Quantum monodromy and semi-classical trace formulæ

We consider an h-pseudodifferential operator whose symbol has a closed Hamiltonian trajectory. There exists a Fourier integral operator which quantizes in a natural way the Poincaré map. With the help of this monodromy operator, we give a trace formula which leads to a new proof of the trace formula of Duistermaat–Guillemin and Gutzwiller.     

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.     

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.     

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.     

Δ-graphs of polytopes in Bruns and Gubeladze K-theory

W. Bruns and J. Gubeladze introduced a new version of algebraic K-theory where K-groups are additionally parameterized by polytopes of some type. In this paper we propose a concept of stable E-equivalence which can be used to calculate K-groups for high-dimensional polytopes. Polytopes which are stable E-equivalent have similar inner structures and isomorphic K-groups. In addition, for each polytope we define a Δ-graph which is an oriented graph being invariant under a stable E-equivalence.     

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.     

A Periodicity Theorem in K-theory for Finite Covariant Systems

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The differential analytic index in Simons–Sullivan differential K-theory

We define the Simons–Sullivan differential analytic index by translating the Freed–Lott differential analytic index via explicit ring isomorphisms between Freed–Lott differential K-theory and Simons–Sullivan differential K-theory. We prove the differential Grothendieck–Riemann–Roch theorem in Simons–Sullivan differential K-theory using a theorem of Bismut.