Results 1  10
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122
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 101 (24 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 69 (18 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 nonnega ..."
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Cited by 48 (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.
Reducing Randomness Via Irrational Numbers
 In Proceedings of the TwentyNinth Annual ACM Symposium on Theory of Computing
, 1997
"... . We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In contrast to the classical technique of DeMillo, Lipton, Schwart ..."
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Cited by 35 (0 self)
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. We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In contrast to the classical technique of DeMillo, Lipton, Schwartz, and Zippel, this methodology can decrease the error probability by increasing the precision of the approximations instead of using more random bits. Consequently, randomized algorithms that use the classical technique can generally be improved using the new methodology. To demonstrate the methodology, we discuss two nontrivial applications. The first is to decide whether a graph has a perfect matching in parallel. Our new NC algorithm uses fewer random bits while doing less work than the previously best NC algorithm by Chari, Rohatgi, and Srinivasan. The second application is to test the equality of two multisets of integers. Our new algorithm improves upon the previously best algorithms ...
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 31 (4 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
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 19 (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...
On coverings of the Integers associated with an irreducibility theorem of A. Schinzel
, 2000
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The average length of a trajectory in a certain billiard in a flat twotorus
, 2003
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ON THE MAILLET–BAKER CONTINUED FRACTIONS
"... Abstract. We use the Schmidt Subspace Theorem to establish the transcendence of a class of quasiperiodic continued fractions. This improves earlier works of Maillet and of A. Baker. We also improve an old result of Davenport and Roth on the rate of increase of the denominators of the convergents to ..."
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Cited by 13 (6 self)
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Abstract. We use the Schmidt Subspace Theorem to establish the transcendence of a class of quasiperiodic continued fractions. This improves earlier works of Maillet and of A. Baker. We also improve an old result of Davenport and Roth on the rate of increase of the denominators of the convergents to any real algebraic number. 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 12 (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.