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813c72883cf82050c812d0e7cfede2130d723084 | now let x 1 (t), …, x n | now let x 1 (t), …, x n (t) {\displaystyle x^{1}(t),\ldots,x^{n}(t)\,} be n {\displaystyle n\,} linearly independent solutions to the homogeneous equation x ′ = a (t) x {\displaystyle x'=a(t)x\,} and arrange them in columns to form a fundamental matrix: | wikipedia |
78cc93c2c30da1dfcd5be2817548f88a846ff90d | for instance, consider x ′ = a (t) | for instance, consider x ′ = a (t) x + g (t) {\displaystyle x'=a(t)x+g(t)\,} where x {\displaystyle x\,} is a vector and a (t) {\displaystyle a(t)\,} is an n × n {\displaystyle n\times n\,} matrix function of t {\displaystyle t\,}, which is continuous for t ∈ i, a ≤ t ≤ b {\displaystyle t\in i,a\leq t\leq b\,}, where i... | wikipedia |
21570241c39cf56330c8cba31fcadb8226466b73 | hereford road skew bridge is a disused railway | hereford road skew bridge is a disused railway bridge in ledbury, herefordshire. built in 1881 to carry the ledbury and gloucester railway across the hereford road at an angle of approximately 45°, it was built as a ribbed skew arch with stone spandrels and wing walls, and ribs of blue brick. | wikipedia |
ca581f91933c34e454ed8f2851650e6cfe3f99db | the oblique bridge was built in 1881 and | the oblique bridge was built in 1881 and constructed with stone abutments, parapet walls, spandrels and wing walls but to accommodate the skew it was built with 13 separate staggered but overlapping ribs in blue brick. photographs show that each rib forms a separate segmental right arch equal in width to three stretche... | wikipedia |
b0d2c37503d3eb55575d4cc45a6ee9c6cfff2d6a | the skew canal bridge to which reference is | the skew canal bridge to which reference is made is still in place at monkhide, carrying a minor road over one of the few remaining stretches of the old canal. however, now that the railway has been dismantled the canal has become the subject of an active and ambitious restoration scheme. | wikipedia |
56803a960c8acd7842ffd18940d70c78dce119fd | the skew bridge over the hereford road is | the skew bridge over the hereford road is probably one of the most 'skew' railway bridges in the country although the 'skew' canal bridge at monkhide is believed to be the most angled bridge in the country. it is quite a feat of engineering and the brickwork is well worth a look from the road below. | wikipedia |
cc001af742b7b4261260028ca8e678ebb91975e2 | the two smaller companies were amalgamated into the | the two smaller companies were amalgamated into the great western railway in 1892, and on 4 january 1917 the double track between ledbury and dymock was singled to provide materials for the great war. the line closed to passenger traffic on 11 july 1959, with the section between ledbury and dymock closing completely an... | wikipedia |
ae0062ee36304a95dc00d3bbe37fb7738468dacc | the line opened to traffic on 27 july | the line opened to traffic on 27 july 1885, when the existing ledbury station was renamed ledbury junction, the ledbury and gloucester line curving away from the worcester and hereford railway line on an embankment immediately west of the station. just south of the junction the double-track line was carried at an awkwa... | wikipedia |
451ab7668c87a348ddfc8bbbac1dc562b0ad49c7 | the herefordshire and gloucestershire canal opened in two | the herefordshire and gloucestershire canal opened in two phases in 1798 and 1845, but in 1863, after a period of financial difficulty, it was leased to the great western railway and in 1881 work started on converting the southern section into a railway. the ross and ledbury railway company's intention was to build a l... | wikipedia |
b91b0f3b0cdd8506a14e201784501f95cd6a2d20 | hereford road skew bridge is a disused railway | hereford road skew bridge is a disused railway bridge in ledbury, herefordshire. built in 1881 to carry the ledbury and gloucester railway across the hereford road at an angle of approximately 45°, it was built as a ribbed skew arch with stone spandrels and wing walls, and ribs of blue brick. the railway line was close... | wikipedia |
5043a088d53b40186488cd11860a9a991763d4d7 | bruce lincoln is caroline e. haskell distinguished service | bruce lincoln is caroline e. haskell distinguished service professor emeritus of the history of religions in the divinity school of the university of chicago, where he also holds positions in the center for middle eastern studies, committee on the ancient mediterranean world, committee on the history of culture, and in... | wikipedia |
618e0a0d35918d1ee73856d8d53e13379b8d1597 | lincoln graduated from haverford college in 1970 with | lincoln graduated from haverford college in 1970 with a b.a. in religion, and then took his ph.d. in the history of religions from the university of chicago in 1976, where he wrote his dissertation, "priests, warriors, and cattle: a comparative study of east african and indo-iranian religious systems" under mircea elia... | wikipedia |
710e1519d966a0b4bc551cef6f9350ad8c7fd5f5 | for many years his primary scholarly concern was | for many years his primary scholarly concern was the study of indo-european religion, where his work came to criticize the ideological presuppositions of research on purported indo-european origins. since the late 1990s, his work has dealt extensively with methodological problems, and issues concerning religion, power ... | wikipedia |
f0178751c67cde2000aafa546452d6a032a3aaa7 | bruce lincoln (born 1948) is caroline e. haskell | bruce lincoln (born 1948) is caroline e. haskell distinguished service professor emeritus of the history of religions in the divinity school of the university of chicago, where he also holds positions in the center for middle eastern studies, committee on the ancient mediterranean world, committee on the history of cul... | wikipedia |
e9092be28e5b556648ac4a041d59f95ce0897f37 | cécile mourer-chauviré is a french paleontologist specializing in | cécile mourer-chauviré is a french paleontologist specializing in birds of the eocene and the oligocene. in her early career, she discovered with her husband the laang spean cave site of prehistoric humans in cambodia. | wikipedia |
eed9f695d214bf158faf16d2d87dac7c6933ec3d | colleagues have honoured mourer-chauviré by naming fossil bird | colleagues have honoured mourer-chauviré by naming fossil bird species and genera after her. as of 2013, the following were named after her: aythya chauvirae, cypseloides mourerchauvireae, chauvireria balcanica, pica mourerae, oligosylphe mourerchauvireae, tyto mourerchauvireae, afrocygnus chauvireae, asphaltoglaux cec... | wikipedia |
c535d13835d2c9638a939b95f0e973ccf80fc1cc | in 2011, she published with her colleagues on | in 2011, she published with her colleagues on lavocatavis africana, an african fossil that may belong to the phorusrhacidae clade (terror birds). the algerian find is significant as previous finds from the era in africa were not land-dwelling birds and phorusrhacidae was not previously known outside of the americas. | wikipedia |
faea086c16b54630cd7a8f8b5b03646bf79ecd86 | in 1970, at the outbreak of civil war | in 1970, at the outbreak of civil war in cambodia, she returned with her two small children to france. in 1971, she secured an appointment with cnrs at claude bernard university lyon 1. in 1975 she completed her "thèse d’etat", in 1984 her habilitation, and in 1985 she was appointed director of research in cnrs which s... | wikipedia |
5092611cc37545b5413492d0a50529b4ff182b01 | following her marriage in 1964 to roland mourer, | following her marriage in 1964 to roland mourer, she relocated to cambodia where he was assigned by the french military as a "coopérant" in kampong chhnang. in 1965 she was appointed as a geology professor at royal university of phnom penh, a post she held until the civil war in 1970. during this time she discovered wi... | wikipedia |
9098ed30f0b7137de4111ac72c18a0b32231a8bb | cécile chauviré was born on 5 november 1939 | cécile chauviré was born on 5 november 1939 in lyon, france. she studied at university of lyon. her early work was on large quaternary mammals. she then proceeded in 1961 to a doctorate in centre national de la recherche scientifique focusing on pleistocene birds, a topic few at the time studied in france or europe. | wikipedia |
5b3cc4851a6ea48cb20abb68b735a291dab82dc8 | cécile mourer-chauviré (born 1939) is a french paleontologist | cécile mourer-chauviré (born 1939) is a french paleontologist specializing in birds of the eocene and the oligocene. in her early career, she discovered with her husband the laang spean cave site of prehistoric humans in cambodia. | wikipedia |
117306dd643c9957c906d809092864b1fe315c5a | the continental o-300 and the c145 are a | the continental o-300 and the c145 are a family of air-cooled flat-6 aircraft piston engines built by teledyne continental motors. first produced in 1947, versions were still in production as of 2004. it was produced under licence in the united kingdom by rolls-royce in the 1960s. | wikipedia |
1b6ab7bb312f750ee25922cf9f1b9dea4ba5466f | the go-300 engine has a tbo (time between | the go-300 engine has a tbo (time between overhaul) of 1200 hours, while 1800 hours is the standard for ungeared o-300 engines. the go-300 engine suffered reliability problems as a result of pilots mishandling the engine and operating it at too low an engine rpm. this caused the cessna skylark to develop a poor reputat... | wikipedia |
b578d48ef996ee2507ae500f6ccab1fedaf56e85 | the go-300 employs a reduction gearbox, so that | the go-300 employs a reduction gearbox, so that the engine turns at 3200 rpm to produce a propeller rpm of 2400. the go-300 produces 175 hp (130 kw) whereas the ungeared o-300 produces 145 hp (108 kw). | wikipedia |
006c7f3a47aebe5aa9ed91dc69862fca26038a1a | the c-145 was developed from the 125 hp | the c-145 was developed from the 125 hp (93 kw) c-125 engine. both powerplants share the same crankcase, although the c-145 produces an additional 20 hp (15 kw) through a longer piston stroke, higher compression ratio of 7.0:1 and different carburetor jetting. | wikipedia |
2ccea3d9c759c085d4cf1f7c95da14a32e04d512 | a cd ripper, cd grabber, or cd extractor | a cd ripper, cd grabber, or cd extractor is software that rips raw digital audio in compact disc digital audio (cd-da) format tracks on a compact disc to standard computer sound files, such as wav or mp3. a more formal term used for the process of ripping audio cds is digital audio extraction (dae). | wikipedia |
4777f36a677cef2d93383bdd35da84b7f9e75b5a | properties of an optical drive helping in achieving | properties of an optical drive helping in achieving a perfect rip are a small sample offset (at best zero), no jitter, no or deactivatable caching, and a correct implementation and feed-back of the c1 and c2 error states. there are databases listing these features for multiple brands and versions of optical drives. als... | wikipedia |
ab43cc5d8291f6c2b3d39705536974ad355d848e | in the context of digital audio extraction from | in the context of digital audio extraction from compact discs, seek jitter causes extracted audio samples to be doubled-up or skipped entirely if the compact disc drive re-seeks. the problem occurs because the red book does not require block-accurate addressing during seeking. as a result, the extraction process may re... | wikipedia |
c6825d45ab1d052cff668c169d7e5376c7ac3895 | most ripping programs will assist in tagging the | most ripping programs will assist in tagging the encoded files with metadata. the mp3 file format, for example, allows tags with title, artist, album and track number information. some will try to identify the disc being ripped by looking up network services like amg's lasso, freedb, gracenote 's cddb, gd3 or musicbrai... | wikipedia |
34743f4496aa393198824b78ab8eba2912d10f42 | as an intermediate step, some ripping programs save | as an intermediate step, some ripping programs save the extracted audio in a lossless format such as wav, flac, or even raw pcm audio. the extracted audio can then be encoded with a lossy codec like mp3, vorbis, wma or aac. the encoded files are more compact and are suitable for playback on digital audio players. they ... | wikipedia |
11a6fa19bedf0558517c06e627ea1e6e8afb7549 | in the early days of computer cd-rom drives | in the early days of computer cd-rom drives and audio compression mechanisms (such as mp2), cd ripping was considered undesirable by copyright holders, with some attempting to retrofit copy protection into the simple iso9660 standard. as time progressed, most music publishers became more open to the idea that since ind... | wikipedia |
293976dc68d787ac9eae4c3c65d7a3b240cedaad | cadmium tellurite is a colourless solid that is | cadmium tellurite is a colourless solid that is insoluble in water. it is a semiconductor. it is part of the monoclinic crystal system, with space group p2 /c (no. 14). it can also crystallize in the cubic crystal system and hexagonal crystal system at temperatures above 540 °c. | wikipedia |
a04ec2d2a837159db868275a38d32043fe413b2e | a cantilever is a rigid structural element that | a cantilever is a rigid structural element that extends horizontally and is unsupported at one end. typically it extends from a flat vertical surface such as a wall, to which it must be firmly attached. like other structural elements, a cantilever can be formed as a beam, plate, truss, or slab. when subjected to a stru... | wikipedia |
445a3f123e05992bff2632aeebe2612c8151dfe6 | a chemical sensor can be obtained by coating | a chemical sensor can be obtained by coating a recognition receptor layer over the upper side of a microcantilever beam. a typical application is the immunosensor based on an antibody layer that interacts selectively with a particular immunogen and reports about its content in a specimen. in the static mode of operatio... | wikipedia |
8ea9970111234cb1a967357770b919e206bddec2 | the principal advantage of mems cantilevers is their | the principal advantage of mems cantilevers is their cheapness and ease of fabrication in large arrays.the challenge for their practical application lies in the square and cubic dependences of cantilever performance specifications on dimensions. these superlinear dependences mean that cantilevers are quite sensitive to... | wikipedia |
c2a26b44fcaefa900e4ebaf3bd2436aa53f2680d | where f {\displaystyle f} is force and w | where f {\displaystyle f} is force and w {\displaystyle w} is the cantilever width. the spring constant is related to the cantilever resonance frequency ω 0 {\displaystyle \omega _{0}} by the usual harmonic oscillator formula ω 0 = k / m equivalent {\displaystyle \omega _{0}={\sqrt {k/m_{\text{equivalent}}}}}. a change... | wikipedia |
d101ddba3f36b517598f79bda14cfe038a25945d | where ν {\displaystyle \nu } is poisson's ratio, | where ν {\displaystyle \nu } is poisson's ratio, e {\displaystyle e} is young's modulus, l {\displaystyle l} is the beam length and t {\displaystyle t} is the cantilever thickness. very sensitive optical and capacitive methods have been developed to measure changes in the static deflection of cantilever beams used in d... | wikipedia |
28f817ee00c389a196bab0121aed6dfb420f856e | cantilevered beams are the most ubiquitous structures in | cantilevered beams are the most ubiquitous structures in the field of microelectromechanical systems (mems). an early example of a mems cantilever is the resonistor, an electromechanical monolithic resonator. mems cantilevers are commonly fabricated from silicon (si), silicon nitride (si n), or polymers. the fabricatio... | wikipedia |
d66ee3b2ade839982a2da9483b8070f77b53116c | cantilever wings require much stronger and heavier spars | cantilever wings require much stronger and heavier spars than would otherwise be needed in a wire-braced design. however, as the speed of the aircraft increases, the drag of the bracing increases sharply, while the wing structure must be strengthened, typically by increasing the strength of the spars and the thickness ... | wikipedia |
04cbb30c7954780d09220e03c86a26a7febb30ac | to resist horizontal shear stress from either drag | to resist horizontal shear stress from either drag or engine thrust, the wing must also form a stiff cantilever in the horizontal plane. a single-spar design will usually be fitted with a second smaller drag-spar nearer the trailing edge, braced to the main spar via additional internal members or a stressed skin. the w... | wikipedia |
a16b27dd4d4b0f2afa3da6804493a29465fc2fa3 | in the cantilever wing, one or more strong | in the cantilever wing, one or more strong beams, called spars, run along the span of the wing. the end fixed rigidly to the central fuselage is known as the root and the far end as the tip. in flight, the wings generate lift and the spars carry this load through to the fuselage. | wikipedia |
abfe37452d05a0bc85a1da686cd0b2facf9df704 | hugo junkers pioneered the cantilever wing in 1915. | hugo junkers pioneered the cantilever wing in 1915. only a dozen years after the wright brothers ' initial flights, junkers endeavored to eliminate virtually all major external bracing members in order to decrease airframe drag in flight. the result of this endeavor was the junkers j 1 pioneering all-metal monoplane of... | wikipedia |
3602afb758e8902a12a87399ed7087b2db8fe2a6 | the cantilever is commonly used in the wings | the cantilever is commonly used in the wings of fixed-wing aircraft. early aircraft had light structures which were braced with wires and struts. however, these introduced aerodynamic drag which limited performance. while it is heavier, the cantilever avoids this issue and allows the plane to fly faster. | wikipedia |
b9ae890463c862d36a4268664fb19cb6b7befb4f | in an architectural application, frank lloyd wright 's | in an architectural application, frank lloyd wright 's fallingwater used cantilevers to project large balconies. the east stand at elland road stadium in leeds was, when completed, the largest cantilever stand in the world holding 17,000 spectators.the roof built over the stands at old trafford uses a cantilever so tha... | wikipedia |
4d0e94d9fbdc06e988126045fcc5ac186fe059f1 | temporary cantilevers are often used in construction.the partially | temporary cantilevers are often used in construction.the partially constructed structure creates a cantilever, but the completed structure does not act as a cantilever.this is very helpful when temporary supports, or falsework, cannot be used to support the structure while it is being built (e.g., over a busy roadway o... | wikipedia |
d155cf6077b4b4d739cccf594814878ac2d5cdae | cantilevers are widely found in construction, notably in | cantilevers are widely found in construction, notably in cantilever bridges and balconies (see corbel). in cantilever bridges, the cantilevers are usually built as pairs, with each cantilever used to support one end of a central section. the forth bridge in scotland is an example of a cantilever truss bridge. a cantile... | wikipedia |
7b8a505203921033e3cff68527726bd9423b4919 | a cantilever is a rigid structural element that | a cantilever is a rigid structural element that extends horizontally and is unsupported at one end. typically it extends from a flat vertical surface such as a wall, to which it must be firmly attached. like other structural elements, a cantilever can be formed as a beam, plate, truss, or slab. | wikipedia |
3763b042726e841b9551178fdadf1317b4bb9667 | in geometric topology, busemann functions are used to | in geometric topology, busemann functions are used to study the large-scale geometry of geodesics in hadamard spaces and in particular hadamard manifolds. they are named after herbert busemann, who introduced them; he gave an extensive treatment of the topic in his 1955 book "the geometry of geodesics". | wikipedia |
c04bee344b428de1bd47cf47c2e0c0e9d7f3f2c4 | busemann functions can be used to determine special | busemann functions can be used to determine special visual metrics on the class of cat(-1) spaces. these are complete geodesic metric spaces in which the distances between points on the boundary of a geodesic triangle are less than or equal to the comparison triangle in the hyperbolic upper half plane or equivalently t... | wikipedia |
6c82d0cbb2956fbb52be6e26b89974862ffeaf7a | if a, b, c and d are in | if a, b, c and d are in order on the unit circle. hence the same inequalities are valid for the three cyclic of the quadruple a, b, c, d. if a and b are switched then the cross ratios are sent to their inverses, so lie between 0 and 1; similarly if c and d are switched. if both pairs are switched, the cross ratio remai... | wikipedia |
9e54b40dc0b77bbef4f8469f020e702936860851 | to prove this, it suffices to show that | to prove this, it suffices to show that log (f (a), f (b); f (c), f (d)) ≤ b log (a, b; c, d) + c. from the previous section it suffices show d (,) ≤ p d (,) + q. this follows from the fact that the images under f of and lie within h -neighbourhoods of and; the minimal distance can be estimated using the quasi-isometry... | wikipedia |
53ed26cd851ef86413c7bafcff81a1eaad83f1f7 | it has already been checked that f (and | it has already been checked that f (and is inverse) are continuous. composing f, and hence f, with complex conjugation if necessary, it can further be assumed that f preserves the orientation of the circle. in this case, if a, b, c, d are in order on the circle, so too are there images under f; hence both (a, b; c, d) ... | wikipedia |
89777c362336b668180d8095e96c82ebb20ba0a1 | if f is quasisymmetric then it is also | if f is quasisymmetric then it is also quasi-möbius, with c = a and d = b: this follows by multiplying the first inequality for (z, z, z) and (z, z, z). conversely any quasi-möbius homeomorphism f is quasisymmetric. to see this, it can be first be checked that f (and hence f) is hölder continuous. let s be the set of c... | wikipedia |
e70081822d0d84e497f7ad8ec120a9531b8e22c3 | since both sides are invariant under möbius transformations, | since both sides are invariant under möbius transformations, it suffices to check this in the case that a = 0, b = ∞, c = x and d = 1. in this case the geodesic line is the positive imaginary axis, right hand side equals | log | x ||, p = | x | i and q = i. so the left hand side equals | log | x ||. note that p and q a... | wikipedia |
1e508c73ed15aa98d4c981687d26e5190e9eb667 | a more general and precise geometric interpretation of | a more general and precise geometric interpretation of the cross ratio can be given using projections of ideal points on to a geodesic line; it does not depend on the order of the points on the circle and therefore whether or not geodesic lines intersect. | wikipedia |
0071620cc84a1ae7941f6d3d72b38f2776fc0541 | since log = log (1 + exp(–2 x))/2 | since log = log (1 + exp(–2 x))/2 is bounded above and below in x ≥ 0. note that a, b, c, d are in order around the unit circle if and only if (a, b; c, d) > 1. | wikipedia |
b905e43e2d939d5fae87c2caced87e878cab3768 | now let a, b, c, d be points | now let a, b, c, d be points on the unit circle or real axis in that order. then the geodesics and do not intersect and the distance between these geodesics is well defined: there is a unique geodesic line cutting these two geodesics orthogonally and the distance is given by the length of the geodesic segment between t... | wikipedia |
c889ad751fb3556801d65d9446823f0041992209 | since a, b, c and d all appear | since a, b, c and d all appear in the numerator defining the cross ratio, to understand the behaviour of the cross ratio under permutations of a, b, c and d, it suffices to consider permutations that fix d, so only permute a, b and c. the cross ratio transforms according to the anharmonic group of order 6 generated by ... | wikipedia |
168dae15e5013ece591244af0da1d5e55b971f62 | if g is a complex möbius transformation then | if g is a complex möbius transformation then it leaves the cross ratio invariant: (g (a), g (b); g (c), g (d)) = (a, b: c, d). since the möbius group acts simply transitively on triples of points, the cross ratio can alternatively be described as the complex number z in c \{0,1} such that g (a) = 0, g (b) = 1, g (c) = ... | wikipedia |
aaa599a2e4ad2c7df6d75def9a7c7c284ebffd9b | given two distinct points z, w on the | given two distinct points z, w on the unit circle or real axis there is a unique hyperbolic geodesic joining them. it is given by the circle (or straight line) which cuts the unit circle unit circle or real axis orthogonally at those two points. given four distinct points a, b, c, d in the extended complex plane their ... | wikipedia |
ca2fcc250b11a7d00495fcb65433f607ad0d5ff3 | in the other direction, it is straightforward to | in the other direction, it is straightforward to prove that the homomorphism is injective. suppose that f is a quasi-isometry of the unit disk such that ∂ f is the identity. the assumption and the morse lemma implies that if γ(r) is a geodesic line, then f (γ(r)) lies in an h -neighbourhood of γ(r). now take a second g... | wikipedia |
c7c44427e5fbabce846bfb436cac4e97c936ff09 | in the case of the unit disk, teichmüller | in the case of the unit disk, teichmüller theory implies that the homomorphism carries quasiconformal homeomorphisms of the disk onto the group of quasi-möbius homeomorphisms of the circle (using for example the ahlfors–beurling or douady–earle extension): it follows that the homomorphism from the quasi-isometry group ... | wikipedia |
a4e3c24a2323d4c683781f8f6943708d177f7599 | on cat(-1) spaces, a finer version of continuity | on cat(-1) spaces, a finer version of continuity asserts that ∂ f is a quasi-möbius mapping with respect to a natural class of metric on ∂ x, the "visual metrics" generalising the euclidean metric on the unit circle and its transforms under the möbius group. these visual metrics can be defined in terms of busemann func... | wikipedia |
74906c61f8117f9db40e92fe3b30a2590e75f861 | to check that ∂ f is continuous, note | to check that ∂ f is continuous, note that if γ and γ are geodesic rays that are uniformly close on, within a distance η, then f ∘ γ and f ∘ γ lie within a distance λη + ε on, so that δ and δ lie within a distance λη + ε + 2 h (λ,ε); hence on a smaller interval, δ and δ lie within a distance (r / r)⋅ by convexity. | wikipedia |
45090245a2fafda43ab0ccf8b0a9a7e8025d10e4 | fixing a point x in x, given a | fixing a point x in x, given a geodesic ray γ starting at x, the image f ∘ γ under a quasi-isometry f is a quasi-geodesic ray. by the morse-mostow lemma it is within a bounded distance of a unique geodesic ray δ starting at x. this defines a mapping ∂ f on the boundary ∂ x of x, independent of the quasi-equivalence cla... | wikipedia |
3d80d085b8e2b491f480bab75c1588d35b762ed0 | note that quasi-inverses are unique up to quasi-equivalence; | note that quasi-inverses are unique up to quasi-equivalence; that equivalent definition could be given using possibly different right and left-quasi inverses, but they would necessarily be quasi-equivalent; that quasi-isometries are closed under composition which up to quasi-equivalence depends only the quasi-equivalen... | wikipedia |
2666877a76f4121ba276de765ed8b4ae487c68f5 | if x is the poincaré unit disk, or | if x is the poincaré unit disk, or more generally a cat(-1) space, the morse lemma on stability of quasigeodesics implies that every quasi-isometry of x extends uniquely to the boundary. by definition two self-mappings f, g of x are quasi-equivalent if sup d (f (x), g (x)) < ∞, so that corresponding points are at a uni... | wikipedia |
74ea66920991616cc6c9116386c04dc34fb726d6 | before discussing cat(-1) spaces, this section will describe | before discussing cat(-1) spaces, this section will describe the efremovich–tikhomirova theorem for the unit disk d with the poincaré metric. it asserts that quasi-isometries of d extend to quasi-möbius homeomorphisms of the unit disk with the euclidean metric. the theorem forms the prototype for the more general theor... | wikipedia |
00f81e58d4c2a2491f6f6869bd2385a28f2890eb | if Γ(t) is a geodesic say with constant | if Γ(t) is a geodesic say with constant λ and ε, let Γ (t) be the unit speed geodesic for the segment. the estimate above shows that for fixed r > 0 and n sufficiently large, (Γ) is a cauchy sequence in c (, x) with the uniform metric. thus Γ tends to a geodesic ray γ uniformly on compacta the bound on the hausdorff di... | wikipedia |
85cd29bf250442d27e81333644c70314b291ab3e | recall that in a hadamard space if and | recall that in a hadamard space if and are two geodesic segments and the intermediate points c (t) and c (t) divide them in the ratio t:(1 – t), then d (c (t), c (t)) is a convex function of t. in particular if Γ (t) and Γ (t) are geodesic segments of unit speed defined on starting at the same point then | wikipedia |
f08e5c71570646d3587d5f1a38ddb5e91f27b95c | since this is a large-scale phenomenon, it is | since this is a large-scale phenomenon, it is enough to check that any maps Δ from {0, 1, 2,..., n } for any n > 0 to the disk satisfying the inequalities is within a hausdorff distance r of the geodesic segment. for then translating it may be assumed without loss of generality Γ is defined on with r > 1 and then, taki... | wikipedia |
881f3ea6b671a63344f91c349415d7755c9a1697 | the generalisation of morse's lemma to cat(-1) spaces | the generalisation of morse's lemma to cat(-1) spaces is often referred to as the morse–mostow lemma and can be proved by a straightforward generalisation of the classical proof. there is also a generalisation for the more general class of hyperbolic metric spaces due to gromov. gromov's proof is given below for the po... | wikipedia |
0f03ec7a0116aa0316ff13bcb7379f8c399c925b | every point of Γ lies within a distance | every point of Γ lies within a distance h of. thus orthogonal projection p carries each point of Γ onto a point in the closed convex set at a distance less than h. since p is continuous and Γ connected, the map p must be onto since the image contains the endpoints of. but then every point of is within a distance h of a... | wikipedia |
cb1a6fd3c2ae65e9e8d689eae7b88b17f6988343 | let γ(t) be the geodesic line containing the | let γ(t) be the geodesic line containing the geodesic segment. then there is a constant h > 0 depending only on λ and ε such that h -neighbourhood Γ lies within an h -neighbourhood of γ(r). indeed for any s > 0, the subset of for which Γ(t) lies outside the closure of the s -neighbourhood of γ(r) is open, so a countabl... | wikipedia |
f8e160d011b3cf6334db77533183d0ed28f2fa98 | applying an isometry in the upper half plane, | applying an isometry in the upper half plane, it may be assumed that the geodesic line is the positive imaginary axis in which case the orthogonal projection onto it is given by p (z) = i | z | and | z | / im z = cosh d (z, pz). hence the hypothesis implies | γ(t) | ≥ cosh(s) im γ(t), so that | wikipedia |
8dd960e408fec88fc7f737b1e1eba1183fd8eb42 | Γ can be replaced by a continuous piecewise | Γ can be replaced by a continuous piecewise geodesic curve Δ with the same endpoints lying at a finite hausdorff distance from Γ less than c = (2λ + 1)ε: break up the interval on which Γ is defined into equal subintervals of length 2λε and take the geodesics between the images under Γ of the endpoints of the subinterva... | wikipedia |
3d579a79450d87ca9b02e4b7d978070902412d69 | morse's lemma on stability of geodesics. in the | morse's lemma on stability of geodesics. in the hyperbolic disk there is a constant r depending on λ and ε such that any quasigeodesic segment Γ defined on a finite interval is within a hausdorff distance r of the geodesic segment. | wikipedia |
5a426da5bcad715229a76615acdf4d31a8ccbb3b | by definition a quasigeodesic Γ defined on an | by definition a quasigeodesic Γ defined on an interval with −∞ ≤ a < b ≤ ∞ is a map Γ(t) into a metric space, not necessarily continuous, for which there are constants λ ≥ 1 and ε > 0 such that for all s and t: | wikipedia |
743a75b29a4bb85527a3e2c5200b43d73ff2a16d | in the case of spaces of negative curvature, | in the case of spaces of negative curvature, such as the poincaré disk, cat(-1) and hyperbolic spaces, there is a metric structure on their gromov boundary. this structure is preserved by the group of quasi-isometries which carry geodesics rays to quasigeodesic rays. quasigeodesics were first studied for negatively cur... | wikipedia |
04356be76c5dcbfb429604ccedea47fa2b8581cf | when x is a hadamard manifold (or more | when x is a hadamard manifold (or more generally a proper hadamard space), gromov's ideal boundary ∂ x = x \ x can be realised explicitly as "asymptotic limits" of geodesics by using busemann functions. fixing a base point x, there is a unique geodesic γ(t) parametrised by arclength such that γ(0) = x and γ ˙ (0) {\dis... | wikipedia |
88020a92d9581fc92935aed8fc053c7718787c23 | the space x is embedded into y by | the space x is embedded into y by sending x to the function f (y) = d (y, x) – d (x, x). let x be the closure of x in y. then x is compact (metrisable) and contains x as an open subset; moreover compactifications arising from different choices of basepoint are naturally homeomorphic. compactness follows from the arzelà... | wikipedia |
0e4cbc5f974aca9d6d072e828b82f0b3521465e2 | if x is a proper metric space, gromov's | if x is a proper metric space, gromov's compactification can be defined as follows. fix a point x in x and let x = b (x, n). let y = c (x) be the space of lipschitz continuous functions on x,.e. those for which | f (x) – f (y) | ≤ a d (x, y) for some constant a > 0. the space y can be topologised by the seminorms ‖ f ‖... | wikipedia |
a6aeb6fd747acdc1872dbdc3680072ded095f112 | eberlein & o'neill (1973) defined a compactification of | eberlein & o'neill (1973) defined a compactification of a hadamard manifold x which uses busemann functions. their construction, which can be extended more generally to proper (i.e. locally compact) hadamard spaces, gives an explicit geometric realisation of a compactification defined by gromov—by adding an "ideal boun... | wikipedia |
6f01b65d53eee9c4735cbce2d4d1fe4f65489e16 | note that this argument could be shortened using | note that this argument could be shortened using the fact that two busemann functions h and h differ by a constant if and only if the corresponding geodesic rays satisfy sup d (γ(t),δ(t)) < ∞. indeed, all the geodesics defined by the flow α satisfy the latter condition, so differ by constants. since along any of these ... | wikipedia |
8dfa3bf68dc8277886f5a4e627257d1847c43d1d | hence h (y) = 0 on the level | hence h (y) = 0 on the level surface of h containing x. the flow α can be used to transport this result to all the level surfaces of h. for general y take s such that h (α (x)) = h (y) and set x = α (x). then h (y) = 0, where γ (t) = α (x) = γ(s + t). but then h = h – s, so that h (y) = s. hence g (y) = s = h ((α (x)) ... | wikipedia |
2d732f34f788f34a49e7a93126e63740e093a859 | let c (− r) be the closed convex | let c (− r) be the closed convex set of points z with h (z) ≤ − r. since x is a hadamard space for every point y in x there is a unique closest point p (y) to y in c (- r). it depends continuously on y and if y lies outside c (- r), then p (y) lies on the hypersurface h (z) = − r —the boundary ∂ c (– r) of c (– r) —and... | wikipedia |
7e248ab13797d9222fd6e629dd07260594c4cfca | so that d (γ(s),γ(t)) = | s − | so that d (γ(s),γ(t)) = | s − t |. let g (y) = h (y), the busemann function for γ with base point x. in particular g (x) = 0. to prove (1), it suffices to show that g = h – h (x)1. | wikipedia |
d3e7dc945036a9e75d0f5dd7e5bb23c9463e61f1 | it therefore remains to show that (3) implies | it therefore remains to show that (3) implies (1). fix x in x. let α be the gradient flow for h. it follows that h ∘ α (x) = h (x) + t and that γ(t) = α (x) is a geodesic through x parametrised by arclength with γ(0) = x. indeed, if s < t, then | wikipedia |
b87cb7961f503f0d4c553fd0849623c48f0242e0 | hence it follows that h is a c | hence it follows that h is a c function with dh dual to the vector field v, so that ‖ dh (y) ‖ = 1. the vector field v is thus the gradient vector field for h. the geodesics through any point are the flow lines for the flow α for v, so that α is the gradient flow for h. | wikipedia |
3e3e1f69a5dec13b7d9b3d5bd42f91d2b78d2d01 | the assertion on the outer terms follows from | the assertion on the outer terms follows from the first variation formula for arclength, but can be deduced directly as follows. let a = δ ˙ (0) {\displaystyle a={\dot {\delta }}(0)} and b = γ ˙ (0) {\displaystyle b={\dot {\gamma }}(0)}, both unit vectors. then for tangent vectors p and q at y in the unit ball | wikipedia |
5502522b68afa47964ee3dadac2f26a68503f4d8 | the outer terms tend to (δ ˙ (0), | the outer terms tend to (δ ˙ (0), γ ˙ (0)) {\displaystyle ({\dot {\delta }}(0),{\dot {\gamma }}(0))} as s tends to 0, so the middle term has the same limit, as claimed. a similar argument applies for s < 0. | wikipedia |
350ebe76cbf9c8e0c34c7e8fda7cc832b8cc5cfb | from the previous properties of h, for each | from the previous properties of h, for each y there is a unique geodesic γ(t) parametrised by arclength with γ(0) = y such that h ∘ γ(t) = h (y) + t. it has the property that it cuts ∂ b (y, r) at t = ± r: in the previous notation γ(r) = u and γ(– r) = v. the vector field v defined by the unit vector γ ˙ (0) {\displays... | wikipedia |
c72694bac5f572a615e774f9859563fcae045b13 | taking a sequence t tending to ∞ and | taking a sequence t tending to ∞ and h = f, there are points u and v which satisfy these conditions for h for n sufficiently large. passing to a subsequence if necessary, it can be assumed that u and v tend to u and v. by continuity these points satisfy the conditions for h. to prove uniqueness, note that by compactnes... | wikipedia |
7637ffa1800be1b3a9b7de0d7797b882c5bf4187 | suppose that x, y are points in a | suppose that x, y are points in a hadamard manifold and let γ (s) be the geodesic through x with γ (0) = y. this geodesic cuts the boundary of the closed ball b (y, r) at the two points γ(± r). thus if d (x, y) > r, there are points u, v with d (y, u) = d (y, v) = r such that | d (x, u) − d (x, v) | = 2 r. by continuit... | wikipedia |
2fe25c9843ad30768ee00474cfad23d97d76b64f | on the other hand, h tends to b | on the other hand, h tends to b uniformly on bounded sets if and only if d (x, x) tends to ∞ and for t > 0 arbitrarily large the sequence obtained by taking the point on each segment at a distance t from x tends to γ(t). for d (x, x) ≥ t, let x (t) be the point in with d (x, x (t)) = t. suppose first that h tends to b ... | wikipedia |
f872fcae1411c2d1f239e40d9b1048e1951dedae | when x is a hadamard space, gromov's ideal | when x is a hadamard space, gromov's ideal boundary ∂ x = x \ x can be realised explicitly as "asymptotic limits" of geodesic rays using busemann functions. if x is an unbounded sequence in x with h (x) = d (x, x) − d (x, x) tending to h in y, then h vanishes at x, is convex, lipschitz with lipschitz constant 1 and has... | wikipedia |
d7fd83661aa05a73006a121730dac2d5e7cc16be | the space x is embedded into y by | the space x is embedded into y by sending x to the function f (y) = d (y, x) – d (x, x). let x be the closure of x in y. then x is metrisable, since y is, and contains x as an open subset; moreover bordifications arising from different choices of basepoint are naturally homeomorphic. let h (x) = (d (x, x) + 1). then h ... | wikipedia |
6afe32dcd4c0935cdecac054903d352179f9b4e2 | if x is a metric space, gromov's bordification | if x is a metric space, gromov's bordification can be defined as follows. fix a point x in x and let x = b (x, n). let y = c (x) be the space of lipschitz continuous functions on x, i.e. those for which | f (x) – f (y) | ≤ a d (x, y) for some constant a > 0. the space y can be topologised by the seminorms ‖ f ‖ = sup |... | wikipedia |
bcb1557155e52a64c28a6ea0ff028cefbe5ec4f7 | in the previous section it was shown that | in the previous section it was shown that if x is a hadamard space and x is a fixed point in x then the union of the space of busemann functions vanishing at x and the space of functions h (x) = d (x, y) − d (x, y) is closed under taking uniform limits on bounded sets. this result can be formalised in the notion of bor... | wikipedia |
5315e7eecfe8acb64a075e1341fd515b9f32ad17 | now suppose that sup d (γ (t), γ | now suppose that sup d (γ (t), γ (t)) < ∞. let δ (t) be the geodesic ray starting at y associated with h. then sup d (γ (t), δ (t)) < ∞. hence sup d (δ (t), δ (t)) < ∞. since δ and δ both start at y, it follows that δ (t) ≡ δ (t). by the previous result h and h differ by a constant; so h and h differ by a constant. | wikipedia |
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