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56c252e6b4f54f3bb0eb0a637b6c424135256023 | a brahmagupta quadrilateral is a cyclic quadrilateral with | a brahmagupta quadrilateral is a cyclic quadrilateral with integer sides, integer diagonals, and integer area. all brahmagupta quadrilaterals with sides a, b, c, d, diagonals e, f, area k, and circumradius r can be obtained by clearing denominators from the following expressions involving rational parameters t, u, and ... | wikipedia |
1b9fdacbc647df2934dddb418b9041960dc1b1f4 | four line segments, each perpendicular to one side | four line segments, each perpendicular to one side of a cyclic quadrilateral and passing through the opposite side's midpoint, are concurrent. these line segments are called the maltitudes, which is an abbreviation for midpoint altitude. their common point is called the anticenter. it has the property of being the refl... | wikipedia |
0b9a3ed1f23389a4bbae7798840ab0be544bad81 | let b {\displaystyle b} denote the angle between | let b {\displaystyle b} denote the angle between sides a {\displaystyle a} and b {\displaystyle b}, c {\displaystyle c} the angle between b {\displaystyle b} and c {\displaystyle c}, and d {\displaystyle d} the angle between c {\displaystyle c} and d {\displaystyle d}, then: | wikipedia |
c51bac03659541295fa1cbe95d3e0eefa20f3023 | a set of sides that can form a | a set of sides that can form a cyclic quadrilateral can be arranged in any of three distinct sequences each of which can form a cyclic quadrilateral of the same area in the same circumcircle (the areas being the same according to brahmagupta's area formula). any two of these cyclic quadrilaterals have one diagonal leng... | wikipedia |
fff3b4b5d82994fe1d6c6875b56b46378c3c8b0b | in a cyclic quadrilateral with successive vertices a, | in a cyclic quadrilateral with successive vertices a, b, c, d and sides a = ab, b = bc, c = cd, and d = da, the lengths of the diagonals p = ac and q = bd can be expressed in terms of the sides as | wikipedia |
b680ead61ae20bb0c7aca4cec9d9b285fa836770 | four unequal lengths, each less than the sum | four unequal lengths, each less than the sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals, which by brahmagupta's formula all have the same area. specifically, for sides a, b, c, and d, side a could be opposite any of side b, side c, or side d. | wikipedia |
ccef1a7bc01a70afeaa20295f8dbdb2214501545 | where s, the semiperimeter, is s = 1 | where s, the semiperimeter, is s = 1 / 2 (a + b + c + d). this is a corollary of bretschneider's formula for the general quadrilateral, since opposite angles are supplementary in the cyclic case. if also d = 0, the cyclic quadrilateral becomes a triangle and the formula is reduced to heron's formula. | wikipedia |
0ee7af261687689331e436fc23072c49de6ce02a | in a convex quadrilateral abcd, let efg be | in a convex quadrilateral abcd, let efg be the diagonal triangle of abcd and let ω {\displaystyle \omega } be the nine-point circle of efg. abcd is cyclic if and only if the point of intersection of the bimedians of abcd belongs to the nine-point circle ω {\displaystyle \omega }. | wikipedia |
b2152cf702219184c423b31225bf0b422e0734f9 | the intersection e may be internal or external | the intersection e may be internal or external to the circle. in the former case, the cyclic quadrilateral is abcd, and in the latter case, the cyclic quadrilateral is abdc. when the intersection is internal, the equality states that the product of the segment lengths into which e divides one diagonal equals that of th... | wikipedia |
051b6eeb0c7fdf464d4c2ae6b9dad5f327ff7a22 | other necessary and sufficient conditions for a convex | other necessary and sufficient conditions for a convex quadrilateral abcd to be cyclic are: let e be the point of intersection of the diagonals, let f be the intersection point of the extensions of the sides ad and bc, let ω {\displaystyle \omega } be a circle whose diameter is the segment, ef, and let p and q be pasca... | wikipedia |
d414f4a5e8b6f6a1109fb5906e822c23a787029c | in 1836 duncan gregory generalized this result as | in 1836 duncan gregory generalized this result as follows: given any convex cyclic 2 n -gon, then the two sums of alternate interior angles are each equal to (n -1) π {\displaystyle \pi }. this result can be further generalized as follows: lf a1a2...a2n (n > 1) is any cyclic 2 n -gon in which vertex ai->ai+k (vertex ai... | wikipedia |
4088aff4066c191115e556bd7912d4db6fb31e69 | any square, rectangle, isosceles trapezoid, or antiparallelogram is | any square, rectangle, isosceles trapezoid, or antiparallelogram is cyclic. a kite is cyclic if and only if it has two right angles – a right kite. a bicentric quadrilateral is a cyclic quadrilateral that is also tangential and an ex-bicentric quadrilateral is a cyclic quadrilateral that is also ex-tangential. a harmon... | wikipedia |
d9d67be2d8be26ff6e61179dc53c08e776014ab2 | all triangles have a circumcircle, but not all | all triangles have a circumcircle, but not all quadrilaterals do. an example of a quadrilateral that cannot be cyclic is a non-square rhombus. the section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to have a circumcircle. | wikipedia |
dd0c9c6469e8adcd08ab36c0baec89039ff1afcb | in euclidean geometry, a cyclic quadrilateral or inscribed | in euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. this circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. the center of the circle and its radius are called the circumcenter and the ci... | wikipedia |
211e3e519ee02fc56770c078f64052e90efc7bb1 | rai made his debut for the india under-23 | rai made his debut for the india under-23 side on 27 march 2015 against uzbekistan u23 in the 2016 afc u23 championship qualifiers. he played 80 minutes as india u23 fell 2–0. almost a year later, rai made his debut on the bench for the india senior side in their fifa world cup qualifier against iran. he did not come o... | wikipedia |
3e249ef26c8aaa9ef11a7d0012f189ae40c4736d | after helping dempo regain promotion to the i-league, | after helping dempo regain promotion to the i-league, rai signed with indian super league side kerala blasters on 6 september 2016. he made his debut for the team on 1 october 2016 against northeast united. he started the match and played 76 minutes as kerala blasters fell 1–0. | wikipedia |
6662a21da4e6b691a069f9d629bfc1bbfa30f2ac | born in duliajan, assam, rai was scouted by | born in duliajan, assam, rai was scouted by the duliajan football academy and later by the tata football academy at the age of 13 and trained there for four years. after graduating from tata, rai signed with i-league side dempo. on signing rai, dempo head coach at the time, arthur papas said "vinit is a very cultured m... | wikipedia |
d1eb4265612aa54ad90e63cdf378e3c1b067873f | oliver jasen burke is a scottish professional footballer | oliver jasen burke is a scottish professional footballer who plays as a right winger or forward for german bundesliga club werder bremen. he came through the academy at nottingham forest, signing a professional contract in september 2014 and making his debut the following february. in august 2016, burke signed for bund... | wikipedia |
8cfeacaeaee19209f8406c5e2072f71cb414b85d | during the 2017–18 season, burke was selected in | during the 2017–18 season, burke was selected in the under-21 squad. he made his debut at that level on 5 september, and he scored the first goal in a 2–0 win against the netherlands. in the 2018 toulon tournament, burke scored goals against france and south korea as scotland finished in fourth place. in the third-plac... | wikipedia |
0f31c869274f68e27143e6dd8a41568bbe399255 | he was left out of the full squad | he was left out of the full squad in june 2017, and was a late call-up to the under-20 squad for the 2017 toulon tournament. burke scored both goals in a 3–2 defeat against the czech republic. as captain, scotland claimed a historic first ever victory against brazil at any level, in the second match. the team went on t... | wikipedia |
988b68d4a793a7eed87dcf15b6b4ce399f0b5ecd | burke received his first call-up to the senior | burke received his first call-up to the senior scotland squad on 10 march 2016 for their friendly match against denmark. he made his debut as an 82nd-minute substitute for goalscorer matt ritchie in the 1–0 win at hampden park 19 days later. | wikipedia |
26b034f06a68d6a70d7378d611965c27cbeb822c | burke was eligible to play for england in | burke was eligible to play for england in addition to scotland, having been brought up south of the border from a young age, but he opted to play for his country of birth. he was first called up by scotland u19 and scored on his debut for the team, a 3–1 win against mexico u18, on 6 october 2015. he went on to make six... | wikipedia |
582b3d8266cf1bb746c690cf5c675b2fafba0468 | after making two appearances for bremen in august | after making two appearances for bremen in august 2023, burke again returned to the efl championship on 1 september, the last day of the transfer window, when he signed for birmingham city on loan for the 2023–24 season. he failed to scored in 23 appearances in all competitions, of which half were as a starter, and pla... | wikipedia |
c0c4bbbaa024c59aa1385b9df5db6a9992daa3f5 | burke scored his first goal for bremen on | burke scored his first goal for bremen on 13 august 2022, netting late to secure a 2–2 draw with vfb stuttgart. a week later, he scored the decisive goal in a dramatic, come-from-behind 3–2 win at borussia dortmund. | wikipedia |
5f0c9be5b59a3b0fb202d63630f1f0d2b33d2a53 | sheffield united signed burke from west bromwich albion | sheffield united signed burke from west bromwich albion on a three-year contract in september 2020; callum robinson moved in the opposite direction. burke scored his first goal for the club in a 2–1 away league win over manchester united on 27 january 2021, clinching their first league win against manchester united sin... | wikipedia |
e3ca3627e921592202eeec79e91cdd2c858c5260 | on 30 august 2019, burke was loaned to | on 30 august 2019, burke was loaned to la liga club alavés for the season. burke made his debut for his new club in their 1–0 defeat to sevilla on 15 september, and scored his first goal for alavés in a 2–1 la liga win over eibar on 7 february 2020. | wikipedia |
f97fe9e50106968c46b691c6e0c6112b92ce93d7 | burke moved on loan to celtic on 5 | burke moved on loan to celtic on 5 january 2019 until the end of the season. he made his debut two weeks later, playing the full 90 minutes of a 3–0 home win over airdrieonians in the scottish cup. on 23 january, his scottish premiership debut, he played as a striker in the absence of leigh griffiths and odsonne Édouar... | wikipedia |
7a6c5662fce9c9512f3cdd72923b0d603c367b27 | on 14 august 2018, burke scored his first | on 14 august 2018, burke scored his first goal for the baggies, the game's only in a home win over luton town in the first round of the efl cup. he made only six total appearances in the first half of the season, of which three were in the championship, all from the bench. | wikipedia |
6d709e5e2a54f9d10951c2d303e0302109e82c6c | he made his debut two days later, replacing | he made his debut two days later, replacing goalscorer jay rodriguez for the final two minutes of a 1–1 home draw against stoke city. owing to hamstring injury, he made only one other substitute appearance by mid-november and took part with the under-21 team in the efl trophy, scoring in a 2–1 group stage defeat at cov... | wikipedia |
139c0cbc0f1fbafb95972a9ba336e0c8ad86aa15 | burke returned to english football on 25 august | burke returned to english football on 25 august 2017, moving to premier league club west bromwich albion for a reported fee of £15 million, breaking his own record as the most expensive scottish footballer. he signed a five-year contract at the west midlands club. burke said that he chose west bromwich albion as a way ... | wikipedia |
1b03c53dd4c5815e8fa083270a875fc14339da4e | on his debut for leipzig, a home bundesliga | on his debut for leipzig, a home bundesliga game against borussia dortmund on 10 september, burke came on after 69 minutes in place of yussuf poulsen and provided an assist for naby keïta to score the only goal. fifteen days later he opened the scoring in a 1–1 draw at 1. fc köln, becoming the first scot to score in ge... | wikipedia |
fc23d25479feedf43b2f3e9f69b93b79e1412f53 | on 28 august 2016, in a move that | on 28 august 2016, in a move that caused outrage among forest supporters, who criticised the club's owner fawaz al-hasawi, burke joined newly promoted bundesliga club rb leipzig on a five-year contract for an undisclosed fee. the deal was reported by bbc sport to be worth around £13 million, which set a new record for ... | wikipedia |
f51348e51565b3eaa4152491a75897852a904cf5 | ahead of the 2016–17 season burke made an | ahead of the 2016–17 season burke made an impression in the pre-season friendlies, scoring three times in six matches. under the new management of philippe montanier, burke scored in the opening game of the season, a 4–3 win over burton albion, and scored a brace two weeks later against wigan athletic. burke scored on ... | wikipedia |
3ac635e37d9d13a6c6ec599d2d615b83a5a7e183 | ahead of the 2015–16 season, burke signed another | ahead of the 2015–16 season, burke signed another three-year contract with the club on 29 july. burke played for the club's reserves until december when he was called up to the first team by manager dougie freedman, whom burke considered his mentor. on 29 december, burke scored his first goal for forest from a curling ... | wikipedia |
f507d9afc9a76bec4296cd364357ba4a5330d532 | burke was sent on a month-long loan to | burke was sent on a month-long loan to league one club bradford city on 24 february. he made his debut for them on the same day, playing for 75 minutes in a 2–1 loss away to swindon town. burke made another start on 28 february in a 2–0 loss against peterborough united and appeared twice as an unused substitute before ... | wikipedia |
d2f7fac6b7b040dab953a49349e9dae5344b9daa | burke was born in kirkcaldy, scotland, but grew | burke was born in kirkcaldy, scotland, but grew up in melton mowbray, leicestershire, england where he joined local club mowbray rangers aged eight. after one season at the club, he was scouted to join nottingham forest's academy where he progressed for the next nine years. burke signed his first professional contract ... | wikipedia |
0a84750ccee203185a46ab13f0fadddfd85bf825 | he came through the academy at nottingham forest, | he came through the academy at nottingham forest, signing a professional contract in september 2014 and making his debut the following february. in august 2016, burke signed for bundesliga club rb leipzig for a reported fee of £13 million, a record fee for a scottish player. a year later, having helped leipzig to runne... | wikipedia |
c1d9d56243e299b738846593e38028e6d9dbd7d3 | in mathematical analysis, the kakutani fixed-point theorem is | in mathematical analysis, the kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. it provides sufficient conditions for a set-valued function defined on a convex, compact subset of a euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. the kakutani fixe... | wikipedia |
a1f4774d0bc78e13bbe7b715fd79ada12ca05de2 | in his game theory textbook, ken binmore recalls | in his game theory textbook, ken binmore recalls that kakutani once asked him at a conference why so many economists had attended his talk. when binmore told him that it was probably because of the kakutani fixed point theorem, kakutani was puzzled and replied, "what is the kakutani fixed point theorem?" | wikipedia |
412063d1946f2112f120efdf22501ec84dfae6d7 | kakutani's theorem for n-simplices can be used to | kakutani's theorem for n-simplices can be used to prove the theorem for an arbitrary compact, convex s. once again we employ the same technique of creating increasingly finer subdivisions. but instead of triangles with straight edges as in the case of n-simplices, we now use triangles with curved edges. in formal terms... | wikipedia |
ff9d11c9477d3d71d59f211f22f53a69a3fcc940 | in dimensions greater one, n -simplices are the | in dimensions greater one, n -simplices are the simplest objects on which kakutani's theorem can be proved. informally, a n -simplex is the higher-dimensional version of a triangle. proving kakutani's theorem for set-valued function defined on a simplex is not essentially different from proving it for intervals. the ad... | wikipedia |
abc201a2de6353fd948adce416d548d9f0c309d4 | otherwise, we can write the following. recall that | otherwise, we can write the following. recall that we can parameterize a line between two points a and b by (1-t)a + tb. using our finding above that q<x<p, we can create such a line between p and q as a function of x (notice the fractions below are on the unit interval). by a convenient writing of x, and since φ(x) is... | wikipedia |
5aea45bd10f4738edf37cc312dcea5f736dee456 | we have a pair of sequences of intervals, | we have a pair of sequences of intervals, and we would like to show them to converge to a limiting point with the bolzano-weierstrass theorem. to do so, we construe these two interval sequences as a single sequence of points, (a, p, b, q). this lies in the cartesian product × × ×, which is a compact set by tychonoff's ... | wikipedia |
b2def29e2dc4eb5ce5d0ddbaf70606a9834244ad | such a sequence can be constructed as follows. | such a sequence can be constructed as follows. let a = 0 and b = 1. let p be any point in φ(0) and q be any point in φ(1). then, conditions (1)–(4) are immediately fulfilled. moreover, since p ∈ φ(0) ⊂, it must be the case that p ≥ 0 and hence condition (5) is fulfilled. similarly condition (6) is fulfilled by q. | wikipedia |
145ebad975d8350a3a822c65864d92b749b87dc6 | thus, the closed intervals form a sequence of | thus, the closed intervals form a sequence of subintervals of. condition (2) tells us that these subintervals continue to become smaller while condition (3)–(6) tell us that the function φ shifts the left end of each subinterval to its right and shifts the right end of each subinterval to its left. | wikipedia |
b368d3741930a0a3fc88ba37703f0545153a11de | the proof of kakutani's theorem is simplest for | the proof of kakutani's theorem is simplest for set-valued functions defined over closed intervals of the real line. moreover, the proof of this case is instructive since its general strategy can be carried over to the higher-dimensional case as well. | wikipedia |
35a81d0e19b85a94a12e5cabbb00b4e0c035fdc1 | since s {\displaystyle s} is compact, we can | since s {\displaystyle s} is compact, we can take a convergent subsequence x n → x {\displaystyle x_{n}\to x}. then (x, x) ∈ graph (φ) {\displaystyle (x,x)\in \operatorname {graph} (\varphi)} since it is a closed set. | wikipedia |
bbbe027253d3e6a397f3ea97c6f34979a3144f62 | by the approximate selection theorem, there exists a | by the approximate selection theorem, there exists a sequence of continuous f n: s → s {\displaystyle f_{n}:s\to s} such that graph (f n) ⊂ 1 / n {\displaystyle \operatorname {graph} (f_{n})\subset _{1/n}}. by brouwer fixed-point theorem, there exists a sequence x n {\displaystyle x_{n}} such that f n (x n) = x n {\dis... | wikipedia |
ae2340447ff2bf2c88f58baa77fe76d1cc783d2a | in general equilibrium theory in economics, kakutani's theorem | in general equilibrium theory in economics, kakutani's theorem has been used to prove the existence of set of prices which simultaneously equate supply with demand in all markets of an economy. the existence of such prices had been an open question in economics going back to at least walras. the first proof of this res... | wikipedia |
ed05886daab83e84e3e3e584c8b164c00dfc4ca9 | mathematician john nash used the kakutani fixed point | mathematician john nash used the kakutani fixed point theorem to prove a major result in game theory. stated informally, the theorem implies the existence of a nash equilibrium in every finite game with mixed strategies for any finite number of players. this work later earned him a nobel prize in economics. in this cas... | wikipedia |
ef685083df67bf6f746f7b17d25c4da2637fcf6c | we can show this by using the closed | we can show this by using the closed graph theorem for set-valued functions, which says that for a compact hausdorff range space y, a set-valued function φ: x →2 has a closed graph if and only if it is upper hemicontinuous and φ (x) is a closed set for all x. since all euclidean spaces are hausdorff (being metric space... | wikipedia |
0d3dac440ff5fe9451dd75673e945f2cb50fe270 | the function has no fixed point. though it | the function has no fixed point. though it satisfies all other requirements of kakutani's theorem, its graph is not closed; for example, consider the sequences x = 0.5 - 1/ n, y = 3/4. | wikipedia |
8a4e405c97bfa6eff81773f942985e6590673ef0 | φ (x) = { 3 / 4 0 | φ (x) = { 3 / 4 0 ≤ x < 0.5 x = 0.5 1 / 4 0.5 < x ≤ 1 {\displaystyle \varphi (x)={\begin{cases}3/4&0\leq x<0.5\\{}&x=0.5\\1/4&0.5<x\leq 1\end{cases}}} | wikipedia |
e13c5072cdbba307672574f839c8404b810a64f3 | the function: φ (x) = {\displaystyle \varphi (x)=}, | the function: φ (x) = {\displaystyle \varphi (x)=}, shown on the figure at the right, satisfies all kakutani's conditions, and indeed it has many fixed points: any point on the 45° line (dotted line in red) which intersects the graph of the function (shaded in grey) is a fixed point, so in fact there is an infinity of ... | wikipedia |
4bef36b5caac8a272314b1de8b090dbdee1dba36 | in mathematical analysis, the kakutani fixed-point theorem is | in mathematical analysis, the kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. it provides sufficient conditions for a set-valued function defined on a convex, compact subset of a euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. the kakutani fixe... | wikipedia |
439f7cd7984bf0dd203bea51fce46f046054101e | in mathematics and computer science, connectivity is one | in mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements that need to be removed to separate the remaining nodes into two or more isolated subgraphs. it is closely related to the theory of network flow problems. the connectivity of a grap... | wikipedia |
3dbf457f9326a5bcc0ff47ed60b671b59db3f796 | in computational complexity theory, sl is the class | in computational complexity theory, sl is the class of problems log-space reducible to the problem of determining whether two vertices in a graph are connected, which was proved to be equal to l by omer reingold in 2004. hence, undirected graph connectivity may be solved in o(log n) space. | wikipedia |
484c08e99c5390855b9b60ee90b535cc5cf7616c | by menger's theorem, for any two vertices u | by menger's theorem, for any two vertices u and v in a connected graph g, the numbers κ (u, v) and λ (u, v) can be determined efficiently using the max-flow min-cut algorithm. the connectivity and edge-connectivity of g can then be computed as the minimum values of κ (u, v) and λ (u, v), respectively. | wikipedia |
ec5b3ddec3af4a1bdd890611b251985063f6822d | the problem of determining whether two vertices in | the problem of determining whether two vertices in a graph are connected can be solved efficiently using a search algorithm, such as breadth-first search. more generally, it is easy to determine computationally whether a graph is connected (for example, by using a disjoint-set data structure), or to count the number of... | wikipedia |
ae74caf750b3bf3f98e9ea7b9d36b20c68b074bd | menger's theorem asserts that for distinct vertices u, | menger's theorem asserts that for distinct vertices u, v, λ (u, v) equals λ ′(u, v), and if u is also not adjacent to v then κ (u, v) equals κ ′(u, v). this fact is actually a special case of the max-flow min-cut theorem. | wikipedia |
5cef5c40f7d18d992b9d8ed9ff8b3113ea660a0c | if u and v are vertices of a | if u and v are vertices of a graph g, then a collection of paths between u and v is called independent if no two of them share a vertex (other than u and v themselves). similarly, the collection is edge-independent if no two paths in it share an edge. the number of mutually independent paths between u and v is written ... | wikipedia |
8be13c66ad5035b4fb2e1ac6c29ee056d869f763 | a cutset x of g is called a | a cutset x of g is called a non-trivial cutset if x does not contain the neighborhood n(u) of any vertex u ∉ x. then the superconnectivity κ 1 {\displaystyle \kappa _{1}} of g is κ 1 (g) = min { | x |: x is a non-trivial cutset }. {\displaystyle \kappa _{1}(g)=\min\{|x|:x{\text{ is a non-trivial cutset}}\}.} | wikipedia |
5b982432638eb6e2fe542095d78b9be9873042e9 | more precisely: a g connected graph is said | more precisely: a g connected graph is said to be super-connected or super-κ if all minimum vertex-cuts consist of the vertices adjacent with one (minimum-degree) vertex.a g connected graph is said to be super-edge-connected or super-λ if all minimum edge-cuts consist of the edges incident on some (minimum-degree) vert... | wikipedia |
754ee3d0e5dcb412598f19a505043b7ae9910bbc | a graph is said to be super-connected or | a graph is said to be super-connected or super-κ if every minimum vertex cut isolates a vertex. a graph is said to be hyper-connected or hyper-κ if the deletion of each minimum vertex cut creates exactly two components, one of which is an isolated vertex. a graph is semi-hyper-connected or semi-hyper-κ if any minimum v... | wikipedia |
5478027104951bf954dda5a0a77da0effb902e38 | analogous concepts can be defined for edges. in | analogous concepts can be defined for edges. in the simple case in which cutting a single, specific edge would disconnect the graph, that edge is called a bridge. more generally, an edge cut of g is a set of edges whose removal renders the graph disconnected. the edge-connectivity λ (g) is the size of a smallest edge c... | wikipedia |
f030bac9b556d58a1654c8d22e4a56b7de729f5e | a vertex cut for two vertices u and | a vertex cut for two vertices u and v is a set of vertices whose removal from the graph disconnects u and v. the local connectivity κ (u, v) is the size of a smallest vertex cut separating u and v. local connectivity is symmetric for undirected graphs; that is, κ (u, v) = κ (v, u). moreover, except for complete graphs,... | wikipedia |
0e2af164c6fdaa54b1d00bfee80c1015018dfb81 | more precisely, any graph g (complete or not) | more precisely, any graph g (complete or not) is said to be k -vertex-connected if it contains at least k + 1 vertices, but does not contain a set of k − 1 vertices whose removal disconnects the graph; and κ (g) is defined as the largest k such that g is k -connected. in particular, a complete graph with n vertices, de... | wikipedia |
53aacbe68c6a25a7609ebdbd42c2e2d9fc980cc7 | a vertex cut or separating set of a | a vertex cut or separating set of a connected graph g is a set of vertices whose removal renders g disconnected. the vertex connectivity κ (g) (where g is not a complete graph) is the size of a minimal vertex cut. a graph is called k -vertex-connected or k -connected if its vertex connectivity is k or greater. | wikipedia |
c6f46e5833b99a880554bf85aa052bb7341e0aeb | a directed graph is called weakly connected if | a directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. it is unilaterally connected or unilateral (also called semiconnected) if it contains a directed path from u to v or a directed path from v to u for every pair of vertices u,... | wikipedia |
52d4ee044a91f43b44b2173f642527861bb657e2 | a graph is said to be connected if | a graph is said to be connected if every pair of vertices in the graph is connected. this means that there is a path between every pair of vertices. an undirected graph that is not connected is called disconnected. an undirected graph g is therefore disconnected if there exist two vertices in g such that no path in g h... | wikipedia |
bbaff27089a8f79aa4b124518e7b8d7a3dbee014 | in an undirected graph g, two vertices u | in an undirected graph g, two vertices u and v are called connected if g contains a path from u to v. otherwise, they are called disconnected. if the two vertices are additionally connected by a path of length 1 (that is, they are the endpoints of a single edge), the vertices are called adjacent. | wikipedia |
f73755ad0f707bc5c05e77a868bf9948dda8162b | in mathematics and computer science, connectivity is one | in mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. it is closely related to the theory of network flow problems. the conne... | wikipedia |
2c14a906fd17845f73205f96822501d7e31cda4b | for the 1952–1953 season, ratkovicz joined the milwaukee | for the 1952–1953 season, ratkovicz joined the milwaukee hawks. his playing time increased with the hawks, and he averaged over thirty minutes a game in his first year with the hawks. that season, he averaged nine-and-a-half points and nearly seven-and-a-half rebounds per game. he recorded similar numbers during the 19... | wikipedia |
7f189c30430bf405b1398523c7d439b1513962ec | ratkovicz's role was reduced during the 1951–1952 season; | ratkovicz's role was reduced during the 1951–1952 season; playing in twenty and a half minutes a game, he averaged seven and a half points and five rebounds per game while serving as a backup to dolph schayes. | wikipedia |
9de6653192194daea245fdef2554cb1963d64c2a | statistically, ratkovicz's best season came the following year | statistically, ratkovicz's best season came the following year during the 1950–1951 season. playing in all 66 of the nationals' team games, he averaged thirteen points and over eight rebounds per game. his 41.5% field goal percentage was the sixth highest in the nba that season, and he also ranked in the top ten in fre... | wikipedia |
e93c7398cffc202412c28e4de0f9d95332c2e9ef | when the nbl merged with the basketball association | when the nbl merged with the basketball association of america and formed the national basketball association, ratkovicz joined the syracuse nationals for the 1949–1950 season. playing alongside future basketball hall of famers dolph schayes and al cervi, ratkovicz averaged eight and a half points per game in his first... | wikipedia |
7ae332e16acbb71cc6a7538c441294cf5934c52c | ratkovicz's pro career began when he was 19 | ratkovicz's pro career began when he was 19 years old, playing 13 games for the chicago bruins of the national basketball league during the 1941–1942 season. ratkovicz missed the next three seasons due to military service, then returned to chicago in time for the 1945–1946 season, this time playing for the chicago amer... | wikipedia |
9bab0a7ed74a93edf824c377967228c94e62ebb8 | george ratkovicz (november 13, 1922 – november 10, | george ratkovicz (november 13, 1922 – november 10, 2008) was an american basketball player who played for five seasons in the national basketball league and for six seasons in the national basketball association. he played center and forward during his career. | wikipedia |
4ea3b7c2f866a5a185afe3fda8b7423ce6a6ab2d | keita fujimura (藤村 慶太, fujimura keita, born september | keita fujimura (藤村 慶太, fujimura keita, born september 2, 1993 in iwate) is a japanese football player. he plays for zweigen kanazawa. | wikipedia |
c38343aa1921161e87c5ed8bda0f27ecc8c83304 | the 2018 ncaa division i men's soccer season | the 2018 ncaa division i men's soccer season was the 60th season of ncaa championship men's college soccer. the regular season began on august 24, 2018, and continued into the first weekend of november 2018. the season culminated with the four-team college cup at meredith field at harder stadium in santa barbara, calif... | wikipedia |
ce97a51e92c4b0c4de076e30fc7a4c16c261c209 | note 3: after six years of divisional play, | note 3: after six years of divisional play, with home-and-home series within the divisions, the big west conference teams will be playing a single, eight-team, seven game schedule. this move comes two years before cal state bakersfield moves in to make the big west a nine team league. | wikipedia |
60eda72b228c2ba6aaf96cf7813e53598e30c992 | note 2: the team formerly known as the | note 2: the team formerly known as the fort wayne mastodons officially changed its athletic identity to the purdue fort wayne mastodons on june 18, 2018, shortly before the creation of purdue university fort wayne (pfw). previously, the pfw athletic program represented indiana university – purdue university fort wayne ... | wikipedia |
c38343aa1921161e87c5ed8bda0f27ecc8c83304 | the 2018 ncaa division i men's soccer season | the 2018 ncaa division i men's soccer season was the 60th season of ncaa championship men's college soccer. the regular season began on august 24, 2018, and continued into the first weekend of november 2018. the season culminated with the four-team college cup at meredith field at harder stadium in santa barbara, calif... | wikipedia |
8ad7582bc513aba09ef8e53be07a2a4d65ec2a3d | sarah l. pett is a professor of infectious | sarah l. pett is a professor of infectious diseases at university college london. pett is interested in the immunopathology of infections and the development of optimised treatment pathways for infections. during the covid-19 pandemic, pett led a clinical trial that investigated the efficacy of remdesivir as a treatmen... | wikipedia |
ce0443da51eb15d0237fb1b72d8e136323ce94c4 | during the covid-19 pandemic pett led the adaptive | during the covid-19 pandemic pett led the adaptive covid-19 treatment trial (actt-eu/uk), a clinical trial into the efficacy of remdesivir as a treatment from coronavirus disease. adult inpatients were given remdesivir or a placebo through a drip for up to ten days of their stay in hospital. pett showed that patients t... | wikipedia |
68c1f369e233eb1f9ec64bf7f67ec3a756dbcfe9 | in 2000 pett joined the kirby institute in | in 2000 pett joined the kirby institute in new south wales, where she led international randomized controlled trials. in 2013 pett joined the medical research council clinical trials unit at university college london. she was promoted to the chair of the infectious diseases theme in 2016. | wikipedia |
8ad7582bc513aba09ef8e53be07a2a4d65ec2a3d | sarah l. pett is a professor of infectious | sarah l. pett is a professor of infectious diseases at university college london. pett is interested in the immunopathology of infections and the development of optimised treatment pathways for infections. during the covid-19 pandemic, pett led a clinical trial that investigated the efficacy of remdesivir as a treatmen... | wikipedia |
3c2fe54ac1fa4bf70f6c667d72c30bc7573af46a | john willard "jack" turner was a national basketball | john willard "jack" turner was a national basketball association (nba) player for the chicago packers. he was drafted with the ninth pick in the second round of the 1961 nba draft by the chicago packers. in his one nba season, turner averaged 4.8 points per game, 2.0 rebounds per game and 1.0 assist per game. turner di... | wikipedia |
730e8a5128536858c6f5b48f6a8bd837e7abde34 | john willard "jack" turner (june 5, 1939 – | john willard "jack" turner (june 5, 1939 – may 5, 2013) was a national basketball association (nba) player for the chicago packers. he was drafted with the ninth pick in the second round of the 1961 nba draft by the chicago packers. in his one nba season, turner averaged 4.8 points per game, 2.0 rebounds per game and 1... | wikipedia |
5ca6b122a4d2d74aef3bf4abdf811dd701d42194 | diane b. paul is an american historian of | diane b. paul is an american historian of science who is emeritus professor of political science in the college of liberal arts at the university of massachusetts boston. she taught in the university of massachusetts boston's department of political science for 33 years prior to her retirement in 2003. among the positi... | wikipedia |
ee45caac566b656f306076084b58babc76a517b5 | diane b. paul (born february 4, 1946) is | diane b. paul (born february 4, 1946) is an american historian of science who is emeritus professor of political science in the college of liberal arts at the university of massachusetts boston. she taught in the university of massachusetts boston's department of political science for 33 years prior to her retirement i... | wikipedia |
62241f352da604f05e90c04c5712c608d08d970a | the hanle effect, also known as zero-field level | the hanle effect, also known as zero-field level crossing, is a reduction in the polarization of light when the atoms emitting the light are subject to a magnetic field in a particular direction, and when they have themselves been excited by polarized light. experiments which utilize the hanle effect include measuring ... | wikipedia |
1f79ea62e6423ba8b343f3d02fb243009942e925 | the zero-field hanle level crossings involve magnetic fields, | the zero-field hanle level crossings involve magnetic fields, in which the states which are degenerate at zero magnetic field are split due to the zeeman effect. there is also the closely analogous zero-field stark level crossings with electric fields, in which the states which are degenerate at zero electric field are... | wikipedia |
761dc6546ca80b0cef9bce159640d04b65c0a5b4 | the effect was initially considered in the context | the effect was initially considered in the context of gasses, followed by applications to solid state physics. it has been used to measure both the states of localized electrons and free electrons. for spin-polarized electrical currents, the hanle effect provides a way to measure the effective spin lifetime in a partic... | wikipedia |
13959ea839caafe03de3a9a519f8b0e8fd2f067c | the first experimental evidence for the effect came | the first experimental evidence for the effect came from robert w. wood, and lord rayleigh. the effect is named after wilhelm hanle, who was the first to explain the effect, in terms of classical physics, in zeitschrift für physik in 1924. initially, the causes of the effect were controversial, and many theorists mista... | wikipedia |
5401995dba8f599d42611b3d2849994bb44ad274 | the hanle effect, also known as zero-field level | the hanle effect, also known as zero-field level crossing, is a reduction in the polarization of light when the atoms emitting the light are subject to a magnetic field in a particular direction, and when they have themselves been excited by polarized light. | wikipedia |
53d1ead516ce409e73949948f7935af7369b98c1 | the enzyme phenylalanine ammonia lyase catalyzes the conversion | the enzyme phenylalanine ammonia lyase catalyzes the conversion of l-phenylalanine to ammonia and trans-cinnamic acid.: phenylalanine ammonia lyase (pal) is the first and committed step in the phenyl propanoid pathway and is therefore involved in the biosynthesis of the polyphenol compounds such as flavonoids, phenylpr... | wikipedia |
74876f8cab6267383e0c2588e454187ac1a1673c | as of late 2007, 5 structures have been | as of late 2007, 5 structures have been solved for this class of enzymes, with pdb accession codes 1t6j, 1t6p, 1w27, 1y2m, and 2nyf. | wikipedia |
a7ae0bd04155f7b6cf1926eb0538e5904824136c | analogous to how aspartame is synthesized, pal is | analogous to how aspartame is synthesized, pal is also used to synthesize unnatural amino acids from various substituted cinnamic acids for research purposes. steric hindrance from arene substitution limits pal's utility for this purpose however. for instance, when rhodotorula glutinis was used to affect this biotransf... | wikipedia |
745568f3697d0f0ee91a7f4d2f922a4db031a1ba | the reverse reaction catalyzed by pal has been | the reverse reaction catalyzed by pal has been explored for use to convert trans -cinnamic acid to l -phenylalanine, which is a precursor of the sweetener aspartame. this process was developed by genex corporation but was never commercially adopted. | wikipedia |
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