Critical Norm Research Articles

Blow-up of the critical norm for a supercritical semilinear heat equation

We consider the scaling critical Lebesgue norm of blow-up solutions to the semilinear heat equation u_{t}=\Delta u+|u|^{p-1}u in an arbitrary C^{2+\alpha} domain of \mathbf{R}^{n} . In the range p>p_{S}:=(n+2)/(n-2) , we show that the critical norm must be unbounded near the blow-up time, where the type I blow-up condition is not imposed. The range p>p_{S} is optimal in view of the existence of type II blow-up solutions with bounded critical norm for p=p_{S} .

Teaching the Middle Set: What Counts as Being ‘Able’ to Talk About a Character?

ABSTRACT This essay considers what happens in the English classroom when teaching the same lesson to two classes considered to be of different levels of ‘ability’. It explores what happens during a discussion about the fate of Eva Smith in An Inspector Calls when students’ reading of a text diverges. I consider what teachers do when students’ readings of a text goes against our expectations and accepted critical norms of both the teacher and the National Curriculum. Finally, I explore the potential problems with using retrieval practice as a way to manage divergent readings and interpretations in classroom discussion.

Academic freedom and innovation.

Academic freedom is a critical norm of science. Despite the widely postulated importance of academic freedom, the literature attests to a dearth of research on the topic. Specifically, we know little about how academic freedom relates to indicators of societal progress, such as innovation. We address this research gap by empirically assessing the impact of academic freedom on the quantity (patent applications) and quality (patent citations) of innovation output using a comprehensive sample of 157 countries over the 1900-2015 period. We find that improving academic freedom by one standard deviation increases patent applications by 41% and forward citations by 29%. The results are robust across a range of different specifications. Our findings constitute an alarming plea to policymakers: global academic freedom has declined over the past decade for the first time in the last century and our estimates suggest that this decline poses a substantial threat to the innovation output of countries in terms of both quantity and quality.

Extending science instruction beyond the CER: Use of critical questions in the argumentation of middle school science students

AbstractCritiquing arguments is important for K‐12 science students to learn but not emphasized by the predominant claim‐evidence‐reasoning (CER) argumentation model. Drawing on the work of Yu and Zenker (2020), and Dove and Nussbaum (2018), we developed a tool for supplementing CER with critical questions (CQs) from philosophy that cover most, if not all, the logical dimensions of argument critique. Six middle school science teachers designed lessons involving argumentation, including the use of CQs. We assessed the effects on student self‐efficacy for engaging in argument critique, teacher self‐efficacy for using argument pedagogy, and teachers’ perceptions of the value of CQs. Qualitative data included teacher interviews, lesson transcripts, and student work samples. Quantitative data included surveys of student self‐efficacy administered at the beginning and end of the school year. There was evidence suggesting an increase over time in students’ confidence for engaging in argument critique and teachers’ confidence with argument pedagogy. However, only four of the six teachers were confident and skilled enough to include CQs in their lessons. Those who did use CQs tended to perceive them as providing a helpful structure for critique, prompts for deeper thinking, and a tool for fostering critical classroom norms. Discussion of CQs may have benefitted students’ writing by promoting peer critique and encouragement to elaborate. Overall, CQs afford students with a framework for judging argument strength. Scientific argumentation involving CQs provides a more contemporary philosophical basis for scientific argumentation than CER or the Toulmin model as it emphasizes the critical and dialogic nature of science.

Norms of Composition Operators from Weighted Harmonic Bloch Spaces into Weighted Harmonic Zygmund Spaces

This article examines the norms of composition operators from the weighted harmonic Bloch space BHλ,0<λ<∞ to the weighted harmonic Zygmund space ZHβ,0<β<∞. The critical norm is on the open unit disk. We first give necessary and sufficient conditions where the composition operator between BHλ and ZHβ is bounded. Secondly, we will study the compactness case of the composition operator between BHλ and ZHβ. Finally, we will estimate the essential norms of the composition operator between BHλ and ZHβ.

Global well-posedness for the radial, defocusing, nonlinear wave equation for 3 < p < 5

abstract: In this paper we continue the study of the defocusing, energy-subcritical nonlinear wave equation with radial initial data lying in the critical Sobolev space. In this case we prove scattering in the critical norm when $3<p<5$.

Emergent Normativity: Communities of Practice, Technology, and Lethal Autonomous Weapon Systems

Abstract Lethal autonomous weapon systems (LAWS) are the subject of considerable international debate turning around the extent to which humans remain in control over using force. But what is precisely at stake is less clear as stakeholders have different perspectives on the technologies that animate LAWS. Such differences matter because they shape the substance of the debate, which regulatory options are put on the table, and also normativity on LAWS in the sense of understandings of appropriateness. To understand this process, I draw on practice theories, science and technology studies (STS), and critical norm research. I argue that a constellation of communities of practice (CoPs) shapes the public debate about LAWS and focus on three of these CoPs: diplomats, weapon manufacturers, and journalists. Actors in these CoPs discursively perform practices of boundary-work, in the STS sense, to shape understandings of technologies at the heart of LAWS: automation, autonomy, and AI. I analyze these dynamics empirically in two steps: first, by offering a general-level analysis of practices of boundary-work performed by diplomats at the Group of Governmental Experts on LAWS from 2017 to 2022; and second, through examining such practices performed by weapon manufacturers and journalists in relation to the use of loitering munitions, a particular type of LAWS, in the Second Libyan Civil War (2014–2020).

Strichartz estimates and global well-posedness of the cubic NLS on

Abstract The optimal $L^4$ -Strichartz estimate for the Schrödinger equation on the two-dimensional rational torus $\mathbb {T}^2$ is proved, which improves an estimate of Bourgain. A new method based on incidence geometry is used. The approach yields a stronger $L^4$ bound on a logarithmic time scale, which implies global existence of solutions to the cubic (mass-critical) nonlinear Schrödinger equation in $H^s(\mathbb {T}^2)$ for any $s>0$ and data that are small in the critical norm.

Legal Ideals Pancasila in the Development of a National Environmental Legal System

The discussion of legal ideals Pancasila in Indonesia has become a prolonged discourse and may not be final. This is motivated by the dynamics and problems of the nation and state in various sectors of life, one of which is the environmental sector. This paper aims to examine the potential for developing legal ideals in the national environmental law system. The research method used is normative legal research. The results of the study show that in its implementation the legal ideals of Pancasila are still not running as they should in the environmental sector in Indonesia. This is because there are deviations in values ​​in the preparation of legal policies and action agendas. In the development of a national environmental law system, legal ideals cannot be seen and read explicitly, but must continue to be explored and interpreted in every Pancasila value. The ideals of Pancasila law contain environmental values such as ecological, ecoreligious, ecosocial, and ecocracy values. The legal ideals of Pancasila are positioned and functioned as a way of life, guiding norms, and critical norms in the legal agenda in the environmental sector

Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds

We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schr\"odinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on $L^2$-based critical Sobolev norms imply scattering estimates.

Whose media freedom is being defended? Norm contestation in international media freedom campaigns

AbstractThis article analyses how international advocacy campaigns approach and define media freedom, and what influences this process. It does this through a two-year case study of the Media Freedom Coalition—an intergovernmental partnership of over 50 countries—that included 55 interviews with key stakeholders, observations, and document analysis. This revelatory case sheds light on how norms of media freedom are constructed and contested on the international stage, and their implications for journalists, media freedom and geo-politics. We show that the Coalition adopted a state-centric, accountability-focused, and negative understanding of media freedom. This discourse legitimized a narrow, reactive, and “resource-light” approach to supporting media freedom, focused on “other” countries. We argue that critical norm research provides a helpful prism for understanding this Coalition’s operations, and the global politics of media freedom more generally. These findings have important implications for understandings of “norm entrepreneurship,” “media imperialism,” and “media freedom” itself.

Finite time blow up and concentration phenomena for a solution to drift-diffusion equations in higher dimensions

We show the finite time blow up of a solution to the Cauchy problem of a drift-diffusion equation of a parabolic-elliptic type in higher space dimensions. If the initial data satisfies a certain condition involving the entropy functional, then the corresponding solution to the equation does not exist globally in time and blows up in a finite time for the scaling critical space. Besides there exists a concentration point such that the solution exhibits the concentration in the critical norm. This type of blow up was observed in the scaling critical two dimensions. The proof is based on the profile decomposition and the Shannon inequality in the weighted space.

Defocusing [formula omitted]-critical inhomogeneous nonlinear Schrödinger equations

In this paper, we study the critical norm problem for the defocusing inhomogeneous nonlinear Schrödinger equations (INLS). As a first attempt, we consider the defocusing H˙12-critical (INLS) iut+Δu=|x|−b|u|p−1u with p=1+2(2−b)d−1 and 0<b<12 in dimensions d≥5. We utilize the concentration-compactness/rigidity method as in Kenig-Merle [22] and Murphy [32] to show that if a solution u is uniformly bounded in the critical space H˙x12(Rd) throughout its lifespan, then the solution u must be global and scatter. The key ingredients of the proof include Lin-Strauss Morawetz estimate and inhomogeneous Strichartz estimate.

The ethics of asymmetric politics

Polarization often happens asymmetrically. One political actor radicalizes, and the results reverberate through the political system. This is how the deep divisions in contemporary American politics arose: the Republican Party radicalized. Republican officeholders began to use extreme legislative tactics. Republican voters became animated by contempt for their political rivals and by the defense of their own social superiority. The party as a whole launched a wide-ranging campaign of voter suppression and its members endorsed violence in the face of electoral defeat. This paper is about how such asymmetric polarization affects everyone else’s obligations. My core claim is that two kinds of relationship – civic friendship and non-subordination – underpin critical democratic norms. Republican misbehavior has severed cross-partisan civic friendships. Their authoritarianism forfeits their claim to non-subordination. The former means that non-Republicans need not justify policy on public grounds. The latter undercuts Republicans’ claim to enjoy minority vetoes when out of power and it gives their rivals reason to disobey the laws that Republicans make when they are in power. More generally, when one political actor contravenes the proper norms of democratic politics, their opposition is not bound by those norms.

Freedom and dialectics: On the critical theory of Moishe Postone and Theodor Adorno

This article examines the relation between the critical theory of Moishe Postone and the philosophy of Theodor Adorno. While the former is clearly influenced by the latter, these influences mostly stay implicit. When explicit, he does not so easily put his own thought in line with Adorno’s and is highly critical of him. I will investigate the ways in which ideas from Adorno made their way into Postone’s work, but also where the latter diverts from them. As will be shown, Postone’s critical theory runs into a problem when one examines the immanent ground of his critical norm, freedom. Adorno, on the contrary, does succeed in employing the same norm in his own immanent critique. I will conclude that the problem Postone encounters can be avoided by being more of an ‘Adornian’, and that his theory allows this.

Scattering for the Defocusing, Nonlinear Schrödinger Equation With Initial Data in a Critical Space

Abstract In this note, we prove scattering for a defocusing nonlinear Schrödinger equation with initial data lying in a critical Besov space. In addition, we obtain polynomial bounds on the scattering size as a function of the critical Besov norm.

A Minimum Critical Blowup Rate for the High-Dimensional Navier–Stokes Equations

We prove quantitative regularity and blowup theorems for the incompressible Navier-Stokes equations in $\mathbb R^d$, $d\geq4$ when the solution lies in the critical space $L_t^\infty L_x^d$. Explicit subcritical bounds on the solution are obtained in terms of the critical norm. A consequence is that $\|u(t)\|_{L_x^d(\mathbb R^d)}$ grows at a minimum rate of $(\log\log\log\log(T_*-t)^{-1})^c$ along a sequence of times approaching a hypothetical blowup at $T_*$. These results quantify a theorem of Dong and Du and extend the three-dimensional work of Tao.

Localized smoothing and concentration for the Navier-Stokes equations in the half space

We establish a local-in-space short-time smoothing effect for the Navier-Stokes equations in the half space. The whole space analogue, due to Jia and Šverák [21], is a central tool in two of the authors' recent work on quantitative Lx3 blow-up criteria [7]. The main difficulty is that the non-local effects of the pressure in the half space are much stronger than in the whole space. As an application, we demonstrate that the critical Lx3 norm must concentrate at scales ∼T⁎−t in the presence of a Type I blow-up.

Extent of knowledge and attitudes on plagiarism among undergraduate medical students in South India - a multicentre, cross-sectional study to determine the need for incorporating research ethics in medical undergraduate curriculum

BackgroundUndergraduate medical students in India participate in various research activities However, plagiarism is rampant, and we hypothesize that it is the lack of knowledge on how to avoid plagiarism. This study’s objective was to measure the extent of knowledge and attitudes towards plagiarism among undergraduate medical students in India.MethodsIt was a multicentre, cross-sectional study conducted over a two-year period (January 2018 – December 2019). Undergraduate medical students were given a pre-tested semi-structured questionnaire which contained: (a) Demographic details; (b) A quiz developed by Indiana University, USA to assess knowledge; and (c) Attitudes towards Plagiarism (ATP) questionnaire.ResultsEleven medical colleges (n = 4 government medical colleges [GMCs] and n = 7 private medical colleges [PMCs]) participated. A total of N = 4183 students consented. The mean (SD) knowledge score was 4.54 (1.78) out of 10. The factors (adjusted odds ratio [aOR]; 95% Confidence interval [CI]; p value) that emerged as significant predictors of poor knowledge score were early years of medical education (0.110; 0.063, 0.156; < 0.001) and being enrolled in a GMC (0.348; 0.233, 0.463; < 0.001).The overall mean (SD) scores of the three attitude components namely permissive, critical and submissive norms were 37.56 (5.25), 20.35 (4.20) and 31.20 (4.28) respectively, corresponding to the moderate category.ConclusionThe overall knowledge score was poor. A vast majority of study participants fell in the moderate category of attitude score. These findings warrant the need for incorporating formal training in the medical education curriculum.

Children and Adolescents' Ingroup Biases and Developmental Differences in Evaluations of Peers Who Misinform.

Previous developmental research shows that young children display a preference for ingroup members when it comes to who they accept information from – even when that information is false. However, it is not clear how this ingroup bias develops into adolescence, and how it affects responses about peers who misinform in intergroup contexts, which is important to explore with growing numbers of young people on online platforms. Given that the developmental span from childhood to adolescence is when social groups and group norms are particularly important, the present study took a Social Reasoning Developmental Approach. This study explored whether children and adolescents respond differently to a misinformer spreading false claims about a peer breaking COVID-19 rules, depending on (a) the group membership of the misinformer and their target and (b) whether the ingroup had a “critical” norm that values questioning information before believing it. 354 United Kingdom-based children (8–11 years old) and adolescents (12–16 years old) read about an intergroup scenario in which a peer spreads misinformation on WhatsApp about a competitor. Participants first made moral evaluations, which asked them to judge and decide whether or not to include the misinformer, with follow-up “Why?” questions to capture their reasoning. This was followed by asking them to attribute intentions to the misinformer. Results showed that ingroup preferences emerged both when participants morally evaluated the misinformer, and when they justified those responses. Participants were more likely to evaluate an ingroup compared to an outgroup misinformer positively, and more likely to accuse an outgroup misinformer of dishonesty. Adolescents attributed more positive intentions to the misinformer compared with children, with children more likely to believe an outgroup misinformer was deliberately misinforming. The critical norm condition resulted in children making more positive intentionality attributions toward an ingroup misinformer, but not an outgroup misinformer. This study’s findings highlight the importance of shared group identity with a misinformer when morally evaluating and reasoning about their actions, and the key role age plays in intentionality attributions surrounding a misinformer when their intentions are ambiguous.