|
Limiting Case (mathematics)
In mathematics, a limiting case of a mathematical object is a special case that arises when one or more components of the object take on their most extreme possible values. For example: * In statistics, the limiting case of the binomial distribution is the Poisson distribution. As the number of events tends to infinity in the binomial distribution, the random variable changes from the binomial to the Poisson distribution. *A circle is a limiting case of various other figures, including the Cartesian oval, the ellipse, the superellipse, and the Cassini oval. Each type of figure is a circle for certain values of the defining parameters, and the generic figure appears more like a circle as the limiting values are approached. *Archimedes calculated an approximate value of π by treating the circle as the limiting case of a regular polygon with 3 × 2''n'' sides, as ''n'' gets large. *In electricity and magnetism, the long wavelength limit is the limiting case when the wave ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
 |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
 |
Demand Curve
A demand curve is a graph depicting the inverse demand function, a relationship between the price of a certain commodity (the ''y''-axis) and the quantity of that commodity that is demanded at that price (the ''x''-axis). Demand curves can be used either for the price-quantity relationship for an individual consumer (an individual demand curve), or for all consumers in a particular market (a market demand curve). It is generally assumed that demand curves slope down, as shown in the adjacent image. This is because of the law of demand: for most goods, the quantity demanded falls if the price rises. Certain unusual situations do not follow this law. These include Veblen goods, Giffen goods, and speculative bubbles where buyers are attracted to a commodity if its price rises. Demand curves are used to estimate behaviour in competitive markets and are often combined with supply curves to find the equilibrium price (the price at which sellers together are willing to sell the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
 |
Hyperbola
In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component (topology), connected components or branches, that are mirror images of each other and resemble two infinite bow (weapon), bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane (mathematics), plane and a double cone (geometry), cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Besides being a conic section, a hyperbola can arise as the locus (mathematics), locus of points whose difference of distances to two fixed focus (geometry), foci is constant, as a curve for each point of which the rays to two fix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
Line Segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special case of an ''arc (geometry), arc'', with zero curvature. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using an overline (vinculum (symbol), vinculum) above the symbols for the two endpoints, such as in . Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (geometry), edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. Wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
 |
Parallel Lines
In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. '' Parallel curves'' are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called ''skew lines''. Line segments and Euclidean vectors are parallel if they have the same direction or opposite direction (not necessarily the same length). Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism. Symbol The parallel symbol is \parallel. For example, AB \par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
 |
Parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a Point (geometry), point (the Focus (geometry), focus) and a Line (geometry), line (the Directrix (conic section), directrix). The focus does not lie on the directrix. The parabola is the locus (mathematics), locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane (geometry), plane Parallel (geometry), parallel to another plane that is tangential to the conical surface. The graph of a function, graph of a quadratic function y=ax^2+bx+ c (with a\neq 0 ) is a parabola with its axis parallel to the -axis. Conversely, every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
 |
Radius
In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is the line segment or distance from its center to any of its Vertex (geometry), vertices. The name comes from the Latin ''radius'', meaning ray but also the spoke of a chariot wheel.Definition of Radius at dictionary.reference.com. Accessed on 2009-08-08. The typical abbreviation and mathematical symbol for radius is ''R'' or ''r''. By extension, the diameter ''D'' is defined as twice the radius:Definition of radius at mathwords.com. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
 |
Circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a Disk (mathematics), disc. The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Terminology * Annulus (mathematics), Annulus: a ring-shaped object, the region bounded by two concentric circles. * Circular arc, Arc: any Connected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
Point (geometry)
In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical spaces. As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves, two-dimensional surfaces, and higher-dimensional objects consist. In classical Euclidean geometry, a point is a primitive notion, defined as "that which has no part". Points and other primitive notions are not defined in terms of other concepts, but only by certain formal properties, called axioms, that they must satisfy; for example, ''"there is exactly one straight line that passes through two distinct points"''. As physical diagrams, geometric figures are made with tools such as a compass, scriber, or pen, whose pointed tip can mark a small dot or prick a small hole representing a point, or can be drawn across a surface to represent a curve. A po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
|
Generic Property
In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic square matrix is invertible." As another example, a generic property of a space is a property that holds at "almost all" points of the space, as in the statement, "If is a smooth function between smooth manifolds, then a generic point of is not a critical value of ." (This is by Sard's theorem.) There are many different notions of "generic" (what is meant by "almost all") in mathematics, with corresponding dual notions of "almost none" (negligible set); the two main classes are: * In measure theory, a generic property is one that holds almost everywhere, with the dual concept being null set, meaning "with probability 0". * In topology and algebraic geometry, a generic property is one t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
 |
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class; "degeneracy" is the condition of being a degenerate case. The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a ''degenerate triangle'' if at least one side length or angle is zero. Equivalently, it becomes a "line segment". Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one. This is similar to the cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
 |
Continuous Compounding
Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower. Compound interest is contrasted with simple interest, where previously accumulated interest is not added to the principal amount of the current period. Compounded interest depends on the simple interest rate applied and the frequency at which the interest is compounded. Compounding frequency The ''compounding frequency'' is the number of times per given unit of time the accumulated interest is capitalized, on a regular basis. The frequency could be yearly, half-yearly, quarterly, monthly, weekly, daily, continuously, or not at all until maturity. For example, monthly capitalization with interest expressed as an annual rate means that the compounding frequency is 12, with time periods measured in months. Annual equivalent rate To help consum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |