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

Investigation and characterization of constraint effects on flaw growth during fatigue loading of composite materials

An investigative program is presented in an attempt to add to the current understanding of constraint effects on the response of composite materials under cyclic loading. The objectives were: (1) to use existing data and to develop additional data in order to establish an understanding and quantitative description of flaw growth in unidirectional lamina under cyclic loading at different load direction to fiber direction angles; (2) to establish a similar understanding and description of flaw growth in lamina which are embedded in laminates between other unflawed lamina; (3) to determine the nature of the influence of constraint on flaw growth by quantitatively comparing the results of the tests; and (4) to develop a model and philosophy of constraints effects based on our investigative results

Ordering dynamics of the driven lattice gas model

The evolution of a two-dimensional driven lattice-gas model is studied on an L_x X L_y lattice. Scaling arguments and extensive numerical simulations are used to show that starting from random initial configuration the model evolves via two stages: (a) an early stage in which alternating stripes of particles and vacancies are formed along the direction y of the driving field, and (b) a stripe coarsening stage, in which the number of stripes is reduced and their average width increases. The number of stripes formed at the end of the first stage is shown to be a function of L_x/L_y^\phi, with \phi ~ 0.2. Thus, depending on this parameter, the resulting state could be either single or multi striped. In the second, stripe coarsening stage, the coarsening time is found to be proportional to L_y, becoming infinitely long in the thermodynamic limit. This implies that the multi striped state is thermodynamically stable. The results put previous studies of the model in a more general framework

The 100 Most Cited Papers Concerning the Insular Cortex of the Brain: A Bibliometric Analysis

Background: The insula is one of the most researched brain regions with many highly cited papers. However, unlike the literature of other fields, there is currently no study that has identified the 100 most cited papers within the literature of the insula. The aim of the current study was to fill in the knowledge gap by determining which publications concerning the insula have been cited most often, who contributed to them, and what topics they were dealing with.Methods: The Web of Science online database was searched to identify the 100 most cited publications mentioning the insular cortex in their titles, abstracts or keywords. To systematically exclude irrelevant publications, the search strategy was finalized as: TS = (insula OR insular OR “island of Reil”) NOT TS = (“insular biogeography” OR “insular mammal*” OR “*insular lymphatic*”) NOT WC = (“Geochemistry and Geophysics” OR “Ecology”). The identified publications were sorted in descending order of citation count. The 100 most cited publications concerning the insula of the brain were identified and their bibliometric data was extracted and assessed. The VOSviewer software was used with default parameters to generate a bubble map that analyzes and visualizes the words/phrases used in the titles and abstracts of the publications.Results: There were 67 articles on experiments/lab studies and 33 meta-analyses/reviews but no opinion or methods paper. They had an average of 943.4 citations (or 62.9 citations per year), 93.5 references and 13.4 pages. There were 35 papers published in open access. USA was the major contributing country. The most top-ranked publications were concerning emotion, salience and pain.Conclusion: Two-thirds of the publications concerned the normal brain function/mechanism (n = 67), whereas 20 publications concerned disease/therapeutic intervention and another 13 concerned normal anatomy. For the 67 original articles, 57 used human subjects whereas 10 used animal models. MRI was the commonest modality (n = 37), followed by PET (n = 16). Nine articles investigated by histology, two by multiple modalities and three by other modality

Nontransferable Individual Payoffs in Cooperative Stochastic Dynamic Games

Abstract In cooperative dynamic games with non-transferable payoffs, the players&apos; agreed-upon cooperative actions would determine the resulting payoff that each player receives. This article develops a mechanism for the derivation of individual player&apos;s payoff functions in cooperative stochastic dynamic games with nontransferable payoffs. This is the first time that individual player&apos;s payoff functions are characterized in an analytically derivable form in such a framework. An illustrative example is provided. Mathematics Subject Classifications: 91A12, 91A2

A Unix based VLSI design workstation.

Dept. of Electrical and Computer Engineering. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1988 .Y485. Source: Masters Abstracts International, Volume: 40-07, page: . Thesis (M.A.Sc.)--University of Windsor (Canada), 1988

UTJECAJ RELIGIOZNOSTI I DUHOVNOSTI NA SVAKODNEVNU PRAKSU SOCIJALNIH RADNIKA

Although social work can trace the roots of its establishment as a profession to religion/spirituality, the relationship between social work and religion/spirituality has deteriorated due to the former’s professionalisation and secularisation. Nevertheless, religious/spiritual values are important for many social work services and for the practice of the profession. It is, hence, worth demarcating the role of religion/spirituality for frontline social workers. In the current qualitative study, repeated interviews were conducted with 11 Chinese social workers from diverse social work settings. It was noted that Eastern and Western religion/spirituality can instil meaning and value in the profession’s nature.Iako se uspostavljanje socijalnog rada kao profesije temelji na religIoznosti/duhovnosti, odnos između socijalnog rada i religioznosti/duhovnosti narušen je zbog profesionalizacije i sekularizacije socijalnog rada. Usprkos tome, religiozne/duhovne vrijednosti važne su za mnoge usluge socijalnog rada i za praksu socijalnog rada kao profesije. Stoga je važno utvrditi ulogu religioznosti/duhovnosti za najizloženije socijalne radnike. U ovom kvalitativnom istraživanju provedeni su ponovljeni razgovori s 11 kineskih socijalnih radnika iz različitih područja socijalnog rada. Uočeno je da istočnjačka i zapadnjačka religioznost/duhovnost mogu unijeti značenje i vrijednost u prirodu profesije