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		<id>https://en.formulasearchengine.com/w/index.php?title=Liouville%27s_theorem_(Hamiltonian)&amp;diff=228242</id>
		<title>Liouville&#039;s theorem (Hamiltonian)</title>
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		<updated>2015-01-10T13:38:02Z</updated>

		<summary type="html">&lt;p&gt;46.7.233.225: /* Remarks */&lt;/p&gt;
&lt;hr /&gt;
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		<author><name>46.7.233.225</name></author>
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		<id>https://en.formulasearchengine.com/w/index.php?title=Amputation&amp;diff=272683</id>
		<title>Amputation</title>
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		<updated>2014-01-27T16:53:20Z</updated>

		<summary type="html">&lt;p&gt;46.7.151.79: /* Causes */&lt;/p&gt;
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		<author><name>46.7.151.79</name></author>
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	<entry>
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		<title>Current divider</title>
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		<updated>2013-12-11T12:38:02Z</updated>

		<summary type="html">&lt;p&gt;46.7.137.149: /* Current divider */&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;[[Image:Acceleration.svg|thumb|250px|The green line shows the slope of the velocity-time graph at the particular point where the two lines [[Tangent|touch]]. Its slope is the acceleration at that point.]]&lt;br /&gt;
&lt;br /&gt;
In [[mechanics#Mechanics in science and engineering|mechanics]], the [[derivative]] of the [[position (vector)|position]] vs. [[time]] [[Graph of a function|graph]] of an object is equal to the [[velocity]] of the object. In the [[SI|International System of Units]], the position of the moving object is measured in meters relative to the [[Origin (mathematics)|origin]], while the time is measured in [[second]]s. Placing position on the [[y-axis]] and time on the [[x-axis]], the [[slope]] of the curve is given by: &lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;v = \frac{\Delta y}{\Delta x} = \frac{\Delta s}{\Delta t}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Here &amp;lt;math&amp;gt;s&amp;lt;/math&amp;gt; is the position of the object, and &amp;lt;math&amp;gt;t&amp;lt;/math&amp;gt; is the time. Therefore, the slope of the curve gives the change in position (in metres) divided by the change in time (in seconds), which is the definition of the average velocity (in [[meters per second]] &amp;lt;math&amp;gt;(\begin{matrix} \frac{m}{s} \end{matrix})&amp;lt;/math&amp;gt;) for that interval of time on the graph. If this interval is made to be [[infinitesimal]]ly small, such that &amp;lt;math&amp;gt;{\Delta s}&amp;lt;/math&amp;gt; becomes &amp;lt;math&amp;gt;{ds}&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;{\Delta t}&amp;lt;/math&amp;gt; becomes &amp;lt;math&amp;gt;{dt}&amp;lt;/math&amp;gt;, the result is the instantaneous [[velocity]] at time &amp;lt;math&amp;gt;t&amp;lt;/math&amp;gt;, or the [[derivative]] of the position with respect to time.&lt;br /&gt;
&lt;br /&gt;
A similar fact also holds true for the velocity vs. time graph. The slope of a velocity vs. time graph is [[acceleration]], this time, placing velocity on the y-axis and time on the x-axis. Again the slope of a line is change in &amp;lt;math&amp;gt;y&amp;lt;/math&amp;gt; over change in &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt;: &lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;a = \frac{\Delta y}{\Delta x} = \frac{\Delta v}{\Delta t}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Where &amp;lt;math&amp;gt;v&amp;lt;/math&amp;gt; is the velocity, measured in &amp;lt;math&amp;gt;\begin{matrix} \frac{m}{s} \end{matrix}&amp;lt;/math&amp;gt;, and &amp;lt;math&amp;gt;t&amp;lt;/math&amp;gt; is the time measured in seconds. This slope therefore defines the average acceleration over the interval, and reducing the interval infinitesimally gives &amp;lt;math&amp;gt;\begin{matrix} \frac{dv}{dt} \end{matrix}&amp;lt;/math&amp;gt;, the instantaneous acceleration at time &amp;lt;math&amp;gt;t&amp;lt;/math&amp;gt;, or the derivative of the velocity with respect to time (or the [[second derivative]] of the position with respect to time). The units of this slope or derivative are in [[Metre per second squared|meters per second per second]] (&amp;lt;math&amp;gt;\begin{matrix} \frac{m}{s^2} \end{matrix}&amp;lt;/math&amp;gt;, usually termed &amp;quot;meters per second-squared&amp;quot;), and so, therefore, is the acceleration.&lt;br /&gt;
&lt;br /&gt;
Since the velocity of the object is the [[derivative]] of the position graph, the [[integral|area under the line]] in the velocity vs. time graph is the [[Displacement (vector)|displacement]] of the object. (Velocity is on the y-axis and time on the x-axis. Multiplying the velocity by the time, the seconds cancel out and only meters remain. &amp;lt;math&amp;gt;\begin{matrix} \frac{m}{s} \end{matrix}s = m&amp;lt;/math&amp;gt;.) &lt;br /&gt;
&lt;br /&gt;
The same multiplication rule holds true for acceleration vs. time graphs. When &amp;lt;math&amp;gt;(\begin{matrix} \frac{m}{s^2} \end{matrix})&amp;lt;/math&amp;gt; is multiplied by time (s), velocity is obtained. (&amp;lt;math&amp;gt;\begin{matrix} \frac{m}{s^2} \end{matrix}s = \begin{matrix} \frac{m}{s} \end{matrix}&amp;lt;/math&amp;gt;).&lt;br /&gt;
&lt;br /&gt;
==Variable rates of change==&lt;br /&gt;
[[Image:Velocity vs time graph.svg|thumb|250px|In this example, the yellow [[area]] represents the [[displacement (vector)|displacement]] of the object as it moves. (The [[distance]] can be measured by taking the [[absolute value]] of the function.) The three green lines represent the values for acceleration at different points along the curve.]]&lt;br /&gt;
The expressions given above apply only when the rate of change is constant or when only the average ([[arithmetic mean|mean]]) rate of change is required. If the velocity or positions change non-[[linear]]ly over time, such as in the example shown in the figure, then [[derivative|differentiation]] provides the correct solution. Differentiation reduces the time-spans used above to be extremely small ([[infinitesimal]]) and gives a velocity or acceleration at each point on the graph rather than between a start and end point. The [[derivative]] forms of the above equations are&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;v = \frac{ds}{dt},&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;a = \frac{dv}{dt}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Since acceleration differentiates the expression involving position, it can be rewritten as a [[second derivative]] with respect to position:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;a = \frac{d^2 s}{dt^2}.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Since, for the purposes of mechanics such as this, [[integral|integration]] is the opposite of differentiation, it is also possible to express position as a function of velocity and velocity as a function of acceleration. The process of determining the area under the curve, as described above, can give the [[displacement (vector)|displacement]] and change in velocity over particular time intervals by using [[definite integral]]s:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;s(t_2)-s(t_1) = \int_{t_1}^{t_2}{v}\, dt, &amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;v(t_2)-v(t_1) = \int_{t_1}^{t_2}{a}\, dt. &amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
==See also==&lt;br /&gt;
*[[Displacement (vector)]]&lt;br /&gt;
*[[Velocity]]&lt;br /&gt;
*[[Acceleration]]&lt;br /&gt;
*[[Kinematics]]&lt;br /&gt;
{{Kinematics}}&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
*{{cite book&lt;br /&gt;
 | first = Richard&lt;br /&gt;
 | last = Wolfson&lt;br /&gt;
 | authorlink =&lt;br /&gt;
 | coauthors = Jay M. Pasachoff&lt;br /&gt;
 | year = 1999&lt;br /&gt;
 | title = Physics for Scientists and Engineers &lt;br /&gt;
 | edition = 3rd ed.&lt;br /&gt;
 | pages = 23–38&lt;br /&gt;
 | publisher = [[Addison-Wesley]] &lt;br /&gt;
 | location = [[Reading, Massachusetts]]&lt;br /&gt;
 | isbn = 0-321-03571-2&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
[[Category:Classical mechanics]]&lt;/div&gt;</summary>
		<author><name>46.7.137.149</name></author>
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	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Miraclebet&amp;diff=25012</id>
		<title>Miraclebet</title>
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		<updated>2013-05-19T15:34:11Z</updated>

		<summary type="html">&lt;p&gt;46.7.236.155: rm nonsense&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;{{multiple issues|orphan =August 2010|COI=December 2009|notability =December 2009|primarysources=December 2009}}&lt;br /&gt;
&lt;br /&gt;
&#039;&#039;&#039;Regulatory Feedback Networks&#039;&#039;&#039; describe a class of neural networks &lt;br /&gt;
related to &#039;&#039;&#039;Virtual Lateral Inhibition&#039;&#039;&#039;  (named to distinguish it &lt;br /&gt;
from [[lateral inhibition]]) that perform inference using [[negative &lt;br /&gt;
feedback]].&amp;lt;ref&amp;gt;J. Reggia, “Virtual lateral inhibition in parallel&lt;br /&gt;
 activation models of associative memory,” in Proc. 9th International &lt;br /&gt;
Joint Conference on Artificial Intelligence., Aug. 1985, pp. &lt;br /&gt;
244-248.&amp;lt;/ref&amp;gt;&amp;lt;ref name=mcfadden&amp;gt;Mcfadden, F. E. (1995). &lt;br /&gt;
&amp;quot;Convergence of Competitive Activation Models Based on Virtual Lateral &lt;br /&gt;
Inhibition.&amp;quot; Neural Networks 8(6): 865-875.&amp;lt;/ref&amp;gt;&amp;lt;ref &lt;br /&gt;
name=first&amp;gt;Achler, T. (2002). Input Shunt Networks. Neurocomputing, &lt;br /&gt;
44, 249-255.&amp;lt;/ref&amp;gt; The feedback is implemented during recognition &lt;br /&gt;
and during recognition connectivity parameters are not changed.  Thus &lt;br /&gt;
this is completely separate from learning/training (e.g. [[supervised &lt;br /&gt;
learning]] or [[unsupervised learning]]).  This is also different from &lt;br /&gt;
models of [[attentional shift|spatial attention]].  Instead, these &lt;br /&gt;
networks determine the relevance of inputs through a &amp;quot;conservation of &lt;br /&gt;
information principle&amp;quot;.&lt;br /&gt;
&lt;br /&gt;
== How the network functions ==&lt;br /&gt;
The computational basis of conservation of information is that an input &lt;br /&gt;
should not pass more information than is justified to the next layer. &lt;br /&gt;
Thus inputs are regulated by the outputs they activate. Subsequently, &lt;br /&gt;
each input’s contribution (i.e. &lt;br /&gt;
[http://www.scholarpedia.org/article/Visual_salience salience]) is &lt;br /&gt;
adjusted through feedback regulation by its associated outputs. The &lt;br /&gt;
amplitudes of the adjusted inputs are propagated to the output layer. A &lt;br /&gt;
new salience is re-evaluated based on the new output activity (through &lt;br /&gt;
feedback). This can be iterated until the networks reach steady &lt;br /&gt;
state.&amp;lt;ref name=mcfadden /&amp;gt; At every step, the role of salience is&lt;br /&gt;
 to maintain the relation where: the total activity of outputs connected&lt;br /&gt;
 to an input will be equivalent to the input’s amplitude.&amp;lt;ref &lt;br /&gt;
name=first /&amp;gt;&amp;lt;ref name=agi&amp;gt;Achler T., Amir E., “Input Feedback &lt;br /&gt;
Networks: Classification and Inference Based on Network Structure” &lt;br /&gt;
Artificial General Intelligence 2008 &lt;br /&gt;
[http://reason.cs.uiuc.edu/tsvi/AGI.pdf pdf]&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
== How the network is used ==&lt;br /&gt;
These networks are best suited for nodes with binary connections.&amp;lt;ref&lt;br /&gt;
 name=first /&amp;gt;&amp;lt;ref name=agi /&amp;gt;&amp;lt;ref name=shedding&amp;gt;Achler &lt;br /&gt;
T., Omar C., Amir E., “Shedding Weights: More With Less”, IEEE Proc. &lt;br /&gt;
International Joint Conference on Neural Networks, 2008 &lt;br /&gt;
[http://reason.cs.uiuc.edu/tsvi/IJCNN.pdf pdf]&amp;lt;/ref&amp;gt; Instead of &lt;br /&gt;
weights determining the relevance of connections, input salience is &lt;br /&gt;
adjusted at the time of recognition. For example, a node representing &lt;br /&gt;
car may connect to features wheels, door, and bumper. A node &lt;br /&gt;
representing bicycle may connect to features wheels, pedals, and chain. &lt;br /&gt;
Given wheels, the network will determine how relevant the wheels are to &lt;br /&gt;
either the bicycle or car nodes during recognition.&lt;br /&gt;
&lt;br /&gt;
== Benefits/costs ==&lt;br /&gt;
This model displays unparalleled performance given simultaneous &lt;br /&gt;
patterns, addressing [[Curse of dimensionality|combinatorial &lt;br /&gt;
explosions]]     associated with simultaneous patterns.&amp;lt;ref &lt;br /&gt;
name=shedding /&amp;gt;&amp;lt;ref&amp;gt;Achler T., Vural C., Amir, E., &amp;quot;Counting &lt;br /&gt;
with Biologically Inspired Regulatory Feedback Networks”, IEEE Proc. &lt;br /&gt;
International Joint Conference on Neural Networks, 2009 &lt;br /&gt;
[http://reason.cs.uiuc.edu/tsvi/counting.pdf pdf]&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The model can also generate solutions composed of multiple output nodes &lt;br /&gt;
with minimal overlap.&amp;lt;ref name=agi /&amp;gt;&amp;lt;ref&amp;gt;Achler T., “Using &lt;br /&gt;
Non-Oscillatory Dynamics to Disambiguate Simultaneous Patterns”, IEEE &lt;br /&gt;
Proc. International Joint Conference on Neural Networks, 2009 &lt;br /&gt;
[http://reason.cs.uiuc.edu/tsvi/Dynamics%20for%20Disambiguation%20IJCNN%202009.pdf  pdf]&amp;lt;/ref&amp;gt; This property groups patterns together in a manner &lt;br /&gt;
that suggests a way out of a fundamental recognition conundrum called &lt;br /&gt;
the [[binding problem]] (&#039;unity of perception&#039; version).&lt;br /&gt;
&lt;br /&gt;
In contrast to conventional neural networks or machine learning methods &lt;br /&gt;
these networks cannot be guaranteed to be able to capture any arbitrary &lt;br /&gt;
pattern.  However for the patterns they can capture, they show these &lt;br /&gt;
properties.&lt;br /&gt;
&lt;br /&gt;
== Implementation ==&lt;br /&gt;
Suppose there are [[Fuzzy logic|fuzzy-type]] input features &lt;br /&gt;
&amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; and output nodes &amp;lt;math&amp;gt;y&amp;lt;/math&amp;gt;. &lt;br /&gt;
Each output node &amp;lt;math&amp;gt;y_j&amp;lt;/math&amp;gt; is defined by set of &lt;br /&gt;
feedforward binary connections &amp;lt;math&amp;gt;FF_j&amp;lt;/math&amp;gt; from &lt;br /&gt;
&amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt;&#039;s. It also has a set of symmetrical feedback &lt;br /&gt;
connections &amp;lt;math&amp;gt;FB&amp;lt;/math&amp;gt; that implement [[negative &lt;br /&gt;
feedback]].  Due to the symmetry each member of &lt;br /&gt;
&amp;lt;math&amp;gt;FF_j&amp;lt;/math&amp;gt; (a connection from input to output) has a &lt;br /&gt;
corresponding member in &amp;lt;math&amp;gt;FB&amp;lt;/math&amp;gt; (a connection from &lt;br /&gt;
the same output to same input) that returns and inhibits the input.   &lt;br /&gt;
Lets label &amp;lt;math&amp;gt;FB_i&amp;lt;/math&amp;gt; the set of connections that &lt;br /&gt;
return to an &amp;lt;math&amp;gt;x_i&amp;lt;/math&amp;gt;. &lt;br /&gt;
&amp;lt;math&amp;gt;|FF_j|&amp;lt;/math&amp;gt; is the number of connections to &lt;br /&gt;
&amp;lt;math&amp;gt;y_j&amp;lt;/math&amp;gt;.  Lets label &amp;lt;math&amp;gt;s_i&amp;lt;/math&amp;gt; &lt;br /&gt;
the salience of input &amp;lt;math&amp;gt;x_i&amp;lt;/math&amp;gt;. Then the activity of&lt;br /&gt;
 the output node is determined by:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;y_j(t+\Delta t)=\frac{y_j(t)}{|FF_j|}\sum_{k\in &lt;br /&gt;
FF_j}s_k&amp;lt;/math&amp;gt;.&lt;br /&gt;
&amp;lt;br/&amp;gt;&lt;br /&gt;
The salience value &amp;lt;math&amp;gt;s_i&amp;lt;/math&amp;gt; of a given &lt;br /&gt;
&amp;lt;math&amp;gt;x_i&amp;lt;/math&amp;gt; is determined by:&lt;br /&gt;
&amp;lt;br/&amp;gt;&lt;br /&gt;
&amp;lt;math&amp;gt;s_i =\frac{x_i}{\sum_{r\in{FB_i}}y_r(t)}&amp;lt;/math&amp;gt;.&lt;br /&gt;
&amp;lt;br/&amp;gt;&lt;br /&gt;
These equations can be iterated until the network reaches steady state.&lt;br /&gt;
&lt;br /&gt;
== See also ==&lt;br /&gt;
* [[Visual perception]]&lt;br /&gt;
* [[Visual Object Recognition in Cognitive Neuroscience]]&lt;br /&gt;
* [[Bag of words model in computer vision]]&lt;br /&gt;
* [[Computational neuroscience]]&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
&amp;lt;references/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
{{DEFAULTSORT:Regulatory Feedback Network}}&lt;br /&gt;
[[Category:Control theory]]&lt;br /&gt;
[[Category:Computational neuroscience]]&lt;/div&gt;</summary>
		<author><name>46.7.236.155</name></author>
	</entry>
	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Statistical_hypothesis_testing&amp;diff=221168</id>
		<title>Statistical hypothesis testing</title>
		<link rel="alternate" type="text/html" href="https://en.formulasearchengine.com/w/index.php?title=Statistical_hypothesis_testing&amp;diff=221168"/>
		<updated>2012-08-25T14:08:34Z</updated>

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		<author><name>46.7.98.42</name></author>
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