An advocatus was an advocate in the Middle Ages. It was used in Continental Europe as the title of the lay lord charged with the protection and representation in secular matters of an abbey. The office is traceable as early as the beginning of the 5th century in the Roman Empire, the churches being allowed to choose defensores from the body of advocates to represent them in the courts. In the Frankish Kingdom, under the Merovingians, these lay representatives of the churches appear as agentes, defensores and advocati; and under the Ca ...
In our personal human experiences, we seem to exist in a universe with three spatial dimensions. Some theories in physics, including string theory, include the idea that there are additional spatial dimensions. Such theories suggest that there may be a specific number of spatial dimensions such as 10. The question, "Why 10 dimensions?" arises from these theories.
Why 10 dimensions? - Why 10 11 or 26 physical dimensions in string theory?.
This is one of the questions discussed by Michio Kaku in his book H ...
A codomain in mathematics is the set of "output" values associated with (or mapped to) the domain of "input" arguments in a function. For any given function , the set A, on which f is defined (the arguments), is called the domain, and B (the set of possible values) is called the codomain of f. The set of all actual values or f(A) is called the range of f.
Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish b ...
In mathematics a constant function is a function whose values do not vary and thus are constant. For example, if we have the function f(x) = 4, then f is constant since f maps any value to 4. More formally, a function f : A → B, is a constant function if f(x) = f(y) for all x and y in A.
Notice that every empty function, that is, any function whose domain equals the empty set, is included in the above definition vacuously, sin ...
The term cluster refers to the grouping together of elements within a domain - usually spatial.
Cluster may refer to:
Cluster bomb
Cluster (band)
Cluster (chemistry)
Metal cluster
Other related archivesCluster (band), Cluster (chemistry), Cluster bomb, Metal cluster, domain, elements, spatial
In statistics, mean has two related meanings:
the average in ordinary English, which is more correctly called the arithmetic mean, to distinguish it from geometric mean or harmonic mean. The average is also called sample mean.
the expected value of a random variable, which is also called the population mean.
As well as statistics, means are often used in geometry and analysis; a wide range of means have been developed for these purposes, which are not much used in statistics. See the Other m ...
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually.
Formally, given a partially ordered set (P, ≤), an element u of P is an upper bound of a subset S of P, if
s ≤ u, for all elements s of S.
Using ≥ instead of ≤ leads to ...
A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more of a "weight" than others. They occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be constructed in both discrete and continuous settings.
Weight function - Discrete weights.
In the discrete setting, a weight function is a positive function defined on a discrete set A, which is typically finite or ...
In calculus, the indefinite integral of a given function (i.e. the set of all antiderivatives of the function) is always written with a constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives. If a function f is defined on an interval and F is an antiderivative of f, then the set of all antiderivatives of f is given by the funct ...
In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). Although it is often described by a number of "components", each of which is dependent upon the particular coordinate system being used, a vector is an object with properties which do not depend on the coordinate system used to describe it. ...
In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. If small changes in the input can produce a broken jump in the changes of the output (or the value of the output is not defined), the function is said to be discontinuous (or to have a discontinuity). The context in this entry is real-valued functions on the real domain or on topological or metric spaces other than the complex numbers; for complex-valued functions see comple ...
In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. A convolution is a kind of very general moving average, as one can see by taking one of the functions to be an indicator function of an interval.
Convolution - Uses.
Convolution and related operations are found in ma ...
Carl Richard Woese is an American microbiologist famous for defining the Archaea (a new domain or kingdom of life) in 1976 by phylogenetic taxonomy of 16S ribosomal RNA, a technique pioneered by Woese and which is now standard practice. He was also the originator of the RNA world hypothesis in 1967, although not by that name. He was born in Syracuse, New York, on July 15, 1928. Woese is currently a professor of Microbiology a ...
In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.
A function is injective (one-to-one) if or, equivalently, if . One could also say that every element of the codomain (sometimes called range) is mapped to by at most one element (argument) of the domain; not every element of t ...
Abduction, or abductive reasoning, is the process of reasoning to the best explanations. In other words, it is the reasoning process that starts from a set of facts and derives their most likely explanations. The term abduction is sometimes used to mean just the generation of hypotheses to explain observations or conclusions, but the former definition is more common both in philosophy and computing.
Deduction and abduction differ in the direction in which a rule like “a entail ...
In mathematics, a large number of methods have been proposed for the summation of divergent series. These generally take the form of some linear functional L with domain contained in some space S of numerical sequences. That is, firstly, a useful method for attributing a sum to a series that doesn't converge should at least be linear. Secondly, the sequence of partial sums of the series is considered, which is an equivalent way of presenting it.
For any such L, its abelian theorem is the result that if c ...
The Aquificae phylum is a diverse collection of bacteria that live in harsh environmental settings. They have been found in hot springs, sulfer pools, thermal ocean vents. Members of the genus Aquifex, for example, are productive in water between 85 to 95 °C. They are the dominant members of most terrestrial neutral to alkaline hot springs above 60 degrees celsius. They are autotrophs, and are the primary carbon fixers in these environments. They a ...
In mathematics, an involution, or an involutary function, is a function that is its own inverse, so that
f(f(x)) = x for all x in the domain of f.
Involution - General properties.
The identity map is a trivial example of an involution. Common examples in mathematics of more interesting involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation.
Other exa ...
In mathematics, an analytic function is a function that is locally given by a convergent power series. Analytic functions can be thought of as a bridge between polynomials and general functions. There exist real analytic functions and complex analytic functions, which have similarities as well as differences.
Analytic function - Definitions.
Formally, function f is real analytic on an open set D in the real line if for any x0 in D one can write ...
In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system meets all of a certain set of criteria when there are three or more choices. These criteria are called unrestricted domain, non-imposition, non-dictatorship, monotonicity, and independence of irrelevant alternatives, and are defined below.
The theorem is named after economist Kenneth Arrow, who proved the theorem in his Ph.D. thesis and popular ...
