Air (classical element)Air or Wind is one of the four classical elements along with water, earth and fire in ancient Greek philosophy and in Western alchemy. According to Plato, it is associated with the octahedron; air is considered to be both hot and wet. The ancient Greeks used two words for air: aer meant the dim lower atmosphere, and aether meant the bright upper atmosphere above the clouds. Plato, for instance writes that "So it is with air: there is the brightest variety which we call aether, the muddiest which we call mist and darkness, and other kinds for which we have no name.
Aether (classical element)According to ancient and medieval science, aether (ˈiːθər, alternative spellings include æther, aither, and ether), also known as the fifth element or quintessence, is the material that fills the region of the universe beyond the terrestrial sphere. The concept of aether was used in several theories to explain several natural phenomena, such as the propagation of light and gravity.
Classical elementThe classical elements typically refer to earth, water, air, fire, and (later) aether which were proposed to explain the nature and complexity of all matter in terms of simpler substances. Ancient cultures in Greece, Tibet, and India had similar lists which sometimes referred, in local languages, to "air" as "wind" and the fifth element as "void". These different cultures and even individual philosophers had widely varying explanations concerning their attributes and how they related to observable phenomena as well as cosmology.
Earth (classical element)Earth is one of the classical elements, in some systems being one of the four along with air, fire, and water. Earth is one of the four classical elements in ancient Greek philosophy and science. It was commonly associated with qualities of heaviness, matter and the terrestrial world. Due to the hero cults, and chthonic underworld deities, the element of earth is also associated with the sensual aspects of both life and death in later occultism. Empedocles of Acragas (c. 495 – c.
Water (classical element)Water is one of the classical elements in ancient Greek philosophy along with air, earth and fire, in the Asian Indian system Panchamahabhuta, and in the Chinese cosmological and physiological system Wu Xing. In contemporary esoteric traditions, it is commonly associated with the qualities of emotion and intuition. Water was one of many archai proposed by the Pre-socratics, most of whom tried to reduce all things to a single substance. However, Empedocles of Acragas (c. 495 – c.
Uniform continuityIn mathematics, a real function of real numbers is said to be uniformly continuous if there is a positive real number such that function values over any function domain interval of the size are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number , then there is a positive real number such that at any and in any function interval of the size .
Lipschitz continuityIn mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity).
Absolute continuityIn calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized (by the fundamental theorem of calculus) in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration.
Continuous functionIn mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is .
Semi-continuityIn mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function is upper (respectively, lower) semicontinuous at a point if, roughly speaking, the function values for arguments near are not much higher (respectively, lower) than A function is continuous if and only if it is both upper and lower semicontinuous.
Modulus of continuityIn mathematical analysis, a modulus of continuity is a function ω : [0, ∞] → [0, ∞] used to measure quantitatively the uniform continuity of functions. So, a function f : I → R admits ω as a modulus of continuity if and only if for all x and y in the domain of f. Since moduli of continuity are required to be infinitesimal at 0, a function turns out to be uniformly continuous if and only if it admits a modulus of continuity. Moreover, relevance to the notion is given by the fact that sets of functions sharing the same modulus of continuity are exactly equicontinuous families.
Inequality (mathematics)In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities: The notation a < b means that a is less than b. The notation a > b means that a is greater than b. In either case, a is not equal to b. These relations are known as strict inequalities, meaning that a is strictly less than or strictly greater than b.
SmoothnessIn mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called differentiability class. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or function).
Triangle inequalityIn mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If x, y, and z are the lengths of the sides of the triangle, with no side being greater than z, then the triangle inequality states that with equality only in the degenerate case of a triangle with zero area.
Set theorySet theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory.
Set (mathematics)A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics.
Empty setIn mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called non-empty. In some textbooks and popularizations, the empty set is referred to as the "null set".
Fuzzy setIn mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of L-relations when L is the unit interval [0, 1].
Bump mappingBump mapping is a texture mapping technique in computer graphics for simulating bumps and wrinkles on the surface of an object. This is achieved by perturbing the surface normals of the object and using the perturbed normal during lighting calculations. The result is an apparently bumpy surface rather than a smooth surface although the surface of the underlying object is not changed. Bump mapping was introduced by James Blinn in 1978. Normal mapping is the most common variation of bump mapping used.
Texture mappingTexture mapping is a method for mapping a texture on a . Texture here can be high frequency detail, surface texture, or color. The original technique was pioneered by Edwin Catmull in 1974. Texture mapping originally referred to diffuse mapping, a method that simply mapped pixels from a texture to a 3D surface ("wrapping" the image around the object).
