Distinguish Taurus disambiguation About the surface and mathematical concept of a torus Merge from Standard torus date October 2009 Image Torus.png right thumb 250px A torus In geometry , a torus pl. tori ... in this case the surface has a ring shape and is called a ring torus or simply torus if the ring shape is implicit. Other types of torus include the horn torus, which is generated when the axis is tangent to the circle, and the spindle torus, which is generated when the axis is a chord of the circle ... generates the surface of a sphere . The ring torus bounds a solid known as a toroid . The adjective ... tube s, many lifebuoy s, O ring s and vortex ring s. In topology , a ring torus is Homeomorphism ... torus is one way to embed this space into Euclidean space, but another way to do this is the Cartesian ... the Clifford torus , surface in 4 space . The word torus comes from the Latin word meaning cushion . ref cite book url http books.google.com ?id GGoNQzt3I IC&pg PA16&lpg PA16&dq torus Latin word for a cushion ... ref Geometry Image torus cycles.png thumb right A torus is the product of two circles, in this case ... phi math direction, represented by the blue arrow. A torus can be defined parametric equation parametric ally by ref http www.geom.uiuc.edu zoo toptype torus standard eqns.html ref math x u, v R r ... 0, 2 , R or A is the distance from the center of the tube to the center of the torus .... An implicit function implicit equation in Cartesian coordinates for a torus radially symmetric ... The surface area and interior volume of this torus are easily computed using Pappus s centroid theorem giving ref Mathworld TorusTorus ref math A 4 pi 2 R r left 2 pi r right left 2 pi R right ... on the inner side of the tube exactly cancel out the gains on the outer side. As a torus is the product ... system, and math theta math and math phi math , angles measured from the center point. As a torus ... fusion devices. Topology Topology Topologically , a torus is a closed surface defined as the product ... more details
Torus or tori may refer to In botany Torus, a structure of the xylem a synonym for a sagittal keel , a structure found in crania In mathematics Torus , a surface Torus knot Algebraic torus Genus 2 surface Double torus Umbilic torus In medicine Torus palatinus , a bony growth on the palate Torus mandibularis , a bony growth on the mandible Torus fracture , a term used in radiology to describe an incomplete fracture of the distal radius in children where there is no obvious fracture line on any radiograph. In nuclear physics Torus nuclear physics Joint European Torus , an experimental nuclear fusion reactor In Architecture A semicircular molding see Molding decorative Types See also Toroidal disambiguation Tokamak disambig io Toro homonimo uk ... more details
Triple torus or three torus can refer to one of the two following concepts, both related to a torus . Three dimensional torus The three dimensional torus , or triple torus , is defined as the Cartesian product of three circles, math mathbb T 3 S 1 times S 1 times S 1. math In contrast, the usual torus is the Cartesian product of two circles only. The triple torus is a three dimensional compact space compact manifold with no manifold Manifold with boundary boundary . It can be obtained by gluing the three pairs of opposite faces of a cube . Torus like surface with three holes In the theory of surfaces, a triple torus refers to a smooth function smooth closed surface closed surface with three holes, or, in other words, a surface of Genus mathematics genus three. It can be obtained by attaching three Handle mathematics handle s to a sphere or by gluing taking the connected sum of three torus tori . center gallery caption Several representations of a triple torus widths 150px heights 150px perrow 3 Image Sphere with three handles.png A sphere with three handle mathematics handles Image Triple torus array.png The connected sum of three torus mathematics tori Image Triple torus illustration.png Pretzel style triple torus gallery center External links MathWorld title Triple Torus urlname TripleTorus Category Geometric topology Category Surfaces ... more details
Mergeto torus date October 2009 Image Sphere like degenerate torus.gif right frame 250px As the distance to the axis of revolution decreases, the ring torus becomes a spindle torus and then Degeneracy mathematics degenerates into a sphere. In mathematics , a standard torus is a circular torus of revolution, that is, any surface of revolution generated by rotation rotating a circle in three dimensional space about an axis coplanar with the circle. When the axis passes through the center of the circle, the torus degenerates into a sphere . This case is normally excluded from the definition. With no further restrictions on the location of the axis in this plane, there are three classes of standard tori the ring torus , where the axis is disjoint from the circle the horn torus , where the axis is tangent to the circle and the spindle torus , where the axis meets the circle in two distinct points. ref The terminology here comes from Harvnb Weisstein 2003 . Some authors use the term spindle torus for horn torus and vice versa see e.g. Harvnb Gray Abbena Salamon 2006 . The second edition of this book ... is the z axis, a standard torus can be defined Parametric equation parametrically by math ... to the Centre geometry center of the torus, and r is the radius of the tube. The three different classes ... will be the familiar ring torus. The case R r corresponds to the horn torus, which in effect is a torus with no hole . The case R r describes a self intersecting surface called a spindle torus . When used by itself, the word torus usually refers to the ring torus, but may also refer to any topological space homeomorphic to a ring torus. Johannes Kepler Kepler called the external portion of the spindle torus the apple , and defined the lemon as the interior portion of the surface. The two ... , r R math . gallery caption Bottom halves and cross sections Image Standard torus ring.png Ring torus Image Standard torus horn.png Horn torus Image Standard torus spindle.png Spindle torus gallery Notes ... more details
Image TorusKnot3D.png thumb right A 3,7 3D computer graphics 3D torus knot rendered by Apple Computer Apple Grapher . Image Eurelea.png thumb right EureleA Award showing a 2,3 torus knot. In knot theory , a torus knot is a special kind of knot mathematics knot which lies on the surface of an unknotted torus in R sup 3 sup . Similarly, a torus link is a link knot theory link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integer s p and q . The p , q torus knot winds p times around a circle inside the torus, which goes all the way around the torus, and q times around a line through the hole in the torus, which passes once through the hole, usually drawn as an axis of symmetry . If p and q are not relatively prime, then we have a torus link with more than one component. The p , q torus knot can be given by the parametrization math x r ... phi 2 math and math 0 phi 2 pi math . This lies on the surface of the torus given by math r 2 2 z 2 ... up to continuous deformation. The illustrations for the 2,3 and 3,8 torus knots can be obtained by taking math r cos q phi 4 math , and in the case of the 2,3 torus knot by furthermore subtracting ... knot left.svg thumb right the 2,3 torus knot, also known as the trefoil knot A torus knot is unknot ... torus knot, also known as the trefoil knot . Properties Image TorusKnot 3 8.png thumb right Diagram of a 3,8 torus knot. Each torus knot is prime knot prime and chirality mathematics chiral . Any p , q torus knot can be made from a braid theory closed braid with p strands. The appropriate braid word is math sigma 1 sigma 2 cdots sigma p 1 q. math The Crossing number knot theory crossing number of a torus knot is given by c min p &minus 1 q , q &minus 1 p . The knot genus genus of a torus knot is math g frac 1 2 p 1 q 1 . math The Alexander polynomial of a torus knot is math frac t pq 1 t 1 t p 1 t q 1 math The Jones polynomial of a right handed torus knot is given by math t p 1 q 1 2 frac ... more details
In mathematics , a solid torus is a topological space homeomorphic to math S 1 times D 2 math , i.e. the cartesian product of the circle with a two dimensional ball mathematics disc endowed with the product topology . The solid torus is a connected space connected , compact space compact , Orientation mathematics orientable 3 dimensional manifold with boundary. The boundary is homeomorphic to math S 1 times S 1 math , the ordinary torus . Image Torus illustration.png thumb right Solid torus A standard way to picture a solid torus is as a toroid geometry toroid , embedded in 3 space . Since the disk math D 2 math is contractible , the solid torus has the homotopy type of math S 1 math . Therefore the fundamental group and Homology mathematics homology groups are isomorphism isomorphic to those of the circle math pi 1 S 1 times D 2 cong pi 1 S 1 cong mathbb Z , math math H k S 1 times D 2 cong H k S 1 cong begin cases mathbb Z & mbox if k 0,1 0 & mbox otherwise end cases . math See also Whitehead manifold Hyperbolic Dehn surgery Category Topology topology stub eo Solida toro pl Pe ny torus ru ... more details
File Clifford torus.gif thumb right 256px In geometric topology , the Clifford torus is a special kind of torus sitting inside R sup 4 sup . Alternatively, it can be seen as a torus sitting inside C sup ... of the Clifford torus lies at a fixed distance from the origin therefore, it can also be viewed as sitting inside a 3 sphere . The Clifford torus is also known as a square torus , because it is Isometry ... torus is math S 1 times S 1 cos theta , sin theta , cos phi , sin phi , , 0 leq theta 2 pi, 0 leq ... torus is an embedded torus in R sup 2 sup × R sup 2 sup R sup 4 sup . If R sup 4 sup is given by coordinates x sub 1 sub , y sub 1 sub , x sub 2 sub , y sub 2 sub , then the Clifford torus is given ... the Clifford torus as an embedding embedded torus in C sup 2 sup . In two copies of C , we have the following ... 2 pi math and math S 1 e i phi , , 0 leq phi 2 pi . math Now the Clifford torus appears as math S 1 ... sub 2 sub , then the Clifford torus is given by math left z 1 right 2 1 left z 2 right 2 . math In the Clifford torus as defined above, the distance of any point of the Clifford torus to the origin ... points at a distance of 2 from the origin of C sup 2 sup is a 3 sphere, and so the Clifford torus sits inside this 3 sphere. In fact, the Clifford torus divides this 3 sphere into two congruent Solid torus solid tori . See Heegaard splitting . Instead of defining the Clifford torus as the product ... radius of 1 2, the Clifford torus instead sits in the unit 3 sphere S sup 3 sup . Since ... Clifford torus defined above to other equivalent tori via rigid rotations. The six dimensional group ... directions of a torus preserves the torus as opposed to moving it to a different torus ... In symplectic geometry , the Clifford torus gives an example of an embedded Lagrangian submanifold ... in C gives a Lagrangian torus of C sup 2 sup , so these need not be Clifford tori. The Hsiang Lawson ... states that any minimal surface minimally embedded torus in the 3 sphere with the Metric tensor ... more details
An oral torus is a lesion made of compact bone and occurs along the palate or the Human mandible mandible inside the mouth. The palatal torus or torus palatinus occurs along the palate, close to the midline, whereas the mandibular torus or torus mandibularis occur along the lingual side of the mandible. Occurrences of tori are more frequent in women then they are in men. Tori are associated with adulthood and rarely appear before the age of 15. The palatal version of tori have a higher occurrence in Native Americans in the United States Native American and Inuit populations. Treatment is not necessary unless they become an obstruction to chewing or prosthetic appliances. References reflist Ibsen, Olga A.C. & Joan Anderson Phelan, 2004. Oral Pathology for the Dental Hygienist. 4th edition. Philadelphia, Saunders. ISBN 0 7216 9946 4. dentistry stub musculoskeletal stub disease stub Category Oral pathology ... more details
Unreferenced stub auto yes date December 2009 Image PIA04433 Jupiter Torus Diagram.jpg thumb right Jupiter s gas toruses generated by Io moon Io green and Europa moon Europa blue A gas torus is a torus toroidal cloud of gas or Plasma physics plasma that encircles a planet. In the Solar System , gas tori tend to be produced by the interaction of a satellite s atmosphere with the magnetic field of a planet. The most famous example of this is the Io plasma torus , which is produced by the ionization of roughly 1 ton per second of oxygen and sulfur from the tenuous atmosphere of Jupiter Jupiter s volcanic moon, Io moon Io . Other examples include the largely neutral torus of oxygen and hydrogen produced by Saturn Saturn s moon, Enceladus moon Enceladus , and the proposed though not observationally supported torus of nitrogen produced by Saturn s moon, Titan moon Titan . A notable use of a gas torus in fiction is as the setting for Larry Niven s novels The Integral Trees and The Smoke Ring novel The Smoke Ring , in which a gas giant in orbit around a neutron star generates a gas torus of sufficient density to allow life including humans to survive in it. This arrangement is not particularly plausible in the real world, however. External links http www.agu.org journals ABS 2009 2009GL041030.shtml http www.agu.org journals ABS 2009 2009GL041030.shtml Magnetospherics DEFAULTSORT Gas Torus Category Astrophysics Astronomy stub ca Torus de gas zh ... more details
Image Umbilic Torus.png thumb right Umbilic Torus The umbilic torus is a single edged 3 dimensional figure created by Helaman Ferguson as a mathematical artwork. Ferguson created a 27 inch 69 centimeters bronze sculpture, Umbilic Torus , and it is his most widely known piece of art. The lone edge goes three times around the ring before returning to the starting point. A cross section geometry cross section of the surface taken from an umbilic torus corresponds with a hypocycloid . The torus is defined by the following set of parametric equations . math x sin u left 7 cos left u over 3 2v right 2 cos left u over3 v right right math math y cos u left 7 cos left u over 3 2v right 2 cos left u over 3 v right right math math z sin left u over 3 2v right 2 sin left u over 3 v right math math mbox for pi le u le pi, quad pi le v le pi. , math In 2010 it was announced that James Harris Simons Jim Simons had commissioned an Umbilic Torus sculpture to be constructed outside the Math and Physics buildings at Stony Brook University , in close proximity to the Simons Center for Geometry and Physics . The torus will be made out of cast bronze, and will be mounted on a stainless steel column. The total weight of the sculpture will be convert 65 t MT , and will have a height of convert 28 ft m . The torus will have a diameter of convert 24 ft m , the same diameter as the granite base. Various mathematical equations will be inscribed on the base. Installation should be complete by the Spring or Summer of 2011. See also Torus M bius strip Mathematics and art References Larson, Roland E., et al. Calculus . Ed. Charles Hartford. 6th ed. Boston Houghton Mifflin Company, 1998. External links http www.helasculpt.com gallery umbilictorus4inch Umbilic Torus Category Sculptures Category Mathematics and culture ... more details
torus the subgroup of all diagonal matrices . That is, math T left mathrm diag e i theta ... to the product of n circles, so the unitary group U n has rank n . A maximal torus in the special unitary group SU n U n is just the intersection of T and SU n which is a torus of dimension n &minus 1. A maximal torus in special orthogonal group SO 2 n is a given by the set of all simultaneous rotation s in n pairwise orthogonal 2 planes. This is also a maximal torus in the group SO 2 n 1 where the action ... group Sp n has rank n . A maximal torus is given by the set of all diagonal matrices whose entries ... and let math mathfrak g math be the Lie algebra of G . A maximal torus in G is a maximal abelian subgroup ... Given a maximal torus T in G , every element g G is conjugate to an element in T . Since the conjugate of a maximal torus is a maximal torus, every element of G lies in some maximal torus. All maximal ... group Given a torus T not necessarily maximal , the Weyl group of G with respect to T can be defined ... torus math T T 0 math in G then the corresponding Weyl group is called the Weyl group of G ... Maximal Torus Category Lie groups Category Representation theory of Lie groups ru ... more details
Infobox Anatomy Name Torus tubarius Latin GraySubject 230 GrayPage 1043 Image Gray915.png Caption Auditory tube, laid open by a cut in its long axis. Torus tubarius not labeled. Image2 Caption2 System MeshName MeshNumber DorlandsPre t 14 DorlandsSuf 12813973 The base of the cartilaginous portion of the Eustachian tube pharyngotympanic tube auditory tube lies directly under the mucous membrane of the nasal part of the pharynx , where it forms an elevation, the torus tubarius , the torus of the auditory tube , or cushion, behind the pharyngeal orifice of the tube. Two folds run inferiorly posteriorly, the vertical fold of mucous membrane, the salpingopharyngeal fold , stretches from the lower part of the torus it contains the Salpingopharyngeus muscle . anteriorly, the second and smaller fold, the salpingopalatine fold , stretches from the upper part of the torus to the palate it contains the levator veli palatini muscle. The tensor veli palatini is lateral to the levator and does not contribute to the fold, since the origin is deep to the cartilaginous opening. External links SUNYAnatomyLabs 31 14 01 02 LoyolaMedEd grossanatomy dissector labs h n nasal na4 1.html RocheLexicon 25420.000 1 Gray s anatomy stub Auditory system Nose anatomy ... more details
In mathematics , in the sub field of geometric topology , a torus bundle is a kind of surface bundle over the circle , which in turn are a class of three manifold s. Construction To obtain a torus bundle let math f math be an orientability orientation preserving homeomorphism of the two dimensional torus math T math to itself. Then the three manifold math M f math is obtained by taking the Cartesian product of math T math and the unit interval and gluing one component of the Boundary topology boundary of the resulting manifold to the other boundary component via the map math f math . Then math M f math is the torus bundle with monodromy math f math . Examples For example, if math f math is the identity map i.e., the map which fixes every point of the torus then the resulting torus bundle math M f math is the three torus the Cartesian product of three circle s. Seeing the possible kinds of torus bundles in more detail requires an understanding of William Thurston s Thurston s geometrization conjecture geometrization program. Briefly, if math f math is glossary of group theory finite order , then the manifold math M f math has Euclidean geometry . If math f math is a power of a Dehn twist then math M f math has Nil geometry . Finally, if math f math is an Anosov map then the resulting three manifold has Sol geometry . These three cases exactly correspond to the three possibilities for the absolute value of the trace of the action of math f math on the homology mathematics homology of the torus either less than two, equal to two, or greater than two. References Anyone seeking more information on this subject, presented in an elementary way, may consult Jeffrey Weeks mathematician Jeff Weeks book The Shape of Space . Category Fiber bundles Category Geometric topology Category 3 manifolds ... more details
In differential geometry , a Wente torus is an immersion mathematics immersed torus of constant mean curvature , discovered by harvtxt Wente 1984 . They are counterexamples to the conjecture that every closed, compact space compact , constant mean curvature surface is a sphere though this is true if the surface is embedding embedded . There are similar examples known for every positive genus mathematics genus . References Citation last1 Wente first1 Henry C. title Counterexample to a conjecture of H. Hopf. url http projecteuclid.org euclid.pjm 1102702809 year 1986 journal Pac. J. Math. volume 121 pages 193 243 http www.math.utoledo.edu wente torus.html Wente torus http www.msri.org about sgp jim geom cmc library wente index.html The Wente tori DEFAULTSORT Wente Torus Category Differential geometry ... more details
Image Stanford torus external view by Don Davis AC76 0525.jpg thumb right Exterior view of a Stanford torus. Bottom center is the non rotating primary solar mirror, which reflects sunlight onto the angled ring of secondary mirrors around the hub. Painting by Donald E. Davis. Image Stanford torus under construction.jpg thumb right External view of a Stanford torus with some of the radiation shielding ... right Interior of a Stanford torus, painted by Donald E. Davis The Stanford torus is a proposed design ... residents. ref ibid. NASA Study, pg 1, The Overall System , pg 60, Summary ref The Stanford Torus ... O Neill later proposed his Island One or Bernal sphere as an alternative to the torus. ref Gerard K. O Neil, The High Frontier , William Morrow & Co., 1977, p149 ref Stanford torus refers only to this particular ... of a torus , or Doughnut donut shaped ring, that is 1.8 km in diameter for the proposed 10,000 ..., NASA study, p46 ref Sunlight is provided to the interior of the torus by a system of mirror s. The ring .... 5 ref The interior space of the torus itself is used as living space, and is large enough that a natural environment can be simulated the torus appears similar to a long, narrow, straight glacial valley ... Torus idea The novels of the Gaea Trilogy by John Varley author John Varley are set on an unusual organic satellite of Saturn that is shaped as a Stanford Torus. In the anime series Mobile ... torus. The anime series Mobile Suit Gundam 00 also depicts Stanford tori type space stations. The Laplace Residence in Gundam Unicorn is also an example of a Stanford Torus. Hideo Kojima s PlayStation 2 video game Zone of the Enders is set aboard a Stanford torus type space station orbiting ..., the Alliance capital, however is a Stanford torus, although it is never seen in game. Saturnalia ... novels that included a Stanford torus, including The Two Faces of Tomorrow , Endgame Enigma , and Voyage From Yesteryear . FreeMarket Station , from the FreeMarket RPG is a large Stanford Torus in stationary ... more details
Infobox Disease Name Torus mandibularis Image Torus cropped.jpg Caption Left mandibular torus as visualized in an intraoral mirror DiseasesDB ICD10 ICD10 K 10 0 k 00 ICD9 ICDO OMIM Torus mandibularis pl. tori mandibular or mandibular torus pl. mandibular tori in English is a bone bony exostosis growth in the Human mandible mandible along the surface nearest to the tongue . Mandibular tori are usually present near the premolar s and above the location of the mylohyoid muscle s attachment to the mandible. ref name neville21 Neville, B.W., D. Damm, C. Allen, J. Bouquot. Oral & Maxillofacial Pathology . Second edition. 2002. Page 21. ISBN 0 7216 9003 3. ref In 90 of cases, there is a torus on both the left and right sides, making this finding an overwhelmingly Symmetry biology Bilateral symmetry bilateral condition. The prevalence of mandibular tori ranges from 5 40 . It is less common than bony growths occurring on the palate , known as torus palatinus . Mandibular tori are more common in Asian and Inuit populations, and slightly more common in male s. In the United States, the prevalence is 7 10 of the population with similar findings between blacks and whites. It is believed that mandibular tori are caused by several factors. ref name neville21 Neville, B.W., D. Damm, C. Allen, J. Bouquot. Oral & Maxillofacial Pathology . Second edition. 2002. Page 21. ISBN 0 7216 9003 3. ref They are more common in early adult life and are associated with bruxism . The size of the tori may fluctuate throughout life, and in some cases the tori can be large enough to touch each other in the midline of mouth. Consequently, it is believed that mandibular tori are the result of local stresses and not solely .... The tori may also complicate the fabrication of dentures . If removal of the tori is needed, torus ... 1360589 Consultant Photoclinic December 1, 2008, Torus mandibularis Dentofacial anomalies and jaw disease Category Oral pathology es Torus mandibularis ... more details
Refimprove date May 2009 Infobox disease Name Torus palatinus Image 06 06 06palataltori.jpg Caption An example of palatal torus. DiseasesDB ICD10 ICD10 K 10 0 k 00 ICD9 ICDO OMIM MedlinePlus eMedicineSubj eMedicineTopic MeshID Torus palatinus pl. tori palatinus palatinus torus pl. palatal tori in English is a bone bony protrusion on the palate . Palatal tori are usually present on the midline of the hard palate. ref name neville20 Neville, B.W., D. Damm, C. Allen, J. Bouquot. Oral & Maxillofacial Pathology . Second edition. 2002. Page 20. ISBN 0 7216 9003 3. ref Most palatal tori are less than 2 cm in diameter, but their size can change throughout life. Prevalence of palatal tori ranges from 9 60 and are more common than bony growths occurring on the Human mandible mandible , known as Torus Mandibularis torus mandibularis . Palatal tori are more common in Asian and Inuit populations, and twice more common in female s. In the United States, the prevalence is 20 35 of the population with similar findings between blacks and whites. Although some research suggest palatal tori to be an autosomal dominant trait, it is generally believed that palatal tori are caused by several factors. ref name neville20 They are more common in early adult life and can increase in size. In some older people, the size of the tori may decrease due to bone resorption. Consequently, it is believed that mandibular tori are the result of local stresses and not solely on Genetics genetic influences. Sometimes, the tori are categorized by their appearance. ref name neville20 Arising as a broad base and a smooth surface, flat tori are located on the midline of the palate and extend symmetrically to either side. Spindle tori have a ridge located at their midline. Nodular tori have multiple bony growths ... the fabrication of dentures . If removal of the tori is needed, Torus removal surgery surgery ... top Radiology and Pathology MedPix Teaching File DEFAULTSORT Torus Palatinus Category Oral pathology ... more details
In mathematics , an algebraic torus is a type of commutative affine algebraic group . These groups were named by analogy with the theory of tori in Lie group theory see maximal torus . The theory of tori ... but no deformations. Definition Given a base Scheme mathematics scheme S , an algebraic torus over ... particularly important case is when S is the spectrum of a field K , making a torus over S an algebraic ... the rank mathematics rank of the torus, and it is a locally constant function on S . If a torus is isomorphic to a product of multiplicative groups G sub m sub S , the torus is said to be split . All ... torus given by Weil restriction restriction of scalars over a separable extension. Restriction of scalars over an inseparable field extension will yield a commutative group scheme that is not a torus. Weights Over a separably closed field, a torus T admits two primary invariants. The weight lattice ... abelian groups whose rank is that of the torus, and they have a canonical nondegenerate pairing math ... by linear transformations on weights or coweights, and the automorphism group of a torus ... lattices of a torus over K are defined as the respective lattices over the separable closure. This induces ... Galois group of K . Given a finite separable field extension L K and a torus T over K , we ... of fundamental groupoids of the base with respect the fpqc topology. If the torus is locally ... groupoids. In particular, an etale sheaf gives rise to a quasi isotrivial torus, and if S is locally noetherian and normal more generally, Unibranch local ring geometrically unibranched , the torus is isotrivial. As a partial converse, a theorem of Grothendieck asserts that any torus of finite type is quasi isotrivial, i.e., split by an etale surjection. Given a rank n torus T over S , a twisted form is a torus over S for which there exists a fpqc covering of S for which their base extensions are isomorphic, i.e., it is a torus of the same rank. Isomorphism classes of twisted forms of a split ... more details
Image Pinched torus.jpg thumb A Pinched Torus In mathematics , and especially topology and differential geometry , a pinched torus or croissant surface is a kind of two dimensional surface . It gets its name from its resemblance to a torus that has been pinched at a single point. A pinched torus is an example of an orientable surface orientable , compact surface compact 2 dimensional pseudomanifold . ref cite journal last1 Brasselet first1 J. P. year 1996 title Intersection of Algebraic Cycles journal Journal of Mathematical Sciences publisher Springer New York volume 82 issue 5 pages 3625 3632 url http www.springerlink.com content ju28j2wqm174hx10 ref Parametrisation A pinched torus is easily parametrisable. Let us write nowrap 1 g x , y 2 sin x 2 .cos y . An example of such a parametrisation which was used to plot the picture is given by nowrap 1 0,2&pi sup small 2 small sup R sup small 3 small sup where math f x,y left g x,y cos x , g x,y sin x , sin left frac x 2 right sin y right math Topology Topologically, the pinched torus is homotopy equivalent to the Wedge sum wedge of a sphere and a circle. ref name TOP Citation first Allen last Hatcher title Algebraic Topology publisher Cambridge University Press year 2001 ISBN 0521795400 ref ref name TOP0 citeweb url http www.math.cornell.edu hatcher AT ATch0.pdf title Chapter 0 Algebraic Topology author Allen Hatcher accessdate August 6, 2010 ref It is homeomorphic to a sphere with two distinct points being quotient space identified . ref name TOP ref name TOP0 Homology Let P denote the pinched torus. The homology group s of P over the integer s can be calculated. They are given by math H 0 P, Z cong Z, H 1 P, Z cong Z, text and H 2 P, Z cong Z. math Cohomology The cohomology group s of P over the integer s can be calculated. They are given by math H 0 P, Z cong Z, H 1 P, Z cong Z, text and H 2 P, Z cong Z. math References reflist Category Surfaces ... more details
In mathematics , a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense i.e. the cartesian product of some number N circle s . Here N must be the even number 2 n , where n is the complex dimension of M . All such complex structures can be obtained as follows take a lattice group lattice in C sup n sup considered as real vector space then the quotient group C sup n sup &Lambda is a compact space compact complex manifold. All complex tori, up to isomorphism, are obtained in this way. For n 1 this is the classical period lattice construction of elliptic curve s. For n 1 Bernhard Riemann found necessary and sufficient conditions for a complex torus to be an algebraic variety those that are varieties can be embedded into complex projective space , and are the abelian varieties . The actual projective embeddings are complicated see equations defining abelian varieties when n 1, and are really coextensive with the theory of theta function s of several complex variable s with fixed modulus . There is nothing as simple as the cubic curve description for n 1. Computer algebra can handle cases for small n reasonably well. By Chow s theorem , no complex torus other than the abelian varieties can fit into projective space. References Citation last1 Birkenhake first1 Christina last2 Lange first2 Herbert title Complex tori publisher Birkh user Boston location Boston, MA series Progress in Mathematics isbn 978 0 8176 4103 0 id MathSciNet id 1713785 year 1999 volume 177 Category Complex manifolds Category Complex surfaces Category Abelian varieties ... more details
advertisement date August 2009 unreferenced date November 2008 Image Torus games logo.png thumb 130px right Torus Games company logo. Torus Games Pty. Ltd. is a video games developer founded in 1994 and is growing to be one of Australia s largest developers. The company has released over 60 titles with most being on handheld game console s. Torus Games is currently located in Bayswater, Melbourne . Torus is a family business, with the current managing director being Bill McIntosh. Their latest release is Scooby Doo And the Spooky Swamp for Wii, PS2, and DS. Torus Games began developing their first game in 1994, a Game Boy and Game Gear game based on the film Stargate , published by Acclaim Entertainment . Torus is a licensed developer for Nintendo s Wii console, Nintendo DS , Nintendo DSi , Sony PSP , PlayStation 2 , Xbox , Xbox360 , Personal computer PC , Nokia N Gage , Leapfrog Enterprises Leapster , and Game Boy Advance . Torus is currently working on a number of secret in house projects based on their own design. Prior to this, all of the games have been funded by third party publishers. Games Xbox 360 Monster Jam game Monster Jam Activision 2007 Wii Scooby Doo And the Spooky Swamp Warner Bros. Interactive 2010 Kid Adventures Sky Captain D3Publisher 2010 Scooby Doo First Frights Warner Bros. Interactive 2009 Monster Jam Urban Assault Activision 2008 Zoo Hospital Majesco 2008 Monster Jam game Monster Jam Activision 2007 Indianapolis 500 Legends Destineer 2007 PlayStation Portable Monster Jam Urban Assault Activision 2008 Shrek Smash and Crash Racing Activision 2006 PlayStation 2 Scooby Doo And the Spooky Swamp Warner Bros. Interactive 2010 Scooby Doo First Frights Warner Bros. Interactive 2009 Monster Jam Urban Assault Activision 2008 Monster Jam game Activision 2007 ... Leapster L Max Nascar 2007 External links http www.torusgames.com Official Torus Games website http ... established in 1994 Category Video game developers ca Torus Games ... more details
In mathematics , the mapping torus in topology of a homeomorphism f of some topological space X to itself is a particular geometric construction with f . Take the cartesian product of X with a closed interval I , and glue the boundary components together by the static homeomorphism math M f frac I times X 1,x sim 0,f x math The result is a fiber bundle whose base is a circle and whose fiber is the original space X . If X is a manifold , M sub f sub will be a manifold of dimension one higher, and it is said to fiber bundle fiber over the circle . Mapping tori of surface homeomorphisms play a key role in the theory of 3 manifold s and have been intensely studied. If S is a closed surface of surface Classification of closed surfaces genus g 2 and if f is a self homeomorphism of S , the mapping torus M sub f sub is a Closed manifold closed 3 manifold that fiber bundle fibers over the circle with fiber S . A deep result of William Thurston Thurston states that in this case the 3 manifold M sub f sub is a hyperbolic manifold hyperbolic if and only if f is a pseudo Anosov map pseudo Anosov homeomorphism of S ref W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces , Bulletin of the American Mathematical Society , vol. 19 1988 , pp. 417&ndash 431 ref . References reflist Category General topology Category Geometric topology Category Homeomorphisms fr Tore d application ... more details
Orphan date February 2009 BLP sources date May 2008 Torus Tammer born 1969 is an Australia n film maker . Career Tammer was born in Melbourne, Australia ref name News.com cite news url http www.news.com.au entertainment story 0,23663,19987137 5007181,00.html title Red hot and smoking date 2 August 2006 publisher News Limited accessdate 2009 10 13 ref and moved to Los Angeles in 1992 and began working odd jobs on low budget films. During this period, he met film producers Mike Erwin and Max Kirishima . Erwin and Kirishima hired Tammer to work for their production company, Den Pictures . While at Den, Tammer worked on the development of a remake of the classic cult film Easy Rider the rights to which Den Pictures owned . With the encouragement of Erwin and Kirishima, Tammer began developing two pet projects Golgo 13 and Preacher . Both projects were adaptations of comic books Golgo 13 AKA The Professional by the legendary Japanese comic book artist Takao Saito and Preacher which at the time, was a cutting edge new series for Vertigo DC comics created by Garth Ennis. Both projects never made it through the development phase and Tammer embarked on a solo career. In 1995, Tammer worked in development for Valerie Kearns and screenwriter James V. Hart at the now defunct HBI Pictures . After a short stint, Tammer in 1996 decided to begin his own independent project so he wrote and subsequently directed Lone Greasers . It was here that he met close friend and future producing partner Daniel Dubiecki . Dubiecki produced Lone Greasers which showcased an ensemble cast of veteran character actors including former X front man John Doe, Peter Dobson and Mariah O brien. Tammer continued writing and directing until 1999 at which point, he decided to focus solely on his writing career. From ... Wikipedia Persondata . NAME Tammer, Torus ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 1969 PLACE OF BIRTH DATE OF DEATH PLACE OF DEATH DEFAULTSORT Tammer, Torus Category Australian film producers ... more details
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 ... 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 Asymmetric key cryptosystems Crypto stub ... more details
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