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, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface.

Informally, the theorem says that the curvature of a surface can be determined entirely by measuring distances along paths on the surface.

Informally, the second axiom says it doesn't matter whether we restrict to W in one step or restrict first to V, then to W.

Informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs.

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.

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

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

Informally, it says that all sufficiently long words in a regular language may be pumped — that is, have a middle section of the word repeated an arbitrary number of times — to produce a new word which also lies within the same language.

** Informally, any GPS navigation device, especially an automotive navigation system

Informally, and primarily in fiction, a detective is any licensed or unlicensed person who solves crimes, including historical crimes, or looks into records.

Informally, the term is often applied in a joking manner to describe any horse that acts up and bucks with or without a rider.

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, any sloping line is called diagonal.

Informally, software developers may also use zeroise to mean any overwriting of sensitive data, not necessarily of a cryptographic nature.

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, 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 ".

: 4: Informally, any whaler of any size or any vessel engaged in whale watching.

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 following titles of nobility, and any holders thereof:

Informally, a Euclidean plane isometry is any way of transforming the plane without " deforming " it.

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 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, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer.

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 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, input positions can be thought of as locations at token boundaries.

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 " 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, a conservative force can be thought of as a force that conserves mechanical energy.

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, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values.

Informally, we can consider these products on the left hand side as being elements of the free group F =

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

Informally, the capacity of a classification model is related to how complicated it can be.

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, C can be regarded as the *- algebra of continuous complex-valued functions over the compact matrix quantum group, and u can be regarded as a finite-dimensional representation of the compact matrix quantum group.

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, eigenfaces can be considered a set of " standardized face ingredients ", derived from statistical analysis of many pictures of faces.