Let R be an o-minimal expansion of a real closed field. Given definable continuous functions f : U → R and : U → (0,+∞), where U is an open subset of Rn, we construct a definable Cm-function g : U → R with |g(x)− f(x)| < (x) for all x ∈ U . Moreover, we show that if f is uniformly continuous, then g can also chosen to be uniformly continuous.

For G = SL(3, R) and G = SO(2, n), we give explicit, practical conditions that determine whether or not a closed, connected subgroup H of G has the property that there exists a compact subset C of G with CHC = G. To do this, we fix a Cartan decomposition G = KA + K of G, and then carry out an approximate calculation of (KHK) ∩ A + for each closed, connected subgroup H of G.

A quadriculated disk is a juxtaposition of finitely many squares along sides forming a closed topological disk such that vertices of squares which lie in the interior of the disk belong to precisely four squares. In this paper, by a square we mean a topological disk with four privileged boundary points, the vertices. A simple example of a quadriculated disk is the n ×m rectangle divided into un...

Given a subset S of R, the Helly number h(S) is the largest size of an inclusionwise minimal family of convex sets whose intersection is disjoint from S. A convex set is S-free if its interior contains no point of S. The parameter f(S) is the largest number of maximal faces in an inclusionwise maximal S-free convex set. We study the relation between the parameters h(S) and f(S). Our main result...

In this paper we deal with the following generalized vector quasiequilibrium problem: given a closed convex set K in a normed space X, a subset D in a Hausdorff topological vector space Y , and a closed convex cone C in R. Let Γ : K → 2, Φ : K → 2 be two multifunctions and f : K×D×K → R be a single-valued mapping. Find a point (x̂, ŷ) ∈ K×D such that (x̂, ŷ) ∈ Γ(x̂)× Φ(x̂), and {f(x̂, ŷ, z) : z ∈ Γ(...

Some problems concerning the additive properties of subsets of R are investigated. From a result of G. G . Lorentz in additive number theory, we show that if P is a nonempty perfect subset of R, then there is a perfect set M with Lebesgue measure zero so that P+M = R. In contrast to this, it is shown that (1) if S is a subset of R is concentrated about a countable set C, then A(S+R) = 0, for ev...

The stereographic projection is constructed in topological modules. Let A be an additively symmetric closed subset of a R-module M such that 0∈int(A). If there exists continuous functional m*:M→R the dual module M*, invertible s∈U(R) and element boundary bd(A) way m*−1({s})∩int(A)=⌀, a∈m*−1({s})∩bd(A), s+m*bd(A)\{−a}⊆U(R), then following function b↦−a+2s(m*(b)+s)−1(b+a), from bd(A)\{−a} to (m*)...

We investigate two density questions for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) an open set $\Omega\subset\mathbb R^n$, $\mathcal{D}(\Omega)$ is dense $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \overline{\Omega}\}$ whenever $\partial\Omega$ has zero Lebesgue measure $\Omega$ "thick" (in sense o...

A subset S of R is said to be polyhedral if it is the intersection of a finite number of closed halfspaces, i.e., if there exist J ∈ N and collections {y1, . . . , yJ} ⊂ R, {α1, . . . , αJ} ⊂ R such that S = ∩j=1{x ∈ R : y jx ≤ αj}. A function f : R → [−∞,+∞] is polyhedral if its epigraph epi f ⊂ R is a polyhedral set. Clearly, any polyhedral set is automatically convex and closed. Consequently...

Let A be a commutative unital R-algebra and let ρ be a seminorm on A which satisfies ρ(ab) ≤ ρ(a)ρ(b). We apply T. Jacobi’s representation theorem [10] to determine the closure of a ∑A-module S of A in the topology induced by ρ, for any integer d ≥ 1. We show that this closure is exactly the set of all elements a ∈ A such that α(a) ≥ 0 for every ρ-continuous R-algebra homomorphism α ∶ AÐ→ R wit...