Besov regularization,thresholding and wavelets for smoothing data

Smoothing of data is a problem very important for many applications ranging from 1-D signals (e.g., speech) to 2-D and 3-D signals (e.g., images). Many methods exist in the literature for facing the problem; in the present paper we point our attention on regularization. We shall treat regularization methods in a general framework which is well suited for wavelet analysis; in particular we shall investigate on the relation existing between thresholding methods and regularization. We shall also introduce a new regularization method (Besov regularization), which includes some known regularization and thresholding methods as particular cases. Numerical experiments based on some test problems will be performed in order to compare the performance of some methods of smoothing data.

AMS (MOS) Subject Classifications: 65R30, 41A60.     

Invariant generators for generalized distributions

The existence of invariant generators for distributions satisfying a compatibility condition with the symmetry algebra is proved.     

A Class of Integrable Flows on the Space of Symmetric Matrices

For a given skew symmetric real n × n matrix N, the bracket [X, Y] _N = XNY − YNX defines a Lie algebra structure on the space Sym(n, N) of symmetric n × n real matrices and hence a corresponding Lie-Poisson structure. The purpose of this paper is to investigate the geometry, integrability, and linearizability of the Hamiltonian system , or equivalently in Lax form, the equation on this space along with a detailed study of the Poisson geometry itself. If N has distinct eigenvalues, it is proved that this system is integrable on a generic symplectic leaf of the Lie-Poisson structure of Sym(n, N). This is established by finding another compatible Poisson structure. If N is invertible, several remarkable identifications can be implemented. First, (Sym(n, N), [·, ·]) is Lie algebra isomorphic with the symplectic Lie algebra associated to the symplectic form on given by N ⁻¹. In this case, the system is the reduction of the geodesic flow of the left invariant Frobenius metric on the underlying symplectic group Sp(n, N ⁻¹). Second, the trace of the product of matrices defines a non-invariant non-degenerate inner product on Sym(n, N) which identifies it with its dual. Therefore Sym(n, N) carries a natural Lie-Poisson structure as well as a compatible “frozen bracket” structure. The Poisson diffeomorphism from Sym(n, N) to maps our system to a Mischenko-Fomenko system, thereby providing another proof of its integrability if N is invertible with distinct eigenvalues. Third, there is a second ad-invariant inner product on Sym(n, N); using it to identify Sym(n, N) with itself and composing it with the dual of the Lie algebra isomorphism with , our system becomes a Mischenko- Fomenko system directly on Sym(n, N). If N is invertible and has distinct eigenvalues, it is shown that this geodesic flow on Sym(n, N) is linearized on the Prym subvariety of the Jacobian of the spectral curve associated to a Lax pair formulation with parameter of the system. If, on the other hand, N has nullity one and distinct eigenvalues, in spite of the fact that the system is completely integrable, it is shown that the flow does not linearize on the Jacobian of the spectral curve. Research partially supported by NSF grants CMS-0408542 and DMS-0604307. Research partially supported by the Swiss SCOPES grant IB7320-110721/1, 2005-2008, and MEdC Contract 2-CEx 06-11-22/25.07.2006. Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932. Research partially supported by the Swiss NSF and the Swiss SCOPES grant IB7320-110721/1.     

Three-dimensional quantitative structure-activity relationships of steroid aromatase inhibitors

Summary Inhibition of aromatase, a cytochrome P450 that converts androgens to estrogens, is relevant in the therapeutic control of breast cancer. We investigate this inhibition using a three-dimensional quantitative structure-activity relationship (3D QSAR) method known as Comparative Molecular Field Analysis, CoMFA [Cramer III, R.D. et al., J. Am. Chem. Soc., 110 (1988) 5959]. We analyzed the data for 50 steroid inhibitors [Numazawa, M. et al., J. Med. Chem., 37 (1994) 2198, and references cited therein] assayed against androstenedione on human placental microsomes. An initial CoMFA resulted in a three-component model for log(1/K_i), with an explained variance r² of 0.885, and a cross-validated q² of 0.673. Chemometric studies were performed using GOLPE [Baroni, M. et al., Quant. Struct.-Act. Relatsh., 12 (1993) 9]. The CoMFA/GOLPE model is discussed in terms of robustness, predictivity, explanatory power and simplicity. After randomized exclusion of 25 or 10 compounds (repeated 25 times), the q² for one component was 0.62 and 0.61, respectively, while r² was 0.674. We demonstrate that the predictive r² based on the mean activity (Y_m) of the training set is misleading, while the test set Y_m-based predictive r² index gives a more accurate estimate of external predictivity. Using CoMFA, the observed differences in aromatase inhibition among C6-substituted steroids are rationalized at the atomic level. The CoMFA fields are consistent with known, potent inhibitors of aromatase, not included in the model. When positioned in the same alignment, these compounds have distinct features that overlap with the steric and electrostatic fields obtained in the CoMFA model. The presence of two hydrophobic binding pockets near the aromatase active site is discussed: a steric bulk tolerant one, common for C4, C6-alpha and C7-alpha substitutents, and a smaller one at the C6-beta region.     

On the generators of semisimple lie algebras

It will be shown that given any element X in a simple Lie algebra $\text{Q}$ over C, there exists a Y ∈ $\text{Q}$ such that the Lie algebra generated by X and Y is $\text{Q}$. The result is extended to the real semisimple Lie algebras. In some sense the main theorem of this paper can be regarded as an extension of Morozov-Jacobson theorem concerning three dimensional simple Lie algebras (see the remark at the end of Sec. 4). A new property of a special class of regular elements, known as the cyclic elements, is given.     

Canonical maps between semidirect products with applications to elasticity and superfluids

It is shown that two canonical maps arising in the Poisson bracket formulations of elasticity and superfluids are particular instances of general canonical maps between duals of semidirect product Lie algebras.