Background: The removable dentures or fixed can be used as a replacement for missing teeth and restore the aesthetics functional condition of patien. Dentures with good adaptation obtained from accurate impressions. Dimensional accuracy adequate mechanical properties are one main requirements working model making materials. addition sodium borate help gypsum type III achieve balance dimentional...

Within this paper we study the Minkowski sum of prisms (" Cephoids ") in a finite dimensional vector space. For a vector a ∈ R n with positive components we write ¯ a = (1 ¯ a 1 1 ¯ an) and denote by = ¯ a = {x ∈ R n || ¯ a, x 1 , x 0} the associated prism. We provide a representation of a finite sum of prisms in terms of inequalities.

Let SH denote the homothety class generated by a convex set S ⊂ R: SH = {a + λS | a ∈ R, λ > 0}. We determine conditions for the Minkowski sum BH + CH or the Minkowski difference BH ∼ CH of homothety classes BH and CH generated by closed convex sets B,C ⊂ R to lie in a homothety class generated by a closed convex set (more generally, in the union of countably many homothety classes generated by...

Fakhruddin has proved that for two lattice polygons P and Q any lattice point in their Minkowski sum can be written as a sum of a lattice point in P and one in Q, provided P is smooth and the normal fan of P is a subdivision of the normal fan of Q. We give a shorter combinatorial proof of this fact that does not need the smoothness assumption on P .

Under the extra assumption that the boundary is of class C this was proven by S. Nakajima (= A. Matsumura) in 1926 Theorem 1 solves this problem. For convex bodies with C boundaries and positive curvature Nakajima’s result was generalized by Chakerian [?] in 1967 to “relative geometry” where the width and brightness are measured with with respect to some convex body K0 symmetric about the origi...

3,3′-Diaminobenzidine was applied at doses of 5, 10, 25, and 50 mg kg−1 of soil and their effects were evaluated on indigenous soil microbial communities (viable aerobic bacteria, fungal populations, aerobic N2-fixing bacteria, denitrifying, and nitrifying bacteria), and soil enzymatic activities (acid and alkaline phosphatases, arylsulfatase, and dehydrogenase). At doses of 5 or 10 mg kg−1, 3,...