Pell Equations and Weak Regularity Principles

L 1 and L ∞ stability of transition densities of perturbed diffusions

Random Operators and Stochastic Equations ◽

10.1515/rose-2021-2067 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Ilya Bitter ◽

Valentin Konakov

Keyword(s):

Linear Growth ◽

Stability Result ◽

Regularity Conditions ◽

Initial Condition ◽

Weak Regularity ◽

Transition Densities ◽

The Stability ◽

Drift Terms

Abstract In this paper, we derive a stability result for L 1 {L_{1}} and L ∞ {L_{\infty}} perturbations of diffusions under weak regularity conditions on the coefficients. In particular, the drift terms we consider can be unbounded with at most linear growth, and the estimates reflect the transport of the initial condition by the unbounded drift through the corresponding flow. Our approach is based on the study of the distance in L 1 {L_{1}} - L 1 {L_{1}} metric between the transition densities of a given diffusion and the perturbed one using the McKean–Singer parametrix expansion. In the second part, we generalize the well-known result on the stability of diffusions with bounded coefficients to the case of at most linearly growing drift.

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Solution of certain Pell equations

Asian-European Journal of Mathematics ◽

10.1142/s1793557118500560 ◽

2018 ◽

Vol 11 (04) ◽

pp. 1850056 ◽

Cited By ~ 1

Author(s):

Zahid Raza ◽

Hafsa Masood Malik

Keyword(s):

Positive Integer ◽

Fundamental Solution ◽

Positive Solutions ◽

Continued Fraction ◽

Pell Equation ◽

Lucas Sequences ◽

Pell Equations ◽

Integer Solutions ◽

Positive Integers

Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the Pell form [Formula: see text] including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation [Formula: see text] in terms of [Formula: see text] We extend all the results of the papers [3, 10, 27, 37].

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Reverse mathematics and a Ramsey-type König's Lemma

Journal of Symbolic Logic ◽

10.2178/jsl.7704120 ◽

2012 ◽

Vol 77 (4) ◽

pp. 1272-1280 ◽

Cited By ~ 9

Author(s):

Stephen Flood

Keyword(s):

Reverse Mathematics ◽

Ramsey's Theorem ◽

Weak Regularity ◽

Ramsey’S Theorem ◽

Ramsey Type ◽

Weak König’S Lemma ◽

König’S Lemma

AbstractIn this paper, we propose a weak regularity principle which is similar to both weak König's lemma and Ramsey's theorem. We begin by studying the computational strength of this principle in the context of reverse mathematics. We then analyze different ways of generalizing this principle.

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