Mathematics Professional Development Workshop for Middle School Teachers: Concept Versus Memorization

This article includes professional development topics for middle school mathematics and science teachers from two week-long Urban Teacher Institutes. These Institutes were held at J. Sargeant Reynolds Community College (JSRCC) and its partner institution, Virginia Commonwealth University (VCU), during the summers of 2007 and 2008, and were supported by a grant obtained by Dr. Harriet Morrison (JSRCC). Co-author Dr. Dewey Taylor directed the 2007 workshop, and both authors served as faculty leaders in both workshops. The workshops focused on teaching in an urban environment and community mapping (understanding the details of a certain locale to make the teacher more knowledgeable about the environments of both the students and the schools). The community mapping aspect of the workshops was led by Dr. Shirley Key of the University of Memphis. They featured content teaching and applications led by VCU faculty in mathematics, physics, forensics, engineering, mathematics education, and science education. This article focuses on the mathematics professional development strand in the workshop which featured conceptual learning with graphing calculator support as an alternative to the memorization of formulas

Capabilities and limitations of the Jicamarca radar as an MST radar

The Jicamarca radar (Long. 76.52W, Lat. 11.56S), located at 20 km from Lima at approximately 500 meters over sea level, is surrounded by mountains which provide a good shield from man-made interference. The radio horizon goes from a few hundred meters, across the dry valley where it is located, to 15 km, along the valley in the direction of the continental divide. This limits the clutter to 15 km, except for one high peak at 21 km. It is the most equatorial of all existing MST radars. Its proximity to the Andes, makes its location unique for the study of lee waves and orographic-induced turbulence. Vertical as well as horizontal projections of MST velocities are obtained by simultaneously pointing with different sections of the antenna into three or four different directions. The transmitters, receivers, and systems for data acquisition, processing, and control are included

Spin-2 Amplitudes in Black-Hole Evaporation

Quantum amplitudes for $s=2$ gravitational-wave perturbations of Einstein/scalar collapse to a black hole are treated by analogy with $s=1$ Maxwell perturbations. The spin-2 perturbations split into parts with odd and even parity. We use the Regge-Wheeler gauge; at a certain point we make a gauge transformation to an asymptotically-flat gauge, such that the metric perturbations have the expected falloff behaviour at large radii. By analogy with $s=1$, for $s=2$ natural 'coordinate' variables are given by the magnetic part $H_{ij} (i,j=1,2,3)$ of the Weyl tensor, which can be taken as boundary data on a final space-like hypersurface $\Sigma_F$. For simplicity, we take the data on the initial surface $\Sigma_I$ to be exactly spherically-symmetric. The (large) Lorentzian proper-time interval between $\Sigma_I$ and $\Sigma_F$, measured at spatial infinity, is denoted by $T$. We follow Feynman's $+i\epsilon$ prescription and rotate $T$ into the complex: $T\to{\mid}T{\mid} \exp(-i\theta)$, for $0<\theta\leq\pi/2$. The corresponding complexified {\it classical} boundary-value problem is expected to be well-posed. The Lorentzian quantum amplitude is recovered by taking the limit as $\theta\to 0_+$. For boundary data well below the Planck scale, and for a locally supersymmetric theory, this involves only the semi-classical amplitude $\exp(iS^{(2)}_{\rm class}$, where $S^{(2)}_{\rm class}$ denotes the second-variation classical action. The relations between the $s=1$ and $s=2$ natural boundary data, involving supersymmetry, are investigated using 2-component spinor language in terms of the Maxwell field strength $\phi_{AB}=\phi_{(AB)}$ and the Weyl spinor $\Psi_{ABCD}=\Psi_{(ABCD)}$

Cretaceous-to-recent record of elevated 3He/4He along the Hawaiian-Emperor volcanic chain

Helium isotopes are a robust geochemical tracer of a primordial mantle component in hot spot volcanism. The high 3He/4He (up to 35 RA, where RA is the atmospheric 3He/4He ratio of 1.39 × 10−6) of some Hawaiian Island volcanism is perhaps the classic example. New results for picrites and basalts from the Hawaiian-Emperor seamount chain indicate that the hot spot has produced high 3He/4He lavas for at least the last 76 million years. Picrites erupted at 76 Ma have 3He/4He (10–14 RA), which is at the lower end of the range for the Hawaiian Islands but still above the range of modern mid-ocean ridge basalt (MORB; 6–10 RA). This was at a time when hot spot volcanism was occurring on thin lithosphere close to a spreading ridge and producing lava compositions otherwise nearly indistinguishable from MORB. After the hot spot and spreading center diverged during the Late Cretaceous, the hot spot produced lavas with significantly higher 3He/4He (up to 24 RA). Although 3He/4He ratios stabilized at relatively high values by 65 Ma, other chemical characteristics such as La/Yb and 87Sr/86Sr did not reach and stabilize at Hawaiian-Island-like values until ~45 Ma. Our limited 3He/4He record for the Hawaiian hot spot shows a poor correlation with plume flux estimates (calculated from bathymetry and residual gravity anomalies [Van Ark and Lin, 2004]). If 3He is a proxy for the quantity of primordial mantle material within the plume, then the lack of correlation between 3He/4He and calculated plume flux suggests that variation in primordial mantle flux is not the primary factor controlling total plume flux

Relic Radiation from an Evaporating Black Hole

We present a non-string-theoretic calculation of the microcanonical entropy of relic integer-spin Hawking radiation -- at fixed total energy $E$. The only conserved macroscopic quantity is the total energy $E$ (the total energy of the relic radiation). Data for a boundary-value approach, with massless, integer-spin perturbations, are set on initial and final space-like hypersurfaces. In the resulting 1-dimensional statistical-mechanics problem, the real part of the (complex) time separation at spatial infinity, $T = {\mid}T{\mid}\exp(-i\delta), \delta >0$, is the variable conjugate to the total energy. We count the number of weak-field configurations on the final space-like hypersurface with energy $E$. One recovers the Cardy formula and the Bekenstein-Hawking entropy, if Re(T) is of the order of the black-hole life- time, leading to a statistical interpretation of black-hole entropy. The microcanonical entropy includes a logarithmic correction to the black-hole area law, which is {\it universal} (independent of black-hole parameters). Here, the discreteness of the energy levels is crucial. This approach is compared with that of string theory for the transition to the fundamental-string r\'egime in the final stages of evaporation. The squared coupling, $g^2$, regulating the transition to a highly-excited string state and {\it vice versa}, can be related to the angle, $\delta$, of complex-time rotation above. The strong-coupling r\'egime corresponds to a Euclidean black hole, while the physical limit of a Lorentzian space-time (as $\delta \to 0_+$) corresponds to the weak-coupling r\'egime. This resembles the transition to a highly-excited string-like state which subsequently decays into massless particles, thereby avoiding the naked singularity.Comment: To appear in International Journal of Modern Physics

Ab initio Wannier-function-based correlated calculations of Born effective charges of crystalline Li2_{2}O and LiCl

In this paper we have used our recently developed ab initio Wannier-function-based methodology to perform extensive Hartree-Fock and correlated calculations on Li$_{2}$O and LiCl to compute their Born effective charges. Results thus obtained are in very good agreement with the experiments. In particular, for the case of Li$_{2}$O, we resolve a controversy originating in the experiment of Osaka and Shindo {[}Solid State Commun. 51 (1984) 421] who had predicted the effective charge of Li ions to be in the range 0.58--0.61, a value much smaller compared to its nominal value of unity, thereby, suggesting that the bonding in the material could be partially covalent. We demonstrate that effective charge computed by Osaka and Shindo is the Szigeti charge, and once the Born charge is computed, it is in excellent agreement with our computed value. Mulliken population analysis of Li$_{2}$O also confirms ionic nature of the bonding in the substance.Comment: 11 pages, 1 figure. To appear in Phys. Rev. B (Feb 2008

Black hole evaporation in a spherically symmetric non-commutative space-time

Recent work in the literature has studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space-time and weak radiation at a very late time. The relevant quantum amplitudes have been evaluated for bosonic and fermionic fields, showing that no information is lost in collapse to a black hole. On the other hand, recent developments in noncommutative geometry have shown that, in general relativity, the effects of non-commutativity can be taken into account by keeping the standard form of the Einstein tensor on the left-hand side of the field equations and introducing a modified energy-momentum tensor as a source on the right-hand side. Relying on the recently obtained non-commutativity effect on a static, spherically symmetric metric, we have considered from a new perspective the quantum amplitudes in black hole evaporation. The general relativity analysis of spin-2 amplitudes has been shown to be modified by a multiplicative factor F depending on a constant non-commutativity parameter and on the upper limit R of the radial coordinate. Limiting forms of F have been derived which are compatible with the adiabatic approximation.Comment: 8 pages, Latex file with IOP macros, prepared for the QFEXT07 Conference, Leipzig, September 200

Gravitational amplitudes in black-hole evaporation: the effect of non-commutative geometry

Recent work in the literature has studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space-time and weak radiation at a very late time. The relevant quantum amplitudes have been evaluated for bosonic and fermionic fields, showing that no information is lost in collapse to a black hole. On the other hand, recent developments in noncommutative geometry have shown that, in general relativity, the effects of noncommutativity can be taken into account by keeping the standard form of the Einstein tensor on the left-hand side of the field equations and introducing a modified energy-momentum tensor as a source on the right-hand side. The present paper, relying on the recently obtained noncommutativity effect on a static, spherically symmetric metric, considers from a new perspective the quantum amplitudes in black hole evaporation. The general relativity analysis of spin-2 amplitudes is shown to be modified by a multiplicative factor F depending on a constant non-commutativity parameter and on the upper limit R of the radial coordinate. Limiting forms of F are derived which are compatible with the adiabatic approximation here exploited. Approximate formulae for the particle emission rate are also obtained within this framework.Comment: 14 pages, 2 figures, Latex macros. In the final version, section 5 has been amended, the presentation has been improved, and References 21-24 have been added. Last misprints amended in Section 5 and Ref. 2