Nontrivial Solutions for Resonant Hemivariational Inequalities

We study a resonant semilinear elliptic hemivariational inequality. Under some assumptions of strong resonance on the Clarke subdifferential of the superpotential, and using nonsmooth critical point theory, the existence of a nontrivial solution of the problem is shown. This paper has been partially supported by the State Committee for Scientific Research of Poland (KBN) under research grants no. 2 P03A 003 25 and no. 4 T07A 027 26.     

Structure of 18-cyanoprogesterone,C22H29O2N

The crystal structure of 18-cyanoprogesterone was determined by X-ray diffraction methods:P2₁2₁2₁ a=7.436(2),b=11.322(2),c=22.642(2) Å. The structure was solved usingShelx-86. Final conventionalR=0.054.R _w =0.051 for 1841 reflections. TheA ring has an intermediate sofa-half-chair conformation with asymmetry parameters C _s ^/1 =11.0, C ₂ ^3,4 =14.9. The steroid skeleton exhibits a flattening of theA ring relative to the rest of the molecule. The progesterone side chain has a typical conformation, and the C16-C17-C20-O20 torsion angle is –19.0(6)°.     

The first structurally characterized nonorganometallic titanium(III) alkoxo-bridged dinuclear complexes

The reaction of [Ti4(OMe)14Cl2] (1) with an excess of AlMe3 gave the cocrystallite [Ti2(mu-OMe)2(mu-Cl)Cl3(thf)3].[Ti2(mu-OMe)3Cl3(thf)3] (2.3) species in a 1:1 ratio. Similar to 2, [Ti2(mu-OEt)2(mu-Cl)Cl3-(thf)3] (4) was obtained in the reaction of an equimolar mixture of TiCl4 and Ti(OEt)4 with Al/AlMe3. The short distance [2.543(1)av A in 2.3 and 2.599(1) A in 4] between "Ti(+3)" atoms, their diamagnetism, and ELF analysis indicate the presence of a Ti-Ti bond.     

Cauchy boundary andb-incompleteness of space-time

It is shown that if a space-time (M, g) is time-orientable and its Levi-Civita connection [in the bundle of orthonormal frames over (M, g)] is reducible to anO(3) structure, one can naturally select a nonvanishing timelike vector field and a Riemann metricg ⁺ onM. The Cauchy boundary of the Riemann space (M, g ⁺) consists of endpoints ofb-incomplete curves in (M, g); we call it theCauchy singular boundary. We use the space-time of a cosmic string with a conic singularity to test our method. The Cauchy singular boundary of this space-time is explicitly constructed. It turns out to consist of what should be expected.     

Adams methods for neutral functional differential equations

Summary In this paper Adams type methods for the special case of neutral functional differential equations are examined. It is shown thatk-step methods maintain orderk+1 for sufficiently small step size in a sufficiently smooth situation. However, when these methods are applied to an equation with a non-smooth solution the order of convergence is only one. Some computational considerations are given and numerical experiments are presented.     

Synthesis of Novel Series of 6‐Chloro‐1,1‐dioxo‐1,4,2‐benzodithiazine Derivatives with Potential Biological Activity

A series of 6′‐chloro‐1′,1′‐dioxo‐2′H‐spiro[benzo[d][1,3,7]oxadiazocine‐4,3′‐(1,4,2‐benzodithiazine)]‐2,6(1H,5H)‐dione derivatives 2a , 2b and 3a , 3b have been synthesized starting from 3‐aminobenzodithiazines 1a , 1b and isatoic anhydride. Subsequent reactions of 2a with 3‐chlorophenyl isocyanate gave condensation products 4 and 5 . Compound 2a was also converted into 3‐(2‐aminobenzamido)‐6‐chloro‐7‐methyl‐1,1‐dioxo‐1,4,2‐benzodithiazine derivatives 6 , 7 , 8 , 9 , 10 . The mechanisms of the reactions are discussed.     

On Economic Equilibrium Type Problems with Applications

The existence solution for a class of economic equilibrium type problems in reflexive Banach space is considered. New results concerning the variational inequality approach to Arrow–Debreu model of economic equilibrium introduced in Naniewicz (Math Oper Res 32:436–466, 2007) are found and applied to ensure the existence of Pareto optimal solutions for a class of multiobjective optimization problems with so-called budget-like constraints. To achieve this goal, the theory of pseudo-monotone multivalued mappings combined with some fixed point technique for multivalued mappings with nonconvex values is used.     

Evolution of density perturbations in a realistic universe

Prompted by the recent more precise determination of the basic cosmological parameters and growing evidence that the matter-energy content of the universe is now dominated by dark energy and dark matter we present the general solution of the equation that describes the evolution of density perturbations in the linear approximation. It turns out that as in the standard CDM model the density perturbations grow very slowly during the radiation dominated epoch and their amplitude increases by a factor of about 4000 in the matter and later dark energy dominated epoch of expansion of the universe.