A note on stability of polynomial equations

Motivated by the notion of Ulam’s type stability and some recent results of S.-M. Jung, concerning the stability of zeros of polynomials, we prove a stability result for functional equations that have polynomial forms, considerably improving the results in the literature.     

Stability of the equation of the p-Wright affine functions

We prove some stability results for the equation of the p-Wright affine functions.     

Mild solutions of semilinear evolution equation on an unbounded interval and their applications

The aim of this paper is to give an existence theorem for a semilinear equation of evolution on an unbounded interval in the case when the generator of a semigroup depends on time parameter. The considerations of this paper are based on the concept of a measure of noncompactness in Fréchet space and the classical Tichonov fixed point principle. Moreover, an application of the main theorem to the second-order parabolic PDE is given.     

Remarks on stability of linear recurrence of higher order

We prove some stability results for linear recurrences with constant coefficients in normed spaces. As a consequence we obtain a complete solution of the problem of the Hyers–Ulam stability for such recurrences.     

On stability of a family of functional equations

We prove a stability result for a family of functional equations containing the exponential and the Go??b-Schinzel functional equations. Thus we extend and improve a recent result of J. Chudziak.     

Dispersion-free component of non-covalent interaction via mutual polarization of fragment densities

Comprehensive tests within a diverse set of noncovalently bonded systems are carried out to assess the performance of the recently-developed dispersion-free approach in the framework of density functional theory [?. Rajchel, P. Z?uchowski, M. Szcze?s?niak, and G. Cha?asin?ski, Phys. Rev. Lett. 104, 163001 (2010)]. A numerical algorithm which cures the convergence problems of the previous implementation is presented.     

Sharp weak type estimates for Riesz transforms

Let $d$ be a given positive integer and let $\{R_j\}_{j=1}^d$ denote the collection of Riesz transforms on $\mathbb {R}^d$ . For $1<p<\infty $ , we determine the best constant $C_p$ such that the following holds. For any locally integrable function $f$ on $\mathbb {R}^d$ and any $j\in \{1,\,2,\,\ldots ,\,d\}$ , $$\begin{aligned} ||(R_jf)_+||_{L^{p,\infty }(\mathbb {R}^d)}\le C_p||f||_{L^{p,\infty }(\mathbb {R}^d)}. \end{aligned}$$ A related statement for Riesz transforms on spheres is also established. The proofs exploit Gundy–Varopoulos representation of Riesz transforms and appropriate inequality for orthogonal martingales.     

A density functional theory approach to noncovalent interactions via interacting monomer densities

A recently proposed "DFT + dispersion" treatment (Rajchel et al., Phys. Rev. Lett., 2010, 104, 163001) is described in detail and illustrated by more examples. The formalism derives the dispersion-free density functional theory (DFT) interaction energy and combines it with the dispersion energy from separate DFT calculations. It consists of the self-consistent polarization of DFT monomers restrained by the exclusion principle via the Pauli blockade technique. Within the monomers a complete exchange-correlation potential should be used, but between them only the exact exchange operates. The application to a wide range of molecular complexes from rare-gas dimers to hydrogen-bonds to π-electron interactions shows good agreement with benchmark values.     

High Cytotoxic Activity of Phosphonium Salts and Their Complementary Selectivity towards HeLa and K562 Cancer Cells: Identification of Tri-n-butyl-n-hexadecylphosphonium bromide as a Highly Potent Anti-HeLa Phosphonium Salt

Quaternary ammonium and phosphonium salts have been screened for their toxic effect on HeLa and K562 cancer cell lines, as well as on normal HUVEC cells. Tri-n-butyl-n-hexadecylphosphonium bromide, the first phosphonium salt with a halogen anion tested against HeLa cells, was 12 times more potent (IC₅₀ <5 μm after 24 and 48 h) than the clinically used reference compound cisplatin and 17 times more potent than tri-n-hexyltetradecylphosphonium bis(trifluoromethylsulfonyl)imide, which, to the best of our knowledge, is the first phosphonium salt to be evaluated in HeLa cells. However, it was inactive against K562 cells (24 and 48 h). According to a caspase-3/7 assay, its toxicity has not been connected with the induction of apoptosis. In contrast, triphenylalkylphosphonium iodides with shorter C_1–5 alkyl chains were inactive against HeLa cells but very active against K562 cells (IC₅₀=6–10 μm after 48 h). Phosphonium cations with halide counterions proved to be more potent than those with (CF₃SO₂)₂N⁻ as the anion, as in the anticancer agent NSC 747251, or other anions in molecules with similar alkyl chain lengths. On the other hand, a series of ammonium salts containing a short methylthiomethyl or methoxymethyl side chain revealed low cytotoxicity (IC₅₀ >500 μm after 24 and 48 h) against both HeLa and K562 cancer cell lines as well as normal HUVEC cells, showing that the nontoxic N⁺CH₂YMe (Y=S, O) structural motif in ammonium salts could be suitable for further optimization and development, especially in transfection experiments.