MathematicsMathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them.
Projection (linear algebra)In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself (an endomorphism) such that . That is, whenever is applied twice to any vector, it gives the same result as if it were applied once (i.e. is idempotent). It leaves its unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.
Spin groupIn mathematics the spin group Spin(n) is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2) The group multiplication law on the double cover is given by lifting the multiplication on . As a Lie group, Spin(n) therefore shares its dimension, n(n − 1)/2, and its Lie algebra with the special orthogonal group. For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n).
Mathematical analysisAnalysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).
Mathematics educationIn contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out scholarly research into the transfer of mathematical knowledge. Although research into mathematics education is primarily concerned with the tools, methods, and approaches that facilitate practice or the study of practice, it also covers an extensive field of study encompassing a variety of different concepts, theories and methods.
Spin structureIn differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in the definition of any theory with uncharged fermions. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory.
Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
BeeBees are winged insects closely related to wasps and ants, known for their roles in pollination and, in the case of the best-known bee species, the western honey bee, for producing honey. Bees are a monophyletic lineage within the superfamily Apoidea. They are presently considered a clade, called Anthophila. There are over 16,000 known species of bees in seven recognized biological families. Some species - including honey bees, bumblebees, and stingless bees - live socially in colonies while most species (>90%) - including mason bees, carpenter bees, leafcutter bees, and sweat bees - are solitary.
Honey beeA honey bee (also spelled honeybee) is a eusocial flying insect within the genus Apis of the bee clade, all native to mainland Afro-Eurasia. After bees spread naturally throughout Africa and Eurasia, humans became responsible for the current cosmopolitan distribution of honey bees, introducing multiple subspecies into South America (early 16th century), North America (early 17th century), and Australia (early 19th century).
Queen beeA queen bee is typically an adult, mated female (gyne) that lives in a colony or hive of honey bees. With fully developed reproductive organs, the queen is usually the mother of most, if not all, of the bees in the beehive. Queens are developed from larvae selected by worker bees and specially fed in order to become sexually mature. There is normally only one adult, mated queen in a hive, in which case the bees will usually follow and fiercely protect her.
Bee stingA bee sting is the wound and pain caused by the stinger of a female bee puncturing skin. Bee stings differ from insect bites, with the venom of stinging insects having considerable chemical variation. The reaction of a person to a bee sting may vary according to the bee species. While bee stinger venom is slightly acidic and causes only mild pain in most people, allergic reactions may occur in people with allergies to venom components. A honey bee that is away from the hive foraging for nectar or pollen will rarely sting, except when stepped on or roughly handled.
Africanized beeThe Africanized bee, also known as the Africanized honey bee and colloquially as the "killer bee", is a hybrid of the western honey bee (Apis mellifera), produced originally by crossbreeding of the East African lowland honey bee (A. m. scutellata) with various European honey bee subspecies such as the Italian honey bee (A. m. ligustica) and the Iberian honey bee (A. m. iberiensis). The East African lowland honey bee was first introduced to Brazil in 1956 in an effort to increase honey production, but 26 swarms escaped quarantine in 1957.
