ONE LINEAR ANALYTIC APPROXIMATION FOR STOCHASTIC INTEGRODIFFERENTIAL EQUATIONS

This article concerns the construction of approximate solutions for a general stochastic integrodifferential equation which is not explicitly solvable and whose coeffcients functionally depend on Lebesgue integrals and stochastic integrals with respect to martingales. The approximate equations are linear ordinary stochastic differential equations, the solutions of which are defined on sub-intervals of an arbitrary partition of the time interval and connected at successive division points. The closeness of the initial and approximate solutions is measured in the L^p-th norm, uniformly on the time interval. The convergence with probability one is also given.     

Derivative free two-point methods with and without memory for solving nonlinear equations

Two families of derivative free two-point iterative methods for solving nonlinear equations are constructed. These methods use a suitable parametric function and an arbitrary real parameter. It is proved that the first family has the convergence order four requiring only three function evaluations per iteration. In this way it is demonstrated that the proposed family without memory supports the Kung-Traub hypothesis (1974) on the upper bound 2ⁿ of the order of multipoint methods based on n + 1 function evaluations. Further acceleration of the convergence rate is attained by varying a free parameter from step to step using information available from the previous step. This approach leads to a family of two-step self-accelerating methods with memory whose order of convergence is at least and even in special cases. The increase of convergence order is attained without any additional calculations so that the family of methods with memory possesses a very high computational efficiency. Numerical examples are included to demonstrate exceptional convergence speed of the proposed methods using only few function evaluations.     

Distance spectra and distance energy of integral circulant graphs

The distance energy of a graph G is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of G. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the circulant group, i.e. its adjacency matrix is circulant. A graph is called integral if all eigenvalues of its adjacency matrix are integers. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer. In this paper, we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix D. Furthermore, we calculate the distance spectra and distance energy of unitary Cayley graphs. In conclusion, we present two families of pairs (G₁,G₂) of integral circulant graphs with equal distance energy - in the first family G₁ is subgraph of G₂, while in the second family the diameter of both graphs is three.     

Bromophilic substitution/carbophilic substitution cascade reactions of α,α-dibromo-2-methoxyacetophenone with C-, N- and O-nucleophiles

α,α-Dibromo-2-methoxyacetophenone reacts, under mild reaction conditions, with C-, N- and O-nucleophiles via a bromophilic substitution/protonation/carbophilic substitution cascade process to afford α-monosubstituted-2-methoxyacetophenones in moderate to good yield. The only exception from this reaction pathway is the reaction with the anion derived from malononitrile in which 2-aroyl-1,1,3,3-tetracyanopropene is obtained.     

Paragraded Rings and Their Ideals

The notions of a paragraded ring and a homogeneous ideal, which are at the same time a generalization of the classical graduation, as defined by Bourbaki, and an extension of the earlier work done by M. Krasner, were introduced by M. Krasner and M. Vukovi?. After recalling the notion of paragraded rings, we list and prove several facts about them. One of the most important properties is that the homogeneous part of the direct product and the direct sum of paragraded rings are the direct product and the direct sum of the corresponding homogeneous parts, respectively. Next we give the notion of a homogeneous ideal of a paragraded ring and prove that the factor ring obtained from a paragraded ring and its homogeneous ideal is also a paragraded ring. After that, we deal with basic facts about homogeneous ideals.     

Improvements on the reverse order laws

In many papers concerning properties of generalized inverses in different settings, we can find the results with many redundant instances of assuming regularity of certain elements. We have made an effort to widen the range of applicability of concrete results by considering more general cases of the problems without imposing any such additional assumptions. This was the main motivation for this paper, and we present several significant improvements of the reverse order laws on $\{1,3\}$ and $\{1,2,3\}$-inverses in the ring setting.