Spectral Properties of Image Measures Under the Infinite Conflict Interaction

We introduce the conflict interaction with two positions between a couple of image probability measures and consider the associated dynamical system. We prove the existence of invariant limiting measures and find the criteria for these measures to be a pure point, absolutely continuous, or singular cotinuous as well as to have any topological type and arbitary Hausdorff dimension.     

On fractal properties of non-normal numbers with respect to Rényi f-expansions generated by piecewise linear functions

The paper is devoted to the study of fractal properties of subsets of the set of non-normal numbers with respect to Rényi f  -expansions generated by continuous increasing piecewise linear functions defined on [0,+∞)$[0,+\infty )$. All such expansions are expansions for real numbers generated by infinite linear IFS f={_f0,_f1,…,_fn,…}$f=\{f_0,f_1,\dots ,f_n,\dots \}$ with the following list of ratios _Q∞=(_q0,_q1,…,_qn,…)$Q_{\infty }=(q_0,q_1,\dots ,q_n,\dots )$.     

Homogeneity range of the NaTl-type Zintl phase in the ternary system Li-In-Ag

The remarkably broad homogeneity range of the NaTl-type Zintl phase in the ternary phase diagram Li-In-Ag at room temperature was determined by structure evaluation using X-ray powder diffraction. The colours of the investigated Zintl phases correlate with the valence electron concentration (VEC) as already established for the quasibinary cut Li_0.5(In_xAg_1−x)_0.5 with 0.47?x?1.00, i.e. with decreasing VECs the colour changes from grey over reddish to bright yellow. All compounds in the new quasibinary cut Li_x(In_0.5Ag_0.5)_1−x with 0.47?x?0.60 appear free from vacancies in the Li-sublattice, even for Li-deficient compositions. The partial occupation of Li-sites by excess Ag and In instead is in full agreement with the behaviour of the binary NaTl-type Zintl phases Li_xZn_1−x and Li_xCd_1−x (0.47?x?0.54) with a low VEC about 1.5.     

On fine fractal properties of generalized infinite Bernoulli convolutions

The paper is devoted to the investigation of generalized infinite Bernoulli convolutions, i.e., the distributions μξ of the following random variables: where ak are terms of a given positive convergent series; ξk are independent random variables taking values 0 and 1 with probabilities p0k and p1k correspondingly.We give (without any restriction on {an}) necessary and sufficient conditions for the topological support of ξ to be a nowhere dense set. Fractal properties of the topological support of ξ and fine fractal properties of the corresponding probability measure μξ itself are studied in details for the case where ak?rk:=ak+1+ak+2+? (i.e., rk−1?2rk) for all sufficiently large k. The family of minimal dimensional (in the sense of the Hausdorff–Besicovitch dimension) supports of μξ for the above mentioned case is also studied in details. We describe a series of sets (with additional structural properties) which play the role of minimal dimensional supports of generalized Bernoulli convolutions. We also show how a generalization of M. Cooper's dimensional results on symmetric Bernoulli convolutions can easily be derived from our results.     

Electrophilicity of 5-benzylidene-1,3-dimethylbarbituric and -thiobarbituric acids

The kinetics of reactions of acceptor-stabilized carbanions 2a-m with benzylidenebarbituric and -thiobarbituric acids 1a-e has been determined in a dimethyl sulfoxide solution at 20 degrees C. Second-order rate constants were employed to determine the electrophilicity parameters E of the benzylidenebarbituric and -thiobarbituric acids 1a-e according to the correlation equation log k(20 degrees C) = s(N + E). With E parameters in the range of -10.4 to -13.9, the electrophilicities of 1a-e are comparable to those of analogously substituted benzylidenemalononitriles.     

Crystal,electronic structure and hydrogenation properties of the Mg5.57Ni16Ge7.43 cluster phase with a new type of polyhedron

The ternary germanide Mg_5.57Ni₁₆Ge_7.43 (cubic, space group Fmm, cF116) belongs to the structural family based on the Th₆Mn₂₃-type. The Ge1 and Ge2 atoms fully occupy the 4a (mm symmetry) and 24d (m.mm) sites, respectively. The Ni1 and Ni2 atoms both fully occupy two 32f sites (.3m symmetry). The Mg/Ge statistical mixture occupies the 24e site with 4m.m symmetry. The structure of the title compound contains a three-core-shell cluster. At (0,0,0), there is a Ge1 atom which is surrounded by eight Ni atoms at the vertices of a cube and consequently six Mg atoms at the vertices of an octahedron. These surrounded eight Ni and six Mg atoms form a [Ge1Ni₈(Mg/Ge)₆] rhombic dodecahedron with a coordination number of 14. The [GeNi₈(Mg/Ge)₆] rhombic dodecahedron is encapsulated within the [Ni₂₄] rhombicuboctahedron, which is again encapsulated within an [Ni₃₂(Mg/Ge)₂₄] pentacontatetrahedron; thus, the three-core-shell cluster [GeNi₈(Mg/Ge)₆@Ni₂₄@Ni₃₂(Mg/Ge)₂₄] results. The pentacontatetrahedron is a new representative of Pavlyuk's polyhedra group based on pentagonal, tetragonal and trigonal faces. The dominance of the metallic type of bonding between atoms in the Mg_5.57Ni₁₆Ge_7.43 structure is confirmed by the results of the electronic structure calculations. The hydrogen sorption capacity of this intermetallic at 570 K reaches 0.70 wt% H₂.     

Periodic cracks and temperature-dependent stress in Mo/Si multilayers on Si substrates

This work examines formation of the peculiar periodic crack patterns observed in the thermally loaded Mo/Si multilayers. Using the substrate curvature measurements, the macroscopic film stress evolution during thermal cycling was investigated. Then high-speed microscopic observation of crack propagation in the annealed Mo/Si multilayers was presented providing experimental evidence of the mechanism underlying formation of the periodic crack patterns. The origin of the peculiar periodic crack patterns was determined. They were observed to form by the slow crack propagation under quasi-static conditions as a result of the interaction between the channelling crack propagation and the advance of the delamination front.     

Lanthanum tetrazinc,LaZn4

The structure of lanthanum tetrazinc, LaZn₄, has been determined from single‐crystal X‐ray diffraction data for the first time, approximately 70 years after its discovery. The compound exhibits a new structure type in the space group Cmcm, with one La atom and two Zn atoms occupying sites with m2m symmetry, and one Zn atom occupying a site with 2.. symmetry. The structure is closely related to the BaAl₄, La₃Al₁₁, BaNi₂Si₂ and CaCu₅ structure types, which can be presented as close‐packed arrangements of 18‐vertex clusters, in this case LaZn₁₈. The kindred structure types contain related 18‐vertex clusters around atoms of the rare earth or alkaline earth metal.     

On singularity and fine spectral structure of random continued fractions

We study properties of the distribution of a random variable of the continued fraction form where are independent and not necessarily identically distributed random variables. We prove the singularity of and study the fine spectral structure of such measures.