We study the topological properties of max-plus convex sets in an Archimedean Riesz space $E$ with respect to the topology and the max-plus structure associated to a given order unit $\boldsymbol u$; the definition of max-plus convex sets is algebraic and we do not assume that $E$ has an a priori given topological structure. To a given unit $\boldsymbol{u}$ one can associate two equivalent norms on $E$ one of which, denoted $\|\cdot\|_{\boldsymbol u}$, is classical, the other$\|\cdot\|_{\boldsymbol{hu}}$ is introduced here following a previous unpublished work of Stéphane Gaubert on the geodesic structure of finite dimensional max-plus; it is shown that the distance ${\sf D}_{\boldsymbol h\boldsymbol u}$ on $E$ associated to $\|\cdot\|_{\boldsymbol h\boldsymbol u}$ is a geodesic distance, called the Hilbert affine distance associated to $\boldsymbol u$, for which max-plus convex sets in $E$ are precisely the geodesically closed sets. Under suitable assumptions, we establish max-plus versions of some fixed points and continuous selection theorems that are well known for linear convex sets and we show that hyperspaces of compact max-plus convex sets are Absolute Retracts. We formulate a max-plus version of the Knaster-Kuratowski-Mazurkiewicz Lemma from which, following A. Granas and J. Dugundji, all of the consequences of the classical KKM Lemma can be derived in a max-plus version. P. de la Harpe showed that the interior of the standard simplex $\Delta_n$ equipped with the classical Hilbert metric-defined by the cross-ration of four appropriate points is isometric to a finite dimensional normed space. We give an explicit proof of that result: the norm space in question is $\mathbb R^n$ with the Hilbert affine norm $\|\cdot\|_{\boldsymbol h\boldsymbol u}$ with respect to $\boldsymbol u = (1, \ldots, 1)$.

We study the exact number of stable positive solutions of weakly nonlinear elliptic equations with small diffusion under rather general conditions on the nonlinearity.

Existence and spatio-temporal patterns of periodic solutions to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer $O(2) \times \Gamma \times \mathbb Z_2$-equivariant degree theory, where $O(2)$ is related to the reversing symmetry, $\Gamma$ reflects the symmetric character of the coupling in the corresponding network and $\mathbb Z_2$ is related to the oddness of the right-hand side. Abstract results are supported by a concrete example with $\Gamma = D_6$ - the dihedral group of order 12.

Thanks to a connection between two completely different topics, the classical eigenvalue problem in a finite dimensional real vector space and the Brouwer degree for maps between oriented differentiable real manifolds, we are able to solve, at least in the finite dimensional context, a conjecture regarding global continuation in nonlinear spectral theory that we formulated in some recent papers. The infinite dimensional case seems nontrivial, and is still unsolved.

We study the convergence of inexact iterates of nonexpansive mappings which take a nonempty closed subset of a complete metric space into the space in the case where the errors are sufficiently small.

We establish a variant for spheres of results obtained in [7], [3] for affine space. The principal result, that, if $m$ is a power of $2$ and $k\geq 1$, then $km$ continuous densities on the unit sphere in $\mathbb R^{m+1}$ may be simultaneously bisected by a set of at most $k$ hyperplanes through the origin, is essentially equivalent to the main theorem of Hubard and Karasev in [7]. But the methods used, involving Euler classes of vector bundles over symmetric powers of real projective spaces and an 'orbifold' fixed point theorem for involutions, are substantially different from those in [7], [3].

We study the elliptic system\begin{equation*}\begin{cases}-\Delta u_1 - \kappa_1u_1 = \mu_1|u_1|^{p-2}u_1 + \lambda\alpha|u_1|^{\alpha-2}|u_2|^\beta u_1, \\-\Delta u_2 - \kappa_2u_2 = \mu_2|u_2|^{p-2}u_2 + \lambda\beta|u_1|^\alpha|u_2|^{\beta-2}u_2, \\u_1,u_2\in D^{1,2}_0(\Omega),\end{cases}\end{equation*}where $\Omega$ is a bounded domain in $\mathbb R^N$, $N\geq 3$, $\kappa_1,\kappa_2\in\mathbb R$, $\mu_1,\mu_2,\lambda> 0$, $\alpha,\beta> 1$, and $\alpha + \beta = p\le 2^*:={2N}/({N-2})$. For $p\in (2,2^*)$ we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for $\lambda$ sufficiently large. If $p=2^*$ and $\kappa_1,\kappa_2> 0$ we establish the existence of a ground state for $\lambda$ sufficiently large if, either $N\ge5$, or $N=4$ and neither $\kappa_1$ nor $\kappa_2$ are Dirichlet eigenvalues of $-\Delta$ in $\Omega$.

In 19th century, Fermat's little theorem ``$a^p\equiv a({\rm mod}\;p)$ for $a\in\mathbb Z$, $p$ prime'' was generalized in two directions: Schönemann proved a corresponding congruence for the coefficients of monic polynomials, whereas Gauss found a congruence result with $p$ replaced by any $n\in\mathbb N$. Here, we shall give an elementary proof of the common generalization of these two results.

In a previous paper, we classified the homotopy classes of proper Fredholm maps from an infinite dimensional Hilbert manifold to its model space in terms of a suitable version of framed cobordism. We explicitly computed these homotopy classes for non-positive index. In this paper, we compute the homotopy classes of proper Fredholm maps of index one from a simply connected Hilbert manifold to its model space. This classification uses a new numerical invariant for proper Fredholm maps of index one.

Consider the class QS of all non-degenerate planar quadratic differential systems and its subclass QSE of all systems possessing an invariant ellipse. In this paper we classify the family QSE according to their geometric properties encoded in the configurations of invariant ellipses and invariant straight lines which these systems could possess. The classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of algebraic and geometric invariants and also in terms of invariant polynomials and it yields a total of 35 distinct such configurations. This classification is also an algorithm which makes it possible to verify for any given real quadratic differential system if it has invariant ellipses or not and to specify its configuration of invariant ellipses and straight lines. This work will prove helpful in studying the integrability of the systems in QSE. It is also a stepping stone for studying the topological classification of this family. Since it is known that the maximum number of limit cycles occurring in systems of QSE is 1, this goal is thus not out of reach at the moment.

In the framework of the theory of Equilibrium Manifold of a Pure Exchange Economy, introduced by Balasko, we give a rigorous definition of a crisis transition and establish some infinitesimal criteria which distinguish unavoidable crises from other critical equilibria. Our approach builds on the relationship between the mathematical notions of branching, envelopes, and the intrinsic derivative.