Bad properties of the Bernstein numbers

Albrecht Pietsch

Displaying similar documents to “Bad properties of the Bernstein numbers”

Approximation properties for modified $(p, q)$ -Bernstein-Durrmeyer operators

Mohammad Mursaleen, Ahmed A. H. Alabied (2018)

Mathematica Bohemica

Similarity:

We introduce modified $(p, q)$ -Bernstein-Durrmeyer operators. We discuss approximation properties for these operators based on Korovkin type approximation theorem and compute the order of convergence using usual modulus of continuity. We also study the local approximation property of the sequence of positive linear operators $D_{n, p, q}^{*}$ and compute the rate of convergence for the function $f$ belonging to the class ${Lip}_{M} (γ)$ .

Semiproper ideals

Hiroshi Sakai (2005)

Fundamenta Mathematicae

Similarity:

We say that an ideal I on $_{κ} λ$ is semiproper if the corresponding poset $ℙ_{I}$ is semiproper. In this paper we investigate properties of semiproper ideals on $_{κ} λ$ .

Siciak’s extremal function via Bernstein and Markov constants for compact sets in $ℂ^{N}$

Leokadia Bialas-Ciez (2012)

Annales Polonici Mathematici

Similarity:

The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E \subset ℂ^{N}$ . We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function $Φ_{E}$ . Moreover, we show that one of these extremal-like functions is equal to $Φ_{E}$ if E is a nonpluripolar set with $l i m_{n \to \infty} M ₙ {(E)}^{1 / n} = 1$ where $M ₙ (E) : = s u p {| | | g r a d P | | |}_{E} / {| | P | |}_{E}$ , the supremum is taken over all polynomials P of N variables...

On norm closed ideals in $L (ℓ_{p}, ℓ_{q})$

B. Sari, Th. Schlumprecht, N. Tomczak-Jaegermann, V. G. Troitsky (2007)

Studia Mathematica

Similarity:

It is well known that the only proper non-trivial norm closed ideal in the algebra L(X) for $X = ℓ_{p}$ (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of $L (ℓ_{p} \oplus ℓ_{q})$ for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in $L (ℓ_{p}, ℓ_{q})$ for p < q. This paper shows that for 1 < p < 2 < q < ∞ there are at least four distinct proper closed ideals in $L (ℓ_{p}, ℓ_{q})$ , including one that has not been studied before. The proofs use various methods...

On domains with ACC on invertible ideals

Stefania Gabelli (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

If $A$ is a domain with the ascending chain condition on (integral) invertible ideals, then the group $I(A)$ of its invertible ideals is generated by the set $I_{m}(A)$ of maximal invertible ideals. In this note we study some properties of $I_{m}(A)$ and we prove that, if $I(A)$ is a free group on $I_{m}(A)$ , then $A$ is a locally factorial Krull domain.

Bernstein and De Giorgi type problems: new results via a geometric approach

Alberto Farina, Berardino Sciunzi, Enrico Valdinoci (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form $div (a (| \nabla u (x) |) \nabla u (x)) + f (u (x)) = 0 .$ Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in $ℝ^{2}$ and $ℝ^{3}$ and of the Bernstein problem on the flatness of minimal area graphs in $ℝ^{3}$ . A one-dimensional symmetry result in the half-space is also obtained as a byproduct...

Effective Nullstellensatz for arbitrary ideals

János Kollár (1999)

Journal of the European Mathematical Society

Similarity:

Let $f_{i}$ be polynomials in $n$ variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials $g_{i}$ such that $\sum g_{i} f_{i} = 1$ . The effective versions of this result bound the degrees of the $g_{i}$ in terms of the degrees of the $f_{j}$ . The aim of this paper is to generalize this to the case when the $f_{i}$ are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.

On quasi $n$ -ideals of commutative rings

Adam Anebri, Najib Mahdou, Emel Aslankarayiğit Uğurlu (2022)

Czechoslovak Mathematical Journal

Similarity:

Let $R$ be a commutative ring with a nonzero identity. In this study, we present a new class of ideals lying properly between the class of $n$ -ideals and the class of $(2, n)$ -ideals. A proper ideal $I$ of $R$ is said to be a quasi $n$ -ideal if $\sqrt{I}$ is an $n$ -ideal of $R .$ Many examples and results are given to disclose the relations between this new concept and others that already exist, namely, the $n$ -ideals, the quasi primary ideals, the $(2, n)$ -ideals and the $p r$ -ideals. Moreover, we use the quasi $n$ -ideals to characterize...

Monomial ideals with tiny squares and Freiman ideals

Ibrahim Al-Ayyoub, Mehrdad Nasernejad (2021)

Czechoslovak Mathematical Journal

Similarity:

We provide a construction of monomial ideals in $R = K [x, y]$ such that $μ (I^{2}) < μ (I)$ , where $μ$ denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring $R$ , we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case of this characterization leads to some further investigations on $μ (I^{k})$ that generalize...

$(δ, 2)$ -primary ideals of a commutative ring

Gülşen Ulucak, Ece Yetkin Çelikel (2020)

Czechoslovak Mathematical Journal

Similarity:

Let $R$ be a commutative ring with nonzero identity, let $ℐ (ℛ)$ be the set of all ideals of $R$ and $δ : ℐ (ℛ) \to ℐ (ℛ)$ an expansion of ideals of $R$ defined by $I \mapsto δ (I)$ . We introduce the concept of $(δ, 2)$ -primary ideals in commutative rings. A proper ideal $I$ of $R$ is called a $(δ, 2)$ -primary ideal if whenever $a, b \in R$ and $a b \in I$ , then $a^{2} \in I$ or $b^{2} \in δ (I)$ . Our purpose is to extend the concept of $2$ -ideals to $(δ, 2)$ -primary ideals of commutative rings. Then we investigate the basic properties of $(δ, 2)$ -primary ideals and also discuss the relations among $(δ, 2)$ -primary, $δ$ -primary...

On domains with ACC on invertible ideals

Stefania Gabelli (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

On Bernstein inequalities for multivariate trigonometric polynomials in $L_{p}$ , $0 \leq p \leq \infty$

Laiyi Zhu, Xingjun Zhao (2022)

Czechoslovak Mathematical Journal

Similarity:

Let $𝕋_{n}$ be the space of all trigonometric polynomials of degree not greater than $n$ with complex coefficients. Arestov extended the result of Bernstein and others and proved that $∥ (1 / n) T_{n}^{'} ∥_{p} \leq {∥ T_{n} ∥}_{p}$ for $0 \leq p \leq \infty$ and $T_{n} \in 𝕋_{n}$ . We derive the multivariate version of the result of Golitschek and Lorentz ${∥{|T_{n} cos α + \frac{1}{n} \nabla T_{n} sin α|}_{l_{\infty}^{(m)}}∥}_{p} \leq {∥ T_{n} ∥}_{p}, 0 \leq p \leq \infty$ for all trigonometric polynomials (with complex coeffcients) in $m$ variables of degree at most $n$ .

$G r$ - $(2, n)$ -ideals in graded commutative rings

Khaldoun Al-Zoubi, Shatha Alghueiri, Ece Y. Celikel (2020)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let $G$ be a group with identity $e$ and let $R$ be a $G$ -graded ring. In this paper, we introduce and study the concept of graded $(2, n)$ -ideals of $R$ . A proper graded ideal $I$ of $R$ is called a graded $(2, n)$ -ideal of $R$ if whenever $r s t \in I$ where $r, s, t \in h (R)$ , then either $r t \in I$ or $r s \in G r (0)$ or $s t \in G r (0)$ . We introduce several results concerning $g r$ - $(2, n)$ -ideals. For example, we give a characterization of graded $(2, n)$ -ideals and their homogeneous components. Also, the relations between graded $(2, n)$ -ideals and others that already exist, namely, the graded prime...

Non-regularity for Banach function algebras

J. Feinstein, D. Somerset (2000)

Studia Mathematica

Similarity:

Let A be a unital Banach function algebra with character space $Φ_{A}$ . For $x \in Φ_{A}$ , let $M_{x}$ and $J_{x}$ be the ideals of functions vanishing at x and in a neighbourhood of x, respectively. It is shown that the hull of $J_{x}$ is connected, and that if x does not belong to the Shilov boundary of A then the set $y \in Φ_{A} : M_{x} \supseteq J_{y}$ has an infinite connected subset. Various related results are given.

An irrational problem

Franklin D. Tall (2002)

Fundamenta Mathematicae

Similarity:

Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define $X_{M}$ to be X ∩ M with topology generated by $U \cap M : U \in \cap M$ . Suppose $X_{M}$ is homeomorphic to the irrationals; must $X = X_{M}$ ? We have partial results. We also answer a question of Gruenhage by showing that if $X_{M}$ is homeomorphic to the “Long Cantor Set”, then $X = X_{M}$ .

Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups

Lingli Zeng, Jizhu Nan (2016)

Czechoslovak Mathematical Journal

Similarity:

Let $F$ be a finite field of characteristic $p$ and $K$ a field which contains a primitive $p$ th root of unity and $char K \neq p$ . Suppose that a classical group $G$ acts on the $F$ -vector space $V$ . Then it can induce the actions on the vector space $V \oplus V$ and on the group algebra $K [V \oplus V]$ , respectively. In this paper we determine the structure of $G$ -invariant ideals of the group algebra $K [V \oplus V]$ , and establish the relationship between the invariant ideals of $K [V]$ and the vector invariant ideals of $K [V \oplus V]$ , if $G$ is a unitary group or orthogonal...

Basic relations valid for the Bernstein spaces $B ²_{σ}$ and their extensions to larger function spaces via a unified distance concept

P. L. Butzer, R. L. Stens, G. Schmeisser (2014)

Banach Center Publications

Similarity:

Some basic theorems and formulae (equations and inequalities) of several areas of mathematics that hold in Bernstein spaces $B_{σ}^{p}$ are no longer valid in larger spaces. However, when a function f is in some sense close to a Bernstein space, then the corresponding relation holds with a remainder or error term. This paper presents a new, unified approach to these errors in terms of the distance of f from $B_{σ}^{p}$ . The difficult situation of derivative-free error estimates is also covered. ...

Generalization of the $S$ -Noetherian concept

Abdelamir Dabbabi, Ali Benhissi (2023)

Archivum Mathematicum

Similarity:

Let $A$ be a commutative ring and $𝒮$ a multiplicative system of ideals. We say that $A$ is $𝒮$ -Noetherian, if for each ideal $Q$ of $A$ , there exist $I \in 𝒮$ and a finitely generated ideal $F \subseteq Q$ such that $I Q \subseteq F$ . In this paper, we study the transfer of this property to the polynomial ring and Nagata’s idealization.

On the mappings $𝒵_{A}$ and $ℑ_{A}$ in intermediate rings of $C (X)$

Mehdi Parsinia (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In this article, we investigate new topological descriptions for two well-known mappings $𝒵_{A}$ and $ℑ_{A}$ defined on intermediate rings $A (X)$ of $C (X)$ . Using this, coincidence of each two classes of $z$ -ideals, $𝒵_{A}$ -ideals and $ℑ_{A}$ -ideals of $A (X)$ is studied. Moreover, we answer five questions concerning the mapping $ℑ_{A}$ raised in [J. Sack, S. Watson, $C$ and $C^{*}$ among intermediate rings, Topology Proc. 43 (2014), 69–82].