On the design of spherical gradient index lenses

Classical spherical gradient index (GRIN) lenses (such as Maxwell Fish Eye lens, Eaton lens, Luneburg lens, etc.) design procedure using the Abel integral equation is reviewed and reorganized. Each lens is fully defined by a function called the angle of flight which describes the ray deflection through the lens. The radial refractive index distribution is obtained by applying a linear integral transformation to the angle of flight. The interest of this formulation is in the linearity of the integral transformation which allows us to derive new solutions from linear combinations of known lenses. Beside the review of the classical GRIN designs, we present a numerical method for GRIN lenses defined by the Abel integral equation with fixed limits, which is an ill-posed problem.

Inverse bremsstrahlung absorption in spherical laser targets

Inverse bremsstrahlung has been incorporated into an analytical model of the expanding corona of a laser-irradiated spherical target. Absorption decreases slowly with increasing intensity, in agreement with some numerical simulations, and contrary to estimates from simple models in use up to now, which are optimistic at low values of intensity and very pessimistic at high values. Present results agree well with experimental data from many laboratories; substantial absorption is found up to moderate intensities,say below IOl5 W cm-2 for 1.06 pm light. Anomalous absorption, wher, included in the analysis, leaves practically unaffected the ablation pressure and mass ablation rate, for given absorbed intensity. Universal results are given in dimensionless fom.

Current collection by an active spherical electrode in an unmagnetized plasma

A theoretical model for the steady-state response of anodic contactors that emit a plasma current Ii and collect electrons from a collisionless, unmagnetized plasma is presented. The use of a (kinetic) monoenergetic population for the attracted species, well known in passive probe theory, gives both accuracy and tractability to the theory. The monoenergetic population is proved to behave like an isentropic fluid with radial plus centripetal motion, allowing direct comparisons with ad hoc fluid models. Also, a modification of the original monoenergetic equations permits analysis of contactors operating in orbit-limited conditions. Besides that, the theory predicts that, only for plasma emissions above certain threshold current a presheath/double layer/core structure for the potential is formed (the core mode), while for emissions below that threshold, a plasma contactor behaves exactly as a positive-ion emitter with a presheath/sheath structure (the no-core mode). Ion emitters are studied as a particular case. Emphasis is placed on obtaining dimensionless charts and approximate asymptotic laws of the current-voltage characteristic.

Non-periodic driving of coupled oscillators:a spherical swing

Nonlinearly coupled, damped oscillators at 1:1 frequency ratio, one oscillator being driven coherently for efficient excitation, are exemplified by a spherical swing with some phase-mismatch between drive and response. For certain damping range, excitation is found to succeed if it lags behind, but to produce a chaotic attractor if it leads the response. Although a period-doubhng sequence, for damping increasing, leads to the attractor, this is actually born as a hard (as regards amplitude) bifurcation at a zero growth-rate parametric line; as damping decreases, an unstable fixed point crosses an invariant plane to enter as saddle-focus a phase-space domain of physical solutions. A second hard bifurcation occurs at the zero mismatch line, the saddle-focus leaving that domain. Times on the attractor diverge when approaching either fine, leading to exactly one-dimensional and noninvertible limit maps, which are analytically determined.

Spherical collectors versus bare tethers for drag, thrust, and power generation

Deorbit, power generation, and thrusting performances of a bare thin-tape tether and an insulated tether with a spherical electron collector are compared for typical conditions in low-Earth orbit and common values of length L = 4−20 km and cross-sectional area of the tether A = 1−5 mm2. The relative performance of moderately large spheres, as compared with bare tapes, improves but still lags as one moves from deorbiting to power generation and to thrusting: Maximum drag in deorbiting requires maximum current and, thus, fully reflects on anodic collection capability, whereas extracting power at a load or using a supply to push current against the motional field requires reduced currents. The relative performance also improves as one moves to smaller A, which makes the sphere approach the limiting short-circuit current, and at greater L, with the higher bias only affecting moderately the already large bare-tape current. For a 4-m-diameter sphere, relative performances range from 0.09 sphere-to-bare tether drag ratio for L = 4 km and A = 5 mm2 to 0.82 thrust–efficiency ratio for L = 20 km and A = 1 mm2. Extremely large spheres collecting the short-circuit current at zero bias at daytime (diameters being about 14 m for A = 1 mm2 and 31 m for A = 5 mm2) barely outperform the bare tape for L = 4 km and are still outperformed by the bare tape for L = 20 km in both deorbiting and power generation; these large spheres perform like the bare tape in thrusting. In no case was sphere or sphere-related hardware taken into account in evaluating system mass, which would have reduced the sphere performances even further.

Spherical collectors versus bare tethers for drag, thrust, and power generation

Performances of ED-tethers using either spherical collectors or bare tethers for drag, thrust, or power generation, are compared. The standard Parker-Murphy model of current to a full sphere, with neither space-charge nor plasmamotion effects considered, but modified to best fit TSS1R results, is used (the Lam, Al'pert/Gurevich space-charge limited model will be used elsewhere) In the analysis, the spherical collector is assumed to collect current well beyond its random-current value (thick-heath). Both average current in the bare-tether and current to the sphere are normalized with the short-circuit current in the absence of applied power, allowing a comparison of performances for all three applications in terms of characteristic dimensionless numbers. The sphere is always substantially outperformed by the bare-tether if ohmic effects are weak, though its performance improves as such effects increase.

Experimental evidence for super-resolution in the spherical geodesic waveguide

The previous publications (Miñano et al, 2011 and Gonzalez et al, 2012) have shown that using a Spherical Geodesic Waveguide (SGW) it can be achieved the super-resolution up to λ/3000, which is far below the classic Abbe diffraction limit, close to a set of discrete microwave frequencies. The SGW was designed and simulated in COMSOL as a thin geodesic waveguide bounded by an ideal and lossless metal. Herein we present the experimental results for a manufactured SGW, slightly modified due to fabrication requirements, showing the super-resolution up to λ/105.

Current-voltage response of a spherical plasma contactor

A theoretical model for a contactor, collecting electrons from an ambient, unmagnetized plasma and emitting a current Iiis discussed. The relation between Ii and the potential bias of the contactor is found to be crucial for the formation of a quasineutral core around the anode and, consequently, for the current colleted. Approximate analytical laws and charts for the current-voltage response are provided.

Inverse kinematics of a humanoid robot with non-spherical hip: a hybrid algorithm approach

This paper describes an approach to solve the inverse kinematics problem of humanoid robots whose construction shows a small but non negligible offset at the hip which prevents any purely analytical solution to be developed. Knowing that a purely numerical solution is not feasible due to variable efficiency problems, the proposed one first neglects the offset presence in order to obtain an approximate “solution” by means of an analytical algorithm based on screw theory, and then uses it as the initial condition of a numerical refining procedure based on the Levenberg‐Marquardt algorithm. In this way, few iterations are needed for any specified attitude, making it possible to implement the algorithm for real‐time applications. As a way to show the algorithm’s implementation, one case of study is considered throughout the paper, represented by the SILO2 humanoid robot.

Activation of a spherical carbon for toluene adsorption at low concentration

This paper complements a previous one [1] about toluene adsorption on a commercial spherical activated carbon and on samples obtained from it by CO2 or steam activation. The present paper deals with the activation of a commercial spherical carbon (SC) having low porosity and high bed density (0.85 g/cm3) using the same procedure. Our results show that SC can be well activated with CO2 or steam. The increase in the burn-off percentage leads to an increase in the gravimetric adsorption capacity (more intensively for CO2) and a decrease in bed density (more intensively for CO2). However, for similar porosity developments similar bed densities are achieved for CO2 and steam. Especial attention is paid to differences between both activating agents, comparing samples having similar or different activation rates, showing that CO2 generates more narrow porosity and penetrates more inside the spherical particles than steam. Steam activates more from the outside to the interior of the spheres and hence produces larger spheres size reductions. With both activation agents and with a suitable combination of porosity development and bed density, quite high volumetric adsorption values of toluene (up to 236 g toluene/L) can be obtained even using a low toluene concentration (200 ppmv).

Spherical activated carbon as an enhanced support for TiO2/AC photocatalysts

Titanium dioxide nanoparticles prepared in situ by sol–gel method were supported on a spherical activated carbon to prepare TiO2/AC hybrid photocatalysts for the oxidation of gaseous organic compounds. Additionally, a granular activated carbon was studied for comparison purposes. In both types of TiO2/AC composites the effect of different variables (i.e., the thermal treatment conditions used during the preparation of these materials) and the UV-light wavelength used during photocatalytic oxidation were analyzed. The prepared materials were deeply characterized (by gas adsorption, TGA, XRD, SEM and photocatalytic propene oxidation). The obtained results show that the carbon support has an important effect on the properties of the deposited TiO2 and, therefore, on the photocatalytic activity of the resulting TiO2/AC composites. Thus, the hybrid materials prepared over the spherical activated carbon show better results than those prepared over the granular one; a good TiO2 coverage with a high crystallinity of the deposited titanium dioxide, which just needs an air oxidation treatment at low-moderate temperature (350–375 °C) to present high photoactivity, without the need of additional inert atmosphere treatments. Additionally, these materials are more active at 365 nm than at 257.7 nm UV radiation, opening the possibility of using solar light for this application.

Spherical carbons: Synthesis, characterization and activation processes

Spherical carbons have been prepared through hydrothermal treatment of three carbohydrates (glucose, saccharose and cellulose). Preparation variables such as treatment time, treatment temperature and concentration of carbohydrate have been analyzed to obtain spherical carbons. These spherical carbons can be prepared with particle sizes larger than 10 μm, especially from saccharose, and have subsequently been activated using different activation processes (H3PO4, NaOH, KOH or physical activation with CO2) to develop their textural properties. All these spherical carbons maintained their spherical morphology after the activation process, except when KOH/carbon ratios higher than 4/1 were used, which caused partial destruction of the spheres. The spherical activated carbons develop interesting textural properties with the four activating agents employed, reaching surface areas up to 3100 m2/g. Comparison of spherical activated carbons obtained with the different activating agents, taking into account the yields obtained after the activation process, shows that phosphoric acid activation produces spherical activated carbons with higher developed surface areas. Also, the spherical activated carbons present different oxygen groups’ content depending on the activating agent employed (higher surface oxygen groups content for chemical activation than for physical activation).

Factors governing the adsorption of ethanol on spherical activated carbons

Ethanol adsorption on different activated carbons (mostly spherical ones) was investigated covering the relative pressure range from 0.001 to 1. Oxygen surface contents of the ACs were modified by oxidation (in HNO3 solution or air) and/or by thermal treatment in N2. To differentiate the concomitant effects of porosity and oxygen surface chemistry on ethanol adsorption, different sets of samples were used to analyze different relative pressure ranges (below 1000 ppmv concentration and close to unity). To see the effect of oxygen surface chemistry, selected samples having similar porosity but different oxygen contents were studied in the low relative pressure range. At low ethanol concentration (225 ppmv) adsorption is favored in oxidized samples, remarking the effect of the oxidizing treatment used (HNO3 is more effective than air) and the type of oxygen functionalities created (carboxyl and anhydride groups are more effective than phenolic, carbonyl and derivatives). To analyze the high relative pressure range, spherical and additional ACs were used. As the relative pressure of ethanol increases, the effect of oxygen-containing surface groups decreases and microporosity becomes the most important variable affecting the adsorption of ethanol.

Student mathematical notebook of Thomas Noyes, 1793

This sewn volume contains Noyes’ mathematical exercises in geometry; trigonometry; surveying; measurement of heights and distances; plain, oblique, parallel, middle latitude, and mercator sailing; and dialing. Many of the exercises are illustrated by carefully hand-drawn diagrams, including a mariners’ compass and moon dials.