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.     

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.     

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.     

Solvable covers of the boolean variety of unital ℓ-groups

The varieties of solvable lattice-ordered groups covering the abelian variety were shown independently by Gurchenkov, Reilly, and Darnel to be the Scrimger varieties of ?-groups and the three Medvedev representable covers. In this article, the authors give a parallel characterization of varieties of solvable unital ?-groups which cover the minimal nontrivial variety of boolean unital ?-groups.     

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 average crosscap number II: Bounds for a graph

The bounds are obtained for the average crosscap number. Let G be a graph which is not a tree. It is shown that the average crosscap number of G is not less thanβ(G)-1/2β(G)-1β(G) and not larger thanβ(G). Furthermore, we also describe the structure of the graphs which attain the bounds of the average crosscap number.     

On the distribution of

Suppose that α is an irrational number and β is a real number. It is proved that there are infinitely many prime numbers p such that ‖ αp-β‖ <p ^-9/28.     

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.     

Solving the Ku–Wales conjecture on the eigenvalues of the derangement graph

We give a new recurrence formula for the eigenvalues of the derangement graph. Consequently, we provide a simpler proof of the Alternating Sign Property of the derangement graph. Moreover, we prove that the absolute value of the eigenvalue decreases whenever the corresponding partition with a fixed first part decreases in the dominance order. In particular, this settles affirmatively a conjecture of Ku and Wales [C.Y. Ku, D.B. Wales, Eigenvalues of the derangement graph, J. Combin. Theory Ser. A 117 (2010) 289–312] regarding the lower and upper bounds for the absolute values of these eigenvalues.     

Kazhdan–Lusztig polynomials of boolean elements

We give closed combinatorial product formulas for Kazhdan–Lusztig polynomials and their parabolic analogue of type q in the case of boolean elements, introduced in (Marietti in J. Algebra 295:1–26, 2006), in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of the group. In the case of classical Weyl groups, this combinatorial interpretation can be restated in terms of statistics of (signed) permutations. As an application of the formulas, we compute the intersection homology Poincaré polynomials of the Schubert varieties of boolean elements.     

On the Average of Exponents

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