In this case, is the smallest σ-algebra that contains the open intervals of R. While there are many Borel measures μ, the choice of Borel measure which assigns for every interval is sometimes called " the " Borel measure on R. In practice, even " the " Borel measure is not the most useful measure defined on the σ-algebra of Borel sets ; indeed, the Lebesgue measure is an extension of " the " Borel measure which possesses the crucial property that it is a complete measure ( unlike the Borel measure ).

The electric field was formally defined as the force exerted per unit charge, but the concept of potential allows for a more useful and equivalent definition: the electric field is the local gradient of the electric potential.

Electronic noise is defined as unwanted disturbances superposed on a useful signal that tend to obscure its information content.

Modern customs and institutions offer few useful parallels to the legal and social context which defined the gladiatoria munera Under law, anyone condemned to the arena or the gladiator schools ( ad ludum ) was a servus poenae under sentence of death unless manumitted.

This version is useful because most well-studied complexity classes are defined as classes of decision problems, not function problems.

The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed ( distance per unit of time ).

Since organizations can be viewed as systems, management can also be defined as human action, including design, to facilitate the production of useful outcomes from a system.

For example, components and modules are useful for large-scale programming, but the interaction of macros and these other constructs must be defined for their use together.

A related inversion formula more useful in combinatorics is as follows: suppose F ( x ) and G ( x ) are complex-valued functions defined on the interval < nowiki >

Exponentiation to the power of zero can also be defined if the operation has an identity element, so the existence of identity elements becomes especially useful in power-associative contexts.

Beyond those modern-use characters, all others may be defined to be obsolete or rare ; these are better candidates for private-use registration than for congesting the public list of generally useful Unicodes.

By Legendre-transforming this expression, he defined the concepts of enthalpy and " free energy " ( now universally known as the " Gibbs free energy "), a thermodynamic potential which is especially useful to chemists since it determines whether a reaction will proceed spontaneously at a fixed temperature and pressure.

Beyond the ability of two or more computer systems to exchange information, semantic interoperability is the ability to automatically interpret the information exchanged meaningfully and accurately in order to produce useful results as defined by the end users of both systems.

An approach using reduced protein representation ( pseudo-atoms representing groups of atoms are defined ) and statistical potential are useful in protein structure prediction and modeling of the folding pathways.

It can be broadly defined as anything that children read, but a more useful definition may be fiction, poetry and drama intended for and used by children and young people, a list to which many add non-fiction.

Chomsky argued that the notions " grammatical " and " ungrammatical " could be defined in a meaningful and useful way.

We say that the base generates the topology T. Bases are useful because many properties of topologies can be reduced to statements about a base generating that topology, and because many topologies are most easily defined in terms of a base which generates them.

This WAIS measurement would only be useful to the extent it could be shown to be related to other operationally defined measurements, e. g., to the measured probability of graduation from university.

( Note that the contact X on the second rung serves no useful purpose, as X is already defined in the ' AND ' function of S from the 1st rung.

Automata are defined to study useful machines under mathematical formalism.

The Protestant reformer John Calvin said that repentance " may be justly defined to be “ a true conversion of our life to God, proceeding from a serious fear of God, and consisting in the mortification of the flesh and of the old man, and in the vivification of the Spirit .” He further said that " it will be useful to amplify and explain the definition we have given ; in which there are three points to be particularly considered.

Natural experiments are most useful when there has been a clearly defined and large change in the treatment ( or exposure ) to a clearly defined subpopulation ( and no change to a comparable subpopulation ), so that changes in responses may be plausibly attributed to the change in treatments ( or exposure ).< ref name =" DiNardo " >

There is a useful geometrical way of looking at the adjoint of an operator A on H as follows: we consider the graph G ( A ) of A defined by

Although this notation is compact and convenient for some purposes, the most useful notation is the Legendre symbol, also called the quadratic character, which is defined for all integers a and positive odd prime numbers p as

Technology is defined here as a creative invention, either in the form of an object or a methodology.

Until the invention of the pendulum and the development of reliable clocks during the 17th century, the equation of time as defined by Ptolemy remained a curiosity, of importance only to astronomers.

The scope of the patented invention or the extent of protection is defined in the claims of the granted patent.

There is no official symbol for the currency but the Iranian standard ISIRI 820 defined a symbol for use on typewriters ( mentioning that it is an invention of the standards committee itself ) and the two Iranian standards ISIRI 2900 and ISIRI 3342 define a character code to be used for it.

The word " glagolitic " is defined as an adjective meaning " of, relating to, or denoting a Slavic alphabet whose invention is attributed to Saint Cyril, preserved only in certain Roman Catholic liturgical books found in Dalmatia " from New Latin ' glagoliticus ', from Serbo-Croat ' glagolica ' the Glagolitic alphabet ; related to Old Church Slavonic ' glagolŭ ' word.

A subject invention is defined as " any invention of the contractor that is conceived or first actually reduced to practice in the performance of work under a funding agreement.

Invention in the U. S. is generally defined to comprise two steps: ( 1 ) conception of the invention and ( 2 ) reduction to practice of the invention.

It was emphatically rural and egalitarian in its orientation, and it trended to glorify the simple ways of life of the countryside rather than revel in urban grandeur of the military pride of the state … The irony of this invention of nationhood is the fact that those individuals who most strongly promoted the idea of Norwegianness as a rural form of life, were themselves urban and highly educated people – their daily life was very far removed from that of the simple peasants who they defined as the carriers of national identity .”

In some jurisdictions, such as the United States, the patent holder may even seek to broaden the scope of the invention defined in the claims by filing a reissue application, although a broadening reissue in the USA must be filed within 2 years from grant.

Patent protection was first excluded by Loi n ° 68-1 du 2 janvier 1968 sur les brevets d ' invention and defined in copyright by Loi n ° 85-660 du 3 juillet 1985 relative aux droits d ' auteur et aux droits des artistes-interprètes, des producteurs de phonogrammes et de vidéogrammes et des entreprises de communication audiovisuelle.

A small business is defined as one which is independently owned and operated and which is not dominant in its field.

Where boundary maintenance describes the boundaries or limits of the group, systemic linkage is defined `` as the process whereby one or more of the elements of at least two social systems is articulated in such a manner that the two systems in some ways and on some occasions may be viewed as a single unit.

Even those who appear in only one or two scenes are full personalities, defined with economical precision.

Used in this sense, it usually contrasted to egoism, which is defined as acting to the benefit of one self.

The ampere was originally defined as one tenth of the CGS system electromagnetic unit of current ( now known as the abampere ), the amount of current that generates a force of two dynes per centimetre of length between two wires one centimetre apart.

Most narrowly, one may consider numbers explicitly defined in terms of polynomials, exponentials, and logarithms – this does not include algebraic numbers, but does include some simple transcendental numbers such as e or log ( 2 ).

However, this school was not one of the " Ten Schools " defined by Hanshu.

In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

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

In this case, the absolute magnitude is defined as the apparent magnitude that the object would have if it were one astronomical unit ( AU ) from both the Sun and the observer.

This is one of a series of articles about the differences between British English and American English, which, for the purposes of these articles, are defined as follows:

Tolstoy defined art ( and by no coincidence also characterized its value ) as the following: " Art is a human activity consisting in this, that one man consciously, by means of certain external signs, hands on to others feelings he has lived through, and that other people are infected by these feelings and also experience them.

Since one degree is defined as one three hundred and sixtieth () of a rotation, one minute of arc is of a rotation.

Critics of the SAP argue in favor of a weak anthropic principle ( WAP ) similar to the one defined by Brandon Carter, which states that the universe's ostensible fine tuning is the result of selection bias: i. e., only in a universe capable of eventually supporting life will there be living beings capable of observing any such fine tuning, while a universe less compatible with life will go unbeheld.

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 ).

To holders of this view, one's orientation is defined by whom one is predisposed to fall in love with, whether or not one desires that person sexually.

The acute withdrawal phase can be defined as lasting between one to three weeks.

Moderate use is defined by The Dietary Guidelines for Americans as no more than two alcoholic beverages a day for men and no more than one alcoholic beverage a day for women.

In information theory, one bit is typically defined as the uncertainty of a binary random variable that is 0 or 1 with equal probability, or the information that is gained when the value of such a variable becomes known.

For instance, division of real numbers is a partial function, because one can't divide by zero: a / 0 is not defined for any real a.