Character formulæ and GKRS multiplets in equivariant K-theory

Let $G$ be a compact Lie group, $H$ a closed subgroup of maximal rank and $X$ a topological $G$ -space. We obtain a variety of results concerning the structure of the $H$ -equivariant K-ring $K_H^*(X)$ viewed as a module over the $G$ -equivariant K-ring $K_G^*(X)$ . One result is that the module has a nonsingular bilinear pairing; another is that the module contains multiplets which are analogous to the Gross–Kostant–Ramond–Sternberg multiplets of representation theory.     

On the capacity of boolean graph formulæ

In a recent paper we have solved several well-known combinatorial problems treating them as special cases of our generalization of Shannon's notion of graph capacity. We present a new simple formalism to deal with all such problems in a unified manner, considering graphs or families of graphs as special formulæ the variables of which are pairs of vertices of their common vertex sets. In all of these problems, the maximum size of a set ofn-length sequences from a fixed alphabet is to be determined under various restrictions on the element pairs present in the same coordinate of any two sequences from the set. For sufficiently homogeneous formulæ capacity becomes computable.New applications include generalizations of our result on the maximum number of pairwise qualitatively independentk-partitions of ann-set from independence to various forms of qualitative dependence.Work partially supported by the Italian Ministry of the University and Scientific Research, Project: Algoritmi, Modelli di Calcolo e Strutture Informative.     

Multi-resurgence and connection-to-Stokes formulæ for some linear meromorphic differential systems

In this article, we consider a linear meromorphic differential system with several levels $r_1<...<r_p$. For any k, we prove that the Borel transforms of its $r_k$-reduced formal solutions are resurgent and we give the general form of all their singularities. Next, under some convenient hypotheses on the geometric configuration of singular points, we display exact formulæ to express some Stokes multipliers of level $r_k$ of initial system in terms of connection constants in the Borel plane, generalizing thus formulæ already obtained by M. Loday-Richaud and the author for systems with a single level. As an illustration, we develop one numerical example.     

Some Gaussian binomial sum formulæ with applications

We introduce and compute some Gaussian q-binomial sums formulæ. In order to prove these sums, our approach is to use q-analysis, in particular a formula of Rothe, and computer algebra. We present some applications of our results.     

Inequalities for differentiable mappings with application to special means and quadrature formulæ

Improvements are obtained to some recent error estimates of Dragomir and Agarwal, based on convexity, for the trapezoidal formula. Corresponding estimates are established for the midpoint formula. A parallel development is made based on concavity.     

Poincaré series and monodromy of the simple and unimodal boundary singularities

A boundary singularity is a singularity of a function on a manifold with boundary. The simple and unimodal boundary singularities were classified by V.I. Arnold and V.I. Matov. The McKay correspondence can be generalized to the simple boundary singularities. We consider the monodromy of the simple, parabolic, and exceptional unimodal boundary singularities. We show that the characteristic polynomial of the monodromy is related to the Poincaré series of the coordinate algebra of the ambient singularity.     

Poincaré-Steklov integral equations and the Riemann monodromy problem

We consider the Poincaré-Steklov singular integral equation obtained by reducing a boundary value problem for the Laplace operator with a spectral parameter in the boundary condition to the boundary. It is shown that this equation can be restated equivalently in terms of the classical Riemann monodromy problem. Several equations of this type are solved in elliptic functions. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow Institute of Physical Engineering (Physical-Technical). Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 9–22, April–June, 2000. Translated by V. E. Nazaikinskii     

Knauf’s degree and monodromy in planar potential scattering

We consider Hamiltonian systems on (T*?², dq ∧ dp) defined by a Hamiltonian function H of the “classical” form H = p ²/2 + V(q). A reasonable decay assumption V(q) → 0, ‖q‖ → ∞, allows one to compare a given distribution of initial conditions at t = ?∞ with their final distribution at t = +∞. To describe this Knauf introduced a topological invariant deg(E), which, for a nontrapping energy E > 0, is given by the degree of the scattering map. For rotationally symmetric potentials V(q) = W(‖q‖), scattering monodromy has been introduced independently as another topological invariant. In the present paper we demonstrate that, in the rotationally symmetric case, Knauf’s degree deg(E) and scattering monodromy are related to one another. Specifically, we show that scattering monodromy is given by the jump of the degree deg(E), which appears when the nontrapping energy E goes from low to high values.     

Min–max formulæ for the speeds of pulsating travelling fronts in periodic excitable media

This paper is concerned with some nonlinear propagation phenomena for reaction–advection–diffusion equations in a periodic framework. It deals with travelling wave solutions of the equation $u_t =\nabla\cdot(A(z)\nabla u)\;+q(z)\cdot\nabla u+\,f(z,u),\qquad t\in\mathbb{R},\quad z\in\Omega,$ propagating with a speed c. In the case of a “combustion” nonlinearity, the speed c exists and it is unique, while the front u is unique up to a translation in t. We give a min–max and a max–min formula for this speed c. On the other hand, in the case of a “ZFK” or a “KPP” nonlinearity, there exists a minimal speed of propagation c*. In this situation, we give a min–max formula for c*. Finally, we apply this min–max formula to prove a variational formula involving eigenvalue problems for the minimal speed c* in the “KPP” case.     

Pseudoisotopies and the Bökstedt trace

In this paper we study the group of isotopy classes of pseudoisotopies of a compact manifold of high dimension. We set up Igusa’s exact sequence from the point of view of A-theory and use the Bökstedt trace map to detect elements coming from the \({Wh^+_1}\) -term. The main result relates the non-vanishing of \({\pi_0\mathcal{P}(M)}\) to a curious question in group theory.     

Measure-Preserving Transformations, Copulæ and Compatibility

We study the relationship between copulas and measure-preserving transformations on the Borel sets of the unit interval. This also allows to investigate the connection with a restricted compatibility problem for copulas. To this end, in order to construct a 3-copula from two given 2-copulas A and B, we modify the *-operation introduced in Darsow et al., show that A * B is always compatible with A and B, and study the set of all copulas compatible with A and B.