The central heights of stability groups of series in vector spaces

Bertram A. F. Wehrfritz

Displaying similar documents to “The central heights of stability groups of series in vector spaces”

Quantitative stability for sumsets in $ℝ^{n}$

Alessio Figalli, David Jerison (2015)

Journal of the European Mathematical Society

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Given a measurable set $A \subset ℝ^{n}$ of positive measure, it is not difficult to show that $| A + A | = | 2 A |$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $(| A + A | - | 2 A |) / | A |$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $(| A + A | - | 2 A |) / | A |$ .

On Noether and strict stability, Hilbert exponent, and relative Nullstellensatz

Chia-chi Tung (2013)

Annales Polonici Mathematici

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Conditions characterizing the membership of the ideal of a subvariety arising from (effective) divisors in a product complex space Y × X are given. For the algebra $_{Y} [V]$ of relative regular functions on an algebraic variety V, the strict stability is proved, in the case where Y is a normal space, and the Noether stability is established under a weakened condition. As a consequence (for both general and complete intersections) a global Nullstellensatz is derived for divisors in $Y \times ℂ^{N}$ , respectively,...

$𝒟_{n, r}$ is not potentially nilpotent for $n \geq 4 r - 2$

Yan Ling Shao, Yubin Gao, Wei Gao (2016)

Czechoslovak Mathematical Journal

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An $n \times n$ sign pattern $𝒜$ is said to be potentially nilpotent if there exists a nilpotent real matrix $B$ with the same sign pattern as $𝒜$ . Let $𝒟_{n, r}$ be an $n \times n$ sign pattern with $2 \leq r \leq n$ such that the superdiagonal and the $(n, n)$ entries are positive, the $(i, 1)$ $(i = 1, \dots, r)$ and $(i, i - r + 1)$ $(i = r + 1, \dots, n)$ entries are negative, and zeros elsewhere. We prove that for $r \geq 3$ and $n \geq 4 r - 2$ , the sign pattern $𝒟_{n, r}$ is not potentially nilpotent, and so not spectrally arbitrary.

Certain decompositions of matrices over Abelian rings

Nahid Ashrafi, Marjan Sheibani, Huanyin Chen (2017)

Czechoslovak Mathematical Journal

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A ring $R$ is (weakly) nil clean provided that every element in $R$ is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let $R$ be abelian, and let $n \in ℕ$ . We prove that $M_{n} (R)$ is nil clean if and only if $R / J (R)$ is Boolean and $M_{n} (J (R))$ is nil. Furthermore, we prove that $R$ is weakly nil clean if and only if $R$ is periodic; $R / J (R)$ is $ℤ_{3}$ , $B$ or $ℤ_{3} \oplus B$ where $B$ is a Boolean ring, and that $M_{n} (R)$ is weakly nil clean if and only if $M_{n} (R)$ is nil clean for all $n \geq 2$ .

A note on infinite $a S$ -groups

Reza Nikandish, Babak Miraftab (2015)

Czechoslovak Mathematical Journal

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Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $a S$ -group. We study some properties of $a S$ -groups. For instance, it is shown that a nilpotent group $G$ is an $a S$ -group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $a S$ -group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...

A subclass of strongly clean rings

Orhan Gurgun, Sait Halicioglu and Burcu Ungor (2015)

Communications in Mathematics

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In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$ , and let $J^{#}$ denote the set of all elements of $R$ which are nilpotent in $R / J$ . An element $a \in R$ is called provided that there exists an idempotent $e \in R$ such that $a e = e a$ and $a - e$ or $a + e$ is an element of $J^{#}$ . A ring $R$ is said to be in case every element in $R$ is very $J^{#}$ -clean. We prove that every very $J^{#}$ -clean ring is strongly $π$ -rad clean and has stable range one. It is shown that for a...

Partial differential equations in Banach spaces involving nilpotent linear operators

Antonia Chinnì, Paolo Cubiotti (1996)

Annales Polonici Mathematici

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Let E be a Banach space. We consider a Cauchy problem of the type ⎧ $D_{t}^{k} u + \sum_{j = 0}^{k - 1} \sum_{| α | \leq m} A_{j, α} (D_{t}^{j} D_{x}^{α} u) = f$ in $ℝ^{n + 1}$ , ⎨ ⎩ $D_{t}^{j} u (0, x) = φ_{j} (x)$ in $ℝ^{n}$ , j=0,...,k-1, where each $A_{j, α}$ is a given continuous linear operator from E into itself. We prove that if the operators $A_{j, α}$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u \in C^{\infty} (ℝ^{n + 1}, E)$ whose derivatives are equi-bounded on each bounded subset of $ℝ^{n + 1}$ .

Semicommutativity of the rings relative to prime radical

Handan Kose, Burcu Ungor (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called $P$ -semicommutative. We prove that a ring $R$ is $P$ -semicommutative if and only if $R [x]$ is $P$ -semicommutative if and only if $R [x, x^{- 1}]$ is $P$ -semicommutative. Also, if $R [[x]]$ is $P$ -semicommutative, then $R$ is $P$ -semicommutative. The converse holds provided that $P (R)$ is nilpotent and $R$ is power serieswise Armendariz. For each positive integer $n$ , $R$ is $P$ -semicommutative...

Small and large time stability of the time taken for a Lévy process to cross curved boundaries

Philip S. Griffin, Ross A. Maller (2013)

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

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This paper is concerned with the small time behaviour of a Lévy process $X$ . In particular, we investigate theof the times, ${\bar{T}}_{b} (r)$ and $T_{b}^{*} (r)$ , at which $X$ , started with $X_{0} = 0$ , first leaves the space-time regions ${(t, y) \in ℝ^{2} : y \leq r t^{b}, t \geq 0}$ (one-sided exit), or ${(t, y) \in ℝ^{2} : | y | \leq r t^{b}, t \geq 0}$ (two-sided exit), $0 \leq b l t; 1$ , as $r ↓ 0$ . Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $L^{p}$ . In many instances these are...

On Lindenstrauss-Pełczyński spaces

Jesús M. F. Castillo, Yolanda Moreno, Jesús Suárez (2006)

Studia Mathematica

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We consider some stability aspects of the classical problem of extension of C(K)-valued operators. We introduce the class ℒ of Banach spaces of Lindenstrauss-Pełczyński type as those such that every operator from a subspace of c₀ into them can be extended to c₀. We show that all ℒ-spaces are of type $ℒ_{\infty}$ but not conversely. Moreover, $ℒ_{\infty}$ -spaces will be characterized as those spaces E such that E-valued operators from w*(l₁,c₀)-closed subspaces of l₁ extend to l₁. Regarding examples we will...

On feebly nil-clean rings

Marjan Sheibani Abdolyousefi, Neda Pouyan (2024)

Czechoslovak Mathematical Journal

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A ring $R$ is feebly nil-clean if for any $a \in R$ there exist two orthogonal idempotents $e, f \in R$ and a nilpotent $w \in R$ such that $a = e - f + w$ . Let $R$ be a 2-primal feebly nil-clean ring. We prove that every matrix ring over $R$ is feebly nil-clean. The result for rings of bounded index is also obtained. These provide many classes of rings over which every matrix is the sum of orthogonal idempotent and nilpotent matrices.

On the structural theory of ${II}_{1}$ factors of negatively curved groups

Ionut Chifan, Thomas Sinclair (2013)

Annales scientifiques de l'École Normale Supérieure

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Ozawa showed in [21] that for any i.c.c. hyperbolic group, the associated group factor $L Γ$ is solid. Developing a new approach that combines some methods of Peterson [29], Ozawa and Popa [27, 28], and Ozawa [25], we strengthen this result by showing that $L Γ$ is strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana [12], we show that profinite actions of lattices in $Sp (n, 1)$ , $n \geq 2$ , are virtually $W^{*}$ -superrigid.

A note on normal generation and generation of groups

Andreas Thom (2015)

Communications in Mathematics

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In this note we study sets of normal generators of finitely presented residually $p$ -finite groups. We show that if an infinite, finitely presented, residually $p$ -finite group $G$ is normally generated by $g_{1}, \dots, g_{k}$ with order $n_{1}, \dots, n_{k} \in {1, 2, \dots} \cup {\infty}$ , then $β_{1}^{(2)} (G) \leq k - 1 - \sum_{i = 1}^{k} \frac{1}{n_{i}},$ where $β_{1}^{(2)} (G)$ denotes the first $ℓ^{2}$ -Betti number of $G$ . We also show that any $k$ -generated group with $β_{1}^{(2)} (G) \geq k - 1 - ε$ must have girth greater than or equal $1 / ε$ .