Geometry, Coordinatization and Cardinality of the Rational Numbers from Physical Perspective

In this article, we will first discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric continuum. According to this definition, points are not regarded as the basic constituents of a line segment and a line segment is considered to be a fundamental geometric object. This definition is in particular suitable to coordinatize different points on the straight line preserving the order properties of real numbers. Geometrically fundamental nature of line segments are required in physical theories like the string theory. We will construct a new topology suitable for this alternate definition of the straight line as a geometric continuum. We will discuss the cardinality of rational numbers in the later half of the article. We will first discuss what we do in an actual process of counting and define functions well-defined on the set of all positive integers. We will follow an alternate approach that depends on the Hausdorff topology of real numbers to demonstrate that the set of positive rationals can have a greater cardinality than the set of positive integers. This approach is more consistent with an actual act of counting. We will illustrate this aspect further using well-behaved functionals of convergent functions defined on the finite dimensional Cartezian products of the set of positive integers and non-negative integers. These are similar to the partition functions in statistical physics. This article indicates that the axiom of choice can be a better technique to prove theorems that use second-countability. This is important for the metrization theorems and physics of spacetime.

The F- Topology on Space of Zero Dimensional rings

Let R be a subring of a ring T, and let F be a non-principal ultrafilter on the natural numbers IN. We consider properties and applications of a countably compact, Hausdorff topology called the "F-topology" defined on space of all zero-dimensional subring of T that contains a fixed subring R. We show that the F-topology is strictly finer than the Zariski topology. We extend results regarding distinguished spectral topologies on the space of zero-dimensional subring.

Metric topology on the moduli space

<p>We define the smooth Lipschitz topology on the moduli space and show that each conformal class is dense in the moduli space endowed with Gromov-Hausdorff topology, which offers an answer to Tuschmann’s question.</p>

COVID-19, bitcoin market efficiency, herd behaviour

PurposeUnlike previous crisis where investors tend to put their assets in safe havens like gold, the recent coronavirus pandemic is characterised by an increase in the Bitcoin purchasing described as risk heaven. This paper aims to analyse the Bitcoin dynamics and the investor response by focusing on herd biases. Therefore, the main objective of this work is to study the degree of efficiency through multifractal analysis in order to detect herd behaviour leading to build the best predictions and strategies.Design/methodology/approachThis paper develops a novel methodology that detects the presence of herding biases and assesses the inefficiency of Bitcoin through an inefficiency index (MLM) by using statistical indicators defined by measures of persistence. This study, also, investigates the nonlinear dynamical properties of Bitcoin by estimating the Multifractal Detrended Fluctuation Analysis (MFDFA) leading to deduce the effect of COVID-19 on the Bitcoin performance. Besides, this work performs an event study to capture abnormal changes created by COVID-19 related events capable to analyse the Bitcoin market response.FindingsThe empirical results of the generalized Hurst exponent GHE estimation indicates that Bitcoin is multifractal before this pandemic and becomes less fractal after the outbreak. Using an efficiency index (MLM), Bitcoin is found to be more efficient after the pandemic. Based on the Hausdorff topology, the authors showed that this pandemic has reduced the herd bias.Research limitations/implicationsThe uncertainty of COVID-19 disease and the lasting of its duration make it difficult to make the best prediction.Practical implicationsThe main contribution of this study is the evaluation of the Bitcoin value after the COVID19 outbreak. This work has practical implications as it provides new insights on trading opportunities and social reactions.Originality/valueTo the authors’ knowledge, this work represents the first study that analyses the Bitcoin response to different events related to COVID-19 and detects the presence of herding behaviour in such a crisis.

A metrizable semitopological semilattice with non-closed partial order

We construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X : xy = x} is a non-closed dense subset of X × X. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergent sequences.

On the lattice of weak topologies on the bicyclic monoid with adjoined zero

A Hausdorff topology τ on the bicyclic monoid with adjoined zero C0 is called weak if it is contained in the coarsest inverse semigroup topology on C0. We show that the lattice W of all weak shift-continuous topologies on C0 is isomorphic to the lattice SIF1×SIF1 where SIF1 is the set of all shift-invariant filters on ω with an attached element 1 endowed with the following partial order: F≤G if and only if G=1 or F⊂G. Also, we investigate cardinal characteristics of the lattice W. In particular, we prove that W contains an antichain of cardinality 2c and a well-ordered chain of cardinality c. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type t.

A note on compact-like semitopological groups

We present a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is provided a semiregular semitopological group $G$ which is not $T_3$. We show that each weakly semiregular compact semitopological group is a topological group. On the other hand, constructed examples of quasiregular $T_1$ compact and $T_2$ sequentially compact quasitopological groups, which are not paratopological groups. Also we prove that a semitopological group $(G,\tau)$ is a topological group provided there exists a Hausdorff topology $\sigma\supset\tau$ on $G$ such that $(G,\sigma)$ is a precompact topological group and $(G,\tau)$ is weakly semiregular or $(G,\sigma)$ is a feebly compact paratopological group and $(G,\tau)$ is $T_3$.

The structure of spaces with Bakry–Émery Ricci curvature bounded below

We explore the structure of limit spaces of sequences of Riemannian manifolds with Bakry–Émery Ricci curvature bounded below in the Gromov–Hausdorff topology. By extending the techniques established by Cheeger and Cloding for Riemannian manifolds with Ricci curvature bounded below, we prove that each tangent space at a point of the limit space is a metric cone. We also analyze the singular structure of the limit space as in a paper of Cheeger, Colding and Tian. Our results will be applied to study the limit spaces for a sequence of Kähler metrics arising from solutions of certain complex Monge–Ampère equations for the existence problem of Kähler–Ricci solitons on a Fano manifold via the continuity method.

Convergence of Riemannian 4-manifolds with L 2 L^{2} -curvature bounds

In this work we prove convergence results of sequences of Riemannian 4-manifolds with almost vanishing {L^{2}} -norm of a curvature tensor and a non-collapsing bound on the volume of small balls. In Theorem 1.1 we consider a sequence of closed Riemannian 4-manifolds, whose {L^{2}} -norm of the Riemannian curvature tensor tends to zero. Under the assumption of a uniform non-collapsing bound and a uniform diameter bound, we prove that there exists a subsequence that converges with respect to the Gromov–Hausdorff topology to a flat manifold. In Theorem 1.2 we consider a sequence of closed Riemannian 4-manifolds, whose {L^{2}} -norm of the Riemannian curvature tensor is uniformly bounded from above, and whose {L^{2}} -norm of the traceless Ricci-tensor tends to zero. Here, under the assumption of a uniform non-collapsing bound, which is very close to the Euclidean situation, and a uniform diameter bound, we show that there exists a subsequence which converges in the Gromov–Hausdorff sense to an Einstein manifold. In order to prove Theorem 1.1 and Theorem 1.2, we use a smoothing technique, which is called {L^{2}} -curvature flow. This method was introduced by Jeffrey Streets. In particular, we use his “tubular averaging technique” in order to prove distance estimates of the {L^{2}} -curvature flow, which only depend on significant geometric bounds. This is the content of Theorem 1.3.

On dimensions of tangent cones in limit spaces with lower Ricci curvature bounds

We show that if X is a limit of n-dimensional Riemannian manifolds with Ricci curvature bounded below and γ is a limit geodesic in X, then along the interior of γ same scale measure metric tangent cones {T_{\gamma(t)}X} are Hölder continuous with respect to measured Gromov–Hausdorff topology and have the same dimension in the sense of Colding–Naber.