In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as "Property Q characterises Y up to

Up to

In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...

 isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

". The first type of statement says in different words that the extension

Extension (semantics)

In any of several studies that treat the use of signs - for example, in linguistics, logic, mathematics, semantics, and semiotics - the extension of a concept, idea, or sign consists of the things to which it applies, in contrast with its comprehension or intension, which consists very roughly of...

 of P is a singleton set. 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

Equivalence relation

In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

 might be involved).

Examples

• "Among probability distribution
  Probability distribution
  In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
  
  s on the interval from 0 to ∞ on the real line, memorylessness
  Memorylessness
  In probability and statistics, memorylessness is a property of certain probability distributions: the exponential distributions of non-negative real numbers and the geometric distributions of non-negative integers....
  
   characterizes the exponential distribution
  Exponential distribution
  In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...
  
  s." This statement means that the exponential distributions are the only such probability distributions that are memoryless.

• "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
  Gamma function
  In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
  
  ." This means that among all such functions, the gamma function is the only one that is log-convex. (A function f is log-convex iff
  IFF
  IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
  
   log(f) is a convex function
  Convex function
  In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
  
  . The base of the logarithm does not matter as long as it is more than 1, but conventionally mathematicians take "log" with no subscript to mean the natural logarithm
  Natural logarithm
  The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
  
  , whose base is e.)

• The circle is characterized as a manifold
  Manifold
  In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
  
   by being one-dimensional, compact
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
   and connected
  Connected space
  In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
  
  ; here the characterization, as a smooth manifold, is up to
  Up to
  In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
  
   diffeomorphism
  Diffeomorphism
  In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
  
  .