The set of intersections of Af, the rotated curve, with the original curve C consists of just the set of forward corner points on C corresponding to the vertex at Af, plus the vertex itself.

In the C-plane we construct a set of rectangular Cartesian coordinates u, V with the origin at Q and such that both C and Af have finite slope at Q.

The fact that there can not be any limit points of the set except in closed intervals follows from the argument used in Lemma 1, namely, that near any tangent point in the C-plane the curves C and Af are analytic, and therefore the difference between them must be a monotone function in some neighborhood on either side of the tangent point.

We first define a function b{t} as follows: given the set of squares such that each has three corners on C and vertex at t, b{t} is the corresponding set of positive parametric differences between T and the backward corner points.

Compound comparisons typically compare two sets of groups means where one set has two or more groups ( e. g., compare average group means of group A, B and C with group D ).

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

This guarantees for any partition of a set X the existence of a subset C of X containing exactly one element from each part of the partition.

We will abbreviate " Zermelo-Fraenkel set theory plus the negation of the axiom of choice " by ZF ¬ C.

Science fiction set in what was the future but is now the past, like Arthur C. Clarke's 2001: A Space Odyssey or Nineteen Eighty-Four, are not alternate history because the author has not made the conscious choice to change the past.

* Consider a set with three elements, A, B, and C. The following operation:

* 1814 – British troops invade Washington, D. C. and during the Burning of Washington the White House is set ablaze, though not burned to the ground ; as well as several other buildings.

C. S. Lewis supported this argument and challenged the evolutionary naturalistic view of morality – that morality evolved and is a human construct – by arguing that without objective moral truths, moral scepticism would set in, leading to moral anarchy.

Both the highest and lowest temperature records for the state were set in the Interior, with 100 ° F ( 38 ° C ) in Fort Yukon and − 80 ° F (− 64 ° C ) in Prospect Creek.

The C and C ++ programming languages, for example, define byte as an " addressable unit of data storage large enough to hold any member of the basic character set of the execution environment " ( clause 3. 6 of the C standard ).

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

Note that a locally finite Borel measure automatically satisfies μ ( C ) < ∞ for every compact set C.

* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

" Analog " indicates something that is mathematically represented by a set of continuous values ; for example, the analog clock uses constantly-moving hands on a physical clock face, where moving the hands directly alters the information that clock is providing.

Bandwidth is the difference between the upper and lower frequencies in a continuous set of frequencies.

If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B ( X, Y ).

Besides living together under continuous observation, which is the major attraction of the contest, the program relies on four basic props: The stripped-bare, back-to-basics environment in which they live, the evictions, the weekly tasks and competitions set by Big Brother and the " Diary / Confession Room ", in which the housemates individually convey their thoughts, feelings, and frustrations and reveal their nominees for eviction.

For a rigorous definition of basis with a continuous set of indices and consequently for a rigorous definition of position and momentum basis see.

The set of invertible elements in any unital Banach algebra is an open set, and the inversion operation on this set is continuous, ( and hence homeomorphism ) so that it forms a topological group under multiplication.

For instance, any continuous function defined on a compact space into an ordered set ( with the order topology ) such as the real line is bounded.

In fact, every compact metric space is a continuous image of the Cantor set.

* Every continuous map from a compact space to a Hausdorff space is closed and proper ( i. e., the pre-image of a compact set is compact.

If we set, then η is continuous at 0.

A function from the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane ; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no " holes " or " jumps ".

If is continuous in an open set Ω and the partial derivatives of ƒ with respect to x and y exist in Ω, and satisfies the Cauchy – Riemann equations throughout Ω, then ƒ is holomorphic ( and thus analytic ).

If X is a topological space and M is a complete metric space, then the set C < sub > b </ sub >( X, M ) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B ( X, M ) and hence also complete.

Each station transmitted a continuous wave signal that, by comparing the phase difference of the signals from the Master and one of the Slaves, resulted in a set of hyperbolic lines of position called a pattern.

For instance, by assigning to every open set U the associative algebra of real-valued continuous functions on U, one obtains a presheaf of algebras on X.

We are interested in the following set of continuous functions called loops with base point x < sub > 0 </ sub >.

Numbers could be represented in a continuous " analog " form, for instance a voltage or some other physical property was set to be proportional to the number.

If one identifies C with R < sup > 2 </ sup >, then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the Cauchy-Riemann equations, a set of two partial differential equations.

One set of ploughs was right-handed, and the other left-handed, allowing continuous ploughing along the field, as with the turnwrest and reversible ploughs.

Quantization is the procedure of constraining something from a relatively large or continuous set of values ( such as the real numbers ) to a relatively small discrete set ( such as the integers ).

A continuous random variable maps outcomes to values of an uncountable set ( e. g., the real numbers ).

The symbol μ denotes a positive measure: that is, a real-valued positive set function defined on a σ-algebra which is countably additive.

If X is a real-valued random variable and a is a number then the event X ≤ a is the set of outcomes whose corresponding value of X is less than or equal to a.

* The set of all real-valued continuous functions on a separable space has a cardinality less than or equal to c. This follows since such functions are determined by their values on dense subsets.

Let be the space of real-valued continuous functions on X which vanish at infinity ; that is, a continuous function f is in if, for every, there exists a compact set such that on

For instance, a real-valued measurable function is a function for which the preimage of each Borel set is measurable.

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity.

The multivariate normal distribution is often used to describe, at least approximately, any set of ( possibly ) correlated real-valued random variables each of which clusters around a mean value.

For real-valued functions of several real variables, this holds if and only if the absolute value of the slopes of all secant lines are bounded by K. The set of lines of slope K passing through a point on the graph of the function forms a circular cone, and a function is Lipschitz if and only if the graph of the function everywhere lies completely outside of this cone ( see figure ).

* Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.

In particular the set of all real-valued Lipschitz functions on a compact metric space X having Lipschitz constant ≤ K is a locally compact convex subset of the Banach space C ( X ).

A vector field V defined on a set S is called a gradient field or a conservative field if there exists a real-valued function ( a scalar field ) f on S such that

A pseudometric space is a set together with a non-negative real-valued function ( called a pseudometric ) such that, for every,

) Typically, features are either categorical ( also known as nominal, i. e. consisting of one of a set of unordered items, such as a gender of " male " or " female ", or a blood type of " A ", " B ", " AB " or " O "), ordinal ( consisting of one of a set of ordered items, e. g. " large ", " medium " or " small "), integer-valued ( e. g. a count of the number of occurrences of a particular word in an email ) or real-valued ( e. g. a measurement of blood pressure ).

* algebras of functions, such as the R-algebra of all real-valued continuous functions defined on the interval, or the C-algebra of all holomorphic functions defined on some fixed open set in the complex plane.

Let U be an open set in R < sup > n </ sup > and φ: U → R < sup > n </ sup > an injective differentiable function with continuous partial derivatives, the Jacobian of which is nonzero for every x in U. Then for any real-valued, compactly supported, continuous function f, with support contained in φ ( U ),

For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set.

A cardinal κ is called real-valued measurable if there is an atomless κ-additive measure on the power set of κ.

A real-valued function f on an interval ( or, more generally, a convex set in vector space ) is said to be concave if, for any x and y in the interval and for any t in,

Some books say that range of this function is its codomain, the set of all real numbers, reflecting that the function is real-valued.

In systems governed by classical mechanics, any experimentally observable value can be shown to be given by a real-valued function on the set of all possible system states.

In mathematics, the zero set of a real-valued function f: X → R ( or more generally, a function taking values in some additive group ) is the subset of X ( the inverse image of

A real-valued function, whose domain is any set, can have a global maximum and minimum.