Split (1)

train · 485k rows

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.

id stringlengths 40 40	title stringlengths 15 120	text stringlengths 41 3.14k	source stringclasses 2 values
9172e48eb256058ab2aa9b9ef8480907b9ceb264	in the semi-discrete case, x = y =	in the semi-discrete case, x = y = r d {\textstyle x=y=\mathbb {r} ^{d}} and μ {\textstyle \mu } is a continuous distribution over r d {\textstyle \mathbb {r} ^{d}}, while ν = ∑ j = 1 j ν j δ y i {\textstyle \nu =\sum _{j=1}^{j}\nu _{j}\delta _{y_{i}}} is a discrete distribution which assigns probability mass ν j {\tex...	wikipedia
43791957c96cb81b5e0bfeb790cf10d68b692c3a	in the semi-discrete case, x = y =	d μ (x) + ∑ j = 1 j ψ j ν j: ψ j + φ (x) ≤ c (x, y j) } {\displaystyle \sup \left\{\int _{x}\varphi (x)d\mu (x)+\sum _{j=1}^{j}\psi _{j}\nu _{j}:\psi _{j}+\varphi (x)\leq c\left(x,y_{j}\right)\right\}} for the dual, which can be rewritten as: sup ψ ∈ r j { ∫ x inf j { c (x, y j) − ψ j } d μ (x) + ∑ j = 1 j ψ j ν j } {...	wikipedia
e53939dcfc4219abaa05da9f1a4f54c41878d2db	where ⊗ {\textstyle \otimes } is the kronecker	where ⊗ {\textstyle \otimes } is the kronecker product, 1 n × m {\textstyle 1_{n\times m}} is a matrix of size n × m {\textstyle n\times m} with all entries of ones, and i n {\textstyle i_{n}} is the identity matrix of size n {\textstyle n}. as a result, setting z = vec (γ) {\textstyle z=\operatorname {vec} \left(\gamm...	wikipedia
82c8504e3a9a341f06f74d76da75f2c15ffad9f9	in order to input this in a linear	in order to input this in a linear programming problem, we need to vectorize the matrix γ x y {\textstyle \gamma _{xy}} by either stacking its columns or its rows, we call vec {\textstyle \operatorname {vec} } this operation. in the column-major order, the constraints above rewrite as	wikipedia
6ebec818c7b954879848af2b576b451395c1ca59	in the case where the margins μ {\textstyle	in the case where the margins μ {\textstyle \mu } and ν {\textstyle \nu } are discrete, let μ x {\textstyle \mu _{x}} and ν y {\textstyle \nu _{y}} be the probability masses respectively assigned to x ∈ x {\textstyle x\in \mathbf {x} } and y ∈ y {\textstyle y\in \mathbf {y} }, and let γ x y {\textstyle \gamma _{xy}} be...	wikipedia
c57cfb37397b5aeec5e48a2d9475b548940d1df7	for 1 ≤ p < ∞ {\displaystyle 1\leq	for 1 ≤ p < ∞ {\displaystyle 1\leq p<\infty }, let p p (r) {\displaystyle {\mathcal {p}}_{p}(\mathbb {r})} denote the collection of probability measures on r {\displaystyle \mathbb {r} } that have finite p {\displaystyle p} -th moment. let μ, ν ∈ p p (r) {\displaystyle \mu,\nu \in {\mathcal {p}}_{p}(\mathbb {r})} and l...	wikipedia
a185791a860706c8d2950cec38eb28a112011d1a	the economic interpretation is clearer if signs are	the economic interpretation is clearer if signs are flipped. let x ∈ x {\textstyle x\in x} stand for the vector of characteristics of a worker, y ∈ y {\textstyle y\in y} for the vector of characteristics of a firm, and Φ (x, y) = − c (x, y) {\textstyle \phi (x,y)=-c(x,y)} for the economic output generated by worker x {...	wikipedia
59781db311cecc684b1e300e96d7b19aab8510f5	the economic interpretation is clearer if signs are	mathbb {r} }. if the dual problem has a solution, one can see that: v (y) = sup x { Φ (x, y) − u (x) } {\displaystyle v(y)=\sup _{x}\left\{\phi (x,y)-u(x)\right\}} so that u (x) {\textstyle u(x)} interprets as the equilibrium wage of a worker of type x {\textstyle x}, and v (y) {\textstyle v(y)} interprets as the equil...	wikipedia
6a28dd334418222f3507e6460ba260aa140b0e20	where the supremum runs over all pairs of	where the supremum runs over all pairs of bounded and continuous functions φ: x → r {\displaystyle \varphi:x\rightarrow \mathbb {r} } and ψ: y → r {\displaystyle \psi:y\rightarrow \mathbb {r} } such that	wikipedia
9ffcee5a78802b5d4c3a14ab0d8eca69f2b3d5fb	where Γ (μ, ν) {\displaystyle \gamma (\mu,\nu)} denotes	where Γ (μ, ν) {\displaystyle \gamma (\mu,\nu)} denotes the collection of all probability measures on x × y {\displaystyle x\times y} with marginals μ {\displaystyle \mu } on x {\displaystyle x} and ν {\displaystyle \nu } on y {\displaystyle y}. it can be shown that a minimizer for this problem always exists when the c...	wikipedia
205716c1d97abf19504b9e7eb6f3892b329ddde9	we can improve on this by adopting kantorovich's	we can improve on this by adopting kantorovich's formulation of the optimal transportation problem, which is to find a probability measure γ {\displaystyle \gamma } on x × y {\displaystyle x\times y} that attains the infimum	wikipedia
19559cb5570f6812064688280eb7de3f35eb596b	monge's formulation of the optimal transportation problem can	monge's formulation of the optimal transportation problem can be ill-posed, because sometimes there is no t {\displaystyle t} satisfying t ∗ (μ) = ν {\displaystyle t_{*}(\mu)=\nu }: this happens, for example, when μ {\displaystyle \mu } is a dirac measure but ν {\displaystyle \nu } is not.	wikipedia
3134db8859f04055f2f6952bb66487e64b5d7633	where t ∗ (μ) {\displaystyle t_{*}(\mu)} denotes the	where t ∗ (μ) {\displaystyle t_{*}(\mu)} denotes the push forward of μ {\displaystyle \mu } by t {\displaystyle t}. a map t {\displaystyle t} that attains this infimum (i.e. makes it a minimum instead of an infimum) is called an "optimal transport map".	wikipedia
8659fdbaa41eb5faa8d45e3c4e85501428174443	let x {\displaystyle x} and y {\displaystyle y}	let x {\displaystyle x} and y {\displaystyle y} be two separable metric spaces such that any probability measure on x {\displaystyle x} (or y {\displaystyle y}) is a radon measure (i.e. they are radon spaces). let c: x × y → {\displaystyle c:x\times y\to } be a borel- measurable function. given probability measures μ {...	wikipedia
4a60a007c8ab6b0c41c4df32049d561d31ccf961	the transportation problem as it is stated in	the transportation problem as it is stated in modern or more technical literature looks somewhat different because of the development of riemannian geometry and measure theory. the mines-factories example, simple as it is, is a useful reference point when thinking of the abstract case. in this setting, we allow the pos...	wikipedia
6ff6080449037d16ff83f26326c48c410ac6843a	note that the above cost functions consider only	note that the above cost functions consider only the horizontal distance traveled by the books, not the horizontal distance traveled by a device used to pick each book up and move the book into position. if the latter is considered instead, then, of the two transport plans, the second is always optimal for the euclidea...	wikipedia
e7836fd69f6fcdc7a3ef6d06cb103a9b87f065e7	if the cost function is proportional to euclidean	if the cost function is proportional to euclidean distance (c (x, y) = α ‖ x − y ‖ {\displaystyle c(x,y)=\alpha \\|x-y\\|} for some α > 0 {\displaystyle \alpha >0}) then these two candidates are both optimal. if, on the other hand, we choose the strictly convex cost function proportional to the square of euclidean distan...	wikipedia
94a0effe06f2e452128364805386a6795361aecd	the following simple example illustrates the importance of	the following simple example illustrates the importance of the cost function in determining the optimal transport plan. suppose that we have n {\displaystyle n} books of equal width on a shelf (the real line), arranged in a single contiguous block. we wish to rearrange them into another contiguous block, but shifted on...	wikipedia
a674bfe33bd13c26fb56908d51fcb2f0858c1e88	is the least of all possible transport plans	is the least of all possible transport plans from m {\displaystyle m} to f {\displaystyle f}. this motivating special case of the transportation problem is an instance of the assignment problem.more specifically, it is equivalent to finding a minimum weight matching in a bipartite graph.	wikipedia
b5cdf95d69843c0754aa95aec87cc0023e4f6448	suppose that we have a collection of m	suppose that we have a collection of m {\displaystyle m} mines mining iron ore, and a collection of n {\displaystyle n} factories which use the iron ore that the mines produce. suppose for the sake of argument that these mines and factories form two disjoint subsets m {\displaystyle m} and f {\displaystyle f} of the eu...	wikipedia
e498ea83452d47fb86bf362494279e4585d191e3	major advances were made in the field during	major advances were made in the field during world war ii by the soviet mathematician and economist leonid kantorovich. consequently, the problem as it is stated is sometimes known as the monge–kantorovich transportation problem. the linear programming formulation of the transportation problem is also known as the hitc...	wikipedia
e62ef8a425e14e7495c624e8eb98e704d174cbb6	in the 1920s a.n. tolstoi was one of	in the 1920s a.n. tolstoi was one of the first to study the transportation problem mathematically. in 1930, in the collection transportation planning volume i for the national commissariat of transportation of the soviet union, he published a paper "methods of finding the minimal kilometrage in cargo-transportation in ...	wikipedia
ab12efa98af4053ec0f4d8f67a4f111988e3c4b6	stanisław barcewicz was a polish violinist, conductor and	stanisław barcewicz was a polish violinist, conductor and teacher. although his repertoire included almost all of the classical and romantic violin literature, he was valued primarily for his interpretations of works by henryk wieniawski and felix mendelssohn. he also premiered works by his teacher pyotr ilyich tchaiko...	wikipedia
623849aff69f42156d32d826dffe5ff54ab3b21f	as an indication of his standing, other violinists	as an indication of his standing, other violinists in the collection include joseph joachim, eugène ysaÿe, pablo de sarasate, arnold rosé, jan kubelík, willy burmester, jacques thibaud, marie hall, franz von vecsey, joseph szigeti and fritz kreisler.	wikipedia
a88ca2a05940bb4c93ff5216dce0c5807df043e0	in 1902 his former pupil mieczysław karłowicz wrote	in 1902 his former pupil mieczysław karłowicz wrote his violin concerto in a major, op. 8 for barcewicz, who premiered it on 21 march 1903 with the berlin philharmonic under the composer.	wikipedia
6d2cefead9c784be89e7bbd91d714686b406285f	barcewicz also founded and led the warsaw string	barcewicz also founded and led the warsaw string quartet from 1892 until his death, and he also played in a renowned piano trio with the polish pianist aleksander michałowski and the russian cellist aleksandr verzhbilovich.	wikipedia
2f3479c94858f5ff1338868a0f041d7415abfbb1	in 1885, he became the concertmaster and second	in 1885, he became the concertmaster and second conductor of the warsaw opera. he occasionally conducted operas there, and he also conducted the warsaw philharmonic orchestra. in 1886, he was appointed violin and viola professor at the warsaw conservatory, and was the institution's director from 1910 to 1918, succeedin...	wikipedia
7496fa218e1b3cd06914f4cf66ee61b52236bce3	in 1881 barcewicz premiered johan svendsen 's romance	in 1881 barcewicz premiered johan svendsen 's romance for violin and orchestra in kristiania (oslo). on 14 january 1892 he gave the polish premieres of tchaikovsky's violin concerto in d and sérénade mélancolique, in warsaw, under the composer's baton.	wikipedia
c518f5637602308ea8eb6dac049481481e8b1846	on 20 or 21 september 1878, as part	on 20 or 21 september 1878, as part of the 1878 paris world exhibition, he performed at the trocadéro in a concert of works by tchaikovsky, including the first public performance of the valse-scherzo in c, conducted by nikolai rubinstein. he later toured europe extensively, including appearances in leipzig, dresden, ha...	wikipedia
2c115c733ae79f52017942a47b445386341925a9	stanisław barcewicz was born in warsaw in 1858,	stanisław barcewicz was born in warsaw in 1858, and first studied violin at the institute of music there under apollinaire de kontski (apolinary kątski) and władysław gorski. at the age of 11 he publicly performed beriot 's violin concerto no. 7 in g major. he then studied at the moscow conservatory, where his teachers...	wikipedia
88e5cfe049f3342ec717d4bf870139011401e589	stanisław barcewicz (16 april 1858 – 1 september	stanisław barcewicz (16 april 1858 – 1 september 1929) was a polish violinist, conductor and teacher. although his repertoire included almost all of the classical and romantic violin literature, he was valued primarily for his interpretations of works by henryk wieniawski and felix mendelssohn. he also premiered works ...	wikipedia
5d65789f43cd29224fef8c12f04d1e497ad1beda	following two seasons with ik oskarshamn, alm left	following two seasons with ik oskarshamn, alm left the shl and opted to sign a one-year contract in the hockeyallsvenskan, with newly relegated club, djurgårdens if on 30 september 2022.	wikipedia
0f15f2bb868495a6bdac7bc1ebf9a9e6ef997846	on may 30, 2016, he signed a deal	on may 30, 2016, he signed a deal to return to his native sweden with former club, skellefteå aik of the shl. alm played two seasons in his third stint within the organization before leaving as a free agent following the 2018–19 season.	wikipedia
b889f7a67f28e4e2b10331eae894a872f6e3c98f	on may 28, 2014, alm was signed to	on may 28, 2014, alm was signed to a two-year entry-level contract with the nashville predators of the national hockey league (nhl). during his two years in the predators' organization, he did not see any playing time in the nhl and made 81 appearances for nashville's ahl affiliate, the milwaukee admirals.	wikipedia
b9950e3bc28c3193d6ee1f233026db42bb344b0e	undrafted, alm played as a youngster in his	undrafted, alm played as a youngster in his native sweden with skellefteå aik. he made his senior swedish hockey league (shl) debut with frölunda hc during the 2009–10 season, before returning to skellefteå for the 2010–11 campaign. in 2013 and 2014, he won the swedish championship with the club.	wikipedia
540bcf73b0b94d561149661eb0edaa2a560888c1	psr b1620-26 b is an exoplanet located approximately	psr b1620-26 b is an exoplanet located approximately 12,400 light-years from earth in the constellation of scorpius. it bears the unofficial nicknames "methuselah" and "the genesis planet" due to its extreme age. the planet is in a circumbinary orbit around the two stars of psr b1620-26 and is the first circumbinary pl...	wikipedia
bd52ab5107b66e6dcbc43bed186ad6ac1727fa8d	though not officially recognized, the name "methuselah" is	though not officially recognized, the name "methuselah" is commonly used for the planet in popular articles. this name is usually used as the informal name to show the similarities to the planets of the solar system, while the "letter name" is used astronomically. methuselah is the only planet to have received a biblic...	wikipedia
fed1d4357560e46de81f92eb44282ba95f4fd586	while the designation psr b1620-26 b is not	while the designation psr b1620-26 b is not used in any scientific papers, the planet is listed in the simbad database as psr b1620-26 b. some popular sources use the designation psr b1620-26 c to refer to the planet, as it was described as the third member of a triple system (composed of the planet and two stars). thi...	wikipedia
23ed742ffdea629bca4463c036b1c917ca2cd388	the study of the planetary orbit allowed the	the study of the planetary orbit allowed the mass of the white dwarf star to be estimated as well, and theories of the formation of the planet suggested that the white dwarf should be young and hot. on july 10, 2003, the detection of the white dwarf and confirmation of its predicted properties were announced by a team ...	wikipedia
ca91a96c41c22af1b959132427412dda992179b1	like nearly all extrasolar planets discovered prior to	like nearly all extrasolar planets discovered prior to 2008, psr b1620-26 b was originally detected through the doppler shifts its orbit induces on radiation from the star it orbits (in this case, changes in the apparent pulsation period of the pulsar). in the early 1990s, a group of astronomers led by donald backer, w...	wikipedia