Measure (mathematics)In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge.
Beta decayIn nuclear physics, beta decay (β-decay) is a type of radioactive decay in which an atomic nucleus emits a beta particle (fast energetic electron or positron), transforming into an isobar of that nuclide. For example, beta decay of a neutron transforms it into a proton by the emission of an electron accompanied by an antineutrino; or, conversely a proton is converted into a neutron by the emission of a positron with a neutrino in so-called positron emission.
Σ-finite measureIn mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞), and a set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is a countable union of measurable sets each with finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.
Vector mesonIn high energy physics, a vector meson is a meson with total spin 1 and odd parity (usually noted as JP = 1−). Vector mesons have been seen in experiments since the 1960s, and are well known for their spectroscopic pattern of masses. The vector mesons contrast with the pseudovector mesons, which also have a total spin 1 but instead have even parity. The vector and pseudovector mesons are also dissimilar in that the spectroscopy of vector mesons tends to show nearly pure states of constituent quark flavors, whereas pseudovector mesons and scalar mesons tend to be expressed as composites of mixed states.
Radioactive decayRadioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is considered radioactive. Three of the most common types of decay are alpha, beta, and gamma decay, all of which involve emitting particles. The weak force is the mechanism that is responsible for beta decay, while the other two are governed by the electromagnetism and nuclear force.
Complete measureIn mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (X, Σ, μ) is complete if and only if The need to consider questions of completeness can be illustrated by considering the problem of product spaces. Suppose that we have already constructed Lebesgue measure on the real line: denote this measure space by We now wish to construct some two-dimensional Lebesgue measure on the plane as a product measure.
ColliderA collider is a type of particle accelerator that brings two opposing particle beams together such that the particles collide. Colliders may either be ring accelerators or linear accelerators. Colliders are used as a research tool in particle physics by accelerating particles to very high kinetic energy and letting them impact other particles. Analysis of the byproducts of these collisions gives scientists good evidence of the structure of the subatomic world and the laws of nature governing it.
Borel measureIn mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. Let be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of ; this is known as the σ-algebra of Borel sets. A Borel measure is any measure defined on the σ-algebra of Borel sets.
PiThe number pi (paɪ; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number pi appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern.
Electron captureElectron capture (K-electron capture, also K-capture, or L-electron capture, L-capture) is a process in which the proton-rich nucleus of an electrically neutral atom absorbs an inner atomic electron, usually from the K or L electron shells. This process thereby changes a nuclear proton to a neutron and simultaneously causes the emission of an electron neutrino. Proton + Electron → Neutron + Electron Neutrino or when written as a nuclear reaction equation, ^{0}{-1}e + ^{1}{1}p -> ^{1}{0}n + ^{0}{0} ν Since this single emitted neutrino carries the entire decay energy, it has this single characteristic energy.
