A Reorganization in the Continuity of Subject Matter in Mathematics

This thesis considers a reorganization in the order of arrangement of certain topics in elementary and undergraduate mathematics; i.e. , arithmetic, algebra, plane geometry, solid geometry, trigonometry, analytic geometry, and calculus. Two terms important in the discussion are reorganization, the process of changing the relative position of topics or proofs in mathematics to an earlier or later place in the development of subject matter, and continuity, the logical order of topics arranged according to the need of one to explain the other. The purpose of the thesis is two-fold; First, to show what arrangement of topics may be desirable; and, Second, to justify the proposed changes by showing that such a reorganization will make it possible to give a simpler and more complete presentation of mathematics without affecting the logical sequence of topics. The discussion reviews the recent changes in elementary mathematics during the past forty years. These changes, in general, may be thought of as either of a general character indicating a trend or of a special character indicating a rearrangement in the order of particular topics. The general arrangement of the thesis is somewhat as follows. It is observed that propositions in elementary mathematics have been proved by methods of analytic geometry and calculus. Proofs of certain propositions in plane geometry are possible by coordinate methods. When they are presented in algebra, these proofs are not only simple but provide further understanding of topics in algebra, such as graphs, ratio and proportion, and the operations of algebra. Proofs of certain propositions, or formulas, from elementary mathematics are possible by means of integration. Such proofs by calculus are too difficult to be presented in algebra. These proofs should be postponed to calculus where the simple method of integration justifies the omission of any earlier type of proof of these propositions in elementary mathematics. In the conclusion of this discussion a rearrangement of topics in elementary mathematics (seventh year mathematics, eighth year mathematics, first year algebra, second course in algebra, and plane geometry) with special attention to the continuity of subject matter is given. Such a rearrangement, of necessity, implies changes in the order of some of the topics in later mathematics

The place and teaching of calculus in secondary schools

Thesis (M.A.)--Boston Universit

Analytic Continuation for Asymptotically AdS 3D Gravity

We have previously proposed that asymptotically AdS 3D wormholes and black holes can be analytically continued to the Euclidean signature. The analytic continuation procedure was described for non-rotating spacetimes, for which a plane t=0 of time symmetry exists. The resulting Euclidean manifolds turned out to be handlebodies whose boundary is the Schottky double of the geometry of the t=0 plane. In the present paper we generalize this analytic continuation map to the case of rotating wormholes. The Euclidean manifolds we obtain are quotients of the hyperbolic space by a certain quasi-Fuchsian group. The group is the Fenchel-Nielsen deformation of the group of the non-rotating spacetime. The angular velocity of an asymptotic region is shown to be related to the Fenchel-Nielsen twist. This solves the problem of classification of rotating black holes and wormholes in 2+1 dimensions: the spacetimes are parametrized by the moduli of the boundary of the corresponding Euclidean spaces. We also comment on the thermodynamics of the wormhole spacetimes.Comment: 28 pages, 14 figure

Black Hole Thermodynamics and Riemann Surfaces

We use the analytic continuation procedure proposed in our earlier works to study the thermodynamics of black holes in 2+1 dimensions. A general black hole in 2+1 dimensions has g handles hidden behind h horizons. The result of the analytic continuation is a hyperbolic 3-manifold having the topology of a handlebody. The boundary of this handlebody is a compact Riemann surface of genus G=2g+h-1. Conformal moduli of this surface encode in a simple way the physical characteristics of the black hole. The moduli space of black holes of a given type (g,h) is then the Schottky space at genus G. The (logarithm of the) thermodynamic partition function of the hole is the Kaehler potential for the Weil-Peterson metric on the Schottky space. Bekenstein bound on the black hole entropy leads us to conjecture a new strong bound on this Kaehler potential.Comment: 17+1 pages, 9 figure

On the relationship between plane and solid geometry

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned area

Near-Field Microwave Microscopy on nanometer length scales

The Near-Field Microwave Microscope (NSMM) can be used to measure ohmic losses of metallic thin films. We report on the presence of a new length scale in the probe-to- sample interaction for the NSMM. We observe that this length scale plays an important role when the tip to sample separation is less than about 10nm. Its origin can be modeled as a tiny protrusion at the end of the tip. The protrusion causes deviation from a logarithmic increase of capacitance versus decreasing height of the probe above the sample. We model this protrusion as a cone at the end of a sphere above an infinite plane. By fitting the frequency shift of the resonator versus height data (which is directly related to capacitance versus height) for our experimental setup, we find the protrusion size to be 3nm to 5nm. For one particular tip, the frequency shift of the NSMM relative to 2 micrometers away saturates at a value of about -1150 kHz at a height of 1nm above the sample, where the nominal range of sheet resistance values of the sample are 15 ohms to 150 ohms. Without the protrusion, the frequency shift would have followed the logarithmic dependence and reached a value of about -1500 kHz.Comment: 6 pages, 7 figures (included in 6 pages

Accretion Disk Illumination in Schwarzschild and Kerr Geometries: Fitting Formulae

We describe the methodology and compute the illumination of geometrically thin accretion disks around black holes of arbitrary spin parameter $a$ exposed to the radiation of a point-like, isotropic source at arbitrary height above the disk on its symmetry axis. We then provide analytic fitting formulae for the illumination as a function of the source height $h$ and the black hole angular momentum $a$. We find that for a source on the disk symmetry axis and $h/M > 3$, the main effect of the parameter $a$ is allowing the disk to extend to smaller radii (approaching $r/M \to 1$ as $a/M \to 1$) and thus allow the illumination of regions of much higher rotational velocity and redshift. We also compute the illumination profiles for anisotropic emission associated with the motion of the source relative to the accretion disk and present the fractions of photons absorbed by the black hole, intercepted by the disk or escaping to infinity for both isotropic and anisotropic emission for $a/M=0$ and $a/M=0.99$. As the anisotropy (of a source approaching the disk) increases the illumination profile reduces (approximately) to a single power-law, whose index, $q$, because of absorption of the beamed photons by the black hole, saturates to a value no higher than $q \gtrsim 3$. Finally, we compute the fluorescence Fe line profiles associated with the specific illumination and compare them among various cases.Comment: 26 pages, 21 b/w figures, accepted for publication in the Astrophysical Journal as of 4/16/200

General Relativistic Ray-Tracing Method for Estimating the Energy and Momentum Deposition by Neutrino Pair Annihilation in Collapsars

Bearing in mind the application to the collapsar models of gamma-ray bursts (GRBs), we develop a numerical scheme and code for estimating the deposition of energy and momentum due to the neutrino pair annihilation ($\nu + {\bar \nu} \rightarrow e^{-} + e^{+}$) in the vicinity of accretion tori around a Kerr black hole. Our code is designed to solve the general relativistic neutrino transfer by a ray-tracing method. To solve the collisional Boltzmann equation in curved spacetime, we numerically integrate the so-called rendering equation along the null geodesics. For the neutrino opacity, the charged-current $\beta$-processes are taken into account, which are dominant in the vicinity of the accretion tori. The numerical accuracy of the developed code is certificated by several tests, in which we show comparisons with the corresponding analytic solutions. Based on the hydrodynamical data in our collapsar simulation, we estimate the annihilation rates in a post-processing manner. Increasing the Kerr parameter from 0 to 1, it is found that the general relativistic effect can increase the local energy deposition rate by about one order of magnitude, and the net energy deposition rate by several tens of percents. After the accretion disk settles into a stationary state (typically later than $\sim 9$ s from the onset of gravitational collapse), we point out that the neutrino-heating timescale in the vicinity of the polar funnel region can be shorter than the dynamical timescale. Our results suggest the neutrino pair annihilation has a potential importance equal to the conventional magnetohydrodynamic mechanism for igniting the GRB fireballs.Comment: 33 pages, 15 figures, accepted to the Ap