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id stringlengths 40 40	title stringlengths 15 120	text stringlengths 41 3.14k	source stringclasses 2 values
513a80d5add75ffd33ff3f30123608d8331a6846	some may argue that matrix planting is just	some may argue that matrix planting is just another term for ground cover, but matrix planting is concerned with successive layers of vegetation, one above the other, through which plants form multi-dimensional communities. few would refer to the stratified vegetation of a wood as ground cover, though seen from a bird’...	wikipedia
0ee752e844d770209786dfd99c3535c1a4df78af	the strongest matrices consist of a succession of	the strongest matrices consist of a succession of layers of vegetation through which sunlight filters, until at ground level there is enough only to support plants that can cope with very little light. the best examples of such matrices occur in deciduous woodlands, but that does not mean all gardens have to become mic...	wikipedia
788065d9aadf850095de64e20089faf4c4302f4f	the aim of matrix planting is 1) encourage	the aim of matrix planting is 1) encourage the plants you do want, and 2) discourage the plants you do not want. the key to success lies in the choice of plants. ill-judged choices result in excessive dominance by one or two species, and the disappearance of those that cannot cope. well judged choices lead to the estab...	wikipedia
84abe8c4fa1979ed7b93bc47eea165ae1a469019	matrix planting is based on choosing and managing	matrix planting is based on choosing and managing plants in ways which enable them to form similar matrices in the garden. the aim is to enable the plants to occupy the ground and the space above it so effectively that no space is left for weeds and to do this in ways that are decorative and sympathetic to the setting ...	wikipedia
454bba272cc71a7c1d3bec82e4d35033f7ccc285	kim hei-sook is a south korean academic and	kim hei-sook is a south korean academic and university administrator. she has been a professor of philosophy at ewha womans university since 1987, and was elected president of the university in 2017. kim began her studies at ewha, graduating with a bachelor of arts in english literature in 1976 and a master of arts in ...	wikipedia
c6879145867d89aa7193e365d94223bca11eeee5	as co-chair of the faculty council, kim was	as co-chair of the faculty council, kim was one of the leaders of protests against choi kyung-hee, who resigned as president of ewha in october 2016 over corruption charges and was later sentenced to imprisonment. in early 2017, kim was elected as choi's replacement as university president, winning the first-ever direc...	wikipedia
b51a89f677a0ecebdf3b6a7c15fb6d94001a937f	kim began her studies at ewha, graduating with	kim began her studies at ewha, graduating with a bachelor of arts in english literature in 1976 and a master of arts in christian philosophy in 1979. she completed a doctorate at the university of chicago in 1987, with a thesis on epistemology. kim subsequently returned to ewha to take up a professorship in the philoso...	wikipedia
1ad844e91db7fbcad212696ebf010768f6401f38	kim hei-sook (korean: 김혜숙; born c. 1955) is	kim hei-sook (korean: 김혜숙; born c. 1955) is a south korean academic and university administrator. she has been a professor of philosophy at ewha womans university since 1987, and was elected president of the university in 2017.	wikipedia
33dab0a999cc3daa64a9b9d53fb5684509655a1a	joan m. redwing is an american materials scientist	joan m. redwing is an american materials scientist known for research on electronic and optoelectronic materials, including the processing of semiconductor thin films and nanomaterials by metalorganic chemical vapor deposition (mocvd). redwing is a distinguished professor of materials science and engineering and electr...	wikipedia
edc04ad256c758bf2f28e65a362f779b83ebcd2b	in 2015, she was named a fellow of	in 2015, she was named a fellow of the materials research society, a distinction conferred to members whose "sustained and distinguished contributions to the advancement of materials research are internationally recognized". in the following year she was named a fellow of the american association for the advancement of...	wikipedia
47c056c4dea19aafcd25f3c7d617f10aa0a9abd2	redwing was elected a fellow of the american	redwing was elected a fellow of the american physical society in 2012, after a nomination from the aps division of materials physics, for "key contributions to the mechanistic understanding of materials synthesis by vapor growth, including si and sige nanowires, group-iii nitrides and boride-based superconductors."	wikipedia
43b080aec8cf7d510cacb184f28fb16079945235	redwing has served as an editor of the	redwing has served as an editor of the journal of crystal growth and on the executive editorial board of 2d materials. she chaired the 2018 materials research society fall meeting with kristen h. brosnan, david lavan, patrycja paruch, and takao someya.	wikipedia
70c940f36003d8832df4bf3393c7f1c3e5e1e037	in 2014, redwing's research group was one of	in 2014, redwing's research group was one of three groups at the university's center for two-dimensional and layered materials (2dlm) that received an award from the national science foundation (nsf) to research 2d materials. as a 2016 fulbright scholar to sweden, redwing spent three months at lund university with the ...	wikipedia
1ab7b44d4d75f8612ceb6da1ec65a07854d369ba	redwing joined advanced technology materials inc., in connecticut,	redwing joined advanced technology materials inc., in connecticut, as a research engineer, where she researched group iii - nitride semiconductors grown by mocvd, especially aluminum gallium arsenide (algan) and gallium nitride (gan) two-dimensional electron gas heterostructures, and co-authored several patents. she mo...	wikipedia
d44b575967ed4ae8d2780c159e6e6f6e6816896b	joan m. redwing attended the university of pittsburgh,	joan m. redwing attended the university of pittsburgh, pennsylvania, where she received a bachelor of science in chemical engineering in 1986. after graduation, from 1986 to 1988, she worked at general electric corporate research & development in new york where her work was focused on tungsten-coated x-ray targets prod...	wikipedia
33dab0a999cc3daa64a9b9d53fb5684509655a1a	joan m. redwing is an american materials scientist	joan m. redwing is an american materials scientist known for research on electronic and optoelectronic materials, including the processing of semiconductor thin films and nanomaterials by metalorganic chemical vapor deposition (mocvd). redwing is a distinguished professor of materials science and engineering and electr...	wikipedia
0b7e7a0862a566bbbab3349543e96688e1e1aabe	in abstract algebra, kaplansky's theorem on projective modules,	in abstract algebra, kaplansky's theorem on projective modules, first proven by irving kaplansky, states that a projective module over a local ring is free; where a not-necessarily-commutative ring is called local if for each element x, either x or 1 − x is a unit element. the theorem can also be formulated so to chara...	wikipedia
f39d72c2a7335c8502aab741d419c396c0f94826	show that a countably generated projective module over	show that a countably generated projective module over a local ring is free. the idea of the proof of the theorem was also later used by hyman bass to show big projective modules are free. according to, kaplansky's theorem "is very likely the inspiration for a major portion of the results" in the theory of semiperfect ...	wikipedia
e2e524832ea88e3b1c703226b35692cbae238fb6	(⇒) {\displaystyle (\rightarrow)} is by azumaya's theorem as	(⇒) {\displaystyle (\rightarrow)} is by azumaya's theorem as in the proof of 1. ⇒ 2. {\displaystyle 1.\rightarrow 2.}. conversely, suppose r 2 {\displaystyle r^{2}} has the above property and that an element x in r is given. consider the linear map σ: r 2 → r, σ (a, b) = a − b {\displaystyle \sigma:r^{2}\to r,\,\sigma ...	wikipedia
9776b48f8d10efcfd848a1e893d636eee703ff32	the implication 1. ⇒ 2. {\displaystyle 1.\rightarrow 2.}	the implication 1. ⇒ 2. {\displaystyle 1.\rightarrow 2.} is exactly (usual) kaplansky's theorem and azumaya's theorem. the converse 2. ⇒ 1. {\displaystyle 2.\rightarrow 1.} follows from the following general fact, which is interesting itself:	wikipedia
9cceaa9efc96d180574742670cd772fa63901921	proof of the theorem: let n {\displaystyle n}	proof of the theorem: let n {\displaystyle n} be a projective module over a local ring. then, by definition, it is a direct summand of some free module f {\displaystyle f}. this f {\displaystyle f} is in the family f {\displaystyle {\mathfrak {f}}} in lemma 1; thus, n {\displaystyle n} is a direct sum of countably gene...	wikipedia
1383cf1bd5b8c67aee11d8209fdd125c45de7bea	indeed, assume the claim is valid. then choose	indeed, assume the claim is valid. then choose a sequence x 1, x 2, … {\displaystyle x_{1},x_{2},\dots } in n that is a generating set. then using the claim, write n = h 1 ⊕ n 1 {\displaystyle n=h_{1}\oplus n_{1}} where x 1 ∈ h 1 ∈ g {\displaystyle x_{1}\in h_{1}\in {\mathcal {g}}}. then we write x 2 = y + z {\displays...	wikipedia
f55ddb484914b5ab6a7d1e4fbfbe3a4623676960	proof: let g {\displaystyle {\mathcal {g}}} denote the	proof: let g {\displaystyle {\mathcal {g}}} denote the family of modules that are isomorphic to modules of the form ⨁ i ∈ f m i {\displaystyle \bigoplus _{i\in f}m_{i}} for some finite subset f ⊂ i {\displaystyle f\subset i}. the assertion is then implied by the following claim:	wikipedia
7f02b06617b5457e9e08ac0af3a1eac6789bfff2	lemma 2 — if m i, i ∈	lemma 2 — if m i, i ∈ i {\displaystyle m_{i},i\in i} are countably generated modules with local endomorphism rings and if n {\displaystyle n} is a countably generated module that is a direct summand of ⨁ i ∈ i m i {\displaystyle \bigoplus _{i\in i}m_{i}}, then n {\displaystyle n} is isomorphic to ⨁ i ∈ i ′ m i {\displa...	wikipedia
94ffbcc71f4279af634c6c050a524d38d122dbe1	is countably generated as j ′ − j	is countably generated as j ′ − j ⊂ i ′ {\displaystyle j'-j\subset i'}. this contradicts the maximality of (j, b, c) {\displaystyle (j,b,c)}. ◻ {\displaystyle \square }	wikipedia
6e97f5b82fa084c7d80153a13510f5e06b4965d8	now, n j {\displaystyle n_{j}} is a direct	now, n j {\displaystyle n_{j}} is a direct summand of m {\displaystyle m} (since it is a summand of m j {\displaystyle m_{j}}, which is a summand of m {\displaystyle m}); i.e., n j ⊕ m ′ = m {\displaystyle n_{j}\oplus m'=m} for some m ′ {\displaystyle m'}. then, by modular law, n j ′ = n j ⊕ (m ′ ∩ n j ′) {\displaystyl...	wikipedia
740c0d2a6ef2389a4d25e40bff1bfd2828ce50bd	the inclusion ⊂ {\displaystyle \subset } is trivial.	the inclusion ⊂ {\displaystyle \subset } is trivial. conversely, n j ′ {\displaystyle n_{j'}} is the image of n j + l j + m i ′ ⊂ n j + m i ′ {\displaystyle n_{j}+l_{j}+m_{i'}\subset n_{j}+m_{i'}} and so n j ′ ⊂ m j ′ {\displaystyle n_{j'}\subset m_{j'}}. the same is also true for l j ′ {\displaystyle l_{j'}}. hence, t...	wikipedia
c010ffae4bb0dd4d23026137b2437b5bbf0cc721	let i ′ = ⋃ 0 ∞ i	let i ′ = ⋃ 0 ∞ i n {\displaystyle i'=\bigcup _{0}^{\infty }i_{n}} and j ′ = j ∪ i ′ {\displaystyle j'=j\cup i'}. we claim:	wikipedia
b7233e6fdbbd22f652cf52b8a6b7daa63105bd27	proof: let n be a direct summand; i.e.,	proof: let n be a direct summand; i.e., m = n ⊕ l {\displaystyle m=n\oplus l}. using the assumption, we write m = ⨁ i ∈ i m i {\displaystyle m=\bigoplus _{i\in i}m_{i}} where each m i {\displaystyle m_{i}} is a countably generated submodule. for each subset a ⊂ i {\displaystyle a\subset i}, we write m a = ⨁ i ∈ a m i, ...	wikipedia
8a19f1666c066f531820ee5e29dcb18e6f8c26bd	proof: let n be a direct summand; i.e.,	′, c ⊂ c ′ {\displaystyle b\subset b',c\subset c'}. by zorn's lemma, the set contains a maximal element (j, b, c) {\displaystyle (j,b,c)}. we shall show that j = i {\displaystyle j=i}; i.e., n = n j = ⨁ n ′ ∈ b n ′ ∈ f {\displaystyle n=n_{j}=\bigoplus _{n'\in b}n'\in {\mathfrak {f}}}. suppose otherwise. then we can in...	wikipedia
896723b12420101a3df5245996a43a4991484421	lemma 1 — let f {\displaystyle {\mathfrak {f}}}	lemma 1 — let f {\displaystyle {\mathfrak {f}}} denote the family of modules that are direct sums of some countably generated submodules (here modules can be those over a ring, a group or even a set of endomorphisms). if m {\displaystyle m} is in f {\displaystyle {\mathfrak {f}}}, then each direct summand of m {\displa...	wikipedia
47bc6b2f202c1b9d6faafd72f482002bb46a5ed9	the idea of the proof of the theorem	the idea of the proof of the theorem was also later used by hyman bass to show big projective modules (under some mild conditions) are free. according to (anderson & fuller 1992), kaplansky's theorem "is very likely the inspiration for a major portion of the results" in the theory of semiperfect rings.	wikipedia
a58019fd2e160646b4e6c35408bae1e063cf36d5	for a finite projective module over a commutative	for a finite projective module over a commutative local ring, the theorem is an easy consequence of nakayama's lemma. for the general case, the proof (both the original as well as later one) consists of the following two steps:	wikipedia
269ec99e6fc890bef164975f355cd69a68e51199	in abstract algebra, kaplansky's theorem on projective modules,	in abstract algebra, kaplansky's theorem on projective modules, first proven by irving kaplansky, states that a projective module over a local ring is free; where a not-necessarily-commutative ring is called local if for each element x, either x or 1 − x is a unit element. the theorem can also be formulated so to chara...	wikipedia
ec7d471539d0486a2b1a17ac4145c87ef5e41b28	lutetium phthalocyanine (lupc2) is a coordination compound derived	lutetium phthalocyanine (lupc2) is a coordination compound derived from lutetium and two phthalocyanines. it was the first known example of a molecule that is an intrinsic semiconductor. it exhibits electrochromism, changing color when subject to a voltage.	wikipedia
6c12a8c2b5fbd394b65d8b5d67f68578d9ced9cd	lupc, along with many substituted derivatives like the	lupc, along with many substituted derivatives like the alkoxy -methyl derivative lu, can be deposited as a thin film with intrinsic semiconductor properties; said properties arise due to its radical nature and its low reduction potential compared to other metal phthalocyanines. this initially green film exhibits electr...	wikipedia
4eab21979a86eb31cad7ee22be047bcf821a439d	lupc is a double-decker sandwich compound consisting of	lupc is a double-decker sandwich compound consisting of a lu ion coordinated to two the conjugate base of two phthalocyanines. the rings are arranged in a staggered conformation. the extremities of the two ligands are slightly distorted outwards. the complex features a non-innocent ligand, in the sense that the macrocy...	wikipedia
8d9c8b4626f6f86e4e5b50aeeabcc750dbd726e6	lutetium phthalocyanine (lupc) is a coordination compound derived	lutetium phthalocyanine (lupc) is a coordination compound derived from lutetium and two phthalocyanines. it was the first known example of a molecule that is an intrinsic semiconductor. it exhibits electrochromism, changing color when subject to a voltage.	wikipedia
f0bb07aeafed23fefcffa8e52379ffb2ec16b232	on 26 june 2014, following odejayi's departure from	on 26 june 2014, following odejayi's departure from rotherham united, he signed a one-year deal with tranmere rovers. at the end of the season, that saw tranmere rovers relegate out of league two, odejayi was released by the club.	wikipedia
881dfff77362e9a9977e00feb6b8ed07927261b0	on 10 september 2013, odejayi joined league two	on 10 september 2013, odejayi joined league two side accrington stanley on a three-month loan. on 1 january 2014, the loan was extended till the end of the 2013–14 league two season.	wikipedia