CZ:Featured article/Current: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Chunbum Park
(→‎Papacy: Four color theorem)
imported>Chunbum Park
(→‎Four color theorem: Osteoporosis)
Line 1: Line 1:
== '''[[Four color theorem]]''' ==
== '''[[Osteoporosis]]''' ==
----
----
The '''four color theorem''', sometimes known as the '''four color map theorem''' or '''Guthrie's problem''', is a [[problem]] in [[cartography]] and [[mathematics]]. It had been noticed that it only required four [[color]]s to fill in the different [[contiguous]] [[shape]]s on a [[map]] of regions or [[country|countries]] or [[province]]s in a flat surface known as a [[plane (geometry)|plane]] such that no two [[adjacent]] regions with a common [[boundary]] had the same color. But proving this [[proposition]] proved extraordinarily difficult, and it required [[analysis]] by high-powered [[computer]]s before the problem could be solved. In mathematical history, there had been numerous attempts to prove the supposition, but these so-called [[proof (mathematics)|proofs]] turned out to be flawed. There had been accepted proofs that a map could be colored in using more colors than four, such as six or five, but proving that only four colors were required was not done successfully until 1976 by mathematicians Appel and Haken, although some mathematicians do not accept it since parts of the proof consisted of an analysis of [[discrete]] cases by a computer.<ref name=Math1>{{cite news
As defined by the National Osteoporosis Foundation:
|title= Four-Color Theorem
{|align="center" style="width:90%;font-size:98%;"
|publisher= Wolfram MathWorld
|
|quote= Six colors can be proven to suffice for the g=0 case, and this number can easily be reduced to five, but reducing the number of colors all the way to four proved very difficult. This result was finally obtained by Appel and Haken (1977), who constructed a computer-assisted proof that four colors were sufficient. However, because part of the proof consisted of an exhaustive analysis of many discrete cases by a computer, some mathematicians do not accept it. However, no flaws have yet been found, so the proof appears valid. A shorter, independent proof was constructed by Robertson et al. (1996; Thomas 1998).
<font face="Gill Sans MT">Osteoporosis, or porous bone, is a disease characterized by low bone mass and [microarchitectural] structural deterioration of bone tissue, leading to bone fragility and an increased susceptibility to fractures, especially of the hip, spine and wrist, although any bone can be affected.</font><ref name=fastfacts>National Osteoporosis Foundation (2005) [https://docs.google.com/viewer?url=http%3A%2F%2Fmedschool.creighton.edu%2Ffileadmin%2Fuser%2Fmedicine%2Fimages%2FCreighton_FIRST%2FOsteo_Spotlight%2FFast_Facts.pdf].</ref>
|date= 2010-04-18
|}
|url= http://mathworld.wolfram.com/Four-ColorTheorem.html
|accessdate= 2010-04-18
}}</ref> But, at the present time, the proof remains viable, and was confirmed independently by Robertson and Thomas in association with other mathematicians in 1996&ndash;1998 who have offered a simpler version of the proof, but it is still complex, even for advanced mathematicians.<ref name=Math1/> It is possible that an even simpler, more elegant, proof will someday be discovered, but many mathematicians think that a shorter, more elegant and simple proof is impossible.


''[[Four color theorem|.... (read more)]]''
The increase in fragility results from both low bone mass and impaired bone quality.
 
Emphasizing the factor of ‘bone strength’, a factor in addition to and distinct from low bone mass, and the consequent increase risk of sustaining a fracture, a [[National Institutes of Health]] (NIH) Consensus Development Panel on Osteoporosis Prevention, Diagnosis, and Therapy has defined osteoporosis in 2001 as
 
{|align="center" style="width:90%;font-size:98%;"
|
<font face="Gill Sans MT">a skeletal disease characterized by compromised bone strength predisposing a person to an increased risk of fracture. Bone strength primarily reflects the integration of bone density and bone quality...</font><ref name=NIH2001> NIH Consensus Development Panel on Osteoporosis Prevention, Diagnosis, and Therapy. (2001)  [http://dx.doi.org/10.1001/jama.285.6.785 Osteoporosis prevention, diagnosis, and therapy]. ''JAMA'' 285(6):785-795. PMID 11176917.</ref>
|}
 
Primary osteoporosis can be of two major types: postmenopausal osteoporosis (osteoporosis, postmenopausal) and age-related or senile osteoporosis."
{|align="left" cellpadding="10" style="background:lightgray; width:35%; border: 1px solid #aaa; margin:20px; font-size: 93%; font-family: Gill Sans MT;"
| Osteoporosis, which literally means "porous bone", is a disease in which the density and quality of bone are reduced. As the bones become more porous and fragile, the risk of fracture is greatly increased. The loss of bone occurs "silently" and progressively. Often there are no symptoms until the first fracture occurs.
: [http://www.iofbonehealth.org/patients-public/about-osteoporosis/what-is-osteoporosis.html &mdash;The International Osteoporosis Foundation (IOF)]
|-
| Osteoporosis means “porous bone.” If you look at healthy bone under a microscope, you will see that parts of it look like a honeycomb. If you have osteoporosis, the holes and spaces in the honeycomb are much bigger than they are in healthy bone. This means your bones have lost density or mass. It also means that the structure of your bone tissues has become abnormal. As your bones become less dense, they become weaker.
: [http://www.nof.org/aboutosteoporosis/bonebasics/whatisosteoporosis &mdash;The National Osteoporosis Foundation]
|}
Although more common in women, osteoporosis may occur in males.<ref name="pmid18385499">{{cite journal |author=Ebeling PR |title=Clinical practice. Osteoporosis in men |journal=N. Engl. J. Med. |volume=358 |issue=14 |pages=1474–82 |year=2008 |month=April |pmid=18385499 |doi=10.1056/NEJMcp0707217 |url=http://content.nejm.org/cgi/pmidlookup?view=short&pmid=18385499&promo=ONFLNS19 |issn=}}</ref>
 
''[[Osteoporosis|.... (read more)]]''


{| class="wikitable collapsible collapsed" style="width: 90%; float: center; margin: 0.5em 1em 0.8em 0px;"
{| class="wikitable collapsible collapsed" style="width: 90%; float: center; margin: 0.5em 1em 0.8em 0px;"
|-
|-
! style="text-align: center;" | &nbsp;[[Four color theorem#References|notes]]
! style="text-align: center;" | &nbsp;[[Osteoporosis#References|notes]]
|-
|-
|
|
{{reflist|2}}
{{reflist|2}}
|}
|}

Revision as of 02:12, 30 March 2013

Osteoporosis

As defined by the National Osteoporosis Foundation:

Osteoporosis, or porous bone, is a disease characterized by low bone mass and [microarchitectural] structural deterioration of bone tissue, leading to bone fragility and an increased susceptibility to fractures, especially of the hip, spine and wrist, although any bone can be affected.[1]

The increase in fragility results from both low bone mass and impaired bone quality.

Emphasizing the factor of ‘bone strength’, a factor in addition to and distinct from low bone mass, and the consequent increase risk of sustaining a fracture, a National Institutes of Health (NIH) Consensus Development Panel on Osteoporosis Prevention, Diagnosis, and Therapy has defined osteoporosis in 2001 as

a skeletal disease characterized by compromised bone strength predisposing a person to an increased risk of fracture. Bone strength primarily reflects the integration of bone density and bone quality...[2]

Primary osteoporosis can be of two major types: postmenopausal osteoporosis (osteoporosis, postmenopausal) and age-related or senile osteoporosis."

Osteoporosis, which literally means "porous bone", is a disease in which the density and quality of bone are reduced. As the bones become more porous and fragile, the risk of fracture is greatly increased. The loss of bone occurs "silently" and progressively. Often there are no symptoms until the first fracture occurs.
—The International Osteoporosis Foundation (IOF)
Osteoporosis means “porous bone.” If you look at healthy bone under a microscope, you will see that parts of it look like a honeycomb. If you have osteoporosis, the holes and spaces in the honeycomb are much bigger than they are in healthy bone. This means your bones have lost density or mass. It also means that the structure of your bone tissues has become abnormal. As your bones become less dense, they become weaker.
—The National Osteoporosis Foundation

Although more common in women, osteoporosis may occur in males.[3]

.... (read more)

Retrieved from "https://citizendium.org/wiki/index.php?title=CZ:Featured_article/Current&oldid=829686"