Volume 11, Issue 7

We generalize Colliot-Thélène’s construction of flasque resolutions of reductive group schemes over a field to a broad class of base schemes.

New criteria of existence and conjugacy of Hall subgroups of finite groups are given.

We show that a barely transitive group is totally imprimitive if and only if it is locally graded. Moreover, we obtain the description of a barely transitive group G for the case G has a cyclic subgroup 〈x〉 which intersects non-trivially with all subgroups and for the case a point stabilizer H of G has a subgroup H 1 of finite index in H satisfying the identity χ(H 1) = 1, where χ is a multi-linear commutator of weight w.

Let (X, d X) and (Y,d Y) be pointed compact metric spaces with distinguished base points e X and e Y. The Banach algebra of all $\mathbb{K}$-valued Lipschitz functions on X — where $\mathbb{K}$ is either‒or ℝ — that map the base point e X to 0 is denoted by Lip0(X). The peripheral range of a function f ∈ Lip0(X) is the set Ranµ(f) = {f(x): |f(x)| = ‖f‖∞} of range values of maximum modulus. We prove that if T 1, T 2: Lip0(X) → Lip0(Y) and S 1, S 2: Lip0(X) → Lip0(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip0(X), then there are mappings φ1φ2: Y → $\mathbb{K}$ with φ1(y)φ2(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T j(f)(y) = φ j(y)S j(f)(ψ(y)) for all f ∈ Lip0(X), y ∈ Y, and j = 1, 2. In particular, if S 1 and S 2 are identity functions, then T 1 and T 2 are weighted composition operators.

Recently, Jiménez-Melado et al. [Jiménez-Melado A., Llorens-Fuster E., Mazcuñán-Navarro E.M., The Dunkl-Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl., 2008, 342(1), 298–310] defined the Dunkl-Williams constant DW(X) of a normed linear space X. In this paper we present some characterizations of this constant. As an application, we calculate DW(ℓ2-ℓ∞) in the Day-James space ℓ2-ℓ∞.

Assume that L p,q, $L^{p_1 ,q_1 } ,...,L^{p_n ,q_n } $ are Lorentz spaces. This article studies the question: what is the size of the set $E = \{ (f_1 ,...,f_n ) \in L^{p_{1,} q_1 } \times \cdots \times L^{p_n ,q_n } :f_1 \cdots f_n \in L^{p,q} \} $. We prove the following dichotomy: either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is σ-porous in $L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $, provided 1/p ≠ 1/p 1 + … + 1/p n. In general case we obtain that either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is meager. This is a generalization of the results for classical L p spaces.

We investigate boundary regularity of solutions of generalized Stokes equations. The problem is complemented with perfect slip boundary conditions and we assume that the nonlinear elliptic operator satisfies non-standard ϕ-growth conditions. We show the existence of second derivatives of velocity and their optimal regularity.

We consider the so-called Ladyzhenskaya model of incompressible fluid, with an additional artificial smoothing term ɛΔ3. We establish the global existence, uniqueness, and regularity of solutions. Finally, we show that there exists an exponential attractor, whose dimension we estimate in terms of the relevant physical quantities, independently of ɛ > 0.

We show that if the Lyapunov exponents of a linear delay equation x′ = L(t)x t are limits, then the same happens with the exponential growth rates of the solutions to the equation x′ = L(t)x t + f(t, x t) for any sufficiently small perturbation f.

We study the Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability for a delay differential equation. Some examples are given.

For a metrizable space X and a finite measure space (Ω, $\mathfrak{M}$, µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $\mathfrak{M}$-measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.

A given set W = {W X } of n-variable class C 1 functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum $\sum\limits_\chi {(\int_{\mathbb{R}^n } {\nabla f \cdot } \nabla W_\chi ^* )} W_\chi $ converges to f with respect to the norm \(\left\| {\nabla ( \cdot )} \right\|_{L^2 (\mathbb{R}^n )} \) . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = {W x } of compactly supported class C 2−ɛ functions on ℝn such that where X has the structure \(\chi = (\tilde \chi ,\nu )\) , ν ∈ ℤ. W is a wavelet set in the sense that the functions indexed by \(\tilde \chi \) are generated by an averaging of lattice translations with wave propagations, and there are two additional discrete parameters associated with the latter. These functions indexed by \(\tilde \chi \) are the unit-scale wavelets. The support volumes of our unit-scale wavelets are not uniformly bounded, however. While the practical value of this construction is doubtful, our motivation is theoretical. The point is that a gradient-orthonormal basis of compactly supported wavelets has never been constructed in dimension n > 1. (In one dimension the construction of such a basis is easy — just anti-differentiate the Haar functions.)

A dominating broadcast on a graph G = (V, E) is a function f: V → {0, 1, ..., diam G} such that f(v) ≤ e(v) (the eccentricity of v) for all v ∈ V and such that each vertex is within distance f(v) from a vertex v with f(v) > 0. The cost of a broadcast f is σ(f) = Σv∈V f(v), and the broadcast number λ b (G) is the minimum cost of a dominating broadcast. A set X ⊆ V(G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; possibly y = x. The irredundance number ir (G) is the cardinality of a smallest maximal irredundant set of G. We prove the bound λb(G) ≤ 3 ir(G)/2 for any graph G and show that equality is possible for all even values of ir (G). We also consider broadcast domination as an integer programming problem, the dual of which provides a lower bound for λb.

Let G = (V, E) be a simple graph of order n and i be an integer with i ≥ 1. The set X i ⊆ V(G) is called an i-packing if each two distinct vertices in X i are more than i apart. A packing colouring of G is a partition X = {X 1, X 2, …, X k} of V(G) such that each colour class X i is an i-packing. The minimum order k of a packing colouring is called the packing chromatic number of G, denoted by χρ(G). In this paper we show, using a theoretical proof, that if q = 4t, for some integer t ≥ 3, then 9 ≤ χρ(C 4 □ C q). We will also show that if t is a multiple of four, then χρ(C 4 □ C q) = 9.