A Gravitational Lens need not produce an Odd Number of Images

Given any space-time $M$ without singularities and any event $O$, there is a natural continuous mapping $f$ of a two dimensional sphere into any space-like slice $T$ not containing $O$. The set of future null geodesics (or the set of past null geodesics) forms a 2-sphere $S^2$ and the map $f$ sends a point in $S^2$ to the point in $T$ which is the intersection of the corresponding geodesic with $T$. To require that $f$, which maps a two dimensional space into a three dimensional space, satisfy the condition that any point in the image of $f$ has an odd number of preimages, is to place a very strong condition on $f$. This is exactly what happens in any case where the odd image theorem holds for a transparent gravitational lens. It is argued here that this condition on $f$ is probably too restrictive to occur in general; and if it appears to hold in a specific example, then some $f$ should be calculated either analytically or numerically to provide either an illustrative example or counterexample.Comment: 8 pages, amste

Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes

We present a systematic method for constructing boundary conditions (numerical and physical) of the required accuracy, for compact (Pade-like) high-order finite-difference schemes for hyperbolic systems. First, a roper summation-by-parts formula is found for the approximate derivative. A 'simultaneous approximation term' (SAT) is then introduced to treat the boundary conditions. This procedure leads to time-stable schemes even in the system case. An explicit construction of the fourth-order compact case is given. Numerical studies are presented to verify the efficacy of the approach

The stability of numerical boundary treatments for compact high-order finite-difference schemes

The stability characteristics of various compact fourth and sixth order spatial operators are assessed using the theory of Gustafsson, Kreiss and Sundstrom (G-K-S) for the semi-discrete Initial Boundary Value Problem (IBVP). These results are then generalized to the fully discrete case using a recently developed theory of Kreiss. In all cases, favorable comparisons are obtained between the G-K-S theory, eigenvalue determination, and numerical simulation. The conventional definition of stability is then sharpened to include only those spatial discretizations that are asymptotically stable. It is shown that many of the higher order schemes which are G-K-S stable are not asymptotically stable. A series of compact fourth and sixth order schemes, which are both asymptotically and G-K-S stable for the scalar case, are then developed

The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme

The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem a randomized polynomial-time algorithm that computes a (1+eps)-approximation to the optimal tour, for any fixed eps>0, in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar (T-04)

Entropy Change through Rayleigh-B\'enard Convective Transition with Rigid Boundaries

The previous investigation on Rayleigh-B\'enard convection of a dilute classical gas [T. Kita: J. Phys. Soc. Jpn. {\bf 75} (2006) 124005] is extended to calculate entropy change of the convective transition with the rigid boundaries. We obtain results qualitatively similar to those of the stress-free boundaries. Above the critical Rayleigh number, the roll convection is realized among possible steady states with periodic structures, carrying the highest entropy as a function of macroscopic mechanical variables.Comment: 5 pages, 4 figure

Effects of Chemical Potential on Hadron Masses in the Phase Transition Region

We study the response of hadron masses with respect to chemical potential at $\mu=0$. Our preliminary results of the pion channel show that $\partial m/\partial \mu$ in the confinement phase is significantly larger than that in the deconfinement phase, which is consistent with the chiral restoration.Comment: LATTICE99 (finite temperature and density), 3 pages, 3 figure

A review of the health impact of smoking control at the workplace

Purpose. To summarize and provide a critical review of worksite health promotion program evaluations published between 1968 and 1994 that addressed the health impact of worksite smoking cessation programs and smoking policies. Methods. A comprehensive literature search conducted under the auspices of the Centers for Disease Control and Prevention identified 53 smoking cessation program evaluation reports, of which 41 covered worksite single-topic cessation programs. Nine additional reports were located through manual search of citations from published reports and reviews. These 50 reports covered 52 original data-based studies of cessation programs. The search produced 19 reports for tobacco policy evaluations, of which 12 addressed health impact. An additional 17 reports were located by the authors. These 29 reports covered 29 studies of policy impact. Summary of Important Findings. Smoking cessation group programs were found to be more effective than minimal treatment programs, although less intensive treatment, when combined with high participation rates, can influence the total population. Tobacco policies were found to reduce cigarette consumption at work and worksite environmental tobacco smoke (RTS) exposure. Conclusions. The literature is rated suggestive for group and incentive interventions; indicative for minimal interventions, competitions, and medical interventions; and acceptable for the testing of incremental effects. Because of the lack of experimental control, the smoking policy literature is rated as weak, although there is strong consistency in results for reduced cigarette consumption and decreased exposure to ETS at work. (Am J Health Promot 1998;13[2]:83-104.

Exceptional type-I superconductivity of the layered silver oxide Ag5_5Pb2_2O6_6

We report zero-resistivity transition and the details of magnetic transition of a layered silver oxide Ag$_5$Pb$_2$O$_6$ single crystal, which make definitive evidence of superconductivity in this compound. In the AC susceptibility of a mono-crystal, we observed large supercooling, as well as positive peaks in the real part of the susceptibility indicating the reversibility of magnetic process. These observations reveal that Ag$_5$Pb$_2$O$_6$ is probably the first oxide that shows type-I superconductivity. Evaluation of the superconducting parameters not only gives confirming evidence of type-I superconductivity, but also indicates that it is a dirty-limit superconductor. We also analyze supercooling to determine the upper limit of the Ginzburg-Landau parameter.Comment: v2: PACS numbers are adde