Formally given to bishop Philip of Senj, the permission to use the Glagolitic liturgy ( the Roman Rite conducted in Slavic language instead of Latin, not the Byzantine rite ), actually extended to all Croatian lands, mostly along the Adriatic coast.

Formally, we are given a set of hypotheses and a set of manifestations ; they are related by the domain knowledge, represented by a function that takes as an argument a set of hypotheses and gives as a result the corresponding set of manifestations.

Formally, the two parts are given by the following expression, where is the number being encoded:

Formally the self-inductance of a wire loop would be given by the above equation with i

Formally, the statement that " value decreases over time " is given by defining the linear differential operator as:

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

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 Secretary-General is given the same rank as a High Commissioner or ambassador.

Formally, given two categories C and D, an equivalence of categories consists of a functor F: C → D, a functor G: D → C, and two natural isomorphisms ε: FG → I < sub > D </ sub > and η: I < sub > C </ sub >→ GF.

Formally, given a vector field v, a vector potential is a vector field A such that

Formally, an absolute coequalizer of a pair in a category C is a coequalizer as defined above but with the added property that given any functor F ( Q ) together with F ( q ) is the coequalizer of F ( f ) and F ( g ) in the category D. Split coequalizers are examples of absolute coequalizers.

Formally, given complex-valued functions f and g of a natural number variable n, one writes

Formally, given a partially ordered set ( P, ≤), then an element g of a subset S of P is the greatest element of S if

Formally, a deterministic algorithm computes a mathematical function ; a function has a unique value for any given input, and the algorithm is a process that produces this particular value as output.

Formally, given two partially ordered sets ( S, ≤) and ( T, ≤), a function f: S → T is an order-embedding if f is both order-preserving and order-reflecting, i. e. for all x and y in S, one has

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, given a G-bundle B and a map H → G ( which need not be an inclusion ),

Formally, given a graph G, a vertex labeling is a function mapping vertices of G to a set of labels.

Formally, if and only if, for any given, there exists an such that

Formally, given two mappings

Formally this mission was declared to be only in pursuit of the Russian vessels and ammunition taken to Anzali by the White Russian counter-revolutionary General Denikin, who had been given asylum by British forces in Anzali.

Formally, a married or widowed woman can be called by the given name of her husband ( Madame ( given name of husband ) family name or Madame veuve ( given name of husband ) family name ); this is now slightly out of fashion.

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, the theorem is stated as follows: There exist unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d.

Formally, collective noun forms such as “ a group of people ” are represented by second-order variables, or by first-order variables standing for sets ( which are well-defined objects in mathematics and logic ).

Formally, such genes should increase in frequency when

Formally, an elliptic function is a function meromorphic on for which there exist two non-zero complex numbers and with ( in other words, not parallel ), such that and for all.

Formally, if there exists some B ≥ 0 such that

Formally, cl ( S ) denotes the smallest subset Y of M that contains S such that for each reaction ( A, B )

Formally, the word is applied to persons who are publicly accepted in a recognised capacity, such as professional employment, graduation from a course of study, etc., to give critical commentaries in one or any of a number of specific fields of public or private achievement or endeavour.

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 measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ.

Formally, a ring is an Abelian group ( R, +), together with a second binary operation * such that for all a, b and c in R,

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, a Hopf algebra is a ( associative and coassociative ) bialgebra H over a field K together with a K-linear map S: H → H ( called the antipode ) such that the following diagram commutes:

Formally, assuming the axiom of choice, cardinality of a set X is the least ordinal α such that there is a bijection between X and α.

Formally a congruence subgroup is one that contains Γ ( N ) for some N, and the least such N is the level or Stufe of the subgroup.

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 vertex cover of a graph G is a set C of vertices such that each edge of G is incident to at least one vertex in C. The set C is said to cover the edges of G. The following figure shows examples of vertex covers in two graphs ( and the set C is marked with red ).

Formally, choose such that

Formally, a string is a finite sequence of symbols such as letters or digits.

Formally, a cardinal number κ is λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with the critical point of j being κ and j ( κ ) ≥ λ.

Formally, we define a bad field as a structure of the form ( K, T ), where K is an algebraically closed field and T is an infinite, proper, distinguished subgroup of K, such that ( K, T ) is of finite Morley rank in its full language.