A module M is called square-free if it contains nonon-zero omorphic submodules and B with A∩B= 0. Dually, Mis dual-square-free has no proper M=A+B M/A∼=M/B. In this paper we show that M=⊕i∈I Mi, then iff each Mi Mj ⊕j=i∈I are orthogonal. M=⊕ni=1Mi, dual-square-free, 1⩽i⩽n, ⊕ni=jMi factor-orthogonal. Moreover, in the finite case, weshow M=⊕i∈ISi a direct sum of non-is simple modules, dual-square...

[b(oh)3]0.78[b(oh)2(oso3h)]0.22 or (bsa) was simply prepared by addition of chlorosulfonic acid to commercial boric acid at room temperature. this cheap and green solid acid was used as an efficient catalyst for synthesis of acylals from aldehydes under solvent-free conditions at room temperature.

In this paper we study the dynamics of Hepatitis B virus (HBV) infection under administration of a vaccine and treatment, where the disease is transmitted directly from the parents to the offspring  and also through contact with infective individuals. Stability of the disease-free steady state is investigated. The basic reproductive rate, $R_0$, is derived. The results show that the dynamics of...

The energy E(G) of a graph G is equal to the sum of the absolute values of the eigenvalues of G. Several classes of graphs are known that satisfy the condition E(G) > n , where n is the number of vertices. We now show that the same property holds for (i) biregular graphs of degree a b , with q quadrangles, if q<= abn/4 and 5<=a < b = 0 (iii) triregular graphs of degree 1, a, b that are quadran...

We provide a simplified version of a construction of Charles Read. For any n ≥ 2, there are n isometries with orthogonal ranges with the property that the nonself-adjoint weak-∗-closed algebra that they generate is all of B(H). A free semigroup algebra is the weak operator topology closed algebra generated by an n-tuple of isometries with pairwise orthogonal ranges. The C*-algebra generated by ...