An Analysis of Zeno’s Paradox and Non-measurable Sets Based on Dialectical Infinity

The Problem of Continuity and Discreteness is the basic problem of philosophy and mathematics. For a long time, there is no clear understanding of this problem, which leads to the stagnation of the problem, because the essence of the problem is a problem of finity and infinity. The essence of the philosophical thought on which the mathematical definition of “line segment is composed of dots” is the idea of actual infinity, and geometric dot is equivalent to algebraic zero in terms of measure properties. In view of the above contradictions, this paper presents two solutions satisfying both the philosophical and mathematical circles based on the view of dialectical infinity, and the authors make a deep analysis of Zeno’s paradox and the non-measurable set based on both solutions.

Around Taylor’s Theorem on the Convergence of Sequences of Functions

Egoroff’s classical theorem shows that from a pointwise convergence we can get a uniform convergence outside the set of an arbitrary small measure. Taylor’s theorem shows the possibility of controlling the convergence of the sequences of functions on the set of the full measure. Namely, for every sequence of real-valued measurable factions |fn } n∈ℕ pointwise converging to a function f on a measurable set E, there exist a decreasing sequence |δn } n∈ℕ of positive reals converging to 0 and a set A ⊆ E such that E \ A is a nullset and lim n → + ∞ | f n ( x ) − f ( x ) | δ n = 0 for all x ∈ A . Let J ( A , { f n } ) {\lim _{n \to + \infty }}\frac{{|{f_n}(x) - f(x)|}}{{{\delta _n}}} = 0\,{\rm{for}}\,{\rm{all}}\,x \in A.\,{\rm{Let}}\,J(A,\,\{ {f_n}\} ) denote the set of all such sequences |δn } n∈ℕ. The main results of the paper concern basic properties of sets of all such sequences for a given set A and a given sequence of functions. A relationship between pointwise convergence, uniform convergence and the Taylor’s type of convergence is considered.

Lorentz and Gale–Ryser theorems on general measure spaces

Based on the Gale–Ryser theorem [2, 6], for the existence of suitable $(0,1)$ -matrices for different partitions of a natural number, we revisit the classical result of Lorentz [4] regarding the characterization of a plane measurable set, in terms of its cross-sections, and extend it to general measure spaces.

Density Estimates of 1-Avoiding Sets via Higher Order Correlations

We improve the best known upper bound on the density of a planar measurable set A containing no two points at unit distance to 0.25442. We use a combination of Fourier analytic and linear programming methods to obtain the result. The estimate is achieved by means of obtaining new linear constraints on the autocorrelation function of A utilizing triple-order correlations in A, a concept that has not been previously studied.

Non-Parametric Probability Distributions Embedded Inside of a Linear Space Provided with a Quadratic Metric

There exist uncertain situations in which a random event is not a measurable set, but it is a point of a linear space inside of which it is possible to study different random quantities characterized by non-parametric probability distributions. We show that if an event is not a measurable set then it is contained in a closed structure which is not a σ-algebra but a linear space over R. We think of probability as being a mass. It is really a mass with respect to problems of statistical sampling. It is a mass with respect to problems of social sciences. In particular, it is a mass with regard to economic situations studied by means of the subjective notion of utility. We are able to decompose a random quantity meant as a geometric entity inside of a metric space. It is also possible to decompose its prevision and variance inside of it. We show a quadratic metric in order to obtain the variance of a random quantity. The origin of the notion of variability is not standardized within this context. It always depends on the state of information and knowledge of an individual. We study different intrinsic properties of non-parametric probability distributions as well as of probabilistic indices summarizing them. We define the notion of α-distance between two non-parametric probability distributions.

Lipschitz one sets modulo sets of measure zero

We denote the local “little” and “big” Lipschitz functions of a function f : ℝ → ℝ by lip f and Lip f. In this paper we continue our research concerning the following question. Given a set E ⊂ ℝ is it possible to find a continuous function f such that lip f = 1E or Lip f = 1E?In giving some partial answers to this question uniform density type (UDT) and strong uniform density type (SUDT) sets play an important role.In this paper we show that modulo sets of zero Lebesgue measure any measurable set coincides with a Lip 1 set.On the other hand, we prove that there exists a measurable SUDT set E such that for any Gδ set E͠ satisfying ∣EΔE͠∣ = 0 the set E͠ does not have UDT. Combining these two results we obtain that there exist Lip 1 sets not having UDT, that is, the converse of one of our earlier results does not hold.

On sets of points of approximate continuity and ϱ-upper continuity

In the paper some properties of sets of points of approximate continuity and ϱ-upper continuity are presented. We will show that for every Lebesgue measurable set E ⊂ ℝ there exists a function f : ℝ → ℝ which is approximately (ϱ-upper) continuous exactly at points from E. We also study properties of sets of points at which real function has Denjoy property. Some other related topics are discussed.

Reconstruction of the One-Dimensional Lebesgue Measure

SummaryIn the Mizar system ([1], [2]), Józef Białas has already given the one-dimensional Lebesgue measure [4]. However, the measure introduced by Białas limited the outer measure to a field with finite additivity. So, although it satisfies the nature of the measure, it cannot specify the length of measurable sets and also it cannot determine what kind of set is a measurable set. From the above, the authors first determined the length of the interval by the outer measure. Specifically, we used the compactness of the real space. Next, we constructed the pre-measure by limiting the outer measure to a semialgebra of intervals. Furthermore, by repeating the extension of the previous measure, we reconstructed the one-dimensional Lebesgue measure [7], [3].

Torsion Discriminance for Stability of Linear Time-Invariant Systems

This paper extends the former approaches to describe the stability of n-dimensional linear time-invariant systems via the torsion τ ( t ) of the state trajectory. For a system r ˙ ( t ) = A r ( t ) where A is invertible, we show that (1) if there exists a measurable set E 1 with positive Lebesgue measure, such that r ( 0 ) ∈ E 1 implies that lim t → + ∞ τ ( t ) ≠ 0 or lim t → + ∞ τ ( t ) does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set E 2 with positive Lebesgue measure, such that r ( 0 ) ∈ E 2 implies that lim t → + ∞ τ ( t ) = + ∞ , then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the ith curvature ( i = 1 , 2 , ⋯ ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix.

Intersections of Projections and Slicing Theorems for the Isotropic Grassmannian and the Heisenberg group

This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of ℝ2n, as well as dimension of intersections of sets with isotropic planes. It is shown that if A and B are Borel subsets of ℝ2n of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of A and B under orthogonal projections onto these planes have positive Hausdorff m-measure. In addition, if A is a measurable set of Hausdorff dimension greater than m, then there is a set B ⊂ ℝ2n with dim B ⩽ m such that for all x ∈ ℝ2n\B there is a positive measure set of isotropic m-planes for which the translate by x of the orthogonal complement of each such plane, intersects A on a set of dimension dim A – m. These results are then applied to obtain analogous results on the nth Heisenberg group.