Inverse image functorIn mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map , the inverse image functor is a functor from the of sheaves on Y to the category of sheaves on X. The is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features. Suppose we are given a sheaf on and that we want to transport to using a continuous map .
Pontryagin dualityIn mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive group of the integers (also with the discrete topology), the real numbers, and every finite dimensional vector space over the reals or a p-adic field.
MIT LicenseThe MIT License is a permissive free software license originating at the Massachusetts Institute of Technology (MIT) in the late 1980s. As a permissive license, it puts only very limited restriction on reuse and has, therefore, high license compatibility. Unlike copyleft software licenses, the MIT License also permits reuse within proprietary software, provided that all copies of the software or its substantial portions include a copy of the terms of the MIT License and also a copyright notice.
ElsevierElsevier (ˈɛlzəviːr) is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as The Lancet, Cell, the ScienceDirect collection of electronic journals, Trends, the Current Opinion series, the online citation database Scopus, the SciVal tool for measuring research performance, the ClinicalKey search engine for clinicians, and the ClinicalPath evidence-based cancer care service.
Sigma-additive set functionIn mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of k disjoint sets (where k is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely additive set function (the terms are equivalent).
Geometric latticeIn the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumption of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of flats of finite, or finite and infinite, matroids, and every geometric or matroid lattice comes from a matroid in this way. A lattice is a poset in which any two elements and have both a least upper bound, called the join or supremum, denoted by , and a greatest lower bound, called the meet or infimum, denoted by .
Stone dualityIn mathematics, there is an ample supply of categorical dualities between certain of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation theorem for Boolean algebras. These concepts are named in honor of Marshall Stone. Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics.
Large cardinalIn the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=ωα). The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results.
Poincaré dualityIn mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the ()th homology group of M, for all integers k Poincaré duality holds for any coefficient ring, so long as one has taken an orientation with respect to that coefficient ring; in particular, since every manifold has a unique orientation mod 2, Poincaré duality holds mod 2 without any assumption of orientation.
Apache LicenseThe Apache License is a permissive free software license written by the Apache Software Foundation (ASF). It allows users to use the software for any purpose, to distribute it, to modify it, and to distribute modified versions of the software under the terms of the license, without concern for royalties. The ASF and its projects release their software products under the Apache License. The license is also used by many non-ASF projects. Beginning in 1995, the Apache Group (later the Apache Software Foundation) released successive versions of the Apache HTTP Server.
