The attractive nonlinear delta-function potential

We solve the continuous one-dimensional Schr\"{o}dinger equation for the case of an inverted {\em nonlinear} delta-function potential located at the origin, obtaining the bound state in closed form as a function of the nonlinear exponent. The bound state probability profile decays exponentially away from the origin, with a profile width that increases monotonically with the nonlinear exponent, becoming an almost completely extended state when this approaches two. At an exponent value of two, the bound state suffers a discontinuous change to a delta-like profile. Further increase of the exponent increases again the width of the probability profile, although the bound state is proven to be stable only for exponents below two. The transmission of plane waves across the nonlinear delta potential increases monotonically with the nonlinearity exponent and is insensitive to the sign of its opacity.Comment: submitted to Am. J. of Phys., sixteen pages, three figure

Beyond the Standard Model for Montaneros

These notes cover (i) electroweak symmetry breaking in the Standard Model (SM) and the Higgs boson, (ii) alternatives to the SM Higgs boson including an introduction to composite Higgs models and Higgsless models that invoke extra dimensions, (iii) the theory and phenomenology of supersymmetry, and (iv) various further beyond topics, including Grand Unification, proton decay and neutrino masses, supergravity, superstrings and extra dimensions.Comment: Based on lectures by John Ellis at the 5th CERN-Latin-American School of High-Energy Physics, Recinto Quirama, Colombia, 15 - 28 Mar 2009, 84 pages, 35 figure

Barriers to Pursuing STEM-Related Careers: Perceptions of Hispanic Girls Enrolled in Advanced High School STEM Courses

Researchers indicate that the United States has fallen behind other nations in science, technology, engineering, and mathematics (STEM) education (President\u27s Council of Advisors on Science and Technology, 2010, 2012). A declining interest in the field of engineering as demonstrated by students who pursue degrees in STEM fields also threatens the U.S. competitive edge (National Science Foundation, 2013; Schneider, Judy, & Mazuca, 2012). Although some students perform successfully in STEM courses, an achievement gap between school-aged boys and girls is well-documented in the literature (e.g., College Board, 2007). Moreover, Hispanic students are underrepresented in science-related courses and careers (Hanley & Noblit, 2009) and even fewer Hispanic girls are attracted to the STEM areas despite the increase in the Hispanic population in general and in higher education (Dolan, 2009). In fact, few studies were located that addressed perspectives of Hispanic girls about their experiences and perceptions related to science and engineering (Crisp, Nora, & Taggart, 2009; Moller et al., 2015; O\u27Shea, Heilbronner, & Reis, 2010). Specifically, there is a need to attract girls and Hispanic students to mathematics and science coursework and careers. Therefore, the purpose of this collective case study was to explore and identify potential barriers and supports related to select Hispanic high school girls\u27 decisions to pursue advanced coursework and future careers in STEM. By increasing awareness of these potential barriers, school leaders will be better positioned to develop strategies and support systems to encourage Hispanic girls to take advanced science courses and seek out postsecondary studies and careers in STEM fields

Atypical late-time singular regimes accurately diagnosed in stagnation-point-type solutions of 3D Euler flows

We revisit, both numerically and analytically, the finite-time blowup of the infinite-energy solution of 3D Euler equations of stagnation-point-type introduced by Gibbon et al. (1999). By employing the method of mapping to regular systems, presented in Bustamante (2011) and extended to the symmetry-plane case by Mulungye et al. (2015), we establish a curious property of this solution that was not observed in early studies: before but near singularity time, the blowup goes from a fast transient to a slower regime that is well resolved spectrally, even at mid-resolutions of $512^2.$ This late-time regime has an atypical spectrum: it is Gaussian rather than exponential in the wavenumbers. The analyticity-strip width decays to zero in a finite time, albeit so slowly that it remains well above the collocation-point scale for all simulation times $t < T^* - 10^{-9000}$, where $T^*$ is the singularity time. Reaching such a proximity to singularity time is not possible in the original temporal variable, because floating point double precision ($\approx 10^{-16}$) creates a `machine-epsilon' barrier. Due to this limitation on the \emph{original} independent variable, the mapped variables now provide an improved assessment of the relevant blowup quantities, crucially with acceptable accuracy at an unprecedented closeness to the singularity time: $T^*- t \approx 10^{-140}.

Quadratic invariants for discrete clusters of weakly interacting waves

We consider discrete clusters of quasi-resonant triads arising from a Hamiltonian three-wave equation. A cluster consists of N modes forming a total of M connected triads. We investigate the problem of constructing a functionally independent set of quadratic constants of motion. We show that this problem is equivalent to an underlying basic linear problem, consisting of finding the null space of a rectangular M × N matrix with entries 1, −1 and 0. In particular, we prove that the number of independent quadratic invariants is equal to J ≡ N − M* ≥ N − M, where M* is the number of linearly independent rows in Thus, the problem of finding all independent quadratic invariants is reduced to a linear algebra problem in the Hamiltonian case. We establish that the properties of the quadratic invariants (e.g., locality) are related to the topological properties of the clusters (e.g., types of linkage). To do so, we formulate an algorithm for decomposing large clusters into smaller ones and show how various invariants are related to certain parts of a cluster, including the basic structures leading to M* < M. We illustrate our findings by presenting examples from the Charney–Hasegawa–Mima wave model, and by showing a classification of small (up to three-triad) clusters