Strong and Weak Solutions to Second Order Differential Inclusions Governed by Monotone Operators

In this paper we introduce the concept of a weak solution for second order differential inclusions of the form u″(t) ∈ Au(t) + f(t), where A is a maximal monotone operator in a Hilbert space H. We prove existence and uniqueness of weak solutions to two point boundary value problems associated with such kind of equations. Furthermore, existence of (strong and weak) solutions to the equation above which are bounded on the positive half axis is proved under the optimal condition tf(t) ∈ L ¹(0, ∞; H), thus solving a long-standing open problem (for details, see our comments in Section 3 of the paper). Our treatment regarding weak solutions is similar to the corresponding theory related to the first order differential inclusions of the form f(t) ∈ u′(t) + Au(t) which has already been well developed.     

Row–column interaction models,with an R implementation

We propose a family of models called row–column interaction models (RCIMs) for two-way table responses. RCIMs apply some link function to a parameter (such as the cell mean) to equal a row effect plus a column effect plus an optional interaction modelled as a reduced-rank regression. What sets this work apart from others is that our framework incorporates a very wide range of statistical models, e.g., (1) log-link with Poisson counts is Goodman’s RC model, (2) identity-link with a double exponential distribution is median polish, (3) logit-link with Bernoulli responses is a Rasch model, (4) identity-link with normal errors is two-way ANOVA with one observation per cell but allowing semi-complex modelling of interactions of the form  \(\mathbf{A}\mathbf{C}^T\) , (5) exponential-link with normal responses are quasi-variances. Proposed here also is a least significant difference plot augmentation of quasi-variances. Being a special case of RCIMs, quasi-variances are naturally extended from the \(M=1\) linear/additive predictor  \(\eta \) case (within the exponential family) to the \(M>1\) case (vector generalized linear model families). A rank-1 Goodman’s RC model is also shown to estimate the site scores and optimums of an equal-tolerances Poisson unconstrained quadratic ordination. New functions within the VGAM R package are described with examples. Altogether, RCIMs facilitate the analysis of matrix responses of many data types, therefore are potentially useful to many areas of applied statistics.     

A new approach based on the surrogating method in the project time compression problems

This paper develops a mathematical model for project time compression problems in CPM/PERT type networks. It is noted this formulation of the problem will be an adequate approximation for solving the time compression problem with any continuous and non-increasing time-cost curve. The kind of this model is Mixed Integer Linear Program (MILP) with zero-one variables, and the Benders' decomposition procedure for analyzing this model has been developed. Then this paper proposes a new approach based on the surrogating method for solving these problems. In addition, the required computer programs have been prepared by the author to execute the algorithm. An illustrative example solved by the new algorithm, and two methods are compared by several numerical examples. Computational experience with these data shows the superiority of the new approach.     

Sensitive and selective fluorescence detection of aqueous uranyl ions using water-soluble CdTe quantum dots

Herein we report a fluorescent method for sensitive and selective detection of uranyl ions using CdTe quantum dots functionalized with mercaptopropionic acid, which the fluorescence of the quantum dots could be quantitatively quenched through electron transfer mechanism. The detection limit of the method was estimated to be 4 nM, less than the maximum allowed content of 130 nM for uranyl in drinking water defined by the U.S. Environmental Protection Agency. Furthermore, the probe was successfully applied in detection of uranyl ions in real samples, demonstrating its potential practical applications for monitoring of uranyl ions in environment.     

Rapid and Efficient Synthesis of Imidazolines and Bisimidazolines Under Microwave and Ultrasonic Irradiation

Small assemblies of 2-imidazolines and bisimidazolines from appropriate nitriles and ethylenediamine with catalytic amounts of P₂S₅ employing a microwave assisted protocol were prepared. Sonication of this system also led to successful synthesis of 2-imidazolines and bisimidazolines. Another advantage of these systems is the ability to carry out large scale reactions.     

Positive definite superfunctions and unitary representations of lie supergroups

For a broad class of Fréchet-Lie supergroups $ \mathcal{G} $ , we prove that there exists a correspondence between positive definite smooth (resp., analytic) superfunctions on $ \mathcal{G} $ and matrix coefficients of smooth (resp., analytic) unitary representations of the Harish-Chandra pair (G, $ \mathfrak{g} $ ) associated to $ \mathcal{G} $ . As an application, we prove that a smooth positive definite superfunction on $ \mathcal{G} $ is analytic if and only if it restricts to an analytic function on the underlying manifold of $ \mathcal{G} $ . When the underlying manifold of $ \mathcal{G} $ is 1-connected we obtain a necessary and sufficient condition for a linear functional on the universal enveloping algebra U( $ {{\mathfrak{g}}_{\mathbb{C}}} $ ) to correspond to a matrix coefficient of a unitary representation of (G, $ \mathfrak{g} $ ). The class of Lie supergroups for which the aforementioned results hold is characterised by a condition on the convergence of the Trotter product formula. This condition is strictly weaker than assuming that the underlying Lie group of $ \mathcal{G} $ is a locally exponential Fréchet-Lie group. In particular, our results apply to examples of interest in representation theory such as mapping supergroups and diffeomorphism supergroups.     

Investigation of magnetic field effect on surface and finite-site free energy in one-dimensional Ising model of nanosystems

We investigate a one-dimensional (1-D) Ising model for finite-site systems. The finite-site free energy and the surface free energy are calculated via the transfer matrix method. We show that, at high magnetic fields, the surface free energy has an asymptotic limit. The absolute surface energy increases when the value of f (the ratio of magnetic field to nearest-neighbor interactions) increases, and for f?≥?10 approaches a constant value. For the values of f?≥?0.2, the finite-site free energy also increases, but slowly. The thermodynamic limit in which physical properties approach the bulk value is also explored.     

Ab initio molecular dynamics simulation of ionic liquids

Ab initio Car-Parinnello molecular dynamics is used to simulate the structure and the dynamics of 1-butyl-3-methylimidazolium iodide ([bmim]I) ionic liquid at 300 K. Site-site pair correlation functions reveal that the anion has a strong interaction with any three C-H's of the imidazolium ring. The ring bends over and wraps around the anion such that the two nitrogen atoms take a distance to the anion. Electron donating butyl group contributes the electronic polarization in addition to geometrical (out-of-plane) polarization of the ring due to the liquid environment. This facilitates bending of the ring along the axis passing through nitrogen atoms. The average bending angle depends largely on the alkyl chain length and slightly on the anion type. Redistribution of electron density over the ring caused by the electron donating alkyl group provides additional independent evidence to the instability of lattice structure, hence the low melting point of the ionic liquid. Simulated viscosity and diffusion coefficients of [bmim]I are in quite agreement with the experiments.     

Ionic‐liquid‐mediated poly(dimethylsiloxane)‐ grafted carbon nanotube fiber prepared by the sol–gel technique for the head space solid‐phase microextraction of methyl tert‐butyl ether using GC

A headspace solid‐phase microextraction method was developed for the preconcentration and extraction of methyl tert‐butyl ether. An ionic‐liquid‐mediated multiwalled carbon nanotube–poly(dimethylsiloxane) hybrid coating, which was prepared by covalent functionalization of multiwalled carbon nanotubes with hydroxyl‐terminated poly(dimethylsiloxane) using the sol–gel technique, was used as solid‐phase microextraction adsorbent. This innovative fiber exhibited a highly porous surface structure, high thermal stability (at least 320°C) and long lifespan (over 210 uses). Potential factors affecting the extraction efficiency were optimized. Under the optimum conditions, the method LOD (S/N = 3) was 0.007 ng/mL and the LOQ (S/N = 10) was 0.03 ng/mL. The calibration curve was linear in the range of 0.03–200 ng/mL. The RSDs for one fiber (repeatability, n = 5) at three different concentrations (0.05, 1, and 150 ng/mL) were 5.1, 4.2, and 4.6% and for the fibers obtained from different batches (reproducibility, n = 3) were 6.5, 5.9, and 6.3%, respectively. The developed method was successfully applied to the determination of methyl tert‐butyl ether in different real water samples on three consecutive days. The relative recoveries for the spiked samples with 0.05, 1, and 150 ng/mL were between 94–104%.     

Stability analysis of Caputo fractional-order nonlinear systems revisited

In this paper stability analysis of fractional-order nonlinear systems is studied. An extension of Lyapunov direct method for fractional-order systems using Bihari’s and Bellman–Gronwall’s inequality and a proof of comparison theorem for fractional-order systems are proposed.