Asymptotic properties of ground states of scalar field equations with a vanishing parameter

We study the leading order behaviour of positive solutions of the equation -\Delta u +\varepsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\R^N, where $N\ge 3$, $q>p>2$ and when $\varepsilon>0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$, $q$ and $N$. The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $p^\ast=\frac{2N}{N-2}$. For $p<p^\ast$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p>p^\ast$ the solution asymptotically coincides with the solution of the equation with $\varepsilon=0$. In the most delicate case $p=p^\ast$ the asymptotic behaviour of the solutions is given by a particular solution of the critical Emden--Fowler equation, whose choice depends on $\varepsilon$ in a nontrivial way

Removing opportunities to calculate improves students' performance on subsequent word problems.

BackgroundIn two studies we investigated whether removing opportunities to calculate could improve students' subsequent ability to solve similar word problems. Students were first asked to write explanations for three word-problems that they thought would help another student understand the problems. Half of the participants explained typical word problems (i.e., problems with enough information to make calculating an answer possible), while the other half explained the same problems with numbers removed, making calculating an answer impossible. We hypothesized that removing opportunities to calculate would induce students to think relationally about the word problems, which would result in higher levels of performance on subsequent transfer problems.ResultsIn both studies, participants who explained the non-calculable problems performed significantly better on the transfer test than participants who explained the typical (i.e., calculable) problems. This was so in spite of the manipulation not fully suppressing students' desire to calculate. Many students in the non-calculable group explicitly stated that they needed numbers in order to answer the question or made up numbers with which to calculate. There was a significant, positive relationship between the frequency with which students made up numbers and their self-reported mathematics anxiety.ConclusionsWe hypothesized that the mechanism at play was a reduction in instrumental thinking (and an increase in relational thinking). Interventions designed to help students remediate prior mathematical failure should perhaps focus less on the specific skills students are lacking, and more on the dispositions they bring to the task of "doing mathematics.

On Pauli Pairs

The state of a system in classical mechanics can be uniquely reconstructed if we know the positions and the momenta of all its parts. In 1958 Pauli has conjectured that the same holds for quantum mechanical systems. The conjecture turned out to be wrong. In this paper we provide a new set of examples of Pauli pairs, being the pairs of quantum states indistinguishable by measuring the spatial location and momentum. In particular, we construct a new set of spatially localized Pauli pairs.Comment: submitted to JM

Photonic crystals of coated metallic spheres

It is shown that simple face-centered-cubic (fcc) structures of both metallic and coated metallic spheres are ideal candidates to achieve a tunable complete photonic bandgap (CPBG) for optical wavelengths using currently available experimental techniques. For coated microspheres with the coating width to plasma wavelength ratio $l_c/\lambda_p \leq 10$ and the coating and host refractive indices $n_c$ and $n_h$, respectively, between 1 and 1.47, one can always find a sphere radius $r_s$ such that the relative gap width $g_w$ (gap width to the midgap frequency ratio) is larger than 5% and, in some cases, $g_w$ can exceed 9%. Using different coatings and supporting liquids, the width and midgap frequency of a CPBG can be tuned considerably.Comment: 14 pages, plain latex, 3 ps figures, to appear in Europhys. Lett. For more info on this subject see http://www.amolf.nl/research/photonic_materials_theory/moroz/moroz.htm

Preliminary results of the Vega 1 and Vega 2 optical investigation of aerosol in the atmosphere of Venus at 30-60 KM

Aerosol concentration profiles were measured by an aerosol spectrometer above the landing sites of the Vega 1 and Vega 2 landers. Approximately the same altitude zones were found as in previous experiments: a three-layered basic cloud cover, an intermediate zone and subcloud haze. There were significant quantitative differences in the concentrations of particles, however, and especially in the spectra of their dimensions. Nightglow was found in the troposphere of Venus at a wavelength of about 1 micron. The backscatter coefficient and the extinction coefficient change very little between 32 and 63 km. Large numbers of submicron particles apparently exist in the atmosphere above the landing sites

Role of the middle ear muscle apparatus in mechanisms of speech signal discrimination

A method of impedance reflexometry was used to examine 101 students with hearing impairment in order to clarify the interrelation between speech discrimination and the state of the middle ear muscles. Ability to discriminate speech signals depends to some extent on the functional state of intraaural muscles. Speech discrimination was greatly impaired in the absence of stapedial muscle acoustic reflex, in the presence of low thresholds of stimulation and in very small values of reflex amplitude increase. Discrimination was not impeded in positive AR, high values of relative thresholds and normal increase of reflex amplitude in response to speech signals with augmenting intensity

Spectrophotometric experiment on the Verera-11 and Venera-12 descent vehicles: Some results of the analysis of the spectrum of the daytime sky of Venus

The spectra of the daytime sky of Venus were recorded on the Venera-11 and Venera-12 descent vehicles at various altitudes above the planet's surface, within the interval of 4500 to 12,000 Angstroms. The angular distribution of the brightness of the scattered radiation was recorded and the ratio of water and carbon dioxide were studied, with respect to the cloud cover boundaries

Metallo-dielectric diamond and zinc-blende photonic crystals

It is shown that small inclusions of a low absorbing metal can have a dramatic effect on the photonic band structure. In the case of diamond and zinc-blende photonic crystals, several complete photonic band gaps (CPBG's) can open in the spectrum, between the 2nd-3rd, 5th-6th, and 8th-9th bands. Unlike in the purely dielectric case, in the presence of small inclusions of a low absorbing metal the largest CPBG for a moderate dielectric constant (epsilon<=10) turns out to be the 2nd-3rd CPBG. The 2nd-3rd CPBG is the most important CPBG, because it is the most stable against disorder. For a diamond and zinc-blende structure of nonoverlapping dielectric and metallo-dielectric spheres, a CPBG begins to decrease with an increasing dielectric contrast roughly at the point where another CPBG starts to open--a kind of gap competition. A CPBG can even shrink to zero when the dielectric contrast increases further. Metal inclusions have the biggest effect for the dielectric constant 2<=epsilon<=12, which is a typical dielectric constant at near infrared and in the visible for many materials, including semiconductors and polymers. It is shown that one can create a sizeable and robust 2nd-3rd CPBG at near infrared and visible wavelengths even for a photonic crystal which is composed of more than 97% low refractive index materials (n<=1.45, i.e., that of silica glass or a polymer). These findings open the door for any semiconductor and polymer material to be used as genuine building blocks for the creation of photonic crystals with a CPBG and significantly increase the possibilities for experimentalists to realize a sizeable and robust CPBG in the near infrared and in the visible. One possibility is a construction method using optical tweezers, which is analyzed here.Comment: 25 pp, 23 figs, RevTex, to appear in Phys Rev B. For more information look at http://www.amolf.nl/research/photonic_materials_theory/moroz/moroz.htm