Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour

Classical Jacobi polynomials $P_{n}^{(\alpha,\beta)}$, with $\alpha, \beta>-1$, have a number of well-known properties, in particular the location of their zeros in the open interval $(-1,1)$. This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong asymptotics of Jacobi polynomials where the real parameters $\alpha_n,\beta_n$ depend on $n$ in such a way that $$\lim_{n\to\infty}\frac{\alpha_{n}}{n}=A, \quad \lim_{n\to\infty}\frac{\beta_{n}}{n}=B,$$ with $A,B \in \mathbb{R}$. We restrict our attention to the case where the limits $A,B$ are not both positive and take values outside of the triangle bounded by the straight lines A=0, B=0 and $A+B+2=0$. As a corollary, we show that in the limit the zeros distribute along certain curves that constitute trajectories of a quadratic differential. The non-hermitian orthogonality relations for Jacobi polynomials with varying parameters lie in the core of our approach; in the cases we consider, these relations hold on a single contour of the complex plane. The asymptotic analysis is performed using the Deift-Zhou steepest descent method based on the Riemann-Hilbert reformulation of Jacobi polynomials.Comment: 37 pages, 10 figure

Psicología cognoscitiva y esquemas conceptuales de los alumnos

There are three main sections in this paper. The first section presents an overview of the research that has been undertaken into students' conceptions relating to mechanics. One central feature of students' aintuitive mechanics» is the association between their notions of force and motion. The consequences of such ideas for students' reasoning about various static and dynamic systems are described. A number of general features of students' alternative conceptions are then outlined. The second section outlines current perspectives on learning with particular emphasis being paid to the constructivist perspective; a perspective which emphasises the active role of learners in the construction of their knowledge. The pedagogical consequences of adopting this perspective are outlined in the third section

The cutaneous 'rabbit' illusion affects human primary sensory cortex somatopically

We used functional magnetic resonance imaging (fMRI) to study neural correlates of a robust somatosensory illusion that can dissociate tactile perception from physical stimulation. Repeated rapid stimulation at the wrist, then near the elbow, can create the illusion of touches at intervening locations along the arm, as if a rabbit hopped along it. We examined brain activity in humans using fMRI, with improved spatial resolution, during this version of the classic cutaneous rabbit illusion. As compared with control stimulation at the same skin sites (but in a different order that did not induce the illusion), illusory sequences activated contralateral primary somatosensory cortex, at a somatotopic location corresponding to the filled-in illusory perception on the forearm. Moreover, the amplitude of this somatosensory activation was comparable to that for veridical stimulation including the intervening position on the arm. The illusion additionally activated areas of premotor and prefrontal cortex. These results provide direct evidence that illusory somatosensory percepts can affect primary somatosensory cortex in a manner that corresponds somatotopically to the illusory percept

Study of critical defects in ablative heat shield systems for the space shuttle

Results are presented from an investigation to determine the effects of fabrication-induced defects on the performance of an ablative heat shield material in a simulated space shuttle reentry environment. Nondestructive methods for detecting the defects were investigated. The material considered is a fiber-filled, honeycomb-reinforced, low-density elastomer. Results were obtained for density variations, voids, fiber bundles, crushed honeycomb, undercut honeycomb, unbonded areas, face sheet delaminations, and cure variations. The data indicate that, within reasonable tolerances, the fabrication defects investigated are not critical in terms of reentry performance of the heat shield

All-Electron Path Integral Monte Carlo Simulations of Warm Dense Matter: Application to Water and Carbon Plasmas

We develop an all-electron path integral Monte Carlo (PIMC) method with free-particle nodes for warm dense matter and apply it to water and carbon plasmas. We thereby extend PIMC studies beyond hydrogen and helium to elements with core electrons. PIMC pressures, internal energies, and pair-correlation functions compare well with density functional theory molecular dynamics (DFT-MD) at temperatures of (2.5-7.5)$\times10^5$ K and both methods together form a coherent equation of state (EOS) over a density-temperature range of 3--12 g/cm$^3$ and 10$^4$--10$^9$ K

The cluster galaxy luminosity function at z=0.3z=0.3: a recent origin for the faint-end upturn ?

We derive deep luminosity functions (to $M_z=-15$) for galaxies in Abell 1835 ($z=0.25$) and AC 114 ($z=0.31$) and compare these with the local $z'$ luminosity function for 69 clusters. The data show that the faint-end upturn, the excess of galaxies above a single Schechter function at $M_z < -17$, does not exist in the higher redshift clusters. This suggests that the faint-end upturn galaxies have been created recently, by infall into clusters of star-forming field populations or via tidal disruption of brighter objects.^MComment: 6 pages, MNRAS main journal, accepted for publicatio

Can the Future Influence the Present?

One widely accepted model of classical electrodynamics assumes that a moving charged particle produces both retarded and advanced fields. This formulation first appeared at least 75 years ago. It was popularized in the 1940\u27s by work of Wheeler and Feynman. But the most fundamental question associated with the model has remained unanswered: When (if ever) does the two-body problem have a unique solution? The present paper gives an answer in one special case. Imagine two identical charged particles alone in the universe moving symmetrically along the x axis. One is at x(t) and the other is at −x(t). Their motion is then governed by a system of functional differential equations involving both retarded and advanced arguments. This system together with the Newtonian initial data x(0)=x0\u3e0 and x′(0)=0 has a unique solution for all time provided x0 is sufficiently large. Perhaps the existence and uniqueness proof given for this special case will pave the way for more general results on this curious two-body problem