Radicals and one-sided ideals: an addendum

1988 ◽  
Vol 108 (1-2) ◽  
pp. 89-90 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Affirmative Answer ◽  
Hereditary Radical ◽  
Open Question

SynopsisIn the paper referred to in the title [2] an open question was raised concerning the equality of the largest left hereditary radical and the largest right hereditary radical contained in each of certain radicals. In this addendum an affirmative answer is provided to this question.

DERIVATIONS OF FRÉCHET NUCLEAR GB-ALGEBRAS

2015 ◽  
Vol 92 (2) ◽  
pp. 290-301 ◽  
Author(s):  
M. WEIGT ◽  
I. ZARAKAS
Keyword(s):  
Affirmative Answer ◽  
Open Question

It is an open question whether every derivation of a Fréchet GB$^{\ast }$-algebra $A[{\it\tau}]$ is continuous. We give an affirmative answer for the case where $A[{\it\tau}]$ is a smooth Fréchet nuclear GB$^{\ast }$-algebra. Motivated by this result, we give examples of smooth Fréchet nuclear GB$^{\ast }$-algebras which are not pro-C$^{\ast }$-algebras.

Open questions concerning antiautomorphisms of division rings with quasi-generalized Engel conditions

2019 ◽  
Vol 18 (09) ◽  
pp. 1950167 ◽  
Author(s):  
M. Chacron ◽  
T.-K. Lee
Keyword(s):  
Finite Order ◽  
Division Ring ◽  
Affirmative Answer ◽  
Major Result ◽  
Division Rings ◽  
Open Questions ◽  
Engel Condition ◽  
Open Question ◽  
Positive Integers

Let [Formula: see text] be a noncommutative division ring with center [Formula: see text], which is algebraic, that is, [Formula: see text] is an algebraic algebra over the field [Formula: see text]. Let [Formula: see text] be an antiautomorphism of [Formula: see text] such that (i) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers depending on [Formula: see text]. If, further, [Formula: see text] has finite order, it was shown in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] that [Formula: see text] is commuting, that is, [Formula: see text], all [Formula: see text]. Posed in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] is the question which asks as to whether the finite order requirement on [Formula: see text] can be dropped. We provide here an affirmative answer to the question. The second major result of this paper is concerned with a nonnecessarily algebraic division ring [Formula: see text] with an antiautomorphism [Formula: see text] satisfying the stronger condition (ii) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are fixed positive integers. It was shown in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036] that if, further, [Formula: see text] has finite order then [Formula: see text] is commuting. We show here, that again the finite order assumption on [Formula: see text] can be lifted answering thus in the affirmative the open question (see Question 2.11 in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036]).

scholarly journals An extension of the Kegel–Wielandt theorem to locally finite groups

1996 ◽  
Vol 38 (2) ◽  
pp. 171-176
Author(s):  
Silvana Franciosi ◽  
Francesco de Giovanni ◽  
Yaroslav P. Sysak
Keyword(s):  
Finite Group ◽  
Affirmative Answer ◽  
Locally Finite Group ◽  
Linear Groups ◽  
Arbitrary Group ◽  
Famous Theorem ◽  
Locally Finite ◽  
Locally Nilpotent ◽  
Finite Products ◽  
Open Question

A famous theorem of Kegel and Wielandt states that every finite group which is the product of two nilpotent subgroups is soluble (see [1], Theorem 2.4.3). On the other hand, it is an open question whether an arbitrary group factorized by two nilpotent subgroups satisfies some solubility condition, and only a few partial results are known on this subject. In particular, Kegel [6] obtained an affirmative answer in the case of linear groups, and in the same article he also proved that every locally finite group which is the product of two locally nilpotent FC-subgroups is locally soluble. Recall that a group G is said to be an FC-group if every element of G has only finitely many conjugates. Moreover, Kazarin [5] showed that if the locally finite group G = AB is factorized by an abelian subgroup A and a locally nilpotent subgroup B, then G is locally soluble. The aim of this article is to prove the following extension of the Kegel–Wielandt theorem to locally finite products of hypercentral groups.

Free Will, Resiliency and Flip-flopping

2019 ◽  
Vol 35 (1) ◽  
pp. 91-98
Author(s):  
James Cain ◽  
Keyword(s):  
Free Will ◽  
A Priori ◽  
Affirmative Answer ◽  
John Martin Fischer ◽  
Open Question

Many philosophers accept with certainty that we are morally responsible but take it to be an open question whether determinism holds. They treat determinism as epistemically compatible with responsibility. Should one who accepts this form of epistemic compatibilism also hold that determinism is metaphysically compatible with responsibility—that it is metaphysically possible for determinism and responsibility to coexist? John Martin Fischer gives two arguments that appear to favor an affirmative answer to this question. He argues that accounts of responsibility, such as his, that are neutral with respect to whether responsible actions are determined have a “resiliency” that counts in their favor. Furthermore, he criticizes libertarians who argue on a priori grounds that determinism cannot coexist with responsibility and who admit that they would retract their argument if determinism were shown to hold; this “metaphysical fl ip-flopping” is said to render their positions implausible. I assess the merits of these arguments.

scholarly journals COMMUTATIVITY DEGREE OF A CLASS OF FINITE GROUPS AND CONSEQUENCES

2013 ◽  
Vol 88 (3) ◽  
pp. 448-452 ◽  
Author(s):  
RAJAT KANTI NATH
Keyword(s):  
Finite Group ◽  
Semidirect Product ◽  
Finite Groups ◽  
Abelian Groups ◽  
Affirmative Answer ◽  
Finite Abelian Groups ◽  
A Finite Group ◽  
Open Question

AbstractThe commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The object of this paper is to compute the commutativity degree of a class of finite groups obtained by semidirect product of two finite abelian groups. As a byproduct of our result, we provide an affirmative answer to an open question posed by Lescot.

scholarly journals Private Stochastic Non-convex Optimization with Improved Utility Rates

2021 ◽  
Author(s):  
Qiuchen Zhang ◽  
Jing Ma ◽  
Jian Lou ◽  
Li Xiong
Keyword(s):  
Convex Optimization ◽  
Nonconvex Optimization ◽  
Deep Neural Networks ◽  
Differential Privacy ◽  
Affirmative Answer ◽  
Empirical Risk ◽  
Convex Case ◽  
Open Question ◽  
Theoretical Results ◽  
Utility Measures

We study the differentially private (DP) stochastic nonconvex optimization with a focus on its under-studied utility measures in terms of the expected excess empirical and population risks. While the excess risks are extensively studied for convex optimization, they are rarely studied for nonconvex optimization, especially the expected population risk. For the convex case, recent studies show that it is possible for private optimization to achieve the same order of excess population risk as to the nonprivate optimization under certain conditions. It still remains an open question for the nonconvex case whether such ideal excess population risk is achievable. In this paper, we progress towards an affirmative answer to this open problem: DP nonconvex optimization is indeed capable of achieving the same excess population risk as to the nonprivate algorithm in most common parameter regimes, under certain conditions (i.e., well-conditioned nonconvexity). We achieve such improved utility rates compared to existing results by designing and analyzing the stagewise DP-SGD with early momentum algorithm. We obtain both excess empirical risk and excess population risk to achieve differential privacy. Our algorithm also features the first known results of excess and population risks for DP-SGD with momentum. Experiment results on both shallow and deep neural networks when respectively applied to simple and complex real datasets corroborate the theoretical results.

scholarly journals On An Open Question in Controlled Rectangular b-Metric Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2239
Author(s):  
Reny George ◽  
Abdelkader Belhenniche ◽  
Sfya Benahmed ◽  
Zoran D. Mitrović ◽  
Nabil Mlaiki ◽  
...  
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Metric Spaces ◽  
Affirmative Answer ◽  
Existence Of A Solution ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
Open Question

In this paper, we give an affirmative answer to an open question posed recently by Mlaiki et al. As a consequence of our results, we get some known results in the literature. We also give an application of our results to the existence of a solution of nonlinear fractional differential equations.

scholarly journals COMPOSITION OPERATORS ON WIENER AMALGAM SPACES

2019 ◽  
Vol 240 ◽  
pp. 257-274
Author(s):  
DIVYANG G. BHIMANI
Keyword(s):  
Composition Operator ◽  
Composition Operators ◽  
Complex Function ◽  
Affirmative Answer ◽  
Modulation Spaces ◽  
Wiener Amalgam Spaces ◽  
Real Analytic ◽  
Amalgam Spaces ◽  
Open Question ◽  
Funct Anal

For a complex function $F$ on $\mathbb{C}$, we study the associated composition operator $T_{F}(f):=F\circ f=F(f)$ on Wiener amalgam $W^{p,q}(\mathbb{R}^{d})\;(1\leqslant p<\infty ,1\leqslant q<2)$. We have shown $T_{F}$ maps $W^{p,1}(\mathbb{R}^{d})$ to $W^{p,q}(\mathbb{R}^{d})$ if and only if $F$ is real analytic on $\mathbb{R}^{2}$ and $F(0)=0$. Similar result is proved in the case of modulation spaces $M^{p,q}(\mathbb{R}^{d})$. In particular, this gives an affirmative answer to the open question proposed in Bhimani and Ratnakumar (J. Funct. Anal. 270(2) (2016), 621–648).

A new correctness criterion for the proof nets of non-commutative multiplicative linear logics

2001 ◽  
Vol 66 (4) ◽  
pp. 1524-1542 ◽  
Author(s):  
Misao Nagayama ◽  
Mitsuhiro Okada
Keyword(s):  
Linear Time ◽  
Linear Logic ◽  
Time Algorithm ◽  
Affirmative Answer ◽  
Correctness Criterion ◽  
Correctness Checking ◽  
Proof Nets ◽  
Open Question

Abstract.This paper presents a new correctness criterion for marked Danos-Reginer graphs (D-R graphs, for short) of Multiplicative Cyclic Linear Logic MCLL and Abrusci's non-commutative Linear Logic MNLL.As a corollary we obtain an affirmative answer to the open question whether a known quadratic-time algorithm for the correctness checking of proof nets for MCLL and MNLL can be improved to linear-time.

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