User:Adam majewski
"The answer is no, although it doesn't seem so easy to give a rigorous counter-example." Glougloubarbaki[1] " Mathematics takes place at different time-scales. If you can solve a problem in 55 minutes that others need an hour to solve, you can probably get a good job. If you can solve a problem in a month that others might need a year to solve, you will probably do well as a graduate student. But if you can solve a problem in 10 years that nobody else can solve in a lifetime, you could be a great mathematician." Robert Israel
" anyone who wishes to study this topic should earn a Ph.D. in number theory and spend several years researching the relevant topics in depth, with the guidance of a world class expert." Alon Amit, PhD in Mathematics; Mathcircler.
"The author apologises wholeheartedly to those who dare read the source code." Freddie R. Exall
A BELIEF IS NOT A PROOF.
"Category theory is ... the most, abstract fields of mathematics" Robb Seaton[2] For the sake of completeness, here is the entire code I used:"
My home page - dead (:-(
Wikipedia - Adam majewski
Commons - Adam majewski
Links :
- mathjax live demo ( mathjax render)
- SyntaxHighlight
- [SyntaxHighlighter]
- [source]
- [detect spoken language]
- Help:Variables
- Help:Editing
- Help:Editing#Inserting_references
- @Username:
- List_of_Wikibooks languages
- formula
- WIKIBOOKS SPECIAL
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- http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference
- https://en.wikibooks.org/wiki/Template:ISBN
rigorous studies of nonlinear systems
- computing enclosures of trajectories
- finding and proving the existence of symbolic dynamics
- obtaining rigorous bounds for the topological entropy
- methods for finding accurate enclosures of chaotic attractor
- interval operators for proving the existence of fixed points and periodic orbits
- methods for finding all short cycles
https://rd.springer.com/chapter/10.1007/978-3-540-95972-4_2
methods
graphic software
- https://www.generic-mapping-tools.org/documentation/
- http://gwyddion.net/documentation/user-guide-en/color-map.html#color-gradient-editor*
text color
some notes
Some notes from | wikipedia
- ,
A Gentle Introduction to the Art of Mathematics by Joe Fields,
parabolic/hyperbolic/elliptic
The meaning of the terms "elliptic, hyperbolic, parabolic" in different disciplines in mathematics[3]
- PDE ( Linear Second Order PDE’s in two Independent Variables) : https://en.wikipedia.org/wiki/Partial_differential_equation
- Moebius transformations = Classification of Isometries ( https://www.mathi.uni-heidelberg.de/~alessandrini/Arith_Reports/1-hyperbolic%20geometry.pdf)
- dicrete local complex dynamics
- Conic section
- Quadratic form
- probability distributions.
- coordinate
hyperbolic
- hyperbolic". usually) means that |f′(t)|≠1| , https://math.stackexchange.com/questions/2172002/is-indeterminate-a-better-name-than-indifferent-for-neutral-fixed-points
- http://www.scholarpedia.org/article/Hyperbolic_dynamics
curves
- Biarc
- folium
- trifolium
https://www.mathcurve.com/courbes2d.gb/rosace/rosace.shtml rose curve = n-folium: The curve is composed of a n base patterns. The pattern is called : the petal or branch / leaf / lobe - symmetrical about Ox obtained for angle between -pi/(2n) and pi/(2n)
osculating circle of a sufficiently smooth plane curve
curvature
Interesting curves involning the curvature concept by Xah Lee:
- Evolute curve (the centers of osculating circles)
- Radial curve (locus of osculating circle normals)
- circle = curve with constant curvature everywhere
- line = curve with curvature of 0 everywhere)
- Clothoid = spiral cirve of linearly increasing curvature)
topology
number theory
algorithms
geometry
- https://www.ics.uci.edu/~eppstein/161/syl.html
- https://www.cs.cmu.edu/~kmcrane/
- http://blancosilva.github.io/post/2014/10/28/Computational-Geometry-in-Python.html
- digital
Moore-Neighbor Tracing
- https://en.wikipedia.org/wiki/Moore_neighborhood
- http://www.imageprocessingplace.com/downloads_V3/root_downloads/tutorials/contour_tracing_Abeer_George_Ghuneim/mmain.html
- https://www.codeproject.com/Articles/1105045/Tracing-Boundary-in-D-Image-Using-Moore-Neighborho
- https://stackoverflow.com/questions/26830697/moore-neighbourhood-in-python
- https://py.checkio.org/en/mission/count-neighbours/
see also:
- https://en.wikipedia.org/wiki/User:TerribleTadpole/sandbox
- https://cs.wikibooks.org/wiki/Geometrie/Vypl%C5%88ov%C3%A1n%C3%AD
- https://commons.wikimedia.org/wiki/Category:Pathfinding
- https://en.wikipedia.org/wiki/Pathfinding
chain code
- http://islab.ulsan.ac.kr/files/announcement/301/20091132.pdf
- https://stackoverflow.com/questions/12885055/chain-code-infinite-loop?rq=1
- https://stackoverflow.com/questions/47001899/freeman-chain-code-infinite-loop-4-adjacency?rq=1
- https://stackoverflow.com/questions/6718525/understanding-freeman-chain-codes-for-ocr?rq=1
- https://www.e-olymp.com/en/problems/1803
- http://airccse.org/journal/ijcga/papers/4214ijcga02.pdf
- http://appliedmaths.sun.ac.za/TW793/slides/slides_11_1.pdf
- http://www.aass.oru.se/Research/Learning/courses/dip/2011/lectures/DIP_2011_L14.pdf
test
( land on the root point of period 267 component : c267 = 0.250137369683480-0.000003221184145 i with angled internal adress :
( land on the root point of period 268 component c268 = 0.250137369683480-0.000003221184145i period = 10000 i with angled internal adress :
mandelbrot set
the Mandelbrot set for the function 1/z - z∙(1 + 0.001∙z)/(1 - 0.002∙z + 0.001∙z2) = "1 -0.002 - 0.999 -0.001 0 1 -0.002 0.001":
http://www.juliasets.dk/UFP.htm
video
- https://www.college-de-france.fr/site/en-pierre-louis-lions/symposium-2017-05-30-15h30.htm
- https://www.college-de-france.fr/site/en-pierre-louis-lions/symposium-2017-05-29-11h30.htm
- https://www.math.stonybrook.edu/jackfest/Talks/
- http://www.math.vt.edu/netmaps/index.php
programs
- http://jsxgraph.uni-bayreuth.de/wp/index.html
- http://www.singsurf.org/index.php
- http://www.javaview.de/index.html
gradient line of the 2D scalar field
- http://hplgit.github.io/
- https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/gradient-and-directional-derivatives/v/gradient
- https://stackoverflow.com/questions/25342072/computing-and-drawing-vector-fields
- https://stackoverflow.com/questions/33059710/how-to-plot-gradfx-y?noredirect=1&lq=1
- https://math.stackexchange.com/questions/462592/trouble-with-gradient-intuition
- https://metacademy.org/graphs/concepts/finite_difference_approximations_to_derivatives
- https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-b-chain-rule-gradient-and-directional-derivatives/session-35-gradient-definition-perpendicular-to-level-curves/
- https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-b-chain-rule-gradient-and-directional-derivatives/session-35-gradient-definition-perpendicular-to-level-curves/MIT18_02SC_notes_18.pdf
- https://metacademy.org/browse
- https://www.desmos.com/calculator/el5syd7ugj
- http://freymanart.com/VectorFields/index.htm
Key words:
- "gradient line" 2d "scalar field"
flow :
- level curves and gradient vector
- flow across continuously-spaced level curves
- The flow’s derivative is the gradient – the flow will follow the gradient vectors
- gradient is the direction of steepest ascent in the zz-direction, the reverse of the flow is the path of an object as it rolls on the surface, starting from a high place and rolling down to a lower place (in the exact opposite direction as the gradient vectors point).
- def from Streamline Tracing on Irregular Grids by H˚akon Hægland
- "The instantaneous curves that are at every point tangent to the direction of the velocity at that point are called streamlines of the flow"
- "A pathline of a fluid particle is the locus of its position in space as time passes. It is thus the trajectory of a particle of fixed identity"
Khan
the gradient points in the direction which increases the value of f most quickly. There are two ways to think about this direction:
- Choose a fixed step size, and find the direction such that a step of that size increases fff the most. Given steps of a constant size away from a particular point, the gradient is the one which increases f the most.
- Choose a fixed increase in fff, and find the direction such that it takes the shortest step to increase fff by that amount. Given steps which increase f by a given size, the gradient direction is the shortest among these.
Either way, you're trying to :
- maximize the rise over run,
- either by maximizing the rise, or minimizing the run.
potential flow
spiral
- https://math.stackexchange.com/questions/2254493/logarithmic-spiral-transform-function
- https://math.stackexchange.com/questions/1232346/find-gradient-of-a-equi-angular-spiral-log-spiral
The Golden Ratio and the Golden Angle
In disc phyllotaxis, as in the sunflower and daisy, the mesh of spirals occurs in Fibonacci numbers because divergence (angle of succession in a single spiral arrangement) approaches the golden ratio. The shape of the spirals depends on the growth of the elements generated sequentially. In mature-disc phyllotaxis, when all the elements are the same size, the shape of the spirals is that of Fermat spirals—ideally. That is because Fermat's spiral traverses equal annuli in equal turns. The full model proposed by H Vogel in 1979[4] is
where θ is the angle, r is the radius or distance from the center, and n is the index number of the floret and c is a constant scaling factor. The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers.[5]
Illustration of Vogel's formula of the pattern of sunflower florets (see article) for n from 1 to 500, using the polar coordinates equations and . Can be produced using the following MATLAB code:
n=1:500; r=sqrt(n); t=2*pi/((sqrt(5)+1)/2+1)*n; plot(r.*cos(t),-r.*sin(t),'o')
https://www.shadertoy.com/view/4lGfDd https://www.youtube.com/watch?v=sj8Sg8qnjOg
intersection of polar curves
- https://math.stackexchange.com/questions/1221260/intersecting-polar-curves-r-1cos%CE%B8-and-r-1-cos%CE%B8
- https://math.stackexchange.com/questions/1064268/suppose-that-two-polar-curves-are-given-by-r-1-cos2-theta-and-r-2-si
- https://math.stackexchange.com/questions/851190/finding-self-intersections-on-a-polar-curve
- https://math.stackexchange.com/questions/2038456/intersection-points-of-two-polar-curves?rq=1
- https://math.stackexchange.com/questions/1281068/intersection-of-polar-curve-with-line?rq=1
- https://math.stackexchange.com/questions/2358269/points-of-intersection-for-two-logarithmic-spirals?rq=1
iteration
- https://math.stackexchange.com/questions/2571345/iterating-quadratic-polynomials/2571349#2571349
- https://math.stackexchange.com/questions/140819/general-mandelbrot-iteration-formulas?rq=1
atan2
- https://stackoverflow.com/questions/17574424/how-to-use-atan2-in-combination-with-other-radian-angle-systems
- http://patrikstas.com/2015/11/05/what-is-difference-between-atan-and-atan2/
- https://mrl.nyu.edu/~perlin/courses/spring2012/texture.html
- https://www.youtube.com/watch?v=mGEtMc20eeE
- https://www.youtube.com/watch?v=Zh6cmQcT1YI
- https://www.youtube.com/watch?v=LHzgW9aQUV8
- https://www.youtube.com/watch?v=Idxeo49szW0
conformal
- " Conformal transformation: One can find solutions to the Dirichlet problem for a wider class of regions by conformal transformations. These are transformations which preserve the angles and have the property that the transplanted function is harmonic if the original function is harmonic. One can obtain conformal transformations using complex function (x,y) -> (u,v): if z=x+iy and F(z) is a polynomial in z and F(z)=u+i v, then the map is conformal. For example, to solve the Dirichlet problem outside a wing, one can transform the circle into the wing using a conformal transformation. Such methods are relevant in engeneering." http://www.math.harvard.edu/archive/21b_fall_03/laplace/index.html
- https://www3.cs.stonybrook.edu/~gu/tutorial/RiemannMapping.html
- https://www.chebfun.org/docs/guide/guide16.html
- https://www.cs.cmu.edu/~kmcrane/Projects/Other/OverviewConformalGeometryProcessing.pdf
- http://www-users.math.umn.edu/~olver/ln_/cml.pdf
- The Euler conformal map, a special case of the M¨obius transformation, maps circles to circles
- https://gregjavens.com/2016/05/11/conformal-map/
- https://gregjavens.com/2016/03/12/laplace-equation-finite-element-method-3/
- http://mathfaculty.fullerton.edu/mathews/c2003/ConformalMapDictionary.2.html
- https://www3.cs.stonybrook.edu/~gu/
- https://arxiv.org/abs/1306.1162
- http://www.eng.biu.ac.il/weberof/publications/
- http://wwwf.imperial.ac.uk/~dgcrowdy/GuTalk.pdf
numerical conformal mapping=
Orthogonal
- https://www.mathcurve.com/courbes2d.gb/orthogonale/orthogonale.shtml
- https://en.wikipedia.org/wiki/Orthogonal_trajectory
- http://www.emathhelp.net/calculators/calculus-1/tangent-line-calculator/
- https://www.whitman.edu/mathematics/calculus_online/section10.02.html
- http://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx
- http://www.emathhelp.net/calculators/calculus-1/online-graphing-calculator/
- https://www.math24.net/tangent-normal-lines/
- https://emathhelp.net/calculators/calculus-1/online-graphing-calculator/?uid=nwd4ldjtge
- https://emathhelp.net/plot/nwd4ldjtge.png
- https://math.stackexchange.com/questions/1919319/how-do-i-get-the-tangent-of-a-curve?rq=1
- https://math.stackexchange.com/questions/992847/finding-the-equation-of-a-tangent-line-to-a-polar-curve?rq=1
tangent to circle in the polar form
z= r(t)
is line :
y = x*tan(t)
Tangent to cardioid in polar form :
z = 2a( 1 - cos(t))
is line :
// y = (x*(-1+2*cos(t))*sin(t)+(-3+3*cos(t))*sin(t))/(-1+2*cos(t)^2-cos(t)) y = (x*(-1+2*cos(t))*sin(t)+(-2a+2a*cos(t))*sin(t))/(-1-cos(t)+2*cos(t)^2)
where t is changing from 0 to 2*pi
Compare :
- circle to cardioid mapping ( w = z^2)
- The objects shown here are the inverse images of the map w = z4 where the point w is on the cricle and on the line in the complex plane.
ellipse
slope m
The equation of the tangent at point has the form
https://www.math24.net/tangent-normal-lines-page-2/#example11 https://www.math24.net/implicit-differentiation/
Implicit
- function
- implicit differentiation = differentiation of the implicit function
sqare root
- http://mathlets.org/mathlets/complex-roots/
- https://flothesof.github.io/branch-cuts-with-square-roots.html
- http://phantomgraphs.weebly.com/
- https://math.stackexchange.com/questions/1797223/graphically-solving-for-complex-roots-how-to-visualize?noredirect=1&lq=1
- Reciprocal Function as a Mapping
- http://functions.wolfram.com/ElementaryFunctions/Sqrt/visualizations/4/
- http://functions.wolfram.com/ElementaryFunctions/Sqrt/visualizations/7/
- http://mathfaculty.fullerton.edu/mathews/c2003/ComplexFunPowerRootMod.html
- https://sites.oxy.edu/ron/math/312/16/ws/08.pdf
- https://usamo.wordpress.com/2017/02/16/holomorphic-logarithms-and-roots/
- http://laussy.org/wiki/MMII/Complex_functions_of_complex_numbers
- https://www.cs.cmu.edu/Groups/AI/html/cltl/clm/_24769_figure12567.gif
- branch cuts
- https://www.cs.cmu.edu/Groups/AI/html/cltl/clm/node129.html
- sqrt : The branch cut for square root lies along the negative real axis, continuous with quadrant II. The range consists of the right half-plane, including the non-negative imaginary axis and excluding the negative imaginary axis.
- https://math.stackexchange.com/questions/923931/problem-identifying-branch-cuts-of-a-square-root-function?noredirect=1&lq=1
repeated
- https://math.stackexchange.com/questions/3283/why-do-i-always-get-1-when-i-keep-hitting-the-square-root-button-on-my-calculat
- https://www.geeksforgeeks.org/sqrt-square-root-decomposition-technique-set-1-introduction/
- http://edspi31415.blogspot.com/2015/08/repeated-presses-of-square-root-button.html
smooth curve from points
"curve fitting is a set of techniques used to fit a curve to data points "
- https://www.quora.com/Whats-the-difference-between-curve-fitting-and-regression
- https://www.codeproject.com/Articles/31859/Draw-a-Smooth-Curve-through-a-Set-of-2D-Points-wit
- https://www.codeproject.com/Articles/25237/Bezier-Curves-Made-Simple
- https://mycurvefit.com/
- https://www.particleincell.com/2012/bezier-splines/
- http://www.mvps.org/directx/articles/catmull/
- https://stackoverflow.com/questions/tagged/curve-fitting?sort=votes&pageSize=50
- https://web.cs.wpi.edu/~matt/courses/cs563/talks/curves.html
- http://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/INT-APP/CURVE-INT-global.html
- https://web.cs.wpi.edu/~matt/courses/cs563/talks/curves.html
- https://gis.stackexchange.com/questions/138881/finding-the-center-line-from-a-set-of-3d-points
Fit method
- linear
- join points with segments = concatenated linear segments
- straight line using linear regression
- nonlinear
- polynomial
- cubic spline
- Smooth Bézier Spline Through Prescribed Points
trace a curve
- To trace the curve we evaluate successive points on the curve
- https://stackoverflow.com/questions/31464345/fitting-a-closed-curve-to-a-set-of-points
- https://stackoverflow.com/questions/14631776/calculate-turning-points-pivot-points-in-trajectory-path
- https://www.quora.com/Whats-the-difference-between-curve-fitting-and-regression
- http://user.engineering.uiowa.edu/~dip/lecture/Segmentation2.html
- http://alice.loria.fr/publications/papers/2014/STREAM/RobustStreamlines.pdf
- https://link.springer.com/article/10.1007/s40819-015-0067-1
- https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/82596/1/0787-12.pdf
- ADAPTIVE MULTIPRECISION PATH TRACKING: https://www.semanticscholar.org/paper/Adaptive-Multiprecision-Path-Tracking-Bates-Hauenstein/6517744e4b68d3e648166448a88bda63e6a597e3
- http://www.math.colostate.edu/~bates/preprints/BHS_ODE_21apr10.pdf
sketch a curve
boundary trace
- http://paulbourke.net/papers/conrec/
- http://user.engineering.uiowa.edu/~dip/lecture/Segmentation2.html
- http://sijoo.tistory.com/251
- http://www.imageprocessingplace.com/downloads_V3/root_downloads/tutorials/contour_tracing_Abeer_George_Ghuneim/moore.html
- https://www.ibiblio.org/e-notes/MSet/big_m.htm
level sets
tree
People
petal
"An attracting petal, P + , for a map M at zero is an open simply connected forward invariant region with 0 ∈ ∂P + , that shrinks down to the origin under iteration of M . More precisely, P + is an attracting petal if M (P + ) ⊂ P + ∪ {0} and n≥0 M n (P + ) = {0}. "[6]
example
http://mathoverflow.net/questions/104482/parabolic-immediate-basins-always-simply-connected?rq=1 "An example is f(z)=z+1−1/zf(z)=z+1−1/z. There is one petal for the neutral point at infinity. Let AA be the dmain of attraction of ∞∞. Critical points are ±i±i. Everything is symmetric with respect to the real line, because the function is real. One critical point is in AA, so by symmetry the other one is also in AA. The map f:A→Af:A→A is 2-to-1 (because ff is of degree 22), so Riemann and Hurwitz tell us that AA is infinitely connected."
shareciteeditflag answered Aug 12 '12 at 13:38
Alexandre Eremenko
cylinder
What is the difference between the cylinder and cylinder ?
topology
"A topologist is someone who doesn't know the difference between a cup of coffee and a donut."
references
- Computer Methods and Borel Summability Applied to Feigenbaum's Equation By Jean Pierre Eckmann
Format Hardback | 297 pages, Publication date 01 May 1985, Publisher Springer , Publication City/Country United States , ISBN10 0387152156, ISBN13 9780387152158
- http://mathoverflow.net/questions/157309/power-series-expansion-of-the-koenigs-function?rq=1
- T. M. CHERRY, A singular case of iteration of analytic functions: A contribution to the small divisor problem. In: Nonlinear Problems of Engineering, W. F. AMES (Ed.), New York, 1964, 29–50.
- Cherry, T. M., "A Singular Case of Iteration of Analytic Functions: A Contribution to the Small-Divisor Problem," in Nonlinear Problems of Engineering (edited by W. F. Ames), Academic Press, New York, 1964, 29-50.
- MR178125 30.40 (57.48) Cherry, T. M. A singular case of iteration of analytic functions: A contribution to the small-divisor problem. 1964 Nonlinear Problems of Engineering pp. 29–50 Academic Press, New York
pl-N | Polski jest językiem ojczystym tego użytkownika. |
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en-2 | This user can read and write intermediate English. |
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.
- ↑ math SE question: is-the-basin-of-attraction-of-a-p-starshaped-wrt-to-p
- ↑ Why Category Theory Matters by Robb Seaton
- ↑ quora : Where-is-the-best-summary-on-the-meaning-of-the-terms-elliptic-hyperbolic-parabolic-as-used-in-different-disciplines-in-mathematics
- ↑ Vogel, H (1979). "A better way to construct the sunflower head". Mathematical Biosciences 44 (44): 179–189. doi:10.1016/0025-5564(79)90080-4
- ↑ Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0-387-97297-8. http://algorithmicbotany.org/papers/#webdocs.
- ↑ NEWTON’S METHOD ON THE COMPLEX EXPONENTIAL FUNCTION MAKO E. HARUTA