Bad properties of the Bernstein numbers

Albrecht Pietsch

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

Semiproper ideals

Hiroshi Sakai (2005)

Fundamenta Mathematicae

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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

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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

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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

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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

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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

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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 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

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Non-regularity for Banach function algebras

J. Feinstein, D. Somerset (2000)

Studia Mathematica

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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

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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

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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

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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. ...

$α$ -ideals in $0$ -distributive posets

Khalid A. Mokbel (2015)

Mathematica Bohemica

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The concept of $α$ -ideals in posets is introduced. Several properties of $α$ -ideals in $0$ -distributive posets are studied. Characterization of prime ideals to be $α$ -ideals in $0$ -distributive posets is obtained in terms of minimality of ideals. Further, it is proved that if a prime ideal $I$ of a $0$ -distributive poset is non-dense, then $I$ is an $α$ -ideal. Moreover, it is shown that the set of all $α$ -ideals $α Id (P)$ of a poset $P$ with $0$ forms a complete lattice. A result analogous to separation theorem for...

Transitive Properties of Ideals on Generalized Cantor Spaces

Jan Kraszewski (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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We compute transitive cardinal coefficients of ideals on generalized Cantor spaces. In particular, we show that there exists a null set $A \subseteq 2^{ω ₁}$ such that for every null set $B \subseteq 2^{ω ₁}$ we can find $x \in 2^{ω ₁}$ such that A ∪ (A+x) cannot be covered by any translation of B.

Base change for Bernstein centers of depth zero principal series blocks

Thomas J. Haines (2012)

Annales scientifiques de l'École Normale Supérieure

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Let $G$ be an unramified group over a $p$ -adic field. This article introduces a base change homomorphism for Bernstein centers of depth-zero principal series blocks for $G$ and proves the corresponding base change fundamental lemma. This result is used in the approach to Shimura varieties with $Γ_{1} (p)$ -level structure initiated by M. Rapoport and the author in [15].

Erratum to “Can $ℬ (ℓ^{p})$ ever be amenable?” (Studia Math. 188 (2008), 151-174)

Matthew Daws, Volker Runde (2009)

Studia Mathematica

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Two ideals connected with strong right upper porosity at a point

Viktoriia Bilet, Oleksiy Dovgoshey, Jürgen Prestin (2015)

Czechoslovak Mathematical Journal

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Let $SP$ be the set of upper strongly porous at $0$ subsets of $ℝ^{+}$ and let $\hat{I} (SP)$ be the intersection of maximal ideals $I \subseteq SP$ . Some characteristic properties of sets $E \in \hat{I} (SP)$ are obtained. We also find a characteristic property of the intersection of all maximal ideals contained in a given set which is closed under subsets. It is shown that the ideal generated by the so-called completely strongly porous at $0$ subsets of $ℝ^{+}$ is a proper subideal of $\hat{I} (SP) .$ Earlier, completely strongly porous sets and some of their properties...

Remarks on $L B I$ -subalgebras of $C (X)$

Mehdi Parsinia (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $A (X)$ denote a subalgebra of $C (X)$ which is closed under local bounded inversion, briefly, an $L B I$ -subalgebra. These subalgebras were first introduced and studied in Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151–163. By characterizing maximal ideals of $A (X)$ , we generalize the notion of $z_{A}^{β}$ -ideals, which was first introduced in Acharyya S.K., De D., An interesting class of ideals in subalgebras of $C (X)$ ...

The prime ideals intersection graph of a ring

M. J. Nikmehr, B. Soleymanzadeh (2017)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a commutative ring with unity and $U (R)$ be the set of unit elements of $R$ . In this paper, we introduce and investigate some properties of a new kind of graph on the ring $R$ , namely, the prime ideals intersection graph of $R$ , denoted by $G_{p} (R)$ . The $G_{p} (R)$ is a graph with vertex set $R^{*} - U (R)$ and two distinct vertices $a$ and $b$ are adjacent if and only if there exists a prime ideal $𝔭$ of $R$ such that $a, b \in 𝔭$ . We obtain necessary and sufficient conditions on $R$ such that $G_{p} (R)$ is disconnected. We find the diameter and...

Squarefree Lexsegment Ideals with Linear Resolution

Vittoria Bonanzinga, Loredana Sorrenti (2008)

Bollettino dell'Unione Matematica Italiana

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In this paper we determine all squarefree completely lexsegment ideals which have a linear resolution. Let $M_{d}$ denote the set of all squarefree monomials of degree $d$ in a polynomial ring $k[x_{1},\ldots,x_{n}]$ in $n$ variables over a field $k$ . We order the monomials lexicographically such that $x_{1}>x_{2}>\ldots>x_{n}$ , thus a lexsegment (of degree $d$ ) is a subset of $M_{d}$ of the form $L(u,v)=\{w\in M_{d}:u\geq w\geq v\}$ for some $u,v\in M_{d}$ con $u\geq v$ . An ideal generated by a lexsegment is called a lexsegment ideal. We describe the procedure to determine when such an ideal has a linear...

The strong persistence property and symbolic strong persistence property

Mehrdad Nasernejad, Kazem Khashyarmanesh, Leslie G. Roberts, Jonathan Toledo (2022)

Czechoslovak Mathematical Journal

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Let $I$ be an ideal in a commutative Noetherian ring $R$ . Then the ideal $I$ has the strong persistence property if and only if $(I^{k + 1} :_{R} I) = I^{k}$ for all $k$ , and $I$ has the symbolic strong persistence property if and only if $(I^{(k + 1)} :_{R} I^{(1)}) = I^{(k)}$ for all $k$ , where $I^{(k)}$ denotes the $k$ th symbolic power of $I$ . We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial...