Uncountable Research Articles

Countability constraints in order-theoretic approaches to computability

AbstractComputability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate such notions from Turing machines to uncountable spaces. Since these machines are used as a baseline for computability in these approaches, countability restrictions on the ordered structures are fundamental. Here, we show several relations between the usual countability restrictions in order-theoretic theories of computability and some more common order-theoretic countability constraints, like order density properties and functional characterizations of the order structure in terms of multi-utilities. As a result, we show how computability can be introduced in some order structures via countability order density and multi-utility constraints.

Global solvability of a two-species chemotaxis–fluid system with Lotka–Volterra type competitive kinetics

In this paper, we study a two-species chemotaxis–fluid system with Lotka–Volterra type competitive kinetics in a bounded and smooth domain [Formula: see text] with no-flux/Dirichlet boundary conditions. We present the global existence of weak energy solution to a two-species chemotaxis Navier–Stokes system, and then the global weak energy solution which coincides with a smooth function throughout [Formula: see text], where [Formula: see text] represents a countable union of open intervals which is such that [Formula: see text]. In such two-species chemotaxis–fluid setting, our existence improves known blow-up prevention by logistic source to blow-up prevention by sub-logistic source, indicating standard logistic source is not the weakest damping source to prevent blow-up. This finding significantly extends previously known ones.

Order boundedness and order continuity properties of positive operator semigroups

Relatively uniformly continuous (ruc) semigroups were recently introduced and studied by Kandic, Kramar-Fijavž, and the second-named author, in order to make the theory of one-parameter operator semigroups available in the setting of vector lattices, where no norm is present in general. In this article, we return to the more standard Banach lattice setting – where both ruc semigroups and C 0 -semigroups are well-defined concepts – and compare both notions. We show that the ruc semigroups are precisely those positive C 0 -semigroups whose orbits are order bounded for small times. We then relate this result to three different topics: (i) equality of the spectral and the growth bound for positive C 0 -semigroups; (ii) a uniform order boundedness principle which holds for all operator families between Banach lattices; and (iii) a description of unbounded order convergence in terms of almost everywhere convergence for nets which have an uncountable index set containing a co-final sequence.

Delta‐convex structure of the singular set of distance functions

AbstractFor the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta‐convex hypersurfaces up to an exceptional set of codimension two. In addition, in dimension two, the whole singular set is equal to a countable union of delta‐convex Jordan arcs up to isolated points. These results are new even in the standard Euclidean space and shown to be optimal in view of regularity.

A geometrical approach for consensus security

Guaranteeing security against eavesdroppers while allowing agents to reach an agreement on some shared variables is an essential feature to foster the adoption of distributed protocols. In the literature, a wide variety of methodologies have been developed in the discrete-time case, and most of them rely on encryption and undirected graphs, often exhibiting quantization or accuracy issues. In this paper, we develop a continuous-time secure consensus algorithm for agents that communicate via a broadcast-only protocol over directed and strongly connected graphs. In detail, the legitimate agents interact by exchanging an extended state, which can be converted into the hidden consensus state via a pre-deployed shared vector, thus preventing eavesdropping. In this paper, we take a geometrical perspective, and we implement our encryption via coordinate shifts. This choice guarantees, on one side, the existence of an uncountable infinity of possible vectors and, on the other side, provides a control-theoretical viewpoint on encryption that naturally blends with a dynamical system. We complement our framework by providing an additional layer of protection against honest but curious legitimate nodes, preventing them from gaining insights into the initial condition of their peers.

Residually continuous pullback attractors for stochastic discrete plate equations

For a nonautonomous stochastic discrete plate equation driven by multiplicative noise, we prove the unique existence of a pullback random attractor, which is a family of pullback‐attracting random compact sets parameterized by time and samples. We then establish the residual dense continuity of the pullback random attractor on the time‐sample plane with respect to the Hausdorff metric. Even without the Cantor continuum hypothesis, we show that the set of all points of continuity of the pullback random attractor is an uncountable set with the continuity cardinality. These results explain both nonexplosive and nonimplosive phenomena for the random plate lattice system.

C 1,α-rectifiability in low codimension in Heisenberg groups

Abstract A natural higher-order notion of C 1 , α {C}^{1,\alpha } -rectifiability, 0 < α ≤ 1 0\lt \alpha \le 1 , is introduced for subsets of the Heisenberg groups H n {{\mathbb{H}}}^{n} in terms of covering a set almost everywhere with a countable union of ( C H 1 , α , H ) \left({{\bf{C}}}_{H}^{1,\alpha },{\mathbb{H}}) -regular surfaces. Using this, we prove a geometric characterization of C 1 , α {C}^{1,\alpha } -rectifiable sets of low codimension in Heisenberg groups H n {{\mathbb{H}}}^{n} in terms of an almost everywhere existence of suitable approximate tangent paraboloids.

ON THE LIMIT SET OF A COMPLEX HYPERBOLIC TRIANGLE GROUP

Abstract Let $\Gamma =\langle I_{1}, I_{2}, I_{3}\rangle $ be the complex hyperbolic $(4,4,\infty )$ triangle group with $I_1I_3I_2I_3$ being unipotent. We show that the limit set of $\Gamma $ is connected and the closure of a countable union of $\mathbb {R}$ -circles.

Tracial oscillation zero and [formula omitted]-stability

Let A be a (not necessarily unital) separable non-elementary simple amenable C⁎-algebra whose tracial basis may not have finite covering dimension and may not be compact but satisfies a certain condition (C). We show that A is Z-stable if and only if A has strict comparison for positive elements. Extremal boundaries of simplexes which satisfy condition (C) may contain countable disjoint unions of n-dimensional cubes (n∈N) as a subset.

On n-scrambled sets

This paper investigates n-scrambled sets in the sense of distribution and Li-Yorke, focusing on the equivalence of n-scrambled sets and the existence of uncountable n-scrambled sets. For the equivalence, n-scrambled sets can imply m-scrambled sets in both senses for any 2≤m<n, but their inverses require certain conditions to hold. A method is provided for constructing an uncountable n-scrambled set in the distribution and Li-Yorke senses, which also verifies the equivalence between n-scrambled sets. For the existence, it is shown that distributionally n-scrambled tuples can be derived from the weak specification property (WSP), a 1-periodic and a non-1-periodic point. And the Li-Yorke (n+1)-scrambled tuple can be characterized by an n-scrambled tuple utilizing the SP. Finally, the uncountable Li-Yorke n-scrambled set is established based on the weak mixing, and the uncountable mean Li-Yorke n-scrambled set is induced by the shadowing and Devaney chaos.

Are almost all trajectories dense in a chaotic set?

Chaos is the most complex dynamic behavior a system can show. Yet, it is produced by two simple, though nonlinear, deterministic mechanisms, hence the more proper name Deterministic Chaos. The first mechanism, stretching, acts locally and linearly, expanding small sets in the system's state space in (at least) one direction and contracting in the others. The second mechanism, folding, acts globally and is the nonlinear operation of bending the elongated sets to form the forward image of the original sets. These two simple operations are invertible and form a chaotic set for the system's forward and backward map, containing an uncountable infinity of astonishingly complex system's trajectories. The simplest explanation of chaos I ever heard is James A. Yorke “pizza machine” (plenary lecture at the SIAM DS, Snowbird, 2003): kneading the pizza dough is nothing but stretching and folding. The mathematical formalization is Stephen Smale's horseshoe map. Using this nice tool, I revisit the hallmarks of chaos, one of which (in the title) still misses a constructive proof.

Quantum Physics in the Context of Countable and Uncountable Infinite Sets from the Perspective of Real Analysis

This study aims to investigate countable and uncountable infinite sets from the perspective of real analysis. Key theorems and definitions related to this topic are presented, along with some specific applications in quantum physics, such as the quantization of energy, the relationships between the discrete and the continuous, and the hypothesis of the linearity of the Schrödinger wave equation.

Some classes of topological spaces extending the class of Δ-spaces

A study of the class Δ \Delta consisting of topological Δ \Delta -spaces was originated by Jerzy Ka̧kol and Arkady Leiderman [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99; Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 267–280]. The main purpose of this paper is to introduce and investigate new classes Δ 2 ⊂ Δ 1 \Delta _2 \subset \Delta _1 properly containing Δ \Delta . We observe that for every first-countable X X the following equivalences hold: X ∈ Δ 1 X\in \Delta _1 iff X ∈ Δ 2 X\in \Delta _2 iff each countable subset of X X is G δ G_{\delta } . Thus, new proposed concepts provide a natural extension of the family of all λ \lambda -sets beyond the separable metrizable spaces. We prove that (1) A pseudocompact space X X belongs to the class Δ 1 \Delta _1 iff countable subsets of X X are scattered. (2) Every regular scattered space belongs to the class Δ 2 \Delta _2 . We investigate whether the classes Δ 1 \Delta _1 and Δ 2 \Delta _2 are invariant under the basic topological operations. Similarly to Δ \Delta , both classes Δ 1 \Delta _1 and Δ 2 \Delta _2 are invariant under the operation of taking countable unions of closed subspaces. In contrast to Δ \Delta , they are not preserved by closed continuous images. Let Y Y be l l -dominated by X X , i.e. C p ( X ) C_p(X) admits a continuous linear map onto C p ( Y ) C_p(Y) . We show that Y ∈ Δ 1 Y \in \Delta _1 whenever X ∈ Δ 1 X \in \Delta _1 . Moreover, we establish that if Y Y is l l -dominated by a compact scattered space X X , then Y Y is a pseudocompact space such that its Stone–Čech compactification β Y \beta Y is scattered.

A Decomposition for Borel Measures \(\mu \le \mathcal{H}^{s}\)

We prove that every finite Borel measure \( \mu \) in \( \mathbb R^N \) that is bounded from above by the Hausdorff measure \( \mathcal{H}^s \) can be split in countable many parts \( \mu{\lfloor_{_{{E_k}}}} \) that are bounded from above by the Hausdorff content \( \mathcal{H}^s_\infty \). Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to give a simpler proof for the existence of solutions of a Dirichlet problem involving an exponential nonlinearity.

A Probabilistic Version of Eneström–Kakeya Theorem for Certain Random Polynomials

This paper presents a comprehensive exploration of a probabilistic adaptation of the Eneström–Kakeya theorem, applied to random polynomials featuring various coefficient distributions. Unlike the deterministic rendition of the theorem, our study dispenses with the necessity of any specific coefficient order. Instead, we consider coefficients drawn from a spectrum of sets with diverse probability distributions, encompassing finite, countable, and uncountable sets. Furthermore, we provide a result concerning the probability of failure of Schur stability for a random polynomial with coefficients distributed independently and identically as standard normal variates. We also provide simulations to corroborate our results.

Gibbs measures of models with uncountable set of spin values on lattice systems

Gibbs measures of models with uncountable set of spin values on lattice systems

Capacity-Achieving Input Distributions of Additive Vector Gaussian Noise Channels: Even-Moment Constraints and Unbounded or Compact Support.

We investigate the support of a capacity-achieving input to a vector-valued Gaussian noise channel. The input is subjected to a radial even-moment constraint and is either allowed to take any value in Rn or is restricted to a given compact subset of Rn. It is shown that the support of the capacity-achieving distribution is composed of a countable union of submanifolds, each with a dimension of n-1 or less. When the input is restricted to a compact subset of Rn, this union is finite. Finally, the support of the capacity-achieving distribution is shown to have Lebesgue measure 0 and to be nowhere dense in Rn.

On the 1-prevalent continuous functions on compact sets and dimensions

Let be a non-empty compact set, and denote by C(K) the family of all real valued continuous functions on K. The main goal of this paper is to consider the generic properties of continuous functions in C(K) in terms of upper box dimension and packing dimension in the sense of prevalence. Precisely, we show that, the function Δ:C(K)→ℝ,Δ(f)=dim(Gf), where dim denotes the upper box dimension or packing dimension, satisfies the intertwining condition, i.e. for all and Lebesgue almost all , we have . This yields that a 1-prevalent continuous function satisfies which partially answers a question of Balka (2016 J. Fractal Geom. 3 407–28). In particular, if K is an uncountable compact set with at most finitely many isolated points then the upper box dimension of the graph of a 1-prevalent is . This result with 1-prevalent replaced by prevalent is due to Balka (2016 J. Fractal Geom. 3 407–28) and complements a result of Gruslys et al (2012 Monatsh. Math. 166 153–80) under the restriction that K satisfies some connected condition. Under the condition that there exists a compact subset such that dimP(F∩I)=dimP(K) for any open interval with , we show that a 1-prevalent continuous function satisfies dimP(Gf)=dimP(K)+1.

Revisiting Quantum Contextuality in an Algebraic Framework.

Within the framework of quantum contextuality, we discuss the ideas of extracontextuality and extravalence, that allow one to relate Kochen-Specker’s and Gleason’s theorems. We emphasize that whereas Kochen-Specker’s is essentially a no-go theorem, Gleason’s provides a mathematical justification of Born’s rule. Our extracontextual approach requires however a way to describe the “Heisenberg cut”. Following an article by John von Neumann on infinite tensor products, this can be done by noticing that the usual formalism of quantum mechanics, associated with unitary equivalence of representations, stops working when countable infinities of particles (or degrees of freedom) are encountered. This is because the dimension of the corresponding Hilbert space becomes uncountably infinite, leading to the loss of unitary equivalence, and to sectorisation. Such an intrinsically contextual approach provides a unified mathematical model including both quantum and classical physics, that appear as required incommensurable facets in the description of nature.

A Self-Similar Infinite Binary Tree Is a Solution to the Steiner Problem

We consider a general metric Steiner problem, which involves finding a set S with the minimal length, such that S∪A is connected, where A is a given compact subset of a given complete metric space X; a solution is called the Steiner tree. Paolini, Stepanov, and Teplitskaya in 2015 provided an example of a planar Steiner tree with an infinite number of branching points connecting an uncountable set of points. We prove that such a set can have a positive Hausdorff dimension, which was an open question (the corresponding tree exhibits self-similar fractal properties).