Existence of positive solutions of delay dynamic equations

In this article we study the existence of positive solutions for a dynamic equations on time scales. The main tool employed here is the Schauder’s fixed point theorem. The asymptotic properties of solutions are also treated. The results obtained here extend the work of Dorociakova and Olach (Tatra Mt Math Publ 43:63–70, 2009). Three examples are also given to illustrate this work.     

Stability and convergence of difference scheme for nonlinear evolutionary type equations

A finite difference scheme is derived for the initial-boundary problem for the nonlinear equation system $$\frac{\partial u}{\partial t}=A\frac{\partial^{2}u}{\partial x^{2}}+f(u),$$ where A is a complex diagonal matrix, f is a complex vector function. The stability and convergence in discrete L ^∞-norm of proposed Crank-Nicolson type finite difference schemes is proved. No restrictions on the ratio of time and space grid steps are assumed. Some numerical experiments have been conducted in order to validate the theoretical results.     

Methods for the numerical solution of the Benjamin‐Bona‐Mahony‐Burgers equation

L^∞‐error estimates for finite element for Galerkin solutions for the Benjamin‐Bona‐Mahony‐Burgers (BBMB) equation are considered. A priori bound and the semidiscrete Galerkin scheme are studied using appropriate projections. For fully discrete Galerkin schemes, we consider the backward Euler method and analyze the corresponding error estimates. For a second order accuracy in time, we propose a three‐level backward method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Towards Magnetic Carbo‐meric Molecular Materials

Numerous studies have underlined the putative diradical character of π‐conjugated molecules that can be described by closed‐shell Lewis structures, for instance, p‐dimethylene p–n phenylenes, or long polyacenes. In the latter compounds, the only way to save the aromaticity of the six‐membered rings is to give up the Lewis electron pairing in the singlet biradical ground state. The present work considers the possibility of doing the same by using the basic C₂ units of carbo‐meric architectures. A series of acyclic and cyclic carbo‐meric architectures is studied by using UB3LYP DFT broken‐symmetry calculations, including spin decontaminations and subsequent geometry optimization of the singlet diradical. The C₂ units are shown to stabilize the singlet biradical by spin delocalization, two of them playing approximately the same role as one radical‐insulating 1,4 phenylene moiety. The results are generalized to the investigation of open‐shell polyradical singlet states of rigid hydrocarbon structures, the symmetry and rigidity of which can assist cooperativity and self spin polarization effect. Several synthesis targets with challenging magnetic/spin properties are suggested in the carbo‐mer series.     

Magnetogenesis in Water Induced by a Chemical Analyte

This Minireview aims to shed light on the emergent field of inducing a change in the magnetic properties of a solution‐phase sample by exposing it to a chemical analyte. A considerable body of knowledge exists on materials that alter their magnetic characteristics after a change in the surrounding physical conditions and a number of cases even exist of solution‐phase samples that do so under these same circumstances. However, examples of dissolved molecules or particles that react in this fashion under constant conditions and in response to an analyte are limited. Although some cases in organic solvents are discussed, the emphasis of this Minireview is on water. Our aim is to provide the reader with guidelines for designing new magnetogenic probes for the detection of the desired chemical analyte.     

Gas-phase protonation of unsaturated ethers: Experimental and theoretical study of 2,3-and 2,5-dihydrofuran and related compounds

The protonation of 2,3- and 2,5-dihydrofuran is examined in gas-phase equilibrium proton transfer reactions conducted in an ion cyclotron resonance spectrometer. The thermodynamically favoured site of protonation in the two compounds is seen to be different: whereas the first isomer forms a carbocation upon protonation, the second isomer protonates on the oxygen atom to form an oxonium ion. The results obtained with substituted derivatives and with linear analogues confirm these conclusions. Molecular orbital calculations on the various structures for protonated bases are performed at the 4–31G level with correction for configuration interaction effects and at the 4–31G* level. The latter basis set provides the best results owing to the introduction of d-type orbitals on the oxygen atom. The calculation results substantiate the experimental observations and provide details on the molecular structure of the protonated species.     

Mesogenic 3,6-bis(4-hydroxyphenyl)-1,2,4,5-tetrazine alkanoate esters

A novel series of 3,6-bis(4-hdroxyphenyl)-1,2,4,5-tetrazine alkanoate esters were synthesized and their mesogenic properties were studied using differential scanning calorimetry (DSC) and polarizing optical microscopy (POM). The impact of changing the tail-core linkage from alkyl or alkoxy to ester is profound. Compared to the alkyl or alkoxy linkages, the ester linkage reduced mesogenic properties. Short-tailed compounds are non mesogenic (4a-4e), while long-tailed compounds (4f-4r) exhibit nematic phases. Unlike the alkyl or alkoxy tail series, none of the 18 presented esters in this series exhibits a smectic phase.