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, 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, 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 speaking, every matrix satisfies its own characteristic equation.

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, a Lyapunov function is a function that takes positive values everywhere except at the equilibrium in question, and decreases ( or is non-increasing ) along every trajectory of the ODE.

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

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, a loop invariant is a statement of the conditions that should be true on entry into a loop and that are guaranteed to remain true on every iteration of the loop.

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 >.

Informally, the Council advises the Fijian government on every aspect of political life.

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, a computational problem consists of problem instances and solutions to these problem instances.

Informally it solves the following problem: Given an integer N, find its prime factors.

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, to estimate the area of a domain D, first pick a simple domain E whose area is easily calculated and which contains D. Now pick a sequence of random points that fall within E. Some fraction of these points will also fall within D. The area of D is then estimated as this fraction multiplied by the area of E.

Informally, a superkey is a set of attributes within a table whose values can be used to uniquely identify a tuple.

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

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

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.

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, the name " NFL Europe " continued to be used in the United States, including for the league's English-language Web site, nfleurope. com.

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, free abelian groups or formal sums may also be seen as signed multisets with elements in B.

Informally, dogs may be studied by those with no specific scientific training, such as publicists and authors, breeders, trainers, police dog handlers, animal communicators and others, through literature, history, and personal experience.

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