Asymptotic representations of solutions of the nonautonomous ordinary differential $n$-th order equations

Mousa Jaber Abu Elshour; Vjacheslav Evtukhov

Displaying similar documents to “Asymptotic representations of solutions of the nonautonomous ordinary differential $n$ -th order equations”

Asymptotic behavior of a sequence defined by iteration with applications

Stevo Stević (2002)

Colloquium Mathematicae

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We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let f: (0,∞)² → (0,∞) be a continuous function such that (a) 0 < f(x,y) < px + (1-p)y for some p ∈ (0,1) and for all x,y ∈ (0,α), where α > 0; (b) $f (x, y) = p x + (1 - p) y - \sum_{s = m}^{\infty} s (x, y)$ uniformly in a neighborhood of the origin, where m > 1, $_{s} (x, y) = \sum_{i = 0}^{s} a_{i, s} x^{s - i} y^{i}$ ; (c) $ₘ (1, 1) = \sum_{i = 0}^{m} a_{i, m} > 0$ . Let x₀,x₁ ∈ (0,α) and $x_{n + 1} = f (x ₙ, x_{n - 1})$ , n ∈ ℕ. Then the sequence (xₙ) satisfies the following asymptotic formula: $x ₙ \sim {((2 - p) / ((m - 1) \sum_{i = 0}^{m} a_{i, m}))}^{1 / (m - 1)} 1 / \sqrt[m - 1]{n}$ .

Asymptotic integration of differential equations with singular $p$ -Laplacian

Milan Medveď, Eva Pekárková (2016)

Archivum Mathematicum

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In this paper we deal with the problem of asymptotic integration of nonlinear differential equations with $p -$ Laplacian, where $1 < p < 2$ . We prove sufficient conditions under which all solutions of an equation from this class are converging to a linear function as $t \to \infty$ .

Envelope functions and asymptotic structures in Banach spaces

Bünyamin Sarı (2004)

Studia Mathematica

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We introduce a notion of disjoint envelope functions to study asymptotic structures of Banach spaces. The main result gives a new characterization of asymptotic- $ℓ_{p}$ spaces in terms of the $ℓ_{p}$ -behavior of “disjoint-permissible” vectors of constant coefficients. Applying this result to Tirilman spaces we obtain a negative solution to a conjecture of Casazza and Shura. Further investigation of the disjoint envelopes leads to a finite-representability result in the spirit of the Maurey-Pisier...

Generalised regular variation of arbitrary order

Edward Omey, Johan Segers (2010)

Banach Center Publications

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Let f be a measurable, real function defined in a neighbourhood of infinity. The function f is said to be of generalised regular variation if there exist functions h ≢ 0 and g > 0 such that f(xt) - f(t) = h(x)g(t) + o(g(t)) as t → ∞ for all x ∈ (0,∞). Zooming in on the remainder term o(g(t)) eventually leads to the relation f(xt) - f(t) = h₁(x)g₁(t) + ⋯ + hₙ(x)gₙ(t) + o(gₙ(t)), each $g_{i}$ being of smaller order than its predecessor $g_{i - 1}$ . The function f is said to be generalised regularly...

Note on distortion and Bourgain ℓ₁-index

Anna Maria Pelczar (2009)

Studia Mathematica

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Relations between different notions measuring proximity to ℓ₁ and distortability of a Banach space are studied. The main result states that a Banach space all of whose subspaces have Bourgain ℓ₁-index greater than $ω^{α}$ , α < ω₁, contains either an arbitrarily distortable subspace or an $ℓ ₁^{α}$ -asymptotic subspace.

Existence and asymptotic behavior of positive solutions for elliptic systems with nonstandard growth conditions

Honghui Yin, Zuodong Yang (2012)

Annales Polonici Mathematici

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Our main purpose is to establish the existence of a positive solution of the system ⎧ $- ∆_{p (x)} u = F (x, u, v)$ , x ∈ Ω, ⎨ $- ∆_{q (x)} v = H (x, u, v)$ , x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where $Ω \subset ℝ^{N}$ is a bounded domain with C² boundary, $F (x, u, v) = λ^{p (x)} [g (x) a (u) + f (v)]$ , $H (x, u, v) = λ^{q (x}) [g (x) b (v) + h (u)]$ , λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and $- ∆_{p (x)} u = - {d i v (| \nabla u |}^{p (x) - 2} \nabla u)$ is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.

Lyapunov functions and $L^{p}$ -estimates for a class of reaction-diffusion systems

Dirk Horstmann (2001)

Colloquium Mathematicae

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We give a sufficient condition for the existence of a Lyapunov function for the system aₜ = ∇(k(a,c)∇a - h(a,c)∇c), x ∈ Ω, t > 0, $ε c ₜ = k_{c} Δ c - f (c) c + g (a, c)$ , x ∈ Ω, t > 0, for $Ω \subset ℝ^{N}$ , completed with either a = c = 0, or ∂a/∂n = ∂c/∂n = 0, or k(a,c) ∂a/∂n = h(a,c) ∂c/∂n, c = 0 on ∂Ω × t > 0. Furthermore we study the asymptotic behaviour of the solution and give some uniform $L^{p}$ -estimates.

Exceptional sets in Waring's problem: two squares and s biquadrates

Lilu Zhao (2014)

Acta Arithmetica

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Let $R_{s} (n)$ denote the number of representations of the positive number n as the sum of two squares and s biquadrates. When $s = 3$ or 4, it is established that the anticipated asymptotic formula for $R_{s} (n)$ holds for all $n \leq X$ with at most $O (X^{(9 - 2 s) / 8 + ε})$ exceptions.

Asymptotic values of modular multiplicities for ${GL}_{2}$

Sandra Rozensztajn (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the irreducible constituents of the reduction modulo $p$ of irreducible algebraic representations $V$ of the group ${Res}_{K / ℚ_{p}} {GL}_{2}$ for $K$ a finite extension of $ℚ_{p}$ . We show that asymptotically, the multiplicity of each constituent depends only on the dimension of $V$ and the central character of its reduction modulo $p$ . As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

Stacks of group representations

Paul Balmer (2015)

Journal of the European Mathematical Society

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We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$ , the derived and the stable categories of representations of a subgroup $H$ can be constructed out of the corresponding category for $G$ by a purely triangulated-categorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods...

Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

Francesca De Marchis, Isabella Ianni, Filomena Pacella (2015)

Journal of the European Mathematical Society

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We consider the semilinear Lane–Emden problem where $p > 1$ and $Ω$ is a smooth bounded domain of $ℝ^{2}$ . The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of $(_{p})$ , as $p \to + \infty$ . Among other results we show, under some symmetry assumptions on $Ω$ , that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p \to + \infty$ , and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of...

A density estimate for the $3 x + 1$ problem

Ivan Korec (1994)

Mathematica Slovaca

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The $n$ -th prime asymptotically

Juan Arias de Reyna, Jérémy Toulisse (2013)

Journal de Théorie des Nombres de Bordeaux

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A new derivation of the classic asymptotic expansion of the $n$ -th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with ${li}^{- 1} (n)$ , after having retained the first $m$ terms, for $1 \leq m \leq 11$ , are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible $r_{3}$ such that, for $n \geq r_{3}$ , we have $p_{n} > s_{3} (n)$ where $s_{3} (n)$ is the sum of the first four terms of the asymptotic...

Asymptotic nature of higher Mahler measure

(2014)

Acta Arithmetica

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We consider Akatsuka’s zeta Mahler measure as a generating function of the higher Mahler measure $m_{k} (P)$ of a polynomial $P,$ where $m_{k} (P)$ is the integral of $l o g^{k} | P |$ over the complex unit circle. Restricting ourselves to P(x) = x - r with |r| = 1 we show some new asymptotic results regarding $m_{k} (P)$ , in particular $| m_{k} (P) | / k! \to 1 / π$ as k → ∞.

Asymptotic properties of ground states of scalar field equations with a vanishing parameter

Vitaly Moroz, Cyrill B. Muratov (2014)

Journal of the European Mathematical Society

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We study the leading order behaviour of positive solutions of the equation $- Δ u + ϵ u - {| u |}^{p - 2} u + {| u |}^{q - 2} u = 0, x \in ℝ^{N}$ , where $N \geq 3$ , $q > p > 2$ and when $ϵ > 0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$ , $q$ and $N$ . The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $2^{*} = \frac{2 N}{N - 2}$ . For $p < 2^{*}$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p > 2^{*}$ the solution asymptotically...

On a sum involving the Möbius function

I. Kiuchi, M. Minamide, Y. Tanigawa (2015)

Acta Arithmetica

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Let $c_{q} (n)$ be the Ramanujan sum, i.e. $c_{q} (n) = \sum_{d | (q, n)} d μ (q / d)$ , where μ is the Möbius function. In a paper of Chan and Kumchev (2012), asymptotic formulas for $\sum_{n \leq y} {(\sum_{q \leq x} c_{q} (n))}^{k}$ (k = 1,2) are obtained. As an analogous problem, we evaluate $\sum_{n \leq y} {(\sum_{n \leq x} c ̂_{q} (n))}^{k}$ (k = 1,2), where $c ̂_{q} (n) : = \sum_{d | (q, n)} d | μ (q / d) |$ .

Tests for the presence of trends in linear processes

S. K. Zaremba

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CONTENTS1. Introduction.............................................................................................................................................52. Assumptions and notations................................................................................................................63. An important asymptotic distribution..................................................................................................84. The asymptotic distribution of $K *_{μ, ν}$ under the...

Spreading and vanishing in nonlinear diffusion problems with free boundaries

Yihong Du, Bendong Lou (2015)

Journal of the European Mathematical Society

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We study nonlinear diffusion problems of the form $u_{t} = u_{x x} + f (u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f (u)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f (u)$ which is $C^{1}$ and satisfies $f (0) = 0$ , we show that the omega limit set $ω (u)$ of every bounded positive solution is determined by a stationary...

Irreducibility of automorphic Galois representations of $G L (n)$ , $n$ at most $5$

Frank Calegari, Toby Gee (2013)

Annales de l’institut Fourier

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Let $π$ be a regular, algebraic, essentially self-dual cuspidal automorphic representation of ${GL}_{n} (𝔸_{F})$ , where $F$ is a totally real field and $n$ is at most $5$ . We show that for all primes $l$ , the $l$ -adic Galois representations associated to $π$ are irreducible, and for all but finitely many primes $l$ , the mod $l$ Galois representations associated to $π$ are also irreducible. We also show that the Lie algebras of the Zariski closures of the $l$ -adic representations are independent of $l$ .

Displaying similar documents to “Asymptotic representations of solutions of the nonautonomous ordinary differential n -th order equations”

Displaying similar documents to “Asymptotic representations of solutions of the nonautonomous ordinary differential $n$ -th order equations”