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 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, the Levenshtein distance between two words is equal to the number of single-character edits required to change one word into the other.

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, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event.

Informally, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event.

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 stated, with certain technical assumptions, every representation of the Heisenberg group is equivalent to the position operators and momentum operators on R < sup > n </ sup >.

In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W < sup >⊥</ sup > of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement.

) Informally, the elements of V < sup > B </ sup > are " Boolean-valued sets ".

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 known as " The Big Book " ( with its first 164 pages virtually unchanged since the 1939 edition ), it suggests a twelve-step program in which members admit that they are powerless over alcohol and need help from a " higher power "; seek guidance and strength through prayer and meditation from God or Higher Power of their own understanding ; take a moral inventory with care to include resentments ; list and become ready to remove character defects ; list and make amends to those harmed, and then try to help other alcoholics recover.

Informally, the expression " infinitesimal calculus " became commonly used to refer to Weierstrass ' approach but has become something of a dead metaphor.

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

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, 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 speaking, a step function is a piecewise constant function having only finitely many pieces.

Informally known as LaGuardia Arts, or LaGuardia High School, Fiorello H. LaGuardia High School of Music & Art and Performing Arts is the only school among the nine specialized high schools in New York City that receives special funding from the New York State legislature through the Hecht Calandra Act.

Informally, a skein relation gives a linear relation between the values of a knot polynomial on a collection of three links which differ from each other only in a small region.

Informally, a free object over a set A can be thought as being a " generic " algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure.

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, 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, lowness means that problems in B are not only solvable by machines which can solve problems in A, but are " easy to solve.

Informally: a formula in prenex form containing existential quantifiers only is provable ( valid ) in first-order logic if and only if a disjunction composed of substitution instances of the quantifier-free subformula of is a tautology ( propositionally derivable ).

Informally, a loss of $ 1 million or more on this portfolio is expected on 1 day in 20.

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

( Informally, this reversal occurs because the refined type is " more liberal " in the types it accepts and " more conservative " in the type it returns.

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.

Informally, such an object can be thought of as an assemblage of one or more chunks of rubber ( cells ), each shaped like a convex polytope, which are glued to each other by their facets — possibly with much stretching and twisting.