A variational problem determined by probability measures

ABSTRACT

An optimization problem of maximizing an integral of a function over a family of probability measures is considered. The problem is a generalization of a well-studied variational problem in mathematical economics, concerning optimal allocations. The specific generalization that we examine arises also in the limit of singularly perturbed optimal control problems. We examine the mathematical problem and allude to the singular perturbation motivation.     

Surface modified SPIONs‐Cr(VI) ions‐immobilized organic‐inorganic hybrid as a magnetically recyclable nanocatalyst for rapid synthesis of polyhydroquinolines under solvent‐free conditions at room temperature

In this work, a magnetic hybrid dichromate nanocomposite with triphenylphosphine surface modified superparamagnetic iron oxide nanoparticles (SPIONs) as a recyclable nanocatalyst was designed, prepared and characterized by Fourier transform infrared spectroscopy (FT‐IR) spectra, X‐ray diffraction (XRD) pattern analysis, vibrating sample magnetometer (VSM) curves, X‐ray fluorescence (XRF) analysis, thermogravimetric analysis (TGA), scanning electron microscopy (SEM) and transmission electron microscopy (TEM) images and dynamic light scattering (DLS) analysis. Then, it was used in a green and efficient procedure for one‐pot multicomponent synthesis of polyhydroquinoline derivatives by the condensation of aldehydes, dimedone or 1,3‐cyclohexadione, ethyl acetoacetate and ammonium acetate. This protocol includes some new and exceptional advantages such as short reaction times, low catalyst loading, high yields, solvent‐free and room temperature conditions, easy separation and reusability of the catalyst.     

Classroom note: Using the binomial series to prove the arithmetic mean–geometric mean inequality

In 1968, Leon Gerber compared (1 + x) ^a to its kth partial sum as a binomial series. His result is stated and, as an application of this result, a proof of the arithmetic mean–geometric mean inequality is presented.     

Stochastic control of helicopter suspended load position

Modelling of the motion of a helicopter with a suspended load is described. The mathematical model takes into account stochastic disturbances acting on the load suspension point. The proposed approach allows solution of the problem of optimal control with minimization of oscillation and control power expenditure. The evolution of the system solution with time is investigated for various disturbance intensities. Computer calculation results are presented as a function of the suspension length and the intensity of stochastic disturbances.     

Existence and multiplicity results for periodic solutions of nonlinear difference equations

We use Brouwer degree to prove existence and multiplicity results for the periodic solutions of some nonlinear second-order and first-order difference equations. We obtain, in particular upper and lower solutions theorems, Ambrosetti–Prodi type results and sharp existence conditions for nonlinearities which are bounded from below or above.     

On a second-order nonlinear discrete multi-point eigenvalue problem

We study the existence and non-existence of positive solutions of a system of nonlinear second-order difference equations with eigenvalues subject to multi-point boundary conditions.     

Comparison of Multivariate Matching Methods: Structures,Distances, and Algorithms

Abstract

A comparison and evaluation is made of recent proposals for multivariate matched sampling in observational studies, where the following three questions are answered: (1) Algorithms: In current statistical practice, matched samples are formed using “nearest available” matching, a greedy algorithm. Greedy matching does not minimize the total distance within matched pairs, though good algorithms exist for optimal matching that do minimize the total distance. How much better is optimal matching than greedy matching? We find that optimal matching is sometimes noticeably better than greedy matching in the sense of producing closely matched pairs, sometimes only marginally better, but it is no better than greedy matching in the sense of producing balanced matched samples. (2) Structures: In common practice, treated units are matched to one control, called pair matching or 1–1 matching, or treated units are matched to two controls, called 1–2 matching, and so on. It is known, however, that the optimal structure is a full matching in which a treated unit may have one or more controls or a control may have one or more treated units. Optimal 1 — k matching is compared to optimal full matching, finding that optimal full matching is often much better. (3) Distances: Matching involves defining a distance between covariate vectors, and several such distances exist. Three recent proposals are compared. Practical advice is summarized in a final section.     

Microwave-assisted Cannizzaro reaction—Optimisation of reaction conditions

The microwave-assisted Cannizzaro reaction was studied in order to develop fully reproducible synthetic protocols for transformation of aldehydes to carboxylic acid and alcohols. Optimised were the following process parameters: power, temperature, and time. Aromatic, heteroaromatic and aliphatic aldehydes were used in the studies. It was found that furfural, thiophene-2-carbaldehyde, pyridinecarboxaldehyde and aromatic aldehydes react under mild conditions, while 1-methyl-pyrrole-2-carboxaldehyde derivatives and aliphatic aldehydes require more drastic reaction conditions and a longer exposure time to microwave radiation.     

An Improved Synthesis of α,β Methylenic Cyclic Ketones

This communication describes a convenient two steps procedure for the preparation of α,β methylenic cyclic ketones from arylmethyl ketones involving a preformed Mannich reagent.