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

Ivan Korec

Displaying similar documents to “A density estimate for the $3 x + 1$ problem”

Mean value densities for temperatures

N. Suzuki, N. A. Watson (2003)

Colloquium Mathematicae

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A positive measurable function K on a domain D in $ℝ^{n + 1}$ is called a mean value density for temperatures if $u (0, 0) = \int \int_{D} K (x, t) u (x, t) d x d t$ for all temperatures u on D̅. We construct such a density for some domains. The existence of a bounded density and a density which is bounded away from zero on D is also discussed.

Construction of an Uncountable Difference between Φ(B) and $Φ_{f} (B)$

Josh Campbell, David Swanson (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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We construct a set B and homeomorphism f where f and $f^{- 1}$ have property N such that the symmetric difference between the sets of density points and of f-density points of B is uncountable.

Maximal upper asymptotic density of sets of integers with missing differences from a given set

Ram Krishna Pandey (2015)

Mathematica Bohemica

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Let $M$ be a given nonempty set of positive integers and $S$ any set of nonnegative integers. Let $\bar{δ} (S)$ denote the upper asymptotic density of $S$ . We consider the problem of finding $μ (M) : = sup_{S} \bar{δ} (S),$ where the supremum is taken over all sets $S$ satisfying that for each $a, b \in S$ , $a - b \notin M .$ In this paper we discuss the values and bounds of $μ (M)$ where $M = {a, b, a + n b}$ for all even integers and for all sufficiently large odd integers $n$ with $a < b$ and $gcd (a, b) = 1 .$

Estimation of the density of a determinantal process

Yannick Baraud (2013)

Confluentes Mathematici

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We consider the problem of estimating the density $Π$ of a determinantal process $N$ from the observation of $n$ independent copies of it. We use an aggregation procedure based on robust testing to build our estimator. We establish non-asymptotic risk bounds with respect to the Hellinger loss and deduce, when $n$ goes to infinity, uniform rates of convergence over classes of densities $Π$ of interest.

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 $t \to \infty$ .

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: $x ₙ \sim {((2 - p) / ((m - 1) \sum_{i = 0}^{m} a_{i, m}))}^{1 / (m - 1)} 1 / \sqrt[m - 1]{n}$ .

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.

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.

A density version of the Carlson–Simpson theorem

Pandelis Dodos, Vassilis Kanellopoulos, Konstantinos Tyros (2014)

Journal of the European Mathematical Society

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We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k \geq 2$ and every set $A$ of words over $k$ satisfying $\lim \sup_{n \to \infty} | A \cap {[k]}^{n} | / k^{n} > 0$ there exist a word $c$ over $k$ and a sequence $(w_{n})$ of left variable words over $k$ such that the set $c \cup {c^{} w_{0} {(a_{0})}^{} . . .^{} w_{n} (a_{n}) : n \in ℕ and a_{0}, . . ., a_{n} \in [k]}$ is contained in $A$ . While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.

Kempisty's theorem for the integral product quasicontinuity

Zbigniew Grande (2006)

Colloquium Mathematicae

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A function f: ℝⁿ → ℝ satisfies the condition $Q_{i} (x)$ (resp. $Q_{s} (x)$ , $Q_{o} (x)$ ) at a point x if for each real r > 0 and for each set U ∋ x open in the Euclidean topology of ℝⁿ (resp. strong density topology, ordinary density topology) there is an open set I such that I ∩ U ≠ ∅ and $| (1 / μ (U \cap I)) \int_{U \cap I} f (t) d t - f (x) | < r$ . Kempisty’s theorem concerning the product quasicontinuity is investigated for the above notions.

Continuity of halo functions associated to homothecy invariant density bases

Oleksandra Beznosova, Paul Hagelstein (2014)

Colloquium Mathematicae

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Let be a collection of bounded open sets in ℝⁿ such that, for any x ∈ ℝⁿ, there exists a set U ∈ of arbitrarily small diameter containing x. The collection is said to be a density basis provided that, given a measurable set A ⊂ ℝⁿ, for a.e. x ∈ ℝⁿ we have $l i m_{k \to \infty} 1 / | R_{k} | \int_{R_{k}} χ_{A} = χ_{A} (x)$ for any sequence $R_{k}$ of sets in containing x whose diameters tend to 0. The geometric maximal operator $M_{}$ associated to is defined on L¹(ℝⁿ) by $M_{} f (x) = s u p_{x \in R \in} 1 / | R | \int_{R} | f |$ . The halo function ϕ of is defined on (1,∞) by $ϕ (u) = s u p 1 / | A | | x \in ℝ ⁿ : M_{} χ_{A} (x) > 1 / u | : 0 < | A | < \infty$ and on [0,1] by ϕ(u) = u. It is shown...

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...

Density estimation via best $L^{2}$ -approximation on classes of step functions

Dietmar Ferger, John Venz (2017)

Kybernetika

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We establish consistent estimators of jump positions and jump altitudes of a multi-level step function that is the best $L^{2}$ -approximation of a probability density function $f$ . If $f$ itself is a step-function the number of jumps may be unknown.

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

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...

Integers not of the form $c (2^{a} + 2^{b}) + p^{α}$

Zhi-Wei Sun, Mao-Hua Le (2001)

Acta Arithmetica

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Localization techniques in $L^{p}$ spaces

A. Pełczyński, H. Rosenthal (1975)

Studia Mathematica

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Displaying similar documents to “A density estimate for the 3 x + 1 problem”

Displaying similar documents to “A density estimate for the $3 x + 1$ problem”