formulasearchengine - User contributions [en]

Microfiltration

2014-08-05T16:14:31Z

140.247.0.35: /* Sterilisation */

My name is Kaitlyn and I am studying Physics and Educational Policy Studies at Coburg / Germany. Feel free to visit my website :: [http://www.gorod-nadym.ru/easy-solutions-assist-you-get-high-quality-website-hosting Hostgator Vouchers]

Jensen's formula

2013-11-18T03:58:04Z

140.247.0.29: minor grammatical improvement; inserting an O-umlaut special character in "Möbius"

<ref>[http://www.theory.caltech.edu/~preskill/ph229]</ref>{{Refimprove|date=October 2009}}

In [[physics]], a '''superoperator''' is a [[linear operator]] acting on a [[vector space]] of [[Linear map|linear operators]].
Sometimes the term refers more specially to a [[completely positive map]] which does not increase or preserves the [[trace (linear algebra)|trace]] of its [[Parameter|argument]].

This specialized meaning is used extensively in the field of [[quantum computing]], especially [[quantum programming]], as they characterise mappings between [[density matrix|density matrices]].

The use of the '''super-''' prefix here is in no way related to its other use in mathematical physics. That is to say superoperators have no connection to [[supersymmetry]] and [[superalgebra]] which are extensions of the usual mathematical concepts defined by extending the [[Ring_(mathematics)|ring]] of numbers to include [[Grassmann number]]s. Since superoperators are themselves operators the use of the '''super-''' prefix is used to distinguish them from the operators upon which they act.
==Example von Neumann Equation==
In [[quantum mechanics]] the [[Schrödinger Equation]], <math>i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi</math> expresses the time evolution of the state vector <math>\psi</math> by the action of the Hamiltonian <math>\hat{H}</math> which is an operator mapping state vectors to state vectors.

In the more general formulation of [[John von Neumann]], statistical states and ensembles are expressed by [[density operator]]s rather than state vectors.
In this context the time evolution of the density operator is expressed via the [[von Neumann equation]] in which density operator is acted upon by a '''superoperator''' <math>\mathcal{H}</math> mapping operators to operators. It is defined by taking the [[commutator]] with respect to the Hamiltonian operator:

<math>i \hbar \frac{\partial}{\partial t}\rho = \mathcal{H}[\rho]</math>

where

<math>\mathcal{H}[\rho] = [\hat{H},\rho] \equiv \hat{H}\rho - \rho\hat{H}</math>

As commutator brackets are used extensively in QM this explicit superoperator presentation of the Hamiltonian's action is typically omitted.

==Example Derivatives of Functions on the Space of Operators==
When considering an operator valued function of operators <math>\hat{H} = \hat{H}(\hat{P})</math> as for example when we define the quantum mechanical Hamiltonian of a particle as a function of the position and momentum operators, we may (for whatever reason) define an “Operator Derivative” <math> \frac{\Delta \hat{H}}{\Delta \hat{P}} </math>
as a '''superoperator''' mapping an operator to an operator.

For example if <math> H(P) = P^3 = PPP</math> then its operator derivative is the superoperator defined by:

<math> \frac{\Delta H}{\Delta P}[X] = X P^2 + PXP + P^2X</math>

This “operator derivative” is simply the [[Jacobian matrix]] of the function (of operators) where one simply treats the operator input and output as vectors and expands the space of operators in some basis. The Jacobian matrix is then an operator (at one higher level of abstraction) acting on that vector space (of operators).
==See also==
[[Lindblad superoperator]]
==References==
{{Reflist}}

[[Category:Quantum information theory]]

Cap product

2013-11-13T23:42:39Z

140.247.0.116: /* The slant product */

{{Unreferenced|date=December 2009}}
In [[mathematics]], the '''regular part''' of a [[Laurent series]] consists of the series of terms with positive powers. That is, if

:<math>f(z) = \sum_{n=-\infty}^{\infty} a_n (z - c)^n,</math>

then the regular part of this Laurent series is

:<math>\sum_{n=0}^{\infty} a_n (z - c)^n.</math>

In contrast, the series of terms with negative powers is the [[principal part]].

{{DEFAULTSORT:Regular Part}}
[[Category:Complex analysis]]

Pauli–Villars regularization

2013-11-12T18:43:31Z

140.247.0.107: grammar (incomplete sentence)

{{Use British (Oxford) English|date=January 2012}}
'''ISO 31-0''' is the introductory part of [[International Organization for Standardization|international standard]] [[ISO 31]] on [[physical quantity|quantities]] and [[physical unit|units]]. It provides guidelines for using physical quantities, quantity and unit symbols, and coherent unit systems, especially the [[SI]]. It is intended for use in all fields of science and technology and is augmented by more specialized conventions defined in other parts of the [[ISO 31]] standard. It is superseded by [[ISO 80000-1]].

==Scope==
ISO 31 covers only [[physical quantities]] used for the quantitative description of physical phenomena. It does not cover [[conventional]] scales (e.g., [[Beaufort scale]], [[Richter scale]], colour intensity scales), results of conventional tests, currencies, or information content. The presentation here is only a brief summary of some of the detailed guidelines and examples given in the standard.

==Quantities and units==
Physical quantities can be grouped into mutually comparable categories. For example, length, width, diameter and wavelength are all in the same category, that is they are all ''quantities of the same kind''. One particular example of such a quantity can be chosen as a reference quantity, called the ''unit'', and then all other quantities in the same category can be expressed in terms of this unit, multiplied by a number called the ''numerical value''. For example, if we write

: the wavelength is ''λ'' = 6.982 × 10−7 m

then "''λ''" is the symbol for the physical quantity (wavelength), "m" is the symbol for the unit (metre), and "6.982 × 10−7" is the numerical value of the wavelength in metres.

More generally, we can write

: ''A'' = {''A''} · [''A'']

where ''A'' is the symbol for the quantity, {''A''} symbolizes the numerical value of ''A'', and [''A''] represents the corresponding unit in which ''A'' is expressed here. Both the numerical value and the unit symbol are factors, and their product is the quantity. A quantity itself has no inherent particular numerical value or unit; as with any product, there are many different combinations of numerical value and unit that lead to the same quantity (e.g., ''A'' = 300 · m = 0.3 · km = ...). This ambiguity makes the {''A''} and [''A''] notations useless, unless they are used together.

The value of a quantity is independent of the unit chosen to represent it. It must be distinguished from the numerical value of the quantity that occurs when the quantity is expressed in a particular unit. The above curly-bracket notation could be extended with a unit-symbol index to clarify this dependency, as in {''λ''}m = 6.982 × 10−7 or equivalently {''λ''}nm = 698.2. In practice, where it is necessary to refer to the numerical value of a quantity expressed in a particular unit, it is notationally more convenient to simply divide the quantity through that unit, as in

: ''λ''/m = 6.982 × 10−7

or equivalently

: ''λ''/nm = 698.2.

This is a particularly useful and widely used notation for labelling the axes of graphs or for the headings of table columns, where repeating the unit after each numerical value can be typographically inconvenient.

==Typographic conventions==
===Symbols for quantities===
* Quantities are generally represented by a symbol formed from single letters of the Latin or Greek alphabet.
* Symbols for quantities are set in [[italic type]], independent of the type used in the rest of the text.
* If in a text different quantities use the same letter symbol, they can be distinguished via subscripts.
* A subscript is only set in italic type if it consists of a symbol for a quantity or a variable. Other subscripts are set in upright ([[roman type|roman]]) type. For example, write ''V''n for a "nominal volume" (where "n" is just an abbreviation for the word "nominal"), but write ''Vn'' if ''n'' is a running index number.

===Names and symbols for units===
{{main|SI|SI base unit|SI derived unit|SI prefix}}
* If an internationally standardized symbol exists for a unit, then only that symbol should be used. See the [[SI]] articles for the list of standard symbols defined by the International System of Units. Note that the distinction between uppercase and lowercase letters is significant for SI unit symbols. For example, "k" is the prefix kilo and "K" stands for the unit kelvin. The symbols of all SI units named after a person or a place start with an uppercase letter, as do the symbols of all prefixes from mega on upwards. All other symbols are lowercase; the only exception is [[litre]], where both l and L are allowed. However, it is stated that the CIPM will examine whether one of the two may be suppressed.
* Symbols for units should be printed in an upright ([[roman type|roman]]) typeface.

===Numbers===
See Sect. 3.3 of the Standard text.
* Numbers should be printed in upright ([[roman type|roman]]) type.
* ISO 31-0 (after Amendment 2) specifies that "the decimal sign is either the comma on the line or the point on the line". This follows [http://www.bipm.org/en/CGPM/db/22/10/ resolution 10] of the 22nd [[General Conference on Weights and Measures|CGPM]], 2003; there is a brief reference to the history of this in [http://www.nist.gov/public_affairs/techbeat/tb2006_1122.htm#decimal].
* Numbers consisting of long sequences of digits can be made more readable by separating them into groups, preferably groups of three, separated by a small space. For this reason, ISO 31-0 specifies that such groups of digits should never be separated by a comma or point, as these are reserved for use as the decimal sign.
* For numbers whose magnitude is less than 1, the decimal sign should be preceded by a zero.
* The multiplication sign is either a cross or a half-height dot, though the latter should not be used when the dot is the decimal separator.

===Expressions===
* Unit symbols follow the numerical value in the expression of a quantity.
* Numerical value and unit symbol are separated by a space. This rule also applies to the symbol "°C" for degrees Celsius, as in "25 °C". The only exception are the symbols for the units of plane angle degree, minute and second, which follow the numerical value without a space in between (for example "30°").
* Where quantities are added or subtracted, parenthesis can be used to distribute a unit symbol over several numerical values, as in
: ''T'' = 25 °C − 3 °C = (25 − 3) °C
: ''P'' = 100 kW ± 5 kW = (100 ± 5) kW
: (but not: 100 ± 5 kW)
: ''d'' = 12 × (1 ± 10−4) m
* Products can be written as ''ab'', ''a b'', ''a''⋅''b'', or ''a''×''b''. The sign for multiplying numbers is a cross (×) or a half-height dot (⋅). The cross should be used adjacent to numbers if a dot on the line is used as the decimal separator, to avoid confusion between a decimal dot and a multiplication dot.
* Division can be written as <math>\frac ab</math>, ''a''/''b'', or by writing the product of ''a'' and ''b''−1, for example ''a''⋅''b''−1. Numerator or denominator can themselves be products or quotients, but in this case, a solidus (/) should not be followed by a multiplication sign or division sign on the same line, unless parentheses are used to avoid ambiguity.

===Mathematical signs and symbols===
A comprehensive list of internationally standardized mathematical symbols and notations can be found in [[ISO 31-11]].

==See also==
* [[ISO 31]]
* [[ISO 1000]]
* [[SI]]
* [[ISO 31-11]]

==References==
* [[International standard]] ISO 31-0: Quantities and units — Part 0: General principles. [[International Organization for Standardization]], Geneva, 1992.
* [http://www.bipm.org/en/si/si_brochure/ SI brochure]. Bureau International des Poids et Mesures.
* I. M. Mills and W. V. Metanomski: [http://www.iupac.org/standing/idcns/fonts_for_symbols.html On the use of italic and roman fonts for symbols in scientific text]. Interdivisional Committee on Nomenclature and Symbols, [[IUPAC]], December 1999.
* T. Cvitas: [http://www.iupac.org/standing/ictns/quantity_and_percents.html Quantity calculus]. Interdivisional Committee on Terminology, Nomenclature and Symbols, [[IUPAC]], February 2002.
* [http://www.physics.nist.gov/cuu/Units/checklist.html Unit rules and style conventions – Check list for reviewing manuscripts]. US [[National Institute for Standards and Technology]], 1998.
* B. N. Taylor and A. Thompson: [http://physics.nist.gov/Pubs/SP330/contents.html The International System of Units (SI)]. NIST Special Publication 330. US National Institute for Standards and Technology, 2008.
* A. Thompson and B. N. Taylor: [http://www.physics.nist.gov/Pubs/SP811/contents.html Guide for the use of the International System of Units (SI)]. NIST Special Publication 811. US National Institute for Standards and Technology, 2008.

{{ISO standards}}

[[Category:ISO 31|#00031-0]]

Infinitely near point

2013-10-13T03:08:22Z

140.247.0.12: /* Definition */

== Coincidence factor ==

The coincidence factor is the reciprocal of the diversity factor. However, differing sources define the simultaneity factor to be identical to either the coincidence factor or the diversity factor. The [[International Electrotechnical Commission]] defines the coincidence and simultaneity factors identically with the diversity factor being the [[reciprocal]]{{dn|date=July 2013}}. Since the only change in definition is to take the inverse, all one needs to know is if the factor is greater than or less than one.

== Diversity ==

The (unofficial) term diversity, as distinguished from diversity factor refers to the percent of time available that a machine, piece of equipment, or facility has its maximum or nominal load or demand (a 70% diversity means that the device in question operates at its nominal or maximum load level 70% of the time that it is connected and turned on).

== Diversified load and diversification factor==

The diversified load is the total expected load (power) to be drawn during a peak period by a device or system of devices. The diversified load is the combination of each devices full load capacity, Utilization Factor, Diversity Factor, Demand Factor and the Load factor(electrical)|load factor. This process is referred to as load diversification. The diversification factor is then defined as:

<math> f_{Diversification} = \frac{\text{Diversified Load}}{\text{Maximum system load}}</math>

== In mathematics ==

Diversity factor is commonly used for a number of mathematics-related topics. One such instance is when completing a coordination study for a system. This diversity factor is used to estimate the load of a particular node in the system.

== See also ==
{{portal|Energy}}
*[[Energy storage]]
*[[Intermittent power source]]
*[[Demand factor]]
*[[Load factor (electrical)|Load Factor]]
*[[Utilization factor]]

==References==

# IEEE Standard 141(TM)-1993, ''IEEE Recommended Practice for Electric Power Distribution for Industrial Plants'', Red Book.
# ''Handbook for Electricity Metering'', Edison Electric Institute, Tenth Edition.
simple word in diversity factor which means ratio sum of the individual max demand to total max demand of the power station.

==External links==
*[http://www.ise.ufl.edu/capehart/papers/diversity.html ise.ufl.edu/capehart/papers/diversity.html]
*[http://www.nfpa.org/assets/files//PDF/necdigest/CodeIssues072704.pdf nfpa.org/assets/files//PDF/necdigest/CodeIssues072704.pdf]

{{Electricity delivery}}



{{DEFAULTSORT:Diversity Factor}}
[[Category:Electricity]]