We consider T f = ∫ x1 0 ∫ x2 0 f (t1, t2)dt1dt2 and a corresponding geometric mean operator G f = exp(1/x1x2) ∫ x1 0 ∫ x2 0 log f (t1, t2)dt1dt2. E. T. Sawyer showed that the Hardy-type inequality ‖T f ‖Lq u ≤ C‖ f ‖Lp v could be characterized by three independent conditions on the weights. We give a simple proof of the fact that if the weight v is of product type, then in fact only one condit...

We show that the triple product defined by the spin domain (Bounded Symmetric Domain of type 4 in Cartan’s classification) is closely related to the geometric product in Clifford algebras. We present the properties of this tri-product and compare it with the geometric product. The spin domain can be used to construct a model in which spin 1 and spin1/2 particles coexist. Using the geometric tri...

In this paper, the (infinite) direct product of fields is investigated. particular, finiteness a given set characterized in terms some ring-theoretic observations. Next, certain localization (whose multiplicative formed by cofinite sets) studied. Finally, it shown that every X can be made into separated scheme, and scheme an affine if only finite set.

Currently, most of CAD/CAM systems are feature-based. Features are important to maintain geometric and non-geometric relations across product models. Often, during data translation some useful geometric and non-geometric information such as features and their associated semantics are stripped off. In this paper, we introduce a new method of feature-based modeling in product data exchange using ...