this paper were announced in [12]

Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finite-dimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose representations form a braided tensor category. However, this intriguing problem is extremely hard and is still widely open. Triangular Hopf algebras are the quasitriangular Hopf algebras whose representations form a symmetric tensor category. In that sense they are the closest to group algebras. The structure of triangular Hopf algebras is far from trivial, and yet is more tractable than that of general Hopf algebras, due to their proximity to groups. This makes triangular Hopf algebras an excellent testing ground for general Hopf algebraic ideas, methods and conjectures. A general classification of triangular Hopf algebras is not known yet. However, the problem was solved in the semisimple case, in the minimal triangular pointed case, and more generally for triangular Hopf algebras with the Chevalley property. In this paper we report on all of this, and explain in full details the mathematics and ideas involved in this theory. The classification in the semisimple case relies on Deligne’s theorem on Tannakian categories and on Movshev’s theory in an essential way. We explain Movshev’s theory in details, and refer to [G5] for a detailed discussion of the first aspect. We also discuss the existence of grouplike elements in quasitriangular semisimple Hopf algebras, and the representation theory of cotriangular semisimple Hopf algebras. We conclude the paper with a list of open problems; in particular with the question whether any finitedimensional triangular Hopf algebra over C has the Chevalley property. 1.

Nous proposons une approche analytique de la théorie de Galois des systèmes aux q-différences linéaires singuliers réguliers. Nous combinons la dualité de Tannaka avec la méthode de classification de Birkhoff à l’aide de la matrice de connexion pour définir et décrire leurs groupes de Galois. Puis nous décrivons des sous-groupes fondamentaux qui donnent lieu à une correspondance de Riemann-Hilbert et à un théorème de densité de type Schlesinger. We propose an analytical approach to the Galois theory of singular regular linear q-difference systems. We use Tannaka duality along with Birkhoff’s classification scheme with the connection matrix to define and describe their Galois groups. Then we describe fundamental subgroups that give rise to a Riemann-Hilbert correspondence and to a density theorem of Schlesinger’s type.

Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.

Neural networks are now one of the most successful learning formalisms. Neurons transform inputs x 1 ; :::; x n into an output f(w 1 x 1 + ::: +w n x n ), where f is a non-linear function and w i are adjustable weights. What f to choose? Usually the logistic function is chosen, but sometimes the use of different functions improves the practical efficiency of the network. We formulate

. We show that there is a generalization of Rodrigues's formula for computing the exponential map exp: so(n) ! SO(n) from skew symmetric matrices to orthogonal matrices when n 4, and we give a method for computing the function log: SO(n) ! so(n). The case where \Gamma1 is an eigenvalue of R 2 SO(n) requires a special treatment. The key idea is the decomposition of a skew symmetric n \Theta n matrix B in terms of skew symmetric matrices B 1 ; : : : ; Bm such that B = ` 1 B 1 + \Delta \Delta \Delta + ` mBm B i B j = B j B i = 0 n (i 6= j); B 3 i = \GammaB i ; where (i` 1 ; \Gammai` 1 ; : : : ; i` m ; \Gammai` m ) are the nonnull eigenvalues of B. We also consider the exponential map exp: se(n) ! SE(n), where se(n) is the Lie algebra of the Lie group SE(n) of (affine) rigid motions. We show that there is a Rodrigues-like formula for computing this exponential map, and we give a method for computing log: SE(n) ! se(n). This yields a direct proof of the surjectivity of exp: se(n) ! ...

Since Chevalley's seminal work [12], the definition of Lie group has been universally agreed. Namely, a Lie group G is an analytic manifold G on which

this paper can be summarised roughly as follows. Recall that the stars of a groupoid are the fibres of the source map.

Pattern theory provides a unified way to represent knowledge in complex systems. We take a pattern theoretic approach to recognizing and tracking ground-based targets in sequences of forward-looking infrared images acquired from an airborne platform. A rich set of transformations on objects represented by three-dimensional faceted models are formulated to accommodate the variability found in FLIR imagery. Our approach seeks the configuration of templates and transformations that provides the best match with the collected sensor data. An hypothesized scene, simulated from the emissive characteristics of the hypothesized scene elements, is compared with the collected data by a likelihood function based on sensor statistics. This likelihood is combined with a prior distribution defined over the set of possible scenes to form a posterior distribution. A jump-diffusion process empirically generates the posterior distribution. The jumps accommodate the discrete aspects of the estimation prob...

. In this work, we introduce the notion of algebraic subgroups of complex Lie groups, and prove that every faithfully representable complex analytic group G admits an algebraic subgroup T (G) which is the largest in the sense that it contains all algebraic subgroups of G . Moreover, the rational representations of the algebraic subgroup T (G) are exactly the restrictions to T (G) of all complex analytic representations of G . This enables us to single out a certain subgroup of a faithfully representable real analytic group G with which the Tannaka duality theorem is restated. Let K be a complex Lie subgroup of a complex analytic group G. We are first concerned with the question of when K admits the structure of an affine algebraic group which is compatible with the analytic structure of the ambient group G in the sense that the restriction to K of every complex analytic representation of G is rational. In [5], Hochschild and Mostow studied this question for the entire group G, i.e....