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...

TheMinkowski sum of two setsA,B ∈ IR, denotedA⊕B, is defined as {a+ b | a ∈ A, b ∈ B}. We describe an efficient and robust implementation for the construction of Minkowski sums of polygons in IR using the convolution of the polygon boundaries. This method allows for faster computation of the sum of non-convex polygons in comparison to the widely-used methods for Minkowski-sum computation that d...

the capitalism free market system is a system that has gone beyond the ideal level and has realized in all over the world. although it has some achievement for mankind, it has lost its function due to its role in destruction of environment. the theoretical bases of this system are lead to this negative operation. the islam universal system is an ideal system that its theoretical bases, that are...

The separability problem for languages from a class C by languages of a class S asks, for two given word languages I and E from C, whether there exists a language S from S which includes I and excludes E, that is, I ⊆ S and S ∩ E = ∅. It is known that separability for context-free languages by any class containing all definite languages (such as regular languages) is undecidable. We show that s...

AIM This 12-week, product-evaluation registry study aimed to compare the effects of supplementation with French Pine Bark Extract (Pycnogenol®) on cognitive function, attention, and mental performance in healthy professionals with increased oxidative stress in a professional context. METHODS Professionals were screened for increased oxidative stress: 60 subjects (range 35-55 years, no risk co...