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 ).

Intuitively, a good separation is achieved by the hyperplane that has the largest distance to the nearest training data point of any class ( so-called functional margin ), since in general the larger the margin the lower the generalization error of the classifier.

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, φ is a self-referential sentence saying that φ has the property ψ.

Intuitively, one can reason that Hick's Law has a logarithmic form because people subdivide the total collection of choices into categories, eliminating about half of the remaining choices at each step, rather than considering each and every choice one-by-one, requiring linear time.

Intuitively, the algorithm has been promised that the input does indeed belong to set of yes instances or no instances.

Intuitively, as an example, consider a function ƒ whose singular support is concentrated on a smooth curve in the plane at which the function has a jump discontinuity.

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.

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?

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 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, a minimal sufficient statistic most efficiently captures all possible information about the parameter θ.

Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region.

Intuitively we would expect it to be even more unlikely for all 5 marbles to be white.

Intuitively all dependencies are the result of keys.

Intuitively, one can think of the radical of I as obtained by taking all the possible roots of elements of I. Rad ( I ) turns out to be an ideal itself, containing I.

Intuitively, this is allowed in the antecedent because we can always restrict the scope of our proof ( if all cars have wheels, then it's safe to say that all black cars have wheels ); and in the succedent because we can always allow for alternative conclusions ( if all cars have wheels, then it's safe to say that all cars have either wheels or wings ).

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 speaking, part of the graph of a function is rotated around an axis, and is modelled by an infinite number of hollow pipes, all infinitely thin.

Intuitively, a system simulates another system if it can match all of its moves.

Intuitively, the algorithm follows all chains of inference after making each of its choices ; this either leads to a contradiction and a backtracking step, or, if no contradiction is derived, it follows that the choice was a correct one that leads to a satisfying assignment.

Intuitively, the marginal probability of X is computed by examining the conditional probability of X given a particular value of Y, and then averaging this conditional probability over the distribution of all values of Y.

Intuitively, in a forward flow problem, it would be fastest if all

Intuitively, the theorem states that to build a VMM it is sufficient that all instructions that could affect the correct functioning of the VMM ( sensitive instructions ) always trap and pass control to the VMM.

Intuitively, the distinction is that the wavefronts of plane waves are truly planar ; all points on a given two-dimensional wavefront are equivalent.

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

Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is no greater than a definite real number ; this bound is called the function's " Lipschitz constant " ( or " modulus of uniform continuity ").

Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its length, and in so doing will minimize its energy.

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 speaking, this means that although may be large, its image must be small in the sense of Lebesgue measure: while may have many critical points in the domain, it may have few critical values in the image.

Intuitively, a ray consists of those points on a line passing through A and proceeding indefinitely, starting at A, in one direction only along the line.

Intuitively, maximizing the function above is equivalent to pulling the points as far away from each other as possible and therefore " unfold " the manifold.

Intuitively, we make X into a cylinder and collapse both ends to two points.

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.

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

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, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.

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.