formulasearchengine - User contributions [en]

Template:Despicable Me

2014-03-01T03:52:48Z

67.174.173.239:

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2014-02-16T21:47:09Z

67.174.155.31: grammer changes

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Edge detection

2014-02-01T19:57:35Z

67.174.255.201: /* See also */

A '''homomorphism''' between two algebras, ''A'' and ''B'', [[Algebra over a field|over a field]] (or [[Algebra (ring theory)|ring]]) ''K'', is a [[Function (mathematics)|map]] <math>F:A\rightarrow B</math> such that for all ''k'' in ''K'' and ''x'',''y'' in ''A'',

* ''F''(''kx'') = ''kF''(''x'')

* ''F''(''x'' + ''y'') = ''F''(''x'') + ''F''(''y'')

* ''F''(''xy'') = ''F''(''x'')''F''(''y'')<ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref><ref>{{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref>

If ''F'' is [[bijective]] then ''F'' is said to be an '''isomorphism''' between ''A'' and ''B''.

A common abbreviation for "homomorphism between algebras" is "algebra homomorphism" or "algebra map". Every algebra homomorphism is a homomorphism of ''K''-modules.

== Unital algebra homomorphisms ==

If ''A'' and ''B'' are two unital algebras, then an algebra homomorphism <math>F:A\rightarrow B</math> is said to be ''unital'' if it maps the unity of ''A'' to the unity of ''B''. Often the words "algebra homomorphism" are actually used in the meaning of "unital algebra homomorphism", so non-unital algebra homomorphisms are excluded.

==Examples==
Let ''A'' = ''K''[''x''] be the set of all polynomials over a field ''K'' and ''B'' be the set of all polynomial functions over ''K''. Both ''A'' and ''B'' are algebras over ''K'' given by the standard multiplication and addition of polynomials and functions, respectively. We can map each <math>f\,</math> in ''A'' to <math>\hat{f}\,</math> in ''B'' by the rule <math>\hat{f}(t) = f(t) \, </math>. A routine check shows that the mapping <math>f \mapsto \hat{f}\,</math> is a homomorphism of the algebras ''A'' and ''B''. This homomorphism is an isomorphism if and only if ''K'' is an infinite field.

''Proof.'' If ''K'' is a finite field then let

:<math>p(x) = \prod\limits_{t \in K} (x-t).\,</math>

''p'' is a nonzero polynomial in ''K''[''x''], however <math>p(t) = 0\,</math> for all ''t'' in ''K'', so <math>\hat{p} = 0\,</math> is the zero function and our homomorphism is not an isomorphism (and, actually, the algebras are not isomorphic, since the algebra of polynomials is infinite while that of polynomial functions is finite).

If ''K'' is infinite then choose a polynomial ''f'' such that <math>\hat{f} = 0\,</math>. We want to show this implies that <math>f = 0\,</math>. Let <math>\deg f = n\,</math> and let <math>t_0,t_1,\dots,t_n\,</math> be ''n'' + 1 distinct elements of ''K''. Then <math>f(t_i) = 0\,</math> for <math>0 \le i \le n</math> and by [[Lagrange interpolation]] we have <math>f = 0\,</math>. Hence the mapping <math>f \mapsto \hat{f}\,</math> is injective. Since this mapping is clearly surjective, it is bijective and thus an algebra isomorphism of ''A'' and ''B''.

If ''A'' is a [[subalgebra]] of ''B'', then for every [[group of units|invertible]] ''b'' in ''B'' the function that takes every ''a'' in ''A'' to ''b''−1 ''a'' ''b'' is an algebra homomorphism (in case <math>A=B</math>, this is called an inner automorphism of ''B''). If ''A'' is also [[simple algebra|simple]] and ''B'' is a [[central simple algebra]], then every homomorphism from ''A'' to ''B'' is given in this way by some ''b'' in ''B''; this is the [[Skolem-Noether theorem]].

==References==
{{reflist}}

{{DEFAULTSORT:Algebra Homomorphism}}
[[Category:Algebras]]
[[Category:Ring theory]]
[[Category:Morphisms]]

Image segmentation

2014-01-25T04:47:02Z

67.174.255.201: /* Clustering methods */

{{multiple issues|
{{expert-subject|1=Cardiology|date=March 2013}}
{{original research|date=March 2013}}
{{primary sources|date=March 2013}}
{{Refimprove|date=March 2013}}
{{unreliable sources|date=March 2013}}
}}
In [[circulatory system|cardiovascular physiology]], '''ejection fraction''' ('''EF''') represents the volumetric fraction of [[blood]] pumped out of the [[ventricle (heart)]] with each heartbeat or [[cardiac cycle]]. In finite mathematics allowed by medical imaging, EF is applied to both the [[right ventricle]], which ejects blood via the [[pulmonary valve]] into the [[pulmonary circulation]], or the [[left ventricle]], which ejects blood via the [[aortic valve]] into the cerebral and [[systemic circulation]].

Imaging of the physiology of the mammalian heart is the art that allows meaningful mathematical expression defining EF. Noninvasive cardiac imaging has become a worldwide utility enabling study of cardiac performance reproducibly and inexpensively.
Simplified, Ejection fraction is a mathematical product allowed by cardiac imaging. As a volumetric mathematical term, Ejection Fraction is an extension of the work of [[Adolph Fick]] in [[cardiac output]]. Dedicated technology such as [[echocardiography]], [[computed tomography]] (CT Scan), [[magnetic resonance imaging]] (MRI) and [[Radionuclide angiography]] (MUGA) scanning have definitively allowed clinically relevant mathematics regarding [[ischemia]], [[congenital heart disease]], and [[heart failure]].

==Overview==
{{unreferenced section|date=March 2013}}
By definition, the volume of blood within a ventricle immediately before a contraction is known as the [[end-diastolic volume]] (EDV). Likewise, the volume of blood left in a ventricle at the end of contraction is [[end-systolic volume]] (ESV). The difference between EDV and ESV represents many variables such as [[stroke volume]] (SV). SV describes a dated volumetric of blood ejected from the right and left ventricles with each heartbeat. Ejection fraction (Ef) is the fraction of the end-diastolic volume that is ejected with each beat; that is, it is stroke volume (SV) divided by end-diastolic volume (EDV):<ref>Morton Kern 5th edition page 180</ref>

<math>E_f (\%) = \frac{SV}{EDV}\times100</math>

Where the stroke volume is given by:

<math>SV = EDV - ESV</math>

==Normal values==
<div style="float:right">
{{Cardiovascular worksheet}}
</div>

In a healthy {{convert|70|kg|lb|adj=on}} man, the SV is approximately 70 mL and the left ventricular EDV is 120 mL, giving an ejection fraction of {{frac|70|120}}, or 0.58 (58%).

Right ventricular volumes being roughly equal to those of the left ventricle, the ejection fraction of the right ventricle physiologically matches that of the left ventricle within mathematically narrow beat-to-beat limits.

Healthy individuals typically have ejection fractions between 50% and 65%.<ref name="isbn0-7216-0187-1">{{cite book |author=Kumar, Vinay; Abbas, Abul K; Aster, Jon. |title=Robbins and Cotran pathologic basis of disease |edition=8th |publisher=Elsevier Saunders |location=St. Louis, Mo |year=2009 |page=574 |isbn=1-4160-3121-9 |oclc= |doi= |accessdate=}}</ref> However, normal values depend upon the modality being used to calculate the ejection fraction, and some sources consider an ejection fraction of 55–75% to be normal. Damage to the muscle of the heart ([[myocardium]]), such as that sustained during [[myocardial infarction]] or in [[atrial fibrillation]] or a plurality of etiologies of [[cardiomyopathy]], compromises the heart's ability to perform as an efficient pump (ejecting blood) and, therefore, reduces ejection fraction. This reduction in the ejection fraction can manifest itself clinically as [[heart failure]]. A low ejection fraction has its cutoff below 40% with symptomatic manifestations constant at 25%.<ref>{{cite web|title=Heart2008;94:426-428 doi:10.1136/hrt.2007.123877|url=http://heart.bmj.com/content/94/4/426.extract}}</ref> In the USA, a chronically low ejection fraction less than 30% is qualifying support for eligibility of disability benefits from the [[Social Security Administration]].<ref>{{cite web|title=Ejection fraction and SSA disability benefit eligibility.|url=http://www.disabilitysecrets.com/win-can-you-get-disability-for-low-poor-ejection-fraction.html}}</ref>

Healthy older adults favorably adapt as the ventricles become less compliant and are routinely echocardiographically proven to have an EF from 55–85% with the help of good genetics and a healthy lifestyle. Compliance [[changevolume /changepressure]] is a property of the heart that allows [[contractility]]. Encyclopedic documentation of the commonly documented "Hyperdynamic" ventricle remains sparse.

The ejection fraction is one of the most important predictors of [[prognosis]]; those with significantly reduced ejection fractions typically have poorer prognoses. However, recent studies have indicated that a preserved ejection fraction does not mean freedom from risk.<ref name="pmid16855265">{{cite journal |author=Owan TE, Hodge DO, Herges RM, Jacobsen SJ, Roger VL, Redfield MM |title=Trends in prevalence and outcome of heart failure with preserved ejection fraction |journal=N. Engl. J. Med. |volume=355 |issue=3 |pages=251–9 |date=July 2006 |pmid=16855265 |doi=10.1056/NEJMoa052256 |url=http://content.nejm.org/cgi/pmidlookup?view=short&pmid=16855265&promo=ONFLNS19}}</ref>{{primary source-inline|date=March 2013}}<ref name="pmid16855266">{{cite journal |author=Bhatia RS, Tu JV, Lee DS, ''et al.'' |title=Outcome of heart failure with preserved ejection fraction in a population-based study |journal=N. Engl. J. Med. |volume=355 |issue=3 |pages=260–9 |date=July 2006 |pmid=16855266 |doi=10.1056/NEJMoa051530 |url=http://content.nejm.org/cgi/pmidlookup?view=short&pmid=16855266&promo=ONFLNS19}}</ref>{{primary source-inline|date=March 2013}}

The [[QT interval]] as recorded on a standard [[electrocardiogram]] or "EKG" represents ventricular depolarazation and ventricular repolarazation and is rate-dependent.<ref>{{cite journal |last=Bazett |first=H. C. |year=1920 |title=An analysis of the time-relations of electrocardiograms |journal=[[Heart (journal)|Heart]] |volume=7 |issue= |pages=353–370 |pmid= }}</ref>{{primary source-inline|date=March 2013}}

==Measurement==
{{unreferenced section|date=March 2013}}
Ejection fraction is commonly measured by [[echocardiography]], in which the volumes of the heart's chambers are measured during the [[cardiac cycle]]. Ejection fraction can then be obtained by dividing stroke volume by end-diastolic volume as described above.

Accurate volumetric measurement of performance of the right and left ventricles of the heart is inexpensively and routinely echocardiographically interpreted worldwide as a ratio of [[dimension]] between the ventricles in [[Systole (medicine)|systole]] and [[diastole]]. For example, a ventricle in greatest dimension could measure 6 cm while in least dimension 4 cm. Measured and easily reproduced beat to beat for ten or more cycles, this ratio may represent a physiologically normal EF of 50-60%. Mathematical expression of this [[Time]]-dependent ratio can then be interpreted as the greater half as [[cardiac output]] and the lesser half as [[cardiac input]].

Other methods of measuring ejection fraction include cardiac MRI, fast-scan cardiac computed axial tomography (CT) imaging, [[Cardiac ventriculography|ventriculography]], [[Gated SPECT]], and the MUGA scan. A MUGA scan involves the injection of a [[radioisotope]] into the blood and detecting its flow through the left ventricle. The historical [[gold standard (test)|gold standard]] for the measurement of ejection fraction is [[ventriculography]].

==Improving EF==
Depending on the burden of systolic heart failure, a physician may make recommendations to help improve EF. Medication for systolic heart failure is commonly prescribed under several ongoing protocols. Other things that could be done to improve how well the heart pumps include:
Limiting Salt – Limiting salt (sodium) to 2,000 mg a day is an important part of maintaining a healthy heart and treating heart failure. With a low EF, the kidneys get less blood than they should. This makes them unable to rid the body of excess water and salt. Eating too much salt can lead to even more fluid buildup. It also increases blood pressure, which makes an already-weakened heart work harder.<ref>"Heart failure with preserved ejection fraction: is this diastolic heart failure?". Retrieved February 16, 2012.</ref>
Fluid Management – With a low EF, blood can back up in the lungs and force fluid into the breathing spaces. The fluid then builds up, making it difficult to breathe. Excess fluid can also cause weight gain and swelling. Depending on the EF, a doctor may limit the amount of daily fluid intake.
Physical Activity – Exercise can help strengthen the heart and improve how well it pumps blood to the rest of the body. All it takes is 30 minutes a day of activity, even if that activity is walking. It is always recommended that patients consult their doctors about an exercise program that is right for them.<ref>{{cite web|last=Heart Rhythm Society|title=Ejection Fraction|url=http://www.hrsonline.org/Patient-Resources/The-Normal-Heart/Ejection-Fraction#axzz2OJu7oZVJ|accessdate=23 March 2013}}</ref>

==Treating Low EF==
Many people having survived a heart attack can benefit from a medical device called an implantable cardiac defibrillator (ICD). An ICD is a pacemaker-like device that treats ventricular fibrillation (VF), the deadly heart rhythm that causes sudden cardiac arrest (SCA).<ref name="medterms.com">http://www.medterms.com/script/main/art.asp?articlekey=7520</ref>
Several large clinical studies have been conducted in recent years to see whether ICDs could help prevent SCA in those people whose heart muscle, and its pumping ability, is damaged by a heart attack. People in the studies had an ejection fraction (EF) of 40 or below.<ref name="medterms.com"/> In these studies, survival rates were significantly higher for people with ICDs compared to those that received traditional medical care.<ref name="medterms.com"/>

==Medication Intervention==
Certain medications help reduce the heart's workload, increase blood flow, widen vessels or eliminate excess water from the body, all of which may help treating low ejection fraction. Prescribed medications may include:
* Inotropes (such as digoxin): Helps the heart to contract more vigorously and effectively, and helps to reduce symptoms.<ref name="cpmc.org">http://www.cpmc.org/services/heart/tx/ejtreatment.html</ref>

* Angiotensin II receptor blockers: Similar to ACE inhibitors, these medications reduce the stress on the heart muscle and may benefit patients with diabetes and heart disease. The medication protects the kidneys from diabetes-related complications.<ref name="cpmc.org"/>

* Betablockers: These medications may improve symptoms by slowing the heart's contraction rate and reducing its pumping action, thus lessening the heart's workload.<ref name="cpmc.org"/>

==See also==
*[[Cardiac output]]
*[[Heart failure]]
*[[QT interval]]

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

{{Cardiovascular physiology}}

[[Category:Cardiovascular physiology]]

Contrast (vision)

2014-01-25T04:30:37Z

67.174.255.201:

[[File:060618 conductor magnet.svg|thumb|300px|right|Conductor moving in a magnetic field.]]

The '''moving magnet and conductor problem''' is a famous [[thought experiment]], originating in the 19th century, concerning the intersection of [[classical electromagnetism and special relativity]]. In it, the current in a [[Electrical conductor|conductor]] moving with constant velocity, ''v'', with respect to a [[magnet]] is calculated in the [[inertial frame|frame of reference]] of the magnet and in the frame of reference of the conductor. The observable quantity in the experiment, the current, is the same in either case, in accordance with the basic ''principle of relativity'', which states: "Only ''relative'' motion is observable; there is no absolute standard of rest".<ref>The ''Laws of Physics'' are the same in all [[inertial frames]].</ref> However, according to Maxwell's equations, the charges in the conductor experience a '''magnetic force''' in the frame of the magnet and an '''electric force''' in the frame of the conductor. The same phenomenon would seem to have two different descriptions depending on the frame of reference of the observer.

This problem, along with the [[Fizeau experiment]], the [[aberration of light]], and more indirectly the [[Tests of special relativity|negative aether drift tests]] such as the [[Michelson–Morley experiment]], formed the basis of Einstein's development of the theory of relativity.<ref name="norton">{{Citation|last=Norton, John D.|year=2004|first1=John D.|journal=Archive for History of Exact Sciences|title= Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905|pages= 45–105|volume=59|url=http://philsci-archive.pitt.edu/archive/00001743/|doi=10.1007/s00407-004-0085-6|bibcode=2004AHES...59...45N}}</ref>

==Introduction==

[[Albert Einstein|Einstein's]] 1905 paper that introduced the world to relativity opens with a description of the magnet/conductor problem.[http://www.fourmilab.ch/etexts/einstein/specrel/www/]

{{Quotation
|It is known that Maxwell's electrodynamics – as usually understood at the present time – when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighborhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighborhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise – assuming equality of relative motion in the two cases discussed – to electric currents of the same path and intensity as those produced by the electric forces in the former case.
|A. Einstein|On the electrodynamics of moving bodies (1905)
}}

An overriding requirement on the descriptions in different frameworks is that they be [[consistency|consistent]]. Consistency is an issue because [[Newton's laws of motion|Newtonian mechanics]] predicts one transformation (so-called [[Galilean invariance]]) for the ''forces'' that drive the charges and cause the current, while electrodynamics as expressed by [[Maxwell's equations]] predicts that the ''fields'' that give rise to these forces transform differently (according to [[Lorentz invariance]]). Observations of the aberration of light, culminating in the [[Michelson–Morley experiment]], established the validity of Lorentz invariance, and the development of [[special relativity]] resolved the resulting disagreement with Newtonian mechanics. Special relativity revised the transformation of forces in moving reference frames to be consistent with Lorentz invariance. The details of these transformations are discussed below.

In addition to consistency, it would be nice to consolidate the descriptions so they appear to be frame-independent. A clue to a framework-independent description is the observation that magnetic fields in one reference frame become electric fields in another frame. Likewise, the [[solenoidal field|solenoidal]] portion of electric fields (the portion that is not originated by electric charges) becomes a magnetic field in another frame: that is, the solenoidal electric fields and magnetic fields are aspects of the same thing.<ref>There are ''two'' constituents of electric field: a [[solenoidal field]] (or ''incompressible field'') and a [[conservative field]] (or ''irrotational field''). The first is transformable to a magnetic field by changing the frame of reference, the second originates in electric charge, and transforms always into an electric field, albeit of different magnitude.</ref> That means the paradox of different descriptions may be only [[semantic gap|semantic]]. A description that uses scalar and vector potentials φ and '''''A''''' instead of '''''B''''' and '''''E''''' avoids the semantical trap. A Lorentz-invariant [[four vector]] ''A''α = (φ / ''c''0, '''''A''''' ) replaces '''E''' and '''B'''<ref>The symbol ''c''0 represents the [[speed of light]] in [[free space]].</ref> and provides a frame-independent description (albeit less visceral than the '''E'''– '''B'''–description).<ref>However, φ and '''''A''''' are not completely disentangled, so the two types of ''E''-field are not separated completely. See Jackson [http://arxiv.org/abs/physics/0204034 ''From Lorenz to Coulomb and other explicit gauge transformations''] The author stresses that ''Lorenz'' is ''not'' a typo.</ref> An alternative unification of descriptions is to think of the physical entity as the [[electromagnetic field tensor]], as described later on. This tensor contains both '''E''' and '''B''' fields as components, and has the same form in all frames of reference.

==Background==
Electromagnetic fields are not directly observable. The existence of [[Classical physics|classical]] electromagnetic fields can be inferred from the motion of charged particles, whose trajectories are observable. Electromagnetic fields do explain the observed motions of classical charged particles.

A strong requirement in [[physics]] is that all observers of the motion of a particle agree on the trajectory of the particle. For instance, if one observer notes that a particle collides with the center of a bullseye, then all observers must reach the same conclusion. This requirement places constraints on the nature of electromagnetic fields and on their transformation from one reference frame to another. It also places constraints on the manner in which fields affect the acceleration and, hence, the trajectories of charged particles.

Perhaps the simplest example, and one that Einstein referenced in his 1905 paper introducing [[special relativity]], is the problem of a conductor moving in the field of a magnet. In the frame of the magnet, a conductor experiences a ''magnetic'' force. In the frame of a conductor moving relative to the magnet, the conductor experiences a force due to an ''electric'' field. The magnetic field in the magnet frame and the electric field in the conductor frame must generate consistent results in the conductor. At the time of Einstein in 1905, the field equations as represented by [[Maxwell's equations]] were properly consistent. Newton's law of motion, however, had to be modified to provide consistent particle trajectories.<ref name=Penrose>{{cite book |title=The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics |author=Roger Penrose (Martin Gardner: foreword) |page= 248 |url=http://books.google.com/books?id=oI0grArWHUMC&pg=PA248&dq=reference+%22laws+of+physics%22
|isbn=0-19-286198-0 |year=1999 |publisher=Oxford University Press }}</ref>

==Transformation of fields, assuming Galilean transformations==

Assuming that the magnet frame and the conductor frame are related by a [[Galilean transformation]], it is straightforward to compute the fields and forces in both frames. This will demonstrate that the induced current is indeed the same in both frames. As a byproduct, this argument will ''also'' yield a general formula for the electric and magnetic fields in one frame in terms of the fields in another frame.<ref>See Jackson, ''Classical Electrodynamics'', Section 5.15.</ref>

In reality, the frames are ''not'' related by a Galilean transformation, but by a [[Lorentz transformation]]. Nevertheless, it will be a Galilean transformation ''to a very good approximation'', at velocities much less than the speed of light.

Unprimed quantities correspond to the rest frame of the magnet, while primed quantities correspond to the rest frame of the conductor. Let '''v''' be the velocity of the conductor, as seen from the magnet frame.

===Magnet frame===

In the rest frame of the magnet, the magnetic field is some fixed field '''B'''('''r'''), determined by the structure and shape of the magnet. The electric field is zero.

In general, the force exerted upon a particle of charge ''q'' in the conductor by the [[electric field]] and [[magnetic field]] is given by (SI units):

:<math>\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}),</math>

where <math>q</math> is the charge on the particle, <math>\mathbf{v}</math> is the particle velocity and '''F''' is the [[Lorentz force]]. Here, however, the electric field is zero, so the force on the particle is

:<math>\mathbf{F} = q \mathbf{v} \times \mathbf{B}.</math>

===Conductor frame===

In the conductor frame, the magnetic field '''B'''' will be related to the magnetic field '''B''' in the magnet frame according to:<ref>This expression can be thought of as an assumption based on our experience with magnets, that their fields are independent of their velocity. At relativistic velocities, or in the presence of an electric field in the magnet frame, this equation would not be correct.</ref>

:<math>\mathbf{B}'(\mathbf{x}',t) = \mathbf{B}(\mathbf{x}'+\mathbf{v}t).</math>

In this frame, there ''is'' an electric field, generated by the [[Faraday's law of induction#The Maxwell-Faraday equation|Maxwell-Faraday equation]]:

:<math>\mathbf{\nabla \times E}' = -\frac{\partial \mathbf{B}'}{\partial t}.</math>

Using the above expression for '''B'''',

:<math>\mathbf{\nabla \times E}' = -(\mathbf{v} \cdot \nabla) \mathbf{B} = -\nabla\times(\mathbf{B} \times \mathbf{v}) - \mathbf{v}(\nabla\cdot \mathbf{B}) = -\nabla\times(\mathbf{B} \times \mathbf{v})</math>

(using the [[chain rule]] and [[Gauss's law for magnetism]]). This has the solution:

:<math>\mathbf{E}' = -\mathbf{B} \times \mathbf{v} = \mathbf{v}\times \mathbf{B}.</math>

A charge ''q'' in the conductor will be at rest in the conductor frame. Therefore, the magnetic force term of the [[Lorentz force]] has no effect, and the force on the charge is given by

:<math>\mathbf{F}' = q\mathbf{E}' = q\mathbf{v} \times \mathbf{B}.</math>

This demonstrates that ''the force is the same in both frames'' (as would be expected), and therefore any observable consequences of this force, such as the induced current, would also be the same in both frames. This is despite the fact that the force is seen to be an electric force in the conductor frame, but a magnetic force in the magnet's frame.

===Galilean transformation formula for fields===

A similar sort of argument can be made if the magnet's frame also contains electric fields. (The [[Ampere-Maxwell equation]] also comes into play, explaining how, in the conductor's frame, this moving electric field will contribute to the magnetic field.) The end result is that, in general,

:<math>\mathbf{E}' = \mathbf{E} + \mathbf{v}\times \mathbf{B}</math>

:<math>\mathbf{B}' = \mathbf{B} - \frac{1}{{c_0}^2} \mathbf{v} \times \mathbf{E},</math>

with ''c''0 the [[speed of light]] in [[free space]].

By plugging these transformation rules into the full [[Maxwell's equations]], it can be seen that if Maxwell's equations are true in one frame, then they are ''almost'' true in the other, but contain incorrect terms pro by the [[Lorentz transformation]], and the field transformation equations also must be changed, according to the expressions given below.

==Transformation of fields as predicted by Maxwell's equations==
{{see also|Classical electromagnetism and special relativity}}

In a frame moving at velocity '''v''', the '''E'''-field in the moving frame when there is no '''E'''-field in the stationary magnet frame [[Relativistic_electromagnetism#More_rigorous_analysis|Maxwell's equations]] transform as:<ref name=Chow>
{{cite book
|author=Tai L. Chow
|title=Electromagnetic theory
|year= 2006
|publisher=Jones and Bartlett
|location=Sudbury MA
|isbn=0-7637-3827-1
|url=http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0763738271&sig=PgEEBA6TQEZ5fD_AhJQ8dd7MGHo#PPA368,M1
|nopp=true
|pages =Chapter 10.21; p. 402–403 ff}}
</ref>

:<math>\mathbf{E}' = \gamma \mathbf{v} \times \mathbf{B}</math>

where

:<math>\gamma = \frac{1}{\sqrt{1 - {(v/c_0)}^2}}</math>

is called the [[Lorentz factor]] and ''c''0 is the [[speed of light]] in [[free space]]. This result is a consequence of requiring that observers in all [[inertial frames]] arrive at the same form for Maxwell's equations. In particular, all observers must see the same speed of light ''c''0. That requirement leads to the [[Lorentz transformation]] for space and time. Assuming a Lorentz transformation, invariance of Maxwell's equations then leads to the above transformation of the fields for this example.

Consequently, the force on the charge is
:<math>\mathbf{F}' = q \mathbf{E}' = q \gamma \mathbf{v} \times \mathbf{B}.</math>

This expression differs from the expression obtained from the nonrelativistic Newton's law of motion by a factor of [[Lorentz factor|<math>\gamma </math>]]. Special relativity modifies space and time in a manner such that the forces and fields transform consistently.

==Modification of dynamics for consistency with Maxwell's equations==
[[File:Moving magnet.PNG|thumb|400px| Figure 1: Conducting bar seen from two inertial frames; in one frame the bar moves with velocity '''v'''; in the ''primed'' frame the bar is stationary because the primed frame moves at the same velocity as the bar. The '''B'''-field varies with position in the ''x''-direction]]
The Lorentz force has the same ''form'' in both frames, though the fields differ, namely:

:<math>\mathbf{F} = q \left[\mathbf{E} + \mathbf{v} \times \mathbf{B} \right].</math>

See Figure 1. To simplify, let the magnetic field point in the ''z''-direction and vary with location ''x'', and let the conductor translate in the positive ''x''-direction with velocity ''v''. Consequently, in the magnet frame where the conductor is moving, the Lorentz force points in the negative ''y''-direction, perpendicular to both the velocity, and the ''B''-field. The force on a charge, here due only to the ''B''-field, is

:<math>F_y = -qvB,</math>

while in the conductor frame where the magnet is moving, the force is also in the negative ''y''-direction, and now due only to the '''E'''-field with a value:

:<math>{F_y}' = qE' = -q\gamma vB.</math>

The two forces differ by the Lorentz factor γ. This difference is expected in a relativistic theory, however, due to the change in space-time between frames, as discussed next.

Relativity takes the Lorentz transformation of space-time suggested by invariance of Maxwell's equations and imposes it upon [[Dynamics (physics)|dynamics]] as well (a revision of [[Newton's laws of motion]]). In this example, the Lorentz transformation affects the ''x''-direction only (the relative motion of the two frames is along the ''x''-direction). The relations connecting time and space are ( ''primes'' denote the moving conductor frame ) :<ref name=Chow2>
{{cite book
|author=Tai L. Chow
|title=Electromagnetic theory
|year= 2006
|publisher=Jones and Bartlett
|location=Sudbury MA
|isbn=0-7637-3827-1
|url=http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0763738271#PPA368,M1
|nopp=true
|pages =Chapter 10.5; p. 368 ff}}
</ref>

:<math>x' = \gamma (x - vt), \quad x = \gamma(x' + vt'),</math>

:<math>t' = \gamma (t - \frac{vx}{c_0^2}), \quad t = \gamma(t' + \frac{vx'}{c_0^2}).</math>

These transformations lead to a change in the ''y''-component of a [[Special relativity#Force|force]]:

:<math>{F_y}' = \gamma F_y.</math>

That is, within [[Lorentz invariance]], force is '''''not''''' the same in all frames of reference, unlike Galilean invariance. But, from the earlier analysis based upon the Lorentz force law:

:<math>\gamma F_y = -q\gamma vB, \quad {F_y}' = -q\gamma v B,</math>

which agrees completely. So the force on the charge is '''''not''''' the same in both frames, but it transforms as expected according to relativity.

==Newton's law of motion in modern notation==
{{main| Formulation of Maxwell's equations in special relativity }}

The modern approach to obtaining the relativistic version of Newton's law of motion can be obtained by writing Maxwell's equations in [[Covariant transformation|covariant form]] and identifying a covariant form that is a generalization of Newton's law of motion.

Newton's law of motion can be written in modern covariant notation in terms of the field strength tensor as (cgs units):

:<math>mc\frac{du^\alpha}{d\tau} = F^{\alpha\beta} q u_\beta,</math>

where ''m'' is the particle [[mass]], ''q'' is the [[Electric charge|charge]], and

:<math>u_\beta = \eta_{\beta\alpha } u^\alpha = \eta_{\beta\alpha} \frac{dx^\alpha}{d\tau}</math>

is the [[four-velocity|4-velocity]] of the particle. Here, <math>\tau</math> is ''c'' times the [[proper time]] of the particle and <math>\eta</math> is the [[Minkowski metric]] tensor.

The field strength tensor is written in terms of fields as:

:<math>F^{\alpha\beta} = \left(\begin{matrix}
0 & {E_x} & {E_y} & {E_z} \\
-{E_x} & 0 & cB_z & -cB_y \\
-{E_y} & -cB_z & 0 & cB_x \\
-{E_z} & cB_y & -cB_x & 0
\end{matrix}\right).</math>

Alternatively, using the four vector:

:<math>A^\alpha = \left(\phi/c, A_x, A_y, A_z \right), </math>

related to the electric and magnetic fields by:

:<math>\mathbf{E} = -\nabla\phi - \partial_t \mathbf{A}, \quad \mathbf{B} = \nabla \times \mathbf{A},</math>

the field tensor becomes:<ref name =DJ_Griffiths>{{cite book |author=DJ Griffiths |title=Introduction to electrodynamics |publisher=Pearson/Addison-Wesley |year=1999 |location=Saddle River NJ |page=541 |isbn =0-13-805326-X}}</ref>

:<math>F^{\alpha\beta} = \frac{\partial A^\beta}{\partial x_\alpha} - \frac{\partial A^\alpha}{\partial x_\beta},</math>

where:

:<math>x_\alpha = \left(-ct, x, y, z \right).</math>

The fields are transformed to a frame moving with constant relative velocity by:

:<math>\acute{F}^{\mu\nu} = {\Lambda^\mu}_\alpha {\Lambda^\nu}_\beta F^{\alpha\beta},</math>

where <math>{\Lambda^\mu}_\alpha</math> is a [[Lorentz transformation]].

In the magnet/conductor problem this gives

:<math>\mathbf{E}' = \gamma \frac{\mathbf{v}}{c} \times \mathbf{B},</math>

which agrees with the traditional transformation when one takes into account the difference between SI and cgs units. Thus, the relativistic modification to Newton's law of motion using the traditional Lorentz force yields predictions for the motion of particles that are consistent in all frames of reference with Maxwell's equations.

==References and notes==

<references/>

==External links==
* [http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/special_relativity/special_relativity.html Magnets and conductors in special relativity]

==Further reading==

* {{cite book | author=Einstein, A. | author-link = Albert Einstein | title=Relativity: The Special and General Theory | location= New York | publisher=Crown| year=1961 | isbn=0-517-02961-8}}

* {{Cite book| last1 = Feynman| first1 = Richard P. |author-link = Richard Feynman |first2 = Robert B. |last2 = Leighton |author2-link = Robert B. Leighton |first3 = Matthew |last3 = Sands |author3-link = Matthew Sands |title = [[The Feynman Lectures on Physics]] |volume = Vol 2 |year = 2006 |isbn = 0-8053-9045-6
|page= 13-6 Chapter 13}} (The relativity of magnetic and electric fields)

* {{cite book | author=Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald | title=Gravitation | location=San Francisco | publisher=W. H. Freeman | year=1973 | isbn=0-7167-0344-0}}

* {{cite book | author=Landau, L. D. and Lifshitz, E. M.| title=Classical Theory of Fields (Fourth Revised English Edition) | location=Oxford | publisher=Pergamon | year=1975 | isbn=0-08-018176-7}}

* {{cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|isbn=0-471-30932-X}}

* {{cite book |author=C Møller |title=The Theory of Relativity |publisher=Oxford University Press |location=Oxford UK |isbn=0-19-560539-X |year=1976 |url=http://worldcat.org/oclc/220221617&referer=brief_results |edition=Second Edition}}

==See also==
{{multicol}}
* [[Principle of relativity]]
* [[Galilean invariance]]
* [[Lorentz transformation]]
* [[Theory of special relativity]]

{{multicol-break}}
* [[Faraday's law of induction|Faraday's law]]
* [[Lenz's law]]
* [[Inertial frame]]
* [[Annus Mirabilis Papers]]

{{multicol-break}}
* [[Electric motor]]
* [[Eddy current]]
* [[Faraday paradox]]
* [[Darwin Lagrangian]]
{{multicol-end}}

[[Category:Electromagnetism]]
[[Category:Special relativity]]

Models of neural computation

2014-01-25T04:17:17Z

67.174.255.201: /* Gain control */

{{unreferenced|date=December 2009}}

In [[optics]], '''differential group delay''' is the [[Difference (mathematics)|difference]] in [[phase velocity|propagation time]] between the two [[eigenmode]]s ''X'' and ''Y'' [[polarizations]]. Consider two [[eigenmodes]] that are the 0° and 90° [[Linearity|linear]] [[polarized light|polarization]] states. If the state of polarization of the input signal is the linear state at 45° between the two eigenmodes, the input signal is divided equally into the two eigenmodes. The power of the [[Transmitter|transmitted signal]] ''E''''T'',total is the combination of the transmitted signals of both ''x'' and ''y'' modes.

: <math>E_T = (E_{i,x} \cdot t_x)^2 + (E_{i,y} \cdot t_y)^2 \, </math>

The differential group delay ''D''''t'' is defined as the difference in propagation time between the eigenmodes: ''D''''t'' = |''t''''t'',''x'' − ''t''''t'',''y''|.

{{DEFAULTSORT:Differential Group Delay}}
[[Category:Optics]]

Goos–Hänchen effect

2012-08-10T18:55:58Z

67.174.35.27: added the word 'phase' to shift, because otherwise it is not entierly what shifts, e.g. could also mean the polarization

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