Related papers: Geometric visualizations of $b^{e}<e^{b}$ when $e<…
For any three nonzero vectors $a,b,c$ in $\mathbb R^2$, we obtain a necessary and sufficient condition for the sum of the three pairwise angles between these vectors to equal $2\pi$. As an easy consequence of this, a proof of Euclid's…
In this expository paper, we discuss a unified framework for proving various geometric inequalities, based on the so-called Alexandrov-Bakelman-Pucci technique. Examples include Cabr\'e's proof of the classical isoperimetric inequality in…
Equivalencies of many basic elementary inequalities are given
We present a complete system of inequalities for the inradius, circumradius, and diameter in the $3$-dimensional Euclidean space. To do so, we prove quasiconcavity of the inradius evaluated over $n$-simplices with a common facet…
Visual representation learning has been a cornerstone in computer vision, involving typical forms such as visual embeddings, structural symbols, and text-based representations. Despite the success of CLIP-type visual embeddings, they often…
This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of…
In quantum geometry, we consider a set of loops, a compact orientable surface and a solid compact spatial region, all inside $\mathbb{R} \times \mathbb{R}^3 \equiv \mathbb{R}^4$, which forms a triple. We want to define an ambient isotopic…
We improve the constant $\frac{\pi}{2}$ in $L^1$-Poincar\'e inequality on Hamming cube. For Gaussian space the sharp constant in $L^1$ inequality is known, and it is $\sqrt{\frac{\pi}{2}}$. For Hamming cube the sharp constant is not known,…
We show one can use classical fields to modify a quantum optics experiment so that Bell's inequalities will be violated. This happens with continuous random variables that are local, but we need to use the correlation matrix to prove there…
We present encube $-$ a qualitative, quantitative and comparative visualisation and analysis system, with application to high-resolution, immersive three-dimensional environments and desktop displays. encube extends previous comparative…
The predictions of quantum mechanics cannot be resolved with a completely classical view of the world. In particular, the statistics of space-like separated measurements on entangled quantum systems violate a Bell inequality. We put forward…
Consider a unital C*-algebra A, a von Neumann algebra M, a unital sub-C*-algebra C of A and a unital *-homomorphism $\pi$ from C to M. Let u: A --> M be a decomposable map (i.e. a linear combination of completely positive maps) which is a…
The aim of this note is to give two new conceptual proofs of Ionescu-Weitzenb\"ock's inequality. The first one, which is a vector proof, provides us a geometric interpretation of the difference between the two sides of this inequality and…
We discuss several classical and recent proofs of the isoperimetric inequality and the Sobolev inequality.
Approximate relations between $e$ and $\pi$ are reviewed, some new connections being established. Nilakantha's series expansion for $\pi$ is transformed to accelerate its convergence. Its comparison with the standard inverse-factorial…
Gravitation as a fundamental interaction that governs all phenomena at large and very small scales, but still not well understood at a quantum level, is a missing cardinal link to unification of all physical interactions. Problems of the…
A phenomenological analysis based on the published branching fractions and $CP$ asymmetry observables of the $B\to K\pi$, $B\to\pi\pi$ and $B\to KK$ dataset is performed. The amplitude decomposition by the topological diagrams and the…
The main result of this paper is Theorem. For every integer $d\geqslant 2$ the set of biLipschitz classes in $\mathbb{E}^d$ has cardinality continuum.
The aim of this work is to extend Becker-Stark inequalities near the origin and {\pi}/2.
Three different representation of the proper Euclidean geometry are considered. They differ in the number of basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point,…