Blobbed topological recursion

We describe the formalism and the properties of the blobbed topological recursion, which provides the general solution of a set of abstract loop equations. This procedure extends the topological recursion by introducing extra terms (blobs) in the initial conditions for each multidifferential ω_g,n. We apply this formalism to topological expansion of formal Hermitian matrix models (the blobs are necessary to include arbitrary interactions) and pose some open questions.     

Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group

We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian structure of step three. We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with the caustic (the set of conjugate points along all geodesics). The group of symmetries of the cut locus is described: it is generated by a one-parameter group of dilations R₊ and a discrete group of reflections Z₂ × Z₂ × Z₂. The cut locus admits a stratification with 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimensional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Two-dimensional strata of the cut locus consist of conjugate points. Finally, one-dimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well. Projections of sub-Riemannian geodesics to the 2-dimensional plane of the distribution are Euler elasticae. For each point of the cut locus, we describe the Euler elasticae corresponding to minimizers coming to this point. Finally, we describe the structure of the optimal synthesis, i. e., the set of minimizers for each terminal point in the Engel group.     

Some Remarks on the Combinatorics of ISn

\noindent We describe the asymptotic behavior of the cardinalities of the finite symmetric inverse semigroup IS_n and its endomorphism semigroup. This is applied to show that the ratio |IS_n|/|End(IS_n)| is asymptotically 0, answering a question of Schein and Teclezghi. We also apply our results to compute the distributions of elements from IS_n with respect to certain combinatorial properties, and to compute the generating functions for |IS_n| and for the number of nilpotent elements in IS_n.     

Simplicial K(G,1)'s

We describe a subcomplex S_* (G) of a K(G,1) which depends on a given presentation of the group G. We prove that under a certain condition S_* (G) is a K(G,1). Complexes of the type S_* (G) arise in the homotopy theory of the classifying space for foliations.This research was partially supported by the N.S.F.     

Realizations of the Associahedron and Cyclohedron

We describe many different realizations with integer coordinates for the associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the Bott-Taubes polytope) and compare them with the permutahedron of type A and B, respectively. The coordinates are obtained by an algorithm which uses an oriented Coxeter graph of type A_n or B_n as the only input data and which specializes to a procedure presented by J.-L. Loday for a certain orientation of A_n. The described realizations have cambrian fans of type A and B as normal fans. This settles a conjecture of N. Reading for cambrian lattices of these types.     

Mappings with finite distortion in l loc p : modulus of continuity and compression of Hausdorff Measure

We study mappings of finite distortion whose distortion functions belong to the Lebesgue space L _loc ^p . We establish a local modulus of continuity estimate for the inverse of such a mapping. As an application, we describe the possible compression of Hausdorff measure under such mappings. We also exhibit examples that describe the extent to which our results are sharp.     

Représentations induites d'algèbres de Hecke affines et singularités de R-matrices

We give an explicit criterion for the irreducibility of some induction products of evaluation modules of affine Hecke algebras of type A. This allows to describe the form of the zeroes and poles of the trigonometric R-matrix associated to any evaluation module of U_ν(sl_N).     

Compact Packings of the Plane with Two Sizes of Discs

We consider packings of the plane using discs of radius 1 and r. A packing is compact if every disc D is tangent to a sequence of discs D₁, D₂, ..., D_n such that D_i is tangent to D_i+1. We prove that there are only nine values of r with r < 1 for which such packings are possible. For each of the nine values we describe the possible compact packings.     

Tensor Products and Whittaker Vectors for Quantum Groups

We describe the action of the center of the quantum group U_q () on the tensor product V ? L(λ) of an infinite-dimensional representation V having an infinitesimal character χ_τ and an irreducible finite-dimensional U_q () representation L(λ) of highest weight λ. We apply this result in order to describe the tensor product of a Whittaker module and a finite-dimensional simple module for the algebra U_q(l₂).     

Sur les algèbres de Lie nilpotentes admettant un tore de dérivations

Summary We describe the class of complex filiform nilpotent Lie algebras provided with a not trivial external torus of derivations. We prove also that, for dimensions greater than 8, any algebraic irreducible component of the variety of complex nilpotent filiform laws of Lie algebra contains an open set whose elements are characteristically nilpotent laws.

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This article was processed by the author using the Springer-Verlag T_EXP Jour1g macro package 1991.     

Involutions for upper triangular matrix algebras

In this paper we describe completely the involutions of the first kind of the algebra UT_n(F) of n×n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F) when n is even and a single class in the odd case.Furthermore we consider the algebra UT₂(F) of the 2×2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution *, we describe the *-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras.Then we consider the *-polynomial identities for the algebra UT₃(F) over a field of characteristic zero. We describe a finite generating set of the ideal of *-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the *-identities for the algebra UT_n(F) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.     

The Validity of the Multifractal Formalism: Results and Examples

By obtaining a new sufficient condition for a valid multifractal formalism, we improve in this paper a result developed by L. Olsen (1995, Adv. Math.116, 82-196). In particular, we describe a large class of measures satisfying the multifractal formalism and for which the construction of Gibbs measures is not possible. Some of these measures are not unidimensional but have a nontrivial multifractal spectrum, giving a negative answer to a question asked by S. J. Taylor (1995, J. Fourier Anal. Appl., special issue). We also describe a necessary condition of validity for the formalism which is very close to the sufficient one. This necessary condition allows us to describe a measure μ for which the multifractal packing dimension function B_μ(q) is a nontrivial real analytic function but the multifractal formalism is nowhere satisfied. This example gives also a solution to a problem posed by Taylor (cited above).     

Boundedly complete basic sequences,c 0-subspaces,and injections of Banach spaces

We study the connection between topological properties of subsets of a given Banach space and their images under linear, continuous one-to-one mappings on the one hand and the existence in a given Banach space of either a boundedly complete basic sequence (BCBS) or an isomorphic copy ofc _o (c _o -subspace) on the other hand. We present criteria for the existence of a BCBS. They are deduced from new characterisations ofG _δ-embeddings which we also present. We obtain a necessary and sufficient condition for separability of a dual Banach space in terms of saturation by BCBS. Criteria for the existence in a Banach space of ac _o -subspace are also presented. We describe the class of separable Banach spaces which contains either a BCBS or ac _o -subspace. This research was supported by the Rashi Foundation.     

Quasidiagonal Solutions of the Yang–Baxter Equation, Quantum Groups and Quantum Super Groups

This paper answers a few questions about algebraic aspects of bialgebras, associated with the family of solutions of the quantum Yang–Baxter equation in Acta Appl. Math. 41 (1995), pp. 57–98. We describe the relations of the bialgebras associated with these solutions and the standard deformations of GL_n and of the supergroup GL(m|n). We also show how the existence of zero divisors in some of these algebras are related to the combinatorics of their related matrix, providing a necessary and sufficient condition for the bialgebras to be a domain. We consider their Poincaré series, and we provide a Hopf algebra structure to quotients of these bialgebras in an explicit way. We discuss the problems involved with the lift of the Hopf algebra structure, working only by localization.     

On the lattice of matric-extensible radicals

A radical α in the universal class of associative rings is called matric-extensible if α (R _n) = (α (R))_n for any ring R, and natural number n, where R _n denotes the nxn matrix ring with entries from R. We investigate matric-extensibility of the lower radical determined by a simple ring S. This enables us to find necessary and sufficient conditions for the lower radical determined by S to be an atom in the lattice of hereditary matric-extensible radicals. We also show that this lattice has atoms which are not of this form. We then describe all atoms of the lattice, and show that it is atomic.     

The Einstein-Kähler metric on Hua construction of the first type

Let HC_I be the Hua construction of the first type. We describe the Einstein-Kähler metric for HC_I. We reduce the Monge-Ampè re equation for the metric to an ordinary differential equation in the auxiliary function\(X\left( {z,w,\zeta } \right)\) This differential equation can be solved to give an implicit function in\(X\left( {z,w,\zeta } \right)\). For some cases, we obtained the solution of the differential equation and the explicit forms of the complete Einstein-Kähler metrics on HC_I which are the non-homogeneous domains.     

Feature selection for high-dimensional data

This paper focuses on feature selection for problems dealing with high-dimensional data. We discuss the benefits of adopting a regularized approach with L ₁ or L ₁–L ₂ penalties in two different applications—microarray data analysis in computational biology and object detection in computer vision. We describe general algorithmic aspects as well as architecture issues specific to the two domains. The very promising results obtained show how the proposed approach can be useful in quite different fields of application.     

Varieties of two-dimensional cylindric algebras.¶Part I: Diagonal-free case

We investigate the lattice of all subvarieties of the variety Df ₂ of two-dimensional diagonal-free cylindric algebras. We prove that a Df ₂-algebra is finitely representable if it is finitely approximable, characterize finite projective Df ₂-algebras, and show that there are no non-trivial injectives and absolute retracts in Df ₂. We prove that every proper subvariety of Df ₂ is locally finite, and hence Df ₂ is hereditarily finitely approximable. We describe all six critical varieties in , which leads to a characterization of finitely generated subvarieties of Df ₂. Finally, we describe all square representable and rectangularly representable subvarieties of Df ₂. Received May 25, 2000; accepted in final form November 2, 2001.     

Minimal hyperspheres in rank two compact symmetric spaces

We describe a method to construct embedded, minimal hyperspheres in rank two compact symmetric spaces which are equivariant under the isotropy action of the symmetric space, and we supply the details of the construction for the exceptional Lie groupG ₂.Partially supported by CNPq (brazil)     

Geometric description of L 1-Spaces

We describe strongly facially symmetric spaces which are isometrically isomorphic to L ₁-space.