Disjoint Families Research Articles

On an 𝑛-manifold in 𝐂ⁿ near an elliptic complex tangent

In this paper, we study the local biholomorphic property of a real n n -manifold M ⊂ C n M\subset \mathbf C^n near an elliptic complex tangent point p ∈ M p\in M . In particular, we are interested in the regularity and the unique disk-filling problem of the local hull of holomorphy M ~ \widetilde {M} of M M near p p , first considered in a paper of Bishop. When M M is a C ∞ C^{\infty } -smooth submanifold, using a result established by Kenig-Webster, we show that near p p , M ~ \widetilde {M} is a smooth Levi-flat ( n + 1 ) (n+1) -manifold with a neighborhood of p p in M M as part of its C ∞ C^{\infty } boundary. Moreover, near p p , M ~ \widetilde {M} is foliated by a family of disjoint embedded complex analytic disks. We also prove a uniqueness theorem for the analytic disks attached to M M . This result was proved in the previous work of Kenig-Webster when n = 2 n=2 . When M M is real analytic, we show that M ~ \widetilde {M} is real analytic with a neighborhood of p p in M M as part of its real analytic boundary. Equivalently, we prove the convergence of the formal solutions of a certain functional equation. When n = 2 n=2 or when n > 2 n>2 but the Bishop invariant does not vanish at the point under study, the analyticity was then previously obtained in the work of Moser-Webster, Moser, and in the author’s joint work with Krantz.

Extensions of functions in Mrówka-Isbell spaces

For an almost disjoint family (a.d.f.) ∑ of subsets of ω, let Ψ(∑) be the Mrówka-Isbell space on ∑. In this article we will analyze the following problem: given an a.d.f. ∑ and a function φ: ∑ → {0, 1} (respectively φ: ∑ → R ) is it possible to extend φ continuously to a big enough subspace ∑ ∪ N of Ψ(∑) for which l Ψ(∑) N ⊃ ∑? Such an extension is called essential. We will prove that: 1. (i) for every a.d.f. ∑ of cardinality 2 ℵ 0 we can find a function φ: ∑ → {0, 1} without essential extensions; 2. (ii) for every m.a.d. family ∑ there exists a function φ: ∑ → R that has no essential extension; and 3. (iii) there exists a Mrówka-Isbell space Ψ(∑) of cardinality ℵ 1 such that every function φ: ϵ → R with at least two different uncountable fibers, has no full extension. On the other hand, under Martin's Axiom every function φ: ∑ → {0, 1} (respectively φ: ∑ → R ) has an essential extension if ¦∑¦ < 2 ℵ 0 . Finally, we analyze these questions under CH and by adding new Cohen reals to a ground model M showing that the existence of an uncountable a.d.f. ∑ for which every onto function φ: ∑ → {0, 1} with infinite fibers has no essential extensions is consistent with ZFC.

Intersecting Systems

An intersecting system of type (∃, ∀, k, n) is a collection [ ]={[Fscr ]1, ..., [Fscr ]m} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair [Fscr ]i and [Fscr ]j of distinct members of [ ] there exists an A∈[Fscr ]i that intersects every B∈[Fscr ]j. Let In (∃, ∀, k) denote the maximum possible cardinality of an intersecting system of type (∃, ∀, k, n). Ahlswede, Cai and Zhang conjectured that for every k≥1, there exists an n0(k) so that In (∃, ∀, k)=(n−1/k−1) for all n&gt;n0(k). Here we show that this is true for k≤3, but false for all k≥8. We also prove some related results.

A New Look at Euler's Theorem for Polyhedra: A Comment

In their interesting paper [6] authors, Branko Grunbaum and Geoffrey Shephard mention (page 125/126) . . . work of [20-25], which our note parallels to some extent. Nefns definition of polyhedral sets differs from ours, but is equivalent to it; his approach to Euler characteristic is same as ours. Actually, this parallelism concerns a major part of article and comes partly into conflict with my own work in field. The main reason is authors' inexplicable statement (p. 126) Nef [1978] defines as faces of a polyhedral set P any family of disjoint, relatively open convex sets that is a dissection of P. ... However, these 'faces' are, in general, not uniquely determined, and have only a limited geometric In fact my definition is one in [8, p. 6.2], or in [1, p. 98]. According to this definition faces of a polyhedron turn out to be relative interiors of faces in intuitive sense. They are uniquely determined and have a clear geometric significance. (Unfortunately, in [1] a typing error slipped in: On p. 98 in formula (06) U0 should be replaced by P n ao, and U by P n U). In a thorough discussion, several further points would have to be critically looked at. I confine myself to two of them: On page 122, Grunbaum and Shephard present their Theorem 2* concerning Euler characteristic of (not necessarily bounded) closed convex polyhedra. They overlook that I have published same result previously in [9, Satz 4, p. 68]. On page 117, authors define Euler characteristic X(P) (in same way as I have done in [10, Satz 2, pp. 44-45]): For a cell C (a nonempty relatively open convex polyhedron) we put x(C) = (-l)dimC, furthermore, X(0) = 0. If P Uc E c C represents P as a finite disjoint union of cells, then X(P) = Ec E c x(c) The definition is followed by three theorems, first of which asserts that X(P) does not depend on partition of P into cells. (For a proof of this Theorem see [10, Satz 2, pp. 44-45]). The second theorem states that X(P) = 1 if P is bounded, closed, and convex. The proofs are omitted since they follow usual techniques. From discovery of his theorem by Euler (in 1750) more than 200 years were needed to find a complete proof ([5, p. 134], [7, p. 94], [2]); for an extended discussion of historical development see [6, pp. 122-127]. It is therefore not evident what is meant by the usual techniques, at least in case of second theorem, which is an essential part of theorem of Euler-Schlafli (Leonhard Euler [3] and [4], and extension to higher dimensions in 1850 by Ludwig Schlafli [12]). In [11] I have published a particularly short and simple proof which I repeat here: First of all, every closed polyhedron P is disjoint union of its faces (without extP) [8, Satz 3, p. 6.7]. If P is closed and convex, faces are cells [8, Satz 8, p. 7.12]. Therefore X(P) = Lin=0( l)iti, where n = dim P and fi is number of faces of dimension i (with fn = 1 for relative interior of P). Now let P be bounded, closed, and convex, and let 4) denote family of all faces F of P with dim F < n. We choose an arbitrary point z E relintP. For each face F E ¢ we define F* = {x = z + A(F-z) with 0 < A < 1}, so F* is

The Arkhangel’skiĭ–Tall problem: a consistent counterexample

We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel'skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in $[ω]^ω$, and a versi

On Haar null sets

We prove that in Polish, abelian, non-locally-compact groups the family of Haar null sets of Christensen does not fulfil the countable chain condition, that is, there exists an uncountable family of pairwise disjoint universally measurable sets which are

𝑝-sequentiality and 𝑝-Fréchet-Urysohn property of Franklin compact spaces

Franklin compact spaces defined by maximal almost disjoint families of subsets of ω \omega are considered from the view of its p p -sequentiality and p p -Fréchet-Urysohn-property for ultrafilters p ∈ ω ∗ p\in \omega ^* . Our principal results are the following: CH implies that for every P P -point p ∈ ω ∗ p\in \omega ^* there are a Franklin compact p p -Fréchet-Urysohn space and a Franklin compact space which is not p p -Fréchet-Urysohn; and, assuming CH, for every Franklin compact space there is a P P -point q ∈ ω ∗ q\in \omega ^* such that it is not q q -Fréchet-Urysohn. Some new problems are raised.

A special class of almost disjoint families

AbstractThe collection of branches (maximal linearly ordered sets of nodes) of the tree &lt;ωω (ordered by inclusion) forms an almost disjoint family (of sets of nodes). This family is not maximal — for example, any level of the tree is almost disjoint from all of the branches. How many sets must be added to the family of branches to make it maximal? This question leads to a series of definitions and results: a set of nodes is off-branch if it is almost disjoint from every branch in the tree; an off-branch family is an almost disjoint family of off-branch sets; and is the minimum cardinality of a maximal off-branch family.Results concerning include: (in ZFC) , and (consistent with ZFC) is not equal to any of the standard small cardinal invariants or = 2ω. Most of these consistency results use standard forcing notions—for example, in the Cohen model.Many interesting open questions remain, though—for example, whether .

Strong meager properties for filters

We analyze several ``strong meager'' properties for filters on the natural numbers between the classical Baire property and a filter being $F_\sigma$. Two such properties have been studied by Talagrand and a few more combinatorial ones are investigated. In particular, we define the notion of a P$^+$-filter, a generalization of the traditional concept of P-filter, and prove the existence of a non-meager P$^+$-filter. Our motivation lies in understanding the structure of filters generated by complements of members of a maximal almost disjoint family.

Disjoint systems

AbstractA disjoint system of type (∀, ∃, k, n) is a collection 𝒞 = {𝒜1,…, 𝒜m} of pairwise disjoint families of k‐subsets of an n‐element set satisfying the following condition. For every ordered pair 𝒜i and 𝒜j of distinct members of 𝒞 and for every A ϵ 𝒜i there exists a B ϵ 𝒜j that does not intersect A. Let Dn (∀, ∃, k) denote the maximum possible cardinality of a disjoint system of type (∀, ∃, k, n). It is shown that for every fixed k ⩾ 2,.magnified imageThis settles a problem of Ahlswede, Cai, and Zhang. Several related problems are considered as well.

Exact Algorithms For The Setup Knapsack Problem

The Setup Knapsack problem consists in selecting items from a set of disjoint families of items, to enter a knapsack while maximizing its value. An item can be selected only if the knapsack is set up for items of its family. We give a 0-1 programming formulation and we propose a reformulation in which setup variables are replaced by equivalent Boolean unions of item assignment variables. We present a Dynamic Programming algorithm and two versions of a two-phase enunierative scheme. These algorithms solve this NP-hard problem to optimality in time that is at worst pseudo-polynomial. We test the algorithms on randomly generated test problems

Homogeneous almost disjoint families

An almost disjoint family is constructed which is isomorphic to any almost disjoint family which can be constructed from it by taking subsets and finite unions. This is applied to the construction of a Boolean algebra with related properties.

On free products of completely regular semigroups

AbstractThe free product*CRSiof an arbitrary family of disjoint completely simple semigroups {Si}i∈i, within the variety CR of completely regular semigroups, is described by means of a theorem generalizing that of Kaďourek and Polák for free completely regular semigroups. A notable consequence of the description is that all maximal subgroups of*CRSiare free, except for those in the factorsSithemselves. The general theorem simplifies in the case of free CR-products of groups and, in particular, free idempotent-generated completely regular semigroups.

Can ideals without ccc be interesting?

We study those ideals I of sets in a perfect Polish space X which admit a Borel measurable f: X→ X with f -1[\\s{ x\\s}]∉ I for each xϵ X. A stronger version of that property (when, additionally, X is a group and I an invariant ideal) states that there exist a Borel set B∉ I and a perfect P⊆ X, such that \\s{ B + x: x ϵ P\\s} forms a disjoint family.

The Axiom of Choice in Second‐Order Predicate Logic

AbstractThe present article deals with the power of the axiom of choice (AC) within the second‐order predicate logic. We investigate the relationship between several variants of AC and some other statements, known as equivalent to AC within the set theory of Zermelo and Fraenkel with atoms, in Henkin models of the one‐sorted second‐order predicate logic with identity without operation variables. The construction of models follows the ideas of Fraenkel and Mostowski. It is e. g. shown that the well‐ordering theorem for unary predicates is independent from AC for binary predicates and from the trichotomy law for unary predicates. Moreover, we show that the AC for binary predicates follows neither from the trichotomy law for unary predicates nor from Zorn's lemma for unary predicates nor from the formalization of the axiom of choice for disjoint families of sets for binary predicates, and that the trichotomy law for unary predicates does not follow from AC for binary predicates.Mathematics Subject Classification: 03B15, 03E25, 04A25.

More on cardinal arithmetic

This paper deals with variety of problems in pcf theory and infinitary combinatorics. We look at normal filters and prc, measures of the size of [lambda]^{ with |B_i|<mu_i, and finally we deal with the existence of strongly almost disjoint families.

Weakly normal closures of filters on Pκλ

The study of filters on Pκλ started by Jech [5] as a natural generalization of that of filters on an uncountable regular cardinal κ. Several notions including weak normality have been generalized. However, there are two versions proposed as weak normality for filters on Pκλ. One is due to Abe [1] as a straightforward generalization of weak normality for filters on κ due to Kanamori [6] and the other is due to Mignone [8]. While Mignone's version is weaker than normality, Kanamori-Abe's version is not in general. In fact, Abe [2] has proved, generalizing Kanamori [6], that a filter is weakly normal in the sense of Abe iff it is weakly normal in the sense of Mignone and there exists no disjoint family of cfλ-many positive sets. Therefore Kanamori-Abe's version is essentially a large cardinal property and Mignone's version seems to be the most natural formulation of “weak” normality.In this paper, we study weak normality in the sense of Mignone. In [8], Mignone studies weak normality of canonically defined filters. We complement his chart and try to find the weakly normal closures of these filters (i.e., the minimal weakly normal filters extending them). Therefore our result is a natural refinement of Carr [4].It is now well known that combinatorics on Pκλ is not a naive generalization of that on κ. For example, Menas [7] showed that stationarity on Pκλ can be characterized by 2-dimensional regressive functions, but not by 1-dimensional ones when λ is strictly larger than κ. We show in terms of weak normality that combinatorics on Pκλ vary drastically with respect to cfλ.

The different ways of stabbing disjoint convex sets

We construct a family ofn disjoint convex set in ℝd having (n/(d−1))d−1 geometric permutations. As well, we complete the enumeration problem for geometric permutations of families of disjoint translates of a convex set in the plane, settle the case for cubes in ℝd, and construct a family ofd+1 translates in ℝd admitting (d+1)!/2 geometric permutations.

Additivity of vector gleason measures

We study the degree of additivity of orthogonal Hilbert-space-valued measures on the latticeL(H) of all projections acting on a Hilbert spaceH. We present criteria for such measures to be completely additive and we establish the connection between the additivity of orthogonal measures and the size of almost disjoint families on dimH. [For example, we show that everyH-valued orthogonal measure is weaklyσ-additive iff (dimH)ω > dim H]. As a corollary we see that finitely additive orthogonal measures distinguish dimensions of Hilbert spaces (this can be viewed as a generalization of a theorem by Kruszynski). As a further corollary, we obtain that, for cardinalsκ, ν with κ>ν,κ≥3, there is no Jordan homomorphism from a typeI κ -factor into a typeI λ -factor. Finally, we show that every latticeL(H) with (dimH)ω = dimH admits a nonzero free orthogonal measure with values inH. Our results contribute to the noncommutative probability theory and also may find applications in the theory of the representation ofC *-algebras.

Stone–Wales pyracylene transformations of the isomers of C84

Under the Stone–Wales (pyracylene) transformation, the 24 isolated-pentagon isomers of C84 fall into two disjoint families, each containing one leapfrog closed shell, suggesting that the experimental product will consist of a mixture of two isomers.