A double torus knot is a closed curve drawn on the surface called a double torus think of the surface of two doughnut s stuck together . More technically, a double torus knot is the homeomorphic image of a circle in S which can be realized as a subset of a genus mathematics genus two handlebody in S . If a link knot theory link is a subset of a genus two handlebody, it is a double torus link . ref Dale Rolfsen, Knots and Links , Publish or Perish, Inc., 1976, ISBN 0 914098 16 0 ref The simplest example of a double torus knot that is not a torus knot is the figure eight knot mathematics figure eight knot . While torus knot s and links are well understood and completely classified, there are many open questions about double torus knots. Two different notations exist for describing double torus knots . The T I notation is given in Rick Norwood F. Norwood , Curves on Surfaces ref Topology and its Applications 33 1989 241 246. ref and a different notation is given in P. Hill, On double torus knots I . ref Journal of Knot Theory and its Ramifications, 1999. ref The big problem, solved in the case of the torus, still open in the case of the double torus, is when do two different notations describe the same knot? References div class references small references div Category Knot theory Category Algebraic topology knottheory stub ... more details
Torus based cryptography involves using algebraic torus algebraic tori to construct a group mathematics group for use in cipher s based on the discrete logarithm problem . This idea was first introduced by Alice Silverberg and Karl Rubin in 2003 in the form of a public key algorithm by the name of CEILIDH . It improves on conventional cryptosystems by representing some elements of large finite fields compactly and therefore transmitting fewer bits. See also Portal Cryptography References Karl Rubin, Alice Silverberg Torus Based Cryptography. CRYPTO 2003 349&ndash 365 External links http www.math.uci.edu asilverb bibliography ceilidh.pdf Torus Based Cryptography &mdash the paper introducing the concept in PDF . Category Public key cryptography Crypto stub ... more details
Deleted image removed File Danis Torus removal.jpg 300px thumb right wikt bilateral Bilateral torus mandibularis mandibular tori UR . The gingiva soft tissue is reflected after anesthetizing the patient to reveal the bony protuberances UL . These protuberances are removed with dental drills and hand instruments LL and the soft tissue is Surgical suture sutured back into place LR . Torus removal surgery is a surgery surgical procedure performed to remove one or more extra protuberances of bone either on the Torus palatinus palate or the Torus mandibularis mandible . Although such segments of extra bone are not harmful in any way in and of themselves, their presence may present a problem for those patients who require certain types of dental prosthesis dental prostheses , such as denture complete or removable partial denture partial dentures . ref name neville21 Neville, BW Damm, D Allen, C Bouquot, J. Oral & Maxillofacial Pathology . 2nd Ed. Philadelphia Saunders, 2002. ISBN 0 7216 9003 3 pages 20 22. ref References Reflist Category Oral surgery ... more details
In mathematics , especially in the area of mathematical analysis known as dynamical systems theory , a linear flow on the torus is a flow mathematics flow on the n dimensional torus math mathbb T n underbrace S 1 times S 1 times cdots times S 1 n math which is represented by the following differential equations with respect to the standard angular coordinates sub 1 sub , sub 2 sub , ..., sub n sub math frac d theta 1 dt omega 1, quad frac d theta 2 dt omega 2, quad cdots, quad frac d theta n dt omega n. math The solution of these equations can explicitly be expressed as math Phi omega t theta 1, theta 2, dots, theta n theta 1 omega 1 t, theta 2 omega 2 t, dots, theta n omega n t mod 2 pi. math If we respesent the torus as R sup n sup Z sup n sup we see that a starting point is moved by the flow in the direction sub 1 sub , sub 2 sub , ..., sub n sub at constant speed and when it reaches the border of the unitary n cube it jumps to the opposite face of the cube. For a linear flow on the torus either all orbits are periodic orbit periodic or all orbits are dense set dense on a subset of the n torus which is a k torus. When the components of are rational dependence rationally independent all the orbits are dense on the whole space. This can be easily seen in the two dimensional case if the two components of are rationally independent then the Poincare section of the flow on an edge of the unit square is an irrational rotation on a circle and therefore its orbits are dense on the circle, as a consequence the orbits of the flow must be dense on the torus. See also Completely integrable system Ergodic theory Quasiperiodic motion Bibliography cite book author Anatole Katok and Boris Hasselblatt title Introduction to the modern theory of dynamical systems publisher Cambridge year 1996 isbn 0 521 57557 5 Category Dynamical systems Category Ergodic theory mathanalysis stub ... more details
on the torus will not be a closed curve, and the restriction of math pi math on this line is injective ..., called the irrational winding of a torus, is dense subspace dense in the torus. Applications Irrational winding of a torus is used to set up a few counter examples related to monomorphism ... regular submanifold of the torus, which shows that the image of a manifold under a continuous ..., the torus can be considered as a Lie group math U 1 times U 1 math , and the line can be considered ... in the torus while, of course, it is still a group. It is also used to show that if a subgroup ... space . See also Torus knot Notes cnote a As a topological Subspace topology subspace of the torus, the irrational winding is not a manifold at all, because it is not locally homeomorphic ... more details
image 2 2 4 4 de Bruijn torus.svg right thumb A De Bruijn torus. Each 2 by 2 binary matrix can be found within it exactly once. In combinatorics combinatorial mathematics, a De Bruijn torus , named after Nicolaas Govert de Bruijn , is an matrix mathematics array of symbols from an alphabet often just 0 and 1 that contains every m by n matrix mathematics matrix exactly once. It is a torus because the edges are considered wraparound for the purpose of finding matrices. Its name comes from the De Bruijn sequence , which can be considered a special case where n is 1 one dimension . One of the main open questions regarding De Bruijn tori is whether a De Bruijn torus for a particular alphabet size can be constructed for a given m and n . It is known that these always exist when n 1, since then we simply get the De Bruijn sequences, which always exist. It is also known that square tori exist whenever m n . ref cite journal title Research problems on Gray codes and universal cycles author1 Jackson first1 Brad last2 Stevens first2 Brett last3 Hurlbert first3 Glenn journal Discrete Mathematics volume 309 issue 17 date Sept. 2009 pages 5341 5348 doi 10.1016 j.disc.2009.04.002 ref References reflist Category Combinatorics ... more details
Image Loewner63.jpg right thumb 200px Charles Loewner in 1963 In differential geometry , Loewner s torus inequality is an inequality mathematics inequality due to Charles Loewner . It relates the Systolic geometry systole and the area of an arbitrary Riemannian metric on the 2 torus . Statement Image TorusSystoleLoop.png right thumb 200px Shortest loop on a torus In 1949 Charles Loewner proved that every metric on the 2 torus math mathbb T 2 math satisfies the optimal inequality math operatorname sys 2 leq frac 2 sqrt 3 operatorname area mathbb T 2 , math where sys is its Systolic geometry systole , i.e. least length of a noncontractible loop. The constant appearing on the right hand side is the Hermite constant math gamma 2 math in dimension 2, so that Loewner s torus inequality can be rewritten as math operatorname sys 2 leq gamma 2 operatorname area mathbb T 2 . math The inequality was first mentioned in the literature in harvtxt Pu 1952 . Case of equality The boundary case of equality is attained if and only if the metric is flat and homothetic to the so called equilateral torus , i.e. torus whose group of deck transformations is precisely the hexagonal lattice spanned by the cube roots of unity in math mathbb C math . Alternative formulation Given a doubly periodic metric on math mathbb R 2 math e.g. an imbedding in math mathbb R 3 math which is invariant by a math mathbb Z 2 math isometric action , there is a nonzero element math g in mathbb Z 2 math and a point math p in mathbb ... of Loewner s torus inequality Loewner s torus inequality can be proved most easily by using the computational ... to the probability measure defined by the measure of the unit area flat torus in the conformal class of the given torus. For the random variable X , one takes the conformal factor of the given ... produces the following version of Loewner s torus inequality with isosystolic defect math mathrm ... Mikhail Katz first3 Mikhail G. last3 Katz year 2009 title Loewner s torus inequality with isosystolic ... more details
The Clifton Pohl torus is an example of a compact space compact Lorentzian manifold that is not geodesically complete . While every compact Riemannian manifold is also geodesically complete, this property does not generalize to pseudo Riemannian manifolds. Definition Consider the manifold math mathrm M mathbb R 2 0 math with the metric math g frac dx otimes dy dy otimes dx x 2 y 2 math Multiplication by any real number is an isometry Riemannian geometry isometry of math M math , in particular the map math lambda x,y 2x,2y math Let math Gamma math be the subgroup of the isometry group generated by math lambda math . Then math Gamma math has a proper, discontinuous action on math M math . Hence the quotient math T M Gamma math , which is topologically the torus , is a Lorentz surface. Geodesic incompleteness It can be verified that the curve math sigma t frac 1 1 t ,0 math is a geodesic of M that is not complete since it is not defined at math t 1 math . Consequently, math M math hence also math T math is geodesically incomplete, despite the fact that math T math is compact. Similarly, the curve math sigma t tan t, 1 math is a null mathematics null geodesic that is incomplete. In fact, every null geodesic on math M math or math T math is incomplete. differential geometry stub Category Lorentzian manifolds ... more details
Use dmy dates date April 2011 Refimprove date March 2011 About the Joint European Torus the Japanese government initiative JET Programme other uses Jet disambiguation Fusion devices name JET image JointEuropeanTorus external.jpg imagetitle JET in 1991 type Tokamak operation 1983 major radius 2.96 m minor radius 1.25 2.10 m volume 100 m sup 3 sup field 3.45 tesla unit T toroidal heating 38 megawatt MW JET , the Joint European Torus , is a magnetic confinement plasma physics experiment located in Oxfordshire, UK. It is currently the largest facility of its kind in operation. Its main purpose is to open the way to future nuclear fusion experimental tokamak reactors such as ITER and DEMO . Construction The reactor is situated on a retired Royal Navy Navy airfield near Culham , Oxfordshire RNAS Culham HMS Hornbill , in the UK the construction of the buildings which house the project was undertaken by Carillion Tarmac Construction , ref Berry Ritchie, The Story of Tarmac Page 100, Published by James & James Publishers Ltd, 1999 ref starting in 1978 with the Torus Hall being completed in January 1982. Construction of the experiment itself began immediately after the completion of the Torus Hall, with the first experiments beginning in 1983. The components for the JET experiment came from manufacturers all over Europe, with these components transported to the site. Because of the extremely high power requirements for the tokamak , and the fact that power draw from the main grid is limited ... and control capabilities were improved. In total, over 86,000 components were changed in the torus ... r133.html IAEA s information about JET http skippy.org.uk jet torus hall Photos from JET torus hall ... European Torus es Joint European Torus fr Joint European Torus it Joint European Torus nl Joint European Torus pl Joint European Torus ru Joint European Torus sv Joint European Torus tr Birle ik Avrupa Torusu uk Joint European Torus ... more details
cite web url http www.pppl.gov nationalsphericaltorus.cfm title National Spherical Torus Experiment ... The National Spherical Torus Experiment NSTX official site http www.pppl.gov nationalsphericaltorus.cfm Overview of The National Spherical Torus Experiment from PPPL fusion experiments Category Fusion ... Spherical Torus Experiment ... more details
Refimprove date July 2007 The Columbia Non neutral Torus CNT is a small stellarator at the Columbia University Plasma Physics Laboratory designed by Thomas Sunn Pedersen with the aid of Wayne Reiersen and Fred Dahlgren of the Princeton Plasma Physics Laboratory to conduct the first investigation of non neutral plasmas confined on magnetic surfaces. The experiment, which began operation in November 2004, is funded by the National Science Foundation and the United States Department of Energy in the form of a Faculty Early Career Development CAREER award. ref name Lyda cite web url http www.columbia.edu cu news 05 07 story.html title Columbia s Stellarator Project Gets Jolt from National Science Foundation. author Alex Lyda work Columbia News date 29 July 2005 quote For his efforts, Pedersen has been recognized by the National Science Foundation s Faculty Early Career Development CAREER Program for ground breaking work in plasma physics. CAREER offers the National Science Foundation s most prestigious award to the early career development activities of teacher scholars who most effectively integrate research and education within the context of the mission of their organization. The award translates to United States dollar 800,000 over five years and is expected to begin in September. ref Technical Design CNT, which is housed in a cylindrical vacuum chamber made of 316 stainless steel , measures 60 inches in diameter and stands 75 inches tall. The empty chamber is capable ... plasmas in the Columbia Non neutral Torus Experiment author Thomas Sunn Pedersen et al. archiveurl http ... The Status of the Design and Construction of the Columbia Non neutral Torus author Jason P. Kremer ... pdfs FST2006.pdf title Construction and Initial Operation of the Columbia Nonneutral Torus work Fusion ... Non neutral Torus work Physics of Plasmas author Thomas Sunn Pedersen et al. Dead link ... Plasma physics Category Stellarators de Columbia Non Neutral Torus ... more details
Infobox album See Wikipedia WikiProject Albums Name Torus Shock Trio Sessions Type studio Artist Conelrad band Conelrad Released September 30, 2004 U.S. br Genre Metal Recorded Pittsburgh, Pennsylvania Pittsburgh Label Lost Tundra U.S. Length 30 00 Producer Conelrad band Conelrad A limited edition collaboration between Conelrad band Conelrad and Steve Moore. Released on September 30, 2004 by Lost Tundra Recordings. Currently out of print the release featured hand screened hand numbered covers made by poster artist Mike Budai. The tracks are instrumental improvisations and were recorded live with no edits or overdubs. ref http www.losttundra.com www.losttundra.com Bot generated title ref Track listing Steve s Chronic Pica small 9 43 small Just Sitting On A Wall, Watching The World Go Bye small 3 25 small Shiny Chromaly Human Skeleton small 7 51 small Obeah Man Vs. Dolemite small 8 50 small Personnel Jeff Gretz Drums Adam MacGregor Guitar Steve Moore Alto Sax References references Category 2004 albums Category Conelrad albums 2000s rock album stub ... more details
in the plasma. Because the magnetic field inside the torus is bent into a circle, the fast ions are hoped ... the torus near the back of the machine. When the injector is pulsed, 20,000 volts accelerates the beam .... It has been shown both theoretically and by experiments in the Madison Symmetric Torus that driving ... through sputtering. In the Madison Symmetric Torus MST , properties of the impurity ions e.g. ... meters per second. See also Nuclear fusion Fusion Tokamak Toroid Torus References references External links http plasma.physics.wisc.edu mst Madison Symmetric Torus fusion experiments University of Wisconsin ... more details
wiktionary toroidal Toroidal describes something which resembles or relates to a torus or toroid Torus Toroid , a doughnut shaped object which resembles a torus Toroidal inductors and transformers , a type of electrical device Toroidal and poloidal , directions in magnetohydrodynamics Toroidal polyhedron Toroidal ring model Toroidal coordinates , a three dimensional orthogonal coordinate system Toroidal engine , an internal combustion engine with pistons that rotate within a toroidal space Toroidal graph , a graph whose vertices can be placed on a torus such that no edges cross Toroidal reflector , a parabolic reflector which has a different focal distance depending on the angle of the mirror Grid network , known as a toroidal network when an n dimensional grid network is connected circularly in more than one dimension See also Atoroidal Torus disambiguation Toroid disambiguation Disambig ... more details
Image Double torus illustration.png right thumb A genus 2 surface. In mathematics, a genus 2 surface also known as a double torus or two holed torus is a surface formed by the connected sum of two torus tori . That is to say, from each of two tori the interior of a disk is removed, and the boundaries of the two disks are identified glued together , forming a double torus. This is the simplest case of the connected sum of n tori. A connected sum of tori is an example of a two dimensional manifold mathematics manifold . According to the surface classification theorem for 2 manifolds, every compact space compact connected space connected 2 manifold is either a sphere, a connected sum of tori, or a connected sum of real projective plane s. Double torus knot s are studied in knot theory . Example The Bolza surface is the most symmetric hyperbolic surface of Genus mathematics genus 2. See also Triple torus References James R. Munkres, Topology, Second Edition , Prentice Hall, 2000, ISBN 0 13 181629 2. William S. Massey, Algebraic Topology An Introduction , Harbrace, 1967. External links MathWorld title Double Torus urlname DoubleTorus Category Topology eo Duopa toro ... more details
NSTX may mean National Spherical Torus Experiment NSTX software Internet protocol IP over domain name system DNS tunneling protocol tunneling software disambig ... more details
Villarceau may refer to Yvon Villarceau , a 19th century French astronomer, mathematician and engineer Villarceau circles , a pair of circles found in an obliquely cut torus, which are named after him Disambiguation ... more details
Incoming links date March 2012 Mandibular means related to the mandible lower jaw bone . Terms containing mandibular include Mandibular nerve Mandibular prominence Mandibular fossa Torus mandibularis Disambiguation ... more details
Fusion devices name Pegasus Toroidal Experiment type Spherical torus Tokamak major radius 45 cm minor radius 40 cm location Madison, Wisconsin , United States of American United States The Pegasus Toroidal Experiment is a Plasma physics plasma confinement experiment relevant to fusion power production, run by the Department of Engineering Physics of the University of Wisconsin Madison . It is a spherical tokamak , a very low aspect ratio version of the tokamak configuration, i.e. the minor radius of the torus is comparable to the major radius. See also Other spherical torus experiments National Spherical Torus Experiment NSTX Mega Ampere Spherical Tokamak MAST Small Tight Aspect Ratio Tokamak START External links http pegasus.ep.wisc.edu Pegasus Toroidal Experiment physics stub fusion experiments University of Wisconsin Madison Category Nuclear fusion Category Fusion reactors Category Fusion power Category University of Wisconsin Madison ... more details
A buccal exostosis is the formation of an exostosis bone mass on the outer, cheek facing side of the maxilla just above the teeth or the cheek facing side of the human mandible mandible . Formation on the Human mandible lower jaw occurs much less commonly than on the upper jaw . They are painless, but may contribute to periodontal disease if they become too large. They can be removed with surgery. Buccal exostoses have no malignancy malignant potential. Related conditions Surfer s ear Exostosis of the ear canal Hereditary Multiple Exostoses HME Subungal exostosis Torus mandibularis Torus palatinus References reflist External links http www.maxillofacialcenter.com BondBook bone exostosis.html Thomas Bond s Book Palatal & Mandibular Torus, & Exostosis Category Oral pathology ... more details
In mathematics , an affine group is said to be diagonalizable if it is group isomorphism isomorphic to a subgroup of D sub n sub , the group of diagonal matrices. A diagonalizable group defined over k is said to split over k or k split if the isomorphism is defined over k . This coincides with the usual notion of splitting lemma split for an algebraic group. Every diagonalizable group splits over k sub s sub . ref The separable closure of k . ref Any closed subgroup and image of diagonalizable groups are diagonalizable. The torsion subgroup of a diagonalizable group is dense. The category of diagonalizable groups defined over k is equivalent to the category of finitely generated abelian group with math Gamma math ref math Gamma operatorname Gal k s k math ref equivariant morphisms without p torsion. This is an analog of Poincar duality and motivated the terminology. A diagonalizable k group is said to be anisotropic if it has no nontrivial k valued character. The so called rigidity theorem on an abelian variety rigidity states that the identity component of the centralizer of a diagonalizable group coincides with the identity component of the normalizer of the group. The fact plays a crucial role in the structure theory of solvable groups. A connected diagonalizable group is called an algebraic torus which is not necessarily compact, in contrast to a complex torus . A k torus is a torus defined over k . The centralizer of a maximal torus is called a Cartan subgroup . Notes reflist References Borel, A. Linear algebraic groups , 2nd ed. Category Algebraic groups Abstract algebra stub ... more details
File Toric lens surface 2.png thumb right Toric lens surface as cap top right from a torus here with R 1.2 r . A toric lens is a lens optics lens with two different optical power powers in two orientations perpendicular to each other. One of the lens surfaces is shaped like a cap from a torus see figure at right , while the other one usually is Sphere spherical . Toric lenses are primarily used in Glasses eyeglass es, contact lens es and intraocular lens es, to correct for Astigmatism eye astigmatism . Torus File Superficie t rica.svg thumb right upright A torus results when a circle with radius r rotates around an axis lying in the same plane as the circle here the z axis at a distance R from the centre of the circle. A torus is the spatial body resulting when a circle with radius r rotates around an axis lying within the same plane as the circle, at a distance R from the circle s centre see figure at right . If R r , a ring torus is produced. If R r , a horn torus is produced, where the opening is contracted into a single point. R r results in a spindle torus , where only two dips remain from the opening these dips become less deep as R approaches 0. When R 0, the torus degenerates into a sphere with radius r . File Les trois types de tores.PNG thumb left When the major radius R approaches 0 here from right to left , the torus becomes a sphere. left The toric lens The greatest radius of curvature of the toric lens surface, nowrap R r , corresponds to the smallest Optical power refractive power , nowrap 1 S n 1 R r , where n is the index of refraction of the lens material. The smallest radius of curvature, r , corresponds to the greatest refractive power, nowrap 1 s n 1 r . Since nowrap 1 R r r , nowrap S s . The lens ... optics Light ray s within the x , y plane of the torus as defined in the figure above are Refraction ... power, nowrap S n 1 R r . Light rays within a plane through the axis of revolution the z axis of the torus ... more details
wiktionary toroid Toroid may refer to Toroid geometry , a doughnut like solid whose surface is a torus . Toroidal inductors and transformers which have wire windings on circular ring shaped magnetic cores. Vortex ring , a toroidal flow in fluid mechanics. See also Toroidal disambiguation Atoroidal disambig sk Toroid ... more details
Unreferenced date December 2009 In mathematics , a surface bundle is a fiber bundle bundle in which the fiber is a surface . When the base space is a circle the total space is 3 manifold three dimensional and is often called a surface bundle over the circle . See also Mapping torus DEFAULTSORT Surface Bundle Category Geometric topology es Surface Bundle ... more details
In mathematics , the Reeb foliation is a particular foliation of the 3 sphere , introduced by the French mathematician Georges Reeb 1920 1992 . It is based on dividing the sphere into two solid tori , along a 2 torus see Clifford torus . Each of the solid tori is then foliated internally, in codimension 1, and the dividing torus surface forms one more leaf. Illustrations File Reebfoliation ring 2d 2.svg 300px thumb 2 dimensional section of Reeb foliation File Reeb foliation half torus POV Ray.png 350px thumb 3 dimensional model of Reeb foliation References cite book author G. Reeb title Sur certaines propri t s toplogiques des vari t s feuill t es series Actualit s Sci. Indust. volume 1183 publisher Hermann location Paris year 1952 cite book title Foliations author Alberto Candel coauthors Lawrence Conlon publisher American Mathematical Society year 2000 isbn 0821808095 page 93 cite book title Introduction to Foliations and Lie Groupoids author Ieke Moerdijk coauthors J. Mrcun series Cambridge studies in advanced mathematics volume 91 publisher Cambridge University Press year 2003 isbn 0 521 83197 0 page 8 External links MathWorld urlname ReebFoliation title Reeb Foliation topology stub Category Foliations es Foliaci n de Reeb ru ... more details
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