Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0402288

The second part of on duality triads' paper

A. K. Kwasniewski
View original source ↗Read source document ↗Export metadata ↓

Abstract

The duality triads were defined in the preceding paper.(ArXiv: math.GM/0402260 v 1 Feb. 2004). Notation, enumeration of formulas and references is therefore to be continued hereby. In this paper Fibonomial triangle and further Pascal-like triangles including q-Gaussian one are given explicit interpretation as discrete time dynamical systems as it is the case with all duality triads.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

A. K. Kwasniewski. 2004-02-18. The second part of on duality triads' paper. https://arxiv.org/abs/math/0402288

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

KEEP EXPLORING

Related papers

arXiv2026

The Hardy-Littlewood conjectures on the twin primes and the binary Goldbach problem are true

A celebrated conjecture of Hardy and Littlewood provides with an asymptotic formula for the counting function of the twin primes. We give an unconditional proof of such a formula by means of a finite Ramanujan expansion of the counting function expressed in terms of the von Mangoldt function and its incomplete form. In a completely analogous way, we solve the conjugate conjecture on the representations of any even integer as the sum of two prime numbers.

math.GM↗
arXiv2026

Equating the Area Sums of Alternate Sectors of a Disc Cut by Concurrent Chords

A disc is cut by $n$ chords through a common interior point $O$, which produces $2n$ curvilinear sectors. We determine when the two sums of alternate sectors have equal area for arbitrary chord directions, which need not be equally spaced. We first derive a closed form for the area of a single sector. We then show that the answer depends on the parity of $n$. For even $n$, at fixed orientation the condition is affine in $r_0^2$, where $r_0$ is the distance from $O$ to the centre; it involves only the total angle of the odd sectors and one trigonometric moment of the directions. For odd $n$, the condition is a nonlinear alternating sum of circular-segment functions of the distances of the chords from the centre. We treat four, six and eight pieces in detail. We also hold the chord directions fixed and translate the whole pencil to an arbitrary interior point, and we characterise the direction sets that give equal sums at every such point. For four chords only equally spaced directions have this property, which gives a converse of the pizza theorem; for an odd number of chords no direction set has it. A fair two-chord cut requires the acute angle between the chords to lie within $(r_0/a)^2$ radians of a right angle, where $a$ is the radius. For equally spaced chords the results reduce to the known theorems of Goldberg and of Mabry and Deiermann.

math.GM↗
arXiv2026

Null Cartan Normal Helices in Minkowski Space-Time

A complete theory of null Cartan normal helices in Minkowski space-time $\mathbb{E}^4_1$ is developed. Two algebraic conditions, obtained by successive differentiation of the helix invariant along a unit $C$-constant normal field, fully characterize null Cartan helices; the quadratic condition yields two mutually orthogonal helix axes in the Lorentzian metric. Special field types are analyzed and null Cartan cubics are shown to be normal helices. On a timelike hypersurface, a Darboux frame with six curvature functions is constructed from first principles, the normal isophotic condition is shown to reduce to a linear first-order ODE, and the existence of normal silhouettes in $\mathbb{E}^4_1$ is established.

math.GM↗