W e consider the rational linear relations between real numbers whose squared trigonometric functions have rational values, angles we call “geodetic.” We construct a convenient basis for the vector space over generated by these angles. Geodetic angles and rational linear combinations of geodetic angles appear naturally in Euclidean geometry; for illustration we apply our results to equidecompos...

Given graphs G and H, and a colouring of the edges of G with k colours, a monochromatic H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic graph isomorphic to H. Let φk(n,H) be the smallest number φ such that any k-edge-coloured graph G of order n, admits a monochromatic H-decomposition with at most φ parts. Here we stud...

This paper presents a review on edge-preserving decomposition based image haze removal. When the background has dust particles or hazes its image quality and color is degraded. It is well-known that local filtering-based edge preserving smoothing method suffers from halo artifacts. In this paper a survey of many different methods for haze removal just like Laplacian-based visibility restoration...

Given graphs G and H, and a colouring of the edges of G with k colours, a monochromatic H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic graph isomorphic to H. Let φk(n,H) be the smallest number φ such that any k-edge-coloured graph G of order n, admits a monochromatic H-decomposition with at most φ parts. Here we stud...

We propose a discrete weighted Helmholtz decomposition for edge element functions. The decomposition is orthogonal in a weighted L inner product and stable uniformly with respect to the jumps in the discontinuous weight function. As an application, the new Helmholtz decomposition is applied to demonstrate the quasi-optimality of a preconditioned edge element system for solving a saddle-point Ma...

We study the decomposition of multigraphs with a constant edge multiplicity into copies of a fixed starH =K1,t :We present necessary and sufficient conditions for such a decomposition to exist where t = 2 and prove NP-completeness of the corresponding decision problem for any t 3.We also prove NP-completeness when the edge multiplicity function is not restricted either on the input G or on the ...

A P -decomposition of a graph G is a set of pairwise edge-disjoint paths of G with edges that cover the edge set of G. Kotzig (1957) proved that a 3-regular graph admits a P3-decomposition if and only if it contains a perfect matching, and also asked what are the necessary and sufficient conditions for an -regular graph to admit a P -decomposition, for odd . Let g, and m be positive integers wi...