Endogenous Money - A Structural Model of Monetary Base

In this paper, a structural monetary base model is developed. An important feature of this approach is that the model combines three parts of the determinants of the monetary base. The three parts are the commercial bank, the public and the central bank. Bank behaviour relies on an explicit specification of a maximum profit-seeking and risk-averse model which describes the determinants of the supply of deposits by banks as well as their demands for earning assets and (free) reserves. The behaviours of the public and central bank are set up exogeneously. According to the structural model, we derive the monetary base equation which is determined by various financial and real variables endogenously.

Atmospheres and Spectra of Strongly Magnetized Neutron Stars II: Effect of Vacuum Polarization

We study the effect of vacuum polarization on the atmosphere structure and radiation spectra of neutron stars with surface magnetic fields B=10^14-10^15 G, as appropriate for magnetars. Vacuum polarization modifies the dielectric property of the medium and gives rise to a resonance feature in the opacity; this feature is narrow and occurs at a photon energy that depends on the plasma density. Vacuum polarization can also induce resonant conversion of photon modes via a mechanism analogous to the MSW mechanism for neutrino oscillation. We construct atmosphere models in radiative equilibrium with an effective temperature of a few \times 10^6 K by solving the full radiative transfer equations for both polarization modes in a fully ionized hydrogen plasma. We discuss the subtleties in treating the vacuum polarization effects in the atmosphere models and present approximate solutions to the radiative transfer problem which bracket the true answer. We show from both analytic considerations and numerical calculations that vacuum polarization produces a broad depression in the X-ray flux at high energies (a few keV \la E \la a few tens of keV) as compared to models without vacuum polarization; this arises from the density dependence of the vacuum resonance feature and the large density gradient present in the atmosphere. Thus the vacuum polarization effect softens the high energy tail of the thermal spectrum, although the atmospheric emission is still harder than the blackbody spectrum because of the non-grey opacities. We also show that the depression of continuum flux strongly suppresses the equivalent width of the ion cyclotron line and therefore makes the line more difficult to observe.Comment: 21 pages, 21 figures; MNRAS; corrected minor typo

Some characterizations of spheres and elliptic paraboloids II

We show some characterizations of hyperspheres in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$ with intrinsic and extrinsic properties such as the $n$-dimensional area of the sections cut off by hyperplanes, the $(n+1)$-dimensional volume of regions between parallel hyperplanes, and the $n$-dimensional surface area of regions between parallel hyperplanes. We also establish two characterizations of elliptic paraboloids in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$ with the $n$-dimensional area of the sections cut off by hyperplanes and the $(n+1)$-dimensional volume of regions between parallel hyperplanes. For further study, we suggest a few open problems.Comment: 10 page

Atmospheres and Spectra of Strongly Magnetized Neutron Stars

We construct atmosphere models for strongly magnetized neutron stars with surface fields $B\sim 10^{12}-10^{15}$ G and effective temperatures $T_{\rm eff}\sim 10^6-10^7$ K. The atmospheres directly determine the characteristics of thermal emission from isolated neutron stars, including radio pulsars, soft gamma-ray repeaters, and anomalous X-ray pulsars. In our models, the atmosphere is composed of pure hydrogen or helium and is assumed to be fully ionized. The radiative opacities include free-free absorption and scattering by both electrons and ions computed for the two photon polarization modes in the magnetized electron-ion plasma. Since the radiation emerges from deep layers in the atmosphere with \rho\ga 10^2 g/cm$^3$, plasma effects can significantly modify the photon opacities by changing the properties of the polarization modes. In the case where the magnetic field and the surface normal are parallel, we solve the full, angle-dependent, coupled radiative transfer equations for both polarization modes. We also construct atmosphere models for general field orientations based on the diffusion approximation of the transport equations and compare the results with models based on full radiative transport. In general, the emergent thermal radiation exhibits significant deviation from blackbody, with harder spectra at high energies. The spectra also show a broad feature (\Delta E/\Ebi\sim 1) around the ion cyclotron resonance \Ebi=0.63 (Z/A)(B/10^{14}{G}) keV, where $Z$ and $A$ are the atomic charge and atomic mass of the ion, respectively; this feature is particularly pronounced when \Ebi\ga 3k\Teff. Detection of the resonance feature would provide a direct measurement of the surface magnetic fields on magnetars.Comment: 29 pages, 11 figures; corrected factor of 2 in He models: minor changes to figs 4 and 9 as a result; other very minor change

Nonlocal Flow of Convex Plane Curves and Isoperimetric Inequalities

In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric difference. We also study a new nonlocal flow of convex plane curves and discuss its evolution behavior. In the second part of the paper we discuss necessary and sufficient conditions (in terms of the (mixed) isoperimetric ratio or (mixed) isoperimetric difference) for two convex closed curves to be homothetic or parallel.Comment: 23 page

On Area-preserving and Length-preserving Nonlocal Flow of Convex Closed Plane Curves

For any $\alpha>0,$ we study $k^{\alpha}$-type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in $C^{\infty}$-norm.$\$Other relevant $k^{\alpha}$-type nonlocal flow is also discussed when $\alpha\geq1.\$Comment: Published in 'Calculus of Variations and Partial Differential Equations