Dynkin system - Revision history

en>Darencline: /* Dynkin's π-λ theorem */ actually, D is larger than the Borel sets, though this is seldom a relevant issue

2014-03-10T16:02:18Z

Dynkin's π-λ theorem: actually, D is larger than the Borel sets, though this is seldom a relevant issue

← Older revision		Revision as of 18:02, 10 March 2014
Line 1:		Line 1:
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en>Darencline: /* Definitions */

2014-03-04T14:28:28Z

Definitions

← Older revision		Revision as of 16:28, 4 March 2014
Line 1:		Line 1:
	~~{{lead rewrite\|date=September 2012}}~~		Alyson is the title people use to call me and I think it sounds quite good when you say it. As a woman what she truly likes is style and she's been doing [http://www.khuplaza.com/dent/14869889 psychic phone] it for quite a while. Office supervising is exactly where her primary earnings comes from but she's currently utilized for another one. Some time in the past he selected to live in [http://www.aseandate.com/index.php?m=member_profile&p=profile&id=13352970 psychics online] North Carolina and he doesn't strategy on altering it.<br><br>My web site :: real psychics, [http://afeen.fbho.net/v2/index.php?do=/profile-210/info/ please click the following page],
	~~[[File:Spherometer.jpg\|thumb\|alt=A common spherometer.\|A common spherometer.]]~~
	~~A '''spherometer'''~~ is ~~an instrument for~~ the ~~precise measurement of the [[radius]] of a [[sphere]]. Originally, these instruments were primarily used by [[optician]]s~~ to ~~measure the [[curvature]] of the surface of a [[lens (optics)\|lens]].<ref>{{1911}}</ref>~~

	The usual form consists of a fine screw moving in a nut carried on the centre of a small three-legged table or frame; the feet forming the vertices of an [[equilateral triangle]]. The lower end of the screw and those of the table legs are finely tapered and terminate in hemispheres, so that each rests on a point. If the screw has two turns of the thread to the millimetre the head is usually divided into 50 equal parts, so that differences of 0.01 millimetre may be measured without using a [[Vernier scale\|vernier]]. A lens, however, may be fitted, in order to magnify the scale divisions. A vertical scale fastened to the table indicates the number of whole turns of the screw and serves as an index for reading the divisions on the head.

	~~A contact-lever, delicate level or electric contact arrangement may be attached to the spherometer in order to indicate the moment of touching more precisely than is possible by the sense of touch~~. ~~To measure the radius of~~ a ~~sphere—e.g. the curvature of a [[lens (optics)\|lens]]—the spherometer~~ is ~~levelled and read, then placed on the sphere, adjusted until the four points exert equal pressure,~~ and ~~read again. The difference gives the thickness of that portion of the sphere cut off by a plane passing through the three feet.~~

	~~==Principles of operation==~~

	~~The spherometer directly measures a [[Sagitta (optics)\|sagittal]],~~ '~~'h''.~~ ~~Using the mean length between two outer legs, ''a'', the spherical radius ''R'' is given by the formula~~:
	~~:::<math>R = (h/2) + (a^2~~/~~6h)<~~/~~math>~~

	~~A spherometer basically is a precision instrument to measure very small lengths. Its name reflects the way it is used to measure the radii of curvature of spherical surfaces~~. ~~It is based on the principle of screw~~. ~~In general the spherometer consists of:~~

	*A base circle of three outer legs, a ring, or the equivalent, having a known radius of the base circle. (The outer legs of some spherometers can be moved to a set of inner holes in order to accommodate a smaller surface.)
	*A central leg, which can be raised or lowered.
	*A reading device for measuring the distance the central leg is moved. On the new spherometer, the vertical scale is marked off in units of 0.5 mm. One complete turn of the dial also corresponds to 0.5 mm and each small graduation on this dial represents 0.005 mm. The small graduations on the old spherometer are 0.001 mm.

	~~==Alternative uses==~~

	~~Since the spherometer is essentially a type of [[micrometer~~]], it ~~can be employed~~ for ~~purposes other than measuring the curvature of a spherical surface. For example, it can be used to measure the thickness of~~ a ~~thin plate~~.

	~~To do so, the instrument~~ is placed on a perfectly level plane surface and the screw turned until the point just touches; the exact instant when it does so is defined by a sudden diminution of resistance succeeded by a considerable increase. The divided head and scale are read; the screw is raised; the thin plate slipped under it; and the process is repeated. The difference between the two readings gives the required thickness.

	~~Similarly, the instrument can measure the depression in an otherwise flat plate. The method would be as~~ for ~~measuring the thickness of a plate, except that the micrometer portion is placed over the depression and the measurement is taken below the surface instead of above~~.

	~~This type of instrument is commonly used~~ in inspections of oil field tool pipe for metal surface pits, fractures, and roundness before being shipped to drilling sites for exploratory wells. The process of inspections are designed to remove weakened [[drill pipe]], so that pipe will not fracture during drilling.<ref>{{cite book \|last=Anderson \|first=Robert O. \|title=Fundamentals of the Petroleum Industry \| location=Norman, Okla. \|publisher=University of Oklahoma Press \|year=1984}}</ref> Tool pipe with thicker than 1" walls for a 4" diameter tube of hardened steel, fitted with tapered thread collars are re-used after drilling is complete, and thinner-walled tubular oil-well casing is in place. Electronic instrumentation similar to ~~the spherometer~~ in ~~design are modified at inspection plants for casing, tubing, and drill pipe.~~ ~~The equivalent measurements in optics would be for a cylinder, or lens with a cylindrical component having an optical axis, where a plane through the lens would produce an oval circumference.~~

	An alternate approach using [[coordinate geometry]] was developed recently. This approach reproduces the well-known result for the spherometer and also leads to a scheme to study [[Aspherical lens\|aspherical]] surfaces.

	A related device is the Cylindrometer (also known as Cylindro-Spherometer and Sphero-Cylindrometer), which can additionally measure the [[Radius of curvature (mathematics)\|radius of curvature]] of a [[Right circular cylinder\|right-circular cylinder]]

	~~==See also==~~
	*[[Lapidary]]
	*[[Lens clock]]
	*[[Lens (optics)]]
	*[[Lensometer]]

	~~==References==~~

	~~{{Reflist}}~~

	~~==External links==~~
	*[http://~~inspirehep~~.~~net/author/S~~.~~A.Khan.5~~/ Sameen Ahmed Khan], [http://indapt.org/images/stories/bulletin2009/bulleting_january_2009.pdf Cylindro-Spherometer] Bulletin of the IAPT, 26(1), 4-6 (January 2009) (Publication of the [[Indian Association of Physics Teachers]]).
	*[http://scholar.~~google.com/citations~~?~~user~~=~~hZvL5eYAAAAJ~~&~~hl Sameen Ahmed Khan], [http://dx.doi.org/10.1119/1.3517029 Cylindrometer], [[The Physics Teacher]], Volume 48, Issue 9, pp.~~&~~nbsp;607 (December 2010) (Publication of the [[American Association of Physics Teachers\|American Association of Physics Teacher~~]]).
	*[http:~~//arxiv.org/a/khan_s_1 Sameen Ahmed Khan]~~,[http://~~indapt~~.~~org/images/stories/bulletin2013/buletin_june_2013~~.~~pdf Coordinate Geometric Approach to Spherometer], Bulletin of the IAPT, Volume 5, No. 6, 139-142 (June 2013). (Publication of the [[Indian Association of Physics Teachers]]). {{arXiv\|1309.1951}}~~
	*[http://~~arxiv~~.~~org~~/~~a/khan_s_1 Sameen Ahmed Khan], [http:~~//~~arxiv.org/abs/1311.3602/ Coordinate Geometric Generalization of~~ the ~~Spherometer and Cylindrometer~~], ~~{{arXiv\|1311.3602}}~~



	~~[[Category:Dimensional instruments]]~~
	~~[[Category:Optics]]~~

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q1935689

2013-03-30T18:33:58Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q1935689

← Older revision		Revision as of 20:33, 30 March 2013
Line 1:		Line 1:
	~~I would like~~ to ~~introduce myself~~ to ~~you~~, ~~I am Jayson Simcox but I don~~'~~t like~~ when ~~people use my full title~~. ~~Doing ballet~~ is ~~something she~~ would ~~never give up~~. ~~Her family members lives~~ in ~~Alaska but her spouse desires them~~ to ~~transfer~~. ~~Distributing production is how he tends~~ to ~~make a residing~~.<br><br>~~Look into my blog post~~; [http://~~netwk~~.~~hannam~~.ac.kr/xe/~~data_2~~/~~85669 free psychic readings~~]		{{lead rewrite\|date=September 2012}}
			[[File:Spherometer.jpg\|thumb\|alt=A common spherometer.\|A common spherometer.]]
			A '''spherometer''' is an instrument for the precise measurement of the [[radius]] of a [[sphere]]. Originally, these instruments were primarily used by [[optician]]s to measure the [[curvature]] of the surface of a [[lens (optics)\|lens]].<ref>{{1911}}</ref>

			The usual form consists of a fine screw moving in a nut carried on the centre of a small three-legged table or frame; the feet forming the vertices of an [[equilateral triangle]]. The lower end of the screw and those of the table legs are finely tapered and terminate in hemispheres, so that each rests on a point. If the screw has two turns of the thread to the millimetre the head is usually divided into 50 equal parts, so that differences of 0.01 millimetre may be measured without using a [[Vernier scale\|vernier]]. A lens, however, may be fitted, in order to magnify the scale divisions. A vertical scale fastened to the table indicates the number of whole turns of the screw and serves as an index for reading the divisions on the head.

			A contact-lever, delicate level or electric contact arrangement may be attached to the spherometer in order to indicate the moment of touching more precisely than is possible by the sense of touch. To measure the radius of a sphere—e.g. the curvature of a [[lens (optics)\|lens]]—the spherometer is levelled and read, then placed on the sphere, adjusted until the four points exert equal pressure, and read again. The difference gives the thickness of that portion of the sphere cut off by a plane passing through the three feet.

			==Principles of operation==

			The spherometer directly measures a [[Sagitta (optics)\|sagittal]], ''h''. Using the mean length between two outer legs, ''a'', the spherical radius ''R'' is given by the formula:
			:::<math>R = (h/2) + (a^2/6h)</math>

			A spherometer basically is a precision instrument to measure very small lengths. Its name reflects the way it is used to measure the radii of curvature of spherical surfaces. It is based on the principle of screw. In general the spherometer consists of:

			*A base circle of three outer legs, a ring, or the equivalent, having a known radius of the base circle. (The outer legs of some spherometers can be moved to a set of inner holes in order to accommodate a smaller surface.)
			*A central leg, which can be raised or lowered.
			*A reading device for measuring the distance the central leg is moved. On the new spherometer, the vertical scale is marked off in units of 0.5 mm. One complete turn of the dial also corresponds to 0.5 mm and each small graduation on this dial represents 0.005 mm. The small graduations on the old spherometer are 0.001 mm.

			==Alternative uses==

			Since the spherometer is essentially a type of [[micrometer]], it can be employed for purposes other than measuring the curvature of a spherical surface. For example, it can be used to measure the thickness of a thin plate.

			To do so, the instrument is placed on a perfectly level plane surface and the screw turned until the point just touches; the exact instant when it does so is defined by a sudden diminution of resistance succeeded by a considerable increase. The divided head and scale are read; the screw is raised; the thin plate slipped under it; and the process is repeated. The difference between the two readings gives the required thickness.

			Similarly, the instrument can measure the depression in an otherwise flat plate. The method would be as for measuring the thickness of a plate, except that the micrometer portion is placed over the depression and the measurement is taken below the surface instead of above.

			This type of instrument is commonly used in inspections of oil field tool pipe for metal surface pits, fractures, and roundness before being shipped to drilling sites for exploratory wells. The process of inspections are designed to remove weakened [[drill pipe]], so that pipe will not fracture during drilling.<ref>{{cite book \|last=Anderson \|first=Robert O. \|title=Fundamentals of the Petroleum Industry \| location=Norman, Okla. \|publisher=University of Oklahoma Press \|year=1984}}</ref> Tool pipe with thicker than 1" walls for a 4" diameter tube of hardened steel, fitted with tapered thread collars are re-used after drilling is complete, and thinner-walled tubular oil-well casing is in place. Electronic instrumentation similar to the spherometer in design are modified at inspection plants for casing, tubing, and drill pipe. The equivalent measurements in optics would be for a cylinder, or lens with a cylindrical component having an optical axis, where a plane through the lens would produce an oval circumference.

			An alternate approach using [[coordinate geometry]] was developed recently. This approach reproduces the well-known result for the spherometer and also leads to a scheme to study [[Aspherical lens\|aspherical]] surfaces.

			A related device is the Cylindrometer (also known as Cylindro-Spherometer and Sphero-Cylindrometer), which can additionally measure the [[Radius of curvature (mathematics)\|radius of curvature]] of a [[Right circular cylinder\|right-circular cylinder]]

			==See also==
			*[[Lapidary]]
			*[[Lens clock]]
			*[[Lens (optics)]]
			*[[Lensometer]]

			==References==

			{{Reflist}}

			==External links==
			*[http://inspirehep.net/author/S.A.Khan.5/ Sameen Ahmed Khan], [http://indapt.org/images/stories/bulletin2009/bulleting_january_2009.pdf Cylindro-Spherometer] Bulletin of the IAPT, 26(1), 4-6 (January 2009) (Publication of the [[Indian Association of Physics Teachers]]).
			*[http://scholar.google.com/citations?user=hZvL5eYAAAAJ&hl Sameen Ahmed Khan], [http://dx.doi.org/10.1119/1.3517029 Cylindrometer], [[The Physics Teacher]], Volume 48, Issue 9, pp. 607 (December 2010) (Publication of the [[American Association of Physics Teachers\|American Association of Physics Teacher]]).
			*[http://arxiv.org/a/khan_s_1 Sameen Ahmed Khan],[http://indapt.org/images/stories/bulletin2013/buletin_june_2013.pdf Coordinate Geometric Approach to Spherometer], Bulletin of the IAPT, Volume 5, No. 6, 139-142 (June 2013). (Publication of the [[Indian Association of Physics Teachers]]). {{arXiv\|1309.1951}}
			*[http://arxiv.org/a/khan_s_1 Sameen Ahmed Khan], [http://arxiv.org/abs/1311.3602/ Coordinate Geometric Generalization of the Spherometer and Cylindrometer], {{arXiv\|1311.3602}}



			[[Category:Dimensional instruments]]
			[[Category:Optics]]

en>ChrisGualtieri: TypoScan Project / General Fixes, typos fixed: , → , using AWB

2012-05-28T19:44:29Z

TypoScan Project / General Fixes, typos fixed: , → , using AWB

New page

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