A filtering problem with a small nonlinear term

In this paper, we consider a filtering problem where the signal X _t satisfies a slightly nonlinear stochastic differential equation and we want to obtain estimates of X _t. To this end, we decompose the nonlinearity with two techniques—a deterministic one and a stochastic one—and this leads us to two sequences of estimates which can be computed by solving finite dimensional equations. We want to compare their performances: we solve this problem in most cases if we restrict ourselves to sufficiently small times t and we give conditions which permit to conclude also for larger times     

P-distances, q-distances and a generalized Ekeland’s variational principle in uniform spaces

In this paper, we attempt to give a unified approach to the existing several versions of Ekeland’s variational principle. In the framework of uniform spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance (particularly, a p-distance), which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized Takahashi’s nonconvex minimization theorem, a generalized Ekeland’s variational principle and a generalized Caristi’s fixed point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland’s variational principle, we deduce a number of particular versions of Ekeland’s principle, which include many known versions of the principle and their improvements.     

A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes

The aim of this paper is to present a kinetic formulation of a model for the coupling of transient free surface and pressurised flows. Firstly, we revisit the system of Saint-Venant equations for free surface flow: we state some properties of Saint-Venant equations, we propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises (in some general case) an energy and preserves the still water steady state. Secondly, we propose a model for pressurised flows in a Saint-Venant-like conservative formulation. We then propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises in any case an energy and preserves the still water steady state. Finally, we propose a dual model that couples these two types of flow.     

Sub-semigroups determined by the zero-divisor graph

In this paper we study sub-semigroups of a finite or an infinite zero-divisor semigroup S determined by properties of the zero-divisor graph Γ(S). We use these sub-semigroups to study the correspondence between zero-divisor semigroups and zero-divisor graphs. In particular, we discover a class of sub-semigroups of reduced semigroups and we study properties of sub-semigroups of finite or infinite semilattices with the least element. As an application, we provide a characterization of the graphs which are zero-divisor graphs of Boolean rings. We also study how local property of Γ(S) affects global property of the semigroup S, and we discover some interesting applications. In particular, we find that no finite or infinite two-star graph has a corresponding nil semigroup.     

Filling polygonal holes with minimal energy surfaces on Powell-Sabin type triangulations

In this paper we present two different methods for filling in a hole in an explicit 3D surface, defined by a smooth function f in a part of a polygonal domain D⊂R². We obtain the final reconstructed surface over the whole domain D. We do the filling in two different ways: discontinuous and continuous. In the discontinuous case, we fill the hole with a function in a Powell-Sabin spline space that minimizes a linear combination of the usual seminorms in an adequate Sobolev space, and approximates (in the least squares sense) the values of f and those of its normal derivatives at an adequate set of points. In the continuous case, we will first replace f outside the hole by a smoothing bivariate spline s_f, and then we fill the hole also with a Powell-Sabin spline minimizing a linear combination of given seminorms. In both cases, we obtain existence and uniqueness of solutions and we present some graphical examples, and, in the continuous case, we also give a local convergence result.     

Sur des notions de n—categorie et n—groupoide non strictes via des ensembles multi-simpliciaux (On the notions of a nonstrict n–category and n–groupoid via multisimplicial sets)

In this paper we first give a simplicial approach to the definition of a nonstrict n–category that we call a n–nerve following the idea that a category could be interpreted as a simplicial set (its nerve). Then we prove that for n=2 our construction is equivalent to the usual nonstrict 2–category (bicategory). Next,we give a simplicial definition of a nonstrict n–groupoïd, and we associate to any topological space a n–groupoïd n (X) which generalises the famous Poincaré groupoïd 1 (X) and embodies the n–truncated homotopy type of . Conversely, we construct for each n–groupoïd a geometric realisation and we show that the functors geometric realisation and Poincaré n–groupoïd induce an equivalence between the category of n–groupoids and the category of n–truncated topological spaces, when we localise both categories by weak equivalence.     

A new predator-prey model with a profitless delay of digestion and impulsive perturbation on the prey

In this paper, we formulate a robust prey-dependent consumption predator-prey model with a delay of digestion and impulsive perturbation on the prey. Using the discrete dynamical system determined by the stroboscopic map, we obtain a ‘predator-eradication’ periodic solution and show that the ‘predator-eradication’ periodic solution is globally attractive when harvesting for the prey is over certain value. Using a new qualitative analysis method for impulsive and delay differential equations, we prove the system is uniformly persistent when harvesting for the prey is under certain value. Further, we show the delay of digestion is a “profitless” time delay. Moreover, we show our theoretical results by numerical simulation. In this paper, the main feature is that we introduce a delay of digestion and impulsive effects into the predator-prey model and exhibit a new mathematical method which is applied to investigate the system which is governed by both impulsive and delay differential equations.     

Un modéle d'lnspections optimales

We consider the following model: we inspect the motion of a Markov process with which an “evolution cost” is associated. We inspect the process at times T ₁…, T _n ,…. If when we inspect, its value is in a given set A, it continues its evolution, otherwise we kill it. At each inspection we associate an "inspection cost" and a "killing cost". The problem consists of finding a sequence of optimal inspections. After the modelization we construct the value function by an iterative procedure as in impulse control theory, by using the theory of analytic functions and theorems of section. Thanks to the criteria of optimality we get a sequence of optimal inspections under very general hypotheses.     

On the Haezendonck-Goovaerts risk measure for extreme risks

In this paper, we are interested in the calculation of the Haezendonck-Goovaerts risk measure, which is defined via a convex Young function and a parameter q∈(0,1) representing the confidence level. We mainly focus on the case in which the risk variable follows a distribution function from a max-domain of attraction. For this case, we restrict the Young function to be a power function and we derive exact asymptotics for the Haezendonck-Goovaerts risk measure as q↑1. As a subsidiary, we also consider the case with an exponentially distributed risk variable and a general Young function, and we obtain an analytical expression for the Haezendonck-Goovaerts risk measure.     

Affine Weyl group symmetries of Frobenius Painlevé equations

In this paper, we introduce a Frobenius Painlevé IV equation and the corresponding Hamilton system, and we give the symmetric form of the Frobenius Painlevé IV equation. Then, we construct the Lax pair of the Frobenius Painlevé IV equation. Furthermore, we recall the Frobenius modified KP hierarchy and the Frobenius KP hierarchy by bilinear equations, then we show how to get Frobenius Painlevé IV equation from the Frobenius modified KP hierarchy. In order to study the different aspects of the Frobenius Painlevé IV equation, we give the similarity reduction and affine Weyl group symmetry of the equation. Similarly, we introduce a Frobenius Painlevé II equation and show the connection between the Frobenius modified KP hierarchy and the Frobenius Painlevé II equation.     

On the first eigenvalue of a fourth order Steklov problem

We prove some results about the first Steklov eigenvalue d ₁ of the biharmonic operator in bounded domains. Firstly, we show that Fichera’s principle of duality (Fichera in Atti Accad Naz Lincei 19:411–418, 1955) may be extended to a wide class of nonsmooth domains. Next, we study the optimization of d ₁ for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problems. Finally, we prove several properties of the ball.     

Coleman automorphisms of finite groups and their minimal normal subgroups

In this paper, we show that all Coleman automorphisms of a finite group with self-central minimal non-trivial characteristic subgroup are inner; therefore the normalizer property holds for these groups. Using our methods we show that the holomorph and wreath product of finite simple groups, among others, have no non-inner Coleman automorphisms. As a further application of our theorems, we provide partial answers to questions raised by M. Hertweck and W. Kimmerle. Furthermore, we characterize the Coleman automorphisms of extensions of a finite nilpotent group by a cyclic p-group. Finally, we note that class-preserving Coleman automorphisms of p-power order of some nilpotent-by-nilpotent groups are inner, extending a result by J. Hai and J. Ge, where p is a prime number.     

Implicative hyper <Emphasis Type="Italic">K</Emphasis>-algebras

In this note we first define the notions of (weak, strong) implicative hyper K-algebras. Then we show by examples that these notions are different. After that we state and prove some theorems which determine the relationship between these notions and (weak) hyper K-ideals. Also we obtain some relations between these notions and (weak) implicative hyper K-ideals. Finally, we study the implicative hyper K-algebras of order 3, in particular we obtain a relationship between the positive implicative hyper K-algebras and (weak, strong) implicative hyper K-algebras under a simple condition.     

Strong solution of Itô type set-valued stochastic differential equation

In this paper, we shall firstly illustrate why we should introduce an It5 type set-valued stochastic differential equation and why we should notice the almost everywhere problem. Secondly we shall give a clear definition of Aumann type Lebesgue integral and prove the measurability of the Lebesgue integral of set-valued stochastic processes with respect to time t. Then we shall present some new properties, especially prove an important inequality of set-valued Lebesgue integrals. Finally we shall prove the existence and the uniqueness of a strong solution to the It5 type set-valued stochastic differential equation.     

The generalized order-k Fibonacci–Pell sequence by matrix methods

In this paper, we consider the usual and generalized order-k Fibonacci and Pell recurrences, then we define a new recurrence, which we call generalized order-k F–P sequence. Also we present a systematic investigation of the generalized order-k F–P sequence. We give the generalized Binet formula, some identities and an explicit formula for sums of the generalized order-k F–P sequence by matrix methods. Further, we give the generating function and combinatorial representations of these numbers. Also we present an algorithm for computing the sums of the generalized order-k Pell numbers, as well as the Pell numbers themselves.     

Higher order Hessian structures on manifolds

In this paper we define nth order Hessian structures on manifolds and study them. In particular, whenn = 3, we make a detailed study and establish a one-to-one correspondence betweenthird-order Hessian structures and acertain class of connections on the second-order tangent bundle of a manifold. Further, we show that a connection on the tangent bundle of a manifold induces a connection on the second-order tangent bundle. Also we define second-order geodesics of special second-order connection which gives a geometric characterization of symmetric third-order Hessian structures.     

Variable metric bundle methods: From conceptual to implementable forms

To minimize a convex function, we combine Moreau-Yosida regularizations, quasi-Newton matrices and bundling mechanisms. First we develop conceptual forms using “reversal” quasi-Newton formulae and we state their global and local convergence. Then, to produce implementable versions, we incorporate a bundle strategy together with a “curve-search”. No convergence results are given for the implementable versions; however some numerical illustrations show their good behaviour even for large-scale problems.     

New extension phenomena for solutions of tangential Cauchy–Riemann equations

In our recent work, we showed that C^∞ CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in ?² are not analytic in general. This result raised again the question on the nature of CR-maps of a real-analytic hypersurfaces.

In this paper, we give a complete picture of what CR-maps actually are. First, we discover an analytic continuation phenomenon for CR-diffeomorphisms which we call the sectorial analyticity property. It appears to be the optimal regularity property for CR-diffeomorphisms in general. We emphasize that such type of extension never appeared previously in the literature. Second, we introduce the class of Fuchsian type hypersurfaces and prove that (infinitesimal generators of) CR-automorphisms of a Fuchsian type hypersurface are still analytic. In particular, this solves a problem formulated earlier by Shafikov and the first author.

Finally, we prove a regularity result for formal CR-automorphisms of Fuchsian type hypersurfaces.     

Nonlocal vertex algebras generated by formal vertex operators

This is the first paper in a series to study vertex algebra-like objects arising from infinite-dimensional quantum groups (quantum affine algebras and Yangians). In this paper we lay the foundations for this study. For any vector space W, we study what we call quasi compatible subsets of Hom (W,W((x))) and we prove that any maximal quasi compatible subspace has a natural nonlocal (namely noncommutative) vertex algebra structure with W as a natural faithful quasi module in a certain sense, and that any quasi compatible subset generates a nonlocal vertex algebra with W as a quasi module. In particular, taking W to be a highest weight module for a quantum affine algebra we obtain a nonlocal vertex algebra with W as a quasi module. We also formulate and study a notion of quantum vertex algebra and we give general constructions of nonlocal vertex algebras, quantum vertex algebras and their modules.     

Integral quadratic forms and Dirichlet series

A Dirichlet series with multiplicative coefficients has an Euler product representation. In this paper we consider the special case where these coefficients are derived from the numbers of representations of an integer by an integral quadratic form. At first we suppose this quadratic form to be positive definite. In general the representation numbers are not multiplicative. Instead we consider the average number of representations over all classes in the genus of the quadratic form. And we consider only representations of integers of the form tk ² with t square-free. If we divide the average representation number for these integers by a suitable factor, we do get a multiplicative function. Using results from Siegel (Ann. Math. 36:527–606, 1935), we derive a uniform expression for the Euler product expansion of the corresponding Dirichlet series. As a special case, we consider the standard quadratic form in n variables corresponding to the identity matrix. Here we use results from Shimura (Am. J. Math. 124:1059–1081, 2002). For 2≤n≤8, the genus of this particular quadratic form contains only one class, and this leads to a rather simple expression for the Dirichlet series, where the coefficients are just the number of representations of a square as the sum of n squares. Finally we consider the indefinite case, where we can get results similar to the definite case.