Related papers: Looking through newly to the amazing irrationals
A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a…
This work is a continuation of [1]. As in the previous article, here we will describe some interesting ideas and a lot of new theorems in plane geometry related to them.
We review the current status of dimensions, as the result of a long and controversial history that includes input from philosophy and physics. Our conclusion is that they are subjective but essential concepts which provide a kind of…
This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent r{\^o}le. This selection is largely…
We prove many new results about interacting Fock spaces. We pose many open problems; for most of them we prove that their solutions have no choice but being nontrivial. We ask the kind reader to consult the extended abstract in the paper.
Relativistic length contraction is revisited and a simple but new thought experiment is proposed in which an apparent asymmetric situation is developed between two different inertial frames regarding detection of light that comes from a…
While describing the results of our recent work on exceptional Lie and Jordan algebras, so tightly intertwined in their connection with elementary particles, we will try to stimulate a critical discussion on the nature of spacetime and…
Rationals are known to form interesting and computationally rich structures, such as Farey sequences and infinite trees. Little attention is being paid to more general, systematic exposition of the basic properties of fractions as a set.…
We prove the irrationality of some factorial series. To do so we combine methods from elementary and analytic number theory with methods from the theory of uniform distribution.
Conventional thinking says the universe is infinite. But it could be finite and relatively small, merely giving the illusion of a greater one, like a hall of mirrors. Recent astronomical measurements add support to a finite space with a…
We exhibit a family of metrizable manifolds such that any finite group appears as the fundamental group of one of them. These spaces are especially interesting as they can be easily visualized, as opposed to classical examples of spaces…
Resultants are getting increasingly important in modern theoretical physics: they appear whenever one deals with non-linear (polynomial) equations, with non-quadratic forms or with non-Gaussian integrals. Being a subject of more than…
The basic formalism of a novel scale invarinat nonlinear analysis is presented. A few analytic number theoretic results are derived independent of standard approaches.
There are versions of "calculus" in many settings, with various mixtures of algebra and analysis. In these informal notes we consider a few examples that suggest a lot of interesting questions.
We establish new and different kinds of proofs of properties that arise due to the orthogonal decomposition of the Hilbert space, including projections, over the unit interval of one dimension. We also see angles between functions,…
We obtain several new characterizations of ultrametric spaces in terms of roundness, generalized roundness, strict p-negative type, and p-polygonal equalities (p > 0). This allows new insight into the isometric embedding of ultrametric…
Ivan Niven's succinct proof that pi is irrational is easy to verify, but it begins with a magical formula that appears to come out of nowhere, and whose origin remains mysterious even after one goes through the proof. The goal of this…
This paper is concerned with analysis on metric spaces in a variety of settings and with several kinds of structure.
The uncertainty associated with probing the quantum state is expressed as the effective abundance (measure) of possibilities for its collapse. New kinds of uncertainty limits entailed by quantum description of the physical system arise in…
Direct techniques for cosmic ray observations have reached an unprecedented level of precision, unveiling fine-details of the energy spectra. I will introduce the evidence for new spectral features which has been accumulated by new…