Measurement of Large Dipolar Couplings of a Liquid Crystal with Terminal Phenyl Rings and Estimation of the Order Parameters

NMR spectroscopy is a powerful means of studying liquid‐crystalline systems at atomic resolutions. Of the many parameters that can provide information on the dynamics and order of the systems, ¹H–¹³C dipolar couplings are an important means of obtaining such information. Depending on the details of the molecular structure and the magnitude of the order parameters, the dipolar couplings can vary over a wide range of values. Thus the method employed to estimate the dipolar couplings should be capable of estimating both large and small dipolar couplings at the same time. For this purpose, we consider here a two‐dimensional NMR experiment that works similar to the insensitive nuclei enhanced by polarization transfer (INEPT) experiment in solution. With the incorporation of a modification proposed earlier for experiments with low radio frequency power, the scheme is observed to enable a wide range of dipolar couplings to be estimated at the same time. We utilized this approach to obtain dipolar couplings in a liquid crystal with phenyl rings attached to either end of the molecule, and estimated its local order parameters.     

The Effect of C4H and C5H on the Microstructure of Aqueous Solutions of 1‐Alkyl‐3‐methylimidazolium Tetrafluoroborate Ionic Liquids

As is well‐known, the C2?H proton of 1‐ethyl‐3‐methylimidazolium tetrafluoroborate ([Emim]BF₄) and 1‐butyl‐3‐methylimidazolium tetrafluoroborate ([Bmim]BF₄) has a strong ability to form hydrogen bonds. The purpose of this work is to evaluate the effect of the interactions of the C4?H and C5?H protons on the microstructure of [Emim]BF₄ and [Bmim]BF₄ with water by using ¹H NMR spectroscopy. The differences between the relative ¹H NMR chemical shifts of C2?H, C4?H, and C5?H and between the interaction‐energy parameters obtained from these chemical shifts are minor, thus suggesting that the interactions of C4?H and C5?H may have a considerable effect on the microstructure. To confirm this, the viscosities of the systems are estimated by using the interaction‐energy parameters obtained from the ¹H NMR chemical shifts of the three studied aromatic protons and water, showing that the interactions of C4?H and C5?H also play an important role in the microstructure.     

Elucidating the Structure of Chiral Molecules by using Amplified Vibrational Circular Dichroism: From Theory to Experimental Realization

Recent experimental observations of enhanced vibrational circular dichroism (VCD) in molecular systems with low‐lying electronically excited states suggest interesting new applications of VCD spectroscopy. The theory describing VCD enhancement through vibronic coupling schemes was derived by Nafie in 1983, but only recently experimental evidence of VCD amplification has demonstrated the extent to which this effect can be exploited as a structure elucidation tool to probe local structure. In this Concept paper, we give an overview of the physics behind vibrational circular dichroism, in particular the equations governing the VCD amplification effect, and review the latest experimental developments with a prospective view on the application of amplified VCD to locally probe biomolecular structure.     

Existence and Exponential Decay of Solutions for a Wave Equation with Integral Nonlocal Boundary Conditions of Memory Type

In this paper, we consider a wave equation with integral nonlocal boundary conditions of memory type. First, we establish two local existence theorems by using Faedo–Galerkin method and standard arguments of density. Next, we give a su?cient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.     

On the relation between formal grade and depth with a view toward vanishing of Lyubeznik numbers

Let (R,𝔪) be a local ring and I an ideal. The aim of the present paper is twofold. At first we continue the investigation to compare fgrade(I,R) with depth R∕I and further we derive some results on the vanishing of Lyubeznik numbers.     

A new method for exact solutions of variant types of time‐fractional Korteweg‐de Vries equations in shallow water waves

The current article devoted on the new method for finding the exact solutions of some time‐fractional Korteweg–de Vries (KdV) type equations appearing in shallow water waves. We employ the new method here for time‐fractional equations viz. time‐fractional KdV‐Burgers and KdV‐mKdV equations for finding the exact solutions. We use here the fractional complex transform accompanied by properties of local fractional calculus for reduction of fractional partial differential equations to ordinary differential equations. The obtained results are demonstrated by graphs for the new solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and multiplicity of systems of Kirchhoff‐type equations with general potentials

This paper is concerned with the following systems of Kirchhoff‐type equations: Under more relaxed assumptions on V(x) and F(x,u,v), we first prove the existence of at least two nontrivial solutions for the aforementioned system by using Morse theory in combination with local linking arguments. Then by using the Clark theorem, the existence results of at least 2k distinct pairs of solutions are obtained. Some recent results from the literature are extended. Copyright © 2016 John Wiley & Sons, Ltd.     

Blow‐up for the b‐family of equations

In this paper, we consider the b‐family of equations on the torus u _t ?u _{t x x} +(b + 1)u u _x =b u _x u _{x x} +u u _{x x x} , which for appropriate values of b reduces to well‐known models, such as the Camassa–Holm equation or the Degasperis–Procesi equation. We establish a local‐in‐space blow‐up criterion. Copyright © 2016 John Wiley & Sons, Ltd.