formulasearchengine - User contributions [en]

Consequentia mirabilis

2014-05-22T15:14:29Z

152.78.90.212: Minor change for clarity

The author is called Irwin Wunder but it's not the most masucline name out there. The factor she adores most is body building and now she is trying to earn cash with it. I am a meter reader but I strategy on changing it. Years in the past we moved to Puerto Rico and my family members loves it. my weblog: [http://premium.asslikethat.com/blog/9567 std testing at home]

Modularity (networks)

2014-02-25T14:15:48Z

152.78.40.207:

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Faraday cup

2013-11-28T12:16:19Z

152.78.41.92:

{{Refimprove|date=July 2013}}
'''Inverse synthetic aperture radar''' ('''ISAR''') is a technique to generate a two-dimensional high resolution image of a target.
ISAR technology utilizes the movement of the target rather than the emitter to create the synthetic aperture. ISAR radars have a significant role aboard maritime patrol aircraft to provide them with radar image of sufficient quality to allow it to be used for target recognition purposes. In situations where other radars display only a single unidentifiable bright moving pixel, the ISAR image is often adequate to discriminate between various missiles, military aircraft, and civilian aircraft.

==Radar Cross Section (RCS) Imaging==
Images of the target region produced by ISAR can be a useful tool in locating scattering regions on the target. ISAR images are produced by rotating the target and processing the resultant [[Doppler effect|Doppler histories]] of the scattering centers. If the target rotates in [[azimuth]] at a constant rate through a small angle, scatters will approach or recede from the radar at a rate depending only on the cross range position- the distance normal to the radar line of sight with the origin at the target axis of rotation. The rotation will result in the generation of cross range dependent Doppler [[frequency|frequencies]] which can be sorted by a [[Fourier transform]]. This operation is equivalent to the generation of a large [[synthetic aperture]] [[phased array]] [[antenna (radio)|antenna]] formed by the coherent summation of the receiver outputs for varying target / antenna geometries. For small angles, an ISAR image is the 2-dimensional Fourier transform of the received signal as a function of frequency and target aspect angle.

If the target is rotated through large angles, the Doppler frequency history of a scatter becomes non linear, following a [[sine-wave]] trajectory. This Doppler history can not be processed directly by a Fourier transform because of the smeared Doppler frequency history resulting in the loss of cross range resolution. The maximum rotation angle which can be processed by an unmodified Fourier transform is determined by the constraint that the aperture phase error across the synthesized aperture should vary by less than a specified arbitrary amount, usually 45 degrees. This occurs when the synthetic aperture to the target range is less than required by the <math>\frac{2 D^2}{\lambda}</math> limit where <math>D</math> is the required lateral extent of the target. At this point the synthetic aperture is within the target nearfield region and requires focusing. The focusing is accomplished by applying a phase correction to the synthetic aperture.

==ISAR Applications==

ISAR is utilized in maritime surveillance for the classification of ships and other objects. In these applications the motion of the object due to wave action often plays a greater role than object rotation. For instance a feature which extends far over the surface of a ship such as a mast will provide a high sinusoidal response which is clearly identifiable in a two dimensional image. Images sometimes produce an uncanny similarity to a visual profile with the interesting effect that as the object rocks towards or away from the receiver the alternating doppler returns cause the profile to cycle between upright and inverted. ISAR for maritime surveillance was pioneered by Texas Instruments in collaboration with the Naval Research Laboratory and became an important capability of the P-3 Orion and the S-3B Viking US Navy aircraft.

Research has also been done with land based ISAR. The difficulty in utilizing this capability is that the object motion is far less in magnitude and usually less periodic than in the maritime case.

Perhaps the most visually striking and scientifically compelling application of ISAR is in the deep space imaging of asteroids. A particularly beautiful example of this is the so-called "dog's bone" [[216 Kleopatra]] asteroid, which lies roughly 20% further away from the earth than does the sun. The asteroid is only 60 miles wide at its midpoint. Yet the imagery is crisp and "feels" like an optical image. This has been cited as akin to using a Los Angeles telescope the size of the human eye's lens to image a car in New York. Of course the "trick" here is that the asteroid is presented amongst a very sparse background, allowing for substantial disambiguation.
Nowadays this technology is also used in Wi-Vi technology systems. The Wi-Vi technology stands for "Wi-Fi and vision". this technology lets us to see through the wall by the use of signals and LCD display.

In February 2013, Indra Sistemas, a Spanish leading-technology corporation announced the first passive ISAR radar. A passive radar is characterised by not emitting any form of radiation, i.e., it uses the signals present in the environment. In this case, the radar uses Digital Terrestrial Television signals as the non-cooperative sources of illumination in the environment.<ref>{{cite web|title=Indra develops the first high-resolution passive radar system|url=http://www.indracompany.com/en/noticia/indra-develops-the-first-high-resolution-passive-radar-system|accessdate=2013-02-11}}</ref>

==Errors in ISAR==
Errors in the ISAR imaging process generally result in defocusing and geometry errors in the image. ISAR transform errors include:
*'''Unknown target or antenna motion:''' Unmodeled motion will cause the target image to defocus and be at an incorrect location. This error is controlled by suitable mechanical design or by the use of [[auto-focus]] techniques. This error can be measured by the analytic [[signal phase]] measurement method described earlier.
*'''Vertical nearfield errors:''' Unless 3D ISAR is performed, the vertical target extent at right angles to the horizontal synthetic aperture must fit within the vertical far field limit. Tall targets will defocus and move to incorrect positions. The 2D ISAR representation of a target region is a planar surface.
*'''Integrated sidelobe return:''' ISAR image quality is degraded by [[Slant range|range]] and [[azimuth compression]] [[side lobe]]s. The sidelobes are due to data [[truncation]] and can be reduced by the application of appropriate window functions. The sidelobes can cause significant image degradation. First, the peaks of the stronger sidelobes may cause a string of progressively weaker targets to appear on either side of a strong target. Second, the combined power of all sidelobes tends to fog or washout detail in low RCS areas. The integrated sidelobe level can under poor conditions reach a level 10 [[Decibel|dB]] below the peak target return.
*'''Frequency and azimuth sampling errors:''' Incorrectly selected frequency or aspect deltas will result in [[aliased images]], creating spurious targets. The SIM program described earlier specifically monitors for aliening errors effectively eliminating this error source.
*'''Antenna aberrations:''' Aberrations in the geometry result when the antenna phase center position is dependent upon the antenna aspect or RF frequency. This error source is normally controlled by using small, simple antennas over narrow frequency bands at long ranges. First order corrections to frequency dispersive antennas such as log periodic can be handled by phase correcting the received signal. Full correction of the aberrations can be accomplished by a direct integration of the ISAR transform using the aberrated geometry.
*'''Target dispersion:''' Dispersive targets have a non-minimum phase response, appearing to shift in position with [[Radio Frequency|RF]] frequency. Examples of dispersive targets include RF absorbers in which the absorption depth is a function of frequency and various antenna in which the phase center position is frequency dependent. [[Continuous Wavelength|CW]] ISAR imaging or in some cases preprocessing prior to a FMCW ISAR transform an eliminate dispersive defocusing of the target image.
*'''Multipath:''' Multiple reflections can result in ISAR imaging distortions such as the classic ghost image trails from [[jet aircraft]] [[tail pipes]].
Errors in the 2D planar Inverse ISAR transform include:
*'''Image blocking modeling errors:''' The Inverse ISAR transform currently assumes that scatters are on a planar surface and cannot block other scatters.
*'''Image multipath modeling errors:''' The Inverse ISAR transform currently does not model the multipath environment. Note the current ISAR transforms also do not correctly process multipath.


== See also ==

* [[Synthetic aperture radar]]
* [[aperture synthesis]]
* [[beamforming]]
* [[phased array]]
* [[Optical heterodyne detection]]

== References ==

{{reflist}}

== External links ==
* [http://ww2.nearfield.com/amta/amta85ds.htm Inverse Synthetic Aperture Imaging Radar by Dan Slater 1985] 
* [http://www.fas.org/man/dod-101/navy/docs/es310/advradrsys/AdvRadr.htm Advanced Radar Systems] 

{{DEFAULTSORT:Inverse Synthetic Aperture Radar}}
[[Category:Radar]]

Gravitational anomaly

2013-11-20T10:03:34Z

152.78.132.250: Undid revision 581695267 by 71.95.39.29 (talk)

{{Redirect|LSZ|other uses|LSZ (disambiguation)}}
{{Quantum field theory|cTopic=Tools}}
In [[quantum field theory]], the '''LSZ reduction formula''' is a method to calculate [[S-matrix]] elements (the [[scattering amplitude]]s) from the [[time ordered|time-ordered]] [[correlation function (quantum field theory)|correlation functions]] of a quantum field theory. It is a step of the path that starts from the [[Lagrangian]] of some quantum field theory and leads to prediction of measurable quantities. It is named after the three German physicists [[Harry Lehmann]], [[Kurt Symanzik]] and [[Wolfhart Zimmermann]].

Although the LSZ reduction formula cannot handle [[bound state]]s, [[massless particle]]s and [[topological soliton]]s, it can be generalized to cover bound states, by use of [[composite field]]s which are often nonlocal. Furthermore, the method, or variants thereof, have turned out to be also fruitful in other fields of theoretical physics. For example in [[statistical physics]] they can be used to get a particularly general formulation of the [[fluctuation-dissipation theorem]].

==In and Out fields==

[[S matrix|S-matrix elements]] are amplitudes of [[Transition of state|transitions]] between ''in'' states and ''out'' states. An ''in'' state <math>|\{p\}\ \mathrm{in}\rangle</math> describes the state of a system of particles which, in a far away past, before interacting, were moving freely with definite momenta <math>\{p\}</math>, and, conversely, an ''out'' state <math>|\{p\}\ \mathrm{out}\rangle</math> describes the state of a system of particles which, long after interaction, will be moving freely with definite momenta <math>\{p\}</math>.

''In'' and ''out'' states are states in [[Heisenberg picture]] so they should not be thought to describe particles at a definite time, but rather to describe the system of particles in its entire evolution, so that the S-matrix element:

:<math>
S_{fi}=\langle \{q\}\ \mathrm{out}| \{p\}\ \mathrm{in}\rangle
</math>

is the [[probability amplitude]] for a set of particles which were prepared with definite momenta <math>\{p\}</math> to interact and be measured later as a new set of particles with momenta <math>\{q\}</math>.

The easy way to build ''in'' and ''out'' states is to seek appropriate field operators that provide the right [[creation and annihilation operators]]. These fields are called respectively ''in'' and ''out'' fields.

Just to fix ideas, suppose we deal with a [[Scalar field (quantum field theory)|Klein-Gordon field]] that interacts in some way which doesn't concern us:

:<math>
\mathcal L= \frac 1 2 \part_\mu \varphi\part^\mu \varphi - \frac 1 2 m_0^2 \varphi^2 +\mathcal L_{\mathrm{int}}
</math>

<math>\mathcal L_{\mathrm{int}}</math> may contain a [[Nonlinear scalar field theory|self interaction]] <math>g\ \varphi^3</math> or interaction with other fields, like a [[Yukawa interaction]] <math>g\ \varphi\bar\psi\psi</math>. From this [[Lagrangian]], using [[Euler-Lagrange equation]]s, the equation of motion follows:

:<math>
\left(\part^2+m_0^2\right)\varphi(x)=j_0(x)
</math>

where, if <math>\mathcal L_{\mathrm{int}}</math> does not contain derivative couplings:

:<math>
j_0=\frac{\part\mathcal L_{\mathrm{int}}}{\part \varphi}
</math>

We may expect the ''in'' field to resemble the asymptotic behaviour of the interacting field as <math>x^0\rightarrow-\infty</math>, making the assumption that in the far away past interaction described by the current <math>j_0</math> is negligible, as particles are far from each other. This hypothesis is named the '[[adiabatic theorem|adiabatic hypothesis]]'. However [[self-energy|self interaction]] never fades away and, besides many other effects, it causes a difference between the Lagrangian mass <math>m_0</math> and the physical mass <math>m</math> of the <math>\varphi</math> [[boson]]. This fact must be taken into account by rewriting the equation of motion as follows:{{Citation needed|date=August 2011}}

:<math>
\left(\part^2+m^2\right)\varphi(x)=j_0(x)+\left(m^2-m_0^2\right)\varphi(x)=j(x)
</math>

This equation can be solved formally using the retarded [[Green's function]] of the Klein-Gordon operator <math>\partial^2+m^2</math>:

:<math>
\Delta_{\mathrm{ret}}(x)=i\theta\left(x^0\right)
\int \frac{\mathrm{d}^3k}{(2\pi)^3 2\omega_k}
\left(e^{-ik\cdot x}-e^{ik\cdot x}\right)_{k^0=\omega_k};\quad
\omega_k=\sqrt{\mathbf{k}^2+m^2}
</math>

allowing us to split interaction from asymptotic behaviour. The solution is:

:<math>
\varphi(x)=\sqrt Z \varphi_{\mathrm{in}}(x) +\int \mathrm{d}^4y \Delta_{\mathrm{ret}}(x-y)j(y)
</math>

The factor <math>\sqrt Z</math> is a normalization factor that will come handy later, the field <math>\varphi_{\mathrm{in}}</math> is a solution of the [[Homogeneous differential equation|homogeneous equation]] associated with the equation of motion:

:<math>
\left(\part^2+m^2\right) \varphi_{\mathrm{in}}(x)=0
</math>,

and hence is a [[free field]] which describes an incoming unperturbed wave, while the last term of the solution gives the [[Perturbation theory (quantum mechanics)|perturbation]] of the wave due to interaction.

The field <math>\varphi_{\mathrm{in}}</math> is indeed the ''in'' field we were seeking, as it describes the asymptotic behaviour of the interacting field as <math>x^0\rightarrow-\infty</math>, though this statement will be made more precise later. It is a free scalar field so it can be expanded in flat waves:

:<math>
\varphi_{\mathrm{in}}(x)=\int \mathrm{d}^3k \left\{f_k(x) a_{\mathrm{in}}(\mathbf{k})+f^*_k(x) a^\dagger_{\mathrm{in}}(\mathbf{k})\right\}
</math>

where:
:<math>
f_k(x)=\left.\frac{e^{-ik\cdot x}}{(2\pi)^{3/2}(2\omega_k)^{1/2}}\right|_{k^0=\omega_k}
</math>

The inverse function for the coefficients in terms of the field can be easily obtained and put in the elegant form:
:<math>
a_{\mathrm{in}}(\mathbf{k})=i\int \mathrm{d}^3x f^*_k(x)\overleftrightarrow\partial_0\varphi_{\mathrm{in}}(x)
</math>
where:

<math>
{\mathrm{g}}{\overleftrightarrow\partial}_0 f = \mathrm{g}\partial_0 f -f\partial_0 \mathrm{g}.
</math>

The [[Fourier coefficient]]s satisfy the algebra of [[creation and annihilation operators]]:

:<math>
[a_{\mathrm{in}}(\mathbf{p}),a_{\mathrm{in}}(\mathbf{q})]=0;\quad [a_{\mathrm{in}}(\mathbf{p}),a^\dagger_{\mathrm{in}}(\mathbf{q})]=\delta^3(\mathbf{p}-\mathbf{q});
</math>

and they can be used to build ''in'' states in the usual way:

:<math>
\left|k_1,\ldots,k_n\ \mathrm{in}\right\rangle=\sqrt{2\omega_{k_1}}a_{\mathrm{in}}^\dagger(\mathbf{k}_1)\ldots \sqrt{2\omega_{k_n}}a_{\mathrm{in}}^\dagger(\mathbf{k}_n)|0\rangle
</math>

The relation between the interacting field and the ''in'' field is not very simple to use, and the presence of the retarded Green's function tempts us to write something like:

:<math>
\varphi(x)\sim\sqrt Z\varphi_{\mathrm{in}}(x)\quad \mathrm{as}\quad x^0\rightarrow-\infty
</math>

implicitly making the assumption that all interactions become negligible when particles are far away from each other. Yet the current <math>j(x)</math> contains also self interactions like those producing the mass shift from <math>m_0</math> to <math>m</math>. These interactions do not fade away as particles drift apart, so much care must be used in establishing asymptotic relations between the interacting field and the ''in'' field.

The correct prescription, as developed by Lehmann, Symanzik and Zimmermann, requires two normalizable states <math>|\alpha\rangle</math> and <math>|\beta\rangle</math>, and a normalizable solution <math>f(x)</math> of the Klein-Gordon equation <math>(\part^2+m^2)f(x)=0</math>. With these pieces one can state a correct and useful but very weak asymptotic relation:

:<math>
\lim_{x^0\rightarrow-\infty}
\int \mathrm{d}^3x \langle\alpha|f(x)\overleftrightarrow\part_0\varphi(x)|\beta\rangle=
\sqrt Z \int \mathrm{d}^3x \langle\alpha|f(x)\overleftrightarrow\part_0\varphi_{\mathrm{in}}(x)|\beta\rangle
</math>

The second member is indeed independent of time as can be shown by deriving and remembering that both <math>\varphi_{\mathrm{in}}</math> and <math>f</math> satisfy the Klein-Gordon equation.

With appropriate changes the same steps can be followed to construct an ''out'' field that builds ''out'' states. In particular the definition of the ''out'' field is:

:<math>
\varphi(x)=\sqrt Z \varphi_{\mathrm{out}}(x) +\int \mathrm{d}^4y \Delta_{\mathrm{adv}}(x-y)j(y)
</math>

where <math>\Delta_{\mathrm{adv}}(x-y)</math> is the advanced Green's function of the Klein-Gordon operator. The weak asymptotic relation between ''out'' field and interacting field is:

:<math>
\lim_{x^0\rightarrow+\infty}
\int \mathrm{d}^3x \langle\alpha|f(x)\overleftrightarrow\part_0\varphi(x)|\beta\rangle=
\sqrt Z \int \mathrm{d}^3x
\langle\alpha|f(x)\overleftrightarrow\part_0\varphi_{\mathrm{out}}(x)|\beta\rangle
</math>

==The reduction formula for scalars==

The asymptotic relations are all that is needed to obtain the LSZ reduction formula. For future convenience we start with the matrix element:

:<math>
\mathcal M=\langle \beta\ \mathrm{out}|\mathrm T\ \varphi(y_1)\ldots\varphi(y_n)|\alpha\ p\ \mathrm{in}\rangle
</math>

which is slightly more general than an S-matrix element. Indeed, <math>\mathcal M</math> is the expectation value of the [[Path-ordering|time-ordered product]] of a number of fields <math>\varphi(y_1)\ldots\varphi(y_n)</math> between an ''out'' state and an ''in'' state. The ''out'' state can contain anything from the vacuum to an undefined number of particles, whose momenta are summarized by the index <math>\beta</math>. The ''in'' state contains at least a particle of momentum <math>p</math>, and possibly many others, whose momenta are summarized by the index <math>\alpha</math>. If there are no fields in the time-ordered product, then <math>\mathcal M</math> is obviously an S-matrix element. The particle with momentum <math>p</math> can be 'extracted' from the ''in'' state by use of a creation operator:

:<math>
\mathcal M=\sqrt{2\omega_p}\
\langle \beta\ \mathrm{out}|
\mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
a_{\mathrm{in}}^\dagger(\mathbf p)
|\alpha\ \mathrm{in}\rangle
</math>

With the assumption that no particle with momentum ''p'' is present in the ''out'' state, that is, we are ignoring forward scattering, we can write:

:<math>
\mathcal M=\sqrt{2\omega_p}\
\langle \beta\ \mathrm{out}|
\mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
a_{\mathrm{in}}^\dagger(\mathbf p)-
a_{\mathrm{out}}^\dagger(\mathbf p)
\mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
|\alpha\ \mathrm{in}\rangle
</math>

because <math>a_{\mathrm{out}}^\dagger</math> acting on the left gives zero. Expressing the construction operators in terms of ''in'' and ''out'' fields, we have:

:<math>
\mathcal M=-i\sqrt{2\omega_p}\
\int \mathrm{d}^3x f_p(x)\overleftrightarrow\part_0
\langle \beta\ \mathrm{out}|
\mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
\varphi_{\mathrm{in}}(x)-
\varphi_{\mathrm{out}}(x)
\mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
|\alpha\ \mathrm{in}\rangle
</math>

Now we can use the asymptotic condition to write:

:<math>
\mathcal M=-i\sqrt{\frac{2\omega_p}{Z}}\left\{
\lim_{x^0\rightarrow-\infty}
\int \mathrm{d}^3x f_p(x)\overleftrightarrow\part_0
\langle \beta\ \mathrm{out}|
\mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
\varphi(x)
|\alpha\ \mathrm{in}\rangle-
\right.
</math>
:<math>
\left. -\lim_{x^0\rightarrow+\infty}
\int \mathrm{d}^3x f_p(x)\overleftrightarrow\part_0
\langle \beta\ \mathrm{out}|
\varphi(x)
\mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
|\alpha\ \mathrm{in}\rangle
\right\}
</math>

Then we notice that the field <math>\varphi(x)</math> can be brought inside the time-ordered product, since it appears on the right when <math>x^0\rightarrow -\infty</math> and on the left when <math>x^0\rightarrow +\infty</math>:

:<math>
\mathcal M=-i\sqrt{\frac{2\omega_p}{Z}}
\left(\lim_{x^0\rightarrow-\infty}-\lim_{x^0\rightarrow+\infty}\right)
\int \mathrm{d}^3x f_p(x)\overleftrightarrow\part_0
\langle \beta\ \mathrm{out}|
\mathrm T\left[\varphi(x)\varphi(y_1)\ldots\varphi(y_n)\right]
|\alpha\ \mathrm{in}\rangle
</math>

In the following, <math>x</math> dependence in the time-ordered product is what matters, so we set:
:<math>
\langle \beta\ \mathrm{out}|
\mathrm T\left[\varphi(x)\varphi(y_1)\ldots\varphi(y_n)\right]
|\alpha\ \mathrm{in}\rangle=
\eta(x)
</math>

It's easy to show by explicitly carrying out the time integration that:

:<math>
\mathcal M=i\sqrt{\frac{2\omega_p}{Z}}
\int \mathrm{d}(x^0)\part_0
\int \mathrm{d}^3x f_p(x)\overleftrightarrow\part_0
\eta(x)
</math>

so that, by explicit time derivation, we have:

:<math>
\mathcal M=i\sqrt{\frac{2\omega_p}{Z}}
\int \mathrm{d}^4 x\left\{f_p(x)\part_0^2\eta(x)-\eta(x)\part_0^2 f_p(x)\right\}
</math>

By its definition we see that <math>f_p(x)</math> is a solution of the Klein-Gordon equation, which can be written as:

:<math>
\part_0^2f_p(x)=\left(\Delta-m^2\right) f_p(x)
</math>

Substituting into the expression for <math>\mathcal M</math> and integrating by parts, we arrive at:

:<math>
\mathcal M=i\sqrt{\frac{2\omega_p}{Z}}
\int \mathrm{d}^4 x f_p(x)\left(\part_0^2-\Delta+m^2\right)\eta(x)
</math>

That is:

:<math>
\mathcal M=\frac{i}{(2\pi)^{3/2} Z^{1/2}}
\int \mathrm{d}^4 x e^{-ip\cdot x} \left(\Box+m^2\right)
\langle \beta\ \mathrm{out}|
\mathrm T\left[\varphi(x)\varphi(y_1)\ldots\varphi(y_n)\right]
|\alpha\ \mathrm{in}\rangle
</math>

Starting from this result, and following the same path another particle can be extracted from the ''in'' state, leading to the insertion of another field in the time-ordered product. A very similar routine can extract particles from the ''out'' state, and the two can be iterated to get vacuum both on right and on left of the time-ordered product, leading to the general formula:

:<math>
\langle p_1,\ldots,p_n\ \mathrm{out}|q_1,\ldots,q_m\ \mathrm{in}\rangle=\int
\prod_{i=1}^{m}
\left\{
\mathrm{d}^4x_i\
i\frac{e^{-iq_i\cdot x_i}}{(2\pi)^{3/2} Z^{1/2}}
\left(\Box_{x_i}+m^2\right)
\right\}\times
</math>
:<math>
\times
\prod_{j=1}^{n}
\left\{
\mathrm{d}^4y_j\
i\frac{e^{+ip_j\cdot y_j}}{(2\pi)^{3/2} Z^{1/2}}
\left(\Box_{y_j}+m^2\right)
\right\}
\langle 0|\mathrm{T}\ \varphi(x_1)\ldots\varphi(x_m)\varphi(y_1)\ldots\varphi(y_n)|0\rangle
</math>

Which is the LSZ reduction formula for Klein-Gordon scalars. It gains a much better looking aspect if it is written using the Fourier transform of the correlation function:

:<math>
\Gamma(p_1,\ldots,p_n)=\int
\prod_{i=1}^{n}
\left\{
\mathrm{d}^4x_i\
e^{i p_i\cdot x_i}
\right\}
\langle 0|\mathrm{T}\ \varphi(x_1)\ldots\varphi(x_n)|0\rangle
</math>

Using the inverse transform to substitute in the LSZ reduction formula, with some effort, the following result can be obtained:

:<math>
\langle p_1,\ldots,p_n\ \mathrm{out}|q_1,\ldots,q_m\ \mathrm{in}\rangle=
\prod_{i=1}^{m}
\left\{
(-i)(2\pi)^{-3/2} Z^{-1/2}
\left(p_i^2-m^2\right)
\right\}\times
</math>
:<math>
\times
\prod_{j=1}^{n}
\left\{
(-i)(2\pi)^{-3/2} Z^{-1/2}
\left(q_j^2-m^2\right)
\right\}
\Gamma(p_1,\ldots,p_n;-q_1,\ldots,-q_m)
</math>

Leaving aside normalization factors, this formula asserts that S-matrix elements are the residues of the poles that arise in the Fourier transform of the correlation functions as four-moments are put on-shell.

==Reduction formula for fermions==
{{Empty section|date=July 2010}}
==Field strength normalization==
The reason of the normalization factor <math>Z</math> in the definition of ''in'' and ''out'' fields can be understood by taking that relation between the vacuum and a single particle state <math>|p\rangle</math> with four-moment on-shell:

:<math>
\langle 0|\varphi(x)|p\rangle=
\sqrt Z \langle 0|\varphi_{\mathrm{in}}(x)|p\rangle +
\int \mathrm{d}^4y \Delta_{\mathrm{ret}}(x-y)
\langle 0|j(y)|p\rangle
</math>

Remembering that both <math>\varphi</math> and <math>\varphi_{\mathrm{in}}</math> are scalar fields with their lorentz transform according to:

:<math>
\varphi(x)=e^{iP\cdot x}\varphi(0)e^{-iP\cdot x}
</math>

where <math>P^\mu</math> is the four-moment operator, we can write:

:<math>
e^{-ip\cdot x}\langle 0|\varphi(0)|p\rangle=
\sqrt Z e^{-ip\cdot x} \langle 0|\varphi_{\mathrm{in}}(0)|p\rangle +
\int \mathrm{d}^4y \Delta_{\mathrm{ret}}(x-y)
\langle 0|j(y)|p\rangle
</math>

Applying the Klein-Gordon operator <math>\part^2+m^2</math> on both sides, remembering that the four-moment <math>p</math> is on-shell and that <math>\Delta_{\mathrm{ret}}</math> is the Green's function of the operator, we obtain:

:<math>
0=0 +
\int \mathrm{d}^4y \delta^4(x-y)
\langle 0|j(y)|p\rangle;
\quad\Leftrightarrow\quad
\langle 0|j(x)|p\rangle=0
</math>

So we arrive to the relation:

:<math>
\langle 0|\varphi(x)|p\rangle=
\sqrt Z \langle 0|\varphi_{\mathrm{in}}(x)|p\rangle
</math>

which accounts for the need of the factor <math>Z</math>. The ''in'' field is a free field, so it can only connect one-particle states with the vacuum. That is, its expectation value between the vacuum and a many-particle state is null. On the other hand, the interacting field can also connect many-particle states to the vacuum, thanks to interaction, so the expectation values on the two sides of the last equation are different, and need a normalization factor in between. The right hand side can be computed explicitly, by expanding the ''in'' field in creation and annihilation operators:

:<math>
\langle 0|\varphi_{\mathrm{in}}(x)|p\rangle=
\int \frac{\mathrm{d}^3q}{(2\pi)^{3/2}(2\omega_q)^{1/2}}
e^{-iq\cdot x} \langle 0|a_{\mathrm{in}}(\mathbf q)|p\rangle=
\int \frac{\mathrm{d}^3q}{(2\pi)^{3/2}}
e^{-iq\cdot x}
\langle 0|a_{\mathrm{in}}(\mathbf q)a^\dagger_{\mathrm{in}}(\mathbf p)|0\rangle
</math>

Using the commutation relation between <math>a_{\mathrm{in}}</math> and <math>a^\dagger_{\mathrm{in}}</math> we obtain:

:<math>
\langle 0|\varphi_{\mathrm{in}}(x)|p\rangle= \frac{e^{-ip\cdot x}}{(2\pi)^{3/2}}
</math>

leading to the relation:

:<math>
\langle 0|\varphi(0)|p\rangle= \sqrt \frac{Z}{(2\pi)^3}
</math>

by which the value of <math>Z</math> may be computed, provided that one knows how to compute <math>\langle 0|\varphi(0)|p\rangle</math>.

==See also==
{{Portal|Mathematics}}

==References==
* The original paper is H. Lehmann, K. Symanzik, and W. Zimmerman, ''Nuovo Cimento'' '''1''', 205 (1955).
* A pedagogical derivation of the LSZ reduction formula can be found in M.E. Peskin and D.V. Schroeder, ''An Introduction to Quantum Field Theory'', Addison-Wesley, Reading, Massachusetts, 1995, Section 7.2.

{{DEFAULTSORT:Lsz Reduction Formula}}
[[Category:Quantum field theory]]

Rayleigh (unit)

2013-06-25T14:48:38Z

152.78.192.25:

In [[constraint satisfaction]], '''local search''' is an incomplete method for finding a solution to a [[constraint satisfaction problem|problem]]. It is based on iteratively improving an assignment of the variables until all constraints are satisfied. In particular, local search algorithms typically modify the value of a variable in an assignment at each step. The new assignment is close to the previous one in the space of assignment, hence the name ''local search''.

All local search algorithms use a function that evaluates the quality of assignment, for example the number of constraints violated by the assignment. This amount is called the ''cost'' of the assignment. The aim of local search is that of finding an assignment of minimal cost, which is a solution if any exists.

[[Image:Local-search-plateau.svg|thumb|Point A is not a solution, but no local move from there decreases cost. However, a solution exists at point B.]]
Two classes of local search algorithms exist. The first one is that of greedy or non-randomized algorithms. These algorithms proceed by changing the current assignment by always trying to decrease (or at least, non-increase) its cost. The main problem of these algorithms is the possible presence of ''plateau''s, which are regions of the space of assignments where no local move decreases cost. The second class of local search algorithm have been invented to solve this problem. They escape these plateaus by doing random moves, and are called randomized local search algorithms.

==Greedy algorithms==
===Hill climbing===
{{main|Hill climbing}}

The most basic form of local search is based on choosing the change that maximally decreases the cost of the solution. This method, called ''hill climbing'', proceeds as follows: first, a random assignment is chosen; then, a value is changed so as to maximally improve the quality of the resulting assignment. If no solution has been found after a given number of changes, a new random assignment is selected. Hill climbing algorithms can only escape a plateau by doing changes that do not change the quality of the assignment. As a result, they can be stuck in a plateau where the quality of assignment has a local maxima.

[[WalkSAT|GSAT]] (greedy sat) was the first local search algorithm for satisfiability, and is a form of hill climbing.

===Constraint weighting or breakout method===

A method for escaping from a local minimum is that of using a weighted sum of violated constraints as a measure of cost, and changing some weights when no improving move is available. More precisely, if no change reduces the cost of the assignment, the algorithm increases the weight of constraints violated by the current assignment.

This way, every move that would not otherwise change the cost of the solution decreases it. Moreover, the weight of constraints that remain violated for a large number of moves keeps increasing. Therefore, during a number of moves not satisfying a constraint, the cost of moves to assignments satisfying that constraint keeps increasing.

===Tabu search===
{{main|Tabu search}}

A drawback of hill climbing with moves that do not decrease cost, is that it may cycle over assignments of the same cost. ''Tabu search'' overcomes this problem by maintaining a list of "forbidden" assignments, called the ''tabu list''. In particular, the tabu list typically contains the list of the last changes. More precisely, it contains the last variable/value such that the variable has been recently assigned to the value.

This list is updated every time the assignment is changed. If a variable is assigned to a value, the pair variable/value is added to the list, and the oldest pair is removed from it. This way, the list only contains the most recent assignments to a variable. If a variable/value pair is in the tabu list, changing the current assignment by setting the variable to the value is forbidden. The algorithm can only choose the best move among the ones that are not forbidden. This way, it cannot cycle over the same solution unless the number of moves in this cycle is larger than the length of the tabu list.

==Random walk==

A random walk algorithm sometimes moves like a greedy algorithm but sometimes moves randomly. It depends on a parameter <math>p</math>, which is a real number between 0 and 1. At every move, with probability <math>p</math> the algorithm proceeds like a greedy algorithm, trying to maximally decrease the cost of the assignment. With probability <math>1-p</math>, however, the solution is changed in some other way, which involves some degree of randomness.

===WalkSAT===
{{main|WalkSAT}}

The random move of WalkSAT is changing the value of a random variable of a random violated constraint. For [[propositional satisfiability]] of [[conjunctive normal form]] formulae, which is the original settings of this algorithm, every such a move changes the value of the variable from true to false or vice versa, and produce the satisfiability of the violated constraint. As for all random walk strategies, a random move is only done with a given probability, and a move maximally decreasing the cost is done otherwise.

===Simulated annealing===
{{main|Simulated annealing}}

The technique of simulated annealing is based on changing the probability of doing a random move over one that maximally decreasing the cost. In particular, the name originates from the strategy of decreasing the probability of doing random moves during the execution of the algorithm, thus virtually "freezing" the space of search.

In particular, if the improvement of cost <math>d</math> of a move is negative (the move increases cost), this move is done with probability <math>e^{-d \cdot T}</math>, where <math>T</math> is real number. Since the probability of doing this move increases with <math>T</math>, this parameter is called the ''temperature''. Simulated annealing decreases this temperature over time, thus allowing more random moves at the beginning and less after time.

==Local search on a cycle cutset==

Local search usually works on all variables, improving a complete assignment to them. However, local search can also be run on a subset of variables, using some other mechanism for the other variables. A proposed algorithm works on a ''cycle cutset'', which is a set of variables that, if removed from the problem, makes it acyclic.

For any assignment of the variables of the cutset, the remaining problem has a [[Forest (graph theory)|forest]] as primal graph. As a result, it can be solved efficiently. In order to guide local search, an algorithm detecting the minimal number of constraints that can be violated is used in place of a satisfiability algorithm on the for forest part of the problem.

This minimal number is found by determining the cost of each variable assignment. This cost is the minimal number of constraints violated by an assignment of the variables in the subtree rooted at the variable, when the variable takes the given value. This cost can be calculated as follows. If <math>Cost(x=a)</math> denotes the cost of the assignment <math>x=a</math> and <math>y_1,\ldots,y_n</math> are the children of <math>x</math>, the following formula holds. In this formula, <math>Violates(x=a, y_i=b)</math> is the 0 or 1 depending on whether the assignment <math>x=a, y_i=b</math> violates the constraint between <math>x</math> and <math>y</math>.

:<math>Cost(x=a) = \sum_{i=1,\ldots,n} \min_{y_i=b} ( Cost(y_i=b) + Violates(x=a, y_i=b) ) </math>

The cost for variables in the cutset is zero, and these variables are assumed to be allowed to take only their given value. With these assumptions, the above formula allows computing the cost of all variable evaluations by iteratively proceeding bottom-up from the leaves to the root(s) of the forest.

The cost of variable evaluations can be used by local search for computing the cost of a solution. The cost of values of the roots of the forest is indeed the minimal number of violated constraints in the forest for these given values. These costs can therefore used to evaluate the cost of the assignment to the cutset variables and to estimate the cost of similar assignments on the cutset variables.

==External links==
* [http://www.nlsde.buaa.edu.cn/~kexu/benchmarks/benchmarks.htm Forced Satisfiable CSP and SAT Benchmarks of Model RB]

==References==

*{{cite book
| first=Rina
| last=Dechter
| title=Constraint Processing
| publisher=Morgan Kaufmann
| url=http://www.ics.uci.edu/~dechter/books/index.html
| year=2003
| isbn=978-1-55860-890-0
}}

[[Category:Constraint programming]]

Savonius wind turbine

2012-08-21T09:07:21Z

152.78.41.59: /* Power and rotational speed */ Fix parsing errors in math formulae

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2012-05-30T12:03:31Z

152.78.205.19:

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2012-05-18T14:12:08Z

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In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won. The tensions in the country had been building. When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set. But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America. That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King. This tremendous struggle for freedom was never easy and was often marked with violence. 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Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King. * 1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech. * 1964 - President Lyndon Johnson signed the bill that was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964. * 1965 - The assignation of Malcolm X and the Watts race rights. * 1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246. This short list is just a few of the highlights of this troubled time in which the rights of all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government. In the years to come there would be great steps forward. 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When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set. But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America. That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King. This tremendous struggle for freedom was never easy and was often marked with violence. Over the next ten years some of the most important milestone in black history took place including... * 1957 - President Eisenhower had to send federal troops to Arkansas to secure admission to Central High School by nine black students. * 1960 - The sit-in at Woolworths lunch counter in Greensboro North Carolina set the stage for nonviolent protest that was used with great success for the rest of the struggle. Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King. * 1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech. * 1964 - President Lyndon Johnson signed the bill that [http://tinyurl.com/ku6vjks http://tinyurl.com/ku6vjks] was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964. * 1965 - The assignation of Malcolm X and the Watts race rights. * 1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246. This short list is just a few of the highlights of this troubled time in which the rights of all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government. In the years to come there would be great steps forward. One by one, every area of American life would see breakthroughs by African Americans in the areas of sports, entertainment, education and politics. There were many proud moments and there were moments of tremendous shame and heinous acts committed by both white and black people. But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture. The struggle is far from over. Discrimination and hate speech continue to be a problem to this day. And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride. We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life. And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few. 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In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won. The tensions in the country had been building. When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set. But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America. That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King. This tremendous struggle for freedom was never easy and was often marked with violence. Over the next ten years some of the most important milestone in black history took place including... * 1957 - President Eisenhower had to send federal troops to Arkansas to secure admission to Central High School by nine black students. * 1960 - The sit-in at Woolworths lunch counter in Greensboro North Carolina set the stage for nonviolent protest that was used with great success for the rest of the struggle. Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King. * 1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech. * 1964 - President Lyndon Johnson signed the bill that was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964. * 1965 - The assignation of Malcolm X and the Watts race rights. * 1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246. This short list is just a few of the highlights of this troubled time in which the rights of all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government. In the years to come there would be great steps forward. One by one, every area of American life would see breakthroughs by African Americans in the areas of sports, entertainment, education and politics. There were many proud moments and there were moments of tremendous shame and heinous acts committed by both white and black people. But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture. The struggle is far from over. Discrimination and hate speech continue to be a problem to this day. And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride. We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life. And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few. A Troubled Time From 1955 to 1965 there was a war right in the middle of America. 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When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set. But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America. That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King. This tremendous struggle for freedom was never easy and was often marked with violence. 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One by one, every area of American life would see breakthroughs by African Americans in the areas of sports, entertainment, education and politics. There were many proud moments and there were moments of tremendous shame and heinous acts committed by both white and black people. But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture. The struggle is far from over. Discrimination and hate speech continue to be a problem to this day. And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride. We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life. And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few. PPPPP 716 A Troubled Time From 1955 to 1965 there was a war right in the middle of America. No, it wasn't a war like World War II or the Revolutionary War. It was a war for the heart and soul of this country to determine once and for all if America was really going to be a land of equal opportunity for all. It is a war that eventually took on the name of "The Civil Rights Movement." We must make no mistake, this was not just a shouting match. Some of the events that we even remember today became quite brutal and deadly. Those who fought in this war on both sides were deadly serious about the causes they represented and willing to fight and even die to see their cause succeed. The war waged for years and steady progress was made but not without tremendous sacrifice by the leaders of the movement who were committed to a giving a new meaning to the phrase "set my people free." In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won. The tensions in the country had been building. When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set. But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America. That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King. This tremendous struggle for freedom was never easy and was often marked with violence. Over the next ten years some of the most important milestone in black history took place including... * 1957 - President Eisenhower had to send federal troops to Arkansas to secure admission to Central High School by nine black students. * 1960 - The sit-in at Woolworths lunch counter in Greensboro North Carolina set the stage for nonviolent protest that was used with great success for the rest of the struggle. Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King. * 1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech. * 1964 - President Lyndon Johnson signed the bill that was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964. * 1965 - The assignation of Malcolm X and the Watts race rights. * 1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246. This short list is just a few of the highlights of this troubled time in which the rights of all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government. In the years to come there would be great steps forward. One by one, every area of American life would see breakthroughs by African Americans in the areas of sports, entertainment, education and politics. There were many proud moments and there were moments of tremendous shame and heinous acts committed by both white and black people. But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture. The struggle is far from over. Discrimination and hate speech continue to be a problem to this day. And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride. We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life. And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few. 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In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won. The tensions in the country had been building. When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set. But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America. That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King. This tremendous struggle for freedom was never easy and was often marked with violence. 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One by one, every area of American life would see breakthroughs by African Americans in the areas of sports, entertainment, education and politics. There were many proud moments and there were moments of tremendous shame and heinous acts committed by both white and black people. But through all that struggle, the society continued to grow and adapt to the will of the people as has always been the tradition in American culture. The struggle is far from over. Discrimination and hate speech continue to be a problem to this day. And while it is easy to reflect on those days of struggle with regret, we can also look at them with pride. We can be proud of the great leaders who demonstrated tremendous courage and wisdom to lead this nation to a better way of life. And we can be proud of America because it is here where such a struggle can result in equality and freedom for all citizens, not just a few. 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In all of black history, there may be no more significant a time since the Civil War when the rights of African Americans were so deeply fought and won. The tensions in the country had been building. When the Supreme Court mandated desegregation in the schools in the historic case Brown versus the Board of Education, the stage was set. But it was on December 1, 1955 when Rosa Parks refused to give up her seat on a bus in Montgomery, Alabama to a white man that the movement finally took shape and became a titanic struggle for the rights of African Americans in America. That first battle brought to the front line one of the most important figures to fight for Civil Rights of that era, the Reverend Martin Luther King. This tremendous struggle for freedom was never easy and was often marked with violence. 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Nonviolent protest and civil disobedience became a staple of the civil rights movement because of the influence of Martin Luther King. * 1963 - The historic March on Washington in which over 200,000 people gathered to hear Dr. Kings famous "I Have a Dream" speech. * 1964 - President Lyndon Johnson signed the bill that was the most significant event of his presidency and one he believed deeply in, the Civil Rights Act of 1964. * 1965 - The assignation of Malcolm X and the Watts race rights. * 1965 - President Johnson takes another bold step to accelerate the civil rights movement implementing Affirmative Action when he issues Executive Order 11246. This short list is just a few of the highlights of this troubled time in which the rights of all citizens of American, black and white and of all colors were being redefined both on the streets, in the courts and in the different branches of government. In the years to come there would be great steps forward. 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2012-04-03T10:37:32Z

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2012-02-13T11:28:58Z

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