Coordinates for vertices of regular honeycombs in hyperbolic space

1966 ◽  
Vol 293 (1432) ◽  
pp. 94-107 ◽  
Keyword(s):  
Symmetry Group ◽  
Hyperbolic Space ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Hyperbolic Spaces ◽  
Fundamental Region ◽  
Homogeneous Coordinates ◽  
Quadratic Diophantine Equation ◽  
Integral Solutions

Schlegel (1883) enumerated all regular honeycombs of hyperbolic spaces of three or more dimensions, having finite cells and vertex figures. Coxeter (1954) extended this enumeration to include honeycombs with infinite cells and/or infinite vertex figures, the fundamental region of the symmetry group still being finite. One of these, {4, 4, 3}, was shown to have for its vertices the points whose coordinates are proportional to the integral solutions of a quadratic Diophantine equation (Coxeter & Whitrow 1950). In the present paper, certain quadratic Diophantine equations are found whose solutions provide homogeneous coordinates for the vertices of {6, 3, 3}, {6, 3, 4}, {4, 3, 4, 3} and {3, 4, 3, 3, 3}. A method is also given for finding coordinates for the vertices of the remaining honeycombs (with finite vertex figures), and the simplest of these are listed.

scholarly journals On a class of insoluble binary quadratic diophantine equations

1991 ◽  
Vol 123 ◽  
pp. 141-151 ◽  
Author(s):  
Franz Halter-Koch
Keyword(s):  
Diophantine Equation ◽  
Class Number ◽  
Diophantine Equations ◽  
Number Fields ◽  
Quadratic Number ◽  
Number Problem ◽  
Quadratic Number Fields ◽  
Quadratic Diophantine Equation ◽  
Real Quadratic

The binary quadratic diophantine equationis of interest in the class number problem for real quadratic number fields and was studied in recent years by several authors (see [4], [5], [2] and the literature cited there).

scholarly journals A Peer Search on Integer Solutions to Quadratic Diophantine Equation with Three Unknowns (Equation)

2020 ◽  
pp. 166-171
Author(s):  
A. Vijayasankar ◽  
Sharadha Kumar ◽  
M. A. Gopalan
Keyword(s):  
Diophantine Equation ◽  
Quadratic Diophantine Equation ◽  
Integer Solutions ◽  
Integral Solutions

The non- homogeneous ternary quadratic diophantine (Equation) is analyzed for its patterns of non-zero distinct integral solutions. Various interesting relations between the solutions and special numbers namely polygonal, Pronic and Gnomonic numbers are exhibited.

scholarly journals On the Diophantine equations z2 = k(k2 + 3) and z2 = k(k2 + 12)

2021 ◽  
Vol 45 (1) ◽  
pp. 127-129
Author(s):  
Shah Mohammad Shahidul Islam ◽  
Abdullah Al Kafi Majumdar
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Analytical Method ◽  
Diophantine Equations ◽  
Integer Solution ◽  
Positive Integer Solution ◽  
Integral Solutions

This paper provides an analytical method of finding all the (positive, integral) solutions of the Diophantine equation z2 = k(k2+3). We also prove analytically that the Diophantine equation z2 = k(k2+12) has no positive, integer solution. J. Bangladesh Acad. Sci. 45(1); 127-129: June 2021

Certain Diophantine Equations Linear In One Unknown

1956 ◽  
Vol 8 ◽  
pp. 5-12 ◽  
Author(s):  
W. H. Mills
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Integral Solutions

1. Introduction. A. Brauer and R. Brauer (2) and Barnes (1) (following a method of Mordell (6)) have solved the Diophantine equation x2+y2+c = xyz subject to the condition (x, y) = 1. Independently, but using the same methods, I treated (4) the equationx2+y2+ax+ay+l = xyz,and subsequently (5) gave a method of obtaining all integral solutions ofx2±y2+ax+by+c = xyz,thereby generalizing (2), (1), and (4).

The Diophantine equation Fn = P(x)

2020 ◽  
Vol 16 (09) ◽  
pp. 2095-2111
Author(s):  
Szabolcs Tengely ◽  
Maciej Ulas
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Infinitely Many Solutions ◽  
Polynomial Formula ◽  
Positive Integers ◽  
Integral Solutions

We consider equations of the form [Formula: see text], where [Formula: see text] is a polynomial with integral coefficients and [Formula: see text] is the [Formula: see text]th Fibonacci number that is, [Formula: see text] and [Formula: see text] for [Formula: see text] In particular, for each [Formula: see text], we prove the existence of a polynomial [Formula: see text] of degree [Formula: see text] such that the Diophantine equation [Formula: see text] has infinitely many solutions in positive integers [Formula: see text]. Moreover, we present results of our numerical search concerning the existence of even degree polynomials representing many Fibonacci numbers. We also determine all integral solutions [Formula: see text] of the Diophantine equations [Formula: see text] for [Formula: see text] and [Formula: see text].

Secure Watermarking using Diophantine Equations for Authentication and Recovery

2015 ◽  
Vol 3 (2) ◽  
Author(s):  
Jayashree Nair ◽  
T. Padma
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Jpeg Compression ◽  
Authentication Scheme ◽  
Original Image ◽  
Invariant Features ◽  
Jpeg2000 Compression ◽  
Frequency Properties ◽  
Visual Authentication

This paper describes an authentication scheme that uses Diophantine equations based generation of the secret locations to embed the authentication and recovery watermark in the DWT sub-bands. The security lies in the difficulty of finding a solution to the Diophantine equation. The scheme uses the content invariant features of the image as a self-authenticating watermark and a quantized down sampled approximation of the original image as a recovery watermark for visual authentication, both embedded securely using secret locations generated from solution of the Diophantine equations formed from the PQ sequences. The scheme is mildly robust to Jpeg compression and highly robust to Jpeg2000 compression. The scheme also ensures highly imperceptible watermarked images as the spatio –frequency properties of DWT are utilized to embed the dual watermarks.

Convergence theorems for total asymptotically nonexpansive single-valued and quasi nonexpansive multi-valued mappings in hyperbolic spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Preeyalak Chuadchawna ◽  
Ali Farajzadeh ◽  
Anchalee Kaewcharoen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Hyperbolic Space ◽  
Hyperbolic Spaces ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive ◽  
Uniformly Convex Hyperbolic Space

Abstract In this paper, we discuss the Δ-convergence and strong convergence for the iterative sequence generated by the proposed scheme to approximate a common fixed point of a total asymptotically nonexpansive single-valued mapping and a quasi nonexpansive multi-valued mapping in a complete uniformly convex hyperbolic space. Finally, by giving an example, we illustrate our result.

Sign in / Sign up

Export Citation Format

Share Document