Related papers: Answering Hilbert's 1st Problem
A choice of first-order variables for the characteristic problem of the linearized Einstein equations is found which casts the system into manifestly well-posed form. The concept of well-posedness for characteristic problems invoked is that…
We consider initial value problems for differential-algebraic equations in a possibly infinite-dimensional Hilbert space. Assuming a growth condition for the associated operator pencil, we prove existence and uniqueness of solutions for…
The necessity of complex numbers in quantum mechanics has long been debated. This paper develops a real Kahler space formulation of quantum mechanics [19], asserting equivalence to the standard complex Hilbert space framework. By mapping…
In this paper, we consider natural Hilbert-space representations $\left\{ \left(\mathbb{C}^{2},\pi_{t}\right)\right\} _{t\in\mathbb{R}}$ of the hypercomplex system $\left\{ \mathbb{H}_{t}\right\} _{t\in\mathbb{R}}$, and study the…
This note describes a representation of the real numbers due to Schanuel. The representation lets us construct the real numbers from first principles. Like the well-known construction of the real numbers using Dedekind cuts, the idea is…
Considered will be properties of the set of real numbers $\Re$ generated by an operator that has form of an exponential function of Gelfond-Schneider type with rational arguments. It will be shown that such created set has cardinal number…
We formulate a division problem for a class of overdetermined systems introduced by L. H{\"o}rmander, and establish an effective divisibility criterion. In addition, we prove a coherence theorem which extends Nadel's coherence theorem from…
Our starting point is Mumford's conjecture, on representations of Chevalley groups over fields, as it is phrased in the preface of "Geometric Invariant Theory". After extending the conjecture appropriately, we show that it holds over an…
This is a work in two parts devoted to solutions of the so-called {\em four Landau's problems} in Number Theory, listed by Edmund Landau at the 1912 International Congress of Mathematics. In Part I the {\em Goldbach's conjecture} is proved.…
A natural number N is said to be palindromic if its binary representation reads the same forwards and backwards. In this paper we study the quotients of two palindromic numbers and answer some basic questions about the resulting sets of…
In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…
The main purpose of this paper is to show that there exists a positive number $\lambda_{1}$, the first eigenvalue, such that some $p(x)$-Laplace equation admits a solution if $\lambda=\lambda_{1}$ and that $\lambda_{1}$ is simple, i.e.,…
Remarks on the Cantor's nondenumerability proof of 1891 that the real numbers are noncountable will be given. By the Cantor's diagonal procedure, it is not possible to build numbers that are different from all numbers in a general assumed…
Proposed in 1937, the Collatz conjecture has remained in the spotlight for mathematicians and computer scientists alike due to its simple proposal, yet intractable proof. In this paper, we propose several novel theorems, corollaries, and…
Discussions surrounding the nature of the infinite in mathematics have been underway for two millennia. Mathematicians, philosophers, and theologians have all taken part. The basic question has been whether the infinite exists only in…
This article critically reappraises arguments in support of Cantor's theory of transfinite numbers. The following results are reported: i) Cantor's proofs of nondenumerability are refuted by analyzing the logical inconsistencies in…
We show: 1) The existence of the first twisted Hilbert space that is not isomorphic to its dual; this solves a problem posed by Cabello in [Nonlinear centralizers in homology, Math. Ann. 358 (2014), no. 3-4, 779-798]. 2) The existence of a…
Given a single (differential-algebraic) input-output equation, we present a method for finding different representations of the associated system in the form of rational realizations; these are dynamical systems with rational right-hand…
We show how imaginary numbers in quantum physics can be eliminated by enlarging the Hilbert Space followed by an imposition of - what effectively amounts to - a superselection rule. We illustrate this procedure with a qubit and apply it to…
The Hilbert program was actually a specific approach for proving consistency. Quantifiers were supposed to be replaced by $\epsilon$-terms. $\epsilon{x}A(x)$ was supposed to denote a witness to $\exists{x}A(x)$, arbitrary if there is none.…