Σ-algebraIn mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair is called a measurable space. The σ-algebras are a subset of the set algebras; elements of the latter only need to be closed under the union or intersection of finitely many subsets, which is a weaker condition.
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.
Inflation (cosmology)In physical cosmology, cosmic inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the early universe. The inflationary epoch is believed to have lasted from seconds to between and seconds after the Big Bang. Following the inflationary period, the universe continued to expand, but at a slower rate. The acceleration of this expansion due to dark energy began after the universe was already over 7.7 billion years old (5.4 billion years ago).
De Sitter universeA de Sitter universe is a cosmological solution to the Einstein field equations of general relativity, named after Willem de Sitter. It models the universe as spatially flat and neglects ordinary matter, so the dynamics of the universe are dominated by the cosmological constant, thought to correspond to dark energy in our universe or the inflaton field in the early universe.
De Sitter spaceIn mathematical physics, n-dimensional de Sitter space (often abbreviated to dSn) is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an n-sphere (with its canonical Riemannian metric). The main application of de Sitter space is its use in general relativity, where it serves as one of the simplest mathematical models of the universe consistent with the observed accelerating expansion of the universe.
Willem de SitterWillem de Sitter (6 May 1872 – 20 November 1934) was a Dutch mathematician, physicist, and astronomer. Born in Sneek, de Sitter studied mathematics at the University of Groningen and then joined the Groningen astronomical laboratory. He worked at the Cape Observatory in South Africa (1897–1899). Then, in 1908, de Sitter was appointed to the chair of astronomy at Leiden University. He was director of the Leiden Observatory from 1919 until his death. De Sitter made major contributions to the field of physical cosmology.
InflationIn economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reduction in the purchasing power of money. The opposite of inflation is deflation, a decrease in the general price level of goods and services. The common measure of inflation is the inflation rate, the annualized percentage change in a general price index.
Inflation targetingIn macroeconomics, inflation targeting is a monetary policy where a central bank follows an explicit target for the inflation rate for the medium-term and announces this inflation target to the public. The assumption is that the best that monetary policy can do to support long-term growth of the economy is to maintain price stability, and price stability is achieved by controlling inflation. The central bank uses interest rates as its main short-term monetary instrument.
Monetary inflationMonetary inflation is a sustained increase in the money supply of a country (or currency area). Depending on many factors, especially public expectations, the fundamental state and development of the economy, and the transmission mechanism, it is likely to result in price inflation, which is usually just called "inflation", which is a rise in the general level of prices of goods and services.Michael F. Bryan, On the Origin and Evolution of the Word "Inflation", clevelandfed.
Cosmic microwave backgroundThe cosmic microwave background (CMB, CMBR) is microwave radiation that fills all space in the observable universe. It is a remnant that provides an important source of data on the primordial universe. With a standard optical telescope, the background space between stars and galaxies is almost completely dark. However, a sufficiently sensitive radio telescope detects a faint background glow that is almost uniform and is not associated with any star, galaxy, or other object.
Eternal inflationEternal inflation is a hypothetical inflationary universe model, which is itself an outgrowth or extension of the Big Bang theory. According to eternal inflation, the inflationary phase of the universe's expansion lasts forever throughout most of the universe. Because the regions expand exponentially rapidly, most of the volume of the universe at any given time is inflating. Eternal inflation, therefore, produces a hypothetically infinite multiverse, in which only an insignificant fractal volume ends inflation.
Field (physics)In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. For example, on a weather map, the surface temperature is described by assigning a number to each point on the map; the temperature can be considered at a certain point in time or over some interval of time, to study the dynamics of temperature change. A surface wind map, assigning an arrow to each point on a map that describes the wind speed and direction at that point, is an example of a vector field, i.
OscillationOscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current. Oscillations can be used in physics to approximate complex interactions, such as those between atoms.
Asset price inflationAsset price inflation is the economic phenomenon whereby the price of assets rise and become inflated. A common reason for higher asset prices is low interest rates. When interest rates are low, investors and savers cannot make easy returns using low-risk methods such as government bonds or savings accounts. To still get a return on their money, investors instead have to buy up other assets such as stocks and real estate, thereby bidding up the price and creating asset price inflation.
Demand-pull inflationDemand-pull inflation occurs to arise when aggregate demand in an economy is more than aggregate supply. It involves inflation rising as real gross domestic product rises and unemployment falls, as the economy moves along the Phillips curve. This is commonly described as "too much money chasing too few goods". More accurately, it should be described as involving "too much money spent chasing too few goods", since only money that is spent on goods and services can cause inflation.
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.
Harmonic oscillatorIn classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x: where k is a positive constant. If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
Classical field theoryA classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically intended to describe electromagnetism and gravitation, two of the fundamental forces of nature. A physical field can be thought of as the assignment of a physical quantity at each point of space and time.
IP addressAn Internet Protocol address (IP address) is a numerical label such as 192.0.2.1 that is connected to a computer network that uses the Internet Protocol for communication. An IP address serves two main functions: network interface identification, and location addressing. Internet Protocol version 4 (IPv4) defines an IP address as a 32-bit number. However, because of the growth of the Internet and the depletion of available IPv4 addresses, a new version of IP (IPv6), using 128 bits for the IP address, was standardized in 1998.
IP PBXAn IP PBX ("Internet Protocol private branch exchange") is a system that connects telephone extensions to the public switched telephone network (PSTN) and provides internal communication for a business. An IP PBX is a PBX system with IP connectivity and may provide additional audio, video, or instant messaging communication utilizing the TCP/IP protocol stack. Voice over IP (VoIP) gateways can be combined with traditional PBX functionality to allow businesses to use their managed intranet to help reduce long distance expenses and take advantage of the benefits of a single network for voice and data (converged network).
