Ordinary differential equationIn mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a_0(x), .
Linear differential equationIn mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x. Such an equation is an ordinary differential equation (ODE).
Differential equationIn mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Homogeneous differential equationA differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written where f and g are homogeneous functions of the same degree of x and y. In this case, the change of variable y = ux leads to an equation of the form which is easy to solve by integration of the two members. Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives.
Numerical methods for ordinary differential equationsNumerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.
Stochastic differential equationA stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices, random growth models or physical systems that are subjected to thermal fluctuations. SDEs have a random differential that is in the most basic case random white noise calculated as the derivative of a Brownian motion or more generally a semimartingale.
Partial differential equationIn mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x2 − 3x + 2 = 0. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations.
Socialist calculation debateThe socialist calculation debate, sometimes known as the economic calculation debate, was a discourse on the subject of how a socialist economy would perform economic calculation given the absence of the law of value, money, financial prices for capital goods and private ownership of the means of production. More specifically, the debate was centered on the application of economic planning for the allocation of the means of production as a substitute for capital markets and whether or not such an arrangement would be superior to capitalism in terms of efficiency and productivity.
Bessel functionBessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation for an arbitrary complex number , which represents the order of the Bessel function. Although and produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of . The most important cases are when is an integer or half-integer.
Function (mathematics)In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).
Calculation in kindNOTOC Calculation in kind or calculation in-natura is a way of valuating resources and a system of accounting that uses disaggregated physical magnitudes as opposed to a common unit of calculation. As the basis for a socialist economy, it was proposed to replace money and financial calculation. In an in-kind economy products are produced for their use values (their utility) and accounted in physical terms. By contrast, in money-based economies, commodities are produced for their exchange value and accounted in monetary terms.
Inverse functionIn mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by For a function , its inverse admits an explicit description: it sends each element to the unique element such that f(x) = y. As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result.
Economic calculation problemThe economic calculation problem (sometimes abbreviated ECP) is a criticism of using economic planning as a substitute for market-based allocation of the factors of production. It was first proposed by Ludwig von Mises in his 1920 article "Economic Calculation in the Socialist Commonwealth" and later expanded upon by Friedrich Hayek. In his first article, Mises described the nature of the price system under capitalism and described how individual subjective values (while criticizing other theories of value) are translated into the objective information necessary for rational allocation of resources in society.
EdoEdo (), also romanized as Jedo, Yedo or Yeddo, is the former name of Tokyo. Edo, formerly a jōkamachi (castle town) centered on Edo Castle located in Musashi Province, became the de facto capital of Japan from 1603 as the seat of the Tokugawa shogunate. Edo grew to become one of the largest cities in the world under the Tokugawa. After the Meiji Restoration in 1868 the Meiji government renamed Edo as Tokyo (, "Eastern Capital") and relocated the Emperor from the historic capital of Kyoto to the city.
Edo periodThe Edo period or Tokugawa period is the period between 1603 and 1867 in the history of Japan, when Japan was under the rule of the Tokugawa shogunate and the country's 300 regional daimyo. Emerging from the chaos of the Sengoku period, the Edo period was characterized by economic growth, strict social order, isolationist foreign policies, a stable population, perpetual peace, and popular enjoyment of arts and culture, colloquially referred to as Oedo.
Edo CastleEdo Castle is a flatland castle that was built in 1457 by Ōta Dōkan in Edo, Toshima District, Musashi Province. In modern times it is part of the Tokyo Imperial Palace in Chiyoda, Tokyo, and is therefore also known as Chiyoda Castle. Tokugawa Ieyasu established the Tokugawa shogunate there, and it was the residence of the shōgun and the headquarters of the military government during the Edo period (1603–1867) in Japanese history. After the resignation of the shōgun and the Meiji Restoration, it became the Tokyo Imperial Palace.
Tokugawa shogunateThe Tokugawa shogunate (ˌtɒkuːˈgɑːwə ; Tokugawa bakufu, tokɯŋawa baꜜkɯ̥ɸɯ), also known as the Edo shogunate, was the military government of Japan during the Edo period from 1603 to 1868. The Tokugawa shogunate was established by Tokugawa Ieyasu after victory at the Battle of Sekigahara, ending the civil wars of the Sengoku period following the collapse of the Ashikaga shogunate. Ieyasu became the shōgun, and the Tokugawa clan governed Japan from Edo Castle in the eastern city of Edo (Tokyo) along with the daimyō lords of the samurai class.
Edo societyEdo society refers to the society of Japan under the rule of the Tokugawa Shogunate during the Edo period from 1603 to 1868. Edo society was a feudal society with strict social stratification, customs, and regulations intended to promote political stability. Japanese people were assigned into a hierarchy of social classes based on the Four Occupations that were hereditary. The Emperor of Japan and the kuge were the official ruling class of Japan but had no power.
Cours-la-ReineThe Cours-la-Reine, also spelled Cours la Reine (without hyphens), is a public park and garden promenade located along the River Seine, between the Place de la Concorde and the Place du Canada, in the 8th arrondissement of Paris. It is one of the oldest parks in Paris, created in 1616 by Queen Marie de Medicis. The further extension of the garden between Place du Canada to Place d'Alma is called the Cours Albert Premier. Queen Marie de Medicis, nostalgic for the gardens of her native Florence, created the Cours-la-Reine not long after she began making the Luxembourg Garden (1612-1630).
2019 Maharashtra Legislative Assembly electionThe 2019 Maharashtra Legislative Assembly election was held on 21 October 2019 to elect all 288 members of the state's Legislative Assembly. After a 61.4% turnout in the election, the ruling National Democratic Alliance (NDA) of the Bharatiya Janata Party (BJP) and Shiv Sena (SHS) won a majority. Following differences over the government formation, the alliance was dissolved, precipitating a political crisis. Since a council of ministers had not been formed after no party could manage to form the government, President's rule was imposed in the state.
