Well-posedness of the spatially homogeneous Landau equation for soft potentials

We consider the spatially homogeneous Landau equation of kinetic theory, and provide a differential inequality for the Wasserstein distance with quadratic cost between two solutions. We deduce some well-posedness results. The main difficulty is that this equation presents a singularity for small relative velocities. Our uniqueness result is the first one in the important case of soft potentials. Furthermore, it is almost optimal for a class of moderately soft potentials, that is for a moderate singularity. Indeed, in such a case, our result applies for initial conditions with finite mass, energy, and entropy. For the other moderately soft potentials, we assume additionally some moment conditions on the initial data. For very soft potentials, we obtain only a local (in time) well-posedness result, under some integrability conditions. Our proof is probabilistic, and uses a stochastic version of the Landau equation, in the spirit of Tanaka [H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Geb. 46 (1) (1978-1979) 67-105].     

A semi-algebraic approach for asymptotic stability analysis

This paper deals with the problem of computing Lyapunov functions for asymptotic stability analysis of autonomous polynomial systems of differential equations. We propose a new semi-algebraic approach by making advantage of the local property of the Lyapunov function as well as its derivative. This is done by first constructing a semi-algebraic system and then solving this semi-algebraic system in an adaptive way. Experiment results show that our semi-algebraic approach is more efficient in practice, especially for low-order systems.     

An algorithmic approach to finding factorial designs with generalized minimum aberration

Factorial designs are arguably the most widely used designs in scientific investigations. Generalized minimum aberration (GMA) and uniformity are two important criteria for evaluating both regular and non-regular designs. The generation of GMA designs is a non-trivial problem due to the sequential optimization nature of the criterion. Based on an analytical expression between the generalized wordlength pattern and a uniformity measure, this paper converts the generation of GMA designs to a constrained optimization problem, and provides effective algorithms for solving this particular problem. Moreover, many new designs with GMA or near-GMA are reported, which are also (nearly) optimal under the uniformity measure.     

Spectral inclusion for unbounded block operator matrices

In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W²(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W²(A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W²(A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given.     

The Loewner driving function of trajectory arcs of quadratic differentials

We obtain a first order differential equation for the driving function of the chordal Loewner differential equation in the case where the domain is slit by a curve which is a trajectory arc of a certain type of quadratic differential. In particular this includes the case when the curve is a path on the square, triangle or hexagonal lattice in the upper half-plane or, indeed, in any domain with boundary on the lattice. We also demonstrate how we use this to calculate the driving function numerically. Equivalent results for other variants of the Loewner differential equation are also obtained.     

Warmstarting for interior point methods applied to the long-term power planning problem

The long-term planning of electricity generation in a liberalised market using the Bloom and Gallant model can be posed as a quadratic programming (QP) problem with an exponential number of linear inequality constraints called load-matching constraints (LMCs) and several other linear non-LMCs. Direct solution methods are inefficient at handling such problems and a heuristic procedure has been devised to generate only those LMCs that are likely to be active at the optimiser. The problem is then solved as a finite succession of QP problems with an increasing, though still limited, number of LMCs, which can be solved efficiently using a direct method, as would be the case with a QP interior-point algorithm. Warm starting between successive QP solutions helps then in reducing the number of iterations necessary to reach the optimiser.     

Knowledge based proximal support vector machines

We propose a proximal version of the knowledge based support vector machine formulation, termed as knowledge based proximal support vector machines (KBPSVMs) in the sequel, for binary data classification. The KBPSVM classifier incorporates prior knowledge in the form of multiple polyhedral sets, and determines two parallel planes that are kept as distant from each other as possible. The proposed algorithm is simple and fast as no quadratic programming solver needs to be employed. Effectively, only the solution of a structured system of linear equations is needed.     

Mapping of an ultrasonic bath for ultrasound assisted extraction of mangiferin from Mangifera indica leaves

The present work deals with the mapping of an ultrasonic bath for the maximum extraction of mangiferin from Mangifera indica leaves. I₃⁻ liberation experiments (chemical transformations) and extraction (physical transformations) were carried out at different locations in an ultrasonic bath and compared. The experimental findings indicated a similar trend in variation in an ultrasonic bath by both these methods. Various parameters such as position and depth of vessel in an ultrasonic bath, diameter and shape of a vessel, frequency and input power which affect the extraction yield have been studied in detail. Maximum yield of mangiferin obtained was approximately 31 mg/g at optimized parameters: distance of 2.54 cm above the bottom of the bath, 7 cm diameter of vessel, flat bottom vessel, 6.35 cm liquid height, 122 W input power and 25 kHz frequency. The present work indicates that the position and depth of vessel in an ultrasonic bath, diameter and shape of a vessel, frequency and input power have significant effect on the extraction yield. This work can be used as a base for all ultrasonic baths to obtain maximum efficiency for ultrasound assisted extraction.     

Growth of Sn(O,S)2 buffer layers and its application to Cu(In,Ga)Se2 solar cells

Sn-based thin films as new buffer layer for Cd-free Cu(In,Ga)Se₂ (CIGS) solar cells were developed. The Sn(O,S)₂ films were formed on CIGS substrates by chemical bath deposition from an alkaline ammonia solution by reacting tin(IV) chloride with thiourea. Optimization of the growth process allowed the smooth and conformal coverage of the films on the CIGS substrates with a thickness of 20 nm that was a self-limited thickness in the chemical bath deposition process. XPS analysis revealed that the as-deposited films contained Sn–O, Sn–OH, and Sn–S bondings and the ratio of Sn–S bonding to Sn–O bonding was 0.3. The CIGS solar cell fabricated with a 20-nm thick Sn(O,S)₂ buffer layer had the best efficiency of 11.5% without AR coating. The open circuit voltage, short circuit current, and fill factor were 0.55 V, 34.4 mA/cm², and FF = 0.61, respectively. The open circuit voltage and fill factor were low compared to the conventional CIGS solar cell with a 50-nm thick CdS buffer due to too thin Sn(O,S)₂ buffer layer.