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id stringlengths 40 40	title stringlengths 15 120	text stringlengths 41 3.14k	source stringclasses 2 values
002f7b238666396489e855316fa943dc12a864f0	suppose firstly that γ and δ are two	suppose firstly that γ and δ are two geodesic rays with busemann functions differing by a constant. shifting the argument of one of the geodesics by a constant, it may be assumed that b = b = b, say. let c be the closed convex set on which b (x) ≤ − r. then b (γ(t)) = b (γ(t)) = − t and similarly b (δ(t)) = − t. then f...	wikipedia
6b4b272adaee10d09e95485aeb7924e0519bbfa2	hence h (y) = 0. similarly h (x)	hence h (y) = 0. similarly h (x) = 0. hence h (y) = 0 on the level surface of h containing x. now for t ≥ 0 and z in x, let α (z) = γ (t) the geodesic ray starting at z. then α = α ∘ α and h ∘ α = h − t. moreover, by boundedness, d (α (u),α (v)) ≤ d (u, v). the flow α can be used to transport this result to all the lev...	wikipedia
ab716735c16fd1a335868f2b3bf306b6fea0d64d	fix x in x. let γ (s) be	fix x in x. let γ (s) be the geodesic ray starting at x. let g (z) = h (z), the busemann function for γ with base point x. in particular g (x) = 0. it suffices to show that g = h – h (y)1. now take y with h (x) = h (y) and let δ (t) be the geodesic ray starting at y corresponding to h. then	wikipedia
da079fce593aa3b97a915034fefa807e16bae063	let c be the closed convex set of	let c 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. it depends continuously on y and if y lies outside c, then p (y) lies on the hypersurface h (z) = − r —the boundary ∂ c of c —and p (y) satisfies the inequalit...	wikipedia
f0f0bc4fe1e7bae7b7c14695ea86f64f533123e5	to prove the first assertion, it is enough	to prove the first assertion, it is enough to check this for t sufficiently large. in that case γ(t) and δ (t − h (y)) are the projections of x and y onto the closed convex set h ≤ − t. therefore, d (γ (t), δ (t − h (y))) ≤ d (x, y). hence d (γ (t), δ (t)) ≤ d (γ (t), δ (t − h (y))) + d (δ (t − h (y)), δ (t)) ≤ d (x, y...	wikipedia
58187a5e1376fd8444d84b0fded7c2fdac0ebc3d	the third condition implies that v is the	the third condition implies that v is the closest point to y in the closed convex set c of points u {\displaystyle u} such that h (u) ≤ h (y) – r. let δ (t) for 0 ≤ t ≤ r be the geodesic joining y to v. then k (t) = h (δ (t)) - h (y) is a convex lipschitz function on with lipschitz constant 1 satisfying k (t) ≤ – t and...	wikipedia
7f6a45094cea5b9d74ce663c1c6f91123e625b24	comment. note that, since any closed convex subset	comment. note that, since any closed convex subset of a hadamard subset of a hadamard space is also a hadamard space, any closed ball in a hadamard space is a hadamard space. in particular it need not be the case that every geodesic segment is contained in a geodesic defined on the whole of r or even a semi-infinite in...	wikipedia
3c2e0e86a062234b5eb4b4d4c826a5b7d9b717f1	uniform limits. the above argument proves more generally	uniform limits. the above argument proves more generally that if d (x, x) tends to infinity and the functions h (x) = d (x, x) – d (x, x) tend uniformly on bounded sets to h (x), then h is convex, lipschitz with lipschitz constant 1 and, given y in x and r > 0, there is a unique point v with d (y, v) = r such that h (v...	wikipedia
7a8b2eb701abf485c236cf749621a367ab794c3e	so that u is a cauchy sequence. if	so that u is a cauchy sequence. if u is its limit, then d (y, u) = r and h (u) = h (y) − r. by uniqueness it follows that u = v and hence d (y, v) = r, as required.	wikipedia
a8b9faf3850e928a1ec5db7a24261e5f88bcced3	let v be the closest point to y	let v be the closest point to y in c. then h (v) = h (y) − r and so h is minimised by v in b (y, r) where r = d (y, v) and v is the unique point where h is minimised. by the lipschitz condition r = \| h (y) − h (v)\| ≤ r. to prove the assertion, it suffices to show that r = r, i.e. d (y, v) = r. on the other hand, h is t...	wikipedia
eac7c0b434f063fa383036a4dfb9ffbcc86cb01a	this condition persists for busemann functions. the statement	this condition persists for busemann functions. the statement and proof of the property for busemann functions relies on a fundamental theorem on closed convex subsets of a hadamard space, which generalises orthogonal projection in a hilbert space: if c is a closed convex set in a hadamard space x, then every point x i...	wikipedia
3a7562af550b24fd2d6861dc8d78eb9c07a6c7d2	next suppose that x, y are points in	next suppose that x, y are points in a hadamard space, and let δ (s) be the geodesic through x with δ (0) = y and δ (t) = x, where t = d (x, y). this geodesic cuts the boundary of the closed ball b (y, r) at the point δ (r). thus if d (x, y) > r, there is a point v with d (y, v) = r such that d (x, v) = d (x, y) − r.	wikipedia
75bb690bf489387b12a94a626e2a6b5bf21b1877	note that the inequality above for \| f	note that the inequality above for \| f s (y) − f t (y) \| {\displaystyle \|f_{s}(y)-f_{t}(y)\|} (together with its proof) also holds for geodesic segments: if γ (t) is a geodesic segment starting at x and parametrised by arclength then	wikipedia
311c6903d813a9d8b42e013ba49a1d4e0f456123	let h (t) = d (y,γ(t)) − t	let h (t) = d (y,γ(t)) − t = f (y). since γ (t) {\displaystyle \gamma (t)} is parametrised by arclength, alexandrov's first comparison theorem for hadamard spaces implies that the function g (t) = d (y,γ(t)) − t is convex. hence for 0< s < t	wikipedia
f0cf3dab39cd16981adac5c1621a27a2443ecb2f	in a hadamard space, where any two points	in a hadamard space, where any two points are joined by a unique geodesic segment, the function f = f t {\displaystyle f=f_{t}} is convex, i.e. convex on geodesic segments {\displaystyle }. explicitly this means that if z (s) {\displaystyle z(s)} is the point which divides {\displaystyle } in the ratio s : (1 − s), the...	wikipedia
c79cb1782c625060ad8b5dd8fdb5845c3528348a	where the term in brackets on the right	where the term in brackets on the right hand side is the poisson kernel for the unit disk and ζ {\displaystyle \zeta } corresponds to the radial geodesic γ {\displaystyle \gamma } from the origin towards ζ {\displaystyle \zeta }, γ (t) = ζ tanh (t / 2) {\displaystyle \gamma (t)=\zeta \tanh(t/2)}. the computation of d (...	wikipedia
fad45188ff373924c31882a810033d8424227fc9	by dini's theorem, the functions f t (x)	by dini's theorem, the functions f t (x) = d (x, γ (t)) − t {\displaystyle f_{t}(x)=d(x,\gamma (t))-t} tend to b γ (x) {\displaystyle b_{\gamma }(x)} uniformly on compact sets as t tends to infinity.	wikipedia
17eb8593465869597c2b80aae3462a1062414a95	thus, when t is very large, the distance	thus, when t is very large, the distance d (γ (t), x) {\displaystyle d{\big (}\gamma (t),x{\big)}} is approximately equal to b γ (x) + t {\displaystyle b_{\gamma }(x)+t}. given a ray γ, its busemann function is always well-defined: indeed the right hand side above f t (x) = def d (γ (t), x) − t {\displaystyle f_{t}(x){...	wikipedia
90c57b1506e09681869861e75b0da844dd47ca59	let (x, d) {\displaystyle (x,d)} be a metric	let (x, d) {\displaystyle (x,d)} be a metric space. a geodesic ray is a path γ: [ 0, ∞) → x {\displaystyle \gamma :[0,\infty)\to x} which minimizes distance everywhere along its length. i.e., for all t, t ′ ∈ [ 0, ∞) {\displaystyle t,t'\in [0,\infty)},	wikipedia
d8acb88d694e8c674713f5a5f285b2059b6bc9c8	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 (simply connected complete riemannian manifolds of nonpositive curvature). they are named after herbert busemann, who introduced them; he gave an extensive treatment ...	wikipedia
21183507adcaf7430da0b661ba0a02a558d2db4e	the bridge at nimreh is a roman bridge	the bridge at nimreh is a roman bridge in the vicinity of shahba, syria, dating to the 3rd or 4th century ad. its transversal arch construction derives from old building traditions of the hauran region and is arguably unique in roman bridge building.	wikipedia
38a4441b9c345239fa39561ec377252c3ad0aaa1	presumably, the bridge was built directly onto the	presumably, the bridge was built directly onto the solid stone. its almost unadorned masonry walls consist of rectangular, roughly smoothed basalt blocks of varying size without mortar (opus quasi-quadratum). since roman bridges rarely featured basalt, its use is undoubtedly due to the availability of the material in s...	wikipedia
7f94ce36a21686e6bf52f2eb4dfeec110309271a	the vault was made of three parallel, transversal	the vault was made of three parallel, transversal arches standing at intervals of 1.20 m; the arches were built of 60 cm long, 30 cm wide and just as high ashlar, while the spaces in between are covered with longish basalt blocks. this design principle – transversal arches with lintels – seems to be unique among roman ...	wikipedia
994e912cd4064bf7da9bff52051764cf5f2b808f	the bridge is situated 10 km southeast from	the bridge is situated 10 km southeast from shahba, close to the modern road at the foot of the mountain village nimreh, which was known in the onomastikon of eusebius by the name namara and regarded as an important place (vicus grandis). here, the bridge crosses the wadi al-liwa almost right-angled, making a small ben...	wikipedia
0f9a34356b6841f93300598ebf5d6591e646ae0b	after the annexation by emperor trajan in 106	after the annexation by emperor trajan in 106 ad, the infrastructure of the newly established provincia arabia was greatly expanded by the construction of roman forts and military roads to ward off nomadic incursions; in bostra the legio iii cyrenaica was stationed. the rugged terrain, with many wadis crisscrossing it,...	wikipedia
9d804d7c456751473044c4304758d89b2c05f80b	the bridge at nimreh is located in the	the bridge at nimreh is located in the hauran (ancient name: auranitis), a mountainous volcanic region c. 80 km southeast from damascus in the transitional zone between agricultural land and the desert. due to fertile soils and sufficient rainfalls, the area yielded in roman times particularly high returns, which could...	wikipedia
580f5c44d307c3709cf639e9fb6d84c7d4856560	the bridge at nimreh is a roman bridge	the bridge at nimreh is a roman bridge in the vicinity of shahba (ancient philippopolis), syria, dating to the 3rd or 4th century ad. its transversal arch construction derives from old building traditions of the hauran region and is arguably unique in roman bridge building.	wikipedia
ec3467bc51522a3720290b43d8e720f593429ba9	brochosomes are intricately structured microscopic granules secreted by	brochosomes are intricately structured microscopic granules secreted by leafhoppers and typically found on their body surface and, more rarely, eggs. brochosomes were first described in 1952 with the aid of an electron microscope. brochosomes are hydrophobic and help keep the insect cuticle clean. these particles have ...	wikipedia
877ec8086e86bf6a63d792b35ba87975a84235c2	in several new world genera of the leafhopper	in several new world genera of the leafhopper subfamily cicadellinae (including the glassy-winged sharpshooter and related species) brochosomes are also used as a coating on egg masses. in gravid females from these genera, the malpighian tubules switch over from production of regular brochosomes, described above, to pr...	wikipedia
f11ddb48c9b827e90a62a822152e543cc20d39fc	after each molt, most leafhopper species release droplets	after each molt, most leafhopper species release droplets of the brochosome-containing fluid through the anus and actively spread them over the newly formed integument. this behavior is called anointing. dry brochosomes are further distributed across the body and appendages in repeated bouts of grooming, in which leafh...	wikipedia
04f8de0f60dadd33fef400f716c7ded89c996b90	brochosomes are produced within cells of specialized glandular	brochosomes are produced within cells of specialized glandular segments of the malpighian tubules – the primary excretory organs of insects, which often serve additional functions. each cell simultaneously manufactures a large number of brochosomes within its golgi complexes and eventually releases them into the lumen ...	wikipedia
d43d97267065776374b54521685ce232a76876b9	most species of leafhoppers produce hollow spherical brochosomes,	most species of leafhoppers produce hollow spherical brochosomes, 0.2–0.7 micrometres in diameter, with a honeycombed outer wall. they often consist of 20 hexagonal and 12 pentagonal cells, making the outline of each brochosome approximating a truncated icosahedron – the geometry of a soccer ball and a c buckminsterful...	wikipedia
af1ea011e8dce091402571e325e77e66935c7d87	brochosomes are intricately structured microscopic granules secreted by	brochosomes are intricately structured microscopic granules secreted by leafhoppers (the family cicadellidae of the insect order hemiptera) and typically found on their body surface and, more rarely, eggs. brochosomes were first described in 1952 with the aid of an electron microscope. brochosomes are hydrophobic and h...	wikipedia
be18f4819b069027c3c259816413cc4e018a44d1	cleidson rodrigues dos santos or simply cleidson is	cleidson rodrigues dos santos or simply cleidson is a brazilian footballer who plays primarily as a left back. he currently plays for itumbiara esporte clube. a strong, versatile player who can deputise at centre back should the need arise, cleidson was linked with several european clubs during the 2012 summer transfer...	wikipedia
e12c5ff675562275887eb9fabc7218100beded80	a strong, versatile player who can deputise at	a strong, versatile player who can deputise at centre back should the need arise, cleidson was linked with several european clubs during the 2012 summer transfer window: most notably vfb stuttgart of the german bundesliga as well as queens park rangers and swansea city, then both of the english premier league.	wikipedia
f39ce10e9e358b28dd11b75139e6b5e21f394c62	cleidson rodrigues dos santos or simply cleidson (born	cleidson rodrigues dos santos or simply cleidson (born 13 august 1988) is a brazilian footballer who plays primarily as a left back. he currently plays for itumbiara esporte clube.	wikipedia
85739f92f29b7e855a8472bc6c617d50a2653b93	dean edward martin is an english football manager,	dean edward martin is an english football manager, coach and former player who is assistant coach of Ía. a midfielder, martin spent much of his playing career in iceland with ka and Ía and he also played in england, hong kong and the republic of ireland. he moved into coaching and management whilst still a player and h...	wikipedia
74a3801b0687cf8707151715187ba0604c5a9bef	in international football, martin has assisted and coached	in international football, martin has assisted and coached iceland at youth level and in january 2017 he was appointed the ksÍ's head of talent identification. he departed his roles to become assistant manager of china women in november 2017, but he left the setup after manager sigurður ragnar eyjólfsson was sacked in ...	wikipedia
a59d902f45d63cad94d555d450253ccdcb3c698c	martin completed his ksÍ 'a' coaching badges in	martin completed his ksÍ 'a' coaching badges in 2009 and has a bsc in sports science from reykjavík university. he has held management roles at ka, breiðablik u19 and hk u19. he has worked as an assistant manager at club sides Ía, Íbv and hk. on 26 september 2018, martin was appointed as manager of newly-relegated 2. d...	wikipedia
51f1b330ba19e95f3e23e415b82593173a6848ab	between 1995 and 2001, martin spent the autumn	between 1995 and 2001, martin spent the autumn and winter months playing for clubs outside iceland. he returned to england to play for brentford (1995–96), hereford united (1998–99) and stevenage borough during the 1999–00 and 2000–01 seasons. he also played in hong kong for eastern aa and hong kong rangers and in the ...	wikipedia