01625 finally we discuss some group theoretical aspects of the classification of quantum cellular automata and consider these automata on higher dimensional real projective spaces

quantum cellular automata tensor networks and area laws quantum cellular automata are unitary maps that preserve locality and respect causality we identify them in any dimension with simple tensor networks projected entangled pair unitary whose bond dimension does not grow with the system size as a result they satisfy an area law for the entanglement entropy they can create we define other classes of nonunitary maps the so called quantum channels that either respect causality or preserve locality we show that whereas the latter obey an area law for the number of quantum correlations they can create as measured by the quantum mutual information the former may violate it we also show that neither of them can be expressed as tensor networks with a bond dimension that is independent of the system size

monotonicity based fast algorithms for map estimation of markov sequences over noisy channels in this correspondence we study algorithmic approach to solving the problem of maximum a posteriori map estimation of markov sequences transmitted over noisy channels which is also known as the map decoding problem for the class of memoryless binary channels that produce independent substitution and erasure errors the map sequence estimation problem can be formulated and solved as one of the longest path in a weighted directed acyclic graph but for algorithm efficiency we transform the graph problem to one of matrix search if the underlying matrix is totally monotone then the complexity of map sequence estimation can be greatly reduced we give a sufficient condition for the matrix induced by map sequence estimation to be totally monotone which is indeed the case if the input sequence is gaussian markov under this condition the complexity of map decoding can be reduced from o n sup 2 m to o nm where n is the size of source alphabet and m is the length of input sequence furthermore for markov sequences of fixed length code we propose a block parsing strategy to reduce the complexity of map sequence estimation to o m n sup 2 m logm or to o m nm logm depending on if the total monotonicity holds another significance of this correspondence lies in the applicability of the presented algorithmic approach which has been thoroughly studied in computer science literature to many other discrete optimization problems encountered in both source and channel coding ranging from optimal multiresolution and multiple description quantizer design to context quantization for minimum conditional entropy and to optimal packetization with uneven error protection

the regularized chapman enskog expansion for scalar conservation laws rosenao r has recently proposed a regularized version of the chapmanenskog expansion of hydrodynamics this regularized expansion resembles the usual navier stokes viscosity terms at low wave numbers but unlike the latter it has the advantage of being a bounded macroscopic approximation to the linearized collision operator in this paper we study the behavior of the rosenau regularization of the chapman enskog expansion r c e in the context of scalar conservation laws we show that this r c e model retains the essential properties of the usual viscosity approximation e g existence of travelling waves monotonicity upper lipschitz continuity etc and at the same time it sharpens the standard viscous shock layers we prove that the regularized r c e approximation converges to the underlying inviscid entropy solution as its mean free path e 0 and we estimate the convergence rate

essentially commuting projections abstract let h h h be a fixed orthogonal decomposition of a hilbert space with both subspaces of infinite dimension and let e e be the projections onto h and h we study the set p c c of orthogonal projections p in h which essentially commute with e or equivalently with e i e p e p e e p is compact by means of the projection onto the calkin algebra one sees that these projections p p c c fall into nine classes four discrete classes which correspond to p being 0 1 e or e and five essential classes which we describe below the discrete classes are respectively the finite rank projections finite co rank projections the sato grassmannian of h and the sato grassmannian of h thus the connected components of each of these classes are parametrized by the integers via de rank the co rank or the fredholm index respectively the essential classes are shown to be connected we are interested in the geometric structure of p c c being the set of selfadjoint projections of the c algebra b c c of operators in b h which essentially commute with e in particular we study the problem of existence of minimal geodesics joining two given projections in the same component we show that the hopf rinow theorem holds in the discrete classes but not in the essential classes conditions for the existence and uniqueness of geodesics in these latter classes are found

on pairs of projections in a hilbert space we give an algebraic derivation of the canonical form of a generic pair of projections the result is used to determine the spectral shift of a pair of projections and various properties of fredholm pairs

the spectral shift function and the invariance principle abstract the new representation formula for the spectral shift function due to f gesztesy and k a makarov is considered this formula is extended to the case of relatively trace class perturbations the proof is based on the analysis of a certain new unitary invariant for a pair of self adjoint operators

high energy asymptotics of the magnetic spectral shift function we consider the three dimensional schrodinger operators h0 and h where h0 i a 2 b a is a magnetic potential generating a constant magnetic field of strength b 0 and h h0 v where v l1 r3 r satisfies certain regularity conditions then the spectral shift function e h h0 for the pair of operators h h0 is well defined for energies e 2qb q z we study the asymptotic behavior of e h h0 as e e or r 0 b where or e 0 dist e 2bz r we obtain a weyl type formula lime e ore 1 2 e h h0 1 4 2 r3v x dx

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

periodic jacobi matrices on trees abstract we begin the systematic study of the spectral theory of periodic jacobi matrices on trees including a formal definition the most significant result that appears here for the first time is that these operators have no singular continuous spectrum we review important previous results of sunada and aomoto and present several illuminating examples we present many open problems and conjectures that we hope will stimulate further work

index theory of one dimensional quantum walks and cellular automata if a one dimensional quantum lattice system is subject to one step of a reversible discrete time dynamics it is intuitive that as much quantum information as moves into any given block of cells from the left has to exit that block to the right for two types of such systems namely quantum walks and cellular automata we make this intuition precise by defining an index a quantity that measures the net flow of quantum information through the system the index supplies a complete characterization of two properties of the discrete dynamics first two systems s1 s2 can be pieced together in the sense that there is a system s which acts like s1 in one region and like s2 in some other region if and only if s1 and s2 have the same index second the index labels connected components of such systems equality of the index is necessary and sufficient for the existence of a continuous deformation of s1 into s2 in the case of quantum walks the index is integer valued whereas for cellular automata it takes values in the group of positive rationals in both cases the map s mapsto rm ind s is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems systems with trivial index are precisely those which can be realized by partitioned unitaries and the prototypes of systems with non trivial index are shifts

truncation of scales by time relaxation abstract we study a time relaxation regularization of flow problems proposed and tested extensively by stolz and adams the aim of the relaxation term is to drive the unresolved fluctuations in a computational simulation to zero exponentially fast by an appropriate and often problem dependent choice of its coefficient this relaxation term is thus intermediate between a tunable numerical stabilization and a continuum modeling term our aim herein is to understand how this term by itself acts to truncate solution scales and to use this understanding to give insight into parameter selection

asymptotic evolution of quantum walks with random coin we study the asymptotic position distribution of general quantum walks on a lattice including walks with a random coin which is chosen from step to step by a general markov chain in the unitary i e nonrandom case we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other for example a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense with any lattice dimension and coin dimension we find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time namely as the distribution of the discrete time analog of the group velocity in the random case we let a markov chain control process pick in each step one of finitely many unitary walks in the sense described above in ballistic order we find a nonrandom drift which depends only on the mean of the control process and not on the initial state in diffusive scaling the limiting distribution is asymptotically gaussian with a covariance matrix diffusion matrix depending on momentum the diffusion matrix depends not only on the mean but also on the transition rates of the control process in the nonrandom limit i e when the coins chosen are all very close or the transition rates of the control process are small leading to long intervals of ballistic evolution the diffusion matrix diverges our method is based on spatial fourier transforms and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

pairs of orthogonal projections with a fixed difference abstract let a be a self adjoint contraction on a hilbert space h in 2014 e andruchow proved that if a is a difference of two orthogonal projections with non trivial generic part then there exist infinitely many pairs p q of orthogonal projections such that a p q in the present paper we give a sufficient and necessary condition for a to be a difference of a pair of orthogonal projections then we give a characterization of all pairs p q of orthogonal projections such that a p q moreover we characterize the von neumann algebra generated by such pairs p q and consider the connected components of the set p q a p q

parametrizing projections with selfadjoint operators abstract let h h h be an orthogonal decomposition of a hilbert space with e e the corresponding projections let a be a selfadjoint operator in h which is codiagonal with respect to this decomposition i e a h h and a h h we consider three maps which assign a selfadjoint projection to a 1 the graph map a projection onto the graph of a h 2 the exponential map of the grassmann manifold p of h the space of selfadjoint projections in h at e exp a e i 2 a e e i 2 a 3 the map p called here the davis map based on a result by chandler davis characterizing the selfadjoint contractions which are the difference of two projections the ranges of these maps are studied and compared using davis map one can solve the following operator matrix completion problem given a contraction a h h complete the matrix a 2 a 2 to a projection p in order that p e is minimal

anti selfadjoint operators as commutators of projections abstract let a be an anti selfadjoint operator on a hilbert space h with a 1 2 we give a sufficient and necessary condition for a to be a commutator of a pair of orthogonal projections and establish the general representation of all pairs p q of orthogonal projections such that a p q q p then we discuss the path components of the set c a p q a p q q p we prove that the action of unitary group u a is transitive in each path component of c a when a is in generic position moreover we characterize the von neumann algebra generated by all projections in c a as an application we obtain that the set of all commutators of pairs of orthogonal projections is connected

critical lieb thirring bounds in gaps and the generalized nevai conjecture for finite gap jacobi matrices we prove bounds of the form e i d h dist e e h 1 2 l 1 norm of a perturbation where i is a gap included are gaps in continuum one dimensional periodic schrodinger operators and finite gap jacobi matrices where we get a generalized nevai conjecture about an l 1 condition implying a szeg condition one key is a general new form of the birman schwinger bound in gaps

decoherent quantum walks driven by a generic coin operation abstract we consider the effect of different unitary noise mechanisms on the evolution of a quantum walk qw on a linear chain with a generic coin operation i bit flip channel noise restricted to the coin subspace of the qw and ii topological noise caused by randomly broken links in the linear chain similarities and differences in the respective decoherent dynamics of the walker as a function of the probability per unit time of a decoherent event taking place are discussed

stability and analyticity in maximum norm for simplicial lagrange finite element semidiscretizations of parabolic equations with dirichlet boundary conditions stability and analyticity estimates in maximum norm are shown for spatially discrete finite element approximations based on simplicial lagrange elements for the model heat equation with dirichlet boundary conditions the bounds are logarithm free and valid in arbitrary dimension and for arbitrary polynomial degree the work continues an earlier study by schatz et al 5 in which neumann boundary conditions were considered

discrete time adiabatic theorems for quantum mechanical systems the theory of adiabatic asymptotics is adapted to systems with discrete time evolution the corresponding theorems about the approximation of physical time evolution by the adiabatic time evolution are shown to hold true in a discrete setting

the number theoretical spin chain and the riemann zeroes it is an empirical observation that the riemann zeta function can be well approximated in its critical strip using the number theoretical spin chain a proof of this would imply the riemann hypothesis here we relate that question to the one of spectral radii of a family of markov chains this in turn leads to the question whether certain graphs are ramanujan

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

spectral theory of discontinuous functions of self adjoint operators and scattering theory in the smooth scattering theory framework we consider a pair of self adjoint operators h0 h and discuss the spectral projections of these operators corresponding to the interval 1 the purpose of the paper is to study the spectral properties of the dierence d of these spectral projections we completely describe the absolutely continuous spectrum of the operator d in terms of the eigenvalues of the scattering matrix s for the operators h0 and h we also prove that the singular continuous spectrum of the operator d is empty and that its eigenvalues may accumulate only at thresholds in the absolutely continuous spectrum

general explicit descriptions for intertwining operators and direct rotations of two orthogonal projections abstract in this paper based on the block operator technique and operator spectral theory general explicit descriptions for intertwining operators and direct rotations of two orthogonal projections are established as a consequence kato s result kato 1996 14 is improved so are j avron r seiler and b simon s theorem 2 3 avron et al 1994 6 and c davis w m kahan s result davis and kahan 1970 11

column space equalities for orthogonal projectors in their paper y tian g p h styan rank equalities for idempotent and involutory matrices linear algebra appl 335 2001 101 117 tian and styan established several rank equalities involving a pair of idempotent matrices p and q subsequently these results are reinvestigated from the point of view of the following question provided that idempotent p q are hermitian which relationships given in the aforementioned paper remain valid when ranks are replaced with column spaces simultaneously some related results are established which shed additional light on the links between subspaces attributed to various functions of a pair of orthogonal projectors

the index formula and the spectral shift function for relatively trace class perturbations we compute the fredholm index index da of the operator da d dt a on l2 r h associated with the operator path a t t where af t a t f t for a e t r and appropriate f l2 r h via the spectral shift function a a associated with the pair a a of asymptotic operators a a on the separable complex hilbert space h in the case when a t is generally an unbounded relatively trace class perturbation of the unbounded self adjoint operator a we derive a formula an extension of a formula due to pushnitski relating the spectral shift function a a for the pair a a and the corresponding spectral shift function h2 h1 for the pair of operators h2 h1 dada da da in this relative trace class context h2 h1 1 1 2 1 2 a a d 2 1 2for a e 0 this formula is then used to identify the fredholm index of da with 0 a a in addition we prove that index da coincides with the spectral flow spflow a t t of the family a t t r and also relate it to the fredholm perturbation determinant for the pair a a index da spflow a t t 0 a a 1lime 0im ln deth a iei a iei 1 0 h2 h1 with the choice of the branch of ln deth on c such that limim z ln deth a zi a zi 1 0 we also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta invariant

unitaries permuting two orthogonal projections let p and q be two orthogonal projections on a separable hilbert space h wang du and dou proved that there exists a unitary u with upu 1 q uqu 1 p if and only if dim ker p ker 1 q dim ker q ker 1 p both may be infinite we provide a new proof using the supersymmetric machinery of avron seiler and simon

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

the birman schwinger principle on the essential spectrum let h0 and h be self adjoint operators in a hilbert space we consider the spectral projections of h0 and h corresponding to a semi infinite interval of the real line we discuss the index of this pair of spectral projections and prove an identity which extends the birman schwinger principle onto the essential spectrum we also relate this index to the spectrum of the scattering matrix for the pair h0 h

eigenvalues of functions of orthogonal projectors by representing two orthogonal projectors in a finite dimensional vector space as partitioned matrices several characterizations concerning eigenvalues of various functions of the pair are obtained these results substantially extend the ones already available in the literature additionally some related results dealing with the functions of a pair of orthogonal projectors are provided with the emphasis laying on the problem of invertibility

on boundary value problems for dirac type operators i regularity and self adjointness in a series of papers we will develop systematically the basic spectral theory of self adjoint boundary value problems for operators of dirac type we begin in this paper with the characterization of self adjoint boundary conditions with optimal regularity for which we will derive the heat asymptotics and index theorems in subsequent publications along with a number of new results we extend and simplify the proofs of many known theorems our point of departure is the simple structure which is displayed by dirac type operators near the boundary thus our proofs are given in an abstract functional analytic setting generalizing considerably the framework of compact manifolds with boundary the results of this paper have been announced previously by the authors j bruning and m lesch in geometric aspects of partial differential equations b booss bavnbek and k p wojciechowski eds contemporary mathematics vol 242 pp 203 205 amer math soc providence ri 1999

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

numerical analysis of a nonlinear time relaxation model of fluids abstract this report investigates a time relaxation model for the regularization of the strong solution of the navier stokes equations following the earlier works of adams stolz and kleiser 37 38 ervin layton and neda 11 layton and neda 26 the nonlinear term investigated herein aims to better control the generalized turbulent fluctuations thus providing better stabilization effect on the underlying model we study the numerical properties of the resulting model and prove that the addition of the nonlinear term does not affect the overall order of convergence to the strong solution of the navier stokes equation our theoretical results based on the finite element method are confirmed by the numerical tests performed in the last section of the paper

a study of vector quantization for noisy channels several issues related to vector quantization for noisy channels are discussed an algorithm based on simulated annealing is developed for assigning binary codewords to the vector quantizer code vectors it is shown that this algorithm could result in dramatic performance improvements as compared to randomly selected codewords a modification of the simulated annealing algorithm for binary codeword assignment is developed for the case where the bits in the codeword are subjected to unequal error probabilities resulting from unequal levels of error protection an algorithm for the design of an optimal vector quantizer for a noisy channel is briefly discussed and its robustness under channel mismatch conditions is studied numerical results for a stationary first order gauss markov source and a binary symmetric channel are provided it is concluded that the channel optimized vector quantizer design algorithm if used carefully can result in a fairly robust system with no additional delay the case in which the communication channel is nonstationary as in mobile radio channels is studied and some preliminary ideas for quantizer design are presented

pseudo gray coding a pseudo gray code is an assignment of n bit binary indexes to 2 points in a euclidean space so that the hamming distance between two points corresponds closely to the euclidean distance pseudo gray coding provides a redundancy free error protection scheme for vector quantization vq of analog signals when the binary indexes are used as channel symbols on a discrete memoryless channel and the points are signal codevectors binary indexes are assigned to codevectors in a way that reduces the average quantization distortion introduced in the reproduced source vectors when a transmitted index is corrupted by channel noise a globally optimal solution to this problem is generally intractable due to an inherently large computational complexity a locally optimal solution the binary switching algorithm is introduced based on the objective of minimizing a useful upper bound on the average system distortion the algorithm yields a significant reduction in average distortion and converges in reasonable running times the sue of pseudo gray coding is motivated by the increasing need for low bit rate vq based encoding systems that operate on noisy channels such as in mobile radio speech communications

the index of a pair of projections we discuss pairs of self adjoint projections with p q j 2n 1 the trace ideal and prove that for m n tr p q 2m 1 tr p q 2n 1 dim ker q ran p dim ker p ran q is an integer we also prove that there exists a unitary v interchanging p and q if and only if this integer is 0

on the algebra generated by two projections the von neumann algebra generated by a pair of projections on a separable hilbert space occurs naturally in the study of quantum hall systems by a systematic use of kato s dual pair of self adjoint operators corresponding to the two projections the argument referring to the structure of the algebra can be simplified and the existence of an action of the group su 2 on the algebra via inner automorphisms can be shown also the algebra can be given an r structure with respect to which it is r commutative and an extension of the classical commutative differential calculus becomes possible

holder estimates for green s functions on convex polyhedral domains and their applications to finite element methods a model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered the aim of this paper is twofold first sharp holder estimates for the corresponding green s function are obtained as an applications of these estimates to finite element methods we show the best approximation property of the error in w 1 infty in contrast to previously known results w p 2 regularity for p 3 which does not hold for general convex polyhedral domains is not required furthermore the new green s function estimates allow us to obtain localized error estimates at a point

spectral flow and bifurcation of critical points of strongly indefinite functionals part i general theory abstract spectral flow is a well known homotopy invariant of paths of selfadjoint fredholm operators we describe here a new construction of this invariant and prove the following theorem let i u rbe ac2function on the product of a real intervali a b with a neighborhood u of the origin of a real separable hilbert space h and such that for each in i 0is a critical point of the functional assume that the hessian l of at0is fredholm and moreover that laand lbare nonsingular if the spectral flow of the path l idoes not vanish then the interval i contains at least one bifurcation point from the trivial branch for solutions of the equation x x 0 equivalently every neighborhood of i 0 contains points of the form x wherex 0 is a critical point of

operators which are the difference of two projections abstract we study the set d of differences d a p q p q p where p denotes the set of orthogonal projections in h we describe models and factorizations for elements in d which are related to the geometry of p the study of d throws new light on the geodesic structure of p we show that two projections in generic position are joined by a unique minimal geodesic the topology of d is examined particularly its connected components are studied also we study the subsets d c d f where d c are the compact elements in d and d f are the differences a p q such that the pair p q is a fredholm pair p q is a fredholm pair if q p r p r p r q is a fredholm operator

twisted duality of the car algebra we give a complete proof of the twisted duality property m q zm q z of the self dual car algebra in any fock representation the proof is based on the natural halmos decomposition of the reference hilbert space when two suitable closed subspaces have been distinguished we use modular theory and techniques developed by kato concerning pairs of projections in some essential steps of the proof as a byproduct of the proof we obtain an explicit and simple formula for the graph of the modular operator this formula can be also applied to fermionic free nets hence giving a formula of the modular operator for any double cone

quantizers with uniform encoders and channel optimized decoders scalar quantizers with uniform encoders and channel optimized decoders are studied for uniform sources and binary symmetric channels it is shown that the natural binary code nbc and folded binary code fbc induce point density functions that are uniform on proper subintervals of the source support whereas the gray code gc does not induce a point density function the mean squared errors mse for the nbc fbc gc and for randomly chosen index assignments are calculated and the nbc is shown to be mean squared optimal among all possible index assignments for all bit error rates and all quantizer transmission rates in contrast it is shown that almost all index assignments perform poorly and have degenerate codebooks

dpcm picture transmission over noisy channels with the aid of a markov model the problem of dpcm picture transmission over noisy channels is considered it is well known that dpcm systems are very sensitive to channel errors the goal in this work is to build robustness against channel errors three methods are proposed in this paper and are obtained by modeling the encoded signal as a markov sequence first an optimum method for decoding correlated sequences is derived and it is shown to require viterbi decoding then a modified map method mmap for markov sequences is described a maximal signal to noise msnr receiver for dpcm systems is also developed that minimizes the distortion power due to channel errors the appropriate cost matrix for this receiver is computed these methods are applied to dpcm picture transmission over noisy channels and are compared with a another method the snr graphs as well as subjective examination of the received pictures demonstrate that the proposed procedures are quite effective and superior to that method among the proposed methods the msnr receiver was found to be more effective than the others for a given order of the markov model it is observed that the proposed methods are most beneficial for low detail pictures

spectral shift function in the large coupling constant limit given two selfadjoint operators h0 and v v v we study the spectrum of the operator h h0 v 0 we consider the quantity h h0 r which coincides with krein s spectral shift function of the pair h h0 if v is of trace class and study its asymptotic behavior as applications to differential operators are given

the birman schwinger principle in von neumann algebras of finite type abstract we introduce a relative index for a pair of dissipative operators in a von neumann algebra of finite type and prove an analog of the birman schwinger principle in this setting as an application of this result revisiting the birman krein formula in the abstract scattering theory we represent the de la harpe skandalis determinant of the characteristic function of dissipative operators in the algebra in terms of the relative index