Congruence (geometry)In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the of the other. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object.
Algorithmic artAlgorithmic art or algorithm art is art, mostly visual art, in which the design is generated by an algorithm. Algorithmic artists are sometimes called algorists. Algorithmic art, also known as computer-generated art, is a subset of generative art (generated by an autonomous system) and is related to systems art (influenced by systems theory). Fractal art is an example of algorithmic art. For an image of reasonable size, even the simplest algorithms require too much calculation for manual execution to be practical, and they are thus executed on either a single computer or on a cluster of computers.
Trial and errorTrial and error is a fundamental method of problem-solving characterized by repeated, varied attempts which are continued until success, or until the practicer stops trying. According to W.H. Thorpe, the term was devised by C. Lloyd Morgan (1852–1936) after trying out similar phrases "trial and failure" and "trial and practice". Under Morgan's Canon, animal behaviour should be explained in the simplest possible way. Where behavior seems to imply higher mental processes, it might be explained by trial-and-error learning.
Meta-optimizationIn numerical optimization, meta-optimization is the use of one optimization method to tune another optimization method. Meta-optimization is reported to have been used as early as in the late 1970s by Mercer and Sampson for finding optimal parameter settings of a genetic algorithm. Meta-optimization and related concepts are also known in the literature as meta-evolution, super-optimization, automated parameter calibration, hyper-heuristics, etc.
Mutually orthogonal Latin squaresIn combinatorial mathematics, two Latin squares of the same size (order) are said to be orthogonal if when superimposed the ordered paired entries in the positions are all distinct. A set of Latin squares, all of the same order, all pairs of which are orthogonal is called a set of mutually orthogonal Latin squares. This concept of orthogonality in combinatorics is strongly related to the concept of blocking in statistics, which ensures that independent variables are truly independent with no hidden confounding correlations.
Orthogonal arrayIn mathematics, an orthogonal array (more specifically, a fixed-level orthogonal array) is a "table" (array) whose entries come from a fixed finite set of symbols (for example, {1,2,...,v}), arranged in such a way that there is an integer t so that for every selection of t columns of the table, all ordered t-tuples of the symbols, formed by taking the entries in each row restricted to these columns, appear the same number of times. The number t is called the strength of the orthogonal array.
No free lunch in search and optimizationIn computational complexity and optimization the no free lunch theorem is a result that states that for certain types of mathematical problems, the computational cost of finding a solution, averaged over all problems in the class, is the same for any solution method. The name alludes to the saying "no such thing as a free lunch", that is, no method offers a "short cut". This is under the assumption that the search space is a probability density function. It does not apply to the case where the search space has underlying structure (e.
ParallelogramIn Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.
Pontryagin classIn mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Given a real vector bundle E over M, its k-th Pontryagin class is defined as where: denotes the -th Chern class of the complexification of E, is the -cohomology group of M with integer coefficients. The rational Pontryagin class is defined to be the image of in , the -cohomology group of M with rational coefficients.
Euler classIn mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic. It is named after Leonhard Euler because of this. Throughout this article is an oriented, real vector bundle of rank over a base space . The Euler class is an element of the integral cohomology group constructed as follows.
