Cardinality Of The Continuum Research Articles

ON possible Turán densities

The Turan density π(F) of a family F of k-graphs is the limit as n → ∞ of the maximum edge density of an F-free k-graph on n vertices. Let Π ∞ (k) consist of all possible Turan densities and let Π fin (k) ⊆ Π ∞ (k) be the set of Turan densities of finite k-graph families. Here we prove that Π fin (k) contains every density obtained from an arbitrary finite construction by optimally blowing it up and using recursion inside the specified set of parts. As an application, we show that Π fin (k) contains an irrational number for each k ≥ 3. Also, we show that Π ∞ (k) has cardinality of the continuum. In particular, Π ∞ (k) ≠ Π fin (k) .

A Solution of a Problem of I. P. Natanson Concerning the Decomposition of an Interval into Disjoint Perfect Sets

In a previous paper published in this journal, it was demonstrated that any bounded, closed interval of the real line can, except for a set of Lebesgue measure 0, be expressed as a union of c pairwise disjoint perfect sets, where c is the cardinality of the continuum. It turns out that the methodology presented there cannot be used to show that such an interval is actually decomposable into c nonoverlapping perfect sets without the exception of a set of Lebesgue measure 0. We shall show, utilizing a Hilbert-type space-filling curve, that such a decomposition is possible. Furthermore, we prove that, in fact, any interval, bounded or not, can be so expressed.

A semi-invertible operator Oseledets theorem

AbstractSemi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we establish a semi-invertible multiplicative ergodic theorem that for the first time can be applied to the study of transfer operators associated to the composition of piecewise expanding interval maps randomly chosen from a set of cardinality of the continuum. We also give an application of the theorem to random compositions of perturbations of an expanding map in higher dimensions.

Monotone Smoothing Splines using General Linear Systems

AbstractIn this paper, a method is proposed to solve the problem of monotone smoothing splines using general linear systems. This problem, also called monotone control theoretic splines, has been solved only when the curve generator is modeled by the second‐order integrator, but not for other cases. The difficulty in the problem is that the monotonicity constraint should be satisfied over an interval which has the cardinality of the continuum. To solve this problem, we first formulate the problem as a semi‐infinite quadratic programming problem, and then we adopt a discretization technique to obtain a finite‐dimensional quadratic programming problem. It is shown that the solution of the finite‐dimensional problem always satisfies the infinite‐dimensional monotonicity constraint. It is also proved that the approximated solution converges to the exact solution as the discretization grid‐size tends to zero. An example is presented to show the effectiveness of the proposed method.

A compact metric space that is universal for orbit spectra of homeomorphisms

We say a space X with property P is a universal space for orbit spectra of homeomorphisms with propertyP provided that if Y is any space with property P and the same cardinality as X and h:Y→Y is any (auto)homeomorphism then there is a homeomorphism g:X→X such that the orbit equivalence classes for h and g are isomorphic. We construct a compact metric space X that is universal for homeomorphisms of compact metric spaces of cardinality of the continuum c and prove that there is no such space that is countably infinite. In the presence of some set theoretic assumptions we also give a separable metric space of size c that is universal for homeomorphisms on separable metric spaces.

A Large Group of Absolutely Nonmeasurable Additive Functions

By assuming the Continuum Hypothesis, it is proved that there exists a subgroup of \(\mathbb{R}^\mathbb{R}\) of cardinality strictly greater than the cardinality of the continuum, all nonzero members of which are absolutely nonmeasurable additive functions.

The family of bi-Lipschitz classes of delone sets in euclidean space has the cardinality of the continuum

It is proved that the family of bi-Lipschitz classes of Delone sets in Euclidean space of dimension at least 2 has the cardinality of the continuum.

The lattice of extensions of the modal logic of two equivalence relations has the cardinality of the continuum

A continuum of different logics over S5⋇S5 is constructed. This proves that the cardinality of the lattice of all normal extensions of the logic of two equivalence relations Ext(S5⋇S5) is continuum.

Ordinal remainders of ψ-spaces

We consider which ordinals, with the order topology, can be Stone–Čech remainders of which spaces of the form ψ ( κ , M ) , where ω ⩽ κ is a cardinal number and M ⊂ [ κ ] ω is a maximal almost disjoint family of countable subsets of κ (MADF). The cardinality of the continuum, denoted c , and its successor cardinal, c + , play important roles. We show that if κ > c + , then no ψ ( κ , M ) has any ordinal as a Stone–Čech remainder. If κ ⩽ c then for every ordinal δ < κ + there exists M δ ⊂ [ κ ] ω , a MADF, such that β ψ ( κ , M δ ) ∖ ψ ( κ , M δ ) is homeomorphic to δ + 1 . For κ = c + , β ψ ( κ , M δ ) ∖ ψ ( κ , M δ ) is homeomorphic to δ + 1 if and only if c + ⩽ δ < c + ⋅ ω .

Involutions on Zilber fields

In this paper, we briefly outline the definition of Zilber field, which is a structure analogue to the complex field with the exponential function. An open conjecture, including Schanuel’s Conjecture, is whether the complex field is itself one of these structure. In view of this conjecture, a natural question raised by Zilber, Kirby, Macintyre and others is whether they have an automorphism of order two akin to complex conjugation. We announce, without proof, the positive answer: for cardinality up to the continuum there exists an involution of the field commuting with the exponential function. Moreover, in the case of cardinality of the continuum, the automorphism can be taken such that its fixed field is exactly ℝ , and the kernel of the exponential function is 2\pi i \mathbb Z .

A Kronecker–Weyl theorem for subsets of abelian groups

Let N be the set of non-negative integer numbers, T the circle group and c the cardinality of the continuum. Given an abelian group G of size at most 2c and a countable family E of infinite subsets of G, we construct “Baire many” monomorphisms π:G→Tc such that π(E) is dense in {y∈Tc:ny=0} whenever n∈N, E∈E, nE={0} and {x∈E:mx=g} is finite for all g∈G and m∈N∖{0} such that n=mk for some k∈N∖{1}. We apply this result to obtain an algebraic description of countable potentially dense subsets of abelian groups, thereby making a significant progress towards a solution of a problem of Markov going back to 1944. A particular case of our result yields a positive answer to a problem of Tkachenko and Yaschenko (2002) [22, Problem 6.5]. Applications to group actions and discrete flows on Tc, Diophantine approximation, Bohr topologies and Bohr compactifications are also provided.

Additivity of the two-dimensional Miller ideal

Let J(M^2) denote the sigma-ideal associated with two-dimensional Miller forcing. We show that it is relatively consistent with ZFC that the additivity of J(M^2) is bigger than the covering number of the ideal of the meager subsets of the Baire space. We also show that Martin's Axiom implies that the additivity of J(M^2) is the cardinality of the continuum. Finally we prove that there are no analytical infinite maximal antichains in any finite product of the power set of the natural numbers modulo the ideal of the finite subsets of the natural numbers.

On a semilattice of numberings

We study some properties of a \( \mathfrak{c} \)-universal semilattice \( \mathfrak{A} \) with the cardinality of the continuum, i.e., of an upper semilattice of m-degrees. In particular, it is shown that the quotient semilattice of such a semilattice modulo any countable ideal will be also \( \mathfrak{c} \)-universal. In addition, there exists an isomorphism Open image in new window onto some ideal of the semilattice \( \mathfrak{A} \) such that \( {\mathfrak{A} \mathord{\left/ {\vphantom {\mathfrak{A} {\iota \left( \mathfrak{A} \right)}}} \right. \kern-\nulldelimiterspace} {\iota \left( \mathfrak{A} \right)}} \) will be also \( \mathfrak{c} \)-universal. Furthermore, a property of the group of its automorphisms is obtained. To study properties of this semilattice, the technique and methods of admissible sets are used. More exactly, it is shown that the semilattice of mΣ-degrees \( L_{m\Sigma }^{\mathbb{H}\mathbb{F}\left( S \right)} \) on the hereditarily finite superstructure \( \mathbb{H}\mathbb{F} \)(S) over a countable set S will be a \( \mathfrak{c} \)-universal semilattice with the cardinality of the continuum.

Prime-representing functions

We construct prime-representing functions. In particular we show that there exist real numbers α > 1 such that ⌈α 2 n ⌉ is prime for all n ∈ ℕ. Indeed the set consisting of such numbers α has the cardinality of the continuum.

Almost Isometry-Invariant Sets and Shadings

An almost isometry-invariant set $A\subset \mathbb{R}$ satisfies $|gA \Delta A| < \mathbf{c}$ for any isometry $g$ acting on $\mathbb{R}$, where ${\bf c}$ is the cardinality of the continuum. A shading is any set $S\subseteq \mathbb{R}$ in which $\frac{\mu(S\cap I)}{\mu(I)}$ has the same constant value for every finite interval $I$, for any Banach measure $\mu$. (A Banach measure is a finitely additive, isometry-invariant extension of the Lebesgue measure to $2^{\mathbb{R}}$.) In this paper we prove several theorems that show how these two types of sets are related. We also prove several sum and difference set results for almost isometry-invariant sets. Finally, we completely solve a problem involving subsets of Archimedean sets first posed by R. Mabry and partially solved by K. Neu.

Discrete orbits in topologically transitive cylindrical transformations

In this paper we provide a few recipes how to construct a topologically transitive cocycle over an arbitrary odometer possessing discrete orbits.It is shown that for every odometer, there exists a topologicallytransitive cocycle such that the set of points with discrete orbitsstarting form zero level has the cardinality of the continuum.

The Uncountable Lattice of All Subfunctors of the Functor of Rational Representations

In set theory the cardinality of the continuum \(|{\mathbb R}|\) is the cardinal number of some interesting sets, like the Cantor set or the transcendental numbers. We will prove that the cardinal number of all subfunctors of the functor of rational representations \(k \otimes_{{\mathbb Z}} R_{{\mathbb Q}}\), taking values on odd order groups over the field k of characteristic 2, is equal to \(|{\mathbb R}|\). When the characteristic q > 0 of the field k is not necessarily even, we will present a formula giving the dimension of the evaluations SC,k(G), of the simple functor SC,k, at any group G of order prime to q and being associated to a suitable cyclic group C.

On diversity and stability of unit bases for the Euclidean metric

We prove the existence of a family Ω(n) of 2 c (where c is the cardinality of the continuum) subgraphs of the unit distance graph (E n , 1) of the Euclidean space E n , n ≥ 2, such that (a) for each graph G ϵ Ω(n), any homomorphism of G to (E n , 1) is an isometry of E n ; moreover, for each subgraph G 0 of the graph G obtained from G by deleting less than c vertices, less than c stars, and less than c edges (we call such a subgraph reduced), any homomorphism of G 0 to (E n , 1) is an isometry (of the set of the vertices of G 0); (b) each graph G ϵ Ω(n) cannot be homomorphically mapped to any other graph of the family Ω(n), and the same is true for each reduced subgraph of G.

On effective presentations of formal concept lattices

We study the isomorphism types of ordered structures of concepts for computable formal contexts from the point of view of computable model theory. We show that, although these structures could have the cardinality of the continuum or—if they are countable—could have an arbitrary high hyperarithmetical complexity, they are in some sense very close to computable orderings. We give some sufficient conditions for these orderings to have computable presentations. A full description is given of the isomorphism types of discrete concept lattices. We also give some counterexamples.

Minimal pseudocompact group topologies on free abelian groups

A Hausdorff topological group G is minimal if every continuous isomorphism f : G → H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence { σ n : n ∈ N } of cardinals such that w ( G ) = sup { σ n : n ∈ N } and sup { 2 σ n : n ∈ N } ⩽ | G | ⩽ 2 w ( G ) , where w ( G ) is the weight of G. If G is an infinite minimal abelian group, then either | G | = 2 σ for some cardinal σ, or w ( G ) = min { σ : | G | ⩽ 2 σ } ; moreover, the equality | G | = 2 w ( G ) holds whenever cf ( w ( G ) ) > ω . For a cardinal κ, we denote by F κ the free abelian group with κ many generators. If F κ admits a pseudocompact group topology, then κ ⩾ c , where c is the cardinality of the continuum. We show that the existence of a minimal pseudocompact group topology on F c is equivalent to the Lusin's Hypothesis 2 ω 1 = c . For κ > c , we prove that F κ admits a (zero-dimensional) minimal pseudocompact group topology if and only if F κ has both a minimal group topology and a pseudocompact group topology. If κ > c , then F κ admits a connected minimal pseudocompact group topology of weight σ if and only if κ = 2 σ . Finally, we establish that no infinite torsion-free abelian group can be equipped with a locally connected minimal group topology.