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Topic: Upper half-plane

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Upper half-plane - Wikipedia, the free encyclopedia The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative sectional curvature. In this terminology, the upper half-plane is H In mathematics, the upper half-plane H is the set of complex numbers en.wikipedia.org /wiki/Upper_half_plane   (273 words)

Half Half crown The half-crown was a denomination of 1971. Half iterate The half iterate of a function f (denoted by f The half iterate is a special case of fractional iteration o... Half Moon, North Carolina Half Moon is a town located in 2000 census, the town had a total population of 6,645. www.brainyencyclopedia.com /topics/half.html   (1219 words)

Body Show that the hyperbolic geodesics in the upper half plane are transformed by this inversion into circular arcs (or line segments) perpendicular to the boundary of the disk. In particular, consider the plane determined by X and the North and South Poles, the plane tangent to the sphere at X, and the planes tangent to the sphere at the North and South Poles. Note that only the lower open hemisphere is projected onto the plane; that is, if X is a point in the lower open hemisphere, then its gnomic projection is the point, g(X), where the ray from the center through X intersects the plane. www.math.cornell.edu /~dwh/books/eg99/Ch16/Ch16.html   (1497 words)

Hyperbolic Geometry Seminar After studying isometries of the upper half plane we were ready to deal with covering spaces and show that the hyperbolic plane is the universal cover of closed orientable 2 manifolds with positive genus. Triangles on the upper half plane are delta thin. This included lectures on the geometry of the hyperbolic plane. www.math.columbia.edu /~pinkham/teaching/seminars/hyperbolic_sem.html   (210 words)

MATH3401 Complex Analysis  transforms the upper half of the z-plane onto the interior of a square in the w-plane. maps the upper half of the z-plane to an n-sided polygon in the w-plane. A transformation that maps the upper half of the z-plane onto the unit circle in the www.maths.uq.edu.au /courses/MATH3401/MATH3401_WorkedSolns7.htm   (460 words)

Elliptic and Modular Functions Given a lattice L = [a, b] in the complex plane, this function returns the value of the elliptic j-invariant of L. This is the j-invariant of tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane. Given a point t in the upper half plane and a positive integer p, return the normalized q-series expansion of the discriminant Delta(q) evaluated at t to precision p. Let z be a point in the upper half-plane and let L be a lattice in C. The Eisenstein series are defined as the coefficients of the Laurent Series expansion of the Weierstrass wp-function: wp(z, L) = ((1)/(z^2)) + sum_(2 <= k) G_k(L)(2k - 1)z^(2k - 2) where G_k(L) are the Eisenstein series. www.math.wisc.edu /help/magma/text467.html   (1285 words)

Body This is the usual upper half plane model of the hyperbolic plane thought of as a map of the hyperbolic plane in the same way that we use planar maps of the spherical surface of the earth. Thus, we have established that the annular hyperbolic plane is the same as the usual upper half plane model of the hyperbolic plane. In the upper half plane model an ideal triangle is a triangle with all three vertices either on the x-axis or at infinity. www.math.cornell.edu /~dwh/papers/crochet/crochet.html   (3801 words)

PlanetMath: upper half plane This is version 4 of upper half plane, born on 2003-05-12, modified 2003-09-09. The upper half plane in the complex plane, abbreviated UHP, is defined as See Also: unit disk, complex, unit disk upper half plane conformal equivalence theorem planetmath.org /encyclopedia/UpperHalfPlane.html   (57 words)

Siegel upper half-plane - Wikipedia, the free encyclopedia In mathematics, a Siegel upper half-plane is the set of n×n symmetric matrices over the complex number field whose imaginary part is positive definite. For example, when n = 1, the Siegel upper half-plane is the upper half-plane. There is a group action of the symplectic group en.wikipedia.org /wiki/Siegel_upper_half_plane   (81 words)

Schwarz-Christoffel Maps One quarter of the surface is conformally the upper half plane. The Schwarz-Christoffel formulas gives an integral expression which maps the upper half plane conformally onto a polygonal shaped domain. If we map this upper half plane not into euclidean space but instead to a polygonal domain, using first G dh as a Schwarz-Christoffel integrand, we get the following zigzag shaped domain: www.indiana.edu /~minimal/essays/christoffel   (212 words)

5.html near the real axis viewed from the upper half-plane. The right graphic is a contour plot of the scaled absolute value of the imaginary part, meaning the height values of the left graphic translate into color values in the right graphic. The right graphic is a contour plot of the scaled absolute value of the real part, meaning the height values of the left graphic translate into color values in the right graphic. functions.wolfram.com /ElementaryFunctions/Log/visualizations/5.html   (954 words)

Comparing Analog and Digital Complex Planes The upper-half plane corresponds to positive frequencies (counterclockwise circular or corkscrew motion) while the lower-half plane corresponds to negative frequencies (clockwise motion). plane, the upper-half plane corresponds to positive frequencies while the lower-half plane corresponds to negative frequencies. In the left-half plane we have decaying (stable) exponential envelopes, while in the right-half plane we have growing (unstable) exponential envelopes. ccrma-www.stanford.edu /~jos/mdft/Comparing_Analog_Digital_Complex.html   (388 words)

18.013A Calculus with Applications, Fall 2001, Online Textbook where C is the semicircle of radius R in the upper half plane, and again the integral on C goes to zero as R increases. Thus this function has a singularity at i in the upper half plane and a singularity at -i in the lower half plane, with residues and the second terms in the numerator and denominator will dominate in the upper half plane making the integrand approach -i and the first terms will dominate in the lower half plane so that it approaches i as y ocw.mit.edu /ans7870/18/18.013a/textbook/chapter27/section02.html   (627 words)

X3 Mathematical Methods: Conformal Mappings This region should be extendend to infinity in all directions in the upper half plane. This region should be extendend to infinity in the vertical direction to give a vertical strip in the upper half plane with x between -1 and 1. This region should be extendend to infinity in the horizontal direction to give a horizontal strip in the right half plane with y between 0 and 2. www.staff.city.ac.uk /~ra933/X3MathsMethods/Conformal.html   (149 words)

AMERICAN MATHEMATICAL MONTHLY - February 2001 The eigenfunctions for finite upper half planes have contours that roughly show the same short of chaos as those for arithmetical quantum chaos, at least for Maass wave forms for the modular group of 2x2 integer matrices of determinant one. C.L. Chai and W.-C. Li have shown that the histograms of the spectra of the finite upper half plane graphs approach the Wigner semi-circle distribution as the number of vertices of the graph approaches infinity. The histograms for differences of adjacent eigenvalues for finite upper half plane graphs appear to approach the Poisson density, as do the level spacings for arithmetical quantum chaos. www.maa.org /pubs/monthly_feb02_toc.html   (708 words)

PlanetMath: UHP (in upper half plane) owned by brianbirgen unit disk upper half plane conformal equivalence theorem owned by brianbirgen upper and lower bounds to binomial coefficient owned by rspuzio planetmath.org /encyclopedia/U   (716 words)

Upper Half-Plane Model The points are Euclidean points, (x, y), from half of the plane without the border (the border is the x axis). Notice that the strict equality ensures that the x axis is NOT included. No comment about x ensures x is any real number. www.uh.edu /~hollyer/Module7/m7ppt/sld044.htm   (44 words)

Upper half plane -- Facts, Info, and Encyclopedia article In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, the upper half plane H is the set of (A number of the form a+bi where a and b are real numbers and i is the square root of -1) complex numbers The multi-dimensional analog of the upper half-plane is the (Click link for more info and facts about Siegel upper half-space) Siegel upper half-space. The set is called the Siegel upper half-space of (Click link for more info and facts about genus) genus n. www.absoluteastronomy.com /encyclopedia/u/up/upper_half_plane.htm   (338 words)

The Upper Half Plane Thus we define Hh^ * to be the upper half plane union the cusps. The upper half complex plane is defined by Hh := {z in C For x an element of the upper half plane, if x is a cusp, returns the value of x as an object of type www.umich.edu /~gpcc/scs/magma/text461.htm   (340 words)

Complex Analysis Furthermore, the upper semicircular portion of the boundary is mapped onto the line negative u-axis, and the segment Furthermore, the upper semicircle of the disk is mapped onto the line is a conformal mapping at each point z in the complex plane. math.fullerton.edu /mathews/c2002/ca0903.html   (177 words)

Disk and Upper Half-Plane Models of Hyperbolic Geomtry In the Upper Half-Plane model, a line is defined as a semicircle with center on the x-axis. Disk and Upper Half-Plane Models of Hyperbolic Geometry We assume, without loss of generality, that the radius of C is 1, and that its center is at the origin of the Euclidean plane. cs.unm.edu /~joel/NonEuclid/model.html   (1249 words)

The Poincaré Disk and the Upper Half Plane The result is the Upper Half Plane model of hyperbolic 2-space. Another involves stereographically mapping the Poincaré disc directly onto half of a unit sphere, turning the sphere on its side, and then stereographically projecting onto the plane z=0. The geodesics are the intersections of the hemisphere with vertical planes. www.geom.uiuc.edu /~crobles/hyperbolic/hypr/ibm/puhp   (332 words)

Interactive Java Applet Simulation of the Upper Half Plane The blue point is free -- you can move it about with the mouse to observe the behavior of lines in the Upper Half Plane. Interactive Java Applet Simulation of the Upper Half Plane There are two points in the model below which determine a unique line. www.geom.uiuc.edu /~crobles/hyperbolic/hypr/modl/uhp/uhpjava.html   (47 words)

Hyperbolic Geometry Although we first present the upper half-plane model and prove most of the fundamental facts there, we will generally after that use the unit disc. We will describe two models, the upper half-plane model, which we denote by U and the unit disc model, which we initially denote by D. Since we know that angular excess corresponds to negative curvature, we see that the hyperbolic plane is a negatively curved space. geom.math.uiuc.edu /docs/education/institute91/handouts/node37.html   (760 words)

Helena A. Verrill A Fundamental domain for a congruence subgroup is a region in the upper half plane which tells you about the quotient of the upper half plane by that group. If you take all the translates of a fundamental domain under the action of its congruence subgroup you get a tessellation of the upper half plane, similar to the pictures Escher drew on the disk. They are very important in number theory because of the associated modular forms. web.usna.navy.mil /~wdj/colloq/talk01_10.htm   (195 words)

Integrals from to The trick is to complete the integral along the real axis by a very large semicircle in the upper or lower half-plane (either works in this case), such that the integral along this semicircle vanishes. In this case we cannot immediately complete the contour by a semicircle in the upper half-plane, because We then do the integral on the right by completing the contour with the semicircle in the upper half-plane: www.astro.cf.ac.uk /undergrad/module/PX3211/com/node12.html   (265 words)

On the \Gamma-equivariant form of the Berezin's quantization of the upper half plane (ResearchIndex) On the \Gamma-equivariant form of the Berezin's quantization of the upper half plane Abstract: this paper we consider the \Gammaequivariant form of the Berezin's quantization of the upper half plane which will correspond to a deformation quantization of the (singular) space =\Gamma. On the \Gamma-equivariant form of the Berezin's quantization of the upper half plane (ResearchIndex) citeseer.ist.psu.edu /602518.html   (629 words)

Fundamental domains and other pictures in the upper half plane, with Magma In the third picture, a tiling of the upper half plane is created by translating these 6 triangles by elements of Gamma(2). The second shows a tiling of the upper half plane using images of this domain, and random colouring. To complete the modular curves defined by quotienting by congruence subgroups, the cusps (rationals and infinity) are adjoined to the upper half plane. www.math.lsu.edu /~verrill/fundomain/magmaFD.html   (794 words)

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