Disjoint Families Research Articles

A classification of automorphic Lie algebras on complex tori

Abstract We classify the automorphic Lie algebras of equivariant maps from a complex torus to $\mathfrak{sl}_2(\mathbb{C})$ . For each case, we compute a basis in a normal form. The automorphic Lie algebras correspond precisely to two disjoint families of Lie algebras parametrised by the modular curve of $\mathrm{PSL}_2({\mathbb{Z}})$ , apart from four cases, which are all isomorphic to Onsager’s algebra.

Almost disjoint families under determinacy

Almost disjoint families under determinacy

Almost disjoint families and the geometry of nonseparable spheres

We consider uncountable almost disjoint families of subsets of N, the Johnson-Lindenstrauss Banach spaces (XA,‖‖∞) induced by them, and their natural equivalent renormings (XA,‖‖∞,2). We introduce a partial order PA and characterize some geometric properties of the spheres of (XA,‖‖∞) and of (XA,‖‖∞,2) in terms of combinatorial properties of PA. This provides a range of independence and absolute results concerning the above mentioned geometric properties via combinatorial properties of almost disjoint families. Exploiting the extreme behavior of some known and some new almost disjoint families we show the existence of Banach spaces where the unit spheres display surprising geometry:(1)There is a Banach space of density continuum whose unit sphere is the union of countably many sets of diameters strictly less than 1.(2)It is consistent that for every ρ>0 there is a nonseparable Banach space, where for every δ>0 there is ε>0 such that every uncountable (1−ε)-separated set of elements of the unit sphere contains two elements distant by less than 1 and two elements distant at least by 2−ρ−δ. It should be noted that for every ε>0 every nonseparable Banach space has a plenty of uncountable (1−ε)-separated sets by the Riesz Lemma.We also obtain a consistent dichotomy for the spaces of the form (XA,‖‖∞,2): The Open Coloring Axiom implies that the unit sphere of every Banach space of the form (XA,‖‖∞,2) either is the union of countably many sets of diameter strictly less than 1 or it contains an uncountable (2−ε)-separated set for every ε>0.

(Invited) Quiescent solitary waves in dual-core systems with separated Bragg grating and cubic–quintic nonlinearity

The existence and stability of zero-velocity solitons in a dual-core optical fiber, where one core possesses cubic–quintic nonlinearity and the other core is linear with a Bragg grating, are investigated. The frequency spectrum for the model consists of three bandgaps, of which only the central band gap is a genuine one. There are no soliton solutions in the central gap. On the other hand, the upper and lower gaps are filled with two distinct and disjoint families of soliton solutions, termed Type-1 and Type-2 solitons, that differ in their shapes and phase characteristics. When the relative group velocity in the linear core is zero, implicit analytical solutions for Type-1 and Type-2 quiescent solitons are found. For nonzero values of the relative group velocity, soliton solutions can only be found using numerical techniques. The stability of solitons has been analyzed systematically by means of numerical stability analysis. It is found that Type-2 solitons are always unstable. In the case of Type-1 solitons, stability regions are identified in the upper and lower bandgaps. The effects and interplay of the relative group velocity and the Bragg grating-induced linear coupling coefficient on the stability of the solitons are also analyzed.

On P-like ideals induced by disjoint families

We consider a combinatorial property isolated in the field of ideal convergence, a P-property for two ideals on natural numbers. We show that among selected ideals induced by disjoint families, not all pairs satisfy the P-property for two ideals. In many cases we specify the inducing partitions for which the corresponding ideals possess the property. Regarding selector ideals, a useful coloring-like necessary condition is provided.

Forcing theory and combinatorics of the real line

AbstractThe main purpose of this dissertation is to apply and develop new forcing techniques to obtain models where several cardinal characteristics are pairwise different as well as force many (even more, continuum many) different values of cardinal characteristics that are parametrized by reals. In particular, we look at cardinal characteristics associated with strong measure zero, Yorioka ideals, and localization and anti-localization cardinals.In this thesis we introduce the property “F-linked” of subsets of posets for a given free filter F on the natural numbers, and define the properties “ $\mu $ -F-linked” and “ $\theta $ -F-Knaster” for posets in a natural way. We show that $\theta $ -F-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. These kinds of posets led to the development of a general technique to construct $\theta $ - $\textrm {Fr}$ -Knaster posets (where $\textrm {Fr}$ is the Frechet ideal) via matrix iterations of ${<}\theta $ -ultrafilter-linked posets (restricted to some level of the matrix). The latter technique allows proving consistency results about Cichoń’s diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal characteristics associated with it are pairwise different. Another important application is to show that three strongly compact cardinals are enough to force that Cichoń’s diagram can be separated into 10 different values. Later on, it was shown by Goldstern, Kellner, Mejía, and Shelah that no large cardinals are needed for Cichoń’s maximum (J. Eur. Math. Soc. 24 (2022), no. 11, p. 3951–3967).On the other hand, we deal with certain types of tree forcings including Sacks forcing, and show that these increase the covering of the strong measure zero ideal $\mathcal {SN}$ . As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal characteristics of the continuum. Even more, Sacks forcing can be used to force that $\operatorname {\mathrm{non}}(\mathcal {SN})<\operatorname {\mathrm{cov}}(\mathcal {SN})<\operatorname {\mathrm{cof}}(\mathcal {SN})$ , which is the first consistency result where more than two cardinal characteristics associated with $\mathcal {SN}$ are pairwise different. To obtain another result in this direction, we provide bounds for $\operatorname {\mathrm{cof}}(\mathcal {SN})$ , which generalizes Yorioka’s characterization of $\mathcal {SN}$ (J. Symbolic Logic 67.4 (2002), p. 1373–1384). As a consequence, we get the consistency of $\operatorname {\mathrm{add}}(\mathcal {SN})=\operatorname {\mathrm{cov}}(\mathcal {SN})<\operatorname {\mathrm{non}}(\mathcal {SN})<\operatorname {\mathrm{cof}}(\mathcal {SN})$ with ZFC (via a matrix iteration forcing construction).We conclude this thesis by combining creature forcing approaches by Kellner and Shelah (Arch. Math. Logic 51.1–2 (2012), p. 49–70) and by Fischer, Goldstern, Kellner, and Shelah (Arch. Math. Logic 56.7–8 (2017), p. 1045–1103) to show that, under CH, there is a proper $\omega ^\omega $ -bounding poset with $\aleph _2$ -cc that forces continuum many pairwise different cardinal characteristics, parametrized by reals, for each one of the following six types: uniformity and covering numbers of Yorioka ideals as well as both kinds of localization and anti-localization cardinals, respectively. This answers several open questions from Klausner and Mejía (Arch. Math. Logic 61 (2022), pp. 653–683).Abstract prepared by Miguel Antonio Cardona-MontoyaE-mail: miguel.cardona@upjs.skURL: https://repositum.tuwien.at/handle/20.500.12708/19629

TIGHT EVENTUALLY DIFFERENT FAMILIES

AbstractGeneralizing the notion of a tight almost disjoint family, we introduce the notions of a tight eventually different family of functions in Baire space and a tight eventually different set of permutations of $\omega $ . Such sets strengthen maximality, exist under $\mathsf {MA} (\sigma \mathrm {-centered})$ and come with a properness preservation theorem. The notion of tightness also generalizes earlier work on the forcing indestructibility of maximality of families of functions. As a result we compute the cardinals $\mathfrak {a}_e$ and $\mathfrak {a}_p$ in many known models by giving explicit witnesses and therefore obtain the consistency of several constellations of cardinal characteristics of the continuum including $\mathfrak {a}_e = \mathfrak {a}_p = \mathfrak {d} < \mathfrak {a}_T$ , $\mathfrak {a}_e = \mathfrak {a}_p < \mathfrak {d} = \mathfrak {a}_T$ , $\mathfrak {a}_e = \mathfrak {a}_p =\mathfrak {i} < \mathfrak {u}$ , and $\mathfrak {a}_e=\mathfrak {a}_p = \mathfrak {a} < non(\mathcal N) = cof(\mathcal N)$ . We also show that there are $\Pi ^1_1$ tight eventually different families and tight eventually different sets of permutations in L thus obtaining the above inequalities alongside $\Pi ^1_1$ witnesses for $\mathfrak {a}_e = \mathfrak {a}_p = \aleph _1$ .Moreover, we prove that tight eventually different families are Cohen indestructible and are never analytic.

-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

Abstract Given an uncountable cardinal $\kappa $ , we consider the question of whether subsets of the power set of $\kappa $ that are usually constructed with the help of the axiom of choice are definable by $\Sigma _1$ -formulas that only use the cardinal $\kappa $ and sets of hereditary cardinality less than $\kappa $ as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets definable in this way and use it to generalize two classical nondefinability results to higher cardinals. First, we show that a classical result of Mathias on the complexity of maximal almost disjoint families of sets of natural numbers can be generalized to measurable limits of measurables. Second, we prove that for a limit of countably many measurable cardinals, the existence of a simply definable well-ordering of subsets of $\kappa $ of length at least $\kappa ^+$ implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the nonexistence of $\Sigma _1$ -definitions of certain objects at singular strong limit cardinals. Finally, we show that both large cardinal assumptions and forcing axioms cause analogs of these statements to hold at the first uncountable cardinal $\omega _1$ .

BANACH SPACES IN WHICH LARGE SUBSETS OF SPHERES CONCENTRATE

AbstractWe construct a nonseparable Banach space $\mathcal {X}$ (actually, of density continuum) such that any uncountable subset $\mathcal {Y}$ of the unit sphere of $\mathcal {X}$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon $ for some $\varepsilon>0$ ). This solves in the negative the central problem of the search for a nonseparable version of Kottman’s theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets of the unit sphere. The property of our space is strong enough to imply that it contains neither an uncountable Auerbach system nor an uncountable equilateral set. The space is a strictly convex renorming of the Johnson–Lindenstrauss space induced by an $\mathbb {R}$ -embeddable almost disjoint family of subsets of $\mathbb {N}$ . We also show that this special feature of the almost disjoint family is essential to obtain the above properties.

Ordinal sums of triangular norms on sublattices of a bounded lattice

In this paper, we present several necessary and sufficient conditions to ensure an ordinal sum of t-norms is still a t-norm on bounded lattices. For a fixed or an arbitrary bounded sublattice of a bounded lattice, we provide a direct characterization theorem, respectively. Meanwhile, we extend these results to the ordinal sum for a family of pairwise disjoint bounded sublattices. Furthermore, we investigate ordinal sum of t-norms on product lattices and reveal that there exist ordinal sum t-norms on non-trivial bounded sublattices of product lattices, which is different from the previous results of ordinal sum of t-norms defined on subintervals.

How to Tee a Hyperplane

A letter tee () is a geometric figure consisting of two line segments meeting at a right angle. We construct a covering of a hyperplane by a disjoint family of tees.

Backward Extensions of Weighted Shifts on Directed Trees

The weighted shifts are long known and form an important class of operators. One of generalisations of this class are weighted shifts on directed trees, where the linear order of coordinates in $$\ell ^2$$ is replaced by a more involved graph structure. In this paper we focus on the question of joint backward extending of a given family of weighted shifts on directed trees to a weighted shift on an enveloping directed tree that preserves subnormality or power hyponormality of considered operators. One of the main results shows that the existence of such a “joint backward extension” for a family of weighted shifts on directed trees depends only on the possibility of backward extending of single weighted shifts that are members of the family. We introduce a generalised framework of weighted shifts on directed forests (disjoint families of directed trees) which seems to be more convenient to work with. A characterisation of leafless directed forests on which all hyponormal weighted shifts are power hyponormal is given.

P-points, MAD families and Cardinal Invariants

P-points, MAD families and Cardinal Invariants

Support lines for disjoint families of unit disks

Support lines for disjoint families of unit disks

Vietoris hyperspaces over scattered Priestley spaces

We study Vietoris hyperspaces of closed and closed final sets of Priestley spaces. We are particularly interested in Skula topologies. A topological space is \emph{Skula} if its topology is generated by differences of open sets of another topology. A compact Skula space is scattered and moreover has a natural well-founded ordering compatible with the topology, namely, it is a Priestley space. One of our main objectives is investigating Vietoris hyperspaces of general Priestley spaces, addressing the question when their topologies are Skula and computing the associated ordinal ranks. We apply our results to scattered compact spaces based on certain almost disjoint families, in particular, Lusin families and ladder systems.

Towards resolving Keller’s cube tiling conjecture in dimension seven

Abstract A cube tiling of ℝ d is a family of pairwise disjoint cubes [0, 1) d + T = {[0, 1) d + t: t ∈ T} such that ∪ t∈T ([0, 1) d + t) = ℝ d . Two cubes [0, 1) d + t, [0, 1) d + s are called a twin pair if |t j −s j | = 1 for some j ∈ [d] = {1, ⋅, d} and t i = s i for every i ∈ [d]∖{j}. In 1930, Keller conjectured that in every cube tiling of ℝ d there is a twin pair. For x ∈ ℝ d and i ∈ [d], let L(T, x, i) be the set of all ith coordinates t i of vectors t ∈ T such that ([0, 1) d + t)∩([0, 1] d + x)≠∅ and t i ≤ x i . Let r − ( T ) = min x ∈ R d max 1 ≤ i ≤ d | L ( T , x , i ) | $r^-(T)=\min_{x\in \mathbb{R}^d} \max_{1\leq i\leq d}|L(T,x,i)|$ and r + ( T ) = max x ∈ R d max 1 ≤ i ≤ d | L ( T , x , i ) | $r^ + (T)=\max_{x\in \mathbb{R}^d} \max_{1\leq i\leq d}|L(T,x,i)|$ . Before 2019 it was known that Keller’s conjecture is true for dimensions d ≤ 6 and false for all dimensions d = 8. Moreover, in dimension 7 it was known to be true if r −(T) ≤ 2 or r +(T) = 5. The present paper resolves the case r +(T) = 4. At the end of 2019, when the paper was still under review, Brakensiek et al. resolved the cases r +(T) ∈ {3, 4, 6}, proving thereby Keller’s conjecture in dimension 7.

On the multiple recurrence properties for disjoint systems

We consider a mutually disjoint family of measure preserving transformations T1, …, Tk on a probability space \((X,{\cal B},\mu )\). We obtain the multiple recurrence property of T1, …, Tk and this result is utilized to derive multiple recurrence of Poincaré type in metric spaces. We also present the multiple recurrence property of Khintchine type. Further, we study multiple ergodic averages of disjoint systems and we show that T1, …, Tk are uniformly jointly ergodic if each Ti is ergodic.

ALMOST DISJOINT AND MAD FAMILIES IN VECTOR SPACES AND CHOICE PRINCIPLES

AbstractIn set theory without the Axiom of Choice ( $\mathsf {AC}$ ), we investigate the open problem of the deductive strength of statements which concern the existence of almost disjoint and maximal almost disjoint (MAD) families of infinite-dimensional subspaces of a given infinite-dimensional vector space, as well as the extension of almost disjoint families in infinite-dimensional vector spaces to MAD families.

Maximal almost disjoint families and pseudocompactness of hyperspaces

We show that all Ψ-spaces associated to maximal almost disjoint families have pseudocompact Vietoris hyperspace if and only if MAc(P(ω)/fin) holds. We further study the question whether there is a maximal almost disjoint family whose hyperspace is pseudocompact and construct a consistent example of a maximal almost disjoint family of size ω2<c whose hyperspace is not pseudocompact.

Norm-attaining lattice homomorphisms

In this paper we study the structure of the set ${\\rm Hom}(X,\\mathbb{R})$ of all lattice homomorphisms from a Banach lattice $X$ into $\\mathbb{R}$. Using the relation among lattice homomorphisms and disjoint families, we prove that the topological dual of the free Banach lattice ${\\rm FBL}(A)$ generated by a set $A$ contains a disjoint family of cardinality $2^{|A|}$, answering a question of B. de Pagter and A. W. Wickstead. We also deal with norm-attaining lattice homomorphisms. For classical Banach lattices, as $c_0$, $L_p$- and $C(K)$-spaces, every lattice homomorphism on it attains its norm, which shows, in particular, that there is no James theorem for this class of functions. We prove that, indeed, every lattice homomorphism on $X$ and $C(K,X)$ attains its norm whenever $X$ has order continuous norm. On the other hand, we provide what seems to be the first example in the literature of a lattice homomorphism which does not attain its norm. In general, we study the existence and characterization of lattice homomorphisms not attaining their norm in free Banach lattices. As a consequence, it is shown that no Bishop–Phelps type theorem holds true in the Banach lattice setting, i.e., not every lattice homomorphism can be approximated by norm-attaining lattice homomorphisms.