scholarly journals A note on the Diophantine Equation x^2-kxy+ky^2+ly=0

Mathematica ◽  
2021 ◽  
Vol 63 (86) (2) ◽  
pp. 151-157
Author(s):  
Sakha A. Alkabouss ◽  
◽  
Boualem Benseba ◽  
Nacira Berbara ◽  
Simon Earp-Lynch ◽  
...  
Keyword(s):  
Positive Solutions ◽  
Diophantine Equation

We investigate the Diophantine equation x^2 −kxy + ky^2 + ly = 0 for integers k and l with k even. We give a characterization of the positive solutions of this equation in terms of k and l. We also consider the same equation for other values of k and l.

scholarly journals Solvability of the diophantine equation x2 − Dy2 = ± 2 and new invariants for real quadratic fields

1994 ◽  
Vol 134 ◽  
pp. 137-149 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Real Quadratic Field ◽  
Real Quadratic Fields ◽  
Real Quadratic

In our recent papers [3, 4, 5], we defined some new D-invariants for any square-free positive integer D and considered their properties and interrelations among them. Especially, as an application of it, we discussed in [5] the characterization of real quadratic field Q() of so-called Richaud-Degert type in terms of these new D-invariants.

scholarly journals Positive solutions of the diophantine equation

1982 ◽  
Vol 5 (2) ◽  
pp. 311-314 ◽  
Author(s):  
W. R. Utz
Keyword(s):  
Positive Solutions ◽  
Diophantine Equation ◽  
Positive Integers ◽  
Integral Solutions

Integral solutions ofx3+λy+1−xyz=0are observed for all integralλ. Forλ=2the 13 solutions of the equation in positive integers are determined. Solutions of the equation in positive integers were previously determined for the caseλ=1.

scholarly journals On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1796
Author(s):  
Štěpán Hubálovský ◽  
Eva Trojovská
Keyword(s):  
Positive Integer ◽  
Fixed Points ◽  
Natural Number ◽  
Positive Solutions ◽  
Diophantine Equation ◽  
Fibonacci Number ◽  
Euler Totient Function

Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk≡0(modn). In this paper, we shall find all positive solutions of the Diophantine equation z(φ(n))=n, where φ is the Euler totient function.

scholarly journals Existence of Positive Solutions for a Discrete Three-Point Boundary Value Problem

2007 ◽  
Vol 2007 ◽  
pp. 1-14
Author(s):  
Huting Yuan ◽  
Guang Zhang ◽  
Hongliang Zhao
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Positive Parameter ◽  
Point Boundary ◽  
Constant Number ◽  
Fixed Integer ◽  
Negative Numbers ◽  
Existence Of Positive Solutions

A discrete three-point boundary value problemΔ2xk−1+λfk(xk)=0,k=1,2,…,n, x0=0,axl=xn+1, is considered, where1≤l≤nis a fixed integer,ais a real constant number, andλis a positive parameter. A characterization of the values ofλis carried out so that the boundary value problem has the positive solutions. Particularly, in this paper the constantacan be negative numbers. The similar results are not valid for the three-point boundary value problem of differential equations.

Morphological Characterization of Mycobacteriophage R1

1974 ◽  
Vol 32 ◽  
pp. 254-255
Author(s):  
B. L. Soloff ◽  
T. A. Rado
Keyword(s):  
Carbon Source ◽  
Stationary Phase ◽  
Growth Phase ◽  
Minimal Medium ◽  
Morphological Characterization ◽  
Logarithmic Growth Phase ◽  
Complete Lysis ◽  
Lysogenic Culture ◽  
Logarithmic Growth

Mycobacteriophage R1 was originally isolated from a lysogenic culture of M. butyricum. The virus was propagated on a leucine-requiring derivative of M. smegmatis, 607 leu−, isolated by nitrosoguanidine mutagenesis of typestrain ATCC 607. Growth was accomplished in a minimal medium containing glycerol and glucose as carbon source and enriched by the addition of 80 μg/ ml L-leucine. Bacteria in early logarithmic growth phase were infected with virus at a multiplicity of 5, and incubated with aeration for 8 hours. The partially lysed suspension was diluted 1:10 in growth medium and incubated for a further 8 hours. This permitted stationary phase cells to re-enter logarithmic growth and resulted in complete lysis of the culture.

AEM analysis of crystallization in Ti-based amorphous alloys

1983 ◽  
Vol 41 ◽  
pp. 270-271
Author(s):  
A.R. Pelton ◽  
A.F. Marshall ◽  
Y.S. Lee
Keyword(s):  
Magnetic Properties ◽  
Amorphous Alloys ◽  
Amorphous Materials ◽  
Isothermal Annealing ◽  
Amorphous Matrix ◽  
Nucleation And Growth ◽  
Current Interest ◽  
Amorphous Films ◽  
Electrical And Magnetic Properties

Amorphous materials are of current interest due to their desirable mechanical, electrical and magnetic properties. Furthermore, crystallizing amorphous alloys provides an avenue for discerning sequential and competitive phases thus allowing access to otherwise inaccessible crystalline structures. Previous studies have shown the benefits of using AEM to determine crystal structures and compositions of partially crystallized alloys. The present paper will discuss the AEM characterization of crystallized Cu-Ti and Ni-Ti amorphous films.Cu60Ti40: The amorphous alloy Cu60Ti40, when continuously heated, forms a simple intermediate, macrocrystalline phase which then transforms to the ordered, equilibrium Cu3Ti2 phase. However, contrary to what one would expect from kinetic considerations, isothermal annealing below the isochronal crystallization temperature results in direct nucleation and growth of Cu3Ti2 from the amorphous matrix.

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