Equicontinuity and Convergent Sequences in the Spaces $ ^{\prime }_C$ and $_M$

Jan Kisyński

Displaying similar documents to “Equicontinuity and Convergent Sequences in the Spaces $_{C}^{'}$ and $_{M}$ ”

Primefree shifted Lucas sequences

Lenny Jones (2015)

Acta Arithmetica

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We say a sequence $= {(s ₙ)}_{n \geq 0}$ is primefree if |sₙ| is not prime for all n ≥ 0, and to rule out trivial situations, we require that no single prime divides all terms of . In this article, we focus on the particular Lucas sequences of the first kind, $_{a} = {(u ₙ)}_{n \geq 0}$ , defined by u₀ = 0, u₁ = 1, and uₙ = aun-1 + un-2 for n≥2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k such that both of the shifted sequences $_{a} \pm k$ are simultaneously primefree. This...

On Fourier asymptotics of a generalized Cantor measure

Bérenger Akon Kpata, Ibrahim Fofana, Konin Koua (2010)

Colloquium Mathematicae

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Let d be a positive integer and μ a generalized Cantor measure satisfying $μ = \sum_{j = 1}^{m} a_{j} μ \circ S_{j}^{- 1}$ , where $0 < a_{j} < 1$ , $\sum_{j = 1}^{m} a_{j} = 1$ , $S_{j} = ρ R + b_{j}$ with 0 < ρ < 1 and R an orthogonal transformation of $ℝ^{d}$ . Then ⎧1 < p ≤ 2 ⇒ ⎨ $s u p_{r > 0} r^{d (1 / α^{'} - 1 / p^{'})} (\int_{J_{x}^{r}} {| μ ̂ (y) |}^{p^{'}} {d y)}^{1 / p^{'}} \leq D ₁ ρ^{- d / α^{'}}$ , $x \in ℝ^{d}$ , ⎩ p = 2 ⇒ infr≥1 rd(1/α’-1/2) (∫J₀r|μ̂(y)|² dy)1/2 ≥ D₂ρd/α’ $,$ where $J_{x}^{r} = \prod_{i = 1}^{d} (x_{i} - r / 2, x_{i} + r / 2)$ , α’ is defined by $ρ^{d / α^{'}} = {(\sum_{j = 1}^{m} a_{j}^{p})}^{1 / p}$ and the constants D₁ and D₂ depend only on d and p.

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

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Given an integer $n \geq 3$ , let $u_{1}, ..., u_{n}$ be pairwise coprime integers $\geq 2$ , $𝒟$ a family of nonempty proper subsets of ${1, ..., n}$ with “enough” elements, and $ε$ a function $𝒟 \to {\pm 1}$ . Does there exist at least one prime $q$ such that $q$ divides $\prod_{i \in I} u_{i} - ε (I)$ for some $I \in 𝒟$ , but it does not divide $u_{1} \dots u_{n}$ ? We answer this question in the positive when the $u_{i}$ are prime powers and $ε$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if $ε_{0} \in {\pm 1}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

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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Representation functions for binary linear forms

Fang-Gang Xue (2024)

Czechoslovak Mathematical Journal

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Let $ℤ$ be the set of integers, $ℕ_{0}$ the set of nonnegative integers and $F (x_{1}, x_{2}) = u_{1} x_{1} + u_{2} x_{2}$ be a binary linear form whose coefficients $u_{1}$ , $u_{2}$ are nonzero, relatively prime integers such that $u_{1} u_{2} \neq \pm 1$ and $u_{1} u_{2} \neq - 2$ . Let $f : ℤ \to ℕ_{0} \cup {\infty}$ be any function such that the set $f^{- 1} (0)$ has asymptotic density zero. In 2007, M. B. Nathanson (2007) proved that there exists a set $A$ of integers such that $r_{A, F} (n) = f (n)$ for all integers $n$ , where $r_{A, F} (n) = | {(a, a^{'}) : n = u_{1} a + u_{2} a^{'} : a, a^{'} \in A} |$ . We add the structure of difference for the binary linear form $F (x_{1}, x_{2})$ .

Towards Bauer's theorem for linear recurrence sequences

Mariusz Skałba (2003)

Colloquium Mathematicae

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Consider a recurrence sequence ${(x_{k})}_{k \in ℤ}$ of integers satisfying $x_{k + n} = a_{n - 1} x_{k + n - 1} + . . . + a ₁ x_{k + 1} + a ₀ x_{k}$ , where $a ₀, a ₁, . . ., a_{n - 1} \in ℤ$ are fixed and a₀ ∈ -1,1. Assume that $x_{k} > 0$ for all sufficiently large k. If there exists k₀∈ ℤ such that $x_{k ₀} < 0$ then for each negative integer -D there exist infinitely many rational primes q such that $q | x_{k}$ for some k ∈ ℕ and (-D/q) = -1.

Earnshaw’s Theorem in Electrostatics and a Conditional Converse to Dirichlet’s Theorem

Jeffrey Rauch (2013-2014)

Séminaire Laurent Schwartz — EDP et applications

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For the dynamics $x^{''} = - \nabla_{x} V (x)$ , an equilibrium point $\underset{̲}{x}$ are always unstable when on a neighborhood of $\underset{̲}{x}$ the non constant $V$ satisfies $P (x, \partial) V = 0$ for a real second order elliptic $P$ . The proof uses a result of Kozlov [6].

On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

On the structure of sequences with forbidden zero-sum subsequences

W. D. Gao, R. Thangadurai (2003)

Colloquium Mathematicae

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We study the structure of longest sequences in $ℤ ₙ^{d}$ which have no zero-sum subsequence of length n (or less). We prove, among other results, that for $n = 2^{a}$ and d arbitrary, or $n = 3^{a}$ and d = 3, every sequence of c(n,d)(n-1) elements in $ℤ ₙ^{d}$ which has no zero-sum subsequence of length n consists of c(n,d) distinct elements each appearing n-1 times, where $c (2^{a}, d) = 2^{d}$ and $c (3^{a}, 3) = 9$ .

Differences of two semiconvex functions on the real line

Václav Kryštof, Luděk Zajíček (2016)

Commentationes Mathematicae Universitatis Carolinae

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It is proved that real functions on $ℝ$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^{1}$ -functions, or of two strongly paraconvex functions) coincide with semismooth functions on $ℝ$ (i.e. those locally Lipschitz functions on $ℝ$ for which $f_{+}^{'} (x) = {lim}_{t \to x +} f_{+}^{'} (t)$ and $f_{-}^{'} (x) = {lim}_{t \to x -} f_{-}^{'} (t)$ for each $x$ ). Further, for each modulus $ω$ , we characterize the class $D S C_{ω}$ of functions on $ℝ$ which can be written as $f = g - h$ , where $g$ and $h$ are semiconvex with modulus $C ω$ (for some $C > 0$ ) using a new...

On the equation $ϕ (x + m) = ϕ (x) + m$

A. Makowski (1964)

Matematički Vesnik

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On C $^{*}$ -spaces

P. Srivastava, K. K. Azad (1981)

Matematički Vesnik

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$ℵ_{1}$ -incompactness of Z

Ralph McKenzie (1971)

Colloquium Mathematicae

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On a characterization of $L^{p}$ -norm

Janusz Matkowski (1989)

Annales Polonici Mathematici

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On the $p^{λ}$ problem

Stephan Baier (2004)

Acta Arithmetica

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On the proximate order of $f^{(s)} (z) * g^{(s)} (z)$ and ${[f (z) * g (z)]}^{(s)}$

M. K. Sen (1971)

Annales Polonici Mathematici

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The existence of $P (α)$ -points of N* for $ℵ_{0} α < 𝔠$

A. Szymański (1977)

Colloquium Mathematicae

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On spirallike functions of order $α$ and type $β$ with fixed second coefficient

M. K. Aouf (1988)

Matematički Vesnik

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An application of the method wherein the $p h i$ - and $c h i$ - methods are combined for the determination of the grating constant. [II.]

Swami Jnanananda (1936)

Časopis pro pěstování matematiky a fysiky

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Геометрическая теория состояний, граница фон Неймана, двойственность $C^{*}$ -алгерб

А.М. Вершик (1972)

Zapiski naucnych seminarov Leningradskogo

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On a fixobject property for $l^{c} β$ -semigroupoids

M. Đurić (1973)

Matematički Vesnik

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Kernel theorems in spaces of generalized functions

Antoine Delcroix (2010)

Banach Center Publications

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In analogy to the classical isomorphism between ((ℝⁿ), $^{'} (ℝ^{m}))$ and $^{'} (ℝ^{m + n})$ (resp. $((ℝ ⁿ),^{'} (ℝ^{m}))$ and $^{'} (ℝ^{m + n})$ ), we show that a large class of moderate linear mappings acting between the space $_{C} (ℝ ⁿ)$ of compactly supported generalized functions and (ℝⁿ) of generalized functions (resp. the space $_{} (ℝ ⁿ)$ of Colombeau rapidly decreasing generalized functions and the space $_{τ} (ℝ ⁿ)$ of temperate ones) admits generalized integral representations, with kernels belonging to specific regular subspaces of $(ℝ^{m + n})$ (resp. $_{τ} (ℝ^{m + n})$ ). The main novelty is to use...

Displaying similar documents to “Equicontinuity and Convergent Sequences in the Spaces C ' and M ”

Displaying similar documents to “Equicontinuity and Convergent Sequences in the Spaces $_{C}^{'}$ and $_{M}$ ”