In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.
Filter, filtering or filters may refer to:
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements.
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective.
The Mohs scale of mineral hardness is a qualitative ordinal scale characterizing scratch resistance of various minerals through the ability of harder material to scratch softer material. Created in 1812 by German geologist and mineralogist Friedrich Mohs, it is one of several definitions of hardness in materials science, some of which are more quantitative. The method of comparing hardness by observing which minerals can scratch others is of great antiquity, having been mentioned by Theophrastus in his treatise On Stones, c. 300 BC, followed by Pliny the Elder in his Naturalis Historia, c. 77 AD. While greatly facilitating the identification of minerals in the field, the Mohs scale does not show how well hard materials perform in an industrial setting.
Mathematical analysis is the branch of mathematics dealing with limits
and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups
such that
. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading.
In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A is a collection of finitary operations on A; the set A with this structure is also called an algebra.
Phi is the 21st letter of the Greek alphabet.
In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space is a linear functional f on V so that for all positive elements v of V, that is v≥0, it holds that
In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra.
The ProgramByDesign project is an outreach effort of the PLT research group. The goal is to train college faculty, high school teachers, and possibly even middle school teachers, in programming and computing.
Connected Mathematics is a comprehensive mathematics program intended for U.S. students in grades 6-8. The curriculum design, text materials for students, and supporting resources for teachers were created and have been progressively refined by the Connected Mathematics Project (CMP) at Michigan State University with advice and contributions from many mathematics teachers, curriculum developers, mathematicians, and mathematics education researchers.
Functional Skills are qualifications that have been developed by the UK Government as part of an initiative to improve England’s literacy, numeracy and ICT skills. They are replacing Key Skills Level 1 and 2 qualifications from September 2012; in the short-term Entry Level Key Skills will remain as there is no current replacement – this is still in review.
Multi-function material is a composite material. The traditional approach to the development of structures is to address the loadcarrying function and other functional requirements separately. Recently, however, there has been increased interest in the development of load-bearing materials and structures which have integral non-load-bearing functions, guided by recent discoveries about how multifunctional biological systems work.
Millwork building materials are historically any woodmill-produced products for building construction. Stock profiled and patterned millwork building components fabricated by milling at a planing mill can usually be installed with minimal alteration. Today, millwork also encompasses items that are made using alternatives to wood, including synthetics, plastics, and wood-adhesive composites.
Bootstrap is based at Brown University (USA), and builds on the research and development done there. Bootstrap curriculum consists of 4 research-based curricular computer science modules for grades 6-12. The 4 modules are Bootstrap:Algebra, Bootstrap:Reactive, Bootstrap:Data Science, and Bootstrap:Physics. Bootstrap materials reinforce core concepts from mainstream subjects like Math, Physics and more, enabling non-CS teachers to adopt the introductory materials while delivering rigorous and engaging computing content drawn from Computer Science classes at universities like Brown, WPI, and Northeastern.
