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In this article, two possible applications of prism-based probes for liquid-switching, level-sensing, and as a refractometer are described. Theoretical formulation is developed for the emergent intensity for each system and possible range of operation for each system is reported. By comparing the theoretical investigations, some hints are given in optimum usage of prism in each case. Variation of the beam divergence, incident angle, and prism glass index are major parameters that are considered in this investigation. The obtained results are compared with some available experimental results and practical points concerning the effective application of each geometry is reported. Theoretical expressions developed here are in good agreement with the experimental measurements.

In this paper we study the problems of separability of two disjoint point sets in 3D by multiple criteria extending some notions on separability of two disjoint point sets in the plane.

Various angle characteristics are used (e.g. in finite element methods or computer graphics) when evaluating the quality of computational meshes which may consist, in the three-dimensional case, of tetrahedra, prisms, hexahedra and pyramids. Thus, it is of interest to derive (preferably tight) bounds for dihedral angle sums, i.e. sums of angles between faces, of such mesh elements. For tetrahedra this task was solved by Gaddum in 1952. For pyramids, this was resolved by Korotov, Lund and Vatne in 2022. In this paper, we compute tight bounds for the remaining two cases, hexahedra and prisms.

Science Academies' Role in Development: Presidents Reveal It All.

Protein Nanocages: The Versatile Molecular Shell.

A red-white coloring of a nontrivial connected graph $G$ of diameter $d$ is an assignment of red and white colors to the vertices of $G$ where at least one vertex is colored red. Associated with each vertex $v$ of $G$ is a $d$-vector, called the code of $v$, whose $i$th coordinate is the number of red vertices at distance $i$ from $v$. A red-white coloring of $G$ for which distinct vertices have distinct codes is called an identification coloring or ID-coloring of $G$. A graph $G$ possessing an ID-coloring is an ID-graph. The minimum number of red vertices among all ID-colorings of an ID-graph $G$ is the identification number or ID-number of $G$. It is shown that the grid $P_m\square P_n$ is an ID-graph if and only $(m,n)\ne (2,2)$ and the prism $C_n\square K_2$ is an ID-graph if and only if $n\ge 6$.

We consider prisms over complete 3-uniform hypergraphs and decompose them into Hamiltonian cycles of Katona–Kierstead type. We show that prisms over complete 3-uniform hypergraphs of order 4, 5 and 8 can be decomposed into Hamiltonian cycles and we suggest that prisms over complete 3-uniform hypergraphs of order in the form 6t + 2 may have Hamiltonian decompositions.