Pointlike sets with respect to R and J

We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety of all finite R-trivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute -pointlike sets, where denotes the pseudovariety of all finite J-trivial semigroups. We finally show that, in contrast with the situation for , the natural adaptation of Henckell’s algorithm to computes pointlike sets, but not all of them.     

A decomposition of R into two homeomorphic rigid parts

Let X be separable, completely metrizable, and dense in itself. We show that if X admits a triple (D₁, D₂, h) of two countable dense subsets D₁ and D₂ and a homeomorphism h: X?D₁ → X?D₂, satisfying some special properties, then there is a rigid subspace A of X such that A is homeomorphic to X?A = h[A]; for $\text{X =}\text{R}$, such atriple is shown to exist.     

The zeros of a quadratic form at square-free points

Let F(x₁,…,xn) be a nonsingular indefinite quadratic form, n=3 or 4. For n=4, suppose the determinant of F is a square. Results are obtained on the number of solutions of

F(x₁,…,xn)=0     

A remark on renormalized volume and Euler characteristic for ache 4-manifolds

This note computes a renormalized volume and a renormalized Gauss-Bonnet-Chern formula for asymptotically complex hyperbolic Einstein (so-called ache) 4-manifolds.     

A generalization of the primitive normal basis theorem

The primitive normal basis theorem asks whether every finite field extension has a primitive normal basis of this extension. The proof of this problem has recently been completed by Lenstra and Schoof (1987) [6], and another proof is given by Cohen and Huczynska (2003) [3]. We present a more general result, where the primitive element generating a normal basis is replaced by a primitive element generating the finite Carlitz module. Such generators always exist except for finitely many cases which might not exist.     

A constructive proof of Ky Fan's generalization of Tucker's lemma

We present a proof of Ky Fan's combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fan's lemma to allow for triangulations of Sⁿ that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tucker's lemma that holds for a more general class of triangulations than the usual version.     

Formal paths, iterated integrals and the center problem for ordinary differential equations

We continue the study of the center problem for the ordinary differential equation started in [A. Brudnyi, An explicit expression for the first return map in the center problem, J. Differential Equations 206 (2004) 306-314; A. Brudnyi, On the center problem for ordinary differential equations, Amer. J. Math. 128 (2006) 419-451; A. Brudnyi, An algebraic model for the center problem, Bull. Sci. Math. 128 (2004) 839-857; A. Brudnyi, On center sets of ODEs determined by moments of their coefficients, Bull. Sci. Math. 130 (2006) 33-48; A. Brudnyi, Vanishing of higher-order moments on Lipschitz curves, Bull. Sci. Math. 132 (3) (2008) 165-181]. In this paper we present the highlights of the algebraic theory of centers.     

On a generalization of absolute neighborhood retracts

In this paper we generalize the concept of absolute neighborhood retract by introducing the notion of absolute neighborhood multi-retract. Furthermore, the Lefschetz fixed point theorem for admissible maps defined on absolute neighborhood multi-retracts is proved.     

On preimages of a class of generalized monotone operators

In this paper we first provide a geometric interpretation of the Minty-Browder monotonicity which allows us to extend this concept to the so called h-monotonicity, still formulated in an analytic way. A topological concept of monotonicity is also known in the literature: it requires the connectedness of all preimages of the operator involved. This fact is important since combined with the local injectivity, it ensures global injectivity. When a linear structure is present on the source space, one can ask for the preimages to even be convex. In an earlier paper, the authors have shown that Minty-Browder monotone operators defined on convex open sets do have convex preimages, obtaining as a by-product global injectivity theorems. In this paper we study the preimages of h-monotone operators, by showing that they are not divisible by closed connected hypersurfaces, and investigate them from the dimensional point of view. As a consequence we deduce that h-monotone local homeomorphisms are actually global homeomorphisms, as the proved properties of their preimages combined with local injectivity still produce global injectivity.     

From iterated tilted algebras to cluster-tilted algebras

In this paper the relationship between iterated tilted algebras and cluster-tilted algebras and relation extensions is studied. In the Dynkin case, it is shown that the relationship is very strong and combinatorial.     

Functional equations for higher logarithms

Following earlier work by Abel and others, Kummer gave in 1840 functional equations for the polylogarithm function Li _m (z) up to m = 5, but no example for larger m was known until recently. We give the first genuine 2-variable functional equation for the 7-logarithm. We investigate and relate identities for the 3-logarithm given by Goncharov and Wojtkowiak and deduce a certain family of functional equations for the 4-logarithm.     

Moments of products of elliptic integrals

We consider the moments of products of complete elliptic integrals of the first and second kinds. In particular, we derive new results using elementary means, aided by computer experimentation and a theorem of W. Zudilin. Diverse related evaluations, and two conjectures, are also given.     

A framization of the Hecke algebra of type B

We introduce a framization of the Hecke algebra of type B. For this framization, we construct a faithful tensorial representation and two linear bases. We also construct a Markov trace on such an algebra, and from this trace we derive isotopy invariants for framed and classical knots and links in the solid torus.     

Algebraic independence of arithmetic gamma values and Carlitz zeta values

We consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for Fq[θ] and provide complete algebraic independence results for them.     

Differential approximation of min sat, max sat and related problems

We present differential approximation results (both positive and negative) for optimal satisfiability, optimal constraint satisfaction, and some of the most popular restrictive versions of them. As an important corollary, we exhibit an interesting structural difference between the landscapes of approximability classes in standard and differential paradigms.     

6 j symbols for $U_q (\mathfrak{s}\mathfrak{l}_2 )$U_q (\mathfrak{s}\mathfrak{l}_2 ) and non-Euclidean tetrahedra

We relate the semiclassical asymptotics of the 6j symbols for the quantized enveloping algebra at q a root of unity (resp. q real positive) to the geometry of spherical (resp. hyperbolic) tetrahedra.     

A categorification of finite-dimensional irreducible representations of quantum \mathfraksl2{\mathfrak{sl}_2} and their tensor products

The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for . The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra . For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed.We also give a categorical version of the quantised Schur–Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules.