We present a result which is obtained by combining a result of Carlson with the Finitary Dual Ramsey Theorem of Graham-Rothschild. We start by introducing some notation. We conform to the usual practice of identifying the least infinite ordinal ω with the set of non-negative integers. Given α, β ≤ ω, a partition of α into β blocks is an onto function X : α→ β such that min ( X−1({n}) ) < min ( ...

For many control systems in real life, impulses and delays are intrinsic phenomena that do not modify their controllability. So we conjecture that under certain conditions the abrupt changes and delays as perturbations of a system do not destroy its controllability. There are many practical examples of impulsive control systems with delays, such as a chemical reactor system, a financial system ...

Ω f (x, δw(x), Dw(x)) dx, on the space W loc (Ω;R N ), N > 1, where Ω ⊂ R is an open bounded domain and f : Ω × R × R nN×. . .×RN(n+m−1 m ) → R a Carathéodory function. δw ≡ (w,Dw, . . . , Dw) denotes the vector containing the lower order derivatives. For k = 1, . . . ,m we use the notation Du ≡ {Dui} i=1,...,N for the derivative of order k. Note that Du is an element of the space ⊙k(Rn;RN ) of...

Consider the N -coupled nonlinear elliptic system (P) >< >: −∆Uj + Uj = μU j + βUj X k 6=j U k in Ω, Uj > 0 in Ω, Uj = 0 on ∂Ω, j = 1, . . . , N. where Ω is a smooth and bounded (or unbounded if Ω is radially symmetric) domain in Rn, n ≤ 3. By using a ZN index theory, we prove the existence of multiple solutions of (P) and show the dependence of multiplicity results on the coupling constant β.

Let H denote the class of positive harmonic functions on a bounded domain Ω in R . Let S be a sphere contained in Ω, and let σ denote the (N − 1)-dimensional measure. We give a condition on Ω which guarantees that there exists a constant K, depending only on Ω and S, such that R S u dσ ≤ K R ∂Ω udσ for every u ∈ H∩C(Ω). If this inequality holds for every such u, then it also holds for a large c...

Let a0, . . . , ak−1 be analytic functions on a domain Ω. Let F be a family of meromorphic functions f on Ω such that f 6= 0 and f (k) + ak−1f (k−1) + . . . + a0f 6= 0 on Ω, for all f ∈ F . Then {f ′/f : f ∈ F} is a normal family. Furthermore, let a0, . . . , ak−1 be meromorphic functions on a domain Ω. Let F be a family of meromorphic functions f on Ω such that f 6= 0, f ′ 6= 0 and f (k) + ak−...

We show that the iterated images of a Jacobian pair f : C 2 → C 2 stabilize; that is, all the sets f k (C 2) are equal for k sufficiently large. More generally, let X be a closed algebraic subset of C N , and f : X → X an open polynomial map with X − f (X) a finite set. We show that the sets f k (X) stabilize, and for any cofinite subset Ω ⊆ X with f (Ω) ⊆ Ω, the sets f k (Ω) stabilize. We appl...

We show that the vorticity of a viscous flow in R 3 admits an atomic decomposition of the form ω(x, t) = ∞ k=1 ω k (x − x k , t), with localized and oscillating building blocks ω k , if such a property is satisfied at the beginning of the evolution. We also study the long time behavior of an isolated coherent structure and the special behavior of flows with highly oscillating vorticities.

For a positive integer k, a k-relation on a set Ω is a non-empty subset ∆ of the k-fold Cartesian product Ω; ∆ is called a k-relation for a permutation group H on Ω if H leaves ∆ invariant setwise. The k-closure H of H, in the sense of Wielandt, is the largest permutation group K on Ω such that the set of k-relations for K is equal to the set of k-relations for H. We study k-relations for finit...

In this paper we prove higher integrability results for vector fields B,E, (B,E) ∈ L2− (Ω, R) × L2−ε(Ω, R), ε small, such that div B = 0, curl E = 0 satisfying a “reverse” inequality of the type |B| + |E| ≤ ( K + 1 K ) 〈B,E〉+ |F | with K ≥ 1 and F ∈ L(Ω, R), r > 2 − ε. Applications to the theory of quasiconformal mappings and partial differential equations are given. In particular, we prove reg...