Defect Distributions Related to Weakly Convergent Sequences in Bessel-Type Spaces $${H_{\Lambda }^{-s,p}}$$

We consider microlocal defect distributions associated to a weakly convergent sequences \(u_n\) in \(H^{-s,p}_{\Lambda }\) and \(v_n\) in \(H^{s+m,q}_{\Lambda }\) through the space of pseudo-differential operators with the symbols in \((s^{m,N+1}_\Lambda )_0\). Symbols correspond to a weight function \(\Lambda \) determining a quasi-elliptic symbol. Results are applied to partial differential equations with symbols related to weights of the type \(\Lambda \).     

Synthesis, structure and reactivity of palladium(II) complexes of chiral N-heterocyclic carbene-imine and -amine hybrid ligands

The synthesis and structures of chiral N-heterocyclic carbene (NHC)-N-donor complexes of silver(I) and palladium(II) are reported. The X-ray structure of an NHC-imine silver(I) complex [((nPr)CN(CHPh))AgBr](2) exhibits an Ag(2)Br(2) dimer motif where the imine group is not coordinated to the silver atom. Reaction between 2 and [PdCl(2)(MeCN)(2)] gives the palladium(II) complex [(kappa(2)-(nPr)CN(CHPh))PdCl(2)](3) that contains a chelating NHC-imine ligand as shown by single-crystal X-ray diffraction. Slow hydrolysis of related complexes [(kappa(2)-(nPr)CN(CHPh))PdCl(2)](3) and [(kappa(2)-((Ph)(2)CH)CN(CHPh))PdCl(2)](4) using triethylammonium chloride and water lead to the precipitation of single crystals of insoluble NHC-amino palladium(II) complexes [(kappa(2)-(nPr)CN(H(2)))PdCl(2)](6) and [(kappa(2)-((Ph)(2)CH)CN(H(2)))PdCl(2)](7), respectively. In the solid state, complexes 6 and 7 both exhibit intermolecular hydrogen bonding between chlorine and an amino-hydrogen atom resulting in an infinite chain structure. Substitution of an amino hydrogen for an ethyl group gives the soluble complex [(kappa(2)-(iPr)CN((H)Et))PdCl(2)](12). Reaction between two equivalents of 2 and [PdCl(2)(MeCN)(2)] gives the di-NHC complex [(kappa(1)-(nPr)CN(CHPh))(2)PdCl(2)](5) that does not contain a coordinated imine as shown by single crystal X-ray diffraction. Conproportionation between 5 and an equivalent of [PdCl(2)(MeCN)(2)] to does not occur at temperatures up to 100 degrees C in CD(3)CN.     

Homogeneity in generalized function algebras

We investigate homogeneity in the special Colombeau algebra on Rd as well as on the pierced space Rd?{0}. It is shown that strongly scaling invariant functions on Rd are simply the constants. On the pierced space, strongly homogeneous functions of degree α admit tempered representatives, whereas on the whole space, such functions are polynomials with generalized coefficients. We also introduce weak notions of homogeneity and show that these are consistent with the classical notion on the distributional level. Moreover, we investigate the relation between generalized solutions of the Euler differential equation and homogeneity.     

Divergent Type Quasilinear Dirichlet Problem with Singularities

A quasilinear equation of divergent type with singular data and singular coefficients is approximated by a net of equations of the same type with enough regular coefficients and data. Solutions of the net of equations are obtained by the classical methods. Known a priory estimates are improved so that a net of solutions can be considered as a solution in an appropriate algebra of generalized functions.     

Distributed-order fractional wave equation on a finite domain: creep and forced oscillations of a rod

We study waves in a rod of finite length with a viscoelastic constitutive equation of distributed fractional order type for the special choice of weight functions. Prescribing boundary conditions on displacement and stress, we obtain, as special solutions, cases corresponding to creep and forced oscillations. In solving system of differential and integro-differential equations, we use the Laplace transformation in the time domain.     

Structure theorems for generalized random processes

Generalized random processes are classified by various types of continuity. Representation theorems of a generalized random process on {M _p } on a set with arbitrary large probability, as well as representations of a correlation operator of a generalized random process on {M _p } and L ^r (R), r > 1, are given. Especially, Gaussian generalized random processes are proven to be representable as a sum of derivatives of classical Gaussian processes with appropriate growth rate at infinity. Examples show the essence of all the proposed assumptions. In order to emphasize the differences in the concept of generalized random processes defined by various conditions of continuity, the stochastic differential equation y′(ω; t) = f(ω; t) is considered, where y is a generalized random process having a point value at t = 0 in the sense of Lojasiewicz. This paper was supported by the project Functional analysis, ODEs and PDEs with singularities, No. 144016, financed by the Ministry of Science, Republic of Serbia.