1,355,562 research outputs found
Approximation diophantienne (théorie de Markoff)
No full textTowards 1880, A. Markoff gave precisions about structure of the set of approximation constants greater than 1/3 for irrational numbers. This theory establishes links between constants, arithmetical minima for quadratic forms, and the solutions of the diophantine equation x2 + y2 +z2 = 3xyz. The present dissertation generalizes the original formalism built by Markoff. It introduces the notion of (a, r,E)-theory of Markoff, among which the (2,0,-1) theory is the original Markoff theory. The corresponding diophantine equation is given with an interpretation for the whole calculus. From that is derives the resolution of the diophantine equation x2 + y2 +z2 = (a +1)xyz and some arborescent constructions. For the systematic research of holes in the Markoff's spectra, the author gives confirmation for the results of Schecker and Freiman, concerning the Hall's ray. He gives examples and gives confirmation for some results of Kinney and PitcherVers 1880, A. Markoff a précisé la structure de l'ensemble des constantes d'approximations des nombres irrationnels plus grandes que 1/3. Sa théorie établit un lien entre ces constantes, des minima arithmétiques de formes quadratiques, et les solutions entières de l'équation diophantienne x2 + y2 +z2 = 3xyz. La présente thèse généralise le formalisme original de Markoff. Ceci introduit la notion de (a,r, E) - théorie de Markoff, dont la (2,0,-1) -théorie recouvre les calculs originaux. L'équation diophantienne correspondante est donnée ainsi qu'une interprétation des calculs. Il en résulte la résolution de l'équation diophantienne x2 + y2 +z2 = (a +1)xyz et diverses constructions arborescentes. Pour la recherche systématique des trous des spectres de Markoff et Perron, l'auteur confirme les résultats de Scheker et Freiman concernant le rayon de Hall. Il donne des exemples et confirme certains résultats de Kinney et Pitche
A note on the Markoff condition and central words
No full textWe define Markoff words as certain factors appearing in bi-infinite words satisfying the Markoff condition. We prove that these words coincide with central words, yielding a new characterization of Christoffel words
Relativistic hydrodynamic jets: Mixing effects, radial structure and synchrotron emission for continuous and episodic AGN jets
No full textContains fulltext :
319698.pdf (Publisher’s version ) (Open Access)Radboud University, 19 juni 2025Promotores : Achterberg, A., Markoff, S.B.viii, 239 p
Conditions of positivity on a shadow Markoff tree.
No full textAn analogue of the Markoff equation has recently been introduced by the author and Valentin Ovsienko. A conjecture about the necessary and sufficient conditions for positivity of solutionsto this equation is formulated and discussed
Conditions of positivity on a shadow Markoff Tree
No full textAn analogue of the Markoff equation has recently been introduced by the
author and Valentin Ovsienko. A conjecture about the necessary and sufficient
conditions for positivity of solutions to this equation is formulated and
discussed
Some limit theorems for nonhomogeneous Markoff processes
No full textWe intend to study some problems related to the asymptotic behaviour of a physical system the evolution of which is markovian. The typical example of such an evolution is furnished by an homogeneous discrete chain with a finite number of possible states considered first by A. A. Markoff. In §1 we recall briefly the main results of this theory and in §2 we treat its obvious generalization to the continuous parameter case. In §3 we pass to the proper object of this paper and we establish a limit theorem for time-homogeneous Markoff processes. This limit theorem is then extended to the nonhomogeneous case under some supplementary conditions (§4). Finally we give an application of this theory to random functions connected with a Markoff process (§5).</p
Markoff phenomena
No full textThe purpose of this thesis is to discuss Markoff Numbers, their associated binary quadratic forms, together with the units in the associated real quadratic field. Relations betweeen the Markoff Numbers, the explicit structure of the automorph group of the forms, generators of the Commutator Subgroup Γ’ of SL₂(Z) = Γ and the lengths of geodesies on certain Riemann Surfaces are conveyed. A conjecture combining these relations is formulated and expressed at the end of this paper.Science, Faculty ofMathematics, Department ofGraduat
On the Markoff Equation
No full textIn recent work by Bourgain, Gamburd and Sarnak, they prove that almost all Markoff numbers (counted with multiplicity) are highly composite. For any fixed , they conclude that a (natural) density-1 subset of Markoff numbers have at least distinct prime factors.
We present a modified version of their argument to conclude our main theorem that the set of Markoff numbers such that has at least prime factors has (natural) density equal to 1, where is the function given by a composition of six natural logarithms.
We also present the work of Zagier on the count of Markoff numbers (with multiplicity) below a given bound and Meiri and Puder's recent result on the transitivity of action of the Markoff group on triples of solutions to the Markoff equation on composite moduli. We restrict ourselves to Meiri and Puder's treatment only for the products of primes , as necessary for the proof of our main theorem
The uniqueness of the Markoff numbers
No full textA Markoff triple is a set of three positive integers satisfying the diophantine equation
x
2
+
y
2
+
z
2
=
3
x
y
z
{x^2} + {y^2} + {z^2} = 3xyz
. The maximum of the three numbers is called a Markoff number. We show: If there are Markoff triples
(
x
1
,
y
1
,
z
)
({x_1},{y_1},z)
and
(
x
2
,
y
2
,
z
)
({x_2},{y_2},z)
with the same Markoff number z, then
x
1
=
x
2
{x_1} = {x_2}
or
x
1
=
y
2
{x_1} = {y_2}
.</p
Markoff triples and quasifuchsian groups
No full textWe study the global behaviour of trees of Markoff triples over the complex numbers. We relate this to the space of type-preserving representations of the punctured torus group into PSL(2, C). In particular, we explore which Markoff triples correspond to quasifuchsian representations. We derive a variation of McShane's identity for quasifuchsian groups. In the case of non-discrete representations, we attempt to relate the asymptotic behaviour of Markoff triples to the realisability of laminations in hyperbolic 3-space. We also consider how some of these issues might be related for more general surfaces
- …
