formulasearchengine - User contributions [en]

Vincent's theorem

2014-02-23T22:18:10Z

75.0.183.103: Minor- punctuation, syntax, extra words

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Half-value layer

2014-01-23T21:48:27Z

75.0.185.130:

{{multiple issues|
{{Refimprove|date=June 2009}}
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An '''Energy Slave''' is that quantity of energy (ability to do work) which, when used to construct and drive non-human [[infrastructure]] ([[machine]]s, [[road]]s, [[Electric power|power grids]], [[fuel]], draft animals, wind-driven pumps, etc.) replaces a unit of human labor (actual work). An energy slave does the work of a person, through the consumption of [[energy]] in the non-human infrastructure.<ref>{{cite book|title=The first measured century: an illustrated guide to trends in America, 1900-2000|first1=Theodore|last1= Caplow|first2= Louis |last2=Hicks|first3= Ben J. |last3=Wattenberg|publisher= [[American Enterprise Institute]] |year= 2001|ISBN=0-8447-4138-8}}</ref>

==History==
The term was first used by [[R. Buckminster Fuller]] in the caption of an illustration for the cover of the February 1940 issue of [[Fortune Magazine]], entitled "World Energy". Alfred Ubbelohde also coined the term, apparently independently, in his 1955 book, "Man and Energy", but the term did not come to be widely used until the 1960s, and is generally credited to Fuller.

==Usage==
An Energy Slave is used to compare the productivity of a person and the energy that would be required to produce that work in the modern, [[oil|oil fueled]] industrial [[economy]], although it could be applied anywhere that labor is produced with non-human sourced energy. It does not include the ancillary costs of damage to the environment or social structures. Formally, one Energy Slave produces one unit of human labor through the non-human tools and energy supplied by the industrial economy, and therefore 1 ES times a constant that converts to work accomplished = 1 human labor unit.

The choice to “employ” Energy Slaves is only at the margins of their total impact, so are called slaves because users receive the value produced by them as an entitlement of the society.

===Macro view===
One way to look at an Energy Slave might be called the “[[macroscopic scale|macro]]” view. This is to look at the total flow through of energy divided by the number of persons being supported by the [[infrastructure]] where that energy is being used. It is a number that can change instantaneously as the flow through of energy changes. Although this formulation is challenged by the need to decide whom to include in the count, and the massive data-keeping it would require, it is intuitively simple because we merely divide one number by the other, and guides us in thinking about the other perspective. It is suited to large blocks of a given economy and to comparisons of economies.

===Micro view===
Another way to look at the Energy Slave might be called the “[[microeconomics|micro]]” view. This is to look in detail at the substitution of [[human labor]] by non-human sources of [[productivity]], in particular the modern industrial infrastructure of machines and services.

==Energy expenditure==
As a very simple example, ten [[apple]] pickers descend from their trees and walk to the processing shed with their apples, and then return to their trees. They have produced some number of units of work. Ten other apple pickers unload their apples into an empty box, and then return to picking. The box, now full, is carried by a field [[tractor]] to the processing shed. The work of these ten pickers plus the driver of the tractor plus all of the energy inputs have also produced that number of units of work.

The energy inputs include the life cycle share of the energy required to build and maintain that tractor and the box (called "[[embodied energy]]"), plus the [[fuel]] required to run it for the time occupied by bringing, placing, idling, and returning that box to the shed, plus the energy required to acquire, process, transport, and distribute that energy (more embedded energy).

The [[energy]] used in the two systems is not defined to be [[Equality (mathematics)|equal]]. The ratio of the energy used to produce an energy slave’s volume of work, through [[machine]] labor, as opposed to the energy used to produce a unit of human labor, is one of the most salient questions implicitly raised by this concept.

If we let ''Lw'' the Labor of walkers, ''Ln'' the labor of non-walkers, ''Ld'' the labor of driver, ''Ei'' the Non-human energy inputs, ''C'' the Constant to convert units of energy into units of work. Further, let 1 ''Ph'' be one person-hour.

Then given what has been said above
:<math>Lw = Ln + Ld + CEi</math>

And therefore, since the human labor inputs equate to the energy slave units
:<math>Lw - Ln - Ld = CEi</math>

Supposing then that work is measured in Person-hours, and supposing further that the walkers require half an hour each to go to the shed and back, that both groups take 6 minutes to fill their apple pouches, and the driver takes 30 minutes to go to and return from the shed:
:<math>Lw = 10\, Laborers \cdot 0.6\,\frac{Ph}{Laborer} = 6\,Ph</math>
:<math>Ln = 10\, Laborers \cdot 0.1\,\frac{Ph}{Laborer} = 1\,Ph</math>
:<math>Ld = 1\, Laborer \cdot 0.5\,\frac{Ph}{Laborer} = 0.5\,Ph</math>

Therefore
:<math>CEi = 6 Ph - 1 Ph - 0.5 Ph = 4.5 Ph</math>

The Energy inputs (times the constant) replace 4.5 person-hours of labor. Returning to the original definition, an energy slave is the energy required to produce a unit of human labor otherwise than organically, so we need to convert these 4.5 person-hours into energy slaves.

In the original example, ten laborers produced their all-human work output in six hours. The other laborers plus their machines produced the same work in 1.5 hours of human labor. Therefore the energy slaves replaced 4.5/6.0 hours of human productivity, and there are 7.5 Energy Slaves.

The question “How many energy slaves do I have?” (<math>{}_{Es}</math>) is answered by looking at the amount of energy required to build and drive the infrastructure to support your life style (<math>{}_{Ei^{*}}</math>), multiplied by Slaves per unit of energy (<math>{}_\frac{Es^{*}}{Ei}</math>). This would be expressed as

:<math>Es = Ei^{*} \cdot \frac{Es^{*}}{Ei}</math>

===Questions===

One of the questions that arises is:

Was the [[calorie|caloric]] expenditure of the laborers producing the person-hours less than, equal to, or greater than, the caloric energy inputs to the system? In terms of the variables used thus far, is the question if <math>{}_{7.5\,Es \cdot \frac{calories}{Es}}</math> is less or equal ''Ei'', or bigger?

In meaningful terms, did the economy use more or less energy, by using energy slaves, than it would if it had used actual human labor? This value might be a kind of benchmark, but an economy crossing this benchmark won’t notice a qualitative difference.

==Implications==

The implication of the energy slave unit is that each of the workers who did not walk were able to return to picking apples, and therefore increase their personal productivity. Doing the labor of 10 persons with 2.5 persons worth of work (10-7.5), the laborers with Energy Slaves can produce 10/2.5, or 4 times, as much work, in the same amount of time. Their personal wealth, and/or that of their employer, can be expected to increase; however, because of the huge energy investment behind the [[infrastructure]], the margin of benefit for the employer and the workers must be less than 4 times the value of the work of the unassisted workers.

If energy slaves were actually free, then we would seek to shunt off as much labor as possible onto them. However, they are not free, and the cost of an energy slave, compared to the cost of human labor, may decide when to use an energy slave and when to use a person. A more interesting question than that about the calories used by the different systems is the question of the cost of each. The cost of human labor trends downward as the number of workers grows faster than the work available to support them, and as the number of energy slaves decreases per person. Meanwhile, as the cost of energy increases, the investment required to use energy slaves instead of people may become greater than the cost of people.

When someone discusses the amount of energy used to produce, harvest, transport and distribute a head of [[broccoli]] to a store three thousand miles away, the energy used can be expressed in terms of the number of energy slaves required to do that work. Since there are so many such deeply nested costs associated with the industrial infrastructure, we need some way to resolve the energy used to build the truck, to smelt the [[steel]] and convert [[petroleum]] into plastics, into units of human labor. Would we directly substitute an actual person walking across the country with the head of broccoli for the truck that actually carries it for the sake of comparative productivity? Or do we just divide the units of energy used by the industrial infrastructure by the number of calories used by one person to accomplish the same task, to get energy-slaves?

People who use this term want to convey in human terms the amount of energy required to support our modern [[United States|American]] lifestyle. Another way to articulate this ratio is in terms of the energy required to grow food and transport it, as compared to the energy that food provides to a person. In a society with only human labor, you could not consume more energy than you produce in food. What does it mean when the energy required to produce food exceeds the food value it provides to a person? Just how much more energy than is contained in the food is acceptable? These are questions like those when using the "energy slave" unit, that need to be answered by people seeking understanding of these units.

==References==
{{reflist}}

==External links==
* [http://www.manicore.com/anglais/documentation_a/slaves.html How much of a slave master am I ?] by Jean-Marc Jancovici
* [http://www.fulltable.com/vts/f/fortune/xb/50.jpg] Energy Map by R. Buckminster Fuller

[[Category:Energy economics]]

Hund's rules

2014-01-23T19:23:27Z

75.0.185.130: Minor-

{{Unreferenced|date=February 2008}}

In [[mathematics]], the '''monster Lie algebra''' is an infinite
dimensional [[generalized Kac–Moody algebra]] acted on by the [[monster group]], which was used to prove the [[monstrous moonshine]] conjectures.

== Structure ==
The monster Lie algebra ''m'' is a ''Z2''-graded Lie algebra.
The piece of degree ''(m,n)'' has dimension ''cmn'' if
''(m,n)'' is nonzero, and dimension 2 if ''(m,n)'' is (0,0).
The integers ''cn'' are the coefficients
of ''qn'' of the [[j-invariant]] as [[elliptic modular function]]
::<math>j(q) -744 = {1 \over q} + 196884 q + 21493760 q^2 + \cdots.</math>


The [[Cartan subalgebra]] is the 2-dimensional subspace of degree
(0,0), so the monster Lie algebra has rank 2.

The monster Lie algebra has just one real [[Simple root (root system)|simple root]], given by the vector
(1,-1), and the [[Weyl group]] has order 2, and acts by mapping
''(m,n)'' to ''(n,m)''. The imaginary simple roots are the vectors

:(1,''n'') for ''n'' = 1,2,3,...,

and they have multiplicities ''cn''.

The [[denominator formula]] for the monster Lie algebra is the product formula
for the ''j''-invariant:

::<math>j(p)-j(q) = \left({1 \over p} - {1 \over q}\right) \prod_{n,m=1}^{\infty}(1-p^n q^m)^{c_{nm}}.</math>

== Construction ==

There are two ways to construct the monster Lie algebra. As it is a generalized Kac–Moody algebra whose simple roots are known, it can be defined by explicit generators and relations; however, this presentation does not give an action of the monster group on it.

It can also be constructed from the [[monster vertex algebra]] by using the [[Goddard–Thorn theorem]] of [[string theory]]. This construction is much harder, but has the advantage of proving that the [[monster group]] acts naturally on it.

== References ==

* Richard Borcherds, "Vertex algebras, Kac-Moody algebras, and the Monster", ''Proc. Natl. Acad. Sci. USA.'' '''83''' (1986) 3068-3071
* Igor Frenkel, James Lepowsky, Arne Meurman, "Vertex operator algebras and the Monster". ''Pure and Applied Mathematics, 134.'' Academic Press, Inc., Boston, MA, 1988. liv+508 pp. ISBN 0-12-267065-5
* [[Victor Kac]], "Vertex algebras for beginners". ''University Lecture Series, 10.'' American Mathematical Society, 1998. viii+141 pp. ISBN 0-8218-0643-2
* R. W. Carter, "Lie Algebras of Finite and Affine Type", Cambridge Studies No. 96, 2005, ISBN 0-521-85138-6 (Introductory study text with a brief account of Borcherds algebra in Ch. 21)

[[Category:Lie algebras]]
[[Category:Moonshine theory]]

Box–Jenkins

2014-01-13T23:54:54Z

75.0.179.192: Minor-

[[File:Margolus block neighborhood.svg|thumb|240px|The Margolus neighborhood for a two-dimensional block cellular automaton. The partition of the cells alternates between the set of {{math|2 × 2}} blocks indicated by the solid blue lines, and the set of blocks indicated by the dashed red lines.]]
A '''block cellular automaton''' or '''partitioning cellular automaton''' is a special kind of [[cellular automaton]] in which the lattice of cells is divided into non-overlapping blocks (with different partitions at different time steps) and the transition rule is applied to a whole block at a time rather than a single cell. Block cellular automata are useful for simulations of physical quantities, because it is straightforward to choose transition rules that obey physical constraints such as [[reversible cellular automaton|reversibility]] and [[conservation law]]s.<ref name="schiff">{{citation|title=Cellular Automata: A Discrete View of the World|first=Joel L.|last=Schiff|publisher=Wiley|contribution=4.2.1 Partitioning Cellular Automata|pages=115–116|year=2008}}</ref>

==Definition==
A block cellular automaton consists of the following components:<ref name="schiff"/><ref name="tm">{{citation|title=Cellular Automata Machines: A New Environment for Modeling|first1=Tommaso|last1=Toffoli|author1-link=Tommaso Toffoli|first2=Norman|last2=Margolus|author2-link=Norman Margolus|publisher=MIT Press|year=1987|chapter=II.12 The Margolus neighborhood|pages=119–138}}</ref>
*A regular [[Lattice (group)|lattice]] of cells
*A finite set of the states that each cell may be in
*A partition of the cells into a uniform [[tessellation]] in which each tile of the partition has the same size and shape
*A rule for shifting the partition after each time step
*A transition rule, a function that takes as input an assignment of states for the cells in a single tile and produces as output another assignment of states for the same cells.
In each time step, the transition rule is applied simultaneously and synchronously to all of the tiles in the partition. Then, the partition is shifted and the same operation is repeated in the next time step, and so forth. In this way, as with any cellular automaton, the pattern of cell states changes over time to perform some nontrivial computation or simulation.

==Neighborhoods==
The simplest partitioning scheme is probably the '''Margolus neighborhood''', named after [[Norman Margolus]], who first studied block cellular automata using this neighborhood structure. In the Margolus neighborhood, the lattice is divided into {{math|2}}-cell blocks (or {{math|2 × 2}} squares in two dimensions, or {{math|2 × 2 × 2}} cubes in three dimensions, etc.) which are shifted by one cell (along each dimension) on alternate timesteps.<ref name="schiff"/><ref name="tm"/><ref name="m84">{{citation|first=N.|last=Margolus|authorlink=Norman Margolus|title=Physics-like models of computation|journal=Physica D|volume=10|year=1984|pages=81–95|doi=10.1016/0167-2789(84)90252-5}}. Reprinted in {{citation|editor-first=Stephen|editor-last=Wolfram|editor-link=Stephen Wolfram|title=Theory and Applications of Cellular Automata|series=Advanced series on complex systems|volume=1|publisher=World Scientific|year=1986|pages=232–246}}</ref>

A closely related technique due to K. Morita and M. Harao<ref>{{citation|first1=K.|last1=Morita|first2=M.|last2=Harao|title=Computation universality of 1 dimensional reversible (injective) cellular automata|journal=Transactions Institute of Electronics, Information and Communication Engineers, E|volume=72|pages=758–762|year=1989}}</ref> consists in partitioning each cell into a finite number of parts, each part being devoted to some neighbor. The evolution proceeds by exchanging the corresponding parts between neighbors and then applying on each cell a purely local transformation <math>F</math> depending only on the state of the cell (and not on the states of its neighbors). With such a construction scheme, the cellular automaton is guaranteed to be reversible if the local transformation <math>F</math> is itself a [[bijection]]. This technique may be viewed as a block cellular automaton on a finer lattice of cells, formed by the parts of each larger cell; the blocks of this finer lattice alternate between the sets of parts within a single large cell and the sets of parts in neighboring cells that share parts with each other.

==Reversibility and conservation==
As long as the rule for evolving each block is [[reversible cellular automaton|reversible]], the entire automaton will also be. More strongly, in this case, the time-reversed behavior of the automaton can also be described as a block cellular automaton, with the same block structure and with a transition rule that inverts the original automaton's rule within each block. The converse is also true: if the blocks are not individually reversible, the global evolution cannot be reversible: if two different configurations ''x'' and ''y'' of a block lead to the same result state ''z'', then a global configuration with ''x'' in one block would be indistinguishable after one step from the configuration in which the ''x'' is replaced by ''y''. That is, a cellular automaton is reversible globally if and only if it is reversible at the block level.<ref name="bbm"/>

The ease of designing reversible block cellular automata, and of testing block cellular automata for reversibility, is in strong contrast to cellular automata with other non-block neighborhood structures, for which it is [[undecidable problem|undecidable]] whether the automaton is reversible and for which the reverse dynamics may not be describable as an automaton with the same neighborhood.<ref>{{citation|first=Jarkko|last=Kari|authorlink=Jarkko Kari|title=Reversibility of 2D cellular automata is undecidable|journal=Physica D|volume=45|year=1990|pages=379–385|doi=10.1016/0167-2789(90)90195-U}}</ref> Any reversible cellular automaton may be simulated by a reversible block cellular automaton with a larger number of states; however, because of the undecidability of reversibility for non-block cellular automata, there is no computable bound on the radius of the regions in the non-block automaton that correspond to blocks in the simulation, and the translation from a non-block rule to a block rule is also not computable.<ref>{{citation|title=On the circuit depth of structurally reversible cellular automata|first=Jarkko|last=Kari|authorlink=Jarkko Kari|journal=Fundamenta Informaticae|volume=38|year=1999|pages=93–107}}; {{citation|title=Representing reversible cellular automata with reversible block cellular automata|last=Durand-Lose|first=Jérôme|journal=Discrete Mathematics and Theoretical Computer Science|volume=AA|year=2001|pages=145–154|url=http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/download/264/855}}</ref>

Block cellular automata are also a convenient formalism in which to design rules that, in addition to reversibility, implement [[conservation laws]] such as the conservation of particle number, conservation of momentum, etc.. For instance, if the rule within each block preserves the number of live cells in the block, then the global evolution of the automaton will also preserve the same number. This property is useful in the applications of cellular automata to physical simulation.<ref name="nkos">{{citation|last=Wolfram|first=Stephen|authorlink=Stephen Wolfram|year=2002|title=[[A New Kind of Science]]|pages=459–464|publisher=Wolfram Media|isbn=1-57955-008-8}}</ref>

==Simulation by conventional cellular automata==
As Toffoli and Margolus write,<ref name="tm"/> the block cellular automaton model does not introduce any additional power compared to a conventional cellular automaton that uses the same neighborhood structure at each time step: any block cellular automaton may be simulated on a conventional cellular automaton by using more states and a larger neighborhood. Specifically, let the two automata use the same lattice of cells, but let each state of the conventional automaton specify the state of the block automaton, the phase of its partition shifting pattern, and the position of the cell within its block. For instance, with the Margolus neighborhood, this would increase the number of states by a factor of eight: there are four possible positions that a cell may take in its {{math|2 × 2}} block, and two phases to the partition. Additionally, let the neighborhood of the conventional automaton be the union of the blocks containing the given cell in the block cellular automaton. Then with this neighborhood and state structure, each update to the block automaton may be simulated by a single update to the conventional cellular automaton.

==Applications==
Block cellular automata are commonly used to implement [[lattice gas]]es and other quasi-physical simulations, due to the ease of simulating physical constraints such as conservation laws in these systems.<ref name="schiff"/><ref name="nkos"/>
For instance, the Margolus model may be used to simulate the HPP lattice gas model, in which particles move in two perpendicular directions and scatter at right angles when they collide with each other. In the block cellular simulation of this model, the update rule moves each cell to the cell diagonally opposite in its block, except in the case that a cell contains two diagonally opposite particles, in which case they are replaced by the complementary pair of diagonally opposite particles. In this way, particles move diagonally and scatter according to the HPP model.<ref name="tm"/><ref name="schiff2"/> An alternative rule that simulates the HPP lattice gas model with horizontal and vertical motion of particles, rather than with diagonal motion, involves rotating the contents of each block clockwise or counterclockwise in alternating phases, except again in the case that a cell contains two diagonally opposite particles, in which case it remains unchanged.<ref name="tm"/>
In either of these models, momentum (the sum of the [[velocity|velocity vectors]] of the moving particles) is conserved, as well as their number, an essential property for simulating physical gases. However, the HPP models are somewhat unrealistic as a model of gas dynamics, because they have additional non-physical conservation rules: the total momentum within each line of motion, as well as the total momentum of the overall system, is conserved. More complex models based on the hexagonal grid avoid this problem.<ref name="schiff2">"5.5.4 Lattice Gases", in {{harvtxt|Schiff|2008}}, pp. 165–169.</ref>

These automata may also be used to model the motion of grains of [[sand]] in sand piles and [[hourglass]]es. In this application, one may use a Margolus neighborhood with an update rule that preserves the number of grains within each {{math|2 × 2}} block but that moves each grain as far down within its block as possible. If a block includes two grains that are stacked vertically on top of each other, the transition function of the automaton replaces it by a block in which the grains are side-by-side, in effect allowing tall sand piles to topple and spread. This model is not reversible, but it still obeys a conservation law on the number of particles.<ref name="sandpile"/> A modified rule, using the same neighborhood but moving the particles sideways to the extent possible as well as down, allows the simulated sandpiles to spread even when they are not very steep.<ref>{{citation|title=Cellular gravity|first1=Frédéric|last1=Gruau|first2=John|last2=Tromp|journal=Parallel Processing Letters|volume=10|issue=4|pages=383–393|year=2000|url=http://oai.cwi.nl/oai/asset/1132/1132A.pdf}}</ref> More sophisticated cellular automaton sand pile models are also possible, incorporating phenomena such as wind transport and friction.<ref name="sandpile">{{citation|title=Cellular Automata Modeling of Physical Systems|first1=Bastien|last1=Chopard|first2=Michael|last2=Droz|publisher=Cambridge University Press|year=1998|contribution=2.2.6 The sand pile rule|pages=42–46}}</ref>

Margolus' original application for the block cellular automaton model was to the [[Billiard-ball computer|billiard ball model]] of reversible computation, in which [[Boolean logic]] signals are simulated by moving particles and logic gates are simulated by [[elastic collision]]s of those particles. It is possible, for instance, to perform billiard-ball computations in the two-dimensional Margolus model, with two states per cell, and with the number of live cells conserved by the evolution of the model. In the "BBM" rule that simulates the billiard-ball model in this way, signals consist of single live cells, moving diagonally. To accomplish this motion, the block transition function replaces a block containing a single live cell with another block in which the cell has been moved to the opposite corner of the block. Similarly, elastic collisions may be performed by a block transition function that replaces two diagonally opposite live cells by the other two cells of the block. In all other configurations of a block, the block transition function makes no change to its state. In this model, {{math|2 × 4}} rectangles of live cells (carefully aligned with respect to the partition) remain stable, and may be used as mirrors to guide the paths of the moving particles. For instance, the illustration of the Margolus neighborhood shows four particles and a mirror; if the next step uses the blue partition, then two particles are moving towards the mirror while the other two are about to collide, whereas if the next step uses the red partition, then two particles are moving away from the mirror and the other two have just collided and will move apart from each other.<ref name="m84"/><ref name="bbm">{{citation|first=Jérôme|last=Durand-Lose|contribution=Computing inside the billiard ball model|title=Collision-Based Computing|editor-first=Andrew|editor-last=Adamatzky|editor-link=Andrew Adamatzky|publisher=Springer-Verlag|year=2002|pages=135–160}}</ref><ref name="cc"/>

==Additional rules==
[[File:Critters block automaton.png|thumb|Gliders escape a central random seed, past the debris of earlier glider crashes, in the Critters rule.]]
Toffoli and Margolus<ref name="tm"/> suggest two more reversible rules for the Margolus neighborhood with two-state cells that, while not motivated by physical considerations, lead to interesting dynamics.

===Critters===
In the "Critters" rule, the transition function reverses the state of every cell in a block, except for a block with exactly two live cells which remains unchanged. Additionally, blocks with three live cells undergo a 180-degree rotation as well as the state reversal.<ref name="tm"/> This is a reversible rule, and it obeys conservation laws on the number of particles (counting a particle as a live cell in even phases and as a dead cell in odd phases) and on the parity of the number of particles along diagonal lines.<ref name="cc">{{citation|first=Norman|last=Margolus|authorlink=Norman Margolus|contribution=Crystalline Computation|title=Feynman and Computation|publisher=Perseus Books|year=1999|pages=267–305|editor-first=Anthony J. G.|editor-last=Hey|arxiv=comp-gas/9811002 }}</ref> Because it is reversible, initial states in which all cells take randomly chosen states remain unstructured throughout their evolution. However, when started with a smaller field of random cells centered within a larger region of dead cells, this rule leads to complex dynamics similar to those in [[Conway's Game of Life]] in which many small patterns similar to life's [[Glider (Conway's Life)|glider]] escape from the central random area and interact with each other.<ref name="tm"/><ref name="cc"/> Unlike the gliders in Life, reversibility and the conservation of particles together imply that when gliders crash together in Critters, at least one must escape, and often these crashes allow both incoming gliders to reconstitute themselves on different outgoing tracks. By means of such collisions, this rule can also simulate the billiard ball model of computing, although in a more complex way than the BBM rule.<ref name="cc"/> The Critters rule can also support more complex [[Spaceship (cellular automaton)|spaceships]] of varying speeds as well as [[Oscillator (cellular automaton)|oscillators]] with infinitely many different periods.<ref name="marotta">{{citation|title=Living in Critters' world|first=Sebastian M.|last=Marotta|url=http://web01.unicentro.br/revistas/index.php/RECEN/article/viewFile/385/537|year=2005|journal=Revista Ciências Exatas e Naturais|volume=7|issue=1}}</ref>

===Tron===
[[File:Trip-a-Tron.png|frame|The rectilinear shapes generated by the Tron rule.]]
In the "Tron" rule, the transition function leaves each block unchanged except when all four of its cells have the same state, in which case their states are all reversed. Running this rule from initial conditions in the form of a rectangle of live cells, or from similar simple straight-edged shapes, leads to complex rectilinear patterns. Toffoli and Margolus also suggest that this rule can be used to implement a local synchronization rule that allows any Margolus-neighborhood block cellular automaton to be simulated using an [[asynchronous cellular automaton]]. In this simulation, each cell of an asynchronous automaton stores both a state for the simulated automaton and a second bit representing the [[parity (mathematics)|parity]] of a timestamp for that cell; therefore, the resulting asynchronous automaton has twice as many states as the automaton it simulates. The timestamps are constrained to differ by at most one between adjacent cells, and any block of four cells whose timestamps all have the correct parity may be updated according to the block rule being simulated. When an update of this type is performed, the timestamp parities should also be updated according to the Tron rule, which necessarily preserves the constraint on adjacent timestamps. By performing local updates in this way, the evolution of each cell in the asynchronous automaton is identical to its evolution in the synchronous block automaton being simulated.<ref name="tm"/><ref>{{citation|contribution=Modeling and Analysis of Margolus Quantum Cellular Automata Using Net-Theoretical Methods|first1=Leo|last1=Ojala|first2=Olli-Matti|last2=Penttinen|first3=Elina|last3=Parviainen|title=Applications and Theory of Petri Nets 2004|year=2004|series=Lecture Notes in Computer Science|publisher=Springer-Verlag|volume=3099|pages=331–350|doi=10.1007/978-3-540-27793-4_19}}</ref>

==See also==
*[[Toothpick sequence]], a fractal pattern that can be emulated by cellular automata with the Margolus neighborhood

==References==
{{reflist|colwidth=30em}}

==External links==
*[http://www.cise.ufl.edu/~skoehler/critters/index.html Critters simulation], Seth Koehler, Univ. of Florida

[[Category:Cellular automata]]

Silverman's game

2013-12-24T12:27:13Z

75.0.178.254: Remove irrelevant constraint

{{lowercase}}
{{Electronic structure methods}}
In [[solid-state physics]], '''k·p [[perturbation theory (quantum mechanics)|perturbation theory]]''' is an approximation scheme for calculating the [[band structure]] (particularly [[Effective mass (solid-state physics)|effective mass]]) and optical properties of crystalline solids.<ref name=Yu2.6/><ref name=Kittel/><ref name=Harrison>
{{cite book
|author=W.P. Harrison
|year=1989 |origyear=1980
|title=Electronic Structure and the Properties of Solids
|edition=Reprint
|pages=158 ''ff''
|publisher=[[Dover Publications]]
|isbn=0-486-66021-4
}}</ref> It is pronounced "k dot p", and is also called the "k·p method". This theory has been applied specifically in the framework of the '''Luttinger–Kohn''' model (after [[Joaquin Mazdak Luttinger]] and [[Walter Kohn]]), and of the '''Kane model''' (after [[Evan O. Kane]]).

==Background and derivation==

===Bloch's theorem and wavevectors===
{{see also|Bloch wave}}

According to [[quantum mechanics]] (in the [[Hartree-Fock|single-electron approximation]]), the [[electron]]s in any material have [[wavefunction]]s which can be described by the following [[Schrödinger equation]]:
:<math>\left(\frac{p^2}{2m}+V\right)\psi = E\psi</math>
where '''p''' is the [[momentum operator|quantum-mechanical momentum operator]], ''V'' is the [[electrostatic potential|potential]], and ''m'' is the mass of an electron. (This equation neglects the [[spin-orbit effect]]; see below.)

In a [[crystalline solid]], ''V'' is a [[periodic function]], with the same periodicity as the [[crystal lattice]]. [[Bloch wave|Bloch's theorem]] proves that the solutions to this differential equation can be written as follows:
:<math>\psi_{n,\mathbf{k}}(\mathbf{x}) = e^{i\mathbf{k}\cdot\mathbf{x}} u_{n,\mathbf{k}}(\mathbf{x})</math>
where '''k''' is a vector (called the ''wavevector''), ''n'' is a discrete index (called the ''[[band structure|band]] index''), and ''u''''n'','''k''' is a function with the same periodicity as the crystal lattice.

For any given ''n'', the associated states are called a [[band structure|band]]. In each band, there will be a relation between the wavevector '''k''' and the energy of the state ''E''''n'','''k''', called the [[dispersion relation|band dispersion]]. Calculating this dispersion is one of the primary applications of ''k''·''p'' perturbation theory.

===Perturbation theory===
{{see also|Perturbation theory (quantum mechanics)}}

The periodic function ''u''''n'','''k''' satisfies the following Schrödinger-type equation:<ref name=Yu2.6>
{{cite book
|author=P. Yu, M. Cardona
|year=2005
|title=Fundamentals of Semiconductors: Physics and Materials Properties
|url=http://books.google.com/books?id=W9pdJZoAeyEC&pg=PA244&dq=isbn=3540254706#PPA68,M1
|edition=3rd
|page=Section 2.6, pp. 68 ''ff' |nopp=yes
|publisher=[[Springer (publisher)|Springer]]
|isbn=3-540-25470-6
}}</ref>
:<math>H_{\mathbf{k}} u_{n,\mathbf{k}}=E_{n,\mathbf{k}}u_{n,\mathbf{k}}</math>
where the [[Hamiltonian (quantum mechanics)|Hamiltonian]] is
:<math>H_{\mathbf{k}} = \frac{p^2}{2m} + \frac{\hbar \mathbf{k}\cdot\mathbf{p}}{m} + \frac{\hbar^2 k^2}{2m} + V </math>
Note that '''k''' is a vector consisting of three real numbers with units of [[inverse length]], while '''p''' is a vector of operators; to be explicit,
:<math>\mathbf{k}\cdot\mathbf{p} = k_x (-i\hbar \frac{\partial}{\partial x}) + k_y (-i\hbar \frac{\partial}{\partial y}) + k_z (-i\hbar \frac{\partial}{\partial z})</math>
In any case, we write this Hamiltonian as the sum of two terms:
:<math>H=H_0+H_{\mathbf{k}}', \;\; H_0 = \frac{p^2}{2m}+V, \;\; H_{\mathbf{k}}' = \frac{\hbar^2 k^2}{2m} + \frac{\hbar \mathbf{k}\cdot\mathbf{p}}{m}</math>
This expression is the basis for [[perturbation theory (quantum mechanics)|perturbation theory]]. The "unperturbed Hamiltonian" is ''H''0, which in fact equals the exact Hamiltonian at '''k'''=0 (i.e., at the [[Gamma point]]). The "perturbation" is the term <math>H_{\mathbf{k}}'</math>. The analysis that results is called "k·p perturbation theory", due to the term proportional to ''k''·''p''. The result of this analysis is an expression for ''E''''n'','''k''' and ''u''''n'','''k''' in terms of the energies and wavefunctions at '''k'''=0.

Note that the "perturbation" term <math>H_{\mathbf{k}}'</math> gets progressively smaller as '''k''' approaches zero. Therefore, k·p perturbation theory is most accurate for small values of '''k'''. However, if enough terms are included in the [[perturbation theory (quantum mechanics)|perturbative expansion]], then the theory can in fact be reasonably accurate for any value of '''k''' in the entire [[Brillouin zone]].

===Expression for a nondegenerate band===

For a nondegenerate band (i.e., a band which has a different energy at '''k'''=0 from any other band), with an [[extremum]] at '''k'''=0, and with no [[spin-orbit coupling]], the result of ''k''·''p'' perturbation theory is (to [[perturbation theory|lowest nontrivial order]]):<ref name=Yu2.6/>
:<math>u_{n,\mathbf{k}} = u_{n,0}+\frac{\hbar}{m}\sum_{n' \neq n}\frac{\langle u_{n,0} | \mathbf{k}\cdot\mathbf{p} | u_{n',0} \rangle}{E_{n,0}-E_{n',0}} u_{n',0}</math>
:<math>E_{n,\mathbf{k}} = E_{n,0}+\frac{\hbar^2 k^2}{2m} + \frac{\hbar^2}{m^2} \sum_{n'\neq n} \frac{|\langle u_{n,0} | \mathbf{k}\cdot\mathbf{p} | u_{n',0} \rangle |^2}{E_{n,0}-E_{n',0}}</math>

The parameters that are required to do these calculations, namely ''E''''n'',0 and <math>\langle u_{n,0} | \mathbf{p} | u_{n',0} \rangle</math>, are typically inferred from experimental data. (The latter are called "optical matrix elements".)

In practice, the sum over ''n''' often includes only the nearest one or two bands, since these tend to be the most important (due to the denominator). However, for improved accuracy, especially at larger '''k''', more bands must be included, as well as more terms in the perturbative expansion than the ones written above.

====Effective mass====
{{main|Effective mass (solid-state physics)}}

Using the expression above for the energy dispersion relation, a simplified expression for the effective mass in the conduction band of a semiconductor can be found.<ref name=Harrison/> To approximate the dispersion relation in the case of the conduction band, take the energy ''En0'' as the minimum conduction band energy ''Ec0'' and include in the summation only terms with energies near the valence band maximum, where the energy difference in the denominator is smallest. (These terms are the largest contributions to the summation.) This denominator is then approximated as the band gap ''Eg'', leading to an energy expression:
:<math>E_c(\boldsymbol k ) \approx E_{c0} +\frac{(\hbar k)^2}{2m} +\frac{\hbar ^2}{{E_g}m^2}\sum_n {|\langle u_{c,0}|\mathbf{k}\cdot\mathbf{p}| u_{n,0} \rangle |^2} </math>
The effective mass in direction ℓ is then:
:<math> \frac{1} {m}_{\ell} = {{1} \over {\hbar^2}} \sum_{ m} \cdot {{\partial^{\ 2} E_{c} (\boldsymbol{k})} \over {\partial k_{\ell} \partial k_ m}} \approx \frac{1}{m}+\frac{2}{E_gm^2}\sum_{m,\ n} {\langle u_{c,0}|p_{\ell}| u_{n,0} \rangle }{\langle u_{n,0}|p_{m}| u_{c,0} \rangle } </math>
Ignoring the details of the matrix elements, the key consequences are that the effective mass varies with the smallest bandgap and goes to zero as the gap goes to zero.<ref name=Harrison/> A useful approximation for the matrix elements in [[direct gap]] semiconductors is:<ref name=DirectGap>A ''direct gap'' semiconductor is one where the valence band maximum and conduction band minimum occur at the same position in '''k'''-space, usually the so-called Γ-point where '''k''' = 0.</ref>
:<math>\frac{2}{E_gm^2}\sum_{m,\ n} {|\langle u_{c,0}|p_{\ell}| u_{n,0} \rangle |}{|\langle u_{c,0}|p_{m}| u_{n,0} \rangle |} \approx 20\mathrm{eV} \frac{1}{mE_{g}} \ , </math>
which applies within about 15% or better to most group-IV, III-V and II-VI semiconductors.<ref name=Table2.22>See [http://books.google.com/books?id=W9pdJZoAeyEC&pg=PA244&dq=isbn:3540254706#PPA71,M1 Table 2.22] in Yu & Cardona, ''op. cit.''</ref>

In contrast to this simple approximation, in the case of valence band energy the ''spin-orbit'' interaction must be introduced (see below) and many more bands must be individually considered. The calculation is provided in Yu and Cardona.<ref name=valence>See Yu & Cardona, ''op. cit.'' pp. 75-82</ref> In the valence band the mobile carriers are ''[[Electron hole|holes]]''. One finds there are two types of hole, named ''heavy'' and ''light'', with anisotropic masses.

===''k''·''p'' model with spin-orbit interaction===

Including the [[spin-orbit interaction]], the Schrödinger equation for ''u'' is:<ref name=Kittel>
{{cite book
|author=C. Kittel
|year=1987
|title=Quantum Theory of Solids
|edition=Second Revised Printing
|pages=186–190
|isbn=0-471-62412-8
|publisher=[[John Wiley & Sons|Wiley]]
|location=New York
}}</ref>
:<math>H_{\mathbf{k}} u_{n,\mathbf{k}}=E_{n,\mathbf{k}}u_{n,\mathbf{k}}</math>
where
:<math>H_{\mathbf{k}} = \frac{p^2}{2m} + \frac{\hbar \mathbf{k}\cdot\mathbf{p}}{m} + \frac{\hbar^2 k^2}{2m} + V + \frac{1}{4 m^2 c^2} (\vec \sigma \times \nabla V)\cdot (\hbar\mathbf{k}+\mathbf{p})</math>
where <math>\vec \sigma=(\sigma_x,\sigma_y,\sigma_z)</math> is a vector consisting of the three [[Pauli matrix|Pauli matrices]]. This Hamiltonian can be subjected to the same sort of perturbation-theory analysis as above.

===Calculation in degenerate case===

For degenerate or nearly degenerate bands, in particular the [[valence band]]s in certain materials such as [[gallium arsenide]], the equations can be analyzed by the methods of [[perturbation theory (quantum mechanics)|degenerate perturbation theory]].<ref name=Yu2.6/><ref name=Kittel/> Models of this type include the "[[Luttinger-Kohn model]]" (a.k.a. "Kohn-Luttinger model"),<ref>
{{cite journal
|author=J. M. Luttinger, W. Kohn
|year=1955
|title=Motion of Electrons and Holes in Perturbed Periodic Fields
|journal=[[Physical Review]]
|volume=97 |issue= |pages=869
|doi=10.1103/PhysRev.97.869
|bibcode = 1955PhRv...97..869L }}</ref> and the "[[Kane model]]".<ref>
{{cite journal
|author=Evan O. Kane
|year=1957
|title=Band Structure of Indium Antimonide
|journal=[[Journal of Physics and Chemistry of Solids]]
|volume=1 |pages=249
|doi=10.1016/0022-3697(57)90013-6
|bibcode = 1957JPCS....1..249K }}</ref>

==Notes and references==
{{reflist}}

==See also==
{{Col-begin}}
{{Col-1-of-2}}
'''Electronic band structure'''
* [[Electronic band structure]]
* [[Nearly free electron model]]
* [[Kronig-Penney model]]
'''Band properties'''
* [[Band gap]]
* [[Effective mass (solid-state physics)|Effective mass]]
* [[Density of states]]
* [[Fermi surface]]
{{Col-2-of-2}}
'''Wavefunctions'''
* [[Wannier functions]]
* [[Bloch waves]]
'''Fundamental theory'''
* [[Kohn-Sham equations]]
* [[Local-density approximation]]

{{col-end}}

{{DEFAULTSORT:K P Perturbation Theory}}
[[Category:Electronic structure methods]]

Poset game

2013-12-20T12:53:37Z

75.0.178.254: Use canonical capitalization

In [[statistics]], the '''predicted residual sums of squares (PRESS) statistic''' is a form of [[cross-validation (statistics)|cross-validation]] used in [[regression analysis]] to provide a summary measure of the fit of a model to a sample of observations that were not themselves used to estimate the model. It is calculated as the sums of squares of the prediction residuals for those observations.<ref>{{cite web |url=http://www.statsoft.com/textbook/statistics-glossary/p/button/p/ |title=Statsoft:StatSoft.com Electronic Statistics Textbook - Statistics Glossary |accessdate=August 2012}}</ref><ref>Allen, D. M. (1974), "The Relationship Between Variable Selection and Data Augmentation and a Method for Prediction," ''Technometrics'', 16, 125–127</ref><ref>Tarpey, Thaddeus (2000) "A Note on the Prediction Sum of Squares Statistic for Restricted Least Squares", ''The American Statistician'', Vol. 54, No. 2, May, pp. 116–118</ref>

A ''fitted model'' having been produced, each observation in turn is removed and the model is refitted using the remaining observations. The out-of-sample predicted value is calculated for the omitted observation in each case, and the PRESS statistic is calculated as the sum of the squares of all the resulting prediction errors:<ref>{{cite web |url=http://www.oga-lab.net/RGM2/func.php?rd_id=qpcR:PRESS |title=R Graphical Manual:Allen's PRESS (Prediction Sum-Of-Squares) statistic, aka P-square |accessdate=August 2012}}</ref>

: <math>\operatorname{PRESS} =\sum_{i=1}^n (y_i - \hat{y}_{i, -i})^2 </math>

Given this procedure, the PRESS statistic can be calculated for a number of candidate model structures for the same dataset, with the lowest values of PRESS indicating the best structures. Models that are over-parameterised would tend to give small residuals for observations included in the model-fitting but large residuals for observations that are excluded.

== References ==
{{Reflist}}

[[Category:Regression diagnostics]]
[[Category:Model selection]]

{{statistics-stub}}

Almost integer

2013-11-28T18:42:57Z

75.0.187.132: Minor- (copy edit)

{{enzyme
| Name = progesterone 5-alpha-reductase
| EC_number = 1.3.1.30
| CAS_number = 72412-84-1
| IUBMB_EC_number = 1/3/1/30
| GO_code = 0050213
| image =
| width =
| caption =
}}
In [[enzymology]], a '''progesterone 5alpha-reductase''' ({{EC number|1.3.1.30}}) is an [[enzyme]] that [[catalysis|catalyzes]] the [[chemical reaction]]

:5alpha-pregnan-3,20-dione + NADP+ <math>\rightleftharpoons</math> progesterone + NADPH + H+

Thus, the two [[substrate (biochemistry)|substrates]] of this enzyme are [[5alpha-pregnan-3,20-dione]] and [[nicotinamide adenine dinucleotide phosphate|NADP+]], whereas its 3 [[product (chemistry)|products]] are [[progesterone]], [[nicotinamide adenine dinucleotide phosphate|NADPH]], and [[hydrogen ion|H+]].

This enzyme belongs to the family of [[oxidoreductase]]s, specifically those acting on the CH-CH group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is '''5alpha-pregnan-3,20-dione:NADP+ 5-oxidoreductase'''. Other names in common use include '''steroid 5-alpha-reductase''', and '''Delta4-steroid 5alpha-reductase (progesterone)'''. This enzyme participates in [[c21-steroid hormone metabolism]].

==References==
{{reflist|1}}
* {{cite journal | author = Cheng YJ, Karavolas HJ | date = 1975 | title = Subcellular distribution and properties of progesterone (delta4-steroid) 5alpha-reductase in rat medial basal hypothalamus | journal = J. Biol. Chem. | volume = 250 | pages = 7997–8003 | pmid = 240847 | issue = 20 }}
* {{cite journal | author = Cheng YJ, Karavolas HJ | date = 1975 | title = Properties and subcellular distribution of delta4-steroid (progesterone) 5alpha-reductase in rat anterior pituitary | journal = Steroids. | volume = 26 | pages = 57–71 | pmid = 1166484 | doi = 10.1016/0039-128X(75)90006-9 | issue = 1 }}

{{1.3-enzyme-stub}}

[[Category:EC 1.3.1]]
[[Category:NADPH-dependent enzymes]]
[[Category:Enzymes of unknown structure]]

Incompressible flow

2013-11-24T22:17:00Z

75.0.180.243: Minor- punctuation, syntax, extra words, changed 'which' to 'that' in restrictive clause for U.S readers (makes no difference in UK, etc.)

{{about|a general description of a function used in mathematics and physics to describe conservative fields|the scalar potential of electromagnetism|electric potential|all other uses|potential}}

'''Scalar potential''', simply stated, describes the situation where the difference in the [[potential energy|potential energies]] of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity.

[[File:Mass potential well increasing mass.gif|thumb|gravitational potential well of an increasing mass where <math> \mathbf{F} = -\nabla P </math>]]A '''scalar [[potential]]''' is a fundamental concept in [[vector analysis]] and [[physics]] (the adjective ''scalar'' is frequently omitted if there is no danger of confusion with [[vector potential]]). The scalar potential is an example of a [[scalar field]]. Given a [[vector field]] '''F''', the scalar potential ''P'' is defined such that:
:<math> \mathbf{F} = -\nabla P = - \left(
\frac{\partial P}{\partial x},
\frac{\partial P}{\partial y},
\frac{\partial P}{\partial z}
\right), </math><ref>{{cite book|author=Herbert Goldstein|title=Classical Mechanics|edition=2|pages=3–4|ISBN=978-0-201-02918-5}}</ref>
where '''∇'''P is the [[gradient]] of P and the second part of the equation is minus the gradient for a function of the [[Cartesian coordinate system|Cartesian coordinates]] x,y,z.<ref>The second part of this equation is ONLY valid for Cartesian coordinates, other coordinate systems such as cylindrical or spherical coordinates will have more complicated representations. derived from the [[gradient theorem|fundamental theorem of the gradient]].</ref> In some cases, mathematicians may use a positive sign in front of the gradient to define the potential.<ref>See [http://www.math.umn.edu/~nykamp/m2374/readings/findpot/] for an example where the potential is defined without a negative. Other references such as {{citation | title=The Calculus with Analytic Geometry|edition=5|author=Louis Leithold|page=1199}} avoid using the term ''potential'' when solving for a function from its gradient.</ref> Because of this definition of P in terms of the gradient, the direction of '''F''' at any point is the direction of the steepest decrease of P at that point, its magnitude is the rate of that decrease per unit length.

In order for '''F''' to be described in terms of a scalar potential only, the following have to be true:
#<math>-\int_a^b \mathbf{F}\cdot d\mathbf{l} = P(\mathbf{b})-P(\mathbf{a})</math>, where the integration is over a [[Jordan arc]] passing from location '''a''' to location '''b''' and P('''b''') is P evaluated at location '''b''' .
#<math>\oint \mathbf{F}\cdot d\mathbf{l}=0</math>, where the integral is over any simple closed path, otherwise known as a [[Jordan curve]].
#<math>{\nabla}\times{\mathbf{F}} =0. </math>
The first of these conditions represents the [[gradient theorem|fundamental theorem of the gradient]] and is true for any vector field that is a gradient of a [[differentiable]] [[single-valued function|single valued]] scalar field P. The second condition is a requirement of '''F''' so that it can be expressed as the gradient of a scalar function. The third condition re-expresses the second condition in terms of the [[curl (mathematics)|curl]] of '''F''' using the [[stokes' theorem|fundamental theorem of the curl]]. A vector field '''F''' that satisfies these conditions is said to be [[irrotational vector field|irrotational]] (Conservative).

Scalar potentials play a prominent role in many areas of physics and engineering. The [[gravity potential]] is the scalar potential associated with the gravity per unit mass, i.e., the [[acceleration]] due to the field, as a function of position. The gravity potential is the gravitational [[potential energy]] per unit mass. In [[electrostatics]] the [[electric potential]] is the scalar potential associated with the [[electric field]], i.e., with the [[electrostatic force]] per unit [[Electric charge|charge]]. The electric potential is in this case the electrostatic potential energy per unit charge. In [[fluid dynamics]], irrotational [[lamellar field]]s have a scalar potential only in the special case when it is a [[Laplacian field]]. Certain aspects of the [[nuclear force]] can be described by a [[Yukawa potential]]. The potential play a prominent role in the [[Lagrangian mechanics|Lagrangian]] and [[Hamiltonian mechanics|Hamiltonian]] formulations of [[classical mechanics]]. Further, the scalar potential is the fundamental quantity in [[quantum mechanics]].

Not every vector field has a scalar potential. Those that do are called '''[[conservative vector field|conservative]]''', corresponding to the notion of [[conservative force]]
in physics. Examples of non-conservative forces include frictional forces, magnetic forces, and in fluid mechanics a [[solenoidal|solenoidal field]] velocity field. By the [[Helmholtz decomposition]] theorem however, all vector fields can be describable in terms of a scalar potential and corresponding [[vector potential]]. In electrodynamics the electromagnetic scalar and vector potentials are known together as the [[electromagnetic four-potential]].

==Integrability conditions==
If '''F''' is a [[conservative vector field]] (also called ''irrotational'', ''[[Curl (mathematics)|curl]]-free'', or ''potential''), and its components have [[continuous function|continuous]] [[partial derivative]]s, the '''potential''' of '''F''' with respect to a reference point <math>\mathbf r_0</math> is defined in terms of the [[line integral]]:

:<math>V(\mathbf r) = -\int_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r} = -\int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt,</math>

where ''C'' is a parametrized path from <math>\mathbf r_0</math> to <math>\mathbf r,</math>

: <math>\mathbf{r}(t), a\leq t\leq b, \mathbf{r}(a)=\mathbf{r_0}, \mathbf{r}(b)=\mathbf{r}.</math>

The fact that the line integral depends on the path ''C'' only through its terminal points <math>\mathbf r_0</math> and <math>\mathbf r</math> is, in essence, the '''path independence property''' of a conservative vector field. The [[fundamental theorem of calculus]] for line integrals implies that if ''V'' is defined in this way, then <math> \mathbf{F}= -\nabla V,</math> so that ''V'' is a scalar potential of the conservative vector field '''F'''. Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it. If ''V'' is defined in terms of the line integral, the ambiguity of ''V'' reflects the freedom in the choice of the reference point <math>\mathbf r_0.</math>

==Altitude as gravitational potential energy==
[[File:Gravity field near earth.gif|thumb|uniform gravitational field near the Earth's surface]][[File:GravityPotential.jpg|thumb|300px|Plot of a two-dimensional slice of the gravitational potential in and around a uniform spherical body. The [[inflection point]]s of the cross-section are at the surface of the body.]]
An example is the (nearly) uniform [[gravitational field]] near the Earth's surface. It has a potential energy
:<math> U = m g h </math>
where ''U'' is the gravitational potential energy and ''h'' is the height above the surface. This means that gravitational potential energy on a [[contour map]] is proportional to altitude. On a contour map, the two-dimensional negative gradient of the altitude is a two-dimensional vector field, whose vectors are always perpendicular to the contours and also perpendicular to the direction of gravity. But on the hilly region represented by the contour map, the three-dimensional negative gradient of ''U'' always points straight downwards in the direction of gravity; '''F'''. However, a ball rolling down a hill cannot move directly downwards due to the normal force of the hill's surface, which cancels out the component of gravity perpendicular to the hill's surface. The component of gravity that remains to move the ball is parallel to the surface:
:<math> F_S = - m g \ \sin \theta </math>
where ''θ'' is the angle of inclination, and the component of ''FS'' perpendicular to gravity is
:<math> F_P = - m g \ \sin \theta \ \cos \theta = - {1 \over 2} m g \sin 2 \theta. </math>
This force ''FP'', parallel to the ground, is greatest when ''θ'' is 45 degrees.

Let Δ''h'' be the uniform interval of altitude between contours on the contour map, and let Δ''x'' be the distance between two contours. Then
:<math> \theta = \tan^{-1}\frac{\Delta h}{\Delta x} </math>
so that
:<math> F_P = - m g { \Delta x \, \Delta h \over \Delta x^2 + \Delta h^2 }. </math>
However, on a contour map, the gradient is inversely proportional to Δ''x'', which is not similar to force ''FP'': altitude on a contour map is not exactly a two-dimensional potential field. The magnitudes of forces are different, but the directions of the forces are the same on a contour map as well as on the hilly region of the Earth's surface represented by the contour map.

==Pressure as buoyant potential==
In [[fluid mechanics]], a fluid in equilibrium, but in the presence of a uniform gravitational field is permeated by a uniform buoyant force that cancels out the gravitational force: that is how the fluid maintains its equilibrium. This [[buoyancy|buoyant force]] is the negative gradient of [[pressure]]:

:<math> \mathbf{f_B} = - \nabla p. \, </math>

Since buoyant force points upwards, in the direction opposite to gravity, then pressure in the fluid increases downwards. Pressure in a static body of water increases proportionally to the depth below the surface of the water. The surfaces of constant pressure are planes parallel to the ground. The surface of the water can be characterized as a plane with zero pressure.

If the liquid has a vertical [[vortex]] (whose axis of rotation is perpendicular to the ground), then the vortex causes a depression in the pressure field. The surfaces of constant pressure are parallel to the ground far away from the vortex, but near and inside the vortex the surfaces of constant pressure are pulled downwards, closer to the ground. This also happens to the surface of zero pressure. Therefore, inside the vortex, the top surface of the liquid is pulled downwards into a depression, or even into a tube (a solenoid).

The buoyant force due to a fluid on a solid object immersed and surrounded by that fluid can be obtained by integrating the negative pressure gradient along the surface of the object:

:<math> F_B = - \oint_S \nabla p \cdot \, d\mathbf{S}. </math>

A moving airplane wing makes the air pressure above it decrease relative to the air pressure below it. This creates enough buoyant force to counteract gravity.

==Calculating the scalar potential==
Given a vector field '''E''', its scalar potential ''Φ'' can be calculated to be
:<math> \phi(\vec{r}) = {1 \over 4 \pi} \iiint_{\vec{r}'} {\vec{\nabla}_{\vec{r}'} \bullet \vec{E}(\vec{r}') \over \| \vec{r} - \vec{r}' \|} \, d\tau' </math>
where dτ' is an infinitesimal volume element with respect to r'. Then, if '''E''' is [[irrotational vector field|irrotational]] (Conservative),
:<math> \vec{E} = -\vec{\nabla} \phi = - {1 \over 4 \pi} \vec{\nabla} \iiint_{\vec{r}'} {\vec{\nabla}_{\vec{r}'} \bullet \vec{E}(\vec{r}') \over \| \vec{r} - \vec{r}' \|} \, d\tau' </math>
This formula is known to be correct if '''E''' is [[continuous function|continuous]] and vanishes asymptotically to zero towards infinity, decaying faster than 1/''r'' and if the [[divergence]] of '''E''' likewise vanishes towards infinity, decaying faster than 1/''r''2.

Proof:

Consider the equation <math> \vec{\nabla}\phi = -\vec{E} </math>.

Take the divergence of both sides to get:

<math> \vec{\nabla} \bullet \vec{\nabla}\phi = -\vec{\nabla} \bullet \vec{E} \iff \nabla^2\phi = -\vec{\nabla} \bullet \vec{E} </math>

The Green's function solution for the above Poisson's equation is:

<math> \phi(\vec{r}) = \frac{1}{4\pi}\iiint_{\vec{r}'} \frac{\vec{\nabla}_{\vec{r}'} \bullet \vec{E}(\vec{r}')}{\|\vec{r}-\vec{r}'\|}d\tau' </math>

Another formula can be derived from the above formula as follows:

<math> \phi(\vec{r}) = \frac{1}{4\pi}\iiint_{\vec{r}'} \frac{\vec{\nabla}_{\vec{r}'} \bullet \vec{E}(\vec{r}')}{\|\vec{r}-\vec{r}'\|}d\tau' = \frac{1}{4\pi}\left(\iiint_{\vec{r}'} \vec{\nabla}_{\vec{r}'} \bullet \frac{\vec{E}(\vec{r}')}{\|\vec{r}-\vec{r}'\|}d\tau' - \iiint_{\vec{r}'} \left(\vec{E}(\vec{r}') \bullet \vec{\nabla}_{\vec{r}'}\frac{1}{\|\vec{r} - \vec{r}'\|}\right)d\tau'\right) </math>

<math> = \frac{1}{4\pi}\left(\iint_{\vec{r}' \; \text{is at infinity}} \frac{\vec{E}(\vec{r}')}{\|\vec{r}-\vec{r}'\|} \bullet d\vec{A}' - \iiint_{\vec{r}'} \left(\vec{E}(\vec{r}') \bullet -\frac{\vec{r}' - \vec{r}}{\|\vec{r} - \vec{r}'\|^3}\right)d\tau'\right) </math>

<math> = -\frac{1}{4\pi}\iiint_{\vec{r}'} \frac{\vec{E}(\vec{r}') \bullet (\vec{r} - \vec{r}')}{\|\vec{r} - \vec{r}'\|^3}d\tau' </math>

Therefore, the scalar potential can also be computed using:

<math> \phi(\vec{r}) = -\frac{1}{4\pi}\iiint_{\vec{r}'} \frac{\vec{E}(\vec{r}') \bullet (\vec{r} - \vec{r}')}{\|\vec{r} - \vec{r}'\|^3}d\tau' </math>

== Calculating the Scalar Potential for n dimensions ==

A general formula for the scalar potential in <math>n \geq 3</math> dimensions can be derived using an approach similar to the derivation of the Helmholtz decomposition theorem.

Let <math>A_n</math> denote the "surface area" of a solid n-dimensional unit sphere embedded in <math>\mathbb{R}^n</math>.

Let <math>\vec{F}</math> denote an irrotational/conservative vector field in <math>\mathbb{R}^n</math>. For any <math>i, j \in \{1, 2, \dots, n\}</math>, it is the case that <math>\frac{\partial}{\partial x_i}F_j - \frac{\partial}{\partial x_j}F_i = 0</math>.

<math>\vec{F}(\vec{r}) = \int_{\vec{r}' \in \mathbb{R}^n}^{} \delta^n(\vec{r} - \vec{r}')\vec{F}(\vec{r}')d\tau'</math>
<math> = \frac{1}{A_n}\int_{\vec{r}' \in \mathbb{R}^n}^{} \left(\vec{\nabla}_{\vec{r}} \bullet{} \frac{\vec{r}-\vec{r}'}{|\vec{r}-\vec{r}'|^n}\right)\vec{F}(\vec{r}')d\tau'</math>

<math> = \frac{-1}{(n-2)A_n}\int_{\vec{r}' \in \mathbb{R}^n}^{} \left(\nabla^2_{\vec{r}}\frac{1}{|\vec{r}-\vec{r}'|^{n-2}}\right)\vec{F}(\vec{r}')d\tau'</math>
<math> = \frac{-1}{(n-2)A_n}\nabla^2_{\vec{r}}\int_{\vec{r}' \in \mathbb{R}^n}^{} \frac{1}{|\vec{r}-\vec{r}'|^{n-2}}\vec{F}(\vec{r}')d\tau'</math>

We will now prove that the vector field <math>\int_{\vec{r}' \in \mathbb{R}^n}^{} {\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}\vec{F}(\vec{r}')d\tau'} </math> is irrotational.

Let <math>i, j \in \{1, 2, \dots, n\}</math> be arbitrary and assume that <math>i \neq j</math>. We need to prove that

<math>\frac{\partial}{\partial x_i} \int_{\vec{r}' \in \mathbb{R}^n}^{} {\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_j(\vec{r}')d\tau'} - \frac{\partial}{\partial x_j} \int_{\vec{r}' \in \mathbb{R}^n}^{} {\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_i(\vec{r}')d\tau'} = 0</math>

<math>\frac{\partial}{\partial x_i} \int_{\vec{r}' \in \mathbb{R}^n}^{} {\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_j(\vec{r}')d\tau'} - \frac{\partial}{\partial x_j} \int_{\vec{r}' \in \mathbb{R}^n}^{} {\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_i(\vec{r}')d\tau'}</math>

<math> = \int_{\vec{r}' \in \mathbb{R}^n}^{} {\left(\frac{\partial}{\partial x_i}\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}\right)F_j(\vec{r}')d\tau'} - \int_{\vec{r}' \in \mathbb{R}^n}^{} {\left(\frac{\partial}{\partial x_j}\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}\right)F_i(\vec{r}')d\tau'}</math>

<math> = \int_{\vec{r}' \in \mathbb{R}^n}^{} {-\left(\frac{\partial}{{\partial x_i'}}\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}\right)F_j(\vec{r}')d\tau'} - \int_{\vec{r}' \in \mathbb{R}^n}^{} {-\left(\frac{\partial}{{\partial x_j'}}\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}\right)F_i(\vec{r}')d\tau'}</math>

<math> = \int_{\vec{r}' \in \mathbb{R}^n}^{} {\bigg(-\left(\frac{\partial}{{\partial x_i'}}\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}\right)F_j(\vec{r}') + \left(\frac{\partial}{{\partial x_j'}}\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}\right)F_i(\vec{r}')\bigg)d\tau'}</math>

<math> = \int_{\vec{r}' \in \mathbb{R}^n}^{} {\bigg(-\frac{\partial}{{\partial x_i'}}\left(\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_j(\vec{r}')\right) + \frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}\frac{\partial}{{\partial x_i'}}F_j(\vec{r}') + \frac{\partial}{{\partial x_j'}}\left(\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_i(\vec{r}')\right) - \frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}\frac{\partial}{{\partial x_j'}}F_i(\vec{r}')\bigg)d\tau'}</math>

<math> = \int_{\vec{r}' \in \mathbb{R}^n}^{} {\bigg(-\frac{\partial}{{\partial x_i'}}\left(\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_j(\vec{r}')\right) + \frac{\partial}{{\partial x_j'}}\left(\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_i(\vec{r}')\right) + \frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}\left(\frac{\partial}{{\partial x_i'}}F_j(\vec{r}') - \frac{\partial}{{\partial x_j'}}F_i(\vec{r}')\right)\bigg)d\tau'}</math>

Since <math>\vec{F}</math> is irrotational, the above expression equals:

<math> = \int_{\vec{r}' \in \mathbb{R}^n}^{} {\bigg(-\frac{\partial}{{\partial x_i'}}\left(\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_j(\vec{r}')\right) + \frac{\partial}{{\partial x_j'}}\left(\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_i(\vec{r}')\right)\bigg)d\tau'}</math>

<math> = -\int_{\vec{r}' \in \mathbb{R}^n}^{} {\bigg(\frac{\partial}{{\partial x_i'}}\left(\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_j(\vec{r}')\right) - \frac{\partial}{{\partial x_j'}}\left(\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_i(\vec{r}')\right)\bigg)d\tau'}</math>

If we hold constant all co-ordinates except <math>x_i</math> and <math>x_j</math>, integrating over <math>x_i</math> and <math>x_j</math> using Green's theorem yields the following curve integral embedded in the <math>x_i, x_j</math> plane:

<math>-{\oint_{\vec{r}' \text{ is at infinity}} {\bigg(\left(\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_i(\vec{r}')\right)\hat{x}_i + \left(\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}F_j(\vec{r}')\right)\hat{x}_j\bigg) \bullet{} d\vec{r}'}}</math>

For the total integral to approach 0 at infinity, it must be the case that the above curve integral is <math>o(|\vec{r}'|^{-(n-2)})</math> as the boundary approaches infinity. This is the case if <math>\vec{F}(\vec{r})</math> is <math>o(|\vec{r}|^{-1})</math> as <math>\vec{r}</math> approaches infinity. Therefore <math>\int_{\vec{r}' \in \mathbb{R}^n}^{} {\frac{1}{|{\vec{r}-\vec{r}'}|^{n-2}}\vec{F}(\vec{r}')d\tau'} </math> is irrotational.

For an irrotational vector field <math>\vec{G}</math>, it can be shown that

<math>\nabla^2\vec{G} = \vec{\nabla}(\vec{\nabla} \bullet{} \vec{G})</math>

Therefore:

<math>\vec{F}(\vec{r}) = \vec{\nabla}_{\vec{r}}\bigg(\frac{-1}{(n-2)A_n}\vec{\nabla}_{\vec{r}} \bullet{} \int_{\vec{r}' \in \mathbb{R}^n}^{} {\frac{1}{|\vec{r}-\vec{r}'|^{n-2}}\vec{F}(\vec{r}')d\tau'}\bigg)</math>

<math> = \vec{\nabla}_{\vec{r}}\bigg(\frac{-1}{(n-2)A_n}\int_{\vec{r}' \in \mathbb{R}^n}^{} {\vec{\nabla}_{\vec{r}} \bullet{} \bigg(\frac{1}{|\vec{r}-\vec{r}'|^{n-2}}\vec{F}(\vec{r}')\bigg)d\tau'}\bigg)</math>

<math> = \vec{\nabla}_{\vec{r}}\bigg(\frac{-1}{(n-2)A_n}\int_{\vec{r}' \in \mathbb{R}^n}^{} {\bigg(\vec{\nabla}_{\vec{r}} \frac{1}{|\vec{r}-\vec{r}'|^{n-2}}\bigg) \bullet{} \vec{F}(\vec{r}')d\tau'}\bigg)</math>

<math> = \vec{\nabla}_{\vec{r}}\bigg(\frac{1}{A_n}\int_{\vec{r}' \in \mathbb{R}^n}^{} {\frac{\vec{r}-\vec{r}'}{|\vec{r}-\vec{r}'|^n} \bullet{} \vec{F}(\vec{r}')d\tau'}\bigg)</math>

<math> = \vec{\nabla}_{\vec{r}}\bigg(\frac{1}{A_n}\int_{\vec{r}' \in \mathbb{R}^n}^{} {\frac{{(\vec{r}-\vec{r}') \bullet{} \vec{F}(\vec{r}')}}{|\vec{r}-\vec{r}'|^n}d\tau'}\bigg)</math>

In conclusion:

<math> \phi(\vec{r}) = -{\frac{1}{A_n}\int_{\vec{r}' \in \mathbb{R}^n}^{} {\frac{{(\vec{r}-\vec{r}') \bullet{} \vec{F}(\vec{r}')}}{|\vec{r}-\vec{r}'|^n}d\tau'}}</math>

is a valid scalar potential for <math>\vec{F}</math>.

== See also ==
* [[Gradient theorem]]
* [[Fundamental theorem of vector analysis]]

==References==
<references/>

{{DEFAULTSORT:Scalar Potential}}
[[Category:Potentials]]
[[Category:Vector calculus]]

[[fr:Champ de vecteurs#Champ de gradient]]

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