Intuitively, a space is complete if there are no " points missing " from it ( inside or at the boundary ).

Intuitively, the distinction says merely that there is no canonical choice of where the origin should go in the space, because it can be translated anywhere.

Intuitively, it records information about the basic shape, or holes, of the topological space.

In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are " near " S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.

Intuitively, the divergence ( denoted ∇•) of a vector field is a measure of flux diverging radially outwards from a point, so the negative is the amount piling up at a point, hence the rate of change of density in a region of space must be the amount of flux leaving or collecting in some region ( see main article for details ).

Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.

Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces.

Intuitively, this has the consequence that all points of the space are " lumped together " and cannot be distinguished by topological means ; it belongs to a pseudometric space in which the distance between any two points is zero.

Intuitively spoken, singular homology counts, for each dimension n, the n-dimensional holes of a space.

In mathematics, an extreme point of a convex set S in a real vector space is a point in S which does not lie in any open line segment joining two points of S. Intuitively, an extreme point is a " vertex " of S.

Intuitively, the fundamental group measures how the holes behave on a space ; if there are no holes, the fundamental group is trivial — equivalently, the space is simply connected.

Intuitively, the polar decomposition separates A into a component that stretches the space along a set of orthogonal axes, represented by P, and a rotation represented by U. The decomposition of the complex conjugate of is given by.

Intuitively, a contractible space is one that can be continuously shrunk to a point.

Intuitively, an algebraic space is a scheme modulo a " nice " equivalence relation ; the resulting category of algebraic spaces extends the category of schemes and allows to carry out several natural constructions that are needed for example in deformation theory or in the construction of moduli spaces but are not possible in the smaller category of schemes.

Intuitively, for two sets S and T to have the same cardinality means that it is possible to " pair off " elements of S with elements of T in such a fashion that every element of S is paired off with exactly one element of T and vice versa.

Intuitively, it is the " area so far " function of the probability distribution.

Intuitively, an expander is a finite, undirected multigraph in which every subset of the vertices " which is not too large " has a " large " boundary.

Intuitively, a field is a set F that is a commutative group with respect to two compatible operations, addition and multiplication, with " compatible " being formalized by distributivity, and the caveat that the additive identity ( 0 ) has no multiplicative inverse ( one cannot divide by 0 ).

Weather and GW Bridge traffic screen off other influences. Intuitively, this reduction in representation size is achieved simply because each variable depends only on a subset of the other variables.

Intuitively, one can say that the air follows the curve of the foil, but this is not very rigorous or precise.

Intuitively, a lambda abstraction represents an anonymous function that takes a single input, and the is said to bind in, and an application represents the application of input to some function.

Intuitively, NP is the set of all decision problems for which the instances where the answer is " yes " have efficiently verifiable proofs of the fact that the answer is indeed " yes ".

Intuitively then, the oracle machine can perform all of the usual operations of a Turing machine, and can also query the oracle for an answer to a specific question of the form " is x in A?

Intuitively, the probability that any single number is divisible by a prime ( or any integer ), p is 1 / p.

) Intuitively, is a quine, a function that returns its own source code ( Gödel number ), except that rather than returning it directly, passes its Gödel number to and returns the result.

#: Intuitively, minimisation seeks -- beginning the search from 0 and proceeding upwards -- the smallest argument that causes the function to return zero ; if there is no such argument, the search never terminates.

Intuitively, multiplying by a scalar r stretches a vector out by a factor of r. Geometrically, this can be visualized ( at least in the case when r is an integer ) as placing r copies of the vector in a line where the endpoint of one vector is the initial point of the next vector.

Intuitively, the reasoning is that, as web crawlers have a limit to how many pages they can crawl in a given time frame, ( 1 ) they will allocate too many new crawls to rapidly changing pages at the expense of less frequently updating pages, and ( 2 ) the freshness of rapidly changing pages lasts for shorter period than that of less frequently changing pages.

Intuitively then, a meromorphic function is a ratio of two well-behaved ( holomorphic ) functions.

Intuitively, we can think of this as being all meromorphic functions whose poles at every point are no worse than the corresponding coefficient in D ; if the coefficient in D at z is negative, then we require that h has a zero of at least that multiplicity at z – if the coefficient in D is positive, h can have a pole of at most that order.

Intuitively, non-strict functions correspond to control structures.

Intuitively, this says that if one writes a function which is polynomial-time assuming that function calls are constant-time, and if those called functions themselves require polynomial time, then the entire algorithm takes polynomial time.

Intuitively this notation groups functions according to their growth respective to some parameter ; as such, the notation is abusive in two respects:

Intuitively, the process can be pictured as follows: first shrink the real line to the open interval (- π, π ) on the x-axis ; then bend the ends of this interval upwards ( in positive y-direction ) and move them towards each other, until you get a circle with one point ( the topmost one ) missing.

Intuitively this is because it required a choice, rigorously because any such choice of isomorphisms will not commute with all linear maps ; see for detailed discussion.

Intuitively, the kernel of the morphism f: X → Y is the " most general " morphism k: K → X that yields zero when composed with ( followed by ) f.

Intuitively, an index falls into this set if and only if for every " there is an such that the Turing machine with index halts on input after steps ”.

Intuitively, a stochastic matrix represents a Markov chain with no sink states, this implies that the application of the stochastic matrix to a probability distribution would redistribute the probability mass of the original distribution while preserving its total mass.

Intuitively, this reflects the fact that reliability involves freedom from random error and random errors do not correlate with one another.

If, for example, we simply look at a curve in the real affine plane there might be singular P modulo the stalk, or alternatively as the sum of m ( m − 1 )/ 2, where m is the multiplicity, over all infinitely near singular points Q lying over the singular point P. Intuitively, a singular point with delta invariant δ concentrates δ ordinary double points at P. For an irreducible and reduced curve and a point P we can define δ algebraically as the length of where is the local ring at P and is its integral closure.

Intuitively, a vector field is best visualized as an ' arrow ' attached to each point of a region, with variable length and direction.

Intuitively, one can think of the m constraints as reducing the problem to one with n-m free variables.

Intuitively, since propellant is by far the largest part of a rocket, propellant costs would be expected to be significant, but it turns out that with hydrocarbon fuel these costs can be under $ 50 per kg of payload.

Intuitively one then naturally expects that the constant C is itself positive, and with some work this can be proved.

Intuitively, in such a case, starting from what we know about the parameter prior to observing the data point, we then can update our knowledge based on the data point and end up with a new distribution of the same form as the old one.

Intuitively, the procedure proceeds step by step, with a specific rule to cover what to do at each step of the calculation.

Intuitively, a dessin d ' enfant is simply a graph, with its vertices colored alternating black and white, embedded onto an oriented surface that, in many cases, is simply a plane.

Intuitively, given a function which is rather irregular, by convolving it with a mollifier the function gets " mollified ", that is, its sharp features are smoothed, while still remaining close to the original nonsmooth ( generalized ) function.

Intuitively, the cost function encourages factories with high flows between each other to be placed close together.

Intuitively, an infinite graph has arbitrarily large finite subgraphs with any density less than its upper density, and does not have arbitrarily large finite subgraphs with density greater than its upper density.