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

A Radio--Optical Reference Frame VIII. CCD observations from KPNO and CTIO: internal calibration and first results

In this pilot investigation, precise optical positions in the FK5 system are presented for a set of 16 compact extragalactic radio sources, which will be part of the future radio--optical reference frame. The 0.9 m KPNO and CTIO telescopes equipped with 2K CCD's have been used for this project. The astrometric properties of these instruments are investigated in detail. New techniques of using wide field CCD observations for astrometry in general are developed. An internal precision of 5 to 31 mas in position per single exposure is found, depending on the brightness of the object. The tie to the primary optical reference system is established by photographic astrometry using dedicated astrographs on both hemispheres. An accuracy of $\approx 30$ mas per source is estimated for the multi--step reduction procedure when based on the future Hipparcos catalog, while the FK5--based positions suffer from system errors of 100 to 200 mas as compared to the radio positions. This work provides a contribution to the international effort to link the Hipparcos instrumental coordinate system to the quasi--inertial VLBI radio reference frame. Precise radio and optical astrometry of a large sample of compact extragalactic sources will also contribute to the astrophysics of these objects by comparing the respective centers of emission at the optical and radio wavelengths.Comment: AAS v.4 LaTeX, 2 parts on 1 file (main text + deluxetable), accepted by AJ, Dec.95, fig. with reprint

Interpretation of neutrino flux limits from neutrino telescopes on the Hillas plot

We discuss the interplay between spectral shape and detector response beyond a simple E^-2 neutrino flux at neutrino telescopes, at the example of time-integrated point source searches using IceCube-40 data. We use a self-consistent model for the neutrino production, in which protons interact with synchrotron photons from co-accelerated electrons, and we fully take into account the relevant pion and kaon production modes, the flavor composition at the source, flavor mixing, and magnetic field effects on the secondaries (pions, muon, and kaons). Since some of the model parameters can be related to the Hillas parameters R (size of the acceleration region) and B (magnetic field), we relate the detector response to the Hillas plane. In order to compare the response to different spectral shapes, we use the energy flux density as a measure for the pion production efficiency times luminosity of the source. We demonstrate that IceCube has a very good reach in this quantity for AGN nuclei and jets for all source declinations, while the spectra of sources with strong magnetic fields are found outside the optimal reach. We also demonstrate where neutrinos from kaon decays and muon tracks from tau decays can be relevant for the detector response. Finally, we point out the complementarity between IceCube and other experiments sensitive to high-energy neutrinos, at the example of 2004-2008 Earth-skimming neutrino data from Auger. We illustrate that Auger, in principle, is better sensitive to the parameter region in the Hillas plane from which the highest-energetic cosmic rays may be expected in this model.Comment: 28 pages, 10 figures. Substantial clarifications, such as on definition of "sensitivity" and model descriptio

Half-Life of 14^{14}O

We have measured the half-life of $^{14}$O, a superallowed $(0^{+} \to 0^{+})$ $\beta$ decay isotope. The $^{14}$O was produced by the $^{12}$C($^{3}$He,n)$^{14}$O reaction using a carbon aerogel target. A low-energy ion beam of $^{14}$O was mass separated and implanted in a thin beryllium foil. The beta particles were counted with plastic scintillator detectors. We find $t_{1/2} = 70.696\pm 0.052$ s. This result is $1.5\sigma$ higher than an average value from six earlier experiments, but agrees more closely with the most recent previous measurement.Comment: 10 pages, 5 figure

Fat transforms ascorbic acid from inhibiting to promoting acid-catalysed N-nitrosation

&lt;b&gt;Background&lt;/b&gt;: The major potential site of acid nitrosation is the proximal stomach, an anatomical site prone to a rising incidence of metaplasia and adenocarcinoma. Nitrite, a pre-carcinogen present in saliva, can be converted to nitrosating species and N-nitroso compounds by acidification at low gastric pH in the presence of thiocyanate. &lt;b&gt;Aims&lt;/b&gt;: To assess the effect of lipid and ascorbic acid on the nitrosative chemistry under conditions simulating the human proximal stomach. &lt;b&gt;Methods&lt;/b&gt;: The nitrosative chemistry was modelled in vitro by measuring the nitrosation of four secondary amines under conditions simulating the proximal stomach. The N-nitrosamines formed were measured by gas chromatography–ion-trap tandem mass spectrometry, while nitric oxide and oxygen levels were measured amperometrically. &lt;b&gt;Results&lt;/b&gt;: In absence of lipid, nitrosative stress was inhibited by ascorbic acid through conversion of nitrosating species to nitric oxide. Addition of ascorbic acid reduced the amount of N-nitrosodimethylamine formed by fivefold, N-nitrosomorpholine by .1000-fold, and totally prevented the formation of N-nitrosodiethylamine and N-nitrosopiperidine. In contrast, when 10% lipid was present, ascorbic acid increased the amount of Nnitrosodimethylamine, N-nitrosodiethylamine and N-nitrosopiperidine formed by approximately 8-, 60- and 140-fold, respectively, compared with absence of ascorbic acid. &lt;b&gt;Conclusion&lt;/b&gt;: The presence of lipid converts ascorbic acid from inhibiting to promoting acid nitrosation. This may be explained by nitric oxide, formed by ascorbic acid in the aqueous phase, being able to regenerate nitrosating species by reacting with oxygen in the lipid phase

A Role for the Vacuolating Cytotoxin, VacA, in Colonization and Helicobacter pylori-Induced Metaplasia in the Stomach

Carriage of Helicobacter pylori strains producing more active (s1/i1) forms of VacA is strongly associated with gas-tric adenocarcinoma. To our knowledge, we are the first to determine effects of different polymorphic forms of VacA on inflammation and metaplasia in the mouse stomach. Bacteria producing the less active s2/i2 form of VacA colonized mice more efficiently than mutants null for VacA or producing more active forms of it, providing the first evidence of a positive role for the minimally active s2/i2 toxin. Strains producing more active toxin forms induced more severe and extensive metaplasia and in flammation in the mouse stomach than strains producing weakly active (s2/i2) toxin. We also examined the association in humans, controlling for cag PAI status. In human gastric biopsy specimens, the vacA i1 allele was strongly associated with precancerous intestinal metaplasia, with almost complete absence of intestinal metaplasia in subjects infected with i2-type strains, even in a vacA s1, cagA+ background

Stability of spikes in the shadow Gierer-Meinhardt system with Robin boundary conditions

We consider the shadow system of the Gierer-Meinhardt system in a smooth bounded domain RN,At=2A−A+,x, t>0, ||t=−||+Ardx, t>0 with the Robin boundary condition +aAA=0, x, where aA>0, the reaction rates (p,q,r,s) satisfy 1<p<()+, q>0, r>0, s0, 1<<+, the diffusion constant is chosen such that 1, and the time relaxation constant is such that 0. We rigorously prove the following results on the stability of one-spike solutions: (i) If r=2 and 1<p<1+4/N or if r=p+1 and 1<p<, then for aA>1 and sufficiently small the interior spike is stable. (ii) For N=1 if r=2 and 1<p3 or if r=p+1 and 1<p<, then for 0<aA<1 the near-boundary spike is stable. (iii) For N=1 if 3<p<5 and r=2, then there exist a0(0,1) and µ0>1 such that for a(a0,1) and µ=2q/(s+1)(p−1)(1,µ0) the near-boundary spike solution is unstable. This instability is not present for the Neumann boundary condition but only arises for the Robin boundary condition. Furthermore, we show that the corresponding eigenvalue is of order O(1) as 0. ©2007 American Institute of Physic

3-D multiobservable probabilistic inversion for the compositional and thermal structure of the lithosphere and upper mantle: III. Thermochemical tomography in the Western-Central U.S.

Acknowledgments We are indebted to F. Darbyshire and J. von Hunen for useful comments on earlier versions of this work. This manuscript benefited from thorough and constructive reviews by W. Levandowski and an anonymous reviewer. We also thank J. Connolly, M. Sambridge, B. Kennett, S. Lebedev, B. Shan, U. Faul, and M. Qashqai for insightful discussions about, and contributions to, some of the concepts presented in this paper. The work of J.C.A. has been supported by two Australian Research Council Discovery grants (DP120102372 and DP110104145). Seismic data are from the IRIS DMS. D.L.S. acknowledges support from NSF grant EAR-135866. This is contribution 848 from the ARC Centre of Excellence for Core to Crust Fluid Systems (http://www.ccfs.mq.edu.au) and 1106 in the GEMOC Key Centre (http://www.gemoc.mq.edu.au).Peer reviewedPublisher PD