Related papers: Point-Dimension Theory (Part I): The Point-Extende…
We associate with any finite subset of a metric space an infinite sequence of scale invariant numbers $\rho_1,\rho_2,\dots$ derived from a variant of differential entropy called the genial entropy. As statistics for point processes, these…
In this paper, we answer a question of Nussbaum, Priyadarshi, and Lunel [Positive operators and Hausdorff dimension of invariant sets, Trans. Amer. Math. Soc. 364(2) (2012) 1029-1066.]. We also show that the Hausdorff dimension and box…
Rovelli's relational interpretation of quantum mechanics tells us that the description of a system in the formalism of quantum mechanics is not an absolute, but it is relative to the observer itself. The interpretation goes further and…
In the current paper we present a new proof of the small ball inequality in two dimensions. More importantly, this new argument, based on an approach inspired by lacunary Fourier series, reveals the first formal connection between this…
We explore the properties of physical theories in space-times with two time dimensions. We show that the common arguments used to rule such theories out do not apply if the dynamics associated with the additional time dimension is thermal…
In this paper, we are going to put in a single consistent framework apparently unrelated pieces of information, i.e. zero-point length, extra-dimensions, string T-duality. More in details we are going to introduce a modified Kaluza-Klein…
In the field theories in physics, any particular region of the presumed space-time continuum and all interactions between elementary objects therein can be objectively measured and/or accounted for mathematically. Since this does not apply…
The statement of the Gauss-Bonnet theorem brings up an unexpected form of reflexivity (major concept of philosophy of mathematics), so that geometry contemplates itself in it. It is therefore the revolutionary and multifaceted concept of…
We pedagogically present the information theory as originally established, explaining its essential ideas and paying attention to the expression employed to measure the amount of information. Also we discussed relationships between…
If one takes seriously the postulate of quantum mechanics in which physical states are rays in the standard Hilbert space of the theory, one is naturally lead to a geometric formulation of the theory. Within this formulation of quantum…
In this Ph.D. thesis several topics in doubly special relativity are explored. The starting point of this theory is very different from other perspectives: it is not a fundamental theory, but it is considered a low energy limit of a quantum…
In this article, we present new comments to the article On Kant's First Insight Into The Problem of Space Dimensionality and Its Physical Foundations. In particular, we discuss how the space concept is designed in the first writing of Kant.…
We propose a heuristic model of the universe as a growing quasicrystal projected from a higher-dimensional lattice. This quasicrystalline framework offers a novel perspective on cosmic expansion, where the intrinsic growth dynamics…
As indicated by Einstein's general relativity, matter and geometry are two faces of a single nature. In our point of view, extra dimensions, as a member of the {\em geometry face}, will be treated as a part of the {\em matter face} when…
It is argued that quantum theory is best understood as requiring an ontological duality of res extensa and res potentia, where the latter is understood per Heisenberg's original proposal, and the former is roughly equivalent to Descartes'…
Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct…
The idea that possible configurations of a physical system can be represented as points in a multidimensional configuration space ${\cal C}$ is explored. The notion of spacetime, without ${\cal C}$, does not exist in this theory. Spacetime…
Physical dimensions are {\it not numbers}, but used as {\it numbers} to perform dimensional analysis by the physicist. The law of excluded middle falls short of explaining the contradictory meanings of the same symbols. The statements like…
We provide exposition into the field of projection theory, which lies at the intersection of incidence geometry and geometric measure theory. We first give the necessary preliminaries in Chapter 2, focusing on incidences between points and…
The starting point of this work is the principle that all movement of particles and photons in the observable Universe must follow geodesics of a 4-dimensional space where time intervals are always a measure of geodesic arc lengths, i.e.…