Continuum Hypothesis Research Articles

Metrizability of CHART groups

For compact Hausdorff admissible right topological (CHART) group G, we prove w(G)=πχ(G). This equality is well known for compact topological groups. This implies the criteria for the metrizability of CHART groups: if G is first-countable (2013, Moors, Namioka) or G is Fréchet (2013, Glasner, Megrelishvili), or G has countable π-character (2022, Reznichenko) then G is metrizable. Under the continuum hypothesis (CH) assumption, a sequentially compact CHART group is metrizable. Namioka's theorem that metrizable CHART groups are topological groups extends to CHART groups with small weight.

Sparse analytic systems

Abstract Erdős [7] proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family $\mathcal {F}$ of (real or complex) analytic functions, such that $\big \{ f(x) \ : \ f \in \mathcal {F} \big \}$ is countable for every x. We strengthen Erdős’ result by proving that CH is equivalent to the existence of what we call sparse analytic systems of functions. We use such systems to construct, assuming CH, an equivalence relation $\sim $ on $\mathbb {R}$ such that any ‘analytic-anonymous’ attempt to predict the map $x \mapsto [x]_\sim $ must fail almost everywhere. This provides a consistently negative answer to a question of Bajpai-Velleman [2].

The Keisler–Shelah isomorphism theorem and the continuum hypothesis

We show that if for any two elementary equivalent structures $\mathbf M, \mathbf N$ of size at most continuum in a countable language, $\mathbf M^{\omega }/ \mathcal U \simeq \mathbf N^\omega / \mathcal U$ for some ultrafilter $\mathcal U$ on $\omega ,$ t

Accommodating the continuum hypothesis with the déjà vu/déjà vécu distinction.

On Barzykowski and Moulin's continuum hypothesis, déjà vu and involuntary autobiographical memories (IAMs) share their underpinning neurocognitive processes. A discontinuity issue for them is that familiarity and episodic recollection exhibit different neurocognitive signatures. This issue can be overcome, I say, provided the authors are ready to distinguish a déjà vécu/episodic IAM continuity and a déjà vu/semantic IAM continuity.

Understanding of P vs NP Problem: With Extension of the Continuum Hypothesis

Understanding of P vs NP Problem: With Extension of the Continuum Hypothesis

The Historical Logic on the Basic Theory of Physics—A Summary on the Cosmic Continuum Theory

Any scientific system has a unified basic theory. But physics has no unified basic theory in the modern sense. Classical mechanics, relativity and quantum mechanics have their own basic concepts, categories and principles, so none of them can be regarded as true basic theories of physics. Cosmic Continuum Theory holds that the continuity and discreteness of the universe are fundamental issues related to the unification of physics. Because the contradiction between quantum non-locality and local reality is the fundamental obstacle to the unification of physics, while locality and non-locality correspond to the continuity and discreteness of physical reality respectively. The cosmic continuum theory introduces mathematical continuum and axiomatic ideas to reconstruct the basic theory of physics, and by the correspondence of existence and its dimensions to achieve the unification of the essence of physical reality, by introducing the cosmic continuum hypothesis to achieve the unification of the continuity and discreteness of physical reality, by introducing axiomatic methods to achieve formal unification of the foundations on physics. From the perspective of Cosmic Continuum, classical mechanics, relativity and quantum mechanics are no longer the basic theories of physics, but three branch theories of physics that are respectively applicable to macroscopic, cosmoscopic and microcosmic systems.

Erratum to Chance and the Continuum Hypothesis

Erratum to Chance and the Continuum Hypothesis

On the additivity of strong homology for locally compact separable metric spaces

We show that it is consistent relative to a weakly compact cardinal that strong homology is additive and compactly supported within the class of locally compact separable metric spaces. This complements work of Mardešić and Prasolov [14] showing that the Continuum Hypothesis implies that a countable sum of Hawaiian earrings witnesses the failure of strong homology to possess either of these properties. Our results build directly on work of Lambie-Hanson and the second author [3] which establishes the consistency, relative to a weakly compact cardinal, of limsA = 0 for all s ≥ 1 for a certain pro-abelian group A; we show that that work’s arguments carry implications for the vanishing and additivity of the lims functors over a substantially more general class of pro-abelian groups indexed by ℕℕ.

On the commutant of B(H) in its ultrapower

Let $${\cal B}\left( H \right)$$ be the algebra of bounded linear operators on a separable infinite-dimensional Hilbert space H. In 2004 Kirchberg asked whether the relative commutant of $${\cal B}\left( H \right)$$ in its ultrapower is trivial. In [13] the authors have shown that under the Continuum Hypothesis the commutant of $${\cal B}\left( H \right)$$ in its ultrapower depends on the choice of the ultrafilter. We here give a combinatorial characterization of the class of non-principal ultrafilters for which this commutant is non-trivial, answering Question 5.2 of [13]. This reduces Kirchberg’s question to a purely set-theoretic question: Can the existence of non-flat ultrafilters be proven in ZFC? In addition, we introduce the notion of quasi P-points and show that for such ultrafilters, and for ultrafilters satisfying the three functions property, the relative commutant of $${\cal B}\left( H \right)$$ in its ultrapower is trivial.

Rasterize the Lidar Point Cloud on The Ground Out Method Optimization Analysis

This paper is mainly aimed at the over-segmentation or under-segmentation phenomenon of the ground point cloud in the moving scene of the unmanned platform to identify the target. This paper proposes an optimized rasterization method to remove the ground point clouds appropriately. First, the partitioned data by quadrant is processed based on the asymmetric distribution of front-to-back and left-to-right point cloud data. Then, the ground estimation of the lowest point is carried out by drilling into the laser layer data, and the connection between the estimated points and the ground detection point cloud data is segmented using the road reflection intensity correction. Next, the ground point cloud detection filtering is optimized using the raster elevation difference. Finally, an obstacle continuum hypothesis model is used to improve the under-segmentation phenomenon that occurs inside the raster. The overall ground point cloud detection filtering effect, the algorithm has a certain degree of universality and ideal ground detection filtering effect within the road, the overall achieve the desired goal.

Resolving the Singularity by Looking at the Dot and Demonstrating the Undecidability of the Continuum Hypothesis

Einsteinian gravity, of which Newtonian gravity is a part, is fraught with the problem of singularity that has been established as a theorem by Hawking and Penrose. The hypothesis that founds the basis of both Einsteinian and Newtonian theories of gravity is that bodies with unequal magnitudes of masses fall with the same acceleration under the gravity of a source object. Since, the Einstein’s equations is one of the assumptions that underlies the proof of the singularity theorem, therefore, the above hypothesis is implicitly one of the founding pillars of the same. In this work, I demonstrate how one can possibly write a non-singular theory of gravity which manifests that the above mentioned hypothesis is only valid in an approximate sense in the “large distance” scenario. To mention a specific instance, under the gravity of the earth, a 5 kg and a 500 kg fall with accelerations which differ by approximately \(113.148\times 10^{-32}\) meter/sec\(^2\) and the more massive object falls with less acceleration. Further, I demonstrate why the concept of gravitational field is not definable in the “small distance” regime which automatically justifies why the Einstein’s and Newton’s theories fail to provide any “small distance” analysis. In course of writing down this theory, I demonstrate why the continuum hypothesis as spelled out by Goedel, is undecidable. The theory has several aspects which provide the following realizations: (i) Descartes’ self-skepticism concerning exact representation of numbers by drawing lines (ii) Born’s wish of taking into account “natural uncertainty in all observations” while describing “a physical situation” by means of “real numbers” (iii) Klein’s vision of having “a fusion of arithmetic and geometry” where “a point is replaced by a small spot” (iv) Goedel’s assertion about “non-standard analysis, in some version” being “the analysis of the future”.

Between reduced powers and ultrapowers

One of our results is a transfer principle between ultrapowers and reduced powers associated with the Fréchet ideal. Although motivated by the Elliott classification programme, this result applies to any axiomatizable category. We also show that there exists a nonprincipal ultrafilter \mathcal{U} on \mathbb{N} such that for every countable (or separable metric) structure B in a countable language the quotient map from the reduced power associated with the Fréchet ideal onto an ultrapower has a right inverse. While the transfer principle is proved without appealing to additional set-theoretic axioms, the conclusion of the latter theorem relies on the Continuum Hypothesis and it is independent of the standard axioms of set theory. We also prove that in the category of C^* -algebras, tensoring with the C^* -algebra of all continuous functions on the Cantor space preserves elementary equivalence. As a side note, neither the Jiang–Su algebra \mathcal{Z} nor any UHF algebra share this property.

Investigation of feed flow effect using CFD-DSMC method in a gas centrifuge

In this study, the effect of rarefied region in the rotor of the centrifuge was investigated. In a rotor of the centrifuge, the feed inlet is positioned in the rarefied area. The continuum hypothesis is not valid in the rarefied area; therefore, it desires to be analyzed by probabilistic methods like Direct Simulation Monte Carlo (DSMC). In the present study, Computational Fluid Dynamic (CFD) method is used to simulate the continuum area, and the DSMC method is employed in the rarefied area. An implicit coupled density-based scheme was performed for CFD method, and Variable Hard Sphere (VHS) and diffuse model were employed in the DSMC method. Also, the local Knudsen number was defined to determine the interface location between the continuum-rarefied regions (r=0.086 m). The comparison results of pure CFD and CFD-DSMC methods illustrated large differences between the flow properties in the rarefied regions. The results showed that the value of separation power obtained from pure CFD and CFD-DSMC solution is 10.5% different.

Orderings of ultrafilters on Boolean algebras

We study two generalizations of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras. To highlight the difference between them, we develop new techniques to construct incomparable ultrafilters in this setting. Furthermore, we discuss the relation with Tukey reducibility and prove that, assuming the Continuum Hypothesis, there exist ultrafilters on the Cohen algebra which are RK-equivalent in the generalized sense but Tukey-incomparable, in stark contrast with the classical setting.

A study by the lattice discrete element method for exploring the fractal nature of scale effects

Nowadays, there are many applications in the field of Engineering related to quasi-brittle materials such as ceramics, natural stones, and concrete, among others. When damage is produced, two phenomena can take place: the damage produced governs the collapse process when working with this type of material, and its random nature rules the nonlinear behavior up to the collapse. The interaction among clouds of micro-cracks generates the localization process that implies transforming a continuum domain into a discontinue one. This process also governs the size effect, that is, the changes of the global parameters as the strength and characteristic strain and energies when the size of the structure changes. Some aspects of the scaling law based on the fractal concepts proposed by Prof Carpinteri are analyzed in this work. On the other hand, the Discrete Method is an interesting option to be used in the simulation collapse process of quasi-brittle materials. This method can allow failures with relative ease. Moreover, it can also help to relax the continuum hypothesis. In the present work, a version of the Discrete Element Method is used to simulate the mechanical behavior of different size specimens until collapse by analyzing the size effect represented by this method. This work presents two sets of examples. Its results allow the researchers to see the connection between the numerical results regarding the size effect and the theoretical law based on the fractal dimension of the parameter studied. Two main aspects appear as a result of the analysis presented here. Understand better some aspects of the size effect using the numerical tool and show that the Lattice Discrete Element Method has enough robustness to be applied in the nonlinear analysis of structures built by quasi-brittle materials.

Between reduced powers and ultrapowers, II.

We prove that, consistently with ZFC, no ultraproduct of countably infinite (or separable metric, non-compact) structures is isomorphic to a reduced product of countable (or separable metric) structures associated to the Fréchet filter. Since such structures are countably saturated, the Continuum Hypothesis implies that they are isomorphic when elementarily equivalent.

The finer selection properties

For a function f from ω to ω, a topological space X satisfies ⋃f(Γ,Γ) if for each sequence (Un:n∈ω,Unhas no finite subcover) of elements of Γ, select for each n a finite subset Fn⊆Un such that |Fn|≤f(n) for all n and {⋃Fn:n∈ω} is an element of Γ, where Γ denotes the family of all open γ-covers of X. In this paper, we prove the following results.(1)Assume the Continuum Hypothesis. There is a set of real numbers that satisfies ⋃fin(Γ,Γ) and S1(Γ,O) but not ⋃id(Γ,Γ), where id is the identity function from ω to ω.(2)Assume b=c. There is a set of real numbers that satisfies ⋃id(Γ,Γ) but not ⋃k(Γ,Γ) for all natural numbers k≥1.(3)Assume the Continuum Hypothesis. For each natural number k≥2, there is a set of real numbers that satisfies ⋃k+1(Γ,Γ) but not ⋃k(Γ,Γ). These results answer an open problem proposed by Zdomskyy and a conjecture proposed by Tsaban.

The Price of Mathematical Scepticism

ABSTRACT This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions. Underlying this argument is the following philosophical view. Mathematical belief springs from certain intuitions, each of which can be either accepted or doubted in its entirety, but not half-accepted. Therefore, our beliefs about reality, bivalence, choice and consistency should all be aligned.

Determination of pressure resistance of a partially stoppered vial by using a coupled CFD-0D model of lyophilization

Modeling lyophilization in a vial is frequently done on a single vial level. When setting up a numerical model, the main focus is on heat and mass transfer inside the lyophilizate, whereas the vapor dynamics in the headspace of the vial is taken into account simply through imposing the system pressure as a pressure boundary condition. The present paper offers a deeper insight into the interaction of the sublimated vapor flow and the corresponding vapor pressure conditions inside the headspace of a partially stoppered vial. This is achieved through a coupled numerical solution of the heat and mass transfer inside the product by means of a 0D model describing the frozen domain (ice) and the 3D fluid flow inside the vial geometry with the partially opened stopper, computed by means of Computational Fluid Dynamics. Due to low pressures, the slip flow regime within the continuum hypothesis has to be considered, leading to imposing velocity slip conditions at the solid walls. The 0D model is used for the computation of sublimation mass flow rate as well as heat transfer rate to the vial, with the results of the water vapor mass flow rate and the temperature communicated to the 3D CFD model as a new inlet boundary conditions for computation of compressible fluid flow dynamics inside the vial. The obtained CFD pressure field solution allows derivation of a pressure resistance model for a targeted vial stopper combination, which is then used in calculating the corresponding pressure drop in the headspace of the partially stoppered vial. The coupled CFD-0D model results are validated based on the results of dedicated experimental water runs on several vial and stopper geometries and show, that the vial geometry, but especially the installed stopper, alter the pressure field conditions inside the vial. The increased in-vial local vapor pressure values lead to a decrease of the mass flow rates and an increase of temperatures at the bottom of the product, which range from 0.6 K for the highest system pressure and up to 5.4 K for the lowest system pressure tested. The presented coupled model is suitable for the use in further studies of the impact of various vial forms as well as stoppers on the lyophilization dynamics in a vial.

Universal autohomeomorphisms of ℕ*

We study the existence of universal autohomeomorphisms of N ∗ \mathbb {N}^* . We prove that the Continuum Hypothesis ( C H \mathsf {CH} ) implies there is such an autohomeomorphism and show that there are none in any model where all autohomeomorphisms of N ∗ \mathbb {N}^* are trivial.