Proof by contradictionIn logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved.
Lipschitz continuityIn mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity).
Constructive proofIn mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.
Proof by exhaustionProof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. This is a method of direct proof. A proof by exhaustion typically contains two stages: A proof that the set of cases is exhaustive; i.e.
Polish spaceIn the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations.
Metric mapIn the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance. These maps are the morphisms in the , Met. Such functions are always continuous functions. They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps. Specifically, suppose that and are metric spaces and is a function from to . Thus we have a metric map when, for any points and in , Here and denote the metrics on and respectively.
Compact operatorIn functional analysis, a branch of mathematics, a compact operator is a linear operator , where are normed vector spaces, with the property that maps bounded subsets of to relatively compact subsets of (subsets with compact closure in ). Such an operator is necessarily a bounded operator, and so continuous. Some authors require that are Banach, but the definition can be extended to more general spaces. Any bounded operator that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalization of the class of finite-rank operators in an infinite-dimensional setting.
Contraction mappingIn mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number such that for all x and y in M, The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for k ≤ 1, then the mapping is said to be a non-expansive map. More generally, the idea of a contractive mapping can be defined for maps between metric spaces.
History of the SATThe SAT is a standardized test commonly used for the purpose of admission to colleges and universities in the United States. The test, owned by the College Board and originally developed by Carl Brigham, was first administered on June 23, 1926, to about 8,000 students. The test was introduced as a supplement to the College Board essay exams already in use for college admissions, but ease of administration of the SAT and other factors led to the discontinuation of the essay exams during World War II.
North Hollywood High SchoolNorth Hollywood High School (NHHS) is a public high school in the North Hollywood neighborhood of Los Angeles, California, United States. It is in the San Fernando Valley and enrolls approximately 2,500 students. Several neighborhoods, including most of North Hollywood, Valley Village, Studio City and Sun Valley, send students to it. It is accredited by the Western Association of Schools and Colleges. Its principal is Ricardo Rosales. As of 2023, the campus is undergoing construction, and many buildings are in transition.
NK Lokomotiva ZagrebNogometni klub Lokomotiva Zagreb (Lokomotiva Zagreb Football Club), commonly known as Lokomotiva Zagreb or simply Lokomotiva, is a Croatian professional football club based in Zagreb. It competes in the Croatian First Football League, the country's top division. Founded in 1914, the club's only period of success came in the late 1940s and early 1950s before spending most of the following five decades in lower-level leagues. Between 2007 and 2009 they won three consecutive promotions to rise from the fourth level to the first in the Croatian football league system.
Bensonhurst, BrooklynBensonhurst is a residential neighborhood in the southwestern section of the New York City borough of Brooklyn. The neighborhood is bordered on the northwest by 14th Avenue, on the northeast by 60th Street, on the southeast by Avenue P and 22nd Avenue (Bay Parkway) and on the southwest by 86th Street. It is adjacent to the neighborhoods of Dyker Heights to the northwest, Borough Park and Mapleton to the northeast, Bath Beach to the southwest, and Gravesend to the southeast. Bensonhurst contains several major ethnic enclaves.
