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67
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 81 (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
Stability and linear independence associated with wavelet decompositions
 Proc. Amer. Math. Soc
, 1993
"... Wavelet decompositions are based on basis functions satisfying refinement equations. The stability, linear independence and orthogonality of the integer translates of basis functions play an essential role in the study of wavelets. In this paper we characterize these properties in terms of the mask ..."
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Cited by 59 (14 self)
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Wavelet decompositions are based on basis functions satisfying refinement equations. The stability, linear independence and orthogonality of the integer translates of basis functions play an essential role in the study of wavelets. In this paper we characterize these properties in terms of the mask sequence in the refinement equation satisfied by the basis function.
On vanishing sums of roots of unity
 J. Algebra
, 1995
"... Abstract. Consider the mth roots of unity in C, where m> 0 is an integer. We address the following question: For what values of n can one find n such mth roots of unity (with repetitions allowed) adding up to zero? We prove that the answer is exactly the set of linear combinations with nonnegativ ..."
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Cited by 20 (0 self)
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Abstract. Consider the mth roots of unity in C, where m> 0 is an integer. We address the following question: For what values of n can one find n such mth roots of unity (with repetitions allowed) adding up to zero? We prove that the answer is exactly the set of linear combinations with nonnegative integer coefficients of the prime factors of m. 1.
qseries identities and values of certain Lfunctions
 Duke Math. J
"... As usual, define Dedekind’s etafunction η(z) by the infinite product η(z): = q 1/24 n 1 − q) ( q: = e 2πiz throughout). n=1 In a recent paper, D. Zagier proved that (note: empty products equal 1 throughout) n=0 η(24z) − q ( 1 − q 24) ( 1 − q 48) ·· · ( 1 − q 24n)) = η(24z)D(q) + E(q), where the ..."
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Cited by 19 (2 self)
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As usual, define Dedekind’s etafunction η(z) by the infinite product η(z): = q 1/24 n 1 − q) ( q: = e 2πiz throughout). n=1 In a recent paper, D. Zagier proved that (note: empty products equal 1 throughout) n=0 η(24z) − q ( 1 − q 24) ( 1 − q 48) ·· · ( 1 − q 24n)) = η(24z)D(q) + E(q), where the series D(q) and E(q) are defined by D(q) = − 1 2 + E(q) = 1
On Duadic Codes
, 1986
"... We define a class of qary cyclic codes, the socalled duadic codes. These codes are a direct generalization of QR codes. The results of Leon, Masley and Pless on binary duadic codes are generalized. Duadic codes of composite length and a low minimum distance are constructed. We consider duadic code ..."
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Cited by 10 (0 self)
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We define a class of qary cyclic codes, the socalled duadic codes. These codes are a direct generalization of QR codes. The results of Leon, Masley and Pless on binary duadic codes are generalized. Duadic codes of composite length and a low minimum distance are constructed. We consider duadic codes of length a prime power, and we give an existence test for cyclic projective planes. Furthermore, we give bounds for the minimum distance of all binary duadic codes of length <=241.
On coverings of the Integers associated with an irreducibility theorem of A. Schinzel
 A K Peters
, 2000
"... this paper is to give a partially expository account of results related to coverings of the integers (defined below) while at the same time making some new observations concerning a related polynomial problem. The polynomial problem we will consider is to determine whether for a given positive integ ..."
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Cited by 9 (2 self)
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this paper is to give a partially expository account of results related to coverings of the integers (defined below) while at the same time making some new observations concerning a related polynomial problem. The polynomial problem we will consider is to determine whether for a given positive integer
Normal Bases over Finite Fields
, 1993
"... Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to repr ..."
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Cited by 9 (0 self)
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Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...
Bijective arithmetic codings of hyperbolic automorphisms of the 2torus, and binary quadratic forms
 J. Dynam. Control Systems
, 1998
"... Abstract. We study the arithmetic codings of hyperbolic automorphisms of the 2torus, i.e. the continuous mappings acting from a certain symbolic space of sequences with a finite alphabet endowed with an appropriate structure of additive group onto the torus which preserve this structure and turn th ..."
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Cited by 7 (4 self)
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Abstract. We study the arithmetic codings of hyperbolic automorphisms of the 2torus, i.e. the continuous mappings acting from a certain symbolic space of sequences with a finite alphabet endowed with an appropriate structure of additive group onto the torus which preserve this structure and turn the twosided shift into a given automorphism of the torus. This group is uniquely defined by an automorphism, and such an arithmetic coding is a homomorphism of that group onto T 2. The necessary and sufficient condition of the existence of a bijective arithmetic coding is obtained; it is formulated in terms of a certain binary quadratic form constructed by means of a given automorphism. Furthermore, we describe all bijective arithmetic codings in terms the Dirichlet group of the corresponding quadratic field. The minimum of that quadratic form over the nonzero elements of the lattice coincides with the minimal possible order of the kernel of a homomorphism described above. In this work we continue studying the symbolic dynamics of ergodic automorphisms of the 2torus. The dynamics of automorphisms of the torus is related more
Topological Conjugacy of Linear Endomorphisms of the 2Torus
 Trans. Amer. Math. Soc
, 1997
"... . We describe two complete sets of numerical invariants of topological conjugacy for linear endomorphisms of the twodimensional torus, i.e., continuous maps from the torus to itself which are covered by linear maps of the plane. The trace and determinant are part of both complete sets, and two can ..."
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Cited by 6 (1 self)
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. We describe two complete sets of numerical invariants of topological conjugacy for linear endomorphisms of the twodimensional torus, i.e., continuous maps from the torus to itself which are covered by linear maps of the plane. The trace and determinant are part of both complete sets, and two candidates are proposed for a third (and last) invariant which, in both cases, can be understood from the topological point of view. One of our invariants is in fact the ideal class of the LatimerMacDuffeeTaussky theory, reformulated in more elementary terms and interpreted as describing some topology. Merely, one has to look at how closed curves on the torus intersect their image under the endomorphism. Part of the intersection information (the intersection number counted with multiplicity) can be captured by a binary quadratic form associated to the map, so that we can use the classical theories initiated by Lagrange and Gauss. To go beyond the intersection number, and shortcut the classifi...