Existence of Nondecreasing and Continuous Solutions of an Integral Equation with Linear Modification of the Argument

We use a technique associated with measures of noncompactness to prove the existence of nondecreasing solutions to an integral equation with linear modification of the argument in the space C[0, 1]. In the last thirty years there has been a great deal of work in the field of differential equations with a modified argument. A special class is represented by the differential equation with affine modification of the argument which can be delay differential equations or differential equations with linear modifications of the argument. In this case we study the following integral equation x（t） = a（t） ＋ （Tx）（t） ∫0^σ（t） u（t, s, x（s）, x（λs））ds 0 〈 λ 〈 1 which can be considered in connection with the following Cauchy problem x＇（t） = u（t, s, x（t）, x（λt））, t ∈ [0, 1], 0 〈 λ 〈 1 x（0） = u0.     

Bilinear biorthogonal expansions and the Dunkl kernel on the real line

We study an extension of the classical Paley–Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier–Neumann type series as special cases, and it also provides a bilinear expansion for the Dunkl kernel (in the rank 1 case) which is a Dunkl analogue of Gegenbauer’s expansion of the plane wave and the corresponding sampling expansions. In fact, we show how to derive sampling and Fourier–Neumann type expansions from the results related to the bilinear expansion for the Dunkl kernel.     

Über korrespondierende Punkte der Steinerschen Fläche vierter Ordnung und die Hauptpunkte derselben

Ohne Zusammenfassung     

On Global Lipshitz-continuous Solutions of Isentropic Gas Dynamics

Building on work in [Yunguang Lu (1993). The global Hölder continuous solution of isentropic gas dynamics. Proc. Roy. Soc. Edinburgh Sect ., 123A , 231-238.] we consider the Cauchy problem for the isentropic equations of gas dynamics in Eulerian coordinates. For a certain class of initial data we prove existence of Lipshitz-continuous solution for a wide class of the pressure functions.     

Synthesis of Dendrimers with Porphyrine Core and their Supramolecular Complexes with Fullerene C60

Dendritic branches of poly(arylether) with peripheral butyl chains have been attached to a porphyrine core. Dendrimers of first, second and third generation were synthesized. Viability to form supramolecular complexes with fullerene C₆₀ was studied with two dendrimers of second and third generation. The supramolecular complexes were characterized by ¹H, ¹³C NMR in solution, FTIR, UV-vis spectroscopy and elemental analysis.     

Focused microwave irradiation-assisted synthesis of N-cyclohexyl-1,2,4-oxadiazole derivatives with antitumor activity

A facile synthesis of 3,5-disubstituted 1,2,4-oxadiazole derivatives under focused microwave irradiation (FMWI) is reported. Arylamidoximes 1a–i and dicyclohexylcarbodiimide (DCC) were carried out in DMF under FMWI to obtain 1,2,4-oxadiazoles 2a–i in 61–81% yields. All compounds exhibited antiproliferative activities in vitro against three human cancer cell lines HCT-116, PC-3, and SNB-19.