Projected explicit lawson methods for the integration of Schrödinger equation

In this article, it is proved that explicit Lawson methods, when projected onto one of the invariants of nonlinear Schrödinger equation (norm) are also automatically projected onto another invariant (momentum) for many solutions. As this procedure is very cheap and geometric because two invariants are conserved, it offers an efficient tool to integrate some solutions of this equation till long times. On the other hand, we show a detailed study on the numerical performance of these methods against splitting ones, with fixed and variable stepsize implementation. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 78–104, 2015     

Threefolds in ℙ5 with One Apparent Quadruple Point

Abstract

In this article we classify all the smooth threefolds in ?⁵ with an apparent quadruple point provided that the family of its 4-secant lines is an irreducible (first order) congruence. This is sufficient to conclude the classification of all the smooth codimension two varieties in ? ⁿ with one apparent (n ? 1)-point and with irreducible family of (n ? 1)-secant lines.     

Statistical evaluation of non‐profile analyses for the in vitro bioequivalence

For locally acting drug products such as nasal aerosols and nasal sprays, the 2003 US Food and Drug Administration (FDA) draft guidance suggests that bioequivalence between generic and brand‐name products be established through in vitro tests. In addition, for non‐profile analyses based on spray content uniformity, droplet size distribution, spray pattern, priming, and re‐priming, the draft US FDA guidance recommends that the population bioequivalence (PBE) between generic and innovator's products be demonstrated. However, the linearized criterion recommended in the draft FDA guidance does not take into consideration the variations due to batches, samples, and life stages. Hence, under a two‐stage nested random effects model, we apply the methods of modified large‐sample (MLS) and generalized pivotal quantities (GPQs) to construct the upper 95% confidence limit for in vitro PBE criterion with consideration of variance components as the statistical testing procedures for establishing the in vitro BE. A simulation study was conducted to compare empirical size and empirical power among the three methods. A numerical example illustrates the proposed methods. Copyright © 2010 John Wiley & Sons, Ltd.     

Effects of gravity on the shape of liquid droplets

We report the gravity effects on the shape and focal length changes of liquid droplets. As the droplet size decreases, the gravity effect is gradually weakened. Moreover, if the outside space of the droplet is filled with another immiscible liquid rather than air, the filled liquid helps to offset the gravity effect even though their densities do not match well. Good agreement between experiment and simulation is obtained. The negligible gravity effect enables us to improve the optical performances of the liquid lens by choosing suitable liquids without worrying their density mismatch.     

Four-Dimensional Geometric Quantities versus the Usual Three-Dimensional Quantities: The Resolution of Jackson’s Paradox

In this paper we present definitions of different four-dimensional (4D) geometric quantities (Clifford multivectors). New decompositions of the torque N and the angular momentum M (bivectors) into 1-vectors N_s, N_t and M_s, M_t, respectively, are given. The torques N_s, N_t (the angular momentums M_s, M_t), taken together, contain the same physical information as the bivector N (the bivector M). The usual approaches that deal with the 3D quantities etc. and their transformations are objected from the viewpoint of the invariant special relativity (ISR). In the ISR, it is considered that 4D geometric quantities are well-defined both theoretically and experimentally in the 4D spacetime. This is not the case with the usual 3D quantities. It is shown that there is no apparent electrodynamic paradox with the torque, and that the principle of relativity is naturally satisfied, when the 4D geometric quantities are used instead of the 3D quantities.     

New explicit solutions to the p-Laplace equation based on isoparametric foliations

In contrast to an infinite family of explicit examples of two-dimensional p-harmonic functions obtained by G. Aronsson in the late 80s, there is very little known about the higher-dimensional case. In this paper, we show how to use isoparametric polynomials to produce diverse examples of p-harmonic and biharmonic functions. Remarkably, for some distinguished values of p and the ambient dimension n this yields first examples of rational and algebraic p-harmonic functions. Moreover, we show that there are no p-harmonic polynomials of the isoparametric type. This supports a negative answer to a question proposed in 1980 by J. Lewis.     

Some minimal bimolecular mass-action systems with limit cycles

We discuss three examples of bimolecular mass-action systems with three species, due to Feinberg, Berner, Heinrich, and Wilhelm. Each system has a unique positive equilibrium which is unstable for certain rate constants and then exhibits stable limit cycles, but no chaotic behaviour. For some rate constants in the Feinberg–Berner system, a stable equilibrium, an unstabe limit cycle, and a stable limit cycle coexist. All three networks are minimal in some sense.By way of homogenising these three examples, we construct bimolecular mass-conserving mass-action systems with four species that admit a stable limit cycle. The homogenised Feinberg–Berner system and the homogenised Wilhelm–Heinrich system admit the coexistence of a stable equilibrium, an unstable limit cycle, and a stable limit cycle.     

On the preservation of invariants in the simulation of solitary waves in some nonlinear dispersive equations

In this paper we generalize some results in the literature concerning the structure of numerical approximations to solitary wave solutions of some nonlinear, dispersive equations is studied. We prove that those time discretizations with the property of preserving, exactly or approximately up to certain order, some invariants of the problems, have a better propagation of the error and provide a more suitable simulation of the solitary waves. The generalization involves the treatment of nonlocal operators and two different kinds of equations.     

Parallel asynchronous focal plane array processing

A new approach to focal plane processing based on silicon injection mode devices is suggested. These devices provide a natural basis for parallel asynchronous focal plane image preprocessing. The simplicity and novel properties of the devices would permit an independent analog processing channel to be dedicated to every pixel. A laminar architecture built from arrays of the devices would form a two-dimensional (2-D) array processor with a 2-D array of inputs located directly behind the focal plane detector array. A 2-D image data stream would propagate in neuronlike asynchronous pulse coded form through the laminar processor. No multiplexing, digitization or serial processing would occur in the preprocessing stage. High performance is expected because approximately linear pulse coding has already been observed for input currents ranging over six orders of magnitude down to one picoampere with noise referred to input of about 10 femtoamperes. Very low power requirements suggest utility in space and in conjunction with very large arrays. Multispectral processing is possible because of compatibility with the cryogenic environment of high performance infrared detectors.     

Focal shift of three-portion concentric piecewise cylindrical vector beam

Focal shift of a concentric piecewise cylindrical vector beam is investigated by means of vector diffractive theory. A section of the beam consists of three concentric portions. The center circle portion and outer annular portion are radial polarized, and the inner annular portion is generalized polarized. The phase of the inner annular portion is tunable. For certain geometric parameters of the beam, the focal shift occurs and can be adjusted by phase shift and polarized direction of the inner annular portion. For some cases, a three-peak focal pattern may occur, and peaks shift in the same direction, and one local minimum also shifts. For some other cases, a two-peak focal pattern may occur with shifting peaks. The tunable quantitative focal shift is investigated in detail.