Limit (mathematics)In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to and direct limit in . In formulas, a limit of a function is usually written as (although a few authors use "Lt" instead of "lim") and is read as "the limit of f of x as x approaches c equals L".
Limit inferior and limit superiorIn mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.
Geometric seriesIn mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series is geometric, because each successive term can be obtained by multiplying the previous term by . In general, a geometric series is written as , where is the coefficient of each term and is the common ratio between adjacent terms.
Japanese calligraphyJapanese calligraphy, also called 習字, is a form of calligraphy, or artistic writing, of the Japanese language. Written Japanese was originally based on Chinese characters only, but the advent of the hiragana and katakana Japanese syllabaries resulted in intrinsically Japanese calligraphy styles. The term shodō (書道, "way of writing") is of Chinese origin and is widely used to describe the art of Chinese calligraphy during the medieval Tang dynasty. Early Japanese calligraphy was originated from Chinese calligraphy.
Ink wash paintingInk wash painting (; is a type of Chinese ink brush painting which uses washes of black ink, such as that used in East Asian calligraphy, in different concentrations. It emerged during the Tang dynasty of China (618–907), and overturned earlier, more realistic techniques. It is typically monochrome, using only shades of black, with a great emphasis on virtuoso brushwork and conveying the perceived "spirit" or "essence" of a subject over direct imitation.
PaintingPainting is the practice of applying paint, pigment, color or other medium to a solid surface (called the "matrix" or "support"). The medium is commonly applied to the base with a brush, but other implements, such as knives, sponges, and airbrushes, can be used. In art, the term "painting" describes both the act and the result of the action (the final work is called "a painting").
DrawingDrawing is a visual art that uses an instrument to mark paper or another two-dimensional surface. The instrument might be pencils, crayons, pens with inks, brushes with paints, or combinations of these, and in more modern times, computer styluses with graphics tablets. A drawing instrument releases a small amount of material onto a surface, leaving a visible mark. The most common support for drawing is paper, although other materials, such as cardboard, vellum, wood, plastic, leather, canvas, and board, have been used.
Limit of a functionAlthough the function \tfrac{\sin x}{x} is not defined at zero, as x becomes closer and closer to zero, \tfrac{\sin x}{x} becomes arbitrarily close to 1. In other words, the limit of \tfrac{\sin x}{x}, as x approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x.
One-sided limitIn calculus, a one-sided limit refers to either one of the two limits of a function of a real variable as approaches a specified point either from the left or from the right. The limit as decreases in value approaching ( approaches "from the right" or "from above") can be denoted: The limit as increases in value approaching ( approaches "from the left" or "from below") can be denoted: If the limit of as approaches exists then the limits from the left and from the right both exist and are equal.
Uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every point in .
Pointwise convergenceIn mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Suppose that is a set and is a topological space, such as the real or complex numbers or a metric space, for example. A net or sequence of functions all having the same domain and codomain is said to converge pointwise to a given function often written as if (and only if) The function is said to be the pointwise limit function of the Sometimes, authors use the term bounded pointwise convergence when there is a constant such that .
Compact convergenceIn mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology. Let be a topological space and be a metric space. A sequence of functions is said to converge compactly as to some function if, for every compact set , uniformly on as . This means that for all compact , If and with their usual topologies, with , then converges compactly to the constant function with value 0, but not uniformly.
Radius of convergenceIn mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or . When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges.
Taylor seriesIn mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.
Divergence (statistics)In information geometry, a divergence is a kind of statistical distance: a binary function which establishes the separation from one probability distribution to another on a statistical manifold. The simplest divergence is squared Euclidean distance (SED), and divergences can be viewed as generalizations of SED. The other most important divergence is relative entropy (also called Kullback–Leibler divergence), which is central to information theory.
Series (mathematics)In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.
F-divergenceIn probability theory, an -divergence is a function that measures the difference between two probability distributions and . Many common divergences, such as KL-divergence, Hellinger distance, and total variation distance, are special cases of -divergence. These divergences were introduced by Alfréd Rényi in the same paper where he introduced the well-known Rényi entropy. He proved that these divergences decrease in Markov processes.
DivergenceIn vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region.
Geometric progressionIn mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. Examples of a geometric sequence are powers rk of a fixed non-zero number r, such as 2k and 3k.
Kullback–Leibler divergenceIn mathematical statistics, the Kullback–Leibler divergence (also called relative entropy and I-divergence), denoted , is a type of statistical distance: a measure of how one probability distribution P is different from a second, reference probability distribution Q. A simple interpretation of the KL divergence of P from Q is the expected excess surprise from using Q as a model when the actual distribution is P.
