The geometry of some two-character sets

A projective (n, d, w ₁, w ₂) _q set (or a two-character set for short) is a set of n points of PG(d − 1, q) with the properties that the set generates PG(d − 1, q) and that every hyperplane meets the set in either n − w ₁ or n − w ₂ points. Here geometric constructions of some two-character sets are given. The constructions mainly involve commuting polarities, symplectic polarities and normal line-spreads of projective spaces. Some information about the automorphism groups of such sets is provided.      

Some cubic Julia sets

We figure out geometric properties of the Julia setJ _a of cubic complex polynomialC _a(z) =z ³ +az(a ∈ ?) and the smallest ellipse which surroundsJ _a.     

Some sets obeying harmonic synthesis

LetX be a (not necessarily closed) subspace of the dual spaceB ^* of a separable Banach spaceB. LetX ₁ denote the set of all weak ^* limits of sequences inX. DefineX _a , for every ordinal numbera, by the inductive rule:X _a = (U _{b < a} X _b ) ₁ .There is always a countable ordinala such thatX _a is the weak ^* closure ofX; the first sucha is called theorder ofX inB ^* . LetE be a closed subset of a locally compact abelian group. LetPM(E) be the set of pseudomeasures, andM(E) the set of measures, whose supports are contained inE. The setE obeys synthesis if and only ifM(E) is weak ^* dense inPM(E). Varopoulos constructed an example in which the order ofM(E) is 2. The authors construct, for every countable ordinala, a setE inR that obeys synthesis, and such that the order ofM(E) inPM(E) isa. This work was done in Jerusalem, when the second-named author was a visitor at the Institute of Mathematics of the Hebrew University of Jerusalem, with the support of an NSF International Travel Grant and of NSF Grant GP33583.     

Some new Kirkman signal sets

A decomposition of the blocks of an \(\textsf {STS}(v)\) into partial parallel classes of size m is equivalent to a Kirkman signal set \(\textsf {KSS}(v,m)\). We give decompositions of \(\textsf {STS}(4v-3)\) into classes of size \(v-1\) when \(v \equiv 3 \pmod {6}\), \(v \not = 3\). We also give decompositions of \(\textsf {STS}(v)\) into classes of various sizes when v is a product of two arbitrary integers that are both congruent to \(3 \pmod {6}\). These results produce new families of \(\textsf {KSS}(v,m)\).     

Some theorems on Chebyshev sets

We consider the class of -suns which is used in the study of Chebyshev sets. We give sufficient conditions for a set to be a -sun. We prove that, in a uniformly smooth Banach space, a weakly closed Chebyshev set is convex.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 135–144, February, 1972.The author expresses his thanks to V. I. Berdyshev and N. I. Chernykh for a lively discussion of this note.     

Some notes on convex sets

In [1] Tukey proves that if A and B are closed convex subsets in a Banach space, so that A is bounded and A-B is dense in the open unit ball U then A-BU. We shall give here a more general result than the former one which contains the Banach's isomorphism theorem as particular case. Other results over convex sets are also given.The author is indebted to the referee for the several very valuable comments.     

Some remarks on good sets

It is shown that (1) if a good set has finitely many related components, then they are full, (2) loops correspond one-to-one to extreme points of a convex set. Some other properties of good sets are discussed.     

Some properties of Baire sets and sets of positive measure

In un lavoro classico Steinhaus ha dimostrato undici teoremi interessanti nei quali si considerano insiemi distanziali di sottoinsiemi della linea reale,aventi misura di Lebesgue positiva. Majumder ha dimostrato che sono validi teoremi analoghi a quelli di Steinhaus per insiemi del tipoR(E)={x/y: x,y∈E} doveE∪R; 0?E. In questo lavoro si considerano funzioni generali e si investiga sui quali degli undici teoremi di Steinhaus possono essere generalizzati. Nell'articolo, inoltre, usando una funzione generalef, vengono dati risultati analoghi a quelli di Steinhaus per insiemi di Baire.     

Some new two-character sets in PG and a distance-2 ovoid in the generalized hexagon H

In this paper, we construct a new infinite class of two-character sets in and determine their automorphism groups. From this construction arise new infinite classes of two-weight codes and strongly regular graphs, and a new distance-2 ovoid of the split Cayley hexagon of order 4.     

Some aspects of intuitionistic fuzzy sets

We first discuss the significant role that duality plays in many aggregation operations involving intuitionistic fuzzy subsets. We then consider the extension to intuitionistic fuzzy subsets of a number of ideas from standard fuzzy subsets. In particular we look at the measure of specificity. We also look at the problem of alternative selection when decision criteria satisfaction is expressed using intuitionistic fuzzy subsets. We introduce a decision paradigm called the method of least commitment. We briefly look at the problem of defuzzification of intuitionistic fuzzy subsets.     

Some Lowenheim-Skolem results for admissible sets

For any admissible setA, there is anA-recursive set of sentences of ℒ _A which has a model but noA-finite model.A countable admissible set has the Lowenheim-Skolem property iff it is recurisvely inaccessible and locally countable.     

Some further remarks on good sets

We show that in n-fold cartesian product, n ≥ 4, a related component need not be a full component. We also prove that when n ≥ 4, uniform boundedness of lengths of geodesics is not a necessary condition for boundedness of solutions of (1) for bounded function f.     

Some remarks on totally imperfect sets

We prove the following two theorems.

Theorem 1. Let be a strongly meager subset of . Then it is dual Ramsey null.

We will say that a -ideal of subsets of satisfies the condition iff for every , if

then .

Theorem 2. The -ideals of perfectly meager sets, universally meager sets and perfectly meager sets in the transitive sense satisfy the condition .

     

Some new large sets of t-designs

We investigate the existence of some large sets of size nine. We take advantage of the recursive and direct constructing method to show that in cases $LS\left[9\right](2,4,v)$ the trivial necessary condition is also sufficient. In particular $LS\left[9\right](2,4,29)$ is constructed. This fills a gap.