On Singularities of Certain Non-linear Second-Order Ordinary Differential Equations

The method of blowing up points of indeterminacy of certain systems of two ordinary differential equations is applied to obtain information about the singularity structure of the solutions of the corresponding non-linear differential equations. We first deal with the so-called Painlevé example, which passes the Painlevé test, but the solutions have more complicated singularities. Resolving base points in the equivalent system of equations we can explain the complicated structure of singularities of the original equation. The Smith example has a solution with non-isolated singularity, which is an accumulation point of algebraic singularities. Smith’s equation can be written as a system in two ways. We show that the sequence of blow-ups for both systems can be infinite. Another example that we consider is the Painlevé-Ince equation. When the usual Painlevé analysis is applied, it possesses both positive and negative resonances. We show that for three equivalent systems there is an infinite sequence of blow-ups and another one that terminates, which further gives a Laurent expansion of the solution around a movable pole. Moreover, for one system it is even possible to obtain the general solution after a sequence of blow-ups.

Ground States for Mass Critical Two Coupled Semi-Relativistic Hartree Equations with Attractive Interactions

We prove the existence and nonexistence of $L^{2}(\mathbb R^3)$-normalized solutions of two coupled semi-relativistic Hartree equations, which arisen from the studies of boson stars and multi-component Bose–Einstein condensates. Under certain condition on the strength of intra-specie and inter-specie interactions, by proving some delicate energy estimates, we give a precise description on the concentration behavior of ground state solutions of the system. Furthermore, an optimal blowing up rate for the ground state solutions of the system is also proved.

“It’s Cool, Modifying and All, but I Don’t Want Anything Blowing Up on Me:” A Focus Group Study of Motivations to Modify Electronic Nicotine Delivery Systems (ENDS)

Introduction: Modifications to electronic nicoti ne delivery systems (ENDS) can pose health risks to users. This study explored users’ motivations for modifying ENDS devices and how perceived risks of modifications influenced modification behaviors as product availability and device characteristics changed over time. Method: We conducted nine focus groups (February–June 2020) with 32 current ENDS users (18+, used ENDS in the past 30 days, and had been using ENDS for more than 2 months). Results: Participants primarily modified ENDS devices to improve their experiences, such as experimenting with flavor, controlling nicotine levels, or using cannabis products with ENDS. Another reason for modifying was routine maintenance to ensure a satisfactory experience, including maintaining coils and keeping batteries charged. The broader availability of ENDS products shifted modification behaviors over time, with newer devices making some modifications (e.g., coil replacement) easier and making more intricate modifications (e.g., building coil from scratch) less common. Participants were aware of modification dangers and cited perceived risk as the reason for avoiding certain modifications, such as battery alterations. Conclusions: Modifications of ENDS are ongoing and evolving among users and should be considered by the Food and Drug Administration (FDA) and other regulatory decision-makers as product authorization reviews are conducted and product standards are developed.

Explicit Blowing Up Solutions for a Higher Order Parabolic Equation with Hessian Nonlinearity

In this work we consider a nonlinear parabolic higher order partial differential equation that has been proposed as a model for epitaxial growth. This equation possesses both global-in-time solutions and solutions that blow up in finite time, for which this blow-up is mediated by its Hessian nonlinearity. Herein, we further analyze its blow-up behaviour by means of the construction of explicit solutions in the square, the disc, and the plane. Some of these solutions show complete blow-up in either finite or infinite time. Finally, we refine a blow-up criterium that was proved for this evolution equation. Still, existent blow-up criteria based on a priori estimates do not completely reflect the singular character of these explicit blowing up solutions.

Blowing-up Solutions for 2nd-Order Critical Elliptic Equations: The Impact of the Scalar Curvature

Given a closed manifold $(M^n,g)$, $n\geq 3$, Druet [5, 7] proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation $$ \begin{align*} &\Delta_gu+h_0u=u^{\frac{n+2}{n-2}},\ u>0 \ \textrm{in }M\end{align*}$$is that $h_0\in C^1(M)$ touches the Yamabe potential somewhere when $n\geq 4$ (the condition is different for $n=6$). In this paper, we prove that Druet’s condition is also sufficient provided we add its natural differentiable version. For $n\geq 6$, our arguments are local. For the low dimensions $n\in \{4,5\}$, our proof requires to introduce a suitable mass that is defined only where Druet’s condition holds. This mass carries global information both on $h_0$ and $(M,g)$.

Bubble bag end: a bubbly resolution of curvature singularity

We construct a family of smooth charged bubbling solitons in $$ \mathbbm{M} $$ M 4×T2, four-dimensional Minkowski with a two-torus. The solitons are characterized by a degeneration pattern of the torus along a line in $$ \mathbbm{M} $$ M 4 defining a chain of topological cycles. They live in the same parameter regime as non-BPS non-extremal four-dimensional black holes, and are ultracompact with sizes ranging from miscroscopic to macroscopic scales. The six-dimensional framework can be embedded in type IIB supergravity where the solitons are identified with geometric transitions of non-BPS D1-D5-KKm bound states. Interestingly, the geometries admit a minimal surface that smoothly opens up to a bubbly end of space. Away from the solitons, the solutions are indistinguishable from a new class of singular geometries. By taking a limit of large number of bubbles, the soliton geometries can be matched arbitrarily close to the singular spacetimes. This provides the first classical resolution of a curvature singularity beyond the framework of supersymmetry and supergravity by blowing up topological cycles wrapped by fluxes at the vicinity of the singularity.

BLOWING UP THE FASHION BUBBLE, OR NINE THINGS WRONG WITH FASHION: AN OUTSIDER’S COMMENT. A CRITICAL ESSAY ON FASHION AS A CREATIVE INDUSTRY

The world of fashion has lived in a bubble long before the concept found its way into social networks. Well before the social networks themselves, in fact. The very emergence of fashion as an idea occurred within the bubble of the social life at Palace of Versailles, France, where the notorious Louis XIV sported great interest in the looks of his court in addition to those of his own. The article is an outsider’s attempt to have a sober look at the fashion industry that until recently seemed to have maintained the “structure of feeling” of the 17th century Palace of Versailles. Today’s social realities, however, put fashion in the state of a shock that probably equals that of the Storming of the Bastille in 1789, even though it is presumably much less sudden. Written in the manner of the most popular contemporary fashion media format – a bullet list, the text presents a conceptual analysis of the world’s second most wasteful yet poisonously attractive industry, critically reflecting on such characteristic values of the fashion field as concept and features, hierarchy, arrogance, resources and philosophy.