Characterization (mathematics)

Last updated April 03, 2024

In mathematics, a characterization of an object is a set of conditions that, while different from the definition of the object, is logically equivalent to it.^[1] To say that "Property P characterizes object X" is to say that not only does X have property P, but that X is the only thing that has property P (i.e., P is a defining property of X). Similarly, a set of properties P is said to characterize X, when these properties distinguish X from all other objects. Even though a characterization identifies an object in a unique way, several characterizations can exist for a single object. Common mathematical expressions for a characterization of X in terms of P include "P is necessary and sufficient for X", and "X holds if and only if P".

It is also common to find statements such as "Property Q characterizes Y up to isomorphism". The first type of statement says in different words that the extension of P is a singleton set, while the second says that the extension of Q is a single equivalence class (for isomorphism, in the given example — depending on how up to is being used, some other equivalence relation might be involved).

A reference on mathematical terminology notes that characteristic originates from the Greek term kharax, "a pointed stake":

From Greek kharax came kharakhter, an instrument used to mark or engrave an object. Once an object was marked, it became distinctive, so the character of something came to mean its distinctive nature. The Late Greek suffix -istikos converted the noun character into the adjective characteristic, which, in addition to maintaining its adjectival meaning, later became a noun as well.^[2]

Just as in chemistry, the characteristic property of a material will serve to identify a sample, or in the study of materials, structures and properties will determine characterization, in mathematics there is a continual effort to express properties that will distinguish a desired feature in a theory or system. Characterization is not unique to mathematics, but since the science is abstract, much of the activity can be described as "characterization". For instance, in Mathematical Reviews , as of 2018, more than 24,000 articles contain the word in the article title, and 93,600 somewhere in the review.

In an arbitrary context of objects and features, characterizations have been expressed via the heterogeneous relation aRb, meaning that object a has feature b. For example, b may mean abstract or concrete. The objects can be considered the extensions of the world, while the features are expression of the intensions. A continuing program of characterization of various objects leads to their categorization.

Examples

A rational number, generally defined as a ratio of two integers, can be characterized as a number with finite or repeating decimal expansion.^[1]
A parallelogram is a quadrilateral whose opposing sides are parallel. One of its characterizations is that its diagonals bisect each other. This means that the diagonals in all parallelograms bisect each other, and conversely, that any quadrilateral whose diagonals bisect each other must be a parallelogram. The latter statement is only true if inclusive definitions of quadrilaterals are used (so that, for example, rectangles count as parallelograms), which is the dominant way of defining objects in mathematics nowadays.
"Among probability distributions on the interval from 0 to ∞ on the real line, memorylessness characterizes the exponential distributions." This statement means that the exponential distributions are the only probability distributions that are memoryless, provided that the distribution is continuous as defined above (see Characterization of probability distributions for more).
"According to Bohr–Mollerup theorem, among all functions f such that f(1) = 1 and x f(x) = f(x + 1) for x > 0, log-convexity characterizes the gamma function." This means that among all such functions, the gamma function is the only one that is log-convex.^[3]
The circle is characterized as a manifold by being one-dimensional, compact and connected; here the characterization, as a smooth manifold, is up to diffeomorphism.

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<span class="mw-page-title-main">Exponential function</span> Mathematical function, denoted exp(x) or e^x

The exponential function is a mathematical function denoted by $or . Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the operation of taking powers of a number, but various modern definitions allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics".$

<span class="mw-page-title-main">Isomorphism</span> In mathematics, invertible homomorphism

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσοςisos "equal", and μορφήmorphe "form" or "shape".

In mathematics, the logarithm is the inverse function to exponentiation. That means that the logarithm of a number $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$ . For example, since $1000 = 10 3$ , the logarithm base 10 of $1000$ is $3$ , or $log 10 (1000) = 3$ . The logarithm of $x$ to base $b$ is denoted as $log b (x)$ , or without parentheses, $log b x$ , or even without the explicit base, $log x$ , when no confusion is possible, or when the base does not matter such as in big O notation.

In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices $,, and is sometimes denoted as .$

In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time between production errors, or length along a roll of fabric in the weaving manufacturing process. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term "oblong" is used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD.

In mathematics, equality is a relationship between two quantities or, more generally, two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. Equality between $A$ and $B$ is written $A = B$ , and pronounced " $A$ equals $B$ ". The symbol " $=$ " is called an "equals sign". Two objects that are not equal are said to be distinct.

<span class="mw-page-title-main">Parallelogram</span> Quadrilateral with two pairs of parallel sides

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

<span class="mw-page-title-main">Rhombus</span> Quadrilateral with sides of equal length

In plane Euclidean geometry, a rhombus is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

<span class="mw-page-title-main">Trapezoid</span> Convex quadrilateral with at least one pair of parallel sides

In geometry, a trapezoid in North American English, or trapezium in British English, is a quadrilateral that has at least one pair of parallel sides.

In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term exponential class is sometimes used in place of "exponential family", or the older term Koopman–Darmois family. Sometimes loosely referred to as "the" exponential family, this class of distributions is distinct because they all possess a variety of desirable properties, most importantly the existence of a sufficient statistic.

<span class="mw-page-title-main">Midpoint</span> Point on a line segment which is equidistant from both endpoints

In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four sides of equal length and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted $ABCD .$

In probability and statistics, memorylessness is a property of certain probability distributions. It usually refers to the cases when the distribution of a "waiting time" until a certain event occurs does not depend on how much time has elapsed already. To model memoryless situations accurately, we must constantly 'forget' which state the system is in: the probabilities would not be influenced by the history of the process.

In convex analysis, a non-negative function $f : R n \to R +$ is logarithmically concave if its domain is a convex set, and if it satisfies the inequality

In Euclidean geometry, Varignon's theorem holds that the midpoints of the sides of an arbitrary quadrilateral form a parallelogram, called the Varignon parallelogram. It is named after Pierre Varignon, whose proof was published posthumously in 1731.

In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside the quadrilateral. It has also been called an exscriptible quadrilateral. The circle is called its excircle, its radius the exradius and its center the excenter. The excenter lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect. The ex-tangential quadrilateral is closely related to the tangential quadrilateral.

In Euclidean geometry, an equidiagonal quadrilateral is a convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics, where quadrilaterals were classified first according to whether they were equidiagonal and then into more specialized types.

In mathematics in general, a characterization theorem says that a particular object – a function, a space, etc. – is the only one that possesses properties specified in the theorem. A characterization of a probability distribution accordingly states that it is the only probability distribution that satisfies specified conditions. More precisely, the model of characterization of probability distribution was described by V.M. Zolotarev in such manner. On the probability space we define the space $of random variables with values in measurable metric space and the space of random variables with values in measurable metric space . By characterizations of probability distributions we understand general problems of description of some set in the space by extracting the sets and which describe the properties of random variables and their images , obtained by means of a specially chosen mapping :{\mathcal {X}}\to {\mathcal {Y}}} . The description of the properties of the random variables and of their images is equivalent to the indication of the set from which must be taken and of the set into which its image must fall. So, the set which interests us appears therefore in the following form:$

References

1 2 Weisstein, Eric W. "Characterization". mathworld.wolfram.com. Retrieved 2019-11-21.
↑ Steven Schwartzmann (1994) The Words of Mathematics: An etymological dictionary of mathematical terms used in English, page 43, The Mathematical Association of America ISBN 0-88385-511-9
↑ A function f is log-convex if and only if log(f) is a convex function. The base of the logarithm does not matter as long as it is more than 1, but mathematicians generally take "log" with no subscript to mean the natural logarithm, whose base is e.

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[:0-1] 1 2 Weisstein, Eric W. "Characterization". mathworld.wolfram.com. Retrieved 2019-11-21.

[2] Steven Schwartzmann (1994) The Words of Mathematics: An etymological dictionary of mathematical terms used in English, page 43, The Mathematical Association of America ISBN 0-88385-511-9

[3] A function f is log-convex if and only if log(f) is a convex function. The base of the logarithm does not matter as long as it is more than 1, but mathematicians generally take "log" with no subscript to mean the natural logarithm, whose base is e.

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Characterization (mathematics)

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