Rest point theorems for autonomous control systems

The classical Poincaré-Bendixson theory in the plane is generalized to multivalued vector fields. For control systems $\text{x}\text{?}\text{= f(x, u)}$ where the state x is in Rⁿ and the control u is valued in a compact subset of R^m we study the existence (and nonexistence) of rest states. Some special emphasis is focused on the case n = 2.     

Cantor aperiodic systems and Bratteli diagrams

We prove that every Cantor aperiodic system is homeomorphic to the Vershik map acting on the space of infinite paths of an ordered Bratteli diagram and give several corollaries of this result. To cite this article: K. Medynets, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the Schur–Szegö composition of polynomials

The Schur–Szegö composition of two polynomials of degree ?n introduces an interesting semigroup structure on polynomial spaces and is one of the basic tools in the analytic theory of polynomials. In the present Note we show how it interacts with the stratification of polynomials according to the multiplicities of their zeros and we present the induced semigroup structure on the set of all ordered partitions of n. To cite this article: V. Kostov, B. Shapiro, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Quadratic mappings of spheres

In this paper we obtain a basis-free method for determining the general form of quadratic maps over R between spheres. We show that all quadratic maps (over certain R-lattices) between spheres are Hopf maps, and that the classical Hopf fibrations, S^2m?1→S^m, for m=2, 4, 8, are the unique nontrivial maps over Z, up to action by the orthogonal group.     

Note sur les corps différentiellement clos valués

In the paper by Guzy and Point, Differential topological fields, the model-completion ${(\mathit{OVF})}_D^1$ of the theory of ordered valued differential fields ${\mathit{OVF}}_D$ is established. Models of this theory are closed ordered differential fields (the theory CODF was studied by Singer) which have a non-trivial convex (for the order) subring as valuation ring. Here we prove the valued analogue of a result of Singer: if K is a model of ${(\mathit{OVF})}_D^1$ then $K(\mathrm{i})$ (${\mathrm{i}}^2=?1$) is a model of the theory of differentially closed valued fields which is the model-completion of the theory of non-trivially valued differential fields of characteristic zero. To cite this article: N. Guzy, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On holomorphic isometric embeddings of the unit n-ball into products of two unit m-balls

We study holomorphic isometric embeddings of the complex unit n-ball into products of two complex unit m-balls with respect to their Bergman metrics up to normalization constants (the isometric constant). There are two trivial holomorphic isometric embeddings for m ?? n, given by F ₁(z)?=?(0, I _n;m (z)) with the isometric constant equal to (m?+?1)/(n?+?1) and F ₂(z)?=?(I _n;m (z), I _n;m (z)) with the isometric constant equal to 2(m?+?1)/(n?+?1). Here ${I_{n;m}:\mathbb{C}^n \longrightarrow \mathbb{C}^m}$ is the canonical embedding. We prove that when m < 2n, these are the only holomorphic isometric embeddings up to unitary transformations.     

Heterogeneous wires made of martensitic materials

In this paper we present a direct derivation of a theory of heterogeneous wires starting from three-dimensional nonlinear hyperelasticity augmented by an interfacial energy term. The derivation involves no a priori choice of asymptotic expansion or ansatz. It yields a wire theory with two Cosserat vector fields. The theory is applied to multiwell energy functions appropriate for martensitic materials. A formal derivation of higher theories of homogeneous wires is given, which yields three additional Cosserat vector fields and an explicit form for the bending and torsion energy. To cite this article: H. Le Dret, N. Meunier, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

The Tutte–Potts connection in the presence of an external magnetic field

The classical relationship between the Tutte polynomial of graph theory and the Potts model of statistical mechanics has resulted in valuable interactions between the disciplines. Unfortunately, it does not include the external magnetic fields that appear in most Potts model applications. Here we define the V-polynomial, which lifts the classical relationship between the Tutte polynomial and the zero field Potts model to encompass external magnetic fields. The V-polynomial generalizes Noble and Welshʼs W-polynomial, which extends the Tutte polynomial by incorporating vertex weights and adapting contraction to accommodate them. We prove that the variable field Potts model partition function (with its many specializations) is an evaluation of the V-polynomial, and hence a polynomial with deletion–contraction reduction and Fortuin–Kasteleyn type representation. This unifies an important segment of Potts model theory and brings previously successful combinatorial machinery, including complexity results, to bear on a wider range of statistical mechanics models.     

Sur l'équivalence des théories de Galois différentielles générales

We show that general differential Galois theory of Malgrange (2001) and ours (1996) are equivalent. To cite this article: H. Umemura, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Une méthode par éléments finis mixte préservant la positivité pour le problème de contact en élasticité

We consider a mixed finite element method for the contact problem in elasticity that furnishes approximated fields (displacements and constraints) satisfying the sign conditions of the continuous problem. We prove that the method verifies similar a priori error estimates as the standard method. To cite this article: P. Hild, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A generalization of Turán's theorem to directed graphs

We consider an extremal problem for directed graphs which is closely related to Turán's theorem giving the maximum number of edges in a graph on n vertices which does not contain a complete subgraph on m vertices. For an integer n?2, let T_n denote the transitive tournament with vertex set X_n={1,2,3,…,n} and edge set {(i,j):1?i<j?n}. A subgraph H of T_n is said to be m-locally unipathic when the restriction of H to each m element subset of X_n consisting of m consecutive integers is unipathic. We show that the maximum number of edges in a m-locally unipathic subgraph of T_n is ($\text{q}\text{2}\text{)(m?1)}{}^2\text{+q(m?1)r+?}\text{1}\text{4}\text{r}{}^2\text{?}$ where n= q(m?1+r and $\text{?}\text{1}\text{2}\text{(m?1)??r<?}\text{3}\text{2}\text{(m?1)?}$. As is the case with Turán's theorem, the extremal graphs for our problem are complete multipartite graphs. Unlike Turán's theorem, the part sizes will not be uniform. The proof of our principal theorem rests on a combinatorial theory originally developed to investigate the rank of partially ordered sets.     

The model theory of m‐ordered differential fields

In his Ph.D. thesis [7], L. van den Dries studied the model theory of fields (more precisely domains) with finitely many orderings and valuations where all open sets according to the topology defined by an order or a valuation is globally dense according with all other orderings and valuations. Van den Dries proved that the theory of these fields is companionable and that the theory of the companion is decidable (see also [8]). In this paper we study the case where the fields are expanded with finitely many orderings and an independent derivation. We show that the theory of these fields still admits a model companion in the language L = {+, –, ·, D, <₁, …, <_m, 1, 0}. We denote this model companion by CODF_m and give a geometric axiomatization of this theory which uses basic notions of algebraic geometry and some generalized open subsets which appear naturally in this context. This axiomatization allows to recover (just by putting m = 1) the one given in [4] for the theory CODF of closed ordered differential fields. Most of the technics we use here are already present in [2] and [4]. Finally, we prove that it is possible to describe the completions of CODF_m and to obtain quantifier elimination in a slightly enriched (infinite) language. This generalizes van den Dries' results in the “derivation free” case. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

New formulations of linearized elasticity problems,based on extensions of Donati's theorem

The classical Donati theorem is used for characterizing smooth matrix fields as linearized strain tensor fields. In this Note, we give several generalizations of this theorem, notably to matrix fields whose components are only in $H^{\mathrm{?1}}$. We then show that our extensions of Donati's theorem allow to reformulate in a novel fashion linearized three-dimensional elasticity problems as quadratic minimization problems with the strains as the primary unknowns. To cite this article: C. Amrouche et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the flat remainder in normal forms of families of analytic planar saddles

We give an explicit expression for the (finitely) flat remainder after analytic normal form reduction of a family of planar saddles of diffeomorphisms or vector fields. We distinguish between a rational or irrational ratio of the moduli of the eigenvalues at the saddle for a certain value of the parameter. To cite this article: P. Bonckaert, F. Verstringe, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Scheduling precedence graphs of bounded height

The existence of a schedule for a partially ordered set of unit length tasks on m identical processors is known to be NP-complete (J. D. Ullman, NP-complete scheduling problems, J. Comput. System Sci., 10 (1975), 384–393). The problem remains NP-complete even if we restrict the precedence graph to be of height bounded by a constant. (J. K. Lenkstra and A. H. G. Rinnooy Kan, Complexity of scheduling under precedence constraints, Operations Res., 26 (1978), 22–35; D. Dolev and M. K. Warmuth, “Scheduling Flat Graphs,” IBM Research Report RJ 3398, 1982). In these NP-completeness proofs the upper bound on the number of available processors varies with the problem instance. We present a polynomial algorithm for the case where the upper bound on the number of available processors and the height of the precedence graph are both constants.     

The ordered field property and a finite algorithm for the Nash bargaining solution

This note proves that the two person Nash bargaining theory with polyhedral bargaining regions needs only an ordered field (which always includes the rational number field) as its scalar field. The existence of the Nash bargaining solution is the main part of this result and the axiomatic characterization can be proved in the standard way with slight modifications. We prove the existence by giving a finite algorithm to calculate the Nash solution for a polyhedral bargaining problem, whose speed is of orderBm(m-1) (m is the number of extreme points andB is determined by the extreme points).     

On the fundamental theorem of surface theory under weak regularity assumptions

We consider a symmetric, positive definite matrix field of order two and a symmetric matrix field of order two that together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply connected open subset of $\text{R}{}^2$. If these fields are of class C² and C¹ respectively, the fundamental theorem of surface theory asserts that there exists a surface immersed in the three-dimensional Euclidean space with the given matrix fields as its first and second fundamental forms. The purpose of this Note is to prove that this theorem still holds true under the weaker regularity assumptions that these fields are of class W^1,∞_loc and L^∞_loc respectively, the Gauss and Codazzi–Mainardi equations being then understood in a distributional sense. To cite this article: S. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Classification projective des espaces d’opérateurs différentiels agissant sur les densités

One classifies the representation D_λμ of sl(m + 1, R) obtained by letting its projective embedding in the Lie algebra of vector fields of R^m, m > 1, act by Lie derivatives on the space of differential operators between densities of weight λ and μ. For each μ - λ, there is only finitely many isomorphism classes, most frequently one, in which case D_λμ is isomorphic to its graded space relative to the order of differentiation.     

Partition,Construction, and Enumeration of M–P Invertible Matrices over Finite Fields

A necessary and sufficient condition for an m×n matrix A over F_q having a Moor–Penrose generalized inverse (M–P inverse for short) was given in (C. K. Wu and E. Dawson, 1998, Finite Fields Appl. 4, 307–315). In the present paper further necessary and sufficient conditions are obtained, which make clear the set of m×n matrices over F_q having an M–P inverse and reduce the problem of constructing M–P invertible matrices to that of constructing subspaces of certain type with respect to some classical groups. Moreover, an explicit formula for the M–P inverse of a matrix which is M–P invertible is also given. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing results in geometry of classical groups over finite fields (Z. X. Wan, 1993, “Geometry of Classical Groups over Finite Fields”, Studentlitteratur, Chatwell Bratt).