We consider now the graph of the function f{t} on Af.

The number of ordinary values of the function f{t} at T will be called its multiplicity at T.

The values Af are the ordinary values at Af of a multi-valued function g{t} which has components corresponding to those of f{t}.

In some neighborhood of an isolated tangent point in the f-plane, say Af, the function Af is either double-valued or has no values defined, except at the tangent point itself, where it is single-valued.

In the second category the function Af has no values defined in a neighborhood Af.

A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:

: For any set X of nonempty sets, there exists a choice function f defined on X.

Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers.

Area can be defined as a function from a collection M of special kind of plane figures ( termed measurable sets ) to the set of real numbers which satisfies the following properties:

Each gate consisted of one inverting vacuum tube amplifier, preceded by a resistor divider input network that defined the logical function.

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

One common version, the two-argument Ackermann – Péter function, is defined as follows for nonnegative integers m and n:

Ackermann's three-argument function,, is defined such that for p = 0, 1, 2, it reproduces the basic operations of addition, multiplication, and exponentiation as

Ackermann's original three-argument function is defined recursively as follows for nonnegative integers m, n, and p:

Of the various two-argument versions, the one developed by Péter and Robinson ( called " the " Ackermann function by some authors ) is defined for nonnegative integers m and n as follows:

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

In a category with exponentials, using the isomorphism ( in computer science, this is called currying ), the Ackermann function may be defined via primitive recursion over higher-order functionals as follows:

where Succ is the usual successor function and Iter is defined by primitive recursion as well:

The function ƒ ( x ) may or may not be defined at a, and its precise value at the point x = a does not affect the asymptote.

In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function ƒ ( n ) defined on the set of natural numbers ( i. e. positive integers ) that " expresses some arithmetical property of n ."

An example of an arithmetic function is the non-principal character ( mod 4 ) defined by

`` Mr. Gross, your report says that ' our function is investigative and advisory and does not in any way derogate from or prejudice Mr. Bang-Jensen's rights as a staff member.

The relation between osmotic pressure and the Gibbs function may also be developed in an analogous way.

In a way the rules of evidence can function to give a judge limited inquisitorial powers as the judge may exclude evidence he / she believes is not trustworthy or irrelevant to the legal issue at hand.

Careful study of the different function and processes of the eyes, ears, and other organs paved the way for improved cameras, television, radio transmitters and receivers, and many other useful tools.

His own view is that consciousness has subjective, first-person causal powers by being essentially intentional due simply to the way human brains function biologically ; conscious persons can perform computations, but consciousness is not inherently computational the way computer programs are.

All physical quantities are defined this way at each instant of time, in the current configuration, as a function of the vector position.

Another way of computing this derivative is to view the composite function as the composite of and h. Applying the chain rule to this situation gives:

One of these, Itō's lemma, expresses the composite of an Itō process ( or more generally a semimartingale ) dX < sub > t </ sub > with a twice-differentiable function f. In Itō's lemma, the derivative of the composite function depends not only on dX < sub > t </ sub > and the derivative of f but also on the second derivative of f. The dependence on the second derivative is a consequence of the non-zero quadratic variation of the stochastic process, which broadly speaking means that the process can move up and down in a very rough way.

In mathematics and computer science, currying is the technique of transforming a function that takes multiple arguments ( or an n-tuple of arguments ) in such a way that it can be called as a chain of functions each with a single argument ( partial application ).

Gurevich adds the pointer machine model of Kolmogorov and Uspensky ( 1953, 1958 ): "... they just wanted to ... convince themselves that there is no way to extend the notion of computable function.

All of these drugs affect cell division or DNA synthesis and function in some way.

The equations are one way of looking at the condition on a function to be differentiable in the sense of complex analysis: in other words they encapsulate the notion of function of a complex variable by means of conventional differential calculus.

The manner in which Xavier's powers function indicates that his telepathy is physical in some way, as it can be enhanced by physical means ( for example, Cerebro ) and can be disrupted by physical means ( for example, Magneto's alteration of the Earth's magnetic field ).

If the limit exists, meaning that there is a way of choosing a value for Q ( 0 ) that makes the graph of Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q ( 0 ).

Linking a particular generic function to a method in a class is accomplished this way:

An equivalent way to view an extractor is as a bivariate function

A variety of definitions exist: F. Stuart Chapin and coauthors define it as " the application of ecological science to resource management to promote long-term sustainability of ecosystems and the delivery of essential ecosystem goods and services ", while Norman Christensen and coauthors defined it as " management driven by explicit goals, executed by policies, protocols, and practices, and made adaptable by monitoring and research based on our best understanding of the ecological interactions and processes necessary to sustain ecosystem structure and function " and Peter Brussard and colleagues defined it as " managing areas at various scales in such a way that ecosystem services and biological resources are preserved while appropriate human use and options for livelihood are sustained ".

Functional decomposition refers broadly to the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed ( i. e., recomposed ) from those parts by function composition.

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.

Unconcerned with the practical function of his actions, the dancer is engrossed exclusively in their `` motional content ''.

This is the primary function of the imagination operating in the absence of the original experiential stimulus by which the images were first appropriated.

But because it is the function of the mind to turn the one into the other by means of the capacities with which words endow it, we do not unwisely examine the type of distinction, in the sphere of politics, on which decisions hang.

It is, however, a disarming disguise, or perhaps a shield, for not only has Mercer proved himself to be one of the few great lyricists over the years, but also one who can function remarkably under pressure.

Men seem almost universally to want a sense of function, that is, a feeling that their existence makes a difference to someone, living or unborn, close and immediate or generalized.

And there is one other point in the Poetics that invites moral evaluation: Aristotle's notion that the distinctive function of tragedy is to purge one's emotions by arousing pity and fear.

It is possible that international organization will ultimately supplant the multi-state system, but its proper function for the immediate future is to reform and supplement that system in order to render pluralism more compatible with an interdependent world.

A primary function is the operation of a Government Bid Center, which receives bids daily from the Federal Government's principal purchasing agencies.

The education function of the Institute is carried on by the staff in the departments of pathology and its consultants.

The responsibility for taking the initiative in generating ideas is that of every officer in the Department who has a policy function, regardless of rank.

This function is staffed by engineers chosen for their technical competence and who have the title, member of the technical staff.

If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.

This is interesting for it combines both the thermodynamic concept of a minimum Gibbs function for equilibrium and minimum mechanical potential energy for equilibrium.

The concept of the strain energy as a Gibbs function difference Af and exerting a force normal to the shearing face is compatible with the information obtained from optical birefringence studies of fluids undergoing shear.

A proton magnetic resonance study of polycrystalline Af as a function of magnetic field and temperature is presented.

Within certain wide limits anatomy dictates function and, if one is permitted to speculate, potential pathology should be included in this statement as well.

When each number of successes X is paired with its probability of occurrence Af, the set of pairs Af, is a probability function called a binomial distribution.

In some neighborhood in the f-plane of any ordinary point of the graph, the function f is a single-valued, continuous function.