Sur les arborescences dans un graphe oriente

The main result of this paper is the following theorem: Let G = (X,E) be a digraph without loops or multiple edges, |X| ?3, and h be an integer ?1, if G contains a spanning arborescence and if d⁺_G(x)+d^?_G(x)+d^?_G(y)+d^?_G(y)? 2|X |?2h?1 for all x, y?X, x ≠ y, non adjacent in G, then G contains a spanning arborescence with ?h terminal vertices. A strengthening of Gallai-Milgram's theorem is also proved.     

A generalization of the cosine-sine functional equation on groups

The functional equation f(xy)=f(x)g(y)+g(x)f(y)+h(x)h(y) is solved where f, g, h are complex functions defined on a group.     

Finite metrics in switching classes

Let g:D×D→R be a symmetric function on a finite set D satisfying g(x,x)=0 for all x∈D. A switch g^σ of g w.r.t. a local valuation σ:D→R is defined by g^σ(x,y)=σ(x)+g(x,y)+σ(y) for x≠y and g^σ(x,x)=0 for all x. We show that every symmetric function g has a unique minimal semimetric switch, and, moreover, there is a switch of g that is isometric to a finite Manhattan metric. Also, for each metric on D, we associate an extension metric on the set of all nonempty subsets of D, and we show that this extended metric inherits the switching classes on D.     

Stability of a simple Levi–Civitá functional equation on non-unital commutative semigroups

In this paper, we study the Hyers–Ulam stability of a simple Levi–Civitá functional equation f(x+y)=f(x)h(y)+f(y) and its pexiderization f(x+y)= g(x) h(y)+k(y) on non-unital commutative semigroups by investigating the functional inequalities |f(x+y)?f(x)h(y)?f(y)|≤?? and |f(x+y)?g(x)h(y)?k(y)|≤??, respectively. We also study the bounded solutions of the simple Levi–Civitá functional inequality.     

Dirac and Plateau billiards in domains with corners

Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C ²-smooth Riemannian metrics g on a smooth manifold X, such that scal _g (x) ≥ κ(x), is closed under C ⁰-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry, rather than topology of manifolds with their scalar curvatures bounded from below.     

Generalized functional equation and its applications in information theory

The functional equationf(x,y)+g(x)h(y)F(u/1?x,ν/1?y)=f(u,ν)+g(u)h(ν)F(x/1?u,y/1?ν) ... (1) forx, y, u, ν ∈ [0, 1) andx+u,y+ν ∈ [0,1) whereg andh satisfy the functional equationφ (x+y?xy)=φ(x)φ(y)... (2) has been solved for some non-constant solution of (2) in [0, 1] withφ (0)=1,φ(1)=0 and the solution is used in characterising some measures of information.     

Applications of utility functions defined on quasi-metric spaces

A quasi-metric space (X,d) is called sup-separable if (X,d^s) is a separable metric space, where d^s(x,y)=max{d(x,y),d(y,x)} for all x,y∈X. We characterize those preferences, defined on a sup-separable quasi-metric space, for which there is a semi-Lipschitz utility function. We deduce from our results that several interesting examples of quasi-metric spaces which appear in different fields of theoretical computer science admit semi-Lipschitz utility functions. We also apply our methods to the study of certain kinds of dynamical systems defined on quasi-metric spaces.     

Subpancyclicity of line graphs and degree sums along paths

A graph is called subpancyclic if it contains a cycle of length ? for each ? between 3 and the circumference of the graph. We show that if G is a connected graph on n?146 vertices such that d(u)+d(v)+d(x)+d(y)>(n+10/2) for all four vertices u,v,x,y of any path P=uvxy in G, then the line graph L(G) is subpancyclic, unless G is isomorphic to an exceptional graph. Moreover, we show that this result is best possible, even under the assumption that L(G) is hamiltonian. This improves earlier sufficient conditions by a multiplicative factor rather than an additive constant.     

Eigenvalue estimate for the weighted p-Laplacian

Let M be an n-dimensional closed manifold with metric g, dμ = e ^h(x) dV(x) be the weighted measure and ? _{μ, p} be the weighted p-Laplacian. In this article, we get the lower bound estimate of the first nonzero eigenvalue for the weighted p-Laplacian when the m-dimensional Bakry-émery curvature has a positive lower bound.     

A functional equation having monomials as solutions

For each n=1,2,3, we obtain the general solution and the stability of the functional equation

f(2x+y)+f(2x-y)=2^n-2[f(x+y)+f(x-y)+6f(x)].     

Variations on the Drygas equation and its stability

We study the stability of the Drygas functional equation:

g(xy)+g(xy⁻¹)=2g(x)+g(y)+g(y⁻¹)     

A universal Cauchy functional equation over the positive reals

The functional equation af(xy)+bf(x)f(y)+cf(x+y)+d(f(x)+f(y))=0 whose shape contains all the four well-known forms of Cauchy's functional equation is solved for solutions which are functions having the positive reals as their domain. This complements an earlier work of Dhombres in 1988 where the same functional equation was solved for solutions whose domains contain zero, which leaves out the logarithmic function. Here not only the logarithmic function is recovered but the analysis is entirely different and is based on solving appropriate difference equations.     

Stability for quadratic functional equation in the spaces of generalized functions

In this paper, we consider the general solution of quadratic functional equation

f(ax+y)+f(ax−y)=f(x+y)+f(x−y)+2(a²−1)f(x)     

On the Obstruction to Linearizability of Second-Order Ordinary Differential Equations

In this paper, we investigate the action of the pseudogroup of all point transformations on the bundle of equations y″=u ⁰(x,y)+u ¹(x,y)y′+u ²(x,y)(y′)²+u ³(x,y)(y′)³. We calculate the 1st nontrivial differential invariant of this action. It is a horizontal differential 2-form with values in some algebra, it is defined on the bundle of 2-jets of sections of the bundle under consideration. We prove that this form is a unique obstruction to linearizability of these equations by point transformations.     

Note on the holonomy groups of pseudo-Riemannian manifolds

For an arbitrary subalgebra h ? so(r, s) a polynomial pseudo-Riemannian metric of signature (r + 2, s + 2) is constructed, the holonomy algebra of this metric contains h as a subalgebra. This result shows the essential distinction between the holonomy algebras of pseudo-Riemannian manifolds of index greater than or equal to 2 and the holonomy algebras of Riemannian and Lorentzian manifolds.     

Homogeneous tri-additive forms and derivations

Gleason [A.M. Gleason, The definition of a quadratic form, Amer. Math. Monthly 73 (1966) 1049-1066] determined all functionals Q on K-vector spaces satisfying the parallelogram law Q(x+y)+Q(x-y)=2Q(x)+2Q(y) and the homogeneity Q(λx)=λ²Q(x). Associated with Q is a unique symmetric bi-additive form S such that Q(x)=S(x,x) and 4S(x,y)=Q(x+y)-Q(x-y). Homogeneity of Q corresponds to that of S: S(λx,λy)=λ²S(x,y). The associated S is not necessarily bi-linear.Let V be a vector space over a field K, char(K)≠2,3. A tri-additive form T on V is a map of V³ into K that is additive in each of its three variables. T is homogeneous of degree 3 if T(λx,λy,λz)=λ³T(x,y,z) for all .We determine the structure of tri-additive forms that are homogeneous of degree 3. One of the keys to this investigation is to find the general solution of the functional equation

F(t)+t³G(1/t)=0,     

The stability of a cubic type functional equation with the fixed point alternative

In this note we investigate the generalized Hyers-Ulam-Rassias stability for the new cubic type functional equation f(x+y+2z)+f(x+y−2z)+f(2x)+f(2y)=2[f(x+y)+2f(x+z)+2f(x−z)+2f(y+z)+2f(y−z)] by using the fixed point alternative. The first systematic study of fixed point theorems in nonlinear analysis is due to G. Isac and Th.M. Rassias [Internat. J. Math. Math. Sci. 19 (1996) 219-228].     

Metric generalized inverse for linear manifolds and extremal solutions of linear inclusion in Banach spaces

Let X,Y be Banach spaces and M a linear manifold in X×Y={{x,y}∣x∈X,y∈Y}. The central problem which motivates many of the concepts and results of this paper is the problem of characterization and construction of all extremal solutions of a linear inclusion y∈M(x). First of all, concept of metric operator parts and metric generalized inverses for linear manifolds are introduced and investigated, and then, characterizations of the set of all extremal or least extremal solutions in terms of metric operator parts and metric generalized inverses of linear manifolds are given by the methods of geometry of Banach spaces. The principal tool in this paper is the generalized orthogonal decomposition theorem in Banach spaces.     

A functional equation originating from quadratic forms

In this paper, we obtain the general solution and the stability of the 2-variable quadratic functional equation

f(x+y,z+w)+f(x−y,z−w)=2f(x,z)+2f(y,w).