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634
A proof of Alon’s second eigenvalue conjecture
, 2003
"... A dregular graph has largest or first (adjacency matrix) eigenvalue λ1 = d. Consider for an even d ≥ 4, a random dregular graph model formed from d/2 uniform, independent permutations on {1,...,n}. We shall show that for any ɛ>0 we have all eigenvalues aside from λ1 = d are bounded by 2 √ d − 1 ..."
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Cited by 100 (1 self)
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A dregular graph has largest or first (adjacency matrix) eigenvalue λ1 = d. Consider for an even d ≥ 4, a random dregular graph model formed from d/2 uniform, independent permutations on {1,...,n}. We shall show that for any ɛ>0 we have all eigenvalues aside from λ1 = d are bounded by 2 √ d − 1 +ɛwith probability 1 − O(n−τ), where τ = ⌈ � √ d − 1+1 � /2⌉−1. We also show that this probability is at most 1 − c/nτ ′, for a constant c and a τ ′ that is either τ or τ +1 (“more often ” τ than τ + 1). We prove related theorems for other models of random graphs, including models with d odd. These theorems resolve the conjecture of Alon, that says that for any ɛ>0andd, the second largest eigenvalue of “most ” random dregular graphs are at most 2 √ d − 1+ɛ (Alon did not specify precisely what “most ” should mean or what model of random graph one should take). 1
Iterated function systems and permutation representations of the Cuntz algebra
, 1996
"... We study a class of representations of the Cuntz algebras ON, N = 2, 3,..., acting on L 2 (T) where T = R�2πZ. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into irreducibles, and show how the ONirreducibles decompose when rest ..."
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Cited by 75 (19 self)
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We study a class of representations of the Cuntz algebras ON, N = 2, 3,..., acting on L 2 (T) where T = R�2πZ. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into irreducibles, and show how the ONirreducibles decompose when restricted to the subalgebra UHFN ⊂ ON of gaugeinvariant elements; and we show that the whole structure is accounted for by arithmetic and combinatorial properties of the integers Z. We have general results on a class of representations of ON on Hilbert space H such that the generators Si as operators permute the elements in some orthonormal basis for H. We then use this to extend our results from L 2 (T) to L 2 ( T d) , d> 1; even to L 2 (T) where T is some fractal version of the torus which carries more of the algebraic
Geometric Models for Quasicrystals I. Delone Sets of Finite Type
, 1998
"... This paper studies three classes of discrete sets X in R n which have a weak translational order imposed by increasingly strong restrictions on their sets of interpoint vectors X \Gamma X . A finitely generated Delone set is one such that the abelian group [X \Gamma X ] generated by X \Gamma X i ..."
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Cited by 60 (6 self)
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This paper studies three classes of discrete sets X in R n which have a weak translational order imposed by increasingly strong restrictions on their sets of interpoint vectors X \Gamma X . A finitely generated Delone set is one such that the abelian group [X \Gamma X ] generated by X \Gamma X is finitely generated, so that [X \Gamma X ] is a lattice or a quasilattice. For such sets the abelian group [X ] is finitely generated, and by choosing a basis of [X ] one obtains a homomorphism OE : [X ]!Z s . A Delone set of finite type is a Delone set X such that X \Gamma X is a discrete closed set. A Meyer set is a Delone set X such that X \Gamma X is a Delone set. Delone sets of finite type form a natural class for modelling quasicrystalline structures, because the property of being a Delone set of finite type is determined by "local rules." That is, a Delone set X is of finite type if and only if it has a 20 finite number of neighborhoods of radius 2R, up to translation, where R is ...
On the Capacity of TwoDimensional RunLength Constrained Channels
 IEEE Trans. Inform. Theory
, 1999
"... Twodimensional binary patterns that satisfy onedimensional (d; k) runlength constraints both horizontally and vertically are considered. For a given d and k, the capacity C d; k is defined as C d; k =lim m;n!1 log 2 N m;n =mn, where N m;n denotes the number of m 2 n rectangular patterns that s ..."
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Cited by 60 (11 self)
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Twodimensional binary patterns that satisfy onedimensional (d; k) runlength constraints both horizontally and vertically are considered. For a given d and k, the capacity C d; k is defined as C d; k =lim m;n!1 log 2 N m;n =mn, where N m;n denotes the number of m 2 n rectangular patterns that satisfy the twodimensional (d; k) runlength constraint. Bounds on C d; k are given and it is proven for every d 1 and every k?dthat C d; k =0if and only if k = d +1. Encoding algorithms are also discussed.
Dynamical Sources in Information Theory: A General Analysis of Trie Structures
 ALGORITHMICA
, 1999
"... Digital trees, also known as tries, are a general purpose flexible data structure that implements dictionaries built on sets of words. An analysis is given of three major representations of tries in the form of arraytries, list tries, and bsttries ("ternary search tries"). The size an ..."
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Cited by 55 (7 self)
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Digital trees, also known as tries, are a general purpose flexible data structure that implements dictionaries built on sets of words. An analysis is given of three major representations of tries in the form of arraytries, list tries, and bsttries ("ternary search tries"). The size and the search costs of the corresponding representations are analysed precisely in the average case, while a complete distributional analysis of height of tries is given. The unifying data model used is that of dynamical sources and it encompasses classical models like those of memoryless sources with independent symbols, of finite Markovchains, and of nonuniform densities. The probabilistic behaviour of the main parameters, namely size, path length, or height, appears to be determined by two intrinsic characteristics of the source: the entropy and the probability of letter coincidence. These characteristics are themselves related in a natural way to spectral properties of specific transfer operators of the Ruelle type.
Fifty Years of Shannon Theory
, 1998
"... A brief chronicle is given of the historical development of the central problems in the theory of fundamental limits of data compression and reliable communication. ..."
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Cited by 38 (0 self)
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A brief chronicle is given of the historical development of the central problems in the theory of fundamental limits of data compression and reliable communication.
Arithmetic and Growth of Periodic Orbits
, 2001
"... Two natural properties of integer sequences are introduced and studied. The first, exact realizability, is the property that the sequence coincides with the number of periodic points under some map. This is shown to impose a strong inner structure on the sequence. The second, realizability in rate, ..."
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Cited by 30 (11 self)
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Two natural properties of integer sequences are introduced and studied. The first, exact realizability, is the property that the sequence coincides with the number of periodic points under some map. This is shown to impose a strong inner structure on the sequence. The second, realizability in rate, is the property that the sequence asympototically approximates the number of periodic points under some map. In both cases we discuss when a sequence can have that property. For exact realizability, this amounts to examining the range and domain among integer sequences of the paired transformations
Polynomial invariants for fibered 3manifolds and Teichmüller geodesics for foliations
, 1999
"... ..."
Numeration systems and Markov partitions from self similar tilings
 Trans. Amer. Math. Soc
, 1999
"... Abstract. Using self similar tilings we represent the elements of R n as digit expansions with digits in R n being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the int ..."
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Abstract. Using self similar tilings we represent the elements of R n as digit expansions with digits in R n being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the integer lattice. We use the digit expansions inherited from these tilings to give a symbolic representation for the toral automorphisms. Fractals and fractal tilings have captured the imaginations of a wide spectrum of disciplines. Computer generated images of fractal sets are displayed in public science centers, museums, and on the covers of scientific journals. Fractal tilings which have interesting properties are finding applications in many areas of mathematics. For example, number theorists have linked fractal tilings of R 2 with numeration systems for R 2 in complex bases [16], [8]. We will see that fractal self similar tilings of R n provide natural building blocks for numeration systems of R n. These numeration systems generalize the 1dimensional cases in [14],[10],[11] as well as the 2dimensional cases mentioned above.