number of ordered triples (a, b, c) such that ab ∈ E(G) and bc / ∈ E(G). Observe that each vertex v has deg v = 0, 1, 2, 3, 4, or 5. If deg v is 0 or 5, then v is the second vertex in 0 such triples. If deg v is 1 or 4, then v is the second vertex in 4 such triples. If deg v is 2 or 3, then v is the second vertex in 6 such triples. Thus, there are at most 36 such ordered triples. Note that each...

In the spirit of the HSV color space, we introduce a simple transform of the RGB color cube into a light/dark dimension and a 2D hue. The hue is a normalized (barycentric) triangle with pure red, green, and blue at the vertices, often called the Maxwell Color Triangle. Each cross section of the space is the same barycentric triangle, and the light/dark dimension runs zero to one for each hue so...

This article describes a new hierarchical model for representing a terrain. The model, called a Hierarchical Triangulated Irregular Network (HTIN), is a method for compression of spatial data and representation of a topographic surface at successively finer levels of detail. A HTIN is a hierarchy of triangle-based surface approximations, where each node, except for the root, is a triangulated i...

Let A', B', C' be the feet of the altitudes of an acute-angled triangle ABC (A' E BC, B' E AC, C' E AB). Let a',b',b' denote the sides of the podaire triangle A' B' C'. Smaranda.che's Podaire theorem [2] (see [1]) sta.tes tha.t La'b' < ~ L a (1) where a, b, c are the sides of the triangle ABC. Our aim is to improve (1) in the following form: (2) First we need the following a.uxiliary propositio...

The famous Morley theorem says that the adjacent angle trisectors of a triangle form an equilateral triangle. We recall some known proofs and provide several new. In hyperbolic geometry, we compute the side lengths of the associated Morley triangle and show that the limit is the Euclidean (flat) equilateral case. The perspectivity properties hold also in general. We introduce new invariants suc...

The current mode of ocean governance in the biogeographically defined space Coral Triangle emerged due to framing marine degradation as a de-bounded risk with transboundary nature. This justified rescaling issue’s from national regional. article will explore how form Initiative on Reefs, Fisheries, and Food Security (CTI-CFF) is an example regional multi-level regulatory arrangement based disag...

This research investigated the source of an ostensible reflexivity effect in pigeons reported by Sweeney and Urcuioli (2010). In Experiment 1, pigeons learned two symmetrically reinforced symbolic successive matching tasks (hue-form and form-hue) using red-green and triangle-horizontal line stimuli. They differed in their third concurrently trained baseline task: form-form matching with stimuli...

In this paper, we enlarge the language of triangle algebra by addinga unary operation that describes properties of a state. Thesestructure algebras are called state triangle algebra. The vitalproperties of these algebras are given. The notion of state interval-valued residuated lattice (IVRL)-filters are introduced and givesome examples and properties of them are given. ...