More formally, it is the theoretical temperature at which entropy reaches its minimum value.

More formally, the complexity of a string is the length of the shortest possible description of the string in some fixed universal description language ( the sensitivity of complexity relative to the choice of description language is discussed below ).

More formally, a block cipher is specified by an encryption function

More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient

More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets ( sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers ).

More formally design has been defined as follows.

More formally, it is " the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.

More formally, such coordinates can be obtained from three-dimensional coordinates using the artifice of a map projection.

More formally, a " representation " means a homomorphism from the group to the automorphism group of an object.

More formally, case has been defined as " a system of marking dependent nouns for the type of relationship they bear to their heads.

More formally, if a, b, c, and d are all integers, then the fraction < sup > a </ sup >⁄< sub > b </ sub > is irreducible if and only if there is no other equal fraction < sup > c </ sup >⁄< sub > d </ sub > such that | c | < | a | or | d | < | b |, where | a | means the absolute value of a.

More formally, the Klein bottle is the quotient space described as the square × with sides identified by the relations for and for:

More formally, a lexicon is a language's inventory of lexemes.

More formally, the environment is modeled as a Markov decision process ( MDP ) with states and actions with the following probability distributions: the instantaneous cost distribution, the observation distribution and the transition, while a policy is defined as conditional distribution over actions given the observations.

( More work is required to derive this result formally.

More formally, statistical theory defines a statistic as a function of a sample where the function itself is independent of the sample's distribution ; that is, the function can be stated before realization of the data.

More formally, # P is the class of function problems of the form " compute ƒ ( x )," where ƒ is the number of accepting paths of a nondeterministic Turing machine running in polynomial time.

More formally, the output must satisfy two conditions:

More formally, we can state the Transfinite Recursion Theorem as follows.

More formally, ubiquitous computing is defined as " machines that fit the human environment instead of forcing humans to enter theirs.

More formally, complex systems often exhibit hysteresis.

More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i. e., a combination of translations, rotations and reflections.

More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.

More formally, we can think of a truth condition as what makes for the truth of a sentence in an inductive definition of truth ( for details, see the semantic theory of truth ).

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More unusual configurations such as the H, U, X, and W have also been used.

More generally, if X < sub > 1 </ sub >, ..., X < sub > n </ sub > are independent random variables, not necessarily identically distributed, but all having the same variance, then

More precisely, for every Tychonoff space X, there exists a compact Hausdorff space K such that X is homeomorphic to a subspace of K.

More generally, if F is a subset of the algebraic dual space, then the initial topology of X with respect to F, denoted by σ ( X, F ), is the weak topology with respect to F.

More generally, if a family of seminorms Q defines the topology on Y, then the seminorms p < sub > q, x </ sub > on L ( X, Y ) defining the strong topology are given by

More recent studies have served to produce a reportedly fairer and more honest opinion of Leo X.

More strongly, if one wishes to prove that X and G ( X ) are not naturally isomorphic, without reference to a particular isomorphism, this requires showing that for any isomorphism η, there is some A with which it does not commute ; in some cases a single automorphism A works for all candidate isomorphisms η, while in other cases one must show how to construct a different A < sub > η </ sub > for each isomorphism.

More formally, a multivariate random variable is a column vector X = ( X < sub > 1 </ sub >, ..., X < sub > n </ sub >)< sup > T </ sup > ( or its transpose, which is a row vector ) whose components are scalar-valued random variables on the same probability space ( Ω,, P ), where Ω is the sample space, is the sigma-algebra ( the collection of all events ), and P is the probability measure ( a function returning every event's probability ).

More formally, a set R X Y is called a ( combinatorial ) rectangle if whenever R and R then R. Equivalently, R can also be viewed as a submatrix of the input matrix A such that R = M N where M X and N Y.

More generally, a function f defined on X is said to be Hölder continuous or to satisfy a Hölder condition of order α > 0 on X if there exists a constant M > 0 such that

More explicitly, an injective continuous map f: X → Y between topological spaces X and Y is a topological embedding if f yields a homeomorphism between X and f ( X ) ( where f ( X ) carries the subspace topology inherited from Y ).

More precisely, suppose that a representative π is chosen for each isomorphism class of irreducible unitary representation, and denote the collection of all such π by Σ.

More generally, in the n-dimensional Euclidean space R < sup > n </ sup >, the Radon transform of a compactly supported continuous function ƒ is a function Rƒ on the space Σ < sub > n </ sub > of all hyperplanes in R < sup > n </ sup >.

More realistically, though, the hat matrix would involve an observation covariance matrix Σ indicating the non-zero correlation among observations.

More precisely, a right Haar measure on a locally compact group G is a countably additive measure μ defined on the Borel sets of G which is right invariant in the sense that μ ( A x ) = μ ( A ) for x an element of G and A a Borel subset of G and also satisfies some regularity conditions ( spelled out in detail in the article on Haar measure ).

More precisely put, an observable is a real valued function f on the phase space Γ that is integrable with respect to the microcanonical ensemble measure μ.

More precisely, Feld-Tai reciprocity requires the Hermitian ( or rather, complex-symmetric ) symmetry of the electromagnetic operators as above, but also relies on the assumption that the operator relating and is a constant scalar multiple of the operator relating and, which is true when ε is a constant scalar multiple of μ ( the two operators generally differ by an interchange of ε and μ ).