On minimal sets and strictly weaker topologies

In [1], [2] Farrag characterized the stirictly weaker principal topologies than any given principal topology on a nonempty set by using the minimal open sets which are defined by Steiner [3]. This paper mainly generalizes this result by using the minimal sets, which are defined in the paper with respect to the given topology τ on a nonempty set.     

On generalized perfect difference sets constructed from Sidon sets

Let $\mathbb{N}$ be the set of positive integers. For a nonempty set A of integers and every integer u, denote by $d_A(u)$ the number of $(a,a^{\prime })$ with $a,a^{\prime }\in A$ such that $u=a-a^{\prime }$. For a sequence S of positive integers, let $S(x)$ be the counting function of S. The set $A\subseteq \mathbb{N}$ is called a perfect difference set if $d_A(u)=1$ for every positive integer u. In 2008, Cilleruelo and Nathanson (2008) [4] constructed dense perfect difference sets from dense Sidon sets. In this paper, as a main result, we prove that: let $f:\mathbb{N}\to \mathbb{N}$ be an increasing function satisfying $f(n)\ge 2$ for any positive integer n, then for every Sidon set B and every function $\omega (x)\to \infty$, there exists a set $A\subseteq \mathbb{N}$ such that $d_A(u)=f(u)$ for every positive integer u and $B(x/3)-\omega (x)\le A(x)\le B(x/3)+\omega (x)$ for all $x\ge C_{f,B,\omega }$.     

On complete partially ordered sets and compatible topologies

We describe two complete partially ordered sets which are the intersection of complete linear orderings but which have no compatible Hausdorff topology. One is two-dimensional, while the second is countable, and leads to an example of a countable, compact, T ₁ space with a countable base which is not the continuous image of any compact Hausdorff space.     

Generalized open sets in generalized topologies

Summary We study the even power means of a sum analogous to Dedekind sums, and give a sharp asymptotic formula.     

On the structure of convex sets with symmetries

Asymmetry of a compact convex body L ì Rⁿ{\mathcal L \subset {\bf R}^n} viewed from an interior point O{\mathcal O} can be measured by considering how far L{\mathcal L} is from its inscribed simplices that contain O{\mathcal O}. This leads to a measure of symmetry s(L, O){\sigma(\mathcal L, \mathcal O)}. The interior of L{\mathcal L} naturally splits into regular and singular sets, where the singular set consists of points O{\mathcal O} with largest possible s(L, O){\sigma(\mathcal L, \mathcal O)}. In general, to calculate the singular set of a compact convex body is difficult. In this paper we determine a large subset of the singular set in centrally symmetric compact convex bodies truncated by hyperplane cuts. As a function of the interior point O{\mathcal O}, s(L, .){\sigma(\mathcal L, .)} is concave on the regular set. It is natural to ask to what extent does concavity of s(L, .){\sigma(\mathcal L, .)} extend to the whole (interior) of L{\mathcal L}. It has been shown earlier that in dimension two, s(L, .){\sigma(\mathcal L, .)} is concave on L{\mathcal L}. In this paper, we show that in dimensions greater than two, for a centrally symmetric compact convex body L{\mathcal L}, s(L, .){\sigma(\mathcal L, .)} is a non-concave function provided that L{\mathcal L} has a codimension one simplicial intersection. This is the case, for example, for the n-dimensional cube, n ≥ 3. This non-concavity result relies on the fact that a centrally symmetric compact convex body has no regular points.     

On the structure of sets with positive reach

We give a complete characterization of compact sets with positive reach (proximally C ¹ sets) in the plane and of one‐dimensional sets with positive reach in . Further, we prove that if is a set of positive reach of topological dimension , then A has its “k‐dimensional regular part” which is a k‐dimensional “uniform” C ^{1, 1} manifold open in A and can be locally covered by finitely many ‐dimensional DC surfaces. We also show that if has positive reach, then can be locally covered by finitely many semiconcave hypersurfaces.     

On the structure of bull-free perfect graphs

A bull is a graph obtained by adding a pendant vertex at two vertices of a triangle. Chvátal and Sbihi showed that the Strong Perfect Graph Conjecture holds for bull-free graphs. We show that bull-free perfect graphs are quasi-parity graphs, and that bull-free perfect graphs with no antihole are perfectly contractile. Our proof yields a polynomial algorithm for coloring bull-free strict quasi-parity graphsPartially supported by CNPq, grant 30 1160/91.0     

On the structure ofn-point sets

Letn be an integer greater than one. Our main result, called the “Structure Theorem” is that a set that containsn−1 disjoint continua that are cut by a single line cannot be ann-point set, that is, a set that meets every line in preciselyn points. This theorem unifies and significantly improves upon a number of known theorems. The second part of the paper is devoted to several theorems that address the question when a set that meets every line in at mostn points can be extended to ann-point set. These theorems also highlight the sharpness of the Structure Theorem.     

Preserving levels of projective determinacy by tree forcings

We prove that various classical tree forcings—for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing—preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes are added to thin projective transitive relations by these forcings.     

On the structure of Sidon sets

LetG be any connected compact group with dual object. We give in this paper a new proof that the union of any two Sidon sets in is again a Sidon set. We also show that any Sidon subset of is the union of a set whose elements have bounded degree with a finite union of sets which satisfy a quasi-independence condition.     

On the structure of stationary sets

We isolate several classes of stationary sets of [k]ωand investigate implications among them. Under a large cardinal assumption, we prove a structure theorem for stationary sets.     

On the structure of WDC sets

WDC sets in were recently defined as sublevel sets of DC functions (differences of convex functions) at weakly regular values. They form a natural and substantial generalization of sets with positive reach and still admit the definition of curvature measures. Using results on singularities of convex functions, we obtain regularity results on the boundaries of WDC sets. In particular, the boundary of a compact WDC set can be covered by finitely many DC surfaces. More generally, we prove that any compact WDC set M of topological dimension can be decomposed into the union of two sets, one of them being a k‐dimensional DC manifold open in M, and the other can be covered by finitely many DC surfaces of dimension . We also characterize locally WDC sets among closed Lipschitz domains and among lower‐dimensional Lipschitz manifolds. Finally, we find a full characterization of locally WDC sets in the plane.     

On the multiplicative structure of odd perfect numbers

Starting with Euler's theorem that any odd perfect number n has the form n = p^ep_i^2e_i … p_k^2e_k, where p, p₁,…,p_k are distinct odd primes and p ≡ e ≡ 1 (mod 4), we show that extensive subsets of these numbers (so described) can be eliminated from consideration. A typical result says: if p^e, p_i^2e_i,…,p_r^2e_r are all of the prime-power divisors of such an n with p ≡ p_i ≡ 1 (mod 4), then the ordered set {e₁,…,e_r} contains an even number or odd number of odd numbers according as e ≡ pore ≡ p (mod 8).     

Nonstandard topologies with bases that consist only of standard sets

Let be an infinite set, a set of pseudo-metrics on and If is limited (finite) for every and every then, for each we can define a pseudo-metric on by writing st We investigate the conditions under which the topology induced on by has a basis consisting only of standard sets. This investigation produces a theory with a variety of applications in functional analysis. For example, a specialization of some of our general results will yield such classical compactness theorems as Schauder's theorem, Mazur's theorem, and Gelfand-Philips's theorem.

     

On the measure of scrambled sets of tree maps with zero topological entropy

In this paper, we show that if f is a tree map of zero topological entropy and μ is an f-invariant Borel measure then any scrambled set S has zero outer μ-measure (hence μ-measurable). In particular, if S is measurable, it has zero μ-measure.