Approximate （m,n）-Cauchy-Jensen Additive Mappings in C^＊-algebras

Concerning the stability problem of functional equations, we introduce a general (m, n)-Cauchy-Jensen functional equation and establish new theorems about the Hyers-Ulam stability of general (m, n)-Cauchy-Jensen additive mappings in C*-algebras, which generalize the results obtained for Cauchy-Jensen type additive mappings.     

On a Hilbert-Type Integral Inequality Related to the Extended Hurwitz Zeta Function in the Whole Plane

By using techniques of real analysis and weight functions, a few equivalent statements of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The constant factor related the extended Hurwitz zeta function is proved to be the best possible. As applications, a few equivalent statements of a Hilbert-type integral inequality with the homogeneous kernel in the whole plane are deduced. We also consider the operator expressions and some corollaries.

     

A facile gold(I)-catalysed intramolecular alkyne hydroarylation approach to methyl 5-amino-2H-1-benzopyran-8-carboxylate derivatives

A high yielding and selective method for producing methyl 5-amino-2H-1-benzopyran-8-carboxylate derivatives via gold(I)-catalysed intramolecular alkyne hydroarylation has been developed.     

Asymptotic behavior of alternative Jensen and Jensen type functional equations

In 1941 D.H. Hyers solved the well-known Ulam stability problem for linear mappings. In 1951 D.G. Bourgin was the second author to treat the Ulam problem for additive mappings. In 1982-2005 we established the Hyers-Ulam stability for the Ulam problem of linear and nonlinear mappings. In 1998 S.-M. Jung and in 2002-2005 the authors of this paper investigated the Hyers-Ulam stability of additive and quadratic mappings on restricted domains. In this paper we improve our bounds and thus our results obtained, in 2003 for Jensen type mappings and establish new theorems about the Ulam stability of additive mappings of the second form on restricted domains. Besides we introduce alternative Jensen type functional equations and investigate pertinent stability results for these alternative equations. Finally, we apply our recent research results to the asymptotic behavior of functional equations of these alternative types. These stability results can be applied in stochastic analysis, financial and actuarial mathematics, as well as in psychology and sociology.     

Complex formation processes of terminally protected peptides containing two or three histidyl residues. Characterization of the mixed metal complexes of peptides

Copper(ii), nickel(ii) and zinc(ii) complexes of the peptides Ac-HVVH-NH(2) and Ac-HAAHVVH-NH(2) have been studied by potentiometric, UV-vis, CD, EPR and NMR spectroscopic measurements. Both tetra and heptapeptides can form relatively stable macrochelates with copper(ii), nickel(ii) and zinc(ii) ions, in which the ligands are coordinated via the side-chain imidazole functions. Formation of the macrochelates slightly suppresses, but cannot prevent the copper(ii) and nickel(ii) ion promoted deprotonation and coordination of the amide functionalities. The overall stoichiometry of the major species is [MH(-3)L](-) with a 4N (= N(-),N(-),N(-),N(im)) coordination mode. In the case of Ac-HAAHVVH-NH(2), coordination isomers of this species can exist with a preference for copper(ii) or nickel(ii) binding at the internal histidyl residue. In the copper(ii)-Ac-HAAHVVH-NH(2) system, the presence of the two anchoring sites results in the formation of dinuclear complexes. The existence of these species requires the involvement of amide functions in metal binding. Both equilibrium and spectroscopic data support the fact that the copper(ii) ions of the dinuclear species are independent from each other providing a good chance for the formation of various mixed metal complexes. It was found that zinc(ii) is not able to significantly alter the copper(ii) binding of the heptapeptide, but it can occupy the uncoordinated histidyl sites. The formation of the copper(ii)-nickel(ii) mixed species was obtained in alkaline solutions and CD spectra suggest the statistical distribution of the two metal ions among the histidyl residues. The binding of HAAHVVH to palladium(ii) is exclusive below pH 8 and the mixed metal species of palladium(ii) and copper(ii) ions are formed only in slightly basic solutions.     

Interaction of Cu(II) and Ni(II) with the 63-93 fragment of histone H2B

Chromatin proteins are believed to represent reactive sites for metal ion binding. We have synthesized the 31 amino acid peptide Ac-NSFVNDIFERIAGEASRLAHYNKRSTITSRE-NH2, corresponding to the 63-93 fragment of the histone H2B and studied its interaction with Cu(II) and Ni(II). Potentiometric and spectroscopic studies (UV-vis, CD, NMR and EPR) showed that histidine 21 acts as an anchoring binding site for the metal ion. Complexation of the studied peptide with Cu(II) starts at pH 4 with the formation of the monodentate species CuH2L. At physiological pH values, the 3N complex (N(Im), 2N(-)), CuL is favoured while at basic pH values the 4N (N(Im), 3N(-)) coordination mode is preferred. Ni(II) forms several complexes with the peptide starting from the distorted octahedral NiH2L at about neutral pH, to a square planar complex where the peptide is bound through a (N(Im), 3N(-)) mode in an equatorial plane at basic pH values. These results could be important in revealing more information about the mechanism of metal induced toxicity and carcinogenesis.     

A generalized mixed type of quartic–cubic–quadratic–additive functional equations

We determine the general solution of the functional equation f(x + ky) + f(x-ky) = g(x + y) + g(x-y) + h(x) + h(y) for fixed integers with k ≠ 0; ±1 without assuming any regularity conditions for the unknown functions f, g, h, and0020[(h)\tilde] \tilde{h} . The method used for solving these functional equations is elementary but it exploits an important result due to Hosszú. The solution of this functional equation can also be obtained in groups of certain type by using two important results due to Székelyhidi.     

Some properties of a hypergeometric function which appear in an approximation problem

In this paper we consider properties and power expressions of the functions $f:(-1,1)\rightarrow \mathbb{R }$ and $f_L:(-1,1)\rightarrow \mathbb{R }$ , defined by $$\begin{aligned} f(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma }{\sqrt{1-t^2}}\,\mathrm{d}t \quad \text{ and}\quad f_L(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma \log (1+x t)}{\sqrt{1-t^2}}\,\mathrm{d}t, \end{aligned}$$ respectively, where $\gamma $ is a real parameter, as well as some properties of a two parametric real-valued function $D(\,\cdot \,;\alpha ,\beta ) :(-1,1) \rightarrow \mathbb{R }$ , defined by $$\begin{aligned} D(x;\alpha ,\beta )= f(x;\beta )f(x;-\alpha -1)- f(x;-\alpha )f(x;\beta -1),\quad \alpha ,\beta \in \mathbb{R }. \end{aligned}$$ The inequality of Turán type $$\begin{aligned} D(x;\alpha ,\beta )>0,\quad -1<x<1, \end{aligned}$$ for $\alpha +\beta >0$ is proved, as well as an opposite inequality if $\alpha +\beta <0$ . Finally, for the partial derivatives of $D(x;\alpha ,\beta )$ with respect to $\alpha $ or $\beta $ , respectively $A(x;\alpha ,\beta )$ and $B(x;\alpha ,\beta )$ , for which $A(x;\alpha ,\beta )=B(x;-\beta ,-\alpha )$ , some results are obtained. We mention also that some results of this paper have been successfully applied in various problems in the theory of polynomial approximation and some “truncated” quadrature formulas of Gaussian type with an exponential weight on the real semiaxis, especially in a computation of Mhaskar–Rahmanov–Saff numbers.     

Fixed Points and Stability for Quadratic Mappings in β–Normed Left Banach Modules on Banach Algebras

The goal of the present paper is to investigate some new stability results by applying the alternative fixed point of generalized quadratic functional equation $$\begin{array}{ll}f\left(\sum\limits_{i=1}^{n}a_ix_i\right)+{\sum\limits_{i=1}^{n-1}}{\sum\limits_{j=i+1}^{n}}\left[f(a_ix_i+a_jx_j)+2f(a_ix_i-a_jx_j)\right]\\ \qquad \quad = (3n-2){\sum\limits_{i=1}^{n}}a^2_{i}f(x_{i})\end{array}$$ in β–Banach modules on Banach algebras, where ${a_{1},\dots,a_{n}\in \mathbb{Z}{\setminus}\{0\}}$ and some ${\ell\in\{1 , 2 ,\dots, n-1\},}$ a _? ?≠ ±1 and a _n ?=?1, where n is a positive integer greater or at least equal to two.