Formally, oxidation state is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100 % ionic.

Formally, these reside in a complex separable Hilbert space-variously called the " state space " or the " associated Hilbert space " of the system-that is well defined up to a complex number of norm 1 ( the phase factor ).

Formally the " head of state " can also personally be the " head of government " ( ex officio or by ad hoc cumulation such as an absolute monarch nominating himself ) but otherwise has formal precedence over the Head of Government and other ministers, whether he is their actual political superior ( absolute monarch, executive president ) or rather theoretical or ceremonial in character.

Formally, a detailed balanced process is the one on which the probability to be in the state x and transit to x ' must be the same as to be in the state x ' and transit to x.

Formally declared on October 9, 1970, the Khmer Republic was a right-wing pro-United States military-led government headed by General Lon Nol and Prince Sisowath Sirik Matak that took power in the March 18, 1970 coup against Prince Norodom Sihanouk, then the country's head of state.

Formally, the problem can be stated as follows: given a desired property, expressed as a temporal logic formula p, and a structure M with initial state s, decide if.

Formally, the Cisalpine Republic was an independent state allied with France, but the treaty of alliance established the effective subalternity of the new republic to France.

Formally, an unlabelled state transition system is a tuple ( S, →) where S is a set ( of states ) and → ⊆ S × S is a binary relation over S ( of transitions ).

Formally a member of the country's collective head of state and, as president of parliament, holder of the third-highest office, he was left with little political influence.

Formally, knowing the step response of a dynamical system gives information on the stability of such a system, and on its ability to reach one stationary state when starting from another.

Formally, a system is said to be observable if, for any possible sequence of state and control vectors, the current state can be determined in finite time using only the outputs ( this definition is slanted towards the state space representation ).

Formally, a deterministic finite automaton may be defined by the tuple where is the set of states of the automaton, is the set of input symbols, is the transition function that takes a state and an input symbol to a new state, is the initial state of the automaton, and is the set of accepting or final states of the automaton.

Formally known as the Constitution of the Republic of Poland (), it replaced the temporary amendments put into place in 1992 designed to reverse the effects of Communism, establishing the nation as " a democratic state ruled by law and implementing the principles of social justice ".

Formally, it was owned by the PWS state aircraft manufacturer, in fact it was subordinated to the PZL.

Formally, a state is a set of conditions: a state is represented by the set of conditions that are true in it.

Formally, it remained a state of the Holy Roman Empire until its dissolution in 1806.

Formally, a topological space X is called compact if each of its open covers has a finite subcover.

Formally, an inner product space is a vector space V over the field together with an inner product, i. e., with a map

Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space.

Formally, this means symmetry under a sub-group of the Euclidean group of isometries in two or three dimensional Euclidean space.

Formally, the question of whether the universe is infinite or finite is whether it is an unbounded or bounded metric space.

Formally, we start with a metric space M and a subset X.

Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature < tt >(−,+,+,+)</ tt > ( Some may also prefer the alternative signature < tt >(+,−,−,−)</ tt >; in general, mathematicians and general relativists prefer the former while particle physicists tend to use the latter.

Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.

Formally, a ringed space ( X, O < sub > X </ sub >) is a topological space X together with a sheaf of rings O < sub > X </ sub > on X.

Formally, a coalgebra over a field K is a vector space C over K together with K-linear maps Δ: C → C ⊗ C and ε: C → K such that

Formally, an ultrametric space is a set of points with an associated distance function ( also called a metric )

Formally, a rigged Hilbert space consists of a Hilbert space H, together with a subspace Φ which carries a finer topology, that is one for which the natural inclusion

Formally, a frame on a homogeneous space G / H consists of a point in the tautological bundle G → G / H.

Formally, an iterated function system is a finite set of contraction mappings on a complete metric space.

Formally, rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space.

Formally, a complex projective space is the space of complex lines through the origin of an ( n + 1 )- dimensional complex vector space.

Formally, arbitrage transactions have negative skew – prices can get a small amount closer ( but often no closer than 0 ), while they can get very far apart.

Formally the algorithm can be described as:

Formally, it is expected that information can not travel faster than the speed of light.

Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by a matrix A and the translation as the addition of a vector, an affine map acting on a vector can be represented as

Formally, the theorem can be stated as follows:

Formally, a decision problem is P-complete ( complete for the complexity class P ) if it is in P and that every problem in P can be reduced to it by using an appropriate reduction.

Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero and non-unit x of R can be written as a product ( including an empty product ) of irreducible elements p < sub > i </ sub > of R and a unit u:

Formally, a function ƒ is real analytic on an open set D in the real line if for any x < sub > 0 </ sub > in D one can write

Formally, the problem of supervised pattern recognition can be stated as follows: Given an unknown function ( the ground truth ) that maps input instances to output labels, along with training data assumed to represent accurate examples of the mapping, produce a function that approximates as closely as possible the correct mapping.

Formally, a Menger sponge can be defined as follows:

Formally, the case where only a subset of parameters is defined is still a composite hypothesis ; nonetheless, the term point hypothesis is often applied in such cases, particularly where the hypothesis test can be structured in such a way that the distribution of the test statistic ( the distribution under the null hypothesis ) does not depend on the parameters whose values have not been specified under the point null hypothesis.

Formally, a Coxeter group can be defined as a group with the presentation

Formally, the poem can be understood as the combination of two sonnets, though the spacing of the stanzas is irregular.

Formally, the mutual information of two discrete random variables X and Y can be defined as:

Formally, Mbenga – Mbuti Pygmy music consists of at most only four parts, and can be described as an, " ostinato with variations ," or similar to a passacaglia, in that it is cyclical.

Formally, the complexity class MA is the set of decision problems that can be decided in polynomial time by an Arthur – Merlin protocol where Merlin's only move precedes any computation by Arthur.

Formally, the definition only requires some invertibility, so we can substitute for Q any matrix M whose eigenvalues do not include − 1.

Formally, the Calvinist position can be expressed this way:

Formally, given a finite set X, a collection C of subsets of X, all of size n, has Property B if we can partition X into two disjoint subsets Y and Z such that every set in C meets both Y and Z.

Formally, the issue is that interfertile " able to interbreed " is not a transitive relation – if A can breed with B, and B can breed with C, it does not follow that A can breed with C – and thus does not define an equivalence relation.