One-pot synthesis of 3,4-Dihydropyrimidin-2(1<Emphasis Type="Italic">H</Emphasis>)-ones (thiones) promoted by nano-BF<Subscript>3</Subscript>.SiO<Subscript>2</Subscript>

3,4-Dihydropyrimidin-2(1H)-ones and 3,4-dihydropyrimidin-2(1H)-thione were synthesized under solvent free condition in the presence of nano-silica supported boron trifluoride (nano-BF₃.SiO₂). The reactions were carried out at 80 °C for 15 min under solvent free condition. This methods have some advantages such as good to excellent yield, mild reaction condition, ease of operation and workup, short reaction time and high product purity.     

Nano-TiO<Subscript>2</Subscript>: An eco-friendly and re-usable catalyst for the synthesis of 14-Aryl or alkyl-14<Emphasis Type="Italic">H</Emphasis>-dibenzo[<Emphasis Type="Italic">a,j</Emphasis>]xanthenes

Synthesis of 14-aryl or alkyl-14H-dibenzo[a,j]xanthenes using nano-TiO₂ as eco-friendly and efficient catalyst is reported. Short reaction times, high yields, a clean process, simple methodology, easy work-up and green conditions are advantages of this protocol.     

On edge star sets in trees

Let A be a Hermitian matrix whose graph is G (i.e. there is an edge between the vertices i and j in G if and only if the (i,j) entry of A is non-zero). Let λ be an eigenvalue of A with multiplicity m_A(λ). An edge e=ij is said to be Parter (resp., neutral, downer) for λ,A if m_A(λ)−m_A−e(λ) is negative (resp., 0, positive ), where A−e is the matrix resulting from making the (i,j) and (j,i) entries of A zero. For a tree T with adjacency matrix A a subset S of the edge set of G is called an edge star set for an eigenvalue λ of A, if |S|=m_A(λ) and A−S has no eigenvalue λ. In this paper the existence of downer edges and edge star sets for non-zero eigenvalues of the adjacency matrix of a tree is proved. We prove that neutral edges always exist for eigenvalues of multiplicity more than 1. It is also proved that an edge e=uv is a downer edge for λ,A if and only if u and v are both downer vertices for λ,A; and e=uv is a neutral edge if u and v are neutral vertices. Among other results, it is shown that any edge star set for each eigenvalue of a tree is a matching.     

Equimatchable claw-free graphs

A graph is equimatchable if all of its maximal matchings have the same size. A graph is claw-free if it does not have a claw as an induced subgraph. In this paper, we provide the first characterization of claw-free equimatchable graphs by identifying the equimatchable claw-free graph families. This characterization implies an efficient recognition algorithm.     

Division Algebras with Left Algebraic Commutators

Let D be a division algebra with center F and K a (not necessarily central) subfield of D. An element a ∈ D is called left algebraic (resp. right algebraic) over K, if there exists a non-zero left polynomial a ₀ + a ₁ x + ? + a _n x ⁿ (resp. right polynomial a ₀ + x a ₁ + ? + x ⁿ a _n ) over K such that a ₀ + a ₁ a + ? + a _n a ⁿ = 0 (resp. a ₀ + a a ₁ + ? + a ⁿ a _n ). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite provided that the center of division algebra is infinite. Also, we show that every division algebra whose multiplicative group of commutators is left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. Among other results we present similar result regarding additive commutators under certain conditions.     

On the Finiteness of Noetherian Rings with Finitely Many Regular Elements

Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if R \ ZD(R) is finite, then R is finite.     

On the role of anisotropic turbomachinery flow structures in inter-scale turbulence energy flux as deduced from SPIV measurements

The present study uses stereoscopic particle image velocimetry in the rotor exit of a centrifugal turbomachine to analyse anisotropy and geometrical characteristics of tensorial flow quantities. The purpose is to identify dominant topologies of turbulence stress tensor and principal directions of flow structures. The misalignment between principal directions of strain and turbulence stress tensors is more evident in the jet–wake interaction regions and questions the eddy-viscosity models which assume an exact alignment between stress/strain eigenvectors. Anisotropy analysis based on the barycentric approach shows that the disk-like structure and/or the rod-like structure limiting states of turbulence are the most frequent topologies of turbulence stress. Additionally, planar straining is the dominant deformation characteristic in the measurement area. These anisotropic behaviours considerably attribute to the turbulence energy cascade. Conditional isolation of flow structures based on inter-scale energy flux shows that a larger extent of turbulence stress anisotropy results in a larger energy flux and therefore significantly affects the dynamics of turbulent flow structures.     

The Inclusion Ideal Graph of Rings

Let R be a ring with unity. The inclusion ideal graph of a ring R, denoted by In(R), is a graph whose vertices are all nontrivial left ideals of R and two distinct left ideals I and J are adjacent if and only if I ? J or J ? I. In this paper, we show that In(R) is not connected if and only if R ? M ₂(D) or D ₁ × D ₂, for some division rings, D, D ₁ and D ₂. Moreover, we prove that if In(R) is connected, then diam(In(R)) ≤3. It is shown that if In(R) is a tree, then In(R) is a caterpillar with diam(In(R)) ≤3. Also, we prove that the girth of In(R) belongs to the set {3, 6, ∞}. Finally, we determine the clique number and the chromatic number of the inclusion ideal graph for some classes of rings.