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

Axisymmetric bifurcations of thick spherical shells under inflation and compression

Incremental equilibrium equations and corresponding boundary conditions for an isotropic, hyperelastic and incompressible material are summarized and then specialized to a form suitable for the analysis of a spherical shell subject to an internal or an external pressure. A thick-walled spherical shell during inflation is analyzed using four different material models. Specifically, one and two terms in the Ogden energy formulation, the Gent model and an I1 formulation recently proposed by Lopez-Pamies. We investigate the existence of local pressure maxima and minima and the dependence of the corresponding stretches on the material model and on shell thickness. These results are then used to investigate axisymmetric bifurcations of the inflated shell. The analysis is extended to determine the behavior of a thick-walled spherical shell subject to an external pressure. We find that the results of the two terms Ogden formulation, the Gent and the Lopez-Pamies models are very similar, for the one term Ogden material we identify additional critical stretches, which have not been reported in the literature before

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

Two-phase stretching of molecular chains

While stretching of most polymer chains leads to rather featureless force-extension diagrams, some, notably DNA, exhibit non-trivial behavior with a distinct plateau region. Here we propose a unified theory that connects force-extension characteristics of the polymer chain with the convexity properties of the extension energy profile of its individual monomer subunits. Namely, if the effective monomer deformation energy as a function of its extension has a non-convex (concave up) region, the stretched polymer chain separates into two phases: the weakly and strongly stretched monomers. Simplified planar and 3D polymer models are used to illustrate the basic principles of the proposed model. Specifically, we show rigorously that when the secondary structure of a polymer is mostly due to weak non-covalent interactions, the stretching is two-phase, and the force-stretching diagram has the characteristic plateau. We then use realistic coarse-grained models to confirm the main findings and make direct connection to the microscopic structure of the monomers. We demostrate in detail how the two-phase scenario is realized in the \alpha-helix, and in DNA double helix. The predicted plateau parameters are consistent with single molecules experiments. Detailed analysis of DNA stretching demonstrates that breaking of Watson-Crick bonds is not necessary for the existence of the plateau, although some of the bonds do break as the double-helix extends at room temperature. The main strengths of the proposed theory are its generality and direct microscopic connection.Comment: 16 pges, 22 figure

The A-B Signal Detection Theory Model

The purpose of this research was threefold: (1) Present the a-b SDT model as an alternative framework to overcome the limitations of the underlying SDT model and the traditional measures of sensitivity and criterion setting, (2) Provide empirical support to validate the adequacy of the a-b SDT model, and (3) Conduct a Monte Carlo Study to compare and contrast the strengths and weaknesses of both the traditional and the a-b SDT models across the full spectrum of response values with the goal of providing researchers and practitioners with recommendations regarding the adequacy of each model. The results from this research have both theoretical implications and practical applications. The findings from the empirical study suggest that Green and Swets (1966)\u27s contention that the detection and response processes are independent from each other is questionable. Furthermore, the findings from the Monte Carlo Study suggest that the a-b SDT model provides more accurate measures to capture the dependency between these two processes. This is particularly important for researchers and practitioners who are interested in studying human-automation interaction factors and how sensory and perceptual factors may affect humans\u27 response biases while interacting with automated systems

Using Signal Detection Theory to Examine the Effects of Workload and Likelihood Displays on Human Responses to Alarms

The purpose of this study was to examine the effects of workload and likelihood displays on human responses to alarms through the use of Signal Detection Theory (SDT). Thirty students performed compensatory tracking, resource management, and system monitoring tasks. They interacted with a Binary Alarm System (BAS) or a Likelihood Alarm System (LAS) during varying workload levels. Overall Response Rate (ORR), False Alarm Response Rate (FARR), True Alarm Response Rate (TARR), Sensitivity (d\u27), and Bias (c), were used as alarm response measures. Root Mean Squared Tracking Error (RMSTE) and Root Mean Squared Resource Management Error (RMSRMF) reflected primary-task performance. Proportion of Correct Resets (PCR) reflected secondary-task performance. As expected, participants responded significantly more often when they used the BAS during low workload. However, they responded significantly more often to true alarms when they used the LAS when compared to the BAS regardless of workload level. Participants were significantly more sensitive when they used the LAS, and they were significantly more biased during high workload. Last, participants performed the system monitoring task significantly better when they used the LAS. The main theoretical contribution of this study was to demonstrate the usefulness of adapting SDT to study human response to alarm signals. The main practical contribution of this study was to show the ability to enhance knowledge-based databases and fault diagnosis methods used by alarm systems by incorporating likelihood information as a fundamental aspect of their decision-making algorithms. Future research should focus on refining the methods used to develop LASs, which may ultimately serve as an adequate way to mitigate the cry-wolf effect