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Topic: Upper half plane

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Hyperbolic geometry

	Model Plane HQ - Half Plane Model
		half plane model is a conformal model attributed to Henri Poincaré.
		The Upper-half plane model of hyperbolic space Hyperbolic space is a fascinating placefull of surprises.
		It is a proper subset of the Euclidean plane.
	www.modelplane-hq.com /halfplanemodel (785 words)

	The Upper Half Plane (Site not responding. Last check: 2007-10-08)
		The upper half complex plane is defined by Hh := {z in C
		Thus we define Hh^ * to be the upper half plane union the cusps.
		For x an element of the upper half place, returns the value of x as an element of the complex field.
	www.umich.edu /~gpcc/scs/magma/text461.htm (340 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)

	Upper half plane (Site not responding. Last check: 2007-10-08)
		In geometry, the Half-plane model of hyperbolic geometry is described by hyperbolic motions.
		In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0.
		It is also a model of the hyperbolic plane.
	www.xasa.com /wiki/en/wikipedia/u/up/upper_half_plane.html (109 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).
		In the left-half plane we have decaying (stable) exponential envelopes, while in the right-half plane we have growing (unstable) exponential envelopes.
		plane, the upper-half plane corresponds to positive frequencies while the lower-half plane corresponds to negative frequencies.
	www-ccrma.stanford.edu /~jos/mdft/Comparing_Analog_Digital_Complex.html (400 words)

	PlanetMath: Poincaré upper half plane model
		The Poincaré upper half plane model has the drawback that lines in the model do not necessarily resemble Euclidean lines; however, it has the advantage that it is angle preserving.
		That is, the Euclidean measure of an angle within the model is the angle measure in hyperbolic geometry.
		This is version 8 of Poincaré upper half plane model, born on 2007-05-21, modified 2007-06-03.
	planetmath.org /PoincareUpperHalfPlaneModel.html (250 words)

	How to make the Tao Glider Paper Airplane: Online plane designs.
		Fold the upper half of the paper airplane in half again...
		Its tempting to just slap the fold together, but the final weight here needs to be balanced or the plane will fly crooked as a politician later in life.
		This way when the rocket tanks fail (don't forget we were drinking back a few steps ago) we are safely in the control booth away from the whole issue.
	www.personaltao.com /tao/paperairplane.htm (599 words)

	Body
		Note that in the construction of a hyperbolic plane is dependent on the r (the radius of the annuli) which is often called the radius of the hyperbolic plane.
		The upper half plane model is a convenient way to study the hyperbolic plane -- think of it as a map of the hyperbolic plane in the same way that we use planar maps of the spherical surface of the earth.
		(0) to vectors in the tangent plane at z(p).
	www.math.cornell.edu /~dwh/books/eg99/Ch05/Ch05.html (4021 words)

	Upper Half-plane (1) (Site not responding. Last check: 2007-10-08)
		It does not show us a panoramic view of a hyperbolic plane, but it is quite useful for making simple calculations, to see the relationship with horocycles or Pseudo-spheres, and especially to think about infinity.
		In the upper half-plane corresponding points on a semicircle are not symmetrical unless the semicircle itself is symmetry with respect to Y-axis.
		And it is the upper half-plane that is deformed from the disk to another open plane.
	www1.kcn.ne.jp /~iittoo/us21a_uppe.htm (1853 words)

	AMERICAN MATHEMATICAL MONTHLY - February 2001
		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.
		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.
	www.maa.org /pubs/monthly_feb02_toc.html (697 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).
		In the left-half plane we have decaying (stable) exponential envelopes, while in the right-half plane we have growing (unstable) exponential envelopes.
		plane, the upper-half plane corresponds to positive frequencies while the lower-half plane corresponds to negative frequencies.
	www.technick.net /public/code/cp_dpage.php?aiocp_dp=guide_dft_comparing_analog_digital (447 words)

	The Upper Half Plane (Site not responding. Last check: 2007-10-08)
		The upper half complex plane is defined by Hh := {z in C
		Thus we define Hh^ * to be the upper half plane union the cusps.
		For x an element of the upper half place, returns the value of x as an element of the complex field.
	magma.maths.usyd.edu.au /magma/htmlhelp/text519.htm (340 words)

	H(+) plane and Euclidean plane (Site not responding. Last check: 2007-10-08)
		The hyperbolic length in the upper half-plane and the length in the Euclidean plane are the identical.
		The way of transforming a triangle is the same as we usually do with a pair of compasses in the Euclidean plane, except that the center of a circle in the upper half-plane shits up.
		In the hyperbolic plane, one end of the plane is at the intersection of the red and green hyperbolic parallel lines and the other end is perpendicular to the yellow base line.
	www1.kcn.ne.jp /~iittoo/us31_tool.htm (1017 words)

	Cabinet Magazine Online - Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina
		A hyperbolic plane is a surface in which the space curves away from itself at every point.
		The hyperbolic plane is sometimes described as a surface in which the space expands.
		DH: The discovery of the hyperbolic plane came from the attempt to prove Euclid's fifth postulate, which is also known as the parallel postulate.
	www.cabinetmagazine.org /issues/16/crocheting.php (2537 words)

	Erlangen program Summary
		For example, in the plane, a rotation (say, a quarter-turn) about a fixed point is a transformation, since it maps the plane onto itself, and no two different points get rotated to the same point.
		On the other hand, a map that sends the plane onto the upper-half plane by folding it along the x-axis like a sheet of paper is not a transformation, since it does not map the plane onto the entire plane, and two different points can get mapped to the same point.
		The transformations that preserve Euclidean space are the rotations, translations (maps that shift the plane in some fixed direction, by some fixed amount), reflections across a line, and glide reflections (maps that consist of a reflection across a line, followed by a shift in the direction of the line).
	www.bookrags.com /Erlangen_program (2093 words)

	Upper half-plane - (Site not responding. Last check: 2007-10-08)
		In mathematics, the upper half-plane H is the set of complex numbers
		Other names are hyperbolic plane, Poincaré plane and Lobachevsky plane, particularly in texts by Russian authors.
		The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative sectional curvature.
	www.gurgaongrid.com /mediawiki/index.php/Upper_half-plane (264 words)

	Elliptic Curves and Modular Functions
		For instance, the set of all rotations of the plane about the origin is a group, and the orbit of any particular point in the plane is a circle whose radius is the distance of the point from the origin.
		It is an algebraic object that encodes geometric information about the half plane H. Note that while H admits a symmetry of translation by a real number, it does not admit a translation by any non-real number.
		Modular functions are, then, the automorphic functions on the upper half plane under the action of the modular group.
	www.mbay.net /~cgd/flt/flt05.htm (2994 words)

	The Mandelbrot Set as a Modular Form
		Modular forms are a particular kind of function on the complex upper half-plane studied in analytic number theory and the theory of elliptic curves.
		This mapping is curious because it is not infrequent in the literature, and because a periodic function on the upper half-plane takes the appearance of a self-similar function on the disk.
		This map is a conformal map that takes points in the upper half-plane to points in the interior of a unit disk.
	linas.org /math/dedekind/dedekind.html (3929 words)

	The Complex Plane
		If we consider each point on the plane as having an x and a y value, both being real numbers, then we can locate any point in the plane by plotting the x value along the horizontal axis and the y value along the vertical axis.
		In the complex plane, the same region is defined by Re Likewise, the upper half-plane is denoted as Im We can also define a vertical strip by, for example, 0 < Re This would be in the right half-plane between 0 and 2 and extending infinitely up and down.
		Now that we've defined what we mean by the complex plane, there are a few basic ideas that you must know to help you in your study of the Riemann Hypothesis.
	www.math.unl.edu /~jcliber/sectionIV.html (884 words)

	Reference.com/Encyclopedia/Hyperfunction
		If f is holomorphic on the whole complex plane, then it should have the same boundary values when considered as a function on either the upper or lower half-plane.
		be the upper half-plane and lower half-plane respectively.
		If f is any holomorphic function on the whole complex plane, then the restriction of f to the real axis is a hyperfunction, represented by either (f, 0) or (0, f).
	www.reference.com /browse/wiki/Hyperfunction (535 words)

	NonEuclid: 10: Disk and Upper Half-Plane Models
		In the disk model, a line is defined as an arc of a circle that is orthogonal to the unit circle.
		In the Upper Half-Plane model, a line is defined as a semicircle with center on the x-axis.
		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.
	www.cs.unm.edu /~joel/NonEuclid/model.html (1232 words)

	page two of trailers
		There are cut-outs in the floor of the trailer for cabame/interplane strut attach tangs and aileron bellcrank- push rod.
		The stab/elevator is captured between the padded boards which are screwed into saddles on the side of the box.
		At the U.S. Air Force Museum's bi-annual WWI fly-in we assembled the planes in the restoration facility's hangar.
	www.kcdawnpatrol.org /trailers_2.htm (496 words)

	Hint: Consider y = the least upper bound of...
		Prove of cube root using least upper bound - How do I define the upper bound, the particular coefficient and epsilon to solve the cube function in order to obtain my goal of proofing the cube root.
		Upper support was measured using a rating fr...
		integral - Using the maxima and minima of the function, produce upper and lower estimates of the integral.
	www.brainmass.com /homework-help/math/other/24114 (185 words)

	Under
		sle kappa=2.0 This shows a conformal map of the half plane onto the half plane with a fractal curve removed.
		sle kappa=6.0 This shows a conformal map of the half plane onto the half plane with yet another fractal curve removed.
		tilted This illustrates the conformal map which takes the upper half plane onto the upper half plane with a tilted slit removed.
	math.arizona.edu /~tgk/under (165 words)

	Mutual relations among models (Site not responding. Last check: 2007-10-08)
		The semicircle on the upper half-plane is drawn large by the amount of translatation from point O to point O'.
		The upper half-plane we see is the reverse side, and the positive direction of its X-axis is leftward.
		It agrees with the half-line on the upper half-plane.
	www1.kcn.ne.jp /~iittoo/us40_mode.htm (1924 words)

	18.013A Calculus with Applications, Fall 2001, Online Textbook
		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
		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.
		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)

	Poincaré half-plane model - Education - Information - Educational Resources - Encyclopedia - Music
		In non-Euclidean geometry, the Poincaré model model of the hyperbolic plane regards it as a homogeneous space for the group of Möbius transformations.
		In the half plane model, the geodesics are either circular arcs or straight lines, orthogonal to the boundary (in both cases).
		It is also possible to see quite plainly the negative curvature of the hyperbolic plane, via its effect on the sum of angles in triangles and squares.
	www.music.us /education/P/Poincar%e9-half-plane-model.htm (460 words)

	Reply to Popular Mechanics re 9/11
		This is particularly worrisome since the first plane struck one side of the north tower, causing (you would think) a weakening on that side where the exterior columns were struck, and a more intense fire on that side than on the other side.
		The explanation furnished by the FBI for the mystery plane, whose existence it initially denied, serves less to reassure than to reinforce suspicions that a cover-up of sorts is under way, that the government is manipulating the truth in a manner it considers to be palatable to the broader US public.
		The fl boxes from the planes which allegedly hit the Twin Towers conveniently disappeared completely, in contrast to the passport of Mohammed Atta, which (according to the official 9/11 myth) miraculously survived the first impact to waft down and be discovered two blocks away by someone unknown and turned into the FBI.
	www.serendipity.li /wot/pop_mech/reply_to_popular_mechanics.htm (16917 words)

	math lessons - Upper half plane
		In mathematics, the upper half plane H is the set of complex numbers
		The lower half plane, defined by y < 0, is equally good, but less used by convention.
		The open unit disk D is equivalent by a conformal mapping, meaning that it is usually possible to pass between H and D.
	www.mathdaily.com /lessons/Upper_half_plane (182 words)

	Math Forum Discussions - self maps of the upper half plane
		upper half plane is mapped into a disk entirely contained in the upper
		Re: self maps of the upper half plane
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /kb/thread.jspa?forumID=253&threadID=1310606&messageID=4135115 (131 words)

	Scattering Theory
		This case is very reminiscent of one-dimensional scattering, in which a plane wave from the left generates outgoing waves in both directions, and the amplitudes can be calculated from the Schrödinger equation for a single energy eigenstate.
		goes to zero in the upper half k¢ plane, but diverges in the lower half, so we must close the contour in the upper half plane to ensure no contribution from the semicircle at infinity.
		E is along the real axis, and the contour is closed in the half plane where the integrand goes to zero for in the imaginary direction, that is, in the lower half plane for
	galileo.phys.virginia.edu /classes/752.mf1i.spring03/ScatteringTheory.htm (3304 words)

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