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Diophantine condition and real or complex Brjuno functions
 Proc. Int. Conf. on Noise of frequencies in oscillators and the dynamics of algebraic numbers, La Chapelle des Bois
, 1999
"... Abstract. The continued fraction expansion of the real number x = a0 + x0, a0 ∈ Z, is given by 0 ≤ xn < 1, x −1 n = an+1 + xn+1, an+1 ∈ IN, for n ≥ 0. The Brjuno function is then B(x) = ∑∞ x0x1... xn−1 ln(x−1 n=0 n), and the number x satisfies the Brjuno diophantine condition whenever B(x) is bo ..."
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Cited by 2 (1 self)
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Abstract. The continued fraction expansion of the real number x = a0 + x0, a0 ∈ Z, is given by 0 ≤ xn < 1, x −1 n = an+1 + xn+1, an+1 ∈ IN, for n ≥ 0. The Brjuno function is then B(x) = ∑∞ x0x1... xn−1 ln(x−1 n=0 n), and the number x satisfies the Brjuno diophantine condition whenever B
DIOPHANTINE CONDITIONS IN WELLPOSEDNESS THEORY OF COUPLED KDVTYPE SYSTEMS: LOCAL THEORY
, 2009
"... We consider the local wellposedness problem of a oneparameter family of coupled KdVtype systems both in the periodic and nonperiodic setting. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter α when α = 1. In the periodic setting, we u ..."
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Cited by 5 (2 self)
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use the Diophantine conditions to characterize the resonances, and establish sharp local wellposedness of the system in H s (Tλ), s ≥ s ∗, where s ∗ = s ∗ (α) ∈ ( 1,1] is determined by the Diophantine characterization of certain 2 constants derived from the coupling parameter α. We also present a
ftp ejde.math.txstate.edu (login: ftp) DIOPHANTINE CONDITIONS IN GLOBAL WELLPOSEDNESS FOR COUPLED KDVTYPE SYSTEMS
"... Abstract. We consider the global wellposedness problem of a oneparameter family of coupled KdVtype systems both in the periodic and nonperiodic setting. When the coupling parameter α = 1, we prove the global wellposedness in Hs(R) for s> 3/4 and Hs(T) for s ≥ −1/2 via the Imethod developed ..."
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method developed by CollianderKeelStaffilaniTakaokaTao [5]. When α 6 = 1, as in the local theory [14], certain resonances occur, closely depending on the value of α. We use the Diophantine conditions to characterize the resonances. Then, via the second iteration of the Imethod, we establish a global well
Factoring wavelet transforms into lifting steps
 J. Fourier Anal. Appl
, 1998
"... ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This dec ..."
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Cited by 573 (8 self)
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ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is wellknown to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a selfcontained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e, nonunitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a waveletlike transform that maps integers to integers. 1.
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