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This chapter begins with a discussion of spherically symmetric spacetimes, the Schwarzschild metric, and other coordinates. It then covers Schwarzschild spacetime, the motion of the planets and perihelion precession, stability of circular orbits, deflection of light rays, red shift and time delay, spherically symmetric interior solutions, the Schwarzschild black hole, spherically symmetric gravitational collapse, the Reissner–Nordström solution, and Schwarzschild spacetime in dimension n + 1.

The hippocampus and associated structures are organized in multiple loops, with reciprocal connections to the neocortex. The most prominent collective pattern of hippocampal neurons is theta oscillation. In one-dimensional tasks, pyramidal cells fire maximally at particular positions, signifying the place field center. The assembly members that define the current location also contribute spikes to the representation of past and future positions in multiple theta cycles. Similarly, positional “distances” among items of an episodic list can be coded by the synaptic strengths between the cell assemblies. In two-dimensional environments, exploration leads to crossing the same positions from different directions. These junctions serve to establish a map and subsequent landmark (map-based) navigation. The hallmark of the cognitive map is the presence of omnidirectional place cells in the hippocampus and tessellating “grid cells” in the entorhinal cortex. Neuron members of an omnidirectional or explicit assembly collectively define or symbolize the semantic “meaning” of an item.

It reviews Newton’s argument for the claim that the moon is maintained in its orbit by an inverse-square force directed toward the earth. It introduces Newton’s moon-test argument for identifying the force that maintains the moon in its orbit with terrestrial gravity. Newton shows that inverse-square adjusting the centripetal acceleration exhibited by the lunar orbit agrees with Huygens’ measurement of the strength of terrestrial gravity at the surface of the earth. This chapter includes a somewhat detailed account of Newton’s moon-test calculation, including his dubious precession correction. It discusses his first two Rules for reasoning in natural philosophy, and his two-body correction. It examines the informative moon-test argument presented in Newton’s scholium to proposition 4. It argues that the agreement between the moon-test measurements and the pendulum measurements of the strength of terrestrial gravity is an example of empirical success. It also reviews the lunar precession problem.

Part I argues that the precision of Newton’s moon-test calculation goes beyond what modern least squares assessment can support from his cited data and that his data afford no support for his precession correction to offset the action of the sun; but, that Newton is innocent of Westfall’s main accusation of data fudging in the moon-test. Part II argues that Newton’s inference does not depend on his precession correction or on his selection of which lunar distance estimates to include. It argues that a correction for syzygy distances can defend the larger lunar distance Newton assigns in his moon-test of corollary 7 of proposition 37. Appendix 1 discusses the details of Newton’s moon-test calculation from corollary 7 of proposition 37 of book 3. It shows that Newton’s moon-test inference continues to hold up when simplifying assumptions of his basic calculation are replaced by more realistic approximations.

Deliberation entails the sequential, serial search through possible options. This means that deliberation requires a mechanism to represent the structure of the world, from which predictions can be generated concerning these options and the expectations of the consequences of taking those options. Deliberation requires a mechanism to move mentally through those predictions as well as a mechanism to evaluate and compare those predictions. Neural signals for each of these factors have been found in the rat.

This chapter presents simple ideas for understanding how spin-polarized currents can be used to exert spin-transfer torques in magnetic devices. The chapter reviews recent progress for measuring the magnetic dynamics that result from spin-transfer torques in 100-nm-scale metallic spin valves and magnetic tunnel junctions. The chapter also discusses how the transport of spin and charge in magnetic devices changes when the structures are made even smaller, extending from magnetic particles a micron in diameter, to a few nanometers, and down to a single molecule. As the size of the magnet shrinks, effects such as Coulomb blockade and energy-level quantization can become dominant, and it becomes necessary to move beyond simple independent-electron models to consider the true correlated many-electron quantum states at the root of ferromagnetism.

As was discussed in the previous chapter, the precession of the equinoxes discovered by Hipparchus was the basis of the Stoics' claim that there was a divinity of great power accountable for such phenomenon. This precession was symbolized by the death of the bull being killed by Perseus—whom they referred to as their native god. His power was shown not just over the bull, but also over other constellations resting on the celestial equator when the spring equinox was in Taurus. The Perseus cult then spread to the pirates of Cilicia who had close associations with Mithridates Eupator. This led the Cilician pirates to adopt the name Mithras as their new god.

A spherically symmetric metric has two unknown scalar metric functions. The Schwarzschild solution to the GR field equation is presented. An embedding diagram is used to visualize such a warp space. GR predicts a solar deflection of a light ray that is twice as large as that implied by the equivalence principle. We present a brief discussion of gravitational lensing, with the lens equation derived. The precession of Mercury's perihelion and the Shapiro time delay of a light ray are worked out as successful tests of general relativity.

The nonequilibrium character of magnetisation processes means that magnetisation processes are time-dependent, even if the external magnetic field is kept constant. Intrinsic magnetic properties, such as magnetisation and magnetic anisotropy, are realised on very small length scales, typically less than 1 nanometer, and correspond to very fast processes of quantum-mechanical origin. They can be considered as equilibrium properties, described by the Boltzmann distribution. By contrast, extrinsic phenomena, such as hysteresis, are realised on length scales of several nanometers or more. By atomic standards, they require considerable equilibration times, and many methods familiar from equilibrium statistical mechanics become inapplicable. The simplest model of magnetisation dynamics is a single spin in a magnetic field. This chapter discusses the dynamics of magnetisation, quantum dynamics and resonance, spin precession, uniform magnetic resonance, spin waves, spin dynamics in inhomogeneous magnets, relaxation and its physical origin, coarse-grained models, Langevin models, slow magnetisation dynamics, magnetic viscosity and sweep-rate dependence, superposition model of magnetic viscosity, energy-barrier models, and superparamagnetism.

The oldest extant medieval globe made in the Latin West dates from ca. 1320-1340. It is a precession globe built in keeping with Ptolemy's description and does not fit in any known medieval tradition. The oldest celestial maps in the mathematical tradition were made around 1425 by Conrad of Dyffenbach. His trapezoidal projection appears to be completely new. Another of his maps is based on the polar equidistant projection. This latter projection was also used in ca. 1435 for a pair of celestial maps attributed to Reinardus Gensfelder and closely connected to the Vienna globe making enterprise. The stereographic projection was used for celestial maps other than astrolabes only in the second half of the fifteenth century. The globe by Hans Dorn of 1480 exemplifies the Viennese tradition in globe making. This globe and the other extant celestial globe of 1492 by Johannes Stöffler were designed to help the astrologer to fix the mundane houses. The constellations on Stöffler's globe show the impact of Michael Scot's iconography.

Electron diffraction comes from the combined scattering from many atoms arranged in a crystal. Bragg’s law and the Ewald sphere explain where the diffraction spots will fall on a detector and when the different reflections will be excited. Phase identification and zone-axis orientation is done from one or a few ED patterns. Determination of the unit-cell dimensions come from three main zone axes, high-order Laue zones (HOLZ) or a tilt series. Precession geometry is explained with ray paths and snap-shots from the collection of precession patterns in the TEM. Digital (by software) and analogue (by hardware) precession are compared. The geometrical factors that influence intensities, i.e. the Lorentz factor are given. The very latest techniques for collecting complete 3D ED data is shown, with the two existing geometries, automated diffraction tomography and the rotation method. Convergent-beam electron diffraction (CBED) is briefly introduced.

In the central field, the energy and angular momentum are conserved. It allows for the reduction of this problem to the problem of the motion of the particle in the effective one-dimensional field. Here the motion of a particle in Coulomb field or in the field of the isotropic harmonic oscillation with small perturbations are the most important ones. The authors discuss how the motion of a particle in the given central field can be described qualitatively for different values of the angular momentum and of the energy. Several problems deal with the motion of a particle in the Coulomb field under influence of weak constant uniform electric or magnetic fields (the classical analog of the Stark or Zeeman effect). In addition, the authors consider the motion of a charged particle in the field of the magnetic monopole and magnetic dipole. The motion of the Earth–Moon system in the field of the Sun is considered in some approximation. The displacement of the Coulomb orbit under the influence of a small force of radiation damping.

This chapter discusses the mass, size and shape of the Earth and consequent gravitational field, including the important effects due to the Earth not being a perfect sphere. A brief discussion of the pressure at the Earth’s center is also included. The relation between the gravitational potential of the Earth and its moments of inertia is then derived and a formula for the gravitational acceleration as a function of polar angle is presented. It is shown how the difference in these moments caused by the Earth’s non-sphericity leads to torques being exerted on the Earth by the Sun and the Moon. This leads to a very slow precession of the Earth’s angular momentum in the ecliptic plane, the so-called precession of the equinoxes: its magnitude is calculated. The chapter concludes with a discussion of the precession of the Earth’s angular momentum. It discusses the Earth’ axis of symmetry, a small effect caused by the two axes not exactly coinciding. Appendix A discusses at greater length the difference between lunar and sidereal day and Appendix B provides more details on the torques acting on the Earth due to the Sun and the Moon.

We imagine a group of people living on the inner surface of a huge rotating cylinder in flat spacetime. Their experiences are described and calculated. Thus we introduce gravimagnetic effects and the connection between gravitational time dilation and gravitational acceleration. Gravimagnetic effects such as the force on moving particles and the precession of gyroscopes are derived. The Thomas precession is obtained. These observations illustrate GR ideas that are applicable more generally. Some properties of the general stationary metric are introduced.

The linearized theory is applied to sources such as ordinary stars whose speed is small compared to the speed of light. This yields the “gravitoelectromagnetic” theory. The gravitoelectromagnetic field equations are obtained, along with their general solution via scalar and vector potentials. It is shown how to calculate the metric perturbation, and hence the field, due to a rotating ring or a ball, and thus how to calculate orbits, timing, and the Lense-Thirring precession.

This chapter addresses a rich variety of effects in spin dynamics arising under the conditions of pump-probe experiments. Here we consider the case where the electron spin is injected by a periodic train of circularly polarized pump pulses and precesses between the pulses in an external magnetic field. Nontrivial effects such as resonant spin amplification and spin coherence mode-locking take place due to commensurability of the repetition period of pump pulses and the charge carrier spin precession period. Theoretical approaches to describing the electron and nuclear spin coherence and experimental manifestations of these unusual regimes of spin dynamics are discussed in detail.

Celestial cartography developed according to two distinct traditions characterized by the way the stars are being mapped. One tradition uses descriptions of locations of stars within constellations, such as found in the literary works of the classical authors Eudoxus (Phaenomena), Aratus (Phaenomena) and Eratosthenes (Epitome Catasterismorum). The other method uses mathematical coordinates such as listed in the well-known Ptolemaic star catalogue. Both traditions are based on a spherical geometry of the cosmos. Concepts such as signs, conventions and precession, and the prerequisites to globe making are introduced. Also the problems involved in drawing constellation on the outside of a sphere are discussed. With the help of Hipparchus's rule the left and right side of a constellation figure is fixed, but the consequences of this rule in marking the constellation on a sphere was not always respected in antiquity.

The oldest artefact in Islamic celestial cartography is the ceiling painting in Quṣayr ^cAmra. All other mappings are found on celestial globes. In the ninth century Arabic astronomers wrote the first treatises on the use of globes, an activity underlining the significance of globes in education. A highlight in Islamic celestial cartography is the Book on the Constellations of the fixed Stars which the Persian astronomer al-Ṣūfī wrote for his patron ^cAḍud al-Dawla. The illustrations of this treatise in Oxford, MS Marsh 144 may have been copied from a globe, possibly one designed by al-Sūfi himself. The earliest extant mathematical celestial globes were made in Muslim Spain in ca. 1080. Although made in the west these globes show glimpses of an eastern tradition in globe making predating the work of al-Ṣūfī. The remaining eight early globes are made between 1145/46 and 1383/84 and show all the impact of al-Ṣūfī's work.