Envelope functions and asymptotic structures in Banach spaces

Bünyamin Sarı

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Displaying similar documents to “Envelope functions and asymptotic structures in Banach spaces”

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.

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 $t \to \infty$ .

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.

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...

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.

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...

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 → ∞.

On operators from separable reflexive spaces with asymptotic structure

Bentuo Zheng (2008)

Studia Mathematica

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Let 1 < q < p < ∞ and q ≤ r ≤ p. Let X be a reflexive Banach space satisfying a lower- $ℓ_{q}$ -tree estimate and let T be a bounded linear operator from X which satisfies an upper- $ℓ_{p}$ -tree estimate. Then T factors through a subspace of ${(\sum F ₙ)}_{ℓ_{r}}$ , where (Fₙ) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an $(ℓ_{p}, ℓ_{q})$ FDD. Similarly, let 1 < q < r < p < ∞ and let X be a separable reflexive Banach space satisfying an asymptotic...

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

Ivan Korec (1994)

Mathematica Slovaca

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Existence and asymptotic behaviour of some time-inhomogeneous diffusions

Mihai Gradinaru, Yoann Offret (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Let us consider a solution of a one-dimensional stochastic differential equation driven by a standard Brownian motion with time-inhomogeneous drift coefficient ${ρ sgn (x) | x |}^{α} / t^{β}$ . This process can be viewed as a Brownian motion evolving in a potential, possibly singular, depending on time. We prove results on the existence and uniqueness of solution, study its asymptotic behaviour and made a precise description, in terms of parameters $ρ$ , $α$ and $β$ , of the recurrence, transience and convergence. More precisely,...

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...

Asymptotic behaviour of averages of k-dimensional marginals of measures on ℝⁿ

Jesús Bastero, Julio Bernués (2009)

Studia Mathematica

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We study the asymptotic behaviour, as n → ∞, of the Lebesgue measure of the set $x \in K : | P_{E} (x) | \leq t$ for a random k-dimensional subspace E ⊂ ℝⁿ and an isotropic convex body K ⊂ ℝⁿ. For k growing slowly to infinity, we prove it to be close to the suitably normalised Gaussian measure in $ℝ^{k}$ of a t-dilate of the Euclidean unit ball. Some of the results hold for a wider class of probabilities on ℝⁿ.

An invariance method for constructing fundamental solutions for $P (□_{m} n)$

Zofia Szmydt, Bogdan Ziemian (1985)

Annales Polonici Mathematici

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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...

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 $c ̂_{q} (n) : = \sum_{d | (q, n)} d | μ (q / d) |$ .

Behaviour of the first eigenvalue of the p-Laplacian in a domain with a hole

M. Sango (2001)

Colloquium Mathematicae

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We investigate the behaviour of a sequence $λ_{s}$ , s = 1,2,..., of eigenvalues of the Dirichlet problem for the p-Laplacian in the domains $Ω_{s}$ , s = 1,2,..., obtained by removing from a given domain Ω a set $E_{s}$ whose diameter vanishes when s → ∞. We estimate the deviation of $λ_{s}$ from the eigenvalue of the limit problem. For the derivation of our results we construct an appropriate asymptotic expansion for the sequence of solutions of the original eigenvalue problem.

On asymptotically symmetric Banach spaces

M. Junge, D. Kutzarova, E. Odell (2006)

Studia Mathematica

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A Banach space X is asymptotically symmetric (a.s.) if for some C < ∞, for all m ∈ ℕ, for all bounded sequences ${(x_{j}^{i})}_{j = 1}^{\infty} \subseteq X$ , 1 ≤ i ≤ m, for all permutations σ of 1,...,m and all ultrafilters ₁,...,ₘ on ℕ, $l i m_{n ₁, ₁} . . . l i m_{n ₘ, ₘ} | | \sum_{i = 1}^{m} x_{n_{i}}^{i} | | \leq C l i m_{n_{σ (1)},_{σ (1)}} . . . l i m_{n_{σ (m)},_{σ (m)}} | | \sum_{i = 1}^{m} x_{n_{i}}^{i} | |$ . We investigate a.s. Banach spaces and several natural variations. X is weakly a.s. (w.a.s.) if the defining condition holds when restricted to weakly convergent sequences ${(x_{j}^{i})}_{j = 1}^{\infty}$ . Moreover, X is w.n.a.s. if we restrict the condition further to normalized weakly null sequences. If X is a.s. then...

Separated sequences in asymptotically uniformly convex Banach spaces

Sylvain Delpech (2010)

Colloquium Mathematicae

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We prove that the unit sphere of every infinite-dimensional Banach space X contains an α-separated sequence, for every $0 < α < 1 + δ ̅_{X} (1)$ , where $δ ̅_{X}$ denotes the modulus of asymptotic uniform convexity of X.