Derivations with invertible values in flexible algebras

Derivations with invertible values of 0 – torsion flexible algebras satisfying x(yz) = (xz)y over an algebraically closed field are described. For this class of algebra with unit element 1 and derivation with invertible value d is either a Cayley – Dickson algebra over its center Z(A) or a factor algebra of polynomial algebra C[a]/(a2) over a Cayley – Dickson division algebra; also C is 2 – torsion, d(C) = 0 and d(a) = 1+ua for some u in center of C and d is an outer derivation. Moreover, C is a split Cayley – Dickson algebra over its center Z having a derivation with invertible value d if and only if C is obtained by means of Cayley – Dickson process from its associative division subalgebra and can be represented as a direct sum C = V ⊕ aV.

Tilting and Cluster Tilting for Preprojective Algebras and Coxeter Groups

We study the stable category of the graded Cohen–Macaulay modules of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\boldsymbol{w})$ of this category associated with each reduced expression $\boldsymbol{w}$ of $w$ and give a sufficient condition on $\boldsymbol{w}$ such that $M(\boldsymbol{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\boldsymbol{w})$. Moreover, we compare it with a triangle equivalence given by Amiot–Reiten–Todorov for a cluster category.

On Stanley Depths of Certain Monomial Factor Algebras

Let S = K[x1 , . . . , xn] be the polynomial ring in n-variables over a ûeld K and I a monomial ideal of S. According to one standard primary decomposition of I, we get a Stanley decomposition of the monomial factor algebra S/I. Using this Stanley decomposition, one can estimate the Stanley depth of S/I. It is proved that sdepthS(S/I) ≤ sizeS(I). When I is squarefree and bigsizeS(I) ≤ 2, the Stanley conjecture holds for S/I, i.e., sdepthS(S/I) ≥ depthS(S/I).

Calculation of the Shape Factor From a Small Rectangular Plane to a Triangular Surface Perpendicular to the Rectangular Plane Without a Common Edge

In developing an analytical model of radiative heat transfer inside attics, the necessity of calculating the shape factor from a small rectangle to a triangular plane may arise. This paper describes a methodology of calculation for such a shape factor from simple relations using shape-factor algebra. The shape factor is calculated using a mirror image of the triangular surface, in such a way that the small rectangular plane is positioned symmetrical between the two triangular surfaces. Numerical results are presented in graphical form.