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

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

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

A Computational Interpretation of Context-Free Expressions

We phrase parsing with context-free expressions as a type inhabitation problem where values are parse trees and types are context-free expressions. We first show how containment among context-free and regular expressions can be reduced to a reachability problem by using a canonical representation of states. The proofs-as-programs principle yields a computational interpretation of the reachability problem in terms of a coercion that transforms the parse tree for a context-free expression into a parse tree for a regular expression. It also yields a partial coercion from regular parse trees to context-free ones. The partial coercion from the trivial language of all words to a context-free expression corresponds to a predictive parser for the expression

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

FUSE Spectra of the Black Hole Binary LMC X-3

Far-ultraviolet spectra of LMC X-3 were taken covering photometric phases 0.47 to 0.74 in the 1.7-day orbital period of the black-hole binary (phase zero being superior conjunction of the X-ray source). The continuum is faint and flat, but appears to vary significantly during the observations. Concurrent RXTE/ASM observations show the system was in its most luminous X-ray state during the FUSE observations. The FUV spectrum contains strong terrestrial airglow emission lines, while the only stellar lines clearly present are emissions from the O VI resonance doublet. Their flux does not change significantly during the FUSE observations. These lines are modelled as two asymmetrical profiles, including the local ISM absorptions due to C II and possibly O VI. Velocity variations of O VI emission are consistent with the orbital velocity of the black hole and provide a new constraint on its mass.Comment: 12 pages including 1 table, 4 diagrams To appear in A

Analytical model of brittle destruction based on hypothesis of scale similarity

The size distribution of dust particles in nuclear fusion devices is close to the power function. A function of this kind can be the result of brittle destruction. From the similarity assumption it follows that the size distribution obeys the power law with the exponent between -4 and -1. The model of destruction has much in common with the fractal theory. The power exponent can be expressed in terms of the fractal dimension. Reasonable assumptions on the shape of fragments concretize the power exponent, and vice versa possible destruction laws can be inferred on the basis of measured size distributions.Comment: 10 pages, 3 figure