Let (An)n∈N be a stationary sequence of topical (i.e., isotone and additively homogeneous) operators. Let x(n,x0) be defined by x(0, x0) = x0 and x(n+1, x0) = Anx(n,x0). It can model a wide range of systems including train or queuing networks, job-shop, timed digital circuits or parallel processing systems. When (An)n∈N has the memory loss property, (x(n,x0))n∈N satisfies a strong law of large ...

Lower semicontinuity properties of multiple integrals u ∈W (Ω;R) 7→ ∫ Ω f(x, u(x), · · · ,∇u(x)) dx are studied when f may grow linearly with respect to the highest-order derivative, ∇ku, and admissible W k,1(Ω;Rd) sequences converge strongly in W k−1,1(Ω;Rd). It is shown that under certain continuity assumptions on f, convexity, 1-quasiconvexity or k-polyconvexity of ξ 7→ f(x0, u(x0), · · · ,∇...

For simplicity, we use the following convention: n denotes a natural number, i denotes an integer, p, x, x0, y denote real numbers, q denotes a rational number, and f denotes a partial function from R to R. Let q be an integer. The functor qZ yields a function from R into R and is defined as follows: (Def. 1) For every real number x holds (qZ)(x) = x q Z. Next we state a number of propositions:...

A k-nacci sequence in a finite group is a sequence of group elements x0, x1, x2, . . . , xn, . . . for which, given an initial seed set x0, x1, x2, . . . , xj−1 , each element is defined by xn x0x1 . . . xn−1, for j ≤ n < k, and xn xn−kxn−k 1 . . . xn−1, for n ≥ k. We also require that the initial elements of the sequence, x0, x1, x2, . . . , xj−1, generate the group, thus forcing the k-nacci s...

ion An abstraction (V , ψ,F) is given by: • V : a finite set of observables, • ψ: a mapping from V → R into V ] → R, • F: a C∞ mapping from V ] → R into V ] → R; such that: • ψ is linear with positive coefficients, and for any sequence (xn) ∈ (V → R+)N such that (||xn||) diverges towards +∞, then (||ψ(xn)||) diverges as well (for arbitrary norms || · || and || · ||), • the following diagram com...

Let g0(N) be the genus of the modular curve X0(N). We record several properties of the sequence {g0(N)}. Even though the average size of g0(N) is (1.25/π)N , a random positive integer has probability zero of being a value of g0(N). Also, if N is a random positive integer then g0(N) is odd with probability one.

Proposition 1.3. Let X be a subset of R, let x0 be a limit point of X, let f : X → R be a function, and let L be a real number. Then the following two statements are equivalent. • f is differentiable at x0 on X with derivative L. • For every ε > 0, there exists a δ = δ(ε) > 0 such that, if x ∈ X satisfies |x− x0| < δ, then |f(x)− [f(x0) + L(x− x0)]| ≤ ε |x− x0| . Corollary 1.4 (Mean Value Theor...

A deformation lemma for functionals which are the sum of a locally Lipschitz continuous function and of a concave, proper and upper semicontinuous function is established. Some critical point theorems are then deduced and an application to a class of elliptic variational-hemivariational inequalities is presented. Introduction It is by now well known that the Mountain Pass Theorem of Ambrosetti ...

For some given positive γ, a function f is called outer γ-convex if it satisfies the Jensen inequality f(zi) ≤ (1 − λi)f(x0) + λif(x1) for some z0 : = x0, z1, ..., zk : = x1 ∈ [x0, x1] satisfying ‖zi − zi+1‖ ≤ γ, where λi : = ‖x0 − zi‖/‖x0 − x1‖, i = 1, 2, ..., k − 1. Though the Jensen inequality is only required to hold true at some points (although the location of these points is uncertain) o...