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id stringlengths 40 40	title stringlengths 15 120	text stringlengths 41 3.14k	source stringclasses 2 values
4b031278c571ef0e6a4dac399ac951b314d6a915	for convenience, in this section we treat only	for convenience, in this section we treat only the case where Ω = r n {\displaystyle \omega =\mathbb {r} ^{n}}, i.e. the set e {\displaystyle e} has (globally) finite perimeter. de giorgi's theorem provides geometric intuition for the notion of reduced boundaries and confirms that it is the more natural definition for ...	wikipedia
fec53d29b98bc4cef219aa4213da5701ce7b9d7b	the inclusions above are not necessarily equalities as	the inclusions above are not necessarily equalities as the previous example shows. in that example, ∂ e {\displaystyle \partial e} is the square with the segment sticking out, support d χ e {\displaystyle \operatorname {support} d\chi _{e}} is the square, and ∂ ∗ e {\displaystyle \partial ^{*}e} is the square without i...	wikipedia
3977062a7346be3a4994c8e2659414c62ceafd70	one can remark that by the radon-nikodym theorem	one can remark that by the radon-nikodym theorem the reduced boundary ∂ ∗ e {\displaystyle \partial ^{*}e} is necessarily contained in the support of d χ e {\displaystyle d\chi _{e}}, which in turn is contained in the topological boundary ∂ e {\displaystyle \partial e} as explained in the section above. that is:	wikipedia
f8a15eff79ee54e265a9ba30fb258822304c5f77	definition 4. the reduced boundary of a caccioppoli	definition 4. the reduced boundary of a caccioppoli set e ⊂ r n {\displaystyle e\subset \mathbb {r} ^{n}} is denoted by ∂ ∗ e {\displaystyle \partial ^{*}e} and is defined to be equal to be the collection of points x {\displaystyle x} at which the limit:	wikipedia
96020847ba85aa4e0829fd8ac20dfe58df54fb2d	the topological boundary ∂ e {\displaystyle \partial e}	the topological boundary ∂ e {\displaystyle \partial e} turns out to be too crude for caccioppoli sets because its hausdorff measure overcompensates for the perimeter p (e) {\displaystyle p(e)} defined above. indeed, the caccioppoli set	wikipedia
492b8d70e1e1791aa83362709308fc355276f1d3	with ϕ ∈ c c 1 (a, r	with ϕ ∈ c c 1 (a, r n) {\displaystyle \phi \in c_{c}^{1}(a,\mathbb {r} ^{n})} arbitrary it follows that x 0 {\displaystyle x_{0}} is outside the support of d χ e {\displaystyle d\chi _{e}}.	wikipedia
495ceb08d851a0e59dbae9c7237610deee166c9e	likewise, if a ⊆ e ∘ {\displaystyle a\subseteq	likewise, if a ⊆ e ∘ {\displaystyle a\subseteq e^{\circ }} then χ e (x) = 1 {\displaystyle \chi _{e}(x)=1} for x ∈ a {\displaystyle x\in a} so	wikipedia
6368b9bc62422118cdd739a43f19ae5e6449deae	proof. to see this choose x 0 ∉	proof. to see this choose x 0 ∉ ∂ e {\displaystyle x_{0}\notin \partial e}: then x 0 {\displaystyle x_{0}} belongs to the open set r n ∖ ∂ e {\displaystyle \mathbb {r} ^{n}\setminus \partial e} and this implies that it belongs to an open neighborhood a {\displaystyle a} contained in the interior of e {\displaystyle e} ...	wikipedia
7495990190fa4a54fc8a17e65d31a2a9443eed7c	lemma. the support of the vector-valued radon measure	lemma. the support of the vector-valued radon measure d χ e {\displaystyle d\chi _{e}} is a subset of the topological boundary ∂ e {\displaystyle \partial e} of e {\displaystyle e}.	wikipedia
5262d73d4fd0e4f89033dba36c751d86a426ef94	it is natural to try to understand the	it is natural to try to understand the relationship between the objects d χ e {\displaystyle d\chi _{e}}, \| d χ e \| {\displaystyle \|d\chi _{e}\|}, and the topological boundary ∂ e {\displaystyle \partial e}. there is an elementary lemma that guarantees that the support (in the sense of distributions) of d χ e {\displays...	wikipedia
28755295b48329bac9ecca41b437e470d92d6133	for any given caccioppoli set e ⊂ r	for any given caccioppoli set e ⊂ r n {\displaystyle e\subset \mathbb {r} ^{n}} there exist two naturally associated analytic quantities: the vector-valued radon measure d χ e {\displaystyle d\chi _{e}} and its total variation measure \| d χ e \| {\displaystyle \|d\chi _{e}\|}. given that	wikipedia
7cf3fca133656a891c1a33dd0e9a02ab0c40a610	actually de giorgi considered the case Ω =	actually de giorgi considered the case Ω = r n {\displaystyle \omega =\mathbb {r} ^{n}}: however, the extension to the general case is not difficult. it can be proved that the two definitions are exactly equivalent: for a proof see the already cited de giorgi's papers or the book (giusti 1984). now having defined what ...	wikipedia
8c0a95b667852fa8aacbeb6915e8716413d57c34	definition 3. let Ω {\displaystyle \omega } be	definition 3. let Ω {\displaystyle \omega } be an open subset of r n {\displaystyle \mathbb {r} ^{n}} and let e {\displaystyle e} be a borel set. the perimeter of e {\displaystyle e} in Ω {\displaystyle \omega } is the value	wikipedia
df50103512306eb07b3fe0a50bff08132c22b97e	as one can easily prove, w λ χ	as one can easily prove, w λ χ (x) {\displaystyle w_{\lambda }\chi (x)} is a smooth function for all x ∈ r n {\displaystyle x\in \mathbb {r} ^{n}}, such that	wikipedia
895cb626eb670f2023780f7c5705c55b42a5cb51	as noted for the case of general functions	as noted for the case of general functions of bounded variation, this vector measure d χ e {\displaystyle d\chi _{e}} is the distributional or weak gradient of χ e {\displaystyle \chi _{e}}. the total variation measure associated with d χ e {\displaystyle d\chi _{e}} is denoted by \| d χ e \| {\displaystyle \|d\chi _{e}\|}...	wikipedia
2fd9c5aa6e298733c892e8548d6e0a75787d4c50	therefore, a caccioppoli set has a characteristic function	therefore, a caccioppoli set has a characteristic function whose total variation is locally bounded. from the theory of functions of bounded variation it is known that this implies the existence of a vector-valued radon measure d χ e {\displaystyle d\chi _{e}} such that	wikipedia
50f3c14f1ecc214348ba8f19aedefb583ce2c257	definition 2. the borel set e {\displaystyle e}	definition 2. the borel set e {\displaystyle e} is a caccioppoli set if and only if it has finite perimeter in every bounded open subset Ω {\displaystyle \omega } of r n {\displaystyle \mathbb {r} ^{n}}, i.e.	wikipedia
fb7102912cf8a8099f4c04519c65a1a8e7caaf7c	where χ e {\displaystyle \chi _{e}} is the	where χ e {\displaystyle \chi _{e}} is the characteristic function of e {\displaystyle e}. that is, the perimeter of e {\displaystyle e} in an open set Ω {\displaystyle \omega } is defined to be the total variation of its characteristic function on that open set. if Ω = r n {\displaystyle \omega =\mathbb {r} ^{n}}, the...	wikipedia
8ed9b588e8c3be06551c9d8fc1aa0642b3909f2f	definition 1. let Ω {\displaystyle \omega } be	definition 1. let Ω {\displaystyle \omega } be an open subset of r n {\displaystyle \mathbb {r} ^{n}} and let e {\displaystyle e} be a borel set. the perimeter of e {\displaystyle e} in Ω {\displaystyle \omega } is defined as follows	wikipedia
f5b4f360b6af3bdbac335ed1f069c4b5442f5bf7	the last paper of de giorgi on the	the last paper of de giorgi on the theory of perimeters was published in 1958: in 1959, after the death of caccioppoli, he started to call sets of finite perimeter "caccioppoli sets". two years later herbert federer and wendell fleming published their paper (federer & fleming 1960), changing the approach to the theory....	wikipedia
6927665fefd1170bc88b73b8613446b01b487b0c	in 1952 ennio de giorgi presented his first	in 1952 ennio de giorgi presented his first results, developing the ideas of caccioppoli, on the definition of the measure of boundaries of sets at the salzburg congress of the austrian mathematical society: he obtained this results by using a smoothing operator, analogous to a mollifier, constructed from the gaussian ...	wikipedia
d19e3e30a31cf2e36597a4cc3e606fc914d54daf	lamberto cesari introduced the "right" generalization of functions	lamberto cesari introduced the "right" generalization of functions of bounded variation to the case of several variables only in 1936: perhaps, this was one of the reasons that induced caccioppoli to present an improved version of his theory only nearly 24 years later, in the talk (caccioppoli 1953) at the iv umi congr...	wikipedia
7b86f29d84acc1e37b9289b611f15ec85ee15fc0	in the paper (caccioppoli 1928), he precised by	in the paper (caccioppoli 1928), he precised by using a triangular mesh as an increasing net approximating the open domain, defining positive and negative variations whose sum is the total variation, i.e. the area functional. his inspiring point of view, as he explicitly admitted, was those of giuseppe peano, as expres...	wikipedia
edf300bc8380c65f23c76353fc32170c19da2596	the basic concept of a caccioppoli set was	the basic concept of a caccioppoli set was first introduced by the italian mathematician renato caccioppoli in the paper (caccioppoli 1927): considering a plane set or a surface defined on an open set in the plane, he defined their measure or area as the total variation in the sense of tonelli of their defining functio...	wikipedia
b49fca448fc4bff1490120a57a10d5dfdffa7379	in mathematics, a caccioppoli set is a set	in mathematics, a caccioppoli set is a set whose boundary is measurable and has (at least locally) a finite measure. a synonym is set of (locally) finite perimeter. basically, a set is a caccioppoli set if its characteristic function is a function of bounded variation.	wikipedia
9feaa9b421ae6b24362d02184d526085b0121cc9	the montagu whaler was the standard seaboat of	the montagu whaler was the standard seaboat of the royal navy between 1910–1970, it was a clinker built 27 by 6 feet open boat, which could be pulled by oars or powered by sail – a shorter version of 25 feet (7.6 m) was also built. it was double-ended; having a pointed stem and stern. retired rear admiral the honourabl...	wikipedia
a2ff0b5963e2368c94bb64de2d4f5e0511387b4f	steve evans, of the bristol charity all aboard	steve evans, of the bristol charity all aboard watersports, came across her while doing internet research, and obtained her for a few pounds and brought her to bristol to be fully restored at underfall yard then to join the all aboard watersports fleet.	wikipedia
7140ae10f5657736a26a76fa5736bcc0ea1a86ef	the swan was a typical montagu, she was	the swan was a typical montagu, she was badly burnt then was acquired by bernie bruing in the 1960s and towed to cornwall beach where she was restored and used for family boat trips. it was sold to brown of falmouth and consequently donated to a ferry company but left to rot. falmouth boat builders recovered her and re...	wikipedia
8f897a7cd6fb35503e0dce148ee05d22e5e370c2	when whalers were replaced by rigid inflatables, some	when whalers were replaced by rigid inflatables, some were passed down to auxiliary reserves and sea cadet units as training vessels. others were abandoned and some rescued by enthusiasts, two were seen for many years decaying at sunderland point, lancaster. the swan took part in the 2019 great river race in london, co...	wikipedia
8f5d14eedc2b8a4644d4b94b3ddd808f2e9265a8	preparing the whaler to sail was a co-operative	preparing the whaler to sail was a co-operative process involving all members of the crew, at the command up mast, rig the boat, the mast had to be set and secured, the centreboard lowered. the sails have to be raised and trimmed.	wikipedia
66cb17fa4c3b9e44d5d40786cf1d6e21241e60f6	in port, an important function was to maintain	in port, an important function was to maintain communication with the shore when a vessel was anchored off. the desire to hook up to the shore electricity has caused warships go alongside a jetty, wharf, or quay. the whaler was used to help set or weigh anchor, and fetch water. in addition it was used for training and ...	wikipedia
da41ced632423c9107c9e6a19a757b577146eb1f	the role of the seaboat was varied, including	the role of the seaboat was varied, including transfer of personnel between ships whilst at sea and recovering a man overboard. during wartime, this list extended with the rescue of survivors from sunken ships and crash-landed aircraft, gathering the (sometimes gory) evidence of the destruction of a u-boat and, in seve...	wikipedia
2c63f4594a36196c969826598522e1893e883f4d	the whaler was often the largest boat on	the whaler was often the largest boat on many of the smaller warships. therefore it usually had the job of seaboat. this is a boat kept ready for immediate lowering whilst at sea. it is equipped with water, food and other stores in case the boat should be separated from the ship for any length of time. there is one sea...	wikipedia
bd2131e8ede714d7202d3fa03cfcec44893ab8af	he saw active service during the crimean war,	he saw active service during the crimean war, 1854–1856, the 2nd china war where he survived the shipwreck of frigate hms raleigh (1845), and the indian mutiny in 1857, winning five service medals he retired as a captain in 1886. and was promoted to rear admiral on the retired list on 14 february 1892	wikipedia
12be9e574a3bb6a78c93b05839041bccd810dc06	victor alexander entered naval college in september 1853	victor alexander entered naval college in september 1853 at the age of twelve. in december 1853 he joined hms princess royal, a 91-gun screw-propelled second-rate, as a midshipman junior officer, where he became a.d.c. to his uncle. he was given his first command, his uncle's 12-oared cutter.	wikipedia
83a3ebc3d288b7b646b954b4460d62dd964f65dd	victor alexander montagu was the second son of	victor alexander montagu was the second son of the seventh earl of sandwich. his mother, a daughter of the marquis of anglesea, who commanded cavalry at waterloo. his ancestor, the fourth earl of sandwich, john montagu, served as first lord of the admiralty and sponsored the voyages of captain james cook.	wikipedia
62af44e44540acee3035284a7aeffc6363817d54	mainsail: 142 square feet (13.2 m) this is	mainsail: 142 square feet (13.2 m) this is loose footed and laced to the yard which is raised on a traveller on the mainmast. the sheets are rove to the rear thwart. throat brails are fitted, and used when gybing.	wikipedia
2a5f3a88052d0cb197aec320c7c23db65de81370	this is a yawl rig with a (standing	this is a yawl rig with a (standing lug) mainsail. the main mast is stepped on the keelson and it is secured by an iron clamp to the second thwart. it is held by a forestay and two shrouds. the mizzen is stepped abaft the stern benches in a shoe on the hog.	wikipedia
f7ac88d18281c340b2e6a503a3966e6c91c62da2	the montagu whaler is a 27 feet (8.2	the montagu whaler is a 27 feet (8.2 m) clinker built boat with a generous 6 feet (1.8 m) beam, fitted with a drop keel. in england they were built from wych elm or sand elm. in malta they used mahogany.	wikipedia
d0ad8dc36e740af3b1f85bfade0c8ffb707a5f65	after 1956 the montagu was gradually replaced with	after 1956 the montagu was gradually replaced with the 27-foot motor whaler, a three-in-one whaler with an inboard petrol engine: this could also be pulled or sailed. they were heavy and handled poorly, and were superseded by the motor whaler mod 1. which abandoned the sailing rig. they continued until the 1990s.	wikipedia