Dual matroid - Revision history

en>Oliver Kullmann: The cycles, not the "circuits", are the complements of a flat of the dual matroid (as stated in https://en.wikipedia.org/wiki/Matroid#Sums_and_unions )

2014-08-09T15:42:26Z

The *cycles*, not the "circuits", are the complements of a flat of the dual matroid (as stated in https://en.wikipedia.org/wiki/Matroid#Sums_and_unions )

← Older revision		Revision as of 17:42, 9 August 2014
Line 1:		Line 1:
	~~The '''Tetrahedral hypothesis''' is an [[Superseded scientific theories\|obsolete scientific theory]] attempting~~ to ~~explain~~ the ~~arrangement~~ of ~~the [[Earth]]'s [[continent]]s and [[ocean]]s by referring~~ to the ~~geometry~~ of a ~~[[tetrahedron]]~~. ~~Although it was a historically interesting theory~~ in ~~the late 19th~~ and ~~early 20th century, it was superseded by~~ the ~~concepts of [[continental drift]] and modern [[plate tectonics]]~~.		CMS provides the best platform to create websites that fulfill all the specifications of SEO. It is very easy to customize plugins according to the needs of a particular business. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. After activating, you will find their website link and get the activation code from their website. <br><br>

	~~==Theory==~~		purcase and download - WPZOOM Tribune wordpress Theme, find and use the WPZOOM Discount Code. You do not catch a user's attention through big and large pictures that usually takes a millennium to load up. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. If you enjoyed this information and you would certainly such as to receive more facts relating to [http://dinky.in/?WordpressDropboxBackup327633 wordpress dropbox backup] kindly browse through the web-page. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. You can also get a free keyword tool that is to determine how strong other competing sites are and number of the searches on the most popular search sites. <br><br>It is also popular because willing surrogates,as well as egg and sperm donors,are plentiful. By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Thousands of plugins are available in Word - Press plugin's library which makes the task of selecting right set of plugins for your website a very tedious task. " Thus working with a Word - Press powered web application, making any changes in the website design or website content is really easy and self explanatory. <br><br>The disadvantage is it requires a considerable amount of time to set every thing up. * Robust CRM to control and connect with your subscribers. A higher percentage of women are marrying at older ages,many are delaying childbearing until their careers are established, the divorce rate is high and many couples remarry and desire their own children. If you choose a blog then people will be able to post articles on your site and people will be able to make comments on your posts (unless you turn comments off). It does take time to come up having a website that gives you the much needed results hence the web developer must be ready to help you along the route. <br><br>Many developers design websites and give them to the clients, but still the client faces problems to handle the website. s ability to use different themes and skins known as Word - Press Templates or Themes. The days of spending a lot of time and money to have a website built are long gone. Word - Press is an open source content management system which is easy to use and offers many user friendly features. Article Source: Hostgator discount coupons for your Wordpress site here.

	~~This idea~~, ~~described~~ as ~~‘"ingenious" by geologist~~ [~~[Arthur Holmes]],<ref>{{cite book \|author=[[Arthur Holmes]] \|title=Principles of physical geology \|url=~~http://~~books~~.~~google.com~~/~~books~~?~~id=XUJRAAAAMAAJ \|year=1965 \|publisher=Nelson \|~~page~~=32 \|isbn=0~~-~~17-448020-2}}</ref> is now~~ of ~~historical interest only, being finally refuted by~~ that ~~same Holmes.<ref>''Nature'' magazine, June 6th. 1925, p.873.</ref> It attempted~~ to ~~explain apparent anomalies in~~ the ~~distribution of land and water~~ on the ~~Earth's surface:~~<~~ref~~>{{cite book \|author= A. Z. Bukhari \|title=Encyclopedia of nature of geography \|url=http://books.google.com/books?id=vef_5GExuUgC&pg=PA109 \|year=2005 \|publisher=Anmol Publications PVT. Limited \|isbn=978-81-261-2443-5 \|chapter= Continental Drift \|pages=109–113 }}<~~/ref~~>
	~~#More than 75% of the Earth's land area~~ is ~~in the [[northern hemisphere]].~~
	~~#Continents~~ are ~~roughly triangular~~.
	~~#Oceans are roughly triangular.~~
	~~#The north pole is surrounded by water, the south pole by land.~~
	~~#Exactly opposite the Earth from land is almost always water.~~
	~~#The [[Pacific Ocean]] occupies about one third of the Earth's surface.~~

	~~To understand its appeal, consider the "regular solids": the [[sphere]] and the 5~~-~~member set of [[Platonic Solids]]. The solid with the lowest number of sides is the [[tetrahedron]] (four equilateral triangles); progressing through the [[cube]]~~, and the ~~[[icosahedron]] (20 sides), the sphere~~ can be ~~considered to have an infinite number of sides~~. ~~All six regular solids share many symmetries.~~

	~~Now, for each regular solid, we may relate its surface area and volume by the equation:~~
	~~: <math>V = k\times A \times\sqrt{A}</math>~~
	~~where'' k'' is~~ a ~~characteristic~~ of ~~each solid~~, ~~''V'' its volume~~, ~~and~~ ''~~A'' its area. As we traverse~~ the set ~~in order~~ of ~~increasing number of faces, we find that k increases for each member; it is 0.0227~~ for a ~~tetrahedron and 0.0940 for a sphere~~. Thus ~~the tetrahedron is the regular solid~~ with ~~the largest surface area for~~ a ~~given volume, and makes a reasonable endpoint for a shrinking spherical Earth.<ref>{{cite book \|title=Principles of Physical Geography \|publisher=APH Publishing \|year=2004 \|isbn= 978~~-~~81-7648-731-3 \|url=http://books.google.com/?id=TXkdMyGF_wgC&pg=PA84&lpg=PA84 \|pages= 84–85 \|author9=S. A. Qazi }}</ref>~~

	~~==History==~~
	~~[[Image:Tetrahedral hypothesis.jpg\|thumb\|alt=view of earth above Antarctica\|Illustration of Tetrahedral hypothesis by [[William Lowthian Green]] from 1875]]~~
	~~The theory was first proposed by [[William Lowthian Green]] in 1875.<ref>{{cite book \|title=Vestiges of the molten globe~~, ~~as exhibited~~ in the ~~figure of the earth, volcanic action~~ and ~~physiography \|author=[[William Lowthian Green]] \|publisher= E~~. ~~Stanford \|place=London \|year=1875 \|oclc=3571917 \|url= http://books.google.com/books?id=9DkDAAAAQAAJ }}~~<~~/ref~~>
	~~It was still popular in 1917 when summarized as:~~

	<~~blockquote~~>"The ~~law~~ of ~~least action … demands that the somewhat rigid crustal portion of the earth keep in contact with the lessening interior~~ with ~~the least possible readjustment of its surface~~. ~~… a shrinking sphere tends by the law~~ of ~~least action to collapse into a tetrahedron~~, ~~or a tetra-hedroid~~, ~~a sphere marked by four equal and equidistant triangular projections; and~~ the ~~earth with its three about equal~~ and ~~equidistant double continental masses triangular southward with three intervening depressed oceans triangular northward, its northern ocean~~ and ~~southern continent, with land everywhere antipodal to water, realizes the tetrahedroid status remarkably~~.“<ref>{{cite news \|title= Tetrahedral Deformations and Intercontinental Torsions \|author = [[Benjamin Kendall Emerson]] \|work= Proceedings, American Philosophical Society \|volume= 56 \|number= 6 \|year= 1917 \|url=http://books.google.com/books?id=4lULAAAAIAAJ&pg=PA445 \|publisher=American Philosophical Society \|isbn=978-1-4223-7253-1 \|pages=445–472 }}</ref></blockquote>
	~~This is suggesting that~~ a ~~cooling spherical Earth might have shrunk~~ to ~~form a tetrahedron, with its vertices~~ and ~~edges forming the continents, and four oceans~~ (~~[[Pacific Ocean]], [[Atlantic Ocean]], [[Indian Ocean]] and [[Arctic Ocean]]~~) ~~on its faces~~.

	~~By 1915 German [[Alfred Wegener]] (1880–1930) had proposed in his [[continental drift]] theory~~ that ~~land masses moved great distances over~~ the ~~Earth's history. Wegener was also at first met with hostile reactions.<ref>{{cite~~ web \|title= Alfred Wegener (1880-1930) \|url= http://www.ucmp.berkeley.edu/history/wegener.html \|work= [[University of California Museum of Paleontology]] web site \|publisher= University of California \|year= 2006 \|accessdate= August 4, 2010 }}</ref>
	~~By the mid-1920s the same Holmes developed theories on what could cause~~ the ~~drift~~.<~~ref>{{cite journal \|title= The Origin of the Continents \|author= [[Arthur Holmes]] \|journal= Nature \|date= June 6, 1925 \|doi= 10.1038/115873a0 \|pages= 873–874 \|volume= 115 \|issue=2901}}</ref~~><~~ref~~>~~{{cite journal \|title= Theory of Continental Drift: a Symposium on the Origin~~ and Movement of Land Masses, both Inter-Continental and Intra-Continental, as proposed by Alfred Wegener \|author= [[Arthur Holmes]] \|journal= Nature \|date= September 22, 1928 \|doi= 10.1038/122431a0 \|pages= 431–433 \|volume= 122 \|issue=3073}}</ref>
	~~The [[plate tectonics]] theory is now generally accepted~~ to ~~explain~~ the ~~dynamic nature of~~ the ~~Earth's surface;~~ the ~~tetrahedral shape plays no special role in modern theories~~.<ref>{{cite web \|title= Plate Tectonics: The Rocky History of an Idea \|url= http://www.ucmp.berkeley.edu/geology/techist.html \|work= [[University of California Museum of Paleontology]] web site \|publisher= University of California \|year= 2006 \|accessdate= August 4, 2010 }}</ref>
	~~Explanations of details such as water~~ to ~~land ratios, the precise shape of continents~~ and ~~their sizes continue to be developed~~.

	~~==References==~~
	~~{{Reflist}}~~

	~~==Further reading==~~
	* {{cite book \|title= The ~~Tetrahedral Earth \|work= The Book~~ of ~~Earths: Hollow Earth, Ancient Maps, Atlantis,~~ and ~~Other Theories \|url= http://www~~.~~sacred~~-~~texts~~.~~com/earth/boe/boe34~~.~~htm \|author= Edna Kenton \|publisher= Forgotten Books\|isbn= 978-1-60506-415-4 \|accessdate= August 4, 2010 }}~~

	~~[[Category:Obsolete scientific theories]]~~
	~~[[Category:Plate tectonics]]~~

128.227.35.87: /* Basic properties */ added link for flats

2013-10-22T21:11:49Z

Basic properties: added link for flats

← Older revision		Revision as of 23:11, 22 October 2013
Line 1:		Line 1:
			The '''Tetrahedral hypothesis''' is an [[Superseded scientific theories\|obsolete scientific theory]] attempting to explain the arrangement of the [[Earth]]'s [[continent]]s and [[ocean]]s by referring to the geometry of a [[tetrahedron]]. Although it was a historically interesting theory in the late 19th and early 20th century, it was superseded by the concepts of [[continental drift]] and modern [[plate tectonics]].

			==Theory==

	~~By investing in a premium Word~~ - ~~Press theme~~, ~~you~~'~~re investing~~ in the ~~future~~ of ~~your website. This means you can setup your mailing list~~ and ~~auto-responder~~ on ~~your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like~~. ~~Your parishioners and certainly interested audience can come~~ in ~~to you for further information from~~ the ~~group and sometimes even approaching happenings and systems with~~ the ~~church~~. In the ~~recent years~~, ~~there has been a notable rise in~~ the ~~number~~ of ~~companies hiring Indian Word - Press developers~~. ~~You can easily customize~~ the ~~titles~~ of the ~~posts in Word - Press blog in a way that only title comes in~~ the ~~new post link~~ and ~~not~~ the ~~date or category~~ of ~~posts~~. <br><br>~~Generally~~, ~~for my private income-making market websites~~, ~~I will thoroughly research~~ and ~~discover~~ the ~~leading 10 most worthwhile niches to venture into. You will have to invest some money into tuning up your own blog but~~, ~~if done wisely, your investment will pay off in the long run~~. ~~Some plugins ask users to match pictures or add numbers,~~ and ~~although effective, they appear unprofessional and unnecessary~~. ~~t need to use~~ the ~~back button or~~ the ~~URL to get to your home page~~. ~~Word~~ - ~~Press makes it possible to successfully and manage your website~~. <br><br>~~ve labored so hard to publish~~ and ~~put up on their website~~. The ~~following piece~~ of ~~content is meant to make your choice easier and reassure you~~ that the ~~decision to go ahead~~ with ~~this conversion is requited~~ with ~~rich benefits:~~. ~~Whether or not it's an viewers on your web page, your social media pages~~, or ~~your web page~~, ~~those who have~~ a ~~present~~ and ~~effective viewers of "fans" are best best for provide provides~~, ~~reductions~~, ~~and deals to help re-invigorate their viewers and add~~ to ~~their main point here. Our skilled expertise~~, ~~skillfulness and excellence have been well known all across~~ the ~~world~~. ~~Article Source~~: ~~Stevens works in Internet and Network Marketing~~. <br><br>~~You can add keywords but it~~ is ~~best~~ to ~~leave this alone. I have compiled~~ a ~~few tips on how you can start a food blog~~ and ~~hopefully~~ the ~~following information~~ and ~~tips can help you to get started~~ on ~~your food blogging creative journey~~. ~~Some examples of its additional features include; code inserter~~ (~~for use with adding Google Analytics, Adsense section targeting etc~~) ~~Webmaster verification assistant, Link Mask Generator, Robots. It supports backup scheduling and allows you to either download~~ the ~~backup file or email it to you~~. Fortunately, Word - Press Customization Service is available these days, right from custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project. <br>~~<br>If you cherished this posting and you would like to receive extra data with regards to [~~http://l4.vc/~~wordpress_backup_plugin_966310 backup plugin~~] ~~kindly stop by our own~~ web site. ~~More it extends numerous opportunities where your firm is at comfort and rest assured~~ of ~~no risks & errors~~. ~~Mahatma Gandhi is known as one~~ of the ~~most prominent personalities~~ and ~~symbols~~ of ~~peace~~, ~~non~~-~~violence~~ and ~~freedom~~. ~~As a result, it~~ is ~~really crucial~~ to ~~just take aid~~ of ~~some experience when searching for superior quality totally free Word - Press themes, Word - Press Premium Themes for your~~ web ~~site. )~~ ~~Remote Login~~: ~~With the process~~ of ~~PSD~~ to ~~Wordpress conversion comes~~ the ~~advantage~~ of ~~flexibility~~. ~~Get started today so that people searching for your type~~ of ~~business will be directed to you~~.		This idea, described as ‘"ingenious" by geologist [[Arthur Holmes]],<ref>{{cite book \|author=[[Arthur Holmes]] \|title=Principles of physical geology \|url=http://books.google.com/books?id=XUJRAAAAMAAJ \|year=1965 \|publisher=Nelson \|page=32 \|isbn=0-17-448020-2}}</ref> is now of historical interest only, being finally refuted by that same Holmes.<ref>''Nature'' magazine, June 6th. 1925, p.873.</ref> It attempted to explain apparent anomalies in the distribution of land and water on the Earth's surface:<ref>{{cite book \|author= A. Z. Bukhari \|title=Encyclopedia of nature of geography \|url=http://books.google.com/books?id=vef_5GExuUgC&pg=PA109 \|year=2005 \|publisher=Anmol Publications PVT. Limited \|isbn=978-81-261-2443-5 \|chapter= Continental Drift \|pages=109–113 }}</ref>
			#More than 75% of the Earth's land area is in the [[northern hemisphere]].
			#Continents are roughly triangular.
			#Oceans are roughly triangular.
			#The north pole is surrounded by water, the south pole by land.
			#Exactly opposite the Earth from land is almost always water.
			#The [[Pacific Ocean]] occupies about one third of the Earth's surface.

			To understand its appeal, consider the "regular solids": the [[sphere]] and the 5-member set of [[Platonic Solids]]. The solid with the lowest number of sides is the [[tetrahedron]] (four equilateral triangles); progressing through the [[cube]], and the [[icosahedron]] (20 sides), the sphere can be considered to have an infinite number of sides. All six regular solids share many symmetries.

			Now, for each regular solid, we may relate its surface area and volume by the equation:
			: <math>V = k\times A \times\sqrt{A}</math>
			where'' k'' is a characteristic of each solid, ''V'' its volume, and ''A'' its area. As we traverse the set in order of increasing number of faces, we find that k increases for each member; it is 0.0227 for a tetrahedron and 0.0940 for a sphere. Thus the tetrahedron is the regular solid with the largest surface area for a given volume, and makes a reasonable endpoint for a shrinking spherical Earth.<ref>{{cite book \|title=Principles of Physical Geography \|publisher=APH Publishing \|year=2004 \|isbn= 978-81-7648-731-3 \|url=http://books.google.com/?id=TXkdMyGF_wgC&pg=PA84&lpg=PA84 \|pages= 84–85 \|author9=S. A. Qazi }}</ref>

			==History==
			[[Image:Tetrahedral hypothesis.jpg\|thumb\|alt=view of earth above Antarctica\|Illustration of Tetrahedral hypothesis by [[William Lowthian Green]] from 1875]]
			The theory was first proposed by [[William Lowthian Green]] in 1875.<ref>{{cite book \|title=Vestiges of the molten globe, as exhibited in the figure of the earth, volcanic action and physiography \|author=[[William Lowthian Green]] \|publisher= E. Stanford \|place=London \|year=1875 \|oclc=3571917 \|url= http://books.google.com/books?id=9DkDAAAAQAAJ }}</ref>
			It was still popular in 1917 when summarized as:

			<blockquote>"The law of least action … demands that the somewhat rigid crustal portion of the earth keep in contact with the lessening interior with the least possible readjustment of its surface. … a shrinking sphere tends by the law of least action to collapse into a tetrahedron, or a tetra-hedroid, a sphere marked by four equal and equidistant triangular projections; and the earth with its three about equal and equidistant double continental masses triangular southward with three intervening depressed oceans triangular northward, its northern ocean and southern continent, with land everywhere antipodal to water, realizes the tetrahedroid status remarkably.“<ref>{{cite news \|title= Tetrahedral Deformations and Intercontinental Torsions \|author = [[Benjamin Kendall Emerson]] \|work= Proceedings, American Philosophical Society \|volume= 56 \|number= 6 \|year= 1917 \|url=http://books.google.com/books?id=4lULAAAAIAAJ&pg=PA445 \|publisher=American Philosophical Society \|isbn=978-1-4223-7253-1 \|pages=445–472 }}</ref></blockquote>
			This is suggesting that a cooling spherical Earth might have shrunk to form a tetrahedron, with its vertices and edges forming the continents, and four oceans ([[Pacific Ocean]], [[Atlantic Ocean]], [[Indian Ocean]] and [[Arctic Ocean]]) on its faces.

			By 1915 German [[Alfred Wegener]] (1880–1930) had proposed in his [[continental drift]] theory that land masses moved great distances over the Earth's history. Wegener was also at first met with hostile reactions.<ref>{{cite web \|title= Alfred Wegener (1880-1930) \|url= http://www.ucmp.berkeley.edu/history/wegener.html \|work= [[University of California Museum of Paleontology]] web site \|publisher= University of California \|year= 2006 \|accessdate= August 4, 2010 }}</ref>
			By the mid-1920s the same Holmes developed theories on what could cause the drift.<ref>{{cite journal \|title= The Origin of the Continents \|author= [[Arthur Holmes]] \|journal= Nature \|date= June 6, 1925 \|doi= 10.1038/115873a0 \|pages= 873–874 \|volume= 115 \|issue=2901}}</ref><ref>{{cite journal \|title= Theory of Continental Drift: a Symposium on the Origin and Movement of Land Masses, both Inter-Continental and Intra-Continental, as proposed by Alfred Wegener \|author= [[Arthur Holmes]] \|journal= Nature \|date= September 22, 1928 \|doi= 10.1038/122431a0 \|pages= 431–433 \|volume= 122 \|issue=3073}}</ref>
			The [[plate tectonics]] theory is now generally accepted to explain the dynamic nature of the Earth's surface; the tetrahedral shape plays no special role in modern theories.<ref>{{cite web \|title= Plate Tectonics: The Rocky History of an Idea \|url= http://www.ucmp.berkeley.edu/geology/techist.html \|work= [[University of California Museum of Paleontology]] web site \|publisher= University of California \|year= 2006 \|accessdate= August 4, 2010 }}</ref>
			Explanations of details such as water to land ratios, the precise shape of continents and their sizes continue to be developed.

			==References==
			{{Reflist}}

			==Further reading==
			* {{cite book \|title= The Tetrahedral Earth \|work= The Book of Earths: Hollow Earth, Ancient Maps, Atlantis, and Other Theories \|url= http://www.sacred-texts.com/earth/boe/boe34.htm \|author= Edna Kenton \|publisher= Forgotten Books\|isbn= 978-1-60506-415-4 \|accessdate= August 4, 2010 }}

			[[Category:Obsolete scientific theories]]
			[[Category:Plate tectonics]]

en>David Eppstein: self-dual matroids and their algorithmic complexity

2012-08-31T00:11:09Z

self-dual matroids and their algorithmic complexity

New page

By investing in a premium Word - Press theme, you're investing in the future of your website. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. You can easily customize the titles of the posts in Word - Press blog in a way that only title comes in the new post link and not the date or category of posts. Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. Some plugins ask users to match pictures or add numbers, and although effective, they appear unprofessional and unnecessary. t need to use the back button or the URL to get to your home page. Word - Press makes it possible to successfully and manage your website. ve labored so hard to publish and put up on their website. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. Whether or not it's an viewers on your web page, your social media pages, or your web page, those who have a present and effective viewers of "fans" are best best for provide provides, reductions, and deals to help re-invigorate their viewers and add to their main point here. Our skilled expertise, skillfulness and excellence have been well known all across the world. Article Source: Stevens works in Internet and Network Marketing. You can add keywords but it is best to leave this alone. I have compiled a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. Some examples of its additional features include; code inserter (for use with adding Google Analytics, Adsense section targeting etc) Webmaster verification assistant, Link Mask Generator, Robots. It supports backup scheduling and allows you to either download the backup file or email it to you. Fortunately, Word - Press Customization Service is available these days, right from custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project. If you cherished this posting and you would like to receive extra data with regards to [http://l4.vc/wordpress_backup_plugin_966310 backup plugin] kindly stop by our own web site. More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. As a result, it is really crucial to just take aid of some experience when searching for superior quality totally free Word - Press themes, Word - Press Premium Themes for your web site. ) Remote Login: With the process of PSD to Wordpress conversion comes the advantage of flexibility. Get started today so that people searching for your type of business will be directed to you.

Dual matroid - Revision history

en>Oliver Kullmann: The *cycles*, not the "circuits", are the complements of a flat of the dual matroid (as stated in https://en.wikipedia.org/wiki/Matroid#Sums_and_unions )

128.227.35.87: /* Basic properties */ added link for flats

en>David Eppstein: self-dual matroids and their algorithmic complexity

en>Oliver Kullmann: The cycles, not the "circuits", are the complements of a flat of the dual matroid (as stated in https://en.wikipedia.org/wiki/Matroid#Sums_and_unions )