Informally, other members of the higher hierarchic echelons of the Catholic Church are in recent times also occasionally called " Princes of the church ", in which case this title may occasionally carry ironic connotations.

Informally, many Mechlinians ( in Dutch Mechelaars and locally pronounced Mecheleirs, people from this city ) daily speak Mechlinian ( in Dutch as well as locally Mechels ), a dialect by Dutch-speakers of Belgium pin-pointingly distincted from other Brabantic dialects by three different vowels of Dutch ( as in zout or rauw, in kei or bijl, in nu ) being typically pronounced as a same vowel – close to the one in English ' raw ' which does not appear in other dialects of the Flemish Region or in standard Dutch.

Informally, generalized Kac – Moody algebras are the Lie algebras that

Informally, a set of natural numbers A is Turing reducible to a set B if there is an oracle machine that correctly tells whether numbers are in A when run with B as the oracle set ( in this case, the set A is also said to be ( relatively ) computable from B and recursive in B ).

Informally, these theorems say that given more time, a Turing machine can solve more problems.

Informally, given a problem, the Turing jump returns the set of Turing machines which halt when given access to an oracle that solves that problem.

Informally speaking, the prime number theorem states that if a random integer is selected in the range of zero to some large integer N, the probability that the selected integer is prime is about 1 / ln ( N ), where ln ( N ) is the natural logarithm of N. For example, among the positive integers up to and including N = 10 < sup > 3 </ sup > about one in seven numbers is prime, whereas up to and including N = 10 < sup > 10 </ sup > about one in 23 numbers is prime ( where ln ( 10 < sup > 3 </ sup >)= 6. 90775528. and ln ( 10 < sup > 10 </ sup >)= 23. 0258509 ).

Informally, the theorem states that arithmetical truth cannot be defined in arithmetic.

Informally, it states that " A random variable that depends in a Lipschitz way on many independent variables ( but not too much on any of them ) is essentially constant ".

Such a definition can be formulated in terms of equivalence classes of smooth functions on M. Informally, we will say that two smooth functions f and g are equivalent at a point x if they have the same first-order behavior near x.

Informally, an object is reachable if it is referenced by at least one variable in the program, either directly or through references from other reachable objects.

Informally, a relational database table is often described as " normalized " if it is in the Third Normal Form.

Informally, a graph is a good expander if it has low degree and high expansion parameters.

Informally, a measure has the property of being monotone function | monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B.

Informally, a set of strategies is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy.

In other words, L can be solved in polynomial time by an oracle machine with an oracle for H. Informally, we can think of an algorithm that can call such an oracle machine as a subroutine for solving H, and solves L in polynomial time, if the subroutine call takes only one step to compute.

Informally, two sequences ( a < sub > n </ sub >) and ( b < sub > n </ sub >) become closer and closer if and only if more and more of their terms agree exactly.

Informally, G has the above presentation if it is the " freest group " generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R.

Informally, this means that arguments of this form do not give good reason to establish their conclusions, even if their premises are true.

Informally, if P is any polyhedron or polytope, and tP is the polytope formed by expanding P by a factor of t in each dimension, then L ( int P, t ) is the number of integer lattice points in tP.

Informally, subvarieties are in general position if they cannot be described more simply than others.

Informally, A probabilistically causes B if As occurrence increases the probability of B.

Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set.

Informally, two crystals tend to be in the same crystal system if they have similar symmetries, though there are many exceptions to this.

Informally, given the prime factorization of m, take b to be the product of the prime factors of m that have an odd exponent ( if there are none, then take b to be 1 ).

Informally, a thick object in our space is simply connected if it consists of one piece and does not have any " holes " that pass all the way through it.

Informally, a language L is in MA if for all strings in the language, there is a polynomial sized proof that Merlin can send Arthur to convince him of this fact with high probability, and for all strings not in the language there is no proof that convinces Arthur with high probability.

# From conclude: Informally, this says that if A is a theorem, then it is provable.

Informally put, the idea behind the proof of the no-trade theorem is that if there is common knowledge about the structure of a market, then any bid or offer ( i. e. attempt to initiate a trade ) will reveal the bidder's private knowledge and will be incorporated into market prices even before anyone accepts the bid or offer, so no profit will result.

Informally, the term measure word is also sometimes used to refer to numeral classifiers, which are used with nouns that are countable in some languages.

A pure or homogeneous simplicial k-complex is a simplicial complex where every simplex of dimension less than k is a face of some simplex of dimension exactly k. Informally, a pure 1-complex " looks " like it's made of a bunch of lines, a 2-complex " looks " like it's made of a bunch of triangles, etc.

Informally it can be considered more widely: from a post-colonial form of amical protection, or protectorate, to confederation of unequal members when the lesser partner ( s ) delegate to the major one ( often the former colonial power ) some authority normally exclusively retained by a self-governing state, often in such fields as defence and foreign relations, while often enjoying favorable economic terms such as market access.

In the mathematical field of graph theory, a spanning tree T of a connected, undirected graph G is a tree composed of all the vertices and some ( or perhaps all ) of the edges of G. Informally, a spanning tree of G is a selection of edges of G that form a tree spanning every vertex.

Informally, the theorem says that given some formal arithmetic, the concept of truth in that arithmetic is not definable using the expressive means that arithmetic affords.

Informally, a weakly simple polygon is a polygon in which some sides can " touch " but cannot " cross over ".

Informally, in attempting to estimate the causal effect of some variable x on another y, an instrument is a third variable z which affects y only through its effect on x.

Informally, Nakayama's lemma says that one can still regard a coherent sheaf as coming from a vector bundle in some sense.

Informally, this means that whenever we are given the unit ball of some seminorm, we can find a " much smaller " unit ball of another seminorm inside it, or that any neighborhood of 0 contains a " much smaller " neighborhood.

Informally, the " P vs. NP " question asks whether every optimization problem whose answers can be efficiently verified for correctness / optimality can be solved optimally with an efficient algorithm.

" Informally, hypothesis boosting problem asks whether an efficient learning algorithm that outputs an hypothesis whose performance is only slightly better than random guessing a weak learner implies the existence of an efficient algorithm that outputs an hypothesis of arbitrary accuracy a strong learner.

Informally, this means that given a constructive proof that an object exists, then that constructive proof can be turned into an algorithm for generating an example of it.

Informally, the algorithm constructs a finite state machine that resembles a trie with additional links between the various internal nodes.

Informally, this is true because a polynomial-time algorithm can make polynomially many queries to other polynomial-time algorithms, retaining its polynomial running time.