formulasearchengine - User contributions [en]

Fanno flow

2014-02-24T05:06:50Z

132.216.49.125: Undid revision 544675687 by Addbot (talk)

The name of the writer is Figures. My day occupation is a meter reader. Years ago he moved to North Dakota and his family members loves it. Doing ceramics is what adore performing. My web page; [http://richlinked.com/index.php?do=/profile-32092/info/ home std test kit]

Troland

2014-02-19T00:43:03Z

132.216.57.243: /* Unit status */ The cited source says nothing about trolands being deprecated.

Book or Software Editor Crosser from Kelowna, has hobbies and interests which includes model trains, property developers in [http://taiwannews.somee.com/member.asp?action=view&memName=TodLeMessurier437 singapore Property Listing] and texting. Advocates that you just visit Kutná Hora: Historical Town Centre.

Sigma-algebra

2013-12-08T05:55:24Z

132.216.227.214:

{{refimprove|date=May 2009}}
[[File:Sierpinski triangle.svg|thumb|Sierpinski triangle]]
[[File:Random Sierpinski Triangle animation.gif|thumb|Generated using a random algorithm]]
[[File:Multigrade operator AND.svg|thumb|Sierpinski triangle in logic: The first 16 [[Logical conjunction|conjunctions]] of [[Lexicographical order|lexicographically]] ordered arguments The columns interpreted as binary numbers give 1, 3, 5, 15, 17, 51... {{OEIS|A001317}}]]
The '''Sierpinski triangle''' (also with the original orthography ''Sierpiński''), also called the '''Sierpinski gasket''' or the '''Sierpinski Sieve''', is a [[fractal]] and [[attractive fixed set]] named after the [[Poland|Polish]] [[mathematician]] [[Wacław Sierpiński]] who described it in 1915. However, similar patterns appear already in the 13th-century [[Cosmati]] [[mosaic]]s in the cathedral of [[Anagni]], [[Italy]],<ref>{{citation|first=Stephen|last=Wolfram|authorlink=Stephen Wolfram|title=[[A New Kind of Science]]|year=2002|publisher=Wolfram Media|pages=43, 873}}</ref> and other places, such as in the nave of the Roman Basilica of [[Santa Maria in Cosmedin]].<ref>[http://www.flickr.com/photos/mymuk/6304896451 "Geometric floor mosaic (Sierpinski triangles), nave of Santa Maria in Cosmedin, Forum Boarium, Rome"], 5 September 2011, [[Flickr]]</ref>

Originally constructed as a curve, this is one of the basic examples of [[self-similarity|self-similar]] sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction.

Comparing the Sierpinski triangle or the [[Sierpinski carpet]] to equivalent repetitive tiling arrangements, it is evident that similar structures can be built into any [[rep-tile]] arrangements.

==Construction==

An algorithm for obtaining arbitrarily close approximations to the Sierpinski triangle is as follows:

Note: each removed triangle (a ''trema'') is [[topology|topologically]] an [[open set]].<ref>[http://www.cut-the-knot.org/triangle/Tremas.shtml "Sierpinski Gasket by Trema Removal"]</ref>

:[[Image:Sierpinski triangle evolution.svg|512px|The evolution of the Sierpinski triangle]]

#Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpinski triangle uses an [[equilateral triangle]] with a base parallel to the horizontal axis (first image).
#Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner (image 2). Note the emergence of the central hole - because the three shrunken triangles can between them cover only 3/4 of the area of the original. (Holes are an important feature of Sierpinski's triangle.)
#Repeat step 2 with each of the smaller triangles (image 3 and so on).

This process of recursively removing triangles is an example of a [[finite subdivision rule]].

Note that this infinite process is not dependent upon the starting shape being a triangle—it is just clearer that way. The first few steps starting, for example, from a square also tend towards a Sierpinski triangle. [[Michael Barnsley]] used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals."<ref>[[Michael Barnsley]], ''et al.''{{PDF|[http://www.maths.anu.edu.au/~barnsley/pdfs/V-var_super_fractals.pdf "V-variable fractals and superfractals"]|2.22 MB}}</ref>

:[[Image:Sierpinski triangle evolution square.svg|512px|Iterating from a square]]

The actual fractal is what would be obtained after an infinite number of iterations. More formally, one describes it in terms of functions on closed sets of points. If we let <math>d_a</math> note the dilation by a factor of ½ about a point a, then the Sierpinski triangle with corners a, b, and c is the fixed set of the transformation <math>d_a</math> U <math>d_b</math> U <math>d_c</math>.

This is an [[attractive fixed set]], so that when the operation is applied to any other set repeatedly, the images converge on the Sierpinski triangle. This is what is happening with the triangle above, but any other set would suffice.

If one takes a point and applies each of the transformations <math>d_a</math>, <math>d_b</math>, and <math>d_c</math> to it randomly, the resulting points will be dense in the Sierpinski triangle, so the following algorithm will again generate arbitrarily close approximations to it:

Start by labeling '''p'''1, '''p'''2 and '''p'''3 as the corners of the Sierpinski triangle, and a random point '''v'''1. Set '''v'''n+1 = ½ ( '''v'''n + '''p'''rn ), where rn is a random number 1, 2 or 3. Draw the points '''v'''1 to '''v'''∞. If the first point '''v'''1 was a point on the Sierpiński triangle, then all the points '''v'''n lie on the Sierpinski triangle. If the first point '''v'''1 to lie within the perimeter of the triangle is not a point on the Sierpinski triangle, none of the points '''v'''n will lie on the Sierpinski triangle, however they will converge on the triangle. If '''v'''1 is outside the triangle, the only way '''v'''n will land on the actual triangle, is if '''v'''n is on what would be part of the triangle, if the triangle was infinitely large.

[[Image:Sierpinski chaos animated.gif|thumb|right|200px|Animated creation of a Sierpinski triangle using the chaos game]]

[[Image:Animated construction of Sierpinski Triangle.gif|thumb|left|250px|Animated construction of a Sierpinski triangle]]
'''Or more simply:'''
# Take 3 points in a plane to form a triangle, you need not draw it.
# Randomly select any point inside the triangle and consider that your current position.
# Randomly select any one of the 3 vertex points.
# Move half the distance from your current position to the selected vertex.
# Plot the current position.
# Repeat from step 3.

''Note: This method is also called the [[Chaos game]]. You can start from any point outside or inside the triangle, and it would eventually form the Sierpinski Gasket with a few leftover points. It is interesting to do this with pencil and paper. A brief outline is formed after placing approximately one hundred points, and detail begins to appear after a few hundred.

[[Image:Sierpinski1.png|thumb|right|250px|Sierpinski triangle using IFS]]
'''Or using an Iterated function system'''

An alternative way of computing the Sierpinski triangle uses an [[Iterated function system]] and starts by a point at the origin (''x''0 = 0, ''y''0 = 0). The new points are iteratively computed by randomly applying (with equal probability) one of the following three coordinate transformations (using the so-called [[chaos game]]): 
''x''''n''+1 = 0.5 ''x''''n'' 
''y''''n''+1 = 0.5 ''y''''n''; a half-size copy 
This coordinate transformation is drawn in yellow in the [[:File:Sierpinski1.png|figure]].

''x''''n''+1 = 0.5 ''x''''n'' + 0.25 
''y''''n''+1 = 0.5 ''y''''n'' + 0.5 <math>\sqrt{3}\over 2</math>; a half-size copy shifted right and up 
This coordinate transformation is drawn using red color in the [[:File:Sierpinski1.png|figure]].

''x''''n''+1 = 0.5 ''x''''n'' + 0.5 
''y''''n''+1 = 0.5 ''y''''n''; a half-size copy doubled shifted to the right

This coordinate transformation is drawn using blue color in the [[:File:Sierpinski1.png|figure]].

'''Or using an L-system''' — The Sierpinski triangle drawn using an [[L-system#Example 6: Sierpiński triangle|L-system]].

'''bitwise AND''' - The 2D AND function, z=AND(x,y) can also produce a white on black right angled Sierpinski triangle if all pixels of which z=0 are white, and all other values of z are black.

'''bitwise XOR''' - The values of the discrete, 2D XOR function, z=XOR(x,y) also exhibit structures related to the Sierpinski triangle. For example, one could generate the Sierpinski triangle by setting up a 2 dimensional matrix, [rows][columns] placing the uppermost point on [1][n/2], then cycling through the remaining cells row by row the value of the cell being XOR([i-1][j-1],[i-1][j+1])

'''Other means''' — The Sierpinski triangle also appears in certain [[cellular automata]] (such as [[Rule 90]]), including those relating to [[Conway's Game of Life]]. The automaton "12/1" when applied to a single cell will generate four approximations of the Sierpinski triangle.

If one takes [[Pascal's triangle]] with 2''n'' rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpinski triangle. More precisely, the [[limit of a sequence|limit]] as ''n'' approaches infinity of this parity-colored 2''n''-row Pascal triangle is the Sierpinski triangle.

==Properties==
For integer number of dimensions ''d'', when doubling a side of an object, ''2'' ''d'' copies of it are created, i.e. 2 copies for 1 dimensional object, 4 copies for 2 dimensional object and 8 copies for 3 dimensional object. For Sierpinski triangle doubling its side creates 3 copies of itself. Thus Sierpinski triangle has [[Hausdorff dimension]] log(3)/log(2) ≈ 1.585, which follows from solving ''2'' ''d'' = ''3'' for ''d''.<ref name=FFG120>{{cite book | zbl=0689.28003 | last=Falconer | first=Kenneth | title=Fractal geometry: mathematical foundations and applications | location=Chichester | publisher=John Wiley | year=1990 | isbn=0-471-92287-0 | page=120 }}</ref>

The area of a Sierpinski triangle is zero (in [[Lebesgue measure]]). The area remaining after each iteration is clearly 3/4 of the area from the previous iteration, and an infinite number of iterations results in zero. {{Citation needed|date=August 2007}}

The points of a Sierpinski triangle have a simple characterization in [[Barycentric coordinates (mathematics)#Barycentric coordinates on triangles|Barycentric coordinates]].<ref>http://www.cut-the-knot.org/ctk/Sierpinski.shtml</ref> If a point has coordinates (0.''u''1''u''2''u''3…,0.''v''1''v''2''v''3…,0.''w''1''w''2''w''3…), expressed as [[Binary number]]s, then the point is in Sierpinski's triangle if and only if ''u''''i''+''v''''i''+''w''''i''=1 for all ''i''.

==Analogues in higher dimensions==
[[Image:Sierpinski pyramid.png|thumb|333px|right|A Sierpinski square-based pyramid and its 'inverse']][[Image:Sierpiński Pyramid from Above.PNG|thumb|A Sierpiński triangle-based pyramid as seen from above (4 main sections highlighted). Note the self-similarity in this 2-dimensional projected view, so that the resulting triangle could be a 2D fractal in itself.]]

The tetrix is the three-dimensional analogue of the Sierpinski triangle, formed by repeatedly shrinking a regular [[tetrahedron]] to one half its original height, putting together four copies of this tetrahedron with corners touching, and then repeating the process. This can also be done with a square [[Pyramid (geometry)|pyramid]] and five copies instead.
A tetrix constructed from an initial tetrahedron of side-length L has the property that the total surface area remains constant with each iteration.

The initial surface area of the (iteration-0) tetrahedron of side-length L is <math>L^2 \sqrt{3}</math>. At the next iteration, the side-length is halved

:<math>L \rightarrow { L \over 2 }</math>

and there are 4 such smaller tetrahedra. Therefore, the total surface area after the first iteration is:

:<math>4 \left( \left( {L \over 2} \right)^2 \sqrt{3} \right) = 4 { {L^2} \over 4 } \sqrt{3} = L^2 \sqrt{3}.</math>

This remains the case after each iteration. Though the surface area of each subsequent tetrahedron is 1/4 that of the tetrahedron in the previous iteration, there are 4 times as many—thus maintaining a constant total surface area.

The total enclosed volume, however, is geometrically decreasing (factor of 0.5) with each iteration and asymptotically approaches 0 as the number of iterations increases. In fact, it can be shown that, while having fixed area, it has no 3-dimensional character. The [[Hausdorff dimension]] of such a construction is <math>\textstyle\frac{\ln 4}{\ln 2}=2</math> which agrees with the finite area of the figure. (A Hausdorff dimension strictly between 2 and 3 would indicate 0 volume and infinite area.)

==See also==
* [[Apollonian gasket]]
* [[Chaos game]]
* [[Koch snowflake]]
* [[List of fractals by Hausdorff dimension]]
* [[Pascal's triangle]]
* [[Rule 90]]
* [[Sierpinski carpet]]
* [[Sierpiński arrowhead curve]]
* [[Triforce]]

==References==
<references/>

==External links==
{{Commons category|Sierpinski triangles}}
* {{MathWorld|title=Sierpinski Sieve|urlname=SierpinskiSieve}}
* Paul W. K. Rothemund, Nick Papadakis, and Erik Winfree, [http://biology.plosjournals.org/perlserv/?request=get-document&doi=10.1371/journal.pbio.0020424 Algorithmic Self-Assembly of DNA Sierpinski Triangles], ''PLoS Biology'', volume 2, issue 12, 2004.
* [http://www.cut-the-knot.org/Curriculum/Geometry/Tremas.shtml Sierpinski Gasket by Trema Removal] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/triangle/Hanoi.shtml Sierpinski Gasket and Tower of Hanoi] at [[cut-the-knot]]
*[http://www.shapeways.com/model/119919/sierpinski_tetrahedron.html 3D printed Stage 5 Sierpinski Tetrahedron]

{{Fractals}}

[[Category:Fractals]]
[[Category:Factorial and binomial topics]]
[[Category:Curves]]
[[Category:Topological spaces]]
[[Category:Triangles]]
[[Category:Cellular automaton patterns]]
[[Category:Science and technology in Poland]]

{{Link GA|de}}

Entropy (classical thermodynamics)

2013-11-05T17:29:56Z

132.216.24.103: /* Refrigerators */

:{{For|a lemma on Lie algebras|Whitehead's lemma (Lie algebras)}}
'''Whitehead's lemma''' is a technical result in [[abstract algebra]] used in [[algebraic K-theory]]. It states that a [[matrix (mathematics)|matrix]] of the form

:<math>
\begin{bmatrix}
u & 0 \\
0 & u^{-1} \end{bmatrix}</math>

is equivalent to the [[identity matrix]] by [[elementary matrices|elementary transformations]] (that is, transvections):

:<math>
\begin{bmatrix}
u & 0 \\
0 & u^{-1} \end{bmatrix} = e_{21}(u^{-1}) e_{12}(1-u) e_{21}(-1) e_{12}(1-u^{-1}). </math>

Here, <math>e_{ij}(s)</math> indicates a matrix whose diagonal block is <math>1</math> and <math>ij^{th}</math> entry is <math>s</math>.

The name "Whitehead's lemma" also refers to the closely related result that the [[derived group]] of the [[stable general linear group]] is the group generated by [[elementary matrices]].<ref name=Mil31>{{cite book | last1=Milnor | first1=John Willard | author1-link= John Milnor | title=Introduction to algebraic K-theory | publisher=[[Princeton University Press]] | location=Princeton, NJ | mr=0349811 | year=1971 | zbl=0237.18005 | series=Annals of Mathematics Studies | volume=72 | at=Section 3.1 }}</ref><ref name=Sn164>{{cite book | title=Explicit Brauer Induction: With Applications to Algebra and Number Theory | volume=40 | series=Cambridge Studies in Advanced Mathematics | first=V. P. | last=Snaith | authorlink= | publisher=[[Cambridge University Press]] | year=1994 | isbn=0-521-46015-8 | zbl=0991.20005 | page=164 }}</ref> In symbols,
:<math>\operatorname{E}(A) = [\operatorname{GL}(A),\operatorname{GL}(A)]</math>.

This holds for the stable group (the [[direct limit]] of matrices of finite size) over any ring, but not in general for the unstable groups, even over a field. For instance for
:<math>\operatorname{GL}(2,\mathbb{Z}/2\mathbb{Z})</math>

one has:
:<math>\operatorname{Alt}(3) \cong [\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}),\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z})] < \operatorname{E}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{SL}_2(\mathbb{Z}/2\mathbb{Z}) = \operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z}) \cong \operatorname{Sym}(3),</math>
where Alt(3) and Sym(3) denote the [[alternating group|alternating]] resp. [[symmetric group]] on 3 letters.

==See also==
*[[Special linear group#Relations to other subgroups of GL(n,A)]]

==References==
<references/>

[[Category:Matrix theory]]
[[Category:Lemmas]]
[[Category:K-theory]]
[[Category:Theorems in abstract algebra]]

{{Abstract-algebra-stub}}

D'Alembert's paradox

2013-10-27T13:34:37Z

132.216.48.162:

'''Verbal arithmetic''', also known as '''alphametics''', '''cryptarithmetic''', '''crypt-arithmetic''', '''cryptarithm''' or '''word addition''', is a type of [[mathematical game]] consisting of a mathematical [[equation]] among unknown [[number]]s, whose [[numerical digit|digit]]s are represented by [[Letter (alphabet)|letter]]s. The goal is to identify the value of each letter. The name can be extended to puzzles that use non-alphabetic symbols instead of letters.

The equation is typically a basic operation of [[arithmetic]], such as [[addition]], [[multiplication]], or [[division (mathematics)|division]]. The classic example, published in the July 1924 issue of Strand Magazine by [[Henry Dudeney]],<ref>[[Henry Dudeney|H. E. Dudeney]], in ''[[Strand Magazine]]'' vol. 68 (July 1924), pp. 97 and 214.</ref> is:

<math>\begin{matrix}
& & \text{S} & \text{E} & \text{N} & \text{D} \\
+ & & \text{M} & \text{O} & \text{R} & \text{E} \\
\hline
= & \text{M} & \text{O} & \text{N} & \text{E} & \text{Y} \\
\end{matrix}</math>

The solution to this puzzle is O = 0, M = 1, Y = 2, E = 5, N = 6, D = 7, R = 8, and S = 9.

Traditionally, each letter should represent a different digit, and (as in ordinary arithmetic notation) the leading digit of a multi-digit number must not be zero. A good puzzle should have a unique solution, and the letters should make up a cute phrase (as in the example above).

Verbal arithmetic can be useful as a motivation and source of exercises in the [[education|teaching]] of [[algebra]].

==History==
Verbal arithmetic puzzles are quite old and their inventor is not known. An 1864 example in The American Agriculturist<ref name="agriculturist">{{Cite news | newspaper = American Agriculturist | pages = 349 | volume = 23 | issue = 12 | date = December 1864}}
</ref> disproves the popular notion that it was invented by [[Sam Loyd]]. The name "cryptarithmie" was coined by puzzlist Minos (pseudonym of [[Simon Vatriquant]]) in the May 1931 issue of Sphinx, a Belgian magazine of recreational mathematics, and was translated as "cryptarithmetic" by [[Maurice Kraitchik]] in 1942.<ref>[[Maurice Kraitchik]], Mathematical Recreations (1953), pp. 79-80.</ref> In 1955, J. A. H. Hunter introduced the word "alphametic" to designate cryptarithms, such as Dudeney's, whose letters form meaningful [[word]]s or phrases.<ref>J. A. H. Hunter, in the [[Toronto]] ''Globe and Mail'' (27 October 1955), p. 27.</ref>

==Solving cryptarithms==
Solving a cryptarithm by hand usually involves a mix of deductions and exhaustive tests of possibilities. For instance, the following sequence of deductions solves Dudeney's SEND + MORE = MONEY puzzle above (columns are numbered from right to left):

<math>\begin{matrix}
& & \text{S} & \text{E} & \text{N} & \text{D} \\
+ & & \text{M} & \text{O} & \text{R} & \text{E} \\
\hline
= & \text{M} & \text{O} & \text{N} & \text{E} & \text{Y} \\
\end{matrix}</math>

#From column 5, '''M = 1''' since it is the only carry-over possible from the sum of two single digit numbers in column 4.
#Since there is a carry in column 5, O must be less than or equal to M (from column 4). But O cannot be equal to M, so O is less than M. Therefore '''O = 0'''.
#Since O is 1 less than M, S is either 8 or 9 depending on whether there is a carry in column 4. But if there were a carry in column 4, N would be less than or equal to O (from column 3). This is impossible since O = 0. Therefore there is no carry in column 4 and '''S = 9'''.
#If there were no carry in column 3 then E = N, which is impossible. Therefore there is a carry and N = E + 1.
#If there were no carry in column 2, then ( N + R ) mod 10 = E, and N = E + 1, so ( E + 1 + R ) mod 10 = E which means ( 1 + R ) mod 10 = 0, so R = 9. But S = 9, so there must be a carry in column 2 so '''R = 8'''.
#To produce a carry in column 2, we must have D + E = 10 + Y.
#Y is at least 2 so D + E is at least 12.
#The only two pairs of available numbers that sum to at least 12 are (5,7) and (6,7) so either E = 7 or D = 7.
#Since N = E + 1, E can't be 7 because then N = 8 = R so '''D = 7'''.
#E can't be 6 because then N = 7 = D so '''E = 5''' and '''N = 6'''.
#D + E = 12 so '''Y = 2'''.

The use of [[modular arithmetic]] often helps. For example, use of mod-10 arithmetic allows the columns of an addition problem to be treated as [[simultaneous equations]], while the use of mod-2 arithmetic allows inferences based on the [[parity (mathematics)|parity]] of the variables.

In [[computer science]], cryptarithms provide good examples to illustrate the [[brute force search|brute force]] method, and algorithms that generate all [[permutation]]s of ''m'' choices from ''n'' possibilities. For example, the Dudeney puzzle above can be solved by testing all assignments of eight values among the digits 0 to 9 to the eight letters S,E,N,D,M,O,R,Y, giving 1,814,400 possibilities. They provide also good examples for [[backtracking]] paradigm of [[algorithm]] design.

==Other information==
When generalized to arbitrary bases, the problem of determining if a cryptarithm has a solution is [[NP-complete]].<ref>{{cite journal | author = [[David Eppstein]] | title = On the NP-completeness of cryptarithms | journal = SIGACT News | volume = 18 | issue = 3 | pages = 38–40 | year = 1987 | url = http://www.ics.uci.edu/~eppstein/pubs/Epp-SN-87.pdf | doi = 10.1145/24658.24662}}</ref> (The generalization is necessary for the hardness result because in base 10, there are only 10! possible assignments of digits to letters, and these can be checked against the puzzle in linear time.)

Alphametics can be combined with other number puzzles such as Sudoku and Kakuro to create cryptic [[Sudoku]] and [[Kakuro]].

==See also==
* [[Diophantine equation]]
* [[Mathematical puzzle]]s
* [[Permutation]]
* [[Puzzle]]s

==References==
{{Reflist}}
{{More footnotes|date=July 2010}}
* [[Martin Gardner]], ''Mathematics, Magic, and Mystery''. Dover (1956)
* [[Journal of Recreational Mathematics]], has a regular alphametics column.
* Jack van der Elsen, ''Alphametics''. Maastricht (1998)
* Kahan S., Have some sums to solve: The complete alphametics book, Baywood Publishing, (1978)
* Brooke M. One Hundred & Fifty Puzzles in Crypt-Arithmetic. New York: Dover, (1963)

==External links==
* [http://code.activestate.com/recipes/576615/ Alphametic Solver written in Python]
* [http://www.cut-the-knot.org/cryptarithms/st_crypto.shtml Cryptarithms] at [[cut-the-knot]]
* {{MathWorld | urlname=Alphametic| title=Alphametic}}
* {{MathWorld | urlname=Cryptarithmetic | title=Cryptarithmetic}}
* [http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/ALPHAMETIC/ Alphametics and Cryptarithms]
* [http://www.iread.it/cryptarithms.php An on-line tool to create Alphametics and Cryptarithms]

[[Category:Articles with inconsistent citation formats]]
[[Category:Logic puzzles]]

Sieverts' law

2013-04-21T01:13:29Z

132.216.49.30: /* Justification */

'''Differential dynamic microscopy'''
Differential dynamic microscopy (DDM) is an optical technique that allows performing [[light scattering]] experiments by means of a simple [[optical microscope]].<ref name="cerbino08">R. Cerbino, V. Trappe, "Differential dynamic microscopy: Probing wavevector-dependent dynamics with a microscope", Phys. Rev. Lett. 100, 188102 (2008), http://dx.doi.org/10.1103/PhysRevLett.100.188102</ref><ref name="giavazzi09">F. Giavazzi, D. Brogioli, V. Trappe, T. Bellini, R. Cerbino, "Scattering information obtained by optical microscopy: Differential Dynamic Microscopy and beyond", Phys. Rev. E 80, 031403 (2009), http://dx.doi.org/10.1103/PhysRevE.80.031403</ref> DDM is suitable for typical [[Soft matter|soft materials]] such as for instance [[liquid]]s or [[gels]] made of [[colloids]], [[polymers]] and [[liquid crystal]]s but also for biological materials like bacteria and [[Cell (biology)|cells]].

==The basic idea==
The typical DDM data is a time sequence of microscope images (movie) acquired at some height within the sample (typically at its mid-plane). If the image intensity is locally proportional to the concentration of particles or molecules to be studied (possibly convoluted with the microscope [[Point spread function|point spread function (PSF)]]), each movie can be analyzed in the Fourier space to obtain information about the dynamics of concentration Fourier modes, '''independent on the fact that the particles/molecules can be individually optically resolved or not'''. After suitable calibration also information about the Fourier amplitude of the concentration modes can be extracted.

==Applicability and working principle==
The concentration-intensity proportionality is valid at least in two very important cases that distinguish two corresponding classes of DDM methods:

# '''scattering-based DDM''': where the image is the result of the superposition of the strong transmitted beam with the weakly scattered light from the particles. Typical cases where this condition can be obtained are [[Bright field microscopy|bright field]], [[phase contrast]], [[Polarized light microscopy|polarized]] microscopes.
# '''fluorescence-based DDM''': where the image is the result of the incoherent addition of the intensity emitted by the particles ([[Fluorescent microscope|fluorescence]], [[Confocal microscopy|confocal]]) microscopes

In both cases the convolution with the [[Point spread function|PSF]] in the [[real space]] corresponds to a simple product in the [[Reciprocal space|Fourier space]], which guarantees that studying a given Fourier mode of the image intensity provides information about the corresponding Fourier mode of the concentration field. In contrast with [[Single particle tracking|particle tracking]], there is no need of resolving the individual particles, which allows DDM to characterize the dynamics of particles or other moving entities whose size is much smaller than the wavelength of light. Still, the images are acquired in the real space, which provides several advantages with respect to traditional (far field) scattering methods.

==Data analysis==
DDM is based on an algorithm proposed in<ref name="croccolo06">F. Croccolo, D. Brogioli, A. Vailati, M. Giglio and D. S. Cannell, "Use of dynamic schlieren interferometry to study fluctuations during free diffusion" , Applied Optics 45, 2166 (2006)</ref> and,<ref name="alaimo06">M. Alaimo, D. Magatti, F. Ferri, and M.A.C. Potenza, " Heterodyne speckle velocimetry", Appl. Phys. Lett. 88, 191101 (2006)</ref> which is conveniently named [[Differential Dynamic Algorithm|Differential Dynamic Algorithm (DDA)]]. DDA works by subtracting images acquired at different times and taking advantage that, as the delay <math>\Delta t</math> between two subtracted images gets large, the energy content of the difference image increases correspondingly. A two-dimensional [[Fast Fourier transform|Fast Fourier Transform (FFT)]] analysis of the difference images allows to quantify the growth of the signal contains for each wave vector <math>q</math> and one can calculate the Fourier power spectrum of the difference images for different delays <math>\Delta t</math> to obtain the so-called ''image structure function'' <math>D(q;\Delta t)</math>. Calculation shows that for both scattering- and fluorescence-based DDM
{{NumBlk|:|<math>D(q;\Delta t) =B(q)+T(q)I(q)[1-f(q;\Delta t)] \,</math>|{{EquationRef|1}}}}
where <math>f(q;\Delta t)</math> is the normalized [[intermediate scattering function]] that would be measured in a [[Dynamic light scattering|Dynamic Light Scattering (DLS)]] experiment, <math>I(q)</math> the sample scattering intensity that would be measured in a [[Static light scattering|Static Light Scattering (SLS)]] experiment, <math>B(q)</math> a background term due to the noise along the detection chain <math>T(q)</math> a transfer function that depends on the microscope details.<ref name="giavazzi09" /> Equation ({{EquationNote|1}}) shows that DDM can be used for [[Dynamic light scattering|DLS]] experiments, provided that a model for the normalized [[intermediate scattering function]] is available.<ref name="giavazzi09" /> For instance, in the case of [[Brownian motion]] one has <math>f(q;\Delta t)=e^{-Dq^{2}\Delta t},</math> where <math>D</math> is the [[Mass diffusivity|diffusion coefficient]] of the Brownian particles. If the transfer function <math>T(q)</math> is determined by calibrating the microscope with a suitable sample, DDM can be employed also for [[Static light scattering|SLS]] experiments. Alternative algorithms for data analysis are suggested in.<ref name="giavazzi09" />

==Relationship with other imaging-based scattering methods==
Scattering-based DDM belongs to the so-called '''near-field (or deep Fresnel) scattering''' family,<ref name="cerbino09">R. Cerbino, A. Vailati, "Near-field scattering techniques: Novel instrumentation and results from time and spatially resolved investigations of soft matter systems", Curr. Op. Coll. Int. Science 14, 416 (2009), http://dx.doi.org/10.1016/j.cocis.2009.07.003</ref> a recently introduced family of imaging-based scattering methods.<ref name="giglio00">M. Giglio, M. Carpineti, and A. Vailati, "Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g(r)", Phys. Rev. Lett. 85, 1416 (2000), http://dx.doi.org/10.1103/PhysRevLett.85.1416</ref><ref name="brogioli02">D. Brogioli, A. Vailati and M. Giglio,"Heterodyne near-field scattering", Appl. Phys. Lett. 81, 4109 (2002), http://dx.doi.org/10.1063/1.1524702</ref> ''Near field'' is used here in a similar way to what is used for [[Speckle_pattern#Near-field_speckles|near field speckles]] i.e. as a particular case of Fresnel region as opposed to the ''far field'' or Fraunhofer region. The near field scattering family includes also quantitative [[shadowgraph]]y<ref name="wu95">M. Wu, G. Ahlers and D. S. Cannell, "Thermally induced fluctuations below the onset of Rayleigh-Bénard convection", Phys. Rev. Lett. 75, 1743 (1995), http://dx.doi.org/10.1103/PhysRevLett.75.1743</ref> and [[Schlieren]].<ref name="croccolo06" />

==Applications of DDM==
DDM was introduced in 2008 and it was applied for characterizing the dynamics of [[colloidal particle]]s in [[Brownian motion]].<ref name="cerbino08" /> More recently it has been successfully applied also to the study of aggregation processes of colloidal nanoparticles,<ref name="ferri11">F. Ferri, A. D’Angelo, M. Lee, A. Lotti, M.C. Pigazzini, K. Singh and R. Cerbino, "Kinetics of colloidal fractal aggregation by differential dynamic microscopy", Eur. Phys. J. Special Topics, 199, 139-148 (2011), http://dx.doi.org/10.1140/epjst/e2011-01509-9</ref> of bacterial motions<ref name="wilson11">L. G. Wilson, V. A. Martinez, J. Schwarz-Linek, J. Tailleur, G. Bryant, P. N. Pusey, W. C. K. Poon, "Differential dynamic microscopy of bacterial motility" Phys. Rev. Lett. 2011, 106, 018101, http://dx.doi.org/10.1103/PhysRevLett.106.018101</ref><ref name="martinez12">V. A. Martinez, R. Besseling, O. A. Croze, J. Tailleur, M. Reufer, J. Schwarz-Linek, L. G. Wilson, M. A. Bees, W. C. K. Poon "Differential dynamic microscopy: A high-throughput method for characterizing the motility of microorganisms", available from arXiv:1202.1702v1, http://arxiv.org/abs/1202.1702v1</ref> and of the dynamics of anisotropic colloids.<ref name="reufer12">M. Reufer, V. A. Martinez, P. Schurtenberger, and W. C. K. Poon, "Differential Dynamic Microscopy for Anisotropic Colloidal Dynamics", Langmuir, Article ASAP, http://dx.doi.org/10.1021/la204904a</ref>

== References ==
{{Reflist}}

== External links ==
* [https://sites.google.com/site/cerbino/research/ddm DDM page on the personal website of Roberto Cerbino]



[[Category:Scientific techniques]]
[[Category:Microscopy]]
[[Category:Scattering, absorption and radiative transfer (optics)]]
[[Category:Biochemistry methods]]
[[Category:Physical chemistry]]
[[Category:Spectroscopy]]

Obstacle problem

2012-08-19T06:11:34Z

132.216.88.105: /* Level surfaces and the free boundary */

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