The gradient flow motion of boundary vortices

We consider the gradient flow of a family of energy functionals describing the formation of boundary vortices in thin magnetic films. We obtain motion laws for the singularities in all time scalings by using the method of Γ-convergence of gradient flows.     

On the cuspidal modular forms for the Fuchsian groups of the first kind

In this paper we study the construction and non-vanishing of cuspidal modular forms of weight m?3 for arbitrary Fuchsian groups of the first kind. We give a spanning set for the space of cuspidal modular forms Sm(Γ) of weight m?3 in a uniform way which does not depend on the fact that Γ has cusps or not.     

Configurations of limit cycles in Liénard equations

We show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial Liénard equation. The related vector field X is Morse–Smale. Moreover it has the minimum number of singularities required for realizing the configuration in a Liénard equation. We provide an explicit upper bound on the degree of X, which is lower than the results obtained before, obtained in the context of general polynomial vector fields.     

An explicit spine for the Picard modular group over the Gaussian integers

Let Γ\D be an arithmetic quotient of a symmetric space of non-compact type. A spine D₀ is a Γ-equivariant deformation retraction of D with dimension equal to the virtual cohomological dimension of Γ. We explicitly construct a spine for the case of Γ=SU(2,1;Z[i]). The spine is then used to compute the cohomology of Γ\D with various local coefficients.     

The configuration space of the n-arms machine in the Euclidean space

As a typical kind of mechanical linkage, we consider the n-arms machine in Rd. The machine consists of n 2-legs with equal length such that the initial point of each 2-leg is fixed to a circle, while all tips of the 2-legs are combined to a central joint. We determine the homotopy type of the configuration space of the n-arms machine.     

Hecke operators on Drinfeld cusp forms

In this paper, we study the Drinfeld cusp forms for Γ₁(T) and Γ(T) using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for Γ₁(T) of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that the Hecke operators are not diagonalizable in general for Γ₁(T) of large weights, and not for Γ(T) even of small weights. The Hecke eigenvalues on cusp forms for Γ(T) with small weights are determined and the eigenspaces characterized.     

Embedded resolutions of non necessarily normal affine toric varietiesRésolution plongée d'une variété torique non nécessairement normale

We give a method to construct a partial embedded resolution of a nonnecessarily normal affine toric variety Z^Γ equivariantly embedded in a normal affine toric variety Z_ρ. This partial resolution is an embedded normalization inside a normal toric ambient space and a resolution of singularities of the ambient space, which always exists, provides an embedded resolution. The advantage is that this partial resolution is completely determined by the embedding Z^Γ?Z_ρ. As a by-product, the construction of the normalization is made without an explicit computation of the saturation of the semigroup Γ of the toric variety (see [3]). This result is valid for a base field k algebraically closed of arbitrary characteristic. To cite this article: P.D. González Pérez, B. Teissier, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 379–382.     

Boundary amenability of relatively hyperbolic groups

Let K be a fine hyperbolic graph and Γ be a group acting on K with finite quotient. We prove that Γ is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups are exact. We prove this by showing that the group Γ acts amenably on a compact topological space. We include some applications to the theories of group von Neumann algebras and of measurable orbit equivalence relations.     

A note on stable equivalences of Morita type

We investigate when an exact functor --Γ which induces a stable equivalence is part of a stable equivalence of Morita type. If Λ and Γ are finite dimensional algebras over a field k whose semisimple quotients are separable, we give a necessary and sufficient condition for this to be the case. This generalizes a result of Rickard’s for self-injective algebras. As a corollary, we see that the two functors given by tensoring with the bimodules in a stable equivalence of Morita type are right and left adjoints of one another, provided that these bimodules are indecomposable. This fact has many interesting consequences for stable equivalences of Morita type. In particular, we show that a stable equivalence of Morita type induces another stable equivalence of Morita type between certain self-injective algebras associated to the original algebras. We further show that when there exists a stable equivalence of Morita type between Λ and Γ, it is possible to replace Λ by a Morita equivalent k-algebra Δ such that Γ is a subring of Δ and the induction and restriction functors induce inverse stable equivalences.     

On deformations of crossed products

Let A∗Γ be a crossed product algebra, where A is semisimple, finitely generated over its center and Γ is a finite group. We give a necessary and sufficient condition in terms of the outer action of Γ on A for the existence of a multi-parametric semisimple deformation of the form A((t₁,…,t_n))∗Γ (with the induced outer action). The main tool in the proof is the solution of the so-called twisting problem. We also give an example which shows that the condition is not sufficient if one drops the condition on the finite generation of A over its center.     

On interpolation of Banach algebras and factorization of weakly compact homomorphisms

We show a necessary and sufficient condition on the lattice Γ for the general real method to preserve the Banach-algebra structure. As an application we derive factorization of weakly compact homomorphisms through interpolation properties of weakly compact operators.     

Free and free abelian Euler-Satake characteristics of nonorientable 2-orbifolds

We compute the Γ-sectors and Γ-Euler-Satake characteristic of a closed, effective 2-dimensional orbifold Q where Γ is a free or free abelian group. Using this information, we determine a family of orbifolds such that the complete collection of Γ-Euler-Satake characteristics associated to free and free abelian groups determines the number and type of singular points of Q as well as the Euler characteristic of the underlying space. Additionally, we show that any collection of these groups whose Euler-Satake characteristics determine this information contains both free and free abelian groups of arbitrarily large rank. It follows that the collection of Euler-Satake characteristics associated to free and free abelian groups constitute a finer orbifold invariant than the collection of Euler-Satake characteristics associated to free groups or free abelian groups alone.     

Connections on modules over singularities of finite CM representation type

Let A be a commutative k-algebra, where k is an algebraically closed field of characteristic 0, and let M be an A-module. We consider the following question: Under what conditions is it possible to find a connection on M?We consider the maximal Cohen-Macaulay (MCM) modules over complete CM algebras that are isolated singularities, and usually assume that the singularities have finite CM representation type. It is known that any MCM module over a simple singularity of dimension d≤2 admits an integrable connection. We prove that an MCM module over a simple singularity of dimension d≥3 admits a connection if and only if it is free. Among singularities of finite CM representation type, we find examples of curves with MCM modules that do not admit connections, and threefolds with non-free MCM modules that admit connections.Let A be a singularity not necessarily of finite CM representation type, and consider the condition that A is a Gorenstein curve or a -Gorenstein singularity of dimension d≥2. We show that this condition is sufficient for the canonical module ω_A to admit an integrable connection, and conjecture that it is also necessary. In support of the conjecture, we show that if A is a monomial curve singularity, then the canonical module ω_A admits an integrable connection if and only if A is Gorenstein.     

Iyama’s finiteness theorem via strongly quasi-hereditary algebras

Let Λ be an artin algebra and X a finitely generated Λ-module. Iyama has shown that there exists a module Y such that the endomorphism ring Γ of X⊕Y is quasi-hereditary, with a heredity chain of length n, and that the global dimension of Γ is bounded by this n. In general, one only knows that a quasi-hereditary algebra with a heredity chain of length n must have global dimension at most 2n−2. We want to show that Iyama’s better bound is related to the fact that the ring Γ he constructs is not only quasi-hereditary, but even left strongly quasi-hereditary. By definition, the left strongly quasi-hereditary algebras are the quasi-hereditary algebras with all standard left modules of projective dimension at most 1.     

Geometric configurations of singularities for quadratic differential systems with three distinct real simple finite singularities

In this work we classify, with respect to the geometric equivalence relation, the global configurations of singularities, finite and infinite, of quadratic differential systems possessing exactly three distinct finite simple singularities. This relation is finer than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci (or saddles) of different orders. Such distinctions are, however, important in the production of limit cycles close to the foci (or loops) in perturbations of the systems. The notion of geometric equivalence relation of configurations of singularities allows us to incorporate all these important geometric features which can be expressed in purely algebraic terms. The geometric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in a work published in 2013 when the classification was done for systems with total multiplicity m _f of finite singularities less than or equal to one. That work was continued in an article which is due to appear in 2014 where the geometric classification of configurations of singularities was done for the case m _f = 2. In this article we go one step further and obtain the geometric classification of singularities, finite and infinite, for the subclass mentioned above. We obtain 147 geometrically distinct configurations of singularities for this family. We give here the global bifurcation diagram of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for this class of systems. The bifurcation set of this diagram is algebraic. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of polynomial invariants, a fact which gives us an algorithm for determining the geometric configuration of singularities for any quadratic system in this particular class.     

Completeness for flat modal fixpoint logics

This paper exhibits a general and uniform method to prove axiomatic completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ(x,p₁,…,p_n), where x occurs only positively in γ, we obtain the flat modal fixpoint language L_?(Γ) by adding to the language of polymodal logic a connective ?_γ for each γ∈Γ. The term ?_γ(φ₁,…,φ_n) is meant to be interpreted as the least fixed point of the functional interpretation of the term γ(x,φ₁,…,φ_n). We consider the following problem: given Γ, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L_?(Γ) on Kripke structures. We prove two results that solve this problem.First, let be the logic obtained from the basic polymodal by adding a Kozen-Park style fixpoint axiom and a least fixpoint rule, for each fixpoint connective ?_γ. Provided that each indexing formula γ satisfies a certain syntactic criterion, we prove this axiom system to be complete.Second, addressing the general case, we prove the soundness and completeness of an extension of . This extension is obtained via an effective procedure that, given an indexing formula γ as input, returns a finite set of axioms and derivation rules for ?_γ, of size bounded by the length of γ. Thus the axiom system is finite whenever Γ is finite.     

Asymptotic formulas for the triple Gamma function Γ3 by means of its integral representation

There is an abundant literature on inequalities for the Gamma function Γ and its various related functions as well as their approximations. Only very recently, several authors began to investigate various inequalities for the double Gamma function Γ₂ and its approximation. Here, in this sequel to some of these recent works, we aim at presenting an integral representation of the triple Gamma function Γ₃, which is then used to derive an asymptotic formula for Γ₃. As a by-product of the results presented here, integral representations and asymptotic formulas for the Gamma function Γ and the double Gamma function Γ₂ are also given. The methods and techniques used in this paper can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple Gamma functions Γ_n (n ≧ 4).     

The method of boundary integral equations for a mixed scattering problem

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R² as cross section. To this end, we solve a scattering problem for the Helmholtz equation in R² where the scattering object is a combination of a crack Γ and a bounded obstacle D, and we have Dirichlet-impedance type boundary condition on Γ and Dirichlet boundary condition on ∂D (∂D∈C²). Applying potential theory, the problem can be reformulated as a boundary integral system. We establish the existence and uniqueness of a solution to the system by using the Fredholm theory.     

Representing abstract groups by powers of a graph

We deal with the problem of representing several abstract groups simultaneously by one graph as automorphism groups of its powers. We call subgroups Γ₁,…, Γ_n of a finite group Γ representable iff there is a graph G and an injective mapping φ from ∪_i=1ⁿΓ_i into the symmetric group on V(G) such that for i=1,…, n φ|_Γi is a monomorphism onto Aut Gⁱ. We give a necessary and a sufficient condition for groups being representable, the latter implying, e.g., that finite groups Γ₁≤…≤Γ_n are representable.