Finitely generated moduleIn mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R, or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide.
Free moduleIn mathematics, a free module is a module that has a basis, that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set S and ring R, there is a free R-module with basis S, which is called the free module on S or module of formal R-linear combinations of the elements of S. A free abelian group is precisely a free module over the ring Z of integers.
Projective moduleIn mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains.
Cyclic moduleIn mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z-module) that is generated by one element. A left R-module M is called cyclic if M can be generated by a single element i.e. M = (x) = Rx = {rx r ∈ R} for some x in M. Similarly, a right R-module N is cyclic if N = yR for some y ∈ N. 2Z as a Z-module is a cyclic module.
Olive oilOlive oil is a liquid fat obtained by pressing whole olives, the fruit of Olea europaea, a traditional tree crop of the Mediterranean Basin, and extracting the oil. It is commonly used in cooking for frying foods or as a salad dressing. It can also be found in some cosmetics, pharmaceuticals, soaps, and fuels for traditional oil lamps. It also has additional uses in some religions. The olive is one of three core food plants in Mediterranean cuisine, together with wheat and grapes.
ViscosityThe viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity is defined scientifically as a force multiplied by a time divided by an area. Thus its SI units are newton-seconds per square metre, or pascal-seconds. Viscosity quantifies the internal frictional force between adjacent layers of fluid that are in relative motion.
1973 in FranceEvents from the year 1973 in France. President: Georges Pompidou Prime Minister: Pierre Messmer 4 March – Legislative Election held. 11 March – Legislative Election held. 3 June – A Tupolev Tu-144 supersonic aircraft crashes at the Paris air show; 14 are killed. 11 July – Varig Flight 820 Boeing 707 crashes near Orly Airport, Paris, resulting in 123 deaths, with 11 survivors. 20 July – France resumes nuclear bomb tests in Mururoa Atoll, over the protests of Australia and New Zealand. 23 September – Cantonales Elections held.
Cinema of FranceFrench cinema consists of the film industry and its film productions, whether made within the nation of France or by French film production companies abroad. It is the oldest and largest precursor of national cinemas in Europe; with primary influence also on the creation of national cinemas in Asia. France continues to have a particularly strong film industry, due in part to protections afforded by the French government. In 2013, France was the second largest exporter of films in the world after the United States.
Alpes-de-Haute-ProvenceAlpes-de-Haute-Provence or sometimes abbreviated as AHP (alp də ot pʁɔvɑ̃s; Aups d'Auta Provença; Alps of Upper Provence) is a department in the Provence-Alpes-Côte d'Azur region of France, bordering Alpes-Maritimes and Italy to the east, Var to the south, Vaucluse to the west, Drôme and Hautes-Alpes to the north. Formerly part of the province of Provence, it had a population of 164,308 in 2019, which makes it the 94th most populated French department.
Barneville-CarteretBarneville-Carteret (baʁnəvil kaʁtʁɛ) is a commune in the Manche department in the Normandy region of north-western France. For many years it has been a popular seaside resort destination. The commune resulted from a merger of two communes in 1964: Barneville-sur-Mer and Carteret whose port has ferry connections to the Channel Islands. The inhabitants of the commune are known as Barnevillais or Barnevillaises and Carteretais or Carteretaises.
Differential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative.
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).
