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...

LIDIA S. SZCZEPANIAK,1,2 EVELYN E. BABCOCK,3 FRITZ SCHICK,4 ROBERT L. DOBBINS,1,2 ABHIMANYU GARG,2 DENNIS K. BURNS,5 J. DENIS MCGARRY,1,2,6 AND DANIEL T. STEIN1,2 1Center for Diabetes Research, Departments of 2Internal Medicine, 3Radiology, 5Pathology, and 6Biochemistry, University of Texas Southwestern Medical Center, Dallas, Texas 75235; and 4Department of Radiology, University of Tübingen, 7...

Vacuum.-By creating a vacuum the process of decalcification can be characterized as follows: Insoluble calciurn-salts+acid o-soluble calciumsalts+CO2. The carbon dioxide, rapidly extracted, will disturb the balance of the chemical equilibrium, and, according to Le Chatelier, result in an acceleration of the reaction. Waerhaug (1949), however, points out that there will be a more intensive conta...

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...

Services are available to help support existing employment for individuals with psychiatric disabilities; however, there is a gap in services targeting job interview skills that can help obtain employment. We assessed the feasibility and efficacy of Virtual Reality Job Interview Training (VR-JIT) in a randomized controlled trial. Participants were randomized to VR-JIT (n = 25) or treatment-as-u...

Efforts for drug free sport include developing a better understanding of the behavioural determinants that underline doping with an increased interest in developing anti-doping prevention and intervention programmes. Empirical testing of both is dominated by self-report questionnaires, which is the most widely used method in psychological assessments and sociology polls. Disturbingly, the poten...

We present a sample of 171 candidate groups and clusters of galaxies at intermediate redshift over an area of ≈ 550 sq. degrees at the South Galactic Pole, selected by using optically identified radio sources from the NRAO VLA Sky Survey as tracers of dense environments. Out of these 171 candidates, 76 have a counterpart in the literature while 95 of them are previously unknown clusters. This p...

This paper presents a new algorithm for computing the hit point and normal of a ray and a general convex object. The algorithm is loosely based on the the Gilbert-Johnson-Keerthi algorithm for computing the distance between convex objects in the sense that it is applicable to the same family of convex objects and uses the same subalgorithm for computing the closest point of a simplex. Since thi...