Optically active polymers containing side-chain photochromic moieties: Synthesis and chiroptical properties of copolymers of trans-4-(phenylazo)-1-naphthyl acrylate and trans-4-(1-naphthylazo)-phenyl acrylate with (?)-menthyl acrylate

The synthesis of photochromic optically active copolymers from trans-4-(phenylazo)-1-naphthyl acrylate (PANA), or trans-4-(1-naphthylazo)-phenyl acrylate (NAPA), and (?)-menthyl acrylate (MtA) is described. The copolymers prepared, having different contents of trans-phenylazonaphthalene moieties, have been characterized by IR, ¹H-NMR, UV, and GPC techniques. The chiroptical properties have been investigated by circular dichroism (CD) and the induced optical activity on the side-chain trans-phenylazonaphthalene chromophores discussed in terms of different conformational situations of the macromolecules in both the copolymer series. © 1994 John Wiley & Sons, Inc.     

Study on the nature of interaction of BrCl with HF,H2O,and NH3

By counterpoise‐corrected optimization method, the interactions of BrCl with the first‐row hydrides (HF, H₂O, NH₃) have been investigated at the MP2/6–311++G(3d,3p) level. To understand that the X? Br‐type (X = F, O, N) structure is more stable than the corresponding hydrogen‐bonded structure in these complexes, the electronic properties were also investigated. Symmetry‐adapted perturbation theory (SAPT) analysis has been carried out to understand the nature of the weak hydrogen bond and X? Br‐type interactions. On the other hand, for the weak hydrogen‐bonded complexes and the X? Br‐type complexes charges transfer is well correlated with the total induction energies. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

A new approach for the derivation of bounds for the Jensen difference

There are many useful applications of Jensen's inequality in several fields of science, and due to this reason, a lot of results are devoted to this inequality in the literature. The main theme of this article is to present a new method of finding estimates of the Jensen difference for differentiable functions. By applying definition of convex function, and integral Jensen's inequality for concave function in the identity pertaining the Jensen difference, we derive bounds for the Jensen difference. We present integral version of the bounds in Riemann sense as well. The sharpness of the proposed bounds through examples are discussed, and we conclude that the proposed bounds are better than some existing bounds even with weaker conditions. Also, we present some new variants of the Hermite–Hadamard and Hölder inequalities and some new inequalities for geometric, quasi-arithmetic, and power means. Finally, we give some applications in information theory.     

Asymptotic Behavior of a Stochastic Predator-prey Model with Beddington-DeAngelis Functional Response and L\'{e}vy Jumps

A stochastic two-prey-one-predator model with Beddington-DeAngelis functional response and L\''{e}vy jumps is proposed and investigated in this paper. First of all, we prove the existence and uniqueness of the global positive solution, and stochastic ultimate boundedness of the solution. Next, under a simple assumption, by using It\^{o} formula and other important inequalities, some sufficient conditions are established to ensure the extinction and persistence in the mean of the system. The results show that neither strong white noise nor L\''{e}vy noise is conducive to the persistence of the population. Finally, the theoretical results are verified by numerical simulations.     

Wilson's functional equations on groups

We study properties of solutionsf, g, h C(G) of the functional equation

A counterexample to subexponential growth of orthogonal polynomials

We construct an example of polynomialsp _n orthonormal with respect to a measure such that the sequencep _n (x) has an exponential lower bound for a pointx in the support of .Communicated by Paul Nevai.     

On a characterization of polynomials by divided differences

Summary We consider the functional equationf[x ₁,x ₂,, x _n ] =h(x ₁ + +x _n ) (x ₁,,x _n K, x _j x _k forj k), (D) wheref[x ₁,x ₂,,x _n ] denotes the (n – 1)-st divided difference off and prove Theorem. Let n be an integer, n 2, let K be a field, char(K) 2, with # K 8(n – 2) + 2. Let, furthermore, f, h: K K be functions. Then we have that f, h fulfil (D) if, and only if, there are constants a_j K, 0 j n (a := a_n, b := a_{n – 1}) such thatf = ax ⁿ +bx ^{n – 1} + +a ₀ and h = ax + b.     

An integral representation for decomposable measures of measurable functions

Summary We start with a measurem on a measurable space (,A), decomposable with respect to an Archimedeant-conorm on a real interval [0,M], which generalizes an additive measure. Using the integral introduced by the second author, a Radon-Nikodym type theorem, needed in what follows, is given.The integral naturally leads to a -decomposable measurem on the space of all measurable functions from to [0, 1]. The main result of the present paper is the converse of this, namely that, under natural conditions, any -decomposable measurem on can be represented as an integral of a certain Markov-kernelK. We extend this representation to measures on which have values in a set of distribution functions.These results generalize the work done by the first author in the case of additive measures.     

On Lipschitz property of solutions of functional equations

In this work it is proved that under certain conditions the continuous vector-valued solutionsf of the functional equation

De Rham systems and the solution of a class of functional equations

The main objective of this paper is to solve the functional equation