kadison singer algebras ii general case a new class of operator algebras kadison singer ks algebras is introduced these highly noncommutative non self adjoint algebras generalize triangular matrix algebras they are determined by certain minimally generating lattices of projections in the von neumann algebras corresponding to the commutant of the diagonals of the ks algebras it is shown that these lattices and their reduced forms are often homeomorphic to classical manifolds such as the sphere

arimoto renyi conditional entropy and bayesian m ary hypothesis testing this paper gives upper and lower bounds on the minimum error probability of bayesian m ary hypothesis testing in terms of the arimoto renyi conditional entropy of an arbitrary order alpha the improved tightness of these bounds over their specialized versions with the shannon conditional entropy alpha 1 is demonstrated in particular in the case where m is finite we show how to generalize fano s inequality under both the conventional and list decision settings as a counterpart to the generalized fano s inequality allowing m to be infinite a lower bound on the arimoto renyi conditional entropy is derived as a function of the minimum error probability explicit upper and lower bounds on the minimum error probability are obtained as a function of the arimoto renyi conditional entropy for both positive and negative alpha furthermore we give upper bounds on the minimum error probability as functions of the renyi divergence in the setup of discrete memoryless channels we analyze the exponentially vanishing decay of the arimoto renyi conditional entropy of the transmitted codeword given the channel output when averaged over a random coding ensemble

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

bayesian m ary hypothesis testing the meta converse and verdu han bounds are tight two alternative exact characterizations of the minimum error probability of bayesian m ary hypothesis testing are derived the first expression corresponds to the error probability of an induced binary hypothesis test and implies the tightness of the meta converse bound by polyanskiy et al the second expression is a function of an information spectrum measure and implies the tightness of a generalized verdu han lower bound the formulas characterize the minimum error probability of several problems in information theory and help to identify the steps where existing converse bounds are loose

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

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

isomorphisms of certain csl algebras abstract we show that an isomorphism between two reflexive operator algebras on hilbert space with commutative subspace lattices csl is automatically continuous and induces an isomorphism between the lattices for such algebras with completely distributive subspace lattices cdc algebras the quasi spatially implemented isomorphisms are shown to be exactly those that preserve the rank of all finite rank operators we present some examples of nonspatial isomorphisms and discuss some sufficient conditions on the lattices that ensure that all isomorphisms be spatially implemented the relationship between isomorphisms and derivations of csl algebras is also investigated

derivations of a class of kadison singer algebras abstract let l be a double triangle lattice of projections in a finite von neumann algebra acting on a separable and complex hilbert space k we show that every derivation from the reflexive algebra determined by l into b k is quasi spatial and automatically continuous we also obtain that every local derivation on the reflexive algebra is a derivation

singular continuous spectrum on a cantor set of zero lebesgue measure for the fibonacci hamiltonian it is rigorously proven that the spectrum of the tight binding fibonacci hamiltonian h mn m n 1 m n 1 m n n 1 n where 5 1 2 and means integer part is a cantor set of zero lebesgue measure for all real nonzero and the spectral measures are purely singular continuous this follows from a recent result by kotani coupled with the vanishing of the lyapunov exponent in the spectrum

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

a generalized poor verdu error bound for multihypothesis testing a lower bound on the minimum error probability for multihypothesis testing is established the bound which is expressed in terms of the cumulative distribution function of the tilted posterior hypothesis distribution given the observation with tilting parameter generalizes an earlier bound due the poor and verdu 1995 a sufficient condition is established under which the new bound minus a multiplicative factor provides the exact error probability asymptotically in examples illustrating the new bound are also provided

algebraic isomorphisms and strongly double triangle subspace lattices abstract let d 0 k l m x be a strongly double triangle subspace lattice on a non zero complex reflexive banach space x which means that at least one of three sums k l l m and m k is closed it is proved that a non zero element s of alg d is single in the sense that for any a b alg d either as 0 or sb 0 whenever asb 0 if and only if s is of rank two we also show that every algebraic isomorphism between two strongly double triangle subspace lattice algebras is quasi spatial

spectral properties of one dimensional quasi crystals in this paper we prove that the one dimensional schrodinger operator onl 2 with potential given by upsilon n lambda chi 1 alpha 1 x n alpha alpha notin mathbb q has a cantor spectrum of zero lebesgue measure for any irrational and any 0 we can thus extend the kotani result on the absence of absolutely continuous spectrum for this model to all mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpepge9fr xfr x frpewzqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiaadihaiiaacq wfiiiztuudjxwak1uy0hmmaehbfv3yslgzg0uy0hgiud3bagqbbiab nj8ubaa 4628 x in mathbb t

a note on the poor verdu upper bound for the channel reliability function in an earlier work poor and verdu 1995 established an upper bound for the reliability function of arbitrary single user discrete time channels with memory they also conjectured that their bound is tight for all coding rates we demonstrate via a counterexample involving memoryless binary erasure channels becs that the poor verdu upper bound is not tight at low rates we conclude by examining possible improvements to this bound

spectral properties of one dimensional schrodinger operators with potentials generated by substitutions we investigate one dimensional discrete schrodinger operators whose potentials are invariant under a substitution rule the spectral properties of these operators can be obtained from the analysis of a dynamical system called the trace map we give a careful derivation of these maps in the general case and exhibit some specific properties under an additional easily verifiable ypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying schrodinger operator is singular and supported on a set of zero lebesgue measure a condition allowing to exclude point spectrum is also given the application of our theorems is explained on a series of examples

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

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

bayesian error based sequences of statistical information bounds the relation between statistical information and bayesian error is sharpened by deriving finite sequences of upper and lower bounds on equivocation entropy ee in terms of the minimum probability of error mpe and related bayesian quantities the well known fano upper bound and feder merhav lower bound on ee are tightened by including a succession of posterior probabilities starting at the largest which directly controls the mpe and proceeding to successively lower ones a number of other interesting results are also derived including a sequence of upper bounds on the mpe in terms of a previously introduced sequence of generalized posterior distributions the tightness of the various bounds is numerically evaluated for a simple example

kadison singer algebras hyperfinite case a new class of operator algebras kadison singer algebras ks algebras is introduced these highly noncommutative non self adjoint algebras generalize triangular matrix algebras they are determined by certain minimally generating lattices of projections in the von neumann algebras corresponding to the commutant of the diagonals of the ks algebras a new invariant for the lattices is introduced to classify these algebras

on some automorphisms of a class of kadison singer algebras abstract let l be the kadison singer lattice generated by a nontrivial nest n on an infinite dimensional separable hilbert space h and a rank one projection p determined by a separating vector for the von neumann algebra n and alg l be the corresponding kadison singer algebra in this paper we study the problem on the weak density of the subalgebra generated all the rank one operators characterize the single elements in alg l and give the quasi spatiality of an automorphism of alg l depending on whether i has an immediate predecessor in n or not

a lower bound on the probability of error in multihypothesis testing consider two random variables x and y where x is finitely or countably infinitely valued and where y is arbitrary let spl epsiv denote the minimum probability of error incurred in estimating x from y it is shown that spl epsiv spl ges sub 0 spl les spl alpha spl les 1 sup sup 1 spl alpha p spl pi x y spl les spl alpha where spl pi x y denotes the posterior probability of x given y this bound finds information theoretic applications in the proof of converse channel coding theorems it generalizes and strengthens previous lower bounds due to shannon and to verdu and han 1994