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Cylinder sets are often used to define a measure, using the Kolmogorov extension theorem; for example, the measure of a cylinder set of length m might be given by 1/m or by 1/2 m. Cylinder sets may be used to define a metric on the space: for example, one says that two strings are ε-close if a fraction 1−ε of the letters in the strings match.
The generation of a bicylinder Calculating the volume of a bicylinder. A bicylinder generated by two cylinders with radius r has the volume =, and the surface area [1] [6] =.. The upper half of a bicylinder is the square case of a domical vault, a dome-shaped solid based on any convex polygon whose cross-sections are similar copies of the polygon, and analogous formulas calculating the volume ...
In some elementary treatments, a cylinder always means a circular cylinder. [2] An open cylinder is a cylindrical surface without the bases. The height (or altitude) of a cylinder is the perpendicular distance between its bases. The cylinder obtained by rotating a line segment about a fixed line that it is parallel to is a cylinder of ...
MapReduce is a programming model and an associated implementation for processing and generating big data sets with a parallel and distributed algorithm on a cluster. [1] [2] [3]A MapReduce program is composed of a map procedure, which performs filtering and sorting (such as sorting students by first name into queues, one queue for each name), and a reduce method, which performs a summary ...
For a product space, the cylinder σ-algebra is the one that is generated by cylinder sets. In the context of a Banach space X , {\displaystyle X,} the cylindrical σ-algebra A ( X , X ′ ) {\displaystyle {\mathfrak {A}}(X,X')} is defined to be the coarsest σ-algebra (that is, the one with the fewest measurable sets) such that every ...
This is a list of two-dimensional geometric shapes in Euclidean and other geometries. For mathematical objects in more dimensions, see list of mathematical shapes. For a broader scope, see list of shapes.
Simple examples. A simple example of a regular surface is given by the 2-sphere {(x, y, z) | x 2 + y 2 + z 2 = 1}; this surface can be covered by six Monge patches (two of each of the three types given above), taking h(u, v) = ± (1 − u 2 − v 2) 1/2. It can also be covered by two local parametrizations, using stereographic projection.
One can use the mapping cylinder to construct homotopy colimits: [citation needed] this follows from the general statement that any category with all pushouts and coequalizers has all colimits. That is, given a diagram, replace the maps by cofibrations (using the mapping cylinder) and then take the ordinary pointwise limit (one must take a bit ...