Let $R$ be a ring‎, ‎and let $n‎, ‎d$ be non-negative integers‎. ‎A right $R$-module $M$ is called $(n‎, ‎d)$-projective if $Ext^{d+1}_R(M‎, ‎A)=0$ for every $n$-copresented right $R$-module $A$‎. ‎$R$ is called right $n$-cocoherent if every $n$-copresented right $R$-module is $(n+1)$-coprese-nted‎, ‎it is called a right co-$(n,d)$-ring if every right $R$-module is $(n‎, ‎d)$-projective‎. ‎$R$...

Two types of tests are available for studying personality. These are the paper-pencil tests and the projective tests. Projective techniques which reveal the subject's personality through his treatment of the test material are generally more valid and more revealing than the paper-pencil tests which require the underlining of words or the answering of "Yes" or "No" to a list of questions. This i...

Let X be a smooth, complete toric variety. Let A1(X) be the group of algebraic 1-cycles on X modulo numerical equivalence and N1(X) = A1(X) ⊗Z Q . Consider in N1(X) the cone NE(X) generated by classes of curves on X. It is a well-known result due to M. Reid [11] that NE(X) is closed, polyhedral and generated by classes of invariant curves on X. The variety X is projective if and only if NE(X) i...

Up to now, the shortest invariant representation of a quadric has 138 summands and there has been no invariant representation of a twisted cubic in 3D projective space, which limit to some extent the applications of invariants in 3D space. In this paper, we give a very short invariant representation of a quadric cone, a special quadric, which has only two summands similar to the invariant repre...