In mathematics, an abstract polytope is an algebraic partially ordered set or poset which captures the combinatorial properties of a traditional polytope, but not any purely geometric properties such as angles, edge lengths, etc. An ordinary geometric polytope is said to be a realization in some real N-dimensional space, typically Euclidean, of the corresponding abstract polytope. The abstract definition allows some more general combinatorial structures than traditional definitions of a polytope, thus allowing many new objects that have no counterpart in traditional theory. The term polytope is...

In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. == Definition == The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of Henri Lebesgue.A modern definition is as follows. An open cover of a topological space X is a family of open sets whose union contains X. The ply or order of a cover is the smallest number n (if it exists) such that each point of the space belongs to, at most, n sets in the cover...

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube...

In topology, a discipline within mathematics, a negative-dimensional space is an extension of the usual notion of space, allowing for negative dimensions. The concept of negative-dimensional spaces is applied, for example, to analyze linguistic statistics. == Definition == Suppose that Mt0 is a compact space of Hausdorff dimension t0, which is an element of a scale of compact spaces embedded in each other and parametrized by t (0 < t < ∞). Such scales are considered equivalent with respect to Mt0 if the compact spaces constituting them coincide for t ≥ t0. It is said that the compact space Mt0...

African trypanosomiasis, also known as African sleeping sickness or simply sleeping sickness, is an insect-borne parasitic disease of humans and other animals. It is caused by protozoa of the species Trypanosoma brucei. Humans are infected by two types, Trypanosoma brucei gambiense (TbG) and Trypanosoma brucei rhodesiense (TbR). TbG causes over 98% of reported cases. Both are usually transmitted by the bite of an infected tsetse fly and are most common in rural areas.Initially, the first stage of the disease is characterized by fevers, headaches, itchiness, and joint pains, beginning one to three...