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Topic: Fundamental region

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Modular group

	Fundamental domain - Wikipedia, the free encyclopedia
		In geometry, the fundamental domain of a symmetry group of an object or pattern is a part of the pattern, as small as possible, which, based on the symmetry, determines the whole object or pattern.
		In practice the main use of a fundamental domain may be to compute integrals on G/Γ, in which case the set of measure zero is mentioned only to keep straight the pedantic assertion that D is exactly a set of coset representatives, and may quickly be forgotten.
		Each triangular region is a free regular set of H/Γ; the grey one (with the third point of the triangle at infinity) is the canonical fundamental domain.
	en.wikipedia.org /wiki/Fundamental_domain (883 words)

	The Seventeen Wallpaper Groups
		The fundamental region of this group is also the translation region, hence the unit lengths of the translation vectors correspond to the lengths of two intersecting edges of the fundamental region.
		The length of the glide vector is equal to the length of one of the edges parallel to the mirror of reflection.
		The fundamental region is an isosceles right triangle with a 180 rotation center at the 90 vertex and 90 degree rotation centers at the other two vertices.
	jwilson.coe.uga.edu /EMT668/EMAT6680.F99/McCallum/WALLPA~1/SEVENT~1.HTM (3603 words)

	Wallpaper Groups: the 17 plane symmetry groups
		A fundamental region for the symmetry group is half of a parallelogram that is a fundamental region for the translation group.
		A fundamental region for the translation group is a rectangle, and one can be chosen that is split by an axis of reflection so that one of the half rectangles forms a fundamental region for the symmetry group.
		The lattice is square, and an eighth, a triangle, of a fundamental region for the translation group is a fundamental region for the symmetry group.
	www.clarku.edu /~djoyce/wallpaper/seventeen.html (970 words)

	Chapter 2.6
		A fundamental region of the symmetry group p6m is a right-angled triangle defined by the sides and altitude of an equilateral triangle that corresponds to a regular tessellation {3,6}.
		Usually, a fundamental region of the symmetry group p4g is an isosceles right-angled triangle defined by the centers of adjacent sides and by the corresponding vertex of a square of the regular tessellation {4,4}.
		A fundamental region of the symmetry group p4m is an isosceles right-angled triangle defined by the center of a side, its belonging vertex, and the center of a square that corresponds to the regular tessellation {4,4}.
	www.emis.de /monographs/jablan/chap26.htm (12081 words)

	Time For Making Historic Decisions In The Middle East
		A fundamental truth about the current situation is that it came about through a combination of the passivity and complacency of the Arab Middle East towards solving its regional problems on the one hand, and the aggressiveness and activism of Israel and the US neoconservative ideologues on the other.
		At the regional level, the challenge is to come up with a positive vision that rebuilds the region economically and rehabilitates it politically, not destroys it through war.
		The fundamental fact remains that while the last Gulf war was fought in defense of the status quo in the region, the next will be fought to destroy it.
	www.mafhoum.com /press4/127P57.htm (5414 words)

	Trellis shaping for modulation systems (US5150381)
		The region occupied by the available sequences in sequence space is shaped to minimize the required power, e.g., the region may consist of a fundamental region of a trellis code, and more particularly, an approximation to the Voronoi region of a trellis code (where the trellis code is not just a repeated lattice code).
		For example, the fundamental region comprises the set of possible signal point sequences that are decoded to the zero sequence in the trellis code by an approximation to a minimum distance decoder for the code having delay M, M greater than or equal to 1.
		The fundamental region comprises the set of possible signal point sequences that are decoded to a common error region by a fair, exhaustive decoder for the trellis code.
	www.delphion.com /details?&pn10=US05150381 (2115 words)

	Triply Periodic Minimal Surfaces (Site not responding. Last check: 2007-10-08)
		The fundamental regions usually are one of Coxeter's kaleidoscopic cells.
		The fundamental region is a tetrahedron which is 1/48 of a cube.
		The fundamental region is a tetrahedron which is 1/96 of a cube.
	susqu.edu /facstaff/b/brakke/evolver/examples/periodic/periodic.html (1166 words)

	Specifications of QDBM Version 1
		Because the region pointed to by `dptr' is allocated with the `malloc' call, it should be released with the `free' call if it is no longer in use.
		`ptr' specifies the pointer to the region to be appended.
		Because the region of the return value is allocated with the `malloc' call, it should be released with the `free' call if it is no longer in use.
	qdbm.sourceforge.net /spex.html (16812 words)

	Generators and fundamental regions
		Fundamental regions in Kaleidomania are chosen to be as simple as possible.
		For example, a fundamental region for the stip pattern above is the yellow strip (thought of as extending to infinity) illustrated below.
		This is the fundamental region used by Kaleidomania for this type of strip pattern.
	www.math.uga.edu /~clint/2005/5210/gens.htm (508 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The fundamental region contains a basic "unit" of the pattern - sometimes called "motif." It is the replication of this motif that produces the repeating hyperbolic pattern.
		A fundamental (polygonal) region can be specified given: (1) the number of vertices, and (2) the valences or "foldedness" of each vertex.
		The next step, after the fundamental region has been specified, is to replicate the region throughout the circle model to produce the pattern.
	www.d.umn.edu /~mohs0025/urop/proposal.txt (906 words)

	Chapter 2.4
		By using different elementary asymmetric figures belonging to the fundamental region and different directed forms with acute angles oriented toward the direction of the axis, it is possible to intensify the already existing impression of motion.
		A fundamental region of the symmetry group of friezes m1 or mm is rectilinear, with boundaries incident withthe reflection lines.
		Since all friezes with the polar axis may have a curvilinear fundamental region, their visual dynamism may be emphasized by choosing an acuteangular fundamental region with the acute angle oriented toward the direction of the axis or by choosing an acuteangular elementary asymmetric figure belonging to the fundamental region and directed in the same way.
	www.zblmath.fiz-karlsruhe.de /exx/monographs/jablan/chap24.htm (3852 words)

	Chapter 4.2
		Hence, a varying of the form of a fundamental region is restricted to a change in the shape of boundaries that do not belong to the reflection lines or inversion circles.
		The form of a fundamental region of conformal symmetry groups is defined by the invariance of all the points of inversion circles and reflection lines.
		In conformal symmetry groups that do not require the constancy of the form of a fundamental region, a variety of corresponding conformal symmetry rosettes in ornamental art is achieved by varying the boundaries of a fundamental region or the form of an elementary asymmetric figure belonging to a fundamental region.
	www.emis.de /monographs/jablan/chap42.htm (7801 words)

	Patterns
		Asymmetry is both the absence of symmetry, and a fundamental basis for symmetry.
		Symmetry analysis may result in the identification of a fundamental region that is the smallest element required to explain the repetition that forms a pattern.
		The fundamental region, repeated to form a pattern, consists of a motif or design, or portion thereof, and the surrounding area that defines the unit of repeat.
	www.msu.edu /~checkle2/redesign/asymm.htm (154 words)

	[No title]
		Perhaps more importantly, the community of investors in the region now appears to be better networked (both internally and externally), more mature, and more aware of and attuned to the region's fundamental sources of innovation.
		The region's technology clusters are spread throughout: the Ventura Freeway Corridor, Santa Monica and the West Side, Orange County, Pasadena and the San Gabriel Valley, and the South Bay, to name just some of the major clusters.
		In addition, while the region's economic diversity is a strength that insulates it during times of market volatility, the relative lack of resident core competencies often prevents it from being considered seriously by investors outside the region.
	www.larta.org /LAVox/ArticleLinks/2002/020311_sandollar.asp (1471 words)

	J-invariant - Wikipedia, the free encyclopedia
		, there is a τ in the fundamental region such that c=j(τ).
		Thus, j has the property of mapping the fundamental region to the entire complex plane, and vice-versa.
		As a Riemann surface, the fundamental region has genus 0, and every (level one) modular function is a rational function in j; and, conversely, every rational function in j is a modular function.
	en.wikipedia.org /wiki/J-invariant (844 words)

	Euclidean Manifolds (Site not responding. Last check: 2007-10-08)
		To obtain an outsider's view of the manifold, copies of the fundamental region are translated via straight lines until they tile the plane and then edges that are adjacent are glued together.
		This is constructed using a rectangular block as the fundamental region.
		In this illustration the objects inside the fundamental region are a cube with a light and dark side, and a paper airplane representing the camera.
	www.cs.brown.edu /~tor/vr/doc/node7.html (851 words)

	Time For Making Historic Decisions In The Middle East
		The objective of forcing a regime change in Iraq, and the downgrading of the Gulf region’s strategic role in global energy markets are important cornerstones of this vision.
		Today the region is experiencing the cumulative effect of delayed and incomplete responses to a long series of what were “present” dangers in the course of the last few decades.
		As the past two decades have demonstrated, leaving any part of the region as a festering sore, whether Lebanon, Algeria or Afghanistan, leaves the entire region open to revolution, terrorism, economic disruption and external interference.
	www.mees.com /postedarticles/oped/a46n01d01.htm (5473 words)

	tile caption (Site not responding. Last check: 2007-10-08)
		It is a fundamental domain of the square lattice, in the sense that every point of the plane has a representative within the red region, where point and representative are related by a unique lattice translation.
		One of the geometrically simplest fundamental domains is the Voronoi domain, defined to be the set of points of the plane for which the origin is the closest lattice point.
		The analog for the triangular lattice, a twelve-fold symmetric fundamental region, also exists and was discovered in the context of quasiperiodic tilings by my former student Eric Cockayne.
	people.ccmr.cornell.edu /~veit/tile.html (269 words)

	Chapter 3.2
		A fundamental region of the similarity symmetry group K is a part of the plane, bounded by two homologous lines of the dilatation K.
		A fundamental region of the symmetry group L is a part of the plane defined by two homologous lines of the transformation L.
		A fundamental region of the symmetry group M is a part of the plane defined by two homologous lines of the dilative reflection M.
	www.matem.unam.mx /EMIS/monographs/jablan/chap32.htm (8768 words)

	User Interface (Site not responding. Last check: 2007-10-08)
		Geomview does this by calculating at the end of each draw cycle where in the fundamental region the camera is. It then moves the camera back to the fundamental region at the origin.
		The airplane also appears in all the other copies of the fundamental region, so the user can see how the space is glued together.
		The lets the user see the shape of fundamental region that manifold is based upon and the location of gluing boundaries.
	www.cs.brown.edu /~tor/vr/doc/node14.html (958 words)

	My UROP Proposal (Site not responding. Last check: 2007-10-08)
		Even though replications of the same region may not appear to be the same Euclidean size, they are the same hyperbolic size.
		The main idea is to traverse the graphs formed by the edges and vertices of the “tilting” of the hyperbolic place by copies of the fundamental region.
		I proposed to use the tasks of fundamental region specification and replication, where the program will also support some basic operations that are useful in adding sub-patterns to the fundamental region.
	www.d.umn.edu /~mohs0025/urop/urop_proposal.html (800 words)

	project
		Look at each fundamental region shown to the left of the pattern and perform the appropriate isometries on it.
		In fact, each fundamental region, no matter how fanciful the design, is really a simple geometric shape (such as a parallelogram or triangle).
		For instance, the fundamental region of the pattern at the top and bottom of this page is actually one eighth of each square (a triangle).
	www.stanford.edu /dept/sme/jnmm/project.html (710 words)

	UNIS/Press Release
		At the regional level, I believe that developing countries could enhance the effectiveness of their preparatory work through closer cooperation among each other.
		However, the region's openness and rapid growth that continue to serve as growth engines, are key stimuli for higher quality and closer economic cooperation.
		I hope that this gathering today will facilitate a free exchange of views, on issues that are highly complex and often difficult to agree on, so that when all is said and done, we will have contributed in a small but constructive way to a strengthened multilateral trading system.
	www.unescap.org /unis/press/2002/jul/l_19_02.htm (1371 words)

	PBS Cyber School: Tessel Your Way to Nagano! (Site not responding. Last check: 2007-10-08)
		Students create their own fundamental regions by cutting out a shape from one side of the Post-It Note and taping it to the side directly opposite it.
		Students now have a fundamental region that should tessellate (if they have not overlapped their pieces of Post-It Note, or left tape sticking over the outside of the region, or inadvertently cut off a piece of the paper).
		For example: a protrusion on the fundamental region might remind one student of a nose while another student might say it is the wheel on a car, or still another student might volunteer that the shape looks like a frog and that the part sticking out is one of the eyes.
	www.pbs.org /ocs/teach/lesson30_2.html (1069 words)

	Nat' Academies Press, Free Electron Lasers and Other Advanced Sources of Light: Scientific Research Opportunities (1994)
		The fingerprint region containing fundamental molecular vibrations is in the infrared portion of the spectrum, and excitation of valence electrons in chemical bonds requires light of ultraviolet wavelengths.
		These solid-state devices, which produce tunable fundamental radiation in the range of 700 nm to 1100 nm, are excellent for mode-locking to create the short pulses that are efficient for producing other wavelengths by nonlinear processes.
		In the near-infrared, visible, and ultraviolet regions, support for additional free electron lasers should receive relatively low priority with respect to support of conventional lasers and with respect to support for free electron lasers in other wavelength regions.
	www.nap.edu /openbook/NI000099/html/32.html (2232 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		> >By fundamental region I mean any region that under transformation by >the members of the group just covers the space without overlapping and >without interstices.
		I need to describe the >bounds of the fundamental region in Eulerian or Polar angles or something >similar...
		Then you need only look at the region of points which are closer to the first point than to any of its translates.
	www.math.niu.edu /~rusin/known-math/97/fund.region (289 words)

	Extended Glossary
		Each triangular shaped region is the image of a single triangular shaped fundamental region that is acted on, in succession, by each modular group element.
		Since the right-angled vertex of each region is the fixed point of an order two element, the order two action can be transferred, by conjugation, to any other order two action.
		Full clusters (sometimes referred to as flowers) of fundamental regions have their common point fixed at a rational number.
	www.xula.edu /math/faculty/McCreary/modularGarden/extendedGlossary.htm (1586 words)

	Andrea, Stephen Alfred (1964-04-06) On the embedding of homeomorphisms on the plane in flows. ...
		As a consequence, it is shown that if T is embedded in a flow then [...] is a proper subset of the plane for any compact set A. The author suspects that this property might be shared by all homeomorphisms in the general class.
		It is found that for an arbitrary T there exists a natural partition of the plane into a collection of "fundamental regions", with the property that the restriction of T to any fundamental region must be embedded in a flow within that region whenever T over the whole plane is embedded in a flow.
		Finally, it is shown by an example that even if the restrictions of T to its fundamental regions are all equivalent to translations, it might still be impossible to create a flow for T over all of [...].
	etd.caltech.edu /etd/available/etd-09062002-104631 (346 words)

	18.02 at ESG - Fundamental Theorems (Site not responding. Last check: 2007-10-08)
		Here, the region of integration is bounded by the two points a and b, and the function F(x), an antiderivative of f(x), is evaluated on these boundary points.
		The region of integration has been generalized to any path in the plane (the extension to three or more independent variables is straightforward, but the figures are not).
		The integrand over the region R seems to be some sort of derivative of the integrand along the contour that bounds the region.
	web.mit.edu /18.02-esg/www/fths/main.html (565 words)

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