Results 1  10
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28
Classical Limit Of The Quantized Hyperbolic Toral Automorphisms
 COMM. MATH. PHYS
, 1995
"... The canonical quantization of any hyperbolic symplectomorphism A of the 2torus yields a periodic unitary operator on a Ndimensional Hilbert space, N = 1 h . We prove that this quantum system becomes ergodic and mixing at the classical limit (N !1, N prime) which can be interchanged with the tim ..."
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Cited by 48 (5 self)
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The canonical quantization of any hyperbolic symplectomorphism A of the 2torus yields a periodic unitary operator on a Ndimensional Hilbert space, N = 1 h . We prove that this quantum system becomes ergodic and mixing at the classical limit (N !1, N prime) which can be interchanged with the timeaverage limit. The recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the Lebesgue measure: the Wigner functions of the eigenstates, supported on the classical periodic orbits, are indeed proved to become uniformly spread in phase space.
Invariants of 3–manifolds and projective representations of mapping class groups via quantum groups at roots of unity
 Comm. Math. Phys
, 1995
"... Abstract. An example of a finite dimensional factorizable ribbon Hopf Calgebra is given by a quotient H = uq(g) of the quantized universal enveloping algebra Uq(g) at a root of unity q of odd degree. The mapping class group Mg,1 of a surface of genus g with one hole projectively acts by automorphis ..."
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Cited by 22 (1 self)
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Abstract. An example of a finite dimensional factorizable ribbon Hopf Calgebra is given by a quotient H = uq(g) of the quantized universal enveloping algebra Uq(g) at a root of unity q of odd degree. The mapping class group Mg,1 of a surface of genus g with one hole projectively acts by automorphisms in the Hmodule H ∗⊗g, if H ∗ is endowed with the coadjoint Hmodule structure. There exists a projective representation of the mapping class group Mg,n of a surface of genus g with n holes labelled by finite dimensional Hmodules X1,..., Xn in the vector space HomH(X1 ⊗ · · · ⊗ Xn, H ∗⊗g). An invariant of closed oriented 3manifolds is constructed. Modifications of these constructions for a class of ribbon Hopf algebras satisfying weaker conditions than factorizability (including most of uq(g) at roots of unity q of even degree) are described. After works of Moore and Seiberg [44], Witten [62], Reshetikhin and Turaev [51], Walker [61], Kohno [22, 23] and Turaev [59] it became clear that any semisimple abelian ribbon category with finite number of simple objects satisfying some nondegeneracy condition gives rise to projective representations of mapping class groups
The determination of Gauss sums
 Bull. Amer. Math. Soc. (New Series
, 1981
"... geometric series can show that P\ 2 e 2 ™ / p 0, where/? is any integer exceeding one. Suppose that we replace n by n k ..."
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Cited by 18 (0 self)
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geometric series can show that P\ 2 e 2 ™ / p 0, where/? is any integer exceeding one. Suppose that we replace n by n k
Amplitude scale estimation for quantizationbased watermarking
 IEEE Transactions on Signal Processing
, 2004
"... Abstract—In this paper we propose a maximum likelihood technique to combat amplitude scaling attacks within a quantizationbased watermarking context. We concentrate on operations that are common in many applications and at the same time devastating to this class of watermarking schemes, namely, ampl ..."
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Cited by 12 (2 self)
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Abstract—In this paper we propose a maximum likelihood technique to combat amplitude scaling attacks within a quantizationbased watermarking context. We concentrate on operations that are common in many applications and at the same time devastating to this class of watermarking schemes, namely, amplitude scaling in combination with additive noise. First we derive the probability density function of the watermarked and attacked data in the absence of subtractive dither. Next we extend these models to incorporate subtractive dither in the encoder. The dither sequence is primarily used for security purposes, and the dither is assumed to be known also to the decoder. We design the dither signal statistics such that an attacker having no knowledge of the dither cannot decode the watermark. Using an approximation of the probability density function in the presence of subtractive dither, we derive a maximum likelihood procedure for estimating amplitude scaling factors. Experiments are performed with synthetic and real audio signals, showing the feasibility of the proposed approach under realistic conditions. Index Terms—Maximum likelihood estimation, probability of error, quantization, statistics, subtractive dither, watermarking. I.
Towards Practical Deterministic WriteAll Algorithms
 IN PROC., 13TH ACM SYMP. ON PARALLEL ALGORITHMS AND ARCHITECTURES, 2001
, 2001
"... The problem of performing t tasks on n asynchronous or undependable processors is a basic problem in parallel and distributed computing. We consider an abstraction of this problem called the WriteAl l problemusing n processors write 1's into all locations of an array of size t. The most e#cient ..."
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Cited by 9 (4 self)
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The problem of performing t tasks on n asynchronous or undependable processors is a basic problem in parallel and distributed computing. We consider an abstraction of this problem called the WriteAl l problemusing n processors write 1's into all locations of an array of size t. The most e#cient known deterministic asynchronous algorithms for this problem are due to Anderson and Woll. The first class of algorithms has work complexity of O(t ), for n t and any #>0, and they are the best known for the full range of processors (n = t). To schedule the work of the processors, the algorithms use lists of q permutations on [q](q n) that have certain combinatorial properties. Instantiating such an algorithm for a specific # either requires substantial preprocessing (exponential in 1/# )to find the requisite permutations, or imposes a prohibitive constant (exponential in 1/# ) hidden by the asymptotic analysis. The second class deals with the specific case of t = n 2, and these algorithms have work complexity of O(t log t). They also use lists of permutations with the same combinatorial properties. However instantiating these algorithms requires exponential in n preprocessing to find the permutations. To alleviate this costly instantiation Kanellakis and Shvartsman proposed a simple way of computing the permutations. They conjectured that their construction has the desired properties but they provided no analysis. In this paper
The number of irreducible factors of a polynomial
 809–834; II, Acta Arith. 78 (1996), 125–142; III, Number theory in progress
, 1993
"... Abstract. Let F(x) be a polynomial with coefficients in an algebraic number field k. We estimate the number of irreducible cyclotomic factors of F in k[x], the number of irreducible noncyclotomic factors of F, the number of nth roots of unity among the roots of F, and the number of primitive nth roo ..."
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Cited by 7 (1 self)
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Abstract. Let F(x) be a polynomial with coefficients in an algebraic number field k. We estimate the number of irreducible cyclotomic factors of F in k[x], the number of irreducible noncyclotomic factors of F, the number of nth roots of unity among the roots of F, and the number of primitive nth roots of unity among the roots of F. All of these quantities are counted with multiplicity and estimated by expressions which depend explicitly on k, on the degree of F and height of F, and (when appropriate) on n. We show by constructing examples that some of our results are essentially sharp. 1.
Zeros and orthogonality of the AskeyWilson polynomials for q a root of unity
 Duke Math. J
, 1997
"... We study some properties of the AskeyWilson polynomials (AWP) when q is a primitive Nth root of unity. For general fourparameter AWP, zeros of the Nth polynomial and the orthogonality measure are found explicitly. Special subclasses of the AWP, e.g., the continuous qJacobi and big qJacobi polyno ..."
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Cited by 3 (0 self)
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We study some properties of the AskeyWilson polynomials (AWP) when q is a primitive Nth root of unity. For general fourparameter AWP, zeros of the Nth polynomial and the orthogonality measure are found explicitly. Special subclasses of the AWP, e.g., the continuous qJacobi and big qJacobi polynomials, are considered in detail. A set of discrete weight functions positive on a real interval is described. Some new trigonometric identities related to the AWP are obtained. Normalization conditions of some polynomials are expressed in terms of the Gauss sums.