Computer-assisted proofA computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program.
Field extensionIn mathematics, particularly in algebra, a field extension is a pair of fields such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry.
One-sided limitIn calculus, a one-sided limit refers to either one of the two limits of a function of a real variable as approaches a specified point either from the left or from the right. The limit as decreases in value approaching ( approaches "from the right" or "from above") can be denoted: The limit as increases in value approaching ( approaches "from the left" or "from below") can be denoted: If the limit of as approaches exists then the limits from the left and from the right both exist and are equal.
RiceRice is the seed of the grass species Oryza sativa (Asian rice) or, less commonly, O. glaberrima (African rice). The name wild rice is usually used for species of the genera Zizania and Porteresia, both wild and domesticated, although the term may also be used for primitive or uncultivated varieties of Oryza. As a cereal grain, domesticated rice is the most widely consumed staple food for over half of the world's human population, particularly in Asia and Africa.
Proofs and RefutationsProofs and Refutations: The Logic of Mathematical Discovery is a 1976 book by philosopher Imre Lakatos expounding his view of the progress of mathematics. The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron. A central theme is that definitions are not carved in stone, but often have to be patched up in the light of later insights, in particular failed proofs. This gives mathematics a somewhat experimental flavour.
Analytic functionIn mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about converges to the function in some neighborhood for every in its domain.
Proof theoryProof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
Brown riceBrown rice is a whole grain rice with the inedible outer hull removed. This kind of rice sheds its outer hull or husk but the bran and germ layer remain on, constituting the brown or tan colour of rice. White rice is the same grain without the hull, the bran layer, and the cereal germ. Red rice, gold rice, and black rice (also called purple rice) are all whole rices with differently pigmented outer layers. Brown rice generally needs longer cooking times than white rice, unless it is broken or flour blasted (which perforates the bran without removing it).
Limit inferior and limit superiorIn mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.
Differentiable manifoldIn mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.
Wild riceWild rice, also called manoomin, mnomen, Canada rice, Indian rice, or water oats, is any of four species of grasses that form the genus Zizania, and the grain that can be harvested from them. The grain was historically gathered and eaten in both North America and China, but eaten less in China, where the plant's stem is used as a vegetable. Wild rice is not directly related to domesticated rice (Oryza sativa and Oryza glaberrima), although they are close cousins, all belonging to the tribe Oryzeae.
Rice hullsRice hulls (or rice husks) are the hard protecting coverings of grains of rice. In addition to protecting rice during the growing season, rice hulls can be put to use as building material, fertilizer, insulation material, or fuel. Rice hulls are part of the chaff of the rice. Rice hulls are part of the rice seed. The hull protects the grain during the growing season from pests. The hull is formed from hard materials, including opaline silica and lignin.
