Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0401039

Invariant and evolutionary properties of the skew-symmetric differential forms

L. I. Petrova
View original source ↗Read source document ↗Export metadata ↓

Abstract

The present work pursues the aim to draw attention to unique possibilities of the skew-symmetric differential forms. At present the theory of skew-symmetric exterior differential forms that possess invariant properties has been developed. In the work the readers are introduced to the skew-symmetric differential forms that were called evolutionary ones because they possess evolutionary properties. The combined mathematical apparatus of exterior and evolutionary skew-symmetric differential forms in essence is a new mathematical language. This apparatus can describe transitions from nonconjugated operators to conjugated ones. There are no such possibilities in any mathematical formalism. In the present work it has been shown that the properties of exterior or evolutionary forms are explicitly or implicitly accounted for in all mathematical (and physical) formalisms.

Explore related subjects

math.GM

Keep this discovery

BibTeXRIS

L. I. Petrova. 2004-01-06. Invariant and evolutionary properties of the skew-symmetric differential forms. https://arxiv.org/abs/math/0401039

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗