A Higher-Order Energy Expansion to Two-Dimensional Singularly Neumann Problems

Of concern is the following singularly perturbed semilinear elliptic problem \begin{equation*} \left\{ \begin{array}{c} \mbox{${\epsilon}^2\Delta u -u+u^p =0$ in $\Omega$}\\ \mbox{$u>0$ in $\Omega$ and $\frac{\partial u}{\partial \nu}=0$ on $\partial \Omega$}, \end{array} \right. \end{equation*} where $\Omega$ is a bounded domain in ${\mathbf{R}}^N$ with smooth boundary $\partial \Omega$, $\epsilon>0$ is a small constant and $1< p<\left(\frac{N+2}{N-2}\right)_+$. Associated with the above problem is the energy functional $J_{\epsilon}$ defined by \begin{equation*} J_{\epsilon}[u]:=\int_{\Omega}\left(\frac{\epsilon^2}{2}{|\nabla u|}^2 +\frac{1}{2}u^2 -F(u)\right)dx \end{equation*} for $u\in H^1(\Omega)$, where $F(u)=\int_{0}^{u}s^p ds$. Ni and Takagi (\cite{nt1}, \cite{nt2}) proved that for a single boundary spike solution $u_{\epsilon}$, the following asymptotic expansion holds: \begin{equation*} (1) \ \ \ \ \ \ \ \ J_{\epsilon}[u_{\epsilon}]=\epsilon^{N} \left[\frac{1}{2}I[w]-c_1 \epsilon H(P_{\epsilon})+o(\epsilon)\right], \end{equation*} where $I[w]$ is the energy of the ground state, $c_1 >0$ is a generic constant, $P_{\epsilon}$ is the unique local maximum point of $u_{\epsilon}$ and $H(P_{\epsilon})$ is the boundary mean curvature function at $P_{\epsilon}\in \partial \Omega$. Later, Wei and Winter (\cite{ww3}, \cite{ww4}) improved the result and obtained a higher-order expansion of $J_{\epsilon}[u_{\epsilon}]$: \begin{equation*} (2) \ \ \ \ \ \ J_{\epsilon}[u_{\epsilon}]=\epsilon^{N} \left[\frac{1}{2}I[\omega]-c_{1} \epsilon H(P_{\epsilon})+\epsilon^2 [c_2(H(P_\epsilon))^2 +c_{3} R(P_\epsilon)]+o(\epsilon^2)\right], \end{equation*} where $c_2$ and $c_3>0$ are generic constants and $R(P_\epsilon)$ is the scalar curvature at $P_\epsilon$. However, if $N=2$, the scalar curvature is always zero. The expansion (2) is no longer sufficient to distinguish spike locations with same mean curvature. In this paper, we consider this case and assume that $2 \leq p <+\infty$. Without loss of generality, we may assume that the boundary near P\in\partial\Om is represented by the graph $\{ x_2 = \rho_{P} (x_1) \}$. Then we have the following higher order expansion of $J_\epsilon[u_\epsilon]:$ \begin{equation*} (3) \ \ \ \ \ J_\epsilon [u_\epsilon] =\epsilon^N \left[\frac{1}{2}I[w]-c_1 \epsilon H({P_\epsilon})+c_2 \epsilon^2(H({P_\epsilon}))^2 ] +\epsilon^3 [P(H({P_\epsilon}))+c_3S({P_\epsilon})]+o(\epsilon^3)\right], \end{equation*} where H(P_\ep)= \rho_{P_\ep}^{''} (0) is the curvature, $P(t)=A_1 t+A_2 t^2+A_3 t^3$ is a polynomial, $c_1$, $c_2$, $c_3$ and $A_1$, $A_2$,$A_3$ are generic real constants and S(P_\epsilon)= \rho_{P_\ep}^{(4)} (0). In particular $c_3<0$. Some applications of this expansion are given

Spinodal Decomposition and the Tomita Sum Rule

The scaling properties of a phase-ordering system with a conserved order parameter are studied. The theory developed leads to scaling functions satisfying certain general properties including the Tomita sum rule. The theory also gives good agreement with numerical results for the order parameter scaling function in three dimensions. The values of the associated nonequilibrium decay exponents are given by the known lower bounds.Comment: 15 pages, 6 figure

Phase-ordering of conserved vectorial systems with field-dependent mobility

The dynamics of phase-separation in conserved systems with an O(N) continuous symmetry is investigated in the presence of an order parameter dependent mobility M(\phi)=1-a \phi^2. The model is studied analytically in the framework of the large-N approximation and by numerical simulations of the N=2, N=3 and N=4 cases in d=2, for both critical and off-critical quenches. We show the existence of a new universality class for a=1 characterized by a growth law of the typical length L(t) ~ t^{1/z} with dynamical exponent z=6 as opposed to the usual value z=4 which is recovered for a<1.Comment: RevTeX, 8 pages, 13 figures, to be published in Phys. Rev.

Macrorealism from entropic Leggett-Garg inequalities

We formulate entropic Leggett-Garg inequalities, which place constraints on the statistical outcomes of temporal correlations of observables. The information theoretic inequalities are satisfied if macrorealism holds. We show that the quantum statistics underlying correlations between time-separated spin component of a quantum rotor mimics that of spin correlations in two spatially separated spin-$s$ particles sharing a state of zero total spin. This brings forth the violation of the entropic Leggett-Garg inequality by a rotating quantum spin-$s$ system in similar manner as does the entropic Bell inequality (Phys. Rev. Lett. 61, 662 (1988)) by a pair of spin-$s$ particles forming a composite spin singlet state.Comment: 5 pages, RevTeX, 2 eps figures, Accepted for publication in Phys. Rev.

A solution for estimating the tensile yield strength from small specimens

The small punch test is an innovative test that utilises small disc-shaped specimens to assess the mechanical behaviour of materials. The main advantage is the relatively small specimen size. In this article, a modified analytical solution for the small punch maximum bend strength is proposed that is based on classical plate theory. A clear linear relationship is observed between the tensile yield strength σYS and the small punch maximum bend strength σy for both alloys and metal matrix composites. Copyright by ASTM Int'l (all rights reserved)

Ethical dilemmas are not simply black and white

This is an Accepted Manuscript of an article published by Taylor & Francis Group in Ethics and Social Practice on 1 July 2008, available online at: http://dx.doi.org/10.1080/17496530801948838.This article aims to highlight some of the ethical issues that arise when social work educators plan to involve service users and carers from black and minority ethnic (BME) communities in the teaching of social work students. Between 2005 and 2007, the authors carried out a two-part project that involved working with service users and carers from BME communities in the area around Liverpool in Britain. The article first discusses the background for this two-part project, highlighting two themes relating to the ethical dilemmas we experienced. The first of these themes concerned conducting a project in a political context based on short and intermittent funding and intransigent bureaucracy. Our second theme concerned how to reconcile bringing together a group of people because they were recognized as having a shared experience while at the same time there were a myriad differences within the group. We then discuss these issues in light of the ethical approach we adopted, based on being open and honest, flexible in a respectful and meaningful way, and on anti-oppressive ethics and shared responsibility.Peer reviewe

Fremanezumab for the Preventive Treatment of Chronic Migraine.

BACKGROUND: Fremanezumab, a humanized monoclonal antibody targeting calcitonin gene-related peptide (CGRP), is being investigated as a preventive treatment for migraine. We compared two fremanezumab dose regimens with placebo for the prevention of chronic migraine. METHODS: In this phase 3 trial, we randomly assigned patients with chronic migraine (defined as headache of any duration or severity on ≥15 days per month and migraine on ≥8 days per month) in a 1:1:1 ratio to receive fremanezumab quarterly (a single dose of 675 mg at baseline and placebo at weeks 4 and 8), fremanezumab monthly (675 mg at baseline and 225 mg at weeks 4 and 8), or matching placebo. Both fremanezumab and placebo were administered by means of subcutaneous injection. The primary end point was the mean change from baseline in the average number of headache days (defined as days in which headache pain lasted ≥4 consecutive hours and had a peak severity of at least a moderate level or days in which acute migraine-specific medication [triptans or ergots] was used to treat a headache of any severity or duration) per month during the 12 weeks after the first dose. RESULTS: Of 1130 patients enrolled, 376 were randomly assigned to fremanezumab quarterly, 379 to fremanezumab monthly, and 375 to placebo. The mean number of baseline headache days (as defined above) per month was 13.2, 12.8, and 13.3, respectively. The least-squares mean (±SE) reduction in the average number of headache days per month was 4.3±0.3 with fremanezumab quarterly, 4.6±0.3 with fremanezumab monthly, and 2.5±0.3 with placebo (P CONCLUSIONS: Fremanezumab as a preventive treatment for chronic migraine resulted in a lower frequency of headache than placebo in this 12-week trial. Injection-site reactions to the drug were common. The long-term durability and safety of fremanezumab require further study. (Funded by Teva Pharmaceuticals; ClinicalTrials.gov number, NCT02621931 .)

Computational models for inferring biochemical networks

Biochemical networks are of great practical importance. The interaction of biological compounds in cells has been enforced to a proper understanding by the numerous bioinformatics projects, which contributed to a vast amount of biological information. The construction of biochemical systems (systems of chemical reactions), which include both topology and kinetic constants of the chemical reactions, is NP-hard and is a well-studied system biology problem. In this paper, we propose a hybrid architecture, which combines genetic programming and simulated annealing in order to generate and optimize both the topology (the network) and the reaction rates of a biochemical system. Simulations and analysis of an artificial model and three real models (two models and the noisy version of one of them) show promising results for the proposed method.The Romanian National Authority for Scientific Research, CNDI–UEFISCDI, Project No. PN-II-PT-PCCA-2011-3.2-0917

Competence self-perceptions

This chapter discusses the different operationalizations of competence self-perceptions and the implications for advancing theory, research, and practice