Pick's theorem - Factbites

Pick's theorem - Factbites
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Topic: Pick's theorem

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	Schwarz-Ahlfors-Pick theorem - Wikipedia, the free encyclopedia
		The theorem states that every holomorphic automorphism of the
	en.wikipedia.org /wiki/Schwarz-Alhfors-Pick_theorem

	Newton.txt
		In the statement of the theorem, we have ignored the fact that the number of circles and number of points must be integers, so the actual number of points can be slightly larger due to rounding up.
		Otherwise, pick any root with $k\geq 2$ two canals, and let $m_2,\ldots,m_{k+1}$ be the total numbers of roots between adjacent canals.
		This formula is an immediate corollary to the residue theorem for the map $1/(f(z)-z)$ (evaluated best in coordinates such that $\infty$ is not a fixed point).
	www.mathlab.sunysb.edu /~scott/Papers/Newton/Newton.txt

	Search Results for theorem*
		Theorem 2 of Euclid's Phaenomena consists of four propositions with proofs for only three of them while the missing one is replaced by the remark "that this is the case has been shown elsewhere"; indeed theorem and proof are found as Theorem 10 in Autolycus's 'Rotating Sphere'.
		The theorem relating convergence almost everywhere and uniform convergence by D F Egorov, one of Bugaev's pupils, in 1911 is seen as marking the beginning of the Moscow school of the theory of functions of a real variable.
		This means that a theorem in Autolycus's work has first a general statement, then a construction related to a particular figure with points in the figure denoted by letters, next comes the demonstration of the theorem, and finally a conclusion relating to the general statement is sometimes drawn.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=theorem*&CONTEXT=1

	Introduction
		For such a class of mappings, it is possible to find a Bloch theorem in
		3 ] proved the famous theorem which bears his name:
		Since confirmation of the Bieberbach conjecture by de Branges, perhaps the outstanding open problem in complex analysis is that of finding the exact value of the Bloch constant.
	www.3dfractals.com /bloch/node1.html

	Read This: Complex Analysis: The Geometric Viewpoint, Second Edition
		In the introductory chapter, the author presents the main results in standard complex analysis: the Maximum Principle, the Schwarz lemma (and the Schwarz-Pick lemma), the Riemann Mapping theorem and the theorems of Picard.
		the Schwarz's lemma, Liouville's theorem or Montel's theorem) has a very good chance to understand standard complex analysis as it is studied in graduate school.
		The result is that one can learn the flavor and some of the methodology of differential geometry without being encumbered by its notation and machinery." We can see how this principle works in writing by reading the proof of K. Hahn's theorem, where Krantz chooses the most useful (and visually effective) viewpoint (p.
	www.maa.org /reviews/complexgeometricviewpoint.html

	Proc 97 Abstracts
		For the special case of the hyperbolic metric on a Nehari region which is not M\"obius equivalent to a strip, this property gives an elegant geometrical interpretation for the Ahlfors-Weill quasiconformal reflection in the boundary of the region.
		We first discuss some motivations from system theory and $n$-th root asymptotics for the error when the function to be approximated is analytic in a neighborhood of the circle.
		The interpretation reveals that Ahlfors-Weill reflection is a natural extension of the classical notion of Schwarz reflection in a circle or line.
	www.heldermann.de /CMF/cmfproc97abs.htm

	An Ahlfors Celebration
		Ahlfors, who would have been 90 this year, died late last fall.
		Below is the list of speakers and titles for the Ahlfors Celebration to be held at Stanford University, September 19-21, 1997.
		All talks will be held in room 380-C of the Department of Mathematics, Building 380, except for the public lecture by Steven Krantz, to be held in Room 420-041 of the Psychology Building, which is entered through the lower level of the Mathematics Building.
	www.msri.org /activities/events/9798/ahlfors

	Selections from MSRI's Video Archive, Volume 1
		Dennis Sullivan The Ahlfors-Bers measurable Riemann mapping theorem in higher dimensions
	math.rice.edu /MSRIlectures/ln

	Search Results for Paris
		It was a monument resplendent in its simple lines - the main theorem from his Paris memoir, formulated in few words.
		While completing the work for his doctoral dissertation he went to Gottingen where he was influenced by Schwarz and Klein; he received a doctorate in mathematics from Paris in 1887 for this thesis.
		Ahlfors went to Paris with Nevanlinna for three months before returning to Finland.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Paris&CONTEXT=1

	ho_wei
		As a warning, there was a bit of confusion about 'Taylor's theorem' -- I was referring to the theorem in Ahlfors, but he thought I meant the real-variable version...
		I defined the former and mentioned the fundamental theorem of calculus.
		(I asked if they wanted me to define things like Artin symbol, but they declined.) T: What classical theorem...
	www.math.princeton.edu /graduate/generals/ho_wei

	List of mathematical topics (S) - Wikipedia, the free encyclopedia
		Simple theorems in the algebra of sets -- Simplex --
	en.wikipedia.org /wiki/List_of_mathematical_topics_%28S%29

	Descriptions of spring 2005 courses in the Rutgers-New Brunswick Math Graduate Program
		Topics to be covered: the Riesz representation theorem for positive linear functionals on C(X), the Birkhoff ergodic theorem, the Marcinkiewicz interpolation theorem, the Riesz-Thorin interpolation theorem, the Fourier transform and elementary theory of singular integrals.
		We outline the proofs of these results, including the positive mass theorem of Schoen and Yau on which the proof of the last case relies.
		This course will be an introduction to Lie groups, beginning with the general linear group and the other classical groups (the unitary, orthogonal and symplectic groups) and finishing with the Weyl character formula and the Borel-Weil theorem for the irreducible representations of a compact, connected Lie group such as U(n).
	www.math.rutgers.edu /grad/courses/spring_2005_descriptions.html

	New preprints in 1998
		OGUISO K. - An equivariant Torelli theorem for K3 surfaces with finite group action and its application to fibered Calabi-Yau threefolds Djursholm 1997 s.
		POENARU V. - Le theoreme de la compactification etrange processus infinis et la conjecture de Poincare en dimension trois VI, Partie C Bures-sur-Yvette 1997 s.
		ZORICH V.A. - Quasiconformal immersions of Riemannian manifolds and a Picard-type theorem Bonn 1998 s.
	www.impan.gov.pl /LIB/pr98.html

	Transactions of the American Mathematical Society
		N. Lakic, An isometry theorem for quadraric differentials on Riemann surfaces of finite genus, Trans.
		M. Knopp, A corona theorem for automorphic forms and related results, Amer.
		H. Royden, Automorphisms and isometries of Teichmüller space, Advances in the Theory of Riemann Surfaces (L. Ahlfors et al., eds.), Ann.
	www.ams.org /tran/1996-348-03/S0002-9947-96-01490-0/home.html

	Content Frame for the Finding Aid to the Papers of Norbert Wiener, 1898-1966
		NW attended the International Congress of Mathematicians in Oslo, Norway, and lectured on Tauberian Gap Theorems.
		NW collaborated with Harry Ray Pitt at M.I.T. NW gave the Dohme lecture at Johns Hopkins on Tauberian Theorems.
		NW lectured on analysis at the Semi-centennial of the A.M.S. NW served as chief consultant in the field of mechanical and electrical aids to computation for the National Defence Research Committee.
	www.aip.org /history/ead/mit_wiener/19990053_content.html

	Douady, Hubbard: On the dynamics of polynomial-like mappings
		BERS, The Riemann Mappings Theorem for Variable Metrics (Annals of Math., Vol.
	www.numdam.org /numdam-bin/item?id=ASENS_1985_4_18_2_287_0

	CP-bib.bib
		J.}, VOLUME = {43}, YEAR = {1991}, PAGES = {27--36}) %CP @ARTICLE(BSt90, AUTHOR = {Alan F. Beardon and Kenneth Stephenson}, TITLE = {The uniformization theorem for circle packings}, JOURNAL = {Indiana Univ. Math.
		Math}, VOLUME = {117}, PAGES = {653--667}, YEAR = {1995}) %CP @ARTICLE(HS95c, AUTHOR = {Zheng-Xu He and Oded Schramm}, TITLE = {Inverse {Riemann} mapping theorem for relative circle domains}, JOURNAL = {Pac.
		G\'eom.}, YEAR = {1996--97}, PAGES = {153--161}, VOLUME = {15}) %CP @ARTICLE(tD97, AUTHOR = {Tomasz Dubejko}, TITLE = {Recurrent random walks, {Liouville's} theorem, and circle packings}, JOURNAL = {Math.
	www.math.utk.edu /~kens/CP-bib.bib

	Pick_1bis.doc
		Some applications of Pick’s theorem (In preparation) Generalizations of Pick’s theorem There are various generalizations of the Pick’s theorem, to more general polygons [3], [4], [8], to higher-dimensional polyhedra [5], [6], [7] and to lattices other than square lattices [1], [2].
		The original form of the Pick’s theorem, that was first published in 1899 [2], is stated for simple polygons P, i.e.
		H. Steinhaus,- Mathematical Snapshots, Oxford University Press, 1969 A proof of Pick’s theorem First we prove the theorem for rectangles R with edges parallel to the lattice, i.
	web.unife.it /progetti/geometria/divulg/Pick_1bis.doc

	Pick's Theorem: An Interactive Activity
		With Pick's theorem one may determine area of a (polygonal) portion of a map.
		Pick's Theorem on three dimensional regular rectangular solids
		The area measurement application of Pick's theorem I mentioned above comes from the real world experience.
	www.cut-the-knot.org /ctk/Pick.shtml (758 words)

	Archimedes' Stomachion (Pick's Theorem)
		Using Pick's theorem the areas of the fourteen pieces can be determined as in the above example; e.g., the blue piece in the upper right-hand corner has area
		proof of Pick's Theorem as presented in the above paper by Dale E. Varberg can be found on a
		ecent proofs and extensions of Pick's theorem can be found in
	www.mcs.drexel.edu /~crorres/Archimedes/Stomachion/Pick.html (154 words)

	Archimedes' Stomachion (Pick's Theorem)
		Using Pick's theorem the areas of the fourteen pieces can be determined as in the above example; e.g., the blue piece in the upper right-hand corner has area
		proof of Pick's Theorem as presented in the above paper by Dale E. Varberg can be found on a
		ecent proofs and extensions of Pick's theorem can be found in
	www.mcs.drexel.edu /~crorres/Archimedes/Stomachion/Pick.html (154 words)

	Pick
		Pick's theorem Therefore, if the theorem is true for polygons constructed from n triangles, the theorem is also true for...
		Pick City, North Dakota Pick City is a city located in 2000 census, the city had a total population of 166.
		Pick A pick, a tool used for manual labour, consists of a hard spike attached perpendicular to a handle.
	www.brainyencyclopedia.com /topics/pick.html (257 words)

	Pick_1bis.doc
		Some applications of Pick’s theorem (In preparation) Generalizations of Pick’s theorem There are various generalizations of the Pick’s theorem, to more general polygons [3], [4], [8], to higher-dimensional polyhedra [5], [6], [7] and to lattices other than square lattices [1], [2].
		H. Steinhaus,- Mathematical Snapshots, Oxford University Press, 1969 A proof of Pick’s theorem First we prove the theorem for rectangles R with edges parallel to the lattice, i.
		Many proofs of this delightful theorem are known, but we are inviting the reader to give a proof by himself, and there are also various generalizations.
	web.unife.it /progetti/geometria/divulg/Pick_1bis.doc (257 words)

	TopCoder Features
		Pick's theorem is useful when we need to find the number of lattice points inside a large polygon.
		In order to show that Pick's theorem holds for all lattice polygons we have to prove it in 4 separate parts.
		The above formula is called Pick's Theorem due to Georg Alexander Pick (1859- 1943).
	www.topcoder.com /index?t=features&c=feat_010505 (257 words)

	Archimedes' Stomachion (Pick's Theorem)
		Using Pick's theorem the areas of the fourteen pieces can be determined as in the above example; e.g., the blue piece in the upper right-hand corner has area
		proof of Pick's Theorem as presented in the above paper by Dale E. Varberg can be found on a
		ecent proofs and extensions of Pick's theorem can be found in
	www.mcs.drexel.edu /~crorres/Archimedes/Stomachion/Pick.html (257 words)

	PlanetMath: proof of Pick's theorem
		Again by additivity, it suffices to prove Pick's theorem for rectangles and rectangular triangles which have no lattice points on the hypotenuse and whose other two sides are parallel to the coordinate axes.
		This is version 2 of proof of Pick's theorem, born on 2002-11-19, modified 2002-11-19.
		"proof of Pick's theorem" is owned by giri.
	planetmath.org /encyclopedia/ProofOfPicksTheorem.html (257 words)

	picks_thm2
		It's known as Pick's Theorem (G. Pick, 1900; Jbuch 31, 215).
		MR 56 #2854 Gaskell, R. W.; Klamkin, M. S.; Watson, P. Triangulations and Pick's theorem.
		However, we can represent them if we are willing to use additive and subtractive core triangles - the proof that the Theorem remains valid under that extension is left as an exercise for the reader.
	www.math.niu.edu /~rusin/known-math/98/picks_thm2 (257 words)

	Belmont High School
		This theorem is named for Georg Alexander Pick, who was born in Vienna, Austria, in 1859 and died about 1943 in the Nazi concentration camp at Theresienstadt.
		For the three figures labeled A, B, and C(in your handout)determine their areas using Pick's Theorem, and verify their areas by dividing up the figures and using traditional formulas for the area of the smaller figures.
		Determine its area by using Pick's Theorem, and then verify the area by using traditional area formulas.
	myschoolonline.com /page/0,1871,0-32619-36-6904,00.html (257 words)

	Pick's Theorem: An Interactive Activity
		The area measurement application of Pick's theorem I mentioned above comes from the real world experience.
		With Pick's theorem one may determine area of a (polygonal) portion of a map.
		Polygons covered by the theorem have their vertices located at nodes of a square grid or lattice whose nodes are spaced at distance 1 from their immediate neighbors.
	www.cut-the-knot.org /ctk/Pick.shtml (758 words)

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In the News (Fri 25 Dec 09)