Intuitively, the continuous operator A never " lengthens " any vector more than by a factor of c. Thus the image of a bounded set under a continuous operator is also bounded.

Intuitively, a continuous curve in 2 or 3 ( or higher ) dimensions can be thought of as the path of a continuously moving point.

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

Intuitively, partial function application says " if you fix the first arguments of the function, you get a function of the remaining arguments ".

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

Intuitively, represents a small perturbation in the index of T. By noting that and that is always between 0 and 1, can be used to ignore the floor function in the index.

Intuitively the flow rate is a function of the cross section area of the tube bore.

Intuitively, an inverse function is obtained from by interchanging the roles of the dependent and independent variables.

Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not.

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, the sentence " for every x there exists a y such that R ( x, y )" is converted into the equivalent form " there exists a function f mapping every x into a y such that, for every x it holds R ( x, f ( x ))".

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, 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, a submodular function over the subsets demonstrates " diminishing returns ".

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

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, 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, 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, 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, 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.

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 seems to mean that the number of inputs and the number of outputs can always be equal in each subswitch, but intuition does not prove this can be done nor does it tell us how to do so.

Intuitively, mutual information measures the information that X and Y share: it measures how much knowing one of these variables reduces uncertainty about the other.

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 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 means there are computational limits on what can be proven by computer programs.

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.

Intuitively, this finds the filter that is as close as you can get to the desired response given that you can use only N coefficients.

Intuitively, one can view a unit as the smallest testable part of an application.

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