Some variations of the Bohman–Korovkin theorem

In this paper we develop the main aspects of the Bohman–Korovkin theorem on approximation of continuous functions with the use of A-statistical convergence and matrix summability method which includes both convergence and almost convergence. Since statistical convergence and almost convergence methods are incompatible we conclude that these methods can be used alternatively to get some approximation results.     

Approximate solution for two stage open networks with Markov-modulated queues minimizing the state space explosion problem

Analytical solutions for two-dimensional Markov processes suffer from the state space explosion problem. Two stage tandem networks are effectively used for analytical modelling of various communication and computer systems which have tandem system behaviour. Performance evaluation of tandem systems with feedbacks can be handled with these models. However, because of the numerical difficulties caused by large state spaces, considering server failures and repairs at the second stage employing multiple servers has not been possible. The solution proposed in this paper is approximate with a high degree of accuracy. Using this approach, two stage open networks with multiple servers, break downs, and repairs at the second stage as well as feedback can be modelled as three-dimensional Markov processes and solved for performability measures. Results show that, unlike other approaches such as spectral expansion, the steady state solution is possible regardless of the number of servers employed.     

1,3‐Bis(4‐methyl­phen­yl)triazene, 1‐(4‐chloro­phen­yl)‐3‐(4‐fluoro­phen­yl)triazene and 1‐(4‐fluoro­phen­yl)‐3‐(4‐methyl­phen­yl)triazene

The title 4,4′‐disubstituted diphen­yl‐1,3‐triazines, C₁₄H₁₅N₃, (I), C₁₂H₉ClFN₃, (II), and C₁₃H₁₂FN₃, (III), each contain a triazene group (–N=N—NH–) having an extended conformation. The dihedral angles between the two benzene rings in (I), (II) and (III) are 4.3, 3.4 and 6.5°, respectively. The mol­ecules are almost entirely planar, with maximum deviations from the mean planes of 0.1087 (2), −0.1072 (7) and 0.1401 (3) Å, respectively. In each compound, the molecules are linked by N—H⋯N hydrogen bonds to form chains and pack similarly in the crystal structures.     

N1‐(2,6‐Di­methyl­phenyl)‐N2‐hydroxy‐α‐oxo‐α‐phenyl­acet­amidine

The structure of the title compound, C₁₆H₁₆N₂O₂, consists of a dimeric arrangement around an inversion centre of acet­amidine mol­ecules linked via O—H⋯N hydrogen bonds. There are also H⋯π‐ring interactions. All these interactions result in the formation of infinite chains parallel to the (101) axis. The oxime group has an E conformation.     

2‐Meth­oxy‐4‐[(5‐oxo‐2‐phenyl‐4,5‐dihydro‐1,3‐oxazol‐4‐yl­idene)meth­yl]­phenyl 4‐methyl­benzene­sulfonate

Mol­ecules of the title compound, C₂₄H₁₉NO₆S, adopt the Z configuration and have a distorted tetra­hedral geometry around the S atom. The oxazolone, 2‐phenyl and methoxy­phenyl rings are approximately coplanar. The C atom between the methoxy­phenyl and oxazolone rings displays a distorted trigonal bonding geometry. Pairs of mol­ecules are linked into dimers through weak C—H⋯O hydrogen bonds.     

N—H⋯N and C—H⋯π inter­actions in 4‐amino‐3‐methyl‐5‐(p‐tol­yl)‐4H‐1,2,4‐triazole and 4‐amino‐3‐methyl‐5‐phenyl‐4H‐1,2,4‐triazole

The title compounds, C₁₀H₁₂N₄, (I), and C₉H₁₀N₄, (II), have been synthesized and characterized both spectroscopically and structurally. The dihedral angles between the triazole and benzene ring planes are 26.59 (9) and 42.34 (2)°, respectively. In (I), mol­ecules are linked principally by N—H⋯N hydrogen bonds involving the amino NH₂ group and a triazole N atom, forming R₄⁴(20) and R₂⁴(10) rings which link to give a three‐dimensional network of mol­ecules. The hydrogen bonding is supported by two different C—H⋯π inter­actions from the tolyl ring to either a triazole ring or a tolyl ring in neighboring mol­ecules. In (II), inter­molecular hydrogen bonds and C—H⋯π inter­actions produce R₃⁴(15) and R₄⁴(21) rings.     

X-ray structure of {[μ-N,N′-Bis(salicylidene)-1, 3-propane-diamine] (dimethylformamide) zinc(II)} dibromo cadmium(ii)

The title compound, forms crystals which are the orthorhombic system, space group P2 ₁2₁2₁, with unit cell dimensions a = 10.3928(12) Å, b = 14.6298(13) Å, c = 15.9514(11) Å, V = 2425.3(4) ų. The cell contains four molecules. The structure is a heteronuclear dimeric complex with Zn^II and Cd^II ions. The coordination around the Zn^II ion is distorted square–pyramid and the Cd^II ion is distorted tetrahedral coordination. The Zn·sCd distance is 3.3594(7) Å.     

4‐[(3‐Chloro­phenyl)­diazenyl]‐2‐{[tris(hydroxy­methyl)­methyl]amino­methyl­ene}cyclo­hexa‐3,5‐dien‐1(2H)‐one

The title compound, C₁₇H₁₈ClN₃O₄, adopts the keto–amine tautomeric form and displays an intramolecular N—H⋯O hydrogen bond [N⋯O = 2.639 (2) Å]. The configuration around the azo N=N double bond is trans, and the dihedral angle between the planes of the two aromatic rings is 20.5 (2)°. The mol­ecules are linked by O—H⋯O hydrogen bonds to form a three‐dimensional network.