Intuitively, one would expect systems which collect more than one type of data to be more useful than systems which collect only one type of information ( such as single-purpose laboratory or 911 call-center based systems ), and be less prone to false alarms, and this appears to be the case.

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

Intuitively, if one pulled and stretched out the paths in the maze in the proper way, the result could be made to resemble a tree.

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, one would not expect this pattern from firing a single particle at the slits, because the particle should pass through one slit or the other, not a complex overlap of both.

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, one should think of a + b as the " union " or the " least upper bound " of a and b and of ab as some multiplication which is monotonic, in the sense that a ≤ b implies ax ≤ bx.

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

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

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, given an equation that one is seeking to solve,

Intuitively one expects that the waves in the system return to the initial state, as characterized by the amplitudes and phases ( and accounting for the passage of time ).

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

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, this is caused by the storm simply blowing the water towards one side of the basin in the direction of its winds.

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, one may think that the larger the weight for a criterion is, the more critical that criterion should be.

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, if we placed a small paddle wheel there, the larger " current " on its right side would cause the paddlewheel to rotate clockwise, which corresponds to a curl in the negative z direction.

Intuitively, this chance may seem small.

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 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 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, 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, 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 ).

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

#: 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, there are many cases where people do want to take the numbers involved into account.

Intuitively, it seems clear that the answer should be more than a half, since there are more plain cookies in bowl # 1.

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 ”.

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 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, 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, it is thought that there are more positive integers than perfect squares, since every perfect square is already positive, and many other positive integers exist besides.

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.

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, 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, 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.