Uncountable Research Articles

Basic properties of 𝑋 for which the space 𝐶_{𝑝}(𝑋) is distinguished

In our paper [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99] we showed that a Tychonoff spaceXXis aΔ\Delta-space (in the sense of R. W. Knight [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60], G. M. Reed [Fund. Math. 110 (1980), pp. 145–152]) if and only if the locally convex spaceCp(X)C_{p}(X)is distinguished. Continuing this research, we investigate whether the classΔ\DeltaofΔ\Delta-spaces is invariant under the basic topological operations.We prove that ifX∈ΔX \in \Deltaandφ:X→Y\varphi :X \to Yis a continuous surjection such thatφ(F)\varphi (F)is anFσF_{\sigma }-set inYYfor every closed setF⊂XF \subset X, then alsoY∈ΔY\in \Delta. As a consequence, ifXXis a countable union of closed subspacesXiX_isuch that eachXi∈ΔX_i\in \Delta, then alsoX∈ΔX\in \Delta. In particular,σ\sigma-product of any family of scattered Eberlein compact spaces is aΔ\Delta-space and the product of aΔ\Delta-space with a countable space is aΔ\Delta-space. Our results give answers to several open problems posed by us [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99].LetT:Cp(X)⟶Cp(Y)T:C_p(X) \longrightarrow C_p(Y)be a continuous linear surjection. We observe thatTTadmits an extension to a linear continuous operatorT^\widehat {T}fromRX\mathbb {R}^XontoRY\mathbb {R}^Yand deduce thatYYis aΔ\Delta-space wheneverXXis. Similarly, assuming thatXXandYYare metrizable spaces, we show thatYYis aQQ-set wheneverXXis.Making use of obtained results, we provide a very short proof for the claim that every compactΔ\Delta-space has countable tightness. As a consequence, under Proper Forcing Axiom every compactΔ\Delta-space is sequential.In the article we pose a dozen open questions.

New Condition on Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

New Condition on Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

The only Dance in Town: Unique Equilibrium in a Generalized Model of Price Competition*

We study a model of simultaneous price competition that subsumes many employed in the literature over the last several decades. Firms sell a homogeneous good to consumers characterized by the number of prices they (exogenously) consider. We show there is a unique equilibrium if and only if some consumers consider exactly two prices. The equilibrium is in symmetric mixed strategies. When no consumer considers exactly two prices, there is, in addition, an uncountable infinity of asymmetric equilibria. Our results show that the paradigm generically produces a unique equilibrium, and only the commonly sought symmetric equilibrium is robust to perturbations in consumer behavior.

Sufficient Turing instability conditions for the Schnakenberg system

A classical reaction-diffusion system, the Schnakenberg system, is under consideration in a bounded domain $\Omega\subset\mathbb{R}^m$ with Neumann boundary conditions. We study diffusion-driven instability of a stationary spatially homogeneous solution of this system, also called the Turing instability, which arises when the diffusion coefficient $d$ changes. An analytical description of the region of necessary and sufficient conditions for the Turing instability in the parameter plane is obtained by analyzing the linearized system in diffusionless and diffusion approximations. It is shown that one of the boundaries of the region of necessary conditions is an envelope of the family of curves that bound the region of sufficient conditions. Moreover, the intersection points of two consecutive curves of this family lie on a straight line whose slope depends on the eigenvalues of the Laplace operator and does not depend on the diffusion coefficient. We find an analytical expression for the critical diffusion coefficient at which the stability of the equilibrium position of the system is lost. We derive conditions under which the set of wavenumbers corresponding to neutral stability modes is countable, finite, or empty. It is shown that the semiaxis $d>1$ can be represented as a countable union of half-intervals with split points expressed in terms of the eigenvalues of the Laplace operator; each half-interval is characterized by the minimum wavenumber of loss of stability.

NOTES ON SOME ERDŐS–HAJNAL PROBLEMS

AbstractWe make comments on some problems Erdős and Hajnal posed in their famous problem list. Let X be a graph on $\omega _1$ with the property that every uncountable set A of vertices contains a finite set s such that each element of $A-s$ is joined to one of the elements of s. Does then X contain an uncountable clique? (Problem 69) We prove that both the statement and its negation are consistent. Do there exist circuitfree graphs $\{X_n:n<\omega \}$ on $\omega _1$ such that if $A\in [\omega _1]^{\aleph _1}$ , then $\{n<\omega :X_n\cap [A]^2=\emptyset \}$ is finite? (Problem 61) We show that the answer is yes under CH, and no under Martin’s axiom. Does there exist $F:[\omega _1]^2\to 3$ with all three colors appearing in every uncountable set, and with no triangle of three colors. (Problem 68) We give a different proof of Todorcevic’ theorem that the existence of a $\kappa $ -Suslin tree gives $F:[\kappa ]^2\to \kappa $ establishing $\kappa \not \to [\kappa ]^2_{\kappa }$ with no three-colored triangles. This statement in turn implies the existence of a $\kappa $ -Aronszajn tree.

Divergent predictive states: The statistical complexity dimension of stationary, ergodic hidden Markov processes.

Even simply defined, finite-state generators produce stochastic processes that require tracking an uncountable infinity of probabilistic features for optimal prediction. For processes generated by hidden Markov chains, the consequences are dramatic. Their predictive models are generically infinite state. Until recently, one could determine neither their intrinsic randomness nor structural complexity. The prequel to this work introduced methods to accurately calculate the Shannon entropy rate (randomness) and to constructively determine their minimal (though, infinite) set of predictive features. Leveraging this, we address the complementary challenge of determining how structured hidden Markov processes are by calculating their statistical complexity dimension-the information dimension of the minimal set of predictive features. This tracks the divergence rate of the minimal memory resources required to optimally predict a broad class of truly complex processes.

Direct delta mush skinning compression with continuous examples

Direct Delta Mush (DDM) is a high-quality, direct skinning method with a low setup cost. However, its storage and run-time computing cost are relatively high for two reasons: its skinning weights are 4 X 4 matrices instead of scalars like other direct skinning methods, and its computation requires one 3 X 3 Singular Value Decomposition per vertex. In this paper, we introduce a compression method that takes a DDM model and splits it into two layers: the first layer is a smaller DDM model that computes a set of virtual bone transformations and the second layer is a Linear Blend Skinning model that computes per-vertex transformations from the output of the first layer. The two-layer model can approximate the deformation of the original DDM model with significantly lower costs. Our main contribution is a novel problem formulation for the DDM compression based on a continuous example-based technique, in which we minimize the compression error on an uncountable set of example poses. This formulation provides an elegant metric for the compression error and simplifies the problem to the common linear matrix factorization. Our formulation also takes into account the skeleton hierarchy of the model, the bind pose, and the range of motions. In addition, we propose a new update rule to optimize DDM weights of the first layer and a modification to resolve the floating-point cancellation issue of DDM.

Unit representation of semiorders II: The general case

Necessary and sufficient conditions under which semiorders on uncountable sets can be represented by a real-valued function and a constant threshold are known. We show that the proof strategy that we used for constructing representations in the case of denumerable semiorders can be adapted to the uncountable case. We use it to give an alternative proof of the existence of strict unit representations. In contrast to the countable case, semiorders on uncountable sets that admit a strict unit representation do not necessarily admit a nonstrict unit representation, and conversely. By adapting the proof strategy used for strict unit representations, we establish a characterization of the semiorders that admit a nonstrict representation. Conditions for the existence of other special unit representations are also obtained. • New constructive proof of existence of unit representation of uncountable semiorders. • Proof idea as for countable semiorders. • Gives control on the representation. • Allows to prove the existence of strict and nonstrict representations. • Allows to prove the existence of special representations.

Interface and connection model in the railway traffic control system

The article presents a model of connection of ETCS application and classical base layer equipment. The model distinguishes three layers: physical, logic and data, which require different modelling techniques and at the same time must be consistent. The model will form the basis for the digital mapping in the Digital Twin of the ETCS application. Layer division is a natural way to represent the structure of a device and its operating rules. It allows a detailed and structured representation of the interfaces of a connection and then an analysis of the connection both with respect to the layer of interest and from the point of view of the interaction between features in the different layers. The S-interface of the LEU encoder of the ETCS is described, taking into account different solutions encountered in practice. The conditions of the connection between the LEU encoder and the environment form a description of one of the two boundaries between the ETCS application, i.e. the implemented ERTMS/ETCS on a specific area of the railway network, and the environment. A general connection model and definitions of a connection and an interface are presented. As an example, the electrical connection with signals transmitted through galvanic connections has been assumed to be typical for LEU encoder and track-side signalling control circuits found in base layer equipment. The physical layer is described in terms of physical parameters and their values. The parameters are divided into electrical (current, voltage and frequency) and mechanical ones (number of leads, conductor thickness, etc.). The values of the electrical parameters are expressed in terms of a uncountable set with defined limits. The logic layer was described in a vector-matrix form. Logic signals are assigned to electrical signals with specific physical parameters. The data layer contains information about the assignment of specific telegrams to specific electrical signals.

Approximate tangents, harmonic measure, and domains with rectifiable boundaries

Let Ω ⊂ R n + 1 \Omega \subset \mathbb {R}^{n+1} , n ≥ 1 n \geq 1 , be an open and connected set. Set T n \mathcal {T}_n to be the set of points ξ ∈ ∂ Ω \xi \in \partial \Omega so that there exists an approximate tangent n n -plane for ∂ Ω \partial \Omega at ξ \xi and ∂ Ω \partial \Omega satisfies the weak lower Ahlfors-David n n -regularity condition at ξ \xi . We first show that T n \mathcal {T}_n can be covered by a countable union of boundaries of bounded Lipschitz domains. Then, letting ∂ ⋆ Ω \partial ^\star \Omega be a subset of T n \mathcal {T}_n where Ω \Omega satisfies an appropriate thickness condition, we prove that ∂ ⋆ Ω \partial ^\star \Omega can be covered by a countable union of boundaries of bounded Lipschitz domains contained in Ω \Omega . As a corollary we obtain that if Ω \Omega has locally finite perimeter, ∂ Ω \partial \Omega is weakly lower Ahlfors-David n n -regular, and the measure-theoretic boundary coincides with the topological boundary of Ω \Omega up to a set of H n \mathcal {H}^n -measure zero, then ∂ Ω \partial \Omega can be covered, up to a set of H n \mathcal {H}^n -measure zero, by a countable union of boundaries of bounded Lipschitz domains that are contained in Ω \Omega . This implies that in such domains, H n | ∂ Ω \mathcal {H}^n|_{\partial \Omega } is absolutely continuous with respect to harmonic measure.

Interactive Information Design

We study the interaction between multiple information designers who try to influence the behavior of a set of agents. When each designer can choose information policies from a compact set of statistical experiments with countable support, such games always admit subgame-perfect equilibria. When designers produce public information, every equilibrium of the simple game in which the set of messages coincides with the set of states is robust in the sense that it is an equilibrium with larger and possibly infinite and uncountable message sets. The converse is true for a class of Markovian equilibria only. When designers produce information for their own corporation of agents, robust pure-strategy equilibria exist and are characterized via an auxiliary normal-form game in which the set of strategies of each designer is the set of outcomes induced by Bayes correlated equilibria in her corporation.

A Misconception About the Hardy-Weinberg Law.

The Hardy-Weinberg law of population genetics is usually associated with the notion of random mating of parents. A numerical example for a triallelic autosomal locus shows that an uncountable set of mating combinations can maintain Hardy-Weinberg proportions. Therefore, one cannot infer random mating in a population from the observation of Hardy-Weinberg equilibrium. The mating system which ensures that the genotypic distribution of offspring is the same as that of the parents is specified.

A note on optimal degree-three spanners of the square lattice

In this short note, we prove that the degree-three dilation of the square lattice [Formula: see text] is [Formula: see text]. This disproves a conjecture of Dumitrescu and Ghosh. We give a computer-assisted proof of a local-global property for the uncountable set of geometric graphs achieving the optimal dilation.

What’s in the Bag?

The Ross-Littlewood paradox describes a process of repeatedly adding and then removing chips from a bag. During the process, the size of the bag grows at each step; but at the end of the process, the bag is mysteriously found to be empty. This paper explores some new questions about which sets of chips could remain in the bag at the end of the process as well as some stochastic question not pursued in Ross' work. The results presented in this paper were all proven by undergraduate students at the author's institution as the students learned to work with quantifiers, uncountable sets, perfect subsets of the real line, probability, recurrence relations, and measure theory for the first time.

Robust Finite-State Controllers for Uncertain POMDPs

Uncertain partially observable Markov decision processes (uPOMDPs) allow the probabilistic transition and observation functions of standard POMDPs to belong to a so-called uncertainty set. Such uncertainty, referred to as epistemic uncertainty, captures uncountable sets of probability distributions caused by, for instance, a lack of data available. We develop an algorithm to compute finite-memory policies for uPOMDPs that robustly satisfy specifications against any admissible distribution. In general, computing such policies is theoretically and practically intractable. We provide an efficient solution to this problem in four steps. (1) We state the underlying problem as a nonconvex optimization problem with infinitely many constraints. (2) A dedicated dualization scheme yields a dual problem that is still nonconvex but has finitely many constraints. (3) We linearize this dual problem and (4) solve the resulting finite linear program to obtain locally optimal solutions to the original problem. The resulting problem formulation is exponentially smaller than those resulting from existing methods. We demonstrate the applicability of our algorithm using large instances of an aircraft collision-avoidance scenario and a novel spacecraft motion planning case study.

INFINITE NUMBER OF PARAMETER REGIONS WITH FRACTAL NONCHAOTIC ATTRACTORS IN A PIECEWISE MAP

We identify a countable infinity of parameter regimes with strange nonchaotic attractors (SNAs). At the edge of each arc parameter area, there is an uncountable infinity of SNAs with torus intermittency. The mechanism for the creation of SNAs in different regime is induced by an [Formula: see text]-frequency quasiperiodic orbit through a quasiperiodic analog of saddle-node bifurcation (Type-[Formula: see text] intermittent route). We describe the transition between tori and SNAs by the largest Lyapunov exponent and phase diagram. These SNAs are characterized by the phase sensitivity exponents, rational approximations, singular-continuous spectra, and distribution of finite-time Lyapunov exponents.

Panentheism in the Light of Mathematical Understandings of Infinity and Connectedness

ABSTRACT This paper investigates some consequences of a mathematical understanding of infinity and connectedness for a panentheistic conception of God. Given the existence of God and an understanding of the world in terms of the finite, countable infinity, or uncountable infinity I argue, (a) that a panentheistic conception of God is supported given a mathematical understanding of the infinite, (b) that by applying the notion of uncountable infinity and the mathematical concept of connectedness a panentheistic God can be seen as unifying irrespective of whether the world includes a finite, countably infinite or an uncountably infinite number of entities.

Second order rectifiability of varifolds of bounded mean curvature

We prove that the support of an m dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered {mathscr {H}}^{m} almost everywhere by a countable union of m dimensional submanifolds of class {mathcal {C}}^{2} . The {mathcal {C}}^{2} -regularity of the submanifolds is optimal.

On the closure of the Hodge locus of positive period dimension

Given {{mathbb {V}}} a polarizable variation of {{mathbb {Z}}}-Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for {{mathbb {V}}}^otimes is the set of closed points s of S where the fiber {{mathbb {V}}}_s has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for {{mathbb {V}}}^otimes is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for {{mathbb {V}}}. Under the assumption that the adjoint group of the generic Mumford–Tate group of {{mathbb {V}}} is simple we prove that the union of the special subvarieties for {{mathbb {V}}} whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space {{mathcal {A}}}_g of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of {{mathcal {A}}}_g is either a closed algebraic subvariety of S or is Zariski-dense in S.

On (monotonically) metacompact subspaces of GO-spaces and related conclusions

In the second part of this article, we show that if Z is a metacompact subspace of a GO-space X and U is a family of open subsets of X such that Z ⊂ ⋃ U then there exists a point-finite family V of open subsets of X such that Z ⊂ ⋃ V and V ≺ U . Thus every subspace Y of a GO-space X is metacompact in X if and only if Y is a metacompact subspace of X . In the third part of this article, we get the following conclusions. We show that if ( X , τ , < ) is a GO-space and Z is a monotonically (countably) metacompact subspace of X , then Z is monotonically (countably) metacompact in X . By this conclusion we show that if ( X , τ , < ) is a GO-space with property ( B ) such that the maximal dense in itself set Z of X is a monotonically (countably) metacompact subspace of X , then X is monotonically (countably) metacompact. This gives a partial answer to [8, Question] . In the fourth part of this article, we show that if X is in PIGO and X is a countable unions of D -spaces, then X is a D -space, where PIGO is the class of perfect images of GO-spaces. In the last part of this article we point out that there is an error in the proof of Theorem 10 in (2018) [23] . Then we finally give a new proof for it.