CohomologyIn mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
Group cohomologyIn mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups .
Lie algebra cohomologyIn mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by to coefficients in an arbitrary Lie module. If is a compact simply connected Lie group, then it is determined by its Lie algebra, so it should be possible to calculate its cohomology from the Lie algebra.
LegA leg is a weight-bearing and locomotive anatomical structure, usually having a columnar shape. During locomotion, legs function as "extensible struts". The combination of movements at all joints can be modeled as a single, linear element capable of changing length and rotating about an omnidirectional "hip" joint. As an anatomical animal structure it is used for locomotion. The distal end is often modified to distribute force (such as a foot). Most animals have an even number of legs.
Newtonian fluidA Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of change of the fluid's velocity vector. A fluid is Newtonian only if the tensors that describe the viscous stress and the strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow.
Non-Newtonian fluidA non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, that is, it has variable viscosity dependent on stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for example, becomes runnier when shaken and is thus a non-Newtonian fluid. Many salt solutions and molten polymers are , as are many commonly found substances such as custard, toothpaste, starch suspensions, corn starch, paint, blood, melted butter, and shampoo.
Human legThe human leg is the entire lower limb of the human body, including the foot, thigh or sometimes even the hip or buttock region. The major bones of the leg are the femur (thigh bone), tibia (shin bone), and adjacent fibula. The thigh is between the hip and knee, while the calf (rear) and shin (front) are between the knee and foot. Legs are used for standing, many forms of human movement, recreation such as dancing, and constitute a significant portion of a person's mass.
EvolutionismEvolutionism is a term used (often derogatorily) to denote the theory of evolution. Its exact meaning has changed over time as the study of evolution has progressed. In the 19th century, it was used to describe the belief that organisms deliberately improved themselves through progressive inherited change (orthogenesis). The teleological belief went on to include cultural evolution and social evolution. In the 1970s, the term "Neo-Evolutionism" was used to describe the idea that "human beings sought to preserve a familiar style of life unless change was forced on them by factors that were beyond their control.
Theistic evolutionTheistic evolution (also known as theistic evolutionism or God-guided evolution) is a view that God acts and creates through laws of nature. It posits that the concept of God is compatible with the findings of modern science, including evolution. Theistic evolution is not in itself a scientific theory, but includes a range of views about how science relates to religious beliefs and the extent to which God intervenes. It rejects the strict creationist doctrines of special creation, but can include beliefs such as creation of the human soul.
Time evolutionTime evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called stateful systems). In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies is governed by the principles of classical mechanics. In their most rudimentary form, these principles express the relationship between forces acting on the bodies and their acceleration given by Newton's laws of motion.
Molecular evolutionMolecular evolution is the process of change in the sequence composition of cellular molecules such as DNA, RNA, and proteins across generations. The field of molecular evolution uses principles of evolutionary biology and population genetics to explain patterns in these changes. Major topics in molecular evolution concern the rates and impacts of single nucleotide changes, neutral evolution vs. natural selection, origins of new genes, the genetic nature of complex traits, the genetic basis of speciation, the evolution of development, and ways that evolutionary forces influence genomic and phenotypic changes.
EvolutionIn biology, evolution is the change in heritable characteristics of biological populations over successive generations. Evolution occurs when evolutionary processes such as natural selection (including sexual selection) and genetic drift act on genetic variation, resulting in certain characteristics becoming more or less common within a population over successive generations. The process of evolution has given rise to biodiversity at every level of biological organisation.
Objections to evolutionObjections to evolution have been raised since evolutionary ideas came to prominence in the 19th century. When Charles Darwin published his 1859 book On the Origin of Species, his theory of evolution (the idea that species arose through descent with modification from a single common ancestor in a process driven by natural selection) initially met opposition from scientists with different theories, but eventually came to receive overwhelming acceptance in the scientific community.
Quantum field theoryIn theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles.
Sociocultural evolutionSociocultural evolution, sociocultural evolutionism or social evolution are theories of sociobiology and cultural evolution that describe how societies and culture change over time. Whereas sociocultural development traces processes that tend to increase the complexity of a society or culture, sociocultural evolution also considers process that can lead to decreases in complexity (degeneration) or that can produce variation or proliferation without any seemingly significant changes in complexity (cladogenesis).
Rejection of evolution by religious groupsRecurring cultural, political, and theological rejection of evolution by religious groups exists regarding the origins of the Earth, of humanity, and of other life. In accordance with creationism, species were once widely believed to be fixed products of divine creation, but since the mid-19th century, evolution by natural selection has been established by the scientific community as an empirical scientific fact.
Sequence spaceIn functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication.
Banach spaceIn mathematics, more specifically in functional analysis, a Banach space (pronounced ˈbanax) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly.
Fréchet spaceIn functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically Banach spaces.
Algebraic quantum field theoryAlgebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by . The axioms are stated in terms of an algebra given for every open set in Minkowski space, and mappings between those. Let be the set of all open and bounded subsets of Minkowski space. An algebraic quantum field theory is defined via a net of von Neumann algebras on a common Hilbert space satisfying the following axioms: Isotony: implies .
