Formation Of Singularities Research Articles

Lagrangian dynamics and regularity of the spin Euler equation

We derive the spin Euler equation for ideal flows by applying the spherical Clebsch mapping. This equation is based on the spin vector, a unit vector field encoding vortex lines, instead of the velocity. The spin Euler equation enables a feasible Lagrangian study of fluid dynamics, as the isosurface of a spin-vector component is a vortex surface and material surface in ideal flows. We establish a non-blowup criterion for the spin Euler equation, suggesting that the Laplacian of the spin vector must diverge if the solution forms a singularity at some finite time. The direct numerical simulations (DNS) of three ideal flows – the vortex knot, the vortex link and the modified Taylor–Green flow – are conducted by solving the spin Euler equation. The evolution of the Lagrangian vortex surface illustrates that the regions with large vorticity are rapidly stretched into spiral sheets. The DNS result exhibits a pronounced double-exponential growth of the maximum norm of Laplacian of the spin vector, showing no evidence of the finite-time singularity formation if the double-exponential growth holds at later times. Moreover, the present criterion with Lagrangian nature appears to be more sensitive than the Beale–Kato–Majda criterion in detecting the flows that are incapable of producing finite-time singularities.

Harmonic Flow of Quaternion-Kähler Structures

We formulate the gradient Dirichlet flow of Sp(2)Sp(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Sp}(2)\ extrm{Sp}(1)$$\\end{document}-structures on 8-manifolds, as the first systematic study of a geometric quaternion-Kähler (QK) flow. Its critical condition of harmonicity is especially relevant in the QK setting, since torsion-free structures are often topologically obstructed. We show that the conformally parallel property implies harmonicity, extending a result of Grigorian in the G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document} case. We also draw several comparisons with Spin(7)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Spin}(7)$$\\end{document}-structures. Analysing the QK harmonic flow, we prove an almost-monotonicity formula, which implies to long-time existence under small initial energy, via ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document}-regularity.We set up a theory of harmonic QK solitons, constructing a non-trivial steady example. We produce explicit long-time solutions: one, converging to a torsion-free limit on the hyperbolic plane; and another, converging to a limit which is harmonic but not torsion-free, on the manifold SU(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SU}(3)$$\\end{document}. We also study compactness and the formation of singularities.

The Acquisition of Jordanian Arabic Plural Morphology by Children with Autism Spectrum Disorder

Aim. This study investigates the acquisition of plural morphology by native Jordanian children with autism spectrum disorder (ASD). The primary objective of the study is to assess children’s capacity to use plural morphology in Jordanian Arabic, aiming to identify the most frequently utilized form of the plural and the reasons behind its prevalence. 
 Methods. The study involved a sample of 10 male and female children with ASD, averaging twelve years old, from Tawasul Center for Autism in Amman, Jordan. Utilizing twenty representative pictures featuring familiar plural items such as chairs, televisions, and children, the participants were tasked with identifying the depicted elements. 
 Results. The findings indicated that the children faced challenges in employing plurals in Arabic, with the feminine morpheme being the most consistently used. However, instances of overgeneralization in the use of the feminine morpheme were noted, leading to inaccuracies. Moreover, children avoided plural forms in several cases and resorted to singular forms paired with words that indicate duality or plurality.
 Conclusion. The research results indicate that, by the age of 12, individuals with ASD experience challenges in acquiring proficiency in plural forms in Jordanian Arabic. The study underscores the difficulties these individuals encounter in understanding linguistic elements related to plurality within the specific linguistic context. It also demonstrates that comprehending the complexities of how autistic children acquire plural morphemes is not solely an academic interest; it holds practical significance for designing interventions to enhance their language proficiency and improve overall communication skills.

Existence and stability of shrinkers for the harmonic map heat flow in higher dimensions

We study singularity formation for the heat flow of harmonic maps from Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document}. For each d≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 4$$\\end{document}, we construct a compact, d-dimensional, rotationally symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution (shortly shrinker) that represents a stable blowup mechanism for the corresponding Cauchy problem.

The rhetoric of torture in Henri Alleg’s La Question

ABSTRACT Drawing essentially on David Spurr’s The Rhetoric of Empire (1993), this article discusses the discursive mechanisms that accompany the practice of torture as described in Henri Alleg’s La Question (1958). Examining the relation between these mechanisms and the broader discourses of colonialism and subjection, it shows that the inscription of colonial power on the injured and dehumanised body of the tortured is paralleled by the verbal denigration of these victims, reduced to a state of infantilism or, worse, animality, through verbal reflexes such as animalistic adjectives and tutoiement – a form of speech which consists in the use of the singular form of the second person (you) and which is usually reserved for children and those deemed not worthy of respect. More importantly, it argues that, rejecting both the victim status and the dehumanisation to which his jailers attempt to reduce him, Alleg shapes his own empowering rhetorical strategies, in turn debasing the torturers, ridiculing them, or mocking the colonial myths of chivalry and racial superiority. Instead of the latter, Alleg proposes a discourse of humanism and interracial brotherhood that conquers the violence he endures.

Syntactic variation in Hong Kong English

Kirkpatrick and Lixun (2021) maintain that two significant morphosyntactic processes have been at play in early Englishes. These are simplification and regularization. Simplification refers to the relatively simplified inflectional morphology in English today. Kirkpatrick and Lixun (2021) provide an example for the word stan (i.e., stone in Old English) that showed great differences in the singular and plural form in nominative, accusative, genitive and dative case in Old English. Another process is regularization, through which some of the strong verb forms for past tense in English have changed to take the weak or the regular form. To illustrate, the past tense of work was wrought but over time, it has changed to worked.

Formation of Singularity for Isentropic Irrotational Compressible Euler Equations

The domain of science and engineering relies heavily on an in-depth comprehension of fluid dynamics, given the prevalence of fluids such as water, air, and interstellar gas in the universe. Euler equations form the basis for the study of fluid motion. This paper is concerned with the Cauchy problem of isentropic compressible Euler equations away from the vacuum. We use the integration method with the general test function f=f(r), proving that there exist the corresponding blowup results of C1 irrotational solutions for Euler equations and Euler equations with time-dependent damping in Rn (n≥2), provided the density-independent initial functional is sufficiently large. We also provide two simple and explicit test functions f(r)=r and f(r)=1+r, to demonstrate the blowup phenomenon in the one-dimensional case. In particular, our results are applicable to the non-radial system.

The role of dimension and electric charge on a collapsing geometry in Einstein–Gauss–Bonnet gravity

The analysis of the continual gravitational contraction of a spherically symmetric shell of charged radiation is extended to higher dimensions in Einstein–Gauss–Bonnet gravity. The spacetime metric, which is of Boulware–Deser type, is real only up to a maximum electric charge and thus collapse terminates with the formation of a branch singularity. This branch singularity divides the higher dimensional spacetime into two regions, a real and physical one, and a complex region. This is not the case in neutral Einstein–Gauss–Bonnet gravity as well as general relativity. The charged gravitational collapse process is also similar for all dimensions N≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 5$$\\end{document} unlike in the neutral scenario where there is a marked difference between the N=5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=5$$\\end{document} and N>5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N>5$$\\end{document} cases. In the case where N=5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=5$$\\end{document} uncharged collapse ceases with the formation of a weaker, conical singularity which remains naked for a time depending on the Gauss–Bonnet invariant, before succumbing to an event horizon. The similarity of charged collapse for all higher dimensions is a unique feature in the theory. The sufficient conditions for the formation of a naked singularity are studied for the higher dimensional charged Boulware–Deser spacetime. For particular choices of the mass and charge functions, naked branch singularities are guaranteed and indeed inevitable in higher dimensional Einstein–Gauss–Bonnet gravity. The strength of the naked branch singularities is also tested and it is found that these singularities become stronger with increasing dimension, and no extension of spacetime through them is possible.

Singularity formation for the cylindrically symmetric rotating relativistic Euler equations of Chaplygin gases

This paper studies the formation of singularities in smooth solutions of the relativistic Euler equations of Chaplygin gases with cylindrically symmetric rotating structures. This is a nonhomogeneous hyperbolic system with highly nonlinear structures and fully linearly degenerating characteristic fields. We introduce a pair of auxiliary functions and use the characteristic decomposition technique to overcome the influence of the rotating structures in the system. It is verified that smooth solutions develop into a singularity in finite time and the mass-energy density tends to infinity at the blowup point for a type of rotating initial data.

Numerical study on how advection delays and removes singularity formation in the Navier–Stokes equations

We numerically study a distorted version of the Euler and Navier–Stokes equations, which are obtained by depleting the advection term systematically. It is known that in the inviscid case some solutions blow up in finite time when advection is totally discarded, Constantin (1986 Commun. Math. Phys. 104 311–26). Taking a pair of orthogonally offset vortex tubes and the Taylor–Green vortex as initial data, we show the following. (1) Blowup persists even with viscosity when advection is discarded, and (2) for small viscosity, the time of blowup increases logarithmically as we reinstate advection using a continuous parameter, which would be consistent with the regularity of the Navier–Stokes equations. A tiny mismatch in the coefficient of the advection term, as minute as parts per trillion, throws the system out of compactness and leads to blowup.

Web-Based Google Translate Inconsistencies in Bahasa-Arabic Translations from the ‎Arabic Thesis Writer's Perspective

This research aims to reveal the inconsistencies of machine translation from the perspective of students who use web-based Google Translate in their Arabic thesis. This research has an impact on translation machine users from Arabic teachers and researchers to be more careful in using Google Translate when translating from Bahasa to Arabic, especially in inconsistent aspects. The researcher used a mixed method with an Explanatory Sequential Design‎ because after distributing questionnaires in collecting quantitative data to collect patterns of inconsistencies that occurred, it was important to explain the initial findings from the quantitative data qualitatively through interviews. ‎Quantitative data is analyzed using simple percentage statistics, then qualitative data is analyzed through four steps, namely data collection, reduction, presentation, and conclusion. The results of this research found that inconsistencies in web-based Google Translate translation results were found in four patterns. First, Google Translate is very inconsistent in translating terms that have the same meaning (mutarodifat), but the terms that appear in the translation results are often different, even within the same paragraph. Second, inconsistencies very often also occur in the aspects of mufrad, mutsanna and jamak, the Google translation machine cannot differentiate between the singularity and plurality aspects of the object being translated, so translation errors often occur where a singular word that should be translated as mufrad will instead become plural or on the contrary. Third, the translation machine does not recognize the gender of the subject in the translated sentence, so translation inconsistencies in the mudzakkar and mu'annats aspects occur very often. Inconsistencies in the gender aspect, as well as the mufrad, mutsanna, and jamak aspects, have an impact on the fourth error pattern, namely grammatical inconsistencies, namely errors in using the verb (fi'il) in a sentence. Of the four patterns of inconsistency, the term inconsistency is a new thing that has not been revealed much. The grammatical aspect is closely related to aspects of the singular or plural form as well as the masculine or feminine form which is the subject of the translated sentence and needs to be discussed further by other researchers.

Formation of singularities in plasma ion dynamics

We study the formation of singularity for the Euler–Poisson system equipped with the Boltzmann relation, which describes the dynamics of ions in an electrostatic plasma. In general, it is known that smooth solutions to nonlinear hyperbolic equations fail to exist globally in time. We establish criteria for C 1 blow-up of the Euler–Poisson system, both for the isothermal and pressureless cases. In particular, our blow-up condition for the pressureless model does not require that the gradient of velocity is negatively large. In fact, our result particularly implies that the smooth solutions can break down even if the gradient of initial velocity is trivial. For the isothermal case, we prove that smooth solutions leave C 1 class in a finite time when the gradients of the Riemann functions are initially large.

On radially symmetric oscillations of a collisional cold plasma

We study the influence of the friction term on the radially symmetric solutions of the repulsive Euler–Poisson equations with a non‐zero background, corresponding to cold plasma oscillations in multiple spatial dimensions. It is shown that for any arbitrarily small non‐negative constant friction coefficient, there exists a neighborhood of the zero equilibrium in the norm such that the solution of the Cauchy problem with initial data belonging to this neighborhood remains globally smooth in time. Moreover, this solution converges to zero as . This result contrasts with the situation of zero friction, where any small deviation from the zero equilibrium generally leads to a blow‐up. Our method allows us to estimate the lifetime of smooth solutions. Further, we prove that for any initial data, one can find such a coefficient of friction that the respective solution to the Cauchy problem keeps smoothness for all and converges to zero. We also present the results of numerical experiments for physically reasonable situations, which allows us to estimate the value of the friction coefficient, which makes it possible to suppress the formation of singularities of solutions.

Ancient Solutions of Ricci Flow with Type I Curvature Growth

Ancient solutions of the Ricci flow arise naturally as models for singularity formation. There has been significant progress towards the classification of such solutions under natural geometric assumptions. Nonnegatively curved solutions in dimensions 2 and 3, and uniformly PIC solutions in higher dimensions are now well understood. We consider ancient solutions of arbitrary dimension which are complete and have Type I curvature growth. We show that a simply connected, noncompact, κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-noncollapsed Type I ancient solution with nonnegative sectional curvature necessarily splits a Euclidean factor. It follows that a κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-noncollapsed Type I ancient solution which is weakly PIC2 is a locally symmetric space.

Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow

For any manifold N p N^p admitting an Einstein metric with positive Einstein constant, we study the behavior of the Ricci flow on high-dimensional products M = N p × S q + 1 M = N^p \times S^{q+1} with doubly warped product metrics. In particular, we provide a rigorous construction of local, type II, conical singularity formation on such spaces. It is shown that for any k > 1 k > 1 there exists a solution with curvature blow-up rate ‖ R m ‖ ∞ ( t ) ≳ ( T − t ) − k \| Rm \|_{\infty } (t) \gtrsim (T-t)^{-k} with singularity modeled on a Ricci-flat cone at parabolic scales.

A Sociolinguistic Examination of the Dual Usted in Medellín, Colombia: Evidence from Semi-spontaneous Speech and Implicit Language Attitudes

Abstract: The system of second person singular forms of address (2PS) in Medellín, Colombia is tripartite consisting of tú, vos , and usted , while also including the existence of a dual usted . The current study compares usage of the intimate usted versus the distant usted with data from an oral discourse completion task (DCT) while also investigating listeners' implicit language attitudes toward the usage of usted utilizing the matched guise technique. A sample of speakers from Medellín (N=72) stratified by age, sex, and education level completed an oral DCT, and a subset (N=38) also completed a matched guise task. For usage, it was found that intimate usted occurred 36.7% of the time with distant usted occurring 63.3% of the time. A multivariate mixed effects regression selected all fixed effects as significant. Specifically, intimate usted was favored by commands, statements, indirect commands, male interlocutors, same age interlocutors, male speakers, younger speakers, and speakers with high school education. These findings show that although the default usted usage is with distant interlocutors, the intimate usted is extremely vital, especially between men and with younger speakers. The current study also demonstrates the importance of comparing usage data and attitudinal data.

Finite-time blow-up in hyperbolic Keller–Segel system of consumption type with logarithmic sensitivity

This paper deals with finite-time blow-up of a hyperbolic Keller–Segel system of consumption type with the logarithmic sensitivity 0\\right)$?> ∂tρ=−χ∇⋅ρ∇logc,∂tc=−μcρχ,μ>0 in Rd(d⩾1) for nonvanishing initial data. This system is closely related to tumor angiogenesis, an important example of chemotaxis. Our singularity formation is not because c touches zero (which makes logc diverge) but due to the blowup of C1×C2 -norm of (ρ,c) . As a corollary, we also construct initial data near any constant equilibrium state which blows up in finite time for any d⩾1 .

Culturally-responsive devising as performance

ABSTRACT Drawing on a recent case study from Singapore, this essay examines how devising can be expanded beyond notions of “improvisational” and “fixed” into a more emergent, speculative practice that embodies both singular and collective forms of creative performance, and suggests ways in which practitioners may blur the boundaries between being-with as a lived approach to devising, and intercultural performance as a crucial contemporary set of skills for everyday life.

Gravitational collapse without singularity formation in Brans–Dicke theory

Gravitational collapse without singularity formation in Brans–Dicke theory

Singularity formation for the relativistic Euler‐Poisson equations with repulsive force and damping

AbstractIn this paper, we consider the blowup mechanism of regular solutions to the spherically symmetry relativistic Euler‐Poisson equations with repulsive force and damping. By means of a weighted function and the structured functional, we show the regular solution will blow up in finite time.