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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.
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Informally and G
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, this reads as follows: something exists with the property F, there is only one such thing, and this unique thing also has the property G.
Informally, this means that if G acts unitarily on a Hilbert space and has " almost invariant vectors ", then it has a nonzero invariant vector.
Informally, if F is '= y ', G is ' is Pegasus ', and we substitute ' Pegasus ' for y, then ( 4 ) appears to allow us to infer from ' everything identical with Pegasus is Pegasus ' that something is identical with Pegasus.
Informally and has
Informally, the expression " infinitesimal calculus " became commonly used to refer to Weierstrass ' approach but has become something of a dead metaphor.
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 this has been called the " Step Boundary ", although the step-like shape of the boundary was not apparent for several decades — the source of the Arkansas, believed to be near the 42nd parallel, was not known until John C. Frémont located it in the 1840s, hundreds of miles south of the 42nd parallel.
Informally, the market spilled into the surrounding neighborhood, both in terms of street vendors and in terms of the nearby Magazin Universal (" Universal Store ") named Bucur Obor, a large commercial building that has been parcelled up into hundreds of small, independent retail stores.
Informally, one might say that the Dirac string carries away the " excess curvature " that would otherwise prevent F from being a closed form, as one has that everywhere except at the location of the monopole.
Informally and above
Informally, flak jacket may refer to a ballistic vest, particularly Type III and above, which is reinforced with either steel, titanium, ceramic, or polyethylene plates.
Informally and if
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 the Church – Turing thesis states that if some method ( algorithm ) exists to carry out a calculation, then the same calculation can also be carried out by a Turing machine ( as well as by a recursively definable function, and by a λ-function ).
Informally, a relational database table is often described as " normalized " if it is in the Third Normal Form.
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, 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, this means that arguments of this form do not give good reason to establish their conclusions, even if their premises are true.
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, 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, 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.
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 and is
Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin.
Informally, it is the similarity between observations as a function of the time separation between them.
Informally we can think of elements of the Lie algebra as elements of the group that are " infinitesimally close " to the identity, and the Lie bracket is something to do with the commutator of two such infinitesimal elements.
Informally, word formation rules form " new words " ( that is, new lexemes ), while inflection rules yield variant forms of the " same " word ( lexeme ).
Informally, he may have been known as " Dickon ", according to a sixteenth-century legend of a note, warning of treachery, that was sent to the Duke of Norfolk on the eve of Bosworth: " Jack of Norffolke be not to bolde ,/ For Dyckon thy maister is bought and solde ".
Informally, a permutation of a set of objects is an arrangement of those objects into a particular order.