spectral asymptotics for the third order operator with periodic coefficients abstract we consider the self adjoint third order operator with 1 periodic coefficients on the real line the spectrum of the operator is absolutely continuous and covers the real line we determine the high energy asymptotics of the periodic antiperiodic eigenvalues and of the branch points of the lyapunov function furthermore in the case of small coefficients we show that either whole spectrum has multiplicity one or the spectrum has multiplicity one except for a small spectral nonempty interval with multiplicity three in the last case the asymptotics of this small interval is determined

tunstall adaptive coding and miscoding in the first part of this paper we tackle the case where a variable length to block tunstall code is used to encode the wrong source miscoding it turns out that exactly as happens in the case with huffman coding the asymptotic excess rate is given by the informational divergence between the probability distribution ruling the source and the probability distribution for which the tunstall code had been devised we also prove asymptotic equality between the individual rate of a codeword and the corresponding self information this allows us to bound the maximal and the minimal length in a tunstall tree in the second part of the paper we study the case where the probability distribution of the source changes in time if this happens it is necessary to frequently update the current code to ensure that the optimality conditions concerning its rate are met this coding procedure is known as adaptive coding we propose some schemes for adaptive tunstall coding based on the structure of the coding tree on the ordering property analogous to gallager s 1978 sibling property of the huffman code and on the tunstall regions analogous to the attraction regions due to longo and galasso 1982

a universal variable to fixed length source code based on lawrence s algorithm it is shown that the modified lawrence algorithm is universal over the class of binary memoryless sources and that the rate converges asymptotically optimally fast to the source entropy it is proven that no codes exist that have a better asymptotic performance the asymptotic bounds show that universal variable to fixed length codes can have a significantly lower redundancy than universal fixed to variable length codes with the same number of codewords

variable to fixed length codes for markov sources petry s efficient and optimal variable to fixed length source code for discrete memoryless sources was described by schalkwijk by extending this coding technique we are able to give an algorithm for markov sources that is easy to implement we can bound the loss of efficiency as a function of the code complexity and the mismatch between the source and the code rates arbitrarily close to the source entropy are shown to be achievable in this sense the codes introduced are optimal

the estimates of periodic potentials in terms of effective masses letgn an an n 1 denote the gaps mn be the effective masses and n an 1 an a0 0 be the spectral bands of the hill operatort d2 dx2 v x inl2 r wherev is a 1 periodic real potential froml2 0 1 let the length gapln gn hn be the height of the corresponding slit on the quasimomentum domain and n 2 2n 1 n 0 be the band reduction let l n sqrt a n sqrt a n n geqq 1 n 1 denote the gap length for the operator sqrt t geqq 0 introduce the sequencesl ln h hn l ln n m mn and the norms left f right m 2 sum nolimits n 0 left 2 pi n right 2m f n 2 m geqq 0 m 0 the following results are obtained i the estimates of v l h 1 l 1 in terms of m 2 ii identities for the dirichlet integral of quasimomentum and integral of potentials and so on iii the generation of i ii for more general potentials

borg type theorems for matrix valued schrodinger operators abstract a borg type uniqueness theorem for matrix valued schrodinger operators is proved more precisely assuming a reflectionless potential matrix and its spectrum a half line 0 we derive the triviality of the potential matrix our approach is based on trace formulas and matrix valued herglotz representation theorems as a by product of our techniques we obtain an extension of borg s classical result from the class of periodic scalar potentials to the class of reflectionless matrix valued potentials

a note on approximation of uniform distributions from variable to fixed length codes in this correspondence we prove that the probability distribution induced on the leaves of a tunstall parse tree for a given source is a unique lower bound in the partially ordered set of the probability distributions induced by all possible parse trees with a same number of leaves and ordered according to the majorization partial order we apply this result to the problem of optimally approximating a uniform distribution with flips of a biased coin

distribution of poles for scattering on the real line abstract the density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential in the case of a potential with finite singularities at the endpoints of the support asymptotic formulae for the poles are given while in the c 0 case an example of a potential with infinitely many scattering poles on i r is constructed the scattering amplitude of a compactly supported potential is also characterized

a fast rigorous technique for computing the regulator of a real quadratic field we present a new algorithm for computing the regulator of a real quadratic field q d based on an algorithm for unconditionally verifying the correctness of the regulator produced by a subexponential algorithm that runs in expected time o d 1 6 e under the generalized riemann hypothesis the correctness of our algorithm relies on no unproven hypotheses and is currently the fastest known unconditional algorithm for computing the regulator a number of implementation issues and performance enhancements are discussed and we present the results of computations demonstrating the efficiency of the new algorithm

on the parallel generation of the residues for the continued fraction factoring algorithm etude des proprietes du developpement en fraction continue d un irrationnel quadratique application au developpement d un algorithme efficace de factorisation

the inverse problem for the hill operator a direct approach we consider the hill operator t yd2 dx2 v x on l2 r where v 2 l2 0 1 is an even 1 periodic real potential and r 1 0 v x dx 0 it is wellknown that the spectrum of t is absolutely continuous and consists of intervals s1 s2 such that sn any1 an any1 an an n 1 these intervals are separated by gaps g1 g2 where gn an an if a gap degenerates that is gn then the corresponding segments sn sn 1 merge let u x e v x e v be the solutions of the equation yf 00 vf ef e 2 c 1 1

bounds on the redundancy of huffman codes corresp new upper bounds on the redundancy of huffman codes are provided a bound that for 2 9 leq p 1 leq 0 4 is sharper than the bound of gallager when the probability of the most likely source letter p 1 is the only known probability is presented the improved bound is the tightest possible for 1 3 leq p 1 leq 0 4 upper bounds are presented on the redundancy of huffman codes when the extreme probabilities p 1 and p n are known

computer verification of the ankeny artin chowla conjecture for all primes less than 100 000 000 000 let p be a prime congruent to 1 modulo 4 and let t u be rational integers such that t u p 2 is the fundamental unit of the real quadratic field q p the ankeny artin chowla conjecture aac conjecture asserts that p will not divide u this is equivalent to the assertion that p will not divide b p 1 2 where b n denotes the nth bernoulli number although first published in 1952 this conjecture still remains unproved today indeed it appears to be most difficult to prove even testing the conjecture can be quite challenging because of the size of the numbers t u for example when p 40094470441 then both t and u exceed 10 330 000 in 1988 the aac conjecture was verified by computer for all p 10 9 in this paper we describe a new technique for testing the aac conjecture and we provide some results of a computer run of the method for all primes p up to 10 11

a numerical investigation into the length of the period of the continued fraction expansion of p d cd 1 21ogd o d l 2 where c 7 22t2 2 3546 the brief numerical study on the size of p d in beach and williams 1 seems to indicate that these bounds are not very good for larger values of d the purpose of this note is to suggest why this should be the case and to provide numerical evidence for the belief that p d can be bounded by an expression of the form cd 1 21og log d where c is a constant

an application of informational divergence to huffman codes a classification of all probability distributions over the finite alphabet of an information source is given where the classes are the sets of distributions sharing the same binary huffman code such a classification can be used in noiseless coding when the distribution of the finite memoryless source varies in time or becomes gradually known instead of applying the huffman algorithm to each new estimate of the probability distribution if a simple test based on the above classification is passed then the huffman code used previously is optimal also for the new distribution

a rigorous subexponential algorithm for computation of class groups let c d denote the gauss class group of quadratic forms of a negative discriminant d or equivalently the class group of the imaginary quadratic field q a we give a rigorous proof that there exists a las ve gas algorithm that will compute the structure of c d with an expected running time of l d vf2 1 bit operations where l d exp vlog d log log d thus of course also includes the computation of the class number h d the cardinality of c d ibm research division almaden research center k53 802 650 harry road san jose california 95 120 6099 current address k mccurley organization 1423 sandia national laboratories albuquerque new mexico 87185 this content downloaded from 207 46 13 180 on thu 08 sep 2016 05 47 29 utc all use subject to http about jstor org terms

the lyapunov function for schrodinger operators with a periodic 2 2 matrix potential abstract we consider the schrodinger operator on the real line with a 2 2 matrix valued 1 periodic potential the spectrum of this operator is absolutely continuous and consists of intervals separated by gaps we define a lyapunov function which is analytic on a two sheeted riemann surface on each sheet the lyapunov function has the same properties as in the scalar case but it has branch points which we call resonances we prove the existence of real as well as non real resonances for specific potentials we determine the asymptotics of the periodic and the anti periodic spectrum and of the resonances at high energy we show that there exist two type of gaps 1 stable gaps where the endpoints are the periodic and the anti periodic eigenvalues 2 unstable resonance gaps where the endpoints are resonances i e real branch points of the lyapunov function we also show that periodic and anti periodic spectrum together determine the spectrum of the matrix hill operator

on the composition of tunstall messages we study the composition of messages in an encoding tree for a tunstall code and more generally in a tree whose skewness is bounded for such trees a sort of law of large numbers holds true actually we provide a direct and converse coding theorem for variable length to block length source codes when a vanishing error probability is allowed

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

eigenvalue estimates and trace formulas for the matrix hill s equation abstract estimates for floquet multipliers and periodic eigenvalues are developed for the matrix hill s equation basic estimates and a description of the periodic eigenvalues as roots of an entire function are adequate for establishing trace formulas with a residue computation refined estimates and power sum formulas are available for restricted pairs of hill s operators

conformal spectral theory for the monodromy matrix for any n x n monodromy matrix we define the lyapunov function which is analytic on an associated n sheeted riemann surface on each sheet the lyapunov function has the standard properties of the lyapunov function for the hill operator the lyapunov function has real or complex branch points which we call resonances we determine the asymptotics of the periodic anti periodic spectrum and of the resonances at high energy we show that the endpoints of each gap are periodic anti periodic eigenvalues or resonances real branch points moreover the following results are obtained 1 we define the quasimomentum as an analytic function on the riemann surface of the lyapunov function various properties and estimates of the quasimomentum are obtained 2 we construct the conformal mapping with imaginary part given by the lyapunov exponent and we obtain various properties of this conformal mapping which are similar to the case of the hill operator 3 we determine various new trace formulae for potentials and the lyapunov exponent 4 we obtain a priori estimates of gap lengths in terms of the dirichlet integral we apply these results to the schrodinger operators and to first order periodic systems on the real line with a matrix valued complex self adjoint periodic potential