scholarly journals Sharp Moser–Trudinger inequality on the Heisenberg group at the critical case and applications

2012 ◽  
Vol 231 (6) ◽  
pp. 3259-3287 ◽  
Author(s):  
Nguyen Lam ◽  
Guozhen Lu
Keyword(s):  
Heisenberg Group ◽  
Critical Case ◽  
Trudinger Inequality

Extremal functions for the Moser–Trudinger inequality of Adimurthi–Druet type in W1,N(ℝN)

2019 ◽  
Vol 21 (04) ◽  
pp. 1850023 ◽  
Author(s):  
Van Hoang Nguyen
Keyword(s):  
Critical Case ◽  
Blow Up ◽  
Sharp Inequality ◽  
Extremal Functions ◽  
Analysis Method ◽  
Subcritical Case ◽  
Best Constant ◽  
Blow Up Analysis ◽  
Nirenberg Inequality ◽  
Trudinger Inequality

We study the existence and nonexistence of maximizers for variational problem concerning the Moser–Trudinger inequality of Adimurthi–Druet type in [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text] both in the subcritical case [Formula: see text] and critical case [Formula: see text] with [Formula: see text] and [Formula: see text] denotes the surface area of the unit sphere in [Formula: see text]. We will show that MT[Formula: see text] is attained in the subcritical case if [Formula: see text] or [Formula: see text] and [Formula: see text] with [Formula: see text] being the best constant in a Gagliardo–Nirenberg inequality in [Formula: see text]. We also show that MT[Formula: see text] is not attained for [Formula: see text] small which is different from the context of bounded domains. In the critical case, we prove that MT[Formula: see text] is attained for [Formula: see text] small enough. To prove our results, we first establish a lower bound for MT[Formula: see text] which excludes the concentrating or vanishing behaviors of their maximizer sequences. This implies the attainability of MT[Formula: see text] in the subcritical case. The proof in the critical case is based on the blow-up analysis method. Finally, by using the Moser sequence together with the scaling argument, we show that MT[Formula: see text]. Our results settle the questions left open in [J. M. do Ó and M. de Souza, A sharp inequality of Trudinger–Moser type and extremal functions in [Formula: see text], J. Differential Equations 258 (2015) 4062–4101; Trudinger–Moser inequality on the whole plane and extremal functions, Commun. Contemp. Math. 18 (2016) 32 pp.].

The Moser–Trudinger inequality in unbounded domains of Heisenberg group and sub-elliptic equations

2012 ◽  
Vol 75 (12) ◽  
pp. 4483-4495 ◽  
Author(s):  
William S. Cohn ◽  
Nguyen Lam ◽  
Guozhen Lu ◽  
Yunyan Yang
Keyword(s):  
Heisenberg Group ◽  
Elliptic Equations ◽  
Unbounded Domains ◽  
Trudinger Inequality

Weighted estimates for commutators of Hausdorff operators on the Heisenberg group

2020 ◽  
pp. 39-62
Author(s):  
Nguyen Minh Chuong ◽  
◽  
Dao Van Duong ◽  
Nguyen Duc Duyet ◽  
◽  
...  
Keyword(s):  
Heisenberg Group ◽  
Weighted Estimates ◽  
Hausdorff Operators

Qualities of a good essay: an assessment of the writings of Nigerian undergraduates

2020 ◽  
Vol 3 (7) ◽  
pp. 82-88
Author(s):  
Usoro Mark Okono
Keyword(s):  
Critical Case ◽  
State University ◽  
Descriptive Linguistics ◽  
Purposive Sampling ◽  
Writing Test ◽  
Units Of Analysis ◽  
Regular Practice

This research sought to discover the capabilities of Nigerian undergraduates in handling the salient characteristics of essay in English. Such qualities as clarity, economy, simplicity, unity and coherence were the variables in the assessment. The study was conducted within the framework of the theory of descriptive linguistics and its sub-discipline of stylistics. Four topics representing argumentative, descriptive, expository and narrative essays were given to students for each of them to voluntarily choose one and write on in a strictly supervised writing test. All the essays were marked on the above stated variables. Critical case sampling strand of the purposive sampling was used to select four outstanding essays each representing one of the four departments of the Akwa Ibom State University of Nigeria. Paragraph and sentence formed some of the units of analysis. It was found out that the four subjects whose essays were analyzed proved their mettle in producing readable and creative prose in the four genres with some room for improvement. It is suggested that the Use of English programme in Nigerian universities should be extended from one to two years in addition to regular practice in writing by students and feedback from lecturers.

The horofunction boundary of the Heisenberg group

2009 ◽  
Vol 242 (2) ◽  
pp. 299-310 ◽  
Author(s):  
Tom Klein ◽  
Andrew Nicas
Keyword(s):  
Heisenberg Group

scholarly journals Publications about Women, Science, and Engineering: Use of Sex and Gender in Titles over a Forty-six-year Period

2021 ◽  
pp. 016224392110276
Author(s):  
Mary Frank Fox ◽  
Diana Roldan Rueda ◽  
Gerhard Sonnert ◽  
Amanda Nabors ◽  
Sarah Bartel
Keyword(s):  
Social Stratification ◽  
Critical Case ◽  
Social Inequalities ◽  
Sex And Gender ◽  
Science And Engineering ◽  
Analytical Issue ◽  
Key Features ◽  
And Gender ◽  
Science Policies

This article focuses on key features of the use of sex and gender in titles of articles about women, science, and engineering over an important forty-six-year period (1965–2010). The focus is theoretically and empirically consequential. Theoretically, the paper addresses science as a critical case that connects femininity/masculinity to social stratification; and the use of sex and gender as an enduring, analytical issue that reveals perspectives on hierarchies of femininity/masculinity. Empirically, this article identifies the emergence, development, and stabilization of published articles about women, science, and engineering that use sex and gender in their titles. The distinctive method involves search, retrieval, and review of 23,430 articles, using intercoder reliabilities for inclusion/exclusion. This results in a uniquely specified and comprehensive set of articles on our subject and the identification of titles with sex and gender. Findings point to (1) the growth of gender titles, (2) their increase in every field, (3) differing concentrations of sex and gender titles in journals, (4) a span of telling topic areas, and (5) higher citation rates of gender, compared to sex, titles. Broader implications appear in reasons for the growth of gender titles, meanings of topic areas that occur, insights into social inequalities and science policies, and emerging complexities of nonbinary categories of sex/gender.

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